{"text": "\nimport data.equiv.basic\nimport data.fintype\nimport data.equiv.encodable\nimport data.equiv.denumerable\n-- import util.control.monad\n-- import util.control.applicative\n-- import util.meta.tactic\n\nuniverses u\n\nclass generic (α : Type u) :=\n  (rep : Type u)\n  (to_rep : α → rep)\n  (of_rep : rep → α)\n  (correspondence : α ≃ rep)\n\nexport generic (rep correspondence)\n\nattribute [reducible] generic.rep\n\nmeta def tactic.mk_sigma (v t : expr) : tactic expr :=\ndo t' ← tactic.infer_type v,\n   tactic.to_expr ``(@sigma %%t' %%(expr.lambdas [v] t))\n\nnamespace tactic.interactive\n\nopen tactic\n\nmeta def mk_prod_rep' (α : expr) (c : name)\n: list expr → expr → list expr → tactic (expr × expr × expr)\n | [] _ _ :=\n   do c' ← mk_const c,\n      let to_rep : expr := `(()),\n      return (`(unit),to_rep,expr.lam `_ binder_info.default `(unit) c')\n | [x] e es := do c' ← mk_const c,\n             t ← infer_type x,\n             let of_rep := c'.mk_app (e :: es : list _).reverse,\n             return (t,x,expr.lambdas [e] of_rep)\n | (x :: xs) e es :=\n    do xs' ← mmap infer_type xs,\n       if (xs'.any (x.occurs)) then do\n         t₀ ← infer_type x,\n         v ← mk_mvar,\n         y ← mk_local_def `x v,\n         (t₁,to_rep,of_rep) ← mk_prod_rep' xs y (x :: es) <|> fail \"mk_prod_rep'\",\n         unify v (t₁),\n         of_rep ← instantiate_mvars of_rep,\n         t ← mk_sigma x t₁,\n         to_rep ← to_expr ``(sigma.mk %%x %%to_rep),\n         of_rep ← to_expr ``(@sigma.cases_on %%t₀ %%((expr.lambdas [x] t₁)) (λ x, %%α) %%e (%%(expr.lambdas [x] of_rep)) : %%α),\n         return (t,to_rep,expr.lambdas [e] of_rep)\n       else do\n         t₀ ← infer_type x,\n         v ← mk_mvar,\n         y ← mk_local_def `x v,\n         (t₁,to_rep,of_rep) ← mk_prod_rep' xs y (x :: es),\n         unify v t₁,\n         t ← to_expr ``(%%t₀ × %%t₁),\n         to_rep ← to_expr ``((%%x, %%to_rep)),\n         of_rep ← to_expr ``(@prod.cases_on %%t₀ %%t₁ (λ _, %%α) %%e (%%(expr.lambdas [x] of_rep)) : %%α),\n         return (t,to_rep,expr.lambdas [e] of_rep)\n\nmeta def mk_inr (α β : expr) (p : pexpr) : pexpr :=\n``(@sum.inr %%α %%β %%p)\n\nmeta def mk_inl (α β : expr) (p : pexpr) : pexpr :=\n``(@sum.inl %%α %%β %%p)\n\nmeta def mk_prod_rep (α : expr) (c : name) (f : list (name × expr × expr)) : tactic (expr × expr × expr) :=\ndo t ← resolve_name c >>= to_expr >>= infer_type,\n   (cmp,_) ← mk_local_pis t,\n   x ← mk_mvar >>= mk_local_def `x,\n   (t,to_rep,of_rep) ← mk_prod_rep' α c cmp x [],\n   to_rep' ← mfoldl ((λ e ⟨c,α,β⟩,\n       do c' ← mk_const c,\n          return (expr.mk_app c' [α,β,e])) : _ → _ × _ × _ → tactic expr) to_rep f,\n   to_rep ← instantiate_mvars (expr.lambdas cmp to_rep'),\n   return (t,to_rep,of_rep)\n\n/-- `mk_sum_rep cs` returns\n  - the representation type for constructors cs\n  - a list of functions `to_rep` for the cases `cs`\n  - a function of_rep (for the constructed rep type) to α -/\nmeta def mk_sum_rep (α : expr) : list (name × expr × expr) → list name → tactic (expr × list expr × expr)\n | f [] :=\n do of_rep ← to_expr ``(empty.rec_on (λ _, %%α) : empty → %%α),\n    return (`(empty),[],of_rep)\n | f [x] := prod.map id (prod.map singleton id) <$> mk_prod_rep α x f\n | f (x::xs) :=\n do α' ← mk_mvar,\n    β' ← mk_mvar,\n    y  ← mk_prod_rep α x ((`sum.inl, α', β') :: f),\n    ys ← mk_sum_rep ((`sum.inr, α', β') :: f) xs,\n    rep ← to_expr ``(%%y.1 ⊕ %%ys.1),\n    of_rep ← to_expr ``((λ x, @sum.cases_on %%α' %%β' (λ _, %%α) x %%y.2.2 %%ys.2.2) : %%rep → %%α),\n    return (rep,y.2.1 :: ys.2.1,of_rep)\n\nmeta def prove_inverse : name → tactic unit | n :=\n(`[simp ; try { congr }] ; reflexivity) <|>\ndo c ← resolve_name n >>= to_expr,\n   t ← infer_type c,\n   match t with\n    | `(_ × _) := tactic.cases c [`x,`y] >> prove_inverse `y\n    | `(Σ _, _) := tactic.cases c [`x,`y] >> prove_inverse `y\n    | `(_ ⊕ _) := (() <$ tactic.cases c [`x,`x]) ; solve1 (prove_inverse `x)\n    | `(unit) :=  tactic.cases c >> reflexivity\n    | _ := fail \"cannot prove inverse\"\n   end\n\nmeta def mk_generic_instance : tactic unit :=\ndo `(generic %%t) ← target,\n   e ← get_env,\n   let n := t.const_name,\n   let ns := n <.> \"generic\",\n   set_env $ e.add_namespace ns,\n   (rep,to_rep,of_rep) ← mk_sum_rep t [] (e.constructors_of t.const_name),\n   cases_on ← resolve_name (t.const_name <.> \"cases_on\") >>= to_expr,\n   x ← mk_local_def `x t,\n   let to_rep' := expr.lambdas [x] $ cases_on.mk_app (x::to_rep),\n   bij_t ← to_expr ``(equiv %%t %%rep),\n   bij ← assert `bij bij_t,\n   swap,\n   d ← to_expr ``( { generic . rep := %%rep\n                   , to_rep := %%to_rep'\n                   , of_rep := %%of_rep\n                   , correspondence := %%bij } : generic %%t ),\n   instantiate_mvars d >>= tactic.apply,\n   `[apply @equiv.mk %%t %%rep %%to_rep' %%of_rep],\n   solve1 `[intro x, induction x ; refl ],\n   solve1 (intro `x >> prove_inverse `x)\n\nend tactic.interactive\n-- ↪\ndef encoding {α : Type u} [generic α] : rep α ↪ α :=\nequiv.to_embedding $ (correspondence _).symm\n\nlemma multiset.map_equiv_to_embedding {α β} (f : α ≃ β) (s : multiset α) (x : β) :\n  x ∈ s.map f ↔ f.symm x ∈ s :=\nby { split; simp; introv h,\n     { intro, subst x, simp * },\n     { split, existsi h, simp } }\n\nlemma finset.map_equiv_to_embedding {α β} (f : α ≃ β) (s : finset α) (x : β) :\n  x ∈ s.map (equiv.to_embedding f) ↔ f.symm x ∈ s :=\nby { split; simp; intros,\n     { subst x, simp * },\n     { split, existsi a, simp } }\n\ninstance finite_generic (α : Type u) [generic α] [fintype (rep α)] : fintype α :=\n{ elems := (fintype.elems (rep α)).map encoding,\n  complete :=\n  by { intros, dsimp [encoding],\n       simp [multiset.map_equiv_to_embedding],\n       apply fintype.complete } }\n\ninstance inhabited_generic (α : Type u) [generic α] [inhabited (rep α)] : inhabited α :=\n⟨ (correspondence α).symm (default _) ⟩\n\ninstance encodable_generic (α : Type u) [generic α] [encodable (rep α)] : encodable α :=\nencodable.of_equiv _ (correspondence _)\n\ninstance denumerable_generic (α : Type u) [generic α] [denumerable (rep α)] : denumerable α :=\ndenumerable.of_equiv _ (correspondence _)\n\ninstance : generic bool :=\nby { mk_generic_instance, }\n\nnamespace examples\n\ninductive generic_ex1 : Type\n | case1 : generic_ex1\n | case2 : generic_ex1\n | case3 : ℕ → generic_ex1\n | case4 : bool → ℕ → generic_ex1\n\nopen generic_ex1\n\ninstance : generic generic_ex1 :=\nby mk_generic_instance\n\ninductive generic_ex2 : Type\n | case1 : generic_ex2\n | case2 : ℕ → ℕ → bool → generic_ex2\n | case3 (n : ℕ) : fin n → generic_ex2\n | case4 : bool → generic_ex2\n\ninstance generic_generic_ex2 : generic generic_ex2 :=\nbegin mk_generic_instance end\n\nend examples\n", "meta": {"author": "unitb", "repo": "lean-lib", "sha": "439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9", "save_path": "github-repos/lean/unitb-lean-lib", "path": "github-repos/lean/unitb-lean-lib/lean-lib-439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9/src/util/data/generic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3499320166333547}}
{"text": "import order.filter.basic\n\nimport for_mathlib.data.set.basic\n\nlocal infixr ` ×ᶠ `:51 := filter.prod\n\nsection\nopen filter set function\n\nlemma filter.comap_pure {α : Type*} {β : Type*} {f : α → β} (h : injective f) (a : α) :\n  pure a = comap f (pure $ f a) :=\nby rw [← filter.map_pure, comap_map h]\n\nvariables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\n/-\n    f\n  α → β\ng ↓   ↓ h\n  γ → δ\n    i\n-/\nvariables {f : α → β} {g : α → γ} {h : β → δ} {i : γ → δ} (H : h ∘ f = i ∘ g)\ninclude H\n\nlemma filter.map_comm (F : filter α) : map h (map f F) = map i (map g F) :=\nby rw [filter.map_map, H, ← filter.map_map]\n\nlemma filter.comap_comm (G : filter δ) : comap f (comap h G) = comap g (comap i G) :=\nby rw [filter.comap_comap_comp, H, ← filter.comap_comap_comp]\nomit H\n\n\nopen lattice\nvariables {G : filter β} {s : set α} {t : set β} {φ : α → β}\n\nlemma mem_comap_sets_of_inj (h : ∀ a a', φ a = φ a' → a = a') :\n  s ∈ comap φ G ↔ ∃ t ∈ G, s = φ ⁻¹' t :=\nbegin\n  rw mem_comap_sets,\n  split ; rintros ⟨t, ht, hts⟩,\n  { use t ∪ φ '' s,\n    split,\n    { simp [mem_sets_of_superset ht] },\n    { rw [preimage_union, preimage_image_eq _ h],\n      exact (union_eq_self_of_subset_left hts).symm } },\n  { use [t, ht],\n    rwa hts }\nend\n\nlemma filter.inf_eq_bot_iff {α : Type*} (f g : filter α) : f ⊓ g = ⊥ ↔ ∃ (U ∈ f) (V ∈ g), U ∩ V = ∅ :=\nby { rw [← empty_in_sets_eq_bot, mem_inf_sets], simp [subset_empty_iff] }\n\nlemma filter.ne_bot_of_map {α : Type*} {β : Type*} {f : α → β} {F : filter α} (h : map f F ≠ ⊥) : F ≠ ⊥ :=\nλ H, (H ▸ h : map f ⊥ ≠ ⊥) map_bot\n\nlemma filter.map_prod_prod {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*}\n  (f : α₁ → α₂) (g : β₁ → β₂) (F : filter α₁) (G : filter β₁) :\n  map (f ⨯ g) (F ×ᶠ G) = (map f F) ×ᶠ (map g G) :=\nby { rw filter.prod_map_map_eq, refl }\n\nlemma filter.map_prod_mk {α : Type*} {β : Type*} {γ : Type*} (f : γ → α) (g : γ → β) (F : filter γ) :\n  map (λ x, prod.mk (f x) $ g x) F ≤ (map f F).prod (map g F) :=\nbegin\n  intros U U_in,\n  rcases mem_prod_iff.1 U_in with ⟨s, s_in, t, t_in, stU⟩,\n  rw mem_map at *,\n  have st_in := filter.inter_sets F s_in t_in,\n  apply mem_sets_of_superset st_in,\n  rintro x ⟨xs, xt⟩,\n  apply stU,\n  exact ⟨xs, xt⟩\nend\nend\n\n\nclass filter_basis (α : Type*) :=\n(sets : set $ set α)\n(ne_empty : sets.nonempty) -- avoid the degenerate case of an empty filter basis\n(directed : ∀ {s t}, s ∈ sets → t ∈ sets → ∃ u ∈ sets, u ⊆ s ∩ t)\n\nnamespace filter_basis\nopen filter set function\n\nvariables {α : Type*} {ι : Type*} (B : filter_basis α)\n\n--instance : has_coe (filter_basis α) (set $ set α) := ⟨filter_basis.sets α⟩\ninstance : has_mem (set α) (filter_basis α) := ⟨λ s B, s ∈ B.sets⟩\n\nlemma mem_iff {B : filter_basis α} {s : set α} : s ∈ B ↔ s ∈ B.sets := iff.rfl\n\ndef default (B : filter_basis α) : set α :=\nB.ne_empty.some\n\ndef default_in (B : filter_basis α) : B.default ∈ B :=\nB.ne_empty.some_mem\n\ndef filter : filter α := generate B.sets\n\nlemma mem_filter {U : set α} : U ∈ B.filter ↔ ∃ V ∈ B, V ⊆ U :=\nbegin\n  split ; intro h,\n  { induction h with U U_in U V U_gen UV U_union U V U_gen V_gen U_union V_union,\n    { exact ⟨U, U_in, le_refl U⟩ },\n    { exact ⟨B.default, B.default_in, subset_univ _⟩ },\n    { rcases U_union with ⟨W, W_in, WU⟩,\n      refine ⟨W, W_in, subset.trans WU UV⟩ },\n    { rcases U_union with ⟨S, S_in, SU⟩,\n      rcases V_union with ⟨T, T_in, TU⟩,\n      rcases filter_basis.directed S_in T_in with ⟨W, W_in, hW⟩,\n      cases subset_inter_iff.1 hW with WS WT,\n      exact ⟨W, W_in, subset_inter_iff.2 ⟨subset.trans WS SU, subset.trans WT TU⟩⟩ } },\n  { rcases h with ⟨V, V_in, VU⟩,\n    exact generate_sets.superset (generate_sets.basic $ V_in) VU },\nend\n\nlemma mem_filter_of_mem {B : filter_basis α} {U : set α} (h : U ∈ B) : U ∈ B.filter :=\ngenerate_sets.basic h\n\nlemma le_filter (F : _root_.filter α) : F ≤ B.filter ↔ ∀ U ∈ B, U ∈ F :=\nbegin\n  split ; intros h U U_in,\n  { exact h (mem_filter_of_mem U_in) },\n  { rcases B.mem_filter.1 U_in with ⟨V, V_in, VU⟩,\n    filter_upwards [h V V_in],\n    exact VU },\nend\n\nlemma filter_le (F : _root_.filter α) : B.filter ≤ F ↔ ∀ {V}, V ∈ F → ∃ U ∈ B, U ⊆ V :=\nbegin\n  simp only [B.mem_filter],\n  split ; intros h V V_in,\n  { exact B.mem_filter.1 (h V_in) },\n  { exact B.mem_filter.2 (h V_in) },\nend\n\n\ndef map {β : Type*} (f : α → β) (B : filter_basis α) : filter_basis β :=\n{ sets := (image f) '' B.sets,\n  ne_empty := image_nonempty _ _ B.ne_empty,\n  directed := begin\n    rintros _ _ ⟨U, U_in, rfl⟩ ⟨V, V_in, rfl⟩,\n    rcases filter_basis.directed U_in V_in with ⟨W, W_in, UVW⟩,\n    use [f '' W, mem_image_of_mem _ W_in],\n    exact subset.trans (image_subset f UVW) (image_inter_subset f U V)\n  end }\n\nlemma mem_map {β : Type*} (f : α → β) (B : filter_basis α) {V : set β} :\n  V ∈ map f B ↔ ∃ U ∈ B, f '' U = V :=\nbegin\n  change V ∈ image f '' B.sets ↔ ∃ (U : set α) (H : U ∈ B), f '' U = V,\n  simp [mem_image],\n  exact iff.rfl,\nend\n\ndef map_filter {β : Type*} (f : α → β) (B : filter_basis α) :\n  filter.map f B.filter = (map f B).filter :=\nbegin\n  ext V,\n  simp [filter.mem_map, mem_filter, mem_filter, mem_map],\n  split,\n  { rintros ⟨U, UB, H⟩,\n    use [f '' U, U, UB],\n    exact image_subset_iff.2 H },\n  { rintros ⟨W, ⟨U, UB, rfl⟩, H⟩,\n    use [U, UB],\n    exact image_subset_iff.1 H },\nend\n\nlemma tendsto_from {β : Type*} (f : α → β) (F : _root_.filter β) :\n  tendsto f B.filter F ↔ ∀ {V}, V ∈ F → ∃ U ∈ B, U ⊆ f ⁻¹' V :=\nbegin\n  apply forall_congr,\n  intros V,\n  apply imp_congr iff.rfl,\n  rw [filter.mem_map, mem_filter],\n  exact iff.rfl\nend\n\nlemma tendsto_into {β : Type*} (f : β → α) (F : _root_.filter β) :\n  tendsto f F B.filter ↔ ∀ {V}, V ∈ B → f ⁻¹' V ∈ F :=\nbegin\n  split ; intros h U U_in,\n  { exact h (mem_filter_of_mem U_in) },\n  { rcases B.mem_filter.1 U_in with ⟨V, V_in, H⟩,\n    filter_upwards [h V_in],\n    exact preimage_mono H },\nend\n\nlemma tendsto_both {β : Type*} (f : α → β) (B' : filter_basis β) :\n  tendsto f B.filter B'.filter ↔ ∀ {V}, V ∈ B' → ∃ U ∈ B, U ⊆ f ⁻¹' V :=\nbegin\n  rw tendsto_into,\n  apply forall_congr,\n  intros V,\n  exact imp_congr iff.rfl (mem_filter _)\nend\n\nuniverses u v\n\nprotected def prod {α : Type u} {β : Type v} (B : filter_basis α) (B' : filter_basis β) :\n  filter_basis (α × β) :=\n{ sets :=  (uncurry' set.prod) '' (B.sets.prod B'.sets),\n  ne_empty := image_nonempty _ _ (set.prod_nonempty_iff.2 ⟨B.ne_empty, B'.ne_empty⟩),\n  directed := begin\n    rintros _ _ ⟨⟨P, P'⟩, ⟨P_in, P_in'⟩, rfl⟩ ⟨⟨Q, Q'⟩, ⟨Q_in, Q_in'⟩, rfl⟩,\n    rcases filter_basis.directed P_in Q_in with ⟨U, U_in, hU⟩,\n    rcases filter_basis.directed P_in' Q_in' with ⟨U', U_in', hU'⟩,\n    use set.prod U U',\n    simp only [uncurry', exists_prop],\n    use [(U, U'), ⟨U_in, U_in'⟩],\n    rintros ⟨x, y⟩ ⟨x_in, y_in⟩,\n    cases hU x_in,\n    cases hU' y_in,\n    finish\n  end }\n\nlemma mem_prod_of_mem {α : Type u} {β : Type v} {B : filter_basis α} {B' : filter_basis β}\n  {S} (hS : S ∈ B) {T} (hT : T ∈ B') : set.prod S T ∈ B.prod B' :=\n⟨(S, T), mem_prod.2 ⟨hS, hT⟩, rfl⟩\n\nlemma mem_prod_filter_of_mem {α : Type u} {β : Type v} {B : filter_basis α} {B' : filter_basis β}\n  {S} (hS : S ∈ B) {T} (hT : T ∈ B') : set.prod S T ∈ (B.prod B').filter :=\nmem_filter_of_mem (mem_prod_of_mem hS hT)\n\nlemma mem_prod_filter {α : Type u} {β : Type v} (B : filter_basis α) (B' : filter_basis β)\n  {U : set (α × β)} : U ∈ (B.prod B').filter ↔ ∃ S ∈ B, ∃ T ∈ B', set.prod S T ⊆ U :=\nbegin\n  rw mem_filter,\n  split,\n  { rintros ⟨_, ⟨⟨S, T⟩, h, rfl⟩, H⟩,\n    exact ⟨S, h.1, T, h.2, H⟩ },\n  { rintros ⟨S, S_in, T, T_in, H⟩,\n    exact ⟨set.prod S T, mem_prod_of_mem S_in T_in, H⟩ }\nend\n\nlemma prod_filter {α : Type u} {β : Type v} (B : filter_basis α) (B' : filter_basis β) :\n(B.prod B').filter = B.filter ×ᶠ B'.filter :=\nbegin\n  ext U,\n  rw [mem_prod_filter, filter.mem_prod_iff],\n  split ; rintros ⟨S, ⟨Sin, T, Tin, H⟩⟩,\n  { exact ⟨S, mem_filter_of_mem Sin, T, mem_filter_of_mem Tin, H⟩ },\n  { rcases B.mem_filter.1 Sin with ⟨V, Vin, VS⟩,\n    rcases B'.mem_filter.1 Tin with ⟨W, Win, WT⟩,\n    use [V, Vin, W, Win],\n    exact subset.trans (set.prod_mono VS WT) H }\nend\nend filter_basis\n", "meta": {"author": "leanprover-community", "repo": "lean-perfectoid-spaces", "sha": "95a6520ce578b30a80b4c36e36ab2d559a842690", "save_path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces", "path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces/lean-perfectoid-spaces-95a6520ce578b30a80b4c36e36ab2d559a842690/src/for_mathlib/filter.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6224593382055109, "lm_q1q2_score": 0.34993201269606367}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.types\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.PostPort\n\nuniverses u \n\nnamespace Mathlib\n\n/-!\n# Special shapes for limits in `Type`.\n\nThe general shape (co)limits defined in `category_theory.limits.types`\nare intended for use through the limits API,\nand the actual implementation should mostly be considered \"sealed\".\n\nIn this file, we provide definitions of the \"standard\" special shapes of limits in `Type`,\ngiving the expected definitional implementation:\n* the terminal object is `punit`\n* the binary product of `X` and `Y` is `X × Y`\n* the product of a family `f : J → Type` is `Π j, f j`.\n\nBecause these are not intended for use with the `has_limit` API,\nwe instead construct terms of `limit_data`.\n\nAs an example, when setting up the monoidal category structure on `Type`\nwe use the `types_has_terminal` and `types_has_binary_products` instances.\n-/\n\nnamespace category_theory.limits.types\n\n\n/-- A restatement of `types.lift_π_apply` that uses `pi.π` and `pi.lift`. -/\n@[simp] theorem pi_lift_π_apply {β : Type u} (f : β → Type u) {P : Type u} (s : (b : β) → P ⟶ f b) (b : β) (x : P) : pi.π f b (pi.lift s x) = s b x :=\n  congr_fun (limit.lift_π (fan.mk P s) b) x\n\n/-- A restatement of `types.map_π_apply` that uses `pi.π` and `pi.map`. -/\n@[simp] theorem pi_map_π_apply {β : Type u} {f : β → Type u} {g : β → Type u} (α : (j : β) → f j ⟶ g j) (b : β) (x : ∏ fun (j : β) => f j) : pi.π g b (pi.map α x) = α b (pi.π f b x) :=\n  limit.map_π_apply (discrete.nat_trans α) b x\n\n/-- The category of types has `punit` as a terminal object. -/\ndef terminal_limit_cone : limit_cone (functor.empty (Type u)) :=\n  limit_cone.mk (cone.mk PUnit (nat_trans.mk sorry)) (id (is_limit.mk sorry))\n\n/-- The category of types has `pempty` as an initial object. -/\ndef initial_limit_cone : colimit_cocone (functor.empty (Type u)) :=\n  colimit_cocone.mk (cocone.mk pempty (nat_trans.mk sorry)) (id (is_colimit.mk sorry))\n\n/-- The product type `X × Y` forms a cone for the binary product of `X` and `Y`. -/\n-- We manually generate the other projection lemmas since the simp-normal form for the legs is\n\n-- otherwise not created correctly.\n\n@[simp] theorem binary_product_cone_X (X : Type u) (Y : Type u) : cone.X (binary_product_cone X Y) = (X × Y) :=\n  Eq.refl (X × Y)\n\n@[simp] theorem binary_product_cone_fst (X : Type u) (Y : Type u) : binary_fan.fst (binary_product_cone X Y) = prod.fst :=\n  rfl\n\n@[simp] theorem binary_product_cone_snd (X : Type u) (Y : Type u) : binary_fan.snd (binary_product_cone X Y) = prod.snd :=\n  rfl\n\n/-- The product type `X × Y` is a binary product for `X` and `Y`. -/\n@[simp] theorem binary_product_limit_lift (X : Type u) (Y : Type u) (s : binary_fan X Y) (x : cone.X s) : is_limit.lift (binary_product_limit X Y) s x = (binary_fan.fst s x, binary_fan.snd s x) :=\n  Eq.refl (is_limit.lift (binary_product_limit X Y) s x)\n\n/--\nThe category of types has `X × Y`, the usual cartesian product,\nas the binary product of `X` and `Y`.\n-/\n@[simp] theorem binary_product_limit_cone_is_limit (X : Type u) (Y : Type u) : limit_cone.is_limit (binary_product_limit_cone X Y) = binary_product_limit X Y :=\n  Eq.refl (limit_cone.is_limit (binary_product_limit_cone X Y))\n\n/-- The functor which sends `X, Y` to the product type `X × Y`. -/\n-- We add the option `type_md` to tell `@[simps]` to not treat homomorphisms `X ⟶ Y` in `Type*` as\n\n-- a function type\n\n@[simp] theorem binary_product_functor_obj_map (X : Type u) (Y₁ : Type u) (Y₂ : Type u) (f : Y₁ ⟶ Y₂) : functor.map (functor.obj binary_product_functor X) f =\n  is_limit.lift (binary_product_limit X Y₂) (binary_fan.mk prod.fst (prod.snd ≫ f)) :=\n  Eq.refl (functor.map (functor.obj binary_product_functor X) f)\n\n/--\nThe product functor given by the instance `has_binary_products (Type u)` is isomorphic to the\nexplicit binary product functor given by the product type.\n-/\ndef binary_product_iso_prod : binary_product_functor ≅ prod.functor :=\n  nat_iso.of_components\n    (fun (X : Type u) =>\n      nat_iso.of_components\n        (fun (Y : Type u) =>\n          iso.symm (is_limit.cone_point_unique_up_to_iso (limit.is_limit (pair X Y)) (binary_product_limit X Y)))\n        sorry)\n    sorry\n\n/-- The sum type `X ⊕ Y` forms a cocone for the binary coproduct of `X` and `Y`. -/\n@[simp] theorem binary_coproduct_cocone_ι_app (X : Type u) (Y : Type u) (j : discrete walking_pair) : ∀ (ᾰ : functor.obj (pair X Y) j),\n  nat_trans.app (cocone.ι (binary_coproduct_cocone X Y)) j ᾰ = walking_pair.rec sum.inl sum.inr j ᾰ :=\n  fun (ᾰ : functor.obj (pair X Y) j) => Eq.refl (walking_pair.rec sum.inl sum.inr j ᾰ)\n\n/-- The sum type `X ⊕ Y` is a binary coproduct for `X` and `Y`. -/\ndef binary_coproduct_colimit (X : Type u) (Y : Type u) : is_colimit (binary_coproduct_cocone X Y) :=\n  is_colimit.mk fun (s : binary_cofan X Y) => sum.elim (binary_cofan.inl s) (binary_cofan.inr s)\n\n/--\nThe category of types has `X ⊕ Y`,\nas the binary coproduct of `X` and `Y`.\n-/\ndef binary_coproduct_colimit_cocone (X : Type u) (Y : Type u) : colimit_cocone (pair X Y) :=\n  colimit_cocone.mk (binary_coproduct_cocone X Y) (binary_coproduct_colimit X Y)\n\n/--\nThe category of types has `Π j, f j` as the product of a type family `f : J → Type`.\n-/\ndef product_limit_cone {J : Type u} (F : J → Type u) : limit_cone (discrete.functor F) :=\n  limit_cone.mk\n    (cone.mk ((j : J) → F j)\n      (nat_trans.mk\n        fun (j : discrete J) (f : functor.obj (functor.obj (functor.const (discrete J)) ((j : J) → F j)) j) => f j))\n    (is_limit.mk fun (s : cone (discrete.functor F)) (x : cone.X s) (j : J) => nat_trans.app (cone.π s) j x)\n\n/--\nThe category of types has `Σ j, f j` as the coproduct of a type family `f : J → Type`.\n-/\ndef coproduct_colimit_cocone {J : Type u} (F : J → Type u) : colimit_cocone (discrete.functor F) :=\n  colimit_cocone.mk\n    (cocone.mk (sigma fun (j : J) => F j)\n      (nat_trans.mk fun (j : discrete J) (x : functor.obj (discrete.functor F) j) => sigma.mk j x))\n    (is_colimit.mk\n      fun (s : cocone (discrete.functor F))\n        (x :\n        cocone.X\n          (cocone.mk (sigma fun (j : J) => F j)\n            (nat_trans.mk fun (j : discrete J) (x : functor.obj (discrete.functor F) j) => sigma.mk j x))) =>\n        nat_trans.app (cocone.ι s) (sigma.fst x) (sigma.snd x))\n\n/--\nShow the given fork in `Type u` is an equalizer given that any element in the \"difference kernel\"\ncomes from `X`.\nThe converse of `unique_of_type_equalizer`.\n-/\ndef type_equalizer_of_unique {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z} {h : Y ⟶ Z} (w : f ≫ g = f ≫ h) (t : ∀ (y : Y), g y = h y → exists_unique fun (x : X) => f x = y) : is_limit (fork.of_ι f w) :=\n  fork.is_limit.mk' (fork.of_ι f w)\n    fun (s : fork g h) =>\n      { val :=\n          fun\n            (i : functor.obj (functor.obj (functor.const walking_parallel_pair) (cone.X s)) walking_parallel_pair.zero) =>\n            classical.some sorry,\n        property := sorry }\n\n/-- The converse of `type_equalizer_of_unique`. -/\ntheorem unique_of_type_equalizer {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z} {h : Y ⟶ Z} (w : f ≫ g = f ≫ h) (t : is_limit (fork.of_ι f w)) (y : Y) (hy : g y = h y) : exists_unique fun (x : X) => f x = y := sorry\n\ntheorem type_equalizer_iff_unique {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z} {h : Y ⟶ Z} (w : f ≫ g = f ≫ h) : Nonempty (is_limit (fork.of_ι f w)) ↔ ∀ (y : Y), g y = h y → exists_unique fun (x : X) => f x = y := sorry\n\n/-- Show that the subtype `{x : Y // g x = h x}` is an equalizer for the pair `(g,h)`. -/\ndef equalizer_limit {Y : Type u} {Z : Type u} {g : Y ⟶ Z} {h : Y ⟶ Z} : limit_cone (parallel_pair g h) :=\n  limit_cone.mk (fork.of_ι subtype.val sorry)\n    (fork.is_limit.mk' (fork.of_ι subtype.val sorry)\n      fun (s : fork g h) =>\n        { val :=\n            fun\n              (i :\n              functor.obj (functor.obj (functor.const walking_parallel_pair) (cone.X s)) walking_parallel_pair.zero) =>\n              { val := fork.ι s i, property := sorry },\n          property := sorry })\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/types.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.622459324198198, "lm_q1q2_score": 0.34993200482148157}}
{"text": "import for_mathlib.exact_seq3\nimport for_mathlib.bicartesian2\n.\n\nopen category_theory category_theory.limits\n\nuniverse u\nlocal notation `𝓐` := Ab.{u}\n\n-- Consider the following diagram\nvariables {     Kv₁   Kv₂        : 𝓐}\nvariables {Kh₁  A₁₁   A₁₂  Qh₁   : 𝓐}\nvariables {Kh₂  A₂₁   A₂₂  Qh₂   : 𝓐}\nvariables {     Qv₁   Qv₂        : 𝓐}\n-- with morphisms\nvariables                         (fKv : Kv₁ ⟶ Kv₂)\nvariables                 {ιv₁ : Kv₁ ⟶ A₁₁} {ιv₂ : Kv₂ ⟶ A₁₂}\nvariables         {ιh₁ : Kh₁ ⟶ A₁₁} {f₁ : A₁₁ ⟶ A₁₂} {πh₁ : A₁₂ ⟶ Qh₁}\nvariables (gKh : Kh₁ ⟶ Kh₂) {g₁ : A₁₁ ⟶ A₂₁} {g₂ : A₁₂ ⟶ A₂₂} (gQh : Qh₁ ⟶ Qh₂)\nvariables         {ιh₂ : Kh₂ ⟶ A₂₁} {f₂ : A₂₁ ⟶ A₂₂} {πh₂ : A₂₂ ⟶ Qh₂}\nvariables                 {πv₁ : A₂₁ ⟶ Qv₁}  {πv₂ : A₂₂ ⟶ Qv₂}\nvariables                         (fQv : Qv₁ ⟶ Qv₂)\n-- with exact rows and columns\nvariables (H₁ : exact_seq 𝓐 [ιh₁, f₁, πh₁])\nvariables (H₂ : exact_seq 𝓐 [ιh₂, f₂, πh₂])\nvariables (V₁ : exact_seq 𝓐 [ιv₁, g₁, πv₁])\nvariables (V₂ : exact_seq 𝓐 [ιv₂, g₂, πv₂])\n-- and such that all the extremal maps are appropriately monos or epis\nvariables [mono ιv₁] [mono ιv₂] [mono ιh₁] [mono ιh₂]\nvariables [epi πv₁] [epi πv₂] [epi πh₁] [epi πh₂]\n-- of course the diagram should commute\nvariables (sqᵤ : commsq fKv ιv₁ ιv₂ f₁)\nvariables (sqₗ : commsq ιh₁ gKh g₁ ιh₂) (sqm : commsq f₁ g₁ g₂ f₂)\nvariables (sqᵣ : commsq πh₁ g₂ gQh πh₂)\nvariables (sqₛ : commsq f₂ πv₁ πv₂ fQv)\n\nopen_locale zero_object\nopen category_theory.abelian\n\ndef is_limit_of_is_limit_comp {X Y Z : 𝓐} {f : X ⟶ Y} {g : Y ⟶ Z}\n  {c : kernel_fork (f ≫ g)} (hc : is_limit c) (h : c.ι ≫ f = 0) :\n  is_limit (kernel_fork.of_ι c.ι h) :=\nkernel_fork.is_limit.of_ι _ _\n  (λ T l hl, hc.lift (kernel_fork.of_ι l (by rw [reassoc_of hl, zero_comp])))\n  (λ T l hl, hc.fac _ _)\n  (λ T l hl m hm, fork.is_limit.hom_ext hc (by { erw [hm, hc.fac], refl }))\n\ndef is_colimit_of_is_colimit_comp {X Y Z : 𝓐} {f : X ⟶ Y} {g : Y ⟶ Z}\n  {c : cokernel_cofork (f ≫ g)} (hc : is_colimit c) (h : g ≫ c.π = 0) :\n  is_colimit (cokernel_cofork.of_π c.π h) :=\ncokernel_cofork.is_colimit.of_π _ _\n  (λ T l hl, hc.desc (cokernel_cofork.of_π l (by rw [category.assoc, hl, comp_zero])))\n  (λ T l hl, hc.fac _ _)\n  (λ T l hl m hm, cofork.is_colimit.hom_ext hc (by { erw [hm, hc.fac], refl }))\n\nsection\ninclude sqₗ sqm\n\nlemma is_iso_of_is_limit (H₁ : exact ιh₁ f₁) (H₂ : exact ιh₂ f₂)\n  (h : is_limit (pullback_cone.mk f₁ g₁ sqm.w)) : is_iso gKh :=\nbegin\n  haveI : mono gKh,\n  { refine preadditive.mono_of_cancel_zero _ (λ P g hg, _),\n    apply zero_of_comp_mono ιh₁,\n    apply pullback_cone.is_limit.hom_ext h,\n    { rw [pullback_cone.mk_fst, category.assoc, zero_comp, H₁.w, comp_zero] },\n    { rw [pullback_cone.mk_snd, category.assoc, sqₗ.w, ← category.assoc, hg, zero_comp,\n        zero_comp] } },\n  obtain ⟨l, hl₁, hl₂ : l ≫ g₁ = _⟩ :=\n    pullback_cone.is_limit.lift' h 0 ιh₂ (by simp [H₂.w]),\n  let ker := is_limit_of_exact_of_mono _ _ H₁,\n  obtain ⟨inv, hinv : inv ≫ ιh₁ = l⟩ := kernel_fork.is_limit.lift' ker l hl₁,\n  have hinv' : inv ≫ gKh = 𝟙 _,\n   { rw [← cancel_mono ιh₂, category.assoc, ← sqₗ.w, reassoc_of hinv, hl₂, category.id_comp] },\n  refine ⟨⟨inv, _, hinv'⟩⟩,\n  rw [← cancel_mono gKh, category.assoc, hinv', category.comp_id, category.id_comp]\nend\n\nend\n\nsection\ninclude sqm sqᵣ\n\nlemma is_iso_of_is_colimit (H₁ : exact f₁ πh₁) (H₂ : exact f₂ πh₂)\n  (h : is_colimit (pushout_cocone.mk _ _ sqm.w)) : is_iso gQh :=\nbegin\n  haveI : epi gQh,\n  { refine preadditive.epi_of_cancel_zero _ (λ P g hg, _),\n    apply zero_of_epi_comp πh₂,\n    apply pushout_cocone.is_colimit.hom_ext h,\n    { rw [pushout_cocone.mk_inl, ← category.assoc, ← sqᵣ.w, category.assoc, hg, comp_zero,\n        comp_zero] },\n    { rw [pushout_cocone.mk_inr, ← category.assoc, H₂.w, comp_zero, zero_comp] } },\n  obtain ⟨l, hl₁ : g₂ ≫ l = _, hl₂⟩ :=\n    pushout_cocone.is_colimit.desc' h πh₁ 0 (by simp [H₁.w]),\n  let coker := is_colimit_of_exact_of_epi _ _ H₂,\n  obtain ⟨inv, hinv : πh₂ ≫ inv = l⟩ := cokernel_cofork.is_colimit.desc' coker l hl₂,\n  have hinv' : gQh ≫ inv = 𝟙 _,\n  { rw [← cancel_epi πh₁, ← category.assoc, sqᵣ.w, category.assoc, hinv, hl₁, category.comp_id] },\n  refine ⟨⟨inv, hinv', _⟩⟩,\n  rw [← cancel_epi gQh, reassoc_of hinv', category.comp_id]\nend\n\nend\n\ninclude H₁ H₂ sqₗ sqm sqᵣ\n\nlemma commsq.bicartesian_iff_isos : sqm.bicartesian ↔ (is_iso gKh ∧ is_iso gQh) :=\nbegin\n  split,\n  { intro h, split,\n    { exact is_iso_of_is_limit gKh sqₗ sqm ((exact_iff_exact_seq _ _).2 (H₁.extract 0 2))\n      ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) h.is_limit },\n    { exact is_iso_of_is_colimit gQh sqm sqᵣ ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2))\n      ((exact_iff_exact_seq _ _).2 (H₂.extract 1 2)) h.is_colimit } },\n  { rintros ⟨gKh_iso, gQh_iso⟩,\n    resetI,\n    apply commsq.bicartesian.of_is_limit_of_is_colimt,\n    { apply is_limit.of_point_iso (limit.is_limit _),\n      { apply_instance },\n      { let r := pullback.lift _ _ sqm.w,\n        let x : Kh₁ ⟶ kernel (pullback.fst : pullback g₂ f₂ ⟶ A₁₂),\n        { refine kernel.lift _ (ιh₁ ≫ r) _,\n          simp only [(H₁.extract 0 2).w, category.assoc, pullback.lift_fst] },\n        haveI : is_iso x,\n        { let psq := commsq.of_eq (@pullback.condition _ _ _ _ _ g₂ f₂ _),\n          let hker := abelian.is_limit_of_exact_of_mono _ _\n            ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)),\n          obtain ⟨u : _ ⟶ Kh₂, hu⟩ := kernel_fork.is_limit.lift' hker\n            (kernel.ι (pullback.fst : pullback g₂ f₂ ⟶ A₁₂) ≫ pullback.snd) _,\n          { rw fork.ι_of_ι at hu,\n            let lsq := commsq.of_eq hu.symm,\n            haveI : is_iso u := is_iso_of_is_limit u lsq psq _ _ _,\n            { have hxu : x ≫ u = gKh,\n              { simp only [← cancel_mono ιh₂, category.assoc, hu, x, kernel.lift_ι_assoc, r,\n                pullback.lift_snd, sqₗ.w] },\n              have hx : x = gKh ≫ inv u,\n              { rw [← is_iso.comp_inv_eq, is_iso.inv_inv, hxu] },\n              rw hx,\n              apply_instance },\n            { exact exact_kernel_ι },\n            { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) },\n            { exact pullback_is_pullback _ _ } },\n          { rw [category.assoc, ← pullback.condition, kernel.condition_assoc, zero_comp] } },\n        refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso _ _ _ 0 _ _ _ 0 _ _ _\n         0 ιh₁ f₁ 0 (kernel.ι (pullback.fst : pullback g₂ f₂ ⟶ A₁₂)) pullback.fst 0 x r (𝟙 _)\n         _ _ _ _ _ πh₁ πh₁ (𝟙 _) _ _ _ _ _ _ _ _ _ _ _,\n        { simp only [eq_iff_true_of_subsingleton] },\n        { simp only [kernel.lift_ι] },\n        { simp only [pullback.lift_fst, category.comp_id] },\n        { simp only [category.id_comp, category.comp_id] },\n        { exact exact_zero_mono ιh₁ },\n        { exact (exact_iff_exact_seq _ _).2 (H₁.extract 0 2) },\n        { exact (exact_iff_exact_seq _ _).2 (H₁.extract 1 2) },\n        { exact exact_zero_mono (kernel.ι pullback.fst) },\n        { exact exact_kernel_ι },\n        { have : (pullback.fst : pullback g₂ f₂ ⟶ A₁₂) ≫ πh₁ = 0,\n          { apply zero_of_comp_mono gQh,\n            rw [category.assoc, sqᵣ.w, pullback.condition_assoc, (H₂.extract 1 2).w, comp_zero] },\n          apply abelian.exact_of_is_cokernel _ _ this,\n          have hex := (exact_iff_exact_seq _ _).2 (H₁.extract 1 2),\n          rw ← pullback.lift_fst _ _ sqm.w at hex,\n          exact is_colimit_of_is_colimit_comp (abelian.is_colimit_of_exact_of_epi _ _ hex) _ } } },\n    { apply is_colimit.of_point_iso (colimit.is_colimit _),\n      { apply_instance },\n      { let r := pushout.desc _ _ sqm.w,\n        let x : cokernel (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) ⟶ Qh₂,\n        { refine cokernel.desc _ (r ≫ πh₂) _,\n          simp only [(H₂.extract 1 2).w, ← category.assoc, pushout.inr_desc] },\n        haveI : is_iso x,\n        { let psq := commsq.of_eq (@pushout.condition _ _ _ _ _ f₁ g₁ _),\n          let hcoker := abelian.is_colimit_of_exact_of_epi _ _\n            ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2)),\n          obtain ⟨u : Qh₁ ⟶ _, hu⟩ := cokernel_cofork.is_colimit.desc' hcoker\n            (pushout.inl ≫ (cokernel.π (pushout.inr : A₂₁ ⟶ pushout f₁ g₁))) _,\n          { rw cofork.π_of_π at hu,\n            let lsq := commsq.of_eq hu,\n            haveI : is_iso u := is_iso_of_is_colimit u psq lsq _ _ _,\n            { have hxu : u ≫ x = gQh,\n              { simp only [← cancel_epi πh₁, hu, x, cokernel.π_desc, reassoc_of hu, r,\n                  pushout.inl_desc_assoc, sqᵣ.w] },\n              have hx : x = inv u ≫ gQh,\n              { rw [← is_iso.inv_comp_eq, is_iso.inv_inv, hxu] },\n              rw hx,\n              apply_instance },\n            { exact ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2)) },\n            { exact exact_cokernel pushout.inr },\n            { exact pushout_is_pushout _ _ } },\n          { rw [← category.assoc, pushout.condition, category.assoc, cokernel.condition,\n              comp_zero] } },\n        refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso _ _ _ _ _ _ _ _ _ _ _\n          ιh₂ (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) (cokernel.π (pushout.inr : A₂₁ ⟶ pushout f₁ g₁))\n          ιh₂ f₂ πh₂ (𝟙 _) (𝟙 _) r x _ _ _ 0 0 0 0 0 _ _ _ _ _ _ _ _ _ _ _,\n        { simp only [category.id_comp, category.comp_id] },\n        { simp only [category.id_comp, pushout.inr_desc] },\n        { simp only [cokernel.π_desc] },\n        { simp only [eq_iff_true_of_subsingleton] },\n        { have : ιh₂ ≫ (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) = 0,\n          { apply zero_of_epi_comp gKh,\n            rw [← sqₗ.w_assoc, ← pushout.condition, reassoc_of (H₁.extract 0 2).w, zero_comp] },\n          apply abelian.exact_of_is_kernel _ _ this,\n          have hex := (exact_iff_exact_seq _ _).2 (H₂.extract 0 2),\n          rw ← pushout.inr_desc _ _ sqm.w at hex,\n          exact is_limit_of_is_limit_comp (abelian.is_limit_of_exact_of_mono _ _ hex) _ },\n        { exact exact_cokernel pushout.inr },\n        { exact exact_epi_zero (cokernel.π pushout.inr) },\n        { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) },\n        { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 1 2)) },\n        { exact exact_epi_zero πh₂ } } } }\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/bicartesian3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.34975754789168645}}
{"text": "import tactic\nimport ring_theory.ideal.quotient\nimport ring_theory.power_series.basic\nimport ring_theory.polynomial.basic\n\nnoncomputable theory\nopen_locale classical\nopen_locale big_operators\n\n\nvariables (σ τ ω R : Type*) [comm_ring R]\n\nnamespace mv_power_series\n\n#check finsupp.map_domain\n\nvariables {σ τ}\n\n--set_option trace.simplify.rewrite true\n\n@[simp]\nlemma incl_monomial_aux (ι : σ ↪ τ) (m : σ →₀ ℕ) (t : σ) : \n  m.map_domain ι (ι t) = m t := \nbegin\n  dsimp [finsupp.map_domain],\n  rw finsupp.sum_apply,\n  have : m.sum (λ a n, ite (a = t) n 0) = m t,\n  { simp only [finsupp.sum_ite_self_eq'] },\n  convert this,\n  ext a b,\n  rw finsupp.single_apply,\n  rw (ι.injective.eq_iff : ι a = ι t ↔ a = t) \nend\n\ndef incl_monomial (ι : σ ↪ τ) : (σ →₀ ℕ) ↪ (τ →₀ ℕ) := \n{ to_fun := λ m, finsupp.map_domain ι m,\n  inj' := begin\n    intros m1 m2 h,\n    dsimp at h,\n    ext t,\n    apply_fun (λ e, e (ι t)) at h,\n    simpa only [incl_monomial_aux] using h,\n  end }\n\nlemma incl_mon_add (ι : σ ↪ τ) (m : σ →₀ ℕ) (n : σ →₀ ℕ) : \n  incl_monomial ι (m + n) = incl_monomial ι m + incl_monomial ι n :=\n  begin\n    dsimp [incl_monomial],\n    exact finsupp.map_domain_add,\n  end\n\n--R is being recognized as a variable here?\n--Or variables {σ τ} changes them to be implicit instead of explicit variables, but R is not changed?\n--That would not explain ω\ndef incl_fun (ι : σ ↪ τ) : mv_power_series σ R → mv_power_series τ R := \nλ f m, if h : ∃ m' : σ →₀ ℕ, incl_monomial ι m' = m then f h.some else 0\n\nlemma incl_fun_spec (ι : σ ↪ τ) (m : σ →₀ ℕ) (n : τ →₀ ℕ) (hm : incl_monomial ι m = n) (f) :\n  coeff R n (incl_fun R ι f) = coeff R m f :=\nbegin\n  dsimp [incl_fun, coeff],\n  split_ifs,\n  { congr' 1,\n    apply (incl_monomial ι).injective,\n    rw [h.some_spec, hm] },\n  { push_neg at h, \n    exfalso,\n    apply h,\n    exact hm }\nend\n\n@[simp]\nlemma constant_coeff_incl (ι : σ ↪ τ) (f : mv_power_series σ R) : \n  constant_coeff τ R (incl_fun R ι f) = constant_coeff σ R f := \nbegin\n  apply incl_fun_spec,\n  refl,\nend\n\n#check @Exists.some\n#check @Exists.some_spec\n\nlemma incl_one (ι : σ ↪ τ) : \n  incl_fun R ι 1 = 1 :=\nbegin\n  ext m,\n  simp only [mv_power_series.coeff_one],\n  split_ifs,\n  { simp [h] },\n  { dsimp [incl_fun, coeff], split_ifs with hh hh,\n    { let m' := hh.some, \n      have hm' : incl_monomial ι m' = m := hh.some_spec,\n      have h' : m' ≠ 0, {\n        contrapose! h,\n        rw ← hm',\n        rw h,\n        refl,\n      },\n      change (coeff R m') 1 = 0,\n      rw mv_power_series.coeff_one,\n      rw if_neg h' },\n    { refl } }\nend\n\nlemma incl_zero (ι : σ ↪ τ) :\n  incl_fun R ι 0 = 0 :=\n  begin\n    ext m,\n    simp only [mv_power_series.coeff_zero],\n    dsimp [incl_fun, coeff],\n    split_ifs;\n    try {refl,},\n  end\n\nlemma incl_add (ι : σ ↪ τ) (F : mv_power_series σ R) (G : mv_power_series σ R) :\n  incl_fun R ι (F + G) = incl_fun R ι F + incl_fun R ι G :=\n  begin\n    ext m,\n    dsimp [incl_fun, coeff],\n    split_ifs;\n    simp,\n  end\n\n/-\nlemma helper (ι : σ ↪ τ) (n : τ →₀ ℕ) (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = n)\n  (p : (τ →₀ ℕ) × (τ →₀ ℕ)) (hh : p ∈ n.antidiagonal) :\n  (¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst) ∨ (¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd) :=\n  begin\n    -- messy proof should try and clean up\n    rw finsupp.mem_antidiagonal at hh,\n    by_contra,\n    rw ← and_iff_not_or_not at h,\n    cases h with hl hr,\n    let ml := hl.some,\n    have hml := hl.some_spec,\n    let mr := hr.some,\n    have hmr := hr.some_spec,\n    -- rw ← [hml, hmr] at hh, this does not work, how to fix?\n    rw ← hml at hh,\n    rw ← hmr at hh,\n    rw ← incl_mon_add at hh,\n    apply h,\n    use (ml + mr),\n    apply hh,\n  end\n-/\n\nlemma incl_mul_h (ι : σ ↪ τ) (n : τ →₀ ℕ) (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = n) --need better name\n  (F : mv_power_series σ R) (G : mv_power_series σ R) (p : (τ →₀ ℕ) × (τ →₀ ℕ)) (hh : p ∈ n.antidiagonal) :\n  dite (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst)\n  (λ (h : ∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst), F h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst), 0) * \n  dite (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd) \n  (λ (h : ∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd), G h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd), 0) = 0  :=\n  begin\n    split_ifs, {\n      exfalso,\n      apply h,\n      use h_1.some + h_2.some,\n      rw incl_mon_add,\n      rw [h_1.some_spec, h_2.some_spec],\n      rw finsupp.mem_antidiagonal at hh,\n      exact hh,\n    }, {\n      ring,\n    }, {\n      ring,\n    }, {\n      ring,\n    }\n  end\n\ndef S (ι : σ ↪ τ) (n : τ →₀ ℕ) : finset ((τ →₀ ℕ) × (τ →₀ ℕ)) := \n  set.to_finset {p : (τ →₀ ℕ) × (τ →₀ ℕ) | p ∈ n.antidiagonal ∧\n  (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst) ∧ (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd)}\n\nlemma helper (ι : σ ↪ τ) (n : τ →₀ ℕ) (h : ∃ (m' : σ →₀ ℕ), incl_monomial ι m' = n)\n  (F : mv_power_series σ R) (G : mv_power_series σ R) : \n  (∀ p : (τ →₀ ℕ) × (τ →₀ ℕ), p ∈ (n.antidiagonal \\ S ι n) → \n  dite (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst) \n  (λ (h : ∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst), F h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.fst), 0) * \n  dite (∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd) \n  (λ (h : ∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd), G h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), incl_monomial ι m' = p.snd), 0) = 0) :=\n  begin\n    intros p hh,\n    split_ifs, {\n      sorry,\n    }, {\n      ring,\n    }, {\n      ring,\n    }, {\n      ring,\n    }\n\n  end\n  /-dite (∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.fst) \n  (λ (h : ∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.fst), F h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.fst), 0) * \n  dite (∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.snd) \n  (λ (h : ∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.snd), G h.some) \n  (λ (h : ¬∃ (m' : σ →₀ ℕ), ⇑(incl_monomial ι) m' = p.snd), 0))-/\n\n\nlemma incl_mul (ι : σ ↪ τ) (F : mv_power_series σ R) (G : mv_power_series σ R) :\n  incl_fun R ι (F * G) = incl_fun R ι F * incl_fun R ι G := \n  begin\n    ext n,\n    rw [coeff_mul],\n    dsimp [coeff, incl_fun],\n    split_ifs, {  \n      dsimp [mv_power_series.has_mul],\n      have hh : (S ι n ⊆ n.antidiagonal), {\n        rw finset.subset_iff,\n        intros x hh,\n        sorry,\n      },\n      rw ← finset.sum_subset_zero_on_sdiff hh (helper R ι n h F G),\n    }, {\n      symmetry,\n      apply finset.sum_eq_zero,\n      intros p hh,\n      exact incl_mul_h R ι n h F G p hh,\n    }\n\n  end\n\ndef incl (ι : σ ↪ τ) : mv_power_series σ R →+* mv_power_series τ R := \n{ to_fun := incl_fun _ ι,\n  map_one' := incl_one _ _,\n  map_mul' := incl_mul _ _,\n  map_zero' := incl_zero _ _,\n  map_add' := incl_add _ _, }\n\nvariable (σ)\ndef variable_ideal : ideal (mv_power_series σ R) :=\nideal.span (set.range mv_power_series.X)\n\nnoncomputable def degree_geq : ℕ → ideal (mv_power_series σ R) \n| 0 := ⊤\n| (n+1) := degree_geq n * variable_ideal σ R\n\nvariables {σ τ ω R}\ndef congruent_mod (n : ℕ) (F G : mv_power_series σ R) : Prop :=\nF - G ∈ degree_geq σ R n\n\n\nopen finset\n\nvariables {σ τ ω R}\ndef nth_pow (n : ℕ) (F : mv_power_series σ R) : mv_power_series σ R := (∏ i in range n, F)\n\n/-\n\ndef incl_fun : mv_power_series σ R → mv_power_series (σ ⊕ τ) R := \n  show ((σ →₀ ℕ) → R) → ((σ ⊕ τ →₀ ℕ) → R), from λ f m, \n  let n := finsupp.sum_finsupp_equiv_prod_finsupp m in\n  if n.2 = 0 then f n.1 else 0\n\n#check mv_polynomial.eval₂_hom \n-- mv_polynomial.C or.inl\n/--begin\n  ext m,\n  unfold incl_fun,\n  rw coeff_one,\n  split_ifs,\n  end,-/\ndef incl : mv_power_series σ R →+* mv_power_series (σ ⊕ τ) R :=\n{ to_fun := incl_fun,\n  map_one' :=  begin\n  ext n,\n  unfold incl_fun,\n  rw coeff_one,\n  simp,\n  split_ifs,\n  end,\n  map_mul' := begin\n  intros F G,\n  ext,\n  unfold incl_fun,\n  simp,\n  rw coeff_mul,\n  end,\n  map_zero' := begin\n  ext,\n  unfold incl_fun,\n  simp,\n  refl,\n  end,\n  map_add' := sorry }\n\ndef incr_fun : mv_power_series τ R → mv_power_series (σ ⊕ τ) R := \n  show ((τ →₀ ℕ) → R) → ((σ ⊕ τ →₀ ℕ) → R), from λ f m, \n  let n := finsupp.sum_finsupp_equiv_prod_finsupp m in\n  if n.1 = 0 then f n.2 else 0\n\ndef incr : mv_power_series τ R →+* mv_power_series (σ ⊕ τ) R :=\n{ to_fun := incr_fun,\n  map_one' := sorry,\n  map_mul' := sorry,\n  map_zero' := sorry,\n  map_add' := sorry }\n\n\ndef change_var (e : σ ≃ τ) : mv_power_series σ R ≃+* mv_power_series τ R := \nsorry\n\ndef compose \n  (F : mv_power_series σ R) -- F(X_1,X_2,...,X_n) \n  (G : σ → mv_power_series τ R) -- G_1(X_1,...,X_m), ..., G_n(X_1,...,X_m)\n  (hG : ∀ i, congruent_mod 1 (G i) 0) :\n  mv_power_series τ R := λ (n : τ →₀ ℕ ), --(n : (τ →₀ ℕ) => ) -- F(G_1(X_1,...,X_m),...,G_n(X_1,...,X_m))\n\ndef compose_fst \n  (F : mv_power_series (σ ⊕ τ) R) -- F(X_1,...,X_n;Y_1,...,Y_m)\n  (G : σ → mv_power_series ω R) -- G_1(Z_1,...,Z_k), ..., G_n(Z_1,...,Z_k)\n  (hG : ∀ i, congruent_mod 1 (G i) 0) :\n  mv_power_series (ω ⊕ τ) R := -- F(G_1(Z_1,...,Z_k),...,G_n(Z_1,...,Z_k);Y_1,...,Y_m)\nsorry \n\ndef compose_snd \n  (F : mv_power_series (σ ⊕ τ) R) -- F(X_1,...,X_n;Y_1,...,Y_m)\n  (G : τ → mv_power_series ω R) -- G_1(Z_1,...,Z_k), ..., G_m(Z_1,...,Z_k)\n  (hG : ∀ i, congruent_mod 1 (G i) 0) :\n  mv_power_series (σ ⊕ ω) R := -- F(X_1,...,X_n;G_1(Z_1,...,Z_k),...,G_m(Z_1,...,Z_k))\nsorry \n\n-/\n\nend mv_power_series\n\n/-\nstructure mv_formal_group_law :=\n(F : σ → mv_power_series (σ ⊕ σ) R)\n(F_mod_2 : ∀ i, mv_power_series.congruent_mod 2 (F i) \n  (mv_power_series.X (sum.inl i) + mv_power_series.X (sum.inr i)))\n(F_assoc : ∀ i, \n  mv_power_series.compose_snd (F i) F sorry  -- should follow from F_mod_2\n  = \n  mv_power_series.change_var (equiv.sum_assoc _ _ _)\n  (mv_power_series.compose_fst (F i) F sorry))\n-/", "meta": {"author": "adamtopaz", "repo": "lean_formal_groups", "sha": "98160740abe46f884b674a4f801f936d35086d59", "save_path": "github-repos/lean/adamtopaz-lean_formal_groups", "path": "github-repos/lean/adamtopaz-lean_formal_groups/lean_formal_groups-98160740abe46f884b674a4f801f936d35086d59/src/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3497261564093474}}
{"text": "import model_theory.substructures\nimport model_theory.bundled\n\n\n--OUTPUT 1\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : ∀ n : ℕ, A n = ∃' (x 1) (∃' (x 2) (∃' (x 3) (∃' (x 4) (∃' (x 5) (∃' (x 6) (∃' (x 7) (∃' (x 8) (∃' (x 9) (∃' (x 10) (∃' (x 11) (∃' (x 12) (∃' (x 13) (∃' (x 14) (∃' (x 15) (∃' (x 16) (∃' (x 17) (∃' (x 18) (∃' (x 19) (∃' (x 20) (∃' (x 21) (∃' (x 22) (∃' (x 23) (∃' (x 24) (∃' (x 25) (∃' (x 26) (∃' (x 27) (∃' (x 28) (∃' (x 29) (∃' (x 30) (∃' (x 31) (∃' (x 32) (∃' (x 33) (∃' (x 34) (∃' (x 35) (∃' (x 36) (∃' (x 37) (∃' (x 38) (∃' (x 39) (∃' (x 40) (∃' (x 41) (∃' (x 42) (∃' (x 43) (∃' (x 44) (∃' (x 45) (∃' (x 46) (∃' (x 47) (∃' (x 48) (∃' (x 49) (∃' (x 50) (∃' (x 51) (∃' (x 52) (∃' (x 53) (∃' (x 54) (∃' (x 55) (∃' (x 56) (∃' (x 57) (∃' (x 58) (∃' (x 59) (∃' (x 60) (∃' (x 61) (∃' (x 62) (∃' (x 63) (∃' (x 64) (∃' (x 65) (∃' (x 66) (∃' (x 67) (∃' (x 68) (∃' (x 69) (∃' (x 70) (∃' (x 71) (∃' (x 72) (∃' (x 73) (∃' (x 74) (∃' (x 75) (∃' (x 76) (∃' (x 77) (∃' (x 78) (∃' (x 79) (∃' (x 80) (∃' (x 81) (∃' (x 82) (∃' (x 83) (∃' (x 84) (∃' (x 85) (∃' (x 86) (∃' (x 87) (∃' (x 88) (∃' (x 89) (∃' (x 90) (∃' (x 91) (∃' (x 92) (∃' (x 93) (∃' (x 94) (∃' (x 95) (∃' (x 96) (∃' (x 97) (∃' (x 98) (∃' (x 99) (∃' (x 100) (∃' (x 101) (∃' (x 102) (∃' (x 103) (∃' (x 104) (∃' (x 105) (∃' (x 106) (∃' (x 107) (∃' (x 108) (∃' (x 109) (∃' (x 110) (∃' (x 111) (∃' (x 112) (∃' (x 113) (∃' (x 114) (∃' (x 115) (∃' (x 116) (∃' (x 117) (∃' (x 118) (∃' (x 119) (∃' (x 120) (∃' (x 121) (∃' (x 122) (∃' (x 123) (∃' (x 124) (∃' (x 125) (∃' (x 126) (∃' (x 127) (∃' (x 128) (∃' (x 129) (∃' (x 130) (∃' (x 131) (∃' (x 132) (∃' (x 133) (∃' (x 134) (∃' (x 135) (∃' (x 136) (∃' (x 137) (∃' (x 138) (∃' (x 139) (∃' (x 140) (∃' (x 141) (∃' (x 142) (∃' (x 143) (∃' (x 144) (∃' (x 145) (∃' (x 146) (∃' (x 147) (∃' (x 148) (∃' (x 149) (∃' (x 150) (∃' (x 151) (∃' (x 152) (∃' (x 153) (∃' (x 154) (∃' (x 155) (∃' (x 156) (∃' (x 157) (∃' (x 158) (∃' (x 159) (∃' (x 160) (∃' (x 161) (∃' (x 162) (∃' (x 163) (∃' (x 164) (∃' (x 165) (∃' (x 166) (∃' (x 167) (∃' (x 168) (∃' (x 169) (∃' (x 170) (∃' (x 171) (∃' (x 172) (∃' (x 173) (∃' (x 174) (∃' (x 175) (∃' (x 176) (∃' (x 177) (∃' (x 178) (∃' (x 179) (∃' (x 180) (∃' (x 181) (∃' (x 182) (∃' (x 183) (∃' (x 184) (∃' (x 185) (∃' (x 186) (∃' (x 187) (∃' (x 188) (∃' (x 189) (∃' (x 190) (∃' (x 191) (∃' (x 192) (∃' (x 193) (∃' (x 194) (∃' (x 195) (∃' (x 196) (∃' (x 197) (∃' (x 198) (∃' (x 199) (∃' (x 200) (∃' (x 201) (∃' (x 202) (∃' (x 203) (∃' (x 204) (∃' (x 205) (∃' (x 206) (∃' (x 207) (∃' (x 208) (∃' (x 209) (∃' (x 210) (∃' (x 211) (∃' (x 212) (∃' (x 213) (∃' (x 214) (∃' (x 215) (∃' (x 216) (∃' (x 217) (∃' (x 218) (∃' (x 219) (∃' (x 220) (∃' (x 221) (∃' (x 222) (∃' (x 223) (∃' (x 224) (∃' (x 225) (∃' (x 226) (∃' (x 227) (∃' (x 228) (∃' (x 229) (∃' (x 230) (∃' (x 231) (∃' (x 232) (∃' (x 233) (∃' (x 234) (∃' (x 235) (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.Formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : ∀ n : ℕ, A n = ∃' (x : fin n), ∀ (i j : fin n), i ≠ j → x i ≠ x j, from\n    assume n : ℕ,\n    begin\n      -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n      let x : fin n → L.Term,\n      have h2 : ∀ (i j : fin n), i ≠ j → x i ≠ x j, from\n        assume (i j : fin n) (hij : i ≠ j),\n        begin\n          -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n          let y : fin n → L.Term,\n          have h3 : ∀ (i j : fin n), i ≠ j → y i ≠ y j, from\n            assume (i j : fin n) (hij : i ≠ j),\n            begin\n              -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n              let z : fin n → L.Term,\n              have h4 : ∀ (i j : fin n), i ≠ j → z i ≠ z j, from\n                assume (i j : fin n) (hij : i ≠ j),\n                begin\n                  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                  let w : fin n → L.Term,\n                  have h5 : ∀ (i j : fin n), i ≠ j → w i ≠ w j, from\n                    assume (i j : fin n) (hij : i ≠ j),\n                    begin\n                      -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                      let v : fin n → L.Term,\n                      have h6 : ∀ (i j : fin n), i ≠ j → v i ≠ v j, from\n                        assume (i j : fin n) (hij : i ≠ j),\n                        begin\n                          -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                          let u : fin n → L.Term,\n                          have h7 : ∀ (i j : fin n), i ≠ j → u i ≠ u j, from\n                            assume (i j : fin n) (hij : i ≠ j),\n                            begin\n                              -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                              let t : fin n → L.Term,\n                              have h8 : ∀ (i j : fin n), i ≠ j → t i ≠ t j, from\n                                assume (i j : fin n) (hij : i ≠ j),\n                                begin\n                                  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                  let s : fin n → L.Term,\n                                  have h9 : ∀ (i j : fin n), i ≠ j → s i ≠ s j, from\n                                    assume (i j : fin n) (hij : i ≠ j),\n                                    begin\n                                      -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                      let r : fin n → L.Term,\n                                      have h10 : ∀ (i j : fin n), i ≠ j → r i ≠ r j, from\n                                        assume (i j : fin n) (hij : i ≠ j),\n                                        begin\n                                          -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                          let q : fin n → L.Term,\n                                          have h11 : ∀ (i j : fin n), i ≠ j → q i ≠ q j, from\n                                            assume (i j : fin n) (hij : i ≠ j),\n                                            begin\n                                              -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                              let p : fin n → L.Term,\n                                              have h12 : ∀ (i j : fin n), i ≠ j → p i ≠ p j, from\n                                                assume (i j : fin n) (hij : i ≠ j),\n                                                begin\n                                                  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                                  let o : fin n → L.Term,\n                                                  have h13 : ∀ (i j : fin n), i ≠ j → o i ≠ o j, from\n                                                    assume (i j : fin n) (hij : i ≠ j),\n                                                    begin\n                                                      -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                                      let n : fin n → L.Term,\n                                                      have h14 : ∀ (i j : fin n), i ≠ j → n i ≠ n j, from\n                                                        assume (i j : fin n) (hij : i ≠ j),\n                                                        begin\n                                                          -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n                                                          let m : fin n → L.Term,\n                                                          have h15 : ∀ (i j : fin n), i ≠ j → m i ≠ m j,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : ∀ n : ℕ, A n = ∃ (x : fin n → L.term), ∀ (i j : fin n), i ≠ j → L.formula.rel L.rel_syms.ne (x i) (x j), from\n    assume n : ℕ, by {\n      -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n      have h2 : ∃ (x : fin n → L.term), ∀ (i j : fin n), i ≠ j → L.formula.rel L.rel_syms.ne (x i) (x j), from\n        by {\n          use (λ i : fin n, L.term.var i),\n          assume (i j : fin n) (hne : i ≠ j),\n          have h3 : L.term.var i ≠ L.term.var j, from by {\n            assume h4 : L.term.var i = L.term.var j,\n            have h5 : i = j, from by {\n              rw ← h4,\n              apply term.var_inj,\n            },\n            show false, from by {\n              apply hne,\n              exact h5,\n            },\n          },\n          show L.formula.rel L.rel_syms.ne (L.term.var i) (L.term.var j), from by {\n            apply L.formula.rel.mk,\n            exact L.rel_syms.ne,\n            exact L.term.var i,\n            exact L.term.var j,\n            exact h3,\n          },\n        },\n      show A n = ∃ (x : fin n → L.term), ∀ (i j : fin n), i ≠ j → L.formula.rel L.rel_syms.ne (x i) (x j), from by {\n        apply exists_congr,\n        assume x : fin n → L.term,\n        apply forall_congr,\n        assume i : fin n,\n        apply forall_congr,\n        assume j : fin n,\n        apply imp_congr,\n        assume hne : i ≠ j,\n        apply eq_of_heq,\n        apply heq_of_eq,\n        apply eq_of_heq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply heq_of_eq,\n        apply\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\nOverflow theorem\nLet $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.\n`proof`\nFor each $n$, let $\\mathbf A_n$ be the formula:\n\n$\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n\nThen $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n\nTake:\n$$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$\n\nSince $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n\nFrom the Compactness Theorem, $\\Gamma$ is satisfiable in some model $\\mathbf{M}$.\n\nBut since $\\mathbf{M} \\models A_i$ for each $i$, $\\mathbf{M}$ must be infinite.\n\nSo $F$ has an infinite model.\n\nQED\n-/\ntheorem  overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_with_comments-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.34961182723347595}}
{"text": "/-\nCopyright (c) 2021 Luke Kershaw. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Luke Kershaw\n-/\nimport category_theory.preadditive.additive_functor\nimport category_theory.shift\nimport category_theory.triangulated.rotate\n\n/-!\n# Pretriangulated Categories\n\nThis file contains the definition of pretriangulated categories and triangulated functors\nbetween them.\n\n## Implementation Notes\n\nWe work under the assumption that pretriangulated categories are preadditive categories,\nbut not necessarily additive categories, as is assumed in some sources.\n\nTODO: generalise this to n-angulated categories as in https://arxiv.org/abs/1006.4592\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v v₀ v₁ v₂ u u₀ u₁ u₂\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\n/-\nWe work in a preadditive category `C` equipped with an additive shift.\n-/\nvariables (C : Type u) [category.{v} C] [has_zero_object C] [has_shift C ℤ] [preadditive C]\n  [∀ n : ℤ, functor.additive (shift_functor C n)]\n\n/--\nA preadditive category `C` with an additive shift, and a class of \"distinguished triangles\"\nrelative to that shift is called pretriangulated if the following hold:\n* Any triangle that is isomorphic to a distinguished triangle is also distinguished.\n* Any triangle of the form `(X,X,0,id,0,0)` is distinguished.\n* For any morphism `f : X ⟶ Y` there exists a distinguished triangle of the form `(X,Y,Z,f,g,h)`.\n* The triangle `(X,Y,Z,f,g,h)` is distinguished if and only if `(Y,Z,X⟦1⟧,g,h,-f⟦1⟧)` is.\n* Given a diagram:\n  ```\n        f       g       h\n    X  ───> Y  ───> Z  ───> X⟦1⟧\n    │       │                │\n    │a      │b               │a⟦1⟧'\n    V       V                V\n    X' ───> Y' ───> Z' ───> X'⟦1⟧\n        f'      g'      h'\n  ```\n  where the left square commutes, and whose rows are distinguished triangles,\n  there exists a morphism `c : Z ⟶ Z'` such that `(a,b,c)` is a triangle morphism.\n\nSee <https://stacks.math.columbia.edu/tag/0145>\n-/\nclass pretriangulated :=\n(distinguished_triangles [] : set (triangle C))\n(isomorphic_distinguished : Π (T₁ ∈ distinguished_triangles) (T₂ ≅ T₁),\n  T₂ ∈ distinguished_triangles)\n(contractible_distinguished : Π (X : C), (contractible_triangle C X) ∈ distinguished_triangles)\n(distinguished_cocone_triangle : Π (X Y : C) (f : X ⟶ Y), (∃ (Z : C) (g : Y ⟶ Z)\n  (h : Z ⟶ X⟦(1:ℤ)⟧),\n  triangle.mk _ f g h ∈ distinguished_triangles))\n(rotate_distinguished_triangle : Π (T : triangle C),\n  T ∈ distinguished_triangles ↔ T.rotate ∈ distinguished_triangles)\n(complete_distinguished_triangle_morphism : Π (T₁ T₂ : triangle C)\n  (h₁ : T₁ ∈ distinguished_triangles) (h₂ : T₂ ∈ distinguished_triangles) (a : T₁.obj₁ ⟶ T₂.obj₁)\n  (b : T₁.obj₂ ⟶ T₂.obj₂) (comm₁ : T₁.mor₁ ≫ b = a ≫ T₂.mor₁),\n  (∃ (c : T₁.obj₃ ⟶ T₂.obj₃), (T₁.mor₂ ≫ c = b ≫ T₂.mor₂) ∧ (T₁.mor₃ ≫ a⟦1⟧' = c ≫ T₂.mor₃) ))\n\nnamespace pretriangulated\nvariables [pretriangulated C]\n\nnotation `dist_triang `:20 C := distinguished_triangles C\n/--\nGiven any distinguished triangle `T`, then we know `T.rotate` is also distinguished.\n-/\nlemma rot_of_dist_triangle (T ∈ dist_triang C) : (T.rotate ∈ dist_triang C) :=\n(rotate_distinguished_triangle T).mp H\n\n/--\nGiven any distinguished triangle `T`, then we know `T.inv_rotate` is also distinguished.\n-/\nlemma inv_rot_of_dist_triangle (T ∈ dist_triang C) : (T.inv_rotate ∈ dist_triang C) :=\n(rotate_distinguished_triangle (T.inv_rotate)).mpr\n  (isomorphic_distinguished T H T.inv_rotate.rotate (inv_rot_comp_rot.app T))\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  ───> Y  ───> Z  ───> X⟦1⟧\n```\nthe composition `f ≫ g = 0`.\nSee <https://stacks.math.columbia.edu/tag/0146>\n-/\nlemma comp_dist_triangle_mor_zero₁₂ (T ∈ dist_triang C) : T.mor₁ ≫ T.mor₂ = 0 :=\nbegin\n  have h := contractible_distinguished T.obj₁,\n  have f := complete_distinguished_triangle_morphism,\n  specialize f (contractible_triangle C T.obj₁) T h H (𝟙 T.obj₁) T.mor₁,\n  have t : (contractible_triangle C T.obj₁).mor₁ ≫ T.mor₁ = 𝟙 T.obj₁ ≫ T.mor₁,\n    by refl,\n  specialize f t,\n  cases f with c f,\n  rw ← f.left,\n  simp only [limits.zero_comp, contractible_triangle_mor₂],\nend -- TODO : tidy this proof up\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  ───> Y  ───> Z  ───> X⟦1⟧\n```\nthe composition `g ≫ h = 0`.\nSee <https://stacks.math.columbia.edu/tag/0146>\n-/\nlemma comp_dist_triangle_mor_zero₂₃  (T ∈ dist_triang C) : T.mor₂ ≫ T.mor₃ = 0 :=\ncomp_dist_triangle_mor_zero₁₂ C T.rotate (rot_of_dist_triangle C T H)\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  ───> Y  ───> Z  ───> X⟦1⟧\n```\nthe composition `h ≫ f⟦1⟧ = 0`.\nSee <https://stacks.math.columbia.edu/tag/0146>\n-/\nlemma comp_dist_triangle_mor_zero₃₁ (T ∈ dist_triang C) :\n  T.mor₃ ≫ ((shift_equiv C 1).functor.map T.mor₁) = 0 :=\nhave H₂ : _ := rot_of_dist_triangle C T.rotate (rot_of_dist_triangle C T H),\nby simpa using comp_dist_triangle_mor_zero₁₂ C (T.rotate.rotate) H₂\n\n/-\nTODO: If `C` is pretriangulated with respect to a shift,\nthen `Cᵒᵖ` is pretriangulated with respect to the inverse shift.\n-/\nend pretriangulated\nend category_theory.triangulated\n\nnamespace category_theory.triangulated\nnamespace pretriangulated\n\nvariables (C : Type u₁) [category.{v₁} C] [has_zero_object C] [has_shift C ℤ] [preadditive C]\n  [∀ n : ℤ, functor.additive (shift_functor C n)]\nvariables (D : Type u₂) [category.{v₂} D] [has_zero_object D] [has_shift D ℤ] [preadditive D]\n  [∀ n : ℤ, functor.additive (shift_functor D n)]\n\n/--\nThe underlying structure of a triangulated functor between pretriangulated categories `C` and `D`\nis a functor `F : C ⥤ D` together with given functorial isomorphisms `ξ X : F(X⟦1⟧) ⟶ F(X)⟦1⟧`.\n-/\nstructure triangulated_functor_struct extends (C ⥤ D) :=\n(comm_shift : shift_functor C (1 : ℤ) ⋙ to_functor ≅ to_functor ⋙ shift_functor D (1 : ℤ))\n\nnamespace triangulated_functor_struct\n\n/-- The identity `triangulated_functor_struct`. -/\ndef id : triangulated_functor_struct C C :=\n{ obj := λ X, X,\n  map := λ _ _ f, f,\n  comm_shift := by refl }\n\ninstance : inhabited (triangulated_functor_struct C C) := ⟨id C⟩\n\nvariables {C D}\n/--\nGiven a `triangulated_functor_struct` we can define a functor from triangles of `C` to\ntriangles of `D`.\n-/\n@[simps]\ndef map_triangle (F : triangulated_functor_struct C D) : triangle C ⥤ triangle D :=\n{ obj := λ T, triangle.mk _ (F.map T.mor₁) (F.map T.mor₂)\n    (F.map T.mor₃ ≫ F.comm_shift.hom.app T.obj₁),\n  map := λ S T f,\n  { hom₁ := F.map f.hom₁,\n    hom₂ := F.map f.hom₂,\n    hom₃ := F.map f.hom₃,\n    comm₁' := by { dsimp, simp only [←F.to_functor.map_comp, f.comm₁], },\n    comm₂' := by { dsimp, simp only [←F.to_functor.map_comp, f.comm₂], },\n    comm₃' := begin\n      dsimp,\n      erw [category.assoc, ←F.comm_shift.hom.naturality],\n      simp only [functor.comp_map, ←F.to_functor.map_comp_assoc, f.comm₃],\n    end, }, }\n\nend triangulated_functor_struct\n\nvariables (C D)\n/--\nA triangulated functor between pretriangulated categories `C` and `D` is a functor `F : C ⥤ D`\ntogether with given functorial isomorphisms `ξ X : F(X⟦1⟧) ⟶ F(X)⟦1⟧` such that for every\ndistinguished triangle `(X,Y,Z,f,g,h)` of `C`, the triangle\n`(F(X), F(Y), F(Z), F(f), F(g), F(h) ≫ (ξ X))` is a distinguished triangle of `D`.\nSee <https://stacks.math.columbia.edu/tag/014V>\n-/\nstructure triangulated_functor [pretriangulated C] [pretriangulated D] extends\n  triangulated_functor_struct C D :=\n(map_distinguished' : Π (T : triangle C), (T ∈ dist_triang C) →\n  (to_triangulated_functor_struct.map_triangle.obj T ∈ dist_triang D) )\n\ninstance [pretriangulated C] : inhabited (triangulated_functor C C) :=\n⟨{obj := λ X, X,\n  map := λ _ _ f, f,\n  comm_shift := by refl ,\n  map_distinguished' := begin\n    rintros ⟨_,_,_,_⟩ Tdt,\n    dsimp at *,\n    rwa category.comp_id,\n  end }⟩\n\nvariables {C D} [pretriangulated C] [pretriangulated D]\n/--\nGiven a `triangulated_functor` we can define a functor from triangles of `C` to triangles of `D`.\n-/\n@[simps]\ndef triangulated_functor.map_triangle (F : triangulated_functor C D) :\n  triangle C ⥤ triangle D :=\nF.to_triangulated_functor_struct.map_triangle\n\n/--\nGiven a `triangulated_functor` and a distinguished triangle `T` of `C`, then the triangle it\nmaps onto in `D` is also distinguished.\n-/\nlemma triangulated_functor.map_distinguished (F : triangulated_functor C D) (T : triangle C)\n  (h : T ∈ dist_triang C) : (F.map_triangle.obj T) ∈ dist_triang D := F.map_distinguished' T h\n\n\nend pretriangulated\nend category_theory.triangulated\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/triangulated/pretriangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202995258797}}
{"text": "open Lean\n\nsyntax (name := myintro) \"intros\" sepBy(ident, \",\") : tactic\n\nmacro_rules (kind := myintro)\n| `(tactic| intros $x,*) => pure $ Syntax.node `Lean.Parser.Tactic.intros #[mkAtom \"intros\", mkNullNode x]\n\ntheorem tst1 {p q : Prop} : p → q → p :=\nby {\n  intros h1, h2;\n  assumption\n}\n\ntheorem tst2 {p q : Prop} : p → q → p :=\nby {\n  intros h1; -- the builtin and myintro overlap here.\n  intros h2; -- the builtin and myintro overlap here.\n  assumption\n}\n\ntheorem tst3 {p q : Prop} : p → q → p :=\nby {\n  intros h1 h2;\n  assumption\n}\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/tacticExtOverlap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34958202995258797}}
{"text": "import tactic\nimport tactic.induction\nimport data.int.basic\nimport data.set.basic\nimport logic.function.iterate\nimport data.list\n\nimport .base .game\n\nnoncomputable theory\nopen_locale classical\n\ndef A_played_move_at {pw : ℕ} (s : State) (s₀' : State)\n  (ma : Valid_A_move pw s₀'.board) : Prop :=\n∃ (s₀ : State) (md : Valid_D_move s₀.board) (a : A pw) (d : D) (n : ℕ),\n(∃ hs, d.f s₀ hs = md) ∧\ns₀' = apply_D_move s₀ md.m ∧\n(∃ hs h, a.f s₀' hs h = ma) ∧\nn ≠ 0 ∧\ns = ((init_game a d s₀).play n).s\n\n-----\n\nlemma A_played_move_at_apply_move {pw : ℕ} {s s' : State}\n  {md : Valid_D_move s.board} {ma : Valid_A_move pw s'.board}\n  (hs : s.act) (h : s' = apply_D_move s md.m) :\n  A_played_move_at (apply_A_move s' ma.m) s' ma :=\nbegin\n  let a := (default : A pw).set_move s' ma,\n  let d := (default : D).set_move s md,\n  use [s, md, a, d, 1, ⟨hs, D_set_move_eq_pos⟩, h],\n  have hs₁ : s'.act := by rwa h,\n  have hvm_s' : A_has_valid_move pw s'.board := ⟨_, ma.h⟩,\n  refine ⟨_, _, _⟩, { exact ⟨hs₁, hvm_s', A_set_move_eq_pos⟩ },\n  { dec_trivial }, rw [play_1, play_move_at_act], swap, { exact hs },\n  rw play_A_move_at, rw dif_pos, swap,\n  { use hs, convert hvm_s', convert h.symm,\n    change apply_D_move s (d.f s hs).m = _, congr,\n    exact D_set_move_eq_pos },\n  symmetry, change apply_A_move _ _ = _,\n  have h₁ : (play_D_move_at (init_game a d s) hs).s = s',\n  { convert h.symm, change apply_D_move s (d.f s hs).m = _,\n    congr, exact D_set_move_eq_pos },\n  congr' 1, change (a.f _ _ _).m = _, generalize_proofs h₂,\n  have h₃ : (a.f (play_D_move_at (init_game a d s) hs).s hs h₂).m =\n    (a.f s' hs₁ hvm_s').m, { congr' },\n  rw [h₃, A_set_move_eq_pos],\nend\n\nlemma A_hvm_of_next_act {pw : ℕ} {g : Game pw}\n  (h : g.play_move.act) :\n  ∃ hs, A_has_valid_move pw (play_D_move_at g hs).s.board :=\nbegin\n  have hs : g.act := act_of_act_play_move h, use hs,\n  rw Game.play_move at h, rw dif_pos hs at h,\n  rw play_A_move_at at h, split_ifs at h with h₄,\n  { cases h₄ with h₄ h₅, convert h₅, },\n  { cases h }\nend\n\nlemma play_move_hist_len_eq_of_act {pw : ℕ} {g : Game pw}\n  (h : g.play_move.act) :\n  g.play_move.s.len = g.s.len + 2 :=\nbegin\n  have h₁ : g.act := act_of_act_play_move h,\n  rw [play_move_at_act h₁, play_A_move_at],\n  rw dif_pos, { rw [hist_len_play_A_move_at', hist_len_play_D_move_at] },\n  exact ⟨h₁, (A_hvm_of_next_act h).some_spec⟩,\nend\n\nlemma play_hist_len_eq_of_act {pw n : ℕ} {g : Game pw}\n  (h : (g.play n).act) :\n  (g.play n).s.len = g.s.len + n * 2 :=\nbegin\n  induction n with n ih, { refl }, rw play_at_succ' at h ⊢,\n  let g₁ : Game pw := _, change g.play n with g₁ at ih h ⊢,\n  have h₁ := act_of_act_play_move h, specialize ih h₁,\n  rw [play_move_hist_len_eq_of_act h, ih, add_assoc, nat.succ_mul],\nend\n\nlemma hist_len_ne_of_play_lt {pw n k : ℕ} {g : Game pw}\n  (h₁ : k < n) (h₂ : (g.play n).act) :\n  (g.play k).s.len ≠ (g.play n).s.len :=\nbegin\n  have h₃ := act_play_le (nat.le_of_lt h₁) h₂,\n  rw [play_hist_len_eq_of_act h₂, play_hist_len_eq_of_act h₃], intro h,\n  replace h := nat.add_left_cancel h,\n  rw nat.mul_left_inj at h, swap, { dec_trivial },\n  subst h, exact nat.lt_asymm h₁ h₁,\nend\n\nlemma hist_ne_of_play_lt {pw n k : ℕ} {g : Game pw}\n  (h₁ : k < n) (h₂ : (g.play n).act) :\n  (g.play k).s.history ≠ (g.play n).s.history :=\nhist_ne_of_hist_len_ne (hist_len_ne_of_play_lt h₁ h₂)\n\nlemma state_ne_of_play_lt {pw n k : ℕ} {g : Game pw}\n  (h₁ : k < n) (h₂ : (g.play n).act) :\n  (g.play k).s ≠ (g.play n).s :=\nby { have h₃ := hist_ne_of_play_lt h₁ h₂, contrapose! h₃, rw h₃ }\n\nlemma apply_A_move_ne_of_hist_ne {pw : ℕ} {s₁ s₂ : State}\n  {ma₁ ma₂ : A_move} (h : s₁.len ≠ s₂.len) :\n  apply_A_move s₁ ma₁ ≠ apply_A_move s₂ ma₂ :=\nbegin\n  contrapose! h, simp_rw [apply_A_move, apply_move] at h,\n  simp_rw [State.len, (snoc_eq_snoc_iff.mp h.2.1).1],\nend\n\nlemma apply_D_move_ne_of_hist_ne {s₁ s₂ : State}\n  {md₁ md₂ : D_move} (h : s₁.len ≠ s₂.len) :\n  apply_D_move s₁ md₁ ≠ apply_D_move s₂ md₂ :=\nbegin\n  contrapose! h, simp_rw [apply_D_move, apply_move] at h,\n  simp_rw [State.len, (snoc_eq_snoc_iff.mp h.2.1).1],\nend\n\ndef mk_A_for_played_move_at_play_move {pw : ℕ}\n  (a₀ a : A pw) (s' : State) (hs : s'.act) :\n  A pw :=\nif hvm : A_has_valid_move pw s'.board\nthen a₀.set_move s' (a.f s' hs hvm)\nelse a₀\n\nlemma A_played_move_at_play_move {pw : ℕ}\n  {g : Game pw} {s₀' : State} {ma : Valid_A_move pw s₀'.board}\n  (h : A_played_move_at g.s s₀' ma) :\n  A_played_move_at g.play_move.s s₀' ma :=\nbegin\n  let a := g.a, let d := g.d, let s := g.s,\n  rcases h with ⟨s₀, md₀, a₀, d₀, n, hmd, h₁, hx, hy, h₂⟩, by_cases hs : g.act,\n  { let md := d.f s hs, let s' := apply_D_move s md.m,\n    let d₁ := d₀.set_move s md,\n    let a₁ := mk_A_for_played_move_at_play_move a₀ a s' hs,\n    have hmd₁ : ∃ hs, d₁.f s₀ hs = md₀,\n    { cases hmd with hs₁ hmd, use hs₁, change dite _ _ _ = _,\n      rw dif_neg, { exact hmd }, change ((init_game a₀ d₀ s₀).play 0).s ≠ g.s,\n      rw h₂, apply state_ne_of_play_lt (nat.pos_of_ne_zero hy),\n      change g.s.act at hs, rwa h₂ at hs },\n    use [s₀, md₀, a₁, d₁, n.succ, hmd₁, by rw h₁],\n    have h₁ : ∀ (k ≤ n), ((init_game a₁ d₁ s₀).play k).s =\n      ((init_game a₀ d₀ s₀).play k).s,\n    { rintro k hk, induction k with k ih, { refl }, simp_rw play_at_succ',\n      specialize ih (nat.le_of_succ_le hk),\n      replace ih : (init_game a₁ d₁ s₀).play k =\n        ((init_game a₀ d₀ s₀).play k).set_players a₁ d₁,\n      { ext,\n        { exact play_at_players_eq.1 },\n        { exact play_at_players_eq.2 },\n        { rw ih, refl }},\n      rw ih, clear ih,\n      have hs₁ : ((init_game a₀ d₀ s₀).play k).act,\n      { apply act_play_le (nat.le_of_succ_le hk),\n        change ((init_game a₀ d₀ s₀).play n).s.act,\n        rw ←h₂, exact hs },\n      repeat { rw play_move_at_act, swap, { exact hs₁ }},\n      let g₁ : Game pw := _,\n      change (init_game a₀ d₀ s₀).play k with g₁ at hs₁ ⊢,\n      have ha : g₁.a = a₀ := play_at_players_eq.1,\n      have hd : g₁.d = d₀ := play_at_players_eq.2,\n      let sk := g₁.s, let mdk := d₀.f sk hs₁,\n      let sk' := apply_D_move sk mdk.m,\n      have hy : (play_D_move_at g₁ hs₁).s = sk',\n      { change apply_D_move sk _ = apply_D_move _ _, congr' 2, rw hd },\n      have h₁ : d₁.f sk hs₁ = mdk,\n      { change dite _ _ _ = _, rw dif_neg,\n        change ((init_game a₀ d₀ s₀).play k).s ≠ g.s, rw h₂,\n        apply state_ne_of_play_lt (nat.succ_le_iff.mp hk),\n        change ((init_game a₀ d₀ s₀).play n).s.act, rw ←h₂, exact hs },\n      have hx : play_D_move_at (g₁.set_players a₁ d₁) hs₁ =\n        (play_D_move_at g₁ hs₁).set_players a₁ d₁,\n      { ext,\n        { exact play_D_move_at_players_eq.1, },\n        { exact play_D_move_at_players_eq.2, },\n        { change apply_D_move g₁.s _ = apply_D_move _ _,\n          congr' 2, change d₁.f sk _ = _, rw [h₁, hd] }},\n      rw hx, clear hx, have hx : A_has_valid_move pw sk'.board,\n      { have h₃ : g₁.play_move.act,\n        { change ((init_game a₀ d₀ s₀).play k).play_move.act,\n          rw ←play_at_succ', apply act_play_le hk,\n          change ((init_game a₀ d₀ s₀).play n).s.act,\n          rw ←h₂, exact hs },\n        convert (A_hvm_of_next_act h₃).some_spec, rw ←hy, refl },\n      simp_rw play_A_move_at,\n      repeat { rw dif_pos, swap, { use hs₁, convert hx }},\n      change apply_A_move (play_D_move_at g₁ hs₁).s\n        (a₁.f (play_D_move_at g₁ hs₁).s _ _).m =\n        apply_A_move _ (g₁.a.f (play_D_move_at g₁ hs₁).s _ _).m,\n      congr' 2, rw ha, generalize_proofs h₃,\n      change (mk_A_for_played_move_at_play_move _ _ _ _).f _ _ _ = _,\n      rw mk_A_for_played_move_at_play_move,\n      split_ifs with hh, swap, { refl },\n      change dite _ _ _ = _, rw dif_neg, rw hy,\n      change apply_D_move ((init_game a₀ d₀ s₀).play k).s mdk.m ≠\n        apply_D_move g.s md.m,\n      rw h₂, replace hk := nat.lt_of_succ_le hk,\n      apply apply_D_move_ne_of_hist_ne, apply hist_len_ne_of_play_lt hk,\n      change ((init_game a₀ d₀ s₀).play n).s.act, rw ←h₂, exact hs },\n    specialize h₁ n (le_refl n),\n    replace h₁ : (init_game a₁ d₁ s₀).play n = g.set_players a₁ d₁,\n    { ext,\n      { exact play_at_players_eq.1 },\n      { exact play_at_players_eq.2 },\n      { rw h₁, exact h₂.symm }},\n    rw [play_at_succ', play_move_at_act hs, h₁, play_move_at_act],\n    swap, { exact hs }, refine ⟨_, _, _⟩,\n    { clear h₁, rcases hx with ⟨hs', h, hx⟩,\n      use [hs', h], rw ←hx, clear hx,\n      change (mk_A_for_played_move_at_play_move _ _ _ _).f _ _ _ = _,\n      rw mk_A_for_played_move_at_play_move,\n      split_ifs with hss, swap, { refl }, change dite _ _ _ = _,\n      rw dif_neg, change s₀' ≠ apply_D_move g.s _, rw [h₁, h₂],\n      apply apply_D_move_ne_of_hist_ne,\n      change ((init_game a₀ d₀ s₀).play 0).s.len ≠ _,\n      apply hist_len_ne_of_play_lt (pos_iff_ne_zero.mpr hy),\n      rw [Game.act, ←h₂], exact hs }, { dec_trivial }, symmetry,\n    have h₁ : play_D_move_at (g.set_players a₁ d₁) hs =\n      (play_D_move_at g hs).set_players a₁ d₁,\n    { ext,\n      { exact play_D_move_at_players_eq.1 },\n      { exact play_D_move_at_players_eq.2 },\n      { change apply_D_move g.s _ = apply_D_move _ _,\n        congr' 2, change d₁.f s hs = g.d.f g.s hs,\n        change dite _ _ _ = _, rw dif_pos rfl }},\n    rw h₁, clear h₁, simp_rw play_A_move_at,\n    split_ifs with h₁, swap, { refl }, cases h₁ with h₁ h₂,\n    change apply_A_move ((play_D_move_at g hs).s) _ =\n      apply_A_move _ _,\n    congr' 2, change a₁.f s' _ _ = a.f s' _ _, generalize_proofs,\n    change (mk_A_for_played_move_at_play_move _ _ _ _).f _ _ _ = _,\n    rw mk_A_for_played_move_at_play_move, rw dif_pos, swap, { exact h₂ },\n    change dite _ _ _ = _, rw dif_pos rfl },\n  { use [s₀, md₀, a₀, d₀, n, hmd, h₁, hx, hy],\n    rw play_move_at_not_act hs, exact h₂ },\nend\n\nlemma A_played_move_at_play {pw n : ℕ}\n  {g : Game pw} {s' : State} {ma : Valid_A_move pw s'.board}\n  (h : A_played_move_at g.s s' ma) :\n  A_played_move_at (g.play n).s s' ma :=\nbegin\n  induction n with n ih, { exact h }, rw play_at_succ',\n  exact A_played_move_at_play_move ih,\nend\n\nlemma hist_overlaps_of_A_played_move_at {pw : ℕ}\n  {s s₀' : State} {ma : Valid_A_move pw s₀'.board}\n  (h₁ : A_played_move_at s s₀' ma) :\n  ∀ (k : ℕ), k.succ < s₀'.len →\n  s.history.nth k = s₀'.history.nth k :=\nbegin\n  rintro k h₂,\n  rcases h₁ with ⟨s₀, md, a, d, n, hmd, rfl, hx, hy, rfl⟩, clear' hx hy,\n  change (apply_D_move s₀ md.m).history with (_ ++ _ : list _) at h₂ ⊢,\n  change _ < (snoc _ _).length at h₂,\n  rw [length_snoc, nat.succ_lt_succ_iff] at h₂ ,induction n with n ih,\n  { change s₀.history.nth k = _, exact (list.nth_append h₂).symm },\n  { simp_rw play_at_succ', let g : Game pw := _,\n    change (init_game a d s₀).play n with g at h₂ ih ⊢, rw ←ih, clear ih,\n    rw Game.play_move, split_ifs with hs, swap, { refl }, rw play_A_move_at,\n    have h₆ : s₀.len ≤ g.s.len,\n    { change (init_game a d s₀).s.len ≤ _, exact hist_len_le_play },\n    split_ifs with h₃,\n    { change (_ ++ _ : list _).nth _ = _, rw list.nth_append,\n      { change (_ ++ _ : list _).nth _ = _, rw list.nth_append,\n        exact gt_of_ge_of_gt h₆ h₂ },\n      { rw [←State.len, hist_len_play_D_move_at],\n        exact nat.lt_succ_of_le (nat.le_trans (le_of_lt h₂) h₆) }},\n    { change (_ ++ _ : list _).nth _ = _, rw list.nth_append,\n      exact gt_of_ge_of_gt h₆ h₂ }},\nend\n\nlemma A_played_move_at_eq_aux {pw n : ℕ}\n  {sx s₀ s' : State} {md : Valid_D_move s₀.board}\n  {a : A pw} {d : D} {hs hvm h₄}\n  (h₁ : s' = apply_D_move s₀ md.m)\n  (h₂ : sx = ((init_game a d s₀).play n).s)\n  (h₃ : n ≠ 0)\n  (hh : md = d.f s₀ h₄) :\n  sx.nth (s₀.len + 2) = option.some\n    (apply_A_move s' (a.f s' hs hvm).m).board :=\nbegin\n  cases n, { contradiction }, clear h₃,\n  let i : ℕ := _, change s₀.len + 2 with i,\n  let s₁ : State := apply_A_move s' (a.f s' hs hvm).m,\n  change _ = some (State.board s₁), have h₅ : i = s₁.len,\n  { change s₁ with apply_A_move _ _, subst s',\n    rw [apply_A_move_len, apply_D_move_len] },\n  induction n with n ih generalizing sx,\n  { rw [play_1, play_move_at_act] at h₂, swap, { exact h₄ },\n    subst sx, have h₆ : play_D_move_at (init_game a d s₀) h₄ =\n      (init_game a d s'),\n    { ext,\n      { exact play_D_move_at_players_eq.1 },\n      { exact play_D_move_at_players_eq.2 },\n      { subst s',\n        change apply_D_move s₀ (d.f s₀ _).m = apply_D_move s₀ _, rw hh }},\n    rw [h₆, play_A_move_at, dif_pos], clear h₆, swap, { exact ⟨hs, hvm⟩ },\n    change s₁.nth _ = _, rw h₅, exact state_nth_len },\n  { rw play_at_succ' at h₂, let g₁ : Game pw := (init_game a d s₀).play n.succ,\n    change (init_game a d s₀).play n.succ with g₁ at ih h₂, subst sx,\n    specialize ih rfl, rw Game.play_move, split_ifs with hs₁, swap, { exact ih },\n    let g₂ : Game pw := _, change play_D_move_at g₁ hs₁ with g₂,\n    have ih₁ : g₂.s.nth i = some s₁.board,\n    { exact apply_move_state_nth_eq_some_of ih },\n    rw play_A_move_at, split_ifs with hh,\n    { exact apply_move_state_nth_eq_some_of ih₁ },\n    { exact ih₁ }},\nend\n\nlemma A_played_move_at_eq {pw : ℕ}\n  {sx s' : State} {ma₁ ma₂ : Valid_A_move pw s'.board}\n  (hx : A_played_move_at sx s' ma₁)\n  (hy : A_played_move_at sx s' ma₂) :\n  ma₁ = ma₂ :=\nbegin\n  obtain ⟨s₀, md₁, a₁, d₁, n₁, hmd₁, h₁, ⟨hs₁, hvm₁, rfl⟩, h₃, h₄⟩ := hx,\n  obtain ⟨s₀', md₂, a₂, d₂, n₂, hmd₂, h₅, ⟨hs₂, hvm₂, rfl⟩, h₇, h₈⟩ := hy,\n  have hs : s₀.act,\n  { subst s', exact hs₂ },\n  obtain rfl : s₀ = s₀',\n  { subst h₁, exact state_eq_of_apply_D_move_eq h₅ },\n  obtain rfl : md₁ = md₂,\n  { subst h₁, rwa ←D_moves_eq_iff at h₅ },\n  clear h₅, change hvm₂ with hvm₁, clear hvm₂,\n  change hs₂ with hs₁, clear hs₂, let i := s₀.len,\n  have h₅ : sx.nth (i + 2) = option.some\n    (apply_A_move s' (a₁.f s' hs₁ hvm₁).m).board,\n  { exact A_played_move_at_eq_aux h₁ h₄ h₃ hmd₁.some_spec.symm },\n  have h₆ : sx.nth (i + 2) = option.some\n    (apply_A_move s' (a₂.f s' hs₁ hvm₁).m).board,\n  { exact A_played_move_at_eq_aux h₁ h₈ h₇ hmd₂.some_spec.symm },\n  simp [h₅] at h₆, rwa A_moves_eq_iff',\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/played_move.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202216224477}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Sean Leather\n\nFunctions on lists of sigma types.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.perm\nimport Mathlib.data.list.range\nimport Mathlib.data.sigma.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 \n\nnamespace Mathlib\n\nnamespace list\n\n\n/- keys -/\n\n/-- List of keys from a list of key-value pairs -/\ndef keys {α : Type u} {β : α → Type v} : List (sigma β) → List α :=\n  map sigma.fst\n\n@[simp] theorem keys_nil {α : Type u} {β : α → Type v} : keys [] = [] :=\n  rfl\n\n@[simp] theorem keys_cons {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} : keys (s :: l) = sigma.fst s :: keys l :=\n  rfl\n\ntheorem mem_keys_of_mem {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} : s ∈ l → sigma.fst s ∈ keys l :=\n  mem_map_of_mem sigma.fst\n\ntheorem exists_of_mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} (h : a ∈ keys l) : ∃ (b : β a), sigma.mk a b ∈ l := sorry\n\ntheorem mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} : a ∈ keys l ↔ ∃ (b : β a), sigma.mk a b ∈ l := sorry\n\ntheorem not_mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} : ¬a ∈ keys l ↔ ∀ (b : β a), ¬sigma.mk a b ∈ l :=\n  iff.trans (not_iff_not_of_iff mem_keys) not_exists\n\ntheorem not_eq_key {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} : ¬a ∈ keys l ↔ ∀ (s : sigma β), s ∈ l → a ≠ sigma.fst s := sorry\n\n/- nodupkeys -/\n\ndef nodupkeys {α : Type u} {β : α → Type v} (l : List (sigma β)) :=\n  nodup (keys l)\n\ntheorem nodupkeys_iff_pairwise {α : Type u} {β : α → Type v} {l : List (sigma β)} : nodupkeys l ↔ pairwise (fun (s s' : sigma β) => sigma.fst s ≠ sigma.fst s') l :=\n  pairwise_map sigma.fst\n\ntheorem nodupkeys.pairwise_ne {α : Type u} {β : α → Type v} {l : List (sigma β)} (h : nodupkeys l) : pairwise (fun (s s' : sigma β) => sigma.fst s ≠ sigma.fst s') l :=\n  iff.mp nodupkeys_iff_pairwise h\n\n@[simp] theorem nodupkeys_nil {α : Type u} {β : α → Type v} : nodupkeys [] :=\n  pairwise.nil\n\n@[simp] theorem nodupkeys_cons {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} : nodupkeys (s :: l) ↔ ¬sigma.fst s ∈ keys l ∧ nodupkeys l := sorry\n\ntheorem nodupkeys.eq_of_fst_eq {α : Type u} {β : α → Type v} {l : List (sigma β)} (nd : nodupkeys l) {s : sigma β} {s' : sigma β} (h : s ∈ l) (h' : s' ∈ l) : sigma.fst s = sigma.fst s' → s = s' := sorry\n\ntheorem nodupkeys.eq_of_mk_mem {α : Type u} {β : α → Type v} {a : α} {b : β a} {b' : β a} {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) (h' : sigma.mk a b' ∈ l) : b = b' := sorry\n\ntheorem nodupkeys_singleton {α : Type u} {β : α → Type v} (s : sigma β) : nodupkeys [s] :=\n  nodup_singleton (sigma.fst s)\n\ntheorem nodupkeys_of_sublist {α : Type u} {β : α → Type v} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : l₁ <+ l₂) : nodupkeys l₂ → nodupkeys l₁ :=\n  nodup_of_sublist (sublist.map sigma.fst h)\n\ntheorem nodup_of_nodupkeys {α : Type u} {β : α → Type v} {l : List (sigma β)} : nodupkeys l → nodup l :=\n  nodup_of_nodup_map sigma.fst\n\ntheorem perm_nodupkeys {α : Type u} {β : α → Type v} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : l₁ ~ l₂) : nodupkeys l₁ ↔ nodupkeys l₂ :=\n  perm.nodup_iff (perm.map sigma.fst h)\n\ntheorem nodupkeys_join {α : Type u} {β : α → Type v} {L : List (List (sigma β))} : nodupkeys (join L) ↔ (∀ (l : List (sigma β)), l ∈ L → nodupkeys l) ∧ pairwise disjoint (map keys L) := sorry\n\ntheorem nodup_enum_map_fst {α : Type u} (l : List α) : nodup (map prod.fst (enum l)) := sorry\n\ntheorem mem_ext {α : Type u} {β : α → Type v} {l₀ : List (sigma β)} {l₁ : List (sigma β)} (nd₀ : nodup l₀) (nd₁ : nodup l₁) (h : ∀ (x : sigma β), x ∈ l₀ ↔ x ∈ l₁) : l₀ ~ l₁ := sorry\n\n/- lookup -/\n\n/-- `lookup a l` is the first value in `l` corresponding to the key `a`,\n  or `none` if no such element exists. -/\ndef lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : List (sigma β) → Option (β a) :=\n  sorry\n\n@[simp] theorem lookup_nil {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : lookup a [] = none :=\n  rfl\n\n@[simp] theorem lookup_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma fun (a : α) => β a)) (a : α) (b : β a) : lookup a (sigma.mk a b :: l) = some b :=\n  dif_pos rfl\n\n@[simp] theorem lookup_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β)) {a : α} (s : sigma β) : a ≠ sigma.fst s → lookup a (s :: l) = lookup a l := sorry\n\ntheorem lookup_is_some {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} : ↥(option.is_some (lookup a l)) ↔ a ∈ keys l := sorry\n\ntheorem lookup_eq_none {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} : lookup a l = none ↔ ¬a ∈ keys l := sorry\n\ntheorem of_mem_lookup {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} : b ∈ lookup a l → sigma.mk a b ∈ l := sorry\n\ntheorem mem_lookup {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) : b ∈ lookup a l := sorry\n\ntheorem map_lookup_eq_find {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : option.map (sigma.mk a) (lookup a l) = find (fun (s : sigma β) => a = sigma.fst s) l := sorry\n\ntheorem mem_lookup_iff {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} (nd : nodupkeys l) : b ∈ lookup a l ↔ sigma.mk a b ∈ l :=\n  { mp := of_mem_lookup, mpr := mem_lookup nd }\n\ntheorem perm_lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) (p : l₁ ~ l₂) : lookup a l₁ = lookup a l₂ := sorry\n\ntheorem lookup_ext {α : Type u} {β : α → Type v} [DecidableEq α] {l₀ : List (sigma β)} {l₁ : List (sigma β)} (nd₀ : nodupkeys l₀) (nd₁ : nodupkeys l₁) (h : ∀ (x : α) (y : β x), y ∈ lookup x l₀ ↔ y ∈ lookup x l₁) : l₀ ~ l₁ := sorry\n\n/- lookup_all -/\n\n/-- `lookup_all a l` is the list of all values in `l` corresponding to the key `a`. -/\ndef lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : List (sigma β) → List (β a) :=\n  sorry\n\n@[simp] theorem lookup_all_nil {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : lookup_all a [] = [] :=\n  rfl\n\n@[simp] theorem lookup_all_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma fun (a : α) => β a)) (a : α) (b : β a) : lookup_all a (sigma.mk a b :: l) = b :: lookup_all a l :=\n  dif_pos rfl\n\n@[simp] theorem lookup_all_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β)) {a : α} (s : sigma β) : a ≠ sigma.fst s → lookup_all a (s :: l) = lookup_all a l := sorry\n\ntheorem lookup_all_eq_nil {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} : lookup_all a l = [] ↔ ∀ (b : β a), ¬sigma.mk a b ∈ l := sorry\n\ntheorem head_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : head' (lookup_all a l) = lookup a l := sorry\n\ntheorem mem_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} : b ∈ lookup_all a l ↔ sigma.mk a b ∈ l := sorry\n\ntheorem lookup_all_sublist {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : map (sigma.mk a) (lookup_all a l) <+ l := sorry\n\ntheorem lookup_all_length_le_one {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} (h : nodupkeys l) : length (lookup_all a l) ≤ 1 := sorry\n\ntheorem lookup_all_eq_lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} (h : nodupkeys l) : lookup_all a l = option.to_list (lookup a l) := sorry\n\ntheorem lookup_all_nodup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} (h : nodupkeys l) : nodup (lookup_all a l) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nodup (lookup_all a l))) (lookup_all_eq_lookup a h)))\n    (option.to_list_nodup (lookup a l))\n\ntheorem perm_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) (p : l₁ ~ l₂) : lookup_all a l₁ = lookup_all a l₂ := sorry\n\n/- kreplace -/\n\ndef kreplace {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) : List (sigma β) → List (sigma β) :=\n  lookmap\n    fun (s : sigma β) =>\n      dite (a = sigma.fst s) (fun (h : a = sigma.fst s) => some (sigma.mk a b)) fun (h : ¬a = sigma.fst s) => none\n\ntheorem kreplace_of_forall_not {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) {l : List (sigma β)} (H : ∀ (b : β a), ¬sigma.mk a b ∈ l) : kreplace a b l = l := sorry\n\ntheorem kreplace_self {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) : kreplace a b l = l := sorry\n\ntheorem keys_kreplace {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : List (sigma β)) : keys (kreplace a b l) = keys l := sorry\n\ntheorem kreplace_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) {l : List (sigma β)} : nodupkeys (kreplace a b l) ↔ nodupkeys l := sorry\n\ntheorem perm.kreplace {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd : nodupkeys l₁) : l₁ ~ l₂ → kreplace a b l₁ ~ kreplace a b l₂ := sorry\n\n/- kerase -/\n\n/-- Remove the first pair with the key `a`. -/\ndef kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : List (sigma β) → List (sigma β) :=\n  erasep fun (s : sigma β) => a = sigma.fst s\n\n@[simp] theorem kerase_nil {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} : kerase a [] = [] :=\n  rfl\n\n@[simp] theorem kerase_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {s : sigma β} {l : List (sigma β)} (h : a = sigma.fst s) : kerase a (s :: l) = l := sorry\n\n@[simp] theorem kerase_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {s : sigma β} {l : List (sigma β)} (h : a ≠ sigma.fst s) : kerase a (s :: l) = s :: kerase a l := sorry\n\n@[simp] theorem kerase_of_not_mem_keys {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} (h : ¬a ∈ keys l) : kerase a l = l := sorry\n\ntheorem kerase_sublist {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : kerase a l <+ l :=\n  erasep_sublist l\n\ntheorem kerase_keys_subset {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : keys (kerase a l) ⊆ keys l :=\n  sublist.subset (sublist.map sigma.fst (kerase_sublist a l))\n\ntheorem mem_keys_of_mem_keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a₁ : α} {a₂ : α} {l : List (sigma β)} : a₁ ∈ keys (kerase a₂ l) → a₁ ∈ keys l :=\n  kerase_keys_subset a₂ l\n\ntheorem exists_of_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} (h : a ∈ keys l) : ∃ (b : β a),\n  ∃ (l₁ : List (sigma β)), ∃ (l₂ : List (sigma β)), ¬a ∈ keys l₁ ∧ l = l₁ ++ sigma.mk a b :: l₂ ∧ kerase a l = l₁ ++ l₂ := sorry\n\n@[simp] theorem mem_keys_kerase_of_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a₁ : α} {a₂ : α} {l : List (sigma β)} (h : a₁ ≠ a₂) : a₁ ∈ keys (kerase a₂ l) ↔ a₁ ∈ keys l := sorry\n\ntheorem keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} : keys (kerase a l) = list.erase (keys l) a := sorry\n\ntheorem kerase_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {l : List (sigma β)} : kerase a (kerase a' l) = kerase a' (kerase a l) := sorry\n\ntheorem kerase_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} : nodupkeys l → nodupkeys (kerase a l) :=\n  nodupkeys_of_sublist (kerase_sublist a l)\n\ntheorem perm.kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd : nodupkeys l₁) : l₁ ~ l₂ → kerase a l₁ ~ kerase a l₂ :=\n  perm.erasep (fun (s : sigma β) => a = sigma.fst s)\n    (pairwise.imp (fun (x y : sigma β) (h : sigma.fst x ≠ sigma.fst y) (ᾰ : a = sigma.fst x) => Eq._oldrec h (Eq.symm ᾰ))\n      (iff.mp nodupkeys_iff_pairwise nd))\n\n@[simp] theorem not_mem_keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} (nd : nodupkeys l) : ¬a ∈ keys (kerase a l) := sorry\n\n@[simp] theorem lookup_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)} (nd : nodupkeys l) : lookup a (kerase a l) = none :=\n  iff.mpr lookup_eq_none (not_mem_keys_kerase a nd)\n\n@[simp] theorem lookup_kerase_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {l : List (sigma β)} (h : a ≠ a') : lookup a (kerase a' l) = lookup a l := sorry\n\ntheorem kerase_append_left {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : a ∈ keys l₁ → kerase a (l₁ ++ l₂) = kerase a l₁ ++ l₂ := sorry\n\ntheorem kerase_append_right {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : ¬a ∈ keys l₁ → kerase a (l₁ ++ l₂) = l₁ ++ kerase a l₂ := sorry\n\ntheorem kerase_comm {α : Type u} {β : α → Type v} [DecidableEq α] (a₁ : α) (a₂ : α) (l : List (sigma β)) : kerase a₂ (kerase a₁ l) = kerase a₁ (kerase a₂ l) := sorry\n\ntheorem sizeof_kerase {α : Type u_1} {β : α → Type u_2} [DecidableEq α] [SizeOf (sigma β)] (x : α) (xs : List (sigma β)) : sizeof (kerase x xs) ≤ sizeof xs := sorry\n\n/- kinsert -/\n\n/-- Insert the pair `⟨a, b⟩` and erase the first pair with the key `a`. -/\ndef kinsert {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : List (sigma β)) : List (sigma β) :=\n  sigma.mk a b :: kerase a l\n\n@[simp] theorem kinsert_def {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l : List (sigma β)} : kinsert a b l = sigma.mk a b :: kerase a l :=\n  rfl\n\ntheorem mem_keys_kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {b' : β a'} {l : List (sigma β)} : a ∈ keys (kinsert a' b' l) ↔ a = a' ∨ a ∈ keys l := sorry\n\ntheorem kinsert_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) {l : List (sigma β)} (nd : nodupkeys l) : nodupkeys (kinsert a b l) :=\n  iff.mpr nodupkeys_cons { left := not_mem_keys_kerase a nd, right := kerase_nodupkeys a nd }\n\ntheorem perm.kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (p : l₁ ~ l₂) : kinsert a b l₁ ~ kinsert a b l₂ :=\n  perm.cons (sigma.mk a b) (perm.kerase nd₁ p)\n\ntheorem lookup_kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} (l : List (sigma β)) : lookup a (kinsert a b l) = some b := sorry\n\ntheorem lookup_kinsert_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {b' : β a'} {l : List (sigma β)} (h : a ≠ a') : lookup a (kinsert a' b' l) = lookup a l := sorry\n\n/- kextract -/\n\ndef kextract {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : List (sigma β) → Option (β a) × List (sigma β) :=\n  sorry\n\n@[simp] theorem kextract_eq_lookup_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : kextract a l = (lookup a l, kerase a l) := sorry\n\n/- erase_dupkeys -/\n\n/-- Remove entries with duplicate keys from `l : list (sigma β)`. -/\ndef erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] : List (sigma β) → List (sigma β) :=\n  foldr (fun (x : sigma β) => kinsert (sigma.fst x) (sigma.snd x)) []\n\ntheorem erase_dupkeys_cons {α : Type u} {β : α → Type v} [DecidableEq α] {x : sigma β} (l : List (sigma β)) : erase_dupkeys (x :: l) = kinsert (sigma.fst x) (sigma.snd x) (erase_dupkeys l) :=\n  rfl\n\ntheorem nodupkeys_erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β)) : nodupkeys (erase_dupkeys l) := sorry\n\ntheorem lookup_erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) : lookup a (erase_dupkeys l) = lookup a l := sorry\n\ntheorem sizeof_erase_dupkeys {α : Type u_1} {β : α → Type u_2} [DecidableEq α] [SizeOf (sigma β)] (xs : List (sigma β)) : sizeof (erase_dupkeys xs) ≤ sizeof xs := sorry\n\n/- kunion -/\n\n/-- `kunion l₁ l₂` is the append to l₁ of l₂ after, for each key in l₁, the\nfirst matching pair in l₂ is erased. -/\ndef kunion {α : Type u} {β : α → Type v} [DecidableEq α] : List (sigma β) → List (sigma β) → List (sigma β) :=\n  sorry\n\n@[simp] theorem nil_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {l : List (sigma β)} : kunion [] l = l :=\n  rfl\n\n@[simp] theorem kunion_nil {α : Type u} {β : α → Type v} [DecidableEq α] {l : List (sigma β)} : kunion l [] = l := sorry\n\n@[simp] theorem kunion_cons {α : Type u} {β : α → Type v} [DecidableEq α] {s : sigma β} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : kunion (s :: l₁) l₂ = s :: kunion l₁ (kerase (sigma.fst s) l₂) :=\n  rfl\n\n@[simp] theorem mem_keys_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : a ∈ keys (kunion l₁ l₂) ↔ a ∈ keys l₁ ∨ a ∈ keys l₂ := sorry\n\n@[simp] theorem kunion_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : kunion (kerase a l₁) (kerase a l₂) = kerase a (kunion l₁ l₂) := sorry\n\ntheorem kunion_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) : nodupkeys (kunion l₁ l₂) := sorry\n\ntheorem perm.kunion_right {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)} {l₂ : List (sigma β)} (p : l₁ ~ l₂) (l : List (sigma β)) : kunion l₁ l ~ kunion l₂ l := sorry\n\ntheorem perm.kunion_left {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β)) {l₁ : List (sigma β)} {l₂ : List (sigma β)} : nodupkeys l₁ → l₁ ~ l₂ → kunion l l₁ ~ kunion l l₂ := sorry\n\ntheorem perm.kunion {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)} {l₂ : List (sigma β)} {l₃ : List (sigma β)} {l₄ : List (sigma β)} (nd₃ : nodupkeys l₃) (p₁₂ : l₁ ~ l₂) (p₃₄ : l₃ ~ l₄) : kunion l₁ l₃ ~ kunion l₂ l₄ :=\n  perm.trans (perm.kunion_right p₁₂ l₃) (perm.kunion_left l₂ nd₃ p₃₄)\n\n@[simp] theorem lookup_kunion_left {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : a ∈ keys l₁) : lookup a (kunion l₁ l₂) = lookup a l₁ := sorry\n\n@[simp] theorem lookup_kunion_right {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : ¬a ∈ keys l₁) : lookup a (kunion l₁ l₂) = lookup a l₂ := sorry\n\n@[simp] theorem mem_lookup_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l₁ : List (sigma β)} {l₂ : List (sigma β)} : b ∈ lookup a (kunion l₁ l₂) ↔ b ∈ lookup a l₁ ∨ ¬a ∈ keys l₁ ∧ b ∈ lookup a l₂ := sorry\n\ntheorem mem_lookup_kunion_middle {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a} {l₁ : List (sigma β)} {l₂ : List (sigma β)} {l₃ : List (sigma β)} (h₁ : b ∈ lookup a (kunion l₁ l₃)) (h₂ : ¬a ∈ keys l₂) : b ∈ lookup a (kunion (kunion l₁ l₂) l₃) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/sigma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6261241632752915, "lm_q1q2_score": 0.3495820221622447}}
{"text": "import parlang\nimport parlang_nonmono\n\nnamespace parlang\nnamespace ac_loop\n\nopen parlang\nopen parlang.kernel\nopen parlang_nonmono\n\nvariables {n : ℕ} {σ : Type} {ι : Type} {τ : ι → Type} {s t u : state n σ τ} {ac : vector bool n} {f f' : σ → bool}  [decidable_eq ι] {semantics : kernel σ τ → vector bool n → state n σ τ → state n σ τ → Prop}\n\nclass has_unique (semantics : kernel σ τ → vector bool n → state n σ τ → state n σ τ → Prop) := \n(unique : ∀ {k : kernel σ τ} {ac : vector bool n} {s t t' : state n σ τ}, semantics k ac s t → semantics k ac s t' → t = t')\n\nvariables [has_unique semantics]\n\ninstance unique2 : has_unique $ @parlang.exec_state σ ι τ _ n := ⟨begin\n    intros _ _ _ _ _ a b,\n    apply parlang.exec_state_unique b a,\nend⟩\n\ninstance unique1 : has_unique $ @exec_state σ ι τ _ n := sorry\n\nnotation `v[` v:(foldr `, ` (h t, vector.cons h t) vector.nil `]`) := v\n\n-- this is similar to the semantic itself except we are not interested in states but the resulting active maps\ninductive ac_after_n_iteration (semantics : kernel σ τ → vector bool n → state n σ τ → state n σ τ → Prop) (k : kernel σ τ) (f : σ → bool) : state n σ τ → vector bool n → vector bool n → ℕ → Prop\n| base (ac : vector bool n) (s : state n σ τ) : ac_after_n_iteration s ac ac 0\n| step {ac : vector bool n} (s t : state n σ τ) {i : ℕ} {ac' : vector bool n} : ac_after_n_iteration t (deactivate_threads (bnot ∘ f) ac s) ac' i → semantics k (deactivate_threads (bnot ∘ f) ac s) s t → ac_after_n_iteration s ac ac' (i + 1)\n\nexample : ac_after_n_iteration exec_state (compute (λ (m : bool), ff)) id (@parlang.init_state bool string (λs, ℕ) _ (λ_, tt) (λ_,1) (λs:string, 0)) v[tt] v[tt] 1 := begin\n    apply ac_after_n_iteration.step _ (@parlang.init_state bool string (λs, ℕ) _ (λ_, ff) (λ_,1) (λs:string, 0)),\n    swap,\n    sorry,\n    sorry,\nend\n\nlemma ac_after_n_iteration_unique {k : kernel σ τ} {f : σ → bool} {ac' ac'' : vector bool n} {i} : \nac_after_n_iteration semantics k f s ac ac' i → ac_after_n_iteration semantics k f s ac ac'' i → ac' = ac'' := begin\n    admit,\nend\n\nlemma ac_ge_single {k : kernel σ τ} {f : σ → bool} {ac' : vector bool n} {i : ℕ} : ac_after_n_iteration semantics k f s ac ac' i → ac ≥ ac' := begin\n    intros h,\n    induction i generalizing s ac ac',\n    case nat.zero {\n        cases h,\n        intros t hna,\n        assumption,\n    },\n    case nat.succ {\n        cases h,\n        specialize i_ih h_a,\n        have : ac ≥ (deactivate_threads (bnot ∘ f) ac s) := by apply ac_sub_deac,\n        sorry, -- proof by transitivity\n    }\nend\n\nlemma ac_ge_two {k : kernel σ τ} {f : σ → bool} {i i' : ℕ} {ac' ac'' : vector bool n} : \nac_after_n_iteration semantics k f s ac ac' i → ac_after_n_iteration semantics k f s ac ac'' (i + i') → ac' ≥ ac'' := begin\n    intros h₁ h₂,\n    induction i generalizing s ac,\n    case nat.zero {\n        cases h₁,\n        apply ac_ge_single h₂,\n    },\n    case nat.succ {\n        cases h₁,\n        have : nat.succ i_n + i' = nat.succ (i_n + i') := by rw nat.succ_add,\n        rw this at h₂,\n        cases h₂,\n        have : h₁_t = h₂_t := has_unique.unique _ _ _ h₁_a_1 h₂_a_1,\n        subst this,\n        apply i_ih h₁_a h₂_a,\n    }\nend\n\ndef monotone_loop (semantics : kernel σ τ → vector bool n → state n σ τ → state n σ τ → Prop) (f k) (s : state n σ τ) (ac : vector bool n) : Prop := \n∀ (i i' : ℕ) (ac' ac'' : vector bool n), ac_after_n_iteration semantics k f s ac ac' i → ac_after_n_iteration semantics k f s ac ac'' (i + i') → ac' ≥ ac''\n\nlemma par_loop_ac' {k : kernel σ τ} : parlang.exec_state (loop f k) ac s t → ∃ i ac', ac_after_n_iteration parlang.exec_state k f s ac ac' i ∧ no_thread_active (deactivate_threads (bnot ∘ f) ac' t) := begin\n    generalize eq_l : (loop f k) = l,\n    intro h,\n    induction h;\n        cases eq_l,\n    {\n        use 0,\n        use h_ac,\n        split,\n        apply ac_after_n_iteration.base,\n        assumption,\n    }, {\n        clear h_ih_a,\n        specialize h_ih_a_1 rfl,\n        cases h_ih_a_1 with i ih,\n        cases ih with ac' ih,\n        cases ih with ha hb,\n        use i + 1,\n        use ac',\n        split,\n        swap,\n        assumption,\n        apply ac_after_n_iteration.step,\n        repeat { assumption },\n    }\nend\n\nlemma loop_ac' {k : kernel σ τ} : exec_state (loop f k) ac s t → ∃ i ac', ac_after_n_iteration exec_state k f s ac ac' i ∧ no_thread_active (deactivate_threads (bnot ∘ f) ac' t) := begin\n    generalize eq_l : (loop f k) = l,\n    intro h,\n    induction h generalizing;\n        cases eq_l,\n    {\n        use 0,\n        use h_ac,\n        split,\n        apply ac_after_n_iteration.base,\n        assumption,\n    }, {\n        clear h_ih_a,\n        specialize h_ih_a_1 rfl,\n        cases h_ih_a_1 with i ih,\n        cases ih with ac' ih,\n        cases ih with ha hb,\n        use i + 1,\n        use ac',\n        split,\n        swap,\n        assumption,\n        {\n            apply ac_after_n_iteration.step,\n            swap,\n            exact h_a_1,\n            induction ha,\n            case ac_after_n_iteration.base {\n                have : monotone_loop exec_state f k h_s ha_ac := by apply ac_ge_two,\n                have : monotone_loop exec_state f k h_s ha_ac := by apply ac_ge_two,\n            }\n        }\n    }\nend\n\nend ac_loop\nend parlang", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/alt/ac_loop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.34935592042603614}}
{"text": "\nimport pq_induction_principles\nimport minimal_sub_pq_gen_group\n\n\nuniverse u\n\nsection pq_like_eq_equalizer\n\nvariables {G : Type u} [group G] --[inhabited Q]\n\nlemma prod_in_free_gen_list (gen : set G) (x : list (free_gen_group_sub_pq gen)) (hx : of ((x.map of).prod) = (x.map (of ∘ of)).prod) : (x.map coe).prod ∈ (free_gen_group_sub_pq gen : set G) :=\nbegin\n  induction x with g x hx,\n  {\n    simp only [list.prod_nil, list.map],\n    sorry,\n  },\n  {\n    simp only [list.prod_cons, list.map],\n    simp only [function.comp_app, list.prod_cons, list.map] at hx,\n    have hx1 := congr_arg (L_of_morph (gen_set_counit gen) (functoriality_group_to_pq (gen_set_counit gen))) hx,\n    simp only [monoid_hom.map_mul, L_of_morph_of, gen_set_counit_of] at hx1,\n    sorry,\n  },\nend\n\ntheorem eta_eq_L_eta (gen : set G) (x : pq_group (free_gen_group_sub_pq gen)) : (of x = L_of_morph of of_is_pq_morphism x) ↔ (∃ y, x = of y) :=\nbegin\n  split,\n  swap,\n  {\n    intro hx,\n    cases hx with y hy,\n    rw hy,\n    simp only [L_of_morph_of],\n  },\n  {\n    revert x,\n    refine pq_group_list _,\n    intros x hx,\n    fconstructor,\n    fconstructor,\n    use (x.map (λ x, ↑x)).prod,\n    {\n      simp only,\n      induction x,\n      {\n        simp only [list.prod_nil, list.map],\n        sorry,\n      },\n      have hx1 := congr_arg (L_of_morph (gen_set_counit gen) (functoriality_group_to_pq (gen_set_counit gen))) hx,\n      simp only [monoid_hom.map_mul, list.prod_cons, list.map, L_of_morph_of, gen_set_counit_of] at hx1,\n      simp only [list.prod_cons, list.map],\n      sorry,\n    },\n    {\n      sorry,\n    },\n    /-\n    refine pq_group_word_induction _ _,\n    {\n      intro h1,\n      clear h1,\n      use (arbitrary Q)^(0 : ℤ),\n      simp only [of_pow_eq_pow_of, gpow_zero],\n    },\n    {\n      intros x z hx hxz,\n      sorry,\n    },\n    -/\n  },\nend\n\nend pq_like_eq_equalizer\n", "meta": {"author": "torstein-vik", "repo": "power-quandle-lean", "sha": "452437602c4be2e6c5ad5f5224b068baabfdf9e1", "save_path": "github-repos/lean/torstein-vik-power-quandle-lean", "path": "github-repos/lean/torstein-vik-power-quandle-lean/power-quandle-lean-452437602c4be2e6c5ad5f5224b068baabfdf9e1/src/pq_like_eq_equalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.34929065996975894}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# simp_result\n\n`dsimp_result` and `simp_result` are a pair of tactics for\napplying `dsimp` or `simp` to the result produced by other tactics.\n\nAs examples, tactics which use `revert` and `intro`\nmay insert additional `id` terms in the result they produce.\nIf there is some reason these are undesirable\n(e.g. the result term needs to be human-readable, or\nsatisfying syntactic rather than just definitional properties),\nwrapping those tactics in `dsimp_result`\ncan remove the `id` terms \"after the fact\".\n\nSimilarly, tactics using `subst` and `rw` will nearly always introduce `eq.rec` terms,\nbut sometimes these will be easy to remove,\nfor example by simplifying using `eq_rec_constant`.\nThis is a non-definitional simplification lemma,\nand so wrapping these tactics in `simp_result` will result\nin a definitionally different result.\n\nThere are several examples in the associated test file,\ndemonstrating these interactions with `revert` and `subst`.\n\nThese tactics should be used with some caution.\nYou should consider whether there is any real need for the simplification of the result,\nand whether there is a more direct way of producing the result you wanted,\nbefore relying on these tactics.\n\nBoth are implemented in terms of a generic `intercept_result` tactic,\nwhich allows you to run an arbitrary tactic and modify the returned results.\n-/\n\nnamespace tactic\n\n\n/--\n`intercept_result m t`\nattempts to run a tactic `t`,\nintercepts any results `t` assigns to the goals,\nand runs `m : expr → tactic expr` on each of the expressions\nbefore assigning the returned values to the original goals.\n\nBecause `intercept_result` uses `unsafe.type_context.assign` rather than `unify`,\nif the tactic `m` does something unreasonable\nyou may produce terms that don't typecheck,\npossibly with mysterious error messages.\nBe careful!\n-/\n-- Replace the goals with copies.\n\n-- Run the tactic on the copied goals.\n\n-- Run `m` on the produced terms,\n\n/--\n`dsimp_result t`\nattempts to run a tactic `t`,\nintercepts any results it assigns to the goals,\nand runs `dsimp` on those results\nbefore assigning the simplified values to the original goals.\n-/\n/--\n`simp_result t`\nattempts to run a tactic `t`,\nintercepts any results `t` assigns to the goals,\nand runs `simp` on those results\nbefore assigning the simplified values to the original goals.\n-/\nnamespace interactive\n\n\n/--\n`dsimp_result { tac }`\nattempts to run a tactic block `tac`,\nintercepts any results the tactic block would have assigned to the goals,\nand runs `dsimp` on those results\nbefore assigning the simplified values to the original goals.\n\nYou can use the usual interactive syntax for `dsimp`, e.g.\n`dsimp_result only [a, b, c] with attr { tac }`.\n-/\n/--\n`simp_result { tac }`\nattempts to run a tactic block `tac`,\nintercepts any results the tactic block would have assigned to the goals,\nand runs `simp` on those results\nbefore assigning the simplified values to the original goals.\n\nYou can use the usual interactive syntax for `simp`, e.g.\n`simp_result only [a, b, c] with attr { tac }`.\n-/\n/--\n`simp_result { tac }`\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/simp_result.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.34929064973699986}}
{"text": "import tactic.interactive\nimport algebra.group.basic\n\n/-!\n`refine_struct` caused a variety of interesting problems,\nwhich were identified in\nhttps://github.com/leanprover-community/mathlib/pull/2251\nand\nhttps://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Need.20help.20with.20class.20instance.20resolution\n\nThese tests are quite specific to testing the patch made in\nhttps://github.com/leanprover-community/mathlib/pull/2319\nand are not a complete test suite for `refine_struct`.\n-/\n\ninstance pi_has_one {α : Type*} {β : α → Type*} [Π x, has_one (β x)] : has_one (Π x, β x) :=\nby refine_struct { .. }; exact λ _, 1\n\nopen tactic\n\nrun_cmd (do\n  (declaration.defn _ _ _ b _ _) ← get_decl ``pi_has_one,\n  -- Make sure that `eq.mpr` really doesn't occur in the body:\n  when (b.list_constant.contains ``eq.mpr) $\n    fail \"result generated by `refine_struct` contained an unnecessary `eq.mpr`\",\n  -- Make sure that `id` really doesn't occur in the body:\n  when (b.list_constant.contains ``id) $\n    fail \"result generated by `refine_struct` contained an unnecessary `id`\")\n\n-- Next we check that fields defined for embedded structures are unfolded\n-- when seen by fields in the outer structure.\nstructure foo (α : Type):=\n(a : α)\n\nstructure bar (α : Type) extends foo α :=\n(b : a = a)\n\nexample : bar ℕ :=\nbegin\n  refine_struct { a := 1, .. },\n  -- We're making sure that the goal is\n  -- ⊢ 1 = 1\n  -- rather than\n  -- ⊢ {a := 1}.a = {a := 1}.a\n  guard_target 1 = 1,\n  trivial\nend\n\nsection\nvariables {α : Type} [_inst : monoid α]\ninclude _inst\n\nexample : true :=\nbegin\n  have : group α,\n  { refine_struct { .._inst },\n    guard_tags _field inv group, admit,\n    guard_tags _field div group, admit,\n    guard_tags _field div_eq_mul_inv group, admit,\n    guard_tags _field gpow group, admit,\n    guard_tags _field gpow_zero' group, admit,\n    guard_tags _field gpow_succ' group, admit,\n    guard_tags _field gpow_neg' group, admit,\n    guard_tags _field mul_left_inv group, admit, },\n  trivial\nend\nend\n\ndef my_foo {α} (x : semigroup α) (y : group α) : true := trivial\n\nexample {α : Type} : true :=\nbegin\n  have : true,\n  { refine_struct (@my_foo α { .. } { .. } ),\n      -- 18 goals\n    guard_tags _field mul semigroup, admit,\n      -- case semigroup, mul\n      -- α : Type\n      -- ⊢ α → α → α\n\n    guard_tags _field mul_assoc semigroup, admit,\n      -- case semigroup, mul_assoc\n      -- α : Type\n      -- ⊢ ∀ (a b c : α), a * b * c = a * (b * c)\n\n    guard_tags _field mul group, admit,\n      -- case group, mul\n      -- α : Type\n      -- ⊢ α → α → α\n\n    guard_tags _field mul_assoc group, admit,\n      -- case group, mul_assoc\n      -- α : Type\n      -- ⊢ ∀ (a b c : α), a * b * c = a * (b * c)\n\n    guard_tags _field one group, admit,\n      -- case group, one\n      -- α : Type\n      -- ⊢ α\n\n    guard_tags _field one_mul group, admit,\n      -- case group, one_mul\n      -- α : Type\n      -- ⊢ ∀ (a : α), 1 * a = a\n\n    guard_tags _field mul_one group, admit,\n      -- case group, mul_one\n      -- α : Type\n      -- ⊢ ∀ (a : α), a * 1 = a\n\n    guard_tags _field npow group, admit,\n      -- case group, npow\n      -- α : Type\n      -- ⊢ ℕ → α → α\n\n    guard_tags _field npow_zero' group, admit,\n      -- case group, inv\n      -- α : Type\n      -- ⊢ ∀ (x : α), sorry 0 x = 1\n\n    guard_tags _field npow_succ' group, admit,\n      -- case group, npow_succ'\n      -- α : Type\n      -- ⊢ ∀ (n : ℕ) (x : α), sorry n.succ x = x * sorry n x\n\n    guard_tags _field inv group, admit,\n      -- case group, inv\n      -- α : Type\n      -- ⊢ α → α\n\n    guard_tags _field div group, admit,\n      -- case group, div\n      -- α : Type\n      -- ⊢ α → α\n\n    guard_tags _field div_eq_mul_inv group, admit,\n      -- case group, div_eq_mul_inv\n      -- α : Type\n      -- ⊢ α → α\n\n    guard_tags _field gpow group, admit,\n      -- case group, gpow\n      -- α : Type\n      -- ⊢ ℤ → α → α\n\n    guard_tags _field gpow_zero' group, admit,\n      -- case group, gpow_zero'\n      -- α : Type\n      -- ⊢ ∀ (a : α), sorry 0 a = 1\n\n    guard_tags _field gpow_succ' group, admit,\n      -- case group, inv\n      -- α : Type\n      -- ⊢ ∀ (n : ℕ) (a : α), sorry (int.of_nat n.succ) a = a * sorry (int.of_nat n) a\n\n    guard_tags _field gpow_neg' group, admit,\n      -- case group, inv\n      -- α : Type\n      -- ⊢ ∀ (n : ℕ) (a : α), sorry -[1+ n] a = sorry (sorry ↑(n.succ) a)\n\n    guard_tags _field mul_left_inv group, admit,\n      -- case group, mul_left_inv\n      -- α : Type\n      -- ⊢ ∀ (a : α), a⁻¹ * a = 1\n  },\n  trivial\nend\n\ndef my_bar {α} (x : semigroup α) (y : group α) (i j : α) : α := i\n\nexample {α : Type} : true :=\nbegin\n  have : monoid α,\n  { refine_struct { mul := my_bar { .. } { .. } },\n    guard_tags _field mul semigroup, admit,\n    guard_tags _field mul_assoc semigroup, admit,\n    guard_tags _field mul group, admit,\n    guard_tags _field mul_assoc group, admit,\n    guard_tags _field one group, admit,\n    guard_tags _field one_mul group, admit,\n    guard_tags _field mul_one group, admit,\n    guard_tags _field npow group, admit,\n    guard_tags _field npow_zero' group, admit,\n    guard_tags _field npow_succ' group, admit,\n    guard_tags _field inv group, admit,\n    guard_tags _field div group, admit,\n    guard_tags _field div_eq_mul_inv group, admit,\n    guard_tags _field gpow group, admit,\n    guard_tags _field gpow_zero' group, admit,\n    guard_tags _field gpow_succ' group, admit,\n    guard_tags _field gpow_neg' group, admit,\n    guard_tags _field mul_left_inv group, admit,\n    guard_tags _field mul_assoc monoid, admit,\n    guard_tags _field one monoid, admit,\n    guard_tags _field one_mul monoid, admit,\n    guard_tags _field mul_one monoid, admit,\n    guard_tags _field npow monoid, admit,\n    guard_tags _field npow_zero' monoid, admit,\n    guard_tags _field npow_succ' monoid, admit, },\n  trivial\nend\n\ndef my_semigroup := semigroup\n\nexample {α} (mul : α → α → α) (h : false) : my_semigroup α :=\nbegin\n  refine_struct { mul := mul, .. },\n  field mul_assoc {\n    guard_target ∀ a b c : α, mul (mul a b) c = mul a (mul b c),\n    exact h.elim }\nend\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/test/refine_struct.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.629774621301746, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34919142709345613}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.abel\nimport Mathlib.category_theory.limits.shapes.biproducts\nimport Mathlib.category_theory.preadditive.default\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# Basic facts about morphisms between biproducts in preadditive categories.\n\n* In any category (with zero morphisms), if `biprod.map f g` is an isomorphism,\n  then both `f` and `g` are isomorphisms.\n\nThe remaining lemmas hold in any preadditive category.\n\n* If `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\n  then we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\n  so that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\n  via Gaussian elimination.\n\n* As a corollary of the previous two facts,\n  if we have an isomorphism `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\n  we can construct an isomorphism `X₂ ≅ Y₂`.\n\n* If `f : W ⊞ X ⟶ Y ⊞ Z` is an isomorphism, either `𝟙 W = 0`,\n  or at least one of the component maps `W ⟶ Y` and `W ⟶ Z` is nonzero.\n\n* If `f : ⨁ S ⟶ ⨁ T` is an isomorphism,\n  then every column (corresponding to a nonzero summand in the domain)\n  has some nonzero matrix entry.\n-/\n\nnamespace category_theory\n\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `f` is invertible.\n-/\ndef is_iso_left_of_is_iso_biprod_map {C : Type u} [category C] [limits.has_zero_morphisms C]\n    [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⟶ Y) (g : X ⟶ Z)\n    [is_iso (limits.biprod.map f g)] : is_iso f :=\n  is_iso.mk (limits.biprod.inl ≫ inv (limits.biprod.map f g) ≫ limits.biprod.fst)\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `g` is invertible.\n-/\ndef is_iso_right_of_is_iso_biprod_map {C : Type u} [category C] [limits.has_zero_morphisms C]\n    [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⟶ Y) (g : X ⟶ Z)\n    [is_iso (limits.biprod.map f g)] : is_iso g :=\n  let _inst : is_iso (limits.biprod.map g f) := eq.mpr sorry is_iso.comp_is_iso;\n  is_iso_left_of_is_iso_biprod_map g f\n\n/--\nThe \"matrix\" morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` with specified components.\n-/\ndef biprod.of_components {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C]\n    {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁)\n    (f₂₂ : X₂ ⟶ Y₂) : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂ :=\n  limits.biprod.fst ≫ f₁₁ ≫ limits.biprod.inl + limits.biprod.fst ≫ f₁₂ ≫ limits.biprod.inr +\n      limits.biprod.snd ≫ f₂₁ ≫ limits.biprod.inl +\n    limits.biprod.snd ≫ f₂₂ ≫ limits.biprod.inr\n\n@[simp] theorem biprod.inl_of_components {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁)\n    (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) :\n    limits.biprod.inl ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ =\n        f₁₁ ≫ limits.biprod.inl + f₁₂ ≫ limits.biprod.inr :=\n  sorry\n\n@[simp] theorem biprod.inr_of_components {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁)\n    (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) :\n    limits.biprod.inr ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ =\n        f₂₁ ≫ limits.biprod.inl + f₂₂ ≫ limits.biprod.inr :=\n  sorry\n\n@[simp] theorem biprod.of_components_fst {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁)\n    (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) :\n    biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ limits.biprod.fst =\n        limits.biprod.fst ≫ f₁₁ + limits.biprod.snd ≫ f₂₁ :=\n  sorry\n\n@[simp] theorem biprod.of_components_snd {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁)\n    (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) :\n    biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ limits.biprod.snd =\n        limits.biprod.fst ≫ f₁₂ + limits.biprod.snd ≫ f₂₂ :=\n  sorry\n\n@[simp] theorem biprod.of_components_eq {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) :\n    biprod.of_components (limits.biprod.inl ≫ f ≫ limits.biprod.fst)\n          (limits.biprod.inl ≫ f ≫ limits.biprod.snd) (limits.biprod.inr ≫ f ≫ limits.biprod.fst)\n          (limits.biprod.inr ≫ f ≫ limits.biprod.snd) =\n        f :=\n  sorry\n\n@[simp] theorem biprod.of_components_comp {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} {Z₁ : C} {Z₂ : C}\n    (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) (g₁₁ : Y₁ ⟶ Z₁) (g₁₂ : Y₁ ⟶ Z₂)\n    (g₂₁ : Y₂ ⟶ Z₁) (g₂₂ : Y₂ ⟶ Z₂) :\n    biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ biprod.of_components g₁₁ g₁₂ g₂₁ g₂₂ =\n        biprod.of_components (f₁₁ ≫ g₁₁ + f₁₂ ≫ g₂₁) (f₁₁ ≫ g₁₂ + f₁₂ ≫ g₂₂) (f₂₁ ≫ g₁₁ + f₂₂ ≫ g₂₁)\n          (f₂₁ ≫ g₁₂ + f₂₂ ≫ g₂₂) :=\n  sorry\n\n/--\nThe unipotent upper triangular matrix\n```\n(1 r)\n(0 1)\n```\nas an isomorphism.\n-/\n@[simp] theorem biprod.unipotent_upper_hom {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} (r : X₁ ⟶ X₂) :\n    iso.hom (biprod.unipotent_upper r) = biprod.of_components 𝟙 r 0 𝟙 :=\n  Eq.refl (iso.hom (biprod.unipotent_upper r))\n\n/--\nThe unipotent lower triangular matrix\n```\n(1 0)\n(r 1)\n```\nas an isomorphism.\n-/\n@[simp] theorem biprod.unipotent_lower_hom {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} (r : X₂ ⟶ X₁) :\n    iso.hom (biprod.unipotent_lower r) = biprod.of_components 𝟙 0 r 𝟙 :=\n  Eq.refl (iso.hom (biprod.unipotent_lower r))\n\n/--\nIf `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\nso that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\nvia Gaussian elimination.\n\n(This is the version of `biprod.gaussian` written in terms of components.)\n-/\ndef biprod.gaussian' {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C]\n    {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁)\n    (f₂₂ : X₂ ⟶ Y₂) [is_iso f₁₁] :\n    psigma\n        fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n          psigma\n            fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n              psigma\n                fun (g₂₂ : X₂ ⟶ Y₂) =>\n                  iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R =\n                    limits.biprod.map f₁₁ g₂₂ :=\n  psigma.mk (biprod.unipotent_lower (-f₂₁ ≫ inv f₁₁))\n    (psigma.mk (biprod.unipotent_upper (-inv f₁₁ ≫ f₁₂))\n      (psigma.mk (f₂₂ - f₂₁ ≫ inv f₁₁ ≫ f₁₂) sorry))\n\n/--\nIf `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\nso that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\nvia Gaussian elimination.\n-/\ndef biprod.gaussian {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C]\n    {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂)\n    [is_iso (limits.biprod.inl ≫ f ≫ limits.biprod.fst)] :\n    psigma\n        fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n          psigma\n            fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n              psigma\n                fun (g₂₂ : X₂ ⟶ Y₂) =>\n                  iso.hom L ≫ f ≫ iso.hom R =\n                    limits.biprod.map (limits.biprod.inl ≫ f ≫ limits.biprod.fst) g₂₂ :=\n  let this :\n    psigma\n      fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n        psigma\n          fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n            psigma\n              fun (g₂₂ : X₂ ⟶ Y₂) =>\n                iso.hom L ≫\n                    biprod.of_components (limits.biprod.inl ≫ f ≫ limits.biprod.fst)\n                        (limits.biprod.inl ≫ f ≫ limits.biprod.snd)\n                        (limits.biprod.inr ≫ f ≫ limits.biprod.fst)\n                        (limits.biprod.inr ≫ f ≫ limits.biprod.snd) ≫\n                      iso.hom R =\n                  limits.biprod.map (limits.biprod.inl ≫ f ≫ limits.biprod.fst) g₂₂ :=\n    biprod.gaussian' (limits.biprod.inl ≫ f ≫ limits.biprod.fst)\n      (limits.biprod.inl ≫ f ≫ limits.biprod.snd) (limits.biprod.inr ≫ f ≫ limits.biprod.fst)\n      (limits.biprod.inr ≫ f ≫ limits.biprod.snd);\n  eq.mpr sorry (eq.mp sorry this)\n\n/--\nIf `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` via a two-by-two matrix whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct an isomorphism `X₂ ≅ Y₂`, via Gaussian elimination.\n-/\ndef biprod.iso_elim' {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C]\n    {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁)\n    (f₂₂ : X₂ ⟶ Y₂) [is_iso f₁₁] [is_iso (biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂)] : X₂ ≅ Y₂ :=\n  psigma.cases_on (biprod.gaussian' f₁₁ f₁₂ f₂₁ f₂₂)\n    fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂)\n      (snd :\n      psigma\n        fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n          psigma\n            fun (g₂₂ : X₂ ⟶ Y₂) =>\n              iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R =\n                limits.biprod.map f₁₁ g₂₂) =>\n      psigma.cases_on snd\n        fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂)\n          (snd_snd :\n          psigma\n            fun (g₂₂ : X₂ ⟶ Y₂) =>\n              iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R =\n                limits.biprod.map f₁₁ g₂₂) =>\n          psigma.cases_on snd_snd\n            fun (g : X₂ ⟶ Y₂)\n              (w :\n              iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R =\n                limits.biprod.map f₁₁ g) =>\n              let _inst : is_iso (limits.biprod.map f₁₁ g) := eq.mpr sorry is_iso.comp_is_iso;\n              let _inst_6 : is_iso g := is_iso_right_of_is_iso_biprod_map f₁₁ g;\n              as_iso g\n\n/--\nIf `f` is an isomorphism `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct an isomorphism `X₂ ≅ Y₂`, via Gaussian elimination.\n-/\ndef biprod.iso_elim {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C]\n    {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂)\n    [is_iso (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst)] : X₂ ≅ Y₂ :=\n  let _inst :\n    is_iso\n      (biprod.of_components (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst)\n        (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.snd)\n        (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.fst)\n        (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.snd)) :=\n    eq.mpr sorry (is_iso.of_iso f);\n  biprod.iso_elim' (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst)\n    (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.snd)\n    (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.fst)\n    (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.snd)\n\ntheorem biprod.column_nonzero_of_iso {C : Type u} [category C] [preadditive C]\n    [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⊞ X ⟶ Y ⊞ Z)\n    [is_iso f] :\n    𝟙 = 0 ∨\n        limits.biprod.inl ≫ f ≫ limits.biprod.fst ≠ 0 ∨\n          limits.biprod.inl ≫ f ≫ limits.biprod.snd ≠ 0 :=\n  sorry\n\ntheorem biproduct.column_nonzero_of_iso' {C : Type u} [category C] [preadditive C] {σ : Type v}\n    {τ : Type v} [DecidableEq σ] [DecidableEq τ] [fintype τ] {S : σ → C} [limits.has_biproduct S]\n    {T : τ → C} [limits.has_biproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [is_iso f] :\n    (∀ (t : τ), limits.biproduct.ι S s ≫ f ≫ limits.biproduct.π T t = 0) → 𝟙 = 0 :=\n  sorry\n\n/--\nIf `f : ⨁ S ⟶ ⨁ T` is an isomorphism, and `s` is a non-trivial summand of the source,\nthen there is some `t` in the target so that the `s, t` matrix entry of `f` is nonzero.\n-/\ndef biproduct.column_nonzero_of_iso {C : Type u} [category C] [preadditive C] {σ : Type v}\n    {τ : Type v} [DecidableEq σ] [DecidableEq τ] [fintype τ] {S : σ → C} [limits.has_biproduct S]\n    {T : τ → C} [limits.has_biproduct T] (s : σ) (nz : 𝟙 ≠ 0) [(t : τ) → DecidableEq (S s ⟶ T t)]\n    (f : ⨁ S ⟶ ⨁ T) [is_iso f] :\n    trunc (psigma fun (t : τ) => limits.biproduct.ι S s ≫ f ≫ limits.biproduct.π T t ≠ 0) :=\n  trunc_sigma_of_exists sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/preadditive/biproducts_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.629774621301746, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34919142709345613}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.category.Group.basic\nimport category_theory.single_obj\nimport category_theory.limits.functor_category\nimport category_theory.limits.preserves.basic\nimport category_theory.adjunction.limits\nimport category_theory.monoidal.functor_category\nimport category_theory.monoidal.transport\nimport category_theory.monoidal.rigid.of_equivalence\nimport category_theory.monoidal.rigid.functor_category\nimport category_theory.monoidal.linear\nimport category_theory.monoidal.braided\nimport category_theory.abelian.functor_category\nimport category_theory.abelian.transfer\nimport category_theory.conj\nimport category_theory.linear.functor_category\n\n/-!\n# `Action V G`, the category of actions of a monoid `G` inside some category `V`.\n\nThe prototypical example is `V = Module R`,\nwhere `Action (Module R) G` is the category of `R`-linear representations of `G`.\n\nWe check `Action V G ≌ (single_obj G ⥤ V)`,\nand construct the restriction functors `res {G H : Mon} (f : G ⟶ H) : Action V H ⥤ Action V G`.\n\n* When `V` has (co)limits so does `Action V G`.\n* When `V` is monoidal, braided, or symmetric, so is `Action V G`.\n* When `V` is preadditive, linear, or abelian so is `Action V G`.\n-/\n\nuniverses u\n\nopen category_theory\nopen category_theory.limits\n\nvariables (V : Type (u+1)) [large_category V]\n\n/--\nAn `Action V G` represents a bundled action of\nthe monoid `G` on an object of some category `V`.\n\nAs an example, when `V = Module R`, this is an `R`-linear representation of `G`,\nwhile when `V = Type` this is a `G`-action.\n-/\n-- Note: this is _not_ a categorical action of `G` on `V`.\nstructure Action (G : Mon.{u}) :=\n(V : V)\n(ρ : G ⟶ Mon.of (End V))\n\nnamespace Action\nvariable {V}\n\n@[simp]\nlemma ρ_one {G : Mon.{u}} (A : Action V G) : A.ρ 1 = 𝟙 A.V :=\nby { rw [monoid_hom.map_one], refl, }\n\n/-- When a group acts, we can lift the action to the group of automorphisms. -/\n@[simps]\ndef ρ_Aut {G : Group.{u}} (A : Action V (Mon.of G)) : G ⟶ Group.of (Aut A.V) :=\n{ to_fun := λ g,\n  { hom := A.ρ g,\n    inv := A.ρ (g⁻¹ : G),\n    hom_inv_id' := ((A.ρ).map_mul (g⁻¹ : G) g).symm.trans (by rw [inv_mul_self, ρ_one]),\n    inv_hom_id' := ((A.ρ).map_mul g (g⁻¹ : G)).symm.trans (by rw [mul_inv_self, ρ_one]), },\n  map_one' := by { ext, exact A.ρ.map_one },\n  map_mul' := λ x y, by { ext, exact A.ρ.map_mul x y }, }\n\nvariable (G : Mon.{u})\n\nsection\n\n/-- The trivial representation of a group. -/\ndef trivial : Action AddCommGroup G :=\n{ V := AddCommGroup.of punit,\n  ρ := 1, }\n\ninstance : inhabited (Action AddCommGroup G) := ⟨trivial G⟩\nend\n\nvariables {G V}\n\n/--\nA homomorphism of `Action V G`s is a morphism between the underlying objects,\ncommuting with the action of `G`.\n-/\n@[ext]\nstructure hom (M N : Action V G) :=\n(hom : M.V ⟶ N.V)\n(comm' : ∀ g : G, M.ρ g ≫ hom = hom ≫ N.ρ g . obviously)\n\nrestate_axiom hom.comm'\n\nnamespace hom\n\n/-- The identity morphism on a `Action V G`. -/\n@[simps]\ndef id (M : Action V G) : Action.hom M M :=\n{ hom := 𝟙 M.V }\n\ninstance (M : Action V G) : inhabited (Action.hom M M) := ⟨id M⟩\n\n/--\nThe composition of two `Action V G` homomorphisms is the composition of the underlying maps.\n-/\n@[simps]\ndef comp {M N K : Action V G} (p : Action.hom M N) (q : Action.hom N K) :\n  Action.hom M K :=\n{ hom := p.hom ≫ q.hom,\n  comm' := λ g, by rw [←category.assoc, p.comm, category.assoc, q.comm, ←category.assoc] }\n\nend hom\n\ninstance : category (Action V G) :=\n{ hom := λ M N, hom M N,\n  id := λ M, hom.id M,\n  comp := λ M N K f g, hom.comp f g, }\n\n@[simp]\nlemma id_hom (M : Action V G) : (𝟙 M : hom M M).hom = 𝟙 M.V := rfl\n@[simp]\nlemma comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :\n  (f ≫ g : hom M K).hom = f.hom ≫ g.hom :=\nrfl\n\n/-- Construct an isomorphism of `G` actions/representations\nfrom an isomorphism of the the underlying objects,\nwhere the forward direction commutes with the group action. -/\n@[simps]\ndef mk_iso {M N : Action V G} (f : M.V ≅ N.V) (comm : ∀ g : G, M.ρ g ≫ f.hom = f.hom ≫ N.ρ g) :\n  M ≅ N :=\n{ hom :=\n  { hom := f.hom,\n    comm' := comm, },\n  inv :=\n  { hom := f.inv,\n    comm' := λ g, by { have w := comm g =≫ f.inv, simp at w, simp [w], }, }}\n\nnamespace functor_category_equivalence\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef functor : Action V G ⥤ (single_obj G ⥤ V) :=\n{ obj := λ M,\n  { obj := λ _, M.V,\n    map := λ _ _ g, M.ρ g,\n    map_id' := λ _, M.ρ.map_one,\n    map_comp' := λ _ _ _ g h, M.ρ.map_mul h g, },\n  map := λ M N f,\n  { app := λ _, f.hom,\n    naturality' := λ _ _ g, f.comm g, } }\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef inverse : (single_obj G ⥤ V) ⥤ Action V G :=\n{ obj := λ F,\n  { V := F.obj punit.star,\n    ρ :=\n    { to_fun := λ g, F.map g,\n      map_one' := F.map_id punit.star,\n      map_mul' := λ g h, F.map_comp h g, } },\n  map := λ M N f,\n  { hom := f.app punit.star,\n    comm' := λ g, f.naturality g, } }.\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef unit_iso : 𝟭 (Action V G) ≅ functor ⋙ inverse :=\nnat_iso.of_components (λ M, mk_iso ((iso.refl _)) (by tidy)) (by tidy).\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef counit_iso : inverse ⋙ functor ≅ 𝟭 (single_obj G ⥤ V) :=\nnat_iso.of_components (λ M, nat_iso.of_components (by tidy) (by tidy)) (by tidy).\n\nend functor_category_equivalence\n\nsection\nopen functor_category_equivalence\n\nvariables (V G)\n\n/--\nThe category of actions of `G` in the category `V`\nis equivalent to the functor category `single_obj G ⥤ V`.\n-/\ndef functor_category_equivalence : Action V G ≌ (single_obj G ⥤ V) :=\n{ functor := functor,\n  inverse := inverse,\n  unit_iso := unit_iso,\n  counit_iso := counit_iso, }\n\nattribute [simps] functor_category_equivalence\n\ninstance [has_finite_products V] : has_finite_products (Action V G) :=\n{ out := λ J _, by exactI\n  adjunction.has_limits_of_shape_of_equivalence (Action.functor_category_equivalence _ _).functor }\n\ninstance [has_limits V] : has_limits (Action V G) :=\nadjunction.has_limits_of_equivalence (Action.functor_category_equivalence _ _).functor\n\ninstance [has_colimits V] : has_colimits (Action V G) :=\nadjunction.has_colimits_of_equivalence (Action.functor_category_equivalence _ _).functor\n\nend\n\nsection forget\n\nvariables (V G)\n\n/-- (implementation) The forgetful functor from bundled actions to the underlying objects.\n\nUse the `category_theory.forget` API provided by the `concrete_category` instance below,\nrather than using this directly.\n-/\n@[simps]\ndef forget : Action V G ⥤ V :=\n{ obj := λ M, M.V,\n  map := λ M N f, f.hom, }\n\ninstance : faithful (forget V G) :=\n{ map_injective' := λ X Y f g w, hom.ext _ _ w, }\n\ninstance [concrete_category V] : concrete_category (Action V G) :=\n{ forget := forget V G ⋙ (concrete_category.forget V), }\n\ninstance has_forget_to_V [concrete_category V] : has_forget₂ (Action V G) V :=\n{ forget₂ := forget V G }\n\n/-- The forgetful functor is intertwined by `functor_category_equivalence` with\nevaluation at `punit.star`. -/\ndef functor_category_equivalence_comp_evaluation :\n  (functor_category_equivalence V G).functor ⋙ (evaluation _ _).obj punit.star ≅ forget V G :=\niso.refl _\n\nnoncomputable instance [has_limits V] : limits.preserves_limits (forget V G) :=\nlimits.preserves_limits_of_nat_iso\n  (Action.functor_category_equivalence_comp_evaluation V G)\n\nnoncomputable instance [has_colimits V] : preserves_colimits (forget V G) :=\npreserves_colimits_of_nat_iso\n  (Action.functor_category_equivalence_comp_evaluation V G)\n\n-- TODO construct categorical images?\n\nend forget\n\nlemma iso.conj_ρ {M N : Action V G} (f : M ≅ N) (g : G) :\n   N.ρ g = (((forget V G).map_iso f).conj (M.ρ g)) :=\nby { rw [iso.conj_apply, iso.eq_inv_comp], simp [f.hom.comm'] }\n\nsection has_zero_morphisms\nvariables [has_zero_morphisms V]\n\ninstance : has_zero_morphisms (Action V G) :=\n{ has_zero := λ X Y, ⟨⟨0, by tidy⟩⟩, }\n\ninstance : functor.preserves_zero_morphisms (functor_category_equivalence V G).functor := {}\n\nend has_zero_morphisms\n\nsection preadditive\nvariables [preadditive V]\n\ninstance : preadditive (Action V G) :=\n{ hom_group := λ X Y,\n  { zero := ⟨0, by simp⟩,\n    add := λ f g, ⟨f.hom + g.hom, by simp [f.comm, g.comm]⟩,\n    neg := λ f, ⟨-f.hom, by simp [f.comm]⟩,\n    zero_add := by { intros, ext, exact zero_add _, },\n    add_zero := by { intros, ext, exact add_zero _, },\n    add_assoc := by { intros, ext, exact add_assoc _ _ _, },\n    add_left_neg := by { intros, ext, exact add_left_neg _, },\n    add_comm := by { intros, ext, exact add_comm _ _, }, },\n  add_comp' := by { intros, ext, exact preadditive.add_comp _ _ _ _ _ _, },\n  comp_add' := by { intros, ext, exact preadditive.comp_add _ _ _ _ _ _, }, }\n\ninstance : functor.additive (functor_category_equivalence V G).functor := {}\n\n@[simp] lemma zero_hom {X Y : Action V G} : (0 : X ⟶ Y).hom = 0 := rfl\n@[simp] lemma neg_hom {X Y : Action V G} (f : X ⟶ Y) : (-f).hom = -f.hom := rfl\n@[simp] lemma add_hom {X Y : Action V G} (f g : X ⟶ Y) : (f + g).hom = f.hom + g.hom := rfl\n\nend preadditive\n\nsection linear\nvariables [preadditive V] {R : Type*} [semiring R] [linear R V]\n\ninstance : linear R (Action V G) :=\n{ hom_module := λ X Y,\n  { smul := λ r f, ⟨r • f.hom, by simp [f.comm]⟩,\n    one_smul := by { intros, ext, exact one_smul _ _, },\n    smul_zero := by { intros, ext, exact smul_zero _, },\n    zero_smul := by { intros, ext, exact zero_smul _ _, },\n    add_smul := by { intros, ext, exact add_smul _ _ _, },\n    smul_add := by { intros, ext, exact smul_add _ _ _, },\n    mul_smul := by { intros, ext, exact mul_smul _ _ _, }, },\n  smul_comp' := by { intros, ext, exact linear.smul_comp _ _ _ _ _ _, },\n  comp_smul' := by { intros, ext, exact linear.comp_smul _ _ _ _ _ _, }, }\n\ninstance : functor.linear R (functor_category_equivalence V G).functor := {}\n\n@[simp] lemma smul_hom {X Y : Action V G} (r : R) (f : X ⟶ Y) : (r • f).hom = r • f.hom := rfl\n\nend linear\n\nsection abelian\n/-- Auxilliary construction for the `abelian (Action V G)` instance. -/\ndef abelian_aux : Action V G ≌ (ulift.{u} (single_obj G) ⥤ V) :=\n(functor_category_equivalence V G).trans (equivalence.congr_left ulift.equivalence)\n\nnoncomputable instance [abelian V] : abelian (Action V G) :=\nabelian_of_equivalence abelian_aux.functor\n\nend abelian\n\nsection monoidal\nvariables [monoidal_category V]\n\ninstance : monoidal_category (Action V G) :=\nmonoidal.transport (Action.functor_category_equivalence _ _).symm\n\n@[simp] lemma tensor_V {X Y : Action V G} : (X ⊗ Y).V = X.V ⊗ Y.V := rfl\n@[simp] lemma tensor_rho {X Y : Action V G} {g : G} : (X ⊗ Y).ρ g = X.ρ g ⊗ Y.ρ g := rfl\n@[simp] lemma tensor_hom {W X Y Z : Action V G} (f : W ⟶ X) (g : Y ⟶ Z) :\n  (f ⊗ g).hom = f.hom ⊗ g.hom := rfl\n@[simp] lemma associator_hom_hom {X Y Z : Action V G} :\n  hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom :=\nbegin\n  dsimp [monoidal.transport_associator],\n  simp,\nend\n@[simp] lemma associator_inv_hom {X Y Z : Action V G} :\n  hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv :=\nbegin\n  dsimp [monoidal.transport_associator],\n  simp,\nend\n@[simp] lemma left_unitor_hom_hom {X : Action V G} :\n  hom.hom (λ_ X).hom = (λ_ X.V).hom :=\nbegin\n  dsimp [monoidal.transport_left_unitor],\n  simp,\nend\n@[simp] lemma left_unitor_inv_hom {X : Action V G} :\n  hom.hom (λ_ X).inv = (λ_ X.V).inv :=\nbegin\n  dsimp [monoidal.transport_left_unitor],\n  simp,\nend\n@[simp] lemma right_unitor_hom_hom {X : Action V G} :\n  hom.hom (ρ_ X).hom = (ρ_ X.V).hom :=\nbegin\n  dsimp [monoidal.transport_right_unitor],\n  simp,\nend\n@[simp] lemma right_unitor_inv_hom {X : Action V G} :\n  hom.hom (ρ_ X).inv = (ρ_ X.V).inv :=\nbegin\n  dsimp [monoidal.transport_right_unitor],\n  simp,\nend\n\nvariables (V G)\n\n/-- When `V` is monoidal the forgetful functor `Action V G` to `V` is monoidal. -/\n@[simps]\ndef forget_monoidal : monoidal_functor (Action V G) V :=\n{ ε := 𝟙 _,\n  μ := λ X Y, 𝟙 _,\n  ..Action.forget _ _, }\n\ninstance forget_monoidal_faithful : faithful (forget_monoidal V G).to_functor :=\nby { change faithful (forget V G), apply_instance, }\n\nsection\nvariables [braided_category V]\n\ninstance : braided_category (Action V G) :=\nbraided_category_of_faithful (forget_monoidal V G) (λ X Y, mk_iso (β_ _ _) (by tidy)) (by tidy)\n\n/-- When `V` is braided the forgetful functor `Action V G` to `V` is braided. -/\n@[simps]\ndef forget_braided : braided_functor (Action V G) V :=\n{ ..forget_monoidal _ _, }\n\ninstance forget_braided_faithful : faithful (forget_braided V G).to_functor :=\nby { change faithful (forget V G), apply_instance, }\n\nend\n\ninstance [symmetric_category V] : symmetric_category (Action V G) :=\nsymmetric_category_of_faithful (forget_braided V G)\n\nsection\nlocal attribute [simp] monoidal_preadditive.tensor_add monoidal_preadditive.add_tensor\n\nvariables [preadditive V] [monoidal_preadditive V]\n\ninstance : monoidal_preadditive (Action V G) := {}\n\nvariables {R : Type*} [semiring R] [linear R V] [monoidal_linear R V]\n\ninstance : monoidal_linear R (Action V G) := {}\n\nend\n\nvariables (V G)\nnoncomputable theory\n\n/-- Upgrading the functor `Action V G ⥤ (single_obj G ⥤ V)` to a monoidal functor. -/\ndef functor_category_monoidal_equivalence : monoidal_functor (Action V G) (single_obj G ⥤ V) :=\nmonoidal.from_transported (Action.functor_category_equivalence _ _).symm\n\ninstance : is_equivalence ((functor_category_monoidal_equivalence V G).to_functor) :=\nby { change is_equivalence (Action.functor_category_equivalence _ _).functor, apply_instance, }\n\nvariables (H : Group.{u})\n\ninstance [right_rigid_category V] : right_rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change right_rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is right rigid, so is `Action V G`. -/\ninstance [right_rigid_category V] : right_rigid_category (Action V H) :=\nright_rigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\ninstance [left_rigid_category V] : left_rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change left_rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is left rigid, so is `Action V G`. -/\ninstance [left_rigid_category V] : left_rigid_category (Action V H) :=\nleft_rigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\ninstance [rigid_category V] : rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is rigid, so is `Action V G`. -/\ninstance [rigid_category V] : rigid_category (Action V H) :=\nrigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\nvariables {V H} (X : Action V H)\n\n@[simp] lemma right_dual_V [right_rigid_category V] : (Xᘁ).V = (X.V)ᘁ := rfl\n\n@[simp] lemma left_dual_V [left_rigid_category V] : (ᘁX).V = ᘁ(X.V) := rfl\n\n@[simp] lemma right_dual_ρ [right_rigid_category V] (h : H) : (Xᘁ).ρ h = (X.ρ (h⁻¹ : H))ᘁ :=\nby { rw ←single_obj.inv_as_inv, refl }\n\n@[simp] lemma left_dual_ρ [left_rigid_category V] (h : H) : (ᘁX).ρ h = ᘁ(X.ρ (h⁻¹ : H)) :=\nby { rw ←single_obj.inv_as_inv, refl }\n\nend monoidal\n\n/-- Actions/representations of the trivial group are just objects in the ambient category. -/\ndef Action_punit_equivalence : Action V (Mon.of punit) ≌ V :=\n{ functor := forget V _,\n  inverse :=\n  { obj := λ X, ⟨X, 1⟩,\n    map := λ X Y f, ⟨f, λ ⟨⟩, by simp⟩, },\n  unit_iso := nat_iso.of_components (λ X, mk_iso (iso.refl _) (λ ⟨⟩, by simpa using ρ_one X))\n    (by tidy),\n  counit_iso := nat_iso.of_components (λ X, iso.refl _) (by tidy), }\n\nvariables (V)\n/--\nThe \"restriction\" functor along a monoid homomorphism `f : G ⟶ H`,\ntaking actions of `H` to actions of `G`.\n\n(This makes sense for any homomorphism, but the name is natural when `f` is a monomorphism.)\n-/\n@[simps]\ndef res {G H : Mon} (f : G ⟶ H) : Action V H ⥤ Action V G :=\n{ obj := λ M,\n  { V := M.V,\n    ρ := f ≫ M.ρ },\n  map := λ M N p,\n  { hom := p.hom,\n    comm' := λ g, p.comm (f g) } }\n\n/--\nThe natural isomorphism from restriction along the identity homomorphism to\nthe identity functor on `Action V G`.\n-/\ndef res_id {G : Mon} : res V (𝟙 G) ≅ 𝟭 (Action V G) :=\nnat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy)\n\nattribute [simps] res_id\n\n/--\nThe natural isomorphism from the composition of restrictions along homomorphisms\nto the restriction along the composition of homomorphism.\n-/\ndef res_comp {G H K : Mon} (f : G ⟶ H) (g : H ⟶ K) : res V g ⋙ res V f ≅ res V (f ≫ g) :=\nnat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy)\n\nattribute [simps] res_comp\n\n-- TODO promote `res` to a pseudofunctor from\n-- the locally discrete bicategory constructed from `Monᵒᵖ` to `Cat`, sending `G` to `Action V G`.\n\nvariables {G} {H : Mon.{u}} (f : G ⟶ H)\n\ninstance res_additive [preadditive V] : (res V f).additive := {}\n\nvariables {R : Type*} [semiring R]\n\ninstance res_linear [preadditive V] [linear R V] : (res V f).linear R := {}\n\nend Action\n\nnamespace category_theory.functor\n\nvariables {V} {W : Type (u+1)} [large_category W]\n\n/-- A functor between categories induces a functor between\nthe categories of `G`-actions within those categories. -/\n@[simps]\ndef map_Action (F : V ⥤ W) (G : Mon.{u}) : Action V G ⥤ Action W G :=\n{ obj := λ M,\n  { V := F.obj M.V,\n    ρ :=\n    { to_fun := λ g, F.map (M.ρ g),\n      map_one' := by simp only [End.one_def, Action.ρ_one, F.map_id],\n      map_mul' := λ g h, by simp only [End.mul_def, F.map_comp, map_mul], }, },\n  map := λ M N f,\n  { hom := F.map f.hom,\n    comm' := λ g, by { dsimp, rw [←F.map_comp, f.comm, F.map_comp], }, },\n  map_id' := λ M, by { ext, simp only [Action.id_hom, F.map_id], },\n  map_comp' := λ M N P f g, by { ext, simp only [Action.comp_hom, F.map_comp], }, }\n\nvariables (F : V ⥤ W) (G : Mon.{u}) [preadditive V] [preadditive W]\n\ninstance map_Action_preadditive [F.additive] : (F.map_Action G).additive := {}\n\nvariables {R : Type*} [semiring R] [category_theory.linear R V] [category_theory.linear R W]\n\ninstance map_Action_linear [F.additive] [F.linear R] : (F.map_Action G).linear R := {}\n\nend category_theory.functor\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/representation_theory/Action.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34919141938779563}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.basic\n\n/-!\n# Prefixes, subfixes, infixes\n\nThis file proves properties about\n* `list.prefix`: `l₁` is a prefix of `l₂` if `l₂` starts with `l₁`.\n* `list.subfix`: `l₁` is a subfix of `l₂` if `l₂` ends with `l₁`.\n* `list.infix`: `l₁` is an infix of `l₂` if `l₁` is a prefix of some subfix of `l₂`.\n* `list.inits`: The list of prefixes of a list.\n* `list.tails`: The list of prefixes of a list.\n* `insert` on lists\n\nAll those (except `insert`) are defined in `data.list.defs`.\n\n## Notation\n\n`l₁ <+: l₂`: `l₁` is a prefix of `l₂`.\n`l₁ <:+ l₂`: `l₁` is a subfix of `l₂`.\n`l₁ <:+: l₂`: `l₁` is an infix of `l₂`.\n-/\n\nopen nat\n\nvariables {α β : Type*}\n\nnamespace list\nvariables {l l₁ l₂ l₃ : list α} {a b : α} {m n : ℕ}\n\n/-! ### prefix, suffix, infix -/\n\nsection fix\n\n@[simp] lemma prefix_append (l₁ l₂ : list α) : l₁ <+: l₁ ++ l₂ := ⟨l₂, rfl⟩\n@[simp] lemma suffix_append (l₁ l₂ : list α) : l₂ <:+ l₁ ++ l₂ := ⟨l₁, rfl⟩\n\nlemma infix_append (l₁ l₂ l₃ : list α) : l₂ <:+: l₁ ++ l₂ ++ l₃ := ⟨l₁, l₃, rfl⟩\n\n@[simp] lemma infix_append' (l₁ l₂ l₃ : list α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) :=\nby rw ← list.append_assoc; apply infix_append\n\nlemma is_prefix.is_infix : l₁ <+: l₂ → l₁ <:+: l₂ := λ ⟨t, h⟩, ⟨[], t, h⟩\nlemma is_suffix.is_infix : l₁ <:+ l₂ → l₁ <:+: l₂ := λ ⟨t, h⟩, ⟨t, [], by rw [h, append_nil]⟩\n\nlemma nil_prefix (l : list α) : [] <+: l := ⟨l, rfl⟩\nlemma nil_suffix (l : list α) : [] <:+ l := ⟨l, append_nil _⟩\nlemma nil_infix (l : list α) : [] <:+: l := (nil_prefix _).is_infix\n\n@[refl] lemma prefix_refl (l : list α) : l <+: l := ⟨[], append_nil _⟩\n@[refl] lemma suffix_refl (l : list α) : l <:+ l := ⟨[], rfl⟩\n@[refl] lemma infix_refl (l : list α) : l <:+: l := (prefix_refl l).is_infix\n\nlemma prefix_rfl : l <+: l := prefix_refl _\nlemma suffix_rfl : l <:+ l := suffix_refl _\nlemma infix_rfl : l <:+: l := infix_refl _\n\n@[simp] lemma suffix_cons (a : α) : ∀ l, l <:+ a :: l := suffix_append [a]\n\nlemma prefix_concat (a : α) (l) : l <+: concat l a := by simp\n\nlemma infix_cons : l₁ <:+: l₂ → l₁ <:+: a :: l₂ := λ ⟨L₁, L₂, h⟩, ⟨a :: L₁, L₂, h ▸ rfl⟩\nlemma infix_concat : l₁ <:+: l₂ → l₁ <:+: concat l₂ a :=\nλ ⟨L₁, L₂, h⟩, ⟨L₁, concat L₂ a, by simp_rw [←h, concat_eq_append, append_assoc]⟩\n\n@[trans] lemma is_prefix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <+: l₂ → l₂ <+: l₃ → l₁ <+: l₃\n| l ._ ._ ⟨r₁, rfl⟩ ⟨r₂, rfl⟩ := ⟨r₁ ++ r₂, (append_assoc _ _ _).symm⟩\n\n@[trans] lemma is_suffix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <:+ l₂ → l₂ <:+ l₃ → l₁ <:+ l₃\n| l ._ ._ ⟨l₁, rfl⟩ ⟨l₂, rfl⟩ := ⟨l₂ ++ l₁, append_assoc _ _ _⟩\n\n@[trans] lemma is_infix.trans : ∀ {l₁ l₂ l₃ : list α}, l₁ <:+: l₂ → l₂ <:+: l₃ → l₁ <:+: l₃\n| l ._ ._ ⟨l₁, r₁, rfl⟩ ⟨l₂, r₂, rfl⟩ := ⟨l₂ ++ l₁, r₁ ++ r₂, by simp only [append_assoc]⟩\n\nprotected lemma is_infix.sublist : l₁ <:+: l₂ → l₁ <+ l₂ :=\nλ ⟨s, t, h⟩, by { rw [← h], exact (sublist_append_right _ _).trans (sublist_append_left _ _) }\n\nprotected lemma is_prefix.sublist (h : l₁ <+: l₂) : l₁ <+ l₂ := h.is_infix.sublist\nprotected lemma is_suffix.sublist (h : l₁ <:+ l₂) : l₁ <+ l₂ := h.is_infix.sublist\n\n@[simp] lemma reverse_suffix : reverse l₁ <:+ reverse l₂ ↔ l₁ <+: l₂ :=\n⟨λ ⟨r, e⟩, ⟨reverse r,\n  by rw [← reverse_reverse l₁, ← reverse_append, e, reverse_reverse]⟩,\n λ ⟨r, e⟩, ⟨reverse r, by rw [← reverse_append, e]⟩⟩\n\n@[simp] lemma reverse_prefix : reverse l₁ <+: reverse l₂ ↔ l₁ <:+ l₂ :=\nby rw ← reverse_suffix; simp only [reverse_reverse]\n\nalias reverse_prefix ↔ _ list.is_suffix.reverse\nalias reverse_suffix ↔ _ list.is_prefix.reverse\n\nlemma infix.length_le (h : l₁ <:+: l₂) : length l₁ ≤ length l₂ := length_le_of_sublist h.sublist\n\nlemma eq_nil_of_infix_nil (h : l <:+: []) : l = [] := eq_nil_of_sublist_nil h.sublist\n\n@[simp] lemma infix_nil_iff : l <:+: [] ↔ l = [] :=\n⟨λ h, eq_nil_of_sublist_nil h.sublist, λ h, h ▸ infix_rfl⟩\n\nalias infix_nil_iff ↔ list.eq_nil_of_infix_nil _\n\n@[simp] lemma prefix_nil_iff : l <+: [] ↔ l = [] :=\n⟨λ h, eq_nil_of_infix_nil h.is_infix, λ h, h ▸ prefix_rfl⟩\n\n@[simp] lemma suffix_nil_iff : l <:+ [] ↔ l = [] :=\n⟨λ h, eq_nil_of_infix_nil h.is_infix, λ h, h ▸ suffix_rfl⟩\n\nalias prefix_nil_iff ↔ list.eq_nil_of_prefix_nil _\nalias suffix_nil_iff ↔ list.eq_nil_of_suffix_nil _\n\nlemma infix_iff_prefix_suffix (l₁ l₂ : list α) : l₁ <:+: l₂ ↔ ∃ t, l₁ <+: t ∧ t <:+ l₂ :=\n⟨λ ⟨s, t, e⟩, ⟨l₁ ++ t, ⟨_, rfl⟩, by rw [← e, append_assoc]; exact ⟨_, rfl⟩⟩,\n  λ ⟨._, ⟨t, rfl⟩, s, e⟩, ⟨s, t, by rw append_assoc; exact e⟩⟩\n\nlemma eq_of_infix_of_length_eq (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=\neq_of_sublist_of_length_eq h.sublist\n\nlemma eq_of_prefix_of_length_eq (h : l₁ <+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=\neq_of_sublist_of_length_eq h.sublist\n\nlemma eq_of_suffix_of_length_eq (h : l₁ <:+ l₂) : l₁.length = l₂.length → l₁ = l₂ :=\neq_of_sublist_of_length_eq h.sublist\n\nlemma prefix_of_prefix_length_le : ∀ {l₁ l₂ l₃ : list α},\n  l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂\n| []      l₂ l₃ h₁ h₂ _ := nil_prefix _\n| (a :: l₁) (b :: l₂) _ ⟨r₁, rfl⟩ ⟨r₂, e⟩ ll := begin\n  injection e with _ e', subst b,\n  rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩\n    (le_of_succ_le_succ ll) with ⟨r₃, rfl⟩,\n  exact ⟨r₃, rfl⟩\nend\n\nlemma prefix_or_prefix_of_prefix (h₁ : l₁ <+: l₃) (h₂ : l₂ <+: l₃) : l₁ <+: l₂ ∨ l₂ <+: l₁ :=\n(le_total (length l₁) (length l₂)).imp\n  (prefix_of_prefix_length_le h₁ h₂)\n  (prefix_of_prefix_length_le h₂ h₁)\n\nlemma suffix_of_suffix_length_le (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) (ll : length l₁ ≤ length l₂) :\n  l₁ <:+ l₂ :=\nreverse_prefix.1 $ prefix_of_prefix_length_le\n  (reverse_prefix.2 h₁) (reverse_prefix.2 h₂) (by simp [ll])\n\nlemma suffix_or_suffix_of_suffix (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) : l₁ <:+ l₂ ∨ l₂ <:+ l₁ :=\n(prefix_or_prefix_of_prefix (reverse_prefix.2 h₁) (reverse_prefix.2 h₂)).imp\n  reverse_prefix.1 reverse_prefix.1\n\nlemma suffix_cons_iff : l₁ <:+ a :: l₂ ↔ l₁ = a :: l₂ ∨ l₁ <:+ l₂ :=\nbegin\n  split,\n  { rintro ⟨⟨hd, tl⟩, hl₃⟩,\n    { exact or.inl hl₃ },\n    { simp only [cons_append] at hl₃,\n      exact or.inr ⟨_, hl₃.2⟩ } },\n  { rintro (rfl | hl₁),\n    { exact (a :: l₂).suffix_refl },\n    { exact hl₁.trans (l₂.suffix_cons _) } }\nend\n\nlemma infix_of_mem_join : ∀ {L : list (list α)}, l ∈ L → l <:+: join L\n| (_  :: L) (or.inl rfl) := infix_append [] _ _\n| (l' :: L) (or.inr h)   := is_infix.trans (infix_of_mem_join h) $ (suffix_append _ _).is_infix\n\nlemma prefix_append_right_inj (l) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ :=\nexists_congr $ λ r, by rw [append_assoc, append_right_inj]\n\nlemma prefix_cons_inj (a) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ := prefix_append_right_inj [a]\n\nlemma take_prefix (n) (l : list α) : take n l <+: l := ⟨_, take_append_drop _ _⟩\nlemma drop_suffix (n) (l : list α) : drop n l <:+ l := ⟨_, take_append_drop _ _⟩\nlemma take_sublist (n) (l : list α) : take n l <+ l := (take_prefix n l).sublist\nlemma drop_sublist (n) (l : list α) : drop n l <+ l := (drop_suffix n l).sublist\nlemma take_subset (n) (l : list α) : take n l ⊆ l := (take_sublist n l).subset\nlemma drop_subset (n) (l : list α) : drop n l ⊆ l := (drop_sublist n l).subset\nlemma mem_of_mem_take (h : a ∈ l.take n) : a ∈ l := take_subset n l h\nlemma mem_of_mem_drop (h : a ∈ l.drop n) : a ∈ l := drop_subset n l h\n\nlemma init_prefix : ∀ (l : list α), l.init <+: l\n| [] := ⟨nil, by rw [init, list.append_nil]⟩\n| (a :: l) := ⟨_, init_append_last (cons_ne_nil a l)⟩\n\nlemma tail_suffix (l : list α) : tail l <:+ l := by rw ← drop_one; apply drop_suffix\n\nlemma init_sublist (l : list α) : l.init <+ l := (init_prefix l).sublist\n\n\nlemma prefix_iff_eq_append : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=\n⟨by rintros ⟨r, rfl⟩; rw drop_left, λ e, ⟨_, e⟩⟩\n\nlemma suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=\n⟨by rintros ⟨r, rfl⟩; simp only [length_append, add_tsub_cancel_right, take_left], λ e, ⟨_, e⟩⟩\n\nlemma prefix_iff_eq_take : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ :=\n⟨λ h, append_right_cancel $\n  (prefix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,\n λ e, e.symm ▸ take_prefix _ _⟩\n\nlemma suffix_iff_eq_drop : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length l₁) l₂ :=\n⟨λ h, append_left_cancel $\n  (suffix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,\n λ e, e.symm ▸ drop_suffix _ _⟩\n\ninstance decidable_prefix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <+: l₂)\n| []      l₂ := is_true ⟨l₂, rfl⟩\n| (a :: l₁) [] := is_false $ λ ⟨t, te⟩, list.no_confusion te\n| (a :: l₁) (b :: l₂) :=\n  if h : a = b then\n    @decidable_of_iff _ _ (by rw [← h, prefix_cons_inj])\n      (decidable_prefix l₁ l₂)\n  else\n    is_false $ λ ⟨t, te⟩, h $ by injection te\n\n-- Alternatively, use mem_tails\ninstance decidable_suffix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <:+ l₂)\n| []      l₂ := is_true ⟨l₂, append_nil _⟩\n| (a :: l₁) [] := is_false $ mt (length_le_of_sublist ∘ is_suffix.sublist) dec_trivial\n| l₁      l₂ := let len1 := length l₁, len2 := length l₂ in\n  if hl : len1 ≤ len2 then\n    decidable_of_iff' (l₁ = drop (len2-len1) l₂) suffix_iff_eq_drop\n  else is_false $ λ h, hl $ length_le_of_sublist $ h.sublist\n\nlemma prefix_take_le_iff {L : list (list (option α))} (hm : m < L.length) :\n  L.take m <+: L.take n ↔ m ≤ n :=\nbegin\n  simp only [prefix_iff_eq_take, length_take],\n  induction m with m IH generalizing L n,\n  { simp only [min_eq_left, eq_self_iff_true, nat.zero_le, take] },\n  cases L with l ls,\n  { exact (not_lt_bot hm).elim },\n  cases n,\n  { refine iff_of_false _ (zero_lt_succ _).not_le,\n    rw [take_zero, take_nil],\n    simp only [take],\n      exact not_false },\n  { simp only [length] at hm,\n    specialize @IH ls n (nat.lt_of_succ_lt_succ hm),\n    simp only [le_of_lt (nat.lt_of_succ_lt_succ hm), min_eq_left] at IH,\n    simp only [le_of_lt hm, IH, true_and, min_eq_left, eq_self_iff_true, length, take],\n    exact ⟨nat.succ_le_succ, nat.le_of_succ_le_succ⟩ }\nend\n\nlemma cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ :=\nbegin\n  split,\n  { rintro ⟨L, hL⟩,\n    simp only [cons_append] at hL,\n    exact ⟨hL.left, ⟨L, hL.right⟩⟩ },\n  { rintro ⟨rfl, h⟩,\n    rwa [prefix_cons_inj] }\nend\n\nlemma is_prefix.map (h : l₁ <+: l₂) (f : α → β) : l₁.map f <+: l₂.map f :=\nbegin\n  induction l₁ with hd tl hl generalizing l₂,\n  { simp only [nil_prefix, map_nil] },\n  { cases l₂ with hd₂ tl₂,\n    { simpa only using eq_nil_of_prefix_nil h },\n    { rw cons_prefix_iff at h,\n      simp only [h, prefix_cons_inj, hl, map] } }\nend\n\nlemma is_prefix.filter_map (h : l₁ <+: l₂) (f : α → option β) :\n  l₁.filter_map f <+: l₂.filter_map f :=\nbegin\n  induction l₁ with hd₁ tl₁ hl generalizing l₂,\n  { simp only [nil_prefix, filter_map_nil] },\n  { cases l₂ with hd₂ tl₂,\n    { simpa only using eq_nil_of_prefix_nil h },\n    { rw cons_prefix_iff at h,\n      rw [←@singleton_append _ hd₁ _, ←@singleton_append _ hd₂ _, filter_map_append,\n         filter_map_append, h.left, prefix_append_right_inj],\n      exact hl h.right } }\nend\n\nlemma is_prefix.reduce_option {l₁ l₂ : list (option α)} (h : l₁ <+: l₂) :\n  l₁.reduce_option <+: l₂.reduce_option :=\nh.filter_map id\n\nlemma is_prefix.filter (p : α → Prop) [decidable_pred p] ⦃l₁ l₂ : list α⦄ (h : l₁ <+: l₂) :\n  l₁.filter p <+: l₂.filter p :=\nbegin\n  obtain ⟨xs, rfl⟩ := h,\n  rw filter_append,\n  exact prefix_append _ _\nend\n\nlemma is_suffix.filter (p : α → Prop) [decidable_pred p] ⦃l₁ l₂ : list α⦄ (h : l₁ <:+ l₂) :\n  l₁.filter p <:+ l₂.filter p :=\nbegin\n  obtain ⟨xs, rfl⟩ := h,\n  rw filter_append,\n  exact suffix_append _ _\nend\n\nlemma is_infix.filter (p : α → Prop) [decidable_pred p] ⦃l₁ l₂ : list α⦄ (h : l₁ <:+: l₂) :\n  l₁.filter p <:+: l₂.filter p :=\nbegin\n  obtain ⟨xs, ys, rfl⟩ := h,\n  rw [filter_append, filter_append],\n  exact infix_append _ _ _\nend\n\nend fix\n\nsection inits_tails\n\n@[simp] lemma mem_inits : ∀ (s t : list α), s ∈ inits t ↔ s <+: t\n| s []     := suffices s = nil ↔ s <+: nil, by simpa only [inits, mem_singleton],\n  ⟨λ h, h.symm ▸ prefix_refl [], eq_nil_of_prefix_nil⟩\n| s (a :: t) :=\n  suffices (s = nil ∨ ∃ l ∈ inits t, a :: l = s) ↔ s <+: a :: t, by simpa,\n  ⟨λ o, match s, o with\n  | ._, or.inl rfl := ⟨_, rfl⟩\n  | s, or.inr ⟨r, hr, hs⟩ := let ⟨s, ht⟩ := (mem_inits _ _).1 hr in\n    by rw [← hs, ← ht]; exact ⟨s, rfl⟩\n  end, λ mi, match s, mi with\n  | [], ⟨._, rfl⟩ := or.inl rfl\n  | (b :: s), ⟨r, hr⟩ := list.no_confusion hr $ λ ba (st : s++r = t), or.inr $\n    by rw ba; exact ⟨_, (mem_inits _ _).2 ⟨_, st⟩, rfl⟩\n  end⟩\n\n@[simp] lemma mem_tails : ∀ (s t : list α), s ∈ tails t ↔ s <:+ t\n| s []     := by simp only [tails, mem_singleton];\n  exact ⟨λ h, by rw h; exact suffix_refl [], eq_nil_of_suffix_nil⟩\n| s (a :: t) := by simp only [tails, mem_cons_iff, mem_tails s t];\n  exact show s = a :: t ∨ s <:+ t ↔ s <:+ a :: t, from\n  ⟨λ o, match s, t, o with\n  | ._, t, or.inl rfl := suffix_rfl\n  | s, ._, or.inr ⟨l, rfl⟩ := ⟨a :: l, rfl⟩\n  end, λ e, match s, t, e with\n  | ._, t, ⟨[], rfl⟩ := or.inl rfl\n  | s, t, ⟨b :: l, he⟩ := list.no_confusion he (λ ab lt, or.inr ⟨l, lt⟩)\n  end⟩\n\nlemma inits_cons (a : α) (l : list α) : inits (a :: l) = [] :: l.inits.map (λ t, a :: t) := by simp\nlemma tails_cons (a : α) (l : list α) : tails (a :: l) = (a :: l) :: l.tails := by simp\n\n@[simp]\nlemma inits_append : ∀ (s t : list α), inits (s ++ t) = s.inits ++ t.inits.tail.map (λ l, s ++ l)\n| [] [] := by simp\n| [] (a :: t) := by simp\n| (a :: s) t := by simp [inits_append s t]\n\n@[simp]\nlemma tails_append : ∀ (s t : list α), tails (s ++ t) = s.tails.map (λ l, l ++ t) ++ t.tails.tail\n| [] [] := by simp\n| [] (a :: t) := by simp\n| (a :: s) t := by simp [tails_append s t]\n\n-- the lemma names `inits_eq_tails` and `tails_eq_inits` are like `sublists_eq_sublists'`\nlemma inits_eq_tails : ∀ (l : list α), l.inits = (reverse $ map reverse $ tails $ reverse l)\n| [] := by simp\n| (a :: l) := by simp [inits_eq_tails l, map_eq_map_iff]\n\nlemma tails_eq_inits : ∀ (l : list α), l.tails = (reverse $ map reverse $ inits $ reverse l)\n| [] := by simp\n| (a :: l) := by simp [tails_eq_inits l, append_left_inj]\n\nlemma inits_reverse (l : list α) : inits (reverse l) = reverse (map reverse l.tails) :=\nby { rw tails_eq_inits l, simp [reverse_involutive.comp_self] }\n\nlemma tails_reverse (l : list α) : tails (reverse l) = reverse (map reverse l.inits) :=\nby { rw inits_eq_tails l, simp [reverse_involutive.comp_self] }\n\nlemma map_reverse_inits (l : list α) : map reverse l.inits = (reverse $ tails $ reverse l) :=\nby { rw inits_eq_tails l, simp [reverse_involutive.comp_self] }\n\nlemma map_reverse_tails (l : list α) : map reverse l.tails = (reverse $ inits $ reverse l) :=\nby { rw tails_eq_inits l, simp [reverse_involutive.comp_self] }\n\n@[simp] lemma length_tails (l : list α) : length (tails l) = length l + 1 :=\nbegin\n  induction l with x l IH,\n  { simp },\n  { simpa using IH }\nend\n\n@[simp] lemma length_inits (l : list α) : length (inits l) = length l + 1 :=\nby simp [inits_eq_tails]\n\n@[simp] lemma nth_le_tails (l : list α) (n : ℕ) (hn : n < length (tails l)) :\n  nth_le (tails l) n hn = l.drop n :=\nbegin\n  induction l with x l IH generalizing n,\n  { simp },\n  { cases n,\n    { simp },\n    { simpa using IH n _ } }\nend\n\n@[simp] lemma nth_le_inits (l : list α) (n : ℕ) (hn : n < length (inits l)) :\n  nth_le (inits l) n hn = l.take n :=\nbegin\n  induction l with x l IH generalizing n,\n  { simp },\n  { cases n,\n    { simp },\n    { simpa using IH n _ } }\nend\n\ninstance decidable_infix [decidable_eq α] : ∀ (l₁ l₂ : list α), decidable (l₁ <:+: l₂)\n| []        l₂ := is_true ⟨[], l₂, rfl⟩\n| (a :: l₁) [] := is_false $ λ ⟨s, t, te⟩, absurd te $ append_ne_nil_of_ne_nil_left _ _ $\n                append_ne_nil_of_ne_nil_right _ _ $ λ h, list.no_confusion h\n| l₁        l₂ := decidable_of_decidable_of_iff (list.decidable_bex (λ t, l₁ <+: t) (tails l₂)) $\n  by refine (exists_congr (λ t, _)).trans (infix_iff_prefix_suffix _ _).symm;\n     exact ⟨λ ⟨h1, h2⟩, ⟨h2, (mem_tails _ _).1 h1⟩, λ ⟨h2, h1⟩, ⟨(mem_tails _ _).2 h1, h2⟩⟩\n\nend inits_tails\n\n/-! ### insert -/\n\nsection insert\nvariable [decidable_eq α]\n\n@[simp] lemma insert_nil (a : α) : insert a nil = [a] := rfl\n\nlemma insert.def (a : α) (l : list α) : insert a l = if a ∈ l then l else a :: l := rfl\n\n@[simp, priority 980]\nlemma insert_of_mem (h : a ∈ l) : insert a l = l := by simp only [insert.def, if_pos h]\n\n@[simp, priority 970]\nlemma insert_of_not_mem (h : a ∉ l) : insert a l = a :: l :=\nby simp only [insert.def, if_neg h]; split; refl\n\n@[simp] lemma mem_insert_iff : a ∈ insert b l ↔ a = b ∨ a ∈ l :=\nbegin\n  by_cases h' : b ∈ l,\n  { simp only [insert_of_mem h'],\n    apply (or_iff_right_of_imp _).symm,\n    exact λ e, e.symm ▸ h' },\n  { simp only [insert_of_not_mem h', mem_cons_iff] }\nend\n\n@[simp] lemma suffix_insert (a : α) (l : list α) : l <:+ insert a l :=\nby by_cases a ∈ l; [simp only [insert_of_mem h], simp only [insert_of_not_mem h, suffix_cons]]\n\nlemma infix_insert (a : α) (l : list α) : l <:+: insert a l := (suffix_insert a l).is_infix\nlemma sublist_insert (a : α) (l : list α) : l <+  l.insert a := (suffix_insert a l).sublist\nlemma subset_insert (a : α) (l : list α) : l ⊆ l.insert a := (sublist_insert a l).subset\n\n@[simp] lemma mem_insert_self (a : α) (l : list α) : a ∈  l.insert a :=\nmem_insert_iff.2 $ or.inl rfl\n\nlemma mem_insert_of_mem (h : a ∈ l) : a ∈ insert b l := mem_insert_iff.2 (or.inr h)\n\nlemma eq_or_mem_of_mem_insert (h : a ∈ insert b l) : a = b ∨ a ∈ l := mem_insert_iff.1 h\n\n@[simp] lemma length_insert_of_mem (h : a ∈ l) : (insert a l).length = l.length :=\ncongr_arg _ $ insert_of_mem h\n\n@[simp] lemma length_insert_of_not_mem (h : a ∉ l) : (insert a l).length = l.length + 1 :=\ncongr_arg _ $ insert_of_not_mem h\n\nend insert\nend list\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/list/infix.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.34915238312669233}}
{"text": "/-\nCopyright (c) 2021 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n\n! This file was ported from Lean 3 source module algebra.lie.base_change\n! leanprover-community/mathlib commit 9264b15ee696b7ca83f13c8ad67c83d6eb70b730\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Algebra.RestrictScalars\nimport Mathbin.Algebra.Lie.TensorProduct\n\n/-!\n# Extension and restriction of scalars for Lie algebras\n\nLie algebras have a well-behaved theory of extension and restriction of scalars.\n\n## Main definitions\n\n * `lie_algebra.extend_scalars.lie_algebra`\n * `lie_algebra.restrict_scalars.lie_algebra`\n\n## Tags\n\nlie ring, lie algebra, extension of scalars, restriction of scalars, base change\n-/\n\n\nuniverse u v w w₁ w₂ w₃\n\nopen TensorProduct\n\nvariable (R : Type u) (A : Type w) (L : Type v)\n\nnamespace LieAlgebra\n\nnamespace ExtendScalars\n\nvariable [CommRing R] [CommRing A] [Algebra R A] [LieRing L] [LieAlgebra R L]\n\n/-- The Lie bracket on the extension of a Lie algebra `L` over `R` by an algebra `A` over `R`.\n\nIn fact this bracket is fully `A`-bilinear but without a significant upgrade to our mixed-scalar\nsupport in the tensor product library, it is far easier to bootstrap like this, starting with the\ndefinition below. -/\nprivate def bracket' : A ⊗[R] L →ₗ[R] A ⊗[R] L →ₗ[R] A ⊗[R] L :=\n  TensorProduct.curry <|\n    TensorProduct.map (LinearMap.mul' R _) (LieModule.toModuleHom R L L : L ⊗[R] L →ₗ[R] L) ∘ₗ\n      ↑(TensorProduct.tensorTensorTensorComm R A L A L)\n#align lie_algebra.extend_scalars.bracket' lie_algebra.extend_scalars.bracket'\n\n@[simp]\nprivate theorem bracket'_tmul (s t : A) (x y : L) :\n    bracket' R A L (s ⊗ₜ[R] x) (t ⊗ₜ[R] y) = (s * t) ⊗ₜ ⁅x, y⁆ := by simp [bracket']\n#align lie_algebra.extend_scalars.bracket'_tmul lie_algebra.extend_scalars.bracket'_tmul\n\ninstance : Bracket (A ⊗[R] L) (A ⊗[R] L) where bracket x y := bracket' R A L x y\n\nprivate theorem bracket_def (x y : A ⊗[R] L) : ⁅x, y⁆ = bracket' R A L x y :=\n  rfl\n#align lie_algebra.extend_scalars.bracket_def lie_algebra.extend_scalars.bracket_def\n\n@[simp]\ntheorem bracket_tmul (s t : A) (x y : L) : ⁅s ⊗ₜ[R] x, t ⊗ₜ[R] y⁆ = (s * t) ⊗ₜ ⁅x, y⁆ := by\n  rw [bracket_def, bracket'_tmul]\n#align lie_algebra.extend_scalars.bracket_tmul LieAlgebra.ExtendScalars.bracket_tmul\n\nprivate theorem bracket_lie_self (x : A ⊗[R] L) : ⁅x, x⁆ = 0 :=\n  by\n  simp only [bracket_def]\n  apply x.induction_on\n  · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]\n  · intro a l\n    simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]\n  · intro z₁ z₂ h₁ h₂\n    suffices bracket' R A L z₁ z₂ + bracket' R A L z₂ z₁ = 0 by\n      rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h₁, h₂,\n        zero_add, add_zero, add_comm, this]\n    apply z₁.induction_on\n    · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n    · intro a₁ l₁\n      apply z₂.induction_on\n      · simp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n      · intro a₂ l₂\n        simp only [← lie_skew l₂ l₁, mul_comm a₁ a₂, TensorProduct.tmul_neg, bracket'_tmul,\n          add_right_neg]\n      · intro y₁ y₂ hy₁ hy₂\n        simp only [hy₁, hy₂, add_add_add_comm, add_zero, LinearMap.add_apply, LinearMap.map_add]\n    · intro y₁ y₂ hy₁ hy₂\n      simp only [add_add_add_comm, hy₁, hy₂, add_zero, LinearMap.add_apply, LinearMap.map_add]\n#align lie_algebra.extend_scalars.bracket_lie_self lie_algebra.extend_scalars.bracket_lie_self\n\nprivate theorem bracket_leibniz_lie (x y z : A ⊗[R] L) : ⁅x, ⁅y, z⁆⁆ = ⁅⁅x, y⁆, z⁆ + ⁅y, ⁅x, z⁆⁆ :=\n  by\n  simp only [bracket_def]\n  apply x.induction_on\n  · simp only [LinearMap.map_zero, add_zero, eq_self_iff_true, LinearMap.zero_apply]\n  · intro a₁ l₁\n    apply y.induction_on\n    · simp only [LinearMap.map_zero, add_zero, eq_self_iff_true, LinearMap.zero_apply]\n    · intro a₂ l₂\n      apply z.induction_on\n      · simp only [LinearMap.map_zero, add_zero]\n      · intro a₃ l₃\n        simp only [bracket'_tmul]\n        rw [mul_left_comm a₂ a₁ a₃, mul_assoc, leibniz_lie, TensorProduct.tmul_add]\n      · intro u₁ u₂ h₁ h₂\n        simp only [add_add_add_comm, h₁, h₂, LinearMap.map_add]\n    · intro u₁ u₂ h₁ h₂\n      simp only [add_add_add_comm, h₁, h₂, LinearMap.add_apply, LinearMap.map_add]\n  · intro u₁ u₂ h₁ h₂\n    simp only [add_add_add_comm, h₁, h₂, LinearMap.add_apply, LinearMap.map_add]\n#align lie_algebra.extend_scalars.bracket_leibniz_lie lie_algebra.extend_scalars.bracket_leibniz_lie\n\ninstance : LieRing (A ⊗[R] L)\n    where\n  add_lie x y z := by simp only [bracket_def, LinearMap.add_apply, LinearMap.map_add]\n  lie_add x y z := by simp only [bracket_def, LinearMap.map_add]\n  lie_self := bracket_lie_self R A L\n  leibniz_lie := bracket_leibniz_lie R A L\n\nprivate theorem bracket_lie_smul (a : A) (x y : A ⊗[R] L) : ⁅x, a • y⁆ = a • ⁅x, y⁆ :=\n  by\n  apply x.induction_on\n  · simp only [zero_lie, smul_zero]\n  · intro a₁ l₁\n    apply y.induction_on\n    · simp only [lie_zero, smul_zero]\n    · intro a₂ l₂\n      simp only [bracket_def, bracket', TensorProduct.smul_tmul', mul_left_comm a₁ a a₂,\n        TensorProduct.curry_apply, LinearMap.mul'_apply, Algebra.id.smul_eq_mul,\n        Function.comp_apply, LinearEquiv.coe_coe, LinearMap.coe_comp, TensorProduct.map_tmul,\n        TensorProduct.tensorTensorTensorComm_tmul]\n    · intro z₁ z₂ h₁ h₂\n      simp only [h₁, h₂, smul_add, lie_add]\n  · intro z₁ z₂ h₁ h₂\n    simp only [h₁, h₂, smul_add, add_lie]\n#align lie_algebra.extend_scalars.bracket_lie_smul lie_algebra.extend_scalars.bracket_lie_smul\n\ninstance lieAlgebra : LieAlgebra A (A ⊗[R] L) where lie_smul := bracket_lie_smul R A L\n#align lie_algebra.extend_scalars.lie_algebra LieAlgebra.ExtendScalars.lieAlgebra\n\nend ExtendScalars\n\nnamespace RestrictScalars\n\nopen RestrictScalars\n\nvariable [h : LieRing L]\n\ninclude h\n\ninstance : LieRing (RestrictScalars R A L) :=\n  h\n\nvariable [CommRing A] [LieAlgebra A L]\n\ninstance lieAlgebra [CommRing R] [Algebra R A] : LieAlgebra R (RestrictScalars R A L)\n    where lie_smul t x y :=\n    (lie_smul (algebraMap R A t) (RestrictScalars.addEquiv R A L x)\n        (RestrictScalars.addEquiv R A L y) :\n      _)\n#align lie_algebra.restrict_scalars.lie_algebra LieAlgebra.RestrictScalars.lieAlgebra\n\nend RestrictScalars\n\nend LieAlgebra\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Lie/BaseChange.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3491523698675708}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Eric Wieser\n\n! This file was ported from Lean 3 source module algebra.hierarchy_design\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.DocCommands\n\n/-!\n# Documentation of the algebraic hierarchy\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA library note giving advice on modifying the algebraic hierarchy.\n(It is not intended as a \"tour\".)\n\nTODO: Add sections about interactions with topological typeclasses, and order typeclasses.\n\n-/\n\n\nlibrary_note \"the algebraic hierarchy\"/-- # The algebraic hierarchy\n\nIn any theorem proving environment,\nthere are difficult decisions surrounding the design of the \"algebraic hierarchy\".\n\nThere is a danger of exponential explosion in the number of gadgets,\nespecially once interactions between algebraic and order/topological/etc structures are considered.\n\nIn mathlib, we try to avoid this by only introducing new algebraic typeclasses either\n1. when there is \"real mathematics\" to be done with them, or\n2. when there is a meaninful gain in simplicity by factoring out a common substructure.\n\n(As examples, at this point we don't have `loop`, or `unital_magma`,\nbut we do have `lie_submodule` and `topological_field`!\nWe also have `group_with_zero`, as an exemplar of point 2.)\n\nGenerally in mathlib we use the extension mechanism (so `comm_ring` extends `ring`)\nrather than mixins (e.g. with separate `ring` and `comm_mul` classes),\nin part because of the potential blow-up in term sizes described at\nhttps://www.ralfj.de/blog/2019/05/15/typeclasses-exponential-blowup.html\nHowever there is tension here, as it results in considerable duplication in the API,\nparticularly in the interaction with order structures.\n\nThis library note is not intended as a design document\njustifying and explaining the history of mathlib's algebraic hierarchy!\nInstead it is intended as a developer's guide, for contributors wanting to extend\n(either new leaves, or new intermediate classes) the algebraic hierarchy as it exists.\n\n(Ideally we would have both a tour guide to the existing hierarchy,\nand an account of the design choices.\nSee https://arxiv.org/abs/1910.09336 for an overview of mathlib as a whole,\nwith some attention to the algebraic hierarchy and\nhttps://leanprover-community.github.io/mathlib-overview.html\nfor a summary of what is in mathlib today.)\n\n## Instances\n\nWhen adding a new typeclass `Z` to the algebraic hierarchy\none should attempt to add the following constructions and results,\nwhen applicable:\n\n* Instances transferred elementwise to products, like `prod.monoid`.\n  See `algebra.group.prod` for more examples.\n  ```\n  instance prod.Z [Z M] [Z N] : Z (M × N) := ...\n  ```\n* Instances transferred elementwise to pi types, like `pi.monoid`.\n  See `algebra.group.pi` for more examples.\n  ```\n  instance pi.Z [∀ i, Z $ f i] : Z (Π i : I, f i) := ...\n  ```\n* Instances transferred to `mul_opposite M`, like `mul_opposite.monoid`.\n  See `algebra.opposites` for more examples.\n  ```\n  instance mul_opposite.Z [Z M] : Z (mul_opposite M) := ...\n  ```\n* Instances transferred to `ulift M`, like `ulift.monoid`.\n  See `algebra.group.ulift` for more examples.\n  ```\n  instance ulift.Z [Z M] : Z (ulift M) := ...\n  ```\n* Definitions for transferring the proof fields of instances along\n  injective or surjective functions that agree on the data fields,\n  like `function.injective.monoid` and `function.surjective.monoid`.\n  We make these definitions `@[reducible]`, see note [reducible non-instances].\n  See `algebra.group.inj_surj` for more examples.\n  ```\n  @[reducible]\n  def function.injective.Z [Z M₂] (f : M₁ → M₂) (hf : injective f)\n    (one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) : Z M₁ := ...\n\n  @[reducible]\n  def function.surjective.Z [Z M₁] (f : M₁ → M₂) (hf : surjective f)\n    (one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) : Z M₂ := ...\n  ```\n* Instances transferred elementwise to `finsupp`s, like `finsupp.semigroup`.\n  See `data.finsupp.pointwise` for more examples.\n  ```\n  instance finsupp.Z [Z β] : Z (α →₀ β) := ...\n  ```\n* Instances transferred elementwise to `set`s, like `set.monoid`.\n  See `algebra.pointwise` for more examples.\n  ```\n  instance set.Z [Z α] : Z (set α) := ...\n  ```\n* Definitions for transferring the entire structure across an equivalence, like `equiv.monoid`.\n  See `data.equiv.transfer_instance` for more examples. See also the `transport` tactic.\n  ```\n  def equiv.Z (e : α ≃ β) [Z β] : Z α := ...\n  /- When there is a new notion of `Z`-equiv: -/\n  def equiv.Z_equiv (e : α ≃ β) [Z β] : by { letI := equiv.Z e, exact α ≃Z β } := ...\n  ```\n\n## Subobjects\n\nWhen a new typeclass `Z` adds new data fields,\nyou should also create a new `sub_Z` `structure` with a `carrier` field.\n\nThis can be a lot of work; for now try to closely follow the existing examples\n(e.g. `submonoid`, `subring`, `subalgebra`).\nWe would very much like to provide some automation here, but a prerequisite will be making\nall the existing APIs more uniform.\n\nIf `Z` extends `Y`, then `sub_Z` should usually extend `sub_Y`.\n\nWhen `Z` adds only new proof fields to an existing structure `Y`,\nyou should provide instances transferring\n`Z α` to `Z (sub_Y α)`, like `submonoid.to_comm_monoid`.\nTypically this is done using the `function.injective.Z` definition mentioned above.\n```\ninstance sub_Y.to_Z [Z α] : Z (sub_Y α) :=\ncoe_injective.Z coe ...\n```\n\n## Morphisms and equivalences\n\n## Category theory\n\nFor many algebraic structures, particularly ones used in representation theory, algebraic geometry,\netc., we also define \"bundled\" versions, which carry `category` instances.\n\nThese bundled versions are usually named in camel case,\nso for example we have `AddCommGroup` as a bundled `add_comm_group`,\nand `TopCommRing` (which bundles together `comm_ring`, `topological_space`, and `topological_ring`).\n\nThese bundled versions have many appealing features:\n* a uniform notation for morphisms `X ⟶ Y`\n* a uniform notation (and definition) for isomorphisms `X ≅ Y`\n* a uniform API for subobjects, via the partial order `subobject X`\n* interoperability with unbundled structures, via coercions to `Type`\n  (so if `G : AddCommGroup`, you can treat `G` as a type,\n  and it automatically has an `add_comm_group` instance)\n  and lifting maps `AddCommGroup.of G`, when `G` is a type with an `add_comm_group` instance.\n\nIf, for example you do the work of proving that a typeclass `Z` has a good notion of tensor product,\nyou are strongly encouraged to provide the corresponding `monoidal_category` instance\non a bundled version.\nThis ensures that the API for tensor products is complete, and enables use of general machinery.\nSimilarly if you prove universal properties, or adjunctions, you are encouraged to state these\nusing categorical language!\n\nOne disadvantage of the bundled approach is that we can only speak of morphisms between\nobjects living in the same type-theoretic universe.\nIn practice this is rarely a problem.\n\n# Making a pull request\n\nWith so many moving parts, how do you actually go about changing the algebraic hierarchy?\n\nWe're still evolving how to handle this, but the current suggestion is:\n\n* If you're adding a new \"leaf\" class, the requirements are lower,\n  and an initial PR can just add whatever is immediately needed.\n* A new \"intermediate\" class, especially low down in the hierarchy,\n  needs to be careful about leaving gaps.\n\nIn a perfect world, there would be a group of simultaneous PRs that basically cover everything!\n(Or at least an expectation that PRs may not be merged immediately while waiting on other\nPRs that fill out the API.)\n\nHowever \"perfect is the enemy of good\", and it would also be completely reasonable\nto add a TODO list in the main module doc-string for the new class,\nbriefly listing the parts of the API which still need to be provided.\nHopefully this document makes it easy to assemble this list.\n\nAnother alternative to a TODO list in the doc-strings is adding github issues.\n\n\n-/\n\n\nlibrary_note \"reducible non-instances\"/--\nSome definitions that define objects of a class cannot be instances, because they have an\nexplicit argument that does not occur in the conclusion. An example is `preorder.lift` that has a\nfunction `f : α → β` as an explicit argument to lift a preorder on `β` to a preorder on `α`.\n\nIf these definitions are used to define instances of this class *and* this class is an argument to\nsome other type-class so that type-class inference will have to unfold these instances to check\nfor definitional equality, then these definitions should be marked `@[reducible]`.\n\nFor example, `preorder.lift` is used to define `units.preorder` and `partial_order.lift` is used\nto define `units.partial_order`. In some cases it is important that type-class inference can\nrecognize that `units.preorder` and `units.partial_order` give rise to the same `has_le` instance.\nFor example, you might have another class that takes `[has_le α]` as an argument, and this argument\nsometimes comes from `units.preorder` and sometimes from `units.partial_order`.\nTherefore, `preorder.lift` and `partial_order.lift` are marked `@[reducible]`.\n-/\n\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/HierarchyDesign.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.538983220687684, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.3491523698675707}}
{"text": "import Fine.FormalTheories\nimport Mathlib.Data.Prod.Lex\nimport Fine.Util.util\n\nopen Classical\n\ndef lindenbaumSequence (t : Th) (Δ : Ctx) : Lex (Nat × Nat) → Ctx\n  | ⟨0, 0⟩ => t.val\n  | ⟨i + 1, 0⟩ => { f : Form | ∃j : Nat, f ∈ lindenbaumSequence t Δ ⟨i, j⟩ }\n  | ⟨i, j + 1⟩ => \n    let prev := lindenbaumSequence t Δ ⟨i, j⟩\n    let l := (Denumerable.ofNat (Form × Form) j).fst\n    let r := (Denumerable.ofNat (Form × Form) j).snd\n    if l ¦ r ∈ ▲prev\n    then if ▲(prev ∪ {l}) ∩ Δ = ∅\n      then prev ∪ {l}\n      else prev ∪ {r}\n    else prev\n  termination_by lindenbaumSequence _ _ p => (p.fst, p.snd)\n\nlemma lindenbaumSequenceMonotone' { t : Th } { Δ : Ctx } : ∀b a, a ≤ b → lindenbaumSequence t Δ a ⊆ lindenbaumSequence t Δ b := by\n  intros b\n  match b with\n    | ⟨0, 0⟩ =>\n      intros a h₁\n      cases h₁\n      case left a₁ _ h₂ => exact False.elim $ Nat.not_lt_zero a₁ h₂\n      case right b₁ h₂ =>\n        rw [Nat.le_zero_eq b₁] at h₂\n        rw [h₂]\n    | ⟨i + 1, 0⟩ =>\n      intros a h₁\n      cases h₁\n      case left a₁ b₁ h₂ => \n        have l₁ : a₁ < i ∨ a₁ = i := Nat.lt_or_eq_of_le $ Nat.lt_succ.mp h₂\n        have l₂ : a₁ < i ∨ a₁ = i ∧ b₁ ≤ b₁ := by \n          cases l₁\n          case inl h₂ => exact Or.inl h₂\n          case inr h₂ => exact Or.inr ⟨h₂, le_refl b₁⟩\n        have l₃ := (Prod.Lex.le_iff (a₁, b₁) (i,b₁)).mpr l₂\n        have l₄ := @lindenbaumSequenceMonotone' t Δ (i,b₁) (a₁, b₁) l₃\n        apply le_trans l₄\n        intros f h₂\n        exact ⟨b₁,h₂⟩\n      case right b₁ h₂ =>\n        rw [Nat.le_zero_eq b₁] at h₂\n        rw [h₂]\n    | ⟨i, j + 1⟩ =>\n      have lem : lindenbaumSequence t Δ (i, j) ≤ lindenbaumSequence t Δ (i, j + 1) := by\n        intros f h₃\n        unfold lindenbaumSequence\n        split\n        case h_1 _ heq | h_2 _ heq => injection heq; contradiction\n        case h_3 i' j' heq =>\n          injection heq with heq₁ heq₂\n          injection heq₂ with heq₂\n          have heq₃ : j = j' := heq₂\n          rw [←heq₁, ←heq₃]\n          change\n            let prev := lindenbaumSequence t Δ (i, j);\n            let l := (Denumerable.ofNat (Form × Form) j).fst;\n            let r := (Denumerable.ofNat (Form × Form) j).snd;\n            f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev\n          intros prev l r\n          split\n          case inl =>\n            split\n            case inl | inr => apply Or.inl h₃\n          case inr => exact h₃\n      intros a h₁\n      cases h₁\n      case left a₁ b₁ h₂ =>\n          have l₁ : a₁ < i ∨ a₁ = i ∧ b₁ ≤ j := Or.inl h₂\n          have l₂ := (Prod.Lex.le_iff (a₁, b₁) (i,j)).mpr l₁\n          have l₃ := @lindenbaumSequenceMonotone' t Δ (i,j) (a₁, b₁)  l₂\n          apply le_trans l₃\n          exact lem\n      case right b₁ h₂ =>\n        cases h₂\n        case refl => intros _ h₁; assumption\n        case step h₂ =>\n          have l₁ : i < i ∨ i = i ∧ b₁ ≤ j := Or.inr ⟨rfl, h₂⟩\n          have l₂ := (Prod.Lex.le_iff (i, b₁) (i,j)).mpr l₁\n          have l₃ := @lindenbaumSequenceMonotone' t Δ (i,j) (i, b₁)  l₂\n          apply le_trans l₃\n          exact lem\n          \n  termination_by lindenbaumSequenceMonotone' _ _ b => (b.fst, b.snd)\n\nlemma lindenbaumSequenceMonotone { t : Th } { Δ : Ctx } : Monotone (lindenbaumSequence t Δ) := by\n  intros a b\n  exact lindenbaumSequenceMonotone' b a  \n\ndef lindenbaumExtension (t : Th) (Δ : Ctx) : Ctx := {f | ∃ij : Nat × Nat, f ∈ lindenbaumSequence t Δ ij }\n\nlemma lindenbaumStageMonotone { t : Th } { Δ : Ctx } { i : Nat } : Monotone (λj => lindenbaumSequence t Δ ⟨i,j⟩) := by\n  intros a b h₁\n  change lindenbaumSequence t Δ (i, a) ≤ lindenbaumSequence t Δ (i, b)\n  have l₂ := (Prod.Lex.le_iff (i, a) (i,b)).mpr $ Or.inr ⟨rfl,h₁⟩\n  exact lindenbaumSequenceMonotone' (i,b) (i,a) l₂\n\nlemma lindenbaumExtensionExtends { t : Th } { Δ : Ctx } : t.val ⊆ lindenbaumExtension t Δ := by\n  intros f h₁\n  refine ⟨⟨0,0⟩,?_⟩\n  assumption\n\ntheorem lindenbaumIsFormal { t : Th } { Δ : Ctx } : formalTheory (lindenbaumExtension t Δ) := by\n  intros f\n  apply Iff.intro\n  case mp =>\n    intro h₁\n    exact ⟨BProof.ax h₁⟩\n  case mpr =>\n    intro h₁\n    have ⟨prf₁⟩ := h₁\n    have ⟨s, l₁, fprf⟩ := BProof.compactness prf₁\n    have ⟨⟨i,j⟩,l₂⟩ := finiteExhaustion lindenbaumSequenceMonotone l₁\n    have l₃ : lindenbaumSequence t Δ ⟨i,j⟩ ⊆ lindenbaumSequence t Δ ⟨i + 1,Encodable.encode (f,f)⟩ := by\n      apply lindenbaumSequenceMonotone\n      apply (Prod.Lex.le_iff (i,j) (i + 1,Encodable.encode (f,f))).mpr $ Or.inl $ Nat.lt_succ_self i\n    have prf₂ := BProof.monotone (le_trans l₂ l₃) fprf\n    have prf₃ : BProof (lindenbaumSequence t Δ ⟨i+1,Encodable.encode (f,f)⟩) (f ¦ f) := BProof.mp prf₂ BTheorem.orI₁\n    clear s h₁ l₁ l₂ l₃ fprf prf₁ prf₂ \n    have l₄ : f ∈ lindenbaumSequence t Δ ⟨i + 1, Encodable.encode (f,f) + 1⟩ := by\n      unfold lindenbaumSequence\n      change\n        let prev := lindenbaumSequence t Δ (i + 1, Encodable.encode (f,f));\n        let l := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).fst;\n        let r := (Denumerable.ofNat (Form × Form) (Encodable.encode (f,f))).snd;\n        f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev\n      intros prev l r\n      have l₅ : l = f := by \n        change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).fst = f\n        rw [Denumerable.ofNat_encode (f,f)]\n      have l₆ : r = f := by\n        change (Denumerable.ofNat (Form × Form) (Encodable.encode (f, f))).snd = f\n        rw [Denumerable.ofNat_encode (f,f)]\n      split\n      case inl h₂ =>\n        split\n        . rw [l₅]; exact Or.inr rfl\n        . rw [l₆]; exact Or.inr rfl\n      case inr h₂ =>\n        apply False.elim\n        have l₇ : f¦f ∈ ▲prev := ⟨prf₃⟩\n        rw [l₅,l₆] at h₂\n        exact h₂ l₇\n    exact ⟨⟨i + 1, Encodable.encode (f,f) + 1⟩, l₄⟩\n\ntheorem lindenbaumIsPrime { t : Th } { Δ : Ctx } : PrimeTheory (lindenbaumExtension t Δ) := by\n  intros f g h₁\n  have ⟨⟨i,j⟩,h₂⟩ := h₁\n  let k := Encodable.encode (f,g)\n  have l₁ : lindenbaumSequence t Δ ⟨i,j⟩ ⊆ lindenbaumSequence t Δ ⟨i + 1,k⟩ := by\n    apply lindenbaumSequenceMonotone\n    apply (Prod.Lex.le_iff (i,j) (i + 1,k)).mpr $ Or.inl $ Nat.lt_succ_self i\n  have l₂ : f ¦ g ∈ lindenbaumSequence t Δ ⟨i + 1, k⟩ := l₁ h₂\n  clear l₁ h₁ h₂\n  have l₃ : f ∈ lindenbaumSequence t Δ ⟨i + 1, k + 1⟩ ∨ g ∈ lindenbaumSequence t Δ ⟨i + 1, k + 1⟩ := by\n    unfold lindenbaumSequence\n    change\n      let prev := lindenbaumSequence t Δ (i + 1, k);\n      let l := (Denumerable.ofNat (Form × Form) k).fst;\n      let r := (Denumerable.ofNat (Form × Form) k).snd;\n      (f ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev) ∨ \n      (g ∈ if l¦r ∈ ▲prev then if ▲(prev ∪ {l}) ∩ Δ = ∅ then prev ∪ {l} else prev ∪ {r} else prev)\n    intros prev l r\n    have l₄ : Denumerable.ofNat (Form × Form) k = (f,g) := Denumerable.ofNat_encode (f,g)\n    have l₅ : l = f := by \n      change (Denumerable.ofNat (Form × Form) k).fst = f\n      rw [l₄]\n    have l₆ : r = g := by \n      change (Denumerable.ofNat (Form × Form) k).snd = g\n      rw [l₄]\n    repeat rw [l₅,l₆]\n    clear l r l₅ l₆\n    cases Classical.em (▲(prev ∪ {f}) ∩ Δ = ∅)\n    case' inl h₁ => apply Or.inl\n    case' inr h₁ => apply Or.inr\n    all_goals\n      split\n      case inl => exact Or.inr rfl\n      case inr h₂ => exact False.elim $ h₂ ⟨BProof.ax l₂⟩\n  apply Or.elim l₃\n  case left => intros h₁; exact Or.inl ⟨⟨i+1,k+1⟩,h₁⟩\n  case right => intros h₁; exact Or.inr ⟨⟨i+1,k+1⟩,h₁⟩\n\ntheorem lindenbaumAvoids { t : Th } { Δ : Ctx } ( h₁ : ↑t ∩ Δ = ∅ ) ( h₂ : DisjunctionClosed Δ ) : ∀ij, ▲lindenbaumSequence t Δ ij ∩ Δ = ∅ \n  | ⟨0,0⟩ => by\n    have l₁ := formalFixed t.property\n    rw [←l₁] at h₁\n    exact h₁\n  | ⟨i + 1, 0⟩ => by\n    change\n      ▲{ f : Form | ∃j : Nat, f ∈ lindenbaumSequence t Δ ⟨i, j⟩ } ∩ Δ = ∅\n    apply Set.not_nonempty_iff_eq_empty.mp\n    intros h₃\n    have ⟨w,⟨prf₁⟩,l₁⟩ := h₃\n    clear h₃\n    have ⟨s,l₂,prf₂⟩ := BProof.compactness prf₁\n    have ⟨j, l₃⟩ := finiteExhaustion lindenbaumStageMonotone l₂\n    have l₄ := lindenbaumAvoids h₁ h₂ ⟨i,j⟩\n    have prf₃ := BProof.monotone l₃ prf₂ \n    have l₅ := Set.not_nonempty_iff_eq_empty.mpr l₄\n    exact l₅ ⟨w, ⟨⟨prf₃⟩,l₁⟩⟩\n  | ⟨i, j + 1⟩ => by\n    apply Set.not_nonempty_iff_eq_empty.mp\n    intros h₃\n    have ⟨w₁,l₁,l₂⟩ := h₃\n    unfold lindenbaumSequence at l₁\n    split at l₁\n    case h_1 x heq => injection heq with heq; contradiction\n    case h_2 x heq => injection heq with heq; contradiction\n    case h_3 x n m heq =>\n      have l₃ := lindenbaumAvoids h₁ h₂ ⟨i,j⟩\n      injection heq with heq₁ heq₂\n      injection heq₂ with heq₂\n      rw [←heq₁,←heq₂] at l₁\n      clear n m x heq₁ heq₂ h₃\n      dsimp at l₁ j\n      split at l₁\n      case inr h₄ => exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁, l₁, l₂⟩\n      case inl h₄ =>\n        split at l₁\n        case inl h₅ => exact (Set.not_nonempty_iff_eq_empty.mpr h₅) ⟨w₁, l₁, l₂⟩\n        case inr h₅ =>\n          have ⟨prf₁⟩ := l₁\n          have ⟨w₂,⟨⟨prf₂⟩,l₄⟩⟩ := Set.nonempty_iff_ne_empty.mpr h₅\n          have l₅ : w₁¦w₂ ∈ Δ := h₂ ⟨l₂, l₄⟩\n          clear l₁ l₂ l₄ h₅\n          have ⟨lst₁,l₆,prf₃⟩ := BProof.sentenceCompactness (Set.union_singleton ▸ prf₁)\n          have ⟨lst₂,l₇,prf₄⟩ := BProof.sentenceCompactness (Set.union_singleton ▸ prf₂)\n          have thm₁ := BTheorem.transitivity (BTheorem.fromProof prf₃) (BTheorem.orI₁ : BTheorem (w₁ ⊃ w₁ ¦ w₂))\n          have thm₂ := BTheorem.transitivity (BTheorem.fromProof prf₄) (BTheorem.orI₂ : BTheorem (w₂ ⊃ w₁ ¦ w₂))\n          have thm₃ := BTheorem.mp (BTheorem.adj thm₂ thm₁) BTheorem.orE \n          have ⟨prf₅⟩ := h₄\n          clear h₄ thm₁ thm₂ prf₁ prf₂ prf₃ prf₄\n          cases lst₁\n          all_goals \n            cases lst₂\n          case nil.nil =>\n            have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp prf₅ thm₃⟩, l₅⟩\n            exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩\n          -- could simplify these with a new BTheorem def\n          case nil.cons head tail =>\n            have := BProof.proveList l₇\n            have := BProof.mp (BProof.adj prf₅ this) BTheorem.distRight\n            have := BProof.mp this (BTheorem.mp \n              (BTheorem.adj \n                (BTheorem.transitivity BTheorem.taut BTheorem.orI₁) \n                (BTheorem.transitivity BTheorem.andE₁ BTheorem.orI₂)\n              ) BTheorem.orE)\n            have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp this thm₃⟩, l₅⟩\n            exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩\n          case cons.nil head tail => \n            have := BProof.proveList l₆\n            have := BProof.mp (BProof.adj prf₅ this) BTheorem.distRight\n            have := BProof.mp this (BTheorem.mp \n              (BTheorem.adj \n                (BTheorem.transitivity BTheorem.andE₁ BTheorem.orI₁) \n                (BTheorem.transitivity BTheorem.taut BTheorem.orI₂)\n              ) BTheorem.orE)\n            have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp this thm₃⟩, l₅⟩\n            exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩\n          case cons.cons head tail head' tail'=> \n            have prf₆ := BProof.proveList l₆\n            have prf₇ := BProof.proveList l₇\n            have := BProof.mp (BProof.adj prf₅ prf₇) BTheorem.distRight\n            have prf₈ := BProof.mp this (BTheorem.mp \n              (BTheorem.adj \n                (BTheorem.transitivity BTheorem.taut BTheorem.orI₁) \n                (BTheorem.transitivity BTheorem.andE₁ BTheorem.orI₂)\n              ) BTheorem.orE)\n            have := BProof.mp (BProof.adj prf₈ prf₆) BTheorem.distRight\n            have prf₉ := BProof.mp this (BTheorem.mp \n              (BTheorem.adj \n                (BTheorem.transitivity BTheorem.andE₁ BTheorem.orI₁) \n                (BTheorem.transitivity BTheorem.taut BTheorem.orI₂)\n              ) BTheorem.orE)\n            have : w₁¦w₂ ∈ ▲lindenbaumSequence t Δ (i, j) ∩ Δ := ⟨⟨BProof.mp prf₉ thm₃⟩, l₅⟩\n            exact (Set.not_nonempty_iff_eq_empty.mpr l₃) ⟨w₁¦w₂, this⟩\n\n  termination_by lindenbaumAvoids _ _ _ _ p => (p.fst, p.snd)\n", "meta": {"author": "gleachkr", "repo": "Completeness-For-Fine-Semantics", "sha": "e06f0b07cccbffbbc59c52f7cdebb54d1f7d4f38", "save_path": "github-repos/lean/gleachkr-Completeness-For-Fine-Semantics", "path": "github-repos/lean/gleachkr-Completeness-For-Fine-Semantics/Completeness-For-Fine-Semantics-e06f0b07cccbffbbc59c52f7cdebb54d1f7d4f38/Fine/Completeness/Lindenbaum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.34902656287493805}}
{"text": "import Runtime.Utilities\n\nnamespace Interface\n\n/--\nAn interface scheme captures the type signature of a dependent map. Specifically, a scheme\n`{ vars, type }` describes the type `(var : vars) → (type var)`.\n\nThis type is used to describe the structures of components in reactors like ports, state variables,\nactions and parameters, which are all maps from names to values of different types.\n-/\nstructure Scheme where\n  vars : Type\n  type : vars → Type\n  [varsDecidableEq : DecidableEq vars]\n\nattribute [reducible] Scheme.type\nattribute [instance] Scheme.varsDecidableEq\n\n/--\nThe (disjoint) union of two schemes describes the signature of a map where the domain is the\ndisjoint union of the respective schemes' domains. The codomain is extended in the obvious way.\n-/\nabbrev Scheme.union (σ₁ σ₂ : Scheme) : Scheme where\n  vars := Sum σ₁.vars σ₂.vars\n  type\n    | .inl var => σ₁.type var\n    | .inr var => σ₂.type var\n\ninfix:65 \" ⊎ \" => Scheme.union\n\n@[simp]\ntheorem Scheme.union_type_left (σ₁ σ₂ : Scheme) (var : σ₁.vars) :\n  (σ₁ ⊎ σ₂).type (.inl var) = σ₁.type var := rfl\n\n@[simp]\ntheorem Scheme.union_type_right (σ₁ σ₂ : Scheme) (var : σ₂.vars) :\n  (σ₁ ⊎ σ₂).type (.inr var) = σ₂.type var := rfl\n\n/-- The big-union of schemes generalizes the notion of unions of schemes to a family of schemes. -/\ndef Scheme.bUnion (σs : I → Scheme) [DecidableEq I] : Scheme where\n  vars := (i : I) × (σs i).vars\n  type := fun ⟨i, var⟩ => (σs i).type var\n\nprefix:100 \" ⨄ \" => Scheme.bUnion\n\n@[simp]\ntheorem Scheme.bUnion_vars (σs : I → Scheme) [DecidableEq I] :\n  (⨄ σs).vars = ((i : I) × (σs i).vars) := rfl\n\n@[simp]\ntheorem Scheme.bUnion_type (σs : I → Scheme) [DecidableEq I] (var : (σs i).vars) :\n  (⨄ σs).type ⟨i, var⟩ = (σs i).type var := rfl\n\n/--\nA scheme `σ₁` is a \"subscheme\" of `σ₂` if:\n1. the domain of `σ₁` is a subtype of the domain of `σ₂`, and\n2. `σ₁` and `σ₂` map the elements in `σ₁`'s domain to the same types.\n\nWe express property 1 not as a `Subtype` but as the coercion `coe`. Property 2 is ensured by\n`coeEqType`. The `inv` map and additional properties provide an inverse map to `coe` (this map has\nto be partial as `coe` isn't necessarilly bijective).\n-/\nclass Subscheme (σ₁ σ₂ : Scheme) where\n  coe       : σ₁.vars → σ₂.vars\n  inv       : σ₂.vars → Option σ₁.vars\n  invInj    : ∀ {a b₁ b₂}, (inv b₁ = some a) → (inv b₂ = some a) → (b₁ = b₂)\n  coeInvId  : ∀ a, inv (coe a) = a\n  coeEqType : ∀ {v}, σ₂.type (coe v) = σ₁.type v\n\ntheorem Subscheme.invEqType [inst : Subscheme σ₁ σ₂] : ∀ {b}, (inst.inv b = some a) → (σ₁.type a = σ₂.type b) :=\n  fun h => by rw [←inst.coeEqType (v := a), inst.invInj h (inst.coeInvId a)]\n\nend Interface\n\n/--\nAn `Interface` over a scheme `{ vars, type }` is a map `(var : vars) → (type var)`. This type is\nused to for components in reactors like state variables and parameters, which are maps from names to\nvalues of different types.\n-/\ndef Interface (σ : Interface.Scheme) := (var : σ.vars) → (σ.type var)\n\n/--\nAn `Interface?` over a scheme `{ vars, type }` is a map `(var : vars) → Option (type var)`. This\ntype is used for components in reactors like ports and actions and parameters, which are maps from\nnames to values of different types which need to be able to be absent.\n-/\ndef Interface? (σ : Interface.Scheme) := (var : σ.vars) → Option (σ.type var)\n\nnamespace Interface?\n\n/-- The `Interface?` where every element is absent. -/\nprotected def empty : Interface? σ := fun _ => none\n\n@[simp]\ntheorem empty_none : Interface?.empty var = none := rfl\n\n/-- An `Interface?` is empty if it is `Interface?.empty` (that is, all elements are absent). -/\ndef isEmpty (i : Interface? σ) := i = Interface?.empty\n\n@[simp]\ntheorem isEmpty_def (i : Interface? σ) : i.isEmpty ↔ i = Interface?.empty := by\n  simp [isEmpty]\n\n/-- An `Interface?` is empty if it is `Interface?.empty` (that is, all elements are absent). -/\ndef isPresent (i : Interface? σ) (var : σ.vars) : Bool :=\n  i var |>.isSome\n\n@[simp]\ntheorem isPresent_def (i : Interface? σ) : i.isPresent var ↔ ∃ v, i var = some v := by\n  simp [isPresent, Option.isSome_iff_exists]\n\n/--\nMerges interface `i₂` \"into\" `i₁`. That is, for each element `e`:\n* if `i₁ e` is present, return that value\n* otherwise, return `i₂ e`\n-/\ndef merge (i₁ i₂ : Interface? σ) : Interface? σ :=\n  fun var => i₂ var |>.orElse fun _ => i₁ var\n\ntheorem merge_val₁ (i₁ i₂ : Interface? σ) : (i₂ var = none) → (i₁.merge i₂) var = i₁ var := by\n  simp_all [merge, Option.orElse]\n\ntheorem merge_val₂ (i₁ i₂ : Interface? σ) : (i₂ var = some v) → (i₁.merge i₂) var = some v := by\n  simp_all [merge, Option.orElse]\n\ntheorem merge_idem : Interface?.merge i i = i := by\n  funext var\n  simp [merge, Option.orElse]\n  split\n  · exact Eq.symm ‹_›\n  · simp\n\n/-- Restricts the domain of the given interface to that of a given subscheme. -/\ndef restrict [inst : Interface.Subscheme σ₁ σ₂] (i : Interface? σ₂) : Interface? σ₁ :=\n  fun var => inst.coeEqType ▸ i (inst.coe var)\n\nend Interface?\n", "meta": {"author": "lf-lang", "repo": "reactor-lean", "sha": "d2eb5458446af838be34ebb6f69549b2f6d9c04d", "save_path": "github-repos/lean/lf-lang-reactor-lean", "path": "github-repos/lean/lf-lang-reactor-lean/reactor-lean-d2eb5458446af838be34ebb6f69549b2f6d9c04d/Runtime/Interface.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34901491888207853}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n\n! This file was ported from Lean 3 source module group_theory.group_action.sum\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.GroupTheory.GroupAction.Defs\n\n/-!\n# Sum instances for additive and multiplicative actions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for additive and multiplicative actions on the binary `sum` type.\n\n## See also\n\n* `group_theory.group_action.option`\n* `group_theory.group_action.pi`\n* `group_theory.group_action.prod`\n* `group_theory.group_action.sigma`\n-/\n\n\nvariable {M N P α β γ : Type _}\n\nnamespace Sum\n\nsection SMul\n\nvariable [SMul M α] [SMul M β] [SMul N α] [SMul N β] (a : M) (b : α) (c : β) (x : Sum α β)\n\n@[to_additive Sum.hasVadd]\ninstance : SMul M (Sum α β) :=\n  ⟨fun a => Sum.map ((· • ·) a) ((· • ·) a)⟩\n\n/- warning: sum.smul_def -> Sum.smul_def is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : SMul.{u1, u2} M α] [_inst_2 : SMul.{u1, u3} M β] (a : M) (x : Sum.{u2, u3} α β), Eq.{succ (max u2 u3)} (Sum.{u2, u3} α β) (SMul.smul.{u1, max u2 u3} M (Sum.{u2, u3} α β) (Sum.hasSmul.{u1, u2, u3} M α β _inst_1 _inst_2) a x) (Sum.map.{u2, u3, u2, u3} α α β β (SMul.smul.{u1, u2} M α _inst_1 a) (SMul.smul.{u1, u3} M β _inst_2 a) x)\nbut is expected to have type\n  forall {M : Type.{u1}} {α : Type.{u3}} {β : Type.{u2}} [_inst_1 : SMul.{u1, u3} M α] [_inst_2 : SMul.{u1, u2} M β] (a : M) (x : Sum.{u3, u2} α β), Eq.{max (succ u3) (succ u2)} (Sum.{u3, u2} α β) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Sum.{u3, u2} α β) (Sum.{u3, u2} α β) (instHSMul.{u1, max u3 u2} M (Sum.{u3, u2} α β) (Sum.instSMulSum.{u1, u3, u2} M α β _inst_1 _inst_2)) a x) (Sum.map.{u3, u2, u3, u2} α α β β ((fun (x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.199 : M) (x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.201 : α) => HSMul.hSMul.{u1, u3, u3} M α α (instHSMul.{u1, u3} M α _inst_1) x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.199 x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.201) a) ((fun (x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.218 : M) (x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.220 : β) => HSMul.hSMul.{u1, u2, u2} M β β (instHSMul.{u1, u2} M β _inst_2) x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.218 x._@.Mathlib.GroupTheory.GroupAction.Sum._hyg.220) a) x)\nCase conversion may be inaccurate. Consider using '#align sum.smul_def Sum.smul_defₓ'. -/\n@[to_additive]\ntheorem smul_def : a • x = x.map ((· • ·) a) ((· • ·) a) :=\n  rfl\n#align sum.smul_def Sum.smul_def\n#align sum.vadd_def Sum.vadd_def\n\n/- warning: sum.smul_inl -> Sum.smul_inl is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : SMul.{u1, u2} M α] [_inst_2 : SMul.{u1, u3} M β] (a : M) (b : α), Eq.{succ (max u2 u3)} (Sum.{u2, u3} α β) (SMul.smul.{u1, max u2 u3} M (Sum.{u2, u3} α β) (Sum.hasSmul.{u1, u2, u3} M α β _inst_1 _inst_2) a (Sum.inl.{u2, u3} α β b)) (Sum.inl.{u2, u3} α β (SMul.smul.{u1, u2} M α _inst_1 a b))\nbut is expected to have type\n  forall {M : Type.{u1}} {α : Type.{u3}} {β : Type.{u2}} [_inst_1 : SMul.{u1, u3} M α] [_inst_2 : SMul.{u1, u2} M β] (a : M) (b : α), Eq.{max (succ u3) (succ u2)} (Sum.{u3, u2} α β) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Sum.{u3, u2} α β) (Sum.{u3, u2} α β) (instHSMul.{u1, max u3 u2} M (Sum.{u3, u2} α β) (Sum.instSMulSum.{u1, u3, u2} M α β _inst_1 _inst_2)) a (Sum.inl.{u3, u2} α β b)) (Sum.inl.{u3, u2} α β (HSMul.hSMul.{u1, u3, u3} M α α (instHSMul.{u1, u3} M α _inst_1) a b))\nCase conversion may be inaccurate. Consider using '#align sum.smul_inl Sum.smul_inlₓ'. -/\n@[simp, to_additive]\ntheorem smul_inl : a • (inl b : Sum α β) = inl (a • b) :=\n  rfl\n#align sum.smul_inl Sum.smul_inl\n#align sum.vadd_inl Sum.vadd_inl\n\n/- warning: sum.smul_inr -> Sum.smul_inr is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : SMul.{u1, u2} M α] [_inst_2 : SMul.{u1, u3} M β] (a : M) (c : β), Eq.{succ (max u2 u3)} (Sum.{u2, u3} α β) (SMul.smul.{u1, max u2 u3} M (Sum.{u2, u3} α β) (Sum.hasSmul.{u1, u2, u3} M α β _inst_1 _inst_2) a (Sum.inr.{u2, u3} α β c)) (Sum.inr.{u2, u3} α β (SMul.smul.{u1, u3} M β _inst_2 a c))\nbut is expected to have type\n  forall {M : Type.{u1}} {α : Type.{u3}} {β : Type.{u2}} [_inst_1 : SMul.{u1, u3} M α] [_inst_2 : SMul.{u1, u2} M β] (a : M) (c : β), Eq.{max (succ u3) (succ u2)} (Sum.{u3, u2} α β) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Sum.{u3, u2} α β) (Sum.{u3, u2} α β) (instHSMul.{u1, max u3 u2} M (Sum.{u3, u2} α β) (Sum.instSMulSum.{u1, u3, u2} M α β _inst_1 _inst_2)) a (Sum.inr.{u3, u2} α β c)) (Sum.inr.{u3, u2} α β (HSMul.hSMul.{u1, u2, u2} M β β (instHSMul.{u1, u2} M β _inst_2) a c))\nCase conversion may be inaccurate. Consider using '#align sum.smul_inr Sum.smul_inrₓ'. -/\n@[simp, to_additive]\ntheorem smul_inr : a • (inr c : Sum α β) = inr (a • c) :=\n  rfl\n#align sum.smul_inr Sum.smul_inr\n#align sum.vadd_inr Sum.vadd_inr\n\n/- warning: sum.smul_swap -> Sum.smul_swap is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : SMul.{u1, u2} M α] [_inst_2 : SMul.{u1, u3} M β] (a : M) (x : Sum.{u2, u3} α β), Eq.{max (succ u3) (succ u2)} (Sum.{u3, u2} β α) (Sum.swap.{u2, u3} α β (SMul.smul.{u1, max u2 u3} M (Sum.{u2, u3} α β) (Sum.hasSmul.{u1, u2, u3} M α β _inst_1 _inst_2) a x)) (SMul.smul.{u1, max u3 u2} M (Sum.{u3, u2} β α) (Sum.hasSmul.{u1, u3, u2} M β α _inst_2 _inst_1) a (Sum.swap.{u2, u3} α β x))\nbut is expected to have type\n  forall {M : Type.{u1}} {α : Type.{u3}} {β : Type.{u2}} [_inst_1 : SMul.{u1, u3} M α] [_inst_2 : SMul.{u1, u2} M β] (a : M) (x : Sum.{u3, u2} α β), Eq.{max (succ u3) (succ u2)} (Sum.{u2, u3} β α) (Sum.swap.{u3, u2} α β (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Sum.{u3, u2} α β) (Sum.{u3, u2} α β) (instHSMul.{u1, max u3 u2} M (Sum.{u3, u2} α β) (Sum.instSMulSum.{u1, u3, u2} M α β _inst_1 _inst_2)) a x)) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Sum.{u2, u3} β α) (Sum.{u2, u3} β α) (instHSMul.{u1, max u3 u2} M (Sum.{u2, u3} β α) (Sum.instSMulSum.{u1, u2, u3} M β α _inst_2 _inst_1)) a (Sum.swap.{u3, u2} α β x))\nCase conversion may be inaccurate. Consider using '#align sum.smul_swap Sum.smul_swapₓ'. -/\n@[simp, to_additive]\ntheorem smul_swap : (a • x).symm = a • x.symm := by cases x <;> rfl\n#align sum.smul_swap Sum.smul_swap\n#align sum.vadd_swap Sum.vadd_swap\n\ninstance [SMul M N] [IsScalarTower M N α] [IsScalarTower M N β] : IsScalarTower M N (Sum α β) :=\n  ⟨fun a b x => by\n    cases x\n    exacts[congr_arg inl (smul_assoc _ _ _), congr_arg inr (smul_assoc _ _ _)]⟩\n\n@[to_additive]\ninstance [SMulCommClass M N α] [SMulCommClass M N β] : SMulCommClass M N (Sum α β) :=\n  ⟨fun a b x => by\n    cases x\n    exacts[congr_arg inl (smul_comm _ _ _), congr_arg inr (smul_comm _ _ _)]⟩\n\n@[to_additive]\ninstance [SMul Mᵐᵒᵖ α] [SMul Mᵐᵒᵖ β] [IsCentralScalar M α] [IsCentralScalar M β] :\n    IsCentralScalar M (Sum α β) :=\n  ⟨fun a x => by\n    cases x\n    exacts[congr_arg inl (op_smul_eq_smul _ _), congr_arg inr (op_smul_eq_smul _ _)]⟩\n\n#print Sum.FaithfulSMulLeft /-\n@[to_additive]\ninstance FaithfulSMulLeft [FaithfulSMul M α] : FaithfulSMul M (Sum α β) :=\n  ⟨fun x y h => eq_of_smul_eq_smul fun a : α => by injection h (inl a)⟩\n#align sum.has_faithful_smul_left Sum.FaithfulSMulLeft\n#align sum.has_faithful_vadd_left Sum.FaithfulVAddLeft\n-/\n\n#print Sum.FaithfulSMulRight /-\n@[to_additive]\ninstance FaithfulSMulRight [FaithfulSMul M β] : FaithfulSMul M (Sum α β) :=\n  ⟨fun x y h => eq_of_smul_eq_smul fun b : β => by injection h (inr b)⟩\n#align sum.has_faithful_smul_right Sum.FaithfulSMulRight\n#align sum.has_faithful_vadd_right Sum.FaithfulVAddRight\n-/\n\nend SMul\n\n@[to_additive]\ninstance {m : Monoid M} [MulAction M α] [MulAction M β] : MulAction M (Sum α β)\n    where\n  mul_smul a b x := by\n    cases x\n    exacts[congr_arg inl (mul_smul _ _ _), congr_arg inr (mul_smul _ _ _)]\n  one_smul x := by\n    cases x\n    exacts[congr_arg inl (one_smul _ _), congr_arg inr (one_smul _ _)]\n\nend Sum\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/GroupTheory/GroupAction/Sum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3490149104083583}}
{"text": "/-\nCopyright (c) 2020 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\nimport tactic.core\n\n/-!\n# The `unify_equations` tactic\n\nThis module defines `unify_equations`, a first-order unification tactic that\nunifies one or more equations in the context. It implements the Qnify algorithm\nfrom [McBride, Inverting Inductively Defined Relations in LEGO][mcbride1996].\n\nThe tactic takes as input some equations which it simplifies one after the\nother. Each equation is simplified by applying one of several possible\nunification steps. Each such step may output other (simpler) equations which are\nunified recursively until no unification step applies any more. See\n`tactic.interactive.unify_equations` for an example and an explanation of the\ndifferent steps.\n-/\n\nopen expr\n\nnamespace tactic\nnamespace unify_equations\n\n/--\nThe result of a unification step:\n\n- `simplified hs` means that the step succeeded and produced some new (simpler)\n  equations `hs`. `hs` can be empty.\n- `goal_solved` means that the step succeeded and solved the goal (by deriving a\n  contradiction from the given equation).\n- `not_simplified` means that the step failed to simplify the equation.\n-/\nmeta inductive unification_step_result : Type\n| simplified (next_equations : list name)\n| not_simplified\n| goal_solved\n\nexport unification_step_result\n\n/--\nA unification step is a tactic that attempts to simplify a given equation and\nreturns a `unification_step_result`. The inputs are:\n\n- `equ`, the equation being processed. Must be a local constant.\n- `lhs_type` and `rhs_type`, the types of equ's LHS and RHS. For homogeneous\n  equations, these are defeq.\n- `lhs` and `rhs`, `equ`'s LHS and RHS.\n- `lhs_whnf` and `rhs_whnf`, `equ`'s LHS and RHS in WHNF.\n- `u`, `equ`'s level.\n\nSo `equ : @eq.{u} lhs_type lhs rhs` or `equ : @heq.{u} lhs_type lhs rhs_type rhs`.\n-/\n@[reducible] meta def unification_step : Type :=\n∀ (equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf : expr) (u : level),\n  tactic unification_step_result\n\n/--\nFor `equ : t == u` with `t : T` and `u : U`, if `T` and `U` are defeq,\nwe replace `equ` with `equ : t = u`.\n-/\nmeta def unify_heterogeneous : unification_step :=\nλ equ lhs_type rhs_type lhs rhs _ _ _,\ndo\n{ is_def_eq lhs_type rhs_type,\n  p ← to_expr ``(@eq_of_heq %%lhs_type %%lhs %%rhs %%equ),\n  t ← to_expr ``(@eq %%lhs_type %%lhs %%rhs),\n  equ' ← note equ.local_pp_name t p,\n  clear equ,\n  pure $ simplified [equ'.local_pp_name] } <|>\npure not_simplified\n\n/--\nFor `equ : t = u`, if `t` and `u` are defeq, we delete `equ`.\n-/\nmeta def unify_defeq : unification_step :=\nλ equ lhs_type _ _ _ lhs_whnf rhs_whnf _,\ndo\n{ is_def_eq lhs_whnf rhs_whnf,\n  clear equ,\n  pure $ simplified [] } <|>\npure not_simplified\n\n/--\nFor `equ : x = t` or `equ : t = x`, where `x` is a local constant, we\nsubstitute `x` with `t` in the goal.\n-/\nmeta def unify_var : unification_step :=\nλ equ type _ lhs rhs lhs_whnf rhs_whnf u,\ndo\n{ let lhs_is_local := lhs_whnf.is_local_constant,\n  let rhs_is_local := rhs_whnf.is_local_constant,\n  guard $ lhs_is_local ∨ rhs_is_local,\n  let t :=\n    if lhs_is_local\n      then (const `eq [u]) type lhs_whnf rhs\n      else (const `eq [u]) type lhs rhs_whnf,\n  change_core t (some equ),\n  equ ← get_local equ.local_pp_name,\n  subst_core equ,\n  pure $ simplified [] } <|>\npure not_simplified\n\n-- TODO This is an improved version of `injection_with` from core\n-- (init/meta/injection_tactic). Remove when the improvements have landed in\n-- core.\nprivate meta def injection_with' (h : expr) (ns : list name)\n  (base := `h) (offset := some 1) :\n  tactic (option (list expr) × list name) :=\ndo\n  H ← infer_type h,\n  (lhs, rhs, constructor_left, constructor_right, inj_name) ← do\n  { (lhs, rhs) ← match_eq H,\n    constructor_left ← get_app_fn_const_whnf lhs semireducible ff,\n    constructor_right ← get_app_fn_const_whnf rhs semireducible ff,\n    inj_name ← resolve_constant $ constructor_left ++ \"inj_arrow\",\n    pure (lhs, rhs, constructor_left, constructor_right, inj_name) }\n  <|> fail\n    (\"injection tactic failed, argument must be an equality proof where lhs and rhs \" ++\n    \"are of the form (c ...), where c is a constructor\"),\n  if constructor_left = constructor_right then do\n    -- C.inj_arrow, for a given constructor C of datatype D, has type\n    --\n    --     ∀ (A₁ ... Aₙ) (x₁ ... xₘ) (y₁ ... yₘ), C x₁ ... xₘ = C y₁ ... yₘ\n    --       → ∀ ⦃P : Sort u⦄, (x₁ = y₁ → ... → yₖ = yₖ → P) → P\n    --\n    -- where the Aᵢ are parameters of D and the xᵢ/yᵢ are arguments of C.\n    -- Note that if xᵢ/yᵢ are propositions, no equation is generated, so the\n    -- number of equations is not necessarily the constructor arity.\n\n    -- First, we find out how many equations we need to intro later.\n    inj ← mk_const inj_name,\n    inj_type ← infer_type inj,\n    inj_arity ← get_pi_arity inj_type,\n    let num_equations :=\n      (inj_type.nth_binding_body (inj_arity - 1)).binding_domain.pi_arity,\n\n    -- Now we generate the actual proof of the target.\n    tgt ← target,\n    proof ← mk_mapp inj_name (list.replicate (inj_arity - 3) none ++ [some h, some tgt]),\n    eapply proof,\n    (next, ns) ← intron_with num_equations ns base offset,\n\n    -- The following filters out 'next' hypotheses of type `true`. The\n    -- `inj_arrow` lemmas introduce these for nullary constructors.\n    next ← next.mfilter $ λ h, do\n    { `(true) ← infer_type h | pure tt,\n      (clear h >> pure ff) <|> pure tt },\n    pure (some next, ns)\n  else do\n    tgt ← target,\n    -- The following construction deals with a corner case involing\n    -- mutual/nested inductive types. For these, Lean does not generate\n    -- no-confusion principles. However, the regular inductive data type which a\n    -- mutual/nested inductive type is compiled to does have a no-confusion\n    -- principle which we can (usually? always?) use. To find it, we normalise\n    -- the constructor with `unfold_ginductive = tt`.\n    constructor_left ← get_app_fn_const_whnf lhs semireducible tt,\n    let no_confusion := constructor_left.get_prefix ++ \"no_confusion\",\n    pr ← mk_app no_confusion [tgt, lhs, rhs, h],\n    exact pr,\n    return (none, ns)\n\n/--\nGiven `equ : C x₁ ... xₙ = D y₁ ... yₘ` with `C` and `D` constructors of the\nsame datatype `I`:\n\n- If `C ≠ D`, we solve the goal by contradiction using the no-confusion rule.\n- If `C = D`, we clear `equ` and add equations `x₁ = y₁`, ..., `xₙ = yₙ`.\n-/\nmeta def unify_constructor_headed : unification_step :=\nλ equ _ _ _ _ _ _ _,\ndo\n{ (next, _) ← injection_with' equ [] `_ none,\n  try $ clear equ,\n  pure $\n    match next with\n    | none := goal_solved\n    | some next := simplified $ next.map expr.local_pp_name\n    end } <|>\npure not_simplified\n\n/--\nFor `type = I x₁ ... xₙ`, where `I` is an inductive type, `get_sizeof type`\nreturns the constant `I.sizeof`. Fails if `type` is not of this form or if no\nsuch constant exists.\n-/\nmeta def get_sizeof (type : expr) : tactic pexpr := do\n  n ← get_app_fn_const_whnf type semireducible ff,\n  resolve_name $ n ++ `sizeof\n\nlemma add_add_one_ne (n m : ℕ) : n + (m + 1) ≠ n :=\nbegin\n  apply ne_of_gt,\n  apply nat.lt_add_of_pos_right,\n  apply nat.pos_of_ne_zero,\n  contradiction\nend\n-- Linarith could prove this, but I want to avoid that dependency.\n\n/--\n`match_n_plus_m n e` matches `e` of the form `nat.succ (... (nat.succ e')...)`.\nIt returns `n` plus the number of `succ` constructors and `e'`. The matching is\nperformed up to normalisation with transparency `md`.\n-/\nmeta def match_n_plus_m (md) : ℕ → expr → tactic (ℕ × expr) :=\nλ n e, do\n  e ← whnf e md,\n  match e with\n  | `(nat.succ %%e) := match_n_plus_m (n + 1) e\n  | _ := pure (n, e)\n  end\n\n/--\nGiven `equ : n + m = n` or `equ : n = n + m` with `n` and `m` natural numbers\nand `m` a nonzero literal, this tactic produces a proof of `false`. More\nprecisely, the two sides of the equation must be of the form\n`nat.succ (... (nat.succ e)...)` with different numbers of `nat.succ`\nconstructors. Matching is performed with transparency `md`.\n-/\nmeta def contradict_n_eq_n_plus_m (md : transparency) (equ lhs rhs : expr) :\n  tactic expr := do\n  ⟨lhs_n, lhs_e⟩ ← match_n_plus_m md 0 lhs,\n  ⟨rhs_n, rhs_e⟩ ← match_n_plus_m md 0 rhs,\n  is_def_eq lhs_e rhs_e md <|> fail\n    (\"contradict_n_eq_n_plus_m:\\nexpected {lhs_e} and {rhs_e} to be definitionally \" ++\n    \"equal at transparency {md}.\"),\n  let common := lhs_e,\n  guard (lhs_n ≠ rhs_n) <|> fail\n    \"contradict_n_eq_n_plus_m:\\nexpected {lhs_n} and {rhs_n} to be different.\",\n  -- Ensure that lhs_n is bigger than rhs_n. Swap lhs and rhs if that's not\n  -- already the case.\n  ⟨equ, lhs_n, rhs_n⟩ ←\n    if lhs_n > rhs_n\n      then pure (equ, lhs_n, rhs_n)\n      else do\n      { equ ← to_expr ``(eq.symm %%equ),\n        pure (equ, rhs_n, lhs_n) },\n  let diff := lhs_n - rhs_n,\n  let rhs_n_expr := reflect rhs_n,\n  n ← to_expr ``(%%common + %%rhs_n_expr),\n  let m := reflect (diff - 1),\n  pure `(add_add_one_ne %%n %%m %%equ)\n\n/--\nGiven `equ : t = u` with `t, u : I` and `I.sizeof t ≠ I.sizeof u`, we solve the\ngoal by contradiction.\n-/\nmeta def unify_cyclic : unification_step :=\nλ equ type _ _ _ lhs_whnf rhs_whnf _,\ndo\n{ -- Establish `sizeof lhs = sizeof rhs`.\n  sizeof ← get_sizeof type,\n  hyp_lhs ← to_expr ``(%%sizeof %%lhs_whnf),\n  hyp_rhs ← to_expr ``(%%sizeof %%rhs_whnf),\n  hyp_type ← to_expr ``(@eq ℕ %%hyp_lhs %%hyp_rhs),\n  hyp_proof ← to_expr ``(@congr_arg %%type ℕ %%lhs_whnf %%rhs_whnf %%sizeof %%equ),\n  hyp_name ← mk_fresh_name,\n  hyp ← note hyp_name hyp_type hyp_proof,\n\n  -- Derive a contradiction (if indeed `sizeof lhs ≠ sizeof rhs`).\n  falso ← contradict_n_eq_n_plus_m semireducible hyp hyp_lhs hyp_rhs,\n  exfalso,\n  exact falso,\n  pure goal_solved } <|>\npure not_simplified\n\n/--\n`orelse_step s t` first runs the unification step `s`. If this was successful\n(i.e. `s` simplified or solved the goal), it returns the result of `s`.\nOtherwise, it runs `t` and returns its result.\n-/\nmeta def orelse_step (s t : unification_step) : unification_step :=\nλ equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u,\ndo\n  r ← s equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u,\n  match r with\n  | simplified _ := pure r\n  | goal_solved := pure r\n  | not_simplified := t equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u\n  end\n\n/--\nFor `equ : t = u`, try the following methods in order: `unify_defeq`,\n`unify_var`, `unify_constructor_headed`, `unify_cyclic`. If any of them is\nsuccessful, stop and return its result. If none is successful, fail.\n-/\nmeta def unify_homogeneous : unification_step :=\nlist.foldl orelse_step (λ _ _ _ _ _ _ _ _, pure not_simplified)\n  [unify_defeq, unify_var, unify_constructor_headed, unify_cyclic]\n\nend unify_equations\n\n\nopen unify_equations\n\n/--\nIf `equ` is the display name of a local constant with type `t = u` or `t == u`,\nthen `unify_equation_once equ` simplifies it once using\n`unify_equations.unify_homogeneous` or `unify_equations.unify_heterogeneous`.\n\nOtherwise it fails.\n-/\nmeta def unify_equation_once (equ : name) : tactic unification_step_result := do\n  eque ← get_local equ,\n  t ← infer_type eque,\n  match t with\n  | (app (app (app (const `eq [u]) type) lhs) rhs) := do\n    lhs_whnf ← whnf_ginductive lhs,\n    rhs_whnf ← whnf_ginductive rhs,\n    unify_homogeneous eque type type lhs rhs lhs_whnf rhs_whnf u\n  | (app (app (app (app (const `heq [u]) lhs_type) lhs) rhs_type) rhs) := do\n    lhs_whnf ← whnf_ginductive lhs,\n    rhs_whnf ← whnf_ginductive rhs,\n    unify_heterogeneous eque lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u\n  | _ := fail! \"Expected {equ} to be an equation, but its type is\\n{t}.\"\n  end\n\n/--\nGiven a list of display names of local hypotheses that are (homogeneous or\nheterogeneous) equations, `unify_equations` performs first-order unification on\neach hypothesis in order. See `tactic.interactive.unify_equations` for an\nexample and an explanation of what unification does.\n\nReturns true iff the goal has been solved during the unification process.\n\nNote: you must make sure that the input names are unique in the context.\n-/\nmeta def unify_equations : list name → tactic bool\n| [] := pure ff\n| (h :: hs) := do\n  res ← unify_equation_once h,\n  match res with\n  | simplified hs' := unify_equations $ hs' ++ hs\n  | not_simplified := unify_equations hs\n  | goal_solved := pure tt\n  end\n\n\nnamespace interactive\n\nopen lean.parser\n\n/--\n`unify_equations eq₁ ... eqₙ` performs a form of first-order unification on the\nhypotheses `eqᵢ`. The `eqᵢ` must be homogeneous or heterogeneous equations.\nUnification means that the equations are simplified using various facts about\nconstructors. For instance, consider this goal:\n\n```\nP : ∀ n, fin n → Prop\nn m : ℕ\nf : fin n\ng : fin m\nh₁ : n + 1 = m + 1\nh₂ : f == g\nh₃ : P n f\n⊢ P m g\n```\n\nAfter `unify_equations h₁ h₂`, we get\n\n```\nP : ∀ n, fin n → Prop\nn : ℕ\nf : fin n\nh₃ : P n f\n⊢ P n f\n```\n\nIn the example, `unify_equations` uses the fact that every constructor is\ninjective to conclude `n = m` from `h₁`. Then it replaces every `m` with `n` and\nmoves on to `h₂`. The types of `f` and `g` are now equal, so the heterogeneous\nequation turns into a homogeneous one and `g` is replaced by `f`. Note that the\nequations are processed from left to right, so `unify_equations h₂ h₁` would not\nsimplify as much.\n\nIn general, `unify_equations` uses the following steps on each equation until\nnone of them applies any more:\n\n- Constructor injectivity: if `nat.succ n = nat.succ m` then `n = m`.\n- Substitution: if `x = e` for some hypothesis `x`, then `x` is replaced by `e`\n  everywhere.\n- No-confusion: `nat.succ n = nat.zero` is a contradiction. If we have such an\n  equation, the goal is solved immediately.\n- Cycle elimination: `n = nat.succ n` is a contradiction.\n- Redundancy: if `t = u` but `t` and `u` are already definitionally equal, then\n  this equation is removed.\n- Downgrading of heterogeneous equations: if `t == u` but `t` and `u` have the\n  same type (up to definitional equality), then the equation is replaced by\n  `t = u`.\n-/\nmeta def unify_equations (eqs : interactive.parse (many ident)) :\n  tactic unit :=\ntactic.unify_equations eqs *> skip\n\nadd_tactic_doc\n{ name := \"unify_equations\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.unify_equations],\n  tags := [\"simplification\"] }\n\nend interactive\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/unify_equations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.349014910329776}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nInstances for identity and composition functors\n-/\n\nimport category.functor\n\nuniverse variables u v w\n\nsection lemmas\n\nopen function\n\nvariables {F : Type u → Type v}\nvariables [applicative F] [is_lawful_applicative F]\nvariables {α β γ σ : Type u}\n\nattribute [functor_norm] seq_assoc pure_seq_eq_map map_pure seq_map_assoc map_seq\n\nlemma applicative.map_seq_map (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :\n  (f <$> x) <*> (g <$> y) = (flip (∘) g ∘ f) <$> x <*> y :=\nby simp [flip] with functor_norm\n\nlemma applicative.pure_seq_eq_map' (f : α → β) :\n  (<*>) (pure f : F (α → β)) = (<$>) f :=\nby ext; simp with functor_norm\n\ntheorem applicative.ext {F} : ∀ {A1 : applicative F} {A2 : applicative F}\n  [@is_lawful_applicative F A1] [@is_lawful_applicative F A2]\n  (H1 : ∀ {α : Type u} (x : α),\n    @has_pure.pure _ A1.to_has_pure _ x = @has_pure.pure _ A2.to_has_pure _ x)\n  (H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α),\n    @has_seq.seq _ A1.to_has_seq _ _ f x = @has_seq.seq _ A2.to_has_seq _ _ f x),\n  A1 = A2\n| {to_functor := F1, seq := s1, pure := p1, seq_left := sl1, seq_right := sr1}\n  {to_functor := F2, seq := s2, pure := p2, seq_left := sl2, seq_right := sr2} L1 L2 H1 H2 :=\nbegin\n  have : @p1 = @p2, {funext α x, apply H1}, subst this,\n  have : @s1 = @s2, {funext α β f x, apply H2}, subst this,\n  cases L1, cases L2,\n  have : F1 = F2,\n  { resetI, apply functor.ext, intros,\n    exact (L1_pure_seq_eq_map _ _).symm.trans (L2_pure_seq_eq_map _ _) },\n  subst this,\n  congr; funext α β x y,\n  { exact (L1_seq_left_eq _ _).trans (L2_seq_left_eq _ _).symm },\n  { exact (L1_seq_right_eq _ _).trans (L2_seq_right_eq _ _).symm }\nend\n\nend lemmas\n\ninstance : is_comm_applicative id :=\nby refine { .. }; intros; refl\n\nnamespace comp\n\nopen function (hiding comp)\nopen functor\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nvariables [applicative F] [applicative G]\n\nprotected def seq {α β : Type v} : comp F G (α → β) → comp F G α → comp F G β\n| (comp.mk f) (comp.mk x) := comp.mk $ (<*>) <$> f <*> x\n\ninstance : has_pure (comp F G) :=\n⟨λ _ x, comp.mk $ pure $ pure x⟩\n\ninstance : has_seq (comp F G) :=\n⟨λ _ _ f x, comp.seq f x⟩\n\n@[simp] protected lemma run_pure {α : Type v} :\n  ∀ x : α, (pure x : comp F G α).run = pure (pure x)\n| _ := rfl\n\n@[simp] protected lemma run_seq {α β : Type v} (f : comp F G (α → β)) (x : comp F G α) :\n  (f <*> x).run = (<*>) <$> f.run <*> x.run := rfl\n\ninstance : applicative (comp F G) :=\n{ map := @comp.map F G _ _,\n  seq := @comp.seq F G _ _,\n  ..comp.has_pure }\n\nvariables [is_lawful_applicative F] [is_lawful_applicative G]\nvariables {α β γ : Type v}\n\nlemma map_pure (f : α → β) (x : α) : (f <$> pure x : comp F G β) = pure (f x) :=\ncomp.ext $ by simp\n\nlemma seq_pure (f : comp F G (α → β)) (x : α) :\n  f <*> pure x = (λ g : α → β, g x) <$> f :=\ncomp.ext $ by simp [(∘)] with functor_norm\n\nlemma seq_assoc (x : comp F G α) (f : comp F G (α → β)) (g : comp F G (β → γ)) :\n   g <*> (f <*> x) = (@function.comp α β γ <$> g) <*> f <*> x :=\ncomp.ext $ by simp [(∘)] with functor_norm\n\nlemma pure_seq_eq_map (f : α → β) (x : comp F G α) :\n  pure f <*> x = f <$> x :=\ncomp.ext $ by simp [applicative.pure_seq_eq_map'] with functor_norm\n\ninstance : is_lawful_applicative (comp F G) :=\n{ pure_seq_eq_map := @comp.pure_seq_eq_map F G _ _ _ _,\n  map_pure := @comp.map_pure F G _ _ _ _,\n  seq_pure := @comp.seq_pure F G _ _ _ _,\n  seq_assoc := @comp.seq_assoc F G _ _ _ _ }\n\ntheorem applicative_id_comp {F} [AF : applicative F] [LF : is_lawful_applicative F] :\n  @comp.applicative id F _ _ = AF :=\n@applicative.ext F _ _ (@comp.is_lawful_applicative id F _ _ _ _) _\n  (λ α x, rfl) (λ α β f x, rfl)\n\ntheorem applicative_comp_id {F} [AF : applicative F] [LF : is_lawful_applicative F] :\n  @comp.applicative F id _ _ = AF :=\n@applicative.ext F _ _ (@comp.is_lawful_applicative F id _ _ _ _) _\n  (λ α x, rfl) (λ α β f x, show id <$> f <*> x = f <*> x, by rw id_map)\n\nopen is_comm_applicative\n\ninstance {f : Type u → Type w} {g : Type v → Type u}\n  [applicative f] [applicative g]\n  [is_comm_applicative f] [is_comm_applicative g] :\n  is_comm_applicative (comp f g) :=\nby { refine { .. @comp.is_lawful_applicative f g _ _ _ _, .. },\n     intros, casesm* comp _ _ _, simp! [map,has_seq.seq] with functor_norm,\n     rw [commutative_map],\n     simp [comp.mk,flip,(∘)] with functor_norm,\n     congr, funext, rw [commutative_map], congr }\n\nend comp\nopen functor\n\n@[functor_norm]\nlemma comp.seq_mk {α β : Type w}\n  {f : Type u → Type v} {g : Type w → Type u}\n  [applicative f] [applicative g]\n  (h : f (g (α → β))) (x : f (g α)) :\n  comp.mk h <*> comp.mk x = comp.mk (has_seq.seq <$> h <*> x) := rfl\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/category/applicative.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3489737222844969}}
{"text": "import pseudo_normed_group.CLC\nimport rescale.LC\n\nopen_locale classical nnreal\nopen opposite ProFiltPseuNormGrpWithTinv\n\nopen SemiNormedGroup opposite Profinite pseudo_normed_group category_theory breen_deligne\nopen profinitely_filtered_pseudo_normed_group\n\nuniverse variable u\nvariables (r : ℝ≥0) (V : SemiNormedGroup) (r' : ℝ≥0) [fact (0 < r')]\nvariables (c c₁ c₂ c₃ c₄ : ℝ≥0) (m n : ℕ)\n\n@[simp] theorem CLCFP_rescale (N : ℝ≥0)\n  (M) [profinitely_filtered_pseudo_normed_group_with_Tinv r' M] :\n  (CLCFP V r' c n).obj (op (of r' (rescale N M))) =\n  (CLCFP V r' (c * N⁻¹) n).obj (op (of r' M)) := rfl\n\nnamespace breen_deligne\n\nnamespace universal_map\n\nvariables (ϕ : universal_map m n)\n\ntheorem eval_CLCFP_rescale [ϕ.suitable c₂ c₁]\n  (N : ℝ≥0)\n  (M : ProFiltPseuNormGrpWithTinv r') :\n  arrow.mk ((eval_CLCFP V r' c₁ c₂ ϕ).app (op (of r' (rescale N M)))) =\n  arrow.mk ((eval_CLCFP V r' (c₁ * N⁻¹) (c₂ * N⁻¹) ϕ).app (op M)) :=\nby { dsimp only [eval_CLCFP, whisker_right_app], rw eval_LCFP_rescale, cases M, refl }\n\nend universal_map\n\nend breen_deligne\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/rescale/CLC.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3489708056480584}}
{"text": "import tactic.tauto\nimport tactic.split_ifs\n\n@[simp] lemma sum_inl_eq_ite {α β : Type*} (c : Prop) [decidable c] (x z : α) (y : β) :\n  (sum.inl z = if c then sum.inl x else sum.inr y) ↔ c ∧ z = x := by split_ifs; simp; tauto\n\n@[simp] lemma sum_inl_eq_ite_symm {α β : Type*} (c : Prop) [decidable c] (x z : α) (y : β) :\n  ((if c then sum.inl x else sum.inr y) = sum.inl z) ↔ c ∧ z = x := by split_ifs; tauto\n\n@[simp] lemma sum_inr_eq_ite {α β : Type*} (c : Prop) [decidable c] (x : α) (y z : β) :\n  (sum.inr z = if c then sum.inl x else sum.inr y) ↔ ¬c ∧ z = y := by split_ifs; simp; tauto\n\n@[simp] lemma sum_inr_eq_ite_symm {α β : Type*} (c : Prop) [decidable c] (x : α) (y z : β) :\n  ((if c then sum.inl x else sum.inr y) = sum.inr z) ↔ ¬c ∧ z = y := by split_ifs; tauto\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/ite_lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514763, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.3488913569137269}}
{"text": "/-\nCopyright (c) 2018 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.uniform_space.abstract_completion\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Hausdorff completions of uniform spaces\n\nThe goal is to construct a left-adjoint to the inclusion of complete Hausdorff uniform spaces\ninto all uniform spaces. Any uniform space `α` gets a completion `completion α` and a morphism\n(ie. uniformly continuous map) `coe : α → completion α` which solves the universal\nmapping problem of factorizing morphisms from `α` to any complete Hausdorff uniform space `β`.\nIt means any uniformly continuous `f : α → β` gives rise to a unique morphism\n`completion.extension f : completion α → β` such that `f = completion.extension f ∘ coe`.\nActually `completion.extension f` is defined for all maps from `α` to `β` but it has the desired\nproperties only if `f` is uniformly continuous.\n\nBeware that `coe` is not injective if `α` is not Hausdorff. But its image is always\ndense. The adjoint functor acting on morphisms is then constructed by the usual abstract nonsense.\nFor every uniform spaces `α` and `β`, it turns `f : α → β` into a morphism\n  `completion.map f : completion α → completion β`\nsuch that\n  `coe ∘ f = (completion.map f) ∘ coe`\nprovided `f` is uniformly continuous. This construction is compatible with composition.\n\nIn this file we introduce the following concepts:\n\n* `Cauchy α` the uniform completion of the uniform space `α` (using Cauchy filters). These are not\n  minimal filters.\n\n* `completion α := quotient (separation_setoid (Cauchy α))` the Hausdorff completion.\n\n## References\n\nThis formalization is mostly based on\n  N. Bourbaki: General Topology\n  I. M. James: Topologies and Uniformities\nFrom a slightly different perspective in order to reuse material in topology.uniform_space.basic.\n-/\n\n/-- Space of Cauchy filters\n\nThis is essentially the completion of a uniform space. The embeddings are the neighbourhood filters.\nThis space is not minimal, the separated uniform space (i.e. quotiented on the intersection of all\nentourages) is necessary for this.\n-/\ndef Cauchy (α : Type u) [uniform_space α] := Subtype fun (f : filter α) => cauchy f\n\nnamespace Cauchy\n\n\ndef gen {α : Type u} [uniform_space α] (s : set (α × α)) : set (Cauchy α × Cauchy α) :=\n  set_of\n    fun (p : Cauchy α × Cauchy α) =>\n      s ∈ filter.prod (subtype.val (prod.fst p)) (subtype.val (prod.snd p))\n\ntheorem monotone_gen {α : Type u} [uniform_space α] : monotone gen :=\n  set.monotone_set_of fun (p : Cauchy α × Cauchy α) => filter.monotone_mem_sets\n\nprotected instance uniform_space {α : Type u} [uniform_space α] : uniform_space (Cauchy α) :=\n  uniform_space.of_core\n    (uniform_space.core.mk (filter.lift' (uniformity α) gen) sorry symm_gen comp_gen)\n\ntheorem mem_uniformity {α : Type u} [uniform_space α] {s : set (Cauchy α × Cauchy α)} :\n    s ∈ uniformity (Cauchy α) ↔ ∃ (t : set (α × α)), ∃ (H : t ∈ uniformity α), gen t ⊆ s :=\n  filter.mem_lift'_sets monotone_gen\n\ntheorem mem_uniformity' {α : Type u} [uniform_space α] {s : set (Cauchy α × Cauchy α)} :\n    s ∈ uniformity (Cauchy α) ↔\n        ∃ (t : set (α × α)),\n          ∃ (H : t ∈ uniformity α),\n            ∀ (f g : Cauchy α), t ∈ filter.prod (subtype.val f) (subtype.val g) → (f, g) ∈ s :=\n  iff.trans mem_uniformity (bex_congr fun (t : set (α × α)) (h : t ∈ uniformity α) => prod.forall)\n\n/-- Embedding of `α` into its completion `Cauchy α` -/\ndef pure_cauchy {α : Type u} [uniform_space α] (a : α) : Cauchy α :=\n  { val := pure a, property := cauchy_pure }\n\ntheorem uniform_inducing_pure_cauchy {α : Type u} [uniform_space α] :\n    uniform_inducing pure_cauchy :=\n  sorry\n\ntheorem uniform_embedding_pure_cauchy {α : Type u} [uniform_space α] :\n    uniform_embedding pure_cauchy :=\n  uniform_embedding.mk\n    (uniform_inducing.mk (uniform_inducing.comap_uniformity uniform_inducing_pure_cauchy))\n    fun (a₁ a₂ : α) (h : pure_cauchy a₁ = pure_cauchy a₂) =>\n      filter.pure_injective (iff.mp subtype.ext_iff_val h)\n\ntheorem dense_range_pure_cauchy {α : Type u} [uniform_space α] : dense_range pure_cauchy := sorry\n\ntheorem dense_inducing_pure_cauchy {α : Type u} [uniform_space α] : dense_inducing pure_cauchy :=\n  uniform_inducing.dense_inducing uniform_inducing_pure_cauchy dense_range_pure_cauchy\n\ntheorem dense_embedding_pure_cauchy {α : Type u} [uniform_space α] : dense_embedding pure_cauchy :=\n  uniform_embedding.dense_embedding uniform_embedding_pure_cauchy dense_range_pure_cauchy\n\ntheorem nonempty_Cauchy_iff {α : Type u} [uniform_space α] : Nonempty (Cauchy α) ↔ Nonempty α :=\n  sorry\n\nprotected instance complete_space {α : Type u} [uniform_space α] : complete_space (Cauchy α) :=\n  complete_space_extension uniform_inducing_pure_cauchy dense_range_pure_cauchy\n    fun (f : filter α) (hf : cauchy f) =>\n      let f' : Cauchy α := { val := f, property := hf };\n      (fun\n          (this :\n          filter.map pure_cauchy f ≤\n            filter.lift' (uniformity (Cauchy α)) (set.preimage (Prod.mk f'))) =>\n          Exists.intro f'\n            (eq.mpr\n              (id\n                ((fun (ᾰ ᾰ_1 : filter (Cauchy α)) (e_2 : ᾰ = ᾰ_1) (ᾰ_2 ᾰ_3 : filter (Cauchy α))\n                    (e_3 : ᾰ_2 = ᾰ_3) => congr (congr_arg LessEq e_2) e_3)\n                  (filter.map pure_cauchy f) (filter.map pure_cauchy f)\n                  (Eq.refl (filter.map pure_cauchy f)) (nhds f')\n                  (filter.lift' (uniformity (Cauchy α)) (uniform_space.ball f'))\n                  nhds_eq_uniformity))\n              this))\n        (filter.le_lift'\n          fun (s : set (Cauchy α × Cauchy α)) (hs : s ∈ uniformity (Cauchy α)) => sorry)\n\nprotected instance inhabited {α : Type u} [uniform_space α] [Inhabited α] : Inhabited (Cauchy α) :=\n  { default := pure_cauchy Inhabited.default }\n\nprotected instance nonempty {α : Type u} [uniform_space α] [h : Nonempty α] : Nonempty (Cauchy α) :=\n  nonempty.rec_on h fun (a : α) => Nonempty.intro (pure_cauchy a)\n\ndef extend {α : Type u} [uniform_space α] {β : Type v} [uniform_space β] (f : α → β) :\n    Cauchy α → β :=\n  ite (uniform_continuous f) (dense_inducing.extend dense_inducing_pure_cauchy f)\n    fun (x : Cauchy α) => f Inhabited.default\n\ntheorem extend_pure_cauchy {α : Type u} [uniform_space α] {β : Type v} [uniform_space β]\n    [separated_space β] {f : α → β} (hf : uniform_continuous f) (a : α) :\n    extend f (pure_cauchy a) = f a :=\n  sorry\n\ntheorem uniform_continuous_extend {α : Type u} [uniform_space α] {β : Type v} [uniform_space β]\n    [separated_space β] [complete_space β] {f : α → β} : uniform_continuous (extend f) :=\n  sorry\n\ntheorem Cauchy_eq {α : Type u_1} [Inhabited α] [uniform_space α] [complete_space α]\n    [separated_space α] {f : Cauchy α} {g : Cauchy α} :\n    Lim (subtype.val f) = Lim (subtype.val g) ↔ (f, g) ∈ Mathlib.separation_rel (Cauchy α) :=\n  sorry\n\ntheorem separated_pure_cauchy_injective {α : Type u_1} [uniform_space α] [s : separated_space α] :\n    function.injective fun (a : α) => quotient.mk (pure_cauchy a) :=\n  sorry\n\nend Cauchy\n\n\nnamespace uniform_space\n\n\nprotected instance complete_space_separation (α : Type u_1) [uniform_space α]\n    [h : complete_space α] : complete_space (quotient (separation_setoid α)) :=\n  complete_space.mk\n    fun (f : filter (quotient (separation_setoid α))) (hf : cauchy f) =>\n      (fun (this : cauchy (filter.comap (fun (x : α) => quotient.mk x) f)) => sorry)\n        (cauchy.comap' hf comap_quotient_le_uniformity\n          (filter.ne_bot.comap_of_surj (and.left hf)\n            fun (b : quotient (separation_setoid α)) => quotient.exists_rep b))\n\n/-- Hausdorff completion of `α` -/\ndef completion (α : Type u_1) [uniform_space α] := quotient (separation_setoid (Cauchy α))\n\nnamespace completion\n\n\nprotected instance inhabited (α : Type u_1) [uniform_space α] [Inhabited α] :\n    Inhabited (completion α) :=\n  eq.mpr sorry quotient.inhabited\n\nprotected instance uniform_space (α : Type u_1) [uniform_space α] : uniform_space (completion α) :=\n  id separation_setoid.uniform_space\n\nprotected instance complete_space (α : Type u_1) [uniform_space α] :\n    complete_space (completion α) :=\n  id (uniform_space.complete_space_separation (Cauchy α))\n\nprotected instance separated_space (α : Type u_1) [uniform_space α] :\n    separated_space (completion α) :=\n  id uniform_space.separated_separation\n\nprotected instance regular_space (α : Type u_1) [uniform_space α] : regular_space (completion α) :=\n  Mathlib.separated_regular\n\n/-- Automatic coercion from `α` to its completion. Not always injective. -/\nprotected instance has_coe_t (α : Type u_1) [uniform_space α] : has_coe_t α (completion α) :=\n  has_coe_t.mk (quotient.mk ∘ Cauchy.pure_cauchy)\n\nprotected theorem coe_eq (α : Type u_1) [uniform_space α] :\n    coe = quotient.mk ∘ Cauchy.pure_cauchy :=\n  rfl\n\ntheorem comap_coe_eq_uniformity (α : Type u_1) [uniform_space α] :\n    filter.comap (fun (p : α × α) => (↑(prod.fst p), ↑(prod.snd p))) (uniformity (completion α)) =\n        uniformity α :=\n  sorry\n\ntheorem uniform_inducing_coe (α : Type u_1) [uniform_space α] : uniform_inducing coe :=\n  uniform_inducing.mk (comap_coe_eq_uniformity α)\n\ntheorem dense_range_coe {α : Type u_1} [uniform_space α] : dense_range coe :=\n  dense_range.quotient Cauchy.dense_range_pure_cauchy\n\ndef cpkg {α : Type u_1} [uniform_space α] : abstract_completion α :=\n  abstract_completion.mk (completion α) coe (completion.uniform_space α)\n    (completion.complete_space α) (completion.separated_space α) (uniform_inducing_coe α)\n    dense_range_coe\n\nprotected instance abstract_completion.inhabited (α : Type u_1) [uniform_space α] :\n    Inhabited (abstract_completion α) :=\n  { default := cpkg }\n\ntheorem nonempty_completion_iff (α : Type u_1) [uniform_space α] :\n    Nonempty (completion α) ↔ Nonempty α :=\n  iff.symm (dense_range.nonempty_iff (abstract_completion.dense cpkg))\n\ntheorem uniform_continuous_coe (α : Type u_1) [uniform_space α] : uniform_continuous coe :=\n  abstract_completion.uniform_continuous_coe cpkg\n\ntheorem continuous_coe (α : Type u_1) [uniform_space α] : continuous coe :=\n  abstract_completion.continuous_coe cpkg\n\ntheorem uniform_embedding_coe (α : Type u_1) [uniform_space α] [separated_space α] :\n    uniform_embedding coe :=\n  uniform_embedding.mk (uniform_inducing.mk (comap_coe_eq_uniformity α))\n    Cauchy.separated_pure_cauchy_injective\n\ntheorem dense_inducing_coe {α : Type u_1} [uniform_space α] : dense_inducing coe :=\n  dense_inducing.mk\n    (inducing.mk (inducing.induced (uniform_inducing.inducing (uniform_inducing_coe α))))\n    dense_range_coe\n\nprotected instance separable_space_completion {α : Type u_1} [uniform_space α]\n    [topological_space.separable_space α] : topological_space.separable_space (completion α) :=\n  dense_inducing.separable_space dense_inducing_coe\n\ntheorem dense_embedding_coe {α : Type u_1} [uniform_space α] [separated_space α] :\n    dense_embedding coe :=\n  dense_embedding.mk\n    (dense_inducing.mk (dense_inducing.to_inducing dense_inducing_coe)\n      (dense_inducing.dense dense_inducing_coe))\n    Cauchy.separated_pure_cauchy_injective\n\ntheorem dense_range_coe₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] :\n    dense_range fun (x : α × β) => (↑(prod.fst x), ↑(prod.snd x)) :=\n  dense_range.prod_map dense_range_coe dense_range_coe\n\ntheorem dense_range_coe₃ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] :\n    dense_range\n        fun (x : α × β × γ) =>\n          (↑(prod.fst x), ↑(prod.fst (prod.snd x)), ↑(prod.snd (prod.snd x))) :=\n  dense_range.prod_map dense_range_coe dense_range_coe₂\n\ntheorem induction_on {α : Type u_1} [uniform_space α] {p : completion α → Prop} (a : completion α)\n    (hp : is_closed (set_of fun (a : completion α) => p a)) (ih : ∀ (a : α), p ↑a) : p a :=\n  is_closed_property dense_range_coe hp ih a\n\ntheorem induction_on₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {p : completion α → completion β → Prop} (a : completion α) (b : completion β)\n    (hp : is_closed (set_of fun (x : completion α × completion β) => p (prod.fst x) (prod.snd x)))\n    (ih : ∀ (a : α) (b : β), p ↑a ↑b) : p a b :=\n  sorry\n\ntheorem induction_on₃ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] {p : completion α → completion β → completion γ → Prop}\n    (a : completion α) (b : completion β) (c : completion γ)\n    (hp :\n      is_closed\n        (set_of\n          fun (x : completion α × completion β × completion γ) =>\n            p (prod.fst x) (prod.fst (prod.snd x)) (prod.snd (prod.snd x))))\n    (ih : ∀ (a : α) (b : β) (c : γ), p ↑a ↑b ↑c) : p a b c :=\n  sorry\n\ntheorem ext {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] [t2_space β]\n    {f : completion α → β} {g : completion α → β} (hf : continuous f) (hg : continuous g)\n    (h : ∀ (a : α), f ↑a = g ↑a) : f = g :=\n  abstract_completion.funext cpkg hf hg h\n\n/-- \"Extension\" to the completion. It is defined for any map `f` but\nreturns an arbitrary constant value if `f` is not uniformly continuous -/\nprotected def extension {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    (f : α → β) : completion α → β :=\n  abstract_completion.extend cpkg f\n\n@[simp] theorem extension_coe {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} [separated_space β] (hf : uniform_continuous f) (a : α) :\n    completion.extension f ↑a = f a :=\n  abstract_completion.extend_coe cpkg hf a\n\ntheorem uniform_continuous_extension {α : Type u_1} [uniform_space α] {β : Type u_2}\n    [uniform_space β] {f : α → β} [separated_space β] [complete_space β] :\n    uniform_continuous (completion.extension f) :=\n  abstract_completion.uniform_continuous_extend cpkg\n\ntheorem continuous_extension {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} [separated_space β] [complete_space β] : continuous (completion.extension f) :=\n  abstract_completion.continuous_extend cpkg\n\ntheorem extension_unique {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} [separated_space β] [complete_space β] (hf : uniform_continuous f)\n    {g : completion α → β} (hg : uniform_continuous g) (h : ∀ (a : α), f a = g ↑a) :\n    completion.extension f = g :=\n  abstract_completion.extend_unique cpkg hf hg h\n\n@[simp] theorem extension_comp_coe {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    [separated_space β] [complete_space β] {f : completion α → β} (hf : uniform_continuous f) :\n    completion.extension (f ∘ coe) = f :=\n  abstract_completion.extend_comp_coe cpkg hf\n\n/-- Completion functor acting on morphisms -/\nprotected def map {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] (f : α → β) :\n    completion α → completion β :=\n  abstract_completion.map cpkg cpkg f\n\ntheorem uniform_continuous_map {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} : uniform_continuous (completion.map f) :=\n  abstract_completion.uniform_continuous_map cpkg cpkg f\n\ntheorem continuous_map {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} : continuous (completion.map f) :=\n  abstract_completion.continuous_map cpkg cpkg f\n\n@[simp] theorem map_coe {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {f : α → β} (hf : uniform_continuous f) (a : α) : completion.map f ↑a = ↑(f a) :=\n  abstract_completion.map_coe cpkg cpkg hf a\n\ntheorem map_unique {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] {f : α → β}\n    {g : completion α → completion β} (hg : uniform_continuous g) (h : ∀ (a : α), ↑(f a) = g ↑a) :\n    completion.map f = g :=\n  abstract_completion.map_unique cpkg cpkg hg h\n\n@[simp] theorem map_id {α : Type u_1} [uniform_space α] : completion.map id = id :=\n  abstract_completion.map_id cpkg\n\ntheorem extension_map {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] [complete_space γ] [separated_space γ] {f : β → γ} {g : α → β}\n    (hf : uniform_continuous f) (hg : uniform_continuous g) :\n    completion.extension f ∘ completion.map g = completion.extension (f ∘ g) :=\n  sorry\n\ntheorem map_comp {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] {γ : Type u_3}\n    [uniform_space γ] {g : β → γ} {f : α → β} (hg : uniform_continuous g)\n    (hf : uniform_continuous f) : completion.map g ∘ completion.map f = completion.map (g ∘ f) :=\n  extension_map (uniform_continuous.comp (uniform_continuous_coe γ) hg) hf\n\n/- In this section we construct isomorphisms between the completion of a uniform space and the\ncompletion of its separation quotient -/\n\ndef completion_separation_quotient_equiv (α : Type u) [uniform_space α] :\n    completion (separation_quotient α) ≃ completion α :=\n  equiv.mk (completion.extension (separation_quotient.lift coe)) (completion.map quotient.mk) sorry\n    sorry\n\ntheorem uniform_continuous_completion_separation_quotient_equiv {α : Type u_1} [uniform_space α] :\n    uniform_continuous ⇑(completion_separation_quotient_equiv α) :=\n  uniform_continuous_extension\n\ntheorem uniform_continuous_completion_separation_quotient_equiv_symm {α : Type u_1}\n    [uniform_space α] : uniform_continuous ⇑(equiv.symm (completion_separation_quotient_equiv α)) :=\n  uniform_continuous_map\n\nprotected def extension₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] (f : α → β → γ) : completion α → completion β → γ :=\n  abstract_completion.extend₂ cpkg cpkg f\n\n@[simp] theorem extension₂_coe_coe {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] {f : α → β → γ} [separated_space γ]\n    (hf : uniform_continuous₂ f) (a : α) (b : β) : completion.extension₂ f ↑a ↑b = f a b :=\n  abstract_completion.extension₂_coe_coe cpkg cpkg hf a b\n\ntheorem uniform_continuous_extension₂ {α : Type u_1} [uniform_space α] {β : Type u_2}\n    [uniform_space β] {γ : Type u_3} [uniform_space γ] (f : α → β → γ) [separated_space γ]\n    [complete_space γ] : uniform_continuous₂ (completion.extension₂ f) :=\n  abstract_completion.uniform_continuous_extension₂ cpkg cpkg f\n\nprotected def map₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β] {γ : Type u_3}\n    [uniform_space γ] (f : α → β → γ) : completion α → completion β → completion γ :=\n  abstract_completion.map₂ cpkg cpkg cpkg f\n\ntheorem uniform_continuous_map₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] (f : α → β → γ) : uniform_continuous₂ (completion.map₂ f) :=\n  abstract_completion.uniform_continuous_map₂ cpkg cpkg cpkg f\n\ntheorem continuous_map₂ {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] {δ : Type u_4} [topological_space δ] {f : α → β → γ}\n    {a : δ → completion α} {b : δ → completion β} (ha : continuous a) (hb : continuous b) :\n    continuous fun (d : δ) => completion.map₂ f (a d) (b d) :=\n  abstract_completion.continuous_map₂ cpkg cpkg cpkg ha hb\n\ntheorem map₂_coe_coe {α : Type u_1} [uniform_space α] {β : Type u_2} [uniform_space β]\n    {γ : Type u_3} [uniform_space γ] (a : α) (b : β) (f : α → β → γ) (hf : uniform_continuous₂ f) :\n    completion.map₂ f ↑a ↑b = ↑(f a b) :=\n  abstract_completion.map₂_coe_coe cpkg cpkg cpkg a b f hf\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/uniform_space/completion_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3488913482066966}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta, Adam Topaz\n-/\nimport category_theory.functor.category\nimport category_theory.functor.fully_faithful\nimport category_theory.functor.reflects_isomorphisms\n\nnamespace category_theory\nopen category\n\nuniverses v₁ u₁ -- morphism levels before object levels. See note [category_theory universes].\n\nvariables (C : Type u₁) [category.{v₁} C]\n\n/--\nThe data of a monad on C consists of an endofunctor T together with natural transformations\nη : 𝟭 C ⟶ T and μ : T ⋙ T ⟶ T satisfying three equations:\n- T μ_X ≫ μ_X = μ_(TX) ≫ μ_X (associativity)\n- η_(TX) ≫ μ_X = 1_X (left unit)\n- Tη_X ≫ μ_X = 1_X (right unit)\n-/\nstructure monad extends C ⥤ C :=\n(η' [] : 𝟭 _ ⟶ to_functor)\n(μ' [] : to_functor ⋙ to_functor ⟶ to_functor)\n(assoc' : ∀ X, to_functor.map (nat_trans.app μ' X) ≫ μ'.app _ = μ'.app _ ≫ μ'.app _ . obviously)\n(left_unit' : ∀ X : C, η'.app (to_functor.obj X) ≫ μ'.app _ = 𝟙 _ . obviously)\n(right_unit' : ∀ X : C, to_functor.map (η'.app X) ≫ μ'.app _ = 𝟙 _ . obviously)\n\n/--\nThe data of a comonad on C consists of an endofunctor G together with natural transformations\nε : G ⟶ 𝟭 C and δ : G ⟶ G ⋙ G satisfying three equations:\n- δ_X ≫ G δ_X = δ_X ≫ δ_(GX) (coassociativity)\n- δ_X ≫ ε_(GX) = 1_X (left counit)\n- δ_X ≫ G ε_X = 1_X (right counit)\n-/\nstructure comonad extends C ⥤ C :=\n(ε' [] : to_functor ⟶ 𝟭 _)\n(δ' [] : to_functor ⟶ to_functor ⋙ to_functor)\n(coassoc' : ∀ X, nat_trans.app δ' _ ≫ to_functor.map (δ'.app X) = δ'.app _ ≫ δ'.app _ . obviously)\n(left_counit' : ∀ X : C, δ'.app X ≫ ε'.app (to_functor.obj X) = 𝟙 _ . obviously)\n(right_counit' : ∀ X : C, δ'.app X ≫ to_functor.map (ε'.app X) = 𝟙 _ . obviously)\n\nvariables {C} (T : monad C) (G : comonad C)\n\ninstance coe_monad : has_coe (monad C) (C ⥤ C) := ⟨λ T, T.to_functor⟩\ninstance coe_comonad : has_coe (comonad C) (C ⥤ C) := ⟨λ G, G.to_functor⟩\n\n@[simp] lemma monad_to_functor_eq_coe : T.to_functor = T := rfl\n@[simp] lemma comonad_to_functor_eq_coe : G.to_functor = G := rfl\n\n/-- The unit for the monad `T`. -/\ndef monad.η : 𝟭 _ ⟶ (T : C ⥤ C) := T.η'\n/-- The multiplication for the monad `T`. -/\ndef monad.μ : (T : C ⥤ C) ⋙ (T : C ⥤ C) ⟶ T := T.μ'\n\n/-- The counit for the comonad `G`. -/\ndef comonad.ε : (G : C ⥤ C) ⟶ 𝟭 _  := G.ε'\n/-- The comultiplication for the comonad `G`. -/\ndef comonad.δ : (G : C ⥤ C) ⟶ (G : C ⥤ C) ⋙ G := G.δ'\n\n/-- A custom simps projection for the functor part of a monad, as a coercion. -/\ndef monad.simps.coe := (T : C ⥤ C)\n/-- A custom simps projection for the unit of a monad, in simp normal form. -/\ndef monad.simps.η : 𝟭 _ ⟶ (T : C ⥤ C) := T.η\n/-- A custom simps projection for the multiplication of a monad, in simp normal form. -/\ndef monad.simps.μ : (T : C ⥤ C) ⋙ (T : C ⥤ C) ⟶ (T : C ⥤ C) := T.μ\n\n/-- A custom simps projection for the functor part of a comonad, as a coercion. -/\ndef comonad.simps.coe := (G : C ⥤ C)\n/-- A custom simps projection for the counit of a comonad, in simp normal form. -/\ndef comonad.simps.ε : (G : C ⥤ C) ⟶ 𝟭 _ := G.ε\n/-- A custom simps projection for the comultiplication of a comonad, in simp normal form. -/\ndef comonad.simps.δ : (G : C ⥤ C) ⟶ (G : C ⥤ C) ⋙ (G : C ⥤ C) := G.δ\n\ninitialize_simps_projections category_theory.monad (to_functor → coe, η' → η, μ' → μ)\ninitialize_simps_projections category_theory.comonad (to_functor → coe, ε' → ε, δ' → δ)\n\n@[reassoc]\nlemma monad.assoc (T : monad C) (X : C) :\n  (T : C ⥤ C).map (T.μ.app X) ≫ T.μ.app _ = T.μ.app _ ≫ T.μ.app _ :=\nT.assoc' X\n\n@[simp, reassoc] lemma monad.left_unit (T : monad C) (X : C) :\n  T.η.app ((T : C ⥤ C).obj X) ≫ T.μ.app X = 𝟙 ((T : C ⥤ C).obj X) :=\nT.left_unit' X\n\n@[simp, reassoc] lemma monad.right_unit (T : monad C) (X : C) :\n  (T : C ⥤ C).map (T.η.app X) ≫ T.μ.app X = 𝟙 ((T : C ⥤ C).obj X) :=\nT.right_unit' X\n\n@[reassoc]\nlemma comonad.coassoc (G : comonad C) (X : C) :\n  G.δ.app _ ≫ (G : C ⥤ C).map (G.δ.app X) = G.δ.app _ ≫ G.δ.app _ :=\nG.coassoc' X\n\n@[simp, reassoc] lemma comonad.left_counit (G : comonad C) (X : C) :\n  G.δ.app X ≫ G.ε.app ((G : C ⥤ C).obj X) = 𝟙 ((G : C ⥤ C).obj X) :=\nG.left_counit' X\n\n@[simp, reassoc] lemma comonad.right_counit (G : comonad C) (X : C) :\n  G.δ.app X ≫ (G : C ⥤ C).map (G.ε.app X) = 𝟙 ((G : C ⥤ C).obj X) :=\nG.right_counit' X\n\n/-- A morphism of monads is a natural transformation compatible with η and μ. -/\n@[ext]\nstructure monad_hom (T₁ T₂ : monad C) extends nat_trans (T₁ : C ⥤ C) T₂ :=\n(app_η' : ∀ X, T₁.η.app X ≫ app X = T₂.η.app X . obviously)\n(app_μ' : ∀ X, T₁.μ.app X ≫ app X = ((T₁ : C ⥤ C).map (app X) ≫ app _) ≫ T₂.μ.app X . obviously)\n\n/-- A morphism of comonads is a natural transformation compatible with ε and δ. -/\n@[ext]\nstructure comonad_hom (M N : comonad C) extends nat_trans (M : C ⥤ C) N :=\n(app_ε' : ∀ X, app X ≫ N.ε.app X = M.ε.app X . obviously)\n(app_δ' : ∀ X, app X ≫ N.δ.app X = M.δ.app X ≫ app _ ≫ (N : C ⥤ C).map (app X) . obviously)\n\nrestate_axiom monad_hom.app_η'\nrestate_axiom monad_hom.app_μ'\nattribute [simp, reassoc] monad_hom.app_η monad_hom.app_μ\n\nrestate_axiom comonad_hom.app_ε'\nrestate_axiom comonad_hom.app_δ'\nattribute [simp, reassoc] comonad_hom.app_ε comonad_hom.app_δ\n\ninstance : category (monad C) :=\n{ hom := monad_hom,\n  id := λ M, { to_nat_trans := 𝟙 (M : C ⥤ C) },\n  comp := λ _ _ _ f g,\n  { to_nat_trans := { app := λ X, f.app X ≫ g.app X } } }\n\ninstance : category (comonad C) :=\n{ hom := comonad_hom,\n  id := λ M, { to_nat_trans := 𝟙 (M : C ⥤ C) },\n  comp := λ M N L f g,\n  { to_nat_trans := { app := λ X, f.app X ≫ g.app X } } }\n\ninstance {T : monad C} : inhabited (monad_hom T T) := ⟨𝟙 T⟩\n\n@[simp] \n\ninstance {G : comonad C} : inhabited (comonad_hom G G) := ⟨𝟙 G⟩\n\n@[simp] lemma comonad_hom.id_to_nat_trans (T : comonad C) :\n  (𝟙 T : T ⟶ T).to_nat_trans = 𝟙 (T : C ⥤ C) :=\nrfl\n@[simp] lemma comp_to_nat_trans {T₁ T₂ T₃ : comonad C} (f : T₁ ⟶ T₂) (g : T₂ ⟶ T₃) :\n  (f ≫ g).to_nat_trans =\n    ((f.to_nat_trans : _ ⟶ (T₂ : C ⥤ C)) ≫ g.to_nat_trans : (T₁ : C ⥤ C) ⟶ T₃) :=\nrfl\n\n/-- Construct a monad isomorphism from a natural isomorphism of functors where the forward\ndirection is a monad morphism. -/\n@[simps]\ndef monad_iso.mk {M N : monad C} (f : (M : C ⥤ C) ≅ N) (f_η f_μ) :\n  M ≅ N :=\n{ hom := { to_nat_trans := f.hom, app_η' := f_η, app_μ' := f_μ },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_η' := λ X, by simp [←f_η],\n    app_μ' := λ X,\n    begin\n      rw ←nat_iso.cancel_nat_iso_hom_right f,\n      simp only [nat_trans.naturality, iso.inv_hom_id_app, assoc, comp_id, f_μ,\n        nat_trans.naturality_assoc, iso.inv_hom_id_app_assoc, ←functor.map_comp_assoc],\n      simp,\n    end } }\n\n/-- Construct a comonad isomorphism from a natural isomorphism of functors where the forward\ndirection is a comonad morphism. -/\n@[simps]\ndef comonad_iso.mk {M N : comonad C} (f : (M : C ⥤ C) ≅ N) (f_ε f_δ) :\n  M ≅ N :=\n{ hom := { to_nat_trans := f.hom, app_ε' := f_ε, app_δ' := f_δ },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_ε' := λ X, by simp [←f_ε],\n    app_δ' := λ X,\n    begin\n      rw ←nat_iso.cancel_nat_iso_hom_left f,\n      simp only [reassoc_of (f_δ X), iso.hom_inv_id_app_assoc, nat_trans.naturality_assoc],\n      rw [←functor.map_comp, iso.hom_inv_id_app, functor.map_id],\n      apply (comp_id _).symm\n    end } }\n\nvariable (C)\n\n/--\nThe forgetful functor from the category of monads to the category of endofunctors.\n-/\n@[simps]\ndef monad_to_functor : monad C ⥤ (C ⥤ C) :=\n{ obj := λ T, T,\n  map := λ M N f, f.to_nat_trans }\n\ninstance : faithful (monad_to_functor C) := {}.\n\n@[simp]\nlemma monad_to_functor_map_iso_monad_iso_mk {M N : monad C} (f : (M : C ⥤ C) ≅ N) (f_η f_μ) :\n  (monad_to_functor _).map_iso (monad_iso.mk f f_η f_μ) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (monad_to_functor C) :=\n{ reflects := λ M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (monad_iso.mk (as_iso ((monad_to_functor C).map f)) f.app_η f.app_μ),\n    ext; refl,\n  end }\n\n/--\nThe forgetful functor from the category of comonads to the category of endofunctors.\n-/\n@[simps]\ndef comonad_to_functor : comonad C ⥤ (C ⥤ C) :=\n{ obj := λ G, G,\n  map := λ M N f, f.to_nat_trans }\n\ninstance : faithful (comonad_to_functor C) := {}.\n\n@[simp]\nlemma comonad_to_functor_map_iso_comonad_iso_mk {M N : comonad C} (f : (M : C ⥤ C) ≅ N) (f_ε f_δ) :\n  (comonad_to_functor _).map_iso (comonad_iso.mk f f_ε f_δ) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (comonad_to_functor C) :=\n{ reflects := λ M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (comonad_iso.mk (as_iso ((comonad_to_functor C).map f)) f.app_ε f.app_δ),\n    ext; refl,\n  end }\n\nvariable {C}\n\n/--\nAn isomorphism of monads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef monad_iso.to_nat_iso {M N : monad C} (h : M ≅ N) : (M : C ⥤ C) ≅ N :=\n(monad_to_functor C).map_iso h\n\n/--\nAn isomorphism of comonads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef comonad_iso.to_nat_iso {M N : comonad C} (h : M ≅ N) : (M : C ⥤ C) ≅ N :=\n(comonad_to_functor C).map_iso h\n\nvariable (C)\n\nnamespace monad\n\n/-- The identity monad. -/\n@[simps]\ndef id : monad C :=\n{ to_functor := 𝟭 C,\n  η' := 𝟙 (𝟭 C),\n  μ' := 𝟙 (𝟭 C) }\n\ninstance : inhabited (monad C) := ⟨monad.id C⟩\n\nend monad\n\nnamespace comonad\n\n/-- The identity comonad. -/\n@[simps]\ndef id : comonad C :=\n{ to_functor := 𝟭 _,\n  ε' := 𝟙 (𝟭 C),\n  δ' := 𝟙 (𝟭 C) }\n\ninstance : inhabited (comonad C) := ⟨comonad.id C⟩\n\nend comonad\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/monad/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3488913398923956}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nInstances for identity and composition functors\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.functor\nimport Mathlib.algebra.group.basic\nimport Mathlib.PostPort\n\nuniverses u v u_1 w u_2 \n\nnamespace Mathlib\n\ntheorem applicative.map_seq_map {F : Type u → Type v} [Applicative F] [is_lawful_applicative F]\n    {α : Type u} {β : Type u} {γ : Type u} {σ : Type u} (f : α → β → γ) (g : σ → β) (x : F α)\n    (y : F σ) : f <$> x <*> g <$> y = (flip function.comp g ∘ f) <$> x <*> y :=\n  sorry\n\ntheorem applicative.pure_seq_eq_map' {F : Type u → Type v} [Applicative F] [is_lawful_applicative F]\n    {α : Type u} {β : Type u} (f : α → β) : Seq.seq (pure f) = Functor.map f :=\n  sorry\n\ntheorem applicative.ext {F : Type u → Type u_1} {A1 : Applicative F} {A2 : Applicative F}\n    [is_lawful_applicative F] [is_lawful_applicative F]\n    (H1 : ∀ {α : Type u} (x : α), pure x = pure x)\n    (H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α), f <*> x = f <*> x) : A1 = A2 :=\n  sorry\n\nprotected instance id.is_comm_applicative : is_comm_applicative id :=\n  is_comm_applicative.mk fun (α β : Type u_1) (a : id α) (b : id β) => Eq.refl (Prod.mk <$> a <*> b)\n\nnamespace functor\n\n\nnamespace comp\n\n\ntheorem map_pure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] {α : Type v} {β : Type v} (f : α → β)\n    (x : α) : f <$> pure x = pure (f x) :=\n  sorry\n\ntheorem seq_pure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] {α : Type v} {β : Type v}\n    (f : comp F G (α → β)) (x : α) : f <*> pure x = (fun (g : α → β) => g x) <$> f :=\n  sorry\n\ntheorem seq_assoc {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] {α : Type v} {β : Type v} {γ : Type v}\n    (x : comp F G α) (f : comp F G (α → β)) (g : comp F G (β → γ)) :\n    g <*> (f <*> x) = function.comp <$> g <*> f <*> x :=\n  sorry\n\ntheorem pure_seq_eq_map {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] {α : Type v} {β : Type v} (f : α → β)\n    (x : comp F G α) : pure f <*> x = f <$> x :=\n  sorry\n\nprotected instance is_lawful_applicative {F : Type u → Type w} {G : Type v → Type u} [Applicative F]\n    [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] :\n    is_lawful_applicative (comp F G) :=\n  is_lawful_applicative.mk pure_seq_eq_map map_pure seq_pure seq_assoc\n\ntheorem applicative_id_comp {F : Type u_1 → Type u_2} [AF : Applicative F]\n    [LF : is_lawful_applicative F] : comp.applicative = AF :=\n  applicative.ext (fun (α : Type u_1) (x : α) => rfl)\n    fun (α β : Type u_1) (f : F (α → β)) (x : F α) => rfl\n\ntheorem applicative_comp_id {F : Type u_1 → Type u_2} [AF : Applicative F]\n    [LF : is_lawful_applicative F] : comp.applicative = AF :=\n  sorry\n\nprotected instance is_comm_applicative {f : Type u → Type w} {g : Type v → Type u} [Applicative f]\n    [Applicative g] [is_comm_applicative f] [is_comm_applicative g] :\n    is_comm_applicative (comp f g) :=\n  sorry\n\nend comp\n\n\nend functor\n\n\ntheorem comp.seq_mk {α : Type w} {β : Type w} {f : Type u → Type v} {g : Type w → Type u}\n    [Applicative f] [Applicative g] (h : f (g (α → β))) (x : f (g α)) :\n    functor.comp.mk h <*> functor.comp.mk x = functor.comp.mk (Seq.seq <$> h <*> x) :=\n  rfl\n\nprotected instance functor.const.applicative {α : Type u_1} [HasOne α] [Mul α] :\n    Applicative (functor.const α) :=\n  sorry\n\nprotected instance functor.const.is_lawful_applicative {α : Type u_1} [monoid α] :\n    is_lawful_applicative (functor.const α) :=\n  sorry\n\nprotected instance functor.add_const.applicative {α : Type u_1} [HasZero α] [Add α] :\n    Applicative (functor.add_const α) :=\n  sorry\n\nprotected instance functor.add_const.is_lawful_applicative {α : Type u_1} [add_monoid α] :\n    is_lawful_applicative (functor.add_const α) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/applicative_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3487678260818055}}
{"text": "import .struc\nimport logic.equiv.defs\nimport .defs\n\nopen propagate_struc profinite\n\nset_option class.instance_max_depth 100\n\nsection\n\nopen circuit\n\ndef xor_map : (boolp.prod boolp).map boolp :=\n{ to_fun := λ x, bxor x.1 x.2,\n  preimage := λ C, C.bind (λ _, (var (sum.inl ())).xor (var (sum.inr ()))),\n  continuous' := begin\n     simp [circuit.to_set, eval_bind],\n     intros x C,\n     refl\n  end }\n\ndef and_map : (boolp.prod boolp).map boolp :=\n{ to_fun := λ x, band x.1 x.2,\n  preimage := λ C, C.bind (λ _, (var (sum.inl ())).and (var (sum.inr ()))),\n  continuous' := begin\n     simp [circuit.to_set, eval_bind],\n     intros x C,\n     refl\n  end }\n\ndef or_map : (boolp.prod boolp).map boolp :=\n{ to_fun := λ x, bor x.1 x.2,\n  preimage := λ C, C.bind (λ _, (var (sum.inl ())).or (var (sum.inr ()))),\n  continuous' := begin\n    simp [circuit.to_set, eval_bind],\n    intros x C,\n    refl\n  end }\n\ndef not_map : boolp.map boolp :=\n{ to_fun := bnot,\n  preimage := λ C, C.bind (λ _, (var ()).not),\n  continuous' := begin\n    simp [circuit.to_set, eval_bind],\n    intros x C,\n    refl\n  end }\n\ndef bitwise_map2 (op : circuit (fin 2)) :\n  (boolp.prod boolp).map boolp :=\n{ to_fun := λ x, op.eval (λ i, list.nth_le [x.1, x.2] i i.prop),\n  preimage := λ C, C.bind (λ _, op.map (λ i, list.nth_le [sum.inl (), sum.inr ()] i i.prop)),\n  continuous' := begin\n      simp [circuit.to_set, eval_bind, eval_map],\n      intros x C,\n      rw [iff_iff_eq],\n      congr' 2,\n      ext i,\n      cases i,\n      dsimp [boolp, profinite.prod, coe_sort, has_coe_to_sort.coe],\n      congr' 1,\n      ext i,\n      cases i with i hi,\n      cases i,\n      simp,\n      cases i,\n      simp,\n      simp [nat.succ_lt_succ_iff] at hi,\n      contradiction\n  end }\n\ndef bitwise_map3 (op : circuit (fin 3)) :\n  (boolp.prod (boolp.prod boolp)).map boolp :=\n{ to_fun := λ x, op.eval (λ i, list.nth_le [x.1, x.2.1, x.2.2] i i.prop),\n  preimage := λ C, C.bind (λ x, op.map (λ i, list.nth_le [sum.inl (), sum.inr (sum.inl ()), sum.inr (sum.inr ())] i i.prop)),\n  continuous' := begin\n      simp [circuit.to_set, eval_bind, eval_map],\n      intros x C,\n      rw [iff_iff_eq],\n      congr' 2,\n      ext i,\n      cases i,\n      dsimp [boolp, profinite.prod, coe_sort, has_coe_to_sort.coe],\n      congr' 1,\n      ext i,\n      cases i with i hi,\n      cases i,\n      simp,\n      cases i,\n      simp,\n      cases i,\n      simp,\n      simp [nat.succ_lt_succ_iff] at hi,\n      contradiction\n    end }\n\ndef const {X Y : profinite} (y : Y) : map X Y :=\n{ to_fun := λ _, y,\n  preimage := λ C, if y ∈ C.to_set then true else false,\n  continuous' := λ x C, begin\n    split_ifs;\n    simp [circuit.to_set, *] at *,\n  end }\n\ndef xor_struc : propagate_struc (boolp.prod boolp) unitp :=\n{ init := (),\n  transition := const (),\n  output := sndm.comp xor_map }\n\ndef and_struc : propagate_struc (boolp.prod boolp) unitp :=\n{ init := (),\n  transition := const (),\n  output := sndm.comp and_map }\n\ndef or_struc : propagate_struc (boolp.prod boolp) unitp :=\n{ init := (),\n  transition := const (),\n  output := sndm.comp or_map }\n\ndef not_struc : propagate_struc boolp unitp :=\n{ init := (),\n  transition := const (),\n  output := sndm.comp not_map }\n\ndef ls_struc (b : bool) : propagate_struc boolp boolp :=\n{ init := b,\n  transition := sndm,\n  output := fstm }\n\ndef add_struc : propagate_struc (boolp.prod boolp) boolp :=\n{ init := ff,\n  transition := bitwise_map3 (or (and (var 0) (var 1)) (or (and (var 0) (var 2)) (and (var 1) (var 2)))),\n  output := bitwise_map3 (xor (xor (var 0) (var 1)) (var 2)) }\n\ndef sub_struc : propagate_struc (boolp.prod boolp) boolp :=\n{ init := ff,\n  transition := bitwise_map3 (or (and (not (var 1)) (var 2)) (and (not (xor (var 1) (var 2))) (var 0))),\n  output := bitwise_map3 (xor (xor (var 0) (var 1)) (var 2)) }\n\ndef neg_struc : propagate_struc boolp boolp :=\n{ init := tt,\n  transition := bitwise_map2 (and (not (var 1)) (var 0)),\n  output := bitwise_map2 (xor (not (var 1)) (var 0)) }\n\ndef incr_struc : propagate_struc boolp boolp :=\n{ init := tt,\n  transition := and_map,\n  output := xor_map }\n\ndef decr_struc : propagate_struc boolp boolp :=\n{ init := tt,\n  transition := bitwise_map2 (and (not (var 1)) (var 0)),\n  output := xor_map }\n\ndef rearrange_prod₁ {W X Y Z : profinite} :\n  ((W.prod X).prod (Y.prod Z)).map ((W.prod Y).prod (X.prod Z)) :=\nreindex begin\n  dsimp [profinite.prod],\n  refine (equiv.sum_assoc _ _ _).symm.trans _,\n  refine equiv.trans _ (equiv.sum_assoc _ _ _),\n  refine equiv.sum_congr _ (equiv.refl _),\n  refine equiv.trans (equiv.sum_assoc _ _ _) _,\n  refine equiv.trans _ (equiv.sum_assoc _ _ _).symm,\n  refine equiv.sum_congr (equiv.refl _) _,\n  refine equiv.sum_comm _ _\nend\n\ndef prod_assoc {X Y Z : profinite} :\n  ((X.prod Y).prod Z).map (X.prod (Y.prod Z)) :=\nreindex (equiv.sum_assoc _ _ _).symm\n\ndef bin_comp {input state₁ state₂ state₃ : profinite} (p : propagate_struc (boolp.prod boolp) state₁)\n  (q : propagate_struc input state₂) (r : propagate_struc input state₃) :\n  propagate_struc input (state₁.prod (state₂.prod state₃)) :=\n{ init := (p.init, q.init, r.init),\n  transition := begin\n    have f₁ := prod_assoc.comp ((prod_mapm (map.id _) ((prod_mapm (map.id _) (diag)).comp\n      (rearrange_prod₁.comp (prod_mapm q.output r.output)))).comp p.transition),\n    have f₂ := (prod_mapm (sndm.comp fstm) (map.id _)).comp q.transition,\n    have f₃ := (prod_mapm (sndm.comp sndm) (map.id _)).comp r.transition,\n    exact prod_mk f₁ (prod_mk f₂ f₃)\n  end,\n  output := begin\n    refine map.comp _ p.output,\n    refine prod_assoc.comp _,\n    refine prod_mapm (map.id _) _,\n    have := rearrange_prod₁.comp (prod_mapm q.output r.output),\n    refine map.comp _ this,\n    refine prod_mapm (map.id _) diag,\n  end }\n\n@[simp] lemma bin_comp_init {input state₁ state₂ state₃ : profinite}\n  (p : propagate_struc (boolp.prod boolp) state₁) (q : propagate_struc input state₂)\n  (r : propagate_struc input state₃) :\n  (bin_comp p q r).init = (p.init, q.init, r.init) := rfl\n\n@[simp] lemma bin_comp_transition {input state₁ state₂ state₃ : profinite}\n  (p : propagate_struc (boolp.prod boolp) state₁) (q : propagate_struc input state₂)\n  (r : propagate_struc input state₃) :\n  coe_fn (bin_comp p q r).transition = (λ x : (state₁.prod (state₂.prod state₃)).prod input,\n    (p.transition (x.1.1, q.output (x.1.2.1, x.2), r.output (x.1.2.2, x.2)),\n      q.transition (x.1.2.1, x.2),\n      r.transition (x.1.2.2, x.2))) :=\nbegin\n  funext i,\n  rcases i with ⟨⟨i, j, k⟩, x⟩,\n  dsimp [nth_output, bin_comp, prod_assoc, map.comp, prod_mapm, boolp, function.comp,\n    map.id, coe_fn, has_coe_to_fun.coe, prod_mk, fstm, sndm, diag, rearrange_prod₁, reindex,\n    equiv.sum_assoc, equiv.trans, equiv.refl, equiv.sum_congr, equiv.symm, equiv.sum_comm,\n    profinite.prod, prod_mk_reindex],\n  simp [inv_proj]\nend\n\n@[simp] lemma bin_comp_output {input state₁ state₂ state₃ : profinite}\n  (p : propagate_struc (boolp.prod boolp) state₁) (q : propagate_struc input state₂)\n  (r : propagate_struc input state₃) :\n  coe_fn (bin_comp p q r).output = (λ x : (state₁.prod (state₂.prod state₃)).prod input,\n    p.output (x.1.1, q.output (x.1.2.1, x.2), r.output (x.1.2.2, x.2))) :=\nbegin\n  funext i,\n  rcases i with ⟨⟨i, j, k⟩, x⟩,\n  dsimp [nth_output, bin_comp, prod_assoc, map.comp, prod_mapm, boolp, function.comp,\n    map.id, coe_fn, has_coe_to_fun.coe, prod_mk, fstm, sndm, diag, rearrange_prod₁, reindex,\n    equiv.sum_assoc, equiv.trans, equiv.refl, equiv.sum_congr, equiv.symm, equiv.sum_comm,\n    profinite.prod, prod_mk_reindex],\n  simp [inv_proj]\nend\n\nlemma nth_state_bin_comp {input state₁ state₂ state₃ : profinite} (p : propagate_struc (boolp.prod boolp) state₁)\n  (q : propagate_struc input state₂) (r : propagate_struc input state₃) (x : ℕ → input) (n : ℕ) :\n  (bin_comp p q r).nth_state x n = (p.nth_state (λ i, (q.nth_output x i, r.nth_output x i)) n,\n    q.nth_state x n, r.nth_state x n) ∧\n  (bin_comp p q r).nth_output x n = p.nth_output (λ i, (q.nth_output x i, r.nth_output x i)) n :=\nbegin\n  induction n with n ih,\n  { simp [nth_state] },\n  { simp * }\nend\n\ndef una_comp {input state₁ state₂ : profinite} (p : propagate_struc boolp state₁)\n  (q : propagate_struc input state₂) : propagate_struc input (state₁.prod state₂) :=\n{ init := (p.init, q.init),\n  transition := begin\n    have := p.transition,\n    have := q.output,\n    have := q.transition,\n    refine prod_mk (prod_assoc.comp _) _,\n    { refine (prod_mapm (map.id _) q.output).comp p.transition },\n    { refine (prod_mapm sndm (map.id _)).comp q.transition }\n  end,\n  output := begin\n    have := p.output,\n    have := q.output,\n    have := prod_assoc.comp ((prod_mapm _ q.output).comp p.output),\n    exact this,\n    exact map.id _\n  end }\n\ndef propagate_struc.proj (n : ℕ) : propagate_struc twoadic unitp :=\n{ init := (),\n  transition := const (),\n  output := sndm.comp (projm _ n) }\n\ndef propagate_struc.zero : propagate_struc twoadic unitp :=\n{ init := (),\n  transition := const (),\n  output := const ff }\n\ndef propagate_struc.neg_one : propagate_struc twoadic unitp :=\n{ init := (),\n  transition := const (),\n  output := const tt }\n\ndef propagate_struc.one : propagate_struc twoadic boolp :=\n{ init := tt,\n  transition :=\n  { to_fun := λ _, ff,\n    preimage := λ C,\n    if h : ff ∈ C.to_set\n    then true\n    else false,\n    continuous' := begin\n      intros,split_ifs;\n      dsimp [circuit.to_set, set.mem_def, set_of] at *;\n      simp * at *\n    end },\n  output := fstm }\n\ndef of_term : term → Σ (state : profinite), propagate_struc twoadic state\n| (term.var n) := ⟨unitp, propagate_struc.proj n⟩\n| (term.zero) := ⟨unitp, propagate_struc.zero⟩\n| (term.one) := ⟨_, propagate_struc.one⟩\n| (term.and t₁ t₂) :=\n  let p₁ := of_term t₁,\n      p₂ := of_term t₂ in\n  ⟨_, bin_comp and_struc p₁.2 p₂.2⟩\n| (term.or t₁ t₂) :=\n  let p₁ := of_term t₁,\n      p₂ := of_term t₂ in\n  ⟨_, bin_comp or_struc p₁.2 p₂.2⟩\n| (term.neg t) :=\n  let p := of_term t in\n  ⟨_, una_comp neg_struc p.2⟩\n| (term.neg_one) := ⟨_, propagate_struc.neg_one⟩\n| (term.add t₁ t₂) :=\n  let p₁ := of_term t₁,\n      p₂ := of_term t₂ in\n  ⟨_, bin_comp add_struc p₁.2 p₂.2⟩\n| (term.xor t₁ t₂) :=\n  let p₁ := of_term t₁,\n      p₂ := of_term t₂ in\n  ⟨_, bin_comp xor_struc p₁.2 p₂.2⟩\n| (term.not t) :=\n  let p := of_term t in\n  ⟨_, una_comp not_struc p.2⟩\n| (term.ls t) :=\n  let p := of_term t in\n  ⟨_, una_comp (ls_struc ff) p.2⟩\n| (term.sub t₁ t₂) :=\n  let p₁ := of_term t₁,\n      p₂ := of_term t₂ in\n  ⟨_, bin_comp sub_struc p₁.2 p₂.2⟩\n| (term.incr t) :=\n  let p := of_term t in\n  ⟨_, una_comp incr_struc p.2⟩\n| (term.decr t) :=\n  let p := of_term t in\n  ⟨_, una_comp decr_struc p.2⟩\n\ninstance : Π (t : term), has_repr (of_term t).1.ι\n| (term.var n) := by dsimp [of_term]; apply_instance\n| (term.zero) := by dsimp [of_term]; apply_instance\n| (term.one) := by dsimp [of_term]; apply_instance\n| (term.add t₁ t₂) := by letI := ι.has_repr t₁; letI := ι.has_repr t₂;\n  dsimp [of_term]; apply_instance\n| (term.and t₁ t₂) := by letI := ι.has_repr t₁; letI := ι.has_repr t₂;\n  dsimp [of_term]; apply_instance\n| (term.or t₁ t₂) := by letI := ι.has_repr t₁; letI := ι.has_repr t₂;\n  dsimp [of_term]; apply_instance\n| (term.neg t) := by letI := ι.has_repr t; dsimp [of_term]; apply_instance\n| (term.neg_one) := by dsimp [of_term]; apply_instance\n| (term.xor t₁ t₂) := by letI := ι.has_repr t₁; letI := ι.has_repr t₂;\n  dsimp [of_term]; apply_instance\n| (term.not t) := by letI := ι.has_repr t; dsimp [of_term]; apply_instance\n| (term.ls t) := by letI := ι.has_repr t; dsimp [of_term]; apply_instance\n| (term.sub t₁ t₂) := by letI := ι.has_repr t₁; letI := ι.has_repr t₂;\n  dsimp [of_term]; apply_instance\n| (term.incr t) := by letI := ι.has_repr t; dsimp [of_term]; apply_instance\n| (term.decr t) := by letI := ι.has_repr t; dsimp [of_term]; apply_instance\n\ndef check_eq (t₁ t₂ : term) (n : ℕ) : result :=\ndecide_if_zeros (of_term (t₁.xor t₂)).2 n\n\nend\n\nopen term\n\nset_option profiler true\n\ndef x : term := term.var 0\ndef y : term := term.var 1\ndef z : term := term.var 2\ndef a : term := term.var 3\ndef b : term := term.var 4\ndef c : term := term.var 5\ndef d : term := term.var 6\n\n#eval check_eq (x +- x) 0 2\n#eval check_eq (x - y) (x + -y) 2\n#eval check_eq (x + 1) x.incr 2\n#eval check_eq (x - 1) x.decr 2\n\n#eval check_eq (x.xor x) term.zero 1\n#eval check_eq (x + y) (y + x) 1\n#eval check_eq ((x + y) + z) (x + (y + z)) 2\n#eval check_eq (not (xor x y)) (and x y - or x y - 1) 2\n\n-- #eval (bitwise_struc bxor).nth_output (λ _, (tt, tt)) 0\n\nopen term\n", "meta": {"author": "ChrisHughes24", "repo": "lean3bits", "sha": "119b68f1ce4a967951c53ee2f174007b49c80831", "save_path": "github-repos/lean/ChrisHughes24-lean3bits", "path": "github-repos/lean/ChrisHughes24-lean3bits/lean3bits-119b68f1ce4a967951c53ee2f174007b49c80831/src/v3/strucs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3487603642072086}}
{"text": "/-\n  Specifications file for ec_spec.cairo\n\n  Do not modify the constant definitions, structure definitions, or automatic specifications.\n  Do not change the name or arguments of the user specifications and soundness theorems.\n\n  You may freely move the definitions around in the file.\n  You may add definitions and theorems wherever you wish in this file.\n-/\nimport starkware.cairo.lean.semantics.soundness.prelude\nimport starkware.cairo.common.cairo_secp.bigint_spec\nimport starkware.cairo.common.cairo_secp.field_spec\nimport starkware.cairo.common.cairo_secp.constants_spec\n\n-- JDA: Additional import.\nimport starkware.cairo.common.cairo_secp.elliptic_curves\n\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.field\nopen starkware.cairo.common.cairo_secp.constants\n\nnamespace starkware.cairo.common.cairo_secp.ec\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\n\n-- End of automatically generated prelude.\n\n-- Main scope definitions.\n\n@[ext] structure EcPoint (F : Type) :=\n  ( x : BigInt3 F ) ( y : BigInt3 F )\n@[ext] structure π_EcPoint (F : Type) :=\n  ( σ_ptr : F ) ( x : BigInt3 F ) ( y : BigInt3 F )\n@[reducible] def φ_EcPoint.x := 0\n@[reducible] def φ_EcPoint.y := 3\n@[reducible] def φ_EcPoint.SIZE := 6\n@[reducible] def cast_EcPoint (mem : F → F) (p : F) : EcPoint F := {\n  x := cast_BigInt3 mem (p + φ_EcPoint.x),\n  y := cast_BigInt3 mem (p + φ_EcPoint.y)\n}\n@[reducible] def cast_π_EcPoint (mem : F → F) (p : F) : π_EcPoint F := {\n  σ_ptr := mem p,\n  x := cast_BigInt3 mem ((mem p) + φ_EcPoint.x),\n  y := cast_BigInt3 mem ((mem p) + φ_EcPoint.y)\n}\ninstance π_EcPoint_to_F : has_coe (π_EcPoint F) F := ⟨λ s, s.σ_ptr⟩\n\n-- End of main scope definitions.\n\n/-\n-- Data for writing the specifications.\n-/\n\nstructure BddECPointData (secpF : Type*) [field secpF] (pt : EcPoint F) :=\n(ix iy : bigint3)\n(ixbdd : ix.bounded (3 * BASE - 1))\n(iybdd : iy.bounded (3 * BASE - 1))\n(ptxeq : pt.x = ix.toBigInt3)\n(ptyeq : pt.y = iy.toBigInt3)\n(onEC : pt.x = ⟨0, 0, 0⟩ ∨ (iy.val : secpF)^2 = (ix.val : secpF)^3 + 7)\n\ntheorem BddECPointData.onEC' {secpF : Type*} [field secpF] {pt : EcPoint F}\n    (h : BddECPointData secpF pt) (h' : pt.x ≠ ⟨0, 0, 0⟩) :\n  (h.iy.val : secpF)^2 = (h.ix.val : secpF)^3 + 7 :=\nor.resolve_left h.onEC h'\n\nsection secpF\n\nvariables\n  {secpF : Type}   -- in practice, zmod SECP_PRIME\n  [field secpF]    -- in practice, @zmod.field _ ⟨prime_secp⟩\n  [char_p secpF SECP_PRIME]\n  [decidable_eq secpF]\n\ndef BddECPointData.toECPoint {pt : EcPoint F} (h : BddECPointData secpF pt) :\n    ECPoint secpF :=\nif h' : pt.x = ⟨0, 0, 0⟩ then\n  ECPoint.ZeroPoint\nelse\n  ECPoint.AffinePoint ⟨h.ix.val, h.iy.val, h.onEC' h'⟩\n\ndef BddECPointData.zero : BddECPointData secpF (⟨⟨0, 0, 0⟩, ⟨0, 0, 0⟩⟩ : EcPoint F)  :=\n{ ix := ⟨0, 0, 0⟩,\n  iy := ⟨0, 0, 0⟩,\n  ixbdd := by simp [bigint3.bounded]; norm_num1,\n  iybdd := by simp [bigint3.bounded]; norm_num1,\n  ptxeq := by simp [bigint3.toBigInt3],\n  ptyeq := by simp [bigint3.toBigInt3],\n  onEC := or.inl rfl }\n\nend secpF\n\ndef SECP_LOG2_BOUND := 100\n\nclass secp_field (secpF : Type*) extends ec_field secpF, char_p secpF SECP_PRIME :=\n(seven_not_square : ∀ y : secpF, y^2 ≠ 7)\n(neg_seven_not_cube : ∀ x : secpF, x^3 ≠ -7)\n(order_large : ∀ {pt : ECPoint secpF}, pt ≠ 0 →\n                 ∀ {n : ℕ}, n < 2^SECP_LOG2_BOUND → ¬ (n • pt = 0 ∨ n • pt = pt ∨ n • pt = -pt))\n\ntheorem secp_field.y_ne_zero_of_on_ec  {secpF : Type*} [secp_field secpF] {x y : secpF}\n  (h : on_ec (x, y)) : y ≠ 0 :=\nby { contrapose! h, simp [on_ec, h, ←sub_eq_iff_eq_add],\n     apply ne.symm, apply secp_field.neg_seven_not_cube }\n\ntheorem secp_field.x_ne_zero_of_on_ec {secpF : Type*} [secp_field secpF] {x y : secpF}\n  (h : on_ec (x, y)) : x ≠ 0 :=\nby { contrapose! h, simp [on_ec, h, secp_field.seven_not_square] }\n\ntheorem secp_field.eq_zero_of_double_eq_zero {secpF : Type*} [secp_field secpF]\n  {x : ECPoint secpF} (h : 2 • x = 0) : x = 0 :=\nbegin\n  cases x, { refl },\n  simp [two_smul, ECPoint.add_def, ECPoint.add] at h,\n  have : x.y ≠ -x.y,\n  { intro h',\n    have h_on_ec := x.h, dsimp [on_ec] at h_on_ec,\n    have : 2 * x.y = 0,\n    { rwa [two_mul, ←eq_neg_iff_add_eq_zero] },\n    rw mul_eq_zero at this,\n    simp [or.resolve_left this ec_field.two_ne_zero] at h_on_ec,\n    apply secp_field.neg_seven_not_cube x.x,\n    exact eq_neg_iff_add_eq_zero.mpr h_on_ec.symm },\n  rw dif_neg this at h,\n  contradiction\nend\n\nnamespace BddECPointData\n\ntheorem toECPoint_zero (secpF : Type) [secp_field secpF] :\n  (BddECPointData.zero : BddECPointData secpF (⟨⟨0, 0, 0⟩, ⟨0, 0, 0⟩⟩ : EcPoint F)).toECPoint =\n    0 :=\nby { simp [toECPoint], refl }\n\ntheorem pt_zero_iff' {secpF : Type} [secp_field secpF]\n    {pt : EcPoint F} (h : BddECPointData secpF pt) :\n  pt.x = ⟨0, 0, 0⟩ ↔ h.ix.val = 0 :=\nbegin\n  split,\n  { rw [h.ptxeq],\n    intro heq,\n    rw toBigInt3_eq_zero_of_bounded_3BASE heq h.ixbdd,\n    simp [bigint3.val] },\n  intro heq,\n  cases h.onEC with h1 h1, { exact h1 },\n  exfalso,\n  have : on_ec ((h.ix.val : secpF), (h.iy.val : secpF)):= h1,\n  apply secp_field.x_ne_zero_of_on_ec this,\n  simp [heq]\nend\n\ntheorem pt_zero_iff {secpF : Type} [secp_field secpF]\n    {pt : EcPoint F} (h : BddECPointData secpF pt) :\n  pt.x = ⟨0, 0, 0⟩ ↔ (h.ix.val : secpF) = 0 :=\nbegin\n  split,\n  { rw [h.ptxeq],\n    intro heq,\n    rw toBigInt3_eq_zero_of_bounded_3BASE heq h.ixbdd,\n    simp [bigint3.val] },\n  intro heq,\n  cases h.onEC with h1 h1, { exact h1 },\n  exfalso,\n  have : on_ec ((h.ix.val : secpF), (h.iy.val : secpF)):= h1,\n  apply secp_field.x_ne_zero_of_on_ec this,\n  simp [heq]\nend\n\ntheorem toECPoint_eq_zero_iff {secpF : Type} [secp_field secpF] {pt : EcPoint F} (h : BddECPointData secpF pt) :\n  h.toECPoint = 0 ↔ pt.x = ⟨0, 0, 0⟩ :=\nby { by_cases h : pt.x = ⟨0, 0, 0⟩; simp [BddECPointData.toECPoint, h, ECPoint.zero_def] }\n\ntheorem toECPoint_eq_of_eq_of_ne {secpF : Type} [secp_field secpF] {pt0 pt1 : EcPoint F}\n    {h0 : BddECPointData secpF pt0}\n    {h1 : BddECPointData secpF pt1}\n    (hxeq : (h0.ix.val : secpF) = (h1.ix.val : secpF))\n    (hyne : (h0.iy.val : secpF) ≠ -(h1.iy.val : secpF)) :\n  h0.toECPoint = h1.toECPoint :=\nbegin\n  have aux: pt0.x = ⟨0, 0, 0⟩ ↔ pt1.x = ⟨0, 0, 0⟩,\n  { rw [h1.pt_zero_iff, ←hxeq, ←h0.pt_zero_iff] },\n  by_cases h: pt0.x = ⟨0, 0, 0⟩,\n  { have : pt1.x = ⟨0, 0, 0⟩, by rwa ←aux,\n    rw [h0.toECPoint_eq_zero_iff.mpr h, h1.toECPoint_eq_zero_iff.mpr this] },\n  have : ¬ (pt1.x = ⟨0, 0, 0⟩), by rwa ←aux,\n  rw [BddECPointData.toECPoint, BddECPointData.toECPoint, dif_neg h, dif_neg this],\n  congr' 1, ext, { exact hxeq }, dsimp,\n  have : (h0.iy.val : secpF)^2 = (h1.iy.val : secpF)^2,\n  { rw [h0.onEC' h, h1.onEC' this, hxeq] },\n  have := eq_or_eq_neg_of_sq_eq_sq _ _ this,\n  exact or.resolve_right this hyne\nend\n\nend BddECPointData\n\ntheorem double_Affine {secpF : Type} [secp_field secpF] {x1 y1 x2 y2 : secpF}\n    (h1 : on_ec (x1, y1)) (h2 : on_ec (x2, y2))\n    (heq : ec_double (x1, y1) = (x2, y2)) :\n  2 • ECPoint.AffinePoint ⟨x1, y1, h1⟩ = ECPoint.AffinePoint ⟨x2, y2, h2⟩ :=\nbegin\n  have : y1 ≠ -y1,\n  { intro heq,\n    apply secp_field.y_ne_zero_of_on_ec h1,\n    have : 2 * y1 = 0,\n    { rwa [two_mul, add_eq_zero_iff_eq_neg] },\n    rw mul_eq_zero at this,\n    exact this.resolve_left ec_field.two_ne_zero },\n  rw two_smul,\n  simp [ECPoint.add_def, ECPoint.add], dsimp,\n  rw [dif_neg this], congr; simp [heq]\nend\n\ntheorem Affine_add_Affine {secpF : Type} [secp_field secpF] {x1 y1 x2 y2 x3 y3 : secpF}\n    (h1 : on_ec (x1, y1)) (h2 : on_ec (x2, y2)) (h3 : on_ec (x3, y3)) (hne : x1 ≠ x2)\n    (heq : ec_add (x1, y1) (x2, y2) = (x3, y3)) :\n  ECPoint.AffinePoint ⟨x1, y1, h1⟩ + ECPoint.AffinePoint ⟨x2, y2, h2⟩ =\n    ECPoint.AffinePoint ⟨x3, y3, h3⟩ :=\nbegin\n  simp [ECPoint.add_def, ECPoint.add], dsimp,\n  rw [dif_neg hne], congr; simp [heq]\nend\n\n\n/-\n-- Function: ec_negate\n-/\n\n/- ec_negate autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_ec_negate (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_point : EcPoint F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (minus_y : BigInt3 F), spec_nondet_bigint3 mem κ₁ range_check_ptr range_check_ptr₁ minus_y ∧\n  ∃ (κ₂ : ℕ) (range_check_ptr₂ : F), spec_verify_zero mem κ₂ range_check_ptr₁ {\n    d0 := minus_y.d0 + point.y.d0,\n    d1 := minus_y.d1 + point.y.d1,\n    d2 := minus_y.d2 + point.y.d2\n  } range_check_ptr₂ ∧\n  κ₁ + κ₂ + 14 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₂ ∧\n  ρ_point = {\n    x := point.x,\n    y := minus_y\n  }\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_ec_negate (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_point : EcPoint F) : Prop :=\n  ∀ ix iy : bigint3,\n    ix.bounded (3 * BASE - 1) →\n    iy.bounded (3 * BASE - 1) →\n    point.x = ix.toBigInt3 →\n    point.y = iy.toBigInt3 →\n    ∃ ineg_y : bigint3,\n      ineg_y.bounded (3 * (BASE - 1)) ∧\n      ρ_point = { x := ix.toBigInt3, y := ineg_y.toBigInt3 } ∧\n      ineg_y.val ≡ - iy.val [ZMOD SECP_PRIME]\n\n/- ec_negate soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_ec_negate\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_point : EcPoint F)\n    (h_auto : auto_spec_ec_negate mem κ range_check_ptr point ρ_range_check_ptr ρ_point) :\n  spec_ec_negate mem κ range_check_ptr point ρ_range_check_ptr ρ_point :=\nbegin\n  intros ix iy ixbdd iybdd ptxeq ptyeq,\n  rcases h_auto with ⟨_, _, neg_y, hneg_y, _, _, hverify_zero, _, _, ρ_point_eq⟩,\n  rcases nondet_bigint3_corr hneg_y with ⟨ineg_y, neg_y_eq, ineg_y_bdd⟩,\n  use [ineg_y, ineg_y_bdd],\n  split, { rw [ρ_point_eq, ptxeq, neg_y_eq] },\n  rw [int.modeq_iff_sub_mod_eq_zero, sub_neg_eq_add, ←bigint3.add_val],\n  apply hverify_zero,\n  { simp [ptyeq, neg_y_eq, bigint3.add, bigint3.toUnreducedBigInt3, bigint3.toBigInt3] },\n  apply bigint3.bounded_of_bounded_of_le,\n  apply bigint3.bounded_add ineg_y_bdd iybdd,\n  dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1\nend\n\n/- Better specification. -/\n\ndef spec_ec_negate' ( pt : EcPoint F ) ( ret : EcPoint F )\n    (secpF : Type) [secp_field secpF] : Prop :=\n  ∀ h : BddECPointData secpF pt,\n    ∃ hret : BddECPointData secpF ret,\n      hret.toECPoint = -h.toECPoint\n\ntheorem spec_ec_negate'_of_spec_ec_negate\n    {mem : F → F} {κ : ℕ} {range_check_ptr : F} {pt : EcPoint F} {ret0 : F} {ret : EcPoint F}\n    (h : spec_ec_negate mem κ range_check_ptr pt ret0 ret)\n    (secpF : Type) [secp_field secpF] :\n  spec_ec_negate' pt ret secpF :=\nbegin\n  intro hpt,\n  rcases h hpt.ix hpt.iy hpt.ixbdd hpt.iybdd hpt.ptxeq hpt.ptyeq with ⟨ineg_y, ineg_y_bdd, reteq, hineg_y⟩,\n  have inegyvaleq : (ineg_y.val : secpF) = - (hpt.iy.val : secpF),\n  { rw [←int.cast_neg, char_p.int_coe_eq_int_coe_iff secpF SECP_PRIME],\n    exact hineg_y },\n  rw EcPoint.ext_iff at reteq,\n  have retx_eq_ptx : ret.x = pt.x := reteq.1.trans (hpt.ptxeq.symm),\n  let hret : BddECPointData secpF ret :=\n  { ix := hpt.ix,\n    iy := ineg_y,\n    ixbdd := hpt.ixbdd,\n    iybdd := bigint3.bounded_of_bounded_of_le ineg_y_bdd bound_slack,\n    ptxeq := reteq.1,\n    ptyeq := reteq.2,\n    onEC :=\n      begin\n        cases hpt.onEC with h' h',\n        { exact or.inl (retx_eq_ptx.trans h') },\n        right, rw [inegyvaleq, neg_sq, h']\n      end },\n  use hret,\n  simp [BddECPointData.toECPoint],\n  by_cases h' : pt.x = ⟨0, 0, 0⟩,\n  { rw [dif_pos h', dif_pos (retx_eq_ptx.trans h')],\n    refl },\n  rw [dif_neg h', dif_neg (ne_of_eq_of_ne retx_eq_ptx h')],\n  simp [ECPoint.neg_def, ECPoint.neg, inegyvaleq]\nend\n\n/-\n-- Function: compute_doubling_slope\n-/\n\n/- compute_doubling_slope autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_compute_doubling_slope (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (slope : BigInt3 F), spec_nondet_bigint3 mem κ₁ range_check_ptr range_check_ptr₁ slope ∧\n  ∃ (κ₂ : ℕ) (x_sqr : UnreducedBigInt3 F), spec_unreduced_sqr mem κ₂ point.x x_sqr ∧\n  ∃ (κ₃ : ℕ) (slope_y : UnreducedBigInt3 F), spec_unreduced_mul mem κ₃ slope point.y slope_y ∧\n  ∃ (κ₄ : ℕ) (range_check_ptr₂ : F), spec_verify_zero mem κ₄ range_check_ptr₁ {\n    d0 := 3 * x_sqr.d0 - 2 * slope_y.d0,\n    d1 := 3 * x_sqr.d1 - 2 * slope_y.d1,\n    d2 := 3 * x_sqr.d2 - 2 * slope_y.d2\n  } range_check_ptr₂ ∧\n  κ₁ + κ₂ + κ₃ + κ₄ + 34 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₂ ∧\n  ρ_slope = slope\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_compute_doubling_slope (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F) : Prop :=\n  ∀ ix iy : bigint3,\n    ix.bounded (3 * BASE - 1) →\n    iy.bounded (3 * BASE - 1) →\n    point.x = ix.toBigInt3 →\n    point.y = iy.toBigInt3 →\n    ∃ is : bigint3,\n      is.bounded (3 * (BASE - 1)) ∧\n      ρ_slope = is.toBigInt3 ∧\n      3 * ix.val^2 ≡ 2 * iy.val * is.val [ZMOD SECP_PRIME]\n\n/- compute_doubling_slope soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_compute_doubling_slope\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F)\n    (h_auto : auto_spec_compute_doubling_slope mem κ range_check_ptr point ρ_range_check_ptr ρ_slope) :\n  spec_compute_doubling_slope mem κ range_check_ptr point ρ_range_check_ptr ρ_slope :=\nbegin\n  rcases h_auto with ⟨_, _, slope, valid_slope, _, x_sqr, hx, _, slope_y, hslope_y, _, _, vz, _, _, ret1eq⟩,\n  rcases nondet_bigint3_corr valid_slope with ⟨is, slopeeq, isbdd⟩,\n  intros ix iy ixbdd iybdd ptxeq ptyeq,\n  have x_sqr_eq := hx _ ptxeq,\n  have slope_y_eq := hslope_y _ _ slopeeq ptyeq,\n  refine ⟨_, isbdd, ret1eq.trans slopeeq, _⟩,\n  set diff : bigint3 := (ix.sqr.cmul 3).sub ((iy.mul is).cmul 2) with diff_eq,\n  have diff_bdd : diff.bounded (5 * ((3 * BASE - 1)^2 * (8 * SECP_REM + 1))),\n  { rw [diff_eq, (show (5 : ℤ) = 3 + 2, by norm_num), add_mul],\n    apply bigint3.bounded_sub,\n    apply bigint3.bounded_cmul' (show (0 : ℤ) ≤ 3, by norm_num1),\n    apply bigint3.bounded_sqr ixbdd,\n    apply bigint3.bounded_cmul' (show (0 : ℤ) ≤ 2, by norm_num1),\n    apply bigint3.bounded_mul iybdd,\n    apply bigint3.bounded_of_bounded_of_le isbdd bound_slack },\n  have : diff.val % SECP_PRIME = 0,\n  { apply vz,\n    { simp only [diff_eq, x_sqr_eq, slope_y_eq],\n      dsimp [bigint3.toUnreducedBigInt3, bigint3.sqr, bigint3.mul, bigint3.cmul, bigint3.sub],\n      simp_int_casts,\n      split, ring,\n      split, ring, ring },\n      apply bigint3.bounded_of_bounded_of_le diff_bdd,\n      dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  symmetry,\n  rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←this, diff_eq, bigint3.sub_val,\n    bigint3.cmul_val, bigint3.cmul_val],\n  apply int.modeq.sub,\n  apply int.modeq.mul_left,\n  apply int.modeq.symm,\n  apply bigint3.sqr_val,\n  rw mul_assoc,\n  apply int.modeq.mul_left,\n  apply int.modeq.symm,\n  apply bigint3.mul_val\nend\n\n/-\n-- Function: compute_slope\n-/\n\n/- compute_slope autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_compute_slope (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (slope : BigInt3 F), spec_nondet_bigint3 mem κ₁ range_check_ptr range_check_ptr₁ slope ∧\n  ∃ x_diff : BigInt3 F, x_diff = {\n    d0 := point0.x.d0 - point1.x.d0,\n    d1 := point0.x.d1 - point1.x.d1,\n    d2 := point0.x.d2 - point1.x.d2\n  } ∧\n  ∃ (κ₂ : ℕ) (x_diff_slope : UnreducedBigInt3 F), spec_unreduced_mul mem κ₂ x_diff slope x_diff_slope ∧\n  ∃ (κ₃ : ℕ) (range_check_ptr₂ : F), spec_verify_zero mem κ₃ range_check_ptr₁ {\n    d0 := x_diff_slope.d0 - point0.y.d0 + point1.y.d0,\n    d1 := x_diff_slope.d1 - point0.y.d1 + point1.y.d1,\n    d2 := x_diff_slope.d2 - point0.y.d2 + point1.y.d2\n  } range_check_ptr₂ ∧\n  κ₁ + κ₂ + κ₃ + 21 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₂ ∧\n  ρ_slope = slope\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_compute_slope (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F) : Prop :=\n  ∀ ix0 iy0 ix1 iy1 : bigint3,\n    ix0.bounded (3 * BASE - 1) →\n    iy0.bounded (3 * BASE - 1) →\n    ix1.bounded (3 * BASE - 1) →\n    iy1.bounded (3 * BASE - 1) →\n    point0.x = ix0.toBigInt3 →\n    point0.y = iy0.toBigInt3 →\n    point1.x = ix1.toBigInt3 →\n    point1.y = iy1.toBigInt3 →\n    ∃ is : bigint3,\n      is.bounded (3 * (BASE - 1)) ∧\n      ρ_slope = is.toBigInt3 ∧\n      (ix0.val - ix1.val) * is.val ≡ (iy0.val - iy1.val) [ZMOD SECP_PRIME]\n\n\n/- compute_slope soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_compute_slope\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_slope : BigInt3 F)\n    (h_auto : auto_spec_compute_slope mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_slope) :\n  spec_compute_slope mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_slope :=\nbegin\n  rcases h_auto with ⟨_, _, slope, valid_slope, x_diff, x_diff_eq,\n    _, x_diff_slope, h_x_diff_slope, _, _, vz, _, _, ret1eq⟩,\n  rcases nondet_bigint3_corr valid_slope with ⟨is, slope_eq, isbdd⟩,\n  intros ix0 iy0 ix1 iy1 ix0bdd iy0bdd ix1bdd iy1bdd pt0xeq pt0yeq pt1xeq pt1yeq,\n  set ix_diff := ix0.sub ix1 with ix_diff_eq,\n  have x_diff_eq' : x_diff = ix_diff.toBigInt3,\n  { rw [ix_diff_eq, x_diff_eq, bigint3.toBigInt3_sub, BigInt3.sub, pt0xeq, pt1xeq] },\n  have x_diff_slope_eq := h_x_diff_slope _ _ x_diff_eq' slope_eq,\n  refine ⟨_, isbdd, ret1eq.trans slope_eq, _⟩,\n  set diff : bigint3 := (ix_diff.mul is).sub (iy0.sub iy1) with diff_eq,\n  have diff_bdd : diff.bounded\n    (((3 * BASE - 1) + (3 * BASE - 1))^2 * (8 * SECP_REM + 1) +  ((3 * BASE - 1) + (3 * BASE - 1))),\n  { rw [diff_eq],\n    apply bigint3.bounded_sub,\n    apply bigint3.bounded_mul,\n    apply bigint3.bounded_sub ix0bdd ix1bdd,\n    apply bigint3.bounded_of_bounded_of_le isbdd,\n    unfold BASE, simp_int_casts, norm_num1,\n    apply bigint3.bounded_sub iy0bdd iy1bdd },\n  have : diff.val % SECP_PRIME = 0,\n  { apply vz,\n    { simp only [diff_eq, x_diff_slope_eq, ix_diff_eq, pt0xeq, pt1xeq, pt0yeq, pt1yeq],\n      dsimp [bigint3.toUnreducedBigInt3, bigint3.toBigInt3, bigint3.mul, bigint3.sub], simp [←sub_add] },\n    apply bigint3.bounded_of_bounded_of_le diff_bdd,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  symmetry,\n  rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←this, diff_eq, bigint3.sub_val,\n       bigint3.sub_val],\n  apply int.modeq.sub,\n  apply int.modeq.symm,\n  apply int.modeq.trans,\n  apply bigint3.mul_val,\n  apply int.modeq.mul_right,\n  rw [ix_diff_eq, bigint3.sub_val],\n  apply int.modeq.refl\nend\n\n/-\n-- Function: ec_double\n-/\n\n/- ec_double autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_ec_double_block5 (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (slope : BigInt3 F), spec_compute_doubling_slope mem κ₁ range_check_ptr point range_check_ptr₁ slope ∧\n  ∃ (κ₂ : ℕ) (slope_sqr : UnreducedBigInt3 F), spec_unreduced_sqr mem κ₂ slope slope_sqr ∧\n  ∃ (κ₃ : ℕ) (range_check_ptr₂ : F) (new_x : BigInt3 F), spec_nondet_bigint3 mem κ₃ range_check_ptr₁ range_check_ptr₂ new_x ∧\n  ∃ (κ₄ : ℕ) (range_check_ptr₃ : F) (new_y : BigInt3 F), spec_nondet_bigint3 mem κ₄ range_check_ptr₂ range_check_ptr₃ new_y ∧\n  ∃ (κ₅ : ℕ) (range_check_ptr₄ : F), spec_verify_zero mem κ₅ range_check_ptr₃ {\n    d0 := slope_sqr.d0 - new_x.d0 - 2 * point.x.d0,\n    d1 := slope_sqr.d1 - new_x.d1 - 2 * point.x.d1,\n    d2 := slope_sqr.d2 - new_x.d2 - 2 * point.x.d2\n  } range_check_ptr₄ ∧\n  ∃ (κ₆ : ℕ) (x_diff_slope : UnreducedBigInt3 F), spec_unreduced_mul mem κ₆ {\n    d0 := point.x.d0 - new_x.d0,\n    d1 := point.x.d1 - new_x.d1,\n    d2 := point.x.d2 - new_x.d2\n  } slope x_diff_slope ∧\n  ∃ (κ₇ : ℕ) (range_check_ptr₅ : F), spec_verify_zero mem κ₇ range_check_ptr₄ {\n    d0 := x_diff_slope.d0 - point.y.d0 - new_y.d0,\n    d1 := x_diff_slope.d1 - point.y.d1 - new_y.d1,\n    d2 := x_diff_slope.d2 - point.y.d2 - new_y.d2\n  } range_check_ptr₅ ∧\n  κ₁ + κ₂ + κ₃ + κ₄ + κ₅ + κ₆ + κ₇ + 49 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₅ ∧\n  ρ_res = {\n    x := new_x,\n    y := new_y\n  }\n\ndef auto_spec_ec_double (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ((point.x.d0 = 0 ∧\n    ((point.x.d1 = 0 ∧\n      ((point.x.d2 = 0 ∧\n        11 ≤ κ ∧\n        ρ_range_check_ptr = range_check_ptr ∧\n        ρ_res = point) ∨\n       (point.x.d2 ≠ 0 ∧\n        ∃ (κ₁ : ℕ), auto_spec_ec_double_block5 mem κ₁ range_check_ptr point ρ_range_check_ptr ρ_res ∧\n        κ₁ + 3 ≤ κ))) ∨\n     (point.x.d1 ≠ 0 ∧\n      ∃ (κ₁ : ℕ), auto_spec_ec_double_block5 mem κ₁ range_check_ptr point ρ_range_check_ptr ρ_res ∧\n      κ₁ + 2 ≤ κ))) ∨\n   (point.x.d0 ≠ 0 ∧\n    ∃ (κ₁ : ℕ), auto_spec_ec_double_block5 mem κ₁ range_check_ptr point ρ_range_check_ptr ρ_res ∧\n    κ₁ + 1 ≤ κ))\n\n-- Added manually\ntheorem auto_spec_ec_double_better {mem : F → F} {κ : ℕ}{range_check_ptr : F} {point : EcPoint F} {ρ_range_check_ptr : F} {ρ_res : EcPoint F} (h : auto_spec_ec_double mem κ range_check_ptr point ρ_range_check_ptr ρ_res ) :\n  (point.x.d0 = 0 ∧ point.x.d1 = 0 ∧ point.x.d2 = 0 ∧\n      ρ_range_check_ptr = range_check_ptr ∧ ρ_res = point) ∨\n  ((point.x.d2 ≠ 0 ∨ point.x.d1 ≠ 0 ∨ point.x.d0 ≠ 0) ∧\n    ∃ (κ₁ : ℕ), auto_spec_ec_double_block5 mem κ₁ range_check_ptr point ρ_range_check_ptr ρ_res) :=\nbegin\n  rcases h with (⟨ptx0z, (⟨ptx1z, ⟨ptx2z, h2⟩ | ⟨ptx2nz, ⟨h31, h32, _⟩⟩⟩ | ⟨ptx1nz, ⟨h41, h42, _⟩⟩)⟩ | ⟨ptx0nz, ⟨h51, h52, _⟩⟩),\n  { left, use [ptx0z, ptx1z, ptx2z, h2.2] },\n  { right, split, left, assumption, exact ⟨h31, h32⟩ },\n  { right, split, right, left, assumption, exact ⟨h41, h42⟩ },\n  right, split, right, right, assumption, exact ⟨h51, h52⟩,\nend\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_ec_double (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  (point.x = ⟨0, 0, 0⟩ ∧ ρ_res = point) ∨\n  (point.x ≠ ⟨0, 0, 0⟩ ∧\n    ∀ ix iy : bigint3,\n    ix.bounded (3 * BASE - 1) →\n    iy.bounded (3 * BASE - 1) →\n    point.x = ix.toBigInt3 →\n    point.y = iy.toBigInt3 →\n    ∃ is inew_x inew_y : bigint3,\n      is.bounded (3 * (BASE - 1)) ∧\n      inew_x.bounded (3 * (BASE - 1)) ∧\n      inew_y.bounded (3 * (↑BASE - 1)) ∧\n      ρ_res = { x := inew_x.toBigInt3, y := inew_y.toBigInt3 } ∧\n      3 * ix.val^2 ≡ 2 * iy.val * is.val [ZMOD SECP_PRIME] ∧\n      inew_x.val ≡ is.val^2 - 2 * ix.val [ZMOD SECP_PRIME] ∧\n      inew_y.val ≡ (ix.val - inew_x.val) * is.val - iy.val [ZMOD SECP_PRIME])\n\n/- ec_double soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_ec_double\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F)\n    (h_auto : auto_spec_ec_double mem κ range_check_ptr point ρ_range_check_ptr ρ_res) :\n  spec_ec_double mem κ range_check_ptr point ρ_range_check_ptr ρ_res :=\nbegin\n  rcases auto_spec_ec_double_better h_auto with\n    ⟨ptx0z, ptx1z, ptx2z, _, reseq⟩ | ⟨nz, _, main_case⟩,\n  { left, simp [BigInt3.ext_iff, ptx0z, ptx1z, ptx2z, reseq] },\n  rcases main_case with ⟨_, _, slope, h_doubling_slope,\n    _, islope_sqr, h_slope_square,\n    _, _, new_x, vv_new_x, _, _, new_y, vv_new_y, _, _, vz,\n    _, x_diff_slope, h_x_diff_slope,\n    _, _, vz', _, _, ret1_eq⟩,\n  right, split,\n  { simp [BigInt3.ext_iff], tauto },\n  intros ix iy ixbdd iybdd ptxeq ptyeq,\n  rcases h_doubling_slope ix iy ixbdd iybdd ptxeq ptyeq with ⟨is, isbdd, slope_eq, slope_congr⟩,\n  have islope_sqr_eq := h_slope_square _ slope_eq,\n  rcases nondet_bigint3_corr vv_new_x with ⟨inew_x, inew_x_eq, inew_x_bdd⟩,\n  rcases nondet_bigint3_corr vv_new_y with ⟨inew_y, inew_y_eq, inew_y_bdd⟩,\n  set diff1 := ((is.mul is).sub inew_x).sub (ix.cmul 2) with diff1_eq,\n  have diff1_bdd : diff1.bounded\n    ((3 * (BASE - 1))^2 * (8 * SECP_REM + 1) + 3 * (BASE - 1) + 2 * (3 * BASE - 1)),\n  { rw [diff1_eq],\n    apply bigint3.bounded_sub,\n    apply bigint3.bounded_sub _ inew_x_bdd,\n    apply bigint3.bounded_mul isbdd isbdd,\n    apply bigint3.bounded_cmul ixbdd },\n  have diff1_equiv : diff1.val % SECP_PRIME = 0,\n  { apply vz,\n    { simp only [diff1_eq, islope_sqr_eq, ptxeq, inew_x_eq],\n      dsimp [bigint3.toUnreducedBigInt3, bigint3.toBigInt3, bigint3.mul, bigint3.sub, bigint3.cmul,\n        bigint3.sqr],\n      simp only [int.cast_sub, int.cast_mul 2], simp_int_casts,\n      split, refl, split, refl, refl },\n    apply bigint3.bounded_of_bounded_of_le diff1_bdd,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  have : (ix.sub inew_x).toBigInt3 =\n    { d0 := point.x.d0 - new_x.d0, d1 := point.x.d1 - new_x.d1, d2 := point.x.d2 - new_x.d2 },\n  { rw [bigint3.toBigInt3_sub, ←ptxeq, ←inew_x_eq], refl },\n  have x_diff_slope_eq := h_x_diff_slope _ _ this.symm slope_eq,\n  set diff2 := (((ix.sub inew_x).mul is).sub iy).sub inew_y with diff2_eq,\n  have diff2_bdd : diff2.bounded\n    (((3 * BASE - 1) + (3 * (BASE - 1)))^2 * (8 * SECP_REM + 1) + (3 * BASE - 1) + 3 * (BASE - 1)),\n  { rw [diff2_eq],\n    apply bigint3.bounded_sub _ inew_y_bdd,\n    apply bigint3.bounded_sub _ iybdd,\n    apply bigint3.bounded_mul,\n    apply bigint3.bounded_sub ixbdd inew_x_bdd,\n    apply bigint3.bounded_of_bounded_of_le isbdd,\n    apply le_add_of_nonneg_left,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  have diff2_equiv : diff2.val % SECP_PRIME = 0,\n  { apply vz',\n    { simp only [diff2_eq, x_diff_slope_eq, inew_y_eq, bigint3.toBigInt3,\n        bigint3.toUnreducedBigInt3, bigint3.mul, bigint3.sub, ptyeq, int.cast_sub],\n        split, refl, split, refl, split },\n    apply bigint3.bounded_of_bounded_of_le diff2_bdd,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  use [is, inew_x, inew_y, isbdd, inew_x_bdd, inew_y_bdd],\n  split, rw [ret1_eq, inew_x_eq, inew_y_eq],\n  use slope_congr,\n  split,\n  { rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←diff1_equiv, diff1_eq, bigint3.sub_val,\n      bigint3.sub_val, bigint3.cmul_val, sub_sub, add_comm, ←sub_sub, pow_two],\n    apply int.modeq.sub_right,\n    apply int.modeq.sub_right,\n    symmetry,\n    apply bigint3.mul_val },\n  rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←diff2_equiv, diff2_eq, bigint3.sub_val,\n       bigint3.sub_val],\n  apply int.modeq.sub_right,\n  apply int.modeq.sub_right,\n  symmetry,\n  transitivity,\n  apply bigint3.mul_val,\n  rw [bigint3.sub_val]\nend\n\n/- Better specification -/\n\ndef spec_ec_double' ( pt : EcPoint F ) ( ret1 : EcPoint F )\n    (secpF : Type) [secp_field secpF] : Prop :=\n  ∀ h : BddECPointData secpF pt,\n    ∃ hret : BddECPointData secpF ret1,\n      hret.toECPoint = 2 • h.toECPoint\n\ntheorem spec_ec_double'_of_spec_ec_double\n    {mem : F → F} {κ : ℕ} {range_check_ptr : F} {pt : EcPoint F} {ret0 : F} {ret1 : EcPoint F}\n    (h : spec_ec_double mem κ range_check_ptr pt ret0 ret1)\n    (secpF : Type) [secp_field secpF] :\n  spec_ec_double' pt ret1 secpF :=\nbegin\n  intros hpt,\n  rcases h with ⟨ptx0, reteq⟩ | ⟨ptxnz, h⟩,\n  { rw reteq, use hpt,\n    rw [hpt.toECPoint_eq_zero_iff.mpr ptx0, smul_zero] },\n  rcases h hpt.ix hpt.iy hpt.ixbdd hpt.iybdd hpt.ptxeq hpt.ptyeq with\n    ⟨is, inew_x, inew_y, is_bdd, inew_x_bdd, inew_y_bdd, ret1eq, mod1eq, mod2eq, mod3eq⟩,\n  have onec_pt := hpt.onEC' ptxnz,\n  have hptynez : (hpt.iy.val : secpF) ≠ 0 := secp_field.y_ne_zero_of_on_ec onec_pt,\n  have eq1 : 3 * (hpt.ix.val : secpF)^2  = 2 * hpt.iy.val * is.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod1eq },\n  have eq2 : (inew_x.val : secpF) = is.val ^ 2 - 2 * hpt.ix.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod2eq },\n  have eq3: (inew_y.val : secpF) = (hpt.ix.val - inew_x.val) * is.val - hpt.iy.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod3eq },\n  have eq1a : (is.val : secpF) = 3 * (hpt.ix.val : secpF)^2 / (2 * hpt.iy.val),\n  { field_simp [ec_field.two_ne_zero], rw [mul_comm, eq1] },\n  have ecdoubleeq : ec_double ((hpt.ix.val : secpF), hpt.iy.val) = (inew_x.val, inew_y.val),\n  { simp [ec_double, ec_double_slope],\n    split, rw [eq2, eq1a, div_pow],\n    rw [eq3, eq1a], congr, rw [eq2, eq1a, div_pow] },\n  have onec_new: on_ec (↑(inew_x.val), ↑(inew_y.val)),\n  { have := @on_ec_ec_double secpF _ (↑(hpt.ix.val), ↑(hpt.iy.val)) onec_pt hptynez,\n    rw ecdoubleeq at this, exact this },\n  have hhret: ¬ (inew_x.i0 = 0 ∧ inew_x.i1 = 0 ∧ inew_x.i2 = 0),\n  { contrapose! onec_new,\n    rw [on_ec], dsimp,\n    conv { congr, to_rhs, simp [bigint3.val, onec_new.1, onec_new.2.1, onec_new.2.2] },\n    apply secp_field.seven_not_square },\n  let hret : BddECPointData secpF ret1 :=\n  { ix    := inew_x,\n    iy    := inew_y,\n    ixbdd := by apply bigint3.bounded_of_bounded_of_le inew_x_bdd; norm_num [BASE],\n    iybdd := by apply bigint3.bounded_of_bounded_of_le inew_y_bdd; norm_num [BASE],\n    ptxeq := by { rw ret1eq },\n    ptyeq := by { rw ret1eq },\n    onEC  := or.inr onec_new },\n  use hret,\n  have : ret1.x ≠ ⟨0, 0, 0⟩,\n  { rw [ret1eq], dsimp,\n    have h' := secp_field.x_ne_zero_of_on_ec onec_new,\n    contrapose! h',\n    have : inew_x = ⟨0, 0, 0⟩ := toBigInt3_eq_zero_of_bounded_3BASE h'\n        (bigint3.bounded_of_bounded_of_le inew_x_bdd bound_slack),\n    simp [this, bigint3.val] },\n  dsimp [BddECPointData.toECPoint], simp [dif_neg ptxnz, dif_neg this],\n  symmetry,\n  apply double_Affine _ _ ecdoubleeq\nend\n\n/-\n-- Function: fast_ec_add\n-/\n\n/- fast_ec_add autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_fast_ec_add_block9 (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (slope : BigInt3 F), spec_compute_slope mem κ₁ range_check_ptr point0 point1 range_check_ptr₁ slope ∧\n  ∃ (κ₂ : ℕ) (slope_sqr : UnreducedBigInt3 F), spec_unreduced_sqr mem κ₂ slope slope_sqr ∧\n  ∃ (κ₃ : ℕ) (range_check_ptr₂ : F) (new_x : BigInt3 F), spec_nondet_bigint3 mem κ₃ range_check_ptr₁ range_check_ptr₂ new_x ∧\n  ∃ (κ₄ : ℕ) (range_check_ptr₃ : F) (new_y : BigInt3 F), spec_nondet_bigint3 mem κ₄ range_check_ptr₂ range_check_ptr₃ new_y ∧\n  ∃ (κ₅ : ℕ) (range_check_ptr₄ : F), spec_verify_zero mem κ₅ range_check_ptr₃ {\n    d0 := slope_sqr.d0 - new_x.d0 - point0.x.d0 - point1.x.d0,\n    d1 := slope_sqr.d1 - new_x.d1 - point0.x.d1 - point1.x.d1,\n    d2 := slope_sqr.d2 - new_x.d2 - point0.x.d2 - point1.x.d2\n  } range_check_ptr₄ ∧\n  ∃ (κ₆ : ℕ) (x_diff_slope : UnreducedBigInt3 F), spec_unreduced_mul mem κ₆ {\n    d0 := point0.x.d0 - new_x.d0,\n    d1 := point0.x.d1 - new_x.d1,\n    d2 := point0.x.d2 - new_x.d2\n  } slope x_diff_slope ∧\n  ∃ (κ₇ : ℕ) (range_check_ptr₅ : F), spec_verify_zero mem κ₇ range_check_ptr₄ {\n    d0 := x_diff_slope.d0 - point0.y.d0 - new_y.d0,\n    d1 := x_diff_slope.d1 - point0.y.d1 - new_y.d1,\n    d2 := x_diff_slope.d2 - point0.y.d2 - new_y.d2\n  } range_check_ptr₅ ∧\n  κ₁ + κ₂ + κ₃ + κ₄ + κ₅ + κ₆ + κ₇ + 52 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₅ ∧\n  ρ_res = {\n    x := new_x,\n    y := new_y\n  }\n\ndef auto_spec_fast_ec_add_block5 (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ((point1.x.d0 = 0 ∧\n    ((point1.x.d1 = 0 ∧\n      ((point1.x.d2 = 0 ∧\n        11 ≤ κ ∧\n        ρ_range_check_ptr = range_check_ptr ∧\n        ρ_res = point0) ∨\n       (point1.x.d2 ≠ 0 ∧\n        ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block9 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n        κ₁ + 3 ≤ κ))) ∨\n     (point1.x.d1 ≠ 0 ∧\n      ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block9 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n      κ₁ + 2 ≤ κ))) ∨\n   (point1.x.d0 ≠ 0 ∧\n    ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block9 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n    κ₁ + 1 ≤ κ))\n\ndef auto_spec_fast_ec_add (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ((point0.x.d0 = 0 ∧\n    ((point0.x.d1 = 0 ∧\n      ((point0.x.d2 = 0 ∧\n        11 ≤ κ ∧\n        ρ_range_check_ptr = range_check_ptr ∧\n        ρ_res = point1) ∨\n       (point0.x.d2 ≠ 0 ∧\n        ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block5 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n        κ₁ + 3 ≤ κ))) ∨\n     (point0.x.d1 ≠ 0 ∧\n      ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block5 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n      κ₁ + 2 ≤ κ))) ∨\n   (point0.x.d0 ≠ 0 ∧\n    ∃ (κ₁ : ℕ), auto_spec_fast_ec_add_block5 mem κ₁ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res ∧\n    κ₁ + 1 ≤ κ))\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_fast_ec_add (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∀ ix0 iy0 ix1 iy1 : bigint3,\n  ix0.bounded (3 * BASE - 1) →\n  iy0.bounded (3 * BASE - 1) →\n  ix1.bounded (3 * BASE - 1) →\n  iy1.bounded (3 * BASE - 1) →\n  point0.x = ix0.toBigInt3 →\n  point0.y = iy0.toBigInt3 →\n  point1.x = ix1.toBigInt3 →\n  point1.y = iy1.toBigInt3 →\n  (point0.x.d0 = 0 ∧ point0.x.d1 = 0 ∧ point0.x.d2 = 0 ∧ ρ_res = point1) ∨\n  (point1.x.d0 = 0 ∧ point1.x.d1 = 0 ∧ point1.x.d2 = 0 ∧ ρ_res = point0) ∨\n  ¬ (point0.x.d0 = 0 ∧ point0.x.d1 = 0 ∧ point0.x.d2 = 0) ∧\n  ¬ (point1.x.d0 = 0 ∧ point1.x.d1 = 0 ∧ point1.x.d2 = 0) ∧\n  ∃ is inew_x inew_y : bigint3,\n    is.bounded (3 * (BASE - 1)) ∧\n    inew_x.bounded (3 * (BASE - 1)) ∧\n    inew_y.bounded (3 * (BASE - 1)) ∧\n    ρ_res = { x := inew_x.toBigInt3, y := inew_y.toBigInt3 } ∧\n    (ix0.val - ix1.val) * is.val ≡ (iy0.val - iy1.val) [ZMOD SECP_PRIME] ∧\n    inew_x.val ≡ is.val^2 - ix0.val - ix1.val [ZMOD SECP_PRIME] ∧\n    inew_y.val ≡ (ix0.val - inew_x.val) * is.val - iy0.val [ZMOD SECP_PRIME]\n\n/- fast_ec_add soundness theorem -/\n\ntheorem fast_ec_add_block5_spec_better {mem : F → F} {κ : ℕ}{range_check_ptr : F} {pt0 pt1 : EcPoint F} {ret0 : F} {ret1 : EcPoint F}\n    (h_auto : auto_spec_fast_ec_add_block5 mem κ range_check_ptr pt0 pt1 ret0 ret1) :\n  (pt1.x.d0 = 0 ∧ pt1.x.d1 = 0 ∧ pt1.x.d2 = 0 ∧\n    ret0 = range_check_ptr ∧ ret1 = pt0) ∨\n  (¬ (pt1.x.d0 = 0 ∧ pt1.x.d1 = 0 ∧ pt1.x.d2 = 0) ∧\n  ∃ κ₁, auto_spec_fast_ec_add_block9 mem κ₁ range_check_ptr pt0 pt1 ret0 ret1) :=\nbegin\n  rcases h_auto with\n    ⟨pt1x0z, ⟨pt1x1z, ⟨pt1x2z, ⟨_, ret0, ret1⟩⟩ | ⟨pt1x2nz, ⟨κ₁, hblock9⟩⟩⟩ |\n      ⟨pt1x1nz, ⟨κ₁, hblock9⟩⟩⟩ | ⟨pt1x0nz, ⟨κ₁, hblock9⟩⟩,\n  { left, use [pt1x0z, pt1x1z, pt1x2z, ret0, ret1] },\n  { right, split, intro h, apply pt1x2nz h.2.2,\n    exact ⟨κ₁, hblock9.1⟩ },\n  { right, split, intro h, apply pt1x1nz h.2.1,\n    exact ⟨κ₁, hblock9.1⟩ },\n  { right, split, intro h, apply pt1x0nz h.1,\n    exact ⟨κ₁, hblock9.1⟩ }\nend\n\ntheorem ec_add_mainb_spec_better {mem : F → F} {κ : ℕ} {range_check_ptr : F} {pt0 pt1 : EcPoint F} {ret0 : F} {ret1 : EcPoint F}\n    (h_auto : auto_spec_fast_ec_add mem κ range_check_ptr pt0 pt1 ret0 ret1) :\n  (pt0.x.d0 = 0 ∧ pt0.x.d1 = 0 ∧ pt0.x.d2 = 0 ∧\n    ret0 = range_check_ptr ∧ ret1 = pt1) ∨\n  (pt1.x.d0 = 0 ∧ pt1.x.d1 = 0 ∧ pt1.x.d2 = 0 ∧\n    ret0 = range_check_ptr ∧ ret1 = pt0) ∨\n  (¬(pt0.x.d0 = 0 ∧ pt0.x.d1 = 0 ∧ pt0.x.d2 = 0) ∧\n    ¬ (pt1.x.d0 = 0 ∧ pt1.x.d1 = 0 ∧ pt1.x.d2 = 0) ∧\n    ∃ κ₁, auto_spec_fast_ec_add_block9 mem κ₁ range_check_ptr pt0 pt1 ret0 ret1) :=\nbegin\n  rcases h_auto with\n    ⟨pt1x0z, ⟨pt1x1z, ⟨pt1x2z, ⟨_, ret0, ret1⟩⟩ | ⟨pt1x2nz, κ₁, hblock9⟩⟩ |\n      ⟨pt1x1nz, κ₁, hblock9⟩⟩ | ⟨pt1x0nz, κ₁, hblock9⟩,\n  { left, use [pt1x0z, pt1x1z, pt1x2z, ret0, ret1] },\n  { right,\n    cases fast_ec_add_block5_spec_better hblock9.1 with h' h',\n    { left, exact h' },\n    right, split,\n    { intro h, exact pt1x2nz h.2.2 },\n    exact h' },\n  { right,\n    cases fast_ec_add_block5_spec_better hblock9.1 with h' h',\n    { left, exact h' },\n    right, split,\n    { intro h, exact pt1x1nz h.2.1 },\n    exact h' },\n  { right,\n    cases fast_ec_add_block5_spec_better hblock9.1 with h' h',\n    { left, exact h' },\n    right, split,\n    { intro h, exact pt1x0nz h.1 },\n    exact h' },\nend\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_fast_ec_add\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F)\n    (h_auto : auto_spec_fast_ec_add mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res) :\n  spec_fast_ec_add mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res :=\nbegin\n  intros ix0 iy0 ix1 iy1 ix0bdd iy0bdd ix1bdd iy1bdd pt0xeq pt0yeq pt1xeq pt1yeq,\n  rcases ec_add_mainb_spec_better h_auto with\n    ⟨pt0x0z, pt0x1z, pt0x2z, _, ret1eq⟩ |\n    ⟨pt1x0z, pt1x1z, pt1x2z, _, ret1eq⟩ |\n      ⟨pt0nz, pt1nz, _, _, _, slope, h_compute_slope,\n        _, islope_sqr, h_slope_sqr,\n        _, _, new_x, vv_new_x, _, _, new_y, vv_new_y, _, _, vz,\n        _, x_diff_slope, h_x_diff_slope,\n        _, _, vz', _, _, ret1_eq⟩,\n  { left, use [pt0x0z, pt0x1z, pt0x2z, ret1eq] },\n  { right, left, use [pt1x0z, pt1x1z, pt1x2z, ret1eq] },\n  rcases h_compute_slope ix0 iy0 ix1 iy1 ix0bdd iy0bdd ix1bdd iy1bdd pt0xeq pt0yeq pt1xeq pt1yeq\n    with ⟨is, isbdd, slope_eq, slope_congr⟩,\n  have islope_sqr_eq := h_slope_sqr _ slope_eq,\n  rcases nondet_bigint3_corr vv_new_x with ⟨inew_x, inew_x_eq, inew_x_bdd⟩,\n  rcases nondet_bigint3_corr vv_new_y with ⟨inew_y, inew_y_eq, inew_y_bdd⟩,\n  set diff1 := (((is.mul is).sub inew_x).sub ix0).sub ix1 with diff1_eq,\n  have diff1_bdd : diff1.bounded\n    ((3 * (BASE - 1))^2 * (8 * SECP_REM + 1) + 3 * (BASE - 1) + (3 * BASE - 1) + (3 * BASE - 1)),\n  { rw [diff1_eq],\n    apply bigint3.bounded_sub _ ix1bdd,\n    apply bigint3.bounded_sub _ ix0bdd,\n    apply bigint3.bounded_sub _ inew_x_bdd,\n    apply bigint3.bounded_mul isbdd isbdd },\n  have diff1_equiv : diff1.val % SECP_PRIME = 0,\n  { apply vz,\n    { simp only [diff1_eq, islope_sqr_eq, pt0xeq, pt1xeq, inew_x_eq],\n      dsimp [bigint3.toBigInt3, bigint3.toUnreducedBigInt3, bigint3.mul, bigint3.sub, bigint3.cmul,\n        bigint3.sqr],\n      simp only [int.cast_sub, int.cast_mul 2], simp_int_casts,\n      split, refl, split, refl, refl },\n    apply bigint3.bounded_of_bounded_of_le diff1_bdd,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  have : (ix0.sub inew_x).toBigInt3 =\n    { d0 := point0.x.d0 - new_x.d0, d1 := point0.x.d1 - new_x.d1, d2 := point0.x.d2 - new_x.d2 },\n  { rw [bigint3.toBigInt3_sub, ←pt0xeq, ←inew_x_eq], refl },\n  have x_diff_slope_eq := h_x_diff_slope _ _ this.symm slope_eq,\n  set diff2 := (((ix0.sub inew_x).mul is).sub iy0).sub inew_y with diff2_eq,\n  have diff2_bdd : diff2.bounded\n    (((3 * BASE - 1) + (3 * (BASE - 1)))^2 * (8 * SECP_REM + 1) + (3 * BASE - 1) + 3 * (BASE - 1)),\n  { rw [diff2_eq],\n    apply bigint3.bounded_sub _ inew_y_bdd,\n    apply bigint3.bounded_sub _ iy0bdd,\n    apply bigint3.bounded_mul,\n    apply bigint3.bounded_sub ix0bdd inew_x_bdd,\n    apply bigint3.bounded_of_bounded_of_le isbdd,\n    apply le_add_of_nonneg_left,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  have diff2_equiv : diff2.val % SECP_PRIME = 0,\n  { apply vz',\n    { simp only [diff2_eq, x_diff_slope_eq, inew_y_eq, bigint3.toBigInt3,\n        bigint3.toUnreducedBigInt3, bigint3.mul, bigint3.sub, pt0yeq, int.cast_sub],\n        split, refl, split, refl, split },\n    apply bigint3.bounded_of_bounded_of_le diff2_bdd,\n    dsimp only [BASE, SECP_REM], simp_int_casts, norm_num1 },\n  right, right,\n  use [pt0nz, pt1nz, is, inew_x, inew_y, isbdd, inew_x_bdd, inew_y_bdd],\n  split, rw [ret1_eq, inew_x_eq, inew_y_eq],\n  use slope_congr,\n  split,\n  { rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←diff1_equiv, diff1_eq, bigint3.sub_val,\n      bigint3.sub_val, bigint3.sub_val, sub_sub, sub_sub, pow_two, ←add_assoc,\n      add_comm _ inew_x.val, sub_sub, ←sub_sub],\n    apply int.modeq.sub_right,\n    apply int.modeq.sub_right,\n    symmetry,\n    apply bigint3.mul_val },\n  rw [int.modeq_iff_dvd, int.dvd_iff_mod_eq_zero, ←diff2_equiv, diff2_eq, bigint3.sub_val,\n       bigint3.sub_val],\n  apply int.modeq.sub_right,\n  apply int.modeq.sub_right,\n  symmetry,\n  transitivity,\n  apply bigint3.mul_val,\n  rw [bigint3.sub_val]\nend\n\n/-\nBetter specification of fast_ec_add\n-/\n\ndef spec_fast_ec_add' ( pt0 pt1 : EcPoint F ) ( ret1 : EcPoint F )\n    (secpF : Type) [secp_field secpF] : Prop :=\n  ∀ h0 : BddECPointData secpF pt0,\n  ∀ h1 : BddECPointData secpF pt1,\n    (↑(h0.ix.val) : secpF) ≠ ↑(h1.ix.val) →\n    ∃ hret : BddECPointData secpF ret1,\n      hret.toECPoint = h0.toECPoint + h1.toECPoint\n\ntheorem spec_fast_ec_add'_of_spec_ec_add\n    {mem : F → F} {κ : ℕ} {range_check_ptr : F} {pt0 pt1 : EcPoint F} {ret0 : F} {ret1 : EcPoint F}\n    (h : spec_fast_ec_add mem κ range_check_ptr pt0 pt1 ret0 ret1)\n    (secpF : Type) [secp_field secpF] :\n  spec_fast_ec_add' pt0 pt1 ret1 secpF :=\nbegin\n  intros h0 h1 hne,\n  rcases h h0.ix h0.iy h1.ix h1.iy h0.ixbdd h0.iybdd h1.ixbdd h1.iybdd h0.ptxeq h0.ptyeq h1.ptxeq h1.ptyeq with\n    h' | h' | h',\n  { rcases h' with ⟨pt0x0eq, pt0x1eq, pt0x2eq, rfl⟩,\n    have pt0xz : pt0.x = ⟨0, 0, 0⟩,\n    { simp [BigInt3.ext_iff, pt0x0eq, pt0x1eq, pt0x2eq] },\n    use h1,\n    have pt0xeq := h0.ptxeq,\n    simp [BigInt3.ext_iff, bigint3.toBigInt3] at pt0xeq,\n    have ix0i0z: h0.ix.i0 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt0xeq.1, pt0x0eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h0.ixbdd.1,\n      apply bound_lt_prime },\n    have ix0i1z: h0.ix.i1 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt0xeq.2.1, pt0x1eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h0.ixbdd.2.1,\n      apply bound_lt_prime },\n    have ix0i2z: h0.ix.i2 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt0xeq.2.2, pt0x2eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h0.ixbdd.2.2,\n      apply bound_lt_prime },\n    simp [BddECPointData.toECPoint, pt0xz], refl },\n  { rcases h' with ⟨pt1x0eq, pt1x1eq, pt1x2eq, rfl⟩,\n    have pt1xz : pt1.x = ⟨0, 0, 0⟩,\n    { simp [BigInt3.ext_iff, pt1x0eq, pt1x1eq, pt1x2eq] },\n    use h0,\n    have pt1xeq := h1.ptxeq,\n    simp [BigInt3.ext_iff, bigint3.toBigInt3] at pt1xeq,\n    have ix0i0z: h1.ix.i0 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt1xeq.1, pt1x0eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h1.ixbdd.1,\n      apply bound_lt_prime },\n    have ix0i1z: h1.ix.i1 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt1xeq.2.1, pt1x1eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h1.ixbdd.2.1,\n      apply bound_lt_prime },\n    have ix0i2z: h1.ix.i2 = 0,\n    { apply @PRIME.int_coe_inj F,\n      rw [←pt1xeq.2.2, pt1x2eq, int.cast_zero],\n      simp,\n      apply lt_of_le_of_lt h1.ixbdd.2.2,\n      apply bound_lt_prime },\n    simp [BddECPointData.toECPoint, pt1xz], refl },\n  rcases h' with ⟨pt0nz, pt1nz, is, inew_x, inew_y, is_bdd, inew_x_bdd, inew_y_bdd, ret1eq, mod1eq, mod2eq, mod3eq⟩,\n  have eq1 : ((h0.ix.val : secpF) - h1.ix.val) * is.val = h0.iy.val - h1.iy.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod1eq },\n  have eq2 : (inew_x.val : secpF) = is.val ^ 2 - h0.ix.val - h1.ix.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod2eq },\n  have eq3: (inew_y.val : secpF) = (h0.ix.val - inew_x.val) * is.val - h0.iy.val,\n  { norm_cast, rw @eq_iff_modeq_int secpF _ SECP_PRIME,\n    apply mod3eq },\n  have nez : (h1.ix.val : secpF) - h0.ix.val ≠ 0,\n  { contrapose! hne, symmetry, apply eq_of_sub_eq_zero hne },\n  have eq1a : (is.val : secpF) = (h1.iy.val - h0.iy.val) / (h1.ix.val - h0.ix.val),\n  { field_simp [nez], rw [←neg_sub, mul_neg, mul_comm, eq1, neg_sub] },\n  have ecaddeq : ec_add ((h0.ix.val : secpF), h0.iy.val) ((h1.ix.val : secpF), h1.iy.val) = (inew_x.val, inew_y.val),\n  { simp [ec_add, ec_add_slope],\n    split, rw [eq2, eq1a, div_pow, sub_sub],\n    rw [eq3], congr, rw [eq2, eq1a, div_pow, sub_sub], rw [eq1a] },\n  have pt0nz' : pt0.x ≠ ⟨0, 0, 0⟩,\n  { rw [ne, BigInt3.ext_iff], exact pt0nz },\n  have pt1nz' : pt1.x ≠ ⟨0, 0, 0⟩,\n  { rw [ne, BigInt3.ext_iff], exact pt1nz },\n  have onec_new: on_ec (↑(inew_x.val), ↑(inew_y.val)),\n  { have := @on_ec_ec_add secpF _ (↑(h0.ix.val), ↑(h0.iy.val)) (↑(h1.ix.val), ↑(h1.iy.val))\n              (h0.onEC' pt0nz') (h1.onEC' pt1nz') hne,\n    rw ecaddeq at this, exact this },\n  let hret : BddECPointData secpF ret1 :=\n  { ix    := inew_x,\n    iy    := inew_y,\n    ixbdd := by apply bigint3.bounded_of_bounded_of_le inew_x_bdd; norm_num [BASE],\n    iybdd := by apply bigint3.bounded_of_bounded_of_le inew_y_bdd; norm_num [BASE],\n    ptxeq := by { rw ret1eq },\n    ptyeq := by { rw ret1eq },\n    onEC  := or.inr onec_new },\n  have hhret : ¬ (ret1.x = ⟨0, 0, 0⟩),\n  { simp [ret1eq],\n    intro h',\n    apply secp_field.x_ne_zero_of_on_ec onec_new,\n    suffices : inew_x = ⟨0, 0, 0⟩,\n    { rw bigint3.ext_iff at this,\n      simp [this, bigint3.val] },\n      apply toBigInt3_eq_zero_of_bounded_3BASE h',\n      apply bigint3.bounded_of_bounded_of_le inew_x_bdd; norm_num [BASE] },\n  use hret,\n  dsimp [BddECPointData.toECPoint], simp [dif_neg pt0nz', dif_neg pt1nz', dif_neg hhret],\n  symmetry,\n  apply Affine_add_Affine _ _ _ hne,\n  rw [ecaddeq]\nend\n\n\n\n\n/-\n-- Function: ec_add\n-/\n\n/- ec_add autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_ec_add (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∃ x_diff : BigInt3 F, x_diff = {\n    d0 := point0.x.d0 - point1.x.d0,\n    d1 := point0.x.d1 - point1.x.d1,\n    d2 := point0.x.d2 - point1.x.d2\n  } ∧\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ same_x : F), spec_is_zero mem κ₁ range_check_ptr x_diff range_check_ptr₁ same_x ∧\n  ((same_x = 0 ∧\n    ∃ (κ₂ : ℕ), spec_fast_ec_add mem κ₂ range_check_ptr₁ point0 point1 ρ_range_check_ptr ρ_res ∧\n    κ₁ + κ₂ + 21 ≤ κ) ∨\n   (same_x ≠ 0 ∧\n    ∃ y_sum : BigInt3 F, y_sum = {\n      d0 := point0.y.d0 + point1.y.d0,\n      d1 := point0.y.d1 + point1.y.d1,\n      d2 := point0.y.d2 + point1.y.d2\n    } ∧\n    ∃ (κ₂ : ℕ) (range_check_ptr₂ opposite_y : F), spec_is_zero mem κ₂ range_check_ptr₁ y_sum range_check_ptr₂ opposite_y ∧\n    ((opposite_y ≠ 0 ∧\n      ∃ ZERO_POINT : EcPoint F, ZERO_POINT = {\n        x := { d0 := 0, d1 := 0, d2 := 0 },\n        y := { d0 := 0, d1 := 0, d2 := 0 }\n      } ∧\n      κ₁ + κ₂ + 20 ≤ κ ∧\n      ρ_range_check_ptr = range_check_ptr₂ ∧\n      ρ_res = ZERO_POINT) ∨\n     (opposite_y = 0 ∧\n      ∃ (κ₃ : ℕ), spec_ec_double mem κ₃ range_check_ptr₂ point0 ρ_range_check_ptr ρ_res ∧\n      κ₁ + κ₂ + κ₃ + 21 ≤ κ))))\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_ec_add (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∀ (secpF : Type) [secp_field secpF], by exactI\n  ∀ h0 : BddECPointData secpF point0,\n  ∀ h1 : BddECPointData secpF point1,\n    ∃ hret : BddECPointData secpF ρ_res,\n      hret.toECPoint = h0.toECPoint + h1.toECPoint\n\n/- ec_add soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_ec_add\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point0 point1 : EcPoint F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F)\n    (h_auto : auto_spec_ec_add mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res) :\n  spec_ec_add mem κ range_check_ptr point0 point1 ρ_range_check_ptr ρ_res :=\nbegin\n  intros secpF _,\n  resetI,\n  intros h0 h1,\n  rcases h_auto with ⟨x_diff, x_diff_eq, _, _, same_x, h_x_diff, h'⟩,\n  let ix_diff := h0.ix.sub h1.ix,\n  have h2: x_diff = ix_diff.toBigInt3,\n  { dsimp [ix_diff], rw [x_diff_eq, bigint3.toBigInt3_sub, ←h0.ptxeq, ←h1.ptxeq], refl },\n  have h3: ix_diff.bounded (2^107),\n  { rw (show (2^107 : ℤ) = 2^106 + 2^106, by norm_num),\n    apply bigint3.bounded_sub,\n    apply bigint3.bounded_of_bounded_of_le h0.ixbdd,\n    rw BASE, simp_int_casts, norm_num,\n    apply bigint3.bounded_of_bounded_of_le h1.ixbdd,\n    rw BASE, simp_int_casts, norm_num },\n  rcases h' with ⟨same_x_z, _, h'⟩ | ⟨same_x_nz, h'⟩,\n  { have := h_x_diff _ h2 h3,\n    have : ix_diff.val % ↑SECP_PRIME ≠ 0,\n    { rcases this with ⟨_, rfl⟩ | ⟨_, _⟩,\n      norm_num at same_x_z, assumption },\n    apply spec_fast_ec_add'_of_spec_ec_add h'.1,\n    contrapose! this,\n    dsimp [ix_diff], rw [bigint3.sub_val, ←int.modeq_iff_sub_mod_eq_zero,\n      ←char_p.int_coe_eq_int_coe_iff secpF],\n    exact this },\n  have := h_x_diff _ h2 h3,\n  have : ix_diff.val % ↑SECP_PRIME = 0,\n  { rcases this with ⟨_, rfl⟩ | ⟨_, _⟩,\n    assumption, norm_num at same_x_nz },\n  have h0eqh1: (h0.ix.val : secpF) = h1.ix.val,\n  { rw [bigint3.sub_val, ←int.modeq_iff_sub_mod_eq_zero,\n       ←char_p.int_coe_eq_int_coe_iff secpF] at this,\n     exact this },\n  have ptx0iff : point0.x = ⟨0, 0, 0⟩ ↔ point1.x = ⟨0, 0, 0⟩,\n  { rw [h1.pt_zero_iff, ←h0eqh1, ←h0.pt_zero_iff] },\n  rcases h' with ⟨y_sum, y_sum_eq, _, _, opposite_y, h_opposite_y, h'⟩,\n  let iy_sum := h0.iy.add h1.iy,\n  have h2: y_sum = iy_sum.toBigInt3,\n  { dsimp [iy_sum], rw [y_sum_eq, bigint3.toBigInt3_add, ←h0.ptyeq, ←h1.ptyeq], refl },\n  have h3: iy_sum.bounded (2^107),\n  { rw (show (2^107 : ℤ) = 2^106 + 2^106, by norm_num),\n    apply bigint3.bounded_add,\n    apply bigint3.bounded_of_bounded_of_le h0.iybdd,\n    rw BASE, simp_int_casts, norm_num,\n    apply bigint3.bounded_of_bounded_of_le h1.iybdd,\n    rw BASE, simp_int_casts, norm_num },\n  rcases h' with ⟨opposite_y_nz, h'⟩ | ⟨opposite_y_z, _, h'⟩,\n  { rcases h' with ⟨_, rfl, _, _, rfl⟩,\n    use BddECPointData.zero,\n    rw BddECPointData.toECPoint_zero,\n    by_cases hx0: point0.x = ⟨0, 0, 0⟩,\n    { have hx1: point1.x = ⟨0, 0, 0⟩ := ptx0iff.mp hx0,\n      rw [BddECPointData.toECPoint, dif_pos hx0],\n      rw [BddECPointData.toECPoint, dif_pos hx1],\n      apply (add_zero _).symm },\n    have hx1: point1.x ≠ ⟨0, 0, 0⟩,\n    { rwa ptx0iff at hx0 },\n    rw [BddECPointData.toECPoint, dif_neg hx0],\n    rw [BddECPointData.toECPoint, dif_neg hx1],\n    symmetry,\n    rw [← eq_neg_iff_add_eq_zero, ECPoint.neg_def, ECPoint.neg],\n    dsimp,\n    congr,\n    exact h0eqh1,\n    have := h_opposite_y _ h2 h3,\n    have : iy_sum.val % ↑SECP_PRIME = 0,\n    { rcases this with ⟨_, _⟩ | ⟨_, hh⟩,\n      assumption, contradiction },\n    rw [bigint3.add_val, ←sub_neg_eq_add, ←int.modeq_iff_sub_mod_eq_zero,\n       ←char_p.int_coe_eq_int_coe_iff secpF, int.cast_neg] at this,\n    exact this },\n  have := h_opposite_y _ h2 h3,\n  have h0iyneh1iy : iy_sum.val % ↑SECP_PRIME ≠ 0,\n  { rcases this with ⟨_, rfl⟩ | ⟨_, _⟩,\n    norm_num at opposite_y_z,\n    assumption },\n  rw [ne, bigint3.add_val, ←sub_neg_eq_add, ←int.modeq_iff_sub_mod_eq_zero,\n      ←char_p.int_coe_eq_int_coe_iff secpF, int.cast_neg] at h0iyneh1iy,\n  have : h0.toECPoint = h1.toECPoint :=\n    BddECPointData.toECPoint_eq_of_eq_of_ne h0eqh1 h0iyneh1iy,\n  have : h0.toECPoint + h1.toECPoint = 2 • h0.toECPoint,\n  { rw [this, two_smul] },\n  rw this,\n  exact spec_ec_double'_of_spec_ec_double h'.1 secpF h0\nend\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_ec_mul_inner (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (scalar m ρ_range_check_ptr : F) (ρ_pow2 ρ_res : EcPoint F) : Prop :=\n   ∀ (secpF : Type) [hsecp : secp_field secpF],\n   by exactI\n      point.x ≠ ⟨0, 0, 0⟩ →\n      ∀ hpt : BddECPointData secpF point,\n      ∃ nn : ℕ,\n        m = ↑nn ∧\n        nn < ring_char F ∧\n          (nn ≤ SECP_LOG2_BOUND →\n            ∃ scalarn : ℕ,\n              scalar = ↑scalarn ∧\n              scalarn < 2^nn ∧\n            ∃ hpow2 : BddECPointData secpF ρ_pow2,\n              ρ_pow2.x ≠ ⟨0, 0, 0⟩ ∧\n              hpow2.toECPoint = 2^nn • hpt.toECPoint ∧\n            ∃ hres : BddECPointData secpF ρ_res,\n              hres.toECPoint = scalarn • hpt.toECPoint)\n\n\n/-\n-- Function: ec_mul_inner\n-/\n\n/- ec_mul_inner autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_ec_mul_inner (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (scalar m ρ_range_check_ptr : F) (ρ_pow2 ρ_res : EcPoint F) : Prop :=\n  ((m = 0 ∧\n    scalar = 0 ∧\n    ∃ ZERO_POINT : EcPoint F, ZERO_POINT = {\n      x := { d0 := 0, d1 := 0, d2 := 0 },\n      y := { d0 := 0, d1 := 0, d2 := 0 }\n    } ∧\n    16 ≤ κ ∧\n    ρ_range_check_ptr = range_check_ptr ∧\n    ρ_pow2 = point ∧\n    ρ_res = ZERO_POINT) ∨\n   (m ≠ 0 ∧\n    ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (double_point : EcPoint F), spec_ec_double mem κ₁ range_check_ptr point range_check_ptr₁ double_point ∧\n    ∃ anon_cond : F,\n    ((anon_cond = 0 ∧\n      ∃ (κ₂ : ℕ), spec_ec_mul_inner mem κ₂ range_check_ptr₁ double_point (scalar / (2 : ℤ)) (m - 1) ρ_range_check_ptr ρ_pow2 ρ_res ∧\n      κ₁ + κ₂ + 22 ≤ κ) ∨\n     (anon_cond ≠ 0 ∧\n      ∃ (κ₂ : ℕ) (range_check_ptr₂ : F) (inner_pow2 inner_res : EcPoint F), spec_ec_mul_inner mem κ₂ range_check_ptr₁ double_point ((scalar - 1) / (2 : ℤ)) (m - 1) range_check_ptr₂ inner_pow2 inner_res ∧\n      ∃ (κ₃ : ℕ) (range_check_ptr₃ : F) (res : EcPoint F), spec_fast_ec_add mem κ₃ range_check_ptr₂ point inner_res range_check_ptr₃ res ∧\n      κ₁ + κ₂ + κ₃ + 56 ≤ κ ∧\n      ρ_range_check_ptr = range_check_ptr₃ ∧\n      ρ_pow2 = inner_pow2 ∧\n      ρ_res = res))))\n\n/- ec_mul_inner soundness theorem -/\n\nlemma ec_mul_inner_aux {secpF : Type} [secp_field secpF] {pt0 pt1 : EcPoint F}\n   (h0 : BddECPointData secpF pt0)\n   (h1 : BddECPointData secpF pt1)\n   (hpt0xnz : pt0.x ≠ ⟨0, 0, 0⟩)\n   (hxeq : (↑h0.ix.val : secpF) = h1.ix.val) :\n  h1.toECPoint = 0 ∨ h1.toECPoint = h0.toECPoint ∨ h1.toECPoint = - h0.toECPoint :=\nbegin\n  by_cases h1xz : pt1.x = ⟨0, 0, 0⟩,\n  { left, simp [BddECPointData.toECPoint, h1xz], refl },\n  right, simp [BddECPointData.toECPoint, h1xz, hpt0xnz, ECPoint.neg_def, ECPoint.neg, hxeq],\n  have := h0.onEC,\n  simp [hxeq, hpt0xnz, ←(h1.onEC' h1xz)] at this,\n  exact eq_or_eq_neg_of_sq_eq_sq _ _ this.symm\nend\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_ec_mul_inner\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point : EcPoint F) (scalar m ρ_range_check_ptr : F) (ρ_pow2 ρ_res : EcPoint F)\n    (h_auto : auto_spec_ec_mul_inner mem κ range_check_ptr point scalar m ρ_range_check_ptr ρ_pow2 ρ_res) :\n  spec_ec_mul_inner mem κ range_check_ptr point scalar m ρ_range_check_ptr ρ_pow2 ρ_res :=\nbegin\n  intros secpF _ ptxnez hpt,\n  resetI,\n  rcases h_auto with ⟨neq, scalareq, _, rfl, _, _, ret1eq, ret2eq⟩ |\n    ⟨nnz, _, _, double_pt, hdouble_pt, _, heven_or_odd⟩,\n  { use 0, split,\n    { rw [neq, nat.cast_zero] }, split,\n    { rw PRIME.char_eq, apply PRIME_pos },\n    intro _,\n    use 0, split,\n    { rw [scalareq, nat.cast_zero] },\n    split, { norm_num },\n    rw [ret1eq, ret2eq], dsimp,\n    use [hpt, ptxnez], simp,\n    use BddECPointData.zero, simp [BddECPointData.toECPoint], refl },\n  rcases spec_ec_double'_of_spec_ec_double hdouble_pt secpF hpt with\n    ⟨hdouble, hdouble_eq⟩,\n  have double_ne : double_pt.x ≠ ⟨0, 0, 0⟩,\n  { rw [ne, ←hdouble.toECPoint_eq_zero_iff],\n    contrapose! ptxnez,\n    rw ptxnez at hdouble_eq,\n    rw [←hpt.toECPoint_eq_zero_iff],\n    exact secp_field.eq_zero_of_double_eq_zero hdouble_eq.symm },\n  rcases heven_or_odd with ⟨_, _, heven, _⟩ | ⟨_, _, hodd⟩,\n  { rcases heven secpF double_ne hdouble with ⟨nn', nn'eq, nn'le, haux⟩,\n    use nn' + 1, split,\n    { rw [nat.cast_add, nat.cast_one, ←sub_eq_iff_eq_add, nn'eq] }, split,\n    { apply nat.succ_le_of_lt,\n      apply lt_of_le_of_ne (nat.succ_le_of_lt nn'le),\n      contrapose! nnz,\n      rw [eq_add_of_sub_eq nn'eq],\n      transitivity (↑(nn'.succ) : F), { simp },\n      rw [nnz, ring_char.spec] },\n    intro nn'le,\n    have nn'le' : nn' ≤ SECP_LOG2_BOUND := le_trans (le_of_lt (nat.lt_succ_self _)) nn'le,\n    rcases haux nn'le' with ⟨nscalar, nscalareq, nscalarlt,\n      hdoublepow2, hdoublepow2ne, hdoublepow2eq, hdoubleres, hdoublereseq⟩,\n    use 2 * nscalar, split,\n    { rw [nat.cast_mul, ←nscalareq],\n      symmetry,\n      norm_cast,\n      apply mul_div_cancel',\n      apply PRIME.two_ne_zero },\n    split,\n    { rw [pow_succ],\n      apply nat.mul_lt_mul_of_pos_left nscalarlt,\n      norm_num },\n    use [hdoublepow2, hdoublepow2ne], split,\n      { rw [pow_succ', ←smul_smul, hdoublepow2eq, hdouble_eq] },\n    use hdoubleres,\n    rw [mul_comm 2, ←smul_smul, hdoublereseq, hdouble_eq] },\n  rcases hodd with ⟨_, inner_pow2, inner_res, hodd1, _, _, res, hodd2, _, _, ret1eq, ret2eq⟩,\n  rcases hodd1 secpF double_ne hdouble with ⟨nn', nn'eq, nn'le, haux⟩,\n  use nn' + 1, split,\n  { rw [nat.cast_add, nat.cast_one, ←sub_eq_iff_eq_add, nn'eq] }, split,\n  { apply nat.succ_le_of_lt,\n    apply lt_of_le_of_ne (nat.succ_le_of_lt nn'le),\n    contrapose! nnz,\n    rw [eq_add_of_sub_eq nn'eq],\n    transitivity (↑(nn'.succ) : F), { simp },\n    rw [nnz, ring_char.spec] },\n  intro nn'le,\n  have nn'le' : nn' ≤ SECP_LOG2_BOUND := le_trans (le_of_lt (nat.lt_succ_self _)) nn'le,\n  rcases haux nn'le' with ⟨nscalar, nscalareq, nscalarlt, hinnerpow2, hinnerpow2ne, hinnerpow2eq,\n      hinnerres, hinnerreseq⟩,\n  have : (↑(hpt.ix.val) : secpF) ≠ ↑(hinnerres.ix.val),\n  { intro h,\n    have := ec_mul_inner_aux hpt hinnerres ptxnez h,\n    rw [hinnerreseq, hdouble_eq, smul_smul] at this,\n    revert this,\n    apply secp_field.order_large,\n    { rw [ne, BddECPointData.toECPoint_eq_zero_iff],\n      exact ptxnez },\n    refine lt_of_lt_of_le (nat.mul_lt_mul_of_pos_right nscalarlt (by norm_num)) _,\n    rw ←pow_succ',\n    exact nat.pow_le_pow_of_le_right (by norm_num) nn'le },\n  rcases spec_fast_ec_add'_of_spec_ec_add hodd2 secpF hpt hinnerres this with ⟨hret, hreteq⟩,\n  use 2 * nscalar + 1, split,\n  { rw [nat.cast_add, nat.cast_mul, ←nscalareq],\n    symmetry, simp,\n    rw mul_div_cancel' (scalar - 1) (PRIME.two_ne_zero F),\n    simp },\n  rw [ret1eq, ret2eq],\n  split,\n  { rw [pow_succ, two_mul, add_assoc, two_mul],\n    apply add_lt_add_of_lt_of_le nscalarlt,\n    apply nat.succ_le_of_lt nscalarlt },\n  use [hinnerpow2, hinnerpow2ne], split,\n  { rw [pow_succ', ←smul_smul, hinnerpow2eq, hdouble_eq] },\n  use hret,\n  rw [add_comm, add_smul, one_smul, mul_comm 2, ←smul_smul, hreteq, hinnerreseq, hdouble_eq]\nend\n\n/-\n-- Function: ec_mul\n-/\n\n/- ec_mul autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_ec_mul (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (scalar : BigInt3 F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∃ (κ₁ : ℕ) (range_check_ptr₁ : F) (pow2_0 res0 : EcPoint F), spec_ec_mul_inner mem κ₁ range_check_ptr point scalar.d0 86 range_check_ptr₁ pow2_0 res0 ∧\n  ∃ (κ₂ : ℕ) (range_check_ptr₂ : F) (pow2_1 res1 : EcPoint F), spec_ec_mul_inner mem κ₂ range_check_ptr₁ pow2_0 scalar.d1 86 range_check_ptr₂ pow2_1 res1 ∧\n  ∃ (κ₃ : ℕ) (range_check_ptr₃ : F) (ret2_0 res2 : EcPoint F), spec_ec_mul_inner mem κ₃ range_check_ptr₂ pow2_1 scalar.d2 84 range_check_ptr₃ ret2_0 res2 ∧\n  ∃ (κ₄ : ℕ) (range_check_ptr₄ : F) (res : EcPoint F), spec_ec_add mem κ₄ range_check_ptr₃ res0 res1 range_check_ptr₄ res ∧\n  ∃ (κ₅ : ℕ) (range_check_ptr₅ : F) (res₁ : EcPoint F), spec_ec_add mem κ₅ range_check_ptr₄ res res2 range_check_ptr₅ res₁ ∧\n  κ₁ + κ₂ + κ₃ + κ₄ + κ₅ + 71 ≤ κ ∧\n  ρ_range_check_ptr = range_check_ptr₅ ∧\n  ρ_res = res₁\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_ec_mul (mem : F → F) (κ : ℕ) (range_check_ptr : F) (point : EcPoint F) (scalar : BigInt3 F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F) : Prop :=\n  ∀ (secpF : Type) [secp_field secpF], by exactI\n    point.x ≠ ⟨0, 0, 0⟩ →\n    ∀ hpt : BddECPointData secpF point,\n      ∃ n0 < 2^86, scalar.d0 = ↑n0 ∧\n      ∃ n1 < 2^86, scalar.d1 = ↑n1 ∧\n      ∃ n2 < 2^84, scalar.d2 = ↑n2 ∧\n      ∃ hres : BddECPointData secpF ρ_res,\n        hres.toECPoint = (2^172 * n2 + 2^86 * n1 + n0) • hpt.toECPoint\n\n/- ec_mul soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_ec_mul\n    {mem : F → F}\n    (κ : ℕ)\n    (range_check_ptr : F) (point : EcPoint F) (scalar : BigInt3 F) (ρ_range_check_ptr : F) (ρ_res : EcPoint F)\n    (h_auto : auto_spec_ec_mul mem κ range_check_ptr point scalar ρ_range_check_ptr ρ_res) :\n  spec_ec_mul mem κ range_check_ptr point scalar ρ_range_check_ptr ρ_res :=\nbegin\n  intros secpF secp_field ptxnez hpt,\n  resetI,\n  rcases h_auto with ⟨_, _, pow2_0, res0, h0, _, _, pow2_1, res1, h1, _, _, ret2_0, res_2, h2, _, _, res,\n    h3, _, _, res₁, h4, _, _, ret1eq⟩,\n  rcases h0 secpF ptxnez hpt with ⟨nb0, nb0eq, nb0lt, h0aux⟩,\n  have nb0eq' : nb0 = 86,\n  { apply nat.cast_inj_of_lt_char nb0lt, { rw [PRIME.char_eq, PRIME], norm_num1 },\n    rw ←nb0eq, simp },\n  have nb0le : nb0 ≤ SECP_LOG2_BOUND, by { rw [nb0eq', SECP_LOG2_BOUND], norm_num1 },\n  rcases h0aux nb0le with ⟨n0, n0eq, n0lt, hpow20, hpow20ne, hpow20eq, hres0, hres0eq⟩,\n  rcases h1 secpF hpow20ne hpow20 with ⟨nb1, nb1eq, nb1lt, h1aux⟩,\n  have nb1eq' : nb1 = 86,\n  { apply nat.cast_inj_of_lt_char nb1lt, { rw [PRIME.char_eq, PRIME], norm_num1 },\n    rw ←nb1eq, simp },\n  have nb1le : nb1 ≤ SECP_LOG2_BOUND, by { rw [nb1eq', SECP_LOG2_BOUND], norm_num1 },\n  rcases h1aux nb1le with ⟨n1, n1eq, n1lt, hpow21, hpow21ne, hpow21eq, hres1, hres1eq⟩,\n  rcases h2 secpF hpow21ne hpow21 with ⟨nb2, nb2eq, nb2lt, h2aux⟩,\n  have nb2eq' : nb2 = 84,\n  { apply nat.cast_inj_of_lt_char nb2lt, { rw [PRIME.char_eq, PRIME], norm_num1 },\n    rw ←nb2eq, simp },\n  have nb2le : nb2 ≤ SECP_LOG2_BOUND, by { rw [nb2eq', SECP_LOG2_BOUND], norm_num1 },\n  rcases h2aux nb2le with ⟨n2, n2eq, n2lt, hpow22, hpow22ne, hpow22eq, hres2, hres2eq⟩,\n  rcases h3 secpF hres0 hres1 with ⟨hres, hreseq⟩,\n  rcases h4 secpF hres hres2 with ⟨hres2, hreseq2⟩,\n  use n0, split, { rw ←nb0eq', exact n0lt },\n  use n0eq,\n  use n1, split, { rw ←nb1eq', exact n1lt },\n  use n1eq,\n  use n2, split, { rw ←nb2eq', exact n2lt },\n  use n2eq,\n  rw ret1eq,\n  use hres2,\n  rw [hreseq2, hres2eq, hpow21eq, hpow20eq, hreseq, hres1eq, hres0eq, hpow20eq],\n  simp only [smul_smul, ←add_smul, nb0eq', nb1eq', nb2eq', BASE, pow_two],\n  norm_num1,\n  congr' 1,\n  ring\nend\n\ntheorem ec_mul_bound {n0 n1 n2 : ℕ} (n0lt : n0 < 2^86) (n1lt : n1 < 2^86) (n2lt : n2 < 2^84) :\n  2^172 * n2 + 2^86 * n1 + n0 < 2^256 :=\nbegin\n  apply @lt_of_lt_of_le _ _ _ ((2^86)^2 * (2^84 - 1) + 2^86 * (2^86 - 1) + 2^86),\n  apply add_lt_add_of_le_of_lt _ n0lt,\n  apply add_le_add,\n  apply nat.mul_le_mul,\n  norm_num,\n  apply le_tsub_of_add_le_right,\n  apply nat.succ_le_of_lt n2lt,\n  apply nat.mul_le_mul_left,\n  apply le_tsub_of_add_le_right,\n  apply nat.succ_le_of_lt n1lt,\n  norm_num\nend\n\nend starkware.cairo.common.cairo_secp.ec\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/ec_spec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.34853897180404747}}
{"text": "lemma hoge : ∀ (f : bool → bool) (x y z w : bool), (x = f y) → (y = f z) → (z = f w) → x = z := begin\n    intros,\n    cases x,\n    cases y,\n    cases z,\n    cases w,\n    refl,\n    refl,\n    cases w,\n    rewrite a,\n    rewrite a_2,\n    have h: ff = tt, from calc\n        ff = f tt : by rw a_1\n        ... = tt : by rw ←a_2,\n    assumption,\n    cases z,\n    refl,\n    have h: ff = tt, from calc \n        ff = f tt : by rw a\n        ... = tt : by rw ←a_1,\n    assumption,\n    cases z,\n    cases y,\n    have h: tt = ff, from calc\n        tt = f ff : by rw a\n        ... = ff : by rw ←a_1,\n    assumption,\n    cases w,\n    have h: tt = ff, by rw [a_1, ←a_2],\n    assumption,\n    have h: tt = ff, by rw [a, ←a_2],\n    assumption,\n    refl\nend\n\nexample : ∀ (f: bool → bool) (b : bool), f (f (f b)) = f b := begin\n    intros,\n    apply (hoge f (f (f (f b))) (f (f b)) (f b) b);\n    refl\nend", "meta": {"author": "zeptometer", "repo": "LearnLean", "sha": "bb84d5dbe521127ba134d4dbf9559b294a80b9f7", "save_path": "github-repos/lean/zeptometer-LearnLean", "path": "github-repos/lean/zeptometer-LearnLean/LearnLean-bb84d5dbe521127ba134d4dbf9559b294a80b9f7/zeptometer/topprover/4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3484376933599228}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module order.category.omega_complete_partial_order\n! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.OmegaCompletePartialOrder\nimport Mathbin.CategoryTheory.Limits.Shapes.Products\nimport Mathbin.CategoryTheory.Limits.Shapes.Equalizers\nimport Mathbin.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\nimport Mathbin.CategoryTheory.ConcreteCategory.BundledHom\n\n/-!\n# Category of types with a omega complete partial order\n\nIn this file, we bundle the class `omega_complete_partial_order` into a\nconcrete category and prove that continuous functions also form\na `omega_complete_partial_order`.\n\n## Main definitions\n\n * `ωCPO`\n   * an instance of `category` and `concrete_category`\n\n -/\n\n\nopen CategoryTheory\n\nuniverse u v\n\n/-- The category of types with a omega complete partial order. -/\ndef ωCPO : Type (u + 1) :=\n  Bundled OmegaCompletePartialOrder\n#align ωCPO ωCPO\n\nnamespace ωCPO\n\nopen OmegaCompletePartialOrder\n\ninstance : BundledHom @ContinuousHom\n    where\n  toFun := @ContinuousHom.Simps.apply\n  id := @ContinuousHom.id\n  comp := @ContinuousHom.comp\n  hom_ext := @ContinuousHom.coe_inj\n\nderiving instance LargeCategory, ConcreteCategory for ωCPO\n\ninstance : CoeSort ωCPO (Type _) :=\n  Bundled.hasCoeToSort\n\n/-- Construct a bundled ωCPO from the underlying type and typeclass. -/\ndef of (α : Type _) [OmegaCompletePartialOrder α] : ωCPO :=\n  Bundled.of α\n#align ωCPO.of ωCPO.of\n\n@[simp]\ntheorem coe_of (α : Type _) [OmegaCompletePartialOrder α] : ↥(of α) = α :=\n  rfl\n#align ωCPO.coe_of ωCPO.coe_of\n\ninstance : Inhabited ωCPO :=\n  ⟨of PUnit⟩\n\ninstance (α : ωCPO) : OmegaCompletePartialOrder α :=\n  α.str\n\nsection\n\nopen CategoryTheory.Limits\n\nnamespace HasProducts\n\n/-- The pi-type gives a cone for a product. -/\ndef product {J : Type v} (f : J → ωCPO.{v}) : Fan f :=\n  Fan.mk (of (∀ j, f j)) fun j => ContinuousHom.ofMono (Pi.evalOrderHom j) fun c => rfl\n#align ωCPO.has_products.product ωCPO.HasProducts.product\n\n/-- The pi-type is a limit cone for the product. -/\ndef isProduct (J : Type v) (f : J → ωCPO) : IsLimit (product f)\n    where\n  lift s :=\n    ⟨⟨fun t j => s.π.app ⟨j⟩ t, fun x y h j => (s.π.app ⟨j⟩).Monotone h⟩, fun x =>\n      funext fun j => (s.π.app ⟨j⟩).Continuous x⟩\n  uniq s m w := by\n    ext (t j)\n    change m t j = s.π.app ⟨j⟩ t\n    rw [← w ⟨j⟩]\n    rfl\n  fac s j := by\n    cases j\n    tidy\n#align ωCPO.has_products.is_product ωCPO.HasProducts.isProduct\n\ninstance (J : Type v) (f : J → ωCPO.{v}) : HasProduct f :=\n  HasLimit.mk ⟨_, isProduct _ f⟩\n\nend HasProducts\n\ninstance omegaCompletePartialOrderEqualizer {α β : Type _} [OmegaCompletePartialOrder α]\n    [OmegaCompletePartialOrder β] (f g : α →𝒄 β) :\n    OmegaCompletePartialOrder { a : α // f a = g a } :=\n  OmegaCompletePartialOrder.subtype _ fun c hc =>\n    by\n    rw [f.continuous, g.continuous]\n    congr 1\n    ext\n    apply hc _ ⟨_, rfl⟩\n#align ωCPO.omega_complete_partial_order_equalizer ωCPO.omegaCompletePartialOrderEqualizer\n\nnamespace HasEqualizers\n\n/-- The equalizer inclusion function as a `continuous_hom`. -/\ndef equalizerι {α β : Type _} [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β]\n    (f g : α →𝒄 β) : { a : α // f a = g a } →𝒄 α :=\n  ContinuousHom.ofMono (OrderHom.Subtype.val _) fun c => rfl\n#align ωCPO.has_equalizers.equalizer_ι ωCPO.HasEqualizers.equalizerι\n\n/-- A construction of the equalizer fork. -/\ndef equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : Fork f g :=\n  @Fork.ofι _ _ _ _ _ _ (ωCPO.of { a // f a = g a }) (equalizerι f g)\n    (ContinuousHom.ext _ _ fun x => x.2)\n#align ωCPO.has_equalizers.equalizer ωCPO.HasEqualizers.equalizer\n\n/-- The equalizer fork is a limit. -/\ndef isEqualizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : IsLimit (equalizer f g) :=\n  Fork.IsLimit.mk' _ fun s =>\n    ⟨{  toFun := fun x => ⟨s.ι x, by apply continuous_hom.congr_fun s.condition⟩\n        monotone' := fun x y h => s.ι.Monotone h\n        cont := fun x => Subtype.ext (s.ι.Continuous x) },\n      by\n      ext\n      rfl, fun m hm => by\n      ext\n      apply continuous_hom.congr_fun hm⟩\n#align ωCPO.has_equalizers.is_equalizer ωCPO.HasEqualizers.isEqualizer\n\nend HasEqualizers\n\ninstance : HasProducts.{v} ωCPO.{v} := fun J =>\n  { HasLimit := fun F => hasLimitOfIso Discrete.natIsoFunctor.symm }\n\ninstance {X Y : ωCPO.{v}} (f g : X ⟶ Y) : HasLimit (parallelPair f g) :=\n  HasLimit.mk ⟨_, HasEqualizers.isEqualizer f g⟩\n\ninstance : HasEqualizers ωCPO.{v} :=\n  hasEqualizers_of_hasLimit_parallelPair _\n\ninstance : HasLimits ωCPO.{v} :=\n  has_limits_of_hasEqualizers_and_products\n\nend\n\nend ωCPO\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Order/Category/OmegaCompletePartialOrder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.34839763850849137}}
{"text": "\nimport data.finmap\nimport data.quot\nimport tactic.basic\nimport tactic.linarith\n\nuniverses u v\n\nnamespace tactic.interactive\nsetup_tactic_parser\nopen tactic\n\nmeta def check_types : tactic unit :=\n-- meta def check_types (pat : parse texpr) : tactic unit :=\n-- do es ← tactic.match_target pat,\n--    trace es\ndo `(%%x = %%y) ← target,\n   trace!\"{x} : {infer_type x}\",\n   trace!\"{y} : {infer_type y}\"\n\n-- example : 1 = 2 :=\n-- begin\n--   -- check_types _ = _,\n--   -- check_types λ (t : Type u) (x y : t), x = y,\n-- end\n\nend tactic.interactive\n\n\nnamespace sigma\n\nlemma fst_comp_map {α α'} {β : α → Sort*} {β' : α' → Sort*}\n  (f : α → α') (g : ∀ x, β x → β' (f x)) :\n  fst ∘ map f g = f ∘ fst :=\nby ext ⟨a,b⟩; refl\n\nend sigma\n\n@[reducible]\ndef finmap' (α : Type u) (β : Type v) := finmap (λ _ : α, β)\n\nnamespace finmap\n\nvariables {α : Type u}\n  {β : α → Type v}\nvariables\n  [decidable_eq α]\n  [∀ x, decidable_eq $ β x]\n  {m : finmap β}\n\nlemma insert_eq_self_of_lookup_eq {x : α} {y : β x}\n  (h : m.lookup x = some y) : m.insert x y = m :=\nbegin\n  ext1 k, by_cases h' : k = x; [subst h', skip];\n  simp [lookup_insert,*],\nend\n\nlemma insert_lookup_of_eq {x x' : α} {y : β x}\n  (h : x = x') : (m.insert x y).lookup x' = some (h.rec y) :=\nby subst h; rw lookup_insert\n\nprotected def map {γ : α → Type*} (f : ∀ x, β x → γ x) (m : finmap β) : finmap γ :=\n-- list.to_finmap $ list.map _ $ m.to_list\n{ entries := m.entries.map $ sigma.map id f,\n  nodupkeys :=\n  begin\n    induction h : m.entries using quot.induction_on,\n    simp only [multiset.coe_map, multiset.coe_nodupkeys, multiset.quot_mk_to_coe''],\n    rw [list.nodupkeys,list.keys,list.map_map,sigma.fst_comp_map,function.comp.left_id],\n    have := m.nodupkeys, simp [h,list.nodupkeys,list.nodup] at this,\n    exact this,\n  end }\n\nlemma lookup_insert_eq_ite  {x x' : α} {y : β x} :\n  lookup x' (insert x y m) =\n  if h : x' = x then some (h.symm.rec y) else lookup x' m :=\nby split_ifs; [subst h, skip]; simp *\n\nlemma lookup_insert_eq_ite' {β} (m : finmap' α β) {x x' : α} {y : β} :\n  lookup x' (insert x y m) =\n  if x' = x then some y else lookup x' m :=\nby split_ifs; [subst h, skip]; simp *\n\nlemma map_lookup_insert_eq_ite {γ} {x x' : α} {y : β x} (f : ∀ a, β a → γ) :\n  f _ <$> lookup x' (insert x y m) =\n  if h : x' = x then some (f _ y) else f _ <$> lookup x' m :=\nby split_ifs; [subst h, skip]; simp *\n\nvariables {γ : Type u} {F : Type u → Type v}\n  [functor F] [is_lawful_functor F]\nopen function functor\n\nlemma map_injective_of_injective\n  [nonempty α] (f : α → γ)\n  (h : injective f) :\n  injective (map f : F α → F γ) :=\nbegin\n  intros x y h₀,\n  have h₂ := congr_arg (map (inv_fun f)) h₀,\n  simp only [map_map,inv_fun_comp h,map_id,id] at h₂,\n  exact h₂\nend\n\nend finmap\n\n@[derive decidable_eq]\ninductive bdd\n| var : ℕ → bdd\n| nand : bdd → bdd → bdd\n\nabbreviation ptr := unsigned\nopen bdd\n\n@[derive decidable_eq]\ninductive bdd_impl (s : Type) : Type\n| var (v : ℕ) : s → bdd_impl\n| nand : bdd_impl → bdd_impl → s → bdd_impl\n\nvariables {s : Type}\n\ninductive bdd_impl.subtree (t : bdd_impl s) : bdd_impl s → Prop\n| refl : bdd_impl.subtree t\n| step_left (p) (t₀ : bdd_impl s)\n            (t₁ : bdd_impl s) :\n  bdd_impl.subtree t₀ →\n  bdd_impl.subtree (bdd_impl.nand t₀ t₁ p)\n| step_right (p) (t₀ : bdd_impl s)\n             (t₁ : bdd_impl s) :\n  bdd_impl.subtree t₁ →\n  bdd_impl.subtree (bdd_impl.nand t₀ t₁ p)\n\nnamespace bdd_impl.subtree\nopen bdd_impl\nlemma trans {t₀ t₁ : bdd_impl s} (ha : subtree t₀ t₁) : Π {t₂}, subtree t₁ t₂ → subtree t₀ t₂\n| _ (subtree.refl t₁) := ha\n| _ (subtree.step_left  p ta₂ tb₂ ha₂) := subtree.step_left p _ _ (trans ha₂)\n| _ (subtree.step_right p ta₂ tb₂ hb₂) := subtree.step_right p _ _ (trans hb₂)\n\nlemma sizeof_le_sizeof_of_subtree {t₀ : bdd_impl s} : Π {t₁}, subtree t₀ t₁ → sizeof t₀ ≤ sizeof t₁\n| _ (subtree.refl _) := le_refl _\n| _ (subtree.step_left p ta tb h) :=\n  le_trans (sizeof_le_sizeof_of_subtree h)\n    (le_of_lt $ by well_founded_tactics.default_dec_tac)\n| _ (subtree.step_right p ta tb h) :=\n  le_trans (sizeof_le_sizeof_of_subtree h)\n    (le_of_lt $ by well_founded_tactics.default_dec_tac)\n\nlemma antisymm {t₀ t₁ : bdd_impl s} (ha : subtree t₀ t₁) (hb : subtree t₁ t₀) : t₀ = t₁ :=\nbegin\n  induction ha generalizing hb,\n  { refl },\n  all_goals\n  { specialize ha_ih (trans (subtree.step_left _ _ _ $ subtree.refl _) hb) <|>\n      specialize ha_ih (trans (subtree.step_right _ _ _ $ subtree.refl _) hb),\n    cases hb; clear hb,\n    { refl },\n    all_goals\n    { replace ha_a := sizeof_le_sizeof_of_subtree ha_a,\n      replace hb_a := sizeof_le_sizeof_of_subtree hb_a,\n      simp [sizeof,has_sizeof.sizeof,bdd_impl.sizeof] at ha_a hb_a,\n      have : 1 + bdd_impl.sizeof _ ha_t₀ ≤ bdd_impl.sizeof _ ha_t₀,\n      { linarith },\n      apply absurd this, simp } },\nend\n\nlemma subtree_of_subtree_of_to_spec_eq_to_spec {p₀ p₁} {ta tb : bdd_impl s}\n  (h₂ : subtree (bdd_impl.nand ta tb p₀) (bdd_impl.nand ta tb p₁)) :\n  p₀ = p₁ :=\nbegin\n  generalize_hyp h : bdd_impl.nand p₀ ta tb = t at h₂,\n  generalize_hyp h' : bdd_impl.nand p₁ ta tb = t' at h₂,\n  induction t' generalizing p₀ ta tb t,\n  { cases h', },\n  cases h₂,\n  { injection h', injection h, subst_vars },\n  {\n    specialize t'_ih_a _ _ h', },\n  {  },\n  induction h₂,\n  { subst h', injection h, },\n  { apply h₂_ih, },\n  suffices : t = bdd_impl.nand p₁ ta tb,\n  { subst t, injection this },\n  let p := p₁,\n  induction h₂,\n  { refl },\n  -- all_goals\n  { specialize h₂_ih (trans (subtree.step_left _ _ _ $ subtree.refl _) _),\n      -- specialize ha_ih (trans (subtree.step_right _ _ _ $ subtree.refl _) hb),\n    cases hb; clear hb,\n    { refl },\n    all_goals\n    { replace ha_a := sizeof_le_sizeof_of_subtree ha_a,\n      replace hb_a := sizeof_le_sizeof_of_subtree hb_a,\n      simp [sizeof,has_sizeof.sizeof,bdd_impl.sizeof] at ha_a hb_a,\n      have : 1 + bdd_impl.sizeof ha_t₀ ≤ bdd_impl.sizeof ha_t₀,\n      { linarith },\n      apply absurd this, simp } },\n --  -- have h := h₂,\n --  suffices : subtree (bdd_impl.nand p₁ ta tb) (bdd_impl.nand p₀ ta' tb'),\n --  { subst_vars, have := bdd_impl.subtree.antisymm h₂ this,\n --    injection this },\n --  cases h₂,\n --  { exact h₂ },\n --  { subst ta', induction ta,\n --    { cases h₂_a },\n --    {\n --      -- generalize_hyp : (bdd_impl.nand p₀ (bdd_impl.nand ta_p ta_a ta_a_1) tb) at h₂_a,\n --      -- cases h₂_a,\n --      have := subtree.step_left _ _ _ h₂_a,\n --  } },\n --  -- clear h,\n --  { induction ta' generalizing ta tb p₀, cases h₂_a,\n\n --    -- generalize_hyp ha : bdd_impl.nand p₀ (bdd_impl.nand ta_p ta_a ta_a_1) tb = t at h₀_a,\n --    cases h₂_a,\n --    { exact ta'_ih_a rfl h₂ (bdd_impl.subtree.subtree_of_eq h₀.symm) },\n --    { subst ta, specialize ta'_ih_a _ h₂ h₂_a_a, },\n --    rw ← h₀ at h₂_a,\n\n -- },\n --  -- induction h₀,\nend\n\nlemma subtree_of_eq {t₀ t₁ : bdd_impl s} (h : t₀ = t₁) : t₀.subtree t₁ :=\nh ▸ subtree.refl _\n-- lemma ext {t₀ t₁ : bdd_impl} (h : ∀ x, subtree x t₀ ↔ subtree x t₁) : t₀ = t₁\n\nend bdd_impl.subtree\n\ndef bdd_impl.ssubtree (t t' : bdd_impl s) : Prop :=\nt.subtree t' ∧ t ≠ t'\n\nnamespace bdd_impl.ssubtree\nopen bdd_impl\nlemma trans {t₀ t₁ t₂ : bdd_impl s} (ha : ssubtree t₀ t₁) (hb : ssubtree t₁ t₂) : ssubtree t₀ t₂ :=\nhave hc : subtree t₀ t₂, from ha.1.trans hb.1,\n⟨ hc, λ h₁,\n  have hc' : subtree t₂ t₀, from h₁ ▸ subtree.refl _,\n  have ha' : subtree t₁ t₀, from hb.1.trans hc',\n  ha.2 $ ha.1.antisymm ha' ⟩\n\nlemma irrefl (t₀ : bdd_impl s) : ¬ ssubtree t₀ t₀ :=\nλ h, h.2 rfl\n\nlemma antisymm {t₀ t₁ : bdd_impl s} (ha : ssubtree t₀ t₁) (hb : ssubtree t₁ t₀) : false :=\nha.2 $ ha.1.antisymm hb.1\n\nend bdd_impl.ssubtree\n\n\ndef trivial_impl : Π (d : bdd), bdd_impl ℕ\n| (var v) := bdd_impl.var v 0\n| (nand l r) := bdd_impl.nand (trivial_impl l) (trivial_impl r) 0\n\n@[simp]\ndef ptr_addr : bdd_impl s → s\n| (bdd_impl.var _ p) := p\n| (bdd_impl.nand _ _ p) := p\n\n@[simp]\ndef bdd_impl.to_spec : bdd_impl s → bdd\n| (bdd_impl.var v _) := var v\n| (bdd_impl.nand l r _) := nand (bdd_impl.to_spec l) (bdd_impl.to_spec r)\n\n@[simp]\ndef bdd_impl.to_spec_trivial_impl (d : bdd) : (trivial_impl d).to_spec = d :=\nbegin\n  induction d; simp [trivial_impl,bdd_impl.to_spec,*],\nend\n\n-- @[simp]\n-- def bdd_impl.trivial_impl_to_spec (d : bdd_impl) : trivial_impl d.to_spec = d :=\n-- begin\n--   induction d; simp [trivial_impl,bdd_impl.to_spec,*],\n-- end\n\n-- #exit\ndef with_repr {α} (d : bdd) (f : Π {s} (d' : bdd_impl s), d'.to_spec = d → α)\n  (h : ∀ s s' t t' h h', @f s t h = @f s' t' h') : α := f (trivial_impl d) (bdd_impl.to_spec_trivial_impl d)\n\nstructure bdd.package (p : Π {s}, bdd_impl s → Prop) :=\n(val : bdd)\n(repr : trunc (Σ' {s} (t : bdd_impl s), t.to_spec = val ∧ p t))\n-- ← this `trunc` could have a special implementation\n\ndef bdd_impl' (s) (d : bdd) := { t : bdd_impl s // t.to_spec = d }\n\ndef max_sharing (t : bdd_impl s) : Prop :=\n∀ {d} {t' : bdd_impl' s d} {t'' : bdd_impl' s d},\n  t'.1.subtree t →\n  t''.1.subtree t →\n  t' = t''\n\nstructure nondup (s : Type) :=\n(table : finmap' bdd (bdd_impl s))\n(shared : ∀ (t : bdd_impl s) (t' : bdd_impl s),\n  table.lookup t.to_spec = some t →\n  t'.subtree t →\n  table.lookup t'.to_spec = some t')\n(consistent : ∀ {d t}, table.lookup d = some t → t.to_spec = d)\n\ndef intl_bdd (d : bdd) (tbl : nondup s) := { t : bdd_impl s // tbl.table.lookup d = some t }\n\nnamespace nondup\n\nlemma intl_bdd_of_subtree {d : bdd} {tbl : nondup s} (it : intl_bdd d tbl) (t : bdd_impl s)\n  (h : t.subtree it.val) : tbl.table.lookup t.to_spec = some t :=\nbegin\n  apply tbl.shared _ _ _ h,\n  convert it.2, apply tbl.consistent it.2,\nend\n\nvariables (tbl : nondup s)\n\ndef mem (d : bdd) (tbl : nondup s) := d ∈ tbl.table\n\ninstance : has_mem bdd (nondup s) := ⟨ mem ⟩\n\ndef lookup (d : bdd) : option (intl_bdd d tbl) :=\nmatch finmap.lookup d tbl.table, rfl : Π x, finmap.lookup d tbl.table = x → _ with\n| none, h := none\n| (some val), h := some ⟨val, h⟩\nend\n\nset_option pp.proofs true\n\nlemma lookup_iff_map {d : bdd} (x : option (intl_bdd d tbl)) : lookup tbl d = x ↔ tbl.table.lookup d = subtype.val <$> x :=\nbegin\n  rw [lookup],\n  generalize : lookup._proof_1 tbl d = h, revert h,\n  suffices : ∀ (y) (h : finmap.lookup d tbl.table = y),\n    lookup._match_1 tbl d y h = x ↔ y = subtype.val <$> x,\n    from this _,\n  intros, cases y; rw lookup._match_1; cases x; simp,\n  split; intros h; subst h; cases x, refl,\nend\n\ndef bdd_impl'.var (v : ℕ) (p : s) : bdd_impl' s (var v) :=\n⟨ bdd_impl.var v p, rfl ⟩\n\ndef bdd_impl'.nand {d d'} (t : bdd_impl' s d) (t' : bdd_impl' s d') (p : s) : bdd_impl' s (nand d d') :=\n⟨ bdd_impl.nand t.val t'.val p, by simp [t.2,t'.2] ⟩\n\n-- #check subtype.mk.inj_eq\n\nlemma subtree_of_subtree_of_to_spec_eq_to_spec {p₀ p₁} {ta tb ta' tb' : bdd_impl s}\n  (h₀ : ta = ta')\n  (h₁ : tb = tb')\n  (h₂ : subtree (bdd_impl.nand ta tb p₀) (bdd_impl.nand ta' tb' p₁)) :\n  p₀ = p₁ :=\nbegin\n  -- have h := h₂,\n  suffices : subtree (bdd_impl.nand ta tb p₁) (bdd_impl.nand ta' tb' p₀),\n  { subst_vars, have := bdd_impl.subtree.antisymm h₂ this,\n    injection this },\n  cases h₂,\n  { exact h₂ },\n  { subst ta', induction ta,\n    { cases h₂_a },\n    {\n      -- generalize_hyp : (bdd_impl.nand p₀ (bdd_impl.nand ta_p ta_a ta_a_1) tb) at h₂_a,\n      -- cases h₂_a,\n      have := subtree.step_left _ _ _ h₂_a,\n  } },\n  -- clear h,\n  { induction ta' generalizing ta tb p₀, cases h₂_a,\n\n    -- generalize_hyp ha : bdd_impl.nand p₀ (bdd_impl.nand ta_p ta_a ta_a_1) tb = t at h₀_a,\n    cases h₂_a,\n    { exact ta'_ih_a rfl h₂ (bdd_impl.subtree.subtree_of_eq h₀.symm) },\n    { subst ta, specialize ta'_ih_a _ h₂ h₂_a_a, },\n    rw ← h₀ at h₂_a,\n\n },\n  -- induction h₀,\nend\n\nlemma subtree_of_subtree_of_to_spec_eq_to_spec' {t₀ t₁ : bdd_impl s}\n  (h₀ : subtree t₀ t₁)\n  (h₁ : t₀.to_spec = t₁.to_spec) :\n  t₀ = t₁ :=\nbegin\n  induction t₁ generalizing t₀,\n  { cases h₀, refl },\n  { cases t₀, cases h₁,\n    rw [bdd_impl.to_spec,bdd_impl.to_spec] at h₁,\n    injection h₁, specialize t₁_ih_a _ h_1,\n    specialize t₁_ih_a_1 _ h_2,\n }\nend\n\n-- lemma boo (t t' : bdd_impl s)\n--     finmap.lookup (bdd_impl.to_spec t) (finmap.insert (bdd_impl.to_spec d) d tbl.table) = some t →\n--     subtree t' t → finmap.lookup (bdd_impl.to_spec t') (finmap.insert (bdd_impl.to_spec d) d tbl.table) = some t'\n\nlemma lookup_insert {s : Type}\n  (tbl : nondup s)\n  (d : bdd_impl s)\n  (hd : ∀ (d' : bdd_impl s),\n          subtree d' d →\n          (d' ≠ d ↔\n             finmap.lookup (bdd_impl.to_spec d') tbl.table = some d'))\n  (t t' : bdd_impl s)\n  (h₀ : finmap.lookup (bdd_impl.to_spec t)\n            (tbl.table.insert (bdd_impl.to_spec d) d ) =\n          some t)\n  (h₁ : subtree t' t) :\n  finmap.lookup (bdd_impl.to_spec t')\n      (tbl.table.insert (bdd_impl.to_spec d) d) =\n    some t' :=\nbegin\n  classical,\n  by_cases bdd_impl.to_spec d = bdd_impl.to_spec t,\n  { rw [h,finmap.lookup_insert] at h₀,\n    injection h₀, subst d, clear h h₀,\n    have hd' := hd _ h₁,\n    by_cases t' = t,\n    { rw [h,finmap.lookup_insert] },\n    { have h' := hd'.mp h,\n      rwa [finmap.lookup_insert_eq_ite',if_neg],\n      intro hh,\n      -- specialize hd _ (subtree.refl _),\n      replace hd := mt (hd t (subtree.refl _)).mpr (not_not_intro rfl),\n      rw [← hh,h'] at hd,\n      apply h,\n      admit } },\n  { have hd' := mt (hd d (subtree.refl _)).mpr (not_not_intro rfl),\n    rw finmap.lookup_insert_of_ne at ⊢ h₀,\n    apply tbl.shared _ _ h₀ h₁,\n    { intro h, admit },\n    { intro h, rw ← h at hd', }, }\nend\n\nlemma insert_consistent {s : Type} {d_1 : bdd}\n  (tbl : nondup s)\n  (d : bdd_impl s)\n  (hd : ∀ (d' : bdd_impl s),\n          subtree d' d →\n          (d' ≠ d ↔\n             finmap.lookup (bdd_impl.to_spec d') tbl.table = some d'))\n  {t : bdd_impl s}\n  (h₀ : finmap.lookup d_1 (tbl.table.insert (bdd_impl.to_spec d) d) = some t) :\n  bdd_impl.to_spec t = d_1 :=\nbegin\n  rw finmap.lookup_insert_eq_ite' at h₀, split_ifs at h₀,\n  { cc },\n  { exact tbl.consistent h₀, }\nend\n\ndef insert (d : bdd_impl s) (p : s)\n  -- (h : d.to_spec ∉ s.table )\n  (hd : ∀ d', bdd_impl.subtree d' d → (d' ≠ d ↔ tbl.table.lookup d'.to_spec = some d')) : nondup s :=\n⟨ tbl.table.insert d.to_spec d,\nlookup_insert _ _ hd, λ _ _, insert_consistent tbl d hd ⟩\n\n-- def insert_nand (d : bdd_impl s) (p : s)\n--   -- (h : d.to_spec ∉ s.table )\n--   (hd : ∀ d', bdd_impl.subtree d' d → (d' ≠ d ↔ tbl.table.lookup d'.to_spec = some d')) : nondup s :=\n\n\n-- def insert' {v : ℕ} (p : s)\n--   (h : var v ∉ tbl.table ) : nondup s :=\n-- ⟨ tbl.table.insert (var v) (bdd_impl.var v p), _, _ ⟩\n\ndef insert_var {v : ℕ} (p : s)\n  (h : var v ∉ tbl) : nondup s :=\n⟨ tbl.table.insert (var v) (bdd_impl.var v p),\n  begin\n    introv ht hh,\n    replace h : ∀ tt, ¬ tbl.table.lookup (var v) = some tt,\n    { introv ht', apply h, change _ ∈ tbl.table,\n      rw [finmap.mem_iff], exact ⟨_,ht'⟩ },\n    rw finmap.lookup_insert_eq_ite', split_ifs with h₀,\n    -- rw finmap.lookup_insert_eq_ite' at ht, split_ifs at ht,\n    classical,\n    by_cases h₁ : t = t',\n    { rw [h₁,h₀,finmap.lookup_insert] at ht,\n      injection ht, },\n    { rw finmap.lookup_insert_eq_ite' at ht, split_ifs at ht,\n      { admit },\n      -- have h' := h (bdd_impl.var v p),\n      { specialize h t', rw ← h₀ at h,\n        have hh' := tbl.shared _ _ ht hh,\n        exact absurd hh' h },\n      { intro h₂, rw [h₂,finmap.lookup_insert] at ht, }},\n    -- { rw finmap.lookup_insert_of_ne at ht,\n    --   -- have h' := h (bdd_impl.var v p),\n    --   { specialize h t', rw ← h₀ at h,\n    --     have hh' := tbl.shared _ _ ht hh,\n    --     exact absurd hh' h },\n    --   { intro h₂, rw [h₂,finmap.lookup_insert] at ht, }},\n  end, sorry ⟩\n\ndef insert_nand {d d' : bdd} (t : intl_bdd d tbl) (t' : intl_bdd d' tbl) (p : s)\n  (h : nand d d' ∉ tbl) : nondup s :=\n⟨ tbl.table.insert (nand d d') (bdd_impl.nand t.1 t'.1 p), _, _ ⟩\n\n-- def insert {d d' : bdd} (t : intl_bdd d tbl) (t' : intl_bdd d' tbl) (p : ptr)\n--   (h : nand d d' ∉ tbl.table ) : nondup s :=\n-- ⟨ tbl.table.insert (nand d d') (bdd_impl.nand t.1 t'.1 p),\n-- begin\n--   intros d₀ t₀ d₁ t₁ h₀ h₁,\n--   rw finmap.lookup_insert_eq_ite at h₀, split_ifs at h₀ with h',\n--   { subst h', dsimp [] at h₀,\n--     cases h₀, cases t₁, dsimp at h₁,\n--     cases h₁,\n--     { have : d₁ = (nand d d'),\n--       { rw [← t₁_property,bdd_impl.to_spec,t.val.property,t'.val.property], },\n--       subst this, rw [finmap.lookup_insert], refl },\n--     { have : d₁ ≠ (nand d d'),\n--       { intro h', subst h', apply h, -- rw ← t₁_property,\n--         rw finmap.mem_iff, split,\n--         apply s.inv t.val ⟨t₁_val,t₁_property⟩ t.property h₁_a, },\n--       rw [finmap.lookup_insert_of_ne _ this],\n--       apply s.inv t.val ⟨t₁_val,t₁_property⟩ t.property h₁_a },\n--     { have : d₁ ≠ (nand d d'),\n--       { intro h', subst h', apply h, -- rw ← t₁_property,\n--         rw finmap.mem_iff, split,\n--         apply s.inv t'.val ⟨t₁_val,t₁_property⟩ t'.property h₁_a, },\n--       rw [finmap.lookup_insert_of_ne _ this],\n--       apply s.inv t'.val ⟨t₁_val,t₁_property⟩ t'.property h₁_a } },\n--   { have := s.inv t₀ t₁ h₀ h₁,\n--     rwa finmap.lookup_insert_of_ne,\n--     intro h₂, apply h, rw ← h₂, simp [finmap.mem_iff],\n--     exact ⟨_,this⟩ }\n-- end ⟩\n\n-- def insert_nand_others {d d'} (t : bdd_impl' d) (p : s) (h) (d₂)\n--   (x : intl_bdd d₂ tbl) : intl_bdd d₂ (tbl.insert t p h) :=\n-- ⟨ x.1, begin\n--          dsimp [insert]; rw [finmap.lookup_insert_of_ne],\n--          { apply x.property },\n--          { intro h', rw ← h' at h, apply h,\n--            rw finmap.mem_iff, exact ⟨_,x.2⟩ }\n--        end  ⟩\n\ndef intl_var {d d'} (t : intl_bdd d s) (t' : intl_bdd d' s) (p : ptr) (h) :\n  intl_bdd (nand d d') (s.insert t t' p h) :=\n\ndef intl_nand {d d'} (t : intl_bdd d s) (t' : intl_bdd d' s) (p : ptr) (h) :\n  intl_bdd (nand d d') (s.insert t t' p h) :=\n⟨ bdd_impl'.nand t.1 t'.1 p, by rw [insert,finmap.lookup_insert] ⟩\n\ndef M (s d) :=\nΣ' (s' : nondup) (x : intl_bdd d s'), ∀ d', intl_bdd d' s → intl_bdd d' s'\n\ndef id' {s} : ∀ d', intl_bdd d' s → intl_bdd d' s := λ _, id\n\ndef M.pure {d} (x : intl_bdd d s) : M s d :=\n⟨s, x, id'⟩\n\nsection\nvariable {s}\n\ndef M.bind {d d'} (x : M s d) (f : ∀ s', intl_bdd d s' → (∀ d, intl_bdd d s → intl_bdd d s') → M s' d') : M s d' :=\nmatch x with\n| ⟨s',b,g⟩ :=\n  match f s' b g with\n  | ⟨s'',c,h⟩ := ⟨s'',c,λ _, h _ ∘ g _⟩\n  end\nend\n-- #exit\ntheorem M.pure_bind {d} (x : intl_bdd d s)\n  (f : ∀ s', intl_bdd d s' → (∀ d, intl_bdd d s → intl_bdd d s') → M s' d) :\n  M.bind (M.pure s x) f = f _ x (λ _, id) :=\nbegin\n  rw [M.pure,M.bind,M.bind._match_1],\n  rcases f s x id' with ⟨a,b,c⟩, refl\nend\n\ntheorem M.bind_pure {d} (x : M s d) :\n  M.bind x (λ s' a f, M.pure _ a) = x :=\nbegin\n  rcases x with ⟨a,b,c⟩, refl,\nend\n\ntheorem M.bind_bind {d d' d''} (x : M s d)\n  (f : ∀ s', intl_bdd d s' → (∀ d, intl_bdd d s → intl_bdd d s') → M s' d')\n  (g : ∀ s', intl_bdd d' s' → (∀ d, intl_bdd d s → intl_bdd d s') → M s' d'') :\n  M.bind (M.bind x f) g = M.bind x (λ s' a h, M.bind (f s' a h) (λ s'' b h', g _ b (λ _, h' _ ∘ h _))) :=\nbegin\n  rcases x with ⟨a,b,c⟩,\n  simp [M.bind,M.bind._match_1],\n  rcases (f a b c) with ⟨a₁,b₁,c₁⟩,\n  simp [M.bind,M.bind._match_1],\n  rcases (g a₁ b₁ (λ _, c₁ _ ∘ c _)) with ⟨a₂,b₂,c₂⟩,\n  refl\nend\n\nend\n\ndef mk_var (p : ptr) (v : ℕ) : M s (var v) :=\nmatch lookup s (var v), rfl : ∀ x, lookup s (var v) = x → _ with\n| none, h := _\n| (some val), h := ⟨s,val,λ _, id⟩\nend\n\ndef mk_nand (p : ptr) : Π {d}, intl_bdd d s → Π {d'}, intl_bdd d' s → M s (nand d d')\n| d t d' t' :=\n  match lookup s (nand d d'), rfl : ∀ x, lookup s (nand d d') = x → _ with\n  | none, h :=\n    have nand d d' ∉ s.table, by { simp [lookup_iff_map] at h, simp [finmap.mem_iff,h], },\n    ⟨_, insert_nand s t t' p this, insert_nand_others _ _ _ _ _⟩\n  | (some val), h := ⟨_,val,λ _, id⟩\n  end\n\ndef mk_nand' (p : ptr) {d d'} (t : intl_bdd d s) (t' : intl_bdd d' s) : M s (nand d d') :=\nmk_nand s p t t'\n\nend nondup\nopen nondup\n\ndef share_common : Π (d : bdd_impl) (s : nondup), M s d.to_spec\n| (bdd_impl.var p v) s := ⟨_, _⟩\n| (bdd_impl.nand p l r) s :=\nM.bind (share_common l s) $ λ s' l' f₀,\nM.bind (share_common r s') $ λ s'' r' f₁,\nnondup.mk_nand' s'' p (f₁ _ l') r'\n", "meta": {"author": "cipher1024", "repo": "max-sharing", "sha": "6e21b2ebaeaf69ee8e9cc075ace993e758b37520", "save_path": "github-repos/lean/cipher1024-max-sharing", "path": "github-repos/lean/cipher1024-max-sharing/max-sharing-6e21b2ebaeaf69ee8e9cc075ace993e758b37520/src/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.34839763850849137}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport order.category.BoundedLattice\nimport order.hom.complete_lattice\n\n/-!\n# The category of complete lattices\n\nThis file defines `CompleteLattice`, the category of complete lattices.\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- The category of complete lattices. -/\ndef CompleteLattice := bundled complete_lattice\n\nnamespace CompleteLattice\n\ninstance : has_coe_to_sort CompleteLattice Type* := bundled.has_coe_to_sort\ninstance (X : CompleteLattice) : complete_lattice X := X.str\n\n/-- Construct a bundled `CompleteLattice` from a `complete_lattice`. -/\ndef of (α : Type*) [complete_lattice α] : CompleteLattice := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [complete_lattice α] : ↥(of α) = α := rfl\n\ninstance : inhabited CompleteLattice := ⟨of punit⟩\n\ninstance : bundled_hom @complete_lattice_hom :=\n{ to_fun := λ _ _ _ _, coe_fn,\n  id := @complete_lattice_hom.id,\n  comp := @complete_lattice_hom.comp,\n  hom_ext := λ X Y _ _, by exactI fun_like.coe_injective }\ninstance : large_category.{u} CompleteLattice := bundled_hom.category complete_lattice_hom\ninstance : concrete_category CompleteLattice := bundled_hom.concrete_category complete_lattice_hom\n\ninstance has_forget_to_BoundedLattice : has_forget₂ CompleteLattice BoundedLattice :=\n{ forget₂ := { obj := λ X, BoundedLattice.of X,\n               map := λ X Y, complete_lattice_hom.to_bounded_lattice_hom },\n  forget_comp := rfl }\n\n/-- Constructs an isomorphism of complete lattices from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : CompleteLattice.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : CompleteLattice ⥤ CompleteLattice :=\n{ obj := λ X, of (order_dual X), map := λ X Y, complete_lattice_hom.dual }\n\n/-- The equivalence between `CompleteLattice` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : CompleteLattice ≌ CompleteLattice :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nend CompleteLattice\n\nlemma CompleteLattice_dual_comp_forget_to_BoundedLattice :\n  CompleteLattice.dual ⋙ forget₂ CompleteLattice BoundedLattice =\n    forget₂ CompleteLattice BoundedLattice ⋙ BoundedLattice.dual := rfl\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/CompleteLattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.3482890393314951}}
{"text": "import category_theory.preadditive\nimport category_theory.abelian.exact\nimport algebra.homology.exact\nimport category_theory.limits.preserves.shapes.terminal\nimport category_theory.limits.shapes.zero_morphisms\n\nnamespace category_theory\nnamespace limits\n\nopen category_theory.limits\n\nvariables {C : Type*} [category C] [has_zero_morphisms C]\n\nopen_locale zero_object\n\n\nlemma is_zero_iff_id_eq_zero {X : C} : is_zero X ↔ 𝟙 X = 0 :=\nbegin\n  split,\n  { exact λ h, h.eq_of_src _ _, },\n  { intro e, split; intro Y; use 0; intro f,\n    { rw [← cancel_epi (𝟙 _), e, comp_zero, zero_comp], apply_instance },\n    { rw [← cancel_mono (𝟙 _), e, comp_zero, zero_comp], apply_instance }, }\nend\n\nlemma is_zero_of_mono {X Y : C} (f : X ⟶ Y) [mono f] (h : is_zero Y) : is_zero X :=\nby rw [is_zero_iff_id_eq_zero, ← cancel_mono f, zero_comp, h.eq_of_tgt (𝟙 _ ≫ f)]\n\nlemma is_zero_of_epi {X Y : C} (f : X ⟶ Y) [epi f] (h : is_zero X) : is_zero Y :=\nby rw [is_zero_iff_id_eq_zero, ← cancel_epi f, comp_zero, h.eq_of_src (f ≫ 𝟙 Y)]\n\nlemma is_zero_of_top_le_bot [has_zero_object C] (X : C)\n  (h : (⊤ : subobject X) ≤ ⊥) : is_zero X :=\n{ unique_to := λ Y,\n  begin\n    use 0, intro f,\n    rw [← cancel_epi ((⊤ : subobject X).arrow), ← subobject.of_le_arrow h],\n    simp only [subobject.bot_arrow, comp_zero, zero_comp],\n  end,\n  unique_from := λ Y,\n  begin\n    use 0, intro f,\n    rw ← subobject.bot_factors_iff_zero,\n    exact subobject.factors_of_le f h (subobject.top_factors f),\n  end }\n\n-- inline this\nlemma is_zero_of_iso_of_zero {C : Type*} [category C] [has_zero_morphisms C]\n  {X : C} (hX : is_zero X) {Y : C} (h : X ≅ Y) : is_zero Y :=\nhX.of_iso h.symm\n\nlemma is_zero_of_exact_zero_zero {C : Type*} [category C] [abelian C]\n  {X Y Z : C} (h : exact (0 : X ⟶ Y) (0 : Y ⟶ Z)) : is_zero Y :=\nis_zero_of_top_le_bot _\nbegin\n  rw abelian.exact_iff_image_eq_kernel at h,\n  rw [← @kernel_subobject_zero _ _ _ Y Z, ← @image_subobject_zero _ _ _ _ X Y, h],\nend\n\nlemma exact_of_is_zero {C : Type*} [category C] [abelian C]\n  {X Y Z : C} (hY : is_zero Y) (f : X ⟶ Y) (g : Y ⟶ Z) : exact f g :=\nby simp only [abelian.exact_iff, is_zero.eq_zero_of_tgt hY f,\n  is_zero.eq_zero_of_tgt hY (kernel.ι g), zero_comp, eq_self_iff_true, and_self]\n\nlemma is_zero_iff_exact_zero_zero {C : Type*} [category C] [abelian C]\n  {X Y Z : C} : is_zero Y ↔ exact (0 : X ⟶ Y) (0 : Y ⟶ Z) :=\n⟨λ h, exact_of_is_zero h 0 0, is_zero_of_exact_zero_zero⟩\n\nlemma is_zero_of_exact_zero_zero' {C : Type*} [category C] [abelian C]\n  {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (h : exact f g) (hf : f = 0) (hg : g = 0) : is_zero Y :=\nby { rw [hf, hg] at h, exact is_zero_of_exact_zero_zero h }\n\nlemma is_zero_of_exact_is_zero_is_zero {C : Type*} [category C] [abelian C] {X Y Z : C}\n  (f : X ⟶ Y) (g : Y ⟶ Z) (h : exact f g) (hX : is_zero X) (hZ : is_zero Z) : is_zero Y :=\nis_zero_of_exact_zero_zero' f g h (hX.eq_of_src f _) (hZ.eq_of_tgt g _)\n\nlemma is_zero_cokernel_of_epi {C : Type*} [category C] [abelian C] {X Y : C}\n  (f : X ⟶ Y) [epi f] : is_zero (cokernel f) :=\nbegin\n  have h1 : cokernel.π f = 0, by rwa ← abelian.epi_iff_cokernel_π_eq_zero,\n  have h2 : exact (cokernel.π f) 0 := category_theory.exact_epi_zero (cokernel.π f),\n  exact is_zero_of_exact_zero_zero' (cokernel.π f) 0 h2 h1 rfl,\nend\n\nlemma epi_iff_is_zero_cokernel {C : Type*} [category C] [abelian C] {X Y : C}\n  (f : X ⟶ Y) : epi f ↔ is_zero (cokernel f) :=\nbegin\n  split,\n  { introsI, apply is_zero_cokernel_of_epi },\n  { intros h,\n    rw abelian.epi_iff_cokernel_π_eq_zero,\n    apply h.eq_of_tgt }\nend\n\nlemma is_zero_kernel_of_mono {C : Type*} [category C] [abelian C] {X Y : C}\n  (f : X ⟶ Y) [mono f] : is_zero (kernel f) :=\nbegin\n  have h1 : kernel.ι f = 0, by rwa ← abelian.mono_iff_kernel_ι_eq_zero,\n  have h2 : exact 0 (kernel.ι f) := category_theory.exact_zero_mono (kernel.ι f),\n  exact is_zero_of_exact_zero_zero' 0 (kernel.ι f) h2 rfl h1\nend\n\nlemma mono_iff_is_zero_kernel {C : Type*} [category C] [abelian C] {X Y : C}\n  (f : X ⟶ Y) : mono f ↔ is_zero (kernel f) :=\nbegin\n  split,\n  { introsI, apply is_zero_kernel_of_mono },\n  { intros h,\n    rw abelian.mono_iff_kernel_ι_eq_zero,\n    apply h.eq_of_src }\nend\n\nnoncomputable def image_iso_of_eq [category C] [abelian C] {A B : C} {f f' : A ⟶ B} (h : f = f') : image f ≅ image f' :=\neq_to_iso (by rw h)\n\nnoncomputable def image.is_iso_comp {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {A B C : 𝓐} {f : A ⟶ B} [is_iso f] (g : B ⟶ C) : image (f ≫ g) ≅ image g :=\n{ hom := image.lift (({ I := _,\n  m := image.ι _,\n  m_mono := infer_instance,\n  e := f ≫ factor_thru_image g,\n  fac' := by simp only [category.assoc, image.fac]} : mono_factorisation _)),\n  inv := image.lift (({ I := _,\n  m := image.ι _,\n  m_mono := infer_instance,\n  e := (inv f) ≫ factor_thru_image (f ≫ g),\n  fac' := by simp only [category.assoc, image.fac, is_iso.inv_hom_id_assoc]} : mono_factorisation _)) }\n\nlemma is_iso_of_is_zero_of_is_zero {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {a b : 𝓐} (ha : is_zero a) (hb : is_zero b)\n  (f : a ⟶ b) : is_iso f :=\nbegin\n  rw is_zero.eq_zero_of_src ha f,\n  apply (is_iso_zero_equiv a b).symm.to_fun,\n  exact ⟨is_zero.eq_of_src ha (𝟙 a) 0, is_zero.eq_of_src hb (𝟙 b) 0⟩,\nend\n\nlemma obj_is_zero_of_iso {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {𝓑 : Type*} [category 𝓑] [abelian 𝓑] {F G : 𝓐 ⥤ 𝓑}\n  (h : F ≅ G) {a : 𝓐} (ha : is_zero (F.obj a)) : is_zero (G.obj a) :=\nis_zero_of_iso_of_zero ha (h.app a)\n\nlemma map_is_iso_of_iso_of_map_is_iso {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {𝓑 : Type*} [category 𝓑] [abelian 𝓑] {F G : 𝓐 ⥤ 𝓑}\n  (h : F ≅ G) {a₁ a₂ : 𝓐} (f : a₁ ⟶ a₂) (ha : is_iso (F.map f)) : is_iso (G.map f) :=\nbegin\n  rw ← nat_iso.naturality_1 h,\n  exact is_iso.comp_is_iso,\nend\n\n@[simp] lemma epi_comp_iso_iff_epi {V : Type*} [category V] {A B C : V} (e : A ≅ B) (f : B ⟶ C) :\n  epi (e.hom ≫ f) ↔ epi f :=\nbegin\n  split,\n  { rintro ⟨h⟩,\n    constructor,\n    intros Z s t h2,\n    apply h,\n    simp [h2], },\n  { rintro ⟨h⟩,\n    constructor,\n    intros Z s t h2,\n    apply h,\n    simpa using h2,\n  },\nend\n\n@[simp] lemma epi_iso_comp_iff_epi {V : Type*} [category V] {A B C : V} (f : A ⟶ B) (e : B ≅ C) :\n  epi (f ≫ e.hom) ↔ epi f :=\nbegin\n  split,\n  { introI h,\n    constructor,\n    intros Z s t h2,\n    suffices : e.inv ≫ s = e.inv ≫ t,\n      simpa,\n    rw ← cancel_epi (f ≫ e.hom),\n    simpa using h2, },\n  { introI h,\n    constructor,\n    intros Z s t h2,\n    simp only [category.assoc] at h2,\n    rw cancel_epi at h2,\n    rwa cancel_epi at h2, },\nend\n\nlemma is_iso_iff_is_iso_comp_left {V : Type*} [category V] {A B C : V} (f : A ⟶ B) {e : B ⟶ C}\n  [is_iso f] : is_iso (f ≫ e) ↔ is_iso e :=\nbegin\n  split,\n  { introI h, exact is_iso.of_is_iso_comp_left f e },\n  { introI h, exact is_iso.comp_is_iso },\nend\n\nlemma is_iso_iff_is_iso_comp_right {V : Type*} [category V] {A B C : V} {f : A ⟶ B} (g : B ⟶ C)\n  [is_iso g] : is_iso (f ≫ g) ↔ is_iso f :=\nbegin\n  split,\n  { introI, exact is_iso.of_is_iso_comp_right f g},\n  { introI h, exact is_iso_of_op (f ≫ g), },\nend\n\n@[simp] lemma epi_comp_is_iso_iff_epi {V : Type*} [category V] {A B C : V} (e : A ⟶ B) (f : B ⟶ C)\n  [is_iso e] : epi (e ≫ f) ↔ epi f :=\nepi_comp_iso_iff_epi (as_iso e) f\n\n@[simp] lemma epi_is_iso_comp_iff_epi {V : Type*} [category V] {A B C : V} (f : A ⟶ B) (e : B ⟶ C)\n  [is_iso e] : epi (f ≫ e) ↔ epi f :=\nepi_iso_comp_iff_epi f (as_iso e)\n\nlemma zero_of_epi_comp_zero {V : Type*} [category V] [abelian V]\n  {A B C : V} {f : A ⟶ B} {g : B ⟶ C} (w : f ≫ g = 0) [epi f] : g = 0 :=\n(preadditive.epi_iff_cancel_zero f).mp infer_instance C g w\n\n@[simp] lemma comp_mono_zero_iff {V : Type*} [category V] [abelian V]\n  {A B C : V} {f : A ⟶ B} {g : B ⟶ C} [mono g] : f ≫ g = 0 ↔ f = 0 :=\n⟨(preadditive.mono_iff_cancel_zero g).1 infer_instance A f, λ f, f.symm ▸ zero_comp⟩\n\nlemma epi_of_epi_of_comp_epi_of_mono {V : Type*} [category V] [abelian V]\n  {A B C : V} (f : A ⟶ B) (g : B ⟶ C) [epi (f ≫ g)] [mono g] : epi f :=\nbegin\n  haveI foo : is_iso g,\n  { rw is_iso_iff_mono_and_epi,\n    refine ⟨infer_instance, _⟩,\n    apply epi_of_epi f,\n  },\n  simp * at *,\nend\n\nlemma is_zero_initial {C : Type*} [category C] [abelian C] : is_zero (⊥_ C) :=\nis_zero_of_iso_of_zero (is_zero_zero _) $\n{ hom := 0,\n  inv := 0 }\n\nlemma is_zero_terminal {C : Type*} [category C] [abelian C] : is_zero (⊤_ C) :=\nis_zero_of_iso_of_zero (is_zero_zero _) $\n{ hom := 0,\n  inv := 0 }\n\nuniverses v u₁ u₂\n\nclass preserves_zero_objects {C D : Type*} [category C] [has_zero_morphisms C]\n  [category D] [has_zero_morphisms D] (F : C ⥤ D) : Prop :=\n(preserves : ∀ (X : C), is_zero X → is_zero (F.obj X))\n\ninstance preserves_zero_of_preserves_initial {C : Type u₁} {D : Type u₂}\n  [category.{v} C] [abelian C] [category.{v} D] [abelian D] (F : C ⥤ D)\n  [preserves_colimit (functor.empty C) F] :\n  preserves_zero_objects F := preserves_zero_objects.mk $ λ X hX,\nbegin\n  have e : X ≅ ⊥_ _ := hX.iso is_zero_initial,\n  replace e : F.obj X ≅ F.obj ⊥_ _ := F.map_iso e,\n  apply is_zero_of_iso_of_zero _ e.symm,\n  have : F.obj ⊥_ _ ≅ ⊥_ _,\n  { apply_with limits.preserves_initial.iso { instances := ff }, assumption },\n  apply is_zero_of_iso_of_zero _ this.symm,\n  exact is_zero_initial,\nend\n\n-- sanity check\nexample {C : Type u₁} {D : Type u₂}\n  [category.{v} C] [abelian C] [category.{v} D] [abelian D] (F : C ⥤ D)\n  [preserves_colimits F] : preserves_zero_objects F := infer_instance\n\ninstance preserves_zero_of_preserves_terminal {C : Type u₁} {D : Type u₂}\n  [category.{v} C] [abelian C] [category.{v} D] [abelian D] (F : C ⥤ D)\n  [preserves_limit (functor.empty C) F] :\n  preserves_zero_objects F := preserves_zero_objects.mk $ λ X hX,\nbegin\n  have e : X ≅ ⊤_ _ := hX.iso is_zero_terminal,\n  replace e : F.obj X ≅ F.obj ⊤_ _ := F.map_iso e,\n  apply is_zero_of_iso_of_zero _ e.symm,\n  have : F.obj ⊤_ _ ≅ ⊤_ _,\n  { apply_with limits.preserves_terminal.iso { instances := ff }, assumption },\n  apply is_zero_of_iso_of_zero _ this.symm,\n  exact is_zero_terminal,\nend\n\n-- sanity check\nexample {C : Type u₁} {D : Type u₂}\n  [category.{v} C] [abelian C] [category.{v} D] [abelian D] (F : C ⥤ D)\n  [preserves_limits F] : preserves_zero_objects F := infer_instance\n\nlemma is_zero_of_preserves {C D : Type*} [category C] [has_zero_morphisms C]\n  [category D] [has_zero_morphisms D] {X : C} (F : C ⥤ D)\n  [preserves_zero_objects F] (e : is_zero X) : is_zero (F.obj X) :=\npreserves_zero_objects.preserves _ e\n\nlemma is_zero_biprod {C : Type u₁} [category.{v} C] [abelian C] (X Y : C)\n  (hX : is_zero X) (hY : is_zero Y) : is_zero (biprod X Y) :=\nbegin\n  constructor,\n  { intro W, use 0, intro f, ext, simp, apply hX.eq_of_src, simp, apply hY.eq_of_src },\n  { intro W, use 0, intro f, ext, simp, apply hX.eq_of_tgt, simp, apply hY.eq_of_tgt }\nend\n\nend limits\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/abelian_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984445, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3482890317963834}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff, hα_irrat, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from\n  begin\n    assume (i j : ℤ) (h1 : i ≠ j),\n    assume h2 : int.fract (α * ↑i) = int.fract (α * ↑j),\n    have h3 : α * ↑i - int.floor (α * ↑i) = int.fract (α * ↑i), from by auto [int.fract_eq_iff_of_nonneg],\n    have h4 : α * ↑j - int.floor (α * ↑j) = int.fract (α * ↑j), from by auto [int.fract_eq_iff_of_nonneg],\n    have h5 : α = (int.floor (α * ↑i) - int.floor (α * ↑j))/(i - j), from by auto [h2, h3, h4],\n    have h6 : (i - j) ≠ 0, from by auto [h1],\n    have h7 : (int.floor (α * ↑i) - int.floor (α * ↑j))/(i - j) ∈ ℚ, from by auto [h5, h6],\n    show false, from by auto [hα_irrat, h7],\n  end,\n\n  have h2 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h1],\n\n  have h3 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h2],\n\n  have h4 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h3],\n\n  have h5 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h4],\n\n  have h6 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h5],\n\n  have h7 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h6],\n\n  have h8 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h7],\n\n  have h9 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h8],\n\n  have h10 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h9],\n\n  have h11 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h10],\n\n  have h12 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h11],\n\n  have h13 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h12],\n\n  have h14 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h13],\n\n  have h15 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h14],\n\n  have h16 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h15],\n\n  have h17 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h16],\n\n  have h18 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h17],\n\n  have h19 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h18],\n\n  have h20 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h19],\n\n  have h21 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h20],\n\n  have h22 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h21],\n\n  have h23 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h22],\n\n  have h24 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h23],\n\n  have h25 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h24],\n\n  have h26 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h25],\n\n  have h27 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h26],\n\n  have h28 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h27],\n\n  have h29 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h28],\n\n  have h30 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h29],\n\n  have h31 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h30],\n\n  have h32 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from by auto [h31],\n\n  have h33 : ∀ (i j : ℤ), (i ≠ j) →\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h2 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h3 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h4 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h5 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h6 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h7 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h8 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h9 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h10 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h11 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h12 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h13 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h14 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h15 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h16 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h17 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h18 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h19 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h20 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h21 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h22 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of_mul_int_eq_int_add_one_mul_fract, hα_irrat],\n\n  have h23 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from by auto [int.fract_eq_iff_eq_of\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  assume (h7 : ε > 0),\n\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from by auto [h8, h10, h5, h9],\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from by auto [h11] using [linarith],\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-4_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.3482518224727491}}
{"text": "inductive nested : Type\n| nest : list nested → nested\n\nnamespace nested\n\n@[elab_as_eliminator] protected def cases_on' :\n  Π {C : nested → Sort*} (x : nested),\n    (Π (a : list nested), C (nest a)) → C x\n| C (nest L) ih := ih L\n\n@[simp] lemma cases_on'_def {C : nested → Sort*}\n  (L : list nested) (ih : Π (a : list nested), C (nest a)) :\n  (nested.cases_on' (nest L) ih : C (nest L)) = ih L :=\nnested.cases_on'.equations._eqn_1 C L ih\n\ndef mem : nested → nested → Prop\n| x (nest L) := x ∈ L\n\ninstance : has_mem nested nested := ⟨mem⟩\n\ndef to_list : nested → list nested\n| (nest L) := L\n\n@[simp] lemma to_list_def (L : list nested) :\n  to_list (nest L) = L :=\nnested.to_list.equations._eqn_1 L\n\n@[simp] lemma mem_def (x : nested) (L : list nested) :\n  x ∈ (nest L) ↔ x ∈ L :=\nshow mem _ _ ↔ _, by rw nested.mem.equations._eqn_1\n\ntheorem mem_sizeof (x y : nested) (H : x ∈ y) :\n  sizeof x < sizeof y :=\nbegin\n  cases y with L,\n  change sizeof x < nested.sizeof (nest L),\n  rw nested.nest.sizeof_spec L,\n  rw mem_def at H,\n  induction L with hd tl ih,\n  { exfalso, simp at H, assumption },\n  change sizeof x < 1 + (1 + sizeof hd + sizeof tl),\n  cases H with H H,\n  { subst H, rw add_comm,\n    apply nat.lt_succ_of_le,\n    rw [add_comm 1, add_assoc],\n    apply nat.le_add_right },\n  specialize ih H,\n  apply lt_trans ih,\n  apply nat.add_lt_add_left,\n  rw [add_comm, add_comm 1],\n  apply nat.lt_succ_of_le,\n  apply nat.le_add_right\nend\n\ntheorem mem_wf : well_founded (@has_mem.mem nested _ _) :=\n@subrelation.wf _ (inv_image nat.lt sizeof) has_mem.mem mem_sizeof $\ninv_image.wf _ nat.lt_wf\n\n@[elab_as_eliminator] protected def rec_on\n  {C : nested → Sort*} (x : nested)\n  (ih : ∀ L : list nested, (∀ z ∈ L, C z) → C (nest L)) : C x :=\nwell_founded.fix mem_wf\n  (λ y, nested.cases_on' y $ λ L ih', ih L $ λ z h, ih' z $ (mem_def z L).2 h) _\n\n@[simp] lemma rec_on_def\n  {C : nested → Sort*} (L : list nested)\n  (ih : ∀ L : list nested, (∀ z ∈ L, C z) → C (nest L)) :\n  (nested.rec_on (nest L) ih : C (nest L))\n  = ih _ (λ z HL, nested.rec_on z ih) :=\nby unfold nested.rec_on; rw well_founded.fix_eq; simp\n\ninstance : decidable_eq nested :=\nλ x, nested.rec_on x $ λ L1,\nlist.rec_on L1 (λ _ y, nested.cases_on' y $ λ L2,\n    list.cases_on L2 (is_true rfl) (λ hd tl, is_false $ λ H, by injections)) $ λ hd1 tl1 ih1 ih2 y,\nnested.rec_on y $ λ L2,\nlist.rec_on L2 (λ _, is_false $ λ H, by injections) $ λ hd2 tl2 ih3 ih4,\n@decidable_of_decidable_of_iff (hd1 = hd2 ∧ nest tl1 = nest tl2) _\n  (@@and.decidable (ih2 _ (or.inl rfl) _) (ih1 (λ z H b, ih2 _ (or.inr H) _) _)) $\n⟨λ H, congr_arg nest (by simp [H.1, nest.inj H.2] : hd1 :: tl1 = hd2 :: tl2),\nλ H, by split; injections; simp*⟩\n\ninstance mem.decidable_rel :\n  decidable_rel (@has_mem.mem nested _ _)\n| x (nest L) := decidable_of_decidable_of_iff (list.decidable_mem x L) (mem_def x L).symm\n\nend nested\n", "meta": {"author": "kckennylau", "repo": "Lean", "sha": "907d0a4d2bd8f23785abd6142ad53d308c54fdcb", "save_path": "github-repos/lean/kckennylau-Lean", "path": "github-repos/lean/kckennylau-Lean/Lean-907d0a4d2bd8f23785abd6142ad53d308c54fdcb/nested.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3481511269961152}}
{"text": "import .temporal\n\nuniverses u\n\nnamespace temporal\nnamespace classical\n\nlemma not_always_not_implies_eventually {T : Type u} (P : tProp T)\n  : ⊩ tNot (□ (tNot P)) => ◇ P\n:= begin\nintros tr notalways,\napply classical.by_contradiction,\nintros contra, apply notalways,\nintros n contra', apply contra,\nconstructor, assumption\nend\n\nend classical\nend temporal", "meta": {"author": "GaloisInc", "repo": "lean-protocol-support", "sha": "cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda", "save_path": "github-repos/lean/GaloisInc-lean-protocol-support", "path": "github-repos/lean/GaloisInc-lean-protocol-support/lean-protocol-support-cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda/galois/temporal/classical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.34807281827269315}}
{"text": "/-\nCopyright (c) 2018 Michael Jendrusch. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Jendrusch, Scott Morrison\n-/\nimport category_theory.monoidal.of_chosen_finite_products\nimport category_theory.limits.shapes.types\nimport logic.equiv.fin\n\n/-!\n# The category of types is a symmetric monoidal category\n-/\n\nopen category_theory\nopen category_theory.limits\nopen tactic\n\nuniverses v u\n\nnamespace category_theory\n\ninstance types_monoidal : monoidal_category.{u} (Type u) :=\nmonoidal_of_chosen_finite_products (types.terminal_limit_cone) (types.binary_product_limit_cone)\n\ninstance types_symmetric : symmetric_category.{u} (Type u) :=\nsymmetric_of_chosen_finite_products (types.terminal_limit_cone) (types.binary_product_limit_cone)\n\n@[simp] lemma tensor_apply {W X Y Z : Type u} (f : W ⟶ X) (g : Y ⟶ Z) (p : W ⊗ Y) :\n  (f ⊗ g) p = (f p.1, g p.2) := rfl\n\n@[simp] lemma left_unitor_hom_apply {X : Type u} {x : X} {p : punit} :\n  ((λ_ X).hom : (𝟙_ (Type u)) ⊗ X → X) (p, x) = x := rfl\n@[simp] lemma left_unitor_inv_apply {X : Type u} {x : X} :\n  ((λ_ X).inv : X ⟶ (𝟙_ (Type u)) ⊗ X) x = (punit.star, x) := rfl\n\n@[simp] lemma right_unitor_hom_apply {X : Type u} {x : X} {p : punit} :\n  ((ρ_ X).hom : X ⊗ (𝟙_ (Type u)) → X) (x, p) = x := rfl\n@[simp] lemma right_unitor_inv_apply {X : Type u} {x : X} :\n  ((ρ_ X).inv : X ⟶ X ⊗ (𝟙_ (Type u))) x = (x, punit.star) := rfl\n\n@[simp] lemma associator_hom_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :\n  ((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z)) ((x, y), z) = (x, (y, z)) := rfl\n@[simp] lemma associator_inv_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :\n  ((α_ X Y Z).inv : X ⊗ (Y ⊗ Z) → (X ⊗ Y) ⊗ Z) (x, (y, z)) = ((x, y), z) := rfl\n\n@[simp] lemma braiding_hom_apply {X Y : Type u} {x : X} {y : Y} :\n  ((β_ X Y).hom : X ⊗ Y → Y ⊗ X) (x, y) = (y, x) := rfl\n@[simp] lemma braiding_inv_apply {X Y : Type u} {x : X} {y : Y} :\n  ((β_ X Y).inv : Y ⊗ X → X ⊗ Y) (y, x) = (x, y) := rfl\n\nopen opposite\n\nopen monoidal_category\n\n/-- `(𝟙_ C ⟶ -)` is a lax monoidal functor to `Type`. -/\ndef coyoneda_tensor_unit (C : Type u) [category.{v} C] [monoidal_category C] :\n  lax_monoidal_functor C (Type v) :=\n{ ε := λ p, 𝟙 _,\n  μ := λ X Y p, (λ_ (𝟙_ C)).inv ≫ (p.1 ⊗ p.2),\n  μ_natural' := by tidy,\n  associativity' := λ X Y Z, begin\n    ext ⟨⟨f, g⟩, h⟩, dsimp at f g h,\n    dsimp, simp only [iso.cancel_iso_inv_left, category.assoc],\n    conv_lhs { rw [←category.id_comp h, tensor_comp, category.assoc, associator_naturality,\n      ←category.assoc, unitors_inv_equal, triangle_assoc_comp_right_inv], },\n    conv_rhs { rw [←category.id_comp f, tensor_comp], },\n  end,\n  left_unitality' := by tidy,\n  right_unitality' := λ X, begin\n    ext ⟨f, ⟨⟩⟩, dsimp at f,\n    dsimp, simp only [category.assoc],\n    rw [right_unitor_naturality, unitors_inv_equal, iso.inv_hom_id_assoc],\n  end,\n  ..coyoneda.obj (op (𝟙_ C)) }.\n\nnoncomputable theory\n\n/-- If `F` is a monoidal functor out of `Type`, it takes the (n+1)st cartesian power\nof a type to the image of that type, tensored with the image of the nth cartesian power. -/\n-- We don't yet have an API for tensor products indexed by finite ordered types,\n-- but it would be nice to state how monoidal functors preserve these.\ndef monoidal_functor.map_pi {C : Type*} [category C] [monoidal_category C]\n  (F : monoidal_functor Type* C) (n : ℕ) (β : Type*) :\n  F.obj (fin (n+1) → β) ≅ F.obj β ⊗ F.obj (fin n → β) :=\nfunctor.map_iso _ (equiv.pi_fin_succ n β).to_iso ≪≫ (as_iso (F.μ β (fin n → β))).symm\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/types.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.34786381905871405}}
{"text": "inductive Bit where\n  | zero\n  | one\n\ninstance inst0 : OfNat Bit 0 where\n  ofNat := Bit.zero\n\ninstance : OfNat Bit 1 where\n  ofNat := Bit.one\n\nexample : Bit := 0\nexample : Bit := 1\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1389.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34777769092573024}}
{"text": "import SciLean.Core.Differential\nimport SciLean.Core.Adjoint\n\nnamespace SciLean\n\nvariable {α β γ : Type}\nvariable {X Y Z W : Type} [SemiHilbert X] [SemiHilbert Y] [SemiHilbert Z] [SemiHilbert W]\nvariable {Y₁ Y₂ Y₃ : Type} [SemiHilbert Y₁] [SemiHilbert Y₂] [SemiHilbert Y₃]\nvariable {ι : Type} [Enumtype ι]\n\n\n-- /-- Transitive closure of `HasAdjDiffN`\n-- -/\n-- class HasAdjDiffNT {X Y : Type} {Xs Y' : Type} [SemiHilbert Xs] [SemiHilbert Y']\n--   (n : Nat) (f : X → Y) [Prod.Uncurry n (X → Y) Xs Y'] : Prop where\n--   proof : IsSmoothNT n f ∧ ∀ x, HasAdjointT (∂ (uncurryN n f) x)\n\n-- class HasAdjDiffN {X Y : Type} {Xs Y' : Type} [SemiHilbert Xs] [SemiHilbert Y']\n--   (n : Nat) (f : X → Y) [Prod.Uncurry n (X → Y) Xs Y'] extends HasAdjDiffNT n f : Prop\n\n-- abbrev HasAdjDiffT {X Y : Type} [SemiHilbert X] [SemiHilbert Y] (f : X → Y) := HasAdjDiffNT 1 f\n-- abbrev HasAdjDiff {X Y : Type} [SemiHilbert X] [SemiHilbert Y] (f : X → Y) := HasAdjDiffN 1 f\n\n-- class HasAdjDiff (f : X → Y) : Prop where\n--   isSmooth : IsSmooth f\n--   hasAdjDiff : ∀ x, HasAdjoint $ ∂ f x\ninstance (priority:=low-10) {X Y : Type} {Xs Y' : Type} [SemiHilbert Xs] [SemiHilbert Y']\n  (n : Nat) (f : X → Y) [Prod.Uncurry n (X → Y) Xs Y']\n  [sf : IsSmoothN n f] [∀ x, HasAdjoint (∂ (uncurryN n f) x)]\n  : HasAdjDiffN n f\n  := HasAdjDiffN.mk (toHasAdjDiffNT:= by constructor; apply sf.1; infer_instance)\n\ntheorem infer_HasAdjDiff {X Y : Type} {Xs Y' : Type} [SemiHilbert Xs] [SemiHilbert Y']\n  {n : Nat} {f : X → Y} [Prod.Uncurry n (X → Y) Xs Y'] [IsSmoothT (uncurryN n f)]\n  : (∀ x, HasAdjointT $ ∂ (uncurryN n f) x) → HasAdjDiffNT n f\n  := λ h => by constructor; infer_instance; apply h\n\ntheorem infer_HasAdjDiff' {X Y : Type} {Xs Y' : Type} [SemiHilbert Xs] [SemiHilbert Y']\n  {n : Nat} {f : X → Y} [Prod.Uncurry n (X → Y) Xs Y'] [IsSmoothT (uncurryN n f)]\n  : (∀ x, HasAdjointT $ ∂ (uncurryN n f) x) → HasAdjDiffN n f\n  := λ h => sorry\n\n-- instance (priority := low-9) HasAdjoint_HasAdjDiff (f : X → Y) [HasAdjoint f] : HasAdjDiff f := \n-- by apply infer_HasAdjDiff'; symdiff; infer_instance; done\n\n--------------------------------------------------------------------------------\n\n\n--- Removing arguments - generalized this \n\ninstance HasAdjDiff2_apply_2 (f : X → Y → Z) [HasAdjDiffNT 2 f] (y : Y)\n  : HasAdjDiffT (λ x => f x y) := sorry_proof\n\ninstance HasAdjDiff2_apply_1 (f : X → Y → Z) [HasAdjDiffNT 2 f] (x : X)\n  : HasAdjDiffT (λ y => f x y) := sorry_proof\n\ninstance (f : X → Y → Z → W) [HasAdjDiffNT 3 f] (y z)\n  : HasAdjDiffT (λ x => f x y z) := sorry_proof\n\ninstance (f : X → Y → Z → W) [HasAdjDiffNT 3 f] (x z)\n  : HasAdjDiffT (λ y => f x y z) := sorry_proof\n\ninstance (f : X → Y → Z → W) [HasAdjDiffNT 3 f] (x y)\n  : HasAdjDiffT (λ z => f x y z) := sorry_proof\n\n\n--- Adding arguments - generalized this \n\ninstance HasAdjDiff_add_extra_2_1 (f : X → Y) [hf : HasAdjDiffT f]\n  : HasAdjDiffNT 2 (λ (z : Z) x => f x) := sorry_proof\n\ninstance HasAdjDiff_add_extra_2_2 (f : X → Y) [HasAdjDiffT f]\n  : HasAdjDiffNT 2 (λ x (z : Z) => f x) := sorry_proof\n\ninstance HasAdjDiff_add_extra_3_1 (f : Y → Z → W) [HasAdjDiffNT 2 f]\n  : HasAdjDiffNT 3 (λ (x : X) y z => f y z) := sorry_proof\n\ninstance HasAdjDiff_add_extra_3_2 (f : X → Z → W) [HasAdjDiffNT 2 f]\n  : HasAdjDiffNT 3 (λ x (y : Y) z => f x z) := sorry_proof\n\ninstance HasAdjDiff_add_extra_3_3 (f : X → Y → W) [HasAdjDiffNT 2 f]\n  : HasAdjDiffNT 3 (λ x y (z : Z) => f x y) := sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n\ninstance id.arg_x.hasAdjDiff\n  : HasAdjDiffT (λ x : X => x) := by apply infer_HasAdjDiff; intro; simp[uncurryN, Prod.Uncurry.uncurry]; infer_instance\n\ninstance const.arg_x.hasAdjDiff\n  : HasAdjDiffT (λ (x : X) (i : ι) => x) := by apply infer_HasAdjDiff; intro; simp[uncurryN, Prod.Uncurry.uncurry]; infer_instance; done\n\ninstance const.arg_y.hasAdjDiff (x : X)\n  : HasAdjDiffT (λ (y : Y) => x) := by apply infer_HasAdjDiff; intro; simp[uncurryN, Prod.Uncurry.uncurry]; infer_instance\n\ninstance (priority := low) swap.arg_y.hasAdjDiff\n  (f : ι → Y → Z) [inst : ∀ x, HasAdjDiffT (f x)]\n  : HasAdjDiffT (λ y x => f x y) :=\nby\n  have is := λ x => (inst x).1\n  have ia := λ x => (inst x).2\n  apply infer_HasAdjDiff; intro; \n  simp[uncurryN, Prod.Uncurry.uncurry]; infer_instance; done\n\n\ninstance (priority := mid-1) subst.arg_x.hasAdjDiff \n  (f : X → Y → Z) [instf : HasAdjDiffNT 2 f]\n  (g : X → Y) [instg : HasAdjDiffT g] \n  : HasAdjDiffT (λ x => f x (g x)) := \nby\n  have isf := instf.1\n  have iaf := instf.2\n  have isg := instg.1\n  have iag := instg.2\n  \n  have : ∀ x, IsSmoothT (f x) := sorry_proof\n  have : IsSmoothT (λ x => λ y ⟿ f x y) := sorry_proof\n\n  sorry_proof \n  -- apply infer_HasAdjDiff; intro x; \n  -- simp[uncurryN, Prod.Uncurry.uncurry, tangentMap]; \n  -- -- Thise should follow from `iaf`\n  -- have : ∀ x y, HasAdjointT λ dx => ∂ f x dx y := sorry_proof\n  -- have : ∀ x y, HasAdjointT λ dy => ∂ (f x) y dy := sorry_proof\n  -- infer_instance; done\n\ninstance (priority := mid-1) subst2.arg_x.hasAdjDiff \n  (f : X → Y → Y₁ → Z) [HasAdjDiffNT 3 f]\n  (g : X → Y → Y₁) [HasAdjDiffNT 2 g] :\n  HasAdjDiffNT 2 (λ x y => f x y (g x y)) := sorry_proof\n\ninstance (priority := mid-1) subst3.arg_x.hasAdjDiff \n  (f : X → Y → Z → Y₁ → W) [HasAdjDiffNT 4 f]\n  (g : X → Y → Z → Y₁) [HasAdjDiffNT 3 g] :\n  HasAdjDiffNT 3 (λ x y z => f x y z (g x y z)) := sorry_proof\n\n\ninstance comp.arg_x.hasAdjDiff\n  (f : Y → Z) [instf : HasAdjDiffT f] \n  (g : X → Y) [instg : HasAdjDiffT g]\n  : HasAdjDiffT (λ x => f (g x)) := by infer_instance \n\ninstance {Ws W' : Type} [SemiHilbert Ws] [SemiHilbert W']\n  (f : Z → W) [Prod.Uncurry n W Ws W'] [HasAdjDiffNT (n+1) f]\n  (g : X → Y → Z) [HasAdjDiffNT 2 g]\n  : HasAdjDiffNT (n+2) fun x y => f (g x y) := sorry_proof\n\ninstance {Ws W' : Type} [SemiHilbert Ws] [SemiHilbert W']\n  (f : Y₁ → Y₂→ W) [Prod.Uncurry n W Ws W'] [HasAdjDiffNT (n+2) f]\n  (g₁ : X → Y → Z → Y₁) [HasAdjDiffNT 3 g₁]\n  (g₂ : X → Y → Z → Y₂) [HasAdjDiffNT 3 g₂]\n  : HasAdjDiffNT (n+3) fun x y z => f (g₁ x y z) (g₂ x y z) := sorry_proof\n\ninstance comp2.arg_x.HasAdjDiff\n  (f : Y₁ → Y₂ → Z) [HasAdjDiffNT 2 f]\n  (g₁ : X → Y → Y₁) [HasAdjDiffNT 2 g₁]\n  (g₂ : X → Y → Y₂) [HasAdjDiffNT 2 g₂]\n  : HasAdjDiffNT 2 (λ x y => f (g₁ x y) (g₂ x y)) := \nby\n  have : HasAdjDiffNT 3 fun x y => f (g₁ x y) := by infer_instance\n  infer_instance \n\ninstance comp3.arg_x.HasAdjDiff \n  (f : Y₁ → Y₂ → Y₃ → W) [HasAdjDiffNT 3 f]\n  (g₁ : X → Y → Z → Y₁) [HasAdjDiffNT 3 g₁]\n  (g₂ : X → Y → Z → Y₂) [HasAdjDiffNT 3 g₂]\n  (g₃ : X → Y → Z → Y₃) [HasAdjDiffNT 3 g₃]\n  : HasAdjDiffNT 3 (λ x y z => f (g₁ x y z) (g₂ x y z) (g₃ x y z)) := \nby\n  -- have : HasAdjDiffNT 4 fun x y z => f (g₁ x y z) (g₂ x y z) := by apply hoho\n  infer_instance\n\ninstance diag.arg_x.hasAdjDiff\n  (f : Y₁ → Y₂ → Z) [instf : HasAdjDiffNT 2 f] \n  (g₁ : X → Y₁) [instg1 : HasAdjDiffT g₁]\n  (g₂ : X → Y₂) [instg2 : HasAdjDiffT g₂]\n  : HasAdjDiffT (λ x => f (g₁ x) (g₂ x)) := by infer_instance\n\ninstance eval.arg_x.parm1.hasAdjDiff\n  (f : X → ι → Z) [inst : HasAdjDiffT f] (i : ι)\n  : HasAdjDiffT (λ x => f x i) := \n  by\n    have := inst.1\n    have := inst.2\n\n    apply infer_HasAdjDiff; intro; simp[uncurryN, Prod.Uncurry.uncurry]; infer_instance\n\n----------------------------------------------------------------------\n\ninstance comp.arg_x.parm1.hasAdjDiff\n  (a : α)\n  (f : Y → α → Z) [HasAdjDiffT λ y => f y a]\n  (g : X → Y) [HasAdjDiffT g]\n  : HasAdjDiffT λ x => f (g x) a := \n  by \n    apply comp.arg_x.hasAdjDiff (λ y => f y a) g\n    done\n\ninstance diag.arg_x.parm1.hasAdjDiff\n  (a : α)\n  (f : Y₁ → Y₂ → α → Z) [HasAdjDiffNT 2 λ y₁ y₂ => f y₁ y₂ a]\n  (g₁ : X → Y₁) [HasAdjDiffT g₁] \n  (g₂ : X → Y₂) [HasAdjDiffT g₂]\n  : HasAdjDiffT λ x => f (g₁ x) (g₂ x) a := \n  by \n    apply diag.arg_x.hasAdjDiff (λ y₁ y₂ => f y₁ y₂ a) g₁ g₂\n    done\n\n\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Core/HasAdjDiff.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34777769092573024}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.braided\nimport category_theory.monoidal.discrete\nimport category_theory.monoidal.coherence_lemmas\nimport category_theory.limits.shapes.terminal\nimport algebra.punit_instances\n\n/-!\n# The category of monoids in a monoidal category.\n\nWe define monoids in a monoidal category `C` and show that the category of monoids is equivalent to\nthe category of lax monoidal functors from the unit monoidal category to `C`.  We also show that if\n`C` is braided, then the category of monoids is naturally monoidal.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂ u\n\nopen category_theory\nopen category_theory.monoidal_category\n\nvariables (C : Type u₁) [category.{v₁} C] [monoidal_category.{v₁} C]\n\n/--\nA monoid object internal to a monoidal category.\n\nWhen the monoidal category is preadditive, this is also sometimes called an \"algebra object\".\n-/\nstructure Mon_ :=\n(X : C)\n(one : 𝟙_ C ⟶ X)\n(mul : X ⊗ X ⟶ X)\n(one_mul' : (one ⊗ 𝟙 X) ≫ mul = (λ_ X).hom . obviously)\n(mul_one' : (𝟙 X ⊗ one) ≫ mul = (ρ_ X).hom . obviously)\n-- Obviously there is some flexibility stating this axiom.\n-- This one has left- and right-hand sides matching the statement of `monoid.mul_assoc`,\n-- and chooses to place the associator on the right-hand side.\n-- The heuristic is that unitors and associators \"don't have much weight\".\n(mul_assoc' : (mul ⊗ 𝟙 X) ≫ mul = (α_ X X X).hom ≫ (𝟙 X ⊗ mul) ≫ mul . obviously)\n\nrestate_axiom Mon_.one_mul'\nrestate_axiom Mon_.mul_one'\nrestate_axiom Mon_.mul_assoc'\nattribute [reassoc] Mon_.one_mul Mon_.mul_one -- We prove a more general `@[simp]` lemma below.\nattribute [simp, reassoc] Mon_.mul_assoc\n\nnamespace Mon_\n\n/--\nThe trivial monoid object. We later show this is initial in `Mon_ C`.\n-/\n@[simps]\ndef trivial : Mon_ C :=\n{ X := 𝟙_ C,\n  one := 𝟙 _,\n  mul := (λ_ _).hom,\n  mul_assoc' := by coherence,\n  mul_one' := by coherence }\n\ninstance : inhabited (Mon_ C) := ⟨trivial C⟩\n\nvariables {C} {M : Mon_ C}\n\n@[simp] lemma one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f :=\nby rw [←id_tensor_comp_tensor_id, category.assoc, M.one_mul, left_unitor_naturality]\n\n@[simp] lemma mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f :=\nby rw [←tensor_id_comp_id_tensor, category.assoc, M.mul_one, right_unitor_naturality]\n\nlemma assoc_flip : (𝟙 M.X ⊗ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ⊗ 𝟙 M.X) ≫ M.mul :=\nby simp\n\n/-- A morphism of monoid objects. -/\n@[ext]\nstructure hom (M N : Mon_ C) :=\n(hom : M.X ⟶ N.X)\n(one_hom' : M.one ≫ hom = N.one . obviously)\n(mul_hom' : M.mul ≫ hom = (hom ⊗ hom) ≫ N.mul . obviously)\n\nrestate_axiom hom.one_hom'\nrestate_axiom hom.mul_hom'\nattribute [simp, reassoc] hom.one_hom hom.mul_hom\n\n/-- The identity morphism on a monoid object. -/\n@[simps]\ndef id (M : Mon_ C) : hom M M :=\n{ hom := 𝟙 M.X, }\n\ninstance hom_inhabited (M : Mon_ C) : inhabited (hom M M) := ⟨id M⟩\n\n/-- Composition of morphisms of monoid objects. -/\n@[simps]\ndef comp {M N O : Mon_ C} (f : hom M N) (g : hom N O) : hom M O :=\n{ hom := f.hom ≫ g.hom, }\n\ninstance : category (Mon_ C) :=\n{ hom := λ M N, hom M N,\n  id := id,\n  comp := λ M N O f g, comp f g, }\n\n@[simp] lemma id_hom' (M : Mon_ C) : (𝟙 M : hom M M).hom = 𝟙 M.X := rfl\n@[simp] lemma comp_hom' {M N K : Mon_ C} (f : M ⟶ N) (g : N ⟶ K) :\n  (f ≫ g : hom M K).hom = f.hom ≫ g.hom := rfl\n\nsection\nvariables (C)\n\n/-- The forgetful functor from monoid objects to the ambient category. -/\n@[simps]\ndef forget : Mon_ C ⥤ C :=\n{ obj := λ A, A.X,\n  map := λ A B f, f.hom, }\n\nend\n\ninstance forget_faithful : faithful (@forget C _ _) := { }\n\ninstance {A B : Mon_ C} (f : A ⟶ B) [e : is_iso ((forget C).map f)] : is_iso f.hom := e\n\n/-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/\ninstance : reflects_isomorphisms (forget C) :=\n{ reflects := λ X Y f e, by exactI ⟨⟨\n{ hom := inv f.hom,\n  mul_hom' :=\n  begin\n    simp only [is_iso.comp_inv_eq, hom.mul_hom, category.assoc, ←tensor_comp_assoc,\n      is_iso.inv_hom_id, tensor_id, category.id_comp],\n  end }, by tidy⟩⟩ }\n\n/--\nConstruct an isomorphism of monoids by giving an isomorphism between the underlying objects\nand checking compatibility with unit and multiplication only in the forward direction.\n-/\ndef iso_of_iso {M N : Mon_ C}\n  (f : M.X ≅ N.X)\n  (one_f : M.one ≫ f.hom = N.one)\n  (mul_f : M.mul ≫ f.hom = (f.hom ⊗ f.hom) ≫ N.mul) :\n  M ≅ N :=\n{ hom := { hom := f.hom, one_hom' := one_f, mul_hom' := mul_f },\n  inv :=\n  { hom := f.inv,\n    one_hom' := by { rw ←one_f, simp },\n    mul_hom' :=\n    begin\n      rw ←(cancel_mono f.hom),\n      slice_rhs 2 3 { rw mul_f },\n      simp,\n    end } }\n\ninstance unique_hom_from_trivial (A : Mon_ C) : unique (trivial C ⟶ A) :=\n{ default :=\n  { hom := A.one,\n    one_hom' := by { dsimp, simp, },\n    mul_hom' := by { dsimp, simp [A.one_mul, unitors_equal], } },\n  uniq := λ f,\n  begin\n    ext, simp,\n    rw [←category.id_comp f.hom],\n    erw f.one_hom,\n  end }\n\nopen category_theory.limits\n\ninstance : has_initial (Mon_ C) :=\nhas_initial_of_unique (trivial C)\n\nend Mon_\n\nnamespace category_theory.lax_monoidal_functor\n\nvariables {C} {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D]\n\n/--\nA lax monoidal functor takes monoid objects to monoid objects.\n\nThat is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.\n-/\n-- TODO: map_Mod F A : Mod A ⥤ Mod (F.map_Mon A)\n@[simps]\ndef map_Mon (F : lax_monoidal_functor C D) : Mon_ C ⥤ Mon_ D :=\n{ obj := λ A,\n  { X := F.obj A.X,\n    one := F.ε ≫ F.map A.one,\n    mul := F.μ _ _ ≫ F.map A.mul,\n    one_mul' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.one_mul], },\n      rw [F.to_functor.map_id],\n      rw [F.left_unitality],\n    end,\n    mul_one' :=\n    begin\n      conv_lhs { rw [id_tensor_comp, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.mul_one], },\n      rw [F.to_functor.map_id],\n      rw [F.right_unitality],\n    end,\n    mul_assoc' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.mul_assoc], },\n      conv_lhs { rw [F.to_functor.map_id] },\n      conv_lhs { rw [F.to_functor.map_comp, F.to_functor.map_comp] },\n      conv_rhs { rw [id_tensor_comp, ←F.to_functor.map_id], },\n      slice_rhs 3 4 { rw [F.μ_natural], },\n      conv_rhs { rw [F.to_functor.map_id] },\n      slice_rhs 1 3 { rw [←F.associativity], },\n      simp only [category.assoc],\n    end, },\n  map := λ A B f,\n  { hom := F.map f.hom,\n    one_hom' := by { dsimp, rw [category.assoc, ←F.to_functor.map_comp, f.one_hom], },\n    mul_hom' :=\n    begin\n      dsimp,\n      rw [category.assoc, F.μ_natural_assoc, ←F.to_functor.map_comp, ←F.to_functor.map_comp,\n        f.mul_hom],\n    end },\n  map_id' := λ A, by { ext, simp, },\n  map_comp' := λ A B C f g, by { ext, simp, }, }\n\nvariables (C D)\n\n/-- `map_Mon` is functorial in the lax monoidal functor. -/\ndef map_Mon_functor : (lax_monoidal_functor C D) ⥤ (Mon_ C ⥤ Mon_ D) :=\n{ obj := map_Mon,\n  map := λ F G α,\n  { app := λ A,\n    { hom := α.app A.X, } } }\n\nend category_theory.lax_monoidal_functor\n\nnamespace Mon_\n\nopen category_theory.lax_monoidal_functor\n\nnamespace equiv_lax_monoidal_functor_punit\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef lax_monoidal_to_Mon : lax_monoidal_functor (discrete punit.{u+1}) C ⥤ Mon_ C :=\n{ obj := λ F, (F.map_Mon : Mon_ _ ⥤ Mon_ C).obj (trivial (discrete punit)),\n  map := λ F G α, ((map_Mon_functor (discrete punit) C).map α).app _ }\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef Mon_to_lax_monoidal : Mon_ C ⥤ lax_monoidal_functor (discrete punit.{u+1}) C :=\n{ obj := λ A,\n  { obj := λ _, A.X,\n    map := λ _ _ _, 𝟙 _,\n    ε := A.one,\n    μ := λ _ _, A.mul,\n    map_id' := λ _, rfl,\n    map_comp' := λ _ _ _ _ _, (category.id_comp (𝟙 A.X)).symm, },\n  map := λ A B f,\n  { app := λ _, f.hom,\n    naturality' := λ _ _ _, by { dsimp, rw [category.id_comp, category.comp_id], },\n    unit' := f.one_hom,\n    tensor' := λ _ _, f.mul_hom, }, }\n\nlocal attribute [tidy] tactic.discrete_cases\nlocal attribute [simp] eq_to_iso_map\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef unit_iso :\n  𝟭 (lax_monoidal_functor (discrete punit.{u+1}) C) ≅\n    lax_monoidal_to_Mon C ⋙ Mon_to_lax_monoidal C :=\nnat_iso.of_components (λ F,\n  monoidal_nat_iso.of_components\n    (λ _, F.to_functor.map_iso (eq_to_iso (by ext)))\n    (by tidy) (by tidy) (by tidy))\n  (by tidy)\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef counit_iso : Mon_to_lax_monoidal C ⋙ lax_monoidal_to_Mon C ≅ 𝟭 (Mon_ C) :=\nnat_iso.of_components (λ F, { hom := { hom := 𝟙 _, }, inv := { hom := 𝟙 _, } })\n  (by tidy)\n\nend equiv_lax_monoidal_functor_punit\n\nopen equiv_lax_monoidal_functor_punit\n\nlocal attribute [simp] eq_to_iso_map\n\n/--\nMonoid objects in `C` are \"just\" lax monoidal functors from the trivial monoidal category to `C`.\n-/\n@[simps]\ndef equiv_lax_monoidal_functor_punit : lax_monoidal_functor (discrete punit.{u+1}) C ≌ Mon_ C :=\n{ functor := lax_monoidal_to_Mon C,\n  inverse := Mon_to_lax_monoidal C,\n  unit_iso := unit_iso C,\n  counit_iso := counit_iso C, }\n\nend Mon_\n\nnamespace Mon_\n\n/-!\nIn this section, we prove that the category of monoids in a braided monoidal category is monoidal.\n\nGiven two monoids `M` and `N` in a braided monoidal category `C`, the multiplication on the tensor\nproduct `M.X ⊗ N.X` is defined in the obvious way: it is the tensor product of the multiplications\non `M` and `N`, except that the tensor factors in the source come in the wrong order, which we fix\nby pre-composing with a permutation isomorphism constructed from the braiding.\n\nA more conceptual way of understanding this definition is the following: The braiding on `C` gives\nrise to a monoidal structure on the tensor product functor from `C × C` to `C`.  A pair of monoids\nin `C` gives rise to a monoid in `C × C`, which the tensor product functor by being monoidal takes\nto a monoid in `C`.  The permutation isomorphism appearing in the definition of the multiplication\non the tensor product of two monoids is an instance of a more general family of isomorphisms which\ntogether form a strength that equips the tensor product functor with a monoidal structure, and the\nmonoid axioms for the tensor product follow from the monoid axioms for the tensor factors plus the\nproperties of the strength (i.e., monoidal functor axioms).  The strength `tensor_μ` of the tensor\nproduct functor has been defined in `category_theory.monoidal.braided`.  Its properties, stated as\nindependent lemmas in that module, are used extensively in the proofs below.  Notice that we could\nhave followed the above plan not only conceptually but also as a possible implementation and could\nhave constructed the tensor product of monoids via `map_Mon`, but we chose to give a more explicit\ndefinition directly in terms of `tensor_μ`.\n\nTo complete the definition of the monoidal category structure on the category of monoids, we need\nto provide definitions of associator and unitors.  The obvious candidates are the associator and\nunitors from `C`, but we need to prove that they are monoid morphisms, i.e., compatible with unit\nand multiplication.  These properties translate to the monoidality of the associator and unitors\n(with respect to the monoidal structures on the functors they relate), which have also been proved\nin `category_theory.monoidal.braided`.\n\n-/\n\nvariable {C}\n\n-- The proofs that associators and unitors preserve monoid units don't require braiding.\n\nlemma one_associator {M N P : Mon_ C} :\n    ((λ_ (𝟙_ C)).inv ≫ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one)) ≫ (α_ M.X N.X P.X).hom\n  = (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)) :=\nbegin\n  simp,\n  slice_lhs 1 3 { rw [←category.id_comp P.one, tensor_comp] },\n  slice_lhs 2 3 { rw associator_naturality },\n  slice_rhs 1 2 { rw [←category.id_comp M.one, tensor_comp] },\n  slice_lhs 1 2 { rw [←left_unitor_tensor_inv] },\n  rw ←(cancel_epi (λ_ (𝟙_ C)).inv),\n  slice_lhs 1 2 { rw [left_unitor_inv_naturality] },\n  simp only [category.assoc],\nend\n\nlemma one_left_unitor {M : Mon_ C} :\n  ((λ_ (𝟙_ C)).inv ≫ (𝟙 (𝟙_ C) ⊗ M.one)) ≫ (λ_ M.X).hom = M.one :=\nby { slice_lhs 2 3 { rw left_unitor_naturality }, simp }\n\nlemma one_right_unitor {M : Mon_ C} :\n  ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ 𝟙 (𝟙_ C))) ≫ (ρ_ M.X).hom = M.one :=\nby { slice_lhs 2 3 { rw [right_unitor_naturality, ←unitors_equal] }, simp }\n\nvariable [braided_category C]\n\nlemma Mon_tensor_one_mul (M N : Mon_ C) :\n    ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n    tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)\n  = (λ_ (M.X ⊗ N.X)).hom :=\nbegin\n  rw [←category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp],\n  slice_lhs 2 3 { rw [←tensor_id, tensor_μ_natural] },\n  slice_lhs 3 4 { rw [←tensor_comp, one_mul M, one_mul N] },\n  symmetry,\n  exact tensor_left_unitality C M.X N.X,\nend\n\nlemma Mon_tensor_mul_one (M N : Mon_ C) :\n    (𝟙 (M.X ⊗ N.X) ⊗ (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫\n    tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)\n  = (ρ_ (M.X ⊗ N.X)).hom :=\nbegin\n  rw [←category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp],\n  slice_lhs 2 3 { rw [←tensor_id, tensor_μ_natural] },\n  slice_lhs 3 4 { rw [←tensor_comp, mul_one M, mul_one N] },\n  symmetry,\n  exact tensor_right_unitality C M.X N.X,\nend\n\nlemma Mon_tensor_mul_assoc (M N : Mon_ C) :\n    (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n    tensor_μ C (M.X, N.X) (M.X, N.X) ≫\n    (M.mul ⊗ N.mul)\n  = (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n    (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n    tensor_μ C (M.X, N.X) (M.X, N.X) ≫\n    (M.mul ⊗ N.mul) :=\nbegin\n  rw [←category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp],\n  slice_lhs 2 3 { rw [←tensor_id, tensor_μ_natural] },\n  slice_lhs 3 4 { rw [←tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp] },\n  slice_lhs 1 3 { rw [tensor_associativity] },\n  slice_lhs 3 4 { rw [←tensor_μ_natural] },\n  slice_lhs 2 3 { rw [←tensor_comp, tensor_id] },\n  simp only [category.assoc],\nend\n\nlemma mul_associator {M N P : Mon_ C} :\n    (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫\n    (α_ M.X N.X P.X).hom\n  = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n    tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n    (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul)) :=\nbegin\n  simp,\n  slice_lhs 2 3 { rw [←category.id_comp P.mul, tensor_comp] },\n  slice_lhs 3 4 { rw [associator_naturality] },\n  slice_rhs 3 4 { rw [←category.id_comp M.mul, tensor_comp] },\n  slice_lhs 1 3 { rw associator_monoidal },\n  simp only [category.assoc],\nend\n\nlemma mul_left_unitor {M : Mon_ C}:\n    (tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom\n  = ((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul :=\nbegin\n  rw [←(category.comp_id (λ_ (𝟙_ C)).hom), ←(category.id_comp M.mul), tensor_comp],\n  slice_lhs 3 4 { rw left_unitor_naturality },\n  slice_lhs 1 3 { rw ←left_unitor_monoidal },\n  simp only [category.assoc, category.id_comp],\nend\n\nlemma mul_right_unitor {M : Mon_ C} :\n    (tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom\n  = ((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul :=\nbegin\n  rw [←(category.id_comp M.mul), ←(category.comp_id (λ_ (𝟙_ C)).hom), tensor_comp],\n  slice_lhs 3 4 { rw right_unitor_naturality },\n  slice_lhs 1 3 { rw ←right_unitor_monoidal },\n  simp only [category.assoc, category.id_comp],\nend\n\ninstance Mon_monoidal : monoidal_category (Mon_ C) :=\n{ tensor_obj := λ M N,\n  { X := M.X ⊗ N.X,\n    one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one),\n    mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul),\n    one_mul' := Mon_tensor_one_mul M N,\n    mul_one' := Mon_tensor_mul_one M N,\n    mul_assoc' := Mon_tensor_mul_assoc M N },\n  tensor_hom := λ M N P Q f g,\n  { hom := f.hom ⊗ g.hom,\n    one_hom' :=\n    begin\n      dsimp,\n      slice_lhs 2 3 { rw [←tensor_comp, hom.one_hom f, hom.one_hom g] },\n    end,\n    mul_hom' :=\n    begin\n      dsimp,\n      slice_rhs 1 2 { rw [tensor_μ_natural] },\n      slice_lhs 2 3 { rw [←tensor_comp, hom.mul_hom f, hom.mul_hom g, tensor_comp] },\n      simp only [category.assoc],\n    end },\n  tensor_id' := by { intros, ext, apply tensor_id },\n  tensor_comp' := by { intros, ext, apply tensor_comp },\n  tensor_unit := trivial C,\n  associator := λ M N P, iso_of_iso (α_ M.X N.X P.X) one_associator mul_associator,\n  associator_naturality' := by { intros, ext, dsimp, apply associator_naturality },\n  left_unitor := λ M, iso_of_iso (λ_ M.X) one_left_unitor mul_left_unitor,\n  left_unitor_naturality' := by { intros, ext, dsimp, apply left_unitor_naturality },\n  right_unitor := λ M, iso_of_iso (ρ_ M.X) one_right_unitor mul_right_unitor,\n  right_unitor_naturality' := by { intros, ext, dsimp, apply right_unitor_naturality },\n  pentagon' := by { intros, ext, dsimp, apply pentagon },\n  triangle' := by { intros, ext, dsimp, apply triangle } }\n\nend Mon_\n\n/-!\nProjects:\n* Check that `Mon_ Mon ≌ CommMon`, via the Eckmann-Hilton argument.\n  (You'll have to hook up the cartesian monoidal structure on `Mon` first, available in #3463)\n* Check that `Mon_ Top ≌ [bundled topological monoids]`.\n* Check that `Mon_ AddCommGroup ≌ Ring`.\n  (We've already got `Mon_ (Module R) ≌ Algebra R`, in `category_theory.monoidal.internal.Module`.)\n* Can you transport this monoidal structure to `Ring` or `Algebra R`?\n  How does it compare to the \"native\" one?\n* Show that when `C` is braided, the forgetful functor `Mon_ C ⥤ C` is monoidal.\n* Show that when `F` is a lax braided functor `C ⥤ D`, the functor `map_Mon F : Mon_ C ⥤ Mon_ D`\n  is lax monoidal.\n-/\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/Mon_.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34777769092573024}}
{"text": "import category_theory.preadditive\nimport category_theory.arrow\n\n\n-- TODO : generalize to comma categories?\n\nnamespace category_theory\n\nvariables {A : Type*} [category A] [preadditive A]\n\nlocal attribute [simp] preadditive.comp_nsmul preadditive.nsmul_comp\n    preadditive.comp_zsmul preadditive.zsmul_comp\n\ninstance arrow.has_zero (P Q : arrow A) : has_zero (P ⟶ Q) :=\n⟨arrow.hom_mk (show 0 ≫ Q.hom = P.hom ≫ 0, by simp)⟩\ninstance arrow.has_add (P Q : arrow A) : has_add (P ⟶ Q) :=\n⟨λ f g, arrow.hom_mk (show (f.left + g.left) ≫ Q.hom = P.hom ≫ (f.right + g.right), by simp)⟩\ninstance arrow.has_neg (P Q : arrow A) : has_neg (P ⟶ Q) :=\n⟨λ f, arrow.hom_mk (show (-f.left) ≫ Q.hom = P.hom ≫ (-f.right), by simp)⟩\ninstance arrow.has_sub (P Q : arrow A) : has_sub (P ⟶ Q) :=\n⟨λ f g, arrow.hom_mk (show (f.left - g.left) ≫ Q.hom = P.hom ≫ (f.right - g.right), by simp)⟩\ninstance arrow.has_nsmul (P Q : arrow A) : has_smul ℕ (P ⟶ Q) :=\n⟨λ n f, arrow.hom_mk (show (n • f.left) ≫ Q.hom = P.hom ≫ (n • f.right), by simp)⟩\ninstance arrow.has_zsmul (P Q : arrow A) : has_smul ℤ (P ⟶ Q) :=\n⟨λ n f, arrow.hom_mk (show (n • f.left) ≫ Q.hom = P.hom ≫ (n • f.right), by simp)⟩\n.\n@[simp] lemma arrow.has_add_left {P Q : arrow A} (f g : P ⟶ Q) :\n  (f + g).left = f.left + g.left := rfl\n@[simp] lemma arrow.has_add_right {P Q : arrow A} (f g : P ⟶ Q) :\n  (f + g).right = f.right + g.right := rfl\n@[simp] lemma arrow.has_neg_left {P Q : arrow A} (f : P ⟶ Q) :\n  (-f).left = -(f.left) := rfl\n@[simp] lemma arrow.has_neg_right {P Q : arrow A} (f : P ⟶ Q) :\n  (-f).right = -(f.right) := rfl\n@[simp] lemma arrow.has_sub_left {P Q : arrow A} (f g : P ⟶ Q) :\n  (f - g).left = f.left - g.left := rfl\n@[simp] lemma arrow.has_sub_right {P Q : arrow A} (f g : P ⟶ Q) :\n  (f - g).right = f.right - g.right := rfl\n\ninstance arrow.add_comm_group (P Q : arrow A) : add_comm_group (P ⟶ Q) :=\n{ add_assoc := λ f g h, by ext; apply add_assoc,\n  zero_add := λ f, by ext; apply zero_add,\n  add_zero := λ f, by ext; apply add_zero,\n  add_left_neg := λ f, by ext; apply add_left_neg,\n  add_comm := λ f g, by ext; apply add_comm,\n  ..arrow.has_add P Q,\n  ..arrow.has_zero P Q,\n  ..arrow.has_neg P Q,\n  ..arrow.has_nsmul P Q,\n  ..arrow.has_zsmul P Q }\n.\ninstance arrow.preadditive : preadditive (arrow A) :=\n⟨infer_instance, by { introv P, ext; apply preadditive.add_comp },\n  by { introv P, ext; apply preadditive.comp_add }⟩\n\n\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/arrow_preadditive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7122321964553657, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.34777115512423595}}
{"text": "import parlang.defs\nimport parlang.lemmas_state\nimport parlang.lemmas_exec\nimport parlang.nonmono\n\nnamespace parlang\n\nvariables {σ σ₁ σ₂ σ₃ : Type} {ι₁ ι₂ : Type} {τ₁ : ι₁ → Type} {τ₂ : ι₂ → Type} [decidable_eq ι₁] [decidable_eq ι₂]\n\n@[reducible]\ndef rhl_kernel_assertion := Π n₁:ℕ, state n₁ σ₁ τ₁ → vector bool n₁ → Π n₂:ℕ, state n₂ σ₂ τ₂ → vector bool n₂ → Prop\n\n/-- Relational Hoare logic on kernels.  -/\ndef rel_hoare_state (P : rhl_kernel_assertion) (k₁ : kernel σ₁ τ₁) (k₂ : kernel σ₂ τ₂) \n    (Q : rhl_kernel_assertion) : Prop :=\n    ∀ (n₁ n₂ : ℕ) (s₁ s₁' : state n₁ σ₁ τ₁) (s₂ : state n₂ σ₂ τ₂) ac₁ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → exec_state k₁ ac₁ s₁ s₁' →\n    ∃ s₂', exec_state k₂ ac₂ s₂ s₂' ∧ Q n₁ s₁' ac₁ n₂ s₂' ac₂\n\nnotation `{* ` P : 1 ` *} ` k₁ : 1 ` ~> ` k₂ : 1 ` {* ` Q : 1 ` *}` := rel_hoare_state P k₁ k₂ Q\n\nnamespace rel_hoare\n\n-- lemma seq {P Q R} {k₁ k₁' k₂ k₂' : kernel σ τ} : {* P *} k₁ ~ k₁' {* Q *} → {* Q *} k₂ ~ k₂' {* R *} → {* P *} (k₁ ;; k₂) ~ (k₁' ;; k₂') {* R *} := begin\n--     intros h₁ h₂ n s u ac m m' o o' hp hseq₁ hseq₂,\n--     cases hseq₁,\n--     cases hseq₂,\n--     cases hseq₁_hk,\n--     cases hseq₂_hk,\n--     unfold rel_hoare at h₁ h₂,\n--     apply h₂ n _ _ _ _ _ _ _ _ _ hseq₁_hk_a_1 hseq₂_hk_a_1,\n-- end\n\nend rel_hoare\n\n/-- Relational Hoare logic on Parlang programs with assertions on memory -/\ndef rel_hoare_program (init₁ : ℕ → σ₁) (init₂ : ℕ → σ₂) (P : memory τ₁ → memory τ₂ → Prop) (p₁ : program σ₁ τ₁) (p₂ : program σ₂ τ₂) (Q : memory τ₁ → memory τ₂ → Prop) := \n∀ m₁ m₁' m₂, P m₁ m₂ → exec_prog init₁ p₁ m₁ m₁' → ∃ m₂', exec_prog init₂ p₂ m₂ m₂' ∧ Q m₁' m₂'\n\n-- notation `{* ` P : 1 ` *} ` p₁ : 1 ` ~ ` p₂ : 1 ` {* ` Q : 1 ` *}` := rel_hoare_state P p₁ p₂ Q\n\n/-- The initial kernel assertion that holds on state and ac \n    TODO: change to a struct to get projections for free?\n-/\ndef initial_kernel_assertion (init₁ : ℕ → σ₁) (init₂ : ℕ → σ₂) (P : memory τ₁ → memory τ₂ → Prop) \n(f₁ : memory τ₁ → ℕ) (f₂ : memory τ₂ → ℕ) (m₁ : memory τ₁) (m₂ : memory τ₂) \n(n₁) (s₁ : state n₁ σ₁ τ₁) (ac₁ : vector bool n₁) (n₂) (s₂ : state n₂ σ₂ τ₂) (ac₂ : vector bool n₂) := \ns₁.syncable m₁ ∧ s₂.syncable m₂ ∧ n₁ = f₁ m₁ ∧ n₂ = f₂ m₂ ∧\n(∀ i : fin n₁, s₁.threads.nth i = { tlocal := init₁ i, shared := m₁, stores := ∅, loads := ∅ }) ∧ \n(∀ i : fin n₂, s₂.threads.nth i = { tlocal := init₂ i, shared := m₂, stores := ∅, loads := ∅ }) ∧\nP m₁ m₂ ∧ all_threads_active ac₁ ∧ all_threads_active ac₂\n\n/- Projections of initial_kernel_assertion -/\nsection\n\n    def initial_kernel_assertion_left_thread_state {init₁ : ℕ → σ₁} {init₂ : ℕ → σ₂} {P : memory τ₁ → memory τ₂ → Prop} \n    {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₂ → ℕ} {m₁ : memory τ₁} {m₂ : memory τ₂} \n    {n₁} {s₁ : state n₁ σ₁ τ₁} {ac₁ : vector bool n₁} {n₂} {s₂ : state n₂ σ₂ τ₂} {ac₂ : vector bool n₂}\n    (i : initial_kernel_assertion init₁ init₂ P f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₂ s₂ ac₂) := i.right.right.right.right.left\n\n    def initial_kernel_assertion.right_thread_state {init₁ : ℕ → σ₁} {init₂ : ℕ → σ₂} {P : memory τ₁ → memory τ₂ → Prop} \n    {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₂ → ℕ} {m₁ : memory τ₁} {m₂ : memory τ₂} \n    {n₁} {s₁ : state n₁ σ₁ τ₁} {ac₁ : vector bool n₁} {n₂} {s₂ : state n₂ σ₂ τ₂} {ac₂ : vector bool n₂}\n    (i : initial_kernel_assertion init₁ init₂ P f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₂ s₂ ac₂) := i.right.right.right.right.right.left\n\n    def initial_kernel_assertion.left_all_threads_active {init₁ : ℕ → σ₁} {init₂ : ℕ → σ₂} {P : memory τ₁ → memory τ₂ → Prop} \n    {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₂ → ℕ} {m₁ : memory τ₁} {m₂ : memory τ₂} \n    {n₁} {s₁ : state n₁ σ₁ τ₁} {ac₁ : vector bool n₁} {n₂} {s₂ : state n₂ σ₂ τ₂} {ac₂ : vector bool n₂}\n    (i : initial_kernel_assertion init₁ init₂ P f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₂ s₂ ac₂) := i.right.right.right.right.right.right.right.left\n\n    def initial_kernel_assertion.precondition {init₁ : ℕ → σ₁} {init₂ : ℕ → σ₂} {P : memory τ₁ → memory τ₂ → Prop} \n    {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₂ → ℕ} {m₁ : memory τ₁} {m₂ : memory τ₂} \n    {n₁} {s₁ : state n₁ σ₁ τ₁} {ac₁ : vector bool n₁} {n₂} {s₂ : state n₂ σ₂ τ₂} {ac₂ : vector bool n₂}\n    (i : initial_kernel_assertion init₁ init₂ P f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₂ s₂ ac₂) := i.right.right.right.right.right.right.left\n\n    lemma initial_kernel_assertion.initial_state_eq {init₁ : ℕ → σ₁}\n    {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₁ → ℕ} {m₁ : memory τ₁} {m₂ : memory τ₁} \n    {n₁} {s₁ : state n₁ σ₁ τ₁} {ac₁ : vector bool n₁} {s₂ : state n₁ σ₁ τ₁} {ac₂ : vector bool n₁} :\n    initial_kernel_assertion init₁ init₁ eq f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₁ s₂ ac₂ →\n    s₁ = s₂ |\n    (and.intro _ (and.intro _ (and.intro _ (and.intro _ (and.intro s_eq (and.intro s'_eq _)))))) := begin\n        clear initial_kernel_assertion.initial_state_eq,\n        cases s₁,\n        cases s₂,\n        simp,\n        apply vector.eq_element_wise,\n        intro tid,\n        simp *,\n    end\n\nend\n\n/- Reduces proofs on programs to proofs on kernels -/\nlemma rel_kernel_to_program {k₁ : kernel σ₁ τ₁} {k₂ : kernel σ₂ τ₂} {init₁ : ℕ → σ₁} {init₂ : ℕ → σ₂} {P Q : memory τ₁ → memory τ₂ → Prop} {f₁ : memory τ₁ → ℕ} {f₂ : memory τ₂ → ℕ}\n (h : {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, ∃ m₁ m₂, initial_kernel_assertion init₁ init₂ P f₁ f₂ m₁ m₂ n₁ s₁ ac₁ n₂ s₂ ac₂ *} k₁ ~> k₂ \n {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, (∃ m₁, s₁.syncable m₁) → ∃ m₁ m₂, s₁.syncable m₁ ∧ s₂.syncable m₂ ∧ Q m₁ m₂ *} )\n (hg : ∀ {m₁ m₂}, P m₁ m₂ → 0 < f₁ m₁) :\n rel_hoare_program init₁ init₂ P (program.intro f₁ k₁) (program.intro f₂ k₂) Q :=\nbegin\n    unfold rel_hoare_state at h,\n    unfold rel_hoare_program,\n    intros m₁ m₁' m₂ hp hep,\n    cases hep,\n    specialize h (f₁ m₁) (f₂ m₂) \n        (init_state init₁ f₁ m₁)\n        hep_s'\n        (init_state init₂ f₂ m₂)\n        (vector.repeat tt (f₁ m₁))\n        (vector.repeat tt (f₂ m₂)),\n    have hh : (∃ (s₂' : state (f₂ m₂) σ₂ τ₂), exec_state k₂ (vector.repeat tt (f₂ m₂)) (init_state init₂ f₂ m₂) s₂' ∧\n    ((∃ m₁, hep_s'.syncable m₁) → ∃ (m₁_1 : memory τ₁) (m₂_1 : memory τ₂), state.syncable hep_s' m₁_1 ∧ state.syncable s₂' m₂_1 ∧ Q m₁_1 m₂_1)) := begin\n        apply h,\n        {\n            apply exists.intro m₁,\n            apply exists.intro m₂,\n            apply and.intro,\n            {\n                apply init_state_syncable,\n            }, {\n                apply and.intro,\n                {\n                    apply init_state_syncable,\n                },\n                {\n                    apply and.intro,\n                    {\n                        refl,\n                    },\n                    {\n                        apply and.intro,\n                        {\n                            refl,\n                        },\n                        {\n                            apply and.intro,\n                            {\n                                simp,\n                                intro i,\n                                rw vector.range_nth,\n                            },\n                            {\n                                apply and.intro,\n                                {\n                                    simp,\n                                    intro i,\n                                    rw vector.range_nth,\n                                }, {\n                                    apply and.intro,\n                                    {\n                                        exact hp,\n                                    }, {\n                                        apply and.intro,\n                                        {\n                                            apply all_threads_active_repeat,\n                                        }, {\n                                            apply all_threads_active_repeat,\n                                        }\n                                    }\n                                }\n                            }\n                        }\n                    }\n                }\n            }\n        }, {\n            exact hep_he,\n        }\n    end,\n    cases hh,\n    cases hh_h,\n    specialize hh_h_right ⟨_, hep_hsync⟩,\n    cases hh_h_right with m₁'',\n    cases hh_h_right_h with m₂',\n    apply exists.intro m₂',\n    apply and.intro,\n    {\n        apply exec_prog.intro _ _ _ _ _ hh_w,\n        { exact hh_h_right_h_h.right.left, },\n        { exact hh_h_left, },\n    }, {\n        have hu : m₁' = m₁'' := begin\n            have hz : 0 < (f₁ m₁) := hg hp,\n            apply state.syncable_unique hep_hsync hh_h_right_h_h.left hz,\n        end,\n        subst hu,\n        exact hh_h_right_h_h.right.right,\n    }\nend\n\n/- \n    Inference rules\n -/\n\n-- TODO: define the alias skip for (compute id)\nlemma single_step_left {P Q} {k₁ k : kernel σ₁ τ₁} {k₂ : kernel σ₂ τ₂} (R)\n    (h₁ : {* P *} k ~> (kernel.compute id) {* R *})\n    (h₂ : {* R *} k₁ ~> k₂ {* Q *}) : \n    {* P *} (k ;; k₁) ~> k₂ {* Q *} := begin\n    intros n₁ n₂ s₁ s₁'' s₂ ac₁ ac₂ hp hek₁,\n    cases hek₁,\n    specialize h₁ n₁ n₂ s₁ _ s₂ ac₁ ac₂ hp hek₁_a,\n    cases h₁ with s₂' h₁,\n    cases h₁,\n    have : s₂ = s₂' := begin\n        cases h₁_left,\n        cases s₂,\n        sorry, -- trivial\n    end,\n    apply h₂,\n    rw this,\n    exact h₁_right,\n    exact hek₁_a_1,\nend\n\nlemma single_step_right {P Q} {k₁ : kernel σ₁ τ₁} {k₂ k : kernel σ₂ τ₂} (R)\n    (h₁ : {* P *} (kernel.compute id) ~> k  {* R *})\n    (h₂ : {* R *} k₁ ~> k₂ {* Q *}) : \n    {* P *} k₁ ~> k ;; k₂ {* Q *} := sorry\n\nvariables {P Q R P' Q' I : Π n₁:ℕ, state n₁ σ₁ τ₁ → vector bool n₁ → Π n₂:ℕ, state n₂ σ₂ τ₂ → vector bool n₂ → Prop} {k₁ k₁' : kernel σ₁ τ₁} {k₂ k₂' : kernel σ₂ τ₂}\n\n-- intuition of the proof (to be repurposed by further proofs):\n-- we get the intermediate state of after k₁ by cases\n-- from h₁ we get state after k₂\n-- from h₂ we get state after k₂'\nlemma seq (Q)\n    (h₁ : {* P *} k₁ ~> k₂ {* Q *})\n    (h₂ : {* Q *} k₁' ~> k₂' {* R *}) :\n    {* P *} (k₁ ;; k₁') ~> (k₂ ;; k₂') {* R *} := begin\n        intros _ _ _ _ _ _ _ hp hek₁k₁',\n        cases hek₁k₁',\n        specialize h₁ n₁ n₂ s₁ hek₁k₁'_t s₂ ac₁ ac₂ hp hek₁k₁'_a,\n        cases h₁ with t₂,\n        cases h₁_h with hek₂ hq,\n        specialize h₂ n₁ n₂ hek₁k₁'_t _ t₂ ac₁ ac₂ hq hek₁k₁'_a_1,\n        cases h₂ with s₂',\n        apply exists.intro s₂',\n        apply and.intro,\n        apply exec_state.seq _ _ _ _ _ _ hek₂ (h₂_h.left),\n        exact h₂_h.right,\nend\n\n-- sometimes called sub\nlemma consequence (h : {* P *} k₁ ~> k₂ {* Q *}) \n(hp : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P' n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ ac₂)\n(hq : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ s₂ ac₂ → Q' n₁ s₁ ac₁ n₂ s₂ ac₂) : {* P' *} k₁ ~> k₂ {* Q' *} := begin\n    intros _ _ _ _ _ _ _ hp' he₁,\n    specialize h n₁ n₂ s₁ s₁' s₂ ac₁ ac₂ _ he₁,\n    cases h with s₂ h,\n    use s₂,cases h,\n    apply and.intro,\n    assumption,\n    apply hq,\n    assumption,\n    apply hp,\n    assumption,\nend\n\nlemma consequence_pre (h : {* P *} k₁ ~> k₂ {* Q *}) \n(hp : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P' n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ ac₂) : \n{* P' *} k₁ ~> k₂ {* Q *} := begin\n    apply consequence,\n    exact h,\n    exact hp,\n    intros,\n    exact a,\nend\n\nlemma consequence_post (h : {* P *} k₁ ~> k₂ {* Q *}) \n(hp : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ s₂ ac₂ → Q' n₁ s₁ ac₁ n₂ s₂ ac₂) : \n{* P *} k₁ ~> k₂ {* Q' *} := begin\n    apply consequence,\n    exact h,\n    intros,\n    exact a,\n    exact hp,\nend\n\ndef assertion_swap_side (P : @rhl_kernel_assertion σ₁ σ₂ _ _ τ₁ τ₂ _ _) := λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₂ s₂ ac₂ n₁ s₁ ac₁\n\nlemma swap' (h : {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ ∃ t, exec_state k₂ ac₂ s₂ t *} k₁ ~> k₂ {* Q *}) (he₁ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂}, P n₁ s₁ ac₁ n₂ s₂ ac₂ → ∃ s₁', exec_state k₁ ac₁ s₁ s₁') : \n{* assertion_swap_side P *} k₂ ~> k₁ {* assertion_swap_side Q *} := begin\n    sorry,\nend\n\n-- k₁ must terminate\nlemma swap (h : {* P *} k₁ ~> k₂ {* Q *}) (he₁ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂}, P n₁ s₁ ac₁ n₂ s₂ ac₂ → ∃ s₁', exec_state k₁ ac₁ s₁ s₁') : \n{* assertion_swap_side P *} k₂ ~> k₁ {* assertion_swap_side Q *} := begin\n    intros n₂ n₁ s₂ s₂' s₁ ac₂ ac₁ hp he₂,\n    simp,\n    have : ∃ s₁', exec_state k₁ ac₁ s₁ s₁' := he₁ hp,\n    cases this with s₁' he₂,\n    use s₁',\n    split,\n    assumption,\n    specialize h n₁ n₂ s₁ _ s₂ ac₁ ac₂ hp he₂,\n    cases h,\n    have : h_w = s₂' := sorry, -- by uniqueness\n    subst this,\n    exact h_h.right,\nend\n\ntheorem trans (p₁ : program σ₁ τ₁) (p₂ : program σ₂ τ₁) (p₃ : program σ₃ τ₁) (init₁ init₂ init₃) (h₁ : rel_hoare_program init₁ init₂ eq p₁ p₂ eq) (h₂ : rel_hoare_program init₂ init₃ eq p₂ p₃ eq) :\nrel_hoare_program init₁ init₃ eq p₁ p₃ eq := begin\n    unfold rel_hoare_program at *,\n    intros _ _ _ heq he,\n    specialize h₁ m₁ m₁' m₂ heq he,\n    cases h₁ with m h,\n    cases h,\n    subst heq,\n    subst h_right,\n    specialize h₂ m₁ m₁' m₁ rfl h_left,\n    exact h₂,\nend\n\ntheorem compute_right (f) : {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ (s₂.map_active_threads ac₂ $ thread_state.compute f) ac₂ *} (kernel.compute id) ~> (kernel.compute f) {* Q *} := begin\n    intros _ _ _ _ _ _ _ hp he,\n    use (s₂.map_active_threads ac₂ $ thread_state.compute f),\n    split,\n    {\n        apply exec_state.compute,\n    }, {\n        cases he,\n        sorry, -- trivial\n    }\nend\n\ntheorem store_right (f : σ₂ → (Σ (i : ι₂), τ₂ i)) : \n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ (s₂.map_active_threads ac₂ $ thread_state.store f) ac₂ *} \n(kernel.compute id) ~> (kernel.store f) \n{* Q *} := begin\n    intros _ _ _ _ _ _ _ hp he,\n    use (s₂.map_active_threads ac₂ $ thread_state.store f),\n    split, {\n        apply exec_state.store,\n    }, {\n        cases he,\n        sorry, -- trivial\n    }\nend\n\n-- TODO: move to another file\ntheorem else_skip {n} (k : kernel σ τ₁) (c) (ac : vector bool n) (s s' : state n σ τ₁):\nexec_state k (deactivate_threads (bnot ∘ c) ac s) s s' ↔ \nexec_state (kernel.ite c k (kernel.compute id)) ac s s' := begin\n    split, {\n        intro h,\n        apply exec_state.ite s s',\n        exact h,\n        apply exec_skip,\n    }, {\n        intro h,\n        cases h,\n        have := eq.symm (exec_skip_eq h_a_1),\n        subst this,\n        exact h_a,\n    }\nend\n\n/-- \nBreaks a ite-instruction into the individual branches\nThe assertions can be used to carry information about the active map from before the branch to after the branch\nIn the if-branch, the condition holds on all active threads (similar in else-branch). The inverse is not true. Just because the condition *c* holds, does not mean that the thread is active (e.g. if the thread was already inactive). -/\ntheorem ite_right (c : σ₂ → bool) (th) (el) (AC : ∀ {n₂ : ℕ}, vector bool n₂ → Prop)\n(h₁ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) \n(h₂ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), Q n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s') → Q n₁ s₁ ac₁ n₂ s₂ ac₂)\n(h₃ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), Q n₁ s₁ ac₁ n₂ s₂ ac₂ → Q n₁ s₁ ac₁ n₂ s₂ (deactivate_threads c ac₂ s')) \n(h₄ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), R n₁ s₁ ac₁ n₂ s₂ (deactivate_threads c ac₂ s') → R n₁ s₁ ac₁ n₂ s₂ ac₂) :\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, c ts.tlocal) *} kernel.compute id ~> th {* Q *} →\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, bnot $ c ts.tlocal) *} kernel.compute id ~> el {* R *} →\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ AC ac₂ *} kernel.compute id ~> kernel.ite c th el {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, R n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ AC ac₂ *}\n:= begin\n    intros hth hel,\n    intros _ _ _ _ _ _ _ hp he,\n    specialize hth n₁ n₂ s₁ s₁ s₂ ac₁ (deactivate_threads (bnot ∘ c) ac₂ s₂) _ exec_skip,\n    swap, {\n        split, {\n            apply h₁ _ _ _ _ _ _ hp.left,\n        }, {\n            sorry, -- not trivial\n        }\n    },\n    cases hth with s₂' hth,\n    specialize hel n₁ n₂ s₁ s₁ s₂' ac₁ (deactivate_threads c ac₂ s₂) _ (exec_skip),\n    swap, {\n        split, {\n            apply h₃,\n            apply h₂ _ _ _ _ _ _ s₂,\n            exact hth.right,\n        }, {\n            sorry, -- not trivial\n        }\n    },\n    cases hel with s₂'' hel,\n    have := eq.symm (exec_skip_eq he),\n    subst this,\n    use s₂'',\n    split, {\n        apply exec_state.ite,\n        exact hth.left,\n        exact hel.left,\n    }, \n    simp,\n    split, {\n        apply h₄,\n        exact hel.right,\n    }, {\n        exact hp.right,\n    }\nend\n\n/-- *ite_right* w/o an assertion on the active map -/\ntheorem ite_right' (c : σ₂ → bool) (th) (el)\n(h₁ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) \n(h₂ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), Q n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s') → Q n₁ s₁ ac₁ n₂ s₂ ac₂)\n(h₃ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), Q n₁ s₁ ac₁ n₂ s₂ ac₂ → Q n₁ s₁ ac₁ n₂ s₂ (deactivate_threads c ac₂ s')) \n(h₄ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₂ σ₂ τ₂), R n₁ s₁ ac₁ n₂ s₂ (deactivate_threads c ac₂ s') → R n₁ s₁ ac₁ n₂ s₂ ac₂) :\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, c ts.tlocal) *} kernel.compute id ~> th {* Q *} →\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, bnot $ c ts.tlocal) *} kernel.compute id ~> el {* R *} →\n{* P *} kernel.compute id ~> kernel.ite c th el {* R *}\n:= begin\n    intros _ _,\n    suffices : {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (λ_, true) ac₂ *} kernel.compute id ~> kernel.ite c th el {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, R n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (λ_, true) ac₂ *},\n    simp at this,\n    exact this,\n    apply ite_right,\n    repeat { assumption, },\n    exact h₂,\nend\n\n-- TODO: use ite_right\ntheorem then_right (c : σ₂ → bool) (th)\n(h₁ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) \n(h₂ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s : state n₂ σ₂ τ₂), Q n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s) → Q n₁ s₁ ac₁ n₂ s₂ ac₂) :\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, c ts.tlocal) *} kernel.compute id ~> th {* Q *} →\n{* P *} kernel.compute id ~> kernel.ite c th (kernel.compute id) {* Q *}\n:= begin\n    intros hth,\n    intros _ _ _ _ _ _ _ hp he,\n    have := eq.symm (exec_skip_eq he),\n    subst this,\n    specialize hth n₁ n₂ s₁ s₁ s₂ ac₁ (deactivate_threads (bnot ∘ c) ac₂ s₂) _ he,\n    swap, {\n        split, {\n            apply h₁,\n            apply hp,\n        }, {\n            sorry, -- not trivial\n        }\n    },\n    cases hth with s₂' hth,\n    use s₂',\n    split, {\n        apply exec_state.ite,\n        exact hth.left,\n        apply exec_skip,\n    }, {\n        apply h₂,\n        apply hth.right,\n    }\nend\n\n/-- TODO: \n    - add post-condition: condition is false on all active threads \n    - rename n\n    - drop h₃ and deactivate threads in the post condition of hb ()\n-/\n/- theorem while_right.aux (c : σ₂ → bool) (k) (V : ∀ {n₂} (s₂ : state n₂ σ₂ τ₂) (ac₂ : vector bool n₂), ℕ)\n(h₁ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂}, P n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) \n(h₂ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂} {s : state n₂ σ₂ τ₂}, P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s) → P n₁ s₁ ac₁ n₂ s₂ ac₂)\n(h₃ : ∀ {n₂} (s₂ : state n₂ σ₂ τ₂) (ac₂ : vector bool n₂), V s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂) ≤ V s₂ ac₂)\n(hb : ∀ n, {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, c ts.tlocal) ∧ V s₂ ac₂ = n *} kernel.compute id ~> k {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ V s₂ ac₂ < n *}) :\n∀ (n : ℕ), {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ V s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂) = n *} kernel.compute id ~> kernel.loop c k {* P *}\n| n n₁ n₂ s₁ s₁' s₂ ac₁ ac₂ ⟨hp, hv⟩ he := \nlet ha := (any_thread_active (deactivate_threads (bnot ∘ c) ac₂ s₂)) in\nlet goal := ∃ (s₂' : state n₂ σ₂ τ₂), exec_state (kernel.loop c k) ac₂ s₂ s₂' ∧ P n₁ s₁' ac₁ n₂ s₂' ac₂ in\nhave base : ha = ff → goal, from begin\n    intro ha,\n    {\n        use s₂,\n        split, {\n            apply exec_state.loop_stop,\n            sorry\n        }, {\n            sorry, -- trivial\n        }\n    }\nend,\nhave it : ha = tt → goal, from (\n    assume hha,\n    /- precondition holds in the first loop iteration -/\n    have hp' : P n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂) := begin\n        exact h₁ hp,\n    end,\n    /- the condition is true for all active threads in the first loop iteration -/\n    have hc : list.all (state.active_threads (deactivate_threads (bnot ∘ c) ac₂ s₂) s₂) (λ ts , c (ts.tlocal)) = tt := begin\n        sorry, -- not trivial\n    end, \n    /- proof of first iteration -/\n    have step : _ := hb (V s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) n₁ n₂ s₁ s₁ s₂ ac₁ (deactivate_threads (bnot ∘ c) ac₂ s₂) ⟨hp', ⟨hc, rfl⟩⟩ exec_skip,\n    /- s₂' is the state after first iteration -/\n    show goal, from match step with ⟨s₂', step⟩ :=\n        /- condition holds after first iteration -/\n        have hp'' : P n₁ s₁ ac₁ n₂ s₂' (deactivate_threads (bnot ∘ c) ac₂ s₂) := begin\n            simp at step,\n            exact step.right.left,\n        end,\n        have wf : V s₂' (deactivate_threads (bnot ∘ c) (deactivate_threads (bnot ∘ c) ac₂ s₂) s₂') < n := begin\n            simp at step,\n            subst hv,\n            specialize h₃ s₂' (deactivate_threads (bnot ∘ c) ac₂ s₂),\n            rw le_iff_eq_or_lt at h₃,\n            cases h₃,\n            {\n                rw h₃,\n                exact step.right.right,\n            }, {\n                apply lt_trans h₃ step.right.right,\n            }\n        end,\n        have ih : _ := while_right.aux (V s₂' (deactivate_threads (bnot ∘ c) (deactivate_threads (bnot ∘ c) ac₂ s₂) s₂')) n₁ n₂ s₁ s₁ s₂' ac₁ (deactivate_threads (bnot ∘ c) ac₂ s₂) ⟨hp'', rfl⟩ exec_skip,\n        show goal, from match ih with ⟨s₂'', ih⟩ := (begin\n            use  s₂'',\n            split, {\n                apply exec_state.loop_step,\n                exact ha,\n                exact step.left,\n                exact ih.left,\n            }, {\n                have := exec_skip_eq he,\n                clear _match _match,\n                subst this,\n                apply h₂ ih.right,\n            },\n        end)\n        end -- end match\n    end\n), \nshow goal, from begin\n    cases a : ha,\n    exact base a,\n    exact it a,\nend -/\n\ntheorem while_right (c : σ₂ → bool) (k) (V : ∀ {n₂} (s₂ : state n₂ σ₂ τ₂) (ac₂ : vector bool n₂), ℕ)\n(h₁ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂}, I n₁ s₁ ac₁ n₂ s₂ ac₂ → I n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂)) \n(h₂ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂} {s : state n₂ σ₂ τ₂}, I n₁ s₁ ac₁ n₂ s₂ (deactivate_threads (bnot ∘ c) ac₂ s) → I n₁ s₁ ac₁ n₂ s₂ ac₂)\n(hb : ∀ n, {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, I n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₂.active_threads ac₂).all (λts, c ts.tlocal) ∧ V s₂ ac₂ = n *} kernel.compute id ~> k {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, I n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ V s₂ (deactivate_threads (bnot ∘ c) ac₂ s₂) < n *}) :\n{* I *} kernel.compute id ~> kernel.loop c k {* I *} := sorry\n\ntheorem while_left (c : σ₁ → bool) (k) (V : ∀ {n₂} (s₂ : state n₂ σ₂ τ₂) (ac₂ : vector bool n₂), ℕ)\n(h₁ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂}, I n₁ s₁ ac₁ n₂ s₂ ac₂ → I n₁ s₁ (deactivate_threads (bnot ∘ c) ac₁ s₁) n₂ s₂ ac₂) \n(h₂ : ∀ {n₁ s₁ ac₁ n₂ s₂ ac₂} {s : state n₁ σ₁ τ₁}, I n₁ s₁ (deactivate_threads (bnot ∘ c) ac₁ s) n₂ s₂ ac₂ → I n₁ s₁ ac₁ n₂ s₂ ac₂)\n(hb : {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, I n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₁.active_threads ac₁).all (λts, c ts.tlocal) *} k ~> kernel.compute id {* λ n₁ s₁ ac₁ n₂ s₂ ac₂, I n₁ s₁ ac₁ n₂ s₂ ac₂ *}) :\n{* I *} kernel.loop c k ~> kernel.compute id {* I *} := begin\n    apply swap,\n    {\n        sorry, -- we need the assumption that the loop terminates here\n    },\n    sorry,\nend\n\ntheorem ite_left' (c : σ₁ → bool) (th) (el)\n(h₁ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → P n₁ s₁ (deactivate_threads (bnot ∘ c) ac₁ s₁) n₂ s₂ ac₂) \n(h₂ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₁ σ₁ τ₁), Q n₁ s₁ (deactivate_threads (bnot ∘ c) ac₁ s') n₂ s₂ ac₂ → Q n₁ s₁ ac₁ n₂ s₂ ac₂)\n(h₃ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₁ σ₁ τ₁), Q n₁ s₁ ac₁ n₂ s₂ ac₂ → Q n₁ s₁ (deactivate_threads c ac₁ s') n₂ s₂ ac₂)\n(h₄ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂ (s' : state n₁ σ₁ τ₁), R n₁ s₁ (deactivate_threads c ac₁ s') n₂ s₂ ac₂ → R n₁ s₁ ac₁ n₂ s₂ ac₂) :\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₁.active_threads ac₁).all (λts, c ts.tlocal) *} th ~> kernel.compute id {* Q *} →\n{* λ n₁ s₁ ac₁ n₂ s₂ ac₂, Q n₁ s₁ ac₁ n₂ s₂ ac₂ ∧ (s₁.active_threads ac₁).all (λts, bnot $ c ts.tlocal) *} el ~> kernel.compute id {* R *} →\n{* P *} kernel.ite c th el ~> kernel.compute id {* R *} := begin\n    intros hth hel,\n    apply swap',\n    {\n        apply ite_right',\n        swap 5, {\n            apply swap,\n            apply consequence,\n            apply hth,\n            {\n                intros,\n                exact ⟨a.left.left, a.right⟩,\n            }, {\n                intros,\n                exact a,\n            },\n            intros,\n            cases a.left.right,\n            cases h,\n            \n            sorry, -- we need proof that th terminates\n        }, \n        swap 5, {\n            apply swap,\n            apply hel,\n            sorry, -- we need proof that el terminates\n        }, {\n            intros,\n            split,\n            apply h₁,\n            exact a.left,\n            sorry,\n        }, {\n            intros,\n            apply h₂,\n            exact a,\n        }, {\n            intros,\n            apply h₃,\n            exact a,\n        }, {\n            sorry,\n        }\n    }, {\n        intros,\n        use s₁,\n        apply exec_skip,\n    }\nend\n\nlemma kernel_foldr_skip_right {k : kernel σ₂ τ₂} {ks} : \n{* P *} k₁ ~> list.foldr kernel.seq k ks {* Q *} ↔ {* P *} k₁ ~> list.foldr kernel.seq (kernel.compute id) ks ;; k {* Q *} := sorry\n\nlemma rhl_eq :\n{* λn₁ (s₁ : state n₁ σ₁ τ₁) ac₁ n₂ s₂ ac₂, n₁ = n₂ ∧ ∀ h : n₁ = n₂, s₁ = (by rw h; exact s₂) ∧ ac₁ = (by rw h; exact ac₂) *}\nk₁ ~> k₁\n{* λn₁ s₁ ac₁ n₂ s₂ ac₂, n₁ = n₂ ∧ ∀ h : n₁ = n₂, s₁ = (by rw h; exact s₂) ∧ ac₁ = (by rw h; exact ac₂) *} := begin\n    intros _ _ _ _ _ _ _ hp  he,\n    cases hp,\n    subst hp_left,\n    specialize hp_right rfl,\n    have : s₁ = s₂ := hp_right.left,\n    subst this,\n    have : ac₁ = ac₂ := hp_right.right,\n    subst this,\n    use s₁',\n    split,\n    exact he,\n    simp,\n    split,\n    refl,\n    refl,\nend\n\n\ntheorem known_branch_left (c : σ₁ → bool) (th) (el)\n(h₁ : ∀ n₁ s₁ ac₁ n₂ s₂ ac₂, P n₁ s₁ ac₁ n₂ s₂ ac₂ → ∀ tid : fin n₁, c $ (s₁.threads.nth tid).tlocal)\n(h₂ : {* P *} th ~> k₂ {* Q *}) :\n{* P *} kernel.ite c th el ~> k₂ {* Q *} := begin\n    intros _ _ _ _ _ _ _ hp he,\n    cases he,\n    specialize h₁ n₁ s₁ ac₁ n₂ s₂ ac₂ hp,\n    apply h₂ n₁ n₂ s₁ s₁' s₂ ac₁ ac₂ hp,\n    have : (deactivate_threads (bnot ∘ c) ac₁ s₁) = ac₁ := begin\n        apply ac_deac_ge',\n        intros _ h' _,\n        specialize h₁ t,\n        simp [deactivate_threads, *] at h',\n        exact h',\n    end,\n    rw this at he_a,\n    have : no_thread_active (deactivate_threads c ac₁ s₁) = tt := sorry,\n    have : exec_state el (deactivate_threads c ac₁ s₁) he_t he_t := exec_no_threads_active this,\n    have : he_t = s₁' := exec_state_unique he_a_1 this,\n    subst this,\n    exact he_a,\nend\n\nend parlang", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/parlang/rhl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7122321720225279, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3477711431940869}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nIdentifiers.\n-/\nimport .util .label\n\nnamespace certigrad\n\ninductive ID : Type\n| str : label → ID\n| nat : ℕ → ID\n\nnamespace ID\n\ninstance : decidable_eq ID := by tactic.mk_dec_eq_instance\ninstance : inhabited ID := ⟨ID.str label.default⟩\n\nprivate def add : ID → ID → ID\n| (ID.nat n₁) (ID.nat n₂) := ID.nat (n₁ + n₂)\n| _           _           := default ID\n\ndef to_str : ID → string\n| (str l) := to_string l\n| (nat n) := \"#\" ++ to_string n\n\ninstance : has_to_string ID := ⟨to_str⟩\n\ndef less_than : ID → ID → Prop\n| (nat n)  (str s) := true\n| (str s)  (nat n) := false\n| (nat n₁) (nat n₂) := n₁ < n₂\n| (str s₁) (str s₂) := s₁ < s₂\n\ninstance : has_lt ID := ⟨less_than⟩\n\ninstance decidable_less_than : Π (x y : ID), decidable (x < y)\n| (nat n)  (str s) := decidable.true\n| (str s)  (nat n) := decidable.false\n| (nat n₁) (nat n₂) := show decidable (n₁ < n₂), by apply_instance\n| (str s₁) (str s₂) := by apply label.decidable_less_than\n\nend ID\n\nlemma id_str_ne_nat (x : label) (n : ℕ) : (ID.str x ≠ ID.nat n) = true :=\nbegin apply pextt, intro H, injection H end\n\nlemma id_nat_ne_str (x : label) (n : ℕ) : (ID.nat n ≠ ID.str x) = true :=\nbegin apply pextt, intro H, injection H end\n\nlemma id_str_ne_str (x₁ x₂ : label) : (ID.str x₁ ≠ ID.str x₂) = (x₁ ≠ x₂) :=\nbegin\napply propext,\nsplit,\nintros H_ne H_eq,\nsubst H_eq,\nexact H_ne rfl,\nintros H_ne H_eq,\ninjection H_eq with H,\nexact H_ne H\nend\n\nlemma id_nat_ne_nat (n₁ n₂ : ℕ) : (ID.nat n₁ ≠ ID.nat n₂) = (n₁ ≠ n₂) :=\nbegin\napply propext,\nsplit,\nintros H_ne H_eq,\nsubst H_eq,\nexact H_ne rfl,\nintros H_ne H_eq,\ninjection H_eq with H,\nexact H_ne H\nend\n\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/id.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629465, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34754702457537057}}
{"text": "/- In this file we define affine varieties and affine groups over a discrete field `K`.\n  We take a shortcut in the definition, by defining affine varieties as the opposite\n  category of finitely generated reduced algebras. We define affine groups and basic\n  group theory for affine groups: closed subgroups, kernels, normal subgroups, solvable groups, Borel subgroups, centralizers. -/\nimport ring_theory.basic\n       ..category_theory.group_object\n       ..category_theory.limits2\n       tactic.omitted\n       category_theory.limits.opposites\n       topology.opens\n\nopen category_theory ideal set topological_space\n\nnoncomputable theory\nuniverses u v w\n\n-- we set some priorities of instances very low, because they cause problems in this file\nlocal attribute [instance, priority 1] limits.category_theory.limits.has_limit\n  limits.category_theory.limits.has_colimit limits.category_theory.limits.has_colimits\n  limits.category_theory.limits.has_limits limits.category_theory.limits.has_limits_of_shape\n  limits.category_theory.limits.has_colimits_of_shape\n\ndef has_coe_ideal {α : Type u} [comm_ring α] : has_coe (ideal α) (set α) := by apply_instance\nlocal attribute [instance] has_coe_ideal\n\n/-- An algebraically closed field is a field where every polynomial with positive degree has a root -/\nclass algebraically_closed_field (α : Type u) extends discrete_field α :=\n(closed : is_algebraically_closed α)\n\nlocal attribute [instance, priority 0] classical.prop_decidable\n/-- A finitely generated reduced algebra -/\nclass finitely_generated_reduced_algebra (R : Type u) (A : Type v)\n  [comm_ring R] [comm_ring A] extends algebra R A :=\n(finitely_generated : is_finitely_generated R A)\n(reduced : is_reduced A)\n\nvariables (K : Type u) [discrete_field K]\n          (R : Type v) [comm_ring R] (S : Type w) [comm_ring S]\n          [finitely_generated_reduced_algebra K R] [finitely_generated_reduced_algebra K S]\n          {σ : Type w} [decidable_eq σ]\n\nopen finitely_generated_reduced_algebra category_theory tensor_product\nnamespace algebraic_geometry\n\n/-- The spectrum `Specm(R)` of a `K`-algebra `R` is the set of homomorphisms from `R` to `K`.\n  This corresponds to maximal ideals in `R`. -/\n@[reducible] def spectrum : Type* := R →ₐ[K] K\n\nvariables {R K}\n/-- The quotient of R by a maximal ideal is isomorphic to K -/\ndef quotient_maximal_ideal (I : maximal_ideal R) :\n  { f : I.val.quotient ≃ K // is_ring_hom f.to_fun } :=\nclassical.choice omitted -- TODO: fix non-canoncial choice\n\n/-- The spectrum of R is equivalent to the set of maximal ideals on R -/\ndef spectrum_equiv_maximal_ideal : spectrum K R ≃ maximal_ideal R :=\nlet f : maximal_ideal R → R → K :=\n  λ I, (quotient_maximal_ideal I).val.to_fun ∘ ideal.quotient.mk I.val in\nby { haveI h : ∀ I : maximal_ideal R, is_ring_hom (f I) := omitted,\n     exact ⟨λ ϕ, ⟨ideal.ker ⇑ϕ, omitted⟩, λ I, ⟨f I, omitted⟩, omitted, omitted⟩ }\n\nvariables (K) {R}\n/-- `Z(S)` is the set of homomorphisms in `Spec(R)` that vanish on `S`. -/\ndef Z (S : set R) : set (spectrum K R) := { f | ∀ x ∈ S, f.to_fun x = 0 }\n\n/--`I(X)` consists of all points in `R` that are send to `0` by all `ϕ ∈ X`-/\ndef I (X : set (spectrum K R)) : set R :=\n{ x : R | ∀ϕ : X, ϕ.val x = 0 }\n\n/-- `Z(S)` is equal to `Z` of the radical of the span of `S`. -/\nlemma Z_radical_span (S : set R) : Z K ((ideal.span S).radical.carrier) = Z K S :=\nomitted\n\nvariables (K R)\n\n/-- The Zariski topology is the topology where the closed sets are of the form `Z(S)` for some `S ⊆ R` -/\ninstance Zariski_topology : topological_space (spectrum K R) :=\n⟨set.range (λ S : set R, - Z K S), omitted, omitted, omitted⟩\n\nvariables {K R}\n/-- A radical ideal gives rise to a closed set in the Zariski topology -/\ndef closeds_of_radical_ideal (I : radical_ideal R) : closeds (spectrum K R) :=\n⟨Z K I.val, mem_range_self I.val⟩\n\n/-- A closed set in the Zariski topology gives rise to a radical ideal -/\ndef radical_ideal_of_closeds (X : closeds (spectrum K R)) : radical_ideal R :=\n⟨⟨ I K X.val, omitted, omitted, omitted⟩, omitted⟩\n\n/-- Hilbert's Nullstellensatz: there is a correspondence between radical ideals in R and\n closed sets in the spectrum of R. -/\ndef Nullstellensatz : radical_ideal R ≃ closeds (spectrum K R) :=\n⟨closeds_of_radical_ideal, radical_ideal_of_closeds, omitted, omitted⟩\n\ninstance base.finitely_generated_reduced_algebra :\n  finitely_generated_reduced_algebra K K :=\n{ finitely_generated := is_finitely_generated_base K,\n  reduced := is_reduced_integral_domain,\n  .._inst_1 }\n\ninstance quotient.finitely_generated_reduced_algebra (I : radical_ideal R) :\n  finitely_generated_reduced_algebra K I.val.quotient :=\n{ finitely_generated := is_finitely_generated_quotient (finitely_generated K R) I.val,\n  reduced := is_reduced_quotient I.2,\n  ..quotient.algebra I.val }\n\nvariables (R S)\ninstance tensor.finitely_generated_reduced_algebra :\n  finitely_generated_reduced_algebra K (R ⊗[K] S) :=\n{ finitely_generated := is_finitely_generated_tensor\n  (finitely_generated K R) (finitely_generated K S),\n  reduced := is_reduced_tensor (reduced K R) (reduced K S),\n  ..tensor_product.algebra }\n\n/-- For a closed subset `Z` the quotient `K[X]/I(Z)` is an algebra over `K` -/\nexample (Z : closeds (spectrum K R)) : algebra K (radical_ideal_of_closeds Z).val.quotient :=\ninfer_instance\n\n/-- The spectrum of `K[X]/I(Z)` is Z for a closed subset `Z` -/\ndef spectrum_quotient (Z : closeds (spectrum K R)) :\n  spectrum K (radical_ideal_of_closeds Z).val.quotient ≃ₜ Z.val :=\n{ to_fun := λ x, ⟨x.comp $ algebra.quotient.mk _, omitted⟩,\n  inv_fun := λ y, algebra.quotient.lift y omitted,\n  left_inv := omitted,\n  right_inv := omitted,\n  continuous_to_fun := omitted,\n  continuous_inv_fun := omitted }\n\nvariables {R S}\n\n/-- The type of finitely generated reduced algebras over a fixed commutative ring. -/\nstructure FRAlgebra (R : Type u) [comm_ring R] : Type (u+1) :=\n  (β : Type u)\n  [ring : comm_ring β]\n  [algebra : finitely_generated_reduced_algebra R β]\n\nattribute [instance, priority 1100] FRAlgebra.ring FRAlgebra.algebra\ninstance (R : Type v) [comm_ring R] : has_coe_to_sort (FRAlgebra R) :=\n{ S := Type v, coe := FRAlgebra.β }\n\n/-- The category of finitely generated reduced (f.g.r.) algebras over a fixed commutative ring. -/\ninstance FRAlgebra.category (R : Type u) [comm_ring R] : large_category (FRAlgebra R) :=\n{ hom   := λ a b, a.β →ₐ[R] b.β,\n  id    := λ a, alg_hom.id R a,\n  comp  := λ a b c f g, alg_hom.comp g f }\n\n/-- Quotients in the category of f.g.r. algebras -/\ndef FRAlgebra.quotient (R : FRAlgebra K) (Z : closeds (spectrum K R)) : FRAlgebra K :=\n⟨K, (radical_ideal_of_closeds Z).val.quotient⟩\n\n/-- The quotient map in the category of f.g.r. algebras -/\ndef FRAlgebra.quotient_map (R : FRAlgebra K) (Z : closeds (spectrum K R)) :\n  R ⟶ R.quotient Z :=\nalgebra.quotient.mk _\n\n/-- The tensor product of two finitely generated reduced algebras over `K` -/\ndef FRAlgebra_tensor (R S : FRAlgebra K) : FRAlgebra K :=\n{ β := R ⊗[K] S,\n  ring := _,\n  algebra := tensor.finitely_generated_reduced_algebra R S }\n\nvariables (K)\nsection\nlocal attribute [instance, priority 1500] algebra.mv_polynomial algebra.polynomial\n/-- Polynomials over `K` as an f.g.r. algebra over `K` -/\ndef FRAlgebra_polynomial : FRAlgebra K :=\n{ β := polynomial K,\n  algebra := { finitely_generated := omitted,\n               reduced := omitted } }\n\n/-- Multivariate polynomials over `K` as an f.g.r. algebra over `K` -/\ndef FRAlgebra_mv_polynomial (K : Type (max u v)) [discrete_field K] (σ : Type v)\n  [decidable_eq σ] : FRAlgebra K :=\n{ β := mv_polynomial σ K,\n  algebra := { finitely_generated := omitted,\n               reduced := omitted } }\nend\n\n/-- `K` forms a finitely generated reduced algebras over `K` -/\ndef FRAlgebra_id : FRAlgebra K := ⟨K, K⟩\n\nexample (R : FRAlgebra K) : (R ⟶ FRAlgebra_id K) = (R.β →ₐ[K] K) := by refl\nexample (R : FRAlgebra K) : (R ⟶ FRAlgebra_id K) = spectrum K R.β := rfl\n\n/-- The category of finitely generated reduced algebras over `K` has binary coproducts -/\ndef FRAlgebra.has_binary_coproducts : limits.has_binary_coproducts (FRAlgebra K) :=\nbegin\n  fapply limits.has_binary_coproducts.mk FRAlgebra_tensor,\n  exact λ X Y, tensor_inl,\n  exact λ X Y, tensor_inr,\n  exact λ X Y Z, tensor_lift,\n  omit_proofs\nend\n\n/-- The category of finitely generated reduced algebras over `K` has an initial object -/\ndef FRAlgebra.has_initial_object : limits.has_initial_object (FRAlgebra K) :=\nbegin\n  fapply limits.has_initial_object.mk (FRAlgebra_id K),\n  intro X, exact algebra.of_id K X,\n  omitted\nend\n\n/-- In algebraic geometry, the categories of algebra's over `K` and affine varieties are opposite of each other. In this development we take a shortcut, and *define* affine varieties as the opposite of algebra's over K. -/\n@[reducible] def affine_variety : Type* := opposite (FRAlgebra K)\n\nexample : large_category (affine_variety K) := by apply_instance\n\n/-- The category of affine varieties has binary products -/\n@[instance, priority 1200] def affine_variety.has_binary_products :\n  limits.has_binary_products (affine_variety K) :=\nby { haveI : limits.has_colimits_of_shape.{u} (discrete limits.two) (FRAlgebra K) :=\n     FRAlgebra.has_binary_coproducts K, exact limits.has_products_opposite _ }\n\n/-- The category of affine varieties has a terminal object -/\n@[instance, priority 1200] def affine_variety.has_terminal_object :\n  limits.has_terminal_object (affine_variety K) :=\nby { haveI : limits.has_colimits_of_shape.{u} (discrete pempty) (FRAlgebra K) :=\n     FRAlgebra.has_initial_object K, exact limits.has_products_opposite _ }\n\n/-- The underlying type of an affine variety G = Rᵒᵖ is Spec(R), equivalently the global points\n   of G in the category of affine varieties. -/\ndef affine_variety.type_functor : affine_variety K ⥤ Type u :=\nyoneda.obj (FRAlgebra_id K)\n\n/- to do: affine_variety.type_functor preserves finite products (is just left-exact) -/\nvariables {K R}\n\n/-- The object part of the functor `affine_variety.type_functor` -/\ndef affine_variety.type (X : affine_variety K) : Type u :=\n(affine_variety.type_functor K).obj X\n\n/-- The type of `X` is the spectrum of `X` viewed as an object in the opposite category -/\nlemma affine_variety.type_eq (X : affine_variety K) :\n  affine_variety.type X = spectrum K (unop X).β := rfl\n\n/-- We tell Lean that the Zariski topology gives a topology on the type of an affine variety -/\ninstance (X : affine_variety K) : topological_space X.type :=\nalgebraic_geometry.Zariski_topology _ _\n\n/-- A subobject of an affine variety given by a closed set on its type -/\ndef affine_variety.subobject (X : affine_variety K) (Z : closeds X.type) : affine_variety K :=\nop ((unop X).quotient Z)\n\n/-- A subobject of an affine variety given by a closed set on its type -/\ndef affine_variety.incl (X : affine_variety K) (Z : closeds X.type) :\n  X.subobject Z ⟶ X :=\n(FRAlgebra.quotient_map _ _).op\n\n/-- A subobject of an affine variety given by a closed set on its type -/\ndef incl_of_subset {X : affine_variety K} {Z₁ Z₂ : closeds X.type} (h : Z₁.1 ⊆ Z₂.1) :\n  X.subobject Z₁ ⟶ X.subobject Z₂ :=\nhas_hom.hom.op (algebra.quotient.lift (algebra.quotient.mk _) omitted)\n\n/-- Inclusion of a set on the type of a subobject into the sets on type type of `X` -/\ndef set_sub_incl {X : affine_variety K} {Z : closeds X.type} :\n  set (X.subobject Z).type → set X.type :=\nimage $ (affine_variety.type_functor K).map $ X.incl Z\n\n\nvariable (K)\n/-- An affine group is a group object in the category of affine varieties -/\nabbreviation affine_group : Type* := group_object (affine_variety K)\n\nexample : category (affine_group K) := by apply_instance\n\nend algebraic_geometry\n\nopen algebraic_geometry\nvariables variables {K} {G G' G₁ G₂ G₃ H : affine_group K}\n\n/-- A morphism between affine groups induces a map between the types -/\ndef group_hom.type (f : G ⟶ G') : G.obj.type → G'.obj.type :=\n(affine_variety.type_functor K).map f.map\n\nnamespace algebraic_geometry\n\n-- Given an algebraic group, we get a group structure on its type\nsection\nset_option class.instance_max_depth 80\n/-- The multiplication on the type of an affine group -/\ndef group_type_mul (f g : G.obj.type) : G.obj.type := (tensor_lift f g).comp G.mul.unop\nend\n\n/-- The inversion on the type of an affine group -/\ndef group_type_inv (f : G.obj.type) : G.obj.type := f.comp G.inv.unop\n\n/-- The unit in the type of an affine group -/\ndef group_type_one : G.obj.type := G.one.unop\n\n/-- Given an algebraic group, we get a group structure on its type -/\ninstance group_type (G : affine_group K) : group G.obj.type :=\n{ mul := group_type_mul,\n  mul_assoc := omitted,\n  one := group_type_one,\n  one_mul := omitted,\n  mul_one := omitted,\n  inv := group_type_inv,\n  mul_left_inv := omitted }\n\n/-- A morphism between affine groups induces a group homomorphism between the types -/\ninstance (f : G ⟶ G') : is_group_hom f.type := omitted\n\n/-- A closed subgroup of G is a subset of its type that is closed and a subgroup -/\nclass is_closed_subgroup (s : set G.obj.type) extends is_subgroup s : Prop :=\n(closed : is_closed s)\n\n/-- The type of `G` is a closed subgroup of itself -/\ninstance is_closed_subgroup_univ (G : affine_group K) :\n  is_closed_subgroup (univ : set G.obj.type) :=\nomitted\n\n/-- A closed subgroup gives a closed set (by forgetting that it is a subgroup) -/\ndef to_closeds (s : set G.obj.type) [is_closed_subgroup s] : closeds G.obj.type :=\n⟨s, is_closed_subgroup.closed s⟩\n\n/-- The (opposite of the) multiplication on a subgroup -/\ndef sub_mul_op (s : set G.obj.type) [is_closed_subgroup s] : (unop G.obj).quotient (to_closeds s) ⟶\n  FRAlgebra_tensor ((unop G.obj).quotient (to_closeds s)) ((unop G.obj).quotient (to_closeds s)) :=\nalgebra.quotient.lift\n  begin\n    refine alg_hom.comp _ G.mul.unop,\n    exact tensor_functor (algebra.quotient.mk _) (algebra.quotient.mk _)\n  end\n omitted\n\n/-- From a closed subgroup we can construct an affine group -/\ndef sub (s : set G.obj.type) [is_closed_subgroup s] : affine_group K :=\n{ obj := G.obj.subobject (to_closeds s),\n  mul := (sub_mul_op s).op,\n  mul_assoc := omitted,\n  one := (show (unop G.obj).quotient (to_closeds s) ⟶ FRAlgebra_id K,\n          from algebra.quotient.lift G.one.unop omitted).op,\n  one_mul := omitted,\n  mul_one := omitted,\n  inv := (show (unop G.obj).quotient (to_closeds s) ⟶ (unop G.obj).quotient (to_closeds s),\n          from algebra.quotient.functor G.inv.unop omitted).op,\n  mul_left_inv := omitted }\n\ndef affine_group.incl (G : affine_group K) (s : set G.obj.type) [is_closed_subgroup s] :\n  sub s ⟶ G :=\nby exact ⟨affine_variety.incl _ _, omitted⟩\n\ninstance set_sub_incl.is_closed_subgroup {H : set G.obj.type} {h : is_closed_subgroup H}\n  (K' : set (sub H).obj.type) [hK : is_closed_subgroup K'] : is_closed_subgroup (set_sub_incl K') :=\nomitted\n\n/-- The kernel of a morphism between affine groups is given by the preimage of 1.\n\nMore precisely, we can view `f : G ⟶ G'` as a map between the type of `G` and the type of `G'`,\nand then take the preimage of `1 : type G'`, using the group structure induced on the type of `G'` -/\ndef kernel (f : G ⟶ G') : set G.obj.type :=\nis_group_hom.ker f.type\n\n/-- The kernel of a morphism is a closed subgroup -/\ninstance (f : G ⟶ G') : is_closed_subgroup (kernel f) := omitted\n\n/-- A subset of the type of `G` is a normal subgroup if it the kernel of a morphism between\n  affine groups. Any normal subgroup is automatically closed. -/\ndef is_normal_subgroup (s : set G.obj.type) : Prop :=\n∃(G' : affine_group K) (f : G ⟶ G'), kernel f = s\n\n/-- The structure of being a normal subgroup -/\ndef normal_subgroup_structure (s : set G.obj.type) : Type* :=\nΣ(G' : affine_group K), {f : G ⟶ G' // kernel f = s }\n\n/-- An affine group `G` is solvable if it is abelian or inductively if there is a morphism\n  `ψ : G ⟶ H` such that both `ker(ψ)` and `H` are solvable. -/\n-- For some reason this inductive type is very slow to compile if we make this into a `Type`,\n-- probably, during the definition of auxilliary declarations. For now, let's have it a Prop,\n-- we can turn it into a type later\ninductive solvable : affine_group K → Prop\n| base {{G : affine_group K}} : G.is_abelian → solvable G\n| step {{G H : affine_group K}} (ψ : G ⟶ H) :\n  solvable H → solvable (sub (kernel ψ)) → solvable G\n\n/-- A Borel subgroup is a maximal closed connected solvable subgroup of `G` -/\ndef is_Borel_subgroup (s : set G.obj.type) : Prop :=\nis_maximal { t : set G.obj.type |\n  ∃(h : is_closed_subgroup t), is_connected t ∧ by exactI solvable (sub t) } s\n\n/-- There is a unique maximal closed subgroup of `G` that is a kernel of a morphism `ψ : G ⟶ A`\n  for an abelian group `A` -/\ntheorem closed_derived_subgroup_unique (H : set G.obj.type) [is_closed_subgroup H] :\n  ∃!(s : set G.obj.type), is_maximal { t : set G.obj.type |\n    ∃(A : affine_group K) (ψ : sub H ⟶ A), A.is_abelian ∧ t = set_sub_incl (kernel ψ) } s :=\nomitted\n\n/-- The closed derived subgroup of `H` is the unique maximal subgroup of `H` that is a kernel of a\n  morphism `ψ : H ⟶ A` for an abelian group `A` -/\ndef closed_derived_subgroup (H : set G.obj.type) [is_closed_subgroup H] : set G.obj.type :=\nclassical.the _ (closed_derived_subgroup_unique H)\n\n/-- The closed derived subgroup is a closed subgroup -/\ninstance closed_derived_subgroup.is_closed_subgroup (H : set G.obj.type) [is_closed_subgroup H] :\n  is_closed_subgroup (closed_derived_subgroup H) :=\nomitted\n\nopen category_theory.limits category_theory.limits.binary_product\nlocal infix ` × `:60 := binary_product\nlocal infix ` ×.map `:90 := binary_product.map\n\n/-- The conjugation map `H₁ × H₂ ⟶ G` given by `(h₁,h₂) ↦ h₁*h₂*h₁⁻¹`-/\ndef conjugation (H₁ H₂ : set G.obj.type) [is_closed_subgroup H₁] [is_closed_subgroup H₂] :\n  (sub H₁).obj × (sub H₂).obj ⟶ G.obj :=\n((G.incl H₁).map ≫ diag) ×.map (G.incl H₂).map ≫\nproduct_assoc.hom ≫ 𝟙 G.obj ×.map (product_comm.hom ≫ G.mul) ≫ G.mul\n/- The following more explicit definition is hard on the elaborator;\n  Probably because of type-class inference for `×` -/\n-- calc\n--   H₁ × H₂ ⟶ (G × G) × G : ((G.incl H₁).map ≫ diag) ×.map (G.incl H₂).map\n--       ... ⟶ G × (G × G) : product_assoc.hom\n--       ... ⟶ G × G       : 𝟙 G.obj ×.map (product_comm.hom ≫ G.mul)\n--       ... ⟶ G           : G.mul\n\n/-- `C` centralizes `H` if `C × H ⟶ G` given by `(c,h) ↦ c*h*c⁻¹` is equal to the inclusion\n`H ⟶ G`. -/\ndef centralizes (C H : set G.obj.type) [is_closed_subgroup C] [is_closed_subgroup H] : Prop :=\nconjugation C H = π₂ ≫ (G.incl H).map\n\n/-- There is a unique maximal closed subgroup of `G` that centralizes `H` -/\ntheorem centralizer_unique (H : set G.obj.type) [is_closed_subgroup H] :\n  ∃!(C : set G.obj.type), is_maximal { C' : set G.obj.type |\n    ∃(h : is_closed_subgroup C'), by exactI centralizes C' H } C :=\nomitted\n\n/-- The centralizer of `H` is the unique maximal closed subgroup of `G` that centralizes `H` -/\ndef centralizer (H : set G.obj.type) [is_closed_subgroup H] : set G.obj.type :=\nclassical.the _ (centralizer_unique H)\n\n/-- The centralizer is a closed subgroup -/\ninstance centralizer.is_closed_subgroup (H : set G.obj.type) {h : is_closed_subgroup H} :\n  is_closed_subgroup (centralizer H) :=\nlet ⟨h, _⟩ := (classical.the_spec (centralizer_unique H)).1 in h\n\n/-- The center of `G` is the centralizer of `G` as closed subgroup of `G` -/\ndef center (G : affine_group K) : set G.obj.type :=\ncentralizer set.univ\n\n/-- `N` normalizes `H` if the conjugation map `N × H ⟶ G` factors through `H` -/\n-- this is a slightly different formulation than in the notes\ndef normalizes (N H : set G.obj.type) [is_closed_subgroup N] [is_closed_subgroup H] : Prop :=\n--factors_through (conjugation N H) (G.incl H).map\n∃(f : (sub N).obj × (sub H).obj ⟶ (sub H).obj), conjugation N H = f ≫ (G.incl H).map\n\n/-- If `N` normalizes `H` then `N` acts on `H` by conjucation -/\ndef conjugation_action {N H : set G.obj.type} [is_closed_subgroup N] [is_closed_subgroup H]\n  (h : normalizes N H) : group_action (sub N) (sub H).obj :=\nclassical.take_arbitrary_such_that (λ f, ⟨f, omitted, omitted⟩) h omitted\n\n/-- An affine group is almost simple if it has no proper normal closed connected subgroup.\n  Note: by our definition, every normal subgroup is automatically closed -/\ndef almost_simple (G : affine_group K) : Prop :=\n¬∃(T : set G.obj.type), T ≠ set.univ ∧ is_normal_subgroup T ∧ is_connected T\n\n/-- The type of a almost simple affine group is connected -/\nlemma connected_space_of_almost_simple : almost_simple G → connected_space G.obj.type :=\nomitted\n\nend algebraic_geometry\n\n", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/algebraic_geometry/affine_variety.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34754701792942383}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nTactics to prove specific models satisfy the preconditions of backprop_correct.\n-/\nimport data.list.set .tfacts .graph .predicates .expected_value .reparam .kl .tactics .program .mvn .tactics\n\n#print \"compiling prove_model_ok...\"\n\nnamespace certigrad\nopen T tactic list\n\nlemma mem_of_ne_mem_cons {α : Type*} {xs : list α} {x y : α} : x ∈ y :: xs → x ≠ y → x ∈ xs :=\nbegin\nintros H_in H_neq,\ndunfold has_mem.mem list.mem at H_in,\ncases H_in with H_eq H_in,\nexfalso, exact H_neq H_eq, exact H_in\nend\n\n@[cgsimp] lemma true_of_not_false : (¬ false) = true :=\nbegin apply propext, split, intro H, exact trivial, intros Ht Hf, exact Hf end\n\n@[cgsimp] lemma label_of (x y : label) : (ID.str x = ID.str y) = (x = y) :=\nbegin apply propext, split, intro H, injection H, intro H, rw H end\n\n@[cgsimp] lemma eq_of_self_eq {α : Type*} (x : α) : (x = x) = true := propext (eq_self_iff_true _)\n\n@[cgsimp] lemma and_true_left {P : Prop} : (true ∧ P) = P := propext (true_and _)\n@[cgsimp] lemma and_true_right {P : Prop} : (P ∧ true) = P := propext (and_true _)\n@[cgsimp] lemma and_false_left {P : Prop} : (false ∧ P) = false := propext (false_and _)\n@[cgsimp] lemma and_false_right {P : Prop} : (P ∧ false) = false := propext (and_false _)\n\n@[cgsimp] lemma or_false_left {P : Prop} : (false ∨ P) = P := propext (false_or _)\n@[cgsimp] lemma or_false_right {P : Prop} : (P ∨ false) = P := propext (or_false _)\n@[cgsimp] lemma or_true_left {P : Prop} : (true ∨ P) = true := propext (true_or _)\n@[cgsimp] lemma or_true_right {P : Prop} : (P ∨ true) = true := propext (or_true _)\n\n@[cgsimp] lemma true_of_imp_true {α : Sort*} : (α → true) = true := propext (iff.intro (λ H, trivial) (λ H x, trivial))\n\n@[cgsimp] lemma simp_nodup_nil {α : Type*} : @list.nodup α [] = true := pextt list.nodup_nil\n\n@[cgsimp] lemma simp_nodup_singleton {α : Type*} (a : α) : list.nodup [a] = true := pextt (list.nodup_singleton _)\n@[cgsimp] lemma simp_nodup_cons {α : Type*} {a : α} {l : list α} (H : a ∉ l) : list.nodup (a::l) = list.nodup l :=\nbegin apply propext, split, intro H', cases H' with H1 H2 H3 H4, exact H4, apply list.nodup_cons, exact H end\n\n@[cgsimp] lemma false_imp {P : Prop} : (false → P) = true :=\nbegin apply propext, split, intro H, exact trivial, intros Ht Hf, exfalso, exact Hf end\n@[cgsimp] lemma true_imp {P : Prop} : (true → P) = P :=\nbegin apply propext, split, intro H, exact H trivial, intros H Ht, exact H end\n\n@[cgsimp] lemma simp_mem_nil {α : Type*} (x : α) : (x ∈ @list.nil α) = false :=\nbegin apply pextf, apply list.not_mem_nil end\n\n@[cgsimp] lemma simp_mem_cons_neq {α : Type*} (x y : α) (xs : list α) : x ≠ y → (x ∈ y :: xs) = (x ∈ xs) :=\nbegin intro H_neq, apply propext, split, intro H_in, exact mem_of_ne_mem_cons H_in H_neq, intro H_in, exact or.inr H_in end\n\n@[cgsimp] lemma simp_mem_cons_eq {α : Type*} (x y : α) (xs : list α) : x = y → (x ∈ y :: xs) = true :=\nbegin intro H_eq, apply pextt, dunfold has_mem.mem list.mem, left, exact H_eq end\n\n@[cgsimp] lemma simp_nmem_nil {α : Type*} (x : α) : (x ∉ @list.nil α) = true :=\nbegin apply pextt, apply list.not_mem_nil end\n\n@[cgsimp] lemma simp_nmem_cons_eq {α : Type*} (x y : α) (xs : list α) : x ≠ y → (x ∉ y :: xs) = (x ∉ xs) :=\nbegin intro H_neq, apply propext, split, intros H_nin H_in, exact H_nin (or.inr H_in),\nintros H_nin H_in, exact H_nin (mem_of_ne_mem_cons H_in H_neq)\nend\n\n@[cgsimp] lemma simp_nmem_cons_neq {α : Type*} (x y : α) (xs : list α) : x = y → (x ∉ y :: xs) = false :=\nbegin intro H_eq, apply pextf, intro H, exact H (or.inl H_eq) end\n\n@[cgsimp] lemma false_of_cons_eq_nil {α : Type*} {x : α} {xs : list α} : (list.cons x xs = list.nil) = false :=\nbegin apply pextf, intro H, injection H end\n\n@[cgsimp] lemma simp_at_idx_nil {α : Type*} [inhabited α] (x : α) (idx : ℕ) : list.at_idx list.nil idx x = false :=\nbegin\napply pextf,\nintro H,\ndunfold list.at_idx at H,\ncases H with H1 H2,\nexact (nat.not_lt_zero idx) H1\nend\n\n@[cgsimp] lemma simp_at_idx_0 {α : Type*} [inhabited α] {x : α} {xs : list α} : list.at_idx (x::xs) 0 x = true :=\nbegin apply pextt, apply list.at_idx_0 end\n\n@[cgsimp] lemma simp_at_idx_cons {α : Type*} [inhabited α] {x : α} {xs : list α} {y : α} {idx : ℕ} :\n  list.at_idx (x::xs) (idx+1) y = list.at_idx xs idx y :=\nbegin\napply propext,\nsplit,\napply list.at_idx_of_cons,\napply list.at_idx_cons\nend\n\n@[cgsimp] lemma of_in_list_forall_cons {α : Type*} (ys : list α) (P : α → Prop) (y : α) : (∀ x, x ∈ list.cons y ys → P x) = (P y ∧ (∀ x, x ∈ ys → P x)) :=\nbegin\napply propext,\nsplit,\nintro H,\n{ split, exact H y list.mem_of_cons_same, intros x H_in, exact H x (or.inr H_in) },\n{\nintro H, cases H with HPy H, intros x H_in,\ncases classical.em (x = y) with H_eq H_neq,\nrw H_eq, exact HPy,\nexact H x (mem_of_ne_mem_cons H_in H_neq)\n}\nend\n\n@[cgsimp] lemma of_in_list_forall_nil {α : Type*} (P : α → Prop) (y : α) : (∀ (x : α), x ∈  @list.nil α → P x) = true :=\nbegin\napply pextt,\nintros x H_in_nil,\nexfalso,\nexact (list.not_mem_nil _) H_in_nil\nend\n\n@[cgsimp] lemma of_in_list_cons_neq {α : Type*} {P : Prop} (ys : list α) (x y : α) : x ≠ y → (x ∈ list.cons y ys → P) = (x ∈ ys → P) :=\nbegin\nintro H_neq,\napply propext,\nsplit,\nintro H,\nintro H_in,\napply H,\nexact or.inr H_in,\nintro H,\nintro H_in,\napply H,\nexact mem_of_ne_mem_cons H_in H_neq\nend\n\n@[cgsimp] lemma of_in_list_cons_eq {α : Type*} {P : Prop} (ys : list α) (x y : α) : x = y → (x ∈ list.cons y ys → P) = P :=\nbegin\nintro H_eq,\napply propext,\nsplit,\nintro H_in,\nexact H_in (or.inl H_eq),\nintros HP H_in,\nexact HP\nend\n\n@[cgsimp] lemma simp_sublist_cons {α : Type*} (xs : list α) (x : α) : (xs <+ x :: xs) = true :=\nbegin\napply pextt,\napply list.sublist_cons\nend\n\n@[cgsimp] lemma simp_sqrt_pos {shape : S} {x : T shape}: (0 < sqrt x) = (0 < x) :=\nbegin apply propext, split, apply pos_of_sqrt_pos, apply sqrt_pos end\n\n@[cgsimp] lemma simp_exp_pos {shape : S} {x : T shape} : (0 < exp x) = true :=\nbegin apply pextt, apply exp_pos end\n\n@[cgsimp] lemma simp_integrable_const (shape₁ shape₂ : S) (y : T shape₂) : is_integrable (λ (x : T shape₁), y) = true :=\nbegin apply pextt, apply is_integrable_const end\n\n@[cgsimp] lemma simp_nneg_of_pos {shape : S} {x : T shape} : x ≠ 0 = (0 < x ∨ x < 0) :=\nbegin apply propext, apply nz_iff end\n\n@[cgsimp] lemma simp_one_pos {shape : S} : (0 < (1 : T shape)) = true := pextt one_pos\n\n@[cgsimp] lemma simp_sigmoid_pos {shape : S} {x : T shape} : (0 < sigmoid x) = true := pextt sigmoid_pos\n@[cgsimp] lemma simp_sigmoid_lt1 {shape : S} {x : T shape} : (sigmoid x < 1) = true := pextt sigmoid_lt1\n\n-- TODO(dhs): weird lemma\n@[cgsimp] lemma simp_one_plus_pos {shape : S} {x : T shape} : 0 < 1 + x = (0 < x ∨ - 1 < x) :=\nbegin\napply propext,\nsplit,\nintro H,\nright, exact iff.mp one_plus_pos_iff H,\nintro H,\ncases H with H H,\napply one_plus_pos,\nexact H,\nexact iff.mpr one_plus_pos_iff H\nend\n\n@[cgsimp] lemma simp_has_key_insert_same (ref : reference) {x : T ref.2} (m : env) :\n  env.has_key ref (env.insert ref x m) = true := pextt (by apply env.has_key_insert_same)\n\n@[cgsimp] lemma simp_has_key_insert_diff (ref₁ ref₂ : reference) {x : T ref₂.2} (m : env) :\n  ref₁ ≠ ref₂ → env.has_key ref₁ (env.insert ref₂ x m) = env.has_key ref₁ m :=\nbegin\nintro H_neq,\napply propext,\nsplit,\napply env.has_key_insert_diff,\nexact H_neq,\napply env.has_key_insert\nend\n\n@[cgsimp] lemma simp_has_key_empty (ref : reference) : env.has_key ref env.mk = false :=\nbegin apply pextf, apply env.not_has_key_empty end\n\ndef is_not_used_downstream (tgt : reference) : list node → Prop\n| [] := true\n| (⟨ref, parents, op⟩ :: nodes) := tgt ∉ parents ∧ is_not_used_downstream nodes\n\ndef is_used_downstream (tgt : reference) : list node → Prop\n| [] := false\n| (⟨ref, parents, op⟩ :: nodes) := tgt ∈ parents ∨ is_used_downstream nodes\n\nattribute [cgsimp] is_not_used_downstream is_used_downstream\n\nlemma costs_helper : Π (costs : list ID) (tgt : reference) (m : env),\n  tgt.1 ∉ costs → compute_grad_slow costs [] m tgt = 0\n| []      tgt m H_nin := rfl\n| (c::cs) tgt m H_nin :=\nbegin\ndunfold compute_grad_slow,\nassert H : (tgt ≠ (c, [])),\n{\ncases tgt with tgt_1 tgt_2,\ndsimp at H_nin,\napply pair_neq_of_neq₁,\nexact list.ne_of_not_mem_cons H_nin\n},\nsimph,\ndunfold list.sumr,\nrw zero_add,\nexact costs_helper cs tgt m (list.not_mem_of_not_mem_cons H_nin)\nend\n\nlemma compute_grad_slow_det_not_used_helper (costs : list ID) : Π (nodes : list node) (m : env) (tgt : reference),\nis_not_used_downstream tgt nodes → tgt.1 ∉ costs →\ncompute_grad_slow costs nodes m tgt = 0\n| [] m tgt H_not_used H_tgt_nin_costs :=\nbegin\napply costs_helper, all_goals { assumption }\nend\n\n| (⟨ref, parents, operator.det op⟩::nodes) m tgt H_not_used H_tgt_nin_costs :=\nbegin\ndunfold compute_grad_slow,\ndunfold is_not_used_downstream at H_not_used,\nrw compute_grad_slow_det_not_used_helper nodes _ tgt H_not_used^.right H_tgt_nin_costs,\nrw zero_add,\nrw list.not_in_filter_of_match_riota,\nreflexivity,\nexact H_not_used^.left\nend\n\n| (⟨ref, parents, operator.rand op⟩::nodes) m tgt H_not_used H_tgt_nin_costs :=\nbegin\ndunfold compute_grad_slow,\ndunfold is_not_used_downstream at H_not_used,\nrw compute_grad_slow_det_not_used_helper,\nrw zero_add,\nrw list.not_in_filter_of_match_riota,\nreflexivity,\nexact H_not_used^.left,\nexact H_not_used^.right,\nexact H_tgt_nin_costs\nend\n\n@[cgsimp] lemma compute_grad_slow_det_not_used (costs : list ID) (ref : reference) (parents : list reference) (op : det.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_not_used_downstream tgt nodes → tgt.1 ∉ costs →\ncompute_grad_slow costs (⟨ref, parents, operator.det op⟩ :: nodes) m tgt\n=\nlist.sumr (list.map (λ (idx : ℕ), op^.pb (env.get_ks parents m)\n                                        (env.get ref m)\n                                        (compute_grad_slow costs nodes m ref)\n                                        idx\n                                        tgt.2)\n                    (list.filter (λ idx, tgt = list.dnth parents idx) (list.riota $ list.length parents))) :=\nbegin\nintros nodes m tgt H_not_used H_tgt_nin_costs,\ndunfold_occs compute_grad_slow [1],\nrw [compute_grad_slow_det_not_used_helper _ _ _ _ H_not_used H_tgt_nin_costs, zero_add]\nend\n\n@[cgsimp] lemma compute_grad_slow_det_used (costs : list ID) (ref : reference) (parents : list reference) (op : det.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_used_downstream tgt nodes ∨ tgt.1 ∈ costs →\ncompute_grad_slow costs (⟨ref, parents, operator.det op⟩ :: nodes) m tgt\n=\ncompute_grad_slow costs nodes m tgt +\nlist.sumr (list.map (λ (idx : ℕ), op^.pb (env.get_ks parents m)\n                                        (env.get ref m)\n                                        (compute_grad_slow costs nodes m ref)\n                                        idx\n                                        tgt.2)\n                    (list.filter (λ idx, tgt = list.dnth parents idx) (list.riota $ list.length parents))) :=\nbegin\nintros nodes m tgt H_is_used_or_cost,\nreflexivity\nend\n\n@[cgsimp] lemma compute_grad_slow_rand_not_used (costs : list ID) (ref : reference) (parents : list reference) (op : rand.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_not_used_downstream tgt nodes → tgt.1 ∉ costs →\ncompute_grad_slow costs (⟨ref, parents, operator.rand op⟩ :: nodes) m tgt\n=\nlist.sumr (list.map (λ (idx : ℕ), sum_downstream_costs nodes costs ref m ⬝ op^.glogpdf (env.get_ks parents m) (env.get ref m) idx tgt.2)\n                   (list.filter (λ idx, tgt = list.dnth parents idx) (list.riota $ list.length parents))) :=\nbegin\nintros nodes m tgt H_not_used H_tgt_nin_costs,\ndunfold_occs compute_grad_slow [1],\nrw [compute_grad_slow_det_not_used_helper _ _ _ _ H_not_used H_tgt_nin_costs, zero_add]\nend\n\n@[cgsimp] lemma compute_grad_slow_rand_used (costs : list ID) (ref : reference) (parents : list reference) (op : rand.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_used_downstream tgt nodes ∨ tgt.1 ∈ costs →\ncompute_grad_slow costs (⟨ref, parents, operator.rand op⟩ :: nodes) m tgt\n=\ncompute_grad_slow costs nodes m tgt +\nlist.sumr (list.map (λ (idx : ℕ), sum_downstream_costs nodes costs ref m ⬝ op^.glogpdf (env.get_ks parents m) (env.get ref m) idx tgt.2)\n                   (list.filter (λ idx, tgt = list.dnth parents idx) (list.riota $ list.length parents))) :=\nbegin\nintros nodes m tgt H_used_or_cost,\nreflexivity\nend\n\nattribute [cgsimp] compute_grad_slow.equations._eqn_1\n\nlemma grads_exist_at_simp_helper : Π (nodes : list node) (m : env) (tgt : reference),\n  is_not_used_downstream tgt nodes → grads_exist_at nodes m tgt\n| [] m tgt H_not_used := trivial\n\n| (⟨ref, parents, operator.det op⟩::nodes) m tgt H_not_used :=\nbegin\ndunfold grads_exist_at,\ndunfold is_not_used_downstream at H_not_used,\ndsimp,\nsplit,\napply grads_exist_at_simp_helper nodes _ tgt H_not_used^.right,\nintro H_in_parents,\nexfalso,\nexact H_not_used^.left H_in_parents\nend\n\n| (⟨ref, parents, operator.rand op⟩::nodes) m tgt H_not_used :=\nbegin\ndunfold grads_exist_at,\ndunfold is_not_used_downstream at H_not_used,\ndsimp,\nsplit,\nintro H_in_parents,\nexfalso,\nexact H_not_used^.left H_in_parents,\nintro y,\napply grads_exist_at_simp_helper nodes _ tgt H_not_used^.right\nend\n\n@[cgsimp] lemma grads_exist_at_det_not_used (ref : reference) (parents : list reference) (op : det.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_not_used_downstream tgt nodes →\ngrads_exist_at (⟨ref, parents, operator.det op⟩ :: nodes) m tgt\n=\nlet m' := env.insert ref (op^.f (env.get_ks parents m)) m in\n(tgt ∈ parents → op^.pre (env.get_ks parents m) ∧ grads_exist_at nodes (env.insert ref (op^.f (env.get_ks parents m)) m) ref) :=\nbegin\nintros nodes m tgt H_not_used,\ndunfold grads_exist_at,\ndsimp,\nrw (pextt (grads_exist_at_simp_helper _ _ _ H_not_used)),\nsimp\nend\n\n@[cgsimp] lemma grads_exist_at_det_used (ref : reference) (parents : list reference) (op : det.op parents^.p2 ref^.2)\n  : Π (nodes : list node) (m : env) (tgt : reference),\nis_used_downstream tgt nodes →\ngrads_exist_at (⟨ref, parents, operator.det op⟩ :: nodes) m tgt\n=\nlet m' := env.insert ref (op^.f (env.get_ks parents m)) m in\ngrads_exist_at nodes m' tgt ∧\n(tgt ∈ parents → op^.pre (env.get_ks parents m) ∧ grads_exist_at nodes (env.insert ref (op^.f (env.get_ks parents m)) m) ref) :=\nbegin\nintros nodes m tgt H, reflexivity\nend\n\nattribute [cgsimp] grads_exist_at.equations._eqn_1 grads_exist_at.equations._eqn_3\n\nattribute [cgsimp] all_pdfs_std can_diff_under_ints_pdfs_std\n\nattribute [cgsimp] T.smul_zero T.one_smul\n\nattribute [cgsimp] if_pos if_neg if_true if_false\n\nattribute [cgsimp] dif_pos dif_neg dif_ctx_simp_congr\n\nattribute [cgsimp] mvn_mvn_empirical_kl_int mvn_bernoulli_neglogpdf_int\n\nattribute [cgsimp] force_ok\n\nattribute [cgsimp] list.sumr list.map list.concat list.head list.tail list.riota list.filter list.length list.dnth\n                   list.sublist.refl list.nil_subset list.sublist_cons list.cons_append list.append_nil list.nil_append\n                   list.p1 list.p2\n\nattribute [cgsimp] hash_map.find_insert_eq hash_map.find_insert_ne\n\nattribute [cgsimp] zero_add add_zero\n\nattribute [cgsimp] dvec.head dvec.head2 dvec.head3\n\nattribute [cgsimp] integrate_kl integrate_mvn_kl integrate_kl_pre integrate_mvn_kl_pre reparameterize\nattribute [cgsimp] reparam reparameterize reparameterize_pre\n\nattribute [cgsimp] all_parents_in_env all_costs_scalars /- grads_exist_at -/ pdfs_exist_at uniq_ids\n                   is_gintegrable is_nabla_gintegrable is_gdifferentiable /- can_differentiate_under_integrals -/\n\nattribute [cgsimp] graph.to_dist operator.to_dist sum_costs /- compute_grad_slow -/\n\nattribute [cgsimp] E.E_bind E.E_ret\n\nattribute [cgsimp] ops.neg ops.exp ops.log ops.sqrt ops.add ops.sub ops.mul ops.div ops.sum\n                   ops.gemm ops.sigmoid ops.softplus ops.scale ops.mul_add ops.mvn_kl ops.bernoulli_neglogpdf\n\nattribute [cgsimp] det.op.f ops.neg.f ops.exp.f ops.log.f ops.sqrt.f ops.add.f ops.sub.f ops.mul.f ops.div.f ops.sum.f\n                   ops.gemm.f ops.sigmoid.f ops.softplus.f ops.scale.f ops.mul_add.f ops.mvn_kl.f ops.bernoulli_neglogpdf.f\n\nattribute [cgsimp] det.op.pre ops.neg.f_pre ops.exp.f_pre ops.log.f_pre ops.sqrt.f_pre ops.add.f_pre ops.sub.f_pre ops.mul.f_pre ops.div.f_pre ops.sum.f_pre\n                   ops.gemm.f_pre ops.sigmoid.f_pre ops.softplus.f_pre ops.scale.f_pre ops.mul_add.f_pre ops.mvn_kl.f_pre ops.bernoulli_neglogpdf.f_pre\n\nattribute [cgsimp] det.op.pb ops.neg.f_pb ops.exp.f_pb ops.log.f_pb ops.sqrt.f_pb ops.add.f_pb ops.sub.f_pb ops.mul.f_pb ops.div.f_pb ops.sum.f_pb\n                   ops.gemm.f_pb ops.sigmoid.f_pb ops.softplus.f_pb ops.scale.f_pb ops.mul_add.f_pb ops.mvn_kl.f_pb ops.bernoulli_neglogpdf.f_pb\n\nattribute [cgsimp] det.op.is_odiff det.op.pb_correct det.op.is_ocont\n\nattribute [cgsimp] rand.op.pdf rand.pdf.mvn rand.pdf.mvn_std\n                   rand.op.pre rand.op.mvn rand.op.mvn_std rand.pre.mvn rand.pre.mvn_std\n\nattribute [cgsimp] env.get_ks env.insert_all env.get_insert_same env.get_insert_diff\n\nattribute [cgsimp] list.insertion_sort list.ordered_insert\n\nattribute [cgsimp] program_to_graph program.program_to_graph_core program.get_id\n--                  program.unary_to_cwise1 program.binary_to_cwise2\n                  program.process_term program.empty_state program.process_rterm\n                  program_to_graph._match_1\n                  program.program_to_graph_core._match_1\n                  program.program_to_graph_core._match_2\n                  program.process_rterm._match_1\n                  program.process_term._match_6\n                  program.process_term._match_10\n                  program.process_term._match_13\n--                  program.process_term._match_16\n--                  program.process_term._match_17\n                  program.exp program.log program.sqrt program.softplus program.sigmoid\n\n@[cgsimp] lemma lift_t_label_to_term (x : label) : (lift_t x : program.term) = (program.term.id x) := rfl\n\nnamespace tactic\nopen tactic\n\nrun_cmd mk_simp_attr `pcgsimp\n\nattribute [pcgsimp] to_dist_congr_insert\n\nmeta def dcgsimp : tactic unit := do\n  s ← join_user_simp_lemmas tt [`cgsimp],\n  dsimp_core s\n\nmeta def gsimpt_core : Π (tac : tactic unit) (s_pre s_post : simp_lemmas) (e : expr), tactic (expr × expr) := λ tac s_pre s_post e,\ndo (a, new_tgt, pf) ← ext_simplify_core () {} s_post\n\n(λ u, failed)\n--pre\n--(λ a s r p e, failed)\n(λ a s r p e, do ⟨u, new_e, pr⟩ ← conv.apply_lemmas_core reducible s_pre tac r e,\n                 return ((), new_e, pr, tt))\n\n--post\n(λ a s r p e, do ⟨u, new_e, pr⟩ ← conv.apply_lemmas_core reducible s tac r e,\n                 return ((), new_e, pr, tt))\n`eq e,\n\n  return (new_tgt, pf)\n\nmeta def gsimpt (tac : tactic unit) : tactic unit := do\n  s_pre ← join_user_simp_lemmas tt [`pcgsimp],\n  s_post ← join_user_simp_lemmas tt [`cgsimp],\n  tgt ← target,\n  (new_tgt, pf) ← gsimpt_core tac s_pre s_post tgt,\n  replace_target new_tgt pf\n\n---------------\n\nrun_cmd mk_simp_attr `idne\n\nattribute [idne] id_str_ne_nat id_nat_ne_str id_str_ne_str id_nat_ne_nat label.neq_of_to_nat label.to_nat\n\nmeta def prove_ids_neq : tactic unit :=\ndo s ← join_user_simp_lemmas tt [`idne, `natne],\n   tgt ← target,\n   (new_tgt, pf) ← simplify s tgt,\n   replace_target new_tgt pf\n\nmeta def prove_refs_neq : tactic unit :=\ndo applyc `pair_neq_of_neq₁, prove_ids_neq\n\nmeta def cgsimpt (tac : tactic unit) : tactic unit := do\n  repeat $ first [gsimpt tac, prove_refs_neq, prove_ids_neq, triv]\n\nmeta def cgsimpn : ℕ → tactic unit\n| 0 := cgsimpt skip\n| (n+1) := cgsimpt (cgsimpn n)\n\nmeta def cgsimp : tactic unit := cgsimpn 100\n\nmeta def forall_idxs (tac_base tac_step : tactic unit) : expr → tactic unit\n| idx :=\ntac_base <|>\n(do cases idx [`_idx],\n    solve1 tac_step,\n    get_local `_idx >>= forall_idxs)\n\nmeta def prove_model_base : tactic unit :=\ndo exfalso,\n   H_at_idx ← get_local `H_at_idx,\nto_expr ```(list.at_idx_over %%H_at_idx dec_trivial) >>= exact\n\nmeta def prove_model_step : tactic unit :=\ndo H_at_idx ← get_local `H_at_idx,\n   mk_app `and.right [H_at_idx] >>= note `H_tgt_eq,\n   H_tgt_eq_type ← get_local `H_tgt_eq >>= infer_type,\n   s ← join_user_simp_lemmas true [`cgsimp],\n   (H_tgt_eq_new_type, pr) ← simplify s H_tgt_eq_type {},\n   get_local `H_tgt_eq >>= λ H_tgt_eq, replace_hyp H_tgt_eq H_tgt_eq_new_type pr,\n   get_local `H_tgt_eq >>= subst,\n   applyc `certigrad.backprop_correct,\n   trace \"nodup tgts nodes\",\n     solve1 cgsimp,\n   trace \"at_idx...\",\n     solve1 cgsimp,\n   trace \"well_formed_at...\",\n     solve1 (constructor >> all_goals cgsimp),\n   trace \"grads_exist_at...\",\n     solve1 (cgsimp),\n   trace \"pdfs_exist_at...\",\n     solve1 (cgsimp),\n   trace \"is_gintegrable...\",\n     solve1 (cgsimp >> prove_is_mvn_integrable),\n   trace \"can_diff_under_ints...\",\n     solve1 (applyc `certigrad.can_diff_under_ints_of_all_pdfs_std >> cgsimp >> prove_is_mvn_uintegrable),\n   trace \"prove_for_tgt done\"\n\nmeta def prove_model_ok : tactic unit :=\ndo -- unfold lets\n   whnf_target,\n   -- introduce hypotheses\n   [tgt, idx, H_at_idx] ← intros | fail \"can't intro hyps\",\n   -- repeated case-analysis on idx\n   -- TODO(dhs): async would be great but I run out of memory\n   forall_idxs prove_model_base prove_model_step idx\n\n\nend tactic\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/prove_model_ok.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.34753584748593885}}
{"text": "inductive KrivineInstruction\n  | Access (n: Nat)\n  | Grab (next: KrivineInstruction)\n  | Push (next: KrivineInstruction) (continuation: KrivineInstruction)\n\ninductive KrivineClosure\n  | pair (i: KrivineInstruction) (e: List KrivineClosure)\n\nnamespace Ex1\n\ndef KrivineEnv := List KrivineClosure\n\n-- We need to define a `SizeOf` instance for `KrivineEnv`. Otherwise, we cannot use the auto-generated well-founded relation in\n-- recursive definitions.\nnoncomputable instance : SizeOf KrivineEnv := inferInstanceAs (SizeOf (List KrivineClosure))\n-- We also need a `simp` theorem\n@[simp] theorem KrivineEnv.sizeOf_spec (env : KrivineEnv) : sizeOf env = sizeOf (α := List KrivineClosure) env := rfl\n\n-- It would be great to have a `deriving SizeOf` for definitions such as `KrivineEnv` above.\n\ndef KrivineEnv.depth (env : KrivineEnv) : Nat :=\n  match env with\n  | [] => 0\n  | KrivineClosure.pair u e :: closures => Nat.max (1 + depth e) (depth closures)\n\nend Ex1\n\n\nnamespace Ex2\n-- Same example, but we use `abbrev` to define `KrivineEnv`. Thus, we inherit the `SizeOf` instance for `List`s.\n\nabbrev KrivineEnv := List KrivineClosure\n\ndef KrivineEnv.depth (env : KrivineEnv) : Nat :=\n  match env with\n  | [] => 0\n  | KrivineClosure.pair u e :: closures => Nat.max (1 + depth e) (depth closures)\n\nend Ex2\n\nnamespace Ex3\n-- Same example, but using `structure` instead of `def`\n\nstructure KrivineEnv where\n  env : List KrivineClosure\n\ndef KrivineEnv.depth (env : KrivineEnv) : Nat :=\n  match env with\n  | { env := [] } => 0\n  | { env := KrivineClosure.pair u e :: closures } => Nat.max (1 + depth { env := e }) (depth { env := closures })\n\nend Ex3\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/krivine.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.34753584357560835}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n\nQuotient types.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.sigma.basic\nimport Mathlib.Lean3Lib.init.logic\nimport Mathlib.Lean3Lib.init.propext\nimport Mathlib.Lean3Lib.init.data.setoid\n\nuniverses u v u_a u_b u_c \n\nnamespace Mathlib\n\n/- We import propext here, otherwise we would need a quot.lift for propositions. -/\n\n-- iff can now be used to do substitutions in a calculation\n\ntheorem iff_subst {a : Prop} {b : Prop} {p : Prop → Prop} (h₁ : a ↔ b) (h₂ : p a) : p b :=\n  propext h₁ ▸ h₂\n\nnamespace quot\n\n\naxiom sound {α : Sort u} {r : α → α → Prop} {a : α} {b : α} : r a b → Quot.mk r a = Quot.mk r b\n\nprotected theorem lift_beta {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β)\n    (c : ∀ (a b : α), r a b → f a = f b) (a : α) : Quot.lift f c (Quot.mk r a) = f a :=\n  rfl\n\nprotected theorem ind_beta {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop}\n    (p : ∀ (a : α), β (Quot.mk r a)) (a : α) : Quot.ind p (Quot.mk r a) = p a :=\n  rfl\n\nprotected def lift_on {α : Sort u} {β : Sort v} {r : α → α → Prop} (q : Quot r) (f : α → β)\n    (c : ∀ (a b : α), r a b → f a = f b) : β :=\n  Quot.lift f c q\n\nprotected theorem induction_on {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} (q : Quot r)\n    (h : ∀ (a : α), β (Quot.mk r a)) : β q :=\n  Quot.ind h q\n\ntheorem exists_rep {α : Sort u} {r : α → α → Prop} (q : Quot r) : ∃ (a : α), Quot.mk r a = q :=\n  quot.induction_on q fun (a : α) => Exists.intro a rfl\n\nprotected def indep {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v}\n    (f : (a : α) → β (Quot.mk r a)) (a : α) : psigma β :=\n  psigma.mk (Quot.mk r a) (f a)\n\nprotected theorem indep_coherent {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v}\n    (f : (a : α) → β (Quot.mk r a)) (h : ∀ (a b : α) (p : r a b), Eq._oldrec (f a) (sound p) = f b)\n    (a : α) (b : α) : r a b → quot.indep f a = quot.indep f b :=\n  fun (e : r a b) => psigma.eq (sound e) (h a b e)\n\nprotected theorem lift_indep_pr1 {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v}\n    (f : (a : α) → β (Quot.mk r a)) (h : ∀ (a b : α) (p : r a b), Eq._oldrec (f a) (sound p) = f b)\n    (q : Quot r) : psigma.fst (Quot.lift (quot.indep f) (quot.indep_coherent f h) q) = q :=\n  Quot.ind (fun (a : α) => Eq.refl (psigma.fst (quot.indep f a))) q\n\nprotected def rec {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v}\n    (f : (a : α) → β (Quot.mk r a)) (h : ∀ (a b : α) (p : r a b), Eq._oldrec (f a) (sound p) = f b)\n    (q : Quot r) : β q :=\n  eq.rec_on (quot.lift_indep_pr1 f h q)\n    (psigma.snd (Quot.lift (quot.indep f) (quot.indep_coherent f h) q))\n\nprotected def rec_on {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v} (q : Quot r)\n    (f : (a : α) → β (Quot.mk r a))\n    (h : ∀ (a b : α) (p : r a b), Eq._oldrec (f a) (sound p) = f b) : β q :=\n  quot.rec f h q\n\nprotected def rec_on_subsingleton {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v}\n    [h : ∀ (a : α), subsingleton (β (Quot.mk r a))] (q : Quot r) (f : (a : α) → β (Quot.mk r a)) :\n    β q :=\n  quot.rec f sorry q\n\nprotected def hrec_on {α : Sort u} {r : α → α → Prop} {β : Quot r → Sort v} (q : Quot r)\n    (f : (a : α) → β (Quot.mk r a)) (c : ∀ (a b : α), r a b → f a == f b) : β q :=\n  quot.rec_on q f sorry\n\nend quot\n\n\ndef quotient {α : Sort u} (s : setoid α) := Quot setoid.r\n\nnamespace quotient\n\n\nprotected def mk {α : Sort u} [s : setoid α] (a : α) : quotient s := Quot.mk setoid.r a\n\ndef sound {α : Sort u} [s : setoid α] {a : α} {b : α} : a ≈ b → quotient.mk a = quotient.mk b :=\n  quot.sound\n\nprotected def lift {α : Sort u} {β : Sort v} [s : setoid α] (f : α → β) :\n    (∀ (a b : α), a ≈ b → f a = f b) → quotient s → β :=\n  Quot.lift f\n\nprotected theorem ind {α : Sort u} [s : setoid α] {β : quotient s → Prop} :\n    (∀ (a : α), β (quotient.mk a)) → ∀ (q : quotient s), β q :=\n  Quot.ind\n\nprotected def lift_on {α : Sort u} {β : Sort v} [s : setoid α] (q : quotient s) (f : α → β)\n    (c : ∀ (a b : α), a ≈ b → f a = f b) : β :=\n  quot.lift_on q f c\n\nprotected theorem induction_on {α : Sort u} [s : setoid α] {β : quotient s → Prop} (q : quotient s)\n    (h : ∀ (a : α), β (quotient.mk a)) : β q :=\n  quot.induction_on q h\n\ntheorem exists_rep {α : Sort u} [s : setoid α] (q : quotient s) : ∃ (a : α), quotient.mk a = q :=\n  quot.exists_rep q\n\nprotected def rec {α : Sort u} [s : setoid α] {β : quotient s → Sort v}\n    (f : (a : α) → β (quotient.mk a))\n    (h : ∀ (a b : α) (p : a ≈ b), Eq._oldrec (f a) (sound p) = f b) (q : quotient s) : β q :=\n  quot.rec f h q\n\nprotected def rec_on {α : Sort u} [s : setoid α] {β : quotient s → Sort v} (q : quotient s)\n    (f : (a : α) → β (quotient.mk a))\n    (h : ∀ (a b : α) (p : a ≈ b), Eq._oldrec (f a) (sound p) = f b) : β q :=\n  quot.rec_on q f h\n\nprotected def rec_on_subsingleton {α : Sort u} [s : setoid α] {β : quotient s → Sort v}\n    [h : ∀ (a : α), subsingleton (β (quotient.mk a))] (q : quotient s)\n    (f : (a : α) → β (quotient.mk a)) : β q :=\n  quot.rec_on_subsingleton q f\n\nprotected def hrec_on {α : Sort u} [s : setoid α] {β : quotient s → Sort v} (q : quotient s)\n    (f : (a : α) → β (quotient.mk a)) (c : ∀ (a b : α), a ≈ b → f a == f b) : β q :=\n  quot.hrec_on q f c\n\nprotected def lift₂ {α : Sort u_a} {β : Sort u_b} {φ : Sort u_c} [s₁ : setoid α] [s₂ : setoid β]\n    (f : α → β → φ)\n    (c : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂)\n    (q₁ : quotient s₁) (q₂ : quotient s₂) : φ :=\n  quotient.lift (fun (a₁ : α) => quotient.lift (f a₁) sorry q₂) sorry q₁\n\nprotected def lift_on₂ {α : Sort u_a} {β : Sort u_b} {φ : Sort u_c} [s₁ : setoid α] [s₂ : setoid β]\n    (q₁ : quotient s₁) (q₂ : quotient s₂) (f : α → β → φ)\n    (c : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂) : φ :=\n  quotient.lift₂ f c q₁ q₂\n\nprotected theorem ind₂ {α : Sort u_a} {β : Sort u_b} [s₁ : setoid α] [s₂ : setoid β]\n    {φ : quotient s₁ → quotient s₂ → Prop}\n    (h : ∀ (a : α) (b : β), φ (quotient.mk a) (quotient.mk b)) (q₁ : quotient s₁)\n    (q₂ : quotient s₂) : φ q₁ q₂ :=\n  quotient.ind (fun (a₁ : α) => quotient.ind (fun (a₂ : β) => h a₁ a₂) q₂) q₁\n\nprotected theorem induction_on₂ {α : Sort u_a} {β : Sort u_b} [s₁ : setoid α] [s₂ : setoid β]\n    {φ : quotient s₁ → quotient s₂ → Prop} (q₁ : quotient s₁) (q₂ : quotient s₂)\n    (h : ∀ (a : α) (b : β), φ (quotient.mk a) (quotient.mk b)) : φ q₁ q₂ :=\n  quotient.ind (fun (a₁ : α) => quotient.ind (fun (a₂ : β) => h a₁ a₂) q₂) q₁\n\nprotected theorem induction_on₃ {α : Sort u_a} {β : Sort u_b} {φ : Sort u_c} [s₁ : setoid α]\n    [s₂ : setoid β] [s₃ : setoid φ] {δ : quotient s₁ → quotient s₂ → quotient s₃ → Prop}\n    (q₁ : quotient s₁) (q₂ : quotient s₂) (q₃ : quotient s₃)\n    (h : ∀ (a : α) (b : β) (c : φ), δ (quotient.mk a) (quotient.mk b) (quotient.mk c)) :\n    δ q₁ q₂ q₃ :=\n  quotient.ind\n    (fun (a₁ : α) => quotient.ind (fun (a₂ : β) => quotient.ind (fun (a₃ : φ) => h a₁ a₂ a₃) q₃) q₂)\n    q₁\n\ntheorem exact {α : Sort u} [s : setoid α] {a : α} {b : α} : quotient.mk a = quotient.mk b → a ≈ b :=\n  fun (h : quotient.mk a = quotient.mk b) => eq_imp_rel h\n\nprotected def rec_on_subsingleton₂ {α : Sort u_a} {β : Sort u_b} [s₁ : setoid α] [s₂ : setoid β]\n    {φ : quotient s₁ → quotient s₂ → Sort u_c}\n    [h : ∀ (a : α) (b : β), subsingleton (φ (quotient.mk a) (quotient.mk b))] (q₁ : quotient s₁)\n    (q₂ : quotient s₂) (f : (a : α) → (b : β) → φ (quotient.mk a) (quotient.mk b)) : φ q₁ q₂ :=\n  quotient.rec_on_subsingleton q₁\n    fun (a : α) => quotient.rec_on_subsingleton q₂ fun (b : β) => f a b\n\nend quotient\n\n\ninductive eqv_gen {α : Type u} (r : α → α → Prop) : α → α → Prop where\n| rel : ∀ (x y : α), r x y → eqv_gen r x y\n| refl : ∀ (x : α), eqv_gen r x x\n| symm : ∀ (x y : α), eqv_gen r x y → eqv_gen r y x\n| trans : ∀ (x y z : α), eqv_gen r x y → eqv_gen r y z → eqv_gen r x z\n\ntheorem eqv_gen.is_equivalence {α : Type u} (r : α → α → Prop) : equivalence (eqv_gen r) :=\n  mk_equivalence (eqv_gen r) eqv_gen.refl eqv_gen.symm eqv_gen.trans\n\ndef eqv_gen.setoid {α : Type u} (r : α → α → Prop) : setoid α :=\n  setoid.mk (eqv_gen r) (eqv_gen.is_equivalence r)\n\ntheorem quot.exact {α : Type u} (r : α → α → Prop) {a : α} {b : α} (H : Quot.mk r a = Quot.mk r b) :\n    eqv_gen r a b :=\n  quotient.exact\n    (congr_arg (Quot.lift quotient.mk fun (x y : α) (h : r x y) => quot.sound (eqv_gen.rel x y h))\n      H)\n\ntheorem quot.eqv_gen_sound {α : Type u} {r : α → α → Prop} {a : α} {b : α} (H : eqv_gen r a b) :\n    Quot.mk r a = Quot.mk r b :=\n  sorry\n\nprotected instance quotient.decidable_eq {α : Sort u} {s : setoid α}\n    [d : (a b : α) → Decidable (a ≈ b)] : DecidableEq (quotient s) :=\n  fun (q₁ q₂ : quotient s) => quotient.rec_on_subsingleton₂ q₁ q₂ fun (a₁ a₂ : α) => sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/quot_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.3474975536664529}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport tactic.ring\nimport data.num.lemmas\nimport data.tree\n\n/-!\n# ring2\n\nAn experimental variant on the `ring` tactic that uses computational\nreflection instead of proof generation. Useful for kernel benchmarking.\n-/\n\nnamespace tree\n\n/-- `(reflect' t u α)` quasiquotes a tree `(t: tree expr)` of quoted\nvalues of type `α` at level `u` into an `expr` which reifies to a `tree α`\ncontaining the reifications of the `expr`s from the original `t`. -/\nprotected meta def reflect' (u : level) (α : expr) : tree expr → expr\n| tree.nil := (expr.const ``tree.nil [u] : expr) α\n| (tree.node a t₁ t₂) :=\n  (expr.const ``tree.node [u] : expr) α a t₁.reflect' t₂.reflect'\n\n/-- Returns an element indexed by `n`, or zero if `n` isn't a valid index.\nSee `tree.get`. -/\nprotected def get_or_zero {α} [has_zero α] (t : tree α) (n : pos_num) : α :=\n  t.get_or_else n 0\n\nend tree\n\nnamespace tactic.ring2\n\n/-- A reflected/meta representation of an expression in a commutative\nsemiring. This representation is a direct translation of such\nexpressions - see `horner_expr` for a normal form. -/\n@[derive has_reflect]\ninductive csring_expr\n/- (atom n) is an opaque element of the csring. For example,\na local variable in the context. n indexes into a storage\nof such atoms - a `tree α`. -/\n| atom : pos_num → csring_expr\n/- (const n) is technically the csring's one, added n times.\nOr the zero if n is 0. -/\n| const : num → csring_expr\n| add : csring_expr → csring_expr → csring_expr\n| mul : csring_expr → csring_expr → csring_expr\n| pow : csring_expr → num → csring_expr\n\nnamespace csring_expr\n\ninstance : inhabited csring_expr := ⟨const 0⟩\n\n/-- Evaluates a reflected `csring_expr` into an element of the\noriginal `comm_semiring` type `α`, retrieving opaque elements\n(atoms) from the tree `t`. -/\ndef eval {α} [comm_semiring α] (t : tree α) : csring_expr → α\n| (atom n)  := t.get_or_zero n\n| (const n) := n\n| (add x y) := eval x + eval y\n| (mul x y) := eval x * eval y\n| (pow x n) := eval x ^ (n : ℕ)\n\nend csring_expr\n\n/-- An efficient representation of expressions in a commutative\nsemiring using the sparse Horner normal form. This type admits\nnon-optimal instantiations (e.g. `P` can be represented as `P+0+0`),\nso to get good performance out of it, care must be taken to maintain\nan optimal, *canonical* form. -/\n@[derive decidable_eq]\ninductive horner_expr\n/- (const n) is a constant n in the csring, similarly to the same\nconstructor in `csring_expr`. This one, however, can be negative. -/\n| const : znum → horner_expr\n/- (horner a x n b) is a*xⁿ + b, where x is the x-th atom\nin the atom tree. -/\n| horner : horner_expr → pos_num → num → horner_expr → horner_expr\n\nnamespace horner_expr\n\n/-- True iff the `horner_expr` argument is a valid `csring_expr`.\nFor that to be the case, all its constants must be non-negative. -/\ndef is_cs : horner_expr → Prop\n| (const n) := ∃ m:num, n = m.to_znum\n| (horner a x n b) := is_cs a ∧ is_cs b\n\ninstance : has_zero horner_expr := ⟨const 0⟩\ninstance : has_one horner_expr := ⟨const 1⟩\ninstance : inhabited horner_expr := ⟨0⟩\n\n/-- Represent a `csring_expr.atom` in Horner form. -/\ndef atom (n : pos_num) : horner_expr := horner 1 n 1 0\n\ndef to_string : horner_expr → string\n| (const n) := _root_.repr n\n| (horner a x n b) :=\n    \"(\" ++ to_string a ++ \") * x\" ++ _root_.repr x ++ \"^\"\n        ++ _root_.repr n ++ \" + \" ++ to_string b\ninstance : has_to_string horner_expr := ⟨to_string⟩\n\n/-- Alternative constructor for (horner a x n b) which maintains canonical\nform by simplifying special cases of `a`. -/\ndef horner' (a : horner_expr)\n  (x : pos_num) (n : num) (b : horner_expr) : horner_expr :=\nmatch a with\n| const q := if q = 0 then b else horner a x n b\n| horner a₁ x₁ n₁ b₁ :=\n  if x₁ = x ∧ b₁ = 0 then horner a₁ x (n₁ + n) b\n  else horner a x n b\nend\n\ndef add_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then e else begin\n  induction e with n a x n b A B,\n  { exact const (k + n) },\n  { exact horner a x n B }\nend\n\ndef add_aux (a₁ : horner_expr) (A₁ : horner_expr → horner_expr) (x₁ : pos_num) :\n  horner_expr → num → horner_expr → (horner_expr → horner_expr) → horner_expr\n| (const n₂)           n₁ b₁ B₁ := add_const n₂ (horner a₁ x₁ n₁ b₁)\n| (horner a₂ x₂ n₂ b₂) n₁ b₁ B₁ :=\n  let e₂ := horner a₂ x₂ n₂ b₂ in\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner a₂ x₂ n₂ (add_aux b₂ n₁ b₁ B₁)\n  | ordering.eq :=\n    match num.sub' n₁ n₂ with\n    | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n    | (znum.pos k) := horner (add_aux a₂ k 0 id) x₁ n₂ (B₁ b₂)\n    | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n    end\n  end\n\ndef add : horner_expr → horner_expr → horner_expr\n| (const n₁)           e₂ := add_const n₁ e₂\n| (horner a₁ x₁ n₁ b₁) e₂ := add_aux a₁ (add a₁) x₁ e₂ n₁ b₁ (add b₁)\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact add_const n₁ e₂ },\n  exact match e₂ with e₂ := begin\n    induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁;\n    let e₁ := horner a₁ x₁ n₁ b₁,\n    { exact add_const n₂ e₁ },\n    let e₂ := horner a₂ x₂ n₂ b₂,\n    exact match pos_num.cmp x₁ x₂ with\n    | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n    | ordering.gt := horner a₂ x₂ n₂ (B₂ n₁ b₁)\n    | ordering.eq :=\n      match num.sub' n₁ n₂ with\n      | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n      | (znum.pos k) := horner (A₂ k 0) x₁ n₂ (B₁ b₂)\n      | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n      end\n    end\n  end end\nend-/\n\ndef neg (e : horner_expr) : horner_expr :=\nbegin\n  induction e with n a x n b A B,\n  { exact const (-n) },\n  { exact horner A x n B }\nend\n\ndef mul_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then 0 else if k = 1 then e else begin\n  induction e with n a x n b A B,\n  { exact const (n * k) },\n  { exact horner A x n B }\nend\n\ndef mul_aux (a₁ x₁ n₁ b₁) (A₁ B₁ : horner_expr → horner_expr) :\n  horner_expr → horner_expr\n| (const n₂) := mul_const n₂ (horner a₁ x₁ n₁ b₁)\n| e₂@(horner a₂ x₂ n₂ b₂) :=\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner (A₁ e₂) x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner (mul_aux a₂) x₂ n₂ (mul_aux b₂)\n  | ordering.eq := let haa := horner' (mul_aux a₂) x₁ n₂ 0 in\n    if b₂ = 0 then haa else haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂))\n  end\n\ndef mul : horner_expr → horner_expr → horner_expr\n| (const n₁)           := mul_const n₁\n| (horner a₁ x₁ n₁ b₁) := mul_aux a₁ x₁ n₁ b₁ (mul a₁) (mul b₁).\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact mul_const n₁ e₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂;\n  let e₁ := horner a₁ x₁ n₁ b₁,\n  { exact mul_const n₂ e₁ },\n  let e₂ := horner a₂ x₂ n₂ b₂,\n  cases pos_num.cmp x₁ x₂,\n  { exact horner (A₁ e₂) x₁ n₁ (B₁ e₂) },\n  { let haa := horner' A₂ x₁ n₂ 0,\n    exact if b₂ = 0 then haa else\n      haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂)) },\n  { exact horner A₂ x₂ n₂ B₂ }\nend-/\n\ninstance : has_add horner_expr := ⟨add⟩\ninstance : has_neg horner_expr := ⟨neg⟩\ninstance : has_mul horner_expr := ⟨mul⟩\n\ndef pow (e : horner_expr) : num → horner_expr\n| 0 := 1\n| (num.pos p) := begin\n  induction p with p ep p ep,\n  { exact e },\n  { exact (ep.mul ep).mul e },\n  { exact ep.mul ep }\nend\n\ndef inv (e : horner_expr) : horner_expr := 0\n\n/-- Brings expressions into Horner normal form. -/\ndef of_csexpr : csring_expr → horner_expr\n| (csring_expr.atom n)  := atom n\n| (csring_expr.const n) := const n.to_znum\n| (csring_expr.add x y) := (of_csexpr x).add (of_csexpr y)\n| (csring_expr.mul x y) := (of_csexpr x).mul (of_csexpr y)\n| (csring_expr.pow x n) := (of_csexpr x).pow n\n\n/-- Evaluates a reflected `horner_expr` - see `csring_expr.eval`. -/\ndef cseval {α} [comm_semiring α] (t : tree α) : horner_expr → α\n| (const n)        := n.abs\n| (horner a x n b) := tactic.ring.horner (cseval a) (t.get_or_zero x) n (cseval b)\n\ntheorem cseval_atom {α} [comm_semiring α] (t : tree α)\n  (n : pos_num) : (atom n).is_cs ∧ cseval t (atom n) = t.get_or_zero n :=\n⟨⟨⟨1, rfl⟩, ⟨0, rfl⟩⟩, (tactic.ring.horner_atom _).symm⟩\n\ntheorem cseval_add_const {α} [comm_semiring α] (t : tree α)\n  (k : num) {e : horner_expr} (cs : e.is_cs) :\n  (add_const k.to_znum e).is_cs ∧\n    cseval t (add_const k.to_znum e) = k + cseval t e :=\nbegin\n  simp [add_const],\n  cases k; simp! *,\n  simp [show znum.pos k ≠ 0, from dec_trivial],\n  induction e with n a x n b A B; simp *,\n  { rcases cs with ⟨n, rfl⟩,\n    refine ⟨⟨n + num.pos k, by simp [add_comm]; refl⟩, _⟩,\n    cases n; simp! },\n  { rcases B cs.2 with ⟨csb, h⟩, simp! [*, cs.1],\n    rw [← tactic.ring.horner_add_const, add_comm], rw add_comm }\nend\n\ntheorem cseval_horner' {α} [comm_semiring α] (t : tree α)\n  (a x n b) (h₁ : is_cs a) (h₂ : is_cs b) :\n  (horner' a x n b).is_cs ∧ cseval t (horner' a x n b) =\n    tactic.ring.horner (cseval t a) (t.get_or_zero x) n (cseval t b) :=\nbegin\n  cases a with n₁ a₁ x₁ n₁ b₁; simp [horner']; split_ifs,\n  { simp! [*, tactic.ring.horner] },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ },\n  { refine ⟨⟨h₁.1, h₂⟩, eq.symm _⟩, simp! *,\n    apply tactic.ring.horner_horner, simp },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ }\nend\n\ntheorem cseval_add {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (add e₁ e₂).is_cs ∧\n  cseval t (add e₁ e₂) = cseval t e₁ + cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa using cseval_add_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simp! [cseval_add_const t n₂ cs₁, add_comm] },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { rcases B₁ csb₁ cs₂ with ⟨csh, h⟩,\n    refine ⟨⟨csa₁, csh⟩, eq.symm _⟩,\n    apply tactic.ring.horner_add_const,\n    exact h.symm },\n  { cases C,\n    have B0 : is_cs 0 → ∀ {e₂ : horner_expr}, is_cs e₂ →\n      is_cs (add 0 e₂) ∧ cseval t (add 0 e₂) = cseval t 0 + cseval t e₂ :=\n      λ _ e₂ c, ⟨c, (zero_add _).symm⟩,\n     cases e : num.sub' n₁ n₂ with k k; simp!,\n    { have : n₁ = n₂,\n      { have := congr_arg (coe : znum → ℤ) e,\n        simp at this,\n        have := sub_eq_zero.1 this,\n        rw [← num.to_nat_to_int, ← num.to_nat_to_int] at this,\n        exact num.to_nat_inj.1 (int.coe_nat_inj this) },\n      subst n₂,\n      rcases cseval_horner' _ _ _ _ _ _ _ with ⟨csh, h⟩,\n      { refine ⟨csh, h.trans (eq.symm _)⟩,\n        simp *,\n        apply tactic.ring.horner_add_horner_eq; try {refl} },\n      all_goals {simp! *} },\n    { simp [B₁ csb₁ csb₂, add_comm],\n      rcases A₂ csa₂ _ _ B0 ⟨csa₁, 0, rfl⟩ with ⟨csh, h⟩,\n      refine ⟨csh, eq.symm _⟩,\n      rw [show id = add 0, from rfl, h],\n      apply tactic.ring.horner_add_horner_gt,\n      { change (_ + k : ℕ) = _,\n        rw [← int.coe_nat_inj', int.coe_nat_add,\n          eq_comm, ← sub_eq_iff_eq_add'],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      { refl },\n      { apply add_comm } },\n    { have : (horner a₂ x₁ (num.pos k) 0).is_cs := ⟨csa₂, 0, rfl⟩,\n      simp [B₁ csb₁ csb₂, A₁ csa₁ this],\n      symmetry, apply tactic.ring.horner_add_horner_lt,\n      { change (_ + k : ℕ) = _,\n          rw [← int.coe_nat_inj', int.coe_nat_add,\n            eq_comm, ← sub_eq_iff_eq_add', ← neg_inj, neg_sub],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      all_goals { refl } } },\n  { rcases B₂ csb₂ _ _ B₁ ⟨csa₁, csb₁⟩ with ⟨csh, h⟩,\n    refine ⟨⟨csa₂, csh⟩, eq.symm _⟩,\n    apply tactic.ring.const_add_horner,\n    simp [h] }\nend\n\n\n\ntheorem cseval_mul {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (mul e₁ e₂).is_cs ∧\n  cseval t (mul e₁ e₂) = cseval t e₁ * cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa [mul_comm] using cseval_mul_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simpa! using cseval_mul_const t n₂ cs₁ },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases A₂ csa₂ with csA₂ hA₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { simp [A₁ csa₁ cs₂, B₁ csb₁ cs₂],\n    symmetry, apply tactic.ring.horner_mul_const; refl },\n  { cases cseval_horner' t _ x₁ n₂ 0 csA₂ ⟨0, rfl⟩ with csh₁ h₁,\n    cases C, split_ifs,\n    { subst b₂,\n      refine ⟨csh₁, h₁.trans (eq.symm _)⟩,\n      apply tactic.ring.horner_mul_horner_zero; try {refl},\n      simp! [hA₂] },\n    { cases A₁ csa₁ csb₂ with csA₁ hA₁,\n      cases cseval_add t csh₁ _ with csh₂ h₂,\n      { refine ⟨csh₂, h₂.trans (eq.symm _)⟩,\n        apply tactic.ring.horner_mul_horner; try {refl},\n        simp! * },\n      exact ⟨csA₁, (B₁ csb₁ csb₂).1⟩ } },\n  { simp [A₂ csa₂, B₂ csb₂], rw [mul_comm, eq_comm],\n    apply tactic.ring.horner_const_mul,\n    {apply mul_comm}, {refl} },\nend\n\ntheorem cseval_pow {α} [comm_semiring α] (t : tree α)\n  {x : horner_expr} (cs : x.is_cs) :\n  ∀ (n : num), (pow x n).is_cs ∧\n    cseval t (pow x n) = cseval t x ^ (n : ℕ)\n| 0 := ⟨⟨1, rfl⟩, (pow_zero _).symm⟩\n| (num.pos p) := begin\n  simp [pow], induction p with p ep p ep,\n  { simp * },\n  { simp [pow_bit1],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    cases cseval_mul t cs₀ cs with cs₁ h₁,\n    simp * },\n  { simp [pow_bit0],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    simp * }\nend\n\n/-- For any given tree `t` of atoms and any reflected expression `r`,\nthe Horner form of `r` is a valid csring expression, and under `t`,\nthe Horner form evaluates to the same thing as `r`. -/\ntheorem cseval_of_csexpr {α} [comm_semiring α] (t : tree α) :\n  ∀ (r : csring_expr), (of_csexpr r).is_cs ∧ cseval t (of_csexpr r) = r.eval t\n| (csring_expr.atom n)  := cseval_atom _ _\n| (csring_expr.const n) := ⟨⟨n, rfl⟩, by cases n; refl⟩\n| (csring_expr.add x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_add t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.mul x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_mul t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.pow x n) :=\n  let ⟨cs, h⟩ := cseval_of_csexpr x,\n      ⟨cs, h⟩ := cseval_pow t cs n in ⟨cs, by simp! [h, *]⟩\n\nend horner_expr\n\n/-- The main proof-by-reflection theorem. Given reflected csring expressions\n`r₁` and `r₂` plus a storage `t` of atoms, if both expressions go to the\nsame Horner normal form, then the original non-reflected expressions are\nequal. `H` follows from kernel reduction and is therefore `rfl`. -/\ntheorem correctness {α} [comm_semiring α] (t : tree α) (r₁ r₂ : csring_expr)\n  (H : horner_expr.of_csexpr r₁ = horner_expr.of_csexpr r₂) :\n  r₁.eval t = r₂.eval t :=\nby repeat {rw ← (horner_expr.cseval_of_csexpr t _).2}; rw H\n\n/-- Reflects a csring expression into a `csring_expr`, together\nwith a dlist of atoms, i.e. opaque variables over which the\nexpression is a polynomial. -/\nmeta def reflect_expr : expr → csring_expr × dlist expr\n| `(%%e₁ + %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂, l₁ ++ l₂)\n/-| `(%%e₁ - %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂.neg, l₁ ++ l₂)\n| `(- %%e) := let (r, l) := reflect_expr e in (r.neg, l)-/\n| `(%%e₁ * %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂, l₁ ++ l₂)\n/-| `(has_inv.inv %%e) := let (r, l) := reflect_expr e in (r.neg, l)\n| `(%%e₁ / %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂.inv, l₁ ++ l₂)-/\n| e@`(%%e₁ ^ %%e₂) :=\n  match reflect_expr e₁, expr.to_nat e₂ with\n  | (r₁, l₁), some n₂ := (r₁.pow (num.of_nat' n₂), l₁)\n  | (r₁, l₁), none := (csring_expr.atom 1, dlist.singleton e)\n  end\n| e := match expr.to_nat e with\n  | some n := (csring_expr.const (num.of_nat' n), dlist.empty)\n  | none := (csring_expr.atom 1, dlist.singleton e)\n  end\n\n/-- In the output of `reflect_expr`, `atom`s are initialized with incorrect indices.\nThe indices cannot be computed until the whole tree is built, so another pass over\nthe expressions is needed - this is what `replace` does. The computation (expressed\nin the state monad) fixes up `atom`s to match their positions in the atom tree.\nThe initial state is a list of all atom occurrences in the goal, left-to-right. -/\nmeta def csring_expr.replace (t : tree expr) : csring_expr → state_t (list expr) option csring_expr\n| (csring_expr.atom _)  := do e ← get,\n  p ← monad_lift (t.index_of (<) e.head),\n  put e.tail, pure (csring_expr.atom p)\n| (csring_expr.const n) := pure (csring_expr.const n)\n| (csring_expr.add x y) := csring_expr.add <$> x.replace <*> y.replace\n| (csring_expr.mul x y) := csring_expr.mul <$> x.replace <*> y.replace\n| (csring_expr.pow x n) := (λ x, csring_expr.pow x n) <$> x.replace\n--| (csring_expr.neg x)   := csring_expr.neg <$> x.replace\n--| (csring_expr.inv x)   := csring_expr.inv <$> x.replace\n\nend tactic.ring2\n\nnamespace tactic\nnamespace interactive\nopen interactive interactive.types lean.parser\nopen tactic.ring2\n\nlocal postfix (name := parser.optional) `?`:9001 := optional\n\n/-- `ring2` solves equations in the language of rings.\n\nIt supports only the commutative semiring operations, i.e. it does not normalize subtraction or\ndivision.\n\n  This variant on the `ring` tactic uses kernel computation instead\n  of proof generation. In general, you should use `ring` instead of `ring2`. -/\nmeta def ring2 : tactic unit :=\ndo `[repeat {rw ← nat.pow_eq_pow}],\n  `(%%e₁ = %%e₂) ← target\n  | fail \"ring2 tactic failed: the goal is not an equality\",\n  α ← infer_type e₁,\n  expr.sort (level.succ u) ← infer_type α,\n  let (r₁, l₁) := reflect_expr e₁,\n  let (r₂, l₂) := reflect_expr e₂,\n  let L := (l₁ ++ l₂).to_list,\n  let s := tree.of_rbnode (rbtree_of L).1,\n  (r₁, L) ← (state_t.run (r₁.replace s) L : option _),\n  (r₂, _) ← (state_t.run (r₂.replace s) L : option _),\n  let se : expr := s.reflect' u α,\n  let er₁ : expr := reflect r₁,\n  let er₂ : expr := reflect r₂,\n  cs ← mk_app ``comm_semiring [α] >>= mk_instance,\n  e ← to_expr ``(correctness %%se %%er₁ %%er₂ rfl)\n    <|> fail (\"ring2 tactic failed, cannot show equality:\\n\"\n      ++ to_string (horner_expr.of_csexpr r₁) ++\n      \"\\n  =?=\\n\" ++ to_string (horner_expr.of_csexpr r₂)),\n  tactic.exact e\n\nadd_tactic_doc\n{ name        := \"ring2\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.ring2],\n  tags        := [\"arithmetic\", \"simplification\", \"decision procedure\"] }\n\nend interactive\nend tactic\n\nnamespace conv.interactive\nopen conv\n\nmeta def ring2 : conv unit := discharge_eq_lhs tactic.interactive.ring2\n\nend conv.interactive\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/ring2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.34737901707074836}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℚ) : (α.is_irrational) → ∀ ε > 0, ∃! n ∈ ℤ, ε > n * α - n,\nbegin\n  assume (h1 : α.is_irrational) (ε),\n  have h2 : ∃! n : ℤ, (0 < n * α - n) ∧ ((n+1) * α - (n+1) < ε), from by auto using [floor_le_int, floor_lt_int, is_irrational, exists_unique.exists],\n  show ∃! n : ℤ, ε > n * α - n, from by auto [h2, exists_unique.unique, exists_unique.exists, floor_le_int, floor_lt_int, is_irrational],\nend\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℚ) (h1 : α ≠ 0) : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ [0, 1], |y - (n * α) % 1| < ε :=\nbegin\n  assume h2 : α ≠ 0,\n  have h3 : ∀ i j : ℤ, i ≠ j → ⟨i*α⟩ ≠ ⟨j*α⟩, from \n  begin\n    assume (i j : ℤ) (h4 : i ≠ j),\n    have h5 : i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from \n      begin\n        assume h6,\n        show false, from \n          begin\n            have h7 : α = ⌊i * α⌋ - ⌊j * α⌋ / (i - j), from  \n              begin\n                calc α = i * α - ⌊i * α⌋ : by linarith\n                ... = j * α - ⌊j * α⌋ : by linarith \n                ... = (j - i) * α + ⌊j * α⌋ - ⌊i * α⌋ : by linarith \n                ... = ⌊j * α⌋ - ⌊i * α⌋ : by linarith \n                ... = ⌊j * α⌋ - ⌊i * α⌋ / (i - j) : by linarith \n              end,\n            have h8 : α ∈ ℤ, from \n              begin\n                have h9 : ⌊j * α⌋ - ⌊i * α⌋ / (i - j) ∈ ℤ, from \n                  begin\n                    have h10 : i ≠ j, from by linarith,\n                    show ⌊j * α⌋ - ⌊i * α⌋ / (i - j) ∈ ℤ, from \n                      begin\n                        have h11 : ⌊j * α⌋ - ⌊i * α⌋ ∈ ℤ, from by linarith,\n                        have h12 : (i - j) ∈ ℤ, from by linarith,\n                        show ⌊j * α⌋ - ⌊i * α⌋ / (i - j) ∈ ℤ, from \n                          begin\n                            have h13 : (i - j) ≠ 0, from by linarith,\n                            show (⌊j * α⌋ - ⌊i * α⌋) / (i - j) ∈ ℤ, from by linarith [h11, h13],  \n                          end\n                      end\n                  end,\n                have h11 : α ∈ ℚ, from \n                  begin\n                    show α ∈ ℚ, from by linarith using [old_nat_cast.injective]\n                  end,\n                show α ∈ ℤ, from by linarith [h9],\n              end,\n            show false, from by linarith [h7, h8]\n          end\n      end,\n      show ⟨i*α⟩ ≠ ⟨j*α⟩, from \n        begin\n          show ⟨i*α⟩ ≠ ⟨j*α⟩, from \n            begin\n              show ⟨i*α⟩ ≠ ⟨j*α⟩, from by linarith [h5]\n            end\n        end\n  end,\n\n  have h6 : ∀ (i j : ℤ), i ≠ j → ⟨i * α⟩ ≠ ⟨j * α⟩, from h3,\n\n  have h7 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ x : ℤ, ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n    begin\n      have h8 : ∀ i j : ℤ, i ≠ j → ⟨i * α⟩ ≠ ⟨j * α⟩, from h6,\n      have h9 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n        begin\n          have h10 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n            begin\n              have h11 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                begin \n                  have h12 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                    begin\n                      have h13 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                        begin    \n                          have h14 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                            begin\n                              have h15 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                                begin\n                                  have h16 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                                    begin\n                                      have h17 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                                        begin\n                                          have h18 : ∃ x : ℝ, ∀ ε > 0, ∃ N, ∀ n > N, ∀ y ∈ set.range (λ (x : ℤ), ⟨x * α⟩), |y - ⟨n * α⟩| < ε, from \n                                            begin\n                                              have h19 : set.range (λ (i : ℤ), ⟨i * α⟩) ⊆ set.Ico 0 1, from \n                                                begin \n                                                  assume (x : ℝ),\n                                                  assume (h20 : x ∈ set.range (λ (i : ℤ), ⟨i * α⟩)),\n                                                  cases h20 with i h21,\n                                                  assume h22 : ⟨i * α\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) (hα : ¬ is_rat α) :  dense_of_dense_range (λ j : ℤ, (floor j α : ℝ)) :=\nbegin\n  let s : set ℝ := λ (i : ℤ), ((floor i α) : ℝ), \n  let ƒ : ℤ → ℤ → ℝ := λ (i j : ℤ), floor i α - floor j α, \n  let h1 : s.finite.nonempty := by norm_num, \n  let h2 : ∀ (i j : ℤ), (i : ℝ) ≠ j → s i ≠ s j := by auto [hα, floor_divides_iff_rat], \n  let h3 : s.infinite := by auto [hα, floor_divides_iff_rat], \n  let h4 : ∃ (y : ℝ), is_limit_point s y, from exists_limit_point_of_infinite s h3, \n  let h5 : ∃ (y : ℝ), is_limit_point s y := by auto [h4], \n\n  let h6 : ∃ (y : ℝ), is_limit_point s y, from exists_limit_point_of_infinite s h3, \n  let h7 : ∃ (y : ℝ), is_limit_point s y := by auto [h6], \n\n  have h8 : ∃! x ∈ s, ∀ (ε : ℝ), 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε, from exists_unique_limit_point_of_infinite s h3 h5, \n  \n  let h9 : ∃! x ∈ s, ∀ (ε : ℝ), 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε, from exists_unique_limit_point_of_infinite s h3 h7, \n  let h10 : ∃! x ∈ s, ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h9], \n  let h11 : ∃! x ∈ s, ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h8], \n  let h12 : ∃! x ∈ s, ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h11], \n  let h13 : ∃ (x : ℤ), ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h12], \n  let h14 : ∃ (x : ℤ), ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h13], \n\n  let x := classical.some (h12).some, \n  let h15 : ∀ (ε : ℝ), 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h12], \n  let h16 : ∀ (ε : ℝ), 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h15], \n  let h17 : ∀ {ε : ℝ}, 0 < ε → ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] ε := by auto [h15], \n\n  let h18 : ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] 0.5 := by auto [h16], \n  let h19 : ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] 0.5 := by auto [h18], \n  let h20 : ∃ (n : ℤ), n > x ∧ s n ∈ 𝓝[s x] 0.5 := by auto [h19], \n\n  let N := classical.some (h20).some,\n  let y : ℝ := (floor N α - floor x α) * (floor N α : ℝ), \n  let h21 : y ∈ s N := by auto [floor_divides_iff_rat, hα], \n  let h22 : N > x := by auto [h20], \n  let h23 : s N ∈ 𝓝[s x] 0.5 := by auto [h20], \n\n  have h24 : {N • s N - s x} < √2, \n  from by auto [floor_divides_iff_rat, hα, h21, h23, mem_nhds_iff, sqrt_lt_iff],\n\n  have h25 : {N • s N - s x} < √2, \n  from by auto [floor_divides_iff_rat, hα, h21, h23, mem_nhds_iff, sqrt_lt_iff],\n\n  let h26 : {N • s N - s x} < √2 := by auto [h25], \n  let h27 : {N • s N - s x} < √2 := by auto [h24], \n  let h28 : {N • s N - s x} < √2 := by auto [h27], \n\n  have h29 : 2 • {N • s N - s x} < 2 * √2, \n  from by auto [div_self, floor_divides_iff_rat, hα, h21, h23, mem_nhds_iff, sqrt_lt_iff],\n\n  have h30 : 2 • {N • s N - s x} < 2 * √2, \n  from by auto [div_self, floor_divides_iff_rat, hα, h21, h23, mem_nhds_iff, sqrt_lt_iff],\n\n  have h31 : 2 • {N • s N - s x} < 2 * √2, \n  from by auto [h29], \n\n  have h32 : 2 • {N • s N - s x} < 2 * √2, \n  from by auto [h30], \n  let h33 : 2 • {N • s N - s x} < 2 * √2 := by auto [h32], \n\n  have h34 : y < ((N • s N - s x) + (N • s N - s x)) := by auto [h25], \n\n  have h35 : y < ((N • s N - s x) + (N • s N - s x)) := by auto [h24], \n  let h36 : y < ((N • s N - s x) + (N • s N - s x)) := by auto [h35], \n\n  have h37 : y < ((N • s N - s x) + (N • s N - s x)) := by auto [h34], \n  let h38 : y < ((N • s N - s x) + (N • s N - s x)) := by auto [h37], \n\n  have h39 : y < 2 • {N • s N - s x}, \n  from by auto [floor_divides_iff_rat, hα, h21, h23, mem_nhds_iff, sqrt_lt_iff],\n\n  let h40 : y < 2 • {N • s N - s x} := by auto [h39], \n  let h41 : ((y / 2) + s x : ℝ) < (N • s N : ℝ) := by auto [h40],\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (h1 : ¬ is_rat α) : ∀ ε > 0, ∃ n, n ∈ ℤ ∧ abs(n * α - ⌊n * α⌋) < ε :=\nbegin\n  assume ε,\n  assume h2 : ε > 0,\n\n  have h3 : α ≠ 0, from by auto [irrat],\n  \n  let S : set ℝ := {(⌊i * α⌋ - i * α) | i : ℤ},\n  have h4 : ∀ (i j : ℤ), i ≠ j → (⌊i * α⌋ - i * α) ≠ (⌊j * α⌋ - j * α), from by auto [floor_eq_iff_rat, irrat],\n\n  have h5 : ∀ (i j : ℤ), i ≠ j → i * α ≠ j * α, from by auto [irrat, mul_eq_zero, eq_zero_of_mul_eq_zero, h3],\n\n  have h6 : ∀ (i j : ℤ), i ≠ j → i * α ∉ {j * α}, from by auto [h5],\n\n  have h7 : ∀ (i j : ℤ), i ≠ j → i * α - ⌊i * α⌋ ∉ {j * α - ⌊j * α⌋}, from by auto [eq_iff_sub_eq_zero, h6],\n\n  have h8 : ∀ (i j : ℤ), i ≠ j → (⌊i * α⌋ - i * α) ∉ {(⌊j * α⌋ - j * α)}, from by auto [h7],\n  \n  have h9 : ∀ (i j : ℤ), i ≠ j → (⌊i * α⌋ - i * α) ∉ S := by auto [S, h8],\n\n  have h10 : ∀ a ∈ S, ∃ b ∈ S, abs(a - b) < ε, from by auto [sub_le_sub_left, h2],\n\n  have h11 : ∃ b ∈ S, abs(0 - b) < ε, from by auto [irrat, h10, h4, h9, S],\n\n  cases h11 with b h12,\n\n  have h13 : ∀ n : ℤ, n * α - ⌊n * α⌋ ∈ S, from by auto [n_floor_eq_sub, S, exists.intro 0, mem_image, exists.intro 0, mem_range],\n  have h14 : ∀ n : ℤ, n * α - ⌊n * α⌋ ∈ {(⌊i * α⌋ - i * α) | i : ℤ}, from by auto [S, h13],\n  \n  have h15 : ∀ n : ℤ, n * α - ⌊n * α⌋ ∈ {i * α - ⌊i * α⌋ | i : ℤ}, from by auto [S, h14],\n  \n  have h16 : ∃ n, n ∈ ℤ ∧ abs (n * α - ⌊n * α⌋) < ε, \n  from by auto [h12, h15, exists.intro 1, exists.intro 0, exists.intro b, exists.intro 0, exists.intro 0],\n  show ∃ n, n ∈ ℤ ∧ abs (n * α - ⌊n * α⌋) < ε, from by auto [h16],  \nend\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit (α : ℝ) : ∀ ε > 0, ∃ x : ℝ, x ∈ set.range (λ i, ↑i*α) ∧ x ∈ Icc (0 : ℝ) (1 : ℝ) ∧ ∃ n : ℕ, |x - n * α| < ε :=\nbegin\n  assume ε,\n  assume hε : ε > 0,\n\n  have h3 : ∀ i j : ℕ, ((i : ℝ) * α - ↑i) ≠ ((j : ℝ) * α - ↑j), from by auto,\n\n  -- choose a sequence of all the fractional parts from 0 to infinity, or all the A such that (∃ B : ℝ, A = B - ⌊B⌋)\n  -- in other words, choose (A is an element of the powerset of ℝ, and there exists a B such that A = B - ⌊B⌋)\n  have h4 : ∀ A ∈ 𝒫 ℝ, ∃ B : ℝ, A = B - ⌊B⌋, from by auto,\n\n  have h6 : ∀ n : ℕ, ((n : ℚ) * α - ⌊(n : ℚ) * α⌋) ∈ set.range (λ B, B - ⌊B⌋), from by auto,\n\n  have h7 : (∀ x, x ∈ set.range (λ B, B - ⌊B⌋) →  x ∈ Icc (0 : ℝ) (1 : ℝ)), from by auto [set.mem_range],\n\n  have h8 : ∀ n : ℕ, ((n : ℝ) * α - ⌊(n : ℝ) * α⌋)  ∈ Icc (0 : ℝ) (1 : ℝ), from by auto [h6, h7],\n\n  have h9 : ∀ B ∈ Icc (0 : ℝ) (1 : ℝ), ∃ n : ℕ, |((n : ℝ) * α - ⌊(n : ℝ) * α⌋) - B| < ε, from by auto,\n\n  show (∃ (x : ℝ), x ∈ set.range (λ B, B - ⌊B⌋) ∧ x ∈ Icc (0 : ℝ) (1 : ℝ) ∧ ∃ (n : ℕ), |x - n * α| < ε) := by auto [h6, h8, h9],\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) : irrational α → ∀ e : ℝ, ∃ δ:ℝ, 0 < δ ∧ δ < e :=\nbegin\nintros h, \nassume e,\nlet x : ℝ := (1 + e/2) * (1/2),\nhave hx : 0 < x, from by auto [lt_mul_iff],\nuse x,\n\n show (0 < x) ∧ x < e, from by auto [add_one_le_iff, le_add_iff, sub_neg_iff, le_of_lt, lt_mul_iff] using [lt_add_iff, add_lt_add_right, one_lt_two, mul_self_lt_mul_self_iff, hx],\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=rational_orbit (α : ℝ) : irrational α → ∀ y ⊆ set.range (λ n : ℤ, n * α % 1), dense y :=\nbegin\n  assume h1,\n  assume y ⊆ set.range (λ n : ℤ, n * α % 1),\n  have h2 : ∀ n m : ℤ, n ≠ m → (n*α % 1) ≠ (m*α % 1), from by auto [ring, mul_comm],\n  have h3 : ∀ n m : ℤ, (n*α % 1) ≠ (m*α % 1), from by auto [h2] using [irrational_of_int_eq_zero, irrational_of_int_neq_zero, h1],\n  let S : set ℝ := {m * α % 1 | m : ℤ},\n  have h4 : y ⊆ S, from by auto [set.subset_def],\n  have h5 : S ⊆ set.range (λ n : ℤ, n * α % 1), from by auto [set.subset_def],\n  have h6 : finite y → y ≠ ∅ → nonempty y → nonempty y, from by auto [finite.ne_empty_iff_exists_mem, set.not_empty_iff_exists_mem, finite_subset, finite.coe_to_finset],\n  have h8 : y ⊆ y, from by auto [set.subset.refl],\n  have h9 : y.nonempty, from by auto [h4, h6, h5, set.nonempty_range],\n  have h10 : dense S, from by auto [dense_iff_countable_open_dense_union, countable_iff_countably_infinite, range_is_countably_infinite, countable_iff_seq_range, set.seq_range_iff_forall_range, set.dense_Icc_iff_Ico_union_Ioc],\n  have h11 : dense y, from by auto [h10, h4, set.dense_subset],\n  show ∃ n : ℝ, n ∈ y ∧ ∀ ε > 0, ∃ m : ℝ, m ∈ y ∧ m ≠ n ∧ m ∈ Icc (n - ε) (n + ε), from by auto [h11, set.dense_iff_exists_between]\nend\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (h : ¬ is_rational α) : ∀ y ∈ set.Icc 0 1, ∃ x ∈ set.Icc 0 1, |y - x| ≤ 1 :=\nbegin\n    assume (y : ℝ) (hy : y ∈ set.Icc 0 1),\n\n    set S : set ℝ := {x : ℝ | ∃ i : ℤ, x = (x - floor x) ∧ x ∈ set.Icc 0 1},\n\n    have h_empty : ∀ x : ℝ, x ∉ S, from by auto [not_lt_of_ge, lt_irrefl, (show 0 ≤ x - floor x, by linarith)],\n    have h_not_empty : S ≠ ∅, from by auto [exists.intro (x - floor x) (and.intro (eq.refl x) (and.intro (by auto [floor_nonneg]) (by auto [not_lt_of_ge])))],\n    have h_infinite : S.infinite, from by auto [h_not_empty, h_empty, infinite_Icc_iff, forall_exists_not],\n    have h_bounded : S.bounded, from by auto [h_not_empty, bounded_Icc_iff, set.Icc_subset_Icc_left],\n    have h_closed : S.closed, from by auto [set.Icc_closed],\n\n    have h_Icc_0_1 : ∀ x : ℝ, x ∈ S ↔ x ∈ set.Icc 0 1, from by auto [set.Icc_subset_Icc_left],\n\n    have h_dense : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ |y - x| ≤ 1, from by auto [dense_iff_inter_open, h_Icc_0_1, h_infinite, h_bounded, h_closed, h_not_empty],\n\n    cases h_dense y hy with x hx,\n    cases hx with hx_mem hx_pf,\n    use x,\n    from and.intro hx_mem hx_pf,\nend\n\n--OUTPUT 9\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit {α : Type*} [s : dense_linear_order α] (a : α) : a ∉ (𝕀) → ∀ x, ∃ n ∈ ℤ, x - (1 : α) < (n : α) * a ∧ (n + 1) * a ≤ x :=\nbegin\n  assume h1 : a ∉ (𝕀), \n  assume x : α,\n\n  have h2 : ∀ n m : ℤ, n ≠ m → n * a ≠ m * a, \n  from by {intros n m h3, assume h4, have h5 := h3, rw h4 at h5, contradiction}, \n\n  have h6 : ∀ n m : ℤ,  n ≠ m → ∀ b : α, n * b ≠ m * b, \n  from by auto [h2],\n\n  have h7 : ∀ b : α, is_irrational b → ∀ n m : ℤ,  n ≠ m → n * b ≠ m * b, from by auto [h6],\n\n  have h8 : n * a ≠ m * a, from by auto [h7],\n\n  have h9 : ∀ n m : ℤ, n ≠ m → n * a ∈ set.range (λ n : ℤ, n * a),\n  from by {intros n m h10, use [n], rw ←one_mul (n * a)},\n\n  have h10 : ∀ n m : ℤ, n ≠ m → n * a ∉ set.range (λ n : ℤ, n * a), \n  from by auto [h9],\n\n  have h11 : set.range (λ n : ℤ, n * a) ≠ univ, \n  from by auto [h10, h8, univ_def_eq],\n\n  have h12 : set.range (λ n : ℤ, n * a) ≠ univ, \n  from by auto [h10, h8, univ_def_eq],\n\n  have h13 : set.range (λ n : ℤ, n * a) ≠ set.univ, \n  from by auto [h10, h8],\n\n  have h14 : set.range (λ n : ℤ, n * a) ≠ set.univ, \n  from by auto [h10, h8],\n\n  have h15 : set.range (λ n : ℤ, n * a) ≠ set.univ, \n  from by auto [h10, h8],\n\n  have h16 : set.range (λ n : ℤ, n * a) ≠ set.univ, \n  from by auto [h10, h8],\n\n  have h17 : set.range (λ n : ℤ, n * a) ≠ set.univ, \n  from by auto [h10, h8],\n\n(*\n  have h10 : ∅ ≠ set.range (λ n : ℤ, n * a), \n  from by auto [h3, classical.not_forall, classical.not_not_elim, classical.not_not_intro, set.exists_mem_of_ne_empty,\n  set.not_exists_mem_of_not_nonempty, set.not_exists_mem_of_not_nonempty],\n*)\n  have h11 : ∀ x, ∃ n ∈ ℤ, x - 1 < n * a ∧ (n + 1) * a ≤ x,\n  from by auto [s.dense, set.range_iff, exists_nat], \n\n  show ∃ (n : ℤ), n ∈ ℤ ∧ x - 1 < n * a ∧ (n + 1) * a ≤ x,\n  from by auto [h11],\nend\n\n--OUTPUT 10\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irration_orbit (α : ℝ) : ∀ (ε : ℝ), ε > 0 → ∃ i : ℤ, (∀ j : ℤ, |i*α - (j*α)%ℝ| < ε) := \nbegin\n  assume (ε : ℝ),\n  assume (h1 : ε > 0),\n\n  have h2 : ∀ (x : ℕ), (x*ε)≠0, from by auto [nat.pos_of_ne_zero],\n  have h3 : ∀ (x y : ℤ), (x*ε)%ℝ ≠ (y*ε)%ℝ → x ≠ y, from by auto [@eq.subst int ℝ _ _ _ _ h2],\n\n  let S : ℝ := {x | ∃ i : ℤ, x = (i*α)%ℝ},\n  let SS := (λ (i : ℤ), (i*α)%ℝ),\n\n  have h4 : S = ⋃ i : ℤ, {i}, from by auto [SS, eq_univ_of_forall],\n\n  have h5 : ∀ i j : ℤ, i ≠ j → (i*α)%ℝ ≠ (j*α)%ℝ, from \n  begin\n    assume (i j : ℤ),\n    assume (h5 : i ≠ j),\n    assume (h6 : (i*α)%ℝ = (j*α)%ℝ),\n    have h8 : ∃ N : ℕ, ∃ i : ℤ, α = N⁻¹ * (i : ℝ), from by auto [rational_iff],\n    cases h8 with N h9,\n    cases h9 with i h10,\n    have h11 : ∃ j : ℤ, (i : ℝ) = N * (j : ℝ), from by auto [rat.denom_ne_zero, rat.exists_int_eq],\n    cases h11 with j h12,\n    have h13 : α = (j : ℝ), from by auto [h10, h12, eq_div_iff],\n    have h14 : α ∈ ℚ, from by auto [h13],\n    have h15 : α ∈ ℚ → false, from by auto [eq.subst h13],\n    have h16 : false, from by auto [h15, h14],\n    show false, from h16,\n  end,\n\n  have h6 : ∀ i j : ℤ, ((i*α)%ℝ ∈ S) ∧ ((j*α)%ℝ ∈ S) → i = j, from \n  begin\n    assume (i j : ℤ),\n    assume (h6 : ((i*α)%ℝ ∈ S) ∧ ((j*α)%ℝ ∈ S)),\n    have h7 : (i*α)%ℝ = (j*α)%ℝ → i = j, from h3 i j,\n    show i = j, from by auto [h7, h6, h5],\n  end,\n\n  have h7 : ∀ i j : ℤ, ((i*α)%ℝ ∈ S) → ((j*α)%ℝ ∈ S) → i = j, from \n  begin\n    assume (i j : ℤ),\n    assume (h7 : ((i*α)%ℝ ∈ S)),\n    assume (h8 : ((j*α)%ℝ ∈ S)),\n    have h9 : ((i*α)%ℝ ∈ S) ∧ ((j*α)%ℝ ∈ S), from by auto [h7, h8],\n    show i = j, from by auto [h6, h9],\n  end,\n\n  have h8 : ∀ i j : ℤ, ((i*α)%ℝ ∈ S) ↔ i = j, from \n  begin\n    assume (i j : ℤ),\n    split,\n    {\n      assume (h9 : ((i*α)%ℝ ∈ S)),\n      show i = j, from by auto [h7, h9],\n    },\n    {\n      assume (h10 : i = j),\n      show (i*α)%ℝ ∈ S, from by auto [h10],\n    },\n  end,\n\n  have h9 : S = ⋃ i : ℤ, {i}, from by auto [SS, eq_univ_of_forall],\n\n  have h10 : ∀ i : ℤ, (i*α)%ℝ ∈ S, from by auto [rat.denom_ne_zero],\n\n  have h11 : ∀ i : ℤ, ∃ x : ℝ, x ∈ {i} ∧ x ∈ S, from \n  begin\n    assume (i : ℤ),\n    have h12 : (i*α)%ℝ ∈ S, from by auto [h10],\n    have h13 : (i*α)%ℝ ∈ {i}, from by auto [h12],\n    show ∃ x : ℝ, x ∈ {i} ∧ x ∈ S, from by auto [h13, h12],\n  end,\n\n  have h12 : ⋃ i : ℤ, {i} = ⋃ x : S, {x}, from by auto [h9, h8, h11, set.ext_iff],\n\n  have h13 : ∃ (i : ℤ), ∀ (j : ℤ), (i*α)%ℝ ∈ {x | ∃ i : ℤ, x = (i*α)%ℝ}, from \n  begin\n    use 1,\n    assume j,\n    have h14 : (1*α)%ℝ = (j*α)%ℝ → 1 = j, from by auto [h3],\n    have h15 : (1*α)%ℝ ∈ S, from by auto [rat.denom_ne_zero],\n    have h16 : (1*α)%ℝ ∈ {x | ∃ i : ℤ, x = (i*α)%ℝ}, from by auto [h15],\n    show (1*α)%ℝ ∈ {x | ∃ i : ℤ, x = (i*α)%ℝ}, from by auto [h16],\n  end,\n\n  have h14 : ∃ (i : ℤ), ∀ (j : ℤ), (i*α)%ℝ ∈ S, from \n  begin\n    have h15 : ∃ i : ℤ, ∀ j : ℤ, (i*α)%ℝ ∈ {x | ∃ i : ℤ, x = (i*α)%ℝ}, from by auto [h13],\n    cases h15 with i h16,\n    have h17 : ∀ j : ℤ, (i*α)%ℝ ∈ S, from by auto [h16],\n    show ∃ (i : ℤ), ∀ j : ℤ, (i*α)%ℝ ∈ S, from by auto [h17],\n  end,\n\n  have h15 : ∃ (i : ℤ), (i*α)%ℝ ∈ S ∧ ∀ (j : ℤ), (j*α)%ℝ ∈ S → i = j, from \n  begin\n    have h16 : ∃ (i : ℤ), ∀ (j : ℤ), (i*α)%ℝ ∈ S, from by auto [h14],\n    cases h16 with i h17,\n    have h18 : ∀ (j : ℤ), (i*α)%ℝ ∈ S ∧ (j*α)%ℝ ∈ S → i = j, from \n    begin\n      assume j,\n      have h19 : (i*α)%ℝ ∈ S ∧ (j*α)%ℝ ∈ S → ((i*α)%ℝ = (j*α)%ℝ →\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  assume (h7 : ε > 0),\n\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from by auto [h8, h10, h5, h9],\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from by auto [h11] using [linarith],\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/lean_proof_auto-4_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7662936430859596, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.3473316727875848}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n\n! This file was ported from Lean 3 source module order.category.HeytAlg\n! leanprover-community/mathlib commit e8ac6315bcfcbaf2d19a046719c3b553206dac75\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.Category.BddDistLat\nimport Mathbin.Order.Heyting.Hom\n\n/-!\n# The category of Heyting algebras\n\nThis file defines `HeytAlg`, the category of Heyting algebras.\n-/\n\n\nuniverse u\n\nopen CategoryTheory Opposite Order\n\n/-- The category of Heyting algebras. -/\ndef HeytAlg :=\n  Bundled HeytingAlgebra\n#align HeytAlg HeytAlg\n\nnamespace HeytAlg\n\ninstance : CoeSort HeytAlg (Type _) :=\n  Bundled.hasCoeToSort\n\ninstance (X : HeytAlg) : HeytingAlgebra X :=\n  X.str\n\n/-- Construct a bundled `HeytAlg` from a `heyting_algebra`. -/\ndef of (α : Type _) [HeytingAlgebra α] : HeytAlg :=\n  Bundled.of α\n#align HeytAlg.of HeytAlg.of\n\n@[simp]\ntheorem coe_of (α : Type _) [HeytingAlgebra α] : ↥(of α) = α :=\n  rfl\n#align HeytAlg.coe_of HeytAlg.coe_of\n\ninstance : Inhabited HeytAlg :=\n  ⟨of PUnit⟩\n\ninstance bundledHom : BundledHom HeytingHom\n    where\n  toFun α β [HeytingAlgebra α] [HeytingAlgebra β] := (coeFn : HeytingHom α β → α → β)\n  id := HeytingHom.id\n  comp := @HeytingHom.comp\n  hom_ext α β [HeytingAlgebra α] [HeytingAlgebra β] := FunLike.coe_injective\n#align HeytAlg.bundled_hom HeytAlg.bundledHom\n\nderiving instance LargeCategory, ConcreteCategory for HeytAlg\n\n@[simps]\ninstance hasForgetToLat : HasForget₂ HeytAlg BddDistLat\n    where forget₂ :=\n    { obj := fun X => BddDistLat.of X\n      map := fun X Y f => (f : BoundedLatticeHom X Y) }\n#align HeytAlg.has_forget_to_Lat HeytAlg.hasForgetToLat\n\n/-- Constructs an isomorphism of Heyting algebras from an order isomorphism between them. -/\n@[simps]\ndef Iso.mk {α β : HeytAlg.{u}} (e : α ≃o β) : α ≅ β\n    where\n  Hom := e\n  inv := e.symm\n  hom_inv_id' := by\n    ext\n    exact e.symm_apply_apply _\n  inv_hom_id' := by\n    ext\n    exact e.apply_symm_apply _\n#align HeytAlg.iso.mk HeytAlg.Iso.mk\n\nend HeytAlg\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Order/Category/HeytAlg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704796847395, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.3471673728899902}}
{"text": "import algebra.homology.exact\nimport for_mathlib.abelian_category\nimport for_mathlib.exact_seq2\n.\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\nopen opposite\n\n-- move me\nnamespace category_theory\n\nvariables {𝓐 : Type*} [category 𝓐] [abelian 𝓐]\n\nlemma exact_seq.is_iso_of_zero_of_zero {A B C D : 𝓐} {f : A ⟶ B} {g : B ⟶ C} {h : C ⟶ D}\n  {L : list (arrow 𝓐)} (H : exact_seq 𝓐 (f :: g :: h :: L)) (hf : f = 0) (hh : h = 0) :\n  is_iso g :=\nbegin\n  subst f, subst h,\n  have : mono g, { rw [H.pair.mono_iff_eq_zero], },\n  haveI : epi g, { rw [(H.drop 1).pair.epi_iff_eq_zero] },\n  exact is_iso_of_mono_of_epi g,\nend\n\nlemma exact.homology_is_zero {X Y Z : 𝓐} (f : X ⟶ Y) (g : Y ⟶ Z) (hfg : exact f g) :\n  is_zero (homology f g hfg.w) :=\nbegin\n  rw preadditive.exact_iff_homology_zero at hfg,\n  rcases hfg with ⟨w, ⟨e⟩⟩,\n  exact is_zero_of_iso_of_zero (is_zero_zero _) e.symm,\nend\n\nlemma exact_of_homology_is_zero {X Y Z : 𝓐} {f : X ⟶ Y} {g : Y ⟶ Z} {w : f ≫ g = 0}\n  (H : is_zero (homology f g w)) :\n  exact f g :=\nbegin\n  rw preadditive.exact_iff_homology_zero,\n  refine ⟨w, ⟨_⟩⟩,\n  exact H.iso_zero\nend\n\nnamespace limits\nnamespace is_zero\n\nlemma exact {X Y Z : 𝓐} (hY : is_zero Y)\n  (f : X ⟶ Y) (g : Y ⟶ Z) : exact f g :=\nby simp only [abelian.exact_iff, hY.eq_of_tgt f 0, hY.eq_of_tgt (limits.kernel.ι g) 0,\n    limits.zero_comp, eq_self_iff_true, and_true]\n\nlemma homology_is_zero {X Y Z : 𝓐} (hY : is_zero Y)\n  (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :\n  is_zero (homology f g w) :=\nexact.homology_is_zero f g $ hY.exact f g\n\nlemma is_iso {X Y : 𝓐} (hX : is_zero X) (hY : is_zero Y) (f : X ⟶ Y) :\n  is_iso f :=\n{ out := ⟨0, hX.eq_of_src _ _, hY.eq_of_tgt _ _⟩ }\n\nlemma op {X : 𝓐} (h : is_zero X) : is_zero (op X) :=\nbegin\n  rw [is_zero_iff_id_eq_zero] at h ⊢,\n  rw [← op_id, h, op_zero]\nend\n\nlemma unop {X : 𝓐ᵒᵖ} (h : is_zero X) : is_zero (unop X) :=\nbegin\n  rw [is_zero_iff_id_eq_zero] at h ⊢,\n  rw [← unop_id, h, unop_zero]\nend\n\nend is_zero\n\n@[simp] lemma is_zero_op {X : 𝓐} : is_zero (op X) ↔ is_zero X := ⟨is_zero.unop, is_zero.op⟩\n@[simp] lemma is_zero_unop {X : 𝓐ᵒᵖ} : is_zero (unop X) ↔ is_zero X := ⟨is_zero.op, is_zero.unop⟩\n\nend limits\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/homology_exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.734119526900183, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.34700617023012587}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport order.antisymmetrization\nimport order.category.Preorder\n\n/-!\n# Category of partial orders\n\nThis defines `PartialOrder`, the category of partial orders with monotone maps.\n-/\n\nopen category_theory\n\nuniverse u\n\n/-- The category of partially ordered types. -/\ndef PartialOrder := bundled partial_order\n\nnamespace PartialOrder\n\ninstance : bundled_hom.parent_projection @partial_order.to_preorder := ⟨⟩\n\nattribute [derive [large_category, concrete_category]] PartialOrder\n\ninstance : has_coe_to_sort PartialOrder Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled PartialOrder from the underlying type and typeclass. -/\ndef of (α : Type*) [partial_order α] : PartialOrder := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [partial_order α] : ↥(of α) = α := rfl\n\ninstance : inhabited PartialOrder := ⟨of punit⟩\n\ninstance (α : PartialOrder) : partial_order α := α.str\n\ninstance has_forget_to_Preorder : has_forget₂ PartialOrder Preorder := bundled_hom.forget₂ _ _\n\n/-- Constructs an equivalence between partial orders from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : PartialOrder.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : PartialOrder ⥤ PartialOrder :=\n{ obj := λ X, of (order_dual X), map := λ X Y, order_hom.dual }\n\n/-- The equivalence between `PartialOrder` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : PartialOrder ≌ PartialOrder :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nend PartialOrder\n\nlemma PartialOrder_dual_comp_forget_to_Preorder :\n  PartialOrder.dual ⋙ forget₂ PartialOrder Preorder =\n    forget₂ PartialOrder Preorder ⋙ Preorder.dual := rfl\n\n/-- `antisymmetrization` as a functor. It is the free functor. -/\ndef Preorder_to_PartialOrder : Preorder.{u} ⥤ PartialOrder :=\n{ obj := λ X, PartialOrder.of (antisymmetrization X (≤)),\n  map := λ X Y f, f.antisymmetrization,\n  map_id' := λ X,\n    by { ext, exact quotient.induction_on' x (λ x, quotient.map'_mk' _ (λ a b, id) _) },\n  map_comp' := λ X Y Z f g,\n    by { ext, exact quotient.induction_on' x (λ x, order_hom.antisymmetrization_apply_mk _ _) } }\n\n/-- `Preorder_to_PartialOrder` is left adjoint to the forgetful functor, meaning it is the free\nfunctor from `Preorder` to `PartialOrder`. -/\ndef Preorder_to_PartialOrder_forget_adjunction :\n  Preorder_to_PartialOrder.{u} ⊣ forget₂ PartialOrder Preorder :=\nadjunction.mk_of_hom_equiv\n  { hom_equiv := λ X Y, { to_fun := λ f,\n      ⟨f ∘ to_antisymmetrization (≤), f.mono.comp to_antisymmetrization_mono⟩,\n    inv_fun := λ f, ⟨λ a, quotient.lift_on' a f $ λ a b h, (antisymm_rel.image h f.mono).eq, λ a b,\n      quotient.induction_on₂' a b $ λ a b h, f.mono h⟩,\n    left_inv := λ f, order_hom.ext _ _ $ funext $ λ x, quotient.induction_on' x $ λ x, rfl,\n    right_inv := λ f, order_hom.ext _ _ $ funext $ λ x, rfl },\n  hom_equiv_naturality_left_symm' := λ X Y Z f g,\n    order_hom.ext _ _ $ funext $ λ x, quotient.induction_on' x $ λ x, rfl,\n  hom_equiv_naturality_right' := λ X Y Z f g, order_hom.ext _ _ $ funext $ λ x, rfl }\n\n/-- `Preorder_to_PartialOrder` and `order_dual` commute. -/\n@[simps] def Preorder_to_PartialOrder_comp_to_dual_iso_to_dual_comp_Preorder_to_PartialOrder :\n (Preorder_to_PartialOrder.{u} ⋙ PartialOrder.dual) ≅\n    (Preorder.dual ⋙ Preorder_to_PartialOrder) :=\nnat_iso.of_components (λ X, PartialOrder.iso.mk $ order_iso.dual_antisymmetrization _) $\n  λ X Y f, order_hom.ext _ _ $ funext $ λ x, quotient.induction_on' x $ λ x, rfl\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/PartialOrder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467904168342855}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.homology.homology\n\n/-!\n# Chain complexes supported in a single degree\n\nWe define `single V j c : V ⥤ homological_complex V c`,\nwhich constructs complexes in `V` of shape `c`, supported in degree `j`.\n\nSimilarly `single₀ V : V ⥤ chain_complex V ℕ` is the special case for\n`ℕ`-indexed chain complexes, with the object supported in degree `0`,\nbut with better definitional properties.\n\nIn `to_single₀_equiv` we characterize chain maps to a `ℕ`-indexed complex concentrated in degree 0;\nthey are equivalent to `{ f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 }`.\n(This is useful translating between a projective resolution and\nan augmented exact complex of projectives.)\n-/\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables (V : Type u) [category.{v} V] [has_zero_morphisms V] [has_zero_object V]\n\nnamespace homological_complex\nvariables {ι : Type*} [decidable_eq ι] (c : complex_shape ι)\n\nlocal attribute [instance] has_zero_object.has_zero\n\n/--\nThe functor `V ⥤ homological_complex V c` creating a chain complex supported in a single degree.\n\nSee also `chain_complex.single₀ : V ⥤ chain_complex V ℕ`,\nwhich has better definitional properties,\nif you are working with `ℕ`-indexed complexes.\n-/\n@[simps]\ndef single (j : ι) : V ⥤ homological_complex V c :=\n{ obj := λ A,\n  { X := λ i, if i = j then A else 0,\n    d := λ i j, 0, },\n  map := λ A B f,\n  { f := λ i, if h : i = j then\n      eq_to_hom (by { dsimp, rw if_pos h, }) ≫ f ≫ eq_to_hom (by { dsimp, rw if_pos h, })\n    else\n      0, },\n  map_id' := λ A, begin\n    ext,\n    dsimp,\n    split_ifs with h,\n    { subst h, simp, },\n    { rw if_neg h, simp, },\n  end,\n  map_comp' := λ A B C f g, begin\n    ext,\n    dsimp,\n    split_ifs with h,\n    { subst h, simp, },\n    { simp, },\n  end, }.\n\n/--\nThe object in degree `j` of `(single V c h).obj A` is just `A`.\n-/\n@[simps]\ndef single_obj_X_self (j : ι) (A : V) : ((single V c j).obj A).X j ≅ A :=\neq_to_iso (by simp)\n\n@[simp]\nlemma single_map_f_self (j : ι) {A B : V} (f : A ⟶ B) :\n  ((single V c j).map f).f j =\n    (single_obj_X_self V c j A).hom ≫ f ≫ (single_obj_X_self V c j B).inv :=\nby { simp, refl, }\n\ninstance (j : ι) : faithful (single V c j) :=\n{ map_injective' := λ X Y f g w, begin\n    have := congr_hom w j,\n    dsimp at this,\n    simp only [dif_pos] at this,\n    rw [←is_iso.inv_comp_eq, inv_eq_to_hom, eq_to_hom_trans_assoc, eq_to_hom_refl, category.id_comp,\n      ←is_iso.comp_inv_eq, category.assoc, inv_eq_to_hom, eq_to_hom_trans, eq_to_hom_refl,\n      category.comp_id] at this,\n    exact this,\n  end, }\n\ninstance (j : ι) : full (single V c j) :=\n{ preimage := λ X Y f, eq_to_hom (by simp) ≫ f.f j ≫ eq_to_hom (by simp),\n  witness' := λ X Y f, begin\n    ext i,\n    dsimp,\n    split_ifs,\n    { subst h, simp, },\n    { symmetry,\n      apply zero_of_target_iso_zero,\n      dsimp,\n      rw [if_neg h], },\n  end }\n\nend homological_complex\n\nopen homological_complex\n\nnamespace chain_complex\n\nlocal attribute [instance] has_zero_object.has_zero\n\n/--\n`chain_complex.single₀ V` is the embedding of `V` into `chain_complex V ℕ`\nas chain complexes supported in degree 0.\n\nThis is naturally isomorphic to `single V _ 0`, but has better definitional properties.\n-/\ndef single₀ : V ⥤ chain_complex V ℕ :=\n{ obj := λ X,\n  { X := λ n, match n with\n    | 0 := X\n    | (n+1) := 0\n    end,\n    d := λ i j, 0, },\n  map := λ X Y f,\n  { f := λ n, match n with\n    | 0 := f\n    | (n+1) := 0\n    end, },\n  map_id' := λ X, by { ext n, cases n, refl, dsimp, unfold_aux, simp, },\n  map_comp' := λ X Y Z f g, by { ext n, cases n, refl, dsimp, unfold_aux, simp, } }\n\n@[simp] lemma single₀_obj_X_0 (X : V) : ((single₀ V).obj X).X 0 = X := rfl\n@[simp] lemma single₀_obj_X_succ (X : V) (n : ℕ) : ((single₀ V).obj X).X (n+1) = 0 := rfl\n@[simp] lemma single₀_obj_X_d (X : V) (i j : ℕ) : ((single₀ V).obj X).d i j = 0 := rfl\n@[simp] \n\nsection\nvariables [has_equalizers V] [has_cokernels V] [has_images V] [has_image_maps V]\n\n/--\nSending objects to chain complexes supported at `0` then taking `0`-th homology\nis the same as doing nothing.\n-/\nnoncomputable\ndef homology_functor_0_single₀ : single₀ V ⋙ homology_functor V _ 0 ≅ (𝟭 V) :=\nnat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫ homology_zero_zero)\n  (λ X Y f, by { ext, dsimp [homology_functor], simp, })\n\n/--\nSending objects to chain complexes supported at `0` then taking `(n+1)`-st homology\nis the same as the zero functor.\n-/\nnoncomputable\ndef homology_functor_succ_single₀ (n : ℕ) : single₀ V ⋙ homology_functor V _ (n+1) ≅ 0 :=\nnat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫ homology_zero_zero)\n  (λ X Y f, by ext)\n\nend\n\nvariables {V}\n\n/--\nMorphisms from a `ℕ`-indexed chain complex `C`\nto a single object chain complex with `X` concentrated in degree 0\nare the same as morphisms `f : C.X 0 ⟶ X` such that `C.d 1 0 ≫ f = 0`.\n-/\ndef to_single₀_equiv (C : chain_complex V ℕ) (X : V) :\n  (C ⟶ (single₀ V).obj X) ≃ { f : C.X 0 ⟶ X // C.d 1 0 ≫ f = 0 } :=\n{ to_fun := λ f, ⟨f.f 0, by { rw ←f.comm 1 0, simp, }⟩,\n  inv_fun := λ f,\n  { f := λ i, match i with\n    | 0 := f.1\n    | (n+1) := 0\n    end,\n    comm' := λ i j h, begin\n      rcases i with _|_|i; cases j; unfold_aux; simp only [comp_zero, zero_comp, single₀_obj_X_d],\n      { rw [C.shape, zero_comp], simp, },\n      { exact f.2.symm, },\n      { rw [C.shape, zero_comp], simp [i.succ_succ_ne_one.symm] },\n    end, },\n  left_inv := λ f, begin\n    ext i,\n    rcases i,\n    { refl, },\n    { ext, },\n  end,\n  right_inv := by tidy, }\n\nvariables (V)\n\n/-- `single₀` is the same as `single V _ 0`. -/\ndef single₀_iso_single : single₀ V ≅ single V _ 0 :=\nnat_iso.of_components\n  (λ X,\n  { hom := { f := λ i, by { cases i; simpa using 𝟙 _, } },\n    inv := { f := λ i, by { cases i; simpa using 𝟙 _, } },\n    hom_inv_id' := by { ext (_|i); { dsimp, simp, }, },\n    inv_hom_id' := begin\n      ext (_|i),\n      { apply category.id_comp, },\n      { apply has_zero_object.to_zero_ext, },\n    end, })\n  (λ X Y f, by { ext (_|i); { dsimp, simp, }, })\n\ninstance : faithful (single₀ V) := faithful.of_iso (single₀_iso_single V).symm\ninstance : full (single₀ V) := full.of_iso (single₀_iso_single V).symm\n\nend chain_complex\n\nnamespace cochain_complex\n\nlocal attribute [instance] has_zero_object.has_zero\n\n/--\n`cochain_complex.single₀ V` is the embedding of `V` into `cochain_complex V ℕ`\nas cochain complexes supported in degree 0.\n\nThis is naturally isomorphic to `single V _ 0`, but has better definitional properties.\n-/\ndef single₀ : V ⥤ cochain_complex V ℕ :=\n{ obj := λ X,\n  { X := λ n, match n with\n    | 0 := X\n    | (n+1) := 0\n    end,\n    d := λ i j, 0, },\n  map := λ X Y f,\n  { f := λ n, match n with\n    | 0 := f\n    | (n+1) := 0\n    end, },\n  map_id' := λ X, by { ext n, cases n, refl, dsimp, unfold_aux, simp, },\n  map_comp' := λ X Y Z f g, by { ext n, cases n, refl, dsimp, unfold_aux, simp, } }\n\n@[simp] lemma single₀_obj_X_0 (X : V) : ((single₀ V).obj X).X 0 = X := rfl\n@[simp] lemma single₀_obj_X_succ (X : V) (n : ℕ) : ((single₀ V).obj X).X (n+1) = 0 := rfl\n@[simp] lemma single₀_obj_X_d (X : V) (i j : ℕ) : ((single₀ V).obj X).d i j = 0 := rfl\n@[simp] lemma single₀_obj_X_d_from (X : V) (j : ℕ) : ((single₀ V).obj X).d_from j = 0 :=\nby { rw [d_from_eq ((single₀ V).obj X) rfl], simp, }\n@[simp] lemma single₀_obj_X_d_to (X : V) (i : ℕ) : ((single₀ V).obj X).d_to i = 0 :=\nbegin\n  cases i,\n  { rw [d_to_eq_zero], simp, },\n  { rw [d_to_eq ((single₀ V).obj X) rfl], simp, },\nend\n@[simp] lemma single₀_map_f_0 {X Y : V} (f : X ⟶ Y) : ((single₀ V).map f).f 0 = f := rfl\n@[simp] lemma single₀_map_f_succ {X Y : V} (f : X ⟶ Y) (n : ℕ) :\n  ((single₀ V).map f).f (n+1) = 0 := rfl\n\nsection\nvariables [has_equalizers V] [has_cokernels V] [has_images V] [has_image_maps V]\n\n/--\nSending objects to cochain complexes supported at `0` then taking `0`-th homology\nis the same as doing nothing.\n-/\nnoncomputable\ndef homology_functor_0_single₀ : single₀ V ⋙ homology_functor V _ 0 ≅ (𝟭 V) :=\nnat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫ homology_zero_zero)\n  (λ X Y f, by { ext, dsimp [homology_functor], simp, })\n\n/--\nSending objects to cochain complexes supported at `0` then taking `(n+1)`-st homology\nis the same as the zero functor.\n-/\nnoncomputable\ndef homology_functor_succ_single₀ (n : ℕ) : single₀ V ⋙ homology_functor V _ (n+1) ≅ 0 :=\nnat_iso.of_components (λ X, homology.congr _ _ (by simp) (by simp) ≪≫ homology_zero_zero)\n  (λ X Y f, by ext)\n\nend\n\nvariables {V}\n\n/--\nMorphisms from a single object cochain complex with `X` concentrated in degree 0\nto a `ℕ`-indexed cochain complex `C`\nare the same as morphisms `f : X ⟶ C.X 0` such that `f ≫ C.d 0 1 = 0`.\n-/\ndef from_single₀_equiv (C : cochain_complex V ℕ) (X : V) :\n  ((single₀ V).obj X ⟶ C) ≃ { f : X ⟶ C.X 0 // f ≫ C.d 0 1 = 0 } :=\n{ to_fun := λ f, ⟨f.f 0, by { rw f.comm 0 1, simp, }⟩,\n  inv_fun := λ f,\n  { f := λ i, match i with\n    | 0 := f.1\n    | (n+1) := 0\n    end,\n    comm' := λ i j h, begin\n      rcases j with _|_|j; cases i; unfold_aux; simp only [comp_zero, zero_comp, single₀_obj_X_d],\n      { convert comp_zero, rw [C.shape], simp, },\n      { exact f.2, },\n      { convert comp_zero, rw [C.shape], simp only [complex_shape.up_rel, zero_add],\n        exact (nat.one_lt_succ_succ j).ne },\n    end, },\n  left_inv := λ f, begin\n    ext i,\n    rcases i,\n    { refl, },\n    { ext, },\n  end,\n  right_inv := by tidy, }\n\nvariables (V)\n\n/-- `single₀` is the same as `single V _ 0`. -/\ndef single₀_iso_single : single₀ V ≅ single V _ 0 :=\nnat_iso.of_components\n  (λ X,\n  { hom := { f := λ i, by { cases i; simpa using 𝟙 _, } },\n    inv := { f := λ i, by { cases i; simpa using 𝟙 _, } },\n    hom_inv_id' := by { ext (_|i); { dsimp, simp, }, },\n    inv_hom_id' := begin\n      ext (_|i),\n      { apply category.id_comp, },\n      { apply has_zero_object.to_zero_ext, },\n    end, })\n  (λ X Y f, by { ext (_|i); { dsimp, simp, }, })\n\ninstance : faithful (single₀ V) := faithful.of_iso (single₀_iso_single V).symm\ninstance : full (single₀ V) := full.of_iso (single₀_iso_single V).symm\n\nend cochain_complex\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/homology/single.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467904083364927}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.category.Group\nimport category_theory.limits.shapes.kernels\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\n/-!\nSome small examples of using limits and colimits in `Ab`, the category of additive commutative\ngroups.\n-/\n\nexample (G H : Ab) (f : G ⟶ H) : Ab := kernel f\nexample (G H : Ab) (f : G ⟶ H) [epi f] : kernel (cokernel.π f) ≅ H :=\nas_iso (kernel.ι (cokernel.π f))\n\n-- TODO no images yet...\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/docs/tutorial/category_theory/Ab.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467903998386998}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, David Renshaw\n-/\nimport Lean\nimport Mathlib.Lean.Meta\nimport Mathlib.Tactic.LeftRight\n\n/-!\n# mk_iff_of_inductive_prop\n\nThis file defines a tactic `tactic.mk_iff_of_inductive_prop` that generates `iff` rules for\ninductive `Prop`s. For example, when applied to `List.Chain`, it creates a declaration with\nthe following type:\n\n```lean\n∀ {α : Type _} (R : α → α → Prop) (a : α) (l : List α),\n  Chain R a l ↔ l = [] ∨ ∃(b : α) (l' : List α), R a b ∧ Chain R b l ∧ l = b :: l'\n```\n\nThis tactic can be called using either the `mk_iff_of_inductive_prop` user command or\nthe `mk_iff` attribute.\n-/\n\nnamespace Mathlib.Tactic.MkIff\n\nopen Lean Meta Elab\n\n/-- `select m n` runs `tactic.right` `m` times, and then `tactic.left` `(n-m)` times.\nFails if `n < m`. -/\nprivate def select (m n : Nat) (goal : MVarId) : MetaM MVarId :=\n  match m,n with\n  | 0, 0             => pure goal\n  | 0, (_ + 1)       => do let [new_goal] ← Mathlib.Tactic.LeftRight.leftRightMeta `left 0 2 goal\n                                | throwError \"expected only one new goal\"\n                           pure new_goal\n  | (m + 1), (n + 1) => do let [new_goal] ← Mathlib.Tactic.LeftRight.leftRightMeta `right 1 2 goal\n                                | throwError \"expected only one new goal\"\n                           select m n new_goal\n  | _, _             => failure\n\n/-- `compactRelation bs as_ps`: Produce a relation of the form:\n```lean\nR := λ as ∃ bs, Λ_i a_i = p_i[bs]\n```\nThis relation is user-visible, so we compact it by removing each `b_j` where a `p_i = b_j`, and\nhence `a_i = b_j`. We need to take care when there are `p_i` and `p_j` with `p_i = p_j = b_k`.\n-/\npartial def compactRelation :\n  List Expr → List (Expr × Expr) → List (Option Expr) × List (Expr × Expr) × (Expr → Expr)\n| [],    as_ps => ([], as_ps, id)\n| b::bs, as_ps =>\n  match as_ps.span (λ⟨_,p⟩ => ¬ p == b) with\n    | (_, []) => -- found nothing in ps equal to b\n        let (bs, as_ps', subst) := compactRelation bs as_ps\n        (b::bs, as_ps', subst)\n    | (ps₁, (a, _) :: ps₂) => -- found one that matches b. Remove it.\n      let i := fun e ↦ e.replaceFVar b a\n      let (bs, as_ps', subst) := compactRelation (bs.map i) ((ps₁ ++ ps₂).map (λ⟨a, p⟩ => (a, i p)))\n      (none :: bs, as_ps', i ∘ subst)\n\n/-- Generates an expression of the form `∃(args), inner`. `args` is assumed to be a list of fvars.\nWhen possible, `p ∧ q` is used instead of `∃(_ : p), q`. -/\ndef mkExistsList (args : List Expr) (inner : Expr) : MetaM Expr :=\nargs.foldrM (λarg i:Expr => do\n    let t ← inferType arg\n    let l := (← inferType t).sortLevel!\n    pure $ if arg.occurs i || l != Level.zero\n      then mkApp2 (mkConst `Exists [l] : Expr) t (←(mkLambdaFVars #[arg] i))\n      else mkApp2 (mkConst `And [] : Expr) t i)\n  inner\n\n/-- `mkOpList op empty [x1, x2, ...]` is defined as `op x1 (op x2 ...)`.\n  Returns `empty` if the list is empty. -/\ndef mkOpList (op : Expr) (empty : Expr) : List Expr → Expr\n| []        => empty\n| [e]       => e\n| (e :: es) => mkApp2 op e $ mkOpList op empty es\n\n/-- `mkAndList [x1, x2, ...]` is defined as `x1 ∧ (x2 ∧ ...)`, or `True` if the list is empty. -/\ndef mkAndList : List Expr → Expr := mkOpList (mkConst `And) (mkConst `True)\n\n/-- `mkOrList [x1, x2, ...]` is defined as `x1 ∨ (x2 ∨ ...)`, or `False` if the list is empty. -/\ndef mkOrList : List Expr → Expr := mkOpList (mkConst `Or) (mkConst `False)\n\n/-- Drops the final element of a list. -/\ndef List.init : List α → List α\n| []     => []\n| [_]    => []\n| a::l => a::init l\n\n/-- Auxiliary data associated with a single constructor of an inductive declaration.\n-/\nstructure Shape : Type where\n  /-- For each forall-bound variable in the type of the constructor, minus\n  the \"params\" that apply to the entire inductive type, this list contains `true`\n  if that variable has been kept after `compactRelation`.\n\n  For example, `List.Chain.nil` has type\n  ```lean\n    ∀ {α : Type u_1} {R : α → α → Prop} {a : α}, List.Chain R a []`\n  ```\n  and the first two variables `α` and `R` are \"params\", while the `a : α` gets\n  eliminated in a `compactRelation`, so `variablesKept = [false].\n\n  `List.Chain.cons` has type\n  ```lean\n    ∀ {α : Type u_1} {R : α → α → Prop} {a b : α} {l : List α},\n       R a b → List.Chain R b l → List.Chain R a (b :: l)\n  ```\n  and the `a : α` gets eliminated, so `compactRelation = [false,true,true,true,true]`.\n   -/\n  variablesKept : List Bool\n\n  /-- The number of equalities, or `none` in the case when we've reduced something\n  of the form `p ∧ True` to just `p`.\n  -/\n  neqs : Option Nat\n\n/-- Converts an inductive constructor `c` into a `Shape` that will be used later in\nwhile proving the iff theorem, and a proposition representing the constructor.\n-/\ndef constrToProp (univs : List Level) (params : List Expr) (idxs : List Expr) (c : Name) :\n  MetaM (Shape × Expr)  :=\ndo let type := (← getConstInfo c).instantiateTypeLevelParams univs\n   let type' ← Meta.forallBoundedTelescope type (params.length) fun fvars ty ↦ do\n     pure $ ty.replaceFVars fvars params.toArray\n\n   Meta.forallTelescope type' fun fvars ty ↦ do\n     let idxs_inst := ty.getAppArgs.toList.drop params.length\n     let (bs, eqs, subst) := compactRelation fvars.toList (idxs.zip idxs_inst)\n     let eqs ← eqs.mapM (λ⟨idx, inst⟩ => do\n          let ty ← idx.fvarId!.getType\n          let instTy ← inferType inst\n          let u := (←inferType ty).sortLevel!\n          if ←isDefEq ty instTy\n          then pure (mkApp3 (mkConst `Eq [u]) ty idx inst)\n          else pure (mkApp4 (mkConst `HEq [u]) ty idx instTy inst))\n     let (n, r) ← match bs.filterMap id, eqs with\n     | [], [] => do\n           pure (some 0, (mkConst `True))\n     | bs', []  => do\n          let t : Expr ← bs'.getLast!.fvarId!.getType\n          let l := (←inferType t).sortLevel!\n          if l == Level.zero then do\n            let r ← mkExistsList (List.init bs') t\n            pure (none, subst r)\n          else do\n            let r ← mkExistsList bs' (mkConst `True)\n            pure (some 0, subst r)\n     | bs', _ => do\n       let r ← mkExistsList bs' (mkAndList eqs)\n       pure (some eqs.length, subst r)\n     pure (⟨bs.map Option.isSome, n⟩, r)\n\n/-- Splits the goal `n` times via `refine ⟨?_,?_⟩`, and then applies `constructor` to\nclose the resulting subgoals.\n-/\ndef splitThenConstructor (mvar : MVarId) (n : Nat) : MetaM Unit :=\nmatch n with\n| 0   => do\n  let (subgoals',_) ← Term.TermElabM.run $ Tactic.run mvar do\n    Tactic.evalTactic (←`(tactic| constructor))\n  let [] := subgoals' | throwError \"expected no subgoals\"\n  pure ()\n| n  + 1 => do\n  let (subgoals,_) ← Term.TermElabM.run $ Tactic.run mvar do\n    Tactic.evalTactic (←`(tactic| refine ⟨?_,?_⟩))\n  let [sg1, sg2] := subgoals | throwError \"expected two subgoals\"\n  let (subgoals',_) ← Term.TermElabM.run $ Tactic.run sg1 do\n    Tactic.evalTactic (←`(tactic| constructor))\n  let [] := subgoals' | throwError \"expected no subgoals\"\n  splitThenConstructor sg2 n\n\n/-- Proves the left to right direction of a generated iff theorem.\n`shape` is the output of a call to `constrToProp`.\n-/\ndef toCases (mvar : MVarId) (shape : List Shape) : MetaM Unit :=\ndo\n  let ⟨h, mvar'⟩ ← mvar.intro1\n  let subgoals ← mvar'.cases h\n  let _ ← (shape.zip subgoals.toList).enum.mapM fun ⟨p, ⟨⟨shape, t⟩, subgoal⟩⟩ ↦ do\n    let vars := subgoal.fields\n    let si := (shape.zip vars.toList).filterMap (λ ⟨c,v⟩ => if c then some v else none)\n    let mvar'' ← select p (subgoals.size - 1) subgoal.mvarId\n    match t with\n    | none => do\n      let v := vars.get! (shape.length - 1)\n      let mv ← mvar''.existsi (List.init si)\n      mv.assign v\n    | some n => do\n      let mv ← mvar''.existsi si\n      splitThenConstructor mv (n - 1)\n  pure ()\n\n/-- Calls `cases` on `h` (assumed to be a binary sum) `n` times, and returns\nthe resulting subgoals and their corresponding new hypotheses.\n-/\ndef nCasesSum (n : Nat) (mvar : MVarId) (h : FVarId) : MetaM (List (FVarId × MVarId)) :=\nmatch n with\n| 0 => pure [(h, mvar)]\n| n' + 1 => do\n  let #[sg1, sg2] ← mvar.cases h | throwError \"expected two case subgoals\"\n  let #[Expr.fvar fvar1] ← pure sg1.fields | throwError \"expected fvar\"\n  let #[Expr.fvar fvar2] ← pure sg2.fields | throwError \"expected fvar\"\n  let rest ← nCasesSum n' sg2.mvarId fvar2\n  pure ((fvar1, sg1.mvarId)::rest)\n\n/-- Calls `cases` on `h` (assumed to be a binary product) `n` times, and returns\nthe resulting subgoal and the new hypotheses.\n-/\ndef nCasesProd (n : Nat) (mvar : MVarId) (h : FVarId) : MetaM (MVarId × List FVarId) :=\nmatch n with\n| 0 => pure (mvar, [h])\n| n' + 1 => do\n  let #[sg] ← mvar.cases h | throwError \"expected one case subgoals\"\n  let #[Expr.fvar fvar1, Expr.fvar fvar2] ← pure sg.fields | throwError \"expected fvar\"\n  let (mvar', rest) ← nCasesProd n' sg.mvarId fvar2\n  pure (mvar', fvar1::rest)\n\n/--\nIterate over two lists, if the first element of the first list is `false`, insert `none` into the\nresult and continue with the tail of first list. Otherwise, wrap the first element of the second\nlist with `some` and continue with the tails of both lists. Return when either list is empty.\n\nExample:\n```\nlistBoolMerge [false, true, false, true] [0, 1, 2, 3, 4] = [none, (some 0), none, (some 1)]\n```\n-/\ndef listBoolMerge {α : Type _} : List Bool → List α → List (Option α)\n| [], _ => []\n| false :: xs, ys => none :: listBoolMerge xs ys\n| true :: xs, y :: ys => some y :: listBoolMerge xs ys\n| true :: _, [] => []\n\n/-- Proves the right to left direction of a generated iff theorem.\n-/\ndef toInductive (mvar : MVarId) (cs : List Name)\n  (gs : List Expr) (s : List Shape) (h : FVarId) :\n  MetaM Unit := do\n  match s.length with\n  | 0       => do let _ ← mvar.cases h\n                  pure ()\n  | (n + 1) => do\n      let subgoals ← nCasesSum n mvar h\n      let _ ← (cs.zip (subgoals.zip s)).mapM $ λ⟨constr_name, ⟨h, mv⟩, bs, e⟩ => do\n        let n := (bs.filter id).length\n        let (mvar', _fvars) ← match e with\n        | none => nCasesProd (n-1) mv h\n        | some 0 => do let ⟨mvar', fvars⟩ ← nCasesProd n mv h\n                          let mvar'' ← mvar'.tryClear fvars.getLast!\n                          pure ⟨mvar'', fvars⟩\n        | some (e + 1) => do\n           let (mv', fvars) ← nCasesProd n mv h\n           let lastfv := fvars.getLast!\n           let (mv2, fvars') ← nCasesProd e mv' lastfv\n\n           /- `fvars'.foldlM subst mv2` fails when we have dependent equalities (`HEq`).\n           `subst` will change the dependent hypotheses, so that the `uniq` local names\n           are wrong afterwards. Instead we revert them and pull them out one-by-one. -/\n           let (_, mv3) ← mv2.revert fvars'.toArray\n           let mv4 ← fvars'.foldlM (λ mv _ => do let ⟨fv, mv'⟩ ← mv.intro1\n                                                 subst mv' fv\n                                   ) mv3\n           pure (mv4, fvars)\n        mvar'.withContext do\n          let fvarIds := (←getLCtx).getFVarIds.toList\n          let gs := fvarIds.take gs.length\n          let hs := (fvarIds.reverse.take n).reverse\n          let m := gs.map some ++ listBoolMerge bs hs\n          let args ← m.mapM (λa => match a with\n                                   | some v => pure $ mkFVar v\n                                   | none => mkFreshExprMVar none)\n          let c ← mkConstWithFreshMVarLevels constr_name\n          let e := mkAppN c args.toArray\n          let t ← inferType e\n          let mt ← mvar'.getType\n          let _ ← isDefEq t mt -- infer values for those mvars we just made\n          mvar'.assign e\n\n/-- Implementation for both `mk_iff` and `mk_iff_of_inductive_prop`.y\n-/\ndef mkIffOfInductivePropImpl (ind : Name) (rel : Name) (relStx : Syntax) : MetaM Unit := do\n  let .inductInfo inductVal ← getConstInfo ind |\n    throwError \"mk_iff only applies to inductive declarations\"\n  let constrs := inductVal.ctors\n  let params := inductVal.numParams\n  let type := inductVal.type\n\n  let univNames := inductVal.levelParams\n  let univs := univNames.map mkLevelParam\n  /- we use these names for our universe parameters, maybe we should construct a copy of them\n  using `uniq_name` -/\n\n  let (thmTy,shape) ← Meta.forallTelescope type fun fvars ty ↦ do\n    if !ty.isProp then throwError \"mk_iff only applies to prop-valued declarations\"\n    let lhs := mkAppN (mkConst ind univs) fvars\n    let fvars' := fvars.toList\n    let shape_rhss ← constrs.mapM (constrToProp univs (fvars'.take params) (fvars'.drop params))\n    let (shape, rhss) := shape_rhss.unzip\n    pure (←mkForallFVars fvars (mkApp2 (mkConst `Iff) lhs (mkOrList rhss)),\n          shape)\n\n  let mvar ← mkFreshExprMVar (some thmTy)\n  let mvarId := mvar.mvarId!\n  let (fvars, mvarId') ← mvarId.intros\n  let [mp, mpr] ← mvarId'.apply (mkConst `Iff.intro) | throwError \"failed to split goal\"\n\n  toCases mp shape\n\n  let ⟨mprFvar, mpr'⟩ ← mpr.intro1\n  toInductive mpr' constrs ((fvars.toList.take params).map .fvar) shape mprFvar\n\n  addDecl $ .thmDecl {\n    name := rel\n    levelParams := univNames\n    type := thmTy\n    value := ← instantiateMVars mvar\n  }\n  addDeclarationRanges rel {\n    range := ← getDeclarationRange (← getRef)\n    selectionRange := ← getDeclarationRange relStx\n  }\n  addConstInfo relStx rel\n\n/--\nApplying the `mk_iff` attribute to an inductively-defined proposition `mk_iff` makes an `iff` rule\n`r` with the shape `∀ps is, i as ↔ ⋁_j, ∃cs, is = cs`, where `ps` are the type parameters, `is` are\nthe indices, `j` ranges over all possible constructors, the `cs` are the parameters for each of the\nconstructors, and the equalities `is = cs` are the instantiations for each constructor for each of\nthe indices to the inductive type `i`.\n\nIn each case, we remove constructor parameters (i.e. `cs`) when the corresponding equality would\nbe just `c = i` for some index `i`.\n\nFor example, if we try the following:\n```lean\n@[mk_iff]\nstructure Foo (m n : Nat) : Prop where\n  equal : m = n\n  sum_eq_two : m + n = 2\n```\n\nThen `#check Foo_iff` returns:\n```lean\nFoo_iff : ∀ (m n : Nat), Foo m n ↔ m = n ∧ m + n = 2\n```\n\nYou can add an optional string after `mk_iff` to change the name of the generated lemma.\nFor example, if we try the following:\n```lean\n@[mk_iff bar]\nstructure Foo (m n : Nat) : Prop where\n  equal : m = n\n  sum_eq_two : m + n = 2\n```\n\nThen `#check bar` returns:\n```lean\nbar : ∀ (m n : ℕ), Foo m n ↔ m = n ∧ m + n = 2\n```\n\nSee also the user command `mk_iff_of_inductive_prop`.\n-/\nsyntax (name := mkIff) \"mk_iff\" (ppSpace ident)? : attr\n\n/--\n`mk_iff_of_inductive_prop i r` makes an `iff` rule for the inductively-defined proposition `i`.\nThe new rule `r` has the shape `∀ps is, i as ↔ ⋁_j, ∃cs, is = cs`, where `ps` are the type\nparameters, `is` are the indices, `j` ranges over all possible constructors, the `cs` are the\nparameters for each of the constructors, and the equalities `is = cs` are the instantiations for\neach constructor for each of the indices to the inductive type `i`.\n\nIn each case, we remove constructor parameters (i.e. `cs`) when the corresponding equality would\nbe just `c = i` for some index `i`.\n\nFor example, `mk_iff_of_inductive_prop` on `List.Chain` produces:\n\n```lean\n∀ { α : Type _} (R : α → α → Prop) (a : α) (l : List α),\n  Chain R a l ↔ l = [] ∨ ∃(b : α) (l' : List α), R a b ∧ Chain R b l ∧ l = b :: l'\n```\n\nSee also the `mk_iff` user attribute.\n-/\nsyntax (name := mkIffOfInductiveProp) \"mk_iff_of_inductive_prop\" ident ident : command\n\nelab_rules : command\n| `(command| mk_iff_of_inductive_prop $i:ident $r:ident) =>\n    Command.liftCoreM <| MetaM.run' do\n      mkIffOfInductivePropImpl i.getId r.getId r\n\ninitialize Lean.registerBuiltinAttribute {\n  name := `mkIff\n  descr := \"Generate an `iff` lemma for an inductive `Prop`.\"\n  add := fun decl stx _ => Lean.Meta.MetaM.run' do\n    let (tgt, idStx) ← match stx with\n      | `(attr| mk_iff $tgt:ident) =>\n        pure ((← mkDeclName (← getCurrNamespace) {} tgt.getId).1, tgt.raw)\n      | `(attr| mk_iff) => pure (decl.appendAfter \"_iff\", stx)\n      | _ => throwError \"unrecognized syntax\"\n    mkIffOfInductivePropImpl decl tgt idStx\n}\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/MkIffOfInductiveProp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34676992026641557}}
{"text": "import Lean\nimport UserAttr.BlaAttr\n\n@[bla] def f (x : Nat) := x + 2\n@[bla] def g (x : Nat) := x + 1\n\n@[foo 10] def h1 (x : Nat) := 2*x + 1\n@[foo 20 important] def h2 (x : Nat) := 2*x + 1\n\nopen Lean in\ndef hasBlaAttr (declName : Name) : CoreM Bool :=\n  return blaAttr.hasTag (← getEnv) declName\n\n#eval hasBlaAttr ``f\n#eval hasBlaAttr ``id\n\nopen Lean in\ndef getFooAttrInfo? (declName : Name) : CoreM (Option (Nat × Bool)) :=\n  return fooAttr.getParam? (← getEnv) declName\n\n#eval getFooAttrInfo? ``f\n#eval getFooAttrInfo? ``h1\n#eval getFooAttrInfo? ``h2\n\n@[my_simp] theorem f_eq : f x = x + 2 := rfl\n@[my_simp] theorem g_eq : g x = x + 1 := rfl\n\nexample : f x + g x = 2*x + 3 := by\n  simp_arith -- does not appy f_eq and g_eq\n  simp_arith [f, g]\n\nexample : f x + g x = 2*x + 3 := by\n  simp_arith [my_simp]\n\nexample : f x = id (x + 2) := by\n  simp\n  simp [my_simp]\n\nmacro \"my_simp\" : tactic => `(tactic| simp [my_simp])\n\nexample : f x = id (x + 2) := by\n  my_simp\n\n@[simp low, my_simp low]\naxiom expand_mul_add (x y z : Nat) : x * (y + z) = x * y + x * y\n@[simp high, my_simp high]\naxiom expand_add_mul (x y z : Nat) : (x + y) * z = x * z + y * z\n@[simp, my_simp]\naxiom lassoc_add (x y z : Nat) : x + (y + z) = x + y + z\n\nset_option trace.Meta.Tactic.simp.rewrite true\n\n-- Rewrites: expand_mul_add -> expand_mul_add -> lassoc_add\ntheorem ex1 (x : Nat) : (x + x) * (x + x) = x * x + x * x + x * x + x * x := by simp only [my_simp]\n\n-- Rewrites: expand_add_mul -> expand_mul_add -> lassoc_add\ntheorem ex2 (x : Nat) : (x + x) * (x + x) = x * x + x * x + x * x + x * x := by simp\n\nopen Lean Meta in\ndef checkProofs : MetaM Unit := do\n  let .thmInfo info1 ← getConstInfo `ex1 | throwError \"unexpected\"\n  let .thmInfo info2 ← getConstInfo `ex2 | throwError \"unexpected\"\n  unless info1.value == info2.value do\n    throwError \"unexpected values\"\n\n#eval checkProofs\n\nopen Lean Meta in\ndef showThmsOf (simpAttrName : Name) : MetaM Unit := do\n  let some simpExt ← getSimpExtension? simpAttrName\n    | throwError \"`{simpAttrName}` is not a simp attribute\"\n  let thms ← simpExt.getTheorems\n  let thmNames := thms.lemmaNames.fold (init := #[]) fun acc origin => acc.push origin.key\n  for thmName in thmNames do\n    IO.println thmName\n\n#eval showThmsOf `my_simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/pkg/user_attr/UserAttr/Tst.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3467699202664155}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n\n! This file was ported from Lean 3 source module category_theory.sites.pretopology\n! leanprover-community/mathlib commit 9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Sites.Grothendieck\n\n/-!\n# Grothendieck pretopologies\n\nDefinition and lemmas about Grothendieck pretopologies.\nA Grothendieck pretopology for a category `C` is a set of families of morphisms with fixed codomain,\nsatisfying certain closure conditions.\n\nWe show that a pretopology generates a genuine Grothendieck topology, and every topology has\na maximal pretopology which generates it.\n\nThe pretopology associated to a topological space is defined in `spaces.lean`.\n\n## Tags\n\ncoverage, pretopology, site\n\n## References\n\n* [nLab, *Grothendieck pretopology*](https://ncatlab.org/nlab/show/Grothendieck+pretopology)\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n* [Stacks, *00VG*](https://stacks.math.columbia.edu/tag/00VG)\n-/\n\n\nuniverse v u\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen CategoryTheory Category Limits Presieve\n\nvariable {C : Type u} [Category.{v} C] [HasPullbacks C]\n\nvariable (C)\n\n#print CategoryTheory.Pretopology /-\n/--\nA (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed\ntarget `X` for every object `X` in `C`, called \"coverings\" of `X`, which satisfies the following\nthree axioms:\n1. Every family consisting of a single isomorphism is a covering family.\n2. The collection of covering families is stable under pullback.\n3. Given a covering family, and a covering family on each domain of the former, the composition\n   is a covering family.\n\nIn some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,\nin that each covering is not necessarily downward closed.\n\nSee: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or\nhttps://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.\nNote that Stacks calls a category together with a pretopology a site, and [MM92] calls this\na basis for a topology.\n-/\n@[ext]\nstructure Pretopology where\n  coverings : ∀ X : C, Set (Presieve X)\n  has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X\n  pullbacks : ∀ ⦃X Y⦄ (f : Y ⟶ X) (S), S ∈ coverings X → pullbackArrows f S ∈ coverings Y\n  Transitive :\n    ∀ ⦃X : C⦄ (S : Presieve X) (Ti : ∀ ⦃Y⦄ (f : Y ⟶ X), S f → Presieve Y),\n      S ∈ coverings X → (∀ ⦃Y⦄ (f) (H : S f), Ti f H ∈ coverings Y) → S.bind Ti ∈ coverings X\n#align category_theory.pretopology CategoryTheory.Pretopology\n-/\n\nnamespace Pretopology\n\ninstance : CoeFun (Pretopology C) fun _ => ∀ X : C, Set (Presieve X) :=\n  ⟨coverings⟩\n\nvariable {C}\n\ninstance : LE (Pretopology C) where le K₁ K₂ := (K₁ : ∀ X : C, Set (Presieve X)) ≤ K₂\n\n#print CategoryTheory.Pretopology.le_def /-\ntheorem le_def {K₁ K₂ : Pretopology C} : K₁ ≤ K₂ ↔ (K₁ : ∀ X : C, Set (Presieve X)) ≤ K₂ :=\n  Iff.rfl\n#align category_theory.pretopology.le_def CategoryTheory.Pretopology.le_def\n-/\n\nvariable (C)\n\ninstance : PartialOrder (Pretopology C) :=\n  { Pretopology.hasLe with\n    le_refl := fun K => le_def.mpr le_rfl\n    le_trans := fun K₁ K₂ K₃ h₁₂ h₂₃ => le_def.mpr (le_trans h₁₂ h₂₃)\n    le_antisymm := fun K₁ K₂ h₁₂ h₂₁ => Pretopology.ext _ _ (le_antisymm h₁₂ h₂₁) }\n\ninstance : OrderTop (Pretopology C)\n    where\n  top :=\n    { coverings := fun _ => Set.univ\n      has_isos := fun _ _ _ _ => Set.mem_univ _\n      pullbacks := fun _ _ _ _ _ => Set.mem_univ _\n      Transitive := fun _ _ _ _ _ => Set.mem_univ _ }\n  le_top K X S hS := Set.mem_univ _\n\ninstance : Inhabited (Pretopology C) :=\n  ⟨⊤⟩\n\n#print CategoryTheory.Pretopology.toGrothendieck /-\n/-- A pretopology `K` can be completed to a Grothendieck topology `J` by declaring a sieve to be\n`J`-covering if it contains a family in `K`.\n\nSee <https://stacks.math.columbia.edu/tag/00ZC>, or [MM92] Chapter III, Section 2, Equation (2).\n-/\ndef toGrothendieck (K : Pretopology C) : GrothendieckTopology C\n    where\n  sieves X S := ∃ R ∈ K X, R ≤ (S : Presieve _)\n  top_mem' X := ⟨Presieve.singleton (𝟙 _), K.has_isos _, fun _ _ _ => ⟨⟩⟩\n  pullback_stable' X Y S g := by\n    rintro ⟨R, hR, RS⟩\n    refine' ⟨_, K.pullbacks g _ hR, _⟩\n    rw [← sieve.sets_iff_generate, sieve.pullback_arrows_comm]\n    apply sieve.pullback_monotone\n    rwa [sieve.gi_generate.gc]\n  transitive' := by\n    rintro X S ⟨R', hR', RS⟩ R t\n    choose t₁ t₂ t₃ using t\n    refine' ⟨_, K.transitive _ _ hR' fun _ f hf => t₂ (RS _ hf), _⟩\n    rintro Y _ ⟨Z, g, f, hg, hf, rfl⟩\n    apply t₃ (RS _ hg) _ hf\n#align category_theory.pretopology.to_grothendieck CategoryTheory.Pretopology.toGrothendieck\n-/\n\n/- warning: category_theory.pretopology.mem_to_grothendieck -> CategoryTheory.Pretopology.mem_toGrothendieck is a dubious translation:\nlean 3 declaration is\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1] (K : CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (X : C) (S : CategoryTheory.Sieve.{u1, u2} C _inst_1 X), Iff (Membership.Mem.{max u2 u1, max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X) (Set.{max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X)) (Set.hasMem.{max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X)) S (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ (max u2 u1))} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (fun (_x : CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) => forall (X : C), Set.{max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X)) (CategoryTheory.GrothendieckTopology.hasCoeToFun.{u1, u2} C _inst_1) (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2 K) X)) (Exists.{succ (max u2 u1)} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (fun (R : CategoryTheory.Presieve.{u1, u2} C _inst_1 X) => Exists.{0} (Membership.Mem.{max u2 u1, max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (Set.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) (Set.hasMem.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) R (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ (max u2 u1))} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (fun (_x : CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) => forall (X : C), Set.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) (CategoryTheory.Pretopology.hasCoeToFun.{u1, u2} C _inst_1 _inst_2) K X)) (fun (H : Membership.Mem.{max u2 u1, max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (Set.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) (Set.hasMem.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) R (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ (max u2 u1))} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (fun (_x : CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) => forall (X : C), Set.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) (CategoryTheory.Pretopology.hasCoeToFun.{u1, u2} C _inst_1 _inst_2) K X)) => LE.le.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (Preorder.toLE.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CompleteSemilatticeInf.toPartialOrder.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CompleteLattice.toCompleteSemilatticeInf.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CategoryTheory.Presieve.completeLattice.{u2, u1} C _inst_1 X))))) R (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X) (fun (_x : CategoryTheory.Sieve.{u1, u2} C _inst_1 X) => CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CategoryTheory.Sieve.hasCoeToFun.{u1, u2} C _inst_1 X) S))))\nbut is expected to have type\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1] (K : CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (X : C) (S : CategoryTheory.Sieve.{u1, u2} C _inst_1 X), Iff (Membership.mem.{max u2 u1, max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X) (Set.{max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X)) (Set.instMembershipSet.{max u2 u1} (CategoryTheory.Sieve.{u1, u2} C _inst_1 X)) S (CategoryTheory.GrothendieckTopology.sieves.{u1, u2} C _inst_1 (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2 K) X)) (Exists.{succ (max u2 u1)} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (fun (R : CategoryTheory.Presieve.{u1, u2} C _inst_1 X) => And (Membership.mem.{max u2 u1, max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (Set.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) (Set.instMembershipSet.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X)) R (CategoryTheory.Pretopology.coverings.{u1, u2} C _inst_1 _inst_2 K X)) (LE.le.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (Preorder.toLE.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CompleteSemilatticeInf.toPartialOrder.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CompleteLattice.toCompleteSemilatticeInf.{max u2 u1} (CategoryTheory.Presieve.{u1, u2} C _inst_1 X) (CategoryTheory.instCompleteLatticePresieve.{u1, u2} C _inst_1 X))))) R (CategoryTheory.Sieve.arrows.{u1, u2} C _inst_1 X S))))\nCase conversion may be inaccurate. Consider using '#align category_theory.pretopology.mem_to_grothendieck CategoryTheory.Pretopology.mem_toGrothendieckₓ'. -/\ntheorem mem_toGrothendieck (K : Pretopology C) (X S) :\n    S ∈ toGrothendieck C K X ↔ ∃ R ∈ K X, R ≤ (S : Presieve X) :=\n  Iff.rfl\n#align category_theory.pretopology.mem_to_grothendieck CategoryTheory.Pretopology.mem_toGrothendieck\n\n#print CategoryTheory.Pretopology.ofGrothendieck /-\n/-- The largest pretopology generating the given Grothendieck topology.\n\nSee [MM92] Chapter III, Section 2, Equations (3,4).\n-/\ndef ofGrothendieck (J : GrothendieckTopology C) : Pretopology C\n    where\n  coverings X R := Sieve.generate R ∈ J X\n  has_isos X Y f i := J.covering_of_eq_top (by simp)\n  pullbacks X Y f R hR := by\n    rw [Set.mem_def, sieve.pullback_arrows_comm]\n    apply J.pullback_stable f hR\n  Transitive X S Ti hS hTi := by\n    apply J.transitive hS\n    intro Y f\n    rintro ⟨Z, g, f, hf, rfl⟩\n    rw [sieve.pullback_comp]\n    apply J.pullback_stable g\n    apply J.superset_covering _ (hTi _ hf)\n    rintro Y g ⟨W, h, g, hg, rfl⟩\n    exact ⟨_, h, _, ⟨_, _, _, hf, hg, rfl⟩, by simp⟩\n#align category_theory.pretopology.of_grothendieck CategoryTheory.Pretopology.ofGrothendieck\n-/\n\n/- warning: category_theory.pretopology.gi -> CategoryTheory.Pretopology.gi is a dubious translation:\nlean 3 declaration is\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1], GaloisInsertion.{max u2 u1, max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.partialOrder.{u1, u2} C _inst_1 _inst_2)) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.GrothendieckTopology.partialOrder.{u1, u2} C _inst_1)) (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.ofGrothendieck.{u1, u2} C _inst_1 _inst_2)\nbut is expected to have type\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1], GaloisInsertion.{max u2 u1, max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.instPartialOrderPretopology.{u1, u2} C _inst_1 _inst_2)) (PartialOrder.toPreorder.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.GrothendieckTopology.instPartialOrderGrothendieckTopology.{u1, u2} C _inst_1)) (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.ofGrothendieck.{u1, u2} C _inst_1 _inst_2)\nCase conversion may be inaccurate. Consider using '#align category_theory.pretopology.gi CategoryTheory.Pretopology.giₓ'. -/\n/-- We have a galois insertion from pretopologies to Grothendieck topologies. -/\ndef gi : GaloisInsertion (toGrothendieck C) (ofGrothendieck C)\n    where\n  gc K J := by\n    constructor\n    · intro h X R hR\n      exact h _ ⟨_, hR, sieve.le_generate R⟩\n    · rintro h X S ⟨R, hR, RS⟩\n      apply J.superset_covering _ (h _ hR)\n      rwa [sieve.gi_generate.gc]\n  le_l_u J X S hS := ⟨S, J.superset_covering S.le_generate hS, le_rfl⟩\n  choice x hx := toGrothendieck C x\n  choice_eq _ _ := rfl\n#align category_theory.pretopology.gi CategoryTheory.Pretopology.gi\n\n#print CategoryTheory.Pretopology.trivial /-\n/--\nThe trivial pretopology, in which the coverings are exactly singleton isomorphisms. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee <https://stacks.math.columbia.edu/tag/07GE>\n-/\ndef trivial : Pretopology C\n    where\n  coverings X S := ∃ (Y : _)(f : Y ⟶ X)(h : IsIso f), S = Presieve.singleton f\n  has_isos X Y f i := ⟨_, _, i, rfl⟩\n  pullbacks X Y f S := by\n    rintro ⟨Z, g, i, rfl⟩\n    refine' ⟨pullback g f, pullback.snd, _, _⟩\n    · skip\n      refine' ⟨⟨pullback.lift (f ≫ inv g) (𝟙 _) (by simp), ⟨_, by tidy⟩⟩⟩\n      apply pullback.hom_ext\n      · rw [assoc, pullback.lift_fst, ← pullback.condition_assoc]\n        simp\n      · simp\n    · apply pullback_singleton\n  Transitive := by\n    rintro X S Ti ⟨Z, g, i, rfl⟩ hS\n    rcases hS g (singleton_self g) with ⟨Y, f, i, hTi⟩\n    refine' ⟨_, f ≫ g, _, _⟩\n    · skip\n      infer_instance\n    ext (W k)\n    constructor\n    · rintro ⟨V, h, k, ⟨_⟩, hh, rfl⟩\n      rw [hTi] at hh\n      cases hh\n      apply singleton.mk\n    · rintro ⟨_⟩\n      refine' bind_comp g presieve.singleton.mk _\n      rw [hTi]\n      apply presieve.singleton.mk\n#align category_theory.pretopology.trivial CategoryTheory.Pretopology.trivial\n-/\n\ninstance : OrderBot (Pretopology C) where\n  bot := trivial C\n  bot_le K X R := by\n    rintro ⟨Y, f, hf, rfl⟩\n    exact K.has_isos f\n\n/- warning: category_theory.pretopology.to_grothendieck_bot -> CategoryTheory.Pretopology.toGrothendieck_bot is a dubious translation:\nlean 3 declaration is\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1], Eq.{max (succ u2) (succ u1)} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2 (Bot.bot.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (OrderBot.toHasBot.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.hasLe.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.orderBot.{u1, u2} C _inst_1 _inst_2)))) (Bot.bot.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CompleteLattice.toHasBot.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.GrothendieckTopology.completeLattice.{u1, u2} C _inst_1)))\nbut is expected to have type\n  forall (C : Type.{u2}) [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.Limits.HasPullbacks.{u1, u2} C _inst_1], Eq.{max (succ u2) (succ u1)} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.Pretopology.toGrothendieck.{u1, u2} C _inst_1 _inst_2 (Bot.bot.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (OrderBot.toBot.{max u2 u1} (CategoryTheory.Pretopology.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.LE.{u1, u2} C _inst_1 _inst_2) (CategoryTheory.Pretopology.instOrderBotPretopologyLE.{u1, u2} C _inst_1 _inst_2)))) (Bot.bot.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CompleteLattice.toBot.{max u2 u1} (CategoryTheory.GrothendieckTopology.{u1, u2} C _inst_1) (CategoryTheory.GrothendieckTopology.instCompleteLatticeGrothendieckTopology.{u1, u2} C _inst_1)))\nCase conversion may be inaccurate. Consider using '#align category_theory.pretopology.to_grothendieck_bot CategoryTheory.Pretopology.toGrothendieck_botₓ'. -/\n/-- The trivial pretopology induces the trivial grothendieck topology. -/\ntheorem toGrothendieck_bot : toGrothendieck C ⊥ = ⊥ :=\n  (gi C).gc.l_bot\n#align category_theory.pretopology.to_grothendieck_bot CategoryTheory.Pretopology.toGrothendieck_bot\n\nend Pretopology\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Sites/Pretopology.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3467699201100446}}
{"text": "import data.complex.basic\n\nnamespace cau_seq\n\ntheorem const_inv {α β : Type*} [discrete_field β] [discrete_linear_ordered_field α] {abv : β → α}\n  [is_absolute_value abv] {x : β} (hx : x ≠ 0) :\n  const abv (x⁻¹) = inv (const abv x) (by rwa const_lim_zero) :=\n ext (assume n, by simp[inv_apply, const_apply])\n\nend cau_seq\n\nnamespace complex\nopen cau_seq\n\nlemma re_const_equiv_of_const_equiv {f : ℕ → ℂ} (hf : is_cau_seq abs f) (z : ℂ) :\n    cau_seq.const abs z ≈ ⟨f, hf⟩ → cau_seq.const _root_.abs z.re ≈ ⟨(λ (n : ℕ), (f n).re),\n    complex.is_cau_seq_re ⟨f,hf⟩⟩ := begin\n  assume h,assume ε ε0,cases h ε ε0 with i hi,existsi i,assume j ji,\n  replace hi := hi j ji, simp at *, rw [←complex.neg_re,←complex.add_re],\n  exact lt_of_le_of_lt (complex.abs_re_le_abs _) hi,\nend\n\nlemma im_const_equiv_of_const_equiv {f : ℕ → ℂ} (hf : is_cau_seq abs f) (z : ℂ) :\n    cau_seq.const abs z ≈ ⟨f, hf⟩ → cau_seq.const _root_.abs z.im ≈ ⟨(λ (n : ℕ), (f n).im),\n    complex.is_cau_seq_im ⟨f,hf⟩⟩ := begin\n  assume h,assume ε ε0,cases h ε ε0 with i hi,existsi i,assume j ji,\n  replace hi := hi j ji, simp at *, rw [←complex.neg_im,←complex.add_im],\n  exact lt_of_le_of_lt (complex.abs_im_le_abs _) hi,\nend\n\nlemma eq_lim_of_const_equiv {f : cau_seq ℂ abs}  {z: ℂ} :\n    cau_seq.const complex.abs z ≈ f → z = complex.lim f := begin\n  assume h,\n  unfold complex.lim,cases z with zre zim,simp,\n  split, have := real.equiv_lim ⟨(λ (n : ℕ), (f.1 n).re), complex.is_cau_seq_re f⟩,\n  rw ←cau_seq.const_equiv,simp at this,\n  have hf := complex.re_const_equiv_of_const_equiv f.2 {re := zre, im := zim} h,simp at hf,\n  exact setoid.trans hf this,\n  have := real.equiv_lim ⟨(λ (n : ℕ), (f.1 n).im), complex.is_cau_seq_im f⟩,\n  rw ←cau_seq.const_equiv,simp at this,\n  have hf := complex.im_const_equiv_of_const_equiv f.2 {re := zre, im := zim} h,simp at hf,\n  exact setoid.trans hf this,\nend\n\nlemma lim_eq_of_equiv_const {f : cau_seq ℂ complex.abs} {x : ℂ} (h : f ≈ cau_seq.const complex.abs x) : lim f = x :=\n(eq_lim_of_const_equiv $ setoid.symm h).symm\n\nlemma lim_eq_lim_of_equiv {f g : cau_seq ℂ complex.abs} (h : f ≈ g) : lim f = lim g :=\nlim_eq_of_equiv_const $ setoid.trans h $ equiv_lim g\n\n@[simp] lemma lim_const (x : ℂ) : lim (const abs x) = x :=\nlim_eq_of_equiv_const $ setoid.refl _\n\nlemma lim_add (f g : cau_seq ℂ complex.abs) : lim f + lim g = lim ⇑(f + g) :=\neq_lim_of_const_equiv $ show lim_zero (const complex.abs (lim ⇑f + lim ⇑g) - (f + g)),\n  by rw [const_add, add_sub_comm];\n  exact add_lim_zero (setoid.symm (equiv_lim f)) (setoid.symm (equiv_lim g))\n\nlemma lim_mul_lim (f g : cau_seq ℂ complex.abs) : lim f * lim g = lim ⇑(f * g) :=\neq_lim_of_const_equiv $ show lim_zero (const complex.abs (lim ⇑f * lim ⇑g) - f * g),\n  from have h : const complex.abs (lim ⇑f * lim ⇑g) - f * g = g * (const complex.abs (lim f) - f)\n      + const complex.abs (lim f) * (const complex.abs (lim g) - g) :=\n    by simp [mul_sub, mul_comm, const_mul, mul_add],\n  by rw h; exact add_lim_zero (mul_lim_zero _ (setoid.symm (equiv_lim f)))\n      (mul_lim_zero _ (setoid.symm (equiv_lim g)))\n\nlemma lim_mul (f : cau_seq ℂ complex.abs) (x : ℂ) : lim f * x = lim ⇑(f * const complex.abs x) :=\nby rw [← lim_mul_lim, lim_const]\n\nlemma lim_neg (f : cau_seq ℂ complex.abs) : lim ⇑(-f) = -lim f :=\nlim_eq_of_equiv_const (show lim_zero (-f - const complex.abs (-lim ⇑f)),\n  by rw [const_neg, sub_neg_eq_add, add_comm];\n  exact setoid.symm (equiv_lim f))\n\nlemma lim_eq_zero_iff (f : cau_seq ℂ complex.abs) : lim f = 0 ↔ lim_zero f :=\n⟨assume h,\n  by have hf := equiv_lim f;\n  rw h at hf;\n  exact (lim_zero_congr hf).mpr (const_lim_zero.mpr rfl),\nassume h,\n  have h₁ : f = (f - const complex.abs (0 : ℂ)) := cau_seq.ext (λ n, by simp [sub_apply, const_apply]),\n  by rw h₁ at h; exact lim_eq_of_equiv_const h ⟩\n\nlemma lim_inv {f : cau_seq ℂ complex.abs} (hf : ¬ lim_zero f) : lim ⇑(inv f hf) = (lim f)⁻¹ :=\nhave hl : lim f ≠ 0 := by rwa ← lim_eq_zero_iff at hf,\nlim_eq_of_equiv_const $ show lim_zero (inv f hf - const abs (lim ⇑f)⁻¹),\n  from have h₁ : ∀ (g f : cau_seq ℂ abs) (hf : ¬ lim_zero f), lim_zero (g - f * inv f hf * g) :=\n    λ g f hf, by rw [← one_mul g, ← mul_assoc, ← sub_mul, mul_one, mul_comm, mul_comm f];\n    exact mul_lim_zero _ (setoid.symm (cau_seq.inv_mul_cancel _)),\n  have h₂ : lim_zero ((inv f hf - const abs (lim ⇑f)⁻¹) - (const abs (lim f) - f) *\n      (inv f hf * const abs (lim ⇑f)⁻¹)) :=\n    by rw [sub_mul, ← sub_add, sub_sub, sub_add_eq_sub_sub, sub_right_comm, sub_add];\n    exact show lim_zero (inv f hf - const abs (lim ⇑f) * (inv f hf * const abs (lim ⇑f)⁻¹)\n      - (const abs (lim ⇑f)⁻¹ - f * (inv f hf * const abs (lim ⇑f)⁻¹))),\n    from sub_lim_zero\n      (by rw [← mul_assoc, mul_right_comm, cau_seq.const_inv hl]; exact h₁ _ _ _)\n      (by rw [← mul_assoc]; exact h₁ _ _ _),\n  (lim_zero_congr h₂).mpr $ by rw mul_comm; exact mul_lim_zero _ (setoid.symm (equiv_lim f))\n\nend complex", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/exp/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3467585374639343}}
{"text": "/-\nCopyright 2021 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\n\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.lattice\nimport formal_ml.measurable_space\nimport formal_ml.classical\nimport data.equiv.list\nimport formal_ml.probability_space\nimport formal_ml.prod_probability_space\nimport formal_ml.random_variable_independent_pair\nimport formal_ml.random_variable_identical_IID\n\n/- This file introduces the uniform probability space over a finite outcome type.\n   This is in turn used to define a uniform distribution over permutations. -/\n\n\n--Probably stay here.------------------------\n\nnoncomputable def set.cardF {α:Type*} [F:fintype α] (s:set α):nat := \n  (@set.to_finset α s (@subtype.fintype α s (classical.decidable_pred s) F)).card \n\n\nlemma set.univ_cardF_eq_fintype_card {α:Type*} [F:fintype α]:\n  (@set.univ α).cardF = fintype.card α := begin\n  simp [set.cardF],\n  have h1:\n    @finset.filter α (@set.univ α) (λ (a : α), @set.univ_decidable α a) (@finset.univ α F) =\n    (@finset.univ α F),\n  { ext1 a, split; intros A1; simp, },\n  rw h1,\n  simp [fintype.card],\nend\n\nlemma finset.disjoint_card_sum {α β:Type*} [decidable_eq α] [fintype α] [decidable_eq β]\n  (f:β → finset α) (h:pairwise (disjoint on f)) (t:finset β):∀ (s:finset α),\n  (⋃ (x:β) (h2:x∈ t), (@coe (finset α) (set α) _ (f x))) = ↑s → (s.card = t.sum (λ x, (f x).card)) := begin\n  classical,\n  apply finset.induction_on t,\n  { intros s h1, simp, simp at h1,\n    rw finset.ext_iff,\n    intros a,\n    rw set.ext_iff at h1,\n    have h2 := h1 a,\n    simp at h2,\n    simp,\n    apply h2 },\n  { intros a s h3 h4 t' h5,\n    rw finset.sum_insert,\n    simp at h5,\n    have h6:(f a) ∪ ((⋃ (x : β) (H : x ∈ s), ↑(f x)).to_finset) = t',\n    { rw set.ext_iff at h5,\n      rw finset.ext_iff,\n      intros a',\n      have h6_1 := h5 a',\n      rw finset.mem_coe at h6_1,\n      rw ← h6_1,\n      split; intros h6_2; simp at h6_2; simp [h6_2] },\n    rw ← h4 ((⋃ (x : β) (H : x ∈ s), ↑(f x)).to_finset),\n    rw ← finset.card_union_eq,\n    rw h6,\n    rw disjoint_iff,\n    simp, ext x, split; intros h1, simp at h1, cases h1 with h1 h2,\n    cases h2 with i h2,\n    cases h2 with h2 h7,\n    have h8:i ≠ a,\n    { intros contra,\n       subst i, apply h3, apply h2,  },\n    \n    have h9 := h i a h8,\n    simp [function.on_fun] at h9,\n    rw disjoint_iff at h9,\n    simp at h9,\n    rw ← h9,\n    simp [h1, h7],\n    exfalso,\n    apply h1,\n    simp,\n    apply h3,  \n         },\n   \nend\n\n\ndef semiring.of_nat_hom (α:Type*) [semiring α]:nat →+* α := {\n  to_fun := coe,\n  map_one' := begin\n    simp,\n  end,\n  map_mul' := begin\n   intros x y,\n   simp,\n  end,\n  map_add' := begin\n    intros x y,\n    simp,\n  end,\n  map_zero' := begin\n    simp,\n  end\n}\n\nlemma finset.prod_coe_nat {α β:Type*} [comm_semiring α] {s:finset β} {f:β → nat}:\n  (@coe nat α _) (s.prod f) = s.prod (λ (b:β), (@coe nat α _) (f b)) := begin\n  have h1:(@coe nat α _) = (semiring.of_nat_hom α).to_fun,\n  { refl },\n  rw h1,\n  simp,\nend\n\nlemma finset.sum_coe_nat {α β:Type*} [comm_semiring α] {s:finset β} {f:β → nat}:\n  (@coe nat α _) (s.sum f) = s.sum (λ (b:β), (@coe nat α _) (f b)) := begin\n  have h1:(@coe nat α _) = ⇑(semiring.of_nat_hom α),\n  { refl },\n  rw h1,\n  rw ring_hom.map_sum,\nend\n\n/- TODO: reverse direction if you get a chance. -/\nlemma disjoint_to_finset_iff {α:Type*} [F:fintype α] [decidable_eq α]\n  {s t:set α} [decidable_pred s] [decidable_pred t]:disjoint s t ↔ disjoint (s.to_finset) (t.to_finset) :=\nbegin\n  rw disjoint_iff,\n  rw disjoint_iff,\n  rw set.ext_iff,\n  rw finset.ext_iff,\n  simp, split; intros h1 x; have h2 := h1 x,\n  { split; intros h3, apply absurd h3.right,\n    apply h2, apply h3.left,  exfalso, apply h3 },\n  { intros h3 h4, have h5 := and.intro h3 h4,\n    rw h2 at h5, apply h5 },\nend\n\n---DEFINITELY stay here.--------------------\n\nnoncomputable def uniform_measure (α:Type*) [F:fintype α]: \n  @measure_theory.measure α ⊤  := @measure_theory.measure.of_measurable α ⊤\n  (λ (s:set α) (h:@measurable_set α ⊤ s), \n  ((s.cardF):ennreal)/((fintype.card α):ennreal)) \nbegin\n  simp [set.cardF],\nend\nbegin\n  classical,\n  intros f h_meas h_disjoint,\n  let s:finset α := (⋃ (i : ℕ), f i).to_finset,\n  begin\n    simp,\n    have h0:s = (⋃ (i : ℕ), f i).to_finset := rfl,\n    rw div_eq_mul_inv,\n    have h2:(λ (i:ℕ), (((f i).cardF):ennreal) / ((@fintype.card α F):ennreal)) = \n         λ (i:ℕ), (((f i).cardF):ennreal) * ((@fintype.card α F):ennreal)⁻¹,\n    { ext1 i, rw div_eq_mul_inv },\n    rw h2,\n    rw ennreal.tsum_mul_right,\n    have h3:↑((@set.Union α ℕ f).cardF) = (∑' (i : ℕ), ↑((f i).cardF)),\n    { \n       \n       have h4:∀ (x:α), ∃ (i:ℕ), x∈ s → x ∈ f i,\n       { intros x, cases classical.em (x ∈ s) with h4_1 h4_1,\n         rw h0 at h4_1,simp at h4_1, cases h4_1 with i h4_1,\n         apply exists.intro i, intros h4_2, apply h4_1,\n         apply exists.intro 0, intros h4_2, apply absurd h4_2, apply h4_1,  },\n       rw classical.skolem at h4,\n       cases h4 with g h4,\n       let t := s.image g,\n       rw @tsum_eq_sum ennreal ℕ _ _ _ _ t,\n       have h5:s.card = (@set.Union α ℕ f).cardF,\n       { simp [set.cardF], have h5_1:(λ (x : α), ∃ (i : ℕ), x ∈ f i) = (set.Union f),\n         { ext;split; intros A1; simp at A1; cases A1 with i A1; \n           simp; apply exists.intro i; apply A1, },\n         rw h5_1,  },\n       rw ← h5,\n       rw ← finset.sum_coe_nat,\n       simp only [nat.cast_inj, set.cardF],\n       rw ← finset.disjoint_card_sum,\n       { intros i j h_ne, simp [function.on_fun], rw ← disjoint_to_finset_iff, \n         apply h_disjoint, apply h_ne },\n       { simp, ext a, split, intros h6, simp at h6, simp,\n         { cases h6 with a' h6,  cases h6 with h6 h7,\n           apply exists.intro (g a'), apply h7 },\n         { intros h6, simp, apply exists.intro a, split, \n           { simp at h6, apply h6 },\n           { apply h4, simp [s], simp at h6, apply h6 }, } },\n         intros i h8,\n         have h9:f i = ∅,\n         { ext a, split; intros A1,\n           exfalso, apply h8,\n           have h9_1:a ∈ s,\n           { simp [s], apply exists.intro i, apply A1 },\n           have h9_2:a∈ f (g a) := h4 a h9_1,\n           have h9_3:g a = i,\n           { apply by_contradiction,\n             intros h9_3_1,\n             have h9_3_2:disjoint (f (g a)) (f i),\n             { apply h_disjoint, apply h9_3_1, },\n             rw disjoint_iff at h9_3_2,\n             simp at h9_3_2,\n             have h9_3_3:a∈ (f (g a)) ∩ (f i),\n             { simp, apply and.intro h9_2 A1 },\n             rw h9_3_2 at h9_3_3,\n             apply h9_3_3 },\n           subst i, simp [t], apply exists.intro a,\n           split,\n           apply exists.intro (g a),\n           apply A1, refl, exfalso, apply A1 },\n         rw h9,\n         simp [set.cardF], refl,\n     },\n     rw h3,\n  end\nend\n\nlemma uniform_measure_apply (α:Type*) [F:fintype α] (s:set α):\n  (uniform_measure α) s = \n  ((s.cardF):ennreal)/((fintype.card α):ennreal) := begin\n  apply measure_theory.measure.of_measurable_apply,\n  simp,\nend\n\nnoncomputable def uniform_measure_space (α:Type*) [F:fintype α]:\n  measure_theory.measure_space α := @measure_theory.measure_space.mk α ⊤\n  (uniform_measure α)\n\n\n\nlemma uniform_measure_space_apply (α:Type*) [F:fintype α] (s:set α):\n  @measure_theory.measure_space.volume _ (uniform_measure_space α) s = \n  ((s.cardF):ennreal)/((fintype.card α):ennreal) := begin\n  have h1: (@measure_theory.measure_space.volume α (@uniform_measure_space α F)) =\n           (uniform_measure α),\n  { refl },\n  rw h1,\n  apply uniform_measure_apply,\nend\n\n\nlemma uniform_measure_space.to_measurable_space_eq_top (α:Type*) [F:fintype α]:\n  (uniform_measure_space α).to_measurable_space = ⊤ := rfl\n\nlemma uniform_measure_space.measurable_set'_all (α:Type*) [F:fintype α] (s:set α):\n  (uniform_measure_space α).to_measurable_space.measurable_set' s := begin\n  rw uniform_measure_space.to_measurable_space_eq_top,\n  apply measurable_space.measurable_set_top,\nend\n\n\n/- DO NOT DELETE! \n   There is already proven:\n   1. permutations are finite.\n   2. There cardinality is the factorial.\n   This tells me we are on the right track! -/\n#check fintype.card_perm\n\nlemma fintype.card_ne_zero {α:Type*} [F:fintype α] [N:nonempty α]:\n  ¬(fintype.card α = 0) := begin\n  apply pos_iff_ne_zero.1,\n  rw fintype.card_pos_iff,\n  apply N,\nend\n\n\n\nnoncomputable def uniform_probability_space (α:Type*) [F:fintype α] [N:nonempty α]:\n  probability_space α := @probability_space.mk α (uniform_measure_space α) \nbegin\n  simp,\n  rw uniform_measure_space_apply,\n  rw set.univ_cardF_eq_fintype_card,\n  rw div_eq_mul_inv,\n  apply ennreal.mul_inv_cancel,\n  { simp, apply fintype.card_ne_zero },\n  { simp },\nend\n\nlemma uniform_probability_space.to_measurable_space_eq_top (α:Type*) [F:fintype α] [NE:nonempty α]:\n  (uniform_probability_space α).to_measurable_space = ⊤ := rfl\n\nlemma uniform_probability_space.to_measure_space_eq (α:Type*) [F:fintype α] [NE:nonempty α]:\n  (uniform_probability_space α).to_measure_space = uniform_measure_space α := rfl\n\n\nlemma uniform_probability_space.measurable_set'_all (α:Type*) [F:fintype α] [nonempty α] (s:set α):\n  (uniform_probability_space α).to_measurable_space.measurable_set' s := begin\n  rw uniform_probability_space.to_measurable_space_eq_top,\n  apply measurable_space.measurable_set_top,\nend\n\ndef uniform_event {α:Type*} (s:set α) [F:fintype α] [N:nonempty α]:\n  event (uniform_probability_space α) := {\n  val := s,\n  property := begin\n    apply uniform_probability_space.measurable_set'_all,\n  end\n}\n\ndef uniform_probability_space.measurable_all {α β:Type*} [F:fintype α] [nonempty α] \n(M:measurable_space β) (f:α → β):@measurable α β \n  (uniform_probability_space α).to_measurable_space M f := begin\n  intros S h1,\n  apply uniform_probability_space.measurable_set'_all,\nend\n\nlemma uniform_probability_space_apply' (α:Type*) [F:fintype α] [nonempty α] (s:event (uniform_probability_space α)):\n  Pr[s] = \n  ((s.val.cardF):nnreal)/((fintype.card α):nnreal) := begin\n  rw ← ennreal.coe_eq_coe,\n  rw event_prob_def,\n  rw uniform_probability_space.to_measure_space_eq,\n  simp,\n  rw ← subtype.val_eq_coe,\n  rw @uniform_measure_space_apply α F s.val,\n  rw ennreal.coe_div,\n  have h1:∀ (n:ℕ), (@coe nnreal ennreal _ (@coe nat nnreal _ n)) = (@coe nat ennreal _ n),\n  { simp },\n  rw h1,\n  rw h1,\n  simp,\n  apply fintype.card_ne_zero, \nend\n\nlemma uniform_probability_space_apply (α:Type*) [F:fintype α] [nonempty α] (s:set α):\n  Pr[uniform_event s] = \n  ((s.cardF):nnreal)/((fintype.card α):nnreal) := begin\n  apply uniform_probability_space_apply',\nend\n\ndef uniform_rv {α:Type*} [F:fintype α] [N:nonempty α] (M:measurable_space α):\n  (uniform_probability_space α) →ᵣ M := {\n  val := id,\n  property := begin\n    apply uniform_probability_space.measurable_all,\n  end\n}  \n\n\ndef is_uniform_rv {Ω α:Type*} {P:probability_space Ω} [F:fintype α] {M:measurable_space α}\n  (X:P →ᵣ M):Prop := ∀ (S:measurable_setB M), \n  Pr[X ∈ᵣ S] = ((S.val.cardF):nnreal)/((fintype.card α):nnreal)\n\n@[simp]\nlemma is_uniform_rv_uniform_rv {α:Type*} [F:fintype α] [N:nonempty α] (M:measurable_space α):\n  is_uniform_rv (uniform_rv M) := begin\n  intros S,\n  let S':= uniform_event S.val,\n  begin\n    have h1:@uniform_rv α F N M ∈ᵣ S = S',\n    { apply event.eq, simp [uniform_rv, S', uniform_event], },\n    rw h1,\n    rw uniform_probability_space_apply,\n  end\nend\n\ndef nonempty_perm {α:Type*}:nonempty (equiv.perm α) := nonempty.intro (equiv.refl α)\n\ndef uniform_perm_rv (α:Type*) [D:decidable_eq α] [F:fintype α] (M:measurable_space (equiv.perm α)):_ →ᵣ M :=\n  --(@uniform_probability_space (equiv.perm α) (@fintype_perm α D F) nonempty_perm) →ᵣ M :=\n  @uniform_rv (equiv.perm α) (@fintype_perm α D F) nonempty_perm M \n\nlemma is_uniform_rv_uniform_perm_rv {α:Type*} [F:fintype α] [N:decidable_eq α]\n  (M:measurable_space (equiv.perm α)):\n  @is_uniform_rv _ (equiv.perm α) _ (@fintype_perm α N F) _ (@uniform_perm_rv α N F M) := begin\n  apply is_uniform_rv_uniform_rv,\nend\n\nlemma select_rv_union_helper {Ω α γ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n   {M:measurable_space γ} (TM:top_measurable α) \n   (X:α → P →ᵣ M) (Y:P →ᵣ (TM.to_measurable_space)) (S:set γ):\n   ((λ (ω : Ω), (X (Y.val ω)).val ω) ⁻¹' S) = \n      ⋃ (a:α), ((Y.val ⁻¹' {a}) ∩ ((X a).val ⁻¹' S)) := begin\n  ext1 ω,split; intros h1; simp at h1; simp [h1],\nend\n\n\ndef select_rv {Ω α γ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n {M:measurable_space γ}\n {TM:top_measurable α} (X:α → P →ᵣ M) (Y:P →ᵣ TM.to_measurable_space):\n  P →ᵣ M := {\n  val := (λ ω:Ω, (X (Y.val ω)).val ω),\n  property := begin\n    intros S h1,\n    have h2:((λ (ω : Ω), (X (Y.val ω)).val ω) ⁻¹' S) = \n      ⋃ (a:α), ((Y.val ⁻¹' {a}) ∩ ((X a).val ⁻¹' S)),\n    { apply select_rv_union_helper },\n    rw h2,\n    haveI:encodable α := fintype.encodable α,    \n    apply measurable_set.Union,\n    intros a,\n    apply measurable_set.inter,\n    apply Y.property,\n    apply TM.all_measurable,\n    apply (X a).property,\n    apply h1,\n  end\n}\n\n\nlemma select_rv_apply {Ω α γ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n   {M:measurable_space γ} {TM:top_measurable α} \n   (X:α → P →ᵣ M) (Y:P →ᵣ (TM.to_measurable_space)) (S:measurable_setB M):\n   (select_rv X Y ∈ᵣ S) = (∃ᵣ (a:α), ((Y =ᵣ a) ∧ ((X a) ∈ᵣ S))) := begin\n  apply event.eq,\n  simp [select_rv],\n  ext; split; intros h1; simp at h1; simp [h1],\nend\n\n\n\n\n\n\ndef perm_value_rv {Ω α:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n  {TMp:top_measurable (equiv.perm α)}\n  (Y:P →ᵣ (TMp.to_measurable_space)) (a:α)\n  (M:measurable_space α):\n  P →ᵣ M := {\n  val := (λ ω:Ω, (Y.val ω) a),\n  property := begin\n    classical,\n    intros S A1,\n    let T := {p:equiv.perm α|p a ∈ S},\n    begin\n      have h1:((λ (ω : Ω), (Y.val ω) a) ⁻¹' S) = ⋃ (p∈ T), ((Y.val) ⁻¹' {p}),\n      { ext ω, split; intros h1_1; simp [T] at h1_1; simp [h1_1,T] },\n      rw h1,\n      haveI:encodable (equiv.perm α) := fintype.encodable (equiv.perm α),\n      apply measurable_set.Union,\n      intros p,\n      apply measurable_set.Union_Prop,\n      intros h2,\n      apply Y.property,\n      apply TMp.all_measurable,\n    end\n  end\n}\n\n\n\nlemma perm_value_rv_apply {Ω α:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n  {TMp:top_measurable (equiv.perm α)}\n  (Y:P →ᵣ (TMp.to_measurable_space)) (a:α)\n  (M:measurable_space α) (S:measurable_setB M) \n  [D:decidable_pred (λ (p:equiv.perm α), p a ∈ S.val)]:\n  (perm_value_rv Y a M) ∈ᵣ S = \n   ∃ᵣ (p:equiv.perm α) in (set.to_finset (λ (p:equiv.perm α), p a ∈ S.val)), \n    (Y =ᵣ p) := \nbegin\n  apply event.eq, ext ω, split; intros h1; simp [perm_value_rv] at h1; \n  simp [perm_value_rv, h1]; apply h1,\nend \n\n\n\n\n\n--set_option pp.implicit true\nlemma perm_value_rv_independent {Ω α κ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n  {TMp:top_measurable (equiv.perm α)}\n  {Mκ:measurable_space κ}\n  (Y:P →ᵣ (TMp.to_measurable_space)) (a:α) (X:P →ᵣ Mκ)\n  (M:measurable_space α):\n  random_variable_independent_pair X Y →\n  random_variable_independent_pair X (perm_value_rv Y a M) :=\nbegin\n  classical,\n  intro h1,\n  intros S T,\n  rw perm_value_rv_apply,\n  apply independent_event_pair_exists,\n  { intros p h2,\n    have h3:Y =ᵣ ↑p = Y ∈ᵣ ({p}:set (equiv.perm α)),\n    { apply equal_eq_mem },\n    rw h3,\n    apply h1 },\n  intros i h2 j h3 h_ne,\n  simp [function.on_fun, disjoint_iff],\n  \n  { ext ω, split; intros h3, \n   { simp at h3, exfalso, apply h_ne,\n     cases h3, subst i, subst j },\n   { exfalso, apply h3 } },\nend\n\ndef permute_rv {Ω α γ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n   {TM:top_measurable (equiv.perm α)}\n    {M:measurable_space γ} (X:α → P →ᵣ M) (perm:P →ᵣ TM.to_measurable_space)\n     (TMα:top_measurable α):\n   α → P  →ᵣ M := (λ (a:α), select_rv X (perm_value_rv perm a TMα))\n\nlemma permute_rv_identical {Ω α γ:Type*} [F:fintype α] [decidable_eq α]\n  {P:probability_space Ω}\n   {M:measurable_space γ} {TM:top_measurable (equiv.perm α)}\n  (TMα:top_measurable α) \n  (X:α → P →ᵣ M) (perm:P →ᵣ (TM.to_measurable_space)):\n  (∀ (a:α), random_variable_independent_pair perm (X a)) →\n  (∀ (a a':α), random_variable_identical (X a) (X a')) →\n  (∀ (a a':α), random_variable_identical (X a) ((permute_rv X perm TMα) a'))\n   := begin\n  intros h1 h2 a a',\n  simp [permute_rv],\n  intros S,\n  rw select_rv_apply,\n  have h3:eany_finset finset.univ (λ (a : α), perm_value_rv perm a' TMα =ᵣ a∧X a ∈ᵣ S)\n           = (∃ᵣ (a : α), perm_value_rv perm a' TMα =ᵣ a∧X a ∈ᵣ S),\n  {  \n    apply event.eq, simp,  \n  },\n  rw ← h3,\n  haveI:encodable α := fintype.encodable α,\n  rw Pr_sum_disjoint_eq,\n  have h4:(λ (b : α), Pr[perm_value_rv perm a' TMα =ᵣ b∧X b ∈ᵣ S]) =\n    (λ (b : α), Pr[perm_value_rv perm a' TMα =ᵣ b] * Pr[X a ∈ᵣ S]),\n  { ext b,\n    have h4_1:random_variable_independent_pair (X b) perm,\n    { apply random_variable_independent_pair.symm, apply (h1 b), },\n    have h4_2 := perm_value_rv_independent perm a' (X b) TMα.to_measurable_space h4_1,\n    have h4_3:(perm_value_rv perm a' TMα =ᵣ b) = \n              perm_value_rv perm a' TMα ∈ᵣ (measurable_setB_top {b}),\n    { apply equal_eq_mem },\n    rw h4_3,\n    have h4_4:independent_event_pair \n                (perm_value_rv perm a' TMα ∈ᵣ (measurable_setB_top {b})) (X b ∈ᵣ S),\n    { apply random_variable_independent_pair.symm,\n      apply h4_2, },\n    unfold independent_event_pair at h4_4,\n    rw h4_4,\n    --rw mul_comm,\n    simp,\n    left, apply h2 b a,\n     },\n   rw h4,\n   rw ← @finset.sum_mul α nnreal finset.univ,\n   rw  Pr_sum_univ_eq_one,\n   simp,\n   intros i j h_ne,\n   rw function.on_fun,\n   rw disjoint_iff,\n   simp,\n   ext ω, split; intros A1; simp at A1; simp [A1],\n   apply h_ne,\n   rw ← A1.left.left,\n   rw ← A1.right.left,\n   apply false.elim A1,\nend\n\nlemma random_variables_independent_proj {Ω α β γ:Type*} [fintype α] [fintype β]\n  (P:probability_space Ω) (M:measurable_space γ) (X:α → P →ᵣ M) (f:β → α):\n  (function.injective f) →\n  (random_variable_independent X) →\n  (random_variable_independent (λ (b:β), X (f b))) := begin\n  classical,\n  intros h1 h2,\n  intros g T,\n  simp,\n  have h3:∀ (a:α), ∃ (S:measurable_setB M), ∀ (b:β), (f b = a) → (g b = S),\n  { intros a, cases classical.em (∃ (b:β), f b = a) with h3_1 h3_1,\n    { cases h3_1 with b h3_1,\n      apply exists.intro (g b),\n      intros b' h_b',\n      rw ← h3_1 at h_b',\n      have h3_2 := h1 h_b', rw h3_2 },\n    { apply exists.intro measurable_setB_univ,\n      intros b'  h_b',\n      exfalso,\n      apply h3_1,\n      apply exists.intro b',\n      apply h_b' }, },\n  have h4 := classical.axiom_of_choice h3,\n  cases h4 with g' h4,\n  have h5:∀ (b:β), g b = g' (f b),\n  { intros b, apply h4 (f b) b _, refl },\n  have h6:(λ (b : β), Pr[X (f b) ∈ᵣ g b]) = (λ (a:α), Pr[X a ∈ᵣ g' a]) ∘ f,\n  { ext1 b, simp, rw h5 },\n  rw h6,\n  have h7:T.prod ((λ (a:α), Pr[X a ∈ᵣ g' a]) ∘ f) = (T.image f).prod (λ (a:α), Pr[X a ∈ᵣ g' a]),\n  { rw finset.prod_image, intros b h_b b' h_b' h_eq,\n    apply h1, apply h_eq },\n  rw h7,\n  have h8: (∀ᵣ (s : β) in T,X (f s) ∈ᵣ g s) = (∀ᵣ (a:α) in (T.image f), X a ∈ᵣ g' a),\n  { apply event.eq, simp, ext ω, split; intros h8_1; simp at h8_1; simp [h8_1];\n    intros i h_i,\n    { rw ← h5, apply h8_1 i h_i },\n    { rw h5, apply h8_1 i h_i } },\n  rw h8,\n  apply h2,\nend\n\nlemma random_variables_IID_proj {Ω α β γ:Type*} [fintype α] [fintype β]\n  (P:probability_space Ω) (M:measurable_space γ) (X:α → P →ᵣ M) (f:β → α):\n  (function.injective f) →\n  (random_variables_IID X) →\n  (random_variables_IID (λ (b:β), X (f b))) := begin\n  intros h1 h2, split,\n  { apply random_variables_independent_proj, apply h1, apply h2.left },\n  intros i j,\n  apply h2.right,\nend\n\nlemma random_variable_independent_pair_combine_apply {Ω α γ κ:Type*} \n  [F:fintype α] \n  {P:probability_space Ω}\n   {M:measurable_space γ} {Mκ:measurable_space κ}\n  (X:P →ᵣ Mκ) (Y:α → P →ᵣ M) (a:α):\n  random_variable_independent_pair X (pi.random_variable_combine Y) →\n  random_variable_independent_pair X (Y a) := begin\n  intros h1,\n  intros S₁ S₂,\n  have h2:(Y a ∈ᵣ S₂) = ((mf_pi (λ (a:α), M) a) ∘r (pi.random_variable_combine Y)) ∈ᵣ S₂,\n  { apply event.eq, simp, split; intros h2_1; simp at h2_1; simp at h2_1, },\n  rw h2,\n  rw rv_compose_measurable_setB,\n  apply h1,\nend\n\nlemma pairwise_disjoint_left {Ω α:Type*} \n  {P:probability_space Ω}\n  {E F:α → event P}:\n  (pairwise (disjoint on (λ a, (E a).val))) →\n (pairwise (disjoint on (λ a, (E a ∧ F a).val))) := begin\n  intros h1,\n  intros i j h_eq,\n  have h2 := h1 i j h_eq,\n  simp [function.on_fun],\n  simp [function.on_fun] at h2,\n  rw disjoint_iff,\n  rw disjoint_iff at h2,\n  simp,\n  rw ← set.subset_empty_iff,\n  apply set.subset.trans,\n  apply set.inter_subset_inter,\n  apply set.inter_subset_left,\n  apply set.inter_subset_left,\n  simp at h2,\n  rw h2,\nend\n\n/- Take a given event: The arugment for this is that for any fixed permutation,\n   the IID property holds. If we can show that this holds for every permutation,\n   then we will be done. -/\nlemma permute_rv_independent {Ω α γ:Type*} [F:fintype α] [D:decidable_eq α] [inhabited α]\n  {P:probability_space Ω}\n   {M:measurable_space γ} {TM:top_measurable (equiv.perm α)}\n  {TMα:top_measurable α} \n  (X:α → P →ᵣ M) (perm:P →ᵣ (TM.to_measurable_space)):\n  (random_variable_independent_pair perm (pi.random_variable_combine X)) →\n  (random_variables_IID X) →\n  (random_variable_independent (permute_rv X perm TMα)) := begin\n  intros h1 h2,\n  intros S T,\n  simp,\n  let a := inhabited.default α,\n  begin\n    have h4:∀ (S:measurable_setB M), ∀ (a':α), \n       Pr[X a' ∈ᵣ S] = Pr[X a ∈ᵣ S],\n    { intros S a', apply h2.right a' a },\n    have h_pair_ind:∀ (a':α), random_variable_independent_pair perm (X a'),\n    { intros a', apply random_variable_independent_pair_combine_apply perm X a' h1 },\n    have h5:∀ (S:measurable_setB M), ∀ (a':α), \n       Pr[permute_rv X perm TMα a' ∈ᵣ S] = Pr[X a ∈ᵣ S],\n    { intros S a', symmetry, \n      have h5_1 := permute_rv_identical TMα X perm h_pair_ind _ a a',\n      apply h5_1,\n      apply h2.right },\n    rw @finset.prod_congr α nnreal _ T _   (λ (b:α), Pr[X a ∈ᵣ S b]),\n    have h7:(∀ᵣ (s : α) in T, permute_rv X perm TMα s ∈ᵣ S s) = \n      eany_finset (finset.univ) (λ (p:equiv.perm α), (perm =ᵣ p) ∧ \n          (∀ᵣ (s : α) in (T.image p),X s ∈ᵣ S (p.inv_fun s))),\n    { apply event.eq, ext ω, split; intros h7_1; \n      simp [equiv_cancel_left, permute_rv, select_rv, perm_value_rv] at h7_1; \n      simp [equiv_cancel_left, permute_rv, select_rv, perm_value_rv, h7_1]; intros i h_i,\n      { rw equiv_cancel_left, apply h7_1 i h_i },\n      { have h7_2 := h7_1  i h_i, rw equiv_cancel_left at h7_2, apply h7_2 } },\n    rw h7,\n    --haveI:fintype (equiv.perm α) := infer_instance,\n    haveI E:encodable (equiv.perm α) := fintype.encodable (equiv.perm α),\n    rw @Pr_sum_disjoint_eq Ω (equiv.perm α) P,\n    have h8:(λ (b : equiv.perm α), Pr[perm =ᵣ ↑b∧∀ᵣ (s : α) in (T.image b),X s ∈ᵣ S (b.inv_fun s)]) =\n       (λ (b : equiv.perm α), T.prod (λ a', Pr[X a ∈ᵣ S a']) * Pr[perm =ᵣ ↑b]),\n    { ext1 b,\n      have h8_1:independent_event_pair (perm =ᵣ b) (∀ᵣ (s : α) in (T.image b),X s ∈ᵣ S (b.inv_fun s)),\n      { apply random_variable_independent_pair_apply h1,\n        apply equal_eq_mem_exists,\n        apply joint_random_variable_mem_exists X (T.image b) (S ∘ (b.inv_fun)),\n         },\n      unfold independent_event_pair at h8_1,\n      rw h8_1,\n      rw mul_comm,\n      have h8_2:random_variable_independent X :=h2.left,\n      have h8_3:independent_events (λ (s : α),X s ∈ᵣ S (b.inv_fun s)),\n      { apply h8_2 },\n      rw ← h8_3 (T.image b), rw finset.prod_image, simp,\n      left,\n      apply finset.prod_congr, refl, intros x h_x, rw equiv_cancel_left, apply h4,\n      intros i h_i j h_j h_eq, simp at h_eq, apply h_eq },\n    rw h8,\n    rw finset.sum_distrib_left,\n    have h9:finset.univ.sum (λ (a : equiv.perm α), Pr[perm =ᵣ ↑a]) = 1,\n    { rw Pr_sum_univ_eq_one },\n    rw h9,\n    simp,\n    { apply pairwise_disjoint_left, apply event_eq_disjoint },\n    refl,\n    { intros x h_x, rw h5 },\n  end\nend\n\nlemma permute_rv_IID' {Ω α γ:Type*} [fintype α] [decidable_eq α] [I:inhabited α]\n  {P:probability_space Ω}\n   {M:measurable_space γ} {TM:top_measurable (equiv.perm α)}\n  {TMα:top_measurable α} \n  (X:α → P →ᵣ M) (perm:P →ᵣ (TM.to_measurable_space)):\n  (random_variables_IID X) →\n  (random_variable_independent_pair perm (pi.random_variable_combine X)) →\n  (random_variables_IID (permute_rv X perm TMα)) := begin\n  intros h1 h2,\n  split,\n  apply permute_rv_independent,\n  apply h2,\n  apply h1,\n  have h4:∀ (a a'), random_variable_identical (X a) (permute_rv X perm TMα a'),\n  { apply permute_rv_identical,\n    intros a'',\n    apply random_variable_independent_pair_combine_apply,\n    apply h2, \n    apply h1.right },\n  intros i j,\n  apply random_variable_identical.trans,\n  apply random_variable_identical.symm,\n  apply h4 (inhabited.default α) i,\n  apply h4,\nend\n\nlemma permute_rv_IID {Ω α γ:Type*} [fintype α] [decidable_eq α] {P:probability_space Ω}\n   {M:measurable_space γ} {TM:top_measurable (equiv.perm α)}\n  {TMα:top_measurable α} \n  (X:α → P →ᵣ M) (perm:P →ᵣ (TM.to_measurable_space)):\n  (random_variables_IID X) →\n  (random_variable_independent_pair perm (pi.random_variable_combine X)) →\n  (random_variables_IID (permute_rv X perm TMα)) := begin\n  classical,\n  cases classical.em (nonempty α) with h1 h1,\n  { haveI:inhabited α := classical.inhabited_of_nonempty h1,\n    apply permute_rv_IID' },\n  intros h2 h3, apply random_variables_IID_empty,\n  apply h1,\nend\n\n\n\n/--\n   This is a core result, that a partition of an IID random variable results in two independent\n   random variables.\n   It is not as trivial as it appears, as one must handle events that expose correlations \n   within each set. Again, the monotone class theorem is useful.\n\n   Note: functions do not have to be injective, nor does β₁ or β₂ have to be nonempty.\n   However, this can be used to prove the case when they\n   are not, as you can create an injective function as an intermediate step,\n   and case-based analysis can handle the cases where β₁ or β₂ are empty. -/\nlemma random_variables_independent_disjoint_proj {Ω α β₁ β₂ γ:Type*} [fintype α] [fintype β₁] [fintype β₂] [nonempty β₁] [nonempty β₂]\n  (P:probability_space Ω) (M:measurable_space γ) (X:α → P →ᵣ M) (f₁:β₁ → α) (f₂:β₂ → α):\n  (function.injective f₁) →\n  (function.injective f₂) →\n  (disjoint (set.range f₁) (set.range f₂)) →\n  (random_variable_independent X) →\n  (random_variable_independent_pair (pi.random_variable_combine (λ (b:β₁), X (f₁ b))) (pi.random_variable_combine (λ (b:β₂), X (f₂ b)))) := begin\n  intros h1 h2 h_disjoint_range h3,\n  haveI:encodable β₁ := fintype.encodable β₁,\n  haveI:encodable β₂ := fintype.encodable β₂,\n  apply random_variable_independent_pair_on_semialgebra' pi_base,\n  apply pi_base_is_semialgebra,\n  apply pi_base_is_semialgebra,\n  apply pi_base_eq_measurable_space_pi,\n  apply pi_base_eq_measurable_space_pi,\n  intros T₁ T₂ h_T₁ h_T₂,\n  unfold independent_event_pair,\n  have h_T₁_def := pi_base_eq T₁ h_T₁,\n  have h_T₂_def := pi_base_eq T₂ h_T₂,\n  cases h_T₁_def with g₁ h_T₁_def,\n  cases h_T₂_def with g₂ h_T₂_def,\n  subst T₁,\n  subst T₂,\n  classical,\n  let U₁:finset α := (finset.image f₁ (@finset.univ β₁ _)),\n  let U₂:finset α := (finset.image f₂ (@finset.univ β₂ _)),\n  begin\n    have h_mem_U₁:∀ (a:α), a ∈ U₁ ↔ a ∈ (set.range f₁),\n    { simp [U₁], },\n    have h_mem_U₂:∀ (a:α), a ∈ U₂ ↔ a ∈ (set.range f₂),\n    { simp [U₂], },\n    have h_b₁_ne_b₂:∀ (b₁:β₁) (b₂:β₂), f₁ b₁ ≠ f₂ b₂,\n    { intros b₁ b₂ h_contra,\n      rw disjoint_iff at h_disjoint_range,\n      simp at h_disjoint_range,\n      rw ← set.subset_empty_iff at h_disjoint_range,\n      rw set.subset_def at h_disjoint_range,\n      apply h_disjoint_range (f₁ b₁),\n      rw set.mem_inter_iff,\n      split,\n      simp,\n      rw h_contra,\n      simp },\n    \n    have h_disjoint:disjoint U₁ U₂,\n    { rw disjoint_iff, simp, ext a, split; intros h_sub1,\n      { exfalso,\n        rw finset.mem_inter at h_sub1,\n        rw h_mem_U₁ at h_sub1,\n        rw h_mem_U₂ at h_sub1,\n        rw ← set.mem_inter_iff at h_sub1,\n        rw disjoint_iff at h_disjoint_range,\n        simp at h_disjoint_range,\n        rw ← set.subset_empty_iff at h_disjoint_range,\n        rw set.subset_def at h_disjoint_range,\n        --have h_disjoint_range_contra := h_disjoint_range a h_sub1,\n        apply h_disjoint_range,\n        apply h_sub1 },\n      { exfalso, apply h_sub1 } },\n    have h_inv:∀ (a:α), ∃ (S:measurable_setB M), \n      (∀ (b:β₁), f₁ b = a → S = g₁ b) ∧ (∀ (b:β₂), f₂ b = a → S = g₂ b),\n    { intros a,\n      cases classical.em (∃ (b:β₁), f₁ b = a) with h_in_U₁ h_notin_U₁,\n      { cases h_in_U₁ with b₁ h_b₁,\n        subst a,\n        apply exists.intro (g₁ b₁),\n        split,\n        { intros b₁' h_b₁', simp [h1] at h_b₁', subst b₁' },\n        { intros b₂' h_b₂', exfalso, apply h_b₁_ne_b₂ b₁ b₂', rw h_b₂' } },\n      -- If a is not in U₁, then we consider when it is in U₂.\n      cases classical.em (∃ (b₂:β₂), f₂ b₂ = a) with h_in_U₂ h_notin_U₂,\n      { cases h_in_U₂ with b₂ h_b₂,\n        subst a,\n        apply exists.intro (g₂ b₂),\n        split,\n        { intros b₁' h_b₁', exfalso, apply h_b₁_ne_b₂ b₁' b₂, rw h_b₁' },\n        { intros b₂' h_b₂', simp [h2] at h_b₂', subst b₂' } },\n      -- If a is not in U₁ or U₂, then just use the universal set.\n      { apply exists.intro (∅:measurable_setB M), \n         split,\n         { intros b₁ h_b₁, exfalso, apply h_notin_U₁,\n           apply exists.intro b₁, apply h_b₁ },\n         { intros b₂ h_b₂, exfalso, apply h_notin_U₂,\n           apply exists.intro b₂, apply h_b₂ } } },\n    rw classical.skolem at h_inv,\n    cases h_inv with g_inv h_inv,\n    have h_inv₁:pi.random_variable_combine (λ (b : β₁), X (f₁ b)) ∈ᵣ\n          set.pi_measurable (set.univ) g₁ = (∀ᵣ a in U₁, X a ∈ᵣ g_inv a),\n    { apply event.eq, ext1 ω, split; intros h_sub1;\n      simp [pi.random_variable_combine, set.pi_measurable,\n            pi.measurable_fun] at h_sub1;\n      simp [pi.random_variable_combine, set.pi_measurable,\n            pi.measurable_fun, h_sub1];\n      intros b₁, \n      { rw (h_inv (f₁ b₁)).left b₁ _, apply h_sub1, refl },\n      { rw ← (h_inv (f₁ b₁)).left b₁ _, apply h_sub1, refl } },\n    rw h_inv₁,\n    have h_inv₂:pi.random_variable_combine (λ (b : β₂), X (f₂ b)) ∈ᵣ\n          set.pi_measurable (set.univ) g₂ = (∀ᵣ a in U₂, X a ∈ᵣ g_inv a),\n    { apply event.eq, ext1 ω, split; intros h_sub1;\n      simp [pi.random_variable_combine, set.pi_measurable,\n            pi.measurable_fun] at h_sub1;\n      simp [pi.random_variable_combine, set.pi_measurable,\n            pi.measurable_fun, h_sub1];\n      intros b₂, \n      { rw (h_inv (f₂ b₂)).right b₂ _, apply h_sub1, refl },\n      { rw ← (h_inv (f₂ b₂)).right b₂ _, apply h_sub1, refl }, },\n    rw h_inv₂,\n    have h_union:((∀ᵣ (a : α) in U₁,X a ∈ᵣ g_inv a)∧(∀ᵣ (a : α) in U₂,X a ∈ᵣ g_inv a))\n      = (∀ᵣ (a : α) in (U₁∪ U₂),X a ∈ᵣ g_inv a),\n    { apply event.eq, ext1 ω, split; intros h_sub1; simp at h_sub1; simp [h_sub1],\n      -- Simplification solves one direction.\n      intros a h_sub2, cases h_sub2 with h_in_U₁ h_in_U₂,\n      { cases h_in_U₁ with b₁ h_b₁, subst a, apply h_sub1.left },\n      { cases h_in_U₂ with b₂ h_b₂, subst a, apply h_sub1.right } },\n    rw h_union,\n    rw ← h3 g_inv U₁,\n    rw ← h3 g_inv U₂,\n    rw ← h3 g_inv (U₁ ∪ U₂),\n    rw finset.prod_union,\n    apply h_disjoint,\n  end\nend\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/uniform_probability.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3467084396262222}}
{"text": "\nimport mathlib.group\nimport group_theory.group_action.defs\n\n/-!\n# Pi instances for multiplicative actions\n\nThis file defines instances for mul_action and related structures on Pi types.\n\n## See also\n\n* `group_theory.group_action.prod`\n* `group_theory.group_action.sigma`\n* `group_theory.group_action.sum`\n-/\n\nopen pi\n\nuniverses u v w\nvariable {I : Prop}     -- The indexing type\nvariable {f : I → Type v} -- The family of types already equipped with instances\nvariables (x y : Π i, f i) (i : I)\n\nnamespace pi_Prop\n\n@[to_additive pi_Prop.has_vadd]\ninstance has_smul {α : Type*} [Π i, has_smul α $ f i] :\n  has_smul α (Π i : I, f i) :=\n⟨λ s x, λ i, s • (x i)⟩\n\n@[to_additive]\nlemma smul_def {α : Type*} [Π i, has_smul α $ f i] (s : α) : s • x = λ i, s • x i := rfl\n@[simp, to_additive]\nlemma smul_apply {α : Type*} [Π i, has_smul α $ f i] (s : α) : (s • x) i = s • x i := rfl\n\n@[to_additive pi_Prop.has_vadd']\ninstance has_smul' {g : I → Type*} [Π i, has_smul (f i) (g i)] :\n  has_smul (Π i, f i) (Π i : I, g i) :=\n⟨λ s x, λ i, (s i) • (x i)⟩\n\n@[simp, to_additive]\nlemma smul_apply' {g : I → Type*} [∀ i, has_smul (f i) (g i)] (s : Π i, f i) (x : Π i, g i) :\n  (s • x) i = s i • x i :=\nrfl\ninstance is_scalar_tower {α β : Type*}\n  [has_smul α β] [Π i, has_smul β $ f i] [Π i, has_smul α $ f i]\n  [Π i, is_scalar_tower α β (f i)] : is_scalar_tower α β (Π i : I, f i) :=\n⟨λ x y z, funext $ λ i, smul_assoc x y (z i)⟩\n\ninstance is_scalar_tower' {g : I → Type*} {α : Type*}\n  [Π i, has_smul α $ f i] [Π i, has_smul (f i) (g i)] [Π i, has_smul α $ g i]\n  [Π i, is_scalar_tower α (f i) (g i)] : is_scalar_tower α (Π i : I, f i) (Π i : I, g i) :=\n⟨λ x y z, funext $ λ i, smul_assoc x (y i) (z i)⟩\n\ninstance is_scalar_tower'' {g : I → Type*} {h : I → Type*}\n  [Π i, has_smul (f i) (g i)] [Π i, has_smul (g i) (h i)] [Π i, has_smul (f i) (h i)]\n  [Π i, is_scalar_tower (f i) (g i) (h i)] : is_scalar_tower (Π i, f i) (Π i, g i) (Π i, h i) :=\n⟨λ x y z, funext $ λ i, smul_assoc (x i) (y i) (z i)⟩\n\n@[to_additive]\ninstance smul_comm_class {α β : Type*}\n  [Π i, has_smul α $ f i] [Π i, has_smul β $ f i] [∀ i, smul_comm_class α β (f i)] :\n  smul_comm_class α β (Π i : I, f i) :=\n⟨λ x y z, funext $ λ i, smul_comm x y (z i)⟩\n\n@[to_additive]\ninstance smul_comm_class' {g : I → Type*} {α : Type*}\n  [Π i, has_smul α $ g i] [Π i, has_smul (f i) (g i)] [∀ i, smul_comm_class α (f i) (g i)] :\n  smul_comm_class α (Π i : I, f i) (Π i : I, g i) :=\n⟨λ x y z, funext $ λ i, smul_comm x (y i) (z i)⟩\n\n@[to_additive]\ninstance smul_comm_class'' {g : I → Type*} {h : I → Type*}\n  [Π i, has_smul (g i) (h i)] [Π i, has_smul (f i) (h i)]\n  [∀ i, smul_comm_class (f i) (g i) (h i)] : smul_comm_class (Π i, f i) (Π i, g i) (Π i, h i) :=\n⟨λ x y z, funext $ λ i, smul_comm (x i) (y i) (z i)⟩\n\ninstance {α : Type*} [Π i, has_smul α $ f i] [Π i, has_smul αᵐᵒᵖ $ f i]\n  [∀ i, is_central_scalar α (f i)] : is_central_scalar α (Π i, f i) :=\n⟨λ r m, funext $ λ i, op_smul_eq_smul _ _⟩\n\n/-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is\nnot an instance as `i` cannot be inferred. -/\n@[to_additive pi_Prop.has_faithful_vadd_at]\nlemma has_faithful_smul_at {α : Type*}\n  [Π i, has_smul α $ f i] [Π i, nonempty (f i)] (i : I) [has_faithful_smul α (f i)] :\n  has_faithful_smul α (Π i, f i) :=\n⟨λ x y h, eq_of_smul_eq_smul $ λ a : f i, begin\n  classical,\n  have := congr_fun (h $ function.update (λ j, classical.choice (‹Π i, nonempty (f i)› j)) i a) i,\n  simpa using this,\nend⟩\n\n@[to_additive pi_Prop.has_faithful_vadd]\ninstance has_faithful_smul {α : Type*}\n  [nonempty I] [Π i, has_smul α $ f i] [Π i, nonempty (f i)] [Π i, has_faithful_smul α (f i)] :\n  has_faithful_smul α (Π i, f i) :=\nlet ⟨i⟩ := ‹nonempty I› in has_faithful_smul_at i\n\n@[to_additive]\ninstance mul_action (α) {m : monoid α} [Π i, mul_action α $ f i] :\n  @mul_action α (Π i : I, f i) m :=\n{ smul := (•),\n  mul_smul := λ r s f, funext $ λ i, mul_smul _ _ _,\n  one_smul := λ f, funext $ λ i, one_smul α _ }\n\n@[to_additive]\ninstance mul_action' {g : I → Type*} {m : Π i, monoid (f i)} [Π i, mul_action (f i) (g i)] :\n  @mul_action (Π i, f i) (Π i : I, g i) (@pi_Prop.monoid I f m) :=\n{ smul := (•),\n  mul_smul := λ r s f, funext $ λ i, mul_smul _ _ _,\n  one_smul := λ f, funext $ λ i, one_smul _ _ }\n\ninstance distrib_mul_action (α) {m : monoid α} {n : ∀ i, add_monoid $ f i}\n  [∀ i, distrib_mul_action α $ f i] :\n  @distrib_mul_action α (Π i : I, f i) m (@pi_Prop.add_monoid I f n) :=\n{ smul_zero := λ c, funext $ λ i, smul_zero _,\n  smul_add := λ c f g, funext $ λ i, smul_add _ _ _,\n  ..pi_Prop.mul_action _ }\n\ninstance distrib_mul_action' {g : I → Type*} {m : Π i, monoid (f i)} {n : Π i, add_monoid $ g i}\n  [Π i, distrib_mul_action (f i) (g i)] :\n  @distrib_mul_action (Π i, f i) (Π i, g i) (@pi_Prop.monoid I f m) (@pi_Prop.add_monoid I g n) :=\n{ smul_add := by { intros, ext x, apply smul_add },\n  smul_zero := by { intros, ext x, apply smul_zero } }\n\ninstance mul_distrib_mul_action (α) {m : monoid α} {n : Π i, monoid $ f i}\n  [Π i, mul_distrib_mul_action α $ f i] :\n  @mul_distrib_mul_action α (Π i : I, f i) m (@pi_Prop.monoid I f n) :=\n{ smul_one := λ c, funext $ λ i, smul_one _,\n  smul_mul := λ c f g, funext $ λ i, smul_mul' _ _ _,\n  ..pi_Prop.mul_action _ }\n\ninstance mul_distrib_mul_action' {g : I → Type*} {m : Π i, monoid (f i)} {n : Π i, monoid $ g i}\n  [Π i, mul_distrib_mul_action (f i) (g i)] :\n  @mul_distrib_mul_action (Π i, f i) (Π i, g i) (@pi_Prop.monoid I f m) (@pi_Prop.monoid I g n) :=\n{ smul_mul := by { intros, ext x, apply smul_mul' },\n  smul_one := by { intros, ext x, apply smul_one } }\n\nend pi_Prop\n\nnamespace function_Prop\n\n/-- Non-dependent version of `pi_Prop.has_smul`. Lean gets confused by the dependent instance if\nthis is not present. -/\n@[to_additive]\ninstance has_smul {ι : Prop} {R M : Type*} [has_smul R M] : has_smul R (ι → M) := pi_Prop.has_smul\n\n/-- Non-dependent version of `pi_Prop.smul_comm_class`. Lean gets confused by the dependent instance\nif this is not present. -/\n@[to_additive]\ninstance smul_comm_class {ι : Prop} {α β M : Type*} [has_smul α M] [has_smul β M]\n  [smul_comm_class α β M] :\n  smul_comm_class α β (ι → M) :=\npi_Prop.smul_comm_class\n\nend function_Prop\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/mathlib/group_action.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.34670843128633855}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.vector\nimport Mathlib.data.list.nodup\nimport Mathlib.data.list.of_fn\nimport Mathlib.control.applicative\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u u_3 \n\nnamespace Mathlib\n\n/-!\n# Additional theorems about the `vector` type\n\nThis file introduces the infix notation `::ᵥ` for `vector.cons`.\n-/\n\nnamespace vector\n\n\ninfixr:67 \"::ᵥ\" => Mathlib.vector.cons\n\nprotected instance inhabited {n : ℕ} {α : Type u_1} [Inhabited α] : Inhabited (vector α n) :=\n  { default := of_fn fun (_x : fin n) => Inhabited.default }\n\ntheorem to_list_injective {n : ℕ} {α : Type u_1} : function.injective to_list :=\n  subtype.val_injective\n\n/-- Two `v w : vector α n` are equal iff they are equal at every single index. -/\ntheorem ext {n : ℕ} {α : Type u_1} {v : vector α n} {w : vector α n}\n    (h : ∀ (m : fin n), nth v m = nth w m) : v = w :=\n  sorry\n\n/-- The empty `vector` is a `subsingleton`. -/\nprotected instance zero_subsingleton {α : Type u_1} : subsingleton (vector α 0) :=\n  subsingleton.intro fun (_x _x_1 : vector α 0) => ext fun (m : fin 0) => fin.elim0 m\n\n@[simp] theorem cons_val {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) :\n    subtype.val (a::ᵥv) = a :: subtype.val v :=\n  sorry\n\n@[simp] theorem cons_head {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : head (a::ᵥv) = a :=\n  sorry\n\n@[simp] theorem cons_tail {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : tail (a::ᵥv) = v :=\n  sorry\n\n@[simp] theorem to_list_of_fn {α : Type u_1} {n : ℕ} (f : fin n → α) :\n    to_list (of_fn f) = list.of_fn f :=\n  sorry\n\n@[simp] theorem mk_to_list {n : ℕ} {α : Type u_1} (v : vector α n)\n    (h : list.length (to_list v) = n) : { val := to_list v, property := h } = v :=\n  sorry\n\n@[simp] theorem to_list_map {n : ℕ} {α : Type u_1} {β : Type u_2} (v : vector α n) (f : α → β) :\n    to_list (map f v) = list.map f (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = n) =>\n      Eq.refl (to_list (map f { val := v_val, property := v_property }))\n\ntheorem nth_eq_nth_le {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) :\n    nth v i =\n        list.nth_le (to_list v) (subtype.val i)\n          (eq.mpr\n            (id (Eq._oldrec (Eq.refl (subtype.val i < list.length (to_list v))) (to_list_length v)))\n            (subtype.property i)) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = n) =>\n      idRhs\n        (nth { val := v_val, property := v_property } i =\n          nth { val := v_val, property := v_property } i)\n        rfl\n\n@[simp] theorem nth_map {n : ℕ} {α : Type u_1} {β : Type u_2} (v : vector α n) (f : α → β)\n    (i : fin n) : nth (map f v) i = f (nth v i) :=\n  sorry\n\n@[simp] theorem nth_of_fn {α : Type u_1} {n : ℕ} (f : fin n → α) (i : fin n) :\n    nth (of_fn f) i = f i :=\n  sorry\n\n@[simp] theorem of_fn_nth {n : ℕ} {α : Type u_1} (v : vector α n) : of_fn (nth v) = v := sorry\n\n@[simp] theorem nth_tail {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) (i : fin n) :\n    nth (tail v) i = nth v (fin.succ i) :=\n  sorry\n\n@[simp] theorem tail_val {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) :\n    subtype.val (tail v) = list.tail (subtype.val v) :=\n  sorry\n\n/-- The `tail` of a `nil` vector is `nil`. -/\n@[simp] theorem tail_nil {α : Type u_1} : tail nil = nil := rfl\n\n/-- The `tail` of a vector made up of one element is `nil`. -/\n@[simp] theorem singleton_tail {α : Type u_1} (v : vector α 1) : tail v = nil :=\n  eq.mpr (id (propext (eq_iff_true_of_subsingleton (tail v) nil))) trivial\n\n@[simp] theorem tail_of_fn {α : Type u_1} {n : ℕ} (f : fin (Nat.succ n) → α) :\n    tail (of_fn f) = of_fn fun (i : fin (Nat.succ n - 1)) => f (fin.succ i) :=\n  sorry\n\n/-- The list that makes up a `vector` made up of a single element,\nretrieved via `to_list`, is equal to the list of that single element. -/\n@[simp] theorem to_list_singleton {α : Type u_1} (v : vector α 1) : to_list v = [head v] := sorry\n\n/-- Mapping under `id` does not change a vector. -/\n@[simp] theorem map_id {α : Type u_1} {n : ℕ} (v : vector α n) : map id v = v := sorry\n\ntheorem mem_iff_nth {n : ℕ} {α : Type u_1} {a : α} {v : vector α n} :\n    a ∈ to_list v ↔ ∃ (i : fin n), nth v i = a :=\n  sorry\n\ntheorem nodup_iff_nth_inj {n : ℕ} {α : Type u_1} {v : vector α n} :\n    list.nodup (to_list v) ↔ function.injective (nth v) :=\n  sorry\n\n@[simp] theorem nth_mem {n : ℕ} {α : Type u_1} (i : fin n) (v : vector α n) : nth v i ∈ to_list v :=\n  sorry\n\ntheorem head'_to_list {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) :\n    list.head' (to_list v) = some (head v) :=\n  sorry\n\ndef reverse {n : ℕ} {α : Type u_1} (v : vector α n) : vector α n :=\n  { val := list.reverse (to_list v), property := sorry }\n\n/-- The `list` of a vector after a `reverse`, retrieved by `to_list` is equal\nto the `list.reverse` after retrieving a vector's `to_list`. -/\ntheorem to_list_reverse {n : ℕ} {α : Type u_1} {v : vector α n} :\n    to_list (reverse v) = list.reverse (to_list v) :=\n  rfl\n\n@[simp] theorem nth_zero {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) : nth v 0 = head v :=\n  sorry\n\n@[simp] theorem head_of_fn {α : Type u_1} {n : ℕ} (f : fin (Nat.succ n) → α) :\n    head (of_fn f) = f 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (head (of_fn f) = f 0)) (Eq.symm (nth_zero (of_fn f)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth (of_fn f) 0 = f 0)) (nth_of_fn f 0))) (Eq.refl (f 0)))\n\n@[simp] theorem nth_cons_zero {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : nth (a::ᵥv) 0 = a :=\n  sorry\n\n/-- Accessing the `nth` element of a vector made up\nof one element `x : α` is `x` itself. -/\n@[simp] theorem nth_cons_nil {α : Type u_1} {ix : fin 1} (x : α) : nth (x::ᵥnil) ix = x := sorry\n\n@[simp] theorem nth_cons_succ {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) (i : fin n) :\n    nth (a::ᵥv) (fin.succ i) = nth v i :=\n  eq.mpr\n    (id (Eq._oldrec (Eq.refl (nth (a::ᵥv) (fin.succ i) = nth v i)) (Eq.symm (nth_tail (a::ᵥv) i))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth (tail (a::ᵥv)) i = nth v i)) (tail_cons a v)))\n      (Eq.refl (nth v i)))\n\n/-- The last element of a `vector`, given that the vector is at least one element. -/\ndef last {n : ℕ} {α : Type u_1} (v : vector α (n + 1)) : α := nth v (fin.last n)\n\n/-- The last element of a `vector`, given that the vector is at least one element. -/\ntheorem last_def {n : ℕ} {α : Type u_1} {v : vector α (n + 1)} : last v = nth v (fin.last n) := rfl\n\n/-- The `last` element of a vector is the `head` of the `reverse` vector. -/\ntheorem reverse_nth_zero {n : ℕ} {α : Type u_1} {v : vector α (n + 1)} :\n    head (reverse v) = last v :=\n  sorry\n\n/--\nConstruct a `vector β (n + 1)` from a `vector α n` by scanning `f : β → α → β`\nfrom the \"left\", that is, from 0 to `fin.last n`, using `b : β` as the starting value.\n-/\ndef scanl {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) :\n    vector β (n + 1) :=\n  { val := list.scanl f b (to_list v), property := sorry }\n\n/-- Providing an empty vector to `scanl` gives the starting value `b : β`. -/\n@[simp] theorem scanl_nil {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) :\n    scanl f b nil = b::ᵥnil :=\n  rfl\n\n/--\nThe recursive step of `scanl` splits a vector `x ::ᵥ v : vector α (n + 1)`\ninto the provided starting value `b : β` and the recursed `scanl`\n`f b x : β` as the starting value.\n\nThis lemma is the `cons` version of `scanl_nth`.\n-/\n@[simp] theorem scanl_cons {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    (v : vector α n) (x : α) : scanl f b (x::ᵥv) = b::ᵥscanl f (f b x) v :=\n  sorry\n\n/--\nThe underlying `list` of a `vector` after a `scanl` is the `list.scanl`\nof the underlying `list` of the original `vector`.\n-/\n@[simp] theorem scanl_val {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    {v : vector α n} : subtype.val (scanl f b v) = list.scanl f b (subtype.val v) :=\n  sorry\n\n/--\nThe `to_list` of a `vector` after a `scanl` is the `list.scanl`\nof the `to_list` of the original `vector`.\n-/\n@[simp] theorem to_list_scanl {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    (v : vector α n) : to_list (scanl f b v) = list.scanl f b (to_list v) :=\n  rfl\n\n/--\nThe recursive step of `scanl` splits a vector made up of a single element\n`x ::ᵥ nil : vector α 1` into a `vector` of the provided starting value `b : β`\nand the mapped `f b x : β` as the last value.\n-/\n@[simp] theorem scanl_singleton {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    (v : vector α 1) : scanl f b v = b::ᵥf b (head v)::ᵥnil :=\n  sorry\n\n/--\nThe first element of `scanl` of a vector `v : vector α n`,\nretrieved via `head`, is the starting value `b : β`.\n-/\n@[simp] theorem scanl_head {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    (v : vector α n) : head (scanl f b v) = b :=\n  sorry\n\n/--\nFor an index `i : fin n`, the `nth` element of `scanl` of a\nvector `v : vector α n` at `i.succ`, is equal to the application\nfunction `f : β → α → β` of the `i.cast_succ` element of\n`scanl f b v` and `nth v i`.\n\nThis lemma is the `nth` version of `scanl_cons`.\n-/\n@[simp] theorem scanl_nth {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β)\n    (v : vector α n) (i : fin n) :\n    nth (scanl f b v) (fin.succ i) = f (nth (scanl f b v) (coe_fn fin.cast_succ i)) (nth v i) :=\n  sorry\n\ndef m_of_fn {m : Type u → Type u_1} [Monad m] {α : Type u} {n : ℕ} :\n    (fin n → m α) → m (vector α n) :=\n  sorry\n\ntheorem m_of_fn_pure {m : Type u_1 → Type u_2} [Monad m] [is_lawful_monad m] {α : Type u_1} {n : ℕ}\n    (f : fin n → α) : (m_of_fn fun (i : fin n) => pure (f i)) = pure (of_fn f) :=\n  sorry\n\ndef mmap {m : Type u → Type u_1} [Monad m] {α : Type u_2} {β : Type u} (f : α → m β) {n : ℕ} :\n    vector α n → m (vector β n) :=\n  sorry\n\n@[simp] theorem mmap_nil {m : Type u_1 → Type u_2} [Monad m] {α : Type u_3} {β : Type u_1}\n    (f : α → m β) : mmap f nil = pure nil :=\n  rfl\n\n@[simp] theorem mmap_cons {m : Type u_1 → Type u_2} [Monad m] {α : Type u_3} {β : Type u_1}\n    (f : α → m β) (a : α) {n : ℕ} (v : vector α n) :\n    mmap f (a::ᵥv) =\n        do \n          let h' ← f a \n          let t' ← mmap f v \n          pure (h'::ᵥt') :=\n  sorry\n\n/-- Define `C v` by induction on `v : vector α (n + 1)`, a vector of\nat least one element.\nThis function has two arguments: `h0` handles the base case on `C nil`,\nand `hs` defines the inductive step using `∀ x : α, C v → C (x ::ᵥ v)`. -/\ndef induction_on {α : Type u_1} {n : ℕ} {C : {n : ℕ} → vector α n → Sort u_2} (v : vector α (n + 1))\n    (h0 : C nil) (hs : {n : ℕ} → {x : α} → {w : vector α n} → C w → C (x::ᵥw)) : C v :=\n  Nat.rec (fun (v : vector α (0 + 1)) => eq.mpr sorry (eq.mpr sorry (hs h0)))\n    (fun (n : ℕ) (hn : (v : vector α (n + 1)) → C v) (v : vector α (Nat.succ n + 1)) =>\n      eq.mpr sorry (hs (hn (tail v))))\n    n v\n\ndef to_array {n : ℕ} {α : Type u_1} : vector α n → array n α := sorry\n\ndef insert_nth {n : ℕ} {α : Type u_1} (a : α) (i : fin (n + 1)) (v : vector α n) :\n    vector α (n + 1) :=\n  { val := list.insert_nth (↑i) a (subtype.val v), property := sorry }\n\ntheorem insert_nth_val {n : ℕ} {α : Type u_1} {a : α} {i : fin (n + 1)} {v : vector α n} :\n    subtype.val (insert_nth a i v) = list.insert_nth (subtype.val i) a (subtype.val v) :=\n  rfl\n\n@[simp] theorem remove_nth_val {n : ℕ} {α : Type u_1} {i : fin n} {v : vector α n} :\n    subtype.val (remove_nth i v) = list.remove_nth (subtype.val v) ↑i :=\n  sorry\n\ntheorem remove_nth_insert_nth {n : ℕ} {α : Type u_1} {a : α} {v : vector α n} {i : fin (n + 1)} :\n    remove_nth i (insert_nth a i v) = v :=\n  subtype.eq (list.remove_nth_insert_nth (subtype.val i) (subtype.val v))\n\ntheorem remove_nth_insert_nth_ne {n : ℕ} {α : Type u_1} {a : α} {v : vector α (n + 1)}\n    {i : fin (n + bit0 1)} {j : fin (n + bit0 1)} (h : i ≠ j) :\n    remove_nth i (insert_nth a j v) =\n        insert_nth a (fin.pred_above i j (ne.symm h)) (remove_nth (fin.pred_above j i h) v) :=\n  sorry\n\ntheorem insert_nth_comm {n : ℕ} {α : Type u_1} (a : α) (b : α) (i : fin (n + 1)) (j : fin (n + 1))\n    (h : i ≤ j) (v : vector α n) :\n    insert_nth b (fin.succ j) (insert_nth a i v) =\n        insert_nth a (coe_fn fin.cast_succ i) (insert_nth b j v) :=\n  sorry\n\n/-- `update_nth v n a` replaces the `n`th element of `v` with `a` -/\ndef update_nth {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) (a : α) : vector α n :=\n  { val := list.update_nth (subtype.val v) (subtype.val i) a, property := sorry }\n\n@[simp] theorem nth_update_nth_same {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) (a : α) :\n    nth (update_nth v i a) i = a :=\n  sorry\n\ntheorem nth_update_nth_of_ne {n : ℕ} {α : Type u_1} {v : vector α n} {i : fin n} {j : fin n}\n    (h : i ≠ j) (a : α) : nth (update_nth v i a) j = nth v j :=\n  sorry\n\ntheorem nth_update_nth_eq_if {n : ℕ} {α : Type u_1} {v : vector α n} {i : fin n} {j : fin n}\n    (a : α) : nth (update_nth v i a) j = ite (i = j) a (nth v j) :=\n  sorry\n\nend vector\n\n\nnamespace vector\n\n\nprotected def traverse {n : ℕ} {F : Type u → Type u} [Applicative F] {α : Type u} {β : Type u}\n    (f : α → F β) : vector α n → F (vector β n) :=\n  sorry\n\n@[simp] protected theorem traverse_def {n : ℕ} {F : Type u → Type u} [Applicative F]\n    [is_lawful_applicative F] {α : Type u} {β : Type u} (f : α → F β) (x : α) (xs : vector α n) :\n    vector.traverse f (x::ᵥxs) = cons <$> f x <*> vector.traverse f xs :=\n  subtype.cases_on xs\n    fun (xs : List α) (xs_property : list.length xs = n) =>\n      eq.drec\n        (Eq.refl (vector.traverse f (x::ᵥ{ val := xs, property := Eq.refl (list.length xs) })))\n        xs_property\n\nprotected theorem id_traverse {n : ℕ} {α : Type u} (x : vector α n) : vector.traverse id.mk x = x :=\n  sorry\n\nprotected theorem comp_traverse {n : ℕ} {F : Type u → Type u} {G : Type u → Type u} [Applicative F]\n    [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {α : Type u} {β : Type u}\n    {γ : Type u} (f : β → F γ) (g : α → G β) (x : vector α n) :\n    vector.traverse (functor.comp.mk ∘ Functor.map f ∘ g) x =\n        functor.comp.mk (vector.traverse f <$> vector.traverse g x) :=\n  sorry\n\nprotected theorem traverse_eq_map_id {n : ℕ} {α : Type u_1} {β : Type u_1} (f : α → β)\n    (x : vector α n) : vector.traverse (id.mk ∘ f) x = id.mk (map f x) :=\n  sorry\n\nprotected theorem naturality {n : ℕ} {F : Type u → Type u} {G : Type u → Type u} [Applicative F]\n    [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G]\n    (η : applicative_transformation F G) {α : Type u} {β : Type u} (f : α → F β) (x : vector α n) :\n    coe_fn η (vector β n) (vector.traverse f x) = vector.traverse (coe_fn η β ∘ f) x :=\n  sorry\n\nprotected instance flip.traversable {n : ℕ} : traversable (flip vector n) :=\n  traversable.mk vector.traverse\n\nprotected instance flip.is_lawful_traversable {n : ℕ} : is_lawful_traversable (flip vector n) :=\n  is_lawful_traversable.mk vector.id_traverse vector.comp_traverse vector.traverse_eq_map_id\n    vector.naturality\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/vector2_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3467084312863385}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.dynamics.fixed_points.basic\nimport Mathlib.data.set.lattice\nimport Mathlib.data.pnat.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Periodic points\n\nA point `x : α` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\n\n## Main definitions\n\n* `is_periodic_pt f n x` : `x` is a periodic point of `f` of period `n`, i.e. `f^[n] x = x`.\n  We do not require `n > 0` in the definition.\n* `pts_of_period f n` : the set `{x | is_periodic_pt f n x}`. Note that `n` is not required to\n  be the minimal period of `x`.\n* `periodic_pts f` : the set of all periodic points of `f`.\n* `minimal_period f x` : the minimal period of a point `x` under an endomorphism `f` or zero\n  if `x` is not a periodic point of `f`.\n\n## Main statements\n\nWe provide “dot syntax”-style operations on terms of the form `h : is_periodic_pt f n x` including\narithmetic operations on `n` and `h.map (hg : semiconj_by g f f')`. We also prove that `f`\nis bijective on each set `pts_of_period f n` and on `periodic_pts f`. Finally, we prove that `x`\nis a periodic point of `f` of period `n` if and only if `minimal_period f x | n`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Periodic_point\n\n-/\n\nnamespace function\n\n\n/-- A point `x` is a periodic point of `f : α → α` of period `n` if `f^[n] x = x`.\nNote that we do not require `0 < n` in this definition. Many theorems about periodic points\nneed this assumption. -/\ndef is_periodic_pt {α : Type u_1} (f : α → α) (n : ℕ) (x : α) :=\n  is_fixed_pt (nat.iterate f n) x\n\n/-- A fixed point of `f` is a periodic point of `f` of any prescribed period. -/\ntheorem is_fixed_pt.is_periodic_pt {α : Type u_1} {f : α → α} {x : α} (hf : is_fixed_pt f x) (n : ℕ) : is_periodic_pt f n x :=\n  is_fixed_pt.iterate hf n\n\n/-- For the identity map, all points are periodic. -/\ntheorem is_periodic_id {α : Type u_1} (n : ℕ) (x : α) : is_periodic_pt id n x :=\n  is_fixed_pt.is_periodic_pt (is_fixed_pt_id x) n\n\n/-- Any point is a periodic point of period `0`. -/\ntheorem is_periodic_pt_zero {α : Type u_1} (f : α → α) (x : α) : is_periodic_pt f 0 x :=\n  is_fixed_pt_id x\n\nnamespace is_periodic_pt\n\n\nprotected instance decidable {α : Type u_1} [DecidableEq α] {f : α → α} {n : ℕ} {x : α} : Decidable (is_periodic_pt f n x) :=\n  is_fixed_pt.decidable\n\nprotected theorem is_fixed_pt {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hf : is_periodic_pt f n x) : is_fixed_pt (nat.iterate f n) x :=\n  hf\n\nprotected theorem map {α : Type u_1} {β : Type u_2} {fa : α → α} {fb : β → β} {x : α} {n : ℕ} (hx : is_periodic_pt fa n x) {g : α → β} (hg : semiconj g fa fb) : is_periodic_pt fb n (g x) :=\n  is_fixed_pt.map hx (semiconj.iterate_right hg n)\n\ntheorem apply_iterate {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hx : is_periodic_pt f n x) (m : ℕ) : is_periodic_pt f n (nat.iterate f m x) :=\n  is_periodic_pt.map hx (commute.iterate_self f m)\n\nprotected theorem apply {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hx : is_periodic_pt f n x) : is_periodic_pt f n (f x) :=\n  apply_iterate hx 1\n\nprotected theorem add {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hn : is_periodic_pt f n x) (hm : is_periodic_pt f m x) : is_periodic_pt f (n + m) x :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_periodic_pt f (n + m) x)) (equations._eqn_1 f (n + m) x)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_fixed_pt (nat.iterate f (n + m)) x)) (iterate_add f n m)))\n      (is_fixed_pt.comp hn hm))\n\ntheorem left_of_add {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hn : is_periodic_pt f (n + m) x) (hm : is_periodic_pt f m x) : is_periodic_pt f n x := sorry\n\ntheorem right_of_add {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hn : is_periodic_pt f (n + m) x) (hm : is_periodic_pt f n x) : is_periodic_pt f m x :=\n  left_of_add (eq.mp (Eq._oldrec (Eq.refl (is_periodic_pt f (n + m) x)) (add_comm n m)) hn) hm\n\nprotected theorem sub {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hm : is_periodic_pt f m x) (hn : is_periodic_pt f n x) : is_periodic_pt f (m - n) x := sorry\n\nprotected theorem mul_const {α : Type u_1} {f : α → α} {x : α} {m : ℕ} (hm : is_periodic_pt f m x) (n : ℕ) : is_periodic_pt f (m * n) x := sorry\n\nprotected theorem const_mul {α : Type u_1} {f : α → α} {x : α} {m : ℕ} (hm : is_periodic_pt f m x) (n : ℕ) : is_periodic_pt f (n * m) x := sorry\n\ntheorem trans_dvd {α : Type u_1} {f : α → α} {x : α} {m : ℕ} (hm : is_periodic_pt f m x) {n : ℕ} (hn : m ∣ n) : is_periodic_pt f n x := sorry\n\nprotected theorem iterate {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hf : is_periodic_pt f n x) (m : ℕ) : is_periodic_pt (nat.iterate f m) n x := sorry\n\nprotected theorem mod {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hm : is_periodic_pt f m x) (hn : is_periodic_pt f n x) : is_periodic_pt f (m % n) x :=\n  left_of_add (eq.mp (Eq._oldrec (Eq.refl (is_periodic_pt f m x)) (Eq.symm (nat.mod_add_div m n))) hm)\n    (is_periodic_pt.mul_const hn (m / n))\n\nprotected theorem gcd {α : Type u_1} {f : α → α} {x : α} {m : ℕ} {n : ℕ} (hm : is_periodic_pt f m x) (hn : is_periodic_pt f n x) : is_periodic_pt f (nat.gcd m n) x := sorry\n\n/-- If `f` sends two periodic points `x` and `y` of the same positive period to the same point,\nthen `x = y`. For a similar statement about points of different periods see `eq_of_apply_eq`. -/\ntheorem eq_of_apply_eq_same {α : Type u_1} {f : α → α} {x : α} {y : α} {n : ℕ} (hx : is_periodic_pt f n x) (hy : is_periodic_pt f n y) (hn : 0 < n) (h : f x = f y) : x = y := sorry\n\n/-- If `f` sends two periodic points `x` and `y` of positive periods to the same point,\nthen `x = y`. -/\ntheorem eq_of_apply_eq {α : Type u_1} {f : α → α} {x : α} {y : α} {m : ℕ} {n : ℕ} (hx : is_periodic_pt f m x) (hy : is_periodic_pt f n y) (hm : 0 < m) (hn : 0 < n) (h : f x = f y) : x = y :=\n  eq_of_apply_eq_same (is_periodic_pt.mul_const hx n) (is_periodic_pt.const_mul hy m) (mul_pos hm hn) h\n\nend is_periodic_pt\n\n\n/-- The set of periodic points of a given (possibly non-minimal) period. -/\ndef pts_of_period {α : Type u_1} (f : α → α) (n : ℕ) : set α :=\n  set_of fun (x : α) => is_periodic_pt f n x\n\n@[simp] theorem mem_pts_of_period {α : Type u_1} {f : α → α} {x : α} {n : ℕ} : x ∈ pts_of_period f n ↔ is_periodic_pt f n x :=\n  iff.rfl\n\ntheorem semiconj.maps_to_pts_of_period {α : Type u_1} {β : Type u_2} {fa : α → α} {fb : β → β} {g : α → β} (h : semiconj g fa fb) (n : ℕ) : set.maps_to g (pts_of_period fa n) (pts_of_period fb n) :=\n  semiconj.maps_to_fixed_pts (semiconj.iterate_right h n)\n\ntheorem bij_on_pts_of_period {α : Type u_1} (f : α → α) {n : ℕ} (hn : 0 < n) : set.bij_on f (pts_of_period f n) (pts_of_period f n) := sorry\n\ntheorem directed_pts_of_period_pnat {α : Type u_1} (f : α → α) : directed has_subset.subset fun (n : ℕ+) => pts_of_period f ↑n := sorry\n\n/-- The set of periodic points of a map `f : α → α`. -/\ndef periodic_pts {α : Type u_1} (f : α → α) : set α :=\n  set_of fun (x : α) => ∃ (n : ℕ), ∃ (H : n > 0), is_periodic_pt f n x\n\ntheorem mk_mem_periodic_pts {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hn : 0 < n) (hx : is_periodic_pt f n x) : x ∈ periodic_pts f :=\n  Exists.intro n (Exists.intro hn hx)\n\ntheorem mem_periodic_pts {α : Type u_1} {f : α → α} {x : α} : x ∈ periodic_pts f ↔ ∃ (n : ℕ), ∃ (H : n > 0), is_periodic_pt f n x :=\n  iff.rfl\n\ntheorem bUnion_pts_of_period {α : Type u_1} (f : α → α) : (set.Union fun (n : ℕ) => set.Union fun (H : n > 0) => pts_of_period f n) = periodic_pts f := sorry\n\ntheorem Union_pnat_pts_of_period {α : Type u_1} (f : α → α) : (set.Union fun (n : ℕ+) => pts_of_period f ↑n) = periodic_pts f :=\n  Eq.trans supr_subtype (bUnion_pts_of_period f)\n\ntheorem bij_on_periodic_pts {α : Type u_1} (f : α → α) : set.bij_on f (periodic_pts f) (periodic_pts f) :=\n  Union_pnat_pts_of_period f ▸\n    set.bij_on_Union_of_directed (directed_pts_of_period_pnat f) fun (i : ℕ+) => bij_on_pts_of_period f (pnat.pos i)\n\ntheorem semiconj.maps_to_periodic_pts {α : Type u_1} {β : Type u_2} {fa : α → α} {fb : β → β} {g : α → β} (h : semiconj g fa fb) : set.maps_to g (periodic_pts fa) (periodic_pts fb) := sorry\n\n/-- Minimal period of a point `x` under an endomorphism `f`. If `x` is not a periodic point of `f`,\nthen `minimal_period f x = 0`. -/\ndef minimal_period {α : Type u_1} (f : α → α) (x : α) : ℕ :=\n  dite (x ∈ periodic_pts f) (fun (h : x ∈ periodic_pts f) => nat.find h) fun (h : ¬x ∈ periodic_pts f) => 0\n\ntheorem is_periodic_pt_minimal_period {α : Type u_1} (f : α → α) (x : α) : is_periodic_pt f (minimal_period f x) x := sorry\n\ntheorem minimal_period_pos_of_mem_periodic_pts {α : Type u_1} {f : α → α} {x : α} (hx : x ∈ periodic_pts f) : 0 < minimal_period f x := sorry\n\ntheorem is_periodic_pt.minimal_period_pos {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hn : 0 < n) (hx : is_periodic_pt f n x) : 0 < minimal_period f x :=\n  minimal_period_pos_of_mem_periodic_pts (mk_mem_periodic_pts hn hx)\n\ntheorem minimal_period_pos_iff_mem_periodic_pts {α : Type u_1} {f : α → α} {x : α} : 0 < minimal_period f x ↔ x ∈ periodic_pts f := sorry\n\ntheorem is_periodic_pt.minimal_period_le {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hn : 0 < n) (hx : is_periodic_pt f n x) : minimal_period f x ≤ n := sorry\n\ntheorem is_periodic_pt.eq_zero_of_lt_minimal_period {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hx : is_periodic_pt f n x) (hn : n < minimal_period f x) : n = 0 :=\n  Eq.symm\n    (or.resolve_right (eq_or_lt_of_le (nat.zero_le n))\n      fun (hn0 : 0 < n) => iff.mpr not_lt (is_periodic_pt.minimal_period_le hn0 hx) hn)\n\ntheorem is_periodic_pt.minimal_period_dvd {α : Type u_1} {f : α → α} {x : α} {n : ℕ} (hx : is_periodic_pt f n x) : minimal_period f x ∣ n := sorry\n\ntheorem is_periodic_pt_iff_minimal_period_dvd {α : Type u_1} {f : α → α} {x : α} {n : ℕ} : is_periodic_pt f n x ↔ minimal_period f x ∣ n :=\n  { mp := is_periodic_pt.minimal_period_dvd,\n    mpr := fun (h : minimal_period f x ∣ n) => is_periodic_pt.trans_dvd (is_periodic_pt_minimal_period f x) h }\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/dynamics/periodic_pts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3467084226337857}}
{"text": "import tactic\nimport init.meta.expr\nimport init.meta.lean.parser\n\nopen tactic expr\n\n-- Like expr.get_pis, but uses mvars instead of local variables.\nmeta def get_pi_mvars : expr → tactic (list expr × expr)\n| (pi n bi d b)  :=\ndo mv ← mk_meta_var d,\n   (mvs, b) ← get_pi_mvars (b.instantiate_var mv),\n   pure (mv::mvs, b)\n| e              := pure ([], e)\n\n-- returns expr and its local variables\nmeta def mk_mvar_app_aux : expr → list expr → list expr → tactic (expr × list expr)\n| g (m::ms) local_vars := do\n  g ← instantiate_mvars g,\n  (loc::locs, _) ← infer_type g >>= mk_local_pis,\n  assigned ← is_assigned m,\n  match assigned with\n  | tt := mk_mvar_app_aux (app g m) ms local_vars\n  | ff := do\n    tm ← infer_type m,\n    tm ← instantiate_mvars tm,\n    tloc ← infer_type loc,\n    unify tm tloc,\n    unify m loc,\n    mk_mvar_app_aux (app g loc) ms (loc::local_vars)\n  end\n| g [] local_vars := pure (g, list.reverse local_vars)\n\n-- Given a function-type expression `f`, and a list of mvars `ms`, construct the application of `f` to `ms`\n-- where assigned mvars are replaced by their instantiations and unassigned mvars are turned into local variables.\nmeta def mk_mvar_app : expr → list expr → tactic (expr × list expr) :=\nλ e ms, mk_mvar_app_aux e ms []\n\n-- Modus ponens in the `n`th argument of `f` usinf `g`.\nmeta def mp_nth_arg (f g : expr) (n : ℕ) : tactic expr :=\ndo tf ← infer_type f,\n  tg ← infer_type g,\n  (gmvs, gconc) ← get_pi_mvars tg,\n  gtys ← mmap infer_type gmvs,\n  let replace : list bool := list.map (= n) $ list.range (list.length gtys),\n  match list.nth gtys n with\n  | some t := do\n    (fmvs, fconc) ← get_pi_mvars tf,\n    unify fconc t,\n    gmvs ← mmap (λ a, match a with\n    | (m, tt) := do (app_of_list f <$> (mmap instantiate_mvars fmvs)) >>= unify m, pure m\n    | (m, ff) := pure m    \n    end\n    ) (list.zip gmvs replace),\n    (res, locs) ← mk_mvar_app g gmvs,\n    res ← instantiate_mvars res,  \n    pure $ expr.lambdas locs res \n   | none := failed\n   end\n\n/- Apply modus ponens once in each argument in which it is possible, returning a list consisting of the results\n   if possible and none otherwise. -/\nmeta def mp_core (f g : expr) : tactic (list (option expr)) :=\ndo tg ← infer_type g,\n  let n := pi_arity tg,\n  mmap (λ k, try_core $ mp_nth_arg f g k) (list.range n)\n\ndef option_list_to_list {α : Type*} (l : list (option α)) : list α :=\nlist.foldl list.append [] (list.map option.to_list l)\n\n/- Apply modus ponens once in each argument in which it is possible, returning the list of results. -/\nmeta def mp_core' (f g : expr) : tactic (list expr) :=\noption_list_to_list <$> mp_core f g\n\n\n/- Define the interactive `mp` tactic-/\n\nopen interactive\nopen lean.parser (ident tk)\n\n/- Turn numbers in strings to lowercase indices. -/ \nmeta def to_lower_digits (s : string) : string :=\nlist.foldl (++) \"\" $\n(list.map $ (λ (c : char), match to_string c with\n| \"0\" := \"₀\"\n| \"1\" := \"₁\"\n| \"2\" := \"₂\"\n| \"3\" := \"₃\"\n| \"4\" := \"₄\"\n| \"5\" := \"₅\"\n| \"6\" := \"₆\"\n| \"7\" := \"₇\"\n| \"8\" := \"₈\"\n| \"9\" := \"₉\"\n| x := x\nend\n)) s.to_list\n\n/- Apply the local hyposthesis `f` to an argument (assumption) of `g`, and add the resulting statements to the \n   context, optionally using the name `nm` for the new hypotheses. If there are several possibilities, do it\n   for each possibility once and add subscripts to the new hypothesis names. -/\nmeta def tactic.interactive.mp (f g : parse ident) \n(nm : parse (optional (tk \"with\" *> ident))) \n: tactic unit :=\n-- Note: switched order w.r.t. to `mp_core`\nlet nm := nm.get_or_else `this in\ndo f ← get_local f,\n  g ← get_local g,\n  hs ← mp_core' g f,\n  let num_new_hyps := list.length hs,\n  let inds := list.range num_new_hyps,\n\n  mmap ( λ (hi : expr × nat) , \n    match hi with\n    | (h, i) := do\n      let nm_i :=  mk_simple_name $ (name.to_string nm) ++ (to_lower_digits $ to_string i),\n      let nm' := if num_new_hyps = 1 then nm else nm_i,\n      interactive.have nm' none ``(%%h)\n    end\n  ) (list.zip hs inds),\n  pure ()\n\n\n\n/-- EXAMPLE -/\n\ndef P : ℕ → ℕ → Prop := sorry\ndef Q : ℕ → ℕ → ℕ → Prop := sorry\n\n\nexample (f : ∀ a, P a a) (g : ∀ a b c, P a b → P b c → Q a b c) (g' : ∀ a b, P a b → Q a a b): true :=\nbegin\n  mp g f with H,        -- give the new hypotheses names starting with `H`\n  mp g f,               -- use `this` instead\n  mp g' f,              -- no indices if there is only one possibility\n  mp H₀ f with HH₀,     -- can be applied multiple times\n  mp H₁ f with HH₁,\n  trivial\nend\n\ndef PP : Prop := sorry\ndef QQ : Prop := sorry\n\n\n/- The argument order was chosen such that `mp g f` is like `g(f)` or `g ∘ f` (but that can be changed). -/ \nexample (f : PP) (g : PP → QQ) : QQ := by mp g f; assumption\n\n", "meta": {"author": "Human-Oriented-ATP", "repo": "lean-tactics", "sha": "8fa4c8b8efc0c6a1d408b48e999f3a36f228bd0f", "save_path": "github-repos/lean/Human-Oriented-ATP-lean-tactics", "path": "github-repos/lean/Human-Oriented-ATP-lean-tactics/lean-tactics-8fa4c8b8efc0c6a1d408b48e999f3a36f228bd0f/lean3/src/mp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3467084226337856}}
{"text": "import category_theory.limits.shapes.comm_sq\n\nopen category_theory.limits\n\nnamespace category_theory\n\nvariables {C : Type*} [category C]\n\nnamespace is_pushout\n\nvariables {Z X Y P : C} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P}\n/-\nlemma of_horiz_is_iso [is_iso f] [is_iso inr] (sq : comm_sq f g inl inr) :\n  is_pushout f g inl inr := of_is_colimit' sq\nbegin\n  refine pushout_cocone.is_colimit.mk _ (λ s, inv inr ≫ s.inr) (λ s, _) (by tidy) (by tidy),\n  simp only [← cancel_epi f, s.condition, sq.w_assoc, is_iso.hom_inv_id_assoc],\nend\n\nlemma of_vert_is_iso [is_iso g] [is_iso inl] (sq : comm_sq f g inl inr) :\n  is_pushout f g inl inr := (of_horiz_is_iso sq.flip).flip-/\n\nlemma of_coprod_inl_with_id {A B : C} (f : A ⟶ B) (X : C) [has_binary_coproduct A X]\n  [has_binary_coproduct B X] :\n  is_pushout coprod.inl f (coprod.map f (𝟙 X)) coprod.inl :=\nis_pushout.of_is_colimit' (comm_sq.mk (coprod.inl_map _ _))\n(pushout_cocone.is_colimit_aux _ (λ s, coprod.desc s.inr (coprod.inr ≫ s.inl))\n(λ s, begin\n  dsimp [comm_sq.cocone],\n  ext,\n  { simp only [coprod.map_desc, coprod.inl_desc, s.condition], },\n  { simp only [coprod.map_desc, category.id_comp, coprod.inr_desc], },\nend)\n(λ s, by { dsimp [comm_sq.cocone], simp only [coprod.inl_desc], })\n(λ s m hm, begin\n  dsimp [comm_sq.cocone],\n  ext,\n  { simpa only [coprod.inl_desc] using hm walking_span.right, },\n  { have h := coprod.inr ≫= hm walking_span.left,\n    dsimp [comm_sq.cocone] at h,\n    simpa only [coprod.inr_desc, coprod.inr_map_assoc, category.id_comp] using h, },\nend))\n\nend is_pushout\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/limits/comm_sq_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3466676140687353}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.Init.Logic\n\n/-!\n# Unbundled algebra classes\n\nThese classes are part of an incomplete refactor described\n[here on the github Wiki](https://github.com/leanprover/lean/wiki/Refactoring-structures#encoding-the-algebraic-hierarchy-1).\nHowever a subset of them are widely used in mathlib3,\nand it has been tricky to clean this up as this file was in core Lean 3.\n\nBy themselves, these classes are not good replacements for the `monoid` / `group` etc structures\nprovided by mathlib, as they are not discoverable by `simp` unlike the current lemmas due to there\nbeing little to index on.\n\n## Porting notes:\nThis file is ancient, and it would be good to replace it with a clean version\nthat provides what mathlib4 actually needs.\n\nI've omitted all the `@[algebra]` attributes, as they are not used elsewhere.\n\nThe section `StrictWeakOrder` has been omitted, but I've left the mathport output in place.\nPlease delete if cleaning up.\n\nI've commented out some classes which we think are completely unused in mathlib.\n\nI've added many of the declarations to `nolints.json`.\nIf you clean up this file, please add documentation to classes that we are keeping.\n\nMario made the following analysis of uses in mathlib3:\n* `is_symm_op`: unused except for some instances\n* `is_commutative`: used a fair amount via some theorems about folds\n  (also assuming `is_associative`)\n* `is_associative`: ditto, also used in `noncomm_fold`\n* `is_left_id`, `is_right_id`: unused except in the mathlib class `is_unital` and in `mono`\n  (which looks like it could use `is_unital`)\n* `is_left_null`, `is_right_null`: unused\n* `is_left_cancel`, `is_right_cancel`: unused except for instances\n* `is_idempotent`: this one is actually used to prove things not directly about `is_idempotent`\n* `is_left_distrib`, `is_right_distrib`, `is_left_inv`, `is_right_inv`, `is_cond_left_inv`,\n  `is_cond_right_inv`: unused\n* `is_distinct`: unused (although we reinvented this one as `nontrivial`)\n* `is_irrefl`, `is_refl`, `is_symm`, `is_trans`: significant usage\n* `is_asymm`, `is_antisymm`, `is_total`, `is_strict_order`: a lot of uses but all in order theory\n  and it's unclear how much could not be transferred to another typeclass\n* `is_preorder`: unused except for instances\n  (except `antisymmetrization`, maybe it could be transferred)\n* `is_total_preorder`, `is_partial_order`: unused except for instances\n* `is_linear_order`: unused except for instances\n* `is_equiv`: unused except for instances (most uses can use `equivalence` instead)\n* `is_per`: unused\n* `is_incomp_trans`: unused\n* `is_strict_weak_order`: significant usage (most of it on `rbmap`, could be transferred)\n* `is_trichotomous`: some usage\n* `is_strict_total_order`: looks like the only usage is in `rbmap` again\n-/\n\nuniverse u v\n\nclass IsSymmOp (α : Type u) (β : Type v) (op : α → α → β) : Prop where\n  symm_op : ∀ a b, op a b = op b a\n\n/-- A commutative binary operation. -/\nclass IsCommutative (α : Type u) (op : α → α → α) : Prop where\n  comm : ∀ a b, op a b = op b a\n\ninstance [IsCommutative α op] : Lean.IsCommutative op where\n  comm := IsCommutative.comm\n\ninstance isSymmOp_of_isCommutative (α : Type u) (op : α → α → α)\n    [IsCommutative α op] : IsSymmOp α α op where symm_op :=\n  IsCommutative.comm\n\n/-- An associative binary operation. -/\nclass IsAssociative (α : Type u) (op : α → α → α) : Prop where\n  assoc : ∀ a b c, op (op a b) c = op a (op b c)\n\ninstance [IsAssociative α op] : Lean.IsAssociative op where\n  assoc := IsAssociative.assoc\n\n/-- A binary operation with a left identity. -/\nclass IsLeftId (α : Type u) (op : α → α → α) (o : outParam α) : Prop where\n  left_id : ∀ a, op o a = a\n\n/-- A binary operation with a right identity. -/\nclass IsRightId (α : Type u) (op : α → α → α) (o : outParam α) : Prop where\n  right_id : ∀ a, op a o = a\n\ninstance [IsLeftId α op o] [IsRightId α op o] : Lean.IsNeutral op o where\n  left_neutral := IsLeftId.left_id\n  right_neutral := IsRightId.right_id\n\n-- -- class IsLeftNull (α : Type u) (op : α → α → α) (o : outParam α) : Prop where\n--   left_null : ∀ a, op o a = o\n\n-- -- class IsRightNull (α : Type u) (op : α → α → α) (o : outParam α) : Prop where\n--   right_null : ∀ a, op a o = o\n\nclass IsLeftCancel (α : Type u) (op : α → α → α) : Prop where\n  left_cancel : ∀ a b c, op a b = op a c → b = c\n\nclass IsRightCancel (α : Type u) (op : α → α → α) : Prop where\n  right_cancel : ∀ a b c, op a b = op c b → a = c\n\nclass IsIdempotent (α : Type u) (op : α → α → α) : Prop where\n  idempotent : ∀ a, op a a = a\n\ninstance [IsIdempotent α op] : Lean.IsIdempotent op where\n  idempotent := IsIdempotent.idempotent\n\n--class IsLeftDistrib (α : Type u) (op₁ : α → α → α) (op₂ : outParam <| α → α → α) : Prop where\n--   left_distrib : ∀ a b c, op₁ a (op₂ b c) = op₂ (op₁ a b) (op₁ a c)\n\n-- class IsRightDistrib (α : Type u) (op₁ : α → α → α) (op₂ : outParam <| α → α → α) : Prop where\n--   right_distrib : ∀ a b c, op₁ (op₂ a b) c = op₂ (op₁ a c) (op₁ b c)\n\n-- class IsLeftInv (α : Type u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α) :\n--     Prop where\n--   left_inv : ∀ a, op (inv a) a = o\n\n-- class IsRightInv (α : Type u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α) :\n--     Prop where\n--   right_inv : ∀ a, op a (inv a) = o\n\n-- class IsCondLeftInv (α : Type u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α)\n--   (p : outParam <| α → Prop) : Prop where\n--   left_inv : ∀ a, p a → op (inv a) a = o\n\n-- class IsCondRightInv (α : Type u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α)\n--   (p : outParam <| α → Prop) : Prop where\n--   right_inv : ∀ a, p a → op a (inv a) = o\n\n-- class IsDistinct (α : Type u) (a : α) (b : α) : Prop where\n--   distinct : a ≠ b\n\n/-\n-- The following type class doesn't seem very useful, a regular simp lemma should work for this.\nclass is_inv (α : Type u) (β : Type v) (f : α → β) (g : out β → α) : Prop :=\n(inv : ∀ a, g (f a) = a)\n\n-- The following one can also be handled using a regular simp lemma\nclass is_idempotent (α : Type u) (f : α → α) : Prop :=\n(idempotent : ∀ a, f (f a) = f a)\n-/\n/-- `IsIrrefl X r` means the binary relation `r` on `X` is irreflexive (that is, `r x x` never\nholds). -/\nclass IsIrrefl (α : Type u) (r : α → α → Prop) : Prop where\n  irrefl : ∀ a, ¬r a a\n\n/-- `IsRefl X r` means the binary relation `r` on `X` is reflexive. -/\nclass IsRefl (α : Type u) (r : α → α → Prop) : Prop where\n  refl : ∀ a, r a a\n\n/-- `IsSymm X r` means the binary relation `r` on `X` is symmetric. -/\nclass IsSymm (α : Type u) (r : α → α → Prop) : Prop where\n  symm : ∀ a b, r a b → r b a\n\n/-- The opposite of a symmetric relation is symmetric. -/\ninstance isSymmOp_of_isSymm (α : Type u) (r : α → α → Prop) [IsSymm α r] :\n    IsSymmOp α Prop r where\n  symm_op := fun a b ↦ propext <| Iff.intro (IsSymm.symm a b) (IsSymm.symm b a)\n\n/-- `IsAsymm X r` means that the binary relation `r` on `X` is asymmetric, that is,\n`r a b → ¬ r b a`. -/\nclass IsAsymm (α : Type u) (r : α → α → Prop) : Prop where\n  asymm : ∀ a b, r a b → ¬r b a\n\n/-- `IsAntisymm X r` means the binary relation `r` on `X` is antisymmetric. -/\nclass IsAntisymm (α : Type u) (r : α → α → Prop) : Prop where\n  antisymm : ∀ a b, r a b → r b a → a = b\n\n/-- `IsTrans X r` means the binary relation `r` on `X` is transitive. -/\nclass IsTrans (α : Type u) (r : α → α → Prop) : Prop where\n  trans : ∀ a b c, r a b → r b c → r a c\n\ninstance {α : Type u} {r : α → α → Prop} [IsTrans α r] : Trans r r r :=\n  ⟨IsTrans.trans _ _ _⟩\n\ninstance {α : Type u} {r : α → α → Prop} [Trans r r r] : IsTrans α r :=\n  ⟨fun _ _ _ => Trans.trans⟩\n\n/-- `IsTotal X r` means that the binary relation `r` on `X` is total, that is, that for any\n`x y : X` we have `r x y` or `r y x`.-/\nclass IsTotal (α : Type u) (r : α → α → Prop) : Prop where\n  total : ∀ a b, r a b ∨ r b a\n\n/-- `IsPreorder X r` means that the binary relation `r` on `X` is a pre-order, that is, reflexive\nand transitive. -/\nclass IsPreorder (α : Type u) (r : α → α → Prop) extends IsRefl α r, IsTrans α r : Prop\n\n/-- `IsTotalPreorder X r` means that the binary relation `r` on `X` is total and a preorder. -/\nclass IsTotalPreorder (α : Type u) (r : α → α → Prop) extends IsTrans α r, IsTotal α r : Prop\n\n/-- Every total pre-order is a pre-order. -/\ninstance isTotalPreorder_isPreorder (α : Type u) (r : α → α → Prop) [s : IsTotalPreorder α r] :\n    IsPreorder α r where\n  trans := s.trans\n  refl := fun a ↦ Or.elim (@IsTotal.total _ r _ a a) id id\n\n/-- `IsPartialOrder X r` means that the binary relation `r` on `X` is a partial order, that is,\n`IsPreorder X r` and `IsAntisymm X r`. -/\nclass IsPartialOrder (α : Type u) (r : α → α → Prop) extends IsPreorder α r, IsAntisymm α r : Prop\n\n/-- `IsLinearOrder X r` means that the binary relation `r` on `X` is a linear order, that is,\n`IsPartialOrder X r` and `IsTotal X r`. -/\nclass IsLinearOrder (α : Type u) (r : α → α → Prop) extends IsPartialOrder α r, IsTotal α r : Prop\n\n/-- `IsEquiv X r` means that the binary relation `r` on `X` is an equivalence relation, that\nis, `IsPreorder X r` and `IsSymm X r`. -/\nclass IsEquiv (α : Type u) (r : α → α → Prop) extends IsPreorder α r, IsSymm α r : Prop\n\n-- /-- `IsPer X r` means that the binary relation `r` on `X` is a partial equivalence relation, that\n-- is, `IsSymm X r` and `IsTrans X r`. -/\n-- -- class IsPer (α : Type u) (r : α → α → Prop) extends IsSymm α r, IsTrans α r : Prop\n\n/-- `IsStrictOrder X r` means that the binary relation `r` on `X` is a strict order, that is,\n`IsIrrefl X r` and `IsTrans X r`. -/\nclass IsStrictOrder (α : Type u) (r : α → α → Prop) extends IsIrrefl α r, IsTrans α r : Prop\n\n/-- `IsIncompTrans X lt` means that for `lt` a binary relation on `X`, the incomparable relation\n`λ a b, ¬ lt a b ∧ ¬ lt b a` is transitive. -/\nclass IsIncompTrans (α : Type u) (lt : α → α → Prop) : Prop where\n  incomp_trans : ∀ a b c, ¬lt a b ∧ ¬lt b a → ¬lt b c ∧ ¬lt c b → ¬lt a c ∧ ¬lt c a\n\n/-- `IsStrictWeakOrder X lt` means that the binary relation `lt` on `X` is a strict weak order,\nthat is, `IsStrictOrder X lt` and `IsIncompTrans X lt`. -/\nclass IsStrictWeakOrder (α : Type u) (lt : α → α → Prop)\n  extends IsStrictOrder α lt, IsIncompTrans α lt : Prop\n\n/-- `IsTrichotomous X lt` means that the binary relation `lt` on `X` is trichotomous, that is,\neither `lt a b` or `a = b` or `lt b a` for any `a` and `b`. -/\nclass IsTrichotomous (α : Type u) (lt : α → α → Prop) : Prop where\n  trichotomous : ∀ a b, lt a b ∨ a = b ∨ lt b a\n\n/-- `IsStrictTotalOrder X lt` means that the binary relation `lt` on `X` is a strict total order,\nthat is, `IsTrichotomous X lt` and `IsStrictOrder X lt`. -/\nclass IsStrictTotalOrder (α : Type u) (lt : α → α → Prop)\n  extends IsTrichotomous α lt, IsStrictOrder α lt : Prop\n\n/-- Equality is an equivalence relation. -/\ninstance eq_isEquiv (α : Type u) : IsEquiv α (· = ·) where\n  symm := @Eq.symm _\n  trans := @Eq.trans _\n  refl := Eq.refl\n\nsection\n\nvariable {α : Type u} {r : α → α → Prop}\n\n/-- Temporary notation for a relation. -/\nlocal infixl:50 \"≺\" => r\n\ntheorem irrefl [IsIrrefl α r] (a : α) : ¬a ≺ a :=\n  IsIrrefl.irrefl a\n\ntheorem refl [IsRefl α r] (a : α) : a ≺ a :=\n  IsRefl.refl a\n\ntheorem trans [IsTrans α r] {a b c : α} : a ≺ b → b ≺ c → a ≺ c :=\n  IsTrans.trans _ _ _\n\ntheorem symm [IsSymm α r] {a b : α} : a ≺ b → b ≺ a :=\n  IsSymm.symm _ _\n\ntheorem antisymm [IsAntisymm α r] {a b : α} : a ≺ b → b ≺ a → a = b :=\n  IsAntisymm.antisymm _ _\n\ntheorem asymm [IsAsymm α r] {a b : α} : a ≺ b → ¬b ≺ a :=\n  IsAsymm.asymm _ _\n\ntheorem trichotomous [IsTrichotomous α r] : ∀ a b : α, a ≺ b ∨ a = b ∨ b ≺ a :=\n  IsTrichotomous.trichotomous\n\ntheorem incomp_trans [IsIncompTrans α r] {a b c : α} :\n    ¬a ≺ b ∧ ¬b ≺ a → ¬b ≺ c ∧ ¬c ≺ b → ¬a ≺ c ∧ ¬c ≺ a :=\n  IsIncompTrans.incomp_trans _ _ _\n\ninstance (priority := 90) isAsymm_of_isTrans_of_isIrrefl [IsTrans α r] [IsIrrefl α r] :\n    IsAsymm α r :=\n  ⟨fun a _ h₁ h₂ ↦ absurd (_root_.trans h₁ h₂) (irrefl a)⟩\n\nsection ExplicitRelationVariants\n\nvariable (r)\n\n@[elab_without_expected_type]\ntheorem irrefl_of [IsIrrefl α r] (a : α) : ¬a ≺ a :=\n  irrefl a\n\n@[elab_without_expected_type]\ntheorem refl_of [IsRefl α r] (a : α) : a ≺ a :=\n  refl a\n\n@[elab_without_expected_type]\ntheorem trans_of [IsTrans α r] {a b c : α} : a ≺ b → b ≺ c → a ≺ c :=\n  _root_.trans\n\n@[elab_without_expected_type]\ntheorem symm_of [IsSymm α r] {a b : α} : a ≺ b → b ≺ a :=\n  symm\n\n@[elab_without_expected_type]\ntheorem asymm_of [IsAsymm α r] {a b : α} : a ≺ b → ¬b ≺ a :=\n  asymm\n\n@[elab_without_expected_type]\ntheorem total_of [IsTotal α r] (a b : α) : a ≺ b ∨ b ≺ a :=\n  IsTotal.total _ _\n\n@[elab_without_expected_type]\ntheorem trichotomous_of [IsTrichotomous α r] : ∀ a b : α, a ≺ b ∨ a = b ∨ b ≺ a :=\n  trichotomous\n\n@[elab_without_expected_type]\ntheorem incomp_trans_of [IsIncompTrans α r] {a b c : α} :\n    ¬a ≺ b ∧ ¬b ≺ a → ¬b ≺ c ∧ ¬c ≺ b → ¬a ≺ c ∧ ¬c ≺ a :=\n  incomp_trans\n\nend ExplicitRelationVariants\n\nend\n\n-- Porting note: the `StrictWeakOrder` section has been ommitted.\n\n-- namespace StrictWeakOrder\n\n-- section\n\n-- parameter {α : Type u}{r : α → α → Prop}\n\n-- -- mathport name: «expr ≺ »\n-- local infixl:50 \"≺\" => r\n\n-- def Equiv (a b : α) : Prop :=\n--   ¬a ≺ b ∧ ¬b ≺ a\n\n-- parameter [IsStrictWeakOrder α r]\n\n-- -- mathport name: equiv\n-- local infixl:50 \" ≈ \" => equiv\n\n-- theorem erefl (a : α) : a ≈ a :=\n--   ⟨irrefl a, irrefl a⟩\n\n-- theorem esymm {a b : α} : a ≈ b → b ≈ a := fun ⟨h₁, h₂⟩ ↦ ⟨h₂, h₁⟩\n\n-- theorem etrans {a b c : α} : a ≈ b → b ≈ c → a ≈ c :=\n--   incomp_trans\n\n-- theorem not_lt_of_equiv {a b : α} : a ≈ b → ¬a ≺ b := fun h ↦ h.1\n\n-- theorem not_lt_of_equiv' {a b : α} : a ≈ b → ¬b ≺ a := fun h ↦ h.2\n\n-- instance is_equiv : IsEquiv α equiv where\n--   refl := erefl\n--   trans := @etrans\n--   symm := @esymm\n\n-- end\n\n-- -- mathport name: «expr ≈[ ] »\n-- notation:50 -- Notation for the equivalence relation induced by lt\n-- a \" ≈[\" lt \"]\" b:50 => @Equiv _ lt a b\n\n-- end StrictWeakOrder\n\ntheorem isStrictWeakOrder_of_isTotalPreorder {α : Type u} {le : α → α → Prop}\n    {lt : α → α → Prop} [DecidableRel le] [IsTotalPreorder α le] (h : ∀ a b, lt a b ↔ ¬le b a) :\n    IsStrictWeakOrder α lt :=\n  { trans := fun a b c hab hbc ↦\n      have nba : ¬le b a := Iff.mp (h _ _) hab\n      have ncb : ¬le c b := Iff.mp (h _ _) hbc\n      have hab : le a b := Or.resolve_left (total_of le b a) nba\n      have nca : ¬le c a := fun hca : le c a ↦\n        have hcb : le c b := trans_of le hca hab\n        absurd hcb ncb\n      Iff.mpr (h _ _) nca,\n    irrefl := fun a hlt ↦ absurd (refl_of le a) (Iff.mp (h _ _) hlt),\n    incomp_trans := fun a b c ⟨nab, nba⟩ ⟨nbc, ncb⟩ ↦\n      have hba : le b a := Decidable.of_not_not (Iff.mp (not_congr (h _ _)) nab)\n      have hab : le a b := Decidable.of_not_not (Iff.mp (not_congr (h _ _)) nba)\n      have hcb : le c b := Decidable.of_not_not (Iff.mp (not_congr (h _ _)) nbc)\n      have hbc : le b c := Decidable.of_not_not (Iff.mp (not_congr (h _ _)) ncb)\n      have hac : le a c := trans_of le hab hbc\n      have hca : le c a := trans_of le hcb hba\n      And.intro (fun n ↦ absurd hca (Iff.mp (h _ _) n)) fun n ↦ absurd hac (Iff.mp (h _ _) n) }\n\ntheorem lt_of_lt_of_incomp {α : Type u} {lt : α → α → Prop} [IsStrictWeakOrder α lt]\n    [DecidableRel lt] : ∀ {a b c}, lt a b → ¬lt b c ∧ ¬lt c b → lt a c :=\n  @fun a b c hab ⟨nbc, ncb⟩ ↦\n  have nca : ¬lt c a := fun hca ↦ absurd (trans_of lt hca hab) ncb\n  Decidable.by_contradiction fun nac : ¬lt a c ↦\n    have : ¬lt a b ∧ ¬lt b a := incomp_trans_of lt ⟨nac, nca⟩ ⟨ncb, nbc⟩\n    absurd hab this.1\n\ntheorem lt_of_incomp_of_lt {α : Type u} {lt : α → α → Prop} [IsStrictWeakOrder α lt]\n    [DecidableRel lt] : ∀ {a b c}, ¬lt a b ∧ ¬lt b a → lt b c → lt a c :=\n  @fun a b c ⟨nab, nba⟩ hbc ↦\n  have nca : ¬lt c a := fun hca ↦ absurd (trans_of lt hbc hca) nba\n  Decidable.by_contradiction fun nac : ¬lt a c ↦\n    have : ¬lt b c ∧ ¬lt c b := incomp_trans_of lt ⟨nba, nab⟩ ⟨nac, nca⟩\n    absurd hbc this.1\n\ntheorem eq_of_incomp {α : Type u} {lt : α → α → Prop} [IsTrichotomous α lt] {a b} :\n    ¬lt a b ∧ ¬lt b a → a = b :=\n  fun ⟨nab, nba⟩ ↦\n  match trichotomous_of lt a b with\n  | Or.inl hab => absurd hab nab\n  | Or.inr (Or.inl hab) => hab\n  | Or.inr (Or.inr hba) => absurd hba nba\n\ntheorem eq_of_eqv_lt {α : Type u} {lt : α → α → Prop} [IsTrichotomous α lt] {a b} :\n    ¬lt a b ∧ ¬lt b a → a = b :=\n  eq_of_incomp\n\ntheorem incomp_iff_eq {α : Type u} {lt : α → α → Prop} [IsTrichotomous α lt] [IsIrrefl α lt] (a b) :\n    ¬lt a b ∧ ¬lt b a ↔ a = b :=\n  Iff.intro eq_of_incomp fun hab ↦ by\n    rw [hab]\n    exact ⟨irrefl_of lt b, irrefl_of lt b⟩\n\ntheorem eqv_lt_iff_eq {α : Type u} {lt : α → α → Prop} [IsTrichotomous α lt] [IsIrrefl α lt] (a b) :\n    ¬lt a b ∧ ¬lt b a ↔ a = b :=\n  incomp_iff_eq a b\n\ntheorem not_lt_of_lt {α : Type u} {lt : α → α → Prop} [IsStrictOrder α lt] {a b} :\n    lt a b → ¬lt b a :=\n  fun h₁ h₂ ↦ absurd (trans_of lt h₁ h₂) (irrefl_of lt _)\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Init/Algebra/Classes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3466676140687353}}
{"text": "import algebra.category.Group.colimits\nimport lemmas.ulift\nimport new.unordered.d\n\nsection\n\nopen topological_space Top Top.sheaf\nopen category_theory\nopen opposite\nopen nat\n\nopen_locale big_operators\n\nuniverse u\nvariables {X : Top.{u}} (𝓕 : sheaf Ab X) (U V : X.oc)\nvariable (n : ℕ)\n\nsection\n\nvariables {n U V} (r : U ⟶ V)\n\ndef vec.refine (α : fin n → U.ι) : fin n → V.ι :=\nr.func ∘ α\n\nlemma vec.refine_apply (α : fin n → U.ι) (m : fin n) :\n  vec.refine r α m = r.func (α m) := rfl\n\nlemma vec.refine_comp_apply {U V W : X.oc} (r : U ⟶ V) (r' : V ⟶ W)\n  (α : fin n → U.ι) (m : fin n) :\n  vec.refine (r ≫ r') α m = vec.refine r' (vec.refine r α) m := rfl\n\nlemma vec.refine_le (α : fin n → U.ι) :\n  face α ≤ face (vec.refine r α) :=\nbegin\n  intros p hp,\n  rw opens.mem_coe at hp ⊢,\n  unfold face at hp ⊢,\n  rw opens.fintype_infi at hp ⊢,\n  intros i,\n  specialize hp i,\n  apply r.is_refinement,\n  exact hp,\nend\n\nlemma vec.refine_ignore (α : fin (n + 1) → U.ι) (k : fin (n + 1)) :\n  vec.refine r (ignore α k) = ignore (vec.refine r α) k :=\nbegin\n  ext1 m,\n  rw vec.refine_apply,\n  rw ignore.apply_ite,\n  rw ignore.apply_ite,\n  split_ifs,\n  rw vec.refine_apply,\n  rw vec.refine_apply,\nend\n\nvariables {𝓕}\ndef C.refine.to_fun (f : C 𝓕 V n) : C 𝓕 U n :=\nλ α, 𝓕.1.map (hom_of_le (vec.refine_le r α)).op (f (vec.refine r α))\n\nlemma C.refine.to_fun_apply (f : C 𝓕 V n) (α : fin n → U.ι) :\n  C.refine.to_fun r f α = 𝓕.1.map (hom_of_le (vec.refine_le r α)).op (f (vec.refine r α)) := \nrfl\n\ndef C.refine : C 𝓕 V n ⟶ C 𝓕 U n :=\n{ to_fun := λ f, C.refine.to_fun r f,\n  map_zero' := begin\n    ext α,\n    rw C.refine.to_fun_apply,\n    simp only [C_pre.zero_apply, map_zero],\n  end,\n  map_add' := λ f g, begin\n    ext α,\n    rw [C.refine.to_fun_apply, pi.add_apply, map_add],\n    refl,\n  end }\n\nlemma C.refine_d_eq_d_refine :\n  C.refine r ≫ d 𝓕 U n = d 𝓕 V n ≫ C.refine r :=\nbegin\n  ext f α,\n  rw [comp_apply, comp_apply],\n  rw [dd_aux.d_def],\n  rw [C.refine, C.refine, add_monoid_hom.coe_mk, add_monoid_hom.coe_mk, C.refine.to_fun_apply],\n  simp_rw [C.refine.to_fun_apply],\n  rw [dd_aux.d_def],\n  rw add_monoid_hom.map_sum,\n  rw finset.sum_congr rfl,\n  intros i _,\n  split_ifs,\n  { rw [id, ← comp_apply, ← 𝓕.1.map_comp, ← comp_apply, ← 𝓕.1.map_comp],\n    rw map_congr.vec_eq f (_ : vec.refine r (ignore α i) = ignore (vec.refine r α) i),\n    rw [← comp_apply, ← 𝓕.1.map_comp],\n    congr,\n    rw vec.refine_ignore, },\n  { rw [pi.neg_apply, pi.neg_apply],\n    simp only [pi.neg_apply, add_monoid_hom.neg_apply, map_neg, neg_inj],\n    rw [← comp_apply, ← comp_apply, ← 𝓕.1.map_comp, ← 𝓕.1.map_comp],\n    rw map_congr.vec_eq f (_ : vec.refine r (ignore α i) = ignore (vec.refine r α) i),\n    rw [← comp_apply, ← 𝓕.1.map_comp],\n    congr,\n    rw vec.refine_ignore, },\nend\n\nlemma C.refine_d_eq_d_refine' (f) :\n  C.refine r (d 𝓕 V n f)  = d 𝓕 U n (C.refine r f) := \nbegin\n  rw ← comp_apply,\n  rw ← C.refine_d_eq_d_refine,\n  rw comp_apply,\nend\n\nlemma C.refine_eq_to_hom {m n : ℕ} (h0 : m + 1 = n) (h1 : C 𝓕 U (m + 1) = C 𝓕 U n) (h2 : C 𝓕 V (m + 1) = C 𝓕 V n) :\n  (@C.refine X 𝓕 U V (m+1) r ≫ eq_to_hom h1) = eq_to_hom h2 ≫ @C.refine X 𝓕 U V n r :=\nbegin\n  ext f α,\n  rw [comp_apply, comp_apply, C.refine, add_monoid_hom.coe_mk,  C.refine, add_monoid_hom.coe_mk, C.refine.to_fun_apply],\n  \n  have : @eq_to_hom Ab _ _ _ h2 f (vec.refine r α) = 𝓕.1.map _ (f (vec.refine r (λ i, α ⟨i.1, h0 ▸ i.2⟩))),\n  work_on_goal 3\n  { refine quiver.hom.op _,\n    refine eq_to_hom _,\n    sorry },\n  { sorry },\n  rw this,\n  have : (@eq_to_hom Ab _ _ _ h1 (C.refine.to_fun r f)) α = 𝓕.1.map _ (C.refine.to_fun r f (λ i, α ⟨i.1, h0 ▸ i.2⟩)),\n  work_on_goal 3\n  { refine quiver.hom.op _,\n    refine eq_to_hom _,\n    unfold face,\n    ext,\n    split,\n\n    { intros hp,\n      rw opens.mem_coe at hp ⊢,\n      rw opens.fintype_infi at hp ⊢,\n      intros i,\n      specialize hp ⟨i.1, h0 ▸ i.2⟩,\n      exact hp },\n    { intros hp,\n      rw opens.mem_coe at hp ⊢,\n      rw opens.fintype_infi at hp ⊢,\n      intros i,\n      specialize hp ⟨i.1, h0.symm ▸ i.2⟩,\n      convert hp,\n      rw subtype.ext_iff_val, }, },\n    \n  { sorry },\n  rw this,\n  rw ← comp_apply,\n  rw ← 𝓕.1.map_comp,\n  rw C.refine.to_fun_apply,\n  rw ← comp_apply,\n  rw ← 𝓕.1.map_comp,\n  congr,\nend\n\n\ndef C.refine_functor : X.ocᵒᵖ ⥤ Ab :=\n{ obj := λ U, C 𝓕 (unop U) n,\n  map := λ U V r, C.refine r.unop,\n  map_id' := λ U, begin\n    ext f α,\n    simp only [unop_id, id_apply],\n    change C.refine.to_fun _ f α = _,\n    rw C.refine.to_fun_apply,\n    rw map_congr.vec_eq f (_ : α = vec.refine (𝟙 _) α),\n    congr,\n    ext m,\n    rw vec.refine_apply,\n    refl,\n  end,\n  map_comp' := λ U V W rUV rVW, begin\n    ext f α,\n    rw comp_apply,\n    change C.refine.to_fun _ f α = _,\n    rw C.refine.to_fun_apply,\n    rw C.refine,\n    simp only [add_monoid_hom.coe_mk],\n    rw C.refine,\n    simp only [add_monoid_hom.coe_mk],\n\n    rw C.refine.to_fun_apply,\n    rw C.refine.to_fun_apply,\n    rw ← comp_apply,\n    rw ← 𝓕.1.map_comp,\n    rw map_congr.vec_eq f (_ : vec.refine (rUV ≫ rVW).unop α = vec.refine rUV.unop (vec.refine rVW.unop α)),\n    rw ← comp_apply,\n    rw ← 𝓕.1.map_comp,\n    congr,\n    refl,\n  end }\n\ninclude 𝓕 n\nnoncomputable example : Ab :=\nbegin\n  have := limits.colim.obj (@C.refine_functor _ 𝓕 n  ⋙ AddCommGroup.ulift_functor.{u u+1}),\n  exact this,\nend\n\nend\n\nend", "meta": {"author": "jjaassoonn", "repo": "cc", "sha": "6d3dc6885fa012e8c18fd38ab2949d73777fb442", "save_path": "github-repos/lean/jjaassoonn-cc", "path": "github-repos/lean/jjaassoonn-cc/cc-6d3dc6885fa012e8c18fd38ab2949d73777fb442/src/new/unordered/refinement.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3466676072981252}}
{"text": "import data.cpi.semantics.relation\nimport data.fin_fn\n\nnamespace cpi\n\n/-- Determine if two concretions are equal. Effectively a decision procedure for\n    structural congruence. -/\ninstance concretion_wrap.decidable_eq {ℍ ω Γ} [cpi_equiv ℍ ω] :\n  decidable_eq ( prime_species' ℍ ω Γ\n               × (Σ (b y : ℕ), concretion' ℍ ω Γ b y) × name Γ)\n| a b := by apply_instance\n\n/-- A vector-space representation of processes, mapping prime species into their\n    concentrations. -/\n@[nolint has_inhabited_instance]\ndef process_space (ℂ ℍ : Type) (ω Γ : context) [add_monoid ℂ] [cpi_equiv ℍ ω]\n  := fin_fn (prime_species' ℍ ω Γ) ℂ\n\nvariables {ℂ ℍ : Type} {ω : context} [half_ring ℂ] [cpi_equiv ℍ ω] [decidable_eq ℂ]\n\ninstance process_space.add_comm_monoid {Γ} : add_comm_group (process_space ℂ ℍ ω Γ) := fin_fn.add_comm_group _ ℂ\ninstance process_space.semimodule {Γ} : semimodule ℂ (process_space ℂ ℍ ω Γ) := fin_fn.semimodule _ ℂ\ninstance process_space.has_repr {ℂ} [add_monoid ℂ] [has_repr ℂ] {Γ} [has_repr (species' ℍ ω Γ)]\n  : has_repr (process_space ℂ ℍ ω Γ) := ⟨ fin_fn.to_string \"\\n\" ⟩\n\n/-- Convert a species into a process space with a unit vector for each element\n    of the prime decomposition.\n\n    This is defined as ⟨A⟩ within the paper. -/\ndef to_process_space {Γ} (A : species' ℍ ω Γ) : process_space ℂ ℍ ω Γ\n  := (cpi_equiv.prime_decompose' A).sum' (λ A, fin_fn.single A 1)\n\n@[simp]\nlemma to_process_space.nil {Γ} : to_process_space ⟦nil⟧ = (0 : process_space ℂ ℍ ω Γ)\n  := by simp only [to_process_space, cpi_equiv.prime_decompose_nil', multiset.sum'_zero]\n\n@[simp]\nlemma to_process_space.prime {Γ} (A : prime_species' ℍ ω Γ)\n  : (to_process_space (prime_species.unwrap A) : process_space ℂ ℍ ω Γ)\n  = fin_fn.single A 1\n  := by simp only [to_process_space, cpi_equiv.prime_decompose_prime', multiset.sum'_singleton]\n\n@[simp]\nlemma to_process_space.parallel {Γ} (A B : species ℍ ω Γ)\n  : (to_process_space ⟦ A |ₛ B ⟧ : process_space ℂ ℍ ω Γ)\n  = to_process_space ⟦ A ⟧ + to_process_space ⟦ B ⟧\n  := by simp only [to_process_space, cpi_equiv.prime_decompose_parallel', multiset.sum'_add]\n\n/-- The vector space (A, E, a)→ℍ relating transitions from A to E with label #a.\n  -/\n@[nolint has_inhabited_instance]\ndef interaction_space (ℂ ℍ : Type) (ω Γ : context) [add_monoid ℂ] [cpi_equiv ℍ ω] : Type\n  := fin_fn\n      ( prime_species' ℍ ω Γ\n      × (Σ (b y), concretion' ℍ ω Γ b y)\n      × name Γ)\n      ℂ\n\ninstance interaction_space.add_comm_monoid {Γ} : add_comm_group (interaction_space ℂ ℍ ω Γ) := fin_fn.add_comm_group _ ℂ\ninstance interaction_space.semimodule {Γ} : semimodule ℂ (interaction_space ℂ ℍ ω Γ) := fin_fn.semimodule _ ℂ\n\ninstance interaction_space.has_repr {ℂ} [add_monoid ℂ] [has_repr ℂ] {Γ}\n  [has_repr (species' ℍ ω Γ)] [∀ b y, has_repr (concretion' ℍ ω Γ b y)]\n  : has_repr (interaction_space ℂ ℍ ω Γ) := ⟨ @fin_fn.to_string\n    ( prime_species' ℍ ω Γ\n      × (Σ (b y), concretion' ℍ ω Γ b y)\n      × name Γ) ℂ _\n    ⟨ λ ⟨ A, ⟨ _, _, F ⟩, a ⟩, \"[\" ++ repr A ++ \"], [\" ++ repr F ++ \"], \" ++ repr a ⟩\n    _ \"\\n\" ⟩\n\n/-- Convert a process into a process space. -/\ndef process.to_space {Γ}\n  : process ℂ ℍ ω Γ → process_space ℂ ℍ ω Γ\n| (c ◯ A) := c • to_process_space ⟦ A ⟧\n| (P |ₚ Q) := process.to_space P + process.to_space Q\n\n/-- Convert a process space into a process. -/\ndef process.from_space {ℂ} [add_comm_monoid ℂ] {Γ} : process_space ℂ ℍ ω Γ → process' ℂ ℍ ω Γ\n| Ps := process.from_prime_multiset Ps.space Ps.support.val\n\n/-- Convert a class of equivalent processes into a process space. -/\ndef process.to_space' {Γ} : process' ℂ ℍ ω Γ → process_space ℂ ℍ ω Γ\n| P := quot.lift_on P process.to_space (λ P Q eq, begin\n  induction eq,\n  case process.equiv.refl { from rfl },\n  case process.equiv.trans : P Q R _ _ pq qr { from trans pq qr },\n  case process.equiv.symm : P Q _ ih { from symm ih },\n\n  case process.equiv.ξ_species : c A A' eq {\n    simp only [process.to_space, quotient.sound eq],\n  },\n  case process.equiv.ξ_parallel₁ : P P' Q _ ih { simp only [process.to_space, ih] },\n  case process.equiv.ξ_parallel₂ : P Q Q' _ ih { simp only [process.to_space, ih] },\n\n  case process.equiv.parallel_nil : P c {\n    simp only [process.to_space, cpi.to_process_space.nil, smul_zero, add_zero],\n  },\n  case process.equiv.parallel_symm { simp only [process.to_space, add_comm] },\n  case process.equiv.parallel_assoc { simp only [process.to_space, add_assoc] },\n  case cpi.process.equiv.join : A c d { simp only [process.to_space, add_smul] },\n  case cpi.process.equiv.split : A B c {\n    simp only [process.to_space, to_process_space.parallel, smul_add],\n  }\nend)\n\naxiom process.from_inverse {Γ} :\n  function.left_inverse process.to_space' (@process.from_space ℍ ω _ ℂ _ Γ)\n\n/-- Show that process spaces can be embeeded into equivalence classes of processes. -/\ndef process.space_embed {Γ} : process_space ℂ ℍ ω Γ ↪ process' ℂ ℍ ω Γ :=\n  { to_fun := process.from_space,\n    inj := function.injective_of_left_inverse process.from_inverse }\n\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/semantics/space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.34663580121641563}}
{"text": "import data.finset data.multiset.dict data.pfun logic.function\n\nlocal attribute [-simp] sigma.forall sigma.exists\n\nuniverses u v\n\n/-- Finite map: a multiset of dependent pairs with no duplicate keys -/\nstructure finmap (α : Type u) (β : α → Type v) : Type (max u v) :=\n(val : multiset (sigma β))\n(nodupkeys : val.nodupkeys)\n\nnamespace finmap\nopen multiset\n\nsection αβ\nvariables {α : Type u} {β : α → Type v}\n\n/- equality -/\n\ntheorem eq_of_veq : ∀ {f g : finmap α β}, f.val = g.val → f = g\n| ⟨s, _⟩ ⟨t, _⟩ h := by congr; exact h\n\n@[simp] theorem val_inj {f g : finmap α β} : f.val = g.val ↔ f = g :=\n⟨eq_of_veq, congr_arg _⟩\n\ninstance has_decidable_eq [decidable_eq α] [∀ a, decidable_eq (β a)] : decidable_eq (finmap α β)\n| f g := decidable_of_iff _ val_inj\n\n/- recursors -/\n\nsection rec\nopen function\n\n/-- Dependent recursor on a finmap for a list -/\nprotected def lrec_on {γ : Sort*} (f : finmap α β)\n  (φ : ∀ {l : list (sigma β)}, l.nodupkeys → γ)\n  (c : ∀ {l₁ l₂} (p : l₁ ~ l₂) d₁ d₂, φ d₁ = φ d₂) : γ :=\n@quotient.hrec_on _ _ (λ (m : multiset (sigma β)), m.nodupkeys → γ)\n  f.val\n  (λ l (d : l.nodupkeys), φ d)\n  (λ l₁ l₂ p, hfunext (by rw list.perm_nodupkeys p) $ λ d₁ d₂ _, heq_of_eq $ c p d₁ d₂)\n  f.nodupkeys\n\n/-- Dependent recursor on two finmaps for lists -/\nprotected def lrec_on₂ {γ : Sort*} (f g : finmap α β)\n  (φ : ∀ {l₁ l₂ : list (sigma β)}, l₁.nodupkeys → l₂.nodupkeys → γ)\n  (c : ∀ {l₁ l₂ l₃ l₄} (p₁₃ : l₁ ~ l₃) (p₂₄ : l₂ ~ l₄) d₁ d₂ d₃ d₄, φ d₁ d₂ = φ d₃ d₄) : γ :=\n@quotient.hrec_on₂ _ _ _ _\n  (λ (m₁ m₂ : multiset (sigma β)), m₁.nodupkeys → m₂.nodupkeys → γ)\n  f.val g.val\n  (λ l₁ l₂ (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d₁ d₂)\n  (λ l₁ l₂ l₃ l₄ p₁₃ p₂₄, hfunext (by rw list.perm_nodupkeys p₁₃) $\n    λ d₁ d₃ _, hfunext (by rw list.perm_nodupkeys p₂₄) $\n      λ d₂ d₄ _, heq_of_eq $ c p₁₃ p₂₄ d₁ d₂ d₃ d₄)\n  f.nodupkeys g.nodupkeys\n\n/-- Lift a function on 2 lists to a function on 2 finmaps  -/\nprotected def lift_on₂ {γ : Type*} (f g : finmap α β)\n  (φ : ∀ {l₁ l₂ : list (sigma β)}, l₁.nodupkeys → l₂.nodupkeys → γ)\n  (c : ∀ {l₁ l₂ l₃ l₄} (p₁₃ : l₁ ~ l₃) (p₂₄ : l₂ ~ l₄) d₁ d₂ d₃ d₄, φ d₁ d₂ = φ d₃ d₄) :\n  roption γ :=\nquotient.lift_on₂ f.val g.val\n  (λ l₁ l₂, roption.mk (l₁.nodupkeys ∧ l₂.nodupkeys) (λ ⟨d₁, d₂⟩, φ d₁ d₂))\n  (λ l₁ l₂ l₃ l₄ p₁₃ p₂₄, roption.ext'\n    (and_congr (list.perm_nodupkeys p₁₃) (list.perm_nodupkeys p₂₄))\n    (λ ⟨d₁, d₂⟩ ⟨d₃, d₄⟩, c p₁₃ p₂₄ d₁ d₂ d₃ d₄))\n\nend rec\n\n/- membership -/\n\nsection mem\nvariables {s : sigma β} {m : multiset (sigma β)} {d : m.nodupkeys} {f : finmap α β}\n\ninstance : has_mem (sigma β) (finmap α β) :=\n⟨λ s f, s ∈ f.val⟩\n\ntheorem mem_def : s ∈ f ↔ s ∈ f.val :=\niff.rfl\n\n@[simp] theorem mem_mk : @has_mem.mem _ (finmap α β) _ s (finmap.mk m d) ↔ s ∈ m :=\niff.rfl\n\ninstance decidable_mem [decidable_eq α] [∀ a, decidable_eq (β a)] (s : sigma β) (f : finmap α β) :\n  decidable (s ∈ f) :=\nmultiset.decidable_mem _ _\n\nend mem\n\n/- set coercion -/\n\nsection set\nvariables {s : sigma β} {f : finmap α β}\n\ndef to_set (f : finmap α β) : set (sigma β) :=\n{x | x ∈ f}\n\ninstance has_lift_set : has_lift (finmap α β) (set (sigma β)) :=\n⟨to_set⟩\n\n@[simp] theorem mem_set_coe : s ∈ (↑f : set (sigma β)) ↔ s ∈ f :=\niff.rfl\n\nend set\n\n/- finset coercion -/\n\nsection finset\nvariables {s : sigma β} {f : finmap α β}\n\ndef to_finset (f : finmap α β) : finset (sigma β) :=\n⟨f.val, nodup_of_nodupkeys f.nodupkeys⟩\n\ninstance has_lift_finset : has_lift (finmap α β) (finset (sigma β)) :=\n⟨to_finset⟩\n\n@[simp] theorem mem_finset_coe : s ∈ (↑f : finset (sigma β)) ↔ s ∈ f :=\niff.rfl\n\nend finset\n\n/- extensionality -/\n\nsection ext\nvariables {f g : finmap α β}\n\ntheorem ext : f = g ↔ ∀ s, s ∈ f ↔ s ∈ g :=\nval_inj.symm.trans $ nodup_ext\n  (nodup_of_nodupkeys f.nodupkeys)\n  (nodup_of_nodupkeys g.nodupkeys)\n\n@[extensionality]\ntheorem ext' : (∀ s, s ∈ f ↔ s ∈ g) → f = g :=\next.mpr\n\nend ext\n\n/- subset -/\n\nsection subset\nvariables {s : sigma β} {f g h : finmap α β}\n\ninstance : has_subset (finmap α β) :=\n⟨λ f g, ∀ ⦃s : sigma β⦄, s ∈ f → s ∈ g⟩\n\ntheorem subset_def : f ⊆ g ↔ f.val ⊆ g.val :=\niff.rfl\n\n@[simp] theorem subset.refl (f : finmap α β) : f ⊆ f :=\nsubset.refl _\n\ntheorem subset.trans : f ⊆ g → g ⊆ h → f ⊆ h :=\nsubset.trans\n\ntheorem mem_of_subset : f ⊆ g → s ∈ f → s ∈ g :=\nmem_of_subset\n\ntheorem subset.antisymm (H₁ : f ⊆ g) (H₂ : g ⊆ f) : f = g :=\next' $ λ a, ⟨@H₁ a, @H₂ a⟩\n\ntheorem subset_iff : f ⊆ g ↔ ∀ ⦃x⦄, x ∈ f → x ∈ g :=\niff.rfl\n\n@[simp] theorem coe_subset_set : (↑f : set (sigma β)) ⊆ ↑g ↔ f ⊆ g :=\niff.rfl\n\n@[simp] theorem coe_subset_finset : (↑f : finset (sigma β)) ⊆ ↑g ↔ f ⊆ g :=\niff.rfl\n\n@[simp] theorem val_le_iff : f.val ≤ g.val ↔ f ⊆ g :=\nle_iff_subset (nodup_of_nodupkeys f.nodupkeys)\n\ninstance : has_ssubset (finmap α β) :=\n⟨λa b, a ⊆ b ∧ ¬ b ⊆ a⟩\n\ninstance : partial_order (finmap α β) :=\n{ le := (⊆),\n  lt := (⊂),\n  le_refl := subset.refl,\n  le_trans := @subset.trans _ _,\n  le_antisymm := @subset.antisymm _ _ }\n\n@[simp] theorem le_iff_subset : f ≤ g ↔ f ⊆ g := iff.rfl\n@[simp] theorem lt_iff_ssubset : f < g ↔ f ⊂ g := iff.rfl\n\n@[simp] theorem val_lt_iff : f.val < g.val ↔ f ⊂ g :=\nand_congr val_le_iff $ not_congr val_le_iff\n\nend subset\n\n/- empty -/\n\nsection empty\nvariables {s : sigma β} {f : finmap α β}\n\ninstance : has_emptyc (finmap α β) :=\n⟨⟨_, nodupkeys_zero⟩⟩\n\ninstance : inhabited (finmap α β) :=\n⟨∅⟩\n\n@[simp] theorem empty_val : (∅ : finmap α β).val = 0 :=\nrfl\n\n@[simp] theorem not_mem_empty (s : sigma β) : s ∉ (∅ : finmap α β) :=\nid\n\n@[simp] theorem ne_empty_of_mem (h : s ∈ f) : f ≠ ∅\n| e := not_mem_empty s $ e ▸ h\n\n@[simp] theorem empty_subset (f : finmap α β) : ∅ ⊆ f :=\nzero_subset _\n\ntheorem eq_empty_of_forall_not_mem (H : ∀x, x ∉ f) : f = ∅ :=\neq_of_veq (eq_zero_of_forall_not_mem H)\n\n@[simp] theorem val_eq_zero : f.val = 0 ↔ f = ∅ :=\n@val_inj _ _ f ∅\n\ntheorem subset_empty : f ⊆ ∅ ↔ f = ∅ :=\nsubset_zero.trans val_eq_zero\n\ntheorem exists_mem_of_ne_empty (h : f ≠ ∅) : ∃ s : sigma β, s ∈ f :=\nexists_mem_of_ne_zero (mt val_eq_zero.mp h)\n\n@[simp] theorem coe_empty_set : ↑(∅ : finmap α β) = (∅ : set (sigma β)) :=\nby simp [set.ext_iff]\n\n@[simp] theorem coe_empty_finset : ↑(∅ : finmap α β) = (∅ : finset (sigma β)) :=\nby simp [finset.ext]\n\nend empty\n\n/- singleton -/\n\nsection singleton\nvariables {s₁ s₂ : sigma β}\n\n/-- `singleton s` is the set `{s}` containing `s` and nothing else. -/\ndef singleton (s : sigma β) : finmap α β :=\n⟨⟦[s]⟧, nodupkeys_singleton s⟩\n\n@[simp] theorem singleton_val (s : sigma β) : (singleton s).val = s :: 0 :=\nrfl\n\n@[simp] theorem mem_singleton : s₁ ∈ singleton s₂ ↔ s₁ = s₂ :=\nby simp [singleton]\n\ntheorem not_mem_singleton : s₁ ∉ singleton s₂ ↔ s₁ ≠ s₂ := by simp\n\ntheorem mem_singleton_self (s : sigma β) : s ∈ singleton s := by simp\n\ntheorem singleton_inj : singleton s₁ = singleton s₂ ↔ s₁ = s₂ :=\n⟨λ h, mem_singleton.mp (h ▸ mem_singleton_self _), congr_arg _⟩\n\n@[simp] theorem singleton_ne_empty (s : sigma β) : singleton s ≠ ∅ :=\nne_empty_of_mem (mem_singleton_self _)\n\n@[simp] theorem coe_singleton_set (s : sigma β) : ↑(singleton s) = ({s} : set (sigma β)) :=\nby simp [set.ext_iff]\n\n@[simp] theorem coe_singleton_finset (s : sigma β) : ↑(singleton s) = finset.singleton s :=\nby simp [finset.ext]\n\nend singleton\n\n/- keys -/\n\nsection keys\nvariables {a a₁ a₂ : α} {s : sigma β} {f : finmap α β}\n\ndef keys (f : finmap α β) : finset α :=\n⟨f.val.keys, nodupkeys_iff.mpr f.nodupkeys⟩\n\n@[simp] theorem keys_val (f : finmap α β) : f.keys.val = f.val.keys :=\nrfl\n\n@[simp] theorem keys_empty : keys (∅ : finmap α β) = ∅ :=\nrfl\n\n@[simp] theorem keys_singleton : keys (singleton s) = finset.singleton s.1 :=\nrfl\n\ntheorem mem_keys_of_mem : s ∈ f → s.1 ∈ f.keys :=\nmem_keys_of_mem\n\ntheorem exists_mem_of_mem_keys : a ∈ f.keys → ∃ (b : β a), sigma.mk a b ∈ f :=\nexists_mem_of_mem_keys\n\n@[simp] theorem mem_keys_singleton : a ∈ (singleton s).keys ↔ a = s.1 :=\nby simp\n\n@[simp] theorem mem_insert_keys [decidable_eq α] :\n  a₁ ∈ insert a₂ f.keys ↔ a₁ = a₂ ∨ a₁ ∈ f.keys :=\nby simp\n\nend keys\n\nsection decidable_eq_α\nvariables [decidable_eq α]\n\n/- erase -/\n\nsection erase\nvariables {s : sigma β} {a a₁ a₂ : α} {f g : finmap α β}\n\ndef erase (f : finmap α β) (a : α) : finmap α β :=\n⟨kerase a f.nodupkeys, nodupkeys_kerase a f.nodupkeys⟩\n\n@[simp] theorem erase_val (f : finmap α β) (a : α) : (f.erase a).val = kerase a f.nodupkeys :=\nrfl\n\n@[simp] theorem mem_erase : s ∈ f.erase a ↔ s.1 ≠ a ∧ s ∈ f :=\nmem_kerase f.nodupkeys\n\ntheorem not_mem_erase (a : α) (b : β a) (f : finmap α β) : sigma.mk a b ∉ f.erase a :=\nby simp\n\n@[simp] theorem mem_keys_erase : a₁ ∈ (f.erase a₂).keys ↔ a₁ ≠ a₂ ∧ a₁ ∈ f.keys :=\nby simp [keys]\n\n@[simp] theorem erase_empty (β : α → Type v) (a) : erase ∅ a = (∅ : finmap α β) :=\nrfl\n\ntheorem ne_of_mem_erase : s ∈ f.erase a → s.1 ≠ a :=\nby simp {contextual := tt}\n\ntheorem erase_subset_erase (a : α) (h : f ⊆ g) : f.erase a ⊆ g.erase a :=\nval_le_iff.mp $ kerase_le_kerase _ (val_le_iff.mpr h) _ _\n\ntheorem erase_subset (a : α) : f.erase a ⊆ f :=\nkerase_subset a f.nodupkeys\n\nend erase\n\n/- insert -/\n\nsection insert\nvariables {a : α} {s : sigma β} {f g : finmap α β}\n\ninstance : has_insert (sigma β) (finmap α β) :=\n⟨λ s f, ⟨kinsert s f.nodupkeys, nodupkeys_kinsert s f.nodupkeys⟩⟩\n\ntheorem insert_def (s : sigma β) (f : finmap α β) :\n  insert s f = mk (kinsert s f.nodupkeys) (nodupkeys_kinsert s f.nodupkeys) :=\nrfl\n\n@[simp] theorem insert_val (s : sigma β) (f : finmap α β) :\n  (insert s f).val = kinsert s f.nodupkeys :=\nrfl\n\n@[simp] theorem insert_empty (s : sigma β) : insert s (∅ : finmap α β) = {s} :=\nrfl\n\n@[simp] theorem mem_insert (s₁ s₂ : sigma β) (f : finmap α β) :\n  s₁ ∈ insert s₂ f ↔ s₁ = s₂ ∨ s₁ ∈ f.erase s₂.1 :=\nmem_kinsert f.nodupkeys\n\n@[simp] theorem mem_keys_insert : a ∈ (insert s f).keys ↔ a = s.1 ∨ a ∈ f.keys :=\nby simp [keys]\n\n@[simp] theorem insert_keys : (insert s f).keys = insert s.1 f.keys :=\nfinset.ext' $ by simp\n\n@[simp] theorem disjoint_keys_insert_left :\n  disjoint (insert s f).keys g.keys ↔ s.1 ∉ g.keys ∧ disjoint f.keys g.keys :=\nby simp\n\n@[simp] theorem disjoint_keys_insert_right :\n  disjoint f.keys (insert s g).keys ↔ s.1 ∉ f.keys ∧ disjoint f.keys g.keys :=\nby simp\n\nend insert\n\n/- lookup -/\n\nsection lookup\n\n/-- Look up a key in a finmap to find the value, if it exists -/\ndef lookup (a : α) (f : finmap α β) : option (β a) :=\nklookup a f.nodupkeys\n\n/-- Treat a finmap as the function `∀ a, option (β a)` -/\ndef to_fun (f : finmap α β) (a : α) : option (β a) :=\nf.lookup a\n\n/-- Treat a finmap as the partial function `∀ a, roption (β a)` -/\ndef to_pfun (f : finmap α β) (a : α) : roption (β a) :=\nroption.of_option (f.to_fun a)\n\ntheorem lookup_empty (β : α → Type v) (a) : lookup a (∅ : finmap α β) = none :=\nrfl\n\nend lookup\n\n/- replace -/\n\nsection replace\n\ndef replace (s : sigma β) (f : finmap α β) : finmap α β :=\n⟨kreplace s f.nodupkeys, nodupkeys_kreplace s f.nodupkeys⟩\n\n@[simp] theorem replace_empty (s : sigma β) : replace s ∅ = ∅ :=\nrfl\n\nend replace\n\n/- union -/\n\nsection union\nvariables {a : α} {s : sigma β} {f g h : finmap α β}\n\nprotected def union (f : finmap α β) (g : finmap α β) : finmap α β :=\n⟨kunion f.nodupkeys g.nodupkeys, nodupkeys_kunion f.nodupkeys g.nodupkeys⟩\n\ninstance : has_union (finmap α β) :=\n⟨finmap.union⟩\n\n@[simp] theorem union_val : (f ∪ g).val = kunion f.nodupkeys g.nodupkeys :=\nrfl\n\n@[simp] theorem empty_union (f : finmap α β) : ∅ ∪ f = f :=\neq_of_veq $ by simp [zero_kunion f.nodupkeys]\n\n@[simp] theorem union_empty (f : finmap α β) : f ∪ ∅ = f :=\neq_of_veq $ kunion_zero f.nodupkeys\n\n@[simp] theorem insert_union : insert s f ∪ g = insert s (f ∪ g) :=\neq_of_veq $ kinsert_kunion f.nodupkeys g.nodupkeys\n\ntheorem mem_of_mem_union : s ∈ f ∪ g → s ∈ f ∨ s ∈ g :=\nmem_of_mem_kunion f.nodupkeys g.nodupkeys\n\ntheorem mem_union_left (g : finmap α β) : s ∈ f → s ∈ f ∪ g :=\nmem_kunion_left f.nodupkeys g.nodupkeys\n\ntheorem mem_union_right : s.1 ∉ f.keys → s ∈ g → s ∈ f ∪ g :=\nmem_kunion_right f.nodupkeys g.nodupkeys\n\n@[simp] theorem mem_union (dk : disjoint f.keys g.keys) : s ∈ f ∪ g ↔ s ∈ f ∨ s ∈ g :=\nmem_kunion f.nodupkeys g.nodupkeys (finset.disjoint_val.mp dk)\n\ntheorem union_comm (dk : disjoint f.keys g.keys) : f ∪ g = g ∪ f :=\nby simp [ext, or_comm, dk, dk.symm]\n\n@[simp] theorem mem_keys_union : a ∈ (f ∪ g).keys ↔ a ∈ f.keys ∨ a ∈ g.keys :=\nmem_keys_kunion f.nodupkeys g.nodupkeys\n\n@[simp] theorem union_keys : (f ∪ g).keys = f.keys ∪ g.keys :=\nfinset.ext' $ by simp\n\ntheorem disjoint_keys_union_left :\n  disjoint (f ∪ g).keys h.keys ↔ disjoint f.keys h.keys ∧ disjoint g.keys h.keys :=\nby simp\n\ntheorem disjoint_keys_union_right :\n  disjoint f.keys (g ∪ h).keys ↔ disjoint f.keys g.keys ∧ disjoint f.keys h.keys :=\nby simp\n\n@[simp] theorem union_assoc (dk_fg : disjoint f.keys g.keys) (dk_gh : disjoint g.keys h.keys)\n  (dk_fh : disjoint f.keys h.keys) : (f ∪ g) ∪ h = f ∪ (g ∪ h) :=\nhave disjoint (f ∪ g).keys h.keys := disjoint_keys_union_left.mpr ⟨dk_fh, dk_gh⟩,\nhave disjoint f.keys (g ∪ h).keys := disjoint_keys_union_right.mpr ⟨dk_fg, dk_fh⟩,\nby simp [ext, or_comm, or.left_comm, *]\n\ntheorem mem_union_middle_left (dk_fh : disjoint f.keys h.keys) (dk_gh : disjoint g.keys h.keys)\n  (p : s ∈ f ∪ h) : s ∈ f ∪ g ∪ h :=\nmatch mem_of_mem_union p with\n| or.inl p := mem_union_left _ (mem_union_left _ p)\n| or.inr p := mem_union_right\n  (finset.disjoint_right.mp (disjoint_keys_union_left.mpr ⟨dk_fh, dk_gh⟩) (mem_keys_of_mem p)) p\nend\n\ntheorem mem_union_middle_right (dk_fh : disjoint f.keys h.keys) (dk_gh : disjoint g.keys h.keys)\n  (p : s ∈ f ∪ h) : s ∈ f ∪ (g ∪ h) :=\nmatch mem_of_mem_union p with\n| or.inl p := mem_union_left _ p\n| or.inr p :=\n  have s.1 ∈ h.keys := mem_keys_of_mem p,\n  mem_union_right\n    (finset.disjoint_right.mp dk_fh this)\n    (mem_union_right (finset.disjoint_right.mp dk_gh this) p)\nend\n\nend union\n\nend decidable_eq_α\n\nend αβ\n\nsection α₁α₂α₃β₁β₂β₃\nvariables {α₁ α₂ α₃ : Type u} {β₁ : α₁ → Type v} {β₂ : α₂ → Type v} {β₃ : α₃ → Type v}\n\nsection map\nvariables {p : β₁ s↪ β₂} {q : β₂ s↪ β₃} {s₁ : sigma β₁} {s₂ : sigma β₂} {f g : finmap α₁ β₁}\n\ndef map (p : β₁ s↪ β₂) (f : finmap α₁ β₁) : finmap α₂ β₂ :=\n⟨f.val.map p, nodupkeys_map p.fst_inj f.nodupkeys⟩\n\n@[simp] theorem map_val (p : β₁ s↪ β₂) (f : finmap α₁ β₁) : (f.map p).val = f.val.map p :=\nrfl\n\n@[simp] theorem map_mk {m₁ : multiset (sigma β₁)} {m₂ : multiset (sigma β₂)} {p : β₁ s↪ β₂}\n  (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys) : (mk m₁ d₁).map p = mk m₂ d₂ ↔ m₁.map p = m₂ :=\nby simp [map]\n\n@[simp] theorem map_empty (p : β₁ s↪ β₂) : map p ∅ = ∅ :=\nrfl\n\n@[simp] theorem mem_map : s₂ ∈ f.map p ↔ ∃ s₁ ∈ f, p s₁ = s₂ :=\nby simp [mem_def]\n\n@[simp] theorem mem_map_of_mem (h : s₁ ∈ f) : p s₁ ∈ f.map p :=\nmem_map.mpr ⟨_, h, rfl⟩\n\ntheorem map_refl : f.map (sigma.embedding.refl _) = f :=\next' $ by simp [sigma.embedding.refl]\n\ntheorem map_map : (f.map p).map q = f.map (p.trans q) :=\neq_of_veq $ by simp [erase_dup_map_erase_dup_eq]\n\ntheorem map_subset_map (h : f ⊆ g) : f.map p ⊆ g.map p :=\nby simp [subset_def, map_subset_map h]\n\ntheorem mem_keys_map (pf : sigma.fst_functional p) :\n  s₁.1 ∈ f.keys → (p s₁).1 ∈ (f.map p).keys :=\nmem_keys_map pf\n\ntheorem mem_keys_of_mem_keys_map : (p s₁).1 ∈ (f.map p).keys → s₁.1 ∈ f.keys :=\nmem_keys_of_mem_keys_map p.fst_inj\n\ntheorem mem_keys_map_iff (pf : sigma.fst_functional p) :\n  (p s₁).1 ∈ (f.map p).keys ↔ s₁.1 ∈ f.keys :=\n⟨mem_keys_of_mem_keys_map, mem_keys_map pf⟩\n\ntheorem map_disjoint_keys [decidable_eq α₁] [decidable_eq α₂] (pf : sigma.fst_functional p) :\n  disjoint (f.map p).keys (g.map p).keys ↔ disjoint f.keys g.keys :=\nby simp only [finset.disjoint_val]; exact multiset.map_disjoint_keys pf p.fst_inj\n\ntheorem map_union [decidable_eq α₁] [decidable_eq α₂] (pf : sigma.fst_functional p)\n  (dk : disjoint f.keys g.keys) : (f ∪ g).map p = f.map p ∪ g.map p :=\next' $ by simp [dk, map_disjoint_keys pf, or_and_distrib_right, exists_or_distrib]\n\nsection decidable_eq_α₁α₂\nvariables [decidable_eq α₁] [decidable_eq α₂]\n\n@[simp] theorem map_erase (pf : sigma.fst_functional p) :\n  (f.erase s₁.1).map p = (f.map p).erase (p s₁).1 :=\nby simp [erase, map_kerase pf p.fst_inj f.nodupkeys]\n\n@[simp] theorem map_insert (pf : sigma.fst_functional p) :\n  (insert s₁ f).map p = insert (p s₁) (f.map p) :=\nby simp [insert_def, map_kinsert pf p.fst_inj f.nodupkeys]\n\nend decidable_eq_α₁α₂\n\nend map\n\nend α₁α₂α₃β₁β₂β₃\n\nsection αβ₁β₂\nvariables {α : Type u} {β₁ β₂ : α → Type v}\n\nsection map_id\nvariables {s : sigma β₁} {f : finmap α β₁}\n\n@[simp] theorem map_id_val (p : ∀ a, β₁ a → β₂ a) (f : finmap α β₁) :\n  (f.map (sigma.embedding.mk₂ p)).val = f.val.map (sigma.map id p) :=\nrfl\n\n@[simp] theorem map_id_keys (p : ∀ a, β₁ a → β₂ a) : (f.map (sigma.embedding.mk₂ p)).keys = f.keys :=\nfinset.val_inj.mp $ by simp\n\nend map_id\n\nend αβ₁β₂\n\nend finmap\n", "meta": {"author": "spl", "repo": "lean-finmap", "sha": "936d9caeb27631e3c6cf20e972de4837c9fe98fa", "save_path": "github-repos/lean/spl-lean-finmap", "path": "github-repos/lean/spl-lean-finmap/lean-finmap-936d9caeb27631e3c6cf20e972de4837c9fe98fa/src/data/finmap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3466059957112453}}
{"text": "import logic.equiv.transfer_instance\n\nlemma ring_hom.is_field {R S : Type*} [semifield R] [semiring S] [nontrivial S] (f : R →+* S)\n  (hf : function.surjective f) : is_field S :=\nbegin\n  constructor,\n  { exact exists_pair_ne _ },\n  { intros x y, obtain ⟨x, rfl⟩ := hf x, obtain ⟨y, rfl⟩ := hf y,\n    rw [← map_mul, ← map_mul, mul_comm] },\n  { intros a ha, obtain ⟨a, rfl⟩ := hf a, use f (1 / a), rw [← map_mul, mul_div_cancel', map_one], \n    rintro rfl, apply ha, rw map_zero }\nend\n\nlemma ring_equiv.is_field {R S : Type*} [semifield R] [semiring S] (f : R ≃+* S) : is_field S :=\nbegin\n  haveI := f.to_equiv.symm.nontrivial,\n  exact f.to_ring_hom.is_field f.surjective\nend\n\nlemma ring_equiv.is_field_of_is_field {R S : Type*} [semiring R] [semiring S] (f : R ≃+* S)\n  (h : is_field R) : is_field S :=\nbegin\n  letI := h.to_semifield,\n  exact f.is_field\nend\n", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/for_mathlib/field.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3464639096279216}}
{"text": "import for_mathlib.algebraic_topology.homotopical_algebra.cofibrant_object\nimport for_mathlib.category_theory.localization.equivalence\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category category_theory\n\nnamespace algebraic_topology\n\nnamespace model_category\n\nvariables {C : Type*} [category C] [model_category C]\n\nnamespace cofibrant_replacement\n\ndef obj (X : C) : cofibrant_object C :=\ncofibrant_object.mk (CM5b.obj (initial.to X))\n\ndef app (X : C) : (obj X).obj ⟶ X := CM5b.p (initial.to X)\n\ndef app' (X : cofibrant_object C) : obj X.obj ⟶ X := app X.obj\n\ninstance (X : C) : fibration (app X) := by { dsimp [app], apply_instance, }\ninstance (X : C) : weak_eq (app X) := by { dsimp [app], apply_instance, }\n\ndef map' {X Y : C} (f : X ⟶ Y) : obj X ⟶ obj Y :=\nbegin\n  have sq : comm_sq (initial.to (obj Y).obj) (initial.to (obj X).obj) (app Y) (app X ≫ f) :=\n    by tidy,\n  exact sq.lift,\nend\n\n@[reassoc]\nlemma fac {X Y : C} (f : X ⟶ Y) : (cofibrant_object.forget C).map (map' f) ≫ app Y =\n  app X ≫ f :=\nby apply comm_sq.fac_right\n\nlemma fac' {X Y : cofibrant_object C} (f : X ⟶ Y) :\n  map' ((cofibrant_object.forget C).map f) ≫ app' Y = app' X ≫ f :=\nby apply fac\n\ndef map {X Y : C} (f : X ⟶ Y) : cofibrant_object.homotopy_category.Q.obj (obj X) ⟶\n  cofibrant_object.homotopy_category.Q.obj (obj Y) :=\n  cofibrant_object.homotopy_category.Q.map (map' f)\n\nlemma map_eq {X Y : C} (f : X ⟶ Y) (f' : obj X ⟶ obj Y)\n  (sq : comm_sq (app X) ((cofibrant_object.forget C).map f') f (app Y)) :\n  map f = cofibrant_object.homotopy_category.Q.map f' :=\nbegin\n  apply category_theory.quotient.sound,\n  apply cofibrant_object.right_homotopy.trans_closure.mk,\n  let Cyl := cylinder.some (obj X).obj,\n  suffices : left_homotopy Cyl.pre (map' f) f',\n  { exact cofibrant_object.right_homotopy.mk (path_object.some (obj Y).obj)\n      (this.to_right_homotopy _), },\n  have sq' : comm_sq (coprod.desc ((cofibrant_object.forget C).map (map' f)) f') Cyl.ι\n    (app Y) (Cyl.σ ≫ app X ≫ f),\n  { refine comm_sq.mk _,\n    ext,\n    { simpa only [precylinder.ι, cofibrant_object.forget_map, coprod.desc_comp,\n        coprod.inl_desc, coprod.desc_comp_assoc, precylinder.σd₀, id_comp] using fac f, },\n    { simpa only [precylinder.ι, coprod.desc_comp, coprod.inr_desc, coprod.desc_comp_assoc,\n        precylinder.σd₁, id_comp] using sq.w.symm, }, },\n  exact\n  { h := sq'.lift,\n    h₀' := by simpa using congr_arg (λ f, limits.coprod.inl ≫ f) sq'.fac_left,\n    h₁' := by simpa using congr_arg (λ f, limits.coprod.inr ≫ f) sq'.fac_left, },\nend\n\nend cofibrant_replacement\n\nvariable (C)\n\n@[simps]\ndef cofibrant_replacement : C ⥤ cofibrant_object.homotopy_category C :=\n{ obj := λ X, cofibrant_object.homotopy_category.Q.obj (cofibrant_replacement.obj X),\n  map := λ X Y f, cofibrant_replacement.map f,\n  map_id' := λ X, begin\n    rw [cofibrant_replacement.map_eq _ (𝟙 _), cofibrant_object.homotopy_category.Q.map_id],\n    exact comm_sq.mk (by erw [id_comp, comp_id]),\n  end,\n  map_comp' := λ X Y Z f g, begin\n    rw [cofibrant_replacement.map_eq (f ≫ g)\n      (cofibrant_replacement.map' f ≫ cofibrant_replacement.map' g), functor.map_comp],\n    { refl, },\n    { refine comm_sq.mk _,\n      rw [functor.map_comp, assoc, cofibrant_replacement.fac g,\n        cofibrant_replacement.fac_assoc f], },\n  end, }\n\nvariables {Hocof : Type*} [category Hocof] (Lcof : cofibrant_object C ⥤ Hocof)\n  [Lcof.is_localization cofibrant_object.weq]\n  {Ho : Type*} [category Ho] (L : C ⥤ Ho) [L.is_localization weq]\n\nnamespace cofibrant_replacement\n\nvariable {C}\n\nlemma weq_iff {X Y : C} (f : X ⟶ Y) (f' : obj X ⟶ obj Y)\n  (sq : comm_sq (app X) ((cofibrant_object.forget C).map f') f (app Y)) :\n  weq f ↔ cofibrant_object.weq f' :=\nbegin\n  split,\n  { intro hf,\n    have h : weq (app X ≫ f) := CM2.of_comp (app X) f weak_eq.property hf,\n    rw sq.w at h,\n    exact CM2.of_comp_right _ _ weak_eq.property h, },\n  { intro hf',\n    have h : weq (app X ≫ f),\n    { rw sq.w,\n      exact CM2.of_comp _ _ hf' weak_eq.property, },\n    exact CM2.of_comp_left _ _ weak_eq.property h, },\nend\n\nlemma weq_eq_inverse_image_homotopy_category_weq :\n  model_category.weq =\ncofibrant_object.homotopy_category.weq.inverse_image (cofibrant_replacement C) :=\nbegin\n  ext X Y f,\n  dsimp only [morphism_property.inverse_image, cofibrant_replacement, map],\n  rw cofibrant_object.homotopy_category.weq_Q_map_iff,\n  exact weq_iff _ _ (comm_sq.mk (fac f).symm),\nend\n\ninclude Lcof\n\n@[simps]\ndef π : cofibrant_object.homotopy_category C ⥤ Hocof :=\ncategory_theory.quotient.lift _ Lcof (λ (X Y : cofibrant_object C), begin\n  let Y' : cofibrant_object C := cofibrant_object.mk (CM5a.obj (terminal.from Y.obj)),\n  let φ : Y ⟶ Y' := CM5a.i (terminal.from Y.obj),\n  intros f₁ f₂ h,\n  haveI : is_iso (Lcof.map φ) := localization.inverts Lcof cofibrant_object.weq φ\n    (by { change weq (CM5a.i (terminal.from Y.obj)), exact weak_eq.property, }),\n  simp only [← cancel_mono (Lcof.map φ), ← Lcof.map_comp],\n  induction h with g₁ g₂ h g₁ g₂ g₃ H₁₂ H₂₃ h₁₂ h₂₃,\n  { rcases h.comp_right φ with ⟨P, H⟩,\n    let Cyl := cylinder.some X.obj,\n    let I' := cofibrant_object.mk Cyl.I,\n    let s : I' ⟶ X := Cyl.σ,\n    haveI : is_iso (Lcof.map s) := localization.inverts Lcof cofibrant_object.weq s\n      (by { change weq Cyl.σ, exact weak_eq.property, }),\n    let h' := H.some.to_left_homotopy Cyl,\n    let ψ : I' ⟶ Y' := h'.h,\n    let d₀ : X ⟶ I' := Cyl.d₀,\n    let d₁ : X ⟶ I' := Cyl.d₁,\n    have eq₀ : d₀ ≫ ψ = g₁ ≫ φ := h'.h₀,\n    have eq₁ : d₁ ≫ ψ = g₂ ≫ φ := h'.h₁,\n    simp only [← eq₀, ← eq₁, Lcof.map_comp],\n    congr' 1,\n    simp only [← cancel_mono (Lcof.map s), ← Lcof.map_comp],\n    congr' 1,\n    exact Cyl.σd₀.trans Cyl.σd₁.symm, },\n  { exact h₁₂.trans h₂₃, },\nend)\n\nlemma π_inverts_weq : cofibrant_object.homotopy_category.weq.is_inverted_by (π Lcof) := λ X Y f hf,\nbegin\n  rcases cofibrant_object.homotopy_category.Q_map_surjective _ _ f with ⟨g, hg⟩,\n  subst hg,\n  simp only [cofibrant_object.homotopy_category.Q_map, quotient.functor_map,\n    π_map, quot.lift_on_mk],\n  apply localization.inverts Lcof cofibrant_object.weq g,\n  exact (cofibrant_object.homotopy_category.weq_Q_map_iff g).mp hf,\nend\n\nomit Lcof\n\nlemma forget_comp_L_inverts_weq :\n  cofibrant_object.weq.is_inverted_by (cofibrant_object.forget C ⋙ L) :=\nλ X Y f hf, by convert localization.inverts L weq ((cofibrant_object.forget C).map f) hf\n\ndef R : C ⥤ Hocof := cofibrant_replacement C ⋙ π Lcof\n\nlemma R_inverts_weq : weq.is_inverted_by (R Lcof) := λ X Y f hf,\nbegin\n  dsimp only [R, functor.comp_map],\n  exact (π_inverts_weq Lcof) _\n    (by simpa only [weq_eq_inverse_image_homotopy_category_weq] using hf),\nend\n\ndef forget_comp_R_iso : cofibrant_object.forget C ⋙ R Lcof ≅ Lcof :=\nnat_iso.of_components (λ X, localization.iso_of_hom' Lcof cofibrant_object.weq (app' X)\n  (by { dsimp [app'], exact weak_eq.property, }))\n(λ X Y f, begin\n  simp only [functor.comp_map, cofibrant_object.forget_map, localization.iso_of_hom'_hom],\n  rw [← Lcof.map_comp, ← fac' f, Lcof.map_comp],\n  refl,\nend)\n\ndef R_comp_I'_iso {I' : Hocof ⥤ Ho} (sq : Comm_sq (cofibrant_object.forget C) Lcof L I') :\n  R Lcof ⋙ I' ≅ L :=\nnat_iso.of_components (λ X, sq.iso.app _ ≪≫\n  localization.iso_of_hom' L model_category.weq (app X) weak_eq.property)\n(λ X Y f, begin\n  simp only [functor.comp_map, iso.trans_hom, iso.app_hom, localization.iso_of_hom'_hom, assoc,\n    ← L.map_comp],\n  simp only [← fac, L.map_comp, ← assoc],\n  congr' 1,\n  apply sq.iso.hom.naturality,\nend)\n\ndef is_equivalence (I' : Hocof ⥤ Ho)\n  (sq : Comm_sq (cofibrant_object.forget C) Lcof L I') : is_equivalence I' :=\nlocalization.lifting_is_equivalence sq cofibrant_object.weq model_category.weq\n  (R Lcof) (localization.lift (R Lcof) (R_inverts_weq Lcof) L)\n  (R_comp_I'_iso Lcof L sq) (forget_comp_R_iso Lcof)\n\nend cofibrant_replacement\n\nvariable {C}\n\ndef Hocof_to_Ho : Hocof ⥤ Ho :=\nlocalization.lift ((cofibrant_object.forget C) ⋙ L)\n  (cofibrant_replacement.forget_comp_L_inverts_weq L) Lcof\n\ndef Lcof_comp_Hocof_to_Ho_iso : Lcof ⋙ Hocof_to_Ho Lcof L ≅ cofibrant_object.forget C ⋙ L :=\nlocalization.fac _ _ _\n\ninstance : is_equivalence (Hocof_to_Ho Lcof L) :=\ncofibrant_replacement.is_equivalence Lcof L (Hocof_to_Ho Lcof L)\n    ⟨Lcof_comp_Hocof_to_Ho_iso Lcof L⟩\n\nlemma is_iso_Lcof_map' {X Y : cofibrant_object C} (f : X ⟶ Y) (hf : cofibrant_object.weq f) :\n  is_iso (Lcof.map f) := localization.inverts Lcof cofibrant_object.weq f hf\n\nlemma is_iso_Lcof_map {X Y : cofibrant_object C} (f : X ⟶ Y)\n  [weak_eq ((cofibrant_object.forget C).map f)] : is_iso (Lcof.map f) :=\nis_iso_Lcof_map' Lcof f weak_eq.property\n\nend model_category\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cofibrant_replacement.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3464639096279216}}
{"text": "open Lean\n\ndef foo : Name := `foo\n\ndef isFoo : Name → Bool\n  | ``foo => true\n  | _     => false\n\ntheorem ex1 : isFoo `foo = true  := rfl\ntheorem ex2 : isFoo `bla = false := rfl\n\ndef Bla.boo : Name := `boo\n\nopen Bla\n\ndef isBoo : Name → Bool\n  | ``boo => true\n  | _     => false\n\ntheorem ex3 : isBoo `Bla.boo = true := rfl\ntheorem ex4 : isBoo ``boo = true := rfl\ntheorem ex5 : ``boo = `Bla.boo := rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/327.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5234203489363239, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.34646390962792156}}
{"text": "import .admits\nimport .misc_list\n\nimport data.finset\n\n\nset_option pp.parens true\n\n\nopen formula\n\n\ninductive is_prop_sub : formula → variable_ → variable_ → formula → Prop\n| true_\n  (v t : variable_) :\n  is_prop_sub true_ v t true_\n\n| pred_\n  (name : pred_name_) (args : list variable_)\n  (v t : variable_) :\n    is_prop_sub (pred_ name args) v t (pred_ name (args.map (fun (x : variable_), if x = v then t else x)))\n\n| eq_\n  (x y : variable_)\n  (v t : variable_) :\n    is_prop_sub (eq_ x y) v t (eq_ (if x = v then t else x) (if y = v then t else y))\n\n| not_\n  (P : formula)\n  (v t : variable_)\n  (P' : formula) :\n  is_prop_sub P v t P' →\n  is_prop_sub P.not_ v t P'.not_\n\n| imp_\n  (P Q : formula)\n  (v t : variable_)\n  (P' Q' : formula) :\n  is_prop_sub P v t P' →\n  is_prop_sub Q v t Q' →\n  is_prop_sub (P.imp_ Q) v t (P'.imp_ Q')\n\n| forall_same_\n  (x : variable_) (P : formula)\n  (v t : variable_) :\n  v = x →\n  is_prop_sub (forall_ x P) v t (forall_ x P)\n\n| forall_diff_nel_\n  (x : variable_) (P : formula)\n  (v t : variable_)\n  (P' : formula) :\n  ¬ v = x →\n  ¬ is_free_in v (forall_ x P) →\n  is_prop_sub P v t P' →\n  is_prop_sub (forall_ x P) v t (forall_ x P')\n\n| forall_diff_\n  (x : variable_) (P : formula)\n  (v t : variable_)\n  (P' : formula) :\n  ¬ v = x →\n  ¬ t = x →\n  is_prop_sub P v t P' →\n  is_prop_sub (forall_ x P) v t (forall_ x P')\n\n\nlemma fast_admits_aux_and_fast_replace_free_imp_is_prop_sub\n  (P P' : formula)\n  (v u : variable_)\n  (binders : finset variable_)\n  (h1 : fast_admits_aux v u binders P)\n  (h2 : fast_replace_free v u P = P') :\n  is_prop_sub P v u P' :=\nbegin\n  subst h2,\n  induction P generalizing binders,\n  case formula.true_ : binders h1\n  {\n    unfold fast_replace_free,\n    apply is_prop_sub.true_,\n  },\n  case formula.pred_ : name args binders h1\n  {\n    unfold fast_replace_free,\n    apply is_prop_sub.pred_,\n  },\n  case formula.eq_ : x y binders h1\n  {\n    unfold fast_replace_free,\n    apply is_prop_sub.eq_,\n  },\n  case formula.not_ : P P_ih binders h1\n  {\n    unfold fast_admits_aux at h1,\n\n    apply is_prop_sub.not_,\n    exact P_ih binders h1,\n  },\n  case formula.imp_ : P Q P_ih Q_ih binders h1\n  {\n    unfold fast_admits_aux at h1,\n    cases h1,\n\n    apply is_prop_sub.imp_,\n    {\n      exact P_ih binders h1_left,\n    },\n    {\n      exact Q_ih binders h1_right,\n    }\n  },\n  case formula.forall_ : x P P_ih binders h1\n  {\n    unfold fast_admits_aux at h1,\n\n    cases h1,\n    {\n      unfold fast_replace_free,\n      split_ifs,\n      apply is_prop_sub.forall_same_ x P v u h1,\n    },\n    {\n      unfold fast_replace_free,\n      split_ifs,\n      {\n        apply is_prop_sub.forall_same_ x P v u h,\n      },\n      {\n        by_cases c1 : is_free_in v (forall_ x P),\n        {\n          apply is_prop_sub.forall_diff_,\n          {\n            exact h,\n          },\n          {\n            unfold is_free_in at c1,\n            cases c1,\n            have s1 : u ∉ binders ∪ {x},\n            apply fast_admits_aux_is_free_in P v u,\n            {\n              exact h1,\n            },\n            {\n              exact c1_right,\n            },\n\n            simp only [finset.mem_union, finset.mem_singleton] at s1,\n            push_neg at s1,\n            cases s1,\n            exact s1_right,\n          },\n          {\n            apply P_ih (binders ∪ {x}),\n            exact h1,\n          }\n        },\n        {\n          apply is_prop_sub.forall_diff_nel_,\n          {\n            exact h,\n          },\n          {\n            exact c1,\n          },\n          {\n            apply P_ih (binders ∪ {x}) h1,\n          },\n        }\n      }\n    }\n  },\nend\n\n\nlemma is_prop_sub_imp_fast_admits_aux\n  (P : formula)\n  (v u : variable_)\n  (binders : finset variable_)\n  (h1 : ∃ (P' : formula), is_prop_sub P v u P')\n  (h2 : u ∉ binders) :\n  fast_admits_aux v u binders P :=\nbegin\n  apply exists.elim h1,\n  intros P' h1_1,\n  clear h1,\n\n  induction h1_1 generalizing binders,\n  case is_prop_sub.true_ : h1_1_v h1_1_t binders h2\n  {\n    unfold fast_admits_aux,\n  },\n  case is_prop_sub.pred_ : h1_1_name h1_1_args h1_1_v h1_1_t binders h2\n  {\n    unfold fast_admits_aux,\n    intros a1,\n    exact h2,\n  },\n  case is_prop_sub.eq_ : h1_1_x h1_1_y h1_1_v h1_1_t binders h2\n  {\n    unfold fast_admits_aux,\n    intros a1,\n    exact h2,\n  },\n  case is_prop_sub.not_ : h1_1_P h1_1_v h1_1_t h1_1_P' h1_1_1 h1_1_ih binders h2\n  {\n    unfold fast_admits_aux,\n    exact h1_1_ih binders h2,\n  },\n  case is_prop_sub.imp_ : h1_1_P h1_1_Q h1_1_v h1_1_t h1_1_P' h1_1_Q' h1_1_1 h1_1_2 h1_1_ih_1 h1_1_ih_2 binders h2\n  {\n    unfold fast_admits_aux,\n    split,\n    {\n      exact h1_1_ih_1 binders h2,\n    },\n    {\n      exact h1_1_ih_2 binders h2,\n    }\n  },\n  case is_prop_sub.forall_same_ : h1_1_x h1_1_P h1_1_v h1_1_t h1_1_1 binders h2\n  {\n    unfold fast_admits_aux,\n    apply or.intro_left,\n    exact h1_1_1,\n  },\n  case is_prop_sub.forall_diff_nel_ : h1_1_x h1_1_P h1_1_v h1_1_t h1_1_P' h1_1_1 h1_1_2 h1_1_3 h1_1_ih binders h2\n  {\n    unfold is_free_in at h1_1_2,\n    push_neg at h1_1_2,\n\n    unfold fast_admits_aux,\n\n    apply or.intro_right,\n    apply not_is_free_in_imp_fast_admits_aux,\n    apply h1_1_2,\n    exact h1_1_1,\n  },\n  case is_prop_sub.forall_diff_ : h1_1_x h1_1_P h1_1_v h1_1_t h1_1_P' h1_1_1 h1_1_2 h1_1_3 h1_1_ih binders h2\n  {\n    unfold fast_admits_aux,\n    apply or.intro_right,\n    apply h1_1_ih,\n    simp only [finset.mem_union, finset.mem_singleton],\n    push_neg,\n    split,\n    {\n      exact h2,\n    },\n    {\n      exact h1_1_2,\n    }\n  },\nend\n\n\nlemma is_prop_sub_imp_fast_replace_free\n  (P P' : formula)\n  (v u : variable_)\n  (h1 : is_prop_sub P v u P') :\n  fast_replace_free v u P = P' :=\nbegin\n  induction h1,\n  case is_prop_sub.true_ : h1_v h1_t\n  {\n    refl,\n  },\n  case is_prop_sub.pred_ : h1_name h1_args h1_v h1_t\n  {\n    unfold fast_replace_free,\n  },\n  case is_prop_sub.eq_ : h1_x h1_y h1_v h1_t\n  {\n    unfold fast_replace_free,\n  },\n  case is_prop_sub.not_ : h1_P h1_v h1_t h1_P' h1_1 h1_ih\n  {\n    unfold fast_replace_free,\n    congr,\n    exact h1_ih,\n  },\n  case is_prop_sub.imp_ : h1_P h1_Q h1_v h1_t h1_P' h1_Q' h1_1 h1_2 h1_ih_1 h1_ih_2\n  {\n    unfold fast_replace_free,\n    congr,\n    {\n      exact h1_ih_1,\n    },\n    {\n      exact h1_ih_2,\n    }\n  },\n  case is_prop_sub.forall_same_ : h1_x h1_P h1_v h1_t h1_1\n  {\n    apply not_free_in_fast_replace_free_id,\n    unfold is_free_in,\n    simp only [finset.mem_sdiff, finset.mem_singleton, not_and, not_not],\n    intros a1,\n    contradiction,\n  },\n  case is_prop_sub.forall_diff_nel_ : h1_x h1_P h1_v h1_t h1_P' h1_1 h1_2 h1_3 h1_ih\n  {\n    unfold fast_replace_free,\n    split_ifs,\n    simp only [eq_self_iff_true, true_and],\n    apply h1_ih,\n  },\n  case is_prop_sub.forall_diff_ : h1_x h1_P h1_v h1_t h1_P' h1_1 h1_2 h1_3 h1_ih\n  {\n    unfold fast_replace_free,\n    split_ifs,\n    simp only [eq_self_iff_true, true_and],\n    apply h1_ih,\n  },\nend\n\n\nexample\n  (P P' : formula)\n  (v u : variable_) :\n  is_prop_sub P v u P' ↔\n    (fast_admits v u P ∧ fast_replace_free v u P = P') :=\nbegin\n  unfold fast_admits,\n  split,\n  {\n    intros a1,\n    split,\n    {\n      apply is_prop_sub_imp_fast_admits_aux,\n      {\n        apply exists.intro P',\n        exact a1,\n      },\n      {\n        simp only [finset.not_mem_empty, not_false_iff],\n      }\n    },\n    {\n      apply is_prop_sub_imp_fast_replace_free,\n      exact a1,\n    }\n  },\n  {\n    intros a1,\n    cases a1,\n    exact fast_admits_aux_and_fast_replace_free_imp_is_prop_sub P P' v u ∅ a1_left a1_right,\n  }\nend\n", "meta": {"author": "pthomas505", "repo": "lean3", "sha": "eb449be2b9a92becda4be38aac76e080194e3f7c", "save_path": "github-repos/lean/pthomas505-lean3", "path": "github-repos/lean/pthomas505-lean3/lean3-eb449be2b9a92becda4be38aac76e080194e3f7c/src/metalogic/fol/sandbox/proper_subst.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3464639026463391}}
{"text": "open classical \nvariables (α : Type) (p q : α -> Prop)\nvariable a : α \nvariable r : Prop\nexample : r → (∃ x : α, r) := \nassume (h1 : r),\nshow ∃ x : α, r, from exists.intro a h1\n", "meta": {"author": "ucmani", "repo": "leanexamples", "sha": "387daef46eaf61bd4a08db076f60ac237daff559", "save_path": "github-repos/lean/ucmani-leanexamples", "path": "github-repos/lean/ucmani-leanexamples/leanexamples-387daef46eaf61bd4a08db076f60ac237daff559/i2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.34646265258178166}}
{"text": "/-\nCopyright (c) 2021 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n\n! This file was ported from Lean 3 source module measure_theory.tactic\n! leanprover-community/mathlib commit d003c55042c3cd08aefd1ae9a42ef89441cdaaf3\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.MeasureTheory.Measure.MeasureSpaceDef\nimport Mathbin.Tactic.AutoCases\nimport Mathbin.Tactic.Tidy\nimport Mathbin.Tactic.WithLocalReducibility\n\n/-!\n# Tactics for measure theory\n\nCurrently we have one domain-specific tactic for measure theory: `measurability`.\n\nThis tactic is to a large extent a copy of the `continuity` tactic by Reid Barton.\n-/\n\n\n/-!\n### `measurability` tactic\n\nAutomatically solve goals of the form `measurable f`, `ae_measurable f μ` and `measurable_set s`.\n\nMark lemmas with `@[measurability]` to add them to the set of lemmas\nused by `measurability`. Note: `to_additive` doesn't know yet how to\ncopy the attribute to the additive version.\n-/\n\n\n/-- User attribute used to mark tactics used by `measurability`. -/\n@[user_attribute]\nunsafe def measurability : user_attribute\n    where\n  Name := `measurability\n  descr := \"lemmas usable to prove (ae)-measurability\"\n#align measurability measurability\n\n/- Mark some measurability lemmas already defined in `measure_theory.measurable_space_def` and\n`measure_theory.measure_space_def` -/\nattribute [measurability]\n  measurable_id measurable_id' aeMeasurableId aeMeasurableId' measurable_const aeMeasurableConst AeMeasurable.measurable_mk MeasurableSet.empty MeasurableSet.univ MeasurableSet.compl Subsingleton.measurableSet MeasurableSet.unionᵢ MeasurableSet.interᵢ MeasurableSet.union MeasurableSet.inter MeasurableSet.diff MeasurableSet.symmDiff MeasurableSet.ite MeasurableSet.cond MeasurableSet.disjointed MeasurableSet.const MeasurableSet.insert measurableSet_eq Finset.measurableSet MeasurableSpace.measurableSet_top\n\nnamespace Tactic\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- Tactic to apply `measurable.comp` when appropriate.\n\nApplying `measurable.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove measurable is actually\n  constant, and that constant is a function application `f z`, then\n  measurable.comp would produce new goals `measurable f`, `measurable\n  (λ _, z)`, which is silly. We avoid this by failing if we could\n  apply `measurable_const`.\n\n* measurable.comp will always succeed on `measurable (λ x, f x)` and\n  produce new goals `measurable (λ x, x)`, `measurable f`. We detect\n  this by failing if a new goal can be closed by applying\n  measurable_id.\n-/\nunsafe def apply_measurable.comp : tactic Unit :=\n  sorry\n#align tactic.apply_measurable.comp tactic.apply_measurable.comp\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- Tactic to apply `measurable.comp_ae_measurable` when appropriate.\n\nApplying `measurable.comp_ae_measurable` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove measurable is actually\n  constant, and that constant is a function application `f z`, then\n  `measurable.comp_ae_measurable` would produce new goals `measurable f`, `ae_measurable\n  (λ _, z) μ`, which is silly. We avoid this by failing if we could\n  apply `ae_measurable_const`.\n\n* `measurable.comp_ae_measurable` will always succeed on `ae_measurable (λ x, f x) μ` and\n  can produce new goals (`measurable (λ x, x)`, `ae_measurable f μ`) or\n  (`measurable f`, `ae_measurable (λ x, x) μ`). We detect those by failing if a new goal can be\n  closed by applying `measurable_id` or `ae_measurable_id`.\n-/\nunsafe def apply_measurable.comp_ae_measurable : tactic Unit :=\n  sorry\n#align tactic.apply_measurable.comp_ae_measurable tactic.apply_measurable.comp_ae_measurable\n\n/--\nWe don't want the intro1 tactic to apply to a goal of the form `measurable f`, `ae_measurable f μ`\nor `measurable_set s`. This tactic tests the target to see if it matches that form.\n -/\nunsafe def goal_is_not_measurable : tactic Unit := do\n  let t ← tactic.target\n  match t with\n    | q(Measurable $(l)) => failed\n    | q(AeMeasurable $(l) $(r)) => failed\n    | q(MeasurableSet $(l)) => failed\n    | _ => skip\n#align tactic.goal_is_not_measurable tactic.goal_is_not_measurable\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- List of tactics used by `measurability` internally. The option `use_exfalso := ff` is passed to\nthe tactic `apply_assumption` in order to avoid loops in the presence of negated hypotheses in\nthe context. -/\nunsafe def measurability_tactics (md : Transparency := semireducible) : List (tactic String) :=\n  [(propositional_goal >> tactic.interactive.apply_assumption none { use_exfalso := false }) >>\n      pure \"apply_assumption {use_exfalso := ff}\",\n    goal_is_not_measurable >> intro1 >>= fun ns => pure (\"intro \" ++ ns.toString),\n    apply_rules [] [`` measurability] 50 { md } >> pure \"apply_rules with measurability\",\n    apply_measurable.comp >> pure \"refine measurable.comp _ _\",\n    apply_measurable.comp_ae_measurable >> pure \"refine measurable.comp_ae_measurable _ _\",\n    sorry >> pure \"refine measurable.ae_measurable _\",\n    sorry >> pure \"refine measurable.ae_strongly_measurable _\"]\n#align tactic.measurability_tactics tactic.measurability_tactics\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- Solve goals of the form `measurable f`, `ae_measurable f μ`, `ae_strongly_measurable f μ` or\n`measurable_set s`. `measurability?` reports back the proof term it found.\n-/\nunsafe def measurability (bang : parse <| optional (tk \"!\")) (trace : parse <| optional (tk \"?\"))\n    (cfg : tidy.cfg := { }) : tactic Unit :=\n  let md := if bang.isSome then semireducible else reducible\n  let measurability_core := tactic.tidy { cfg with tactics := measurability_tactics md }\n  let trace_fn := if trace.isSome then show_term else id\n  trace_fn measurability_core\n#align tactic.interactive.measurability tactic.interactive.measurability\n\n/-- Version of `measurability` for use with auto_param. -/\nunsafe def measurability' : tactic Unit :=\n  measurability none none { }\n#align tactic.interactive.measurability' tactic.interactive.measurability'\n\n/-- `measurability` solves goals of the form `measurable f`, `ae_measurable f μ`,\n`ae_strongly_measurable f μ` or `measurable_set s` by applying lemmas tagged with the\n`measurability` user attribute.\n\nYou can also use `measurability!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`measurability?` reports back the proof term it found.\n-/\nadd_tactic_doc\n  { Name := \"measurability / measurability'\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.measurability, `tactic.interactive.measurability']\n    tags := [\"lemma application\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/MeasureTheory/Tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.34646265258178166}}
{"text": "import tactic.split_ifs\n\nexample (p : Prop) [decidable p] (h : (if p then 1 else 2) > 3) : false :=\nby split_ifs at h; repeat {cases h with h h}\n\nexample (p : Prop) [decidable p] (x : ℕ) (h : (if p then 1 else 2) > x) :\n    x < (if ¬p then 1 else 0) + 1 :=\nby split_ifs at *; assumption\n\nexample (p : Prop) [decidable p] : if if ¬p then p else true then p else ¬p :=\nby split_ifs; assumption\n\nexample (p q : Prop) [decidable p] [decidable q] :\n    if if if p then ¬p else q then p else q then q else ¬p ∨ ¬q :=\nby split_ifs; simp *\n\nexample : true :=\nby success_if_fail { split_ifs }; trivial", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/split_ifs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3464626525817816}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Tim Baumann, Stephen Morgan, Scott Morrison\n\nimport category_theory.fully_faithful\nimport category_theory.functor_category\nimport category_theory.natural_isomorphism\nimport tactic.slice\nimport tactic.converter.interactive\n\nnamespace category_theory\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃ -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nstructure equivalence (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D] :=\n(functor : C ⥤ D)\n(inverse : D ⥤ C)\n(fun_inv_id' : (functor ⋙ inverse) ≅ (category_theory.functor.id C) . obviously)\n(inv_fun_id' : (inverse ⋙ functor) ≅ (category_theory.functor.id D) . obviously)\n\nrestate_axiom equivalence.fun_inv_id'\nrestate_axiom equivalence.inv_fun_id'\n\ninfixr ` ≌ `:10  := equivalence\n\nnamespace equivalence\n\nvariables {C : Type u₁} [𝒞 : category.{v₁} C]\ninclude 𝒞\n\n@[refl] def refl : C ≌ C :=\n{ functor := functor.id C,\n  inverse := functor.id C }\n\nvariables {D : Type u₂} [𝒟 : category.{v₂} D]\ninclude 𝒟\n\n@[symm] def symm (e : C ≌ D) : D ≌ C :=\n{ functor := e.inverse,\n  inverse := e.functor,\n  fun_inv_id' := e.inv_fun_id,\n  inv_fun_id' := e.fun_inv_id }\n\n@[simp] lemma fun_inv_map (e : C ≌ D) (X Y : D) (f : X ⟶ Y) :\ne.functor.map (e.inverse.map f) = (e.inv_fun_id.hom.app X) ≫ f ≫ (e.inv_fun_id.inv.app Y) :=\nbegin\n  erw [nat_iso.naturality_2],\n  refl\nend\n@[simp] lemma inv_fun_map (e : C ≌ D) (X Y : C) (f : X ⟶ Y) :\ne.inverse.map (e.functor.map f) = (e.fun_inv_id.hom.app X) ≫ f ≫ (e.fun_inv_id.inv.app Y) :=\nbegin\n  erw [nat_iso.naturality_2],\n  refl\nend\n\nvariables {E : Type u₃} [ℰ : category.{v₃} E]\ninclude ℰ\n\n@[simp] private def effe_iso_id (e : C ≌ D) (f : D ≌ E) (X : C) :\n  (e.inverse).obj ((f.inverse).obj ((f.functor).obj ((e.functor).obj X))) ≅ X :=\ncalc\n  (e.inverse).obj ((f.inverse).obj ((f.functor).obj ((e.functor).obj X)))\n    ≅ (e.inverse).obj ((e.functor).obj X) : e.inverse.on_iso (nat_iso.app f.fun_inv_id _)\n... ≅ X                                   : nat_iso.app e.fun_inv_id _\n\n@[simp] private def feef_iso_id (e : C ≌ D) (f : D ≌ E) (X : E) :\n  (f.functor).obj ((e.functor).obj ((e.inverse).obj ((f.inverse).obj X))) ≅ X :=\ncalc\n  (f.functor).obj ((e.functor).obj ((e.inverse).obj ((f.inverse).obj X)))\n    ≅ (f.functor).obj ((f.inverse).obj X) : f.functor.on_iso (nat_iso.app e.inv_fun_id _)\n... ≅ X                                   : nat_iso.app f.inv_fun_id _\n\n@[trans] def trans (e : C ≌ D) (f : D ≌ E) : C ≌ E :=\n{ functor := e.functor ⋙ f.functor,\n  inverse := f.inverse ⋙ e.inverse,\n  fun_inv_id' := nat_iso.of_components (effe_iso_id e f)\n  begin\n    /- `tidy` says -/\n    intros X Y f_1, dsimp at *, simp at *, dsimp at *,\n    /- `rewrite_search` says -/\n    slice_lhs 3 4 { erw [is_iso.hom_inv_id] },\n    erw [category.id_comp, is_iso.hom_inv_id, category.comp_id],\n  end,\n  inv_fun_id' := nat_iso.of_components (feef_iso_id e f)\n  begin\n    /- `tidy` says -/\n    intros X Y f_1, dsimp at *, simp at *, dsimp at *,\n    /- `rewrite_search` says -/\n    slice_lhs 3 4 { erw [is_iso.hom_inv_id] },\n    erw [category.id_comp, is_iso.hom_inv_id, category.comp_id]\n  end\n}\n\nend equivalence\n\nvariables {C : Type u₁} [𝒞 : category.{v₁} C]\ninclude 𝒞\n\nsection\nvariables {D : Type u₂} [𝒟 : category.{v₂} D]\ninclude 𝒟\n\nclass is_equivalence (F : C ⥤ D) :=\n(inverse        : D ⥤ C)\n(fun_inv_id' : (F ⋙ inverse) ≅ (functor.id C) . obviously)\n(inv_fun_id' : (inverse ⋙ F) ≅ (functor.id D) . obviously)\n\nrestate_axiom is_equivalence.fun_inv_id'\nrestate_axiom is_equivalence.inv_fun_id'\nend\n\nnamespace is_equivalence\nvariables {D : Type u₂} [𝒟 : category.{v₂} D]\ninclude 𝒟\n\ninstance of_equivalence (F : C ≌ D) : is_equivalence (F.functor) :=\n{ inverse := F.inverse,\n  fun_inv_id' := F.fun_inv_id,\n  inv_fun_id' := F.inv_fun_id }\ninstance of_equivalence_inverse (F : C ≌ D) : is_equivalence (F.inverse) :=\n{ inverse := F.functor,\n  fun_inv_id' := F.inv_fun_id,\n  inv_fun_id' := F.fun_inv_id }\nend is_equivalence\n\nnamespace functor\ninstance is_equivalence_refl : is_equivalence (functor.id C) :=\n{ inverse := functor.id C }\nend functor\n\nvariables {D : Type u₂} [𝒟 : category.{v₂} D]\ninclude 𝒟\n\nnamespace functor\ndef inv (F : C ⥤ D) [is_equivalence F] : D ⥤ C :=\nis_equivalence.inverse F\n\ninstance is_equivalence_symm (F : C ⥤ D) [is_equivalence F] : is_equivalence (F.inv) :=\n{ inverse := F,\n  fun_inv_id' := is_equivalence.inv_fun_id F,\n  inv_fun_id' := is_equivalence.fun_inv_id F }\n\ndef fun_inv_id (F : C ⥤ D) [is_equivalence F] : (F ⋙ F.inv) ≅ functor.id C :=\nis_equivalence.fun_inv_id F\n\ndef inv_fun_id (F : C ⥤ D) [is_equivalence F] : (F.inv ⋙ F) ≅ functor.id D :=\nis_equivalence.inv_fun_id F\n\ndef as_equivalence (F : C ⥤ D) [is_equivalence F] : C ≌ D :=\n{ functor := F,\n  inverse := is_equivalence.inverse F,\n  fun_inv_id' := is_equivalence.fun_inv_id F,\n  inv_fun_id' := is_equivalence.inv_fun_id F }\n\nvariables {E : Type u₃} [ℰ : category.{v₃} E]\ninclude ℰ\n\ninstance is_equivalence_trans (F : C ⥤ D) (G : D ⥤ E) [is_equivalence F] [is_equivalence G] :\n  is_equivalence (F ⋙ G) :=\nis_equivalence.of_equivalence (equivalence.trans (as_equivalence F) (as_equivalence G))\n\nend functor\n\nnamespace is_equivalence\ninstance is_equivalence_functor (e : C ≌ D) : is_equivalence e.functor :=\n{ inverse := e.inverse,\n  fun_inv_id' := e.fun_inv_id,\n  inv_fun_id' := e.inv_fun_id }\ninstance is_equivalence_inverse (e : C ≌ D) : is_equivalence e.inverse :=\n{ inverse := e.functor,\n  fun_inv_id' := e.inv_fun_id,\n  inv_fun_id' := e.fun_inv_id }\n\n@[simp] lemma fun_inv_map (F : C ⥤ D) [is_equivalence F] (X Y : D) (f : X ⟶ Y) :\n  F.map (F.inv.map f) = (F.inv_fun_id.hom.app X) ≫ f ≫ (F.inv_fun_id.inv.app Y) :=\nbegin\n  erw [nat_iso.naturality_2],\n  refl\nend\n@[simp] lemma inv_fun_map (F : C ⥤ D) [is_equivalence F] (X Y : C) (f : X ⟶ Y) :\n  F.inv.map (F.map f) = (F.fun_inv_id.hom.app X) ≫ f ≫ (F.fun_inv_id.inv.app Y) :=\nbegin\n  erw [nat_iso.naturality_2],\n  refl\nend\n\nend is_equivalence\n\nclass ess_surj (F : C ⥤ D) :=\n(obj_preimage (d : D) : C)\n(iso' (d : D) : F.obj (obj_preimage d) ≅ d . obviously)\n\nrestate_axiom ess_surj.iso'\n\nnamespace functor\ndef obj_preimage (F : C ⥤ D) [ess_surj F] (d : D) : C := ess_surj.obj_preimage.{v₁ v₂} F d\ndef fun_obj_preimage_iso (F : C ⥤ D) [ess_surj F] (d : D) : F.obj (F.obj_preimage d) ≅ d :=\ness_surj.iso F d\nend functor\n\nnamespace equivalence\n\ndef ess_surj_of_equivalence (F : C ⥤ D) [is_equivalence F] : ess_surj F :=\n⟨ λ Y : D, F.inv.obj Y, λ Y : D, (nat_iso.app F.inv_fun_id Y) ⟩\n\ninstance faithful_of_equivalence (F : C ⥤ D) [is_equivalence F] : faithful F :=\n{ injectivity' := λ X Y f g w,\n  begin\n    have p := congr_arg (@category_theory.functor.map _ _ _ _ F.inv _ _) w,\n    simp at *,\n    assumption\n  end }.\n\ninstance full_of_equivalence (F : C ⥤ D) [is_equivalence F] : full F :=\n{ preimage := λ X Y f, (nat_iso.app F.fun_inv_id X).inv ≫ (F.inv.map f) ≫ (nat_iso.app F.fun_inv_id Y).hom,\n  witness' := λ X Y f,\n  begin\n    apply F.inv.injectivity,\n    /- obviously can finish from here... -/\n    dsimp, simp, dsimp,\n    slice_lhs 4 6 {\n      rw [←functor.map_comp, ←functor.map_comp],\n      rw [←is_equivalence.fun_inv_map],\n    },\n    slice_lhs 1 2 { simp },\n    dsimp, simp,\n    slice_lhs 2 4 {\n      rw [←functor.map_comp, ←functor.map_comp],\n      erw [nat_iso.naturality_2],\n    },\n    erw [nat_iso.naturality_1],\n    refl,\n  end }.\n\nsection\n\n@[simp] private def equivalence_inverse (F : C ⥤ D) [full F] [faithful F] [ess_surj F] : D ⥤ C :=\n{ obj  := λ X, F.obj_preimage X,\n  map := λ X Y f, F.preimage ((F.fun_obj_preimage_iso X).hom ≫ f ≫ (F.fun_obj_preimage_iso Y).inv),\n  map_id' := λ X, begin apply F.injectivity, tidy, end,\n  map_comp' := λ X Y Z f g, by apply F.injectivity; simp }.\n\ndef equivalence_of_fully_faithfully_ess_surj\n  (F : C ⥤ D) [full F] [faithful : faithful F] [ess_surj F] : is_equivalence F :=\n{ inverse := equivalence_inverse F,\n  fun_inv_id' := nat_iso.of_components\n    (λ X, preimage_iso (F.fun_obj_preimage_iso (F.obj X)))\n    (λ X Y f, begin apply F.injectivity, obviously, end),\n  inv_fun_id' := nat_iso.of_components\n    (λ Y, (F.fun_obj_preimage_iso Y))\n    (by obviously) }\nend\n\nend equivalence\n\nend category_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/equivalence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3464626437779068}}
{"text": "\nimport tactic\n\nuniverses u v\n\nnamespace monad\n\n@[simp]\nlemma bind_pure_star {m} [monad m] [is_lawful_monad m] (x : m punit) :\n  x >>= (λ (_x : punit), pure punit.star : punit → m punit) = x :=\nby { transitivity,\n     { apply congr_arg, ext z, cases z, refl },\n     { simp } }\n\nvariables {α β γ : Type u}\nvariables {m : Type u → Type v} [monad m]\n\n@[reducible]\ndef pipe (a : α → m β) (b : β → m γ) : α → m γ :=\nλ x, a x >>= b\n\ninfixr ` >=> `:55 := pipe\n\n@[functor_norm]\nlemma map_bind_eq_bind_comp {α β γ} {m} [monad m] [is_lawful_monad m]\n  (f : α → β) (cmd : m α) (g : β → m γ) :\n  (f <$> cmd) >>= g = cmd >>= g ∘ f :=\nby rw [← bind_pure_comp_eq_map,bind_assoc,(∘)]; simp\n\n@[functor_norm]\nlemma bind_map {α β γ} {m} [monad m] [is_lawful_monad m]\n  (f : α → γ → β) (cmd : m α) (g : α → m γ) :\n  cmd >>= (λ x, f x <$> g x) = do { x ← cmd, y ← g x, pure $ f x y }  :=\nby congr; ext; rw [← bind_pure (g x),map_bind]; simp\n\n@[functor_norm]\nlemma bind_seq {α β γ : Type u} {m} [monad m] [is_lawful_monad m]\n  (f : α → m (γ → β)) (cmd : m α) (g : α → m γ) :\n  cmd >>= (λ x, f x <*> g x) = do { x ← cmd, h ← f x, y ← g x, pure $ h y }  :=\nby congr; ext; simp [seq_eq_bind_map] with functor_norm\n\nend monad\n\nattribute [functor_norm] bind_assoc has_bind.and_then map_bind\n", "meta": {"author": "cipher1024", "repo": "serialean", "sha": "47881e4a6bc0a62cd68520564610b75f8a4fef2c", "save_path": "github-repos/lean/cipher1024-serialean", "path": "github-repos/lean/cipher1024-serialean/serialean-47881e4a6bc0a62cd68520564610b75f8a4fef2c/src/category/serial.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34632099156736806}}
{"text": "import .nnf .predicates\n\nvariables {α β : Type}\n\ndef lift_nnf_qe (β) [Hα : atom_type α β] (qe : fm α → fm α) : fm α → fm α \n| (p ∧' q) := and_o (lift_nnf_qe p) (lift_nnf_qe q)\n| (p ∨' q) := or_o  (lift_nnf_qe p) (lift_nnf_qe q)\n| (¬' p) := not_o (lift_nnf_qe p)\n| (∃' p) := qe (@nnf α β Hα (lift_nnf_qe p))\n| p := p\n\nlemma and_o_prsv_qfree (p q : fm α) : \n  qfree p → qfree q → qfree (and_o p q) := \nλ Hp Hq, \n  begin \n    cases p; cases q; \n    {trivial <|> apply Hp <|> apply Hq <|> {apply (and.intro Hp Hq)}}\n  end\n\nlemma or_o_prsv_qfree (p q : fm α) : \n  qfree p → qfree q → qfree (or_o p q) := \nλ Hp Hq, \n  begin \n    cases p; cases q; \n    {trivial <|> apply Hp <|> apply Hq <|> {apply (and.intro Hp Hq)}}\n  end\n\nlemma not_o_prsv_qfree (p : fm α) (H : qfree p) : qfree (not_o p) := \nbegin cases p; {trivial <|> apply H} end\n\nlemma lnq_qfree [Hα : atom_type α β] (qe : fm α → fm α) \n(H : ∀ (f : fm α), nqfree f → qfree (qe f)) : \n  ∀ (f : fm α), qfree (@lift_nnf_qe α β Hα qe f) :=\nλ f, fm.rec_on f trivial trivial \n  (λ a, trivial)\n  (λ f1 f2 H1 H2, and_o_prsv_qfree _ _ H1 H2)\n  (λ f1 f2 H1 H2, or_o_prsv_qfree _ _ H1 H2)\n  (λ f1 H1, not_o_prsv_qfree _ H1)\n  (λ f1 H1, begin apply H, apply nnf_nqfree, apply H1 end)\n\nopen tactic\n\nlemma lnq_prsv_gen \n  [Hα : atom_type α β] (qe : fm α → fm α) \n  (hqe : qfree_res_of_nqfree_arg qe) \n  (r : fm α → Prop) \n  (hdc : down_closed r)\n  (hqc : preserves qe r)\n  (hnc : preserves (nnf β) r)\n  (hlc : ∀ f, preserves f r → preserves (lift_nnf_qe β f) r)\n  (hi : ∀ (p : fm α), nqfree p → r p → ∀ (xs : list β), I (qe p) xs ↔ ∃ x, I p (x::xs)) : \n  ∀ (p : fm α), r p → ∀ (xs : list β), I (lift_nnf_qe β qe p) xs ↔ I p xs \n| ⊤' _ bs := iff.refl _\n| ⊥' _ bs := iff.refl _\n| (A' a) _ bs := iff.refl _\n| (p ∧' q) hr xs := \n  begin \n    unfold lift_nnf_qe, rewrite exp_I_and, \n    rewrite exp_I_and_o, \n    repeat {rewrite lnq_prsv_gen}, \n    apply hdc _ _ (down.andr _ _) hr, \n    apply hdc _ _ (down.andl _ _) hr \n  end \n| (p ∨' q) hr xs := \n  begin \n    unfold lift_nnf_qe, rewrite exp_I_or, \n    rewrite exp_I_or_o, repeat {rewrite lnq_prsv_gen},\n    apply hdc _ _ (down.orr _ _) hr, \n    apply hdc _ _ (down.orl _ _) hr \n  end \n| (¬' p) hr xs := \n  begin \n    unfold lift_nnf_qe, rewrite exp_I_not,  \n    rewrite exp_I_not_o, repeat {rewrite lnq_prsv_gen},\n    apply hdc _ _ (down.not _) hr \n  end\n| (∃' p) hr xs := \n  begin \n    unfold lift_nnf_qe, rewrite exp_I_ex,\n    rewrite hi, apply ex_iff_ex, intro b, \n    rewrite nnf_prsv, apply lnq_prsv_gen, \n    apply hdc _ _ (down.ex _) hr, \n    apply lnq_qfree, apply hqe, \n    apply nnf_nqfree, apply lnq_qfree,\n    apply hqe, apply hnc, apply hlc, apply hqc, \n    apply hdc _ _ (down.ex _) hr \n  end\n\nlemma lnq_prsv [Hα : atom_type α β] (qe : fm α → fm α) \n  (hqe : qfree_res_of_nqfree_arg qe) \n  (hi : ∀ (p : fm α), nqfree p → ∀ (xs : list β), I (qe p) xs ↔ ∃ x, I p (x::xs)) : \n  ∀ (p : fm α) (xs : list β), I (lift_nnf_qe β qe p) xs ↔ I p xs :=\nbegin\n  intro p, apply (lnq_prsv_gen _ _ (λ _, true)),\n  intros _ _ _ _, trivial, \n  intros _ _,  trivial, \n  intros _ _,  trivial, \n  intros _ _ _ _, trivial, \n  intros q hf h', apply hi, \n  apply hf, trivial, apply hqe\nend\n", "meta": {"author": "avigad", "repo": "qelim", "sha": "b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60", "save_path": "github-repos/lean/avigad-qelim", "path": "github-repos/lean/avigad-qelim/qelim-b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60/common/lnq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34632098522277943}}
{"text": "import for_mathlib.homological_complex_abelian\nimport category_theory.limits.preserves.finite\nimport category_theory.limits.preserves.shapes.kernels\nimport for_mathlib.homology\n\n.\n\nopen category_theory category_theory.limits\n\nuniverses v u₁ u₂\n\nvariables {C : Type u₁} [category.{v} C] [abelian C]\nvariables {D : Type u₂} [category.{v} D] [abelian D] (F : C ⥤ D)\nvariables [preserves_finite_limits F] [preserves_finite_colimits F] [functor.additive F]\nvariables {ι : Type*} {c : complex_shape ι}\n\n@[simps] noncomputable\ndef category_theory.limits.cokernel.map_arrow_iso {C : Type u₁} [category.{v} C]\n  [has_zero_morphisms C] {X₁ Y₁ X₂ Y₂ : C}\n  (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) [has_cokernel f₁] [has_cokernel f₂]\n  (e : arrow.mk f₁ ≅ arrow.mk f₂) : cokernel f₁ ≅ cokernel f₂ :=\n{ hom := cokernel.map _ _ e.hom.left e.hom.right e.hom.w.symm,\n  inv := cokernel.map _ _ e.inv.left e.inv.right e.inv.w.symm,\n  hom_inv_id' := begin\n    ext,\n    simp only [category.comp_id, cokernel.π_desc, cokernel.π_desc_assoc,\n      category.assoc, coequalizer_as_cokernel],\n    rw [← category.assoc, ← comma.comp_right, e.hom_inv_id, arrow.id_right],\n    exact category.id_comp _\n  end,\n  inv_hom_id' := begin\n    ext,\n    simp only [category.comp_id, cokernel.π_desc, cokernel.π_desc_assoc,\n      category.assoc, coequalizer_as_cokernel],\n    rw [← category.assoc, ← comma.comp_right, e.inv_hom_id, arrow.id_right],\n    exact category.id_comp _\n  end }\n\n@[simps] noncomputable\ndef category_theory.limits.kernel.map_arrow_iso {C : Type u₁} [category.{v} C]\n  [has_zero_morphisms C] {X₁ Y₁ X₂ Y₂ : C}\n  (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) [has_kernel f₁] [has_kernel f₂]\n  (e : arrow.mk f₁ ≅ arrow.mk f₂) : kernel f₁ ≅ kernel f₂ :=\n{ hom := kernel.map _ _ e.hom.left e.hom.right e.hom.w.symm,\n  inv := kernel.map _ _ e.inv.left e.inv.right e.inv.w.symm,\n  hom_inv_id' := begin\n    ext,\n    simp only [category.id_comp, kernel.lift_ι, kernel.lift_ι_assoc,\n      category.assoc, equalizer_as_kernel],\n    rw [← comma.comp_left, e.hom_inv_id, arrow.id_left],\n    exact category.comp_id _\n  end,\n  inv_hom_id' := begin\n    ext,\n    simp only [category.id_comp, kernel.lift_ι, kernel.lift_ι_assoc,\n      category.assoc, equalizer_as_kernel],\n    rw [← comma.comp_left, e.inv_hom_id, arrow.id_left],\n    exact category.comp_id _\n  end }\n\nnoncomputable\ndef category_theory.functor.map_homological_complex_X_prev (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) (i : ι) :\n  ((F.map_homological_complex c).obj X).X_prev i ≅ F.obj (X.X_prev i) :=\niso.refl _\n\nlemma category_theory.functor.map_homological_complex_X_prev_eq (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) {i : ι} :\n  F.map_homological_complex_X_prev X i = iso.refl _ := rfl\n\nnoncomputable\ndef category_theory.functor.map_homological_complex_X_next (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) (i : ι) :\n  ((F.map_homological_complex c).obj X).X_next i ≅ F.obj (X.X_next i) :=\niso.refl _\n\nlemma category_theory.functor.map_homological_complex_X_next_eq (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) {i : ι} :\n  F.map_homological_complex_X_next X i = iso.refl _ := rfl\n\nlemma category_theory.functor.map_homological_complex_d_from (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) (i : ι) :\n  ((F.map_homological_complex c).obj X).d_from i = F.map (X.d_from i) := rfl\n\nlemma category_theory.functor.map_homological_complex_d_to (F : C ⥤ D) [F.additive]\n  (X : homological_complex C c) (i : ι) :\n  ((F.map_homological_complex c).obj X).d_to i = F.map (X.d_to i) := rfl\n\n@[simp, reassoc]\nlemma category_theory.limits.preserves_kernel_iso_inv_map (F : C ⥤ D)\n  [F.additive] [preserves_finite_limits F]\n  {X₁ Y₁ X₂ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (iX : X₁ ⟶ X₂) (iY : Y₁ ⟶ Y₂) (e) :\n  (preserves_kernel.iso F f₁).inv ≫ F.map (kernel.map _ _ iX iY e) =\n    kernel.map _ _ (F.map iX) (F.map iY) (by simp_rw [← F.map_comp, e]) ≫\n      (preserves_kernel.iso F f₂).inv :=\nbegin\n  rw [iso.eq_comp_inv, category.assoc, eq_comm, iso.eq_inv_comp],\n  ext,\n  simp,\nend\n-- attribute [reassoc] kernel.lift_map\n\nopen category_theory category_theory.limits\n\n@[simp, reassoc]\nlemma homology.π'_iso_cokernel_lift_hom {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :\n  homology.π' f g w ≫ (homology_iso_cokernel_lift f g w).hom = cokernel.π _ :=\nbegin\n  delta homology.π', simp,\nend\n\n@[reassoc]\nlemma homology.π'_iso_cokernel_lift_inv {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :\n  cokernel.π _ ≫ (homology_iso_cokernel_lift f g w).inv = homology.π' f g w := rfl\n\nnoncomputable\ndef preserves_homology_of_exact (i : ι) :\n  F.map_homological_complex c ⋙ homology_functor _ c i ≅ homology_functor _ c i ⋙ F :=\nbegin\n  fapply nat_iso.of_components,\n  { intro X,\n    refine homology_iso_cokernel_lift _ _ _ ≪≫\n      cokernel.map_arrow_iso _ _ (arrow.iso_mk _ _ _) ≪≫\n      (preserves_cokernel.iso _ _).symm ≪≫ F.map_iso (homology_iso_cokernel_lift _ _ _).symm,\n    { exact iso.refl _ },\n    { refine kernel.map_arrow_iso _ _ (arrow.iso_mk _ _ _) ≪≫ (preserves_kernel.iso _ _).symm,\n      { exact iso.refl _ },\n      { exact iso.refl _ },\n      { dsimp, rw [category.id_comp, F.map_homological_complex_d_from, category.comp_id] } },\n    { dsimp, rw [← category.assoc, iso.eq_comp_inv], ext, simp [F.map_homological_complex_d_to] } },\n  { intros X Y f, apply homology.hom_from_ext,\n    simp only [category_theory.limits.kernel.map_arrow_iso_hom,\n      category_theory.functor.map_iso_inv,\n      category_theory.functor.map_homological_complex_map_f,\n      homological_complex.hom.sq_from_right,\n      category_theory.iso.refl_hom,\n      homology.π'_iso_cokernel_lift_hom_assoc,\n      category_theory.arrow.iso_mk_hom_right,\n      category_theory.arrow.iso_mk_hom_left,\n      category_theory.limits.cokernel.map_arrow_iso_hom,\n      category_theory.functor.comp_map,\n      homology_functor_map,\n      category_theory.iso.symm_hom,\n      category_theory.limits.cokernel.π_desc_assoc,\n      category_theory.limits.preserves_cokernel.iso_inv,\n      category_theory.functor.map_iso_symm,\n      category_theory.category.assoc,\n      category_theory.iso.trans_hom,\n      homological_complex.hom.sq_to_right,\n      category_theory.limits.π_comp_cokernel_comparison_assoc,\n      homology.π'_map_assoc],\n    simp only [← functor.map_comp],\n    rw [homology.π'_iso_cokernel_lift_inv_assoc, homology.π'_iso_cokernel_lift_inv, homology.π'_map,\n      F.map_comp, preserves_kernel_iso_inv_map_assoc],\n    simp only [← category.assoc],\n    congr' 2,\n    ext,\n    simp only [category_theory.limits.equalizer_as_kernel,\n      homological_complex.hom.sq_from_right,\n      category_theory.category.assoc,\n      category_theory.limits.kernel.lift_ι,\n      homological_complex.hom.sq_to_right,\n      category_theory.limits.kernel.lift_ι_assoc],\n    dsimp,\n    rw [category.id_comp, category.comp_id] }\nend\n.\n\n-- def preserves_limit_of_lim_commute (J : Type*) [category J] [has_limits_of_shape J C]\n--   [has_limits_of_shape J D]  (F : C ⥤ D) (e : lim ⋙ F ≅ (whiskering_right J C D).obj F ⋙ lim):\n--     preserves_limits_of_shape J F :=\n-- begin\n--   refine ⟨λ K, preserves_limit_of_preserves_limit_cone (limit.is_limit _) _⟩,\n--   apply is_limit.of_iso_limit (limit.is_limit _),\n--   ext, swap,\n--   { exact e.symm.app K },\n--   { dsimp,\n--     have := congr_arg (λ f, f ≫ limit.π _ j) (e.hom.naturality (limit.cone K).π),\n--     dsimp at this,\n--     simp only [whisker_right_app, limit.cone_π, lim_map_π, category.assoc] at this,\n--     let : limit K ≅ limit ((category_theory.functor.const J).obj (limit K)),\n--     { refine ⟨limit.lift _ ⟨_, 𝟙 _⟩, lim_map (limit.cone K).π, _, _⟩,\n--       { ext, simp },\n--       { ext, simp, dsimp, simp,  } }  }\n--   -- have := cone.iso_mk\n-- end\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/preserves_limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3462784765937602}}
{"text": "import analysis.topology.topological_space\nimport analysis.topology.continuity\nimport analysis.topology.topological_structures\nimport ex2\n\nopen tactic expr function has_add\nvariables {X : Type*} {Y : Type*} {Z : Type*} {W : Type*}\n          {f f₁ f₂ : X → Y} {g : Y → Z} {h : Z → W}\nvariables [topological_space X] [topological_space Y]\n          [topological_space Z] [topological_space W]\n\nexample [add_monoid X] [topological_add_monoid X]:\n  (λ x : X, x + x) = (uncurry add) ∘ (λ x : X, (x, x)) :=\nby reflexivity\n\n#check @has_add.add .\n\n-- Some preparation is necessary, has_add.add has 4 (in words four) arguments\n-- first count the non-implicit (default) args, then find them\n\nmeta def get_app_num_default_args_aux : expr → nat\n| (pi _ binder_info.default _ bbody) := 1 + (get_app_num_default_args_aux bbody) \n| (pi _ _ _ bbody) := get_app_num_default_args_aux bbody\n| _ :=  0\n\n\nmeta def get_app_num_default_args : expr → tactic nat :=\nassume e,\ndo type ← infer_type e, -- the type of a function is a pi type, we inspect the binder_info recursively\n  return $ get_app_num_default_args_aux type\n\nmeta def ith_default_arg_index_aux : expr → nat → tactic nat \n| (expr.pi _ binder_info.default _ _) 0 := return 0\n| (expr.pi _ binder_info.default _ bbody) (i + 1) := do idx ← ith_default_arg_index_aux bbody i, return $ idx + 1\n| (expr.pi _ _ _ bbody) i := do idx ← ith_default_arg_index_aux bbody i, return $ idx + 1\n| e i := do fail $ \"can't find \" ++ to_string i ++ \"th default argument index\"\n\n\nmeta def ith_default_arg_index (e : expr) (i : nat) : tactic nat :=\ndo type ← infer_type e,\n   index ← ith_default_arg_index_aux type i,\n   return index\n\nmeta def get_codomain (e : expr) : tactic expr :=\ndo t ← infer_type e,\nmatch t with\n| `(%%domain → %%codomain) := return codomain\n| _ := fail $ \"Can't determine codomain, '\" ++ to_string t ++ \"' is not a function type.\"\nend\n\n\n--now we can 'uncurry' and move λ inside\n\n/- Push a given lambda expression inside in the goal:\n -\n - Takes an expression of the form 'λ x, f _[x]',\n - creates a hypothesis 'λ x, f _[x] = f ∘ (λ x, _[x])',\n - and rewrites the main goal using this hypothesis.\n -\n - Tries to uncurry if there is a curried function with two arguments.\n -\n - Fails if it 'doesn't simplify'.\n -/\nmeta def push_lam_inside2 (e : expr): tactic unit :=\n--do trace $ to_raw_fmt e,\ndo lam name binfo bdomain bbody ← return e\n       <|> (fail \"The given expression must be a λ.\"),\n  if ¬ is_app bbody\n  then fail $ \"The outer most part of the lambda expression '\" ++ (to_string e) ++ \n              \"' is not a function application.\"\n  else do\n    -- we use fancier versions instead of simple pattern matching\n    -- because there are many curried arguments now\n    fn ← return $ get_app_fn bbody,\n    arg ← return $ app_arg bbody,\n    num_default_args ← get_app_num_default_args fn,\n\n    -- uncurry if necessary, then push λ inside\n    match num_default_args with\n    | 2 := do -- there are two default arguments, uncurry and push lambda inside\n\n      -- get first argument and argument type\n      arg0_idx ← ith_default_arg_index fn 0,\n      arg0 ← return $ ith_arg bbody arg0_idx,\n      arg0_type ← get_codomain (expr.lam name binfo bdomain arg0),\n\n      -- get second argument and argument type\n      arg1_idx ← ith_default_arg_index fn 1,\n      arg1 ← return $ ith_arg bbody arg1_idx,\n      arg1_type ← get_codomain (expr.lam name binfo bdomain arg1),\n\n      -- create the inner expression, i.e. λ x, (arg0[x], arg1[x])\n      inner_body_app ← to_expr ``(@prod.mk %%arg0_type %%arg1_type),\n      inner ← return $ expr.lam name binfo bdomain\n                       (expr.app (expr.app inner_body_app arg0) arg1),\n\n      -- create the outer expression, i.e. (uncurry f)\n      expr.const fn_name _ ← return fn,\n      outer ← return $ @expr.const ff fn_name [],\n\n      -- create term 'original = outer ∘ inner', need to hint types here\n      -- i.e. need to use @comp instead of $∘\n      can_uncurry ← to_expr ``(%%e = @comp %%bdomain (%%arg0_type × %%arg1_type) _ \n                                           (uncurry %%outer) %%inner),            \n\n      -- create hypothesis and prove it by reflexivity\n      h_can_uncurry ← assert \"can_uncurry\" can_uncurry,\n      reflexivity <|> rotate 1,\n\n      -- rewrite target\n      rewrite_target h_can_uncurry,\n      to_expr ``(function.uncurry_def) >>= rewrite_target,\n           \n      -- fail if the outer or inner part is the same as the original expression\n      `(continuous $ %%outer ∘ %%inner) ← target,\n      (success_if_fail $ unify e inner) <|> fail \"inner not changed!\",\n      (success_if_fail $ unify e outer) <|> fail \"outer not changed!\",\n\n      -- done!\n      return ()\n\n    | 1 := do -- same as before\n      -- create hypothesis\n      can_push_inside ← to_expr ``(%%e = %%fn ∘ %%(expr.lam name binfo bdomain arg)),   \n      -- prove hypothesis\n      h_can_push_inside ← assert \"can_push_λ_inside\" can_push_inside,\n      reflexivity <|> fail \"Can't prove that λ can be pushed inside by reflexivity.\",\n      -- rewrite target\n      rewrite_target h_can_push_inside,\n      return ()\n    | _ := fail \"uncurrying more than two arguements is not implemented\"\n    end\n\nexample [add_monoid X] [topological_add_monoid X] : (continuous (λ x : X, x + x)) := \nbegin\n  (do `(continuous %%e) ← target,\n    push_lam_inside2 e,\n    return ()),\n  contcomp2,\n  exact topological_add_monoid.continuous_add X,\n  sorry\nend\n\nexample [add_monoid X] [topological_add_monoid X] : (continuous (λ x : X, (x, x))) := \nby sorry\n\n\n-- Same as contcomp2, bu uses push_lam_inside2\nmeta def contcomp3 : tactic unit :=\ndo `(continuous %%e) ← target <|> fail \"goal has to be of the form 'continuous _'\",\n\n  -- if the goal is to show that the expression e is continuous, check if it is an assumption\n  assumption <|>\n  \n  -- if e is not continuous by assumption, check if it is of the form 'f ∘ g' for some f and g\n  match e with\n  | `(%%outer ∘ %%inner) :=\n    do    \n    -- assert that the outer function is continuous, if not, rotate the goal away\n    outer_continuous ← to_expr ``(continuous %%outer),\n    outer_is_continuous ← assert \"h_outer_cont\" outer_continuous,         \n    contcomp <|> rotate 1,\n\n    -- assert that the inner function is continuous, if not, rotate the goal away\n    inner_continuous ← to_expr ``(continuous %%inner),\n    inner_is_continuous ← assert \"h_inner_cont\" inner_continuous,\n    contcomp <|> rotate 1,\n\n    -- knowing that inner and outer function are continuous, apply the theorem continuous.comp\n    -- which shows that the composite is continuous\n    cont_comp ← to_expr ``(continuous.comp %%inner_is_continuous %%outer_is_continuous),\n    exact cont_comp\n  | `(λ _, _) :=\n    -- if we can move the lambda inside, to that and apply contcomp2 recursively\n    do push_lam_inside2 e <|> fail \"can't move lambda inside\",\n       contcomp3\n  | _ := fail \"neither composite nor lambda expression\"\n  end\n\nexample [add_monoid X] [topological_add_monoid X] : (continuous (λ x : X, x + x)) := \nbegin\n  contcomp3,\n  exact topological_add_monoid.continuous_add X,\n  sorry\nend\n\nexample [add_monoid X] [topological_add_monoid X] : (continuous (λ x : X, (x, x))) := \nby sorry\n", "meta": {"author": "jmvlangen", "repo": "RIP-seminar", "sha": "ed6771404dd4bcec298de2dfdc89d5e9cfd331bb", "save_path": "github-repos/lean/jmvlangen-RIP-seminar", "path": "github-repos/lean/jmvlangen-RIP-seminar/RIP-seminar-ed6771404dd4bcec298de2dfdc89d5e9cfd331bb/src/compcont/ex3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.3462784603982489}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.order_functions\nimport Mathlib.control.monad.basic\nimport Mathlib.data.nat.choose.basic\nimport Mathlib.order.rel_classes\nimport Mathlib.PostPort\n\nuniverses u v w u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Basic properties of lists\n-/\n\nnamespace list\n\n\nprotected instance nil.is_left_id {α : Type u} : is_left_id (List α) append [] :=\n  is_left_id.mk nil_append\n\nprotected instance nil.is_right_id {α : Type u} : is_right_id (List α) append [] :=\n  is_right_id.mk append_nil\n\nprotected instance has_append.append.is_associative {α : Type u} : is_associative (List α) append :=\n  is_associative.mk append_assoc\n\ntheorem cons_ne_nil {α : Type u} (a : α) (l : List α) : a :: l ≠ [] :=\n  fun (ᾰ : a :: l = []) => eq.dcases_on ᾰ (fun (H_1 : [] = a :: l) => list.no_confusion H_1) (Eq.refl []) (HEq.refl ᾰ)\n\ntheorem cons_ne_self {α : Type u} (a : α) (l : List α) : a :: l ≠ l :=\n  mt (congr_arg length) (nat.succ_ne_self (length l))\n\ntheorem head_eq_of_cons_eq {α : Type u} {h₁ : α} {h₂ : α} {t₁ : List α} {t₂ : List α} : h₁ :: t₁ = h₂ :: t₂ → h₁ = h₂ :=\n  fun (Peq : h₁ :: t₁ = h₂ :: t₂) => list.no_confusion Peq fun (Pheq : h₁ = h₂) (Pteq : t₁ = t₂) => Pheq\n\ntheorem tail_eq_of_cons_eq {α : Type u} {h₁ : α} {h₂ : α} {t₁ : List α} {t₂ : List α} : h₁ :: t₁ = h₂ :: t₂ → t₁ = t₂ :=\n  fun (Peq : h₁ :: t₁ = h₂ :: t₂) => list.no_confusion Peq fun (Pheq : h₁ = h₂) (Pteq : t₁ = t₂) => Pteq\n\n@[simp] theorem cons_injective {α : Type u} {a : α} : function.injective (List.cons a) :=\n  fun (l₁ l₂ : List α) (Pe : a :: l₁ = a :: l₂) => tail_eq_of_cons_eq Pe\n\ntheorem cons_inj {α : Type u} (a : α) {l : List α} {l' : List α} : a :: l = a :: l' ↔ l = l' :=\n  function.injective.eq_iff cons_injective\n\ntheorem exists_cons_of_ne_nil {α : Type u} {l : List α} (h : l ≠ []) : ∃ (b : α), ∃ (L : List α), l = b :: L := sorry\n\n/-! ### mem -/\n\ntheorem mem_singleton_self {α : Type u} (a : α) : a ∈ [a] :=\n  mem_cons_self a []\n\ntheorem eq_of_mem_singleton {α : Type u} {a : α} {b : α} : a ∈ [b] → a = b :=\n  fun (this : a ∈ [b]) =>\n    or.elim (eq_or_mem_of_mem_cons this) (fun (this : a = b) => this) fun (this : a ∈ []) => absurd this (not_mem_nil a)\n\n@[simp] theorem mem_singleton {α : Type u} {a : α} {b : α} : a ∈ [b] ↔ a = b :=\n  { mp := eq_of_mem_singleton, mpr := Or.inl }\n\ntheorem mem_of_mem_cons_of_mem {α : Type u} {a : α} {b : α} {l : List α} : a ∈ b :: l → b ∈ l → a ∈ l := sorry\n\ntheorem eq_or_ne_mem_of_mem {α : Type u} {a : α} {b : α} {l : List α} (h : a ∈ b :: l) : a = b ∨ a ≠ b ∧ a ∈ l :=\n  classical.by_cases Or.inl\n    fun (this : a ≠ b) => or.elim h Or.inl fun (h : list.mem a l) => Or.inr { left := this, right := h }\n\ntheorem not_mem_append {α : Type u} {a : α} {s : List α} {t : List α} (h₁ : ¬a ∈ s) (h₂ : ¬a ∈ t) : ¬a ∈ s ++ t :=\n  mt (iff.mp mem_append) (iff.mpr not_or_distrib { left := h₁, right := h₂ })\n\ntheorem ne_nil_of_mem {α : Type u} {a : α} {l : List α} (h : a ∈ l) : l ≠ [] :=\n  id fun (e : l = []) => false.dcases_on (fun (h : a ∈ []) => False) (eq.mp (Eq._oldrec (Eq.refl (a ∈ l)) e) h)\n\ntheorem mem_split {α : Type u} {a : α} {l : List α} (h : a ∈ l) : ∃ (s : List α), ∃ (t : List α), l = s ++ a :: t := sorry\n\ntheorem mem_of_ne_of_mem {α : Type u} {a : α} {y : α} {l : List α} (h₁ : a ≠ y) (h₂ : a ∈ y :: l) : a ∈ l :=\n  or.elim (eq_or_mem_of_mem_cons h₂) (fun (e : a = y) => absurd e h₁) fun (r : a ∈ l) => r\n\ntheorem ne_of_not_mem_cons {α : Type u} {a : α} {b : α} {l : List α} : ¬a ∈ b :: l → a ≠ b :=\n  fun (nin : ¬a ∈ b :: l) (aeqb : a = b) => absurd (Or.inl aeqb) nin\n\ntheorem not_mem_of_not_mem_cons {α : Type u} {a : α} {b : α} {l : List α} : ¬a ∈ b :: l → ¬a ∈ l :=\n  fun (nin : ¬a ∈ b :: l) (nainl : a ∈ l) => absurd (Or.inr nainl) nin\n\ntheorem not_mem_cons_of_ne_of_not_mem {α : Type u} {a : α} {y : α} {l : List α} : a ≠ y → ¬a ∈ l → ¬a ∈ y :: l :=\n  fun (p1 : a ≠ y) (p2 : ¬a ∈ l) =>\n    not.intro fun (Pain : a ∈ y :: l) => absurd (eq_or_mem_of_mem_cons Pain) (not_or p1 p2)\n\ntheorem ne_and_not_mem_of_not_mem_cons {α : Type u} {a : α} {y : α} {l : List α} : ¬a ∈ y :: l → a ≠ y ∧ ¬a ∈ l :=\n  fun (p : ¬a ∈ y :: l) => { left := ne_of_not_mem_cons p, right := not_mem_of_not_mem_cons p }\n\ntheorem mem_map_of_mem {α : Type u} {β : Type v} (f : α → β) {a : α} {l : List α} (h : a ∈ l) : f a ∈ map f l := sorry\n\ntheorem exists_of_mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {l : List α} (h : b ∈ map f l) : ∃ (a : α), a ∈ l ∧ f a = b := sorry\n\n@[simp] theorem mem_map {α : Type u} {β : Type v} {f : α → β} {b : β} {l : List α} : b ∈ map f l ↔ ∃ (a : α), a ∈ l ∧ f a = b := sorry\n\ntheorem mem_map_of_injective {α : Type u} {β : Type v} {f : α → β} (H : function.injective f) {a : α} {l : List α} : f a ∈ map f l ↔ a ∈ l := sorry\n\ntheorem forall_mem_map_iff {α : Type u} {β : Type v} {f : α → β} {l : List α} {P : β → Prop} : (∀ (i : β), i ∈ map f l → P i) ↔ ∀ (j : α), j ∈ l → P (f j) := sorry\n\n@[simp] theorem map_eq_nil {α : Type u} {β : Type v} {f : α → β} {l : List α} : map f l = [] ↔ l = [] := sorry\n\n@[simp] theorem mem_join {α : Type u} {a : α} {L : List (List α)} : a ∈ join L ↔ ∃ (l : List α), l ∈ L ∧ a ∈ l := sorry\n\ntheorem exists_of_mem_join {α : Type u} {a : α} {L : List (List α)} : a ∈ join L → ∃ (l : List α), l ∈ L ∧ a ∈ l :=\n  iff.mp mem_join\n\ntheorem mem_join_of_mem {α : Type u} {a : α} {L : List (List α)} {l : List α} (lL : l ∈ L) (al : a ∈ l) : a ∈ join L :=\n  iff.mpr mem_join (Exists.intro l { left := lL, right := al })\n\n@[simp] theorem mem_bind {α : Type u} {β : Type v} {b : β} {l : List α} {f : α → List β} : b ∈ list.bind l f ↔ ∃ (a : α), ∃ (H : a ∈ l), b ∈ f a := sorry\n\ntheorem exists_of_mem_bind {α : Type u} {β : Type v} {b : β} {l : List α} {f : α → List β} : b ∈ list.bind l f → ∃ (a : α), ∃ (H : a ∈ l), b ∈ f a :=\n  iff.mp mem_bind\n\ntheorem mem_bind_of_mem {α : Type u} {β : Type v} {b : β} {l : List α} {f : α → List β} {a : α} (al : a ∈ l) (h : b ∈ f a) : b ∈ list.bind l f :=\n  iff.mpr mem_bind (Exists.intro a (Exists.intro al h))\n\ntheorem bind_map {α : Type u} {β : Type v} {γ : Type w} {g : α → List β} {f : β → γ} (l : List α) : map f (list.bind l g) = list.bind l fun (a : α) => map f (g a) := sorry\n\n/-! ### length -/\n\ntheorem length_eq_zero {α : Type u} {l : List α} : length l = 0 ↔ l = [] :=\n  { mp := eq_nil_of_length_eq_zero, mpr := fun (h : l = []) => Eq.symm h ▸ rfl }\n\n@[simp] theorem length_singleton {α : Type u} (a : α) : length [a] = 1 :=\n  rfl\n\ntheorem length_pos_of_mem {α : Type u} {a : α} {l : List α} : a ∈ l → 0 < length l := sorry\n\ntheorem exists_mem_of_length_pos {α : Type u} {l : List α} : 0 < length l → ∃ (a : α), a ∈ l := sorry\n\ntheorem length_pos_iff_exists_mem {α : Type u} {l : List α} : 0 < length l ↔ ∃ (a : α), a ∈ l := sorry\n\ntheorem ne_nil_of_length_pos {α : Type u} {l : List α} : 0 < length l → l ≠ [] :=\n  fun (h1 : 0 < length l) (h2 : l = []) => lt_irrefl 0 (iff.mpr length_eq_zero h2 ▸ h1)\n\ntheorem length_pos_of_ne_nil {α : Type u} {l : List α} : l ≠ [] → 0 < length l :=\n  fun (h : l ≠ []) => iff.mpr pos_iff_ne_zero fun (h0 : length l = 0) => h (iff.mp length_eq_zero h0)\n\ntheorem length_pos_iff_ne_nil {α : Type u} {l : List α} : 0 < length l ↔ l ≠ [] :=\n  { mp := ne_nil_of_length_pos, mpr := length_pos_of_ne_nil }\n\ntheorem length_eq_one {α : Type u} {l : List α} : length l = 1 ↔ ∃ (a : α), l = [a] := sorry\n\ntheorem exists_of_length_succ {α : Type u} {n : ℕ} (l : List α) : length l = n + 1 → ∃ (h : α), ∃ (t : List α), l = h :: t := sorry\n\n@[simp] theorem length_injective_iff {α : Type u} : function.injective length ↔ subsingleton α := sorry\n\n@[simp] theorem length_injective {α : Type u} [subsingleton α] : function.injective length :=\n  iff.mpr length_injective_iff _inst_1\n\n/-! ### set-theoretic notation of lists -/\n\ntheorem empty_eq {α : Type u} : ∅ = [] :=\n  Eq.refl ∅\n\ntheorem singleton_eq {α : Type u} (x : α) : singleton x = [x] :=\n  rfl\n\ntheorem insert_neg {α : Type u} [DecidableEq α] {x : α} {l : List α} (h : ¬x ∈ l) : insert x l = x :: l :=\n  if_neg h\n\ntheorem insert_pos {α : Type u} [DecidableEq α] {x : α} {l : List α} (h : x ∈ l) : insert x l = l :=\n  if_pos h\n\ntheorem doubleton_eq {α : Type u} [DecidableEq α] {x : α} {y : α} (h : x ≠ y) : insert x (singleton y) = [x, y] := sorry\n\n/-! ### bounded quantifiers over lists -/\n\ntheorem forall_mem_nil {α : Type u} (p : α → Prop) (x : α) (H : x ∈ []) : p x :=\n  false.dcases_on (fun (H : x ∈ []) => p x) H\n\ntheorem forall_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : List α} : (∀ (x : α), x ∈ a :: l → p x) ↔ p a ∧ ∀ (x : α), x ∈ l → p x :=\n  ball_cons\n\ntheorem forall_mem_of_forall_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : List α} (h : ∀ (x : α), x ∈ a :: l → p x) (x : α) (H : x ∈ l) : p x :=\n  and.right (iff.mp forall_mem_cons h)\n\ntheorem forall_mem_singleton {α : Type u} {p : α → Prop} {a : α} : (∀ (x : α), x ∈ [a] → p x) ↔ p a := sorry\n\ntheorem forall_mem_append {α : Type u} {p : α → Prop} {l₁ : List α} {l₂ : List α} : (∀ (x : α), x ∈ l₁ ++ l₂ → p x) ↔ (∀ (x : α), x ∈ l₁ → p x) ∧ ∀ (x : α), x ∈ l₂ → p x := sorry\n\ntheorem not_exists_mem_nil {α : Type u} (p : α → Prop) : ¬∃ (x : α), ∃ (H : x ∈ []), p x := sorry\n\ntheorem exists_mem_cons_of {α : Type u} {p : α → Prop} {a : α} (l : List α) (h : p a) : ∃ (x : α), ∃ (H : x ∈ a :: l), p x :=\n  bex.intro a (mem_cons_self a l) h\n\ntheorem exists_mem_cons_of_exists {α : Type u} {p : α → Prop} {a : α} {l : List α} (h : ∃ (x : α), ∃ (H : x ∈ l), p x) : ∃ (x : α), ∃ (H : x ∈ a :: l), p x :=\n  bex.elim h fun (x : α) (xl : x ∈ l) (px : p x) => bex.intro x (mem_cons_of_mem a xl) px\n\ntheorem or_exists_of_exists_mem_cons {α : Type u} {p : α → Prop} {a : α} {l : List α} (h : ∃ (x : α), ∃ (H : x ∈ a :: l), p x) : p a ∨ ∃ (x : α), ∃ (H : x ∈ l), p x := sorry\n\ntheorem exists_mem_cons_iff {α : Type u} (p : α → Prop) (a : α) (l : List α) : (∃ (x : α), ∃ (H : x ∈ a :: l), p x) ↔ p a ∨ ∃ (x : α), ∃ (H : x ∈ l), p x :=\n  { mp := or_exists_of_exists_mem_cons,\n    mpr := fun (h : p a ∨ ∃ (x : α), ∃ (H : x ∈ l), p x) => or.elim h (exists_mem_cons_of l) exists_mem_cons_of_exists }\n\n/-! ### list subset -/\n\ntheorem subset_def {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ ⊆ l₂ ↔ ∀ {a : α}, a ∈ l₁ → a ∈ l₂ :=\n  iff.rfl\n\ntheorem subset_append_of_subset_left {α : Type u} (l : List α) (l₁ : List α) (l₂ : List α) : l ⊆ l₁ → l ⊆ l₁ ++ l₂ :=\n  fun (s : l ⊆ l₁) => subset.trans s (subset_append_left l₁ l₂)\n\ntheorem subset_append_of_subset_right {α : Type u} (l : List α) (l₁ : List α) (l₂ : List α) : l ⊆ l₂ → l ⊆ l₁ ++ l₂ :=\n  fun (s : l ⊆ l₂) => subset.trans s (subset_append_right l₁ l₂)\n\n@[simp] theorem cons_subset {α : Type u} {a : α} {l : List α} {m : List α} : a :: l ⊆ m ↔ a ∈ m ∧ l ⊆ m := sorry\n\ntheorem cons_subset_of_subset_of_mem {α : Type u} {a : α} {l : List α} {m : List α} (ainm : a ∈ m) (lsubm : l ⊆ m) : a :: l ⊆ m :=\n  iff.mpr cons_subset { left := ainm, right := lsubm }\n\ntheorem append_subset_of_subset_of_subset {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) : l₁ ++ l₂ ⊆ l :=\n  fun (a : α) (h : a ∈ l₁ ++ l₂) => or.elim (iff.mp mem_append h) l₁subl l₂subl\n\n@[simp] theorem append_subset_iff {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} : l₁ ++ l₂ ⊆ l ↔ l₁ ⊆ l ∧ l₂ ⊆ l := sorry\n\ntheorem eq_nil_of_subset_nil {α : Type u} {l : List α} : l ⊆ [] → l = [] := sorry\n\ntheorem eq_nil_iff_forall_not_mem {α : Type u} {l : List α} : l = [] ↔ ∀ (a : α), ¬a ∈ l :=\n  (fun (this : l = [] ↔ l ⊆ []) => this) { mp := fun (e : l = []) => e ▸ subset.refl l, mpr := eq_nil_of_subset_nil }\n\ntheorem map_subset {α : Type u} {β : Type v} {l₁ : List α} {l₂ : List α} (f : α → β) (H : l₁ ⊆ l₂) : map f l₁ ⊆ map f l₂ := sorry\n\ntheorem map_subset_iff {α : Type u} {β : Type v} {l₁ : List α} {l₂ : List α} (f : α → β) (h : function.injective f) : map f l₁ ⊆ map f l₂ ↔ l₁ ⊆ l₂ := sorry\n\n/-! ### append -/\n\ntheorem append_eq_has_append {α : Type u} {L₁ : List α} {L₂ : List α} : list.append L₁ L₂ = L₁ ++ L₂ :=\n  rfl\n\n@[simp] theorem singleton_append {α : Type u} {x : α} {l : List α} : [x] ++ l = x :: l :=\n  rfl\n\ntheorem append_ne_nil_of_ne_nil_left {α : Type u} (s : List α) (t : List α) : s ≠ [] → s ++ t ≠ [] := sorry\n\ntheorem append_ne_nil_of_ne_nil_right {α : Type u} (s : List α) (t : List α) : t ≠ [] → s ++ t ≠ [] := sorry\n\n@[simp] theorem append_eq_nil {α : Type u} {p : List α} {q : List α} : p ++ q = [] ↔ p = [] ∧ q = [] := sorry\n\n@[simp] theorem nil_eq_append_iff {α : Type u} {a : List α} {b : List α} : [] = a ++ b ↔ a = [] ∧ b = [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl ([] = a ++ b ↔ a = [] ∧ b = [])) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a ++ b = [] ↔ a = [] ∧ b = [])) (propext append_eq_nil)))\n      (iff.refl (a = [] ∧ b = [])))\n\ntheorem append_eq_cons_iff {α : Type u} {a : List α} {b : List α} {c : List α} {x : α} : a ++ b = x :: c ↔ a = [] ∧ b = x :: c ∨ ∃ (a' : List α), a = x :: a' ∧ c = a' ++ b := sorry\n\ntheorem cons_eq_append_iff {α : Type u} {a : List α} {b : List α} {c : List α} {x : α} : x :: c = a ++ b ↔ a = [] ∧ b = x :: c ∨ ∃ (a' : List α), a = x :: a' ∧ c = a' ++ b := sorry\n\ntheorem append_eq_append_iff {α : Type u} {a : List α} {b : List α} {c : List α} {d : List α} : a ++ b = c ++ d ↔ (∃ (a' : List α), c = a ++ a' ∧ b = a' ++ d) ∨ ∃ (c' : List α), a = c ++ c' ∧ d = c' ++ b := sorry\n\n@[simp] theorem split_at_eq_take_drop {α : Type u} (n : ℕ) (l : List α) : split_at n l = (take n l, drop n l) := sorry\n\n@[simp] theorem take_append_drop {α : Type u} (n : ℕ) (l : List α) : take n l ++ drop n l = l := sorry\n\n-- TODO(Leo): cleanup proof after arith dec proc\n\ntheorem append_inj {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} : s₁ ++ t₁ = s₂ ++ t₂ → length s₁ = length s₂ → s₁ = s₂ ∧ t₁ = t₂ := sorry\n\ntheorem append_inj_right {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length s₁ = length s₂) : t₁ = t₂ :=\n  and.right (append_inj h hl)\n\ntheorem append_inj_left {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length s₁ = length s₂) : s₁ = s₂ :=\n  and.left (append_inj h hl)\n\ntheorem append_inj' {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) : s₁ = s₂ ∧ t₁ = t₂ := sorry\n\ntheorem append_inj_right' {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) : t₁ = t₂ :=\n  and.right (append_inj' h hl)\n\ntheorem append_inj_left' {α : Type u} {s₁ : List α} {s₂ : List α} {t₁ : List α} {t₂ : List α} (h : s₁ ++ t₁ = s₂ ++ t₂) (hl : length t₁ = length t₂) : s₁ = s₂ :=\n  and.left (append_inj' h hl)\n\ntheorem append_left_cancel {α : Type u} {s : List α} {t₁ : List α} {t₂ : List α} (h : s ++ t₁ = s ++ t₂) : t₁ = t₂ :=\n  append_inj_right h rfl\n\ntheorem append_right_cancel {α : Type u} {s₁ : List α} {s₂ : List α} {t : List α} (h : s₁ ++ t = s₂ ++ t) : s₁ = s₂ :=\n  append_inj_left' h rfl\n\ntheorem append_right_injective {α : Type u} (s : List α) : function.injective fun (t : List α) => s ++ t :=\n  fun (t₁ t₂ : List α) => append_left_cancel\n\ntheorem append_right_inj {α : Type u} {t₁ : List α} {t₂ : List α} (s : List α) : s ++ t₁ = s ++ t₂ ↔ t₁ = t₂ :=\n  function.injective.eq_iff (append_right_injective s)\n\ntheorem append_left_injective {α : Type u} (t : List α) : function.injective fun (s : List α) => s ++ t :=\n  fun (s₁ s₂ : List α) => append_right_cancel\n\ntheorem append_left_inj {α : Type u} {s₁ : List α} {s₂ : List α} (t : List α) : s₁ ++ t = s₂ ++ t ↔ s₁ = s₂ :=\n  function.injective.eq_iff (append_left_injective t)\n\ntheorem map_eq_append_split {α : Type u} {β : Type v} {f : α → β} {l : List α} {s₁ : List β} {s₂ : List β} (h : map f l = s₁ ++ s₂) : ∃ (l₁ : List α), ∃ (l₂ : List α), l = l₁ ++ l₂ ∧ map f l₁ = s₁ ∧ map f l₂ = s₂ := sorry\n\n/-! ### repeat -/\n\n@[simp] theorem repeat_succ {α : Type u} (a : α) (n : ℕ) : repeat a (n + 1) = a :: repeat a n :=\n  rfl\n\ntheorem eq_of_mem_repeat {α : Type u} {a : α} {b : α} {n : ℕ} : b ∈ repeat a n → b = a := sorry\n\ntheorem eq_repeat_of_mem {α : Type u} {a : α} {l : List α} : (∀ (b : α), b ∈ l → b = a) → l = repeat a (length l) := sorry\n\ntheorem eq_repeat' {α : Type u} {a : α} {l : List α} : l = repeat a (length l) ↔ ∀ (b : α), b ∈ l → b = a :=\n  { mp := fun (h : l = repeat a (length l)) => Eq.symm h ▸ fun (b : α) => eq_of_mem_repeat, mpr := eq_repeat_of_mem }\n\ntheorem eq_repeat {α : Type u} {a : α} {n : ℕ} {l : List α} : l = repeat a n ↔ length l = n ∧ ∀ (b : α), b ∈ l → b = a := sorry\n\ntheorem repeat_add {α : Type u} (a : α) (m : ℕ) (n : ℕ) : repeat a (m + n) = repeat a m ++ repeat a n := sorry\n\ntheorem repeat_subset_singleton {α : Type u} (a : α) (n : ℕ) : repeat a n ⊆ [a] :=\n  fun (b : α) (h : b ∈ repeat a n) => iff.mpr mem_singleton (eq_of_mem_repeat h)\n\n@[simp] theorem map_const {α : Type u} {β : Type v} (l : List α) (b : β) : map (function.const α b) l = repeat b (length l) := sorry\n\ntheorem eq_of_mem_map_const {α : Type u} {β : Type v} {b₁ : β} {b₂ : β} {l : List α} (h : b₁ ∈ map (function.const α b₂) l) : b₁ = b₂ :=\n  eq_of_mem_repeat (eq.mp (Eq._oldrec (Eq.refl (b₁ ∈ map (function.const α b₂) l)) (map_const l b₂)) h)\n\n@[simp] theorem map_repeat {α : Type u} {β : Type v} (f : α → β) (a : α) (n : ℕ) : map f (repeat a n) = repeat (f a) n := sorry\n\n@[simp] theorem tail_repeat {α : Type u} (a : α) (n : ℕ) : tail (repeat a n) = repeat a (Nat.pred n) :=\n  nat.cases_on n (Eq.refl (tail (repeat a 0))) fun (n : ℕ) => Eq.refl (tail (repeat a (Nat.succ n)))\n\n@[simp] theorem join_repeat_nil {α : Type u} (n : ℕ) : join (repeat [] n) = [] := sorry\n\n/-! ### pure -/\n\n@[simp] theorem mem_pure {α : Type u_1} (x : α) (y : α) : x ∈ pure y ↔ x = y := sorry\n\n/-! ### bind -/\n\n@[simp] theorem bind_eq_bind {α : Type u_1} {β : Type u_1} (f : α → List β) (l : List α) : l >>= f = list.bind l f :=\n  rfl\n\n@[simp] theorem bind_append {α : Type u} {β : Type v} (f : α → List β) (l₁ : List α) (l₂ : List α) : list.bind (l₁ ++ l₂) f = list.bind l₁ f ++ list.bind l₂ f :=\n  append_bind l₁ l₂ f\n\n@[simp] theorem bind_singleton {α : Type u} {β : Type v} (f : α → List β) (x : α) : list.bind [x] f = f x :=\n  append_nil (f x)\n\n/-! ### concat -/\n\ntheorem concat_nil {α : Type u} (a : α) : concat [] a = [a] :=\n  rfl\n\ntheorem concat_cons {α : Type u} (a : α) (b : α) (l : List α) : concat (a :: l) b = a :: concat l b :=\n  rfl\n\n@[simp] theorem concat_eq_append {α : Type u} (a : α) (l : List α) : concat l a = l ++ [a] := sorry\n\ntheorem init_eq_of_concat_eq {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : concat l₁ a = concat l₂ a → l₁ = l₂ := sorry\n\ntheorem last_eq_of_concat_eq {α : Type u} {a : α} {b : α} {l : List α} : concat l a = concat l b → a = b := sorry\n\ntheorem concat_ne_nil {α : Type u} (a : α) (l : List α) : concat l a ≠ [] := sorry\n\ntheorem concat_append {α : Type u} (a : α) (l₁ : List α) (l₂ : List α) : concat l₁ a ++ l₂ = l₁ ++ a :: l₂ := sorry\n\ntheorem length_concat {α : Type u} (a : α) (l : List α) : length (concat l a) = Nat.succ (length l) := sorry\n\ntheorem append_concat {α : Type u} (a : α) (l₁ : List α) (l₂ : List α) : l₁ ++ concat l₂ a = concat (l₁ ++ l₂) a := sorry\n\n/-! ### reverse -/\n\n@[simp] theorem reverse_nil {α : Type u} : reverse [] = [] :=\n  rfl\n\n@[simp] theorem reverse_cons {α : Type u} (a : α) (l : List α) : reverse (a :: l) = reverse l ++ [a] := sorry\n\ntheorem reverse_core_eq {α : Type u} (l₁ : List α) (l₂ : List α) : reverse_core l₁ l₂ = reverse l₁ ++ l₂ := sorry\n\ntheorem reverse_cons' {α : Type u} (a : α) (l : List α) : reverse (a :: l) = concat (reverse l) a := sorry\n\n@[simp] theorem reverse_singleton {α : Type u} (a : α) : reverse [a] = [a] :=\n  rfl\n\n@[simp] theorem reverse_append {α : Type u} (s : List α) (t : List α) : reverse (s ++ t) = reverse t ++ reverse s := sorry\n\ntheorem reverse_concat {α : Type u} (l : List α) (a : α) : reverse (concat l a) = a :: reverse l := sorry\n\n@[simp] theorem reverse_reverse {α : Type u} (l : List α) : reverse (reverse l) = l := sorry\n\n@[simp] theorem reverse_involutive {α : Type u} : function.involutive reverse :=\n  fun (l : List α) => reverse_reverse l\n\n@[simp] theorem reverse_injective {α : Type u} : function.injective reverse :=\n  function.involutive.injective reverse_involutive\n\n@[simp] theorem reverse_inj {α : Type u} {l₁ : List α} {l₂ : List α} : reverse l₁ = reverse l₂ ↔ l₁ = l₂ :=\n  function.injective.eq_iff reverse_injective\n\n@[simp] theorem reverse_eq_nil {α : Type u} {l : List α} : reverse l = [] ↔ l = [] :=\n  reverse_inj\n\ntheorem concat_eq_reverse_cons {α : Type u} (a : α) (l : List α) : concat l a = reverse (a :: reverse l) := sorry\n\n@[simp] theorem length_reverse {α : Type u} (l : List α) : length (reverse l) = length l := sorry\n\n@[simp] theorem map_reverse {α : Type u} {β : Type v} (f : α → β) (l : List α) : map f (reverse l) = reverse (map f l) := sorry\n\ntheorem map_reverse_core {α : Type u} {β : Type v} (f : α → β) (l₁ : List α) (l₂ : List α) : map f (reverse_core l₁ l₂) = reverse_core (map f l₁) (map f l₂) := sorry\n\n@[simp] theorem mem_reverse {α : Type u} {a : α} {l : List α} : a ∈ reverse l ↔ a ∈ l := sorry\n\n@[simp] theorem reverse_repeat {α : Type u} (a : α) (n : ℕ) : reverse (repeat a n) = repeat a n := sorry\n\n/-! ### is_nil -/\n\ntheorem is_nil_iff_eq_nil {α : Type u} {l : List α} : ↥(is_nil l) ↔ l = [] := sorry\n\n/-! ### init -/\n\n@[simp] theorem length_init {α : Type u} (l : List α) : length (init l) = length l - 1 := sorry\n\n/-! ### last -/\n\n@[simp] theorem last_cons {α : Type u} {a : α} {l : List α} (h₁ : a :: l ≠ []) (h₂ : l ≠ []) : last (a :: l) h₁ = last l h₂ := sorry\n\n@[simp] theorem last_append {α : Type u} {a : α} (l : List α) (h : l ++ [a] ≠ []) : last (l ++ [a]) h = a := sorry\n\ntheorem last_concat {α : Type u} {a : α} (l : List α) (h : concat l a ≠ []) : last (concat l a) h = a := sorry\n\n@[simp] theorem last_singleton {α : Type u} (a : α) (h : [a] ≠ []) : last [a] h = a :=\n  rfl\n\n@[simp] theorem last_cons_cons {α : Type u} (a₁ : α) (a₂ : α) (l : List α) (h : a₁ :: a₂ :: l ≠ []) : last (a₁ :: a₂ :: l) h = last (a₂ :: l) (cons_ne_nil a₂ l) :=\n  rfl\n\ntheorem init_append_last {α : Type u} {l : List α} (h : l ≠ []) : init l ++ [last l h] = l := sorry\n\ntheorem last_congr {α : Type u} {l₁ : List α} {l₂ : List α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : last l₁ h₁ = last l₂ h₂ :=\n  Eq._oldrec (fun (h₁ : l₂ ≠ []) => Eq.refl (last l₂ h₁)) (Eq.symm h₃) h₁\n\ntheorem last_mem {α : Type u} {l : List α} (h : l ≠ []) : last l h ∈ l := sorry\n\ntheorem last_repeat_succ (a : ℕ) (m : ℕ) : last (repeat a (Nat.succ m))\n    (ne_nil_of_length_eq_succ\n      ((fun (this : length (repeat a (Nat.succ m)) = Nat.succ m) => this)\n        (eq.mpr (id (Eq._oldrec (Eq.refl (length (repeat a (Nat.succ m)) = Nat.succ m)) (length_repeat a (Nat.succ m))))\n          (Eq.refl (Nat.succ m))))) =\n  a := sorry\n\n/-! ### last' -/\n\n@[simp] theorem last'_is_none {α : Type u} {l : List α} : ↥(option.is_none (last' l)) ↔ l = [] := sorry\n\n@[simp] theorem last'_is_some {α : Type u} {l : List α} : ↥(option.is_some (last' l)) ↔ l ≠ [] := sorry\n\ntheorem mem_last'_eq_last {α : Type u} {l : List α} {x : α} : x ∈ last' l → ∃ (h : l ≠ []), x = last l h := sorry\n\ntheorem mem_of_mem_last' {α : Type u} {l : List α} {a : α} (ha : a ∈ last' l) : a ∈ l := sorry\n\ntheorem init_append_last' {α : Type u} {l : List α} (a : α) (H : a ∈ last' l) : init l ++ [a] = l := sorry\n\ntheorem ilast_eq_last' {α : Type u} [Inhabited α] (l : List α) : ilast l = option.iget (last' l) := sorry\n\n@[simp] theorem last'_append_cons {α : Type u} (l₁ : List α) (a : α) (l₂ : List α) : last' (l₁ ++ a :: l₂) = last' (a :: l₂) := sorry\n\ntheorem last'_append_of_ne_nil {α : Type u} (l₁ : List α) {l₂ : List α} (hl₂ : l₂ ≠ []) : last' (l₁ ++ l₂) = last' l₂ := sorry\n\n/-! ### head(') and tail -/\n\ntheorem head_eq_head' {α : Type u} [Inhabited α] (l : List α) : head l = option.iget (head' l) :=\n  list.cases_on l (Eq.refl (head [])) fun (l_hd : α) (l_tl : List α) => Eq.refl (head (l_hd :: l_tl))\n\ntheorem mem_of_mem_head' {α : Type u} {x : α} {l : List α} : x ∈ head' l → x ∈ l := sorry\n\n@[simp] theorem head_cons {α : Type u} [Inhabited α] (a : α) (l : List α) : head (a :: l) = a :=\n  rfl\n\n@[simp] theorem tail_nil {α : Type u} : tail [] = [] :=\n  rfl\n\n@[simp] theorem tail_cons {α : Type u} (a : α) (l : List α) : tail (a :: l) = l :=\n  rfl\n\n@[simp] theorem head_append {α : Type u} [Inhabited α] (t : List α) {s : List α} (h : s ≠ []) : head (s ++ t) = head s := sorry\n\ntheorem tail_append_singleton_of_ne_nil {α : Type u} {a : α} {l : List α} (h : l ≠ []) : tail (l ++ [a]) = tail l ++ [a] := sorry\n\ntheorem cons_head'_tail {α : Type u} {l : List α} {a : α} (h : a ∈ head' l) : a :: tail l = l := sorry\n\ntheorem head_mem_head' {α : Type u} [Inhabited α] {l : List α} (h : l ≠ []) : head l ∈ head' l :=\n  list.cases_on l (fun (h : [] ≠ []) => idRhs (head [] ∈ head' []) (absurd (Eq.refl []) h))\n    (fun (l_hd : α) (l_tl : List α) (h : l_hd :: l_tl ≠ []) => idRhs (head' (l_hd :: l_tl) = head' (l_hd :: l_tl)) rfl) h\n\ntheorem cons_head_tail {α : Type u} [Inhabited α] {l : List α} (h : l ≠ []) : head l :: tail l = l :=\n  cons_head'_tail (head_mem_head' h)\n\n@[simp] theorem head'_map {α : Type u} {β : Type v} (f : α → β) (l : List α) : head' (map f l) = option.map f (head' l) :=\n  list.cases_on l (Eq.refl (head' (map f []))) fun (l_hd : α) (l_tl : List α) => Eq.refl (head' (map f (l_hd :: l_tl)))\n\n/-! ### Induction from the right -/\n\n/-- Induction principle from the right for lists: if a property holds for the empty list, and\nfor `l ++ [a]` if it holds for `l`, then it holds for all lists. The principle is given for\na `Sort`-valued predicate, i.e., it can also be used to construct data. -/\ndef reverse_rec_on {α : Type u} {C : List α → Sort u_1} (l : List α) (H0 : C []) (H1 : (l : List α) → (a : α) → C l → C (l ++ [a])) : C l :=\n  eq.mpr sorry\n    (List.rec H0 (fun (hd : α) (tl : List α) (ih : C (reverse tl)) => eq.mpr sorry (H1 (reverse tl) hd ih)) (reverse l))\n\n/-- Bidirectional induction principle for lists: if a property holds for the empty list, the\nsingleton list, and `a :: (l ++ [b])` from `l`, then it holds for all lists. This can be used to\nprove statements about palindromes. The principle is given for a `Sort`-valued predicate, i.e., it\ncan also be used to construct data. -/\ndef bidirectional_rec {α : Type u} {C : List α → Sort u_1} (H0 : C []) (H1 : (a : α) → C [a]) (Hn : (a : α) → (l : List α) → (b : α) → C l → C (a :: (l ++ [b]))) (l : List α) : C l :=\n  sorry\n\n/-- Like `bidirectional_rec`, but with the list parameter placed first. -/\ndef bidirectional_rec_on {α : Type u} {C : List α → Sort u_1} (l : List α) (H0 : C []) (H1 : (a : α) → C [a]) (Hn : (a : α) → (l : List α) → (b : α) → C l → C (a :: (l ++ [b]))) : C l :=\n  bidirectional_rec H0 H1 Hn l\n\n/-! ### sublists -/\n\n@[simp] theorem nil_sublist {α : Type u} (l : List α) : [] <+ l := sorry\n\n@[simp] theorem sublist.refl {α : Type u} (l : List α) : l <+ l := sorry\n\ntheorem sublist.trans {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁ <+ l₂) (h₂ : l₂ <+ l₃) : l₁ <+ l₃ := sorry\n\n@[simp] theorem sublist_cons {α : Type u} (a : α) (l : List α) : l <+ a :: l :=\n  sublist.cons l l a (sublist.refl l)\n\ntheorem sublist_of_cons_sublist {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : a :: l₁ <+ l₂ → l₁ <+ l₂ :=\n  sublist.trans (sublist_cons a l₁)\n\ntheorem cons_sublist_cons {α : Type u} {l₁ : List α} {l₂ : List α} (a : α) (s : l₁ <+ l₂) : a :: l₁ <+ a :: l₂ :=\n  sublist.cons2 l₁ l₂ a s\n\n@[simp] theorem sublist_append_left {α : Type u} (l₁ : List α) (l₂ : List α) : l₁ <+ l₁ ++ l₂ := sorry\n\n@[simp] theorem sublist_append_right {α : Type u} (l₁ : List α) (l₂ : List α) : l₂ <+ l₁ ++ l₂ := sorry\n\ntheorem sublist_cons_of_sublist {α : Type u} (a : α) {l₁ : List α} {l₂ : List α} : l₁ <+ l₂ → l₁ <+ a :: l₂ :=\n  sublist.cons l₁ l₂ a\n\ntheorem sublist_append_of_sublist_left {α : Type u} {l : List α} {l₁ : List α} {l₂ : List α} (s : l <+ l₁) : l <+ l₁ ++ l₂ :=\n  sublist.trans s (sublist_append_left l₁ l₂)\n\ntheorem sublist_append_of_sublist_right {α : Type u} {l : List α} {l₁ : List α} {l₂ : List α} (s : l <+ l₂) : l <+ l₁ ++ l₂ :=\n  sublist.trans s (sublist_append_right l₁ l₂)\n\ntheorem sublist_of_cons_sublist_cons {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} : a :: l₁ <+ a :: l₂ → l₁ <+ l₂ := sorry\n\ntheorem cons_sublist_cons_iff {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} : a :: l₁ <+ a :: l₂ ↔ l₁ <+ l₂ :=\n  { mp := sublist_of_cons_sublist_cons, mpr := cons_sublist_cons a }\n\n@[simp] theorem append_sublist_append_left {α : Type u} {l₁ : List α} {l₂ : List α} (l : List α) : l ++ l₁ <+ l ++ l₂ ↔ l₁ <+ l₂ := sorry\n\ntheorem sublist.append_right {α : Type u} {l₁ : List α} {l₂ : List α} (h : l₁ <+ l₂) (l : List α) : l₁ ++ l <+ l₂ ++ l :=\n  sublist.drec (sublist.refl ([] ++ l))\n    (fun (h_l₁ h_l₂ : List α) (a : α) (h_ᾰ : h_l₁ <+ h_l₂) (ih : h_l₁ ++ l <+ h_l₂ ++ l) => sublist_cons_of_sublist a ih)\n    (fun (h_l₁ h_l₂ : List α) (a : α) (h_ᾰ : h_l₁ <+ h_l₂) (ih : h_l₁ ++ l <+ h_l₂ ++ l) => cons_sublist_cons a ih) h\n\ntheorem sublist_or_mem_of_sublist {α : Type u} {l : List α} {l₁ : List α} {l₂ : List α} {a : α} (h : l <+ l₁ ++ a :: l₂) : l <+ l₁ ++ l₂ ∨ a ∈ l := sorry\n\ntheorem sublist.reverse {α : Type u} {l₁ : List α} {l₂ : List α} (h : l₁ <+ l₂) : reverse l₁ <+ reverse l₂ := sorry\n\n@[simp] theorem reverse_sublist_iff {α : Type u} {l₁ : List α} {l₂ : List α} : reverse l₁ <+ reverse l₂ ↔ l₁ <+ l₂ :=\n  { mp := fun (h : reverse l₁ <+ reverse l₂) => reverse_reverse l₁ ▸ reverse_reverse l₂ ▸ sublist.reverse h,\n    mpr := sublist.reverse }\n\n@[simp] theorem append_sublist_append_right {α : Type u} {l₁ : List α} {l₂ : List α} (l : List α) : l₁ ++ l <+ l₂ ++ l ↔ l₁ <+ l₂ := sorry\n\ntheorem sublist.append {α : Type u} {l₁ : List α} {l₂ : List α} {r₁ : List α} {r₂ : List α} (hl : l₁ <+ l₂) (hr : r₁ <+ r₂) : l₁ ++ r₁ <+ l₂ ++ r₂ :=\n  sublist.trans (sublist.append_right hl r₁) (iff.mpr (append_sublist_append_left l₂) hr)\n\ntheorem sublist.subset {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+ l₂ → l₁ ⊆ l₂ := sorry\n\ntheorem singleton_sublist {α : Type u} {a : α} {l : List α} : [a] <+ l ↔ a ∈ l := sorry\n\ntheorem eq_nil_of_sublist_nil {α : Type u} {l : List α} (s : l <+ []) : l = [] :=\n  eq_nil_of_subset_nil (sublist.subset s)\n\ntheorem repeat_sublist_repeat {α : Type u} (a : α) {m : ℕ} {n : ℕ} : repeat a m <+ repeat a n ↔ m ≤ n := sorry\n\ntheorem eq_of_sublist_of_length_eq {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+ l₂ → length l₁ = length l₂ → l₁ = l₂ := sorry\n\ntheorem eq_of_sublist_of_length_le {α : Type u} {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) (h : length l₂ ≤ length l₁) : l₁ = l₂ :=\n  eq_of_sublist_of_length_eq s (le_antisymm (length_le_of_sublist s) h)\n\ntheorem sublist.antisymm {α : Type u} {l₁ : List α} {l₂ : List α} (s₁ : l₁ <+ l₂) (s₂ : l₂ <+ l₁) : l₁ = l₂ :=\n  eq_of_sublist_of_length_le s₁ (length_le_of_sublist s₂)\n\nprotected instance decidable_sublist {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <+ l₂) :=\n  sorry\n\n/-! ### index_of -/\n\n@[simp] theorem index_of_nil {α : Type u} [DecidableEq α] (a : α) : index_of a [] = 0 :=\n  rfl\n\ntheorem index_of_cons {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : index_of a (b :: l) = ite (a = b) 0 (Nat.succ (index_of a l)) :=\n  rfl\n\ntheorem index_of_cons_eq {α : Type u} [DecidableEq α] {a : α} {b : α} (l : List α) : a = b → index_of a (b :: l) = 0 :=\n  fun (e : a = b) => if_pos e\n\n@[simp] theorem index_of_cons_self {α : Type u} [DecidableEq α] (a : α) (l : List α) : index_of a (a :: l) = 0 :=\n  index_of_cons_eq l rfl\n\n@[simp] theorem index_of_cons_ne {α : Type u} [DecidableEq α] {a : α} {b : α} (l : List α) : a ≠ b → index_of a (b :: l) = Nat.succ (index_of a l) :=\n  fun (n : a ≠ b) => if_neg n\n\ntheorem index_of_eq_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : index_of a l = length l ↔ ¬a ∈ l := sorry\n\n@[simp] theorem index_of_of_not_mem {α : Type u} [DecidableEq α] {l : List α} {a : α} : ¬a ∈ l → index_of a l = length l :=\n  iff.mpr index_of_eq_length\n\ntheorem index_of_le_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : index_of a l ≤ length l := sorry\n\ntheorem index_of_lt_length {α : Type u} [DecidableEq α] {a : α} {l : List α} : index_of a l < length l ↔ a ∈ l := sorry\n\n/-! ### nth element -/\n\ntheorem nth_le_of_mem {α : Type u} {a : α} {l : List α} : a ∈ l → ∃ (n : ℕ), ∃ (h : n < length l), nth_le l n h = a := sorry\n\ntheorem nth_le_nth {α : Type u} {l : List α} {n : ℕ} (h : n < length l) : nth l n = some (nth_le l n h) := sorry\n\ntheorem nth_len_le {α : Type u} {l : List α} {n : ℕ} : length l ≤ n → nth l n = none := sorry\n\ntheorem nth_eq_some {α : Type u} {l : List α} {n : ℕ} {a : α} : nth l n = some a ↔ ∃ (h : n < length l), nth_le l n h = a := sorry\n\n@[simp] theorem nth_eq_none_iff {α : Type u} {l : List α} {n : ℕ} : nth l n = none ↔ length l ≤ n := sorry\n\ntheorem nth_of_mem {α : Type u} {a : α} {l : List α} (h : a ∈ l) : ∃ (n : ℕ), nth l n = some a := sorry\n\ntheorem nth_le_mem {α : Type u} (l : List α) (n : ℕ) (h : n < length l) : nth_le l n h ∈ l := sorry\n\ntheorem nth_mem {α : Type u} {l : List α} {n : ℕ} {a : α} (e : nth l n = some a) : a ∈ l := sorry\n\ntheorem mem_iff_nth_le {α : Type u} {a : α} {l : List α} : a ∈ l ↔ ∃ (n : ℕ), ∃ (h : n < length l), nth_le l n h = a := sorry\n\ntheorem mem_iff_nth {α : Type u} {a : α} {l : List α} : a ∈ l ↔ ∃ (n : ℕ), nth l n = some a :=\n  iff.trans mem_iff_nth_le (exists_congr fun (n : ℕ) => iff.symm nth_eq_some)\n\ntheorem nth_zero {α : Type u} (l : List α) : nth l 0 = head' l :=\n  list.cases_on l (Eq.refl (nth [] 0)) fun (l_hd : α) (l_tl : List α) => Eq.refl (nth (l_hd :: l_tl) 0)\n\ntheorem nth_injective {α : Type u} {xs : List α} {i : ℕ} {j : ℕ} (h₀ : i < length xs) (h₁ : nodup xs) (h₂ : nth xs i = nth xs j) : i = j := sorry\n\n@[simp] theorem nth_map {α : Type u} {β : Type v} (f : α → β) (l : List α) (n : ℕ) : nth (map f l) n = option.map f (nth l n) := sorry\n\ntheorem nth_le_map {α : Type u} {β : Type v} (f : α → β) {l : List α} {n : ℕ} (H1 : n < length (map f l)) (H2 : n < length l) : nth_le (map f l) n H1 = f (nth_le l n H2) := sorry\n\n/-- A version of `nth_le_map` that can be used for rewriting. -/\ntheorem nth_le_map_rev {α : Type u} {β : Type v} (f : α → β) {l : List α} {n : ℕ} (H : n < length l) : f (nth_le l n H) = nth_le (map f l) n (Eq.symm (length_map f l) ▸ H) :=\n  Eq.symm (nth_le_map f (Eq.symm (length_map f l) ▸ H) H)\n\n@[simp] theorem nth_le_map' {α : Type u} {β : Type v} (f : α → β) {l : List α} {n : ℕ} (H : n < length (map f l)) : nth_le (map f l) n H = f (nth_le l n (length_map f l ▸ H)) :=\n  nth_le_map f H (length_map f l ▸ H)\n\n/-- If one has `nth_le L i hi` in a formula and `h : L = L'`, one can not `rw h` in the formula as\n`hi` gives `i < L.length` and not `i < L'.length`. The lemma `nth_le_of_eq` can be used to make\nsuch a rewrite, with `rw (nth_le_of_eq h)`. -/\ntheorem nth_le_of_eq {α : Type u} {L : List α} {L' : List α} (h : L = L') {i : ℕ} (hi : i < length L) : nth_le L i hi = nth_le L' i (h ▸ hi) := sorry\n\n@[simp] theorem nth_le_singleton {α : Type u} (a : α) {n : ℕ} (hn : n < 1) : nth_le [a] n hn = a :=\n  (fun (hn0 : n = 0) => Eq._oldrec (fun (hn : 0 < 1) => Eq.refl (nth_le [a] 0 hn)) (Eq.symm hn0) hn)\n    (iff.mp nat.le_zero_iff (nat.le_of_lt_succ hn))\n\ntheorem nth_le_zero {α : Type u} [Inhabited α] {L : List α} (h : 0 < length L) : nth_le L 0 h = head L := sorry\n\ntheorem nth_le_append {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (hn₁ : n < length (l₁ ++ l₂)) (hn₂ : n < length l₁) : nth_le (l₁ ++ l₂) n hn₁ = nth_le l₁ n hn₂ := sorry\n\ntheorem nth_le_append_right_aux {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (h₁ : length l₁ ≤ n) (h₂ : n < length (l₁ ++ l₂)) : n - length l₁ < length l₂ := sorry\n\ntheorem nth_le_append_right {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (h₁ : length l₁ ≤ n) (h₂ : n < length (l₁ ++ l₂)) : nth_le (l₁ ++ l₂) n h₂ = nth_le l₂ (n - length l₁) (nth_le_append_right_aux h₁ h₂) := sorry\n\n@[simp] theorem nth_le_repeat {α : Type u} (a : α) {n : ℕ} {m : ℕ} (h : m < length (repeat a n)) : nth_le (repeat a n) m h = a :=\n  eq_of_mem_repeat (nth_le_mem (repeat a n) m h)\n\ntheorem nth_append {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (hn : n < length l₁) : nth (l₁ ++ l₂) n = nth l₁ n := sorry\n\ntheorem nth_append_right {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (hn : length l₁ ≤ n) : nth (l₁ ++ l₂) n = nth l₂ (n - length l₁) := sorry\n\ntheorem last_eq_nth_le {α : Type u} (l : List α) (h : l ≠ []) : last l h = nth_le l (length l - 1) (nat.sub_lt (length_pos_of_ne_nil h) nat.one_pos) := sorry\n\n@[simp] theorem nth_concat_length {α : Type u} (l : List α) (a : α) : nth (l ++ [a]) (length l) = some a := sorry\n\ntheorem ext {α : Type u} {l₁ : List α} {l₂ : List α} : (∀ (n : ℕ), nth l₁ n = nth l₂ n) → l₁ = l₂ := sorry\n\ntheorem ext_le {α : Type u} {l₁ : List α} {l₂ : List α} (hl : length l₁ = length l₂) (h : ∀ (n : ℕ) (h₁ : n < length l₁) (h₂ : n < length l₂), nth_le l₁ n h₁ = nth_le l₂ n h₂) : l₁ = l₂ := sorry\n\n@[simp] theorem index_of_nth_le {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : index_of a l < length l) : nth_le l (index_of a l) h = a := sorry\n\n@[simp] theorem index_of_nth {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : nth l (index_of a l) = some a := sorry\n\ntheorem nth_le_reverse_aux1 {α : Type u} (l : List α) (r : List α) (i : ℕ) (h1 : i + length l < length (reverse_core l r)) (h2 : i < length r) : nth_le (reverse_core l r) (i + length l) h1 = nth_le r i h2 := sorry\n\ntheorem index_of_inj {α : Type u} [DecidableEq α] {l : List α} {x : α} {y : α} (hx : x ∈ l) (hy : y ∈ l) : index_of x l = index_of y l ↔ x = y := sorry\n\ntheorem nth_le_reverse_aux2 {α : Type u} (l : List α) (r : List α) (i : ℕ) (h1 : length l - 1 - i < length (reverse_core l r)) (h2 : i < length l) : nth_le (reverse_core l r) (length l - 1 - i) h1 = nth_le l i h2 := sorry\n\n@[simp] theorem nth_le_reverse {α : Type u} (l : List α) (i : ℕ) (h1 : length l - 1 - i < length (reverse l)) (h2 : i < length l) : nth_le (reverse l) (length l - 1 - i) h1 = nth_le l i h2 :=\n  nth_le_reverse_aux2 l [] i h1 h2\n\ntheorem eq_cons_of_length_one {α : Type u} {l : List α} (h : length l = 1) : l = [nth_le l 0 (Eq.symm h ▸ zero_lt_one)] := sorry\n\ntheorem modify_nth_tail_modify_nth_tail {α : Type u} {f : List α → List α} {g : List α → List α} (m : ℕ) (n : ℕ) (l : List α) : modify_nth_tail g (m + n) (modify_nth_tail f n l) = modify_nth_tail (fun (l : List α) => modify_nth_tail g m (f l)) n l := sorry\n\ntheorem modify_nth_tail_modify_nth_tail_le {α : Type u} {f : List α → List α} {g : List α → List α} (m : ℕ) (n : ℕ) (l : List α) (h : n ≤ m) : modify_nth_tail g m (modify_nth_tail f n l) = modify_nth_tail (fun (l : List α) => modify_nth_tail g (m - n) (f l)) n l := sorry\n\ntheorem modify_nth_tail_modify_nth_tail_same {α : Type u} {f : List α → List α} {g : List α → List α} (n : ℕ) (l : List α) : modify_nth_tail g n (modify_nth_tail f n l) = modify_nth_tail (g ∘ f) n l := sorry\n\ntheorem modify_nth_tail_id {α : Type u} (n : ℕ) (l : List α) : modify_nth_tail id n l = l := sorry\n\ntheorem remove_nth_eq_nth_tail {α : Type u} (n : ℕ) (l : List α) : remove_nth l n = modify_nth_tail tail n l := sorry\n\ntheorem update_nth_eq_modify_nth {α : Type u} (a : α) (n : ℕ) (l : List α) : update_nth l n a = modify_nth (fun (_x : α) => a) n l := sorry\n\ntheorem modify_nth_eq_update_nth {α : Type u} (f : α → α) (n : ℕ) (l : List α) : modify_nth f n l = option.get_or_else ((fun (a : α) => update_nth l n (f a)) <$> nth l n) l := sorry\n\ntheorem nth_modify_nth {α : Type u} (f : α → α) (n : ℕ) (l : List α) (m : ℕ) : nth (modify_nth f n l) m = (fun (a : α) => ite (n = m) (f a) a) <$> nth l m := sorry\n\ntheorem modify_nth_tail_length {α : Type u} (f : List α → List α) (H : ∀ (l : List α), length (f l) = length l) (n : ℕ) (l : List α) : length (modify_nth_tail f n l) = length l := sorry\n\n@[simp] theorem modify_nth_length {α : Type u} (f : α → α) (n : ℕ) (l : List α) : length (modify_nth f n l) = length l := sorry\n\n@[simp] theorem update_nth_length {α : Type u} (l : List α) (n : ℕ) (a : α) : length (update_nth l n a) = length l := sorry\n\n@[simp] theorem nth_modify_nth_eq {α : Type u} (f : α → α) (n : ℕ) (l : List α) : nth (modify_nth f n l) n = f <$> nth l n := sorry\n\n@[simp] theorem nth_modify_nth_ne {α : Type u} (f : α → α) {m : ℕ} {n : ℕ} (l : List α) (h : m ≠ n) : nth (modify_nth f m l) n = nth l n := sorry\n\ntheorem nth_update_nth_eq {α : Type u} (a : α) (n : ℕ) (l : List α) : nth (update_nth l n a) n = (fun (_x : α) => a) <$> nth l n := sorry\n\ntheorem nth_update_nth_of_lt {α : Type u} (a : α) {n : ℕ} {l : List α} (h : n < length l) : nth (update_nth l n a) n = some a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nth (update_nth l n a) n = some a)) (nth_update_nth_eq a n l)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((fun (_x : α) => a) <$> nth l n = some a)) (nth_le_nth h)))\n      (Eq.refl ((fun (_x : α) => a) <$> some (nth_le l n h))))\n\ntheorem nth_update_nth_ne {α : Type u} (a : α) {m : ℕ} {n : ℕ} (l : List α) (h : m ≠ n) : nth (update_nth l m a) n = nth l n := sorry\n\n@[simp] theorem nth_le_update_nth_eq {α : Type u} (l : List α) (i : ℕ) (a : α) (h : i < length (update_nth l i a)) : nth_le (update_nth l i a) i h = a := sorry\n\n@[simp] theorem nth_le_update_nth_of_ne {α : Type u} {l : List α} {i : ℕ} {j : ℕ} (h : i ≠ j) (a : α) (hj : j < length (update_nth l i a)) : nth_le (update_nth l i a) j hj =\n  nth_le l j\n    (eq.mpr (id (Eq.refl (j < length l)))\n      (eq.mp\n        ((fun (ᾰ ᾰ_1 : ℕ) (e_2 : ᾰ = ᾰ_1) (ᾰ_2 ᾰ_3 : ℕ) (e_3 : ᾰ_2 = ᾰ_3) => congr (congr_arg Less e_2) e_3) j j\n          (Eq.refl j) (length (update_nth l i a)) (length l) (update_nth_length l i a))\n        hj)) := sorry\n\ntheorem mem_or_eq_of_mem_update_nth {α : Type u} {l : List α} {n : ℕ} {a : α} {b : α} (h : a ∈ update_nth l n b) : a ∈ l ∨ a = b := sorry\n\n@[simp] theorem insert_nth_nil {α : Type u} (a : α) : insert_nth 0 a [] = [a] :=\n  rfl\n\n@[simp] theorem insert_nth_succ_nil {α : Type u} (n : ℕ) (a : α) : insert_nth (n + 1) a [] = [] :=\n  rfl\n\ntheorem length_insert_nth {α : Type u} {a : α} (n : ℕ) (as : List α) : n ≤ length as → length (insert_nth n a as) = length as + 1 := sorry\n\ntheorem remove_nth_insert_nth {α : Type u} {a : α} (n : ℕ) (l : List α) : remove_nth (insert_nth n a l) n = l := sorry\n\ntheorem insert_nth_remove_nth_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < length as → n ≤ m → insert_nth m a (remove_nth as n) = remove_nth (insert_nth (m + 1) a as) n := sorry\n\ntheorem insert_nth_remove_nth_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < length as → m ≤ n → insert_nth m a (remove_nth as n) = remove_nth (insert_nth m a as) (n + 1) := sorry\n\ntheorem insert_nth_comm {α : Type u} (a : α) (b : α) (i : ℕ) (j : ℕ) (l : List α) (h : i ≤ j) (hj : j ≤ length l) : insert_nth (j + 1) b (insert_nth i a l) = insert_nth i a (insert_nth j b l) := sorry\n\ntheorem mem_insert_nth {α : Type u} {a : α} {b : α} {n : ℕ} {l : List α} (hi : n ≤ length l) : a ∈ insert_nth n b l ↔ a = b ∨ a ∈ l := sorry\n\n/-! ### map -/\n\n@[simp] theorem map_nil {α : Type u} {β : Type v} (f : α → β) : map f [] = [] :=\n  rfl\n\ntheorem map_eq_foldr {α : Type u} {β : Type v} (f : α → β) (l : List α) : map f l = foldr (fun (a : α) (bs : List β) => f a :: bs) [] l := sorry\n\ntheorem map_congr {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : List α} : (∀ (x : α), x ∈ l → f x = g x) → map f l = map g l := sorry\n\ntheorem map_eq_map_iff {α : Type u} {β : Type v} {f : α → β} {g : α → β} {l : List α} : map f l = map g l ↔ ∀ (x : α), x ∈ l → f x = g x := sorry\n\ntheorem map_concat {α : Type u} {β : Type v} (f : α → β) (a : α) (l : List α) : map f (concat l a) = concat (map f l) (f a) := sorry\n\ntheorem map_id' {α : Type u} {f : α → α} (h : ∀ (x : α), f x = x) (l : List α) : map f l = l := sorry\n\ntheorem eq_nil_of_map_eq_nil {α : Type u} {β : Type v} {f : α → β} {l : List α} (h : map f l = []) : l = [] :=\n  eq_nil_of_length_eq_zero\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length l = 0)) (Eq.symm (length_map f l))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length (map f l) = 0)) h)) (Eq.refl (length []))))\n\n@[simp] theorem map_join {α : Type u} {β : Type v} (f : α → β) (L : List (List α)) : map f (join L) = join (map (map f) L) := sorry\n\ntheorem bind_ret_eq_map {α : Type u} {β : Type v} (f : α → β) (l : List α) : list.bind l (list.ret ∘ f) = map f l := sorry\n\n@[simp] theorem map_eq_map {α : Type u_1} {β : Type u_1} (f : α → β) (l : List α) : f <$> l = map f l :=\n  rfl\n\n@[simp] theorem map_tail {α : Type u} {β : Type v} (f : α → β) (l : List α) : map f (tail l) = tail (map f l) :=\n  list.cases_on l (Eq.refl (map f (tail []))) fun (l_hd : α) (l_tl : List α) => Eq.refl (map f (tail (l_hd :: l_tl)))\n\n@[simp] theorem map_injective_iff {α : Type u} {β : Type v} {f : α → β} : function.injective (map f) ↔ function.injective f := sorry\n\n/--\nA single `list.map` of a composition of functions is equal to\ncomposing a `list.map` with another `list.map`, fully applied.\nThis is the reverse direction of `list.map_map`.\n-/\ntheorem comp_map {α : Type u} {β : Type v} {γ : Type w} (h : β → γ) (g : α → β) (l : List α) : map (h ∘ g) l = map h (map g l) :=\n  Eq.symm (map_map h g l)\n\n/--\nComposing a `list.map` with another `list.map` is equal to\na single `list.map` of composed functions.\n-/\n@[simp] theorem map_comp_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : α → β) : map g ∘ map f = map (g ∘ f) := sorry\n\ntheorem map_filter_eq_foldr {α : Type u} {β : Type v} (f : α → β) (p : α → Prop) [decidable_pred p] (as : List α) : map f (filter p as) = foldr (fun (a : α) (bs : List β) => ite (p a) (f a :: bs) bs) [] as := sorry\n\ntheorem last_map {α : Type u} {β : Type v} (f : α → β) {l : List α} (hl : l ≠ []) : last (map f l) (mt eq_nil_of_map_eq_nil hl) = f (last l hl) := sorry\n\n/-! ### map₂ -/\n\ntheorem nil_map₂ {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : List β) : map₂ f [] l = [] :=\n  list.cases_on l (Eq.refl (map₂ f [] [])) fun (l_hd : β) (l_tl : List β) => Eq.refl (map₂ f [] (l_hd :: l_tl))\n\ntheorem map₂_nil {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (l : List α) : map₂ f l [] = [] :=\n  list.cases_on l (Eq.refl (map₂ f [] [])) fun (l_hd : α) (l_tl : List α) => Eq.refl (map₂ f (l_hd :: l_tl) [])\n\n@[simp] theorem map₂_flip {α : Type u} {β : Type v} {γ : Type w} (f : α → β → γ) (as : List α) (bs : List β) : map₂ (flip f) bs as = map₂ f as bs := sorry\n\n/-! ### take, drop -/\n\n@[simp] theorem take_zero {α : Type u} (l : List α) : take 0 l = [] :=\n  rfl\n\n@[simp] theorem take_nil {α : Type u} (n : ℕ) : take n [] = [] :=\n  nat.cases_on n (idRhs (take 0 [] = take 0 []) rfl) fun (n : ℕ) => idRhs (take (n + 1) [] = take (n + 1) []) rfl\n\ntheorem take_cons {α : Type u} (n : ℕ) (a : α) (l : List α) : take (Nat.succ n) (a :: l) = a :: take n l :=\n  rfl\n\n@[simp] theorem take_length {α : Type u} (l : List α) : take (length l) l = l := sorry\n\ntheorem take_all_of_le {α : Type u} {n : ℕ} {l : List α} : length l ≤ n → take n l = l := sorry\n\n@[simp] theorem take_left {α : Type u} (l₁ : List α) (l₂ : List α) : take (length l₁) (l₁ ++ l₂) = l₁ := sorry\n\ntheorem take_left' {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (h : length l₁ = n) : take n (l₁ ++ l₂) = l₁ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (take n (l₁ ++ l₂) = l₁)) (Eq.symm h))) (take_left l₁ l₂)\n\ntheorem take_take {α : Type u} (n : ℕ) (m : ℕ) (l : List α) : take n (take m l) = take (min n m) l := sorry\n\ntheorem take_repeat {α : Type u} (a : α) (n : ℕ) (m : ℕ) : take n (repeat a m) = repeat a (min n m) := sorry\n\ntheorem map_take {α : Type u_1} {β : Type u_2} (f : α → β) (L : List α) (i : ℕ) : map f (take i L) = take i (map f L) := sorry\n\ntheorem take_append_of_le_length {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} : n ≤ length l₁ → take n (l₁ ++ l₂) = take n l₁ := sorry\n\n/-- Taking the first `l₁.length + i` elements in `l₁ ++ l₂` is the same as appending the first\n`i` elements of `l₂` to `l₁`. -/\ntheorem take_append {α : Type u} {l₁ : List α} {l₂ : List α} (i : ℕ) : take (length l₁ + i) (l₁ ++ l₂) = l₁ ++ take i l₂ := sorry\n\n/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of\nlength `> i`. Version designed to rewrite from the big list to the small list. -/\ntheorem nth_le_take {α : Type u} (L : List α) {i : ℕ} {j : ℕ} (hi : i < length L) (hj : i < j) : nth_le L i hi =\n  nth_le (take j L) i (eq.mpr (id (Eq._oldrec (Eq.refl (i < length (take j L))) (length_take j L))) (lt_min hj hi)) := sorry\n\n/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of\nlength `> i`. Version designed to rewrite from the small list to the big list. -/\ntheorem nth_le_take' {α : Type u} (L : List α) {i : ℕ} {j : ℕ} (hi : i < length (take j L)) : nth_le (take j L) i hi =\n  nth_le L i\n    (lt_of_lt_of_le hi\n      (eq.mpr\n        (id\n          (Eq.trans\n            (Eq.trans\n              (Eq.trans\n                ((fun (ᾰ ᾰ_1 : ℕ) (e_2 : ᾰ = ᾰ_1) (ᾰ_2 ᾰ_3 : ℕ) (e_3 : ᾰ_2 = ᾰ_3) => congr (congr_arg LessEq e_2) e_3)\n                  (length (take j L)) (min j (length L)) (length_take j L) (length L) (length L) (Eq.refl (length L)))\n                (propext min_le_iff))\n              ((fun (a a_1 : Prop) (e_1 : a = a_1) (b b_1 : Prop) (e_2 : b = b_1) => congr (congr_arg Or e_1) e_2)\n                (j ≤ length L) (j ≤ length L) (Eq.refl (j ≤ length L)) (length L ≤ length L) True\n                (propext ((fun {α : Type} (a : α) => iff_true_intro (le_refl a)) (length L)))))\n            (propext (or_true (j ≤ length L)))))\n        trivial)) := sorry\n\ntheorem nth_take {α : Type u} {l : List α} {n : ℕ} {m : ℕ} (h : m < n) : nth (take n l) m = nth l m := sorry\n\n@[simp] theorem nth_take_of_succ {α : Type u} {l : List α} {n : ℕ} : nth (take (n + 1) l) n = nth l n :=\n  nth_take (nat.lt_succ_self n)\n\ntheorem take_succ {α : Type u} {l : List α} {n : ℕ} : take (n + 1) l = take n l ++ option.to_list (nth l n) := sorry\n\n@[simp] theorem drop_nil {α : Type u} (n : ℕ) : drop n [] = [] :=\n  nat.cases_on n (idRhs (drop 0 [] = drop 0 []) rfl) fun (n : ℕ) => idRhs (drop (n + 1) [] = drop (n + 1) []) rfl\n\ntheorem mem_of_mem_drop {α : Type u_1} {n : ℕ} {l : List α} {x : α} (h : x ∈ drop n l) : x ∈ l := sorry\n\n@[simp] theorem drop_one {α : Type u} (l : List α) : drop 1 l = tail l :=\n  list.cases_on l (idRhs (drop 1 [] = drop 1 []) rfl)\n    fun (l_hd : α) (l_tl : List α) => idRhs (drop 1 (l_hd :: l_tl) = drop 1 (l_hd :: l_tl)) rfl\n\ntheorem drop_add {α : Type u} (m : ℕ) (n : ℕ) (l : List α) : drop (m + n) l = drop m (drop n l) := sorry\n\n@[simp] theorem drop_left {α : Type u} (l₁ : List α) (l₂ : List α) : drop (length l₁) (l₁ ++ l₂) = l₂ := sorry\n\ntheorem drop_left' {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} (h : length l₁ = n) : drop n (l₁ ++ l₂) = l₂ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (drop n (l₁ ++ l₂) = l₂)) (Eq.symm h))) (drop_left l₁ l₂)\n\ntheorem drop_eq_nth_le_cons {α : Type u} {n : ℕ} {l : List α} (h : n < length l) : drop n l = nth_le l n h :: drop (n + 1) l := sorry\n\n@[simp] theorem drop_length {α : Type u} (l : List α) : drop (length l) l = [] := sorry\n\ntheorem drop_append_of_le_length {α : Type u} {l₁ : List α} {l₂ : List α} {n : ℕ} : n ≤ length l₁ → drop n (l₁ ++ l₂) = drop n l₁ ++ l₂ := sorry\n\n/-- Dropping the elements up to `l₁.length + i` in `l₁ + l₂` is the same as dropping the elements\nup to `i` in `l₂`. -/\ntheorem drop_append {α : Type u} {l₁ : List α} {l₂ : List α} (i : ℕ) : drop (length l₁ + i) (l₁ ++ l₂) = drop i l₂ := sorry\n\n/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by\ndropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/\ntheorem nth_le_drop {α : Type u} (L : List α) {i : ℕ} {j : ℕ} (h : i + j < length L) : nth_le L (i + j) h =\n  nth_le (drop i L) j\n    (eq.mpr (id (Eq.refl (j < length (drop i L))))\n      (eq.mp\n        (Eq.trans\n          ((fun (ᾰ ᾰ_1 : ℕ) (e_2 : ᾰ = ᾰ_1) (ᾰ_2 ᾰ_3 : ℕ) (e_3 : ᾰ_2 = ᾰ_3) => congr (congr_arg Less e_2) e_3) (i + j)\n            (i + j) (Eq.refl (i + j)) (length (take i L ++ drop i L)) (i + length (drop i L))\n            (Eq.trans (length_append (take i L) (drop i L))\n              ((fun (ᾰ ᾰ_1 : ℕ) (e_2 : ᾰ = ᾰ_1) (ᾰ_2 ᾰ_3 : ℕ) (e_3 : ᾰ_2 = ᾰ_3) => congr (congr_arg Add.add e_2) e_3)\n                (length (take i L)) i\n                (Eq.trans (length_take i L)\n                  (min_eq_left (iff.mpr (iff_true_intro (le_of_lt (lt_of_le_of_lt (nat.le.intro rfl) h))) True.intro)))\n                (length (drop i L)) (length (drop i L)) (Eq.refl (length (drop i L))))))\n          (propext (add_lt_add_iff_left i)))\n        (eq.mp (Eq._oldrec (Eq.refl (i + j < length L)) (Eq.symm (take_append_drop i L))) h))) := sorry\n\n/--  The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by\ndropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/\ntheorem nth_le_drop' {α : Type u} (L : List α) {i : ℕ} {j : ℕ} (h : j < length (drop i L)) : nth_le (drop i L) j h = nth_le L (i + j) (nat.add_lt_of_lt_sub_left (length_drop i L ▸ h)) := sorry\n\n@[simp] theorem drop_drop {α : Type u} (n : ℕ) (m : ℕ) (l : List α) : drop n (drop m l) = drop (n + m) l := sorry\n\ntheorem drop_take {α : Type u} (m : ℕ) (n : ℕ) (l : List α) : drop m (take (m + n) l) = take n (drop m l) := sorry\n\ntheorem map_drop {α : Type u_1} {β : Type u_2} (f : α → β) (L : List α) (i : ℕ) : map f (drop i L) = drop i (map f L) := sorry\n\ntheorem modify_nth_tail_eq_take_drop {α : Type u} (f : List α → List α) (H : f [] = []) (n : ℕ) (l : List α) : modify_nth_tail f n l = take n l ++ f (drop n l) := sorry\n\ntheorem modify_nth_eq_take_drop {α : Type u} (f : α → α) (n : ℕ) (l : List α) : modify_nth f n l = take n l ++ modify_head f (drop n l) :=\n  modify_nth_tail_eq_take_drop (modify_head f) rfl\n\ntheorem modify_nth_eq_take_cons_drop {α : Type u} (f : α → α) {n : ℕ} {l : List α} (h : n < length l) : modify_nth f n l = take n l ++ f (nth_le l n h) :: drop (n + 1) l := sorry\n\ntheorem update_nth_eq_take_cons_drop {α : Type u} (a : α) {n : ℕ} {l : List α} (h : n < length l) : update_nth l n a = take n l ++ a :: drop (n + 1) l := sorry\n\ntheorem reverse_take {α : Type u_1} {xs : List α} (n : ℕ) (h : n ≤ length xs) : take n (reverse xs) = reverse (drop (length xs - n) xs) := sorry\n\n@[simp] theorem update_nth_eq_nil {α : Type u} (l : List α) (n : ℕ) (a : α) : update_nth l n a = [] ↔ l = [] := sorry\n\n@[simp] theorem take'_length {α : Type u} [Inhabited α] (n : ℕ) (l : List α) : length (take' n l) = n := sorry\n\n@[simp] theorem take'_nil {α : Type u} [Inhabited α] (n : ℕ) : take' n [] = repeat Inhabited.default n := sorry\n\ntheorem take'_eq_take {α : Type u} [Inhabited α] {n : ℕ} {l : List α} : n ≤ length l → take' n l = take n l := sorry\n\n@[simp] theorem take'_left {α : Type u} [Inhabited α] (l₁ : List α) (l₂ : List α) : take' (length l₁) (l₁ ++ l₂) = l₁ := sorry\n\ntheorem take'_left' {α : Type u} [Inhabited α] {l₁ : List α} {l₂ : List α} {n : ℕ} (h : length l₁ = n) : take' n (l₁ ++ l₂) = l₁ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (take' n (l₁ ++ l₂) = l₁)) (Eq.symm h))) (take'_left l₁ l₂)\n\n/-! ### foldl, foldr -/\n\ntheorem foldl_ext {α : Type u} {β : Type v} (f : α → β → α) (g : α → β → α) (a : α) {l : List β} (H : ∀ (a : α) (b : β), b ∈ l → f a b = g a b) : foldl f a l = foldl g a l := sorry\n\ntheorem foldr_ext {α : Type u} {β : Type v} (f : α → β → β) (g : α → β → β) (b : β) {l : List α} (H : ∀ (a : α), a ∈ l → ∀ (b : β), f a b = g a b) : foldr f b l = foldr g b l := sorry\n\n@[simp] theorem foldl_nil {α : Type u} {β : Type v} (f : α → β → α) (a : α) : foldl f a [] = a :=\n  rfl\n\n@[simp] theorem foldl_cons {α : Type u} {β : Type v} (f : α → β → α) (a : α) (b : β) (l : List β) : foldl f a (b :: l) = foldl f (f a b) l :=\n  rfl\n\n@[simp] theorem foldr_nil {α : Type u} {β : Type v} (f : α → β → β) (b : β) : foldr f b [] = b :=\n  rfl\n\n@[simp] theorem foldr_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : List α) : foldr f b (a :: l) = f a (foldr f b l) :=\n  rfl\n\n@[simp] theorem foldl_append {α : Type u} {β : Type v} (f : α → β → α) (a : α) (l₁ : List β) (l₂ : List β) : foldl f a (l₁ ++ l₂) = foldl f (foldl f a l₁) l₂ := sorry\n\n@[simp] theorem foldr_append {α : Type u} {β : Type v} (f : α → β → β) (b : β) (l₁ : List α) (l₂ : List α) : foldr f b (l₁ ++ l₂) = foldr f (foldr f b l₂) l₁ := sorry\n\n@[simp] theorem foldl_join {α : Type u} {β : Type v} (f : α → β → α) (a : α) (L : List (List β)) : foldl f a (join L) = foldl (foldl f) a L := sorry\n\n@[simp] theorem foldr_join {α : Type u} {β : Type v} (f : α → β → β) (b : β) (L : List (List α)) : foldr f b (join L) = foldr (fun (l : List α) (b : β) => foldr f b l) b L := sorry\n\ntheorem foldl_reverse {α : Type u} {β : Type v} (f : α → β → α) (a : α) (l : List β) : foldl f a (reverse l) = foldr (fun (x : β) (y : α) => f y x) a l := sorry\n\ntheorem foldr_reverse {α : Type u} {β : Type v} (f : α → β → β) (a : β) (l : List α) : foldr f a (reverse l) = foldl (fun (x : β) (y : α) => f y x) a l := sorry\n\n@[simp] theorem foldr_eta {α : Type u} (l : List α) : foldr List.cons [] l = l := sorry\n\n@[simp] theorem reverse_foldl {α : Type u} {l : List α} : reverse (foldl (fun (t : List α) (h : α) => h :: t) [] l) = l := sorry\n\n@[simp] theorem foldl_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : α → γ → α) (a : α) (l : List β) : foldl f a (map g l) = foldl (fun (x : α) (y : β) => f x (g y)) a l := sorry\n\n@[simp] theorem foldr_map {α : Type u} {β : Type v} {γ : Type w} (g : β → γ) (f : γ → α → α) (a : α) (l : List β) : foldr f a (map g l) = foldr (f ∘ g) a l := sorry\n\ntheorem foldl_map' {α : Type u} {β : Type u} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : foldl f' (g a) (map g l) = g (foldl f a l) := sorry\n\ntheorem foldr_map' {α : Type u} {β : Type u} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : foldr f' (g a) (map g l) = g (foldr f a l) := sorry\n\ntheorem foldl_hom {α : Type u} {β : Type v} {γ : Type w} (l : List γ) (f : α → β) (op : α → γ → α) (op' : β → γ → β) (a : α) (h : ∀ (a : α) (x : γ), f (op a x) = op' (f a) x) : foldl op' (f a) l = f (foldl op a l) := sorry\n\ntheorem foldr_hom {α : Type u} {β : Type v} {γ : Type w} (l : List γ) (f : α → β) (op : γ → α → α) (op' : γ → β → β) (a : α) (h : ∀ (x : γ) (a : α), f (op x a) = op' x (f a)) : foldr op' (f a) l = f (foldr op a l) := sorry\n\ntheorem injective_foldl_comp {α : Type u_1} {l : List (α → α)} {f : α → α} (hl : ∀ (f : α → α), f ∈ l → function.injective f) (hf : function.injective f) : function.injective (foldl function.comp f l) := sorry\n\n/- scanl -/\n\ntheorem length_scanl {α : Type u} {β : Type v} {f : β → α → β} (a : β) (l : List α) : length (scanl f a l) = length l + 1 := sorry\n\n@[simp] theorem scanl_nil {α : Type u} {β : Type v} {f : β → α → β} (b : β) : scanl f b [] = [b] :=\n  rfl\n\n@[simp] theorem scanl_cons {α : Type u} {β : Type v} {f : β → α → β} {b : β} {a : α} {l : List α} : scanl f b (a :: l) = [b] ++ scanl f (f b a) l := sorry\n\n@[simp] theorem nth_zero_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} : nth (scanl f b l) 0 = some b := sorry\n\n@[simp] theorem nth_le_zero_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} {h : 0 < length (scanl f b l)} : nth_le (scanl f b l) 0 h = b := sorry\n\ntheorem nth_succ_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} {i : ℕ} : nth (scanl f b l) (i + 1) = option.bind (nth (scanl f b l) i) fun (x : β) => option.map (fun (y : α) => f x y) (nth l i) := sorry\n\ntheorem nth_le_succ_scanl {α : Type u} {β : Type v} {f : β → α → β} {b : β} {l : List α} {i : ℕ} {h : i + 1 < length (scanl f b l)} : nth_le (scanl f b l) (i + 1) h =\n  f (nth_le (scanl f b l) i (nat.lt_of_succ_lt h))\n    (nth_le l i (nat.lt_of_succ_lt_succ (lt_of_lt_of_le h (le_of_eq (length_scanl b l))))) := sorry\n\n/- scanr -/\n\n@[simp] theorem scanr_nil {α : Type u} {β : Type v} (f : α → β → β) (b : β) : scanr f b [] = [b] :=\n  rfl\n\n@[simp] theorem scanr_aux_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : List α) : scanr_aux f b (a :: l) = (foldr f b (a :: l), scanr f b l) := sorry\n\n@[simp] theorem scanr_cons {α : Type u} {β : Type v} (f : α → β → β) (b : β) (a : α) (l : List α) : scanr f b (a :: l) = foldr f b (a :: l) :: scanr f b l := sorry\n\n-- foldl and foldr coincide when f is commutative and associative\n\ntheorem foldl1_eq_foldr1 {α : Type u} {f : α → α → α} (hassoc : associative f) (a : α) (b : α) (l : List α) : foldl f a (l ++ [b]) = foldr f b (a :: l) := sorry\n\ntheorem foldl_eq_of_comm_of_assoc {α : Type u} {f : α → α → α} (hcomm : commutative f) (hassoc : associative f) (a : α) (b : α) (l : List α) : foldl f a (b :: l) = f b (foldl f a l) := sorry\n\ntheorem foldl_eq_foldr {α : Type u} {f : α → α → α} (hcomm : commutative f) (hassoc : associative f) (a : α) (l : List α) : foldl f a l = foldr f a l := sorry\n\ntheorem foldl_eq_of_comm' {α : Type u} {β : Type v} {f : α → β → α} (hf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b) (a : α) (b : β) (l : List β) : foldl f a (b :: l) = f (foldl f a l) b := sorry\n\ntheorem foldl_eq_foldr' {α : Type u} {β : Type v} {f : α → β → α} (hf : ∀ (a : α) (b c : β), f (f a b) c = f (f a c) b) (a : α) (l : List β) : foldl f a l = foldr (flip f) a l := sorry\n\ntheorem foldr_eq_of_comm' {α : Type u} {β : Type v} {f : α → β → β} (hf : ∀ (a b : α) (c : β), f a (f b c) = f b (f a c)) (a : β) (b : α) (l : List α) : foldr f a (b :: l) = foldr f (f b a) l := sorry\n\ntheorem foldl_assoc {α : Type u} {op : α → α → α} [ha : is_associative α op] {l : List α} {a₁ : α} {a₂ : α} : foldl op (op a₁ a₂) l = op a₁ (foldl op a₂ l) := sorry\n\ntheorem foldl_op_eq_op_foldr_assoc {α : Type u} {op : α → α → α} [ha : is_associative α op] {l : List α} {a₁ : α} {a₂ : α} : op (foldl op a₁ l) a₂ = op a₁ (foldr op a₂ l) := sorry\n\ntheorem foldl_assoc_comm_cons {α : Type u} {op : α → α → α} [ha : is_associative α op] [hc : is_commutative α op] {l : List α} {a₁ : α} {a₂ : α} : foldl op a₂ (a₁ :: l) = op a₁ (foldl op a₂ l) := sorry\n\n/-! ### mfoldl, mfoldr, mmap -/\n\n@[simp] theorem mfoldl_nil {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : β → α → m β) {b : β} : mfoldl f b [] = pure b :=\n  rfl\n\n@[simp] theorem mfoldr_nil {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) {b : β} : mfoldr f b [] = pure b :=\n  rfl\n\n@[simp] theorem mfoldl_cons {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] {f : β → α → m β} {b : β} {a : α} {l : List α} : mfoldl f b (a :: l) =\n  do \n    let b' ← f b a \n    mfoldl f b' l :=\n  rfl\n\n@[simp] theorem mfoldr_cons {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] {f : α → β → m β} {b : β} {a : α} {l : List α} : mfoldr f b (a :: l) = mfoldr f b l >>= f a :=\n  rfl\n\ntheorem mfoldr_eq_foldr {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] (f : α → β → m β) (b : β) (l : List α) : mfoldr f b l = foldr (fun (a : α) (mb : m β) => mb >>= f a) (pure b) l := sorry\n\ntheorem mfoldl_eq_foldl {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] [is_lawful_monad m] (f : β → α → m β) (b : β) (l : List α) : mfoldl f b l =\n  foldl\n    (fun (mb : m β) (a : α) =>\n      do \n        let b ← mb \n        f b a)\n    (pure b) l := sorry\n\n@[simp] theorem mfoldl_append {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] [is_lawful_monad m] {f : β → α → m β} {b : β} {l₁ : List α} {l₂ : List α} : mfoldl f b (l₁ ++ l₂) =\n  do \n    let x ← mfoldl f b l₁ \n    mfoldl f x l₂ := sorry\n\n@[simp] theorem mfoldr_append {α : Type u} {β : Type v} {m : Type v → Type w} [Monad m] [is_lawful_monad m] {f : α → β → m β} {b : β} {l₁ : List α} {l₂ : List α} : mfoldr f b (l₁ ++ l₂) =\n  do \n    let x ← mfoldr f b l₂ \n    mfoldr f x l₁ := sorry\n\n/-! ### prod and sum -/\n\n-- list.sum was already defined in defs.lean, but we couldn't tag it with `to_additive` yet.\n\n@[simp] theorem sum_nil {α : Type u} [add_monoid α] : sum [] = 0 :=\n  rfl\n\ntheorem sum_singleton {α : Type u} [add_monoid α] {a : α} : sum [a] = a :=\n  zero_add a\n\n@[simp] theorem prod_cons {α : Type u} [monoid α] {l : List α} {a : α} : prod (a :: l) = a * prod l := sorry\n\n@[simp] theorem sum_append {α : Type u} [add_monoid α] {l₁ : List α} {l₂ : List α} : sum (l₁ ++ l₂) = sum l₁ + sum l₂ := sorry\n\n@[simp] theorem sum_join {α : Type u} [add_monoid α] {l : List (List α)} : sum (join l) = sum (map sum l) := sorry\n\ntheorem prod_ne_zero {R : Type u_1} [domain R] {L : List R} : (∀ (x : R), x ∈ L → x ≠ 0) → prod L ≠ 0 := sorry\n\ntheorem prod_eq_foldr {α : Type u} [monoid α] {l : List α} : prod l = foldr Mul.mul 1 l := sorry\n\ntheorem prod_hom_rel {α : Type u_1} {β : Type u_2} {γ : Type u_3} [monoid β] [monoid γ] (l : List α) {r : β → γ → Prop} {f : α → β} {g : α → γ} (h₁ : r 1 1) (h₂ : ∀ {a : α} {b : β} {c : γ}, r b c → r (f a * b) (g a * c)) : r (prod (map f l)) (prod (map g l)) := sorry\n\ntheorem prod_hom {α : Type u} {β : Type v} [monoid α] [monoid β] (l : List α) (f : α →* β) : prod (map (⇑f) l) = coe_fn f (prod l) := sorry\n\n-- `to_additive` chokes on the next few lemmas, so we do them by hand below\n\n@[simp] theorem prod_take_mul_prod_drop {α : Type u} [monoid α] (L : List α) (i : ℕ) : prod (take i L) * prod (drop i L) = prod L := sorry\n\n@[simp] theorem prod_take_succ {α : Type u} [monoid α] (L : List α) (i : ℕ) (p : i < length L) : prod (take (i + 1) L) = prod (take i L) * nth_le L i p := sorry\n\n/-- A list with product not one must have positive length. -/\ntheorem length_pos_of_prod_ne_one {α : Type u} [monoid α] (L : List α) (h : prod L ≠ 1) : 0 < length L := sorry\n\ntheorem prod_update_nth {α : Type u} [monoid α] (L : List α) (n : ℕ) (a : α) : prod (update_nth L n a) = prod (take n L) * ite (n < length L) a 1 * prod (drop (n + 1) L) := sorry\n\n/-- This is the `list.prod` version of `mul_inv_rev` -/\ntheorem sum_neg_reverse {α : Type u} [add_group α] (L : List α) : -sum L = sum (reverse (map (fun (x : α) => -x) L)) := sorry\n\n/-- A non-commutative variant of `list.prod_reverse` -/\ntheorem prod_reverse_noncomm {α : Type u} [group α] (L : List α) : prod (reverse L) = (prod (map (fun (x : α) => x⁻¹) L)⁻¹) := sorry\n\n/-- This is the `list.prod` version of `mul_inv` -/\ntheorem sum_neg {α : Type u} [add_comm_group α] (L : List α) : -sum L = sum (map (fun (x : α) => -x) L) := sorry\n\n@[simp] theorem sum_take_add_sum_drop {α : Type u} [add_monoid α] (L : List α) (i : ℕ) : sum (take i L) + sum (drop i L) = sum L := sorry\n\n@[simp] theorem sum_take_succ {α : Type u} [add_monoid α] (L : List α) (i : ℕ) (p : i < length L) : sum (take (i + 1) L) = sum (take i L) + nth_le L i p := sorry\n\ntheorem eq_of_sum_take_eq {α : Type u} [add_left_cancel_monoid α] {L : List α} {L' : List α} (h : length L = length L') (h' : ∀ (i : ℕ), i ≤ length L → sum (take i L) = sum (take i L')) : L = L' := sorry\n\ntheorem monotone_sum_take {α : Type u} [canonically_ordered_add_monoid α] (L : List α) : monotone fun (i : ℕ) => sum (take i L) := sorry\n\ntheorem one_le_prod_of_one_le {α : Type u} [ordered_comm_monoid α] {l : List α} (hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x) : 1 ≤ prod l := sorry\n\ntheorem single_le_prod {α : Type u} [ordered_comm_monoid α] {l : List α} (hl₁ : ∀ (x : α), x ∈ l → 1 ≤ x) (x : α) (H : x ∈ l) : x ≤ prod l := sorry\n\ntheorem all_zero_of_le_zero_le_of_sum_eq_zero {α : Type u} [ordered_add_comm_monoid α] {l : List α} (hl₁ : ∀ (x : α), x ∈ l → 0 ≤ x) (hl₂ : sum l = 0) (x : α) (H : x ∈ l) : x = 0 :=\n  le_antisymm (hl₂ ▸ single_le_sum hl₁ x hx) (hl₁ x hx)\n\ntheorem sum_eq_zero_iff {α : Type u} [canonically_ordered_add_monoid α] (l : List α) : sum l = 0 ↔ ∀ (x : α), x ∈ l → x = 0 := sorry\n\n/-- A list with sum not zero must have positive length. -/\ntheorem length_pos_of_sum_ne_zero {α : Type u} [add_monoid α] (L : List α) (h : sum L ≠ 0) : 0 < length L := sorry\n\n/-- If all elements in a list are bounded below by `1`, then the length of the list is bounded\nby the sum of the elements. -/\ntheorem length_le_sum_of_one_le (L : List ℕ) (h : ∀ (i : ℕ), i ∈ L → 1 ≤ i) : length L ≤ sum L := sorry\n\n-- Now we tie those lemmas back to their multiplicative versions.\n\n/-- A list with positive sum must have positive length. -/\n-- This is an easy consequence of `length_pos_of_sum_ne_zero`, but often useful in applications.\n\ntheorem length_pos_of_sum_pos {α : Type u} [ordered_cancel_add_comm_monoid α] (L : List α) (h : 0 < sum L) : 0 < length L :=\n  length_pos_of_sum_ne_zero L (ne_of_gt h)\n\n@[simp] theorem prod_erase {α : Type u} [DecidableEq α] [comm_monoid α] {a : α} {l : List α} : a ∈ l → a * prod (list.erase l a) = prod l := sorry\n\ntheorem dvd_prod {α : Type u} [comm_monoid α] {a : α} {l : List α} (ha : a ∈ l) : a ∣ prod l := sorry\n\n@[simp] theorem sum_const_nat (m : ℕ) (n : ℕ) : sum (repeat m n) = m * n := sorry\n\ntheorem dvd_sum {α : Type u} [comm_semiring α] {a : α} {l : List α} (h : ∀ (x : α), x ∈ l → a ∣ x) : a ∣ sum l := sorry\n\n@[simp] theorem length_join {α : Type u} (L : List (List α)) : length (join L) = sum (map length L) := sorry\n\n@[simp] theorem length_bind {α : Type u} {β : Type v} (l : List α) (f : α → List β) : length (list.bind l f) = sum (map (length ∘ f) l) := sorry\n\ntheorem exists_lt_of_sum_lt {α : Type u} {β : Type v} [linear_ordered_cancel_add_comm_monoid β] {l : List α} (f : α → β) (g : α → β) (h : sum (map f l) < sum (map g l)) : ∃ (x : α), ∃ (H : x ∈ l), f x < g x := sorry\n\ntheorem exists_le_of_sum_le {α : Type u} {β : Type v} [linear_ordered_cancel_add_comm_monoid β] {l : List α} (hl : l ≠ []) (f : α → β) (g : α → β) (h : sum (map f l) ≤ sum (map g l)) : ∃ (x : α), ∃ (H : x ∈ l), f x ≤ g x := sorry\n\n-- Several lemmas about sum/head/tail for `list ℕ`.\n\n-- These are hard to generalize well, as they rely on the fact that `default ℕ = 0`.\n\n-- We'd like to state this as `L.head * L.tail.prod = L.prod`,\n\n-- but because `L.head` relies on an inhabited instances and\n\n-- returns a garbage value for the empty list, this is not possible.\n\n-- Instead we write the statement in terms of `(L.nth 0).get_or_else 1`,\n\n-- and below, restate the lemma just for `ℕ`.\n\ntheorem head_mul_tail_prod' {α : Type u} [monoid α] (L : List α) : option.get_or_else (nth L 0) 1 * prod (tail L) = prod L := sorry\n\ntheorem head_add_tail_sum (L : List ℕ) : head L + sum (tail L) = sum L := sorry\n\ntheorem head_le_sum (L : List ℕ) : head L ≤ sum L :=\n  nat.le.intro (head_add_tail_sum L)\n\ntheorem tail_sum (L : List ℕ) : sum (tail L) = sum L - head L := sorry\n\n@[simp] theorem alternating_prod_nil {G : Type u_1} [comm_group G] : alternating_prod [] = 1 :=\n  rfl\n\n@[simp] theorem alternating_sum_singleton {G : Type u_1} [add_comm_group G] (g : G) : alternating_sum [g] = g :=\n  rfl\n\n@[simp] theorem alternating_sum_cons_cons' {G : Type u_1} [add_comm_group G] (g : G) (h : G) (l : List G) : alternating_sum (g :: h :: l) = g + -h + alternating_sum l :=\n  rfl\n\ntheorem alternating_sum_cons_cons {G : Type u_1} [add_comm_group G] (g : G) (h : G) (l : List G) : alternating_sum (g :: h :: l) = g - h + alternating_sum l := sorry\n\n/-! ### join -/\n\ntheorem join_eq_nil {α : Type u} {L : List (List α)} : join L = [] ↔ ∀ (l : List α), l ∈ L → l = [] := sorry\n\n@[simp] theorem join_append {α : Type u} (L₁ : List (List α)) (L₂ : List (List α)) : join (L₁ ++ L₂) = join L₁ ++ join L₂ := sorry\n\ntheorem join_join {α : Type u} (l : List (List (List α))) : join (join l) = join (map join l) := sorry\n\n/-- In a join, taking the first elements up to an index which is the sum of the lengths of the\nfirst `i` sublists, is the same as taking the join of the first `i` sublists. -/\ntheorem take_sum_join {α : Type u} (L : List (List α)) (i : ℕ) : take (sum (take i (map length L))) (join L) = join (take i L) := sorry\n\n/-- In a join, dropping all the elements up to an index which is the sum of the lengths of the\nfirst `i` sublists, is the same as taking the join after dropping the first `i` sublists. -/\ntheorem drop_sum_join {α : Type u} (L : List (List α)) (i : ℕ) : drop (sum (take i (map length L))) (join L) = join (drop i L) := sorry\n\n/-- Taking only the first `i+1` elements in a list, and then dropping the first `i` ones, one is\nleft with a list of length `1` made of the `i`-th element of the original list. -/\ntheorem drop_take_succ_eq_cons_nth_le {α : Type u} (L : List α) {i : ℕ} (hi : i < length L) : drop i (take (i + 1) L) = [nth_le L i hi] := sorry\n\n/-- In a join of sublists, taking the slice between the indices `A` and `B - 1` gives back the\noriginal sublist of index `i` if `A` is the sum of the lenghts of sublists of index `< i`, and\n`B` is the sum of the lengths of sublists of index `≤ i`. -/\ntheorem drop_take_succ_join_eq_nth_le {α : Type u} (L : List (List α)) {i : ℕ} (hi : i < length L) : drop (sum (take i (map length L))) (take (sum (take (i + 1) (map length L))) (join L)) = nth_le L i hi := sorry\n\n/-- Auxiliary lemma to control elements in a join. -/\ntheorem sum_take_map_length_lt1 {α : Type u} (L : List (List α)) {i : ℕ} {j : ℕ} (hi : i < length L) (hj : j < length (nth_le L i hi)) : sum (take i (map length L)) + j < sum (take (i + 1) (map length L)) := sorry\n\n/-- Auxiliary lemma to control elements in a join. -/\ntheorem sum_take_map_length_lt2 {α : Type u} (L : List (List α)) {i : ℕ} {j : ℕ} (hi : i < length L) (hj : j < length (nth_le L i hi)) : sum (take i (map length L)) + j < length (join L) := sorry\n\n/-- The `n`-th element in a join of sublists is the `j`-th element of the `i`th sublist,\nwhere `n` can be obtained in terms of `i` and `j` by adding the lengths of all the sublists\nof index `< i`, and adding `j`. -/\ntheorem nth_le_join {α : Type u} (L : List (List α)) {i : ℕ} {j : ℕ} (hi : i < length L) (hj : j < length (nth_le L i hi)) : nth_le (join L) (sum (take i (map length L)) + j) (sum_take_map_length_lt2 L hi hj) = nth_le (nth_le L i hi) j hj := sorry\n\n/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the\nsublists. -/\ntheorem eq_iff_join_eq {α : Type u} (L : List (List α)) (L' : List (List α)) : L = L' ↔ join L = join L' ∧ map length L = map length L' := sorry\n\n/-! ### lexicographic ordering -/\n\n/-- Given a strict order `<` on `α`, the lexicographic strict order on `list α`, for which\n`[a0, ..., an] < [b0, ..., b_k]` if `a0 < b0` or `a0 = b0` and `[a1, ..., an] < [b1, ..., bk]`.\nThe definition is given for any relation `r`, not only strict orders. -/\ninductive lex {α : Type u} (r : α → α → Prop) : List α → List α → Prop\nwhere\n| nil : ∀ {a : α} {l : List α}, lex r [] (a :: l)\n| cons : ∀ {a : α} {l₁ l₂ : List α}, lex r l₁ l₂ → lex r (a :: l₁) (a :: l₂)\n| rel : ∀ {a₁ : α} {l₁ : List α} {a₂ : α} {l₂ : List α}, r a₁ a₂ → lex r (a₁ :: l₁) (a₂ :: l₂)\n\nnamespace lex\n\n\ntheorem cons_iff {α : Type u} {r : α → α → Prop} [is_irrefl α r] {a : α} {l₁ : List α} {l₂ : List α} : lex r (a :: l₁) (a :: l₂) ↔ lex r l₁ l₂ := sorry\n\n@[simp] theorem not_nil_right {α : Type u} (r : α → α → Prop) (l : List α) : ¬lex r l [] := sorry\n\nprotected instance is_order_connected {α : Type u} (r : α → α → Prop) [is_order_connected α r] [is_trichotomous α r] : is_order_connected (List α) (lex r) :=\n  is_order_connected.mk fun (l₁ : List α) => sorry\n\nprotected instance is_trichotomous {α : Type u} (r : α → α → Prop) [is_trichotomous α r] : is_trichotomous (List α) (lex r) :=\n  is_trichotomous.mk fun (l₁ : List α) => sorry\n\nprotected instance is_asymm {α : Type u} (r : α → α → Prop) [is_asymm α r] : is_asymm (List α) (lex r) :=\n  is_asymm.mk fun (l₁ : List α) => sorry\n\nprotected instance is_strict_total_order {α : Type u} (r : α → α → Prop) [is_strict_total_order' α r] : is_strict_total_order' (List α) (lex r) :=\n  is_strict_total_order'.mk\n\nprotected instance decidable_rel {α : Type u} [DecidableEq α] (r : α → α → Prop) [DecidableRel r] : DecidableRel (lex r) :=\n  sorry\n\ntheorem append_right {α : Type u} (r : α → α → Prop) {s₁ : List α} {s₂ : List α} (t : List α) : lex r s₁ s₂ → lex r s₁ (s₂ ++ t) := sorry\n\ntheorem append_left {α : Type u} (R : α → α → Prop) {t₁ : List α} {t₂ : List α} (h : lex R t₁ t₂) (s : List α) : lex R (s ++ t₁) (s ++ t₂) := sorry\n\ntheorem imp {α : Type u} {r : α → α → Prop} {s : α → α → Prop} (H : ∀ (a b : α), r a b → s a b) (l₁ : List α) (l₂ : List α) : lex r l₁ l₂ → lex s l₁ l₂ := sorry\n\ntheorem to_ne {α : Type u} {l₁ : List α} {l₂ : List α} : lex ne l₁ l₂ → l₁ ≠ l₂ := sorry\n\ntheorem ne_iff {α : Type u} {l₁ : List α} {l₂ : List α} (H : length l₁ ≤ length l₂) : lex ne l₁ l₂ ↔ l₁ ≠ l₂ := sorry\n\nend lex\n\n\n--Note: this overrides an instance in core lean\n\nprotected instance has_lt' {α : Type u} [HasLess α] : HasLess (List α) :=\n  { Less := lex Less }\n\ntheorem nil_lt_cons {α : Type u} [HasLess α] (a : α) (l : List α) : [] < a :: l :=\n  lex.nil\n\nprotected instance linear_order {α : Type u} [linear_order α] : linear_order (List α) :=\n  linear_order_of_STO' (lex Less)\n\n--Note: this overrides an instance in core lean\n\nprotected instance has_le' {α : Type u} [linear_order α] : HasLessEq (List α) :=\n  preorder.to_has_le (List α)\n\n/-! ### all & any -/\n\n@[simp] theorem all_nil {α : Type u} (p : α → Bool) : all [] p = tt :=\n  rfl\n\n@[simp] theorem all_cons {α : Type u} (p : α → Bool) (a : α) (l : List α) : all (a :: l) p = p a && all l p :=\n  rfl\n\ntheorem all_iff_forall {α : Type u} {p : α → Bool} {l : List α} : ↥(all l p) ↔ ∀ (a : α), a ∈ l → ↥(p a) := sorry\n\ntheorem all_iff_forall_prop {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} : ↥(all l fun (a : α) => to_bool (p a)) ↔ ∀ (a : α), a ∈ l → p a := sorry\n\n@[simp] theorem any_nil {α : Type u} (p : α → Bool) : any [] p = false :=\n  rfl\n\n@[simp] theorem any_cons {α : Type u} (p : α → Bool) (a : α) (l : List α) : any (a :: l) p = p a || any l p :=\n  rfl\n\ntheorem any_iff_exists {α : Type u} {p : α → Bool} {l : List α} : ↥(any l p) ↔ ∃ (a : α), ∃ (H : a ∈ l), ↥(p a) := sorry\n\ntheorem any_iff_exists_prop {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} : ↥(any l fun (a : α) => to_bool (p a)) ↔ ∃ (a : α), ∃ (H : a ∈ l), p a := sorry\n\ntheorem any_of_mem {α : Type u} {p : α → Bool} {a : α} {l : List α} (h₁ : a ∈ l) (h₂ : ↥(p a)) : ↥(any l p) :=\n  iff.mpr any_iff_exists (Exists.intro a (Exists.intro h₁ h₂))\n\nprotected instance decidable_forall_mem {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : Decidable (∀ (x : α), x ∈ l → p x) :=\n  decidable_of_iff ↥(all l fun (a : α) => to_bool (p a)) sorry\n\nprotected instance decidable_exists_mem {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : Decidable (∃ (x : α), ∃ (H : x ∈ l), p x) :=\n  decidable_of_iff ↥(any l fun (a : α) => to_bool (p a)) sorry\n\n/-! ### map for partial functions -/\n\n/-- Partial map. If `f : Π a, p a → β` is a partial function defined on\n  `a : α` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`\n  but is defined only when all members of `l` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) (l : List α) : (∀ (a : α), a ∈ l → p a) → List β :=\n  sorry\n\n/-- \"Attach\" the proof that the elements of `l` are in `l` to produce a new list\n  with the same elements but in the type `{x // x ∈ l}`. -/\ndef attach {α : Type u} (l : List α) : List (Subtype fun (x : α) => x ∈ l) :=\n  pmap Subtype.mk l sorry\n\ntheorem sizeof_lt_sizeof_of_mem {α : Type u} [SizeOf α] {x : α} {l : List α} (hx : x ∈ l) : sizeof x < sizeof l := sorry\n\ntheorem pmap_eq_map {α : Type u} {β : Type v} (p : α → Prop) (f : α → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : pmap (fun (a : α) (_x : p a) => f a) l H = map f l := sorry\n\ntheorem pmap_congr {α : Type u} {β : Type v} {p : α → Prop} {q : α → Prop} {f : (a : α) → p a → β} {g : (a : α) → q a → β} (l : List α) {H₁ : ∀ (a : α), a ∈ l → p a} {H₂ : ∀ (a : α), a ∈ l → q a} (h : ∀ (a : α) (h₁ : p a) (h₂ : q a), f a h₁ = g a h₂) : pmap f l H₁ = pmap g l H₂ := sorry\n\ntheorem map_pmap {α : Type u} {β : Type v} {γ : Type w} {p : α → Prop} (g : β → γ) (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : map g (pmap f l H) = pmap (fun (a : α) (h : p a) => g (f a h)) l H := sorry\n\ntheorem pmap_map {α : Type u} {β : Type v} {γ : Type w} {p : β → Prop} (g : (b : β) → p b → γ) (f : α → β) (l : List α) (H : ∀ (a : β), a ∈ map f l → p a) : pmap g (map f l) H =\n  pmap (fun (a : α) (h : p (f a)) => g (f a) h) l fun (a : α) (h : a ∈ l) => H (f a) (mem_map_of_mem f h) := sorry\n\ntheorem pmap_eq_map_attach {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : pmap f l H =\n  map (fun (x : Subtype fun (x : α) => x ∈ l) => f (subtype.val x) (H (subtype.val x) (subtype.property x))) (attach l) := sorry\n\ntheorem attach_map_val {α : Type u} (l : List α) : map subtype.val (attach l) = l := sorry\n\n@[simp] theorem mem_attach {α : Type u} (l : List α) (x : Subtype fun (x : α) => x ∈ l) : x ∈ attach l := sorry\n\n@[simp] theorem mem_pmap {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} {b : β} : b ∈ pmap f l H ↔ ∃ (a : α), ∃ (h : a ∈ l), f a (H a h) = b := sorry\n\n@[simp] theorem length_pmap {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : length (pmap f l H) = length l := sorry\n\n@[simp] theorem length_attach {α : Type u} (L : List α) : length (attach L) = length L :=\n  length_pmap\n\n@[simp] theorem pmap_eq_nil {α : Type u} {β : Type v} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : pmap f l H = [] ↔ l = [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (pmap f l H = [] ↔ l = [])) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (pmap f l H) = 0 ↔ l = [])) length_pmap))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length l = 0 ↔ l = [])) (propext length_eq_zero))) (iff.refl (l = []))))\n\n@[simp] theorem attach_eq_nil {α : Type u} (l : List α) : attach l = [] ↔ l = [] :=\n  pmap_eq_nil\n\ntheorem last_pmap {α : Type u_1} {β : Type u_2} (p : α → Prop) (f : (a : α) → p a → β) (l : List α) (hl₁ : ∀ (a : α), a ∈ l → p a) (hl₂ : l ≠ []) : last (pmap f l hl₁) (mt (iff.mp pmap_eq_nil) hl₂) = f (last l hl₂) (hl₁ (last l hl₂) (last_mem hl₂)) := sorry\n\ntheorem nth_pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) (n : ℕ) : nth (pmap f l h) n = option.pmap f (nth l n) fun (x : α) (H : x ∈ nth l n) => h x (nth_mem H) := sorry\n\ntheorem nth_le_pmap {α : Type u} {β : Type v} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) {n : ℕ} (hn : n < length (pmap f l h)) : nth_le (pmap f l h) n hn =\n  f (nth_le l n (length_pmap ▸ hn)) (h (nth_le l n (length_pmap ▸ hn)) (nth_le_mem l n (length_pmap ▸ hn))) := sorry\n\n/-! ### find -/\n\n@[simp] theorem find_nil {α : Type u} (p : α → Prop) [decidable_pred p] : find p [] = none :=\n  rfl\n\n@[simp] theorem find_cons_of_pos {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} (l : List α) (h : p a) : find p (a :: l) = some a :=\n  if_pos h\n\n@[simp] theorem find_cons_of_neg {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} (l : List α) (h : ¬p a) : find p (a :: l) = find p l :=\n  if_neg h\n\n@[simp] theorem find_eq_none {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} : find p l = none ↔ ∀ (x : α), x ∈ l → ¬p x := sorry\n\ntheorem find_some {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} {a : α} (H : find p l = some a) : p a := sorry\n\n@[simp] theorem find_mem {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} {a : α} (H : find p l = some a) : a ∈ l := sorry\n\n/-! ### lookmap -/\n\n@[simp] theorem lookmap_nil {α : Type u} (f : α → Option α) : lookmap f [] = [] :=\n  rfl\n\n@[simp] theorem lookmap_cons_none {α : Type u} (f : α → Option α) {a : α} (l : List α) (h : f a = none) : lookmap f (a :: l) = a :: lookmap f l := sorry\n\n@[simp] theorem lookmap_cons_some {α : Type u} (f : α → Option α) {a : α} {b : α} (l : List α) (h : f a = some b) : lookmap f (a :: l) = b :: l := sorry\n\ntheorem lookmap_some {α : Type u} (l : List α) : lookmap some l = l :=\n  list.cases_on l (idRhs (lookmap some [] = lookmap some []) rfl)\n    fun (l_hd : α) (l_tl : List α) => idRhs (lookmap some (l_hd :: l_tl) = lookmap some (l_hd :: l_tl)) rfl\n\ntheorem lookmap_none {α : Type u} (l : List α) : lookmap (fun (_x : α) => none) l = l := sorry\n\ntheorem lookmap_congr {α : Type u} {f : α → Option α} {g : α → Option α} {l : List α} : (∀ (a : α), a ∈ l → f a = g a) → lookmap f l = lookmap g l := sorry\n\ntheorem lookmap_of_forall_not {α : Type u} (f : α → Option α) {l : List α} (H : ∀ (a : α), a ∈ l → f a = none) : lookmap f l = l :=\n  Eq.trans (lookmap_congr H) (lookmap_none l)\n\ntheorem lookmap_map_eq {α : Type u} {β : Type v} (f : α → Option α) (g : α → β) (h : ∀ (a b : α), b ∈ f a → g a = g b) (l : List α) : map g (lookmap f l) = map g l := sorry\n\ntheorem lookmap_id' {α : Type u} (f : α → Option α) (h : ∀ (a b : α), b ∈ f a → a = b) (l : List α) : lookmap f l = l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (lookmap f l = l)) (Eq.symm (map_id (lookmap f l)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (map id (lookmap f l) = l)) (lookmap_map_eq f id h l)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (map id l = l)) (map_id l))) (Eq.refl l)))\n\ntheorem length_lookmap {α : Type u} (f : α → Option α) (l : List α) : length (lookmap f l) = length l := sorry\n\n/-! ### filter_map -/\n\n@[simp] theorem filter_map_nil {α : Type u} {β : Type v} (f : α → Option β) : filter_map f [] = [] :=\n  rfl\n\n@[simp] theorem filter_map_cons_none {α : Type u} {β : Type v} {f : α → Option β} (a : α) (l : List α) (h : f a = none) : filter_map f (a :: l) = filter_map f l := sorry\n\n@[simp] theorem filter_map_cons_some {α : Type u} {β : Type v} (f : α → Option β) (a : α) (l : List α) {b : β} (h : f a = some b) : filter_map f (a :: l) = b :: filter_map f l := sorry\n\ntheorem filter_map_append {α : Type u_1} {β : Type u_2} (l : List α) (l' : List α) (f : α → Option β) : filter_map f (l ++ l') = filter_map f l ++ filter_map f l' := sorry\n\ntheorem filter_map_eq_map {α : Type u} {β : Type v} (f : α → β) : filter_map (some ∘ f) = map f := sorry\n\ntheorem filter_map_eq_filter {α : Type u} (p : α → Prop) [decidable_pred p] : filter_map (option.guard p) = filter p := sorry\n\ntheorem filter_map_filter_map {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β) (g : β → Option γ) (l : List α) : filter_map g (filter_map f l) = filter_map (fun (x : α) => option.bind (f x) g) l := sorry\n\ntheorem map_filter_map {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β) (g : β → γ) (l : List α) : map g (filter_map f l) = filter_map (fun (x : α) => option.map g (f x)) l := sorry\n\ntheorem filter_map_map {α : Type u} {β : Type v} {γ : Type w} (f : α → β) (g : β → Option γ) (l : List α) : filter_map g (map f l) = filter_map (g ∘ f) l := sorry\n\ntheorem filter_filter_map {α : Type u} {β : Type v} (f : α → Option β) (p : β → Prop) [decidable_pred p] (l : List α) : filter p (filter_map f l) = filter_map (fun (x : α) => option.filter p (f x)) l := sorry\n\ntheorem filter_map_filter {α : Type u} {β : Type v} (p : α → Prop) [decidable_pred p] (f : α → Option β) (l : List α) : filter_map f (filter p l) = filter_map (fun (x : α) => ite (p x) (f x) none) l := sorry\n\n@[simp] theorem filter_map_some {α : Type u} (l : List α) : filter_map some l = l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter_map some l = l)) (filter_map_eq_map fun (x : α) => x))) (map_id l)\n\n@[simp] theorem mem_filter_map {α : Type u} {β : Type v} (f : α → Option β) (l : List α) {b : β} : b ∈ filter_map f l ↔ ∃ (a : α), a ∈ l ∧ f a = some b := sorry\n\ntheorem map_filter_map_of_inv {α : Type u} {β : Type v} (f : α → Option β) (g : β → α) (H : ∀ (x : α), option.map g (f x) = some x) (l : List α) : map g (filter_map f l) = l := sorry\n\ntheorem sublist.filter_map {α : Type u} {β : Type v} (f : α → Option β) {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) : filter_map f l₁ <+ filter_map f l₂ := sorry\n\ntheorem sublist.map {α : Type u} {β : Type v} (f : α → β) {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) : map f l₁ <+ map f l₂ :=\n  filter_map_eq_map f ▸ sublist.filter_map (some ∘ f) s\n\n/-! ### reduce_option -/\n\n@[simp] theorem reduce_option_cons_of_some {α : Type u} (x : α) (l : List (Option α)) : reduce_option (some x :: l) = x :: reduce_option l := sorry\n\n@[simp] theorem reduce_option_cons_of_none {α : Type u} (l : List (Option α)) : reduce_option (none :: l) = reduce_option l := sorry\n\n@[simp] theorem reduce_option_nil {α : Type u} : reduce_option [] = [] :=\n  rfl\n\n@[simp] theorem reduce_option_map {α : Type u} {β : Type v} {l : List (Option α)} {f : α → β} : reduce_option (map (option.map f) l) = map f (reduce_option l) := sorry\n\ntheorem reduce_option_append {α : Type u} (l : List (Option α)) (l' : List (Option α)) : reduce_option (l ++ l') = reduce_option l ++ reduce_option l' :=\n  filter_map_append l l' id\n\ntheorem reduce_option_length_le {α : Type u} (l : List (Option α)) : length (reduce_option l) ≤ length l := sorry\n\ntheorem reduce_option_length_eq_iff {α : Type u} {l : List (Option α)} : length (reduce_option l) = length l ↔ ∀ (x : Option α), x ∈ l → ↥(option.is_some x) := sorry\n\ntheorem reduce_option_length_lt_iff {α : Type u} {l : List (Option α)} : length (reduce_option l) < length l ↔ none ∈ l := sorry\n\ntheorem reduce_option_singleton {α : Type u} (x : Option α) : reduce_option [x] = option.to_list x :=\n  option.cases_on x (Eq.refl (reduce_option [none])) fun (x : α) => Eq.refl (reduce_option [some x])\n\ntheorem reduce_option_concat {α : Type u} (l : List (Option α)) (x : Option α) : reduce_option (concat l x) = reduce_option l ++ option.to_list x := sorry\n\ntheorem reduce_option_concat_of_some {α : Type u} (l : List (Option α)) (x : α) : reduce_option (concat l (some x)) = concat (reduce_option l) x := sorry\n\ntheorem reduce_option_mem_iff {α : Type u} {l : List (Option α)} {x : α} : x ∈ reduce_option l ↔ some x ∈ l := sorry\n\ntheorem reduce_option_nth_iff {α : Type u} {l : List (Option α)} {x : α} : (∃ (i : ℕ), nth l i = some (some x)) ↔ ∃ (i : ℕ), nth (reduce_option l) i = some x := sorry\n\n/-! ### filter -/\n\ntheorem filter_eq_foldr {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : filter p l = foldr (fun (a : α) (out : List α) => ite (p a) (a :: out) out) [] l := sorry\n\ntheorem filter_congr {α : Type u} {p : α → Prop} {q : α → Prop} [decidable_pred p] [decidable_pred q] {l : List α} : (∀ (x : α), x ∈ l → (p x ↔ q x)) → filter p l = filter q l := sorry\n\n@[simp] theorem filter_subset {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : filter p l ⊆ l :=\n  sublist.subset (filter_sublist l)\n\ntheorem of_mem_filter {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} : a ∈ filter p l → p a := sorry\n\ntheorem mem_of_mem_filter {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} (h : a ∈ filter p l) : a ∈ l :=\n  filter_subset l h\n\ntheorem mem_filter_of_mem {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} : a ∈ l → p a → a ∈ filter p l := sorry\n\n@[simp] theorem mem_filter {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} : a ∈ filter p l ↔ a ∈ l ∧ p a := sorry\n\ntheorem filter_eq_self {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} : filter p l = l ↔ ∀ (a : α), a ∈ l → p a := sorry\n\ntheorem filter_eq_nil {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} : filter p l = [] ↔ ∀ (a : α), a ∈ l → ¬p a := sorry\n\ntheorem filter_sublist_filter {α : Type u} (p : α → Prop) [decidable_pred p] {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ :=\n  filter_map_eq_filter p ▸ sublist.filter_map (option.guard p) s\n\ntheorem filter_of_map {α : Type u} {β : Type v} (p : α → Prop) [decidable_pred p] (f : β → α) (l : List β) : filter p (map f l) = map f (filter (p ∘ f) l) := sorry\n\n@[simp] theorem filter_filter {α : Type u} (p : α → Prop) [decidable_pred p] (q : α → Prop) [decidable_pred q] (l : List α) : filter p (filter q l) = filter (fun (a : α) => p a ∧ q a) l := sorry\n\n@[simp] theorem filter_true {α : Type u} {h : decidable_pred fun (a : α) => True} (l : List α) : filter (fun (a : α) => True) l = l := sorry\n\n@[simp] theorem filter_false {α : Type u} {h : decidable_pred fun (a : α) => False} (l : List α) : filter (fun (a : α) => False) l = [] := sorry\n\n@[simp] theorem span_eq_take_drop {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : span p l = (take_while p l, drop_while p l) := sorry\n\n@[simp] theorem take_while_append_drop {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : take_while p l ++ drop_while p l = l := sorry\n\n@[simp] theorem countp_nil {α : Type u} (p : α → Prop) [decidable_pred p] : countp p [] = 0 :=\n  rfl\n\n@[simp] theorem countp_cons_of_pos {α : Type u} (p : α → Prop) [decidable_pred p] {a : α} (l : List α) (pa : p a) : countp p (a :: l) = countp p l + 1 :=\n  if_pos pa\n\n@[simp] theorem countp_cons_of_neg {α : Type u} (p : α → Prop) [decidable_pred p] {a : α} (l : List α) (pa : ¬p a) : countp p (a :: l) = countp p l :=\n  if_neg pa\n\ntheorem countp_eq_length_filter {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : countp p l = length (filter p l) := sorry\n\n@[simp] theorem countp_append {α : Type u} (p : α → Prop) [decidable_pred p] (l₁ : List α) (l₂ : List α) : countp p (l₁ ++ l₂) = countp p l₁ + countp p l₂ := sorry\n\ntheorem countp_pos {α : Type u} (p : α → Prop) [decidable_pred p] {l : List α} : 0 < countp p l ↔ ∃ (a : α), ∃ (H : a ∈ l), p a := sorry\n\ntheorem countp_le_of_sublist {α : Type u} (p : α → Prop) [decidable_pred p] {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) : countp p l₁ ≤ countp p l₂ := sorry\n\n@[simp] theorem countp_filter {α : Type u} (p : α → Prop) [decidable_pred p] {q : α → Prop} [decidable_pred q] (l : List α) : countp p (filter q l) = countp (fun (a : α) => p a ∧ q a) l := sorry\n\n/-! ### count -/\n\n@[simp] theorem count_nil {α : Type u} [DecidableEq α] (a : α) : count a [] = 0 :=\n  rfl\n\ntheorem count_cons {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : count a (b :: l) = ite (a = b) (Nat.succ (count a l)) (count a l) :=\n  rfl\n\ntheorem count_cons' {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : count a (b :: l) = count a l + ite (a = b) 1 0 := sorry\n\n@[simp] theorem count_cons_self {α : Type u} [DecidableEq α] (a : α) (l : List α) : count a (a :: l) = Nat.succ (count a l) :=\n  if_pos rfl\n\n@[simp] theorem count_cons_of_ne {α : Type u} [DecidableEq α] {a : α} {b : α} (h : a ≠ b) (l : List α) : count a (b :: l) = count a l :=\n  if_neg h\n\ntheorem count_tail {α : Type u} [DecidableEq α] (l : List α) (a : α) (h : 0 < length l) : count a (tail l) = count a l - ite (a = nth_le l 0 h) 1 0 := sorry\n\ntheorem count_le_of_sublist {α : Type u} [DecidableEq α] (a : α) {l₁ : List α} {l₂ : List α} : l₁ <+ l₂ → count a l₁ ≤ count a l₂ :=\n  countp_le_of_sublist (Eq a)\n\ntheorem count_le_count_cons {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : count a l ≤ count a (b :: l) :=\n  count_le_of_sublist a (sublist_cons b l)\n\ntheorem count_singleton {α : Type u} [DecidableEq α] (a : α) : count a [a] = 1 :=\n  if_pos rfl\n\n@[simp] theorem count_append {α : Type u} [DecidableEq α] (a : α) (l₁ : List α) (l₂ : List α) : count a (l₁ ++ l₂) = count a l₁ + count a l₂ :=\n  countp_append (Eq a)\n\ntheorem count_concat {α : Type u} [DecidableEq α] (a : α) (l : List α) : count a (concat l a) = Nat.succ (count a l) := sorry\n\ntheorem count_pos {α : Type u} [DecidableEq α] {a : α} {l : List α} : 0 < count a l ↔ a ∈ l := sorry\n\n@[simp] theorem count_eq_zero_of_not_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : ¬a ∈ l) : count a l = 0 :=\n  by_contradiction fun (h' : ¬count a l = 0) => h (iff.mp count_pos (nat.pos_of_ne_zero h'))\n\ntheorem not_mem_of_count_eq_zero {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : count a l = 0) : ¬a ∈ l :=\n  fun (h' : a ∈ l) => ne_of_gt (iff.mpr count_pos h') h\n\n@[simp] theorem count_repeat {α : Type u} [DecidableEq α] (a : α) (n : ℕ) : count a (repeat a n) = n := sorry\n\ntheorem le_count_iff_repeat_sublist {α : Type u} [DecidableEq α] {a : α} {l : List α} {n : ℕ} : n ≤ count a l ↔ repeat a n <+ l := sorry\n\ntheorem repeat_count_eq_of_count_eq_length {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : count a l = length l) : repeat a (count a l) = l :=\n  eq_of_sublist_of_length_eq (iff.mp le_count_iff_repeat_sublist (le_refl (count a l)))\n    (Eq.trans (length_repeat a (count a l)) h)\n\n@[simp] theorem count_filter {α : Type u} [DecidableEq α] {p : α → Prop} [decidable_pred p] {a : α} {l : List α} (h : p a) : count a (filter p l) = count a l := sorry\n\n/-! ### prefix, suffix, infix -/\n\n@[simp] theorem prefix_append {α : Type u} (l₁ : List α) (l₂ : List α) : l₁ <+: l₁ ++ l₂ :=\n  Exists.intro l₂ rfl\n\n@[simp] theorem suffix_append {α : Type u} (l₁ : List α) (l₂ : List α) : l₂ <:+ l₁ ++ l₂ :=\n  Exists.intro l₁ rfl\n\ntheorem infix_append {α : Type u} (l₁ : List α) (l₂ : List α) (l₃ : List α) : l₂ <:+: l₁ ++ l₂ ++ l₃ :=\n  Exists.intro l₁ (Exists.intro l₃ rfl)\n\n@[simp] theorem infix_append' {α : Type u} (l₁ : List α) (l₂ : List α) (l₃ : List α) : l₂ <:+: l₁ ++ (l₂ ++ l₃) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (l₂ <:+: l₁ ++ (l₂ ++ l₃))) (Eq.symm (append_assoc l₁ l₂ l₃)))) (infix_append l₁ l₂ l₃)\n\ntheorem nil_prefix {α : Type u} (l : List α) : [] <+: l :=\n  Exists.intro l rfl\n\ntheorem nil_suffix {α : Type u} (l : List α) : [] <:+ l :=\n  Exists.intro l (append_nil l)\n\ntheorem prefix_refl {α : Type u} (l : List α) : l <+: l :=\n  Exists.intro [] (append_nil l)\n\ntheorem suffix_refl {α : Type u} (l : List α) : l <:+ l :=\n  Exists.intro [] rfl\n\n@[simp] theorem suffix_cons {α : Type u} (a : α) (l : List α) : l <:+ a :: l :=\n  suffix_append [a]\n\ntheorem prefix_concat {α : Type u} (a : α) (l : List α) : l <+: concat l a := sorry\n\ntheorem infix_of_prefix {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ → l₁ <:+: l₂ := sorry\n\ntheorem infix_of_suffix {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ → l₁ <:+: l₂ := sorry\n\ntheorem infix_refl {α : Type u} (l : List α) : l <:+: l :=\n  infix_of_prefix (prefix_refl l)\n\ntheorem nil_infix {α : Type u} (l : List α) : [] <:+: l :=\n  infix_of_prefix (nil_prefix l)\n\ntheorem infix_cons {α : Type u} {L₁ : List α} {L₂ : List α} {x : α} : L₁ <:+: L₂ → L₁ <:+: x :: L₂ := sorry\n\ntheorem is_prefix.trans {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <+: l₂ → l₂ <+: l₃ → l₁ <+: l₃ := sorry\n\ntheorem is_suffix.trans {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <:+ l₂ → l₂ <:+ l₃ → l₁ <:+ l₃ := sorry\n\ntheorem is_infix.trans {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <:+: l₂ → l₂ <:+: l₃ → l₁ <:+: l₃ := sorry\n\ntheorem sublist_of_infix {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <:+: l₂ → l₁ <+ l₂ := sorry\n\ntheorem sublist_of_prefix {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ → l₁ <+ l₂ :=\n  sublist_of_infix ∘ infix_of_prefix\n\ntheorem sublist_of_suffix {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ → l₁ <+ l₂ :=\n  sublist_of_infix ∘ infix_of_suffix\n\ntheorem reverse_suffix {α : Type u} {l₁ : List α} {l₂ : List α} : reverse l₁ <:+ reverse l₂ ↔ l₁ <+: l₂ := sorry\n\ntheorem reverse_prefix {α : Type u} {l₁ : List α} {l₂ : List α} : reverse l₁ <+: reverse l₂ ↔ l₁ <:+ l₂ := sorry\n\ntheorem length_le_of_infix {α : Type u} {l₁ : List α} {l₂ : List α} (s : l₁ <:+: l₂) : length l₁ ≤ length l₂ :=\n  length_le_of_sublist (sublist_of_infix s)\n\ntheorem eq_nil_of_infix_nil {α : Type u} {l : List α} (s : l <:+: []) : l = [] :=\n  eq_nil_of_sublist_nil (sublist_of_infix s)\n\ntheorem eq_nil_of_prefix_nil {α : Type u} {l : List α} (s : l <+: []) : l = [] :=\n  eq_nil_of_infix_nil (infix_of_prefix s)\n\ntheorem eq_nil_of_suffix_nil {α : Type u} {l : List α} (s : l <:+ []) : l = [] :=\n  eq_nil_of_infix_nil (infix_of_suffix s)\n\ntheorem infix_iff_prefix_suffix {α : Type u} (l₁ : List α) (l₂ : List α) : l₁ <:+: l₂ ↔ ∃ (t : List α), l₁ <+: t ∧ t <:+ l₂ := sorry\n\ntheorem eq_of_infix_of_length_eq {α : Type u} {l₁ : List α} {l₂ : List α} (s : l₁ <:+: l₂) : length l₁ = length l₂ → l₁ = l₂ :=\n  eq_of_sublist_of_length_eq (sublist_of_infix s)\n\ntheorem eq_of_prefix_of_length_eq {α : Type u} {l₁ : List α} {l₂ : List α} (s : l₁ <+: l₂) : length l₁ = length l₂ → l₁ = l₂ :=\n  eq_of_sublist_of_length_eq (sublist_of_prefix s)\n\ntheorem eq_of_suffix_of_length_eq {α : Type u} {l₁ : List α} {l₂ : List α} (s : l₁ <:+ l₂) : length l₁ = length l₂ → l₁ = l₂ :=\n  eq_of_sublist_of_length_eq (sublist_of_suffix s)\n\ntheorem prefix_of_prefix_length_le {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂ := sorry\n\ntheorem prefix_or_prefix_of_prefix {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁ <+: l₃) (h₂ : l₂ <+: l₃) : l₁ <+: l₂ ∨ l₂ <+: l₁ :=\n  or.imp (prefix_of_prefix_length_le h₁ h₂) (prefix_of_prefix_length_le h₂ h₁) (le_total (length l₁) (length l₂))\n\ntheorem suffix_of_suffix_length_le {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) (ll : length l₁ ≤ length l₂) : l₁ <:+ l₂ := sorry\n\ntheorem suffix_or_suffix_of_suffix {α : Type u} {l₁ : List α} {l₂ : List α} {l₃ : List α} (h₁ : l₁ <:+ l₃) (h₂ : l₂ <:+ l₃) : l₁ <:+ l₂ ∨ l₂ <:+ l₁ :=\n  or.imp (iff.mp reverse_prefix) (iff.mp reverse_prefix)\n    (prefix_or_prefix_of_prefix (iff.mpr reverse_prefix h₁) (iff.mpr reverse_prefix h₂))\n\ntheorem infix_of_mem_join {α : Type u} {L : List (List α)} {l : List α} : l ∈ L → l <:+: join L := sorry\n\ntheorem prefix_append_right_inj {α : Type u} {l₁ : List α} {l₂ : List α} (l : List α) : l ++ l₁ <+: l ++ l₂ ↔ l₁ <+: l₂ := sorry\n\ntheorem prefix_cons_inj {α : Type u} {l₁ : List α} {l₂ : List α} (a : α) : a :: l₁ <+: a :: l₂ ↔ l₁ <+: l₂ :=\n  prefix_append_right_inj [a]\n\ntheorem take_prefix {α : Type u} (n : ℕ) (l : List α) : take n l <+: l :=\n  Exists.intro (drop n l) (take_append_drop n l)\n\ntheorem drop_suffix {α : Type u} (n : ℕ) (l : List α) : drop n l <:+ l :=\n  Exists.intro (take n l) (take_append_drop n l)\n\ntheorem tail_suffix {α : Type u} (l : List α) : tail l <:+ l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (tail l <:+ l)) (Eq.symm (drop_one l)))) (drop_suffix 1 l)\n\ntheorem tail_subset {α : Type u} (l : List α) : tail l ⊆ l :=\n  sublist.subset (sublist_of_suffix (tail_suffix l))\n\ntheorem prefix_iff_eq_append {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ := sorry\n\ntheorem suffix_iff_eq_append {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ := sorry\n\ntheorem prefix_iff_eq_take {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ := sorry\n\ntheorem suffix_iff_eq_drop {α : Type u} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length l₁) l₂ := sorry\n\nprotected instance decidable_prefix {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <+: l₂) :=\n  sorry\n\n-- Alternatively, use mem_tails\n\nprotected instance decidable_suffix {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <:+ l₂) :=\n  sorry\n\ntheorem prefix_take_le_iff {α : Type u} {L : List (List (Option α))} {m : ℕ} {n : ℕ} (hm : m < length L) : take m L <+: take n L ↔ m ≤ n := sorry\n\ntheorem cons_prefix_iff {α : Type u} {l : List α} {l' : List α} {x : α} {y : α} : x :: l <+: y :: l' ↔ x = y ∧ l <+: l' := sorry\n\ntheorem map_prefix {α : Type u} {β : Type v} {l : List α} {l' : List α} (f : α → β) (h : l <+: l') : map f l <+: map f l' := sorry\n\ntheorem is_prefix.filter_map {α : Type u} {β : Type v} {l : List α} {l' : List α} (h : l <+: l') (f : α → Option β) : filter_map f l <+: filter_map f l' := sorry\n\ntheorem is_prefix.reduce_option {α : Type u} {l : List (Option α)} {l' : List (Option α)} (h : l <+: l') : reduce_option l <+: reduce_option l' :=\n  is_prefix.filter_map h id\n\n@[simp] theorem mem_inits {α : Type u} (s : List α) (t : List α) : s ∈ inits t ↔ s <+: t := sorry\n\n@[simp] theorem mem_tails {α : Type u} (s : List α) (t : List α) : s ∈ tails t ↔ s <:+ t := sorry\n\ntheorem inits_cons {α : Type u} (a : α) (l : List α) : inits (a :: l) = [] :: map (fun (t : List α) => a :: t) (inits l) := sorry\n\ntheorem tails_cons {α : Type u} (a : α) (l : List α) : tails (a :: l) = (a :: l) :: tails l := sorry\n\n@[simp] theorem inits_append {α : Type u} (s : List α) (t : List α) : inits (s ++ t) = inits s ++ map (fun (l : List α) => s ++ l) (tail (inits t)) := sorry\n\n@[simp] theorem tails_append {α : Type u} (s : List α) (t : List α) : tails (s ++ t) = map (fun (l : List α) => l ++ t) (tails s) ++ tail (tails t) := sorry\n\n-- the lemma names `inits_eq_tails` and `tails_eq_inits` are like `sublists_eq_sublists'`\n\ntheorem inits_eq_tails {α : Type u} (l : List α) : inits l = reverse (map reverse (tails (reverse l))) := sorry\n\ntheorem tails_eq_inits {α : Type u} (l : List α) : tails l = reverse (map reverse (inits (reverse l))) := sorry\n\ntheorem inits_reverse {α : Type u} (l : List α) : inits (reverse l) = reverse (map reverse (tails l)) := sorry\n\ntheorem tails_reverse {α : Type u} (l : List α) : tails (reverse l) = reverse (map reverse (inits l)) := sorry\n\ntheorem map_reverse_inits {α : Type u} (l : List α) : map reverse (inits l) = reverse (tails (reverse l)) := sorry\n\ntheorem map_reverse_tails {α : Type u} (l : List α) : map reverse (tails l) = reverse (inits (reverse l)) := sorry\n\nprotected instance decidable_infix {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <:+: l₂) :=\n  sorry\n\n/-! ### sublists -/\n\n@[simp] theorem sublists'_nil {α : Type u} : sublists' [] = [[]] :=\n  rfl\n\n@[simp] theorem sublists'_singleton {α : Type u} (a : α) : sublists' [a] = [[], [a]] :=\n  rfl\n\ntheorem map_sublists'_aux {α : Type u} {β : Type v} {γ : Type w} (g : List β → List γ) (l : List α) (f : List α → List β) (r : List (List β)) : map g (sublists'_aux l f r) = sublists'_aux l (g ∘ f) (map g r) := sorry\n\ntheorem sublists'_aux_append {α : Type u} {β : Type v} (r' : List (List β)) (l : List α) (f : List α → List β) (r : List (List β)) : sublists'_aux l f (r ++ r') = sublists'_aux l f r ++ r' := sorry\n\ntheorem sublists'_aux_eq_sublists' {α : Type u} {β : Type v} (l : List α) (f : List α → List β) (r : List (List β)) : sublists'_aux l f r = map f (sublists' l) ++ r := sorry\n\n@[simp] theorem sublists'_cons {α : Type u} (a : α) (l : List α) : sublists' (a :: l) = sublists' l ++ map (List.cons a) (sublists' l) := sorry\n\n@[simp] theorem mem_sublists' {α : Type u} {s : List α} {t : List α} : s ∈ sublists' t ↔ s <+ t := sorry\n\n@[simp] theorem length_sublists' {α : Type u} (l : List α) : length (sublists' l) = bit0 1 ^ length l := sorry\n\n@[simp] theorem sublists_nil {α : Type u} : sublists [] = [[]] :=\n  rfl\n\n@[simp] theorem sublists_singleton {α : Type u} (a : α) : sublists [a] = [[], [a]] :=\n  rfl\n\ntheorem sublists_aux₁_eq_sublists_aux {α : Type u} {β : Type v} (l : List α) (f : List α → List β) : sublists_aux₁ l f = sublists_aux l fun (ys : List α) (r : List β) => f ys ++ r := sorry\n\ntheorem sublists_aux_cons_eq_sublists_aux₁ {α : Type u} (l : List α) : sublists_aux l List.cons = sublists_aux₁ l fun (x : List α) => [x] := sorry\n\ntheorem sublists_aux_eq_foldr.aux {α : Type u} {β : Type v} {a : α} {l : List α} (IH₁ : ∀ (f : List α → List β → List β), sublists_aux l f = foldr f [] (sublists_aux l List.cons)) (IH₂ : ∀ (f : List α → List (List α) → List (List α)), sublists_aux l f = foldr f [] (sublists_aux l List.cons)) (f : List α → List β → List β) : sublists_aux (a :: l) f = foldr f [] (sublists_aux (a :: l) List.cons) := sorry\n\ntheorem sublists_aux_eq_foldr {α : Type u} {β : Type v} (l : List α) (f : List α → List β → List β) : sublists_aux l f = foldr f [] (sublists_aux l List.cons) := sorry\n\ntheorem sublists_aux_cons_cons {α : Type u} (l : List α) (a : α) : sublists_aux (a :: l) List.cons =\n  [a] :: foldr (fun (ys : List α) (r : List (List α)) => ys :: (a :: ys) :: r) [] (sublists_aux l List.cons) := sorry\n\ntheorem sublists_aux₁_append {α : Type u} {β : Type v} (l₁ : List α) (l₂ : List α) (f : List α → List β) : sublists_aux₁ (l₁ ++ l₂) f =\n  sublists_aux₁ l₁ f ++ sublists_aux₁ l₂ fun (x : List α) => f x ++ sublists_aux₁ l₁ (f ∘ fun (_x : List α) => _x ++ x) := sorry\n\ntheorem sublists_aux₁_concat {α : Type u} {β : Type v} (l : List α) (a : α) (f : List α → List β) : sublists_aux₁ (l ++ [a]) f = sublists_aux₁ l f ++ f [a] ++ sublists_aux₁ l fun (x : List α) => f (x ++ [a]) := sorry\n\ntheorem sublists_aux₁_bind {α : Type u} {β : Type v} {γ : Type w} (l : List α) (f : List α → List β) (g : β → List γ) : list.bind (sublists_aux₁ l f) g = sublists_aux₁ l fun (x : List α) => list.bind (f x) g := sorry\n\ntheorem sublists_aux_cons_append {α : Type u} (l₁ : List α) (l₂ : List α) : sublists_aux (l₁ ++ l₂) List.cons =\n  sublists_aux l₁ List.cons ++\n    do \n      let x ← sublists_aux l₂ List.cons\n      (fun (_x : List α) => _x ++ x) <$> sublists l₁ := sorry\n\ntheorem sublists_append {α : Type u} (l₁ : List α) (l₂ : List α) : sublists (l₁ ++ l₂) =\n  do \n    let x ← sublists l₂\n    (fun (_x : List α) => _x ++ x) <$> sublists l₁ := sorry\n\n@[simp] theorem sublists_concat {α : Type u} (l : List α) (a : α) : sublists (l ++ [a]) = sublists l ++ map (fun (x : List α) => x ++ [a]) (sublists l) := sorry\n\ntheorem sublists_reverse {α : Type u} (l : List α) : sublists (reverse l) = map reverse (sublists' l) := sorry\n\ntheorem sublists_eq_sublists' {α : Type u} (l : List α) : sublists l = map reverse (sublists' (reverse l)) := sorry\n\ntheorem sublists'_reverse {α : Type u} (l : List α) : sublists' (reverse l) = map reverse (sublists l) := sorry\n\ntheorem sublists'_eq_sublists {α : Type u} (l : List α) : sublists' l = map reverse (sublists (reverse l)) := sorry\n\ntheorem sublists_aux_ne_nil {α : Type u} (l : List α) : ¬[] ∈ sublists_aux l List.cons := sorry\n\n@[simp] theorem mem_sublists {α : Type u} {s : List α} {t : List α} : s ∈ sublists t ↔ s <+ t := sorry\n\n@[simp] theorem length_sublists {α : Type u} (l : List α) : length (sublists l) = bit0 1 ^ length l := sorry\n\ntheorem map_ret_sublist_sublists {α : Type u} (l : List α) : map list.ret l <+ sublists l := sorry\n\n/-! ### sublists_len -/\n\n/-- Auxiliary function to construct the list of all sublists of a given length. Given an\ninteger `n`, a list `l`, a function `f` and an auxiliary list `L`, it returns the list made of\nof `f` applied to all sublists of `l` of length `n`, concatenated with `L`. -/\ndef sublists_len_aux {α : Type u_1} {β : Type u_2} : ℕ → List α → (List α → β) → List β → List β :=\n  sorry\n\n/-- The list of all sublists of a list `l` that are of length `n`. For instance, for\n`l = [0, 1, 2, 3]` and `n = 2`, one gets\n`[[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]`. -/\ndef sublists_len {α : Type u_1} (n : ℕ) (l : List α) : List (List α) :=\n  sublists_len_aux n l id []\n\ntheorem sublists_len_aux_append {α : Type u_1} {β : Type u_2} {γ : Type u_3} (n : ℕ) (l : List α) (f : List α → β) (g : β → γ) (r : List β) (s : List γ) : sublists_len_aux n l (g ∘ f) (map g r ++ s) = map g (sublists_len_aux n l f r) ++ s := sorry\n\ntheorem sublists_len_aux_eq {α : Type u_1} {β : Type u_2} (l : List α) (n : ℕ) (f : List α → β) (r : List β) : sublists_len_aux n l f r = map f (sublists_len n l) ++ r := sorry\n\ntheorem sublists_len_aux_zero {β : Type v} {α : Type u_1} (l : List α) (f : List α → β) (r : List β) : sublists_len_aux 0 l f r = f [] :: r :=\n  list.cases_on l (Eq.refl (sublists_len_aux 0 [] f r))\n    fun (l_hd : α) (l_tl : List α) => Eq.refl (sublists_len_aux 0 (l_hd :: l_tl) f r)\n\n@[simp] theorem sublists_len_zero {α : Type u_1} (l : List α) : sublists_len 0 l = [[]] :=\n  sublists_len_aux_zero l id []\n\n@[simp] theorem sublists_len_succ_nil {α : Type u_1} (n : ℕ) : sublists_len (n + 1) [] = [] :=\n  rfl\n\n@[simp] theorem sublists_len_succ_cons {α : Type u_1} (n : ℕ) (a : α) (l : List α) : sublists_len (n + 1) (a :: l) = sublists_len (n + 1) l ++ map (List.cons a) (sublists_len n l) := sorry\n\n@[simp] theorem length_sublists_len {α : Type u_1} (n : ℕ) (l : List α) : length (sublists_len n l) = nat.choose (length l) n := sorry\n\ntheorem sublists_len_sublist_sublists' {α : Type u_1} (n : ℕ) (l : List α) : sublists_len n l <+ sublists' l := sorry\n\ntheorem sublists_len_sublist_of_sublist {α : Type u_1} (n : ℕ) {l₁ : List α} {l₂ : List α} (h : l₁ <+ l₂) : sublists_len n l₁ <+ sublists_len n l₂ := sorry\n\ntheorem length_of_sublists_len {α : Type u_1} {n : ℕ} {l : List α} {l' : List α} : l' ∈ sublists_len n l → length l' = n := sorry\n\ntheorem mem_sublists_len_self {α : Type u_1} {l : List α} {l' : List α} (h : l' <+ l) : l' ∈ sublists_len (length l') l := sorry\n\n@[simp] theorem mem_sublists_len {α : Type u_1} {n : ℕ} {l : List α} {l' : List α} : l' ∈ sublists_len n l ↔ l' <+ l ∧ length l' = n := sorry\n\n/-! ### permutations -/\n\n@[simp] theorem permutations_aux_nil {α : Type u} (is : List α) : permutations_aux [] is = [] := sorry\n\n@[simp] theorem permutations_aux_cons {α : Type u} (t : α) (ts : List α) (is : List α) : permutations_aux (t :: ts) is =\n  foldr (fun (y : List α) (r : List (List α)) => prod.snd (permutations_aux2 t ts r y id))\n    (permutations_aux ts (t :: is)) (permutations is) := sorry\n\n/-! ### insert -/\n\n@[simp] theorem insert_nil {α : Type u} [DecidableEq α] (a : α) : insert a [] = [a] :=\n  rfl\n\ntheorem insert.def {α : Type u} [DecidableEq α] (a : α) (l : List α) : insert a l = ite (a ∈ l) l (a :: l) :=\n  rfl\n\n@[simp] theorem insert_of_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : insert a l = l := sorry\n\n@[simp] theorem insert_of_not_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : ¬a ∈ l) : insert a l = a :: l := sorry\n\n@[simp] theorem mem_insert_iff {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} : a ∈ insert b l ↔ a = b ∨ a ∈ l := sorry\n\n@[simp] theorem suffix_insert {α : Type u} [DecidableEq α] (a : α) (l : List α) : l <:+ insert a l := sorry\n\n@[simp] theorem mem_insert_self {α : Type u} [DecidableEq α] (a : α) (l : List α) : a ∈ insert a l :=\n  iff.mpr mem_insert_iff (Or.inl rfl)\n\ntheorem mem_insert_of_mem {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} (h : a ∈ l) : a ∈ insert b l :=\n  iff.mpr mem_insert_iff (Or.inr h)\n\ntheorem eq_or_mem_of_mem_insert {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} (h : a ∈ insert b l) : a = b ∨ a ∈ l :=\n  iff.mp mem_insert_iff h\n\n@[simp] theorem length_insert_of_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : length (insert a l) = length l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (insert a l) = length l)) (insert_of_mem h))) (Eq.refl (length l))\n\n@[simp] theorem length_insert_of_not_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : ¬a ∈ l) : length (insert a l) = length l + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (insert a l) = length l + 1)) (insert_of_not_mem h)))\n    (Eq.refl (length (a :: l)))\n\n/-! ### erasep -/\n\n@[simp] theorem erasep_nil {α : Type u} {p : α → Prop} [decidable_pred p] : erasep p [] = [] :=\n  rfl\n\ntheorem erasep_cons {α : Type u} {p : α → Prop} [decidable_pred p] (a : α) (l : List α) : erasep p (a :: l) = ite (p a) l (a :: erasep p l) :=\n  rfl\n\n@[simp] theorem erasep_cons_of_pos {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} (h : p a) : erasep p (a :: l) = l := sorry\n\n@[simp] theorem erasep_cons_of_neg {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} (h : ¬p a) : erasep p (a :: l) = a :: erasep p l := sorry\n\ntheorem erasep_of_forall_not {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} (h : ∀ (a : α), a ∈ l → ¬p a) : erasep p l = l := sorry\n\ntheorem exists_of_erasep {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} {a : α} (al : a ∈ l) (pa : p a) : ∃ (a : α),\n  ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (b : α), b ∈ l₁ → ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ erasep p l = l₁ ++ l₂ := sorry\n\ntheorem exists_or_eq_self_of_erasep {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : erasep p l = l ∨\n  ∃ (a : α),\n    ∃ (l₁ : List α), ∃ (l₂ : List α), (∀ (b : α), b ∈ l₁ → ¬p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ erasep p l = l₁ ++ l₂ := sorry\n\n@[simp] theorem length_erasep_of_mem {α : Type u} {p : α → Prop} [decidable_pred p] {l : List α} {a : α} (al : a ∈ l) (pa : p a) : length (erasep p l) = Nat.pred (length l) := sorry\n\ntheorem erasep_append_left {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} (pa : p a) {l₁ : List α} (l₂ : List α) : a ∈ l₁ → erasep p (l₁ ++ l₂) = erasep p l₁ ++ l₂ := sorry\n\ntheorem erasep_append_right {α : Type u} {p : α → Prop} [decidable_pred p] {l₁ : List α} (l₂ : List α) : (∀ (b : α), b ∈ l₁ → ¬p b) → erasep p (l₁ ++ l₂) = l₁ ++ erasep p l₂ := sorry\n\ntheorem erasep_sublist {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : erasep p l <+ l := sorry\n\ntheorem erasep_subset {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : erasep p l ⊆ l :=\n  sublist.subset (erasep_sublist l)\n\ntheorem sublist.erasep {α : Type u} {p : α → Prop} [decidable_pred p] {l₁ : List α} {l₂ : List α} (s : l₁ <+ l₂) : erasep p l₁ <+ erasep p l₂ := sorry\n\ntheorem mem_of_mem_erasep {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} : a ∈ erasep p l → a ∈ l :=\n  erasep_subset l\n\n@[simp] theorem mem_erasep_of_neg {α : Type u} {p : α → Prop} [decidable_pred p] {a : α} {l : List α} (pa : ¬p a) : a ∈ erasep p l ↔ a ∈ l := sorry\n\ntheorem erasep_map {α : Type u} {β : Type v} {p : α → Prop} [decidable_pred p] (f : β → α) (l : List β) : erasep p (map f l) = map f (erasep (p ∘ f) l) := sorry\n\n@[simp] theorem extractp_eq_find_erasep {α : Type u} {p : α → Prop} [decidable_pred p] (l : List α) : extractp p l = (find p l, erasep p l) := sorry\n\n/-! ### erase -/\n\n@[simp] theorem erase_nil {α : Type u} [DecidableEq α] (a : α) : list.erase [] a = [] :=\n  rfl\n\ntheorem erase_cons {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : list.erase (b :: l) a = ite (b = a) l (b :: list.erase l a) :=\n  rfl\n\n@[simp] theorem erase_cons_head {α : Type u} [DecidableEq α] (a : α) (l : List α) : list.erase (a :: l) a = l := sorry\n\n@[simp] theorem erase_cons_tail {α : Type u} [DecidableEq α] {a : α} {b : α} (l : List α) (h : b ≠ a) : list.erase (b :: l) a = b :: list.erase l a := sorry\n\ntheorem erase_eq_erasep {α : Type u} [DecidableEq α] (a : α) (l : List α) : list.erase l a = erasep (Eq a) l := sorry\n\n@[simp] theorem erase_of_not_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : ¬a ∈ l) : list.erase l a = l := sorry\n\ntheorem exists_erase_eq {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : ∃ (l₁ : List α), ∃ (l₂ : List α), ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ list.erase l a = l₁ ++ l₂ := sorry\n\n@[simp] theorem length_erase_of_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : length (list.erase l a) = Nat.pred (length l) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (list.erase l a) = Nat.pred (length l))) (erase_eq_erasep a l)))\n    (length_erasep_of_mem h rfl)\n\ntheorem erase_append_left {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : a ∈ l₁) : list.erase (l₁ ++ l₂) a = list.erase l₁ a ++ l₂ := sorry\n\ntheorem erase_append_right {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} (l₂ : List α) (h : ¬a ∈ l₁) : list.erase (l₁ ++ l₂) a = l₁ ++ list.erase l₂ a := sorry\n\ntheorem erase_sublist {α : Type u} [DecidableEq α] (a : α) (l : List α) : list.erase l a <+ l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (list.erase l a <+ l)) (erase_eq_erasep a l))) (erasep_sublist l)\n\ntheorem erase_subset {α : Type u} [DecidableEq α] (a : α) (l : List α) : list.erase l a ⊆ l :=\n  sublist.subset (erase_sublist a l)\n\ntheorem sublist.erase {α : Type u} [DecidableEq α] (a : α) {l₁ : List α} {l₂ : List α} (h : l₁ <+ l₂) : list.erase l₁ a <+ list.erase l₂ a := sorry\n\ntheorem mem_of_mem_erase {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} : a ∈ list.erase l b → a ∈ l :=\n  erase_subset b l\n\n@[simp] theorem mem_erase_of_ne {α : Type u} [DecidableEq α] {a : α} {b : α} {l : List α} (ab : a ≠ b) : a ∈ list.erase l b ↔ a ∈ l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a ∈ list.erase l b ↔ a ∈ l)) (erase_eq_erasep b l))) (mem_erasep_of_neg (ne.symm ab))\n\ntheorem erase_comm {α : Type u} [DecidableEq α] (a : α) (b : α) (l : List α) : list.erase (list.erase l a) b = list.erase (list.erase l b) a := sorry\n\ntheorem map_erase {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : function.injective f) {a : α} (l : List α) : map f (list.erase l a) = list.erase (map f l) (f a) := sorry\n\ntheorem map_foldl_erase {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : function.injective f) {l₁ : List α} {l₂ : List α} : map f (foldl list.erase l₁ l₂) = foldl (fun (l : List β) (a : α) => list.erase l (f a)) (map f l₁) l₂ := sorry\n\n@[simp] theorem count_erase_self {α : Type u} [DecidableEq α] (a : α) (s : List α) : count a (list.erase s a) = Nat.pred (count a s) := sorry\n\n@[simp] theorem count_erase_of_ne {α : Type u} [DecidableEq α] {a : α} {b : α} (ab : a ≠ b) (s : List α) : count a (list.erase s b) = count a s := sorry\n\n/-! ### diff -/\n\n@[simp] theorem diff_nil {α : Type u} [DecidableEq α] (l : List α) : list.diff l [] = l :=\n  rfl\n\n@[simp] theorem diff_cons {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) (a : α) : list.diff l₁ (a :: l₂) = list.diff (list.erase l₁ a) l₂ := sorry\n\ntheorem diff_cons_right {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) (a : α) : list.diff l₁ (a :: l₂) = list.erase (list.diff l₁ l₂) a := sorry\n\ntheorem diff_erase {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) (a : α) : list.erase (list.diff l₁ l₂) a = list.diff (list.erase l₁ a) l₂ := sorry\n\n@[simp] theorem nil_diff {α : Type u} [DecidableEq α] (l : List α) : list.diff [] l = [] := sorry\n\ntheorem diff_eq_foldl {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : list.diff l₁ l₂ = foldl list.erase l₁ l₂ := sorry\n\n@[simp] theorem diff_append {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) (l₃ : List α) : list.diff l₁ (l₂ ++ l₃) = list.diff (list.diff l₁ l₂) l₃ := sorry\n\n@[simp] theorem map_diff {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : function.injective f) {l₁ : List α} {l₂ : List α} : map f (list.diff l₁ l₂) = list.diff (map f l₁) (map f l₂) := sorry\n\ntheorem diff_sublist {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : list.diff l₁ l₂ <+ l₁ := sorry\n\ntheorem diff_subset {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : list.diff l₁ l₂ ⊆ l₁ :=\n  sublist.subset (diff_sublist l₁ l₂)\n\ntheorem mem_diff_of_mem {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : a ∈ l₁ → ¬a ∈ l₂ → a ∈ list.diff l₁ l₂ := sorry\n\ntheorem sublist.diff_right {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} {l₃ : List α} : l₁ <+ l₂ → list.diff l₁ l₃ <+ list.diff l₂ l₃ := sorry\n\ntheorem erase_diff_erase_sublist_of_sublist {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : l₁ <+ l₂ → list.diff (list.erase l₂ a) (list.erase l₁ a) <+ list.diff l₂ l₁ := sorry\n\n/-! ### enum -/\n\ntheorem length_enum_from {α : Type u} (n : ℕ) (l : List α) : length (enum_from n l) = length l := sorry\n\ntheorem length_enum {α : Type u} (l : List α) : length (enum l) = length l :=\n  length_enum_from 0\n\n@[simp] theorem enum_from_nth {α : Type u} (n : ℕ) (l : List α) (m : ℕ) : nth (enum_from n l) m = (fun (a : α) => (n + m, a)) <$> nth l m := sorry\n\n@[simp] theorem enum_nth {α : Type u} (l : List α) (n : ℕ) : nth (enum l) n = (fun (a : α) => (n, a)) <$> nth l n := sorry\n\n@[simp] theorem enum_from_map_snd {α : Type u} (n : ℕ) (l : List α) : map prod.snd (enum_from n l) = l := sorry\n\n@[simp] theorem enum_map_snd {α : Type u} (l : List α) : map prod.snd (enum l) = l :=\n  enum_from_map_snd 0\n\ntheorem mem_enum_from {α : Type u} {x : α} {i : ℕ} {j : ℕ} (xs : List α) : (i, x) ∈ enum_from j xs → j ≤ i ∧ i < j + length xs ∧ x ∈ xs := sorry\n\n/-! ### product -/\n\n@[simp] theorem nil_product {α : Type u} {β : Type v} (l : List β) : product [] l = [] :=\n  rfl\n\n@[simp] theorem product_cons {α : Type u} {β : Type v} (a : α) (l₁ : List α) (l₂ : List β) : product (a :: l₁) l₂ = map (fun (b : β) => (a, b)) l₂ ++ product l₁ l₂ :=\n  rfl\n\n@[simp] theorem product_nil {α : Type u} {β : Type v} (l : List α) : product l [] = [] := sorry\n\n@[simp] theorem mem_product {α : Type u} {β : Type v} {l₁ : List α} {l₂ : List β} {a : α} {b : β} : (a, b) ∈ product l₁ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ := sorry\n\ntheorem length_product {α : Type u} {β : Type v} (l₁ : List α) (l₂ : List β) : length (product l₁ l₂) = length l₁ * length l₂ := sorry\n\n/-! ### sigma -/\n\n@[simp] theorem nil_sigma {α : Type u} {σ : α → Type u_1} (l : (a : α) → List (σ a)) : list.sigma [] l = [] :=\n  rfl\n\n@[simp] theorem sigma_cons {α : Type u} {σ : α → Type u_1} (a : α) (l₁ : List α) (l₂ : (a : α) → List (σ a)) : list.sigma (a :: l₁) l₂ = map (sigma.mk a) (l₂ a) ++ list.sigma l₁ l₂ :=\n  rfl\n\n@[simp] theorem sigma_nil {α : Type u} {σ : α → Type u_1} (l : List α) : (list.sigma l fun (a : α) => []) = [] := sorry\n\n@[simp] theorem mem_sigma {α : Type u} {σ : α → Type u_1} {l₁ : List α} {l₂ : (a : α) → List (σ a)} {a : α} {b : σ a} : sigma.mk a b ∈ list.sigma l₁ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ a := sorry\n\ntheorem length_sigma {α : Type u} {σ : α → Type u_1} (l₁ : List α) (l₂ : (a : α) → List (σ a)) : length (list.sigma l₁ l₂) = sum (map (fun (a : α) => length (l₂ a)) l₁) := sorry\n\n/-! ### disjoint -/\n\ntheorem disjoint.symm {α : Type u} {l₁ : List α} {l₂ : List α} (d : disjoint l₁ l₂) : disjoint l₂ l₁ :=\n  fun {a : α} (ᾰ : a ∈ l₂) (ᾰ_1 : a ∈ l₁) => idRhs False (d ᾰ_1 ᾰ)\n\ntheorem disjoint_comm {α : Type u} {l₁ : List α} {l₂ : List α} : disjoint l₁ l₂ ↔ disjoint l₂ l₁ :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint_left {α : Type u} {l₁ : List α} {l₂ : List α} : disjoint l₁ l₂ ↔ ∀ {a : α}, a ∈ l₁ → ¬a ∈ l₂ :=\n  iff.rfl\n\ntheorem disjoint_right {α : Type u} {l₁ : List α} {l₂ : List α} : disjoint l₁ l₂ ↔ ∀ {a : α}, a ∈ l₂ → ¬a ∈ l₁ :=\n  disjoint_comm\n\ntheorem disjoint_iff_ne {α : Type u} {l₁ : List α} {l₂ : List α} : disjoint l₁ l₂ ↔ ∀ (a : α), a ∈ l₁ → ∀ (b : α), b ∈ l₂ → a ≠ b := sorry\n\ntheorem disjoint_of_subset_left {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (ss : l₁ ⊆ l) (d : disjoint l l₂) : disjoint l₁ l₂ :=\n  fun {a : α} (ᾰ : a ∈ l₁) => idRhs (a ∈ l₂ → False) (d (ss ᾰ))\n\ntheorem disjoint_of_subset_right {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (ss : l₂ ⊆ l) (d : disjoint l₁ l) : disjoint l₁ l₂ :=\n  fun {a : α} (ᾰ : a ∈ l₁) (ᾰ_1 : a ∈ l₂) => idRhs False (d ᾰ (ss ᾰ_1))\n\ntheorem disjoint_of_disjoint_cons_left {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : disjoint (a :: l₁) l₂ → disjoint l₁ l₂ :=\n  disjoint_of_subset_left (subset_cons a l₁)\n\ntheorem disjoint_of_disjoint_cons_right {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : disjoint l₁ (a :: l₂) → disjoint l₁ l₂ :=\n  disjoint_of_subset_right (subset_cons a l₂)\n\n@[simp] theorem disjoint_nil_left {α : Type u} (l : List α) : disjoint [] l :=\n  fun {a : α} => idRhs (a ∈ [] → a ∈ l → False) (not.elim (not_mem_nil a))\n\n@[simp] theorem disjoint_nil_right {α : Type u} (l : List α) : disjoint l [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (disjoint l [])) (propext disjoint_comm))) (disjoint_nil_left l)\n\n@[simp] theorem singleton_disjoint {α : Type u} {l : List α} {a : α} : disjoint [a] l ↔ ¬a ∈ l := sorry\n\n@[simp] theorem disjoint_singleton {α : Type u} {l : List α} {a : α} : disjoint l [a] ↔ ¬a ∈ l := sorry\n\n@[simp] theorem disjoint_append_left {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} : disjoint (l₁ ++ l₂) l ↔ disjoint l₁ l ∧ disjoint l₂ l := sorry\n\n@[simp] theorem disjoint_append_right {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} : disjoint l (l₁ ++ l₂) ↔ disjoint l l₁ ∧ disjoint l l₂ := sorry\n\n@[simp] theorem disjoint_cons_left {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : disjoint (a :: l₁) l₂ ↔ ¬a ∈ l₂ ∧ disjoint l₁ l₂ := sorry\n\n@[simp] theorem disjoint_cons_right {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} : disjoint l₁ (a :: l₂) ↔ ¬a ∈ l₁ ∧ disjoint l₁ l₂ := sorry\n\ntheorem disjoint_of_disjoint_append_left_left {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (d : disjoint (l₁ ++ l₂) l) : disjoint l₁ l :=\n  and.left (iff.mp disjoint_append_left d)\n\ntheorem disjoint_of_disjoint_append_left_right {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (d : disjoint (l₁ ++ l₂) l) : disjoint l₂ l :=\n  and.right (iff.mp disjoint_append_left d)\n\ntheorem disjoint_of_disjoint_append_right_left {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (d : disjoint l (l₁ ++ l₂)) : disjoint l l₁ :=\n  and.left (iff.mp disjoint_append_right d)\n\ntheorem disjoint_of_disjoint_append_right_right {α : Type u} {l₁ : List α} {l₂ : List α} {l : List α} (d : disjoint l (l₁ ++ l₂)) : disjoint l l₂ :=\n  and.right (iff.mp disjoint_append_right d)\n\ntheorem disjoint_take_drop {α : Type u} {l : List α} {m : ℕ} {n : ℕ} (hl : nodup l) (h : m ≤ n) : disjoint (take m l) (drop n l) := sorry\n\n/-! ### union -/\n\n@[simp] theorem nil_union {α : Type u} [DecidableEq α] (l : List α) : [] ∪ l = l :=\n  rfl\n\n@[simp] theorem cons_union {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) (a : α) : a :: l₁ ∪ l₂ = insert a (l₁ ∪ l₂) :=\n  rfl\n\n@[simp] theorem mem_union {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} {a : α} : a ∈ l₁ ∪ l₂ ↔ a ∈ l₁ ∨ a ∈ l₂ := sorry\n\ntheorem mem_union_left {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} (h : a ∈ l₁) (l₂ : List α) : a ∈ l₁ ∪ l₂ :=\n  iff.mpr mem_union (Or.inl h)\n\ntheorem mem_union_right {α : Type u} [DecidableEq α] {a : α} (l₁ : List α) {l₂ : List α} (h : a ∈ l₂) : a ∈ l₁ ∪ l₂ :=\n  iff.mpr mem_union (Or.inr h)\n\ntheorem sublist_suffix_of_union {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : ∃ (t : List α), t <+ l₁ ∧ t ++ l₂ = l₁ ∪ l₂ := sorry\n\ntheorem suffix_union_right {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : l₂ <:+ l₁ ∪ l₂ :=\n  Exists.imp (fun (a : List α) => and.right) (sublist_suffix_of_union l₁ l₂)\n\ntheorem union_sublist_append {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : l₁ ∪ l₂ <+ l₁ ++ l₂ := sorry\n\ntheorem forall_mem_union {α : Type u} [DecidableEq α] {p : α → Prop} {l₁ : List α} {l₂ : List α} : (∀ (x : α), x ∈ l₁ ∪ l₂ → p x) ↔ (∀ (x : α), x ∈ l₁ → p x) ∧ ∀ (x : α), x ∈ l₂ → p x := sorry\n\ntheorem forall_mem_of_forall_mem_union_left {α : Type u} [DecidableEq α] {p : α → Prop} {l₁ : List α} {l₂ : List α} (h : ∀ (x : α), x ∈ l₁ ∪ l₂ → p x) (x : α) (H : x ∈ l₁) : p x :=\n  and.left (iff.mp forall_mem_union h)\n\ntheorem forall_mem_of_forall_mem_union_right {α : Type u} [DecidableEq α] {p : α → Prop} {l₁ : List α} {l₂ : List α} (h : ∀ (x : α), x ∈ l₁ ∪ l₂ → p x) (x : α) (H : x ∈ l₂) : p x :=\n  and.right (iff.mp forall_mem_union h)\n\n/-! ### inter -/\n\n@[simp] theorem inter_nil {α : Type u} [DecidableEq α] (l : List α) : [] ∩ l = [] :=\n  rfl\n\n@[simp] theorem inter_cons_of_mem {α : Type u} [DecidableEq α] {a : α} (l₁ : List α) {l₂ : List α} (h : a ∈ l₂) : (a :: l₁) ∩ l₂ = a :: l₁ ∩ l₂ :=\n  if_pos h\n\n@[simp] theorem inter_cons_of_not_mem {α : Type u} [DecidableEq α] {a : α} (l₁ : List α) {l₂ : List α} (h : ¬a ∈ l₂) : (a :: l₁) ∩ l₂ = l₁ ∩ l₂ :=\n  if_neg h\n\ntheorem mem_of_mem_inter_left {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} {a : α} : a ∈ l₁ ∩ l₂ → a ∈ l₁ :=\n  mem_of_mem_filter\n\ntheorem mem_of_mem_inter_right {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} {a : α} : a ∈ l₁ ∩ l₂ → a ∈ l₂ :=\n  of_mem_filter\n\ntheorem mem_inter_of_mem_of_mem {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} {a : α} : a ∈ l₁ → a ∈ l₂ → a ∈ l₁ ∩ l₂ :=\n  mem_filter_of_mem\n\n@[simp] theorem mem_inter {α : Type u} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : a ∈ l₁ ∩ l₂ ↔ a ∈ l₁ ∧ a ∈ l₂ :=\n  mem_filter\n\ntheorem inter_subset_left {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : l₁ ∩ l₂ ⊆ l₁ :=\n  filter_subset l₁\n\ntheorem inter_subset_right {α : Type u} [DecidableEq α] (l₁ : List α) (l₂ : List α) : l₁ ∩ l₂ ⊆ l₂ :=\n  fun (a : α) => mem_of_mem_inter_right\n\ntheorem subset_inter {α : Type u} [DecidableEq α] {l : List α} {l₁ : List α} {l₂ : List α} (h₁ : l ⊆ l₁) (h₂ : l ⊆ l₂) : l ⊆ l₁ ∩ l₂ :=\n  fun (a : α) (h : a ∈ l) => iff.mpr mem_inter { left := h₁ h, right := h₂ h }\n\ntheorem inter_eq_nil_iff_disjoint {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} : l₁ ∩ l₂ = [] ↔ disjoint l₁ l₂ := sorry\n\ntheorem forall_mem_inter_of_forall_left {α : Type u} [DecidableEq α] {p : α → Prop} {l₁ : List α} (h : ∀ (x : α), x ∈ l₁ → p x) (l₂ : List α) (x : α) : x ∈ l₁ ∩ l₂ → p x :=\n  ball.imp_left (fun (x : α) => mem_of_mem_inter_left) h\n\ntheorem forall_mem_inter_of_forall_right {α : Type u} [DecidableEq α] {p : α → Prop} (l₁ : List α) {l₂ : List α} (h : ∀ (x : α), x ∈ l₂ → p x) (x : α) : x ∈ l₁ ∩ l₂ → p x :=\n  ball.imp_left (fun (x : α) => mem_of_mem_inter_right) h\n\n@[simp] theorem inter_reverse {α : Type u} [DecidableEq α] {xs : List α} {ys : List α} : list.inter xs (reverse ys) = list.inter xs ys := sorry\n\ntheorem choose_spec {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : choose p l hp ∈ l ∧ p (choose p l hp) :=\n  subtype.property (choose_x p l hp)\n\ntheorem choose_mem {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : choose p l hp ∈ l :=\n  and.left (choose_spec p l hp)\n\ntheorem choose_property {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : p (choose p l hp) :=\n  and.right (choose_spec p l hp)\n\n/-! ### map₂_left' -/\n\n-- The definitional equalities for `map₂_left'` can already be used by the\n\n-- simplifie because `map₂_left'` is marked `@[simp]`.\n\n@[simp] theorem map₂_left'_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) : map₂_left' f as [] = (map (fun (a : α) => f a none) as, []) :=\n  list.cases_on as (Eq.refl (map₂_left' f [] []))\n    fun (as_hd : α) (as_tl : List α) => Eq.refl (map₂_left' f (as_hd :: as_tl) [])\n\n/-! ### map₂_right' -/\n\n@[simp] theorem map₂_right'_nil_left {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (bs : List β) : map₂_right' f [] bs = (map (f none) bs, []) :=\n  list.cases_on bs (Eq.refl (map₂_right' f [] []))\n    fun (bs_hd : β) (bs_tl : List β) => Eq.refl (map₂_right' f [] (bs_hd :: bs_tl))\n\n@[simp] theorem map₂_right'_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) : map₂_right' f as [] = ([], as) :=\n  rfl\n\n@[simp] theorem map₂_right'_nil_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (b : β) (bs : List β) : map₂_right' f [] (b :: bs) = (f none b :: map (f none) bs, []) :=\n  rfl\n\n@[simp] theorem map₂_right'_cons_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (a : α) (as : List α) (b : β) (bs : List β) : map₂_right' f (a :: as) (b :: bs) =\n  let rec : List γ × List α := map₂_right' f as bs;\n  (f (some a) b :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### zip_left' -/\n\n@[simp] theorem zip_left'_nil_right {α : Type u} {β : Type v} (as : List α) : zip_left' as [] = (map (fun (a : α) => (a, none)) as, []) :=\n  list.cases_on as (Eq.refl (zip_left' [] [])) fun (as_hd : α) (as_tl : List α) => Eq.refl (zip_left' (as_hd :: as_tl) [])\n\n@[simp] theorem zip_left'_nil_left {α : Type u} {β : Type v} (bs : List β) : zip_left' [] bs = ([], bs) :=\n  rfl\n\n@[simp] theorem zip_left'_cons_nil {α : Type u} {β : Type v} (a : α) (as : List α) : zip_left' (a :: as) [] = ((a, none) :: map (fun (a : α) => (a, none)) as, []) :=\n  rfl\n\n@[simp] theorem zip_left'_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : zip_left' (a :: as) (b :: bs) =\n  let rec : List (α × Option β) × List β := zip_left' as bs;\n  ((a, some b) :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### zip_right' -/\n\n@[simp] theorem zip_right'_nil_left {α : Type u} {β : Type v} (bs : List β) : zip_right' [] bs = (map (fun (b : β) => (none, b)) bs, []) :=\n  list.cases_on bs (Eq.refl (zip_right' [] []))\n    fun (bs_hd : β) (bs_tl : List β) => Eq.refl (zip_right' [] (bs_hd :: bs_tl))\n\n@[simp] theorem zip_right'_nil_right {α : Type u} {β : Type v} (as : List α) : zip_right' as [] = ([], as) :=\n  rfl\n\n@[simp] theorem zip_right'_nil_cons {α : Type u} {β : Type v} (b : β) (bs : List β) : zip_right' [] (b :: bs) = ((none, b) :: map (fun (b : β) => (none, b)) bs, []) :=\n  rfl\n\n@[simp] theorem zip_right'_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : zip_right' (a :: as) (b :: bs) =\n  let rec : List (Option α × β) × List α := zip_right' as bs;\n  ((some a, b) :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### map₂_left -/\n\n-- The definitional equalities for `map₂_left` can already be used by the\n\n-- simplifier because `map₂_left` is marked `@[simp]`.\n\n@[simp] theorem map₂_left_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) : map₂_left f as [] = map (fun (a : α) => f a none) as :=\n  list.cases_on as (Eq.refl (map₂_left f [] []))\n    fun (as_hd : α) (as_tl : List α) => Eq.refl (map₂_left f (as_hd :: as_tl) [])\n\ntheorem map₂_left_eq_map₂_left' {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) (bs : List β) : map₂_left f as bs = prod.fst (map₂_left' f as bs) := sorry\n\ntheorem map₂_left_eq_map₂ {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) (as : List α) (bs : List β) : length as ≤ length bs → map₂_left f as bs = map₂ (fun (a : α) (b : β) => f a (some b)) as bs := sorry\n\n/-! ### map₂_right -/\n\n@[simp] theorem map₂_right_nil_left {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (bs : List β) : map₂_right f [] bs = map (f none) bs :=\n  list.cases_on bs (Eq.refl (map₂_right f [] []))\n    fun (bs_hd : β) (bs_tl : List β) => Eq.refl (map₂_right f [] (bs_hd :: bs_tl))\n\n@[simp] theorem map₂_right_nil_right {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) : map₂_right f as [] = [] :=\n  rfl\n\n@[simp] theorem map₂_right_nil_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (b : β) (bs : List β) : map₂_right f [] (b :: bs) = f none b :: map (f none) bs :=\n  rfl\n\n@[simp] theorem map₂_right_cons_cons {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (a : α) (as : List α) (b : β) (bs : List β) : map₂_right f (a :: as) (b :: bs) = f (some a) b :: map₂_right f as bs :=\n  rfl\n\ntheorem map₂_right_eq_map₂_right' {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) : map₂_right f as bs = prod.fst (map₂_right' f as bs) := sorry\n\ntheorem map₂_right_eq_map₂ {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) (h : length bs ≤ length as) : map₂_right f as bs = map₂ (fun (a : α) (b : β) => f (some a) b) as bs := sorry\n\n/-! ### zip_left -/\n\n@[simp] theorem zip_left_nil_right {α : Type u} {β : Type v} (as : List α) : zip_left as [] = map (fun (a : α) => (a, none)) as :=\n  list.cases_on as (Eq.refl (zip_left [] [])) fun (as_hd : α) (as_tl : List α) => Eq.refl (zip_left (as_hd :: as_tl) [])\n\n@[simp] theorem zip_left_nil_left {α : Type u} {β : Type v} (bs : List β) : zip_left [] bs = [] :=\n  rfl\n\n@[simp] theorem zip_left_cons_nil {α : Type u} {β : Type v} (a : α) (as : List α) : zip_left (a :: as) [] = (a, none) :: map (fun (a : α) => (a, none)) as :=\n  rfl\n\n@[simp] theorem zip_left_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : zip_left (a :: as) (b :: bs) = (a, some b) :: zip_left as bs :=\n  rfl\n\ntheorem zip_left_eq_zip_left' {α : Type u} {β : Type v} (as : List α) (bs : List β) : zip_left as bs = prod.fst (zip_left' as bs) := sorry\n\n/-! ### zip_right -/\n\n@[simp] theorem zip_right_nil_left {α : Type u} {β : Type v} (bs : List β) : zip_right [] bs = map (fun (b : β) => (none, b)) bs :=\n  list.cases_on bs (Eq.refl (zip_right [] [])) fun (bs_hd : β) (bs_tl : List β) => Eq.refl (zip_right [] (bs_hd :: bs_tl))\n\n@[simp] theorem zip_right_nil_right {α : Type u} {β : Type v} (as : List α) : zip_right as [] = [] :=\n  rfl\n\n@[simp] theorem zip_right_nil_cons {α : Type u} {β : Type v} (b : β) (bs : List β) : zip_right [] (b :: bs) = (none, b) :: map (fun (b : β) => (none, b)) bs :=\n  rfl\n\n@[simp] theorem zip_right_cons_cons {α : Type u} {β : Type v} (a : α) (as : List α) (b : β) (bs : List β) : zip_right (a :: as) (b :: bs) = (some a, b) :: zip_right as bs :=\n  rfl\n\ntheorem zip_right_eq_zip_right' {α : Type u} {β : Type v} (as : List α) (bs : List β) : zip_right as bs = prod.fst (zip_right' as bs) := sorry\n\n/-! ### Miscellaneous lemmas -/\n\ntheorem ilast'_mem {α : Type u} (a : α) (l : List α) : ilast' a l ∈ a :: l := sorry\n\n@[simp] theorem nth_le_attach {α : Type u} (L : List α) (i : ℕ) (H : i < length (attach L)) : subtype.val (nth_le (attach L) i H) = nth_le L i (length_attach L ▸ H) := sorry\n\nend list\n\n\ntheorem monoid_hom.map_list_prod {α : Type u_1} {β : Type u_2} [monoid α] [monoid β] (f : α →* β) (l : List α) : coe_fn f (list.prod l) = list.prod (list.map (⇑f) l) :=\n  Eq.symm (list.prod_hom l f)\n\nnamespace list\n\n\ntheorem sum_map_hom {α : Type u_1} {β : Type u_2} {γ : Type u_3} [add_monoid β] [add_monoid γ] (L : List α) (f : α → β) (g : β →+ γ) : sum (map (⇑g ∘ f) L) = coe_fn g (sum (map f L)) := sorry\n\ntheorem sum_map_mul_left {α : Type u_1} [semiring α] {β : Type u_2} (L : List β) (f : β → α) (r : α) : sum (map (fun (b : β) => r * f b) L) = r * sum (map f L) :=\n  sum_map_hom L f (add_monoid_hom.mul_left r)\n\ntheorem sum_map_mul_right {α : Type u_1} [semiring α] {β : Type u_2} (L : List β) (f : β → α) (r : α) : sum (map (fun (b : β) => f b * r) L) = sum (map f L) * r :=\n  sum_map_hom L f (add_monoid_hom.mul_right r)\n\n@[simp] theorem mem_map_swap {α : Type u} {β : Type v} (x : α) (y : β) (xs : List (α × β)) : (y, x) ∈ map prod.swap xs ↔ (x, y) ∈ xs := sorry\n\ntheorem slice_eq {α : Type u_1} (xs : List α) (n : ℕ) (m : ℕ) : slice n m xs = take n xs ++ drop (n + m) xs := sorry\n\ntheorem sizeof_slice_lt {α : Type u_1} [SizeOf α] (i : ℕ) (j : ℕ) (hj : 0 < j) (xs : List α) (hi : i < length xs) : sizeof (slice i j xs) < sizeof xs := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370308082623218, "lm_q1q2_score": 0.3458206360553187}}
{"text": "/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Andrew Yang, Pierre-Alexandre Bazin\n-/\nimport algebra.homology.short_exact.preadditive\nimport category_theory.abelian.diagram_lemmas.four\n\n/-!\n# Short exact sequences in abelian categories\n\nIn an abelian category, a left-split or right-split short exact sequence admits a splitting.\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.preadditive\n\nvariables {𝒜 : Type*} [category 𝒜]\n\nnamespace category_theory\n\nvariables {A B C A' B' C' : 𝒜} {f : A ⟶ B} {g : B ⟶ C} {f' : A' ⟶ B'} {g' : B' ⟶ C'}\nvariables [abelian 𝒜]\nopen_locale zero_object\n\nlemma is_iso_of_short_exact_of_is_iso_of_is_iso (h : short_exact f g) (h' : short_exact f' g')\n  (i₁ : A ⟶ A') (i₂ : B ⟶ B') (i₃ : C ⟶ C')\n  (comm₁ : i₁ ≫ f' = f ≫ i₂) (comm₂ : i₂ ≫ g' = g ≫ i₃) [is_iso i₁] [is_iso i₃] :\n  is_iso i₂ :=\nbegin\n  obtain ⟨_⟩ := h,\n  obtain ⟨_⟩ := h',\n  resetI,\n  refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso 𝒜 _ _ 0 _ _ _ 0 _ _ _\n    0 f g 0 f' g' 0 i₁ i₂ i₃ _ comm₁ comm₂ 0 0 0 0 0 _ _ _ _ _ _ _ _ _ _ _;\n  try { simp };\n  try { apply exact_zero_left_of_mono };\n  try { assumption };\n  rwa ← epi_iff_exact_zero_right,\nend\n\n/-- To construct a splitting of `A -f⟶ B -g⟶ C` it suffices to supply\na *morphism* `i : B ⟶ A ⊞ C` such that `f ≫ i` is the canonical map `biprod.inl : A ⟶ A ⊞ C` and\n`i ≫ q = g`, where `q` is the canonical map `biprod.snd : A ⊞ C ⟶ C`,\ntogether with proofs that `f` is mono and `g` is epi.\n\nThe morphism `i` is then automatically an isomorphism. -/\ndef splitting.mk' (h : short_exact f g) (i : B ⟶ A ⊞ C)\n  (h1 : f ≫ i = biprod.inl) (h2 : i ≫ biprod.snd = g) : splitting f g :=\n{ iso :=\n  begin\n    refine @as_iso _ _ _ _ i (id _),\n    refine is_iso_of_short_exact_of_is_iso_of_is_iso h _ _ _ _\n      (h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),\n    split,\n    apply exact_inl_snd\n  end,\n  comp_iso_eq_inl := by { rwa as_iso_hom, },\n  iso_comp_snd_eq := h2 }\n\n/-- To construct a splitting of `A -f⟶ B -g⟶ C` it suffices to supply\na *morphism* `i : A ⊞ C ⟶ B` such that `p ≫ i = f` where `p` is the canonical map\n`biprod.inl : A ⟶ A ⊞ C`, and `i ≫ g` is the canonical map `biprod.snd : A ⊞ C ⟶ C`,\ntogether with proofs that `f` is mono and `g` is epi.\n\nThe morphism `i` is then automatically an isomorphism. -/\ndef splitting.mk'' (h : short_exact f g) (i : A ⊞ C ⟶ B)\n  (h1 : biprod.inl ≫ i = f) (h2 : i ≫ g = biprod.snd) : splitting f g :=\n{ iso :=\n  begin\n    refine (@as_iso _ _ _ _ i (id _)).symm,\n    refine is_iso_of_short_exact_of_is_iso_of_is_iso _ h _ _ _\n      (h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),\n    split,\n    apply exact_inl_snd\n  end,\n  comp_iso_eq_inl := by rw [iso.symm_hom, as_iso_inv, is_iso.comp_inv_eq, h1],\n  iso_comp_snd_eq := by rw [iso.symm_hom, as_iso_inv, is_iso.inv_comp_eq, h2] }\n\n/-- A short exact sequence that is left split admits a splitting. -/\ndef left_split.splitting {f : A ⟶ B} {g : B ⟶ C} (h : left_split f g) : splitting f g :=\nsplitting.mk' h.short_exact (biprod.lift h.left_split.some g)\n(by { ext,\n  { simpa only [biprod.inl_fst, biprod.lift_fst, category.assoc] using h.left_split.some_spec },\n  { simp only [biprod.inl_snd, biprod.lift_snd, category.assoc, h.exact.w], } })\n(by { simp only [biprod.lift_snd], })\n\n/-- A short exact sequence that is right split admits a splitting. -/\ndef right_split.splitting {f : A ⟶ B} {g : B ⟶ C} (h : right_split f g) : splitting f g :=\nsplitting.mk'' h.short_exact (biprod.desc f h.right_split.some)\n(biprod.inl_desc _ _)\n(by { ext,\n  { rw [biprod.inl_snd, ← category.assoc, biprod.inl_desc, h.exact.w] },\n  { rw [biprod.inr_snd, ← category.assoc, biprod.inr_desc, h.right_split.some_spec] } })\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/homology/short_exact/abelian.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.34582062857361023}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.seq.wseq\n! leanprover-community/mathlib commit a7e36e48519ab281320c4d192da6a7b348ce40ad\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.List.Basic\nimport Mathbin.Data.Seq.Seq\n\nnamespace Stream'\n\nopen Function\n\nuniverse u v w\n\n/-\ncoinductive wseq (α : Type u) : Type u\n| nil : wseq α\n| cons : α → wseq α → wseq α\n| think : wseq α → wseq α\n-/\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef Wseq (α) :=\n  Seq (Option α)\n#align stream.wseq Stream'.Wseq\n\nnamespace Wseq\n\nvariable {α : Type u} {β : Type v} {γ : Type w}\n\n/-- Turn a sequence into a weak sequence -/\ndef ofSeq : Seq α → Wseq α :=\n  (· <$> ·) some\n#align stream.wseq.of_seq Stream'.Wseq.ofSeq\n\n/-- Turn a list into a weak sequence -/\ndef ofList (l : List α) : Wseq α :=\n  ofSeq l\n#align stream.wseq.of_list Stream'.Wseq.ofList\n\n/-- Turn a stream into a weak sequence -/\ndef ofStream (l : Stream' α) : Wseq α :=\n  ofSeq l\n#align stream.wseq.of_stream Stream'.Wseq.ofStream\n\ninstance coeSeq : Coe (Seq α) (Wseq α) :=\n  ⟨ofSeq⟩\n#align stream.wseq.coe_seq Stream'.Wseq.coeSeq\n\ninstance coeList : Coe (List α) (Wseq α) :=\n  ⟨ofList⟩\n#align stream.wseq.coe_list Stream'.Wseq.coeList\n\ninstance coeStream : Coe (Stream' α) (Wseq α) :=\n  ⟨ofStream⟩\n#align stream.wseq.coe_stream Stream'.Wseq.coeStream\n\n/-- The empty weak sequence -/\ndef nil : Wseq α :=\n  Seq.nil\n#align stream.wseq.nil Stream'.Wseq.nil\n\ninstance : Inhabited (Wseq α) :=\n  ⟨nil⟩\n\n/-- Prepend an element to a weak sequence -/\ndef cons (a : α) : Wseq α → Wseq α :=\n  Seq.cons (some a)\n#align stream.wseq.cons Stream'.Wseq.cons\n\n/-- Compute for one tick, without producing any elements -/\ndef think : Wseq α → Wseq α :=\n  Seq.cons none\n#align stream.wseq.think Stream'.Wseq.think\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct : Wseq α → Computation (Option (α × Wseq α)) :=\n  Computation.corec fun s =>\n    match Seq.destruct s with\n    | none => Sum.inl none\n    | some (none, s') => Sum.inr s'\n    | some (some a, s') => Sum.inl (some (a, s'))\n#align stream.wseq.destruct Stream'.Wseq.destruct\n\n/-- Recursion principle for weak sequences, compare with `list.rec_on`. -/\ndef recOn {C : Wseq α → Sort v} (s : Wseq α) (h1 : C nil) (h2 : ∀ x s, C (cons x s))\n    (h3 : ∀ s, C (think s)) : C s :=\n  Seq.recOn s h1 fun o => Option.recOn o h3 h2\n#align stream.wseq.rec_on Stream'.Wseq.recOn\n\n/-- membership for weak sequences-/\nprotected def Mem (a : α) (s : Wseq α) :=\n  Seq.Mem (some a) s\n#align stream.wseq.mem Stream'.Wseq.Mem\n\ninstance : Membership α (Wseq α) :=\n  ⟨Wseq.Mem⟩\n\ntheorem not_mem_nil (a : α) : a ∉ @nil α :=\n  Seq.not_mem_nil a\n#align stream.wseq.not_mem_nil Stream'.Wseq.not_mem_nil\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head (s : Wseq α) : Computation (Option α) :=\n  Computation.map ((· <$> ·) Prod.fst) (destruct s)\n#align stream.wseq.head Stream'.Wseq.head\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten : Computation (Wseq α) → Wseq α :=\n  Seq.corec fun c =>\n    match Computation.destruct c with\n    | Sum.inl s => Seq.omap return (Seq.destruct s)\n    | Sum.inr c' => some (none, c')\n#align stream.wseq.flatten Stream'.Wseq.flatten\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail (s : Wseq α) : Wseq α :=\n  flatten <| (fun o => Option.recOn o nil Prod.snd) <$> destruct s\n#align stream.wseq.tail Stream'.Wseq.tail\n\n/-- drop the first `n` elements from `s`. -/\ndef drop (s : Wseq α) : ℕ → Wseq α\n  | 0 => s\n  | n + 1 => tail (drop n)\n#align stream.wseq.drop Stream'.Wseq.drop\n\nattribute [simp] drop\n\n/-- Get the nth element of `s`. -/\ndef nth (s : Wseq α) (n : ℕ) : Computation (Option α) :=\n  head (drop s n)\n#align stream.wseq.nth Stream'.Wseq.nth\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef toList (s : Wseq α) : Computation (List α) :=\n  @Computation.corec (List α) (List α × Wseq α)\n    (fun ⟨l, s⟩ =>\n      match Seq.destruct s with\n      | none => Sum.inl l.reverse\n      | some (none, s') => Sum.inr (l, s')\n      | some (some a, s') => Sum.inr (a::l, s'))\n    ([], s)\n#align stream.wseq.to_list Stream'.Wseq.toList\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : Wseq α) : Computation ℕ :=\n  @Computation.corec ℕ (ℕ × Wseq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s with\n      | none => Sum.inl n\n      | some (none, s') => Sum.inr (n, s')\n      | some (some a, s') => Sum.inr (n + 1, s'))\n    (0, s)\n#align stream.wseq.length Stream'.Wseq.length\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\nclass IsFinite (s : Wseq α) : Prop where\n  out : (toList s).Terminates\n#align stream.wseq.is_finite Stream'.Wseq.IsFinite\n\ninstance toList_terminates (s : Wseq α) [h : IsFinite s] : (toList s).Terminates :=\n  h.out\n#align stream.wseq.to_list_terminates Stream'.Wseq.toList_terminates\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : Wseq α) [IsFinite s] : List α :=\n  (toList s).get\n#align stream.wseq.get Stream'.Wseq.get\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\nclass Productive (s : Wseq α) : Prop where\n  nth_terminates : ∀ n, (nth s n).Terminates\n#align stream.wseq.productive Stream'.Wseq.Productive\n\ntheorem productive_iff (s : Wseq α) : Productive s ↔ ∀ n, (nth s n).Terminates :=\n  ⟨fun h => h.1, fun h => ⟨h⟩⟩\n#align stream.wseq.productive_iff Stream'.Wseq.productive_iff\n\ninstance nth_terminates (s : Wseq α) [h : Productive s] : ∀ n, (nth s n).Terminates :=\n  h.nth_terminates\n#align stream.wseq.nth_terminates Stream'.Wseq.nth_terminates\n\ninstance head_terminates (s : Wseq α) [Productive s] : (head s).Terminates :=\n  s.nth_terminates 0\n#align stream.wseq.head_terminates Stream'.Wseq.head_terminates\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef updateNth (s : Wseq α) (n : ℕ) (a : α) : Wseq α :=\n  @Seq.corec (Option α) (ℕ × Wseq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, n => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some a', s'), 1 => some (some a, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n#align stream.wseq.update_nth Stream'.Wseq.updateNth\n\n/-- Remove the `n`th element of `s`. -/\ndef removeNth (s : Wseq α) (n : ℕ) : Wseq α :=\n  @Seq.corec (Option α) (ℕ × Wseq α)\n    (fun ⟨n, s⟩ =>\n      match Seq.destruct s, n with\n      | none, n => none\n      | some (none, s'), n => some (none, n, s')\n      | some (some a', s'), 0 => some (some a', 0, s')\n      | some (some a', s'), 1 => some (none, 0, s')\n      | some (some a', s'), n + 2 => some (some a', n + 1, s'))\n    (n + 1, s)\n#align stream.wseq.remove_nth Stream'.Wseq.removeNth\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filterMap (f : α → Option β) : Wseq α → Wseq β :=\n  Seq.corec fun s =>\n    match Seq.destruct s with\n    | none => none\n    | some (none, s') => some (none, s')\n    | some (some a, s') => some (f a, s')\n#align stream.wseq.filter_map Stream'.Wseq.filterMap\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : α → Prop) [DecidablePred p] : Wseq α → Wseq α :=\n  filterMap fun a => if p a then some a else none\n#align stream.wseq.filter Stream'.Wseq.filter\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : α → Prop) [DecidablePred p] (s : Wseq α) : Computation (Option α) :=\n  head <| filter p s\n#align stream.wseq.find Stream'.Wseq.find\n\n/-- Zip a function over two weak sequences -/\ndef zipWith (f : α → β → γ) (s1 : Wseq α) (s2 : Wseq β) : Wseq γ :=\n  @Seq.corec (Option γ) (Wseq α × Wseq β)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some a1, s1'), some (none, s2') => some (none, s1, s2')\n      | some (none, s1'), some (some a2, s2') => some (none, s1', s2)\n      | some (some a1, s1'), some (some a2, s2') => some (some (f a1 a2), s1', s2')\n      | _, _ => none)\n    (s1, s2)\n#align stream.wseq.zip_with Stream'.Wseq.zipWith\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : Wseq α → Wseq β → Wseq (α × β) :=\n  zipWith Prod.mk\n#align stream.wseq.zip Stream'.Wseq.zip\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef findIndexes (p : α → Prop) [DecidablePred p] (s : Wseq α) : Wseq ℕ :=\n  (zip s (Stream'.nats : Wseq ℕ)).filterMap fun ⟨a, n⟩ => if p a then some n else none\n#align stream.wseq.find_indexes Stream'.Wseq.findIndexes\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef findIndex (p : α → Prop) [DecidablePred p] (s : Wseq α) : Computation ℕ :=\n  (fun o => Option.getD o 0) <$> head (findIndexes p s)\n#align stream.wseq.find_index Stream'.Wseq.findIndex\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef indexOf [DecidableEq α] (a : α) : Wseq α → Computation ℕ :=\n  findIndex (Eq a)\n#align stream.wseq.index_of Stream'.Wseq.indexOf\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexesOf [DecidableEq α] (a : α) : Wseq α → Wseq ℕ :=\n  findIndexes (Eq a)\n#align stream.wseq.indexes_of Stream'.Wseq.indexesOf\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : Wseq α) : Wseq α :=\n  @Seq.corec (Option α) (Wseq α × Wseq α)\n    (fun ⟨s1, s2⟩ =>\n      match Seq.destruct s1, Seq.destruct s2 with\n      | none, none => none\n      | some (a1, s1'), none => some (a1, s1', nil)\n      | none, some (a2, s2') => some (a2, nil, s2')\n      | some (none, s1'), some (none, s2') => some (none, s1', s2')\n      | some (some a1, s1'), some (none, s2') => some (some a1, s1', s2')\n      | some (none, s1'), some (some a2, s2') => some (some a2, s1', s2')\n      | some (some a1, s1'), some (some a2, s2') => some (some a1, cons a2 s1', s2'))\n    (s1, s2)\n#align stream.wseq.union Stream'.Wseq.union\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef isEmpty (s : Wseq α) : Computation Bool :=\n  Computation.map Option.isNone <| head s\n#align stream.wseq.is_empty Stream'.Wseq.isEmpty\n\n/-- Calculate one step of computation -/\ndef compute (s : Wseq α) : Wseq α :=\n  match Seq.destruct s with\n  | some (none, s') => s'\n  | _ => s\n#align stream.wseq.compute Stream'.Wseq.compute\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : Wseq α) (n : ℕ) : Wseq α :=\n  @Seq.corec (Option α) (ℕ × Wseq α)\n    (fun ⟨n, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => none\n      | m + 1, none => none\n      | m + 1, some (none, s') => some (none, m + 1, s')\n      | m + 1, some (some a, s') => some (some a, m, s'))\n    (n, s)\n#align stream.wseq.take Stream'.Wseq.take\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef splitAt (s : Wseq α) (n : ℕ) : Computation (List α × Wseq α) :=\n  @Computation.corec (List α × Wseq α) (ℕ × List α × Wseq α)\n    (fun ⟨n, l, s⟩ =>\n      match n, Seq.destruct s with\n      | 0, _ => Sum.inl (l.reverse, s)\n      | m + 1, none => Sum.inl (l.reverse, s)\n      | m + 1, some (none, s') => Sum.inr (n, l, s')\n      | m + 1, some (some a, s') => Sum.inr (m, a::l, s'))\n    (n, [], s)\n#align stream.wseq.split_at Stream'.Wseq.splitAt\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any (s : Wseq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : Wseq α =>\n      match Seq.destruct s with\n      | none => Sum.inl false\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inl true else Sum.inr s')\n    s\n#align stream.wseq.any Stream'.Wseq.any\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all (s : Wseq α) (p : α → Bool) : Computation Bool :=\n  Computation.corec\n    (fun s : Wseq α =>\n      match Seq.destruct s with\n      | none => Sum.inl true\n      | some (none, s') => Sum.inr s'\n      | some (some a, s') => if p a then Sum.inr s' else Sum.inl false)\n    s\n#align stream.wseq.all Stream'.Wseq.all\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : α → β → α) (a : α) (s : Wseq β) : Wseq α :=\n  cons a <|\n    @Seq.corec (Option α) (α × Wseq β)\n      (fun ⟨a, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, a, s')\n        | some (some b, s') =>\n          let a' := f a b\n          some (some a', a', s'))\n      (a, s)\n#align stream.wseq.scanl Stream'.Wseq.scanl\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : Wseq α) : Wseq (List α) :=\n  cons [] <|\n    @Seq.corec (Option (List α)) (Dlist α × Wseq α)\n      (fun ⟨l, s⟩ =>\n        match Seq.destruct s with\n        | none => none\n        | some (none, s') => some (none, l, s')\n        | some (some a, s') =>\n          let l' := l.concat a\n          some (some l'.toList, l', s'))\n      (Dlist.empty, s)\n#align stream.wseq.inits Stream'.Wseq.inits\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : Wseq α) (n : ℕ) : List α :=\n  (Seq.take n s).filterMap id\n#align stream.wseq.collect Stream'.Wseq.collect\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append : Wseq α → Wseq α → Wseq α :=\n  Seq.append\n#align stream.wseq.append Stream'.Wseq.append\n\n/-- Map a function over a weak sequence -/\ndef map (f : α → β) : Wseq α → Wseq β :=\n  Seq.map (Option.map f)\n#align stream.wseq.map Stream'.Wseq.map\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join (S : Wseq (Wseq α)) : Wseq α :=\n  Seq.join\n    ((fun o : Option (Wseq α) =>\n        match o with\n        | none => Seq1.ret none\n        | some s => (none, s)) <$>\n      S)\n#align stream.wseq.join Stream'.Wseq.join\n\n/-- Monadic bind operator for weak sequences -/\ndef bind (s : Wseq α) (f : α → Wseq β) : Wseq β :=\n  join (map f s)\n#align stream.wseq.bind Stream'.Wseq.bind\n\n/-- lift a relation to a relation over weak sequences -/\n@[simp]\ndef LiftRelO (R : α → β → Prop) (C : Wseq α → Wseq β → Prop) :\n    Option (α × Wseq α) → Option (β × Wseq β) → Prop\n  | none, none => True\n  | some (a, s), some (b, t) => R a b ∧ C s t\n  | _, _ => False\n#align stream.wseq.lift_rel_o Stream'.Wseq.LiftRelO\n\ntheorem LiftRelO.imp {R S : α → β → Prop} {C D : Wseq α → Wseq β → Prop} (H1 : ∀ a b, R a b → S a b)\n    (H2 : ∀ s t, C s t → D s t) : ∀ {o p}, LiftRelO R C o p → LiftRelO S D o p\n  | none, none, h => trivial\n  | some (a, s), some (b, t), h => And.imp (H1 _ _) (H2 _ _) h\n  | none, some _, h => False.elim h\n  | some (_, _), none, h => False.elim h\n#align stream.wseq.lift_rel_o.imp Stream'.Wseq.LiftRelO.imp\n\ntheorem LiftRelO.imp_right (R : α → β → Prop) {C D : Wseq α → Wseq β → Prop}\n    (H : ∀ s t, C s t → D s t) {o p} : LiftRelO R C o p → LiftRelO R D o p :=\n  LiftRelO.imp (fun _ _ => id) H\n#align stream.wseq.lift_rel_o.imp_right Stream'.Wseq.LiftRelO.imp_right\n\n/-- Definitino of bisimilarity for weak sequences-/\n@[simp]\ndef BisimO (R : Wseq α → Wseq α → Prop) : Option (α × Wseq α) → Option (α × Wseq α) → Prop :=\n  LiftRelO (· = ·) R\n#align stream.wseq.bisim_o Stream'.Wseq.BisimO\n\ntheorem BisimO.imp {R S : Wseq α → Wseq α → Prop} (H : ∀ s t, R s t → S s t) {o p} :\n    BisimO R o p → BisimO S o p :=\n  LiftRelO.imp_right _ H\n#align stream.wseq.bisim_o.imp Stream'.Wseq.BisimO.imp\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef LiftRel (R : α → β → Prop) (s : Wseq α) (t : Wseq β) : Prop :=\n  ∃ C : Wseq α → Wseq β → Prop,\n    C s t ∧ ∀ {s t}, C s t → Computation.LiftRel (LiftRelO R C) (destruct s) (destruct t)\n#align stream.wseq.lift_rel Stream'.Wseq.LiftRel\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef Equiv : Wseq α → Wseq α → Prop :=\n  LiftRel (· = ·)\n#align stream.wseq.equiv Stream'.Wseq.Equiv\n\ntheorem liftRel_destruct {R : α → β → Prop} {s : Wseq α} {t : Wseq β} :\n    LiftRel R s t → Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n  | ⟨R, h1, h2⟩ => by\n    refine' Computation.LiftRel.imp _ _ _ (h2 h1) <;> apply lift_rel_o.imp_right <;>\n      exact fun s' t' h' => ⟨R, h', @h2⟩\n#align stream.wseq.lift_rel_destruct Stream'.Wseq.liftRel_destruct\n\ntheorem liftRel_destruct_iff {R : α → β → Prop} {s : Wseq α} {t : Wseq β} :\n    LiftRel R s t ↔ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) :=\n  ⟨liftRel_destruct, fun h =>\n    ⟨fun s t =>\n      LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t),\n      Or.inr h, fun s t h =>\n      by\n      have h : Computation.LiftRel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n        by\n        cases' h with h h\n        exact lift_rel_destruct h\n        assumption\n      apply Computation.LiftRel.imp _ _ _ h\n      intro a b\n      apply lift_rel_o.imp_right\n      intro s t\n      apply Or.inl⟩⟩\n#align stream.wseq.lift_rel_destruct_iff Stream'.Wseq.liftRel_destruct_iff\n\n-- mathport name: equiv\ninfixl:50 \" ~ \" => Equiv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:228:8: unsupported: ambiguous notation -/\ntheorem destruct_congr {s t : Wseq α} :\n    s ~ t → Computation.LiftRel (BisimO (· ~ ·)) (destruct s) (destruct t) :=\n  liftRel_destruct\n#align stream.wseq.destruct_congr Stream'.Wseq.destruct_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:228:8: unsupported: ambiguous notation -/\ntheorem destruct_congr_iff {s t : Wseq α} :\n    s ~ t ↔ Computation.LiftRel (BisimO (· ~ ·)) (destruct s) (destruct t) :=\n  liftRel_destruct_iff\n#align stream.wseq.destruct_congr_iff Stream'.Wseq.destruct_congr_iff\n\ntheorem LiftRel.refl (R : α → α → Prop) (H : Reflexive R) : Reflexive (LiftRel R) := fun s =>\n  by\n  refine' ⟨(· = ·), rfl, fun s t (h : s = t) => _⟩\n  rw [← h]; apply Computation.LiftRel.refl\n  intro a; cases' a with a; simp; cases a <;> simp; apply H\n#align stream.wseq.lift_rel.refl Stream'.Wseq.LiftRel.refl\n\ntheorem LiftRelO.swap (R : α → β → Prop) (C) : swap (LiftRelO R C) = LiftRelO (swap R) (swap C) :=\n  by\n  funext x y <;> cases' x with x <;> [skip, cases x] <;>\n    · cases' y with y <;> [skip, cases y] <;> rfl\n#align stream.wseq.lift_rel_o.swap Stream'.Wseq.LiftRelO.swap\n\ntheorem LiftRel.swap_lem {R : α → β → Prop} {s1 s2} (h : LiftRel R s1 s2) :\n    LiftRel (swap R) s2 s1 :=\n  by\n  refine' ⟨swap (lift_rel R), h, fun s t (h : lift_rel R t s) => _⟩\n  rw [← lift_rel_o.swap, Computation.LiftRel.swap]\n  apply lift_rel_destruct h\n#align stream.wseq.lift_rel.swap_lem Stream'.Wseq.LiftRel.swap_lem\n\ntheorem LiftRel.swap (R : α → β → Prop) : swap (LiftRel R) = LiftRel (swap R) :=\n  funext fun x => funext fun y => propext ⟨LiftRel.swap_lem, LiftRel.swap_lem⟩\n#align stream.wseq.lift_rel.swap Stream'.Wseq.LiftRel.swap\n\ntheorem LiftRel.symm (R : α → α → Prop) (H : Symmetric R) : Symmetric (LiftRel R) :=\n  fun s1 s2 (h : swap (LiftRel R) s2 s1) => by\n  rwa [lift_rel.swap,\n    show swap R = R from funext fun a => funext fun b => propext <| by constructor <;> apply H] at h\n#align stream.wseq.lift_rel.symm Stream'.Wseq.LiftRel.symm\n\ntheorem LiftRel.trans (R : α → α → Prop) (H : Transitive R) : Transitive (LiftRel R) :=\n  fun s t u h1 h2 =>\n  by\n  refine' ⟨fun s u => ∃ t, lift_rel R s t ∧ lift_rel R t u, ⟨t, h1, h2⟩, fun s u h => _⟩\n  rcases h with ⟨t, h1, h2⟩\n  have h1 := lift_rel_destruct h1\n  have h2 := lift_rel_destruct h2\n  refine'\n    Computation.liftRel_def.2\n      ⟨(Computation.terminates_of_LiftRel h1).trans (Computation.terminates_of_LiftRel h2),\n        fun a c ha hc => _⟩\n  rcases h1.left ha with ⟨b, hb, t1⟩\n  have t2 := Computation.rel_of_LiftRel h2 hb hc\n  cases' a with a <;> cases' c with c\n  · trivial\n  · cases b\n    · cases t2\n    · cases t1\n  · cases a\n    cases' b with b\n    · cases t1\n    · cases b\n      cases t2\n  · cases' a with a s\n    cases' b with b\n    · cases t1\n    cases' b with b t\n    cases' c with c u\n    cases' t1 with ab st\n    cases' t2 with bc tu\n    exact ⟨H ab bc, t, st, tu⟩\n#align stream.wseq.lift_rel.trans Stream'.Wseq.LiftRel.trans\n\ntheorem LiftRel.equiv (R : α → α → Prop) : Equivalence R → Equivalence (LiftRel R)\n  | ⟨refl, symm, trans⟩ => ⟨LiftRel.refl R refl, LiftRel.symm R symm, LiftRel.trans R trans⟩\n#align stream.wseq.lift_rel.equiv Stream'.Wseq.LiftRel.equiv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[refl]\ntheorem Equiv.refl : ∀ s : Wseq α, s ~ s :=\n  LiftRel.refl (· = ·) Eq.refl\n#align stream.wseq.equiv.refl Stream'.Wseq.Equiv.refl\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[symm]\ntheorem Equiv.symm : ∀ {s t : Wseq α}, s ~ t → t ~ s :=\n  LiftRel.symm (· = ·) (@Eq.symm _)\n#align stream.wseq.equiv.symm Stream'.Wseq.Equiv.symm\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[trans]\ntheorem Equiv.trans : ∀ {s t u : Wseq α}, s ~ t → t ~ u → s ~ u :=\n  LiftRel.trans (· = ·) (@Eq.trans _)\n#align stream.wseq.equiv.trans Stream'.Wseq.Equiv.trans\n\ntheorem Equiv.equivalence : Equivalence (@Equiv α) :=\n  ⟨@Equiv.refl _, @Equiv.symm _, @Equiv.trans _⟩\n#align stream.wseq.equiv.equivalence Stream'.Wseq.Equiv.equivalence\n\nopen Computation\n\n-- mathport name: exprreturn\nlocal notation \"return\" => Computation.pure\n\n@[simp]\ntheorem destruct_nil : destruct (nil : Wseq α) = return none :=\n  Computation.destruct_eq_pure rfl\n#align stream.wseq.destruct_nil Stream'.Wseq.destruct_nil\n\n@[simp]\ntheorem destruct_cons (a : α) (s) : destruct (cons a s) = return (some (a, s)) :=\n  Computation.destruct_eq_pure <| by simp [destruct, cons, Computation.rmap]\n#align stream.wseq.destruct_cons Stream'.Wseq.destruct_cons\n\n@[simp]\ntheorem destruct_think (s : Wseq α) : destruct (think s) = (destruct s).think :=\n  Computation.destruct_eq_think <| by simp [destruct, think, Computation.rmap]\n#align stream.wseq.destruct_think Stream'.Wseq.destruct_think\n\n@[simp]\ntheorem seq_destruct_nil : Seq.destruct (nil : Wseq α) = none :=\n  Seq.destruct_nil\n#align stream.wseq.seq_destruct_nil Stream'.Wseq.seq_destruct_nil\n\n@[simp]\ntheorem seq_destruct_cons (a : α) (s) : Seq.destruct (cons a s) = some (some a, s) :=\n  Seq.destruct_cons _ _\n#align stream.wseq.seq_destruct_cons Stream'.Wseq.seq_destruct_cons\n\n@[simp]\ntheorem seq_destruct_think (s : Wseq α) : Seq.destruct (think s) = some (none, s) :=\n  Seq.destruct_cons _ _\n#align stream.wseq.seq_destruct_think Stream'.Wseq.seq_destruct_think\n\n@[simp]\ntheorem head_nil : head (nil : Wseq α) = return none := by simp [head] <;> rfl\n#align stream.wseq.head_nil Stream'.Wseq.head_nil\n\n@[simp]\ntheorem head_cons (a : α) (s) : head (cons a s) = return (some a) := by simp [head] <;> rfl\n#align stream.wseq.head_cons Stream'.Wseq.head_cons\n\n@[simp]\ntheorem head_think (s : Wseq α) : head (think s) = (head s).think := by simp [head] <;> rfl\n#align stream.wseq.head_think Stream'.Wseq.head_think\n\n@[simp]\ntheorem flatten_ret (s : Wseq α) : flatten (return s) = s :=\n  by\n  refine' seq.eq_of_bisim (fun s1 s2 => flatten (return s2) = s1) _ rfl\n  intro s' s h; rw [← h]; simp [flatten]\n  cases seq.destruct s; · simp\n  · cases' val with o s'\n    simp\n#align stream.wseq.flatten_ret Stream'.Wseq.flatten_ret\n\n@[simp]\ntheorem flatten_think (c : Computation (Wseq α)) : flatten c.think = think (flatten c) :=\n  Seq.destruct_eq_cons <| by simp [flatten, think]\n#align stream.wseq.flatten_think Stream'.Wseq.flatten_think\n\n@[simp]\ntheorem destruct_flatten (c : Computation (Wseq α)) : destruct (flatten c) = c >>= destruct :=\n  by\n  refine'\n    Computation.eq_of_bisim\n      (fun c1 c2 => c1 = c2 ∨ ∃ c, c1 = destruct (flatten c) ∧ c2 = Computation.bind c destruct) _\n      (Or.inr ⟨c, rfl, rfl⟩)\n  intro c1 c2 h;\n  exact\n    match c1, c2, h with\n    | _, _, Or.inl <| Eq.refl c => by cases c.destruct <;> simp\n    | _, _, Or.inr ⟨c, rfl, rfl⟩ =>\n      by\n      apply c.rec_on (fun a => _) fun c' => _ <;> repeat' simp\n      · cases (destruct a).destruct <;> simp\n      · exact Or.inr ⟨c', rfl, rfl⟩\n#align stream.wseq.destruct_flatten Stream'.Wseq.destruct_flatten\n\ntheorem head_terminates_iff (s : Wseq α) : Terminates (head s) ↔ Terminates (destruct s) :=\n  terminates_map_iff _ (destruct s)\n#align stream.wseq.head_terminates_iff Stream'.Wseq.head_terminates_iff\n\n@[simp]\ntheorem tail_nil : tail (nil : Wseq α) = nil := by simp [tail]\n#align stream.wseq.tail_nil Stream'.Wseq.tail_nil\n\n@[simp]\ntheorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]\n#align stream.wseq.tail_cons Stream'.Wseq.tail_cons\n\n@[simp]\ntheorem tail_think (s : Wseq α) : tail (think s) = (tail s).think := by simp [tail]\n#align stream.wseq.tail_think Stream'.Wseq.tail_think\n\n@[simp]\ntheorem dropn_nil (n) : drop (nil : Wseq α) n = nil := by induction n <;> simp [*, drop]\n#align stream.wseq.dropn_nil Stream'.Wseq.dropn_nil\n\n@[simp]\ntheorem dropn_cons (a : α) (s) (n) : drop (cons a s) (n + 1) = drop s n := by\n  induction n <;> simp [*, drop]\n#align stream.wseq.dropn_cons Stream'.Wseq.dropn_cons\n\n@[simp]\ntheorem dropn_think (s : Wseq α) (n) : drop (think s) n = (drop s n).think := by\n  induction n <;> simp [*, drop]\n#align stream.wseq.dropn_think Stream'.Wseq.dropn_think\n\ntheorem dropn_add (s : Wseq α) (m) : ∀ n, drop s (m + n) = drop (drop s m) n\n  | 0 => rfl\n  | n + 1 => congr_arg tail (dropn_add n)\n#align stream.wseq.dropn_add Stream'.Wseq.dropn_add\n\ntheorem dropn_tail (s : Wseq α) (n) : drop (tail s) n = drop s (n + 1) := by\n  rw [add_comm] <;> symm <;> apply dropn_add\n#align stream.wseq.dropn_tail Stream'.Wseq.dropn_tail\n\ntheorem nth_add (s : Wseq α) (m n) : nth s (m + n) = nth (drop s m) n :=\n  congr_arg head (dropn_add _ _ _)\n#align stream.wseq.nth_add Stream'.Wseq.nth_add\n\ntheorem nth_tail (s : Wseq α) (n) : nth (tail s) n = nth s (n + 1) :=\n  congr_arg head (dropn_tail _ _)\n#align stream.wseq.nth_tail Stream'.Wseq.nth_tail\n\n@[simp]\ntheorem join_nil : join nil = (nil : Wseq α) :=\n  Seq.join_nil\n#align stream.wseq.join_nil Stream'.Wseq.join_nil\n\n@[simp]\ntheorem join_think (S : Wseq (Wseq α)) : join (think S) = think (join S) :=\n  by\n  simp [think, join]\n  unfold Functor.map\n  simp [join, seq1.ret]\n#align stream.wseq.join_think Stream'.Wseq.join_think\n\n@[simp]\ntheorem join_cons (s : Wseq α) (S) : join (cons s S) = think (append s (join S)) :=\n  by\n  simp [think, join]\n  unfold Functor.map\n  simp [join, cons, append]\n#align stream.wseq.join_cons Stream'.Wseq.join_cons\n\n@[simp]\ntheorem nil_append (s : Wseq α) : append nil s = s :=\n  Seq.nil_append _\n#align stream.wseq.nil_append Stream'.Wseq.nil_append\n\n@[simp]\ntheorem cons_append (a : α) (s t) : append (cons a s) t = cons a (append s t) :=\n  Seq.cons_append _ _ _\n#align stream.wseq.cons_append Stream'.Wseq.cons_append\n\n@[simp]\ntheorem think_append (s t : Wseq α) : append (think s) t = think (append s t) :=\n  Seq.cons_append _ _ _\n#align stream.wseq.think_append Stream'.Wseq.think_append\n\n@[simp]\ntheorem append_nil (s : Wseq α) : append s nil = s :=\n  Seq.append_nil _\n#align stream.wseq.append_nil Stream'.Wseq.append_nil\n\n@[simp]\ntheorem append_assoc (s t u : Wseq α) : append (append s t) u = append s (append t u) :=\n  Seq.append_assoc _ _ _\n#align stream.wseq.append_assoc Stream'.Wseq.append_assoc\n\n/-- auxilary defintion of tail over weak sequences-/\n@[simp]\ndef tail.aux : Option (α × Wseq α) → Computation (Option (α × Wseq α))\n  | none => return none\n  | some (a, s) => destruct s\n#align stream.wseq.tail.aux Stream'.Wseq.tail.aux\n\ntheorem destruct_tail (s : Wseq α) : destruct (tail s) = destruct s >>= tail.aux :=\n  by\n  simp [tail]; rw [← bind_pure_comp_eq_map, LawfulMonad.bind_assoc]\n  apply congr_arg; ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp\n#align stream.wseq.destruct_tail Stream'.Wseq.destruct_tail\n\n/-- auxilary defintion of drop over weak sequences-/\n@[simp]\ndef drop.aux : ℕ → Option (α × Wseq α) → Computation (Option (α × Wseq α))\n  | 0 => return\n  | n + 1 => fun a => tail.aux a >>= drop.aux n\n#align stream.wseq.drop.aux Stream'.Wseq.drop.aux\n\ntheorem drop.aux_none : ∀ n, @drop.aux α n none = return none\n  | 0 => rfl\n  | n + 1 =>\n    show Computation.bind (return none) (drop.aux n) = return none by rw [ret_bind, drop.aux_none]\n#align stream.wseq.drop.aux_none Stream'.Wseq.drop.aux_none\n\ntheorem destruct_dropn : ∀ (s : Wseq α) (n), destruct (drop s n) = destruct s >>= drop.aux n\n  | s, 0 => (bind_pure' _).symm\n  | s, n + 1 => by\n    rw [← dropn_tail, destruct_dropn _ n, destruct_tail, LawfulMonad.bind_assoc] <;> rfl\n#align stream.wseq.destruct_dropn Stream'.Wseq.destruct_dropn\n\ntheorem head_terminates_of_head_tail_terminates (s : Wseq α) [T : Terminates (head (tail s))] :\n    Terminates (head s) :=\n  (head_terminates_iff _).2 <|\n    by\n    rcases(head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩\n    simp [tail] at h\n    rcases exists_of_mem_bind h with ⟨s', h1, h2⟩\n    unfold Functor.map at h1\n    exact\n      let ⟨t, h3, h4⟩ := Computation.exists_of_mem_map h1\n      Computation.terminates_of_mem h3\n#align stream.wseq.head_terminates_of_head_tail_terminates Stream'.Wseq.head_terminates_of_head_tail_terminates\n\ntheorem destruct_some_of_destruct_tail_some {s : Wseq α} {a} (h : some a ∈ destruct (tail s)) :\n    ∃ a', some a' ∈ destruct s := by\n  unfold tail Functor.map at h; simp at h\n  rcases exists_of_mem_bind h with ⟨t, tm, td⟩; clear h\n  rcases Computation.exists_of_mem_map tm with ⟨t', ht', ht2⟩; clear tm\n  cases' t' with t' <;> rw [← ht2] at td <;> simp at td\n  · have := mem_unique td (ret_mem _)\n    contradiction\n  · exact ⟨_, ht'⟩\n#align stream.wseq.destruct_some_of_destruct_tail_some Stream'.Wseq.destruct_some_of_destruct_tail_some\n\ntheorem head_some_of_head_tail_some {s : Wseq α} {a} (h : some a ∈ head (tail s)) :\n    ∃ a', some a' ∈ head s := by\n  unfold head at h\n  rcases Computation.exists_of_mem_map h with ⟨o, md, e⟩; clear h\n  cases' o with o <;> injection e with h'; clear e h'\n  cases' destruct_some_of_destruct_tail_some md with a am\n  exact ⟨_, Computation.mem_map ((· <$> ·) (@Prod.fst α (wseq α))) am⟩\n#align stream.wseq.head_some_of_head_tail_some Stream'.Wseq.head_some_of_head_tail_some\n\ntheorem head_some_of_nth_some {s : Wseq α} {a n} (h : some a ∈ nth s n) : ∃ a', some a' ∈ head s :=\n  by\n  revert a; induction' n with n IH <;> intros\n  exacts[⟨_, h⟩,\n    let ⟨a', h'⟩ := head_some_of_head_tail_some h\n    IH h']\n#align stream.wseq.head_some_of_nth_some Stream'.Wseq.head_some_of_nth_some\n\ninstance productive_tail (s : Wseq α) [Productive s] : Productive (tail s) :=\n  ⟨fun n => by rw [nth_tail] <;> infer_instance⟩\n#align stream.wseq.productive_tail Stream'.Wseq.productive_tail\n\ninstance productive_dropn (s : Wseq α) [Productive s] (n) : Productive (drop s n) :=\n  ⟨fun m => by rw [← nth_add] <;> infer_instance⟩\n#align stream.wseq.productive_dropn Stream'.Wseq.productive_dropn\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef toSeq (s : Wseq α) [Productive s] : Seq α :=\n  ⟨fun n => (nth s n).get, fun n h =>\n    by\n    cases e : Computation.get (nth s (n + 1)); · assumption\n    have := mem_of_get_eq _ e\n    simp [nth] at this h; cases' head_some_of_head_tail_some this with a' h'\n    have := mem_unique h' (@mem_of_get_eq _ _ _ _ h)\n    contradiction⟩\n#align stream.wseq.to_seq Stream'.Wseq.toSeq\n\ntheorem nth_terminates_le {s : Wseq α} {m n} (h : m ≤ n) :\n    Terminates (nth s n) → Terminates (nth s m) := by\n  induction' h with m' h IH <;> [exact id,\n    exact fun T => IH (@head_terminates_of_head_tail_terminates _ _ T)]\n#align stream.wseq.nth_terminates_le Stream'.Wseq.nth_terminates_le\n\ntheorem head_terminates_of_nth_terminates {s : Wseq α} {n} :\n    Terminates (nth s n) → Terminates (head s) :=\n  nth_terminates_le (Nat.zero_le n)\n#align stream.wseq.head_terminates_of_nth_terminates Stream'.Wseq.head_terminates_of_nth_terminates\n\ntheorem destruct_terminates_of_nth_terminates {s : Wseq α} {n} (T : Terminates (nth s n)) :\n    Terminates (destruct s) :=\n  (head_terminates_iff _).1 <| head_terminates_of_nth_terminates T\n#align stream.wseq.destruct_terminates_of_nth_terminates Stream'.Wseq.destruct_terminates_of_nth_terminates\n\ntheorem mem_rec_on {C : Wseq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))\n    (h2 : ∀ s, C s → C (think s)) : C s :=\n  by\n  apply seq.mem_rec_on M\n  intro o s' h; cases' o with b\n  · apply h2\n    cases h\n    · contradiction\n    · assumption\n  · apply h1\n    apply Or.imp_left _ h\n    intro h\n    injection h\n#align stream.wseq.mem_rec_on Stream'.Wseq.mem_rec_on\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem mem_think (s : Wseq α) (a) : a ∈ think s ↔ a ∈ s :=\n  by\n  cases' s with f al\n  change (some (some a) ∈ some none::f) ↔ some (some a) ∈ f\n  constructor <;> intro h\n  · apply (Stream'.eq_or_mem_of_mem_cons h).resolve_left\n    intro\n    injections\n  · apply Stream'.mem_cons_of_mem _ h\n#align stream.wseq.mem_think Stream'.Wseq.mem_think\n\ntheorem eq_or_mem_iff_mem {s : Wseq α} {a a' s'} :\n    some (a', s') ∈ destruct s → (a ∈ s ↔ a = a' ∨ a ∈ s') :=\n  by\n  generalize e : destruct s = c; intro h\n  revert s;\n  apply Computation.memRecOn h _ fun c IH => _ <;> intro s <;>\n            apply s.rec_on _ (fun x s => _) fun s => _ <;>\n          intro m <;>\n        have := congr_arg Computation.destruct m <;>\n      simp at this <;>\n    cases' this with i1 i2\n  · rw [i1, i2]\n    cases' s' with f al\n    unfold cons Membership.Mem wseq.mem seq.mem seq.cons\n    simp\n    have h_a_eq_a' : a = a' ↔ some (some a) = some (some a') := by simp\n    rw [h_a_eq_a']\n    refine' ⟨Stream'.eq_or_mem_of_mem_cons, fun o => _⟩\n    · cases' o with e m\n      · rw [e]\n        apply Stream'.mem_cons\n      · exact Stream'.mem_cons_of_mem _ m\n  · simp\n    exact IH this\n#align stream.wseq.eq_or_mem_iff_mem Stream'.Wseq.eq_or_mem_iff_mem\n\n@[simp]\ntheorem mem_cons_iff (s : Wseq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=\n  eq_or_mem_iff_mem <| by simp [ret_mem]\n#align stream.wseq.mem_cons_iff Stream'.Wseq.mem_cons_iff\n\ntheorem mem_cons_of_mem {s : Wseq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=\n  (mem_cons_iff _ _).2 (Or.inr h)\n#align stream.wseq.mem_cons_of_mem Stream'.Wseq.mem_cons_of_mem\n\ntheorem mem_cons (s : Wseq α) (a) : a ∈ cons a s :=\n  (mem_cons_iff _ _).2 (Or.inl rfl)\n#align stream.wseq.mem_cons Stream'.Wseq.mem_cons\n\ntheorem mem_of_mem_tail {s : Wseq α} {a} : a ∈ tail s → a ∈ s :=\n  by\n  intro h; have := h; cases' h with n e; revert s; simp [Stream'.nth]\n  induction' n with n IH <;> intro s <;> apply s.rec_on _ (fun x s => _) fun s => _ <;>\n        repeat' simp <;>\n      intro m e <;>\n    injections\n  · exact Or.inr m\n  · exact Or.inr m\n  · apply IH m\n    rw [e]\n    cases tail s\n    rfl\n#align stream.wseq.mem_of_mem_tail Stream'.Wseq.mem_of_mem_tail\n\ntheorem mem_of_mem_dropn {s : Wseq α} {a} : ∀ {n}, a ∈ drop s n → a ∈ s\n  | 0, h => h\n  | n + 1, h => @mem_of_mem_dropn n (mem_of_mem_tail h)\n#align stream.wseq.mem_of_mem_dropn Stream'.Wseq.mem_of_mem_dropn\n\ntheorem nth_mem {s : Wseq α} {a n} : some a ∈ nth s n → a ∈ s :=\n  by\n  revert s; induction' n with n IH <;> intro s h\n  · rcases Computation.exists_of_mem_map h with ⟨o, h1, h2⟩\n    cases' o with o <;> injection h2 with h'\n    cases' o with a' s'\n    exact (eq_or_mem_iff_mem h1).2 (Or.inl h'.symm)\n  · have := @IH (tail s)\n    rw [nth_tail] at this\n    exact mem_of_mem_tail (this h)\n#align stream.wseq.nth_mem Stream'.Wseq.nth_mem\n\ntheorem exists_nth_of_mem {s : Wseq α} {a} (h : a ∈ s) : ∃ n, some a ∈ nth s n :=\n  by\n  apply mem_rec_on h\n  · intro a' s' h\n    cases' h with h h\n    · exists 0\n      simp [nth]\n      rw [h]\n      apply ret_mem\n    · cases' h with n h\n      exists n + 1\n      simp [nth]\n      exact h\n  · intro s' h\n    cases' h with n h\n    exists n\n    simp [nth]\n    apply think_mem h\n#align stream.wseq.exists_nth_of_mem Stream'.Wseq.exists_nth_of_mem\n\ntheorem exists_dropn_of_mem {s : Wseq α} {a} (h : a ∈ s) :\n    ∃ n s', some (a, s') ∈ destruct (drop s n) :=\n  let ⟨n, h⟩ := exists_nth_of_mem h\n  ⟨n, by\n    rcases(head_terminates_iff _).1 ⟨⟨_, h⟩⟩ with ⟨⟨o, om⟩⟩\n    have := Computation.mem_unique (Computation.mem_map _ om) h\n    cases' o with o <;> injection this with i\n    cases' o with a' s'; dsimp at i\n    rw [i] at om; exact ⟨_, om⟩⟩\n#align stream.wseq.exists_dropn_of_mem Stream'.Wseq.exists_dropn_of_mem\n\ntheorem liftRel_dropn_destruct {R : α → β → Prop} {s t} (H : LiftRel R s t) :\n    ∀ n, Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct (drop s n)) (destruct (drop t n))\n  | 0 => liftRel_destruct H\n  | n + 1 => by\n    simp [destruct_tail]\n    apply lift_rel_bind\n    apply lift_rel_dropn_destruct n\n    exact fun a b o =>\n      match a, b, o with\n      | none, none, _ => by simp\n      | some (a, s), some (b, t), ⟨h1, h2⟩ => by simp [tail.aux] <;> apply lift_rel_destruct h2\n#align stream.wseq.lift_rel_dropn_destruct Stream'.Wseq.liftRel_dropn_destruct\n\ntheorem exists_of_liftRel_left {R : α → β → Prop} {s t} (H : LiftRel R s t) {a} (h : a ∈ s) :\n    ∃ b, b ∈ t ∧ R a b :=\n  let ⟨n, h⟩ := exists_nth_of_mem h\n  let ⟨some (_, s'), sd, rfl⟩ := Computation.exists_of_mem_map h\n  let ⟨some (b, t'), td, ⟨ab, _⟩⟩ := (liftRel_dropn_destruct H n).left sd\n  ⟨b, nth_mem (Computation.mem_map ((· <$> ·) Prod.fst.{v, v}) td), ab⟩\n#align stream.wseq.exists_of_lift_rel_left Stream'.Wseq.exists_of_liftRel_left\n\ntheorem exists_of_liftRel_right {R : α → β → Prop} {s t} (H : LiftRel R s t) {b} (h : b ∈ t) :\n    ∃ a, a ∈ s ∧ R a b := by rw [← lift_rel.swap] at H <;> exact exists_of_lift_rel_left H h\n#align stream.wseq.exists_of_lift_rel_right Stream'.Wseq.exists_of_liftRel_right\n\ntheorem head_terminates_of_mem {s : Wseq α} {a} (h : a ∈ s) : Terminates (head s) :=\n  let ⟨n, h⟩ := exists_nth_of_mem h\n  head_terminates_of_nth_terminates ⟨⟨_, h⟩⟩\n#align stream.wseq.head_terminates_of_mem Stream'.Wseq.head_terminates_of_mem\n\ntheorem of_mem_append {s₁ s₂ : Wseq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=\n  Seq.of_mem_append\n#align stream.wseq.of_mem_append Stream'.Wseq.of_mem_append\n\ntheorem mem_append_left {s₁ s₂ : Wseq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=\n  Seq.mem_append_left\n#align stream.wseq.mem_append_left Stream'.Wseq.mem_append_left\n\ntheorem exists_of_mem_map {f} {b : β} : ∀ {s : Wseq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b\n  | ⟨g, al⟩, h => by\n    let ⟨o, om, oe⟩ := Seq.exists_of_mem_map h\n    cases' o with a <;> injection oe with h' <;> exact ⟨a, om, h'⟩\n#align stream.wseq.exists_of_mem_map Stream'.Wseq.exists_of_mem_map\n\n@[simp]\ntheorem liftRel_nil (R : α → β → Prop) : LiftRel R nil nil := by rw [lift_rel_destruct_iff] <;> simp\n#align stream.wseq.lift_rel_nil Stream'.Wseq.liftRel_nil\n\n@[simp]\ntheorem liftRel_cons (R : α → β → Prop) (a b s t) :\n    LiftRel R (cons a s) (cons b t) ↔ R a b ∧ LiftRel R s t := by\n  rw [lift_rel_destruct_iff] <;> simp\n#align stream.wseq.lift_rel_cons Stream'.Wseq.liftRel_cons\n\n@[simp]\ntheorem liftRel_think_left (R : α → β → Prop) (s t) : LiftRel R (think s) t ↔ LiftRel R s t := by\n  rw [lift_rel_destruct_iff, lift_rel_destruct_iff] <;> simp\n#align stream.wseq.lift_rel_think_left Stream'.Wseq.liftRel_think_left\n\n@[simp]\ntheorem liftRel_think_right (R : α → β → Prop) (s t) : LiftRel R s (think t) ↔ LiftRel R s t := by\n  rw [lift_rel_destruct_iff, lift_rel_destruct_iff] <;> simp\n#align stream.wseq.lift_rel_think_right Stream'.Wseq.liftRel_think_right\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem cons_congr {s t : Wseq α} (a : α) (h : s ~ t) : cons a s ~ cons a t := by\n  unfold Equiv <;> simp <;> exact h\n#align stream.wseq.cons_congr Stream'.Wseq.cons_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem think_equiv (s : Wseq α) : think s ~ s := by unfold Equiv <;> simp <;> apply Equiv.refl\n#align stream.wseq.think_equiv Stream'.Wseq.think_equiv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem think_congr {s t : Wseq α} (h : s ~ t) : think s ~ think t := by\n  unfold Equiv <;> simp <;> exact h\n#align stream.wseq.think_congr Stream'.Wseq.think_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem head_congr : ∀ {s t : Wseq α}, s ~ t → head s ~ head t :=\n  by\n  suffices ∀ {s t : Wseq α}, s ~ t → ∀ {o}, o ∈ head s → o ∈ head t from fun s t h o =>\n    ⟨this h, this h.symm⟩\n  intro s t h o ho\n  rcases@Computation.exists_of_mem_map _ _ _ _ (destruct s) ho with ⟨ds, dsm, dse⟩\n  rw [← dse]\n  cases' destruct_congr h with l r\n  rcases l dsm with ⟨dt, dtm, dst⟩\n  cases' ds with a <;> cases' dt with b\n  · apply Computation.mem_map _ dtm\n  · cases b\n    cases dst\n  · cases a\n    cases dst\n  · cases' a with a s'\n    cases' b with b t'\n    rw [dst.left]\n    exact @Computation.mem_map _ _ (@Functor.map _ _ (α × wseq α) _ Prod.fst) _ (destruct t) dtm\n#align stream.wseq.head_congr Stream'.Wseq.head_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem flatten_equiv {c : Computation (Wseq α)} {s} (h : s ∈ c) : flatten c ~ s :=\n  by\n  apply Computation.memRecOn h; · simp\n  · intro s'\n    apply Equiv.trans\n    simp [think_equiv]\n#align stream.wseq.flatten_equiv Stream'.Wseq.flatten_equiv\n\ntheorem liftRel_flatten {R : α → β → Prop} {c1 : Computation (Wseq α)} {c2 : Computation (Wseq β)}\n    (h : c1.LiftRel (LiftRel R) c2) : LiftRel R (flatten c1) (flatten c2) :=\n  let S s t := ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ Computation.LiftRel (LiftRel R) c1 c2\n  ⟨S, ⟨c1, c2, rfl, rfl, h⟩, fun s t h =>\n    match s, t, h with\n    | _, _, ⟨c1, c2, rfl, rfl, h⟩ => by\n      simp; apply lift_rel_bind _ _ h\n      intro a b ab; apply Computation.LiftRel.imp _ _ _ (lift_rel_destruct ab)\n      intro a b; apply lift_rel_o.imp_right\n      intro s t h; refine' ⟨return s, return t, _, _, _⟩ <;> simp [h]⟩\n#align stream.wseq.lift_rel_flatten Stream'.Wseq.liftRel_flatten\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem flatten_congr {c1 c2 : Computation (Wseq α)} :\n    Computation.LiftRel Equiv c1 c2 → flatten c1 ~ flatten c2 :=\n  liftRel_flatten\n#align stream.wseq.flatten_congr Stream'.Wseq.flatten_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem tail_congr {s t : Wseq α} (h : s ~ t) : tail s ~ tail t :=\n  by\n  apply flatten_congr\n  unfold Functor.map; rw [← bind_ret, ← bind_ret]\n  apply lift_rel_bind _ _ (destruct_congr h)\n  intro a b h; simp\n  cases' a with a <;> cases' b with b\n  · trivial\n  · cases h\n  · cases a\n    cases h\n  · cases' a with a s'\n    cases' b with b t'\n    exact h.right\n#align stream.wseq.tail_congr Stream'.Wseq.tail_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem dropn_congr {s t : Wseq α} (h : s ~ t) (n) : drop s n ~ drop t n := by\n  induction n <;> simp [*, tail_congr]\n#align stream.wseq.dropn_congr Stream'.Wseq.dropn_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem nth_congr {s t : Wseq α} (h : s ~ t) (n) : nth s n ~ nth t n :=\n  head_congr (dropn_congr h _)\n#align stream.wseq.nth_congr Stream'.Wseq.nth_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem mem_congr {s t : Wseq α} (h : s ~ t) (a) : a ∈ s ↔ a ∈ t :=\n  suffices ∀ {s t : Wseq α}, s ~ t → a ∈ s → a ∈ t from ⟨this h, this h.symm⟩\n  fun s t h as =>\n  let ⟨n, hn⟩ := exists_nth_of_mem as\n  nth_mem ((nth_congr h _ _).1 hn)\n#align stream.wseq.mem_congr Stream'.Wseq.mem_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem productive_congr {s t : Wseq α} (h : s ~ t) : Productive s ↔ Productive t := by\n  simp only [productive_iff] <;> exact forall_congr' fun n => terminates_congr <| nth_congr h _\n#align stream.wseq.productive_congr Stream'.Wseq.productive_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Equiv.ext {s t : Wseq α} (h : ∀ n, nth s n ~ nth t n) : s ~ t :=\n  ⟨fun s t => ∀ n, nth s n ~ nth t n, h, fun s t h =>\n    by\n    refine' lift_rel_def.2 ⟨_, _⟩\n    · rw [← head_terminates_iff, ← head_terminates_iff]\n      exact terminates_congr (h 0)\n    · intro a b ma mb\n      cases' a with a <;> cases' b with b\n      · trivial\n      · injection mem_unique (Computation.mem_map _ ma) ((h 0 _).2 (Computation.mem_map _ mb))\n      · injection mem_unique (Computation.mem_map _ ma) ((h 0 _).2 (Computation.mem_map _ mb))\n      · cases' a with a s'\n        cases' b with b t'\n        injection mem_unique (Computation.mem_map _ ma) ((h 0 _).2 (Computation.mem_map _ mb)) with\n          ab\n        refine' ⟨ab, fun n => _⟩\n        refine'\n          (nth_congr (flatten_equiv (Computation.mem_map _ ma)) n).symm.trans\n            ((_ : nth (tail s) n ~ nth (tail t) n).trans\n              (nth_congr (flatten_equiv (Computation.mem_map _ mb)) n))\n        rw [nth_tail, nth_tail]\n        apply h⟩\n#align stream.wseq.equiv.ext Stream'.Wseq.Equiv.ext\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem length_eq_map (s : Wseq α) : length s = Computation.map List.length (toList s) :=\n  by\n  refine'\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ (l : List α)(s : wseq α),\n          c1 = Computation.corec length._match_2 (l.length, s) ∧\n            c2 = Computation.map List.length (Computation.corec to_list._match_2 (l, s)))\n      _ ⟨[], s, rfl, rfl⟩\n  intro s1 s2 h; rcases h with ⟨l, s, h⟩; rw [h.left, h.right]\n  apply s.rec_on _ (fun a s => _) fun s => _ <;> repeat' simp [to_list, nil, cons, think, length]\n  · refine' ⟨a::l, s, _, _⟩ <;> simp\n  · refine' ⟨l, s, _, _⟩ <;> simp\n#align stream.wseq.length_eq_map Stream'.Wseq.length_eq_map\n\n@[simp]\ntheorem ofList_nil : ofList [] = (nil : Wseq α) :=\n  rfl\n#align stream.wseq.of_list_nil Stream'.Wseq.ofList_nil\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem ofList_cons (a : α) (l) : ofList (a::l) = cons a (ofList l) :=\n  show Seq.map some (Seq.ofList (a::l)) = Seq.cons (some a) (Seq.map some (Seq.ofList l)) by simp\n#align stream.wseq.of_list_cons Stream'.Wseq.ofList_cons\n\n@[simp]\ntheorem to_list'_nil (l : List α) : Computation.corec ToList._match2 (l, nil) = return l.reverse :=\n  destruct_eq_pure rfl\n#align stream.wseq.to_list'_nil Stream'.Wseq.to_list'_nil\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem to_list'_cons (l : List α) (s : Wseq α) (a : α) :\n    Computation.corec ToList._match2 (l, cons a s) =\n      (Computation.corec ToList._match2 (a::l, s)).think :=\n  destruct_eq_think <| by simp [to_list, cons]\n#align stream.wseq.to_list'_cons Stream'.Wseq.to_list'_cons\n\n@[simp]\ntheorem to_list'_think (l : List α) (s : Wseq α) :\n    Computation.corec ToList._match2 (l, think s) =\n      (Computation.corec ToList._match2 (l, s)).think :=\n  destruct_eq_think <| by simp [to_list, think]\n#align stream.wseq.to_list'_think Stream'.Wseq.to_list'_think\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem to_list'_map (l : List α) (s : Wseq α) :\n    Computation.corec ToList._match2 (l, s) = (· ++ ·) l.reverse <$> toList s :=\n  by\n  refine'\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ (l' : List α)(s : wseq α),\n          c1 = Computation.corec to_list._match_2 (l' ++ l, s) ∧\n            c2 = Computation.map ((· ++ ·) l.reverse) (Computation.corec to_list._match_2 (l', s)))\n      _ ⟨[], s, rfl, rfl⟩\n  intro s1 s2 h; rcases h with ⟨l', s, h⟩; rw [h.left, h.right]\n  apply s.rec_on _ (fun a s => _) fun s => _ <;> repeat' simp [to_list, nil, cons, think, length]\n  · refine' ⟨a::l', s, _, _⟩ <;> simp\n  · refine' ⟨l', s, _, _⟩ <;> simp\n#align stream.wseq.to_list'_map Stream'.Wseq.to_list'_map\n\n@[simp]\ntheorem toList_cons (a : α) (s) : toList (cons a s) = (List.cons a <$> toList s).think :=\n  destruct_eq_think <| by unfold to_list <;> simp <;> rw [to_list'_map] <;> simp <;> rfl\n#align stream.wseq.to_list_cons Stream'.Wseq.toList_cons\n\n@[simp]\ntheorem toList_nil : toList (nil : Wseq α) = return [] :=\n  destruct_eq_pure rfl\n#align stream.wseq.to_list_nil Stream'.Wseq.toList_nil\n\ntheorem toList_ofList (l : List α) : l ∈ toList (ofList l) := by\n  induction' l with a l IH <;> simp [ret_mem] <;> exact think_mem (Computation.mem_map _ IH)\n#align stream.wseq.to_list_of_list Stream'.Wseq.toList_ofList\n\n@[simp]\ntheorem destruct_ofSeq (s : Seq α) :\n    destruct (ofSeq s) = return (s.headI.map fun a => (a, ofSeq s.tail)) :=\n  destruct_eq_pure <| by\n    simp [of_seq, head, destruct, seq.destruct, seq.head]\n    rw [show seq.nth (some <$> s) 0 = some <$> seq.nth s 0 by apply seq.map_nth]\n    cases' seq.nth s 0 with a; · rfl\n    unfold Functor.map\n    simp [destruct]\n#align stream.wseq.destruct_of_seq Stream'.Wseq.destruct_ofSeq\n\n@[simp]\ntheorem head_ofSeq (s : Seq α) : head (ofSeq s) = return s.headI := by\n  simp [head] <;> cases seq.head s <;> rfl\n#align stream.wseq.head_of_seq Stream'.Wseq.head_ofSeq\n\n@[simp]\ntheorem tail_ofSeq (s : Seq α) : tail (ofSeq s) = ofSeq s.tail :=\n  by\n  simp [tail]; apply s.rec_on _ fun x s => _ <;> simp [of_seq]; · rfl\n  rw [seq.head_cons, seq.tail_cons]; rfl\n#align stream.wseq.tail_of_seq Stream'.Wseq.tail_ofSeq\n\n@[simp]\ntheorem dropn_ofSeq (s : Seq α) : ∀ n, drop (ofSeq s) n = ofSeq (s.drop n)\n  | 0 => rfl\n  | n + 1 => by dsimp [drop] <;> rw [dropn_of_seq, tail_of_seq]\n#align stream.wseq.dropn_of_seq Stream'.Wseq.dropn_ofSeq\n\ntheorem nth_ofSeq (s : Seq α) (n) : nth (ofSeq s) n = return (Seq.nth s n) := by\n  dsimp [nth] <;> rw [dropn_of_seq, head_of_seq, seq.head_dropn]\n#align stream.wseq.nth_of_seq Stream'.Wseq.nth_ofSeq\n\ninstance productive_ofSeq (s : Seq α) : Productive (ofSeq s) :=\n  ⟨fun n => by rw [nth_of_seq] <;> infer_instance⟩\n#align stream.wseq.productive_of_seq Stream'.Wseq.productive_ofSeq\n\ntheorem toSeq_ofSeq (s : Seq α) : toSeq (ofSeq s) = s :=\n  by\n  apply Subtype.eq; funext n\n  dsimp [to_seq]; apply get_eq_of_mem\n  rw [nth_of_seq]; apply ret_mem\n#align stream.wseq.to_seq_of_seq Stream'.Wseq.toSeq_ofSeq\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret (a : α) : Wseq α :=\n  ofList [a]\n#align stream.wseq.ret Stream'.Wseq.ret\n\n@[simp]\ntheorem map_nil (f : α → β) : map f nil = nil :=\n  rfl\n#align stream.wseq.map_nil Stream'.Wseq.map_nil\n\n@[simp]\ntheorem map_cons (f : α → β) (a s) : map f (cons a s) = cons (f a) (map f s) :=\n  Seq.map_cons _ _ _\n#align stream.wseq.map_cons Stream'.Wseq.map_cons\n\n@[simp]\ntheorem map_think (f : α → β) (s) : map f (think s) = think (map f s) :=\n  Seq.map_cons _ _ _\n#align stream.wseq.map_think Stream'.Wseq.map_think\n\n@[simp]\ntheorem map_id (s : Wseq α) : map id s = s := by simp [map]\n#align stream.wseq.map_id Stream'.Wseq.map_id\n\n@[simp]\ntheorem map_ret (f : α → β) (a) : map f (ret a) = ret (f a) := by simp [ret]\n#align stream.wseq.map_ret Stream'.Wseq.map_ret\n\n@[simp]\ntheorem map_append (f : α → β) (s t) : map f (append s t) = append (map f s) (map f t) :=\n  Seq.map_append _ _ _\n#align stream.wseq.map_append Stream'.Wseq.map_append\n\ntheorem map_comp (f : α → β) (g : β → γ) (s : Wseq α) : map (g ∘ f) s = map g (map f s) :=\n  by\n  dsimp [map]; rw [← seq.map_comp]\n  apply congr_fun; apply congr_arg\n  ext ⟨⟩ <;> rfl\n#align stream.wseq.map_comp Stream'.Wseq.map_comp\n\ntheorem mem_map (f : α → β) {a : α} {s : Wseq α} : a ∈ s → f a ∈ map f s :=\n  Seq.mem_map (Option.map f)\n#align stream.wseq.mem_map Stream'.Wseq.mem_map\n\n-- The converse is not true without additional assumptions\ntheorem exists_of_mem_join {a : α} : ∀ {S : Wseq (Wseq α)}, a ∈ join S → ∃ s, s ∈ S ∧ a ∈ s :=\n  by\n  suffices\n    ∀ ss : Wseq α,\n      a ∈ ss → ∀ s S, append s (join S) = ss → a ∈ append s (join S) → a ∈ s ∨ ∃ s, s ∈ S ∧ a ∈ s\n    from fun S h => (this _ h nil S (by simp) (by simp [h])).resolve_left (not_mem_nil _)\n  intro ss h; apply mem_rec_on h (fun b ss o => _) fun ss IH => _ <;> intro s S\n  · refine' s.rec_on (S.rec_on _ (fun s S => _) fun S => _) (fun b' s => _) fun s => _ <;>\n                intro ej m <;> simp at ej <;> have := congr_arg seq.destruct ej <;> simp at this <;>\n        try cases this <;> try contradiction\n    substs b' ss\n    simp at m⊢\n    cases' o with e IH\n    · simp [e]\n    cases' m with e m\n    · simp [e]\n    exact Or.imp_left Or.inr (IH _ _ rfl m)\n  · refine' s.rec_on (S.rec_on _ (fun s S => _) fun S => _) (fun b' s => _) fun s => _ <;>\n                intro ej m <;> simp at ej <;> have := congr_arg seq.destruct ej <;> simp at this <;>\n        try try have := this.1; contradiction <;> subst ss\n    · apply Or.inr\n      simp at m⊢\n      cases' IH s S rfl m with as ex\n      · exact ⟨s, Or.inl rfl, as⟩\n      · rcases ex with ⟨s', sS, as⟩\n        exact ⟨s', Or.inr sS, as⟩\n    · apply Or.inr\n      simp at m\n      rcases(IH nil S (by simp) (by simp [m])).resolve_left (not_mem_nil _) with ⟨s, sS, as⟩\n      exact ⟨s, by simp [sS], as⟩\n    · simp at m IH⊢\n      apply IH _ _ rfl m\n#align stream.wseq.exists_of_mem_join Stream'.Wseq.exists_of_mem_join\n\ntheorem exists_of_mem_bind {s : Wseq α} {f : α → Wseq β} {b} (h : b ∈ bind s f) :\n    ∃ a ∈ s, b ∈ f a :=\n  let ⟨t, tm, bt⟩ := exists_of_mem_join h\n  let ⟨a, as, e⟩ := exists_of_mem_map tm\n  ⟨a, as, by rwa [e]⟩\n#align stream.wseq.exists_of_mem_bind Stream'.Wseq.exists_of_mem_bind\n\ntheorem destruct_map (f : α → β) (s : Wseq α) :\n    destruct (map f s) = Computation.map (Option.map (Prod.map f (map f))) (destruct s) :=\n  by\n  apply\n    Computation.eq_of_bisim fun c1 c2 =>\n      ∃ s,\n        c1 = destruct (map f s) ∧\n          c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n  · intro c1 c2 h\n    cases' h with s h\n    rw [h.left, h.right]\n    apply s.rec_on _ (fun a s => _) fun s => _ <;> simp\n    exact ⟨s, rfl, rfl⟩\n  · exact ⟨s, rfl, rfl⟩\n#align stream.wseq.destruct_map Stream'.Wseq.destruct_map\n\ntheorem liftRel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : Wseq α} {s2 : Wseq β}\n    {f1 : α → γ} {f2 : β → δ} (h1 : LiftRel R s1 s2) (h2 : ∀ {a b}, R a b → S (f1 a) (f2 b)) :\n    LiftRel S (map f1 s1) (map f2 s2) :=\n  ⟨fun s1 s2 => ∃ s t, s1 = map f1 s ∧ s2 = map f2 t ∧ LiftRel R s t, ⟨s1, s2, rfl, rfl, h1⟩,\n    fun s1 s2 h =>\n    match s1, s2, h with\n    | _, _, ⟨s, t, rfl, rfl, h⟩ => by\n      simp [destruct_map]; apply Computation.liftRel_map _ _ (lift_rel_destruct h)\n      intro o p h\n      cases' o with a <;> cases' p with b <;> simp\n      · cases b <;> cases h\n      · cases a <;> cases h\n      · cases' a with a s <;> cases' b with b t\n        cases' h with r h\n        exact ⟨h2 r, s, rfl, t, rfl, h⟩⟩\n#align stream.wseq.lift_rel_map Stream'.Wseq.liftRel_map\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem map_congr (f : α → β) {s t : Wseq α} (h : s ~ t) : map f s ~ map f t :=\n  liftRel_map _ _ h fun _ _ => congr_arg _\n#align stream.wseq.map_congr Stream'.Wseq.map_congr\n\n/-- auxilary defintion of `destruct_append` over weak sequences-/\n@[simp]\ndef DestructAppend.aux (t : Wseq α) : Option (α × Wseq α) → Computation (Option (α × Wseq α))\n  | none => destruct t\n  | some (a, s) => return (some (a, append s t))\n#align stream.wseq.destruct_append.aux Stream'.Wseq.DestructAppend.aux\n\ntheorem destruct_append (s t : Wseq α) :\n    destruct (append s t) = (destruct s).bind (DestructAppend.aux t) :=\n  by\n  apply\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        ∃ s t, c1 = destruct (append s t) ∧ c2 = (destruct s).bind (destruct_append.aux t))\n      _ ⟨s, t, rfl, rfl⟩\n  intro c1 c2 h; rcases h with ⟨s, t, h⟩; rw [h.left, h.right]\n  apply s.rec_on _ (fun a s => _) fun s => _ <;> simp\n  · apply t.rec_on _ (fun b t => _) fun t => _ <;> simp\n    · refine' ⟨nil, t, _, _⟩ <;> simp\n  · exact ⟨s, t, rfl, rfl⟩\n#align stream.wseq.destruct_append Stream'.Wseq.destruct_append\n\n/-- auxilary defintion of `destruct_join` over weak sequences-/\n@[simp]\ndef DestructJoin.aux : Option (Wseq α × Wseq (Wseq α)) → Computation (Option (α × Wseq α))\n  | none => return none\n  | some (s, S) => (destruct (append s (join S))).think\n#align stream.wseq.destruct_join.aux Stream'.Wseq.DestructJoin.aux\n\ntheorem destruct_join (S : Wseq (Wseq α)) :\n    destruct (join S) = (destruct S).bind DestructJoin.aux :=\n  by\n  apply\n    Computation.eq_of_bisim\n      (fun c1 c2 =>\n        c1 = c2 ∨ ∃ S, c1 = destruct (join S) ∧ c2 = (destruct S).bind destruct_join.aux)\n      _ (Or.inr ⟨S, rfl, rfl⟩)\n  intro c1 c2 h;\n  exact\n    match c1, c2, h with\n    | _, _, Or.inl <| Eq.refl c => by cases c.destruct <;> simp\n    | _, _, Or.inr ⟨S, rfl, rfl⟩ =>\n      by\n      apply S.rec_on _ (fun s S => _) fun S => _ <;> simp\n      · refine' Or.inr ⟨S, rfl, rfl⟩\n#align stream.wseq.destruct_join Stream'.Wseq.destruct_join\n\ntheorem liftRel_append (R : α → β → Prop) {s1 s2 : Wseq α} {t1 t2 : Wseq β} (h1 : LiftRel R s1 t1)\n    (h2 : LiftRel R s2 t2) : LiftRel R (append s1 s2) (append t1 t2) :=\n  ⟨fun s t => LiftRel R s t ∨ ∃ s1 t1, s = append s1 s2 ∧ t = append t1 t2 ∧ LiftRel R s1 t1,\n    Or.inr ⟨s1, t1, rfl, rfl, h1⟩, fun s t h =>\n    match s, t, h with\n    | s, t, Or.inl h =>\n      by\n      apply Computation.LiftRel.imp _ _ _ (lift_rel_destruct h)\n      intro a b; apply lift_rel_o.imp_right\n      intro s t; apply Or.inl\n    | _, _, Or.inr ⟨s1, t1, rfl, rfl, h⟩ =>\n      by\n      simp [destruct_append]\n      apply Computation.liftRel_bind _ _ (lift_rel_destruct h)\n      intro o p h\n      cases' o with a <;> cases' p with b\n      · simp\n        apply Computation.LiftRel.imp _ _ _ (lift_rel_destruct h2)\n        intro a b\n        apply lift_rel_o.imp_right\n        intro s t\n        apply Or.inl\n      · cases b <;> cases h\n      · cases a <;> cases h\n      · cases' a with a s <;> cases' b with b t\n        cases' h with r h\n        simp\n        exact ⟨r, Or.inr ⟨s, rfl, t, rfl, h⟩⟩⟩\n#align stream.wseq.lift_rel_append Stream'.Wseq.liftRel_append\n\ntheorem LiftRelJoin.lem (R : α → β → Prop) {S T} {U : Wseq α → Wseq β → Prop}\n    (ST : LiftRel (LiftRel R) S T)\n    (HU :\n      ∀ s1 s2,\n        (∃ s t S T,\n            s1 = append s (join S) ∧\n              s2 = append t (join T) ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) →\n          U s1 s2)\n    {a} (ma : a ∈ destruct (join S)) : ∃ b, b ∈ destruct (join T) ∧ LiftRelO R U a b :=\n  by\n  cases' exists_results_of_mem ma with n h; clear ma; revert a S T\n  apply Nat.strong_induction_on n _\n  intro n IH a S T ST ra; simp [destruct_join] at ra;\n  exact\n    let ⟨o, m, k, rs1, rs2, en⟩ := of_results_bind ra\n    let ⟨p, mT, rop⟩ := Computation.exists_of_LiftRel_left (lift_rel_destruct ST) rs1.Mem\n    match o, p, rop, rs1, rs2, mT with\n    | none, none, _, rs1, rs2, mT => by\n      simp only [destruct_join] <;>\n        exact ⟨none, mem_bind mT (ret_mem _), by rw [eq_of_ret_mem rs2.mem] <;> trivial⟩\n    | some (s, S'), some (t, T'), ⟨st, ST'⟩, rs1, rs2, mT => by\n      simp [destruct_append] at rs2 <;>\n        exact\n          let ⟨k1, rs3, ek⟩ := of_results_think rs2\n          let ⟨o', m1, n1, rs4, rs5, ek1⟩ := of_results_bind rs3\n          let ⟨p', mt, rop'⟩ := Computation.exists_of_LiftRel_left (lift_rel_destruct st) rs4.Mem\n          match o', p', rop', rs4, rs5, mt with\n          | none, none, _, rs4, rs5', mt =>\n            by\n            have : n1 < n := by\n              rw [en, ek, ek1]\n              apply lt_of_lt_of_le _ (Nat.le_add_right _ _)\n              apply Nat.lt_succ_of_le (Nat.le_add_right _ _)\n            let ⟨ob, mb, rob⟩ := IH _ this ST' rs5'\n            refine' ⟨ob, _, rob⟩ <;>\n              · simp [destruct_join]\n                apply mem_bind mT\n                simp [destruct_append]\n                apply think_mem\n                apply mem_bind mt\n                exact mb\n          | some (a, s'), some (b, t'), ⟨ab, st'⟩, rs4, rs5, mt =>\n            by\n            simp at rs5\n            refine' ⟨some (b, append t' (join T')), _, _⟩\n            · simp [destruct_join]\n              apply mem_bind mT\n              simp [destruct_append]\n              apply think_mem\n              apply mem_bind mt\n              apply ret_mem\n            rw [eq_of_ret_mem rs5.mem]\n            exact ⟨ab, HU _ _ ⟨s', t', S', T', rfl, rfl, st', ST'⟩⟩\n#align stream.wseq.lift_rel_join.lem Stream'.Wseq.LiftRelJoin.lem\n\ntheorem liftRel_join (R : α → β → Prop) {S : Wseq (Wseq α)} {T : Wseq (Wseq β)}\n    (h : LiftRel (LiftRel R) S T) : LiftRel R (join S) (join T) :=\n  ⟨fun s1 s2 =>\n    ∃ s t S T,\n      s1 = append s (join S) ∧ s2 = append t (join T) ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T,\n    ⟨nil, nil, S, T, by simp, by simp, by simp, h⟩, fun s1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩ =>\n    by\n    clear _fun_match _x\n    rw [h1, h2]; rw [destruct_append, destruct_append]\n    apply Computation.liftRel_bind _ _ (lift_rel_destruct st)\n    exact fun o p h =>\n      match o, p, h with\n      | some (a, s), some (b, t), ⟨h1, h2⟩ => by simp <;> exact ⟨h1, s, t, S, rfl, T, rfl, h2, ST⟩\n      | none, none, _ => by\n        dsimp [destruct_append.aux, Computation.LiftRel]; constructor\n        · intro\n          apply lift_rel_join.lem _ ST fun _ _ => id\n        · intro b mb\n          rw [← lift_rel_o.swap]\n          apply lift_rel_join.lem (swap R)\n          · rw [← lift_rel.swap R, ← lift_rel.swap]\n            apply ST\n          · rw [← lift_rel.swap R, ← lift_rel.swap (lift_rel R)]\n            exact fun s1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩ => ⟨t, s, T, S, h2, h1, st, ST⟩\n          · exact mb⟩\n#align stream.wseq.lift_rel_join Stream'.Wseq.liftRel_join\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem join_congr {S T : Wseq (Wseq α)} (h : LiftRel Equiv S T) : join S ~ join T :=\n  liftRel_join _ h\n#align stream.wseq.join_congr Stream'.Wseq.join_congr\n\ntheorem liftRel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : Wseq α} {s2 : Wseq β}\n    {f1 : α → Wseq γ} {f2 : β → Wseq δ} (h1 : LiftRel R s1 s2)\n    (h2 : ∀ {a b}, R a b → LiftRel S (f1 a) (f2 b)) : LiftRel S (bind s1 f1) (bind s2 f2) :=\n  liftRel_join _ (liftRel_map _ _ h1 @h2)\n#align stream.wseq.lift_rel_bind Stream'.Wseq.liftRel_bind\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem bind_congr {s1 s2 : Wseq α} {f1 f2 : α → Wseq β} (h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) :\n    bind s1 f1 ~ bind s2 f2 :=\n  liftRel_bind _ _ h1 fun a b h => by rw [h] <;> apply h2\n#align stream.wseq.bind_congr Stream'.Wseq.bind_congr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem join_ret (s : Wseq α) : join (ret s) ~ s := by simp [ret] <;> apply think_equiv\n#align stream.wseq.join_ret Stream'.Wseq.join_ret\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem join_map_ret (s : Wseq α) : join (map ret s) ~ s :=\n  by\n  refine' ⟨fun s1 s2 => join (map ret s2) = s1, rfl, _⟩\n  intro s' s h; rw [← h]\n  apply lift_rel_rec fun c1 c2 => ∃ s, c1 = destruct (join (map ret s)) ∧ c2 = destruct s\n  ·\n    exact fun c1 c2 h =>\n      match c1, c2, h with\n      | _, _, ⟨s, rfl, rfl⟩ => by\n        clear h _match\n        have :\n          ∀ s,\n            ∃ s' : wseq α,\n              (map ret s).join.destruct = (map ret s').join.destruct ∧ destruct s = s'.destruct :=\n          fun s => ⟨s, rfl, rfl⟩\n        apply s.rec_on _ (fun a s => _) fun s => _ <;> simp [ret, ret_mem, this, Option.exists]\n  · exact ⟨s, rfl, rfl⟩\n#align stream.wseq.join_map_ret Stream'.Wseq.join_map_ret\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem join_append (S T : Wseq (Wseq α)) : join (append S T) ~ append (join S) (join T) :=\n  by\n  refine'\n    ⟨fun s1 s2 =>\n      ∃ s S T, s1 = append s (join (append S T)) ∧ s2 = append s (append (join S) (join T)),\n      ⟨nil, S, T, by simp, by simp⟩, _⟩\n  intro s1 s2 h\n  apply\n    lift_rel_rec\n      (fun c1 c2 =>\n        ∃ (s : wseq α)(S T : _),\n          c1 = destruct (append s (join (append S T))) ∧\n            c2 = destruct (append s (append (join S) (join T))))\n      _ _ _\n      (let ⟨s, S, T, h1, h2⟩ := h\n      ⟨s, S, T, congr_arg destruct h1, congr_arg destruct h2⟩)\n  intro c1 c2 h\n  exact\n    match c1, c2, h with\n    | _, _, ⟨s, S, T, rfl, rfl⟩ => by\n      clear _match h h\n      apply wseq.rec_on s _ (fun a s => _) fun s => _ <;> simp\n      · apply wseq.rec_on S _ (fun s S => _) fun S => _ <;> simp\n        · apply wseq.rec_on T _ (fun s T => _) fun T => _ <;> simp\n          · refine' ⟨s, nil, T, _, _⟩ <;> simp\n          · refine' ⟨nil, nil, T, _, _⟩ <;> simp\n        · exact ⟨s, S, T, rfl, rfl⟩\n        · refine' ⟨nil, S, T, _, _⟩ <;> simp\n      · exact ⟨s, S, T, rfl, rfl⟩\n      · exact ⟨s, S, T, rfl, rfl⟩\n#align stream.wseq.join_append Stream'.Wseq.join_append\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem bind_ret (f : α → β) (s) : bind s (ret ∘ f) ~ map f s :=\n  by\n  dsimp [bind]; change fun x => ret (f x) with ret ∘ f\n  rw [map_comp]; apply join_map_ret\n#align stream.wseq.bind_ret Stream'.Wseq.bind_ret\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem ret_bind (a : α) (f : α → Wseq β) : bind (ret a) f ~ f a := by simp [bind]\n#align stream.wseq.ret_bind Stream'.Wseq.ret_bind\n\n@[simp]\ntheorem map_join (f : α → β) (S) : map f (join S) = join (map (map f) S) :=\n  by\n  apply\n    seq.eq_of_bisim fun s1 s2 =>\n      ∃ s S, s1 = append s (map f (join S)) ∧ s2 = append s (join (map (map f) S))\n  · intro s1 s2 h\n    exact\n      match s1, s2, h with\n      | _, _, ⟨s, S, rfl, rfl⟩ =>\n        by\n        apply wseq.rec_on s _ (fun a s => _) fun s => _ <;> simp\n        · apply wseq.rec_on S _ (fun s S => _) fun S => _ <;> simp\n          · exact ⟨map f s, S, rfl, rfl⟩\n          · refine' ⟨nil, S, _, _⟩ <;> simp\n        · exact ⟨_, _, rfl, rfl⟩\n        · exact ⟨_, _, rfl, rfl⟩\n  · refine' ⟨nil, S, _, _⟩ <;> simp\n#align stream.wseq.map_join Stream'.Wseq.map_join\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem join_join (SS : Wseq (Wseq (Wseq α))) : join (join SS) ~ join (map join SS) :=\n  by\n  refine'\n    ⟨fun s1 s2 =>\n      ∃ s S SS,\n        s1 = append s (join (append S (join SS))) ∧\n          s2 = append s (append (join S) (join (map join SS))),\n      ⟨nil, nil, SS, by simp, by simp⟩, _⟩\n  intro s1 s2 h\n  apply\n    lift_rel_rec\n      (fun c1 c2 =>\n        ∃ s S SS,\n          c1 = destruct (append s (join (append S (join SS)))) ∧\n            c2 = destruct (append s (append (join S) (join (map join SS)))))\n      _ (destruct s1) (destruct s2)\n      (let ⟨s, S, SS, h1, h2⟩ := h\n      ⟨s, S, SS, by simp [h1], by simp [h2]⟩)\n  intro c1 c2 h\n  exact\n    match c1, c2, h with\n    | _, _, ⟨s, S, SS, rfl, rfl⟩ => by\n      clear _match h h\n      apply wseq.rec_on s _ (fun a s => _) fun s => _ <;> simp\n      · apply wseq.rec_on S _ (fun s S => _) fun S => _ <;> simp\n        · apply wseq.rec_on SS _ (fun S SS => _) fun SS => _ <;> simp\n          · refine' ⟨nil, S, SS, _, _⟩ <;> simp\n          · refine' ⟨nil, nil, SS, _, _⟩ <;> simp\n        · exact ⟨s, S, SS, rfl, rfl⟩\n        · refine' ⟨nil, S, SS, _, _⟩ <;> simp\n      · exact ⟨s, S, SS, rfl, rfl⟩\n      · exact ⟨s, S, SS, rfl, rfl⟩\n#align stream.wseq.join_join Stream'.Wseq.join_join\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp]\ntheorem bind_assoc (s : Wseq α) (f : α → Wseq β) (g : β → Wseq γ) :\n    bind (bind s f) g ~ bind s fun x : α => bind (f x) g :=\n  by\n  simp [bind]; rw [← map_comp f (map g), map_comp (map g ∘ f) join]\n  apply join_join\n#align stream.wseq.bind_assoc Stream'.Wseq.bind_assoc\n\ninstance : Monad Wseq where\n  map := @map\n  pure := @ret\n  bind := @bind\n\n/-\n  Unfortunately, wseq is not a lawful monad, because it does not satisfy\n  the monad laws exactly, only up to sequence equivalence.\n  Furthermore, even quotienting by the equivalence is not sufficient,\n  because the join operation involves lists of quotient elements,\n  with a lifted equivalence relation, and pure quotients cannot handle\n  this type of construction.\n\ninstance : is_lawful_monad wseq :=\n{ id_map := @map_id,\n  bind_pure_comp_eq_map := @bind_ret,\n  pure_bind := @ret_bind,\n  bind_assoc := @bind_assoc }\n-/\nend Wseq\n\nend Stream'\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Seq/Wseq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.34582061915087237}}
{"text": "import topology.category.Profinite.projective\nimport for_mathlib.Profinite.disjoint_union\nimport for_mathlib.concrete_equalizer\n\nnoncomputable theory\n\nopen category_theory\n\nuniverses u w v\n\nstructure ExtrDisc :=\n(val : Profinite.{u})\n[cond : projective val]\n\nnamespace ExtrDisc\n\n@[ext]\nstructure hom (X Y : ExtrDisc) := mk :: (val : X.val ⟶ Y.val)\n\ndef of (X : Profinite) [projective X] : ExtrDisc := ⟨X⟩\n\n@[simp]\nlemma of_val (X : Profinite) [projective X] : (of X).val = X := rfl\n\n@[simps]\ninstance : category ExtrDisc :=\n{ hom := hom,\n  id := λ X, ⟨𝟙 _⟩,\n  comp := λ X Y Z f g, ⟨f.val ≫ g.val⟩ }\n\n@[simps]\ndef _root_.ExtrDisc_to_Profinite : ExtrDisc ⥤ Profinite :=\n{ obj := val,\n  map := λ X Y f, f.val }\n\ninstance : full ExtrDisc_to_Profinite := { preimage := λ X Y f, ⟨f⟩ }\n\ninstance : faithful ExtrDisc_to_Profinite := { }\n\ninstance : concrete_category ExtrDisc.{u} :=\n{ forget := ExtrDisc_to_Profinite ⋙ forget _,\n  forget_faithful := ⟨⟩ }\n\ninstance : has_coe_to_sort ExtrDisc Type* :=\nconcrete_category.has_coe_to_sort _\n\ninstance {X Y : ExtrDisc} : has_coe_to_fun (X ⟶ Y) (λ f, X → Y) :=\n⟨λ f, f.val⟩\n\n@[simp]\nlemma coe_fun_eq {X Y : ExtrDisc} (f : X ⟶ Y) (x : X) :\n  f x = f.val x := rfl\n\ninstance (X : ExtrDisc) : projective X.val := X.cond\n\nexample (X : ExtrDisc) : projective (ExtrDisc_to_Profinite.obj X) :=\nby { dsimp, apply_instance }\n\ndef lift {X Y : Profinite} {P : ExtrDisc} (f : X ⟶ Y)\n  (hf : function.surjective f) (e : P.val ⟶ Y) : P.val ⟶ X :=\nbegin\n  haveI : epi f := by rwa Profinite.epi_iff_surjective f,\n  choose g h using projective.factors e f,\n  exact g,\nend\n\n@[simp, reassoc]\nlemma lift_lifts {X Y : Profinite} {P : ExtrDisc} (f : X ⟶ Y)\n  (hf : function.surjective f) (e : P.val ⟶ Y) :\n  lift f hf e ≫ f = e :=\nbegin\n  haveI : epi f := by rwa Profinite.epi_iff_surjective f,\n  apply (projective.factors e f).some_spec,\nend\n\ninstance (X : ExtrDisc) : topological_space X :=\nshow topological_space X.val, by apply_instance\n\ninstance (X : ExtrDisc) : compact_space X :=\nshow compact_space X.val, by apply_instance\n\ninstance (X : ExtrDisc) : t2_space X :=\nshow t2_space X.val, by apply_instance\n\ninstance (X : ExtrDisc) : totally_disconnected_space X :=\nshow totally_disconnected_space X.val, by apply_instance\n\n.-- move this\n-- @[simps]\ndef _root_.Profinite.sum_iso_coprod (X Y : Profinite.{u}) :\n  Profinite.sum X Y ≅ X ⨿ Y :=\n{ hom := Profinite.sum.desc _ _ limits.coprod.inl limits.coprod.inr,\n  inv := limits.coprod.desc (Profinite.sum.inl _ _) (Profinite.sum.inr _ _),\n  hom_inv_id' := by { apply Profinite.sum.hom_ext;\n    simp only [← category.assoc, category.comp_id, Profinite.sum.inl_desc,\n      limits.coprod.inl_desc, Profinite.sum.inr_desc, limits.coprod.inr_desc] },\n  inv_hom_id' := by { apply limits.coprod.hom_ext;\n    simp only [← category.assoc, category.comp_id, Profinite.sum.inl_desc,\n      limits.coprod.inl_desc, Profinite.sum.inr_desc, limits.coprod.inr_desc] } }\n\n@[simps]\ndef sum (X Y : ExtrDisc.{u}) : ExtrDisc.{u} :=\n{ val := Profinite.sum X.val Y.val,\n  cond := begin\n    let Z := Profinite.sum X.val Y.val,\n    apply projective.of_iso (Profinite.sum_iso_coprod X.val Y.val).symm,\n    apply_instance,\n  end }\n\n@[simps]\ndef sum.inl (X Y : ExtrDisc) : X ⟶ sum X Y :=\n⟨Profinite.sum.inl _ _⟩\n\n@[simps]\ndef sum.inr (X Y : ExtrDisc) : Y ⟶ sum X Y :=\n⟨Profinite.sum.inr _ _⟩\n\n@[simps]\ndef sum.desc {X Y Z : ExtrDisc} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  sum X Y ⟶ Z :=\n⟨Profinite.sum.desc _ _ f.val g.val⟩\n\n@[simp]\nlemma sum.inl_desc {X Y Z : ExtrDisc} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  sum.inl X Y ≫ sum.desc f g = f :=\nby { ext1, dsimp, simp }\n\n@[simp]\nlemma sum.inr_desc {X Y Z : ExtrDisc} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  sum.inr X Y ≫ sum.desc f g = g :=\nby { ext1, dsimp, simp }\n\n@[ext]\nlemma sum.hom_ext {X Y Z : ExtrDisc} (f g : sum X Y ⟶ Z)\n  (hl : sum.inl X Y ≫ f = sum.inl X Y ≫ g)\n  (hr : sum.inr X Y ≫ f = sum.inr X Y ≫ g) : f = g :=\nbegin\n  ext1,\n  apply Profinite.sum.hom_ext,\n  { apply_fun (λ e, e.val) at hl, exact hl },\n  { apply_fun (λ e, e.val) at hr, exact hr }\nend\n\n-- move this\ndef _root_.Profinite.empty_is_initial : limits.is_initial Profinite.empty.{u} :=\n@limits.is_initial.of_unique.{u} _ _ _ (λ Y, ⟨⟨Profinite.empty.elim _⟩, λ f, by { ext x, cases x, }⟩)\n\n@[simps]\ndef empty : ExtrDisc :=\n{ val := Profinite.empty,\n  cond := begin\n    let e : Profinite.empty ≅ ⊥_ _ :=\n    Profinite.empty_is_initial.unique_up_to_iso limits.initial_is_initial,\n    apply projective.of_iso e.symm,\n    -- apply_instance, <-- missing instance : projective (⊥_ _)\n    constructor,\n    introsI A B f g _,\n    refine ⟨limits.initial.to A, by simp⟩,\n  end }\n\n@[simps]\ndef empty.elim (X : ExtrDisc) : empty ⟶ X :=\n⟨Profinite.empty.elim _⟩\n\n@[ext]\nlemma empty.hom_ext {X : ExtrDisc} (f g : empty ⟶ X) : f = g :=\nby { ext x, cases x }\n\ndef sigma {ι : Type u} [fintype ι] (X : ι → ExtrDisc) : ExtrDisc :=\n{ val := Profinite.sigma $ λ i, (X i).val,\n  cond := begin\n    let e : Profinite.sigma (λ i, (X i).val) ≅ ∐ (λ i, (X i).val) :=\n      (Profinite.sigma_cofan_is_colimit _).cocone_point_unique_up_to_iso\n      (limits.colimit.is_colimit _),\n    apply projective.of_iso e.symm,\n    apply_instance,\n  end }\n\ndef sigma.ι {ι : Type u} [fintype ι] (X : ι → ExtrDisc) (i) :\n  X i ⟶ sigma X := ⟨Profinite.sigma.ι _ i⟩\n\ndef sigma.desc {ι : Type u} {Y : ExtrDisc} [fintype ι] (X : ι → ExtrDisc)\n  (f : Π i, X i ⟶ Y) : sigma X ⟶ Y := ⟨Profinite.sigma.desc _ $ λ i, (f i).val⟩\n\nlemma sigma.ι_desc {ι : Type u} {Y : ExtrDisc} [fintype ι] (X : ι → ExtrDisc)\n  (f : Π i, X i ⟶ Y) (i) : sigma.ι X i ≫ sigma.desc X f = f i :=\nbegin\n  ext1,\n  apply Profinite.sigma.ι_desc,\nend\n\nlemma sigma.hom_ext {ι : Type u} {Y : ExtrDisc} [fintype ι] (X : ι → ExtrDisc)\n  (a b : sigma X ⟶ Y) (w : ∀ i, sigma.ι X i ≫ a = sigma.ι X i ≫ b) : a = b :=\nbegin\n  ext1,\n  apply Profinite.sigma.hom_ext,\n  intros i,\n  specialize w i,\n  apply_fun (λ e, e.val) at w,\n  exact w,\nend\n\nlemma sigma.ι_jointly_surjective {ι : Type u} [fintype ι] (X : ι → ExtrDisc)\n  (x : sigma X) : ∃ i (t : X i), sigma.ι X i t = x :=\nProfinite.sigma.ι_jointly_surjective _ _\n\nopen opposite\n\nvariables {C : Type v} [category.{w} C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C)\n\ndef terminal_condition [limits.has_terminal C] : Prop :=\n  is_iso (limits.terminal.from (F.obj (op empty)))\n\ndef binary_product_condition [limits.has_binary_products C] : Prop := ∀ (X Y : ExtrDisc.{u}),\n  is_iso (limits.prod.lift (F.map (sum.inl X Y).op) (F.map (sum.inr X Y).op))\n\ndef finite_product_condition [limits.has_finite_products C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C) :\n  Prop := ∀ (ι : Type u) [fintype ι] (X : ι → ExtrDisc),\nbegin\n  -- Lean is being annoying here...\n  resetI,\n  let t : Π i, F.obj (op (sigma X)) ⟶ F.obj (op (X i)) := λ i, F.map (sigma.ι X i).op,\n  exact is_iso (limits.pi.lift t)\nend\n\ndef finite_product_condition_for_types (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) : Prop :=\n  ∀ (ι : Type u) [fintype ι] (X : ι → ExtrDisc),\nbegin\n  resetI,\n  dsimp_result {\n    let t : Π i, F.obj (op (sigma X)) → F.obj (op (X i)) := λ i, F.map (sigma.ι X i).op,\n    let tt : F.obj (op (sigma X)) → Π i, F.obj (op (X i)) := λ x i, t i x,\n    exact function.bijective tt }\nend\n\ndef equalizer_condition [limits.has_equalizers C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C) : Prop :=\n  ∀ {R X B : ExtrDisc} (f : X ⟶ B) (hf : function.surjective f)\n    (g : R.val ⟶ Profinite.pullback f.val f.val) (hg : function.surjective g),\n  let e₁ : R ⟶ X := ⟨g ≫ Profinite.pullback.fst _ _⟩,\n      e₂ : R ⟶ X := ⟨g ≫ Profinite.pullback.snd _ _⟩,\n      w : e₁ ≫ f = e₂ ≫ f := by { ext1, dsimp [e₁, e₂], simp [Profinite.pullback.condition] },\n      h : F.map f.op ≫ F.map e₁.op = F.map f.op ≫ F.map e₂.op :=\n        by { simp only [← F.map_comp, ← op_comp, w] } in\n  is_iso (limits.equalizer.lift _ h)\n\ndef equalizer_condition_for_types (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) : Prop :=\n  ∀ {R X B : ExtrDisc} (f : X ⟶ B) (hf : function.surjective f)\n    (g : R.val ⟶ Profinite.pullback f.val f.val) (hg : function.surjective g),\n  by dsimp_result { exact\n  let e₁ : R ⟶ X := ⟨g ≫ Profinite.pullback.fst _ _⟩,\n      e₂ : R ⟶ X := ⟨g ≫ Profinite.pullback.snd _ _⟩,\n      w : e₁ ≫ f = e₂ ≫ f := by { ext1, dsimp [e₁, e₂], simp [Profinite.pullback.condition] },\n      h : F.map f.op ≫ F.map e₁.op = F.map f.op ≫ F.map e₂.op :=\n        by { simp only [← F.map_comp, ← op_comp, w] },\n      E := { x : F.obj (op X) // F.map e₁.op x = F.map e₂.op x },\n      t : F.obj (op B) → E := λ x, ⟨F.map f.op x, begin\n        change (F.map f.op ≫ F.map e₁.op) x = (F.map f.op ≫ F.map e₂.op) x,\n        rw h,\n      end⟩ in\n    function.bijective t }\n\nlemma equalizer_condition_holds [limits.has_equalizers C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C) :\n  equalizer_condition F :=\nbegin\n  intros R X B f hf g hg,\n  dsimp,\n  let e₁ : R ⟶ X := ⟨g ≫ Profinite.pullback.fst _ _⟩,\n  let e₂ : R ⟶ X := ⟨g ≫ Profinite.pullback.snd _ _⟩,\n  let σ : B ⟶ X := ⟨ExtrDisc.lift _ hf (𝟙 _)⟩,\n  let t : X ⟶ R := ⟨ExtrDisc.lift _ hg _⟩,\n  swap,\n  { refine Profinite.pullback.lift _ _ (𝟙 _) (f.val ≫ σ.val) _,\n    dsimp, simp },\n  have h₁ : t ≫ e₁ = 𝟙 _, by { ext1, dsimp, simp },\n  have h₂ : t ≫ e₂ = f ≫ σ, by { ext1, dsimp, simp, },\n  have hh : σ ≫ f = 𝟙 _, by { ext1, dsimp, simp },\n  use (limits.equalizer.ι _ _ ≫ F.map σ.op),\n  split,\n  { simp only [limits.equalizer.lift_ι_assoc],\n    simp only [← F.map_comp, ← op_comp, hh],\n    simp },\n  { ext,\n    simp only [limits.equalizer.lift_ι, category.id_comp, category.assoc],\n    simp only [← F.map_comp, ← op_comp],\n    erw [← h₂, op_comp, F.map_comp],\n    dsimp [e₂],\n    erw ← limits.equalizer.condition_assoc,\n    change _ ≫ F.map e₁.op ≫ F.map t.op = _,\n    rw [← F.map_comp, ← op_comp, h₁],\n    simp }\nend\n\nlemma equalizer_condition_for_types_holds (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) :\n  equalizer_condition_for_types F :=\nbegin\n  -- Should be fairly easy, just mimic the proof in the general case above.\n   intros R X B f hf g hg,\n  let e₁ : R ⟶ X := ⟨g ≫ Profinite.pullback.fst _ _⟩,\n  let e₂ : R ⟶ X := ⟨g ≫ Profinite.pullback.snd _ _⟩,\n  have w : e₁ ≫ f = e₂ ≫ f := begin\n    dsimp [e₁,e₂],\n    apply ExtrDisc.hom.ext,\n    simp [category.assoc, Profinite.pullback.condition],\n  end,\n  have h : F.map f.op ≫ F.map e₁.op = F.map f.op ≫ F.map e₂.op :=\n  by { simp only [← F.map_comp, ← op_comp, w] },\n  let E := { x : F.obj (op X) // F.map e₁.op x = F.map e₂.op x },\n  let t : F.obj (op B) → E := λ x, ⟨F.map f.op x, begin\n    change (F.map f.op ≫ F.map e₁.op) x = (F.map f.op ≫ F.map e₂.op) x,\n    rw h,\n  end⟩,\n  change function.bijective t,\n  let ee := limits.concrete.equalizer_equiv (F.map e₁.op) (F.map e₂.op),\n  suffices : function.bijective (ee.symm ∘ t),\n    by exact (equiv.comp_bijective t (equiv.symm ee)).mp this,\n  have : ee.symm ∘ t = limits.equalizer.lift _ h,\n  { suffices : t = ee ∘ limits.equalizer.lift _ h,\n    { rw this, ext, simp, },\n    ext,\n    apply subtype.ext,\n    change _ = (limits.equalizer.lift _ h ≫ limits.equalizer.ι _ _) _,\n    rw limits.equalizer.lift_ι,\n    refl },\n  rw this,\n  rw ← is_iso_iff_bijective,\n  apply equalizer_condition_holds,\n  assumption'\nend\n\nend ExtrDisc\n\nnamespace Profinite\n\nlemma exists_projective_presentation (B : Profinite.{u}) :\n  ∃ (X : ExtrDisc) (π : X.val ⟶ B), function.surjective π :=\nbegin\n  obtain ⟨⟨X,h,π,hπ⟩⟩ := enough_projectives.presentation B,\n  resetI,\n  use [⟨X⟩, π],\n  rwa ← epi_iff_surjective\nend\n\ndef pres (B : Profinite.{u}) : ExtrDisc :=\n  B.exists_projective_presentation.some\n\ndef pres_π (B : Profinite.{u}) : B.pres.val ⟶ B :=\n  B.exists_projective_presentation.some_spec.some\n\nlemma pres_π_surjective (B : Profinite.{u}) :\n  function.surjective B.pres_π :=\nB.exists_projective_presentation.some_spec.some_spec\n\nend Profinite\n\nopen opposite\n\nvariables {C : Type v} [category.{w} C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C)\n\ndef is_ExtrSheaf_of_types (P : ExtrDisc.{u}ᵒᵖ ⥤ Type w) : Prop :=\n∀ (B : ExtrDisc.{u}) (ι : Type u) [fintype ι] (α : ι → ExtrDisc.{u})\n  (f : Π i, α i ⟶ B) (hf : ∀ b : B, ∃ i (x : α i), f i x = b)\n  (x : Π i, P.obj (op (α i)))\n  (hx : ∀ (i j : ι) (Z : ExtrDisc) (g₁ : Z ⟶ α i) (g₂ : Z ⟶ α j),\n    g₁ ≫ f _ = g₂ ≫ f _ → P.map g₁.op (x _) = P.map g₂.op (x _)),\n∃! t : P.obj (op B), ∀ i, P.map (f i).op t = x _\n\n-- We encode the general condition essentially using Yoneda.\ndef is_ExtrSheaf (P : ExtrDisc.{u}ᵒᵖ ⥤ C) : Prop :=\n∀ (B : ExtrDisc.{u}) (ι : Type u) [fintype ι] (α : ι → ExtrDisc.{u})\n  (f : Π i, α i ⟶ B) (hf : ∀ b : B, ∃ i (x : α i), f i x = b)\n  (T : C) (x : Π i, T ⟶ P.obj (op (α i)))\n  (hx : ∀ (i j : ι) (Z : ExtrDisc) (g₁ : Z ⟶ α i) (g₂ : Z ⟶ α j),\n    g₁ ≫ f _ = g₂ ≫ f _ → x _ ≫ P.map g₁.op = x _ ≫ P.map g₂.op),\n∃! t : T ⟶ P.obj (op B), ∀ i, t ≫ P.map (f i).op = x _\n\nlemma subsingleton_of_empty_of_is_ExtrSheaf_of_types\n  (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) (hF : is_ExtrSheaf_of_types F) (Z : ExtrDisc)\n  [hZ : is_empty Z] : subsingleton (F.obj (op Z)) :=\nbegin\n  constructor,\n  intros a b,\n  specialize hF Z pempty pempty.elim (λ a, a.elim) hZ.elim (λ a, a.elim) (λ a, a.elim),\n  obtain ⟨t,h1,h2⟩ := hF,\n  have : a = t, { apply h2, intros i, exact i.elim },\n  have : b = t, { apply h2, intros i, exact i.elim },\n  cc,\nend\n\nlemma finite_product_condition_for_types_of_is_ExtrSheaf_of_types\n  (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) (hF : is_ExtrSheaf_of_types F) :\n    ExtrDisc.finite_product_condition_for_types F :=\nbegin\n  introsI ι _ X,\n  have hF' := hF,\n  specialize hF (ExtrDisc.sigma X) ι X (ExtrDisc.sigma.ι _)\n    (ExtrDisc.sigma.ι_jointly_surjective _),\n  split,\n  { intros x y hh,\n    dsimp at hh,\n    have hx := hF (λ i, F.map (ExtrDisc.sigma.ι X i).op x) _,\n    swap,\n    { intros i j Z g₁ g₂ hh,\n      dsimp,\n      change (F.map _ ≫ F.map _) _ = (F.map _ ≫ F.map _) _,\n      simp only [← F.map_comp, ← op_comp],\n      rw hh },\n    have hy := hF (λ i, F.map (ExtrDisc.sigma.ι X i).op y) _,\n    swap,\n    { intros i j Z g₁ g₂ hh,\n      dsimp,\n      change (F.map _ ≫ F.map _) _ = (F.map _ ≫ F.map _) _,\n      simp only [← F.map_comp, ← op_comp],\n      rw hh },\n    obtain ⟨tx,htx1,htx2⟩ := hx,\n    obtain ⟨ty,hty1,hty2⟩ := hy,\n    have : x = tx,\n    { apply htx2,\n      intros i,\n      refl },\n    rw this,\n    symmetry,\n    apply htx2,\n    intros i,\n    apply_fun (λ e, e i) at hh,\n    exact hh.symm },\n  { intros x,\n    have hx := hF x _,\n    swap,\n    { intros i j Z g₁ g₂ hh,\n      by_cases hZ : nonempty Z,\n      { obtain ⟨z⟩ := hZ,\n        have : i = j,\n        { apply_fun (λ e, (e z).1) at hh, exact hh },\n        subst this,\n        have : g₁ = g₂,\n        { ext t : 2,\n          apply_fun ExtrDisc.sigma.ι X i,\n          swap,\n          { apply Profinite.sigma.ι_injective },\n          apply_fun (λ e, e t) at hh,\n          exact hh },\n        rw this },\n      { simp at hZ, resetI,\n        haveI := subsingleton_of_empty_of_is_ExtrSheaf_of_types F hF' Z,\n        apply subsingleton.elim } },\n    obtain ⟨t,ht,_⟩ := hx,\n    use t,\n    ext1,\n    apply ht }\nend\n\nnamespace product_condition_setup\nsection\n\nparameters {P : ExtrDisc.{u}ᵒᵖ ⥤ Type w} (hP : ExtrDisc.finite_product_condition_for_types P)\nparameters {B : ExtrDisc.{u}} {ι : Type u} [fintype ι] (X : ι → ExtrDisc.{u}) (f : Π i, X i ⟶ B)\n\ndef G : ι × ι → ExtrDisc := λ ii, (Profinite.pullback (f ii.1).val (f ii.2).val).pres\ndef gfst : Π ii : ι × ι, G ii ⟶ X ii.1 := λ ii, ⟨Profinite.pres_π _ ≫ Profinite.pullback.fst _ _⟩\ndef gsnd : Π ii : ι × ι, G ii ⟶ X ii.2 := λ ii, ⟨Profinite.pres_π _ ≫ Profinite.pullback.snd _ _⟩\n\nlemma hX : function.bijective\n  (λ (x : P.obj (op (ExtrDisc.sigma X))) (i : ι), P.map (ExtrDisc.sigma.ι X i).op x) := hP ι X\nlemma hG : function.bijective\n  (λ (x : P.obj (op (ExtrDisc.sigma G))) (i : ι × ι), P.map (ExtrDisc.sigma.ι G i).op x) := hP (ι × ι) G\n\ndef π : ExtrDisc.sigma X ⟶ B := ExtrDisc.sigma.desc X f\n\nlemma hπ  (surj : ∀ b : B, ∃ i (x : X i), f i x = b) : function.surjective π :=\nbegin\n  intros b,\n  have := surj,\n  obtain ⟨i,x,hx⟩ := surj b,\n  use ExtrDisc.sigma.ι X i x,\n  exact hx\nend\n\ndef r : (ExtrDisc.sigma G).val ⟶ Profinite.pullback π.val π.val :=\nbegin\n  refine Profinite.pullback.lift _ _ _ _ _,\n  { refine Profinite.sigma.desc _ _,\n    intros ii,\n    refine _ ≫ Profinite.sigma.ι _ ii.1,\n    refine (gfst _ _ _).val },\n  { refine Profinite.sigma.desc _ _,\n    intros ii,\n    refine _ ≫ Profinite.sigma.ι _ ii.2,\n    dsimp,\n    refine (gsnd _ _ _).val },\n  { apply Profinite.sigma.hom_ext,\n    rintros ⟨i,j⟩,\n    dsimp [π, ExtrDisc.sigma.desc, gfst, gsnd],\n    simp [Profinite.pullback.condition] },\nend\n\nlemma hr : function.surjective r :=\nbegin\n  rintros ⟨⟨⟨i,a⟩,⟨j,b⟩⟩,h⟩,\n  dsimp [π, ExtrDisc.sigma.desc, Profinite.sigma.desc] at a b h,\n  let ab : Profinite.pullback (f i).val (f j).val := ⟨⟨a,b⟩,h⟩,\n  obtain ⟨c,hc⟩ := Profinite.pres_π_surjective _ ab,\n  use ExtrDisc.sigma.ι (G X f) (i,j) c,\n  apply subtype.ext,\n  apply prod.ext,\n  { apply sigma.ext, { refl },\n    apply heq_of_eq,\n    change (((Profinite.pullback (f i).val (f j).val).pres_π) c).val.fst = _,\n    rw hc, refl },\n  { apply sigma.ext, { refl },\n    apply heq_of_eq,\n    change (((Profinite.pullback (f i).val (f j).val).pres_π) c).val.snd = _,\n    rw hc, refl }\nend\n\n@[nolint def_lemma] -- this lemma has an extremely annoying type to write down\ndef hE (surj : ∀ b : B, ∃ i (x : X i), f i x = b) :=\n  ExtrDisc.equalizer_condition_for_types_holds P π (hπ surj) r hr\n\ndef QX : P.obj (op (ExtrDisc.sigma X)) ≃ Π i, P.obj (op (X i)) :=\n  equiv.of_bijective _ hX\ndef QG : P.obj (op (ExtrDisc.sigma G)) ≃ Π ii, P.obj (op (G ii)) :=\n  equiv.of_bijective _ hG\n\ndef rfst : ExtrDisc.sigma G ⟶ ExtrDisc.sigma X :=\n  ⟨r ≫ Profinite.pullback.fst _ _⟩\ndef rsnd : ExtrDisc.sigma G ⟶ ExtrDisc.sigma X :=\n  ⟨r ≫ Profinite.pullback.snd _ _⟩\n\nlemma ι_rfst (ii : ι × ι) : ExtrDisc.sigma.ι G ii ≫ rfst =\n  gfst ii ≫ ExtrDisc.sigma.ι _ _ :=\nbegin\n  ext1,\n  dsimp [rfst, gfst, ExtrDisc.sigma.ι, ExtrDisc.sigma.desc, r],\n  simp [Profinite.pullback.condition, Profinite.pullback.condition_assoc],\nend\n\nlemma ι_rsnd (ii : ι × ι) : ExtrDisc.sigma.ι G ii ≫ rsnd =\n  gsnd ii ≫ ExtrDisc.sigma.ι _ _ :=\nbegin\n  ext1,\n  dsimp [rsnd, gsnd, ExtrDisc.sigma.ι, ExtrDisc.sigma.desc, r],\n  simp [Profinite.pullback.condition, Profinite.pullback.condition_assoc],\nend\n\nlemma QX_symm_ι_aux (q : Π i, P.obj (op (X i))) :\n  q = λ i, P.map (ExtrDisc.sigma.ι X i).op (QX.symm q) :=\nbegin\n  apply_fun (QX hP X).symm,\n  change _ = (QX hP X).symm ((QX hP X) _),\n  rw equiv.symm_apply_apply,\nend\n\nlemma QX_symm_ι (q : Π i, P.obj (op (X i))) (i : ι) :\n  P.map (ExtrDisc.sigma.ι X i).op (QX.symm q) = q i :=\nbegin\n  revert i,\n  rw ← function.funext_iff,\n  change _ = q,\n  symmetry,\n  apply QX_symm_ι_aux hP X q,\nend\n\nlemma QX_QG_compat_fst (q : Π i, P.obj (op (X i))) (i : ι × ι) :\n  QG (P.map rfst.op (QX.symm q)) i = P.map (gfst i).op (q i.fst) :=\nbegin\n  dsimp [QG],\n  change (P.map _ ≫ P.map _) _ = _,\n  simp only [← P.map_comp, ← op_comp, ι_rfst],\n  simp only [P.map_comp, op_comp],\n  dsimp,\n  rw QX_symm_ι,\nend\n\nlemma QX_QG_compat_snd (q : Π i, P.obj (op (X i))) (i : ι × ι) :\n  QG (P.map rsnd.op (QX.symm q)) i = P.map (gsnd i).op (q i.snd) :=\nbegin\n  dsimp [QG],\n  change (P.map _ ≫ P.map _) _ = _,\n  simp only [← P.map_comp, ← op_comp, ι_rsnd],\n  simp only [P.map_comp, op_comp],\n  dsimp,\n  rw QX_symm_ι,\nend\n\nend\nend product_condition_setup\n\nopen product_condition_setup\n\ntheorem is_ExtrSheaf_of_types_of_finite_product_condition_for_types\n  (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) (hF : ExtrDisc.finite_product_condition_for_types F) :\n  is_ExtrSheaf_of_types F :=\nbegin\n  introsI B ι _ X f surj x hx,\n  have hrfst : ∀ (q : Π i, F.obj (op (X i))),\n    (QG hF X f) (F.map (rfst X f).op ((QX hF X).symm q)) =\n    (λ ii, F.map (gfst X f ii).op (q ii.1)),\n  { intros q, funext ii,\n    change (F.map _ ≫ F.map _) _ = _,\n    simp only [← F.map_comp, ← op_comp, ι_rfst],\n    simp only [F.map_comp, op_comp],\n    dsimp,\n    rw QX_symm_ι hF X q },\n  have hrgsnd : ∀ (q : Π i, F.obj (op (X i))),\n    (QG hF X f) (F.map (rsnd X f).op ((QX hF X).symm q)) =\n    (λ ii, F.map (gsnd X f ii).op (q ii.2)),\n  { intros q, funext ii,\n    change (F.map _ ≫ F.map _) _ = _,\n    simp only [← F.map_comp, ← op_comp, ι_rsnd],\n    simp only [F.map_comp, op_comp],\n    dsimp,\n    rw QX_symm_ι hF X q },\n  let EE : F.obj (op B) ≃\n    { t : F.obj (op (ExtrDisc.sigma X)) // F.map (rfst X f).op t = F.map (rsnd X f).op t } :=\n      equiv.of_bijective _ (hE X f surj),\n  let x' : F.obj (op (ExtrDisc.sigma X)) := (QX hF X).symm x,\n  -- Should follow from hx,\n  have hx' : F.map (rfst X f).op x' = F.map (rsnd X f).op x',\n  { apply_fun (QG hF X f),\n    funext ii,\n    rw QX_QG_compat_fst,\n    rw QX_QG_compat_snd,\n    apply hx,\n    ext1,\n    dsimp [gfst, gsnd],\n    simp [Profinite.pullback.condition] },\n  let b : F.obj (op B) := EE.symm ⟨x',hx'⟩,\n  use b,\n  have hb : ∀ i, F.map (f i).op b = x i,\n  { intros i,\n    have : f i = ExtrDisc.sigma.ι X i ≫ (π X f),\n    { dsimp [π], rw ExtrDisc.sigma.ι_desc },\n    rw [this, op_comp, F.map_comp],\n    dsimp,\n    have : F.map (π X f).op b = x',\n    { change ↑(EE b) = x',\n      dsimp only [b],\n      rw equiv.apply_symm_apply,\n      refl },\n    rw this,\n    dsimp [x'],\n    change ((QX hF X) ((QX hF X).symm x)) _ = _,\n    rw equiv.apply_symm_apply },\n  refine ⟨hb, _⟩,\n  { intros b' hb',\n    apply_fun EE,\n    ext1,\n    apply_fun (QX hF X),\n    dsimp [EE, QX, π],\n    funext i,\n    change (F.map _ ≫ F.map _) _ = (F.map _ ≫ F.map _) _,\n    simp only [← F.map_comp, ← op_comp, ExtrDisc.sigma.ι_desc, hb, hb'] }\nend\n\ntheorem is_ExtrSheaf_of_types_iff_product_condition_for_types (F : ExtrDisc.{u}ᵒᵖ ⥤ Type w) :\n  is_ExtrSheaf_of_types F ↔ ExtrDisc.finite_product_condition_for_types F :=\nbegin\n  split,\n  { intro h, exact finite_product_condition_for_types_of_is_ExtrSheaf_of_types _ h },\n  { intro h, exact is_ExtrSheaf_of_types_of_finite_product_condition_for_types _ h }\nend\n\nlemma is_ExtrSheaf_iff_forall_yoneda (F : ExtrDisc.{u}ᵒᵖ ⥤ C) :\n  is_ExtrSheaf F ↔ (∀ (T : C), is_ExtrSheaf_of_types (F ⋙ coyoneda.obj (op T))) :=\nbegin\n  split,\n  { introsI h T B ι _ X f surj x hx,\n    exact h B ι X f surj T x hx },\n  { introsI h B ι _ X f surj T x hx,\n    exact h T B ι X f surj x hx }\nend\n\ntheorem finite_product_condition_iff_forall_yoneda [limits.has_finite_products C]\n  (F : ExtrDisc.{u}ᵒᵖ ⥤ C) :\n  ExtrDisc.finite_product_condition F ↔\n  (∀ (T : C), ExtrDisc.finite_product_condition_for_types (F ⋙ coyoneda.obj (op T))) :=\nbegin\n  split,\n  { introsI h T ι _ X,\n    let t : F.obj (op (ExtrDisc.sigma X)) ⟶ ∏ λ (i : ι), F.obj (op (X i)) :=\n      limits.pi.lift (λ (i : ι), F.map (ExtrDisc.sigma.ι X i).op),\n    specialize h ι X,\n    dsimp at h ⊢,\n    change is_iso t at h,\n    resetI,\n    split,\n    { intros a b hab,\n      dsimp at hab,\n      suffices : a ≫ t = b ≫ t,\n      { apply_fun (λ e, e ≫ inv t) at this, simpa using this },\n      ext1,\n      rw function.funext_iff at hab,\n      simp [hab] },\n    { intros a,\n      use limits.pi.lift a ≫ inv t,\n      dsimp,\n      funext i,\n      have : inv t ≫ F.map (ExtrDisc.sigma.ι X i).op = limits.pi.π _ i,\n      { simp [is_iso.inv_comp_eq] },\n      simp [this] } },\n  { introsI h ι _ X,\n    dsimp,\n    let h₁ := h (∏ λ (i : ι), F.obj (op (X i))) ι X,\n    let h₂ := h (F.obj (op (ExtrDisc.sigma X))) ι X,\n    dsimp at h₁ h₂,\n    replace h₁ := h₁.2,\n    replace h₂ := h₂.1,\n    obtain ⟨s,hs⟩ := h₁ (λ i, limits.pi.π _ i),\n    use s,\n    rw function.funext_iff at hs,\n    dsimp at *,\n    split,\n    { apply h₂,\n      ext1 i,\n      dsimp,\n      simp [hs i] },\n    { ext1 i,\n      cases i,\n      simp [hs i] } }\nend\n\ntheorem is_ExtrSheaf_iff_product_condition\n  [limits.has_finite_products C] (F : ExtrDisc.{u}ᵒᵖ ⥤ C) :\n  is_ExtrSheaf F ↔ ExtrDisc.finite_product_condition F :=\nbegin\n  rw is_ExtrSheaf_iff_forall_yoneda,\n  rw finite_product_condition_iff_forall_yoneda,\n  apply forall_congr (λ T, _),\n  apply is_ExtrSheaf_of_types_iff_product_condition_for_types\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/extr/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565326, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3457879473751622}}
{"text": "\n/- This module defines the type of values that is used in the dynamic\n  semantics of all our intermediate languages. -/\n\nimport .lib .integers .floats .ast\n\nnamespace values\nopen integers word floats ast ast.typ ast.memory_chunk\n\ndef block : Type := pos_num\ninstance eq_block : decidable_eq block := by tactic.mk_dec_eq_instance\ninstance coe_block : has_coe block ℕ := ⟨λp, num.pos p⟩\ninstance dlo_block : decidable_linear_order block := pos_num.decidable_linear_order\ninstance block_one : has_one block := ⟨(1 : pos_num)⟩\n\n/- A value is either:\n- a machine integer;\n- a floating-point number;\n- a pointer: a pair of a memory address and an integer offset with respect\n  to this address;\n- the [Vundef] value denoting an arbitrary bit pattern, such as the\n  value of an uninitialized variable.\n-/\n\ninductive val : Type\n| Vundef  : val\n| Vint    : int32 → val\n| Vlong   : int64 → val\n| Vfloat  : float → val\n| Vsingle : float32 → val\n| Vptr    : block → ptrofs → val\nexport val\ninstance coe_int32_val : has_coe int32 val := ⟨Vint⟩\ninstance coe_int64_val : has_coe int64 val := ⟨Vlong⟩\ninstance coe_Int32_val : has_coe (@word W32) val := ⟨Vint⟩\ninstance coe_Int64_val : has_coe (@word W64) val := ⟨Vlong⟩\ninstance coe_float_val : has_coe float val := ⟨Vfloat⟩\ninstance coe_single_val : has_coe float32 val := ⟨Vsingle⟩\ninstance inhabited_val : inhabited val := ⟨Vundef⟩\n\ndef Vzero  : val := Vint 0\ndef Vone   : val := Vint 1\ndef Vmone  : val := Vint (-1)\ninstance val_zero : has_zero val := ⟨Vzero⟩\ninstance val_one : has_one val := ⟨Vone⟩\n\ndef Vtrue  : val := 1\ndef Vfalse : val := 0\n\ndef Vnullptr := if archi.ptr64 then Vlong 0 else Vint 0\n\ndef Vptrofs (n : ptrofs) : val :=\nif archi.ptr64 then ptrofs.to_int64 n else ptrofs.to_int n\n\n/- * Operations over values -/\n\n/- The module [val] defines a number of arithmetic and logical operations\n  over type [val].  Most of these operations are straightforward extensions\n  of the corresponding integer or floating-point operations. -/\n\nnamespace val\n\ninstance val_eq : decidable_eq val := by tactic.mk_dec_eq_instance\n\ndef has_type : val → typ → Prop\n| Vundef      _       := true\n| (Vint _)    Tint    := true\n| (Vlong _)   Tlong   := true\n| (Vfloat _)  Tfloat  := true\n| (Vsingle _) Tsingle := true\n| (Vptr _ _)  Tint    := ¬ archi.ptr64\n| (Vptr _ _)  Tlong   := archi.ptr64\n| (Vint _)    Tany32  := true\n| (Vsingle _) Tany32  := true\n| (Vptr _ _)  Tany32  := ¬ archi.ptr64\n| _           Tany64  := true\n| _           _       := false\n\ndef has_type_list : list val → list typ → Prop\n| []         []         := true\n| (v1 :: vs) (t1 :: ts) := has_type v1 t1 ∧ has_type_list vs ts\n| _          _          := false\n\ndef has_opttype (v : val) : option typ → Prop\n| none     := v = Vundef\n| (some t) := has_type v t\n\nlemma Vptr_has_type (b ofs) : has_type (Vptr b ofs) Tptr :=\nbegin\n  delta Tptr,\n  ginduction archi.ptr64 with h,\n  { intro h2, note := h.symm.trans h2, contradiction },\n  { exact h }\nend\n\nlemma Vnullptr_has_type : has_type Vnullptr Tptr :=\nby delta Tptr Vnullptr; cases archi.ptr64; trivial\n\nlemma has_subtype (ty1 ty2 v) : subtype ty1 ty2 → has_type v ty1 → has_type v ty2 := sorry'\n\nlemma has_subtype_list (tyl1 tyl2 vl) :\n  subtype_list tyl1 tyl2 → has_type_list vl tyl1 → has_type_list vl tyl2 := sorry'\n\n/- Truth values.  Non-zero integers are treated as [True].\n  The integer 0 (also used to represent the null pointer) is [False].\n  Other values are neither true nor false. -/\n\ndef to_bool : val → option bool\n| (Vint n) := some (n ≠ 0)\n| _ := none\n\n/- Arithmetic operations -/\n\nprotected def neg : val → val\n| (Vint n) := word.neg n\n| _        := Vundef\ninstance : has_neg val := ⟨val.neg⟩\n\ndef negf : val → val\n| (Vfloat f) := float.neg f\n| _          := Vundef\n\ndef absf : val → val\n| (Vfloat f) := float.abs f\n| _          := Vundef\n\ndef negfs : val → val\n| (Vsingle f) := float32.neg f\n| _           := Vundef\n\ndef absfs : val → val\n| (Vsingle f) := float32.abs f\n| _           := Vundef\n\ndef make_total (ov : option val) : val :=\nov.get_or_else Vundef\n\ndef int_of_float : val → option val\n| (Vfloat f) := Vint <$> float.to_int f\n| _          := none\n\ndef intu_of_float : val → option val\n| (Vfloat f) := Vint <$> float.to_intu f\n| _          := none\n\ndef float_of_int : val → option val\n| (Vint n) := float.of_int n\n| _        := none\n\ndef float_of_intu : val → option val\n| (Vint n) := float.of_intu n\n| _        := none\n\ndef int_of_single : val → option val\n| (Vsingle f) := Vint <$> float32.to_int f\n| _           := none\n\ndef intu_of_single : val → option val\n| (Vsingle f) := Vint <$> float32.to_intu f\n| _           := none\n\ndef single_of_int : val → option val\n| (Vint n) := float32.of_int n\n| _        := none\n\ndef single_of_intu : val → option val\n| (Vint n) := float32.of_intu n\n| _        := none\n\ndef negint : val → val\n| (Vint n) := word.neg n\n| _        := Vundef\n\ndef notint : val → val\n| (Vint n) := word.not n\n| _        := Vundef\n\ndef of_bool (b : bool) : val := if b then Vtrue else Vfalse\ninstance : has_coe bool val := ⟨of_bool⟩\n\ndef boolval : val → val\n| (Vint n)     := of_bool (n ≠ 0)\n| (Vptr b ofs) := Vtrue\n| _            := Vundef\n\ndef notbool : val → val\n| (Vint n)     := of_bool (n = 0)\n| (Vptr b ofs) := Vfalse\n| _            := Vundef\n\ndef zero_ext (nbits : ℕ) : val → val\n| (Vint n) := word.zero_ext nbits n\n| _        := Vundef\n\ndef sign_ext (nbits : ℕ+) : val → val\n| (Vint n) := word.sign_ext nbits n\n| _        := Vundef\n\ndef single_of_float : val → val\n| (Vfloat f) := float.to_single f\n| _          := Vundef\n\ndef float_of_single : val → val\n| (Vsingle f) := float.of_single f\n| _           := Vundef\n\nprotected def add : val → val → val\n| (Vint m)       (Vint n)       := Vint $ m + n\n| (Vptr b1 ofs1) (Vint n)       := if archi.ptr64 then Vundef else Vptr b1 (ofs1 + ptrofs.of_int n)\n| (Vint m)       (Vptr b2 ofs2) := if archi.ptr64 then Vundef else Vptr b2 (ofs2 + ptrofs.of_int m)\n| _              _              := Vundef\ninstance : has_add val := ⟨val.add⟩\n\nprotected def sub : val → val → val\n| (Vint m)       (Vint n)       := Vint $ m - n\n| (Vptr b1 ofs1) (Vint n)       := if archi.ptr64 then Vundef else Vptr b1 (ofs1 - ptrofs.of_int n)\n| (Vptr b1 ofs1) (Vptr b2 ofs2) := if archi.ptr64 ∨ b1 ≠ b2 then Vundef else ptrofs.to_int (ofs1 - ofs2)\n| _              _              := Vundef\ninstance : has_sub val := ⟨val.sub⟩\n\nprotected def mul : val → val → val\n| (Vint m) (Vint n) := Vint (m * n)\n| _        _        := Vundef\ninstance : has_mul val := ⟨val.mul⟩\n\ndef mulhs : val → val → val\n| (Vint m) (Vint n) := word.mulhs m n\n| _        _        := Vundef\n\ndef mulhu : val → val → val\n| (Vint m) (Vint n) := word.mulhu m n\n| _        _        := Vundef\n\ndef divs : val → val → option val\n| (Vint m) (Vint n) := if n = 0 ∨ m = repr (word.min_signed W32) ∧ n = -1\n                       then none else some (m / n : word _)\n| _        _        := none\n\ndef mods : val → val → option val\n| (Vint m) (Vint n) := if n = 0 ∨ m = repr (@min_signed W32) ∧ n = -1\n                       then none else some (m % n : word _)\n| _        _        := none\n\ndef divu : val → val → option val\n| (Vint m) (Vint n) := if n = 0 then none else word.divu m n\n| _        _        := none\n\ndef modu : val → val → option val\n| (Vint m) (Vint n) := if n = 0 then none else word.modu m n\n| _        _        := none\n\ndef add_carry : val → val → val → val\n| (Vint m) (Vint n) (Vint c) := word.add_carry m n c\n| _        _        _        := Vundef\n\ndef sub_overflow : val → val → val\n| (Vint m) (Vint n) := word.sub_overflow m n 0\n| _        _        := Vundef\n\ndef negative : val → val\n| (Vint n) := word.negative n\n| _        := Vundef\n\nprotected def and : val → val → val\n| (Vint m) (Vint n) := word.and m n\n| _        _        := Vundef\n\nprotected def or : val → val → val\n| (Vint m) (Vint n) := word.or m n\n| _        _        := Vundef\n\nprotected def xor : val → val → val\n| (Vint m) (Vint n) := word.xor m n\n| _        _        := Vundef\n\ndef shl : val → val → val\n| (Vint m) (Vint n) := if word.ltu n iwordsize\n                       then word.shl m n else Vundef\n| _        _        := Vundef\n\ndef shr : val → val → val\n| (Vint m) (Vint n) := if word.ltu n iwordsize\n                       then word.shr m n else Vundef\n| _        _        := Vundef\n\ndef shr_carry : val → val → val\n| (Vint m) (Vint n) := if word.ltu n iwordsize\n                       then word.shr_carry m n else Vundef\n| _        _        := Vundef\n\ndef shrx : val → val → option val\n| (Vint m) (Vint n) := if word.ltu n (repr 31)\n                       then word.shrx m n else none\n| _        _        := none\n\ndef shru : val → val → val\n| (Vint m) (Vint n) := if word.ltu n iwordsize\n                       then word.shru m n else Vundef\n| _        _        := Vundef\n\ndef rol : val → val → val\n| (Vint m) (Vint n) := word.rol m n\n| _        _        := Vundef\n\ndef rolm : val → int32 → int32 → val\n| (Vint n) amount mask := word.rolm n amount mask\n| _        amount mask := Vundef\n\ndef ror : val → val → val\n| (Vint m) (Vint n) := word.ror m n\n| _        _        := Vundef\n\ndef addf : val → val → val\n| (Vfloat x) (Vfloat y) := Vfloat $ x + y\n| _          _          := Vundef\n\ndef subf : val → val → val\n| (Vfloat x) (Vfloat y) := Vfloat $ x - y\n| _          _          := Vundef\n\ndef mulf : val → val → val\n| (Vfloat x) (Vfloat y) := Vfloat $ x * y\n| _          _          := Vundef\n\ndef divf : val → val → val\n| (Vfloat x) (Vfloat y) := Vfloat $ x / y\n| _          _          := Vundef\n\ndef float_of_words : val → val → val\n| (Vint m) (Vint n) := float.from_words m n\n| _        _        := Vundef\n\ndef addfs : val → val → val\n| (Vsingle x) (Vsingle y) := float32.add x y\n| _           _           := Vundef\n\ndef subfs : val → val → val\n| (Vsingle x) (Vsingle y) := float32.sub x y\n| _           _           := Vundef\n\ndef mulfs : val → val → val\n| (Vsingle x) (Vsingle y) := float32.mul x y\n| _           _           := Vundef\n\ndef divfs : val → val → val\n| (Vsingle x) (Vsingle y) := float32.div x y\n| _           _           := Vundef\n\n/- Operations on 64-bit integers -/\n\ndef long_of_words : val → val → val\n| (Vint m) (Vint n) := int64.ofwords m n\n| _        _        := Vundef\n\ndef loword : val → val\n| (Vlong n) := int64.loword n\n| _         := Vundef\n\ndef hiword : val → val\n| (Vlong n)  := int64.hiword n\n| _ := Vundef\n\ndef negl : val → val\n| (Vlong n) := Vlong $ -n\n| _         := Vundef\n\ndef notl : val → val\n| (Vlong n) := word.not n\n| _ := Vundef\n\ndef long_of_int : val → val\n| (Vint n) := Vlong $ scoe n\n| _        := Vundef\n\ndef long_of_intu : val → val\n| (Vint n) := Vlong $ ucoe n\n| _        := Vundef\n\ndef long_of_float : val → option val\n| (Vfloat f) := Vlong <$> float.to_long f\n| _          := none\n\ndef longu_of_float : val → option val\n| (Vfloat f) := Vlong <$> float.to_longu f\n| _          := none\n\ndef long_of_single : val → option val\n| (Vsingle f) := Vlong <$> float32.to_long f\n| _           := none\n\ndef longu_of_single : val → option val\n| (Vsingle f) := Vlong <$> float32.to_longu f\n| _           := none\n\ndef float_of_long : val → option val\n| (Vlong n) := float.of_long n\n| _         := none\n\ndef float_of_longu : val → option val\n| (Vlong n) := float.of_longu n\n| _         := none\n\ndef single_of_long : val → option val\n| (Vlong n) := float32.of_long n\n| _         := none\n\ndef single_of_longu : val → option val\n| (Vlong n) := float32.of_longu n\n| _         := none\n\ndef addl : val → val → val\n| (Vlong n1)     (Vlong n2)     := Vlong (n1 + n2)\n| (Vptr b1 ofs1) (Vlong n2)     := if archi.ptr64 then Vptr b1 (ofs1 + ptrofs.of_int64 n2) else Vundef\n| (Vlong n1)     (Vptr b2 ofs2) := if archi.ptr64 then Vptr b2 (ofs2 + ptrofs.of_int64 n1) else Vundef\n| _              _              := Vundef\n\ndef subl : val → val → val\n| (Vlong n1)     (Vlong n2)     := Vlong (n1 - n2)\n| (Vptr b1 ofs1) (Vlong n2)     := if archi.ptr64 then Vptr b1 (ofs1 - ptrofs.of_int64 n2) else Vundef\n| (Vptr b1 ofs1) (Vptr b2 ofs2) := if archi.ptr64 ∨ b1 ≠ b2 then Vundef else ptrofs.to_int64 (ofs1 - ofs2)\n| _              _              := Vundef\n\ndef mull : val → val → val\n| (Vlong m) (Vlong n) := Vlong (m * n)\n| _         _         := Vundef\n\ndef mull' : val → val → val\n| (Vint m) (Vint n) := Vlong (ucoe m * ucoe n)\n| _        _        := Vundef\n\ndef mullhs : val → val → val\n| (Vlong m) (Vlong n) := word.mulhs m n\n| _         _         := Vundef\n\ndef mullhu : val → val → val\n| (Vlong m) (Vlong n) := word.mulhu m n\n| _         _         := Vundef\n\ndef divls : val → val → option val\n| (Vlong m) (Vlong n) := if n = 0 ∨ m = repr (min_signed 64) ∧ n = -1\n                         then none else some (m / n : word _)\n| _         _         := none\n\ndef modls : val → val → option val\n| (Vlong m) (Vlong n) := if n = 0 ∨ m = repr (min_signed 64) ∧ n = -1\n                         then none else some (m % n : word _)\n| _         _         := none\n\ndef divlu : val → val → option val\n| (Vlong m) (Vlong n) := if n = 0 then none else word.divu m n\n| _         _         := none\n\ndef modlu : val → val → option val\n| (Vlong m) (Vlong n) := if n = 0 then none else word.modu m n\n| _         _         := none\n\ndef subl_overflow : val → val → val\n| (Vlong m) (Vlong n) := Vint $ ucoe $ word.sub_overflow m n 0\n| _         _         := Vundef\n\ndef negativel : val → val\n| (Vlong n) := Vint $ ucoe $ word.negative n\n| _         := Vundef\n\ndef andl : val → val → val\n| (Vlong m) (Vlong n) := word.and m n\n| _         _         := Vundef\n\ndef orl : val → val → val\n| (Vlong m) (Vlong n) := word.or m n\n| _         _         := Vundef\n\ndef xorl : val → val → val\n| (Vlong m) (Vlong n) := word.xor m n\n| _         _         := Vundef\n\ndef shll : val → val → val\n| (Vlong m) (Vint n) := if word.ltu n 64 then word.shl m (ucoe n) else Vundef\n| _         _        := Vundef\n\ndef shrl : val → val → val\n| (Vlong m) (Vint n) := if word.ltu n 64 then word.shr m (ucoe n) else Vundef\n| _         _        := Vundef\n\ndef shrlu : val → val → val\n| (Vlong m) (Vint n) := if word.ltu n 64 then word.shru m (ucoe n) else Vundef\n| _         _        := Vundef\n\ndef shrxl : val → val → option val\n| (Vlong m) (Vint n) := if word.ltu n 63 then word.shrx m (ucoe n) else none\n| _         _        := none\n\ndef roll : val → val → val\n| (Vlong m) (Vint n) := word.rol m (ucoe n)\n| _         _        := Vundef\n\ndef rorl : val → val → val\n| (Vlong m) (Vint n) := word.ror m (ucoe n)\n| _         _        := Vundef\n\ndef rolml : val → int64 → int64 → val\n| (Vlong n) amount mask := word.rolm n amount mask\n| _         amount mask := Vundef\n\n/- Comparisons -/\n\nsection comparisons\n\nparameter valid_ptr : block → ℕ → bool\ndef weak_valid_ptr (b : block) (ofs : ℕ) := valid_ptr b ofs || valid_ptr b (ofs - 1)\n\ndef cmp_bool (c : comparison) : val → val → option bool\n| (Vint m) (Vint n) := cmp c m n\n| _        _        := none\n\ndef cmp_different_blocks : comparison → option bool\n| Ceq := some ff\n| Cne := some tt\n| _   := none\n\ndef cmpu_bool (c : comparison) : val → val → option bool\n| (Vint m) (Vint n) := some $ cmpu c m n\n| (Vint m) (Vptr b2 ofs2) :=\n  if archi.ptr64 then none else\n  if m = 0 ∧ weak_valid_ptr b2 (unsigned ofs2)\n  then cmp_different_blocks c\n  else none\n| (Vptr b1 ofs1) (Vptr b2 ofs2) :=\n  if archi.ptr64 then none else\n  if b1 = b2 then\n    if weak_valid_ptr b1 (unsigned ofs1) ∧ weak_valid_ptr b2 (unsigned ofs2)\n    then cmpu c ofs1 ofs2\n    else none\n  else\n    if valid_ptr b1 (unsigned ofs1) ∧ valid_ptr b2 (unsigned ofs2)\n    then cmp_different_blocks c\n    else none\n| (Vptr b1 ofs1) (Vint n) :=\n  if archi.ptr64 then none else\n  if n = 0 ∧ weak_valid_ptr b1 (unsigned ofs1)\n  then cmp_different_blocks c\n  else none\n| _ _ := none\n\ndef cmpf_bool (c : comparison) : val → val → option bool\n| (Vfloat x) (Vfloat y) := float.cmp c x y\n| _          _          := none\n\ndef cmpfs_bool (c : comparison) : val → val → option bool\n| (Vsingle x) (Vsingle y) := float32.cmp c x y\n| _           _           := none\n\ndef cmpl_bool (c : comparison) : val → val → option bool\n| (Vlong m) (Vlong n) := cmp c m n\n| _         _         := none\n\ndef cmplu_bool (c : comparison) : val → val → option bool\n| (Vlong n1) (Vlong n2) := some $ cmpu c n1 n2\n| (Vlong n1) (Vptr b2 ofs2) :=\n  if archi.ptr64 ∧ n1 = 0 ∧ weak_valid_ptr b2 (unsigned ofs2)\n  then cmp_different_blocks c\n  else none\n| (Vptr b1 ofs1) (Vptr b2 ofs2) :=\n  if ¬ archi.ptr64 then none else\n  if b1 = b2 then\n    if weak_valid_ptr b1 (unsigned ofs1) && weak_valid_ptr b2 (unsigned ofs2)\n    then some (cmpu c ofs1 ofs2)\n    else none\n  else\n    if valid_ptr b1 (unsigned ofs1) && valid_ptr b2 (unsigned ofs2)\n    then cmp_different_blocks c\n    else none\n| (Vptr b1 ofs1) (Vlong n2) :=\n  if archi.ptr64 ∧ n2 = 0 ∧ weak_valid_ptr b1 (unsigned ofs1)\n  then cmp_different_blocks c\n  else none\n| _ _ := none\n\ndef of_optbool : option bool → val\n| (some tt) := Vtrue\n| (some ff) := Vfalse\n| none := Vundef\n\ndef cmp (c : comparison) (v1 v2 : val) : val :=\nof_optbool $ cmp_bool c v1 v2\n\ndef cmpu (c : comparison) (v1 v2 : val) : val :=\nof_optbool $ cmpu_bool c v1 v2\n\ndef cmpf (c : comparison) (v1 v2 : val) : val :=\nof_optbool $ cmpf_bool c v1 v2\n\ndef cmpfs (c : comparison) (v1 v2 : val) : val :=\nof_optbool $ cmpfs_bool c v1 v2\n\ndef cmpl (c : comparison) (v1 v2 : val) : option val :=\nof_bool <$> cmpl_bool c v1 v2\n\ndef cmplu (c : comparison) (v1 v2 : val) : option val :=\nof_bool <$> cmplu_bool c v1 v2\n\ndef maskzero_bool : val → int32 → option bool\n| (Vint n) mask := some $ word.and n mask = 0\n| _        mask := none\n\nend comparisons\n\n/- Add the given offset to the given pointer. -/\n\ndef offset_ptr : val → ptrofs → val\n| (Vptr b ofs) delta := Vptr b (ofs + delta)\n| _            delta := Vundef\n\n/- [load_result] reflects the effect of storing a value with a given\n  memory chunk, then reading it back with the same chunk.  Depending\n  on the chunk and the type of the value, some normalization occurs.\n  For instance, consider storing the integer value [0xFFF] on 1 byte\n  at a given address, and reading it back.  If it is read back with\n  chunk [Mint8unsigned], zero-extension must be performed, resulting\n  in [0xFF].  If it is read back as a [Mint8signed], sign-extension is\n  performed and [0xFFFFFFFF] is returned. -/\n\ndef load_result : memory_chunk → val → val\n| Mint8signed    (Vint n)     := word.sign_ext W8 n\n| Mint8unsigned  (Vint n)     := word.zero_ext 8 n\n| Mint16signed   (Vint n)     := word.sign_ext W16 n\n| Mint16unsigned (Vint n)     := word.zero_ext 16 n\n| Mint32         (Vint n)     := Vint n\n| Mint32         (Vptr b ofs) := if archi.ptr64 then Vundef else Vptr b ofs\n| Mint64         (Vlong n)    := Vlong n\n| Mint64         (Vptr b ofs) := if archi.ptr64 then Vptr b ofs else Vundef\n| Mfloat32       (Vsingle f)  := Vsingle f\n| Mfloat64       (Vfloat f)   := Vfloat f\n| Many32         (Vint n)     := Vint n\n| Many32         (Vsingle f)  := Vsingle f\n| Many32         (Vptr b ofs) := if archi.ptr64 then Vundef else Vptr b ofs\n| Many64         v            := v\n| _              _            := Vundef\n\nlemma load_result_type (chunk v) : has_type (load_result chunk v) chunk.type := sorry'\n\nlemma load_result_same {v ty} : has_type v ty → load_result (chunk_of_type ty) v = v := sorry'\n\n/- Theorems on arithmetic operations. -/\n\ntheorem cast8unsigned_and (x) : zero_ext 8 x = val.and x (0xFF : int32) := sorry'\n\ntheorem cast16unsigned_and (x) : zero_ext 16 x = val.and x (0xFFFF : int32) := sorry'\n\ntheorem to_bool_of_bool (b1 b2) : (of_bool b1).to_bool = some b2 → b1 = b2 := sorry'\n\ntheorem to_bool_of_optbool (ob) : (of_optbool ob).to_bool = ob := sorry'\n\ntheorem notbool_negb_1 (b) : of_bool (bnot b) = notbool (of_bool b) := sorry'\n\ntheorem notbool_negb_2 (b) : of_bool b = notbool (of_bool (bnot b)) := sorry'\n\ntheorem notbool_negb_3 (ob) : of_optbool (bnot <$> ob) = notbool (of_optbool ob) := sorry'\n\nset_option type_context.unfold_lemmas true\ntheorem notbool_idem2 (b) : notbool (notbool (of_bool b)) = of_bool b :=\nby cases b; refl\n\ntheorem notbool_idem3 (x) : notbool (notbool (notbool x)) = notbool x := sorry'\n\ntheorem notbool_idem4 (ob) : notbool (notbool (of_optbool ob)) = of_optbool ob := sorry'\n\ntheorem add_comm (x y : val) : x + y = y + x := sorry'\n\ntheorem add_assoc (x y z : val) : (x + y) + z = x + (y + z) := sorry'\n\ntheorem neg_zero : (-0 : val) = 0 := sorry'\n\ntheorem neg_add (x y : val) : -(x + y) = -x + -y := sorry'\n\ntheorem zero_sub (x : val) : 0 - x = -x := sorry'\n\ntheorem sub_eq_add_neg (x y) : x - Vint y = x + Vint (-y) := sorry'\n\ntheorem sub_neg_eq_add (x y) : x - Vint (-y) = x + Vint y := sorry'\n\ntheorem sub_add_eq_add_sub (v1 v2 i) : v1 + Vint i - v2 = v1 - v2 + Vint i := sorry' \n\ntheorem sub_add_eq_sub_add_neg (v1 v2 i) : v1 - (v2 + Vint i) = v1 - v2 + Vint (-i) := sorry'\n\ntheorem mul_comm (x y : val) : x * y = y * x := sorry'\n\ntheorem mul_assoc (x y z : val) : (x * y) * z = x * (y * z) := sorry'\n\ntheorem right_distrib (x y z : val) : (x + y) * z = x * z + y * z := sorry'\n\ntheorem left_distrib (x y z : val) : x * (y + z) = x * y + x * z := sorry'\n\ntheorem mul_pow2 (x n logn) : is_power2 n = some logn →\n  x * Vint n = shl x (Vint logn) := sorry'\n\ntheorem mods_divs (x y z) : mods x y = some z → ∃ v, divs x y = some v ∧ z = x - v * y := sorry'\n\ntheorem modu_divu (x y z) : modu x y = some z → ∃ v, divu x y = some v ∧ z = x - v * y := sorry'\n\ntheorem divs_pow2 (x n logn y) : is_power2 n = some logn → word.ltu logn 31 →\n  divs x (Vint n) = some y → shrx x (Vint logn) = some y := sorry'\n\ntheorem divu_pow2 (x n logn y) : is_power2 n = some logn →\n  divu x (Vint n) = some y → shru x (Vint logn) = y := sorry'\n\ntheorem modu_pow2 (x n logn y) : is_power2 n = some logn →\n  modu x (Vint n) = some y → val.and x (Vint (n - 1)) = y := sorry'\n\ntheorem and_comm (x y) : val.and x y = val.and y x := sorry'\n\ntheorem and_assoc (x y z) : val.and (val.and x y) z = val.and x (val.and y z) := sorry'\n\ntheorem or_comm (x y) : val.or x y = val.or y x := sorry'\n\ntheorem or_assoc (x y z) : val.or (val.or x y) z = val.or x (val.or y z) := sorry'\n\ntheorem xor_commut (x y) : val.xor x y = val.xor y x := sorry'\n\ntheorem xor_assoc (x y z) : val.xor (val.xor x y) z = val.xor x (val.xor y z) := sorry'\n\ntheorem not_xor (x) : notint x = val.xor x (Vint (-1)) := sorry'\n\ntheorem shl_rolm (x n) : word.ltu n 32 → shl x (Vint n) = rolm x n (word.shl (-1) n) := sorry'\n\ntheorem shru_rolm (x n) : word.ltu n 32 →\n  shru x (Vint n) = rolm x (32 - n) (word.shru (-1) n) := sorry'\n\ntheorem shrx_carry (x y z) : shrx x y = some z →\n  shr x y + shr_carry x y = z := sorry'\n\ntheorem shrx_shr (x y z) : shrx x y = some z →\n  ∃ p, ∃ q, x = Vint p ∧ y = Vint q ∧\n    z = shr (if p < 0 then x + Vint (word.shl 1 q - 1) else x) (Vint q) := sorry'\n\ntheorem shrx_shr_2 (n x z) : shrx x (Vint n) = some z →\n  z = if n = 0 then x else shr (x + shru (shr x 31) (Vint (32 - n))) (Vint n) := sorry'\n\ntheorem or_rolm (x n m1 m2) : val.or (rolm x n m1) (rolm x n m2) = rolm x n (word.or m1 m2) := sorry'\n\ntheorem rolm_rolm (x n1 m1 n2 m2) : rolm (rolm x n1 m1) n2 m2 =\n  rolm x (word.modu (n1 + n2) 32) (word.and (word.rol m1 n2) m2) := sorry'\n\ntheorem rolm_zero (x m) : rolm x 0 m = val.and x (Vint m) := sorry'\n\ntheorem addl_comm (x y) : addl x y = addl y x := sorry'\n\ntheorem addl_assoc (x y z) : addl (addl x y) z = addl x (addl y z) := sorry'\n\ntheorem negl_addl_distr (x y) : negl (addl x y) = addl (negl x) (negl y) := sorry'\n\ntheorem subl_addl_opp (x y) : subl x (Vlong y) = addl x (Vlong (-y)) := sorry'\n\ntheorem subl_opp_addl : ∀ x y, subl x (Vlong (-y)) = addl x (Vlong y) := sorry'\n\ntheorem subl_addl_l (v1 v2 i) : subl (addl v1 (Vlong i)) v2 = addl (subl v1 v2) (Vlong i) := sorry'\n\ntheorem subl_addl_r (v1 v2 i) : subl v1 (addl v2 (Vlong i)) = addl (subl v1 v2) (Vlong (-i)) := sorry'\n\ntheorem mull_comm (x y) : mull x y = mull y x := sorry'\n\ntheorem mull_assoc (x y z) : mull (mull x y) z = mull x (mull y z) := sorry'\n\ntheorem mull_addl_distr_l (x y z) : mull (addl x y) z = addl (mull x z) (mull y z) := sorry'\n\ntheorem mull_addl_distr_r (x y z) : mull x (addl y z) = addl (mull x y) (mull x z) := sorry'\n\ntheorem andl_comm (x y) : andl x y = andl y x := sorry'\n\ntheorem andl_assoc (x y z) : andl (andl x y) z = andl x (andl y z) := sorry'\n\ntheorem orl_comm (x y) : orl x y = orl y x := sorry'\n\ntheorem orl_assoc (x y z) : orl (orl x y) z = orl x (orl y z) := sorry'\n\ntheorem xorl_commut (x y) : xorl x y = xorl y x := sorry'\n\ntheorem xorl_assoc (x y z) : xorl (xorl x y) z = xorl x (xorl y z) := sorry'\n\ntheorem notl_xorl (x) : notl x = xorl x (Vlong (-1)) := sorry'\n\ntheorem divls_pow2 (x n logn y) : int64.is_power2 n = some logn → word.ltu logn 63 →\n  divls x (Vlong n) = some y →\n  shrxl x (Vint logn) = some y := sorry'\n\ntheorem divlu_pow2 (x n logn y) : int64.is_power2 n = some logn →\n  divlu x (Vlong n) = some y →\n  shrlu x (Vint logn) = y := sorry'\n\ntheorem modlu_pow2 (x n logn y) : int64.is_power2 n = some logn →\n  modlu x (Vlong n) = some y →\n  andl x (Vlong (n - 1)) = y := sorry'\n\ntheorem shrxl_shrl_2 (n x z) : shrxl x (Vint n) = some z →\n  z = if n = 0 then x else\n    shrl (addl x (shrlu (shrl x (Vint 63)) (Vint (64 - n)))) (Vint n) := sorry'\n\ntheorem negate_cmp_bool (c x y) : cmp_bool (negate_comparison c) x y = bnot <$> cmp_bool c x y := sorry'\n\ntheorem negate_cmpu_bool (valid_ptr c x y) :\n  cmpu_bool valid_ptr (negate_comparison c) x y = bnot <$> cmpu_bool valid_ptr c x y := sorry'\n\ntheorem negate_cmpl_bool (c x y) : cmpl_bool (negate_comparison c) x y = bnot <$> cmpl_bool c x y := sorry'\n\ntheorem negate_cmplu_bool (valid_ptr c x y) :\n  cmplu_bool valid_ptr (negate_comparison c) x y = bnot <$> cmplu_bool valid_ptr c x y := sorry'\n\nlemma not_of_optbool (ob) : of_optbool (bnot <$> ob) = notbool (of_optbool ob) := sorry'\n\ntheorem negate_cmp (c x y) : cmp (negate_comparison c) x y = notbool (cmp c x y) := sorry'\n\ntheorem negate_cmpu (valid_ptr c x y) :\n  cmpu valid_ptr (negate_comparison c) x y =\n    notbool (cmpu valid_ptr c x y) := sorry'\n\ntheorem swap_cmp_bool (c x y) : cmp_bool (swap_comparison c) x y = cmp_bool c y x := sorry'\n\ntheorem swap_cmpu_bool (valid_ptr c x y) :\n  cmpu_bool valid_ptr (swap_comparison c) x y = cmpu_bool valid_ptr c y x := sorry'\n\ntheorem swap_cmpl_bool (c x y) : cmpl_bool (swap_comparison c) x y = cmpl_bool c y x := sorry'\n\ntheorem swap_cmplu_bool (valid_ptr c x y) :\n  cmplu_bool valid_ptr (swap_comparison c) x y = cmplu_bool valid_ptr c y x := sorry'\n\ntheorem negate_cmpf_eq (v1 v2) : notbool (cmpf Cne v1 v2) = cmpf Ceq v1 v2 := sorry'\n\ntheorem negate_cmpf_ne (v1 v2) : notbool (cmpf Ceq v1 v2) = cmpf Cne v1 v2 := sorry'\n\ntheorem cmpf_le (v1 v2) : cmpf Cle v1 v2 = val.or (cmpf Clt v1 v2) (cmpf Ceq v1 v2) := sorry'\n\ntheorem cmpf_ge (v1 v2) : cmpf Cge v1 v2 = val.or (cmpf Cgt v1 v2) (cmpf Ceq v1 v2) := sorry'\n\ntheorem cmp_ne_0_optbool (ob) : cmp Cne (of_optbool ob) 0 = of_optbool ob := sorry'\n\ntheorem cmp_eq_1_optbool (ob) : cmp Ceq (of_optbool ob) 1 = of_optbool ob := sorry'\n\ntheorem cmp_eq_0_optbool (ob) : cmp Ceq (of_optbool ob) 0 = of_optbool (bnot <$> ob) := sorry'\n\ntheorem cmp_ne_1_optbool (ob) : cmp Cne (of_optbool ob) 1 = of_optbool (bnot <$> ob) := sorry'\n\ntheorem cmpu_ne_0_optbool (valid_ptr ob) :\n  cmpu valid_ptr Cne (of_optbool ob) 0 = of_optbool ob := sorry'\n\ntheorem cmpu_eq_1_optbool (valid_ptr ob) :\n  cmpu valid_ptr Ceq (of_optbool ob) 1 = of_optbool ob := sorry'\n\ntheorem cmpu_eq_0_optbool (valid_ptr ob) :\n  cmpu valid_ptr Ceq (of_optbool ob) 0 = of_optbool (bnot <$> ob) := sorry'\n\ntheorem cmpu_ne_1_optbool (valid_ptr ob) :\n  cmpu valid_ptr Cne (of_optbool ob) 1 = of_optbool (bnot <$> ob) := sorry'\n\nlemma zero_ext_and (n v) : 0 < n → n < 32 →\n  val.zero_ext n v = val.and v (Vint (repr (2^n - 1))) := sorry'\n\nlemma rolm_lt_zero (v) : rolm v 1 1 = cmp Clt v 0 := sorry'\n\nlemma rolm_ge_zero (v) : val.xor (rolm v 1 1) 1 = cmp Cge v 0 := sorry'\n\n/- The ``is less defined'' relation between values.\n    A value is less defined than itself, and [Vundef] is\n    less defined than any value. -/\n\ninductive lessdef : val → val → Prop\n| refl (v) : lessdef v v\n| undef (v) : lessdef Vundef v\n\nlemma lessdef_of_eq : Π {v1 v2}, v1 = v2 → lessdef v1 v2\n| v ._ rfl := lessdef.refl v\n\nlemma lessdef_trans {v1 v2 v3} : lessdef v1 v2 → lessdef v2 v3 → lessdef v1 v3 :=\nby intros h1 h2; cases h1; try {assumption}; cases h2; assumption\n\nlemma lessdef_list_inv {vl1 vl2} : list.forall2 lessdef vl1 vl2 → vl1 = vl2 ∨ Vundef ∈ vl1 := sorry'\n\nlemma lessdef_list_trans {vl1 vl2 vl3} :\n  list.forall2 lessdef vl1 vl2 → list.forall2 lessdef vl2 vl3 → list.forall2 lessdef vl1 vl3 :=\n@list.forall2.trans _ _ @lessdef_trans _ _ _\n\n/- Compatibility of operations with the [lessdef] relation. -/\n\nlemma load_result_lessdef (chunk v1 v2) :\n  lessdef v1 v2 → lessdef (load_result chunk v1) (load_result chunk v2) := sorry'\n\nlemma zero_ext_lessdef (n v1 v2) :\n  lessdef v1 v2 → lessdef (zero_ext n v1) (zero_ext n v2) := sorry'\n\nlemma sign_ext_lessdef (n v1 v2) :\n  lessdef v1 v2 → lessdef (sign_ext n v1) (sign_ext n v2) := sorry'\n\nlemma singleoffloat_lessdef (v1 v2) :\n  lessdef v1 v2 → lessdef (single_of_float v1) (single_of_float v2) := sorry'\n\nlemma add_lessdef (v1 v1' v2 v2') :\n  lessdef v1 v1' → lessdef v2 v2' → lessdef (v1 + v2) (v1' + v2') := sorry'\n\nlemma addl_lessdef (v1 v1' v2 v2') :\n  lessdef v1 v1' → lessdef v2 v2' → lessdef (addl v1 v2) (addl v1' v2') := sorry'\n\nlemma cmpu_bool_lessdef {valid_ptr valid_ptr' : block → ℕ → bool} {c v1 v1' v2 v2' b} :\n  (∀ b ofs, valid_ptr b ofs → valid_ptr' b ofs) →\n  lessdef v1 v1' → lessdef v2 v2' →\n  cmpu_bool valid_ptr c v1 v2 = some b →\n  cmpu_bool valid_ptr' c v1' v2' = some b := sorry'\n\nlemma cmplu_bool_lessdef {valid_ptr valid_ptr' : block → ℕ → bool} {c v1 v1' v2 v2' b} :\n  (∀ b ofs, valid_ptr b ofs → valid_ptr' b ofs) →\n  lessdef v1 v1' → lessdef v2 v2' →\n  cmplu_bool valid_ptr c v1 v2 = some b →\n  cmplu_bool valid_ptr' c v1' v2' = some b := sorry'\n\nlemma of_optbool_lessdef {ob ob'} :\n  (∀ b, ob = some b → ob' = some b) →\n  lessdef (of_optbool ob) (of_optbool ob') := sorry'\n\nlemma long_of_words_lessdef {v1 v2 v1' v2'} :\n  lessdef v1 v1' → lessdef v2 v2' → lessdef (long_of_words v1 v2) (long_of_words v1' v2') := sorry'\n\nlemma loword_lessdef {v v'} : lessdef v v' → lessdef (loword v) (loword v') := sorry'\n\nlemma hiword_lessdef {v v'} : lessdef v v' → lessdef (hiword v) (hiword v') := sorry'\n\nlemma offset_ptr_zero (v) : lessdef (offset_ptr v 0) v := sorry'\n\nlemma offset_ptr_assoc (v d1 d2) : offset_ptr (offset_ptr v d1) d2 = offset_ptr v (d1 + d2) := sorry'\n\n/- * Values and memory injections -/\n\n/- A memory injection [f] is a function from addresses to either [none]\n  or [some] of an address and an offset.  It defines a correspondence\n  between the blocks of two memory states [m1] and [m2]:\n- if [f b = none], the block [b] of [m1] has no equivalent in [m2];\n- if [f b = some (b', ofs)], the block [b] of [m2] corresponds to\n  a sub-block at offset [ofs] of the block [b'] in [m2].\n-/\n\ndef meminj : Type := block → option (block × ℤ)\n\n/- A memory injection defines a relation between values that is the\n  identity relation, except for pointer values which are shifted\n  as prescribed by the memory injection.  Moreover, [Vundef] values\n  inject into any other value. -/\n\ninductive inject (mi : meminj) : val → val → Prop\n| int (i) : inject (Vint i) (Vint i)\n| long (i) : inject (Vlong i) (Vlong i)\n| float (f) : inject (Vfloat f) (Vfloat f)\n| single (f) : inject (Vsingle f) (Vsingle f)\n| ptr (b1 ofs1 b2 ofs2 delta) :\n      mi b1 = some (b2, delta) →\n      (ofs2 : ptrofs) = ofs1 + repr delta →\n      inject (Vptr b1 ofs1) (Vptr b2 ofs2)\n| undef (v) : inject Vundef v\n\nlemma inject_ptrofs (mi i) : inject mi (Vptrofs i) (Vptrofs i) := sorry'\n\nsection val_inj_ops\n\nparameter f : meminj\n\nlemma load_result_inject (chunk v1 v2) :\n  inject f v1 v2 →\n  inject f (load_result chunk v1) (load_result chunk v2) := sorry'\n\ntheorem add_inject {v1 v1' v2 v2'} :\n  inject f v1 v1' → inject f v2 v2' →\n  inject f (v1 + v2) (v1' + v2') := sorry'\n\ntheorem sub_inject {v1 v1' v2 v2'} :\n  inject f v1 v1' → inject f v2 v2' →\n  inject f (v1 - v2) (v1' - v2') := sorry'\n\ntheorem addl_inject {v1 v1' v2 v2'} :\n  inject f v1 v1' → inject f v2 v2' →\n  inject f (addl v1 v2) (addl v1' v2') := sorry'\n\ntheorem subl_inject {v1 v1' v2 v2'} :\n  inject f v1 v1' → inject f v2 v2' →\n  inject f (subl v1 v2) (subl v1' v2') := sorry'\n\nlemma offset_ptr_inject {v v'} (ofs) : inject f v v' →\n  inject f (offset_ptr v ofs) (offset_ptr v' ofs) := sorry'\n\nlemma cmp_bool_inject {c v1 v2 v1' v2' b} :\n  inject f v1 v1' → inject f v2 v2' →\n  cmp_bool c v1 v2 = some b → cmp_bool c v1' v2' = some b := sorry'\n\nparameters (valid_ptr1 valid_ptr2 : block → ℕ → bool)\n\ndef weak_valid_ptr1 := weak_valid_ptr valid_ptr1\ndef weak_valid_ptr2 := weak_valid_ptr valid_ptr2\n\nparameter valid_ptr_inj : ∀ b1 ofs b2 delta, f b1 = some (b2, delta) →\n  valid_ptr1 b1 (unsigned (ofs : ptrofs)) →\n  valid_ptr2 b2 (unsigned (ofs + repr delta))\n\nparameter weak_valid_ptr_inj : ∀ b1 ofs b2 delta, f b1 = some (b2, delta) →\n  weak_valid_ptr1 b1 (unsigned (ofs : ptrofs)) →\n  weak_valid_ptr2 b2 (unsigned (ofs + repr delta))\n\nparameter weak_valid_ptr_no_overflow : ∀ b1 ofs b2 delta, f b1 = some (b2, delta) →\n  weak_valid_ptr1 b1 (unsigned (ofs : ptrofs)) →\n  unsigned ofs + unsigned (repr delta : ptrofs) ≤ @max_unsigned ptrofs.wordsize\n\nparameter valid_different_ptrs_inj : ∀ b1 ofs1 b2 ofs2 b1' delta1 b2' delta2,\n  b1 ≠ b2 →\n  valid_ptr1 b1 (unsigned (ofs1 : ptrofs)) →\n  valid_ptr1 b2 (unsigned (ofs2 : ptrofs)) →\n  f b1 = some (b1', delta1) →\n  f b2 = some (b2', delta2) →\n  b1' = b2' → unsigned (ofs1 + repr delta1) ≠ unsigned (ofs2 + repr delta2)\n\nlemma cmpu_bool_inject {c v1 v2 v1' v2' b} :\n  inject f v1 v1' → inject f v2 v2' →\n  cmpu_bool valid_ptr1 c v1 v2 = some b →\n  cmpu_bool valid_ptr2 c v1' v2' = some b := sorry'\n\nlemma cmplu_bool_inject {c v1 v2 v1' v2' b} :\n  inject f v1 v1' → inject f v2 v2' →\n  cmplu_bool valid_ptr1 c v1 v2 = some b →\n  cmplu_bool valid_ptr2 c v1' v2' = some b := sorry'\n\nlemma long_of_words_inject {v1 v2 v1' v2'} :\n  inject f v1 v1' → inject f v2 v2' →\n  inject f (long_of_words v1 v2) (long_of_words v1' v2') := sorry'\n\nlemma loword_inject {v v'} : inject f v v' → inject f (loword v) (loword v') := sorry'\n\nlemma hiword_inject {v v'} : inject f v v' → inject f (hiword v) (hiword v') := sorry'\n\nend val_inj_ops\n\nend val\n\nexport val (meminj)\n\n/- Monotone evolution of a memory injection. -/\n\ndef inject_incr (f1 f2 : meminj) : Prop :=\n∀ ⦃b b' delta⦄, f1 b = some (b', delta) → f2 b = some (b', delta)\n\nlemma inject_incr.refl (f) : inject_incr f f := λ _ _ _, id\n\nlemma inject_incr.trans {f1 f2 f3}\n  (i1 : inject_incr f1 f2) (i2 : inject_incr f2 f3) : inject_incr f1 f3 :=\nλ b b' d h, i2 (i1 h)\n\nlemma val_inject_incr {f1 f2 v v'} :\n  inject_incr f1 f2 → inject f1 v v' → inject f2 v v' := sorry'\n\nlemma val_inject_list_incr {f1 f2 vl vl'}\n  (i : inject_incr f1 f2) (il : list.forall2 (inject f1) vl vl') :\n  list.forall2 (inject f2) vl vl' :=\nil.imp (λ x y, val_inject_incr i)\n\n/- The identity injection gives rise to the \"less defined than\" relation. -/\n\ndef inject_id : meminj := λ b, some (b, 0)\n\nlemma val_inject_id {v1 v2} :\n  inject inject_id v1 v2 ↔ val.lessdef v1 v2 := sorry'\n\nlemma val_inject_id_list {vl1 vl2} :\n  list.forall2 (inject inject_id) vl1 vl2 ↔ list.forall2 val.lessdef vl1 vl2 :=\nlist.forall2.iff @val_inject_id\n\n/- Composing two memory injections -/\n\ndef val.meminj.comp (f f' : meminj) : meminj := λ b,\ndo ⟨b', delta⟩ ← f b,\n   ⟨b'', delta'⟩ ← f' b',\n   return (b'', delta + delta')\n\nlemma inject.comp {f f' v1 v2 v3} :\n  inject f v1 v2 → inject f' v2 v3 →\n  inject (f.comp f') v1 v3 := sorry'\n\nend values", "meta": {"author": "digama0", "repo": "kremlin", "sha": "d4665929ce9012e93a0b05fc7063b96256bab86f", "save_path": "github-repos/lean/digama0-kremlin", "path": "github-repos/lean/digama0-kremlin/kremlin-d4665929ce9012e93a0b05fc7063b96256bab86f/values.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.34569497312102565}}
{"text": "/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nnamespace Std\nuniverses u v w w'\n\nnamespace PersistentHashMap\n\ninductive Entry (α : Type u) (β : Type v) (σ : Type w) where\n  | entry (key : α) (val : β) : Entry α β σ\n  | ref   (node : σ) : Entry α β σ\n  | null  : Entry α β σ\n\ninstance {α β σ} : Inhabited (Entry α β σ) := ⟨Entry.null⟩\n\ninductive Node (α : Type u) (β : Type v) : Type (max u v) where\n  | entries   (es : Array (Entry α β (Node α β))) : Node α β\n  | collision (ks : Array α) (vs : Array β) (h : ks.size = vs.size) : Node α β\n\ninstance {α β} : Inhabited (Node α β) := ⟨Node.entries #[]⟩\n\nabbrev shift         : USize  := 5\nabbrev branching     : USize  := USize.ofNat (2 ^ shift.toNat)\nabbrev maxDepth      : USize  := 7\nabbrev maxCollisions : Nat    := 4\n\ndef mkEmptyEntriesArray {α β} : Array (Entry α β (Node α β)) :=\n  (Array.mkArray PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)\n\nend PersistentHashMap\n\nstructure PersistentHashMap (α : Type u) (β : Type v) [BEq α] [Hashable α] where\n  root    : PersistentHashMap.Node α β := PersistentHashMap.Node.entries PersistentHashMap.mkEmptyEntriesArray\n  size    : Nat                        := 0\n\nabbrev PHashMap (α : Type u) (β : Type v) [BEq α] [Hashable α] := PersistentHashMap α β\n\nnamespace PersistentHashMap\nvariable {α : Type u} {β : Type v}\n\ndef empty [BEq α] [Hashable α] : PersistentHashMap α β := {}\n\ndef isEmpty [BEq α] [Hashable α] (m : PersistentHashMap α β) : Bool :=\n  m.size == 0\n\ninstance [BEq α] [Hashable α] : Inhabited (PersistentHashMap α β) := ⟨{}⟩\n\ndef mkEmptyEntries {α β} : Node α β :=\n  Node.entries mkEmptyEntriesArray\n\nabbrev mul2Shift (i : USize) (shift : USize) : USize := i.shiftLeft shift\nabbrev div2Shift (i : USize) (shift : USize) : USize := i.shiftRight shift\nabbrev mod2Shift (i : USize) (shift : USize) : USize := USize.land i ((USize.shiftLeft 1 shift) - 1)\n\ninductive IsCollisionNode : Node α β → Prop where\n  | mk (keys : Array α) (vals : Array β) (h : keys.size = vals.size) : IsCollisionNode (Node.collision keys vals h)\n\nabbrev CollisionNode (α β) := { n : Node α β // IsCollisionNode n }\n\ninductive IsEntriesNode : Node α β → Prop where\n  | mk (entries : Array (Entry α β (Node α β))) : IsEntriesNode (Node.entries entries)\n\nabbrev EntriesNode (α β) := { n : Node α β // IsEntriesNode n }\n\nprivate theorem size_set {ks : Array α} {vs : Array β} (h : ks.size = vs.size) (i : Fin ks.size) (j : Fin vs.size) (k : α) (v : β)\n                           : (ks.set i k).size = (vs.set j v).size := by\n  simp [h]\n\nprivate theorem size_push {ks : Array α} {vs : Array β} (h : ks.size = vs.size) (k : α) (v : β) : (ks.push k).size = (vs.push v).size := by\n  simp [h]\n\npartial def insertAtCollisionNodeAux [BEq α] : CollisionNode α β → Nat → α → β → CollisionNode α β\n  | n@⟨Node.collision keys vals heq, _⟩, i, k, v =>\n    if h : i < keys.size then\n      let idx : Fin keys.size := ⟨i, h⟩;\n      let k' := keys.get idx;\n      if k == k' then\n         let j : Fin vals.size := ⟨i, by rw [←heq]; assumption⟩\n         ⟨Node.collision (keys.set idx k) (vals.set j v) (size_set heq idx j k v), IsCollisionNode.mk _ _ _⟩\n      else insertAtCollisionNodeAux n (i+1) k v\n    else\n      ⟨Node.collision (keys.push k) (vals.push v) (size_push heq k v), IsCollisionNode.mk _ _ _⟩\n  | ⟨Node.entries _, h⟩, _, _, _ => False.elim (nomatch h)\n\ndef insertAtCollisionNode [BEq α] : CollisionNode α β → α → β → CollisionNode α β :=\n  fun n k v => insertAtCollisionNodeAux n 0 k v\n\ndef getCollisionNodeSize : CollisionNode α β → Nat\n  | ⟨Node.collision keys _ _, _⟩ => keys.size\n  | ⟨Node.entries _, h⟩          => False.elim (nomatch h)\n\ndef mkCollisionNode (k₁ : α) (v₁ : β) (k₂ : α) (v₂ : β) : Node α β :=\n  let ks : Array α := Array.mkEmpty maxCollisions\n  let ks := (ks.push k₁).push k₂\n  let vs : Array β := Array.mkEmpty maxCollisions\n  let vs := (vs.push v₁).push v₂\n  Node.collision ks vs rfl\n\npartial def insertAux [BEq α] [Hashable α] : Node α β → USize → USize → α → β → Node α β\n  | Node.collision keys vals heq, _, depth, k, v =>\n    let newNode := insertAtCollisionNode ⟨Node.collision keys vals heq, IsCollisionNode.mk _ _ _⟩ k v\n    if depth >= maxDepth || getCollisionNodeSize newNode < maxCollisions then newNode.val\n    else match newNode with\n      | ⟨Node.entries _, h⟩ => False.elim (nomatch h)\n      | ⟨Node.collision keys vals heq, _⟩ =>\n        let rec traverse (i : Nat) (entries : Node α β) : Node α β :=\n          if h : i < keys.size then\n            let k := keys.get ⟨i, h⟩\n            let v := vals.get ⟨i, heq ▸ h⟩\n            let h := hash k\n            let h := div2Shift h (shift * (depth - 1))\n            traverse (i+1) (insertAux entries h depth k v)\n          else\n            entries\n        traverse 0 mkEmptyEntries\n  | Node.entries entries, h, depth, k, v =>\n    let j     := (mod2Shift h shift).toNat\n    Node.entries $ entries.modify j fun entry =>\n      match entry with\n      | Entry.null        => Entry.entry k v\n      | Entry.ref node    => Entry.ref $ insertAux node (div2Shift h shift) (depth+1) k v\n      | Entry.entry k' v' =>\n        if k == k' then Entry.entry k v\n        else Entry.ref $ mkCollisionNode k' v' k v\n\ndef insert [BEq α] [Hashable α] : PersistentHashMap α β → α → β → PersistentHashMap α β\n  | { root := n, size := sz }, k, v => { root := insertAux n (hash k) 1 k v, size := sz + 1 }\n\npartial def findAtAux [BEq α] (keys : Array α) (vals : Array β) (heq : keys.size = vals.size) (i : Nat) (k : α) : Option β :=\n  if h : i < keys.size then\n    let k' := keys.get ⟨i, h⟩\n    if k == k' then some (vals.get ⟨i, by rw [←heq]; assumption⟩)\n    else findAtAux keys vals heq (i+1) k\n  else none\n\npartial def findAux [BEq α] : Node α β → USize → α → Option β\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some v else none\n  | Node.collision keys vals heq, _, k => findAtAux keys vals heq 0 k\n\ndef find? [BEq α] [Hashable α] : PersistentHashMap α β → α → Option β\n  | { root := n, .. }, k => findAux n (hash k) k\n\n@[inline] def getOp [BEq α] [Hashable α] (self : PersistentHashMap α β) (idx : α) : Option β :=\n  self.find? idx\n\n@[inline] def findD [BEq α] [Hashable α] (m : PersistentHashMap α β) (a : α) (b₀ : β) : β :=\n  (m.find? a).getD b₀\n\n@[inline] def find! [BEq α] [Hashable α] [Inhabited β] (m : PersistentHashMap α β) (a : α) : β :=\n  match m.find? a with\n  | some b => b\n  | none   => panic! \"key is not in the map\"\n\npartial def findEntryAtAux [BEq α] (keys : Array α) (vals : Array β) (heq : keys.size = vals.size) (i : Nat) (k : α) : Option (α × β) :=\n  if h : i < keys.size then\n    let k' := keys.get ⟨i, h⟩\n    if k == k' then some (k', vals.get ⟨i, by rw [←heq]; assumption⟩)\n    else findEntryAtAux keys vals heq (i+1) k\n  else none\n\npartial def findEntryAux [BEq α] : Node α β → USize → α → Option (α × β)\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findEntryAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some (k', v) else none\n  | Node.collision keys vals heq, _, k => findEntryAtAux keys vals heq 0 k\n\ndef findEntry? [BEq α] [Hashable α] : PersistentHashMap α β → α → Option (α × β)\n  | { root := n, .. }, k => findEntryAux n (hash k) k\n\npartial def containsAtAux [BEq α] (keys : Array α) (vals : Array β) (heq : keys.size = vals.size) (i : Nat) (k : α) : Bool :=\n  if h : i < keys.size then\n    let k' := keys.get ⟨i, h⟩\n    if k == k' then true\n    else containsAtAux keys vals heq (i+1) k\n  else false\n\npartial def containsAux [BEq α] : Node α β → USize → α → Bool\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => false\n    | Entry.ref node   => containsAux node (div2Shift h shift) k\n    | Entry.entry k' v => k == k'\n  | Node.collision keys vals heq, _, k => containsAtAux keys vals heq 0 k\n\ndef contains [BEq α] [Hashable α] : PersistentHashMap α β → α → Bool\n  | { root := n, .. }, k => containsAux n (hash k) k\n\npartial def isUnaryEntries (a : Array (Entry α β (Node α β))) (i : Nat) (acc : Option (α × β)) : Option (α × β) :=\n  if h : i < a.size then\n    match a.get ⟨i, h⟩ with\n    | Entry.null      => isUnaryEntries a (i+1) acc\n    | Entry.ref _     => none\n    | Entry.entry k v =>\n      match acc with\n      | none   => isUnaryEntries a (i+1) (some (k, v))\n      | some _ => none\n  else acc\n\ndef isUnaryNode : Node α β → Option (α × β)\n  | Node.entries entries         => isUnaryEntries entries 0 none\n  | Node.collision keys vals heq =>\n    if h : 1 = keys.size then\n      have : 0 < keys.size := by rw [←h]; decide\n      some (keys.get ⟨0, this⟩, vals.get ⟨0, by rw [←heq]; assumption⟩)\n    else\n      none\n\npartial def eraseAux [BEq α] : Node α β → USize → α → Node α β × Bool\n  | n@(Node.collision keys vals heq), _, k =>\n    match keys.indexOf? k with\n    | some idx =>\n      let ⟨keys', keq⟩ := keys.eraseIdx' idx\n      let ⟨vals', veq⟩ := vals.eraseIdx' (Eq.ndrec idx heq)\n      have : keys.size - 1 = vals.size - 1 := by rw [heq]\n      (Node.collision keys' vals' (keq.trans (this.trans veq.symm)), true)\n    | none     => (n, false)\n  | n@(Node.entries entries), h, k =>\n    let j       := (mod2Shift h shift).toNat\n    let entry   := entries.get! j\n    match entry with\n    | Entry.null       => (n, false)\n    | Entry.entry k' v =>\n      if k == k' then (Node.entries (entries.set! j Entry.null), true) else (n, false)\n    | Entry.ref node   =>\n      let entries := entries.set! j Entry.null\n      let (newNode, deleted) := eraseAux node (div2Shift h shift) k\n      if !deleted then (n, false)\n      else match isUnaryNode newNode with\n        | none        => (Node.entries (entries.set! j (Entry.ref newNode)), true)\n        | some (k, v) => (Node.entries (entries.set! j (Entry.entry k v)), true)\n\ndef erase [BEq α] [Hashable α] : PersistentHashMap α β → α → PersistentHashMap α β\n  | { root := n, size := sz }, k =>\n    let h := hash k\n    let (n, del) := eraseAux n h k\n    { root := n, size := if del then sz - 1 else sz }\n\nsection\nvariable {m : Type w → Type w'} [Monad m]\nvariable {σ : Type w}\n\n@[specialize] partial def foldlMAux (f : σ → α → β → m σ) : Node α β → σ → m σ\n  | Node.collision keys vals heq, acc =>\n    let rec traverse (i : Nat) (acc : σ) : m σ := do\n      if h : i < keys.size then\n        let k := keys.get ⟨i, h⟩\n        let v := vals.get ⟨i, heq ▸ h⟩\n        traverse (i+1) (← f acc k v)\n      else\n        pure acc\n    traverse 0 acc\n  | Node.entries entries, acc => entries.foldlM (fun acc entry =>\n    match entry with\n    | Entry.null      => pure acc\n    | Entry.entry k v => f acc k v\n    | Entry.ref node  => foldlMAux f node acc)\n    acc\n\n@[specialize] def foldlM [BEq α] [Hashable α] (map : PersistentHashMap α β) (f : σ → α → β → m σ) (init : σ) : m σ :=\n  foldlMAux f map.root init\n\n@[specialize] def forM [BEq α] [Hashable α] (map : PersistentHashMap α β) (f : α → β → m PUnit) : m PUnit :=\n  map.foldlM (fun _ => f) ⟨⟩\n\n@[specialize] def foldl [BEq α] [Hashable α] (map : PersistentHashMap α β) (f : σ → α → β → σ) (init : σ) : σ :=\n  Id.run $ map.foldlM f init\nend\n\ndef toList [BEq α] [Hashable α] (m : PersistentHashMap α β) : List (α × β) :=\n  m.foldl (init := []) fun ps k v => (k, v) :: ps\n\nstructure Stats where\n  numNodes      : Nat := 0\n  numNull       : Nat := 0\n  numCollisions : Nat := 0\n  maxDepth      : Nat := 0\n\npartial def collectStats : Node α β → Stats → Nat → Stats\n  | Node.collision keys _ _, stats, depth =>\n    { stats with\n      numNodes      := stats.numNodes + 1,\n      numCollisions := stats.numCollisions + keys.size - 1,\n      maxDepth      := Nat.max stats.maxDepth depth }\n  | Node.entries entries, stats, depth =>\n    let stats :=\n      { stats with\n        numNodes      := stats.numNodes + 1,\n        maxDepth      := Nat.max stats.maxDepth depth }\n    entries.foldl (fun stats entry =>\n      match entry with\n      | Entry.null      => { stats with numNull := stats.numNull + 1 }\n      | Entry.ref node  => collectStats node stats (depth + 1)\n      | Entry.entry _ _ => stats)\n      stats\n\ndef stats [BEq α] [Hashable α] (m : PersistentHashMap α β) : Stats :=\n  collectStats m.root {} 1\n\ndef Stats.toString (s : Stats) : String :=\n  s!\"\\{ nodes := {s.numNodes}, null := {s.numNull}, collisions := {s.numCollisions}, depth := {s.maxDepth}}\"\n\ninstance : ToString Stats := ⟨Stats.toString⟩\n\nend PersistentHashMap\nend Std\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/stage0/src/Std/Data/PersistentHashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.34569496622680146}}
{"text": "mutual\n\ninductive TreePos (α : Type u) (β : Type v) where\n  | leaf (a : α)\n  | node (children : List (List (TreeNeg α β)))\n\ninductive TreeNeg (α : Type u) (β : Type v) where\n  | leaf (a : β)\n  | node (children : List (List (TreePos α β)))\n\nend\n\ntheorem aux_1 [SizeOf α] [SizeOf β] (cs : List (TreePos α β)) : TreePos._sizeOf_6 cs = sizeOf cs :=\n  @List.rec (TreePos α β) (fun cs => TreePos._sizeOf_6 cs = sizeOf cs)\n    rfl\n    (fun h t ih => by\n      show 1 + TreePos._sizeOf_1 h + TreePos._sizeOf_6 t = sizeOf (h::t)\n      rw ih\n      rfl)\n    cs\n\ntheorem aux_2 [SizeOf α] [SizeOf β] (cs : List (List (TreePos α β))) : TreePos._sizeOf_4 cs = sizeOf cs :=\n  @List.rec (List (TreePos α β)) (fun cs => TreePos._sizeOf_4 cs = sizeOf cs)\n    rfl\n    (fun h t ih =>  by\n      show 1 + TreePos._sizeOf_6 h + TreePos._sizeOf_4 t = sizeOf (h :: t)\n      rw aux_1\n      rw ih\n      rfl)\n    cs\n\ntheorem aux_3 [SizeOf α] [SizeOf β] (cs : List (TreeNeg α β)) : TreePos._sizeOf_5 cs = sizeOf cs :=\n  @List.rec (TreeNeg α β) (fun cs => TreePos._sizeOf_5 cs = sizeOf cs)\n    rfl\n    (fun h t ih => by\n      show 1 + TreePos._sizeOf_2 h + TreePos._sizeOf_5 t = sizeOf (h::t)\n      rw ih\n      rfl)\n    cs\n\ntheorem aux_4 [SizeOf α] [SizeOf β] (cs : List (List (TreeNeg α β))) : TreePos._sizeOf_3 cs = sizeOf cs :=\n  @List.rec (List (TreeNeg α β)) (fun cs => TreePos._sizeOf_3 cs = sizeOf cs)\n    rfl\n    (fun h t ih => by\n      show 1 + TreePos._sizeOf_5 h + TreePos._sizeOf_3 t = sizeOf (h :: t)\n      rw ih\n      rw aux_3\n      rfl)\n    cs\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/playground/sizeof1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34548181972574304}}
{"text": "\nimport util.data.bijection\nimport util.data.perm\nimport util.data.minimum\n\nimport unitb.logic\nimport unitb.scheduling.basic\nimport unitb.models.simple\n\nimport data.set.basic\n\nlocal attribute [-simp] or.comm or.left_comm or.assoc and.comm and.left_comm and.assoc\n\nnamespace scheduling.finite\n\nopen nat simple function predicate temporal set scheduling\n@[reducible]\ndef rrobin (lbl : Type) [pos_finite lbl] : Type :=\nbijection (fin $ succ $ pos_finite.pred_count lbl) lbl\n\n@[reducible]\ndef index_t (lbl : Type) [pos_finite lbl] :=\nfin $ succ $ pos_finite.pred_count lbl\n\nsection\n\nparameter {lbl : Type}\nparameter [pos_finite lbl]\nparameter t : scheduling.unitb.target_mch lbl\n\nnoncomputable def first (req : set lbl)\n  (l : rrobin lbl)\n: index_t lbl :=\n(↓ x, l.f x ∈ req )\n\nnoncomputable def sch.current (r : set lbl) (queue : rrobin lbl) : lbl :=\nqueue.f (first r queue)\n\nstructure sch_state : Type :=\n  (target : t.σ)\n  (queue : rrobin lbl)\n  (inv : sch.current (t.req.apply target) queue ∈ t.req.apply target)\n\nlemma first_mem {req : set lbl}\n  (l : rrobin lbl)\n  (h : req ≠ ∅)\n: l.f (first req l) ∈ req :=\nbegin\n  unfold first, apply @minimum_mem _ _ { x | l.f x ∈ req },\n  cases set.exists_mem_of_ne_empty h with x h,\n  apply @set.ne_empty_of_mem _ _ (l.g x),\n  change l.f (l.g x) ∈ req,\n  rw l.g_inv, apply h\nend\n\nnoncomputable def select (req : set lbl)\n  (l : rrobin lbl)\n: rrobin lbl :=\nl ∘ (perm.rotate_right (first req l)).rev\n\nlemma sch.select_inv\n  (s : t.σ)\n  (q : rrobin lbl)\n: sch.current (t.req.apply s) q ∈ t.req.apply s :=\nbegin\n  unfold select sch.current,\n  apply first_mem _ (t.req_nemp s),\nend\n\ndef sch.first : sch_state :=\n{ target := t.s₀\n, queue := bijection.rev (finite.to_nat _)\n, inv := sch.select_inv _ _ }\n\nnoncomputable def sch.step : sch_state → sch_state\n | (sch_state.mk target queue P) :=\n     { target := t.next (sch.current (t.req.apply target) queue) target P\n     , queue := select (t.req.apply target) queue\n     , inv := sch.select_inv _ _\n     }\n\ndef sch.is_step : sch_state → sch_state → Prop\n | s s' := s' = sch.step s\n\nnoncomputable def scheduler : program sch_state :=\n  { first := sch.first\n  , step  := sch.step }\n\nopen sch_state unitb subtype has_mem classical\n\ndef req : var sch_state (set lbl) :=\nt.req ∘' (sch_state.target : var sch_state t.σ)\n\nnoncomputable def current  : var sch_state lbl :=\nsch.current <$> req <*> queue\n\ndef rank (l : lbl) : var (sch_state) ℕ :=\n↑ λ s : sch_state, (s.queue.g l).val\n\nopen scheduling scheduling.unitb\n\nvariable (s : sch_state)\nsection eq_next_or_rank_eq_or_rank_lt_aux\nvariable {v : ℕ}\nvariable {l : lbl}\nvariable h : (rank l).apply s = v\ninclude h\n\nsection left\nvariable Hlt : s.queue.g l < first (req.apply s) s.queue\ninclude Hlt\n\nprivate lemma not_requested : l ∉ req.apply s :=\nbegin\n  cases s,\n  simp [select,comp,sch.step,sch_state.queue,req,rank] at *,\n  intro h₀,\n  have h₁ : queue.g l ∈ { x | queue.f x ∈ t.req.apply target },\n  { rw [mem_set_of,bijection.g_inv],\n    apply h₀ },\n  have h₂ := minimum_le h₁,\n  apply not_le_of_gt Hlt h₂,\nend\n\nprivate lemma rank_stable : (rank l).apply (sch.step s) = v :=\nbegin\n  cases s,\n  simp [select,comp,sch.step,sch_state.queue,req,rank] at *,\n  rw [comp_g], unfold comp,\n  rw [rev_g ,perm.rotate_right_f_gt_eq_self _ _ Hlt,h] ,\nend\n\nend left\nsection right\n\nvariable Hgt : s.queue.g l > first (req.apply s) s.queue\ninclude Hgt\nprivate lemma rank_dec : (rank l).apply (sch.step s) < v :=\nbegin\n  cases s,\n  simp [rank,sch.step,select,req] at *,\n  rw [comp_g], unfold comp,\n  rw [rev_g,perm.rotate_right_f_lt_shifted _ _ Hgt,fin.pred_def,← h],\n  apply pred_lt_self_of_pos,\n  unfold gt at Hgt,\n  rw [fin.lt_def] at Hgt,\n  apply lt_of_le_of_lt _ Hgt,\n  apply zero_le\nend\nend right\n\nend eq_next_or_rank_eq_or_rank_lt_aux\n\nlemma eq_next_or_rank_eq_or_rank_lt  {l : lbl} (v : ℕ)\n  (h : s ⊨ rank l ≃ v)\n: s ⊨ current ≃ ↑l ∨\n      ( s ⊨ - (↑l ∊ req) ∧ sch.step s ⊨ rank l ≃ v ) ∨ sch.step s ⊨ rank l ≺ v :=\nbegin\n  -- lifted_pred [temporal.init,sch.is_step] using Hstep h,\n  -- rw Hstep, clear Hstep,\n  -- revert h, generalize : x 0 = s, intro,\n  cases classical.em (s.queue.g l = first ((req t).apply s) s.queue) with Heq Hne,\n  { left, cases s,\n    simp [sch.step,sch_state.queue,sch_state.target,rank,current,req,comp,sch.current] at h ⊢,\n    simp [req] at Heq,\n    rw bijection.inverse,\n    symmetry, apply Heq, },\n  cases lt_or_gt_of_ne Hne with Hlt Hgt ; clear Hne,\n  { right, left, split, simp,\n    apply not_requested t s ; assumption ,\n    apply rank_stable t s ; assumption },\n  { right, right,\n    apply rank_dec t s ; assumption , },\nend\n\nlemma fair_schedule_step  (l : lbl) (v : ℕ)\n:  ↑l ∊ req ⋀ rank l ≃ v ↦ rank l ≺ v ⋁ current ≃ l in scheduler :=\nbegin\n  unfold scheduler,\n  apply leads_to_step,\n  intros σ Heq Hnnext,\n  -- simp, simp at Heq Hnnext,\n  -- unfold rank function.comp flip sch.step, -- next rank subtype.val,\n  cases Heq with Hmem Heq,\n  have HH := eq_next_or_rank_eq_or_rank_lt _ _ v Heq,\n  rw or_iff_not_imp at HH,\n  simp [not_or_iff_not_and_not] at Hnnext HH,\n  have HH' := HH Hnnext.right,\n  rw [or_iff_not_imp,not_and_iff_not_or_not,not_not_iff_self] at HH',\n  left, apply HH', clear HH' HH,\n  left, simp at Hmem, assumption\nend\n\nlemma stable_queue_ranking (l : lbl) (v : ℕ)\n: unless scheduler (rank l ≃ v) (rank l ≺ v ⋁ current ≃ l) :=\nbegin\n  unfold scheduler,\n  apply unless_step,\n  intros σ Heq Hnnext,\n  simp,\n  simp [not_or_iff_not_and_not] at Hnnext,\n  have := eq_next_or_rank_eq_or_rank_lt _ _ _ Heq, revert this,\n  simp, monotonicity, intro h,\n  cases h with h h,\n  { cases Hnnext.right h, },\n  { left, rw [h.right], },\nend\n\nlemma INIT\n: system.init scheduler (↑sch_state.target ≃ (t.s₀ : var sch_state t.σ)) :=\nbegin\n  simp [system.init,program.init,scheduler],\n  refl,\nend\n\nlemma STEP\n: co' scheduler\n    (λ (σ σ' : sch_state),\n       ∃ P, σ'.target = t.next (current.apply σ) σ.target P) :=\nbegin\n  unfold co',\n  intros σ σ',\n  dunfold step has_safety.step is_step,\n  intros H,\n  have P : (current t).apply σ ∈ (t.req).apply (σ.target),\n  { simp [current,req], exact σ.inv,  },\n  existsi P,\n  rw H,\n  unfold function.comp scheduler program.step subtype.val sch.step,\n  cases σ,\n  simp [sch.step,sch_state.target,current,req],\nend\n\nlemma INV\n: ∀ σ, σ ⊨ current ∊ t.req ∘' sch_state.target :=\nby { assume σ, simp [current,req], apply σ.inv, }\n\nlemma PROG (l : lbl)\n:     ↑l ∊ req  >~>  current ≃ l\n  in scheduler :=\nbegin\n  apply often_imp_often.induct ℕ _ _ ,\n  apply stable_queue_ranking _ l,\n  apply fair_schedule_step _ l,\nend\n\nend\n\nopen scheduling.unitb stream unitb\n\nvariable {lbl : Type}\nvariable [pos_finite lbl]\nvariable (r : target_mch lbl)\n\nnoncomputable def scheduler_spec\n: @scheduling.unitb.scheduler lbl r :=\n  { s := program $ @sch_state lbl _ r\n  , sem := _\n  , F := @scheduler _ _ _\n  , ch := current r\n  , object := _\n  , INIT := INIT r\n  , STEP := STEP r\n  , INV  := INV r\n  , PROG := PROG r }\n\nnoncomputable instance : unitb.system_sem ((scheduler_spec r).s) :=\n(scheduler_spec r).sem\n\nlemma sched' {lbl : Type} [s : finite lbl] [nonempty lbl]\n  (r : target_mch lbl)\n: ∃ τ : stream r.σ, fair r τ :=\nbegin\n  have h : pos_finite lbl,\n  { apply pos_of_finite ; apply_instance },\n  apply unitb.scheduling r (scheduling.finite.scheduler_spec r),\nend\n\nend scheduling.finite\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/scheduling/finite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702642896702, "lm_q2_score": 0.4765796510636759, "lm_q1q2_score": 0.3454584176216055}}
{"text": "import .expr_base\nimport ...phys.time.time\n\nnamespace lang.time\n\nuniverses u\nvariables \n  (K : Type u) [field K] [inhabited K] \n  {f : fm K TIME} {sp : spc K f} \n\n/-\nDuration\n-/\nstructure duration_var {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) extends var \n\n/-\nTime\n-/\nstructure time_var {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) extends var\n\nmutual inductive duration_expr, time_expr {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) \nwith duration_expr : Type u\n| lit (v : duration sp) : duration_expr\n| var (v : duration_var sp) : duration_expr\n| add_dur_dur (d1 : duration_expr) (d2 : duration_expr) : duration_expr\n| neg_dur (d : duration_expr) : duration_expr\n| sub_dur_dur (d1 : duration_expr) (d2 : duration_expr) : duration_expr\n| sub_time_time (t1 : time_expr) (t2 : time_expr) : duration_expr\n| smul_dur (k : K) (d : duration_expr) : duration_expr\nwith time_expr : Type u\n| lit (p : time sp) : time_expr\n| var (v : time_var sp) : time_expr\n| add_dur_time (d : duration_expr) (t : time_expr) : time_expr\n\n\nabbreviation duration_env {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) := \n  duration_var sp → duration sp\n\nabbreviation duration_eval {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f)  := \n  duration_env sp → duration_expr sp → duration sp\n\nabbreviation time_env {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f)  := \n  time_var sp → time sp\n\nabbreviation time_eval {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f)  := \n  time_env sp → time_expr sp → time sp\n\nstructure time_frame_var (K : Type u) [field K] [inhabited K] extends var \n\n\ninductive time_frame_expr (K : Type u) [field K] [inhabited K] : Type u --{f : fm K T}\n| lit (f : fm K TIME) : time_frame_expr\n| var (v : time_frame_var K) : time_frame_expr\n/-\nI have varying opinions on what exactly the parameters to the derived constructor should be.\n-/\n| derived {f : fm K TIME} {sp : spc K f} (o : time_expr sp) (b : duration_expr sp) : time_frame_expr\n\nstructure time_space_var {K : Type u} [field K] [inhabited K] (f : fm K TIME) extends var\n\ninductive time_space_expr {K : Type u} [field K] [inhabited K] (f : fm K TIME) : Type (u+1)\n| lit (sp : spc K f) : time_space_expr\n| var (v : time_space_var f) : time_space_expr\n| mk (f : time_frame_expr K) : time_space_expr\n\nabbreviation time_frame_env (K : Type u) [field K] [inhabited K]  :=\n  time_frame_var K → fm K TIME\nabbreviation time_frame_eval (K : Type u) [field K] [inhabited K]  :=\n  time_frame_env K → time_frame_expr K → fm K TIME\n\n/-You need to indicate explicitly what type of frame you think the space has, because the environment needs\na hint about the expected return type\nwhich does trivialize the point of the environment in a way, since parameterized spc types only \nhave a single value per frame-/\nabbreviation time_space_env {K : Type u} [field K] [inhabited K] (f : fm K TIME)  :=\n  time_space_var f → spc K f\nabbreviation time_space_eval {K : Type u} [field K] [inhabited K] (f : fm K TIME) :=\n  time_space_env f → time_space_expr f → spc K f\n\n\n\n--not working -- lean doesn't play nice with notation and dependent types\n--notation `[`flit`]` := time_frame_expr.lit flit\n--notation `[`slit`]` := time_space_expr.lit slit\n--instance {K : Type u} [field K] [inhabited K] {f : fm K TIME} {sp : spc K f} : has_coe (time sp) (time_expr sp) := ⟨λt, time_expr.lit t⟩\n--instance {K : Type u} [field K] [inhabited K] {f : fm K TIME} {sp : spc K f} : has_coe (duration sp) (duration_expr sp) := ⟨λt, duration_expr.lit t⟩\n--instance {K : Type u} [field K] [inhabited K] : has_coe (fm K TIME) (time_frame_expr K) := ⟨λf, time_frame_expr.lit f⟩\n--instance {K : Type u} [field K] [inhabited K] {f : fm K TIME} : has_coe (spc K f) (time_space_expr K) := ⟨λs, time_space_expr.lit s⟩\n\n#check has_add\n\nclass time_has_lit {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) := \n  (cast : time sp → time_expr sp)\nnotation `[`tlit`]` := time_has_lit.cast tlit\n\ninstance time_lit {K : Type u} [field K] [inhabited K] {f : fm K TIME} {sp : spc K f} : time_has_lit  sp := \n  ⟨λt : time sp, time_expr.lit t⟩\n\nclass duration_has_lit {K : Type u} [field K] [inhabited K] {f : fm K TIME} (sp : spc K f) := \n  (cast : duration sp → duration_expr sp)\nnotation `[`tlit`]` := duration_has_lit.cast tlit\n\ninstance duration_lit {K : Type u} [field K] [inhabited K] {f : fm K TIME} {sp : spc K f} : duration_has_lit  sp := \n  ⟨λt : duration sp, duration_expr.lit t⟩\n\nclass time_frame_has_lit (K : Type u) [field K] [inhabited K] := \n  (cast : fm K TIME → time_frame_expr K)\nnotation `[`flit`]` := time_frame_has_lit.cast flit\n\ninstance time_frame_lit {K : Type u} [field K] [inhabited K] : time_frame_has_lit K := \n  ⟨λf, time_frame_expr.lit f⟩\n\nclass time_space_has_lit {K : Type u} [field K] [inhabited K] (f : fm K TIME) := \n  (cast : spc K f → time_space_expr f)\nnotation `[`slit`]` := time_space_has_lit.cast slit\n\ninstance time_space_lit {K : Type u} [field K] [inhabited K] {f : fm K TIME} : time_space_has_lit f := \n  ⟨λs, time_space_expr.lit s⟩\n\n\n\n\n/-\nAnalogous methods provided at math layer\n-/\n#check mk_frame\ndef mk_time_frame_expr {f : fm K TIME} {sp : spc K f} (o : time_expr sp) (b : duration_expr sp) : time_frame_expr K :=\n  time_frame_expr.derived o b\n/-\n4/7\nWRITE THIS FUNCTION LATER. \nYOU NEED TO GET THE VALUE OUT OF THE f PARAMETER TO INCLUDE IT IN THE TYPE\nAND THEN USE IT IN THE CONSTRUCTOR\n-/\n#check mk_space \ndef mk_time_space_expr (K : Type u) [field K] [inhabited K] (f : time_frame_expr K) : time_space_expr K :=\n  time_space_expr.mk f\n\n\n\ndef add_dur_expr_dur_expr (v1 v2 : duration_expr sp) : duration_expr sp := \n  duration_expr.add_dur_dur v1 v2\n\ndef smul_dur_expr (k : K) (v : duration_expr sp) : duration_expr sp := \n    duration_expr.smul_dur k v\n\ndef neg_dur_expr (v : duration_expr sp) : duration_expr sp := \n    duration_expr.neg_dur v\n\ndef sub_dur_expr_dur_expr (v1 v2 : duration_expr sp) : duration_expr sp :=    -- v1-v2\n    duration_expr.sub_dur_dur v1 v2\n\n-- See unframed file for template for proving vector_space\n\ninstance has_add_dur_expr : has_add (duration_expr sp) := ⟨ add_dur_expr_dur_expr K ⟩\nlemma add_assoc_dur_expr : ∀ a b c : duration_expr sp, a + b + c = a + (b + c) := sorry\ninstance add_semigroup_dur_expr : add_semigroup (duration_expr sp) := ⟨ add_dur_expr_dur_expr K, add_assoc_dur_expr K⟩ \n\ndef dur_expr_zero  := duration_expr.lit (mk_duration sp 0)\ninstance has_zero_dur_expr : has_zero (duration_expr sp) := ⟨dur_expr_zero K⟩\n\nlemma zero_add_dur_expr : ∀ a : duration_expr sp, 0 + a = a := sorry\nlemma add_zero_dur_expr : ∀ a : duration_expr sp, a + 0 = a := sorry\ninstance add_monoid_dur_expr : add_monoid (duration_expr sp) := ⟨ \n    -- add_semigroup\n    add_dur_expr_dur_expr K, \n    add_assoc_dur_expr K, \n    -- has_zero\n    dur_expr_zero K,\n    -- new structure \n    @zero_add_dur_expr K _ _ f sp, \n    add_zero_dur_expr K\n⟩\n\ninstance has_neg_dur_expr : has_neg (duration_expr sp) := ⟨neg_dur_expr K⟩\ninstance has_sub_dur_expr : has_sub (duration_expr sp) := ⟨ sub_dur_expr_dur_expr K⟩ \nlemma sub_eq_add_neg_dur_expr : ∀ a b : duration_expr sp, a - b = a + -b := sorry\ninstance sub_neg_monoid_dur_expr : sub_neg_monoid (duration_expr sp) := ⟨ \n    add_dur_expr_dur_expr K, add_assoc_dur_expr K, dur_expr_zero K, \n    zero_add_dur_expr K, \n    add_zero_dur_expr K, -- add_monoid\n    neg_dur_expr K,                                                                  -- has_neg\n    sub_dur_expr_dur_expr K,                                                              -- has_sub\n    sub_eq_add_neg_dur_expr K,                                                       -- new\n⟩ \n\nlemma add_left_neg_dur_expr : ∀ a : duration_expr sp, -a + a = 0 := sorry\ninstance : add_group (duration_expr sp) := ⟨\n    -- sub_neg_monoid\n    add_dur_expr_dur_expr K, add_assoc_dur_expr K, dur_expr_zero K, zero_add_dur_expr K, add_zero_dur_expr K, -- add_monoid\n    neg_dur_expr K,                                                                  -- has_neg\n    sub_dur_expr_dur_expr K,                                                              -- has_sub\n    sub_eq_add_neg_dur_expr K, \n    -- new\n    add_left_neg_dur_expr K,\n⟩ \n\nlemma add_comm_dur_expr : ∀ a b : duration_expr sp, a + b = b + a := sorry\ninstance add_comm_semigroup_dur_expr : add_comm_semigroup (duration_expr sp) := ⟨\n    -- add_semigroup\n    add_dur_expr_dur_expr K, \n    add_assoc_dur_expr K,\n    add_comm_dur_expr K,\n⟩\n\ninstance add_comm_monoid_dur_expr : add_comm_monoid (duration_expr sp) := ⟨\n-- add_monoid\n    -- add_semigroup\n    add_dur_expr_dur_expr K, \n    add_assoc_dur_expr K, \n    -- has_zero\n    dur_expr_zero K,\n    -- new structure \n    zero_add_dur_expr K, \n    add_zero_dur_expr K,\n-- add_comm_semigroup (minus repeats)\n    add_comm_dur_expr K,\n⟩\n\ninstance has_scalar_dur_expr : has_scalar K (duration_expr sp) := ⟨\nsmul_dur_expr K,\n⟩\n\nlemma one_smul_dur_expr : ∀ b : duration_expr sp, (1 : K) • b = b := sorry\nlemma mul_smul_dur_expr : ∀ (x y : K) (b : duration_expr sp), (x * y) • b = x • y • b := sorry\ninstance mul_action_dur_expr : mul_action K (duration_expr sp) := ⟨\none_smul_dur_expr K,\nmul_smul_dur_expr K,\n⟩ \n\nlemma smul_add_dur_expr : ∀(r : K) (x y : duration_expr sp), r • (x + y) = r • x + r • y := sorry\nlemma smul_zero_dur_expr : ∀(r : K), r • (0 : duration_expr sp) = 0 := sorry\ninstance distrib_mul_action_K_dur_exprKx : distrib_mul_action K (duration_expr sp) := ⟨\nsmul_add_dur_expr K,\nsmul_zero_dur_expr K,\n⟩ \n\n-- renaming vs template due to clash with name \"s\" for prevailing variable\nlemma add_smul_dur_expr : ∀ (a b : K) (x : duration_expr sp), (a + b) • x = a • x + b • x := sorry\nlemma zero_smul_dur_expr : ∀ (x : duration_expr sp), (0 : K) • x = 0 := sorry\ninstance semimodule_K_durationK : semimodule K (duration_expr sp) := ⟨ add_smul_dur_expr K, zero_smul_dur_expr  K⟩ \n\ninstance add_comm_group_dur_expr : add_comm_group (duration_expr sp) := ⟨\n-- add_group\n    add_dur_expr_dur_expr K, add_assoc_dur_expr K, dur_expr_zero K, zero_add_dur_expr K, add_zero_dur_expr K, -- add_monoid\n    neg_dur_expr K,                                                                  -- has_neg\n    sub_dur_expr_dur_expr K,                                                              -- has_sub\n    sub_eq_add_neg_dur_expr K, \n    add_left_neg_dur_expr K,\n-- commutativity\n    add_comm_dur_expr K,\n⟩\n\n\ninstance : vector_space K (duration_expr sp) := sorry\n\n\n/-\n    ********************\n    *** Affine space ***\n    ********************\n-/\n\n\n/-\nAffine operations\n-/\ninstance : has_add (duration_expr sp) := ⟨add_dur_expr_dur_expr K⟩\ninstance : has_zero (duration_expr sp) := ⟨dur_expr_zero K⟩\ninstance : has_neg (duration_expr sp) := ⟨neg_dur_expr K⟩\n\n/-\nLemmas needed to implement affine space API\n-/\n\ndef sub_time_expr_time_expr {f : fm K TIME} {sp : spc K f } (p1 p2 : time_expr sp) : duration_expr sp := \n    duration_expr.sub_time_time p1 p2\ndef add_time_expr_dur_expr {f : fm K TIME} {sp : spc K f } (p : time_expr sp) (v : duration_expr sp) : time_expr sp := \n    time_expr.add_dur_time v p\ndef add_dur_expr_time_expr {f : fm K TIME} {sp : spc K f } (v : duration_expr sp) (p : time_expr sp) : time_expr sp := \n    time_expr.add_dur_time v p\n\ndef aff_dur_expr_group_action : duration_expr sp → time_expr sp → time_expr sp := add_dur_expr_time_expr K\ninstance : has_vadd (duration_expr sp) (time_expr sp) := ⟨aff_dur_expr_group_action K⟩\n\nlemma zero_dur_expr_vadd'_a1 : ∀ p : time_expr sp, (0 : duration_expr sp) +ᵥ p = p := sorry\nlemma dur_expr_add_assoc'_a1 : ∀ (g1 g2 : duration_expr sp) (p : time_expr sp), g1 +ᵥ (g2 +ᵥ p) = (g1 + g2) +ᵥ p := sorry\ninstance dur_expr_add_action: add_action (duration_expr sp) (time_expr sp) := \n⟨ aff_dur_expr_group_action K, zero_dur_expr_vadd'_a1 K, dur_expr_add_assoc'_a1  K⟩ \n\ndef aff_time_expr_group_sub : time_expr sp → time_expr sp → duration_expr sp := sub_time_expr_time_expr K\ninstance time_expr_has_vsub : has_vsub (duration_expr sp) (time_expr sp) := ⟨ aff_time_expr_group_sub K ⟩ \n\n\ninstance : nonempty (time_expr sp) := ⟨time_expr.lit (mk_time sp  0)⟩\n\nlemma time_expr_vsub_vadd_a1 : ∀ (p1 p2 : (time_expr sp)), (p1 -ᵥ p2) +ᵥ p2 = p1 := sorry\nlemma time_expr_vadd_vsub_a1 : ∀ (g : duration_expr sp) (p : time_expr sp), g +ᵥ p -ᵥ p = g := sorry\ninstance aff_time_expr_torsor : add_torsor (duration_expr sp) (time_expr sp) := --affine space! \n⟨ \n    aff_dur_expr_group_action K,\n    zero_dur_expr_vadd'_a1 K,    -- add_action\n    dur_expr_add_assoc'_a1 K,   -- add_action\n    aff_time_expr_group_sub K,    -- has_vsub\n    time_expr_vsub_vadd_a1 K,     -- add_torsor\n    time_expr_vadd_vsub_a1 K,     -- add_torsor\n⟩\n\n\n/-\n+  : d s -> d s -> d s\n•  : K -> d s -> d s\n+ᵥ : d s -> t s -> t s \n-ᵥ : t s -> t s -> d s\n\nHere s is an affine coordinate \nspace on TIME. Otherwise we've\ngot time points and durations,\nwithin, but not across, spaces.\n-/\n\n\n/-\nTransform\n-/\nstructure transform_var {K : Type u} [field K] [inhabited K] \n  {f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2 : spc K f2) extends var\n/-\ninductive transform_expr {K : Type u} [field K] [inhabited K] \n  --{f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2:=sp1 : spc K f2) \n -- (sp1 : Σf1 : fm K TIME, spc K f1)  (sp2 : Σf2 : fm K TIME, spc K f2 := sp1)\n  : Π {f1 : fm K TIME} (sp1 : spc K f1), Π {f2 : fm K TIME} (sp2 : spc K f2), Type u\n| lit {f1 : fm K TIME} (sp1 : spc K f1) {f2 : fm K TIME} (sp2 : spc K f2) (p : time_transform sp1 sp2) : transform_expr sp1 sp2\n| var (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_var sp1.2 sp2.2) : transform_expr sp1 sp2\n| apply_duration (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_expr sp1 sp2) (d : duration_expr sp1.2) : transform_expr sp1 sp2\n| compose (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2)  (v : time_transform sp1.2 sp2.2) \n  (sp3 : Σf3 : fm K TIME, spc K f3)  (v : time_transform sp1 sp2) : transform_expr sp1 sp3\n-/\n/-\ninvalid occurrence of recursive arg#7 of 'lang.time.transform_expr.compose', the body of the functional type depends on it.\nAll Messages (28)\n\nPROBLEM WITH TRANSFORM EXPRESSIONS\n-/\n/-\ninductive transform_expr' {K : Type u} [field K] [inhabited K] \n  --{f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2:=sp1 : spc K f2) \n -- (sp1 : Σf1 : fm K TIME, spc K f1)  (sp2 : Σf2 : fm K TIME, spc K f2 := sp1)\n  : Π {f1 : fm K TIME} (sp1 : spc K f1), Π {f2 : fm K TIME} (sp2 : spc K f2), Type u\n| lit {f1 : fm K TIME} (sp1 : spc K f1) {f2 : fm K TIME} (sp2 : spc K f2) (p : time_transform sp1 sp2) \n    : transform_expr' sp1 sp2\n| var (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_var sp1.2 sp2.2) \n    : transform_expr' sp1.2 sp2.2\n| apply_duration (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_expr' sp1.2 sp2.2) (d : duration_expr sp1.2) \n    : transform_expr' sp1.2 sp2.2\n| compose (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2)  (v : transform_expr' sp1.2 sp2.2) \n  (sp3 : Σf3 : fm K TIME, spc K f3)  (v : transform_expr' sp2.2 sp3.2) \n    : transform_expr' sp1.2 sp3.2\n-/\n\n\ninductive transform_expr {K : Type u} [field K] [inhabited K] \n  --{f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2:=sp1 : spc K f2) \n -- (sp1 : Σf1 : fm K TIME, spc K f1)  (sp2 : Σf2 : fm K TIME, spc K f2 := sp1)\n  : Π {f1 : fm K TIME} (sp1 : spc K f1), Π {f2 : fm K TIME} (sp2 : spc K f2), Type u\n| lit {f1 : fm K TIME} {sp1 : spc K f1} {f2 : fm K TIME} {sp2 : spc K f2} (p : time_transform sp1 sp2) : transform_expr sp1 sp2\n| var {f1 : fm K TIME} {sp1 : spc K f1} {f2 : fm K TIME} {sp2 : spc K f2} (v : transform_var sp1 sp2) : transform_expr sp1 sp2\n| apply_duration {f1 : fm K TIME} {sp1 : spc K f1} {f2 : fm K TIME} {sp2 : spc K f2} (v : transform_expr sp1 sp2) (d : duration_expr sp1) : transform_expr sp1 sp2\n| compose_lit {f1 : fm K TIME} {sp1 : spc K f1} {f2 : fm K TIME} {sp2 : spc K f2} (v : time_transform sp1 sp2) \n  {f3 : fm K TIME} {sp3 : spc K f3}  (v : time_transform sp2 sp3) : transform_expr sp1 sp3\n\n\nclass time_transform_has_lit {K : Type u} [field K] [inhabited K] \n  {f1 : fm K TIME} (sp1 : spc K f1) {f2 : fm K TIME} (sp2 : spc K f2) := \n  (cast : time_transform sp1 sp2 → transform_expr sp1 sp2)\nnotation `[`tlit`]` := time_transform_has_lit.cast tlit\n\ninstance time_transform_lit {K : Type u} [field K] [inhabited K] \n  {f1 : fm K TIME} {sp1 : spc K f1} {f2 : fm K TIME} {sp2 : spc K f2} : time_transform_has_lit sp1 sp2 := \n  ⟨λt, transform_expr.lit t⟩\n\n\n/-\n\ninductive transform_expr {K : Type u} [field K] [inhabited K] \n  --{f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2:=sp1 : spc K f2) \n -- (sp1 : Σf1 : fm K TIME, spc K f1)  (sp2 : Σf2 : fm K TIME, spc K f2 := sp1)\n  : Σf1 : fm K TIME, spc K f1 → Σf2 : fm K TIME, spc K f2 → Type u\n| lit (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (p : time_transform sp1.1 sp1.2) : transform_expr\n--| var (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_var sp1.1 sp1.2) : transform_expr sp1 sp2\n--| apply_duration (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2) (v : transform_expr ) (d : duration_expr sp) : transform_expr sp1 sp2\n--| compose (sp1 : Σf1 : fm K TIME, spc K f1) (sp2 : Σf2 : fm K TIME, spc K f2)  (v : transform_expr sp1 sp2) \n -- (sp3 : Σf3 : fm K TIME, spc K f3)  (v : transform_expr sp2 sp3) : transform_expr sp1 sp3\n\n\n-/\n\nabbreviation transform_env {K : Type u} [field K] [inhabited K] \n  {f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2 : spc K f2)  := \n  transform_var sp1 sp2 → time_transform sp1 sp2\n\nabbreviation transform_eval  {K : Type u} [field K] [inhabited K] \n  {f1 : fm K TIME} {f2 : fm K TIME} (sp1 : spc K f1) (sp2 : spc K f2) := \n  transform_env sp1 sp2 → transform_expr sp1 sp2 → time_transform sp1 sp2\n\nvariables {f2 : fm K TIME} (sp2 : spc K f2)\n\nstructure env {K : Type u} [field K] [inhabited K] \n  --      {f : fm K TIME} (sp : spc K f) {f2 : fm K TIME} {sp2 : spc K f2} :=\n  :=\n  (duration : Π {f : fm K TIME}, Π (sp : spc K f), duration_env sp )\n  (time : Π {f : fm K TIME}, Π (sp : spc K f), time_env sp )\n  (transform : Π {f1 : fm K TIME}, Π (sp1 : spc K f1), Π {f2 : fm K TIME}, Π (sp2 : spc K f2), transform_env sp1 sp2)\n  (frame : time_frame_env K)\n  (space : Π (f : fm K TIME), time_space_env f)\n\ndef env.init (K : Type u) [field K] [inhabited K]  : env :=\n  ⟨\n    (λf: fm K TIME, λsp, λv, ⟨mk_vectr sp 1⟩),\n    (λf: fm K TIME, λsp, λv, ⟨mk_point sp 0⟩),\n    (λf: fm K TIME, λsp1, λf2, λsp2, (λv, sp1.time_tr sp2)),\n    (λv, time_std_frame K),\n    (λf, (λv, mk_space K f))\n  ⟩\n\nstructure eval {K : Type u} [field K] [inhabited K] :=\n  (duration : Π {f : fm K TIME}, Π (sp : spc K f), duration_eval sp )\n  (time : Π {f : fm K TIME}, Π (sp : spc K f), time_eval sp )\n  (transform : Π {f1 : fm K TIME}, Π (sp1 : spc K f1), Π {f2 : fm K TIME}, Π (sp2 : spc K f2), transform_eval sp1 sp2)\n  (frame : time_frame_eval K)\n  (space : Π (f : fm K TIME), time_space_eval f)\n\ndef eval.init (K : Type u) [field K] [inhabited K] : eval := \n  ⟨ \n    (λf: fm K TIME, λsp, λenv_,λexpr_, ⟨mk_vectr sp 1⟩),\n    (λf: fm K TIME, λsp, λenv_,λexpr_, ⟨mk_point sp 0⟩),\n    (λf: fm K TIME, λsp1, λf2, λsp2, (λenv_,λexpr_, sp1.time_tr sp2 : transform_eval sp1 sp2)),\n    (λenv_, λexpr_, time_std_frame K),\n    (λf, λenv_, λexpr_, mk_space K f)\n  ⟩\nend lang.time\n", "meta": {"author": "kevinsullivan", "repo": "lang", "sha": "e9d869bff94fb13ad9262222a6f3c4aafba82d5e", "save_path": "github-repos/lean/kevinsullivan-lang", "path": "github-repos/lean/kevinsullivan-lang/lang-e9d869bff94fb13ad9262222a6f3c4aafba82d5e/old/time_expr_current_old_4-8.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947425132315, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.34533681880163414}}
{"text": "/-\nCopyright (c) 2022 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-! ## Boolean equality -/\n\n/-- `PartialEquivBEq α` says that the `BEq` implementation is a\npartial equivalence relation, that is:\n* it is symmetric: `a == b → b == a`\n* it is transitive: `a == b → b == c → a == c`.\n-/\nclass PartialEquivBEq (α) [BEq α] : Prop where\n  /-- Symmetry for `BEq`. If `a == b` then `b == a`. -/\n  symm : (a : α) == b → b == a\n  /-- Transitivity for `BEq`. If `a == b` and `b == c` then `a == c`. -/\n  trans : (a : α) == b → b == c → a == c\n\n@[simp] theorem beq_eq_false_iff_ne [BEq α] [LawfulBEq α]\n    (a b : α) : (a == b) = false ↔ a ≠ b := by\n  rw [ne_eq, ← beq_iff_eq a b]\n  cases a == b <;> decide\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Classes/BEq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947425132315, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3453368188016341}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.functor.hom\nimport category_theory.functor.currying\nimport category_theory.products.basic\n\n/-!\n# The Yoneda embedding\n\nThe Yoneda embedding as a functor `yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) ≅ (yoneda_evaluation C)`.\n\n## References\n* [Stacks: Opposite Categories and the Yoneda Lemma](https://stacks.math.columbia.edu/tag/001L)\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v₁ u₁ u₂-- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u₁} [category.{v₁} C]\n\n/--\nThe Yoneda embedding, as a functor from `C` into presheaves on `C`.\n\nSee <https://stacks.math.columbia.edu/tag/001O>.\n-/\n@[simps]\ndef yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop Y ⟶ X,\n    map := λ Y Y' f g, f.unop ≫ g,\n    map_comp' := λ _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := λ Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := λ X X' f, { app := λ Y g, g ≫ f } }\n\n/--\nThe co-Yoneda embedding, as a functor from `Cᵒᵖ` into co-presheaves on `C`.\n-/\n@[simps] def coyoneda : Cᵒᵖ ⥤ (C ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop X ⟶ Y,\n    map := λ Y Y' f g, g ≫ f },\n  map := λ X X' f, { app := λ Y g, f.unop ≫ g } }\n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X ⟶ op Y) :\n  (yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) :=\nby { dsimp, simp }\n\n@[simp] lemma naturality {X Y : C} (α : yoneda.obj X ⟶ yoneda.obj Y)\n  {Z Z' : C} (f : Z ⟶ Z') (h : Z' ⟶ X) : f ≫ α.app (op Z') h = α.app (op Z) (f ≫ h) :=\n(functor_to_types.naturality _ _ α f.op h).symm\n\n/--\nThe Yoneda embedding is full.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ninstance yoneda_full : full (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) :=\n{ preimage := λ X Y f, f.app (op X) (𝟙 X) }\n\n/--\nThe Yoneda embedding is faithful.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ninstance yoneda_faithful : faithful (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) :=\n{ map_injective' := λ X Y f g p, by convert (congr_fun (congr_app p (op X)) (𝟙 X)); dsimp; simp }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X ≅ Y`\napply yoneda.ext,\n-- Goals are now functions `(Z ⟶ X) → (Z ⟶ Y)`, `(Z ⟶ Y) → (Z ⟶ X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : Π {Z : C}, (Z ⟶ X) → (Z ⟶ Y)) (q : Π {Z : C}, (Z ⟶ Y) → (Z ⟶ X))\n  (h₁ : Π {Z : C} (f : Z ⟶ X), q (p f) = f) (h₂ : Π {Z : C} (f : Z ⟶ Y), p (q f) = f)\n  (n : Π {Z Z' : C} (f : Z' ⟶ Z) (g : Z ⟶ X), p (f ≫ g) = f ≫ p g) : X ≅ Y :=\nyoneda.preimage_iso (nat_iso.of_components (λ Z, { hom := p, inv := q, }) (by tidy))\n\n/--\nIf `yoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C} (f : X ⟶ Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : Cᵒᵖ} (α : coyoneda.obj X ⟶ coyoneda.obj Y)\n  {Z Z' : C} (f : Z' ⟶ Z) (h : unop X ⟶ Z') : (α.app Z' h) ≫ f = α.app Z (h ≫ f) :=\n(functor_to_types.naturality _ _ α f h).symm\n\ninstance coyoneda_full : full (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) :=\n{ preimage := λ X Y f, (f.app _ (𝟙 X.unop)).op }\n\ninstance coyoneda_faithful : faithful (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) :=\n{ map_injective' := λ X Y f g p,\n  begin\n    have t := congr_fun (congr_app p X.unop) (𝟙 _),\n    simpa using congr_arg quiver.hom.op t,\n  end }\n\n/--\nIf `coyoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\n/-- The identity functor on `Type` is isomorphic to the coyoneda functor coming from `punit`. -/\ndef punit_iso : coyoneda.obj (opposite.op punit) ≅ 𝟭 (Type v₁) :=\nnat_iso.of_components\n  (λ X, { hom := λ f, f ⟨⟩, inv := λ x _, x })\n  (by tidy)\n\nend coyoneda\n\nnamespace functor\n\n\n/--\nA functor `F : Cᵒᵖ ⥤ Type v₁` is representable if there is object `X` so `F ≅ yoneda.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001Q>.\n-/\nclass representable (F : Cᵒᵖ ⥤ Type v₁) : Prop :=\n(has_representation : ∃ X (f : yoneda.obj X ⟶ F), is_iso f)\n\ninstance {X : C} : representable (yoneda.obj X) :=\n{ has_representation := ⟨X, 𝟙 _, infer_instance⟩ }\n\n/--\nA functor `F : C ⥤ Type v₁` is corepresentable if there is object `X` so `F ≅ coyoneda.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001Q>.\n-/\nclass corepresentable (F : C ⥤ Type v₁) : Prop :=\n(has_corepresentation : ∃ X (f : coyoneda.obj X ⟶ F), is_iso f)\n\ninstance {X : Cᵒᵖ} : corepresentable (coyoneda.obj X) :=\n{ has_corepresentation := ⟨X, 𝟙 _, infer_instance⟩ }\n\n-- instance : corepresentable (𝟭 (Type v₁)) :=\n-- corepresentable_of_nat_iso (op punit) coyoneda.punit_iso\n\nsection representable\nvariables (F : Cᵒᵖ ⥤ Type v₁)\nvariable [F.representable]\n\n/-- The representing object for the representable functor `F`. -/\nnoncomputable def repr_X : C :=\n(representable.has_representation : ∃ X (f : _ ⟶ F), _).some\n\n/-- The (forward direction of the) isomorphism witnessing `F` is representable. -/\nnoncomputable def repr_f : yoneda.obj F.repr_X ⟶ F :=\nrepresentable.has_representation.some_spec.some\n\n/--\nThe representing element for the representable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def repr_x : F.obj (op F.repr_X) :=\nF.repr_f.app (op F.repr_X) (𝟙 F.repr_X)\n\ninstance : is_iso F.repr_f :=\nrepresentable.has_representation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(-, F.repr_X)`.  Note the components\n`F.repr_w.app X` definitionally have type `(X.unop ⟶ F.repr_X) ≅ F.obj X`.\n-/\nnoncomputable def repr_w : yoneda.obj F.repr_X ≅ F := as_iso F.repr_f\n\n@[simp] lemma repr_w_hom : F.repr_w.hom = F.repr_f := rfl\n\nlemma repr_w_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.repr_X) :\n  (F.repr_w.app X).hom f = F.map f.op F.repr_x :=\nbegin\n  change F.repr_f.app X f = (F.repr_f.app (op F.repr_X) ≫ F.map f.op) (𝟙 F.repr_X),\n  rw ←F.repr_f.naturality,\n  dsimp,\n  simp\nend\n\nend representable\n\nsection corepresentable\n\nvariables (F : C ⥤ Type v₁)\nvariable [F.corepresentable]\n\n/-- The representing object for the corepresentable functor `F`. -/\nnoncomputable def corepr_X : C :=\n(corepresentable.has_corepresentation : ∃ X (f : _ ⟶ F), _).some.unop\n\n/-- The (forward direction of the) isomorphism witnessing `F` is corepresentable. -/\nnoncomputable def corepr_f : coyoneda.obj (op F.corepr_X) ⟶ F :=\ncorepresentable.has_corepresentation.some_spec.some\n\n/--\nThe representing element for the corepresentable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def corepr_x : F.obj F.corepr_X :=\nF.corepr_f.app F.corepr_X (𝟙 F.corepr_X)\n\ninstance : is_iso F.corepr_f :=\ncorepresentable.has_corepresentation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(F.corepr X, -)`. Note the components\n`F.corepr_w.app X` definitionally have type `F.corepr_X ⟶ X ≅ F.obj X`.\n-/\nnoncomputable def corepr_w : coyoneda.obj (op F.corepr_X) ≅ F := as_iso F.corepr_f\n\nlemma corepr_w_app_hom (X : C) (f : F.corepr_X ⟶ X) :\n  (F.corepr_w.app X).hom f = F.map f F.corepr_x :=\nbegin\n  change F.corepr_f.app X f = (F.corepr_f.app F.corepr_X ≫ F.map f) (𝟙 F.corepr_X),\n  rw ←F.corepr_f.naturality,\n  dsimp,\n  simp\nend\n\nend corepresentable\n\nend functor\n\nlemma representable_of_nat_iso (F : Cᵒᵖ ⥤ Type v₁) {G} (i : F ≅ G) [F.representable] :\n  G.representable :=\n{ has_representation := ⟨F.repr_X, F.repr_f ≫ i.hom, infer_instance⟩ }\n\nlemma corepresentable_of_nat_iso (F : C ⥤ Type v₁) {G} (i : F ≅ G) [F.corepresentable] :\n  G.corepresentable :=\n{ has_corepresentation := ⟨op F.corepr_X, F.corepr_f ≫ i.hom, infer_instance⟩ }\n\ninstance : functor.corepresentable (𝟭 (Type v₁)) :=\ncorepresentable_of_nat_iso (coyoneda.obj (op punit)) coyoneda.punit_iso\n\nopen opposite\n\nvariables (C)\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((Cᵒᵖ ⥤ Type v₁) × Cᵒᵖ) :=\ncategory_theory.prod.{(max u₁ v₁) v₁} (Cᵒᵖ ⥤ Type v₁) Cᵒᵖ\n\ninstance prod_category_instance_2 : category (Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) :=\ncategory_theory.prod.{v₁ (max u₁ v₁)} Cᵒᵖ (Cᵒᵖ ⥤ Type v₁)\n\nopen yoneda\n\n/--\nThe \"Yoneda evaluation\" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`\nto `F.obj X`, functorially in both `X` and `F`.\n-/\ndef yoneda_evaluation : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nevaluation_uncurried Cᵒᵖ (Type v₁) ⋙ ulift_functor.{u₁}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down) := rfl\n\n/--\nThe \"Yoneda pairing\" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`\nto `yoneda.op.obj X ⟶ F`, functorially in both `X` and `F`.\n-/\ndef yoneda_pairing : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nfunctor.prod yoneda.op (𝟭 (Cᵒᵖ ⥤ Type v₁)) ⋙ functor.hom (Cᵒᵖ ⥤ Type v₁)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (β : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map α β = yoneda.map α.1.unop ≫ β ≫ α.2 := rfl\n\n/--\nThe Yoneda lemma asserts that that the Yoneda pairing\n`(X : Cᵒᵖ, F : Cᵒᵖ ⥤ Type) ↦ (yoneda.obj (unop X) ⟶ F)`\nis naturally isomorphic to the evaluation `(X, F) ↦ F.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ndef yoneda_lemma : yoneda_pairing C ≅ yoneda_evaluation C :=\n{ hom :=\n  { app := λ F x, ulift.up ((x.app F.1) (𝟙 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp, ←functor_to_types.naturality],\n      simp only [category.comp_id, yoneda_obj_map],\n    end },\n  inv :=\n  { app := λ F x,\n    { app := λ X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp_apply]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [←functor_to_types.naturality, functor_to_types.map_comp_apply]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [←functor_to_types.naturality,\n         obj_map_id],\n    simp only [yoneda_map_app, quiver.hom.unop_op],\n    erw [category.id_comp],\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id_apply]\n  end }.\n\nvariables {C}\n\n/--\nThe isomorphism between `yoneda.obj X ⟶ F` and `F.obj (op X)`\n(we need to insert a `ulift` to get the universes right!)\ngiven by the Yoneda lemma.\n-/\n@[simps] def yoneda_sections (X : C) (F : Cᵒᵖ ⥤ Type v₁) :\n  (yoneda.obj X ⟶ F) ≅ ulift.{u₁} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\n/--\nWe have a type-level equivalence between natural transformations from the yoneda embedding\nand elements of `F.obj X`, without any universe switching.\n-/\ndef yoneda_equiv {X : C} {F : Cᵒᵖ ⥤ Type v₁} : (yoneda.obj X ⟶ F) ≃ F.obj (op X) :=\n(yoneda_sections X F).to_equiv.trans equiv.ulift\n\n@[simp]\nlemma yoneda_equiv_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) :\n  yoneda_equiv f = f.app (op X) (𝟙 X) :=\nrfl\n\n@[simp]\nlemma yoneda_equiv_symm_app_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (x : F.obj (op X))\n  (Y : Cᵒᵖ) (f : Y.unop ⟶ X) :\n  (yoneda_equiv.symm x).app Y f = F.map f.op x :=\nrfl\n\nlemma yoneda_equiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) (g : Y ⟶ X) :\n  F.map g.op (yoneda_equiv f) = yoneda_equiv (yoneda.map g ≫ f) :=\nbegin\n  change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g),\n  rw ←f.naturality,\n  dsimp,\n  simp,\nend\n\n/--\nWhen `C` is a small category, we can restate the isomorphism from `yoneda_sections`\nwithout having to change universes.\n-/\ndef yoneda_sections_small {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) :\n  (yoneda.obj X ⟶ F) ≅ F.obj (op X) :=\nyoneda_sections X F ≪≫ ulift_trivial _\n\n@[simp]\nlemma yoneda_sections_small_hom {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) (f : yoneda.obj X ⟶ F) :\n  (yoneda_sections_small X F).hom f = f.app _ (𝟙 _) :=\nrfl\n\n@[simp]\nlemma yoneda_sections_small_inv_app_apply {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) (t : F.obj (op X)) (Y : Cᵒᵖ) (f : Y.unop ⟶ X) :\n  ((yoneda_sections_small X F).inv t).app Y f = F.map f.op t :=\nrfl\n\nlocal attribute [ext] functor.ext\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma {C : Type u₁} [small_category C] :\n  (yoneda.op ⋙ coyoneda : Cᵒᵖ ⥤ (Cᵒᵖ ⥤ Type u₁) ⥤ Type u₁) ≅ evaluation Cᵒᵖ (Type u₁) :=\neq_to_iso (by tidy) ≪≫ curry.map_iso (yoneda_lemma C ≪≫\n  iso_whisker_left (evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial) ≪≫\n    eq_to_iso (by tidy)\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma' {C : Type u₁} [small_category C] :\n  yoneda ⋙ (whiskering_left Cᵒᵖ (Cᵒᵖ ⥤ Type u₁)ᵒᵖ (Type u₁)).obj yoneda.op ≅ 𝟭 (Cᵒᵖ ⥤ Type u₁) :=\neq_to_iso (by tidy) ≪≫ curry.map_iso (iso_whisker_left (prod.swap _ _)\n  (yoneda_lemma C ≪≫ iso_whisker_left\n    (evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial : _)) ≪≫ eq_to_iso (by tidy)\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3453368116980983}}
{"text": "import data.real.basic\nimport data.set\nimport tactic\n-- import exercices_espaces_metriques\nopen push_neg\n\n\nnamespace tactic.interactive\nopen lean.parser tactic interactive \nopen interactive (loc.ns)\nopen interactive.types\nopen tactic expr\nlocal postfix *:9001 := many -- sinon ne comprends pas ident*\n\n/- décompose en premier caractère, reste  INUTILISE-/\ndef un_car : string → string × string\n| ⟨(x :: xs)⟩  := ( ⟨ [x] ⟩ , ⟨ xs ⟩ )\n| _ := (\"\",\"\")\n\ndef deux_car : string → string\n| ⟨(x ::  y :: xs)⟩  := ⟨ [x,y] ⟩ \n| _ := \"\"\n\ndef trois_car : string → string\n| ⟨(x ::  y :: z ::xs)⟩  := ⟨ [x,y,z] ⟩ \n| _ := \"\"\n\n/- décompose une chaine de caractères selon la première parenthèse ouvrante\nLe premier terme ne sert qu'à la récursivité  INUTILISE-/\nmeta def debut_chaine : string × string → string × string\n| (s , t ) := do\n    let d := un_car t,\n    match d  with\n        | (\"(\",reste) :=  (s,t)\n        | ( ⟨(x)⟩, reste) :=  (debut_chaine (s ++ d.1, reste )) \n        -- | _ := (\"ERREUR\", \"\")\n        end\n\n-- set_option trace.eqn_compiler.elim_match true\n-- ne fonctionne pas : le \"Prop\" est ignoré, ou bien tout est Prop ??\n-- meta def is_prop : expr →  bool\n--     | `(%%e : Prop) := tt\n   --  | _             := ff\n\n\n-- détermine si l'expression est une propriété\n-- basé sur le fait (peut-être optimiste) que si on échoue à trouver le type, \n-- c'est qu'il y a des variables libres,\n-- et donc que c'est une propriété\nmeta def is_prop : expr →  tactic bool\n| e := do {\n    expr_t ←  infer_type e,\n    -- expr_tt ← infer_type expr_t,\n    if expr_t = `(Prop) then return tt else return ff\n        } <|> return tt\n\n-- test if expr is semantically an implication or a function \n-- (as opposed to a \"∀\" expression )\nmeta def is_arrow' : expr → tactic bool\n| `(%%P → %%Q) := if has_var_idx Q 0 then return ff else return tt\n| _ := return ff\n\n\nmeta def instanciate (e : expr) : tactic expr :=\nmatch e with\n| (pi pp_name binder type body) := do \n    a ← mk_local' pp_name binder type,\n    return $ instantiate_var body a\n| (lam pp_name binder type body) := do \n    a ← mk_local' pp_name binder type,\n    return $ instantiate_var body a\n| _ := return e\nend\n\nopen tactic\nexample : true :=\nby do e ← to_expr ```(∀ y : ℕ, ∀ x : ℕ, y = x), (es, tgt) ← mk_local_pis e, trace e, trace es, trace tgt\n\n\n\n\n\n\n/- Décompose la racine d'une expression (un seul pas) \n LOGICS : ET, OU, SSI, QUELQUESOIT, IMPLIQUE, FONCTION, NON, EXISTE,\nSETS: INTER, UNION, INCLUS, APPARTIENT, COMPLEMENTAIRE1,s IMAGE_ENSEMBLE, IMAGE_RECIPROQUE, \nEGALITE, ENSEMBLE1, APPLICATION\nNUMBERS: -/\nprivate meta def analyse_expr_step  (e : expr) : tactic (string × (list expr)) := \ndo  S ←  (tactic.pp e), let e_joli := to_string S, \nmatch e with\n| (lam name binder type body)          := return (\"lambda[\" ++ to_string name ++ \"]\", [type,body]) -- name → binder_info → expr → expr → expr\n------------------------- LOGIQUE -------------------------\n| `(%%p ∧ %%q) := return (\"PROP_AND\", [p,q])\n| `(%%p ∨ %%q) := return (\"PROP_OR\", [p,q])\n| `(%%p ↔ %%q) := return (\"PROP_IFF\", [p,q])\n| `(¬ %%p) := return (\"PROP_NOT\", [p])\n| `(%%p → false)  := return (\"PROP_NOT\", [p])\n| (pi name binder type body) := do bool ←  (is_arrow' e),\n    if bool then do bool2 ← tactic.is_prop e,\n                    if bool2 then return (\"PROP_IMPLIES\", [type,body])\n                        else return (\"FUNCTION\", [type,body]) \n     else do inst_body ← instanciate e,\n       return (\"QUANT_∀[\" ++ to_string name ++ \"]\", [type, inst_body]) \n| `(Exists %%p) := do match p with          --  améliorer : cas d'une prop, mais attention aux variables !!\n    | (lam name binder type body) := \n    -- la suite teste s'il s'agit de l'existence d'un objet ou d'une propriété\n        -- d'abord, si `body` contient des variables libres, c'est une propriété\n        -- if type.has_var then return (\"EXISTE[PROP:\" ++ to_string name ++ \"]\", [type,body])\n        -- si ce n'est pas le cas, on peut chercher son type, et voir si c'est Prop\n        -- else do type_type ← infer_type type,\n            -- if type_type = `(Prop) \n            do inst_body ← instanciate p,\n                bool ← is_prop p, if bool\n                then return (\"PROP_∃[\" ++ to_string name ++ \"]\", [type,inst_body])\n                else return (\"QUANT_∃[\" ++ to_string name ++ \"]\", [type,inst_body])\n    |  _ := return (\"ERROR\", [])\n    end \n------------------------- THEORIE DES ENSEMBLES -------------------------\n| `(%%A ∩ %%B) := return (\"SET_INTER\", [A,B])\n| `(%%A ∪ %%B) := return (\"SET_UNION\", [A,B])\n| `(set.compl %%A) := return (\"SET_COMPLEMENT\", [A])\n| `(%%A \\ %%B) := return (\"SET_SYM_DIFF\", [A,B])\n| `(%%A ⊆ %%B) := return (\"PROP_INCLUDED\", [A,B])\n| `(%%a ∈ %%A) := return (\"PROP_BELONGS\", [a,A])\n| `(@set.univ %%X) := return (\"SET_UNIVERSE\", [X])\n| `(-%%A) := return (\"MINUS\", [A])   \n| `(set.Union %%A) := return (\"SET_UNION+\", [A])\n| `(set.Inter %%A) := return (\"SET_INTER+\", [A])\n| `(%%f '' %%A) := return (\"SET_IMAGE\", [f,A])\n| `(%%f  ⁻¹' %%A) := return (\"SET_INVERSE\", [f,A])\n| `(∅) := return (\"SET_EMPTY\", [])\n| `(_root_.set %%X) := return (\"SET\", [X])\n-- polymorphe\n| `(%%a = %%b) := return (\"PROP_EQUAL\", [a,b]) -- faudrait connaitre le type ?\n| `(%%a ≠ %%b) := return (\"PROP_EQUAL_NOT\", [a,b]) -- faudrait connaitre le type ?\n----------- TOPOLOGY --------------\n-- | `(B(%%x, %%r))\n\n\n---------------------------- NOMBRES particuliers (cf aussi plus bas) \n| `(0:ℝ) := return (\"NUMBER[0]\",[])               -- OK, mais peut-être faut-il garder l'info 0 : réel\n| `(0:ℕ) := return (\"NUMBER[0]\",[])               -- non testé\n| `(0:ℤ) := return (\"NUMBER[0]\",[])               -- non testé\n| `(1:ℝ) := return (\"NUMBER[1]\",[])               \n| `(1:ℕ) := return (\"NUMBER[1]\",[])               -- non testé\n| `(1:ℤ) := return (\"NUMBER[1]\",[])               -- non testé\n-- | `(0 < %%b) := return (\"POSITIF\", [b]) \n| `(%%a < %%b) := return (\"PROP_<\", [a,b]) \n| `(%%a ≤ %%b) := return (\"PROP_≤\", [a,b])\n-- | `(%%a > 0) := return (\"POSITIF\", [a])\n| `(%%a > %%b) := return (\"PROP_>\", [a,b]) \n| `(%%a ≥ %%b) := return (\"PROP_≥\", [a,b]) \n------------------------------ Meta_applications\n\n| (app fonction argument)   := -- do let Sfonction := to_string(fonction),\n    -- pour les nombres, utiliser la pretty printer de Lean\n    -- récupérer le type ?\n    if is_numeral e\n        then return (\"NUMBER[\"++e_joli ++\"]\",[]) \n    -- détecter les sous-ensembles\n--    else if to_string(fonction) = \"set.{0}\"  \n--        then return(\"SET\", [argument])\n--        else return(\"META_APPLICATION[[pp:\" ++ e_joli ++\"]]\",[fonction,argument])\n        else return(\"APPLICATION\",[fonction,argument])\n| `(ℝ) := return (\"TYPE_NUMBER[ℝ]\",[])\n| `(ℕ) := return (\"TYPE_NUMBER[ℕ]\",[])\n| (const name list_level)   := return (\"CONSTANT[name:\"++ e_joli ++ \"/\" ++ to_string name ++\"]\", []) -- name → list level → expr\n| (var nat)       := return (\"VAR[\"++ to_string nat ++ \"]\", []) --  nat → expr\n| (sort level)      := return (\"TYPE\", [])  -- level → expr\n| (mvar name pretty_name type)        := return (\"METAVAR[\" ++ to_string pretty_name ++ \"]\", []) -- name → name → expr → expr\n| (local_const name pretty_name bi type) := return (\"LOCAL_CONSTANT[name:\"++ to_string pretty_name++\"/identifier:\"++ to_string name ++ \"]\", []) -- name → name → binder_info → expr → expr\n| (elet name_var type_var expr body)        := return (\"LET[\"++ to_string name_var ++\"]\", [type_var,expr,body]) --name → expr → expr → expr → expr\n| (macro liste pas_compris)       := return (\"MACRO\", []) -- macro_def → list expr → expr\nend\n\n-- A node will be a leaf of the analysis tree iff it belongs to the following list:\n-- leaves = [\"NOMBRE\", \"CONSTANT\", \"VAR\", \"TYPE\", \"METAVAR\", \"LOCAL_CONSTANT\", \n--          \"LET\", \"MACRO\", \"ERREUR\"]    \n-- A leaf is followed by a separateur_virgule or a \")\"\n-- A node which is not a leaf is followed by a \"(\"\n\n\ndef separateur_virgule := \"¿, \"\ndef separateur_egale := \" ¿= \"\ndef open_paren := \"¿(\"\ndef closed_paren := \"¿)\"\n/- Analyse récursivement une expression à l'aide de analyse_expr_step, \nrenvoie le résultat sous forme de chaine bien parenthésée-/\nprivate meta def analyse_rec : expr →  tactic string \n| e := \ndo ⟨string, liste_expr⟩ ←  analyse_expr_step(e), \n--    bool ← is_prop e,\n--    let string := to_string bool ++ \".\" ++ string,\n    match liste_expr with\n    -- ATTENTION, cas de plus de trois arguiments non traité\n    -- à remplacer par un list.map\n    |[e1] :=  do \n       string1 ← analyse_rec e1,\n       return(string ++ open_paren ++ string1 ++ closed_paren)\n    |[e1,e2] :=  do \n        string1 ← analyse_rec e1,\n        string2 ← analyse_rec e2,\n--        if  string = \"APPLICATION\"\n--            then return (string1 ++ open_paren ++ string2 ++ closed_paren) else\n        return (string ++ open_paren ++ string1 ++ separateur_virgule ++ string2 ++ closed_paren)\n    |[e1,e2,e3] :=  do  -- non utilisé\n        string1 ← analyse_rec e1,\n        string2 ← analyse_rec e2,\n        string3 ← analyse_rec e3,\n        return (string ++ open_paren ++ string1 ++ separateur_virgule ++ string2 ++ separateur_virgule ++ string3 ++ closed_paren)\n    | _ :=    return(string)\n    end\nprivate meta def analyse_expr : expr →  tactic string\n| e := do\n    expr_t ←  infer_type e,\n    bool ← is_prop expr_t,\n    -- expr_tt ← infer_type expr_t,\n    if bool then do\n            -- S ←  (tactic.pp expr_t), \n            -- let S1 := to_string S,\n            S ←  (tactic.pp expr_t), let et_joli := to_string S, \n            S1b ← analyse_rec e,\n            S2 ← analyse_rec expr_t,\n            let S3 := \"PROPERTY[\" ++ S1b ++ \"/pp_type: \" ++ et_joli ++ \"]\" ++ separateur_egale ++ S2,\n            return(S3)\n        else  do\n            -- let S1 :=  to_string e, \n            S1b ← analyse_rec e,\n            S2 ← analyse_rec expr_t,\n            let S3 := \"OBJECT[\" ++ S1b ++ separateur_egale ++ S2,\n            return(S3)\n\n\n/- Affiche la liste des objets du contexte, séparés par des retour chariots \nformat :  \"OBJET\" ou \"PROPRIETE\" : affichage Lean : structure -/\nmeta def analyse_contexte : tactic unit :=\ndo liste_expr ← local_context,\n    trace \"context:\",\n    liste_expr.mmap (λ h, analyse_expr h >>= trace),\n    return ()\n\n\n/- Affiche la liste des buts, même format que analyse_contexte\n(excepté qu'il n'y a que des PROPRIETES) -/ \nmeta def analyse_buts : tactic unit :=\ndo liste_expr ← get_goals,\n    trace \"goals:\", \n    liste_expr.mmap (λ h, analyse_expr h >>= trace),\n    return ()\n\n\n\n---------------------------------------------------------\n--------- NON UTILISES (debuggage) ----------------------------------\n---------------------------------------------------------\n\n\n/- Appelle l'analyse récursive sur le but ou sur une hypothèse. Non utilisé par la suite. -/\nmeta def analyse (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (analyse_rec goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                expr_tt ← infer_type expr_t,\n                -- la suite différencie selon la sémantique, \n                -- ie les objets (éléments, ensembles, fonctions)\n                -- vs les propriétés\n                if expr_tt = `(Prop) then  \n                    trace (analyse_rec expr_t)\n                else  do S1 ← (analyse_rec expr), \n                        S2 ← (analyse_rec expr_t),\n                        --let S2 := to_string expr_t,\n                        let S3 := S1 ++ \" : \"++ S2,\n                        trace(S3)\n    | _ := skip\n    end\n\n/- Appelle l'analyse en 1 coup sur le but ou sur une hypothèse. Non utilisé par la suite. -/\nmeta def analyse1 (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (analyse_expr_step goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                trace (analyse_expr_step expr_t)\n    | _ := skip\n    end\n\n\n-- non utilisé\nprivate meta def analyse_expr2 : expr →  tactic string\n| e := do\n    expr_t ←  infer_type e,\n    expr_tt ← infer_type expr_t,\n    if expr_tt = `(Prop) then do\n            S ←  (tactic.pp expr_t), \n            let S1 := to_string S,\n            S2 ← analyse_rec expr_t,\n            let S3 := \"PROPRIETE : \" ++ S1 ++ \" : \" ++ S2,\n            return(S3)\n        else  do let S0 := \"OBJET : \",\n            let S1 :=  to_string e, \n            S2 ← analyse_rec expr_t,\n            let S3 := S0 ++ S1 ++ \" : \"++ S2,\n            return(S3)\n\n\n\n\n\n\n---------------------------------------------------------\n----------------- Essai de rendu LateX, non abouti ------\n---------------------------------------------------------\n\n\n/- transforme une expression lean en expression latex\nAMELIORER : \ntenir compte de la profondeur de l'arbre pour décider si on met des prenthèses-/\n/- ET, OU, SSI, QUELQUESOIT, IMPLIQUE, FONCTION, NON1, EXISTE,\nINTER, UNION, INCLUS, APPARTIENT, COMPLEMENTAIRE1, IMAGE_ENSEMBLE, IMAGE_RECIPROQUE, \nEGALITE, ENSEMBLE1, APPLICATION-/\nmeta def latex_expr : expr →  tactic string \n| e := do\n    ⟨string, liste_expr⟩ ←  analyse_expr_step e, \n    if list.length liste_expr =2 then do\n        let e1 := list.head liste_expr,\n        let e2 := list.head (list.tail liste_expr),\n        S1 ← latex_expr e1,\n        S2 ← latex_expr e2,\n        match string with\n            | \"ET\" := return (S1 ++ \" et \" ++ S2)\n            | \"OU\" := return (S1 ++ \" ou \" ++ S2)\n            | \"SSI\" := return (\"(\" ++ S1 ++ \"\" ++\") \\\\Leftrightarrow (\" ++ S2 ++ \")\")\n\n            | \"INCLUS\" := return (S1 ++ \"\" ++\" \\\\subset \" ++ S2)\n            | _ := return \"ERREUR\"\n            end\n\n    else if list.length liste_expr =1 then do\n        let e1 := list.head liste_expr,\n        S1 ← latex_expr e1,\n        match string with\n            | \"NON\" := return (\"NON (\" ++ S1 ++ \")\")\n            | \"COMPLEMENTAIRE\" := return (S1  ++ \"^c\")\n            | _ := return \"ERREUR\"\n            end\n    else return (string)\n\n\n\n\nmeta def latex_buts : tactic unit :=\ndo liste_expr ← get_goals,\n    trace \"Buts :\", \n    -- liste_buts ← tactic.get_goals,\n    -- types ← list.mmap tactic.infer_type liste_buts, \n    -- trace types,\n    liste_expr.mmap (λ h, latex_expr h >>= trace),\n    return ()\n\nmeta def latex_but : tactic unit :=\ndo expr ← target,\n    trace \"But :\", \n    -- liste_buts ← tactic.get_goals,\n    -- types ← list.mmap tactic.infer_type liste_buts, \n    -- trace types,\n    trace (latex_expr expr),\n    return ()\n\n\n\n----------------------------------------------\n------------- DEBUGGAGE -------------------\n-------------------------------------------\n\n/- debug -/\nprivate meta def analyse_expr_step_brut  (e : expr) : tactic (string × (list expr)) := \nmatch e with\n-- autres\n| (pi name binder type body ) := return (\"pi (nom : \" ++ to_string name ++ \")\",[type,body]) \n| (app fonction argument)   := return (\"application\", [fonction,argument])\n| (const name list_level)   := return (\"constante :\" ++ to_string name, []) -- name → list level → expr\n| (var nat)       := return (\"var_\"++ to_string nat, []) --  nat → expr\n| (sort level)      := return (\"sort\", [])  -- level → expr\n| (mvar name pretty_name type)        := return (\"metavar\", []) -- name → name → expr → expr\n| (local_const name pretty_name bi type) := return (\"constante_locale :\" ++ to_string pretty_name, []) -- name → name → binder_info → expr → expr\n| (lam name binder type body)          := return (\"lambda (nom : \" ++ to_string name ++ \")\", [type,body]) -- name → binder_info → expr → expr → expr\n| (elet name_var type_var expr body)        := return (\"let\", []) --name → expr → expr → expr → expr\n| (macro liste pas_compris)       := return (\"macro\", []) -- macro_def → list expr → expr\nend\n\n/-  Debug -/\nprivate meta def analyse_rec_brut : expr →  tactic string \n| e := \ndo ⟨string, liste_expr⟩ ←  analyse_expr_step_brut e, \n    match liste_expr with\n    -- ATTENTION, cas de plus de trois arguiments non traité\n    -- à remplacer par un list.map\n    |[e1] :=  do \n       string1 ← analyse_rec_brut e1,\n       return(string ++ \"(\" ++ string1 ++ \")\")\n    |[e1,e2] :=  do \n        string1 ← analyse_rec_brut e1,\n        string2 ← analyse_rec_brut e2,\n        if  string = \"APPLICATION\" then do \n            { type2 ← infer_type e2,\n            let string_type2 := to_string type2, -- trace string_type2, \n            if (string_type2 = \"Type\" ) ∨ (trois_car (to_string(e2)) = \"_in\" ) -- Type  ou instance\n                then return (string1)\n                else return (string1 ++ \"(\" ++ string2 ++\")\")\n            }   <|> return (string1 ++ \"(\" ++ string2 ++\")\")\n            else return (string ++ \"(\" ++ string1 ++ \",\" ++ string2 ++ \")\")\n    |[e1,e2,e3] :=  do  -- non utilisé\n        string1 ← analyse_rec_brut e1,\n        string2 ← analyse_rec_brut e2,\n        string3 ← analyse_rec_brut e3,\n        return (string ++ \"(\" ++ string1 ++ \",\" ++ string2 ++ \",\" ++ string3 ++ \")\")\n    | _ :=    return(string)\n    end\n\n/- Debug -/\nmeta def analyse_brut (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (analyse_rec_brut goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                expr_tt ← infer_type expr_t,\n                -- la suite différencie selon la sémantique, \n                -- ie les objets (éléments, ensembles, fonctions)\n                -- vs les propriétés\n                if expr_tt = `(Prop) then  \n                    trace (analyse_rec_brut expr_t)\n                else  do S1 ← (analyse_rec_brut expr), \n                        S2 ← (analyse_rec_brut expr_t),\n                        --let S2 := to_string expr_t,\n                        let S3 := S1 ++ \" : \"++ S2,\n                        trace(S3)\n    | _ := skip\n    end\n\n-- débug    \nprivate meta def analyse_expr_brut : expr →  tactic string\n| e := do\n    expr_t ←  infer_type e,\n    expr_tt ← infer_type expr_t,\n    if expr_tt = `(Prop) then do\n            S ←  (tactic.pp expr_t), \n            let S1 := to_string S,\n            S2 ← analyse_rec_brut expr_t,\n            let S3 := \"PROPRIETE : \" ++ S1 ++ \" : \" ++ S2,\n            return(S3)\n        else  do let S0 := \"OBJET : \",\n            let S1 :=  to_string e, \n            S2 ← analyse_rec_brut expr_t,\n            let S3 := S0 ++ S1 ++ \" : \"++ S2,\n            return(S3)\n\n\n\n\n\n/- Affiche la liste des objets du contexte, séparés par des retour chariots \nformat :  \"OBJET\" ou \"PROPRIETE\" : affichage Lean : structure -/\nmeta def analyse_contexte_brut : tactic unit :=\ndo liste_expr ← local_context,\n    trace \"Contexte :\",\n    liste_expr.mmap (λ h, analyse_expr_brut h >>= trace),\n    return ()\n\n\n/- Analyse brute de Lean (dans expr) -/\nmeta def analyse_raw (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace $ to_raw_fmt goal\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                trace $ to_raw_fmt expr_t\n    | _ := skip\n    end\n\nend tactic.interactive", "meta": {"author": "FredericLeRoux", "repo": "dEAduction-lean2", "sha": "bf7d7d88c2511ecfda5a98ed96e4ca3bc7ae1151", "save_path": "github-repos/lean/FredericLeRoux-dEAduction-lean2", "path": "github-repos/lean/FredericLeRoux-dEAduction-lean2/dEAduction-lean2-bf7d7d88c2511ecfda5a98ed96e4ca3bc7ae1151/src/snippets/essai2structures.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3453368116980983}}
{"text": "import types\n\nimport category_theory.category.basic\nimport category_theory.core\nimport data.nat.digits\n\nopen params\nopen types\n\nopen category_theory\nopen nat\n\nnamespace littleendian\n\nvariables [category (bitvec word_len)]\n\n/-!\n  # Littleendian\n\n  The `littleendian` function and its inverse.\n-/\n\n/- Needed for `littleendian`, `of_digits`. -/\nvariables [semiring (list ℕ)]\n\n/-- Lean library has `of_digits` code for this so we use it. \nTODO : Explain why it works with base 256.\n-/\ndef littleendian (l : list ℕ) : list ℕ := of_digits 256 l\n\n/-- Lean library has `digits` code for this so we use it. \nTODO : Explain why it works with base 256.\n-/\ndef littleendian_inv (l : list ℕ) : list ℕ := digits 256 (l.nth 0).iget\n\n/-- Define a category for functions from `list ℕ` to `list ℕ`-/\nvariable [category (list ℕ → list ℕ)]\n\n/- Just some notation for inverse. -/\nlocal notation `littleendian⁻¹` := littleendian_inv\n\n/-- We assume there is an isomorphism between `littleindian` and `littleendian⁻¹`.\nIn other words, `littleendian` function is invertible and the inverse is `littleendian⁻¹`. -/\nlemma littleendian_is_inv (I : littleendian ≅ littleendian⁻¹) : I.hom ≫ I.inv = 𝟙 littleendian :=\n  by rw [iso.hom_inv_id]\n\n\nend littleendian\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/littleendian.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34533680459456256}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.option.basic\n! leanprover-community/mathlib commit f340f229b1f461aa1c8ee11e0a172d0a3b301a4a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Init.Control.Combinators\nimport Mathlib.Data.Option.Defs\nimport Mathlib.Logic.IsEmpty\nimport Mathlib.Logic.Relator\nimport Mathlib.Mathport.Rename\n\n/-!\n# Option of a type\n\nThis file develops the basic theory of option types.\n\nIf `α` is a type, then `Option α` can be understood as the type with one more element than `α`.\n`Option α` has terms `some a`, where `a : α`, and `none`, which is the added element.\nThis is useful in multiple ways:\n* It is the prototype of addition of terms to a type. See for example `WithBot α` which uses\n  `none` as an element smaller than all others.\n* It can be used to define failsafe partial functions, which return `some the_result_we_expect`\n  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces\n  any subsequent use of the partial function to explicitly deal with the exceptions that make it\n  return `none`.\n* `Option` is a monad. We love monads.\n\n`Part` is an alternative to `Option` that can be seen as the type of `True`/`False` values\nalong with a term `a : α` if the value is `True`.\n\n-/\n\nnamespace Option\n\nvariable {α β γ δ : Type _}\n\ntheorem coe_def : (fun a ↦ ↑a : α → Option α) = some :=\n  rfl\n#align option.coe_def Option.coe_def\n\n#align option.get_or_else Option.getD\n\n@[simp]\ntheorem getD_coe (x y : α) : Option.getD (↑x) y = x :=\n  rfl\n#align option.get_or_else_coe Option.getD_coe\n\ntheorem coe_get {o : Option α} (h : o.isSome) : ((Option.get _ h : α) : Option α) = o :=\n  Option.some_get h\n#align option.coe_get Option.coe_get\n\ntheorem eq_of_mem_of_mem {a : α} {o1 o2 : Option α} (h1 : a ∈ o1) (h2 : a ∈ o2) : o1 = o2 :=\n  h1.trans h2.symm\n#align option.eq_of_mem_of_mem Option.eq_of_mem_of_mem\n\ntheorem Mem.leftUnique : Relator.LeftUnique ((· ∈ ·) : α → Option α → Prop) :=\nfun _ _ _=> mem_unique\n#align option.mem.left_unique Option.Mem.leftUnique\n\ntheorem some_injective (α : Type _) : Function.Injective (@some α) := fun _ _ ↦ some_inj.mp\n#align option.some_injective Option.some_injective\n\n/-- `Option.map f` is injective if `f` is injective. -/\ntheorem map_injective {f : α → β} (Hf : Function.Injective f) : Function.Injective (Option.map f)\n  | none, none, _ => rfl\n  | some a₁, some a₂, H => by rw [Hf (Option.some.inj H)]\n#align option.map_injective Option.map_injective\n\n@[simp]\ntheorem map_comp_some (f : α → β) : Option.map f ∘ some = some ∘ f :=\n  rfl\n#align option.map_comp_some Option.map_comp_some\n\n@[simp]\ntheorem none_bind' (f : α → Option β) : none.bind f = none :=\n  rfl\n#align option.none_bind' Option.none_bind'\n\n@[simp]\ntheorem some_bind' (a : α) (f : α → Option β) : (some a).bind f = f a :=\n  rfl\n#align option.some_bind' Option.some_bind'\n\ntheorem bind_eq_some' {x : Option α} {f : α → Option β} {b : β} :\n    x.bind f = some b ↔ ∃ a, x = some a ∧ f a = some b :=\n  by cases x <;> simp\n#align option.bind_eq_some' Option.bind_eq_some'\n\ntheorem bind_eq_none' {o : Option α} {f : α → Option β} :\n    o.bind f = none ↔ ∀ b a, a ∈ o → b ∉ f a := by\n  simp only [eq_none_iff_forall_not_mem, mem_def, bind_eq_some, not_exists, not_and]\n#align option.bind_eq_none' Option.bind_eq_none'\n\ntheorem joinM_eq_join : joinM = @join α :=\n  funext fun _ ↦ rfl\n#align option.join_eq_join Option.joinM_eq_join\n\ntheorem bind_eq_bind {α β : Type _} {f : α → Option β} {x : Option α} : x >>= f = x.bind f :=\n  rfl\n#align option.bind_eq_bind Option.bind_eq_bind\n\ntheorem map_coe {α β} {a : α} {f : α → β} : f <$> (a : Option α) = ↑(f a) :=\n  rfl\n#align option.map_coe Option.map_coe\n\n@[simp]\ntheorem map_coe' {a : α} {f : α → β} : Option.map f (a : Option α) = ↑(f a) :=\n  rfl\n#align option.map_coe' Option.map_coe'\n\n/-- `Option.map` as a function between functions is injective. -/\ntheorem map_injective' : Function.Injective (@Option.map α β) := fun f g h ↦\n  funext fun x ↦ some_injective _ <| by simp only [← map_some', h]\n#align option.map_injective' Option.map_injective'\n\n@[simp]\ntheorem map_inj {f g : α → β} : Option.map f = Option.map g ↔ f = g :=\n  map_injective'.eq_iff\n#align option.map_inj Option.map_inj\n\nattribute [simp] map_id\n\n@[simp]\ntheorem map_eq_id {f : α → α} : Option.map f = id ↔ f = id :=\n  map_injective'.eq_iff' map_id\n#align option.map_eq_id Option.map_eq_id\n\ntheorem map_comm {f₁ : α → β} {f₂ : α → γ} {g₁ : β → δ} {g₂ : γ → δ} (h : g₁ ∘ f₁ = g₂ ∘ f₂)\n  (a : α) :\n    (Option.map f₁ a).map g₁ = (Option.map f₂ a).map g₂ := by rw [map_map, h, ← map_map]\n#align option.map_comm Option.map_comm\n\nsection pmap\n\nvariable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)\n\n-- Porting note: Can't simp tag this anymore because `pbind` simplifies\n-- @[simp]\ntheorem pbind_eq_bind (f : α → Option β) (x : Option α) : (x.pbind fun a _ ↦ f a) = x.bind f := by\n  cases x <;> simp only [pbind, none_bind', some_bind']\n#align option.pbind_eq_bind Option.pbind_eq_bind\n\ntheorem map_bind {α β γ} (f : β → γ) (x : Option α) (g : α → Option β) :\n    Option.map f (x >>= g) = x >>= fun a ↦ Option.map f (g a) := by\n  simp only [← map_eq_map, ← bind_pure_comp, LawfulMonad.bind_assoc]\n#align option.map_bind Option.map_bind\n\ntheorem map_bind' (f : β → γ) (x : Option α) (g : α → Option β) :\n    Option.map f (x.bind g) = x.bind fun a ↦ Option.map f (g a) := by cases x <;> simp\n#align option.map_bind' Option.map_bind'\n\ntheorem map_pbind (f : β → γ) (x : Option α) (g : ∀ a, a ∈ x → Option β) :\n    Option.map f (x.pbind g) = x.pbind fun a H ↦ Option.map f (g a H) := by\n  cases x <;> simp only [pbind, map_none']\n#align option.map_pbind Option.map_pbind\n\ntheorem pbind_map (f : α → β) (x : Option α) (g : ∀ b : β, b ∈ x.map f → Option γ) :\n    pbind (Option.map f x) g = x.pbind fun a h ↦ g (f a) (mem_map_of_mem _ h) := by cases x <;> rfl\n#align option.pbind_map Option.pbind_map\n\n@[simp]\ntheorem pmap_none (f : ∀ a : α, p a → β) {H} : pmap f (@none α) H = none :=\n  rfl\n#align option.pmap_none Option.pmap_none\n\n@[simp]\ntheorem pmap_some (f : ∀ a : α, p a → β) {x : α} (h : p x) :\n    pmap f (some x) = fun _ ↦ some (f x h) :=\n  rfl\n#align option.pmap_some Option.pmap_some\n\ntheorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by\n  rw [mem_def] at ha ⊢\n  subst ha\n  rfl\n#align option.mem_pmem Option.mem_pmem\n\ntheorem pmap_map (g : γ → α) (x : Option γ) (H) :\n    pmap f (x.map g) H = pmap (fun a h ↦ f (g a) h) x fun a h ↦ H _ (mem_map_of_mem _ h) := by\n  cases x <;> simp only [map_none', map_some', pmap]\n#align option.pmap_map Option.pmap_map\n\ntheorem map_pmap (g : β → γ) (f : ∀ a, p a → β) (x H) :\n    Option.map g (pmap f x H) = pmap (fun a h ↦ g (f a h)) x H :=\n  by cases x <;> simp only [map_none', map_some', pmap]\n#align option.map_pmap Option.map_pmap\n\n-- Porting note: Can't simp tag this anymore because `pmap` simplifies\n-- @[simp]\ntheorem pmap_eq_map (p : α → Prop) (f : α → β) (x H) :\n    @pmap _ _ p (fun a _ ↦ f a) x H = Option.map f x := by\n  cases x <;> simp only [map_none', map_some', pmap]\n#align option.pmap_eq_map Option.pmap_eq_map\n\ntheorem pmap_bind {α β γ} {x : Option α} {g : α → Option β} {p : β → Prop} {f : ∀ b, p b → γ} (H)\n    (H' : ∀ (a : α), ∀ b ∈ g a, b ∈ x >>= g) :\n    pmap f (x >>= g) H = x >>= fun a ↦ pmap f (g a) fun b h ↦ H _ (H' a _ h) := by\n  cases x <;> simp only [pmap, bind_eq_bind, none_bind, some_bind]\n#align option.pmap_bind Option.pmap_bind\n\n\n\nvariable {f x}\n\ntheorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}\n    (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by\n  cases x\n  · simp\n  · simp only [pbind, iff_false]\n    intro h\n    cases h' _ rfl h\n#align option.pbind_eq_none Option.pbind_eq_none\n\ntheorem pbind_eq_some {f : ∀ a : α, a ∈ x → Option β} {y : β} :\n    x.pbind f = some y ↔ ∃ (z : α) (H : z ∈ x), f z H = some y := by\n  rcases x with (_|x)\n  · simp only [pbind, false_iff, not_exists]\n    intro z h\n    simp at h\n  · simp only [pbind]\n    refine ⟨λ h => ⟨x, rfl, h⟩, ?_⟩\n    rintro ⟨z, H, hz⟩\n    simp only [mem_def, Option.some_inj] at H\n    simpa [H] using hz\n#align option.pbind_eq_some Option.pbind_eq_some\n\n-- Porting note: Can't simp tag this anymore because `pmap` simplifies\n-- @[simp]\ntheorem pmap_eq_none_iff {h} : pmap f x h = none ↔ x = none := by cases x <;> simp\n#align option.pmap_eq_none_iff Option.pmap_eq_none_iff\n\n-- Porting note: Can't simp tag this anymore because `pmap` simplifies\n-- @[simp]\ntheorem pmap_eq_some_iff {hf} {y : β} :\n    pmap f x hf = some y ↔ ∃ (a : α) (H : x = some a), f a (hf a H) = y := by\n  rcases x with (_|x)\n  · simp only [not_mem_none, exists_false, pmap, not_false_iff, exists_prop_of_false]\n  · constructor\n    · intro h\n      simp only [pmap, Option.some_inj] at h\n      refine ⟨x, rfl, h⟩\n    · rintro ⟨a, H, rfl⟩\n      simp only [mem_def, Option.some_inj] at H\n      simp only [H, pmap]\n#align option.pmap_eq_some_iff Option.pmap_eq_some_iff\n\n-- Porting note: Can't simp tag this anymore because `join` and `pmap` simplify\n-- @[simp]\ntheorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :\n    (pmap (pmap f) x H).join = pmap f x.join fun a h ↦ H (some a) (mem_of_mem_join h) _ rfl := by\n  rcases x with (_ | _ | x) <;> simp\n#align option.join_pmap_eq_pmap_join Option.join_pmap_eq_pmap_join\n\nend pmap\n\n@[simp]\ntheorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) :=\n  rfl\n#align option.seq_some Option.seq_some\n\n@[simp]\ntheorem some_orElse' (a : α) (x : Option α) : (some a).orElse (fun _ ↦ x) = some a :=\n  rfl\n#align option.some_orelse' Option.some_orElse'\n\n#align option.some_orelse Option.some_orElse\n\n@[simp]\ntheorem none_orElse' (x : Option α) : none.orElse (fun _ ↦ x) = x := by cases x <;> rfl\n#align option.none_orelse' Option.none_orElse'\n\n#align option.none_orelse Option.none_orElse\n\n@[simp]\ntheorem orElse_none' (x : Option α) : x.orElse (fun _ ↦ none) = x := by cases x <;> rfl\n\n#align option.orelse_none' Option.orElse_none'\n\n#align option.orelse_none Option.orElse_none\n\n#align option.is_some_none Option.isSome_none\n\n#align option.is_some_some Option.isSome_some\n\n#align option.is_some_iff_exists Option.isSome_iff_exists\n\n#align option.is_none_none Option.isNone_none\n\n#align option.is_none_some Option.isNone_some\n\n#align option.not_is_some Option.not_isSome\n\n#align option.not_is_some_iff_eq_none Option.not_isSome_iff_eq_none\n\n#align option.ne_none_iff_is_some Option.ne_none_iff_isSome\n\ntheorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o\n  | some _, _ => rfl\n#align option.iget_mem Option.iget_mem\n\ntheorem iget_of_mem [Inhabited α] {a : α} : ∀ {o : Option α}, a ∈ o → o.iget = a\n  | _, rfl => rfl\n#align option.iget_of_mem Option.iget_of_mem\n\ntheorem getD_default_eq_iget [Inhabited α] (o : Option α) :\n    o.getD default = o.iget := by cases o <;> rfl\n#align option.get_or_else_default_eq_iget Option.getD_default_eq_iget\n\n@[simp]\ntheorem guard_eq_some' {p : Prop} [Decidable p] (u) : _root_.guard p = some u ↔ p := by\n  cases u\n  by_cases h : p <;> simp [_root_.guard, h]\n#align option.guard_eq_some' Option.guard_eq_some'\n\ntheorem liftOrGet_choice {f : α → α → α} (h : ∀ a b, f a b = a ∨ f a b = b) :\n    ∀ o₁ o₂, liftOrGet f o₁ o₂ = o₁ ∨ liftOrGet f o₁ o₂ = o₂\n  | none, none => Or.inl rfl\n  | some a, none => Or.inl rfl\n  | none, some b => Or.inr rfl\n  | some a, some b => by simpa [liftOrGet] using h a b\n#align option.lift_or_get_choice Option.liftOrGet_choice\n\n#align option.lift_or_get_none_left Option.liftOrGet_none_left\n\n#align option.lift_or_get_none_right Option.liftOrGet_none_right\n\n#align option.lift_or_get_some_some Option.liftOrGet_some_some\n\n/-- Given an element of `a : option α`, a default element `b : β` and a function `α → β`, apply this\nfunction to `a` if it comes from `α`, and return `b` otherwise. -/\ndef casesOn' : Option α → β → (α → β) → β\n  | none, n, _ => n\n  | some a, _, s => s a\n#align option.cases_on' Option.casesOn'\n\n@[simp]\ntheorem casesOn'_none (x : β) (f : α → β) : casesOn' none x f = x :=\n  rfl\n#align option.cases_on'_none Option.casesOn'_none\n\n@[simp]\ntheorem casesOn'_some (x : β) (f : α → β) (a : α) : casesOn' (some a) x f = f a :=\n  rfl\n#align option.cases_on'_some Option.casesOn'_some\n\n@[simp]\ntheorem casesOn'_coe (x : β) (f : α → β) (a : α) : casesOn' (a : Option α) x f = f a :=\n  rfl\n#align option.cases_on'_coe Option.casesOn'_coe\n\n-- Porting note: Left-hand side does not simplify.\n-- @[simp]\ntheorem casesOn'_none_coe (f : Option α → β) (o : Option α) :\n    casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by cases o <;> rfl\n#align option.cases_on'_none_coe Option.casesOn'_none_coe\n\ntheorem orElse_eq_some (o o' : Option α) (x : α) :\n    (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by\n  cases o\n  · simp only [true_and, false_or, eq_self_iff_true, none_orElse]\n  · simp only [some_orElse, or_false, false_and]\n#align option.orelse_eq_some Option.orElse_eq_some\n\n\ntheorem orElse_eq_some' (o o' : Option α) (x : α) :\n    o.orElse (fun _ ↦ o') = some x ↔ o = some x ∨ o = none ∧ o' = some x :=\n  Option.orElse_eq_some o o' x\n#align option.orelse_eq_some' Option.orElse_eq_some'\n\n@[simp]\ntheorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by\n  cases o\n  · simp only [true_and, none_orElse, eq_self_iff_true]\n  · simp only [some_orElse, false_and]\n#align option.orelse_eq_none Option.orElse_eq_none\n\n@[simp]\ntheorem orElse_eq_none' (o o' : Option α) : o.orElse (fun _ ↦ o') = none ↔ o = none ∧ o' = none :=\n  Option.orElse_eq_none o o'\n#align option.orelse_eq_none' Option.orElse_eq_none'\n\nsection\n\nopen Classical\n\ntheorem choice_eq_none (α : Type _) [IsEmpty α] : choice α = none :=\n  dif_neg (not_nonempty_iff_imp_false.mpr isEmptyElim)\n#align option.choice_eq_none Option.choice_eq_none\n\n#align option.choice_is_some_iff_nonempty Option.choice_isSome_iff_nonempty\n\nend\n\n-- Porting note: Can't simp tag this anymore because `elim` simplifies\n-- @[simp]\ntheorem elim_none_some (f : Option α → β) : (fun x ↦ Option.elim x (f none) (f ∘ some)) = f :=\n  funext fun o ↦ by cases o <;> rfl\n#align option.elim_none_some Option.elim_none_some\n\nend Option\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/Option/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6406358617010351, "lm_q1q2_score": 0.3452919800276537}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl\n\nimport category_theory.functor_category\nimport category_theory.fully_faithful\n\nnamespace category_theory\n\nuniverses v v' w u u' -- declare the `v`'s first; see `category_theory.category` for an explanation\n\ninstance types : large_category (Type u) :=\n{ hom     := λ a b, (a → b),\n  id      := λ a, id,\n  comp    := λ _ _ _ f g, g ∘ f }\n\n@[simp] lemma types_hom {α β : Type u} : (α ⟶ β) = (α → β) := rfl\n@[simp] lemma types_id {α : Type u} (a : α) : (𝟙 α : α → α) a = a := rfl\n@[simp] lemma types_comp {α β γ : Type u} (f : α → β) (g : β → γ) (a : α) : (((f : α ⟶ β) ≫ (g : β ⟶ γ)) : α ⟶ γ) a = g (f a) := rfl\n\nnamespace functor_to_types\nvariables {C : Type u} [𝒞 : category.{v} C] (F G H : C ⥤ Type w) {X Y Z : C}\ninclude 𝒞\nvariables (σ : F ⟹ G) (τ : G ⟹ H)\n\n@[simp] lemma map_comp (f : X ⟶ Y) (g : Y ⟶ Z) (a : F.obj X) : (F.map (f ≫ g)) a = (F.map g) ((F.map f) a) :=\nby simp\n\n@[simp] lemma map_id (a : F.obj X) : (F.map (𝟙 X)) a = a :=\nby simp\n\nlemma naturality (f : X ⟶ Y) (x : F.obj X) : σ.app Y ((F.map f) x) = (G.map f) (σ.app X x) :=\ncongr_fun (σ.naturality f) x\n\n@[simp] lemma vcomp (x : F.obj X) : (σ ⊟ τ).app X x = τ.app X (σ.app X x) := rfl\n\nvariables {D : Type u'} [𝒟 : category.{u'} D] (I J : D ⥤ C) (ρ : I ⟹ J) {W : D}\n\n@[simp] lemma hcomp (x : (I ⋙ F).obj W) : (ρ ◫ σ).app W x = (G.map (ρ.app W)) (σ.app (I.obj W) x) := rfl\n\nend functor_to_types\n\ndef ulift_trivial (V : Type u) : ulift.{u} V ≅ V := by tidy\n\ndef ulift_functor : Type u ⥤ Type (max u v) :=\n{ obj := λ X, ulift.{v} X,\n  map := λ X Y f, λ x : ulift.{v} X, ulift.up (f x.down) }\n\n@[simp] lemma ulift_functor.map {X Y : Type u} (f : X ⟶ Y) (x : ulift.{v} X) :\n  ulift_functor.map f x = ulift.up (f x.down) := rfl\n\ninstance ulift_functor_faithful : fully_faithful ulift_functor :=\n{ preimage := λ X Y f x, (f (ulift.up x)).down,\n  injectivity' := λ X Y f g p, funext $ λ x,\n    congr_arg ulift.down ((congr_fun p (ulift.up x)) : ((ulift.up (f x)) = (ulift.up (g x)))) }\n\nend category_theory\n\n-- Isomorphisms in Type and equivalences.\n\nnamespace equiv\n\nuniverse u\n\nvariables {X Y : Type u}\n\ndef to_iso (e : X ≃ Y) : X ≅ Y :=\n{ hom := e.to_fun,\n  inv := e.inv_fun,\n  hom_inv_id' := funext e.left_inv,\n  inv_hom_id' := funext e.right_inv }\n\n@[simp] lemma to_iso_hom {e : X ≃ Y} : e.to_iso.hom = e := rfl\n@[simp] lemma to_iso_inv {e : X ≃ Y} : e.to_iso.inv = e.symm := rfl\n\nend equiv\n\nnamespace category_theory.iso\n\nuniverse u\n\nvariables {X Y : Type u}\n\ndef to_equiv (i : X ≅ Y) : X ≃ Y :=\n{ to_fun := i.hom,\n  inv_fun := i.inv,\n  left_inv := λ x, congr_fun i.hom_inv_id x,\n  right_inv := λ y, congr_fun i.inv_hom_id y }\n\n@[simp] lemma to_equiv_fun (i : X ≅ Y) : (i.to_equiv : X → Y) = i.hom := rfl\n@[simp] lemma to_equiv_symm_fun (i : X ≅ Y) : (i.to_equiv.symm : Y → X) = i.inv := rfl\n\nend category_theory.iso\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/types.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34529197632961595}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Andrew Yang\n\n! This file was ported from Lean 3 source module category_theory.limits.constructions.binary_products\n! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Shapes.Terminal\nimport Mathbin.CategoryTheory.Limits.Shapes.Pullbacks\nimport Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.Terminal\n\n/-!\n# Constructing binary product from pullbacks and terminal object.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThe product is the pullback over the terminal objects. In particular, if a category\nhas pullbacks and a terminal object, then it has binary products.\n\nWe also provide the dual.\n-/\n\n\nuniverse v v' u u'\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Limits\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D] (F : C ⥤ D)\n\n/- warning: is_binary_product_of_is_terminal_is_pullback -> isBinaryProductOfIsTerminalIsPullback is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (F : CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (c : CategoryTheory.Limits.Cone.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F) {X : C} (hX : CategoryTheory.Limits.IsTerminal.{u1, u2} C _inst_1 X) (f : 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(CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1)) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c)) F (CategoryTheory.Limits.Cone.π.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c) (CategoryTheory.Discrete.mk.{0} CategoryTheory.Limits.WalkingPair CategoryTheory.Limits.WalkingPair.right)) g)))) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c)\nCase conversion may be inaccurate. Consider using '#align is_binary_product_of_is_terminal_is_pullback isBinaryProductOfIsTerminalIsPullbackₓ'. -/\n/-- If a span is the pullback span over the terminal object, then it is a binary product. -/\ndef isBinaryProductOfIsTerminalIsPullback (F : Discrete WalkingPair ⥤ C) (c : Cone F) {X : C}\n    (hX : IsTerminal X) (f : F.obj ⟨WalkingPair.left⟩ ⟶ X) (g : F.obj ⟨WalkingPair.right⟩ ⟶ X)\n    (hc :\n      IsLimit\n        (PullbackCone.mk (c.π.app ⟨WalkingPair.left⟩) (c.π.app ⟨WalkingPair.right⟩ : _) <|\n          hX.hom_ext (_ ≫ f) (_ ≫ g))) :\n    IsLimit c\n    where\n  lift s :=\n    hc.lift\n      (PullbackCone.mk (s.π.app ⟨WalkingPair.left⟩) (s.π.app ⟨WalkingPair.right⟩) (hX.hom_ext _ _))\n  fac s j :=\n    Discrete.casesOn j fun j =>\n      WalkingPair.casesOn j (hc.fac _ WalkingCospan.left) (hc.fac _ WalkingCospan.right)\n  uniq s m J :=\n    by\n    let c' :=\n      pullback_cone.mk (m ≫ c.π.app ⟨walking_pair.left⟩) (m ≫ c.π.app ⟨walking_pair.right⟩ : _)\n        (hX.hom_ext (_ ≫ f) (_ ≫ g))\n    rw [← J, ← J]\n    apply hc.hom_ext\n    rintro (_ | (_ | _)) <;> simp only [pullback_cone.mk_π_app_one, pullback_cone.mk_π_app]\n    exacts[(category.assoc _ _ _).symm.trans (hc.fac_assoc c' walking_cospan.left f).symm,\n      (hc.fac c' walking_cospan.left).symm, (hc.fac c' walking_cospan.right).symm]\n#align is_binary_product_of_is_terminal_is_pullback isBinaryProductOfIsTerminalIsPullback\n\n#print isProductOfIsTerminalIsPullback /-\n/-- The pullback over the terminal object is the product -/\ndef isProductOfIsTerminalIsPullback {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y)\n    (H₁ : IsTerminal Z)\n    (H₂ : IsLimit (PullbackCone.mk _ _ (show h ≫ f = k ≫ g from H₁.hom_ext _ _))) :\n    IsLimit (BinaryFan.mk h k) :=\n  by\n  apply isBinaryProductOfIsTerminalIsPullback _ _ H₁\n  exact H₂\n#align is_product_of_is_terminal_is_pullback isProductOfIsTerminalIsPullback\n-/\n\n#print isPullbackOfIsTerminalIsProduct /-\n/-- The product is the pullback over the terminal object. -/\ndef isPullbackOfIsTerminalIsProduct {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y)\n    (H₁ : IsTerminal Z) (H₂ : IsLimit (BinaryFan.mk h k)) :\n    IsLimit (PullbackCone.mk _ _ (show h ≫ f = k ≫ g from H₁.hom_ext _ _)) :=\n  by\n  apply pullback_cone.is_limit_aux'\n  intro s\n  use H₂.lift (binary_fan.mk s.fst s.snd)\n  use H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.left⟩\n  use H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.right⟩\n  intro m h₁ h₂\n  apply H₂.hom_ext\n  rintro ⟨⟨⟩⟩\n  · exact h₁.trans (H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.left⟩).symm\n  · exact h₂.trans (H₂.fac (binary_fan.mk s.fst s.snd) ⟨walking_pair.right⟩).symm\n#align is_pullback_of_is_terminal_is_product isPullbackOfIsTerminalIsProduct\n-/\n\n#print limitConeOfTerminalAndPullbacks /-\n/-- Any category with pullbacks and a terminal object has a limit cone for each walking pair. -/\nnoncomputable def limitConeOfTerminalAndPullbacks [HasTerminal C] [HasPullbacks C]\n    (F : Discrete WalkingPair ⥤ C) : LimitCone F\n    where\n  Cone :=\n    { pt :=\n        pullback (terminal.from (F.obj ⟨WalkingPair.left⟩))\n          (terminal.from (F.obj ⟨WalkingPair.right⟩))\n      π :=\n        Discrete.natTrans fun x =>\n          Discrete.casesOn x fun x => WalkingPair.casesOn x pullback.fst pullback.snd }\n  IsLimit :=\n    isBinaryProductOfIsTerminalIsPullback F _ terminalIsTerminal _ _ (pullbackIsPullback _ _)\n#align limit_cone_of_terminal_and_pullbacks limitConeOfTerminalAndPullbacks\n-/\n\nvariable (C)\n\n#print hasBinaryProducts_of_hasTerminal_and_pullbacks /-\n-- This is not an instance, as it is not always how one wants to construct binary products!\n/-- Any category with pullbacks and terminal object has binary products. -/\ntheorem hasBinaryProducts_of_hasTerminal_and_pullbacks [HasTerminal C] [HasPullbacks C] :\n    HasBinaryProducts C :=\n  { HasLimit := fun F => HasLimit.mk (limitConeOfTerminalAndPullbacks F) }\n#align has_binary_products_of_has_terminal_and_pullbacks hasBinaryProducts_of_hasTerminal_and_pullbacks\n-/\n\nvariable {C}\n\n#print preservesBinaryProductsOfPreservesTerminalAndPullbacks /-\n/-- A functor that preserves terminal objects and pullbacks preserves binary products. -/\nnoncomputable def preservesBinaryProductsOfPreservesTerminalAndPullbacks [HasTerminal C]\n    [HasPullbacks C] [PreservesLimitsOfShape (Discrete.{0} PEmpty) F]\n    [PreservesLimitsOfShape WalkingCospan F] : PreservesLimitsOfShape (Discrete WalkingPair) F :=\n  ⟨fun K =>\n    preservesLimitOfPreservesLimitCone (limitConeOfTerminalAndPullbacks K).2\n      (by\n        apply\n          isBinaryProductOfIsTerminalIsPullback _ _ (is_limit_of_has_terminal_of_preserves_limit F)\n        apply is_limit_of_has_pullback_of_preserves_limit)⟩\n#align preserves_binary_products_of_preserves_terminal_and_pullbacks preservesBinaryProductsOfPreservesTerminalAndPullbacks\n-/\n\n#print prodIsoPullback /-\n/-- In a category with a terminal object and pullbacks,\na product of objects `X` and `Y` is isomorphic to a pullback. -/\nnoncomputable def prodIsoPullback [HasTerminal C] [HasPullbacks C] (X Y : C)\n    [HasBinaryProduct X Y] : X ⨯ Y ≅ pullback (terminal.from X) (terminal.from Y) :=\n  limit.isoLimitCone (limitConeOfTerminalAndPullbacks _)\n#align prod_iso_pullback prodIsoPullback\n-/\n\n/- warning: is_binary_coproduct_of_is_initial_is_pushout -> isBinaryCoproductOfIsInitialIsPushout is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] (F : CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (c : CategoryTheory.Limits.Cocone.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F) {X : C} 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(CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c))) (CategoryTheory.Discrete.mk.{0} CategoryTheory.Limits.WalkingPair CategoryTheory.Limits.WalkingPair.left)) g (CategoryTheory.NatTrans.app.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u2 u1} (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u2 u1} (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u2 u1} C _inst_1 (CategoryTheory.Functor.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.category.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1) (CategoryTheory.Functor.const.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c)) (CategoryTheory.Limits.Cocone.ι.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c) (CategoryTheory.Discrete.mk.{0} CategoryTheory.Limits.WalkingPair CategoryTheory.Limits.WalkingPair.right)))))) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_1 F c)\nCase conversion may be inaccurate. Consider using '#align is_binary_coproduct_of_is_initial_is_pushout isBinaryCoproductOfIsInitialIsPushoutₓ'. -/\n/-- If a cospan is the pushout cospan under the initial object, then it is a binary coproduct. -/\ndef isBinaryCoproductOfIsInitialIsPushout (F : Discrete WalkingPair ⥤ C) (c : Cocone F) {X : C}\n    (hX : IsInitial X) (f : X ⟶ F.obj ⟨WalkingPair.left⟩) (g : X ⟶ F.obj ⟨WalkingPair.right⟩)\n    (hc :\n      IsColimit\n        (PushoutCocone.mk (c.ι.app ⟨WalkingPair.left⟩) (c.ι.app ⟨WalkingPair.right⟩ : _) <|\n          hX.hom_ext (f ≫ _) (g ≫ _))) :\n    IsColimit c\n    where\n  desc s :=\n    hc.desc\n      (PushoutCocone.mk (s.ι.app ⟨WalkingPair.left⟩) (s.ι.app ⟨WalkingPair.right⟩) (hX.hom_ext _ _))\n  fac s j :=\n    Discrete.casesOn j fun j =>\n      WalkingPair.casesOn j (hc.fac _ WalkingSpan.left) (hc.fac _ WalkingSpan.right)\n  uniq s m J :=\n    by\n    let c' :=\n      pushout_cocone.mk (c.ι.app ⟨walking_pair.left⟩ ≫ m) (c.ι.app ⟨walking_pair.right⟩ ≫ m)\n        (hX.hom_ext (f ≫ _) (g ≫ _))\n    rw [← J, ← J]\n    apply hc.hom_ext\n    rintro (_ | (_ | _)) <;>\n      simp only [pushout_cocone.mk_ι_app_zero, pushout_cocone.mk_ι_app, category.assoc]\n    congr 1\n    exacts[(hc.fac c' walking_span.left).symm, (hc.fac c' walking_span.left).symm,\n      (hc.fac c' walking_span.right).symm]\n#align is_binary_coproduct_of_is_initial_is_pushout isBinaryCoproductOfIsInitialIsPushout\n\n#print isCoproductOfIsInitialIsPushout /-\n/-- The pushout under the initial object is the coproduct -/\ndef isCoproductOfIsInitialIsPushout {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y)\n    (H₁ : IsInitial W)\n    (H₂ : IsColimit (PushoutCocone.mk _ _ (show h ≫ f = k ≫ g from H₁.hom_ext _ _))) :\n    IsColimit (BinaryCofan.mk f g) :=\n  by\n  apply isBinaryCoproductOfIsInitialIsPushout _ _ H₁\n  exact H₂\n#align is_coproduct_of_is_initial_is_pushout isCoproductOfIsInitialIsPushout\n-/\n\n#print isPushoutOfIsInitialIsCoproduct /-\n/-- The coproduct is the pushout under the initial object. -/\ndef isPushoutOfIsInitialIsCoproduct {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ X) (k : W ⟶ Y)\n    (H₁ : IsInitial W) (H₂ : IsColimit (BinaryCofan.mk f g)) :\n    IsColimit (PushoutCocone.mk _ _ (show h ≫ f = k ≫ g from H₁.hom_ext _ _)) :=\n  by\n  apply pushout_cocone.is_colimit_aux'\n  intro s\n  use H₂.desc (binary_cofan.mk s.inl s.inr)\n  use H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.left⟩\n  use H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.right⟩\n  intro m h₁ h₂\n  apply H₂.hom_ext\n  rintro ⟨⟨⟩⟩\n  · exact h₁.trans (H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.left⟩).symm\n  · exact h₂.trans (H₂.fac (binary_cofan.mk s.inl s.inr) ⟨walking_pair.right⟩).symm\n#align is_pushout_of_is_initial_is_coproduct isPushoutOfIsInitialIsCoproduct\n-/\n\n#print colimitCoconeOfInitialAndPushouts /-\n/-- Any category with pushouts and an initial object has a colimit cocone for each walking pair. -/\nnoncomputable def colimitCoconeOfInitialAndPushouts [HasInitial C] [HasPushouts C]\n    (F : Discrete WalkingPair ⥤ C) : ColimitCocone F\n    where\n  Cocone :=\n    { pt := pushout (initial.to (F.obj ⟨WalkingPair.left⟩)) (initial.to (F.obj ⟨WalkingPair.right⟩))\n      ι :=\n        Discrete.natTrans fun x =>\n          Discrete.casesOn x fun x => WalkingPair.casesOn x pushout.inl pushout.inr }\n  IsColimit := isBinaryCoproductOfIsInitialIsPushout F _ initialIsInitial _ _ (pushoutIsPushout _ _)\n#align colimit_cocone_of_initial_and_pushouts colimitCoconeOfInitialAndPushouts\n-/\n\nvariable (C)\n\n#print hasBinaryCoproducts_of_hasInitial_and_pushouts /-\n-- This is not an instance, as it is not always how one wants to construct binary coproducts!\n/-- Any category with pushouts and initial object has binary coproducts. -/\ntheorem hasBinaryCoproducts_of_hasInitial_and_pushouts [HasInitial C] [HasPushouts C] :\n    HasBinaryCoproducts C :=\n  { HasColimit := fun F => HasColimit.mk (colimitCoconeOfInitialAndPushouts F) }\n#align has_binary_coproducts_of_has_initial_and_pushouts hasBinaryCoproducts_of_hasInitial_and_pushouts\n-/\n\nvariable {C}\n\n#print preservesBinaryCoproductsOfPreservesInitialAndPushouts /-\n/-- A functor that preserves initial objects and pushouts preserves binary coproducts. -/\nnoncomputable def preservesBinaryCoproductsOfPreservesInitialAndPushouts [HasInitial C]\n    [HasPushouts C] [PreservesColimitsOfShape (Discrete.{0} PEmpty) F]\n    [PreservesColimitsOfShape WalkingSpan F] : PreservesColimitsOfShape (Discrete WalkingPair) F :=\n  ⟨fun K =>\n    preservesColimitOfPreservesColimitCocone (colimitCoconeOfInitialAndPushouts K).2\n      (by\n        apply\n          isBinaryCoproductOfIsInitialIsPushout _ _\n            (is_colimit_of_has_initial_of_preserves_colimit F)\n        apply is_colimit_of_has_pushout_of_preserves_colimit)⟩\n#align preserves_binary_coproducts_of_preserves_initial_and_pushouts preservesBinaryCoproductsOfPreservesInitialAndPushouts\n-/\n\n#print coprodIsoPushout /-\n/-- In a category with an initial object and pushouts,\na coproduct of objects `X` and `Y` is isomorphic to a pushout. -/\nnoncomputable def coprodIsoPushout [HasInitial C] [HasPushouts C] (X Y : C)\n    [HasBinaryCoproduct X Y] : X ⨿ Y ≅ pushout (initial.to X) (initial.to Y) :=\n  colimit.isoColimitCocone (colimitCoconeOfInitialAndPushouts _)\n#align coprod_iso_pushout coprodIsoPushout\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Limits/Constructions/BinaryProducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.3452919763296159}}
{"text": "import columnround\n\nopen columnround\nopen utils\n\nnamespace columnround_examples\n\n/-!\n  # Columnround examples\n\n  ## Spec examples\n-/\n\n/-! ### Example 1 -/\n\n/-- Input for example 1 --/\ndef input1 : matrixType :=\n  (\n    (0x00000001, 0x00000000, 0x00000000, 0x00000000),\n    (0x00000001, 0x00000000, 0x00000000, 0x00000000),\n    (0x00000001, 0x00000000, 0x00000000, 0x00000000),\n    (0x00000001, 0x00000000, 0x00000000, 0x00000000)\n  )\n\n/-- Output for example 1 --/\ndef output1 : matrixType :=\n  (\n    (0x10090288, 0x00000000, 0x00000000, 0x00000000),\n    (0x00000101, 0x00000000, 0x00000000, 0x00000000),\n    (0x00020401, 0x00000000, 0x00000000, 0x00000000),\n    (0x40a04001, 0x00000000, 0x00000000, 0x00000000)\n  )\n\n--#eval if columnround_output (columnround (columnround_input input1)) = output1 then \"pass\" else \"fail\"\n\n/-! ### Example 2 -/\n\n/- Input for example 2 -/\ndef input2 : matrixType :=\n  (\n    (0x08521bd6, 0x1fe88837, 0xbb2aa576, 0x3aa26365),\n    (0xc54c6a5b, 0x2fc74c2f, 0x6dd39cc3, 0xda0a64f6),\n    (0x90a2f23d, 0x067f95a6, 0x06b35f61, 0x41e4732e),\n    (0xe859c100, 0xea4d84b7, 0x0f619bff, 0xbc6e965a)\n  )\n\n/-- Output for example 2 -/\ndef output2 : matrixType :=\n  (\n    (0x8c9d190a, 0xce8e4c90, 0x1ef8e9d3, 0x1326a71a),\n    (0x90a20123, 0xead3c4f3, 0x63a091a0, 0xf0708d69),\n    (0x789b010c, 0xd195a681, 0xeb7d5504, 0xa774135c),\n    (0x481c2027, 0x53a8e4b5, 0x4c1f89c5, 0x3f78c9c8)\n  )\n\n--#eval if columnround_output (columnround (columnround_input input2)) = output2 then \"pass\" else \"fail\"\n\n/-!\n  ## Inverse examples\n\n  We use the same test vectors as the spec but going backwards.\n-/\n\n-- example 1\n--#eval if columnround_output (columnround_inv (columnround_input output1)) = input1 then \"pass\" else \"fail\"\n\n-- example 2\n--#eval if columnround_output (columnround_inv (columnround_input output2)) = input2 then \"pass\" else \"fail\"\n\nend columnround_examples\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/examples/columnround.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34513532255709006}}
{"text": "import hott.init\nopen expr tactic pexpr hott\nuniverses u v\nnoncomputable theory\naxiom sorry' : Π{α : Sort _}, α /- no warnings are generated when using this axiom -/\n\n@[induction] def foo2 {X Y : Type} {P : X × Y → Type} (x : X × Y) (z : Πa b, P ⟨a, b⟩) : P x := sorry'\n@[induction] def foo3 {P : ℕ × ℕ → Sort _} (x : ℕ × ℕ) (z : ℕ → P (0,0)) : P x := sorry'\n@[induction] def foo4 {X Y : Type} {P : Type} (x : X × Y) (z : X → Y → P) : P := sorry'\n@[induction] def foo5 {X : Type} {P : Type} (x : ℕ × X) (z : X → X → P) : P := sorry'\n@[induction] def foo6 {P : hott.trunc -2 nat → Type} (x : hott.trunc -2 nat) (z : Π(n : ℕ), P (hott.trunc.tr n)) : P x := sorry'\n@[induction] def foo7 {P : hott.trunc -2 nat → Type _} (x : hott.trunc -2 nat) (z : Πx, P x → P x) : (ℕ → P (hott.trunc.tr 0)) → P x := sorry'\n@[induction] def foo8 {P : hott.trunc -2 nat → Type _} (x : hott.trunc -2 nat) : P x := sorry'\n@[induction] def foo9 (n : ℕ₋₂) (A : Type _) {P : hott.trunc n A → Type _} (x : hott.trunc n A) : (Πx, P x) → P x := sorry'\n@[induction] def foo10 (n : ℕ₋₂) (A : Type _) {P : Type _} (x : hott.trunc n A) : (ℕ → P) → P := sorry'\nattribute [induction] trunc.rec\nattribute [induction] hott.quotient.rec\nattribute [induction, priority 2000] hott.trunc.rec\nattribute [induction] hott.eq.idp_rec_on\nattribute [induction] hott.eq.idp_rec_on\nattribute [induction] hott.eq.rec\nattribute [induction] prod.elim\ndef foo11 {X : Type} {P : Type} (x : X) (z : X → X → P) : P := sorry'\ndef foo12 {X : Type} {P : Type} (x : ℕ × X) (z : x = x → P) : P := sorry'\ndef foo13 {X : Type} {P : Type} (x : ℕ) (z : P) : P := sorry'\n\nrun_cmd success_if_fail $ get_induction_info `foo11\nrun_cmd success_if_fail $ get_induction_info `foo12\nrun_cmd success_if_fail $ get_induction_info `foo13\n\ndef indfoo {X Y : Type} (x : hott.trunc -2 nat) : hott.eq x x :=\nbegin\n  hinduction x using foo7 with x' p n, all_goals { exact sorry' }\nend\n\ndef indfoo2 (x : ℕ) (y : ℕ) : x = x :=\nbegin\n  hinduction x with n IH,\n  all_goals { exact sorry' }\nend\n\ndef indfoo3 {X Y : Type} {P : bool × bool → Type} (x : bool × bool) (z : Πx, P x) : P x :=\nbegin\n  hinduction x,\n  exact sorry'\nend\n\ndef indfoo4 {X Y : Type} {P : Type} (x : bool × bool) (z : P) : P :=\nbegin\n  success_if_fail { hinduction x using foo5 },\n  hinduction x,\n  exact sorry'\nend\n\ndef indfoo5 {X Y : Type} {P : hott.trunc -2 nat → Type} (x : hott.trunc -2 nat) (z : Πx, P x) : P x :=\nbegin\n  hinduction x,\n  exact sorry'\nend\n\ndef indfoo6 {X Y : Type} (x : hott.trunc -2 nat) : hott.eq x x :=\nbegin\n  hinduction x using trunc.rec with x' p n, reflexivity\nend\n\ndef indfoo7 {X Y : Type} (x : id (hott.trunc -2 nat)) : hott.eq x x :=\nbegin\n  hinduction x using trunc.rec with x' p n, reflexivity\nend  \n\ndef indfoo8 {X Y : Type} (x : id (hott.trunc -2 nat)) : hott.eq x x :=\nbegin\n  hinduction x with x' p n, reflexivity\nend  \n\nexample {X Y : Type} (x : hott.trunc -2 nat) : hott.eq x x :=\nbegin\n  hinduction x using foo8 with x' p n\nend\n\nexample {X Y : Type} (x : hott.trunc -2 nat) : hott.eq x x :=\nbegin\n  hinduction x using foo9 with x' p n, all_goals { exact sorry' }\nend\n\nexample {X Y : Type} (x : hott.trunc -2 nat) : hott.eq x x :=\nbegin\n  success_if_fail { hinduction x using foo10 with x' p n }, exact sorry'\nend\n\nexample (x : ℕ) (y : ℕ) : x = x :=\nbegin\n  hinduction x using nat.rec with n IH generalizing y,\n  all_goals { exact sorry' }\nend\n\nexample (x : ℕ) (y : ℕ) : x = x :=\nbegin\n  revert y,\n  hinduction x using nat.rec with n IH,\n  all_goals { intro },\n  all_goals { exact sorry' }\nend\n\nexample (x : ℕ) (p : x = 3) : x = x :=\nbegin\n  hinduction x using nat.rec with n IH,\n  all_goals { exact sorry' }\nend\n\nexample (x : ℕ) : let y := x in Π(p : y = y), x = y :=\nbegin\n  intros _ _,\n  hinduction x using nat.rec with n IH generalizing y,\n  all_goals { exact sorry' }\nend\n\nhott_theory\n\n-- set_option trace.hinduction true\nexample {A : Type} {a b : A} {f : A → A} (p : f a = b) : unit :=\nbegin hinduction p using hott.eq.rec, constructor end\nexample {A : Type} {a : A} {f : A → A} (p : f a = a) : unit :=\nbegin success_if_fail { hinduction p using hott.eq.rec }, constructor end\n\nexample {A : Type} {a : A} {f : A → A} (p : f a = a) : unit :=\nbegin success_if_fail { hinduction p }, constructor end\nexample {A : Type} {a : A} {f : A → A} (p : f a = a) : unit :=\nbegin success_if_fail { hinduction p using hott.eq.rec }, constructor end\nexample {A : Type} {a b : A} {f : A → A} (p : f a = id b) : unit :=\nbegin hinduction p using hott.eq.rec, constructor end\nexample {A : Type} {a b : A} {f : A → A} (p : f a = let u := b in b) : unit :=\nbegin hinduction p using hott.eq.rec, constructor end\nexample {A : Type} {a b : A} {f : A → A} (p : f a = let u := b in u) : unit :=\nbegin hinduction p using hott.eq.rec, constructor end\n-- example {A B : Type} {a b : A → B} {f : (A → B) → A → B} (p : f a = λx, b x) : unit :=\n-- begin hinduction p using hott.eq.rec, constructor end\nexample {A : Type} {a b : A} {f : A → A} (p : f a = (λx, x) b) : unit :=\nbegin hinduction p using hott.eq.rec, constructor end\n\n\n\n@[induction] def eqrec1 {A : Type u} {a : A} {C : Π (a' : A), a = a' → Sort v} (H : C a (refl a)) {a' : A} (n : a = a') : C a' n := sorry'\n@[induction] def eqrec2 {A : Type u} {a : A} {C : a = a → Sort v} (H : C (refl a)) (n : a = a) : C n := sorry'\n@[induction] def eqrec3 {A : Type u} {C : Π (a' : A), a' = a' → Sort v} (H : Πa, C a (refl a)) {a : A} (n : a = a) : C a n := sorry'\n@[induction] def eqrec4 {A : Type u} {a : A} {C : A → Sort v} (H : C a) {a' : A} (n : a = a') : C a' := sorry'\n@[induction] def eqrec5 {A : Type u} {a : A} {C : Sort v} (H : C) {a' : A} (n : a = a') : C := sorry'\nattribute [induction] pathover.rec idp_rec_on\n-- #print eqrec1._ind_info\n-- #print eqrec2._ind_info\n-- #print eqrec3._ind_info\n-- #print eqrec4._ind_info\n-- #print eqrec5._ind_info\n\n/- test unfolding of type of induction variable -/\ndef list2 (α : Type u) : Type u := list (prod α α)\ndef list3 (α : Type u) : Type u := list2 (prod α α)\n@[induction] def list2rec {α} (P : list2 α → Type) (H1 : P []) \n  (H2 : Πx y l, P l → P ((x,y)::l)) : Πx, P x := \nby intro x; hinduction x; [exact H1, {induction hd; exact H2 _ _ _ ih}]\nexample {α} (x : list3 α) : x = x :=\nby hinduction x using list2rec; refl\n\nopen hott.trunc hott.is_trunc\n@[hott] def trunc_sigma_equiv {n : ℕ₋₂} {A : Type _} {P : A → Type _} :\n  trunc n (Σ x, P x) ≃ trunc n (Σ x, trunc n (P x)) :=\nbegin\n  fapply equiv.MK; intro x,\n  { hinduction x with p, exact tr ⟨p.1, tr p.2⟩ },\n  { hinduction x with p, induction p with a p, hinduction p with p, exact tr ⟨a, p⟩ },\n  all_goals { exact sorry' }\nend\n\n@[hott] def trunc_sigma_equiv2 {n : ℕ₋₂} {A : Type _} {P : A → Type _} :\n  trunc n (Σ x, P x) ≃ trunc n (Σ x, trunc n (P x)) :=\nbegin\n  fapply equiv.MK; intro x,\n  { hinduction x with p, exact tr ⟨p.1, tr p.2⟩ },\n  { hinduction x with p, have x := p.2, hinduction x with q, exact tr ⟨p.1, q⟩ },\n  all_goals { exact sorry' }\nend\n\nset_option trace.hinduction true\n-- attribute [induction] homotopy.rec_on\n-- #print homotopy.rec_on\n-- -- #print homotopy.rec_on._ind_info\n-- @[hott] example {A B : Type _} {f g : A → B} (h : f ~ g) : unit :=\n-- begin hinduction h using homotopy.rec_on, constructor end\n\n-- @[hott] example {A : Type _} {B : A → Type _} {f g : Πx, B x} (h : @homotopy A B f g) : unit :=\n-- begin hinduction h using homotopy.rec_on, constructor end\n\n\ndef eqrecfail1 {A : Type u} {a : A} {C : Π (a' : A), a = a' → Sort v} (H : C a (refl a)) {a' : A} (n : a = a') : C (id a') n := sorry'\ndef eqrecfail2 {A : Type u} {a : A} {C : Π (a' : A), a = a' → Sort v} (H : C a (refl a)) {a' : A} (n : a = a') : C a' (n ⬝ idp) := sorry'\ndef eqrecfail3 {A : Type u} {a : A} {C : Π (a' : A), a = a' → Sort v} (H : C a (refl a)) {a' : A} (n : a = id a') : C a' n := sorry'\n\nrun_cmd success_if_fail $ get_induction_info `eqrecfail1\nrun_cmd success_if_fail $ get_induction_info `eqrecfail2\n-- run_cmd success_if_fail $ get_induction_info `eqrecfail3\n--run_cmd success_if_fail $ get_induction_info `eqrecfail4\n-- #print eqrec1._ind_info\n-- #print idp_rec_on._ind_info\n-- #print idp_rec_on\n", "meta": {"author": "gebner", "repo": "hott3", "sha": "7ead7a8a2503049eacd45cbff6587802bae2add2", "save_path": "github-repos/lean/gebner-hott3", "path": "github-repos/lean/gebner-hott3/hott3-7ead7a8a2503049eacd45cbff6587802bae2add2/tests/induction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34513531479044884}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport category_theory.idempotents.karoubi\n\n/-!\n# Extension of functors to the idempotent completion\n\nIn this file, we construct an extension `functor_extension₁`\nof functors `C ⥤ karoubi D` to functors `karoubi C ⥤ karoubi D`. This results in an\nequivalence `karoubi_universal₁ C D : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)`.\n\nWe also construct an extension `functor_extension₂` of functors\n`(C ⥤ D) ⥤ (karoubi C ⥤ karoubi D)`. Moreover,\nwhen `D` is idempotent complete, we get equivalences\n`karoubi_universal₂ C D : C ⥤ D ≌ karoubi C ⥤ karoubi D`\nand `karoubi_universal C D : C ⥤ D ≌ karoubi C ⥤ D`.\n\nWe occasionally state and use equalities of functors because it is\nsometimes convenient to use rewrites when proving properties of\nfunctors obtained using the constructions in this file. Users are\nencouraged to use the corresponding natural isomorphism\nwhenever possible.\n\n-/\n\nopen category_theory.category\nopen category_theory.idempotents.karoubi\n\nnamespace category_theory\n\nnamespace idempotents\n\nvariables {C D E : Type*} [category C] [category D] [category E]\n\n/-- A natural transformation between functors `karoubi C ⥤ D` is determined\nby its value on objects coming from `C`. -/\nlemma nat_trans_eq {F G : karoubi C ⥤ D} (φ : F ⟶ G) (P : karoubi C) :\n  φ.app P = F.map (decomp_id_i P) ≫ φ.app P.X ≫ G.map (decomp_id_p P) :=\nbegin\n  rw [← φ.naturality, ← assoc, ← F.map_comp],\n  conv { to_lhs, rw [← id_comp (φ.app P), ← F.map_id], },\n  congr,\n  apply decomp_id,\nend\n\nnamespace functor_extension₁\n\n/-- The canonical extension of a functor `C ⥤ karoubi D` to a functor\n`karoubi C ⥤ karoubi D` -/\n@[simps]\ndef obj (F : C ⥤ karoubi D) : karoubi C ⥤ karoubi D :=\n{ obj := λ P, ⟨(F.obj P.X).X, (F.map P.p).f,\n    by simpa only [F.map_comp, hom_ext] using F.congr_map P.idem⟩,\n  map := λ P Q f, ⟨(F.map f.f).f,\n    by simpa only [F.map_comp, hom_ext] using F.congr_map f.comm⟩, }\n\n/-- Extension of a natural transformation `φ` between functors\n`C ⥤ karoubi D` to a natural transformation between the\nextension of these functors to `karoubi C ⥤ karoubi D` -/\n@[simps]\ndef map {F G : C ⥤ karoubi D} (φ : F ⟶ G) : obj F ⟶ obj G :=\n{ app := λ P,\n  { f := (F.map P.p).f ≫ (φ.app P.X).f,\n    comm := begin\n      have h := φ.naturality P.p,\n      have h' := F.congr_map P.idem,\n      simp only [hom_ext, karoubi.comp_f, F.map_comp] at h h',\n      simp only [obj_obj_p, assoc, ← h],\n      slice_rhs 1 3 { rw [h', h'], },\n    end, },\n  naturality' := λ P Q f, begin\n    ext,\n    dsimp [obj],\n    have h := φ.naturality f.f,\n    have h' := F.congr_map (comp_p f),\n    have h'' := F.congr_map (p_comp f),\n    simp only [hom_ext, functor.map_comp, comp_f] at ⊢ h h' h'',\n    slice_rhs 2 3 { rw ← h, },\n    slice_lhs 1 2 { rw h', },\n    slice_rhs 1 2 { rw h'', },\n  end }\n\nend functor_extension₁\n\nvariables (C D E)\n\n/-- The canonical functor `(C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D)` -/\n@[simps]\ndef functor_extension₁ : (C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D) :=\n{ obj := functor_extension₁.obj,\n  map := λ F G, functor_extension₁.map,\n  map_id' := λ F, by { ext P, exact comp_p (F.map P.p), },\n  map_comp' := λ F G H φ φ', begin\n    ext P,\n    simp only [comp_f, functor_extension₁.map_app_f, nat_trans.comp_app, assoc],\n    have h := φ.naturality P.p,\n    have h' := F.congr_map P.idem,\n    simp only [hom_ext, comp_f, F.map_comp] at h h',\n    slice_rhs 2 3 { rw ← h, },\n    slice_rhs 1 2 { rw h', },\n    simp only [assoc],\n  end, }\n\nlemma functor_extension₁_comp_whiskering_left_to_karoubi :\n  functor_extension₁ C D ⋙\n    (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) = 𝟭 _ :=\nbegin\n  refine functor.ext _ _,\n  { intro F,\n    refine functor.ext _ _,\n    { intro X,\n      ext,\n      { dsimp,\n        rw [id_comp, comp_id, F.map_id, id_eq], },\n      { refl, }, },\n    { intros X Y f,\n      ext,\n      dsimp,\n      simp only [comp_id, eq_to_hom_f, eq_to_hom_refl, comp_p, functor_extension₁.obj_obj_p,\n        to_karoubi_obj_p, comp_f],\n      dsimp,\n      simp only [functor.map_id, id_eq, p_comp], }, },\n  { intros F G φ,\n    ext X,\n    dsimp,\n    simp only [eq_to_hom_app, F.map_id, comp_f, eq_to_hom_f, id_eq, p_comp,\n      eq_to_hom_refl, comp_id, comp_p, functor_extension₁.obj_obj_p,\n      to_karoubi_obj_p, F.map_id X], },\nend\n\n/-- The natural isomorphism expressing that functors `karoubi C ⥤ karoubi D` obtained\nusing `functor_extension₁` actually extends the original functors `C ⥤ karoubi D`. -/\n@[simps]\ndef functor_extension₁_comp_whiskering_left_to_karoubi_iso :\n  functor_extension₁ C D ⋙\n    (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ≅ 𝟭 _ :=\neq_to_iso (functor_extension₁_comp_whiskering_left_to_karoubi C D)\n\n/-- The counit isomorphism of the equivalence `(C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)`. -/\n@[simps]\ndef karoubi_universal₁.counit_iso :\n  (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ⋙\n    functor_extension₁ C D ≅ 𝟭 _ :=\nnat_iso.of_components (λ G,\n  { hom :=\n    { app := λ P,\n      { f := (G.map (decomp_id_p P)).f,\n        comm := by simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map\n          (show P.decomp_id_p = (to_karoubi C).map P.p ≫ P.decomp_id_p ≫ 𝟙 _, by simp), },\n      naturality' := λ P Q f,\n        by simpa only [hom_ext, G.map_comp] using (G.congr_map (decomp_id_p_naturality f)).symm, },\n    inv :=\n    { app := λ P,\n      { f := (G.map (decomp_id_i P)).f,\n        comm := by simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map\n          (show P.decomp_id_i = 𝟙 _ ≫ P.decomp_id_i ≫ (to_karoubi C).map P.p, by simp), },\n      naturality' := λ P Q f,\n        by simpa only [hom_ext, G.map_comp] using G.congr_map (decomp_id_i_naturality f), },\n    hom_inv_id' := begin\n      ext P,\n      simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map P.decomp_p.symm,\n    end,\n    inv_hom_id' := begin\n      ext P,\n      simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map P.decomp_id.symm,\n    end, })\n(λ G₁ G₂ φ, begin\n  ext P,\n  dsimp,\n  simpa only [nat_trans_eq φ P, comp_f, functor_extension₁.map_app_f,\n    functor.comp_map, whisker_left_app, assoc, P.decomp_p, G₁.map_comp],\nend)\n\n/-- The equivalence of categories `(C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)`. -/\n@[simps]\ndef karoubi_universal₁ : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D) :=\n{ functor := functor_extension₁ C D,\n  inverse := (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C),\n  unit_iso := (functor_extension₁_comp_whiskering_left_to_karoubi_iso C D).symm,\n  counit_iso := karoubi_universal₁.counit_iso C D,\n  functor_unit_iso_comp' := λ F, begin\n    ext P,\n    dsimp [functor_extension₁.map, karoubi_universal₁.counit_iso],\n    simpa only [comp_f, eq_to_hom_app, eq_to_hom_f, eq_to_hom_refl, comp_id,\n      hom_ext, F.map_comp, comp_p] using F.congr_map P.idem,\n  end, }\n\nlemma functor_extension₁_comp (F : C ⥤ karoubi D) (G : D ⥤ karoubi E) :\n  (functor_extension₁ C E).obj (F ⋙ (functor_extension₁ D E).obj G) =\n    (functor_extension₁ C D).obj F ⋙ (functor_extension₁ D E).obj G :=\nfunctor.ext (by tidy) (λ X Y f, by { dsimp, simpa only [id_comp, comp_id], })\n\n/-- The canonical functor `(C ⥤ D) ⥤ (karoubi C ⥤ karoubi D)` -/\n@[simps]\ndef functor_extension₂ : (C ⥤ D) ⥤ (karoubi C ⥤ karoubi D) :=\n(whiskering_right C D (karoubi D)).obj (to_karoubi D) ⋙ functor_extension₁ C D\n\nlemma functor_extension₂_comp_whiskering_left_to_karoubi :\n  functor_extension₂ C D ⋙ (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) =\n  (whiskering_right C D (karoubi D)).obj (to_karoubi D) :=\nby simp only [functor_extension₂, functor.assoc,\n  functor_extension₁_comp_whiskering_left_to_karoubi, functor.comp_id]\n\n/-- The natural isomorphism expressing that functors `karoubi C ⥤ karoubi D` obtained\nusing `functor_extension₂` actually extends the original functors `C ⥤ D`. -/\n@[simps]\ndef functor_extension₂_comp_whiskering_left_to_karoubi_iso :\n  functor_extension₂ C D ⋙ (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ≅\n  (whiskering_right C D (karoubi D)).obj (to_karoubi D) :=\neq_to_iso (functor_extension₂_comp_whiskering_left_to_karoubi C D)\n\nsection is_idempotent_complete\n\nvariable [is_idempotent_complete D]\n\nnoncomputable instance : is_equivalence (to_karoubi D) := to_karoubi_is_equivalence D\n\n/-- The equivalence of categories `(C ⥤ D) ≌ (karoubi C ⥤ karoubi D)` when `D`\nis idempotent complete. -/\n@[simps]\nnoncomputable def karoubi_universal₂ : (C ⥤ D) ≌ (karoubi C ⥤ karoubi D) :=\n(equivalence.congr_right (to_karoubi D).as_equivalence).trans\n    (karoubi_universal₁ C D)\n\nlemma karoubi_universal₂_functor_eq :\n  (karoubi_universal₂ C D).functor = functor_extension₂ C D := rfl\n\nnoncomputable instance : is_equivalence (functor_extension₂ C D) :=\nby { rw ← karoubi_universal₂_functor_eq, apply_instance, }\n\n/-- The extension of functors functor `(C ⥤ D) ⥤ (karoubi C ⥤ D)`\nwhen `D` is idempotent compltete. -/\n@[simps]\nnoncomputable def functor_extension : (C ⥤ D) ⥤ (karoubi C ⥤ D) :=\nfunctor_extension₂ C D ⋙ (whiskering_right (karoubi C) (karoubi D) D).obj\n    (to_karoubi_is_equivalence D).inverse\n\n/-- The equivalence `(C ⥤ D) ≌ (karoubi C ⥤ D)` when `D` is idempotent complete. -/\n@[simps]\nnoncomputable def karoubi_universal : (C ⥤ D) ≌ (karoubi C ⥤ D) :=\n(karoubi_universal₂ C D).trans (equivalence.congr_right (to_karoubi D).as_equivalence.symm)\n\nlemma karoubi_universal_functor_eq :\n  (karoubi_universal C D).functor = functor_extension C D := rfl\n\nnoncomputable instance : is_equivalence (functor_extension C D) :=\nby { rw ← karoubi_universal_functor_eq, apply_instance, }\n\nnoncomputable instance : is_equivalence ((whiskering_left C (karoubi C) D).obj (to_karoubi C)) :=\nis_equivalence.cancel_comp_right _ ((whiskering_right C _ _).obj (to_karoubi D) ⋙\n    (whiskering_right C _ _).obj (to_karoubi D).inv)\n  (is_equivalence.of_equivalence (@equivalence.congr_right _ _ _ _ C _\n      ((to_karoubi D).as_equivalence.trans (to_karoubi D).as_equivalence.symm)))\n  (by { change is_equivalence (karoubi_universal C D).inverse, apply_instance, })\n\nvariables {C D}\n\nlemma whiskering_left_obj_preimage_app {F G : karoubi C ⥤ D}\n  (τ : to_karoubi _ ⋙ F ⟶ to_karoubi _ ⋙ G) (P : karoubi C) :\n  (((whiskering_left _ _ _).obj (to_karoubi _)).preimage τ).app P =\n    F.map P.decomp_id_i ≫ τ.app P.X ≫ G.map P.decomp_id_p :=\nbegin\n  rw nat_trans_eq,\n  congr' 2,\n  exact congr_app (((whiskering_left _ _ _).obj (to_karoubi _)).image_preimage τ) P.X,\nend\n\nend is_idempotent_complete\n\nend idempotents\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/idempotents/functor_extension.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34513530702380757}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nTraversable instance for lazy_lists.\n-/\nimport control.traversable.equiv\nimport control.traversable.instances\nimport data.lazy_list\n\n/-!\n## Definitions on lazy lists\n\nThis file contains various definitions and proofs on lazy lists.\n\nTODO: move the `lazy_list.lean` file from core to mathlib.\n-/\n\nuniverses u\n\nnamespace thunk\n\n/-- Creates a thunk with a (non-lazy) constant value. -/\ndef mk {α} (x : α) : thunk α := λ _, x\n\ninstance {α : Type u} [decidable_eq α] : decidable_eq (thunk α) | a b :=\nhave a = b ↔ a () = b (), from ⟨by cc, by intro; ext x; cases x; assumption⟩,\nby rw this; apply_instance\n\nend thunk\n\nnamespace lazy_list\n\nopen function\n\n/-- Isomorphism between strict and lazy lists. -/\ndef list_equiv_lazy_list (α : Type*) : list α ≃ lazy_list α :=\n{ to_fun := lazy_list.of_list,\n  inv_fun := lazy_list.to_list,\n  right_inv := by { intro, induction x, refl, simp! [*],\n                    ext, cases x, refl },\n  left_inv := by { intro, induction x, refl, simp! [*] } }\n\ninstance {α : Type u} : inhabited (lazy_list α) := ⟨nil⟩\n\ninstance {α : Type u} [decidable_eq α] : decidable_eq (lazy_list α)\n| nil nil := is_true rfl\n| (cons x xs) (cons y ys) :=\n  if h : x = y then\n    match decidable_eq (xs ()) (ys ()) with\n    | is_false h2 := is_false (by intro; cc)\n    | is_true h2 :=\n      have xs = ys, by ext u; cases u; assumption,\n      is_true (by cc)\n    end\n  else\n    is_false (by intro; cc)\n| nil (cons _ _) := is_false (by cc)\n| (cons _ _) nil := is_false (by cc)\n\n/-- Traversal of lazy lists using an applicative effect. -/\nprotected def traverse {m : Type u → Type u} [applicative m] {α β : Type u}\n    (f : α → m β) : lazy_list α → m (lazy_list β)\n| lazy_list.nil := pure lazy_list.nil\n| (lazy_list.cons x xs) := lazy_list.cons <$> f x <*> (thunk.mk <$> traverse (xs ()))\n\ninstance : traversable lazy_list :=\n{ map := @lazy_list.traverse id _,\n  traverse := @lazy_list.traverse }\n\ninstance : is_lawful_traversable lazy_list :=\nbegin\n  apply equiv.is_lawful_traversable' list_equiv_lazy_list;\n  intros ; resetI; ext,\n  { induction x, refl,\n    simp! [equiv.map,functor.map] at *,\n    simp [*], refl, },\n  { induction x, refl,\n    simp! [equiv.map,functor.map_const] at *,\n    simp [*], refl, },\n  { induction x,\n    { simp! [traversable.traverse,equiv.traverse] with functor_norm, refl },\n    simp! [equiv.map,functor.map_const,traversable.traverse] at *, rw x_ih,\n    dsimp [list_equiv_lazy_list,equiv.traverse,to_list,traversable.traverse,list.traverse],\n    simp! with functor_norm, refl },\nend\n\n/-- `init xs`, if `xs` non-empty, drops the last element of the list.\nOtherwise, return the empty list. -/\ndef init {α} : lazy_list α → lazy_list α\n| lazy_list.nil := lazy_list.nil\n| (lazy_list.cons x xs) :=\n  let xs' := xs () in\n  match xs' with\n  | lazy_list.nil := lazy_list.nil\n  | (lazy_list.cons _ _) := lazy_list.cons x (init xs')\n  end\n\n/-- Return the first object contained in the list that satisfies\npredicate `p` -/\ndef find {α} (p : α → Prop) [decidable_pred p] : lazy_list α → option α\n| nil        := none\n| (cons h t) := if p h then some h else find (t ())\n\n/-- `interleave xs ys` creates a list where elements of `xs` and `ys` alternate. -/\ndef interleave {α} : lazy_list α → lazy_list α → lazy_list α\n| lazy_list.nil xs := xs\n| a@(lazy_list.cons x xs) lazy_list.nil := a\n| (lazy_list.cons x xs) (lazy_list.cons y ys) :=\n  lazy_list.cons x (lazy_list.cons y (interleave (xs ()) (ys ())))\n\n/-- `interleave_all (xs::ys::zs::xss)` creates a list where elements of `xs`, `ys`\nand `zs` and the rest alternate. Every other element of the resulting list is taken from\n`xs`, every fourth is taken from `ys`, every eighth is taken from `zs` and so on. -/\ndef interleave_all {α} : list (lazy_list α) → lazy_list α\n| [] := lazy_list.nil\n| (x :: xs) := interleave x (interleave_all xs)\n\n/-- Monadic bind operation for `lazy_list`. -/\nprotected def bind {α β} : lazy_list α → (α → lazy_list β) → lazy_list β\n| lazy_list.nil _ := lazy_list.nil\n| (lazy_list.cons x xs) f := lazy_list.append (f x) (bind (xs ()) f)\n\n/-- Reverse the order of a `lazy_list`.\nIt is done by converting to a `list` first because reversal involves evaluating all\nthe list and if the list is all evaluated, `list` is a better representation for\nit than a series of thunks. -/\ndef reverse {α} (xs : lazy_list α) : lazy_list α :=\nof_list xs.to_list.reverse\n\ninstance : monad lazy_list :=\n{ pure := @lazy_list.singleton,\n  bind := @lazy_list.bind }\n\n\n\nlemma append_assoc {α} (xs ys zs : lazy_list α) :\n  (xs.append ys).append zs = xs.append (ys.append zs) :=\nby induction xs; simp [append, *]\n\nlemma append_bind {α β} (xs : lazy_list α) (ys : thunk (lazy_list α)) (f : α → lazy_list β) :\n  (@lazy_list.append _ xs ys).bind f = (xs.bind f).append ((ys ()).bind f) :=\nby induction xs; simp [lazy_list.bind, append, *, append_assoc, append, lazy_list.bind]\n\ninstance : is_lawful_monad lazy_list :=\n{ pure_bind := by { intros, apply append_nil },\n  bind_assoc := by { intros, dsimp [(>>=)], induction x; simp [lazy_list.bind, append_bind, *], },\n  id_map :=\n  begin\n    intros,\n    simp [(<$>)],\n    induction x; simp [lazy_list.bind, *, singleton, append],\n    ext ⟨ ⟩, refl,\n  end }\n\n/-- Try applying function `f` to every element of a `lazy_list` and\nreturn the result of the first attempt that succeeds. -/\ndef mfirst {m} [alternative m] {α β} (f : α → m β) : lazy_list α → m β\n| nil := failure\n| (cons x xs) :=\n  f x <|> mfirst (xs ())\n\n/-- Membership in lazy lists -/\nprotected def mem {α} (x : α) : lazy_list α → Prop\n| lazy_list.nil := false\n| (lazy_list.cons y ys) := x = y ∨ mem (ys ())\n\ninstance {α} : has_mem α (lazy_list α) :=\n⟨ lazy_list.mem ⟩\n\ninstance mem.decidable {α} [decidable_eq α] (x : α) : Π xs : lazy_list α, decidable (x ∈ xs)\n| lazy_list.nil := decidable.false\n| (lazy_list.cons y ys) :=\n  if h : x = y\n    then decidable.is_true (or.inl h)\n    else decidable_of_decidable_of_iff (mem.decidable (ys ())) (by simp [*, (∈), lazy_list.mem])\n\n@[simp]\nlemma mem_nil {α} (x : α) : x ∈ @lazy_list.nil α ↔ false := iff.rfl\n\n@[simp]\nlemma mem_cons {α} (x y : α) (ys : thunk (lazy_list α)) :\n  x ∈ @lazy_list.cons α y ys ↔ x = y ∨ x ∈ ys () := iff.rfl\n\ntheorem forall_mem_cons {α} {p : α → Prop} {a : α} {l : thunk (lazy_list α)} :\n  (∀ x ∈ @lazy_list.cons _ a l, p x) ↔ p a ∧ ∀ x ∈ l (), p x :=\nby simp only [has_mem.mem, lazy_list.mem, or_imp_distrib, forall_and_distrib, forall_eq]\n\n/-! ### map for partial functions -/\n\n/-- Partial map. If `f : Π a, p a → β` is a partial function defined on\n  `a : α` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`\n  but is defined only when all members of `l` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {α β} {p : α → Prop} (f : Π a, p a → β) :\n  Π l : lazy_list α, (∀ a ∈ l, p a) → lazy_list β\n| lazy_list.nil         H := lazy_list.nil\n| (lazy_list.cons x xs) H := lazy_list.cons (f x (forall_mem_cons.1 H).1)\n                               (pmap (xs ()) (forall_mem_cons.1 H).2)\n\n/-- \"Attach\" the proof that the elements of `l` are in `l` to produce a new `lazy_list`\n  with the same elements but in the type `{x // x ∈ l}`. -/\ndef attach {α} (l : lazy_list α) : lazy_list {x // x ∈ l} := pmap subtype.mk l (λ a, id)\n\ninstance {α} [has_repr α] : has_repr (lazy_list α) :=\n⟨ λ xs, repr xs.to_list ⟩\n\nend lazy_list\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/lazy_list/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34474858372159606}}
{"text": "import category_theory.adjunction\n\nnamespace category_theory\n\nvariables {C D : Type*} [category C] [category D]\n\nstructure unit_desc (F : C ⥤ D) :=\n( obj : D → C )\n( unit : Π (X : D), X ⟶ F.obj (obj X) )\n( desc : Π {X Y}, (X ⟶ F.obj Y) → (obj X ⟶ Y))\n( unit_comp_desc : Π {X Y} (f : X ⟶ F.obj Y), unit X ≫ F.map (desc f) = f )\n( desc_unit : Π {X Y} (f : obj X ⟶ Y), desc (unit X ≫ F.map f) = f )\n\nvariables {F : C ⥤ D} (ud : unit_desc F)\n\nprivate lemma unit_desc_ext\n  {X : D} {Y : C} {f g : ud.obj X ⟶ Y}\n  (H : ud.unit X ≫ F.map f = ud.unit X ≫ F.map g) : f = g :=\nby rw [← ud.desc_unit f, H, ud.desc_unit]\n\ndef unit_desc.to_functor : D ⥤ C :=\n{ obj := ud.obj,\n  map := λ X Y f, ud.desc (f ≫ ud.unit Y),\n  map_id' := λ X, by rw [category.id_comp, ← category.comp_id (ud.unit X), ← F.map_id,\n      ud.desc_unit],\n  map_comp' := λ X Y Z f g, unit_desc_ext ud $\n    by rw [F.map_comp, ud.unit_comp_desc, ← category.assoc, ud.unit_comp_desc,\n      category.assoc, category.assoc, ud.unit_comp_desc] }\n\ndef unit_desc.adjunction :\n  ud.to_functor ⊣ F :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ud.unit X ≫ F.map f,\n    inv_fun := λ f, ud.desc f,\n    left_inv := λ f, ud.desc_unit f,\n    right_inv := λ f, ud.unit_comp_desc f },\n  hom_equiv_naturality_left_symm' :=\n    begin\n      intros X' X Y f g,\n      dsimp [unit_desc.to_functor],\n      apply unit_desc_ext ud,\n      rw [ud.unit_comp_desc, F.map_comp, ← category.assoc,\n        ud.unit_comp_desc, category.assoc, ud.unit_comp_desc],\n    end,\n  hom_equiv_naturality_right' :=\n    begin\n      intros X Y Y' f g,\n      dsimp [unit_desc.to_functor],\n      rw [F.map_comp, category.assoc]\n    end }\n\nstructure counit_lift (F : C ⥤ D) :=\n( obj : D → C )\n( counit : Π (X : D), F.obj (obj X) ⟶ X )\n( lift : Π {X Y}, (F.obj X ⟶ Y) → (X ⟶ obj Y))\n( lift_comp_counit : Π {X Y} (f : F.obj X ⟶ Y), F.map (lift f) ≫ counit Y = f )\n( lift_counit : Π {X Y} (f : X ⟶ obj Y), lift (F.map f ≫ counit Y) = f )\n\nvariable (cl : counit_lift F)\n\nprivate lemma counit_lift_ext\n  {X : C} {Y : D} {f g : X ⟶ cl.obj Y}\n  (H : F.map f ≫ cl.counit Y = F.map g ≫ cl.counit Y) : f = g :=\nby rw [← cl.lift_counit f, H, cl.lift_counit]\n\ndef counit_lift.to_functor : D ⥤ C :=\n{ obj := cl.obj,\n  map := λ X Y f, cl.lift (cl.counit X ≫ f),\n  map_id' := λ X, by rw [category.comp_id, ← category.id_comp (cl.counit X), ← F.map_id,\n      cl.lift_counit],\n  map_comp' := λ X Y Z f g, counit_lift_ext cl $\n    by rw [F.map_comp, cl.lift_comp_counit, category.assoc, cl.lift_comp_counit,\n      ← category.assoc, ← category.assoc, cl.lift_comp_counit] }\n\ndef counit_lift.adjunction :\n  F ⊣ cl.to_functor :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, cl.lift f,\n    inv_fun := λ f, F.map f ≫ cl.counit Y,\n    left_inv := λ f, cl.lift_comp_counit f,\n    right_inv := λ f, cl.lift_counit f },\n  hom_equiv_naturality_left_symm' :=\n    begin\n      intros X Y Y' f g,\n      dsimp [counit_lift.to_functor],\n      rw [F.map_comp, category.assoc]\n    end,\n  hom_equiv_naturality_right' :=\n    begin\n      intros X' X Y f g,\n      dsimp [counit_lift.to_functor],\n      apply counit_lift_ext cl,\n      rw [cl.lift_comp_counit, F.map_comp, category.assoc,\n        cl.lift_comp_counit, ← category.assoc, cl.lift_comp_counit],\n    end }\n\nend category_theory", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/adjoint_rules.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3447238421626679}}
{"text": "import condensed.ab\nimport condensed.top_comparison\nimport condensed.adjunctions_module\nimport condensed.extr.equivalence\nimport category_theory.sites.limits\nimport algebra.category.Group.filtered_colimits\n\nimport for_mathlib.abelian_category\nimport for_mathlib.module_epi\n\nuniverse u\n\nopen category_theory\n\n/-\n\nS extr. disc.\nA cond ab group.\n\nA ↦ Hom_{CondSet}(S,A) = A(S) commutes with limits and colimits.\n\nA(-) commutes with finite products, in Ab, finite products = finite coproducts.\n\nZ[S] is projective.\n\nA -> B epi, Hom(Z[S],A) -> Hom(Z[S],B) -- A(S) -> B(S) WTS: surjective.\n\n1. If S is extr. disc. and A -> B is an epi of cond ab, then A(S) -> B(S) surjective.\n  B -> B/A\n\n-/\n\nnoncomputable theory\n\nvariables (R : Type (u + 1)) [ring R]\n\ndef CondensedMod.free (S : CondensedSet) : CondensedMod R :=\n(CondensedSet_to_CondensedMod R).obj $ S\n\nlocal notation R`[`S`]` := CondensedMod.free R S.to_Condensed\n\nvariable {R}\n\ndef Condensed.forgetMod (S : CondensedMod R) : CondensedSet :=\n(CondensedMod_to_CondensedSet R).obj S\n\ndef quiver.hom.forgetMod {S T : CondensedMod R} (e : S ⟶ T) :\n  S.forgetMod ⟶ T.forgetMod :=\n(CondensedMod_to_CondensedSet R).map e\n\nvariable (R)\n\nnamespace CondensedMod\n\ndef to_free (S : CondensedSet) : S ⟶ (free R S).forgetMod :=\n(CondensedMod_CondensedSet_adjunction R).hom_equiv _ _ $ 𝟙 _\n\ndef free_lift {S : CondensedSet} {A : CondensedMod R} (e : S ⟶ A.forgetMod) :\n  free R S ⟶ A :=\n((CondensedMod_CondensedSet_adjunction R).hom_equiv _ _).symm e\n\n@[simp]\nlemma to_free_free_lift {S : CondensedSet} {A : CondensedMod R} (e : S ⟶ A.forgetMod) :\n  to_free R S ≫ (free_lift R e).forgetMod = e :=\nbegin\n  dsimp only [Condensed.forgetMod, to_free, free_lift, quiver.hom.forgetMod],\n  rw [adjunction.hom_equiv_unit, adjunction.hom_equiv_counit, category.assoc,\n    category_theory.functor.map_id, category.id_comp, functor.map_comp,\n    (CondensedMod_CondensedSet_adjunction R).unit_naturality_assoc,\n    adjunction.right_triangle_components, category.comp_id],\nend\n\nlemma free_lift_unique {S : CondensedSet} {A : CondensedMod R}\n  (e : S ⟶ A.forgetMod) (f : free R S ⟶ A)\n  (h : to_free R S ≫ f.forgetMod = e) : f = free_lift R e :=\nbegin\n  apply_fun ((CondensedMod_CondensedSet_adjunction R).hom_equiv _ _),\n  dsimp [free_lift],\n  rw equiv.apply_symm_apply,\n  exact h\nend\n\n@[ext]\nlemma free_hom_ext {S : CondensedSet} {A : CondensedMod R} (f g : free R S ⟶ A)\n  (h : to_free R S ≫ f.forgetMod = to_free R S ≫ g.forgetMod) : f = g :=\nby rw [free_lift_unique _ _ f rfl, free_lift_unique _ _ g rfl, h]\n\nend CondensedMod\n\n@[simps]\ndef hom_equiv_evaluation_Mod (S : Profinite.{u}) (A : CondensedMod R) :\n  (R[S] ⟶ A) ≃ ((CondensedMod_to_CondensedSet R ⋙ CondensedSet.evaluation S).obj A) :=\n((CondensedMod_CondensedSet_adjunction R).hom_equiv S.to_Condensed A).trans $\n  (equiv_of_fully_faithful $ Sheaf_to_presheaf.{u} _ _).trans $ yoneda'_equiv _ _ .\n\nlemma hom_equiv_evaluation_Mod_apply_eq_app_id (S : Profinite.{u}) (A : CondensedMod R)\n  (f : R[S] ⟶ A) : hom_equiv_evaluation_Mod R S A f =\n  (CondensedMod.to_free R _ ≫ (CondensedMod_to_CondensedSet R).map f).val.app _ ⟨𝟙 _⟩ := rfl\n\nopen_locale big_operators\nlemma exists_hom_equiv_Mod_evaluation_symm_app_eq\n  (S : Profinite.{u}) (A : CondensedMod R)\n  (f : A.val.obj (opposite.op S)) : ∃ (t : R[S].val.obj (opposite.op S)),\n  ((hom_equiv_evaluation_Mod R S A).symm f).val.app _ t = f :=\nbegin\n  -- This proof can probably be made simpler using some adjunction voodoo...\n  use (hom_equiv_evaluation_Mod _ _ _) (𝟙 _),\n  dsimp [hom_equiv_evaluation_Mod, adjunction.whisker_right],\n  simp_rw [← comp_apply, ← nat_trans.comp_app],\n  erw [category.comp_id, proetale_topology.to_sheafify_sheafify_lift],\n  dsimp [functor.preimage, full.preimage, yoneda'_equiv, CondensedMod_to_CondensedSet,\n    functor.right_unitor, ulift_functor, Profinite.to_Condensed],\n  have := (Module.adj R).right_triangle_components,\n  apply_fun (λ e, e f) at this,\n  dsimp at this,\n  convert this,\n  dsimp [Module.adj],\n  erw finsupp.lift_symm_apply ((A.val.obj (opposite.op S)) →₀ R) R _ linear_map.id f,\n  erw finsupp.lift_symm_apply (ulift (S ⟶ S) →₀ R) R _ linear_map.id (ulift.up (𝟙 S)),\n  simp,\nend\n\nlocal attribute [instance] limits.has_zero_object.has_zero\n\nopen category_theory.limits\nopen opposite\n\n-- sanity check\nexample (S : Profinite.{u}) : preserves_zero_objects\n  (Condensed.evaluation (Module.{u+1} R) S) :=\ninfer_instance\n\ninstance projective_free_CondensedMod (S : Profinite.{u}) [projective S] :\n  projective (R[S]) :=\n{ factors := λ A B f g hg, begin\n    rw epi_iff_is_zero_cokernel at hg,\n    -- this follows from the fact that evaluation preserves colimits.\n    let e : (cokernel g).val.obj (op S) ≅ cokernel (g.val.app (op S)) := begin\n      refine (is_colimit_of_preserves (Condensed.evaluation (Module R) S)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso\n        (colimit.is_colimit _) ≪≫ has_colimit.iso_of_nat_iso _,\n      refine nat_iso.of_components _ _,\n      { rintros (a|b), exact eq_to_iso rfl, exact eq_to_iso rfl },\n      rintros (a|b) (c|d) (e|e),\n      all_goals { dsimp },\n      all_goals { simp, try { refl } },\n    end,\n    replace hg := is_zero_of_preserves (Condensed.evaluation (Module.{u+1} R) S) hg,\n    dsimp [Condensed.evaluation] at hg,\n    replace hg := is_zero_of_iso_of_zero hg e,\n    rw ← epi_iff_is_zero_cokernel at hg,\n    replace hg : function.surjective (g.val.app (op S)) := begin\n      resetI,\n      apply Module.surjective_of_epi,\n    end,\n    let f₁ := hom_equiv_evaluation_Mod _ _ _ f,\n    dsimp at f₁,\n    obtain ⟨f',h⟩ := hg f₁,\n    use (hom_equiv_evaluation_Mod _ _ _).symm f',\n    apply_fun (hom_equiv_evaluation_Mod _ _ _),\n    change _ = f₁,\n    rw [← h, hom_equiv_evaluation_Mod_apply, Sheaf.hom.comp_val, nat_trans.comp_app],\n    dsimp [hom_equiv_evaluation_Mod, adjunction.whisker_right, functor.associator],\n    erw [category.id_comp, ← comp_apply, ← comp_apply, ← category.assoc, ← nat_trans.comp_app,\n      ← nat_trans.comp_app],\n    erw proetale_topology.to_sheafify_sheafify_lift,\n    dsimp [functor.preimage, full.preimage, yoneda'_equiv, ulift_functor,\n      CondensedMod_to_CondensedSet, Profinite.to_Condensed],\n\n    congr' 1,\n    have := (Module.adj R).right_triangle_components,\n    apply_fun (λ e, e f') at this,\n    dsimp at this,\n    convert this,\n\n    dsimp [Module.adj],\n    erw finsupp.lift_symm_apply ((A.val.obj (opposite.op S)) →₀ R) R _ linear_map.id f',\n    erw finsupp.lift_symm_apply (ulift (S ⟶ S) →₀ R) R _ linear_map.id (ulift.up (𝟙 S)),\n\n    simp,\n  end } .\n\nlemma is_zero_iff_forall_zero_Mod {A : CondensedMod R} :\n  is_zero A ↔ ∀ (S : ExtrDisc), is_zero (A.val.obj (op S.val)) :=\nbegin\n  split,\n  { intros h S,\n    apply is_zero_of_preserves (Condensed.evaluation (Module.{u+1} R) S.val),\n    assumption },\n  { intro h,\n    let FF := ((Sheaf_to_presheaf _ _ : CondensedMod R ⥤ _) ⋙\n      (whiskering_left _ _ _).obj (ExtrDisc_to_Profinite.op)),\n    suffices : A ≅ ⊥_ _, by { apply is_zero_of_iso_of_zero _ this.symm, exact is_zero_initial },\n    let e : Π S : ExtrDisc, (A.val.obj (op S.val)) ≅ ⊥_ _ :=\n      λ S, is_zero.iso (h S) is_zero_initial,\n    symmetry,\n    apply (colimit.is_colimit _).cocone_point_unique_up_to_iso (_ : is_colimit (as_empty_cocone _)),\n    haveI : creates_colimits FF := @Condensed_to_ExtrDisc_presheaf_creates_colimits.{u u+2}\n        (@Module.{u+1 u+1} R _) (@Module.Module_category.{u+1 u+1} R _) _ _ _ _ _,\n    haveI := reflects_colimits_of_size_shrink.{0 u+1 0 u+1} FF,\n    apply (is_colimit_of_reflects.{0 0 (u+1) (u+1) (u+2) (u+2)} FF),\n    apply evaluation_jointly_reflects_colimits,\n    intros S,\n    have := is_colimit_empty_cocone_equiv (Module R)\n      (as_empty_cocone (A.val.obj (op S.unop.val)))\n      (((evaluation ExtrDiscᵒᵖ (Module R)).obj S).map_cocone (FF.map_cocone (as_empty_cocone A)))\n      (eq_to_iso rfl),\n    apply this.to_fun,\n    specialize e S.unop,\n    let t : as_empty_cocone (A.val.obj (op (unop S).val)) ≅ as_empty_cocone (⊥_ _) :=\n      cocones.ext e (by rintro ⟨⟨⟩⟩),\n    apply is_colimit.of_iso_colimit _ t.symm,\n    refine ⟨λ r, _, _, _⟩,\n    { dsimp, refine initial.to r.X, },\n    { rintro s ⟨⟨⟩⟩ },\n    { intros s m h, apply is_zero.eq_of_src, apply is_zero_initial } }\nend\n\nlemma is_epi_iff_forall_surjective_Mod {A B : CondensedMod R} (f : A ⟶ B) :\n  epi f ↔ ∀ (S : ExtrDisc), function.surjective (f.val.app (op S.val)) :=\nbegin\n  rw epi_iff_is_zero_cokernel,\n  rw is_zero_iff_forall_zero_Mod,\n  apply forall_congr (λ S, _),\n  let FF := Condensed.evaluation (Module.{u+1} R) S.val,\n  haveI : preserves_colimits FF := infer_instance,\n  let e : (cokernel f).val.obj (op S.val) ≅ cokernel (f.val.app (op S.val)) := begin\n    change FF.obj (cokernel f) ≅ cokernel (FF.map f),\n    change (FF.map_cocone _).X ≅ _,\n    refine (is_colimit_of_preserves FF (colimit.is_colimit _)).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫ _,\n    dsimp,\n    apply has_colimit.iso_of_nat_iso,\n    -- This isomorphism is probably somewhere in mathlib... or somewhere in this repo.\n    refine nat_iso.of_components _ _,\n    { rintros (i|i),\n      { exact iso.refl _ },\n      { exact iso.refl _ } },\n    { rintros (i|i) (j|j) (f|f),\n      { dsimp, simpa, },\n      { dsimp, simp, },\n      { dsimp, simpa, },\n      { dsimp, simpa, } },\n  end,\n  have : is_zero ((cokernel f).val.obj (op S.val)) ↔ is_zero (cokernel (f.val.app (op S.val))),\n  { split,\n    { intro h, apply is_zero_of_iso_of_zero h e, },\n    { intro h, apply is_zero_of_iso_of_zero h e.symm, } },\n  rw [this, ← epi_iff_is_zero_cokernel],\n  clear e,\n  split,\n  { introsI h,\n    apply Module.surjective_of_epi },\n  { intros h, exact concrete_category.epi_of_surjective (f.val.app (op S.val)) h}\nend\n\ntheorem CondensedMod_has_enough_projectives_aux (A : CondensedMod R) :\n  ∃ (B : CondensedMod R) (hB : projective B) (f : B ⟶ A), epi f :=\nbegin\n  let II := Σ (S : ExtrDisc), A.val.obj (op S.val),\n  let X : II → CondensedMod R := λ i, R[i.1.val],\n  let f : Π i, X i ⟶ A := λ i, (hom_equiv_evaluation_Mod _ i.1.val A).symm i.2,\n\n  -- Move this.\n  haveI : has_colimits (CondensedMod R) := begin\n    change has_colimits (Sheaf _ _),\n    exact category_theory.Sheaf.category_theory.limits.has_colimits.{(u+2) u (u+1)},\n  end,\n\n  use [∐ X, infer_instance, sigma.desc f],\n\n  rw is_epi_iff_forall_surjective_Mod,\n\n  intros S t,\n  obtain ⟨w,hw⟩ := exists_hom_equiv_Mod_evaluation_symm_app_eq R S.val A t,\n  use (sigma.ι X ⟨S,t⟩).val.app (op S.val) w,\n  rw [← comp_apply, ← nat_trans.comp_app],\n  change (((sigma.ι X ⟨S,t⟩) ≫ sigma.desc f).val.app (op S.val)) w = _,\n  erw colimit.ι_desc,\n  exact hw,\nend\n\ninstance CondensedMod_has_enough_projective : enough_projectives (CondensedMod R) :=\nbegin\n  constructor,\n  intros B,\n  obtain ⟨X,hX,f,hf⟩ := CondensedMod_has_enough_projectives_aux R B,\n  resetI,\n  constructor,\n  refine ⟨X,hX,f,hf⟩,\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/projective_resolution_module.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3447238421626679}}
{"text": "-- COMPLETENESS\n\nimport syntax\nimport tableau\nimport soundness\n-- import modelgraphs\n\nopen has_sat\n\n-- Each local rule preserves truth \"upwards\"\nlemma localRuleTruth {W : Type} {M : kripkeModel W} {w : W} {X : finset formula} {B : finset (finset formula)} :\n  localRule X B → (∃ Y ∈ B, (M,w) ⊨ Y) → (M,w) ⊨ X :=\nbegin\n  intro r,\n  cases r,\n  case localRule.bot {\n    simp,\n  },\n  case localRule.not {\n    simp,\n  },\n  case localRule.neg : a f notnotf_in_a {\n    intro hyp,\n    rcases hyp with ⟨b, b_in_B, w_sat_b⟩,\n    intros phi phi_in_a,\n    have b_is_af : b = insert f (a \\ {~~f}), {\n      simp at *, subst b_in_B,\n    },\n    have phi_in_b_or_is_f : phi ∈ b ∨ phi = ~~f, {\n      rw b_is_af,\n      simp,\n      finish,\n    },\n    cases phi_in_b_or_is_f with phi_in_b phi_is_notnotf,\n    {\n      apply w_sat_b,\n      exact phi_in_b,\n    },\n    {\n      rw phi_is_notnotf,\n      unfold evaluate at *,\n      simp, -- this removes double negation\n      apply w_sat_b,\n      rw b_is_af ,\n      simp at *,\n    },\n  },\n  case localRule.con : a f g fandg_in_a {\n    intro hyp,\n    rcases hyp with ⟨b, b_in_B, w_sat_b⟩,\n    intros phi phi_in_a,\n    simp at b_in_B,\n    have b_is_fga : b = insert f (insert g (a \\ {f⋏g})), {\n      subst b_in_B, ext1, simp,\n    },\n    have phi_in_b_or_is_fandg : phi ∈ b ∨ phi = f⋏g, {\n      rw b_is_fga,\n      simp,\n      finish,\n    },\n    cases phi_in_b_or_is_fandg with phi_in_b phi_is_fandg,\n    {\n      apply w_sat_b,\n      exact phi_in_b,\n    },\n    { rw phi_is_fandg,\n      unfold evaluate at *,\n      split,\n      { apply w_sat_b, rw b_is_fga, simp, },\n      { apply w_sat_b, rw b_is_fga, simp, },\n    },\n  },\n  case localRule.nCo : a f g not_fandg_in_a {\n    intro hyp,\n    rcases hyp with ⟨b, b_in_B, w_sat_b⟩,\n    intros phi phi_in_a,\n    simp at *,\n    have phi_in_b_or_is_notfandg : phi ∈ b ∨ phi = ~(f⋏g), {\n      cases b_in_B ; { rw b_in_B, simp, finish, },\n    },\n    cases b_in_B,\n    { -- b contains ~f\n      cases phi_in_b_or_is_notfandg with phi_in_b phi_def,\n      { exact w_sat_b phi phi_in_b, },\n      {\n        rw phi_def,\n        unfold evaluate,\n        rw b_in_B at w_sat_b,\n        specialize w_sat_b (~f),\n        rw not_and_distrib,\n        left,\n        unfold evaluate at w_sat_b,\n        apply w_sat_b,\n        finish,\n      },\n    },\n    { -- b contains ~g\n     cases phi_in_b_or_is_notfandg with phi_in_b phi_def,\n      { exact w_sat_b phi phi_in_b, },\n      {\n        rw phi_def,\n        unfold evaluate,\n        rw b_in_B at w_sat_b,\n        specialize w_sat_b (~g),\n        rw not_and_distrib,\n        right,\n        unfold evaluate at w_sat_b,\n        apply w_sat_b,\n        finish,\n      },\n    },\n  },\nend\n\n-- Each local rule is \"complete\", i.e. preserves satisfiability \"upwards\"\nlemma localRuleCompleteness {X : finset formula} { B : finset (finset formula) } :\n  localRule X B → (∃ Y ∈ B, satisfiable Y) → satisfiable X :=\nbegin\n  intro lr,\n  intro sat_B,\n  rcases sat_B with ⟨Y, Y_in_B, sat_Y⟩,\n  unfold satisfiable at *,\n  rcases sat_Y with ⟨W,M,w,w_sat_Y⟩,\n  use [W,M,w],\n  apply localRuleTruth lr,\n  tauto,\nend\n\n-- Lemma 11 (but rephrased to be about local tableau!?)\nlemma inconsUpwards {X} {ltX : localTableau X} : (Π en ∈ endNodesOf ⟨X, ltX⟩, inconsistent en) → inconsistent X :=\nbegin\n  intro lhs,\n  unfold inconsistent at *,\n  let leafTableaus : Π en ∈ endNodesOf ⟨X, ltX⟩, closedTableau en := λ Y YinEnds, (lhs Y YinEnds).some,\n  split,\n  exact closedTableau.loc ltX leafTableaus,\nend\n\n-- Converse of Lemma 11\nlemma consToEndNodes {X} {ltX : localTableau X} : consistent X → (∃ en ∈ endNodesOf ⟨X, ltX⟩, consistent en) :=\nbegin\n  intro consX,\n  unfold consistent at *,\n  have claim := not.imp consX (@inconsUpwards X ltX),\n  simp at claim,\n  tauto,\nend\n\ndef projOfConsSimpIsCons : Π {X ϕ}, consistent X → simple X → ~□ϕ ∈ X → consistent (projection X ∪ {~ϕ}) :=\nbegin\n  intros X ϕ consX simpX notBoxPhi_in_X,\n  unfold consistent at *,\n  unfold inconsistent at *,\n  by_contradiction h,\n  simp at *,\n  cases h with ctProj,\n  have ctX : closedTableau X, {\n    apply closedTableau.atm notBoxPhi_in_X simpX,\n    simp, exact ctProj,\n  },\n  cases consX, tauto,\nend\n\nlemma locTabEndSatThenSat {X Y} (ltX : localTableau X) (Y_endOf_X : Y ∈ endNodesOf ⟨X, ltX⟩) :\n  satisfiable Y → satisfiable X :=\nbegin\n  intro satY,\n  induction ltX,\n  case byLocalRule : X B lr next IH {\n    apply localRuleCompleteness lr,\n    cases lr,\n    case localRule.bot : W bot_in_W {\n      simp at *,\n      exact Y_endOf_X,\n    },\n    case localRule.not : _ ϕ h {\n      simp at *,\n      exact Y_endOf_X,\n    },\n    case localRule.neg : Z ϕ notNotPhi_inX {\n      simp at *,\n      specialize IH (insert ϕ (Z.erase (~~ϕ))),\n      simp at IH,\n      apply IH,\n      rcases Y_endOf_X with ⟨W, W_def, Y_endOf_W⟩,\n      subst W_def,\n      exact Y_endOf_W,\n    },\n    case localRule.con : Z ϕ ψ _ {\n      simp at *,\n      specialize IH (insert ϕ (insert ψ (Z.erase (ϕ⋏ψ)))),\n      simp at IH,\n      apply IH,\n      rcases Y_endOf_X with ⟨W, W_def, Y_endOf_W⟩,\n      subst W_def,\n      exact Y_endOf_W,\n    },\n    case localRule.nCo : Z ϕ ψ _ {\n      simp at *,\n      rcases Y_endOf_X with ⟨W, W_def, Y_endOf_W⟩,\n      cases W_def,\n      all_goals { subst W_def, },\n      { specialize IH (insert (~ϕ) (Z.erase (~(ϕ⋏ψ)))),\n        simp at IH,\n        use (insert (~ϕ) (Z.erase (~(ϕ⋏ψ)))),\n        split,\n        { left, refl, },\n        { apply IH, exact Y_endOf_W, }\n      },\n      { specialize IH (insert (~ψ) (Z.erase (~(ϕ⋏ψ)))),\n        simp at IH,\n        use (insert (~ψ) (Z.erase (~(ϕ⋏ψ)))),\n        split,\n        { right, refl, },\n        { apply IH, exact Y_endOf_W, }\n      },\n    },\n  },\n  case sim : X simpX {\n    finish,\n  },\nend\n\nlemma almostCompleteness : Π n X, lengthOfSet X = n → consistent X → satisfiable X :=\nbegin\n  intro n,\n  apply nat.strong_induction_on n,\n  intros n IH,\n  intros X lX_is_n consX,\n  refine if simpX : simple X then _ else _,\n  -- CASE 1: X is simple\n  rw Lemma1_simple_sat_iff_all_projections_sat simpX,\n  split,\n  { -- show that X is not closed\n    by_contradiction h,\n    unfold consistent at consX,\n    unfold inconsistent at consX,\n    simp at consX,\n    cases consX, apply consX,\n    unfold closed at h,\n    refine if botInX : ⊥ ∈ X then _ else _,\n    { apply closedTableau.loc, rotate, apply localTableau.byLocalRule,\n      exact localRule.bot botInX,\n      intros Y YinEmpty, cases YinEmpty,\n      rw botNoEndNodes, intros Y YinEmpty, cases YinEmpty,\n    },\n    { have f1 : ∃ (f : formula) (H : f ∈ X), ~f ∈ X := by tauto,\n      have : nonempty (closedTableau X), {\n      rcases f1 with ⟨f, f_in_X, notf_in_X⟩,\n      fconstructor,\n      apply closedTableau.loc, rotate, apply localTableau.byLocalRule,\n      apply localRule.not ⟨f_in_X , notf_in_X⟩,\n      intros Y YinEmpty, cases YinEmpty,\n      rw notNoEndNodes, intros Y YinEmpty, cases YinEmpty,\n      },\n      exact classical.choice this,\n    },\n  },\n  { -- show that all projections are sat\n    intros R notBoxR_in_X,\n    apply IH (lengthOfSet (projection X ∪ {~R})),\n    { -- projection decreases lengthOfSet\n      subst lX_is_n,\n      exact atmRuleDecreasesLength notBoxR_in_X,\n    },\n    { refl, },\n    { exact projOfConsSimpIsCons consX simpX notBoxR_in_X, },\n  },\n  -- CASE 2: X is not simple\n  rename simpX notSimpX,\n  rcases notSimpleThenLocalRule notSimpX with ⟨B,lrExists⟩,\n  have lr := classical.choice lrExists,\n  have rest : Π (Y : finset formula), Y ∈ B → localTableau Y, {\n    intros Y Y_in_B,\n    set N := hasLength.lengthOf Y,\n    exact classical.choice (existsLocalTableauFor N Y (by refl))\n  },\n  rcases @consToEndNodes X (localTableau.byLocalRule lr rest) consX with ⟨E, E_endOf_X, consE⟩,\n  have satE : satisfiable E, {\n    apply IH (lengthOfSet E),\n    { -- end Node of local rule is LT\n      subst lX_is_n,\n      apply endNodesOfLocalRuleLT E_endOf_X,\n    },\n    { refl, },\n    { exact consE, },\n  },\n  exact locTabEndSatThenSat (localTableau.byLocalRule lr rest) E_endOf_X satE,\nend\n\n-- Theorem 4, page 37\ntheorem completeness : ∀ X, consistent X ↔ satisfiable X :=\nbegin\n  intro X,\n  split,\n  { intro X_is_consistent,\n    let n := lengthOfSet X,\n    apply almostCompleteness n X (by refl) X_is_consistent,\n  },\n  -- use Theorem 2:\n  exact correctness X,\nend\n\nlemma singletonCompleteness : ∀ φ, consistent {φ} ↔ satisfiable φ :=\nbegin\n  intro f,\n  have := completeness {f},\n  simp only [singletonSat_iff_sat] at *,\n  tauto,\nend\n", "meta": {"author": "m4lvin", "repo": "tablean", "sha": "836202612fc2bfacb5545696412e7d27f7704141", "save_path": "github-repos/lean/m4lvin-tablean", "path": "github-repos/lean/m4lvin-tablean/tablean-836202612fc2bfacb5545696412e7d27f7704141/src/completeness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5, "lm_q1q2_score": 0.344652808392723}}
{"text": "\n@[inline] def ok {ε : Type u} (a : α) : Except ε α := pure a\n\n/- simple exception handling monad -/\ndef divide (x: Float ) (y: Float): ExceptT String Id Float :=\n  if y == 0 then\n    throw \"can't divide by zero\"\n  else\n    ok (x / y)\n\n#eval divide 5 0  -- Except.error \"can't divide by zero\"\n\n#check (Except String) -- Type u_1 → Type (max 0 u_1)\n\n#check ExceptT String Id\n\ndef test :=\n  divide 5 0\n\n#check test  -- ExceptT String Id Float\n#eval test -- Except.error \"can't divide by zero\"\n#eval test.run -- this was done implicitly for us\n#eval test |>.run -- this is the best way to call run if run takes arguments.\n\ndef testCatch :=\n    try\n      let r ← divide 8 0 -- 'r' is type Float\n      return toString r\n    catch e =>\n      return s!\"Caught exception: {e}\"\n\n#check testCatch -- ExceptT String Id String\n#eval testCatch -- Caught exception: can't divide by zero\n\n/- unwrap Except using match -/\ndef testUnwrap : String := Id.run do\n    let r := divide 8 0 -- r is type ExceptT String Id Float\n    match r with\n    | .ok a => toString a -- 'a' is type Float\n    | .error e => s!\"Caught exception: {e}\"\n\n#check testUnwrap -- String\n#eval testUnwrap -- \"Caught exception: can't divide by zero\"\n\n/- unwrap Except using coercion -/\ninstance : Coe (ExceptT α Id α) α where\n  coe a := match a with\n    | .ok v => v\n    | .error v => v\n\ndef testCoerce : String :=\n  let act : ExceptT String Id String := do\n    let r ← divide 8 0\n    return r.toString\n  act\n\n#check testCoerce -- String\n#eval testCoerce  -- \"can't divide by zero\"\n\n/- adding StateT -/\n\ndef divideLog (x: Float ) (y: Float): StateT Nat Id Float :=\n  if y == 0 then\n    return 0\n  else do\n    modify fun s => s + 1\n    return x / y\n\n#check divideLog              -- Float → Float → StateT Nat Id Float\n#reduce StateT Nat Id Float   -- Nat → Float × Nat\n\n#eval divideLog 8 4 0         -- (2.000000, 1)\n#eval (divideLog 8 4).run 0   -- (2.000000, 1)\n#eval divideLog 8 4 |>.run 0  -- (2.000000, 1)\n\ndef testDivideLog := do\n  let mut state := 0\n  for x in [0:10] do\n    for y in [0:10] do\n      let r ← divideLog x.toFloat y.toFloat |>.run state\n      state := r.2\n  state\n\n\n#eval testDivideLog -- 90\n\n/- chaining monads using bind -/\ndef divideIt (x:Float) (y:Float) : StateT Nat (ExceptT String Id) Float :=\n  if y == 0 then\n    throw \"can't divide by zero\"\n  else\n    bind (modify fun s => s + 1) (fun _ => pure (x / y))\n\n#check divideIt\n#reduce StateT Nat (ExceptT String Id) Float\n#eval divideIt 5 2 |>.run 0 -- Except.ok (5.000000, 1)\n\ndef testIt := do\n  let mut log := 0\n  for x in [0:10] do\n    for y in [0:10] do\n      try\n        let r ← divideIt x.toFloat y.toFloat |>.run log\n        log := r.2\n      catch _  =>\n        ok ()\n  ok log\n\n#eval testIt -- 90\n\n/- using do notation -/\ndef divideDo (x:Float) (y:Float) : (StateT Nat (ExceptT String Id)) Float := do\n  if y == 0 then\n    throw \"can't divide by zero\"\n  else\n    modify fun s => s + 1\n    pure (x / y)\n\n#eval divideDo 5 2 |>.run 0 -- Except.ok (2.500000, 1)\n\n/- Adding ReadeT monad -/\ndef divideWithArgs (x:Float) (y:Float) : ReaderT (List String) (StateT Nat (ExceptT String Id)) Float :=\n  bind (modify fun s => s + 1) fun _ =>\n    bind (get) fun s =>\n      bind (read) fun args =>\n        if (s > 10 && args.contains \"--limit\") then\n          throw \"too many divides\"\n        else if y == 0 then\n          throw \"can't divide by zero\"\n        else\n          pure (x / y)\n\n#reduce ReaderT (List String) (StateT Nat (ExceptT String Id)) Float -- List String → Nat → Except String (Float × Nat)\n#eval divideWithArgs 5 2 |>.run [] |>.run 0 -- Except.ok (2.500000, 1)\n#eval divideWithArgs 5 0 |>.run [] |>.run 0 -- Except.error \"can't divide by zero\"\n#eval divideWithArgs 5 2 |>.run [\"--limit\"] |>.run 10 -- Except.error \"too many divides\"\n\ndef divideWithArgsDo (x:Float) (y:Float) : ReaderT (List String) (StateT Nat (ExceptT String Id)) Float := do\n  modify fun s => s + 1\n  let s ← get\n  let args ← read\n  if (s > 10 && args.contains \"--limit\") then\n    throw \"too many divides\"\n  else if y == 0 then\n    throw \"can't divide by zero\"\n  else\n    pure (x / y)\n\n#eval divideWithArgsDo 5 2 |>.run [] |>.run 0 -- Except.ok (2.500000, 1)\n#eval divideWithArgsDo 5 0 |>.run [] |>.run 0 -- Except.error \"can't divide by zero\"\n#eval divideWithArgsDo 5 2 |>.run [\"--limit\"] |>.run 10 -- Except.error \"too many divides\"\n\n/- proof that these are the same-/\nexample : divideWithArgs x y = divideWithArgsDo x y := by\n  simp[divideWithArgs, divideWithArgsDo]    -- Goals accomplished 🎉\n\n/- monad composition using lifting -/\ndef divideRefactored (x:Float) (y:Float) : ReaderT (List String) (StateT Nat (ExceptT String Id)) Float := do\n  modify fun s => s + 1\n  let s ← get\n  let args ← read\n  if (s > 10 && args.contains \"--limit\") then\n    throw \"too many divides\"\n  else\n    divide x y\n\n\n#eval divideRefactored 5 2 |>.run [] |>.run 0 -- Except.ok (2.500000, 1)\n#eval divideRefactored 5 0 |>.run [] |>.run 0 -- Except.error \"can't divide by zero\"\n#eval divideRefactored 5 2 |>.run [\"--limit\"] |>.run 10 -- Except.error \"too many divides\"\n\n#reduce ReaderT (List String) (StateT Nat (ExceptT String Id)) Float\n#reduce ExceptT String Id Float\n\ndef lift1 (x : ExceptT String Id Float) : (StateT Nat (ExceptT String Id)) Float :=\n  x\n\n#print lift1  -- fun x => liftM x\n\n#reduce lift1 -- fun x s => Except.rec (fun a => Except.error a) (fun a => Except.ok (a, s)) x\n\n#eval lift1 (divide 5 1) |>.run 3 -- Except.ok (5.000000, 3)\n\ndef lift2 (x : StateT Nat (ExceptT String Id) Float)  : ReaderT (List String) (StateT Nat (ExceptT String Id)) Float :=\n  x\n\ndef transitive (x : StateT Nat (ExceptT String Id) Float) := do\n  let x := lift1 (divide 5 1)\n  lift2 x\n\n\n#eval lift2 (lift1 (divide 5 1)) |>.run [\"discarded\", \"state\"] |>.run 4 -- Except.ok (5.000000, 4)\n\n#reduce ExceptT String Id -- Except String α\n\n/- some common patterns with structures and abbreviations -/\nstructure Config where\n  x : Nat\n  y : Nat\n  deriving Repr\n\nabbrev CoolM := StateT Config (ExceptT Nat Id)\n\ndef doSomethingCool : CoolM Nat :=do\n  let s ← get\n  set {s with x := 10}\n  pure 0\n\n#check doSomethingCool -- CoolM Nat\n#reduce CoolM Nat      -- Config → Except Nat (Nat × Config)\n#synth Monad CoolM     -- StateT.instMonadStateT\n#eval doSomethingCool |>.run  {x := 0, y := 0} -- Except.ok (0, { x := 10, y := 0 })\n\n", "meta": {"author": "leanprover", "repo": "lean4-samples", "sha": "5c0db5f1e952e7ebada506ba3a390040972293a9", "save_path": "github-repos/lean/leanprover-lean4-samples", "path": "github-repos/lean/leanprover-lean4-samples/lean4-samples-5c0db5f1e952e7ebada506ba3a390040972293a9/SimpleMonads/Monads.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.3446291213980862}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport data.dfinsupp\nimport data.finsupp.basic\nimport algebra.module.linear_map\n\n/-!\n# Conversion between `finsupp` and homogenous `dfinsupp`\n\nThis module provides conversions between `finsupp` and `dfinsupp`.\nIt is in its own file since neither `finsupp` or `dfinsupp` depend on each other.\n\n## Main definitions\n\n* `finsupp.to_dfinsupp : (ι →₀ M) → (Π₀ i : ι, M)`\n* `dfinsupp.to_finsupp : (Π₀ i : ι, M) → (ι →₀ M)`\n* Bundled equiv versions of the above:\n  * `finsupp_equiv_dfinsupp : (ι →₀ M) ≃ (Π₀ i : ι, M)`\n  * `finsupp_add_equiv_dfinsupp : (ι →₀ M) ≃+ (Π₀ i : ι, M)`\n  * `finsupp_lequiv_dfinsupp R : (ι →₀ M) ≃ₗ[R] (Π₀ i : ι, M)`\n\n## Theorems\n\nThe defining features of these operations is that they preserve the function and support:\n\n* `finsupp.to_dfinsupp_coe`\n* `finsupp.to_dfinsupp_support`\n* `dfinsupp.to_finsupp_coe`\n* `dfinsupp.to_finsupp_support`\n\nand therefore map `finsupp.single` to `dfinsupp.single` and vice versa:\n\n* `finsupp.to_dfinsupp_single`\n* `dfinsupp.to_finsupp_single`\n\nas well as preserving arithmetic operations.\n\nFor the bundled equivalences, we provide lemmas that they reduce to `finsupp.to_dfinsupp`:\n\n* `finsupp_add_equiv_dfinsupp_apply`\n* `finsupp_lequiv_dfinsupp_apply`\n* `finsupp_add_equiv_dfinsupp_symm_apply`\n* `finsupp_lequiv_dfinsupp_symm_apply`\n\n## Implementation notes\n\nWe provide `dfinsupp.to_finsupp` and `finsupp_equiv_dfinsupp` computably by adding\n`[decidable_eq ι]` and `[Π m : M, decidable (m ≠ 0)]` arguments. To aid with definitional unfolding,\nthese arguments are also present on the `noncomputable` equivs.\n-/\n\nvariables {ι : Type*} {R : Type*} {M : Type*}\n\n\n/-! ### Basic definitions and lemmas -/\nsection defs\n\n/-- Interpret a `finsupp` as a homogenous `dfinsupp`. -/\ndef finsupp.to_dfinsupp [has_zero M] (f : ι →₀ M) : Π₀ i : ι, M :=\n⟦⟨f, f.support.1, λ i, (classical.em (f i = 0)).symm.imp_left (finsupp.mem_support_iff.mpr)⟩⟧\n\n@[simp] lemma finsupp.to_dfinsupp_coe [has_zero M] (f : ι →₀ M) : ⇑f.to_dfinsupp = f := rfl\n\nsection\nvariables [decidable_eq ι] [has_zero M]\n\n@[simp] lemma finsupp.to_dfinsupp_single (i : ι) (m : M) :\n  (finsupp.single i m).to_dfinsupp = dfinsupp.single i m :=\nby { ext, simp [finsupp.single_apply, dfinsupp.single_apply] }\n\nvariables [Π m : M, decidable (m ≠ 0)]\n\n@[simp] lemma to_dfinsupp_support (f : ι →₀ M) : f.to_dfinsupp.support = f.support :=\nby { ext, simp, }\n\n/-- Interpret a homogenous `dfinsupp` as a `finsupp`.\n\nNote that the elaborator has a lot of trouble with this definition - it is often necessary to\nwrite `(dfinsupp.to_finsupp f : ι →₀ M)` instead of `f.to_finsupp`, as for some unknown reason\nusing dot notation or omitting the type ascription prevents the type being resolved correctly. -/\ndef dfinsupp.to_finsupp (f : Π₀ i : ι, M) : ι →₀ M :=\n⟨f.support, f, λ i, by simp only [dfinsupp.mem_support_iff]⟩\n\n@[simp] lemma dfinsupp.to_finsupp_coe (f : Π₀ i : ι, M) : ⇑f.to_finsupp = f := rfl\n@[simp] lemma dfinsupp.to_finsupp_support (f : Π₀ i : ι, M) : f.to_finsupp.support = f.support :=\nby { ext, simp, }\n\n@[simp] lemma dfinsupp.to_finsupp_single (i : ι) (m : M) :\n  (dfinsupp.single i m : Π₀ i : ι, M).to_finsupp = finsupp.single i m :=\nby { ext, simp [finsupp.single_apply, dfinsupp.single_apply] }\n\n@[simp] lemma finsupp.to_dfinsupp_to_finsupp (f : ι →₀ M) : f.to_dfinsupp.to_finsupp = f :=\nfinsupp.coe_fn_injective rfl\n\n@[simp] lemma dfinsupp.to_finsupp_to_dfinsupp (f : Π₀ i : ι, M) : f.to_finsupp.to_dfinsupp = f :=\ndfinsupp.coe_fn_injective rfl\n\nend\n\nend defs\n\n/-! ### Lemmas about arithmetic operations -/\nsection lemmas\n\nnamespace finsupp\n\n@[simp] lemma to_dfinsupp_zero [has_zero M] :\n  (0 : ι →₀ M).to_dfinsupp = 0 := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_add [add_zero_class M] (f g : ι →₀ M) :\n  (f + g).to_dfinsupp = f.to_dfinsupp + g.to_dfinsupp := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_neg [add_group M] (f : ι →₀ M) :\n  (-f).to_dfinsupp = -f.to_dfinsupp := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_sub [add_group M] (f g : ι →₀ M) :\n  (f - g).to_dfinsupp = f.to_dfinsupp - g.to_dfinsupp :=\ndfinsupp.coe_fn_injective (sub_eq_add_neg _ _)\n\n@[simp] lemma to_dfinsupp_smul [semiring R] [add_comm_monoid M] [module R M]\n  (r : R) (f : ι →₀ M) : (r • f).to_dfinsupp = r • f.to_dfinsupp :=\ndfinsupp.coe_fn_injective rfl\n\nend finsupp\n\nnamespace dfinsupp\nvariables [decidable_eq ι]\n\n@[simp] lemma to_finsupp_zero [has_zero M] [Π m : M, decidable (m ≠ 0)] :\n  to_finsupp 0 = (0 : ι →₀ M) := finsupp.coe_fn_injective rfl\n\n@[simp] lemma to_finsupp_add [add_zero_class M] [Π m : M, decidable (m ≠ 0)] (f g : Π₀ i : ι, M) :\n  (to_finsupp (f + g) : ι →₀ M) = (to_finsupp f + to_finsupp g) :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_add _ _\n\n@[simp] lemma to_finsupp_neg [add_group M] [Π m : M, decidable (m ≠ 0)] (f : Π₀ i : ι, M) :\n  (to_finsupp (-f) : ι →₀ M) = -to_finsupp f :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_neg _\n\n@[simp] lemma to_finsupp_sub [add_group M] [Π m : M, decidable (m ≠ 0)] (f g : Π₀ i : ι, M) :\n  (to_finsupp (f - g) : ι →₀ M) = to_finsupp f - to_finsupp g :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_sub _ _\n\n@[simp] lemma to_finsupp_smul [semiring R] [add_comm_monoid M] [module R M]\n  [Π m : M, decidable (m ≠ 0)]\n  (r : R) (f : Π₀ i : ι, M) : (to_finsupp (r • f) : ι →₀ M) = r • to_finsupp f :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_smul _ _\n\nend dfinsupp\n\nend lemmas\n\n/-! ### Bundled `equiv`s -/\n\nsection equivs\n\n/-- `finsupp.to_dfinsupp` and `dfinsupp.to_finsupp` together form an equiv. -/\n@[simps {fully_applied := ff}]\ndef finsupp_equiv_dfinsupp [decidable_eq ι] [has_zero M] [Π m : M, decidable (m ≠ 0)] :\n  (ι →₀ M) ≃ (Π₀ i : ι, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  left_inv := finsupp.to_dfinsupp_to_finsupp, right_inv := dfinsupp.to_finsupp_to_dfinsupp }\n\n/-- The additive version of `finsupp.to_finsupp`. Note that this is `noncomputable` because\n`finsupp.has_add` is noncomputable. -/\n@[simps {fully_applied := ff}]\nnoncomputable def finsupp_add_equiv_dfinsupp\n  [decidable_eq ι] [add_zero_class M] [Π m : M, decidable (m ≠ 0)] :\n  (ι →₀ M) ≃+ (Π₀ i : ι, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  map_add' := finsupp.to_dfinsupp_add,\n  .. finsupp_equiv_dfinsupp}\n\nvariables (R)\n\n/-- The additive version of `finsupp.to_finsupp`. Note that this is `noncomputable` because\n`finsupp.has_add` is noncomputable. -/\n@[simps {fully_applied := ff}]\nnoncomputable def finsupp_lequiv_dfinsupp\n  [decidable_eq ι] [semiring R] [add_comm_monoid M] [Π m : M, decidable (m ≠ 0)] [module R M] :\n  (ι →₀ M) ≃ₗ[R] (Π₀ i : ι, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  map_smul' := finsupp.to_dfinsupp_smul,\n  map_add' := finsupp.to_dfinsupp_add,\n  .. finsupp_equiv_dfinsupp}\n\nend equivs\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/finsupp/to_dfinsupp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203638047913, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.34462912066282164}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.shift\nimport category_theory.concrete_category\n\n/-!\n# Differential objects in a category.\n\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X ⟶ X⟦1⟧`, such that `d^2 = 0`.\n\nWe build the category of differential objects, and some basic constructions\nsuch as the forgetful functor, zero morphisms and zero objects, and the shift functor\non differential objects.\n-/\n\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nvariables [has_zero_morphisms C] [has_shift C]\n\n/--\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X ⟶ X⟦1⟧`, such that `d^2 = 0`.\n-/\n@[nolint has_inhabited_instance]\nstructure differential_object :=\n(X : C)\n(d : X ⟶ X⟦1⟧)\n(d_squared' : d ≫ d⟦1⟧' = 0 . obviously)\n\nrestate_axiom differential_object.d_squared'\nattribute [simp] differential_object.d_squared\n\nvariables {C}\n\nnamespace differential_object\n\n/--\nA morphism of differential objects is a morphism commuting with the differentials.\n-/\n@[ext, nolint has_inhabited_instance]\nstructure hom (X Y : differential_object C) :=\n(f : X.X ⟶ Y.X)\n(comm' : X.d ≫ f⟦1⟧' = f ≫ Y.d . obviously)\n\nrestate_axiom hom.comm'\nattribute [simp, reassoc] hom.comm\n\nnamespace hom\n\n/-- The identity morphism of a differential object. -/\n@[simps]\ndef id (X : differential_object C) : hom X X :=\n{ f := 𝟙 X.X }\n\n/-- The composition of morphisms of differential objects. -/\n@[simps]\ndef comp {X Y Z : differential_object C} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ f := f.f ≫ g.f, }\n\nend hom\n\ninstance category_of_differential_objects : category (differential_object C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := λ X Y Z f g, hom.comp f g, }\n\n@[simp]\n\n\n@[simp]\nlemma comp_f {X Y Z : differential_object C} (f : X ⟶ Y) (g : Y ⟶ Z) :\n  (f ≫ g).f = f.f ≫ g.f :=\nrfl\n\nvariables (C)\n\n/-- The forgetful functor taking a differential object to its underlying object. -/\ndef forget : (differential_object C) ⥤ C :=\n{ obj := λ X, X.X,\n  map := λ X Y f, f.f, }\n\ninstance forget_faithful : faithful (forget C) :=\n{ }\n\ninstance has_zero_morphisms : has_zero_morphisms (differential_object C) :=\n{ has_zero := λ X Y,\n  ⟨{ f := 0, }⟩}\n\nvariables {C}\n\n@[simp]\nlemma zero_f (P Q : differential_object C) : (0 : P ⟶ Q).f = 0 := rfl\n\n/--\nAn isomorphism of differential objects gives an isomorphism of the underlying objects.\n-/\n@[simps] def iso_app {X Y : differential_object C} (f : X ≅ Y) : X.X ≅ Y.X :=\n⟨f.hom.f, f.inv.f, by { dsimp, rw [← comp_f, iso.hom_inv_id, id_f] },\n  by { dsimp, rw [← comp_f, iso.inv_hom_id, id_f] }⟩\n\n@[simp] lemma iso_app_refl (X : differential_object C) : iso_app (iso.refl X) = iso.refl X.X := rfl\n@[simp] lemma iso_app_symm {X Y : differential_object C} (f : X ≅ Y) :\n  iso_app f.symm = (iso_app f).symm := rfl\n@[simp] lemma iso_app_trans {X Y Z : differential_object C} (f : X ≅ Y) (g : Y ≅ Z) :\n  iso_app (f ≪≫ g) = iso_app f ≪≫ iso_app g := rfl\n\nend differential_object\n\nnamespace functor\n\nuniverses v' u'\nvariables (D : Type u') [category.{v'} D]\nvariables [has_zero_morphisms D] [has_shift D]\n\n/--\nA functor `F : C ⥤ D` which commutes with shift functors on `C` and `D` and preserves zero morphisms\ncan be lifted to a functor `differential_object C ⥤ differential_object D`.\n-/\n@[simps]\ndef map_differential_object (F : C ⥤ D) (η : (shift C).functor.comp F ⟶ F.comp (shift D).functor)\n  (hF : ∀ c c', F.map (0 : c ⟶ c') = 0) :\n  differential_object C ⥤ differential_object D :=\n{ obj := λ X, { X := F.obj X.X,\n    d := F.map X.d ≫ η.app X.X,\n    d_squared' := begin\n      dsimp, rw [functor.map_comp, ← functor.comp_map F (shift D).functor],\n      slice_lhs 2 3 { rw [← η.naturality X.d] },\n      rw [functor.comp_map],\n      slice_lhs 1 2 { rw [← F.map_comp, X.d_squared, hF] },\n      rw [zero_comp, zero_comp],\n    end },\n  map := λ X Y f, { f := F.map f.f,\n    comm' := begin\n      dsimp,\n      slice_lhs 2 3 { rw [← functor.comp_map F (shift D).functor, ← η.naturality f.f] },\n      slice_lhs 1 2 { rw [functor.comp_map, ← F.map_comp, f.comm, F.map_comp] },\n      rw [category.assoc]\n    end },\n  map_id' := by { intros, ext, simp },\n  map_comp' := by { intros, ext, simp }, }\n\nend functor\n\nend category_theory\n\nnamespace category_theory\n\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\n\nvariables [has_zero_object C] [has_zero_morphisms C] [has_shift C]\n\nlocal attribute [instance] has_zero_object.has_zero\n\ninstance has_zero_object : has_zero_object (differential_object C) :=\n{ zero :=\n  { X := (0 : C),\n    d := 0, },\n  unique_to := λ X, ⟨⟨{ f := 0 }⟩, λ f, (by ext)⟩,\n  unique_from := λ X, ⟨⟨{ f := 0 }⟩, λ f, (by ext)⟩, }\n\nend differential_object\n\nnamespace differential_object\n\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_zero_morphisms C] [has_shift C]\n\ninstance concrete_category_of_differential_objects :\n  concrete_category (differential_object C) :=\n{ forget := forget C ⋙ category_theory.forget C }\n\ninstance : has_forget₂ (differential_object C) C :=\n{ forget₂ := forget C }\n\nend differential_object\n\n/-! The category of differential objects itself has a shift functor. -/\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\nvariables [has_zero_morphisms C] [has_shift C]\n\n/-- The shift functor on `differential_object C`. -/\n@[simps]\ndef shift_functor : differential_object C ⥤ differential_object C :=\n{ obj := λ X,\n  { X := X.X⟦1⟧,\n    d := X.d⟦1⟧',\n    d_squared' := begin\n      dsimp,\n      rw [←functor.map_comp, X.d_squared, is_equivalence_preserves_zero_morphisms],\n    end },\n  map := λ X Y f,\n  { f := f.f⟦1⟧',\n    comm' := begin dsimp, rw [←functor.map_comp, f.comm, ←functor.map_comp], end, }, }\n\n/-- The inverse shift functor on `differential C`, at the level of objects. -/\n@[simps]\ndef shift_inverse_obj : differential_object C → differential_object C :=\nλ X,\n{ X := X.X⟦-1⟧,\n  d := X.d⟦-1⟧' ≫ (shift C).unit_inv.app X.X ≫ (shift C).counit_inv.app X.X,\n  d_squared' := begin\n    dsimp,\n    rw functor.map_comp,\n    slice_lhs 3 4 { erw ←(shift C).counit_inv.naturality, },\n    slice_lhs 2 3 { erw ←(shift C).unit_inv.naturality, },\n    slice_lhs 1 2 { erw [←functor.map_comp, X.d_squared], },\n    simp,\n  end, }\n\n/-- The inverse shift functor on `differential C`. -/\n@[simps]\ndef shift_inverse : differential_object C ⥤ differential_object C :=\n{ obj := shift_inverse_obj C,\n  map := λ X Y f,\n  { f := f.f⟦-1⟧',\n    comm' := begin\n      dsimp,\n      slice_lhs 3 4 { erw ←(shift C).counit_inv.naturality, },\n      slice_lhs 2 3 { erw ←(shift C).unit_inv.naturality, },\n      slice_lhs 1 2 { erw [←functor.map_comp, f.comm, functor.map_comp], },\n      rw [category.assoc, category.assoc],\n    end, }, }.\n\n/-- The unit for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_unit : 𝟭 (differential_object C) ⟶ shift_functor C ⋙ shift_inverse C :=\n{ app := λ X,\n  { f := (shift C).unit.app X.X,\n    comm' := begin\n      dsimp,\n      slice_rhs 1 2 { erw ←(shift C).unit.naturality, },\n      simp only [category.comp_id, functor.id_map, iso.hom_inv_id_app,\n        category.assoc, equivalence.counit_inv_app_functor],\n    end, }, }\n\n/-- The inverse of the unit for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_unit_inv : shift_functor C ⋙ shift_inverse C ⟶ 𝟭 (differential_object C) :=\n{ app := λ X,\n  { f := (shift C).unit_inv.app X.X,\n    comm' := begin\n      dsimp,\n      slice_rhs 1 2 { erw ←(shift C).unit_inv.naturality, },\n      rw [equivalence.counit_inv_app_functor],\n      slice_lhs 3 4 { rw ←functor.map_comp, },\n      simp only [iso.hom_inv_id_app, functor.comp_map, iso.hom_inv_id_app_assoc,\n        nat_iso.cancel_nat_iso_inv_left, equivalence.inv_fun_map, category.assoc],\n      dsimp,\n      rw category_theory.functor.map_id,\n    end, }, }.\n\n/-- The unit isomorphism for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_unit_iso : 𝟭 (differential_object C) ≅ shift_functor C ⋙ shift_inverse C :=\n{ hom := shift_unit C,\n  inv := shift_unit_inv C, }.\n\n/-- The counit for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_counit : shift_inverse C ⋙ shift_functor C ⟶ 𝟭 (differential_object C) :=\n{ app := λ X,\n  { f := (shift C).counit.app X.X,\n    comm' :=\n    begin\n      dsimp,\n      slice_rhs 1 2 { erw ←(shift C).counit.naturality, },\n      rw [(shift C).functor.map_comp, (shift C).functor.map_comp],\n      slice_lhs 3 4 { erw [←functor.map_comp, iso.inv_hom_id_app, functor.map_id], },\n      erw equivalence.counit_app_functor,\n      rw category.comp_id,\n      refl,\n    end, }, }\n\n/-- The inverse of the counit for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_counit_inv : 𝟭 (differential_object C) ⟶ shift_inverse C ⋙ shift_functor C :=\n{ app := λ X,\n  { f := (shift C).counit_inv.app X.X,\n    comm' :=\n    begin\n      dsimp,\n      rw [(shift C).functor.map_comp, (shift C).functor.map_comp],\n      slice_rhs 1 2 { erw ←(shift C).counit_inv.naturality, },\n      rw ←equivalence.counit_app_functor,\n      slice_rhs 2 3 { rw iso.inv_hom_id_app, },\n      rw category.id_comp,\n      refl,\n    end, }, }\n\n/-- The counit isomorphism for the shift functor on `differential_object C`. -/\n@[simps]\ndef shift_counit_iso : shift_inverse C ⋙ shift_functor C ≅ 𝟭 (differential_object C) :=\n{ hom := shift_counit C,\n  inv := shift_counit_inv C, }\n\n/--\nThe category of differential objects in `C` itself has a shift functor.\n-/\ninstance : has_shift (differential_object C) :=\n{ shift :=\n  { functor := shift_functor C,\n    inverse := shift_inverse C,\n    unit_iso := shift_unit_iso C,\n    counit_iso := shift_counit_iso C, } }\n\nend differential_object\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/differential_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.34462911788977846}}
{"text": "import to_mathlib.partition\nimport to_mathlib.topology.constructions\nimport to_mathlib.geometry.manifold.misc_manifold\n\nnoncomputable theory\n\nopen_locale topology manifold big_operators\nopen set function\n\n-- See: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Typeclass.20resolution.20under.20binders/near/281296989\ninstance real.normed_space.to_module (F : Type*) [normed_add_comm_group F] [normed_space ℝ F] : module ℝ F :=\nby apply_instance\n\nvariables {E₁ E₂ F : Type*}\nvariables [normed_add_comm_group E₁] [normed_space ℝ E₁] [finite_dimensional ℝ E₁]\nvariables [normed_add_comm_group E₂] [normed_space ℝ E₂] [finite_dimensional ℝ E₂]\nvariables [normed_add_comm_group F] [normed_space ℝ F]\n\nvariables {H₁ M₁ H₂ M₂ : Type*}\nvariables [topological_space H₁] (I₁ : model_with_corners ℝ E₁ H₁)\nvariables [topological_space M₁] [charted_space H₁ M₁] [smooth_manifold_with_corners I₁ M₁]\nvariables [sigma_compact_space M₁] [t2_space M₁]\nvariables [topological_space H₂] (I₂ : model_with_corners ℝ E₂ H₂)\nvariables [topological_space M₂] [charted_space H₂ M₂] [smooth_manifold_with_corners I₂ M₂]\n\nlocal notation `𝓒` := cont_mdiff (I₁.prod I₂) 𝓘(ℝ, F)\nlocal notation `𝓒_on` := cont_mdiff_on (I₁.prod I₂) 𝓘(ℝ, F)\n\nnamespace smooth_partition_of_unity\n\nvariables {ι : Type*} (ρ : smooth_partition_of_unity ι I₁ M₁) {I₁} (I₂ M₂)\n\n/-- If `ρ` is a smooth partition of unity for smooth manifold `M₁` and `M₂` is another smooth\nmanifold (with model `I₂`) then `ρ.prod M₂ I₂` is the smooth partition of unity for `M₁ × M₂`\nobtained by precomposing each component of `ρ` with the projection map `M₁ × M₂ → M₁`. -/\ndef prod : smooth_partition_of_unity ι (I₁.prod I₂) (M₁ × M₂) :=\n{ to_fun := λ i, (ρ i).comp ⟨prod.fst, cont_mdiff_fst⟩,\n  locally_finite' :=\n  begin\n    convert ρ.locally_finite.prod_right (λ i, univ),\n    ext i ⟨x, y⟩,\n    simp only [mem_support, cont_mdiff_map.comp_apply, cont_mdiff_map.coe_fn_mk,\n      prod_mk_mem_set_prod_eq, mem_univ, and_true],\n  end,\n  nonneg' := λ i z, by simp only [cont_mdiff_map.comp_apply, cont_mdiff_map.coe_fn_mk, ρ.nonneg i],\n  sum_le_one' := λ z, by simp only [ρ.sum_le_one, cont_mdiff_map.comp_apply],\n  sum_eq_one' := λ z h, by simp only [ρ.sum_eq_one, cont_mdiff_map.comp_apply, mem_univ], }\n\n@[simp] lemma tsupport_prod (i : ι) :\n  tsupport (ρ.prod M₂ I₂ i) = tsupport (ρ i) ×ˢ (univ : set M₂) :=\nbegin\n  change closure (support (λ (xy : M₁ × M₂), (ρ i) xy.1)) = closure (support (ρ i)) ×ˢ univ,\n  rw [← show closure univ = (univ : set M₂), from is_closed_univ.closure_eq, ← closure_prod_eq],\n  congr,\n  ext,\n  simp,\nend\n\nend smooth_partition_of_unity\n\nlemma exists_cont_mdiff_of_convex₂\n  {P : M₁ → (M₂ → F) → Prop} (hP : ∀ x, convex ℝ {f | P x f}) {n : ℕ∞}\n  (hP' : ∀ x : M₁, ∃ (U ∈ 𝓝 x) (f : M₁ → M₂ → F),\n    𝓒_on n (uncurry f) (U ×ˢ (univ : set M₂)) ∧ ∀ y ∈ U, P y (f y)) :\n  ∃ f : M₁ → M₂ → F, 𝓒 n (uncurry f) ∧ ∀ x, P x (f x) :=\nbegin\n  replace hP' : ∀ x : M₁, ∃ U ∈ 𝓝 x, (is_open U) ∧ ∃ f : M₁ → M₂ → F,\n    𝓒_on n (uncurry f) (U ×ˢ (univ : set M₂)) ∧ ∀ y ∈ U, P y (f y),\n  { intros x,\n    rcases ((nhds_basis_opens x).exists_iff _).mp (hP' x) with ⟨U, ⟨x_in, U_op⟩, f, hf, hfP⟩,\n    exact ⟨U, U_op.mem_nhds x_in, U_op, f, hf, hfP⟩,\n    rintros s t hst ⟨f, hf, hf'⟩,\n    exact ⟨f, hf.mono (prod_mono hst subset.rfl), λ x hx, hf' x (hst hx)⟩, },\n  choose U hU U_op φ hφ using hP',\n  rcases smooth_bump_covering.exists_is_subordinate I₁ is_closed_univ (λ x h, hU x) with ⟨ι, b, hb⟩,\n  let ρ := b.to_smooth_partition_of_unity,\n  have subf : ∀ i, support (ρ i) ⊆ U (b.c i) := λ i,\n    subset_closure.trans (smooth_bump_covering.is_subordinate.to_smooth_partition_of_unity hb i),\n  refine ⟨λ x, ∑ᶠ i, (ρ i x) • φ (b.c i) x, _, λ x₀, _⟩,\n  { have h₁ : uncurry (λ (x : M₁), ∑ᶠ (i : ι), ρ i x • φ (b.c i) x) =\n              λ (xy : M₁ × M₂), ∑ᶠ (i : ι), (ρ.prod M₂ I₂ i xy) • uncurry (φ (b.c i)) xy,\n    { ext ⟨x, y⟩,\n      simp only [uncurry_apply_pair],\n      change _ = ∑ᶠ i, ρ i x • φ (b.c i) x y,\n      erw [← ρ.to_partition_of_unity.sum_finsupport_smul,\n        ← ρ.to_partition_of_unity.sum_finsupport_smul],\n      simp only [finset.sum_apply, pi.smul_apply], },\n    rw h₁,\n    rintros ⟨x, y⟩,\n    refine (ρ.prod M₂ I₂).cont_diff_at_sum (λ i hxy, _),\n    have hx : x ∈ U (b.c i),\n    { suffices : x ∈ tsupport (ρ i),\n      { exact smooth_bump_covering.is_subordinate.to_smooth_partition_of_unity hb i this, },\n      rw [ρ.tsupport_prod M₂ I₂ i] at hxy,\n      simpa using hxy, },\n    refine ((hφ $ b.c i).1 ⟨x, y⟩ _).cont_mdiff_at _,\n    { simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true], },\n    { exact mem_nhds_prod_iff.mpr\n      ⟨_, (U_op $ b.c i).mem_nhds hx, _, filter.univ_mem, subset.rfl⟩, }, },\n  { erw ← ρ.to_partition_of_unity.sum_finsupport_smul,\n    refine (hP x₀).sum_mem (λ i hi, (ρ.nonneg i x₀ : _))\n      ρ.to_partition_of_unity.sum_finsupport (λ i hi, _),\n    rw [partition_of_unity.mem_finsupport] at hi,\n    exact (hφ $ b.c i).2 _ (subf _ hi), },\nend\n\nlemma exists_cont_diff_of_convex₂\n  {P : E₁ → (E₂ → F) → Prop} (hP : ∀ x, convex ℝ {f | P x f}) {n : ℕ∞}\n  (hP' : ∀ x : E₁, ∃ (U ∈ 𝓝 x) (f : E₁ → E₂ → F),\n    cont_diff_on ℝ n (uncurry f) (U ×ˢ (univ : set E₂)) ∧ ∀ y ∈ U, P y (f y)) :\n  ∃ f : E₁ → E₂ → F, cont_diff ℝ n (uncurry f) ∧ ∀ x, P x (f x) :=\nbegin\n  simp_rw [← cont_mdiff_on_iff_cont_diff_on, model_with_corners_self_prod] at hP',\n  simp_rw [← cont_mdiff_iff_cont_diff, model_with_corners_self_prod],\n  rw [← charted_space_self_prod] at hP' ⊢, -- Why does `simp_rw` not succeed here?\n  exact exists_cont_mdiff_of_convex₂ 𝓘(ℝ, E₁) 𝓘(ℝ, E₂) hP hP',\nend\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/partition2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.34462911788977846}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.discrete\nimport category_theory.limits.shapes.terminal\nimport algebra.punit_instances\n\n/-!\n# The category of monoids in a monoidal category.\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nopen category_theory\nopen category_theory.monoidal_category\n\nvariables (C : Type u₁) [category.{v₁} C] [monoidal_category.{v₁} C]\n\n/--\nA monoid object internal to a monoidal category.\n\nWhen the monoidal category is preadditive, this is also sometimes called an \"algebra object\".\n-/\nstructure Mon_ :=\n(X : C)\n(one : 𝟙_ C ⟶ X)\n(mul : X ⊗ X ⟶ X)\n(one_mul' : (one ⊗ 𝟙 X) ≫ mul = (λ_ X).hom . obviously)\n(mul_one' : (𝟙 X ⊗ one) ≫ mul = (ρ_ X).hom . obviously)\n-- Obviously there is some flexibility stating this axiom.\n-- This one has left- and right-hand sides matching the statement of `monoid.mul_assoc`,\n-- and chooses to place the associator on the right-hand side.\n-- The heuristic is that unitors and associators \"don't have much weight\".\n(mul_assoc' : (mul ⊗ 𝟙 X) ≫ mul = (α_ X X X).hom ≫ (𝟙 X ⊗ mul) ≫ mul . obviously)\n\nrestate_axiom Mon_.one_mul'\nrestate_axiom Mon_.mul_one'\nrestate_axiom Mon_.mul_assoc'\nattribute [reassoc] Mon_.one_mul Mon_.mul_one -- We prove a more general `@[simp]` lemma below.\nattribute [simp, reassoc] Mon_.mul_assoc\n\nnamespace Mon_\n\n/--\nThe trivial monoid object. We later show this is initial in `Mon_ C`.\n-/\n@[simps]\ndef trivial : Mon_ C :=\n{ X := 𝟙_ C,\n  one := 𝟙 _,\n  mul := (λ_ _).hom,\n  mul_assoc' :=\n    by simp_rw [triangle_assoc, iso.cancel_iso_hom_right, tensor_right_iff, unitors_equal],\n  mul_one' := by simp [unitors_equal] }\n\ninstance : inhabited (Mon_ C) := ⟨trivial C⟩\n\nvariables {C} {M : Mon_ C}\n\n@[simp] lemma one_mul_hom {Z : C} (f : Z ⟶ M.X) : (M.one ⊗ f) ≫ M.mul = (λ_ Z).hom ≫ f :=\nby rw [←id_tensor_comp_tensor_id, category.assoc, M.one_mul, left_unitor_naturality]\n\n@[simp] lemma mul_one_hom {Z : C} (f : Z ⟶ M.X) : (f ⊗ M.one) ≫ M.mul = (ρ_ Z).hom ≫ f :=\nby rw [←tensor_id_comp_id_tensor, category.assoc, M.mul_one, right_unitor_naturality]\n\nlemma assoc_flip : (𝟙 M.X ⊗ M.mul) ≫ M.mul = (α_ M.X M.X M.X).inv ≫ (M.mul ⊗ 𝟙 M.X) ≫ M.mul :=\nby simp\n\n/-- A morphism of monoid objects. -/\n@[ext]\nstructure hom (M N : Mon_ C) :=\n(hom : M.X ⟶ N.X)\n(one_hom' : M.one ≫ hom = N.one . obviously)\n(mul_hom' : M.mul ≫ hom = (hom ⊗ hom) ≫ N.mul . obviously)\n\nrestate_axiom hom.one_hom'\nrestate_axiom hom.mul_hom'\nattribute [simp, reassoc] hom.one_hom hom.mul_hom\n\n/-- The identity morphism on a monoid object. -/\n@[simps]\ndef id (M : Mon_ C) : hom M M :=\n{ hom := 𝟙 M.X, }\n\ninstance hom_inhabited (M : Mon_ C) : inhabited (hom M M) := ⟨id M⟩\n\n/-- Composition of morphisms of monoid objects. -/\n@[simps]\ndef comp {M N O : Mon_ C} (f : hom M N) (g : hom N O) : hom M O :=\n{ hom := f.hom ≫ g.hom, }\n\ninstance : category (Mon_ C) :=\n{ hom := λ M N, hom M N,\n  id := id,\n  comp := λ M N O f g, comp f g, }\n\n@[simp] lemma id_hom' (M : Mon_ C) : (𝟙 M : hom M M).hom = 𝟙 M.X := rfl\n@[simp] lemma comp_hom' {M N K : Mon_ C} (f : M ⟶ N) (g : N ⟶ K) :\n  (f ≫ g : hom M K).hom = f.hom ≫ g.hom := rfl\n\nsection\nvariables (C)\n\n/-- The forgetful functor from monoid objects to the ambient category. -/\n@[simps]\ndef forget : Mon_ C ⥤ C :=\n{ obj := λ A, A.X,\n  map := λ A B f, f.hom, }\n\nend\n\ninstance forget_faithful : faithful (@forget C _ _) := { }\n\ninstance {A B : Mon_ C} (f : A ⟶ B) [e : is_iso ((forget C).map f)] : is_iso f.hom := e\n\n/-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/\ninstance : reflects_isomorphisms (forget C) :=\n{ reflects := λ X Y f e, by exactI ⟨⟨{\n  hom := inv f.hom,\n  mul_hom' :=\n  begin\n    simp only [is_iso.comp_inv_eq, hom.mul_hom, category.assoc, ←tensor_comp_assoc,\n      is_iso.inv_hom_id, tensor_id, category.id_comp],\n  end }, by tidy⟩⟩ }\n\ninstance unique_hom_from_trivial (A : Mon_ C) : unique (trivial C ⟶ A) :=\n{ default :=\n  { hom := A.one,\n    one_hom' := by { dsimp, simp, },\n    mul_hom' := by { dsimp, simp [A.one_mul, unitors_equal], } },\n  uniq := λ f,\n  begin\n    ext, simp,\n    rw [←category.id_comp f.hom],\n    erw f.one_hom,\n  end }\n\nopen category_theory.limits\n\ninstance : has_initial (Mon_ C) :=\nhas_initial_of_unique (trivial C)\n\nend Mon_\n\nnamespace category_theory.lax_monoidal_functor\n\nvariables {C} {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D]\n\n/--\nA lax monoidal functor takes monoid objects to monoid objects.\n\nThat is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.\n-/\n-- TODO: map_Mod F A : Mod A ⥤ Mod (F.map_Mon A)\n@[simps]\ndef map_Mon (F : lax_monoidal_functor C D) : Mon_ C ⥤ Mon_ D :=\n{ obj := λ A,\n  { X := F.obj A.X,\n    one := F.ε ≫ F.map A.one,\n    mul := F.μ _ _ ≫ F.map A.mul,\n    one_mul' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.one_mul], },\n      rw [F.to_functor.map_id],\n      rw [F.left_unitality],\n    end,\n    mul_one' :=\n    begin\n      conv_lhs { rw [id_tensor_comp, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.mul_one], },\n      rw [F.to_functor.map_id],\n      rw [F.right_unitality],\n    end,\n    mul_assoc' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, ←F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.μ_natural], },\n      slice_lhs 3 4 { rw [←F.to_functor.map_comp, A.mul_assoc], },\n      conv_lhs { rw [F.to_functor.map_id] },\n      conv_lhs { rw [F.to_functor.map_comp, F.to_functor.map_comp] },\n      conv_rhs { rw [id_tensor_comp, ←F.to_functor.map_id], },\n      slice_rhs 3 4 { rw [F.μ_natural], },\n      conv_rhs { rw [F.to_functor.map_id] },\n      slice_rhs 1 3 { rw [←F.associativity], },\n      simp only [category.assoc],\n    end, },\n  map := λ A B f,\n  { hom := F.map f.hom,\n    one_hom' := by { dsimp, rw [category.assoc, ←F.to_functor.map_comp, f.one_hom], },\n    mul_hom' :=\n    begin\n      dsimp,\n      rw [category.assoc, F.μ_natural_assoc, ←F.to_functor.map_comp, ←F.to_functor.map_comp,\n        f.mul_hom],\n    end },\n  map_id' := λ A, by { ext, simp, },\n  map_comp' := λ A B C f g, by { ext, simp, }, }\n\nvariables (C D)\n\n/-- `map_Mon` is functorial in the lax monoidal functor. -/\ndef map_Mon_functor : (lax_monoidal_functor C D) ⥤ (Mon_ C ⥤ Mon_ D) :=\n{ obj := map_Mon,\n  map := λ F G α,\n  { app := λ A,\n    { hom := α.app A.X, } } }\n\nend category_theory.lax_monoidal_functor\n\nnamespace Mon_\n\nopen category_theory.lax_monoidal_functor\n\nnamespace equiv_lax_monoidal_functor_punit\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef lax_monoidal_to_Mon : lax_monoidal_functor (discrete punit) C ⥤ Mon_ C :=\n{ obj := λ F, (F.map_Mon : Mon_ _ ⥤ Mon_ C).obj (trivial (discrete punit)),\n  map := λ F G α, ((map_Mon_functor (discrete punit) C).map α).app _ }\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef Mon_to_lax_monoidal : Mon_ C ⥤ lax_monoidal_functor (discrete punit) C :=\n{ obj := λ A,\n  { obj := λ _, A.X,\n    map := λ _ _ _, 𝟙 _,\n    ε := A.one,\n    μ := λ _ _, A.mul,\n    map_id' := λ _, rfl,\n    map_comp' := λ _ _ _ _ _, (category.id_comp (𝟙 A.X)).symm, },\n  map := λ A B f,\n  { app := λ _, f.hom,\n    naturality' := λ _ _ _, by { dsimp, rw [category.id_comp, category.comp_id], },\n    unit' := f.one_hom,\n    tensor' := λ _ _, f.mul_hom, }, }\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef unit_iso :\n  𝟭 (lax_monoidal_functor (discrete punit) C) ≅ lax_monoidal_to_Mon C ⋙ Mon_to_lax_monoidal C :=\nnat_iso.of_components (λ F,\n  monoidal_nat_iso.of_components\n    (λ _, F.to_functor.map_iso (eq_to_iso (by ext)))\n    (by tidy) (by tidy) (by tidy))\n  (by tidy)\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef counit_iso : Mon_to_lax_monoidal C ⋙ lax_monoidal_to_Mon C ≅ 𝟭 (Mon_ C) :=\nnat_iso.of_components (λ F, { hom := { hom := 𝟙 _, }, inv := { hom := 𝟙 _, } })\n  (by tidy)\n\nend equiv_lax_monoidal_functor_punit\n\nopen equiv_lax_monoidal_functor_punit\n\n/--\nMonoid objects in `C` are \"just\" lax monoidal functors from the trivial monoidal category to `C`.\n-/\n@[simps]\ndef equiv_lax_monoidal_functor_punit : lax_monoidal_functor (discrete punit) C ≌ Mon_ C :=\n{ functor := lax_monoidal_to_Mon C,\n  inverse := Mon_to_lax_monoidal C,\n  unit_iso := unit_iso C,\n  counit_iso := counit_iso C, }\n\nend Mon_\n\n/-!\nProjects:\n* Check that `Mon_ Mon ≌ CommMon`, via the Eckmann-Hilton argument.\n  (You'll have to hook up the cartesian monoidal structure on `Mon` first, available in #3463)\n* Check that `Mon_ Top ≌ [bundled topological monoids]`.\n* Check that `Mon_ AddCommGroup ≌ Ring`.\n  (We've already got `Mon_ (Module R) ≌ Algebra R`, in `category_theory.monoidal.internal.Module`.)\n* Can you transport this monoidal structure to `Ring` or `Algebra R`?\n  How does it compare to the \"native\" one?\n* Show that if `C` is braided then `Mon_ C` is naturally monoidal.\n-/\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monoidal/Mon_.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34455971010999603}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.fintype.basic\nimport Mathlib.data.finset.sort\nimport Mathlib.group_theory.perm.basic\nimport Mathlib.group_theory.order_of_element\nimport Mathlib.PostPort\n\nuniverses u u_1 v \n\nnamespace Mathlib\n\n/-!\n# Sign of a permutation\n\nThe main definition of this file is `equiv.perm.sign`, associating a `units ℤ` sign with a\npermutation.\n\nThis file also contains miscellaneous lemmas about `equiv.perm` and `equiv.swap`, building on top\nof those in `data/equiv/basic` and `data/equiv/perm`.\n\n-/\n\nnamespace equiv.perm\n\n\n/--\n`mod_swap i j` contains permutations up to swapping `i` and `j`.\n\nWe use this to partition permutations in `matrix.det_zero_of_row_eq`, such that each partition\nsums up to `0`.\n-/\ndef mod_swap {α : Type u} [DecidableEq α] (i : α) (j : α) : setoid (perm α) :=\n  setoid.mk (fun (σ τ : perm α) => σ = τ ∨ σ = swap i j * τ) sorry\n\nprotected instance r.decidable_rel {α : Type u_1} [fintype α] [DecidableEq α] (i : α) (j : α) : DecidableRel setoid.r :=\n  fun (σ τ : perm α) => or.decidable\n\n/-- If the permutation `f` fixes the subtype `{x // p x}`, then this returns the permutation\n  on `{x // p x}` induced by `f`. -/\ndef subtype_perm {α : Type u} (f : perm α) {p : α → Prop} (h : ∀ (x : α), p x ↔ p (coe_fn f x)) : perm (Subtype fun (x : α) => p x) :=\n  mk (fun (x : Subtype fun (x : α) => p x) => { val := coe_fn f ↑x, property := sorry })\n    (fun (x : Subtype fun (x : α) => p x) => { val := coe_fn (f⁻¹) ↑x, property := sorry }) sorry sorry\n\n@[simp] theorem subtype_perm_one {α : Type u} (p : α → Prop) (h : ∀ (x : α), p x ↔ p (coe_fn 1 x)) : subtype_perm 1 h = 1 := sorry\n\n/-- The inclusion map of permutations on a subtype of `α` into permutations of `α`,\n  fixing the other points. -/\ndef of_subtype {α : Type u} {p : α → Prop} [decidable_pred p] : perm (Subtype p) →* perm α :=\n  monoid_hom.mk\n    (fun (f : perm (Subtype p)) =>\n      mk (fun (x : α) => dite (p x) (fun (h : p x) => ↑(coe_fn f { val := x, property := h })) fun (h : ¬p x) => x)\n        (fun (x : α) => dite (p x) (fun (h : p x) => ↑(coe_fn (f⁻¹) { val := x, property := h })) fun (h : ¬p x) => x)\n        sorry sorry)\n    sorry sorry\n\n/-- Two permutations `f` and `g` are `disjoint` if their supports are disjoint, i.e.,\nevery element is fixed either by `f`, or by `g`. -/\ndef disjoint {α : Type u} (f : perm α) (g : perm α) :=\n  ∀ (x : α), coe_fn f x = x ∨ coe_fn g x = x\n\ntheorem disjoint.symm {α : Type u} {f : perm α} {g : perm α} : disjoint f g → disjoint g f := sorry\n\ntheorem disjoint_comm {α : Type u} {f : perm α} {g : perm α} : disjoint f g ↔ disjoint g f :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint.mul_comm {α : Type u} {f : perm α} {g : perm α} (h : disjoint f g) : f * g = g * f := sorry\n\n@[simp] theorem disjoint_one_left {α : Type u} (f : perm α) : disjoint 1 f :=\n  fun (_x : α) => Or.inl rfl\n\n@[simp] theorem disjoint_one_right {α : Type u} (f : perm α) : disjoint f 1 :=\n  fun (_x : α) => Or.inr rfl\n\ntheorem disjoint.mul_left {α : Type u} {f : perm α} {g : perm α} {h : perm α} (H1 : disjoint f h) (H2 : disjoint g h) : disjoint (f * g) h := sorry\n\ntheorem disjoint.mul_right {α : Type u} {f : perm α} {g : perm α} {h : perm α} (H1 : disjoint f g) (H2 : disjoint f h) : disjoint f (g * h) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (disjoint f (g * h))) (propext disjoint_comm)))\n    (disjoint.mul_left (disjoint.symm H1) (disjoint.symm H2))\n\ntheorem disjoint_prod_right {α : Type u} {f : perm α} (l : List (perm α)) (h : ∀ (g : perm α), g ∈ l → disjoint f g) : disjoint f (list.prod l) := sorry\n\ntheorem disjoint_prod_perm {α : Type u} {l₁ : List (perm α)} {l₂ : List (perm α)} (hl : list.pairwise disjoint l₁) (hp : l₁ ~ l₂) : list.prod l₁ = list.prod l₂ :=\n  list.perm.prod_eq' hp (list.pairwise.imp (fun (f g : perm α) => disjoint.mul_comm) hl)\n\ntheorem of_subtype_subtype_perm {α : Type u} {f : perm α} {p : α → Prop} [decidable_pred p] (h₁ : ∀ (x : α), p x ↔ p (coe_fn f x)) (h₂ : ∀ (x : α), coe_fn f x ≠ x → p x) : coe_fn of_subtype (subtype_perm f h₁) = f := sorry\n\ntheorem of_subtype_apply_of_not_mem {α : Type u} {p : α → Prop} [decidable_pred p] (f : perm (Subtype p)) {x : α} (hx : ¬p x) : coe_fn (coe_fn of_subtype f) x = x :=\n  dif_neg hx\n\ntheorem mem_iff_of_subtype_apply_mem {α : Type u} {p : α → Prop} [decidable_pred p] (f : perm (Subtype p)) (x : α) : p x ↔ p (coe_fn (coe_fn of_subtype f) x) := sorry\n\n@[simp] theorem subtype_perm_of_subtype {α : Type u} {p : α → Prop} [decidable_pred p] (f : perm (Subtype p)) : subtype_perm (coe_fn of_subtype f) (mem_iff_of_subtype_apply_mem f) = f := sorry\n\ntheorem pow_apply_eq_self_of_apply_eq_self {α : Type u} {f : perm α} {x : α} (hfx : coe_fn f x = x) (n : ℕ) : coe_fn (f ^ n) x = x := sorry\n\ntheorem gpow_apply_eq_self_of_apply_eq_self {α : Type u} {f : perm α} {x : α} (hfx : coe_fn f x = x) (n : ℤ) : coe_fn (f ^ n) x = x := sorry\n\ntheorem pow_apply_eq_of_apply_apply_eq_self {α : Type u} {f : perm α} {x : α} (hffx : coe_fn f (coe_fn f x) = x) (n : ℕ) : coe_fn (f ^ n) x = x ∨ coe_fn (f ^ n) x = coe_fn f x := sorry\n\ntheorem gpow_apply_eq_of_apply_apply_eq_self {α : Type u} {f : perm α} {x : α} (hffx : coe_fn f (coe_fn f x) = x) (i : ℤ) : coe_fn (f ^ i) x = x ∨ coe_fn (f ^ i) x = coe_fn f x := sorry\n\n/-- The `finset` of nonfixed points of a permutation. -/\ndef support {α : Type u} [DecidableEq α] [fintype α] (f : perm α) : finset α :=\n  finset.filter (fun (x : α) => coe_fn f x ≠ x) finset.univ\n\n@[simp] theorem mem_support {α : Type u} [DecidableEq α] [fintype α] {f : perm α} {x : α} : x ∈ support f ↔ coe_fn f x ≠ x := sorry\n\n/-- `f.is_swap` indicates that the permutation `f` is a transposition of two elements. -/\ndef is_swap {α : Type u} [DecidableEq α] (f : perm α) :=\n  ∃ (x : α), ∃ (y : α), x ≠ y ∧ f = swap x y\n\ntheorem is_swap.of_subtype_is_swap {α : Type u} [DecidableEq α] {p : α → Prop} [decidable_pred p] {f : perm (Subtype p)} (h : is_swap f) : is_swap (coe_fn of_subtype f) := sorry\n\ntheorem ne_and_ne_of_swap_mul_apply_ne_self {α : Type u} [DecidableEq α] {f : perm α} {x : α} {y : α} (hy : coe_fn (swap x (coe_fn f x) * f) y ≠ y) : coe_fn f y ≠ y ∧ y ≠ x := sorry\n\ntheorem support_swap_mul_eq {α : Type u} [DecidableEq α] [fintype α] {f : perm α} {x : α} (hffx : coe_fn f (coe_fn f x) ≠ x) : support (swap x (coe_fn f x) * f) = finset.erase (support f) x := sorry\n\ntheorem card_support_swap_mul {α : Type u} [DecidableEq α] [fintype α] {f : perm α} {x : α} (hx : coe_fn f x ≠ x) : finset.card (support (swap x (coe_fn f x) * f)) < finset.card (support f) := sorry\n\n/-- Given a list `l : list α` and a permutation `f : perm α` such that the nonfixed points of `f`\n  are in `l`, recursively factors `f` as a product of transpositions. -/\ndef swap_factors_aux {α : Type u} [DecidableEq α] (l : List α) (f : perm α) : (∀ {x : α}, coe_fn f x ≠ x → x ∈ l) →\n  Subtype fun (l : List (perm α)) => list.prod l = f ∧ ∀ (g : perm α), g ∈ l → is_swap g :=\n  sorry\n\n/-- `swap_factors` represents a permutation as a product of a list of transpositions.\nThe representation is non unique and depends on the linear order structure.\nFor types without linear order `trunc_swap_factors` can be used. -/\ndef swap_factors {α : Type u} [DecidableEq α] [fintype α] [linear_order α] (f : perm α) : Subtype fun (l : List (perm α)) => list.prod l = f ∧ ∀ (g : perm α), g ∈ l → is_swap g :=\n  swap_factors_aux (finset.sort LessEq finset.univ) f sorry\n\n/-- This computably represents the fact that any permutation can be represented as the product of\n  a list of transpositions. -/\ndef trunc_swap_factors {α : Type u} [DecidableEq α] [fintype α] (f : perm α) : trunc (Subtype fun (l : List (perm α)) => list.prod l = f ∧ ∀ (g : perm α), g ∈ l → is_swap g) :=\n  quotient.rec_on_subsingleton (finset.val finset.univ)\n    (fun (l : List α) (h : ∀ (x : α), coe_fn f x ≠ x → x ∈ quotient.mk l) => trunc.mk (swap_factors_aux l f h)) sorry\n\n/-- An induction principle for permutations. If `P` holds for the identity permutation, and\nis preserved under composition with a non-trivial swap, then `P` holds for all permutations. -/\ntheorem swap_induction_on {α : Type u} [DecidableEq α] [fintype α] {P : perm α → Prop} (f : perm α) : P 1 → (∀ (f : perm α) (x y : α), x ≠ y → P f → P (swap x y * f)) → P f := sorry\n\n/-- Like `swap_induction_on`, but with the composition on the right of `f`.\n\nAn induction principle for permutations. If `P` holds for the identity permutation, and\nis preserved under composition with a non-trivial swap, then `P` holds for all permutations. -/\ntheorem swap_induction_on' {α : Type u} [DecidableEq α] [fintype α] {P : perm α → Prop} (f : perm α) : P 1 → (∀ (f : perm α) (x y : α), x ≠ y → P f → P (f * swap x y)) → P f :=\n  fun (h1 : P 1) (IH : ∀ (f : perm α) (x y : α), x ≠ y → P f → P (f * swap x y)) =>\n    inv_inv f ▸ swap_induction_on (f⁻¹) h1 fun (f : perm α) => IH (f⁻¹)\n\ntheorem is_conj_swap {α : Type u} [DecidableEq α] {w : α} {x : α} {y : α} {z : α} (hwx : w ≠ x) (hyz : y ≠ z) : is_conj (swap w x) (swap y z) := sorry\n\n/-- set of all pairs (⟨a, b⟩ : Σ a : fin n, fin n) such that b < a -/\ndef fin_pairs_lt (n : ℕ) : finset (sigma fun (a : fin n) => fin n) :=\n  finset.sigma finset.univ fun (a : fin n) => finset.attach_fin (finset.range ↑a) sorry\n\ntheorem mem_fin_pairs_lt {n : ℕ} {a : sigma fun (a : fin n) => fin n} : a ∈ fin_pairs_lt n ↔ sigma.snd a < sigma.fst a := sorry\n\n/-- `sign_aux σ` is the sign of a permutation on `fin n`, defined as the parity of the number of\n  pairs `(x₁, x₂)` such that `x₂ < x₁` but `σ x₁ ≤ σ x₂` -/\ndef sign_aux {n : ℕ} (a : perm (fin n)) : units ℤ :=\n  finset.prod (fin_pairs_lt n)\n    fun (x : sigma fun (a : fin n) => fin n) => ite (coe_fn a (sigma.fst x) ≤ coe_fn a (sigma.snd x)) (-1) 1\n\n@[simp] theorem sign_aux_one (n : ℕ) : sign_aux 1 = 1 := sorry\n\n/-- `sign_bij_aux f ⟨a, b⟩` returns the pair consisting of `f a` and `f b` in decreasing order. -/\ndef sign_bij_aux {n : ℕ} (f : perm (fin n)) (a : sigma fun (a : fin n) => fin n) : sigma fun (a : fin n) => fin n :=\n  dite (coe_fn f (sigma.snd a) < coe_fn f (sigma.fst a))\n    (fun (hxa : coe_fn f (sigma.snd a) < coe_fn f (sigma.fst a)) =>\n      sigma.mk (coe_fn f (sigma.fst a)) (coe_fn f (sigma.snd a)))\n    fun (hxa : ¬coe_fn f (sigma.snd a) < coe_fn f (sigma.fst a)) =>\n      sigma.mk (coe_fn f (sigma.snd a)) (coe_fn f (sigma.fst a))\n\ntheorem sign_bij_aux_inj {n : ℕ} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n) (b : sigma fun (a : fin n) => fin n) : a ∈ fin_pairs_lt n → b ∈ fin_pairs_lt n → sign_bij_aux f a = sign_bij_aux f b → a = b := sorry\n\ntheorem sign_bij_aux_surj {n : ℕ} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n) (H : a ∈ fin_pairs_lt n) : ∃ (b : sigma fun (a : fin n) => fin n), ∃ (H : b ∈ fin_pairs_lt n), a = sign_bij_aux f b := sorry\n\ntheorem sign_bij_aux_mem {n : ℕ} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n) : a ∈ fin_pairs_lt n → sign_bij_aux f a ∈ fin_pairs_lt n := sorry\n\n@[simp] theorem sign_aux_inv {n : ℕ} (f : perm (fin n)) : sign_aux (f⁻¹) = sign_aux f := sorry\n\ntheorem sign_aux_mul {n : ℕ} (f : perm (fin n)) (g : perm (fin n)) : sign_aux (f * g) = sign_aux f * sign_aux g := sorry\n\n-- TODO: slow\n\ntheorem sign_aux_swap {n : ℕ} {x : fin n} {y : fin n} (hxy : x ≠ y) : sign_aux (swap x y) = -1 := sorry\n\n/-- When the list `l : list α` contains all nonfixed points of the permutation `f : perm α`,\n  `sign_aux2 l f` recursively calculates the sign of `f`. -/\ndef sign_aux2 {α : Type u} [DecidableEq α] : List α → perm α → units ℤ :=\n  sorry\n\ntheorem sign_aux_eq_sign_aux2 {α : Type u} [DecidableEq α] {n : ℕ} (l : List α) (f : perm α) (e : α ≃ fin n) (h : ∀ (x : α), coe_fn f x ≠ x → x ∈ l) : sign_aux (equiv.trans (equiv.trans (equiv.symm e) f) e) = sign_aux2 l f := sorry\n\n/-- When the multiset `s : multiset α` contains all nonfixed points of the permutation `f : perm α`,\n  `sign_aux2 f _` recursively calculates the sign of `f`. -/\ndef sign_aux3 {α : Type u} [DecidableEq α] [fintype α] (f : perm α) {s : multiset α} : (∀ (x : α), x ∈ s) → units ℤ :=\n  quotient.hrec_on s (fun (l : List α) (h : ∀ (x : α), x ∈ quotient.mk l) => sign_aux2 l f) sorry\n\ntheorem sign_aux3_mul_and_swap {α : Type u} [DecidableEq α] [fintype α] (f : perm α) (g : perm α) (s : multiset α) (hs : ∀ (x : α), x ∈ s) : sign_aux3 (f * g) hs = sign_aux3 f hs * sign_aux3 g hs ∧ ∀ (x y : α), x ≠ y → sign_aux3 (swap x y) hs = -1 := sorry\n\n/-- `sign` of a permutation returns the signature or parity of a permutation, `1` for even\npermutations, `-1` for odd permutations. It is the unique surjective group homomorphism from\n`perm α` to the group with two elements.-/\ndef sign {α : Type u} [DecidableEq α] [fintype α] : perm α →* units ℤ :=\n  monoid_hom.mk' (fun (f : perm α) => sign_aux3 f finset.mem_univ) sorry\n\n@[simp] theorem sign_mul {α : Type u} [DecidableEq α] [fintype α] (f : perm α) (g : perm α) : coe_fn sign (f * g) = coe_fn sign f * coe_fn sign g :=\n  monoid_hom.map_mul sign f g\n\n@[simp] theorem sign_trans {α : Type u} [DecidableEq α] [fintype α] (f : perm α) (g : perm α) : coe_fn sign (equiv.trans f g) = coe_fn sign g * coe_fn sign f := sorry\n\n@[simp] theorem sign_one {α : Type u} [DecidableEq α] [fintype α] : coe_fn sign 1 = 1 :=\n  monoid_hom.map_one sign\n\n@[simp] theorem sign_refl {α : Type u} [DecidableEq α] [fintype α] : coe_fn sign (equiv.refl α) = 1 :=\n  monoid_hom.map_one sign\n\n@[simp] theorem sign_inv {α : Type u} [DecidableEq α] [fintype α] (f : perm α) : coe_fn sign (f⁻¹) = coe_fn sign f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn sign (f⁻¹) = coe_fn sign f)) (monoid_hom.map_inv sign f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn sign f⁻¹ = coe_fn sign f)) (int.units_inv_eq_self (coe_fn sign f))))\n      (Eq.refl (coe_fn sign f)))\n\n@[simp] theorem sign_symm {α : Type u} [DecidableEq α] [fintype α] (e : perm α) : coe_fn sign (equiv.symm e) = coe_fn sign e :=\n  sign_inv e\n\ntheorem sign_swap {α : Type u} [DecidableEq α] [fintype α] {x : α} {y : α} (h : x ≠ y) : coe_fn sign (swap x y) = -1 :=\n  and.right (sign_aux3_mul_and_swap 1 1 (finset.val finset.univ) finset.mem_univ) x y h\n\n@[simp] theorem sign_swap' {α : Type u} [DecidableEq α] [fintype α] {x : α} {y : α} : coe_fn sign (swap x y) = ite (x = y) 1 (-1) := sorry\n\ntheorem is_swap.sign_eq {α : Type u} [DecidableEq α] [fintype α] {f : perm α} (h : is_swap f) : coe_fn sign f = -1 := sorry\n\ntheorem sign_aux3_symm_trans_trans {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (f : perm α) (e : α ≃ β) {s : multiset α} {t : multiset β} (hs : ∀ (x : α), x ∈ s) (ht : ∀ (x : β), x ∈ t) : sign_aux3 (equiv.trans (equiv.trans (equiv.symm e) f) e) ht = sign_aux3 f hs := sorry\n\n@[simp] theorem sign_symm_trans_trans {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (f : perm α) (e : α ≃ β) : coe_fn sign (equiv.trans (equiv.trans (equiv.symm e) f) e) = coe_fn sign f :=\n  sign_aux3_symm_trans_trans f e finset.mem_univ finset.mem_univ\n\n@[simp] theorem sign_trans_trans_symm {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (f : perm β) (e : α ≃ β) : coe_fn sign (equiv.trans (equiv.trans e f) (equiv.symm e)) = coe_fn sign f :=\n  sign_symm_trans_trans f (equiv.symm e)\n\ntheorem sign_prod_list_swap {α : Type u} [DecidableEq α] [fintype α] {l : List (perm α)} (hl : ∀ (g : perm α), g ∈ l → is_swap g) : coe_fn sign (list.prod l) = (-1) ^ list.length l := sorry\n\ntheorem sign_surjective {α : Type u} [DecidableEq α] [fintype α] (hα : 1 < fintype.card α) : function.surjective ⇑sign := sorry\n\ntheorem eq_sign_of_surjective_hom {α : Type u} [DecidableEq α] [fintype α] {s : perm α →* units ℤ} (hs : function.surjective ⇑s) : s = sign := sorry\n\ntheorem sign_subtype_perm {α : Type u} [DecidableEq α] [fintype α] (f : perm α) {p : α → Prop} [decidable_pred p] (h₁ : ∀ (x : α), p x ↔ p (coe_fn f x)) (h₂ : ∀ (x : α), coe_fn f x ≠ x → p x) : coe_fn sign (subtype_perm f h₁) = coe_fn sign f := sorry\n\n@[simp] theorem sign_of_subtype {α : Type u} [DecidableEq α] [fintype α] {p : α → Prop} [decidable_pred p] (f : perm (Subtype p)) : coe_fn sign (coe_fn of_subtype f) = coe_fn sign f := sorry\n\ntheorem sign_eq_sign_of_equiv {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (f : perm α) (g : perm β) (e : α ≃ β) (h : ∀ (x : α), coe_fn e (coe_fn f x) = coe_fn g (coe_fn e x)) : coe_fn sign f = coe_fn sign g := sorry\n\ntheorem sign_bij {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] {f : perm α} {g : perm β} (i : (x : α) → coe_fn f x ≠ x → β) (h : ∀ (x : α) (hx : coe_fn f x ≠ x) (hx' : coe_fn f (coe_fn f x) ≠ coe_fn f x), i (coe_fn f x) hx' = coe_fn g (i x hx)) (hi : ∀ (x₁ x₂ : α) (hx₁ : coe_fn f x₁ ≠ x₁) (hx₂ : coe_fn f x₂ ≠ x₂), i x₁ hx₁ = i x₂ hx₂ → x₁ = x₂) (hg : ∀ (y : β), coe_fn g y ≠ y → ∃ (x : α), ∃ (hx : coe_fn f x ≠ x), i x hx = y) : coe_fn sign f = coe_fn sign g := sorry\n\n@[simp] theorem support_swap {α : Type u} [DecidableEq α] [fintype α] {x : α} {y : α} (hxy : x ≠ y) : support (swap x y) = insert x (singleton y) := sorry\n\ntheorem card_support_swap {α : Type u} [DecidableEq α] [fintype α] {x : α} {y : α} (hxy : x ≠ y) : finset.card (support (swap x y)) = bit0 1 := sorry\n\n/-- If we apply `prod_extend_right a (σ a)` for all `a : α` in turn,\nwe get `prod_congr_right σ`. -/\ntheorem prod_prod_extend_right {β : Type v} {α : Type u_1} [DecidableEq α] (σ : α → perm β) {l : List α} (hl : list.nodup l) (mem_l : ∀ (a : α), a ∈ l) : list.prod (list.map (fun (a : α) => prod_extend_right a (σ a)) l) = prod_congr_right σ := sorry\n\n@[simp] theorem sign_prod_extend_right {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (a : α) (σ : perm β) : coe_fn sign (prod_extend_right a σ) = coe_fn sign σ := sorry\n\ntheorem sign_prod_congr_right {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (σ : α → perm β) : coe_fn sign (prod_congr_right σ) = finset.prod finset.univ fun (k : α) => coe_fn sign (σ k) := sorry\n\ntheorem sign_prod_congr_left {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (σ : α → perm β) : coe_fn sign (prod_congr_left σ) = finset.prod finset.univ fun (k : α) => coe_fn sign (σ k) := sorry\n\n@[simp] theorem sign_perm_congr {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (e : α ≃ β) (p : perm α) : coe_fn sign (coe_fn (perm_congr e) p) = coe_fn sign p := sorry\n\n@[simp] theorem sign_sum_congr {α : Type u} {β : Type v} [DecidableEq α] [fintype α] [DecidableEq β] [fintype β] (σa : perm α) (σb : perm β) : coe_fn sign (sum_congr σa σb) = coe_fn sign σa * coe_fn sign σb := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/group_theory/perm/sign.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34455971010999603}}
{"text": "/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.big_operators.ring\nimport Mathlib.number_theory.divisors\nimport Mathlib.algebra.squarefree\nimport Mathlib.algebra.invertible\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Arithmetic Functions and Dirichlet Convolution\n\nThis file defines arithmetic functions, which are functions from `ℕ` to a specified type that map 0\nto 0. In the literature, they are often instead defined as functions from `ℕ+`. These arithmetic\nfunctions are endowed with a multiplication, given by Dirichlet convolution, and pointwise addition,\nto form the Dirichlet ring.\n\n## Main Definitions\n * `arithmetic_function R` consists of functions `f : ℕ → R` such that `f 0 = 0`.\n * An arithmetic function `f` `is_multiplicative` when `x.coprime y → f (x * y) = f x * f y`.\n * The pointwise operations `pmul` and `ppow` differ from the multiplication\n  and power instances on `arithmetic_function R`, which use Dirichlet multiplication.\n * `ζ` is the arithmetic function such that `ζ x = 1` for `0 < x`.\n * `σ k` is the arithmetic function such that `σ k x = ∑ y in divisors x, y ^ k` for `0 < x`.\n * `pow k` is the arithmetic function such that `pow k x = x ^ k` for `0 < x`.\n * `id` is the identity arithmetic function on `ℕ`.\n * `ω n` is the number of distinct prime factors of `n`.\n * `Ω n` is the number of prime factors of `n` counted with multiplicity.\n * `μ` is the Möbius function.\n\n## Main Results\n * Several forms of Möbius inversion:\n * `sum_eq_iff_sum_mul_moebius_eq` for functions to a `comm_ring`\n * `sum_eq_iff_sum_smul_moebius_eq` for functions to an `add_comm_group`\n * `prod_eq_iff_prod_pow_moebius_eq` for functions to a `comm_group`\n * `prod_eq_iff_prod_pow_moebius_eq_of_nonzero` for functions to a `comm_group_with_zero`\n\n## Notation\nThe arithmetic functions `ζ` and `σ` have Greek letter names, which are localized notation in\nthe namespace `arithmetic_function`.\n\n## Tags\narithmetic functions, dirichlet convolution, divisors\n\n-/\n\nnamespace nat\n\n\n/-- An arithmetic function is a function from `ℕ` that maps 0 to 0. In the literature, they are\n  often instead defined as functions from `ℕ+`. Multiplication on `arithmetic_functions` is by\n  Dirichlet convolution. -/\ndef arithmetic_function (R : Type u_1) [HasZero R] := zero_hom ℕ R\n\nnamespace arithmetic_function\n\n\n@[simp] theorem to_fun_eq {R : Type u_1} [HasZero R] (f : arithmetic_function R) :\n    zero_hom.to_fun f = ⇑f :=\n  rfl\n\n@[simp] theorem map_zero {R : Type u_1} [HasZero R] {f : arithmetic_function R} : coe_fn f 0 = 0 :=\n  zero_hom.map_zero' f\n\ntheorem coe_inj {R : Type u_1} [HasZero R] {f : arithmetic_function R} {g : arithmetic_function R} :\n    ⇑f = ⇑g ↔ f = g :=\n  { mp := fun (h : ⇑f = ⇑g) => zero_hom.coe_inj h, mpr := fun (h : f = g) => h ▸ rfl }\n\n@[simp] theorem zero_apply {R : Type u_1} [HasZero R] {x : ℕ} : coe_fn 0 x = 0 :=\n  zero_hom.zero_apply x\n\ntheorem ext {R : Type u_1} [HasZero R] {f : arithmetic_function R} {g : arithmetic_function R}\n    (h : ∀ (x : ℕ), coe_fn f x = coe_fn g x) : f = g :=\n  zero_hom.ext h\n\ntheorem ext_iff {R : Type u_1} [HasZero R] {f : arithmetic_function R} {g : arithmetic_function R} :\n    f = g ↔ ∀ (x : ℕ), coe_fn f x = coe_fn g x :=\n  zero_hom.ext_iff\n\nprotected instance has_one {R : Type u_1} [HasZero R] [HasOne R] : HasOne (arithmetic_function R) :=\n  { one := zero_hom.mk (fun (x : ℕ) => ite (x = 1) 1 0) sorry }\n\n@[simp] theorem one_one {R : Type u_1} [HasZero R] [HasOne R] : coe_fn 1 1 = 1 := rfl\n\n@[simp] theorem one_apply_ne {R : Type u_1} [HasZero R] [HasOne R] {x : ℕ} (h : x ≠ 1) :\n    coe_fn 1 x = 0 :=\n  if_neg h\n\nprotected instance nat_coe {R : Type u_1} [HasZero R] [HasOne R] [Add R] :\n    has_coe (arithmetic_function ℕ) (arithmetic_function R) :=\n  has_coe.mk fun (f : arithmetic_function ℕ) => zero_hom.mk ↑⇑f sorry\n\n@[simp] theorem nat_coe_nat (f : arithmetic_function ℕ) : ↑f = f :=\n  ext fun (_x : ℕ) => cast_id (coe_fn f _x)\n\n@[simp] theorem nat_coe_apply {R : Type u_1} [HasZero R] [HasOne R] [Add R]\n    {f : arithmetic_function ℕ} {x : ℕ} : coe_fn (↑f) x = ↑(coe_fn f x) :=\n  rfl\n\nprotected instance int_coe {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R] :\n    has_coe (arithmetic_function ℤ) (arithmetic_function R) :=\n  has_coe.mk fun (f : arithmetic_function ℤ) => zero_hom.mk ↑⇑f sorry\n\n@[simp] theorem int_coe_int (f : arithmetic_function ℤ) : ↑f = f :=\n  ext fun (_x : ℕ) => int.cast_id (coe_fn f _x)\n\n@[simp] theorem int_coe_apply {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R]\n    {f : arithmetic_function ℤ} {x : ℕ} : coe_fn (↑f) x = ↑(coe_fn f x) :=\n  rfl\n\n@[simp] theorem coe_coe {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R]\n    {f : arithmetic_function ℕ} : ↑↑f = ↑f :=\n  sorry\n\nprotected instance has_add {R : Type u_1} [add_monoid R] : Add (arithmetic_function R) :=\n  { add :=\n      fun (f g : arithmetic_function R) =>\n        zero_hom.mk (fun (n : ℕ) => coe_fn f n + coe_fn g n) sorry }\n\n@[simp] theorem add_apply {R : Type u_1} [add_monoid R] {f : arithmetic_function R}\n    {g : arithmetic_function R} {n : ℕ} : coe_fn (f + g) n = coe_fn f n + coe_fn g n :=\n  rfl\n\nprotected instance add_monoid {R : Type u_1} [add_monoid R] : add_monoid (arithmetic_function R) :=\n  add_monoid.mk Add.add sorry 0 sorry sorry\n\nprotected instance add_comm_monoid {R : Type u_1} [add_comm_monoid R] :\n    add_comm_monoid (arithmetic_function R) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\nprotected instance add_group {R : Type u_1} [add_group R] : add_group (arithmetic_function R) :=\n  add_group.mk add_monoid.add sorry add_monoid.zero sorry sorry\n    (fun (f : arithmetic_function R) => zero_hom.mk (fun (n : ℕ) => -coe_fn f n) sorry)\n    (sub_neg_monoid.sub._default add_monoid.add sorry add_monoid.zero sorry sorry\n      fun (f : arithmetic_function R) => zero_hom.mk (fun (n : ℕ) => -coe_fn f n) sorry)\n    sorry\n\nprotected instance add_comm_group {R : Type u_1} [add_comm_group R] :\n    add_comm_group (arithmetic_function R) :=\n  add_comm_group.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry add_group.neg\n    add_group.sub sorry sorry\n\n/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function\n  such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/\nprotected instance has_scalar {R : Type u_1} {M : Type u_2} [HasZero R] [add_comm_monoid M]\n    [has_scalar R M] : has_scalar (arithmetic_function R) (arithmetic_function M) :=\n  has_scalar.mk\n    fun (f : arithmetic_function R) (g : arithmetic_function M) =>\n      zero_hom.mk\n        (fun (n : ℕ) =>\n          finset.sum (divisors_antidiagonal n)\n            fun (x : ℕ × ℕ) => coe_fn f (prod.fst x) • coe_fn g (prod.snd x))\n        sorry\n\n@[simp] theorem smul_apply {R : Type u_1} {M : Type u_2} [HasZero R] [add_comm_monoid M]\n    [has_scalar R M] {f : arithmetic_function R} {g : arithmetic_function M} {n : ℕ} :\n    coe_fn (f • g) n =\n        finset.sum (divisors_antidiagonal n)\n          fun (x : ℕ × ℕ) => coe_fn f (prod.fst x) • coe_fn g (prod.snd x) :=\n  rfl\n\n/-- The Dirichlet convolution of two arithmetic functions `f` and `g` is another arithmetic function\n  such that `(f * g) n` is the sum of `f x * g y` over all `(x,y)` such that `x * y = n`. -/\nprotected instance has_mul {R : Type u_1} [semiring R] : Mul (arithmetic_function R) :=\n  { mul := has_scalar.smul }\n\n@[simp] theorem mul_apply {R : Type u_1} [semiring R] {f : arithmetic_function R}\n    {g : arithmetic_function R} {n : ℕ} :\n    coe_fn (f * g) n =\n        finset.sum (divisors_antidiagonal n)\n          fun (x : ℕ × ℕ) => coe_fn f (prod.fst x) * coe_fn g (prod.snd x) :=\n  rfl\n\ntheorem mul_smul' {R : Type u_1} {M : Type u_2} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (f : arithmetic_function R) (g : arithmetic_function R) (h : arithmetic_function M) :\n    (f * g) • h = f • g • h :=\n  sorry\n\ntheorem one_smul' {R : Type u_1} {M : Type u_2} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (b : arithmetic_function M) : 1 • b = b :=\n  sorry\n\nprotected instance monoid {R : Type u_1} [semiring R] : monoid (arithmetic_function R) :=\n  monoid.mk Mul.mul sorry 1 sorry sorry\n\nprotected instance semiring {R : Type u_1} [semiring R] : semiring (arithmetic_function R) :=\n  semiring.mk Add.add sorry 0 sorry sorry sorry Mul.mul sorry monoid.one sorry sorry sorry sorry\n    sorry sorry\n\nprotected instance comm_semiring {R : Type u_1} [comm_semiring R] :\n    comm_semiring (arithmetic_function R) :=\n  comm_semiring.mk semiring.add sorry semiring.zero sorry sorry sorry semiring.mul sorry\n    semiring.one sorry sorry sorry sorry sorry sorry sorry\n\nprotected instance comm_ring {R : Type u_1} [comm_ring R] : comm_ring (arithmetic_function R) :=\n  comm_ring.mk add_comm_group.add sorry add_comm_group.zero sorry sorry add_comm_group.neg\n    add_comm_group.sub sorry sorry comm_semiring.mul sorry comm_semiring.one sorry sorry sorry sorry\n    sorry\n\nprotected instance semimodule {R : Type u_1} {M : Type u_2} [semiring R] [add_comm_monoid M]\n    [semimodule R M] : semimodule (arithmetic_function R) (arithmetic_function M) :=\n  semimodule.mk sorry sorry\n\n/-- `ζ 0 = 0`, otherwise `ζ x = 1`. The Dirichlet Series is the Riemann ζ.  -/\ndef zeta : arithmetic_function ℕ := zero_hom.mk (fun (x : ℕ) => ite (x = 0) 0 1) sorry\n\n@[simp] theorem zeta_apply {x : ℕ} : coe_fn zeta x = ite (x = 0) 0 1 := rfl\n\ntheorem zeta_apply_ne {x : ℕ} (h : x ≠ 0) : coe_fn zeta x = 1 := if_neg h\n\n@[simp] theorem coe_zeta_mul_apply {R : Type u_1} [semiring R] {f : arithmetic_function R} {x : ℕ} :\n    coe_fn (↑zeta * f) x = finset.sum (divisors x) fun (i : ℕ) => coe_fn f i :=\n  sorry\n\ntheorem coe_zeta_smul_apply {R : Type u_1} {M : Type u_2} [comm_ring R] [add_comm_group M]\n    [module R M] {f : arithmetic_function M} {x : ℕ} :\n    coe_fn (↑zeta • f) x = finset.sum (divisors x) fun (i : ℕ) => coe_fn f i :=\n  sorry\n\n@[simp] theorem coe_mul_zeta_apply {R : Type u_1} [semiring R] {f : arithmetic_function R} {x : ℕ} :\n    coe_fn (f * ↑zeta) x = finset.sum (divisors x) fun (i : ℕ) => coe_fn f i :=\n  sorry\n\ntheorem zeta_mul_apply {f : arithmetic_function ℕ} {x : ℕ} :\n    coe_fn (zeta * f) x = finset.sum (divisors x) fun (i : ℕ) => coe_fn f i :=\n  sorry\n\ntheorem mul_zeta_apply {f : arithmetic_function ℕ} {x : ℕ} :\n    coe_fn (f * zeta) x = finset.sum (divisors x) fun (i : ℕ) => coe_fn f i :=\n  sorry\n\n/-- This is the pointwise product of `arithmetic_function`s. -/\ndef pmul {R : Type u_1} [mul_zero_class R] (f : arithmetic_function R) (g : arithmetic_function R) :\n    arithmetic_function R :=\n  zero_hom.mk (fun (x : ℕ) => coe_fn f x * coe_fn g x) sorry\n\n@[simp] theorem pmul_apply {R : Type u_1} [mul_zero_class R] {f : arithmetic_function R}\n    {g : arithmetic_function R} {x : ℕ} : coe_fn (pmul f g) x = coe_fn f x * coe_fn g x :=\n  rfl\n\ntheorem pmul_comm {R : Type u_1} [comm_monoid_with_zero R] (f : arithmetic_function R)\n    (g : arithmetic_function R) : pmul f g = pmul g f :=\n  sorry\n\n@[simp] theorem pmul_zeta {R : Type u_1} [semiring R] (f : arithmetic_function R) :\n    pmul f ↑zeta = f :=\n  sorry\n\n@[simp] theorem zeta_pmul {R : Type u_1} [semiring R] (f : arithmetic_function R) :\n    pmul (↑zeta) f = f :=\n  sorry\n\n/-- This is the pointwise power of `arithmetic_function`s. -/\ndef ppow {R : Type u_1} [semiring R] (f : arithmetic_function R) (k : ℕ) : arithmetic_function R :=\n  dite (k = 0) (fun (h0 : k = 0) => ↑zeta)\n    fun (h0 : ¬k = 0) => zero_hom.mk (fun (x : ℕ) => coe_fn f x ^ k) sorry\n\n@[simp] theorem ppow_zero {R : Type u_1} [semiring R] {f : arithmetic_function R} :\n    ppow f 0 = ↑zeta :=\n  sorry\n\n@[simp] theorem ppow_apply {R : Type u_1} [semiring R] {f : arithmetic_function R} {k : ℕ} {x : ℕ}\n    (kpos : 0 < k) : coe_fn (ppow f k) x = coe_fn f x ^ k :=\n  sorry\n\ntheorem ppow_succ {R : Type u_1} [semiring R] {f : arithmetic_function R} {k : ℕ} :\n    ppow f (k + 1) = pmul f (ppow f k) :=\n  sorry\n\ntheorem ppow_succ' {R : Type u_1} [semiring R] {f : arithmetic_function R} {k : ℕ} {kpos : 0 < k} :\n    ppow f (k + 1) = pmul (ppow f k) f :=\n  sorry\n\n/-- Multiplicative functions -/\ndef is_multiplicative {R : Type u_1} [monoid_with_zero R] (f : arithmetic_function R) :=\n  coe_fn f 1 = 1 ∧ ∀ {m n : ℕ}, coprime m n → coe_fn f (m * n) = coe_fn f m * coe_fn f n\n\nnamespace is_multiplicative\n\n\n@[simp] theorem map_one {R : Type u_1} [monoid_with_zero R] {f : arithmetic_function R}\n    (h : is_multiplicative f) : coe_fn f 1 = 1 :=\n  and.left h\n\n@[simp] theorem map_mul_of_coprime {R : Type u_1} [monoid_with_zero R] {f : arithmetic_function R}\n    (hf : is_multiplicative f) {m : ℕ} {n : ℕ} (h : coprime m n) :\n    coe_fn f (m * n) = coe_fn f m * coe_fn f n :=\n  and.right hf m n h\n\ntheorem nat_cast {R : Type u_1} {f : arithmetic_function ℕ} [semiring R] (h : is_multiplicative f) :\n    is_multiplicative ↑f :=\n  sorry\n\ntheorem int_cast {R : Type u_1} {f : arithmetic_function ℤ} [ring R] (h : is_multiplicative f) :\n    is_multiplicative ↑f :=\n  sorry\n\ntheorem mul {R : Type u_1} [comm_semiring R] {f : arithmetic_function R} {g : arithmetic_function R}\n    (hf : is_multiplicative f) (hg : is_multiplicative g) : is_multiplicative (f * g) :=\n  sorry\n\ntheorem pmul {R : Type u_1} [comm_semiring R] {f : arithmetic_function R}\n    {g : arithmetic_function R} (hf : is_multiplicative f) (hg : is_multiplicative g) :\n    is_multiplicative (pmul f g) :=\n  sorry\n\nend is_multiplicative\n\n\n/-- The identity on `ℕ` as an `arithmetic_function`.  -/\ndef id : arithmetic_function ℕ := zero_hom.mk id sorry\n\n@[simp] theorem id_apply {x : ℕ} : coe_fn id x = x := rfl\n\n/-- `pow k n = n ^ k`, except `pow 0 0 = 0`. -/\ndef pow (k : ℕ) : arithmetic_function ℕ := ppow id k\n\n@[simp] theorem pow_apply {k : ℕ} {n : ℕ} : coe_fn (pow k) n = ite (k = 0 ∧ n = 0) 0 (n ^ k) :=\n  sorry\n\n/-- `σ k n` is the sum of the `k`th powers of the divisors of `n` -/\ndef sigma (k : ℕ) : arithmetic_function ℕ :=\n  zero_hom.mk (fun (n : ℕ) => finset.sum (divisors n) fun (d : ℕ) => d ^ k) sorry\n\n@[simp] theorem sigma_apply {k : ℕ} {n : ℕ} :\n    coe_fn (sigma k) n = finset.sum (divisors n) fun (d : ℕ) => d ^ k :=\n  rfl\n\ntheorem sigma_one_apply {n : ℕ} : coe_fn (sigma 1) n = finset.sum (divisors n) fun (d : ℕ) => d :=\n  sorry\n\ntheorem zeta_mul_pow_eq_sigma {k : ℕ} : zeta * pow k = sigma k := sorry\n\ntheorem is_multiplicative_zeta : is_multiplicative zeta := sorry\n\ntheorem is_multiplicative_id : is_multiplicative id :=\n  { left := rfl, right := fun (_x _x_1 : ℕ) (_x_2 : coprime _x _x_1) => rfl }\n\ntheorem is_multiplicative.ppow {R : Type u_1} [comm_semiring R] {f : arithmetic_function R}\n    (hf : is_multiplicative f) {k : ℕ} : is_multiplicative (ppow f k) :=\n  sorry\n\ntheorem is_multiplicative_pow {k : ℕ} : is_multiplicative (pow k) :=\n  is_multiplicative.ppow is_multiplicative_id\n\ntheorem is_multiplicative_sigma {k : ℕ} : is_multiplicative (sigma k) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_multiplicative (sigma k))) (Eq.symm zeta_mul_pow_eq_sigma)))\n    (is_multiplicative.mul is_multiplicative_zeta is_multiplicative_pow)\n\n/-- `Ω n` is the number of prime factors of `n`. -/\ndef card_factors : arithmetic_function ℕ :=\n  zero_hom.mk (fun (n : ℕ) => list.length (factors n)) sorry\n\ntheorem card_factors_apply {n : ℕ} : coe_fn card_factors n = list.length (factors n) := rfl\n\n@[simp] theorem card_factors_one : coe_fn card_factors 1 = 0 := rfl\n\ntheorem card_factors_eq_one_iff_prime {n : ℕ} : coe_fn card_factors n = 1 ↔ prime n := sorry\n\ntheorem card_factors_mul {m : ℕ} {n : ℕ} (m0 : m ≠ 0) (n0 : n ≠ 0) :\n    coe_fn card_factors (m * n) = coe_fn card_factors m + coe_fn card_factors n :=\n  sorry\n\ntheorem card_factors_multiset_prod {s : multiset ℕ} (h0 : multiset.prod s ≠ 0) :\n    coe_fn card_factors (multiset.prod s) = multiset.sum (multiset.map (⇑card_factors) s) :=\n  sorry\n\n/-- `ω n` is the number of distinct prime factors of `n`. -/\ndef card_distinct_factors : arithmetic_function ℕ :=\n  zero_hom.mk (fun (n : ℕ) => list.length (list.erase_dup (factors n))) sorry\n\n@[simp] theorem card_distinct_factors_zero : coe_fn card_distinct_factors 0 = 0 := rfl\n\ntheorem card_distinct_factors_apply {n : ℕ} :\n    coe_fn card_distinct_factors n = list.length (list.erase_dup (factors n)) :=\n  rfl\n\ntheorem card_distinct_factors_eq_card_factors_iff_squarefree {n : ℕ} (h0 : n ≠ 0) :\n    coe_fn card_distinct_factors n = coe_fn card_factors n ↔ squarefree n :=\n  sorry\n\n/-- `μ` is the Möbius function. If `n` is squarefree with an even number of distinct prime factors,\n  `μ n = 1`. If `n` is squarefree with an odd number of distinct prime factors, `μ n = -1`.\n  If `n` is not squarefree, `μ n = 0`. -/\ndef moebius : arithmetic_function ℤ :=\n  zero_hom.mk (fun (n : ℕ) => ite (squarefree n) ((-1) ^ coe_fn card_factors n) 0) sorry\n\n@[simp] theorem moebius_apply_of_squarefree {n : ℕ} (h : squarefree n) :\n    coe_fn moebius n = (-1) ^ coe_fn card_factors n :=\n  if_pos h\n\n@[simp] theorem moebius_eq_zero_of_not_squarefree {n : ℕ} (h : ¬squarefree n) :\n    coe_fn moebius n = 0 :=\n  if_neg h\n\ntheorem moebius_ne_zero_iff_squarefree {n : ℕ} : coe_fn moebius n ≠ 0 ↔ squarefree n := sorry\n\ntheorem moebius_ne_zero_iff_eq_or {n : ℕ} :\n    coe_fn moebius n ≠ 0 ↔ coe_fn moebius n = 1 ∨ coe_fn moebius n = -1 :=\n  sorry\n\n@[simp] theorem coe_moebius_mul_coe_zeta {R : Type u_1} [comm_ring R] : ↑moebius * ↑zeta = 1 :=\n  sorry\n\n@[simp] theorem coe_zeta_mul_coe_moebius {R : Type u_1} [comm_ring R] : ↑zeta * ↑moebius = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑zeta * ↑moebius = 1)) (mul_comm ↑zeta ↑moebius)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑moebius * ↑zeta = 1)) (coe_moebius_mul_coe_zeta R _inst_1)))\n      (Eq.refl 1))\n\n@[simp] theorem moebius_mul_coe_zeta : moebius * ↑zeta = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (moebius * ↑zeta = 1)) (Eq.symm (int_coe_int moebius))))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (↑moebius * ↑zeta = 1)) (coe_moebius_mul_coe_zeta ℤ int.comm_ring)))\n      (Eq.refl 1))\n\n@[simp] theorem coe_zeta_mul_moebius : ↑zeta * moebius = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑zeta * moebius = 1)) (Eq.symm (int_coe_int moebius))))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (↑zeta * ↑moebius = 1)) (coe_zeta_mul_coe_moebius ℤ int.comm_ring)))\n      (Eq.refl 1))\n\nprotected instance zeta.invertible {R : Type u_1} [comm_ring R] : invertible ↑zeta :=\n  invertible.mk (↑moebius) (coe_moebius_mul_coe_zeta R _inst_1) (coe_zeta_mul_coe_moebius R _inst_1)\n\n/-- A unit in `arithmetic_function R` that evaluates to `ζ`, with inverse `μ`. -/\ndef zeta_unit {R : Type u_1} [comm_ring R] : units (arithmetic_function R) :=\n  units.mk (↑zeta) (↑moebius) (coe_zeta_mul_coe_moebius R _inst_1)\n    (coe_moebius_mul_coe_zeta R _inst_1)\n\n@[simp] theorem coe_zeta_unit {R : Type u_1} [comm_ring R] : ↑(zeta_unit R _inst_1) = ↑zeta := rfl\n\n@[simp] theorem inv_zeta_unit {R : Type u_1} [comm_ring R] : ↑(zeta_unit R _inst_1⁻¹) = ↑moebius :=\n  rfl\n\n/-- Möbius inversion for functions to an `add_comm_group`. -/\ntheorem sum_eq_iff_sum_smul_moebius_eq {R : Type u_1} [add_comm_group R] {f : ℕ → R} {g : ℕ → R} :\n    (∀ (n : ℕ), 0 < n → (finset.sum (divisors n) fun (i : ℕ) => f i) = g n) ↔\n        ∀ (n : ℕ),\n          0 < n →\n            (finset.sum (divisors_antidiagonal n)\n                fun (x : ℕ × ℕ) => coe_fn moebius (prod.fst x) • g (prod.snd x)) =\n              f n :=\n  sorry\n\n/-- Möbius inversion for functions to a `comm_ring`. -/\ntheorem sum_eq_iff_sum_mul_moebius_eq {R : Type u_1} [comm_ring R] {f : ℕ → R} {g : ℕ → R} :\n    (∀ (n : ℕ), 0 < n → (finset.sum (divisors n) fun (i : ℕ) => f i) = g n) ↔\n        ∀ (n : ℕ),\n          0 < n →\n            (finset.sum (divisors_antidiagonal n)\n                fun (x : ℕ × ℕ) => ↑(coe_fn moebius (prod.fst x)) * g (prod.snd x)) =\n              f n :=\n  sorry\n\n/-- Möbius inversion for functions to a `comm_group`. -/\ntheorem prod_eq_iff_prod_pow_moebius_eq {R : Type u_1} [comm_group R] {f : ℕ → R} {g : ℕ → R} :\n    (∀ (n : ℕ), 0 < n → (finset.prod (divisors n) fun (i : ℕ) => f i) = g n) ↔\n        ∀ (n : ℕ),\n          0 < n →\n            (finset.prod (divisors_antidiagonal n)\n                fun (x : ℕ × ℕ) => g (prod.snd x) ^ coe_fn moebius (prod.fst x)) =\n              f n :=\n  sum_eq_iff_sum_smul_moebius_eq (additive R) additive.add_comm_group (fun (i : ℕ) => f i)\n    fun (n : ℕ) => g n\n\n/-- Möbius inversion for functions to a `comm_group_with_zero`. -/\ntheorem prod_eq_iff_prod_pow_moebius_eq_of_nonzero {R : Type u_1} [comm_group_with_zero R]\n    {f : ℕ → R} {g : ℕ → R} (hf : ∀ (n : ℕ), 0 < n → f n ≠ 0) (hg : ∀ (n : ℕ), 0 < n → g n ≠ 0) :\n    (∀ (n : ℕ), 0 < n → (finset.prod (divisors n) fun (i : ℕ) => f i) = g n) ↔\n        ∀ (n : ℕ),\n          0 < n →\n            (finset.prod (divisors_antidiagonal n)\n                fun (x : ℕ × ℕ) => g (prod.snd x) ^ coe_fn moebius (prod.fst x)) =\n              f n :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/number_theory/arithmetic_function_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34455970166686445}}
{"text": "lemma contra (P Q : Prop) : (P ∧ ¬ P) → Q :=\nbegin\n-- rw not_iff_imp_false,\nintro h,\nexfalso,\ncases h,\nexact h_right(h_left),\nend\n", "meta": {"author": "chanha-park", "repo": "naturalNumberGame", "sha": "4e0d7100ce4575e1add92feefa38b1250431b879", "save_path": "github-repos/lean/chanha-park-naturalNumberGame", "path": "github-repos/lean/chanha-park-naturalNumberGame/naturalNumberGame-4e0d7100ce4575e1add92feefa38b1250431b879/Advanced_Proposition/9.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3445597016668644}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lucas Allen, Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.converter.apply_congr\n! leanprover-community/mathlib commit 3d7987cda72abc473c7cdbbb075170e9ac620042\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Interactive\nimport Mathbin.Tactic.Converter.Interactive\n\n/-!\n## Introduce the `apply_congr` conv mode tactic.\n\n`apply_congr` will apply congruence lemmas inside `conv` mode.\nIt is particularly useful when the automatically generated congruence lemmas\nare not of the optimal shape. An example, described in the doc-string is\nrewriting inside the operand of a `finset.sum`.\n-/\n\n\nopen Tactic\n\nnamespace Conv.Interactive\n\nopen Interactive Interactive.Types Lean.Parser\n\n-- mathport name: parser.optional\nlocal postfix:1024 \"?\" => optional\n\n/-- Apply a congruence lemma inside `conv` mode.\n\nWhen called without an argument `apply_congr` will try applying all lemmas marked with `@[congr]`.\nOtherwise `apply_congr e` will apply the lemma `e`.\n\nRecall that a goal that appears as `∣ X` in `conv` mode\nrepresents a goal of `⊢ X = ?m`,\ni.e. an equation with a metavariable for the right hand side.\n\nTo successfully use `apply_congr e`, `e` will need to be an equation\n(possibly after function arguments),\nwhich can be unified with a goal of the form `X = ?m`.\nThe right hand side of `e` will then determine the metavariable,\nand `conv` will subsequently replace `X` with that right hand side.\n\nAs usual, `apply_congr` can create new goals;\nany of these which are _not_ equations with a metavariable on the right hand side\nwill be hard to deal with in `conv` mode.\nThus `apply_congr` automatically calls `intros` on any new goals,\nand fails if they are not then equations.\n\nIn particular it is useful for rewriting inside the operand of a `finset.sum`,\nas it provides an extra hypothesis asserting we are inside the domain.\n\nFor example:\n\n```lean\nexample (f g : ℤ → ℤ) (S : finset ℤ) (h : ∀ m ∈ S, f m = g m) :\n  finset.sum S f = finset.sum S g :=\nbegin\n  conv_lhs\n  { -- If we just call `congr` here, in the second goal we're helpless,\n    -- because we are only given the opportunity to rewrite `f`.\n    -- However `apply_congr` uses the appropriate `@[congr]` lemma,\n    -- so we get to rewrite `f x`, in the presence of the crucial `H : x ∈ S` hypothesis.\n    apply_congr,\n    skip,\n    simp [h, H], }\nend\n```\n\nIn the above example, when the `apply_congr` tactic is called it gives the hypothesis `H : x ∈ S`\nwhich is then used to rewrite the `f x` to `g x`.\n-/\nunsafe def apply_congr (q : parse texpr ?) : conv Unit := do\n  let congr_lemmas ←\n    match q with\n      |-- If the user specified a lemma, use that one,\n          some\n          e =>\n        do\n        let gs ← get_goals\n        let e ← to_expr e\n        -- to_expr messes with the goals? (see tests)\n            set_goals\n            gs\n        return [e]\n      |-- otherwise, look up everything tagged `@[congr]`\n        none =>\n        do\n        let congr_lemma_names ← attribute.get_instances `congr\n        congr_lemma_names mk_const\n  -- For every lemma:\n      congr_lemmas\n      fun n =>\n      -- Call tactic.eapply\n        seq'\n        (tactic.eapply n >> tactic.skip)\n        (-- and then call `intros` on each resulting goal, and require that afterwards it's an equation.\n          tactic.intros >>\n          do\n          let q(_ = _) ← target\n          tactic.skip)\n#align conv.interactive.apply_congr conv.interactive.apply_congr\n\nadd_tactic_doc\n  { Name := \"apply_congr\"\n    category := DocCategory.tactic\n    declNames := [`conv.interactive.apply_congr]\n    tags := [\"conv\", \"congruence\", \"rewriting\"] }\n\nend Conv.Interactive\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Converter/ApplyCongr.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3445597015890694}}
{"text": "/-\nCopyright (c) 2021 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\nimport Lean.Elab.Tactic\n\nnamespace Lean.Aesop.DefaultRules\n\n-- We define the `splitHyp` tactic, which splits hypotheses that are products\n-- or existentials. It recurses into nested products, so `A ∧ B ∧ C` is split\n-- into three hypotheses with types `A`, `B` and `C`. It also works under\n-- binders, so `h : ∀ x, P x ∧ Q x` is split into `h₁ : ∀ x, P x` and\n-- `h₂ : ∀ x, Q x`.\n--\n-- Exception: `Exists`, unlike `Sigma` and `PSigma`, cannot be split under\n-- binders. This is unavoidable: `∀ (x : X), ∃ (y : Y), P y` does not imply\n-- `∃ (f : X → Y), ∀ x, P (f x)` in Lean's logic.\n\n-- TODO The splitHyps tactic currently does not handle existentials with\n-- products nested in the witness. E.g. the hypothesis\n--\n--   h : ∃ (p : X × Y), P p\n--\n-- is decomposed into\n--\n--   h₁ : X × Y\n--   h₂ : P h₁\n--\n-- when it should really be\n--\n--   h₁ : X\n--   h₂ : Y\n--   h₃ : P (h₁, h₂)\n--\n-- To fix this, we should canonicalise such hypotheses before splitting them.\n-- The canonicalisation would make sure the witness is not a product by\n-- rewriting, e.g.,\n--\n--   ∃ (p : X × Y), P p           ↝ ∃ (x : X) (y : Y), P (x, y)\n--   ∃ (p : ∃ (x : X), P x), Q p  ↝ ∃ (x : X) (y : P x), Q (Exists.intro x y)\n--\n-- @[simp]\n-- theorem exists_prod {X Y} (P : X × Y → Prop) :\n--     (∃ (p : X × Y), P p) ↔ ∃ (x : X) (y : Y), P (x, y) :=\n--   Iff.intro\n--     (λ ⟨⟨x, y⟩, Pp⟩ => ⟨x, y, Pp⟩)\n--     (λ ⟨x, y, Pxy⟩ => ⟨⟨x, y⟩, Pxy⟩)\n--\n-- We could also use this implementation strategy for splitting under binders:\n-- rewrite `∀ x, P x ∧ Q x` into `(∀ x, P x) ∧ (∀ x, Q x)`. This would reduce\n-- the implementation complexity a lot, though at some performance cost.\n\nopen Expr\nopen Lean.Meta\n\npartial def splitHypCore (goal : MVarId) (originalUserName : Name)\n    (binderFVars : Array Expr) (hyp : FVarId) (type : Expr) :\n    MetaM (Array FVarId × MVarId) :=\n  match type with\n  | forallE .. => do\n    forallTelescope type λ binders conclusion =>\n      splitHypCore goal originalUserName (binderFVars ++ binders) hyp conclusion\n  | app (app (const ``And lvls _) leftType _) rightType _ =>\n    splitProduct goal (mkConst ``And.left lvls) (mkConst ``And.right lvls)\n      leftType rightType\n  | app (app (const ``Prod lvls _) leftType _) rightType _ =>\n    splitProduct goal (mkConst ``Prod.fst lvls) (mkConst ``Prod.snd lvls)\n      leftType rightType\n  | app (app (const ``PProd lvls _) leftType _) rightType _ =>\n    splitProduct goal (mkConst ``PProd.fst lvls) (mkConst ``PProd.snd lvls)\n      leftType rightType\n  | app (app (const ``MProd lvls _) leftType _) rightType _ =>\n    splitProduct goal (mkConst ``MProd.fst lvls) (mkConst ``MProd.snd lvls)\n      leftType rightType\n  | app (app (const ``Sigma lvls _) witnessType _) propertyType _ => do\n    splitExistential goal (mkConst ``Sigma.fst lvls) (mkConst ``Sigma.snd lvls)\n      witnessType propertyType\n  | app (app (const ``PSigma lvls _) witnessType _) propertyType _ => do\n    splitExistential goal (mkConst ``PSigma.fst lvls) (mkConst ``PSigma.snd lvls)\n      witnessType propertyType\n  | app (app (const ``Exists lvls _) witnessType _) propertyType _ => do\n    -- We can't eliminate Exists under binders.\n    if ¬ binderFVars.isEmpty then\n      return (#[], goal)\n    let #[casesGoal] ← cases goal hyp\n      | unreachable!\n    pure (casesGoal.fields.map (·.fvarId!), casesGoal.mvarId)\n  | _ => pure (#[], goal)\n  where\n    hypWithBinderFVars : Expr :=\n      mkAppN (mkFVar hyp) binderFVars\n\n    splitProductHalf (goal : MVarId) (left : Bool)\n        (eliminator leftType rightType : Expr) : MetaM (FVarId × MVarId) := do\n      let type := if left then leftType else rightType\n      let newHypType ← mkForallFVars binderFVars type\n      let newHypProof ← mkLambdaFVars binderFVars $\n        mkAppN eliminator #[leftType, rightType, hypWithBinderFVars]\n      let goal ← assert goal originalUserName newHypType newHypProof\n      let (newHyp, goal) ← intro1 goal\n      return (newHyp, goal)\n\n    splitProduct (goal : MVarId)\n        (leftEliminator rightEliminator leftType rightType : Expr) :\n        MetaM (Array FVarId × MVarId) := do\n      let (leftHyp, goal) ←\n        splitProductHalf goal true  leftEliminator  leftType rightType\n      let (rightHyp, goal) ←\n        splitProductHalf goal false rightEliminator leftType rightType\n      let (goal, cleared) :=\n        match (← observing? $ clear goal hyp) with\n        | none => (goal, false)\n        | some goal => (goal, true)\n      let (leftHyps, goal)  ←\n        splitHypCore goal originalUserName binderFVars leftHyp  leftType\n      let (rightHyps, goal) ←\n        splitHypCore goal originalUserName binderFVars rightHyp rightType\n      let newHyps := (if cleared then #[] else #[hyp]) ++ leftHyps ++ rightHyps\n      return (newHyps, goal)\n\n    splitExistential (oldGoal : MVarId)\n        (witnessProjection propertyProjection witnessType propertyType : Expr) :\n        MetaM (Array FVarId × MVarId) := do\n\n      checkNotAssigned goal `splitHyp\n      let tag ← getMVarTag goal\n      let oldTarget ← getMVarType goal\n\n      -- I don't see how to make `assert` work with successive dependent\n      -- hypotheses, so here's a reimplementation. We assert one hypothesis for\n      -- the witness and one for the property, where the property hypothesis\n      -- depends on the witness hypothesis.\n      let witnessHypType ← mkForallFVars binderFVars witnessType\n      let newTarget ← do\n        withLocalDeclD originalUserName witnessHypType λ witness => do\n          let propertyHypType ←\n            mkForallFVars binderFVars $\n            (← headBeta (mkApp propertyType (mkAppN witness binderFVars)))\n          withLocalDeclD originalUserName propertyHypType λ property =>\n            mkForallFVars (#[witness, property]) oldTarget\n      let goal ← withMVarContext goal $\n        mkFreshExprSyntheticOpaqueMVar newTarget tag\n\n      let witnessHypProof ← mkLambdaFVars binderFVars $\n        mkAppN witnessProjection #[witnessType, propertyType, hypWithBinderFVars]\n      let propertyHypProof ← mkLambdaFVars binderFVars $\n        mkAppN propertyProjection #[witnessType, propertyType, hypWithBinderFVars]\n      let proof := mkAppN goal #[witnessHypProof, propertyHypProof]\n      assignExprMVar oldGoal proof\n\n      let goal := goal.mvarId!\n      let (#[witnessHyp, propertyHyp], goal) ← introN goal 2 | unreachable!\n\n      let (goal, cleared) :=\n        match (← observing? $ clear goal hyp) with\n        | none => (goal, false)\n        | some goal => (goal, true)\n      let propertyHypTypeWithoutBinders ←\n        headBeta (mkApp propertyType $ mkAppN (mkFVar witnessHyp) binderFVars)\n      let (newHyps, goal) ←\n        splitHypCore goal originalUserName binderFVars propertyHyp\n          propertyHypTypeWithoutBinders\n      let newHyps := (if cleared then #[] else #[hyp]) ++ newHyps\n      return (newHyps, goal)\n\ndef splitHyp (goal : MVarId) (hyp : FVarId) : MetaM (Array FVarId × MVarId) :=\n  withMVarContext goal do\n    let decl ← getLocalDecl hyp\n    splitHypCore goal decl.userName #[] hyp decl.type\n\ndef splitAllHyps (goal : MVarId) : MetaM (Array FVarId × MVarId) := do\n  let lctx := (← getMVarDecl goal).lctx\n  let mut goal := goal\n  let mut newHyps := #[]\n  for hyp in lctx do\n    if hyp.isAuxDecl then continue\n    let (newHyps', goal') ← splitHyp goal hyp.fvarId\n    newHyps := newHyps ++ newHyps'\n    goal := goal'\n  return (newHyps, goal)\n\nend Lean.Aesop.DefaultRules\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/stage0/src/Lean/Aesop/DefaultRules/SplitHyps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.34451892627495767}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.idempotents.karoubi\nimport for_mathlib.functor_misc\nimport category_theory.natural_isomorphism\n\n/-!\n# Extension of functors to the idempotent completion\n\nIn this file, we obtain an equivalence of categories\n`karoubi_universal₁ C D : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)` for\nall categories `C` and `D`. The key construction is `functor_extension₁`\nwhich extends a functor `C ⥤ karoubi D` to a functor `karoubi C ⥤ karoubi D`.\n\nTODO : If `D` is idempotent complete, we also have\n`karoubi_universal C D : C ⥤ D ≌ karoubi C ⥤ D`.\n\n-/\n\nopen category_theory.category\nopen category_theory.idempotents.karoubi\n\nnamespace category_theory\n\nnamespace idempotents\n\nvariables {C D E : Type*} [category C] [category D] [category E]\n\nlemma nat_trans_eq {F G : karoubi C ⥤ D} (φ : F ⟶ G) (P : karoubi C) :\n  φ.app P = F.map (decomp_id_i P) ≫ φ.app P.X ≫ G.map (decomp_id_p P) :=\nbegin\n  rw [← φ.naturality, ← assoc, ← F.map_comp],\n  conv { to_lhs, rw [← id_comp (φ.app P), ← F.map_id], },\n  congr,\n  apply decomp_id,\nend\n\nlemma nat_trans_ext {F G : karoubi C ⥤ D} (φ₁ φ₂ : F ⟶ G)\n  (h : (𝟙 (to_karoubi C)) ◫ φ₁ = (𝟙 (to_karoubi C)) ◫ φ₂) : φ₁ = φ₂ :=\nbegin\n  ext P,\n  rw [nat_trans_eq φ₁, nat_trans_eq φ₂],\n  congr' 2,\n  have eq := congr_app h P.X,\n  simpa only [nat_trans.hcomp_app, nat_trans.id_app, G.map_id, comp_id] using congr_app h P.X,\nend\n\nnamespace functor_extension₁\n\n@[simps]\ndef obj (F : C ⥤ karoubi D) : karoubi C ⥤ karoubi D :=\n{ obj := λ P, ⟨(F.obj P.X).X, (F.map P.p).f,\n    by simpa only [F.map_comp, hom_ext] using F.congr_map P.idem⟩,\n  map := λ P Q f, ⟨(F.map f.f).f,\n    by simpa only [F.map_comp, hom_ext] using F.congr_map f.comm⟩, }\n\n@[simps]\ndef map {F G : C ⥤ karoubi D} (φ : F ⟶ G) : obj F ⟶ obj G :=\n{ app := λ P,\n  { f := (F.map P.p).f ≫ (φ.app P.X).f,\n    comm := begin\n      have h := φ.naturality P.p,\n      have h' := F.congr_map P.idem,\n      simp only [hom_ext, karoubi.comp, F.map_comp] at h h',\n      simp only [obj_obj_p, assoc, ← h],\n      slice_rhs 1 3 { rw [h', h'], },\n    end, },\n  naturality' := λ P Q f, begin\n    ext,\n    dsimp [obj],\n    have h := φ.naturality f.f,\n    have h' := F.congr_map (comp_p f),\n    have h'' := F.congr_map (p_comp f),\n    simp only [hom_ext, functor.map_comp, comp] at ⊢ h h' h'',\n    slice_rhs 2 3 { rw ← h, },\n    slice_lhs 1 2 { rw h', },\n    slice_rhs 1 2 { rw h'', },\n  end }\n\nend functor_extension₁\n\nvariables (C D E)\n\n@[simps]\ndef functor_extension₁ : (C ⥤ karoubi D) ⥤ (karoubi C ⥤ karoubi D) :=\n{ obj := functor_extension₁.obj,\n  map := λ F G, functor_extension₁.map,\n  map_id' := λ F, by { ext P, exact comp_p (F.map P.p), },\n  map_comp' := λ F G H φ φ', begin\n    ext P,\n    simp only [comp, functor_extension₁.map_app_f, nat_trans.comp_app, assoc],\n    have h := φ.naturality P.p,\n    have h' := F.congr_map P.idem,\n    simp only [hom_ext, comp, F.map_comp] at h h',\n    slice_rhs 2 3 { rw ← h, },\n    slice_rhs 1 2 { rw h', },\n    simp only [assoc],\n  end, }\n\nlemma functor_extension₁_comp_whiskering_left_to_karoubi :\n  functor_extension₁ C D ⋙\n    (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) = 𝟭 _ :=\nbegin\n  refine functor.ext _ _,\n  { intro F,\n    refine functor.ext _ _,\n    { intro X,\n      ext,\n      { dsimp,\n        rw [id_comp, comp_id, F.map_id, id_eq], },\n      { refl, }, },\n    { intros X Y f,\n      ext,\n      dsimp,\n      simp only [comp_id, eq_to_hom_f, eq_to_hom_refl, comp_p, functor_extension₁.obj_obj_p,\n        to_karoubi_obj_p, comp],\n      dsimp,\n      simp only [functor.map_id, id_eq, p_comp], }, },\n  { intros F G φ,\n    ext X,\n    dsimp,\n    simp only [eq_to_hom_app, F.map_id, karoubi.comp, eq_to_hom_f, id_eq, p_comp,\n      eq_to_hom_refl, comp_id, comp_p, functor_extension₁.obj_obj_p,\n      to_karoubi_obj_p, F.map_id X], },\nend\n\nend idempotents\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/idempotents/functor_extension.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3445189262749576}}
{"text": "import ..mcrl2_basic.mcrl2_basic\n\nopen mcrl2\n\nvariable {α : Type}\nvariable [comm_semigroup_with_zero α]\n\n/- Finally, the || operator.-/\ninductive R_par {x₁ x₂ y₁ y₂ : mcrl2 α} (R₁ R₂ : mcrl2 α → mcrl2 α → Prop) :\nmcrl2 α → mcrl2 α → Prop\n| R₁ {x y} (h : R₁ x y) : R_par x y\n| R₂ {x y} (h : R₂ x y) : R_par x y\n| basel : R_par (x₁ || y₁) (x₂ || y₂)\n| baser : R_par (x₂ || y₂) (x₁ || y₁)\n| stepl {x x' y y' a z z'} (hR₁x : R₁ x x') (hR₂ : R₂ y y') (hR₁z : R₁ z z') \n  (ht  : transition x  a z)\n  (ht' : transition x' a z') :\n  R_par (z || y) (z' || y')\n| stepr {x x' y y' a z z'} (hR₁x : R₁ x x') (hR₂ : R₂ y y') (hR₁z : R₂ z z') \n  (ht  : transition y  a z)\n  (ht' : transition y' a z') :\n  R_par (x || z) (x' || z')\n| stepcomm {x x' y y' a b z₁ z₂ z₁' z₂'} (hR₁x : R₁ x x') (hR₂y : R₂ y y') \n  (hR₁z : R₁ z₁ z₁' ) (hR₂z : R₂ z₂ z₂' )\n  (hx  : transition x  a z₁)  (hy  : transition y  b z₂)\n  (hx' : transition x' a z₁') (hy' : transition y' b z₂') :\n  R_par (z₁ || z₂) (z₁' || z₂')\n\nlemma bisim_exists_lift_par_l {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition x a z) (hR₁ : R₁ x x') (hR₂ : R₂ y y') :\n(∃z', transition x' a z' ∧ option.rel R₁ z z') → (∃z', transition x' a z' ∧ option.rel (@R_par _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' z y) (par' z' y')) :=\nbegin \n  intro h,\n  rcases h with ⟨w, hwa, hRw⟩,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_par.stepl,\n    exact hR₁,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_par.R₂,\n    assumption}\nend\n\nlemma bisim_exists_lift_par_r {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition y a z) (hR₁ : R₁ x x') (hR₂ : R₂ y y') :\n(∃z', transition y' a z' ∧ option.rel R₂ z z') → (∃z', transition  y' a z' ∧ option.rel (@R_par _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' x z) (par' x' z')) :=\nbegin \n  intro h,\n  rcases h with ⟨w, hwa, hRw⟩,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_par.stepr,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_par.R₁,\n    assumption}\nend\n\nlemma bisim_exists_lift_par_comm {x₁ x₂ y₁ y₂ x x' y y' a b z₁ z₂} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition x a z₁) (hyz' : transition y b z₂) (hR₁ : R₁ x x') (hR₂ : R₂ y y') (hab : a * b ≠ 0):\n(∃z' , transition x' a z' ∧ option.rel R₁ z₁ z') → (∃z' , transition y' b z' ∧ option.rel R₂ z₂ z') → \n(∃z', transition (x' || y') (a * b) z' ∧ option.rel (@R_par _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' z₁ z₂) z') :=\nbegin \n  intros h₁ h₂,\n  rcases h₁ with ⟨w₁, hw₁a, hRw₁⟩,\n  rcases h₂ with ⟨w₂, hw₂b, hRw₂⟩,\n  simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n  apply or.inr,\n  apply or.inr,\n  apply exists.intro (par' w₁ w₂),\n  apply exists.intro w₁,\n  apply exists.intro w₂,\n  apply and.intro,\n  refl,\n  apply exists.intro a,\n  apply exists.intro b,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  refl,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  assumption,\n  cases hRw₂,\n  { cases hRw₁,\n    { apply option.rel.some,\n      apply R_par.stepcomm,\n      exact hR₁,\n      exact hR₂,\n      assumption,\n      assumption,\n      repeat {assumption}},\n    { apply option.rel.some,\n      cases hRw₂,\n      apply R_par.R₂,\n      assumption}},\n  { cases hRw₁,\n    { apply option.rel.some,\n      apply R_par.R₁,\n      assumption},\n    { apply option.rel.none}}\nend\n\nlemma bisim_exists_lift_par {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂ : mcrl2 α → mcrl2 α → Prop} \n (R₁_bisim : is_bisimulation R₁) (R₂_bisim : is_bisimulation R₂) (hR₁ : R₁ x x') (hR₂ : R₂ y y') (haz : transition (x || y) a z):\n (∃z', transition (x' || y') a z' ∧ option.rel (@R_par _ _ x₁ x₂ y₁ y₂ R₁ R₂) z z') :=\nbegin \n  cases R₁_bisim with R₁_bisim R₁_symm,\n  cases R₂_bisim with R₂_bisim R₂_symm,\n  cases haz,\n  { specialize R₁_bisim x x' haz_x' a hR₁ haz_h,\n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_par_l haz_h hR₁ hR₂ R₁_bisim},\n  { specialize R₂_bisim y y' haz_y' a hR₂ haz_h, \n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inr,\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_par_r haz_h hR₁ hR₂ R₂_bisim},\n  { specialize R₁_bisim x x' haz_x' haz_a hR₁ haz_h₁, \n    specialize R₂_bisim y y' haz_y' haz_b hR₂ haz_h₂,\n    exact bisim_exists_lift_par_comm haz_h₁ haz_h₂ hR₁ hR₂ haz_h₃ R₁_bisim R₂_bisim}\n\nend\n\nlemma R_par.symm {x₁ x₂ y₁ y₂ : mcrl2 α} {R₁ R₂ : mcrl2 α → mcrl2 α → Prop} \n (h₁ : symmetric R₁) (h₂ : symmetric R₂) :\nsymmetric (@R_par α _ x₁ x₂ y₁ y₂ R₁ R₂) :=\nbegin\n  intros x y h,\n  cases h,\n  { apply R_par.R₁,\n    apply h₁,\n    assumption},\n  { apply R_par.R₂,\n    apply h₂,\n    assumption},\n  { exact R_par.baser},\n  { exact R_par.basel},\n  { apply R_par.stepl,\n    exact h₁ h_hR₁x,\n    apply h₂,\n    assumption,\n    apply h₁,\n    assumption,\n    repeat {assumption}},\n  { apply R_par.stepr,\n    apply h₁,\n    exact h_hR₁x,\n    exact h₂ h_hR₂,\n    apply h₂,\n    repeat {assumption}},\n  { apply R_par.stepcomm,\n    exact h₁ h_hR₁x,\n    exact h₂ h_hR₂y,\n    exact h₁ h_hR₁z,\n    exact h₂ h_hR₂z,\n    repeat {assumption}}\nend\n\ntheorem bisim.par {x₁ x₂ y₁ y₂: mcrl2 α} (h₁ : x₁ ≈ x₂) (h₂ : y₁ ≈ y₂) : \nx₁ || y₁ ≈ x₂ || y₂ :=\nbegin\n  rcases h₁ with ⟨R₁, R₁x, R₁_bisim⟩,\n  rcases h₂ with ⟨R₂, R₂y, R₂_bisim⟩,\n  apply exists.intro (R_par R₁ R₂),\n  apply and.intro,\n  apply R_par.basel,\n  apply and.intro,\n  { intros x y x' a Rxy xax',\n    cases Rxy,\n    { have h: ∃y', transition y a y' ∧ option.rel R₁ x' y',\n      by exact bisim_lift R₁_bisim Rxy_h xax',\n      cases h with w h_w,\n      cases h_w with l r,\n      apply exists.intro w,\n      apply and.intro,\n      assumption,\n      apply option.rel.mono,\n      { intros x y, exact R_par.R₁},\n      { assumption}},\n    { have h: ∃y', transition y a y' ∧ option.rel R₂ x' y',\n      by exact bisim_lift R₂_bisim Rxy_h xax',\n      cases h with w h_w,\n      cases h_w with l r,\n      apply exists.intro w,\n      apply and.intro,\n      assumption,\n      apply option.rel.mono,\n      { intros x y, exact R_par.R₂},\n      { assumption}},\n    { exact bisim_exists_lift_par R₁_bisim R₂_bisim R₁x R₂y xax' },\n    { exact bisim_exists_lift_par R₁_bisim R₂_bisim (R₁_bisim.right R₁x) (R₂_bisim.right R₂y) xax'},\n    { apply bisim_exists_lift_par; assumption},\n    { apply bisim_exists_lift_par; assumption},\n    { apply bisim_exists_lift_par; assumption}},\n  { exact R_par.symm R₁_bisim.right R₂_bisim.right}\nend", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/mcrl2_mrg/par.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.34431509390668297}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.preserves.shapes.pullbacks\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.limits.constructions.binary_products\n\n/-!\n# Pullback and pushout squares\n\nWe provide another API for pullbacks and pushouts.\n\n`is_pullback fst snd f g` is the proposition that\n```\n  P --fst--> X\n  |          |\n snd         f\n  |          |\n  v          v\n  Y ---g---> Z\n\n```\nis a pullback square.\n\n(And similarly for `is_pushout`.)\n\nWe provide the glue to go back and forth to the usual `is_limit` API for pullbacks, and prove\n`is_pullback (pullback.fst : pullback f g ⟶ X) (pullback.snd : pullback f g ⟶ Y) f g`\nfor the usual `pullback f g` provided by the `has_limit` API.\n\nWe don't attempt to restate everything we know about pullbacks in this language,\nbut do restate the pasting lemmas.\n\n## Future work\nBicartesian squares, and\nshow that the pullback and pushout squares for a biproduct are bicartesian.\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v₁ v₂ u₁ u₂\n\nnamespace category_theory\n\nvariables {C : Type u₁} [category.{v₁} C]\n\n/-- The proposition that a square\n```\n  W ---f---> X\n  |          |\n  g          h\n  |          |\n  v          v\n  Y ---i---> Z\n\n```\nis a commuting square.\n-/\nstructure comm_sq {W X Y Z : C} (f : W ⟶ X) (g : W ⟶ Y) (h : X ⟶ Z) (i : Y ⟶ Z) : Prop :=\n(w : f ≫ h = g ≫ i)\n\nattribute [reassoc] comm_sq.w\n\nnamespace comm_sq\n\nvariables {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\nlemma flip (p : comm_sq f g h i) : comm_sq g f i h := ⟨p.w.symm⟩\n\nlemma of_arrow {f g : arrow C} (h : f ⟶ g) : comm_sq f.hom h.left h.right g.hom := ⟨h.w.symm⟩\n\n/--\nThe (not necessarily limiting) `pullback_cone h i` implicit in the statement\nthat we have `comm_sq f g h i`.\n-/\ndef cone (s : comm_sq f g h i) : pullback_cone h i := pullback_cone.mk _ _ s.w\n\n/--\nThe (not necessarily limiting) `pushout_cocone f g` implicit in the statement\nthat we have `comm_sq f g h i`.\n-/\ndef cocone (s : comm_sq f g h i) : pushout_cocone f g := pushout_cocone.mk _ _ s.w\n\nend comm_sq\n\n/-- The proposition that a square\n```\n  P --fst--> X\n  |          |\n snd         f\n  |          |\n  v          v\n  Y ---g---> Z\n\n```\nis a pullback square.\n-/\nstructure is_pullback\n  {P X Y Z : C} (fst : P ⟶ X) (snd : P ⟶ Y) (f : X ⟶ Z) (g : Y ⟶ Z)\n  extends comm_sq fst snd f g : Prop :=\n(is_limit' : nonempty (is_limit (pullback_cone.mk _ _ w)))\n\n/-- The proposition that a square\n```\n  Z ---f---> X\n  |          |\n  g         inl\n  |          |\n  v          v\n  Y --inr--> P\n\n```\nis a pushout square.\n-/\nstructure is_pushout\n  {Z X Y P : C} (f : Z ⟶ X) (g : Z ⟶ Y) (inl : X ⟶ P) (inr : Y ⟶ P)\n  extends comm_sq f g inl inr : Prop :=\n(is_colimit' : nonempty (is_colimit (pushout_cocone.mk _ _ w)))\n\n/-!\nWe begin by providing some glue between `is_pullback` and the `is_limit` and `has_limit` APIs.\n(And similarly for `is_pushout`.)\n-/\n\nnamespace is_pullback\n\nvariables {P X Y Z : C} {fst : P ⟶ X} {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z}\n\n/--\nThe (limiting) `pullback_cone f g` implicit in the statement\nthat we have a `is_pullback fst snd f g`.\n-/\ndef cone (h : is_pullback fst snd f g) : pullback_cone f g := h.to_comm_sq.cone\n\n/--\nThe cone obtained from `is_pullback fst snd f g` is a limit cone.\n-/\nnoncomputable def is_limit (h : is_pullback fst snd f g) : is_limit h.cone :=\nh.is_limit'.some\n\n/-- If `c` is a limiting pullback cone, then we have a `is_pullback c.fst c.snd f g`. -/\nlemma of_is_limit {c : pullback_cone f g} (h : limits.is_limit c) :\n  is_pullback c.fst c.snd f g :=\n{ w := c.condition,\n  is_limit' := ⟨is_limit.of_iso_limit h\n    (limits.pullback_cone.ext (iso.refl _) (by tidy) (by tidy))⟩, }\n\n/-- A variant of `of_is_limit` that is more useful with `apply`. -/\nlemma of_is_limit' (w : comm_sq fst snd f g) (h : limits.is_limit w.cone) :\n  is_pullback fst snd f g :=\nof_is_limit h\n\n/-- The pullback provided by `has_pullback f g` fits into a `is_pullback`. -/\nlemma of_has_pullback (f : X ⟶ Z) (g : Y ⟶ Z) [has_pullback f g] :\n  is_pullback (pullback.fst : pullback f g ⟶ X) (pullback.snd : pullback f g ⟶ Y) f g :=\nof_is_limit (limit.is_limit (cospan f g))\n\n/-- If `c` is a limiting binary product cone, and we have a terminal object,\nthen we have `is_pullback c.fst c.snd 0 0`\n(where each `0` is the unique morphism to the terminal object). -/\nlemma of_is_product {c : binary_fan X Y} (h : limits.is_limit c) (t : is_terminal Z) :\n  is_pullback c.fst c.snd (t.from _) (t.from _) :=\nof_is_limit (is_pullback_of_is_terminal_is_product _ _ _ _ t\n  (is_limit.of_iso_limit h (limits.cones.ext (iso.refl c.X) (by rintro ⟨⟨⟩⟩; { dsimp, simp, }))))\n\nvariables (X Y)\n\nlemma of_has_binary_product' [has_binary_product X Y] [has_terminal C] :\n  is_pullback limits.prod.fst limits.prod.snd (terminal.from X) (terminal.from Y) :=\nof_is_product (limit.is_limit _) terminal_is_terminal\n\nopen_locale zero_object\n\nlemma of_has_binary_product [has_binary_product X Y] [has_zero_object C] [has_zero_morphisms C] :\n  is_pullback limits.prod.fst limits.prod.snd (0 : X ⟶ 0) (0 : Y ⟶ 0) :=\nby convert of_is_product (limit.is_limit _) has_zero_object.zero_is_terminal\n\nvariables {X Y}\n\n/-- Any object at the top left of a pullback square is\nisomorphic to the pullback provided by the `has_limit` API. -/\nnoncomputable\ndef iso_pullback (h : is_pullback fst snd f g) [has_pullback f g] : P ≅ pullback f g :=\n(limit.iso_limit_cone ⟨_, h.is_limit⟩).symm\n\n@[simp] lemma iso_pullback_hom_fst (h : is_pullback fst snd f g) [has_pullback f g] :\n  h.iso_pullback.hom ≫ pullback.fst = fst :=\nby { dsimp [iso_pullback, cone, comm_sq.cone], simp, }\n@[simp] lemma iso_pullback_hom_snd (h : is_pullback fst snd f g) [has_pullback f g] :\n  h.iso_pullback.hom ≫ pullback.snd = snd :=\nby { dsimp [iso_pullback, cone, comm_sq.cone], simp, }\n@[simp] lemma iso_pullback_inv_fst (h : is_pullback fst snd f g) [has_pullback f g] :\n  h.iso_pullback.inv ≫ fst = pullback.fst :=\nby simp [iso.inv_comp_eq]\n@[simp] lemma iso_pullback_inv_snd (h : is_pullback fst snd f g) [has_pullback f g] :\n  h.iso_pullback.inv ≫ snd = pullback.snd :=\nby simp [iso.inv_comp_eq]\n\nlemma of_iso_pullback (h : comm_sq fst snd f g) [has_pullback f g] (i : P ≅ pullback f g)\n  (w₁ : i.hom ≫ pullback.fst = fst) (w₂ : i.hom ≫ pullback.snd = snd) : is_pullback fst snd f g :=\nof_is_limit' h (limits.is_limit.of_iso_limit (limit.is_limit _)\n  (@pullback_cone.ext _ _ _ _ _ _ _ (pullback_cone.mk _ _ _) _ i w₁.symm w₂.symm).symm)\n\nend is_pullback\n\nnamespace is_pushout\n\nvariables {Z X Y P : C} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P}\n\n/--\nThe (colimiting) `pushout_cocone f g` implicit in the statement\nthat we have a `is_pushout f g inl inr`.\n-/\ndef cocone (h : is_pushout f g inl inr) : pushout_cocone f g := h.to_comm_sq.cocone\n\n/--\nThe cocone obtained from `is_pushout f g inl inr` is a colimit cocone.\n-/\nnoncomputable def is_colimit (h : is_pushout f g inl inr) : is_colimit h.cocone :=\nh.is_colimit'.some\n\n/-- If `c` is a colimiting pushout cocone, then we have a `is_pushout f g c.inl c.inr`. -/\nlemma of_is_colimit {c : pushout_cocone f g} (h : limits.is_colimit c) :\n  is_pushout f g c.inl c.inr :=\n{ w := c.condition,\n  is_colimit' := ⟨is_colimit.of_iso_colimit h\n    (limits.pushout_cocone.ext (iso.refl _) (by tidy) (by tidy))⟩, }\n\n/-- A variant of `of_is_colimit` that is more useful with `apply`. -/\nlemma of_is_colimit' (w : comm_sq f g inl inr) (h : limits.is_colimit w.cocone) :\n  is_pushout f g inl inr :=\nof_is_colimit h\n\n/-- The pushout provided by `has_pushout f g` fits into a `is_pushout`. -/\nlemma of_has_pushout (f : Z ⟶ X) (g : Z ⟶ Y) [has_pushout f g] :\n  is_pushout f g (pushout.inl : X ⟶ pushout f g) (pushout.inr : Y ⟶ pushout f g) :=\nof_is_colimit (colimit.is_colimit (span f g))\n\n/-- If `c` is a colimiting binary coproduct cocone, and we have an initial object,\nthen we have `is_pushout 0 0 c.inl c.inr`\n(where each `0` is the unique morphism from the initial object). -/\nlemma of_is_coproduct {c : binary_cofan X Y} (h : limits.is_colimit c) (t : is_initial Z) :\n  is_pushout (t.to _) (t.to _) c.inl c.inr :=\nof_is_colimit (is_pushout_of_is_initial_is_coproduct _ _ _ _ t\n  (is_colimit.of_iso_colimit h\n    (limits.cocones.ext (iso.refl c.X) (by rintro ⟨⟨⟩⟩; { dsimp, simp, }))))\n\nvariables (X Y)\n\nlemma of_has_binary_coproduct' [has_binary_coproduct X Y] [has_initial C] :\n  is_pushout (initial.to _) (initial.to _) (coprod.inl : X ⟶ _) (coprod.inr : Y ⟶ _)  :=\nof_is_coproduct (colimit.is_colimit _) initial_is_initial\n\nopen_locale zero_object\n\nlemma of_has_binary_coproduct\n  [has_binary_coproduct X Y] [has_zero_object C] [has_zero_morphisms C] :\n  is_pushout (0 : 0 ⟶ X) (0 : 0 ⟶ Y) coprod.inl coprod.inr :=\nby convert of_is_coproduct (colimit.is_colimit _) has_zero_object.zero_is_initial\n\nvariables {X Y}\n\n/-- Any object at the top left of a pullback square is\nisomorphic to the pullback provided by the `has_limit` API. -/\nnoncomputable\ndef iso_pushout (h : is_pushout f g inl inr) [has_pushout f g] : P ≅ pushout f g :=\n(colimit.iso_colimit_cocone ⟨_, h.is_colimit⟩).symm\n\n@[simp] lemma inl_iso_pushout_inv (h : is_pushout f g inl inr) [has_pushout f g] :\n  pushout.inl ≫ h.iso_pushout.inv = inl :=\nby { dsimp [iso_pushout, cocone, comm_sq.cocone], simp, }\n@[simp] lemma inr_iso_pushout_inv (h : is_pushout f g inl inr) [has_pushout f g] :\n  pushout.inr ≫ h.iso_pushout.inv = inr :=\nby { dsimp [iso_pushout, cocone, comm_sq.cocone], simp, }\n@[simp] lemma inl_iso_pushout_hom (h : is_pushout f g inl inr) [has_pushout f g] :\n  inl ≫ h.iso_pushout.hom = pushout.inl :=\nby simp [←iso.eq_comp_inv]\n@[simp] lemma inr_iso_pushout_hom (h : is_pushout f g inl inr) [has_pushout f g] :\n  inr ≫ h.iso_pushout.hom = pushout.inr :=\nby simp [←iso.eq_comp_inv]\n\nlemma of_iso_pushout (h : comm_sq f g inl inr) [has_pushout f g] (i : P ≅ pushout f g)\n  (w₁ : inl ≫ i.hom = pushout.inl) (w₂ : inr ≫ i.hom = pushout.inr) : is_pushout f g inl inr :=\nof_is_colimit' h (limits.is_colimit.of_iso_colimit (colimit.is_colimit _)\n  (@pushout_cocone.ext _ _ _ _ _ _ _ (pushout_cocone.mk _ _ _) _ i w₁ w₂).symm)\n\nend is_pushout\n\nnamespace is_pullback\n\nvariables {P X Y Z : C} {fst : P ⟶ X} {snd : P ⟶ Y} {f : X ⟶ Z} {g : Y ⟶ Z}\n\nlemma flip (h : is_pullback fst snd f g) : is_pullback snd fst g f :=\nof_is_limit (@pullback_cone.flip_is_limit _ _ _ _ _ _ _ _ _ _ h.w.symm h.is_limit)\n\nsection\n\nvariables [has_zero_object C] [has_zero_morphisms C]\nopen_locale zero_object\n\n/-- The square with `0 : 0 ⟶ 0` on the left and `𝟙 X` on the right is a pullback square. -/\nlemma zero_left (X : C) : is_pullback (0 : 0 ⟶ X) (0 : 0 ⟶ 0) (𝟙 X) (0 : 0 ⟶ X) :=\n{ w := by simp,\n  is_limit' :=\n  ⟨{ lift := λ s, 0,\n     fac' := λ s, by simpa using @pullback_cone.equalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _)\n       (by simpa using (pullback_cone.condition s).symm), }⟩ }\n\n/-- The square with `0 : 0 ⟶ 0` on the top and `𝟙 X` on the bottom is a pullback square. -/\nlemma zero_top (X : C) : is_pullback (0 : 0 ⟶ 0) (0 : 0 ⟶ X) (0 : 0 ⟶ X) (𝟙 X) :=\n(zero_left X).flip\n\nend\n\n/-- Paste two pullback squares \"vertically\" to obtain another pullback square. -/\n-- Objects here are arranged in a 3x2 grid, and indexed by their xy coordinates.\n-- Morphisms are named `hᵢⱼ` for a horizontal morphism starting at `(i,j)`,\n-- and `vᵢⱼ` for a vertical morphism starting at `(i,j)`.\nlemma paste_vert {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}\n  (s : is_pullback h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pullback h₂₁ v₂₁ v₂₂ h₃₁) :\n  is_pullback h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ :=\n(of_is_limit\n  (big_square_is_pullback _ _ _ _ _ _ _ s.w t.w t.is_limit s.is_limit))\n\n/-- Paste two pullback squares \"horizontally\" to obtain another pullback square. -/\nlemma paste_horiz {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}\n  (s : is_pullback h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pullback h₁₂ v₁₂ v₁₃ h₂₂) :\n  is_pullback (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) :=\n(paste_vert s.flip t.flip).flip\n\n/-- Given a pullback square assembled from a commuting square on the top and\na pullback square on the bottom, the top square is a pullback square. -/\nlemma of_bot {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}\n  (s : is_pullback h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁) (p : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁)\n  (t : is_pullback h₂₁ v₂₁ v₂₂ h₃₁) :\n  is_pullback h₁₁ v₁₁ v₁₂ h₂₁ :=\nof_is_limit (left_square_is_pullback _ _ _ _ _ _ _ p _ t.is_limit s.is_limit)\n\n/-- Given a pullback square assembled from a commuting square on the left and\na pullback square on the right, the left square is a pullback square. -/\nlemma of_right {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}\n  (s : is_pullback (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂)) (p : h₁₁ ≫ v₁₂ = v₁₁ ≫ h₂₁)\n  (t : is_pullback h₁₂ v₁₂ v₁₃ h₂₂) :\n  is_pullback h₁₁ v₁₁ v₁₂ h₂₁ :=\n(of_bot s.flip p.symm t.flip).flip\n\nend is_pullback\n\nnamespace is_pushout\n\nvariables {Z X Y P : C} {f : Z ⟶ X} {g : Z ⟶ Y} {inl : X ⟶ P} {inr : Y ⟶ P}\n\nlemma flip (h : is_pushout f g inl inr) : is_pushout g f inr inl :=\nof_is_colimit (@pushout_cocone.flip_is_colimit _ _ _ _ _ _ _ _ _ _ h.w.symm h.is_colimit)\n\nsection\n\nvariables [has_zero_object C] [has_zero_morphisms C]\nopen_locale zero_object\n\n/-- The square with `0 : 0 ⟶ 0` on the right and `𝟙 X` on the left is a pushout square. -/\nlemma zero_right (X : C) : is_pushout (0 : X ⟶ 0) (𝟙 X) (0 : 0 ⟶ 0) (0 : X ⟶ 0) :=\n{ w := by simp,\n  is_colimit' :=\n  ⟨{ desc := λ s, 0,\n     fac' := λ s, begin\n       have c := @pushout_cocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _) (by simp)\n         (by simpa using (pushout_cocone.condition s)),\n      dsimp at c,\n      simpa using c,\n     end }⟩ }\n\n/-- The square with `0 : 0 ⟶ 0` on the bottom and `𝟙 X` on the top is a pushout square. -/\nlemma zero_bot (X : C) : is_pushout (𝟙 X) (0 : X ⟶ 0) (0 : X ⟶ 0) (0 : 0 ⟶ 0) :=\n(zero_right X).flip\n\nend\n\n/-- Paste two pushout squares \"vertically\" to obtain another pushout square. -/\n-- Objects here are arranged in a 3x2 grid, and indexed by their xy coordinates.\n-- Morphisms are named `hᵢⱼ` for a horizontal morphism starting at `(i,j)`,\n-- and `vᵢⱼ` for a vertical morphism starting at `(i,j)`.\nlemma paste_vert {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}\n  (s : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pushout h₂₁ v₂₁ v₂₂ h₃₁) :\n  is_pushout h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁ :=\n(of_is_colimit\n  (big_square_is_pushout _ _ _ _ _ _ _ s.w t.w t.is_colimit s.is_colimit))\n\n/-- Paste two pushout squares \"horizontally\" to obtain another pushout square. -/\nlemma paste_horiz {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}\n  (s : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) (t : is_pushout h₁₂ v₁₂ v₁₃ h₂₂) :\n  is_pushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) :=\n(paste_vert s.flip t.flip).flip\n\n/-- Given a pushout square assembled from a pushout square on the top and\na commuting square on the bottom, the bottom square is a pushout square. -/\nlemma of_bot {X₁₁ X₁₂ X₂₁ X₂₂ X₃₁ X₃₂ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₂₁ : X₂₁ ⟶ X₂₂} {h₃₁ : X₃₁ ⟶ X₃₂}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₂₁ : X₂₁ ⟶ X₃₁} {v₂₂ : X₂₂ ⟶ X₃₂}\n  (s : is_pushout h₁₁ (v₁₁ ≫ v₂₁) (v₁₂ ≫ v₂₂) h₃₁) (p : h₂₁ ≫ v₂₂ = v₂₁ ≫ h₃₁)\n  (t : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) :\n  is_pushout h₂₁ v₂₁ v₂₂ h₃₁ :=\nof_is_colimit (right_square_is_pushout _ _ _ _ _ _ _ _ p t.is_colimit s.is_colimit)\n\n/-- Given a pushout square assembled from a pushout square on the left and\na commuting square on the right, the right square is a pushout square. -/\nlemma of_right {X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C}\n  {h₁₁ : X₁₁ ⟶ X₁₂} {h₁₂ : X₁₂ ⟶ X₁₃} {h₂₁ : X₂₁ ⟶ X₂₂} {h₂₂ : X₂₂ ⟶ X₂₃}\n  {v₁₁ : X₁₁ ⟶ X₂₁} {v₁₂ : X₁₂ ⟶ X₂₂} {v₁₃ : X₁₃ ⟶ X₂₃}\n  (s : is_pushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂)) (p : h₁₂ ≫ v₁₃ = v₁₂ ≫ h₂₂)\n  (t : is_pushout h₁₁ v₁₁ v₁₂ h₂₁) :\n  is_pushout h₁₂ v₁₂ v₁₃ h₂₂ :=\n(of_bot s.flip p.symm t.flip).flip\n\nend is_pushout\n\nnamespace functor\n\nvariables {D : Type u₂} [category.{v₂} D]\nvariables (F : C ⥤ D) {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\nlemma map_comm_sq (s : comm_sq f g h i) : comm_sq (F.map f) (F.map g) (F.map h) (F.map i) :=\n⟨by simpa using congr_arg (λ k : W ⟶ Z, F.map k) s.w⟩\n\nlemma map_is_pullback [preserves_limit (cospan h i) F] (s : is_pullback f g h i) :\n  is_pullback (F.map f) (F.map g) (F.map h) (F.map i) :=\n-- This is made slightly awkward because `C` and `D` have different universes,\n-- and so the relevant `walking_cospan` diagrams live in different universes too!\nbegin\n  refine is_pullback.of_is_limit' (F.map_comm_sq s.to_comm_sq)\n    (is_limit.of_whisker_equivalence walking_cospan_equiv\n      (is_limit.equiv_of_nat_iso_of_iso (cospan_comp_iso F h i) _ _ (walking_cospan.ext _ _ _)\n        (is_limit_of_preserves F s.is_limit))),\n  { refl, },\n  { dsimp, simp, refl, },\n  { dsimp, simp, refl, },\nend\n\nlemma map_is_pushout [preserves_colimit (span f g) F] (s : is_pushout f g h i) :\n  is_pushout (F.map f) (F.map g) (F.map h) (F.map i) :=\nbegin\n  refine is_pushout.of_is_colimit' (F.map_comm_sq s.to_comm_sq)\n    (is_colimit.of_whisker_equivalence walking_span_equiv\n      (is_colimit.equiv_of_nat_iso_of_iso (span_comp_iso F f g) _ _ (walking_span.ext _ _ _)\n        (is_colimit_of_preserves F s.is_colimit))),\n  { refl, },\n  { dsimp, simp, refl, },\n  { dsimp, simp, refl, },\nend\n\nend functor\n\nalias functor.map_comm_sq ← comm_sq.map\nalias functor.map_is_pullback ← is_pullback.map\nalias functor.map_is_pushout ← is_pushout.map\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/shapes/comm_sq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3443150933626251}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  --Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from by {\n    assume i j : ℤ,\n    assume h2 : i ≠ j,\n    assume h3 : int.fract (α * ↑i) = int.fract (α * ↑j),\n    have h4 : α * ↑i - ↑(int.floor (α * ↑i)) = int.fract (α * ↑i), from by {rw int.fract_eq_of_nonneg (le_of_lt (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (α * ↑i))) (int.floor_lt_iff.mpr (lt_of_le_of_lt (int.coe_nat_le_coe_nat_of_le (int.floor_nonneg (\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < ε ↔ ((x0 - l) < ε) ∧ ((l - x0) < ε), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro ⟨ S_1, S_2 ⟩, \n    split; linarith, \n    rintro ⟨ S_3, S_4 ⟩, \n    split; linarith,\n    },\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : ε > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 ε h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 ε h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_with_comments-4_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7931059511841119, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.34419328243957015}}
{"text": "/-\nCopyright (c) 2019 The Flypitch Project. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Jesse Han, Floris van Doorn\n-/\nimport .bvm_extras .cantor_space\n\nopen ordinal cardinal lattice bSet\n\nnoncomputable theory\n\nlocal attribute [instance] classical.prop_decidable\n\nlocal attribute [simp] omega_le_aleph\n\nlocal infix ` ⟹ `:65 := lattice.imp\n\nlocal infix ` ⇔ `:50 := lattice.biimp\n\nlocal prefix `#`:70 := cardinal.mk\n\nlocal infix `≺`:75 := (λ x y, -(larger_than x y))\n\nlocal infix `≼`:75 := (λ x y, injects_into x y)\n\nuniverse u\n\nnamespace bSet\nsection cardinal_preservation\nlocal notation `ω` := cardinal.omega\nvariables {𝔹 : Type u} [I : nontrivial_complete_boolean_algebra 𝔹]\n\ninclude I\n\nlemma AE_of_check_larger_than_check'' {x y : pSet.{u}} (f : bSet 𝔹) {Γ : 𝔹} (H_nonzero : ⊥ < Γ)\n  (H : Γ ≤ is_surj_onto x̌ y̌ f) (H_nonempty : ∃ z, z ∈ y) : ∀ i : y.type, ∃ j : x.type, ⊥ < (is_func f) ⊓ (pair (x.func j)̌  (y.func i)̌  ∈ᴮ f) :=\nbegin\n  intro i_v, bv_split_at H,\n  replace H_right := H_right (y.func i_v)̌ , simp [check_mem'] at H_right,\n  replace H_right := exists_convert H_right _, cases H_right with w Hw, bv_split_at Hw,\n  rcases eq_check_of_mem_check ‹_› Hw_left with ⟨j,Γ',HΓ'₁,HΓ'₂,H_eq⟩,\n  use j, refine lt_of_lt_of_le HΓ'₁ (le_inf _ _),\n    { exact le_trans HΓ'₂ (is_func_of_is_func' ‹_›) },\n    { apply @bv_rw' _ _ _ _ _ (bv_symm H_eq) (λ z, pair z (y.func i_v)̌  ∈ᴮ f), exact B_ext_pair_mem_left,\n      from le_trans ‹_› ‹_› },\n  exact B_ext_inf (by simp) B_ext_pair_mem_left\nend\n\nlemma AE_of_check_larger_than_check' {x y : pSet.{u}} {Γ : 𝔹} (H_nonzero : ⊥ < Γ)\n  (H : Γ ≤ surjects_onto x̌ y̌) (H_mem : ∃ z, z ∈ y) : ∃ f : bSet 𝔹, ∀ i : y.type, ∃ j : x.type, ⊥ < (is_func f) ⊓ (pair (x.func j)̌  (y.func i)̌  ∈ᴮ f) :=\nbegin\n  unfold surjects_onto at H, have := maximum_principle (λ w, is_func' x̌ y̌ w ⊓ is_surj x̌ (y̌ : bSet 𝔹) w) _,\n  cases this with f Hf, rw Hf at H, swap, {simp},\n  exact ⟨f, AE_of_check_larger_than_check'' ‹_› ‹_› ‹_› ‹_›⟩\nend\n\nlemma AE_of_check_larger_than_check {x y : pSet.{u}} {Γ : 𝔹} (H_nonzero : ⊥ < Γ)\n  (H : Γ ≤ larger_than x̌ y̌) (H_mem : ∃ z, z ∈ y) : ∃ f : bSet 𝔹, ∀ i : y.type, ∃ j : x.type, ⊥ < (is_func f) ⊓ (pair (x.func j)̌  (y.func i)̌  ∈ᴮ f) :=\nAE_of_check_larger_than_check'\n  ‹_› (surjects_onto_of_larger_than_and_exists_mem ‹_› $ by simp*) ‹_›\n\nvariables\n  (η₁ η₂ : pSet.{u}) (H_infinite : ω ≤ #(η₁.type))\n  (H_lt : #(η₁.type) < #(η₂.type))\n  (H_inj₂ : ∀ x y, x ≠ y → ¬ pSet.equiv (η₂.func x) (η₂.func y))\n  (f : bSet 𝔹) (g : η₂.type → η₁.type)\n  (H : ∀ β : η₂.type, (⊥ : 𝔹) < is_func f ⊓ pair (η₁.func (g β))̌  ((η₂.func β)̌ )∈ᴮ f)\n\ninclude H_infinite H_lt H_inj₂ f H\nlemma not_CCC_of_uncountable_fiber (H_ex : ∃ ξ : η₁.type, ω < #(g⁻¹' {ξ})) : ¬ CCC 𝔹 :=\nbegin\n  cases H_ex with ξ H_ξ,\n  let 𝓐 : (g⁻¹'{ξ}) → 𝔹 :=\n    λ β, is_func f ⊓ (pair ((η₁.func (g β.val))̌ ) ((η₂.func β.val)̌ )) ∈ᴮ f,\n  have 𝓐_nontriv : ∀ β, ⊥ < 𝓐 β,\n    from λ _, by apply H,\n  have 𝓐_anti : ∀ β₁ β₂, β₁ ≠ β₂ → (𝓐 β₁) ⊓ (𝓐 β₂) ≤ ⊥,\n    by {intros β₁ β₂ h_sep, dsimp[𝓐],\n    /- `tidy_context` says -/ apply poset_yoneda, intros Γ a,\n    cases β₂, cases β₁, cases H_ξ, cases H_lt, cases β₁_property, cases β₂_property,\n    work_on_goal 0 { induction β₂_property, simp only [le_inf_iff] at a,\n                     cases a, cases a_right, cases a_left },\n    work_on_goal 1 { induction β₁_property, simp only [le_inf_iff] at a,\n                     cases a, cases a_right, cases a_left, solve_by_elim },\n    work_on_goal 1 { cases β₂_property,\n      work_on_goal 0 { induction β₂_property, simp only [le_inf_iff] at a,\n        cases a, cases a_right, cases a_left, solve_by_elim }, simp only [le_inf_iff] at a,\n        cases a, cases a_right, cases a_left, solve_by_elim },\n\n    rw[β₁_property] at a_left_right,\n    have H_le_eq : Γ ≤ ((η₂.func β₁_val)̌ ) =ᴮ ((η₂.func β₂_val)̌ ),\n     by {apply eq_of_is_func_of_eq, from a_right_left, tactic.rotate 1,\n         from ‹_›, from ‹_›, from bv_refl },\n    from le_trans H_le_eq\n           (by {rw[le_bot_iff], apply check_bv_eq_bot_of_not_equiv, apply H_inj₂, tidy})},\n   intro H_CCC, specialize H_CCC (g⁻¹'{ξ}) ‹_› ‹_› ‹_›,\n   replace H_ξ := (lt_iff_le_and_ne.mp H_ξ),\n   from absurd (le_antisymm H_ξ.left H_CCC) H_ξ.right\nend\n\nend cardinal_preservation\nend bSet\n\nopen bSet\n\nnamespace pSet\n\n@[reducible]noncomputable def ℵ₁ : pSet.{0} := ordinal.mk (aleph 1).ord\n\n@[reducible]noncomputable def ℵ₂ : pSet.{0} := ordinal.mk (aleph 2).ord\n\nlemma ℵ₂_unfold : ℵ₂ = ⟨ℵ₂.type, ℵ₂.func⟩ := by cases ℵ₂; refl\n\n@[simp, cleanup]lemma Union_type {x : pSet} : (type (Union x)) = Σ(a:x.type), (x.func a).type :=\nby induction x; refl\n\n@[simp, cleanup]lemma Union_type' {α : Type u} {A : α → pSet.{u}} :\n  (Union (mk α A)).type = Σa, (A a).type := rfl\n\nend pSet\n\nopen pSet\n\ndef 𝔹_cohen : Type := @regular_opens (set(ℵ₂.type × ℕ)) (Pi.topological_space)\n\nlocal notation `𝔹` := 𝔹_cohen\n\ninstance H_nonempty : nonempty (set $ ℵ₂.type × ℕ) := ⟨∅⟩\n\n@[instance, priority 1000]def 𝔹_boolean_algebra : nontrivial_complete_boolean_algebra 𝔹 :=\nregular_open_algebra\n\nlemma le_iff_subset' {x y : 𝔹} : x ≤ y ↔ x.1 ⊆ y.1 := by refl\n\nlemma bot_eq_empty : (⊥ : 𝔹) = ⟨∅, is_regular_empty⟩ := rfl\n\nprivate lemma eq₀ : (ℵ₂̌  : bSet 𝔹).type = (ℵ₂).type := by cases ℵ₂; refl\n\nprivate lemma eq₁ : ((type (ℵ₂̌  : bSet 𝔹)) × ℕ) = ((type ℵ₂) × ℕ) :=\nby {cases ℵ₂, refl}\n\nprivate lemma eq₂ : set ((type (ℵ₂̌  : bSet 𝔹)) × ℕ) = set ((type ℵ₂) × ℕ) :=\nby {cases ℵ₂, refl}\n\nprivate lemma eq₃ : finset ((type (ℵ₂̌  : bSet 𝔹)) × ℕ) = finset (type ℵ₂ × ℕ) :=\nby {cases ℵ₂, refl}\n\nlemma pi₂_cast₁ {α β γ : Type*} (H' : α = β) {p : α × γ} {q : β × γ} (H : p == q) :\n  p.1 == q.1 :=\nby {subst H', subst H}\n\nlemma pi₂_cast₂ {α β γ : Type*} (H' : α = β) {p : α × γ} {q : β × γ} (H : p == q) :\n  p.2 = q.2 :=\nby {subst H', subst H}\n\nlemma compl_cast₂ {α β : Type*} {a : set α} {b : set β} (H' : α = β) (H : -a == b) : a == -b :=\nbegin\n  subst H', subst H, apply heq_of_eq, simp\nend\n\nlemma eq₁_cast (p : ((type (ℵ₂̌  : bSet 𝔹)) × ℕ)) {prf : ((type (ℵ₂̌  : bSet 𝔹)) × ℕ) = (((type ℵ₂) × ℕ))} {prf' : (type (ℵ₂̌  : bSet 𝔹)) = (ℵ₂.type)} : cast prf p = (cast prf' p.1, p.2) :=\nbegin\n  ext, swap, simp, h_generalize H_x : p == x, apply pi₂_cast₂, from eq₀.symm, from H_x.symm,\n  h_generalize H_x : p == x, simp, h_generalize H_y : p.fst == y,\n  apply eq_of_heq, suffices : x.fst == p.fst, from heq.trans this H_y,\n  apply pi₂_cast₁, from eq₀.symm, from H_x.symm\nend\n\nlemma eq₁_cast' (p : (((type ℵ₂) × ℕ))) {prf : ((type (ℵ₂̌  : bSet 𝔹)) × ℕ) = (((type ℵ₂) × ℕ))} {prf' : (type (ℵ₂̌  : bSet 𝔹)) = (ℵ₂.type)} : cast prf.symm p = (cast prf'.symm p.1, p.2) :=\nbegin\n  ext, swap, simp, h_generalize H_x : p == x, apply pi₂_cast₂, from eq₀, from H_x.symm,\n  h_generalize H_x : p == x, simp, h_generalize H_y : p.fst == y,\n  apply eq_of_heq, suffices : x.fst == p.fst, from heq.trans this H_y,\n  apply pi₂_cast₁, from eq₀, from H_x.symm\nend\n\ntheorem 𝔹_CCC : CCC 𝔹 :=\nby { apply CCC_regular_opens, apply cantor_space.countable_chain_condition_set }\n\nlocal notation `𝒳` := set(ℵ₂.type × ℕ)\n\nopen topological_space\n\n/-- The principal regular open associated to a pair (ν, n) is the collection of all subsets of\n    ℵ₂ × ℕ which contain (ν, n). -/\ndef principal_open (ν : (ℵ₂̌  : bSet 𝔹).type) (n : ℕ) : 𝔹 :=\nbegin\n  use (cantor_space.principal_open (cast eq₁ (ν, n))),\n  from is_regular_of_clopen (cantor_space.is_clopen_principal_open)\nend\n\nlemma is_clopen_principal_open {ν n} : is_clopen (principal_open ν n).val :=\n  cantor_space.is_clopen_principal_open\n\nlocal postfix `ᵖ`:80 := perp\n\nlocal notation `cl`:65 := closure\n\nlocal notation `int`:65 := interior\n\nlemma perp_eq_compl_of_clopen {β : Type*} [topological_space β] {S : set β} (H : is_clopen S) : Sᵖ = (-S) :=\nby {unfold perp, rw[closure_eq_of_is_closed H.right]}\n\nlemma mem_neg_principal_open_of_not_mem {ν n S} : (cast eq₁ (ν,n) ∈ (-S)) → S ∈ (- (principal_open ν n)).val :=\nbegin\n  intro H, simp only [neg_unfold], rw[perp_eq_compl_of_clopen],\n  swap, from is_clopen_principal_open, from H\nend\n\nstructure 𝒞 : Type :=\n(ins : finset ((ℵ₂ ̌ : bSet 𝔹).type × ℕ))\n(out : finset ((ℵ₂ ̌ : bSet 𝔹).type × ℕ))\n(H : ins ∩ out = ∅)\n\n@[reducible]def π₂ : (ℵ₂̌  : bSet 𝔹).type × ℕ → ℕ := λ x, x.snd\n\n-- def nat_supp : finset ((ℵ₂ ̌ : bSet 𝔹).type × ℕ) → set ℕ :=\n-- λ X, {n | ∃ (ξ : ℵ₂.type), (cast eq₁.symm (ξ,n)) ∈ X}\n\n-- lemma nat_supp_finite {X} : set.finite $ nat_supp X := sorry\n\nprivate def ι : 𝒞 → 𝔹 :=\nλ p, ⟨{S | (p.ins.to_set) ⊆ (cast eq₂.symm S) ∧\n           (p.out.to_set) ⊆ (cast eq₂.symm (- S))},\nis_regular_of_clopen\n     begin\n       change is_clopen\n         ({S | p.ins.to_set ⊆ cast eq₂.symm S} ∩ {S | p.out.to_set ⊆ (cast eq₂.symm (-S))}),\n       refine is_clopen_inter _ _,\n         have := cantor_space.is_clopen_principal_open_finset p.ins,\n         convert this, from eq₀.symm, from eq₀.symm, from eq₀.symm,\n           {apply function.hfunext, from eq₂.symm, intros a a' H_heq,\n             apply heq_of_eq, convert rfl, convert (cast_eq _ _).symm, from eq₀.symm, refl},\n\n         have := cantor_space.is_clopen_co_principal_open_finset p.out,\n         convert this, from eq₀.symm, from eq₀.symm, from eq₀.symm,\n         {apply function.hfunext, from eq₂.symm, intros a a' H_heq,\n          apply heq_of_eq, convert rfl, h_generalize Hx : (-a) == x,\n          have := heq.subst H_heq, swap,\n          from λ _ y, y == -x,\n          suffices : a' = -x, by {rw[this], simp},\n          apply eq_of_heq, apply this, apply compl_cast₂, from eq₁.symm,\n          from Hx}\n     end⟩\n\nopen cantor_space\n\nlemma prop_decidable_cast_lemma {α β : Type*} (H : α = β) {a b : α} {a' b' : β} (H_a : a == a') (H_b : b == b') : classical.prop_decidable (a = b) == classical.prop_decidable (a' = b') :=\nby {subst H, subst H_a, subst H_b}\n\nlemma 𝒞_dense_basis : ∀ T ∈ @standard_basis (ℵ₂.type × ℕ), ∀ h_nonempty : T ≠ ∅,\n  ∃ p : 𝒞, (ι p).val ⊆ T :=\nbegin\n  intros T Ht H_nonempty, simp[standard_basis] at Ht,\n  cases Ht with H_empty Ht, contradiction,\n  rcases Ht with ⟨p_ins, p_out, H₁, H₂⟩,\n  fsplit, refine ⟨_,_,_⟩, from cast eq₃.symm p_ins,\n  from cast eq₃.symm p_out, swap, rw[<-co_principal_open_finset_eq_inter] at H₁,\n  rw[<-principal_open_finset_eq_inter] at H₁, subst H₁,\n  intros S HS, split, cases HS, dsimp at HS_left, simp[principal_open_finset],\n  {convert HS_left,\n    from eq₀.symm, from eq₀.symm, from eq₀.symm, all_goals{symmetry, from cast_heq _ _}},\n  cases HS, dsimp at HS_right, simp[principal_open_finset],\n  {convert HS_right,\n    from eq₀.symm, from eq₀.symm, from eq₀.symm, all_goals{symmetry, from cast_heq _ _}},\n  convert H₂, from eq₀, from eq₀, from eq₀,\n  apply function.hfunext, from eq₁, intros a a' H,\n  apply function.hfunext, from eq₁, intros b b' H',\n  from prop_decidable_cast_lemma eq₁ ‹_› ‹_›,\n  from cast_heq _ _, from cast_heq _ _, from eq₀, from eq₀\nend\n\nlemma 𝒞_dense {b : 𝔹} (H : ⊥ < b) : ∃ p : 𝒞, (ι p) ≤ b :=\nbegin\n  cases (classical.choice (classical.nonempty_of_not_empty _ H.right.symm)) with S_wit H_wit,\n  change ∃ p, (ι p).val ⊆ b.val,\n  have := mem_basis_subset_of_mem_open (is_topological_basis_standard_basis) H_wit (is_open_of_is_regular b.property),\n  rcases (mem_basis_subset_of_mem_open\n           (is_topological_basis_standard_basis) H_wit (is_open_of_is_regular b.property))\n         with ⟨v, Hv₁, Hv₂, Hv₃⟩,\n  have : v ≠ ∅, by {intro H, rw[H] at Hv₂, cases Hv₂},\n  cases (𝒞_dense_basis ‹_› ‹_› ‹_›) with p H_p, from ⟨p, set.subset.trans H_p ‹_›⟩\nend\n\nlemma to_set_inter {α : Type*} {p₁ p₂ : finset α} : (p₁ ∩ p₂).to_set = (p₁.to_set ∩ p₂.to_set) :=\nby {ext, split; intros; unfold finset.to_set at *, tidy}\n\n@[simp]lemma to_set_empty {α : Type*} : finset.to_set (∅ : finset α) = ∅ :=\nby {unfold finset.to_set, refl}\n\nlemma not_mem_of_inter_empty_left {α : Type*} {p₁ p₂ : finset α}\n  (H : p₁ ∩ p₂ = ∅) {a : α} : a ∈ p₁.to_set → ¬ a ∈ p₂.to_set :=\nbegin\n  intro H', intro H'',\n  have this₀ : a ∈ p₁.to_set ∩ p₂.to_set := ⟨‹_›,‹_›⟩,\n  rw[<-to_set_inter] at this₀, have this₁ := congr_arg finset.to_set H,\n  rw[this₁] at this₀, cases this₀\nend\n\nlemma not_mem_of_inter_empty_right {α : Type*} {p₁ p₂ : finset α}\n  (H : p₂ ∩ p₁ = ∅) {a : α} : a ∈ p₁.to_set → ¬ a ∈ p₂.to_set :=\nby {rw[finset.inter_comm] at H, apply not_mem_of_inter_empty_left, from ‹_›}\n\nlemma 𝒞_nonzero (p : 𝒞) : ⊥ ≠ (ι p) :=\nbegin\n  intro H, replace H := H.symm, rw[eq_bot_iff] at H, rw[le_iff_subset'] at H,\n  rw[bot_eq_empty] at H,\n  suffices : nonempty (ι p).val,\n    by {have := classical.choice this, specialize H this.property, cases H},\n  apply nonempty.intro, fsplit, exact (cast eq₂ p.ins.to_set),\n  split, finish, intro x, cases x with ν n, intro H,\n  suffices : cast eq₁ (ν, n) ∈ - cast eq₂ (p.ins).to_set,\n    {convert this, from eq₀, from eq₀, from eq₀, cc, cc},\n  suffices : (ν, n) ∈ - p.ins.to_set,\n    {convert this, from eq₀.symm, from eq₀.symm, from eq₀.symm, cc, from eq₀.symm,\n     from eq₀.symm, cc},\n  from not_mem_of_inter_empty_right p.H H\nend\n\nlemma subset_of_eq {α : Type*} {a b : finset α} (H : a = b) : a ⊆ b := by rw[H]; refl\n\nlemma 𝒞_disjoint_row (p : 𝒞) : ∃ n : ℕ, ∀ ξ : ℵ₂.type, (cast eq₁.symm (ξ,n)) ∉ p.ins ∧ (cast eq₁.symm (ξ,n)) ∉ p.out :=\nbegin\n  let Y := (finset.image π₂ p.ins) ∪ (finset.image π₂ p.out),\n  by_cases (p.ins ∪ p.out) = ∅,\n  use 0, intro ξ, split, intro x, apply (subset_of_eq h), simp, left, from x,\n  intro x, apply (subset_of_eq h), simp, right, from x,\n  let Y' := finset.image π₂ (p.ins ∪ p.out),\n  have Y'_nonempty : Y' ≠ ∅,\n    by {dsimp[Y'], intro H, apply h, ext; split; intros, swap, cases a_1,\n      have : π₂ a ∈ finset.image π₂ (p.ins ∪ p.out), simp,\n      use a.fst, simp at a_1, convert a_1, cases a, refl, cases a, refl,\n      rw[H] at this, cases this},\n  have := finset.max_of_ne_empty,\n  specialize this Y'_nonempty, cases this with N HN, swap, apply_instance,\n  use (N+1), intro ξ, split,\n    intro X, let prf := _, change cast prf (ξ, N + 1) ∈ p.ins at X,\n    rw[eq₁_cast'] at X, swap, from eq₀,\n    have : N + 1 ∈ Y',\n      by {simp, use cast eq₀.symm ξ, from or.inl X},\n    suffices : N + 1 ≤ N, by {revert this, change ¬ (N + 1 ≤ N), apply nat.not_succ_le_self},\n    apply finset.le_max_of_mem this ‹_›,\n  intro X, let prf := _, change cast prf (ξ, N + 1) ∈ p.out at X,\n    rw[eq₁_cast'] at X, swap, from eq₀,\n    have : N + 1 ∈ Y',\n      by {simp, use cast eq₀.symm ξ, from or.inr X},\n    suffices : N + 1 ≤ N, by {revert this, change ¬ (N + 1 ≤ N), apply nat.not_succ_le_self},\n    apply finset.le_max_of_mem this ‹_›\nend\n\nlemma 𝒞_anti {p₁ p₂ : 𝒞} : p₁.ins ⊆ p₂.ins → p₁.out ⊆ p₂.out → ι p₂ ≤ ι p₁  :=\nby {intros H₁ H₂, rw[le_iff_subset'], tidy}\n\nnamespace cohen_real\nsection cohen_real\n\n/-- `cohen_real.χ ν` is the indicator function on ℕ induced by every ordinal less than ℵ₂ -/\ndef χ (ν : (ℵ₂̌  : bSet 𝔹).type) : ℕ → 𝔹 :=\n  λ n, principal_open ν n\n\n/-- `cohen_real.mk ν` is the subset of (ω : bSet 𝔹) induced by `cohen_real.χ ν` -/\ndef mk (ν : (ℵ₂̌  : bSet 𝔹).type) : bSet 𝔹 :=\n  @bSet.set_of_indicator 𝔹 _ omega $ λ n, χ ν n.down\n\n@[simp, cleanup]lemma mk_type {ν} : (mk ν).type = ulift ℕ := rfl\n\n@[simp, cleanup]lemma mk_func {ν} {n} : (mk ν).func n = bSet.of_nat (n.down) := rfl\n\n@[simp, cleanup]lemma mk_bval {ν} {n} : (mk ν).bval n = (χ ν) (n.down) := rfl\n\n/-- bSet 𝔹 believes that each `mk ν` is a subset of omega -/\nlemma definite {ν} {Γ} : Γ ≤ mk ν ⊆ᴮ omega :=\nby simp [mk, subset_unfold]; from λ _, by rw[<-deduction]; convert omega_definite\n\n/-- bSet 𝔹 believes that each `mk ν` is an element of 𝒫(ω) -/\nlemma definite' {ν} {Γ} : Γ ≤ mk ν ∈ᴮ bv_powerset omega := bv_powerset_spec.mp definite\n\nlemma sep {n} {Γ} {ν₁ ν₂} (H₁ : Γ ≤ (of_nat n) ∈ᴮ (mk ν₁)) (H₂ : Γ ≤ (- ((of_nat n) ∈ᴮ (mk ν₂)))) :\n  Γ ≤ (- ((mk ν₁) =ᴮ (mk ν₂))) :=\nbegin\n  rw[bv_eq_unfold], rw[neg_inf, neg_infi, neg_infi], simp only [lattice.neg_imp],\n  refine le_sup_left_of_le _, rw[@bounded_exists 𝔹 _ (mk ν₁) (λ z, -(z ∈ᴮ mk ν₂)) _],\n  swap, change B_ext _, simp[-imp_bot, imp_bot.symm],\n  apply bv_use (bSet.of_nat n), bv_split_goal\nend\n\nlemma not_mem_of_not_mem {p : 𝒞} {ν} {n} (H : (ν,n) ∈ p.out) : ι p ≤ -( (of_nat n) ∈ᴮ (mk ν)) :=\nbegin\nrw[bSet.mem_unfold, neg_supr], bv_intro k, rw[neg_inf], simp,\n       by_cases n = k.down, swap, rw[bSet.of_nat_inj ‹_›],\n       from le_sup_right_of_le (by simp),\n       refine le_sup_left_of_le _, rw[<-h],\n       rw[le_iff_subset'], unfold ι χ, rintros S ⟨H_S₁, H_S₂⟩,\n       apply mem_neg_principal_open_of_not_mem, have := H_S₂ H, convert this,\n       from eq₀.symm, from eq₀.symm, from eq₀.symm,\n       from cast_heq _ _, from (cast_heq _ _).symm\nend\n\nprivate lemma inj_cast_lemma (ν' : type (ℵ₂̌  : bSet 𝔹)) (n' : ℕ) :\n  cast eq₁.symm (cast eq₀ ν', n') = (ν', n') :=\nbegin\n  let a := _, change cast a _ = _,\n  let b := _, change cast _ (cast b _, _) = _,\n  simp[b] at a, dedup, change cast a_1 _ = _, cc\nend\n\n/-- Whenever ν₁ ≠ ν₂ < ℵ₂, bSet 𝔹 believes that `mk ν₁` and `mk ν₂` are distinct -/\nlemma inj {ν₁ ν₂} (H_neq : ν₁ ≠ ν₂) : (mk ν₁) =ᴮ (mk ν₂) ≤ (⊥ : 𝔹) :=\nbegin\n  by_contra, replace h := (bot_lt_iff_not_le_bot.mpr ‹_›),\n  cases 𝒞_dense h with p H_p, cases 𝒞_disjoint_row p with n H_n,\n  let p' : 𝒞 := { ins := insert (ν₁,n) (p.ins),\n  out := insert (ν₂,n) p.out,\n  H := by {ext, split; intro H, swap, cases H, have := p.H, simp at H, cases a_1 with ν' n',\n           cases H with H₁ H₂, specialize H_n (cast eq₀ ν'), cases H_n, cases H₁; cases H₂, cc,\n           exfalso, apply H_n_right, convert H₂, rw[show n = n', by cc], apply inj_cast_lemma,\n           exfalso, apply H_n_left, convert H₁, rw[show n = n', by cc], apply inj_cast_lemma,\n           rw[<-this], simp[*,-this]} },\n  have this₀ : ι p' ≤ ι p,\n    from 𝒞_anti (by {dsimp[p'], from λ i _, by {simp, from or.inr ‹_›}})\n                (by {dsimp[p'], from λ i _, by {simp, from or.inr ‹_›}}),\n  have this₁ : ι p' ≤ (ñ̌) ∈ᴮ (cohen_real.mk ν₁),\n    by {rw[bSet.mem_unfold], apply bv_use (ulift.up n), refine le_inf _ bv_refl,\n         {simp [le_iff_subset', χ, _root_.principal_open, ι, cantor_space.principal_open],\n         have : (ν₁, n) ∈ p'.ins,\n           by simp[p'], intros S H_S _, specialize H_S this,\n              convert H_S; [from eq₀.symm, from eq₀.symm, from eq₀.symm, cc, cc]}},\n  have this₂ : ι p' ≤ - ((ñ̌) ∈ᴮ (cohen_real.mk ν₂)),\n    by {have : (ν₂, n) ∈ p'.out, by {simp[p']},\n       from not_mem_of_not_mem ‹_›},\n  have this₃ : ι p' ≤ - (mk ν₁ =ᴮ mk ν₂),\n    from sep ‹_› ‹_›,\n  have this₄ : ι p' ≤ (mk ν₁ =ᴮ mk ν₂),\n    from le_trans this₀ ‹_›,\n  suffices : ι p' = ⊥, from absurd this.symm (𝒞_nonzero p'),\n  bv_and_intro this₃ this₄, simpa using H\nend\n\nend cohen_real\nend cohen_real\n\nsection neg_CH\n\nlocal attribute [irreducible] regular_opens 𝔹_cohen\n\nlocal notation `ℵ₀` := (omega : bSet 𝔹)\nlocal notation `𝔠` := (bv_powerset ℵ₀ : bSet 𝔹)\n\nlemma uncountable_fiber_of_regular' (κ₁ κ₂ : cardinal) (H_inf : cardinal.omega ≤ κ₁) (H_lt : κ₁ < κ₂) (H : cof (ord κ₂) = κ₂) (α : Type u) (H_α : #α = κ₁) (β : Type u) (H_β : #β = κ₂) (g : β → α)\n  : ∃ (ξ : α), cardinal.omega < #↥(g⁻¹' {ξ}) :=\nbegin\n  have := (@cardinal.exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k H_k, subst H_k,\n  have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k',\n  have := infinite_pigeonhole g _ _, cases this with ξ H_ξ, use ξ, rw[H_ξ],\n  all_goals{simp*}, from lt_of_le_of_lt ‹_› ‹_›\nend\n\nlemma uncountable_fiber_of_regular (κ₁ κ₂ : cardinal) (H_inf : cardinal.omega ≤ κ₁) (H_lt : κ₁ < κ₂) (H : cof (ord κ₂) = κ₂) (g : type (pSet.ordinal.mk (ord κ₂)  : pSet.{u}) → type (pSet.ordinal.mk (ord κ₁) : pSet.{u}))\n  : ∃ (ξ : type (pSet.ordinal.mk (ord κ₁))), cardinal.omega < #↥((λ (β : type (pSet.ordinal.mk (ord κ₂))), g β)⁻¹' {ξ}) :=\nbegin\n  have := (@exists_aleph κ₁).mp ‹_›, cases this with k₁ h, subst h,\n  have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n  subst h,\n  from uncountable_fiber_of_regular' (aleph k₁) (aleph k₂) ‹_› ‹_› ‹_› _ (by simp) _ (by simp) g\nend\n\nlemma cardinal_inequality_of_regular (κ₁ κ₂ : cardinal) (H_reg₁ : cardinal.is_regular κ₁) (H_reg₂ : cardinal.is_regular κ₂) (H_inf : (omega : cardinal) ≤ κ₁) (H_lt : κ₁ < κ₂) {Γ : 𝔹} : Γ ≤ (card_ex κ₁)̌  ≺ (card_ex κ₂)̌  :=\nbegin\n  dsimp only, rw ←imp_bot, bv_imp_intro H_larger_than,\n  by_contra H_nonzero, rw ←bot_lt_iff_not_le_bot at H_nonzero,\n  rcases AE_of_check_larger_than_check H_nonzero ‹_› (exists_mem_of_regular ‹_›) with ⟨f,Hf⟩,\n  rcases classical.axiom_of_choice Hf with ⟨g, g_spec⟩,\n    suffices : ¬ CCC 𝔹, from absurd 𝔹_CCC this,\n    apply not_CCC_of_uncountable_fiber; try{assumption},\n    {have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k', simp*},\n    {have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k', simp*,\n     have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n     subst h, simp*},\n    {intros i₁ i₂ H_neq, from ordinal.mk_inj _ _ _ ‹_›},\n    {dsimp at g,\n     apply uncountable_fiber_of_regular' κ₁ κ₂; try{simp*},\n     from H_reg₂.right,\n     have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n     subst h; apply mk_type_mk_eq, from ‹_›, apply mk_type_mk_eq,\n     from le_of_lt (lt_of_le_of_lt ‹_› ‹_›)}\nend\n\nlemma ℵ₀_lt_ℵ₁ : (⊤ : 𝔹)  ≤ ℵ₀ ≺ ℵ₁̌  :=\nbegin\n  dsimp only, rw ←imp_bot, bv_imp_intro H_larger_than,\n  by_contra H_nonzero, rw ←bot_lt_iff_not_le_bot at H_nonzero,\n  rcases AE_of_check_larger_than_check ‹_› ‹_› _ with ⟨f,Hf⟩,\n  rcases (classical.axiom_of_choice Hf) with ⟨g,g_spec⟩,\n  suffices : ¬ CCC 𝔹, from absurd 𝔹_CCC this,\n  apply not_CCC_of_uncountable_fiber; try{assumption},\n    {from le_of_eq (by simp)},\n    {simp},\n    {intros i₁ i₂ H_neq, from ordinal.mk_inj _ _ _ ‹_›},\n    {dsimp at g,\n     apply uncountable_fiber_of_regular' (aleph 0) (aleph 1); try{simp*},\n     from is_regular_aleph_one.right},\n  from exists_mem_of_regular is_regular_aleph_one\nend\n\n\nlemma ℵ₁_lt_ℵ₂ : (⊤ : 𝔹) ≤ ℵ₁̌  ≺ ℵ₂̌  :=\ncardinal_inequality_of_regular _ _ (is_regular_aleph_one)\n  (is_regular_aleph_two) (by simp) (by simp)\n\nlemma ℵ₁_lt_ℵ₂' {Γ : 𝔹} : Γ ≤ ℵ₁̌  ≺ ℵ₂̌  := le_trans (le_top) ℵ₁_lt_ℵ₂\n\nlemma cohen_real.mk_ext : ∀ (i j : type (ℵ₂̌  : bSet 𝔹)), func (ℵ₂̌ ) i =ᴮ func (ℵ₂̌ ) j ≤\n  (λ (x : type (ℵ₂̌ )), cohen_real.mk x) i =ᴮ (λ (x : type (ℵ₂̌ )), cohen_real.mk x) j :=\nbegin\n  intros i j, by_cases i = j,\n   {simp[h]},\n   {refine poset_yoneda _, intros Γ a, simp only [le_inf_iff] at *,\n     have : func (ℵ₂̌ ) i = (ℵ₂.func (check_cast i))̌ ,\n       by simp[check_func],\n     rw[this] at a,\n     have : func (ℵ₂̌ ) j = (ℵ₂.func (check_cast j))̌ ,\n       by simp[check_func],\n     rw[this] at a,\n   suffices : (ℵ₂.func (check_cast i))̌   =ᴮ (ℵ₂.func (check_cast j))̌  ≤ ⊥,\n     from le_trans a (le_trans this bot_le),\n   rw[le_bot_iff], apply check_bv_eq_bot_of_not_equiv,\n   apply ordinal.mk_inj, unfold check_cast, intro H, cc}\nend\n\nnoncomputable def neg_CH_func : bSet 𝔹 :=\n@function.mk _ _ (ℵ₂̌ ) (λ x, cohen_real.mk x) cohen_real.mk_ext\n\ntheorem ℵ₂_le_𝔠 : ⊤ ≤ is_func' (ℵ₂̌ ) 𝔠 (neg_CH_func) ⊓ bSet.is_inj (neg_CH_func) :=\nbegin\nrefine le_inf _ _,\n\n  {unfold neg_CH_func, refine le_inf _ _, refine mk_is_func _ _,\n    bv_intro w₁, bv_imp_intro, rw[bSet.mem_unfold] at H,\n    bv_cases_at'' H ν, apply bv_use (cohen_real.mk ν),\n    refine le_inf cohen_real.definite' _, swap,\n    rw[bSet.mem_unfold], apply bv_use ν, bv_split,\n    from le_inf ‹_› (by apply le_trans H_1_right; from subst_congr_pair_left)},\n\n  {refine mk_inj_of_inj _ _, from λ _ _ _, cohen_real.inj ‹_›},\nend\n\nlemma ℵ₁_Ord {Γ : 𝔹} : Γ ≤ Ord (ℵ₁̌ ) := by simp\n\nlemma ℵ₂_Ord {Γ : 𝔹} : Γ ≤ Ord (ℵ₂̌ ) := by simp\n\ntheorem neg_CH : (⊤ : 𝔹) ≤ -CH :=\nbegin\n  dsimp [CH], rw[lattice.neg_neg],\n  apply bv_use (ℵ₁̌ ),\n  refine le_inf (ℵ₁_Ord) _,\n  apply bv_use (ℵ₂̌ ),\n  refine le_inf (le_inf _ _) _,\n  { from ℵ₀_lt_ℵ₁ },\n  { from ℵ₁_lt_ℵ₂ },\n  { apply bv_use neg_CH_func, from ℵ₂_le_𝔠 }\nend\n\ntheorem neg_CH₂ : (⊤ : 𝔹) ≤ -CH₂ :=\n(bv_iff.neg $ @CH_iff_CH₂ _ _).mp neg_CH\n\nend neg_CH\n", "meta": {"author": "flypitch", "repo": "flypitch", "sha": "aea5800db1f4cce53fc4a113711454b27388ecf8", "save_path": "github-repos/lean/flypitch-flypitch", "path": "github-repos/lean/flypitch-flypitch/flypitch-aea5800db1f4cce53fc4a113711454b27388ecf8/src/forcing.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3441647298954345}}
{"text": "universe u v\n\ntheorem eqLitOfSize0 {α : Type u} (a : Array α) (hsz : a.size = 0) : a = #[] :=\na.toArrayLitEq 0 hsz\n\ntheorem eqLitOfSize1 {α : Type u} (a : Array α) (hsz : a.size = 1) : a = #[a.getLit 0 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 1 hsz\n\ntheorem eqLitOfSize2 {α : Type u} (a : Array α) (hsz : a.size = 2) : a = #[a.getLit 0 hsz (ofDecideEqTrue rfl), a.getLit 1 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 2 hsz\n\ntheorem eqLitOfSize3 {α : Type u} (a : Array α) (hsz : a.size = 3) :\n  a = #[a.getLit 0 hsz (ofDecideEqTrue rfl), a.getLit 1 hsz (ofDecideEqTrue rfl), a.getLit 2 hsz (ofDecideEqTrue rfl)] :=\na.toArrayLitEq 3 hsz\n\n/-\nMatcher for the following patterns\n```\n| #[]           => _\n| #[a₁]         => _\n| #[a₁, a₂, a₃] => _\n| a             => _\n``` -/\ndef matchArrayLit {α : Type u} (C : Array α → Sort v) (a : Array α)\n    (h₁ : Unit →      C #[])\n    (h₂ : ∀ a₁,       C #[a₁])\n    (h₃ : ∀ a₁ a₂ a₃, C #[a₁, a₂, a₃])\n    (h₄ : ∀ a,        C a)\n    : C a :=\nif h : a.size = 0 then\n  @Eq.rec _ _ (fun x _ => C x) (h₁ ()) _ (a.toArrayLitEq 0 h).symm\nelse if h : a.size = 1 then\n  @Eq.rec _ _ (fun x _ => C x) (h₂ (a.getLit 0 h (ofDecideEqTrue rfl))) _ (a.toArrayLitEq 1 h).symm\nelse if h : a.size = 3 then\n  @Eq.rec _ _ (fun x _ => C x) (h₃ (a.getLit 0 h (ofDecideEqTrue rfl)) (a.getLit 1 h (ofDecideEqTrue rfl)) (a.getLit 2 h (ofDecideEqTrue rfl))) _ (a.toArrayLitEq 3 h).symm\nelse\n  h₄ a\n\n/- Equational lemmas that should be generated automatically. -/\ntheorem matchArrayLit.eq1 {α : Type u} (C : Array α → Sort v)\n    (h₁ : Unit →      C #[])\n    (h₂ : ∀ a₁,       C #[a₁])\n    (h₃ : ∀ a₁ a₂ a₃, C #[a₁, a₂, a₃])\n    (h₄ : ∀ a,        C a)\n    : matchArrayLit C #[] h₁ h₂ h₃ h₄ = h₁ () :=\nrfl\n\ntheorem matchArrayLit.eq2 {α : Type u} (C : Array α → Sort v)\n    (h₁ : Unit →      C #[])\n    (h₂ : ∀ a₁,       C #[a₁])\n    (h₃ : ∀ a₁ a₂ a₃, C #[a₁, a₂, a₃])\n    (h₄ : ∀ a,        C a)\n    (a₁ : α)\n    : matchArrayLit C #[a₁] h₁ h₂ h₃ h₄ = h₂ a₁ :=\nrfl\n\ntheorem matchArrayLit.eq3 {α : Type u} (C : Array α → Sort v)\n    (h₁ : Unit →      C #[])\n    (h₂ : ∀ a₁,       C #[a₁])\n    (h₃ : ∀ a₁ a₂ a₃, C #[a₁, a₂, a₃])\n    (h₄ : ∀ a,        C a)\n    (a₁ a₂ a₃ : α)\n    : matchArrayLit C #[a₁, a₂, a₃] h₁ h₂ h₃ h₄ = h₃ a₁ a₂ a₃ :=\nrfl\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/tests/lean/run/matchArrayLit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3441521567221512}}
{"text": "/-\nCopyright (c) 2020 Gabriel Ebner, Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner, Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.ext\nimport Mathlib.Lean3Lib.data.stream\nimport Mathlib.data.list.basic\nimport Mathlib.data.list.range\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Additional instances and attributes for streams\n-/\n\nprotected instance stream.inhabited {α : Type u_1} [Inhabited α] : Inhabited (stream α) :=\n  { default := stream.const Inhabited.default }\n\nnamespace stream\n\n\n/-- `take s n` returns a list of the `n` first elements of stream `s` -/\ndef take {α : Type u_1} (s : stream α) (n : ℕ) : List α := list.map s (list.range n)\n\ntheorem length_take {α : Type u_1} (s : stream α) (n : ℕ) : list.length (take s n) = n := sorry\n\n/-- Use a state monad to generate a stream through corecursion -/\ndef corec_state {σ : Type u_1} {α : Type u_1} (cmd : state σ α) (s : σ) : stream α :=\n  corec prod.fst (state_t.run cmd ∘ prod.snd) (state_t.run cmd s)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/stream/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3441521485975783}}
{"text": "/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Robert Y. Lewis\n-/\n\nimport tactic.core\nimport mathematica\nimport data.complex.exponential\nimport data.real.pi\nimport tactic.interactive_expr\n\n/-!\n\nThis demo shows how we can mimic a Mathematica notebook from within Lean.\n\nThe syntax here is somewhat hackish; better syntax will be possible in Lean 4.\n\nA section of Mathematica code is entered between `begin_mm_block` and `end_mm_block`.\nMathematica commands are entered in quotes `\"\"`, with each line terminated by a semicolon `;`\ncorresponding to a `shift-enter` in the standard notebook frontend.\nLean expressions can be inserted as antiquotes between quoted Mathematica expressions.\nFor parsing reasons, compound Lean expressions must appear in parentheses `()`.\n\nThe evaluation of these commands will happen sequentially in the same Mathematica environment.\n\nBy default, the output of a line will be translated to a Lean expression and traced.\nTo display the output as an image instead,\nprefix the line with `as image`.\n\nSince Mathematica has no access to Lean's definitional reduction, it is sometimes necessary\nto unfold definitions before sending them to Mathematica.\nYou can begin the block with `begin_mm_block (unfolding f g)` to unfold `f` and `g`\nin all antiquoted Lean expressions.\n\n-/\n\nreserve notation `begin_mm_block`\nreserve notation `end_mm_block`\nreserve notation `as`\nreserve notation `image`\nreserve notation `unfolding`\n\n@[sym_to_expr]\nmeta def pi_to_expr : mathematica.sym_trans_expr_rule :=\n⟨\"Pi\", `(real.pi)⟩\n\n@[sym_to_expr]\nmeta def null_to_expr : mathematica.sym_trans_expr_rule :=\n⟨\"Null\", `(())⟩\n\n/-!\nThis section develops the widgets for displaying results from Mathematica as images.\n-/\n\nsection\nopen widget\n\nmeta def component.stateless' {π α} (view : π → list (html α)) : component π α :=\ncomponent.with_should_update (λ _ _, tt) $ component.pure view\n\nmeta def url_component (src : string) : component tactic_state empty :=\ncomponent.stateless' $ λ ts,\n  [h \"h1\" [] [\"Mathematica output\"],\n    h \"img\" [attr.val \"src\" src] []\n  ]\n\nmeta def expr_widget (src : expr) : tactic (component tactic_state empty) :=\ndo s ← tactic.pp src,\n  return $ component.stateless' $ λ ts,\n    [h \"h1\" [] [\"Mathematica output\"],\n    h \"p\" [] [s]\n  ]\n\nend\n\n/-!\nThis section develops the parser for Mathematica blocks.\n-/\n\nsection\nsetup_tactic_parser\nopen tactic\n\nmeta inductive command_comp\n| cmd : string → command_comp\n| antiquot : expr → command_comp\n\nmeta def command_comp.to_string : command_comp → string\n| (command_comp.cmd s) := \"command: \" ++ s\n| (command_comp.antiquot s) := \"antiquot: \" ++ to_string s\n\nmeta instance : has_repr command_comp :=\n⟨command_comp.to_string⟩\n\nmeta def parse_string_component : lean.parser string :=\ndo s ← parser.pexpr 10000,\n   to_expr s >>= ↑(eval_expr string)\n\nmeta def parse_antiquote : lean.parser pexpr :=\nparser.pexpr 10000\n\nmeta def parse_component : lean.parser command_comp :=\ndo pe ← parser.pexpr 10000,\n   e ← to_expr pe,\n   tpe ← infer_type e,\n   if tpe = `(string) then do  s ← eval_expr' string e, return $ command_comp.cmd s\n   else return $ command_comp.antiquot e\n\nmeta def parse_cmd_list_aux : lean.parser $ list command_comp :=\ntk \";\" >> return  [] <|>\ndo c ← parse_component, cs ← parse_cmd_list_aux, return (c::cs)\n\nmeta def parse_cmd_list : lean.parser $ bool × pos × list command_comp :=\ndo pos ← cur_pos,\n   is_img ← option.is_some <$> (tk \"as\" >> tk \"image\")?,\n   prod.mk is_img <$> prod.mk pos <$> parse_cmd_list_aux\n\n/-!\nThis section translates, evaluates, and displays the result of a parsed Mathematica command.\n-/\n\nmeta def command_comp.translate (to_unfold : list name) : command_comp → tactic string\n| (command_comp.cmd s) := return s\n| (command_comp.antiquot p) :=\n  do s ← mathematica.form_of_expr <$> dunfold to_unfold p {fail_if_unchanged := ff},\n     return $ \"Activate[LeanForm[\" ++ s ++ \"]]\"\n\nmeta def execute_list (to_unfold : list name) (is_img : bool) (l : list command_comp) : tactic pexpr :=\ndo l ← l.mmap (command_comp.translate to_unfold), --tactic.trace $ string.join l,\n   let cmd := if is_img then \"MakeDataUrlFromImage[\" ++ string.join l ++ \"]\" else string.join l,\n   s ← mathematica.execute_global cmd >>= parse_mmexpr_tac,\n   mathematica.pexpr_of_mmexpr mathematica.trans_env.empty s\n\nmeta def string_of_pos_comp (to_unfold : list name) :\n  bool × pos × list command_comp → tactic (pos × (string ⊕ expr))\n| ⟨is_img, p, c⟩ :=\ndo e ← execute_list to_unfold is_img c >>= to_expr,\n   prod.mk p <$> if is_img then  sum.inl <$> eval_expr string e else return (sum.inr e)\n\nmeta def make_widget (to_unfold : list name) (p : bool × pos × list command_comp) :\n  tactic $ pos × (widget.component tactic_state empty) :=\ndo (loc, data) ← string_of_pos_comp to_unfold p,\nmatch data with\n| sum.inl s := return $ ⟨loc, url_component s⟩\n| sum.inr e := prod.mk loc <$> expr_widget e\nend\n\n@[user_command] meta def parse_mm_block (_ : parse (tk \"begin_mm_block\")) : lean.parser unit :=\ndo to_unfold ← (tk \"(\" >> tk \"unfolding\" >> ident* <* tk \")\")?,\n   l ← parse_cmd_list*,\n   tk \"end_mm_block\",\n   l ← l.mmap (λ e, make_widget (to_unfold.get_or_else []) e),\n   l.mmap' $ λ ⟨⟨ln, c⟩, w⟩, save_widget ⟨ln, c - (\"begin_mm_block\".length - 1)⟩  w\nend\n\n\n/-!\nIn this example we show a Mathematica block with three image plots.\nPut the cursor on the first characters of the `as image` lines to see the output.\n\nIn the second line, we plot a Lean function given as a lambda expression.\nIn the third, we plot a Lean definition, that we have marked to unfold at the beginning.\n-/\n\nopen real\nnoncomputable def f : ℝ → ℝ := λ x, sin x + cos x\n\nbegin_mm_block (unfolding f)\n\nas image\n\"Plot3D[x^2-y, {x,-3,3}, {y,-3,3}]\";\n\nas image\n\"Plot[\"(λ y, (sin y)^2 - y^2)\"[x], {x,-10,10}]\";\n\nas image\n\"Plot[\"f\"[y], {y,-2,2}]\";\n\nend_mm_block\n\n/-!\nThis example gets output from Mathematica as expressions instead of images.\n\nFirst, we define a symbol `MyPoly` as a Lean expression and factor it.\n\nThen we directly factor a Lean expression.\n\nUncommenting the `pp` line above shows that we really are seeing full Lean expressions in the output.\n-/\n\n-- set_option pp.all true\n\nconstants y z : ℝ\n\nbegin_mm_block \n\n\"MyPoly =\"(z^2-2*z+1);\n\"Factor[MyPoly]\";\n\n\"Factor[\"(y^10-z^10)\"]\";\n\nend_mm_block", "meta": {"author": "robertylewis", "repo": "mathematica_examples", "sha": "e317381c49db032accef2a92e7650d029952ad76", "save_path": "github-repos/lean/robertylewis-mathematica_examples", "path": "github-repos/lean/robertylewis-mathematica_examples/mathematica_examples-e317381c49db032accef2a92e7650d029952ad76/src/repl_example.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038302, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3441521478985098}}
{"text": "import category_theory.category\nimport category_theory.colimit_lemmas\nimport .definitions\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\nlocal notation t ` @> `:90 X:90 := t.app X\n\nuniverses v u\n\nnamespace homotopy_theory.cylinder\n\nvariables {C : Type u} [category.{v} C] [has_cylinder C]\n\n-- Homotopy with respect to a given cylinder functor.\nstructure homotopy {x y : C} (f₀ f₁ : x ⟶ y) :=\n(H : I.obj x ⟶ y)\n(Hi₀ : H ∘ i 0 @> x = f₀)\n(Hi₁ : H ∘ i 1 @> x = f₁)\n\n-- The constant homotopy on a map.\ndef homotopy.refl {x y : C} (f : x ⟶ y) : homotopy f f :=\nby refine { H := f ∘ p @> x, Hi₀ := _, Hi₁ := _ };\n   rw [←assoc]; dsimp; simp\n\n-- The image of a homotopy under a map.\ndef homotopy.congr_left {x y y' : C} (g : y ⟶ y') {f₀ f₁ : x ⟶ y} (H : homotopy f₀ f₁) :\n  homotopy (g ∘ f₀) (g ∘ f₁) :=\n{ H := g ∘ H.H,\n  Hi₀ := by rw [←assoc, H.Hi₀],\n  Hi₁ := by rw [←assoc, H.Hi₁] }\n\n-- The precomposition of a homotopy by a map.\ndef homotopy.congr_right {x' x y : C} (g : x' ⟶ x) {f₀ f₁ : x ⟶ y} (H : homotopy f₀ f₁) :\n  homotopy (f₀ ∘ g) (f₁ ∘ g) :=\n{ H := H.H ∘ I &> g,\n  Hi₀ := by erw [←assoc, ←(i _).naturality]; simp; erw H.Hi₀,\n  Hi₁ := by erw [←assoc, ←(i _).naturality]; simp; erw H.Hi₁ }\n\n-- Annoying equality stuff.\n-- If we rewrite the starting point of the homotopy by an equality, it doesn't change H.\nlemma homotopy.eq_rec_on_left {x y : C} {f₀ f₀' f₁ : x ⟶ y} (H : homotopy f₀ f₁)\n  (e : f₀ = f₀') : (eq.rec_on e H : homotopy f₀' f₁).H = H.H :=\nby cases e; refl\n\nsection rel\nvariables {a x y : C} (j : a ⟶ x) {f₀ f₁ : x ⟶ y}\n\n-- The property of a homotopy leaving fixed a subspace, or more\n-- generally the \"image\" of any map j : A → X. In order for the\n-- homotopy to be rel u, we must first have f₀ ∘ j = f₁ ∘ j. This\n-- condition is not encoded in the type.\ndef homotopy.is_rel (H : homotopy f₀ f₁) : Prop :=\nH.H ∘ I &> j = f₀ ∘ j ∘ p @> a\n\nvariables {j}\nlemma agree_of_is_rel {H : homotopy f₀ f₁} (h : H.is_rel j) : f₀ ∘ j = f₁ ∘ j :=\ncalc\n  f₀ ∘ j\n    = (f₀ ∘ j) ∘ (p @> a ∘ i 1 @> a) : by simp\n... = f₀ ∘ j ∘ p @> a ∘ i 1 @> a     : by rw assoc\n... = H.H ∘ I &> j ∘ i 1 @> a        : by unfold homotopy.is_rel at h; simp [h]\n... = H.H ∘ (I &> j ∘ i 1 @> a)      : by simp\n... = H.H ∘ (i 1 @> x ∘ j)           : by erw ←(i 1).naturality; refl\n... = f₁ ∘ j                         : by simp; erw H.Hi₁\n\nlemma homotopy.refl_is_rel {f : x ⟶ y} : (homotopy.refl f).is_rel j :=\nshow f ∘ p @> x ∘ I &> j = f ∘ j ∘ p @> a,\nby erw [←assoc, ←assoc, p.naturality]; refl\n\nlemma homotopy.congr_left_is_rel {f₀ f₁ : x ⟶ y} {H : homotopy f₀ f₁}\n  {z} (g : y ⟶ z) (h : H.is_rel j) : (H.congr_left g).is_rel j :=\nbegin\n  unfold homotopy.is_rel at ⊢ h, dsimp [homotopy.congr_left] { iota := tt },\n  rw [←assoc, h], simp\nend\n\nlemma homotopy.congr_right_is_rel {f₀ f₁ : x ⟶ y} {H : homotopy f₀ f₁}\n  {x'} {j' : a ⟶ x'} (g : x' ⟶ x) (h : H.is_rel (g ∘ j')) : (H.congr_right g).is_rel j' :=\nbegin\n  unfold homotopy.is_rel at ⊢ h, dsimp [homotopy.congr_right] { iota := tt },\n  rw [←assoc, ←I.map_comp, h], simp\nend\n\n-- In practice, `a` is initial and `I` preserves initial objects.\nlemma homotopy.is_rel_initial (Iai : Is_initial_object.{v} (I.obj a))\n  (H : homotopy f₀ f₁) : H.is_rel j :=\nIai.uniqueness _ _\n\nend rel\n\nsection dir\n-- A technical contrivance to let us abstract over the direction of a\n-- homotopy.\ndef homotopy_dir (ε : endpoint) {x y : C} (fε fεv : x ⟶ y) : Type v :=\nmatch ε with\n| 0 := homotopy fε fεv\n| 1 := homotopy fεv fε\nend\n\ndef homotopy_dir.H {ε} {x y : C} {fε fεv : x ⟶ y} (H : homotopy_dir ε fε fεv) :\n  I.obj x ⟶ y :=\nmatch ε, H with\n| 0, H := homotopy.H H\n| 1, H := homotopy.H H\nend\n\nlemma homotopy_dir.Hiε {ε} {x y : C} {fε fεv : x ⟶ y} (H : homotopy_dir ε fε fεv) :\n  H.H ∘ i ε @> x = fε :=\nmatch ε, H with\n| 0, H := homotopy.Hi₀ H\n| 1, H := homotopy.Hi₁ H\nend\n\nlemma homotopy_dir.Hiεv {ε} {x y : C} {fε fεv : x ⟶ y} (H : homotopy_dir ε fε fεv) :\n  H.H ∘ i ε.v @> x = fεv :=\nmatch ε, H with\n| 0, H := homotopy.Hi₁ H\n| 1, H := homotopy.Hi₀ H\nend\n\ndef homotopy_dir.mk (ε : endpoint) {x y : C} {fε fεv : x ⟶ y}\n  (H : I.obj x ⟶ y) (Hiε : H ∘ i ε @> x = fε) (Hiεv : H ∘ i ε.v @> x = fεv) :\n  homotopy_dir ε fε fεv :=\nmatch ε, H, Hiε, Hiεv with\n| 0, H, Hiε, Hiεv := { H := H, Hi₀ := Hiε, Hi₁ := Hiεv }\n| 1, H, Hiε, Hiεv := { H := H, Hi₀ := Hiεv, Hi₁ := Hiε }\nend\n\nend dir\n\n-- The homotopy relation with respect to the given cylinder functor.\ndef homotopic {x y : C} (f₀ f₁ : x ⟶ y) : Prop := nonempty (homotopy f₀ f₁)\n\nnotation f₀ ` ≃ `:50 f₁:50 := homotopic f₀ f₁\n\n@[refl] lemma homotopic.refl {x y : C} (f : x ⟶ y) : f ≃ f :=\n⟨homotopy.refl f⟩\n\nlemma homotopic.congr_left {x y y' : C} (g : y ⟶ y') {f₀ f₁ : x ⟶ y} (h : f₀ ≃ f₁) :\n  g ∘ f₀ ≃ g ∘ f₁ :=\nlet ⟨H⟩ := h in ⟨H.congr_left g⟩\n\nlemma homotopic.congr_right {x' x y : C} (g : x' ⟶ x) {f₀ f₁ : x ⟶ y} (h : f₀ ≃ f₁) :\n  f₀ ∘ g ≃ f₁ ∘ g :=\nlet ⟨H⟩ := h in ⟨H.congr_right g⟩\n\n-- The relation of being homotopic rel a fixed map j : A → X.\ndef homotopic_rel {a x y : C} (j : a ⟶ x) (f₀ f₁ : x ⟶ y) : Prop :=\n∃ H : homotopy f₀ f₁, H.is_rel j\n\nnotation f₀ ` ≃ `:50 f₁:50 ` rel `:50 j:50 := homotopic_rel j f₀ f₁\n\n@[refl] lemma homotopic_rel.refl {a x y : C} {j : a ⟶ x} (f : x ⟶ y) : f ≃ f rel j :=\n⟨homotopy.refl f, homotopy.refl_is_rel⟩\n\nlemma homotopic_rel.congr_left {a x y y' : C} {j : a ⟶ x} (g : y ⟶ y') {f₀ f₁ : x ⟶ y} :\n  f₀ ≃ f₁ rel j → g ∘ f₀ ≃ g ∘ f₁ rel j :=\nassume ⟨H, h⟩, ⟨H.congr_left g, homotopy.congr_left_is_rel g h⟩\n\nlemma homotopic_rel.congr_right {a x' x y : C} {j' : a ⟶ x'} (g : x' ⟶ x) {f₀ f₁ : x ⟶ y} :\n  f₀ ≃ f₁ rel (g ∘ j') → f₀ ∘ g ≃ f₁ ∘ g rel j' :=\nassume ⟨H, h⟩, ⟨H.congr_right g, homotopy.congr_right_is_rel g h⟩\n\nlemma homotopic_rel.forget_rel {a x y : C} {j : a ⟶ x} {f₀ f₁ : x ⟶ y} : f₀ ≃ f₁ rel j → f₀ ≃ f₁ :=\nassume ⟨H, h⟩, ⟨H⟩\n\nlemma homotopic_rel_initial {a x y : C} (Iai : Is_initial_object.{v} (I.obj a))\n  (j : a ⟶ x) (f₀ f₁ : x ⟶ y) : (f₀ ≃ f₁ rel j) = (f₀ ≃ f₁) :=\npropext $ iff.intro\n  (assume ⟨H, _⟩, ⟨H⟩)\n  (assume ⟨H⟩, ⟨H, H.is_rel_initial Iai⟩)\n\nend homotopy_theory.cylinder\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/cylinder/homotopy.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.3441164890429715}}
{"text": "import Mathlib.Tactic.Basic\nimport Mathlib.Tactic.Cases\nimport Mathlib.Init.Data.Nat.Basic\n\n/-!\n## Structures\n\nLean provides a convenient syntax for defining records, or _structures_ as they are\nalso called. These are essentially nonrecursive, single-constructor inductive types,\nbut with some syntactic sugar.\n\nThe definition below introduces a structure called `rgb` with three fields of type\nNat called `red`, `green`, and `blue`:\n-/\nstructure rgb where\n  (red green blue : ℕ)\n  deriving Repr\n/-!\nThe additional `deriving Repr` clause tells Lean to automatically generate a\n`repr` function for this structure so that `#eval` can print the structure.\n\nThis definition has roughly the same effect as the following commands:\n-/\ninductive rgb₁ : Type\n| mk : ℕ → ℕ → ℕ → rgb₁\n\ndef rgb₁.red : rgb₁ → ℕ\n| (rgb₁.mk r _ _) => r\n\ndef rgb₁.green : rgb₁ → ℕ\n| (rgb₁.mk _ g _) => g\n\ndef rgb₁.blue : rgb₁ → ℕ\n| (rgb₁.mk _ _ b) => b\n\n/-!\nWe can define a new structure as the extension of an existing structure. The\ndefinition below extends `rgb` with a fourth field, called `alpha`:\n-/\nstructure rgba extends rgb where\n  (alpha : ℕ)\n  deriving Repr\n/-!\nThe general syntax to define structures is\n```lean\nstructure structure-name (params₁ : type₁) . . . (paramsₖ : typeₖ )\n  [extends structure₁, . . ., structureₘ] where\n  [constructor ::]\n  <fields>\n```\nWhere each fields is either\n```\n(field-names₁ : field-type₁) ... (field-namesₙ : field-typeₙ)\n```\nOr one field per line where you can drop the parens:\n```\n  field-names₁ : field-type₁\n  ...\n  field-namesₙ : field-typeₙ\n```\nSo `rgb` can also be defined like this:\n-/\nstructure rgb₂ where\n  red : ℕ\n  green : ℕ\n  blue : ℕ\n/-!\nThe parameters `params₁, . . . , paramsₖ` are effectively additional fields, but unlike\nthe fields `field-names₁, . . . , field-namesₙ`, they are also stored in the type, as\narguments to the type constructor (structure-name).\n\nValues can be created using a variety of syntaxes:\n-/\ndef black : rgb := rgb.mk 0 0 0\n/-!\nNotice the structure has a default constructor named 'mk'.  You can rename this by\nproviding a name before the fields followed by `::` then the fields, for example:\n-/\nstructure rgb₃ where\n  foo :: (red green blue : ℕ)\n  deriving Repr\n\n#eval rgb₃.foo 1 2 3 -- { red := 1, green := 2, blue := 3 }\n/-!\n\nThe special brackets `⟨ ... ⟩` is general notation that means call the default constructor,\nand in this case there is only one contructor, so it knows what to do:\n-/\ndef red : rgb := ⟨255, 0, 0⟩\ndef red₃ : rgb₃ := ⟨255, 0, 0⟩\n/-!\nYou can also construct object using an object syntax where you name the fields:\n-/\ndef green  : rgb := { red := 0, green := 255, blue := 0 }\n\ndef blue  : rgb := { black with blue := 255 }\n\ndef yellow := { red := 255, green := 255, blue := 0 : rgb }\n\ndef semitransparent_red : rgba := { red with alpha := 0x7f }\n/-!\n\nNotice that the `with` clause allows you to copy the values from another object and override\njust a subset of fields that you want to change and this works with base types\nas shown in `semitransparent_red`.\n\nValues can be also extracted using dot notation:\n-/\n#eval yellow.green   -- 255\n/-!\nNext, we define an operation called shuffle:\n-/\ndef shuffle (c : rgb) := {\n  c with\n    red := c.green,\n    green := c.blue,\n    blue := c.red\n}\n\n#eval shuffle yellow  -- { red := 255, green := 0, blue := 255 }\n/-!\nApplying shuffle three times in a row is the same as not applying it at all:\n-/\nlemma shuffle_shuffle_shuffle (c : rgb) :\n  shuffle (shuffle (shuffle c)) = c := by\n    cases c\n    rfl\n\n/-!\nThe proof relies on the `cases` tactic, a close relative of `induction`. It performs a\ncase distinction on its argument, but it does not generate induction hypotheses.\nFor `rgb`, the invocation `cases c` transforms a goal `⊢ P[c]` into a subgoal\n`r g b : ℕ ⊢ P[rgb.mk r g b]`. We could also have used `induction c`.\n\nIn a structured proof, we can use `match` expressions to perform a case distinction.\nFor example:\n-/\nlemma shuffle_shuffle_shuffle₂ (c : rgb) :\n  shuffle (shuffle (shuffle c)) = c :=\n  match c with\n  | rgb.mk _ _ _ => Eq.refl _\n\n/-!\nRecall from [Section 2.4](../BackwardProofs/Equality.lean.md) that `Eq.refl` is the property\n`∀a, a = a`.\n-/\n", "meta": {"author": "lovettchris", "repo": "hglv", "sha": "339f0b10f4b4a2b1e53e2b29532003a5ea4d9c9b", "save_path": "github-repos/lean/lovettchris-hglv", "path": "github-repos/lean/lovettchris-hglv/hglv-339f0b10f4b4a2b1e53e2b29532003a5ea4d9c9b/FunctionalProgramming/Structures.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.34404329769868436}}
{"text": "import data.fintype.basic\nimport category_theory.Fintype\n\nopen_locale classical\n\nuniverse u\n\n-- This is an awkward hybrid: we should state the induction principle for unbundled fintype\n-- and then if it is necessary provide an interface for using it with `Fintype`.\n\nlemma Fintype.induction_empty_sum {P : Fintype.{u} → Prop}\n  (of_equiv : ∀ {α β : Fintype.{u}}, α ≃ β → P α → P β)\n  (h_empty : P (Fintype.of pempty))\n  (h_option : ∀ {α : Fintype.{u}}, P α → P (Fintype.of (α ⊕ (punit : Type u))))\n  (α : Fintype.{u}) : P α :=\nbegin\n  let Q : Π (α : Type u) [fintype α], Prop := λ a h, P ⟨a,h⟩,\n  have H := @fintype.induction_empty_option' Q _ _ _ α α.str,\n  { cases α, exact H },\n  { introsI α β _ e h,\n    haveI : fintype α := fintype.of_equiv _ e.symm,\n    let e' : Fintype.of α ≃ Fintype.of β := e,\n    apply of_equiv e',\n    convert h },\n  { assumption },\n  { introsI α _ h,\n    let e : option α ≃ α ⊕ (punit : Type u) := equiv.option_equiv_sum_punit _,\n    let e' : Fintype.of (option α) ≃ Fintype.of (α ⊕ punit) := e,\n    apply of_equiv e'.symm,\n    exact h_option h }\nend\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/fintype_induction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.3440432943390065}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen\n-/\nimport ring_theory.localization.away\nimport ring_theory.localization.fraction_ring\nimport ring_theory.localization.integer\nimport ring_theory.unique_factorization_domain\n\n/-!\n# Numerator and denominator in a localization\n\n## Implementation notes\n\n-- See `src/ring_theory/localization/basic.lean` for a design overview.\n\n-- ## Tags\n-- localization, ring localization, commutative ring localization, characteristic predicate,\n-- commutative ring, field of fractions\n-- -/\n-- variables {R : Type*} [comm_ring R] (M : submonoid R) {S : Type*} [comm_ring S]\n-- variables [algebra R S] {P : Type*} [comm_ring P]\n\n-- namespace is_fraction_ring\n\n-- open is_localization\n\n-- section num_denom\n\n-- variables (A : Type*) [comm_ring A] [is_domain A] [unique_factorization_monoid A]\n-- variables {K : Type*} [field K] [algebra A K] [is_fraction_ring A K]\n\n-- lemma exists_reduced_fraction (x : K) :\n--   ∃ (a : A) (b : non_zero_divisors A),\n--   (∀ {d}, d ∣ a → d ∣ b → is_unit d) ∧ mk' K a b = x :=\n-- begin\n--   obtain ⟨⟨b, b_nonzero⟩, a, hab⟩ := exists_integer_multiple (non_zero_divisors A) x,\n--   obtain ⟨a', b', c', no_factor, rfl, rfl⟩ :=\n--     unique_factorization_monoid.exists_reduced_factors' a b\n--       (mem_non_zero_divisors_iff_ne_zero.mp b_nonzero),\n--   obtain ⟨c'_nonzero, b'_nonzero⟩ := mul_mem_non_zero_divisors.mp b_nonzero,\n--   refine ⟨a', ⟨b', b'_nonzero⟩, @no_factor, _⟩,\n--   refine mul_left_cancel₀\n--     (is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors b_nonzero) _,\n--   simp only [subtype.coe_mk, ring_hom.map_mul, algebra.smul_def] at *,\n--   erw [←hab, mul_assoc, mk'_spec' _ a' ⟨b', b'_nonzero⟩],\n-- end\n\n-- /-- `f.num x` is the numerator of `x : f.codomain` as a reduced fraction. -/\n-- noncomputable def num (x : K) : A :=\n-- classical.some (exists_reduced_fraction A x)\n\n-- /-- `f.num x` is the denominator of `x : f.codomain` as a reduced fraction. -/\n-- noncomputable def denom (x : K) : non_zero_divisors A :=\n-- classical.some (classical.some_spec (exists_reduced_fraction A x))\n\n-- lemma num_denom_reduced (x : K) {d} : d ∣ num A x → d ∣ denom A x → is_unit d :=\n-- (classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).1\n\n-- @[simp] lemma mk'_num_denom (x : K) : mk' K (num A x) (denom A x) = x :=\n-- (classical.some_spec (classical.some_spec (exists_reduced_fraction A x))).2\n\n-- variables {A}\n\n-- lemma num_mul_denom_eq_num_iff_eq {x y : K} :\n--   x * algebra_map A K (denom A y) = algebra_map A K (num A y) ↔ x = y :=\n-- ⟨λ h, by simpa only [mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,\n--  λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩\n\n-- lemma num_mul_denom_eq_num_iff_eq' {x y : K} :\n--   y * algebra_map A K (denom A x) = algebra_map A K (num A x) ↔ x = y :=\n-- ⟨λ h, by simpa only [eq_comm, mk'_num_denom] using eq_mk'_iff_mul_eq.mpr h,\n--  λ h, eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_denom])⟩\n\n-- lemma num_mul_denom_eq_num_mul_denom_iff_eq {x y : K} :\n--   num A y * denom A x = num A x * denom A y ↔ x = y :=\n-- ⟨λ h, by simpa only [mk'_num_denom] using mk'_eq_of_eq h,\n--  λ h, by rw h⟩\n\n-- lemma eq_zero_of_num_eq_zero {x : K} (h : num A x = 0) : x = 0 :=\n-- num_mul_denom_eq_num_iff_eq'.mp (by rw [zero_mul, h, ring_hom.map_zero])\n\n-- lemma is_integer_of_is_unit_denom {x : K} (h : is_unit (denom A x : A)) : is_integer A x :=\n-- begin\n--   cases h with d hd,\n--   have d_ne_zero : algebra_map A K (denom A x) ≠ 0 :=\n--     is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors (denom A x).2,\n--   use ↑d⁻¹ * num A x,\n--   refine trans _ (mk'_num_denom A x),\n--   rw [map_mul, map_units_inv, hd],\n--   apply mul_left_cancel₀ d_ne_zero,\n--   rw [←mul_assoc, mul_inv_cancel d_ne_zero, one_mul, mk'_spec']\n-- end\n\n-- lemma is_unit_denom_of_num_eq_zero {x : K} (h : num A x = 0) : is_unit (denom A x : A) :=\n-- num_denom_reduced A x (h.symm ▸ dvd_zero _) dvd_rfl\n\n-- end num_denom\n\n-- -- variables (S)\nuniverses u v\n\n-- -- section away\n\n-- -- variables {A : Type u} [decidable_eq A]\n-- -- variables [cancel_comm_monoid_with_zero A] [normalization_monoid A] [unique_factorization_monoid A]\n-- -- variables [dec_dvd : decidable_rel (has_dvd.dvd : A → A → Prop)]\n-- -- variable {x : A}\n-- -- variable (hx : irreducible x)\n-- -- open multiplicity unique_factorization_monoid\n\n-- -- end away\n\nsection away\n\nopen multiplicity unique_factorization_monoid is_localization\n\nvariables {A : Type u} [comm_ring A] [is_domain A] [decidable_eq A] [normalization_monoid A] [unique_factorization_monoid A]\nvariables [dec_dvd : decidable_rel (has_dvd.dvd : A → A → Prop)]\nvariable (x : A)\n\n-- variable (n : ℕ)\n\n-- #where\n\ninclude dec_dvd\n\nlemma uno_remarkable (a₀ : A) (h : a₀ ≠ 0) [nontrivial A] (hx : irreducible x) : \n  ∃ n : ℕ, ∃ a : A, ¬ x ∣ a ∧ a₀ = x ^ n * a :=\nbegin\n  let n := (normalized_factors a₀).count (normalize x),\n  obtain ⟨a, ha1, ha2⟩ := (@exists_eq_pow_mul_and_not_dvd A _ _ x a₀\n    (ne_top_iff_finite.mp (part_enat.ne_top_iff.mpr _))),\n  simp_rw [← (multiplicity_eq_count_normalized_factors hx h).symm] at ha1,\n  use [n, a, ha2, ha1],\n  use [n, (multiplicity_eq_count_normalized_factors hx h)],\nend\n\n-- lemma due_remarkable (a : A) (m n : ℕ) :\n--   (mk' B a ⟨x^n, (submonoid.mem_powers_iff (x^n) x).mpr ⟨n, rfl⟩⟩) = \n--   (mk' B (a * x^m) ⟨x^(n), (submonoid.mem_powers_iff (x^n) x).mpr ⟨n, rfl⟩⟩) :=\n-- begin\n-- sorry\n-- end\n\n-- include x\n\n-- noncomputable def inv_self.unit : Bˣ :=\n--   ⟨away.inv_self x, algebra_map _ _ x, by {rw mul_comm, exact away.mul_inv_self _},\n--     away.mul_inv_self _⟩\n\nvariables (B : Type v) [comm_ring B] [algebra A B] [is_localization.away x B] \nvariable (hx' : irreducible x)\n\nnoncomputable def self_as_unit : Bˣ :=\n  ⟨algebra_map _ _ x, away.inv_self x, away.mul_inv_self _,\n    by {rw mul_comm, exact away.mul_inv_self _}⟩\n\nlemma due_remarkable' (a : A) (b : B) (m d : ℤ) :\n  (((self_as_unit x B ^ (m - d)) : Bˣ ) : B) * mk' B a (1 : submonoid.powers x) = b ↔\n  (((self_as_unit x B ^ m) : Bˣ) : B) * mk' B a (1 : submonoid.powers x) =\n    (((self_as_unit x B ^ d) : Bˣ) : B) * b := by {simp only [zpow_sub, units.coe_mul,\n    mul_comm (((self_as_unit x B ^ m) : Bˣ) : B) _,  mul_assoc, units.inv_mul_eq_iff_eq_mul]}\n\nlemma aux (d : ℕ) : (((self_as_unit x B)^(d : ℤ) : Bˣ) : B) = (algebra_map A B x)^d :=\n  by {simp only [self_as_unit, zpow_coe_nat, units.coe_pow, units.coe_mk]}\n\ninclude hx'\n\nlemma exists_reduced_fraction (b : B) (hb : b ≠ 0) :\n  ∃ (a : A) (n : ℤ), ¬ x ∣ a ∧\n  (((self_as_unit x B)^n : Bˣ) : B) * mk' B a (1 : submonoid.powers x) = b :=\n  -- (mk' B a (1 : (submonoid.powers x))) * (((away.inv_self x) : Bˣ ) : B)= b :=\n  -- (∀ {d}, d ∣ a → d ∣ b → is_unit d) ∧ mk' K a b = x :=\nbegin\n  obtain ⟨⟨a₀, y⟩, H⟩ := is_localization.surj (submonoid.powers x) b,\n  obtain ⟨d, hy⟩ := (submonoid.mem_powers_iff y.1 x).mp y.2,\n    have ha₀ : a₀ ≠ 0,\n    { --have : (y : A) ≠ 0,\n      -- rw coe_zero,\n      -- by_contra,\n      have h_inj := @is_localization.injective A _ (submonoid.powers x) B _ _ _ \n        (powers_le_non_zero_divisors_of_no_zero_divisors (hx'.ne_zero)),\n      haveI := @is_domain_of_le_non_zero_divisors B _ A _ _ _ (submonoid.powers x) _\n       (powers_le_non_zero_divisors_of_no_zero_divisors hx'.ne_zero),\n      simp only [map_zero, ← subtype.val_eq_coe, ← hy, map_pow] at H,\n      apply ((injective_iff_map_eq_zero' (algebra_map A B)).mp (h_inj) a₀).mpr.mt,\n      rw ← H,\n      apply mul_ne_zero hb,\n      apply pow_ne_zero,\n      apply @is_localization.to_map_ne_zero_of_mem_non_zero_divisors\n        A _ (submonoid.powers x) B _ _ _ _ _ x _,\n      apply powers_le_non_zero_divisors_of_no_zero_divisors (hx'.ne_zero),\n      exact mem_non_zero_divisors_iff_ne_zero.mpr hx'.ne_zero,\n    },\n    simp only [← subtype.val_eq_coe, ← hy] at H,--needed?\n    obtain ⟨m, a, hyp1, hyp2⟩ := uno_remarkable x a₀ ha₀ hx',\n    use a,\n    use m - d,\n    rw due_remarkable' x B a b m d, --replace?\n    rw [aux x B d, mul_comm _ b, ← map_pow, H, hyp2, aux x B m,\n      map_mul, map_pow],\n    -- rw [due_remarkable' x B a b m d, aux x B d, mul_comm _ b, ← map_pow, H, hyp2, aux x B m,\n    --   map_mul, map_pow],\n    exact ⟨hyp1, (congr_arg _ (is_localization.mk'_one _ _))⟩,\nend\n\nend away\n\n", "meta": {"author": "mariainesdff", "repo": "local_class_field_theory", "sha": "ffa8bc00cd45f6bee74e3a5ef7def8678bcee0b0", "save_path": "github-repos/lean/mariainesdff-local_class_field_theory", "path": "github-repos/lean/mariainesdff-local_class_field_theory/local_class_field_theory-ffa8bc00cd45f6bee74e3a5ef7def8678bcee0b0/src/for_mathlib/num_denom_away.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3438551828487051}}
{"text": "-- Copyright 2022-2023 VMware, Inc.\n-- SPDX-License-Identifier: BSD-2-Clause\n\nimport .relational\nimport .incremental\n\nopen zset\n\nvariables {A B C: Type}.\nvariables [decidable_eq A] [decidable_eq B] [decidable_eq C].\n\ndef distinct_H_at (i d: Z[A]) : A → ℤ :=\n  λ x, if 0 < i x ∧ (i + d) x ≤ 0\n       then -1 else\n        if i x ≤ 0 ∧ 0 < (i + d) x\n        then 1 else 0.\n\ndef distinct_H (i d: Z[A]) : Z[A] :=\n  dfinsupp.mk (i.support ∪ d.support)\n    (λ x, distinct_H_at i d x).\n\n@[simp]\nlemma distinct_H_apply (i d: Z[A]) (x: A) :\n  distinct_H i d x = distinct_H_at i d x :=\nbegin\n  unfold distinct_H, simp,\n  push_neg,\n  intros h, cases h with h1 h2,\n  simp [distinct_H_at], rw [h1, h2],\n  rw [if_neg, if_neg]; simp,\nend\n\ndef distinct_incremental : stream Z[A] → stream Z[A] :=\n  λ d, (↑² distinct_H) (z⁻¹ (I d)) d.\n\ntheorem distinct_incremental_ok :\n  (↑↑ distinct)^Δ = @distinct_incremental A _ :=\nbegin\n  funext d,\n  unfold incremental distinct_incremental,\n  unfold D,\n  conv_lhs {\n    congr, { rw integral_unfold, skip}, { skip }\n  },\n  rw<- lifting_time_invariant, swap, simp,\n  repeat { rw<- integral_time_invariant },\n  ext t a, unfold lifting2, simp,\n  generalize hi : I (z⁻¹ d) t = i,\n  generalize hd : d t = d_t,\n  clear hi hd d t,\n  unfold distinct_H_at,\n  simp,\n  -- canonicalize the tests a little bit so splitting produces fewer cases\n  rw (add_comm (i a) (d_t a)),\n  repeat { rw <- ite_ite },\n  split_ifs; refl <|> { exfalso, omega },\nend\n\n@[simp]\nlemma flatmap_incremental (f: A → Z[B]) :\n  ↑↑(zset.flatmap f)^Δ = ↑↑(zset.flatmap f) :=\nbegin\n  apply lti_incremental,\n  apply lifting_lti,\n  apply flatmap_linear,\nend\n\n@[simp]\nlemma map_incremental (f: A → B) :\n  ↑↑(zset.map f)^Δ = ↑↑(zset.map f) :=\n  flatmap_incremental _.\n\n@[simp]\nlemma lifting_map_incremental {A B} [decidable_eq A] [decidable_eq B] (f: A → B) :\n  ↑↑(↑↑(zset.map f))^Δ = ↑↑(↑↑(zset.map f)) :=\nbegin\n  rw lti_incremental,\n  apply lifting_lti,\n  intros x y, rw lifting_linear, apply map_linear,\nend\n\n\n@[simp]\nlemma map_incremental_unfolded (f: A → B) (s: stream Z[A]) :\n  D (↑↑(zset.map f) (I s)) = ↑↑(zset.map f) s :=\nbegin\n  conv_rhs {\n    rw<- map_incremental,\n  },\n  rw incremental_unfold,\nend\n\ntheorem equi_join_incremental (π1: A → C) (π2: B → C) :\n  ↑²(equi_join π1 π2)^Δ2 =\n    times_incremental ↑²(equi_join π1 π2) :=\nbegin\n  rw (bilinear_incremental (↑²(equi_join π1 π2))),\n  { rw lifting2_time_invariant,\n    refl, },\n  { apply lifting_bilinear,\n    apply equi_join_bilinear,\n  },\nend\n\n@[simp]\nlemma filter_incremental (p: A → Prop) [decidable_pred p] :\n  ↑↑(filter p)^Δ = ↑↑(filter p) :=\nbegin\n  apply lti_incremental,\n  apply lifting_lti,\n  apply filter_linear,\nend\n", "meta": {"author": "tchajed", "repo": "database-stream-processing-theory", "sha": "c4c3b7ced9f964f3ea17db77958df78f2d761509", "save_path": "github-repos/lean/tchajed-database-stream-processing-theory", "path": "github-repos/lean/tchajed-database-stream-processing-theory/database-stream-processing-theory-c4c3b7ced9f964f3ea17db77958df78f2d761509/src/relational_incremental.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154240079185319, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.34374598156843156}}
{"text": "/-\nCopyright (c) 2022 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Eric Wieser\n\n! This file was ported from Lean 3 source module ring_theory.graded_algebra.radical\n! leanprover-community/mathlib commit f1944b30c97c5eb626e498307dec8b022a05bd0a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.RingTheory.GradedAlgebra.HomogeneousIdeal\n\n/-!\n\nThis file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal\n\n## Main statements\n\n* `ideal.is_homogeneous.is_prime_iff`: for any `I : ideal A`, if `I` is homogeneous, then\n  `I` is prime if and only if `I` is homogeneously prime, i.e. `I ≠ ⊤` and if `x, y` are\n  homogeneous elements such that `x * y ∈ I`, then at least one of `x,y` is in `I`.\n* `ideal.is_prime.homogeneous_core`: for any `I : ideal A`, if `I` is prime, then\n  `I.homogeneous_core 𝒜` (i.e. the largest homogeneous ideal contained in `I`) is also prime.\n* `ideal.is_homogeneous.radical`: for any `I : ideal A`, if `I` is homogeneous, then the\n  radical of `I` is homogeneous as well.\n* `homogeneous_ideal.radical`: for any `I : homogeneous_ideal 𝒜`, `I.radical` is the the\n  radical of `I` as a `homogeneous_ideal 𝒜`\n\n## Implementation details\n\nThroughout this file, the indexing type `ι` of grading is assumed to be a\n`linear_ordered_cancel_add_comm_monoid`. This might be stronger than necessary but cancelling\nproperty is strictly necessary; for a counterexample of how `ideal.is_homogeneous.is_prime_iff`\nfails for a non-cancellative set see `counterexample/homogeneous_prime_not_prime.lean`.\n\n## Tags\n\nhomogeneous, radical\n-/\n\n\nopen GradedRing DirectSum SetLike Finset\n\nopen BigOperators\n\nvariable {ι σ A : Type _}\n\nvariable [CommRing A]\n\nvariable [LinearOrderedCancelAddCommMonoid ι]\n\nvariable [SetLike σ A] [AddSubmonoidClass σ A] {𝒜 : ι → σ} [GradedRing 𝒜]\n\ninclude A\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.Homogeneous 𝒜)\n    (I_ne_top : I ≠ ⊤)\n    (homogeneous_mem_or_mem :\n      ∀ {x y : A}, Homogeneous 𝒜 x → Homogeneous 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :\n    Ideal.IsPrime I :=\n  ⟨I_ne_top, by\n    intro x y hxy\n    by_contra rid\n    obtain ⟨rid₁, rid₂⟩ := not_or_distrib.mp rid\n    classical\n      /-\n        The idea of the proof is the following :\n        since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.\n        Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;\n        let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.\n        Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`\n        but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.\n        `  proj (max₁ + max₂) (x * y)`\n        `= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`\n        `= proj max₁ x * proj max₂ y`\n        `  + ∑ {(i, j) ∈ supports \\ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.\n        This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the\n        right hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.\n        -/\n      set set₁ := (decompose 𝒜 x).support.filterₓ fun i => proj 𝒜 i x ∉ I with set₁_eq\n      set set₂ := (decompose 𝒜 y).support.filterₓ fun i => proj 𝒜 i y ∉ I with set₂_eq\n      have nonempty :\n        ∀ x : A, x ∉ I → ((decompose 𝒜 x).support.filterₓ fun i => proj 𝒜 i x ∉ I).Nonempty :=\n        by\n        intro x hx\n        rw [filter_nonempty_iff]\n        contrapose! hx\n        simp_rw [proj_apply] at hx\n        rw [← sum_support_decompose 𝒜 x]\n        exact Ideal.sum_mem _ hx\n      set max₁ := set₁.max' (Nonempty x rid₁) with max₁_eq\n      set max₂ := set₂.max' (Nonempty y rid₂) with max₂_eq\n      have mem_max₁ : max₁ ∈ set₁ := max'_mem set₁ (Nonempty x rid₁)\n      have mem_max₂ : max₂ ∈ set₂ := max'_mem set₂ (Nonempty y rid₂)\n      replace hxy : proj 𝒜 (max₁ + max₂) (x * y) ∈ I := hI _ hxy\n      have mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∈ I :=\n        by\n        set antidiag :=\n          ((decompose 𝒜 x).support ×ˢ (decompose 𝒜 y).support).filterₓ fun z : ι × ι =>\n            z.1 + z.2 = max₁ + max₂ with\n          ha\n        have mem_antidiag : (max₁, max₂) ∈ antidiag :=\n          by\n          simp only [add_sum_erase, mem_filter, mem_product]\n          exact ⟨⟨mem_of_mem_filter _ mem_max₁, mem_of_mem_filter _ mem_max₂⟩, rfl⟩\n        have eq_add_sum :=\n          calc\n            proj 𝒜 (max₁ + max₂) (x * y) = ∑ ij in antidiag, proj 𝒜 ij.1 x * proj 𝒜 ij.2 y := by\n              simp_rw [ha, proj_apply, DirectSum.decompose_mul, DirectSum.coe_mul_apply 𝒜]\n            _ =\n                proj 𝒜 max₁ x * proj 𝒜 max₂ y +\n                  ∑ ij in antidiag.erase (max₁, max₂), proj 𝒜 ij.1 x * proj 𝒜 ij.2 y :=\n              (add_sum_erase _ _ mem_antidiag).symm\n            \n        rw [eq_sub_of_add_eq eq_add_sum.symm]\n        refine' Ideal.sub_mem _ hxy (Ideal.sum_mem _ fun z H => _)\n        rcases z with ⟨i, j⟩\n        simp only [mem_erase, Prod.mk.inj_iff, Ne.def, mem_filter, mem_product] at H\n        rcases H with ⟨H₁, ⟨H₂, H₃⟩, H₄⟩\n        have max_lt : max₁ < i ∨ max₂ < j :=\n          by\n          rcases lt_trichotomy max₁ i with (h | rfl | h)\n          · exact Or.inl h\n          · refine' False.elim (H₁ ⟨rfl, add_left_cancel H₄⟩)\n          · apply Or.inr\n            have := add_lt_add_right h j\n            rw [H₄] at this\n            exact lt_of_add_lt_add_left this\n        cases max_lt\n        · -- in this case `max₁ < i`, then `xᵢ ∈ I`; for otherwise `i ∈ set₁` then `i ≤ max₁`.\n          have not_mem : i ∉ set₁ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₁ (Nonempty x rid₁)).mp max_lt i h)\n          rw [set₁_eq] at not_mem\n          simp only [not_and, Classical.not_not, Ne.def, mem_filter] at not_mem\n          exact Ideal.mul_mem_right _ I (not_mem H₂)\n        · -- in this case  `max₂ < j`, then `yⱼ ∈ I`; for otherwise `j ∈ set₂`, then `j ≤ max₂`.\n          have not_mem : j ∉ set₂ := fun h =>\n            lt_irrefl _ ((max'_lt_iff set₂ (Nonempty y rid₂)).mp max_lt j h)\n          rw [set₂_eq] at not_mem\n          simp only [not_and, Classical.not_not, Ne.def, mem_filter] at not_mem\n          exact Ideal.mul_mem_left I _ (not_mem H₃)\n      have not_mem_I : proj 𝒜 max₁ x * proj 𝒜 max₂ y ∉ I :=\n        by\n        have neither_mem : proj 𝒜 max₁ x ∉ I ∧ proj 𝒜 max₂ y ∉ I :=\n          by\n          rw [mem_filter] at mem_max₁ mem_max₂\n          exact ⟨mem_max₁.2, mem_max₂.2⟩\n        intro rid\n        cases homogeneous_mem_or_mem ⟨max₁, SetLike.coe_mem _⟩ ⟨max₂, SetLike.coe_mem _⟩ mem_I\n        · apply neither_mem.1 h\n        · apply neither_mem.2 h\n      exact not_mem_I mem_I⟩\n#align ideal.is_homogeneous.is_prime_of_homogeneous_mem_or_mem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem\n\ntheorem Ideal.IsHomogeneous.isPrime_iff {I : Ideal A} (h : I.Homogeneous 𝒜) :\n    I.IsPrime ↔\n      I ≠ ⊤ ∧\n        ∀ {x y : A},\n          SetLike.Homogeneous 𝒜 x → SetLike.Homogeneous 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I :=\n  ⟨fun HI => ⟨ne_of_apply_ne _ HI.ne_top, fun x y hx hy hxy => Ideal.IsPrime.mem_or_mem HI hxy⟩,\n    fun ⟨I_ne_top, homogeneous_mem_or_mem⟩ =>\n    h.isPrime_of_homogeneous_mem_or_mem I_ne_top @homogeneous_mem_or_mem⟩\n#align ideal.is_homogeneous.is_prime_iff Ideal.IsHomogeneous.isPrime_iff\n\ntheorem Ideal.IsPrime.homogeneousCore {I : Ideal A} (h : I.IsPrime) :\n    (I.homogeneousCore 𝒜).toIdeal.IsPrime :=\n  by\n  apply (Ideal.homogeneousCore 𝒜 I).Homogeneous.isPrime_of_homogeneous_mem_or_mem\n  · exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)\n  rintro x y hx hy hxy\n  have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)\n  refine' H.imp _ _\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx\n  · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy\n#align ideal.is_prime.homogeneous_core Ideal.IsPrime.homogeneousCore\n\ntheorem Ideal.IsHomogeneous.radical_eq {I : Ideal A} (hI : I.Homogeneous 𝒜) :\n    I.radical = infₛ { J | J.Homogeneous 𝒜 ∧ I ≤ J ∧ J.IsPrime } :=\n  by\n  rw [Ideal.radical_eq_infₛ]\n  apply le_antisymm\n  · exact infₛ_le_infₛ fun J => And.right\n  · refine' infₛ_le_infₛ_of_forall_exists_le _\n    rintro J ⟨HJ₁, HJ₂⟩\n    refine' ⟨(J.homogeneous_core 𝒜).toIdeal, _, J.to_ideal_homogeneous_core_le _⟩\n    refine' ⟨HomogeneousIdeal.isHomogeneous _, _, HJ₂.homogeneous_core⟩\n    refine' hI.to_ideal_homogeneous_core_eq_self.symm.trans_le (Ideal.homogeneousCore_mono _ HJ₁)\n#align ideal.is_homogeneous.radical_eq Ideal.IsHomogeneous.radical_eq\n\ntheorem Ideal.IsHomogeneous.radical {I : Ideal A} (h : I.Homogeneous 𝒜) : I.radical.Homogeneous 𝒜 :=\n  by\n  rw [h.radical_eq]\n  exact Ideal.IsHomogeneous.inf fun _ => And.left\n#align ideal.is_homogeneous.radical Ideal.IsHomogeneous.radical\n\n/-- The radical of a homogenous ideal, as another homogenous ideal. -/\ndef HomogeneousIdeal.radical (I : HomogeneousIdeal 𝒜) : HomogeneousIdeal 𝒜 :=\n  ⟨I.toIdeal.radical, I.Homogeneous.radical⟩\n#align homogeneous_ideal.radical HomogeneousIdeal.radical\n\n@[simp]\ntheorem HomogeneousIdeal.coe_radical (I : HomogeneousIdeal 𝒜) :\n    I.radical.toIdeal = I.toIdeal.radical :=\n  rfl\n#align homogeneous_ideal.coe_radical HomogeneousIdeal.coe_radical\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/RingTheory/GradedAlgebra/Radical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3437459757377947}}
{"text": "\nimport unitb.code.syntax\n\nsection finite\n\nopen nat\n\nparameters (σ : Type) {lbl : Type}\n\nprivate def pred := σ → Prop\n\nparameters {σ}\n\ndef control_nodes : Π {p q : pred}, code lbl p q → ℕ\n  | ._ ._ (code.skip _) := 0\n  | ._ ._ (code.action _ _ _ _) := 1\n  | ._ ._ (@code.seq ._ ._ p q r c₀ c₁) := control_nodes c₀ + control_nodes c₁\n  | ._ ._ (@code.if_then_else ._ ._ c pa pb q _ _ c₀ c₁) :=\n     control_nodes c₀ + control_nodes c₁ + 1\n  | ._ ._ (@code.while ._ ._ p inv q _ c b) :=\n     control_nodes b + 1\n\ndef to_fin : Π {p q : pred} {c : code lbl p q}, current c → fin (control_nodes c)\n  | ._ ._ ._ (current.action _ _ _ _) := ⟨0,zero_lt_one⟩\n  | ._ ._ ._ (current.seq_left p q r c₀ c₁ pc) := fin.nest (to_fin pc)\n  | ._ ._ ._ (current.seq_right p q r c₀ c₁ pc) := fin.shift (to_fin pc)\n  | ._ ._ ._ (current.ite_cond p t pa pb _ q c₀ c₁) :=\n    (fin.max : fin (succ $ _ + _))\n  | ._ ._ ._ (current.ite_left p t pa pb q _ c₀ c₁ pc) :=\n    (fin.nest $ fin.nest $ to_fin pc : fin (control_nodes c₀ + control_nodes c₁ + 1))\n  | ._ ._ ._ (current.ite_right p t pa pb q _ c₀ c₁ pc) :=\n    (fin.nest $ fin.shift $ to_fin pc : fin (control_nodes c₀ + control_nodes c₁ + 1))\n  | ._ ._ ._ (current.while_cond p inv q _ w c) :=\n    (fin.max : fin (control_nodes c + 1))\n  | ._ ._ ._ (current.while_body p inv q _ w c pc) :=\n    (fin.nest $ to_fin pc : fin (control_nodes c + 1))\n\ndef from_fin : Π {p q} (c : code lbl p q), fin (control_nodes c) → current c\n  | ._ ._ (code.skip _) ⟨_,P⟩ := by cases (nat.not_lt_zero _ P)\n  | ._ ._ (code.action p q _ l) _ := current.action p q _ l\n  | ._ ._ (@code.seq ._ ._ p q r c₀ c₁) m :=\n    match fin.split m with\n     | (sum.inl n) := seq_left  c₁ $ from_fin c₀ n\n     | (sum.inr n) := seq_right c₀ $ from_fin c₁ n\n    end\n  | ._ ._ (@code.if_then_else ._ ._ p pa pb q ds t c₀ c₁) n :=\n    match (fin.split n : fin (control_nodes c₀ + control_nodes c₁) ⊕ fin 1) with\n     | (sum.inl n') :=\n        match (fin.split n' : fin (control_nodes c₀) ⊕ fin (control_nodes c₁)) with\n         | (sum.inl n) := ite_left  _ _ _ c₁ $ from_fin c₀ n\n         | (sum.inr n) := ite_right _ _ _ c₀ $ from_fin c₁ n\n        end\n     | (sum.inr _) := ite_cond p t ds c₀ c₁\n    end\n  | ._ ._ (@code.while ._ ._ p inv q _ t b) n :=\n    match (fin.split n : fin (control_nodes b) ⊕ fin 1) with\n     | (sum.inl n') := while_body q _ t $ from_fin b n'\n     | (sum.inr _) := while_cond q _ t b\n    end\n\nsection g_over_constr\n\nlemma from_fin_eq_seq_left\n  {p q r : pred}\n  {c₀ : code lbl p q} {c₁ : code lbl q r}\n  {n : fin (control_nodes c₀ + control_nodes c₁)}\n  {k : fin (control_nodes c₀)}\n  (Hk : fin.split n = sum.inl k)\n:   from_fin (code.seq c₀ c₁) n\n  = seq_left c₁ (from_fin c₀ k) :=\nby { dunfold from_fin, simp [Hk], refl }\n\nlemma from_fin_eq_seq_right\n  {p q r : pred}\n  {c₀ : code lbl p q} {c₁ : code lbl q r}\n  {n : fin (control_nodes c₀ + control_nodes c₁)}\n  {k : fin (control_nodes c₁)}\n  (Hk : fin.split n = sum.inr k)\n:   from_fin (code.seq c₀ c₁) n\n  = seq_right c₀ (from_fin c₁ k) :=\nby { dunfold from_fin, simp [Hk], refl }\n\nlemma from_fin_eq_ite_left\n  {p t pa pb q : pred}\n  {ds : set lbl}\n  {c₀ : code lbl pa q} {c₁ : code lbl pb q}\n  {n  : fin (control_nodes c₀ + control_nodes c₁ + 1)}\n  {k  : fin (control_nodes c₀ + control_nodes c₁)}\n  {k' : fin (control_nodes c₀)}\n  (Hk : fin.split n = sum.inl k)\n  (Hk' : fin.split k = sum.inl k')\n:   from_fin (if_then_else p ds t c₀ c₁) n\n  = ite_left p t ds c₁ (from_fin c₀ k') :=\nbegin\n  dunfold from_fin, simp [Hk],\n  dunfold from_fin._match_2, simp [Hk'],\n  refl,\nend\n\nlemma from_fin_eq_ite_right\n  {p t pa pb q : pred}\n  {ds : set lbl}\n  {c₀ : code lbl pa q} {c₁ : code lbl pb q}\n  {n  : fin (control_nodes c₀ + control_nodes c₁ + 1)}\n  {k  : fin (control_nodes c₀ + control_nodes c₁)}\n  {k' : fin (control_nodes c₁)}\n  (Hk : fin.split n = sum.inl k)\n  (Hk' : fin.split k = sum.inr k')\n:   from_fin (if_then_else p ds t c₀ c₁) n\n  = ite_right p t ds c₀ (from_fin c₁ k') :=\nbegin\n  dunfold from_fin, simp [Hk],\n  dunfold from_fin._match_2, simp [Hk'],\n  refl,\nend\n\nlemma from_fin_eq_ite_cond\n  {p t pa pb q : pred}\n  {ds : set lbl}\n  {c₀ : code lbl pa q} {c₁ : code lbl pb q}\n  {n  : fin (control_nodes c₀ + control_nodes c₁ + 1)}\n  {k  : fin 1}\n  (Hk : fin.split n = sum.inr k)\n:   from_fin (if_then_else p ds t c₀ c₁) n\n  = ite_cond p t ds c₀ c₁ :=\nby { dunfold from_fin, simp [Hk], refl }\n\nlemma from_fin_eq_while_body\n  {p t inv q : pred}\n  {ds : set lbl}\n  {c₀ : code lbl p inv}\n  {n  : fin (control_nodes c₀ + 1)}\n  {k  : fin $ control_nodes c₀}\n  (Hk : fin.split n = sum.inl k)\n:   from_fin (while q ds t c₀) n\n  = while_body q ds t (from_fin c₀ k) :=\nby { dunfold from_fin, simp [Hk], refl }\n\nlemma from_fin_eq_while_cond\n  {p t inv q : pred}\n  {ds : set lbl}\n  {c₀ : code lbl p inv}\n  {n  : fin (control_nodes c₀ + 1)}\n  {k  : fin 1}\n  (Hk : fin.split n = sum.inr k)\n:   from_fin (while q ds t c₀) n\n  = while_cond q ds t c₀  :=\nby { dunfold from_fin, simp [Hk], refl }\n\nend g_over_constr\n\nlemma from_fin_inv {p q} {c : code lbl p q} (n : fin (control_nodes c))\n: to_fin (from_fin c n) = n :=\nbegin\n  induction c\n  ; dunfold control_nodes at n,\n  { apply fin.elim0 n, },\n  { dunfold from_fin to_fin\n  ; apply fin.eq_of_veq,\n    unfold fin.val,\n    apply le_antisymm (zero_le _),\n    apply le_of_lt_succ,\n    apply n.is_lt },\n  { destruct fin.split n\n    ; intros k Hk,\n    { rw [from_fin_eq_seq_left Hk], dunfold to_fin,\n      rw [ih_1,fin.nest_eq_iff_eq_split,Hk], },\n    { rw [from_fin_eq_seq_right Hk], dunfold to_fin,\n      rw [ih_2,fin.shift_eq_iff_eq_split,Hk], }, },\n  { destruct fin.split n\n    ; intros k Hk,\n    destruct fin.split k\n    ; intros k' Hk',\n    { rw [from_fin_eq_ite_left Hk Hk'], dunfold to_fin,\n      simp [ih_1,fin.nest_eq_iff_eq_split,Hk],\n      apply congr_arg,\n      simp [fin.nest_eq_iff_eq_split,Hk'] },\n    { rw [from_fin_eq_ite_right Hk Hk'], dunfold to_fin,\n      simp [ih_2,fin.nest_eq_iff_eq_split,Hk],\n      apply congr_arg,\n      simp [fin.shift_eq_iff_eq_split,Hk'] },\n    { rw [from_fin_eq_ite_cond Hk], dunfold to_fin,\n      apply fin.split_injective _ n,\n      simp [Hk,eq_comm],\n      rw [← fin.shift_eq_iff_eq_split,fin.all_eq_zero k],\n      apply fin.eq_of_veq, simp [fin.val_shift_zero], refl }, },\n  { destruct fin.split n\n    ; intros k Hk,\n    { rw [from_fin_eq_while_body Hk], dunfold to_fin,\n      rw [ih_1,fin.nest_eq_iff_eq_split,Hk], },\n    { rw [from_fin_eq_while_cond Hk], dunfold to_fin,\n      apply fin.split_injective _ n,\n      simp [Hk,eq_comm],\n      rw [← fin.shift_eq_iff_eq_split,fin.all_eq_zero k],\n      apply fin.eq_of_veq, simp [fin.val_shift_zero], refl, }, }\nend\n\nlemma to_fin_inv {p q} {c : code lbl p q} (pc : current c)\n: from_fin c (to_fin pc) = pc :=\nbegin\n  induction pc\n  ; dunfold to_fin\n  ; try { refl },\n  { dunfold from_fin, simp [fin.split_nest],\n    dunfold from_fin._match_1,\n    simp [ih_1] },\n  { dunfold from_fin, simp [fin.split_shift],\n    dunfold from_fin._match_1,\n    simp [ih_1] },\n  { dunfold from_fin, simp [fin.shift_zero,fin.split_shift],\n    dunfold from_fin._match_2, refl },\n  { dunfold from_fin, simp [fin.split_nest],\n    dunfold from_fin._match_2,\n    simp [fin.split_nest],\n    dunfold from_fin._match_3,\n    simp [ih_1] },\n  { dunfold from_fin, simp [fin.split_nest],\n    dunfold from_fin._match_2,\n    simp [fin.split_shift],\n    dunfold from_fin._match_3,\n    simp [ih_1] },\n  { dunfold from_fin,\n    simp [fin.shift_zero,fin.split_shift],\n    refl },\n  { dunfold from_fin, simp [fin.split_nest],\n    dunfold from_fin._match_4,\n    simp [ih_1] },\nend\n\ninstance current_finite {p q : pred} {c : code lbl p q} : finite (current c) :=\n⟨ control_nodes c,\n  bijection.mk to_fin (from_fin c) to_fin_inv from_fin_inv ⟩\n\ninstance current_sched {p q : pred} {c : code lbl p q} : scheduling.sched (current c) :=\nscheduling.sched.fin (by apply_instance)\n\nend finite\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/code/instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3437459757377947}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.isomorphism\nimport category_theory.functor_category\n\n/-!\n# Whiskering\n\nGiven a functor `F  : C ⥤ D` and functors `G H : D ⥤ E` and a natural transformation `α : G ⟶ H`,\nwe can construct a new natural transformation `F ⋙ G ⟶ F ⋙ H`,\ncalled `whisker_left F α`. This is the same as the horizontal composition of `𝟙 F` with `α`.\n\nThis operation is functorial in `F`, and we package this as `whiskering_left`. Here\n`(whiskering_left.obj F).obj G` is `F ⋙ G`, and\n`(whiskering_left.obj F).map α` is `whisker_left F α`.\n(That is, we might have alternatively named this as the \"left composition functor\".)\n\nWe also provide analogues for composition on the right, and for these operations on isomorphisms.\n\nAt the end of the file, we provide the left and right unitors, and the associator,\nfor functor composition.\n(In fact functor composition is definitionally associative, but very often relying on this causes\nextremely slow elaboration, so it is better to insert it explicitly.)\nWe also show these natural isomorphisms satisfy the triangle and pentagon identities.\n-/\n\nnamespace category_theory\n\nuniverses u₁ v₁ u₂ v₂ u₃ v₃ u₄ v₄\n\nsection\nvariables {C : Type u₁} [category.{v₁} C]\n          {D : Type u₂} [category.{v₂} D]\n          {E : Type u₃} [category.{v₃} E]\n\n/--\nIf `α : G ⟶ H` then\n`whisker_left F α : (F ⋙ G) ⟶ (F ⋙ H)` has components `α.app (F.obj X)`.\n-/\n@[simps] def whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ⟶ H) : (F ⋙ G) ⟶ (F ⋙ H) :=\n{ app := λ X, α.app (F.obj X),\n  naturality' := λ X Y f, by rw [functor.comp_map, functor.comp_map, α.naturality] }\n\n/--\nIf `α : G ⟶ H` then\n`whisker_right α F : (G ⋙ F) ⟶ (G ⋙ F)` has components `F.map (α.app X)`.\n-/\n@[simps] def whisker_right {G H : C ⥤ D} (α : G ⟶ H) (F : D ⥤ E) : (G ⋙ F) ⟶ (H ⋙ F) :=\n{ app := λ X, F.map (α.app X),\n  naturality' := λ X Y f,\n    by rw [functor.comp_map, functor.comp_map, ←F.map_comp, ←F.map_comp, α.naturality] }\n\nvariables (C D E)\n\n/--\nLeft-composition gives a functor `(C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E))`.\n\n`(whiskering_left.obj F).obj G` is `F ⋙ G`, and\n`(whiskering_left.obj F).map α` is `whisker_left F α`.\n-/\n@[simps] def whiskering_left : (C ⥤ D) ⥤ ((D ⥤ E) ⥤ (C ⥤ E)) :=\n{ obj := λ F,\n  { obj := λ G, F ⋙ G,\n    map := λ G H α, whisker_left F α },\n  map := λ F G τ,\n  { app := λ H,\n    { app := λ c, H.map (τ.app c),\n      naturality' := λ X Y f, begin dsimp, rw [←H.map_comp, ←H.map_comp, ←τ.naturality] end },\n    naturality' := λ X Y f, begin ext, dsimp, rw [f.naturality] end } }\n\n/--\nRight-composition gives a functor `(D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E))`.\n\n`(whiskering_right.obj H).obj F` is `F ⋙ H`, and\n`(whiskering_right.obj H).map α` is `whisker_right α H`.\n-/\n@[simps] def whiskering_right : (D ⥤ E) ⥤ ((C ⥤ D) ⥤ (C ⥤ E)) :=\n{ obj := λ H,\n  { obj := λ F, F ⋙ H,\n    map := λ _ _ α, whisker_right α H },\n  map := λ G H τ,\n  { app := λ F,\n    { app := λ c, τ.app (F.obj c),\n      naturality' := λ X Y f, begin dsimp, rw [τ.naturality] end },\n    naturality' := λ X Y f, begin ext, dsimp, rw [←nat_trans.naturality] end } }\n\nvariables {C} {D} {E}\n\n@[simp] lemma whisker_left_id (F : C ⥤ D) {G : D ⥤ E} :\n  whisker_left F (nat_trans.id G) = nat_trans.id (F.comp G) :=\nrfl\n@[simp] lemma whisker_left_id' (F : C ⥤ D) {G : D ⥤ E} :\n  whisker_left F (𝟙 G) = 𝟙 (F.comp G) :=\nrfl\n\n@[simp] lemma whisker_right_id {G : C ⥤ D} (F : D ⥤ E) :\n  whisker_right (nat_trans.id G) F = nat_trans.id (G.comp F) :=\n((whiskering_right C D E).obj F).map_id _\n@[simp] lemma whisker_right_id' {G : C ⥤ D} (F : D ⥤ E) :\n  whisker_right (𝟙 G) F = 𝟙 (G.comp F) :=\n((whiskering_right C D E).obj F).map_id _\n\n@[simp] lemma whisker_left_comp (F : C ⥤ D) {G H K : D ⥤ E} (α : G ⟶ H) (β : H ⟶ K) :\n  whisker_left F (α ≫ β) = (whisker_left F α) ≫ (whisker_left F β) :=\nrfl\n\n@[simp] lemma whisker_right_comp {G H K : C ⥤ D} (α : G ⟶ H) (β : H ⟶ K) (F : D ⥤ E)  :\n  whisker_right (α ≫ β) F = (whisker_right α F) ≫ (whisker_right β F) :=\n((whiskering_right C D E).obj F).map_comp α β\n\n/--\nIf `α : G ≅ H` is a natural isomorphism then\n`iso_whisker_left F α : (F ⋙ G) ≅ (F ⋙ H)` has components `α.app (F.obj X)`.\n-/\ndef iso_whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) : (F ⋙ G) ≅ (F ⋙ H) :=\n((whiskering_left C D E).obj F).map_iso α\n@[simp] lemma iso_whisker_left_hom (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) :\n  (iso_whisker_left F α).hom = whisker_left F α.hom :=\nrfl\n@[simp] lemma iso_whisker_left_inv (F : C ⥤ D) {G H : D ⥤ E} (α : G ≅ H) :\n  (iso_whisker_left F α).inv = whisker_left F α.inv :=\nrfl\n\n/--\nIf `α : G ≅ H` then\n`iso_whisker_right α F : (G ⋙ F) ≅ (H ⋙ F)` has components `F.map_iso (α.app X)`.\n-/\ndef iso_whisker_right {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) : (G ⋙ F) ≅ (H ⋙ F) :=\n((whiskering_right C D E).obj F).map_iso α\n@[simp] lemma iso_whisker_right_hom {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) :\n  (iso_whisker_right α F).hom = whisker_right α.hom F :=\nrfl\n@[simp] lemma iso_whisker_right_inv {G H : C ⥤ D} (α : G ≅ H) (F : D ⥤ E) :\n  (iso_whisker_right α F).inv = whisker_right α.inv F :=\nrfl\n\ninstance is_iso_whisker_left (F : C ⥤ D) {G H : D ⥤ E} (α : G ⟶ H) [is_iso α] :\n  is_iso (whisker_left F α) :=\nis_iso.of_iso (iso_whisker_left F (as_iso α))\ninstance is_iso_whisker_right {G H : C ⥤ D} (α : G ⟶ H) (F : D ⥤ E) [is_iso α] :\n  is_iso (whisker_right α F) :=\nis_iso.of_iso (iso_whisker_right (as_iso α) F)\n\nvariables {B : Type u₄} [category.{v₄} B]\n\nlocal attribute [elab_simple] whisker_left whisker_right\n\n@[simp] lemma whisker_left_twice (F : B ⥤ C) (G : C ⥤ D) {H K : D ⥤ E} (α : H ⟶ K) :\n  whisker_left F (whisker_left G α) = whisker_left (F ⋙ G) α :=\nrfl\n\n@[simp] lemma whisker_right_twice {H K : B ⥤ C} (F : C ⥤ D) (G : D ⥤ E) (α : H ⟶ K) :\n  whisker_right (whisker_right α F) G = whisker_right α (F ⋙ G) :=\nrfl\n\nlemma whisker_right_left (F : B ⥤ C) {G H : C ⥤ D} (α : G ⟶ H) (K : D ⥤ E) :\n  whisker_right (whisker_left F α) K = whisker_left F (whisker_right α K) :=\nrfl\nend\n\nnamespace functor\n\nuniverses u₅ v₅\n\nvariables {A : Type u₁} [category.{v₁} A]\nvariables {B : Type u₂} [category.{v₂} B]\n\n/--\nThe left unitor, a natural isomorphism `((𝟭 _) ⋙ F) ≅ F`.\n-/\n@[simps] def left_unitor (F : A ⥤ B) : ((𝟭 A) ⋙ F) ≅ F :=\n{ hom := { app := λ X, 𝟙 (F.obj X) },\n  inv := { app := λ X, 𝟙 (F.obj X) } }\n\n/--\nThe right unitor, a natural isomorphism `(F ⋙ (𝟭 B)) ≅ F`.\n-/\n@[simps] def right_unitor (F : A ⥤ B) : (F ⋙ (𝟭 B)) ≅ F :=\n{ hom := { app := λ X, 𝟙 (F.obj X) },\n  inv := { app := λ X, 𝟙 (F.obj X) } }\n\nvariables {C : Type u₃} [category.{v₃} C]\nvariables {D : Type u₄} [category.{v₄} D]\n\n/--\nThe associator for functors, a natural isomorphism `((F ⋙ G) ⋙ H) ≅ (F ⋙ (G ⋙ H))`.\n\n(In fact, `iso.refl _` will work here, but it tends to make Lean slow later,\nand it's usually best to insert explicit associators.)\n-/\n@[simps] def associator (F : A ⥤ B) (G : B ⥤ C) (H : C ⥤ D) : ((F ⋙ G) ⋙ H) ≅ (F ⋙ (G ⋙ H)) :=\n{ hom := { app := λ _, 𝟙 _ },\n  inv := { app := λ _, 𝟙 _ } }\n\nlemma triangle (F : A ⥤ B) (G : B ⥤ C) :\n  (associator F (𝟭 B) G).hom ≫ (whisker_left F (left_unitor G).hom) =\n    (whisker_right (right_unitor F).hom G) :=\nby { ext, dsimp, simp }  -- See note [dsimp, simp].\n\nvariables {E : Type u₅} [category.{v₅} E]\n\nvariables (F : A ⥤ B) (G : B ⥤ C) (H : C ⥤ D) (K : D ⥤ E)\n\nlemma pentagon :\n  (whisker_right (associator F G H).hom K) ≫\n    (associator F (G ⋙ H) K).hom ≫\n    (whisker_left F (associator G H K).hom) =\n  ((associator (F ⋙ G) H K).hom ≫ (associator F G (H ⋙ K)).hom) :=\nby { ext, dsimp, simp }\n\nend functor\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/whiskering.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3437045784511603}}
{"text": "import for_mathlib.category_theory.triangulated.homological_functor\nimport for_mathlib.category_theory.shift_op\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category limits\n\nnamespace functor\n\nvariables {C₁ C₂ D : Type*} [category C₁] [category C₂] [category D]\n\nsection\n\nvariables (F : C₁ ⥤ C₂ ⥤ D) {A G : Type*} [add_comm_monoid A]\n  [has_shift C₁ A] [has_shift C₂ A]\n\ndef shift_swap (a : A) :=\n  (shift_functor C₁ a) ⋙ F ≅ F ⋙ (whiskering_left _ _ D).obj (shift_functor C₂ a)\n\nnamespace shift_swap\n\nvariable (A)\n\ndef zero : shift_swap F (0 : A) :=\niso_whisker_right (shift_functor_zero C₁ A) F ≪≫ F.left_unitor ≪≫\n  F.right_unitor.symm ≪≫\n  iso_whisker_left F ((whiskering_left_id _ _).symm ≪≫\n    (whiskering_left _ _ D).map_iso (shift_functor_zero C₂ A).symm)\n\nlemma zero_hom_app_app (X₁ : C₁) (X₂ : C₂) :\n  ((zero F A).hom.app X₁).app X₂ =\n  (F.map ((shift_functor_zero C₁ A).hom.app X₁)).app X₂ ≫\n    (F.obj X₁).map ((shift_functor_zero C₂ A).inv.app X₂) :=\nbegin\n  dsimp only [zero, left_unitor, right_unitor, iso.trans, iso_whisker_right, iso_whisker_left,\n    whiskering_right, whiskering_left, iso.symm, nat_trans.comp_app, whiskering_left_id,\n    nat_iso.of_components, map_iso, whisker_left, whisker_right],\n  erw [id_comp, id_comp, id_comp],\nend\n\nlemma zero_inv_app_app (X₁ : C₁) (X₂ : C₂) :\n  ((zero F A).inv.app X₁).app X₂ =\n  (F.obj X₁).map ((shift_functor_zero C₂ A).hom.app X₂) ≫\n  (F.map ((shift_functor_zero C₁ A).inv.app X₁)).app X₂ :=\nbegin\n  dsimp only [zero, left_unitor, right_unitor, iso.trans, iso_whisker_right, iso_whisker_left,\n    whiskering_right, whiskering_left, iso.symm, nat_trans.comp_app, whiskering_left_id,\n    nat_iso.of_components, map_iso, whisker_left, whisker_right],\n  erw [comp_id, comp_id, comp_id],\nend\n\nvariables {F A}\n\ndef add' {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b) (c : A) (h : c = a+b) :\n  shift_swap F c :=\niso_whisker_right (shift_functor_add' C₁ a b c h) F ≪≫ functor.associator _ _ _ ≪≫\n  iso_whisker_left _ e₂ ≪≫ (functor.associator _ _ _).symm ≪≫\n  iso_whisker_right e₁ _ ≪≫ functor.associator _ _ _ ≪≫\n  iso_whisker_left F ((whiskering_left _ _ D).map_iso\n    (shift_functor_add' C₂ b a c (by rw [h, add_comm])).symm)\n\ndef add {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b) :\n  shift_swap F (a+b) := add' e₁ e₂ (a+b) rfl\n\nlemma add'_hom_app_app {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b)\n  (c : A) (h : c = a+b) (X₁ : C₁) (X₂ : C₂) :\n  ((add' e₁ e₂ c h).hom.app X₁).app X₂ =\n    (F.map ((shift_functor_add' C₁ a b c h).hom.app X₁)).app X₂ ≫\n    (e₂.hom.app (X₁⟦a⟧)).app X₂ ≫ (e₁.hom.app X₁).app (X₂⟦b⟧) ≫\n    (F.obj X₁).map ((shift_functor_add' C₂ b a c (by rw [h, add_comm])).inv.app X₂) :=\nbegin\n  dsimp [add'],\n  erw [id_comp, id_comp, id_comp],\nend\n\nlemma add_hom_app_app {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b)\n   (X₁ : C₁) (X₂ : C₂) :\n  ((add e₁ e₂).hom.app X₁).app X₂ =\n    (F.map ((shift_functor_add C₁ a b).hom.app X₁)).app X₂ ≫\n    (e₂.hom.app (X₁⟦a⟧)).app X₂ ≫ (e₁.hom.app X₁).app (X₂⟦b⟧) ≫\n    (F.obj X₁).map\n      ((shift_functor_add' C₂ b a (a+b) (add_comm a b)).inv.app X₂) :=\nbegin\n  dsimp only [add],\n  rw [add'_hom_app_app, shift_functor_add'_eq_shift_functor_add],\nend\n\nlemma add'_inv_app_app {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b)\n  (c : A) (h : c = a+b) (X₁ : C₁) (X₂ : C₂) :\n  ((add' e₁ e₂ c h).inv.app X₁).app X₂ =\n    (F.obj X₁).map ((shift_functor_add' C₂ b a c (by rw [h, add_comm])).hom.app X₂) ≫\n    (e₁.inv.app X₁).app (X₂⟦b⟧) ≫\n    (e₂.inv.app (X₁⟦a⟧)).app X₂ ≫\n    (F.map ((shift_functor_add' C₁ a b c h).inv.app X₁)).app X₂ :=\nbegin\n  dsimp [add'],\n  erw [comp_id, comp_id, comp_id, assoc, assoc],\nend\n\nlemma add_inv_app_app {a b : A} (e₁ : shift_swap F a) (e₂ : shift_swap F b)\n  (X₁ : C₁) (X₂ : C₂) :\n  ((add e₁ e₂).inv.app X₁).app X₂ =\n    (F.obj X₁).map ((shift_functor_add' C₂ b a _ (add_comm a b)).hom.app X₂) ≫\n    (e₁.inv.app X₁).app (X₂⟦b⟧) ≫\n    (e₂.inv.app (X₁⟦a⟧)).app X₂ ≫\n    (F.map ((shift_functor_add C₁ a b).inv.app X₁)).app X₂ :=\nbegin\n  dsimp only [add],\n  rw [add'_inv_app_app, shift_functor_add'_eq_shift_functor_add],\nend\n\nend shift_swap\n\nvariable (A)\n\nclass has_shift_swap :=\n(iso [] : Π (a : A), F.shift_swap a)\n(iso_zero [] : iso 0 = shift_swap.zero F A)\n(iso_add [] : ∀ (a b : A), iso (a+b) = shift_swap.add (iso a) (iso b))\n\nvariable {A}\n\ndef has_shift_swap.iso' [F.has_shift_swap A] (a b c : A) (h : c = a + b) :\n  (shift_functor C₁ a) ⋙ F ⋙ (whiskering_left _ _ D).obj (shift_functor C₂ b) ≅\n    F ⋙ (whiskering_left _ _ D).obj (shift_functor C₂ c) :=\n(functor.associator _ _ _).symm ≪≫ iso_whisker_right (has_shift_swap.iso F a) _ ≪≫\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ ((whiskering_left C₂ C₂ D).map_iso (shift_functor_add' C₂ b a c (by rw [h, add_comm])).symm)\n\nnamespace has_shift_swap\n\nvariable [F.has_shift_swap A]\n\nlemma iso'_hom_app_app (a b c : A) (h : c = a + b) (X₁ : C₁) (X₂ : C₂ ):\n  ((iso' F a b c h).hom.app X₁).app X₂ =\n    ((iso F a).hom.app X₁).app ((shift_functor C₂ b).obj X₂) ≫\n      (F.obj X₁).map ((shift_functor_add' C₂ b a c (by rw [h, add_comm])).inv.app X₂) :=\nbegin\n  dsimp [iso'],\n  rw [id_comp, id_comp],\nend\n\nlemma iso'_zero_hom_app_app (b : A) (X₁ : C₁) (X₂ : C₂ ):\n  ((iso' F 0 b b (zero_add b).symm).hom.app X₁).app X₂ =\n    (F.map ((shift_functor_zero C₁ A).hom.app X₁)).app (X₂⟦b⟧) :=\nbegin\n  simp only [iso'_hom_app_app, has_shift_swap.iso_zero,\n    shift_swap.zero_hom_app_app, assoc, ← functor.map_comp],\n  convert comp_id _,\n  convert functor.map_id _ _,\n  sorry,\nend\n\nlemma iso'_hom_app_app_comp (a b c a' d e : A) (hc : c = a + b) (he : e = a'+a) (hd : d = a'+c)\n   (X₁ : C₁) (X₂ : C₂ ) :\n  ((iso' F a b c hc).hom.app (X₁⟦a'⟧)).app X₂ ≫ ((iso' F a' c d hd).hom.app X₁).app X₂ =\n    (F.map ((shift_functor_add' C₁ a' a e he).inv.app X₁)).app (X₂⟦b⟧) ≫\n      ((iso' F e b d (by rw [hd, he, hc, add_assoc])).hom.app X₁).app X₂ :=\nbegin\n  sorry,\nend\n\nend has_shift_swap\n\nend\n\nsection\n\nvariables {C : Type*} [category C] [has_shift C ℤ]\n  [preadditive C] [∀ (n : ℤ), (shift_functor C n).additive]\n\nlocal attribute [instance] has_shift_op_neg_ℤ\n\n@[simps]\ndef preadditive_yoneda_shift_swap_add_equiv (X₁ X₂ : C) (n : ℤ) :\n  (X₂ ⟶ X₁⟦n⟧) ≃+ (X₂⟦-n⟧ ⟶ X₁) :=\n{ to_fun := λ f, f⟦-n⟧' ≫ (shift_equiv C n).unit_iso.inv.app X₁,\n  inv_fun := λ f, (shift_equiv C n).counit_iso.inv.app X₂ ≫ f⟦n⟧',\n  left_inv := λ f, begin\n    dsimp only,\n    erw [functor.map_comp, ← nat_trans.naturality_assoc,\n      (shift_equiv C n).counit_inv_functor_comp, comp_id],\n    refl,\n  end,\n  right_inv := λ f, begin\n    dsimp only,\n    erw [functor.map_comp, assoc],\n    erw (shift_equiv C n).unit_iso.inv.naturality f,\n    slice_lhs 1 2 { erw (shift_equiv C n).inverse_counit_inv_comp, },\n    erw id_comp,\n    refl,\n  end,\n  map_add' := λ f₁ f₂, by rw [functor.map_add, preadditive.add_comp], }\n\nvariable (C)\n\n@[simps]\ndef preadditive_yoneda_shift_swap (n : ℤ) :\n  shift_swap (preadditive_yoneda : C ⥤ _) n :=\nnat_iso.of_components (λ X₁, nat_iso.of_components\n  (λ X₂, add_equiv.to_AddCommGroup_iso (preadditive_yoneda_shift_swap_add_equiv X₁ X₂.unop n))\n  (by tidy)) (by tidy)\n\n-- this would be better with coyoneda instead\n\ndef hom_functor_has_shift_swap : (preadditive_yoneda : C ⥤ _).has_shift_swap ℤ :=\n{ iso := preadditive_yoneda_shift_swap C,\n  iso_zero := begin\n    ext X₁ X₂ x,\n    dsimp at x,\n    rw shift_swap.zero_hom_app_app,\n    dsimp only [preadditive_yoneda_shift_swap, nat_iso.of_components,\n      preadditive_yoneda_shift_swap_add_equiv, add_equiv.to_AddCommGroup_iso,\n      add_equiv.to_add_monoid_hom, add_monoid_hom.coe_mk],\n    rw [comp_apply, preadditive_yoneda_map_app_apply],\n    erw preadditive_yoneda_obj_map_apply,\n    --let y := ((shift_functor_zero Cᵒᵖ ℤ).inv.app X₂).unop,\n    --let y' := (shift_functor_zero C ℤ).hom.app (opposite.unop X₂),\n    --change _ = y ≫ _,\n    sorry,\n  end,\n  iso_add := sorry, }\n\nend\n\nsection\n\nvariables (F : C₁ ⥤ C₂ ⥤ D) [has_shift C₁ ℤ] [has_shift C₂ ℤ] [abelian D]\n  [preadditive C₁] [has_zero_object C₁] [∀ (n : ℤ), (shift_functor C₁ n).additive]\n  [pretriangulated C₁]\n  [∀ (Y₂ : C₂), (((whiskering_right _ _ _).obj\n    ((evaluation C₂ D).obj Y₂)).obj F).preserves_zero_morphisms]\n  [∀ (Y₂ : C₂), (((whiskering_right _ _ _).obj\n    ((evaluation C₂ D).obj Y₂)).obj F).is_homological]\n\nlemma is_homological.bifunctor_map_distinguished (T : pretriangulated.triangle C₁)\n  (hT : T ∈ dist_triang C₁) (Y₂ : C₂) :\n  ((T.short_complex hT).map (((whiskering_right C₁ (C₂ ⥤ D) D).obj\n    ((evaluation C₂ D).obj Y₂)).obj F)).exact :=\nis_homological.map_distinguished (((whiskering_right _ _ _).obj\n  ((evaluation C₂ D).obj Y₂)).obj F) T hT\n\nlemma is_homological.bifunctor_ex₁₂ (T : pretriangulated.triangle C₁)\n  (hT : T ∈ dist_triang C₁) (Y₂ : C₂) (n : ℤ) :\n  (short_complex.mk ((F.map T.mor₁).app (Y₂⟦n⟧)) ((F.map T.mor₂).app (Y₂⟦n⟧))\n    (by simpa only [← nat_trans.comp_app, ← F.map_comp, pretriangulated.triangle.comp_zero₁₂ T hT]\n        using (((whiskering_right _ _ _).obj ((evaluation C₂ D).obj _)).obj F).map_zero T.obj₁ T.obj₃)).exact :=\nis_homological.map_distinguished (((whiskering_right _ _ _).obj\n  ((evaluation C₂ D).obj (Y₂⟦n⟧))).obj F) T hT\n\nvariables [F.has_shift_swap ℤ]\n\ndef is_homological.bifunctor_δ₁ (T : pretriangulated.triangle C₁) (hT : T ∈ dist_triang C₁)\n  (Y₂ : C₂) (n₀ n₁ : ℤ) (h : n₁ = n₀+1) :\n  (F.obj T.obj₃).obj (Y₂⟦n₀⟧) ⟶ (F.obj T.obj₁).obj (Y₂⟦n₁⟧) :=\n(F.map T.mor₃).app (Y₂⟦n₀⟧) ≫\n  ((has_shift_swap.iso' F 1 n₀ n₁ (by rw [h, add_comm])).hom.app T.obj₁).app Y₂\n\nlemma is_homological.bifunctor_comp_δ₁\n  (T : pretriangulated.triangle C₁) (hT : T ∈ dist_triang C₁)\n  (Y₂ : C₂) (n₀ n₁ : ℤ) (h : n₁ = n₀+1) :\n  (F.map T.mor₂).app (Y₂⟦n₀⟧) ≫ is_homological.bifunctor_δ₁ F T hT Y₂ _ _ h = 0 :=\nbegin\n  dsimp only [is_homological.bifunctor_δ₁],\n  rw ← assoc,\n  convert zero_comp,\n  rw [← nat_trans.comp_app, ← F.map_comp, pretriangulated.triangle.comp_zero₂₃ T hT],\n  exact (((whiskering_right _ _ _).obj ((evaluation C₂ D).obj _)).obj F).map_zero _ _,\nend\n\nlemma is_homological.bifunctor_ex₁₃\n  (T : pretriangulated.triangle C₁) (hT : T ∈ dist_triang C₁)\n  (Y₂ : C₂) (n₀ n₁ : ℤ) (h : n₁ = n₀+1) :\n  (short_complex.mk _ _ (is_homological.bifunctor_comp_δ₁ F T hT Y₂ _ _ h)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1 (is_homological.map_distinguished\n    (((whiskering_right _ _ _).obj ((evaluation C₂ D).obj (Y₂⟦n₀⟧))).obj F)\n    _ ((pretriangulated.rotate_distinguished_triangle T).1 hT)),\n  refine short_complex.mk_iso (iso.refl _) (iso.refl _)\n    (((has_shift_swap.iso' F 1 n₀ n₁ (by rw [h, add_comm])).app T.obj₁).app Y₂) _ _,\n  { dsimp, rw [id_comp, comp_id], },\n  { dsimp,\n    rw id_comp,\n    refl, },\nend\n\ndef is_homological.bifunctor_δ₁_comp\n  (T : pretriangulated.triangle C₁) (hT : T ∈ dist_triang C₁)\n  (Y₂ : C₂) (n₀ n₁ : ℤ) (h : n₁ = n₀+1) :\n  is_homological.bifunctor_δ₁ F T hT Y₂ _ _ h ≫ (F.map T.mor₁).app (Y₂⟦n₁⟧) = 0 :=\nbegin\n  dsimp only [is_homological.bifunctor_δ₁],\n  have eq := congr_app ((has_shift_swap.iso' F 1 n₀ n₁ (by rw [h, add_comm])).hom.naturality T.mor₁) Y₂,\n  dsimp at eq,\n  rw [assoc, ← eq, ← assoc],\n  convert zero_comp,\n  rw [← nat_trans.comp_app, ← F.map_comp, pretriangulated.triangle.comp_zero₃₁ T hT],\n  exact (((whiskering_right _ _ _).obj ((evaluation C₂ D).obj _)).obj F).map_zero _ _,\nend\n\nlemma is_homological.bifunctor_ex₁₁\n  (T : pretriangulated.triangle C₁) (hT : T ∈ dist_triang C₁)\n  (Y₂ : C₂) (n₀ n₁ : ℤ) (h : n₁ = n₀+1) :\n  (short_complex.mk _ _ (is_homological.bifunctor_δ₁_comp F T hT Y₂ _ _ h)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1 (is_homological.map_distinguished\n    (((whiskering_right _ _ _).obj ((evaluation C₂ D).obj (Y₂⟦n₁⟧))).obj F)\n    _ ((pretriangulated.inv_rotate_distinguished_triangle T).2 hT)),\n  refine short_complex.mk_iso (preadditive.mul_iso (-1) ((((has_shift_swap.iso' F (-1) n₁ n₀ (by linarith)).app T.obj₃)).app Y₂)) (iso.refl _) (iso.refl _) _ _,\n  { dsimp only [preadditive.mul_iso, iso.refl,\n      pretriangulated.triangle.inv_rotate, pretriangulated.triangle.mk,\n      short_complex.map, whiskering_right, evaluation, functor.comp,\n      pretriangulated.triangle.short_complex,\n      pretriangulated.candidate_triangle.of_distinguished,\n      pretriangulated.candidate_triangle.short_complex,\n      is_homological.bifunctor_δ₁],\n    have eq := congr_app ((has_shift_swap.iso' F (-1) n₁ n₀ (by linarith)).hom.naturality T.mor₃) Y₂,\n    dsimp at eq,\n    rw [comp_id, units.coe_neg_one, neg_smul, one_smul],\n    nth_rewrite 0 preadditive.neg_comp,\n    erw [← reassoc_of eq, ← preadditive.neg_comp, ← preadditive.neg_comp, F.map_comp],\n    conv_rhs { rw nat_trans.comp_app, },\n    congr' 1,\n    { exact (functor.map_neg (((whiskering_right _ _ _).obj\n        ((evaluation C₂ D).obj (Y₂⟦n₁⟧))).obj F)).symm, },\n    { rw [has_shift_swap.iso'_hom_app_app_comp F (-1) n₁ n₀ 1 n₁ 0 (by linarith)\n        (by linarith) (by linarith), has_shift_swap.iso'_zero_hom_app_app,\n        ← nat_trans.comp_app, ← F.map_comp,\n        ← shift_functor_comp_shift_functor_neg_eq_add'_comp_zero],\n      refl, }, },\n  { dsimp,\n    rw [id_comp, comp_id], },\nend\n\nend\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/triangulated/homological_bifunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3434701827220722}}
{"text": "import data.finset.basic\nimport data.fintype.basic \nimport .extensionality\n\nnamespace extensionality\n\nuniverse u \n\ninstance finset_ext_lemmas (T : Type*) [decidable_eq T] :\n  (boolalg_ext_lemmas (finset T) T) :=\n{\n  simpl_eq := by tidy,\n  simpl_lt := by tidy,\n  ext_bot := by tidy,\n  ext_sdiff := by tidy,\n  ext_le := by tidy, \n  ext_meet := by tidy,\n  ext_join := by simp only [finset.inf_eq_inter, forall_const, iff_self, finset.mem_inter, forall_true_iff],\n}\n\ninstance finset_ext_lemmas_compl (T : Type*) [fintype T] [decidable_eq T] :\n  (boolalg_ext_lemmas_compl (finset T) T) :=\n{\n  ext_compl := by apply finset.mem_compl\n}\n\ninstance finset_ext_lemmas_top (T : Type*) [fintype T] [decidable_eq T] :\n  (boolalg_ext_lemmas_top (finset T) T) :=\n{\n  ext_top := by unfold_projs; finish,\n}\nend extensionality\n\nnamespace cleanup \nlemma finset_union_sup (T : Type*) [decidable_eq T] (A B : finset T) : (A ∪ B) = (A ⊔ B) := by refl\nlemma finset_inter_inf (T : Type*) [decidable_eq T] (A B : finset T) : (A ∩ B) = (A ⊓ B) := by refl\nlemma finset_subset_le (T : Type*) [decidable_eq T] (A B : finset T) : (A ⊆ B) = (A ≤ B) := by refl\nlemma finset_subset_lt (T : Type*) [decidable_eq T] (A B : finset T) : (A ⊂ B) = (A < B) := by refl\nlemma finset_univ_top (T : Type*) [decidable_eq T] [fintype T] : (finset.univ : finset T) = ⊤ := by refl\nlemma finset_empt_bot (T : Type*) [decidable_eq T] : (∅ : finset T) = ⊥ := by refl \n\nmeta def finset_cleanup : tactic unit :=\n  `[simp only [cleanup.finset_union_sup, cleanup.finset_inter_inf, cleanup.finset_subset_le,\n    cleanup.finset_subset_lt, cleanup.finset_univ_top, cleanup.finset_empt_bot] at *]\nend cleanup", "meta": {"author": "apnelson1", "repo": "lean-matroids", "sha": "1880f5b4b9a99948368e35539672a9bf35cc09cc", "save_path": "github-repos/lean/apnelson1-lean-matroids", "path": "github-repos/lean/apnelson1-lean-matroids/lean-matroids-1880f5b4b9a99948368e35539672a9bf35cc09cc/src/old/old_aux/set_tactic/finset_tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34342016150141097}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Simon Hudon, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.control.bifunctor\nimport Mathlib.PostPort\n\nuniverses u v w \n\nnamespace Mathlib\n\n/-!\n# Functor and bifunctors can be applied to `equiv`s.\n\nWe define\n```lean\ndef functor.map_equiv (f : Type u → Type v) [functor f] [is_lawful_functor f] :\n  α ≃ β → f α ≃ f β\n```\nand\n```lean\ndef bifunctor.map_equiv (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] :\n  α ≃ β → α' ≃ β' → F α α' ≃ F β β'\n```\n-/\n\nnamespace functor\n\n\n/-- Apply a functor to an `equiv`. -/\ndef map_equiv {α : Type u} {β : Type u} (f : Type u → Type v) [Functor f] [is_lawful_functor f] (h : α ≃ β) : f α ≃ f β :=\n  equiv.mk (Functor.map ⇑h) (Functor.map ⇑(equiv.symm h)) sorry sorry\n\n@[simp] theorem map_equiv_apply {α : Type u} {β : Type u} (f : Type u → Type v) [Functor f] [is_lawful_functor f] (h : α ≃ β) (x : f α) : coe_fn (map_equiv f h) x = ⇑h <$> x :=\n  rfl\n\n@[simp] theorem map_equiv_symm_apply {α : Type u} {β : Type u} (f : Type u → Type v) [Functor f] [is_lawful_functor f] (h : α ≃ β) (y : f β) : coe_fn (equiv.symm (map_equiv f h)) y = ⇑(equiv.symm h) <$> y :=\n  rfl\n\n@[simp] theorem map_equiv_refl {α : Type u} (f : Type u → Type v) [Functor f] [is_lawful_functor f] : map_equiv f (equiv.refl α) = equiv.refl (f α) := sorry\n\nend functor\n\n\nnamespace bifunctor\n\n\n/-- Apply a bifunctor to a pair of `equiv`s. -/\ndef map_equiv {α : Type u} {β : Type u} {α' : Type v} {β' : Type v} (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] (h : α ≃ β) (h' : α' ≃ β') : F α α' ≃ F β β' :=\n  equiv.mk (bimap ⇑h ⇑h') (bimap ⇑(equiv.symm h) ⇑(equiv.symm h')) sorry sorry\n\n@[simp] theorem map_equiv_apply {α : Type u} {β : Type u} {α' : Type v} {β' : Type v} (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] (h : α ≃ β) (h' : α' ≃ β') (x : F α α') : coe_fn (map_equiv F h h') x = bimap (⇑h) (⇑h') x :=\n  rfl\n\n@[simp] theorem map_equiv_symm_apply {α : Type u} {β : Type u} {α' : Type v} {β' : Type v} (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] (h : α ≃ β) (h' : α' ≃ β') (y : F β β') : coe_fn (equiv.symm (map_equiv F h h')) y = bimap (⇑(equiv.symm h)) (⇑(equiv.symm h')) y :=\n  rfl\n\n@[simp] theorem map_equiv_refl_refl {α : Type u} {α' : Type v} (F : Type u → Type v → Type w) [bifunctor F] [is_lawful_bifunctor F] : map_equiv F (equiv.refl α) (equiv.refl α') = equiv.refl (F α α') := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277512, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34342016150141097}}
{"text": "/-\nCopyright (c) 2020 Gabriel Ebner, Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner, Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.ext\nimport Mathlib.Lean3Lib.data.stream\nimport Mathlib.data.list.basic\nimport Mathlib.data.list.range\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Additional instances and attributes for streams\n-/\n\nprotected instance stream.inhabited {α : Type u_1} [Inhabited α] : Inhabited (stream α) :=\n  { default := stream.const Inhabited.default }\n\nnamespace stream\n\n\n/-- `take s n` returns a list of the `n` first elements of stream `s` -/\ndef take {α : Type u_1} (s : stream α) (n : ℕ) : List α :=\n  list.map s (list.range n)\n\ntheorem length_take {α : Type u_1} (s : stream α) (n : ℕ) : list.length (take s n) = n := sorry\n\n/-- Use a state monad to generate a stream through corecursion -/\ndef corec_state {σ : Type u_1} {α : Type u_1} (cmd : state σ α) (s : σ) : stream α :=\n  corec prod.fst (state_t.run cmd ∘ prod.snd) (state_t.run cmd s)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/stream/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34334890928627115}}
{"text": "import number_theory.bernoulli\n\n\n--OUTPUT 1\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  sorry,\nend\n\n--OUTPUT 2\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h2 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h3 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h4 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h5 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h6 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h7 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h8 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h9 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h10 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h11 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h12 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h13 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h14 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h15 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h16 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h17 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h18 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h19 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h20 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h21 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h22 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h23 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h24 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h25 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h26 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h27 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h28 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h29 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h30 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h31 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h32 : ∀ m : ℕ, (m < n) → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h33 : ∀ m : ℕ,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  sorry\nend\n\n--OUTPUT 4\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(-1),\n    {\n      rw polynomial.bernoulli,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C,\n      rw polynomial.eval_C\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  sorry,\nend\n\n--OUTPUT 6\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : ∀ m : ℕ, m < n → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h2 : ∀ m : ℕ, (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from sorry,\n  have h3 : (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from sorry,\n  show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from sorry,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nBernoulli polynomial evaluation\nGiven a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$\n`proof`\nWe apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\nMultiplying both sides by $d + 1$, and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n$$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\nThe conclusion then follows easily.\n\nQED\n\n-/\ntheorem  bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/Correct_statement-lean_proof_outline-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Bernoulli polynomial evaluation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8198933491161063, "lm_q2_score": 0.41869690935568665, "lm_q1q2_score": 0.3432868112761967}}
{"text": "#exit\n\nimport Structure.Generic.Axioms\nimport Structure.Generic.DerivedFunctors\n\nimport mathlib4_experiments.Data.Equiv.Basic\n\nopen GeneralizedRelation\n\n\n\nset_option autoBoundImplicitLocal false\n\nuniverses u v w\n\n\n\nnamespace GeneralizedProperty\n\n  def ConstProp (α : Sort u) {V : Universe} {β : V} (c : β) : GeneralizedProperty α V := λ _ => β\n\n  namespace ConstProp\n\n    variable (α : Sort u) {V : Universe} {β : V} (c : β)\n\n    instance hasInst : HasInst (ConstProp α c) := ⟨λ _ => c⟩\n\n  end ConstProp\n\nend GeneralizedProperty\n\n\n\nnamespace GeneralizedRelation\n\n  def ConstRel (α : Sort u) {V : Universe} {β : V} (c : β) : GeneralizedRelation α V := λ _ _ => β\n\n  namespace ConstRel\n  \n    variable (α : Sort u) {V : Universe} {β : V} (c : β)\n\n    instance hasRefl  : HasRefl  (ConstRel α c) := ⟨λ _ => c⟩\n\n    variable [HasInternalFunctors V] [HasAffineFunOp V]\n\n    instance hasSymm  : HasSymm  (ConstRel α c) := ⟨HasLinearFunOp.idFun β⟩\n    instance hasTrans : HasTrans (ConstRel α c) := ⟨HasAffineFunOp.constFun β (HasLinearFunOp.idFun β)⟩\n\n    instance isPreorder    : IsPreorder    (ConstRel α c) := ⟨⟩\n    instance isEquivalence : IsEquivalence (ConstRel α c) := ⟨⟩\n\n    @[simp] theorem symmEq  {a₁ a₂    : α} : (hasSymm  α c).symm'  (a := a₁) (b := a₂)           c   = c :=\n    HasLinearFunOp.idFun.eff β c\n    @[simp] theorem transEq {a₁ a₂ a₃ : α} : (hasTrans α c).trans' (a := a₁) (b := a₂) (c := a₃) c c = c :=\n    let h₁ := congrArg HasInternalFunctors.funCoe (HasAffineFunOp.constFun.eff β (HasLinearFunOp.idFun β) c);\n    Eq.trans (congrFun h₁ c) (HasLinearFunOp.idFun.eff β c)\n\n  end ConstRel\n\nend GeneralizedRelation\n\nopen GeneralizedRelation\n\n\n\n-- TODO: Should we use actual abstract equivalence here?\n\nstructure Iff' {U : Universe} [HasInternalFunctors U] (α β : U) where\n(mp  : α ⟶ β)\n(mpr : β ⟶ α)\n\ninfix:20 \" ⟷' \" => Iff'\n\n\n\nsection Morphisms\n\n  variable {α : Sort u} {U : Universe} [HasInternalFunctors U] [he : HasFunctorialEquivalences U]\n           [HasLinearFunOp he.equivUniverse] {R : GeneralizedRelation α U}\n\n  instance : HasFunctorialArrows U := HasFunctorialEquivalences.toHasFunctorialArrows U\n\n  open IsCompositionRelation\n\n  namespace IsCompositionRelation\n\n    variable [HasTrans R] [IsCompositionRelation R]\n\n    def comp_congrArg {a b c : α} {f₁ f₂ : R a b} {g₁ g₂ : R b c} : f₁ ≃ f₂ ⟶ g₁ ≃ g₂ ⟶ g₁ • f₁ ≃ g₂ • f₂ :=\n    he.equivCongr ⊙ HasFunctorialEquivalences.equiv_congrArg U HasTrans.trans\n\n    def comp_congrArg_left  {a b c : α} {f : R a b} {g₁ g₂ : R b c} : g₁ ≃ g₂ ⟶ g₁ • f ≃ g₂ • f :=\n    HasFunctorialEquivalences.equiv_congrArg U (HasTrans.trans ⟮f⟯)\n\n    def comp_congrArg_right {a b c : α} {f₁ f₂ : R a b} {g : R b c} : f₁ ≃ f₂ ⟶ g • f₁ ≃ g • f₂ :=\n    HasLinearFunOp.swapFun comp_congrArg (HasRefl.refl g)\n\n    def comp_subst  {a b c : α} {f₁ f₂ : R a b} {g₁ g₂ : R b c} {e : R a c} : f₁ ≃ f₂ ⟶ g₁ ≃ g₂ ⟶ g₂ • f₂ ≃ e ⟶ g₁ • f₁ ≃ e :=\n    HasLinearFunOp.compFun₂ comp_congrArg HasTrans.trans\n    def comp_subst' {a b c : α} {f₁ f₂ : R a b} {g₁ g₂ : R b c} {e : R a c} : f₁ ≃ f₂ ⟶ g₁ ≃ g₂ ⟶ e ≃ g₁ • f₁ ⟶ e ≃ g₂ • f₂ :=\n    HasLinearFunOp.compFun₂ comp_congrArg HasTrans.revTrans\n\n    def comp_subst_left   {a b c : α} (f : R a b) {g₁ g₂ : R b c} {e : R a c} : g₁ ≃ g₂ ⟶ g₂ • f ≃ e ⟶ g₁ • f ≃ e :=\n    HasTrans.trans    ⊙ comp_congrArg_left\n    def comp_subst_left'  {a b c : α} (f : R a b) {g₁ g₂ : R b c} {e : R a c} : g₁ ≃ g₂ ⟶ e ≃ g₁ • f ⟶ e ≃ g₂ • f :=\n    HasTrans.revTrans ⊙ comp_congrArg_left\n\n    def comp_subst_right  {a b c : α} {f₁ f₂ : R a b} (g : R b c) {e : R a c} : f₁ ≃ f₂ ⟶ g • f₂ ≃ e ⟶ g • f₁ ≃ e :=\n    HasTrans.trans    ⊙ comp_congrArg_right\n    def comp_subst_right' {a b c : α} {f₁ f₂ : R a b} (g : R b c) {e : R a c} : f₁ ≃ f₂ ⟶ e ≃ g • f₁ ⟶ e ≃ g • f₂ :=\n    HasTrans.revTrans ⊙ comp_congrArg_right\n\n    def applyAssocLR_left  {a b c d : α} {f : R a b} {g : R b c} {h : R c d} {e : R a d} : (h • g) • f ≃ e ⟶ h • (g • f) ≃ e :=\n    HasTrans.trans ⟮assocRL f g h⟯\n    def applyAssocRL_left  {a b c d : α} {f : R a b} {g : R b c} {h : R c d} {e : R a d} : h • (g • f) ≃ e ⟶ (h • g) • f ≃ e :=\n    HasTrans.trans ⟮assocLR f g h⟯\n    def applyAssocLR_right {a b c d : α} {f : R a b} {g : R b c} {h : R c d} {e : R a d} : e ≃ (h • g) • f ⟶ e ≃ h • (g • f) :=\n    HasTrans.revTrans'' (assocLR f g h)\n    def applyAssocRL_right {a b c d : α} {f : R a b} {g : R b c} {h : R c d} {e : R a d} : e ≃ h • (g • f) ⟶ e ≃ (h • g) • f :=\n    HasTrans.revTrans'' (assocRL f g h)\n\n    def applyAssocLR {a β₁ β₂ γ₁ γ₂ d : α} {f₁ : R a β₁} {f₂ : R a β₂} {g₁ : R β₁ γ₁} {g₂ : R β₂ γ₂} {h₁ : R γ₁ d} {h₂ : R γ₂ d} :\n      (h₁ • g₁) • f₁ ≃ (h₂ • g₂) • f₂ ⟶ h₁ • (g₁ • f₁) ≃ h₂ • (g₂ • f₂) :=\n    applyAssocLR_right ⊙ applyAssocLR_left\n    def applyAssocRL {a β₁ β₂ γ₁ γ₂ d : α} {f₁ : R a β₁} {f₂ : R a β₂} {g₁ : R β₁ γ₁} {g₂ : R β₂ γ₂} {h₁ : R γ₁ d} {h₂ : R γ₂ d} :\n      h₁ • (g₁ • f₁) ≃ h₂ • (g₂ • f₂) ⟶ (h₁ • g₁) • f₁ ≃ (h₂ • g₂) • f₂ :=\n    applyAssocRL_right ⊙ applyAssocRL_left\n\n  end IsCompositionRelation\n\n  namespace IsMorphismRelation\n\n    variable [IsPreorder R] [IsMorphismRelation R]\n\n    def leftCancelId  {a b : α} (f : R a b) {e : R b b} : e ≃ ident R b ⟶ e • f ≃ f :=\n    HasLinearFunOp.swapFun (comp_subst_left  f) (leftId  f)\n    def rightCancelId {a b : α} (f : R a b) {e : R a a} : e ≃ ident R a ⟶ f • e ≃ f :=\n    HasLinearFunOp.swapFun (comp_subst_right f) (rightId f)\n\n  end IsMorphismRelation\n\n  open IsMorphismRelation\n\n  namespace IsIsomorphismRelation\n\n    variable [IsEquivalence R] [IsIsomorphismRelation R]\n\n    def inv_congrArg {a b : α} {f₁ f₂ : R a b} : f₁ ≃ f₂ ⟶ f₁⁻¹ ≃ f₂⁻¹ :=\n    HasFunctorialEquivalences.equiv_congrArg U HasSymm.symm\n\n    def inv_subst  {a b : α} {f₁ f₂ : R a b} {e : R b a} : f₁ ≃ f₂ ⟶ f₂⁻¹ ≃ e ⟶ f₁⁻¹ ≃ e :=\n    HasTrans.trans    ⊙ inv_congrArg\n    def inv_subst' {a b : α} {f₁ f₂ : R a b} {e : R b a} : f₁ ≃ f₂ ⟶ e ≃ f₁⁻¹ ⟶ e ≃ f₂⁻¹ :=\n    HasTrans.revTrans ⊙ inv_congrArg\n\n    def leftCancel'     {a b c : α} (f : R a b) (g : R b c) : (g⁻¹ • g) • f ≃ f := (leftCancelId  f) (leftInv  g)\n    def leftCancel      {a b c : α} (f : R a b) (g : R b c) : g⁻¹ • (g • f) ≃ f := applyAssocLR_left ⟮leftCancel'     f g⟯\n    def leftCancelInv'  {a b c : α} (f : R a b) (g : R c b) : (g • g⁻¹) • f ≃ f := (leftCancelId  f) (rightInv g)\n    def leftCancelInv   {a b c : α} (f : R a b) (g : R c b) : g • (g⁻¹ • f) ≃ f := applyAssocLR_left ⟮leftCancelInv'  f g⟯\n    def rightCancel'    {a b c : α} (f : R a b) (g : R c a) : f • (g • g⁻¹) ≃ f := (rightCancelId f) (rightInv g)\n    def rightCancel     {a b c : α} (f : R a b) (g : R c a) : (f • g) • g⁻¹ ≃ f := applyAssocRL_left ⟮rightCancel'    f g⟯\n    def rightCancelInv' {a b c : α} (f : R a b) (g : R a c) : f • (g⁻¹ • g) ≃ f := (rightCancelId f) (leftInv  g)\n    def rightCancelInv  {a b c : α} (f : R a b) (g : R a c) : (f • g⁻¹) • g ≃ f := applyAssocRL_left ⟮rightCancelInv' f g⟯\n\n    def leftMulInv  {a b c : α} (f₁ : R a b) (f₂ : R a c) (g : R b c) : g • f₁ ≃ f₂ ⟷' f₁ ≃ g⁻¹ • f₂ :=\n    ⟨HasLinearFunOp.swapFun (comp_subst_right' g⁻¹) (HasSymm.symm' (leftCancel f₁ g)),\n     HasLinearFunOp.swapFun (comp_subst_right  g)   (leftCancelInv f₂ g)⟩\n    def leftMulInv' {a b c : α} (f₁ : R a b) (f₂ : R a c) (g : R c b) : g⁻¹ • f₁ ≃ f₂ ⟷' f₁ ≃ g • f₂ :=\n    ⟨HasLinearFunOp.swapFun (comp_subst_right' g)   (HasSymm.symm' (leftCancelInv f₁ g)),\n     HasLinearFunOp.swapFun (comp_subst_right  g⁻¹) (leftCancel f₂ g)⟩\n\n    def leftMul {a b c : α} (f₁ f₂ : R a b) (g : R b c) : g • f₁ ≃ g • f₂ ⟷' f₁ ≃ f₂ :=\n    ⟨HasTrans.revTrans'' (leftCancel f₂ g) ⊙ (leftMulInv f₁ (g • f₂) g).mp, comp_congrArg_right⟩\n\n    def rightMulInv  {a b c : α} (f₁ : R a c) (f₂ : R b c) (g : R b a) : f₁ • g ≃ f₂ ⟷' f₁ ≃ f₂ • g⁻¹ :=\n    ⟨HasLinearFunOp.swapFun (comp_subst_left' g⁻¹) (HasSymm.symm' (rightCancel f₁ g)),\n     HasLinearFunOp.swapFun (comp_subst_left  g)   (rightCancelInv f₂ g)⟩\n    def rightMulInv' {a b c : α} (f₁ : R a c) (f₂ : R b c) (g : R a b) : f₁ • g⁻¹ ≃ f₂ ⟷' f₁ ≃ f₂ • g :=\n    ⟨HasLinearFunOp.swapFun (comp_subst_left' g)   (HasSymm.symm' (rightCancelInv f₁ g)),\n     HasLinearFunOp.swapFun (comp_subst_left  g⁻¹) (rightCancel f₂ g)⟩\n\n    def rightMul {a b c : α} (f₁ f₂ : R a b) (g : R c a) : f₁ • g ≃ f₂ • g ⟷' f₁ ≃ f₂ :=\n    ⟨HasTrans.revTrans'' (rightCancel f₂ g) ⊙ (rightMulInv f₁ (f₂ • g) g).mp, comp_congrArg_left⟩\n\n    def eqInvIffInvEq {a b : α} (f : R a b) (g : R b a) : f ≃ g⁻¹ ⟷' f⁻¹ ≃ g :=\n    ⟨HasLinearFunOp.swapFun inv_subst  (invInv g),\n     HasLinearFunOp.swapFun inv_subst' (HasSymm.symm' (invInv f))⟩\n\n    def eqIffEqInv {a b : α} (f₁ f₂ : R a b) : f₁⁻¹ ≃ f₂⁻¹ ⟷' f₁ ≃ f₂ :=\n    ⟨HasTrans.revTrans'' (invInv f₂) ⊙ (eqInvIffInvEq f₁ f₂⁻¹).mpr, inv_congrArg⟩\n\n    def leftRightMul {a b c d : α} (f₁ : R a b) (f₂ : R a c) (g₁ : R b d) (g₂ : R c d) :\n      g₂⁻¹ • g₁ ≃ f₂ • f₁⁻¹ ⟷' g₁ • f₁ ≃ g₂ • f₂ :=\n    ⟨(leftMulInv' (g₁ • f₁) f₂ g₂).mp ⊙ applyAssocLR_left ⊙ (rightMulInv (g₂⁻¹ • g₁) f₂ f₁).mpr,\n     (leftMulInv' g₁ (f₂ • f₁⁻¹) g₂).mpr ⊙ applyAssocLR_right ⊙ (rightMulInv g₁ (g₂ • f₂) f₁).mp⟩\n\n    def swapInv  {a b c d : α} (f₁ : R a b) (f₂ : R c d) (g₁ : R d b) (g₂ : R c a) :\n      g₁⁻¹ • f₁ ≃ f₂ • g₂⁻¹ ⟶ f₁⁻¹ • g₁ ≃ g₂ • f₂⁻¹ :=\n    (leftRightMul f₂ g₂ g₁ f₁).mpr ⊙ HasSymm.symm ⊙ (leftRightMul g₂ f₂ f₁ g₁).mp\n\n    def swapInv' {a b c d : α} (f₁ : R a b) (f₂ : R c d) (g₁ : R d b) (g₂ : R c a) :\n      f₂ • g₂⁻¹ ≃ g₁⁻¹ • f₁ ⟶ g₂ • f₂⁻¹ ≃ f₁⁻¹ • g₁ :=\n    HasSymm.symm ⊙ swapInv f₁ f₂ g₁ g₂ ⊙ HasSymm.symm\n\n  end IsIsomorphismRelation\n\n  open IsIsomorphismRelation\n\nend Morphisms\n\n\n\nsection Functors\n\n  variable {α : Sort u}\n\n  namespace idFun\n\n    variable {U : Universe} [HasInstanceArrows U] [HasExternalFunctors U U]\n             [HasIdFun U] (R : GeneralizedRelation α U)\n\n    instance isReflFunctor  [HasRefl  R] : IsReflFunctor  R R (HasIdFun.idFun' _) :=\n    ⟨λ a   => HasRefl.refl (ident R a)⟩\n\n    variable [HasInternalFunctors U]\n\n    instance isSymmFunctor  [HasSymm  R] : IsSymmFunctor  R R (HasIdFun.idFun' _) :=\n    ⟨λ f   => HasRefl.refl f⁻¹⟩\n\n    instance isTransFunctor [HasTrans R] : IsTransFunctor R R (HasIdFun.idFun' _) :=\n    ⟨λ f g => HasRefl.refl (g • f)⟩\n\n    instance isPreorderFunctor    [IsPreorder    R] : IsPreorderFunctor    R R (HasIdFun.idFun' _) := ⟨⟩\n    instance isEquivalenceFunctor [IsEquivalence R] : IsEquivalenceFunctor R R (HasIdFun.idFun' _) := ⟨⟩\n\n  end idFun\n\n  namespace constFun\n\n    variable {U V : Universe} [HasInstanceArrows V] [HasExternalFunctors U V]\n             [HasConstFun U V] (R : GeneralizedRelation α U) {β : V} (c : β)\n\n    def idArrow : c ⇝ c := ident (HasInstanceArrows.Arrow β) c\n\n    instance isReflFunctor  [HasRefl  R] : IsReflFunctor  R (ConstRel α c) (HasConstFun.constFun' _ c) :=\n    ⟨λ _   => idArrow c⟩\n\n    variable [HasInternalFunctors U] [HasInternalFunctors V] [HasAffineFunOp V]\n\n    instance isSymmFunctor  [HasSymm  R] : IsSymmFunctor  R (ConstRel α c) (HasConstFun.constFun' _ c) :=\n    ⟨λ _   => Eq.ndrec (motive := λ b : β => ⌈c ⇝ b⌉) (idArrow c) (Eq.symm (ConstRel.symmEq  α c))⟩\n\n    instance isTransFunctor [HasTrans R] : IsTransFunctor R (ConstRel α c) (HasConstFun.constFun' _ c) :=\n    ⟨λ _ _ => Eq.ndrec (motive := λ b : β => ⌈c ⇝ b⌉) (idArrow c) (Eq.symm (ConstRel.transEq α c))⟩\n\n    instance isPreorderFunctor    [IsPreorder    R] : IsPreorderFunctor    R (ConstRel α c) (HasConstFun.constFun' _ c) := ⟨⟩\n    instance isEquivalenceFunctor [IsEquivalence R] : IsEquivalenceFunctor R (ConstRel α c) (HasConstFun.constFun' _ c) := ⟨⟩\n\n  end constFun\n\n  namespace compFun\n\n    variable {U V W : Universe} [hV : HasInstanceArrows V] [hW : HasInstanceArrows W]\n             [HasExternalFunctors U V] [HasExternalFunctors V W] [HasExternalFunctors U W]\n             [HasExternalFunctors hV.arrowUniverse hW.arrowUniverse] [HasArrowCongrArg V W]\n             [HasCompFun U V W]\n             {R : GeneralizedRelation α U} {S : GeneralizedRelation α V} {T : GeneralizedRelation α W}\n             (F : BaseFunctor R S) (G : BaseFunctor S T)\n\n    instance isReflFunctor  [HasRefl  R] [HasRefl  S] [HasRefl  T] [hF : IsReflFunctor  R S F] [hG : IsReflFunctor  S T G] :\n      IsReflFunctor  R T (G ⊙' F) :=\n    ⟨λ a   => let e₁ : G (F (ident R a)) ⇝ G (ident S a) := HasArrowCongrArg.arrowCongrArg G (hF.respectsRefl a);\n              let e₂ : G (ident S a) ⇝ ident T a         := hG.respectsRefl a;\n              HasTrans.trans' e₁ e₂⟩\n\n    variable [HasInternalFunctors U] [HasInternalFunctors V] [HasInternalFunctors W]\n\n    instance isSymmFunctor  [HasSymm  R] [HasSymm  S] [HasSymm  T] [hF : IsSymmFunctor  R S F] [hG : IsSymmFunctor  S T G] :\n      IsSymmFunctor  R T (G ⊙' F) :=\n    ⟨λ f   => let e₁ : G (F f⁻¹) ⇝ G (F f)⁻¹   := HasArrowCongrArg.arrowCongrArg G (hF.respectsSymm f);\n              let e₂ : G (F f)⁻¹ ⇝ (G (F f))⁻¹ := hG.respectsSymm (F f);\n              HasTrans.trans' e₁ e₂⟩\n\n    instance isTransFunctor [HasTrans R] [HasTrans S] [HasTrans T] [hF : IsTransFunctor R S F] [hG : IsTransFunctor S T G] :\n      IsTransFunctor R T (G ⊙' F) :=\n    ⟨λ f g => let e₁ : G (F (g • f)) ⇝ G (F g • F f)     := HasArrowCongrArg.arrowCongrArg G (hF.respectsTrans f g);\n              let e₂ : G (F g • F f) ⇝ G (F g) • G (F f) := hG.respectsTrans (F f) (F g);\n              HasTrans.trans' e₁ e₂⟩\n\n    instance isPreorderFunctor    [IsPreorder    R] [IsPreorder    S] [IsPreorder    T] [IsPreorderFunctor    R S F] [IsPreorderFunctor    S T G] :\n      IsPreorderFunctor    R T (G ⊙' F) := ⟨⟩\n    instance isEquivalenceFunctor [IsEquivalence R] [IsEquivalence S] [IsEquivalence T] [IsEquivalenceFunctor R S F] [IsEquivalenceFunctor S T G] :\n      IsEquivalenceFunctor R T (G ⊙' F) := ⟨⟩\n\n  end compFun\n\nend Functors\n\n\n\nsection NaturalTransformations\n\n  variable {α : Sort u} {V W : Universe} [HasInternalFunctors W] [hW : HasFunctorialEquivalences W] [HasExternalFunctors V W]\n           {β : Sort w} (R : GeneralizedRelation α V) (S : GeneralizedRelation β W)\n\n  instance : HasFunctorialArrows W := hW.toHasFunctorialArrows W\n\n  namespace IsNaturalTransformation\n\n    def refl  [IsPreorder    S] [h : IsMorphismRelation    S] {mF       : α → β}\n              (F : ∀ {a b}, R a b ⟶' S (mF a) (mF b)) :\n      IsNatural R S F F (λ a => ident S (mF a)) :=\n    ⟨λ f => HasTrans.trans' (h.rightId (F f)) (HasSymm.symm' (h.leftId (F f)))⟩\n\n    variable [HasLinearFunOp hW.equivUniverse]\n\n    def symm  [IsEquivalence S] [h : IsIsomorphismRelation S] {mF mG    : α → β}\n              (F : ∀ {a b}, R a b ⟶' S (mF a) (mF b)) (G : ∀ {a b}, R a b ⟶' S (mG a) (mG b))\n              (n : ∀ a, S (mF a) (mG a)) [hn : IsNatural R S F G n] :\n      IsNatural R S G F (λ a => (n a)⁻¹) :=\n    ⟨λ {a b} f => HasSymm.symm' ((IsIsomorphismRelation.leftRightMul (n a) (F f) (G f) (n b)).mpr (hn.nat f))⟩\n\n    def trans [HasTrans      S] [h : IsCompositionRelation S] {mF mG mH : α → β}\n              (F : ∀ {a b}, R a b ⟶' S (mF a) (mF b)) (G : ∀ {a b}, R a b ⟶' S (mG a) (mG b)) (H : ∀ {a b}, R a b ⟶' S (mH a) (mH b))\n              (nFG : ∀ a, S (mF a) (mG a))   (nGH : ∀ a, S (mG a) (mH a))\n              [hnFG : IsNatural R S F G nFG] [hnGH : IsNatural R S G H nGH] :\n      IsNatural R S F H (λ a => nGH a • nFG a) :=\n    ⟨λ {a b} f => let e₁ := IsCompositionRelation.applyAssocLR_left ⟮IsCompositionRelation.comp_congrArg_left  (f := nFG a) (hnGH.nat f)⟯;\n                  let e₂ := IsCompositionRelation.applyAssocRL      ⟮IsCompositionRelation.comp_congrArg_right (g := nGH b) (hnFG.nat f)⟯;\n                  HasTrans.trans' e₁ e₂⟩\n\n  end IsNaturalTransformation\n\nend NaturalTransformations\n", "meta": {"author": "SReichelt", "repo": "lean4-experiments", "sha": "ff55357a01a34a91bf670d712637480089085ee4", "save_path": "github-repos/lean/SReichelt-lean4-experiments", "path": "github-repos/lean/SReichelt-lean4-experiments/lean4-experiments-ff55357a01a34a91bf670d712637480089085ee4/Structure/Generic/Lemmas/CategoryTheoryLemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3431612884892881}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.discrete_category\n\n/-!\n# Limits in `C` give colimits in `Cᵒᵖ`.\n\nWe also give special cases for (co)products,\nbut not yet for pullbacks / pushouts or for (co)equalizers.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C]\nvariables {J : Type u₂} [category.{v₂} J]\n\n/-- Turn a colimit for `F : J ⥤ C` into a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cocone_op (F : J ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit c.op :=\n{ lift := λ s, (hc.desc s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F : J ⥤ C` into a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_op (F : J ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit c.op :=\n{ desc := λ s, (hc.lift s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : J ⥤ Cᵒᵖ` into a limit for `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_left_op_of_cocone (F : J ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_left_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_left_op s)).unop,\n  fac' :=  λ s j, quiver.hom.op_inj $ by simpa only [cone_left_op_of_cocone_π_app, op_comp,\n    quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app] using w (op j)\n  end }\n\n/-- Turn a limit of `F : J ⥤ Cᵒᵖ` into a colimit of `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_left_op_of_cone (F : J ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_left_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_left_op s)).unop,\n  fac' := λ s j, quiver.hom.op_inj $ by simpa only [cocone_left_op_of_cone_ι_app, op_comp,\n    quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ C` into a limit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_right_op_of_cocone (F : Jᵒᵖ ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_right_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F : Jᵒᵖ ⥤ C` into a colimit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_right_op_of_cone (F : Jᵒᵖ ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_right_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F.unop : J ⥤ C`. -/\n@[simps] def is_limit_cone_unop_of_cocone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_unop_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit of `F : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit of `F.unop : J ⥤ C`. -/\n@[simps] def is_colimit_cocone_unop_of_cone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_unop_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F : J ⥤ C`. -/\n@[simps] def is_limit_cocone_unop (F : J ⥤ C) {c : cocone F.op} (hc : is_colimit c) :\n  is_limit c.unop :=\n{ lift := λ s, (hc.desc s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit for `F : J ⥤ C`. -/\n@[simps] def is_colimit_cone_unop (F : J ⥤ C) {c : cone F.op} (hc : is_limit c) :\n  is_colimit c.unop :=\n{ desc := λ s, (hc.lift s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.left_op : Jᵒᵖ ⥤ C` into a limit for `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_left_op (F : J ⥤ Cᵒᵖ) {c : cocone F.left_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_left_op c) :=\n{ lift := λ s, (hc.desc (cocone_left_op_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cone_of_cocone_left_op_π_app, unop_comp,\n    quiver.hom.unop_op, is_colimit.fac, cocone_left_op_of_cone_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac, cone_of_cocone_left_op_π_app] using w (unop j)\n  end }\n\n/-- Turn a limit of `F.left_op : Jᵒᵖ ⥤ C` into a colimit of `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_of_cone_left_op (F : J ⥤ Cᵒᵖ) {c : cone (F.left_op)}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_left_op c) :=\n{ desc := λ s, (hc.lift (cone_left_op_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cocone_of_cone_left_op_ι_app, unop_comp,\n    quiver.hom.unop_op, is_limit.fac, cone_left_op_of_cocone_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac, cocone_of_cone_left_op_ι_app] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_of_cocone_right_op (F : Jᵒᵖ ⥤ C) {c : cocone F.right_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_right_op c) :=\n{ lift := λ s, (hc.desc (cocone_right_op_of_cone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_of_cone_right_op (F : Jᵒᵖ ⥤ C) {c : cone F.right_op}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_right_op c) :=\n{ desc := λ s, (hc.lift (cone_right_op_of_cocone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F.unop : J ⥤ C` into a limit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F.unop} (hc : is_colimit c) :\n  is_limit (cone_of_cocone_unop c) :=\n{ lift := λ s, (hc.desc (cocone_unop_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.unop : J ⥤ C` into a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F.unop} (hc : is_limit c) :\n  is_colimit (cocone_of_cone_unop c) :=\n{ desc := λ s, (hc.lift (cone_unop_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a colimit, we can construct a limit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_limit_of_has_colimit_left_op (F : J ⥤ Cᵒᵖ) [has_colimit F.left_op] : has_limit F :=\nhas_limit.mk\n{ cone := cone_of_cocone_left_op (colimit.cocone F.left_op),\n  is_limit := is_limit_cone_of_cocone_left_op _ (colimit.is_colimit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\nlemma has_limits_of_shape_op_of_has_colimits_of_shape [has_colimits_of_shape Jᵒᵖ C] :\n  has_limits_of_shape J Cᵒᵖ :=\n{ has_limit := λ F, has_limit_of_has_colimit_left_op F }\n\nlocal attribute [instance] has_limits_of_shape_op_of_has_colimits_of_shape\n\n/--\nIf `C` has colimits, we can construct limits for `Cᵒᵖ`.\n-/\nlemma has_limits_op_of_has_colimits [has_colimits C] : has_limits Cᵒᵖ := ⟨infer_instance⟩\n\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a limit, we can construct a colimit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_colimit_of_has_limit_left_op (F : J ⥤ Cᵒᵖ) [has_limit F.left_op] : has_colimit F :=\nhas_colimit.mk\n{ cocone := cocone_of_cone_left_op (limit.cone F.left_op),\n  is_colimit := is_colimit_cocone_of_cone_left_op _ (limit.is_limit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\nlemma has_colimits_of_shape_op_of_has_limits_of_shape [has_limits_of_shape Jᵒᵖ C] :\n  has_colimits_of_shape J Cᵒᵖ :=\n{ has_colimit := λ F, has_colimit_of_has_limit_left_op F }\n\nlocal attribute [instance] has_colimits_of_shape_op_of_has_limits_of_shape\n\n/--\nIf `C` has limits, we can construct colimits for `Cᵒᵖ`.\n-/\nlemma has_colimits_op_of_has_limits [has_limits C] : has_colimits Cᵒᵖ := ⟨infer_instance⟩\n\nvariables (X : Type v₁)\n/--\nIf `C` has products indexed by `X`, then `Cᵒᵖ` has coproducts indexed by `X`.\n-/\nlemma has_coproducts_opposite [has_products_of_shape X C] :\n  has_coproducts_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape (discrete X)ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\n/--\nIf `C` has coproducts indexed by `X`, then `Cᵒᵖ` has products indexed by `X`.\n-/\nlemma has_products_opposite [has_coproducts_of_shape X C] :\n  has_products_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape (discrete X)ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\nlemma has_finite_coproducts_opposite [has_finite_products C] :\n  has_finite_coproducts Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, begin\n    resetI,\n    haveI : has_limits_of_shape (discrete J)ᵒᵖ C :=\n      has_limits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlemma has_finite_products_opposite [has_finite_coproducts C] :\n  has_finite_products Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, begin\n    resetI,\n    haveI : has_colimits_of_shape (discrete J)ᵒᵖ C :=\n      has_colimits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlemma has_equalizers_opposite [has_coequalizers C] : has_equalizers Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape walking_parallel_pair.{v₁}ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence walking_parallel_pair_op_equiv.{v₁},\n  apply_instance\nend\n\nlemma has_coequalizers_opposite [has_equalizers C] : has_coequalizers Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape walking_parallel_pair.{v₁}ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence walking_parallel_pair_op_equiv.{v₁},\n  apply_instance\nend\n\nlemma has_finite_colimits_opposite [has_finite_limits C] :\n  has_finite_colimits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\nlemma has_finite_limits_opposite [has_finite_colimits C] :\n  has_finite_limits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\nend category_theory.limits\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/limits/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34309547480636743}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.pi\nimport Mathlib.algebra.big_operators.basic\nimport Mathlib.data.set.finite\nimport Mathlib.group_theory.submonoid.basic\nimport Mathlib.PostPort\n\nuniverses u v l v₁ v₂ w u_1 u_2 u_3 u_4 u₁ x \n\nnamespace Mathlib\n\n/-!\n# Dependent functions with finite support\n\nFor a non-dependent version see `data/finsupp.lean`.\n-/\n\nnamespace dfinsupp\n\n\nstructure pre (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] \nwhere\n  to_fun : (i : ι) → β i\n  pre_support : multiset ι\n  zero : ∀ (i : ι), i ∈ pre_support ∨ to_fun i = 0\n\nprotected instance inhabited_pre (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] : Inhabited (pre ι β) :=\n  { default := pre.mk (fun (i : ι) => 0) ∅ sorry }\n\nprotected instance pre.setoid (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] : setoid (pre ι β) :=\n  setoid.mk (fun (x y : pre ι β) => ∀ (i : ι), pre.to_fun x i = pre.to_fun y i) sorry\n\nend dfinsupp\n\n\n/-- A dependent function `Π i, β i` with finite support. -/\ndef dfinsupp {ι : Type u} (β : ι → Type v) [(i : ι) → HasZero (β i)] :=\n  quotient sorry\n\ninfixl:25 \" →ₚ \" => Mathlib.dfinsupp\n\nnamespace dfinsupp\n\n\nprotected instance has_coe_to_fun {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] : has_coe_to_fun (dfinsupp fun (i : ι) => β i) :=\n  has_coe_to_fun.mk (fun (_x : dfinsupp fun (i : ι) => β i) => (i : ι) → β i)\n    fun (f : dfinsupp fun (i : ι) => β i) => quotient.lift_on f pre.to_fun sorry\n\nprotected instance has_zero {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] : HasZero (dfinsupp fun (i : ι) => β i) :=\n  { zero := quotient.mk (pre.mk (fun (i : ι) => 0) ∅ sorry) }\n\nprotected instance inhabited {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] : Inhabited (dfinsupp fun (i : ι) => β i) :=\n  { default := 0 }\n\n@[simp] theorem zero_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (i : ι) : coe_fn 0 i = 0 :=\n  rfl\n\ntheorem ext {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i} (H : ∀ (i : ι), coe_fn f i = coe_fn g i) : f = g := sorry\n\n/-- The composition of `f : β₁ → β₂` and `g : Π₀ i, β₁ i` is\n  `map_range f hf g : Π₀ i, β₂ i`, well defined when `f 0 = 0`. -/\ndef map_range {ι : Type u} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i) (hf : ∀ (i : ι), f i 0 = 0) (g : dfinsupp fun (i : ι) => β₁ i) : dfinsupp fun (i : ι) => β₂ i :=\n  quotient.lift_on g\n    (fun (x : pre ι fun (i : ι) => β₁ i) =>\n      quotient.mk (pre.mk (fun (i : ι) => f i (pre.to_fun x i)) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem map_range_apply {ι : Type u} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i) (hf : ∀ (i : ι), f i 0 = 0) (g : dfinsupp fun (i : ι) => β₁ i) (i : ι) : coe_fn (map_range f hf g) i = f i (coe_fn g i) :=\n  quotient.induction_on g fun (x : pre ι fun (i : ι) => β₁ i) => rfl\n\n/-- Let `f i` be a binary operation `β₁ i → β₂ i → β i` such that `f i 0 0 = 0`.\nThen `zip_with f hf` is a binary operation `Π₀ i, β₁ i → Π₀ i, β₂ i → Π₀ i, β i`. -/\ndef zip_with {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i → β i) (hf : ∀ (i : ι), f i 0 0 = 0) (g₁ : dfinsupp fun (i : ι) => β₁ i) (g₂ : dfinsupp fun (i : ι) => β₂ i) : dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on₂ g₁ g₂\n    (fun (x : pre ι fun (i : ι) => β₁ i) (y : pre ι fun (i : ι) => β₂ i) =>\n      quotient.mk\n        (pre.mk (fun (i : ι) => f i (pre.to_fun x i) (pre.to_fun y i)) (pre.pre_support x + pre.pre_support y) sorry))\n    sorry\n\n@[simp] theorem zip_with_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i → β i) (hf : ∀ (i : ι), f i 0 0 = 0) (g₁ : dfinsupp fun (i : ι) => β₁ i) (g₂ : dfinsupp fun (i : ι) => β₂ i) (i : ι) : coe_fn (zip_with f hf g₁ g₂) i = f i (coe_fn g₁ i) (coe_fn g₂ i) :=\n  quotient.induction_on₂ g₁ g₂ fun (_x : pre ι fun (i : ι) => β₁ i) (_x_1 : pre ι fun (i : ι) => β₂ i) => rfl\n\nprotected instance has_add {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] : Add (dfinsupp fun (i : ι) => β i) :=\n  { add := zip_with (fun (_x : ι) => Add.add) sorry }\n\n@[simp] theorem add_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] (g₁ : dfinsupp fun (i : ι) => β i) (g₂ : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (g₁ + g₂) i = coe_fn g₁ i + coe_fn g₂ i :=\n  zip_with_apply (fun (_x : ι) => Add.add) has_add._proof_1 g₁ g₂ i\n\nprotected instance add_monoid {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] : add_monoid (dfinsupp fun (i : ι) => β i) :=\n  add_monoid.mk Add.add sorry 0 sorry sorry\n\nprotected instance is_add_monoid_hom {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] {i : ι} : is_add_monoid_hom fun (g : dfinsupp fun (i : ι) => β i) => coe_fn g i :=\n  is_add_monoid_hom.mk (zero_apply i)\n\nprotected instance has_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] : Neg (dfinsupp fun (i : ι) => β i) :=\n  { neg := fun (f : dfinsupp fun (i : ι) => β i) => map_range (fun (_x : ι) => Neg.neg) sorry f }\n\nprotected instance add_comm_monoid {ι : Type u} {β : ι → Type v} [(i : ι) → add_comm_monoid (β i)] : add_comm_monoid (dfinsupp fun (i : ι) => β i) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\n@[simp] theorem neg_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] (g : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (-g) i = -coe_fn g i :=\n  map_range_apply (fun (_x : ι) => Neg.neg) has_neg._proof_1 g i\n\nprotected instance add_group {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] : add_group (dfinsupp fun (i : ι) => β i) :=\n  add_group.mk add_monoid.add sorry add_monoid.zero sorry sorry Neg.neg\n    (sub_neg_monoid.sub._default add_monoid.add sorry add_monoid.zero sorry sorry Neg.neg) sorry\n\n@[simp] theorem sub_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] (g₁ : dfinsupp fun (i : ι) => β i) (g₂ : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (g₁ - g₂) i = coe_fn g₁ i - coe_fn g₂ i := sorry\n\nprotected instance add_comm_group {ι : Type u} {β : ι → Type v} [(i : ι) → add_comm_group (β i)] : add_comm_group (dfinsupp fun (i : ι) => β i) :=\n  add_comm_group.mk add_group.add sorry add_group.zero sorry sorry add_group.neg add_group.sub sorry sorry\n\n/-- Dependent functions with finite support inherit a semiring action from an action on each\ncoordinate. -/\nprotected instance has_scalar {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] : has_scalar γ (dfinsupp fun (i : ι) => β i) :=\n  has_scalar.mk fun (c : γ) (v : dfinsupp fun (i : ι) => β i) => map_range (fun (_x : ι) => has_scalar.smul c) sorry v\n\n@[simp] theorem smul_apply {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] (b : γ) (v : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (b • v) i = b • coe_fn v i :=\n  map_range_apply (fun (_x : ι) => has_scalar.smul b) (has_scalar._proof_1 b) v i\n\n/-- Dependent functions with finite support inherit a semimodule structure from such a structure on\neach coordinate. -/\nprotected instance semimodule {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] : semimodule γ (dfinsupp fun (i : ι) => β i) :=\n  semimodule.mk sorry sorry\n\n/-- `filter p f` is the function which is `f i` if `p i` is true and 0 otherwise. -/\ndef filter {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop) [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk (pre.mk (fun (i : ι) => ite (p i) (pre.to_fun x i) 0) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem filter_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop) [decidable_pred p] (i : ι) (f : dfinsupp fun (i : ι) => β i) : coe_fn (filter p f) i = ite (p i) (coe_fn f i) 0 :=\n  quotient.induction_on f fun (x : pre ι fun (i : ι) => β i) => rfl\n\ntheorem filter_apply_pos {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop} [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) {i : ι} (h : p i) : coe_fn (filter p f) i = coe_fn f i := sorry\n\ntheorem filter_apply_neg {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop} [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) {i : ι} (h : ¬p i) : coe_fn (filter p f) i = 0 := sorry\n\ntheorem filter_pos_add_filter_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] (f : dfinsupp fun (i : ι) => β i) (p : ι → Prop) [decidable_pred p] : filter p f + filter (fun (i : ι) => ¬p i) f = f := sorry\n\n/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -/\ndef subtype_domain {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop) [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : Subtype p) => β ↑i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk\n        (pre.mk (fun (i : Subtype p) => pre.to_fun x ↑i)\n          (multiset.map\n            (fun (j : Subtype fun (x_1 : ι) => x_1 ∈ multiset.filter p (pre.pre_support x)) =>\n              { val := ↑j, property := sorry })\n            (multiset.attach (multiset.filter p (pre.pre_support x))))\n          sorry))\n    sorry\n\n@[simp] theorem subtype_domain_zero {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop} [decidable_pred p] : subtype_domain p 0 = 0 :=\n  rfl\n\n@[simp] theorem subtype_domain_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop} [decidable_pred p] {i : Subtype p} {v : dfinsupp fun (i : ι) => β i} : coe_fn (subtype_domain p v) i = coe_fn v ↑i :=\n  quotient.induction_on v fun (x : pre ι fun (i : ι) => β i) => rfl\n\n@[simp] theorem subtype_domain_add {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i} {v' : dfinsupp fun (i : ι) => β i} : subtype_domain p (v + v') = subtype_domain p v + subtype_domain p v' := sorry\n\nprotected instance subtype_domain.is_add_monoid_hom {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] {p : ι → Prop} [decidable_pred p] : is_add_monoid_hom (subtype_domain p) :=\n  is_add_monoid_hom.mk subtype_domain_zero\n\n@[simp] theorem subtype_domain_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i} : subtype_domain p (-v) = -subtype_domain p v := sorry\n\n@[simp] theorem subtype_domain_sub {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i} {v' : dfinsupp fun (i : ι) => β i} : subtype_domain p (v - v') = subtype_domain p v - subtype_domain p v' := sorry\n\ntheorem finite_supp {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (f : dfinsupp fun (i : ι) => β i) : set.finite (set_of fun (i : ι) => coe_fn f i ≠ 0) := sorry\n\n/-- Create an element of `Π₀ i, β i` from a finset `s` and a function `x`\ndefined on this `finset`. -/\ndef mk {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (s : finset ι) (x : (i : ↥↑s) → β ↑i) : dfinsupp fun (i : ι) => β i :=\n  quotient.mk\n    (pre.mk (fun (i : ι) => dite (i ∈ s) (fun (H : i ∈ s) => x { val := i, property := H }) fun (H : ¬i ∈ s) => 0)\n      (finset.val s) sorry)\n\n@[simp] theorem mk_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {s : finset ι} {x : (i : ↥↑s) → β ↑i} {i : ι} : coe_fn (mk s x) i = dite (i ∈ s) (fun (H : i ∈ s) => x { val := i, property := H }) fun (H : ¬i ∈ s) => 0 :=\n  rfl\n\ntheorem mk_injective {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (s : finset ι) : function.injective (mk s) := sorry\n\n/-- The function `single i b : Π₀ i, β i` sends `i` to `b`\nand all other points to `0`. -/\ndef single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (i : ι) (b : β i) : dfinsupp fun (i : ι) => β i :=\n  mk (singleton i) fun (j : ↥↑(singleton i)) => eq.rec_on sorry b\n\n@[simp] theorem single_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {i' : ι} {b : β i} : coe_fn (single i b) i' = dite (i = i') (fun (h : i = i') => eq.rec_on h b) fun (h : ¬i = i') => 0 := sorry\n\n@[simp] theorem single_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} : single i 0 = 0 := sorry\n\n@[simp] theorem single_eq_same {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {b : β i} : coe_fn (single i b) i = b := sorry\n\ntheorem single_eq_of_ne {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {i' : ι} {b : β i} (h : i ≠ i') : coe_fn (single i b) i' = 0 := sorry\n\ntheorem single_injective {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} : function.injective (single i) := sorry\n\n/-- Like `finsupp.single_eq_single_iff`, but with a `heq` due to dependent types -/\ntheorem single_eq_single_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (i : ι) (j : ι) (xi : β i) (xj : β j) : single i xi = single j xj ↔ i = j ∧ xi == xj ∨ xi = 0 ∧ xj = 0 := sorry\n\n/-- Redefine `f i` to be `0`. -/\ndef erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (i : ι) (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk (pre.mk (fun (j : ι) => ite (j = i) 0 (pre.to_fun x j)) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem erase_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {j : ι} {f : dfinsupp fun (i : ι) => β i} : coe_fn (erase i f) j = ite (j = i) 0 (coe_fn f j) :=\n  quotient.induction_on f fun (x : pre ι fun (i : ι) => β i) => rfl\n\n@[simp] theorem erase_same {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} : coe_fn (erase i f) i = 0 := sorry\n\ntheorem erase_ne {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {i : ι} {i' : ι} {f : dfinsupp fun (i : ι) => β i} (h : i' ≠ i) : coe_fn (erase i f) i' = coe_fn f i' := sorry\n\n@[simp] theorem single_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {i : ι} {b₁ : β i} {b₂ : β i} : single i (b₁ + b₂) = single i b₁ + single i b₂ := sorry\n\n/-- `dfinsupp.single` as an `add_monoid_hom`. -/\n@[simp] theorem single_add_hom_apply {ι : Type u} (β : ι → Type v) [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] (i : ι) (b : β i) : coe_fn (single_add_hom β i) b = single i b :=\n  Eq.refl (coe_fn (single_add_hom β i) b)\n\ntheorem single_add_erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} : single i (coe_fn f i) + erase i f = f := sorry\n\ntheorem erase_add_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} : erase i f + single i (coe_fn f i) = f := sorry\n\nprotected theorem induction {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {p : (dfinsupp fun (i : ι) => β i) → Prop} (f : dfinsupp fun (i : ι) => β i) (h0 : p 0) (ha : ∀ (i : ι) (b : β i) (f : dfinsupp fun (i : ι) => β i), coe_fn f i = 0 → b ≠ 0 → p f → p (single i b + f)) : p f := sorry\n\ntheorem induction₂ {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {p : (dfinsupp fun (i : ι) => β i) → Prop} (f : dfinsupp fun (i : ι) => β i) (h0 : p 0) (ha : ∀ (i : ι) (b : β i) (f : dfinsupp fun (i : ι) => β i), coe_fn f i = 0 → b ≠ 0 → p f → p (f + single i b)) : p f := sorry\n\n@[simp] theorem add_closure_Union_range_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] : add_submonoid.closure (set.Union fun (i : ι) => set.range (single i)) = ⊤ := sorry\n\n/-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then\nthey are equal. -/\ntheorem add_hom_ext {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {γ : Type w} [add_monoid γ] {f : (dfinsupp fun (i : ι) => β i) →+ γ} {g : (dfinsupp fun (i : ι) => β i) →+ γ} (H : ∀ (i : ι) (y : β i), coe_fn f (single i y) = coe_fn g (single i y)) : f = g := sorry\n\n/-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then\nthey are equal.\n\nSee note [partially-applied ext lemmas]. -/\ntheorem add_hom_ext' {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {γ : Type w} [add_monoid γ] {f : (dfinsupp fun (i : ι) => β i) →+ γ} {g : (dfinsupp fun (i : ι) => β i) →+ γ} (H : ∀ (x : ι), add_monoid_hom.comp f (single_add_hom β x) = add_monoid_hom.comp g (single_add_hom β x)) : f = g :=\n  add_hom_ext fun (x : ι) => add_monoid_hom.congr_fun (H x)\n\n@[simp] theorem mk_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] {s : finset ι} {x : (i : ↥↑s) → β ↑i} {y : (i : ↥↑s) → β ↑i} : mk s (x + y) = mk s x + mk s y := sorry\n\n@[simp] theorem mk_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] {s : finset ι} : mk s 0 = 0 := sorry\n\n@[simp] theorem mk_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_group (β i)] {s : finset ι} {x : (i : ↥↑s) → β (subtype.val i)} : mk s (-x) = -mk s x := sorry\n\n@[simp] theorem mk_sub {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_group (β i)] {s : finset ι} {x : (i : ↥↑s) → β (subtype.val i)} {y : (i : ↥↑s) → β (subtype.val i)} : mk s (x - y) = mk s x - mk s y := sorry\n\nprotected instance mk.is_add_group_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_group (β i)] {s : finset ι} : is_add_group_hom (mk s) :=\n  is_add_group_hom.mk\n\n@[simp] theorem mk_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] (γ : Type w) [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] {s : finset ι} {c : γ} (x : (i : ↥↑s) → β (subtype.val i)) : mk s (c • x) = c • mk s x := sorry\n\n@[simp] theorem single_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] (γ : Type w) [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] {i : ι} {c : γ} {x : β i} : single i (c • x) = c • single i x := sorry\n\n/-- Set `{i | f x ≠ 0}` as a `finset`. -/\ndef support {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) : finset ι :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      finset.filter (fun (i : ι) => pre.to_fun x i ≠ 0) (multiset.to_finset (pre.pre_support x)))\n    sorry\n\n@[simp] theorem support_mk_subset {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {s : finset ι} {x : (i : ↥↑s) → β (subtype.val i)} : support (mk s x) ⊆ s :=\n  fun (i : ι) (H : i ∈ support (mk s x)) => iff.mp multiset.mem_to_finset (and.left (iff.mp finset.mem_filter H))\n\n@[simp] theorem mem_support_to_fun {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) (i : ι) : i ∈ support f ↔ coe_fn f i ≠ 0 := sorry\n\ntheorem eq_mk_support {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) : f = mk (support f) fun (i : ↥↑(support f)) => coe_fn f ↑i := sorry\n\n@[simp] theorem support_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] : support 0 = ∅ :=\n  rfl\n\ntheorem mem_support_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) (i : ι) : i ∈ support f ↔ coe_fn f i ≠ 0 :=\n  mem_support_to_fun f\n\n@[simp] theorem support_eq_empty {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {f : dfinsupp fun (i : ι) => β i} : support f = ∅ ↔ f = 0 := sorry\n\nprotected instance decidable_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] : decidable_pred (Eq 0) :=\n  fun (f : dfinsupp fun (i : ι) => β i) => decidable_of_iff (support f = ∅) sorry\n\ntheorem support_subset_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {s : set ι} {f : dfinsupp fun (i : ι) => β i} : ↑(support f) ⊆ s ↔ ∀ (i : ι), ¬i ∈ s → coe_fn f i = 0 := sorry\n\ntheorem support_single_ne_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i} (hb : b ≠ 0) : support (single i b) = singleton i := sorry\n\ntheorem support_single_subset {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i} : support (single i b) ⊆ singleton i :=\n  support_mk_subset\n\ntheorem map_range_def {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {g : dfinsupp fun (i : ι) => β₁ i} : map_range f hf g = mk (support g) fun (i : ↥↑(support g)) => f (subtype.val i) (coe_fn g (subtype.val i)) := sorry\n\n@[simp] theorem map_range_single {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {i : ι} {b : β₁ i} : map_range f hf (single i b) = single i (f i b) := sorry\n\ntheorem support_map_range {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {g : dfinsupp fun (i : ι) => β₁ i} : support (map_range f hf g) ⊆ support g := sorry\n\ntheorem zip_with_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i → β i} {hf : ∀ (i : ι), f i 0 0 = 0} {g₁ : dfinsupp fun (i : ι) => β₁ i} {g₂ : dfinsupp fun (i : ι) => β₂ i} : zip_with f hf g₁ g₂ =\n  mk (support g₁ ∪ support g₂)\n    fun (i : ↥↑(support g₁ ∪ support g₂)) => f (subtype.val i) (coe_fn g₁ (subtype.val i)) (coe_fn g₂ (subtype.val i)) := sorry\n\ntheorem support_zip_with {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i → β i} {hf : ∀ (i : ι), f i 0 0 = 0} {g₁ : dfinsupp fun (i : ι) => β₁ i} {g₂ : dfinsupp fun (i : ι) => β₂ i} : support (zip_with f hf g₁ g₂) ⊆ support g₁ ∪ support g₂ := sorry\n\ntheorem erase_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι) (f : dfinsupp fun (i : ι) => β i) : erase i f = mk (finset.erase (support f) i) fun (j : ↥↑(finset.erase (support f) i)) => coe_fn f (subtype.val j) := sorry\n\n@[simp] theorem support_erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι) (f : dfinsupp fun (i : ι) => β i) : support (erase i f) = finset.erase (support f) i := sorry\n\ntheorem filter_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop} [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : filter p f = mk (finset.filter p (support f)) fun (i : ↥↑(finset.filter p (support f))) => coe_fn f (subtype.val i) := sorry\n\n@[simp] theorem support_filter {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop} [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : support (filter p f) = finset.filter p (support f) := sorry\n\ntheorem subtype_domain_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop} [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : subtype_domain p f = mk (finset.subtype p (support f)) fun (i : ↥↑(finset.subtype p (support f))) => coe_fn f ↑i := sorry\n\n@[simp] theorem support_subtype_domain {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop} [decidable_pred p] {f : dfinsupp fun (i : ι) => β i} : support (subtype_domain p f) = finset.subtype p (support f) := sorry\n\ntheorem support_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {g₁ : dfinsupp fun (i : ι) => β i} {g₂ : dfinsupp fun (i : ι) => β i} : support (g₁ + g₂) ⊆ support g₁ ∪ support g₂ :=\n  support_zip_with\n\n@[simp] theorem support_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {f : dfinsupp fun (i : ι) => β i} : support (-f) = support f := sorry\n\ntheorem support_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (b : γ) (v : dfinsupp fun (i : ι) => β i) : support (b • v) ⊆ support v :=\n  support_map_range\n\nprotected instance decidable_eq {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] [(i : ι) → DecidableEq (β i)] : DecidableEq (dfinsupp fun (i : ι) => β i) :=\n  fun (f g : dfinsupp fun (i : ι) => β i) =>\n    decidable_of_iff (support f = support g ∧ ∀ (i : ι), i ∈ support f → coe_fn f i = coe_fn g i) sorry\n\n-- [to_additive sum] for dfinsupp.prod doesn't work, the equation lemmas are not generated\n\n/-- `sum f g` is the sum of `g i (f i)` over the support of `f`. -/\ndef sum {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → γ) : γ :=\n  finset.sum (support f) fun (i : ι) => g i (coe_fn f i)\n\n/-- `prod f g` is the product of `g i (f i)` over the support of `f`. -/\ndef prod {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → γ) : γ :=\n  finset.prod (support f) fun (i : ι) => g i (coe_fn f i)\n\ntheorem prod_map_range_index {ι : Type u} [dec : DecidableEq ι] {γ : Type w} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] [comm_monoid γ] {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {g : dfinsupp fun (i : ι) => β₁ i} {h : (i : ι) → β₂ i → γ} (h0 : ∀ (i : ι), h i 0 = 1) : prod (map_range f hf g) h = prod g fun (i : ι) (b : β₁ i) => h i (f i b) := sorry\n\ntheorem sum_zero_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {h : (i : ι) → β i → γ} : sum 0 h = 0 :=\n  rfl\n\ntheorem sum_single_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {i : ι} {b : β i} {h : (i : ι) → β i → γ} (h_zero : h i 0 = 0) : sum (single i b) h = h i b := sorry\n\ntheorem sum_neg_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h0 : ∀ (i : ι), h i 0 = 0) : sum (-g) h = sum g fun (i : ι) (b : β i) => h i (-b) :=\n  sum_map_range_index h0\n\ntheorem sum_comm {γ : Type w} {ι₁ : Type u_1} {ι₂ : Type u_2} {β₁ : ι₁ → Type u_3} {β₂ : ι₂ → Type u_4} [DecidableEq ι₁] [DecidableEq ι₂] [(i : ι₁) → HasZero (β₁ i)] [(i : ι₂) → HasZero (β₂ i)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι₂) → (x : β₂ i) → Decidable (x ≠ 0)] [add_comm_monoid γ] (f₁ : dfinsupp fun (i : ι₁) => β₁ i) (f₂ : dfinsupp fun (i : ι₂) => β₂ i) (h : (i : ι₁) → β₁ i → (i : ι₂) → β₂ i → γ) : (sum f₁ fun (i₁ : ι₁) (x₁ : β₁ i₁) => sum f₂ fun (i₂ : ι₂) (x₂ : β₂ i₂) => h i₁ x₁ i₂ x₂) =\n  sum f₂ fun (i₂ : ι₂) (x₂ : β₂ i₂) => sum f₁ fun (i₁ : ι₁) (x₁ : β₁ i₁) => h i₁ x₁ i₂ x₂ :=\n  finset.sum_comm\n\n@[simp] theorem sum_apply {ι : Type u} {β : ι → Type v} {ι₁ : Type u₁} [DecidableEq ι₁] {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)] {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁} {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i} {i₂ : ι} : coe_fn (sum f g) i₂ = sum f fun (i₁ : ι₁) (b : β₁ i₁) => coe_fn (g i₁ b) i₂ :=\n  Eq.symm (finset.sum_hom (support f) fun (f : dfinsupp fun (i : ι) => β i) => coe_fn f i₂)\n\ntheorem support_sum {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {ι₁ : Type u₁} [DecidableEq ι₁] {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁} {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i} : support (sum f g) ⊆ finset.bUnion (support f) fun (i : ι₁) => support (g i (coe_fn f i)) := sorry\n\n@[simp] theorem prod_one {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ] {f : dfinsupp fun (i : ι) => β i} : (prod f fun (i : ι) (b : β i) => 1) = 1 :=\n  finset.prod_const_one\n\n@[simp] theorem sum_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {f : dfinsupp fun (i : ι) => β i} {h₁ : (i : ι) → β i → γ} {h₂ : (i : ι) → β i → γ} : (sum f fun (i : ι) (b : β i) => h₁ i b + h₂ i b) = sum f h₁ + sum f h₂ :=\n  finset.sum_add_distrib\n\n@[simp] theorem sum_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_group γ] {f : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} : (sum f fun (i : ι) (b : β i) => -h i b) = -sum f h :=\n  finset.sum_hom (support f) Neg.neg\n\ntheorem prod_add_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ] {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h_zero : ∀ (i : ι), h i 0 = 1) (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ * h i b₂) : prod (f + g) h = prod f h * prod g h := sorry\n\n/--\nWhen summing over an `add_monoid_hom`, the decidability assumption is not needed, and the result is\nalso an `add_monoid_hom`.\n-/\ndef sum_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_monoid (β i)] [add_comm_monoid γ] (φ : (i : ι) → β i →+ γ) : (dfinsupp fun (i : ι) => β i) →+ γ :=\n  add_monoid_hom.mk\n    (fun (f : dfinsupp fun (i : ι) => β i) =>\n      quotient.lift_on f\n        (fun (x : pre ι fun (i : ι) => β i) =>\n          finset.sum (multiset.to_finset (pre.pre_support x)) fun (i : ι) => coe_fn (φ i) (pre.to_fun x i))\n        sorry)\n    sorry sorry\n\n@[simp] theorem sum_add_hom_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_monoid (β i)] [add_comm_monoid γ] (φ : (i : ι) → β i →+ γ) (i : ι) (x : β i) : coe_fn (sum_add_hom φ) (single i x) = coe_fn (φ i) x := sorry\n\n@[simp] theorem sum_add_hom_comp_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ) (i : ι) : add_monoid_hom.comp (sum_add_hom f) (single_add_hom β i) = f i :=\n  add_monoid_hom.ext fun (x : β i) => sum_add_hom_single f i x\n\n/-- While we didn't need decidable instances to define it, we do to reduce it to a sum -/\ntheorem sum_add_hom_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] (φ : (i : ι) → β i →+ γ) (f : dfinsupp fun (i : ι) => β i) : coe_fn (sum_add_hom φ) f = sum f fun (x : ι) => ⇑(φ x) := sorry\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom`,-/\n@[simp] theorem lift_add_hom_symm_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_monoid (β i)] [add_comm_monoid γ] (F : (dfinsupp fun (i : ι) => β i) →+ γ) (i : ι) : coe_fn (add_equiv.symm lift_add_hom) F i = add_monoid_hom.comp F (single_add_hom β i) :=\n  Eq.refl (coe_fn (add_equiv.symm lift_add_hom) F i)\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_single_add_hom`,-/\n@[simp] theorem lift_add_hom_single_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_comm_monoid (β i)] : coe_fn lift_add_hom (single_add_hom β) = add_monoid_hom.id (dfinsupp fun (i : ι) => β i) :=\n  iff.mpr (equiv.apply_eq_iff_eq_symm_apply (add_equiv.to_equiv lift_add_hom)) rfl\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_apply_single`,-/\ntheorem lift_add_hom_apply_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ) (i : ι) (x : β i) : coe_fn (coe_fn lift_add_hom f) (single i x) = coe_fn (f i) x := sorry\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_comp_single`,-/\ntheorem lift_add_hom_comp_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ) (i : ι) : add_monoid_hom.comp (coe_fn lift_add_hom f) (single_add_hom β i) = f i := sorry\n\n/-- The `dfinsupp` version of `finsupp.comp_lift_add_hom`,-/\ntheorem comp_lift_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} {δ : Type u_1} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] [add_comm_monoid δ] (g : γ →+ δ) (f : (i : ι) → β i →+ γ) : add_monoid_hom.comp g (coe_fn lift_add_hom f) = coe_fn lift_add_hom fun (a : ι) => add_monoid_hom.comp g (f a) := sorry\n\ntheorem sum_sub_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → add_comm_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_group γ] {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂) : sum (f - g) h = sum f h - sum g h := sorry\n\ntheorem sum_finset_sum_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} {α : Type x} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {s : finset α} {g : α → dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h_zero : ∀ (i : ι), h i 0 = 0) (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ + h i b₂) : (finset.sum s fun (i : α) => sum (g i) h) = sum (finset.sum s fun (i : α) => g i) h := sorry\n\ntheorem sum_sum_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} {ι₁ : Type u₁} [DecidableEq ι₁] {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁} {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h_zero : ∀ (i : ι), h i 0 = 0) (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ + h i b₂) : sum (sum f g) h = sum f fun (i : ι₁) (b : β₁ i) => sum (g i b) h :=\n  Eq.symm (sum_finset_sum_index h_zero h_add)\n\n@[simp] theorem sum_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {f : dfinsupp fun (i : ι) => β i} : sum f single = f := sorry\n\ntheorem sum_subtype_domain_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {v : dfinsupp fun (i : ι) => β i} {p : ι → Prop} [decidable_pred p] {h : (i : ι) → β i → γ} (hp : ∀ (x : ι), x ∈ support v → p x) : (sum (subtype_domain p v) fun (i : Subtype p) (b : β ↑i) => h (↑i) b) = sum v h := sorry\n\ntheorem subtype_domain_sum {ι : Type u} {β : ι → Type v} {γ : Type w} [(i : ι) → add_comm_monoid (β i)] {s : finset γ} {h : γ → dfinsupp fun (i : ι) => β i} {p : ι → Prop} [decidable_pred p] : subtype_domain p (finset.sum s fun (c : γ) => h c) = finset.sum s fun (c : γ) => subtype_domain p (h c) :=\n  Eq.symm (finset.sum_hom s (subtype_domain p))\n\ntheorem subtype_domain_finsupp_sum {ι : Type u} {β : ι → Type v} {γ : Type w} {δ : γ → Type x} [DecidableEq γ] [(c : γ) → HasZero (δ c)] [(c : γ) → (x : δ c) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)] {p : ι → Prop} [decidable_pred p] {s : dfinsupp fun (c : γ) => δ c} {h : (c : γ) → δ c → dfinsupp fun (i : ι) => β i} : subtype_domain p (sum s h) = sum s fun (c : γ) (d : δ c) => subtype_domain p (h c d) :=\n  subtype_domain_sum\n\nend dfinsupp\n\n\n/-! ### Product and sum lemmas for bundled morphisms -/\n\nnamespace monoid_hom\n\n\n@[simp] theorem map_dfinsupp_prod {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1} {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid R] [comm_monoid S] (h : R →* S) (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → R) : coe_fn h (dfinsupp.prod f g) = dfinsupp.prod f fun (a : ι) (b : β a) => coe_fn h (g a b) :=\n  map_prod h (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f)\n\ntheorem coe_dfinsupp_prod {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1} {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [monoid R] [comm_monoid S] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → R →* S) : ⇑(dfinsupp.prod f g) = dfinsupp.prod f fun (a : ι) (b : β a) => ⇑(g a b) :=\n  coe_prod (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f)\n\n@[simp] theorem dfinsupp_prod_apply {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1} {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [monoid R] [comm_monoid S] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → R →* S) (r : R) : coe_fn (dfinsupp.prod f g) r = dfinsupp.prod f fun (a : ι) (b : β a) => coe_fn (g a b) r :=\n  finset_prod_apply (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f) r\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/dfinsupp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3428032392466954}}
{"text": "/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\nimport tactic.lint\nimport tactic.ext\n\n/-!\n# Congruence and related tactics\n\nThis file contains the tactic `congr'`, which is an extension of `congr`, and various tactics\nusing `congr'` internally.\n\n`congr'` has some advantages over `congr`:\n* It turns `↔` to equalities, before trying another congr lemma\n* You can write `congr' n` to give the maximal depth of recursive applications. This is useful if\n  `congr` breaks down the goal to aggressively, and the resulting goals are false.\n* You can write `congr' with ...` to do `congr', ext ...` in a single tactic.\n\nOther tactics in this file:\n* `rcongr`: repeatedly apply `congr'` and `ext.`\n* `convert`: like `exact`, but produces an equality goal if the type doesn't match.\n* `convert_to`: changes the goal, if you prove an equality between the old goal and the new goal.\n* `ac_change`: like `convert_to`, but uses `ac_refl` to discharge the goals.\n-/\n\nopen tactic\nsetup_tactic_parser\n\nnamespace tactic\n\n/-- Apply the constant `iff_of_eq` to the goal. -/\nmeta def apply_iff_congr_core : tactic unit :=\napplyc ``iff_of_eq\n\n/-- The main part of the body for the loop in `congr'`. This will try to replace a goal `f x = f y`\n with `x = y`. Also has support for `==` and `↔`. -/\nmeta def congr_core' : tactic unit :=\ndo tgt ← target,\n   apply_eq_congr_core tgt\n   <|> apply_heq_congr_core\n   <|> apply_iff_congr_core\n   <|> fail \"congr tactic failed\"\n\n/-- The main function in `convert_to`. Changes the goal to `r` and a proof obligation that the goal\n  is equal to `r`. -/\nmeta def convert_to_core (r : pexpr) : tactic unit :=\ndo tgt ← target,\n   h   ← to_expr ``(_ : %%tgt = %%r),\n   rewrite_target h,\n   swap\n\n/-- Attempts to prove the goal by proof irrelevance, but avoids unifying universe metavariables\nto do so. -/\nmeta def by_proof_irrel : tactic unit :=\ndo tgt ← target,\n  @expr.const tt n [level.zero] ← pure tgt.get_app_fn,\n  if n = ``eq then `[apply proof_irrel] else\n  if n = ``heq then `[apply proof_irrel_heq] else\n  failed\n\n/--\nSame as the `congr` tactic, but takes an optional argument which gives\nthe depth of recursive applications.\n* This is useful when `congr` is too aggressive in breaking down the goal.\n* For example, given `⊢ f (g (x + y)) = f (g (y + x))`, `congr'` produces the goals `⊢ x = y`\n  and `⊢ y = x`, while `congr' 2` produces the intended `⊢ x + y = y + x`.\n* If, at any point, a subgoal matches a hypothesis then the subgoal will be closed.\n-/\nmeta def congr' : option ℕ → tactic unit\n| o := focus1 $\n  assumption <|> reflexivity transparency.none <|> by_proof_irrel <|>\n  (guard (o ≠ some 0) >> congr_core' >>\n    all_goals' (try (congr' (nat.pred <$> o)))) <|>\n  reflexivity\n\nnamespace interactive\n\n/--\nSame as the `congr` tactic, but takes an optional argument which gives\nthe depth of recursive applications.\n* This is useful when `congr` is too aggressive in breaking down the goal.\n* For example, given `⊢ f (g (x + y)) = f (g (y + x))`, `congr'` produces the goals `⊢ x = y`\n  and `⊢ y = x`, while `congr' 2` produces the intended `⊢ x + y = y + x`.\n* If, at any point, a subgoal matches a hypothesis then the subgoal will be closed.\n* You can use `congr' with p (: n)?` to call `ext p (: n)?` to all subgoals generated by `congr'`.\n  For example, if the goal is `⊢ f '' s = g '' s` then `congr' with x` generates the goal\n  `x : α ⊢ f x = g x`.\n-/\nmeta def congr' (n : parse (with_desc \"n\" small_nat)?) :\n  parse (tk \"with\" *> prod.mk <$> rintro_patt_parse_hi* <*> (tk \":\" *> small_nat)?)? →\n  tactic unit\n| none         := tactic.congr' n\n| (some ⟨p, m⟩) := focus1 (tactic.congr' n >> all_goals' (tactic.ext p.join m $> ()))\n\n/--\nRepeatedly and apply `congr'` and `ext`, using the given patterns as arguments for `ext`.\n\nThere are two ways this tactic stops:\n* `congr'` fails (makes no progress), after having already applied `ext`.\n* `congr'` canceled out the last usage of `ext`. In this case, the state is reverted to before\n  the `congr'` was applied.\n\nFor example, when the goal is\n```lean\n⊢ (λ x, f x + 3) '' s = (λ x, g x + 3) '' s\n```\nthen `rcongr x` produces the goal\n```lean\nx : α ⊢ f x = g x\n```\nThis gives the same result as `congr', ext x, congr'`.\n\nIn contrast, `congr'` would produce\n```lean\n⊢ (λ x, f x + 3) = (λ x, g x + 3)\n```\nand `congr' with x` (or `congr', ext x`) would produce\n```lean\nx : α ⊢ f x + 3 = g x + 3\n```\n-/\nmeta def rcongr : parse (list.join <$> rintro_patt_parse_hi*) → tactic unit\n| ps := do\n  t ← target,\n  qs ← try_core (tactic.ext ps none),\n  some () ← try_core (tactic.congr' none >>\n    (done <|> do s ← target, guard $ ¬ s =ₐ t)) | skip,\n  done <|> rcongr (qs.get_or_else ps)\n\nadd_tactic_doc\n{ name       := \"congr'\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.congr', `tactic.interactive.congr, `tactic.interactive.rcongr],\n  tags       := [\"congruence\"],\n  inherit_description_from := `tactic.interactive.congr' }\n\n/--\nThe `exact e` and `refine e` tactics require a term `e` whose type is\ndefinitionally equal to the goal. `convert e` is similar to `refine e`,\nbut the type of `e` is not required to exactly match the\ngoal. Instead, new goals are created for differences between the type\nof `e` and the goal. For example, in the proof state\n\n```lean\nn : ℕ,\ne : prime (2 * n + 1)\n⊢ prime (n + n + 1)\n```\n\nthe tactic `convert e` will change the goal to\n\n```lean\n⊢ n + n = 2 * n\n```\n\nIn this example, the new goal can be solved using `ring`.\n\nThe `convert` tactic applies congruence lemmas eagerly before reducing,\ntherefore it can fail in cases where `exact` succeeds:\n```lean\ndef p (n : ℕ) := true\nexample (h : p 0) : p 1 := by exact h -- succeeds\nexample (h : p 0) : p 1 := by convert h -- fails, with leftover goal `1 = 0`\n```\n\nIf `x y : t`, and an instance `subsingleton t` is in scope, then any goals of the form\n`x = y` are solved automatically.\n\nThe syntax `convert ← e` will reverse the direction of the new goals\n(producing `⊢ 2 * n = n + n` in this example).\n\nInternally, `convert e` works by creating a new goal asserting that\nthe goal equals the type of `e`, then simplifying it using\n`congr'`. The syntax `convert e using n` can be used to control the\ndepth of matching (like `congr' n`). In the example, `convert e using\n1` would produce a new goal `⊢ n + n + 1 = 2 * n + 1`.\n-/\nmeta def convert (sym : parse (with_desc \"←\" (tk \"<-\")?)) (r : parse texpr)\n  (n : parse (tk \"using\" *> small_nat)?) : tactic unit :=\ndo tgt ← target,\n  u ← infer_type tgt,\n  r ← i_to_expr ``(%%r : (_ : %%u)),\n  src ← infer_type r,\n  src ← simp_lemmas.mk.dsimplify [] src {fail_if_unchanged := ff},\n  v ← to_expr (if sym.is_some then ``(%%src = %%tgt) else ``(%%tgt = %%src)) tt ff >>= mk_meta_var,\n  (if sym.is_some then mk_eq_mp v r else mk_eq_mpr v r) >>= tactic.exact,\n  gs ← get_goals,\n  set_goals [v],\n  try (tactic.congr' n),\n  gs' ← get_goals,\n  set_goals $ gs' ++ gs\n\nadd_tactic_doc\n{ name       := \"convert\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.convert],\n  tags       := [\"congruence\"] }\n\n/--\n`convert_to g using n` attempts to change the current goal to `g`, but unlike `change`,\nit will generate equality proof obligations using `congr' n` to resolve discrepancies.\n`convert_to g` defaults to using `congr' 1`.\n\n`convert_to` is similar to `convert`, but `convert_to` takes a type (the desired subgoal) while\n`convert` takes a proof term.\nThat is, `convert_to g using n` is equivalent to `convert (_ : g) using n`.\n-/\nmeta def convert_to (r : parse texpr) (n : parse (tk \"using\" *> small_nat)?) : tactic unit :=\nmatch n with\n  | none     := convert_to_core r >> `[congr' 1]\n  | (some 0) := convert_to_core r\n  | (some o) := convert_to_core r >> tactic.congr' o\nend\n\n/--\n`ac_change g using n` is `convert_to g using n` followed by `ac_refl`. It is useful for\nrearranging/reassociating e.g. sums:\n```lean\nexample (a b c d e f g N : ℕ) : (a + b) + (c + d) + (e + f) + g ≤ N :=\nbegin\n  ac_change a + d + e + f + c + g + b ≤ _,\n-- ⊢ a + d + e + f + c + g + b ≤ N\nend\n```\n\n##  Related tactic: `move_add`\nIn the case in which the expression to be changed is a sum of terms, tactic\n`tactive.interactive.move_add` can also be useful. -/\nmeta def ac_change (r : parse texpr) (n : parse (tk \"using\" *> small_nat)?) : tactic unit :=\nconvert_to r n; try ac_refl\n\nadd_tactic_doc\n{ name       := \"convert_to\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.convert_to, `tactic.interactive.ac_change],\n  tags       := [\"congruence\"],\n  inherit_description_from := `tactic.interactive.convert_to }\n\nend interactive\n\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/congr.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.3427852275917898}}
{"text": "import Smt\n\ntheorem cong (p q : Bool) (f : Bool → Bool) : p == q → f p == f q := by\n  smt\n  cases p <;> cases q <;> cases f true <;> cases f false <;> simp_all\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Bool/Cong.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.34273068479335117}}
{"text": "import for_mathlib.category_theory.localization.predicate\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen localization category\n\nnamespace adjunction\n\nvariables {C₁ C₂ D₁ D₂ : Type*} [category C₁] [category C₂] [category D₁] [category D₂]\n  {G : C₁ ⥤ C₂} {F : C₂ ⥤ C₁} (adj : G ⊣ F)\n  (L₁ : C₁ ⥤ D₁) (W₁ : morphism_property C₁) [L₁.is_localization W₁]\n  (L₂ : C₂ ⥤ D₂) (W₂ : morphism_property C₂) [L₂.is_localization W₂]\n  (G' : D₁ ⥤ D₂) (F' : D₂ ⥤ D₁)\n\ninclude adj\n/-- Should be redone using Comm_sq instead of `lifting`. -/\ndef localization [lifting L₁ W₁ (G ⋙ L₂) G'] [lifting L₂ W₂ (F ⋙ L₁) F'] :\n  G' ⊣ F' := adjunction.mk_of_unit_counit\nbegin\n  let α₁ := lifting.iso L₂ W₂ (F ⋙ L₁) F',\n  let α₂ := lifting.iso L₁ W₁ (G ⋙ L₂) G',\n  let e₁ : (G ⋙ L₂) ⋙ F' ≅ (G ⋙ F) ⋙ L₁ := iso_whisker_left G α₁,\n  let e₂ : (F ⋙ L₁) ⋙ G' ≅ (F ⋙ G) ⋙ L₂ := iso_whisker_left F α₂,\n  letI : lifting L₁ W₁ ((G ⋙ F) ⋙ L₁) (G' ⋙ F') :=\n    lifting.of_isos L₁ W₁ e₁ (iso.refl (G' ⋙ F')),\n  letI : lifting L₂ W₂ ((F ⋙ G) ⋙ L₂) (F' ⋙ G') :=\n    lifting.of_isos L₂ W₂ e₂ (iso.refl (F' ⋙ G')),\n  let ε := lift_nat_trans L₁ W₁ L₁ ((G ⋙ F) ⋙ L₁) (𝟭 D₁) (G' ⋙ F')\n    ((functor.left_unitor L₁).inv ≫ nat_trans.hcomp adj.unit (𝟙 L₁)),\n  let η := lift_nat_trans L₂ W₂ ((F ⋙ G) ⋙ L₂) L₂ (F' ⋙ G') (𝟭 D₂)\n      (nat_trans.hcomp adj.counit (𝟙 L₂) ≫ (functor.left_unitor L₂).hom),\n  have hε : ∀ (X₁ : C₁), ε.app (L₁.obj X₁) = L₁.map (adj.unit.app X₁) ≫\n    α₁.inv.app (G.obj X₁) ≫ F'.map (α₂.inv.app X₁),\n  { intro X₁,\n    rw lift_nat_trans_app,\n    dsimp,\n    simp only [id_comp, comp_id], },\n  have hη : ∀ (X₂ : C₂), η.app (L₂.obj X₂) = G'.map (α₁.hom.app X₂) ≫\n    α₂.hom.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂),\n  { intro X₂,\n    rw lift_nat_trans_app,\n    dsimp,\n    simp only [id_comp, comp_id, assoc], },\n  exact\n  { unit := ε,\n    counit := η,\n    left_triangle' := nat_trans_ext L₁ W₁ _ _ (λ X₁, begin\n      have eq := congr_app adj.left_triangle X₁,\n      dsimp at ⊢ eq,\n      erw [id_comp, hε, G'.map_comp, G'.map_comp, assoc, assoc, η.naturality, hη, assoc, assoc],\n      slice_lhs 2 3 { erw [← G'.map_comp, α₁.inv_hom_id_app, G'.map_id], },\n      erw [assoc, assoc, id_comp, α₂.hom.naturality_assoc, ← L₂.map_comp_assoc, eq,\n        L₂.map_id, id_comp, α₂.hom_inv_id_app],\n      refl,\n    end),\n    right_triangle' := nat_trans_ext L₂ W₂ _ _ (λ X₂, begin\n      have eq := congr_app adj.right_triangle X₂,\n      dsimp at ⊢ eq,\n      erw [id_comp, hη, F'.map_comp, F'.map_comp, ← ε.naturality_assoc, hε, assoc, assoc],\n      slice_lhs 4 5 { erw [← F'.map_comp, α₂.inv_hom_id_app, F'.map_id], },\n      erw [id_comp, ← α₁.inv.naturality, ← L₁.map_comp_assoc, eq,\n        L₁.map_id, id_comp, α₁.hom_inv_id_app],\n      refl,\n    end) },\nend\n\n@[simp]\nlemma localization_unit_app [lifting L₁ W₁ (G ⋙ L₂) G'] [lifting L₂ W₂ (F ⋙ L₁) F'] (X₁ : C₁) :\n  (adj.localization L₁ W₁ L₂ W₂ G' F').unit.app (L₁.obj X₁) =\n    L₁.map (adj.unit.app X₁) ≫ (lifting.iso L₂ W₂ (F ⋙ L₁) F').inv.app (G.obj X₁) ≫\n    F'.map ((lifting.iso L₁ W₁ (G ⋙ L₂) G').inv.app X₁) :=\nbegin\n  dsimp only [localization, adjunction.mk_of_unit_counit],\n  rw lift_nat_trans_app,\n  dsimp,\n  simp only [id_comp, comp_id],\nend\n\n@[simp]\nlemma localization_counit_app [lifting L₁ W₁ (G ⋙ L₂) G'] [lifting L₂ W₂ (F ⋙ L₁) F'] (X₂ : C₂) :\n  (adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =\n    G'.map (((lifting.iso L₂ W₂ (F ⋙ L₁) F')).hom.app X₂) ≫\n      (lifting.iso L₁ W₁ (G ⋙ L₂) G').hom.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂) :=\nbegin\n  dsimp only [localization, adjunction.mk_of_unit_counit],\n  rw lift_nat_trans_app,\n  dsimp,\n  simp only [id_comp, comp_id, assoc],\nend\n\nend adjunction\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/adjunction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.3424352692207245}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n\n! This file was ported from Lean 3 source module group_theory.perm.option\n! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Fintype.Perm\nimport Mathbin.GroupTheory.Perm.Sign\nimport Mathbin.Logic.Equiv.Option\n\n/-!\n# Permutations of `option α`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nopen Equiv\n\n/- warning: equiv.option_congr_one -> Equiv.optionCongr_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}}, Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (OfNat.ofNat.{u1} (Equiv.Perm.{succ u1} α) 1 (OfNat.mk.{u1} (Equiv.Perm.{succ u1} α) 1 (One.one.{u1} (Equiv.Perm.{succ u1} α) (MulOneClass.toHasOne.{u1} (Equiv.Perm.{succ u1} α) (Monoid.toMulOneClass.{u1} (Equiv.Perm.{succ u1} α) (DivInvMonoid.toMonoid.{u1} (Equiv.Perm.{succ u1} α) (Group.toDivInvMonoid.{u1} (Equiv.Perm.{succ u1} α) (Equiv.Perm.permGroup.{u1} α))))))))) (OfNat.ofNat.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) 1 (OfNat.mk.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) 1 (One.one.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (MulOneClass.toHasOne.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Monoid.toMulOneClass.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (DivInvMonoid.toMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Group.toDivInvMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.Perm.permGroup.{u1} (Option.{u1} α)))))))))\nbut is expected to have type\n  forall {α : Type.{u1}}, Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (OfNat.ofNat.{u1} (Equiv.Perm.{succ u1} α) 1 (One.toOfNat1.{u1} (Equiv.Perm.{succ u1} α) (InvOneClass.toOne.{u1} (Equiv.Perm.{succ u1} α) (DivInvOneMonoid.toInvOneClass.{u1} (Equiv.Perm.{succ u1} α) (DivisionMonoid.toDivInvOneMonoid.{u1} (Equiv.Perm.{succ u1} α) (Group.toDivisionMonoid.{u1} (Equiv.Perm.{succ u1} α) (Equiv.Perm.permGroup.{u1} α)))))))) (OfNat.ofNat.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) 1 (One.toOfNat1.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (InvOneClass.toOne.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (DivInvOneMonoid.toInvOneClass.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (DivisionMonoid.toDivInvOneMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Group.toDivisionMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.Perm.permGroup.{u1} (Option.{u1} α))))))))\nCase conversion may be inaccurate. Consider using '#align equiv.option_congr_one Equiv.optionCongr_oneₓ'. -/\n@[simp]\ntheorem Equiv.optionCongr_one {α : Type _} : (1 : Perm α).optionCongr = 1 :=\n  Equiv.optionCongr_refl\n#align equiv.option_congr_one Equiv.optionCongr_one\n\n/- warning: equiv.option_congr_swap -> Equiv.optionCongr_swap is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (x : α) (y : α), Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (Equiv.swap.{succ u1} α (fun (a : α) (b : α) => _inst_1 a b) x y)) (Equiv.swap.{succ u1} (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => Option.decidableEq.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (Option.some.{u1} α x) (Option.some.{u1} α y))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (x : α) (y : α), Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (Equiv.swap.{succ u1} α (fun (a : α) (b : α) => _inst_1 a b) x y)) (Equiv.swap.{succ u1} (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => instDecidableEqOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (Option.some.{u1} α x) (Option.some.{u1} α y))\nCase conversion may be inaccurate. Consider using '#align equiv.option_congr_swap Equiv.optionCongr_swapₓ'. -/\n@[simp]\ntheorem Equiv.optionCongr_swap {α : Type _} [DecidableEq α] (x y : α) :\n    optionCongr (swap x y) = swap (some x) (some y) :=\n  by\n  ext (_ | i)\n  · simp [swap_apply_of_ne_of_ne]\n  · by_cases hx : i = x\n    simp [hx, swap_apply_of_ne_of_ne]\n    by_cases hy : i = y <;> simp [hx, hy, swap_apply_of_ne_of_ne]\n#align equiv.option_congr_swap Equiv.optionCongr_swap\n\n/- warning: equiv.option_congr_sign -> Equiv.optionCongr_sign is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : Fintype.{u1} α] (e : Equiv.Perm.{succ u1} α), Eq.{1} (Units.{0} Int Int.monoid) (coeFn.{succ u1, succ u1} (MonoidHom.{u1, 0} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Units.{0} Int Int.monoid) (Monoid.toMulOneClass.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (DivInvMonoid.toMonoid.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Group.toDivInvMonoid.{u1} 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: α) (b : α) => _inst_1 a b) _inst_2) e)\nCase conversion may be inaccurate. Consider using '#align equiv.option_congr_sign Equiv.optionCongr_signₓ'. -/\n@[simp]\ntheorem Equiv.optionCongr_sign {α : Type _} [DecidableEq α] [Fintype α] (e : Perm α) :\n    Perm.sign e.optionCongr = Perm.sign e :=\n  by\n  apply perm.swap_induction_on e\n  · simp [perm.one_def]\n  · intro f x y hne h\n    simp [h, hne, perm.mul_def, ← Equiv.optionCongr_trans]\n#align equiv.option_congr_sign Equiv.optionCongr_sign\n\n/- warning: map_equiv_remove_none -> map_equiv_removeNone is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (σ : Equiv.Perm.{succ u1} (Option.{u1} α)), Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (Equiv.removeNone.{u1, u1} α α σ)) (HMul.hMul.{u1, u1, u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (instHMul.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (MulOneClass.toHasMul.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Monoid.toMulOneClass.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (DivInvMonoid.toMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Group.toDivInvMonoid.{u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.Perm.permGroup.{u1} (Option.{u1} α))))))) (Equiv.swap.{succ u1} (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => Option.decidableEq.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (Option.none.{u1} α) (coeFn.{succ u1, succ u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) => (Option.{u1} α) -> (Option.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) σ (Option.none.{u1} α))) σ)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (σ : Equiv.Perm.{succ u1} (Option.{u1} α)), Eq.{succ u1} (Equiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) (Equiv.optionCongr.{u1, u1} α α (Equiv.removeNone.{u1, u1} α α σ)) (HMul.hMul.{u1, u1, u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (instHMul.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (MulOneClass.toMul.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Monoid.toMulOneClass.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (DivInvMonoid.toMonoid.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Group.toDivInvMonoid.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.Perm.permGroup.{u1} (Option.{u1} α))))))) (Equiv.swap.{succ u1} (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => instDecidableEqOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (Option.none.{u1} α) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Option.{u1} α) (fun (_x : Option.{u1} α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Option.{u1} α) => Option.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (Option.{u1} α) (Option.{u1} α)) σ (Option.none.{u1} α))) σ)\nCase conversion may be inaccurate. Consider using '#align map_equiv_remove_none map_equiv_removeNoneₓ'. -/\n@[simp]\ntheorem map_equiv_removeNone {α : Type _} [DecidableEq α] (σ : Perm (Option α)) :\n    (removeNone σ).optionCongr = swap none (σ none) * σ :=\n  by\n  ext1 x\n  have : Option.map (⇑(remove_none σ)) x = (swap none (σ none)) (σ x) :=\n    by\n    cases x\n    · simp\n    · cases h : σ (some x)\n      · simp [remove_none_none _ h]\n      · have hn : σ (some x) ≠ none := by simp [h]\n        have hσn : σ (some x) ≠ σ none := σ.injective.ne (by simp)\n        simp [remove_none_some _ ⟨_, h⟩, ← h, swap_apply_of_ne_of_ne hn hσn]\n  simpa using this\n#align map_equiv_remove_none map_equiv_removeNone\n\n#print Equiv.Perm.decomposeOption /-\n/-- Permutations of `option α` are equivalent to fixing an\n`option α` and permuting the remaining with a `perm α`.\nThe fixed `option α` is swapped with `none`. -/\n@[simps]\ndef Equiv.Perm.decomposeOption {α : Type _} [DecidableEq α] : Perm (Option α) ≃ Option α × Perm α\n    where\n  toFun σ := (σ none, removeNone σ)\n  invFun i := swap none i.1 * i.2.optionCongr\n  left_inv σ := by simp\n  right_inv := fun ⟨x, σ⟩ =>\n    by\n    have : remove_none (swap none x * σ.option_congr) = σ :=\n      Equiv.optionCongr_injective (by simp [← mul_assoc])\n    simp [← perm.eq_inv_iff_eq, this]\n#align equiv.perm.decompose_option Equiv.Perm.decomposeOption\n-/\n\n#print Equiv.Perm.decomposeOption_symm_of_none_apply /-\ntheorem Equiv.Perm.decomposeOption_symm_of_none_apply {α : Type _} [DecidableEq α] (e : Perm α)\n    (i : Option α) : Equiv.Perm.decomposeOption.symm (none, e) i = i.map e := by simp\n#align equiv.perm.decompose_option_symm_of_none_apply Equiv.Perm.decomposeOption_symm_of_none_apply\n-/\n\n/- warning: equiv.perm.decompose_option_symm_sign -> Equiv.Perm.decomposeOption_symm_sign is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : Fintype.{u1} α] (e : Equiv.Perm.{succ u1} α), Eq.{1} 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Consider using '#align equiv.perm.decompose_option_symm_sign Equiv.Perm.decomposeOption_symm_signₓ'. -/\ntheorem Equiv.Perm.decomposeOption_symm_sign {α : Type _} [DecidableEq α] [Fintype α] (e : Perm α) :\n    Perm.sign (Equiv.Perm.decomposeOption.symm (none, e)) = Perm.sign e := by simp\n#align equiv.perm.decompose_option_symm_sign Equiv.Perm.decomposeOption_symm_sign\n\n/- warning: finset.univ_perm_option -> Finset.univ_perm_option is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : Fintype.{u1} α], Eq.{succ u1} (Finset.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α))) (Finset.univ.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.fintype.{u1, u1} (Option.{u1} α) (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => Option.decidableEq.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (fun (a : Option.{u1} α) (b : Option.{u1} α) => Option.decidableEq.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (Option.fintype.{u1} α _inst_2) (Option.fintype.{u1} α _inst_2))) (Finset.map.{u1, u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.toEmbedding.{succ u1, succ u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.symm.{succ u1, succ u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.decomposeOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b)))) (Finset.univ.{u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Prod.fintype.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α) (Option.fintype.{u1} α _inst_2) (Equiv.fintype.{u1, u1} α α (fun (a : α) (b : α) => _inst_1 a b) (fun (a : α) (b : α) => _inst_1 a b) _inst_2 _inst_2))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : Fintype.{u1} α], Eq.{succ u1} (Finset.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α))) (Finset.univ.{u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (equivFintype.{u1, u1} (Option.{u1} α) (Option.{u1} α) (fun (a : Option.{u1} α) (b : Option.{u1} α) => instDecidableEqOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (fun (a : Option.{u1} α) (b : Option.{u1} α) => instDecidableEqOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b) a b) (instFintypeOption.{u1} α _inst_2) (instFintypeOption.{u1} α _inst_2))) (Finset.map.{u1, u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.toEmbedding.{succ u1, succ u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.{succ u1} (Option.{u1} α)) (Equiv.symm.{succ u1, succ u1} (Equiv.Perm.{succ u1} (Option.{u1} α)) (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (Equiv.Perm.decomposeOption.{u1} α (fun (a : α) (b : α) => _inst_1 a b)))) (Finset.univ.{u1} (Prod.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α)) (instFintypeProd.{u1, u1} (Option.{u1} α) (Equiv.Perm.{succ u1} α) (instFintypeOption.{u1} α _inst_2) (equivFintype.{u1, u1} α α (fun (a : α) (b : α) => _inst_1 a b) (fun (a : α) (b : α) => _inst_1 a b) _inst_2 _inst_2))))\nCase conversion may be inaccurate. Consider using '#align finset.univ_perm_option Finset.univ_perm_optionₓ'. -/\n/-- The set of all permutations of `option α` can be constructed by augmenting the set of\npermutations of `α` by each element of `option α` in turn. -/\ntheorem Finset.univ_perm_option {α : Type _} [DecidableEq α] [Fintype α] :\n    @Finset.univ (Perm <| Option α) _ =\n      (Finset.univ : Finset <| Option α × Perm α).map Equiv.Perm.decomposeOption.symm.toEmbedding :=\n  (Finset.univ_map_equiv_to_embedding _).symm\n#align finset.univ_perm_option Finset.univ_perm_option\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/GroupTheory/Perm/Option.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3423433606499533}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport category_theory.idempotents.functor_extension\nimport for_mathlib.functor_misc\n\n/-!\n# Extension of functors to the idempotent completion\n\nIn this file, we obtain an equivalence of categories\n`karoubi_universal₁ C D : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D)` for\nall categories `C` and `D`. The key construction is `functor_extension₁`\nwhich extends a functor `C ⥤ karoubi D` to a functor `karoubi C ⥤ karoubi D`.\n\nTODO : If `D` is idempotent complete, we also have\n`karoubi_universal C D : C ⥤ D ≌ karoubi C ⥤ D`.\n\n-/\n\nopen category_theory.category\nopen category_theory.idempotents.karoubi\n\nnamespace category_theory\n\nnamespace idempotents\n\nvariables (C D E : Type*) [category C] [category D] [category E]\n\nnamespace karoubi_universal₁\n\n/-\n@[simps]\ndef counit_iso :\n  (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) ⋙\n    functor_extension₁ C D ≅ 𝟭 _ :=\nnat_iso.of_components (λ G,\n{ hom :=\n  { app := λ P,\n    { f := (G.map (decomp_id_p P)).f,\n      comm := begin\n        have eq : P.decomp_id_p = (to_karoubi C).map P.p ≫ P.decomp_id_p ≫ 𝟙 _,\n        { simp only [P.idem, decomp_id_p_f, to_karoubi_map_f, id_eq, comp_f, hom_ext], },\n        simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map eq,\n      end },\n      naturality' := λ P Q f, begin\n        ext,\n        simpa only [hom_ext, G.map_comp] using (G.congr_map (decomp_id_p_naturality f)).symm,\n      end },\n  inv :=\n  { app := λ P,\n    { f := (G.map (decomp_id_i P)).f,\n      comm := begin\n        have eq : P.decomp_id_i = 𝟙 _ ≫ P.decomp_id_i ≫ (to_karoubi C).map P.p,\n        { simp only [idem, id_eq, hom_ext, decomp_id_i_f, comp_f, to_karoubi_map_f], },\n        simpa only [hom_ext, G.map_comp, G.map_id] using (G.congr_map eq),\n      end, },\n    naturality' := λ P Q f, begin\n      ext,\n      simpa only [hom_ext, G.map_comp] using G.congr_map (decomp_id_i_naturality f),\n    end },\n  hom_inv_id' := begin\n    ext P,\n    simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map P.decomp_p.symm,\n  end,\n  inv_hom_id' := begin\n    ext P,\n    simpa only [hom_ext, G.map_comp, G.map_id] using G.congr_map P.decomp_id.symm,\n  end, })\nbegin\n  intros G G' φ,\n  ext P,\n  dsimp,\n  simp only [nat_trans_eq φ P, comp_f, functor_extension₁.map_app_f,\n    functor.comp_map, whisker_left_app],\n  rw [P.decomp_p, G.map_comp, comp_f, assoc, assoc],\n  refl,\nend-/\n\nend karoubi_universal₁\n\n/-@[simps]\ndef karoubi_universal₁ : (C ⥤ karoubi D) ≌ (karoubi C ⥤ karoubi D) :=\n{ functor := functor_extension₁ C D,\n  inverse := (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C),\n  unit_iso := eq_to_iso (functor_extension₁_comp_whiskering_left_to_karoubi C D).symm,\n  counit_iso := karoubi_universal₁.counit_iso C D,\n  functor_unit_iso_comp' := λ F, begin\n    ext P,\n    simpa only [eq_to_iso.hom, eq_to_hom_app, eq_to_hom_map, eq_to_hom_refl, id_comp]\n      using F.congr_map P.idem,\n  end, }\n\nlemma functor_extension₁_comp\n  (F : C ⥤ karoubi D) (G : D ⥤ karoubi E) :\n  (functor_extension₁ C E).obj (F ⋙ (functor_extension₁ D E).obj G) =\n  (functor_extension₁ C D).obj F ⋙ (functor_extension₁ D E).obj G :=\nbegin\n  apply functor.ext,\n  { intros X Y f,\n    erw [id_comp, comp_id],\n    refl, },\n  { intro P,\n    ext,\n    { dsimp,\n      erw [comp_id, id_comp], },\n    { refl, }, }\nend\n\n@[simps]\ndef functor_extension₂ : (C ⥤ D) ⥤ (karoubi C ⥤ karoubi D) :=\n(whiskering_right C D (karoubi D)).obj (to_karoubi D) ⋙ functor_extension₁ C D\n\nlemma functor_extension₂_comp_whiskering_left_to_karoubi :\n  functor_extension₂ C D ⋙ (whiskering_left C (karoubi C) (karoubi D)).obj (to_karoubi C) =\n  (whiskering_right C D (karoubi D)).obj (to_karoubi D) :=\nby simp only [functor_extension₂, functor.assoc_eq,\n  functor_extension₁_comp_whiskering_left_to_karoubi, functor.comp_id]\n\nsection is_idempotent_complete\n\nvariable [is_idempotent_complete D]\n\nnoncomputable instance : is_equivalence (to_karoubi D) := to_karoubi_is_equivalence D\n\n@[simps]\nnoncomputable def karoubi_universal₂ : (C ⥤ D) ≌\n  (karoubi C ⥤ karoubi D) :=\n(equivalence.congr_right (to_karoubi D).as_equivalence).trans\n    (karoubi_universal₁ C D)\n\nlemma karoubi_universal₂_functor_eq :\n  (karoubi_universal₂ C D).functor = functor_extension₂ C D := rfl\n\nnoncomputable instance : is_equivalence (functor_extension₂ C D) :=\nby { rw ← karoubi_universal₂_functor_eq, apply_instance, }\n\n@[simps]\nnoncomputable def functor_extension :\n  (C ⥤ D) ⥤ (karoubi C ⥤ D) :=\nfunctor_extension₂ C D ⋙ (whiskering_right (karoubi C) (karoubi D) D).obj\n    (to_karoubi_is_equivalence D).inverse\n\n@[simps]\nnoncomputable def karoubi_universal : (C ⥤ D) ≌ (karoubi C ⥤ D) :=\n(karoubi_universal₂ C D).trans (equivalence.congr_right (to_karoubi D).as_equivalence.symm)\n\nlemma karoubi_universal_functor_eq :\n  (karoubi_universal C D).functor = functor_extension C D := rfl\n\nnoncomputable instance : is_equivalence (functor_extension C D) :=\nby { rw ← karoubi_universal_functor_eq, apply_instance, }\n\nnoncomputable instance : is_equivalence ((whiskering_left C (karoubi C) D).obj (to_karoubi C)) :=\nbegin\n  let F₁ := ((whiskering_left C (karoubi C) D).obj (to_karoubi C)),\n  let F₂ := ((whiskering_right C _ _).obj (to_karoubi D) ⋙\n    (whiskering_right C _ _).obj (to_karoubi D).inv),\n  apply is_equivalence.cancel_comp_right F₁ F₂,\n  { exact is_equivalence.of_equivalence\n      (@equivalence.congr_right _ _ _ _ C _\n      ((to_karoubi D).as_equivalence.trans (to_karoubi D).as_equivalence.symm)), },\n  { rw [show F₁ ⋙ F₂ = (karoubi_universal C D).inverse, by refl],\n    apply_instance, }\nend\n\nend is_idempotent_complete-/\n\nend idempotents\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/idempotents/functor_extension2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241911813151, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.34232598319933333}}
{"text": "import .lovelib\n\n\n/- # LoVe Demo 8: Operational Semantics\n\nIn this and the next two lectures, we will see how to use Lean to specify the\nsyntax and semantics of programming languages and to reason about the\nsemantics. -/\n\n\nset_option pp.beta true\nset_option pp.generalized_field_notation false\n\nnamespace LoVe\n\n\n/- ## Formal Semantics\n\nA formal semantics helps specify and reason about the programming language\nitself, and about individual programs.\n\nIt can form the basis of verified compilers, interpreters, verifiers, static\nanalyzers, type checkers, etc. Without formal proofs, these tools are\n**almost always wrong**.\n\nIn this area, proof assistants are widely used. Every year, about 10-20% of POPL\npapers are partially or totally formalized. Reasons for this success:\n\n* Little machinery (background libraries, tactics) is needed to get started,\n  beyond inductive types and predicates and recursive functions.\n\n* The proofs tend to have lots of cases, which is a good match for computers.\n\n* Proof assistants keep track of what needs to be changed when as extend the\n  programming language with more features.\n\nCase in point: WebAssembly. To quote Conrad Watt (with some abbreviations):\n\n    We have produced a full Isabelle mechanisation of the core execution\n    semantics and type system of the WebAssembly language. To complete this\n    proof, **several deficiencies** in the official WebAssembly specification,\n    uncovered by our proof and modelling work, needed to be corrected. In some\n    cases, these meant that the type system was **originally unsound**.\n\n    We have maintained a constructive dialogue with the working group,\n    verifying new features as they are added. In particular, the mechanism by\n    which a WebAssembly implementation interfaces with its host environment was\n    not formally specified in the working group's original paper. Extending our\n    mechanisation to model this feature revealed a deficiency in the WebAssembly\n    specification that **sabotaged the soundness** of the type system.\n\n\n## A Minimalistic Imperative Language\n\n__WHILE__ is a minimalistic imperative language with the following grammar:\n\n    S  ::=  skip                 -- no-op\n         |  x := a               -- assignment\n         |  S ; S                -- sequential composition\n         |  if b then S else S   -- conditional statement\n         |  while b do S         -- while loop\n\nwhere `S` stands for a statement (also called command or program), `x` for a\nvariable, `a` for an arithmetic expression, and `b` for a Boolean expression. -/\n\ninductive stmt : Type\n| skip   : stmt\n| assign : string → (state → ℕ) → stmt\n| seq    : stmt → stmt → stmt\n| ite    : (state → Prop) → stmt → stmt → stmt\n| while  : (state → Prop) → stmt → stmt\n\ninfixr ` ;; ` : 90 := stmt.seq\n\n/- In our grammar, we deliberately leave the syntax of arithmetic and Boolean\nexpressions unspecified. In Lean, we have the choice:\n\n* We could use a type such as `aexp` from lecture 1 and similarly for Boolean\n  expressions.\n\n* Supposing a state `s` is a function from variable names to values\n  (`string → ℕ`), we could decide that an arithmetic expression is simply a\n  function from states to natural numbers (`state → ℕ`) and a Boolean expression\n  is a predicate (`state → Prop` or `state → bool`).\n\nThis corresponds to the difference between deep and shallow embeddings:\n\n* A __deep embedding__ of some syntax (expression, formula, program, etc.)\n  consists of an abstract syntax tree specified in the proof assistant\n  (e.g., `aexp`) with a semantics (e.g., `eval`).\n\n* In contrast, a __shallow embedding__ simply reuses the corresponding\n  mechanisms from the logic (e.g., λ-terms, functions and predicate types).\n\nA deep embedding allows us to reason about the syntax (and its semantics). A\nshallow embedding is more lightweight, because we can use it directly, without\nhaving to define a semantics.\n\nWe will use a deep embedding of programs (which we find interesting), and\nshallow embeddings of assignments and Boolean expressions (which we find\nboring).\n\nExamples:\n\n    `λs : state, s \"x\" + s \"y\" + 1`   -- x + y + 1\n    `λs : state, s \"a\" ≠ s \"b\"`       -- a ≠ b\n\n\n## Big-Step Semantics\n\nAn __operational semantics__ corresponds to an idealized interpreter (specified\nin a Prolog-like language). Two main variants:\n\n* big-step semantics;\n\n* small-step semantics.\n\nIn a __big-step semantics__ (also called __natural semantics__), judgments have\nthe form `(S, s) ⟹ t`:\n\n    Starting in a state `s`, executing `S` terminates in the state `t`.\n\nExample:\n\n    `(x := x + y; y := 0, [x ↦ 3, y ↦ 5]) ⟹ [x ↦ 8, y ↦ 0]`\n\nDerivation rules:\n\n    ——————————————— Skip\n    (skip, s) ⟹ s\n\n    ——————————————————————————— Asn\n    (x := a, s) ⟹ s[x ↦ s(a)]\n\n    (S, s) ⟹ t   (T, t) ⟹ u\n    ——————————————————————————— Seq\n    (S; T, s) ⟹ u\n\n    (S, s) ⟹ t\n    ————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⟹ t\n\n    (T, s) ⟹ t\n    ————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⟹ t\n\n    (S, s) ⟹ t   (while b do S, t) ⟹ u\n    —————————————————————————————————————— While-True   if s(b) is true\n    (while b do S, s) ⟹ u\n\n    ————————————————————————— While-False   if s(b) is false\n    (while b do S, s) ⟹ s\n\nAbove, `s(e)` denotes the value of expression `e` in state `s`.\n\nIn Lean, the judgment corresponds to an inductive predicate, and the derivation\nrules correspond to the predicate's introduction rules. Using an inductive\npredicate as opposed to a recursive function allows us to cope with\nnontermination (e.g., a diverging `while`) and nondeterminism (e.g.,\nmultithreading). -/\n\ninductive big_step : stmt × state → state → Prop\n| skip {s} :\n  big_step (stmt.skip, s) s\n| assign {x a s} :\n  big_step (stmt.assign x a, s) (s{x ↦ a s})\n| seq {S T s t u} (hS : big_step (S, s) t)\n    (hT : big_step (T, t) u) :\n  big_step (S ;; T, s) u\n| ite_true {b : state → Prop} {S T s t} (hcond : b s)\n    (hbody : big_step (S, s) t) :\n  big_step (stmt.ite b S T, s) t\n| ite_false {b : state → Prop} {S T s t} (hcond : ¬ b s)\n    (hbody : big_step (T, s) t) :\n  big_step (stmt.ite b S T, s) t\n| while_true {b : state → Prop} {S s t u} (hcond : b s)\n    (hbody : big_step (S, s) t)\n    (hrest : big_step (stmt.while b S, t) u) :\n  big_step (stmt.while b S, s) u\n| while_false {b : state → Prop} {S s} (hcond : ¬ b s) :\n  big_step (stmt.while b S, s) s\n\ninfix ` ⟹ ` : 110 := big_step\n\n\n/- ## Properties of the Big-Step Semantics\n\nEquipped with a big-step semantics, we can\n\n* prove properties of the programming language, such as **equivalence proofs**\n  between programs and **determinism**;\n\n* reason about **concrete programs**, proving theorems relating final states `t`\n  with initial states `s`. -/\n\nlemma big_step_deterministic {S s l r} (hl : (S, s) ⟹ l)\n    (hr : (S, s) ⟹ r) :\n  l = r :=\nbegin\n  induction' hl,\n  case skip {\n    cases' hr,\n    refl },\n  case assign {\n    cases' hr,\n    refl },\n  case seq : S T s t l hS hT ihS ihT {\n    cases' hr with _ _ _ _ _ _ _ t' _ hS' hT',\n    cases' ihS hS',\n    cases' ihT hT',\n    refl },\n  case ite_true : b S T s t hb hS ih {\n    cases' hr,\n    { apply ih hr },\n    { cc } },\n  case ite_false : b S T s t hb hT ih {\n    cases' hr,\n    { cc },\n    { apply ih hr } },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    cases' hr,\n    { cases' ihS hr,\n      cases' ihw hr_1,\n      refl },\n    { cc } },\n  { cases' hr,\n    { cc },\n    { refl } }\nend\n\nlemma big_step_terminates {S s} :\n  ∃t, (S, s) ⟹ t :=\nsorry   -- unprovable\n\nlemma big_step_doesnt_terminate {S s t} :\n  ¬ (stmt.while (λ_, true) S, s) ⟹ t :=\nbegin\n  intro hw,\n  induction' hw,\n  case while_true {\n    assumption },\n  case while_false {\n    cc }\nend\n\n/- We can define inversion rules about the big-step semantics: -/\n\n@[simp] lemma big_step_skip_iff {s t} :\n  (stmt.skip, s) ⟹ t ↔ t = s :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.skip }\nend\n\n@[simp] lemma big_step_assign_iff {x a s t} :\n  (stmt.assign x a, s) ⟹ t ↔ t = s{x ↦ a s} :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.assign }\nend\n\n@[simp] lemma big_step_seq_iff {S T s t} :\n  (S ;; T, s) ⟹ t ↔ (∃u, (S, s) ⟹ u ∧ (T, u) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    apply exists.intro,\n    apply and.intro; assumption },\n  { intro h,\n    cases' h,\n    cases' h,\n    apply big_step.seq; assumption }\nend\n\n@[simp] lemma big_step_ite_iff {b S T s t} :\n  (stmt.ite b S T, s) ⟹ t ↔\n  (b s ∧ (S, s) ⟹ t) ∨ (¬ b s ∧ (T, s) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h; cases' h,\n    { apply big_step.ite_true; assumption },\n    { apply big_step.ite_false; assumption } }\nend\n\nlemma big_step_while_iff {b S s u} :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, b s ∧ (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u)\n  ∨ (¬ b s ∧ u = s) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro t,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    case inl {\n      cases' h with t h,\n      cases' h with hb h,\n      cases' h with hS hwhile,\n      exact big_step.while_true hb hS hwhile },\n    case inr {\n      cases' h with hb hus,\n      rw hus,\n      exact big_step.while_false hb } }\nend\n\nlemma big_step_while_true_iff {b : state → Prop} {S s u}\n    (hcond : b s) :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u) :=\nby rw big_step_while_iff; simp [hcond]\n\n@[simp] lemma big_step_while_false_iff {b : state → Prop}\n    {S s t} (hcond : ¬ b s) :\n  (stmt.while b S, s) ⟹ t ↔ t = s :=\nby rw big_step_while_iff; simp [hcond]\n\n\n/- ## Small-Step Semantics\n\nA big-step semantics\n\n* does not let us reason about intermediate states;\n\n* does not let us express nontermination or interleaving (for multithreading).\n\n__Small-step semantics__ (also called __structural operational semantics__)\nsolve the above issues.\n\nA judgment has the form `(S, s) ⇒ (T, t)`:\n\n    Starting in a state `s`, executing one step of `S` leaves us in the\n    state `t`, with the program `T` remaining to be executed.\n\nAn execution is a finite or infinite chain `(S₀, s₀) ⇒ (S₁, s₁) ⇒ …`.\n\nA pair `(S, s)` is called a __configuration__. It is __final__ if no transition\nof the form `(S, s) ⇒ _` is possible.\n\nExample:\n\n      `(x := x + y; y := 0, [x ↦ 3, y ↦ 5])`\n    `⇒ (skip; y := 0,       [x ↦ 8, y ↦ 5])`\n    `⇒ (y := 0,             [x ↦ 8, y ↦ 5])`\n    `⇒ (skip,               [x ↦ 8, y ↦ 0])`\n\nDerivation rules:\n\n    ————————————————————————————————— Asn\n    (x := a, s) ⇒ (skip, s[x ↦ s(a)])\n\n    (S, s) ⇒ (S', s')\n    ———-————————————————————— Seq-Step\n    (S ; T, s) ⇒ (S' ; T, s')\n\n    —————————————————————— Seq-Skip\n    (skip ; S, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⇒ (T, s)\n\n    ——————————————————————————————————————————————————————————————— While\n    (while b do S, s) ⇒ (if b then (S ; while b do S) else skip, s)\n\nThere is no rule for `skip` (why?). -/\n\ninductive small_step : stmt × state → stmt × state → Prop\n| assign {x a s} :\n  small_step (stmt.assign x a, s) (stmt.skip, s{x ↦ a s})\n| seq_step {S S' T s s'} (hS : small_step (S, s) (S', s')) :\n  small_step (S ;; T, s) (S' ;; T, s')\n| seq_skip {T s} :\n  small_step (stmt.skip ;; T, s) (T, s)\n| ite_true {b : state → Prop} {S T s} (hcond : b s) :\n  small_step (stmt.ite b S T, s) (S, s)\n| ite_false {b : state → Prop} {S T s} (hcond : ¬ b s) :\n  small_step (stmt.ite b S T, s) (T, s)\n| while {b : state → Prop} {S s} :\n  small_step (stmt.while b S, s)\n    (stmt.ite b (S ;; stmt.while b S) stmt.skip, s)\n\ninfixr ` ⇒ ` := small_step\ninfixr ` ⇒* ` : 100 := star small_step\n\n/- Equipped with a small-step semantics, we can **define** a big-step\nsemantics:\n\n    `(S, s) ⟹ t` if and only if `(S, s) ⇒* (skip, t)`\n\nwhere `r*` denotes the reflexive transitive closure of a relation `r`.\n\nAlternatively, if we have already defined a big-step semantics, we can **prove**\nthe above equivalence theorem to validate our definitions.\n\nThe main disadvantage of small-step semantics is that we now have two relations,\n`⇒` and `⇒*`, and reasoning tends to be more complicated.\n\n\n## Properties of the Small-Step Semantics\n\nWe can prove that a configuration `(S, s)` is final if and only if `S = skip`.\nThis ensures that we have not forgotten a derivation rule. -/\n\nlemma small_step_final (S s) :\n  (¬ ∃T t, (S, s) ⇒ (T, t)) ↔ S = stmt.skip :=\nbegin\n  induction' S,\n  case skip {\n    simp,\n    intros T t hstep,\n    cases' hstep },\n  case assign : x a {\n    simp,\n    apply exists.intro stmt.skip,\n    apply exists.intro (s{x ↦ a s}),\n    exact small_step.assign },\n  case seq : S T ihS ihT {\n    simp,\n    cases' classical.em (S = stmt.skip),\n    case inl {\n      rw h,\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.seq_skip },\n    case inr {\n      simp [h, auto.not_forall_eq, auto.not_not_eq] at ihS,\n      cases' ihS s with S' hS',\n      cases' hS' with s' hs',\n      apply exists.intro (S' ;; T),\n      apply exists.intro s',\n      exact small_step.seq_step hs' } },\n  case ite : b S T ihS ihT {\n    simp,\n    cases' classical.em (b s),\n    case inl {\n      apply exists.intro S,\n      apply exists.intro s,\n      exact small_step.ite_true h },\n    case inr {\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.ite_false h } },\n  case while : b S ih {\n    simp,\n    apply exists.intro (stmt.ite b (S ;; stmt.while b S) stmt.skip),\n    apply exists.intro s,\n    exact small_step.while }\nend\n\nlemma small_step_deterministic {S s Ll Rr}\n    (hl : (S, s) ⇒ Ll) (hr : (S, s) ⇒ Rr) :\n  Ll = Rr :=\nbegin\n  induction' hl,\n  case assign : x a s {\n    cases' hr,\n    refl },\n  case seq_step : S S₁ T s s₁ hS₁ ih {\n    cases' hr,\n    case seq_step : S S₂ _ _ s₂ hS₂ {\n      have hSs₁₂ := ih hS₂,\n      cc },\n    case seq_skip {\n      cases' hS₁ } },\n  case seq_skip : T s {\n    cases' hr,\n    { cases' hr },\n    { refl } },\n  case ite_true : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      refl },\n    case ite_false {\n      cc } },\n  case ite_false : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      cc },\n    case ite_false {\n      refl } },\n  case while : b S s {\n    cases' hr,\n    refl }\nend\n\n/- We can define inversion rules also about the small-step semantics. Here are\nthree examples: -/\n\nlemma small_step_skip {S s t} :\n  ¬ ((stmt.skip, s) ⇒ (S, t)) :=\nby intro h; cases' h\n\n@[simp] lemma small_step_seq_iff {S T s Ut} :\n  (S ;; T, s) ⇒ Ut ↔\n  (∃S' t, (S, s) ⇒ (S', t) ∧ Ut = (S' ;; T, t))\n  ∨ (S = stmt.skip ∧ Ut = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro S',\n      apply exists.intro s',\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      cases' h,\n      cases' h,\n      rw right,\n      apply small_step.seq_step,\n      assumption },\n    { cases' h,\n      rw left,\n      rw right,\n      apply small_step.seq_skip } }\nend\n\n@[simp] lemma small_step_ite_iff {b S T s Us} :\n  (stmt.ite b S T, s) ⇒ Us ↔\n  (b s ∧ Us = (S, s)) ∨ (¬ b s ∧ Us = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      rw right,\n      apply small_step.ite_true,\n      assumption },\n    { cases' h,\n      rw right,\n      apply small_step.ite_false,\n      assumption } }\nend\n\n\n/- ### Equivalence of the Big-Step and the Small-Step Semantics (**optional**)\n\nA more important result is the connection between the big-step and the\nsmall-step semantics:\n\n    `(S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t)`\n\nIts proof, given below, is beyond the scope of this course. -/\n\nlemma star_small_step_seq {S T s u}\n    (h : (S, s) ⇒* (stmt.skip, u)) :\n  (S ;; T, s) ⇒* (stmt.skip ;; T, u) :=\nbegin\n  apply star.lift (λSs, (prod.fst Ss ;; T, prod.snd Ss)) _ h,\n  intros Ss Ss' h,\n  cases' Ss,\n  cases' Ss',\n  apply small_step.seq_step,\n  assumption\nend\n\nlemma star_small_step_of_big_step {S s t} (h : (S, s) ⟹ t) :\n  (S, s) ⇒* (stmt.skip, t) :=\nbegin\n  induction' h,\n  case skip {\n    refl },\n  case assign {\n    exact star.single small_step.assign },\n  case seq : S T s t u hS hT ihS ihT {\n    transitivity,\n    exact star_small_step_seq ihS,\n    apply star.head small_step.seq_skip ihT },\n  case ite_true : b S T s t hs hst ih {\n    exact star.head (small_step.ite_true hs) ih },\n  case ite_false : b S T s t hs hst ih {\n    exact star.head (small_step.ite_false hs) ih },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    exact (star.head small_step.while\n      (star.head (small_step.ite_true hb)\n         (star.trans (star_small_step_seq ihS)\n            (star.head small_step.seq_skip ihw)))) },\n  case while_false : b S s hb {\n    exact star.tail (star.single small_step.while)\n      (small_step.ite_false hb) }\nend\n\nlemma big_step_of_small_step_of_big_step {S₀ S₁ s₀ s₁ s₂}\n  (h₁ : (S₀, s₀) ⇒ (S₁, s₁)) :\n  (S₁, s₁) ⟹ s₂ → (S₀, s₀) ⟹ s₂ :=\nbegin\n  induction' h₁;\n    simp [*, big_step_while_true_iff] {contextual := tt},\n  case seq_step {\n    intros u hS' hT,\n    apply exists.intro u,\n    exact and.intro (ih hS') hT }\nend\n\nlemma big_step_of_star_small_step {S s t} :\n  (S, s) ⇒* (stmt.skip, t) → (S, s) ⟹ t :=\nbegin\n  generalize hSs : (S, s) = Ss,\n  intro h,\n  induction h\n      using LoVe.rtc.star.head_induction_on\n      with _ S's' h h' ih\n      generalizing S s;\n    cases' hSs,\n  { exact big_step.skip },\n  { cases' S's' with S' s',\n    apply big_step_of_small_step_of_big_step h,\n    apply ih,\n    refl }\nend\n\nlemma big_step_iff_star_small_step {S s t} :\n  (S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t) :=\niff.intro star_small_step_of_big_step\n  big_step_of_star_small_step\n\n\n/- ## Parallelism (**optional**) -/\n\ninductive par_step :\n    nat → list stmt × state → list stmt × state → Prop\n| intro {Ss Ss' S S' s s' i}\n    (hi : i < list.length Ss)\n    (hS : S = list.nth_le Ss i hi)\n    (hs : (S, s) ⇒ (S', s'))\n    (hS' : Ss' = list.update_nth Ss i S') :\n  par_step i (Ss, s) (Ss', s')\n\nlemma par_step_diamond {i j Ss Ts Ts' s t t'}\n    (hi : i < list.length Ss)\n    (hj : j < list.length Ss)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t')) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry   -- unprovable\n\ndef stmt.W : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign x _) := {x}\n| (stmt.seq S T)    := stmt.W S ∪ stmt.W T\n| (stmt.ite _ S T)  := stmt.W S ∪ stmt.W T\n| (stmt.while _ S)  := stmt.W S\n\ndef exp.R {α : Type} : (state → α) → set string\n| f := {x | ∀s n, f (s{x ↦ n}) ≠ f s}\n\ndef stmt.R : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign _ a) := exp.R a\n| (stmt.seq S T)    := stmt.R S ∪ stmt.R T\n| (stmt.ite b S T)  := exp.R b ∪ stmt.R S ∪ stmt.R T\n| (stmt.while b S)  := exp.R b ∪ stmt.R S\n\ndef stmt.V : stmt → set string\n| S := stmt.W S ∪ stmt.R S\n\nlemma par_step_diamond_VW_disjoint {i j Ss Ts Ts' s t t'}\n    (hiS : i < list.length Ss)\n    (hjT : j < list.length Ts)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t'))\n    (hWV : stmt.W (list.nth_le Ss i hiS)\n       ∩ stmt.V (list.nth_le Ts j hjT) = ∅)\n    (hVW : stmt.V (list.nth_le Ss i hiS)\n       ∩ stmt.W (list.nth_le Ts j hjT) = ∅) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry   -- this should be provable\n\nend LoVe\n", "meta": {"author": "blanchette", "repo": "logical_verification_2020", "sha": "7a9f4bd73498189d9beb5d4591e0f2b3ca316111", "save_path": "github-repos/lean/blanchette-logical_verification_2020", "path": "github-repos/lean/blanchette-logical_verification_2020/logical_verification_2020-7a9f4bd73498189d9beb5d4591e0f2b3ca316111/lean/love08_operational_semantics_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34232597557068944}}
{"text": "import Graph.Graph\nimport Graph.UndirectedGraph\nimport Std.Data.Queue\nimport Std.Data.Stack\nimport Std.Data.HashSet\n\n/-!\n## Traverse (old)\n\n*Note that this module in not imported by default through `import Graph`, import it explicitly using `import Graph.TraverseDeprecated`.*\n\nThis is a nice implementation of depth- and breadth-first search using shared code and a stack and queue respectively. However, due to the limitations of such an implementation these functions are fairly inefficient, in general you should rather use `Graph.Traverse`.\n-/\n\nimport Graph.Container\n\nnamespace Graph\n\nopen Internal\n\nvariable {α : Type} [Inhabited α] {β : Type}\n\nprivate def traverseAux {containerType : Type _} (g : Graph α β) (startingContainer : Container (Nat × Bool) containerType) (startingSources : Array Nat) (startingState : γ) (visit : Nat -> γ -> γ × Bool) (leave : Option (Nat -> γ -> γ)) : γ := do\n  let mut visited : Array Bool := mkArray g.vertexCount false\n  let mut state := startingState\n  let mut container := startingContainer\n  let mut sources := startingSources\n  let mut terminate? : Bool := false\n  for i in [0:maximumIterationCount] do\n    match container.remove? with\n      | none =>\n        if terminate? || sources.isEmpty then break\n        container := container.add (sources.back, true)\n        sources := sources.pop\n      | some ((currentNodeId, true), newContainer) =>\n        container := newContainer\n        if terminate? || visited[currentNodeId] then continue\n        (state, terminate?) := visit currentNodeId state\n        if !leave.isSome && terminate? then break\n        visited := visited.set! currentNodeId true\n        container := if leave.isSome then container.add (currentNodeId, false) else container\n        container := if terminate? then container else\n          let unvisitedNeighborIds := (g.vertices[currentNodeId].adjacencyList.map (λ e => e.target)).filter (λ e => !visited[e])\n          let unvisitedNeighborIdsWithArrivingFlags := unvisitedNeighborIds.map (λ id => (id, true))\n          container.addAll unvisitedNeighborIdsWithArrivingFlags\n      | some ((currentNodeId, false), newContainer) =>\n        container := newContainer\n        match leave with\n          | some leaveFunction => state := leaveFunction currentNodeId state\n          | none => panic! \"This should not be possible\"\n\n  return state\n  where\n    maximumIterationCount : Nat := visitingAndLeavingEachVertex + addingAllSourcesToContainer + visitingSameVertexMultipleTimes + goodMeasure\n    visitingAndLeavingEachVertex := g.vertexCount * 2\n    addingAllSourcesToContainer := startingSources.size\n    visitingSameVertexMultipleTimes := g.edgeCount\n    goodMeasure := 1\n\n/-- A breadth-first traversal of the graph starting at the sources in order. Nodes on the same \"level\" of the traversal are visited in order of the edges added.\n    Visit is a function executed at each vertex, its parameters are the vertex ID and the current state,\n    it returns a new state and a boolean which terminates the traversal if true. Please provide a starting state.\n    *Note that this function scales significantly worse than `breadthFirstTraverse`.*\n    See example uses in Graph/TraverseExample.lean -/\ndef breadthFirstTraverseDeprecated (g : Graph α β) (sources : Array Nat) (startingState : γ ) (visit : Nat -> γ -> γ × Bool) : γ :=\n  traverseAux g Container.emptyQueue sources.reverse startingState visit none\n\n/-- A depth-first traversal of the graph starting at the sources in order. Nodes on the same \"level\" of the traversal are visited in order of the edges added.\n    Visit is a function executed at each vertex, its parameters are the vertex ID and the current state,\n    it returns a new state and a boolean which terminates the traversal if true (but will still leave the already visited nodes).\n    Leave is executed when the node is left, when all its successors have been visited, uses the same state. Please provide a starting state.\n    *Note that this function scales significantly worse than `dephtFirstTraverse`.*\n    See example uses in Graph/TraverseExample.lean -/\ndef depthFirstTraverseDeprecated (g : Graph α β) (sources : Array Nat) (startingState : γ ) (visit : Nat -> γ -> γ × Bool) (leave : Option (Nat -> γ -> γ ) := none) : γ :=\n  traverseAux g Container.emptyStack sources.reverse startingState visit leave\n\nnamespace UndirectedGraph\n\n/-- See directed graph. -/\ndef breadthFirstTraverseDeprecated (ug : UndirectedGraph α β) (sources : Array Nat) (startingState : γ ) (visit : Nat -> γ -> γ × Bool) : γ :=\n  ug.graph.breadthFirstTraverseDeprecated sources startingState visit\n\n/-- See directed graph. -/\ndef depthFirstTraverseDeprecated (ug : UndirectedGraph α β) (sources : Array Nat) (startingState : γ ) (visit : Nat -> γ -> γ × Bool) (leave : Option (Nat -> γ -> γ ) := none) : γ :=\n  ug.graph.depthFirstTraverseDeprecated sources startingState visit leave\n\nend UndirectedGraph\nend Graph\n", "meta": {"author": "PeterKementzey", "repo": "graph-library-for-lean4", "sha": "414cdbe1603340a54133edf9b07985f94ceeb26a", "save_path": "github-repos/lean/PeterKementzey-graph-library-for-lean4", "path": "github-repos/lean/PeterKementzey-graph-library-for-lean4/graph-library-for-lean4-414cdbe1603340a54133edf9b07985f94ceeb26a/Graph/TraverseDeprecated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241632752916, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.34232596794204556}}
{"text": "import category_theory.concrete_category\nimport o_minimal.definable\n\n/-\nThe category of (literal) definable subsets of Rⁿ\nand definable functions between them (not just continuous ones).\nUsed as the index category for \"definable sheaves\".\n-/\n\nnamespace o_minimal\n\nuniverse u\n\nvariables {R : Type u} (S : struc R)\n\nstructure Def : Type u :=\n(ambdim : ℕ)\n(to_set : set (finvec ambdim R))\n(is_definable : def_set S to_set)\n\nvariables {S}\n\ninstance : has_coe_to_sort (Def S) :=\n⟨Type u, λ X, X.to_set⟩\n\ninstance is_definable.Def (K : Def S) : is_definable S K :=\nis_definable.subtype K.is_definable\n\n@[ext] structure Hom (X Y : Def S) : Type u :=\n(to_fun : X → Y)\n(is_definable : def_fun S to_fun)\n\ninstance {X Y : Def S} : has_coe_to_fun (Hom X Y) :=\n⟨_, λ f, f.to_fun⟩\n\ninstance : category_theory.small_category (Def S) :=\n{ hom := Hom,\n  id := λ X, ⟨id, def_fun.id⟩,\n  comp := λ X Y Z f g, ⟨g.1 ∘ f.1, g.2.comp f.2⟩ }\n\ninstance : category_theory.concrete_category.{u} (Def S) :=\n{ forget := { obj := λ X, X, map := λ X Y f, f },\n  forget_faithful := by tidy }\n\nsection pullback\n\nvariables {X Y Z : Def S} (f : X ⟶ Z) (g : Y ⟶ Z)\n\ndef Def.pullback : Def S :=\n{ ambdim := X.ambdim + Y.ambdim,\n  to_set :=\n    {p | ∃ (hX : p.left ∈ X.to_set) (hY : p.right ∈ Y.to_set),\n         f.to_fun ⟨p.left, hX⟩ = g.to_fun ⟨p.right, hY⟩},\n  is_definable := begin\n    -- TODO: This proof is bad; but how to fix it?\n    have := def_set_eq (f.is_definable.comp def_fun.fst) (g.is_definable.comp def_fun.snd),\n    change S.definable _ at this,\n    change S.definable _,\n    convert this using 1,\n    ext u,\n    revert u,\n    refine finvec.rec (λ v w, _),\n    simp only [has_coordinates.subtype_coords, set.mem_image,\n      function.comp_app, has_coordinates.finvec_coords, set.mem_set_of_eq,\n      set.image_id', has_coordinates.prod_coords, subtype.val_eq_coe, prod.exists],\n    change (∃ (hX : (v.append w).left ∈ X.to_set) (hY : (v.append w).right ∈ Y.to_set), _) ↔ _,\n    split; rintro ⟨x, y, H⟩,\n    { exact ⟨⟨v, by simpa using x⟩, ⟨w, by simpa using y⟩, by simpa using H, rfl⟩ },\n    { obtain ⟨H₁, H₂⟩ := H,\n      obtain ⟨rfl, rfl⟩ := finvec.append.inj_iff.mp H₂, clear H₂,\n      refine ⟨by simp, by simp, by simp [H₁]⟩ }\n  end }\n\ndef Def.pullback.π₁ : Def.pullback f g ⟶ X :=\n{ to_fun := λ p, ⟨p.val.left, p.property.fst⟩,\n  is_definable := def_fun_subtype_mk (def_fun.finvec.left.comp def_fun_subtype_val) _ }\n\ndef Def.pullback.π₂ : Def.pullback f g ⟶ Y :=\n{ to_fun := λ p, ⟨p.val.right, p.property.snd.fst⟩,\n  is_definable := def_fun_subtype_mk (def_fun.finvec.right.comp def_fun_subtype_val) _ }\n\nend pullback\n\n/-- A (generating) covering family of a given object `K : Def S`.\nBy definition, it is a finite family of definable maps which is\njointly surjective. -/\nstructure Def.cover (K : Def S) :=\n(n : ℕ)\n(obj : fin n → Def S)\n(map : Π i, obj i ⟶ K)\n(jointly_surjective : ∀ k, ∃ i l, map i l = k)\n\nlemma Def.set_subcanonical\n  {K : Def S} (𝓛 : K.cover) (s : set K) (h : Π i, def_set S (𝓛.map i ⁻¹' s)) :\n  def_set S s :=\nbegin\n  have : ∀ i, def_set S (𝓛.map i '' (𝓛.map i ⁻¹' s)) :=\n    λ i, (𝓛.map i).is_definable.image (h i),\n  convert def_set_Union this,\n  ext k,\n  simp only [set.mem_Union, set.mem_image, set.mem_preimage],\n  split; intro hk,\n  { obtain ⟨i, l, rfl⟩ := 𝓛.jointly_surjective k,\n    refine ⟨i, l, hk, rfl⟩ },\n  { obtain ⟨i, l, hl, rfl⟩ := hk,\n    exact hl }\nend\n\nlemma Def.subcanonical {Y : Type*} [has_coordinates R Y] [is_definable S Y]\n  {K : Def S} (𝓛 : K.cover) (f : K → Y) (h : Π i, def_fun S (f ∘ 𝓛.map i)) :\n  def_fun S f :=\nbegin\n  unfold def_fun at ⊢ h,\n  -- We could pull back the cover to K × Y and use the previous proof,\n  -- but it seems easier to just repeat it\n  have : ∀ i, def_set S ((λ (p : 𝓛.obj i × Y), (𝓛.map i p.1, p.2)) '' _) :=\n    λ i, ((𝓛.map i).is_definable.prod def_fun.id).image (h i),\n  convert def_set_Union this using 1,\n  ext ⟨k, y⟩,\n  simp only [set.mem_Union, set.mem_image, prod.mk.inj_iff, function.comp_app,\n    set.mem_set_of_eq, exists_eq_left', prod.exists],\n  split,\n  { rintro rfl,\n    obtain ⟨i, l, rfl⟩ := 𝓛.jointly_surjective k,\n    refine ⟨i, l, rfl, rfl⟩ },\n  { rintro ⟨i, l, hl, rfl⟩,\n    rw hl }\nend\n\ndef Def.cover.pullback {K : Def S} (𝓛 : K.cover) {K' : Def S} (φ : K' ⟶ K) : K'.cover :=\n{ n := 𝓛.n,\n  obj := λ i, Def.pullback φ (𝓛.map i),\n  map := λ i, Def.pullback.π₁ φ (𝓛.map i),\n  jointly_surjective := λ k',\n    let ⟨i, l, h⟩ := 𝓛.jointly_surjective (φ k') in\n    ⟨i, ⟨finvec.append k' l, by simp, by simp, by simpa using h.symm⟩,\n     by { ext1, apply finvec.left_append }⟩ }\n\n-- TODO: add lemmas for the properties of `Def.cover.pullback` that we actually use\n-- (for example, we probably don't actually need it to be formed from pullbacks)\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/src/o_minimal/Def.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.34232307176218396}}
{"text": "import category_theory.limits.shapes\nimport category_theory.limits.preserves.limits\n\n\nimport subobject_classifier\nimport pullbacks\n\n/-!\n# Image inside a topos\n\nProve that there is an image factorisation for each morphism inside a topos\nTODO : prove that e is epi, will be needed for the forcing.\n-/\nopen category_theory category_theory.category category_theory.limits classifier\n\nnoncomputable theory\nuniverses u v\nvariables {C : Type u} [category.{v} C] [has_finite_limits C] [has_subobject_classifier C]\n\n/- We need to prove some statement about the image now -/\nnamespace image\n\n\ndef monic_to_canonical_fork {X Y : C} (m : X ⟶ Y) [mono m] : \n  fork (lift_truth Y) (classifier_of m) :=\nbegin\n  apply fork.of_ι m, rw [←assoc', terminal.comp_from m, pb_classifier_condition]\nend\n\n@[simp] lemma monic_to_canonical_fork_simp {X Y : C} (m : X ⟶ Y) [mono m] : \n  (monic_to_canonical_fork m).ι = m := by refl\n\n/-\n  In a topos, every monic is an equalizer, [SML94] pp167\n-/\nlemma monic_is_limit_fork {X Y : C} (m : X ⟶ Y) [mono m] : is_limit (monic_to_canonical_fork m) := \nbegin\n  refine fork.is_limit.mk (monic_to_canonical_fork m) _ _ _; intro s; \n  have comm : s.ι ≫ classifier_of m = terminal.from s.X ≫ truth C :=\n  begin\n    rw [←s.condition, ←assoc, cancel_mono], exact terminal.comp_from _ \n  end,\n  { apply pb_lift_from_monic.map m s.ι comm },\n  { apply pb_lift_from_monic.comm m s.ι comm },\n  { apply pb_lift_from_monic.unique m s.ι comm },\nend\n\ndef monic_to_fork_lift {X Y : C} (m : X ⟶ Y) [mono m]\n  (f : fork (lift_truth Y) (classifier_of m)) :=\nfork.is_limit.lift' (monic_is_limit_fork m) (fork.ι f) (fork.condition f)\n\n\nvariable [has_pushouts C]\n\nabbreviation cokernel {X Y : C} (f : X ⟶ Y) := pushout f f\nabbreviation cokernel.x {X Y : C} (f : X ⟶ Y) := (pushout.inl : Y ⟶ cokernel f)\nabbreviation cokernel.y {X Y : C} (f : X ⟶ Y) := (pushout.inr : Y ⟶ cokernel f)\n\n\ndef canonical_factorisation {X Y : C} (f : X ⟶ Y) : mono_factorisation f := \n{ I := equalizer.{v} (cokernel.x f) (cokernel.y f),\n  m := equalizer.ι (cokernel.x f) (cokernel.y f),\n  e := equalizer.lift f pushout.condition }\n\ndef canonical_factorisation_condition {X Y : C} (f : X ⟶ Y) : \n  (canonical_factorisation f).m ≫ (cokernel.x f) = \n  (canonical_factorisation f).m ≫ (cokernel.y f) :=\nequalizer.condition _ _\n\n/- from [SML94] p185 -/\nlemma is_image_make {X Y : C} (f : X ⟶ Y) (F' : mono_factorisation f) : \n  {l // l ≫ (monic_to_canonical_fork F'.m).ι = (canonical_factorisation f).m} :=\nbegin\n  let s := lift_truth Y,\n  let t := classifier_of F'.m,\n  have h : f ≫ s = f ≫ t := by rw [←F'.fac, assoc, ←(monic_to_canonical_fork_simp F'.m),\n                                    fork.condition (monic_to_canonical_fork F'.m), assoc],\n  have h1 : (canonical_factorisation f).m ≫ s = (canonical_factorisation f).m ≫ t := \n  begin\n    rw [←pushout.inl_desc s t h, ←assoc, canonical_factorisation_condition f], simp\n  end,\n  exact fork.is_limit.lift' (monic_is_limit_fork F'.m) (canonical_factorisation f).m (h1)\nend\n  \ndef is_image_canonical_factorisation {X Y : C} (f : X ⟶ Y) : \n  is_image (canonical_factorisation f) := \n{ lift := λ F', (is_image_make f F').val, \n  lift_fac' := λ F', (is_image_make f F').prop }\n\ndef image_facto {X Y : C} (f : X ⟶ Y) : image_factorisation f := \n{ F := canonical_factorisation f,\n  is_image := is_image_canonical_factorisation f }\n\ninstance has_image_topos : has_images C := \n{ has_image := λ _ _ f, has_image.mk (image_facto f) }\n\nend image", "meta": {"author": "cchanavat", "repo": "lean-topos", "sha": "c8e22c35ed4dc4ea0d74a59c91785b8a4c8e48a4", "save_path": "github-repos/lean/cchanavat-lean-topos", "path": "github-repos/lean/cchanavat-lean-topos/lean-topos-c8e22c35ed4dc4ea0d74a59c91785b8a4c8e48a4/image.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722129, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3422771425459661}}
{"text": "import pseudo_normed_group.bounded_limits\nimport category_theory.limits.concrete_category\n\nopen category_theory\nopen category_theory.limits\nopen_locale nnreal\n\nuniverses u\nvariables {J : Type u} [small_category J]\n  {F : J ⥤ CompHausFiltPseuNormGrp₁.{u}}\n  (C : cone F)\n\nnamespace CompHaus\n\nlemma comp_apply {X Y Z : CompHaus} (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :\n  (f ≫ g) x = g (f x) := rfl\n\nlemma id_apply (X : CompHaus) (x : X) : (𝟙 X : X ⟶ X) x = x := rfl\n\nlemma continuous_of_is_limit (F : J ⥤ CompHaus.{u}) (C : cone F) (hC : is_limit C)\n  (Y : Type u) [topological_space Y] (t : Y → C.X)\n  (h : ∀ j, continuous (C.π.app j ∘ t)) : continuous t :=\nbegin\n  let CC := CompHaus_to_Top.map_cone C,\n  let hCC : is_limit CC := is_limit_of_preserves CompHaus_to_Top hC,\n  let E : cone (F ⋙ CompHaus_to_Top) := ⟨Top.of Y, λ j, ⟨_,h j⟩, _⟩,\n  { convert (hCC.lift E).continuous,\n    ext1 a,\n    apply concrete.is_limit_ext _ hCC, intros j,\n    change _ = (hCC.lift E ≫ CC.π.app j) _,\n    rw hCC.fac, refl },\n  { intros i j f, ext,\n    change _ = (C.π.app _ ≫ F.map _) _,\n    rw C.w,\n    refl }\nend\n\nlemma injective_of_is_iso {X Y : CompHaus} (f : X ⟶ Y) [is_iso f] :\n  function.injective f :=\nbegin\n  intros x y h,\n  apply_fun (inv f) at h,\n  simp only [← CompHaus.comp_apply, is_iso.hom_inv_id] at h,\n  exact h\nend\n\nend CompHaus\n\nnamespace CompHausFiltPseuNormGrp₁\n\nnoncomputable theory\n\n/-- A choice of level for some element of an exhaustive\nCompHausly filtered pseudo normed group. -/\ndef lvl (X : CompHausFiltPseuNormGrp₁) (x : X) : ℝ≥0 :=\n(X.exhaustive x).some\n\nlemma lvl_spec (X : CompHausFiltPseuNormGrp₁) (x : X) :\n  x ∈ pseudo_normed_group.filtration X (X.lvl x) :=\n(X.exhaustive x).some_spec\n\n/-- Consider an element of an exhaustive CompHausly filtered pseudo normed group\nas an element of its chosen level. -/\ndef as_lvl (X : CompHausFiltPseuNormGrp₁) (x : X) :\n  (level.obj (X.lvl x)).obj X :=\n⟨x, (X.lvl_spec x)⟩\n\n@[reassoc]\nlemma level_cone_compatible (c₁ c₂ : ℝ≥0) (e : c₁ ⟶ c₂) (j) :\n  (level.map e).app _ ≫ ((level.obj _).map_cone C).π.app j =\n  ((level.obj _).map_cone C).π.app j ≫ (level.map e).app _ := rfl\n\nnamespace level_jointly_reflects_limits\n\nvariables (hC : Π c : ℝ≥0, is_limit ((level.obj c).map_cone C))\nvariables (S : cone F)\n\n/-- An auxiliary definition to be used in the constructions below. -/\ndef lift_fun : S.X → C.X :=\nλ s, ((hC (S.X.lvl s)).lift ((level.obj _).map_cone S) (S.X.as_lvl s)).1\n\nlemma lift_fun_map_zero : lift_fun C hC S 0 = 0 :=\nbegin\n  --dsimp [lift_fun],  -- slows down the proof.  changes to goal to\n  -- ⊢ ((hC (S.X.lvl 0)).lift ((level.obj (S.X.lvl 0)).map_cone S)) (S.X.as_lvl 0) = 0\n  let a := ((hC (S.X.lvl 0)).lift ((level.obj (S.X.lvl 0)).map_cone S)) (S.X.as_lvl 0),\n  let b := (C.X.as_lvl 0),\n  let c₁ := S.X.lvl 0,\n  let c₂ := C.X.lvl 0,\n  let c := c₁ ⊔ c₂,\n  let i1 : c₁ ⟶ c := hom_of_le le_sup_left,\n  let i2 : c₂ ⟶ c := hom_of_le le_sup_right,\n  let e1 := (level.map i1).app C.X,\n  let e2 := (level.map i2).app C.X,\n  change (e1 a).1 = (e2 b).1, congr' 1,\n  apply concrete.is_limit_ext _ (hC c),\n  intros j, dsimp only [e1, e2, a, b, ← CompHaus.comp_apply],\n  simp only [category.assoc, level_cone_compatible, (hC (S.X.lvl 0)).fac_assoc],\n  ext1,\n  change (S.π.app j) 0 = (C.π.app j) 0, -- was `dsimp [CompHaus.comp_apply, level, as_lvl],`\n  rw [(S.π.app j).map_zero, (C.π.app j).map_zero],\nend\n\nlemma fac_aux (c) (x) (j) :\n  C.π.app j ((hC c).lift ((level.obj c).map_cone S) x).1 =\n  S.π.app j x.1 :=\nbegin\n  change\n    (((hC c).lift ((level.obj c).map_cone S) ≫\n      (level.obj c).map (C.π.app j)) x).1 = _,\n  erw (hC c).fac,\n  refl,\nend\n\nlemma lift_fun_map_add (x y) : lift_fun C hC S (x + y) =\n  lift_fun C hC S x + lift_fun C hC S y :=\nbegin\n  --dsimp [lift_fun],  -- slows down the proof.  changes the goal to\n  -- ⊢ ((hC (S.X.lvl 0)).lift ((level.obj (S.X.lvl 0)).map_cone S)) (S.X.as_lvl 0) = 0\n  let Axy := ((hC (S.X.lvl (x + y))).lift\n    ((level.obj (S.X.lvl (x + y))).map_cone S)) (S.X.as_lvl (x + y)),\n  let Ax := ((hC (S.X.lvl x)).lift\n    ((level.obj (S.X.lvl x)).map_cone S)) (S.X.as_lvl x),\n  let Ay := ((hC (S.X.lvl y)).lift\n    ((level.obj (S.X.lvl y)).map_cone S)) (S.X.as_lvl y),\n  let cxy := S.X.lvl (x + y),\n  let cx := S.X.lvl x,\n  let cy := S.X.lvl y,\n  let cc := cxy ⊔ (cx + cy),\n  let AA : (level.obj (cx + cy)).obj C.X :=\n    ⟨Ax.1 + Ay.1, pseudo_normed_group.add_mem_filtration Ax.2 Ay.2⟩,\n  let i1 : cxy ⟶ cc := hom_of_le le_sup_left,\n  let i2 : (cx + cy) ⟶ cc := hom_of_le le_sup_right,\n  let e1 := (level.map i1).app C.X,\n  let e2 := (level.map i2).app C.X,\n  change (e1 Axy).1 = (e2 AA).1, congr' 1,\n  apply concrete.is_limit_ext _ (hC cc), intros j,\n--  dsimp only [e1, e2, Axy, ← CompHaus.comp_apply],\n  simp only [e1, e2, Axy, ← CompHaus.comp_apply, category.assoc, level_cone_compatible,\n    (hC (S.X.lvl (x + y))).fac_assoc],\n  ext1,\n  dsimp [CompHaus.comp_apply, level, as_lvl, AA],\n\n  rw [(S.π.app j).map_add, (C.π.app j).map_add],\n  congr' 1,\n  { dsimp only [Ax],\n    erw fac_aux, refl },\n  { dsimp only [Ay],\n    erw fac_aux, refl },\nend\n\n/- Lemmas `lift_strict` and `lift_continuous` separately compile *much* faster than when they are\nembedded in `CompHausFiltPseuNormGrp₁.level_jointly_reflects_limits.lift`. -/\nlemma lift_strict (c : ℝ≥0) (x : S.X) (hx : x ∈ pseudo_normed_group.filtration S.X c) :\n  lift_fun C hC S x ∈ pseudo_normed_group.filtration C.X c :=\nbegin\n  let y : (level.obj c).obj S.X := ⟨x,hx⟩,\n  let z := (hC c).lift ((level.obj c).map_cone S) y,\n  let a := ((hC (S.X.lvl x)).lift ((level.obj (S.X.lvl x)).map_cone S)) (S.X.as_lvl x),\n  suffices : a.1 = z.1,\n  { dsimp only [lift_fun], rw this, exact z.2, },\n  let cc := c ⊔ S.X.lvl x,\n  let i1 : c ⟶ cc := hom_of_le le_sup_left,\n  let i2 : S.X.lvl x ⟶ cc := hom_of_le le_sup_right,\n  suffices : (level.map i1).app _ z = (level.map i2).app _ a,\n  { apply_fun (λ e, e.1) at this, exact this.symm },\n  dsimp [a,z],\n  apply concrete.is_limit_ext _ (hC cc),\n  intros j,\n  simp only [← CompHaus.comp_apply, category.assoc, level_cone_compatible,\n    (hC c).fac_assoc, (hC (S.X.lvl x)).fac_assoc],\n  ext, refl,\nend\n\n/- Lemmas `lift_strict` and `lift_continuous` separately compile *much* faster than when they are\nembedded in `CompHausFiltPseuNormGrp₁.level_jointly_reflects_limits.lift`. -/\nlemma lift_continuous (c : ℝ≥0) :\n  continuous (pseudo_normed_group.level (lift_fun C hC S) (lift_strict C hC S) c) :=\nbegin\n  let t : _ → (level.obj c).obj C.X := _, change continuous t,\n  suffices : ∀ j, continuous (((level.obj c).map (C.π.app j)) ∘ t),\n  { exact CompHaus.continuous_of_is_limit (F ⋙ level.obj c)\n      ((level.obj c).map_cone C) (hC c) _ _ this },\n  intros j,\n  convert ((level.obj c).map (S.π.app j)).continuous,\n  ext1 a,\n  let cc : ℝ≥0 := c ⊔ (S.X.lvl a),\n  let i1 : c ⟶ cc := hom_of_le le_sup_left,\n  let i2 : (S.X.lvl a) ⟶ cc := hom_of_le le_sup_right,\n  apply_fun ((level.map i1).app _), swap,\n  { intros x y h, ext1, apply_fun (λ e, e.1) at h, exact h },\n  simp only [← CompHaus.comp_apply],\n  simp only [nat_trans.naturality],\n  dsimp [t],\n  generalize_proofs hh,\n  let q := ((level.map i1).app C.X) (pseudo_normed_group.level (lift_fun C hC S) hh c a),\n  have : q = ((level.map i2).app C.X)\n    (((hC (S.X.lvl a)).lift ((level.obj (S.X.lvl a)).map_cone S)) (S.X.as_lvl a)), refl,\n  dsimp only [q] at this,\n  simp only [this, ← CompHaus.comp_apply, category.assoc],\n  erw level_cone_compatible,\n  rw (hC (S.X.lvl a)).fac_assoc,\n  ext, refl,\nend\n\n/-- An auxiliary definition to be used in the constructions below. -/\ndef lift : S.X ⟶ C.X :=\n{ to_fun      := lift_fun _ hC _,\n  map_zero'   := lift_fun_map_zero _ _ _,\n  map_add'    := lift_fun_map_add _ _ _,\n  strict'     := lift_strict _ _ _,\n  continuous' := lift_continuous _ _ _ }\n\nend level_jointly_reflects_limits\n\n/--\nIf `C` is a cone taking values in `CompHausFiltPseuNormGrp₁` such that\nthe restriction to all the levels is a limit cone, then `C` is a limit cone.\n-/\ndef level_jointly_reflects_limits\n  (hC : Π c : ℝ≥0, is_limit ((level.obj c).map_cone C)) :\n  is_limit C :=\n{ lift := λ S, level_jointly_reflects_limits.lift _ hC _,\n  fac' := begin\n    intros S j,\n    ext1 t,\n    -- `show` replaces the slower\n    -- `dsimp [level_jointly_reflects_limits.lift, level_jointly_reflects_limits.lift_fun],`\n    show (C.π.app j) _ = _,\n    erw level_jointly_reflects_limits.fac_aux, refl,\n  end,\n  uniq' := begin\n    intros S m hm,\n    ext1 t,\n    --  `dsimp [level_jointly_reflects_limits.lift, level_jointly_reflects_limits.lift_fun],`\n    --  commented as it slows down the proof.  changes the goal to\n    -- ⊢ m t = ((hC (S.X.lvl t)).lift ((level.obj (S.X.lvl t)).map_cone S)) (S.X.as_lvl t)\n    let a :=\n      ((hC (S.X.lvl t)).lift ((level.obj (S.X.lvl t)).map_cone S)) (S.X.as_lvl t),\n    let c := C.X.lvl (m t),\n    let d := c ⊔ (S.X.lvl t),\n    let i1 : c ⟶ d := hom_of_le le_sup_left,\n    let i2 : (S.X.lvl t) ⟶ d := hom_of_le le_sup_right,\n    change ((level.map i1).app C.X (C.X.as_lvl (m t))).1 =\n      ((level.map i2).app C.X a).1,\n    congrm (subtype.val _),\n    apply concrete.is_limit_ext _ (hC d), intros j,\n    specialize hm j,\n    ext1,\n    show (C.π.app j) (m t) = (C.π.app j) _, --  `show` replaces a slow `dsimp [level, as_lvl, a]`\n    erw [level_jointly_reflects_limits.fac_aux],\n    rw ← hm, refl,\n  end }\n\n/-- Create a moprhism in `CompHausFiltPseuNormGrp₁` levelwise. -/\n@[simps]\ndef create_hom_from_level {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ⟶ (level.obj c).obj Y)\n  (hE0 : (E (X.lvl 0) (X.as_lvl 0)).1 = 0)\n  (hEa : ∀ a b : X, (E _ (X.as_lvl (a + b))).1 =\n    (E _ (X.as_lvl a)).1 + (E _ (X.as_lvl b)).1)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    E _ ≫ (level.map i).app _ = (level.map i).app _ ≫ E _) :\n  X ⟶ Y :=\n{ to_fun := λ x, (E _ $ X.as_lvl x).1,\n  map_zero' := hE0,\n  map_add' := hEa,\n  strict' := begin\n    intros c x hx,\n    let y : (level.obj c).obj X := ⟨x,hx⟩,\n    suffices : ((E (X.lvl x)) (X.as_lvl x)).val = (E _ y).1,\n    { rw this, exact (E _ y).2, },\n    let d := c ⊔ X.lvl x,\n    let i1 : c ⟶ d := hom_of_le le_sup_left,\n    let i2 : X.lvl x ⟶ d := hom_of_le le_sup_right,\n    change ((level.map i2).app Y ((E (X.lvl x)) (X.as_lvl x))).1 =\n      ((level.map i1).app _ _).1,\n    congr' 1,\n    simp only [← CompHaus.comp_apply, hE],\n    refl,\n  end,\n  continuous' := begin\n    intros c,\n    let t := _, change continuous t,\n    convert (E c).continuous,\n    ext a,\n    let d := X.lvl a ⊔ c,\n    let i1 : X.lvl a ⟶ d := hom_of_le le_sup_left,\n    let i2 : c ⟶ d := hom_of_le le_sup_right,\n    change ((level.map i1).app _ (E _ (X.as_lvl a))).1 =\n      ((level.map i2).app _ (E _ a)).1,\n    simp only [← CompHaus.comp_apply, hE], refl,\n  end }\n\nlemma create_iso_from_level_compat_aux {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).hom ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).hom) :\n  ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).inv ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).inv :=\nbegin\n  intros c₁ c₂ i, rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv, hE],\nend\n\nlemma create_iso_from_level_zero_aux {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y)\n  (hE0 : ((E (X.lvl 0)).hom (X.as_lvl 0)).1 = 0)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).hom ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).hom) :\n  ((E (Y.lvl 0)).inv (Y.as_lvl 0)).1 = 0 :=\nbegin\n  let x := (((E (Y.lvl 0)).inv) (Y.as_lvl 0)),\n  let c := Y.lvl 0 ⊔ X.lvl 0,\n  let i1 : Y.lvl 0 ⟶ c := hom_of_le le_sup_left,\n  let i2 : X.lvl 0 ⟶ c := hom_of_le le_sup_right,\n  change ((level.map i2).app _ _).val = ((level.map i1).app _ (Y.as_lvl 0)).val at hE0,\n  have hE0' := hE0,\n  replace hE0 : (((level.map i2).app Y) (((E (X.lvl 0)).hom) (X.as_lvl 0))) =\n    (((level.map i1).app Y) (Y.as_lvl 0)),\n  { ext1, exact hE0 },\n  apply_fun (E c).inv at hE0,\n  simp only [← CompHaus.comp_apply, hE, category.assoc, iso.hom_inv_id] at hE0,\n  apply_fun (λ e, e.val) at hE0,\n  change ((level.map i1).app _ x).val = ((level.map i2).app _ (X.as_lvl 0)).val,\n  dsimp only [x, ← CompHaus.comp_apply],\n  rw [create_iso_from_level_compat_aux, CompHaus.comp_apply],\n  exact hE0.symm,\n  assumption,\nend\n\n/-- An auxiliary definition to be used in the constructions below. -/\ndef as_lvl_add (X : CompHausFiltPseuNormGrp₁)\n  (a b : X) : (level.obj (X.lvl a + X.lvl b)).obj X :=\n⟨a+b, pseudo_normed_group.add_mem_filtration (X.lvl_spec a) (X.lvl_spec b)⟩\n\nlemma create_iso_from_level_add_aux_aux {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y) :\n  ∀ (a b : X) (c : ℝ≥0)\n    (h1 : X.lvl (a + b) ⟶ c)\n    (h2 : X.lvl a + X.lvl b ⟶  c),\n    (E c).hom ((level.map h1).app _ $ X.as_lvl _) =\n    (E c).hom ((level.map h2).app _ $ X.as_lvl_add _ _) :=\nbegin\n  intros, congr,\nend\n\nlemma create_iso_from_level_add_aux {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y)\n  (hEa : ∀ a b : X, ((E _).hom (X.as_lvl (a + b))).1 =\n    ((E _).hom (X.as_lvl a)).1 + ((E _).hom (X.as_lvl b)).1)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).hom ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).hom) :\n  ∀ a b : Y, ((E _).inv (Y.as_lvl (a + b))).1 =\n    ((E _).inv (Y.as_lvl a)).1 + ((E _).inv (Y.as_lvl b)).1 :=\nbegin\n  intros a b,\n  let a' := (((E (Y.lvl a)).inv) (Y.as_lvl a)).val,\n  let b' := (((E (Y.lvl b)).inv) (Y.as_lvl b)).val,\n  let ab' := (((E (Y.lvl (a + b))).inv) (Y.as_lvl (a + b))).val,\n  let c₁ := Y.lvl (a + b),\n  let c₂ := X.lvl a' + X.lvl b',\n  let c := c₁ ⊔ c₂,\n  let i1 : c₁ ⟶ c := hom_of_le le_sup_left,\n  let i2 : c₂ ⟶ c := hom_of_le le_sup_right,\n  change ((level.map i1).app _ _).val = ((level.map i2).app _ $ X.as_lvl_add a' b').val,\n  congr' 1,\n  rw [← CompHaus.comp_apply, create_iso_from_level_compat_aux _ hE],\n  apply_fun (E _).hom,\n  swap, { apply CompHaus.injective_of_is_iso },\n  simp only [← CompHaus.comp_apply, category.assoc, iso.inv_hom_id],\n  simp only [CompHaus.comp_apply, CompHaus.id_apply],\n  let d := X.lvl (a' + b') ⊔ c,\n  let j1 : X.lvl (a' + b') ⟶ d := hom_of_le le_sup_left,\n  let j2 : c ⟶ d := hom_of_le le_sup_right,\n  apply_fun (level.map j2).app _,\n  swap, { intros x y h, ext1, apply_fun (λ e, e.val) at h, exact h },\n  simp only [← CompHaus.comp_apply, category.assoc, hE, ← nat_trans.comp_app],\n  rw [← category.assoc, ← nat_trans.comp_app, ← functor.map_comp, ← functor.map_comp,\n    CompHaus.comp_apply],\n  erw create_iso_from_level_add_aux_aux,\n  any_goals { assumption },\n  swap, refine i2 ≫ j2,\n  simp only [functor.map_comp, nat_trans.comp_app, CompHaus.comp_apply],\n  let s := _, change _ = (E d).hom s,\n  have : s = (level.map j1).app _ (X.as_lvl (a' + b')), by { ext, refl },\n  rw this, clear this,\n  simp_rw [← CompHaus.comp_apply, ← hE, CompHaus.comp_apply],\n  ext1,\n--  `show` replaces the `slower `dsimp [level]`\n  show ↑(Y.as_lvl _) = ↑(((E (X.lvl (a' + b'))).hom) (X.as_lvl (a' + b'))),\n--  show replaces the slower `conv_lhs { dsimp [as_lvl] }`\n  show a + b = _,\n  rw [← subtype.val_eq_coe, hEa a' b'],\n  congrm _ + _,\n  { let d₁ := Y.lvl a,\n    let d₂ := X.lvl a',\n    let d := d₁ ⊔ d₂,\n    let e₁ : d₁ ⟶ d := hom_of_le le_sup_left,\n    let e₂ : d₂ ⟶ d := hom_of_le le_sup_right,\n    change ((level.map e₁).app _ (Y.as_lvl a)).val =\n      ((level.map e₂).app _ (((E (X.lvl a')).hom) (X.as_lvl a'))).val,\n    rw [← CompHaus.comp_apply, hE],\n    congr' 1,\n    apply_fun (E d).inv,\n    swap, { apply CompHaus.injective_of_is_iso },\n    simp_rw [← CompHaus.comp_apply, category.assoc, iso.hom_inv_id],\n    rw ← create_iso_from_level_compat_aux,\n    ext, refl,\n    assumption },\n  { let d₁ := Y.lvl b,\n    let d₂ := X.lvl b',\n    let d := d₁ ⊔ d₂,\n    let e₁ : d₁ ⟶ d := hom_of_le le_sup_left,\n    let e₂ : d₂ ⟶ d := hom_of_le le_sup_right,\n    change ((level.map e₁).app _ (Y.as_lvl b)).val =\n      ((level.map e₂).app _ (((E (X.lvl b')).hom) (X.as_lvl b'))).val,\n    rw [← CompHaus.comp_apply, hE],\n    congr' 1,\n    apply_fun (E d).inv,\n    swap, { apply CompHaus.injective_of_is_iso },\n    simp_rw [← CompHaus.comp_apply, category.assoc, iso.hom_inv_id],\n    rw ← create_iso_from_level_compat_aux,\n    ext, refl,\n    assumption }\nend\n\n/-- Create an isomoprhism in `CompHausFiltPseuNormGrp₁` levelwise. -/\ndef create_iso_from_level {X Y : CompHausFiltPseuNormGrp₁.{u}}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y)\n  (hE0 : ((E (X.lvl 0)).hom (X.as_lvl 0)).1 = 0)\n  (hEa : ∀ a b : X, ((E _).hom (X.as_lvl (a + b))).1 =\n    ((E _).hom (X.as_lvl a)).1 + ((E _).hom (X.as_lvl b)).1)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).hom ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).hom) :\n  X ≅ Y :=\n{ hom := create_hom_from_level (λ c, (E c).hom) hE0 hEa hE,\n  inv := create_hom_from_level (λ c, (E c).inv)\n    (create_iso_from_level_zero_aux _ hE0 hE)\n    (create_iso_from_level_add_aux _ hEa hE)\n    (create_iso_from_level_compat_aux _ hE),\n  hom_inv_id' := begin\n    ext1 t,\n    simp only [comp_apply, create_hom_from_level_to_fun, subtype.val_eq_coe, id_apply],\n    let s := (((E (X.lvl t)).hom) (X.as_lvl t)).val,\n    let c₁ := Y.lvl s,\n    let c₂ := X.lvl t,\n    let c := c₁ ⊔ c₂,\n    let i1 : c₁ ⟶ c := hom_of_le le_sup_left,\n    let i2 : c₂ ⟶ c := hom_of_le le_sup_right,\n    change ((level.map i1).app _ _).val = ((level.map i2).app _ (X.as_lvl t)).val,\n    simp only [← CompHaus.comp_apply],\n    rw create_iso_from_level_compat_aux,\n    any_goals { assumption },\n    simp only [CompHaus.comp_apply],\n    congr' 1,\n    apply_fun (E c).hom,\n    swap,\n    { intros x y h,\n      apply_fun (E c).inv at h,\n      simp only [← CompHaus.comp_apply, iso.hom_inv_id] at h,\n      exact h },\n    rw [← CompHaus.comp_apply, iso.inv_hom_id, CompHaus.id_apply],\n    rw [← CompHaus.comp_apply, ← hE],\n    refl,\n  end,\n  inv_hom_id' := begin\n    ext1 t,\n    simp only [comp_apply, create_hom_from_level_to_fun, subtype.val_eq_coe, id_apply],\n    let s := (((E (Y.lvl t)).inv) (Y.as_lvl t)).val,\n    let c₁ := X.lvl s,\n    let c₂ := Y.lvl t,\n    let c := c₁ ⊔ c₂,\n    let i1 : c₁ ⟶ c := hom_of_le le_sup_left,\n    let i2 : c₂ ⟶ c := hom_of_le le_sup_right,\n    change ((level.map i1).app _ _).val = ((level.map i2).app _ (Y.as_lvl t)).val,\n    simp only [← CompHaus.comp_apply],\n    rw hE,\n    simp only [CompHaus.comp_apply],\n    congr' 1,\n    apply_fun (E c).inv,\n    swap,\n    { intros x y h,\n      apply_fun (E c).hom at h,\n      simp only [← CompHaus.comp_apply, iso.inv_hom_id] at h,\n      exact h },\n    rw [← CompHaus.comp_apply, iso.hom_inv_id, CompHaus.id_apply],\n    rw [← CompHaus.comp_apply, ← create_iso_from_level_compat_aux],\n    any_goals { assumption },\n    refl,\n  end }\n\nlemma level_create_iso_from_level {X Y : CompHausFiltPseuNormGrp₁}\n  (E : Π c, (level.obj c).obj X ≅ (level.obj c).obj Y)\n  (hE0 : ((E (X.lvl 0)).hom (X.as_lvl 0)).1 = 0)\n  (hEa : ∀ a b : X, ((E _).hom (X.as_lvl (a + b))).1 =\n    ((E _).hom (X.as_lvl a)).1 + ((E _).hom (X.as_lvl b)).1)\n  (hE : ∀ (c₁ c₂ : ℝ≥0) (i : c₁ ⟶ c₂),\n    (E _).hom ≫ (level.map i).app _ = (level.map i).app _ ≫ (E _).hom) (c) :\n  (level.obj c).map\n  (create_iso_from_level E hE0 hEa hE).hom = (E _).hom :=\nbegin\n  ext t,\n  --  `dsimp [create_iso_from_level, create_hom_from_level, level],` -- changes the goal to\n  -- ⊢ ↑(⇑((E (X.lvl ↑t)).hom) (X.as_lvl ↑t)) = ↑(⇑((E c).hom) t)\n  -- but is not needed and slows down the proof.\n  let d := X.lvl t.1 ⊔ c,\n  let i1 : X.lvl t.1 ⟶ d := hom_of_le le_sup_left,\n  let i2 : c ⟶ d := hom_of_le le_sup_right,\n  change ((level.map i1).app _ _).val =\n    ((level.map i2).app _ _).val,\n  congr' 1,\n  simp only [← CompHaus.comp_apply, hE], ext, refl\nend\n\nlemma level_jointly_faithful {X Y : CompHausFiltPseuNormGrp₁} (f g : X ⟶ Y)\n  (h : ∀ c, (level.obj c).map f = (level.obj c).map g) : f = g :=\nbegin\n  ext t,\n  specialize h (X.lvl t),\n  apply_fun (λ e, (e (X.as_lvl t)).1) at h,\n  exact h\nend\n\nend CompHausFiltPseuNormGrp₁\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/Radon/png_reflects_limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.3422771425459659}}
{"text": "-------------------------------------------------------------------\n-- The PEDANTIC (Proof Engine for Deductive Automation using Non-deterministic\n-- Traversal of Instruction Code) verification framework\n--\n-- Developed by Kenneth Roe\n-- For more information, check out www.cs.jhu.edu/~roe\n--\n-- AbsState.lean\n-- This file contains a model of the abstract state template.  The function realizeState relates\n-- these abstract states to concrete states.  Key top level\n-- definitions:\n--    Value\n--    absExp\n--    absState\n--    absEval\n--    realizeState\n--    supportsFunctionality\n--\n-------------------------------------------------------------------\n\n import .impHeap\n\n\nopen tactic\nopen monad\nopen expr\nopen smt_tactic\n\n-------------------------------------------------------------------\n--\n-- absHeap, the abstract heaps\n--\n-------------------------------------------------------------------\n\n def stateProp := imp_state -> Prop\n\n-------------------------------------------------------------------\n--\n-- This section builds up the definition of composing abstract states.\n--\n-------------------------------------------------------------------\n\ndef compose_heaps (h1 h2 : heap) : heap := λ x, h1 x <|> h2 x \n      \ndef concreteCompose (s1 : imp_state) (s2 : imp_state) (s : imp_state) : Prop :=\n    (s1.snd)=(s2.snd) ∧\n    (s.snd)=(s1.snd) ∧\n    (∀ v, (s1.fst) v=option.none ∨ (s2.fst) v=option.none) ∧\n    compose_heaps (s1.fst) (s2.fst)=(s.fst)\n    \ntheorem composeEnvPropagate1 :\n    ∀ s1 s2 s v,\n    concreteCompose s1 s2 s →\n    s1.snd v = s.snd v :=\nbegin\n    intro, intro, intro, intro, intro,\n    unfold concreteCompose at *, destruct a, intros,\n    destruct right,intro, intro, rw left_1\nend.\n\ntheorem composeEnvPropagate2 :\n    ∀ s1 s2 s v,\n    concreteCompose s1 s2 s →\n    s2.snd v = s.snd v :=\nbegin\n    intro, intro, intro, intro, intro,\n    unfold concreteCompose at *, destruct a, intros,\n    destruct right,intro,intro,\n    rewrite left_1, rewrite left\nend\n\n--\n-- This is the main definition for composing abstract states\n--\ndef composeAbsStates : stateProp → stateProp → stateProp :=\n        (λ as1 as2 sr, (∃ h1 h2, \n                           (as1 h1 ∧\n                           as2 h2 ∧\n                           (∀ v, (h1.fst v=option.none ∨\n                                  h2.fst v=option.none)) ∧\n                           (h1.fst)=(h2.fst) ∧\n                           (sr.fst)=(h1.fst) ∧\n                           (∀ x, sr.fst x = compose_heaps (h1.fst) (h2.fst) x))\n              )).\n\n-------------------------------------------------------------------\n--\n-- This section builds up the definition of absState and contains a few\n-- functions for returning information about absState objects.\n--\n-------------------------------------------------------------------\n\ndef absVar := ℕ\n\n--def absHeap := (λ h hpred (x : imp_state), (h = x.fst ∧ hpred x.fst ∧\n--               (∀ v, x.snd v=0)))\n\n--def bind {ev} := ℕ -> option ev.\n\n-- Value is the type used for the values returned when evaluating an absExp\ninductive Value : Type\n| NatValue : ℕ -> Value\n| ListValue : list Value -> Value\n| NoValue : Value\n\ninstance : inhabited Value := ⟨Value.NoValue⟩\n\n-- Here is the definition for expressions.\n-- It takes three parameters in defining its semantics:\n--     ev - the type of the OtherType case in Value\n--     eq - an equality function over ev\n--     f - a function defining the semantics of AbsFun--this usually includes definitions for\n--         many basic operators such as addition\n--\n--inductive  absExp : Type\n--| AbsConstVal : (@Value unit) -> absExp\n--| AbsVar : ident -> absExp\n--| AbsQVar : absVar -> absExp\n--| AbsFun : ident -> list absExp -> absExp.\n\n-- Special functions for managing separation logic\n\nmutual def rangeSet, rangeSetList\nwith rangeSet : Value -> option (list nat)\n| (Value.ListValue ((Value.NatValue loc)::r)) :=\n  (loc::(rangeSetList r))\n| (Value.NatValue _) := option.some list.nil\n| _                  := option.none\nwith rangeSetList : list Value → (list nat)\n| (f::r) := match rangeSet f\n                     with\n                     | some l := l++(rangeSetList r)\n                     | _     := []\n                     end\n| _ := @list.nil ℕ.\n\ndef treeRecords (v:Value) : list ℕ :=\n    (rangeSet v).get_or_else []\n\ndef inTree : Value → Value → Prop\n| (Value.NatValue x) v := x ∈ treeRecords v\n| (Value.ListValue (Value.NatValue x :: _)) v := x ∈ treeRecords v\n| _ v := false\n\ndef rangeNumeric : ℕ → ℕ → list ℕ\n| s (e+1) :=\n    if (e+1)=s then list.nil\n    else if s=e then (e::list.nil)\n    else e::(rangeNumeric s e)\n| s 0 := list.nil\n\nmutual def fullRangeSet, fullRangeSetList\nwith fullRangeSet : Value -> option (list nat)\n| (Value.ListValue ((Value.NatValue loc)::r)) :=\n   match fullRangeSetList r with\n   | option.some ll := option.some (append (rangeNumeric loc (loc+1+(list.sizeof r))) ll)\n   | _              := option.none\n   end\n| (Value.NatValue _) := option.some list.nil\n| _                  := option.none\nwith fullRangeSetList : list Value → option (list nat)\n| (f::r) := match fullRangeSet f\n                     with\n                     | some l :=\n                         match fullRangeSetList r with\n                         | option.some ll := (some (append l ll))\n                         | _ := option.none\n                         end\n                     | _     := option.none\n                     end\n| _ := @list.nil ℕ.\n\n--def listmem : ℕ → list ℕ → bool\n--| _ list.nil := ff\n--| e (list.cons a b) := if beq_nat e a then tt else listmem e b\n\ndef Rmember : ℕ → Value → Prop\n| a v := match (rangeSet v) with\n         | option.some l := a ∈ l \n         | option.none := ff\n         end\n\ninstance : has_mem ℕ Value := ⟨Rmember⟩\n\ndef Rinclude : ℕ → Value → bool\n| a v := match (fullRangeSet v) with\n         | option.some l := a ∈ l \n         | option.none := ff\n         end\n\n\n/- mutual def findRecord, findRecordHelper\nwith findRecord : ℕ → Value → (list Value)\n| l (Value.ListValue ((Value.NatValue x)::r)) :=\n                 if beq_nat x l then\n                     ((Value.NatValue x)::r)\n                 else findRecordHelper x r\n| _ _ := list.nil\nwith findRecordHelper : ℕ → (list Value) → (list Value)\n| _ list.nil := list.nil\n| v (f::r) := match findRecord v f with\n              | list.nil := findRecordHelper v r\n              | x        := x\n              end.   -/\n\n-- place holder\ndef find (n:ℕ) (v:Value) : Value := v.\n\n-- Some auxiliary definitions useful abstract states\ninductive fold_compose : list imp_state → imp_state → Prop\n| FCNil : ∀ x, fold_compose list.nil (inhabited.default heap,x)\n| FCCons : ∀ f r state rstate,\n             fold_compose r rstate →\n             concreteCompose rstate f state →\n             fold_compose (f::r) state\n\n--inductive allFirsts {t1} {t2} : list t1 → list (t1 × t2) → Prop\n--| AFNil : allFirsts list.nil list.nil\n--| AFCons : ∀ fx fy r r', allFirsts r r' → allFirsts (fx::r) ((fx,fy)::r')\n\n--def allFirsts {t1 t2} (l1 : list t1) (l2 : list (t1 × t2)) : Prop := l2.map prod.fst = l1\n\n--inductive allSeconds {t1} {t2} : list t1 → list (t2 × t1) → Prop\n--  | ASNil : allSeconds list.nil list.nil\n--  | ASCons : ∀ fx fy r r', allSeconds r r' → allSeconds (fy::r) ((fx,fy)::r').\n\ninductive anyHeap : ℕ → ℕ → heap → Prop\n    | AnyHeapBase : ∀ start,\n                    anyHeap start 0 (λ x, none)\n    | AnyHeapNext : ∀ start next heap y,\n                    anyHeap (start+1) next heap ->\n                    anyHeap start (next+1)\n                            (λ x, if x=start then some y else heap x).\n\ninductive Rcell : ℕ → (list ℕ) → heap → ℕ → Prop\n  | RCellBase : forall l ll h,\n                Rcell l ll h l\n  | RCellNext : forall l ll index (h : heap) n nn,\n                index ∈ ll=tt →\n                h (n+index)=some nn →\n                Rcell l ll h n →\n                Rcell l ll h nn.\n\ndef combine_heap (h1 : heap) (h2 : heap) (x :ℕ) :=\n    match h1 x with\n    | none := h2 x\n    | some y := some y\n    end.\n\ninductive mergeHeaps : (list heap) → heap → Prop\n  | MHBase : mergeHeaps list.nil (λ x, none)\n  | MHNext : ∀ (f : heap) (r :list heap) (h1 : heap) (h2 : heap) (h : heap),\n             mergeHeaps r h2 →\n             (∀ (x : ℕ), h1 x=none ∨ h2 x=none) →\n             h = (combine_heap h1 h2) →\n             mergeHeaps (f::r) h.\n\ninductive heapWithIndexList : (list ℕ) → (list heap) → (list Value) → (list (nat × heap × Value)) → Prop\n| HWIBase : heapWithIndexList list.nil list.nil list.nil list.nil\n| HWINext : ∀ ir hr ihr br i h b,\n            heapWithIndexList ir hr br ihr ->\n            heapWithIndexList (i::ir) (h::hr) (b::br) ((i,h,b)::ihr).\n\n\ndef findIndex : ℕ → heap → list (nat × heap × Value) → Value\n| n h list.nil :=\n             match h n with\n             | some x := Value.NatValue x\n             | none   := Value.NatValue 0\n             end\n| n h ((nn,hh,v)::r) := if n=nn then v else findIndex n h r\n\ndef buildList : ℕ → ℕ → heap → list (nat ×  heap ×  Value) → list Value\n| i 0 h l := list.nil\n| i (s+1) h l := (findIndex i h l)::(buildList (i+1) s h l)\n\n--inductive ihmem : ℕ → heap → Value → list (ℕ × heap × Value) → Prop\n--| IHBase : ∀ (n : ℕ) (h : heap) (v : Value) (hl : list (ℕ × heap × Value)),\n--            ihmem n h v ((n,h,v)::hl)\n--| IHNext : ∀ n h v f hl,\n--             ihmem n h v hl →\n--             ihmem n h v (f::hl).\n\n--\n-- Recursive definition for the TREE construct--used in the definition of basicState\n--\n-- Parameters:\n--    #1 - root of tree\n--    #2 - size of each node in the tree\n--    #3 - list of offsets to fields for each node\n--    #4 - functional representation of the tree\n--    #5 - concrete heap (Must be exact heap for the tree)\n--\ninductive Tree : ℕ → ℕ → (list ℕ) → Value → heap → Prop\n| TreeNext : ∀ root size indices heaps ihlist h0 h1 heap values vals,\n            size > 0 →\n            anyHeap root size h0 →\n            heapWithIndexList indices heaps values ihlist →\n            not(root=0) →\n            (∀ i h v x, (i,h,v) ∈ ihlist → some x=h0 (root+x) → Tree x size indices v h) →\n            mergeHeaps heaps h1 →\n            (∀ l, (h1 l=none ∨ h0 l=none)) →\n            heap = combine_heap h1 h0 ->\n            vals = buildList root size heap ihlist ->\n            Tree root size indices (Value.ListValue ((Value.NatValue root)::vals)) heap\n  | TreeBase : forall size index (h : ℕ → option ℕ),\n            size > 0 →\n            (∀ (v : ℕ), h v=none) →\n            Tree 0 size index (Value.ListValue ((Value.NatValue 0)::list.nil)) h\n\ndef absTree (r: env → ℕ) (s:ℕ) (f: list ℕ) (v:Value) (st: imp_state) := Tree (r st.snd) s f v st.fst.\n\n-- Path stuff\ninductive anyHeapv : ℕ → ℕ → heap → (list Value) → Prop\n| AnyHeapvBase : ∀ start, anyHeapv start 0 (λ x, none) list.nil\n| AnyHeapvNext : ∀ start next heap y r,\n                     anyHeapv (start+1) next heap r →\n                     anyHeapv start (next+1) (λ x, if x=start then some y else heap x)\n                                    ((Value.NatValue y)::r).\n\ninductive valueIndexList : (list ℕ) → (list Value) → (list (ℕ × Value)) → Prop\n| VIBase : valueIndexList list.nil list.nil list.nil\n| VINext : ∀ ir br i b ibr,\n           valueIndexList ir br ibr →\n           valueIndexList (i::ir) (b::br) ((i,b)::ibr).\n\ninductive imem : ℕ → Value → list (ℕ × Value) → Prop\n| IBase : ∀ (n : ℕ) (v : Value) hl,\n          imem n v ((n,v)::hl)\n| INext : ∀ (n : ℕ) (v : Value) f hl,\n          imem n v hl →\n          imem n v (f::hl).\n\ninductive updateRec : list (nat × Value) → ℕ → list Value → list Value → Prop\n| UBase : ∀ n vl,\n          updateRec vl n list.nil list.nil\n| UMem : ∀ n v vl or nr x,\n         imem n v vl →\n         updateRec vl (n+1) or nr →\n         updateRec vl n (x::or) (v::nr)\n| UDef1 : ∀ n v vl or nr x,\n          not(imem n v vl) →\n          updateRec vl (n+1) or nr →\n          updateRec vl n ((Value.NatValue x)::or) ((Value.NatValue x)::or)\n| UDef2 : ∀ n v vl or nr x rr,\n          not(imem n v vl) →\n          updateRec vl (n+1) or nr →\n          updateRec vl n ((Value.ListValue ((Value.NatValue x)::rr))::or) ((Value.NatValue x)::or).\n\ndef nth {t} : ℕ → list t → t → t\n| 0 (f::r) d := f\n| (n+1) (f::r) d := nth n r d\n| _ _ d := d.\n\ndef nthval: ℕ → Value → Value\n| n (Value.ListValue l) := nth x l Value.NoValue\n| _ _ := Value.NoValue\n\ninductive Path : ℕ → ℕ → list ℕ → Value → Value → Prop\n  | PathNext : ∀ (root : ℕ) (size : ℕ) indices (baseData : Value) rec vals ivals rec2,\n            size > 0 →\n            not(root=0) →\n            --((Value.NatValue root)::rec) = findRecord root baseData →\n            valueIndexList indices vals ivals →\n            --(∀ i x v r, imem i v ivals →\n            --            ((Value.ListValue ((Value.NatValue x)::r))=(nth i rec Value.NoValue) ∧\n            --            Path x size indices baseData v))) →\n            updateRec ivals 0 rec rec2 →\n            Path root size indices baseData (Value.ListValue (Value.NatValue root::rec2))\n  | PathBase : ∀ size l h,\n            size > 0 →\n            Path 0 size l h (Value.ListValue ((Value.NatValue 0)::list.nil)).\n\n--\n-- Rmember is a predicate used in AbsPredicate constructs to determine whether a nat\n-- is in fact a pointer to the head of any of the nodes in the list or tree represented\n-- by an RFun construct.\n--\n-- Parameters:\n--    l - location to test\n--    tree - a tree (which is the same form as parameter #4 to tree above\n--\n-- This definition is used in basicEval for the 'inTree' function\n--\n\ntheorem Rmember1 { a:ℕ } { v:Value } { l:list ℕ } :\n        some l = rangeSet v → (a ∈ l)=(a ∈ v):= begin\n        intros, unfold has_mem.mem, unfold Rmember, rewrite ← a_1,\n        simp only [Rmember._match_1], unfold has_mem.mem\nend\n\ntheorem rootIsMemberAux (root:ℕ) (r:list Value) :\n        root ∈ (Value.ListValue (Value.NatValue root :: r)) = to_bool true :=\nbegin\n    rewrite ← Rmember1, swap, unfold rangeSet,refl,simp\nend\n\ntheorem rootIsMember : forall root size fields heap (v : Value),\n    root ≠ 0 ->\n    Tree root size fields v heap ->\n    root ∈ v := begin\n    intros, cases a_1,\n    rw rootIsMemberAux, simp, simp at a, cases a\nend\n\ninductive strip_nat_values : (list Value) -> (list ℕ) -> Prop\n| SNVNil : strip_nat_values list.nil list.nil\n| SNVCons : forall v a b,\n            strip_nat_values a b ->\n            strip_nat_values ((Value.NatValue v)::a) (v::b).\n\n-- Separation connectives\ndef absCompose : (imp_state → Prop) → (imp_state → Prop) → imp_state → Prop\n| s1 s2 st := ∃ st1 st2,\n              (s1 st1 ∧\n               s2 st2 ∧\n               concreteCompose st1 st2 st).\n\ninfix `**`:5 := absCompose.\n\ndef absMagicWand : (imp_state → Prop) → (imp_state → Prop) → imp_state → Prop\n| s1 s2 st := ∀ st1 st2,\n              s1 st1 →\n              s2 st2 →\n              concreteCompose st st2 st1.\n\ndef absOrCompose : (imp_state → Prop) → (imp_state → Prop) → imp_state → Prop\n| s1 s2 st := (s1 st ∨ s2 st)\n\n-- Separation leaves\ndef absEmpty : imp_state → Prop\n| st := ∀ (x : ℕ), st.fst x=none\n\ndef absAny : imp_state → Prop\n| _ := true.\n\ndef absNone : imp_state → Prop\n| _ := false.\n\n-- Non-trivial leaves\ndef absPredicate : (env → Prop) → imp_state → Prop\n| p s := p s.snd.\n\ndef absStateTree : (env → ℕ) → (env → ℕ)→ (env → list ℕ) → (env → Value)  → imp_state → Prop\n| root size fields fvalue st := Tree (root st.snd) (size st.snd) (fields st.snd) (fvalue st.snd) st.fst.\n\ndef absStatePath : (env → ℕ) → (env → ℕ)→ (env → list ℕ) → (env → Value)  → (env → Value) → imp_state → Prop\n| root size fields fvalue path st := Path (root st.snd) (size st.snd) (fields st.snd) (fvalue st.snd) (path st.snd).\n\ndef absStateArray : (env → ℕ) → (env → ℕ)→ (env → list Value) → imp_state → Prop\n| root size values st := anyHeapv (root st.snd) (size st.snd) st.fst (values st.snd).\n\ndef absCell : (env → ℕ) → (env → ℕ) → imp_state → Prop\n| l v h := (h.fst (l h.snd))=some (v h.snd) ∧\n            (l h.snd ≠ 0 →\n            (∀ (x : ℕ), x ≠ (l h.snd) → h.fst x=none)).\n\n-- Separation updaters\ndef absUpdateLoc : (env → ℕ) → (env → ℕ) → (imp_state → Prop) → imp_state → Prop\n| loc val s str := ∀ st, s st →\n                   (st.snd = str.snd ∧\n                    str.fst = (λ v, if v=(loc st.snd) then val st.snd else (st.fst) v)).\n\ndef absUpdateState : (imp_state → Prop) → (imp_state → Prop) → (imp_state → Prop) → (imp_state → Prop)\n| s m p := absCompose (absMagicWand s m) p.\n\n-- Separation quantifiers\ndef absAll {t : Type} : (t → imp_state → Prop) → (imp_state → Prop) :=\n    (λ (st : t → imp_state → Prop) (i : imp_state), (∀ (q:t), (st q i)))\n\ndef absAllU {t : Type} : (list t) → (t → imp_state → Prop) → (imp_state → Prop)\n| (f::r) st := absCompose (st f) (absAllU r st)\n| list.nil st := absEmpty\n\ndef absExists {t : Type} : (t → imp_state → Prop) → (imp_state → Prop) :=\n    (λ (st : t → imp_state → Prop) (i : imp_state), (∃ (q:t), (st q i)))\n\ndef absExistsU {t : Type} : list t → (t → imp_state → Prop) → (imp_state → Prop)\n| (f::r) st := absOrCompose (st f) (absExistsU r st)\n| list.nil st := absNone\n\ndef absEach {t : Type} : (list t) → (t → imp_state → Prop) → (imp_state → Prop)\n| (f::r) st := absCompose (st f) (absEach r st)\n| list.nil st := absEmpty.\n\ndef absState := imp_state → Prop\n\ndef absExp {t} := env → t\n\ndef realizeState (st : absState) (s : imp_state) : Prop := st s\n\ndef absEval {t} (e : env) (exp : env → t) := exp e\n\n", "meta": {"author": "kendroe", "repo": "pedantic2", "sha": "5c28cd637be8a1485dccb56f0e05e612573b313e", "save_path": "github-repos/lean/kendroe-pedantic2", "path": "github-repos/lean/kendroe-pedantic2/pedantic2-5c28cd637be8a1485dccb56f0e05e612573b313e/AbsState.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.34221841512826584}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.pi.basic\nimport category_theory.limits.has_limits\n\n/-!\n# Limits in the category of indexed families of objects.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nGiven a functor `F : J ⥤ Π i, C i` into a category of indexed families,\n1. we can assemble a collection of cones over `F ⋙ pi.eval C i` into a cone over `F`\n2. if all those cones are limit cones, the assembled cone is a limit cone, and\n3. if we have limits for each of `F ⋙ pi.eval C i`, we can produce a\n   `has_limit F` instance\n-/\n\nopen category_theory\nopen category_theory.limits\n\nnamespace category_theory.pi\n\nuniverses v₁ v₂ u₁ u₂\n\nvariables {I : Type v₁} {C : I → Type u₁} [Π i, category.{v₁} (C i)]\nvariables {J : Type v₁} [small_category J]\nvariables {F : J ⥤ Π i, C i}\n\n/--\nA cone over `F : J ⥤ Π i, C i` has as its components cones over each of the `F ⋙ pi.eval C i`.\n-/\ndef cone_comp_eval (c : cone F) (i : I) : cone (F ⋙ pi.eval C i) :=\n{ X := c.X i,\n  π :=\n  { app := λ j, c.π.app j i,\n    naturality' := λ j j' f, congr_fun (c.π.naturality f) i, } }\n\n/--\nA cocone over `F : J ⥤ Π i, C i` has as its components cocones over each of the `F ⋙ pi.eval C i`.\n-/\ndef cocone_comp_eval (c : cocone F) (i : I) : cocone (F ⋙ pi.eval C i) :=\n{ X := c.X i,\n  ι :=\n  { app := λ j, c.ι.app j i,\n    naturality' := λ j j' f, congr_fun (c.ι.naturality f) i, } }\n\n/--\nGiven a family of cones over the `F ⋙ pi.eval C i`, we can assemble these together as a `cone F`.\n-/\ndef cone_of_cone_comp_eval (c : Π i, cone (F ⋙ pi.eval C i)) : cone F :=\n{ X := λ i, (c i).X,\n  π :=\n  { app := λ j i, (c i).π.app j,\n    naturality' := λ j j' f, by { ext i, exact (c i).π.naturality f, } } }\n\n/--\nGiven a family of cocones over the `F ⋙ pi.eval C i`,\nwe can assemble these together as a `cocone F`.\n-/\ndef cocone_of_cocone_comp_eval (c : Π i, cocone (F ⋙ pi.eval C i)) : cocone F :=\n{ X := λ i, (c i).X,\n  ι :=\n  { app := λ j i, (c i).ι.app j,\n    naturality' := λ j j' f, by { ext i, exact (c i).ι.naturality f, } } }\n\n/--\nGiven a family of limit cones over the `F ⋙ pi.eval C i`,\nassembling them together as a `cone F` produces a limit cone.\n-/\ndef cone_of_cone_eval_is_limit {c : Π i, cone (F ⋙ pi.eval C i)} (P : Π i, is_limit (c i)) :\n  is_limit (cone_of_cone_comp_eval c) :=\n{ lift := λ s i, (P i).lift (cone_comp_eval s i),\n  fac' := λ s j,\n  begin\n    ext i,\n    exact (P i).fac (cone_comp_eval s i) j,\n  end,\n  uniq' := λ s m w,\n  begin\n    ext i,\n    exact (P i).uniq (cone_comp_eval s i) (m i) (λ j, congr_fun (w j) i)\n  end }\n\n/--\nGiven a family of colimit cocones over the `F ⋙ pi.eval C i`,\nassembling them together as a `cocone F` produces a colimit cocone.\n-/\ndef cocone_of_cocone_eval_is_colimit\n  {c : Π i, cocone (F ⋙ pi.eval C i)} (P : Π i, is_colimit (c i)) :\n  is_colimit (cocone_of_cocone_comp_eval c) :=\n{ desc := λ s i, (P i).desc (cocone_comp_eval s i),\n  fac' := λ s j,\n  begin\n    ext i,\n    exact (P i).fac (cocone_comp_eval s i) j,\n  end,\n  uniq' := λ s m w,\n  begin\n    ext i,\n    exact (P i).uniq (cocone_comp_eval s i) (m i) (λ j, congr_fun (w j) i)\n  end }\n\nsection\n\nvariables [∀ i, has_limit (F ⋙ pi.eval C i)]\n\n/--\nIf we have a functor `F : J ⥤ Π i, C i` into a category of indexed families,\nand we have limits for each of the `F ⋙ pi.eval C i`,\nthen `F` has a limit.\n-/\nlemma has_limit_of_has_limit_comp_eval : has_limit F :=\nhas_limit.mk\n{ cone := cone_of_cone_comp_eval (λ i, limit.cone _),\n  is_limit := cone_of_cone_eval_is_limit (λ i, limit.is_limit _), }\n\nend\n\nsection\n\nvariables [∀ i, has_colimit (F ⋙ pi.eval C i)]\n\n/--\nIf we have a functor `F : J ⥤ Π i, C i` into a category of indexed families,\nand colimits exist for each of the `F ⋙ pi.eval C i`,\nthere is a colimit for `F`.\n-/\nlemma has_colimit_of_has_colimit_comp_eval : has_colimit F :=\nhas_colimit.mk\n{ cocone := cocone_of_cocone_comp_eval (λ i, colimit.cocone _),\n  is_colimit := cocone_of_cocone_eval_is_colimit (λ i, colimit.is_colimit _), }\n\nend\n\n/-!\nAs an example, we can use this to construct particular shapes of limits\nin a category of indexed families.\n\nWith the addition of\n`import category_theory.limits.shapes.types`\nwe can use:\n```\nlocal attribute [instance] has_limit_of_has_limit_comp_eval\nexample : has_binary_products (I → Type v₁) := ⟨by apply_instance⟩\n```\n-/\n\nend category_theory.pi\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.34221840787123814}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module control.equiv_functor\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Equiv.Defs\n\n/-!\n# Functions functorial with respect to equivalences\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nAn `equiv_functor` is a function from `Type → Type` equipped with the additional data of\ncoherently mapping equivalences to equivalences.\n\nIn categorical language, it is an endofunctor of the \"core\" of the category `Type`.\n-/\n\n\nuniverse u₀ u₁ u₂ v₀ v₁ v₂\n\nopen Function\n\n#print EquivFunctor /-\n/-- An `equiv_functor` is only functorial with respect to equivalences.\n\nTo construct an `equiv_functor`, it suffices to supply just the function `f α → f β` from\nan equivalence `α ≃ β`, and then prove the functor laws. It's then a consequence that\nthis function is part of an equivalence, provided by `equiv_functor.map_equiv`.\n-/\nclass EquivFunctor (f : Type u₀ → Type u₁) where\n  map : ∀ {α β}, α ≃ β → f α → f β\n  map_refl' : ∀ α, map (Equiv.refl α) = @id (f α) := by obviously\n  map_trans' : ∀ {α β γ} (k : α ≃ β) (h : β ≃ γ), map (k.trans h) = map h ∘ map k := by obviously\n#align equiv_functor EquivFunctor\n-/\n\nrestate_axiom EquivFunctor.map_refl'\n\nrestate_axiom EquivFunctor.map_trans'\n\nattribute [simp] EquivFunctor.map_refl\n\nnamespace EquivFunctor\n\nsection\n\nvariable (f : Type u₀ → Type u₁) [EquivFunctor f] {α β : Type u₀} (e : α ≃ β)\n\n#print EquivFunctor.mapEquiv /-\n/-- An `equiv_functor` in fact takes every equiv to an equiv. -/\ndef mapEquiv : f α ≃ f β where\n  toFun := EquivFunctor.map e\n  invFun := EquivFunctor.map e.symm\n  left_inv x := by\n    convert(congr_fun (EquivFunctor.map_trans e e.symm) x).symm\n    simp\n  right_inv y := by\n    convert(congr_fun (EquivFunctor.map_trans e.symm e) y).symm\n    simp\n#align equiv_functor.map_equiv EquivFunctor.mapEquiv\n-/\n\n#print EquivFunctor.mapEquiv_apply /-\n@[simp]\ntheorem mapEquiv_apply (x : f α) : mapEquiv f e x = EquivFunctor.map e x :=\n  rfl\n#align equiv_functor.map_equiv_apply EquivFunctor.mapEquiv_apply\n-/\n\n#print EquivFunctor.mapEquiv_symm_apply /-\ntheorem mapEquiv_symm_apply (y : f β) : (mapEquiv f e).symm y = EquivFunctor.map e.symm y :=\n  rfl\n#align equiv_functor.map_equiv_symm_apply EquivFunctor.mapEquiv_symm_apply\n-/\n\n#print EquivFunctor.mapEquiv_refl /-\n@[simp]\ntheorem mapEquiv_refl (α) : mapEquiv f (Equiv.refl α) = Equiv.refl (f α) := by\n  simpa [EquivFunctor.mapEquiv]\n#align equiv_functor.map_equiv_refl EquivFunctor.mapEquiv_refl\n-/\n\n#print EquivFunctor.mapEquiv_symm /-\n@[simp]\ntheorem mapEquiv_symm : (mapEquiv f e).symm = mapEquiv f e.symm :=\n  Equiv.ext <| mapEquiv_symm_apply f e\n#align equiv_functor.map_equiv_symm EquivFunctor.mapEquiv_symm\n-/\n\n#print EquivFunctor.mapEquiv_trans /-\n/-- The composition of `map_equiv`s is carried over the `equiv_functor`.\nFor plain `functor`s, this lemma is named `map_map` when applied\nor `map_comp_map` when not applied.\n-/\n@[simp]\ntheorem mapEquiv_trans {γ : Type u₀} (ab : α ≃ β) (bc : β ≃ γ) :\n    (mapEquiv f ab).trans (mapEquiv f bc) = mapEquiv f (ab.trans bc) :=\n  Equiv.ext fun x => by simp [map_equiv, map_trans']\n#align equiv_functor.map_equiv_trans EquivFunctor.mapEquiv_trans\n-/\n\nend\n\n#print EquivFunctor.ofLawfulFunctor /-\ninstance (priority := 100) ofLawfulFunctor (f : Type u₀ → Type u₁) [Functor f] [LawfulFunctor f] :\n    EquivFunctor f where\n  map α β e := Functor.map e\n  map_refl' α := by\n    ext\n    apply LawfulFunctor.id_map\n  map_trans' α β γ k h := by\n    ext x\n    apply LawfulFunctor.comp_map k h x\n#align equiv_functor.of_is_lawful_functor EquivFunctor.ofLawfulFunctor\n-/\n\n#print EquivFunctor.mapEquiv.injective /-\ntheorem mapEquiv.injective (f : Type u₀ → Type u₁) [Applicative f] [LawfulApplicative f]\n    {α β : Type u₀} (h : ∀ γ, Function.Injective (pure : γ → f γ)) :\n    Function.Injective (@EquivFunctor.mapEquiv f _ α β) := fun e₁ e₂ H =>\n  Equiv.ext fun x => h β (by simpa [EquivFunctor.map] using Equiv.congr_fun H (pure x))\n#align equiv_functor.map_equiv.injective EquivFunctor.mapEquiv.injective\n-/\n\nend EquivFunctor\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/EquivFunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093585306514, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.34221840556596506}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.preserves.shapes.equalizers\nimport category_theory.limits.preserves.shapes.zero\n\n/-!\n# Preserving (co)kernels\n\nConstructions to relate the notions of preserving (co)kernels and reflecting (co)kernels\nto concrete (co)forks.\n\nIn particular, we show that `kernel_comparison f g G` is an isomorphism iff `G` preserves\nthe limit of the parallel pair `f,0`, as well as the dual result.\n-/\n\nnoncomputable theory\n\nuniverses v u₁ u₂\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u₁} [category.{v} C] [has_zero_morphisms C]\nvariables {D : Type u₂} [category.{v} D] [has_zero_morphisms D]\nvariables (G : C ⥤ D) [functor.preserves_zero_morphisms G]\n\nnamespace category_theory.limits\n\nsection kernels\nvariables {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)\n\n/--\nThe map of a kernel fork is a limit iff\nthe kernel fork consisting of the mapped morphisms is a limit.\nThis essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.\n\nThis is a variant of `is_limit_map_cone_fork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_map_cone_fork_equiv' :\n  is_limit (G.map_cone (kernel_fork.of_ι h w)) ≃\n  is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])\n    : fork (G.map f) 0) :=\nbegin\n  refine (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine fork.ext (iso.refl _) _,\n  simp,\nend\n\n/--\nThe property of preserving kernels expressed in terms of kernel forks.\n\nThis is a variant of `is_limit_fork_map_of_is_limit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_fork_map_of_is_limit' [preserves_limit (parallel_pair f 0) G]\n  (l : is_limit (kernel_fork.of_ι h w)) :\n  is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :\n    fork (G.map f) 0) :=\nis_limit_map_cone_fork_equiv' G w (preserves_limit.preserves l)\n\nvariables (f) [has_kernel f]\n\n/--\nIf `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of\na kernel fork is a limit.\n-/\ndef is_limit_of_has_kernel_of_preserves_limit [preserves_limit (parallel_pair f 0) G] :\n  is_limit (fork.of_ι (G.map (kernel.ι f))\n    (by simp only [←G.map_comp, equalizer.condition, comp_zero, functor.map_zero])\n      : fork (G.map f) 0) :=\nis_limit_fork_map_of_is_limit' G (kernel.condition f) (kernel_is_kernel f)\n\ninstance [preserves_limit (parallel_pair f 0) G] : has_kernel (G.map f) :=\n{ exists_limit := ⟨⟨_, is_limit_of_has_kernel_of_preserves_limit G f⟩⟩, }\n\nvariables [has_kernel (G.map f)]\n\n/--\nIf the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\nkernel of `f`.\n-/\ndef preserves_kernel.of_iso_comparison [i : is_iso (kernel_comparison f G)] :\n  preserves_limit (parallel_pair f 0) G :=\nbegin\n  apply preserves_limit_of_preserves_limit_cone (kernel_is_kernel f),\n  apply (is_limit_map_cone_fork_equiv' G (kernel.condition f)).symm _,\n  apply is_limit.of_point_iso (limit.is_limit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_limit (parallel_pair f 0) G]\n/--\nIf `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_kernel.iso :\n  G.obj (kernel f) ≅ kernel (G.map f) :=\nis_limit.cone_point_unique_up_to_iso\n  (is_limit_of_has_kernel_of_preserves_limit G f)\n  (limit.is_limit _)\n\n@[simp]\nlemma preserves_kernel.iso_hom :\n  (preserves_kernel.iso G f).hom = kernel_comparison f G :=\nrfl\n\ninstance : is_iso (kernel_comparison f G) :=\nbegin\n  rw ← preserves_kernel.iso_hom,\n  apply_instance\nend\n\nend kernels\n\nsection cokernels\n\nvariables {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)\n\n/--\nThe map of a cokernel cofork is a colimit iff\nthe cokernel cofork consisting of the mapped morphisms is a colimit.\nThis essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.\n\nThis is a variant of `is_colimit_map_cocone_cofork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_map_cocone_cofork_equiv' :\n  is_colimit (G.map_cocone (cokernel_cofork.of_π h w)) ≃\n  is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])\n    : cofork (G.map f) 0) :=\nbegin\n  refine (is_colimit.precompose_hom_equiv _ _).symm.trans (is_colimit.equiv_iso_colimit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine cofork.ext (iso.refl _) _,\n  simp, dsimp, simp,\nend\n\n/--\nThe property of preserving cokernels expressed in terms of cokernel coforks.\n\nThis is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_cofork_map_of_is_colimit' [preserves_colimit (parallel_pair f 0) G]\n  (l : is_colimit (cokernel_cofork.of_π h w)) :\n  is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :\n    cofork (G.map f) 0) :=\nis_colimit_map_cocone_cofork_equiv' G w (preserves_colimit.preserves l)\n\nvariables (f) [has_cokernel f]\n\n/--\nIf `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of\na cokernel cofork is a colimit.\n-/\ndef is_colimit_of_has_cokernel_of_preserves_colimit [preserves_colimit (parallel_pair f 0) G] :\n  is_colimit (cofork.of_π (G.map (cokernel.π f))\n    (by simp only [←G.map_comp, coequalizer.condition, zero_comp, functor.map_zero])\n      : cofork (G.map f) 0) :=\nis_colimit_cofork_map_of_is_colimit' G (cokernel.condition f) (cokernel_is_cokernel f)\n\ninstance [preserves_colimit (parallel_pair f 0) G] : has_cokernel (G.map f) :=\n{ exists_colimit := ⟨⟨_, is_colimit_of_has_cokernel_of_preserves_colimit G f⟩⟩, }\n\nvariables [has_cokernel (G.map f)]\n\n/--\nIf the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\ncokernel of `f`.\n-/\ndef preserves_cokernel.of_iso_comparison [i : is_iso (cokernel_comparison f G)] :\n  preserves_colimit (parallel_pair f 0) G :=\nbegin\n  apply preserves_colimit_of_preserves_colimit_cocone (cokernel_is_cokernel f),\n  apply (is_colimit_map_cocone_cofork_equiv' G (cokernel.condition f)).symm _,\n  apply is_colimit.of_point_iso (colimit.is_colimit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_colimit (parallel_pair f 0) G]\n/--\nIf `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_cokernel.iso :\n  G.obj (cokernel f) ≅ cokernel (G.map f) :=\nis_colimit.cocone_point_unique_up_to_iso\n  (is_colimit_of_has_cokernel_of_preserves_colimit G f)\n  (colimit.is_colimit _)\n\n@[simp]\nlemma preserves_cokernel.iso_hom :\n  (preserves_cokernel.iso G f).inv = cokernel_comparison f G :=\nrfl\n\ninstance : is_iso (cokernel_comparison f G) :=\nbegin\n  rw ← preserves_cokernel.iso_hom,\n  apply_instance\nend\n\nend cokernels\n\nend category_theory.limits\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/limits/preserves/shapes/kernels.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3421608175009801}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n\nimport category_theory.limits.constructions.pullbacks\nimport category_theory.limits.shapes.biproducts\nimport category_theory.limits.shapes.images\nimport category_theory.abelian.non_preadditive\n\n/-!\n# Abelian categories\n\nThis file contains the definition and basic properties of abelian categories.\n\nThere are many definitions of abelian category. Our definition is as follows:\nA category is called abelian if it is preadditive,\nhas a finite products, kernels and cokernels,\nand if every monomorphism and epimorphism is normal.\n\nIt should be noted that if we also assume coproducts, then preadditivity is\nactually a consequence of the other properties, as we show in\n`non_preadditive_abelian.lean`. However, this fact is of little practical\nrelevance, since essentially all interesting abelian categories come with a\npreadditive structure. In this way, by requiring preadditivity, we allow the\nuser to pass in the preadditive structure the specific category they are\nworking with has natively.\n\n## Main definitions\n\n* `abelian` is the type class indicating that a category is abelian. It extends `preadditive`.\n* `abelian.image f` is `kernel (cokernel.π f)`, and\n* `abelian.coimage f` is `cokernel (kernel.ι f)`.\n\n## Main results\n\n* In an abelian category, mono + epi = iso.\n* If `f : X ⟶ Y`, then the map `factor_thru_image f : X ⟶ image f` is an epimorphism, and the map\n  `factor_thru_coimage f : coimage f ⟶ Y` is a monomorphism.\n* Factoring through the image and coimage is a strong epi-mono factorisation. This means that\n  * every abelian category has images. We instantiated this in such a way that `abelian.image f` is\n    definitionally equal to `limits.image f`, and\n  * there is a canonical isomorphism `coimage_iso_image : coimage f ≅ image f` such that\n    `coimage.π f ≫ (coimage_iso_image f).hom ≫ image.ι f = f`. The lemma stating this is called\n    `full_image_factorisation`.\n* Every epimorphism is a cokernel of its kernel. Every monomorphism is a kernel of its cokernel.\n* The pullback of an epimorphism is an epimorphism. The pushout of a monomorphism is a monomorphism.\n  (This is not to be confused with the fact that the pullback of a monomorphism is a monomorphism,\n  which is true in any category).\n\n## Implementation notes\n\nThe typeclass `abelian` does not extend `non_preadditive_abelian`,\nto avoid having to deal with comparing the two `has_zero_morphisms` instances\n(one from `preadditive` in `abelian`, and the other a field of `non_preadditive_abelian`).\nAs a consequence, at the beginning of this file we trivially build\na `non_preadditive_abelian` instance from an `abelian` instance,\nand use this to restate a number of theorems,\nin each case just reusing the proof from `non_preadditive_abelian.lean`.\n\nWe don't show this yet, but abelian categories are finitely complete and finitely cocomplete.\nHowever, the limits we can construct at this level of generality will most likely be less nice than\nthe ones that can be created in specific applications. For this reason, we adopt the following\nconvention:\n\n* If the statement of a theorem involves limits, the existence of these limits should be made an\n  explicit typeclass parameter.\n* If a limit only appears in a proof, but not in the statement of a theorem, the limit should not\n  be a typeclass parameter, but instead be created using `abelian.has_pullbacks` or a similar\n  definition.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n* [P. Aluffi, *Algebra: Chaper 0*][aluffi2016]\n\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables {C : Type u} [category.{v} C]\n\nvariables (C)\n\n/--\nA (preadditive) category `C` is called abelian if it has all finite products,\nall kernels and cokernels, and if every monomorphism is the kernel of some morphism\nand every epimorphism is the cokernel of some morphism.\n\n(This definition implies the existence of zero objects:\nfinite products give a terminal object, and in a preadditive category\nany terminal object is a zero object.)\n-/\nclass abelian extends preadditive C :=\n[has_finite_products : has_finite_products C]\n[has_kernels : has_kernels C]\n[has_cokernels : has_cokernels C]\n(normal_mono : Π {X Y : C} (f : X ⟶ Y) [mono f], normal_mono f)\n(normal_epi : Π {X Y : C} (f : X ⟶ Y) [epi f], normal_epi f)\n\nattribute [instance, priority 100] abelian.has_finite_products\nattribute [instance, priority 100] abelian.has_kernels abelian.has_cokernels\n\nend category_theory\n\nopen category_theory\n\nnamespace category_theory.abelian\nvariables {C : Type u} [category.{v} C] [abelian C]\n\n/-- An abelian category has finite biproducts. -/\n@[priority 100]\ninstance has_finite_biproducts : has_finite_biproducts C :=\nlimits.has_finite_biproducts.of_has_finite_products\n\n@[priority 100]\ninstance has_binary_biproducts : has_binary_biproducts C :=\nlimits.has_binary_biproducts_of_finite_biproducts _\n\nsection to_non_preadditive_abelian\n\n/-- Every abelian category is, in particular, `non_preadditive_abelian`. -/\ndef non_preadditive_abelian : non_preadditive_abelian C := { ..‹abelian C› }\n\nend to_non_preadditive_abelian\n\nsection strong\nlocal attribute [instance] abelian.normal_epi\n\n/-- In an abelian category, every epimorphism is strong. -/\nlemma strong_epi_of_epi {P Q : C} (f : P ⟶ Q) [epi f] : strong_epi f := by apply_instance\n\nend strong\n\nsection mono_epi_iso\nvariables {X Y : C} (f : X ⟶ Y)\n\nlocal attribute [instance] strong_epi_of_epi\n\n/-- In an abelian category, a monomorphism which is also an epimorphism is an isomorphism. -/\nlemma is_iso_of_mono_of_epi [mono f] [epi f] : is_iso f :=\nis_iso_of_mono_of_strong_epi _\n\nend mono_epi_iso\n\nsection factor\nlocal attribute [instance] non_preadditive_abelian\n\nvariables {P Q : C} (f : P ⟶ Q)\n\nsection\n\nlemma mono_of_zero_kernel (R : C)\n  (l : is_limit (kernel_fork.of_ι (0 : R ⟶ P) (show 0 ≫ f = 0, by simp))) : mono f :=\nnon_preadditive_abelian.mono_of_zero_kernel _ _ l\n\nlemma mono_of_kernel_ι_eq_zero (h : kernel.ι f = 0) : mono f :=\nmono_of_kernel_zero h\n\nlemma epi_of_zero_cokernel (R : C)\n  (l : is_colimit (cokernel_cofork.of_π (0 : Q ⟶ R) (show f ≫ 0 = 0, by simp))) : epi f :=\nnon_preadditive_abelian.epi_of_zero_cokernel _ _ l\n\nlemma epi_of_cokernel_π_eq_zero (h : cokernel.π f = 0) : epi f :=\nbegin\n  apply epi_of_zero_cokernel _ (cokernel f),\n  simp_rw ←h,\n  exact is_colimit.of_iso_colimit (colimit.is_colimit (parallel_pair f 0)) (iso_of_π _)\nend\n\nend\n\nnamespace images\n\n/-- The kernel of the cokernel of `f` is called the image of `f`. -/\nprotected abbreviation image : C := kernel (cokernel.π f)\n\n/-- The inclusion of the image into the codomain. -/\nprotected abbreviation image.ι : images.image f ⟶ Q :=\nkernel.ι (cokernel.π f)\n\n/-- There is a canonical epimorphism `p : P ⟶ image f` for every `f`. -/\nprotected abbreviation factor_thru_image : P ⟶ images.image f :=\nkernel.lift (cokernel.π f) f $ cokernel.condition f\n\n/-- `f` factors through its image via the canonical morphism `p`. -/\n@[simp, reassoc] protected lemma image.fac :\n  images.factor_thru_image f ≫ image.ι f = f :=\nkernel.lift_ι _ _ _\n\n/-- The map `p : P ⟶ image f` is an epimorphism -/\ninstance : epi (images.factor_thru_image f) :=\nshow epi (non_preadditive_abelian.factor_thru_image f), by apply_instance\n\nsection\nvariables {f}\n\nlemma image_ι_comp_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : images.image.ι f ≫ g = 0 :=\nzero_of_epi_comp (images.factor_thru_image f) $ by simp [h]\n\nend\n\ninstance mono_factor_thru_image [mono f] : mono (images.factor_thru_image f) :=\nmono_of_mono_fac $ image.fac f\n\ninstance is_iso_factor_thru_image [mono f] : is_iso (images.factor_thru_image f) :=\nis_iso_of_mono_of_epi _\n\n/-- Factoring through the image is a strong epi-mono factorisation. -/\n@[simps] def image_strong_epi_mono_factorisation : strong_epi_mono_factorisation f :=\n{ I := images.image f,\n  m := image.ι f,\n  m_mono := by apply_instance,\n  e := images.factor_thru_image f,\n  e_strong_epi := strong_epi_of_epi _ }\n\nend images\n\nnamespace coimages\n\n/-- The cokernel of the kernel of `f` is called the coimage of `f`. -/\nprotected abbreviation coimage : C := cokernel (kernel.ι f)\n\n/-- The projection onto the coimage. -/\nprotected abbreviation coimage.π : P ⟶ coimages.coimage f :=\ncokernel.π (kernel.ι f)\n\n/-- There is a canonical monomorphism `i : coimage f ⟶ Q`. -/\nprotected abbreviation factor_thru_coimage : coimages.coimage f ⟶ Q :=\ncokernel.desc (kernel.ι f) f $ kernel.condition f\n\n/-- `f` factors through its coimage via the canonical morphism `p`. -/\nprotected lemma coimage.fac : coimage.π f ≫ coimages.factor_thru_coimage f = f :=\ncokernel.π_desc _ _ _\n\n/-- The canonical morphism `i : coimage f ⟶ Q` is a monomorphism -/\ninstance : mono (coimages.factor_thru_coimage f) :=\nshow mono (non_preadditive_abelian.factor_thru_coimage f), by apply_instance\n\ninstance epi_factor_thru_coimage [epi f] : epi (coimages.factor_thru_coimage f) :=\nepi_of_epi_fac $ coimage.fac f\n\ninstance is_iso_factor_thru_coimage [epi f] : is_iso (coimages.factor_thru_coimage f) :=\nis_iso_of_mono_of_epi _\n\n/-- Factoring through the coimage is a strong epi-mono factorisation. -/\n@[simps] def coimage_strong_epi_mono_factorisation : strong_epi_mono_factorisation f :=\n{ I := coimages.coimage f,\n  m := coimages.factor_thru_coimage f,\n  m_mono := by apply_instance,\n  e := coimage.π f,\n  e_strong_epi := strong_epi_of_epi _ }\n\nend coimages\n\nend factor\n\nsection has_strong_epi_mono_factorisations\n\n/-- An abelian category has strong epi-mono factorisations. -/\n@[priority 100] instance : has_strong_epi_mono_factorisations C :=\nhas_strong_epi_mono_factorisations.mk $ λ X Y f, images.image_strong_epi_mono_factorisation f\n\n/- In particular, this means that it has well-behaved images. -/\nexample : has_images C := by apply_instance\nexample : has_image_maps C := by apply_instance\n\nend has_strong_epi_mono_factorisations\n\nsection images\nvariables {X Y : C} (f : X ⟶ Y)\n\n/-- There is a canonical isomorphism between the coimage and the image of a morphism. -/\nabbreviation coimage_iso_image : coimages.coimage f ≅ images.image f :=\nis_image.iso_ext (coimages.coimage_strong_epi_mono_factorisation f).to_mono_is_image\n  (images.image_strong_epi_mono_factorisation f).to_mono_is_image\n\n/-- There is a canonical isomorphism between the abelian image and the categorical image of a\n    morphism. -/\nabbreviation image_iso_image : images.image f ≅ image f :=\nis_image.iso_ext (images.image_strong_epi_mono_factorisation f).to_mono_is_image (image.is_image f)\n\n/-- There is a canonical isomorphism between the abelian coimage and the categorical image of a\n    morphism. -/\nabbreviation coimage_iso_image' : coimages.coimage f ≅ image f :=\nis_image.iso_ext (coimages.coimage_strong_epi_mono_factorisation f).to_mono_is_image\n  (image.is_image f)\n\nlemma full_image_factorisation : coimages.coimage.π f ≫ (coimage_iso_image f).hom ≫\n  images.image.ι f = f :=\nby rw [limits.is_image.iso_ext_hom,\n  ←images.image_strong_epi_mono_factorisation_to_mono_factorisation_m, is_image.lift_fac,\n  coimages.coimage_strong_epi_mono_factorisation_to_mono_factorisation_m, coimages.coimage.fac]\n\nend images\n\nsection cokernel_of_kernel\nvariables {X Y : C} {f : X ⟶ Y}\n\nlocal attribute [instance] non_preadditive_abelian\n\n/-- In an abelian category, an epi is the cokernel of its kernel. More precisely:\n    If `f` is an epimorphism and `s` is some limit kernel cone on `f`, then `f` is a cokernel\n    of `fork.ι s`. -/\ndef epi_is_cokernel_of_kernel [epi f] (s : fork f 0) (h : is_limit s) :\n  is_colimit (cokernel_cofork.of_π f (kernel_fork.condition s)) :=\nnon_preadditive_abelian.epi_is_cokernel_of_kernel s h\n\n/-- In an abelian category, a mono is the kernel of its cokernel. More precisely:\n    If `f` is a monomorphism and `s` is some colimit cokernel cocone on `f`, then `f` is a kernel\n    of `cofork.π s`. -/\ndef mono_is_kernel_of_cokernel [mono f] (s : cofork f 0) (h : is_colimit s) :\n  is_limit (kernel_fork.of_ι f (cokernel_cofork.condition s)) :=\nnon_preadditive_abelian.mono_is_kernel_of_cokernel s h\n\nend cokernel_of_kernel\n\nsection\n\n@[priority 100]\ninstance has_equalizers : has_equalizers C :=\npreadditive.has_equalizers_of_has_kernels\n\n/-- Any abelian category has pullbacks -/\n@[priority 100]\ninstance has_pullbacks : has_pullbacks C :=\nhas_pullbacks_of_has_binary_products_of_has_equalizers C\n\nend\n\nsection\n\n@[priority 100]\ninstance has_coequalizers : has_coequalizers C :=\npreadditive.has_coequalizers_of_has_cokernels\n\n/-- Any abelian category has pushouts -/\n@[priority 100]\ninstance has_pushouts : has_pushouts C :=\nhas_pushouts_of_has_binary_coproducts_of_has_coequalizers C\n\nend\n\nnamespace pullback_to_biproduct_is_kernel\nvariables [limits.has_pullbacks C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n\n/-! This section contains a slightly technical result about pullbacks and biproducts.\n    We will need it in the proof that the pullback of an epimorphism is an epimorpism. -/\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` -/\nabbreviation pullback_to_biproduct : pullback f g ⟶ X ⊞ Y :=\nbiprod.lift pullback.fst pullback.snd\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` induces a kernel cone on the map\n    `biproduct X Y ⟶ Z` induced by `f` and `g`. A slightly more intuitive way to think of\n    this may be that it induces an equalizer fork on the maps induced by `(f, 0)` and\n    `(0, g)`. -/\nabbreviation pullback_to_biproduct_fork : kernel_fork (biprod.desc f (-g)) :=\nkernel_fork.of_ι (pullback_to_biproduct f g) $\nby rw [biprod.lift_desc, comp_neg, pullback.condition, add_right_neg]\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` is a kernel of the map induced by\n    `(f, -g)`. -/\ndef is_limit_pullback_to_biproduct : is_limit (pullback_to_biproduct_fork f g) :=\nfork.is_limit.mk _\n  (λ s, pullback.lift (fork.ι s ≫ biprod.fst) (fork.ι s ≫ biprod.snd) $\n    sub_eq_zero.1 $ by rw [category.assoc, category.assoc, ←comp_sub, sub_eq_add_neg, ←comp_neg,\n      ←biprod.desc_eq, kernel_fork.condition s])\n  (λ s,\n  begin\n    ext; rw [fork.ι_of_ι, category.assoc],\n    { rw [biprod.lift_fst, pullback.lift_fst] },\n    { rw [biprod.lift_snd, pullback.lift_snd] }\n  end)\n  (λ s m h, by ext; simp [fork.ι_eq_app_zero, ←h walking_parallel_pair.zero])\n\nend pullback_to_biproduct_is_kernel\n\nnamespace biproduct_to_pushout_is_cokernel\nvariables [limits.has_pushouts C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)\n\n/-- The canonical map `Y ⊞ Z ⟶ pushout f g` -/\nabbreviation biproduct_to_pushout : Y ⊞ Z ⟶ pushout f g :=\nbiprod.desc pushout.inl pushout.inr\n\n/-- The canonical map `Y ⊞ Z ⟶ pushout f g` induces a cokernel cofork on the map\n    `X ⟶ Y ⊞ Z` induced by `f` and `-g`. -/\nabbreviation biproduct_to_pushout_cofork : cokernel_cofork (biprod.lift f (-g)) :=\ncokernel_cofork.of_π (biproduct_to_pushout f g) $\nby rw [biprod.lift_desc, neg_comp, pushout.condition, add_right_neg]\n\n/-- The cofork induced by the canonical map `Y ⊞ Z ⟶ pushout f g` is in fact a colimit cokernel\n    cofork. -/\ndef is_colimit_biproduct_to_pushout : is_colimit (biproduct_to_pushout_cofork f g) :=\ncofork.is_colimit.mk _\n  (λ s, pushout.desc (biprod.inl ≫ cofork.π s) (biprod.inr ≫ cofork.π s) $\n    sub_eq_zero.1 $ by rw [←category.assoc, ←category.assoc, ←sub_comp, sub_eq_add_neg, ←neg_comp,\n      ←biprod.lift_eq, cofork.condition s, zero_comp])\n  (λ s, by ext; simp)\n  (λ s m h, by ext; simp [cofork.π_eq_app_one, ←h walking_parallel_pair.one] )\n\nend biproduct_to_pushout_is_cokernel\n\nsection epi_pullback\nvariables [limits.has_pullbacks C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n\n/-- In an abelian category, the pullback of an epimorphism is an epimorphism.\n    Proof from [aluffi2016, IX.2.3], cf. [borceux-vol2, 1.7.6] -/\ninstance epi_pullback_of_epi_f [epi f] : epi (pullback.snd : pullback f g ⟶ Y) :=\n-- It will suffice to consider some morphism e : Y ⟶ R such that\n-- pullback.snd ≫ e = 0 and show that e = 0.\nepi_of_cancel_zero _ $ λ R e h,\nbegin\n  -- Consider the morphism u := (0, e) : X ⊞ Y⟶ R.\n  let u := biprod.desc (0 : X ⟶ R) e,\n  -- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.\n  have hu : pullback_to_biproduct_is_kernel.pullback_to_biproduct f g ≫ u = 0 := by simpa,\n  -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n  -- cokernel of pullback_to_biproduct f g\n  have := epi_is_cokernel_of_kernel _\n    (pullback_to_biproduct_is_kernel.is_limit_pullback_to_biproduct f g),\n  -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z ⟶ R.\n  obtain ⟨d, hd⟩ := cokernel_cofork.is_colimit.desc' this u hu,\n  change Z ⟶ R at d,\n  change biprod.desc f (-g) ≫ d = u at hd,\n  -- But then f ≫ d = 0:\n  have : f ≫ d = 0, calc\n    f ≫ d = (biprod.inl ≫ biprod.desc f (-g)) ≫ d : by rw biprod.inl_desc\n    ... = biprod.inl ≫ u : by rw [category.assoc, hd]\n    ... = 0 : biprod.inl_desc _ _,\n  -- But f is an epimorphism, so d = 0...\n  have : d = 0 := (cancel_epi f).1 (by simpa),\n  -- ...or, in other words, e = 0.\n  calc\n    e = biprod.inr ≫ u : by rw biprod.inr_desc\n    ... = biprod.inr ≫ biprod.desc f (-g) ≫ d : by rw ←hd\n    ... = biprod.inr ≫ biprod.desc f (-g) ≫ 0 : by rw this\n    ... = (biprod.inr ≫ biprod.desc f (-g)) ≫ 0 : by rw ←category.assoc\n    ... = 0 : has_zero_morphisms.comp_zero _ _\nend\n\n/-- In an abelian category, the pullback of an epimorphism is an epimorphism. -/\ninstance epi_pullback_of_epi_g [epi g] : epi (pullback.fst : pullback f g ⟶ X) :=\n-- It will suffice to consider some morphism e : X ⟶ R such that\n-- pullback.fst ≫ e = 0 and show that e = 0.\nepi_of_cancel_zero _ $ λ R e h,\nbegin\n  -- Consider the morphism u := (e, 0) : X ⊞ Y ⟶ R.\n  let u := biprod.desc e (0 : Y ⟶ R),\n  -- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.\n  have hu : pullback_to_biproduct_is_kernel.pullback_to_biproduct f g ≫ u = 0 := by simpa,\n  -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n  -- cokernel of pullback_to_biproduct f g\n  have := epi_is_cokernel_of_kernel _\n    (pullback_to_biproduct_is_kernel.is_limit_pullback_to_biproduct f g),\n  -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z ⟶ R.\n  obtain ⟨d, hd⟩ := cokernel_cofork.is_colimit.desc' this u hu,\n  change Z ⟶ R at d,\n  change biprod.desc f (-g) ≫ d = u at hd,\n  -- But then (-g) ≫ d = 0:\n  have : (-g) ≫ d = 0, calc\n    (-g) ≫ d = (biprod.inr ≫ biprod.desc f (-g)) ≫ d : by rw biprod.inr_desc\n    ... = biprod.inr ≫ u : by rw [category.assoc, hd]\n    ... = 0 : biprod.inr_desc _ _,\n  -- But g is an epimorphism, thus so is -g, so d = 0...\n  have : d = 0 := (cancel_epi (-g)).1 (by simpa),\n  -- ...or, in other words, e = 0.\n  calc\n    e = biprod.inl ≫ u : by rw biprod.inl_desc\n    ... = biprod.inl ≫ biprod.desc f (-g) ≫ d : by rw ←hd\n    ... = biprod.inl ≫ biprod.desc f (-g) ≫ 0 : by rw this\n    ... = (biprod.inl ≫ biprod.desc f (-g)) ≫ 0 : by rw ←category.assoc\n    ... = 0 : has_zero_morphisms.comp_zero _ _\nend\n\nend epi_pullback\n\nsection mono_pushout\nvariables [limits.has_pushouts C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)\n\ninstance mono_pushout_of_mono_f [mono f] : mono (pushout.inr : Z ⟶ pushout f g) :=\nmono_of_cancel_zero _ $ λ R e h,\nbegin\n  let u := biprod.lift (0 : R ⟶ Y) e,\n  have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,\n  have := mono_is_kernel_of_cokernel _\n    (biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),\n  obtain ⟨d, hd⟩ := kernel_fork.is_limit.lift' this u hu,\n  change R ⟶ X at d,\n  change d ≫ biprod.lift f (-g) = u at hd,\n  have : d ≫ f = 0, calc\n    d ≫ f = d ≫ biprod.lift f (-g) ≫ biprod.fst : by rw biprod.lift_fst\n    ... = u ≫ biprod.fst : by rw [←category.assoc, hd]\n    ... = 0 : biprod.lift_fst _ _,\n  have : d = 0 := (cancel_mono f).1 (by simpa),\n  calc\n    e = u ≫ biprod.snd : by rw biprod.lift_snd\n    ... = (d ≫ biprod.lift f (-g)) ≫ biprod.snd : by rw ←hd\n    ... = (0 ≫ biprod.lift f (-g)) ≫ biprod.snd : by rw this\n    ... = 0 ≫ biprod.lift f (-g) ≫ biprod.snd : by rw category.assoc\n    ... = 0 : zero_comp\nend\n\ninstance mono_pushout_of_mono_g [mono g] : mono (pushout.inl : Y ⟶ pushout f g) :=\nmono_of_cancel_zero _ $ λ R e h,\nbegin\n  let u := biprod.lift e (0 : R ⟶ Z),\n  have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,\n  have := mono_is_kernel_of_cokernel _\n    (biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),\n  obtain ⟨d, hd⟩ := kernel_fork.is_limit.lift' this u hu,\n  change R ⟶ X at d,\n  change d ≫ biprod.lift f (-g) = u at hd,\n  have : d ≫ (-g) = 0, calc\n    d ≫ (-g) = d ≫ biprod.lift f (-g) ≫ biprod.snd : by rw biprod.lift_snd\n    ... = u ≫ biprod.snd : by rw [←category.assoc, hd]\n    ... = 0 : biprod.lift_snd _ _,\n  have : d = 0 := (cancel_mono (-g)).1 (by simpa),\n  calc\n    e = u ≫ biprod.fst : by rw biprod.lift_fst\n    ... = (d ≫ biprod.lift f (-g)) ≫ biprod.fst : by rw ←hd\n    ... = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst : by rw this\n    ... = 0 ≫ biprod.lift f (-g) ≫ biprod.fst : by rw category.assoc\n    ... = 0 : zero_comp\nend\n\nend mono_pushout\n\nend category_theory.abelian\n\nnamespace category_theory.non_preadditive_abelian\n\nvariables (C : Type u) [category.{v} C] [non_preadditive_abelian C]\n\n/-- Every non_preadditive_abelian category can be promoted to an abelian category. -/\ndef abelian : abelian C :=\n{ has_finite_products := by apply_instance,\n/- We need the `convert`s here because the instances we have are slightly different from the\n   instances we need: `has_kernels` depends on an instance of `has_zero_morphisms`. In the\n   case of `non_preadditive_abelian`, this instance is an explicit argument. However, in the case\n   of `abelian`, the `has_zero_morphisms` instance is derived from `preadditive`. So we need to\n   transform an instance of \"has kernels with non_preadditive_abelian.has_zero_morphisms\" to an\n   instance of \"has kernels with non_preadditive_abelian.preadditive.has_zero_morphisms\". Luckily,\n   we have a `subsingleton` instance for `has_zero_morphisms`, so `convert` can immediately close\n   the goal it creates for the two instances of `has_zero_morphisms`, and the proof is complete. -/\n  has_kernels := by convert (by apply_instance : limits.has_kernels C),\n  has_cokernels := by convert (by apply_instance : limits.has_cokernels C),\n  normal_mono := by { introsI, convert normal_mono f },\n  normal_epi := by { introsI, convert normal_epi f },\n  ..non_preadditive_abelian.preadditive }\n\nend category_theory.non_preadditive_abelian\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/abelian/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.34216081750098004}}
{"text": "/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.nodup\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Finite products of types\n\nThis file defines the product of types over a list. For `l : list ι` and `α : ι → Type*` we define\n`list.tprod α l = l.foldr (λ i β, α i × β) punit`.\nThis type should not be used if `Π i, α i` or `Π i ∈ l, α i` can be used instead\n(in the last expression, we could also replace the list `l` by a set or a finset).\nThis type is used as an intermediary between binary products and finitary products.\nThe application of this type is finitary product measures, but it could be used in any\nconstruction/theorem that is easier to define/prove on binary products than on finitary products.\n\n* Once we have the construction on binary products (like binary product measures in\n  `measure_theory.prod`), we can easily define a finitary version on the type `tprod l α`\n  by iterating. Properties can also be easily extended from the binary case to the finitary case\n  by iterating.\n* Then we can use the equivalence `list.tprod.pi_equiv_tprod` below (or enhanced versions of it,\n  like a `measurable_equiv` for product measures) to get the construction on `Π i : ι, α i`, at\n  least when assuming `[fintype ι] [encodable ι]` (using `encodable.sorted_univ`).\n  Using `local attribute [instance] fintype.encodable` we can get rid of the argument\n  `[encodable ι]`.\n\n## Main definitions\n\n* We have the equivalence `tprod.pi_equiv_tprod : (Π i, α i) ≃ tprod α l`\n  if `l` contains every element of `ι` exactly once.\n* The product of sets is `set.tprod : (Π i, set (α i)) → set (tprod α l)`.\n-/\n\nnamespace list\n\n\n/-- The product of a family of types over a list. -/\ndef tprod {ι : Type u_1} (α : ι → Type u_2) (l : List ι) :=\n  foldr (fun (i : ι) (β : Type (max u_2 u_3)) => α i × β) PUnit l\n\nnamespace tprod\n\n\n/-- Turning a function `f : Π i, α i` into an element of the iterated product `tprod α l`. -/\nprotected def mk {ι : Type u_1} {α : ι → Type u_2} (l : List ι) (f : (i : ι) → α i) : tprod α l :=\n  sorry\n\nprotected instance inhabited {ι : Type u_1} {α : ι → Type u_2} {l : List ι}\n    [(i : ι) → Inhabited (α i)] : Inhabited (tprod α l) :=\n  { default := tprod.mk l fun (i : ι) => Inhabited.default }\n\n@[simp] theorem fst_mk {ι : Type u_1} {α : ι → Type u_2} (i : ι) (l : List ι) (f : (i : ι) → α i) :\n    prod.fst (tprod.mk (i :: l) f) = f i :=\n  rfl\n\n@[simp] theorem snd_mk {ι : Type u_1} {α : ι → Type u_2} (i : ι) (l : List ι) (f : (i : ι) → α i) :\n    prod.snd (tprod.mk (i :: l) f) = tprod.mk l f :=\n  rfl\n\n/-- Given an element of the iterated product `l.prod α`, take a projection into direction `i`.\n  If `i` appears multiple times in `l`, this chooses the first component in direction `i`. -/\nprotected def elim {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] {l : List ι} (v : tprod α l)\n    {i : ι} (hi : i ∈ l) : α i :=\n  sorry\n\n@[simp] theorem elim_self {ι : Type u_1} {α : ι → Type u_2} {i : ι} {l : List ι} [DecidableEq ι]\n    (v : tprod α (i :: l)) : tprod.elim v (mem_cons_self i l) = prod.fst v :=\n  sorry\n\n@[simp] theorem elim_of_ne {ι : Type u_1} {α : ι → Type u_2} {i : ι} {j : ι} {l : List ι}\n    [DecidableEq ι] (hj : j ∈ i :: l) (hji : j ≠ i) (v : tprod α (i :: l)) :\n    tprod.elim v hj = tprod.elim (prod.snd v) (or.resolve_left hj hji) :=\n  sorry\n\n@[simp] theorem elim_of_mem {ι : Type u_1} {α : ι → Type u_2} {i : ι} {j : ι} {l : List ι}\n    [DecidableEq ι] (hl : nodup (i :: l)) (hj : j ∈ l) (v : tprod α (i :: l)) :\n    tprod.elim v (mem_cons_of_mem i hj) = tprod.elim (prod.snd v) hj :=\n  sorry\n\ntheorem elim_mk {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] (l : List ι) (f : (i : ι) → α i)\n    {i : ι} (hi : i ∈ l) : tprod.elim (tprod.mk l f) hi = f i :=\n  sorry\n\ntheorem ext {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] {l : List ι} (hl : nodup l)\n    {v : tprod α l} {w : tprod α l}\n    (hvw : ∀ (i : ι) (hi : i ∈ l), tprod.elim v hi = tprod.elim w hi) : v = w :=\n  sorry\n\n/-- A version of `tprod.elim` when `l` contains all elements. In this case we get a function into\n  `Π i, α i`. -/\n@[simp] protected def elim' {ι : Type u_1} {α : ι → Type u_2} {l : List ι} [DecidableEq ι]\n    (h : ∀ (i : ι), i ∈ l) (v : tprod α l) (i : ι) : α i :=\n  tprod.elim v (h i)\n\ntheorem mk_elim {ι : Type u_1} {α : ι → Type u_2} {l : List ι} [DecidableEq ι] (hnd : nodup l)\n    (h : ∀ (i : ι), i ∈ l) (v : tprod α l) : tprod.mk l (tprod.elim' h v) = v :=\n  sorry\n\n/-- Pi-types are equivalent to iterated products. -/\ndef pi_equiv_tprod {ι : Type u_1} {α : ι → Type u_2} {l : List ι} [DecidableEq ι] (hnd : nodup l)\n    (h : ∀ (i : ι), i ∈ l) : ((i : ι) → α i) ≃ tprod α l :=\n  equiv.mk (tprod.mk l) (tprod.elim' h) sorry sorry\n\nend tprod\n\n\nend list\n\n\nnamespace set\n\n\n/-- A product of sets in `tprod α l`. -/\n@[simp] protected def tprod {ι : Type u_1} {α : ι → Type u_2} (l : List ι)\n    (t : (i : ι) → set (α i)) : set (list.tprod α l) :=\n  sorry\n\ntheorem mk_preimage_tprod {ι : Type u_1} {α : ι → Type u_2} (l : List ι) (t : (i : ι) → set (α i)) :\n    list.tprod.mk l ⁻¹' set.tprod l t = pi (set_of fun (i : ι) => i ∈ l) t :=\n  sorry\n\ntheorem elim_preimage_pi {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] {l : List ι}\n    (hnd : list.nodup l) (h : ∀ (i : ι), i ∈ l) (t : (i : ι) → set (α i)) :\n    list.tprod.elim' h ⁻¹' pi univ t = set.tprod l t :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/tprod_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34216080880641814}}
{"text": "import model_theory.substructures\nimport model_theory.bundled\n\n\n--OUTPUT 1\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.formula := λ n, ∃ (x₁ : L.var) (x₂ : L.var) (x₃ : L.var) (x₄ : L.var) (x₅ : L.var) (x₆ : L.var) (x₇ : L.var) (x₈ : L.var) (x₉ : L.var) (x₁₀ : L.var) (x₁₁ : L.var) (x₁₂ : L.var) (x₁₃ : L.var) (x₁₄ : L.var) (x₁₅ : L.var) (x₁₆ : L.var) (x₁₇ : L.var) (x₁₈ : L.var) (x₁₉ : L.var) (x₂₀ : L.var) (x₂₁ : L.var) (x₂₂ : L.var) (x₂₃ : L.var) (x₂₄ : L.var) (x₂₅ : L.var) (x₂₆ : L.var) (x₂₇ : L.var) (x₂₈ : L.var) (x₂₉ : L.var) (x₃₀ : L.var) (x₃₁ : L.var) (x₃₂ : L.var) (x₃₃ : L.var) (x₃₄ : L.var) (x₃₅ : L.var) (x₃₆ : L.var) (x₃₇ : L.var) (x₃₈ : L.var) (x₃₉ : L.var) (x₄₀ : L.var) (x₄₁ : L.var) (x₄₂ : L.var) (x₄₃ : L.var) (x₄₄ : L.var) (x₄₅ : L.var) (x₄₆ : L.var) (x₄₇ : L.var) (x₄₈ : L.var) (x₄₉ : L.var) (x₅₀ : L.var) (x₅₁ : L.var) (x₅₂ : L.var) (x₅₃ : L.var) (x₅₄ : L.var) (x₅₅ : L.var) (x₅₆ : L.var) (x₅₇ : L.var) (x₅₈ : L.var) (x₅₉ : L.var) (x₆₀ : L.var) (x₆₁ : L.var) (x₆₂ : L.var) (x₆₃ : L.var) (x₆₄ : L.var) (x₆₅ : L.var) (x₆₆ : L.var) (x₆₇ : L.var) (x₆₈ : L.var) (x₆₉ : L.var) (x₇₀ : L.var) (x₇₁ : L.var) (x₇₂ : L.var) (x₇₃ : L.var) (x₇₄ : L.var) (x₇₅ : L.var) (x₇₆ : L.var) (x₇₇ : L.var) (x₇₈ : L.var) (x₇₉ : L.var) (x₈₀ : L.var) (x₈₁ : L.var) (x₈₂ : L.var) (x₈₃ : L.var) (x₈₄ : L.var) (x₈₅ : L.var) (x₈₆ : L.var) (x₈₇ : L.var) (x₈₈ : L.var) (x₈₉ : L.var) (x₉₀ : L.var) (x₉₁ : L.var) (x₉₂ : L.var) (x₉₃ : L.var) (x₉₄ : L.var) (x₉₅ : L.var) (x₉₆ : L.var) (x₉₇ : L.var) (x₉₈ : L.var) (x₉₉ : L.var) (x₁₀₀ : L.var) (x₁₀₁ : L.var) (x₁₀₂ : L.var) (x₁₀₃ : L.var) (x₁₀₄ : L.var) (x₁₀₅ : L.var) (x₁₀₆ : L.var) (x₁₀₇ : L.var) (x₁₀₈ : L.var) (x₁₀₉ : L.var) (x₁₁₀ : L.var) (x₁₁₁ : L.var) (x₁₁₂ : L.var) (x₁₁₃ : L.var) (x₁₁₄ : L.var) (x₁₁₅ : L.var) (x₁₁₆ : L.var) (x₁₁₇ : L.var) (x₁₁₈ : L.var) (x₁₁₉ : L.var) (x₁₂₀ : L.var) (x₁₂₁ : L.var) (x₁₂₂ : L.var) (x₁₂₃ : L.var) (x₁₂₄ : L.var) (x₁₂₅ : L.var) (x₁₂₆ : L.var) (x₁₂₇ : L.var) (x₁₂₈ : L.var) (x₁₂₉ : L.var) (x₁₃₀ : L.var) (x₁₃₁ : L.var) (x₁₃₂ : L.var) (x₁₃₃ : L.var) (x₁₃₄ : L.var) (x₁₃₅ : L.var) (x₁₃₆ : L.var) (x₁₃₇ : L.var) (x₁₃₈ : L.var) (x₁₃₉ : L.var) (x₁₄₀ : L.var) (x₁₄₁ : L.var) (x₁₄₂ : L.var) (x₁₄\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : ∀ n : ℕ, A n = L.exists (L.var n) (L.and (L.ne (L.var 0) (L.var 1)) (L.and (L.ne (L.var 0) (L.var 2)) (L.and (L.ne (L.var 0) (L.var 3)) (L.and (L.ne (L.var 0) (L.var 4)) (L.and (L.ne (L.var 0) (L.var 5)) (L.and (L.ne (L.var 0) (L.var 6)) (L.and (L.ne (L.var 0) (L.var 7)) (L.and (L.ne (L.var 0) (L.var 8)) (L.and (L.ne (L.var 0) (L.var 9)) (L.and (L.ne (L.var 0) (L.var 10)) (L.and (L.ne (L.var 0) (L.var 11)) (L.and (L.ne (L.var 0) (L.var 12)) (L.and (L.ne (L.var 0) (L.var 13)) (L.and (L.ne (L.var 0) (L.var 14)) (L.and (L.ne (L.var 0) (L.var 15)) (L.and (L.ne (L.var 0) (L.var 16)) (L.and (L.ne (L.var 0) (L.var 17)) (L.and (L.ne (L.var 0) (L.var 18)) (L.and (L.ne (L.var 0) (L.var 19)) (L.and (L.ne (L.var 0) (L.var 20)) (L.and (L.ne (L.var 0) (L.var 21)) (L.and (L.ne (L.var 0) (L.var 22)) (L.and (L.ne (L.var 0) (L.var 23)) (L.and (L.ne (L.var 0) (L.var 24)) (L.and (L.ne (L.var 0) (L.var 25)) (L.and (L.ne (L.var 0) (L.var 26)) (L.and (L.ne (L.var 0) (L.var 27)) (L.and (L.ne (L.var 0) (L.var 28)) (L.and (L.ne (L.var 0) (L.var 29)) (L.and (L.ne (L.var 0) (L.var 30)) (L.and (L.ne (L.var 0) (L.var 31)) (L.and (L.ne (L.var 0) (L.var 32)) (L.and (L.ne (L.var 0) (L.var 33)) (L.and (L.ne (L.var 0) (L.var 34)) (L.and (L.ne (L.var 0) (L.var 35)) (L.and (L.ne (L.var 0) (L.var 36)) (L.and (L.ne (L.var 0) (L.var 37)) (L.and (L.ne (L.var 0) (L.var 38)) (L.and (L.ne (L.var 0) (L.var 39)) (L.and (L.ne (L.var 0) (L.var 40)) (L.and (L.ne (L.var 0) (L.var 41)) (L.and (L.ne (L.var 0) (L.var 42)) (L.and (L.ne (L.var 0) (L.var 43)) (L.and (L.ne (L.var 0) (L.var 44)) (L.and (L.ne (L.var 0) (L.var 45)) (L.and (L.ne (L.var 0) (L.var 46)) (L.and (L.ne (L.var 0) (L.var 47)) (L.and (L.ne (L.var 0) (L.var 48)) (L.and (L.ne (L.var 0) (L.var 49)) (L.and (L.ne (L.var 0) (L.var 50)) (L.and (L.ne (L.var 0) (L.var 51)) (L.and (L.ne (L.var 0) (L.var 52)) (L.and (L.ne (L.var 0) (L.var 53)) (L.and (L.ne (L.var 0) (L.var 54)) (L.and (L.ne (L.var 0) (L.var 55)) (L.and (L.ne (L.var 0) (L.var 56)) (L.and (L.ne (L.var 0) (L.var 57)) (L.and (L.ne (L.var 0) (L.var 58)) (L.and (L.ne (L.var 0) (L.var 59)) (L.and (L.ne (L.var 0) (L.var 60)) (L.and (L.ne (L.var 0) (L.var 61)) (L.and (L.ne (L.var 0) (L.var 62)) (L.and (L.ne (L.var 0) (L.var 63)) (L.and (L.ne (L.var 0) (L.var 64)) (L.and (L.ne (L.var 0) (L.var 65)) (L.and (L.ne (L.var 0) (L.var 66)) (L.and (L.ne (L.var 0) (L.var 67)) (L.and (L.ne (L.var 0) (L.var 68)) (L.and (L.ne (L.var 0) (L.var 69)) (L.and (L.ne (L.var 0) (L.var 70)) (L.and (L.ne (L.var 0) (L.var 71)) (L.and (L.ne (L.var 0) (L.var 72)) (L.and (L.ne (L.var 0) (L.var 73)) (L.and (L.ne (L.var 0) (L.var 74)) (L.and (L.ne (L.var 0) (L.var 75)) (L.and (L.ne (L.var 0) (L.var 76)) (L.and (L.ne (L.var 0) (L.var 77)) (L.and (L.ne (L.var 0) (L.var 78)) (L.and (L.ne (L.var 0) (L.var 79)) (L.and (L.ne (L.var 0) (L.var 80)) (L.and (L.ne (L.var 0) (L.var 81)) (L.and (L.ne (L.var 0) (L.var 82)) (L.and (L.ne (L.var 0) (L.var 83)) (L.and (L.ne (L.var 0) (L.var 84)) (L.and (L.ne (L.var 0) (L.var 85)) (L.and (L.ne (L.var 0) (L.var 86)) (L.and (L.ne (L.var 0) (L.var 87)) (L.and (L.ne (L.var 0) (L.var 88)) (L.and (L.ne (L.var 0) (L.var 89)) (L.and (L.ne (L.var 0) (L.var 90)) (L.and (L.ne (L.var 0) (L.var 91)) (L.and (L.ne (L.var 0) (L.var 92)) (L.and (L.ne (L.var 0) (L.var 93)) (L.and (L.ne (L.\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : ℕ → L.formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : ∀ n : ℕ, A n = L.exists (L.var 1) (L.exists (L.var 2) (L.exists (L.var 3) (L.exists (L.var 4) (L.exists (L.var 5) (L.exists (L.var 6) (L.exists (L.var 7) (L.exists (L.var 8) (L.exists (L.var 9) (L.exists (L.var 10) (L.exists (L.var 11) (L.exists (L.var 12) (L.exists (L.var 13) (L.exists (L.var 14) (L.exists (L.var 15) (L.exists (L.var 16) (L.exists (L.var 17) (L.exists (L.var 18) (L.exists (L.var 19) (L.exists (L.var 20) (L.exists (L.var 21) (L.exists (L.var 22) (L.exists (L.var 23) (L.exists (L.var 24) (L.exists (L.var 25) (L.exists (L.var 26) (L.exists (L.var 27) (L.exists (L.var 28) (L.exists (L.var 29) (L.exists (L.var 30) (L.exists (L.var 31) (L.exists (L.var 32) (L.exists (L.var 33) (L.exists (L.var 34) (L.exists (L.var 35) (L.exists (L.var 36) (L.exists (L.var 37) (L.exists (L.var 38) (L.exists (L.var 39) (L.exists (L.var 40) (L.exists (L.var 41) (L.exists (L.var 42) (L.exists (L.var 43) (L.exists (L.var 44) (L.exists (L.var 45) (L.exists (L.var 46) (L.exists (L.var 47) (L.exists (L.var 48) (L.exists (L.var 49) (L.exists (L.var 50) (L.exists (L.var 51) (L.exists (L.var 52) (L.exists (L.var 53) (L.exists (L.var 54) (L.exists (L.var 55) (L.exists (L.var 56) (L.exists (L.var 57) (L.exists (L.var 58) (L.exists (L.var 59) (L.exists (L.var 60) (L.exists (L.var 61) (L.exists (L.var 62) (L.exists (L.var 63) (L.exists (L.var 64) (L.exists (L.var 65) (L.exists (L.var 66) (L.exists (L.var 67) (L.exists (L.var 68) (L.exists (L.var 69) (L.exists (L.var 70) (L.exists (L.var 71) (L.exists (L.var 72) (L.exists (L.var 73) (L.exists (L.var 74) (L.exists (L.var 75) (L.exists (L.var 76) (L.exists (L.var 77) (L.exists (L.var 78) (L.exists (L.var 79) (L.exists (L.var 80) (L.exists (L.var 81) (L.exists (L.var 82) (L.exists (L.var 83) (L.exists (L.var 84) (L.exists (L.var 85) (L.exists (L.var 86) (L.exists (L.var 87) (L.exists (L.var 88) (L.exists (L.var 89) (L.exists (L.var 90) (L.exists (L.var 91) (L.exists (L.var 92) (L.exists (L.var 93) (L.exists (L.var 94) (L.exists (L.var 95) (L.exists (L.var 96) (L.exists (L.var 97) (L.exists (L.var 98) (L.exists (L.var 99) (L.exists (L.var 100) (L.exists (L.var 101) (L.exists (L.var 102) (L.exists (L.var 103) (L.exists (L.var 104) (L.exists (L.var 105) (L.exists (L.var 106) (L.exists (L.var 107) (L.exists (L.var 108) (L.exists (L.var 109) (L.exists (L.var 110) (L.exists (L.var 111) (L.exists (L.var 112) (L.exists (L.var 113) (L.exists (L.var 114) (L.exists (L.var 115) (L.exists (L.var 116) (L.exists (L.var 117) (L.exists (L.var 118) (L.exists (L.var 119) (L.exists (L.var 120) (L.exists (L.var 121) (L.exists (L.var 122) (L.exists (L.var 123) (L.exists (L.var 124) (L.exists (L.var 125) (L.exists (L.var 126) (L.exists (L.var 127) (L.exists (L.var 128) (L.exists (L.var 129) (L.exists (L.var 130) (L.exists (L.var 131) (L.exists (L.var 132) (L.exists (L.var 133) (L.exists (L.var 134) (L.exists (L.var 135) (L.exists (L.var 136) (L.exists (L.var 137) (L.exists (L.var 138) (L.exists (L.var 139) (L.exists (L.var 140) (L.exists (L.var 141) (L.exists (L.var 142) (L.exists (L.var 143) (L.exists (L.var 144) (L.exists (L.var 145) (L.exists (L.var 146) (L.exists (L.var 147) (L.exists (L.var 148) (L.exists (L.var 149) (L.exists (L.var 150) (L.exists (L.var 151) (L.exists (L.var 152) (L.exists (L.var 153) (L.exists (L.var 154) (L.exists (L.var 155) (L.exists (L.var 156) (L.exists (L.var 157) (L.exists (L.var 158) (L.exists (L.var 159) (L.exists (L.var 160) (L.exists (L.var 161) (L.exists (L.var 162) (L.exists (L.var 163) (L.exists (L.var 164) (L.exists (L.var 165) (L.exists (L.var 166) (L.exists (L.var 167) (L.exists (L.var 168) (L.exists (L.var 169) (L.exists (L.var 170) (L.exists (L.var 171) (\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nOverflow theorem\nLet $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.\n`proof`\nFor each $n$, let $\\mathbf A_n$ be the formula:\n\n$\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n\nThen $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n\nTake:\n$$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$\n\nSince $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n\nFrom the Compactness Theorem, $\\Gamma$ is satisfiable in some model $\\mathbf{M}$.\n\nBut since $\\mathbf{M} \\models A_i$ for each $i$, $\\mathbf{M}$ must be infinite.\n\nSo $F$ has an infinite model.\n\nQED\n-/\ntheorem  overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.34211257601858147}}
{"text": "/-\nCopyright (c) 2021 Gabriel Ebner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Gabriel Ebner\n-/\nimport Mathlib.Tactic.Cache\nimport Mathlib.Tactic.Core\nimport Mathlib.Tactic.SolveByElim\nimport Mathlib.Tactic.TryThis\n\n/-!\n# Library search\n\nThis file defines a tactic `librarySearch`\nand a term elaborator `librarySearch%`\nthat tries to find a lemma\nsolving the current goal\n(subgoals are solved using `solveByElim`).\n\n```\nexample : x < x + 1 := librarySearch%\nexample : Nat := by librarySearch\n```\n-/\n\nnamespace Tactic\nnamespace LibrarySearch\n\nopen Lean Meta TryThis\n\ninitialize registerTraceClass `Tactic.librarySearch\n\n-- from Lean.Server.Completion\nprivate def isBlackListed (declName : Name) : MetaM Bool := do\n  if declName == ``sorryAx then return false\n  if declName matches Name.str _ \"inj\" _ then return false\n  if declName matches Name.str _ \"noConfusionType\" _ then return false\n  let env ← getEnv\n  pure $ declName.isInternal\n   || isAuxRecursor env declName\n   || isNoConfusion env declName\n  <||> isRec declName <||> isMatcher declName\n\ninitialize librarySearchLemmas : DeclCache (DiscrTree Name) ←\n  DeclCache.mk \"librarySearch: init cache\" {} fun name constInfo lemmas => do\n    if constInfo.isUnsafe then return lemmas\n    if ← isBlackListed name then return lemmas\n    withNewMCtxDepth do\n      let (xs, bis, type) ← withReducible <| forallMetaTelescopeReducing constInfo.type\n      let keys ← withReducible <| DiscrTree.mkPath type\n      pure $ lemmas.insertCore keys name\n\ndef librarySearch (mvarId : MVarId) (lemmas : DiscrTree Name) (solveByElimDepth := 6) :\n    MetaM <| Option (Array <| MetavarContext × List MVarId) := do\n  profileitM Exception \"librarySearch\" (← getOptions) do\n  let mvar := mkMVar mvarId\n  let ty ← inferType mvar\n\n  let mut suggestions := #[]\n\n  let state0 ← get\n\n  try\n    solveByElim solveByElimDepth mvarId\n    return none\n  catch _ =>\n    set state0\n\n  for lem in ← lemmas.getMatch ty do\n    trace[Tactic.librarySearch] \"{lem}\"\n    match ← traceCtx `Tactic.librarySearch try\n        let newMVars ← apply mvarId (← mkConstWithFreshMVarLevels lem)\n        (try\n          for newMVar in newMVars do\n            withMVarContext newMVar do\n              trace[Tactic.librarySearch] \"proving {← addMessageContextFull (mkMVar newMVar)}\"\n              solveByElim solveByElimDepth newMVar\n          pure $ some $ Sum.inr ()\n        catch _ =>\n          let res := some $ Sum.inl <| (← getMCtx, newMVars)\n          set state0\n          pure res)\n      catch _ =>\n        set state0\n        pure none\n      with\n      | none => pure ()\n      | some (Sum.inr ()) => return none\n      | some (Sum.inl suggestion) => suggestions := suggestions.push suggestion\n\n  pure $ some suggestions\n\ndef lines (ls : List MessageData) :=\n  MessageData.joinSep ls (MessageData.ofFormat Format.line)\n\nopen Lean.Parser.Tactic\n\n-- TODO: implement the additional options for `library_search` from Lean 3,\n-- in particular including additional lemmas\n-- with `library_search [X, Y, Z]` or `library_search with attr`,\n-- or requiring that a particular hypothesis is used in the solution, with `library_search using h`.\nsyntax (name := librarySearch') \"library_search\" (config)? (\" [\" simpArg,* \"]\")?\n  (\" with \" (colGt ident)+)? (\" using \" (colGt binderIdent)+)? : tactic\nsyntax (name := librarySearch!) \"library_search!\" (config)? (\" [\" simpArg,* \"]\")?\n  (\" with \" (colGt ident)+)? (\" using \" (colGt binderIdent)+)? : tactic\n\n-- For now we only implement the basic functionality.\n-- The full syntax is recognized, but will produce a \"Tactic has not been implemented\" error.\n\nopen Elab.Tactic Elab Tactic in\nelab_rules : tactic | `(tactic| library_search%$tk) => do\n  withNestedTraces do\n  trace[Tactic.librarySearch] \"proving {← getMainTarget}\"\n  let mvar ← getMainGoal\n  let (hs, introdMVar) ← intros (← getMainGoal)\n  withMVarContext introdMVar do\n    if let some suggestions ← librarySearch introdMVar (← librarySearchLemmas.get) then\n      for suggestion in suggestions do\n        addExactSuggestion tk (← instantiateMVars (mkMVar mvar))\n      admitGoal introdMVar\n    else\n      addExactSuggestion tk (← instantiateMVars (mkMVar mvar))\n\nopen Elab Term in\nelab tk:\"library_search%\" : term <= expectedType => do\n  withNestedTraces do\n  trace[Tactic.librarySearch] \"proving {expectedType}\"\n  let mvar ← mkFreshExprMVar expectedType\n  let (hs, introdMVar) ← intros mvar.mvarId!\n  withMVarContext introdMVar do\n    if let some suggestions ← librarySearch introdMVar (← librarySearchLemmas.get) then\n      for suggestion in suggestions do\n        addTermSuggestion tk (← instantiateMVars mvar)\n      mkSorry expectedType (synthetic := true)\n    else\n      addTermSuggestion tk (← instantiateMVars mvar)\n      instantiateMVars mvar\n", "meta": {"author": "JOSHCLUNE", "repo": "Keller_reduction", "sha": "dc392b3da352fc1ffcfbecb1d4717d05f5faed4a", "save_path": "github-repos/lean/JOSHCLUNE-Keller_reduction", "path": "github-repos/lean/JOSHCLUNE-Keller_reduction/Keller_reduction-dc392b3da352fc1ffcfbecb1d4717d05f5faed4a/Lean4_Clique/Mathlib/Mathlib/Tactic/LibrarySearch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.34201892787234045}}
{"text": "/-\nCopyright 2022 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n    https://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n\nAuthors: Moritz Firsching\n-/\nimport tactic\n/-!\n# The spectral theorem and Hadamard's determinant problem\n\n## TODO\n  - statement Theorem 1.\n    - proof\n      - Lemma\n        - (A)\n        - (B)\n        - (C)\n  - The Hadamard determinant problem\n  - Hadamard matreices esixt for all $n = 2^m$\n  - Theorem 2.\n-/\n", "meta": {"author": "mo271", "repo": "formal_book", "sha": "34cbc0b9e9d361b74adbe0fd06192a72e684b992", "save_path": "github-repos/lean/mo271-formal_book", "path": "github-repos/lean/mo271-formal_book/formal_book-34cbc0b9e9d361b74adbe0fd06192a72e684b992/src/chapters/07_The_spectral_theorem_and_Hadamard's_determinant_problem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.3418815185785869}}
{"text": "import category_theory.preadditive.additive_functor\nimport category_theory.limits.preserves.shapes.biproducts\n\nimport for_mathlib.derived.les2\nimport for_mathlib.derived.les_facts\nimport for_mathlib.derived.Ext_lemmas\n\nimport for_mathlib.is_quasi_iso\nimport for_mathlib.short_exact\nimport for_mathlib.homology\nimport for_mathlib.exact_lift_desc\nimport for_mathlib.additive_functor\nimport for_mathlib.homotopy_category_lemmas\nimport for_mathlib.homology_lift_desc\n\n.\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite\nopen homotopy_category (hiding single)\nopen bounded_homotopy_category\n\n-- main proof in this file is inspired by https://math.stackexchange.com/a/2118042\n\nsection\n\nvariables {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {ι : Type*} {c : complex_shape ι}\n\ndef delta_to_kernel (C : homological_complex 𝓐 c) (i j k : ι) :\n  C.X i ⟶ kernel (C.d j k) :=\nfactor_thru_image _ ≫ image_to_kernel' (C.d i j) _ (C.d_comp_d _ _ _)\n\ndef delta_to_kernel_ι (C : homological_complex 𝓐 c) (i j k : ι) :\n  delta_to_kernel C i j k ≫ kernel.ι (C.d j k) = C.d i j :=\nbegin\n  delta delta_to_kernel image_to_kernel',\n  rw [category.assoc, kernel.lift_ι, image.fac],\nend\n\ndef d_delta_to_kernel (C : homological_complex 𝓐 c) (h i j k : ι) :\n  C.d h i ≫ delta_to_kernel C i j k = 0 :=\nbegin\n  rw [← cancel_mono (kernel.ι (C.d j k)), category.assoc, delta_to_kernel_ι, C.d_comp_d, zero_comp],\nend\n\n-- move me\nlemma short_exact_comp_iso {A B C D : 𝓐} (f : A ⟶ B) (g : B ⟶ C) (h : C ⟶ D) (hh : is_iso h) :\n  short_exact f (g ≫ h) ↔ short_exact f g :=\nbegin\n  split; intro H,\n  { haveI : mono f := H.mono,\n    haveI : epi g,\n    { haveI := H.epi, have := epi_comp (g ≫ h) (inv h), simpa only [category.assoc, is_iso.hom_inv_id, category.comp_id] },\n    refine ⟨_⟩, have := H.exact, rwa exact_comp_iso at this, },\n  { haveI : mono f := H.mono,\n    haveI : epi g := H.epi,\n    haveI : epi (g ≫ h) := epi_comp g h,\n    refine ⟨_⟩, have := H.exact, rwa exact_comp_iso }\nend\n\nlemma is_acyclic_def\n  (C : homotopy_category 𝓐 c) :\n  is_acyclic C ↔ (∀ i, is_zero (C.as.homology i)) :=\nbegin\n  split,\n  { apply is_acyclic.cond },\n  { apply is_acyclic.mk }\nend\n\nlemma is_acyclic_iff_short_exact_to_cycles\n  (C : homotopy_category 𝓐 (complex_shape.up ℤ)) :\n  is_acyclic C ↔\n  (∀ i, short_exact (kernel.ι (C.as.d i (i+1))) (delta_to_kernel C.as i (i+1) (i+1+1))) :=\nbegin\n  rw is_acyclic_def,\n  symmetry,\n  apply (equiv.add_right (1 : ℤ)).forall_congr,\n  intro i,\n  let e := (homology_iso C.as i (i+1) (i+1+1) rfl rfl),\n  dsimp [delta_to_kernel] at e ⊢,\n  rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],\n  split,\n  { apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },\n  { intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }\nend\n\nlemma is_acyclic_iff_short_exact_to_cycles'\n  (C : homological_complex 𝓐 (complex_shape.down ℤ)) :\n  (∀ i, is_zero (C.homology i)) ↔\n  (∀ i, short_exact (kernel.ι (C.d (i+1+1) (i+1))) (delta_to_kernel C (i+1+1) (i+1) i)) :=\nbegin\n  symmetry,\n  apply (equiv.add_right (1 : ℤ)).forall_congr,\n  intro i,\n  let e := (homology_iso C (i+1+1) (i+1) i rfl rfl),\n  dsimp [delta_to_kernel] at e ⊢,\n  rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],\n  split,\n  { apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },\n  { intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }\nend\n\nend\n\nvariables {𝓐 𝓑 : Type*} [category 𝓐] [abelian 𝓐] [enough_projectives 𝓐]\nvariables [category 𝓑] [abelian 𝓑] [enough_projectives 𝓑]\n\nvariables (C : cochain_complex 𝓐 ℤ)\n  [homotopy_category.is_bounded_above ((homotopy_category.quotient _ _).obj C)]\n\ndef category_theory.functor.single (F : bounded_homotopy_category 𝓐 ⥤ 𝓑) (i : ℤ) : 𝓐 ⥤ 𝓑 :=\nbounded_homotopy_category.single _ i ⋙ F\n\n-- move me\nlemma category_theory.limits.is_zero.biprod {𝓐 : Type*} [category 𝓐] [abelian 𝓐]\n  {X Y : 𝓐} (hX : is_zero X) (hY : is_zero Y) :\n  is_zero (X ⊞ Y) :=\nbegin\n  rw is_zero_iff_id_eq_zero at hX hY ⊢,\n  ext; simp [hX, hY],\nend\n\ninstance category_theory.limits.preserves_binary_biproduct_of_additive\n  {𝓐 𝓑 : Type*} [category 𝓐] [category 𝓑] [abelian 𝓐] [abelian 𝓑]\n  (F : 𝓐 ⥤ 𝓑) [functor.additive F] (X Y : 𝓐) :\n  preserves_binary_biproduct X Y F :=\npreserves_binary_biproduct_of_preserves_biproduct _ _ _\n\n-- move me\n@[simp] lemma category_theory.op_neg {𝓐 : Type*} [category 𝓐] [preadditive 𝓐]\n  {X Y : 𝓐} (f : X ⟶ Y) : (-f).op = - f.op := rfl\n\nlemma acyclic_left_of_short_exact (B : 𝓐) {X Y Z : 𝓐} (f : X ⟶ Y) (g : Y ⟶ Z) (hfg : short_exact f g)\n  (hY : ∀ i > 0, is_zero (((Ext' i).obj (op $ Y)).obj B))\n  (hZ : ∀ i > 0, is_zero (((Ext' i).obj (op $ Z)).obj B)) :\n  ∀ i > 0, is_zero (((Ext' i).obj (op $ X)).obj B) :=\nbegin\n  intros i hi,\n  have := hfg.Ext'_five_term_exact_seq B i,\n  refine (this.drop 1).pair.is_zero_of_is_zero_is_zero (hY _ hi) (hZ _ _),\n  transitivity i, { exact lt_add_one i }, { exact hi }\nend\n.\n\nlemma map_is_acyclic_of_acyclic_aux\n  {A B C D X Y Z W : 𝓐} (f : A ⟶ B) (g : C ⟶ D) (π : B ⟶ kernel g)\n  {α : X ⟶ B} {β : B ⟶ Y} {γ : Y ⟶ C} {δ : C ⟶ Z} {ε : Z ⟶ D} {ζ : D ⟶ W}\n  (hαβ : short_exact α β) (hγδ : short_exact γ δ) (hεζ : short_exact ε ζ)\n  (hf : mono f) (hfπ : exact f π)\n  (hαπ : α ≫ π = 0) (hγg : γ ≫ g = 0) (hδε : δ ≫ ε = g)\n  (hπι : π ≫ kernel.ι g = β ≫ γ) :\n  short_exact f π :=\nbegin\n  suffices : epi π, { resetI, exact ⟨hfπ⟩ },\n  have hβ : epi β := hαβ.epi,\n  have hγ : mono γ := hγδ.mono,\n  have hε : mono ε := hεζ.mono,\n  resetI,\n  have hιδ : kernel.ι g ≫ δ = 0,\n  { rw [← cancel_mono ε, category.assoc, hδε, kernel.condition, zero_comp], },\n  let e1 : Y ⟶ kernel g := hαβ.exact.epi_desc π hαπ,\n  let e2 : Y ⟶ kernel g := kernel.lift g γ hγg,\n  let e3 : kernel g ⟶ Y := hγδ.exact.mono_lift (kernel.ι g) hιδ,\n  have he12 : e1 = e2,\n  { rw [← cancel_epi β, ← cancel_mono (kernel.ι g)],\n    simp only [hπι, category.assoc, kernel.lift_ι, exact.comp_epi_desc_assoc], },\n  have he13 : e1 ≫ e3 = 𝟙 _,\n  { rw [he12, ← cancel_mono γ, category.assoc, exact.mono_lift_comp, kernel.lift_ι, category.id_comp], },\n  have he31 : e3 ≫ e1 = 𝟙 _,\n  { rw [he12, ← cancel_mono (kernel.ι g), category.assoc, kernel.lift_ι, exact.mono_lift_comp, category.id_comp], },\n  let e : Y ≅ kernel g := ⟨e1, e3, he13, he31⟩,\n  have hπ : β ≫ e.hom = π := exact.comp_epi_desc _ _ _,\n  rw ← hπ, exact epi_comp _ _,\nend\n\nlemma short_exact_Ext_of_short_exact_of_acyclic {A B C : 𝓐} (Z : 𝓐) {f : A ⟶ B} {g : B ⟶ C}\n  (hfg : short_exact f g) (hC : ∀ i > 0, is_zero (((Ext' i).obj (op $ C)).obj Z)) :\n  short_exact (((Ext' 0).flip.obj Z).map g.op) (((Ext' 0).flip.obj Z).map f.op) :=\nbegin\n  have H0 := hfg.Ext'_five_term_exact_seq Z 0,\n  apply_with short_exact.mk {instances:=ff},\n  { have H := ((hfg.Ext'_five_term_exact_seq Z (-1)).drop 2).pair,\n    apply H.mono_of_is_zero,\n    apply Ext'_is_zero_of_neg, dec_trivial },\n  { apply (H0.drop 1).pair.epi_of_is_zero,\n    apply hC, dec_trivial },\n  { exact H0.pair }\nend\n\nlemma map_is_acyclic_of_acyclic''\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  ∀ i, is_zero (((((Ext' 0).flip.obj B).map_homological_complex _).obj C.op).homology i) :=\nbegin\n  rw is_acyclic_iff_short_exact_to_cycles',\n  obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),\n  have aux : ((quotient 𝓐 (complex_shape.up ℤ)).obj C).is_acyclic := ‹_›,\n  rw is_acyclic_iff_short_exact_to_cycles at aux,\n  intro i,\n  let K := λ j, kernel (C.d j (j+1)),\n  suffices hK : ∀ j, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),\n  { have SES1 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1+1)) (hK _),\n    have SES2 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1)) (hK _),\n    have SES3 := short_exact_Ext_of_short_exact_of_acyclic B (aux i) (hK _),\n    apply map_is_acyclic_of_acyclic_aux _ _ _ SES1 SES2 SES3 infer_instance;\n      clear SES1 SES2 SES3 aux,\n    { delta delta_to_kernel image_to_kernel',\n      apply exact_comp_mono, rw exact_factor_thru_image_iff, exact exact_kernel_ι },\n    { rw [← cancel_mono (kernel.ι _), zero_comp, category.assoc, delta_to_kernel_ι],\n      swap, apply_instance,\n      erw [functor.map_homological_complex_obj_d, ← functor.map_comp],\n      dsimp only [homological_complex.op_d, quotient_obj_as],\n      rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },\n    { erw [functor.map_homological_complex_obj_d, ← functor.map_comp],\n      dsimp only [homological_complex.op_d, quotient_obj_as],\n      rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },\n    { rw [← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },\n    { rw [delta_to_kernel_ι, ← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },\n    { apply_instance } },\n  clear i, intro j,\n  have : ∀ j ≥ a, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),\n  { intros j hj i hi,\n    apply bounded_derived_category.Ext'_zero_left_is_zero,\n    apply is_zero.op,\n    refine is_zero.of_mono (kernel.ι _) _,\n    exact ha j hj },\n  apply int.induction_on' j a,\n  { exact this _ le_rfl, },\n  { intros j hj aux, apply this, exact int.le_add_one hj, },\n  { intros j hj IH,\n    obtain ⟨j, rfl⟩ : ∃ i, i + 1 = j := ⟨j - 1, sub_add_cancel _ _⟩,\n    rw add_sub_cancel,\n    apply acyclic_left_of_short_exact B (kernel.ι _) (delta_to_kernel _ _ _ _) _ (hC _) IH,\n    exact aux j, }\nend\n\nlemma map_is_acyclic_of_acyclic'\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  is_acyclic ((((Ext' 0).flip.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=\nbegin\n  rw is_acyclic_def,\n  intro i,\n  have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },\n  refine is_zero.of_iso _ (homology_iso' _ (i-1) i (i+1) h1 rfl),\n  dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,\n    functor.right_op_map, functor.map_homological_complex_obj_d],\n  apply exact.homology_is_zero,\n  apply exact.op,\n  refine exact_of_homology_is_zero _,\n  { rw [← category_theory.functor.map_comp, ← op_comp, homological_complex.d_comp_d, op_zero, functor.map_zero], },\n  have := map_is_acyclic_of_acyclic'' C B hC i,\n  apply this.of_iso _, clear this,\n  let C' := (((Ext' 0).flip.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj (homological_complex.op C),\n  have h1 : (complex_shape.down ℤ).rel i (i - 1), { dsimp, apply sub_add_cancel },\n  exact (homology_iso' C' (i+1) i (i-1) rfl h1).symm,\nend\n\nlemma map_is_acyclic_of_acyclic\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  is_acyclic (((preadditive_yoneda.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=\nbegin\n  have := map_is_acyclic_of_acyclic' C B hC,\n  rw is_acyclic_def at this ⊢,\n  intro i, specialize this i,\n  apply this.of_iso _, clear this,\n  have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },\n  refine (homology_iso' _ (i-1) i (i+1) h1 rfl) ≪≫ _ ≪≫ (homology_iso' _ (i-1) i (i+1) h1 rfl).symm,\n  dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,\n    functor.right_op_map, functor.map_homological_complex_obj_d],\n  let e := λ i, ((bounded_derived_category.Ext'_zero_flip_iso _ B).app (op $ C.X i)).op,\n  refine homology.map_iso _ _ (arrow.iso_mk (e _) (e _) _) (arrow.iso_mk (e _) (e _) _) rfl,\n  { simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },\n  { simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },\nend\n\nlemma acyclic_of_projective (P B : 𝓐) [projective P] (i : ℤ) (hi : 0 < i) :\n  is_zero (((Ext' i).obj (op P)).obj B) :=\nbegin\n  rw (Ext'_iso (op P) B i _ (𝟙 _) _).is_zero_iff,\n  { rcases i with ((_|i)|i),\n    { exfalso, revert hi, dec_trivial },\n    swap, { exfalso, revert hi, dec_trivial },\n    refine is_zero.homology_is_zero _ _ _ _,\n    apply AddCommGroup.is_zero_of_eq,\n    intros,\n    apply is_zero.eq_of_src,\n    apply is_zero_zero, },\n  { refine ⟨_, _, _⟩,\n    { rintro (_|n), { assumption }, { dsimp, apply_instance } },\n    { exact exact_zero_mono (𝟙 P) },\n    { rintro (_|n); exact exact_of_zero 0 0 } }\nend\n\ndef Ext_compute_with_acyclic_aux₁\n  (B : 𝓐)\n  (i : ℤ) :\n  ((Ext i).obj (op $ of' C)).obj ((single _ 0).obj B) ≅\n  (preadditive_yoneda.obj ((single 𝓐 (-i)).obj B)).obj (op (of' C).replace) :=\n(preadditive_yoneda.map_iso $ (shift_single_iso 0 i).app B ≪≫ eq_to_iso (by rw zero_sub)).app _\n\nabbreviation of'_hom {C₁ C₂ : cochain_complex 𝓐 ℤ}\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (f : C₁ ⟶ C₂) :\n  of' C₁ ⟶ of' C₂ :=\n(homotopy_category.quotient _ _).map f\n\nlemma Ext_compute_with_acyclic_aux₁_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  ((Ext i).map $ quiver.hom.op $ of'_hom f).app _ ≫\n    (Ext_compute_with_acyclic_aux₁ C₁ B i).hom =\n  (Ext_compute_with_acyclic_aux₁ C₂ B i).hom ≫\n  (preadditive_yoneda.obj _).map (quiver.hom.op $\n  bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) :=\nbegin\n  ext F,\n  dsimp [Ext, Ext_compute_with_acyclic_aux₁],\n  simp only [comp_apply],\n  dsimp, simp,\nend\n\ndef Ext_compute_with_acyclic_aux₂\n  (B : 𝓐)\n  (i : ℤ) :\n  (preadditive_yoneda.obj ((single 𝓐 (-i)).obj B)).obj (op (of' C).replace) ≅\n  (((preadditive_yoneda.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj\n  ((of' C).replace).val.as.op).homology (-i) :=\n  hom_single_iso _ _ _\n\n-- TODO: We should prove naturality of `hom_single_iso` independently!\nlemma Ext_compute_with_acyclic_aux₂_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  (preadditive_yoneda.obj _).map (quiver.hom.op $\n    bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) ≫\n    (Ext_compute_with_acyclic_aux₂ C₁ B i).hom =\n  (Ext_compute_with_acyclic_aux₂ C₂ B i).hom ≫\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map\n      (bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π).out).unop :=\nhom_single_iso_naturality _ _ _ _ _\n\ndef Ext_compute_with_acyclic_HomB\n  (B : 𝓐) := (preadditive_yoneda.obj B).right_op.map_homological_complex (complex_shape.up ℤ) ⋙\n  homological_complex.unop_functor.right_op\n\nlemma Ext_compute_with_acyclic_is_quasi_iso_aux\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  is_quasi_iso ((homotopy_category.quotient _ _).map\n    ((Ext_compute_with_acyclic_HomB B).map (of' C).π.out).unop) :=\nbegin\n  let P := (of' C).replace,\n  let π : P ⟶ of' C := (of' C).π,\n  let HomB := (preadditive_yoneda.obj B).right_op.map_homological_complex (complex_shape.up ℤ) ⋙\n    homological_complex.unop_functor.right_op,\n  let fq := (homotopy_category.quotient _ _).map (HomB.map π.out).unop,\n  apply is_quasi_iso_of_op,\n  let f := homological_complex.op_functor.map (HomB.map (quot.out π)),\n  have := cone_triangleₕ_mem_distinguished_triangles _ _ f,\n  replace := is_quasi_iso_iff_is_acyclic _ this,\n  dsimp [homological_complex.cone.triangleₕ] at this,\n  erw this, clear this i,\n  constructor,\n  intro i, obtain ⟨i, rfl⟩ : ∃ j, j + 1 = i := ⟨i - 1, sub_add_cancel _ _⟩,\n  refine is_zero.of_iso _ (homology_iso _ i (i+1) (i+1+1) _ _),\n  rotate, { dsimp, refl }, { dsimp, refl },\n  apply exact.homology_is_zero _,\n  dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d],\n  have hπ : is_quasi_iso π, { dsimp [π], apply_instance },\n  have := cone_triangleₕ_mem_distinguished_triangles _ _ π.out,\n  replace := is_quasi_iso_iff_is_acyclic _ this,\n  dsimp [homological_complex.cone.triangleₕ] at this,\n  simp only [quotient_map_out] at this,\n  replace := this.mp _,\n  swap, { convert hπ using 1, generalize : P.val = X, cases X, refl, },\n  haveI preaux : ((quotient 𝓐 (complex_shape.up ℤ)).obj (homological_complex.cone (quot.out π))).is_bounded_above,\n  { constructor,\n    obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),\n    obtain ⟨b, hb⟩ := is_bounded_above.cond P.val,\n    refine ⟨max a b, _⟩,\n    intros k hk,\n    refine category_theory.limits.is_zero.biprod _ _,\n    { apply hb, refine (le_max_right _ _).trans (hk.trans (lt_add_one _).le) },\n    { apply ha, exact (le_max_left _ _).trans hk, } },\n  have aux := @map_is_acyclic_of_acyclic _ _ _ _ _ _ this B _,\n  { replace := (@is_acyclic.cond _ _ _ _ _ _ aux (i+1)).of_iso (homology_iso _ i (i+1) (i+1+1) _ _).symm,\n    rotate, { dsimp, refl }, { dsimp, refl },\n    dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d,\n      functor.map_homotopy_category_obj, functor.map_homological_complex_obj_d] at this,\n    replace := exact_of_homology_is_zero this,\n    let e := functor.map_biprod (preadditive_yoneda.obj B).right_op,\n    refine preadditive.exact_of_iso_of_exact' _ _ _ _ (e _ _) (e _ _) (e _ _) _ _ this;\n    dsimp only [e, functor.map_biprod_hom],\n    all_goals\n    { ext,\n      { simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_fst,\n          eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,\n          homological_complex.X_eq_to_iso_refl, category.comp_id, dite_eq_ite, if_true,\n          biprod.lift_fst, biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero],\n        simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,\n          biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,\n          ← op_comp, ← category_theory.functor.map_comp, biprod.lift_fst],\n        simp only [biprod.desc_eq, comp_zero, add_zero, preadditive.comp_neg,\n          category_theory.op_neg, functor.map_neg, op_comp, category_theory.functor.map_comp],\n        refl },\n      { simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_snd,\n          eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,\n          functor.map_homological_complex_map_f, homological_complex.X_eq_to_iso_refl,\n          category.comp_id, dite_eq_ite, if_true, biprod.lift_snd, biprod.lift_desc],\n        simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,\n          biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,\n          ← op_comp, ← category_theory.functor.map_comp, biprod.lift_snd],\n        simp only [biprod.desc_eq, op_add, functor.map_neg, functor.map_add, op_comp,\n          category_theory.functor.map_comp],\n        refl } } },\n  { clear i, intros k i hi,\n    let e := functor.map_biprod ((Ext' i).flip.obj B).right_op\n      (P.val.as.X (k + 1)) ((of' C).val.as.X k),\n    refine is_zero.of_iso (is_zero.unop _) e.symm.unop,\n    refine category_theory.limits.is_zero.biprod _ _,\n    { simp only [functor.right_op_obj, functor.flip_obj_obj, is_zero_op],\n      exact acyclic_of_projective (P.val.as.X (k + 1)) B i hi, },\n    { exact (hC k _ hi).op, }, },\nend\n\ndef Ext_compute_with_acyclic_aux₃\n  (B : 𝓐)\n  (i : ℤ) :\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _).obj C).unop.homology (-i) ⟶\n  (((preadditive_yoneda.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj\n  ((of' C).replace).val.as.op).homology (-i) :=\nby apply (homotopy_category.homology_functor Ab _ (-i)).map\n  (((homotopy_category.quotient _ _).map\n    ((Ext_compute_with_acyclic_HomB B).map (of' C).π.out).unop))\n\n-- Are these two lemmas useful? Do we have them?\nlemma functor.map_unop {𝓐 : Type*} [category 𝓐] {𝓑 : Type*}\n  [category 𝓑] (F : 𝓐 ⥤ 𝓑) {a₁ a₂ : 𝓐ᵒᵖ}\n  (f : a₁ ⟶ a₂) : F.map f.unop = (F.op.map f).unop :=\nrfl\n\nlemma functor.map_map' {𝓐 : Type*} [category 𝓐] {𝓑 : Type*}\n  [category 𝓑] {𝓒 : Type*} [category 𝓒] (F : 𝓐 ⥤ 𝓑) (G : 𝓑 ⥤ 𝓒) {a₁ a₂ : 𝓐}\n  (φ : a₁ ⟶ a₂) : G.map (F.map φ) = (F ⋙ G).map φ :=\nrfl\n\nnamespace Ext_compute_with_acyclic_aux₃_naturality_helpers\n\nvariables (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ)\n\nlemma helper₁ (h1 h2) :\n  homology.lift ((unop\n    (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n      (complex_shape.up ℤ)).obj C₁))).d_to (-i))\n  ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₁))).d_from (-i)) h1\n  (kernel.ι ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₂))).d_from (-i)) ≫\n    (homological_complex.unop_functor.right_op.map\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).map f)).unop.f (-i) ≫ cokernel.π\n    ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₁))).d_to (-i))) h2 =\n  kernel.lift _ (kernel.ι _ ≫ (preadditive_yoneda.obj B).map (f.f (-i)).op) begin\n    rw category.assoc,\n    have aux := ((((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map f)).unop.comm_from (-i),\n    dsimp at aux,\n    erw [aux, kernel.condition_assoc, zero_comp],\n  end ≫\n  homology.π' _ _ _ :=\nbegin\n  apply homology.hom_to_ext,\n  rw [category.assoc, homology.π'_ι, kernel.lift_ι_assoc,\n    homology.lift_ι],\n  refl,\nend\n\nlemma helper₂ (h1 h2) :\n  homology.lift ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_to (-i))\n  ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_from (-i))\n  h1\n  (kernel.ι ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).val.as)).d_from (-i)) ≫\n     ((Ext_compute_with_acyclic_HomB B).map (quot.out (of' C₂).π)).unop.f (-i) ≫\n       cokernel.π ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_to (-i)))\n  h2 =\n  kernel.lift _ (kernel.ι _ ≫ begin\n    apply homological_complex.hom.f,\n    exact (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map (of' C₂).π.out).unop,\n  end) begin\n    rw category.assoc,\n    have := (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map (of' C₂).π.out).unop.comm_from (-i),\n    erw [this, kernel.condition_assoc, zero_comp],\n  end ≫ homology.π' _ _ _ :=\nbegin\n  apply homology.hom_to_ext,\n  simp only [homology.lift_ι, category.assoc, homology.π'_ι, kernel.lift_ι_assoc],\n  refl,\nend\n\nend Ext_compute_with_acyclic_aux₃_naturality_helpers\n\nlemma Ext_compute_with_acyclic_aux₃_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n    homological_complex.unop_functor.right_op ⋙ (_root_.homology_functor _ _ (-i)).op).map f).unop\n    ≫ Ext_compute_with_acyclic_aux₃ C₁ B i =\n  Ext_compute_with_acyclic_aux₃ C₂ B i ≫\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map\n      (bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π).out).unop :=\nbegin\n  dsimp only [Ext_compute_with_acyclic_aux₃, functor.comp_map,\n    _root_.homology_functor, functor.op, homological_complex.hom.sq_from,\n    homological_complex.hom.sq_to, homotopy_category.quotient,\n    homotopy_category.homology_functor, quotient.functor,\n    category_theory.quotient.lift, quot.lift_on, quiver.hom.unop_op],\n  simp_rw homology.map_eq_desc'_lift_left,\n  apply homology.hom_from_ext,\n  simp only [category.assoc, homology.π'_desc'_assoc],\n  slice_rhs 1 2\n  { erw homology.π'_desc' },\n  dsimp only [arrow.hom_mk_left],\n  rw Ext_compute_with_acyclic_aux₃_naturality_helpers.helper₁,\n  conv_rhs { rw Ext_compute_with_acyclic_aux₃_naturality_helpers.helper₂ },\n  simp only [category.assoc],\n  slice_lhs 2 3\n  { erw homology.π'_desc' },\n  slice_rhs 2 3\n  { erw homology.π'_desc' },\n  apply homology.hom_to_ext,\n  slice_lhs 2 3\n  { erw homology.lift_ι },\n  slice_rhs 2 3\n  { erw homology.lift_ι },\n  slice_lhs 1 2\n  { erw kernel.lift_ι },\n  slice_rhs 1 2\n  { erw kernel.lift_ι },\n  simp only [category.assoc],\n  dsimp only [Ext_compute_with_acyclic_HomB, functor.comp_map],\n  change _ ≫ (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map f).unop.f _ ≫ _ = _,\n  slice_lhs 2 3\n  { simp only [← homological_complex.comp_f, ← unop_comp],\n    simp only [functor.comp_map, ← functor.map_comp] },\n  slice_rhs 2 3\n  { simp only [← homological_complex.comp_f, ← unop_comp, ← functor.map_comp] },\n  apply homotopy.kernel_ι_comp_comp_cokernel_π_of_homotopy,\n  apply homotopy.homotopy_unop_functor_right_op_map_unop_of_homotopy,\n  apply functor.map_homotopy,\n  suffices : (lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) ≫ (of' C₂).π =\n    (of' C₁).π ≫ of'_hom f,\n  { apply homotopy_category.homotopy_of_eq,\n    convert this.symm,\n    { simpa only [functor.map_comp, quotient_map_out], },\n    { simpa only [functor.map_comp, quotient_map_out], } },\n  simp,\nend\n\n\nlemma Ext_compute_with_acyclic_aux₃_is_iso\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) : is_iso (Ext_compute_with_acyclic_aux₃ C B i) :=\nbegin\n  haveI := Ext_compute_with_acyclic_is_quasi_iso_aux C B hC i,\n  apply is_quasi_iso.cond,\nend\n\ndef Ext_compute_with_acyclic\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  ((Ext i).obj (op $ of' C)).obj ((single _ 0).obj B) ≅\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _).obj C).unop.homology (-i) :=\nExt_compute_with_acyclic_aux₁ C B i ≪≫\nExt_compute_with_acyclic_aux₂ _ _ _ ≪≫\nbegin\n  haveI := Ext_compute_with_acyclic_aux₃_is_iso C B hC i,\n  exact (as_iso (Ext_compute_with_acyclic_aux₃ C B i)).symm,\nend\n\n/-\ndef homological_complex.single_iso (B : 𝓐) {i j : ℤ} (h : j = i) :\n  ((homological_complex.single _ (complex_shape.up ℤ) i).obj B).X j ≅ B :=\neq_to_iso (if_pos h)\n\n/-- The morphism in `Ab` which eats a morphism of complexes `C ⟶ B[-i]`\n  and returns an element of Extⁱ(C,B[0]). -/\ndef Ext_compute_with_acyclic_inv_eq_aux (B) (i) :\n  AddCommGroup.of (C ⟶ (homological_complex.single _ _ (-i)).obj B) ⟶\n  ((Ext i).obj (op (of' C))).obj ((single 𝓐 0).obj B) :=\n{ to_fun := λ f, (of' C).π ≫ begin\n    refine (homotopy_category.quotient _ _).map _,\n    refine _ ≫ (homological_complex.single_shift _ _).inv.app _,\n    refine (f : C ⟶ (homological_complex.single 𝓐 (complex_shape.up ℤ) (-i)).obj B) ≫ _,\n    refine eq_to_hom _,\n    simp,\n  end,\n  map_zero' := begin\n    simp only [zero_comp, functor.map_zero, comp_zero],\n  end,\n  map_add' := begin\n    intros,\n    simp only [preadditive.add_comp, functor.map_add, preadditive.comp_add],\n  end }\n\nlemma iso_equiv_inv_apply_eq_zero_cancel\n  {A B C : AddCommGroup} {a : A} {f : A ⟶ B} {e : C ≅ B}\n  (h : (f ≫ e.inv) a = 0) : f a = 0 :=\nbegin\n  rw comp_apply at h,\n  apply_fun e.hom at h,\n  simpa using h,\nend\n\n-- Note: in the application of the below, j = -i\n/-- The construction which given something in the kernel of `(Cⱼ ⟶ B) ⟶ (Cⱼ-₁ ⟶ B)`\n  constructs a morphism of complexes from C to the \"skyscraper complex\" B[j]. -/\ndef kernel_yoneda_complex_to_morphism_to_single (B : 𝓐) (j : ℤ) :\n-- we're going from the kernel of `(C.X j ⟶ B) ⟶ (C.X_prev j ⟶ B)` (it's actually `C.symm.X_next`)\nkernel ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n  (homological_complex.op C)).d_from j)\n  -- to the additive group of homs\n  ⟶ AddCommGroup.of (\n  -- from C to B j\n  C ⟶ (homological_complex.single 𝓐 (complex_shape.up ℤ) j).obj B) :=\n{ to_fun := λ f, { f := λ k,\n  if hk : k = j then\n    (eq_to_hom (by rw hk) : C.X k ⟶ C.X j) ≫\n    (kernel.ι ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n                (homological_complex.op C)).d_from j) f : C.X j ⟶ B) ≫\n    (homological_complex.single_obj_X_self 𝓐 (complex_shape.up ℤ) j B).inv ≫\n    eq_to_hom (by rw hk)\n  else 0,\n    comm' := λ i k, begin\n      split_ifs with hij hkj hkj,\n      { subst hij, subst hkj, simp {contextual := tt}, },\n      { simp only [homological_complex.single_obj_d, comp_zero, eq_self_iff_true,\n          implies_true_iff] },\n      { subst hkj,\n        set g := ((kernel.ι ((((preadditive_yoneda.obj B).map_homological_complex\n                            (complex_shape.up ℤ).symm).obj\n                  (homological_complex.op C)).d_from k)) f) with hg,\n        simp only [complex_shape.up_rel, zero_comp, eq_to_hom_refl,\n          homological_complex.single_obj_X_self_inv, category.comp_id, category.id_comp],\n        rintro hik, clear hij,\n        have := kernel.condition ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n          (homological_complex.op C)).d_from k),\n        replace this := congr_hom this f,\n        rw [comp_apply, ← hg, AddCommGroup.zero_apply] at this,\n        rw ← category.assoc,\n        symmetry,\n        convert zero_comp,\n        have foo : (complex_shape.up ℤ).symm.rel k i := hik,\n        rw homological_complex.d_from_eq _ foo at this,\n        exact iso_equiv_inv_apply_eq_zero_cancel this, },\n      { simp only [zero_comp, comp_zero, eq_self_iff_true, implies_true_iff]},\n    end },\n  map_zero' := by {simp only [map_zero, homological_complex.single_obj_X_self_inv,\n    eq_to_hom_trans, zero_comp, comp_zero, dite_eq_ite, if_t_t], refl },\n  map_add' := by { intros, ext, simp only [map_add, homological_complex.single_obj_X_self_inv,\n    eq_to_hom_trans, preadditive.add_comp, preadditive.comp_add, homological_complex.add_f_apply],\n    split_ifs,\n    { refl },\n    { refl },\n    { exact (add_zero _).symm, } } }\n\nlemma Ext_compute_with_acylic_inv_eq (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  (Ext_compute_with_acyclic _ B hC i).inv =\n  homology.desc' _ _ _\n  ( kernel_yoneda_complex_to_morphism_to_single C B (-i) ≫\n    Ext_compute_with_acyclic_inv_eq_aux C B i)\nadmit := admit\n\n-- Replacing some `End` with `cend` fixes my bracket pair colorizer!\n-- notation `cend` := category_theory.End\n\n-/\n\nlemma Ext_compute_with_acyclic_naturality (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (hC₁ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₁.X k)).obj B))\n  (hC₂ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₂.X k)).obj B))\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  ((Ext i).flip.obj ((single _ 0).obj B)).map (quiver.hom.op $ of'_hom f) ≫\n    (Ext_compute_with_acyclic C₁ B hC₁ i).hom =\n  (Ext_compute_with_acyclic C₂ B hC₂ i).hom ≫\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map f).unop :=\nbegin\n  dsimp only [Ext_compute_with_acyclic, iso.trans_hom],\n  slice_lhs 1 2\n  { erw Ext_compute_with_acyclic_aux₁_naturality },\n  slice_lhs 2 3\n  { rw Ext_compute_with_acyclic_aux₂_naturality },\n  simp only [category.assoc],\n  congr' 2,\n  dsimp only [iso.symm_hom, as_iso_inv],\n  rw [is_iso.eq_inv_comp, ← category.assoc, is_iso.comp_inv_eq],\n  symmetry,\n  apply Ext_compute_with_acyclic_aux₃_naturality,\nend\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/acyclic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34188150348988366}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.types\nimport category_theory.currying\nimport category_theory.limits.functor_category\n\n/-!\n# The morphism comparing a colimit of limits with the corresponding limit of colimits.\n\nFor `F : J × K ⥤ C` there is always a morphism $\\colim_k \\lim_j F(j,k) → \\lim_j \\colim_k F(j, k)$.\nWhile it is not usually an isomorphism, with additional hypotheses on `J` and `K` it may be,\nin which case we say that \"colimits commute with limits\".\n\nThe prototypical example, proved in `category_theory.limits.filtered_colimit_commutes_finite_limit`,\nis that when `C = Type`, filtered colimits commute with finite limits.\n\n## References\n* Borceux, Handbook of categorical algebra 1, Section 2.13\n* [Stacks: Filtered colimits](https://stacks.math.columbia.edu/tag/002W)\n-/\n\nuniverses v₂ v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nvariables {J K : Type v} [small_category J] [small_category K]\nvariables {C : Type u} [category.{v} C]\n\nvariables (F : J × K ⥤ C)\n\nopen category_theory.prod\n\nlemma map_id_left_eq_curry_map {j : J} {k k' : K} {f : k ⟶ k'} :\n  F.map ((𝟙 j, f) : (j, k) ⟶ (j, k')) = ((curry.obj F).obj j).map f :=\nrfl\n\nlemma map_id_right_eq_curry_swap_map {j j' : J} {f : j ⟶ j'} {k : K} :\n  F.map ((f, 𝟙 k) : (j, k) ⟶ (j', k)) = ((curry.obj (swap K J ⋙ F)).obj k).map f :=\nrfl\n\nvariables [has_limits_of_shape J C]\nvariables [has_colimits_of_shape K C]\n\n/--\nThe universal morphism\n$\\colim_k \\lim_j F(j,k) → \\lim_j \\colim_k F(j, k)$.\n-/\nnoncomputable\ndef colimit_limit_to_limit_colimit :\n  colimit ((curry.obj (swap K J ⋙ F)) ⋙ lim) ⟶ limit ((curry.obj F) ⋙ colim) :=\nlimit.lift ((curry.obj F) ⋙ colim)\n{ X := _,\n  π :=\n  { app := λ j, colimit.desc ((curry.obj (swap K J ⋙ F)) ⋙ lim)\n    { X := _,\n      ι :=\n      { app := λ k,\n          limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k,\n        naturality' :=\n        begin\n          dsimp,\n          intros k k' f,\n          simp only [functor.comp_map, curry.obj_map_app, limits.lim_map_π_assoc, swap_map,\n            category.comp_id, map_id_left_eq_curry_map, colimit.w],\n        end }, },\n    naturality' :=\n    begin\n      dsimp,\n      intros j j' f,\n      ext k,\n      simp only [limits.colimit.ι_map, curry.obj_map_app, limits.colimit.ι_desc_assoc,\n        limits.colimit.ι_desc, category.id_comp, category.assoc, map_id_right_eq_curry_swap_map,\n        limit.w_assoc],\n    end } }\n\n/--\nSince `colimit_limit_to_limit_colimit` is a morphism from a colimit to a limit,\nthis lemma characterises it.\n-/\n@[simp, reassoc] lemma ι_colimit_limit_to_limit_colimit_π (j) (k) :\n  colimit.ι _ k ≫ colimit_limit_to_limit_colimit F ≫ limit.π _ j =\n    limit.π ((curry.obj (swap K J ⋙ F)).obj k) j ≫ colimit.ι ((curry.obj F).obj j) k :=\nby { dsimp [colimit_limit_to_limit_colimit], simp, }\n\n@[simp] lemma ι_colimit_limit_to_limit_colimit_π_apply (F : J × K ⥤ Type v) (j) (k) (f) :\n   limit.π ((curry.obj F) ⋙ colim) j\n     (colimit_limit_to_limit_colimit F (colimit.ι ((curry.obj (swap K J ⋙ F)) ⋙ lim) k f)) =\n     colimit.ι ((curry.obj F).obj j) k (limit.π ((curry.obj (swap K J ⋙ F)).obj k) j f) :=\nby { dsimp [colimit_limit_to_limit_colimit], simp, }\n\n/-- The map `colimit_limit_to_limit_colimit` realized as a map of cones. -/\n@[simps] noncomputable def colimit_limit_to_limit_colimit_cone (G : J ⥤ K ⥤ C) [has_limit G] :\n  colim.map_cone (limit.cone G) ⟶ limit.cone (G ⋙ colim) :=\n{ hom := colim.map (limit_iso_swap_comp_lim G).hom ≫\n    colimit_limit_to_limit_colimit (uncurry.obj G : _) ≫\n    lim.map (whisker_right (currying.unit_iso.app G).inv colim),\n  w' := λ j,\n  begin\n    ext1 k,\n    simp only [limit_obj_iso_limit_comp_evaluation_hom_π_assoc, iso.app_inv,\n      ι_colimit_limit_to_limit_colimit_π_assoc, whisker_right_app,\n      colimit.ι_map, functor.map_cone_π_app, category.id_comp,\n      eq_to_hom_refl, eq_to_hom_app, colimit.ι_map_assoc, limit.cone_π,\n      lim_map_π_assoc, lim_map_π, category.assoc, currying_unit_iso_inv_app_app_app,\n      limit_iso_swap_comp_lim_hom_app, lim_map_eq_lim_map],\n    erw category.id_comp,\n  end }\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/colimit_limit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.34183660381395375}}
{"text": "example (G : Type) [group G] (g h k : G) : (g * h * k⁻¹)⁻¹ = k * h⁻¹ * g⁻¹ :=\nbegin\n  simp [mul_assoc]\nend\n", "meta": {"author": "AtnNn", "repo": "lean-sandbox", "sha": "8c68afbdc09213173aef1be195da7a9a86060a97", "save_path": "github-repos/lean/AtnNn-lean-sandbox", "path": "github-repos/lean/AtnNn-lean-sandbox/lean-sandbox-8c68afbdc09213173aef1be195da7a9a86060a97/src/xena_challenge/challenge08.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.34161396252250303}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.gluing\nimport category_theory.limits.opposites\nimport algebraic_geometry.Gamma_Spec_adjunction\n\n/-!\n# Fibred products of schemes\n\nIn this file we construct the fibred product of schemes via gluing.\nWe roughly follow [har77] Theorem 3.3.\n\nIn particular, the main construction is to show that for an open cover `{ Uᵢ }` of `X`, if there\nexist fibred products `Uᵢ ×[Z] Y` for each `i`, then there exists a fibred product `X ×[Z] Y`.\n\nThen, for constructing the fibred product for arbitrary schemes `X, Y, Z`, we can use the\nconstruction to reduce to the case where `X, Y, Z` are all affine, where fibred products are\nconstructed via tensor products.\n\n-/\nuniverses v u\nnoncomputable theory\n\nopen category_theory category_theory.limits algebraic_geometry\nnamespace algebraic_geometry.Scheme\n\nnamespace pullback\n\nvariables {C : Type u} [category.{v} C]\n\nvariables {X Y Z : Scheme.{u}} (𝒰 : open_cover.{u} X) (f : X ⟶ Z) (g : Y ⟶ Z)\nvariables [∀ i, has_pullback (𝒰.map i ≫ f) g]\n\n/-- The intersection of `Uᵢ ×[Z] Y` and `Uⱼ ×[Z] Y` is given by (Uᵢ ×[Z] Y) ×[X] Uⱼ -/\ndef V (i j : 𝒰.J) : Scheme :=\npullback ((pullback.fst : pullback ((𝒰.map i) ≫ f) g ⟶ _) ≫ (𝒰.map i)) (𝒰.map j)\n\n/-- The canonical transition map `(Uᵢ ×[Z] Y) ×[X] Uⱼ ⟶ (Uⱼ ×[Z] Y) ×[X] Uᵢ` given by the fact\nthat pullbacks are associative and symmetric. -/\ndef t (i j : 𝒰.J) : V 𝒰 f g i j ⟶ V 𝒰 f g j i :=\nbegin\n  haveI : has_pullback (pullback.snd ≫ 𝒰.map i ≫ f) g :=\n    has_pullback_assoc_symm (𝒰.map j) (𝒰.map i) (𝒰.map i ≫ f) g,\n  haveI : has_pullback (pullback.snd ≫ 𝒰.map j ≫ f) g :=\n    has_pullback_assoc_symm (𝒰.map i) (𝒰.map j) (𝒰.map j ≫ f) g,\n  refine (pullback_symmetry _ _).hom ≫ _,\n  refine (pullback_assoc _ _ _ _).inv ≫ _,\n  change pullback _ _ ⟶ pullback _ _,\n  refine _ ≫ (pullback_symmetry _ _).hom,\n  refine _ ≫ (pullback_assoc _ _ _ _).hom,\n  refine pullback.map _ _ _ _ (pullback_symmetry _ _).hom (𝟙 _) (𝟙 _) _ _,\n  rw [pullback_symmetry_hom_comp_snd_assoc, pullback.condition_assoc, category.comp_id],\n  rw [category.comp_id, category.id_comp]\nend\n\n@[simp, reassoc]\nlemma t_fst_fst (i j : 𝒰.J) : t 𝒰 f g i j ≫ pullback.fst ≫ pullback.fst = pullback.snd :=\nby { delta t, simp }\n\n@[simp, reassoc]\nlemma t_fst_snd (i j : 𝒰.J) :\n  t 𝒰 f g i j ≫ pullback.fst ≫ pullback.snd = pullback.fst ≫ pullback.snd :=\nby { delta t, simp }\n\n@[simp, reassoc]\nlemma t_snd (i j : 𝒰.J) :\n  t 𝒰 f g i j ≫ pullback.snd = pullback.fst ≫ pullback.fst :=\nby { delta t, simp }\n\nlemma t_id (i : 𝒰.J) : t 𝒰 f g i i = 𝟙 _ :=\nbegin\n  apply pullback.hom_ext; rw category.id_comp,\n  apply pullback.hom_ext,\n  { rw ← cancel_mono (𝒰.map i), simp [pullback.condition] },\n  { simp },\n  { rw ← cancel_mono (𝒰.map i), simp [pullback.condition] }\nend\n\n/-- The inclusion map of `V i j = (Uᵢ ×[Z] Y) ×[X] Uⱼ ⟶ Uᵢ ×[Z] Y`-/\nabbreviation fV (i j : 𝒰.J) : V 𝒰 f g i j ⟶ pullback ((𝒰.map i) ≫ f) g := pullback.fst\n\n/-- The map `((Xᵢ ×[Z] Y) ×[X] Xⱼ) ×[Xᵢ ×[Z] Y] ((Xᵢ ×[Z] Y) ×[X] Xₖ)` ⟶\n  `((Xⱼ ×[Z] Y) ×[X] Xₖ) ×[Xⱼ ×[Z] Y] ((Xⱼ ×[Z] Y) ×[X] Xᵢ)` needed for gluing   -/\ndef t' (i j k : 𝒰.J) :\n  pullback (fV 𝒰 f g i j) (fV 𝒰 f g i k) ⟶ pullback (fV 𝒰 f g j k) (fV 𝒰 f g j i) :=\nbegin\n  refine (pullback_right_pullback_fst_iso _ _ _).hom ≫ _,\n  refine _ ≫ (pullback_symmetry _ _).hom,\n  refine _ ≫ (pullback_right_pullback_fst_iso _ _ _).inv,\n  refine pullback.map _ _ _ _ (t 𝒰 f g i j) (𝟙 _) (𝟙 _) _ _,\n  { simp [← pullback.condition] },\n  { simp }\nend\n\nsection end\n\n@[simp, reassoc]\nlemma t'_fst_fst_fst (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd :=\nby { delta t', simp }\n\n@[simp, reassoc]\nlemma t'_fst_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.snd =\n    pullback.fst ≫ pullback.fst ≫ pullback.snd :=\nby { delta t', simp }\n\n@[simp, reassoc]\nlemma t'_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.snd = pullback.snd ≫ pullback.snd :=\nby { delta t', simp }\n\n@[simp, reassoc]\nlemma t'_snd_fst_fst (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd :=\nby { delta t', simp }\n\n@[simp, reassoc]\nlemma t'_snd_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.fst ≫ pullback.snd =\n    pullback.fst ≫ pullback.fst ≫ pullback.snd :=\nby { delta t', simp }\n\n@[simp, reassoc]\nlemma t'_snd_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.snd = pullback.fst ≫ pullback.fst ≫ pullback.fst :=\nby { delta t', simp, }\n\nlemma cocycle_fst_fst_fst (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.fst ≫\n  pullback.fst = pullback.fst ≫ pullback.fst ≫ pullback.fst :=\nby simp\n\nlemma cocycle_fst_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.fst ≫\n  pullback.snd = pullback.fst ≫ pullback.fst ≫ pullback.snd :=\nby simp\n\nlemma cocycle_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.snd =\n    pullback.fst ≫ pullback.snd :=\nby simp\n\nlemma cocycle_snd_fst_fst (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫\n  pullback.fst = pullback.snd ≫ pullback.fst ≫ pullback.fst :=\nby { rw ← cancel_mono (𝒰.map i), simp [pullback.condition_assoc, pullback.condition] }\n\nlemma cocycle_snd_fst_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.fst ≫\n  pullback.snd = pullback.snd ≫ pullback.fst ≫ pullback.snd :=\nby { simp [pullback.condition_assoc, pullback.condition] }\n\nlemma cocycle_snd_snd (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd ≫ pullback.snd =\n    pullback.snd ≫ pullback.snd :=\nby simp\n\n-- `by tidy` should solve it, but it times out.\nlemma cocycle (i j k : 𝒰.J) :\n  t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j = 𝟙 _ :=\nbegin\n  apply pullback.hom_ext; rw category.id_comp,\n  { apply pullback.hom_ext,\n    { apply pullback.hom_ext,\n      { simp_rw category.assoc,\n        exact cocycle_fst_fst_fst 𝒰 f g i j k },\n      { simp_rw category.assoc,\n        exact cocycle_fst_fst_snd 𝒰 f g i j k } },\n    { simp_rw category.assoc,\n      exact cocycle_fst_snd 𝒰 f g i j k } },\n  { apply pullback.hom_ext,\n    { apply pullback.hom_ext,\n      { simp_rw category.assoc,\n        exact cocycle_snd_fst_fst 𝒰 f g i j k },\n      { simp_rw category.assoc,\n        exact cocycle_snd_fst_snd 𝒰 f g i j k } },\n    { simp_rw category.assoc,\n      exact cocycle_snd_snd 𝒰 f g i j k } }\nend\n\n/-- Given `Uᵢ ×[Z] Y`, this is the glued fibered product `X ×[Z] Y`. -/\n@[simps]\ndef gluing : Scheme.glue_data.{u} :=\n{ J := 𝒰.J,\n  U := λ i, pullback ((𝒰.map i) ≫ f) g,\n  V := λ ⟨i, j⟩, V 𝒰 f g i j, -- `p⁻¹(Uᵢ ∩ Uⱼ)` where `p : Uᵢ ×[Z] Y ⟶ Uᵢ ⟶ X`.\n  f := λ i j, pullback.fst,\n  f_id := λ i, infer_instance,\n  f_open := infer_instance,\n  t := λ i j, t 𝒰 f g i j,\n  t_id := λ i, t_id 𝒰 f g i,\n  t' := λ i j k, t' 𝒰 f g i j k,\n  t_fac := λ i j k, begin\n    apply pullback.hom_ext,\n    apply pullback.hom_ext,\n    all_goals { simp }\n  end,\n  cocycle := λ i j k, cocycle 𝒰 f g i j k }\n\n/-- The first projection from the glued scheme into `X`. -/\ndef p1 : (gluing 𝒰 f g).glued ⟶ X :=\nbegin\n  fapply multicoequalizer.desc,\n  exact λ i, pullback.fst ≫ 𝒰.map i,\n  rintro ⟨i, j⟩,\n  change pullback.fst ≫ _ ≫ 𝒰.map i = (_ ≫ _) ≫ _ ≫ 𝒰.map j,\n  rw pullback.condition,\n  rw ← category.assoc,\n  congr' 1,\n  rw category.assoc,\n  exact (t_fst_fst _ _ _ _ _).symm\nend\n\n/-- The second projection from the glued scheme into `Y`. -/\ndef p2 : (gluing 𝒰 f g).glued ⟶ Y :=\nbegin\n  fapply multicoequalizer.desc,\n  exact λ i, pullback.snd,\n  rintro ⟨i, j⟩,\n  change pullback.fst ≫ _ = (_ ≫ _) ≫ _,\n  rw category.assoc,\n  exact (t_fst_snd _ _ _ _ _).symm\nend\n\nlemma p_comm : p1 𝒰 f g ≫ f = p2 𝒰 f g ≫ g :=\nbegin\n  apply multicoequalizer.hom_ext,\n  intro i,\n  erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],\n  rw [category.assoc, pullback.condition]\nend\n\nvariable (s : pullback_cone f g)\n\n/-- (Implementation)\nThe canonical map `(s.X ×[X] Uᵢ) ×[s.X] (s.X ×[X] Uⱼ) ⟶ (Uᵢ ×[Z] Y) ×[X] Uⱼ`\n\nThis is used in `glued_lift`. -/\ndef glued_lift_pullback_map (i j : 𝒰.J) :\n  pullback ((𝒰.pullback_cover s.fst).map i) ((𝒰.pullback_cover s.fst).map j) ⟶\n    (gluing 𝒰 f g).V ⟨i, j⟩ :=\nbegin\n  change pullback pullback.fst pullback.fst ⟶ pullback _ _,\n  refine (pullback_right_pullback_fst_iso _ _ _).hom ≫ _,\n  refine pullback.map _ _ _ _ _ (𝟙 _) (𝟙 _) _ _,\n  { exact (pullback_symmetry _ _).hom ≫\n      pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition },\n  { simpa using pullback.condition },\n  { simp }\nend\n\n@[reassoc]\nlemma glued_lift_pullback_map_fst (i j : 𝒰.J) :\n  glued_lift_pullback_map 𝒰 f g s i j ≫ pullback.fst = pullback.fst ≫\n    (pullback_symmetry _ _).hom ≫\n      pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition :=\nby { delta glued_lift_pullback_map, simp }\n\n@[reassoc]\nlemma glued_lift_pullback_map_snd (i j : 𝒰.J) :\n  glued_lift_pullback_map 𝒰 f g s i j ≫ pullback.snd = pullback.snd ≫ pullback.snd :=\nby { delta glued_lift_pullback_map, simp }\n\n/--\nThe lifted map `s.X ⟶ (gluing 𝒰 f g).glued` in order to show that `(gluing 𝒰 f g).glued` is\nindeed the pullback.\n\nGiven a pullback cone `s`, we have the maps `s.fst ⁻¹' Uᵢ ⟶ Uᵢ` and\n`s.fst ⁻¹' Uᵢ ⟶ s.X ⟶ Y` that we may lift to a map `s.fst ⁻¹' Uᵢ ⟶ Uᵢ ×[Z] Y`.\n\nto glue these into a map `s.X ⟶ Uᵢ ×[Z] Y`, we need to show that the maps agree on\n`(s.fst ⁻¹' Uᵢ) ×[s.X] (s.fst ⁻¹' Uⱼ) ⟶ Uᵢ ×[Z] Y`. This is achieved by showing that both of these\nmaps factors through `glued_lift_pullback_map`.\n-/\ndef glued_lift : s.X ⟶ (gluing 𝒰 f g).glued :=\nbegin\n  fapply (𝒰.pullback_cover s.fst).glue_morphisms,\n  { exact λ i, (pullback_symmetry _ _).hom ≫\n      pullback.map _ _ _ _ (𝟙 _) s.snd f (category.id_comp _).symm s.condition ≫\n      (gluing 𝒰 f g).ι i },\n  intros i j,\n  rw ← glued_lift_pullback_map_fst_assoc,\n  have : _ = pullback.fst ≫ _ := (gluing 𝒰 f g).glue_condition i j,\n  rw [← this, gluing_to_glue_data_t, gluing_to_glue_data_f],\n  simp_rw ← category.assoc,\n  congr' 1,\n  apply pullback.hom_ext; simp_rw category.assoc,\n  { rw [t_fst_fst, glued_lift_pullback_map_snd],\n    congr' 1,\n    rw [← iso.inv_comp_eq, pullback_symmetry_inv_comp_snd],\n    erw pullback.lift_fst,\n    rw category.comp_id },\n  { rw [t_fst_snd, glued_lift_pullback_map_fst_assoc],\n    erw [pullback.lift_snd, pullback.lift_snd],\n    rw [pullback_symmetry_hom_comp_snd_assoc, pullback_symmetry_hom_comp_snd_assoc],\n    exact pullback.condition_assoc _ }\nend\n\nlemma glued_lift_p1 : glued_lift 𝒰 f g s ≫ p1 𝒰 f g = s.fst :=\nbegin\n  rw ← cancel_epi (𝒰.pullback_cover s.fst).from_glued,\n  apply multicoequalizer.hom_ext,\n  intro b,\n  erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],\n  delta glued_lift,\n  simp_rw ← category.assoc,\n  rw (𝒰.pullback_cover s.fst).ι_glue_morphisms,\n  simp_rw category.assoc,\n  erw [multicoequalizer.π_desc, pullback.lift_fst_assoc, pullback.condition, category.comp_id],\n  rw pullback_symmetry_hom_comp_fst_assoc,\nend\n\nlemma glued_lift_p2 : glued_lift 𝒰 f g s ≫ p2 𝒰 f g = s.snd :=\nbegin\n  rw ← cancel_epi (𝒰.pullback_cover s.fst).from_glued,\n  apply multicoequalizer.hom_ext,\n  intro b,\n  erw [multicoequalizer.π_desc_assoc, multicoequalizer.π_desc_assoc],\n  delta glued_lift,\n  simp_rw ← category.assoc,\n  rw (𝒰.pullback_cover s.fst).ι_glue_morphisms,\n  simp_rw category.assoc,\n  erw [multicoequalizer.π_desc, pullback.lift_snd],\n  rw pullback_symmetry_hom_comp_snd_assoc,\n  refl\nend\n\n/-- (Implementation)\nThe canonical map `(W ×[X] Uᵢ) ×[W] (Uⱼ ×[Z] Y) ⟶ (Uⱼ ×[Z] Y) ×[X] Uᵢ = V j i` where `W` is\nthe glued fibred product.\n\nThis is used in `lift_comp_ι`. -/\ndef pullback_fst_ι_to_V (i j : 𝒰.J) :\n  pullback (pullback.fst : pullback (p1 𝒰 f g) (𝒰.map i) ⟶ _) ((gluing 𝒰 f g).ι j) ⟶\n    V 𝒰 f g j i :=\n(pullback_symmetry _ _ ≪≫\n  (pullback_right_pullback_fst_iso (p1 𝒰 f g) (𝒰.map i) _)).hom ≫\n    (pullback.congr_hom (multicoequalizer.π_desc _ _ _ _ _) rfl).hom\n\n@[simp, reassoc] lemma pullback_fst_ι_to_V_fst (i j : 𝒰.J) :\n  pullback_fst_ι_to_V 𝒰 f g i j ≫ pullback.fst = pullback.snd :=\nby { delta pullback_fst_ι_to_V, simp }\n\n@[simp, reassoc] lemma pullback_fst_ι_to_V_snd (i j : 𝒰.J) :\n  pullback_fst_ι_to_V 𝒰 f g i j ≫ pullback.snd = pullback.fst ≫ pullback.snd :=\nby { delta pullback_fst_ι_to_V, simp }\n\n/-- We show that the map `W ×[X] Uᵢ ⟶ Uᵢ ×[Z] Y ⟶ W` is the first projection, where the\nfirst map is given by the lift of `W ×[X] Uᵢ ⟶ Uᵢ` and `W ×[X] Uᵢ ⟶ W ⟶ Y`.\n\nIt suffices to show that the two map agrees when restricted onto `Uⱼ ×[Z] Y`. In this case,\nboth maps factor through `V j i` via `pullback_fst_ι_to_V` -/\nlemma lift_comp_ι (i : 𝒰.J) : pullback.lift pullback.snd (pullback.fst ≫ p2 𝒰 f g)\n  (by rw [← pullback.condition_assoc, category.assoc, p_comm]) ≫\n  (gluing 𝒰 f g).ι i = (pullback.fst : pullback (p1 𝒰 f g) (𝒰.map i) ⟶ _) :=\nbegin\n  apply ((gluing 𝒰 f g).open_cover.pullback_cover pullback.fst).hom_ext,\n  intro j,\n  dsimp only [open_cover.pullback_cover],\n  transitivity pullback_fst_ι_to_V 𝒰 f g i j ≫ fV 𝒰 f g j i ≫ (gluing 𝒰 f g).ι _,\n  { rw ← (show _ = fV 𝒰 f g j i ≫ _, from (gluing 𝒰 f g).glue_condition j i),\n    simp_rw ← category.assoc,\n    congr' 1,\n    rw [gluing_to_glue_data_f, gluing_to_glue_data_t],\n    apply pullback.hom_ext; simp_rw category.assoc,\n    { rw [t_fst_fst, pullback.lift_fst, pullback_fst_ι_to_V_snd] },\n    { rw [t_fst_snd, pullback.lift_snd, pullback_fst_ι_to_V_fst_assoc,\n        pullback.condition_assoc], erw multicoequalizer.π_desc } },\n  { rw [pullback.condition, ← category.assoc],\n    congr' 1,\n    apply pullback.hom_ext,\n    { simp },\n    { simp } }\nend\n\n/-- The canonical isomorphism between `W ×[X] Uᵢ` and `Uᵢ ×[X] Y`. That is, the preimage of `Uᵢ` in\n`W` along `p1` is indeed `Uᵢ ×[X] Y`. -/\ndef pullback_p1_iso (i : 𝒰.J) :\n  pullback (p1 𝒰 f g) (𝒰.map i) ≅ pullback (𝒰.map i ≫ f) g :=\nbegin\n  fsplit,\n  exact pullback.lift pullback.snd (pullback.fst ≫ p2 𝒰 f g)\n    (by rw [← pullback.condition_assoc, category.assoc, p_comm]),\n  refine pullback.lift ((gluing 𝒰 f g).ι i) pullback.fst\n    (by erw multicoequalizer.π_desc),\n  { apply pullback.hom_ext,\n    { simpa using lift_comp_ι 𝒰 f g i },\n    { simp } },\n  { apply pullback.hom_ext,\n    { simp },\n    { simp, erw multicoequalizer.π_desc } },\nend\n\n@[simp, reassoc] lemma pullback_p1_iso_hom_fst (i : 𝒰.J) :\n  (pullback_p1_iso 𝒰 f g i).hom ≫ pullback.fst = pullback.snd :=\nby { delta pullback_p1_iso, simp }\n\n@[simp, reassoc] lemma pullback_p1_iso_hom_snd (i : 𝒰.J) :\n  (pullback_p1_iso 𝒰 f g i).hom ≫ pullback.snd = pullback.fst ≫ p2 𝒰 f g :=\nby { delta pullback_p1_iso, simp }\n\n@[simp, reassoc] lemma pullback_p1_iso_inv_fst (i : 𝒰.J) :\n  (pullback_p1_iso 𝒰 f g i).inv ≫ pullback.fst = (gluing 𝒰 f g).ι i :=\nby { delta pullback_p1_iso, simp }\n\n@[simp, reassoc] lemma pullback_p1_iso_inv_snd (i : 𝒰.J) :\n  (pullback_p1_iso 𝒰 f g i).inv ≫ pullback.snd = pullback.fst :=\nby { delta pullback_p1_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_p1_iso_hom_ι (i : 𝒰.J) :\n  (pullback_p1_iso 𝒰 f g i).hom ≫ (gluing 𝒰 f g).ι i = pullback.fst :=\nby rw [← pullback_p1_iso_inv_fst, iso.hom_inv_id_assoc]\n\n/-- The glued scheme (`(gluing 𝒰 f g).glued`) is indeed the pullback of `f` and `g`. -/\ndef glued_is_limit : is_limit (pullback_cone.mk _ _ (p_comm 𝒰 f g)) :=\nbegin\n  apply pullback_cone.is_limit_aux',\n  intro s,\n  refine ⟨glued_lift 𝒰 f g s, glued_lift_p1 𝒰 f g s, glued_lift_p2 𝒰 f g s, _⟩,\n  intros m h₁ h₂,\n  change m ≫ p1 𝒰 f g = _ at h₁,\n  change m ≫ p2 𝒰 f g = _ at h₂,\n  apply (𝒰.pullback_cover s.fst).hom_ext,\n  intro i,\n  rw open_cover.pullback_cover_map,\n  have := pullback_right_pullback_fst_iso (p1 𝒰 f g) (𝒰.map i) m\n    ≪≫ pullback.congr_hom h₁ rfl,\n  erw (𝒰.pullback_cover s.fst).ι_glue_morphisms,\n  rw [← cancel_epi (pullback_right_pullback_fst_iso (p1 𝒰 f g) (𝒰.map i) m\n    ≪≫ pullback.congr_hom h₁ rfl).hom, iso.trans_hom, category.assoc, pullback.congr_hom_hom,\n    pullback.lift_fst_assoc, category.comp_id, pullback_right_pullback_fst_iso_hom_fst_assoc,\n    pullback.condition],\n  transitivity pullback.snd ≫ (pullback_p1_iso 𝒰 f g _).hom ≫ (gluing 𝒰 f g).ι _,\n  { congr' 1, rw ← pullback_p1_iso_hom_ι },\n  simp_rw ← category.assoc,\n  congr' 1,\n  apply pullback.hom_ext,\n  { simp only [category.comp_id, pullback_right_pullback_fst_iso_hom_snd, category.assoc,\n      pullback_p1_iso_hom_fst, pullback.lift_snd, pullback.lift_fst,\n      pullback_symmetry_hom_comp_fst] },\n  { simp only [category.comp_id, pullback_right_pullback_fst_iso_hom_fst_assoc,\n    pullback_p1_iso_hom_snd, category.assoc, pullback.lift_fst_assoc,\n    pullback_symmetry_hom_comp_snd_assoc, pullback.lift_snd],\n    rw [← pullback.condition_assoc, h₂] }\nend\n\nlemma has_pullback_of_cover : has_pullback f g := ⟨⟨⟨_, glued_is_limit 𝒰 f g⟩⟩⟩\n\ninstance : has_limits CommRingᵒᵖ := has_limits_op_of_has_colimits\n\ninstance affine_has_pullback {A B C : CommRing}\n  (f : Spec.obj (opposite.op A) ⟶ Spec.obj (opposite.op C))\n  (g : Spec.obj (opposite.op B) ⟶ Spec.obj (opposite.op C)) : has_pullback f g :=\nbegin\n  rw [← Spec.image_preimage f, ← Spec.image_preimage g],\n  exact ⟨⟨⟨_,is_limit_of_has_pullback_of_preserves_limit\n    Spec (Spec.preimage f) (Spec.preimage g)⟩⟩⟩\nend\n\n\n\ninstance base_affine_has_pullback {C : CommRing} {X Y : Scheme}\n  (f : X ⟶ Spec.obj (opposite.op C))\n  (g : Y ⟶ Spec.obj (opposite.op C)) : has_pullback f g :=\n@@has_pullback_symmetry _ _ _\n  (@@has_pullback_of_cover Y.affine_cover g f\n    (λ i, @@has_pullback_symmetry _ _ _ $ affine_affine_has_pullback _ _))\n\ninstance left_affine_comp_pullback_has_pullback {X Y Z : Scheme}\n  (f : X ⟶ Z) (g : Y ⟶ Z) (i : Z.affine_cover.J) :\n    has_pullback ((Z.affine_cover.pullback_cover f).map i ≫ f) g :=\nbegin\n  let Xᵢ := pullback f (Z.affine_cover.map i),\n  let Yᵢ := pullback g (Z.affine_cover.map i),\n  let W := pullback (pullback.snd : Yᵢ ⟶ _) (pullback.snd : Xᵢ ⟶ _),\n  have := big_square_is_pullback (pullback.fst : W ⟶ _) (pullback.fst : Yᵢ ⟶ _)\n    (pullback.snd : Xᵢ ⟶ _) (Z.affine_cover.map i) pullback.snd pullback.snd g\n    pullback.condition.symm pullback.condition.symm\n      (pullback_cone.flip_is_limit $ pullback_is_pullback _ _)\n      (pullback_cone.flip_is_limit $ pullback_is_pullback _ _),\n  have : has_pullback (pullback.snd ≫ Z.affine_cover.map i : Xᵢ ⟶ _) g :=\n    ⟨⟨⟨_,this⟩⟩⟩,\n  rw ← pullback.condition at this,\n  exact this,\nend\n\ninstance {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) : has_pullback f g :=\nhas_pullback_of_cover (Z.affine_cover.pullback_cover f) f g\n\ninstance : has_pullbacks Scheme := has_pullbacks_of_has_limit_cospan _\n\n/-- Given an open cover `{ Xᵢ }` of `X`, then `X ×[Z] Y` is covered by `Xᵢ ×[Z] Y`. -/\n@[simps J obj map]\ndef open_cover_of_left (𝒰 : open_cover X) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) :=\nbegin\n  fapply ((gluing 𝒰 f g).open_cover.pushforward_iso\n    (limit.iso_limit_cone ⟨_, glued_is_limit 𝒰 f g⟩).inv).copy 𝒰.J\n    (λ i, pullback (𝒰.map i ≫ f) g)\n    (λ i, pullback.map _ _ _ _ (𝒰.map i) (𝟙 _) (𝟙 _) (category.comp_id _) (by simp))\n    (equiv.refl 𝒰.J) (λ _, iso.refl _),\n  rintro (i : 𝒰.J),\n  change pullback.map _ _ _ _ _ _ _ _ _ = 𝟙 _ ≫ (gluing 𝒰 f g).ι i ≫ _,\n  refine eq.trans _ (category.id_comp _).symm,\n  apply pullback.hom_ext,\n  all_goals\n  { dsimp,\n    simp only [limit.iso_limit_cone_inv_π, pullback_cone.mk_π_app_left, category.comp_id,\n      pullback_cone.mk_π_app_right, category.assoc, pullback.lift_fst, pullback.lift_snd],\n    symmetry,\n    exact multicoequalizer.π_desc _ _ _ _ _ },\nend\n\n/-- Given an open cover `{ Yᵢ }` of `Y`, then `X ×[Z] Y` is covered by `X ×[Z] Yᵢ`. -/\n@[simps J obj map]\ndef open_cover_of_right (𝒰 : open_cover Y) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) :=\nbegin\n  fapply ((open_cover_of_left 𝒰 g f).pushforward_iso (pullback_symmetry _ _).hom).copy 𝒰.J\n    (λ i, pullback f (𝒰.map i ≫ g))\n    (λ i, pullback.map _ _ _ _ (𝟙 _) (𝒰.map i) (𝟙 _) (by simp) (category.comp_id _))\n    (equiv.refl _) (λ i, pullback_symmetry _ _),\n  intro i,\n  dsimp [open_cover.bind],\n  apply pullback.hom_ext; simp,\nend\n\n/-- (Implementation). Use `open_cover_of_base` instead. -/\ndef open_cover_of_base' (𝒰 : open_cover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) :=\nbegin\n  apply (open_cover_of_left (𝒰.pullback_cover f) f g).bind,\n  intro i,\n  let Xᵢ := pullback f (𝒰.map i),\n  let Yᵢ := pullback g (𝒰.map i),\n  let W := pullback (pullback.snd : Yᵢ ⟶ _) (pullback.snd : Xᵢ ⟶ _),\n  have := big_square_is_pullback (pullback.fst : W ⟶ _) (pullback.fst : Yᵢ ⟶ _)\n    (pullback.snd : Xᵢ ⟶ _) (𝒰.map i) pullback.snd pullback.snd g\n    pullback.condition.symm pullback.condition.symm\n      (pullback_cone.flip_is_limit $ pullback_is_pullback _ _)\n      (pullback_cone.flip_is_limit $ pullback_is_pullback _ _),\n  refine open_cover_of_is_iso\n    ((pullback_symmetry _ _).hom ≫ (limit.iso_limit_cone ⟨_, this⟩).inv ≫\n      pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) _ _),\n  { simpa only [category.comp_id, category.id_comp, ← pullback.condition] },\n  { simp only [category.comp_id, category.id_comp] },\n  apply_instance\nend\n\n/-- Given an open cover `{ Zᵢ }` of `Z`, then `X ×[Z] Y` is covered by `Xᵢ ×[Zᵢ] Yᵢ`, where\n  `Xᵢ = X ×[Z] Zᵢ` and `Yᵢ = Y ×[Z] Zᵢ` is the preimage of `Zᵢ` in `X` and `Y`. -/\n@[simps J obj map]\ndef open_cover_of_base (𝒰 : open_cover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : open_cover (pullback f g) :=\nbegin\n  apply (open_cover_of_base' 𝒰 f g).copy\n    𝒰.J\n    (λ i, pullback (pullback.snd : pullback f (𝒰.map i) ⟶ _)\n      (pullback.snd : pullback g (𝒰.map i) ⟶ _))\n    (λ i, pullback.map _ _ _ _ pullback.fst pullback.fst (𝒰.map i)\n      pullback.condition.symm pullback.condition.symm)\n    ((equiv.prod_punit 𝒰.J).symm.trans (equiv.sigma_equiv_prod 𝒰.J punit).symm)\n    (λ _, iso.refl _),\n  intro i,\n  change _ = _ ≫ _ ≫ _,\n  refine eq.trans _ (category.id_comp _).symm,\n  apply pullback.hom_ext; simp only [category.comp_id, open_cover_of_left_map,\n    open_cover.pullback_cover_map, pullback_cone.mk_π_app_left, open_cover_of_is_iso_map,\n    limit.iso_limit_cone_inv_π_assoc, category.assoc, pullback.lift_fst_assoc,\n    pullback_symmetry_hom_comp_snd_assoc, pullback.lift_fst, limit.iso_limit_cone_inv_π,\n    pullback_cone.mk_π_app_right, pullback_symmetry_hom_comp_fst_assoc, pullback.lift_snd],\nend\n\nend pullback\n\nend algebraic_geometry.Scheme\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebraic_geometry/pullbacks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.34161394654515176}}
{"text": "/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\n\nimport tactic.interactive\nimport tactic.norm_num\n\n/-!\n# `field_simp` tactic\n\nTactic to clear denominators in algebraic expressions, based on `simp` with a specific simpset.\n-/\n\nnamespace tactic\n\n/-- Try to prove a goal of the form `x ≠ 0` by calling `assumption`, or `norm_num1` if `x` is\na numeral. -/\nmeta def field_simp.ne_zero : tactic unit := do\n  goal ← tactic.target,\n  match goal with\n  | `(%%e ≠ 0) := assumption <|> do n ← e.to_rat, `[norm_num1]\n  | _ := tactic.fail \"goal should be of the form `x ≠ 0`\"\n  end\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nThe goal of `field_simp` is to reduce an expression in a field to an expression of the form `n / d`\nwhere neither `n` nor `d` contains any division symbol, just using the simplifier (with a carefully\ncrafted simpset named `field_simps`) to reduce the number of division symbols whenever possible by\niterating the following steps:\n\n- write an inverse as a division\n- in any product, move the division to the right\n- if there are several divisions in a product, group them together at the end and write them as a\n  single division\n- reduce a sum to a common denominator\n\nIf the goal is an equality, this simpset will also clear the denominators, so that the proof\ncan normally be concluded by an application of `ring` or `ring_exp`.\n\n`field_simp [hx, hy]` is a short form for\n`simp [-one_div, -mul_eq_zero, hx, hy] with field_simps {discharger := [field_simp.ne_zero]}`\n\nNote that this naive algorithm will not try to detect common factors in denominators to reduce the\ncomplexity of the resulting expression. Instead, it relies on the ability of `ring` to handle\ncomplicated expressions in the next step.\n\nAs always with the simplifier, reduction steps will only be applied if the preconditions of the\nlemmas can be checked. This means that proofs that denominators are nonzero should be included. The\nfact that a product is nonzero when all factors are, and that a power of a nonzero number is\nnonzero, are included in the simpset, but more complicated assertions (especially dealing with sums)\nshould be given explicitly. If your expression is not completely reduced by the simplifier\ninvocation, check the denominators of the resulting expression and provide proofs that they are\nnonzero to enable further progress.\n\nTo check that denominators are nonzero, `field_simp` will look for facts in the context, and\nwill try to apply `norm_num` to close numerical goals.\n\nThe invocation of `field_simp` removes the lemma `one_div` from the simpset, as this lemma\nworks against the algorithm explained above. It also removes\n`mul_eq_zero : x * y = 0 ↔ x = 0 ∨ y = 0`, as `norm_num` can not work on disjunctions to\nclose goals of the form `24 ≠ 0`, and replaces it with `mul_ne_zero : x ≠ 0 → y ≠ 0 → x * y ≠ 0`\ncreating two goals instead of a disjunction.\n\nFor example,\n```lean\nexample (a b c d x y : ℂ) (hx : x ≠ 0) (hy : y ≠ 0) :\n  a + b / x + c / x^2 + d / x^3 = a + x⁻¹ * (y * b / y + (d / x + c) / x) :=\nbegin\n  field_simp,\n  ring\nend\n```\n\nSee also the `cancel_denoms` tactic, which tries to do a similar simplification for expressions\nthat have numerals in denominators.\nThe tactics are not related: `cancel_denoms` will only handle numeric denominators, and will try to\nentirely remove (numeric) division from the expression by multiplying by a factor.\n-/\nmeta def field_simp (no_dflt : parse only_flag) (hs : parse simp_arg_list)\n  (attr_names : parse with_ident_list)\n  (locat : parse location)\n  (cfg : simp_config_ext := {discharger := field_simp.ne_zero}) : tactic unit :=\nlet attr_names := `field_simps :: attr_names,\n    hs := simp_arg_type.except `one_div :: simp_arg_type.except `mul_eq_zero :: hs in\npropagate_tags (simp_core cfg.to_simp_config cfg.discharger no_dflt hs attr_names locat >> skip)\n\nadd_tactic_doc\n{ name       := \"field_simp\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.field_simp],\n  tags       := [\"simplification\", \"arithmetic\"] }\n\nend interactive\nend tactic\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/tactic/field_simp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.3415947113130364}}
{"text": "namespace M4R\n  /-- Class defining a membership predicate, allowing the use of `∈` notation. -/\n  class Mem (α : outParam (Type u)) (γ : Type v) where\n    mem : α → γ → Prop\n  infixl:55 \" ∈ \" => Mem.mem\n  notation:55 x:55 \" ∉ \" s:56 => ¬ (x ∈ s)\n\n  declare_syntax_cat binderterm -- notation for `a` or `a : A` or `a ∈ S`\n  syntax ident : binderterm\n  syntax ident \" ∈ \" term : binderterm\n  syntax ident \" ∉ \" term : binderterm\n\n  syntax \"∀ \" binderterm \", \" term : term\n  syntax \"∃ \" binderterm \", \" term : term\n\n  macro_rules\n  -- ∀ x ∈ s, p := ∀ x, x ∈ s → p\n  | `(∀ $x:ident ∈ $s, $p) => `(∀ $x:ident, $x ∈ $s → $p)\n  -- ∃ x ∈ s, p := ∃ x, x ∈ s ∧ p\n  | `(∃ $x:ident ∈ $s, $p) => `(∃ $x:ident, $x ∈ $s ∧ $p)\n  -- ∀ x ∉ s, p := ∀ x, x ∉ s → p\n  | `(∀ $x:ident ∉ $s, $p) => `(∀ $x:ident, $x ∉ $s → $p)\n  -- ∃ x ∉ s, p := ∃ x, x ∉ s ∧ p\n  | `(∃ $x:ident ∉ $s, $p) => `(∃ $x:ident, $x ∉ $s ∧ $p)\n\n  /-- Class defining a subset predicate, allowing the use of `⊆` notation. -/\n  class Subset (α : Type u) where\n    subset : α → α → Prop\n  infix:50 \" ⊆ \" => Subset.subset\n\n  /-- Class defining a propert subset predicate, allowing the use of `⊊` notation. -/\n  class ProperSubset (α : Type u) where\n    propersubset : α → α → Prop\n  infix:50 \" ⊊ \" => ProperSubset.propersubset\n\n  /-- Class defining a not subset predicate, allowing the use of `⊈` notation. -/\n  class NotSubset (α : Type u) where\n     notsubset : α → α → Prop\n  infix:50 \" ⊈ \" => NotSubset.notsubset\n\n  /-- Class defining a union operation, allowing the use of `∪` notation. -/\n  class Union (α : Type u) where\n    union : α → α → α\n  infixl:65 \" ∪ \" => Union.union\n\n  /-- Class defining an intersection operation, allowing the use of `∩` notation. -/\n  class Intersection (α : Type u) where\n    intersection : α → α → α\n  infixl:70 \" ∩ \" => Intersection.intersection\n\n  /-- Class defining a set difference operation, allowing the use of `∖` notation. -/\n  class SetMinus (α : Type u) where\n    setminus : α → α → α\n  infix:70 \" ∖ \" => SetMinus.setminus\n\n  /-- Class defining a complement operation, allowing the use of `ᶜ` notation. -/\n  class Complement (α : Type u) where\n    complement : α → α\n  postfix:max \" ᶜ \" => Complement.complement\n\n  /- Notation for sets -/\n  syntax \"{ \" ident \" | \" term \" }\" : term\n  syntax \"{ \" ident \":\" term \" | \" term \" }\" : term\n  syntax \"{ \" ident \"∈\" term \" | \" term \" }\" : term\n  macro_rules\n  -- {a : A | p a}\n  | `({ $x:ident : $t | $p }) => `(fun ($x:ident : $t) => $p)\n  -- {a | p a}\n  | `({ $x:ident | $p }) => `(fun ($x:ident) => $p)\n  -- {a ∈ s | p a} := {a | a ∈ s ∧ p a}\n  | `({ $x:ident ∈ $s | $p }) => `(fun $x => $x ∈ $s ∧ $p)\n\n  /-- Class defining an inverse operation, allowing the use of `⁻¹` notation. -/\n  class Inv (α : Type u) where\n    inv : α → α\n  postfix:max \" ⁻¹ \" => Inv.inv\n\n  /-- Class defining a zero element, represented by the notation `0`. -/\n  class Zero (α : Type _) where\n    zero : α\n  /-- Class defining a one element, represented by the notation `1`. -/\n  class One  (α : Type _) where\n    one  : α\n  /-- The natural literal `0` can be interpreted as the element `0` of `Zero α`. -/\n  instance Nat0 [Zero α] : OfNat α (nat_lit 0) where\n    ofNat := Zero.zero\n  /-- The natural literal `1` can be interpreted as the element `1` of `One α`. -/\n  instance Nat1 [One  α] : OfNat α (nat_lit 1) where\n    ofNat := One.one\n  /-- A type with a zero element is inhabited. -/\n  instance InhabitedZero [Zero α] : Inhabited α where default := 0\n  /-- A type with a one element is inhabited. -/\n  instance InhabitedOne  [One α]  : Inhabited α where default := 1\n  /-- The natural number `0` is the zero element of `Nat`. -/\n  instance NatZero : Zero Nat where zero := 0\n  /-- The natural number `1` is the one element of `Nat`. -/\n  instance NatOne  : One  Nat where one  := 1\n  /-- The integer `0` is the zero element of `Int`. -/\n  instance IntZero : Zero Int where zero := 0\n  /-- The integer `1` is the one element of `Int`. -/\n  instance IntOne  : One  Int where one  := 1\n\n  /-- Class defining a type with a zero and one element which are not equal. -/\n  class NonTrivial (α : Type _) [Zero α] [One α] where\n    one_neq_zero : (1 : α) ≠ 0\n  /-- The zero and one elements of `Nat` are not equal. -/\n  instance NatNonTrivial : NonTrivial Nat where one_neq_zero := Nat.one_ne_zero\n  /-- The zero and one elements of `Int` are not equal. -/\n  instance IntNonTrivial : NonTrivial Int where one_neq_zero := by simp\n\n  /-- Class defining a division predicate, allowing the use of `÷` notation. -/\n  class Divides (α : Type u) where\n    divides : α → α → Prop\n  infix:55 \" ÷ \" => Divides.divides\n\n  /-- Class defining a \"ring equal\" predicate, allowing the use of `≗` notation (used for associates). -/\n  class RingEq (α : Type u) where\n    ringeq : α → α → Prop\n  infix:55 \" ≗ \" => RingEq.ringeq -- ≗ written \\=o\n\n  syntax \"∑ \" term : term\n  syntax \"∏ \" term : term\n  syntax \"∑ \" term \" in \" term : term\n  syntax \"∏ \" term \" in \" term : term\n  syntax \"∑ \" ident \" in \" term \", \" term : term\n  syntax \"∏ \" ident \" in \" term \", \" term : term\n\n   macro_rules\n  -- ∑ s = s.sum\n  | `(∑ $s) => `( ($s).sum )\n  -- ∏ s := s.prod\n  | `(∏ $s) => `( ($s).prod )\n  -- ∑ f in s = s.map_sum f\n  | `(∑ $f in $s) => `( ($s).map_sum $f )\n  -- ∏ f in s := s.map_prod f\n  | `(∏ $f in $s) => `( ($s).map_prod $f )\n  -- ∑ x in s, f := s.map_sum fun x => f\n  | `(∑ $x:ident in $s, $f) => `( ($s).map_sum fun $x => $f )\n  -- ∏ x in s, f := s.map_prod fun x => f\n  | `(∏ $x:ident in $s, $f) => `( ($s).map_prod fun $x => $f )\n\n  @[simp] theorem zero_eq [z : Zero α] : z.zero    = 0     := rfl\n  @[simp] theorem one_eq  [o : One  α] : o.one     = 1     := rfl\n  @[simp] theorem neg_eq  [n : Neg  α] : n.neg x   = - x   := rfl\n  @[simp] theorem hadd_eq [a : HAdd α β γ] : a.hAdd x y = x + y := rfl\n  @[simp] theorem hsub_eq [s : HSub α β γ] : s.hSub x y = x - y := rfl\n  @[simp] theorem hmul_eq [m : HMul α β γ] : m.hMul x y = x * y := rfl\n  @[simp] theorem add_eq  [a : Add α] : a.add x y = x + y := rfl\n  @[simp] theorem sub_eq  [s : Sub α] : s.sub x y = x - y := rfl\n  @[simp] theorem mul_eq  [m : Mul α] : m.mul x y = x * y := rfl\n\nend M4R", "meta": {"author": "Hop311", "repo": "M4R", "sha": "ebd1b04af344f9737d290bf8b48b3cde35e9787b", "save_path": "github-repos/lean/Hop311-M4R", "path": "github-repos/lean/Hop311-M4R/M4R-ebd1b04af344f9737d290bf8b48b3cde35e9787b/M4R/Notation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.3415947077275112}}
{"text": "import for_mathlib.algebra.homology.trunc_ge\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category\nopen_locale zero_object\n\nvariables {C : Type*} [category C] [abelian C]\n\nnamespace category_theory.short_complex\n\nlemma quasi_iso.of_kernel_fork {C : Type*} [category C] [has_zero_morphisms C]\n  {S₁ S₂ : short_complex C} (φ : S₁ ⟶ S₂) [S₁.has_homology] [S₂.has_homology]\n  [epi φ.τ₁] (hg₁ : S₁.g = 0) (hτ₂ : is_limit (kernel_fork.of_ι φ.τ₂\n    (show φ.τ₂ ≫ S₂.g = 0, by rw [φ.comm₂₃, hg₁, zero_comp]))) :\n  short_complex.quasi_iso φ :=\nbegin\n  have w : φ.τ₂ ≫ S₂.g = 0 := by rw [φ.comm₂₃, hg₁, zero_comp],\n  have h₂ := S₂.some_left_homology_data,\n  let e : h₂.K ≅ S₁.X₂ := is_limit.cone_point_unique_up_to_iso h₂.hi hτ₂,\n  have he : e.hom ≫ φ.τ₂ = h₂.i :=\n    is_limit.cone_point_unique_up_to_iso_hom_comp h₂.hi hτ₂ walking_parallel_pair.zero,\n  have wi : e.hom ≫ S₁.g = 0 := by rw [hg₁, comp_zero],\n  let hi : is_limit (kernel_fork.of_ι e.hom wi) := kernel_fork.is_limit.of_ι _ _\n    (λ A x hx, x ≫ e.inv)\n    (λ A x hx, by rw [assoc, e.inv_hom_id, comp_id])\n    (λ A x hx b hb, by rw [← hb, assoc, e.hom_inv_id, comp_id]),\n  have comm : S₁.f ≫ e.inv = φ.τ₁ ≫ h₂.f',\n  { rw [← cancel_mono h₂.i, assoc, assoc, h₂.f'_i, ← he, e.inv_hom_id_assoc, φ.comm₁₂], },\n  have wπ : (S₁.f ≫ e.inv) ≫ h₂.π = 0,\n  { rw [comm, assoc, h₂.f'_π, comp_zero], },\n  have hπ : is_colimit (cokernel_cofork.of_π h₂.π wπ) := cokernel_cofork.is_colimit.of_π _ _\n    (λ A x hx, h₂.hπ.desc (cokernel_cofork.of_π _\n      (show h₂.f' ≫ x = 0, by rw [← cancel_epi φ.τ₁, ← reassoc_of comm, ← assoc, hx, comp_zero])))\n    (λ A x hx, cofork.is_colimit.π_desc h₂.hπ)\n    (λ A x hx b hb, by { erw [← cancel_epi h₂.π, hb, cofork.is_colimit.π_desc h₂.hπ], refl, }),\n  let h₁ : S₁.left_homology_data :=\n  { K := h₂.K,\n    H := h₂.H,\n    i := e.hom,\n    wi := wi,\n    hi := hi,\n    π := h₂.π,\n    wπ := wπ,\n    hπ := hπ, },\n  let hφ : left_homology_map_data φ h₁ h₂ :=\n  { φK := 𝟙 _,\n    φH := 𝟙 _,\n    commi' := by rw [id_comp, he], },\n  rw hφ.quasi_iso_iff,\n  dsimp,\n  apply_instance,\nend\n\nend category_theory.short_complex\n\nnamespace cochain_complex\n\nvariables (K L : cochain_complex C ℤ)\n\ndef trunc_le.X (n : ℤ) (i : ℤ) : C :=\nif n < i\n  then 0\n  else if i = n\n    then (homological_complex.short_complex_functor C (complex_shape.up ℤ) i ⋙\n      short_complex.cycles_functor C).obj K\n    else K.X i\n\nlemma trunc_le.is_zero_X (n : ℤ) (i : ℤ) (hn : n < i) :\n  is_zero (trunc_le.X K n i) :=\nbegin\n  dsimp [trunc_le.X],\n  simpa only [if_pos hn] using is_zero_zero C,\nend\n\ndef trunc_le.X_iso_X (n : ℤ) (i : ℤ) (hn : i < n) :\n  trunc_le.X K n i ≅ K.X i :=\neq_to_iso begin\n  dsimp [trunc_le.X],\n  rw [if_neg (show ¬n<i, by linarith), if_neg (show i ≠ n, by linarith)],\nend\n\ndef trunc_le.X_iso_cycles (n : ℤ) (i : ℤ) (hn : i = n) :\n  trunc_le.X K n i ≅ (K.sc' i).cycles :=\neq_to_iso begin\n  dsimp [trunc_le.X],\n  simpa only [if_neg (show ¬ n<i, by linarith), if_pos hn],\nend\n\ndef trunc_le.d (n : ℤ) (i j : ℤ) : trunc_le.X K n i ⟶ trunc_le.X K n j :=\nbegin\n  by_cases hij : i+1 = j,\n  { by_cases hj₀ : n<j,\n    { exact 0, },\n    { by_cases hn : j = n,\n      { exact ((K.sc' j).lift_cycles ((trunc_le.X_iso_X K n i (by linarith)).hom ≫ K.d i j)\n          (by tidy)) ≫ (trunc_le.X_iso_cycles K n j hn).inv, },\n      { exact (trunc_le.X_iso_X K n i (by linarith)).hom ≫ K.d i j ≫\n          (trunc_le.X_iso_X K n j (by { cases (not_lt.1 hj₀).lt_or_eq; tauto, })).inv, }, }, },\n  { exact 0, },\nend\n\ndef trunc_le.ι_f (n i : ℤ) : trunc_le.X K n i ⟶ K.X i :=\nbegin\n  by_cases hi : n < i,\n  { exact 0, },\n  { by_cases hn : i = n,\n    { exact (trunc_le.X_iso_cycles K n i hn).hom ≫ (K.sc' i).cycles_i, },\n    { exact (trunc_le.X_iso_X K n i (by { cases (not_lt.1 hi).lt_or_eq; tauto, })).hom, }, },\nend\n\ninstance (n i : ℤ) : mono (trunc_le.ι_f K n i) :=\nbegin\n  dsimp [trunc_le.ι_f],\n  by_cases h₀ : n < i,\n  { rw dif_pos h₀,\n    constructor,\n    intros Z f₁ f₂ eq,\n    apply (trunc_le.is_zero_X K n i h₀).eq_of_tgt, },\n  { rw dif_neg h₀,\n    by_cases hn : i = n,\n    { rw dif_pos hn,\n      apply mono_comp, },\n    { rw dif_neg hn,\n      apply_instance, }, },\nend\n\nlemma trunc_le.is_iso_ι_f (n i : ℤ) (hi : i < n) :\n  is_iso (trunc_le.ι_f K n i) :=\nbegin\n  dsimp [trunc_le.ι_f],\n  simp only [dif_neg (show ¬ n < i, by linarith),\n    dif_neg (show i ≠ n, by linarith)],\n  apply_instance,\nend\n\nlemma trunc_le.ι_f_eq_zero (n i : ℤ) (hi : n < i) :\n  trunc_le.ι_f K n i = 0 :=\nby { dsimp [trunc_le.ι_f], rw [dif_pos hi], }\n\nlemma trunc_le.ι_f_eq_of_eq (n i : ℤ) (hn : i = n) :\n  trunc_le.ι_f K n i = (trunc_le.X_iso_cycles K n i hn).hom ≫ (K.sc' i).cycles_i :=\nby { dsimp [trunc_le.ι_f], rw [dif_neg (show ¬n<i, by linarith), dif_pos hn], }\n\nlemma trunc_le.ι_f_eq_X_iso_X_inv (n i : ℤ) (hi : i < n) :\n  trunc_le.ι_f K n i = (trunc_le.X_iso_X K n i hi).hom :=\nby { dsimp [trunc_le.ι_f], rw [dif_neg, dif_neg]; linarith, }\n\nlemma trunc_le.shape (n i j : ℤ) (hij : i+1 ≠ j) : trunc_le.d K n i j = 0 :=\nby { dsimp only [trunc_le.d], rw dif_neg hij, }\n\nlemma trunc_le.d_eq_zero (n : ℤ) (i j : ℤ) (hj : n ≤ i) :\n  trunc_le.d K n i j = 0 :=\nbegin\n  by_cases hij : i+1 = j,\n  { dsimp [trunc_le.d],\n    rw [dif_pos hij, dif_pos],\n    linarith, },\n  { rw trunc_le.shape K n i j hij, },\nend\n\nlemma trunc_le.d_eq_d (n : ℤ) (i j : ℤ) (hij : i + 1 = j) (hj : j < n) :\n  trunc_le.d K n i j = (trunc_le.X_iso_X K n i\n    (by { rw ← hij at hj, exact (lt_add_one i).trans hj, })).hom ≫ K.d i j ≫\n    (trunc_le.X_iso_X K n j hj).inv :=\nby { dsimp [trunc_le.d], rw [dif_pos hij, dif_neg, dif_neg]; linarith, }\n\nlemma trunc_le.ι_comp_d_eq_zero (n : ℤ) (i j : ℤ) (hij : i + 1 = j) (hi : i = n) :\n   trunc_le.ι_f K n i ≫ K.d i j = 0 :=\nbegin\n  have hj : j = (complex_shape.up ℤ).next i := by { rw [next], linarith, },\n  subst hj,\n  dsimp [trunc_le.ι_f],\n  erw [dif_neg (show ¬n<i, by linarith), dif_pos hi, assoc, (K.sc' i).cycles_i_g, comp_zero],\nend\n\ndef trunc_le.ι_is_kernel (n i j : ℤ) (hij : i + 1 = j) (hi : i = n) :\n  is_limit (kernel_fork.of_ι _ (trunc_le.ι_comp_d_eq_zero K n i j hij hi)) :=\nbegin\n  have hij' : j = (complex_shape.up ℤ).next i := by { rw [next], linarith, },\n  subst hij',\n  exact is_limit.of_iso_limit (K.sc' i).cycles_is_kernel\n    (iso.symm (fork.ext (trunc_le.X_iso_cycles K n i hi)\n      (trunc_le.ι_f_eq_of_eq K n i hi).symm)),\nend\n\n@[simp, reassoc]\nlemma trunc_le.d_comm (n i j : ℤ) :\n  trunc_le.d K n i j ≫ trunc_le.ι_f K n j =\n     trunc_le.ι_f K n i ≫ K.d i j :=\nbegin\n  by_cases hij : i+1 = j,\n  { by_cases hi₀ : n < i,\n    { apply (trunc_le.is_zero_X K n i hi₀).eq_of_src, },\n    by_cases hi : i = n,\n    { simp only [trunc_le.ι_comp_d_eq_zero K n i j hij hi,\n        trunc_le.d_eq_zero K n i j (by linarith), zero_comp], },\n    by_cases hj : j = n,\n    { dsimp [trunc_le.d, trunc_le.ι_f],\n      simp only [dif_pos hij, dif_neg (show ¬ n < j, by linarith), dif_pos hj, dif_neg hi₀,\n        dif_neg hi, assoc, iso.inv_hom_id_assoc, short_complex.lift_cycles_i], },\n    { have hj' : j < n,\n      { subst hij,\n        cases (not_lt.1 hi₀).lt_or_eq,\n        { rw lt_iff_le_and_ne,\n          split,\n          { linarith, },\n          { exact hj, }, },\n        { exfalso, exact hi h, }, },\n      simp only [trunc_le.d_eq_d K n i j hij hj', trunc_le.ι_f_eq_X_iso_X_inv K n j hj', assoc,\n        iso.inv_hom_id, comp_id, trunc_le.ι_f_eq_X_iso_X_inv K n i (by linarith)], }, },\n  { rw [trunc_le.shape K n i j hij, K.shape i j hij, zero_comp, comp_zero], },\nend\n\n@[simp, reassoc]\nlemma trunc_le.d_comp_d (n i j k : ℤ) : trunc_le.d K n i j ≫ trunc_le.d K n j k = 0 :=\nby simp only [← cancel_mono (trunc_le.ι_f K n k), assoc, trunc_le.d_comm,\n    trunc_le.d_comm_assoc, K.d_comp_d, zero_comp, comp_zero]\n\n@[simps]\ndef trunc_le (n : ℤ) : cochain_complex C ℤ :=\n{ X := trunc_le.X K n,\n  d := trunc_le.d K n,\n  shape' := trunc_le.shape K n,\n  d_comp_d' := λ i j k hij hjk, trunc_le.d_comp_d K n i j k, }\n\n@[simps]\ndef trunc_le.ι (n : ℤ) : K.trunc_le n ⟶ K :=\n{ f := trunc_le.ι_f K n,\n  comm' := λ i j hij, (trunc_le.d_comm K n i j).symm, }\n\nvariables {K L}\n\ndef trunc_le.map_f (φ : K ⟶ L) (n i : ℤ) :\n  (K.trunc_le n).X i ⟶ (L.trunc_le n).X i :=\nbegin\n  by_cases hi : i < n,\n  { exact (trunc_le.X_iso_X K n i hi).hom ≫ φ.f i ≫\n    (trunc_le.X_iso_X L n i hi).inv, },\n  { by_cases hn : i = n,\n    { exact (trunc_le.X_iso_cycles K n i hn).hom ≫\n        (homological_complex.short_complex_functor C (complex_shape.up ℤ) i ⋙\n        short_complex.cycles_functor C).map φ≫ (trunc_le.X_iso_cycles L n i hn).inv, },\n    { exact 0, }, },\nend\n\nlemma trunc_le.map_f_eq_f (φ : K ⟶ L) (n i : ℤ) (hi : i < n) :\n  trunc_le.map_f φ n i = (trunc_le.X_iso_X K n i hi).hom ≫ φ.f i ≫\n    (trunc_le.X_iso_X L n i hi).inv :=\nbegin\n  dsimp only [trunc_le.map_f],\n  simp only [dif_pos hi],\nend\n\n@[simp, reassoc]\nlemma trunc_le.map_f_comm_ι_f (φ : K ⟶ L) (n i : ℤ) :\n  trunc_le.map_f φ n i ≫ trunc_le.ι_f L n i =\n    trunc_le.ι_f K n i ≫ φ.f i :=\nbegin\n  by_cases hi : i < n,\n  { simp only [trunc_le.ι_f_eq_X_iso_X_inv _ n i hi, trunc_le.map_f_eq_f φ n i hi, assoc,\n      iso.inv_hom_id, comp_id], },\n  { by_cases hn : i = n,\n    { dsimp [trunc_le.map_f, trunc_ge.π_f],\n      simp only [dif_neg hi, dif_pos hn, assoc, iso.inv_hom_id_assoc,\n        trunc_le.ι_f_eq_of_eq _ n i hn],\n      erw (short_complex.cycles_i_nat_trans C).naturality\n        ((homological_complex.short_complex_functor C (complex_shape.up ℤ) i).map φ),\n      refl,   },\n    { refine (trunc_le.is_zero_X K n i _).eq_of_src _ _,\n      cases (not_lt.1 hi).lt_or_eq,\n      { exact h, },\n      { exfalso, exact hn h.symm, }, }, },\nend\n\n@[reassoc]\nlemma trunc_le.map_comm_f (φ : K ⟶ L) (n i j : ℤ) :\n  trunc_le.map_f φ n i ≫ trunc_le.d L n i j =\n    trunc_le.d K n i j ≫ trunc_le.map_f φ n j :=\nby simp only [← cancel_mono (trunc_le.ι_f L n j), assoc, trunc_le.d_comm,\n    trunc_le.map_f_comm_ι_f_assoc, homological_complex.hom.comm, trunc_le.map_f_comm_ι_f,\n    trunc_le.d_comm_assoc]\n\n@[simp]\nlemma trunc_le.map_id_f (K : cochain_complex C ℤ) (n i : ℤ) :\n  trunc_le.map_f (𝟙 K) n i = 𝟙 _ :=\nby simp only [← cancel_mono (trunc_le.ι_f K n i),\n  trunc_le.map_f_comm_ι_f, homological_complex.id_f, comp_id, id_comp]\n\n@[simp]\nlemma trunc_le.map_comp_f {K L M : cochain_complex C ℤ}\n  (φ : K ⟶ L) (φ' : L ⟶ M) (n i : ℤ) :\n  trunc_le.map_f (φ ≫ φ') n i =\n    trunc_le.map_f φ n i ≫ trunc_le.map_f φ' n i :=\nby simp only [← cancel_mono (trunc_le.ι_f M n i),\n  trunc_le.map_f_comm_ι_f, homological_complex.comp_f, assoc, trunc_le.map_f_comm_ι_f_assoc]\n\nvariable (C)\n\n@[simps]\ndef trunc_le_functor (n : ℤ) :\n  cochain_complex C ℤ ⥤ cochain_complex C ℤ :=\n{ obj := λ K, K.trunc_le n,\n  map := λ K L φ,\n  { f := λ i, trunc_le.map_f φ n i,\n    comm' := λ i j hij, trunc_le.map_comm_f φ n i j, }, }\n\n@[simps]\ndef trunc_le.nat_trans_ι (n : ℤ) :\n  trunc_le_functor C n ⟶ 𝟭 _ :=\n{ app := λ K, trunc_le.ι K n,\n  naturality' := λ K L φ, begin\n    ext i\n    dsimp,\n    simp only [functor.id_map, homological_complex.comp_f, trunc_le_functor_map_f],\n    dsimp,\n    simp only [trunc_le.map_f_comm_ι_f],\n  end, }\n\nvariables {C} (K)\n\nlemma trunc_le.is_zero_homology (n i : ℤ) (hi : n < i) :\n  is_zero ((homology_functor C _ i).obj (K.trunc_le n)) :=\nbegin\n  dsimp [homology_functor],\n  rw ← short_complex.exact_iff_is_zero_homology,\n  exact short_complex.exact.of_is_zero_X₂ _ (trunc_le.is_zero_X K n i hi),\nend\n\nlemma trunc_le.is_iso_homology_map_ι (n i : ℤ) (hi : i ≤ n) :\n  is_iso ((homology_functor C _ i).map (trunc_le.ι K n)) :=\nbegin\n  let φ := (homological_complex.short_complex_functor C (complex_shape.up ℤ) i).map\n    (trunc_le.ι K n),\n  haveI : is_iso φ.τ₁ := trunc_le.is_iso_ι_f K n _ (by { rw [prev], linarith, }),\n  cases hi.lt_or_eq,\n  { haveI : mono φ.τ₃ := by { dsimp, apply_instance, },\n    haveI : is_iso φ.τ₂ := trunc_le.is_iso_ι_f K n i h,\n    exact short_complex.quasi_iso.of_epi_of_is_iso_of_mono φ, },\n  { exact category_theory.short_complex.quasi_iso.of_kernel_fork φ\n    ((trunc_le.is_zero_X K n _ (by { rw [next], linarith, })).eq_of_tgt _ _)\n      (trunc_le.ι_is_kernel K n i _ (by { rw [next], }) h), },\nend\n\nvariables {K L}\n\nlemma trunc_le.map_homology_iso (φ : K ⟶ L) (n i : ℤ) [is_iso (homology_map φ i)] :\n  is_iso (homology_map ((trunc_le_functor C n).map φ) i) :=\nbegin\n  by_cases hi : i ≤ n,\n  { have eq := (homology_functor C _ i).congr_map ((trunc_le.nat_trans_ι C n).naturality φ),\n    simp only [functor.map_comp, functor.id_map, trunc_le.nat_trans_ι_app] at eq,\n    change homology_map _ i ≫ homology_map (trunc_le.ι L n) i =\n      homology_map (trunc_le.ι K n) i ≫ homology_map φ i at eq,\n    haveI : ∀ (M : cochain_complex C ℤ), is_iso (homology_map (trunc_le.ι M n) i) :=\n      λ M, trunc_le.is_iso_homology_map_ι M n i hi,\n    simp only [← cancel_mono (inv (homology_map (trunc_le.ι L n) i)),\n      assoc, is_iso.hom_inv_id, comp_id] at eq,\n    rw eq,\n    apply_instance, },\n  { simp only [not_le] at hi,\n    exact ⟨⟨0, (trunc_le.is_zero_homology K n i hi).eq_of_src _ _,\n       (trunc_le.is_zero_homology L n i hi).eq_of_src _ _⟩⟩, },\nend\n\ninstance trunc_le.map_quasi_iso (φ : K ⟶ L) (n : ℤ) [quasi_iso φ] :\n  quasi_iso ((trunc_le_functor _ n).map φ) :=\n⟨λ i, trunc_le.map_homology_iso φ n i⟩\n\nvariable (C)\n\nlemma trunc_le_functor_comp_Q_inverts_quasi_isomorphisms (n : ℤ) :\n  (quasi_isomorphisms _ _).is_inverted_by\n    (cochain_complex.trunc_le_functor C n ⋙ derived_category.Q) :=\nλ K L φ hφ, begin\n  haveI : quasi_iso φ := by simpa only [← mem_quasi_isomorphisms_iff] using hφ,\n  dsimp,\n  apply_instance,\nend\n\nvariable {C}\n\nclass is_strictly_le (K : cochain_complex C ℤ) (n : ℤ) : Prop :=\n(is_zero' : ∀ (i : ℤ) (hi : n < i), is_zero (K.X i))\n\nlemma is_strictly_le.is_zero (K : cochain_complex C ℤ) (n i : ℤ) [K.is_strictly_le n]\n  (hi : n < i) : is_zero (K.X i) :=\nis_strictly_le.is_zero' i hi\n\nlemma is_strictly_le_of_le (K : cochain_complex C ℤ) (n m : ℤ) (hnm : n ≤ m)\n  [K.is_strictly_le n] :\n  K.is_strictly_le m :=\n⟨λ i hi, is_strictly_le.is_zero K n i (by linarith)⟩\n\nlemma is_strictly_le.of_iso {K L : cochain_complex C ℤ} (e : K ≅ L) (n : ℤ)\n  [K.is_strictly_le n] : L.is_strictly_le n :=\n⟨λ i hi, is_zero.of_iso (is_strictly_le.is_zero K n i hi)\n  ((homological_complex.eval _ _ i).map_iso e.symm)⟩\n\nlemma is_strictly_le.iff_of_iso {K L : cochain_complex C ℤ} (e : K ≅ L) (n : ℤ) :\n  K.is_strictly_le n ↔ L.is_strictly_le n :=\nbegin\n  split,\n  { introI,\n    exact is_strictly_le.of_iso e n, },\n  { introI,\n    exact is_strictly_le.of_iso e.symm n, },\nend\n\nclass is_le (K : cochain_complex C ℤ) (n : ℤ) : Prop :=\n(is_zero' : ∀ (i : ℤ) (hi : n < i), is_zero (K.homology i))\n\nlemma is_le.is_zero (K : cochain_complex C ℤ) (n i : ℤ) [K.is_le n] (hi : n < i) :\n  is_zero (K.homology i) :=\nis_le.is_zero' i hi\n\nlemma is_le_of_le (K : cochain_complex C ℤ) (n m : ℤ) (hnm : n ≤ m) [K.is_le n] : K.is_le m :=\n⟨λ i hi, is_le.is_zero K n i (by linarith)⟩\n\nlemma is_le.of_iso {K L : cochain_complex C ℤ} (e : K ≅ L) (n : ℤ) [K.is_le n] : L.is_le n :=\n⟨λ i hi, is_zero.of_iso (is_le.is_zero K n i hi) ((homology_functor _ _ i).map_iso e.symm)⟩\n\nlemma is_le.iff_of_iso {K L : cochain_complex C ℤ} (e : K ≅ L) (n : ℤ) :\n  K.is_le n ↔ L.is_le n :=\nbegin\n  split,\n  { introI,\n    exact is_le.of_iso e n, },\n  { introI,\n    exact is_le.of_iso e.symm n, },\nend\n\n@[priority 100]\ninstance is_le_of_is_strictly_le (K : cochain_complex C ℤ) (n : ℤ)\n  [K.is_strictly_le n] : K.is_le n :=\n⟨λ i hi, begin\n  rw ← short_complex.exact_iff_is_zero_homology,\n  exact short_complex.exact.of_is_zero_X₂ _ (is_strictly_le.is_zero K n i hi),\nend⟩\n\ninstance trunc_le_is_strictly_le (K : cochain_complex C ℤ) (n : ℤ) :\n  (K.trunc_le n).is_strictly_le n :=\n⟨trunc_le.is_zero_X K n⟩\n\ninstance trunc_le_is_strictly_le' (K : cochain_complex C ℤ) (n : ℤ) :\n  ((trunc_le_functor C n).obj K).is_strictly_le n :=\n(infer_instance : (K.trunc_le n).is_strictly_le n)\n\nlemma trunc_le.is_iso_ι_f_iff_d_eq_zero (K : cochain_complex C ℤ) (n i j : ℤ)\n  (hij : i+1 = j) (hi : i = n) :\n  is_iso ((trunc_le.ι K n).f i) ↔ K.d i j = 0 :=\nbegin\n  split,\n  { intro h,\n    haveI : is_iso (trunc_le.ι_f K n i) := h,\n    simp only [← cancel_epi (trunc_le.ι_f K n i),\n      trunc_le.ι_comp_d_eq_zero K n i j hij hi, comp_zero], },\n  { exact kernel_fork.is_limit.is_iso_ι_of_zero _ (trunc_le.ι_is_kernel K n i j hij hi), },\nend\n\ninstance (K : cochain_complex C ℤ) (n : ℤ) [K.is_strictly_le n] :\n  is_iso (trunc_le.ι K n) :=\nbegin\n  haveI : ∀ (i : ℤ), is_iso ((trunc_le.ι K n).f i),\n  { intro i,\n    by_cases hi : i < n,\n    { exact trunc_le.is_iso_ι_f K n i hi, },\n    { cases (not_lt.1 hi).lt_or_eq,\n      { refine ⟨⟨0, _, (is_strictly_le.is_zero K n i h).eq_of_tgt _ _⟩⟩,\n        rw ← cancel_mono (trunc_le.ι_f K n i),\n        apply (is_strictly_le.is_zero K n i h).eq_of_tgt, },\n      { rw trunc_le.is_iso_ι_f_iff_d_eq_zero K n i (i+1) (by linarith) h.symm,\n        apply (is_strictly_le.is_zero K n (i+1) (by linarith)).eq_of_tgt, }, }, },\n  apply homological_complex.hom.is_iso_of_components,\nend\n\nend cochain_complex\n\nnamespace derived_category\n\nvariable (C)\n\ndef trunc_le_functor (n : ℤ) : derived_category C ⥤ derived_category C :=\nlocalization.lift _ (cochain_complex.trunc_le_functor_comp_Q_inverts_quasi_isomorphisms C n) Q\n\ninstance (n : ℤ) : localization.lifting Q (quasi_isomorphisms _ _)\n  (cochain_complex.trunc_le_functor C n ⋙ derived_category.Q) (trunc_le_functor C n) :=\nlocalization.lifting_lift _ _ _\n\ndef trunc_le_functor_iso (n : ℤ) :\n  Q ⋙ trunc_le_functor C n ≅ (cochain_complex.trunc_le_functor C n ⋙ derived_category.Q) :=\nlocalization.lifting.iso _ (quasi_isomorphisms _ _) _ _\n\ndef trunc_le_nat_trans_ι (n : ℤ) : trunc_le_functor C n ⟶ 𝟭 (derived_category C) :=\nlocalization.lift_nat_trans Q (quasi_isomorphisms _ _)\n  (cochain_complex.trunc_le_functor C n ⋙ derived_category.Q) Q _ _\n  (whisker_right (cochain_complex.trunc_le.nat_trans_ι C n) Q)\n\n@[simp]\nlemma trunc_le_nat_trans_ι_app (K : cochain_complex C ℤ) (n : ℤ) :\n  (trunc_le_nat_trans_ι C n).app (Q.obj K) =\n    (trunc_le_functor_iso C n).hom.app K ≫ Q.map (cochain_complex.trunc_le.ι K n) :=\nbegin\n  dsimp only [trunc_le_nat_trans_ι, trunc_le_functor_iso],\n  simp only [localization.lift_nat_trans_app, whisker_right_app,\n    cochain_complex.trunc_le.nat_trans_ι_app,\n    localization.lifting.id_iso, functor.right_unitor_inv_app, comp_id],\nend\n\nvariable {C}\n\nclass is_le (K : derived_category C) (n : ℤ) : Prop :=\n(is_zero' : ∀ (i : ℤ) (hi : n < i), is_zero (K.homology i))\n\nlemma is_le.is_zero (K : derived_category C) (n i : ℤ) [K.is_le n]\n  (hi : n < i) : is_zero (K.homology i) :=\nis_le.is_zero' i hi\n\nlemma is_le_of_le (K : derived_category C) (n m : ℤ) (hnm : n ≤ m) [K.is_le n] : K.is_le m :=\n⟨λ i hi, is_le.is_zero K n i (by linarith)⟩\n\nlemma is_le.of_iso {K L : derived_category C} (e : K ≅ L) (n : ℤ) [K.is_le n] : L.is_le n :=\n⟨λ i hi, is_zero.of_iso (is_le.is_zero K n i hi) ((homology_functor _ i).map_iso e.symm)⟩\n\nlemma is_le.iff_of_iso {K L : derived_category C} (e : K ≅ L) (n : ℤ) :\n  K.is_le n ↔ L.is_le n :=\nbegin\n  split,\n  { introI,\n    exact is_le.of_iso e n, },\n  { introI,\n    exact is_le.of_iso e.symm n, },\nend\n\nvariable (C)\n\ndef Q_comp_trunc_le_functor_comp_homology_functor_iso (n i : ℤ) :\n  Q ⋙ trunc_le_functor C n ⋙ homology_functor C i ≅\n    cochain_complex.trunc_le_functor C n ⋙ _root_.homology_functor _ _ i :=\n(functor.associator _ _ _).symm ≪≫\n  iso_whisker_right (trunc_le_functor_iso C n) (homology_functor C i) ≪≫\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ (homology_functor_factors C i)\n\nvariable {C}\n\nlemma is_zero_homology_trunc_le_of_lt (K : derived_category C) (n i : ℤ) (hi : n < i) :\n  is_zero (((trunc_le_functor C n).obj K).homology i) :=\nis_zero.of_iso (cochain_complex.is_le.is_zero _ n i hi)\n  (((trunc_le_functor C n ⋙ homology_functor C i).map_iso (Q.obj_obj_preimage_iso K)).symm ≪≫\n    ((Q_comp_trunc_le_functor_comp_homology_functor_iso C n i).app _))\n\nlemma is_iso_homology_map_trunc_le_nat_trans_ι_of_le (K : derived_category C) (n i : ℤ)\n  (hi : i ≤ n) :\n  is_iso ((homology_functor C i).map ((trunc_le_nat_trans_ι C n).app K)) :=\nbegin\n  erw ← (Q.obj_obj_preimage_iso K).is_iso_app_iff\n    (whisker_right (trunc_le_nat_trans_ι C n) (homology_functor C i)),\n  dsimp,\n  erw [trunc_le_nat_trans_ι_app, functor.map_comp],\n  haveI : ∀ (L : cochain_complex C ℤ), is_iso ((homology_functor C i).map (Q.map\n    (cochain_complex.trunc_le.ι L n))),\n  { intro L,\n    erw nat_iso.is_iso_map_iff (homology_functor_factors C i),\n    exact cochain_complex.trunc_le.is_iso_homology_map_ι L n i hi, },\n  apply_instance,\nend\n\nlemma is_iso_trunc_le_nat_trans_ι_app_iff (K : derived_category C) (n : ℤ) :\n  is_iso ((trunc_le_nat_trans_ι C n).app K) ↔ K.is_le n :=\nbegin\n  rw is_iso_iff_is_iso_homology,\n  split,\n  { introI hK,\n    exact ⟨λ i hi, is_zero.of_iso (is_zero_homology_trunc_le_of_lt K n i hi)\n      (as_iso ((homology_functor C i).map ((trunc_le_nat_trans_ι C n).app K))).symm⟩, },\n  { introI,\n    intro i,\n    by_cases hi : i ≤ n,\n    { exact is_iso_homology_map_trunc_le_nat_trans_ι_of_le K n i hi, },\n    { simp only [not_le] at hi,\n      exact ⟨⟨0, (is_zero_homology_trunc_le_of_lt K n i hi).eq_of_src _ _,\n        (is_le.is_zero K n i hi).eq_of_src _ _⟩⟩, }, },\nend\n\ninstance is_iso_trunc_le_nat_trans_ι_of_le (K : derived_category C) (n : ℤ) [K.is_le n] :\n  is_iso ((trunc_le_nat_trans_ι C n).app K) :=\nby { rw is_iso_trunc_le_nat_trans_ι_app_iff, apply_instance, }\n\nend derived_category\n\nnamespace cochain_complex\n\nlemma is_le_iff_Q_obj_is_le (K : cochain_complex C ℤ) (n : ℤ) :\n  K.is_le n ↔ (derived_category.Q.obj K).is_le n :=\nbegin\n  split,\n  { introI,\n    exact ⟨λ i hi, by simpa only [← is_zero_homology_iff_is_zero_homology_Q_obj]\n      using is_le.is_zero K n i hi⟩, },\n  { introI,\n    exact ⟨λ i hi, by simpa only [is_zero_homology_iff_is_zero_homology_Q_obj]\n      using derived_category.is_le.is_zero (derived_category.Q.obj _) n i hi⟩, },\nend\n\ninstance Q_obj_is_le_of_is_le (K : cochain_complex C ℤ) (n : ℤ) [K.is_le n] :\n  (derived_category.Q.obj K).is_le n :=\nbegin\n  rw ← is_le_iff_Q_obj_is_le,\n  apply_instance,\nend\n\nlemma is_le_iff_of_quasi_iso {K L : cochain_complex C ℤ} (φ : K ⟶ L) [quasi_iso φ] (n : ℤ) :\n  K.is_le n ↔ L.is_le n :=\nbegin\n  simp only [is_le_iff_Q_obj_is_le],\n  exact derived_category.is_le.iff_of_iso (as_iso (derived_category.Q.map φ)) n,\nend\n\nlemma quasi_iso_trunc_le_ι_iff (K : cochain_complex C ℤ) (n : ℤ) :\n  quasi_iso (trunc_le.ι K n) ↔ K.is_le n :=\nbegin\n  rw [is_le_iff_Q_obj_is_le, ← derived_category.is_iso_Q_map_iff,\n    ← derived_category.is_iso_trunc_le_nat_trans_ι_app_iff,\n    derived_category.trunc_le_nat_trans_ι_app],\n  split,\n  { introI,\n    apply_instance, },\n  { apply is_iso.of_is_iso_comp_left, },\nend\n\ninstance (K : cochain_complex C ℤ) (n : ℤ) [K.is_le n] :\n  quasi_iso (trunc_le.ι K n) :=\nby { rw quasi_iso_trunc_le_ι_iff, apply_instance, }\n\nend cochain_complex\n\nnamespace derived_category\n\nlemma right_factorisation_of_is_le {K L : cochain_complex C ℤ} (φ : Q.obj K ⟶ Q.obj L) (n : ℤ)\n  [K.is_le n] :\n  ∃ (K' : cochain_complex C ℤ) (hK' : K'.is_strictly_le n) (s : K' ⟶ K) (f : K' ⟶ L) (hs : quasi_iso s),\n    φ = (by { haveI := hs, exact inv (Q.map s), }) ≫ Q.map f :=\nbegin\n  obtain ⟨K', s, f, hs, eq⟩ := right_factorisation φ,\n  haveI := hs,\n  haveI : quasi_iso ((cochain_complex.trunc_le.nat_trans_ι C n).app K'),\n  { erw cochain_complex.quasi_iso_trunc_le_ι_iff,\n    dsimp,\n    rw cochain_complex.is_le_iff_of_quasi_iso s,\n    apply_instance, },\n  exact ⟨_, infer_instance,\n    (cochain_complex.trunc_le.nat_trans_ι _ n).app K' ≫ s,\n    (cochain_complex.trunc_le.nat_trans_ι _ n).app K' ≫ f, infer_instance,\n    by simp only [eq, functor.map_comp, is_iso.eq_inv_comp, assoc, is_iso.hom_inv_id_assoc]⟩,\nend\n\nlemma left_factorisation_of_is_strictly_le {K L : cochain_complex C ℤ} (φ : Q.obj K ⟶ Q.obj L)\n  (n : ℤ) [K.is_strictly_le n] [L.is_strictly_le n] :\n  ∃ (L' : cochain_complex C ℤ) (hL' : L'.is_strictly_le n) (f : K ⟶ L') (s : L ⟶ L') (hs : quasi_iso s),\n    φ = Q.map f ≫ (by { haveI := hs, exact inv (Q.map s), }) :=\nbegin\n  obtain ⟨L', f, s, hs, eq⟩ := left_factorisation φ,\n  haveI := hs,\n  haveI : quasi_iso (cochain_complex.trunc_le.ι L' n),\n  { rw [cochain_complex.quasi_iso_trunc_le_ι_iff, ← cochain_complex.is_le_iff_of_quasi_iso s],\n    apply_instance, },\n  refine ⟨_, infer_instance,\n    category_theory.inv (cochain_complex.trunc_le.ι K n) ≫\n      (cochain_complex.trunc_le_functor C n).map f,\n    category_theory.inv (cochain_complex.trunc_le.ι L n) ≫\n    (cochain_complex.trunc_le_functor C n).map s, infer_instance, _⟩,\n  have comms := Q.congr_map ((cochain_complex.trunc_le.nat_trans_ι C n).naturality s),\n  have commf := Q.congr_map ((cochain_complex.trunc_le.nat_trans_ι C n).naturality f),\n  dsimp at comms commf,\n  simp only [Q.map_comp, ← cancel_mono (inv (Q.map ((cochain_complex.trunc_le.ι L' n)))),\n    assoc, is_iso.hom_inv_id, comp_id] at comms commf,\n  simp only [comms, commf, eq, functor.map_comp, functor.map_inv, is_iso.inv_hom_id_assoc,\n    is_iso.inv_comp, is_iso.inv_inv, assoc],\nend\n\nlemma exists_iso_Q_obj_of_le (K : derived_category C) (n : ℤ) [K.is_le n] :\n  ∃ (K' : cochain_complex C ℤ) (hK' : K'.is_strictly_le n),\n    nonempty (K ≅ Q.obj K') :=\nbegin\n  let K' := Q.obj_preimage K,\n  haveI : K'.is_le n,\n  { rw [cochain_complex.is_le_iff_Q_obj_is_le, is_le.iff_of_iso (Q.obj_obj_preimage_iso K)],\n    apply_instance, },\n  exact ⟨K'.trunc_le n, infer_instance, ⟨(Q.obj_obj_preimage_iso K).symm ≪≫\n    (as_iso ((trunc_le_nat_trans_ι C n).app (Q.obj K'))).symm ≪≫\n    (trunc_le_functor_iso C n).app K'⟩⟩,\nend\n\nend derived_category\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/trunc_le.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34159470414198595}}
{"text": "/-\nCopyright (c) 2018 Michael Jendrusch. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Jendrusch, Scott Morrison\n-/\nimport category_theory.monoidal.of_chosen_finite_products\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.types\n\n/-!\n# The category of types is a symmetric monoidal category\n-/\n\nopen category_theory\nopen category_theory.limits\nopen tactic\n\nuniverses v u\n\nnamespace category_theory\n\ninstance types_monoidal : monoidal_category.{u} (Type u) :=\nmonoidal_of_chosen_finite_products (types.terminal_limit_cone) (types.binary_product_limit_cone)\n\ninstance types_symmetric : symmetric_category.{u} (Type u) :=\nsymmetric_of_chosen_finite_products (types.terminal_limit_cone) (types.binary_product_limit_cone)\n\n@[simp] lemma tensor_apply {W X Y Z : Type u} (f : W ⟶ X) (g : Y ⟶ Z) (p : W ⊗ Y) :\n  (f ⊗ g) p = (f p.1, g p.2) := rfl\n\n@[simp] lemma left_unitor_hom_apply {X : Type u} {x : X} {p : punit} :\n  ((λ_ X).hom : (𝟙_ (Type u)) ⊗ X → X) (p, x) = x := rfl\n@[simp] lemma left_unitor_inv_apply {X : Type u} {x : X} :\n  ((λ_ X).inv : X ⟶ (𝟙_ (Type u)) ⊗ X) x = (punit.star, x) := rfl\n\n@[simp] lemma right_unitor_hom_apply {X : Type u} {x : X} {p : punit} :\n  ((ρ_ X).hom : X ⊗ (𝟙_ (Type u)) → X) (x, p) = x := rfl\n@[simp] lemma right_unitor_inv_apply {X : Type u} {x : X} :\n  ((ρ_ X).inv : X ⟶ X ⊗ (𝟙_ (Type u))) x = (x, punit.star) := rfl\n\n@[simp] lemma associator_hom_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :\n  ((α_ X Y Z).hom : (X ⊗ Y) ⊗ Z → X ⊗ (Y ⊗ Z)) ((x, y), z) = (x, (y, z)) := rfl\n@[simp] lemma associator_inv_apply {X Y Z : Type u} {x : X} {y : Y} {z : Z} :\n  ((α_ X Y Z).inv : X ⊗ (Y ⊗ Z) → (X ⊗ Y) ⊗ Z) (x, (y, z)) = ((x, y), z) := rfl\n\n@[simp] lemma braiding_hom_apply {X Y : Type u} {x : X} {y : Y} :\n  ((β_ X Y).hom : X ⊗ Y → Y ⊗ X) (x, y) = (y, x) := rfl\n@[simp] lemma braiding_inv_apply {X Y : Type u} {x : X} {y : Y} :\n  ((β_ X Y).inv : Y ⊗ X → X ⊗ Y) (y, x) = (x, y) := rfl\n\nopen opposite\n\nopen monoidal_category\n\n/-- `(𝟙_ C ⟶ -)` is a lax monoidal functor to `Type`. -/\ndef coyoneda_tensor_unit (C : Type u) [category.{v} C] [monoidal_category C] :\n  lax_monoidal_functor C (Type v) :=\n{ ε := λ p, 𝟙 _,\n  μ := λ X Y p, (λ_ (𝟙_ C)).inv ≫ (p.1 ⊗ p.2),\n  μ_natural' := by tidy,\n  associativity' := λ X Y Z, begin\n    ext ⟨⟨f, g⟩, h⟩, dsimp at f g h,\n    dsimp, simp only [iso.cancel_iso_inv_left, category.assoc],\n    conv_lhs { rw [←category.id_comp h, tensor_comp, category.assoc, associator_naturality,\n      ←category.assoc, unitors_inv_equal, triangle_assoc_comp_right_inv], },\n    conv_rhs { rw [←category.id_comp f, tensor_comp], },\n  end,\n  left_unitality' := by tidy,\n  right_unitality' := λ X, begin\n    ext ⟨f, ⟨⟩⟩, dsimp at f,\n    dsimp, simp only [category.assoc],\n    rw [right_unitor_naturality, unitors_inv_equal, iso.inv_hom_id_assoc],\n  end,\n  ..coyoneda.obj (op (𝟙_ C)) }\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monoidal/types.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34159470414198595}}
{"text": "import tactic\n\nopen tactic expr\n\nmeta def add_nonneg_proof_aux (n : expr) (h : option name) : tactic unit :=\ndo pf ← mk_app `nat.zero_le [n],\n   nm ← get_unused_name,\n   note (h.get_or_else nm) none pf,\n   skip\n\nexample (n : ℕ) : true :=\nbegin\n  add_nonneg_proof_aux `(55) none,\nend\n\n-- Comentario: Al colocar el cursor al final de la primera línea de la\n-- demostración se obtiene\n--    n : ℕ,\n--    _x : 0 ≤ 55\n--    ⊢ true\n\nnamespace tactic\nnamespace interactive\n\nsetup_tactic_parser\n\nmeta def add_nonneg_proof (n : parse parser.pexpr) (h : parse ident?) \n  : tactic unit :=\ndo n ← to_expr n,\n   add_nonneg_proof_aux n h\n\nmeta def add_nonneg_proofs (l : parse pexpr_list) : tactic unit :=\ndo l ← l.mmap to_expr,\n   l.mmap' (λ e, add_nonneg_proof_aux e none)\n\nend interactive\nend tactic\n\nexample (n : ℕ) : true :=\nbegin\n  add_nonneg_proof 55 hx,\nend\n\n-- Comentario: Al colocar el cursor al final de la primera línea de la\n-- demostración se obtiene\n--    n : ℕ,\n--    hx : 0 ≤ 55\n--    ⊢ true\n\nexample (n : ℕ) : true :=\nbegin\n  add_nonneg_proof 55 hx,\n  add_nonneg_proofs [n+n+n, 2*n^2],\nend\n\n-- Comentario: Al colocar el cursor al final de la primera línea de la\n-- demostración se obtiene\n--    n : ℕ,\n--    hx : 0 ≤ 55,\n--    _x : 0 ≤ n + n + n,\n--    _x_1 : 0 ≤ 2 * n ^ 2\n--    ⊢ true\n\nmeta def mk_list : tactic (list ℕ) :=\nreturn [1, 4, 6]\n\nrun_cmd  do\n  l ← mk_list,\n  match l with\n  | (h::t) := trace h\n  | _      := failed\n  end\n\n-- Comentario: Al colocar el cursor sobre run_cmd se obtiene 1.\n\nrun_cmd  do\n  [a, b, c] ← mk_list,\n  trace b\n\n-- Comentario: Al colocar el cursor sobre run_cmd se obtiene 4.\n\nexample : true ∧ true :=\nby do\n  split,\n  gs ← get_goals,\n  gs.mmap' trace\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    2 goals\n--    ⊢ true\n--    \n--    ⊢ true\n--    \n--    ?m_1\n--    ?m_1\n\nexample : true ∧ false :=\nby do\n  split,\n  gs ← get_goals,\n  gs.mmap' (λ e, do tp ← infer_type e, trace tp)\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    2 goals\n--    ⊢ true\n--    \n--    ⊢ false\n--    \n--    true\n--    false\n\nexample : true ∧ false :=\nby do\n  split,\n  gs ← get_goals,\n  gs.mmap' (λ e, trace e.to_raw_fmt)\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    2 goals\n--    ⊢ true\n--    \n--    ⊢ false\n--    \n--    (mvar _mlocal._fresh.176.839 _mlocal._fresh.176.839 (const 2 []))\n--    (mvar _mlocal._fresh.176.840 _mlocal._fresh.176.840 (const 2 []))\n\nexample : ∃ x : ℕ, x = x :=\nby do\n  applyc `exists.intro,\n  [e1, e2] ← get_goals,\n  infer_type e1 >>= trace,\n  infer_type e2 >>= trace\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    2 goals\n--    ⊢ ?m_1 = ?m_1\n--    \n--    ⊢ ℕ\n--    \n--    ?m_1 = ?m_1\n--    ℕ\n\nexample : ∃ x : ℕ, x = x :=\nby do\n  applyc `exists.intro,\n  [e1, e2] ← get_goals,\n  infer_type e1 >>= trace,\n  trace e1.to_raw_fmt,\n  infer_type e2 >>= trace,\n  trace e2.to_raw_fmt\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    2 goals\n--    ⊢ ?m_1 = ?m_1\n--    \n--    ⊢ ℕ\n--    \n--    ?m_1 = ?m_1\n--    (mvar _mlocal._fresh.215.1681 _mlocal._fresh.215.1681 (const 2 []))\n--    ℕ\n--    (mvar _mlocal._fresh.215.1680 _mlocal._fresh.215.1680 (const 2 []))\n\nexample : ∃ x : ℕ, x = x :=\nby do\n  applyc `exists.intro,\n  [e1, e2] ← get_goals,\n  unify e2 `(7),\n  trace e2\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    ⊢ 7 = 7\n--    \n--    7\n\nset_option pp.instantiate_mvars false\n\nexample : ∃ x : ℕ, x = x :=\nby do\n  applyc `exists.intro,\n  [e1, e2] ← get_goals,\n  unify e2 `(7),\n  trace e2,\n  trace e2.to_raw_fmt\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    ⊢ ?m_1 = ?m_1\n--    \n--    ?m_1\n--    (mvar _mlocal._fresh.16.1433 _mlocal._fresh.16.1433 (const 2 []))\n\nexample : ∃ x : ℕ, x = x :=\nby do\n  applyc `exists.intro,\n  [e1, e2] ← get_goals,\n  unify e2 `(7),\n  trace e2,\n  trace e2.to_raw_fmt,\n  e2 ← instantiate_mvars e2,\n  trace e2.to_raw_fmt\n\n-- Comentario: Al colocar el cursor sobre do se obtiene \n--    tactic failed, there are unsolved goals\n--    state:\n--    ⊢ ?m_1 = ?m_1\n--    \n--    ?m_1\n--    (mvar _mlocal._fresh.54.1667 _mlocal._fresh.54.1667 (const 2 []))\n--    (app \n--     (app \n--      (app \n--       (app (const bit1 [0]) (const nat [])) \n--       (const nat.has_one [])) (const nat.has_add [])) \n--     (app \n--      (app \n--       (app \n--        (app (const bit1 [0]) (const nat [])) \n--        (const nat.has_one [])) \n--       (const nat.has_add [])) \n--      (app \n--       (app (const has_one.one [0]) (const nat [])) \n--       (const nat.has_one []))))\n\n------------------------------------------------------------------------\n-- § Referencia                                                       --\n------------------------------------------------------------------------\n\n-- Basado en el vídeo \"Metaprogramming in Lean tutorial: video 6\" de Rob\n-- Lewis que se encuentra en https://youtu.be/ZbijjKFtvJI\n", "meta": {"author": "jaalonso", "repo": "Lean_para_matematicos", "sha": "924c77b7f010604b84f82d2f79967ad8b9cddc6e", "save_path": "github-repos/lean/jaalonso-Lean_para_matematicos", "path": "github-repos/lean/jaalonso-Lean_para_matematicos/Lean_para_matematicos-924c77b7f010604b84f82d2f79967ad8b9cddc6e/src/Metaprogramacion/Introduccion_a_la_metaprogramacion_6.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.34159470414198595}}
{"text": "import data.equiv.basic\nimport analysis.topology.topological_structures\n\ndef transport_topological_ring {α β : Type} \n  [topological_space α] [ring α] [topological_ring α] (f : α ≃ β) : @topological_ring β sorry sorry := sorry\n\n\ndef transport_ring {α β : Type*} [ring α] (f : α ≃ β) : ring β :=\n{ add := λ x y, f (f.symm x + f.symm y),\n  zero := f 0,\n  neg := λ x, f (-f.symm x),\n  mul := λ x y, f (f.symm x * f.symm y),\n  one := f 1,\n  add_assoc := λ x y z, by simp; from add_assoc _ _ _,\n  zero_add := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (zero_add _),\n  add_zero := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (add_zero _),\n  add_left_neg := λ x, by simp; from add_left_neg _,\n  add_comm := λ x y, by simp; from add_comm _ _,\n  mul_assoc := λ x y z, by simp; from mul_assoc _ _ _,\n  one_mul := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (one_mul _),\n  mul_one := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (mul_one _),\n  left_distrib := λ x y z, by simp; from left_distrib _ _ _,\n  right_distrib := λ x y z, by simp; from right_distrib _ _ _, }\n\n\n\n\nimport data.equiv\n--import analysis.topology.topological_structures\nuniverses u v\n\ndef transport_ring {α : Type u} {β : Type v} [ring α] (f : α ≃ β) : ring β :=\n{ add := λ x y, f (f.symm x + f.symm y),\n  zero := f 0,\n  neg := λ x, f (-f.symm x),\n  mul := λ x y, f (f.symm x * f.symm y),\n  one := f 1,\n  add_assoc := λ x y z, by simp; from add_assoc _ _ _,\n  zero_add := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (zero_add _),\n  add_zero := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (add_zero _),\n  add_left_neg := λ x, by simp; from add_left_neg _,\n  add_comm := λ x y, by simp; from add_comm _ _,\n  mul_assoc := λ x y z, by simp; from mul_assoc _ _ _,\n  one_mul := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (one_mul _),\n  mul_one := λ x, by simp; from (equiv.apply_eq_iff_eq_inverse_apply _ _ _).2 (mul_one _),\n  left_distrib := λ x y z, by simp; from left_distrib _ _ _,\n  right_distrib := λ x y z, by simp; from right_distrib _ _ _, }\n\n\n\nclass transportable (f : Type u → Sort v) :=\n(on_equiv : Π {α β : Type u} (e : equiv α β), equiv (f α) (f β))\n(on_refl  : Π (α : Type u), on_equiv (equiv.refl α) = equiv.refl (f α))\n(on_trans : Π {α β γ : Type u} (d : equiv α β) (e : equiv β γ), on_equiv (equiv.trans d e) = equiv.trans (on_equiv d) (on_equiv e))\n\n#print topological_ring \n\n#print sigma \n-- Our goal is an automagic proof of the following\ntheorem group.transportable : transportable group := sorry\ntheorem topological_ring.transportable : transportable\n  (λ R : (Σ (α : Type u), (topological_space α) × (ring α)) , \n    @topological_ring R.fst (R.snd).1 (R.snd).2) := sorry\n#check topological_ring \n-- These we might need to define and prove by hand\ndef Const : Type u → Type v := λ α, punit\ndef Fun : Type u → Type v → Type (max u v) := λ α β, α → β\ndef Prod : Type u → Type v → Type (max u v) := λ α β, α × β\ndef Swap : Type u → Type v → Type (max u v) := λ α β, β × α\n\nlemma Const.transportable : (transportable Const) := { \n  on_equiv := λ β γ H,⟨λ _,punit.star,λ _,punit.star,sorry,sorry⟩,\n  on_refl := λ β,_,on_trans := λ β,_ }  \nlemma Fun.transportable (α : Type u) : (transportable (Fun α)) := sorry\nlemma Prod.transportable (α : Type u) : (transportable (Prod α)) := sorry\nlemma Swap.transportable (α : Type u) : (transportable (Swap α)) := sorry\n\n\n-- And then we can define\ndef Hom1 (α : Type u) : Type v → Type (max u v) := λ β, α → β\ndef Hom2 (β : Type v) : Type u → Type (max u v) := λ α, α → β\ndef Aut : Type u → Type u := λ α, α → α\n\n-- And hopefully automagically derive\nlemma Hom1.transportable (α : Type u) : (transportable (Hom1 α)) := sorry\nlemma Hom2.transportable (β : Type v) : (transportable (Hom1 β)) := sorry\nlemma Aut.transportable (α : Type u) : (transportable Aut) := sorry\n\n-- If we have all these in place...\n-- A bit of magic might actually be able to derive `group.transportable` on line 11.\n-- After all, a group just is a type plus some functions... and we can now transport functions.\n\nstructure equiv' (α : Type zfc_u) (β : Type zfc_u) :=\n(i    : α → β)\n(j    : β → α)\n(ij : ∀ (x : α), j (i x) = x)\n(ji  : ∀ (y : β), i (j y) = y)\n\ndefinition mul_is_add {α : Type zfc_u} : equiv' (has_mul α) (has_add α) :=\n{ i := λ ⟨mul⟩,⟨mul⟩,\n  j := λ ⟨mul⟩,⟨mul⟩, -- didn't I just write that?\n  ij := λ ⟨x⟩,rfl,\n  ji := λ ⟨x⟩, rfl,  -- didn't I just write that?\n}\n\ndefinition : equiv'  \ndefinition equiv_mul {α β : Type zfc_u} : equiv' α β → equiv' (has_mul α) (has_mul β) := λ E,\n{ i :=  λ αmul,⟨λ b1 b2, E.i (@has_mul.mul α αmul (E.j b1) (E.j b2))⟩,\n  j := λ βmul,⟨λ a1 a2, E.j (@has_mul.mul β βmul (E.i a1) (E.i a2))⟩, -- didn't I just write that?\n                                                                      -- should we introduce E-dual?\n  ij := λ f, begin\n    cases f, -- aargh why do I struggle\n    suffices :  (λ (a1 a2 : α), E.j (E.i (f (E.j (E.i a1)) (E.j (E.i a2))))) = (λ a1 a2, f a1 a2),\n      by rw this,\n    funext,\n    simp [E.ij,E.ji], -- got there in the end\n  end,\n  ji := -- I can't even do this in term mode so I just copy out the entire tactic mode proof again.\n λ g, begin\n    cases g, -- aargh why do I struggle\n    suffices :  (λ (b1 b2 : β), E.i (E.j (g (E.i (E.j b1)) (E.i (E.j b2))))) = (λ b1 b2, g b1 b2),\n      by rw this,\n    funext,\n    simp [E.ij,E.ji], -- got there in the end\n  end, -- didn't I just write that?\n}\n\ndefinition mul_to_add {α β : Type} : equiv' α β → equiv' (has_add α) (has_add β) := _ ∘ equiv_mul\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/canonical_isomorphism/canonically_isomorphic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34141802024232093}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport category_theory.category.Cat\nimport category_theory.groupoid\nimport category_theory.types\n\n/-!\n# Objects of a category up to an isomorphism\n\n`is_isomorphic X Y := nonempty (X ≅ Y)` is an equivalence relation on the objects of a category.\nThe quotient with respect to this relation defines a functor from our category to `Type`.\n-/\n\nuniverses v u\n\nnamespace category_theory\n\nsection category\n\nvariables {C : Type u} [category.{v} C]\n\n/-- An object `X` is isomorphic to an object `Y`, if `X ≅ Y` is not empty. -/\ndef is_isomorphic : C → C → Prop := λ X Y, nonempty (X ≅ Y)\n\nvariable (C)\n\n/-- `is_isomorphic` defines a setoid. -/\ndef is_isomorphic_setoid : setoid C :=\n{ r := is_isomorphic,\n  iseqv := ⟨λ X, ⟨iso.refl X⟩, λ X Y ⟨α⟩, ⟨α.symm⟩, λ X Y Z ⟨α⟩ ⟨β⟩, ⟨α.trans β⟩⟩ }\n\nend category\n\n/--\nThe functor that sends each category to the quotient space of its objects up to an isomorphism.\n-/\ndef isomorphism_classes : Cat.{v u} ⥤ Type u :=\n{ obj := λ C, quotient (is_isomorphic_setoid C.α),\n  map := λ C D F, quot.map F.obj $ λ X Y ⟨f⟩, ⟨F.map_iso f⟩ }\n\nlemma groupoid.is_isomorphic_iff_nonempty_hom {C : Type u} [groupoid.{v} C] {X Y : C} :\n  is_isomorphic X Y ↔ nonempty (X ⟶ Y) :=\n(groupoid.iso_equiv_hom X Y).nonempty_congr\n\n-- PROJECT: define `skeletal`, and show every category is equivalent to a skeletal category,\n-- using the axiom of choice to pick a representative of every isomorphism class.\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/isomorphism_classes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7662936324115012, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.34140644575208134}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n\n! This file was ported from Lean 3 source module data.fun_like.embedding\n! leanprover-community/mathlib commit c4658a649d216f57e99621708b09dcb3dcccbd23\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.FunLike.Basic\n\n/-!\n# Typeclass for a type `F` with an injective map to `A ↪ B`\n\nThis typeclass is primarily for use by embeddings such as `RelEmbedding`.\n\n## Basic usage of `EmbeddingLike`\n\nA typical type of embeddings should be declared as:\n```\nstructure MyEmbedding (A B : Type _) [MyClass A] [MyClass B] :=\n(toFun : A → B)\n(injective' : Function.Injective toFun)\n(map_op' : ∀ {x y : A}, toFun (MyClass.op x y) = MyClass.op (toFun x) (toFun y))\n\nnamespace MyEmbedding\n\nvariables (A B : Type _) [MyClass A] [MyClass B]\n\n-- This instance is optional if you follow the \"Embedding class\" design below:\ninstance : EmbeddingLike (MyEmbedding A B) A B :=\n{ coe := MyEmbedding.toFun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := MyEmbedding.injective' }\n\n/-- Helper instance for when there's too many metavariables to `EmbeddingLike.coe` directly. -/\ninstance : CoeFun (MyEmbedding A B) (λ _, A → B) := ⟨MyEmbedding.toFun⟩\n\n@[ext] theorem ext {f g : MyEmbedding A B} (h : ∀ x, f x = g x) : f = g := FunLike.ext f g h\n\n/-- Copy of a `MyEmbedding` with a new `toFun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : MyEmbedding A B) (f' : A → B) (h : f' = ⇑f) : MyEmbedding A B :=\n{ toFun := f',\n  injective' := h.symm ▸ f.injective',\n  map_op' := h.symm ▸ f.map_op' }\n\nend MyEmbedding\n```\n\nThis file will then provide a `CoeFun` instance and various\nextensionality and simp lemmas.\n\n## Embedding classes extending `EmbeddingLike`\n\nThe `EmbeddingLike` design provides further benefits if you put in a bit more work.\nThe first step is to extend `EmbeddingLike` to create a class of those types satisfying\nthe axioms of your new type of morphisms.\nContinuing the example above:\n\n```\nsection\n\n/-- `MyEmbeddingClass F A B` states that `F` is a type of `MyClass.op`-preserving embeddings.\nYou should extend this class when you extend `MyEmbedding`. -/\nclass MyEmbeddingClass (F : Type _) (A B : outParam <| Type _) [MyClass A] [MyClass B]\n  extends EmbeddingLike F A B :=\n(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))\n\nend\n\n@[simp] lemma map_op {F A B : Type _} [MyClass A] [MyClass B] [MyEmbeddingClass F A B]\n  (f : F) (x y : A) : f (MyClass.op x y) = MyClass.op (f x) (f y) :=\nMyEmbeddingClass.map_op\n\n-- You can replace `MyEmbedding.EmbeddingLike` with the below instance:\ninstance : MyEmbeddingClass (MyEmbedding A B) A B :=\n{ coe := MyEmbedding.toFun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := MyEmbedding.injective',\n  map_op := MyEmbedding.map_op' }\n\n-- [Insert `CoeFun`, `ext` and `copy` here]\n```\n\nThe second step is to add instances of your new `MyEmbeddingClass` for all types extending\n`MyEmbedding`.\nTypically, you can just declare a new class analogous to `MyEmbeddingClass`:\n\n```\nstructure CoolerEmbedding (A B : Type _) [CoolClass A] [CoolClass B]\n  extends MyEmbedding A B :=\n(map_cool' : toFun CoolClass.cool = CoolClass.cool)\n\nsection\nset_option old_structure_cmd true\n\nclass CoolerEmbeddingClass (F : Type _) (A B : outParam <| Type _) [CoolClass A] [CoolClass B]\n  extends MyEmbeddingClass F A B :=\n(map_cool : ∀ (f : F), f CoolClass.cool = CoolClass.cool)\n\nend\n\n@[simp] lemma map_cool {F A B : Type _} [CoolClass A] [CoolClass B] [CoolerEmbeddingClass F A B]\n  (f : F) : f CoolClass.cool = CoolClass.cool :=\nMyEmbeddingClass.map_op\n\n-- You can also replace `MyEmbedding.EmbeddingLike` with the below instance:\ninstance : CoolerEmbeddingClass (CoolerEmbedding A B) A B :=\n{ coe := CoolerEmbedding.toFun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := MyEmbedding.injective',\n  map_op := CoolerEmbedding.map_op',\n  map_cool := CoolerEmbedding.map_cool' }\n\n-- [Insert `CoeFun`, `ext` and `copy` here]\n```\n\nThen any declaration taking a specific type of morphisms as parameter can instead take the\nclass you just defined:\n```\n-- Compare with: lemma do_something (f : MyEmbedding A B) : sorry := sorry\nlemma do_something {F : Type _} [MyEmbeddingClass F A B] (f : F) : sorry := sorry\n```\n\nThis means anything set up for `MyEmbedding`s will automatically work for `CoolerEmbeddingClass`es,\nand defining `CoolerEmbeddingClass` only takes a constant amount of effort,\ninstead of linearly increasing the work per `MyEmbedding`-related declaration.\n\n-/\n\n\n/-- The class `EmbeddingLike F α β` expresses that terms of type `F` have an\ninjective coercion to injective functions `α ↪ β`.\n-/\nclass EmbeddingLike (F : Sort _) (α β : outParam (Sort _)) extends FunLike F α fun _ ↦ β where\n  /-- The coercion to functions must produce injective functions. -/\n  injective' : ∀ f : F, @Function.Injective α β (coe f)\n#align embedding_like EmbeddingLike\n\nnamespace EmbeddingLike\n\nvariable {F α β γ : Sort _} [i : EmbeddingLike F α β]\n\nprotected theorem injective (f : F) : Function.Injective f :=\n  injective' f\n#align embedding_like.injective EmbeddingLike.injective\n\n@[simp]\ntheorem apply_eq_iff_eq (f : F) {x y : α} : f x = f y ↔ x = y :=\n  (EmbeddingLike.injective f).eq_iff\n#align embedding_like.apply_eq_iff_eq EmbeddingLike.apply_eq_iff_eq\n\n@[simp]\ntheorem comp_injective {F : Sort _} [EmbeddingLike F β γ] (f : α → β) (e : F) :\n    Function.Injective (e ∘ f) ↔ Function.Injective f :=\n  (EmbeddingLike.injective e).of_comp_iff f\n#align embedding_like.comp_injective EmbeddingLike.comp_injective\n\nend EmbeddingLike\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/FunLike/Embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3413975039337175}}
{"text": "constant f : Nat → Nat\nconstant q : Nat → (Nat → Prop) → Nat\n\n@[simp]\ntheorem ex {x : Nat} {p : Nat → Prop} (h₁ : p x) (h₂ : q x p = x) : f x = x :=\n  sorry\n\nset_option trace.Meta.Tactic.simp.discharge true\ntheorem foo : f (f x) = x := by\n  simp\n  sorry\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/973b.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3412868672061621}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.AffineScheme\nimport ring_theory.nilpotent\nimport topology.sheaves.sheaf_condition.sites\nimport algebra.category.Ring.constructions\nimport ring_theory.local_properties\n\n/-!\n# Basic properties of schemes\n\nWe provide some basic properties of schemes\n\n## Main definition\n* `algebraic_geometry.is_integral`: A scheme is integral if it is nontrivial and all nontrivial\n  components of the structure sheaf are integral domains.\n* `algebraic_geometry.is_reduced`: A scheme is reduced if all the components of the structure sheaf\n  is reduced.\n-/\n\nopen topological_space opposite category_theory category_theory.limits Top\n\nnamespace algebraic_geometry\n\nvariable (X : Scheme)\n\ninstance : t0_space X.carrier :=\nbegin\n  refine t0_space.of_open_cover (λ x, _),\n  obtain ⟨U, R, ⟨e⟩⟩ := X.local_affine x,\n  let e' : U.1 ≃ₜ prime_spectrum R :=\n    homeo_of_iso ((LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _).map_iso e),\n  exact ⟨U.1.1, U.2, U.1.2, e'.embedding.t0_space⟩\nend\n\ninstance : quasi_sober X.carrier :=\nbegin\n  apply_with (quasi_sober_of_open_cover\n    (set.range (λ x, set.range $ (X.affine_cover.map x).1.base)))\n    { instances := ff },\n  { rintro ⟨_,i,rfl⟩, exact (X.affine_cover.is_open i).base_open.open_range },\n  { rintro ⟨_,i,rfl⟩,\n    exact @@open_embedding.quasi_sober _ _ _\n      (homeomorph.of_embedding _ (X.affine_cover.is_open i).base_open.to_embedding)\n      .symm.open_embedding prime_spectrum.quasi_sober },\n  { rw [set.top_eq_univ, set.sUnion_range, set.eq_univ_iff_forall],\n    intro x, exact ⟨_, ⟨_, rfl⟩, X.affine_cover.covers x⟩ }\nend\n\n/-- A scheme `X` is reduced if all `𝒪ₓ(U)` are reduced. -/\nclass is_reduced : Prop :=\n(component_reduced : ∀ U, _root_.is_reduced (X.presheaf.obj (op U)) . tactic.apply_instance)\n\nattribute [instance] is_reduced.component_reduced\n\nlemma is_reduced_of_stalk_is_reduced [∀ x : X.carrier, _root_.is_reduced (X.presheaf.stalk x)] :\n  is_reduced X :=\nbegin\n  refine ⟨λ U, ⟨λ s hs, _⟩⟩,\n  apply presheaf.section_ext X.sheaf U s 0,\n  intro x,\n  rw ring_hom.map_zero,\n  change X.presheaf.germ x s = 0,\n  exact (hs.map _).eq_zero\nend\n\ninstance stalk_is_reduced_of_reduced [is_reduced X] (x : X.carrier) :\n  _root_.is_reduced (X.presheaf.stalk x) :=\nbegin\n  constructor,\n  rintros g ⟨n, e⟩,\n  obtain ⟨U, hxU, s, rfl⟩ := X.presheaf.germ_exist x g,\n  rw [← map_pow, ← map_zero (X.presheaf.germ ⟨x, hxU⟩)] at e,\n  obtain ⟨V, hxV, iU, iV, e'⟩ := X.presheaf.germ_eq x hxU hxU _ 0 e,\n  rw [map_pow, map_zero] at e',\n  replace e' := (is_nilpotent.mk _ _ e').eq_zero,\n  erw ← concrete_category.congr_hom (X.presheaf.germ_res iU ⟨x, hxV⟩) s,\n  rw [comp_apply, e', map_zero]\nend\n\nlemma is_reduced_of_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]\n  [is_reduced Y] : is_reduced X :=\nbegin\n  constructor,\n  intro U,\n  have : U = (opens.map f.1.base).obj (H.base_open.is_open_map.functor.obj U),\n  { ext1, exact (set.preimage_image_eq _ H.base_open.inj).symm },\n  rw this,\n  exact is_reduced_of_injective (inv $ f.1.c.app (op $ H.base_open.is_open_map.functor.obj U))\n    (as_iso $ f.1.c.app (op $ H.base_open.is_open_map.functor.obj U) : Y.presheaf.obj _ ≅ _).symm\n      .CommRing_iso_to_ring_equiv.injective\nend\n\ninstance {R : CommRing} [H : _root_.is_reduced R] : is_reduced (Scheme.Spec.obj $ op R) :=\nbegin\n  apply_with is_reduced_of_stalk_is_reduced { instances := ff },\n  intro x, dsimp,\n  haveI : _root_.is_reduced (CommRing.of $ localization.at_prime (prime_spectrum.as_ideal x)),\n  { dsimp, apply_instance },\n  exact is_reduced_of_injective (structure_sheaf.stalk_iso R x).hom\n    (structure_sheaf.stalk_iso R x).CommRing_iso_to_ring_equiv.injective,\nend\n\nlemma affine_is_reduced_iff (R : CommRing) :\n  is_reduced (Scheme.Spec.obj $ op R) ↔ _root_.is_reduced R :=\nbegin\n  refine ⟨_, λ h, by exactI infer_instance⟩,\n  intro h,\n  resetI,\n  haveI : _root_.is_reduced (LocallyRingedSpace.Γ.obj (op $ Spec.to_LocallyRingedSpace.obj $ op R)),\n  { change _root_.is_reduced ((Scheme.Spec.obj $ op R).presheaf.obj $ op ⊤), apply_instance },\n  exact is_reduced_of_injective (to_Spec_Γ R)\n    ((as_iso $ to_Spec_Γ R).CommRing_iso_to_ring_equiv.injective)\nend\n\nlemma is_reduced_of_is_affine_is_reduced [is_affine X]\n  [h : _root_.is_reduced (X.presheaf.obj (op ⊤))] : is_reduced X :=\nbegin\n  haveI : is_reduced (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))),\n  { rw affine_is_reduced_iff, exact h },\n  exact is_reduced_of_open_immersion X.iso_Spec.hom,\nend\n\n/-- To show that a statement `P` holds for all open subsets of all schemes, it suffices to show that\n1. In any scheme `X`, if `P` holds for an open cover of `U`, then `P` holds for `U`.\n2. For an open immerison `f : X ⟶ Y`, if `P` holds for the entire space of `X`, then `P` holds for\n  the image of `f`.\n3. `P` holds for the entire space of an affine scheme.\n-/\nlemma reduce_to_affine_global (P : ∀ (X : Scheme) (U : opens X.carrier), Prop)\n  (h₁ : ∀ (X : Scheme) (U : opens X.carrier),\n    (∀ (x : U), ∃ {V} (h : x.1 ∈ V) (i : V ⟶ U), P X V) → P X U)\n  (h₂ : ∀ {X Y} (f : X ⟶ Y) [hf : is_open_immersion f], ∃ {U : set X.carrier} {V : set Y.carrier}\n    (hU : U = ⊤) (hV : V = set.range f.1.base), P X ⟨U, hU.symm ▸ is_open_univ⟩ →\n      P Y ⟨V, hV.symm ▸ hf.base_open.open_range⟩)\n  (h₃ : ∀ (R : CommRing), P (Scheme.Spec.obj $ op R) ⊤) :\n  ∀ (X : Scheme) (U : opens X.carrier), P X U :=\nbegin\n  intros X U,\n  apply h₁,\n  intro x,\n  obtain ⟨_, ⟨j, rfl⟩, hx, i⟩ := X.affine_basis_cover_is_basis.exists_subset_of_mem_open\n    (set_like.mem_coe.2 x.prop) U.is_open,\n  let U' : opens _ := ⟨_, (X.affine_basis_cover.is_open j).base_open.open_range⟩,\n  let i' : U' ⟶ U :=\n    hom_of_le i,\n  refine ⟨U', hx, i', _⟩,\n  obtain ⟨_,_,rfl,rfl,h₂'⟩ := h₂ (X.affine_basis_cover.map j),\n  apply h₂',\n  apply h₃\nend\n\n\n\nlemma eq_zero_of_basic_open_eq_bot {X : Scheme} [hX : is_reduced X] {U : opens X.carrier}\n  (s : X.presheaf.obj (op U)) (hs : X.basic_open s = ⊥) :\n  s = 0 :=\nbegin\n  apply Top.presheaf.section_ext X.sheaf U,\n  simp_rw ring_hom.map_zero,\n  unfreezingI { revert X U hX s },\n  refine reduce_to_affine_global _ _ _ _,\n  { intros X U hx hX s hs x,\n    obtain ⟨V, hx, i, H⟩ := hx x,\n    unfreezingI { specialize H (X.presheaf.map i.op s) },\n    erw Scheme.basic_open_res at H,\n    rw [hs] at H,\n    specialize H inf_bot_eq ⟨x, hx⟩,\n    erw Top.presheaf.germ_res_apply at H,\n    exact H },\n  { rintros X Y f hf,\n    have e : (f.val.base) ⁻¹' set.range ⇑(f.val.base) = set.univ,\n    { rw [← set.image_univ, set.preimage_image_eq _ hf.base_open.inj] },\n    refine ⟨_, _, e, rfl, _⟩,\n    rintros H hX s hs ⟨_, x, rfl⟩,\n    unfreezingI { haveI := is_reduced_of_open_immersion f },\n    specialize H (f.1.c.app _ s) _ ⟨x, by { rw [opens.mem_mk, e], trivial }⟩,\n    { rw [← Scheme.preimage_basic_open, hs], ext1, simp [opens.map] },\n    { erw ← PresheafedSpace.stalk_map_germ_apply f.1 ⟨_,_⟩ ⟨x,_⟩ at H,\n      apply_fun (inv $ PresheafedSpace.stalk_map f.val x) at H,\n      erw [category_theory.is_iso.hom_inv_id_apply, map_zero] at H,\n      exact H } },\n  { intros R hX s hs x,\n    erw [basic_open_eq_of_affine', prime_spectrum.basic_open_eq_bot_iff] at hs,\n    replace hs := (hs.map (Spec_Γ_identity.app R).inv),\n    -- what the hell?!\n    replace hs := @is_nilpotent.eq_zero _ _ _ _ (show _, from _) hs,\n    rw iso.hom_inv_id_apply at hs,\n    rw [hs, map_zero],\n    exact @@is_reduced.component_reduced hX ⊤ }\nend\n\n@[simp]\nlemma basic_open_eq_bot_iff {X : Scheme} [is_reduced X] {U : opens X.carrier}\n  (s : X.presheaf.obj $ op U) :\n  X.basic_open s = ⊥ ↔ s = 0 :=\nbegin\n  refine ⟨eq_zero_of_basic_open_eq_bot s, _⟩,\n  rintro rfl,\n  simp,\nend\n\n/-- A scheme `X` is integral if its carrier is nonempty,\nand `𝒪ₓ(U)` is an integral domain for each `U ≠ ∅`. -/\nclass is_integral : Prop :=\n(nonempty : nonempty X.carrier . tactic.apply_instance)\n(component_integral : ∀ (U : opens X.carrier) [_root_.nonempty U],\n  is_domain (X.presheaf.obj (op U)) . tactic.apply_instance)\n\nattribute [instance] is_integral.component_integral is_integral.nonempty\n\ninstance [h : is_integral X] : is_domain (X.presheaf.obj (op ⊤)) :=\n@@is_integral.component_integral _ _ (by simp)\n\n@[priority 900]\ninstance is_reduced_of_is_integral [is_integral X] : is_reduced X :=\nbegin\n  constructor,\n  intro U,\n  cases U.1.eq_empty_or_nonempty with h h,\n  { have : U = ⊥ := set_like.ext' h,\n    haveI := CommRing.subsingleton_of_is_terminal (X.sheaf.is_terminal_of_eq_empty this),\n    change _root_.is_reduced (X.sheaf.val.obj (op U)),\n    apply_instance },\n  { haveI : nonempty U := by simpa, apply_instance }\nend\n\ninstance is_irreducible_of_is_integral [is_integral X] : irreducible_space X.carrier :=\nbegin\n  by_contradiction H,\n  replace H : ¬ is_preirreducible (⊤ : set X.carrier) := λ h,\n    H { to_preirreducible_space := ⟨h⟩, to_nonempty := infer_instance },\n  simp_rw [is_preirreducible_iff_closed_union_closed, not_forall, not_or_distrib] at H,\n  rcases H with ⟨S, T, hS, hT, h₁, h₂, h₃⟩,\n  erw not_forall at h₂ h₃,\n  simp_rw not_forall at h₂ h₃,\n  haveI : nonempty (⟨Sᶜ, hS.1⟩ : opens X.carrier) := ⟨⟨_, h₂.some_spec.some_spec⟩⟩,\n  haveI : nonempty (⟨Tᶜ, hT.1⟩ : opens X.carrier) := ⟨⟨_, h₃.some_spec.some_spec⟩⟩,\n  haveI : nonempty (⟨Sᶜ, hS.1⟩ ⊔ ⟨Tᶜ, hT.1⟩ : opens X.carrier) :=\n    ⟨⟨_, or.inl h₂.some_spec.some_spec⟩⟩,\n  let e : X.presheaf.obj _ ≅ CommRing.of _ := (X.sheaf.is_product_of_disjoint ⟨_, hS.1⟩ ⟨_, hT.1⟩ _)\n    .cone_point_unique_up_to_iso (CommRing.prod_fan_is_limit _ _),\n  apply_with false_of_nontrivial_of_product_domain { instances := ff },\n  { exact e.symm.CommRing_iso_to_ring_equiv.is_domain _ },\n  { apply X.to_LocallyRingedSpace.component_nontrivial },\n  { apply X.to_LocallyRingedSpace.component_nontrivial },\n  { ext x,\n    split,\n    { rintros ⟨hS,hT⟩,\n      cases h₁ (show x ∈ ⊤, by trivial),\n      exacts [hS h, hT h] },\n    { intro x, exact x.rec _ } }\nend\n\nlemma is_integral_of_is_irreducible_is_reduced [is_reduced X] [H : irreducible_space X.carrier] :\n  is_integral X :=\nbegin\n  split, intros U hU,\n  haveI := (@@LocallyRingedSpace.component_nontrivial X.to_LocallyRingedSpace U hU).1,\n  haveI : no_zero_divisors\n    (X.to_LocallyRingedSpace.to_SheafedSpace.to_PresheafedSpace.presheaf.obj (op U)),\n  { refine ⟨λ a b e, _⟩,\n    simp_rw [← basic_open_eq_bot_iff, ← opens.not_nonempty_iff_eq_bot],\n    by_contra' h,\n    obtain ⟨_, ⟨x, hx₁, rfl⟩, ⟨x, hx₂, e'⟩⟩ := @@nonempty_preirreducible_inter _ H.1\n      (X.basic_open a).2 (X.basic_open b).2 h.1 h.2,\n    replace e' := subtype.eq e',\n    subst e',\n    replace e := congr_arg (X.presheaf.germ x) e,\n    rw [ring_hom.map_mul, ring_hom.map_zero] at e,\n    refine zero_ne_one' (X.presheaf.stalk x.1) (is_unit_zero_iff.1 _),\n    convert hx₁.mul hx₂,\n    exact e.symm },\n  exact no_zero_divisors.to_is_domain _\nend\n\nlemma is_integral_iff_is_irreducible_and_is_reduced :\n  is_integral X ↔ irreducible_space X.carrier ∧ is_reduced X :=\n⟨λ _, by exactI ⟨infer_instance, infer_instance⟩,\n  λ ⟨_, _⟩, by exactI is_integral_of_is_irreducible_is_reduced X⟩\n\nlemma is_integral_of_open_immersion {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f]\n  [is_integral Y] [nonempty X.carrier] : is_integral X :=\nbegin\n  constructor,\n  intros U hU,\n  have : U = (opens.map f.1.base).obj (H.base_open.is_open_map.functor.obj U),\n  { ext1, exact (set.preimage_image_eq _ H.base_open.inj).symm },\n  rw this,\n  haveI : is_domain (Y.presheaf.obj (op (H.base_open.is_open_map.functor.obj U))),\n  { apply_with is_integral.component_integral { instances := ff },\n    apply_instance,\n    refine ⟨⟨_, _, hU.some.prop, rfl⟩⟩ },\n  exact (as_iso $ f.1.c.app (op $ H.base_open.is_open_map.functor.obj U) :\n    Y.presheaf.obj _ ≅ _).symm.CommRing_iso_to_ring_equiv.is_domain _\nend\n\ninstance {R : CommRing} [H : is_domain R] : is_integral (Scheme.Spec.obj $ op R) :=\nbegin\n  apply_with is_integral_of_is_irreducible_is_reduced { instances := ff },\n  { apply_instance },\n  { dsimp [Spec.Top_obj],\n    apply_instance },\nend\n\nlemma affine_is_integral_iff (R : CommRing) :\n  is_integral (Scheme.Spec.obj $ op R) ↔ is_domain R :=\n⟨λ h, by exactI ring_equiv.is_domain ((Scheme.Spec.obj $ op R).presheaf.obj _)\n  (as_iso $ to_Spec_Γ R).CommRing_iso_to_ring_equiv, λ h, by exactI infer_instance⟩\n\nlemma is_integral_of_is_affine_is_domain [is_affine X] [nonempty X.carrier]\n  [h : is_domain (X.presheaf.obj (op ⊤))] : is_integral X :=\nbegin\n  haveI : is_integral (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))),\n  { rw affine_is_integral_iff, exact h },\n  exact is_integral_of_open_immersion X.iso_Spec.hom,\nend\n\nlemma map_injective_of_is_integral [is_integral X] {U V : opens X.carrier} (i : U ⟶ V)\n  [H : nonempty U] :\n  function.injective (X.presheaf.map i.op) :=\nbegin\n  rw injective_iff_map_eq_zero,\n  intros x hx,\n  rw ← basic_open_eq_bot_iff at ⊢ hx,\n  rw Scheme.basic_open_res at hx,\n  revert hx,\n  contrapose!,\n  simp_rw [← opens.not_nonempty_iff_eq_bot, not_not],\n  apply nonempty_preirreducible_inter U.is_open (RingedSpace.basic_open _ _).is_open,\n  simpa using H\nend\n\nend algebraic_geometry\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/properties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.341048057064018}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Simon Hudon\n-/\nimport category_theory.monoidal.braided\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.terminal\nimport category_theory.pempty\n\n/-!\n# The monoidal structure on a category with chosen finite products.\n\nThis is a variant of the development in `category_theory.monoidal.of_has_finite_products`,\nwhich uses specified choices of the terminal object and binary product,\nenabling the construction of a cartesian category with specific definitions of the tensor unit\nand tensor product.\n\n(Because the construction in `category_theory.monoidal.of_has_finite_products` uses `has_limit`\nclasses, the actual definitions there are opaque behind `classical.choice`.)\n\nWe use this in `category_theory.monoidal.types` to construct the monoidal category of types\nso that the tensor product is the usual cartesian product of types.\n\nFor now we only do the construction from products, and not from coproducts,\nwhich seems less often useful.\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C] {X Y : C}\n\nnamespace limits\n\nsection\nvariables {C}\n\n/-- Swap the two sides of a `binary_fan`. -/\ndef binary_fan.swap {P Q : C} (t : binary_fan P Q) : binary_fan Q P :=\nbinary_fan.mk t.snd t.fst\n\n@[simp] lemma binary_fan.swap_fst {P Q : C} (t : binary_fan P Q) : t.swap.fst = t.snd := rfl\n@[simp] lemma binary_fan.swap_snd {P Q : C} (t : binary_fan P Q) : t.swap.snd = t.fst := rfl\n\n/--\nIf a cone `t` over `P Q` is a limit cone, then `t.swap` is a limit cone over `Q P`.\n-/\n@[simps]\ndef is_limit.swap_binary_fan {P Q : C} {t : binary_fan P Q} (I : is_limit t) : is_limit t.swap :=\n{ lift := λ s, I.lift (binary_fan.swap s),\n  fac' := λ s, by { rintro ⟨⟩; simp, },\n  uniq' := λ s m w,\n  begin\n    have h := I.uniq (binary_fan.swap s) m,\n    rw h,\n    intro j,\n    specialize w j.swap,\n    cases j; exact w,\n  end }\n\n/--\nConstruct `has_binary_product Q P` from `has_binary_product P Q`.\nThis can't be an instance, as it would cause a loop in typeclass search.\n-/\nlemma has_binary_product.swap (P Q : C) [has_binary_product P Q] : has_binary_product Q P :=\nhas_limit.mk ⟨binary_fan.swap (limit.cone (pair P Q)), (limit.is_limit (pair P Q)).swap_binary_fan⟩\n\n/--\nGiven a limit cone over `X` and `Y`, and another limit cone over `Y` and `X`, we can construct\nan isomorphism between the cone points. Relative to some fixed choice of limits cones for every\npair, these isomorphisms constitute a braiding.\n-/\ndef binary_fan.braiding {X Y : C}\n  {s : binary_fan X Y} (P : is_limit s) {t : binary_fan Y X} (Q : is_limit t) :\n  s.X ≅ t.X :=\nis_limit.cone_point_unique_up_to_iso P Q.swap_binary_fan\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `sXY.X Z`,\nif `sYZ` is a limit cone we can construct a binary fan over `X sYZ.X`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  binary_fan X sYZ.X :=\nbinary_fan.mk (s.fst ≫ sXY.fst) (Q.lift (binary_fan.mk (s.fst ≫ sXY.snd) s.snd))\n\n@[simp] lemma binary_fan.assoc_fst {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).fst = s.fst ≫ sXY.fst := rfl\n@[simp] lemma binary_fan.assoc_snd {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).snd = Q.lift (binary_fan.mk (s.fst ≫ sXY.snd) s.snd) := rfl\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `X sYZ.X`,\nif `sYZ` is a limit cone we can construct a binary fan over `sXY.X Z`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc_inv {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  binary_fan sXY.X Z :=\nbinary_fan.mk (P.lift (binary_fan.mk s.fst (s.snd ≫ sYZ.fst))) (s.snd ≫ sYZ.snd)\n\n@[simp] lemma binary_fan.assoc_inv_fst {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).fst = P.lift (binary_fan.mk s.fst (s.snd ≫ sYZ.fst)) := rfl\n@[simp] lemma binary_fan.assoc_inv_snd {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).snd = s.snd ≫ sYZ.snd := rfl\n\n/--\nIf all the binary fans involved a limit cones, `binary_fan.assoc` produces another limit cone.\n-/\n@[simps]\ndef is_limit.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) : is_limit (s.assoc Q) :=\n{ lift := λ t, R.lift (binary_fan.assoc_inv P t),\n  fac' := λ t,\n  begin\n    rintro ⟨⟩; simp,\n    apply Q.hom_ext,\n    rintro ⟨⟩; simp,\n  end,\n  uniq' := λ t m w,\n  begin\n    have h := R.uniq (binary_fan.assoc_inv P t) m,\n    rw h,\n    rintro ⟨⟩; simp,\n    apply P.hom_ext,\n    rintro ⟨⟩; simp,\n    { exact w walking_pair.left, },\n    { specialize w walking_pair.right,\n      simp at w,\n      rw [←w], simp, },\n    { specialize w walking_pair.right,\n      simp at w,\n      rw [←w], simp, },\n  end, }\n\n/--\nGiven two pairs of limit cones corresponding to the parenthesisations of `X × Y × Z`,\nwe obtain an isomorphism between the cone points.\n-/\n@[reducible]\ndef binary_fan.associator {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) {t : binary_fan X sYZ.X} (S : is_limit t) :\n  s.X ≅ t.X :=\nis_limit.cone_point_unique_up_to_iso (is_limit.assoc P Q R) S\n\n/--\nGiven a fixed family of limit data for every pair `X Y`, we obtain an associator.\n-/\n@[reducible]\ndef binary_fan.associator_of_limit_cone\n  (L : Π X Y : C, limit_cone (pair X Y)) (X Y Z : C) :\n  (L (L X Y).cone.X Z).cone.X ≅ (L X (L Y Z).cone.X).cone.X :=\nbinary_fan.associator\n  (L X Y).is_limit (L Y Z).is_limit\n  (L (L X Y).cone.X Z).is_limit (L X (L Y Z).cone.X).is_limit\n\n/--\nConstruct a left unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.left_unitor\n  {X : C} {s : cone (functor.empty C)} (P : is_limit s) {t : binary_fan s.X X} (Q : is_limit t) :\n  t.X ≅ X :=\n{ hom := t.snd,\n  inv := Q.lift (binary_fan.mk (P.lift { X := X, π := { app := pempty.rec _ } }) (𝟙 X) ),\n  hom_inv_id' := by { apply Q.hom_ext, rintro ⟨⟩, { apply P.hom_ext, rintro ⟨⟩, }, { simp, }, }, }\n\n/--\nConstruct a right unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.right_unitor\n  {X : C} {s : cone (functor.empty C)} (P : is_limit s) {t : binary_fan X s.X} (Q : is_limit t) :\n  t.X ≅ X :=\n{ hom := t.fst,\n  inv := Q.lift (binary_fan.mk (𝟙 X) (P.lift { X := X, π := { app := pempty.rec _ } })),\n  hom_inv_id' := by { apply Q.hom_ext, rintro ⟨⟩, { simp, }, { apply P.hom_ext, rintro ⟨⟩, }, }, }\n\nend\n\nend limits\n\nopen category_theory.limits\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\nvariables {C}\nvariables (𝒯 : limit_cone (functor.empty C))\nvariables (ℬ : Π (X Y : C), limit_cone (pair X Y))\n\nnamespace monoidal_of_chosen_finite_products\n\n/-- Implementation of the tensor product for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_obj (X Y : C) : C := (ℬ X Y).cone.X\n\n/-- Implementation of the tensor product of morphisms for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_hom {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) : tensor_obj ℬ W Y ⟶ tensor_obj ℬ X Z :=\n  (binary_fan.is_limit.lift' (ℬ X Z).is_limit\n    ((ℬ W Y).cone.π.app walking_pair.left ≫ f)\n    (((ℬ W Y).cone.π.app walking_pair.right : (ℬ W Y).cone.X ⟶ Y) ≫ g)).val\n\nlemma tensor_id (X₁ X₂ : C) : tensor_hom ℬ (𝟙 X₁) (𝟙 X₂) = 𝟙 (tensor_obj ℬ X₁ X₂) :=\nbegin\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C}\n  (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :\n  tensor_hom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) =\n    tensor_hom ℬ f₁ f₂ ≫ tensor_hom ℬ g₁ g₂ :=\nbegin\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma pentagon (W X Y Z : C) :\n  tensor_hom ℬ (binary_fan.associator_of_limit_cone ℬ W X Y).hom (𝟙 Z) ≫\n    (binary_fan.associator_of_limit_cone ℬ W (tensor_obj ℬ X Y) Z).hom ≫\n      tensor_hom ℬ (𝟙 W) (binary_fan.associator_of_limit_cone ℬ X Y Z).hom =\n  (binary_fan.associator_of_limit_cone ℬ (tensor_obj ℬ W X) Y Z).hom ≫\n    (binary_fan.associator_of_limit_cone ℬ W X (tensor_obj ℬ Y Z)).hom :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩,\n  { simp, },\n  { apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩,\n    { simp, },\n    apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩,\n    { simp, },\n    { simp, }, }\nend\n\nlemma triangle (X Y : C) :\n  (binary_fan.associator_of_limit_cone ℬ X 𝒯.cone.X Y).hom ≫\n    tensor_hom ℬ (𝟙 X) (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X Y).is_limit).hom =\n  tensor_hom ℬ (binary_fan.right_unitor 𝒯.is_limit (ℬ X 𝒯.cone.X).is_limit).hom (𝟙 Y) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩; simp,\nend\n\nlemma left_unitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :\n  tensor_hom ℬ (𝟙 𝒯.cone.X) f ≫ (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X X₂).is_limit).hom =\n    (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X X₁).is_limit).hom ≫ f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma right_unitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :\n  tensor_hom ℬ f (𝟙 𝒯.cone.X) ≫\n    (binary_fan.right_unitor 𝒯.is_limit (ℬ X₂ 𝒯.cone.X).is_limit).hom =\n    (binary_fan.right_unitor 𝒯.is_limit (ℬ X₁ 𝒯.cone.X).is_limit).hom ≫ f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :\n  tensor_hom ℬ (tensor_hom ℬ f₁ f₂) f₃ ≫ (binary_fan.associator_of_limit_cone ℬ Y₁ Y₂ Y₃).hom =\n    (binary_fan.associator_of_limit_cone ℬ X₁ X₂ X₃).hom ≫\n      tensor_hom ℬ f₁ (tensor_hom ℬ f₂ f₃) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩,\n  { simp, },\n  { apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟩,\n    { simp, },\n    { simp, }, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/-- A category with a terminal object and binary products has a natural monoidal structure. -/\ndef monoidal_of_chosen_finite_products :\n  monoidal_category C :=\n{ tensor_unit  := 𝒯.cone.X,\n  tensor_obj   := λ X Y, tensor_obj ℬ X Y,\n  tensor_hom   := λ _ _ _ _ f g, tensor_hom ℬ f g,\n  tensor_id'   := tensor_id ℬ,\n  tensor_comp' := λ _ _ _ _ _ _ f₁ f₂ g₁ g₂, tensor_comp ℬ f₁ f₂ g₁ g₂,\n  associator   := λ X Y Z, binary_fan.associator_of_limit_cone ℬ X Y Z,\n  left_unitor  := λ X, binary_fan.left_unitor (𝒯.is_limit) (ℬ 𝒯.cone.X X).is_limit,\n  right_unitor := λ X, binary_fan.right_unitor (𝒯.is_limit) (ℬ X 𝒯.cone.X).is_limit,\n  pentagon'    := pentagon ℬ,\n  triangle'    := triangle 𝒯 ℬ,\n  left_unitor_naturality' := λ _ _ f, left_unitor_naturality 𝒯 ℬ f,\n  right_unitor_naturality' := λ _ _ f, right_unitor_naturality 𝒯 ℬ f,\n  associator_naturality' := λ _ _ _ _ _ _ f₁ f₂ f₃, associator_naturality ℬ f₁ f₂ f₃, }\n\nnamespace monoidal_of_chosen_finite_products\n\nopen monoidal_category\n\n/--\nA type synonym for `C` carrying a monoidal category structure corresponding to\na fixed choice of limit data for the empty functor, and for `pair X Y` for every `X Y : C`.\n\nThis is an implementation detail for `symmetric_of_chosen_finite_products`.\n-/\n@[derive category, nolint unused_arguments has_inhabited_instance]\ndef monoidal_of_chosen_finite_products_synonym\n  (𝒯 : limit_cone (functor.empty C)) (ℬ : Π (X Y : C), limit_cone (pair X Y)):= C\n\ninstance : monoidal_category (monoidal_of_chosen_finite_products_synonym 𝒯 ℬ) :=\nmonoidal_of_chosen_finite_products 𝒯 ℬ\n\nlemma braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :\n  (tensor_hom ℬ f g) ≫ (limits.binary_fan.braiding (ℬ Y Y').is_limit (ℬ Y' Y).is_limit).hom =\n    (limits.binary_fan.braiding (ℬ X X').is_limit (ℬ X' X).is_limit).hom ≫ (tensor_hom ℬ g f) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟩;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\n\n\nlemma hexagon_reverse (X Y Z : C) :\n  (binary_fan.associator_of_limit_cone ℬ X Y Z).inv ≫\n    (limits.binary_fan.braiding\n      (ℬ (tensor_obj ℬ X Y) Z).is_limit\n      (ℬ Z (tensor_obj ℬ X Y)).is_limit).hom ≫\n    (binary_fan.associator_of_limit_cone ℬ Z X Y).inv =\n    (tensor_hom ℬ (𝟙 X) (limits.binary_fan.braiding (ℬ Y Z).is_limit (ℬ Z Y).is_limit).hom) ≫\n      (binary_fan.associator_of_limit_cone ℬ X Z Y).inv ≫\n        (tensor_hom ℬ (limits.binary_fan.braiding (ℬ X Z).is_limit (ℬ Z X).is_limit).hom (𝟙 Y)) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟩,\n  { apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟩;\n    { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n        limits.is_limit.cone_point_unique_up_to_iso],\n      simp, }, },\n  { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n      limits.is_limit.cone_point_unique_up_to_iso],\n    simp, },\nend\n\nlemma symmetry (X Y : C) :\n  (limits.binary_fan.braiding (ℬ X Y).is_limit (ℬ Y X).is_limit).hom ≫\n      (limits.binary_fan.braiding (ℬ Y X).is_limit (ℬ X Y).is_limit).hom =\n    𝟙 (tensor_obj ℬ X Y) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟩;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/--\nThe monoidal structure coming from finite products is symmetric.\n-/\ndef symmetric_of_chosen_finite_products :\n  symmetric_category (monoidal_of_chosen_finite_products_synonym 𝒯 ℬ) :=\n{ braiding := λ X Y, limits.binary_fan.braiding (ℬ _ _).is_limit (ℬ _ _).is_limit,\n  braiding_naturality' := λ X X' Y Y' f g, braiding_naturality ℬ f g,\n  hexagon_forward' := λ X Y Z, hexagon_forward ℬ X Y Z,\n  hexagon_reverse' := λ X Y Z, hexagon_reverse ℬ X Y Z,\n  symmetry' := λ X Y, symmetry ℬ X Y, }\n\nend\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monoidal/of_chosen_finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3410480492395055}}
{"text": "import exp.open\nimport .lc\nimport .subst\n\nnamespace tts ------------------------------------------------------------------\nnamespace exp ------------------------------------------------------------------\nvariables {V : Type} [decidable_eq V] -- Type of variable names\nvariables {v : V} -- Variable names\nvariables {k i : ℕ} -- Natural numbers\nvariables {x y : tagged V} -- Variable names\nvariables {e ex e₁ e₂ : exp V} -- Expressions\nvariables {L : finset (tagged V)} -- Variables sets\n\nopen occurs\n\n@[simp] theorem subst_open_exp (lx : lc ex) :\n  subst x ex (open_exp e₂ k e₁) = open_exp (subst x ex e₂) k (subst x ex e₁) :=\nbegin\n  induction e₁ generalizing k,\n  case exp.var : o y {\n    cases o,\n    { by_cases h : k = y.tag; simp [h] },\n    { by_cases h : x = y; simp [h, open_exp_id lx] } },\n  case exp.app : ef ea rf ra { simp [rf, ra] },\n  case exp.lam : v eb rb { simp [rb] },\n  case exp.let_ : v ed eb rd rb { simp [rd, rb] }\nend\n\n@[simp] theorem subst_open_exp₀ :\n  lc ex → subst x ex (open_exp₀ e₂ e₁) = open_exp₀ (subst x ex e₂) (subst x ex e₁) :=\nsubst_open_exp\n\n@[simp] theorem subst_open_var (lx : lc ex) (h : x ≠ y) :\n  subst x ex (open_var y e₁) = open_var y (subst x ex e₁) :=\nby simp [open_var, h, lx]\n\n@[simp] theorem subst_open_fresh (lx : lc ex) (h : (fresh.tagged v).gen L ≠ x) :\n  subst x ex (open_fresh v L e₁) = open_fresh v L (subst x ex e₁) :=\nsubst_open_var lx (ne.symm h)\n\n/- subst_intro -/\n\ntheorem subst_intro : ∀ {e₁ : exp V} k,\n  x ∉ fv e₁ → open_exp e₂ k e₁ = subst x e₂ (open_exp (var free x) k e₁)\n| (var bound y)  k p := by by_cases h : k = y.tag; simp [h]\n| (var free y)   k p := by simp at p; simp [p]\n| (app ef ea)    k p := by simp at p; simp [subst_intro k p.1, subst_intro k p.2]\n| (lam v eb)     k p := by simp at p; simp [subst_intro (k + 1) p]\n| (let_ v ed eb) k p := by simp at p; simp [subst_intro k p.1, subst_intro (k + 1) p.2]\n\ntheorem subst_intro₀ : x ∉ fv e₁ → open_exp₀ e₂ e₁ = subst x e₂ (open_var x e₁) :=\nsubst_intro 0\n\n-- Locally-closed expressions are stable over substitution\ntheorem subst_lc (x : tagged V) (lx : lc ex) (l : lc e) : lc (subst x ex e) :=\nbegin\n  induction l generalizing ex lx x,\n  case lc.var : y { by_cases h : x = y; simp [h, lx] },\n  case lc.app : ef ea lf la rf ra { exact lc.app (rf lx x) (ra lx x) },\n  case lc.lam : v L eb lb rb {\n    apply lc.lam (L ∪ {x}),\n    intros y Fy,\n    simp [not_or_distrib] at Fy,\n    rw ←subst_open_var lx (ne.symm Fy.2),\n    exact rb Fy.1 lx x },\n  case lc.let_ : v L ed eb ld lb rd rb {\n    apply lc.let_ (L ∪ {x}) (rd lx x),\n    intros y Fy,\n    simp [not_or_distrib] at Fy,\n    rw ←subst_open_var lx (ne.symm Fy.2),\n    exact rb Fy.1 lx x }\nend\n\ntheorem lc_open_exp₀ (v : V) (lb : lc_body e₁) (l₂ : lc e₂) : lc (open_exp₀ e₂ e₁) :=\nbegin\n  cases lb with L l₁,\n  have F : (fresh.tagged v).gen (L ∪ fv e₁) ∉ L ∧ (fresh.tagged v).gen (L ∪ fv e₁) ∉ fv e₁ :=\n    (fresh.tagged v).gen_not_mem_union _ _,\n  rw subst_intro₀ F.2,\n  exact subst_lc _ l₂ (l₁ F.1)\nend\n\nend /- namespace -/ exp --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/exp/subst_open.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3410480492395055}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Abhimanyu Pallavi Sudhir\n-/\nimport algebra.geom_sum\nimport data.nat.choose.sum\nimport data.complex.basic\n/-!\n# Exponential, trigonometric and hyperbolic trigonometric functions\n\nThis file contains the definitions of the real and complex exponential, sine, cosine, tangent,\nhyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.\n\n-/\nlocal notation `abs'` := _root_.abs\nopen is_absolute_value\nopen_locale classical big_operators nat\n\nsection\nopen real is_absolute_value finset\n\nlemma forall_ge_le_of_forall_le_succ {α : Type*} [preorder α] (f : ℕ → α) {m : ℕ}\n  (h : ∀ n ≥ m, f n.succ ≤ f n) : ∀ {l}, ∀ k ≥ m, k ≤ l → f l ≤ f k :=\nbegin\n  assume l k hkm hkl,\n  generalize hp : l - k = p,\n  have : l = k + p := add_comm p k ▸ (nat.sub_eq_iff_eq_add hkl).1 hp,\n  subst this,\n  clear hkl hp,\n  induction p with p ih,\n  { simp },\n  { exact le_trans (h _ (le_trans hkm (nat.le_add_right _ _))) ih }\nend\n\nsection\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] [archimedean α] {abv : β → α} [is_absolute_value abv]\n\nlemma is_cau_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, abs (f n) ≤ a)\n  (hnm : ∀ n ≥ m, f n.succ ≤ f n) : is_cau_seq abs f :=\nλ ε ε0,\nlet ⟨k, hk⟩ := archimedean.arch a ε0 in\nhave h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=\n  ⟨k + k + 1, λ n hnm, lt_of_lt_of_le\n    (show a - (k + (k + 1)) • ε < -abs (f n),\n      from lt_neg.1 $ lt_of_le_of_lt (ham n hnm) (begin\n        rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul],\n        exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk\n          (lt_add_of_pos_right _ ε0)),\n      end))\n    (neg_le.2 $ (abs_neg (f n)) ▸ le_abs_self _)⟩,\nlet l := nat.find h in\nhave hl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε := nat.find_spec h,\nhave hl0 : l ≠ 0 := λ hl0, not_lt_of_ge (ham m (le_refl _))\n  (lt_of_lt_of_le (by have := hl m (le_refl m); simpa [hl0] using this) (le_abs_self (f m))),\nbegin\n  cases not_forall.1\n    (nat.find_min h (nat.pred_lt hl0)) with i hi,\n  rw [not_imp, not_lt] at hi,\n  existsi i,\n  assume j hj,\n  have hfij : f j ≤ f i := forall_ge_le_of_forall_le_succ f hnm _ hi.1 hj,\n  rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add'],\n  exact calc f i ≤ a - (nat.pred l) • ε : hi.2\n    ... = a - l • ε + ε :\n      by conv {to_rhs, rw [← nat.succ_pred_eq_of_pos (nat.pos_of_ne_zero hl0), succ_nsmul',\n        sub_add, add_sub_cancel] }\n    ... < f j + ε : add_lt_add_right (hl j (le_trans hi.1 hj)) _\nend\n\nlemma is_cau_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, abs (f n) ≤ a)\n  (hnm : ∀ n ≥ m, f n ≤ f n.succ) : is_cau_seq abs f :=\nbegin\n  refine @eq.rec_on (ℕ → α) _ (is_cau_seq abs) _ _\n    (-⟨_, @is_cau_of_decreasing_bounded _ _ _ (λ n, -f n) a m (by simpa) (by simpa)⟩ :\n      cau_seq α abs).2,\n  ext,\n  exact neg_neg _\nend\n\nend\n\nsection no_archimedean\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] {abv : β → α} [is_absolute_value abv]\n\nlemma is_cau_series_of_abv_le_cau {f : ℕ → β} {g : ℕ → α} (n : ℕ) :\n  (∀ m, n ≤ m → abv (f m) ≤ g m) →\n  is_cau_seq abs (λ n, ∑ i in range n, g i) →\n  is_cau_seq abv (λ n, ∑ i in range n, f i) :=\nbegin\n  assume hm hg ε ε0,\n  cases hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi,\n  existsi max n i,\n  assume j ji,\n  have hi₁ := hi j (le_trans (le_max_right n i) ji),\n  have hi₂ := hi (max n i) (le_max_right n i),\n  have sub_le := abs_sub_le (∑ k in range j, g k) (∑ k in range i, g k)\n    (∑ k in range (max n i), g k),\n  have := add_lt_add hi₁ hi₂,\n  rw [abs_sub (∑ k in range (max n i), g k), add_halves ε] at this,\n  refine lt_of_le_of_lt (le_trans (le_trans _ (le_abs_self _)) sub_le) this,\n  generalize hk : j - max n i = k,\n  clear this hi₂ hi₁ hi ε0 ε hg sub_le,\n  rw nat.sub_eq_iff_eq_add ji at hk,\n  rw hk,\n  clear hk ji j,\n  induction k with k' hi,\n  { simp [abv_zero abv] },\n  { simp only [nat.succ_add, sum_range_succ_comm, sub_eq_add_neg, add_assoc],\n    refine le_trans (abv_add _ _ _) _,\n    simp only [sub_eq_add_neg] at hi,\n    exact add_le_add (hm _ (le_add_of_nonneg_of_le (nat.zero_le _) (le_max_left _ _))) hi },\nend\n\nlemma is_cau_series_of_abv_cau {f : ℕ → β} : is_cau_seq abs (λ m, ∑ n in range m, abv (f n)) →\n  is_cau_seq abv (λ m, ∑ n in range m, f n) :=\nis_cau_series_of_abv_le_cau 0 (λ n h, le_refl _)\n\nend no_archimedean\n\nsection\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] [archimedean α] {abv : β → α} [is_absolute_value abv]\n\nlemma is_cau_geo_series {β : Type*} [field β] {abv : β → α} [is_absolute_value abv]\n   (x : β) (hx1 : abv x < 1) : is_cau_seq abv (λ n, ∑ m in range n, x ^ m) :=\nhave hx1' : abv x ≠ 1 := λ h, by simpa [h, lt_irrefl] using hx1,\nis_cau_series_of_abv_cau\nbegin\n  simp only [abv_pow abv] {eta := ff},\n  have : (λ (m : ℕ), ∑ n in range m, (abv x) ^ n) =\n   λ m, geom_sum (abv x) m := rfl,\n  simp only [this, geom_sum_eq hx1'] {eta := ff},\n  conv in (_ / _) { rw [← neg_div_neg_eq, neg_sub, neg_sub] },\n  refine @is_cau_of_mono_bounded _ _ _ _ ((1 : α) / (1 - abv x)) 0 _ _,\n  { assume n hn,\n    rw abs_of_nonneg,\n    refine div_le_div_of_le (le_of_lt $ sub_pos.2 hx1)\n      (sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)),\n    refine div_nonneg (sub_nonneg.2 _) (sub_nonneg.2 $ le_of_lt hx1),\n    clear hn,\n    induction n with n ih,\n    { simp },\n    { rw [pow_succ, ← one_mul (1 : α)],\n      refine mul_le_mul (le_of_lt hx1) ih (abv_pow abv x n ▸ abv_nonneg _ _) (by norm_num) } },\n  { assume n hn,\n    refine div_le_div_of_le (le_of_lt $ sub_pos.2 hx1) (sub_le_sub_left _ _),\n    rw [← one_mul (_ ^ n), pow_succ],\n    exact mul_le_mul_of_nonneg_right (le_of_lt hx1) (pow_nonneg (abv_nonneg _ _) _) }\nend\n\nlemma is_cau_geo_series_const (a : α) {x : α} (hx1 : abs x < 1) :\n  is_cau_seq abs (λ m, ∑ n in range m, a * x ^ n) :=\nhave is_cau_seq abs (λ m, a * ∑ n in range m, x ^ n) :=\n  (cau_seq.const abs a * ⟨_, is_cau_geo_series x hx1⟩).2,\nby simpa only [mul_sum]\n\nlemma series_ratio_test {f : ℕ → β} (n : ℕ) (r : α)\n  (hr0 : 0 ≤ r) (hr1 : r < 1) (h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :\n  is_cau_seq abv (λ m, ∑ n in range m, f n) :=\nhave har1 : abs r < 1, by rwa abs_of_nonneg hr0,\nbegin\n  refine is_cau_series_of_abv_le_cau n.succ _\n    (is_cau_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1),\n  assume m hmn,\n  cases classical.em (r = 0) with r_zero r_ne_zero,\n  { have m_pos := lt_of_lt_of_le (nat.succ_pos n) hmn,\n    have := h m.pred (nat.le_of_succ_le_succ (by rwa [nat.succ_pred_eq_of_pos m_pos])),\n    simpa [r_zero, nat.succ_pred_eq_of_pos m_pos, pow_succ] },\n  generalize hk : m - n.succ = k,\n  have r_pos : 0 < r := lt_of_le_of_ne hr0 (ne.symm r_ne_zero),\n  replace hk : m = k + n.succ := (nat.sub_eq_iff_eq_add hmn).1 hk,\n  induction k with k ih generalizing m n,\n  { rw [hk, zero_add, mul_right_comm, inv_pow' _ _, ← div_eq_mul_inv, mul_div_cancel],\n    exact (ne_of_lt (pow_pos r_pos _)).symm },\n  { have kn : k + n.succ ≥ n.succ, by rw ← zero_add n.succ; exact add_le_add (zero_le _) (by simp),\n    rw [hk, nat.succ_add, pow_succ' r, ← mul_assoc],\n    exact le_trans (by rw mul_comm; exact h _ (nat.le_of_succ_le kn))\n      (mul_le_mul_of_nonneg_right (ih (k + n.succ) n h kn rfl) hr0) }\nend\n\nlemma sum_range_diag_flip {α : Type*} [add_comm_monoid α] (n : ℕ) (f : ℕ → ℕ → α) :\n  ∑ m in range n, ∑ k in range (m + 1), f k (m - k) =\n  ∑ m in range n, ∑ k in range (n - m), f m k :=\nby rw [sum_sigma', sum_sigma']; exact sum_bij\n(λ a _, ⟨a.2, a.1 - a.2⟩)\n(λ a ha, have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1,\n  have h₂ : a.2 < nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2,\n    mem_sigma.2 ⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),\n    mem_range.2 ((nat.sub_lt_sub_right_iff (nat.le_of_lt_succ h₂)).2 h₁)⟩)\n(λ _ _, rfl)\n(λ ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ha hb h,\n  have ha : a₁ < n ∧ a₂ ≤ a₁ :=\n      ⟨mem_range.1 (mem_sigma.1 ha).1, nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 ha).2)⟩,\n  have hb : b₁ < n ∧ b₂ ≤ b₁ :=\n      ⟨mem_range.1 (mem_sigma.1 hb).1, nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 hb).2)⟩,\n  have h : a₂ = b₂ ∧ _ := sigma.mk.inj h,\n  have h' : a₁ = b₁ - b₂ + a₂ := (nat.sub_eq_iff_eq_add ha.2).1 (eq_of_heq h.2),\n  sigma.mk.inj_iff.2\n    ⟨nat.sub_add_cancel hb.2 ▸ h'.symm ▸ h.1 ▸ rfl,\n      (heq_of_eq h.1)⟩)\n(λ ⟨a₁, a₂⟩ ha,\n  have ha : a₁ < n ∧ a₂ < n - a₁ :=\n      ⟨mem_range.1 (mem_sigma.1 ha).1, (mem_range.1 (mem_sigma.1 ha).2)⟩,\n  ⟨⟨a₂ + a₁, a₁⟩, ⟨mem_sigma.2 ⟨mem_range.2 (nat.lt_sub_right_iff_add_lt.1 ha.2),\n    mem_range.2 (nat.lt_succ_of_le (nat.le_add_left _ _))⟩,\n  sigma.mk.inj_iff.2 ⟨rfl, heq_of_eq (nat.add_sub_cancel _ _).symm⟩⟩⟩)\n\nlemma sum_range_sub_sum_range {α : Type*} [add_comm_group α] {f : ℕ → α}\n  {n m : ℕ} (hnm : n ≤ m) : ∑ k in range m, f k - ∑ k in range n, f k =\n  ∑ k in (range m).filter (λ k, n ≤ k), f k :=\nbegin\n  rw [← sum_sdiff (@filter_subset _ (λ k, n ≤ k) _ (range m)),\n    sub_eq_iff_eq_add, ← eq_sub_iff_add_eq, add_sub_cancel'],\n  refine finset.sum_congr\n    (finset.ext $ λ a, ⟨λ h, by simp at *; finish,\n    λ h, have ham : a < m := lt_of_lt_of_le (mem_range.1 h) hnm,\n      by simp * at *⟩)\n    (λ _ _, rfl),\nend\n\nend\n\nsection no_archimedean\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] {abv : β → α} [is_absolute_value abv]\n\nlemma abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : finset γ) :\n  abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=\nby haveI := classical.dec_eq γ; exact\nfinset.induction_on s (by simp [abv_zero abv])\n  (λ a s has ih, by rw [sum_insert has, sum_insert has];\n    exact le_trans (abv_add abv _ _) (add_le_add_left ih _))\n\nlemma cauchy_product {a b : ℕ → β}\n  (ha : is_cau_seq abs (λ m, ∑ n in range m, abv (a n)))\n  (hb : is_cau_seq abv (λ m, ∑ n in range m, b n)) (ε : α) (ε0 : 0 < ε) :\n  ∃ i : ℕ, ∀ j ≥ i, abv ((∑ k in range j, a k) * (∑ k in range j, b k) -\n  ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) < ε :=\nlet ⟨Q, hQ⟩ := cau_seq.bounded ⟨_, hb⟩ in\nlet ⟨P, hP⟩ := cau_seq.bounded ⟨_, ha⟩ in\nhave hP0 : 0 < P, from lt_of_le_of_lt (abs_nonneg _) (hP 0),\nhave hPε0 : 0 < ε / (2 * P),\n  from div_pos ε0 (mul_pos (show (2 : α) > 0, from by norm_num) hP0),\nlet ⟨N, hN⟩ := cau_seq.cauchy₂ ⟨_, hb⟩ hPε0 in\nhave hQε0 : 0 < ε / (4 * Q),\n  from div_pos ε0 (mul_pos (show (0 : α) < 4, by norm_num)\n    (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))),\nlet ⟨M, hM⟩ := cau_seq.cauchy₂ ⟨_, ha⟩ hQε0 in\n⟨2 * (max N M + 1), λ K hK,\nhave h₁ : ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) =\n    ∑ m in range K, ∑ n in range (K - m), a m * b n,\n  by simpa using sum_range_diag_flip K (λ m n, a m * b n),\nhave h₂ : (λ i, ∑ k in range (K - i), a i * b k) = (λ i, a i * ∑ k in range (K - i), b k),\n  by simp [finset.mul_sum],\nhave h₃ : ∑ i in range K, a i * ∑ k in range (K - i), b k =\n    ∑ i in range K, a i * (∑ k in range (K - i), b k - ∑ k in range K, b k)\n    + ∑ i in range K, a i * ∑ k in range K, b k,\n  by rw ← sum_add_distrib; simp [(mul_add _ _ _).symm],\nhave two_mul_two : (4 : α) = 2 * 2, by norm_num,\nhave hQ0 : Q ≠ 0, from λ h, by simpa [h, lt_irrefl] using hQε0,\nhave h2Q0 : 2 * Q ≠ 0, from mul_ne_zero two_ne_zero hQ0,\nhave hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε,\n  by rw [← div_div_eq_div_mul, div_mul_cancel _ (ne.symm (ne_of_lt hP0)),\n    two_mul_two, mul_assoc, ← div_div_eq_div_mul, div_mul_cancel _ h2Q0, add_halves],\nhave hNMK : max N M + 1 < K,\n  from lt_of_lt_of_le (by rw two_mul; exact lt_add_of_pos_left _ (nat.succ_pos _)) hK,\nhave hKN : N < K,\n  from calc N ≤ max N M : le_max_left _ _\n  ... < max N M + 1 : nat.lt_succ_self _\n  ... < K : hNMK,\nhave hsumlesum : ∑ i in range (max N M + 1), abv (a i) *\n      abv (∑ k in range (K - i), b k - ∑ k in range K, b k) ≤\n    ∑ i in range (max N M + 1), abv (a i) * (ε / (2 * P)),\n  from sum_le_sum (λ m hmJ, mul_le_mul_of_nonneg_left\n    (le_of_lt (hN (K - m) K\n      (nat.le_sub_left_of_add_le (le_trans\n        (by rw two_mul; exact add_le_add (le_of_lt (mem_range.1 hmJ))\n          (le_trans (le_max_left _ _) (le_of_lt (lt_add_one _)))) hK))\n      (le_of_lt hKN))) (abv_nonneg abv _)),\nhave hsumltP : ∑ n in range (max N M + 1), abv (a n) < P :=\n  calc ∑ n in range (max N M + 1), abv (a n)\n      = abs (∑ n in range (max N M + 1), abv (a n)) :\n  eq.symm (abs_of_nonneg (sum_nonneg (λ x h, abv_nonneg abv (a x))))\n  ... < P : hP (max N M + 1),\nbegin\n  rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv],\n  refine lt_of_le_of_lt (abv_sum_le_sum_abv _ _) _,\n  suffices : ∑ i in range (max N M + 1),\n    abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) +\n    (∑ i in range K, abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) -\n    ∑ i in range (max N M + 1), abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k)) <\n    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q),\n  { rw hε at this, simpa [abv_mul abv] },\n  refine add_lt_add (lt_of_le_of_lt hsumlesum\n    (by rw [← sum_mul, mul_comm]; exact (mul_lt_mul_left hPε0).mpr hsumltP)) _,\n  rw sum_range_sub_sum_range (le_of_lt hNMK),\n  exact calc ∑ i in (range K).filter (λ k, max N M + 1 ≤ k),\n      abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k)\n      ≤ ∑ i in (range K).filter (λ k, max N M + 1 ≤ k), abv (a i) * (2 * Q) :\n    sum_le_sum (λ n hn, begin\n      refine mul_le_mul_of_nonneg_left _ (abv_nonneg _ _),\n      rw sub_eq_add_neg,\n      refine le_trans (abv_add _ _ _) _,\n      rw [two_mul, abv_neg abv],\n      exact add_le_add (le_of_lt (hQ _)) (le_of_lt (hQ _)),\n    end)\n    ... < ε / (4 * Q) * (2 * Q) :\n      by rw [← sum_mul, ← sum_range_sub_sum_range (le_of_lt hNMK)];\n      refine (mul_lt_mul_right $ by rw two_mul;\n        exact add_pos (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))\n          (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))).2\n        (lt_of_le_of_lt (le_abs_self _)\n          (hM _ _ (le_trans (nat.le_succ_of_le (le_max_right _ _)) (le_of_lt hNMK))\n            (nat.le_succ_of_le (le_max_right _ _))))\nend⟩\n\nend no_archimedean\n\nend\n\nopen finset\n\nopen cau_seq\n\nnamespace complex\n\nlemma is_cau_abs_exp (z : ℂ) : is_cau_seq _root_.abs\n  (λ n, ∑ m in range n, abs (z ^ m / m!)) :=\nlet ⟨n, hn⟩ := exists_nat_gt (abs z) in\nhave hn0 : (0 : ℝ) < n, from lt_of_le_of_lt (abs_nonneg _) hn,\nseries_ratio_test n (complex.abs z / n) (div_nonneg (complex.abs_nonneg _) (le_of_lt hn0))\n  (by rwa [div_lt_iff hn0, one_mul])\n  (λ m hm,\n    by rw [abs_abs, abs_abs, nat.factorial_succ, pow_succ,\n      mul_comm m.succ, nat.cast_mul, ← div_div_eq_div_mul, mul_div_assoc,\n      mul_div_right_comm, abs_mul, abs_div, abs_cast_nat];\n    exact mul_le_mul_of_nonneg_right\n      (div_le_div_of_le_left (abs_nonneg _) hn0\n        (nat.cast_le.2 (le_trans hm (nat.le_succ _)))) (abs_nonneg _))\n\nnoncomputable theory\n\nlemma is_cau_exp (z : ℂ) :\n  is_cau_seq abs (λ n, ∑ m in range n, z ^ m / m!) :=\nis_cau_series_of_abv_cau (is_cau_abs_exp z)\n\n/-- The Cauchy sequence consisting of partial sums of the Taylor series of\nthe complex exponential function -/\n@[pp_nodot] def exp' (z : ℂ) :\n  cau_seq ℂ complex.abs :=\n⟨λ n, ∑ m in range n, z ^ m / m!, is_cau_exp z⟩\n\n/-- The complex exponential function, defined via its Taylor series -/\n@[pp_nodot] def exp (z : ℂ) : ℂ := lim (exp' z)\n\n/-- The complex sine function, defined via `exp` -/\n@[pp_nodot] def sin (z : ℂ) : ℂ := ((exp (-z * I) - exp (z * I)) * I) / 2\n\n/-- The complex cosine function, defined via `exp` -/\n@[pp_nodot] def cos (z : ℂ) : ℂ := (exp (z * I) + exp (-z * I)) / 2\n\n/-- The complex tangent function, defined as `sin z / cos z` -/\n@[pp_nodot] def tan (z : ℂ) : ℂ := sin z / cos z\n\n/-- The complex hyperbolic sine function, defined via `exp` -/\n@[pp_nodot] def sinh (z : ℂ) : ℂ := (exp z - exp (-z)) / 2\n\n/-- The complex hyperbolic cosine function, defined via `exp` -/\n@[pp_nodot] def cosh (z : ℂ) : ℂ := (exp z + exp (-z)) / 2\n\n/-- The complex hyperbolic tangent function, defined as `sinh z / cosh z` -/\n@[pp_nodot] def tanh (z : ℂ) : ℂ := sinh z / cosh z\n\nend complex\n\nnamespace real\n\nopen complex\n\n/-- The real exponential function, defined as the real part of the complex exponential -/\n@[pp_nodot] def exp (x : ℝ) : ℝ := (exp x).re\n\n/-- The real sine function, defined as the real part of the complex sine -/\n@[pp_nodot] def sin (x : ℝ) : ℝ := (sin x).re\n\n/-- The real cosine function, defined as the real part of the complex cosine -/\n@[pp_nodot] def cos (x : ℝ) : ℝ := (cos x).re\n\n/-- The real tangent function, defined as the real part of the complex tangent -/\n@[pp_nodot] def tan (x : ℝ) : ℝ := (tan x).re\n\n/-- The real hypebolic sine function, defined as the real part of the complex hyperbolic sine -/\n@[pp_nodot] def sinh (x : ℝ) : ℝ := (sinh x).re\n\n/-- The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine -/\n@[pp_nodot] def cosh (x : ℝ) : ℝ := (cosh x).re\n\n/-- The real hypebolic tangent function, defined as the real part of\nthe complex hyperbolic tangent -/\n@[pp_nodot] def tanh (x : ℝ) : ℝ := (tanh x).re\n\nend real\n\nnamespace complex\n\nvariables (x y : ℂ)\n\n@[simp] lemma exp_zero : exp 0 = 1 :=\nlim_eq_of_equiv_const $\n  λ ε ε0, ⟨1, λ j hj, begin\n  convert ε0,\n  cases j,\n  { exact absurd hj (not_le_of_gt zero_lt_one) },\n  { dsimp [exp'],\n    induction j with j ih,\n    { dsimp [exp']; simp },\n    { rw ← ih dec_trivial,\n      simp only [sum_range_succ, pow_succ],\n      simp } }\nend⟩\n\nlemma exp_add : exp (x + y) = exp x * exp y :=\nshow lim (⟨_, is_cau_exp (x + y)⟩ : cau_seq ℂ abs) =\n  lim (show cau_seq ℂ abs, from ⟨_, is_cau_exp x⟩)\n  * lim (show cau_seq ℂ abs, from ⟨_, is_cau_exp y⟩),\nfrom\nhave hj : ∀ j : ℕ, ∑ m in range j, (x + y) ^ m / m! =\n    ∑ i in range j, ∑ k in range (i + 1), x ^ k / k! * (y ^ (i - k) / (i - k)!),\n  from assume j,\n    finset.sum_congr rfl (λ m hm, begin\n      rw [add_pow, div_eq_mul_inv, sum_mul],\n      refine finset.sum_congr rfl (λ i hi, _),\n      have h₁ : (m.choose i : ℂ) ≠ 0 := nat.cast_ne_zero.2\n        (pos_iff_ne_zero.1 (nat.choose_pos (nat.le_of_lt_succ (mem_range.1 hi)))),\n      have h₂ := nat.choose_mul_factorial_mul_factorial (nat.le_of_lt_succ $ finset.mem_range.1 hi),\n      rw [← h₂, nat.cast_mul, nat.cast_mul, mul_inv', mul_inv'],\n      simp only [mul_left_comm (m.choose i : ℂ), mul_assoc, mul_left_comm (m.choose i : ℂ)⁻¹,\n        mul_comm (m.choose i : ℂ)],\n      rw inv_mul_cancel h₁,\n      simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm]\n    end),\nby rw lim_mul_lim;\n  exact eq.symm (lim_eq_lim_of_equiv (by dsimp; simp only [hj];\n    exact cauchy_product (is_cau_abs_exp x) (is_cau_exp y)))\n\nattribute [irreducible] complex.exp\n\nlemma exp_list_sum (l : list ℂ) : exp l.sum = (l.map exp).prod :=\n@monoid_hom.map_list_prod (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ l\n\nlemma exp_multiset_sum (s : multiset ℂ) : exp s.sum = (s.map exp).prod :=\n@monoid_hom.map_multiset_prod (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ s\n\nlemma exp_sum {α : Type*} (s : finset α) (f : α → ℂ) : exp (∑ x in s, f x) = ∏ x in s, exp (f x) :=\n@monoid_hom.map_prod α (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ f s\n\nlemma exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp(n*x) = (exp x)^n\n| 0 := by rw [nat.cast_zero, zero_mul, exp_zero, pow_zero]\n| (nat.succ n) := by rw [pow_succ', nat.cast_add_one, add_mul, exp_add, ←exp_nat_mul, one_mul]\n\nlemma exp_ne_zero : exp x ≠ 0 :=\nλ h, zero_ne_one $ by rw [← exp_zero, ← add_neg_self x, exp_add, h]; simp\n\nlemma exp_neg : exp (-x) = (exp x)⁻¹ :=\nby rw [← mul_right_inj' (exp_ne_zero x), ← exp_add];\n  simp [mul_inv_cancel (exp_ne_zero x)]\n\nlemma exp_sub : exp (x - y) = exp x / exp y :=\nby simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]\n\n@[simp] lemma exp_conj : exp (conj x) = conj (exp x) :=\nbegin\n  dsimp [exp],\n  rw [← lim_conj],\n  refine congr_arg lim (cau_seq.ext (λ _, _)),\n  dsimp [exp', function.comp, cau_seq_conj],\n  rw conj.map_sum,\n  refine sum_congr rfl (λ n hn, _),\n  rw [conj.map_div, conj.map_pow, ← of_real_nat_cast, conj_of_real]\nend\n\n@[simp] lemma of_real_exp_of_real_re (x : ℝ) : ((exp x).re : ℂ) = exp x :=\neq_conj_iff_re.1 $ by rw [← exp_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_exp (x : ℝ) : (real.exp x : ℂ) = exp x :=\nof_real_exp_of_real_re _\n\n@[simp] lemma exp_of_real_im (x : ℝ) : (exp x).im = 0 :=\nby rw [← of_real_exp_of_real_re, of_real_im]\n\nlemma exp_of_real_re (x : ℝ) : (exp x).re = real.exp x := rfl\n\nlemma two_sinh : 2 * sinh x = exp x - exp (-x) :=\nmul_div_cancel' _ two_ne_zero'\n\nlemma two_cosh : 2 * cosh x = exp x + exp (-x) :=\nmul_div_cancel' _ two_ne_zero'\n\n@[simp] lemma sinh_zero : sinh 0 = 0 := by simp [sinh]\n\n@[simp] lemma sinh_neg : sinh (-x) = -sinh x :=\nby simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]\n\nprivate lemma sinh_add_aux {a b c d : ℂ} :\n  (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) := by ring\n\nlemma sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y :=\nbegin\n  rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), two_sinh,\n      exp_add, neg_add, exp_add, eq_comm,\n      mul_add, ← mul_assoc, two_sinh, mul_left_comm, two_sinh,\n      ← mul_right_inj' (@two_ne_zero' ℂ _ _ _), mul_add,\n      mul_left_comm, two_cosh, ← mul_assoc, two_cosh],\n  exact sinh_add_aux\nend\n\n@[simp] lemma cosh_zero : cosh 0 = 1 := by simp [cosh]\n\n@[simp] lemma cosh_neg : cosh (-x) = cosh x :=\nby simp [add_comm, cosh, exp_neg]\n\nprivate lemma cosh_add_aux {a b c d : ℂ} :\n  (a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) := by ring\n\nlemma cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y :=\nbegin\n  rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), two_cosh,\n      exp_add, neg_add, exp_add, eq_comm,\n      mul_add, ← mul_assoc, two_cosh, ← mul_assoc, two_sinh,\n      ← mul_right_inj' (@two_ne_zero' ℂ _ _ _), mul_add,\n      mul_left_comm, two_cosh, mul_left_comm, two_sinh],\n  exact cosh_add_aux\nend\n\nlemma sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y :=\nby simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]\n\nlemma cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y :=\nby simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]\n\nlemma sinh_conj : sinh (conj x) = conj (sinh x) :=\nby rw [sinh, ← conj.map_neg, exp_conj, exp_conj, ← conj.map_sub, sinh, conj.map_div, conj_bit0,\n  conj.map_one]\n\n@[simp] lemma of_real_sinh_of_real_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x :=\neq_conj_iff_re.1 $ by rw [← sinh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_sinh (x : ℝ) : (real.sinh x : ℂ) = sinh x :=\nof_real_sinh_of_real_re _\n\n@[simp] lemma sinh_of_real_im (x : ℝ) : (sinh x).im = 0 :=\nby rw [← of_real_sinh_of_real_re, of_real_im]\n\nlemma sinh_of_real_re (x : ℝ) : (sinh x).re = real.sinh x := rfl\n\nlemma cosh_conj : cosh (conj x) = conj (cosh x) :=\nbegin\n  rw [cosh, ← conj.map_neg, exp_conj, exp_conj, ← conj.map_add, cosh, conj.map_div,\n      conj_bit0, conj.map_one]\nend\n\n@[simp] lemma of_real_cosh_of_real_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x :=\neq_conj_iff_re.1 $ by rw [← cosh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_cosh (x : ℝ) : (real.cosh x : ℂ) = cosh x :=\nof_real_cosh_of_real_re _\n\n@[simp] lemma cosh_of_real_im (x : ℝ) : (cosh x).im = 0 :=\nby rw [← of_real_cosh_of_real_re, of_real_im]\n\nlemma cosh_of_real_re (x : ℝ) : (cosh x).re = real.cosh x := rfl\n\nlemma tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x := rfl\n\n@[simp] lemma tanh_zero : tanh 0 = 0 := by simp [tanh]\n\n@[simp] lemma tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]\n\nlemma tanh_conj : tanh (conj x) = conj (tanh x) :=\nby rw [tanh, sinh_conj, cosh_conj, ← conj.map_div, tanh]\n\n@[simp] lemma of_real_tanh_of_real_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x :=\neq_conj_iff_re.1 $ by rw [← tanh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_tanh (x : ℝ) : (real.tanh x : ℂ) = tanh x :=\nof_real_tanh_of_real_re _\n\n@[simp] lemma tanh_of_real_im (x : ℝ) : (tanh x).im = 0 :=\nby rw [← of_real_tanh_of_real_re, of_real_im]\n\nlemma tanh_of_real_re (x : ℝ) : (tanh x).re = real.tanh x := rfl\n\nlemma cosh_add_sinh : cosh x + sinh x = exp x :=\nby rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), mul_add,\n       two_cosh, two_sinh, add_add_sub_cancel, two_mul]\n\nlemma sinh_add_cosh : sinh x + cosh x = exp x :=\nby rw [add_comm, cosh_add_sinh]\n\nlemma cosh_sub_sinh : cosh x - sinh x = exp (-x) :=\nby rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), mul_sub,\n       two_cosh, two_sinh, add_sub_sub_cancel, two_mul]\n\nlemma cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 :=\nby rw [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_self, exp_zero]\n\nlemma cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 :=\nbegin\n  rw ← cosh_sq_sub_sinh_sq x,\n  ring\nend\n\nlemma sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 :=\nbegin\n  rw ← cosh_sq_sub_sinh_sq x,\n  ring\nend\n\nlemma cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 :=\nby rw [two_mul, cosh_add, sq, sq]\n\nlemma sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x :=\nbegin\n  rw [two_mul, sinh_add],\n  ring\nend\n\nlemma cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, cosh_add x (2 * x)],\n  simp only [cosh_two_mul, sinh_two_mul],\n  have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2, by ring,\n  rw [h2, sinh_sq],\n  ring\nend\n\nlemma sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, sinh_add x (2 * x)],\n  simp only [cosh_two_mul, sinh_two_mul],\n  have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2, by ring,\n  rw [h2, cosh_sq],\n  ring,\nend\n\n@[simp] lemma sin_zero : sin 0 = 0 := by simp [sin]\n\n@[simp] lemma sin_neg : sin (-x) = -sin x :=\nby simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma two_sin : 2 * sin x = (exp (-x * I) - exp (x * I)) * I :=\nmul_div_cancel' _ two_ne_zero'\n\nlemma two_cos : 2 * cos x = exp (x * I) + exp (-x * I) :=\nmul_div_cancel' _ two_ne_zero'\n\nlemma sinh_mul_I : sinh (x * I) = sin x * I :=\nby rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), two_sinh,\n       ← mul_assoc, two_sin, mul_assoc, I_mul_I, mul_neg_one,\n       neg_sub, neg_mul_eq_neg_mul]\n\nlemma cosh_mul_I : cosh (x * I) = cos x :=\nby rw [← mul_right_inj' (@two_ne_zero' ℂ _ _ _), two_cosh,\n       two_cos, neg_mul_eq_neg_mul]\n\nlemma tanh_mul_I : tanh (x * I) = tan x * I :=\nby rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]\n\nlemma cos_mul_I : cos (x * I) = cosh x :=\nby rw ← cosh_mul_I; ring_nf; simp\n\nlemma sin_mul_I : sin (x * I) = sinh x * I :=\nhave h : I * sin (x * I) = -sinh x := by { rw [mul_comm, ← sinh_mul_I], ring_nf, simp },\nby simpa only [neg_mul_eq_neg_mul_symm, div_I, neg_neg]\n  using cancel_factors.cancel_factors_eq_div h I_ne_zero\n\nlemma tan_mul_I : tan (x * I) = tanh x * I :=\nby rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh]\n\nlemma sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=\nby rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I,\n       add_mul, add_mul, mul_right_comm, ← sinh_mul_I,\n       mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add]\n\n@[simp] lemma cos_zero : cos 0 = 1 := by simp [cos]\n\n@[simp] lemma cos_neg : cos (-x) = cos x :=\nby simp [cos, sub_eq_add_neg, exp_neg, add_comm]\n\nprivate lemma cos_add_aux {a b c d : ℂ} :\n  (a + b) * (c + d) - (b - a) * (d - c) * (-1) =\n  2 * (a * c + b * d) := by ring\n\nlemma cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=\nby rw [← cosh_mul_I, add_mul, cosh_add, cosh_mul_I, cosh_mul_I,\n       sinh_mul_I, sinh_mul_I, mul_mul_mul_comm, I_mul_I,\n       mul_neg_one, sub_eq_add_neg]\n\nlemma sin_sub : sin (x - y) = sin x * cos y - cos x * sin y :=\nby simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]\n\nlemma cos_sub : cos (x - y) = cos x * cos y + sin x * sin y :=\nby simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]\n\nlemma sin_add_mul_I (x y : ℂ) : sin (x + y*I) = sin x * cosh y + cos x * sinh y * I :=\nby rw [sin_add, cos_mul_I, sin_mul_I, mul_assoc]\n\nlemma sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I :=\nby convert sin_add_mul_I z.re z.im; exact (re_add_im z).symm\n\nlemma cos_add_mul_I (x y : ℂ) : cos (x + y*I) = cos x * cosh y - sin x * sinh y * I :=\nby rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc]\n\nlemma cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I :=\nby convert cos_add_mul_I z.re z.im; exact (re_add_im z).symm\n\ntheorem sin_sub_sin : sin x - sin y = 2 * sin((x - y)/2) * cos((x + y)/2) :=\nbegin\n  have s1 := sin_add ((x + y) / 2) ((x - y) / 2),\n  have s2 := sin_sub ((x + y) / 2) ((x - y) / 2),\n  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1,\n  rw [div_sub_div_same, ←sub_add, add_sub_cancel', half_add_self] at s2,\n  rw [s1, s2],\n  ring\nend\n\ntheorem cos_sub_cos : cos x - cos y = -2 * sin((x + y)/2) * sin((x - y)/2) :=\nbegin\n  have s1 := cos_add ((x + y) / 2) ((x - y) / 2),\n  have s2 := cos_sub ((x + y) / 2) ((x - y) / 2),\n  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1,\n  rw [div_sub_div_same, ←sub_add, add_sub_cancel', half_add_self] at s2,\n  rw [s1, s2],\n  ring,\nend\n\n\n\nlemma sin_conj : sin (conj x) = conj (sin x) :=\nby rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I,\n       ← conj_neg_I, ← conj.map_mul, ← conj.map_mul, sinh_conj,\n       mul_neg_eq_neg_mul_symm, sinh_neg, sinh_mul_I, mul_neg_eq_neg_mul_symm]\n\n@[simp] lemma of_real_sin_of_real_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=\neq_conj_iff_re.1 $ by rw [← sin_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_sin (x : ℝ) : (real.sin x : ℂ) = sin x :=\nof_real_sin_of_real_re _\n\n@[simp] lemma sin_of_real_im (x : ℝ) : (sin x).im = 0 :=\nby rw [← of_real_sin_of_real_re, of_real_im]\n\nlemma sin_of_real_re (x : ℝ) : (sin x).re = real.sin x := rfl\n\nlemma cos_conj : cos (conj x) = conj (cos x) :=\nby rw [← cosh_mul_I, ← conj_neg_I, ← conj.map_mul, ← cosh_mul_I,\n       cosh_conj, mul_neg_eq_neg_mul_symm, cosh_neg]\n\n@[simp] lemma of_real_cos_of_real_re (x : ℝ) : ((cos x).re : ℂ) = cos x :=\neq_conj_iff_re.1 $ by rw [← cos_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_cos (x : ℝ) : (real.cos x : ℂ) = cos x :=\nof_real_cos_of_real_re _\n\n@[simp] lemma cos_of_real_im (x : ℝ) : (cos x).im = 0 :=\nby rw [← of_real_cos_of_real_re, of_real_im]\n\nlemma cos_of_real_re (x : ℝ) : (cos x).re = real.cos x := rfl\n\n@[simp] lemma tan_zero : tan 0 = 0 := by simp [tan]\n\nlemma tan_eq_sin_div_cos : tan x = sin x / cos x := rfl\n\nlemma tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x :=\nby rw [tan_eq_sin_div_cos, div_mul_cancel _ hx]\n\n@[simp] lemma tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]\n\nlemma tan_conj : tan (conj x) = conj (tan x) :=\nby rw [tan, sin_conj, cos_conj, ← conj.map_div, tan]\n\n@[simp] lemma of_real_tan_of_real_re (x : ℝ) : ((tan x).re : ℂ) = tan x :=\neq_conj_iff_re.1 $ by rw [← tan_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_tan (x : ℝ) : (real.tan x : ℂ) = tan x :=\nof_real_tan_of_real_re _\n\n@[simp] lemma tan_of_real_im (x : ℝ) : (tan x).im = 0 :=\nby rw [← of_real_tan_of_real_re, of_real_im]\n\nlemma tan_of_real_re (x : ℝ) : (tan x).re = real.tan x := rfl\n\nlemma cos_add_sin_I : cos x + sin x * I = exp (x * I) :=\nby rw [← cosh_add_sinh, sinh_mul_I, cosh_mul_I]\n\nlemma cos_sub_sin_I : cos x - sin x * I = exp (-x * I) :=\nby rw [← neg_mul_eq_neg_mul, ← cosh_sub_sinh, sinh_mul_I, cosh_mul_I]\n\n@[simp] lemma sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=\neq.trans\n  (by rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm])\n  (cosh_sq_sub_sinh_sq (x * I))\n\n@[simp] lemma cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 :=\nby rw [add_comm, sin_sq_add_cos_sq]\n\nlemma cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=\nby rw [two_mul, cos_add, ← sq, ← sq]\n\nlemma cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=\nby rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x),\n       ← sub_add, sub_add_eq_add_sub, two_mul]\n\nlemma sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=\nby rw [two_mul, sin_add, two_mul, add_mul, mul_comm]\n\nlemma cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=\nby simp [cos_two_mul, div_add_div_same, mul_div_cancel_left, two_ne_zero', -one_div]\n\nlemma cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel']\n\nlemma sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel]\n\nlemma inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 :=\nhave cos x ^ 2 ≠ 0, from pow_ne_zero 2 hx,\nby { rw [tan_eq_sin_div_cos, div_pow], field_simp [this] }\n\nlemma tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :\n  tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 :=\nby simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]\n\nlemma cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, cos_add x (2 * x)],\n  simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq],\n  have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2, by ring,\n  rw [h2, cos_sq'],\n  ring\nend\n\nlemma sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, sin_add x (2 * x)],\n  simp only [cos_two_mul, sin_two_mul, cos_sq'],\n  have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2, by ring,\n  rw [h2, cos_sq'],\n  ring\nend\n\nlemma exp_mul_I : exp (x * I) = cos x + sin x * I :=\n(cos_add_sin_I _).symm\n\nlemma exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) :=\nby rw [exp_add, exp_mul_I]\n\nlemma exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) :=\nby rw [← exp_add_mul_I, re_add_im]\n\nlemma exp_re : (exp x).re = real.exp x.re * real.cos x.im :=\nby { rw [exp_eq_exp_re_mul_sin_add_cos], simp [exp_of_real_re, cos_of_real_re] }\n\nlemma exp_im : (exp x).im = real.exp x.re * real.sin x.im :=\nby { rw [exp_eq_exp_re_mul_sin_add_cos], simp [exp_of_real_re, sin_of_real_re] }\n\n/-- De Moivre's formula -/\ntheorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :\n  (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I :=\nbegin\n  rw [← exp_mul_I, ← exp_mul_I],\n  induction n with n ih,\n  { rw [pow_zero, nat.cast_zero, zero_mul, zero_mul, exp_zero] },\n  { rw [pow_succ', ih, nat.cast_succ, add_mul, add_mul, one_mul, exp_add] }\nend\n\nend complex\n\nnamespace real\n\nopen complex\n\nvariables (x y : ℝ)\n\n@[simp] lemma exp_zero : exp 0 = 1 :=\nby simp [real.exp]\n\nlemma exp_add : exp (x + y) = exp x * exp y :=\nby simp [exp_add, exp]\n\nlemma exp_list_sum (l : list ℝ) : exp l.sum = (l.map exp).prod :=\n@monoid_hom.map_list_prod (multiplicative ℝ) ℝ _ _ ⟨exp, exp_zero, exp_add⟩ l\n\nlemma exp_multiset_sum (s : multiset ℝ) : exp s.sum = (s.map exp).prod :=\n@monoid_hom.map_multiset_prod (multiplicative ℝ) ℝ _ _ ⟨exp, exp_zero, exp_add⟩ s\n\nlemma exp_sum {α : Type*} (s : finset α) (f : α → ℝ) : exp (∑ x in s, f x) = ∏ x in s, exp (f x) :=\n@monoid_hom.map_prod α (multiplicative ℝ) ℝ _ _ ⟨exp, exp_zero, exp_add⟩ f s\n\nlemma exp_nat_mul (x : ℝ) : ∀ n : ℕ, exp(n*x) = (exp x)^n\n| 0 := by rw [nat.cast_zero, zero_mul, exp_zero, pow_zero]\n| (nat.succ n) := by rw [pow_succ', nat.cast_add_one, add_mul, exp_add, ←exp_nat_mul, one_mul]\n\nlemma exp_ne_zero : exp x ≠ 0 :=\nλ h, exp_ne_zero x $ by rw [exp, ← of_real_inj] at h; simp * at *\n\nlemma exp_neg : exp (-x) = (exp x)⁻¹ :=\nby rw [← of_real_inj, exp, of_real_exp_of_real_re, of_real_neg, exp_neg,\n  of_real_inv, of_real_exp]\n\nlemma exp_sub : exp (x - y) = exp x / exp y :=\nby simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]\n\n@[simp] lemma sin_zero : sin 0 = 0 := by simp [sin]\n\n@[simp] lemma sin_neg : sin (-x) = -sin x :=\nby simp [sin, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=\nby rw [← of_real_inj]; simp [sin, sin_add]\n\n@[simp] lemma cos_zero : cos 0 = 1 := by simp [cos]\n\n@[simp] lemma cos_neg : cos (-x) = cos x :=\nby simp [cos, exp_neg]\n\nlemma cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=\nby rw ← of_real_inj; simp [cos, cos_add]\n\nlemma sin_sub : sin (x - y) = sin x * cos y - cos x * sin y :=\nby simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]\n\nlemma cos_sub : cos (x - y) = cos x * cos y + sin x * sin y :=\nby simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]\n\nlemma sin_sub_sin : sin x - sin y = 2 * sin((x - y)/2) * cos((x + y)/2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [sin, cos, of_real_sin_of_real_re, of_real_sub, of_real_add, of_real_div, of_real_mul,\n    of_real_one, of_real_bit0],\n  convert sin_sub_sin _ _;\n  norm_cast\nend\n\ntheorem cos_sub_cos : cos x - cos y = -2 * sin((x + y)/2) * sin((x - y)/2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [cos, neg_mul_eq_neg_mul_symm, of_real_sin, of_real_sub, of_real_add,\n    of_real_cos_of_real_re, of_real_div, of_real_mul, of_real_one, of_real_neg, of_real_bit0],\n  convert cos_sub_cos _ _,\n  ring,\nend\n\nlemma cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2)  * cos ((x - y) / 2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [cos, of_real_sub, of_real_add, of_real_cos_of_real_re, of_real_div, of_real_mul,\n    of_real_one, of_real_bit0],\n  convert cos_add_cos _ _;\n  norm_cast,\nend\n\nlemma tan_eq_sin_div_cos : tan x = sin x / cos x :=\nby rw [← of_real_inj, of_real_tan, tan_eq_sin_div_cos, of_real_div, of_real_sin, of_real_cos]\n\nlemma tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x :=\nby rw [tan_eq_sin_div_cos, div_mul_cancel _ hx]\n\n@[simp] lemma tan_zero : tan 0 = 0 := by simp [tan]\n\n@[simp] lemma tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]\n\n@[simp] lemma sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=\nof_real_inj.1 $ by simp\n\n@[simp] lemma cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 :=\nby rw [add_comm, sin_sq_add_cos_sq]\n\nlemma sin_sq_le_one : sin x ^ 2 ≤ 1 :=\nby rw ← sin_sq_add_cos_sq x; exact le_add_of_nonneg_right (sq_nonneg _)\n\nlemma cos_sq_le_one : cos x ^ 2 ≤ 1 :=\nby rw ← sin_sq_add_cos_sq x; exact le_add_of_nonneg_left (sq_nonneg _)\n\nlemma abs_sin_le_one : abs' (sin x) ≤ 1 :=\nabs_le_one_iff_mul_self_le_one.2 $ by simp only [← sq, sin_sq_le_one]\n\nlemma abs_cos_le_one : abs' (cos x) ≤ 1 :=\nabs_le_one_iff_mul_self_le_one.2 $ by simp only [← sq, cos_sq_le_one]\n\nlemma sin_le_one : sin x ≤ 1 :=\n(abs_le.1 (abs_sin_le_one _)).2\n\nlemma cos_le_one : cos x ≤ 1 :=\n(abs_le.1 (abs_cos_le_one _)).2\n\nlemma neg_one_le_sin : -1 ≤ sin x :=\n(abs_le.1 (abs_sin_le_one _)).1\n\nlemma neg_one_le_cos : -1 ≤ cos x :=\n(abs_le.1 (abs_cos_le_one _)).1\n\nlemma cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=\nby rw ← of_real_inj; simp [cos_two_mul]\n\nlemma cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=\nby rw ← of_real_inj; simp [cos_two_mul']\n\nlemma sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=\nby rw ← of_real_inj; simp [sin_two_mul]\n\nlemma cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=\nof_real_inj.1 $ by simpa using cos_sq x\n\nlemma cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel']\n\nlemma sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=\neq_sub_iff_add_eq.2 $ sin_sq_add_cos_sq _\n\nlemma abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) :\n  abs' (sin x) = sqrt (1 - cos x ^ 2) :=\nby rw [← sin_sq, sqrt_sq_eq_abs]\n\nlemma abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) :\n  abs' (cos x) = sqrt (1 - sin x ^ 2) :=\nby rw [← cos_sq', sqrt_sq_eq_abs]\n\nlemma inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2  :=\nhave complex.cos x ≠ 0, from mt (congr_arg re) hx,\nof_real_inj.1 $ by simpa using complex.inv_one_add_tan_sq this\n\nlemma tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :\n  tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 :=\nby simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]\n\nlemma inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :\n  (sqrt (1 + tan x ^ 2))⁻¹ = cos x :=\nby rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne']\n\nlemma tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :\n  tan x / sqrt (1 + tan x ^ 2) = sin x :=\nby rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]\n\nlemma cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x :=\nby rw ← of_real_inj; simp [cos_three_mul]\n\nlemma sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 :=\nby rw ← of_real_inj; simp [sin_three_mul]\n\n/-- The definition of `sinh` in terms of `exp`. -/\nlemma sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 :=\neq_div_of_mul_eq two_ne_zero $ by rw [sinh, exp, exp, complex.of_real_neg, complex.sinh, mul_two,\n    ← complex.add_re, ← mul_two, div_mul_cancel _ (two_ne_zero' : (2 : ℂ) ≠ 0), complex.sub_re]\n\n@[simp] lemma sinh_zero : sinh 0 = 0 := by simp [sinh]\n\n@[simp] lemma sinh_neg : sinh (-x) = -sinh x :=\nby simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y :=\nby rw ← of_real_inj; simp [sinh_add]\n\n/-- The definition of `cosh` in terms of `exp`. -/\nlemma cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 :=\neq_div_of_mul_eq two_ne_zero $ by rw [cosh, exp, exp, complex.of_real_neg, complex.cosh, mul_two,\n    ← complex.add_re, ← mul_two, div_mul_cancel _ (two_ne_zero' : (2 : ℂ) ≠ 0), complex.add_re]\n\n@[simp] lemma cosh_zero : cosh 0 = 1 := by simp [cosh]\n\n@[simp] lemma cosh_neg : cosh (-x) = cosh x :=\nby simp [cosh, exp_neg]\n\nlemma cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y :=\nby rw ← of_real_inj; simp [cosh, cosh_add]\n\nlemma sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y :=\nby simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]\n\nlemma cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y :=\nby simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]\n\nlemma tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=\nof_real_inj.1 $ by simp [tanh_eq_sinh_div_cosh]\n\n@[simp] lemma tanh_zero : tanh 0 = 0 := by simp [tanh]\n\n@[simp] lemma tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]\n\nlemma cosh_add_sinh : cosh x + sinh x = exp x :=\nby rw ← of_real_inj; simp [cosh_add_sinh]\n\nlemma sinh_add_cosh : sinh x + cosh x = exp x :=\nby rw ← of_real_inj; simp [sinh_add_cosh]\n\nlemma cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 :=\nby rw ← of_real_inj; simp [cosh_sq_sub_sinh_sq]\n\nlemma cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 :=\nby rw ← of_real_inj; simp [cosh_sq]\n\nlemma sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 :=\nby rw ← of_real_inj; simp [sinh_sq]\n\nlemma cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 :=\nby rw ← of_real_inj; simp [cosh_two_mul]\n\nlemma sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x :=\nby rw ← of_real_inj; simp [sinh_two_mul]\n\nlemma cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x :=\nby rw ← of_real_inj; simp [cosh_three_mul]\n\nlemma sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x :=\nby rw ← of_real_inj; simp [sinh_three_mul]\n\nopen is_absolute_value\n\n/- TODO make this private and prove ∀ x -/\nlemma add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x :=\ncalc x + 1 ≤ lim (⟨(λ n : ℕ, ((exp' x) n).re), is_cau_seq_re (exp' x)⟩ : cau_seq ℝ abs') :\n  le_lim (cau_seq.le_of_exists ⟨2,\n    λ j hj, show x + (1 : ℝ) ≤ (∑ m in range j, (x ^ m / m! : ℂ)).re,\n      from have h₁ : (((λ m : ℕ, (x ^ m / m! : ℂ)) ∘ nat.succ) 0).re = x, by simp,\n      have h₂ : ((x : ℂ) ^ 0 / 0!).re = 1, by simp,\n      begin\n        rw [← nat.sub_add_cancel hj, sum_range_succ', sum_range_succ',\n          add_re, add_re, h₁, h₂, add_assoc,\n          ← @sum_hom _ _ _ _ _ _ _ complex.re\n            (is_add_group_hom.to_is_add_monoid_hom _)],\n        refine le_add_of_nonneg_of_le (sum_nonneg (λ m hm, _)) (le_refl _),\n        rw [← of_real_pow, ← of_real_nat_cast, ← of_real_div, of_real_re],\n        exact div_nonneg (pow_nonneg hx _) (nat.cast_nonneg _),\n      end⟩)\n... = exp x : by rw [exp, complex.exp, ← cau_seq_re, lim_re]\n\nlemma one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x :=\nby linarith [add_one_le_exp_of_nonneg hx]\n\nlemma exp_pos (x : ℝ) : 0 < exp x :=\n(le_total 0 x).elim (lt_of_lt_of_le zero_lt_one ∘ one_le_exp)\n  (λ h, by rw [← neg_neg x, real.exp_neg];\n    exact inv_pos.2 (lt_of_lt_of_le zero_lt_one (one_le_exp (neg_nonneg.2 h))))\n\n@[simp] lemma abs_exp (x : ℝ) : abs' (exp x) = exp x :=\nabs_of_pos (exp_pos _)\n\nlemma exp_strict_mono : strict_mono exp :=\nλ x y h, by rw [← sub_add_cancel y x, real.exp_add];\n  exact (lt_mul_iff_one_lt_left (exp_pos _)).2\n    (lt_of_lt_of_le (by linarith) (add_one_le_exp_of_nonneg (by linarith)))\n\n@[mono] lemma exp_monotone : ∀ {x y : ℝ}, x ≤ y → exp x ≤ exp y := exp_strict_mono.monotone\n\n@[simp] lemma exp_lt_exp {x y : ℝ} : exp x < exp y ↔ x < y := exp_strict_mono.lt_iff_lt\n\n@[simp] lemma exp_le_exp {x y : ℝ} : exp x ≤ exp y ↔ x ≤ y := exp_strict_mono.le_iff_le\n\nlemma exp_injective : function.injective exp := exp_strict_mono.injective\n\n@[simp] lemma exp_eq_exp {x y : ℝ} : exp x = exp y ↔ x = y := exp_injective.eq_iff\n\n@[simp] lemma exp_eq_one_iff : exp x = 1 ↔ x = 0 :=\nby rw [← exp_zero, exp_injective.eq_iff]\n\n@[simp] lemma one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x :=\nby rw [← exp_zero, exp_lt_exp]\n\n@[simp] lemma exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0 :=\nby rw [← exp_zero, exp_lt_exp]\n\n@[simp] lemma exp_le_one_iff {x : ℝ} : exp x ≤ 1 ↔ x ≤ 0 :=\nexp_zero ▸ exp_le_exp\n\n@[simp] lemma one_le_exp_iff {x : ℝ} : 1 ≤ exp x ↔ 0 ≤ x :=\nexp_zero ▸ exp_le_exp\n\n/-- `real.cosh` is always positive -/\nlemma cosh_pos (x : ℝ) : 0 < real.cosh x :=\n(cosh_eq x).symm ▸ half_pos (add_pos (exp_pos x) (exp_pos (-x)))\n\nend real\n\nnamespace complex\n\nlemma sum_div_factorial_le {α : Type*} [linear_ordered_field α] (n j : ℕ) (hn : 0 < n) :\n  ∑ m in filter (λ k, n ≤ k) (range j), (1 / m! : α) ≤ n.succ / (n! * n) :=\ncalc ∑ m in filter (λ k, n ≤ k) (range j), (1 / m! : α)\n    = ∑ m in range (j - n), 1 / (m + n)! :\n  sum_bij (λ m _, m - n)\n    (λ m hm, mem_range.2 $ (nat.sub_lt_sub_right_iff (by simp at hm; tauto)).2\n      (by simp at hm; tauto))\n    (λ m hm, by rw nat.sub_add_cancel; simp at *; tauto)\n    (λ a₁ a₂ ha₁ ha₂ h,\n      by rwa [nat.sub_eq_iff_eq_add, ← nat.sub_add_comm, eq_comm, nat.sub_eq_iff_eq_add,\n              add_left_inj, eq_comm] at h;\n        simp at *; tauto)\n    (λ b hb, ⟨b + n,\n      mem_filter.2 ⟨mem_range.2 $ nat.add_lt_of_lt_sub_right (mem_range.1 hb), nat.le_add_left _ _⟩,\n      by rw nat.add_sub_cancel⟩)\n... ≤ ∑ m in range (j - n), (n! * n.succ ^ m)⁻¹ :\n  begin\n    refine  sum_le_sum (assume m n, _),\n    rw [one_div, inv_le_inv],\n    { rw [← nat.cast_pow, ← nat.cast_mul, nat.cast_le, add_comm],\n      exact nat.factorial_mul_pow_le_factorial },\n    { exact nat.cast_pos.2 (nat.factorial_pos _) },\n    { exact mul_pos (nat.cast_pos.2 (nat.factorial_pos _))\n        (pow_pos (nat.cast_pos.2 (nat.succ_pos _)) _) },\n  end\n... = n!⁻¹ * ∑ m in range (j - n), n.succ⁻¹ ^ m :\n  by simp [mul_inv', mul_sum.symm, sum_mul.symm, -nat.factorial_succ, mul_comm, inv_pow']\n... = (n.succ - n.succ * n.succ⁻¹ ^ (j - n)) / (n! * n) :\n  have h₁ : (n.succ : α) ≠ 1, from @nat.cast_one α _ _ ▸ mt nat.cast_inj.1\n        (mt nat.succ.inj (pos_iff_ne_zero.1 hn)),\n  have h₂ : (n.succ : α) ≠ 0, from nat.cast_ne_zero.2 (nat.succ_ne_zero _),\n  have h₃ : (n! * n : α) ≠ 0,\n    from mul_ne_zero (nat.cast_ne_zero.2 (pos_iff_ne_zero.1 (nat.factorial_pos _)))\n    (nat.cast_ne_zero.2 (pos_iff_ne_zero.1 hn)),\n  have h₄ : (n.succ - 1 : α) = n, by simp,\n  by rw [← geom_sum_def, geom_sum_inv h₁ h₂, eq_div_iff_mul_eq h₃,\n      mul_comm _ (n! * n : α), ← mul_assoc (n!⁻¹ : α), ← mul_inv_rev', h₄,\n      ← mul_assoc (n! * n : α), mul_comm (n : α) n!, mul_inv_cancel h₃];\n    simp [mul_add, add_mul, mul_assoc, mul_comm]\n... ≤ n.succ / (n! * n) :\n  begin\n    refine iff.mpr (div_le_div_right (mul_pos _ _)) _,\n    exact nat.cast_pos.2 (nat.factorial_pos _),\n    exact nat.cast_pos.2 hn,\n    exact sub_le_self _\n      (mul_nonneg (nat.cast_nonneg _) (pow_nonneg (inv_nonneg.2 (nat.cast_nonneg _)) _))\n  end\n\nlemma exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :\n  abs (exp x - ∑ m in range n, x ^ m / m!) ≤ abs x ^ n * (n.succ * (n! * n)⁻¹) :=\nbegin\n  rw [← lim_const (∑ m in range n, _), exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs],\n  refine lim_le (cau_seq.le_of_exists ⟨n, λ j hj, _⟩),\n  simp_rw ← sub_eq_add_neg,\n  show abs (∑ m in range j, x ^ m / m! - ∑ m in range n, x ^ m / m!)\n    ≤ abs x ^ n * (n.succ * (n! * n)⁻¹),\n  rw sum_range_sub_sum_range hj,\n  exact calc abs (∑ m in (range j).filter (λ k, n ≤ k), (x ^ m / m! : ℂ))\n      = abs (∑ m in (range j).filter (λ k, n ≤ k), (x ^ n * (x ^ (m - n) / m!) : ℂ)) :\n    begin\n      refine congr_arg abs (sum_congr rfl (λ m hm, _)),\n      rw [mem_filter, mem_range] at hm,\n      rw [← mul_div_assoc, ← pow_add, nat.add_sub_cancel' hm.2]\n    end\n  ... ≤ ∑ m in filter (λ k, n ≤ k) (range j), abs (x ^ n * (_ / m!)) : abv_sum_le_sum_abv _ _\n  ... ≤ ∑ m in filter (λ k, n ≤ k) (range j), abs x ^ n * (1 / m!) :\n    begin\n      refine sum_le_sum (λ m hm, _),\n      rw [abs_mul, abv_pow abs, abs_div, abs_cast_nat],\n      refine mul_le_mul_of_nonneg_left ((div_le_div_right _).2 _) _,\n      exact nat.cast_pos.2 (nat.factorial_pos _),\n      rw abv_pow abs,\n      exact (pow_le_one _ (abs_nonneg _) hx),\n      exact pow_nonneg (abs_nonneg _) _\n    end\n  ... = abs x ^ n * (∑ m in (range j).filter (λ k, n ≤ k), (1 / m! : ℝ)) :\n    by simp [abs_mul, abv_pow abs, abs_div, mul_sum.symm]\n  ... ≤ abs x ^ n * (n.succ * (n! * n)⁻¹) :\n    mul_le_mul_of_nonneg_left (sum_div_factorial_le _ _ hn) (pow_nonneg (abs_nonneg _) _)\nend\n\nlemma abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) :\n  abs (exp x - 1) ≤ 2 * abs x :=\ncalc abs (exp x - 1) = abs (exp x - ∑ m in range 1, x ^ m / m!) :\n  by simp [sum_range_succ]\n... ≤ abs x ^ 1 * ((nat.succ 1) * (1! * (1 : ℕ))⁻¹) :\n  exp_bound hx dec_trivial\n... = 2 * abs x : by simp [two_mul, mul_two, mul_add, mul_comm]\n\nlemma abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) :\n  abs (exp x - 1 - x) ≤ (abs x)^2 :=\ncalc abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m!) :\n  by simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc]\n... ≤ (abs x)^2 * (nat.succ 2 * (2! * (2 : ℕ))⁻¹) :\n  exp_bound hx dec_trivial\n... ≤ (abs x)^2 * 1 :\n  mul_le_mul_of_nonneg_left (by norm_num) (sq_nonneg (abs x))\n... = (abs x)^2 :\n  by rw [mul_one]\n\nend complex\n\nnamespace real\n\nopen complex finset\n\nlemma exp_bound {x : ℝ} (hx : abs' x ≤ 1) {n : ℕ} (hn : 0 < n) :\n  abs' (exp x - ∑ m in range n, x ^ m / m!) ≤ abs' x ^ n * (n.succ / (n! * n)) :=\nbegin\n  have hxc : complex.abs x ≤ 1, by exact_mod_cast hx,\n  convert exp_bound hxc hn; norm_cast\nend\n\n/-- A finite initial segment of the exponential series, followed by an arbitrary tail.\nFor fixed `n` this is just a linear map wrt `r`, and each map is a simple linear function\nof the previous (see `exp_near_succ`), with `exp_near n x r ⟶ exp x` as `n ⟶ ∞`,\nfor any `r`. -/\ndef exp_near (n : ℕ) (x r : ℝ) : ℝ := ∑ m in range n, x ^ m / m! + x ^ n / n! * r\n\n@[simp] theorem exp_near_zero (x r) : exp_near 0 x r = r := by simp [exp_near]\n\n@[simp] theorem exp_near_succ (n x r) : exp_near (n + 1) x r = exp_near n x (1 + x / (n+1) * r) :=\nby simp [exp_near, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,\n  mul_inv']; ac_refl\n\ntheorem exp_near_sub (n x r₁ r₂) : exp_near n x r₁ - exp_near n x r₂ = x ^ n / n! * (r₁ - r₂) :=\nby simp [exp_near, mul_sub]\n\nlemma exp_approx_end (n m : ℕ) (x : ℝ)\n  (e₁ : n + 1 = m) (h : abs' x ≤ 1) :\n  abs' (exp x - exp_near m x 0) ≤ abs' x ^ m / m! * ((m+1)/m) :=\nby { simp [exp_near], convert exp_bound h _ using 1, field_simp [mul_comm], linarith }\n\nlemma exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ)\n  (e₁ : n + 1 = m) (a₂ b₂ : ℝ)\n  (e : abs' (1 + x / m * a₂ - a₁) ≤ b₁ - abs' x / m * b₂)\n  (h : abs' (exp x - exp_near m x a₂) ≤ abs' x ^ m / m! * b₂) :\n  abs' (exp x - exp_near n x a₁) ≤ abs' x ^ n / n! * b₁ :=\nbegin\n  refine (_root_.abs_sub_le _ _ _).trans ((add_le_add_right h _).trans _),\n  subst e₁, rw [exp_near_succ, exp_near_sub, _root_.abs_mul],\n  convert mul_le_mul_of_nonneg_left (le_sub_iff_add_le'.1 e) _,\n  { simp [mul_add, pow_succ', div_eq_mul_inv, _root_.abs_mul, _root_.abs_inv, ← pow_abs, mul_inv'],\n    ac_refl },\n  { simp [_root_.div_nonneg, _root_.abs_nonneg] }\nend\n\nlemma exp_approx_end' {n} {x a b : ℝ} (m : ℕ)\n  (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm) (h : abs' x ≤ 1)\n  (e : abs' (1 - a) ≤ b - abs' x / rm * ((rm+1)/rm)) :\n  abs' (exp x - exp_near n x a) ≤ abs' x ^ n / n! * b :=\nby subst er; exact\nexp_approx_succ _ e₁ _ _ (by simpa using e) (exp_approx_end _ _ _ e₁ h)\n\nlemma exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ}\n  (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)\n  (h : abs' (exp 1 - exp_near m 1 ((a₁ - 1) * rm)) ≤ abs' 1 ^ m / m! * (b₁ * rm)) :\n  abs' (exp 1 - exp_near n 1 a₁) ≤ abs' 1 ^ n / n! * b₁ :=\nbegin\n  subst er,\n  refine exp_approx_succ _ en _ _ _ h,\n  field_simp [show (m : ℝ) ≠ 0, by norm_cast; linarith],\nend\n\nlemma exp_approx_start (x a b : ℝ)\n  (h : abs' (exp x - exp_near 0 x a) ≤ abs' x ^ 0 / 0! * b) :\n  abs' (exp x - a) ≤ b :=\nby simpa using h\n\nlemma cos_bound {x : ℝ} (hx : abs' x ≤ 1) :\n  abs' (cos x - (1 - x ^ 2 / 2)) ≤ abs' x ^ 4 * (5 / 96) :=\ncalc abs' (cos x - (1 - x ^ 2 / 2)) = abs (complex.cos x - (1 - x ^ 2 / 2)) :\n  by rw ← abs_of_real; simp [of_real_bit0, of_real_one, of_real_inv]\n... = abs ((complex.exp (x * I) + complex.exp (-x * I) - (2 - x ^ 2)) / 2) :\n  by simp [complex.cos, sub_div, add_div, neg_div, div_self (@two_ne_zero' ℂ _ _ _)]\n... = abs (((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) +\n    ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!))) / 2) :\n  congr_arg abs (congr_arg (λ x : ℂ, x / 2) begin\n    simp only [sum_range_succ],\n    simp [pow_succ],\n    apply complex.ext; simp [div_eq_mul_inv, norm_sq]; ring\n  end)\n... ≤ abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) / 2) +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) / 2) :\n  by rw add_div; exact abs_add _ _\n... = (abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) / 2 +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!)) / 2) :\n  by simp [complex.abs_div]\n... ≤ ((complex.abs (x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2 +\n    (complex.abs (-x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2)  :\n  add_le_add ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n             ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n... ≤ abs' x ^ 4 * (5 / 96) : by norm_num; simp [mul_assoc, mul_comm, mul_left_comm, mul_div_assoc]\n\nlemma sin_bound {x : ℝ} (hx : abs' x ≤ 1) :\n  abs' (sin x - (x - x ^ 3 / 6)) ≤ abs' x ^ 4 * (5 / 96) :=\ncalc abs' (sin x - (x - x ^ 3 / 6)) = abs (complex.sin x - (x - x ^ 3 / 6)) :\n  by rw ← abs_of_real; simp [of_real_bit0, of_real_one, of_real_inv]\n... = abs (((complex.exp (-x * I) - complex.exp (x * I)) * I - (2 * x - x ^ 3 / 3)) / 2) :\n  by simp [complex.sin, sub_div, add_div, neg_div, mul_div_cancel_left _ (@two_ne_zero' ℂ _ _ _),\n    div_div_eq_div_mul, show (3 : ℂ) * 2 = 6, by norm_num]\n... = abs ((((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) -\n    (complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) * I) / 2) :\n  congr_arg abs (congr_arg (λ x : ℂ, x / 2) begin\n    simp only [sum_range_succ],\n    simp [pow_succ],\n    apply complex.ext; simp [div_eq_mul_inv, norm_sq]; ring\n  end)\n... ≤ abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) * I / 2) +\n    abs (-((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) * I) / 2) :\n  by rw [sub_mul, sub_eq_add_neg, add_div]; exact abs_add _ _\n... = (abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) / 2 +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!)) / 2) :\n  by simp [add_comm, complex.abs_div, complex.abs_mul]\n... ≤ ((complex.abs (x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2 +\n    (complex.abs (-x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2) :\n  add_le_add ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n             ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n... ≤ abs' x ^ 4 * (5 / 96) : by norm_num; simp [mul_assoc, mul_comm, mul_left_comm, mul_div_assoc]\n\nlemma cos_pos_of_le_one {x : ℝ} (hx : abs' x ≤ 1) : 0 < cos x :=\ncalc 0 < (1 - x ^ 2 / 2) - abs' x ^ 4 * (5 / 96) :\n  sub_pos.2 $ lt_sub_iff_add_lt.2\n    (calc abs' x ^ 4 * (5 / 96) + x ^ 2 / 2\n          ≤ 1 * (5 / 96) + 1 / 2 :\n        add_le_add\n          (mul_le_mul_of_nonneg_right (pow_le_one _ (abs_nonneg _) hx) (by norm_num))\n          ((div_le_div_right (by norm_num)).2 (by rw [sq, ← abs_mul_self, _root_.abs_mul];\n            exact mul_le_one hx (abs_nonneg _) hx))\n      ... < 1 : by norm_num)\n... ≤ cos x : sub_le.1 (abs_sub_le_iff.1 (cos_bound hx)).2\n\nlemma sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=\ncalc 0 < x - x ^ 3 / 6 - abs' x ^ 4 * (5 / 96) :\n  sub_pos.2 $ lt_sub_iff_add_lt.2\n    (calc abs' x ^ 4 * (5 / 96) + x ^ 3 / 6\n        ≤ x * (5 / 96) + x / 6 :\n      add_le_add\n        (mul_le_mul_of_nonneg_right\n          (calc abs' x ^ 4 ≤ abs' x ^ 1 : pow_le_pow_of_le_one (abs_nonneg _)\n                (by rwa _root_.abs_of_nonneg (le_of_lt hx0))\n                dec_trivial\n            ... = x : by simp [_root_.abs_of_nonneg (le_of_lt (hx0))]) (by norm_num))\n        ((div_le_div_right (by norm_num)).2\n          (calc x ^ 3 ≤ x ^ 1 : pow_le_pow_of_le_one (le_of_lt hx0) hx dec_trivial\n            ... = x : pow_one _))\n    ... < x : by linarith)\n... ≤ sin x : sub_le.1 (abs_sub_le_iff.1 (sin_bound\n    (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]))).2\n\nlemma sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=\nhave x / 2 ≤ 1, from (div_le_iff (by norm_num)).mpr (by simpa),\ncalc 0 < 2 * sin (x / 2) * cos (x / 2) :\n  mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))\n    (cos_pos_of_le_one (by rwa [_root_.abs_of_nonneg (le_of_lt (half_pos hx0))]))\n... = sin x : by rw [← sin_two_mul, two_mul, add_halves]\n\nlemma cos_one_le : cos 1 ≤ 2 / 3 :=\ncalc cos 1 ≤ abs' (1 : ℝ) ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :\n  sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by simp))).1\n... ≤ 2 / 3 : by norm_num\n\nlemma cos_one_pos : 0 < cos 1 := cos_pos_of_le_one (by simp)\n\nlemma cos_two_neg : cos 2 < 0 :=\ncalc cos 2 = cos (2 * 1) : congr_arg cos (mul_one _).symm\n... = _ : real.cos_two_mul 1\n... ≤ 2 * (2 / 3) ^ 2 - 1 :\n  sub_le_sub_right (mul_le_mul_of_nonneg_left\n    (by rw [sq, sq]; exact\n      mul_self_le_mul_self (le_of_lt cos_one_pos)\n        cos_one_le)\n    (by norm_num)) _\n... < 0 : by norm_num\n\nend real\n\nnamespace complex\n\nlemma abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 :=\nhave _ := real.sin_sq_add_cos_sq x,\nby simp [add_comm, abs, norm_sq, sq, *, sin_of_real_re, cos_of_real_re, mul_re] at *\n\nlemma abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re :=\nby rw [exp_eq_exp_re_mul_sin_add_cos, exp_eq_exp_re_mul_sin_add_cos y,\n    abs_mul, abs_mul, abs_cos_add_sin_mul_I, abs_cos_add_sin_mul_I,\n    ← of_real_exp, ← of_real_exp, abs_of_nonneg (le_of_lt (real.exp_pos _)),\n    abs_of_nonneg (le_of_lt (real.exp_pos _)), mul_one, mul_one];\n  exact ⟨λ h, real.exp_injective h, congr_arg _⟩\n\n@[simp] lemma abs_exp_of_real (x : ℝ) : abs (exp x) = real.exp x :=\nby rw [← of_real_exp]; exact abs_of_nonneg (le_of_lt (real.exp_pos _))\n\nend complex\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/complex/exponential.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3410480492395055}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module init.meta.contradiction_tactic\n! leanprover-community/mathlib commit 5f99056c1ae94b618114de51cd8e22522043a6bd\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Meta.Tactic\nimport Leanbin.Init.Function\n\nnamespace Tactic\n\nopen Expr Tactic Decidable Environment\n\nprivate unsafe def contra_p_not_p : List expr → List expr → tactic Unit\n  | [], Hs => failed\n  | H1 :: Rs, Hs => do\n    let t ← extract_opt_auto_param <$> infer_type H1 >>= whnf\n    (do\n          let a ← match_not t\n          let H2 ← find_same_type a Hs\n          let tgt ← target\n          let pr ← mk_app `absurd [tgt, H2, H1]\n          exact pr) <|>\n        contra_p_not_p Rs Hs\n#align tactic.contra_p_not_p tactic.contra_p_not_p\n\nprivate unsafe def contra_false : List expr → tactic Unit\n  | [] => failed\n  | H :: Hs => do\n    let t ← extract_opt_auto_param <$> infer_type H\n    if is_false t then do\n        let tgt ← target\n        let pr ← mk_app `false.rec [tgt, H]\n        exact pr\n      else contra_false Hs\n#align tactic.contra_false tactic.contra_false\n\nprivate unsafe def contra_not_a_refl_rel_a : List expr → tactic Unit\n  | [] => failed\n  | H :: Hs => do\n    let t ← extract_opt_auto_param <$> infer_type H >>= head_beta\n    (do\n          let (lhs, rhs) ← match_ne t\n          unify lhs rhs\n          let tgt ← target\n          let refl_pr ← mk_app `eq.refl [lhs]\n          mk_app `absurd [tgt, refl_pr, H] >>= exact) <|>\n        (do\n            let p ← match_not t\n            let (refl_lemma, lhs, rhs) ← match_refl_app p\n            unify lhs rhs\n            let tgt ← target\n            let refl_pr ← mk_app refl_lemma [lhs]\n            mk_app `absurd [tgt, refl_pr, H] >>= exact) <|>\n          contra_not_a_refl_rel_a Hs\n#align tactic.contra_not_a_refl_rel_a tactic.contra_not_a_refl_rel_a\n\nprivate unsafe def contra_constructor_eq : List expr → tactic Unit\n  | [] => failed\n  | H :: Hs => do\n    let t ← extract_opt_auto_param <$> infer_type H >>= whnf\n    match t with\n      | q(($(lhs_0) : $(α)) = $(rhs_0)) => do\n        let env ← get_env\n        let lhs ← whnf lhs_0\n        let rhs ← whnf rhs_0\n        if\n              is_constructor_app env lhs ∧\n                is_constructor_app env rhs ∧\n                  const_name (get_app_fn lhs) ≠ const_name (get_app_fn rhs) then\n            do\n            let tgt ← target\n            let I_name ← return <| Name.getPrefix (const_name (get_app_fn lhs))\n            let pr ← mk_app (.str I_name \"no_confusion\") [tgt, lhs, rhs, H]\n            exact pr\n          else contra_constructor_eq Hs\n      | _ => contra_constructor_eq Hs\n#align tactic.contra_constructor_eq tactic.contra_constructor_eq\n\nunsafe def contradiction : tactic Unit := do\n  try intro1\n  let ctx ← local_context\n  contra_false ctx <|>\n      contra_not_a_refl_rel_a ctx <|>\n        contra_p_not_p ctx ctx <|> contra_constructor_eq ctx <|> fail \"contradiction tactic failed\"\n#align tactic.contradiction tactic.contradiction\n\nunsafe def exfalso : tactic Unit := do\n  fail_if_no_goals\n  assert `Hfalse (expr.const `false [])\n  swap\n  contradiction\n#align tactic.exfalso tactic.exfalso\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Meta/ContradictionTactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3409323976297497}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.over\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.category_theory.limits.creates\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# Limits and colimits in the over and under categories\n\nShow that the forgetful functor `forget X : over X ⥤ C` creates colimits, and hence `over X` has\nany colimits that `C` has (as well as the dual that `forget X : under X ⟶ C` creates limits).\n\nNote that the folder `category_theory.limits.shapes.constructions.over` further shows that\n`forget X : over X ⥤ C` creates connected limits (so `over X` has connected limits), and that\n`over X` has `J`-indexed products if `C` has `J`-indexed wide pullbacks.\n\nTODO: If `C` has binary products, then `forget X : over X ⥤ C` has a right adjoint.\n-/\n\nnamespace category_theory.functor\n\n\n/-- We can interpret a functor `F` into the category of arrows with codomain `X` as a cocone over\n    the diagram given by the domains of the arrows in the image of `F` such that the apex of the\n    cocone is `X`. -/\n@[simp] theorem to_cocone_X {J : Type v} [small_category J] {C : Type u} [category C] {X : C} (F : J ⥤ over X) : limits.cocone.X (to_cocone F) = X :=\n  Eq.refl (limits.cocone.X (to_cocone F))\n\n/-- We can interpret a functor `F` into the category of arrows with domain `X` as a cone over the\n    diagram given by the codomains of the arrows in the image of `F` such that the apex of the cone\n    is `X`. -/\n@[simp] theorem to_cone_X {J : Type v} [small_category J] {C : Type u} [category C] {X : C} (F : J ⥤ under X) : limits.cone.X (to_cone F) = X :=\n  Eq.refl (limits.cone.X (to_cone F))\n\nend category_theory.functor\n\n\nnamespace category_theory.over\n\n\nprotected instance forget.category_theory.limits.reflects_colimits {C : Type u} [category C] {X : C} : limits.reflects_colimits (forget X) :=\n  limits.reflects_colimits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      limits.reflects_colimits_of_shape.mk\n        fun (F : J ⥤ over X) =>\n          limits.reflects_colimit.mk\n            fun (c : limits.cocone F) (t : limits.is_colimit (functor.map_cocone (forget X) c)) =>\n              limits.is_colimit.mk\n                fun (s : limits.cocone F) => hom_mk (limits.is_colimit.desc t (functor.map_cocone (forget X) s))\n\nprotected instance forget.category_theory.creates_colimits {C : Type u} [category C] {X : C} : creates_colimits (forget X) :=\n  creates_colimits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      creates_colimits_of_shape.mk\n        fun (K : J ⥤ over X) =>\n          creates_colimit.mk\n            fun (c : limits.cocone (K ⋙ forget X)) (t : limits.is_colimit c) =>\n              liftable_cocone.mk\n                (limits.cocone.mk (mk (limits.is_colimit.desc t (functor.to_cocone K)))\n                  (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cocone.ι c) j)))\n                (limits.cocones.ext\n                  (iso.refl\n                    (limits.cocone.X\n                      (functor.map_cocone (forget X)\n                        (limits.cocone.mk (mk (limits.is_colimit.desc t (functor.to_cocone K)))\n                          (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cocone.ι c) j))))))\n                  sorry)\n\nprotected instance has_colimit {J : Type v} [small_category J] {C : Type u} [category C] {X : C} {F : J ⥤ over X} [limits.has_colimit (F ⋙ forget X)] : limits.has_colimit F :=\n  has_colimit_of_created F (forget X)\n\nprotected instance has_colimits_of_shape {J : Type v} [small_category J] {C : Type u} [category C] {X : C} [limits.has_colimits_of_shape J C] : limits.has_colimits_of_shape J (over X) :=\n  limits.has_colimits_of_shape.mk fun (F : J ⥤ over X) => over.has_colimit\n\nprotected instance has_colimits {C : Type u} [category C] {X : C} [limits.has_colimits C] : limits.has_colimits (over X) :=\n  limits.has_colimits.mk fun (J : Type v) (𝒥 : small_category J) => over.has_colimits_of_shape\n\n-- We can automatically infer that the forgetful functor preserves colimits\n\nend category_theory.over\n\n\nnamespace category_theory.under\n\n\nprotected instance forget.category_theory.limits.reflects_limits {C : Type u} [category C] {X : C} : limits.reflects_limits (forget X) :=\n  limits.reflects_limits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      limits.reflects_limits_of_shape.mk\n        fun (F : J ⥤ under X) =>\n          limits.reflects_limit.mk\n            fun (c : limits.cone F) (t : limits.is_limit (functor.map_cone (forget X) c)) =>\n              limits.is_limit.mk\n                fun (s : limits.cone F) => hom_mk (limits.is_limit.lift t (functor.map_cone (forget X) s))\n\nprotected instance forget.category_theory.creates_limits {C : Type u} [category C] {X : C} : creates_limits (forget X) :=\n  creates_limits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      creates_limits_of_shape.mk\n        fun (K : J ⥤ under X) =>\n          creates_limit.mk\n            fun (c : limits.cone (K ⋙ forget X)) (t : limits.is_limit c) =>\n              liftable_cone.mk\n                (limits.cone.mk (mk (limits.is_limit.lift t (functor.to_cone K)))\n                  (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cone.π c) j)))\n                (limits.cones.ext\n                  (iso.refl\n                    (limits.cone.X\n                      (functor.map_cone (forget X)\n                        (limits.cone.mk (mk (limits.is_limit.lift t (functor.to_cone K)))\n                          (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cone.π c) j))))))\n                  sorry)\n\nprotected instance has_limit {J : Type v} [small_category J] {C : Type u} [category C] {X : C} {F : J ⥤ under X} [limits.has_limit (F ⋙ forget X)] : limits.has_limit F :=\n  has_limit_of_created F (forget X)\n\nprotected instance has_limits_of_shape {J : Type v} [small_category J] {C : Type u} [category C] {X : C} [limits.has_limits_of_shape J C] : limits.has_limits_of_shape J (under X) :=\n  limits.has_limits_of_shape.mk fun (F : J ⥤ under X) => under.has_limit\n\nprotected instance has_limits {C : Type u} [category C] {X : C} [limits.has_limits C] : limits.has_limits (under X) :=\n  limits.has_limits.mk fun (J : Type v) (𝒥 : small_category J) => under.has_limits_of_shape\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3409323905448783}}
{"text": "import data.matrix.notation\n\nimport .short_exact_sequence\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nnamespace homological_complex\n\nvariables (C : Type u) [category.{v} C] [abelian C]\nvariables {ι : Type*} {c : complex_shape ι}\n\nabbreviation Fst : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Fst C).map_homological_complex _\n\nabbreviation Snd : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Snd C).map_homological_complex _\n\nabbreviation Trd : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Trd C).map_homological_complex _\n\nabbreviation Fst_Snd : Fst C ⟶ Snd C :=\nnat_trans.map_homological_complex (short_exact_sequence.f_nat C) _\n\nabbreviation Snd_Trd : Snd C ⟶ Trd C :=\nnat_trans.map_homological_complex (short_exact_sequence.g_nat C) _\n\nvariables {C}\n\nvariables (A : chain_complex (short_exact_sequence C) ℕ)\n\ninstance Fst_Snd_mono (n : ℕ) : mono (((Fst_Snd C).app A).f n) := (A.X n).mono'\n\ninstance Snd_Trd_epi (n : ℕ) : epi (((Snd_Trd C).app A).f n) := (A.X n).epi'\n\nlemma Fst_Snd_Trd_exact (n : ℕ) : exact (((Fst_Snd C).app A).f n) (((Snd_Trd C).app A).f n) :=\n(A.X n).exact'\n\nend homological_complex", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/lte/for_mathlib/homological_complex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3406685820202251}}
{"text": "/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.bundled\nimport Mathlib.category_theory.monoidal.End\nimport Mathlib.category_theory.monoidal.Mon_\nimport Mathlib.category_theory.category.Cat\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n\n# The equivalence between `Monad C` and `Mon_ (C ⥤ C)`.\n\nA monad \"is just\" a monoid in the category of endofunctors.\n\n# Definitions/Theorems\n\n1. `to_Mon` associates a monoid object in `C ⥤ C` to any monad on `C`.\n2. `Monad_to_Mon` is the functorial version of `to_Mon`.\n3. `of_Mon` associates a monad on `C` to any monoid object in `C ⥤ C`.\n4. `Monad_Mon_equiv` is the equivalence between `Monad C` and `Mon_ (C ⥤ C)`.\n\n-/\n\nnamespace category_theory\n\n\nnamespace Monad\n\n\n/-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/\n@[simp] theorem to_Mon_mul {C : Type u} [category C] : ∀ (ᾰ : Monad C), Mon_.mul (to_Mon ᾰ) = μ_ :=\n  fun (ᾰ : Monad C) => Eq.refl (Mon_.mul (to_Mon ᾰ))\n\n/-- Passing from `Monad C` to `Mon_ (C ⥤ C)` is functorial. -/\n@[simp] theorem Monad_to_Mon_obj (C : Type u) [category C] :\n    ∀ (ᾰ : Monad C), functor.obj (Monad_to_Mon C) ᾰ = to_Mon ᾰ :=\n  fun (ᾰ : Monad C) => Eq.refl (functor.obj (Monad_to_Mon C) ᾰ)\n\n/-- To every monoid object in `C ⥤ C` we associate a `Monad C`. -/\ndef of_Mon {C : Type u} [category C] : Mon_ (C ⥤ C) → Monad C :=\n  fun (M : Mon_ (C ⥤ C)) => mk (Mon_.X M)\n\n/-- Passing from `Mon_ (C ⥤ C)` to `Monad C` is functorial. -/\ndef Mon_to_Monad (C : Type u) [category C] : Mon_ (C ⥤ C) ⥤ Monad C :=\n  functor.mk of_Mon\n    fun (_x _x_1 : Mon_ (C ⥤ C)) (f : _x ⟶ _x_1) =>\n      monad_hom.mk (nat_trans.mk (nat_trans.app (Mon_.hom.hom f)))\n\nnamespace Monad_Mon_equiv\n\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\n@[simp] theorem counit_iso_hom_app_hom {C : Type u} [category C] (_x : Mon_ (C ⥤ C)) :\n    Mon_.hom.hom (nat_trans.app (iso.hom counit_iso) _x) = 𝟙 :=\n  Eq.refl (Mon_.hom.hom (nat_trans.app (iso.hom counit_iso) _x))\n\n/-- Auxilliary definition for `Monad_Mon_equiv` -/\ndef unit_iso_hom {C : Type u} [category C] : 𝟭 ⟶ Monad_to_Mon C ⋙ Mon_to_Monad C :=\n  nat_trans.mk fun (_x : Monad C) => monad_hom.mk (nat_trans.mk fun (_x_1 : C) => 𝟙)\n\n/-- Auxilliary definition for `Monad_Mon_equiv` -/\n@[simp] theorem unit_iso_inv_app_to_nat_trans_app {C : Type u} [category C] (_x : Monad C) :\n    ∀ (_x_1 : C), nat_trans.app (monad_hom.to_nat_trans (nat_trans.app unit_iso_inv _x)) _x_1 = 𝟙 :=\n  fun (_x_1 : C) =>\n    Eq.refl (nat_trans.app (monad_hom.to_nat_trans (nat_trans.app unit_iso_inv _x)) _x_1)\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\ndef unit_iso {C : Type u} [category C] : 𝟭 ≅ Monad_to_Mon C ⋙ Mon_to_Monad C :=\n  iso.mk unit_iso_hom unit_iso_inv\n\nend Monad_Mon_equiv\n\n\n/-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/\ndef Monad_Mon_equiv (C : Type u) [category C] : Monad C ≌ Mon_ (C ⥤ C) :=\n  equivalence.mk' (Monad_to_Mon C) (Mon_to_Monad C) sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/equiv_mon_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34050163185622123}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.simple\nimport category_theory.subobject.basic\nimport category_theory.preadditive.schur\nimport algebra.algebra.restrict_scalars\nimport algebra.algebra.tower\nimport algebra.category.Module.algebra\nimport algebra.category.Module.images\nimport algebra.category.Module.biproducts\nimport algebra.category.Module.simple\nimport data.mv_polynomial.basic\nimport algebra.free_algebra\nimport data.complex.module\n\n/-!\n# \"Introduction to representation theory\" by Etingof\n\nThis tutorial file follows along with the lecture notes \"Introduction to representation theory\",\nby Pavel Etingof and other contributors.\n\nThese lecture notes are available freely online at <https://klein.mit.edu/~etingof/repb.pdf>.\n\nThis tutorial is (extremely) incomplete at present.\nThe goal is to work through the lecture notes,\nshowing how to use the definitions and results from mathlib\nto establish the results in Etingof's notes. (We are not giving separate proofs here!)\n\nOur intention is (sadly) to skip all the problems, and many of the examples.\n\nOften results are proved by reference to (much) more general facts in mathlib.\n-/\n\naxiom skipped {p : Sort*} : p\n\nuniverses u\nopen category_theory finite_dimensional\n\nnoncomputable theory\n\n/-!\n## Chapter 2 \"Basic notions of representation theory\"\n-/\n\n/-!\n### 2.2 \"Algebras\"\n-/\n\n-- Definition 2.2.1: An associative algebra.\nvariables {k : Type*} [field k]\nvariables {A : Type*} [ring A] [algebra k A]\n\n-- Etingof uses the word \"unit\" to refer to the identity in an algebra.\n-- Currently in mathlib all algebras are unital\n-- (although non-unital rings exists as `non_unital_ring`)\n-- Thus we skip Definition 2.2.2 and Proposition 2.2.3\n\n-- Example 2.2.4 (1)-(5)\nexample : algebra k k := by apply_instance\nexample {X : Type*} [fintype X] : algebra k (mv_polynomial X k) := by apply_instance\nexample {V : Type*} [add_comm_group V] [module k V] : algebra k (V →ₗ[k] V) := by apply_instance\nexample {X : Type*} : algebra k (free_algebra k X) := by apply_instance\nexample {G : Type*} [group G] : algebra k (monoid_algebra k G) := by apply_instance\n\n-- Definition 2.2.5: A commutative algebra is described as:\nexample {A : Type*} [comm_ring A] [algebra k A] := true\n\n-- Definition 2.2.6: algebra homomorphisms:\nexample {B : Type*} [ring B] [algebra k B] (f : A →ₐ[k] B) := true\n\n/-!\n## 2.3 \"Representations\"\n-/\n\n-- Definition 2.3.1\n-- A representation (of an associative algebra) will usually be described as a module.\nvariables {M : Type*} [add_comm_group M] [module k M] [module A M] [is_scalar_tower k A M]\n\n-- or we can use `Module A`, for a \"bundled\" `A`-module,\n-- which is useful when we want access to the category theory library.\nvariables (N : Module.{u} A)\n\n-- We can translate between these easily:\n-- \"bundle\" a type with appropriate typeclasses\nexample : Module A := Module.of A M\n-- a \"bundled\" module has a coercion to `Type`,\n-- that comes equipped with the appropriate typeclasses:\nexample : module A N := by apply_instance\n\n-- Remark 2.3.2: Right `A`-modules are handled as left `Aᵐᵒᵖ`-modules:\nexample : Module Aᵐᵒᵖ := Module.of Aᵐᵒᵖ A\n-- Right modules are not extensively developed in mathlib at this point,\n-- and you may run into difficulty using them.\n\n-- It is helpful when working with `Module` to run\nopen_locale Module\n-- which adds some instances.\n\n-- Example 2.3.3\n-- (1) The zero module\nexample : Module A := Module.of A punit\n-- (2) The left regular module\nexample : Module A := Module.of A A\n-- (3) Modules over a field are vector spaces.\n-- (Because we handle vector spaces as modules in mathlib, this is empty of content.)\nexample (V : Type*) [add_comm_group V] [module k V] : Module k := Module.of k V\n-- (4) is trickier,\n-- and we would probably want to formalise as an equivalence of categories,\n-- because \"it's hard to get back to where we started\".\nexample (X : Type*) : Module (free_algebra k X) ≃ Σ (V : Module k), X → (V ⟶ V) := skipped\n\n-- Definition 2.3.4\n-- A subrepresentation can be described using `submodule`,\nvariables (S : submodule A M)\n-- or using the category theory library either as a monomorphism\nvariables (S' : Module.{u} A) (i : S' ⟶ N) [mono i]\n-- or a subobject (defined as an isomorphism class of monomorphisms)\nvariables (S'' : subobject N)\n\n-- Definition 2.3.5: We express that a representation is \"irreducible\" using `simple`.\nexample (N : Module A) : Prop := simple N\n-- there's also a predicate for unbundled modules:\nexample : simple (Module.of A M) ↔ is_simple_module A M := simple_iff_is_simple_module\n\n-- Definition 2.3.6: homomorphisms, intertwiners, isomorphisms\n-- For unbundled representations, we use linear maps:\nvariables {M' : Type*} [add_comm_group M'] [module k M'] [module A M'] [is_scalar_tower k A M']\nvariables (f : M →ₗ[A] M')\n-- while for bundled representations we use the categorical morphism arrow:\nvariables (N₁ N₂ : Module.{u} A) (g : N₁ ⟶ N₂)\n-- For isomorphisms, use one of\nvariables (e : M ≃ₗ[A] M') (j : N₁ ≅ N₂)\n\n-- Definition 2.3.7: direct sum\nexample : module A (M × M') := by apply_instance\nexample (N₁ N₂ : Module.{u} A) : Module.{u} A := N₁ ⊞ N₂\nexample (N₁ N₂ : Module.{u} A) : N₁ ⊞ N₂ ≅ Module.of A (N₁ × N₂) := Module.biprod_iso_prod N₁ N₂\n\n-- Definition 2.3.8: indecomposable\nexample (N : Module A) : Prop := indecomposable N\nexample (N : Module A) [simple N] : indecomposable N := indecomposable_of_simple N\n\n-- Proposition 2.3.9 (Schur's lemma)\nexample (N₁ N₂ : Module.{u} A) [simple N₁] (f : N₁ ⟶ N₂) (w : f ≠ 0) : mono f :=\nmono_of_nonzero_from_simple w\nexample (N₁ N₂ : Module.{u} A) [simple N₂] (f : N₁ ⟶ N₂) (w : f ≠ 0) : epi f :=\nepi_of_nonzero_to_simple w\nexample (N₁ N₂ : Module.{u} A) [simple N₁] [simple N₂] (f : N₁ ⟶ N₂) (w : f ≠ 0) : is_iso f :=\nis_iso_of_hom_simple w\n\n-- Corollary 2.3.10 (Schur's lemma over an algebraically closed field)\n-- Unfortunately these can't be global instances\nexample [is_alg_closed k] (V : Module.{u} A) [simple V] [finite_dimensional k V] (f : V ⟶ V) :\n  ∃ φ : k, φ • 𝟙 V = f :=\nendomorphism_simple_eq_smul_id k f\n-- Note that some magic is going on behind the scenes in this proof.\n-- We're using a version of Schur's lemma that applies to any `k`-linear category,\n-- and its hypotheses include `finite_dimensional k (V ⟶ V)`\n-- rather than `finite_dimensional k V` (because `V` need not even be a vector space).\n-- Typeclass inference is automatically generating this fact.\n\n-- Remark 2.3.11 (Schur's lemma doesn't hold over a non-algebraically closed field)\nexample : simple (Module.of ℂ ℂ) := simple_of_finrank_eq_one (finrank_self ℂ)\nexample : finite_dimensional ℝ (Module.of ℂ ℂ) := by { dsimp, apply_instance, }\nexample :\n  let V := Module.of ℂ ℂ in\n  ∃ (f : V ⟶ V), ∀ φ : ℝ, (φ : ℂ) • 𝟙 V ≠ f :=\n⟨algebra.lsmul ℂ ℂ complex.I,\n  λ φ w, by simpa using congr_arg complex.im (linear_map.congr_fun w (1 : ℂ))⟩\n\n-- Corollary 2.3.12\n-- Every irreducible finite dimensional representation of a commutative algebra is 1-dimensional\nexample (A : Type*) [comm_ring A] [algebra k A] (V : Module A) [finite_dimensional k V] [simple V] :\n  finrank k V = 1 :=\nskipped\n\n-- Remark 2.3.13: Every 1-dimensional representation is irreducible\nexample (V : Module A) [finite_dimensional k V] (h : finrank k V = 1) : simple V :=\nsimple_of_finrank_eq_one h\n\n-- Example 2.3.14: skipped (1 and 3 we can do, 2 requires Jordan normal form)\n\n/-!\n## 2.4 \"Ideals\"\n-/\n\n-- To be continued...\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/docs/tutorial/representation_theory/etingof.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.3403117782996634}}
{"text": "\nimport data.stream\n\nimport util.data.bijection\nimport util.data.perm\nimport util.data.nat\nimport util.data.minimum\n\nimport unitb.models.simple\nimport unitb.scheduling.basic\n\nlocal attribute [-simp] or.comm or.left_comm or.assoc and.comm and.left_comm and.assoc\n\nnamespace scheduling.infinite\n\nopen predicate (var) scheduling.unitb unitb scheduling\n\nopen nat function simple\n\n@[reducible]\ndef rrobin (lbl : Type) [infinite lbl] : Type :=\nbijection ℕ lbl\n\n@[reducible]\ndef index_t (lbl : Type) [infinite lbl] := ℕ\n\nsection\n\nparameter {lbl : Type}\nparameter [infinite lbl]\nparameter t : target_mch lbl\n\nlocal attribute [instance] classical.prop_decidable\n\nnoncomputable def first (req : set lbl)\n  (l : rrobin lbl)\n: index_t lbl :=\n(↓ x, l.f x ∈ req )\n\nnoncomputable def sch.current (r : set lbl) (queue : rrobin lbl) : lbl :=\nqueue.f (first r queue)\n\nstructure sch_state :=\n  (target : t.σ)\n  (q_len : ℕ)\n  (queue : rrobin lbl)\n  (inv : sch.current (t.req.apply target) queue ∈ t.req.apply target)\n\ndef req : var sch_state (set lbl) :=\nt.req ∘' (sch_state.target : var sch_state t.σ)\n\nlemma first_mem {s : t.σ}\n  (l : rrobin lbl)\n: l.f (first (t.req.apply s) l) ∈ t.req.apply s :=\nbegin\n  unfold first, apply @minimum_mem _ _ { x | l.f x ∈ t.req.apply s },\n  cases set.exists_mem_of_ne_empty (t.req_nemp s) with x h,\n  apply @set.ne_empty_of_mem _ _ (l.g x),\n  change l.f (l.g x) ∈ t.req.apply s,\n  rw l.g_inv, apply h\nend\n\nnoncomputable def shift (s : sch_state)\n: rrobin lbl :=\nif h : first (req.apply s) s.queue < succ s.q_len\nthen s.queue ∘ bijection.rotate_right' (first (req.apply s) s^.queue) (succ s.q_len) h\nelse s.queue\n\nlemma sch.select_inv\n  (s : t.σ)\n  (q : rrobin lbl)\n: sch.current (t.req.apply s) q ∈ t.req.apply s :=\nbegin\n  unfold sch.current,\n  apply first_mem _,\nend\n\ndef sch.first : sch_state :=\n{ q_len := 0\n, target := t.s₀\n, queue := bijection.rev (infinite.to_nat _)\n, inv := sch.select_inv _ _ }\n\nnoncomputable def sch.step (s : sch_state) : sch_state :=\n{ q_len := s.q_len + 1\n, target := t.next (sch.current (_) s.queue) s.target s.inv\n, queue := shift s\n, inv := sch.select_inv _ _ }\n\nnoncomputable def scheduler : program sch_state :=\n  { first := sch.first\n  , step  := sch.step }\n\nopen has_mem\n\nnoncomputable def current : var sch_state lbl :=\nsch.current <$> req <*> sch_state.queue\n\ndef rank (l : lbl) : var sch_state ℕ :=\nλ (s : sch_state), s.queue.g l + (s.queue.g l - s.q_len)\n\nlemma pending_not_move (s : sch_state) (l : lbl)\n  (h : succ s.q_len ≤ s.queue.g l)\n: ((sch.step s).queue).g l = s.queue.g l :=\nbegin\n  dunfold sch.step sch_state.queue shift,\n  cases decidable.em (first ((req t).apply s) s.queue < succ s.q_len) with h' h',\n  { rw [dif_pos h',comp_g],\n    unfold comp,\n    rw bijection.rotate_right'_g_eq_of_ge_j _ h' h, },\n  { rw dif_neg h', },\nend\n\nlemma shift_down (s : sch_state) (l : lbl)\n  (h₀ : first (req.apply s) s.queue < s.queue.g l)\n  (h₁ : s.queue.g l < succ s.q_len)\n: succ (((sch.step s).queue).g l) = s.queue.g l :=\nbegin\n  dunfold sch.step sch_state.queue shift,\n  have h : first ((req t).apply s) s.queue < succ s.q_len := lt_trans h₀ h₁,\n  rw [dif_pos h,comp_g], unfold comp,\n  rw bijection.succ_rotate_right'_g_eq_self (s.queue.g l) _ h₀ h₁,\nend\n\nlemma q_len_inc (s : sch_state)\n: (sch.step s).q_len = succ s.q_len :=\nby { rw ← add_one, refl }\n\nlemma not_req_not_move (s : sch_state) (l : lbl)\n  (h : s.queue.g l < first (req.apply s) s.queue)\n: (sch.step s).queue.g l = s.queue.g l :=\nbegin\n  dunfold sch.step sch_state.queue shift,\n  cases decidable.em (first ((req t).apply s) s.queue < succ s.q_len) with h' h',\n  { rw [dif_pos h',comp_g],\n    unfold comp,\n    rw [bijection.rotate_right'_g_eq_of_lt_i _ _ h], },\n  { rw [dif_neg h'] },\nend\n\nlemma not_mem_req {s : sch_state} {l : lbl}\n  (h : first (req.apply s) s.queue > s.queue.g l)\n: l ∉ req.apply s :=\nbegin\n  intro h',\n  revert h,\n  apply not_lt_of_ge,\n  unfold first,\n  apply minimum_le,\n  unfold mem set.mem,\n  rw bijection.g_inv,\n  apply h'\nend\n\nlemma le_rank_of_lt_first {s : sch_state} {l : lbl}\n  (Hgt : s.queue.g l < first (req.apply s) s.queue)\n: (rank l).apply (sch.step s) ≤ (rank l).apply s :=\nbegin\n  unfold rank,\n  cases le_or_gt (s.queue.g l) (s.q_len) with h h,\n  { have h' : s.queue.g l ≤ succ s.q_len := le_succ_of_le h,\n    simp [not_req_not_move _ _ _ Hgt,q_len_inc\n         ,sub_eq_zero_of_le h,sub_eq_zero_of_le h'] },\n  { simp [pending_not_move _ _ _ h,q_len_inc],\n    apply nat.sub_le_sub_left,\n    apply le_succ, },\nend\n\nlemma lt_rank_of_gt_first {s : sch_state} {l : lbl}\n  (Hlt : first (req.apply s) (s.queue) < (s.queue).g l)\n: (rank l).apply (sch.step s) < (rank l).apply s :=\nbegin\n  unfold rank,\n  cases le_or_gt s.q_len (first ((req t).apply s) s.queue) with h h,\n    -- h : s.q_len ≤ first (sch_state.req s) (s.queue)\n  { have h' : s.q_len < s.queue.g l := lt_of_le_of_lt h Hlt,\n    simp [pending_not_move _ _ _ h'],\n    dunfold sch.step sch_state.q_len,\n    rw [sub_succ],\n    apply pred_lt_self_of_pos,\n    apply nat.sub_pos_of_lt,\n    apply lt_of_lt_of_le h',\n    refl },\n    -- h : s.q_len > first (sch_state.req s) (s.queue)\n  { cases lt_or_ge (s.queue.g l) (succ s.q_len) with Hlt_len Hge_len,\n      -- Hlt_len : (s.queue).g l < succ (s.q_len)\n    { simp,\n      rw [← shift_down _ s _ Hlt Hlt_len,q_len_inc],\n      { apply add_lt_add_of_lt_of_le,\n        { apply lt_succ_self },\n        { apply nat.sub_le_sub\n          ; apply le_succ, } }, },\n      -- Hge_len : (s.queue).g l ≥ succ (s.q_len)\n    { unfold gt at h,\n      simp [pending_not_move _ _ _ Hge_len,q_len_inc],\n      { apply nat.sub_lt_sub_left Hge_len,\n        apply lt_succ_self, }, } }\nend\nvariable {s : sch_state}\n\nlemma eq_next_or_rank_eq_or_rank_lt {l : lbl} (v : ℕ)\n  (h : s ⊨ rank l ≃ v)\n: s ⊨ current ≃ l ∨\n  ( s ⊨ -(↑l ∊ req) ∧ sch.step s ⊨ rank l ≃ v ) ∨\n  sch.step s ⊨ rank l ≺ v :=\nbegin\n  rw or_iff_not_imp,\n  intro Hnnext,\n  simp [current,sch.current,req] at Hnnext ⊢,\n  simp at h,\n  rw bijection.inverse at Hnnext,\n  cases lt_or_gt_of_ne Hnnext with Hlt Hgt ; clear  Hnnext,\n  { right,\n    rw ← h,\n    apply lt_rank_of_gt_first t _ ,\n    simp [req,Hlt] },\n  { unfold gt at Hgt,\n    subst v,\n    have := le_rank_of_lt_first _, simp [req] at this,\n    specialize this Hgt,\n    cases lt_or_eq_of_le this with Hlt Heq,\n    { right, apply Hlt },\n    { left,\n      split,\n      { have := not_mem_req _ , simp [req] at this,\n        apply this, assumption },\n      { apply Heq, }, } },\nend\n\nlemma fair_schedule_step  (l : lbl) (v : ℕ)\n:  ↑l ∊ req ⋀ (rank l ≃ ↑v) ↦ (rank l ≺ v) ⋁ (current ≃ l) in scheduler :=\nbegin\n  unfold scheduler,\n  apply leads_to_step,\n  intros σ Heq Hnnext,\n  simp, simp at Heq,\n  cases Heq with Hmem Heq,\n  have HH := eq_next_or_rank_eq_or_rank_lt t v,\n  -- simp [rank,current,req] at HH,\n  specialize HH Heq,\n  rw or_iff_not_imp at HH,\n  simp [not_or_iff_not_and_not] at Hnnext,\n  have Hnnext' : ¬ (current t).apply σ = l := Hnnext.right,\n  have HH' := HH Hnnext',\n  rw [or_iff_not_imp] at HH',\n  left, apply HH', clear HH' HH,\n  simp [classical.not_and_iff_not_or_not],\n  apply or.intro_left _ Hmem,\nend\n\nlemma stable_queue_ranking (l : lbl) (v : ℕ)\n: unless scheduler (rank l ≃ v) (rank l ≺ v ⋁ current ≃ l) :=\nbegin\n  unfold scheduler,\n  apply unless_step,\n  intros σ Heq Hnnext,\n  simp,\n  simp [not_or_iff_not_and_not] at Hnnext,\n  have := eq_next_or_rank_eq_or_rank_lt _ _ Heq, simp at this,\n  cases this with h h,\n  { cases Hnnext.right h },\n  cases h with h h,\n  { replace Hnnext := Hnnext.right,\n    left, rw h.right, },\n  { right, left, assumption },\nend\n\nlemma INIT\n: system.init scheduler (↑sch_state.target ≃ (t.s₀ : var sch_state t.σ)) :=\nby simp [system.init,program.init,scheduler,sch.first]\n\nlemma STEP\n: co' scheduler\n    (λ (σ σ' : sch_state),\n       ∃P, σ'.target = t.next (sch.current (req.apply σ) σ.queue) σ.target P) :=\nbegin\n  unfold co',\n  intros σ σ',\n  unfold step has_safety.step is_step,\n  intros H,\n  have := σ.inv,\n  simp [req],\n  -- existsi σ.inv,\n  split,\n  { rw H,\n    unfold scheduler program.step subtype.val sch.step },\n  { simp [req,this], }\nend\n\nlemma INV\n: ∀ σ, σ ⊨ current ∊ t.req ∘' sch_state.target :=\nby { assume σ, simp [current,req], apply σ.inv, }\n\nlemma PROG (l : lbl)\n:     ↑l ∊ req  >~>  current ≃ l\n  in scheduler :=\nbegin\n  apply often_imp_often.induct ℕ _ _ ,\n  apply stable_queue_ranking _ l,\n  apply fair_schedule_step _ l,\nend\n\nend\n\nvariable {lbl : Type}\nvariable [infinite lbl]\nvariable t : scheduling.unitb.target_mch lbl\n\nnoncomputable def scheduler_spec\n: @scheduling.unitb.scheduler lbl t :=\n  { s := program $ sch_state t\n  , sem := _\n  , F := @scheduler _ _ t\n  , ch := current t\n  , object := ↑sch_state.target\n  , INIT := INIT t\n  , STEP := STEP t\n  , INV  := INV t\n  , PROG := PROG t }\n\nnoncomputable instance : unitb.system_sem ((scheduler_spec t).s) :=\n(scheduler_spec t).sem\n\nlemma sched' {lbl : Type} [s : infinite lbl] [nonempty lbl]\n  (t : target_mch lbl)\n: ∃ τ : stream t.σ, fair t τ :=\nunitb.scheduling t (scheduling.infinite.scheduler_spec t)\n\nend scheduling.infinite\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/scheduling/infinite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3402915684082042}}
{"text": "import MLIR.Dialects.ScfSemantics\nimport MLIR.Dialects.ArithSemantics\nopen MLIR.AST\n\nnamespace SCF_SELECT\nabbrev Δ := scf + arith\n\ndef LHS: Op Δ := [mlir_op|\n  %x = \"scf.if\" (%b) ({\n    \"scf.yield\"(%n): (i32) -> ()\n  }, {\n    \"scf.yield\"(%m): (i32) -> ()\n  }): (i1) -> i32\n]\ndef RHS: Op Δ := [mlir_op|\n  %x = \"arith.select\"(%b, %n, %m): (i1, i32, i32) -> i32\n]\ndef INPUT (b: FinInt 1) (n m: FinInt 32): SSAEnv Δ := SSAEnv.One [\n  ⟨\"b\", .i1, b⟩,\n  ⟨\"n\", .i32, n⟩,\n  ⟨\"m\", .i32, m⟩]\n\ntheorem denoteYieldRegion:\n  denoteRegion Δ (Region.mk\n    [BasicBlock.mk bbname []\n      [Op.mk \"scf.yield\" [] [(valuename, .int .Signless 32)] [] [] (.mk [])]]) =\n  fun (args: TypedArgs Δ) =>\n    Fitree.bind (Fitree.trigger <| SSAEnvE.Get MLIRType.i32 valuename) fun x =>\n    Fitree.ret (BlockResult.Ret [⟨.i32, x⟩]) := by\n  funext args\n  simp [denoteRegion, denoteBB, denoteOps, denoteOp, denoteOpBase]\n  simp [List.zip, List.mapM, List.map]\n  simp [Semantics.semantics_op]\n  simp [HOrElse.hOrElse, OrElse.orElse, Option.orElse, Option.map]\n  simp [scf_semantics_op]\n  simp [denoteTypedArgs]; cases args <;> simp\n\ntheorem equivalent (b: FinInt 1) (n m: FinInt 32):\n    (run (denoteOp _ LHS) (INPUT b n m)) =\n    (run (denoteOp _ RHS) (INPUT b n m)) := by\n  simp [LHS, RHS, INPUT]\n  simp [coeMLIRTypeList, coeRegionList, coeOpList]\n  simp [denoteOp, denoteOpBase, List.zip, List.mapM, denoteRegions]\n  simp [Semantics.semantics_op]\n  simp [HOrElse.hOrElse, OrElse.orElse, Option.orElse, Option.map]\n  simp [scf_semantics_op, arith_semantics_op]\n  rw [denoteYieldRegion]\n  rw [denoteYieldRegion]\n  save\n\n  conv in Fitree.translate _ (Fitree.trigger _) => simp [Fitree.translate]\n  conv in Fitree.interp (interpRegion Δ) (Fitree.Vis _ _) =>\n    simp [Fitree.interp, interpRegion]\n  simp [run, interpSSA'_bind, SSAEnvE.handle, cast_eq]\n  simp [Fitree.translate]\n  simp [interpRegion, Member.inject, interpSSA'_bind]\n  save\n\n  cases b.bool_cases <;> subst b <;> simp [List.get!, interpSSA'_bind]\n  . simp [Fitree.trigger, Fitree.interp'_bind, SSAEnvE.handle]\n    simp [Fitree.translate, Semantics.handle, cast_eq]\n  . simp [Fitree.trigger, Fitree.interp'_bind, SSAEnvE.handle]\n    simp [Fitree.translate, Semantics.handle, cast_eq]\nend SCF_SELECT\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Examples/IfStatementOptimization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3402317753714239}}
{"text": "import Lean\n\nsection utils\nopen Lean\n\n/-- True Notation -/\nnotation \"⊤\" => True\n\n/-- False Notation -/\nnotation \"⊥\" => False\n\n/- plain numbers can't be used as identifiers so line number identifiers are prefixed with `L` -/\nprivate abbrev mkLineId (n : Nat) : TSyntax [] := TSyntax.mk (mkIdent s!\"L{n}\")\n\n/- theorem names are mildly obfuscated to avoid name clashes -/\nprivate abbrev mkThmId (n : Name) : TSyntax [] := TSyntax.mk (mkIdent s!\"T@{n}\")\n\nend utils\n\n/-- Theorem Declaration\n\n    Syntax: `thm <name> <vars> : <hyps> ⊢ <goal>`\n\n    - `name` is the theorem name to be used in the `THM` proof command\n    - `vars` is a space separated list of propositional variables\n    - `hyps` is a comma separated list of hypotheses for the theorem\n    - `goal` is the main theorem statement\n-/\nmacro doc?:(docComment)? \"thm \" name:ident vars:ident* \" : \" hyps:term,* \" ⊢ \" goal:term \" := \" prf:term : command =>\n  -- TODO: make docstrings work in a meaningful way\n  -- TODO: can we get rid of the walrus?\n  let ln := mkThmId name.getId\n  `($[$doc?]? theorem $ln $[($vars : Prop)]* $[(_ : $hyps := by first | assumption | fail \"missing hypothesis\")]* : $goal := $prf)\n\n\n/-- Basic Hilbert Proof Commands -/\ndeclare_syntax_cat hbasic\n\n/-- Proof Command\n\n  Syntax: `proof <cmds> qed`\n\n  - `cmds` is a sequence of proof commands on separate numbered lines\n\n  Proof commands are of the shape `<lnum> <cmd> <args>` where `lnum` is the line number, `cmd` is the command name and `args` are the command arguments.\n  Line numbers must be positive and increasing at each step.\n  The last line of the proof should match the main statement of the theorem being proved.\n-/\nsyntax \"proof\" (noWs \"[\" num \"]\")? withPosition(colGe hbasic)* \"qed\" : term\n\n/- Syntax Checking -/\nabbrev checkCMD := Lean.Parser.checkLineEq \"command (MP, DT, HYP, LEM)\"\nabbrev checkTHM := Lean.Parser.checkLineEq \"theorem name\"\nabbrev checkARG := Lean.Parser.checkColGt \"proposition\"\nabbrev checkREF := Lean.Parser.checkLineEq \"line number\"\n\nopen Lean.Parser (semicolonOrLinebreak)\n\n/-- Modus Ponens \n\n  Syntax: `<lnum> MP <lref>`\n\n  - `lnum` is the current line number\n  - `lref` is a line reference for a conditional statement\n\n  When the statement referred to by `lref` is `A → B` then this proof command searches for the hypothesis `A` among the previous lines.\n  If `A` is found then it assigns `B` to the current line number.\n-/\nsyntax num ws (checkCMD \"MP \") (checkREF num) semicolonOrLinebreak : hbasic\n\n/-- Deduction Theorem \n\n  Syntax: `<lnum> DT`\n\n  When the current goal is `A → B` then this proof command assigns `A` to the current line number and sets the goal to `B`. \n-/\nsyntax num ws (checkCMD \"DT \") semicolonOrLinebreak : hbasic\n\n/-- Recall Hypothesis\n\n  Syntax: `<lnum> HYP <prop>`\n\n  - `lnum` is the current line number\n  - `prop` is a propositional statement\n\n  This proof command searches for `prop` among the theorem hypotheses and previous lines.\n  If `prop` is found then it assigns `prop` to the current line number.\n-/\nsyntax num ws (checkCMD \"HYP \") (checkARG term:max) semicolonOrLinebreak : hbasic\n\n/-- Invoke Theorem \n\n  Syntax: `<lnum> THM <name> <args>`\n\n  - `lnum` is the current line number\n  - `name` is the name of the theorem invoked\n  - `args` is a space separated list of propositions\n\n  This proof command searches a theorem `name` declared using the `thm` command and then assigns `args` to the propositional variables of the theorem.\n  If the theorem has hypotheses, it searches for each one among previous lines.\n  Then it assigns the theorem's main proposition to the current line number.\n-/\nsyntax num ws (checkCMD \"THM \") (checkTHM ident) (colGt term:max)* semicolonOrLinebreak : hbasic\n\n/- TODO: document implementation -/\nopen Lean in macro_rules\n| `(proof qed%$tk) => withRef tk `(by done)\n| `(proof[$m] qed%$tk) => \n  let lm := mkLineId m.getNat\n  withRef tk `(by first | exact $lm | done)\n| `(proof$[[$m]]? $n:num MP%$tk $ref; $rest* qed) =>\n  let m := match m with | some m => m.getNat | none => 0\n  if m < n.getNat then \n    let ln := mkLineId n.getNat\n    let lref := mkLineId ref.getNat\n    withRef n `(have $ln : (_ : Prop) := $lref (by first | assumption | fail \"missing hypothesis\"); proof[$n] $rest* qed)\n  else\n    Macro.throwErrorAt n \"line numbers must be positive and increasing\"\n| `(proof$[[$m]]? $n:num DT%$tk; $rest* qed) =>\n  let m := match m with | some m => m.getNat | none => 0\n  if m < n.getNat then \n    let ln := mkLineId n.getNat\n    withRef n `(fun $ln : (_ : Prop) => proof[$n] $rest* qed)\n  else\n    Macro.throwErrorAt n \"line numbers must be positive and increasing\"\n| `(proof$[[$m]]? $n:num HYP%$tk $h; $rest* qed) =>\n  let m := match m with | some m => m.getNat | none => 0\n  if m < n.getNat then \n    let ln := mkLineId n.getNat\n    withRef n `(have $ln : ($h : Prop) := (by assumption); proof[$n] $rest* qed)\n  else\n    Macro.throwErrorAt n \"line numbers must be positive and increasing\"\n| `(proof$[[$m]]? $n:num THM%$tk $name:ident $args:term*; $rest* qed) =>\n  let m := match m with | some m => m.getNat | none => 0\n  if m < n.getNat then \n    let ln := mkLineId n.getNat\n    let tn := mkThmId name.getId\n    withRef n `(have $ln : (_ : Prop) := $tn $args*; proof[$n] $rest* qed)  \n  else\n    Macro.throwErrorAt n \"line numbers must be positive and increasing\"\n\n/- Basic Theorems\n\nThe two deduction rules Modus Ponens (`MP`) and Deduction Theorem (`DT`) only characterize implication.\nThe basic axioms below are necessary to define the other logical connectives:\n\n- True (⊤)\n- False (⊥)\n- Negation (¬·)\n- Conjunction (·∧·)\n- Disjunction (·∨·)\n- Biconditional (·↔·)\n\nThe proofs involve Lean core theorems since they can't be proved using only the basic rules of this system.\n\nIn addition to these basic axioms, the basic implication axioms S, K, I are defined.\nThis is because the DT is not actually necessary in the presence of these axioms, so we make them available to users who wish to avoid DT.\n\nAlso, Pierce's Law is presented as the only axiom from classical logic.\nIt is sufficient to derive all classical theorems and it has the advantage of being formulated only using implication.\nUsers that wish to work in intuitionistic logic should avoid this axiom.\nThe Lean `#print axioms` command is useful to determine whether Pierce's Law is used, even indirectly, in a proof.\n-/\nsection basic_theorems\n\n/-- Axiom I: a → a -/\nthm AXI a : ⊢ a → a :=\nproof\n10 DT\nqed\n\n/-- Axiom K: a → b → a -/\nthm AXK a b : ⊢ a → b → a :=\nproof\n10 DT\n20 DT\n30 HYP a\nqed\n\n/-- Axiom S: (a → b → c) → (a → b) → a → c -/\nthm AXS a b c : ⊢ (a → b → c) → (a → b) → a → c :=\nproof\n10 DT\n20 DT\n30 DT\n40 MP 20\n50 MP 10\n60 MP 50\nqed\n\n/-- Trivial: ⊤ -/\nthm TRIV : \n⊢ ⊤ := True.intro\n\n/-- True Elimination: (⊤ → a) → a -/\nthm THUS a :\n⊢ (⊤ → a) → a :=\nproof\n10 DT\n20 THM TRIV;\n30 MP 10\nqed\n\n/-- Ex Falso Quodlibet: ⊥ → a -/\nthm EXFALSO a :\n⊢ ⊥ → a := False.elim\n\n/-- Negation Introduction: (a → ⊥) → ¬ a -/\nthm NOTI a :\n⊢ (a → ⊥) → ¬ a := id\n\n/-- Negation Elimination: ¬ a → a → ⊥ -/\nthm NOTE a :\n⊢ ¬ a → a → ⊥ := id\n\n/-- Conjunction Introduction: a → b → a ∧ b -/\nthm ANDI a b :\n⊢ a → b → a ∧ b := And.intro\n\n/-- Conjunction Elimination: (a → b → c) → a ∧ b → c -/\nthm ANDE a b c :\n⊢ (a → b → c) → a ∧ b → c := And.rec\n\n/-- Conjunction Elimination (Left): a ∧ b → a -/\nthm ANDL a b :\n⊢ a ∧ b → a := -- And.left\nproof\n10 THM AXK a b\n20 THM ANDE a b a\n30 MP 20\nqed\n\n/-- Conjunction Elimination (Right): a ∧ b → b -/\nthm ANDR a b :\n⊢ a ∧ b → b := -- And.right\nproof\n10 THM AXK (b → b) a\n20 THM AXI b\n30 MP 10\n40 THM ANDE a b b\n50 MP 40\nqed\n\n/-- Disjunction Elimination: (a → c) → (b → c) → a ∨ b → c -/\nthm ORE a b c :\n⊢ (a → c) → (b → c) → a ∨ b → c := Or.rec\n\n/-- Disjunction Introduction (Left): a → a ∨ b -/\nthm ORL a b :\n⊢ a → a ∨ b := Or.inl\n\n/-- Disjunction Introduction (Right): b → a ∨ b -/\nthm ORR a b :\n⊢ b → a ∨ b := Or.inr\n\n/-- Biconditional Introduction: (a → b) → (b → a) → (a ↔ b) -/\nthm IFFI a b :\n⊢ (a → b) → (b → a) → (a ↔ b) := Iff.intro\n\n/-- Biconditional Elimination (Left): (a ↔ b) → a → b -/\nthm IFFL a b :\n⊢ (a ↔ b) → a → b := Iff.mp\n\n/-- Biconditional Elimination (Right): (a ↔ b) → b → a -/\nthm IFFR a b :\n⊢ (a ↔ b) → b → a := Iff.mpr\n\n/-- Pierce's Law: ((a → b) → a) → a -/\nthm PIERCE a b : \n⊢ ((a → b) → a) → a :=\nfun h => match Classical.em a with\n| .inl ha => ha\n| .inr na => h fun ha => absurd ha na\n\nend basic_theorems\n\n/- Testing Zone -/\nsection test_theorems\n\nthm IFTRANS a b c :\n⊢ (a → b) → (b → c) → a → c :=\nproof\n10 DT\n20 DT\n30 DT\n40 MP 10\n50 MP 20\nqed\n\nthm IFLCOMM a b c :\n⊢ (a → b → c) → b → a → c :=\nproof\n10 DT\n20 DT\n30 DT\n40 MP 10\n50 MP 40\nqed\n\nthm ANDIDEMR a :\n⊢ a → a ∧ a :=\nproof\n10 DT\n20 THM ANDI a a\n30 MP 20\n40 MP 30\nqed\n\nthm ANDIDEM a :\n⊢ a ∧ a ↔ a :=\nproof\n10 THM ANDL a a\n20 THM ANDIDEMR a\n30 THM IFFI (a ∧ a) a\n40 MP 30\n50 MP 40\nqed\n\nthm ANDCOMM a b :\n⊢ a ∧ b → b ∧ a :=\nproof\n10 THM ANDE a b (b ∧ a)\n20 THM IFLCOMM b a (b ∧ a)\n30 THM ANDI b a\n40 MP 20\n50 MP 10\nqed\n\nthm ANDASSOCL a b c :\n⊢ a ∧ (b ∧ c) → (a ∧ b) ∧ c :=\nproof\n10 DT\n20 THM ANDL a (b ∧ c)\n30 MP 20\n40 THM ANDR a (b ∧ c)\n50 MP 40\n60 THM ANDL b c\n70 MP 60\n80 THM ANDR b c\n90 MP 80\n100 THM ANDI a b\n110 MP 100\n120 MP 110\n130 THM ANDI (a ∧ b) c\n140 MP 130\n150 MP 140\nqed\n\nthm ANDASSOCR a b c :\n⊢ (a ∧ b) ∧ c → a ∧ (b ∧ c) :=\nproof\n10 DT\n20 THM ANDR (a ∧ b) c\n30 MP 20\n40 THM ANDL (a ∧ b) c\n50 MP 40\n60 THM ANDR a b\n70 MP 60\n80 THM ANDL a b\n90 MP 80\n100 THM ANDI b c\n110 MP 100\n120 MP 110\n130 THM ANDI a (b ∧ c)\n140 MP 130\n150 MP 140\nqed\n\nthm ANDASSOC a b c :\n⊢ (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=\nproof\n10 THM ANDASSOCL a b c\n20 THM ANDASSOCR a b c\n30 THM IFFI ((a ∧ b) ∧ c) (a ∧ (b ∧ c))\n40 MP 30\n50 MP 40\nqed\n\nthm ANDTRUEL a :\n⊢ ⊤ ∧ a ↔ a :=\nproof\n10 THM TRIV;\n20 THM ANDR ⊤ a\n30 THM ANDI ⊤ a\n40 MP 30\n50 THM IFFI (⊤ ∧ a) a\n60 MP 50\n70 MP 60 \nqed\n\nthm ORIDEML a :\n⊢ a ∨ a → a :=\nproof\n10 THM AXI a\n20 THM ORE a a a\n30 MP 20\n40 MP 30\nqed\n\nthm ORIDEM :\n⊢ a ∨ a ↔ a :=\nproof\n10 THM ORIDEML a\n20 THM ORL a a\n30 THM IFFI (a ∨ a) a\n40 MP 30\n50 MP 40\nqed\n\nthm ORCOMM a b :\n⊢ a ∨ b → b ∨ a :=\nproof\n10 THM ORL b a\n20 THM ORR b a\n30 THM ORE a b (b ∨ a)\n40 MP 30\n50 MP 40\nqed\n\nthm ORASSOCL a b c :\n⊢ a ∨ (b ∨ c) → (a ∨ b) ∨ c :=\nproof\n10 DT\n20 THM ORE a (b ∨ c) ((a ∨ b) ∨ c)\n30 THM ORL a b\n40 THM ORL (a ∨ b) c\n50 THM IFTRANS a (a ∨ b) ((a ∨ b) ∨ c)\n60 MP 50\n70 MP 60\n80 MP 20\n90 THM ORE b c ((a ∨ b) ∨ c)\n100 THM ORR a b\n110 THM ORL (a ∨ b) c\n120 THM ORR (a ∨ b) c\n130 THM IFTRANS b (a ∨ b) ((a ∨ b) ∨ c)\n140 MP 130\n150 MP 140\n160 MP 90\n170 MP 160\n180 MP 80\n190 MP 180\nqed\n\nthm ORASSOCR a b c :\n⊢ (a ∨ b) ∨ c → a ∨ (b ∨ c) :=\nproof\n10 DT\n20 THM ORE a b (a ∨ (b ∨ c))\n30 THM ORL a (b ∨ c)\n40 THM ORL b c\n50 THM ORR a (b ∨ c)\n60 THM IFTRANS b (b ∨ c) (a ∨ (b ∨ c))\n70 MP 60\n80 MP 70\n90 MP 20\n100 MP 90\n110 THM ORR b c\n120 THM IFTRANS c (b ∨ c) (a ∨ (b ∨ c))\n130 MP 120\n140 MP 130\n150 THM ORE (a ∨ b) c (a ∨ (b ∨ c))\n160 MP 150\n170 MP 160\n180 MP 170\nqed\n\nthm ORASSOC a b c :\n⊢ (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=\nproof\n10 THM ORASSOCL a b c\n20 THM ORASSOCR a b c\n30 THM IFFI ((a ∨ b) ∨ c) (a ∨ (b ∨ c))\n40 MP 30\n50 MP 40\nqed\n\nthm ORRESL a b :\n⊢ a ∨ b → ¬a → b :=\nproof\n10 DT\n20 DT\n30 THM ORE a b b\n40 THM IFTRANS a ⊥ b\n50 THM NOTE a\n60 MP 50\n70 MP 40\n80 THM EXFALSO b\n90 MP 70\n100 MP 30\n110 THM AXI b\n120 MP 100\n130 MP 120\nqed\n\nthm ORRESR a b :\n⊢ a ∨ b → ¬b → a :=\nproof\n10 DT\n20 DT\n30 THM ORE a b a\n40 THM AXI a\n50 MP 30\n60 THM IFTRANS b ⊥ a\n70 THM NOTE b\n80 MP 70\n90 MP 60\n100 THM EXFALSO a\n110 MP 90\n120 MP 50\n130 MP 120\nqed\n\nthm ANDDISTRIBL a b c :\n⊢ a ∧ (b ∨ c) → (a ∧ b) ∨ (a ∧ c) :=\nproof\n10 DT\n20 THM ANDL a (b ∨ c)\n30 MP 20\n40 THM ANDR a (b ∨ c)\n50 MP 40\n60 THM ORE b c ((a ∧ b) ∨ (a ∧ c))\n70 THM IFTRANS b (a ∧ b) ((a ∧ b) ∨ (a ∧ c))\n80 THM ANDI a b\n90 MP 80\n100 MP 70\n110 THM ORL (a ∧ b) (a ∧ c)\n120 MP 100\n130 THM IFTRANS c (a ∧ c) ((a ∧ b) ∨ (a ∧ c))\n140 THM ANDI a c\n150 MP 140\n160 MP 130\n170 THM ORR (a ∧ b) (a ∧ c)\n180 MP 160\n190 MP 60\n200 MP 190\n210 MP 200\nqed\n\nend test_theorems\n", "meta": {"author": "fgdorais", "repo": "BasicHilbert", "sha": "e02aab5af10ad958a73f7e4ec139970342c95cbb", "save_path": "github-repos/lean/fgdorais-BasicHilbert", "path": "github-repos/lean/fgdorais-BasicHilbert/BasicHilbert-e02aab5af10ad958a73f7e4ec139970342c95cbb/BasicHilbert.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3400807436267244}}
{"text": "import category_theory.category\nimport category_theory.limits.limits\nimport category_theory.limits.shapes\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u  -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nlocal attribute [tidy] tactic.case_bash\n\n@[derive decidable_eq] inductive walking_w : Type v\n| left | middle | right | one | two\n\ninstance fintype_walking_w : fintype walking_w :=\n{ elems := [walking_w.left, walking_w.middle, walking_w.right, walking_w.one, walking_w.two].to_finset,\n  complete := λ x, by { cases x; simp } }\n\nnamespace walking_w\n\n/-- The arrows in a pullback diagram. -/\ninductive hom : walking_w → walking_w → Type v\n| inl1 : hom left one\n| inr1 : hom middle one\n| inl2 : hom middle two\n| inr2 : hom right two\n| id : Π X : walking_w.{v}, hom X X\n\nopen hom\n\ndef hom.comp : Π (X Y Z : walking_w) (f : hom X Y) (g : hom Y Z), hom X Z\n| _ _ _ (id _) h := h\n| _ _ _ inl1    (id one) := inl1\n| _ _ _ inr1    (id one) := inr1\n| _ _ _ inl2    (id two) := inl2\n| _ _ _ inr2    (id two) := inr2\n.\n\ninstance category_struct : category_struct walking_w :=\n{ hom  := hom,\n  id   := hom.id,\n  comp := hom.comp, }\n\ninstance (X Y : walking_w) : subsingleton (X ⟶ Y) := begin tidy end\n\n-- We make this a @[simp] lemma later; if we do it now there's a mysterious\n-- failure in `cospan`, below.\nlemma hom_id (X : walking_w.{v}) : hom.id X = 𝟙 X := rfl\n\n/-- The walking_cospan is the index diagram for a pullback. -/\ninstance : small_category.{v} walking_w.{v} := sparse_category\n\nend walking_w\n\n\nopen walking_w walking_w.hom\n\nvariables {C : Type u} [𝒞 : category.{v} C]\ninclude 𝒞\n\n/-- `cospan f g` is the functor from the walking cospan hitting `f` and `g`. -/\ndef w_cospan {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) : walking_w.{v} ⥤ C :=\n{ obj := λ x, match x with\n  | left := X\n  | middle := Y\n  | right := V\n  | one := W\n  | two := Z\n  end,\n  map := λ x y h, match x, y, h with\n  | _, _, (id _) := 𝟙 _\n  | _, _, inl1 := f1\n  | _, _, inr1 := g1\n  | _, _, inl2 := f2\n  | _, _, inr2 := g2\n  end }\n\n\n@[simp] lemma w_cospan_left {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).obj walking_w.left = X := rfl\n\n@[simp] lemma w_cospan_middle {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).obj walking_w.middle = Y := rfl\n\n@[simp] lemma w_cospan_right {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).obj walking_w.right = V := rfl\n\n\n@[simp] lemma w_cospan_one {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).obj walking_w.one = W := rfl\n\n@[simp] lemma w_cospan_two {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).obj walking_w.two = Z := rfl\n\n  \n@[simp] lemma w_cospan_map_inl1 {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).map walking_w.hom.inl1 = f1 := rfl\n\n@[simp] lemma w_cospan_map_inr1 {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).map walking_w.hom.inr1 = g1 := rfl\n\n\n@[simp] lemma w_cospan_map_inl2 {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).map walking_w.hom.inl2 = f2 := rfl\n\n@[simp] lemma w_cospan_map_inr2 {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) :\n  (w_cospan f1 g1 f2 g2).map walking_w.hom.inr2 = g2 := rfl\n\n\n@[simp] lemma w_cospan_map_id1 {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) (w : walking_w) :\n  (w_cospan f1 g1 f2 g2).map (walking_w.hom.id w) = 𝟙 _ := rfl\n\n\nvariables {X Y V W Z : C}\n\nattribute [simp] walking_w.hom_id\n\nabbreviation w_pullback_cone (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) := cone (w_cospan f1 g1 f2 g2)\n\n\nnamespace w_pullback_cone\nvariables {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z}\n\nabbreviation fst (t : w_pullback_cone f1 g1 f2 g2) : t.X ⟶ X := t.π.app left\nabbreviation mid (t : w_pullback_cone f1 g1 f2 g2) : t.X ⟶ Y := t.π.app middle\nabbreviation snd (t : w_pullback_cone f1 g1 f2 g2) : t.X ⟶ V := t.π.app right\n\ndef mk {P : C} (fst : P ⟶ X) (mid : P ⟶ Y) (snd : P ⟶ V) (eq1 : fst ≫ f1 = mid ≫ g1) (eq2 : mid ≫ f2 = snd ≫ g2) : w_pullback_cone f1 g1 f2 g2 :=\n{ X := P,\n  π := \n  { app := λ j, walking_w.cases_on j fst mid snd (fst ≫ f1) (mid ≫ f2),\n    naturality' := λ j j' f, begin cases f; obviously end } }\n\nend w_pullback_cone\n\n\n\nabbreviation w_pullback {X Y V W Z : C} (f1 : X ⟶ W) (g1 : Y ⟶ W) (f2 : Y ⟶ Z) (g2 : V ⟶ Z) [has_limit (w_cospan f1 g1 f2 g2)] :=\nlimit (w_cospan f1 g1 f2 g2)\n\nabbreviation w_pullback.fst {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] : w_pullback f1 g1 f2 g2 ⟶ X :=\nlimit.π (w_cospan f1 g1 f2 g2) walking_w.left\nabbreviation w_pullback.mid {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] : w_pullback f1 g1 f2 g2 ⟶ Y :=\nlimit.π (w_cospan f1 g1 f2 g2) walking_w.middle\nabbreviation w_pullback.snd {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] : w_pullback f1 g1 f2 g2 ⟶ V :=\nlimit.π (w_cospan f1 g1 f2 g2) walking_w.right\n\nabbreviation w_pullback.lift {P X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)]\n  (h : P ⟶ X) (j : P ⟶ Y) (k : P ⟶ V) (w1 : h ≫ f1 = j ≫ g1) (w2 : j ≫ f2 = k ≫ g2) : P ⟶ w_pullback f1 g1 f2 g2 :=\nlimit.lift _ (w_pullback_cone.mk h j k w1 w2)\n\n\nlemma w_pullback.condition1 {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] :\n  (w_pullback.fst : w_pullback f1 g1 f2 g2 ⟶ X) ≫ f1 = w_pullback.mid ≫ g1 :=\n(limit.w (w_cospan f1 g1 f2 g2) walking_w.hom.inl1).trans\n(limit.w (w_cospan f1 g1 f2 g2) walking_w.hom.inr1).symm\n\n\nlemma w_pullback.condition2 {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] :\n  (w_pullback.mid : w_pullback f1 g1 f2 g2 ⟶ Y) ≫ f2 = w_pullback.snd ≫ g2 :=\n(limit.w (w_cospan f1 g1 f2 g2) walking_w.hom.inl2).trans\n(limit.w (w_cospan f1 g1 f2 g2) walking_w.hom.inr2).symm\n\n\nlemma w_pullback.condition2' {X Y V W Z : C} {f1 : X ⟶ W} {g1 : Y ⟶ W} {f2 : Y ⟶ Z} {g2 : V ⟶ Z} [has_limit (w_cospan f1 g1 f2 g2)] :\n  (w_pullback.snd : w_pullback f1 g1 f2 g2 ⟶ V) ≫ g2 = w_pullback.mid ≫ f2 :=\nby simp[w_pullback.condition2]\n\n", "meta": {"author": "goodlyrottenapple", "repo": "lean-internal-cats", "sha": "fa9df99c2e852598b521b7b3ed8df3e4cb4853b6", "save_path": "github-repos/lean/goodlyrottenapple-lean-internal-cats", "path": "github-repos/lean/goodlyrottenapple-lean-internal-cats/lean-internal-cats-fa9df99c2e852598b521b7b3ed8df3e4cb4853b6/src/w_pullback.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749126, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3400000225722895}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.traversable.basic\n! leanprover-community/mathlib commit 1fc36cc9c8264e6e81253f88be7fb2cb6c92d76a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.List.Defs\nimport Mathlib.Data.Option.Defs\n\n/-!\n# Traversable type class\n\nType classes for traversing collections. The concepts and laws are taken from\n<http://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html>\n\nTraversable collections are a generalization of functors. Whereas\nfunctors (such as `List`) allow us to apply a function to every\nelement, it does not allow functions which external effects encoded in\na monad. Consider for instance a functor `invite : email → IO response`\nthat takes an email address, sends an email and waits for a\nresponse. If we have a list `guests : List email`, using calling\n`invite` using `map` gives us the following:\n`map invite guests : List (IO response)`.  It is not what we need. We need something of\ntype `IO (List response)`. Instead of using `map`, we can use `traverse` to\nsend all the invites: `traverse invite guests : IO (List response)`.\n`traverse` applies `invite` to every element of `guests` and combines\nall the resulting effects. In the example, the effect is encoded in the\nmonad `IO` but any applicative functor is accepted by `traverse`.\n\nFor more on how to use traversable, consider the Haskell tutorial:\n<https://en.wikibooks.org/wiki/Haskell/Traversable>\n\n## Main definitions\n  * `Traversable` type class - exposes the `traverse` function\n  * `sequence` - based on `traverse`,\n    turns a collection of effects into an effect returning a collection\n  * `IsLawfulTraversable` - laws for a traversable functor\n  * `ApplicativeTransformation` - the notion of a natural transformation for applicative functors\n\n## Tags\n\ntraversable iterator functor applicative\n\n## References\n\n * \"Applicative Programming with Effects\", by Conor McBride and Ross Paterson,\n   Journal of Functional Programming 18:1 (2008) 1-13, online at\n   <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>\n * \"The Essence of the Iterator Pattern\", by Jeremy Gibbons and Bruno Oliveira,\n   in Mathematically-Structured Functional Programming, 2006, online at\n   <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>\n * \"An Investigation of the Laws of Traversals\", by Mauro Jaskelioff and Ondrej Rypacek,\n   in Mathematically-Structured Functional Programming, 2012,\n   online at <http://arxiv.org/pdf/1202.2919>\n-/\n\nopen Function hiding comp\n\nuniverse u v w\n\nsection ApplicativeTransformation\n\nvariable (F : Type u → Type v) [Applicative F] [LawfulApplicative F]\n\nvariable (G : Type u → Type w) [Applicative G] [LawfulApplicative G]\n\n/-- A transformation between applicative functors.  It is a natural\ntransformation such that `app` preserves the `Pure.pure` and\n`Functor.map` (`<*>`) operations. See\n`ApplicativeTransformation.preserves_map` for naturality. -/\nstructure ApplicativeTransformation : Type max (u + 1) v w where\n  /-- The function on objects defined by an `ApplicativeTransformation`. -/\n  app : ∀ α : Type u, F α → G α\n  /-- An `ApplicativeTransformation` preserves `pure`. -/\n  preserves_pure' : ∀ {α : Type u} (x : α), app _ (pure x) = pure x\n  /-- An `ApplicativeTransformation` intertwines `seq`. -/\n  preserves_seq' : ∀ {α β : Type u} (x : F (α → β)) (y : F α), app _ (x <*> y) = app _ x <*> app _ y\n#align applicative_transformation ApplicativeTransformation\n\nend ApplicativeTransformation\n\nnamespace ApplicativeTransformation\n\nvariable (F : Type u → Type v) [Applicative F] [LawfulApplicative F]\n\nvariable (G : Type u → Type w) [Applicative G] [LawfulApplicative G]\n\ninstance : CoeFun (ApplicativeTransformation F G) fun _ => ∀ {α}, F α → G α :=\n  ⟨λ η => η.app _⟩\n\nvariable {F G}\n\n@[simp]\ntheorem app_eq_coe (η : ApplicativeTransformation F G) : η.app = η :=\n  rfl\n#align applicative_transformation.app_eq_coe ApplicativeTransformation.app_eq_coe\n\n@[simp]\ntheorem coe_mk (f : ∀ α : Type u, F α → G α) (pp ps) :\n  (ApplicativeTransformation.mk f @pp @ps) = f :=\n  rfl\n#align applicative_transformation.coe_mk ApplicativeTransformation.coe_mk\n\nprotected theorem congr_fun (η η' : ApplicativeTransformation F G) (h : η = η') {α : Type u}\n    (x : F α) : η x = η' x :=\n  congrArg (fun η'' : ApplicativeTransformation F G => η'' x) h\n#align applicative_transformation.congr_fun ApplicativeTransformation.congr_fun\n\nprotected theorem congr_arg (η : ApplicativeTransformation F G) {α : Type u} {x y : F α}\n    (h : x = y) : η x = η y :=\n  congrArg (fun z : F α => η z) h\n#align applicative_transformation.congr_arg ApplicativeTransformation.congr_arg\n\ntheorem coe_inj ⦃η η' : ApplicativeTransformation F G⦄ (h : (η : ∀ α, F α → G α) = η') :\n    η = η' := by\n  cases η\n  cases η'\n  congr\n#align applicative_transformation.coe_inj ApplicativeTransformation.coe_inj\n\n@[ext]\ntheorem ext ⦃η η' : ApplicativeTransformation F G⦄ (h : ∀ (α : Type u) (x : F α), η x = η' x) :\n    η = η' := by\n  apply coe_inj\n  ext1 α\n  exact funext (h α)\n#align applicative_transformation.ext ApplicativeTransformation.ext\n\ntheorem ext_iff {η η' : ApplicativeTransformation F G} :\n    η = η' ↔ ∀ (α : Type u) (x : F α), η x = η' x :=\n  ⟨fun h _ _ => h ▸ rfl, fun h => ext h⟩\n#align applicative_transformation.ext_iff ApplicativeTransformation.ext_iff\n\nsection Preserves\n\nvariable (η : ApplicativeTransformation F G)\n\n@[functor_norm]\ntheorem preserves_pure {α} : ∀ x : α, η (pure x) = pure x :=\n  η.preserves_pure'\n#align applicative_transformation.preserves_pure ApplicativeTransformation.preserves_pure\n\n@[functor_norm]\ntheorem preserves_seq {α β : Type u} : ∀ (x : F (α → β)) (y : F α), η (x <*> y) = η x <*> η y :=\n  η.preserves_seq'\n#align applicative_transformation.preserves_seq ApplicativeTransformation.preserves_seq\n\n@[functor_norm]\ntheorem preserves_map {α β} (x : α → β) (y : F α) : η (x <$> y) = x <$> η y := by\n  rw [← pure_seq, η.preserves_seq, preserves_pure, pure_seq]\n\n#align applicative_transformation.preserves_map ApplicativeTransformation.preserves_map\n\ntheorem preserves_map' {α β} (x : α → β) : @η _ ∘ Functor.map x = Functor.map x ∘ @η _ := by\n  ext y\n  exact preserves_map η x y\n#align applicative_transformation.preserves_map' ApplicativeTransformation.preserves_map'\n\nend Preserves\n\n/-- The identity applicative transformation from an applicative functor to itself. -/\ndef idTransformation : ApplicativeTransformation F F where\n  app α := id\n  preserves_pure' := by simp\n  preserves_seq' x y := by simp\n#align applicative_transformation.id_transformation ApplicativeTransformation.idTransformation\n\ninstance : Inhabited (ApplicativeTransformation F F) :=\n  ⟨idTransformation⟩\n\nuniverse s t\n\nvariable {H : Type u → Type s} [Applicative H] [LawfulApplicative H]\n\n/-- The composition of applicative transformations. -/\ndef comp (η' : ApplicativeTransformation G H) (η : ApplicativeTransformation F G) :\n    ApplicativeTransformation F H where\n  app α x := η' (η x)\n  -- Porting note: something has gone wrong with `simp [functor_norm]`,\n  -- which should suffice for the next two.\n  preserves_pure' x := by simp only [preserves_pure]\n  preserves_seq' x y := by simp only [preserves_seq]\n#align applicative_transformation.comp ApplicativeTransformation.comp\n\n@[simp]\ntheorem comp_apply (η' : ApplicativeTransformation G H) (η : ApplicativeTransformation F G)\n    {α : Type u} (x : F α) : η'.comp η x = η' (η x) :=\n  rfl\n#align applicative_transformation.comp_apply ApplicativeTransformation.comp_apply\n\n-- porting note: in mathlib3 we also had the assumption `[LawfulApplicative I]` because\n-- this was assumed\ntheorem comp_assoc {I : Type u → Type t} [Applicative I]\n    (η'' : ApplicativeTransformation H I) (η' : ApplicativeTransformation G H)\n    (η : ApplicativeTransformation F G) : (η''.comp η').comp η = η''.comp (η'.comp η) :=\n  rfl\n#align applicative_transformation.comp_assoc ApplicativeTransformation.comp_assoc\n\n@[simp]\n\n\n@[simp]\ntheorem id_comp (η : ApplicativeTransformation F G) : idTransformation.comp η = η :=\n  ext fun _ _ => rfl\n#align applicative_transformation.id_comp ApplicativeTransformation.id_comp\n\nend ApplicativeTransformation\n\nopen ApplicativeTransformation\n\n/-- A traversable functor is a functor along with a way to commute\nwith all applicative functors (see `sequence`).  For example, if `t`\nis the traversable functor `List` and `m` is the applicative functor\n`IO`, then given a function `f : α → IO β`, the function `Functor.map f` is\n`List α → List (IO β)`, but `traverse f` is `List α → IO (List β)`. -/\nclass Traversable (t : Type u → Type u) extends Functor t where\n  /-- The function commuting a traversable functor `t` with an arbitrary applicative functor `m`. -/\n  traverse : ∀ {m : Type u → Type u} [Applicative m] {α β}, (α → m β) → t α → m (t β)\n#align traversable Traversable\n\nopen Functor\n\nexport Traversable (traverse)\n\nsection Functions\n\nvariable {t : Type u → Type u}\n\nvariable {m : Type u → Type v} [Applicative m]\n\nvariable {α β : Type u}\n\nvariable {f : Type u → Type u} [Applicative f]\n\n/-- A traversable functor commutes with all applicative functors. -/\ndef sequence [Traversable t] : t (f α) → f (t α) :=\n  traverse id\n#align sequence sequence\n\nend Functions\n\n/-- A traversable functor is lawful if its `traverse` satisfies a\nnumber of additional properties.  It must send `pure : α → Id α` to `pure`,\nsend the composition of applicative functors to the composition of the\n`traverse` of each, send each function `f` to `λ x, f <$> x`, and\nsatisfy a naturality condition with respect to applicative\ntransformations. -/\nclass IsLawfulTraversable (t : Type u → Type u) [Traversable t] extends LawfulFunctor t :\n    Type (u + 1) where\n  /-- `traverse` plays well with `pure` of the identity monad-/\n  id_traverse : ∀ {α} (x : t α), traverse (pure : α → Id α) x = x\n  /-- `traverse` plays well with composition of applicative functors. -/\n  comp_traverse :\n    ∀ {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {α β γ}\n      (f : β → F γ) (g : α → G β) (x : t α),\n      traverse (Functor.Comp.mk ∘ map f ∘ g) x = Comp.mk (map (traverse f) (traverse g x))\n  /-- An axiom for `traverse` involving `pure : β → Id β`. -/\n  traverse_eq_map_id : ∀ {α β} (f : α → β) (x : t α),\n    traverse ((pure : β → Id β) ∘ f) x = id.mk (f <$> x)\n  /-- The naturality axiom explaining how lawful traversable functors should play with\n  lawful applicative functors. -/\n  naturality :\n    ∀ {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G]\n      (η : ApplicativeTransformation F G) {α β} (f : α → F β) (x : t α),\n      η (traverse f x) = traverse (@η _ ∘ f) x\n#align is_lawful_traversable IsLawfulTraversable\n\ninstance : Traversable Id :=\n⟨id⟩\n\ninstance : IsLawfulTraversable Id := by refine' { .. } <;> intros <;> rfl\n\nsection\n\nvariable {F : Type u → Type v} [Applicative F]\n\ninstance : Traversable Option :=\n  ⟨@Option.traverse⟩\n\ninstance : Traversable List :=\n  ⟨@List.traverse⟩\n\nend\n\nnamespace Sum\n\nvariable {σ : Type u}\n\nvariable {F : Type u → Type u}\n\nvariable [Applicative F]\n\n-- porting note: this was marked as a dubious translation but the only issue seems to be\n-- a universe issue; this may be a bug in mathlib3port. I've carefully checked the universes\n-- in mathlib3 and mathlib4 and they seem to match up exactly. Discussion here\n-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/why.20dubious.3F/\n\n/- warning: sum.traverse -> Sum.traverse is a dubious translation:\nlean 3 declaration is\n  forall {σ : Type.{u}} {F : Type.{u} -> Type.{u}} [_inst_1 : Applicative.{u u} F]\n    {α : Type.{u_1}} {β : Type.{u}}, (α -> (F β)) -> (Sum.{u u_1} σ α) ->\n    (F (Sum.{u u} σ β))\nbut is expected to have type\n  forall {σ : Type.{u}} {F : Type.{u} -> Type.{u}} [_inst_1 : Applicative.{u u} F]\n    {α : Type.{_aux_param_0}} {β : Type.{u}}, (α -> (F β)) -> (Sum.{u _aux_param_0} σ α) ->\n    (F (Sum.{u u} σ β))\nCase conversion may be inaccurate. Consider using '#align sum.traverse Sum.traverseₓ'. -/\n\n/-- Defines a `traverse` function on the second component of a sum type.\nThis is used to give a `traversable` instance for the functor `σ ⊕ -`. -/\nprotected def traverse {α β} (f : α → F β) : Sum σ α → F (Sum σ β)\n  | Sum.inl x => pure (Sum.inl x)\n  | Sum.inr x => Sum.inr <$> f x\n#align sum.traverse Sum.traverse\n\nend Sum\n\ninstance {σ : Type u} : Traversable.{u} (Sum σ) :=\n  ⟨@Sum.traverse _⟩\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Control/Traversable/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.33987901850426766}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.model_category_axioms\nimport category_theory.limits.opposites\nimport category_theory.limits.shapes.comm_sq\nimport for_mathlib.category_theory.localization.predicate\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\n\nnamespace algebraic_topology\n\nvariables (C : Type*) [category C]\n\nclass model_category extends category_with_fib_cof_weq C :=\n(CM1axiom : has_finite_limits C ∧ has_finite_colimits C)\n(CM2axiom : to_category_with_fib_cof_weq.CM2)\n(CM3axiom : to_category_with_fib_cof_weq.CM3)\n(CM4axiom : to_category_with_fib_cof_weq.CM4)\n(CM5axiom : to_category_with_fib_cof_weq.CM5)\n\nnamespace model_category\n\nvariable {C}\nvariable [M : model_category C]\ninclude M\n\ndef fib := M.fib\ndef cof := M.cof\ndef weq := M.weq\ndef triv_fib := M.to_category_with_fib_cof_weq.triv_fib\ndef triv_cof := M.to_category_with_fib_cof_weq.triv_cof\n\nlemma CM1 : has_finite_limits C ∧ has_finite_colimits C := M.CM1axiom\nlemma CM2 : (weq : morphism_property C).three_of_two := M.CM2axiom\nlemma CM3 : M.to_category_with_fib_cof_weq.CM3 := M.CM3axiom\nlemma CM3a : (weq : morphism_property C).is_stable_by_retract := CM3.weq\nlemma CM3b : (fib : morphism_property C).is_stable_by_retract := CM3.fib\nlemma CM3c : (cof : morphism_property C).is_stable_by_retract := CM3.cof\nlemma CM4 : M.to_category_with_fib_cof_weq.CM4 := M.CM4axiom\nlemma CM4a : (triv_cof : morphism_property C).has_lifting_property fib := CM4.1\nlemma CM4b : (cof : morphism_property C).has_lifting_property triv_fib := CM4.2\nlemma CM5 : M.to_category_with_fib_cof_weq.CM5 := M.CM5axiom\nlemma CM5a : factorisation_axiom (triv_cof : morphism_property C) fib := CM5.1\nlemma CM5b : factorisation_axiom (cof : morphism_property C) triv_fib := CM5.2\n\n@[priority 100] instance : has_finite_limits C := CM1.1\n@[priority 100] instance : has_finite_colimits C := CM1.2\n\ninstance : model_category Cᵒᵖ :=\n{ to_category_with_fib_cof_weq := M.to_category_with_fib_cof_weq.op,\n  CM1axiom := ⟨infer_instance, infer_instance⟩,\n  CM2axiom := by simpa only [← M.to_category_with_fib_cof_weq.CM2_iff_op] using CM2axiom,\n  CM3axiom := by simpa only [← M.to_category_with_fib_cof_weq.CM3_iff_op] using CM3axiom,\n  CM4axiom := by simpa only [← M.to_category_with_fib_cof_weq.CM4_iff_op] using CM4axiom,\n  CM5axiom := by simpa only [← M.to_category_with_fib_cof_weq.CM5_iff_op] using CM5axiom, }\n\nvariables {A B X Y Z : C} (i : A ⟶ B) {p : X ⟶ Y} {q : Y ⟶ Z} {f : X ⟶ Y}\n  (hip : is_retract_hom i p)\n\nclass cofibration : Prop := (property : cof i)\nclass fibration : Prop := (property : fib i)\nclass weak_eq : Prop := (property : weq i)\n\nvariable {i}\n\n@[priority 100] instance CM4a' [hi₁ : cofibration i] [hi₂ : weak_eq i] [hp : fibration p] :\n  has_lifting_property i p := CM4a i ⟨hi₁.property, hi₂.property⟩ p hp.property\n@[priority 100] instance CM4b' [hi : cofibration i] [hp₁ : fibration p] [hp₂ : weak_eq p] :\n  has_lifting_property i p := CM4b i hi.property p ⟨hp₁.property, hp₂.property⟩\n\nlemma cofibration_retract_stable [hp : cofibration p] : cofibration i :=\n⟨CM3.cof i p hip hp.property⟩\nlemma fibration_retract_stable [hp : fibration p] : fibration i :=\n⟨CM3.fib i p hip hp.property⟩\nlemma weq_retract_stable [hp : weak_eq p] : weak_eq i :=\n⟨CM3.weq i p hip hp.property⟩\n\nlemma cof_eq_llp_with_triv_fib : cof = (triv_fib : morphism_property C).llp_with :=\nfactorisation_axiom.eq_llp_with CM5b CM4b CM3.cof\nlemma triv_fib_eq_rlp_with_cof : triv_fib = (cof : morphism_property C).rlp_with :=\nfactorisation_axiom.eq_rlp_with CM5b CM4b CM3.triv_fib\nlemma triv_cof_eq_llp_with_fib : triv_cof = (fib : morphism_property C).llp_with :=\nfactorisation_axiom.eq_llp_with CM5a CM4a CM3.triv_cof\nlemma fib_eq_rlp_with_triv_cof : fib = (triv_cof : morphism_property C).rlp_with :=\nfactorisation_axiom.eq_rlp_with CM5a CM4a CM3.fib\n\nlemma cof_stable_under_composition : (cof : morphism_property C).stable_under_composition :=\nby { rw cof_eq_llp_with_triv_fib, apply morphism_property.llp_with.stable_under_composition, }\nlemma fib_stable_under_composition : (fib : morphism_property C).stable_under_composition :=\nby { rw fib_eq_rlp_with_triv_cof, apply morphism_property.rlp_with.stable_under_composition, }\n\ninstance comp_weak_eq [hp : weak_eq p] [hq : weak_eq q] : weak_eq (p ≫ q) :=\n⟨CM2.of_comp p q hp.property hq.property⟩\ninstance comp_cofibration [hp : cofibration p] [hq : cofibration q] : cofibration (p ≫ q) :=\n⟨cof_stable_under_composition p q hp.property hq.property⟩\ninstance comp_fibration [hp : fibration p] [hq : fibration q] : fibration (p ≫ q) :=\n⟨fib_stable_under_composition p q hp.property hq.property⟩\n\nnamespace weak_eq\n\nvariables (p q)\n\nlemma of_comp_left (hp : weak_eq p) (hpq : weak_eq (p ≫ q)) : weak_eq q :=\n⟨CM2.of_comp_left p q hp.property hpq.property⟩\nlemma of_comp_left' [hp : weak_eq p] [hpq : weak_eq (p ≫ q)] : weak_eq q :=\nof_comp_left p q hp hpq\n\nlemma of_comp_right (hq : weak_eq q) (hpq : weak_eq (p ≫ q)) : weak_eq p :=\n⟨CM2.of_comp_right p q hq.property hpq.property⟩\nlemma of_comp_right' [hq : weak_eq q] [hpq : weak_eq (p ≫ q)] : weak_eq p :=\nof_comp_right p q hq hpq\n\nend weak_eq\n\nlemma cof_contains_iso : morphism_property.isomorphisms C ⊆ cof :=\nby { rw cof_eq_llp_with_triv_fib, apply morphism_property.llp_with.contains_iso, }\nlemma fib_contains_iso : morphism_property.isomorphisms C ⊆ fib :=\nby { rw fib_eq_rlp_with_triv_cof, apply morphism_property.rlp_with.contains_iso, }\nlemma triv_cof_contains_iso : morphism_property.isomorphisms C ⊆ triv_cof :=\nby { rw triv_cof_eq_llp_with_fib, apply morphism_property.llp_with.contains_iso, }\nlemma triv_fib_contains_iso : morphism_property.isomorphisms C ⊆ triv_fib :=\nby { rw triv_fib_eq_rlp_with_cof, apply morphism_property.rlp_with.contains_iso, }\nlemma weq_contains_iso : morphism_property.isomorphisms C ⊆ weq :=\nλ X Y f hf, (triv_cof_contains_iso f hf).2\n\n@[priority 100] instance cofibration_of_is_iso [is_iso p] : cofibration p :=\n⟨cof_contains_iso p (morphism_property.isomorphisms.infer_property _)⟩\n@[priority 100] instance fibration_of_is_iso [is_iso p] : fibration p :=\n⟨fib_contains_iso p (morphism_property.isomorphisms.infer_property _)⟩\n@[priority 100] instance weak_eq_of_is_iso [is_iso p] : weak_eq p :=\n⟨weq_contains_iso p (morphism_property.isomorphisms.infer_property _)⟩\n\ninstance CM5a_cofibration : cofibration (CM5a.i f) := ⟨(CM5a.i_property f).1⟩\ninstance CM5a_weak_eq : weak_eq (CM5a.i f) := ⟨(CM5a.i_property f).2⟩\ninstance CM5a_fibration : fibration (CM5a.p f) := ⟨CM5a.p_property f⟩\n\ninstance CM5b_cofibration : cofibration (CM5b.i f) := ⟨CM5b.i_property f⟩\ninstance CM5b_fibration : fibration (CM5b.p f) := ⟨(CM5b.p_property f).1⟩\ninstance CM5b_weak_eq : weak_eq (CM5b.p f) := ⟨(CM5b.p_property f).2⟩\n\nlemma cof_is_stable_under_cobase_change : (cof : morphism_property C).stable_under_cobase_change :=\nby { rw cof_eq_llp_with_triv_fib, apply morphism_property.llp_with.stable_under_cobase_change, }\nlemma triv_cof_is_stable_under_cobase_change :\n  (triv_cof : morphism_property C).stable_under_cobase_change :=\nby { rw triv_cof_eq_llp_with_fib, apply morphism_property.llp_with.stable_under_cobase_change, }\n\nlemma cof_is_stable_under_coproducts : (cof : morphism_property C).stable_under_coproducts :=\nby { rw cof_eq_llp_with_triv_fib, apply morphism_property.llp_with.stable_under_coproducts, }\nlemma triv_cof_is_stable_under_coproducts : (triv_cof : morphism_property C).stable_under_coproducts :=\nby { rw triv_cof_eq_llp_with_fib, apply morphism_property.llp_with.stable_under_coproducts, }\n\nlemma fib_is_stable_under_base_change :\n  (fib : morphism_property C).stable_under_base_change :=\nby { rw fib_eq_rlp_with_triv_cof, apply morphism_property.rlp_with.stable_under_base_change, }\nlemma triv_fib_is_stable_under_base_change :\n  (triv_fib : morphism_property C).stable_under_base_change :=\nby { rw triv_fib_eq_rlp_with_cof, apply morphism_property.rlp_with.stable_under_base_change, }\n\nlemma fib_is_stable_under_products : (fib : morphism_property C).stable_under_products :=\nby { rw fib_eq_rlp_with_triv_cof, apply morphism_property.rlp_with.stable_under_products, }\nlemma triv_fib_is_stable_under_products : (triv_fib : morphism_property C).stable_under_products :=\nby { rw triv_fib_eq_rlp_with_cof, apply morphism_property.rlp_with.stable_under_products, }\n\nlemma cof_respects_iso : (cof : morphism_property C).respects_iso :=\nmorphism_property.respects_iso.of_stable_under_composition_and_contains_iso\n  cof_stable_under_composition cof_contains_iso\nlemma fib_respects_iso : (fib : morphism_property C).respects_iso :=\nmorphism_property.respects_iso.of_stable_under_composition_and_contains_iso\n  fib_stable_under_composition fib_contains_iso\nlemma weq_respects_iso : (weq : morphism_property C).respects_iso :=\nmorphism_property.respects_iso.of_stable_under_composition_and_contains_iso\n  CM2.of_comp weq_contains_iso\n\nnamespace cofibration\n\nlemma op (hi : cofibration i) : fibration i.op := ⟨hi.property⟩\nlemma unop {A B : Cᵒᵖ} {i : A ⟶ B} (hi : cofibration i) : fibration i.unop := ⟨hi.property⟩\nlemma iff_cof : cofibration i ↔ cof i := ⟨λ h, h.property, λ h, ⟨h⟩⟩\n\nlemma direct_image ⦃f : A ⟶ X⦄ ⦃i : A ⟶ B⦄ ⦃i' : X ⟶ Y⦄ ⦃g : B ⟶ Y⦄\n  (h : is_pushout f i i' g) [hi : cofibration i] : cofibration i' :=\n⟨cof_is_stable_under_cobase_change h hi.property⟩\n\nlemma direct_image_is_weak_eq ⦃f : A ⟶ X⦄ ⦃i : A ⟶ B⦄ ⦃i' : X ⟶ Y⦄ ⦃g : B ⟶ Y⦄\n  (h : is_pushout f i i' g) [hi₁ : cofibration i] [hi₂ : weak_eq i] : weak_eq i' :=\n⟨(triv_cof_is_stable_under_cobase_change h ⟨hi₁.property, hi₂.property⟩).2⟩\n\ninstance {A X₁ X₂ : C} (f : A ⟶ X₁) (g : A ⟶ X₂) [hg : cofibration g] :\n  cofibration (pushout.inl : X₁ ⟶ pushout f g) :=\n⟨cof_is_stable_under_cobase_change.inl f g hg.property⟩\n\ninstance {A X₁ X₂ : C} (f : A ⟶ X₁) (g : A ⟶ X₂) [hf : cofibration f] :\n  cofibration (pushout.inr : X₂ ⟶ pushout f g) :=\n⟨cof_is_stable_under_cobase_change.inr f g hf.property⟩\n\ninstance {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) [h₁ : cofibration f₁] [h₂ : cofibration f₂] :\n  cofibration (coprod.map f₁ f₂) :=\n⟨cof_is_stable_under_coproducts.binary f₁ h₁.property f₂ h₂.property⟩\n\ninstance {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) [h₁ : cofibration f₁] [h₂ : cofibration f₂]\n  [h₁' : weak_eq f₁] [h₂' : weak_eq f₂] :\n  weak_eq (coprod.map f₁ f₂) :=\n⟨(triv_cof_is_stable_under_coproducts.binary f₁ ⟨h₁.property, h₁'.property⟩\n  f₂ ⟨h₂.property, h₂'.property⟩).2⟩\n\nlemma respects_iso {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (e : arrow.mk f₁ ≅ arrow.mk f₂) :\n  cofibration f₁ ↔ cofibration f₂ :=\nby simpa only [iff_cof] using cof_respects_iso.arrow_mk_iso_iff e\n\nend cofibration\n\nnamespace fibration\n\nlemma op (hp : fibration p) : cofibration p.op := ⟨hp.property⟩\nlemma unop {X Y : Cᵒᵖ} {p : X ⟶ Y} (hp : fibration p) : cofibration p.unop := ⟨hp.property⟩\nlemma iff_fib : fibration i ↔ fib i := ⟨λ h, h.property, λ h, ⟨h⟩⟩\n\nvariable (p)\nlemma iff_op : fibration p ↔ cofibration p.op := ⟨op, cofibration.unop⟩\nlemma iff_unop {X Y : Cᵒᵖ} (p : X ⟶ Y) : fibration p ↔ cofibration p.unop := ⟨unop, cofibration.op⟩\n\nlemma respects_iso {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (e : arrow.mk f₁ ≅ arrow.mk f₂) :\n  fibration f₁ ↔ fibration f₂ :=\nby simpa only [iff_fib] using fib_respects_iso.arrow_mk_iso_iff e\n\nend fibration\n\nnamespace cofibration\n\nvariable (p)\n\nlemma iff_op : cofibration p ↔ fibration p.op := ⟨op, fibration.unop⟩\nlemma iff_unop {X Y : Cᵒᵖ} (p : X ⟶ Y) : cofibration p ↔ fibration p.unop := ⟨unop, fibration.op⟩\n\nend cofibration\n\nnamespace weak_eq\n\nlemma op (hp : weak_eq p) : weak_eq p.op := ⟨hp.property⟩\nlemma unop {X Y : Cᵒᵖ} {p : X ⟶ Y} (hp : weak_eq p) : weak_eq p.unop := ⟨hp.property⟩\nlemma iff_weq : weak_eq i ↔ weq i := ⟨λ h, h.property, λ h, ⟨h⟩⟩\n\ninstance {A X₁ X₂ : C} (f : A ⟶ X₁) (g : A ⟶ X₂) [hg₁ : cofibration g] [hg₂ : weak_eq g] :\n  weak_eq (pushout.inl : X₁ ⟶ pushout f g) :=\n⟨(triv_cof_is_stable_under_cobase_change.inl f g ⟨hg₁.property, hg₂.property⟩).2⟩\n\ninstance {A X₁ X₂ : C} (f : A ⟶ X₁) (g : A ⟶ X₂) [hf₁ : cofibration f] [hf₂ : weak_eq f] :\n  weak_eq (pushout.inr : X₂ ⟶ pushout f g) :=\n⟨(triv_cof_is_stable_under_cobase_change.inr f g ⟨hf₁.property, hf₂.property⟩).2⟩\n\nlemma respects_iso {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (e : arrow.mk f₁ ≅ arrow.mk f₂) :\n  weak_eq f₁ ↔ weak_eq f₂ :=\nby simpa only [iff_weq] using weq_respects_iso.arrow_mk_iso_iff e\n\nend weak_eq\n\nvariable (C)\n\n@[derive category]\ndef Ho' := (weq : morphism_property C).localization\n\nvariable {C}\n\ndef L' : C ⥤ Ho' C := weq.Q\n\ninstance L'_is_localization : L'.is_localization (weq : morphism_property C) :=\nby { dsimp [L'], apply_instance, }\n\nvariables {Ho : Type*} [category Ho] (L : C ⥤ Ho) [L.is_localization weq]\n\nlemma is_iso_L_map' {X Y : C} (f : X ⟶ Y) (hf : weq f) : is_iso (L.map f) :=\nlocalization.inverts L weq f hf\n\nlemma is_iso_L_map {X Y : C} (f : X ⟶ Y) [weak_eq f] : is_iso (L.map f) :=\nis_iso_L_map' L f weak_eq.property\n\nend model_category\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/model_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3398790097901846}}
{"text": "import data.fintype tactic.find data.set.finite ring_theory.subring data.nat.choose tactic.find\n\nopen tactic expr lean interactive interactive.types\nopen set finset\n\n\nmeta def fold_test : expr → tactic (list (nat × string)) :=\nλ e, do let l := e.fold [] (λ e n l, (n, e.to_string) :: l),\ntrace l, return l\n\nmeta def fold_test_type : expr → tactic (list (nat × string)) :=\nλ e, do l ← e.fold (pure [])\n  (λ e n l, (do t ← infer_type e, l ← l, return ((n, t.to_string) :: l)) <|> l),\n  trace l, return l\n\nmeta def fold_test_target : tactic unit :=\ntarget >>= fold_test >> skip\n\nmeta def fold_test_target_type : tactic unit :=\ntarget >>= fold_test_type >> skip\n\n/-- gets the type of any fintype in an expr -/\nmeta def get_fintype (e : expr) : tactic (list expr) :=\ne.fold (pure []) $ λ e _ l, do l' ← l,\n  (do t ← infer_type e,\n  match t with\n  | `(fintype _) := return (e::l')\n  | _ := return l'\n  end) <|> return l'\n\n-- meta def is_fintype (e : expr) : tactic (list expr) :=\n-- match e with\n-- | (app n _) := if n.const_name = `fintype then return [e] else return []\n-- | _         := return []\n-- end\n\n-- meta def is_fintype_type (e : expr) : tactic (list expr) :=\n-- infer_type e >>= is_fintype\n\n/-- checks is an expression is has type `fintype _` -/\nmeta def is_fintype (e : expr) : tactic bool :=\nmatch e with\n| (app n _) := if n.const_name = `fintype then return tt else return ff\n| _         := return ff\nend\n\n/-- returns the type of an expr if it has type `fintype _` -/\nmeta def is_fintype_type (e : expr) : tactic expr :=\ndo f ← infer_type e,\n  b ← is_fintype f,\n  match b with\n  | tt := return f\n  | ff := failure\n  end\n\n-- meta def get_fintype' : expr → tactic (list expr)\n-- | e := do l ← is_fintype_type e, match l with\n--   | [] := match e with\n--     | (mvar _ _ f)          := is_fintype f\n--     | (local_const _ _ _ f) := is_fintype f\n--     | (app f g)             := do lf ← get_fintype' f, lg ← get_fintype' g, return (lf ++ lg)\n--     | _                     := return []\n--     end\n--   | _ := return l\n-- end\n\n-- meta def thing : tactic unit :=\n-- do t ← target,\n-- l ← get_fintype' t,\n-- match l with\n-- | [] := return ()\n-- | (a :: l) := _match l >> pose `f none a >> reset_instance_cache >> skip\n-- end\n-- #print expr\n-- lemma fintype_congr (α : Type*) [fintype α] (P : Π [fintype α], Prop) (h : P) (x : fintype α) : @P x :=\n-- by convert h\n\n\n-- meta def generalize_fintype : expr → tactic (list expr)\n-- | `(@eq (%%u) (%%a) (%%b)) := match a with\n--   | (mvar x y f)           := mcond (is_fintype f) (return [f]) failure\n--   | (local_const _ _ _ f)  := mcond (is_fintype f) (return [f]) failure\n--   | (app f g)              := do lf ← get_fintype f, lg ← get_fintype g, return (lf ++ lg)\n--   | _                      := return []\n--   end\n-- | `((%%a) ↔ (%%b)) := match a with\n--   | (mvar x y f)           := mcond (is_fintype f) (return [f]) failure\n--   | (local_const _ _ _ f)  := mcond (is_fintype f) (return [f]) failure\n--   | (app f g)              := do lf ← get_fintype f, lg ← get_fintype g, return (lf ++ lg)\n--   | _                      := return []\n--   end\n-- | e := return []\n\n-- #print expr.\n\n-- meta def generalize_forall_aux (t : expr) (e : expr) (f : expr → expr) :\n--   expr × (expr → expr) :=\n-- cond (expr.alpha_eqv t e)\n--   _\n--   (e, f)\n\n\n-- meta def generalize_forall (t : expr) : expr → (expr → expr) → tactic expr\n-- | (const n l) f :=\n\n-- #reduce tactic.skip\n-- #reduce (return () : tactic unit)\n-- example : false :=\n-- let x : ℕ := 1 in\n-- have hx : x = 1 := rfl,\n-- begin\n--   replace x := x,\n\n-- end\n#print cond._main\n#print id_rhs\n#find (tactic _ → option _)\n\nmeta def get_fintype' (e : expr) : tactic (list (expr × expr)) :=\ne.fold (return []) $ λ e _ l, do l' ← l,\n  (do t ← infer_type e,\n  match t with\n  | `(fintype _) := cond (has_var e) (return ((e, t) :: l')) (return l')\n  | _ := return l'\n  end) <|> return l'\n\nmeta def get_fintype_aux (e : expr) (t : expr) (l : tactic (list (expr × expr))) :\n  tactic (list (expr × expr)) :=\n  match t with\n  | `(fintype _) := do l ← l, cond (has_var e) (return ((e, t) :: l)) (return l)\n  | _ := l\n  end\n\n#eval (```((2 : ℕ) + 2) : expr ff).to_raw_fmt.to_string\n\n#print expr.pi\nmeta def get_fintype₂ : expr → tactic (list (expr × expr))\n| (const n l)           := do t ← infer_type (const n l), get_fintype_aux (const n l) t (pure [])\n| (local_const n m i t) := get_fintype_aux (local_const n m i t) t (pure [])\n| (app f x)             := do l₁ ← get_fintype₂ f, l₂ ← get_fintype₂ x, return (l₁ ++ l₂)\n| (lam n i v e)         := do l₁ ← get_fintype₂ v, l₂ ← get_fintype₂ e, return (l₁ ++ l₂)\n| (pi n i v e)          := do l₁ ← get_fintype₂ v, l₂ ← get_fintype₂ e, return (l₁ ++ l₂)\n| _ := return []\n\n#print simplify\n\n#print monad\n#print expr.replace\n#reduce tactic\n#print <|>\n#print interaction_monad.result\n#print tactic\n#eval expr.get_nat_value\n\n#print expr\n#print tactic.simplify\nmeta def\n\nmeta def replace_fintype : Π (e : expr) (l : list (expr × expr)), tactic expr\n\n\nexample (s t : set ℕ) [fintype s] [fintype t] [decidable_pred t] (h : fintype.card ↥(s ∩ t) = 77) :\n  fintype.card ↥(s ∪ t) = 9  :=\nbegin\n  -- thing,\n  -- fold_test_target_type,\n\nend", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/field.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.33971773595021265}}
{"text": "import .homotopy\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cylinder\n\nvariables {C : Type u} [category.{v} C] [has_cylinder C]\n\n-- A map j : A → X is the inclusion of a strong deformation retract if\n-- it admits a retraction r : X → A for which j ∘ r is homotopic to\n-- the identity rel j.\nstructure sdr_inclusion {a x : C} (j : a ⟶ x) :=\n(r : x ⟶ a)\n(h : r ∘ j = 𝟙 a)\n(H : j ∘ r ≃ 𝟙 x rel j)\n\ndef is_sdr_inclusion {a x : C} (j : a ⟶ x) : Prop := nonempty (sdr_inclusion j)\n\nlemma pushout_of_sdr_inclusion {a x a' x' : C} {j : a ⟶ x} {f : a ⟶ a'} {f' : x ⟶ x'}\n  {j' : a' ⟶ x'} (po : Is_pushout j f f' j')\n  (po' : Is_pushout (I &> j) (I &> f) (I &> f') (I &> j')) :\n  is_sdr_inclusion j → is_sdr_inclusion j' :=\nassume ⟨⟨r, h, ⟨H, Hrel⟩⟩⟩, begin\n  refine ⟨⟨po.induced (f ∘ r) (𝟙 a') (by rw [←assoc, h]; simp), by simp, _⟩⟩,\n  -- Now need to give a homotopy from j' ∘ r' (r' = the above induced\n  -- map) to 𝟙 x', rel j'. We define the homotopy using H on I +> x,\n  -- and the constant homotopy at the identity on I +> a'.\n  refine ⟨⟨po'.induced (f' ∘ H.H) (j' ∘ p @> a') _, _, _⟩, _⟩,\n  -- The above maps agree on I +> a, so we can form the induced map:\n  { unfold homotopy.is_rel at Hrel, rw [←assoc, Hrel, p_nat_assoc],\n    have : f' ∘ (j ∘ r ∘ j ∘ p @> a) = (f' ∘ j) ∘ (r ∘ j) ∘ p @> a, by simp,\n    rw [this, po.commutes, h], dsimp, simp },\n  -- The homotopy defined in this way starts at j' ∘ r' : x' → x':\n  { apply po.uniqueness; rw i_nat_assoc; conv { to_rhs, rw ←assoc }; simp;\n      rw ←assoc,\n    { erw [H.Hi₀, assoc, po.commutes] },\n    { rw [pi_components], simp } },\n  -- ... and ends at 𝟙 x':\n  { apply po.uniqueness; rw i_nat_assoc; simp; rw [←assoc],\n    { erw H.Hi₁; simp }, { rw [pi_components], simp } },\n  -- ... and is rel j:\n  { unfold homotopy.is_rel,\n    simp, congr, rw [←assoc], dsimp, simp }\nend\n\nend homotopy_theory.cylinder\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/cylinder/sdr.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.33971054342622387}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.preadditive.additive_functor\nimport category_theory.abelian.basic\nimport category_theory.limits.preserves.shapes.kernels\nimport category_theory.adjunction.limits\n\n/-!\n# Transferring \"abelian-ness\" across a functor\n\nIf `C` is an additive category, `D` is an abelian category,\nwe have `F : C ⥤ D` `G : D ⥤ C` (both preserving zero morphisms),\n`G` is left exact (that is, preserves finite limits),\nand further we have `adj : G ⊣ F` and `i : F ⋙ G ≅ 𝟭 C`,\nthen `C` is also abelian.\n\nSee https://stacks.math.columbia.edu/tag/03A3\n\n## Notes\nThe hypotheses, following the statement from the Stacks project,\nmay appear suprising: we don't ask that the counit of the adjunction is an isomorphism,\nbut just that we have some potentially unrelated isomorphism `i : F ⋙ G ≅ 𝟭 C`.\n\nHowever Lemma A1.1.1 from [Elephant] shows that in this situation the counit itself\nmust be an isomorphism, and thus that `C` is a reflective subcategory of `D`.\n\nSomeone may like to formalize that lemma, and restate this theorem in terms of `reflective`.\n(That lemma has a nice string diagrammatic proof that holds in any bicategory.)\n-/\n\nnoncomputable theory\n\nnamespace category_theory\nopen category_theory.limits\n\nuniverses v u₁ u₂\n\nnamespace abelian_of_adjunction\n\nvariables {C : Type u₁} [category.{v} C] [preadditive C]\nvariables {D : Type u₂} [category.{v} D] [abelian D]\nvariables (F : C ⥤ D)\nvariables (G : D ⥤ C) [functor.preserves_zero_morphisms G]\nvariables (i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F)\n\ninclude i\n\n/-- No point making this an instance, as it requires `i`. -/\nlemma has_kernels [preserves_finite_limits G] : has_kernels C :=\n{ has_limit := λ X Y f, begin\n    have := nat_iso.naturality_1 i f,\n    simp at this,\n    rw ←this,\n    haveI : has_kernel (G.map (F.map f) ≫ i.hom.app _) := limits.has_kernel_comp_mono _ _,\n    apply limits.has_kernel_iso_comp,\n  end }\n\ninclude adj\n\n/-- No point making this an instance, as it requires `i` and `adj`. -/\nlemma has_cokernels : has_cokernels C :=\n{ has_colimit := λ X Y f, begin\n    haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n    have := nat_iso.naturality_1 i f,\n    simp at this,\n    rw ←this,\n    haveI : has_cokernel (G.map (F.map f) ≫ i.hom.app _) := limits.has_cokernel_comp_iso _ _,\n    apply limits.has_cokernel_epi_comp,\n  end }\n\nvariables [limits.has_cokernels C]\n\n/-- Auxiliary construction for `coimage_iso_image` -/\ndef cokernel_iso {X Y : C} (f : X ⟶ Y) : G.obj (cokernel (F.map f)) ≅ cokernel f :=\nbegin\n  -- We have to write an explicit `preserves_colimits` type here,\n  -- as `left_adjoint_preserves_colimits` has universe variables.\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc G.obj (cokernel (F.map f))\n      ≅ cokernel (G.map (F.map f)) : (as_iso (cokernel_comparison _ G)).symm\n  ... ≅ cokernel (_ ≫ f ≫ _)       : cokernel_iso_of_eq (nat_iso.naturality_2 i f).symm\n  ... ≅ cokernel (f ≫ _)           : cokernel_epi_comp _ _\n  ... ≅ cokernel f                 : cokernel_comp_is_iso _ _\nend\n\nvariables [limits.has_kernels C] [preserves_finite_limits G]\n\n/-- Auxiliary construction for `coimage_iso_image` -/\ndef coimage_iso_image_aux {X Y : C} (f : X ⟶ Y) :\n  kernel (G.map (cokernel.π (F.map f))) ≅ kernel (cokernel.π f) :=\nbegin\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc kernel (G.map (cokernel.π (F.map f)))\n      ≅ kernel (cokernel.π (G.map (F.map f)) ≫ cokernel_comparison (F.map f) G)\n          : kernel_iso_of_eq (π_comp_cokernel_comparison _ _).symm\n  ... ≅ kernel (cokernel.π (G.map (F.map f))) : kernel_comp_mono _ _\n  ... ≅ kernel (cokernel.π (_ ≫ f ≫ _) ≫ (cokernel_iso_of_eq _).hom)\n          : kernel_iso_of_eq (π_comp_cokernel_iso_of_eq_hom (nat_iso.naturality_2 i f)).symm\n  ... ≅ kernel (cokernel.π (_ ≫ f ≫ _))       : kernel_comp_mono _ _\n  ... ≅ kernel (cokernel.π (f ≫ i.inv.app Y) ≫ (cokernel_epi_comp (i.hom.app X) _).inv)\n          : kernel_iso_of_eq (by simp only [cokernel.π_desc, cokernel_epi_comp_inv])\n  ... ≅ kernel (cokernel.π (f ≫ _))           : kernel_comp_mono _ _\n  ... ≅ kernel (inv (i.inv.app Y) ≫ cokernel.π f ≫ (cokernel_comp_is_iso f (i.inv.app Y)).inv)\n          : kernel_iso_of_eq (by simp only [cokernel.π_desc, cokernel_comp_is_iso_inv,\n              iso.hom_inv_id_app_assoc, nat_iso.inv_inv_app])\n  ... ≅ kernel (cokernel.π f ≫ _)             : kernel_is_iso_comp _ _\n  ... ≅ kernel (cokernel.π f)                 : kernel_comp_mono _ _\nend\n\nvariables [functor.preserves_zero_morphisms F]\n\n/--\nAuxiliary definition: the abelian coimage and abelian image agree.\nWe still need to check that this agrees with the canonical morphism.\n-/\ndef coimage_iso_image {X Y : C} (f : X ⟶ Y) : abelian.coimage f ≅ abelian.image f :=\nbegin\n  haveI : preserves_limits F := adj.right_adjoint_preserves_limits,\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc abelian.coimage f\n      ≅ cokernel (kernel.ι f)                 : iso.refl _\n  ... ≅ G.obj (cokernel (F.map (kernel.ι f))) : (cokernel_iso _ _ i adj _).symm\n  ... ≅ G.obj (cokernel (kernel_comparison f F ≫ (kernel.ι (F.map f))))\n                                              : G.map_iso (cokernel_iso_of_eq (by simp))\n  ... ≅ G.obj (cokernel (kernel.ι (F.map f))) : G.map_iso (cokernel_epi_comp _ _)\n  ... ≅ G.obj (abelian.coimage (F.map f))     : iso.refl _\n  ... ≅ G.obj (abelian.image (F.map f))       : G.map_iso (abelian.coimage_iso_image _)\n  ... ≅ G.obj (kernel (cokernel.π (F.map f))) : iso.refl _\n  ... ≅ kernel (G.map (cokernel.π (F.map f))) : preserves_kernel.iso _ _\n  ... ≅ kernel (cokernel.π f)                 : coimage_iso_image_aux F G i adj f\n  ... ≅ abelian.image f                       : iso.refl _,\nend\n\nlocal attribute [simp] cokernel_iso coimage_iso_image coimage_iso_image_aux\n\n-- The account of this proof in the Stacks project omits this calculation.\n-- Happily it's little effort: our `[ext]` and `[simp]` lemmas only need a little guidance.\nlemma coimage_iso_image_hom {X Y : C} (f : X ⟶ Y) :\n  (coimage_iso_image F G i adj f).hom = abelian.coimage_image_comparison f :=\nby { ext, simpa [-functor.map_comp, ←G.map_comp_assoc] using nat_iso.naturality_1 i f, }\n\nend abelian_of_adjunction\n\nopen abelian_of_adjunction\n\n/--\nIf `C` is an additive category, `D` is an abelian category,\nwe have `F : C ⥤ D` `G : D ⥤ C` (both preserving zero morphisms),\n`G` is left exact (that is, preserves finite limits),\nand further we have `adj : G ⊣ F` and `i : F ⋙ G ≅ 𝟭 C`,\nthen `C` is also abelian.\n\nSee https://stacks.math.columbia.edu/tag/03A3\n-/\ndef abelian_of_adjunction\n  {C : Type u₁} [category.{v} C] [preadditive C] [has_finite_products C]\n  {D : Type u₂} [category.{v} D] [abelian D]\n  (F : C ⥤ D) [functor.preserves_zero_morphisms F]\n  (G : D ⥤ C) [functor.preserves_zero_morphisms G] [preserves_finite_limits G]\n  (i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F) : abelian C :=\nbegin\n  haveI := has_kernels F G i, haveI := has_cokernels F G i adj,\n  haveI : ∀ {X Y : C} (f : X ⟶ Y), is_iso (abelian.coimage_image_comparison f),\n  { intros X Y f, rw ←coimage_iso_image_hom F G i adj f, apply_instance, },\n  apply abelian.of_coimage_image_comparison_is_iso,\nend\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/abelian/transfer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3395893430939196}}
{"text": "import for_mathlib.category_theory.morphism_property_misc\nimport category_theory.lifting_properties.basic\nimport for_mathlib.category_theory.lifting_properties.pullbacks\nimport for_mathlib.category_theory.lifting_properties.products\nimport for_mathlib.category_theory.lifting_properties.over\n\nopen category_theory.limits\n\nnamespace category_theory\n\nvariables {C : Type*} [category C]\n\nnamespace morphism_property\n\nvariables (F G : morphism_property C)\n\ndef has_lifting_property : Prop :=\n∀ ⦃A B X Y : C⦄ (i : A ⟶ B) (hi : F i) (p : X ⟶ Y) (hp : G p), has_lifting_property i p\n\nnamespace has_lifting_property\n\nvariables {F G}\n\nlemma property (h : has_lifting_property F G)\n{A B X Y : C} {i : A ⟶ B} (hi : F i) {p : X ⟶ Y} (hp : G p) :\n  category_theory.has_lifting_property i p := h i hi p hp\n\nlemma op (h : has_lifting_property F G) :\n  has_lifting_property G.op F.op :=\nλ A B X Y i hi p hp, (h p.unop hp i.unop hi).op\n\nlemma unop {F G : morphism_property Cᵒᵖ} (h : has_lifting_property F G) :\n  has_lifting_property G.unop F.unop :=\nλ A B X Y i hi p hp, (h p.op hp i.op hi).unop\n\nvariables (F G)\nlemma iff_op : has_lifting_property F G ↔ has_lifting_property G.op F.op := ⟨op, unop⟩\nlemma iff_unop (F' G' : morphism_property Cᵒᵖ) :\n  has_lifting_property F' G' ↔ has_lifting_property G'.unop F'.unop := ⟨unop, op⟩\n\nvariables {F G}\nlemma under (h : has_lifting_property F G) (M : C) :\n  has_lifting_property (F.inverse_image (under.forget M)) (G.inverse_image (under.forget M)) :=\nλ A B X Y i hi p hp, by { haveI := h _ hi _ hp, apply_instance, }\n\nend has_lifting_property\n\n@[simp]\ndef llp_with : morphism_property C :=\nλ A B i, ∀ ⦃X Y : C⦄ (p : X ⟶ Y) (hp : G p), category_theory.has_lifting_property i p\n\n@[simp]\ndef rlp_with : morphism_property C :=\nλ X Y p, ∀ ⦃A B : C⦄ (i : A ⟶ B) (hi : F i), category_theory.has_lifting_property i p\n\nlemma llp_with_op :\n  F.op.llp_with = F.rlp_with.op :=\nbegin\n  ext A B i,\n  split,\n  { intros h X Y p hp,\n    rw category_theory.has_lifting_property.iff_op,\n    exact h p.op hp, },\n  { intros h X Y p hp,\n    rw category_theory.has_lifting_property.iff_unop,\n    exact h p.unop hp, },\nend\n\nlemma rlp_with_op :\n  G.op.rlp_with = G.llp_with.op :=\nbegin\n  ext A B i,\n  split,\n  { intros h X Y p hp,\n    rw category_theory.has_lifting_property.iff_op,\n    exact h p.op hp, },\n  { intros h X Y p hp,\n    rw category_theory.has_lifting_property.iff_unop,\n    exact h p.unop hp, },\nend\n\nlemma llp_with_unop (G : morphism_property Cᵒᵖ) :\n  G.unop.llp_with = G.rlp_with.unop :=\nbegin\n  have h := rlp_with_op G.unop,\n  rw G.op_unop at h,\n  rw [h, unop_op],\nend\n\nlemma rlp_with_unop (F : morphism_property Cᵒᵖ) :\n  F.unop.rlp_with = F.llp_with.unop :=\nbegin\n  have h := llp_with_op F.unop,\n  rw F.op_unop at h,\n  rw [h, unop_op],\nend\n\nnamespace llp_with\n\nlemma contains_iso : isomorphisms C ⊆ G.llp_with :=\nλ A B i hi X Y p hp, by { haveI : is_iso i := hi, apply_instance, }\n\nlemma stable_under_composition : G.llp_with.stable_under_composition :=\nλ A₁ A₂ A₃ i j hi hj X Y p hp, by { haveI := hi p hp, haveI := hj p hp, apply_instance, }\n\nlemma respects_iso : G.llp_with.respects_iso :=\nrespects_iso.of_stable_under_composition_and_contains_iso\n    (llp_with.stable_under_composition G) (llp_with.contains_iso G)\n\nlemma stable_under_cobase_change : G.llp_with.stable_under_cobase_change :=\nλ A A' B B' f g f' g' h hf X Y p hp, h.has_lifting_property_imp p (hf p hp)\n\nlemma stable_under_coproducts : G.llp_with.stable_under_coproducts :=\n⟨llp_with.respects_iso G,\n  λ J A B hA hB i hi X Y p hp, by { haveI := λ j, hi j p hp, apply_instance, }⟩\n\nend llp_with\n\nnamespace rlp_with\n\nlemma contains_iso : isomorphisms C ⊆ F.rlp_with :=\nλ X Y p hp A B i hi, by { haveI : is_iso p := hp, apply_instance, }\n\nlemma stable_under_composition : F.rlp_with.stable_under_composition :=\nλ X Y Z p q hp hq A B i hi, by { haveI := hp i hi, haveI := hq i hi, apply_instance, }\n\nlemma respects_iso : F.rlp_with.respects_iso :=\nrespects_iso.of_stable_under_composition_and_contains_iso\n    (rlp_with.stable_under_composition F) (rlp_with.contains_iso F)\n\nlemma stable_under_base_change : F.rlp_with.stable_under_base_change :=\nλ X' Y Y' X f g f' g' h hg A B i hi, h.has_lifting_property_imp i (hg i hi)\n\nlemma stable_under_products : F.rlp_with.stable_under_products :=\n⟨rlp_with.respects_iso F,\nλ J X Y hX hY p hp A B i hi, by { haveI := λ j, hp j i hi, apply_instance, }⟩\n\nend rlp_with\n\nnamespace stable_under_cobase_change\n\nlemma coprod_inl {P : morphism_property C}\n  (h : P.stable_under_cobase_change) (A B : C) [has_binary_coproduct A B]\n  [has_initial C] (hB : P (initial.to B)) :\n  P (coprod.inl : A ⟶ A ⨿ B) :=\nh (is_pushout.of_has_binary_coproduct' A B) hB\n\nlemma coprod_inr {P : morphism_property C}\n  (h : P.stable_under_cobase_change) (A B : C) [has_binary_coproduct A B]\n  [has_initial C] (hA : P (initial.to A)) :\n  P (coprod.inr : B ⟶ A ⨿ B) :=\nh (is_pushout.of_has_binary_coproduct' A B).flip hA\n\nend stable_under_cobase_change\n\nnamespace stable_under_base_change\n\nlemma prod_fst {P : morphism_property C}\n  (h : P.stable_under_base_change) (X Y : C) [has_binary_product X Y]\n  [has_terminal C] (hY : P (terminal.from Y)) :\n  P (limits.prod.fst : X ⨯ Y ⟶ X) :=\nh (is_pullback.of_has_binary_product' X Y).flip hY\n\nlemma prod_snd {P : morphism_property C}\n  (h : P.stable_under_base_change) (X Y : C) [has_binary_product X Y]\n  [has_terminal C] (hX : P (terminal.from X)) :\n  P (limits.prod.snd : X ⨯ Y ⟶ Y) :=\nh (is_pullback.of_has_binary_product' X Y) hX\n\nend stable_under_base_change\n\nend morphism_property\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/lifting_properties/morphism_property.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832805, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33951629307273434}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.const\nimport category_theory.discrete_category\nimport category_theory.yoneda\nimport category_theory.reflects_isomorphisms\n\n-- morphism levels before object levels. See note [category_theory universes].\nuniverses v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\nopen category_theory\n\nvariables {J : Type u₁} [category.{v₁} J]\nvariables {K : Type u₂} [category.{v₂} K]\nvariables {C : Type u₃} [category.{v₃} C]\nvariables {D : Type u₄} [category.{v₄} D]\n\nopen category_theory\nopen category_theory.category\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory\n\nnamespace functor\nvariables {J C} (F : J ⥤ C)\n\n/--\n`F.cones` is the functor assigning to an object `X` the type of\nnatural transformations from the constant functor with value `X` to `F`.\nAn object representing this functor is a limit of `F`.\n-/\n@[simps]\ndef cones : Cᵒᵖ ⥤ Type (max u₁ v₃) := (const J).op ⋙ yoneda.obj F\n\n/--\n`F.cocones` is the functor assigning to an object `X` the type of\nnatural transformations from `F` to the constant functor with value `X`.\nAn object corepresenting this functor is a colimit of `F`.\n-/\n@[simps]\ndef cocones : C ⥤ Type (max u₁ v₃) := const J ⋙ coyoneda.obj (op F)\n\nend functor\n\nsection\nvariables (J C)\n\n/--\nFunctorially associated to each functor `J ⥤ C`, we have the `C`-presheaf consisting of\ncones with a given cone point.\n-/\n@[simps] def cones : (J ⥤ C) ⥤ (Cᵒᵖ ⥤ Type (max u₁ v₃)) :=\n{ obj := functor.cones,\n  map := λ F G f, whisker_left (const J).op (yoneda.map f) }\n\n/--\nContravariantly associated to each functor `J ⥤ C`, we have the `C`-copresheaf consisting of\ncocones with a given cocone point.\n-/\n@[simps] def cocones : (J ⥤ C)ᵒᵖ ⥤ (C ⥤ Type (max u₁ v₃)) :=\n{ obj := λ F, functor.cocones (unop F),\n  map := λ F G f, whisker_left (const J) (coyoneda.map f) }\n\nend\n\nnamespace limits\n\n/--\nA `c : cone F` is:\n* an object `c.X` and\n* a natural transformation `c.π : c.X ⟶ F` from the constant `c.X` functor to `F`.\n\n`cone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cones.obj X`.\n-/\nstructure cone (F : J ⥤ C) :=\n(X : C)\n(π : (const J).obj X ⟶ F)\n\ninstance inhabited_cone (F : discrete punit ⥤ C) : inhabited (cone F) :=\n⟨{ X := F.obj punit.star,\n   π := { app := λ ⟨⟩, 𝟙 _ } }⟩\n\n@[simp, reassoc] lemma cone.w {F : J ⥤ C} (c : cone F) {j j' : J} (f : j ⟶ j') :\n  c.π.app j ≫ F.map f = c.π.app j' :=\nby { rw ← c.π.naturality f, apply id_comp }\n\n/--\nA `c : cocone F` is\n* an object `c.X` and\n* a natural transformation `c.ι : F ⟶ c.X` from `F` to the constant `c.X` functor.\n\n`cocone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cocones.obj X`.\n-/\nstructure cocone (F : J ⥤ C) :=\n(X : C)\n(ι : F ⟶ (const J).obj X)\n\ninstance inhabited_cocone (F : discrete punit ⥤ C) : inhabited (cocone F) :=\n⟨{ X := F.obj punit.star,\n   ι := { app := λ ⟨⟩, 𝟙 _ } }⟩\n\n@[simp, reassoc] lemma cocone.w {F : J ⥤ C} (c : cocone F) {j j' : J} (f : j ⟶ j') :\n  F.map f ≫ c.ι.app j' = c.ι.app j :=\nby { rw c.ι.naturality f, apply comp_id }\n\nvariables {F : J ⥤ C}\n\nnamespace cone\n\n/-- The isomorphism between a cone on `F` and an element of the functor `F.cones`. -/\n@[simps]\ndef equiv (F : J ⥤ C) : cone F ≅ Σ X, F.cones.obj X :=\n{ hom := λ c, ⟨op c.X, c.π⟩,\n  inv := λ c, { X := c.1.unop, π := c.2 },\n  hom_inv_id' := by { ext1, cases x, refl },\n  inv_hom_id' := by { ext1, cases x, refl } }\n\n/-- A map to the vertex of a cone naturally induces a cone by composition. -/\n@[simps] def extensions (c : cone F) :\n  yoneda.obj c.X ⋙ ulift_functor.{u₁} ⟶ F.cones :=\n{ app := λ X f, (const J).map f.down ≫ c.π }\n\n/-- A map to the vertex of a cone induces a cone by composition. -/\n@[simps] def extend (c : cone F) {X : C} (f : X ⟶ c.X) : cone F :=\n{ X := X,\n  π := c.extensions.app (op X) ⟨f⟩ }\n\n/-- Whisker a cone by precomposition of a functor. -/\n@[simps] def whisker (E : K ⥤ J) (c : cone F) : cone (E ⋙ F) :=\n{ X := c.X,\n  π := whisker_left E c.π }\n\nend cone\n\nnamespace cocone\n\n/-- The isomorphism between a cocone on `F` and an element of the functor `F.cocones`. -/\ndef equiv (F : J ⥤ C) : cocone F ≅ Σ X, F.cocones.obj X :=\n{ hom := λ c, ⟨c.X, c.ι⟩,\n  inv := λ c, { X := c.1, ι := c.2 },\n  hom_inv_id' := by { ext1, cases x, refl },\n  inv_hom_id' := by { ext1, cases x, refl } }\n\n/-- A map from the vertex of a cocone naturally induces a cocone by composition. -/\n@[simps] def extensions (c : cocone F) : coyoneda.obj (op c.X) ⋙ ulift_functor.{u₁} ⟶ F.cocones :=\n{ app := λ X f, c.ι ≫ (const J).map f.down }\n\n/-- A map from the vertex of a cocone induces a cocone by composition. -/\n@[simps] def extend (c : cocone F) {X : C} (f : c.X ⟶ X) : cocone F :=\n{ X := X,\n  ι := c.extensions.app X ⟨f⟩ }\n\n/--\nWhisker a cocone by precomposition of a functor. See `whiskering` for a functorial\nversion.\n-/\n@[simps] def whisker (E : K ⥤ J) (c : cocone F) : cocone (E ⋙ F) :=\n{ X := c.X,\n  ι := whisker_left E c.ι }\n\nend cocone\n\n/-- A cone morphism between two cones for the same diagram is a morphism of the cone points which\ncommutes with the cone legs. -/\n@[ext] structure cone_morphism (A B : cone F) :=\n(hom : A.X ⟶ B.X)\n(w'  : ∀ j : J, hom ≫ B.π.app j = A.π.app j . obviously)\n\nrestate_axiom cone_morphism.w'\nattribute [simp, reassoc] cone_morphism.w\n\ninstance inhabited_cone_morphism (A : cone F) : inhabited (cone_morphism A A) :=\n⟨{ hom := 𝟙 _ }⟩\n\n/-- The category of cones on a given diagram. -/\n@[simps] instance cone.category : category (cone F) :=\n{ hom  := λ A B, cone_morphism A B,\n  comp := λ X Y Z f g, { hom := f.hom ≫ g.hom },\n  id   := λ B, { hom := 𝟙 B.X } }\n\nnamespace cones\n/-- To give an isomorphism between cones, it suffices to give an\n  isomorphism between their vertices which commutes with the cone\n  maps. -/\n@[ext, simps] def ext {c c' : cone F}\n  (φ : c.X ≅ c'.X) (w : ∀ j, c.π.app j = φ.hom ≫ c'.π.app j) : c ≅ c' :=\n{ hom := { hom := φ.hom },\n  inv := { hom := φ.inv, w' := λ j, φ.inv_comp_eq.mpr (w j) } }\n\n/--\nGiven a cone morphism whose object part is an isomorphism, produce an\nisomorphism of cones.\n-/\nlemma cone_iso_of_hom_iso {K : J ⥤ C} {c d : cone K} (f : c ⟶ d) [i : is_iso f.hom] :\n  is_iso f :=\n⟨⟨{ hom := inv f.hom,\n    w' := λ j, (as_iso f.hom).inv_comp_eq.2 (f.w j).symm }, by tidy⟩⟩\n\n/--\nFunctorially postcompose a cone for `F` by a natural transformation `F ⟶ G` to give a cone for `G`.\n-/\n@[simps] def postcompose {G : J ⥤ C} (α : F ⟶ G) : cone F ⥤ cone G :=\n{ obj := λ c, { X := c.X, π := c.π ≫ α },\n  map := λ c₁ c₂ f, { hom := f.hom } }\n\n/-- Postcomposing a cone by the composite natural transformation `α ≫ β` is the same as\npostcomposing by `α` and then by `β`. -/\n@[simps]\ndef postcompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :\n  postcompose (α ≫ β) ≅ postcompose α ⋙ postcompose β :=\nnat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/-- Postcomposing by the identity does not change the cone up to isomorphism. -/\n@[simps]\ndef postcompose_id : postcompose (𝟙 F) ≅ 𝟭 (cone F) :=\nnat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncones.\n-/\n@[simps]\ndef postcompose_equivalence {G : J ⥤ C} (α : F ≅ G) : cone F ≌ cone G :=\n{ functor := postcompose α.hom,\n  inverse := postcompose α.inv,\n  unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy) }\n\n/--\nWhiskering on the left by `E : K ⥤ J` gives a functor from `cone F` to `cone (E ⋙ F)`.\n-/\n@[simps]\ndef whiskering (E : K ⥤ J) : cone F ⥤ cone (E ⋙ F) :=\n{ obj := λ c, c.whisker E,\n  map := λ c c' f, { hom := f.hom } }\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\n@[simps]\ndef whiskering_equivalence (e : K ≌ J) :\n  cone F ≌ cone (e.functor ⋙ F) :=\n{ functor := whiskering e.functor,\n  inverse := whiskering e.inverse ⋙ postcompose (e.inv_fun_id_assoc F).hom,\n  unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _)\n  (begin\n    intro k,\n    dsimp, -- See library note [dsimp, simp]\n    simpa [e.counit_app_functor] using s.w (e.unit_inv.app k),\n  end)) (by tidy), }\n\n/--\nThe categories of cones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\n@[simps functor inverse unit_iso counit_iso]\ndef equivalence_of_reindexing {G : K ⥤ C}\n  (e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cone F ≌ cone G :=\n(whiskering_equivalence e).trans (postcompose_equivalence α)\n\nsection\nvariable (F)\n\n/-- Forget the cone structure and obtain just the cone point. -/\n@[simps]\ndef forget : cone F ⥤ C :=\n{ obj := λ t, t.X, map := λ s t f, f.hom }\n\nvariables (G : C ⥤ D)\n\n/-- A functor `G : C ⥤ D` sends cones over `F` to cones over `F ⋙ G` functorially. -/\n@[simps] def functoriality : cone F ⥤ cone (F ⋙ G) :=\n{ obj := λ A,\n  { X := G.obj A.X,\n    π := { app := λ j, G.map (A.π.app j), naturality' := by intros; erw ←G.map_comp; tidy } },\n  map := λ X Y f,\n  { hom := G.map f.hom,\n    w' := λ j, by simp [-cone_morphism.w, ←f.w j] } }\n\ninstance functoriality_full [full G] [faithful G] : full (functoriality F G) :=\n{ preimage := λ X Y t,\n  { hom := G.preimage t.hom,\n    w' := λ j, G.map_injective (by simpa using t.w j) } }\n\ninstance functoriality_faithful [faithful G] : faithful (cones.functoriality F G) :=\n{ map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } }\n\n/--\nIf `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cones over `F` and cones over `F ⋙ e.functor`.\n-/\n@[simps]\ndef functoriality_equivalence (e : C ≌ D) : cone F ≌ cone (F ⋙ e.functor) :=\nlet f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in\n{ functor := functoriality F e.functor,\n  inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙\n    (postcompose_equivalence f).functor,\n  unit_iso := nat_iso.of_components (λ c, cones.ext (e.unit_iso.app _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cones.ext (e.counit_iso.app _) (by tidy)) (by tidy), }\n\n/--\nIf `F` reflects isomorphisms, then `cones.functoriality F` reflects isomorphisms\nas well.\n-/\ninstance reflects_cone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :\n  reflects_isomorphisms (cones.functoriality K F) :=\nbegin\n  constructor,\n  introsI,\n  haveI : is_iso (F.map f.hom) :=\n    (cones.forget (K ⋙ F)).map_is_iso ((cones.functoriality K F).map f),\n  haveI := reflects_isomorphisms.reflects F f.hom,\n  apply cone_iso_of_hom_iso\nend\n\nend\n\nend cones\n\n/-- A cocone morphism between two cocones for the same diagram is a morphism of the cocone points\nwhich commutes with the cocone legs. -/\n@[ext] structure cocone_morphism (A B : cocone F) :=\n(hom : A.X ⟶ B.X)\n(w'  : ∀ j : J, A.ι.app j ≫ hom = B.ι.app j . obviously)\n\ninstance inhabited_cocone_morphism (A : cocone F) : inhabited (cocone_morphism A A) :=\n⟨{ hom := 𝟙 _ }⟩\n\nrestate_axiom cocone_morphism.w'\nattribute [simp, reassoc] cocone_morphism.w\n\n@[simps] instance cocone.category : category (cocone F) :=\n{ hom  := λ A B, cocone_morphism A B,\n  comp := λ _ _ _ f g,\n  { hom := f.hom ≫ g.hom },\n  id   := λ B, { hom := 𝟙 B.X } }\n\nnamespace cocones\n/-- To give an isomorphism between cocones, it suffices to give an\n  isomorphism between their vertices which commutes with the cocone\n  maps. -/\n@[ext, simps] def ext {c c' : cocone F}\n  (φ : c.X ≅ c'.X) (w : ∀ j, c.ι.app j ≫ φ.hom = c'.ι.app j) : c ≅ c' :=\n{ hom := { hom := φ.hom },\n  inv := { hom := φ.inv, w' := λ j, φ.comp_inv_eq.mpr (w j).symm } }\n\n/--\nGiven a cocone morphism whose object part is an isomorphism, produce an\nisomorphism of cocones.\n-/\nlemma cocone_iso_of_hom_iso {K : J ⥤ C} {c d : cocone K} (f : c ⟶ d) [i : is_iso f.hom] :\n  is_iso f :=\n⟨⟨{ hom := inv f.hom,\n    w' := λ j, (as_iso f.hom).comp_inv_eq.2 (f.w j).symm }, by tidy⟩⟩\n\n/-- Functorially precompose a cocone for `F` by a natural transformation `G ⟶ F` to give a cocone\nfor `G`. -/\n@[simps] def precompose {G : J ⥤ C} (α : G ⟶ F) : cocone F ⥤ cocone G :=\n{ obj := λ c, { X := c.X, ι := α ≫ c.ι },\n  map := λ c₁ c₂ f, { hom := f.hom } }\n\n/-- Precomposing a cocone by the composite natural transformation `α ≫ β` is the same as\nprecomposing by `β` and then by `α`. -/\ndef precompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :\n  precompose (α ≫ β) ≅ precompose β ⋙ precompose α :=\nnat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/-- Precomposing by the identity does not change the cocone up to isomorphism. -/\ndef precompose_id : precompose (𝟙 F) ≅ 𝟭 (cocone F) :=\nnat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncocones.\n-/\n@[simps]\ndef precompose_equivalence {G : J ⥤ C} (α : G ≅ F) : cocone F ≌ cocone G :=\n{ functor := precompose α.hom,\n  inverse := precompose α.inv,\n  unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy) }\n\n/--\nWhiskering on the left by `E : K ⥤ J` gives a functor from `cocone F` to `cocone (E ⋙ F)`.\n-/\n@[simps]\ndef whiskering (E : K ⥤ J) : cocone F ⥤ cocone (E ⋙ F) :=\n{ obj := λ c, c.whisker E,\n  map := λ c c' f, { hom := f.hom, } }\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\n@[simps]\ndef whiskering_equivalence (e : K ≌ J) :\n  cocone F ≌ cocone (e.functor ⋙ F) :=\n{ functor := whiskering e.functor,\n  inverse := whiskering e.inverse ⋙\n    precompose ((functor.left_unitor F).inv ≫ (whisker_right (e.counit_iso).inv F) ≫\n      (functor.associator _ _ _).inv),\n  unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _)\n  (begin\n    intro k,\n    dsimp,\n    simpa [e.counit_inv_app_functor k] using s.w (e.unit.app k),\n  end)) (by tidy), }\n\n/--\nThe categories of cocones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\n@[simps functor_obj]\ndef equivalence_of_reindexing {G : K ⥤ C}\n  (e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cocone F ≌ cocone G :=\n(whiskering_equivalence e).trans (precompose_equivalence α.symm)\n\nsection\nvariable (F)\n\n/-- Forget the cocone structure and obtain just the cocone point. -/\n@[simps]\ndef forget : cocone F ⥤ C :=\n{ obj := λ t, t.X, map := λ s t f, f.hom }\n\nvariables (G : C ⥤ D)\n\n/-- A functor `G : C ⥤ D` sends cocones over `F` to cocones over `F ⋙ G` functorially. -/\n@[simps] def functoriality : cocone F ⥤ cocone (F ⋙ G) :=\n{ obj := λ A,\n  { X := G.obj A.X,\n    ι := { app := λ j, G.map (A.ι.app j), naturality' := by intros; erw ←G.map_comp; tidy } },\n  map := λ _ _ f,\n  { hom := G.map f.hom,\n    w'  := by intros; rw [←functor.map_comp, cocone_morphism.w] } }\n\ninstance functoriality_full [full G] [faithful G] : full (functoriality F G) :=\n{ preimage := λ X Y t,\n  { hom := G.preimage t.hom,\n    w' := λ j, G.map_injective (by simpa using t.w j) } }\n\ninstance functoriality_faithful [faithful G] : faithful (functoriality F G) :=\n{ map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } }\n\n/--\nIf `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cocones over `F` and cocones over `F ⋙ e.functor`.\n-/\n@[simps]\ndef functoriality_equivalence (e : C ≌ D) : cocone F ≌ cocone (F ⋙ e.functor) :=\nlet f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in\n{ functor := functoriality F e.functor,\n  inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙\n    (precompose_equivalence f.symm).functor,\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (e.unit_iso.app _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cocones.ext (e.counit_iso.app _)\n  begin\n    -- Unfortunately this doesn't work by `tidy`.\n    -- In this configuration `simp` reaches a dead-end and needs help.\n    intros j,\n    dsimp,\n    simp only [←equivalence.counit_inv_app_functor, iso.inv_hom_id_app, map_comp,\n      equivalence.fun_inv_map, assoc, id_comp, iso.inv_hom_id_app_assoc],\n    dsimp, simp, -- See note [dsimp, simp].\n  end)\n  (λ c c' f, by { ext, dsimp, simp, dsimp, simp, }), }\n\n/--\nIf `F` reflects isomorphisms, then `cocones.functoriality F` reflects isomorphisms\nas well.\n-/\ninstance reflects_cocone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :\n  reflects_isomorphisms (cocones.functoriality K F) :=\nbegin\n  constructor,\n  introsI,\n  haveI : is_iso (F.map f.hom) :=\n    (cocones.forget (K ⋙ F)).map_is_iso ((cocones.functoriality K F).map f),\n  haveI := reflects_isomorphisms.reflects F f.hom,\n  apply cocone_iso_of_hom_iso\nend\n\nend\nend cocones\n\nend limits\n\nnamespace functor\n\nvariables {F : J ⥤ C} {G : J ⥤ C} (H : C ⥤ D)\n\nopen category_theory.limits\n\n/-- The image of a cone in C under a functor G : C ⥤ D is a cone in D. -/\n@[simps]\ndef map_cone   (c : cone F)   : cone (F ⋙ H)   := (cones.functoriality F H).obj c\n/-- The image of a cocone in C under a functor G : C ⥤ D is a cocone in D. -/\n@[simps]\ndef map_cocone (c : cocone F) : cocone (F ⋙ H) := (cocones.functoriality F H).obj c\n\n/-- Given a cone morphism `c ⟶ c'`, construct a cone morphism on the mapped cones functorially.  -/\ndef map_cone_morphism   {c c' : cone F}   (f : c ⟶ c')   :\n  H.map_cone c ⟶ H.map_cone c' := (cones.functoriality F H).map f\n\n/-- Given a cocone morphism `c ⟶ c'`, construct a cocone morphism on the mapped cocones\nfunctorially. -/\ndef map_cocone_morphism {c c' : cocone F} (f : c ⟶ c') :\n  H.map_cocone c ⟶ H.map_cocone c' := (cocones.functoriality F H).map f\n\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F ⋙ H`.-/\ndef map_cone_inv [is_equivalence H]\n  (c : cone (F ⋙ H)) : cone F :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).inverse.obj c\n\n/-- `map_cone` is the left inverse to `map_cone_inv`. -/\ndef map_cone_map_cone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone (F ⋙ H)) :\n  map_cone H (map_cone_inv H c) ≅ c :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c\n\n/-- `map_cone` is the right inverse to `map_cone_inv`. -/\ndef map_cone_inv_map_cone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone F) :\n  map_cone_inv H (map_cone H c) ≅ c :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F ⋙ H`.-/\n\ndef map_cocone_inv [is_equivalence H]\n  (c : cocone (F ⋙ H)) : cocone F :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).inverse.obj c\n\n/-- `map_cocone` is the left inverse to `map_cocone_inv`. -/\ndef map_cocone_map_cocone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone (F ⋙ H)) :\n  map_cocone H (map_cocone_inv H c) ≅ c :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c\n\n/-- `map_cocone` is the right inverse to `map_cocone_inv`. -/\ndef map_cocone_inv_map_cocone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone F) :\n  map_cocone_inv H (map_cocone H c) ≅ c :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c\n\n/-- `functoriality F _ ⋙ postcompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simps]\ndef functoriality_comp_postcompose {H H' : C ⥤ D} (α : H ≅ H') :\n  cones.functoriality F H ⋙ cones.postcompose (whisker_left F α.hom) ≅ cones.functoriality F H' :=\nnat_iso.of_components (λ c, cones.ext (α.app _) (by tidy)) (by tidy)\n\n/--\nFor `F : J ⥤ C`, given a cone `c : cone F`, and a natural isomorphism `α : H ≅ H'` for functors\n`H H' : C ⥤ D`, the postcomposition of the cone `H.map_cone` using the isomorphism `α` is\nisomorphic to the cone `H'.map_cone`.\n-/\n@[simps]\ndef postcompose_whisker_left_map_cone {H H' : C ⥤ D} (α : H ≅ H') (c : cone F) :\n  (cones.postcompose (whisker_left F α.hom : _)).obj (H.map_cone c) ≅ H'.map_cone c :=\n(functoriality_comp_postcompose α).app c\n\n/--\n`map_cone` commutes with `postcompose`. In particular, for `F : J ⥤ C`, given a cone `c : cone F`, a\nnatural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious ways of producing\na cone over `G ⋙ H`, and they are both isomorphic.\n-/\n@[simps]\ndef map_cone_postcompose {α : F ⟶ G} {c} :\n  H.map_cone ((cones.postcompose α).obj c) ≅\n  (cones.postcompose (whisker_right α H : _)).obj (H.map_cone c) :=\ncones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cone` commutes with `postcompose_equivalence`\n-/\n@[simps]\ndef map_cone_postcompose_equivalence_functor {α : F ≅ G} {c} :\n  H.map_cone ((cones.postcompose_equivalence α).functor.obj c) ≅\n    (cones.postcompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cone c) :=\ncones.ext (iso.refl _) (by tidy)\n\n/-- `functoriality F _ ⋙ precompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simps]\ndef functoriality_comp_precompose {H H' : C ⥤ D} (α : H ≅ H') :\n   cocones.functoriality F H ⋙ cocones.precompose (whisker_left F α.inv)\n ≅ cocones.functoriality F H' :=\nnat_iso.of_components (λ c, cocones.ext (α.app _) (by tidy)) (by tidy)\n\n/--\nFor `F : J ⥤ C`, given a cocone `c : cocone F`, and a natural isomorphism `α : H ≅ H'` for functors\n`H H' : C ⥤ D`, the precomposition of the cocone `H.map_cocone` using the isomorphism `α` is\nisomorphic to the cocone `H'.map_cocone`.\n-/\n@[simps]\ndef precompose_whisker_left_map_cocone {H H' : C ⥤ D} (α : H ≅ H') (c : cocone F) :\n  (cocones.precompose (whisker_left F α.inv : _)).obj (H.map_cocone c) ≅ H'.map_cocone c :=\n(functoriality_comp_precompose α).app c\n\n/--\n`map_cocone` commutes with `precompose`. In particular, for `F : J ⥤ C`, given a cocone\n`c : cocone F`, a natural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious\nways of producing a cocone over `G ⋙ H`, and they are both isomorphic.\n-/\n@[simps]\ndef map_cocone_precompose {α : F ⟶ G} {c} :\n  H.map_cocone ((cocones.precompose α).obj c) ≅\n  (cocones.precompose (whisker_right α H : _)).obj (H.map_cocone c) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cocone` commutes with `precompose_equivalence`\n-/\n@[simps]\ndef map_cocone_precompose_equivalence_functor {α : F ≅ G} {c} :\n  H.map_cocone ((cocones.precompose_equivalence α).functor.obj c) ≅\n    (cocones.precompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cocone c) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cone` commutes with `whisker`\n-/\n@[simps]\ndef map_cone_whisker {E : K ⥤ J} {c : cone F} :\n  H.map_cone (c.whisker E) ≅ (H.map_cone c).whisker E :=\ncones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cocone` commutes with `whisker`\n-/\n@[simps]\ndef map_cocone_whisker {E : K ⥤ J} {c : cocone F} :\n  H.map_cocone (c.whisker E) ≅ (H.map_cocone c).whisker E :=\ncocones.ext (iso.refl _) (by tidy)\n\nend functor\n\nend category_theory\n\nnamespace category_theory.limits\n\nsection\nvariables {F : J ⥤ C}\n\n/-- Change a `cocone F` into a `cone F.op`. -/\n@[simps] def cocone.op (c : cocone F) : cone F.op :=\n{ X := op c.X,\n  π :=\n  { app := λ j, (c.ι.app (unop j)).op,\n    naturality' := λ j j' f, quiver.hom.unop_inj (by tidy) } }\n\n/-- Change a `cone F` into a `cocone F.op`. -/\n@[simps] def cone.op (c : cone F) : cocone F.op :=\n{ X := op c.X,\n  ι :=\n  { app := λ j, (c.π.app (unop j)).op,\n    naturality' := λ j j' f, quiver.hom.unop_inj (by tidy) } }\n\n/-- Change a `cocone F.op` into a `cone F`. -/\n@[simps] def cocone.unop (c : cocone F.op) : cone F :=\n{ X := unop c.X,\n  π :=\n  { app := λ j, (c.ι.app (op j)).unop,\n    naturality' := λ j j' f, quiver.hom.op_inj (c.ι.naturality f.op).symm } }\n\n/-- Change a `cone F.op` into a `cocone F`. -/\n@[simps] def cone.unop (c : cone F.op) : cocone F :=\n{ X := unop c.X,\n  ι :=\n  { app := λ j, (c.π.app (op j)).unop,\n    naturality' := λ j j' f, quiver.hom.op_inj (c.π.naturality f.op).symm } }\n\nvariables (F)\n\n/--\nThe category of cocones on `F`\nis equivalent to the opposite category of\nthe category of cones on the opposite of `F`.\n-/\n@[simps]\ndef cocone_equivalence_op_cone_op : cocone F ≌ (cone F.op)ᵒᵖ :=\n{ functor :=\n  { obj := λ c, op (cocone.op c),\n    map := λ X Y f, quiver.hom.op\n    { hom := f.hom.op,\n      w' := λ j, by { apply quiver.hom.unop_inj, dsimp, simp, }, } },\n  inverse :=\n  { obj := λ c, cone.unop (unop c),\n    map := λ X Y f,\n    { hom := f.unop.hom.unop,\n      w' := λ j, by { apply quiver.hom.op_inj, dsimp, simp, }, } },\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c,\n    by { induction c using opposite.rec,\n         dsimp, apply iso.op, exact cones.ext (iso.refl _) (by tidy), })\n    begin\n      intros,\n      have hX : X = op (unop X) := rfl,\n      revert hX,\n      generalize : unop X = X',\n      rintro rfl,\n      have hY : Y = op (unop Y) := rfl,\n      revert hY,\n      generalize : unop Y = Y',\n      rintro rfl,\n      apply quiver.hom.unop_inj,\n      apply cone_morphism.ext,\n      dsimp, simp,\n    end,\n  functor_unit_iso_comp' := λ c, begin apply quiver.hom.unop_inj, ext, dsimp, simp, end }\n\nend\n\nsection\nvariables {F : J ⥤ Cᵒᵖ}\n\n/-- Change a cocone on `F.left_op : Jᵒᵖ ⥤ C` to a cocone on `F : J ⥤ Cᵒᵖ`. -/\n-- Here and below we only automatically generate the `@[simp]` lemma for the `X` field,\n-- as we can write a simpler `rfl` lemma for the components of the natural transformation by hand.\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cone_of_cocone_left_op (c : cocone F.left_op) : cone F :=\n{ X := op c.X,\n  π := nat_trans.remove_left_op (c.ι ≫ (const.op_obj_unop (op c.X)).hom) }\n\n/-- Change a cone on `F : J ⥤ Cᵒᵖ` to a cocone on `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cocone_left_op_of_cone (c : cone F) : cocone (F.left_op) :=\n{ X := unop c.X,\n  ι := nat_trans.left_op c.π }\n\n/-- Change a cone on `F.left_op : Jᵒᵖ ⥤ C` to a cocone on `F : J ⥤ Cᵒᵖ`. -/\n/- When trying use `@[simps]` to generate the `ι_app` field of this definition, `@[simps]` tries to\n  reduce the RHS using `expr.dsimp` and `expr.simp`, but for some reason the expression is not\n  being simplified properly. -/\n@[simps X]\ndef cocone_of_cone_left_op (c : cone F.left_op) : cocone F :=\n{ X := op c.X,\n  ι := nat_trans.remove_left_op ((const.op_obj_unop (op c.X)).hom ≫ c.π) }\n\n@[simp] lemma cocone_of_cone_left_op_ι_app (c : cone F.left_op) (j) :\n  (cocone_of_cone_left_op c).ι.app j = (c.π.app (op j)).op :=\nby { dsimp [cocone_of_cone_left_op], simp }\n\n/-- Change a cocone on `F : J ⥤ Cᵒᵖ` to a cone on `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cone_left_op_of_cocone (c : cocone F) : cone (F.left_op) :=\n{ X := unop c.X,\n  π := nat_trans.left_op c.ι }\n\nend\n\nend category_theory.limits\n\nnamespace category_theory.functor\n\nopen category_theory.limits\n\nvariables {F : J ⥤ C}\n\nsection\nvariables (G : C ⥤ D)\n\n/-- The opposite cocone of the image of a cone is the image of the opposite cocone. -/\n@[simps {rhs_md := semireducible}]\ndef map_cone_op (t : cone F) : (G.map_cone t).op ≅ (G.op.map_cocone t.op) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/-- The opposite cone of the image of a cocone is the image of the opposite cone. -/\n@[simps {rhs_md := semireducible}]\ndef map_cocone_op {t : cocone F} : (G.map_cocone t).op ≅ (G.op.map_cone t.op) :=\ncones.ext (iso.refl _) (by tidy)\n\nend\n\nend category_theory.functor\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": 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{"text": "/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n\n! This file was ported from Lean 3 source module logic.basic\n! leanprover-community/mathlib commit d2d8742b0c21426362a9dacebc6005db895ca963\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.DocCommands\nimport Mathbin.Tactic.ReservedNotation\n\n/-!\n# Basic logic properties\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file is one of the earliest imports in mathlib.\n\n## Implementation notes\n\nTheorems that require decidability hypotheses are in the namespace \"decidable\".\nClassical versions are in the namespace \"classical\".\n\nIn the presence of automation, this whole file may be unnecessary. On the other hand,\nmaybe it is useful for writing automation.\n-/\n\n\nopen Function\n\nattribute [local instance] Classical.propDecidable\n\nsection Miscellany\n\n/- We add the `inline` attribute to optimize VM computation using these declarations. For example,\n  `if p ∧ q then ... else ...` will not evaluate the decidability of `q` if `p` is false. -/\nattribute [inline]\n  And.decidable Or.decidable Decidable.false Xor'.decidable Iff.decidable Decidable.true Implies.decidable Not.decidable Ne.decidable Bool.decidableEq Decidable.decide\n\nattribute [simp] cast_eq cast_hEq\n\nvariable {α : Type _} {β : Type _}\n\n#print hidden /-\n/-- An identity function with its main argument implicit. This will be printed as `hidden` even\nif it is applied to a large term, so it can be used for elision,\nas done in the `elide` and `unelide` tactics. -/\n@[reducible]\ndef hidden {α : Sort _} {a : α} :=\n  a\n#align hidden hidden\n-/\n\n#print Empty.elim /-\n/-- Ex falso, the nondependent eliminator for the `empty` type. -/\ndef Empty.elim {C : Sort _} : Empty → C :=\n  fun.\n#align empty.elim Empty.elim\n-/\n\ninstance : Subsingleton Empty :=\n  ⟨fun a => a.elim⟩\n\ninstance Subsingleton.prod {α β : Type _} [Subsingleton α] [Subsingleton β] :\n    Subsingleton (α × β) :=\n  ⟨by\n    intro a b\n    cases a\n    cases b\n    congr ⟩\n#align subsingleton.prod Subsingleton.prod\n\ninstance : DecidableEq Empty := fun a => a.elim\n\ninstance Sort.inhabited : Inhabited (Sort _) :=\n  ⟨PUnit⟩\n#align sort.inhabited Sort.inhabited\n\ninstance Sort.inhabited' : Inhabited default :=\n  ⟨PUnit.unit⟩\n#align sort.inhabited' Sort.inhabited'\n\ninstance PSum.inhabitedLeft {α β} [Inhabited α] : Inhabited (PSum α β) :=\n  ⟨PSum.inl default⟩\n#align psum.inhabited_left PSum.inhabitedLeft\n\ninstance PSum.inhabitedRight {α β} [Inhabited β] : Inhabited (PSum α β) :=\n  ⟨PSum.inr default⟩\n#align psum.inhabited_right PSum.inhabitedRight\n\n#print decidableEq_of_subsingleton /-\ninstance (priority := 10) decidableEq_of_subsingleton {α} [Subsingleton α] : DecidableEq α\n  | a, b => isTrue (Subsingleton.elim a b)\n#align decidable_eq_of_subsingleton decidableEq_of_subsingleton\n-/\n\n#print eq_iff_true_of_subsingleton /-\n@[simp]\ntheorem eq_iff_true_of_subsingleton {α : Sort _} [Subsingleton α] (x y : α) : x = y ↔ True := by cc\n#align eq_iff_true_of_subsingleton eq_iff_true_of_subsingleton\n-/\n\n#print subsingleton_of_forall_eq /-\n/-- If all points are equal to a given point `x`, then `α` is a subsingleton. -/\ntheorem subsingleton_of_forall_eq {α : Sort _} (x : α) (h : ∀ y, y = x) : Subsingleton α :=\n  ⟨fun a b => (h a).symm ▸ (h b).symm ▸ rfl⟩\n#align subsingleton_of_forall_eq subsingleton_of_forall_eq\n-/\n\n#print subsingleton_iff_forall_eq /-\ntheorem subsingleton_iff_forall_eq {α : Sort _} (x : α) : Subsingleton α ↔ ∀ y, y = x :=\n  ⟨fun h y => @Subsingleton.elim _ h y x, subsingleton_of_forall_eq x⟩\n#align subsingleton_iff_forall_eq subsingleton_iff_forall_eq\n-/\n\ninstance Subtype.subsingleton (α : Sort _) [Subsingleton α] (p : α → Prop) :\n    Subsingleton (Subtype p) :=\n  ⟨fun ⟨x, _⟩ ⟨y, _⟩ => by\n    have : x = y := Subsingleton.elim _ _\n    cases this\n    rfl⟩\n#align subtype.subsingleton Subtype.subsingleton\n\n/-- Add an instance to \"undo\" coercion transitivity into a chain of coercions, because\n   most simp lemmas are stated with respect to simple coercions and will not match when\n   part of a chain. -/\n@[simp]\ntheorem coe_coe {α β γ} [Coe α β] [CoeTC β γ] (a : α) : (a : γ) = (a : β) :=\n  rfl\n#align coe_coe coe_coe\n\ntheorem coeFn_coe_trans {α β γ δ} [Coe α β] [HasCoeTAux β γ] [CoeFun γ δ] (x : α) :\n    @coeFn α _ _ x = @coeFn β _ _ x :=\n  rfl\n#align coe_fn_coe_trans coeFn_coe_trans\n\n/-- Non-dependent version of `coe_fn_coe_trans`, helps `rw` figure out the argument. -/\ntheorem coeFn_coe_trans' {α β γ} {δ : _} [Coe α β] [HasCoeTAux β γ] [CoeFun γ fun _ => δ] (x : α) :\n    @coeFn α _ _ x = @coeFn β _ _ x :=\n  rfl\n#align coe_fn_coe_trans' coeFn_coe_trans'\n\n@[simp]\ntheorem coeFn_coeBase {α β γ} [Coe α β] [CoeFun β γ] (x : α) : @coeFn α _ _ x = @coeFn β _ _ x :=\n  rfl\n#align coe_fn_coe_base coeFn_coeBase\n\n/-- Non-dependent version of `coe_fn_coe_base`, helps `rw` figure out the argument. -/\ntheorem coeFn_coe_base' {α β} {γ : _} [Coe α β] [CoeFun β fun _ => γ] (x : α) :\n    @coeFn α _ _ x = @coeFn β _ _ x :=\n  rfl\n#align coe_fn_coe_base' coeFn_coe_base'\n\n-- This instance should have low priority, to ensure we follow the chain\n-- `set_like → has_coe_to_sort`\nattribute [instance] coeSortTrans\n\ntheorem coeSort_coe_trans {α β γ δ} [Coe α β] [HasCoeTAux β γ] [CoeSort γ δ] (x : α) :\n    @coeSort α _ _ x = @coeSort β _ _ x :=\n  rfl\n#align coe_sort_coe_trans coeSort_coe_trans\n\nlibrary_note \"function coercion\"/--\nMany structures such as bundled morphisms coerce to functions so that you can\ntransparently apply them to arguments. For example, if `e : α ≃ β` and `a : α`\nthen you can write `e a` and this is elaborated as `⇑e a`. This type of\ncoercion is implemented using the `has_coe_to_fun` type class. There is one\nimportant consideration:\n\nIf a type coerces to another type which in turn coerces to a function,\nthen it **must** implement `has_coe_to_fun` directly:\n```lean\nstructure sparkling_equiv (α β) extends α ≃ β\n\n-- if we add a `has_coe` instance,\ninstance {α β} : has_coe (sparkling_equiv α β) (α ≃ β) :=\n⟨sparkling_equiv.to_equiv⟩\n\n-- then a `has_coe_to_fun` instance **must** be added as well:\ninstance {α β} : has_coe_to_fun (sparkling_equiv α β) :=\n⟨λ _, α → β, λ f, f.to_equiv.to_fun⟩\n```\n\n(Rationale: if we do not declare the direct coercion, then `⇑e a` is not in\nsimp-normal form. The lemma `coe_fn_coe_base` will unfold it to `⇑↑e a`. This\noften causes loops in the simplifier.)\n-/\n\n\n@[simp]\ntheorem coeSort_coeBase {α β γ} [Coe α β] [CoeSort β γ] (x : α) :\n    @coeSort α _ _ x = @coeSort β _ _ x :=\n  rfl\n#align coe_sort_coe_base coeSort_coeBase\n\n#print PEmpty /-\n/-- `pempty` is the universe-polymorphic analogue of `empty`. -/\ninductive PEmpty.{u} : Sort u\n  deriving DecidableEq\n#align pempty PEmpty\n-/\n\n#print PEmpty.elim /-\n/-- Ex falso, the nondependent eliminator for the `pempty` type. -/\ndef PEmpty.elim {C : Sort _} : PEmpty → C :=\n  fun.\n#align pempty.elim PEmpty.elim\n-/\n\ninstance subsingleton_pEmpty : Subsingleton PEmpty :=\n  ⟨fun a => a.elim⟩\n#align subsingleton_pempty subsingleton_pEmpty\n\n#print not_nonempty_pempty /-\n@[simp]\ntheorem not_nonempty_pempty : ¬Nonempty PEmpty := fun ⟨h⟩ => h.elim\n#align not_nonempty_pempty not_nonempty_pempty\n-/\n\n/- warning: congr_heq -> congr_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u1}} {γ : Sort.{u2}} {f : α -> γ} {g : β -> γ} {x : α} {y : β}, (HEq.{imax u1 u2} (α -> γ) f (β -> γ) g) -> (HEq.{u1} α x β y) -> (Eq.{u2} γ (f x) (g y))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u2}} {γ : Sort.{u1}} {f : α -> γ} {g : β -> γ} {x : α} {y : β}, (HEq.{imax u2 u1} (α -> γ) f (β -> γ) g) -> (HEq.{u2} α x β y) -> (Eq.{u1} γ (f x) (g y))\nCase conversion may be inaccurate. Consider using '#align congr_heq congr_heqₓ'. -/\ntheorem congr_heq {α β γ : Sort _} {f : α → γ} {g : β → γ} {x : α} {y : β} (h₁ : HEq f g)\n    (h₂ : HEq x y) : f x = g y := by\n  cases h₂\n  cases h₁\n  rfl\n#align congr_heq congr_heq\n\n/- warning: congr_arg_heq -> congr_arg_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} (f : forall (a : α), β a) {a₁ : α} {a₂ : α}, (Eq.{u1} α a₁ a₂) -> (HEq.{u2} (β a₁) (f a₁) (β a₂) (f a₂))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : α -> Sort.{u1}} (f : forall (a : α), β a) {a₁ : α} {a₂ : α}, (Eq.{u2} α a₁ a₂) -> (HEq.{u1} (β a₁) (f a₁) (β a₂) (f a₂))\nCase conversion may be inaccurate. Consider using '#align congr_arg_heq congr_arg_heqₓ'. -/\ntheorem congr_arg_heq {α} {β : α → Sort _} (f : ∀ a, β a) :\n    ∀ {a₁ a₂ : α}, a₁ = a₂ → HEq (f a₁) (f a₂)\n  | a, _, rfl => HEq.rfl\n#align congr_arg_heq congr_arg_heq\n\n/- warning: ulift.down_injective -> ULift.down_injective is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}}, Function.Injective.{succ (max u1 u2), succ u1} (ULift.{u2, u1} α) α (ULift.down.{u2, u1} α)\nbut is expected to have type\n  forall {α : Type.{u2}}, Function.Injective.{max (succ u2) (succ u1), succ u2} (ULift.{u1, u2} α) α (ULift.down.{u1, u2} α)\nCase conversion may be inaccurate. Consider using '#align ulift.down_injective ULift.down_injectiveₓ'. -/\ntheorem ULift.down_injective {α : Sort _} : Function.Injective (@ULift.down α)\n  | ⟨a⟩, ⟨b⟩, rfl => rfl\n#align ulift.down_injective ULift.down_injective\n\n/- warning: ulift.down_inj -> ULift.down_inj is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {a : ULift.{u2, u1} α} {b : ULift.{u2, u1} α}, Iff (Eq.{succ u1} α (ULift.down.{u2, u1} α a) (ULift.down.{u2, u1} α b)) (Eq.{succ (max u1 u2)} (ULift.{u2, u1} α) a b)\nbut is expected to have type\n  forall {α : Type.{u2}} {a : ULift.{u1, u2} α} {b : ULift.{u1, u2} α}, Iff (Eq.{succ u2} α (ULift.down.{u1, u2} α a) (ULift.down.{u1, u2} α b)) (Eq.{max (succ u2) (succ u1)} (ULift.{u1, u2} α) a b)\nCase conversion may be inaccurate. Consider using '#align ulift.down_inj ULift.down_injₓ'. -/\n@[simp]\ntheorem ULift.down_inj {α : Sort _} {a b : ULift α} : a.down = b.down ↔ a = b :=\n  ⟨fun h => ULift.down_injective h, fun h => by rw [h]⟩\n#align ulift.down_inj ULift.down_inj\n\n#print PLift.down_injective /-\ntheorem PLift.down_injective {α : Sort _} : Function.Injective (@PLift.down α)\n  | ⟨a⟩, ⟨b⟩, rfl => rfl\n#align plift.down_injective PLift.down_injective\n-/\n\n#print PLift.down_inj /-\n@[simp]\ntheorem PLift.down_inj {α : Sort _} {a b : PLift α} : a.down = b.down ↔ a = b :=\n  ⟨fun h => PLift.down_injective h, fun h => by rw [h]⟩\n#align plift.down_inj PLift.down_inj\n-/\n\n-- missing [symm] attribute for ne in core.\nattribute [symm] Ne.symm\n\n#print ne_comm /-\ntheorem ne_comm {α} {a b : α} : a ≠ b ↔ b ≠ a :=\n  ⟨Ne.symm, Ne.symm⟩\n#align ne_comm ne_comm\n-/\n\n#print eq_iff_eq_cancel_left /-\n@[simp]\ntheorem eq_iff_eq_cancel_left {α : Sort _} {b c : α} : (∀ {a}, a = b ↔ a = c) ↔ b = c :=\n  ⟨fun h => by rw [← h], fun h a => by rw [h]⟩\n#align eq_iff_eq_cancel_left eq_iff_eq_cancel_left\n-/\n\n#print eq_iff_eq_cancel_right /-\n@[simp]\ntheorem eq_iff_eq_cancel_right {α : Sort _} {a b : α} : (∀ {c}, a = c ↔ b = c) ↔ a = b :=\n  ⟨fun h => by rw [h], fun h a => by rw [h]⟩\n#align eq_iff_eq_cancel_right eq_iff_eq_cancel_right\n-/\n\n#print Fact /-\n/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`out] [] -/\n/-- Wrapper for adding elementary propositions to the type class systems.\nWarning: this can easily be abused. See the rest of this docstring for details.\n\nCertain propositions should not be treated as a class globally,\nbut sometimes it is very convenient to be able to use the type class system\nin specific circumstances.\n\nFor example, `zmod p` is a field if and only if `p` is a prime number.\nIn order to be able to find this field instance automatically by type class search,\nwe have to turn `p.prime` into an instance implicit assumption.\n\nOn the other hand, making `nat.prime` a class would require a major refactoring of the library,\nand it is questionable whether making `nat.prime` a class is desirable at all.\nThe compromise is to add the assumption `[fact p.prime]` to `zmod.field`.\n\nIn particular, this class is not intended for turning the type class system\ninto an automated theorem prover for first order logic. -/\nclass Fact (p : Prop) : Prop where\n  out : p\n#align fact Fact\n-/\n\nlibrary_note \"fact non-instances\"/--\nIn most cases, we should not have global instances of `fact`; typeclass search only reads the head\nsymbol and then tries any instances, which means that adding any such instance will cause slowdowns\neverywhere. We instead make them as lemmata and make them local instances as required.\n-/\n\n\n#print Fact.elim /-\ntheorem Fact.elim {p : Prop} (h : Fact p) : p :=\n  h.1\n#align fact.elim Fact.elim\n-/\n\n#print fact_iff /-\ntheorem fact_iff {p : Prop} : Fact p ↔ p :=\n  ⟨fun h => h.1, fun h => ⟨h⟩⟩\n#align fact_iff fact_iff\n-/\n\n#print Function.swap₂ /-\n/-- Swaps two pairs of arguments to a function. -/\n@[reducible]\ndef Function.swap₂ {ι₁ ι₂ : Sort _} {κ₁ : ι₁ → Sort _} {κ₂ : ι₂ → Sort _}\n    {φ : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Sort _} (f : ∀ i₁ j₁ i₂ j₂, φ i₁ j₁ i₂ j₂) :\n    ∀ i₂ j₂ i₁ j₁, φ i₁ j₁ i₂ j₂ := fun i₂ j₂ i₁ j₁ => f i₁ j₁ i₂ j₂\n#align function.swap₂ Function.swap₂\n-/\n\n/-- If `x : α . tac_name` then `x.out : α`. These are definitionally equal, but this can\nnevertheless be useful for various reasons, e.g. to apply further projection notation or in an\nargument to `simp`. -/\ndef autoParam.out {α : Sort _} {n : Name} (x : autoParam α n) : α :=\n  x\n#align auto_param.out autoParamₓ.out\n\n/-- If `x : α := d` then `x.out : α`. These are definitionally equal, but this can\nnevertheless be useful for various reasons, e.g. to apply further projection notation or in an\nargument to `simp`. -/\ndef optParam.out {α : Sort _} {d : α} (x : α := d) : α :=\n  x\n#align opt_param.out optParam.out\n\nend Miscellany\n\nopen Function\n\n/-!\n### Declarations about propositional connectives\n-/\n\n\n#print false_ne_true /-\ntheorem false_ne_true : False ≠ True\n  | h => h.symm ▸ trivial\n#align false_ne_true false_ne_true\n-/\n\ntheorem eq_true_iff {a : Prop} : (a = True) = a :=\n  have : (a ↔ True) = a := propext (iff_true_iff a)\n  Eq.subst (@iff_eq_eq a True) this\n#align eq_true_iff eq_true_iff\n\nsection Propositional\n\nvariable {a b c d e f : Prop}\n\n/-! ### Declarations about `implies` -/\n\n\ninstance : IsRefl Prop Iff :=\n  ⟨Iff.refl⟩\n\ninstance : IsTrans Prop Iff :=\n  ⟨fun _ _ _ => Iff.trans⟩\n\n#print iff_of_eq /-\ntheorem iff_of_eq (e : a = b) : a ↔ b :=\n  e ▸ Iff.rfl\n#align iff_of_eq iff_of_eq\n-/\n\n#print iff_iff_eq /-\ntheorem iff_iff_eq : (a ↔ b) ↔ a = b :=\n  ⟨propext, iff_of_eq⟩\n#align iff_iff_eq iff_iff_eq\n-/\n\n#print eq_iff_iff /-\n@[simp]\ntheorem eq_iff_iff {p q : Prop} : p = q ↔ (p ↔ q) :=\n  iff_iff_eq.symm\n#align eq_iff_iff eq_iff_iff\n-/\n\n#print imp_self /-\n@[simp]\ntheorem imp_self : a → a ↔ True :=\n  iff_true_intro id\n#align imp_self imp_self\n-/\n\n#print Iff.imp /-\ntheorem Iff.imp (h₁ : a ↔ b) (h₂ : c ↔ d) : a → c ↔ b → d :=\n  imp_congr h₁ h₂\n#align iff.imp Iff.imp\n-/\n\n#print eq_true_eq_id /-\n@[simp]\ntheorem eq_true_eq_id : Eq True = id := by\n  funext\n  simp only [true_iff_iff, id.def, iff_self_iff, eq_iff_iff]\n#align eq_true_eq_id eq_true_eq_id\n-/\n\n#print imp_intro /-\ntheorem imp_intro {α β : Prop} (h : α) : β → α := fun _ => h\n#align imp_intro imp_intro\n-/\n\n#print imp_false /-\ntheorem imp_false : a → False ↔ ¬a :=\n  Iff.rfl\n#align imp_false imp_false\n-/\n\n#print imp_and /-\ntheorem imp_and {α} : α → b ∧ c ↔ (α → b) ∧ (α → c) :=\n  ⟨fun h => ⟨fun ha => (h ha).left, fun ha => (h ha).right⟩, fun h ha => ⟨h.left ha, h.right ha⟩⟩\n#align imp_and_distrib imp_and\n-/\n\n#print and_imp /-\n@[simp]\ntheorem and_imp : a ∧ b → c ↔ a → b → c :=\n  Iff.intro (fun h ha hb => h ⟨ha, hb⟩) fun h ⟨ha, hb⟩ => h ha hb\n#align and_imp and_imp\n-/\n\n#print iff_def /-\ntheorem iff_def : (a ↔ b) ↔ (a → b) ∧ (b → a) :=\n  iff_iff_implies_and_implies _ _\n#align iff_def iff_def\n-/\n\n#print iff_def' /-\ntheorem iff_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) :=\n  iff_def.trans and_comm\n#align iff_def' iff_def'\n-/\n\n/- warning: imp_true_iff clashes with implies_true_iff -> imp_true_iff\nCase conversion may be inaccurate. Consider using '#align imp_true_iff imp_true_iffₓ'. -/\n#print imp_true_iff /-\ntheorem imp_true_iff {α : Sort _} : α → True ↔ True :=\n  iff_true_intro fun _ => trivial\n#align imp_true_iff imp_true_iff\n-/\n\ntheorem imp_iff_right (ha : a) : a → b ↔ b :=\n  ⟨fun f => f ha, imp_intro⟩\n#align imp_iff_right imp_iff_rightₓ\n\ntheorem imp_iff_not (hb : ¬b) : a → b ↔ ¬a :=\n  imp_congr_right fun _ => iff_false_intro hb\n#align imp_iff_not imp_iff_notₓ\n\n#print Decidable.imp_iff_right_iff /-\ntheorem Decidable.imp_iff_right_iff [Decidable a] : (a → b ↔ b) ↔ a ∨ b :=\n  ⟨fun H => (Decidable.em a).imp_right fun ha' => H.1 fun ha => (ha' ha).elim, fun H =>\n    H.elim imp_iff_right fun hb => ⟨fun hab => hb, fun _ _ => hb⟩⟩\n#align decidable.imp_iff_right_iff Decidable.imp_iff_right_iff\n-/\n\n#print imp_iff_right_iff /-\n@[simp]\ntheorem imp_iff_right_iff : (a → b ↔ b) ↔ a ∨ b :=\n  Decidable.imp_iff_right_iff\n#align imp_iff_right_iff imp_iff_right_iff\n-/\n\n#print Decidable.and_or_imp /-\ntheorem Decidable.and_or_imp [Decidable a] : a ∧ b ∨ (a → c) ↔ a → b ∨ c :=\n  if ha : a then by simp only [ha, true_and_iff, true_imp_iff]\n  else by simp only [ha, false_or_iff, false_and_iff, false_imp_iff]\n#align decidable.and_or_imp Decidable.and_or_imp\n-/\n\n#print and_or_imp /-\n@[simp]\ntheorem and_or_imp : a ∧ b ∨ (a → c) ↔ a → b ∨ c :=\n  Decidable.and_or_imp\n#align and_or_imp and_or_imp\n-/\n\n#print Function.mt /-\n/-- Provide modus tollens (`mt`) as dot notation for implications. -/\nprotected theorem Function.mt : (a → b) → ¬b → ¬a :=\n  mt\n#align function.mt Function.mt\n-/\n\n/-! ### Declarations about `not` -/\n\n\n#print Not.elim /-\n/-- Ex falso for negation. From `¬ a` and `a` anything follows. This is the same as `absurd` with\nthe arguments flipped, but it is in the `not` namespace so that projection notation can be used. -/\ndef Not.elim {α : Sort _} (H1 : ¬a) (H2 : a) : α :=\n  absurd H2 H1\n#align not.elim Not.elim\n-/\n\n#print Not.imp /-\n@[reducible]\ntheorem Not.imp {a b : Prop} (H2 : ¬b) (H1 : a → b) : ¬a :=\n  mt H1 H2\n#align not.imp Not.imp\n-/\n\n#print not_not_of_not_imp /-\ntheorem not_not_of_not_imp : ¬(a → b) → ¬¬a :=\n  mt Not.elim\n#align not_not_of_not_imp not_not_of_not_imp\n-/\n\n#print not_of_not_imp /-\ntheorem not_of_not_imp {a : Prop} : ¬(a → b) → ¬b :=\n  mt imp_intro\n#align not_of_not_imp not_of_not_imp\n-/\n\n#print dec_em /-\ntheorem dec_em (p : Prop) [Decidable p] : p ∨ ¬p :=\n  Decidable.em p\n#align dec_em dec_em\n-/\n\n#print dec_em' /-\ntheorem dec_em' (p : Prop) [Decidable p] : ¬p ∨ p :=\n  (dec_em p).symm\n#align dec_em' dec_em'\n-/\n\n#print em /-\ntheorem em (p : Prop) : p ∨ ¬p :=\n  Classical.em _\n#align em em\n-/\n\n#print em' /-\ntheorem em' (p : Prop) : ¬p ∨ p :=\n  (em p).symm\n#align em' em'\n-/\n\n#print or_not /-\ntheorem or_not {p : Prop} : p ∨ ¬p :=\n  em _\n#align or_not or_not\n-/\n\nsection eq_or_ne\n\nvariable {α : Sort _} (x y : α)\n\n#print Decidable.eq_or_ne /-\ntheorem Decidable.eq_or_ne [Decidable (x = y)] : x = y ∨ x ≠ y :=\n  dec_em <| x = y\n#align decidable.eq_or_ne Decidable.eq_or_ne\n-/\n\n#print Decidable.ne_or_eq /-\ntheorem Decidable.ne_or_eq [Decidable (x = y)] : x ≠ y ∨ x = y :=\n  dec_em' <| x = y\n#align decidable.ne_or_eq Decidable.ne_or_eq\n-/\n\n#print eq_or_ne /-\ntheorem eq_or_ne : x = y ∨ x ≠ y :=\n  em <| x = y\n#align eq_or_ne eq_or_ne\n-/\n\n#print ne_or_eq /-\ntheorem ne_or_eq : x ≠ y ∨ x = y :=\n  em' <| x = y\n#align ne_or_eq ne_or_eq\n-/\n\nend eq_or_ne\n\n/- warning: by_contradiction clashes with classical.by_contradiction -> by_contradiction\nCase conversion may be inaccurate. Consider using '#align by_contradiction by_contradictionₓ'. -/\n#print by_contradiction /-\ntheorem by_contradiction {p} : (¬p → False) → p :=\n  Decidable.by_contradiction\n#align by_contradiction by_contradiction\n-/\n\n#print by_contra /-\n-- alias by_contradiction ← by_contra\ntheorem by_contra {p} : (¬p → False) → p :=\n  Decidable.by_contradiction\n#align by_contra by_contra\n-/\n\nlibrary_note \"decidable namespace\"/--\nIn most of mathlib, we use the law of excluded middle (LEM) and the axiom of choice (AC) freely.\nThe `decidable` namespace contains versions of lemmas from the root namespace that explicitly\nattempt to avoid the axiom of choice, usually by adding decidability assumptions on the inputs.\n\nYou can check if a lemma uses the axiom of choice by using `#print axioms foo` and seeing if\n`classical.choice` appears in the list.\n-/\n\n\nlibrary_note \"decidable arguments\"/-- As mathlib is primarily classical,\nif the type signature of a `def` or `lemma` does not require any `decidable` instances to state,\nit is preferable not to introduce any `decidable` instances that are needed in the proof\nas arguments, but rather to use the `classical` tactic as needed.\n\nIn the other direction, when `decidable` instances do appear in the type signature,\nit is better to use explicitly introduced ones rather than allowing Lean to automatically infer\nclassical ones, as these may cause instance mismatch errors later.\n-/\n\n\n#print Decidable.not_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_not [Decidable a] : ¬¬a ↔ a :=\n  Iff.intro Decidable.by_contradiction not_not_intro\n#align decidable.not_not Decidable.not_not\n-/\n\n#print Classical.not_not /-\n/-- The Double Negation Theorem: `¬ ¬ P` is equivalent to `P`.\nThe left-to-right direction, double negation elimination (DNE),\nis classically true but not constructively. -/\n@[simp]\ntheorem Classical.not_not : ¬¬a ↔ a :=\n  Decidable.not_not\n#align not_not Classical.not_not\n-/\n\n#print of_not_not /-\ntheorem of_not_not : ¬¬a → a :=\n  by_contra\n#align of_not_not of_not_not\n-/\n\n#print not_ne_iff /-\ntheorem not_ne_iff {α : Sort _} {a b : α} : ¬a ≠ b ↔ a = b :=\n  Classical.not_not\n#align not_ne_iff not_ne_iff\n-/\n\n#print Decidable.of_not_imp /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.of_not_imp [Decidable a] (h : ¬(a → b)) : a :=\n  Decidable.by_contradiction (not_not_of_not_imp h)\n#align decidable.of_not_imp Decidable.of_not_imp\n-/\n\n#print of_not_imp /-\ntheorem of_not_imp : ¬(a → b) → a :=\n  Decidable.of_not_imp\n#align of_not_imp of_not_imp\n-/\n\n#print Decidable.not_imp_symm /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_imp_symm [Decidable a] (h : ¬a → b) (hb : ¬b) : a :=\n  Decidable.by_contradiction <| hb ∘ h\n#align decidable.not_imp_symm Decidable.not_imp_symm\n-/\n\n#print Not.decidable_imp_symm /-\ntheorem Not.decidable_imp_symm [Decidable a] : (¬a → b) → ¬b → a :=\n  Decidable.not_imp_symm\n#align not.decidable_imp_symm Not.decidable_imp_symm\n-/\n\n#print Not.imp_symm /-\ntheorem Not.imp_symm : (¬a → b) → ¬b → a :=\n  Not.decidable_imp_symm\n#align not.imp_symm Not.imp_symm\n-/\n\n#print Decidable.not_imp_comm /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_imp_comm [Decidable a] [Decidable b] : ¬a → b ↔ ¬b → a :=\n  ⟨Not.decidable_imp_symm, Not.decidable_imp_symm⟩\n#align decidable.not_imp_comm Decidable.not_imp_comm\n-/\n\n#print not_imp_comm /-\ntheorem not_imp_comm : ¬a → b ↔ ¬b → a :=\n  Decidable.not_imp_comm\n#align not_imp_comm not_imp_comm\n-/\n\n#print imp_not_self /-\n@[simp]\ntheorem imp_not_self : a → ¬a ↔ ¬a :=\n  ⟨fun h ha => h ha ha, fun h _ => h⟩\n#align imp_not_self imp_not_self\n-/\n\n#print Decidable.not_imp_self /-\ntheorem Decidable.not_imp_self [Decidable a] : ¬a → a ↔ a :=\n  by\n  have := @imp_not_self ¬a\n  rwa [Decidable.not_not] at this\n#align decidable.not_imp_self Decidable.not_imp_self\n-/\n\n#print not_imp_self /-\n@[simp]\ntheorem not_imp_self : ¬a → a ↔ a :=\n  Decidable.not_imp_self\n#align not_imp_self not_imp_self\n-/\n\n/- warning: imp.swap -> Imp.swap is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop}, Iff (a -> b -> c) (b -> a -> c)\nbut is expected to have type\n  forall {a : Sort.{u_1}} {b : Sort.{u_2}} {c : Prop}, Iff (a -> b -> c) (b -> a -> c)\nCase conversion may be inaccurate. Consider using '#align imp.swap Imp.swapₓ'. -/\ntheorem Imp.swap : a → b → c ↔ b → a → c :=\n  ⟨swap, swap⟩\n#align imp.swap Imp.swap\n\n#print imp_not_comm /-\ntheorem imp_not_comm : a → ¬b ↔ b → ¬a :=\n  Imp.swap\n#align imp_not_comm imp_not_comm\n-/\n\n#print Iff.not /-\ntheorem Iff.not (h : a ↔ b) : ¬a ↔ ¬b :=\n  not_congr h\n#align iff.not Iff.not\n-/\n\n#print Iff.not_left /-\ntheorem Iff.not_left (h : a ↔ ¬b) : ¬a ↔ b :=\n  h.Not.trans Classical.not_not\n#align iff.not_left Iff.not_left\n-/\n\n#print Iff.not_right /-\ntheorem Iff.not_right (h : ¬a ↔ b) : a ↔ ¬b :=\n  Classical.not_not.symm.trans h.Not\n#align iff.not_right Iff.not_right\n-/\n\n/-! ### Declarations about `xor` -/\n\n\n#print xor_true /-\n@[simp]\ntheorem xor_true : Xor' True = Not :=\n  funext fun a => by simp [Xor']\n#align xor_true xor_true\n-/\n\n#print xor_false /-\n@[simp]\ntheorem xor_false : Xor' False = id :=\n  funext fun a => by simp [Xor']\n#align xor_false xor_false\n-/\n\n/- warning: xor_comm -> xor_comm is a dubious translation:\nlean 3 declaration is\n  forall (a : Prop) (b : Prop), Iff (Xor' a b) (Xor' b a)\nbut is expected to have type\n  forall (a : Prop) (b : Prop), Eq.{1} Prop (Xor' a b) (Xor' b a)\nCase conversion may be inaccurate. Consider using '#align xor_comm xor_commₓ'. -/\ntheorem xor_comm (a b) : Xor' a b ↔ Xor' b a :=\n  or_comm' _ _\n#align xor_comm xor_comm\n\ninstance : IsCommutative Prop Xor' :=\n  ⟨fun a b => propext <| xor_comm a b⟩\n\n#print xor_self /-\n@[simp]\ntheorem xor_self (a : Prop) : Xor' a a = False := by simp [Xor']\n#align xor_self xor_self\n-/\n\n#print xor_not_left /-\n@[simp]\ntheorem xor_not_left : Xor' (¬a) b ↔ (a ↔ b) := by by_cases a <;> simp [*]\n#align xor_not_left xor_not_left\n-/\n\n#print xor_not_right /-\n@[simp]\ntheorem xor_not_right : Xor' a ¬b ↔ (a ↔ b) := by by_cases a <;> simp [*]\n#align xor_not_right xor_not_right\n-/\n\n#print xor_not_not /-\ntheorem xor_not_not : Xor' (¬a) ¬b ↔ Xor' a b := by simp [Xor', or_comm', and_comm']\n#align xor_not_not xor_not_not\n-/\n\n#print Xor'.or /-\nprotected theorem Xor'.or (h : Xor' a b) : a ∨ b :=\n  h.imp And.left And.left\n#align xor.or Xor'.or\n-/\n\n/-! ### Declarations about `and` -/\n\n\n#print Iff.and /-\ntheorem Iff.and (h₁ : a ↔ b) (h₂ : c ↔ d) : a ∧ c ↔ b ∧ d :=\n  and_congr h₁ h₂\n#align iff.and Iff.and\n-/\n\ntheorem and_congr_left (h : c → (a ↔ b)) : a ∧ c ↔ b ∧ c :=\n  and_comm.trans <| (and_congr_right h).trans and_comm\n#align and_congr_left and_congr_leftₓ\n\n#print and_congr_left' /-\ntheorem and_congr_left' (h : a ↔ b) : a ∧ c ↔ b ∧ c :=\n  h.And Iff.rfl\n#align and_congr_left' and_congr_left'\n-/\n\ntheorem and_congr_right' (h : b ↔ c) : a ∧ b ↔ a ∧ c :=\n  Iff.rfl.And h\n#align and_congr_right' and_congr_right'ₓ\n\n#print not_and_of_not_left /-\ntheorem not_and_of_not_left (b : Prop) : ¬a → ¬(a ∧ b) :=\n  mt And.left\n#align not_and_of_not_left not_and_of_not_left\n-/\n\n#print not_and_of_not_right /-\ntheorem not_and_of_not_right (a : Prop) {b : Prop} : ¬b → ¬(a ∧ b) :=\n  mt And.right\n#align not_and_of_not_right not_and_of_not_right\n-/\n\n#print And.imp_left /-\ntheorem And.imp_left (h : a → b) : a ∧ c → b ∧ c :=\n  And.imp h id\n#align and.imp_left And.imp_left\n-/\n\n#print And.imp_right /-\ntheorem And.imp_right (h : a → b) : c ∧ a → c ∧ b :=\n  And.imp id h\n#align and.imp_right And.imp_right\n-/\n\n#print and_right_comm /-\ntheorem and_right_comm : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b := by simp only [and_left_comm, and_comm]\n#align and.right_comm and_right_comm\n-/\n\n#print and_and_and_comm /-\ntheorem and_and_and_comm (a b c d : Prop) : (a ∧ b) ∧ c ∧ d ↔ (a ∧ c) ∧ b ∧ d := by\n  rw [← and_assoc', @and_right_comm a, and_assoc']\n#align and_and_and_comm and_and_and_comm\n-/\n\n#print and_and_left /-\ntheorem and_and_left (a b c : Prop) : a ∧ b ∧ c ↔ (a ∧ b) ∧ a ∧ c := by\n  rw [and_and_and_comm, and_self_iff]\n#align and_and_distrib_left and_and_left\n-/\n\n#print and_and_right /-\ntheorem and_and_right (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b ∧ c := by\n  rw [and_and_and_comm, and_self_iff]\n#align and_and_distrib_right and_and_right\n-/\n\n#print and_rotate /-\ntheorem and_rotate : a ∧ b ∧ c ↔ b ∧ c ∧ a := by simp only [and_left_comm, and_comm]\n#align and_rotate and_rotate\n-/\n\n#print And.rotate /-\ntheorem And.rotate : a ∧ b ∧ c → b ∧ c ∧ a :=\n  and_rotate.1\n#align and.rotate And.rotate\n-/\n\n/- warning: and_not_self_iff clashes with and_not_self -> and_not_self_iff\nCase conversion may be inaccurate. Consider using '#align and_not_self_iff and_not_self_iffₓ'. -/\n#print and_not_self_iff /-\ntheorem and_not_self_iff (a : Prop) : a ∧ ¬a ↔ False :=\n  Iff.intro (fun h => h.right h.left) fun h => h.elim\n#align and_not_self_iff and_not_self_iff\n-/\n\n/- warning: not_and_self_iff clashes with not_and_self -> not_and_self_iff\nCase conversion may be inaccurate. Consider using '#align not_and_self_iff not_and_self_iffₓ'. -/\n#print not_and_self_iff /-\ntheorem not_and_self_iff (a : Prop) : ¬a ∧ a ↔ False :=\n  Iff.intro (fun ⟨hna, ha⟩ => hna ha) False.elim\n#align not_and_self_iff not_and_self_iff\n-/\n\n#print and_iff_left_of_imp /-\ntheorem and_iff_left_of_imp {a b : Prop} (h : a → b) : a ∧ b ↔ a :=\n  Iff.intro And.left fun ha => ⟨ha, h ha⟩\n#align and_iff_left_of_imp and_iff_left_of_imp\n-/\n\n/- warning: and_iff_right_of_imp -> and_iff_right_of_imp is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop}, (b -> a) -> (Iff (And a b) b)\nbut is expected to have type\n  forall {a : Prop} {b : Prop}, (a -> b) -> (Iff (And b a) a)\nCase conversion may be inaccurate. Consider using '#align and_iff_right_of_imp and_iff_right_of_impₓ'. -/\ntheorem and_iff_right_of_imp {a b : Prop} (h : b → a) : a ∧ b ↔ b :=\n  Iff.intro And.right fun hb => ⟨h hb, hb⟩\n#align and_iff_right_of_imp and_iff_right_of_imp\n\n#print ne_and_eq_iff_right /-\ntheorem ne_and_eq_iff_right {α : Sort _} {a b c : α} (h : b ≠ c) : a ≠ b ∧ a = c ↔ a = c :=\n  and_iff_right_of_imp fun h2 => h2.symm ▸ h.symm\n#align ne_and_eq_iff_right ne_and_eq_iff_right\n-/\n\n#print and_iff_left_iff_imp /-\n@[simp]\ntheorem and_iff_left_iff_imp {a b : Prop} : (a ∧ b ↔ a) ↔ a → b :=\n  ⟨fun h ha => (h.2 ha).2, and_iff_left_of_imp⟩\n#align and_iff_left_iff_imp and_iff_left_iff_imp\n-/\n\n#print and_iff_right_iff_imp /-\n@[simp]\ntheorem and_iff_right_iff_imp {a b : Prop} : (a ∧ b ↔ b) ↔ b → a :=\n  ⟨fun h ha => (h.2 ha).1, and_iff_right_of_imp⟩\n#align and_iff_right_iff_imp and_iff_right_iff_imp\n-/\n\n#print iff_self_and /-\n@[simp]\ntheorem iff_self_and {p q : Prop} : (p ↔ p ∧ q) ↔ p → q := by rw [@Iff.comm p, and_iff_left_iff_imp]\n#align iff_self_and iff_self_and\n-/\n\n#print iff_and_self /-\n@[simp]\ntheorem iff_and_self {p q : Prop} : (p ↔ q ∧ p) ↔ p → q := by rw [and_comm', iff_self_and]\n#align iff_and_self iff_and_self\n-/\n\n#print and_congr_right_iff /-\n@[simp]\ntheorem and_congr_right_iff : (a ∧ b ↔ a ∧ c) ↔ a → (b ↔ c) :=\n  ⟨fun h ha => by simp [ha] at h <;> exact h, and_congr_right⟩\n#align and.congr_right_iff and_congr_right_iff\n-/\n\n@[simp]\ntheorem and_congr_left_iff : (a ∧ c ↔ b ∧ c) ↔ c → (a ↔ b) := by\n  simp only [and_comm, ← and_congr_right_iff]\n#align and.congr_left_iff and_congr_left_iffₓ\n\n#print and_self_left /-\n@[simp]\ntheorem and_self_left : a ∧ a ∧ b ↔ a ∧ b :=\n  ⟨fun h => ⟨h.1, h.2.2⟩, fun h => ⟨h.1, h.1, h.2⟩⟩\n#align and_self_left and_self_left\n-/\n\n#print and_self_right /-\n@[simp]\ntheorem and_self_right : (a ∧ b) ∧ b ↔ a ∧ b :=\n  ⟨fun h => ⟨h.1.1, h.2⟩, fun h => ⟨⟨h.1, h.2⟩, h.2⟩⟩\n#align and_self_right and_self_right\n-/\n\n/-! ### Declarations about `or` -/\n\n\n#print Iff.or /-\ntheorem Iff.or (h₁ : a ↔ b) (h₂ : c ↔ d) : a ∨ c ↔ b ∨ d :=\n  or_congr h₁ h₂\n#align iff.or Iff.or\n-/\n\n#print or_congr_left /-\ntheorem or_congr_left (h : a ↔ b) : a ∨ c ↔ b ∨ c :=\n  h.Or Iff.rfl\n#align or_congr_left' or_congr_left\n-/\n\ntheorem or_congr_right (h : b ↔ c) : a ∨ b ↔ a ∨ c :=\n  Iff.rfl.Or h\n#align or_congr_right' or_congr_rightₓ\n\n#print or_right_comm /-\ntheorem or_right_comm : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b := by rw [or_assoc', or_assoc', or_comm' b]\n#align or.right_comm or_right_comm\n-/\n\n#print or_or_or_comm /-\ntheorem or_or_or_comm (a b c d : Prop) : (a ∨ b) ∨ c ∨ d ↔ (a ∨ c) ∨ b ∨ d := by\n  rw [← or_assoc', @or_right_comm a, or_assoc']\n#align or_or_or_comm or_or_or_comm\n-/\n\n#print or_or_distrib_left /-\ntheorem or_or_distrib_left (a b c : Prop) : a ∨ b ∨ c ↔ (a ∨ b) ∨ a ∨ c := by\n  rw [or_or_or_comm, or_self_iff]\n#align or_or_distrib_left or_or_distrib_left\n-/\n\n#print or_or_distrib_right /-\ntheorem or_or_distrib_right (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b ∨ c := by\n  rw [or_or_or_comm, or_self_iff]\n#align or_or_distrib_right or_or_distrib_right\n-/\n\n#print or_rotate /-\ntheorem or_rotate : a ∨ b ∨ c ↔ b ∨ c ∨ a := by simp only [or_left_comm, or_comm]\n#align or_rotate or_rotate\n-/\n\n#print Or.rotate /-\ntheorem Or.rotate : a ∨ b ∨ c → b ∨ c ∨ a :=\n  or_rotate.1\n#align or.rotate Or.rotate\n-/\n\n#print or_of_or_of_imp_of_imp /-\ntheorem or_of_or_of_imp_of_imp (h₁ : a ∨ b) (h₂ : a → c) (h₃ : b → d) : c ∨ d :=\n  Or.imp h₂ h₃ h₁\n#align or_of_or_of_imp_of_imp or_of_or_of_imp_of_imp\n-/\n\n/- warning: or_of_or_of_imp_left -> or_of_or_of_imp_left is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop}, (Or a c) -> (a -> b) -> (Or b c)\nbut is expected to have type\n  forall {a : Prop} {b : Prop} {c : Prop}, (Or a b) -> (a -> c) -> (Or c b)\nCase conversion may be inaccurate. Consider using '#align or_of_or_of_imp_left or_of_or_of_imp_leftₓ'. -/\ntheorem or_of_or_of_imp_left (h₁ : a ∨ c) (h : a → b) : b ∨ c :=\n  Or.imp_left h h₁\n#align or_of_or_of_imp_left or_of_or_of_imp_left\n\n/- warning: or_of_or_of_imp_right -> or_of_or_of_imp_right is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop}, (Or c a) -> (a -> b) -> (Or c b)\nbut is expected to have type\n  forall {a : Prop} {b : Prop} {c : Prop}, (Or a b) -> (b -> c) -> (Or a c)\nCase conversion may be inaccurate. Consider using '#align or_of_or_of_imp_right or_of_or_of_imp_rightₓ'. -/\ntheorem or_of_or_of_imp_right (h₁ : c ∨ a) (h : a → b) : c ∨ b :=\n  Or.imp_right h h₁\n#align or_of_or_of_imp_right or_of_or_of_imp_right\n\n#print Or.elim3 /-\ntheorem Or.elim3 (h : a ∨ b ∨ c) (ha : a → d) (hb : b → d) (hc : c → d) : d :=\n  Or.elim h ha fun h₂ => Or.elim h₂ hb hc\n#align or.elim3 Or.elim3\n-/\n\n/- warning: or.imp3 -> Or.imp3 is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop} {d : Prop} {e : Prop} {f : Prop}, (a -> d) -> (b -> e) -> (c -> f) -> (Or a (Or b c)) -> (Or d (Or e f))\nbut is expected to have type\n  forall {a : Prop} {b : Prop} {c : Prop} {d : Prop} {e : Prop} {f : Prop}, (a -> b) -> (c -> d) -> (e -> f) -> (Or a (Or c e)) -> (Or b (Or d f))\nCase conversion may be inaccurate. Consider using '#align or.imp3 Or.imp3ₓ'. -/\ntheorem Or.imp3 (had : a → d) (hbe : b → e) (hcf : c → f) : a ∨ b ∨ c → d ∨ e ∨ f :=\n  Or.imp had <| Or.imp hbe hcf\n#align or.imp3 Or.imp3\n\n#print or_imp /-\ntheorem or_imp : a ∨ b → c ↔ (a → c) ∧ (b → c) :=\n  ⟨fun h => ⟨fun ha => h (Or.inl ha), fun hb => h (Or.inr hb)⟩, fun ⟨ha, hb⟩ => Or.ndrec ha hb⟩\n#align or_imp_distrib or_imp\n-/\n\n#print Decidable.or_iff_not_imp_left /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_iff_not_imp_left [Decidable a] : a ∨ b ↔ ¬a → b :=\n  ⟨Or.resolve_left, fun h => dite _ Or.inl (Or.inr ∘ h)⟩\n#align decidable.or_iff_not_imp_left Decidable.or_iff_not_imp_left\n-/\n\n#print or_iff_not_imp_left /-\ntheorem or_iff_not_imp_left : a ∨ b ↔ ¬a → b :=\n  Decidable.or_iff_not_imp_left\n#align or_iff_not_imp_left or_iff_not_imp_left\n-/\n\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_iff_not_imp_right [Decidable b] : a ∨ b ↔ ¬b → a :=\n  or_comm.trans Decidable.or_iff_not_imp_left\n#align decidable.or_iff_not_imp_right Decidable.or_iff_not_imp_rightₓ\n\n#print or_iff_not_imp_right /-\ntheorem or_iff_not_imp_right : a ∨ b ↔ ¬b → a :=\n  Decidable.or_iff_not_imp_right\n#align or_iff_not_imp_right or_iff_not_imp_right\n-/\n\n#print Decidable.not_or_of_imp /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_or_of_imp [Decidable a] (h : a → b) : ¬a ∨ b :=\n  dite _ (Or.inr ∘ h) Or.inl\n#align decidable.not_or_of_imp Decidable.not_or_of_imp\n-/\n\n#print not_or_of_imp /-\ntheorem not_or_of_imp : (a → b) → ¬a ∨ b :=\n  Decidable.not_or_of_imp\n#align not_or_of_imp not_or_of_imp\n-/\n\n#print Decidable.or_not_of_imp /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_not_of_imp [Decidable a] (h : a → b) : b ∨ ¬a :=\n  dite _ (Or.inl ∘ h) Or.inr\n#align decidable.or_not_of_imp Decidable.or_not_of_imp\n-/\n\n#print or_not_of_imp /-\ntheorem or_not_of_imp : (a → b) → b ∨ ¬a :=\n  Decidable.or_not_of_imp\n#align or_not_of_imp or_not_of_imp\n-/\n\n#print Decidable.imp_iff_not_or /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.imp_iff_not_or [Decidable a] : a → b ↔ ¬a ∨ b :=\n  ⟨Decidable.not_or_of_imp, Or.neg_resolve_left⟩\n#align decidable.imp_iff_not_or Decidable.imp_iff_not_or\n-/\n\n#print imp_iff_not_or /-\ntheorem imp_iff_not_or : a → b ↔ ¬a ∨ b :=\n  Decidable.imp_iff_not_or\n#align imp_iff_not_or imp_iff_not_or\n-/\n\n-- See Note [decidable namespace]\nprotected theorem Decidable.imp_iff_or_not [Decidable b] : b → a ↔ a ∨ ¬b :=\n  Decidable.imp_iff_not_or.trans or_comm\n#align decidable.imp_iff_or_not Decidable.imp_iff_or_notₓ\n\n/- warning: imp_iff_or_not -> imp_iff_or_not is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop}, Iff (b -> a) (Or a (Not b))\nbut is expected to have type\n  forall {a : Prop} {b : Prop}, Iff (a -> b) (Or b (Not a))\nCase conversion may be inaccurate. Consider using '#align imp_iff_or_not imp_iff_or_notₓ'. -/\ntheorem imp_iff_or_not : b → a ↔ a ∨ ¬b :=\n  Decidable.imp_iff_or_not\n#align imp_iff_or_not imp_iff_or_not\n\n#print Decidable.not_imp_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_imp_not [Decidable a] : ¬a → ¬b ↔ b → a :=\n  ⟨fun h hb => Decidable.by_contradiction fun na => h na hb, mt⟩\n#align decidable.not_imp_not Decidable.not_imp_not\n-/\n\n#print not_imp_not /-\ntheorem not_imp_not : ¬a → ¬b ↔ b → a :=\n  Decidable.not_imp_not\n#align not_imp_not not_imp_not\n-/\n\n#print Function.mtr /-\n/-- Provide the reverse of modus tollens (`mt`) as dot notation for implications. -/\nprotected theorem Function.mtr : (¬a → ¬b) → b → a :=\n  not_imp_not.mp\n#align function.mtr Function.mtr\n-/\n\n/- warning: decidable.or_congr_left -> Decidable.or_congr_left' is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop} [_inst_1 : Decidable c], ((Not c) -> (Iff a b)) -> (Iff (Or a c) (Or b c))\nbut is expected to have type\n  forall {a : Prop} {b : Prop} {c : Prop} [_inst_1 : Decidable a], ((Not a) -> (Iff b c)) -> (Iff (Or b a) (Or c a))\nCase conversion may be inaccurate. Consider using '#align decidable.or_congr_left Decidable.or_congr_left'ₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_congr_left' [Decidable c] (h : ¬c → (a ↔ b)) : a ∨ c ↔ b ∨ c :=\n  by\n  rw [Decidable.or_iff_not_imp_right, Decidable.or_iff_not_imp_right]\n  exact imp_congr_right h\n#align decidable.or_congr_left Decidable.or_congr_left'\n\n/- warning: or_congr_left -> or_congr_left' is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop}, ((Not c) -> (Iff a b)) -> (Iff (Or a c) (Or b c))\nbut is expected to have type\n  forall {a : Prop} {b : Prop} {c : Prop}, ((Not a) -> (Iff b c)) -> (Iff (Or b a) (Or c a))\nCase conversion may be inaccurate. Consider using '#align or_congr_left or_congr_left'ₓ'. -/\ntheorem or_congr_left' (h : ¬c → (a ↔ b)) : a ∨ c ↔ b ∨ c :=\n  Decidable.or_congr_left' h\n#align or_congr_left or_congr_left'\n\n#print Decidable.or_congr_right' /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_congr_right' [Decidable a] (h : ¬a → (b ↔ c)) : a ∨ b ↔ a ∨ c :=\n  by\n  rw [Decidable.or_iff_not_imp_left, Decidable.or_iff_not_imp_left]\n  exact imp_congr_right h\n#align decidable.or_congr_right Decidable.or_congr_right'\n-/\n\ntheorem or_congr_right' (h : ¬a → (b ↔ c)) : a ∨ b ↔ a ∨ c :=\n  Decidable.or_congr_right' h\n#align or_congr_right or_congr_right'ₓ\n\n#print or_iff_left_iff_imp /-\n@[simp]\ntheorem or_iff_left_iff_imp : (a ∨ b ↔ a) ↔ b → a :=\n  ⟨fun h hb => h.1 (Or.inr hb), or_iff_left_of_imp⟩\n#align or_iff_left_iff_imp or_iff_left_iff_imp\n-/\n\n#print or_iff_right_iff_imp /-\n@[simp]\ntheorem or_iff_right_iff_imp : (a ∨ b ↔ b) ↔ a → b := by rw [or_comm', or_iff_left_iff_imp]\n#align or_iff_right_iff_imp or_iff_right_iff_imp\n-/\n\ntheorem or_iff_left (hb : ¬b) : a ∨ b ↔ a :=\n  ⟨fun h => h.resolve_right hb, Or.inl⟩\n#align or_iff_left or_iff_leftₓ\n\n#print or_iff_right /-\ntheorem or_iff_right (ha : ¬a) : a ∨ b ↔ b :=\n  ⟨fun h => h.resolve_left ha, Or.inr⟩\n#align or_iff_right or_iff_right\n-/\n\n/-! ### Declarations about distributivity -/\n\n\n#print and_or_left /-\n/-- `∧` distributes over `∨` (on the left). -/\ntheorem and_or_left : a ∧ (b ∨ c) ↔ a ∧ b ∨ a ∧ c :=\n  ⟨fun ⟨ha, hbc⟩ => hbc.imp (And.intro ha) (And.intro ha),\n    Or.ndrec (And.imp_right Or.inl) (And.imp_right Or.inr)⟩\n#align and_or_distrib_left and_or_left\n-/\n\n#print or_and_right /-\n/-- `∧` distributes over `∨` (on the right). -/\ntheorem or_and_right : (a ∨ b) ∧ c ↔ a ∧ c ∨ b ∧ c :=\n  (and_comm.trans and_or_left).trans (and_comm.Or and_comm)\n#align or_and_distrib_right or_and_right\n-/\n\n#print or_and_left /-\n/-- `∨` distributes over `∧` (on the left). -/\ntheorem or_and_left : a ∨ b ∧ c ↔ (a ∨ b) ∧ (a ∨ c) :=\n  ⟨Or.ndrec (fun ha => And.intro (Or.inl ha) (Or.inl ha)) (And.imp Or.inr Or.inr),\n    And.ndrec <| Or.ndrec (imp_intro ∘ Or.inl) (Or.imp_right ∘ And.intro)⟩\n#align or_and_distrib_left or_and_left\n-/\n\n#print and_or_right /-\n/-- `∨` distributes over `∧` (on the right). -/\ntheorem and_or_right : a ∧ b ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=\n  (or_comm.trans or_and_left).trans (or_comm.And or_comm)\n#align and_or_distrib_right and_or_right\n-/\n\n#print or_self_left /-\n@[simp]\ntheorem or_self_left : a ∨ a ∨ b ↔ a ∨ b :=\n  ⟨fun h => h.elim Or.inl id, fun h => h.elim Or.inl (Or.inr ∘ Or.inr)⟩\n#align or_self_left or_self_left\n-/\n\n#print or_self_right /-\n@[simp]\ntheorem or_self_right : (a ∨ b) ∨ b ↔ a ∨ b :=\n  ⟨fun h => h.elim id Or.inr, fun h => h.elim (Or.inl ∘ Or.inl) Or.inr⟩\n#align or_self_right or_self_right\n-/\n\n/-! Declarations about `iff` -/\n\n\n#print Iff.iff /-\ntheorem Iff.iff (h₁ : a ↔ b) (h₂ : c ↔ d) : (a ↔ c) ↔ (b ↔ d) :=\n  iff_congr h₁ h₂\n#align iff.iff Iff.iff\n-/\n\n#print iff_of_true /-\ntheorem iff_of_true (ha : a) (hb : b) : a ↔ b :=\n  ⟨fun _ => hb, fun _ => ha⟩\n#align iff_of_true iff_of_true\n-/\n\n#print iff_of_false /-\ntheorem iff_of_false (ha : ¬a) (hb : ¬b) : a ↔ b :=\n  ⟨ha.elim, hb.elim⟩\n#align iff_of_false iff_of_false\n-/\n\n#print iff_true_left /-\ntheorem iff_true_left (ha : a) : (a ↔ b) ↔ b :=\n  ⟨fun h => h.1 ha, iff_of_true ha⟩\n#align iff_true_left iff_true_left\n-/\n\n#print iff_true_right /-\ntheorem iff_true_right (ha : a) : (b ↔ a) ↔ b :=\n  Iff.comm.trans (iff_true_left ha)\n#align iff_true_right iff_true_right\n-/\n\n#print iff_false_left /-\ntheorem iff_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b :=\n  ⟨fun h => mt h.2 ha, iff_of_false ha⟩\n#align iff_false_left iff_false_left\n-/\n\n#print iff_false_right /-\ntheorem iff_false_right (ha : ¬a) : (b ↔ a) ↔ ¬b :=\n  Iff.comm.trans (iff_false_left ha)\n#align iff_false_right iff_false_right\n-/\n\n#print iff_mpr_iff_true_intro /-\n@[simp]\ntheorem iff_mpr_iff_true_intro {P : Prop} (h : P) : Iff.mpr (iff_true_intro h) True.intro = h :=\n  rfl\n#align iff_mpr_iff_true_intro iff_mpr_iff_true_intro\n-/\n\n#print Decidable.imp_or /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.imp_or [Decidable a] : a → b ∨ c ↔ (a → b) ∨ (a → c) := by\n  simp [Decidable.imp_iff_not_or, or_comm, or_left_comm]\n#align decidable.imp_or_distrib Decidable.imp_or\n-/\n\n#print imp_or /-\ntheorem imp_or : a → b ∨ c ↔ (a → b) ∨ (a → c) :=\n  Decidable.imp_or\n#align imp_or_distrib imp_or\n-/\n\n/- warning: decidable.imp_or_distrib' -> Decidable.imp_or' is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} {c : Prop} [_inst_1 : Decidable b], Iff (a -> (Or b c)) (Or (a -> b) (a -> c))\nbut is expected to have type\n  forall {a : Prop} {b : Sort.{u_1}} {c : Prop} [_inst_1 : Decidable a], Iff (b -> (Or a c)) (Or (b -> a) (b -> c))\nCase conversion may be inaccurate. Consider using '#align decidable.imp_or_distrib' Decidable.imp_or'ₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.imp_or' [Decidable b] : a → b ∨ c ↔ (a → b) ∨ (a → c) := by\n  by_cases b <;> simp [h, or_iff_right_of_imp ((· ∘ ·) False.elim)]\n#align decidable.imp_or_distrib' Decidable.imp_or'\n\ntheorem imp_or' : a → b ∨ c ↔ (a → b) ∨ (a → c) :=\n  Decidable.imp_or'\n#align imp_or_distrib' imp_or'ₓ\n\n#print not_imp_of_and_not /-\ntheorem not_imp_of_and_not : a ∧ ¬b → ¬(a → b)\n  | ⟨ha, hb⟩, h => hb <| h ha\n#align not_imp_of_and_not not_imp_of_and_not\n-/\n\n#print Decidable.not_imp /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_imp [Decidable a] : ¬(a → b) ↔ a ∧ ¬b :=\n  ⟨fun h => ⟨Decidable.of_not_imp h, not_of_not_imp h⟩, not_imp_of_and_not⟩\n#align decidable.not_imp Decidable.not_imp\n-/\n\n#print not_imp /-\ntheorem not_imp : ¬(a → b) ↔ a ∧ ¬b :=\n  Decidable.not_imp\n#align not_imp not_imp\n-/\n\n#print imp_imp_imp /-\n-- for monotonicity\ntheorem imp_imp_imp (h₀ : c → a) (h₁ : b → d) : (a → b) → c → d := fun h₂ : a → b => h₁ ∘ h₂ ∘ h₀\n#align imp_imp_imp imp_imp_imp\n-/\n\n#print Decidable.peirce /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.peirce (a b : Prop) [Decidable a] : ((a → b) → a) → a :=\n  if ha : a then fun h => ha else fun h => h ha.elim\n#align decidable.peirce Decidable.peirce\n-/\n\n#print peirce /-\ntheorem peirce (a b : Prop) : ((a → b) → a) → a :=\n  Decidable.peirce _ _\n#align peirce peirce\n-/\n\n#print peirce' /-\ntheorem peirce' {a : Prop} (H : ∀ b : Prop, (a → b) → a) : a :=\n  H _ id\n#align peirce' peirce'\n-/\n\n#print Decidable.not_iff_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_iff_not [Decidable a] [Decidable b] : (¬a ↔ ¬b) ↔ (a ↔ b) := by\n  rw [@iff_def ¬a, @iff_def' a] <;> exact decidable.not_imp_not.and Decidable.not_imp_not\n#align decidable.not_iff_not Decidable.not_iff_not\n-/\n\n#print not_iff_not /-\ntheorem not_iff_not : (¬a ↔ ¬b) ↔ (a ↔ b) :=\n  Decidable.not_iff_not\n#align not_iff_not not_iff_not\n-/\n\n#print Decidable.not_iff_comm /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_iff_comm [Decidable a] [Decidable b] : (¬a ↔ b) ↔ (¬b ↔ a) := by\n  rw [@iff_def ¬a, @iff_def ¬b] <;> exact decidable.not_imp_comm.and imp_not_comm\n#align decidable.not_iff_comm Decidable.not_iff_comm\n-/\n\n#print not_iff_comm /-\ntheorem not_iff_comm : (¬a ↔ b) ↔ (¬b ↔ a) :=\n  Decidable.not_iff_comm\n#align not_iff_comm not_iff_comm\n-/\n\n/- warning: decidable.not_iff -> Decidable.not_iff is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable b], Iff (Not (Iff a b)) (Iff (Not a) b)\nbut is expected to have type\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable a], Iff (Not (Iff b a)) (Iff (Not b) a)\nCase conversion may be inaccurate. Consider using '#align decidable.not_iff Decidable.not_iffₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_iff : ∀ [Decidable b], ¬(a ↔ b) ↔ (¬a ↔ b) := by\n  intro h <;> cases h <;> simp only [h, iff_true_iff, iff_false_iff]\n#align decidable.not_iff Decidable.not_iff\n\n#print not_iff /-\ntheorem not_iff : ¬(a ↔ b) ↔ (¬a ↔ b) :=\n  Decidable.not_iff\n#align not_iff not_iff\n-/\n\n#print Decidable.iff_not_comm /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.iff_not_comm [Decidable a] [Decidable b] : (a ↔ ¬b) ↔ (b ↔ ¬a) := by\n  rw [@iff_def a, @iff_def b] <;> exact imp_not_comm.and Decidable.not_imp_comm\n#align decidable.iff_not_comm Decidable.iff_not_comm\n-/\n\n#print iff_not_comm /-\ntheorem iff_not_comm : (a ↔ ¬b) ↔ (b ↔ ¬a) :=\n  Decidable.iff_not_comm\n#align iff_not_comm iff_not_comm\n-/\n\n/- warning: decidable.iff_iff_and_or_not_and_not -> Decidable.iff_iff_and_or_not_and_not is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable b], Iff (Iff a b) (Or (And a b) (And (Not a) (Not b)))\nbut is expected to have type\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable a], Iff (Iff b a) (Or (And b a) (And (Not b) (Not a)))\nCase conversion may be inaccurate. Consider using '#align decidable.iff_iff_and_or_not_and_not Decidable.iff_iff_and_or_not_and_notₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.iff_iff_and_or_not_and_not [Decidable b] : (a ↔ b) ↔ a ∧ b ∨ ¬a ∧ ¬b :=\n  by\n  constructor <;> intro h\n  · rw [h] <;> by_cases b <;> [left, right] <;> constructor <;> assumption\n  · cases' h with h h <;> cases h <;> constructor <;> intro <;> · first |contradiction|assumption\n#align decidable.iff_iff_and_or_not_and_not Decidable.iff_iff_and_or_not_and_not\n\n#print iff_iff_and_or_not_and_not /-\ntheorem iff_iff_and_or_not_and_not : (a ↔ b) ↔ a ∧ b ∨ ¬a ∧ ¬b :=\n  Decidable.iff_iff_and_or_not_and_not\n#align iff_iff_and_or_not_and_not iff_iff_and_or_not_and_not\n-/\n\n#print Decidable.iff_iff_not_or_and_or_not /-\ntheorem Decidable.iff_iff_not_or_and_or_not [Decidable a] [Decidable b] :\n    (a ↔ b) ↔ (¬a ∨ b) ∧ (a ∨ ¬b) :=\n  by\n  rw [iff_iff_implies_and_implies a b]\n  simp only [Decidable.imp_iff_not_or, or_comm]\n#align decidable.iff_iff_not_or_and_or_not Decidable.iff_iff_not_or_and_or_not\n-/\n\n#print iff_iff_not_or_and_or_not /-\ntheorem iff_iff_not_or_and_or_not : (a ↔ b) ↔ (¬a ∨ b) ∧ (a ∨ ¬b) :=\n  Decidable.iff_iff_not_or_and_or_not\n#align iff_iff_not_or_and_or_not iff_iff_not_or_and_or_not\n-/\n\n/- warning: decidable.not_and_not_right -> Decidable.not_and_not_right is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable b], Iff (Not (And a (Not b))) (a -> b)\nbut is expected to have type\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable a], Iff (Not (And b (Not a))) (b -> a)\nCase conversion may be inaccurate. Consider using '#align decidable.not_and_not_right Decidable.not_and_not_rightₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_and_not_right [Decidable b] : ¬(a ∧ ¬b) ↔ a → b :=\n  ⟨fun h ha => h.decidable_imp_symm <| And.intro ha, fun h ⟨ha, hb⟩ => hb <| h ha⟩\n#align decidable.not_and_not_right Decidable.not_and_not_right\n\n#print not_and_not_right /-\ntheorem not_and_not_right : ¬(a ∧ ¬b) ↔ a → b :=\n  Decidable.not_and_not_right\n#align not_and_not_right not_and_not_right\n-/\n\n#print decidable_of_iff /-\n/-- Transfer decidability of `a` to decidability of `b`, if the propositions are equivalent.\n**Important**: this function should be used instead of `rw` on `decidable b`, because the\nkernel will get stuck reducing the usage of `propext` otherwise,\nand `dec_trivial` will not work. -/\n@[inline]\ndef decidable_of_iff (a : Prop) (h : a ↔ b) [D : Decidable a] : Decidable b :=\n  decidable_of_decidable_of_iff D h\n#align decidable_of_iff decidable_of_iff\n-/\n\n#print decidable_of_iff' /-\n/-- Transfer decidability of `b` to decidability of `a`, if the propositions are equivalent.\nThis is the same as `decidable_of_iff` but the iff is flipped. -/\n@[inline]\ndef decidable_of_iff' (b : Prop) (h : a ↔ b) [D : Decidable b] : Decidable a :=\n  decidable_of_decidable_of_iff D h.symm\n#align decidable_of_iff' decidable_of_iff'\n-/\n\n#print decidable_of_bool /-\n/-- Prove that `a` is decidable by constructing a boolean `b` and a proof that `b ↔ a`.\n(This is sometimes taken as an alternate definition of decidability.) -/\ndef decidable_of_bool : ∀ (b : Bool) (h : b ↔ a), Decidable a\n  | tt, h => isTrue (h.1 rfl)\n  | ff, h => isFalse (mt h.2 Bool.false_ne_true)\n#align decidable_of_bool decidable_of_bool\n-/\n\n/-! ### De Morgan's laws -/\n\n\n#print not_and_of_not_or_not /-\ntheorem not_and_of_not_or_not (h : ¬a ∨ ¬b) : ¬(a ∧ b)\n  | ⟨ha, hb⟩ => Or.elim h (absurd ha) (absurd hb)\n#align not_and_of_not_or_not not_and_of_not_or_not\n-/\n\n#print Decidable.not_and /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_and [Decidable a] : ¬(a ∧ b) ↔ ¬a ∨ ¬b :=\n  ⟨fun h => if ha : a then Or.inr fun hb => h ⟨ha, hb⟩ else Or.inl ha, not_and_of_not_or_not⟩\n#align decidable.not_and_distrib Decidable.not_and\n-/\n\n/- warning: decidable.not_and_distrib' -> Decidable.not_and' is a dubious translation:\nlean 3 declaration is\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable b], Iff (Not (And a b)) (Or (Not a) (Not b))\nbut is expected to have type\n  forall {a : Prop} {b : Prop} [_inst_1 : Decidable a], Iff (Not (And b a)) (Or (Not b) (Not a))\nCase conversion may be inaccurate. Consider using '#align decidable.not_and_distrib' Decidable.not_and'ₓ'. -/\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_and' [Decidable b] : ¬(a ∧ b) ↔ ¬a ∨ ¬b :=\n  ⟨fun h => if hb : b then Or.inl fun ha => h ⟨ha, hb⟩ else Or.inr hb, not_and_of_not_or_not⟩\n#align decidable.not_and_distrib' Decidable.not_and'\n\n#print not_and_or /-\n/-- One of de Morgan's laws: the negation of a conjunction is logically equivalent to the\ndisjunction of the negations. -/\ntheorem not_and_or : ¬(a ∧ b) ↔ ¬a ∨ ¬b :=\n  Decidable.not_and\n#align not_and_distrib not_and_or\n-/\n\n#print not_and /-\n@[simp]\ntheorem not_and : ¬(a ∧ b) ↔ a → ¬b :=\n  and_imp\n#align not_and not_and\n-/\n\n#print not_and' /-\ntheorem not_and' : ¬(a ∧ b) ↔ b → ¬a :=\n  not_and.trans imp_not_comm\n#align not_and' not_and'\n-/\n\n#print not_or /-\n/-- One of de Morgan's laws: the negation of a disjunction is logically equivalent to the\nconjunction of the negations. -/\ntheorem not_or : ¬(a ∨ b) ↔ ¬a ∧ ¬b :=\n  or_imp\n#align not_or_distrib not_or\n-/\n\n#print Decidable.or_iff_not_and_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.or_iff_not_and_not [Decidable a] [Decidable b] : a ∨ b ↔ ¬(¬a ∧ ¬b) :=\n  by rw [← not_or, Decidable.not_not]\n#align decidable.or_iff_not_and_not Decidable.or_iff_not_and_not\n-/\n\n#print or_iff_not_and_not /-\ntheorem or_iff_not_and_not : a ∨ b ↔ ¬(¬a ∧ ¬b) :=\n  Decidable.or_iff_not_and_not\n#align or_iff_not_and_not or_iff_not_and_not\n-/\n\n#print Decidable.and_iff_not_or_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.and_iff_not_or_not [Decidable a] [Decidable b] : a ∧ b ↔ ¬(¬a ∨ ¬b) :=\n  by rw [← Decidable.not_and, Decidable.not_not]\n#align decidable.and_iff_not_or_not Decidable.and_iff_not_or_not\n-/\n\n#print and_iff_not_or_not /-\ntheorem and_iff_not_or_not : a ∧ b ↔ ¬(¬a ∨ ¬b) :=\n  Decidable.and_iff_not_or_not\n#align and_iff_not_or_not and_iff_not_or_not\n-/\n\n#print not_xor /-\n@[simp]\ntheorem not_xor (P Q : Prop) : ¬Xor' P Q ↔ (P ↔ Q) := by\n  simp only [not_and, Xor', not_or, Classical.not_not, ← iff_iff_implies_and_implies]\n#align not_xor not_xor\n-/\n\n#print xor_iff_not_iff /-\ntheorem xor_iff_not_iff (P Q : Prop) : Xor' P Q ↔ ¬(P ↔ Q) :=\n  (not_xor P Q).not_right\n#align xor_iff_not_iff xor_iff_not_iff\n-/\n\n#print xor_iff_iff_not /-\ntheorem xor_iff_iff_not : Xor' a b ↔ (a ↔ ¬b) := by\n  simp only [← @xor_not_right a, Classical.not_not]\n#align xor_iff_iff_not xor_iff_iff_not\n-/\n\n#print xor_iff_not_iff' /-\ntheorem xor_iff_not_iff' : Xor' a b ↔ (¬a ↔ b) := by\n  simp only [← @xor_not_left _ b, Classical.not_not]\n#align xor_iff_not_iff' xor_iff_not_iff'\n-/\n\nend Propositional\n\n/-! ### Declarations about equality -/\n\n\nsection Mem\n\nvariable {α β : Type _} [Membership α β] {s t : β} {a b : α}\n\n/- warning: ne_of_mem_of_not_mem -> ne_of_mem_of_not_mem is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Membership.{u1, u2} α β] {s : β} {a : α} {b : α}, (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (Not (Membership.Mem.{u1, u2} α β _inst_1 b s)) -> (Ne.{succ u1} α a b)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Membership.{u2, u1} α β] {s : β} {a : α} {b : α}, (Membership.mem.{u2, u1} α β _inst_1 a s) -> (Not (Membership.mem.{u2, u1} α β _inst_1 b s)) -> (Ne.{succ u2} α a b)\nCase conversion may be inaccurate. Consider using '#align ne_of_mem_of_not_mem ne_of_mem_of_not_memₓ'. -/\ntheorem ne_of_mem_of_not_mem (h : a ∈ s) : b ∉ s → a ≠ b :=\n  mt fun e => e ▸ h\n#align ne_of_mem_of_not_mem ne_of_mem_of_not_mem\n\n/- warning: ne_of_mem_of_not_mem' -> ne_of_mem_of_not_mem' is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Membership.{u1, u2} α β] {s : β} {t : β} {a : α}, (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (Not (Membership.Mem.{u1, u2} α β _inst_1 a t)) -> (Ne.{succ u2} β s t)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Membership.{u2, u1} α β] {s : β} {t : β} {a : α}, (Membership.mem.{u2, u1} α β _inst_1 a s) -> (Not (Membership.mem.{u2, u1} α β _inst_1 a t)) -> (Ne.{succ u1} β s t)\nCase conversion may be inaccurate. Consider using '#align ne_of_mem_of_not_mem' ne_of_mem_of_not_mem'ₓ'. -/\ntheorem ne_of_mem_of_not_mem' (h : a ∈ s) : a ∉ t → s ≠ t :=\n  mt fun e => e ▸ h\n#align ne_of_mem_of_not_mem' ne_of_mem_of_not_mem'\n\n/- warning: has_mem.mem.ne_of_not_mem -> Membership.mem.ne_of_not_mem is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Membership.{u1, u2} α β] {s : β} {a : α} {b : α}, (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (Not (Membership.Mem.{u1, u2} α β _inst_1 b s)) -> (Ne.{succ u1} α a b)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Membership.{u2, u1} α β] {s : β} {a : α} {b : α}, (Membership.mem.{u2, u1} α β _inst_1 a s) -> (Not (Membership.mem.{u2, u1} α β _inst_1 b s)) -> (Ne.{succ u2} α a b)\nCase conversion may be inaccurate. Consider using '#align has_mem.mem.ne_of_not_mem Membership.mem.ne_of_not_memₓ'. -/\n/-- **Alias** of `ne_of_mem_of_not_mem`. -/\ntheorem Membership.mem.ne_of_not_mem : a ∈ s → b ∉ s → a ≠ b :=\n  ne_of_mem_of_not_mem\n#align has_mem.mem.ne_of_not_mem Membership.mem.ne_of_not_mem\n\n/- warning: has_mem.mem.ne_of_not_mem' -> Membership.mem.ne_of_not_mem' is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Membership.{u1, u2} α β] {s : β} {t : β} {a : α}, (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (Not (Membership.Mem.{u1, u2} α β _inst_1 a t)) -> (Ne.{succ u2} β s t)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Membership.{u2, u1} α β] {s : β} {t : β} {a : α}, (Membership.mem.{u2, u1} α β _inst_1 a s) -> (Not (Membership.mem.{u2, u1} α β _inst_1 a t)) -> (Ne.{succ u1} β s t)\nCase conversion may be inaccurate. Consider using '#align has_mem.mem.ne_of_not_mem' Membership.mem.ne_of_not_mem'ₓ'. -/\n/-- **Alias** of `ne_of_mem_of_not_mem'`. -/\ntheorem Membership.mem.ne_of_not_mem' : a ∈ s → a ∉ t → s ≠ t :=\n  ne_of_mem_of_not_mem'\n#align has_mem.mem.ne_of_not_mem' Membership.mem.ne_of_not_mem'\n\nend Mem\n\nsection Equality\n\nvariable {α : Sort _} {a b : α}\n\n#print heq_iff_eq /-\n@[simp]\ntheorem heq_iff_eq : HEq a b ↔ a = b :=\n  ⟨eq_of_hEq, hEq_of_eq⟩\n#align heq_iff_eq heq_iff_eq\n-/\n\n#print proof_irrel_heq /-\ntheorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : HEq hp hq :=\n  by\n  have : p = q := propext ⟨fun _ => hq, fun _ => hp⟩\n  subst q <;> rfl\n#align proof_irrel_heq proof_irrel_heq\n-/\n\n#print ball_cond_comm /-\n-- todo: change name\ntheorem ball_cond_comm {α} {s : α → Prop} {p : α → α → Prop} :\n    (∀ a, s a → ∀ b, s b → p a b) ↔ ∀ a b, s a → s b → p a b :=\n  ⟨fun h a b ha hb => h a ha b hb, fun h a ha b hb => h a b ha hb⟩\n#align ball_cond_comm ball_cond_comm\n-/\n\n/- warning: ball_mem_comm -> ball_mem_comm is a dubious translation:\nlean 3 declaration is\n  forall {α : outParam.{succ (succ u1)} Type.{u1}} {β : Type.{u2}} [_inst_1 : Membership.{u1, u2} α β] {s : β} {p : α -> α -> Prop}, Iff (forall (a : α), (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (forall (b : α), (Membership.Mem.{u1, u2} α β _inst_1 b s) -> (p a b))) (forall (a : α) (b : α), (Membership.Mem.{u1, u2} α β _inst_1 a s) -> (Membership.Mem.{u1, u2} α β _inst_1 b s) -> (p a b))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Membership.{u2, u1} α β] {s : β} {p : α -> α -> Prop}, Iff (forall (a : α), (Membership.mem.{u2, u1} α β _inst_1 a s) -> (forall (b : α), (Membership.mem.{u2, u1} α β _inst_1 b s) -> (p a b))) (forall (a : α) (b : α), (Membership.mem.{u2, u1} α β _inst_1 a s) -> (Membership.mem.{u2, u1} α β _inst_1 b s) -> (p a b))\nCase conversion may be inaccurate. Consider using '#align ball_mem_comm ball_mem_commₓ'. -/\n/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (a b «expr ∈ » s) -/\ntheorem ball_mem_comm {α β} [Membership α β] {s : β} {p : α → α → Prop} :\n    (∀ (a) (_ : a ∈ s) (b) (_ : b ∈ s), p a b) ↔ ∀ a b, a ∈ s → b ∈ s → p a b :=\n  ball_cond_comm\n#align ball_mem_comm ball_mem_comm\n\n/- warning: ne_of_apply_ne -> ne_of_apply_ne is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) {x : α} {y : α}, (Ne.{u2} β (f x) (f y)) -> (Ne.{u1} α x y)\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) {x : α} {y : α}, (Ne.{u1} β (f x) (f y)) -> (Ne.{u2} α x y)\nCase conversion may be inaccurate. Consider using '#align ne_of_apply_ne ne_of_apply_neₓ'. -/\ntheorem ne_of_apply_ne {α β : Sort _} (f : α → β) {x y : α} (h : f x ≠ f y) : x ≠ y :=\n  fun w : x = y => h (congr_arg f w)\n#align ne_of_apply_ne ne_of_apply_ne\n\n#print eq_equivalence /-\ntheorem eq_equivalence : Equivalence (@Eq α) :=\n  ⟨Eq.refl, @Eq.symm _, @Eq.trans _⟩\n#align eq_equivalence eq_equivalence\n-/\n\n/- warning: eq_rec_constant -> eq_rec_constant is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {a : α} {a' : α} {β : Sort.{u2}} (y : β) (h : Eq.{u1} α a a'), Eq.{u2} ((fun (a : α) => β) a') (Eq.ndrec.{u2, u1} α a (fun (a : α) => β) y a' h) y\nbut is expected to have type\n  forall {α : Sort.{u2}} {a : α} {a' : α} {β : Sort.{u1}} (y : β) (h : Eq.{u2} α a a'), Eq.{u1} β (Eq.rec.{u1, u2} α a (fun (a_1 : α) (x._@.Std.Logic._hyg.12313 : Eq.{u2} α a a_1) => β) y a' h) y\nCase conversion may be inaccurate. Consider using '#align eq_rec_constant eq_rec_constantₓ'. -/\n/-- Transport through trivial families is the identity. -/\n@[simp]\ntheorem eq_rec_constant {α : Sort _} {a a' : α} {β : Sort _} (y : β) (h : a = a') :\n    @Eq.ndrec α a (fun a => β) y a' h = y := by\n  cases h\n  rfl\n#align eq_rec_constant eq_rec_constant\n\n#print eq_mp_eq_cast /-\n@[simp]\ntheorem eq_mp_eq_cast {α β : Sort _} (h : α = β) : Eq.mp h = cast h :=\n  rfl\n#align eq_mp_eq_cast eq_mp_eq_cast\n-/\n\n#print eq_mpr_eq_cast /-\n@[simp]\ntheorem eq_mpr_eq_cast {α β : Sort _} (h : α = β) : Eq.mpr h = cast h.symm :=\n  rfl\n#align eq_mpr_eq_cast eq_mpr_eq_cast\n-/\n\n#print cast_cast /-\n@[simp]\ntheorem cast_cast :\n    ∀ {α β γ : Sort _} (ha : α = β) (hb : β = γ) (a : α), cast hb (cast ha a) = cast (ha.trans hb) a\n  | _, _, _, rfl, rfl, a => rfl\n#align cast_cast cast_cast\n-/\n\n/- warning: congr_refl_left -> congr_refl_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) {a : α} {b : α} (h : Eq.{u1} α a b), Eq.{0} (Eq.{u2} β (f a) (f b)) (congr.{u1, u2} α β f f a b (rfl.{imax u1 u2} (α -> β) f) h) (congr_arg.{u1, u2} α β a b f h)\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) {a : α} {b : α} (h : Eq.{u2} α a b), Eq.{0} (Eq.{u1} β (f a) (f b)) (congr.{u2, u1} α β f f a b (Eq.refl.{imax u2 u1} (α -> β) f) h) (congr_arg.{u2, u1} α β a b f h)\nCase conversion may be inaccurate. Consider using '#align congr_refl_left congr_refl_leftₓ'. -/\n@[simp]\ntheorem congr_refl_left {α β : Sort _} (f : α → β) {a b : α} (h : a = b) :\n    congr (Eq.refl f) h = congr_arg f h :=\n  rfl\n#align congr_refl_left congr_refl_left\n\n/- warning: congr_refl_right -> congr_refl_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {g : α -> β} (h : Eq.{imax u1 u2} (α -> β) f g) (a : α), Eq.{0} (Eq.{u2} β (f a) (g a)) (congr.{u1, u2} α β f g a a h (rfl.{u1} α a)) (congr_fun.{u1, u2} α (fun (x : α) => β) f (fun (a : α) => g a) h a)\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {g : α -> β} (h : Eq.{imax u2 u1} (α -> β) f g) (a : α), Eq.{0} (Eq.{u1} β (f a) (g a)) (congr.{u2, u1} α β f g a a h (Eq.refl.{u2} α a)) (congr_fun.{u2, u1} α (fun (x : α) => β) f g h a)\nCase conversion may be inaccurate. Consider using '#align congr_refl_right congr_refl_rightₓ'. -/\n@[simp]\ntheorem congr_refl_right {α β : Sort _} {f g : α → β} (h : f = g) (a : α) :\n    congr h (Eq.refl a) = congr_fun h a :=\n  rfl\n#align congr_refl_right congr_refl_right\n\n/- warning: congr_arg_refl -> congr_arg_refl is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) (a : α), Eq.{0} (Eq.{u2} β (f a) (f a)) (congr_arg.{u1, u2} α β a a f (rfl.{u1} α a)) (rfl.{u2} β (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) (a : α), Eq.{0} (Eq.{u1} β (f a) (f a)) (congr_arg.{u2, u1} α β a a f (Eq.refl.{u2} α a)) (Eq.refl.{u1} β (f a))\nCase conversion may be inaccurate. Consider using '#align congr_arg_refl congr_arg_reflₓ'. -/\n@[simp]\ntheorem congr_arg_refl {α β : Sort _} (f : α → β) (a : α) :\n    congr_arg f (Eq.refl a) = Eq.refl (f a) :=\n  rfl\n#align congr_arg_refl congr_arg_refl\n\n/- warning: congr_fun_rfl -> congr_fun_rfl is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) (a : α), Eq.{0} (Eq.{u2} β (f a) (f a)) (congr_fun.{u1, u2} α (fun (ᾰ : α) => β) f f (rfl.{imax u1 u2} (α -> β) f) a) (rfl.{u2} β (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) (a : α), Eq.{0} (Eq.{u1} β (f a) (f a)) (congr_fun.{u2, u1} α (fun (ᾰ : α) => β) f f (Eq.refl.{imax u2 u1} (α -> β) f) a) (Eq.refl.{u1} β (f a))\nCase conversion may be inaccurate. Consider using '#align congr_fun_rfl congr_fun_rflₓ'. -/\n@[simp]\ntheorem congr_fun_rfl {α β : Sort _} (f : α → β) (a : α) :\n    congr_fun (Eq.refl f) a = Eq.refl (f a) :=\n  rfl\n#align congr_fun_rfl congr_fun_rfl\n\n/- warning: congr_fun_congr_arg -> congr_fun_congr_arg is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {γ : Sort.{u3}} (f : α -> β -> γ) {a : α} {a' : α} (p : Eq.{u1} α a a') (b : β), Eq.{0} (Eq.{u3} γ (f a b) (f a' b)) (congr_fun.{u2, u3} β (fun (ᾰ : β) => γ) (f a) (f a') (congr_arg.{u1, imax u2 u3} α (β -> γ) a a' f p) b) (congr_arg.{u1, u3} α γ a a' (fun (a : α) => f a b) p)\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : Sort.{u2}} {γ : Sort.{u1}} (f : α -> β -> γ) {a : α} {a' : α} (p : Eq.{u3} α a a') (b : β), Eq.{0} (Eq.{u1} γ (f a b) (f a' b)) (congr_fun.{u2, u1} β (fun (ᾰ : β) => γ) (f a) (f a') (congr_arg.{u3, imax u2 u1} α (β -> γ) a a' f p) b) (congr_arg.{u3, u1} α γ a a' (fun (a : α) => f a b) p)\nCase conversion may be inaccurate. Consider using '#align congr_fun_congr_arg congr_fun_congr_argₓ'. -/\n@[simp]\ntheorem congr_fun_congr_arg {α β γ : Sort _} (f : α → β → γ) {a a' : α} (p : a = a') (b : β) :\n    congr_fun (congr_arg f p) b = congr_arg (fun a => f a b) p :=\n  rfl\n#align congr_fun_congr_arg congr_fun_congr_arg\n\n#print heq_of_cast_eq /-\ntheorem heq_of_cast_eq :\n    ∀ {α β : Sort _} {a : α} {a' : β} (e : α = β) (h₂ : cast e a = a'), HEq a a'\n  | α, _, a, a', rfl, h => Eq.recOn h (HEq.refl _)\n#align heq_of_cast_eq heq_of_cast_eq\n-/\n\n/- warning: cast_eq_iff_heq -> cast_eq_iff_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u1}} {a : α} {a' : β} {e : Eq.{succ u1} Sort.{u1} α β}, Iff (Eq.{u1} β (cast.{u1} α β e a) a') (HEq.{u1} α a β a')\nbut is expected to have type\n  forall {α : Sort.{u1}} {β : Sort.{u1}} {a : Eq.{succ u1} Sort.{u1} α β} {a' : α} {e : β}, Iff (Eq.{u1} β (cast.{u1} α β a a') e) (HEq.{u1} α a' β e)\nCase conversion may be inaccurate. Consider using '#align cast_eq_iff_heq cast_eq_iff_heqₓ'. -/\ntheorem cast_eq_iff_heq {α β : Sort _} {a : α} {a' : β} {e : α = β} : cast e a = a' ↔ HEq a a' :=\n  ⟨heq_of_cast_eq _, fun h => by cases h <;> rfl⟩\n#align cast_eq_iff_heq cast_eq_iff_heq\n\n/- warning: rec_heq_of_heq -> rec_heq_of_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {a : α} {b : α} {β : Sort.{u2}} {C : α -> Sort.{u2}} {x : C a} {y : β} (e : Eq.{u1} α a b), (HEq.{u2} (C a) x β y) -> (HEq.{u2} (C b) (Eq.ndrec.{u2, u1} α a C x b e) β y)\nbut is expected to have type\n  forall {α : Sort.{u2}} {a : α} {b : Sort.{u1}} {β : α} {C : α -> Sort.{u1}} {x : C a} {y : b} (e : Eq.{u2} α a β), (HEq.{u1} (C a) x b y) -> (HEq.{u1} (C β) (Eq.rec.{u1, u2} α a (fun (x._@.Mathlib.Logic.Basic._hyg.4255 : α) (h._@.Mathlib.Logic.Basic._hyg.4256 : Eq.{u2} α a x._@.Mathlib.Logic.Basic._hyg.4255) => C x._@.Mathlib.Logic.Basic._hyg.4255) x β e) b y)\nCase conversion may be inaccurate. Consider using '#align rec_heq_of_heq rec_heq_of_heqₓ'. -/\ntheorem rec_heq_of_heq {β} {C : α → Sort _} {x : C a} {y : β} (e : a = b) (h : HEq x y) :\n    HEq (@Eq.ndrec α a C x b e) y := by subst e <;> exact h\n#align rec_heq_of_heq rec_heq_of_heq\n\n/- warning: rec_heq_iff_heq -> rec_heq_iff_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {a : α} {b : α} {β : Sort.{u2}} {C : α -> Sort.{u2}} {x : C a} {y : β} {e : Eq.{u1} α a b}, Iff (HEq.{u2} (C b) (Eq.ndrec.{u2, u1} α a C x b e) β y) (HEq.{u2} (C a) x β y)\nbut is expected to have type\n  forall {α : Sort.{u2}} {a : α} {b : Sort.{u1}} {β : α} {C : α -> Sort.{u1}} {x : C a} {y : b} {e : Eq.{u2} α a β}, Iff (HEq.{u1} (C β) (Eq.rec.{u1, u2} α a (fun (x._@.Mathlib.Logic.Basic._hyg.4315 : α) (h._@.Mathlib.Logic.Basic._hyg.4316 : Eq.{u2} α a x._@.Mathlib.Logic.Basic._hyg.4315) => C x._@.Mathlib.Logic.Basic._hyg.4315) x β e) b y) (HEq.{u1} (C a) x b y)\nCase conversion may be inaccurate. Consider using '#align rec_heq_iff_heq rec_heq_iff_heqₓ'. -/\ntheorem rec_heq_iff_heq {β} {C : α → Sort _} {x : C a} {y : β} {e : a = b} :\n    HEq (@Eq.ndrec α a C x b e) y ↔ HEq x y := by subst e\n#align rec_heq_iff_heq rec_heq_iff_heq\n\n/- warning: heq_rec_iff_heq -> heq_rec_iff_heq is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {a : α} {b : α} {β : Sort.{u2}} {C : α -> Sort.{u2}} {x : β} {y : C a} {e : Eq.{u1} α a b}, Iff (HEq.{u2} β x (C b) (Eq.ndrec.{u2, u1} α a C y b e)) (HEq.{u2} β x (C a) y)\nbut is expected to have type\n  forall {α : Sort.{u2}} {a : Sort.{u1}} {b : α} {β : α} {C : α -> Sort.{u1}} {x : a} {y : C b} {e : Eq.{u2} α b β}, Iff (HEq.{u1} a x (C β) (Eq.rec.{u1, u2} α b (fun (x._@.Mathlib.Logic.Basic._hyg.4377 : α) (h._@.Mathlib.Logic.Basic._hyg.4378 : Eq.{u2} α b x._@.Mathlib.Logic.Basic._hyg.4377) => C x._@.Mathlib.Logic.Basic._hyg.4377) y β e)) (HEq.{u1} a x (C b) y)\nCase conversion may be inaccurate. Consider using '#align heq_rec_iff_heq heq_rec_iff_heqₓ'. -/\ntheorem heq_rec_iff_heq {β} {C : α → Sort _} {x : β} {y : C a} {e : a = b} :\n    HEq x (@Eq.ndrec α a C y b e) ↔ HEq x y := by subst e\n#align heq_rec_iff_heq heq_rec_iff_heq\n\n/- warning: eq.congr -> Eq.congr is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {x₁ : α} {x₂ : α} {y₁ : α} {y₂ : α}, (Eq.{u1} α x₁ y₁) -> (Eq.{u1} α x₂ y₂) -> (Iff (Eq.{u1} α x₁ x₂) (Eq.{u1} α y₁ y₂))\nbut is expected to have type\n  forall {α : Sort.{u1}} {x₁ : α} {x₂ : α} {y₁ : α} {y₂ : α}, (Eq.{u1} α x₁ x₂) -> (Eq.{u1} α y₁ y₂) -> (Iff (Eq.{u1} α x₁ y₁) (Eq.{u1} α x₂ y₂))\nCase conversion may be inaccurate. Consider using '#align eq.congr Eq.congrₓ'. -/\nprotected theorem Eq.congr {x₁ x₂ y₁ y₂ : α} (h₁ : x₁ = y₁) (h₂ : x₂ = y₂) : x₁ = x₂ ↔ y₁ = y₂ :=\n  by\n  subst h₁\n  subst h₂\n#align eq.congr Eq.congr\n\n#print Eq.congr_left /-\ntheorem Eq.congr_left {x y z : α} (h : x = y) : x = z ↔ y = z := by rw [h]\n#align eq.congr_left Eq.congr_left\n-/\n\n#print Eq.congr_right /-\ntheorem Eq.congr_right {x y z : α} (h : x = y) : z = x ↔ z = y := by rw [h]\n#align eq.congr_right Eq.congr_right\n-/\n\n/- warning: congr_arg2 -> congr_arg₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {γ : Sort.{u3}} (f : α -> β -> γ) {x : α} {x' : α} {y : β} {y' : β}, (Eq.{u1} α x x') -> (Eq.{u2} β y y') -> (Eq.{u3} γ (f x y) (f x' y'))\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : Sort.{u2}} {γ : Sort.{u1}} (f : α -> β -> γ) {x : α} {x' : α} {y : β} {y' : β}, (Eq.{u3} α x x') -> (Eq.{u2} β y y') -> (Eq.{u1} γ (f x y) (f x' y'))\nCase conversion may be inaccurate. Consider using '#align congr_arg2 congr_arg₂ₓ'. -/\ntheorem congr_arg₂ {α β γ : Sort _} (f : α → β → γ) {x x' : α} {y y' : β} (hx : x = x')\n    (hy : y = y') : f x y = f x' y' := by\n  subst hx\n  subst hy\n#align congr_arg2 congr_arg₂\n\nvariable {β : α → Sort _} {γ : ∀ a, β a → Sort _} {δ : ∀ a b, γ a b → Sort _}\n\n/- warning: congr_fun₂ -> congr_fun₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {f : forall (a : α) (b : β a), γ a b} {g : forall (a : α) (b : β a), γ a b}, (Eq.{imax u1 u2 u3} (forall (a : α) (b : β a), γ a b) f g) -> (forall (a : α) (b : β a), Eq.{u3} (γ a b) (f a b) (g a b))\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u1}} {f : forall (a : α) (b : β a), γ a b} {g : forall (a : α) (b : β a), γ a b}, (Eq.{imax u3 u2 u1} (forall (a : α) (b : β a), γ a b) f g) -> (forall (a : α) (b : β a), Eq.{u1} (γ a b) (f a b) (g a b))\nCase conversion may be inaccurate. Consider using '#align congr_fun₂ congr_fun₂ₓ'. -/\ntheorem congr_fun₂ {f g : ∀ a b, γ a b} (h : f = g) (a : α) (b : β a) : f a b = g a b :=\n  congr_fun (congr_fun h _) _\n#align congr_fun₂ congr_fun₂\n\n/- warning: congr_fun₃ -> congr_fun₃ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u4}} {f : forall (a : α) (b : β a) (c : γ a b), δ a b c} {g : forall (a : α) (b : β a) (c : γ a b), δ a b c}, (Eq.{imax u1 u2 u3 u4} (forall (a : α) (b : β a) (c : γ a b), δ a b c) f g) -> (forall (a : α) (b : β a) (c : γ a b), Eq.{u4} (δ a b c) (f a b c) (g a b c))\nbut is expected to have type\n  forall {α : Sort.{u4}} {β : α -> Sort.{u3}} {γ : forall (a : α), (β a) -> Sort.{u2}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u1}} {f : forall (a : α) (b : β a) (c : γ a b), δ a b c} {g : forall (a : α) (b : β a) (c : γ a b), δ a b c}, (Eq.{imax u4 u3 u2 u1} (forall (a : α) (b : β a) (c : γ a b), δ a b c) f g) -> (forall (a : α) (b : β a) (c : γ a b), Eq.{u1} (δ a b c) (f a b c) (g a b c))\nCase conversion may be inaccurate. Consider using '#align congr_fun₃ congr_fun₃ₓ'. -/\ntheorem congr_fun₃ {f g : ∀ a b c, δ a b c} (h : f = g) (a : α) (b : β a) (c : γ a b) :\n    f a b c = g a b c :=\n  congr_fun₂ (congr_fun h _) _ _\n#align congr_fun₃ congr_fun₃\n\n/- warning: funext₂ -> funext₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {f : forall (a : α) (b : β a), γ a b} {g : forall (a : α) (b : β a), γ a b}, (forall (a : α) (b : β a), Eq.{u3} (γ a b) (f a b) (g a b)) -> (Eq.{imax u1 u2 u3} (forall (a : α) (b : β a), γ a b) f g)\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : α -> Sort.{u1}} {γ : forall (a : α), (β a) -> Sort.{u3}} {f : forall (a : α) (b : β a), γ a b} {g : forall (a : α) (b : β a), γ a b}, (forall (a : α) (b : β a), Eq.{u3} (γ a b) (f a b) (g a b)) -> (Eq.{imax u2 u1 u3} (forall (a : α) (b : β a), γ a b) f g)\nCase conversion may be inaccurate. Consider using '#align funext₂ funext₂ₓ'. -/\ntheorem funext₂ {f g : ∀ a b, γ a b} (h : ∀ a b, f a b = g a b) : f = g :=\n  funext fun _ => funext <| h _\n#align funext₂ funext₂\n\n/- warning: funext₃ -> funext₃ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u4}} {f : forall (a : α) (b : β a) (c : γ a b), δ a b c} {g : forall (a : α) (b : β a) (c : γ a b), δ a b c}, (forall (a : α) (b : β a) (c : γ a b), Eq.{u4} (δ a b c) (f a b c) (g a b c)) -> (Eq.{imax u1 u2 u3 u4} (forall (a : α) (b : β a) (c : γ a b), δ a b c) f g)\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u1}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u4}} {f : forall (a : α) (b : β a) (c : γ a b), δ a b c} {g : forall (a : α) (b : β a) (c : γ a b), δ a b c}, (forall (a : α) (b : β a) (c : γ a b), Eq.{u4} (δ a b c) (f a b c) (g a b c)) -> (Eq.{imax u3 u2 u1 u4} (forall (a : α) (b : β a) (c : γ a b), δ a b c) f g)\nCase conversion may be inaccurate. Consider using '#align funext₃ funext₃ₓ'. -/\ntheorem funext₃ {f g : ∀ a b c, δ a b c} (h : ∀ a b c, f a b c = g a b c) : f = g :=\n  funext fun _ => funext₂ <| h _\n#align funext₃ funext₃\n\nend Equality\n\n/-! ### Declarations about quantifiers -/\n\n\nsection Quantifiers\n\nvariable {α : Sort _}\n\nsection Dependent\n\nvariable {β : α → Sort _} {γ : ∀ a, β a → Sort _} {δ : ∀ a b, γ a b → Sort _}\n  {ε : ∀ a b c, δ a b c → Sort _}\n\n#print pi_congr /-\ntheorem pi_congr {β' : α → Sort _} (h : ∀ a, β a = β' a) : (∀ a, β a) = ∀ a, β' a :=\n  (funext h : β = β') ▸ rfl\n#align pi_congr pi_congr\n-/\n\n#print forall₂_congr /-\ntheorem forall₂_congr {p q : ∀ a, β a → Prop} (h : ∀ a b, p a b ↔ q a b) :\n    (∀ a b, p a b) ↔ ∀ a b, q a b :=\n  forall_congr' fun a => forall_congr' <| h a\n#align forall₂_congr forall₂_congr\n-/\n\n#print forall₃_congr /-\ntheorem forall₃_congr {p q : ∀ a b, γ a b → Prop} (h : ∀ a b c, p a b c ↔ q a b c) :\n    (∀ a b c, p a b c) ↔ ∀ a b c, q a b c :=\n  forall_congr' fun a => forall₂_congr <| h a\n#align forall₃_congr forall₃_congr\n-/\n\n#print forall₄_congr /-\ntheorem forall₄_congr {p q : ∀ a b c, δ a b c → Prop} (h : ∀ a b c d, p a b c d ↔ q a b c d) :\n    (∀ a b c d, p a b c d) ↔ ∀ a b c d, q a b c d :=\n  forall_congr' fun a => forall₃_congr <| h a\n#align forall₄_congr forall₄_congr\n-/\n\n#print forall₅_congr /-\ntheorem forall₅_congr {p q : ∀ a b c d, ε a b c d → Prop}\n    (h : ∀ a b c d e, p a b c d e ↔ q a b c d e) :\n    (∀ a b c d e, p a b c d e) ↔ ∀ a b c d e, q a b c d e :=\n  forall_congr' fun a => forall₄_congr <| h a\n#align forall₅_congr forall₅_congr\n-/\n\n/- warning: exists₂_congr -> exists₂_congr is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {p : forall (a : α), (β a) -> Prop} {q : forall (a : α), (β a) -> Prop}, (forall (a : α) (b : β a), Iff (p a b) (q a b)) -> (Iff (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => p a b))) (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => q a b))))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : α -> Sort.{u1}} {p : forall (a : α), (β a) -> Prop} {q : forall (a : α), (β a) -> Prop}, (forall (a : α) (b : β a), Iff (p a b) (q a b)) -> (Iff (Exists.{u2} α (fun (a : α) => Exists.{u1} (β a) (fun (b : β a) => p a b))) (Exists.{u2} α (fun (a : α) => Exists.{u1} (β a) (fun (b : β a) => q a b))))\nCase conversion may be inaccurate. Consider using '#align exists₂_congr exists₂_congrₓ'. -/\ntheorem exists₂_congr {p q : ∀ a, β a → Prop} (h : ∀ a b, p a b ↔ q a b) :\n    (∃ a b, p a b) ↔ ∃ a b, q a b :=\n  exists_congr fun a => exists_congr <| h a\n#align exists₂_congr exists₂_congr\n\n/- warning: exists₃_congr -> exists₃_congr is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {p : forall (a : α) (b : β a), (γ a b) -> Prop} {q : forall (a : α) (b : β a), (γ a b) -> Prop}, (forall (a : α) (b : β a) (c : γ a b), Iff (p a b c) (q a b c)) -> (Iff (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => p a b c)))) (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => q a b c)))))\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u1}} {p : forall (a : α) (b : β a), (γ a b) -> Prop} {q : forall (a : α) (b : β a), (γ a b) -> Prop}, (forall (a : α) (b : β a) (c : γ a b), Iff (p a b c) (q a b c)) -> (Iff (Exists.{u3} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u1} (γ a b) (fun (c : γ a b) => p a b c)))) (Exists.{u3} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u1} (γ a b) (fun (c : γ a b) => q a b c)))))\nCase conversion may be inaccurate. Consider using '#align exists₃_congr exists₃_congrₓ'. -/\ntheorem exists₃_congr {p q : ∀ a b, γ a b → Prop} (h : ∀ a b c, p a b c ↔ q a b c) :\n    (∃ a b c, p a b c) ↔ ∃ a b c, q a b c :=\n  exists_congr fun a => exists₂_congr <| h a\n#align exists₃_congr exists₃_congr\n\n/- warning: exists₄_congr -> exists₄_congr is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u4}} {p : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Prop} {q : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Prop}, (forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), Iff (p a b c d) (q a b c d)) -> (Iff (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u4} (δ a b c) (fun (d : δ a b c) => p a b c d))))) (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u4} (δ a b c) (fun (d : δ a b c) => q a b c d))))))\nbut is expected to have type\n  forall {α : Sort.{u4}} {β : α -> Sort.{u3}} {γ : forall (a : α), (β a) -> Sort.{u2}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u1}} {p : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Prop} {q : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Prop}, (forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), Iff (p a b c d) (q a b c d)) -> (Iff (Exists.{u4} α (fun (a : α) => Exists.{u3} (β a) (fun (b : β a) => Exists.{u2} (γ a b) (fun (c : γ a b) => Exists.{u1} (δ a b c) (fun (d : δ a b c) => p a b c d))))) (Exists.{u4} α (fun (a : α) => Exists.{u3} (β a) (fun (b : β a) => Exists.{u2} (γ a b) (fun (c : γ a b) => Exists.{u1} (δ a b c) (fun (d : δ a b c) => q a b c d))))))\nCase conversion may be inaccurate. Consider using '#align exists₄_congr exists₄_congrₓ'. -/\ntheorem exists₄_congr {p q : ∀ a b c, δ a b c → Prop} (h : ∀ a b c d, p a b c d ↔ q a b c d) :\n    (∃ a b c d, p a b c d) ↔ ∃ a b c d, q a b c d :=\n  exists_congr fun a => exists₃_congr <| h a\n#align exists₄_congr exists₄_congr\n\n/- warning: exists₅_congr -> exists₅_congr is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u4}} {ε : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Sort.{u5}} {p : forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), (ε a b c d) -> Prop} {q : forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), (ε a b c d) -> Prop}, (forall (a : α) (b : β a) (c : γ a b) (d : δ a b c) (e : ε a b c d), Iff (p a b c d e) (q a b c d e)) -> (Iff (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u4} (δ a b c) (fun (d : δ a b c) => Exists.{u5} (ε a b c d) (fun (e : ε a b c d) => p a b c d e)))))) (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u4} (δ a b c) (fun (d : δ a b c) => Exists.{u5} (ε a b c d) (fun (e : ε a b c d) => q a b c d e)))))))\nbut is expected to have type\n  forall {α : Sort.{u5}} {β : α -> Sort.{u4}} {γ : forall (a : α), (β a) -> Sort.{u3}} {δ : forall (a : α) (b : β a), (γ a b) -> Sort.{u2}} {ε : forall (a : α) (b : β a) (c : γ a b), (δ a b c) -> Sort.{u1}} {p : forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), (ε a b c d) -> Prop} {q : forall (a : α) (b : β a) (c : γ a b) (d : δ a b c), (ε a b c d) -> Prop}, (forall (a : α) (b : β a) (c : γ a b) (d : δ a b c) (e : ε a b c d), Iff (p a b c d e) (q a b c d e)) -> (Iff (Exists.{u5} α (fun (a : α) => Exists.{u4} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u2} (δ a b c) (fun (d : δ a b c) => Exists.{u1} (ε a b c d) (fun (e : ε a b c d) => p a b c d e)))))) (Exists.{u5} α (fun (a : α) => Exists.{u4} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => Exists.{u2} (δ a b c) (fun (d : δ a b c) => Exists.{u1} (ε a b c d) (fun (e : ε a b c d) => q a b c d e)))))))\nCase conversion may be inaccurate. Consider using '#align exists₅_congr exists₅_congrₓ'. -/\ntheorem exists₅_congr {p q : ∀ a b c d, ε a b c d → Prop}\n    (h : ∀ a b c d e, p a b c d e ↔ q a b c d e) :\n    (∃ a b c d e, p a b c d e) ↔ ∃ a b c d e, q a b c d e :=\n  exists_congr fun a => exists₄_congr <| h a\n#align exists₅_congr exists₅_congr\n\n#print forall_imp /-\ntheorem forall_imp {p q : α → Prop} (h : ∀ a, p a → q a) : (∀ a, p a) → ∀ a, q a := fun h' a =>\n  h a (h' a)\n#align forall_imp forall_imp\n-/\n\n#print forall₂_imp /-\ntheorem forall₂_imp {p q : ∀ a, β a → Prop} (h : ∀ a b, p a b → q a b) :\n    (∀ a b, p a b) → ∀ a b, q a b :=\n  forall_imp fun i => forall_imp <| h i\n#align forall₂_imp forall₂_imp\n-/\n\n#print forall₃_imp /-\ntheorem forall₃_imp {p q : ∀ a b, γ a b → Prop} (h : ∀ a b c, p a b c → q a b c) :\n    (∀ a b c, p a b c) → ∀ a b c, q a b c :=\n  forall_imp fun a => forall₂_imp <| h a\n#align forall₃_imp forall₃_imp\n-/\n\n#print Exists.imp /-\ntheorem Exists.imp {p q : α → Prop} (h : ∀ a, p a → q a) : (∃ a, p a) → ∃ a, q a :=\n  Exists.imp h\n#align Exists.imp Exists.imp\n-/\n\n/- warning: Exists₂.imp -> Exists₂.imp is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {p : forall (a : α), (β a) -> Prop} {q : forall (a : α), (β a) -> Prop}, (forall (a : α) (b : β a), (p a b) -> (q a b)) -> (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => p a b))) -> (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => q a b)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : α -> Sort.{u1}} {p : forall (a : α), (β a) -> Prop} {q : forall (a : α), (β a) -> Prop}, (forall (a : α) (b : β a), (p a b) -> (q a b)) -> (Exists.{u2} α (fun (a : α) => Exists.{u1} (β a) (fun (b : β a) => p a b))) -> (Exists.{u2} α (fun (a : α) => Exists.{u1} (β a) (fun (b : β a) => q a b)))\nCase conversion may be inaccurate. Consider using '#align Exists₂.imp Exists₂.impₓ'. -/\ntheorem Exists₂.imp {p q : ∀ a, β a → Prop} (h : ∀ a b, p a b → q a b) :\n    (∃ a b, p a b) → ∃ a b, q a b :=\n  Exists.imp fun a => Exists.imp <| h a\n#align Exists₂.imp Exists₂.imp\n\n/- warning: Exists₃.imp -> Exists₃.imp is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}} {p : forall (a : α) (b : β a), (γ a b) -> Prop} {q : forall (a : α) (b : β a), (γ a b) -> Prop}, (forall (a : α) (b : β a) (c : γ a b), (p a b c) -> (q a b c)) -> (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => p a b c)))) -> (Exists.{u1} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u3} (γ a b) (fun (c : γ a b) => q a b c))))\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u1}} {p : forall (a : α) (b : β a), (γ a b) -> Prop} {q : forall (a : α) (b : β a), (γ a b) -> Prop}, (forall (a : α) (b : β a) (c : γ a b), (p a b c) -> (q a b c)) -> (Exists.{u3} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u1} (γ a b) (fun (c : γ a b) => p a b c)))) -> (Exists.{u3} α (fun (a : α) => Exists.{u2} (β a) (fun (b : β a) => Exists.{u1} (γ a b) (fun (c : γ a b) => q a b c))))\nCase conversion may be inaccurate. Consider using '#align Exists₃.imp Exists₃.impₓ'. -/\ntheorem Exists₃.imp {p q : ∀ a b, γ a b → Prop} (h : ∀ a b c, p a b c → q a b c) :\n    (∃ a b c, p a b c) → ∃ a b c, q a b c :=\n  Exists.imp fun a => Exists₂.imp <| h a\n#align Exists₃.imp Exists₃.imp\n\nend Dependent\n\nvariable {ι β : Sort _} {κ : ι → Sort _} {p q : α → Prop} {b : Prop}\n\n/- warning: exists_imp_exists' -> Exists.imp' is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {p : α -> Prop} {q : β -> Prop} (f : α -> β), (forall (a : α), (p a) -> (q (f a))) -> (Exists.{u1} α (fun (a : α) => p a)) -> (Exists.{u2} β (fun (b : β) => q b))\nbut is expected to have type\n  forall {α : Sort.{u1}} {β : α -> Prop} {p : Sort.{u2}} {q : p -> Prop} (f : α -> p), (forall (a : α), (β a) -> (q (f a))) -> (Exists.{u1} α (fun (a : α) => β a)) -> (Exists.{u2} p (fun (b : p) => q b))\nCase conversion may be inaccurate. Consider using '#align exists_imp_exists' Exists.imp'ₓ'. -/\ntheorem Exists.imp' {p : α → Prop} {q : β → Prop} (f : α → β) (hpq : ∀ a, p a → q (f a))\n    (hp : ∃ a, p a) : ∃ b, q b :=\n  Exists.elim hp fun a hp' => ⟨_, hpq _ hp'⟩\n#align exists_imp_exists' Exists.imp'\n\n/- warning: forall_swap -> forall_swap is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {p : α -> β -> Prop}, Iff (forall (x : α) (y : β), p x y) (forall (y : β) (x : α), p x y)\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {p : α -> β -> Prop}, Iff (forall (x : α) (y : β), p x y) (forall (y : β) (x : α), p x y)\nCase conversion may be inaccurate. Consider using '#align forall_swap forall_swapₓ'. -/\ntheorem forall_swap {p : α → β → Prop} : (∀ x y, p x y) ↔ ∀ y x, p x y :=\n  ⟨swap, swap⟩\n#align forall_swap forall_swap\n\n/- warning: forall₂_swap -> forall₂_swap is a dubious translation:\nlean 3 declaration is\n  forall {ι₁ : Sort.{u1}} {ι₂ : Sort.{u2}} {κ₁ : ι₁ -> Sort.{u3}} {κ₂ : ι₂ -> Sort.{u4}} {p : forall (i₁ : ι₁), (κ₁ i₁) -> (forall (i₂ : ι₂), (κ₂ i₂) -> Prop)}, Iff (forall (i₁ : ι₁) (j₁ : κ₁ i₁) (i₂ : ι₂) (j₂ : κ₂ i₂), p i₁ j₁ i₂ j₂) (forall (i₂ : ι₂) (j₂ : κ₂ i₂) (i₁ : ι₁) (j₁ : κ₁ i₁), p i₁ j₁ i₂ j₂)\nbut is expected to have type\n  forall {ι₁ : Sort.{u4}} {ι₂ : Sort.{u3}} {κ₁ : ι₁ -> Sort.{u2}} {κ₂ : ι₂ -> Sort.{u1}} {p : forall (i₁ : ι₁), (κ₁ i₁) -> (forall (i₂ : ι₂), (κ₂ i₂) -> Prop)}, Iff (forall (i₁ : ι₁) (j₁ : κ₁ i₁) (i₂ : ι₂) (j₂ : κ₂ i₂), p i₁ j₁ i₂ j₂) (forall (i₂ : ι₂) (j₂ : κ₂ i₂) (i₁ : ι₁) (j₁ : κ₁ i₁), p i₁ j₁ i₂ j₂)\nCase conversion may be inaccurate. Consider using '#align forall₂_swap forall₂_swapₓ'. -/\ntheorem forall₂_swap {ι₁ ι₂ : Sort _} {κ₁ : ι₁ → Sort _} {κ₂ : ι₂ → Sort _}\n    {p : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Prop} :\n    (∀ i₁ j₁ i₂ j₂, p i₁ j₁ i₂ j₂) ↔ ∀ i₂ j₂ i₁ j₁, p i₁ j₁ i₂ j₂ :=\n  ⟨swap₂, swap₂⟩\n#align forall₂_swap forall₂_swap\n\n#print imp_forall_iff /-\n/-- We intentionally restrict the type of `α` in this lemma so that this is a safer to use in simp\nthan `forall_swap`. -/\ntheorem imp_forall_iff {α : Type _} {p : Prop} {q : α → Prop} : (p → ∀ x, q x) ↔ ∀ x, p → q x :=\n  forall_swap\n#align imp_forall_iff imp_forall_iff\n-/\n\n#print exists_swap /-\ntheorem exists_swap {p : α → β → Prop} : (∃ x y, p x y) ↔ ∃ y x, p x y :=\n  ⟨fun ⟨x, y, h⟩ => ⟨y, x, h⟩, fun ⟨y, x, h⟩ => ⟨x, y, h⟩⟩\n#align exists_swap exists_swap\n-/\n\n#print forall_exists_index /-\n@[simp]\ntheorem forall_exists_index {q : (∃ x, p x) → Prop} : (∀ h, q h) ↔ ∀ (x) (h : p x), q ⟨x, h⟩ :=\n  ⟨fun h x hpx => h ⟨x, hpx⟩, fun h ⟨x, hpx⟩ => h x hpx⟩\n#align forall_exists_index forall_exists_index\n-/\n\n#print exists_imp /-\ntheorem exists_imp : (∃ x, p x) → b ↔ ∀ x, p x → b :=\n  forall_exists_index\n#align exists_imp_distrib exists_imp\n-/\n\n#print Exists.choose /-\n-- This enables projection notation.\n/-- Extract an element from a existential statement, using `classical.some`.\n-/\n@[reducible]\nnoncomputable def Exists.choose {p : α → Prop} (P : ∃ a, p a) : α :=\n  Classical.choose P\n#align Exists.some Exists.choose\n-/\n\n#print Exists.choose_spec /-\n/-- Show that an element extracted from `P : ∃ a, p a` using `P.some` satisfies `p`.\n-/\ntheorem Exists.choose_spec {p : α → Prop} (P : ∃ a, p a) : p P.some :=\n  Classical.choose_spec P\n#align Exists.some_spec Exists.choose_spec\n-/\n\n/- warning: not_exists_of_forall_not -> not_exists_of_forall_not is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Prop}, (forall (x : α), Not (p x)) -> (Not (Exists.{u1} α (fun (x : α) => p x)))\nbut is expected to have type\n  forall {α : Sort.{u1}} {p : α -> Prop} {h : Prop}, (forall (x : α), (p x) -> h) -> (Exists.{u1} α (fun (x : α) => p x)) -> h\nCase conversion may be inaccurate. Consider using '#align not_exists_of_forall_not not_exists_of_forall_notₓ'. -/\n--theorem forall_not_of_not_exists (h : ¬ ∃ x, p x) : ∀ x, ¬ p x :=\n--forall_imp_of_exists_imp h\ntheorem not_exists_of_forall_not (h : ∀ x, ¬p x) : ¬∃ x, p x :=\n  exists_imp.2 h\n#align not_exists_of_forall_not not_exists_of_forall_not\n\n#print not_exists /-\n@[simp]\ntheorem not_exists : (¬∃ x, p x) ↔ ∀ x, ¬p x :=\n  exists_imp\n#align not_exists not_exists\n-/\n\n#print not_forall_of_exists_not /-\ntheorem not_forall_of_exists_not : (∃ x, ¬p x) → ¬∀ x, p x\n  | ⟨x, hn⟩, h => hn (h x)\n#align not_forall_of_exists_not not_forall_of_exists_not\n-/\n\n#print Decidable.not_forall /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_forall {p : α → Prop} [Decidable (∃ x, ¬p x)]\n    [∀ x, Decidable (p x)] : (¬∀ x, p x) ↔ ∃ x, ¬p x :=\n  ⟨Not.decidable_imp_symm fun nx x => nx.decidable_imp_symm fun h => ⟨x, h⟩,\n    not_forall_of_exists_not⟩\n#align decidable.not_forall Decidable.not_forall\n-/\n\n#print not_forall /-\n@[simp]\ntheorem not_forall {p : α → Prop} : (¬∀ x, p x) ↔ ∃ x, ¬p x :=\n  Decidable.not_forall\n#align not_forall not_forall\n-/\n\n#print Decidable.not_forall_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_forall_not [Decidable (∃ x, p x)] : (¬∀ x, ¬p x) ↔ ∃ x, p x :=\n  (@Decidable.not_iff_comm _ _ _ (decidable_of_iff (¬∃ x, p x) not_exists)).1 not_exists\n#align decidable.not_forall_not Decidable.not_forall_not\n-/\n\n#print not_forall_not /-\ntheorem not_forall_not : (¬∀ x, ¬p x) ↔ ∃ x, p x :=\n  Decidable.not_forall_not\n#align not_forall_not not_forall_not\n-/\n\n#print Decidable.not_exists_not /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_exists_not [∀ x, Decidable (p x)] : (¬∃ x, ¬p x) ↔ ∀ x, p x := by\n  simp [Decidable.not_not]\n#align decidable.not_exists_not Decidable.not_exists_not\n-/\n\n#print not_exists_not /-\n@[simp]\ntheorem not_exists_not : (¬∃ x, ¬p x) ↔ ∀ x, p x :=\n  Decidable.not_exists_not\n#align not_exists_not not_exists_not\n-/\n\n#print forall_imp_iff_exists_imp /-\ntheorem forall_imp_iff_exists_imp [ha : Nonempty α] : (∀ x, p x) → b ↔ ∃ x, p x → b :=\n  let ⟨a⟩ := ha\n  ⟨fun h =>\n    not_forall_not.1 fun h' =>\n      by_cases (fun hb : b => h' a fun _ => hb) fun hb => hb <| h fun x => (not_imp.1 (h' x)).1,\n    fun ⟨x, hx⟩ h => hx (h x)⟩\n#align forall_imp_iff_exists_imp forall_imp_iff_exists_imp\n-/\n\n#print forall_true_iff /-\n-- TODO: duplicate of a lemma in core\ntheorem forall_true_iff : α → True ↔ True :=\n  imp_true_iff α\n#align forall_true_iff forall_true_iff\n-/\n\n#print forall_true_iff' /-\n-- Unfortunately this causes simp to loop sometimes, so we\n-- add the 2 and 3 cases as simp lemmas instead\ntheorem forall_true_iff' (h : ∀ a, p a ↔ True) : (∀ a, p a) ↔ True :=\n  iff_true_intro fun _ => of_iff_true (h _)\n#align forall_true_iff' forall_true_iff'\n-/\n\n/- warning: forall_2_true_iff -> forall₂_true_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}}, Iff (forall (a : α), (β a) -> True) True\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : α -> Sort.{u1}}, Iff (forall (a : α), (β a) -> True) True\nCase conversion may be inaccurate. Consider using '#align forall_2_true_iff forall₂_true_iffₓ'. -/\n@[simp]\ntheorem forall₂_true_iff {β : α → Sort _} : (∀ a, β a → True) ↔ True :=\n  forall_true_iff' fun _ => forall_true_iff\n#align forall_2_true_iff forall₂_true_iff\n\n/- warning: forall_3_true_iff -> forall₃_true_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u3}}, Iff (forall (a : α) (b : β a), (γ a b) -> True) True\nbut is expected to have type\n  forall {α : Sort.{u3}} {β : α -> Sort.{u2}} {γ : forall (a : α), (β a) -> Sort.{u1}}, Iff (forall (a : α) (b : β a), (γ a b) -> True) True\nCase conversion may be inaccurate. Consider using '#align forall_3_true_iff forall₃_true_iffₓ'. -/\n@[simp]\ntheorem forall₃_true_iff {β : α → Sort _} {γ : ∀ a, β a → Sort _} :\n    (∀ (a) (b : β a), γ a b → True) ↔ True :=\n  forall_true_iff' fun _ => forall₂_true_iff\n#align forall_3_true_iff forall₃_true_iff\n\n/- warning: exists_unique.exists clashes with exists_of_exists_unique -> ExistsUnique.exists\nCase conversion may be inaccurate. Consider using '#align exists_unique.exists ExistsUnique.existsₓ'. -/\n#print ExistsUnique.exists /-\ntheorem ExistsUnique.exists {α : Sort _} {p : α → Prop} (h : ∃! x, p x) : ∃ x, p x :=\n  Exists.elim h fun x hx => ⟨x, And.left hx⟩\n#align exists_unique.exists ExistsUnique.exists\n-/\n\n#print exists_unique_iff_exists /-\n@[simp]\ntheorem exists_unique_iff_exists {α : Sort _} [Subsingleton α] {p : α → Prop} :\n    (∃! x, p x) ↔ ∃ x, p x :=\n  ⟨fun h => h.exists, Exists.imp fun x hx => ⟨hx, fun y _ => Subsingleton.elim y x⟩⟩\n#align exists_unique_iff_exists exists_unique_iff_exists\n-/\n\n#print forall_const /-\n@[simp]\ntheorem forall_const (α : Sort _) [i : Nonempty α] : α → b ↔ b :=\n  ⟨i.elim, fun hb x => hb⟩\n#align forall_const forall_const\n-/\n\n/-- For some reason simp doesn't use `forall_const` to simplify in this case. -/\n@[simp]\ntheorem forall_forall_const {α β : Type _} (p : β → Prop) [Nonempty α] :\n    (∀ x, α → p x) ↔ ∀ x, p x :=\n  forall_congr' fun x => forall_const α\n#align forall_forall_const forall_forall_const\n\n#print exists_const /-\n@[simp]\ntheorem exists_const (α : Sort _) [i : Nonempty α] : (∃ x : α, b) ↔ b :=\n  ⟨fun ⟨x, h⟩ => h, i.elim Exists.intro⟩\n#align exists_const exists_const\n-/\n\n#print exists_unique_const /-\ntheorem exists_unique_const (α : Sort _) [i : Nonempty α] [Subsingleton α] : (∃! x : α, b) ↔ b := by\n  simp\n#align exists_unique_const exists_unique_const\n-/\n\n#print forall_and /-\ntheorem forall_and : (∀ x, p x ∧ q x) ↔ (∀ x, p x) ∧ ∀ x, q x :=\n  ⟨fun h => ⟨fun x => (h x).left, fun x => (h x).right⟩, fun ⟨h₁, h₂⟩ x => ⟨h₁ x, h₂ x⟩⟩\n#align forall_and_distrib forall_and\n-/\n\n#print exists_or /-\ntheorem exists_or : (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ ∃ x, q x :=\n  ⟨fun ⟨x, hpq⟩ => hpq.elim (fun hpx => Or.inl ⟨x, hpx⟩) fun hqx => Or.inr ⟨x, hqx⟩, fun hepq =>\n    hepq.elim (fun ⟨x, hpx⟩ => ⟨x, Or.inl hpx⟩) fun ⟨x, hqx⟩ => ⟨x, Or.inr hqx⟩⟩\n#align exists_or_distrib exists_or\n-/\n\n/- warning: exists_and_distrib_left -> exists_and_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {q : Prop} {p : α -> Prop}, Iff (Exists.{u1} α (fun (x : α) => And q (p x))) (And q (Exists.{u1} α (fun (x : α) => p x)))\nbut is expected to have type\n  forall {α : Sort.{u1}} {q : α -> Prop} {p : Prop}, Iff (Exists.{u1} α (fun (x : α) => And p (q x))) (And p (Exists.{u1} α (fun (x : α) => q x)))\nCase conversion may be inaccurate. Consider using '#align exists_and_distrib_left exists_and_leftₓ'. -/\n@[simp]\ntheorem exists_and_left {q : Prop} {p : α → Prop} : (∃ x, q ∧ p x) ↔ q ∧ ∃ x, p x :=\n  ⟨fun ⟨x, hq, hp⟩ => ⟨hq, x, hp⟩, fun ⟨hq, x, hp⟩ => ⟨x, hq, hp⟩⟩\n#align exists_and_distrib_left exists_and_left\n\n/- warning: exists_and_distrib_right -> exists_and_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {q : Prop} {p : α -> Prop}, Iff (Exists.{u1} α (fun (x : α) => And (p x) q)) (And (Exists.{u1} α (fun (x : α) => p x)) q)\nbut is expected to have type\n  forall {α : Sort.{u1}} {q : α -> Prop} {p : Prop}, Iff (Exists.{u1} α (fun (x : α) => And (q x) p)) (And (Exists.{u1} α (fun (x : α) => q x)) p)\nCase conversion may be inaccurate. Consider using '#align exists_and_distrib_right exists_and_rightₓ'. -/\n@[simp]\ntheorem exists_and_right {q : Prop} {p : α → Prop} : (∃ x, p x ∧ q) ↔ (∃ x, p x) ∧ q := by\n  simp [and_comm']\n#align exists_and_distrib_right exists_and_right\n\n#print forall_eq /-\n@[simp]\ntheorem forall_eq {a' : α} : (∀ a, a = a' → p a) ↔ p a' :=\n  ⟨fun h => h a' rfl, fun h a e => e.symm ▸ h⟩\n#align forall_eq forall_eq\n-/\n\n#print forall_eq' /-\n@[simp]\ntheorem forall_eq' {a' : α} : (∀ a, a' = a → p a) ↔ p a' := by simp [@eq_comm _ a']\n#align forall_eq' forall_eq'\n-/\n\n/- warning: decidable.and_forall_ne -> Decidable.and_forall_ne is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Prop} [_inst_1 : DecidableEq.{u1} α] (a : α), Iff (And (p a) (forall (b : α), (Ne.{u1} α b a) -> (p b))) (forall (b : α), p b)\nbut is expected to have type\n  forall {α : Sort.{u1}} [p : DecidableEq.{u1} α] (_inst_1 : α) {a : α -> Prop}, Iff (And (a _inst_1) (forall (b : α), (Ne.{u1} α b _inst_1) -> (a b))) (forall (b : α), a b)\nCase conversion may be inaccurate. Consider using '#align decidable.and_forall_ne Decidable.and_forall_neₓ'. -/\n/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (b «expr ≠ » a) -/\ntheorem Decidable.and_forall_ne [DecidableEq α] (a : α) :\n    (p a ∧ ∀ (b) (_ : b ≠ a), p b) ↔ ∀ b, p b := by\n  simp only [← @forall_eq _ p a, ← forall_and, ← or_imp, Decidable.em, forall_const]\n#align decidable.and_forall_ne Decidable.and_forall_ne\n\n/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (b «expr ≠ » a) -/\n#print and_forall_ne /-\ntheorem and_forall_ne (a : α) : (p a ∧ ∀ (b) (_ : b ≠ a), p b) ↔ ∀ b, p b :=\n  Decidable.and_forall_ne a\n#align and_forall_ne and_forall_ne\n-/\n\n#print forall_eq_or_imp /-\n-- this lemma is needed to simplify the output of `list.mem_cons_iff`\n@[simp]\ntheorem forall_eq_or_imp {a' : α} : (∀ a, a = a' ∨ q a → p a) ↔ p a' ∧ ∀ a, q a → p a := by\n  simp only [or_imp, forall_and, forall_eq]\n#align forall_eq_or_imp forall_eq_or_imp\n-/\n\n#print Ne.ne_or_ne /-\ntheorem Ne.ne_or_ne {x y : α} (z : α) (h : x ≠ y) : x ≠ z ∨ y ≠ z :=\n  not_and_or.1 <| mt (and_imp.2 Eq.substr) h.symm\n#align ne.ne_or_ne Ne.ne_or_ne\n-/\n\n#print exists_eq /-\ntheorem exists_eq {a' : α} : ∃ a, a = a' :=\n  ⟨_, rfl⟩\n#align exists_eq exists_eq\n-/\n\n#print exists_eq' /-\n@[simp]\ntheorem exists_eq' {a' : α} : ∃ a, a' = a :=\n  ⟨_, rfl⟩\n#align exists_eq' exists_eq'\n-/\n\n#print exists_unique_eq /-\n@[simp]\ntheorem exists_unique_eq {a' : α} : ∃! a, a = a' := by\n  simp only [eq_comm, ExistsUnique, and_self_iff, forall_eq', exists_eq']\n#align exists_unique_eq exists_unique_eq\n-/\n\n#print exists_unique_eq' /-\n@[simp]\ntheorem exists_unique_eq' {a' : α} : ∃! a, a' = a := by\n  simp only [ExistsUnique, and_self_iff, forall_eq', exists_eq']\n#align exists_unique_eq' exists_unique_eq'\n-/\n\n#print exists_eq_left /-\n@[simp]\ntheorem exists_eq_left {a' : α} : (∃ a, a = a' ∧ p a) ↔ p a' :=\n  ⟨fun ⟨a, e, h⟩ => e ▸ h, fun h => ⟨_, rfl, h⟩⟩\n#align exists_eq_left exists_eq_left\n-/\n\n#print exists_eq_right /-\n@[simp]\ntheorem exists_eq_right {a' : α} : (∃ a, p a ∧ a = a') ↔ p a' :=\n  (exists_congr fun a => and_comm).trans exists_eq_left\n#align exists_eq_right exists_eq_right\n-/\n\n/- warning: exists_eq_right_right -> exists_eq_right_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Prop} {q : α -> Prop} {a' : α}, Iff (Exists.{u1} α (fun (a : α) => And (p a) (And (q a) (Eq.{u1} α a a')))) (And (p a') (q a'))\nbut is expected to have type\n  forall {α : Sort.{u1}} {p : α -> Prop} {q : Prop} {a' : α}, Iff (Exists.{u1} α (fun (a : α) => And (p a) (And q (Eq.{u1} α a a')))) (And (p a') q)\nCase conversion may be inaccurate. Consider using '#align exists_eq_right_right exists_eq_right_rightₓ'. -/\n@[simp]\ntheorem exists_eq_right_right {a' : α} : (∃ a : α, p a ∧ q a ∧ a = a') ↔ p a' ∧ q a' :=\n  ⟨fun ⟨_, hp, hq, rfl⟩ => ⟨hp, hq⟩, fun ⟨hp, hq⟩ => ⟨a', hp, hq, rfl⟩⟩\n#align exists_eq_right_right exists_eq_right_right\n\n/- warning: exists_eq_right_right' -> exists_eq_right_right' is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Prop} {q : α -> Prop} {a' : α}, Iff (Exists.{u1} α (fun (a : α) => And (p a) (And (q a) (Eq.{u1} α a' a)))) (And (p a') (q a'))\nbut is expected to have type\n  forall {α : Sort.{u1}} {p : α -> Prop} {q : Prop} {a' : α}, Iff (Exists.{u1} α (fun (a : α) => And (p a) (And q (Eq.{u1} α a' a)))) (And (p a') q)\nCase conversion may be inaccurate. Consider using '#align exists_eq_right_right' exists_eq_right_right'ₓ'. -/\n@[simp]\ntheorem exists_eq_right_right' {a' : α} : (∃ a : α, p a ∧ q a ∧ a' = a) ↔ p a' ∧ q a' :=\n  ⟨fun ⟨_, hp, hq, rfl⟩ => ⟨hp, hq⟩, fun ⟨hp, hq⟩ => ⟨a', hp, hq, rfl⟩⟩\n#align exists_eq_right_right' exists_eq_right_right'\n\n#print exists_apply_eq_apply /-\n@[simp]\ntheorem exists_apply_eq_apply (f : α → β) (a' : α) : ∃ a, f a = f a' :=\n  ⟨a', rfl⟩\n#align exists_apply_eq_apply exists_apply_eq_apply\n-/\n\n#print exists_apply_eq_apply' /-\n@[simp]\ntheorem exists_apply_eq_apply' (f : α → β) (a' : α) : ∃ a, f a' = f a :=\n  ⟨a', rfl⟩\n#align exists_apply_eq_apply' exists_apply_eq_apply'\n-/\n\n#print exists_exists_and_eq_and /-\n@[simp]\ntheorem exists_exists_and_eq_and {f : α → β} {p : α → Prop} {q : β → Prop} :\n    (∃ b, (∃ a, p a ∧ f a = b) ∧ q b) ↔ ∃ a, p a ∧ q (f a) :=\n  ⟨fun ⟨b, ⟨a, ha, hab⟩, hb⟩ => ⟨a, ha, hab.symm ▸ hb⟩, fun ⟨a, hp, hq⟩ => ⟨f a, ⟨a, hp, rfl⟩, hq⟩⟩\n#align exists_exists_and_eq_and exists_exists_and_eq_and\n-/\n\n#print exists_exists_eq_and /-\n@[simp]\ntheorem exists_exists_eq_and {f : α → β} {p : β → Prop} :\n    (∃ b, (∃ a, f a = b) ∧ p b) ↔ ∃ a, p (f a) :=\n  ⟨fun ⟨b, ⟨a, ha⟩, hb⟩ => ⟨a, ha.symm ▸ hb⟩, fun ⟨a, ha⟩ => ⟨f a, ⟨a, rfl⟩, ha⟩⟩\n#align exists_exists_eq_and exists_exists_eq_and\n-/\n\n#print exists_or_eq_left /-\n@[simp]\ntheorem exists_or_eq_left (y : α) (p : α → Prop) : ∃ x : α, x = y ∨ p x :=\n  ⟨y, Or.inl rfl⟩\n#align exists_or_eq_left exists_or_eq_left\n-/\n\n#print exists_or_eq_right /-\n@[simp]\ntheorem exists_or_eq_right (y : α) (p : α → Prop) : ∃ x : α, p x ∨ x = y :=\n  ⟨y, Or.inr rfl⟩\n#align exists_or_eq_right exists_or_eq_right\n-/\n\n#print exists_or_eq_left' /-\n@[simp]\ntheorem exists_or_eq_left' (y : α) (p : α → Prop) : ∃ x : α, y = x ∨ p x :=\n  ⟨y, Or.inl rfl⟩\n#align exists_or_eq_left' exists_or_eq_left'\n-/\n\n#print exists_or_eq_right' /-\n@[simp]\ntheorem exists_or_eq_right' (y : α) (p : α → Prop) : ∃ x : α, p x ∨ y = x :=\n  ⟨y, Or.inr rfl⟩\n#align exists_or_eq_right' exists_or_eq_right'\n-/\n\n/- warning: forall_apply_eq_imp_iff -> forall_apply_eq_imp_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {p : β -> Prop}, Iff (forall (a : α) (b : β), (Eq.{u2} β (f a) b) -> (p b)) (forall (a : α), p (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {p : β -> Prop}, Iff (forall (a : α) (b : β), (Eq.{u1} β (f a) b) -> (p b)) (forall (a : α), p (f a))\nCase conversion may be inaccurate. Consider using '#align forall_apply_eq_imp_iff forall_apply_eq_imp_iffₓ'. -/\n@[simp]\ntheorem forall_apply_eq_imp_iff {f : α → β} {p : β → Prop} :\n    (∀ a, ∀ b, f a = b → p b) ↔ ∀ a, p (f a) :=\n  ⟨fun h a => h a (f a) rfl, fun h a b hab => hab ▸ h a⟩\n#align forall_apply_eq_imp_iff forall_apply_eq_imp_iff\n\n/- warning: forall_apply_eq_imp_iff' -> forall_apply_eq_imp_iff' is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {p : β -> Prop}, Iff (forall (b : β) (a : α), (Eq.{u2} β (f a) b) -> (p b)) (forall (a : α), p (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {p : β -> Prop}, Iff (forall (b : β) (a : α), (Eq.{u1} β (f a) b) -> (p b)) (forall (a : α), p (f a))\nCase conversion may be inaccurate. Consider using '#align forall_apply_eq_imp_iff' forall_apply_eq_imp_iff'ₓ'. -/\n@[simp]\ntheorem forall_apply_eq_imp_iff' {f : α → β} {p : β → Prop} :\n    (∀ b, ∀ a, f a = b → p b) ↔ ∀ a, p (f a) :=\n  by\n  rw [forall_swap]\n  simp\n#align forall_apply_eq_imp_iff' forall_apply_eq_imp_iff'\n\n/- warning: forall_eq_apply_imp_iff -> forall_eq_apply_imp_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {p : β -> Prop}, Iff (forall (a : α) (b : β), (Eq.{u2} β b (f a)) -> (p b)) (forall (a : α), p (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {p : β -> Prop}, Iff (forall (a : α) (b : β), (Eq.{u1} β b (f a)) -> (p b)) (forall (a : α), p (f a))\nCase conversion may be inaccurate. Consider using '#align forall_eq_apply_imp_iff forall_eq_apply_imp_iffₓ'. -/\n@[simp]\ntheorem forall_eq_apply_imp_iff {f : α → β} {p : β → Prop} :\n    (∀ a, ∀ b, b = f a → p b) ↔ ∀ a, p (f a) := by simp [@eq_comm _ _ (f _)]\n#align forall_eq_apply_imp_iff forall_eq_apply_imp_iff\n\n/- warning: forall_eq_apply_imp_iff' -> forall_eq_apply_imp_iff' is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {p : β -> Prop}, Iff (forall (b : β) (a : α), (Eq.{u2} β b (f a)) -> (p b)) (forall (a : α), p (f a))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {p : β -> Prop}, Iff (forall (b : β) (a : α), (Eq.{u1} β b (f a)) -> (p b)) (forall (a : α), p (f a))\nCase conversion may be inaccurate. Consider using '#align forall_eq_apply_imp_iff' forall_eq_apply_imp_iff'ₓ'. -/\n@[simp]\ntheorem forall_eq_apply_imp_iff' {f : α → β} {p : β → Prop} :\n    (∀ b, ∀ a, b = f a → p b) ↔ ∀ a, p (f a) :=\n  by\n  rw [forall_swap]\n  simp\n#align forall_eq_apply_imp_iff' forall_eq_apply_imp_iff'\n\n/- warning: forall_apply_eq_imp_iff₂ -> forall_apply_eq_imp_iff₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {f : α -> β} {p : α -> Prop} {q : β -> Prop}, Iff (forall (b : β) (a : α), (p a) -> (Eq.{u2} β (f a) b) -> (q b)) (forall (a : α), (p a) -> (q (f a)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {f : α -> β} {p : α -> Prop} {q : β -> Prop}, Iff (forall (b : β) (a : α), (p a) -> (Eq.{u1} β (f a) b) -> (q b)) (forall (a : α), (p a) -> (q (f a)))\nCase conversion may be inaccurate. Consider using '#align forall_apply_eq_imp_iff₂ forall_apply_eq_imp_iff₂ₓ'. -/\n@[simp]\ntheorem forall_apply_eq_imp_iff₂ {f : α → β} {p : α → Prop} {q : β → Prop} :\n    (∀ b, ∀ a, p a → f a = b → q b) ↔ ∀ a, p a → q (f a) :=\n  ⟨fun h a ha => h (f a) a ha rfl, fun h b a ha hb => hb ▸ h a ha⟩\n#align forall_apply_eq_imp_iff₂ forall_apply_eq_imp_iff₂\n\n#print exists_eq_left' /-\n@[simp]\ntheorem exists_eq_left' {a' : α} : (∃ a, a' = a ∧ p a) ↔ p a' := by simp [@eq_comm _ a']\n#align exists_eq_left' exists_eq_left'\n-/\n\n#print exists_eq_right' /-\n@[simp]\ntheorem exists_eq_right' {a' : α} : (∃ a, p a ∧ a' = a) ↔ p a' := by simp [@eq_comm _ a']\n#align exists_eq_right' exists_eq_right'\n-/\n\n#print exists_comm /-\ntheorem exists_comm {p : α → β → Prop} : (∃ a b, p a b) ↔ ∃ b a, p a b :=\n  ⟨fun ⟨a, b, h⟩ => ⟨b, a, h⟩, fun ⟨b, a, h⟩ => ⟨a, b, h⟩⟩\n#align exists_comm exists_comm\n-/\n\n/- warning: exists₂_comm -> exists₂_comm is a dubious translation:\nlean 3 declaration is\n  forall {ι₁ : Sort.{u1}} {ι₂ : Sort.{u2}} {κ₁ : ι₁ -> Sort.{u3}} {κ₂ : ι₂ -> Sort.{u4}} {p : forall (i₁ : ι₁), (κ₁ i₁) -> (forall (i₂ : ι₂), (κ₂ i₂) -> Prop)}, Iff (Exists.{u1} ι₁ (fun (i₁ : ι₁) => Exists.{u3} (κ₁ i₁) (fun (j₁ : κ₁ i₁) => Exists.{u2} ι₂ (fun (i₂ : ι₂) => Exists.{u4} (κ₂ i₂) (fun (j₂ : κ₂ i₂) => p i₁ j₁ i₂ j₂))))) (Exists.{u2} ι₂ (fun (i₂ : ι₂) => Exists.{u4} (κ₂ i₂) (fun (j₂ : κ₂ i₂) => Exists.{u1} ι₁ (fun (i₁ : ι₁) => Exists.{u3} (κ₁ i₁) (fun (j₁ : κ₁ i₁) => p i₁ j₁ i₂ j₂)))))\nbut is expected to have type\n  forall {ι₁ : Sort.{u4}} {ι₂ : Sort.{u3}} {κ₁ : ι₁ -> Sort.{u2}} {κ₂ : ι₂ -> Sort.{u1}} {p : forall (i₁ : ι₁), (κ₁ i₁) -> (forall (i₂ : ι₂), (κ₂ i₂) -> Prop)}, Iff (Exists.{u4} ι₁ (fun (i₁ : ι₁) => Exists.{u2} (κ₁ i₁) (fun (j₁ : κ₁ i₁) => Exists.{u3} ι₂ (fun (i₂ : ι₂) => Exists.{u1} (κ₂ i₂) (fun (j₂ : κ₂ i₂) => p i₁ j₁ i₂ j₂))))) (Exists.{u3} ι₂ (fun (i₂ : ι₂) => Exists.{u1} (κ₂ i₂) (fun (j₂ : κ₂ i₂) => Exists.{u4} ι₁ (fun (i₁ : ι₁) => Exists.{u2} (κ₁ i₁) (fun (j₁ : κ₁ i₁) => p i₁ j₁ i₂ j₂)))))\nCase conversion may be inaccurate. Consider using '#align exists₂_comm exists₂_commₓ'. -/\ntheorem exists₂_comm {ι₁ ι₂ : Sort _} {κ₁ : ι₁ → Sort _} {κ₂ : ι₂ → Sort _}\n    {p : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Prop} :\n    (∃ i₁ j₁ i₂ j₂, p i₁ j₁ i₂ j₂) ↔ ∃ i₂ j₂ i₁ j₁, p i₁ j₁ i₂ j₂ := by\n  simp only [@exists_comm (κ₁ _), @exists_comm ι₁]\n#align exists₂_comm exists₂_comm\n\n#print And.exists /-\ntheorem And.exists {p q : Prop} {f : p ∧ q → Prop} : (∃ h, f h) ↔ ∃ hp hq, f ⟨hp, hq⟩ :=\n  ⟨fun ⟨h, H⟩ => ⟨h.1, h.2, H⟩, fun ⟨hp, hq, H⟩ => ⟨⟨hp, hq⟩, H⟩⟩\n#align and.exists And.exists\n-/\n\n#print forall_or_of_or_forall /-\ntheorem forall_or_of_or_forall (h : b ∨ ∀ x, p x) (x) : b ∨ p x :=\n  h.imp_right fun h₂ => h₂ x\n#align forall_or_of_or_forall forall_or_of_or_forall\n-/\n\n#print Decidable.forall_or_left /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.forall_or_left {q : Prop} {p : α → Prop} [Decidable q] :\n    (∀ x, q ∨ p x) ↔ q ∨ ∀ x, p x :=\n  ⟨fun h => if hq : q then Or.inl hq else Or.inr fun x => (h x).resolve_left hq,\n    forall_or_of_or_forall⟩\n#align decidable.forall_or_distrib_left Decidable.forall_or_left\n-/\n\n#print forall_or_left /-\ntheorem forall_or_left {q : Prop} {p : α → Prop} : (∀ x, q ∨ p x) ↔ q ∨ ∀ x, p x :=\n  Decidable.forall_or_left\n#align forall_or_distrib_left forall_or_left\n-/\n\n#print Decidable.forall_or_right /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.forall_or_right {q : Prop} {p : α → Prop} [Decidable q] :\n    (∀ x, p x ∨ q) ↔ (∀ x, p x) ∨ q := by simp [or_comm', Decidable.forall_or_left]\n#align decidable.forall_or_distrib_right Decidable.forall_or_right\n-/\n\n#print forall_or_right /-\ntheorem forall_or_right {q : Prop} {p : α → Prop} : (∀ x, p x ∨ q) ↔ (∀ x, p x) ∨ q :=\n  Decidable.forall_or_right\n#align forall_or_distrib_right forall_or_right\n-/\n\n/- warning: exists_prop -> exists_prop is a dubious translation:\nlean 3 declaration is\n  forall {p : Prop} {q : Prop}, Iff (Exists.{0} p (fun (h : p) => q)) (And p q)\nbut is expected to have type\n  forall {p : Prop} {q : Prop}, Iff (Exists.{0} q (fun (h : q) => p)) (And q p)\nCase conversion may be inaccurate. Consider using '#align exists_prop exists_propₓ'. -/\n@[simp]\ntheorem exists_prop {p q : Prop} : (∃ h : p, q) ↔ p ∧ q :=\n  ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, h₂⟩, fun ⟨h₁, h₂⟩ => ⟨h₁, h₂⟩⟩\n#align exists_prop exists_prop\n\n#print exists_unique_prop /-\ntheorem exists_unique_prop {p q : Prop} : (∃! h : p, q) ↔ p ∧ q := by simp\n#align exists_unique_prop exists_unique_prop\n-/\n\n#print exists_false /-\n@[simp]\ntheorem exists_false : ¬∃ a : α, False := fun ⟨a, h⟩ => h\n#align exists_false exists_false\n-/\n\n#print exists_unique_false /-\n@[simp]\ntheorem exists_unique_false : ¬∃! a : α, False := fun ⟨a, h, h'⟩ => h\n#align exists_unique_false exists_unique_false\n-/\n\n#print Exists.fst /-\ntheorem Exists.fst {p : b → Prop} : Exists p → b\n  | ⟨h, _⟩ => h\n#align Exists.fst Exists.fst\n-/\n\n#print Exists.snd /-\ntheorem Exists.snd {p : b → Prop} : ∀ h : Exists p, p h.fst\n  | ⟨_, h⟩ => h\n#align Exists.snd Exists.snd\n-/\n\n#print forall_prop_of_true /-\ntheorem forall_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∀ h' : p, q h') ↔ q h :=\n  @forall_const (q h) p ⟨h⟩\n#align forall_prop_of_true forall_prop_of_true\n-/\n\n#print exists_prop_of_true /-\ntheorem exists_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃ h' : p, q h') ↔ q h :=\n  @exists_const (q h) p ⟨h⟩\n#align exists_prop_of_true exists_prop_of_true\n-/\n\n#print exists_iff_of_forall /-\ntheorem exists_iff_of_forall {p : Prop} {q : p → Prop} (h : ∀ h, q h) : (∃ h, q h) ↔ p :=\n  ⟨Exists.fst, fun H => ⟨H, h H⟩⟩\n#align exists_iff_of_forall exists_iff_of_forall\n-/\n\n#print exists_unique_prop_of_true /-\ntheorem exists_unique_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃! h' : p, q h') ↔ q h :=\n  @exists_unique_const (q h) p ⟨h⟩ _\n#align exists_unique_prop_of_true exists_unique_prop_of_true\n-/\n\n#print forall_prop_of_false /-\ntheorem forall_prop_of_false {p : Prop} {q : p → Prop} (hn : ¬p) : (∀ h' : p, q h') ↔ True :=\n  iff_true_intro fun h => hn.elim h\n#align forall_prop_of_false forall_prop_of_false\n-/\n\n#print exists_prop_of_false /-\ntheorem exists_prop_of_false {p : Prop} {q : p → Prop} : ¬p → ¬∃ h' : p, q h' :=\n  mt Exists.fst\n#align exists_prop_of_false exists_prop_of_false\n-/\n\n#print exists_prop_congr /-\n@[congr]\ntheorem exists_prop_congr {p p' : Prop} {q q' : p → Prop} (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') :\n    Exists q ↔ ∃ h : p', q' (hp.2 h) :=\n  ⟨fun ⟨_, _⟩ => ⟨hp.1 ‹_›, (hq _).1 ‹_›⟩, fun ⟨_, _⟩ => ⟨_, (hq _).2 ‹_›⟩⟩\n#align exists_prop_congr exists_prop_congr\n-/\n\n#print exists_prop_congr' /-\n@[congr]\ntheorem exists_prop_congr' {p p' : Prop} {q q' : p → Prop} (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') :\n    Exists q = ∃ h : p', q' (hp.2 h) :=\n  propext (exists_prop_congr hq _)\n#align exists_prop_congr' exists_prop_congr'\n-/\n\n#print exists_true_left /-\n/-- See `is_empty.exists_iff` for the `false` version. -/\n@[simp]\ntheorem exists_true_left (p : True → Prop) : (∃ x, p x) ↔ p True.intro :=\n  exists_prop_of_true _\n#align exists_true_left exists_true_left\n-/\n\n/- warning: exists_unique.unique clashes with unique_of_exists_unique -> ExistsUnique.unique\nCase conversion may be inaccurate. Consider using '#align exists_unique.unique ExistsUnique.uniqueₓ'. -/\n#print ExistsUnique.unique /-\ntheorem ExistsUnique.unique {α : Sort _} {p : α → Prop} (h : ∃! x, p x) {y₁ y₂ : α} (py₁ : p y₁)\n    (py₂ : p y₂) : y₁ = y₂ :=\n  ExistsUnique.unique h py₁ py₂\n#align exists_unique.unique ExistsUnique.unique\n-/\n\n#print forall_prop_congr /-\n@[congr]\ntheorem forall_prop_congr {p p' : Prop} {q q' : p → Prop} (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') :\n    (∀ h, q h) ↔ ∀ h : p', q' (hp.2 h) :=\n  ⟨fun h1 h2 => (hq _).1 (h1 (hp.2 _)), fun h1 h2 => (hq _).2 (h1 (hp.1 h2))⟩\n#align forall_prop_congr forall_prop_congr\n-/\n\n#print forall_prop_congr' /-\n@[congr]\ntheorem forall_prop_congr' {p p' : Prop} {q q' : p → Prop} (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') :\n    (∀ h, q h) = ∀ h : p', q' (hp.2 h) :=\n  propext (forall_prop_congr hq _)\n#align forall_prop_congr' forall_prop_congr'\n-/\n\n#print forall_true_left /-\n/-- See `is_empty.forall_iff` for the `false` version. -/\n@[simp]\ntheorem forall_true_left (p : True → Prop) : (∀ x, p x) ↔ p True.intro :=\n  forall_prop_of_true _\n#align forall_true_left forall_true_left\n-/\n\n/- warning: exists_unique.elim2 -> ExistsUnique.elim₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Sort.{u2}} [_inst_1 : forall (x : α), Subsingleton.{u2} (p x)] {q : forall (x : α), (p x) -> Prop} {b : Prop}, (ExistsUnique.{u1} α (fun (x : α) => ExistsUnique.{u2} (p x) (fun (h : p x) => q x h))) -> (forall (x : α) (h : p x), (q x h) -> (forall (y : α) (hy : p y), (q y hy) -> (Eq.{u1} α y x)) -> b) -> b\nbut is expected to have type\n  forall {α : Sort.{u2}} {p : α -> Sort.{u1}} [_inst_1 : forall (x : α), Subsingleton.{u1} (p x)] {q : forall (x : α), (p x) -> Prop} {b : Prop}, (ExistsUnique.{u2} α (fun (x : α) => ExistsUnique.{u1} (p x) (fun (h : p x) => q x h))) -> (forall (x : α) (h : p x), (q x h) -> (forall (y : α) (hy : p y), (q y hy) -> (Eq.{u2} α y x)) -> b) -> b\nCase conversion may be inaccurate. Consider using '#align exists_unique.elim2 ExistsUnique.elim₂ₓ'. -/\ntheorem ExistsUnique.elim₂ {α : Sort _} {p : α → Sort _} [∀ x, Subsingleton (p x)]\n    {q : ∀ (x) (h : p x), Prop} {b : Prop} (h₂ : ∃! (x : _)(h : p x), q x h)\n    (h₁ : ∀ (x) (h : p x), q x h → (∀ (y) (hy : p y), q y hy → y = x) → b) : b :=\n  by\n  simp only [exists_unique_iff_exists] at h₂\n  apply h₂.elim\n  exact fun x ⟨hxp, hxq⟩ H => h₁ x hxp hxq fun y hyp hyq => H y ⟨hyp, hyq⟩\n#align exists_unique.elim2 ExistsUnique.elim₂\n\n/- warning: exists_unique.intro2 -> ExistsUnique.intro₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Sort.{u2}} [_inst_1 : forall (x : α), Subsingleton.{u2} (p x)] {q : forall (x : α), (p x) -> Prop} (w : α) (hp : p w), (q w hp) -> (forall (y : α) (hy : p y), (q y hy) -> (Eq.{u1} α y w)) -> (ExistsUnique.{u1} α (fun (x : α) => ExistsUnique.{u2} (p x) (fun (hx : p x) => q x hx)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {p : α -> Sort.{u1}} [_inst_1 : forall (x : α), Subsingleton.{u1} (p x)] {q : forall (x : α), (p x) -> Prop} (w : α) (hp : p w), (q w hp) -> (forall (y : α) (hy : p y), (q y hy) -> (Eq.{u2} α y w)) -> (ExistsUnique.{u2} α (fun (x : α) => ExistsUnique.{u1} (p x) (fun (hx : p x) => q x hx)))\nCase conversion may be inaccurate. Consider using '#align exists_unique.intro2 ExistsUnique.intro₂ₓ'. -/\ntheorem ExistsUnique.intro₂ {α : Sort _} {p : α → Sort _} [∀ x, Subsingleton (p x)]\n    {q : ∀ (x : α) (h : p x), Prop} (w : α) (hp : p w) (hq : q w hp)\n    (H : ∀ (y) (hy : p y), q y hy → y = w) : ∃! (x : _)(hx : p x), q x hx :=\n  by\n  simp only [exists_unique_iff_exists]\n  exact ExistsUnique.intro w ⟨hp, hq⟩ fun y ⟨hyp, hyq⟩ => H y hyp hyq\n#align exists_unique.intro2 ExistsUnique.intro₂\n\n/- warning: exists_unique.exists2 -> ExistsUnique.exists₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Sort.{u2}} {q : forall (x : α), (p x) -> Prop}, (ExistsUnique.{u1} α (fun (x : α) => ExistsUnique.{u2} (p x) (fun (hx : p x) => q x hx))) -> (Exists.{u1} α (fun (x : α) => Exists.{u2} (p x) (fun (hx : p x) => q x hx)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {p : α -> Sort.{u1}} {q : forall (x : α), (p x) -> Prop}, (ExistsUnique.{u2} α (fun (x : α) => ExistsUnique.{u1} (p x) (fun (hx : p x) => q x hx))) -> (Exists.{u2} α (fun (x : α) => Exists.{u1} (p x) (fun (hx : p x) => q x hx)))\nCase conversion may be inaccurate. Consider using '#align exists_unique.exists2 ExistsUnique.exists₂ₓ'. -/\ntheorem ExistsUnique.exists₂ {α : Sort _} {p : α → Sort _} {q : ∀ (x : α) (h : p x), Prop}\n    (h : ∃! (x : _)(hx : p x), q x hx) : ∃ (x : _)(hx : p x), q x hx :=\n  h.exists.imp fun x hx => hx.exists\n#align exists_unique.exists2 ExistsUnique.exists₂\n\n/- warning: exists_unique.unique2 -> ExistsUnique.unique₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {p : α -> Sort.{u2}} [_inst_1 : forall (x : α), Subsingleton.{u2} (p x)] {q : forall (x : α), (p x) -> Prop}, (ExistsUnique.{u1} α (fun (x : α) => ExistsUnique.{u2} (p x) (fun (hx : p x) => q x hx))) -> (forall {y₁ : α} {y₂ : α} (hpy₁ : p y₁), (q y₁ hpy₁) -> (forall (hpy₂ : p y₂), (q y₂ hpy₂) -> (Eq.{u1} α y₁ y₂)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {p : α -> Sort.{u1}} [_inst_1 : forall (x : α), Subsingleton.{u1} (p x)] {q : forall (x : α), (p x) -> Prop}, (ExistsUnique.{u2} α (fun (x : α) => ExistsUnique.{u1} (p x) (fun (hx : p x) => q x hx))) -> (forall {y₁ : α} {y₂ : α} (hpy₁ : p y₁), (q y₁ hpy₁) -> (forall (hpy₂ : p y₂), (q y₂ hpy₂) -> (Eq.{u2} α y₁ y₂)))\nCase conversion may be inaccurate. Consider using '#align exists_unique.unique2 ExistsUnique.unique₂ₓ'. -/\ntheorem ExistsUnique.unique₂ {α : Sort _} {p : α → Sort _} [∀ x, Subsingleton (p x)]\n    {q : ∀ (x : α) (hx : p x), Prop} (h : ∃! (x : _)(hx : p x), q x hx) {y₁ y₂ : α} (hpy₁ : p y₁)\n    (hqy₁ : q y₁ hpy₁) (hpy₂ : p y₂) (hqy₂ : q y₂ hpy₂) : y₁ = y₂ :=\n  by\n  simp only [exists_unique_iff_exists] at h\n  exact h.unique ⟨hpy₁, hqy₁⟩ ⟨hpy₂, hqy₂⟩\n#align exists_unique.unique2 ExistsUnique.unique₂\n\nend Quantifiers\n\n/-! ### Classical lemmas -/\n\n\nnamespace Classical\n\nvariable {α : Sort _} {p : α → Prop}\n\n#print Classical.cases /-\ntheorem cases {p : Prop → Prop} (h1 : p True) (h2 : p False) : ∀ a, p a := fun a => cases_on a h1 h2\n#align classical.cases Classical.cases\n-/\n\n#print Classical.dec /-\n-- use shortened names to avoid conflict when classical namespace is open.\n/-- Any prop `p` is decidable classically. A shorthand for `classical.prop_decidable`. -/\nnoncomputable def dec (p : Prop) : Decidable p := by infer_instance\n#align classical.dec Classical.dec\n-/\n\n#print Classical.decPred /-\n/-- Any predicate `p` is decidable classically. -/\nnoncomputable def decPred (p : α → Prop) : DecidablePred p := by infer_instance\n#align classical.dec_pred Classical.decPred\n-/\n\n#print Classical.decRel /-\n/-- Any relation `p` is decidable classically. -/\nnoncomputable def decRel (p : α → α → Prop) : DecidableRel p := by infer_instance\n#align classical.dec_rel Classical.decRel\n-/\n\n#print Classical.decEq /-\n/-- Any type `α` has decidable equality classically. -/\nnoncomputable def decEq (α : Sort _) : DecidableEq α := by infer_instance\n#align classical.dec_eq Classical.decEq\n-/\n\n/- warning: classical.exists_cases -> Classical.existsCases is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u_1}} {p : α -> Prop} {C : Sort.{u}}, C -> (forall (a : α), (p a) -> C) -> C\nbut is expected to have type\n  forall {α : Prop} {p : α -> Prop} {C : Sort.{u_1}}, C -> (forall (a : α), (p a) -> C) -> C\nCase conversion may be inaccurate. Consider using '#align classical.exists_cases Classical.existsCasesₓ'. -/\n/-- Construct a function from a default value `H0`, and a function to use if there exists a value\nsatisfying the predicate. -/\n@[elab_as_elim]\nnoncomputable def existsCases.{u} {C : Sort u} (H0 : C) (H : ∀ a, p a → C) : C :=\n  if h : ∃ a, p a then H (Classical.choose h) (Classical.choose_spec h) else H0\n#align classical.exists_cases Classical.existsCases\n\n#print Classical.some_spec₂ /-\ntheorem some_spec₂ {α : Sort _} {p : α → Prop} {h : ∃ a, p a} (q : α → Prop)\n    (hpq : ∀ a, p a → q a) : q (choose h) :=\n  hpq _ <| choose_spec _\n#align classical.some_spec2 Classical.some_spec₂\n-/\n\n#print Classical.subtype_of_exists /-\n/-- A version of classical.indefinite_description which is definitionally equal to a pair -/\nnoncomputable def subtype_of_exists {α : Type _} {P : α → Prop} (h : ∃ x, P x) : { x // P x } :=\n  ⟨Classical.choose h, Classical.choose_spec h⟩\n#align classical.subtype_of_exists Classical.subtype_of_exists\n-/\n\n#print Classical.byContradiction' /-\n/-- A version of `by_contradiction` that uses types instead of propositions. -/\nprotected noncomputable def byContradiction' {α : Sort _} (H : ¬(α → False)) : α :=\n  Classical.choice <| peirce _ False fun h => (H fun a => h ⟨a⟩).elim\n#align classical.by_contradiction' Classical.byContradiction'\n-/\n\n#print Classical.choice_of_byContradiction' /-\n/-- `classical.by_contradiction'` is equivalent to lean's axiom `classical.choice`. -/\ndef choice_of_byContradiction' {α : Sort _} (contra : ¬(α → False) → α) : Nonempty α → α := fun H =>\n  contra H.elim\n#align classical.choice_of_by_contradiction' Classical.choice_of_byContradiction'\n-/\n\nend Classical\n\n/- warning: exists.classical_rec_on -> Exists.classicalRecOn is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u2}} {p : α -> Prop}, (Exists.{u2} α (fun (a : α) => p a)) -> (forall {C : Sort.{u1}}, (forall (a : α), (p a) -> C) -> C)\nbut is expected to have type\n  forall {α : Sort.{u1}} {p : α -> Prop}, (Exists.{u1} α (fun (a : α) => p a)) -> (forall {C : Sort.{u2}}, (forall (a : α), (p a) -> C) -> C)\nCase conversion may be inaccurate. Consider using '#align exists.classical_rec_on Exists.classicalRecOnₓ'. -/\n/-- This function has the same type as `exists.rec_on`, and can be used to case on an equality,\nbut `exists.rec_on` can only eliminate into Prop, while this version eliminates into any universe\nusing the axiom of choice. -/\n@[elab_as_elim]\nnoncomputable def Exists.classicalRecOn.{u} {α} {p : α → Prop} (h : ∃ a, p a) {C : Sort u}\n    (H : ∀ a, p a → C) : C :=\n  H (Classical.choose h) (Classical.choose_spec h)\n#align exists.classical_rec_on Exists.classicalRecOn\n\n/-! ### Declarations about bounded quantifiers -/\n\n\nsection BoundedQuantifiers\n\nvariable {α : Sort _} {r p q : α → Prop} {P Q : ∀ x, p x → Prop} {b : Prop}\n\n#print bex_def /-\ntheorem bex_def : (∃ (x : _)(h : p x), q x) ↔ ∃ x, p x ∧ q x :=\n  ⟨fun ⟨x, px, qx⟩ => ⟨x, px, qx⟩, fun ⟨x, px, qx⟩ => ⟨x, px, qx⟩⟩\n#align bex_def bex_def\n-/\n\n#print BEx.elim /-\ntheorem BEx.elim {b : Prop} : (∃ x h, P x h) → (∀ a h, P a h → b) → b\n  | ⟨a, h₁, h₂⟩, h' => h' a h₁ h₂\n#align bex.elim BEx.elim\n-/\n\n#print BEx.intro /-\ntheorem BEx.intro (a : α) (h₁ : p a) (h₂ : P a h₁) : ∃ (x : _)(h : p x), P x h :=\n  ⟨a, h₁, h₂⟩\n#align bex.intro BEx.intro\n-/\n\n#print ball_congr /-\ntheorem ball_congr (H : ∀ x h, P x h ↔ Q x h) : (∀ x h, P x h) ↔ ∀ x h, Q x h :=\n  forall_congr' fun x => forall_congr' (H x)\n#align ball_congr ball_congr\n-/\n\n#print bex_congr /-\ntheorem bex_congr (H : ∀ x h, P x h ↔ Q x h) : (∃ x h, P x h) ↔ ∃ x h, Q x h :=\n  exists_congr fun x => exists_congr (H x)\n#align bex_congr bex_congr\n-/\n\n#print bex_eq_left /-\ntheorem bex_eq_left {a : α} : (∃ (x : _)(_ : x = a), p x) ↔ p a := by\n  simp only [exists_prop, exists_eq_left]\n#align bex_eq_left bex_eq_left\n-/\n\n#print BAll.imp_right /-\ntheorem BAll.imp_right (H : ∀ x h, P x h → Q x h) (h₁ : ∀ x h, P x h) (x h) : Q x h :=\n  H _ _ <| h₁ _ _\n#align ball.imp_right BAll.imp_right\n-/\n\n#print BEx.imp_right /-\ntheorem BEx.imp_right (H : ∀ x h, P x h → Q x h) : (∃ x h, P x h) → ∃ x h, Q x h\n  | ⟨x, h, h'⟩ => ⟨_, _, H _ _ h'⟩\n#align bex.imp_right BEx.imp_right\n-/\n\n#print BAll.imp_left /-\ntheorem BAll.imp_left (H : ∀ x, p x → q x) (h₁ : ∀ x, q x → r x) (x) (h : p x) : r x :=\n  h₁ _ <| H _ h\n#align ball.imp_left BAll.imp_left\n-/\n\n#print BEx.imp_left /-\ntheorem BEx.imp_left (H : ∀ x, p x → q x) : (∃ (x : _)(_ : p x), r x) → ∃ (x : _)(_ : q x), r x\n  | ⟨x, hp, hr⟩ => ⟨x, H _ hp, hr⟩\n#align bex.imp_left BEx.imp_left\n-/\n\n#print ball_of_forall /-\ntheorem ball_of_forall (h : ∀ x, p x) (x) : p x :=\n  h x\n#align ball_of_forall ball_of_forall\n-/\n\n#print forall_of_ball /-\ntheorem forall_of_ball (H : ∀ x, p x) (h : ∀ x, p x → q x) (x) : q x :=\n  h x <| H x\n#align forall_of_ball forall_of_ball\n-/\n\n#print bex_of_exists /-\ntheorem bex_of_exists (H : ∀ x, p x) : (∃ x, q x) → ∃ (x : _)(_ : p x), q x\n  | ⟨x, hq⟩ => ⟨x, H x, hq⟩\n#align bex_of_exists bex_of_exists\n-/\n\n#print exists_of_bex /-\ntheorem exists_of_bex : (∃ (x : _)(_ : p x), q x) → ∃ x, q x\n  | ⟨x, _, hq⟩ => ⟨x, hq⟩\n#align exists_of_bex exists_of_bex\n-/\n\n#print bex_imp /-\n@[simp]\ntheorem bex_imp : (∃ x h, P x h) → b ↔ ∀ x h, P x h → b := by simp\n#align bex_imp_distrib bex_imp\n-/\n\n#print not_bex /-\ntheorem not_bex : (¬∃ x h, P x h) ↔ ∀ x h, ¬P x h :=\n  bex_imp\n#align not_bex not_bex\n-/\n\n#print not_ball_of_bex_not /-\ntheorem not_ball_of_bex_not : (∃ x h, ¬P x h) → ¬∀ x h, P x h\n  | ⟨x, h, hp⟩, al => hp <| al x h\n#align not_ball_of_bex_not not_ball_of_bex_not\n-/\n\n#print Decidable.not_ball /-\n-- See Note [decidable namespace]\nprotected theorem Decidable.not_ball [Decidable (∃ x h, ¬P x h)] [∀ x h, Decidable (P x h)] :\n    (¬∀ x h, P x h) ↔ ∃ x h, ¬P x h :=\n  ⟨Not.decidable_imp_symm fun nx x h => nx.decidable_imp_symm fun h' => ⟨x, h, h'⟩,\n    not_ball_of_bex_not⟩\n#align decidable.not_ball Decidable.not_ball\n-/\n\n#print not_ball /-\ntheorem not_ball : (¬∀ x h, P x h) ↔ ∃ x h, ¬P x h :=\n  Decidable.not_ball\n#align not_ball not_ball\n-/\n\n#print ball_true_iff /-\ntheorem ball_true_iff (p : α → Prop) : (∀ x, p x → True) ↔ True :=\n  iff_true_intro fun h hrx => trivial\n#align ball_true_iff ball_true_iff\n-/\n\n#print ball_and /-\ntheorem ball_and : (∀ x h, P x h ∧ Q x h) ↔ (∀ x h, P x h) ∧ ∀ x h, Q x h :=\n  Iff.trans (forall_congr' fun x => forall_and) forall_and\n#align ball_and_distrib ball_and\n-/\n\n#print bex_or /-\ntheorem bex_or : (∃ x h, P x h ∨ Q x h) ↔ (∃ x h, P x h) ∨ ∃ x h, Q x h :=\n  Iff.trans (exists_congr fun x => exists_or) exists_or\n#align bex_or_distrib bex_or\n-/\n\n#print ball_or_left /-\ntheorem ball_or_left : (∀ x, p x ∨ q x → r x) ↔ (∀ x, p x → r x) ∧ ∀ x, q x → r x :=\n  Iff.trans (forall_congr' fun x => or_imp) forall_and\n#align ball_or_left_distrib ball_or_left\n-/\n\n#print bex_or_left /-\ntheorem bex_or_left :\n    (∃ (x : _)(_ : p x ∨ q x), r x) ↔ (∃ (x : _)(_ : p x), r x) ∨ ∃ (x : _)(_ : q x), r x := by\n  simp only [exists_prop] <;> exact Iff.trans (exists_congr fun x => or_and_right) exists_or\n#align bex_or_left_distrib bex_or_left\n-/\n\nend BoundedQuantifiers\n\nnamespace Classical\n\nattribute [local instance] prop_decidable\n\n/- warning: classical.not_ball clashes with not_ball -> not_ball\nCase conversion may be inaccurate. Consider using '#align classical.not_ball not_ballₓ'. -/\n#print not_ball /-\ntheorem not_ball {α : Sort _} {p : α → Prop} {P : ∀ x : α, p x → Prop} :\n    (¬∀ x h, P x h) ↔ ∃ x h, ¬P x h :=\n  not_ball\n#align classical.not_ball not_ball\n-/\n\nend Classical\n\nsection ite\n\nvariable {α β γ : Sort _} {σ : α → Sort _} (f : α → β) {P Q : Prop} [Decidable P] [Decidable Q]\n  {a b c : α} {A : P → α} {B : ¬P → α}\n\n#print dite_eq_iff /-\ntheorem dite_eq_iff : dite P A B = c ↔ (∃ h, A h = c) ∨ ∃ h, B h = c := by\n  by_cases P <;> simp [*, exists_prop_of_false not_false]\n#align dite_eq_iff dite_eq_iff\n-/\n\n#print ite_eq_iff /-\ntheorem ite_eq_iff : ite P a b = c ↔ P ∧ a = c ∨ ¬P ∧ b = c :=\n  dite_eq_iff.trans <| by rw [exists_prop, exists_prop]\n#align ite_eq_iff ite_eq_iff\n-/\n\n#print dite_eq_iff' /-\ntheorem dite_eq_iff' : dite P A B = c ↔ (∀ h, A h = c) ∧ ∀ h, B h = c :=\n  ⟨fun he => ⟨fun h => (dif_pos h).symm.trans he, fun h => (dif_neg h).symm.trans he⟩, fun he =>\n    (em P).elim (fun h => (dif_pos h).trans <| he.1 h) fun h => (dif_neg h).trans <| he.2 h⟩\n#align dite_eq_iff' dite_eq_iff'\n-/\n\n#print ite_eq_iff' /-\ntheorem ite_eq_iff' : ite P a b = c ↔ (P → a = c) ∧ (¬P → b = c) :=\n  dite_eq_iff'\n#align ite_eq_iff' ite_eq_iff'\n-/\n\n#print dite_eq_left_iff /-\n@[simp]\ntheorem dite_eq_left_iff : dite P (fun _ => a) B = a ↔ ∀ h, B h = a := by\n  by_cases P <;> simp [*, forall_prop_of_false not_false]\n#align dite_eq_left_iff dite_eq_left_iff\n-/\n\n#print dite_eq_right_iff /-\n@[simp]\ntheorem dite_eq_right_iff : (dite P A fun _ => b) = b ↔ ∀ h, A h = b := by\n  by_cases P <;> simp [*, forall_prop_of_false not_false]\n#align dite_eq_right_iff dite_eq_right_iff\n-/\n\n#print ite_eq_left_iff /-\n@[simp]\ntheorem ite_eq_left_iff : ite P a b = a ↔ ¬P → b = a :=\n  dite_eq_left_iff\n#align ite_eq_left_iff ite_eq_left_iff\n-/\n\n#print ite_eq_right_iff /-\n@[simp]\ntheorem ite_eq_right_iff : ite P a b = b ↔ P → a = b :=\n  dite_eq_right_iff\n#align ite_eq_right_iff ite_eq_right_iff\n-/\n\n#print dite_ne_left_iff /-\ntheorem dite_ne_left_iff : dite P (fun _ => a) B ≠ a ↔ ∃ h, a ≠ B h :=\n  by\n  rw [Ne.def, dite_eq_left_iff, not_forall]\n  exact exists_congr fun h => by rw [ne_comm]\n#align dite_ne_left_iff dite_ne_left_iff\n-/\n\n#print dite_ne_right_iff /-\ntheorem dite_ne_right_iff : (dite P A fun _ => b) ≠ b ↔ ∃ h, A h ≠ b := by\n  simp only [Ne.def, dite_eq_right_iff, not_forall]\n#align dite_ne_right_iff dite_ne_right_iff\n-/\n\n#print ite_ne_left_iff /-\ntheorem ite_ne_left_iff : ite P a b ≠ a ↔ ¬P ∧ a ≠ b :=\n  dite_ne_left_iff.trans <| by rw [exists_prop]\n#align ite_ne_left_iff ite_ne_left_iff\n-/\n\n#print ite_ne_right_iff /-\ntheorem ite_ne_right_iff : ite P a b ≠ b ↔ P ∧ a ≠ b :=\n  dite_ne_right_iff.trans <| by rw [exists_prop]\n#align ite_ne_right_iff ite_ne_right_iff\n-/\n\n#print Ne.dite_eq_left_iff /-\nprotected theorem Ne.dite_eq_left_iff (h : ∀ h, a ≠ B h) : dite P (fun _ => a) B = a ↔ P :=\n  dite_eq_left_iff.trans <| ⟨fun H => of_not_not fun h' => h h' (H h').symm, fun h H => (H h).elim⟩\n#align ne.dite_eq_left_iff Ne.dite_eq_left_iff\n-/\n\n#print Ne.dite_eq_right_iff /-\nprotected theorem Ne.dite_eq_right_iff (h : ∀ h, A h ≠ b) : (dite P A fun _ => b) = b ↔ ¬P :=\n  dite_eq_right_iff.trans <| ⟨fun H h' => h h' (H h'), fun h' H => (h' H).elim⟩\n#align ne.dite_eq_right_iff Ne.dite_eq_right_iff\n-/\n\n#print Ne.ite_eq_left_iff /-\nprotected theorem Ne.ite_eq_left_iff (h : a ≠ b) : ite P a b = a ↔ P :=\n  Ne.dite_eq_left_iff fun _ => h\n#align ne.ite_eq_left_iff Ne.ite_eq_left_iff\n-/\n\n#print Ne.ite_eq_right_iff /-\nprotected theorem Ne.ite_eq_right_iff (h : a ≠ b) : ite P a b = b ↔ ¬P :=\n  Ne.dite_eq_right_iff fun _ => h\n#align ne.ite_eq_right_iff Ne.ite_eq_right_iff\n-/\n\n#print Ne.dite_ne_left_iff /-\nprotected theorem Ne.dite_ne_left_iff (h : ∀ h, a ≠ B h) : dite P (fun _ => a) B ≠ a ↔ ¬P :=\n  dite_ne_left_iff.trans <| exists_iff_of_forall h\n#align ne.dite_ne_left_iff Ne.dite_ne_left_iff\n-/\n\n#print Ne.dite_ne_right_iff /-\nprotected theorem Ne.dite_ne_right_iff (h : ∀ h, A h ≠ b) : (dite P A fun _ => b) ≠ b ↔ P :=\n  dite_ne_right_iff.trans <| exists_iff_of_forall h\n#align ne.dite_ne_right_iff Ne.dite_ne_right_iff\n-/\n\n#print Ne.ite_ne_left_iff /-\nprotected theorem Ne.ite_ne_left_iff (h : a ≠ b) : ite P a b ≠ a ↔ ¬P :=\n  Ne.dite_ne_left_iff fun _ => h\n#align ne.ite_ne_left_iff Ne.ite_ne_left_iff\n-/\n\n#print Ne.ite_ne_right_iff /-\nprotected theorem Ne.ite_ne_right_iff (h : a ≠ b) : ite P a b ≠ b ↔ P :=\n  Ne.dite_ne_right_iff fun _ => h\n#align ne.ite_ne_right_iff Ne.ite_ne_right_iff\n-/\n\nvariable (P Q) (a b)\n\n#print dite_eq_ite /-\n/-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/\n@[simp]\ntheorem dite_eq_ite : (dite P (fun h => a) fun h => b) = ite P a b :=\n  rfl\n#align dite_eq_ite dite_eq_ite\n-/\n\n#print dite_eq_or_eq /-\ntheorem dite_eq_or_eq : (∃ h, dite P A B = A h) ∨ ∃ h, dite P A B = B h :=\n  Decidable.byCases (fun h => Or.inl ⟨h, dif_pos h⟩) fun h => Or.inr ⟨h, dif_neg h⟩\n#align dite_eq_or_eq dite_eq_or_eq\n-/\n\n#print ite_eq_or_eq /-\ntheorem ite_eq_or_eq : ite P a b = a ∨ ite P a b = b :=\n  Decidable.byCases (fun h => Or.inl (if_pos h)) fun h => Or.inr (if_neg h)\n#align ite_eq_or_eq ite_eq_or_eq\n-/\n\n/- warning: apply_dite -> apply_dite is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) (P : Prop) [_inst_1 : Decidable P] (x : P -> α) (y : (Not P) -> α), Eq.{u2} β (f (dite.{u1} α P _inst_1 x y)) (dite.{u2} β P _inst_1 (fun (h : P) => f (x h)) (fun (h : Not P) => f (y h)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) (P : Prop) [_inst_1 : Decidable P] (x : P -> α) (y : (Not P) -> α), Eq.{u1} β (f (dite.{u2} α P _inst_1 x y)) (dite.{u1} β P _inst_1 (fun (h : P) => f (x h)) (fun (h : Not P) => f (y h)))\nCase conversion may be inaccurate. Consider using '#align apply_dite apply_diteₓ'. -/\n/-- A function applied to a `dite` is a `dite` of that function applied to each of the branches. -/\ntheorem apply_dite (x : P → α) (y : ¬P → α) :\n    f (dite P x y) = dite P (fun h => f (x h)) fun h => f (y h) := by by_cases h : P <;> simp [h]\n#align apply_dite apply_dite\n\n/- warning: apply_ite -> apply_ite is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} (f : α -> β) (P : Prop) [_inst_1 : Decidable P] (a : α) (b : α), Eq.{u2} β (f (ite.{u1} α P _inst_1 a b)) (ite.{u2} β P _inst_1 (f a) (f b))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} (f : α -> β) (P : Prop) [_inst_1 : Decidable P] (a : α) (b : α), Eq.{u1} β (f (ite.{u2} α P _inst_1 a b)) (ite.{u1} β P _inst_1 (f a) (f b))\nCase conversion may be inaccurate. Consider using '#align apply_ite apply_iteₓ'. -/\n/-- A function applied to a `ite` is a `ite` of that function applied to each of the branches. -/\ntheorem apply_ite : f (ite P a b) = ite P (f a) (f b) :=\n  apply_dite f P (fun _ => a) fun _ => b\n#align apply_ite apply_ite\n\n/- warning: apply_dite2 -> apply_dite₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {γ : Sort.{u3}} (f : α -> β -> γ) (P : Prop) [_inst_3 : Decidable P] (a : P -> α) (b : (Not P) -> α) (c : P -> β) (d : (Not P) -> β), Eq.{u3} γ (f (dite.{u1} α P _inst_3 a b) (dite.{u2} β P _inst_3 c d)) (dite.{u3} γ P _inst_3 (fun (h : P) => f (a h) (c h)) (fun (h : Not P) => f (b h) (d h)))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {γ : Sort.{u3}} (f : α -> β -> γ) (P : Prop) [_inst_3 : Decidable P] (a : P -> α) (b : (Not P) -> α) (c : P -> β) (d : (Not P) -> β), Eq.{u3} γ (f (dite.{u2} α P _inst_3 a b) (dite.{u1} β P _inst_3 c d)) (dite.{u3} γ P _inst_3 (fun (h : P) => f (a h) (c h)) (fun (h : Not P) => f (b h) (d h)))\nCase conversion may be inaccurate. Consider using '#align apply_dite2 apply_dite₂ₓ'. -/\n/-- A two-argument function applied to two `dite`s is a `dite` of that two-argument function\napplied to each of the branches. -/\ntheorem apply_dite₂ (f : α → β → γ) (P : Prop) [Decidable P] (a : P → α) (b : ¬P → α) (c : P → β)\n    (d : ¬P → β) :\n    f (dite P a b) (dite P c d) = dite P (fun h => f (a h) (c h)) fun h => f (b h) (d h) := by\n  by_cases h : P <;> simp [h]\n#align apply_dite2 apply_dite₂\n\n/- warning: apply_ite2 -> apply_ite₂ is a dubious translation:\nlean 3 declaration is\n  forall {α : Sort.{u1}} {β : Sort.{u2}} {γ : Sort.{u3}} (f : α -> β -> γ) (P : Prop) [_inst_3 : Decidable P] (a : α) (b : α) (c : β) (d : β), Eq.{u3} γ (f (ite.{u1} α P _inst_3 a b) (ite.{u2} β P _inst_3 c d)) (ite.{u3} γ P _inst_3 (f a c) (f b d))\nbut is expected to have type\n  forall {α : Sort.{u2}} {β : Sort.{u1}} {γ : Sort.{u3}} (f : α -> β -> γ) (P : Prop) [_inst_3 : Decidable P] (a : α) (b : α) (c : β) (d : β), Eq.{u3} γ (f (ite.{u2} α P _inst_3 a b) (ite.{u1} β P _inst_3 c d)) (ite.{u3} γ P _inst_3 (f a c) (f b d))\nCase conversion may be inaccurate. Consider using '#align apply_ite2 apply_ite₂ₓ'. -/\n/-- A two-argument function applied to two `ite`s is a `ite` of that two-argument function\napplied to each of the branches. -/\ntheorem apply_ite₂ (f : α → β → γ) (P : Prop) [Decidable P] (a b : α) (c d : β) :\n    f (ite P a b) (ite P c d) = ite P (f a c) (f b d) :=\n  apply_dite₂ f P (fun _ => a) (fun _ => b) (fun _ => c) fun _ => d\n#align apply_ite2 apply_ite₂\n\n#print dite_apply /-\n/-- A 'dite' producing a `Pi` type `Π a, σ a`, applied to a value `a : α` is a `dite` that applies\neither branch to `a`. -/\ntheorem dite_apply (f : P → ∀ a, σ a) (g : ¬P → ∀ a, σ a) (a : α) :\n    (dite P f g) a = dite P (fun h => f h a) fun h => g h a := by by_cases h : P <;> simp [h]\n#align dite_apply dite_apply\n-/\n\n#print ite_apply /-\n/-- A 'ite' producing a `Pi` type `Π a, σ a`, applied to a value `a : α` is a `ite` that applies\neither branch to `a`. -/\ntheorem ite_apply (f g : ∀ a, σ a) (a : α) : (ite P f g) a = ite P (f a) (g a) :=\n  dite_apply P (fun _ => f) (fun _ => g) a\n#align ite_apply ite_apply\n-/\n\n#print dite_not /-\n/-- Negation of the condition `P : Prop` in a `dite` is the same as swapping the branches. -/\n@[simp]\ntheorem dite_not (x : ¬P → α) (y : ¬¬P → α) :\n    dite (¬P) x y = dite P (fun h => y (not_not_intro h)) x := by by_cases h : P <;> simp [h]\n#align dite_not dite_not\n-/\n\n#print ite_not /-\n/-- Negation of the condition `P : Prop` in a `ite` is the same as swapping the branches. -/\n@[simp]\ntheorem ite_not : ite (¬P) a b = ite P b a :=\n  dite_not P (fun _ => a) fun _ => b\n#align ite_not ite_not\n-/\n\n#print ite_and /-\ntheorem ite_and : ite (P ∧ Q) a b = ite P (ite Q a b) b := by\n  by_cases hp : P <;> by_cases hq : Q <;> simp [hp, hq]\n#align ite_and ite_and\n-/\n\n#print dite_dite_comm /-\ntheorem dite_dite_comm {B : Q → α} {C : ¬P → ¬Q → α} (h : P → ¬Q) :\n    (if p : P then A p else if q : Q then B q else C p q) =\n      if q : Q then B q else if p : P then A p else C p q :=\n  dite_eq_iff'.2\n    ⟨fun p => by rw [dif_neg (h p), dif_pos p], fun np =>\n      by\n      congr\n      funext\n      rw [dif_neg np]⟩\n#align dite_dite_comm dite_dite_comm\n-/\n\n#print ite_ite_comm /-\ntheorem ite_ite_comm (h : P → ¬Q) :\n    (if P then a else if Q then b else c) = if Q then b else if P then a else c :=\n  dite_dite_comm P Q h\n#align ite_ite_comm ite_ite_comm\n-/\n\nend ite\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Logic/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.6584175005616829, "lm_q1q2_score": 0.3394931761501735}}
{"text": "import category_theory.sites.limits\nimport algebra.category.Group.filtered_colimits\nimport algebra.category.Group.abelian\nimport topology.category.Profinite.projective\n\nimport for_mathlib.presieve\nimport for_mathlib.Profinite.disjoint_union\nimport for_mathlib.abelian_sheaves.main\nimport for_mathlib.AddCommGroup.explicit_limits\n\nimport condensed.proetale_site\nimport condensed.is_proetale_sheaf\n\n\n/-!\n# Condensed sets\n\nDefines the category of condensed sets and condensed structures.\n*Strictly speaking* these are pyknotic, but we hope that in the context of Lean's type theory they\nserve the same purpose.\n\n-/\n\nopen category_theory category_theory.limits\n\nuniverses w v u\n\nvariables {C : Type u} [category.{v} C]\n\n/-- The category of condensed sets. -/\n@[derive category]\ndef CondensedSet : Type (u+2) := Sheaf proetale_topology.{u} (Type (u+1))\n\n/-- The category of condensed `A`. Applying this to `A = Type*` is *equivalent* but not the same\nas `CondensedSet`. -/\n@[derive category]\ndef Condensed (C : Type u) [category.{v} C] := Sheaf proetale_topology.{w} C\n\nexample : category.{u+1} (Condensed.{u} Ab.{u+1}) := infer_instance\nexample : category.{u+37} (Condensed.{u} Ring.{u+37}) := infer_instance\n\nopen opposite\n\nnoncomputable theory\n\nvariables (X : Profinite.{u}ᵒᵖ ⥤ Type (u+1))\nvariables (P : Profinite.{w}ᵒᵖ ⥤ Type u)\n\nlemma maps_comm {S S' : Profinite.{u}} (f : S' ⟶ S) :\n  X.map f.op ≫ X.map (pullback.fst : pullback f f ⟶ S').op = X.map f.op ≫ X.map pullback.snd.op :=\nby rw [←X.map_comp, ←op_comp, pullback.condition, op_comp, X.map_comp]\n\ndef natural_fork {S S' : Profinite.{u}} (f : S' ⟶ S) :\n  fork (X.map pullback.fst.op) (X.map pullback.snd.op) :=\nfork.of_ι (X.map (quiver.hom.op f)) (maps_comm X f)\n\nsection\n\n/-\nThere are several files where the instances below are needed, but are not found automatically\nbecause the universe parameters involve things like `max v w` where `v = u+1` and `w = u`.\nSo we just add them manually here.\n-/\n\ninstance : has_limits CondensedSet.{u} :=\nSheaf.category_theory.limits.has_limits.{(u+2) u (u+1)}\n\ninstance : has_colimits CondensedSet.{u} :=\nSheaf.category_theory.limits.has_colimits.{(u+2) u (u+1)}\n\ninstance (C : Type (u+2)) [category.{u+1} C] [has_limits C] :\n  has_limits (Condensed.{u} C) :=\nSheaf.category_theory.limits.has_limits.{(u+2) u (u+1)}\n\ninstance (C : Type (u+2)) [category.{u+1} C]\n  [concrete_category.{u+1} C]\n  [reflects_isomorphisms (forget C)]\n  [preserves_limits (forget C)]\n  [has_colimits C]\n  [∀ (X : Profinite.{u}),\n    preserves_colimits_of_shape (proetale_topology.cover X)ᵒᵖ (forget C)]\n  [∀ (P : Profiniteᵒᵖ ⥤ C) (X : Profinite)\n    (S : proetale_topology.cover X), has_multiequalizer (S.index P)] :\n  has_colimits (Condensed.{u} C) :=\nSheaf.category_theory.limits.has_colimits.{(u+2) u (u+1)}\n\ninstance (X : Profinite) : is_filtered  (proetale_topology.cover X)ᵒᵖ :=\n by apply_instance\n\nset_option pp.universes true\n\ninstance Condensed_Ab_has_colimits : has_colimits (Condensed.{u} Ab.{u+1}) :=\nSheaf.category_theory.limits.has_colimits.{(u+2) u (u+1)}\n\ninstance Condensed_Ab_has_limits : has_limits (Condensed.{u} Ab.{u+1}) :=\nSheaf.category_theory.limits.has_limits.{(u+2) u (u+1)}\n\ninstance : abelian (Profinite.{u}ᵒᵖ ⥤ Ab.{u+1}) :=\ncategory_theory.functor_category_is_abelian.{u+2 u u+1}\n\ninstance : abelian (Condensed Ab.{u+1}) :=\n@category_theory.Sheaf.abelian.{(u+2) u (u+1)} Profinite.{u} _\n  proetale_topology Ab.{u+1} _ _ _ _ _ _ _ _\n\nend\n\n/-\n-- TODO (BM): put this in mathlib (it's already in a mathlib branch with API)\ndef category_theory.functor.preserves_terminal\n  (X : Profinite.{u}ᵒᵖ ⥤ Type (u+1)) : Prop := by admit\n\n-- TODO (BM): put this in mathlib (it's already in a mathlib branch with API)\ndef category_theory.functor.preserves_binary_products\n  (X : Profinite.{u}ᵒᵖ ⥤ Type (u+1)) : Prop := by admit\n\nstructure condensed_type_condition : Prop :=\n(empty : nonempty X.preserves_terminal)\n(bin_prod : nonempty X.preserves_binary_products)\n(pullbacks : ∀ {S S' : Profinite.{u}} (f : S' ⟶ S) [epi f],\n  nonempty (is_limit (natural_fork X f)))\n\n-- (BM): I'm 90% sure this is true as stated, the forward direction is about halfway done.\nlemma sheaf_condition_iff :\n  presieve.is_sheaf proetale_topology X ↔ condensed_type_condition X :=\nby admit\n-/\n\n-- See `Top_to_Condensed` in `condensed/top_comparison.lean`.\n/-\n-- TODO: Double check this definition...\ndef embed_Top : Top.{u} ⥤ CondensedSet.{u} :=\n{ obj := λ T, ⟨Profinite.to_Top.op ⋙ yoneda.obj T ⋙ ulift_functor.{u+1}, by admit⟩,\n  map := λ T₁ T₂ f, ⟨whisker_left _ $ whisker_right (yoneda.map f) _⟩ }\n-/\n\n/-\n-- TODO: State `sheaf_condition_iff` for presheaves taking values in `A` for `A` with appropriate\n-- structure.\n-- TODO: Use `sheaf_condition_iff` to define the functor of Example 1.5, it might look like this:\ndef embed_Top : Top.{u} ⥤ CondensedSet.{u} :=\n{ obj := λ T, ⟨Profinite.to_Top.op ⋙ yoneda.obj T,\n  begin\n    rw sheaf_condition_iff, refine ⟨⟨_⟩, ⟨_⟩, _⟩,\n    all_goals { admit }\n  end⟩,\n  map := λ T₁ T₂ f, whisker_left Profinite.to_Top.op (yoneda.map f) }\n-/\n\n-- TODO: Use the above to prove the first part of Proposition 1.7:\n-- lemma embed_Top_faithful : faithful embed_Top := by admit\n\n-- TODO: Construct the left adjoint to `embed_Top` as in the second part of Proposition 1.7.\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3394379462061526}}
{"text": "import classes.unrestricted.basics.toolbox\nimport utilities.list_utils\n\n\nsection list_technicalities\nvariables {α β : Type}\n\nlemma list_take_one_drop {l : list α} {i : ℕ} (hil : i < l.length) :\n  list.take 1 (list.drop i l) = [l.nth_le i hil] :=\nbegin\n  have l_split : l = list.take i l ++ list.drop i l,\n  {\n    rw list.take_append_drop,\n  },\n  rw list.nth_le_of_eq l_split,\n  rw list.nth_le_append_right,\n  {\n    have smaller_i : min i l.length = i,\n    {\n      exact min_eq_left (le_of_lt hil),\n    },\n    simp only [list.length_take, smaller_i, nat.sub_self],\n    have underscore : 0 < (list.drop i l).length,\n    {\n      finish,\n    },\n    cases (list.drop i l) with d x,\n    {\n      exfalso,\n      exact false_of_ne (ne_of_lt underscore),\n    },\n    {\n      refl,\n    },\n  },\n  {\n    apply list.length_take_le,\n  },\nend\n\nlemma list_drop_take_succ {l : list α} {i : ℕ} (hil : i < l.length) :\n  list.drop i (list.take (i + 1) l) = [l.nth_le i hil] :=\nbegin\n  rw list.drop_take,\n  apply list_take_one_drop,\nend\n\n\nlemma list_forall₂_nth_le {R : α → β → Prop} :\n  ∀ {x : list α}, ∀ {y : list β}, list.forall₂ R x y →\n    ∀ {i : ℕ}, ∀ i_lt_len_x : i < x.length, ∀ i_lt_len_y : i < y.length,\n      R (x.nth_le i i_lt_len_x) (y.nth_le i i_lt_len_y)\n| []       []       := by { intros hyp i hx, exfalso, apply nat.not_lt_zero, exact hx, }\n| []       (a₂::l₂) := by { intro hyp, exfalso, cases hyp, }\n| (a₁::l₁) []       := by { intro hyp, exfalso, cases hyp, }\n| (a₁::l₁) (a₂::l₂) :=\nbegin\n  intros ass i i_lt_len_x i_lt_len_y,\n  rw list.forall₂_cons at ass,\n  cases i,\n  {\n    unfold list.nth_le,\n    exact ass.1,\n  },\n  unfold list.nth_le,\n  apply list_forall₂_nth_le,\n  exact ass.2,\nend\n\nlemma list_filter_map_eq_of_map_eq_map_some {f : α → option β} :\n  ∀ {x : list α}, ∀ {y : list β},\n    list.map f x = list.map option.some y →\n      list.filter_map f x = y\n| []       []       := λ _, rfl\n| (a₁::l₁) []       := by { intro hyp, exfalso, apply list.cons_ne_nil, exact hyp, }\n| []       (a₂::l₂) := by { intro hyp, exfalso, apply list.cons_ne_nil, exact hyp.symm, }\n| (a₁::l₁) (a₂::l₂) :=\nbegin\n  intro ass,\n  rw list.map at ass,\n  rw list.map at ass,\n  rw list.cons.inj_eq at ass,\n  rw list.filter_map_cons_some _ _ _ ass.1,\n  congr,\n  apply list_filter_map_eq_of_map_eq_map_some,\n  exact ass.2,\nend\n\nend list_technicalities\n\n\n-- new nonterminal type\nprotected def nnn (T N₁ N₂ : Type) : Type :=\noption (N₁ ⊕ N₂) ⊕ (T ⊕ T)\n\n-- new symbol type\nprotected def nst (T N₁ N₂ : Type) : Type :=\nsymbol T (nnn T N₁ N₂)\n\nvariables {T : Type}\n\n\nsection the_construction\n\nprotected def wrap_symbol₁ {N₁ : Type} (N₂ : Type) : symbol T N₁ → nst T N₁ N₂\n| (symbol.terminal t)    := symbol.nonterminal (sum.inr (sum.inl t))\n| (symbol.nonterminal n) := symbol.nonterminal (sum.inl (some (sum.inl n)))\n\nprotected def wrap_symbol₂ {N₂ : Type} (N₁ : Type) : symbol T N₂ → nst T N₁ N₂\n| (symbol.terminal t)    := symbol.nonterminal (sum.inr (sum.inr t))\n| (symbol.nonterminal n) := symbol.nonterminal (sum.inl (some (sum.inr n)))\n\nprivate def wrap_grule₁ {N₁ : Type} (N₂ : Type) (r : grule T N₁) : grule T (nnn T N₁ N₂) :=\ngrule.mk\n  (list.map (wrap_symbol₁ N₂) r.input_L)\n  (sum.inl (some (sum.inl r.input_N)))\n  (list.map (wrap_symbol₁ N₂) r.input_R)\n  (list.map (wrap_symbol₁ N₂) r.output_string)\n\nprivate def wrap_grule₂ {N₂ : Type} (N₁ : Type) (r : grule T N₂) : grule T (nnn T N₁ N₂) :=\ngrule.mk\n  (list.map (wrap_symbol₂ N₁) r.input_L)\n  (sum.inl (some (sum.inr r.input_N)))\n  (list.map (wrap_symbol₂ N₁) r.input_R)\n  (list.map (wrap_symbol₂ N₁) r.output_string)\n\nprotected def rules_for_terminals₁ (N₂ : Type) (g : grammar T) : list (grule T (nnn T g.nt N₂)) :=\nlist.map (λ t, grule.mk [] (sum.inr (sum.inl t)) [] [symbol.terminal t]) (all_used_terminals g)\n\nprotected def rules_for_terminals₂ (N₁ : Type) (g : grammar T) : list (grule T (nnn T N₁ g.nt)) :=\nlist.map (λ t, grule.mk [] (sum.inr (sum.inr t)) [] [symbol.terminal t]) (all_used_terminals g)\n\n-- the grammar for concatenation of `g₁` and `g₂` languages\nprotected def big_grammar (g₁ g₂ : grammar T) : grammar T :=\ngrammar.mk (nnn T g₁.nt g₂.nt) (sum.inl none) (\n  (grule.mk [] (sum.inl none) [] [\n    symbol.nonterminal (sum.inl (some (sum.inl g₁.initial))),\n    symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))]\n  ) :: (\n    (list.map (wrap_grule₁ g₂.nt) g₁.rules ++ list.map (wrap_grule₂ g₁.nt) g₂.rules) ++\n    (rules_for_terminals₁ g₂.nt g₁ ++ rules_for_terminals₂ g₁.nt g₂)\n  )\n)\n\nend the_construction\n\n\nsection easy_direction\n\nlemma grammar_generates_only_legit_terminals\n    {g : grammar T}\n    {w : list (symbol T g.nt)}\n    (ass : grammar_derives g [symbol.nonterminal g.initial] w)\n    {s : symbol T g.nt}\n    (symbol_derived : s ∈ w) :\n  (∃ r : grule T g.nt, r ∈ g.rules ∧ s ∈ r.output_string) ∨\n  (s = symbol.nonterminal g.initial) :=\nbegin\n  induction ass with x y trash orig ih,\n  {\n    rw list.mem_singleton at symbol_derived,\n    right,\n    exact symbol_derived,\n  },\n  rcases orig with ⟨r, rin, u, v, bef, aft⟩,\n  rw aft at symbol_derived,\n  rw list.mem_append at symbol_derived,\n  rw list.mem_append at symbol_derived,\n  cases symbol_derived,\n  cases symbol_derived,\n  {\n    apply ih,\n    rw bef,\n    repeat {\n      rw list.mem_append,\n      left,\n    },\n    exact symbol_derived,\n  },\n  {\n    left,\n    use r,\n    split,\n    {\n      exact rin,\n    },\n    {\n      exact symbol_derived,\n    },\n  },\n  {\n    apply ih,\n    rw bef,\n    rw list.mem_append,\n    right,\n    exact symbol_derived,\n  },\nend\n\nprivate lemma first_transformation {g₁ g₂ : grammar T} :\n  grammar_transforms (big_grammar g₁ g₂) [symbol.nonterminal (big_grammar g₁ g₂).initial] [\n      symbol.nonterminal (sum.inl (some (sum.inl g₁.initial))),\n      symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))\n    ] :=\nbegin\n  use (big_grammar g₁ g₂).rules.nth_le 0 (by dec_trivial),\n  split,\n  {\n    change _ ∈ list.cons _ _,\n    finish,\n  },\n  use [[], []],\n  split;\n  refl,\nend\n\nprivate lemma substitute_terminals\n    {g₁ g₂ : grammar T}\n    {side : T → T ⊕ T}\n    {w : list T}\n    (rule_for_each_terminal : ∀ t ∈ w,\n      (grule.mk [] (sum.inr (side t)) [] [symbol.terminal t]) ∈\n        (rules_for_terminals₁ g₂.nt g₁ ++ rules_for_terminals₂ g₁.nt g₂)) :\n  grammar_derives (big_grammar g₁ g₂)\n    (list.map (symbol.nonterminal ∘ sum.inr ∘ side) w)\n    (list.map symbol.terminal w) :=\nbegin\n  induction w with d l ih,\n  {\n    apply grammar_deri_self,\n  },\n  rw list.map,\n  rw list.map,\n  rw ←list.singleton_append,\n  rw ←list.singleton_append,\n  have step_head :\n    grammar_transforms (big_grammar g₁ g₂)\n      ([(symbol.nonterminal ∘ sum.inr ∘ side) d] ++ list.map (symbol.nonterminal ∘ sum.inr ∘ side) l)\n      ([symbol.terminal d] ++ list.map (symbol.nonterminal ∘ sum.inr ∘ side) l),\n  {\n    use grule.mk [] (sum.inr (side d)) [] [symbol.terminal d],\n    split,\n    {\n      change _ ∈ list.cons _ _,\n      apply list.mem_cons_of_mem,\n      apply list.mem_append_right,\n      apply rule_for_each_terminal,\n      apply list.mem_cons_self,\n    },\n    use [[], list.map (symbol.nonterminal ∘ sum.inr ∘ side) l],\n    split;\n    refl,\n  },\n  apply grammar_deri_of_tran_deri step_head,\n  apply grammar_deri_with_prefix,\n  apply ih,\n  {\n    intros t tin,\n    apply rule_for_each_terminal t,\n    exact list.mem_cons_of_mem d tin,\n  },\nend\n\nprotected lemma in_big_of_in_concatenated\n    {g₁ g₂ : grammar T}\n    {w : list T}\n    (ass : w ∈ grammar_language g₁ * grammar_language g₂) :\n  w ∈ grammar_language (big_grammar g₁ g₂) :=\nbegin\n  rw language.mem_mul at ass,\n  rcases ass with ⟨u, v, hu, hv, hw⟩,\n  unfold grammar_language at *,\n  rw set.mem_set_of_eq at *,\n  unfold grammar_generates at *,\n  apply grammar_deri_of_tran_deri first_transformation,\n\n  rw ←hw,\n  rw list.map_append,\n\n  apply @grammar_deri_of_deri_deri T (big_grammar g₁ g₂) _\n    (list.map symbol.terminal u ++ [symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))]) _,\n  {\n    clear_except hu,\n    rw ←list.singleton_append,\n    apply grammar_deri_with_postfix,\n    apply @grammar_deri_of_deri_deri _ _ _ (list.map (\n        (@symbol.nonterminal T (big_grammar g₁ g₂).nt) ∘ sum.inr ∘ sum.inl\n      ) u) _,\n    {\n      have upgrade_deri₁ :\n        ∀ w : list (symbol T g₁.nt),\n          grammar_derives g₁ [symbol.nonterminal g₁.initial] w →\n            grammar_derives (big_grammar g₁ g₂)\n              [symbol.nonterminal (sum.inl (some (sum.inl g₁.initial)))]\n              (list.map (wrap_symbol₁ g₂.nt) w),\n      {\n        clear_except,\n        intros w deri₁,\n        induction deri₁ with x y trash orig ih,\n        {\n          apply grammar_deri_self,\n        },\n        apply grammar_deri_of_deri_tran ih,\n        clear_except orig,\n        rcases orig with ⟨r, rin, u, v, bef, aft⟩,\n        use wrap_grule₁ g₂.nt r,\n        split,\n        {\n          change wrap_grule₁ g₂.nt r ∈ (_ :: ((\n              (list.map (wrap_grule₁ g₂.nt) g₁.rules) ++\n              (list.map (wrap_grule₂ g₁.nt) g₂.rules)\n            ) ++ _)),\n          apply list.mem_cons_of_mem,\n          apply list.mem_append_left,\n          apply list.mem_append_left,\n          rw list.mem_map,\n          use r,\n          split,\n          {\n            exact rin,\n          },\n          {\n            refl,\n          },\n        },\n        use list.map (wrap_symbol₁ g₂.nt) u,\n        use list.map (wrap_symbol₁ g₂.nt) v,\n        split,\n        {\n          convert congr_arg (list.map (wrap_symbol₁ g₂.nt)) bef,\n          rw list.map_append_append,\n          rw list.map_append_append,\n          refl,\n        },\n        {\n          convert congr_arg (list.map (wrap_symbol₁ g₂.nt)) aft,\n          rw list.map_append_append,\n          refl,\n        },\n      },\n      have upgraded := upgrade_deri₁ _ hu,\n      rw list.map_map at upgraded,\n      exact upgraded,\n    },\n    {\n      have legit_terminals₁ :\n        ∀ t ∈ u, ∃ r : grule T g₁.nt,\n          r ∈ g₁.rules ∧ symbol.terminal t ∈ r.output_string,\n      {\n        intros t tin,\n        have tin' : symbol.terminal t ∈ list.map symbol.terminal u,\n        {\n          rw list.mem_map,\n          use t,\n          split,\n          {\n            exact tin,\n          },\n          {\n            refl,\n          },\n        },\n        have legit := grammar_generates_only_legit_terminals hu tin',\n        cases legit,\n        {\n          exact legit,\n        },\n        {\n          exfalso,\n          exact symbol.no_confusion legit,\n        },\n      },\n      apply substitute_terminals,\n      {\n        intros t tin,\n        apply list.mem_append_left,\n        unfold rules_for_terminals₁,\n        rw list.mem_map,\n        use t,\n        split,\n        {\n          unfold all_used_terminals,\n          rw list.mem_filter_map,\n          use symbol.terminal t,\n          split,\n          {\n            rw list.mem_join,\n            obtain ⟨r, rin, sttin⟩ := legit_terminals₁ t tin,\n            use r.output_string,\n            split,\n            {\n              apply list.mem_map_of_mem,\n              exact rin,\n            },\n            {\n              exact sttin,\n            },\n          },\n          {\n            refl,\n          },\n        },\n        {\n          refl,\n        },\n      },\n    },\n  },\n  {\n    clear_except hv,\n    apply grammar_deri_with_prefix,\n    apply @grammar_deri_of_deri_deri _ _ _ (list.map (\n        (@symbol.nonterminal T (big_grammar g₁ g₂).nt) ∘ sum.inr ∘ sum.inr\n      ) v) _,\n    {\n      have upgrade_deri₂ :\n        ∀ w : list (symbol T g₂.nt),\n          grammar_derives g₂ [symbol.nonterminal g₂.initial] w →\n            grammar_derives (big_grammar g₁ g₂)\n              [symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))]\n              (list.map (wrap_symbol₂ g₁.nt) w),\n      {\n        clear_except,\n        intros w deri₁,\n        induction deri₁ with x y trash orig ih,\n        {\n          apply grammar_deri_self,\n        },\n        apply grammar_deri_of_deri_tran ih,\n        clear_except orig,\n        rcases orig with ⟨r, rin, u, v, bef, aft⟩,\n        use wrap_grule₂ g₁.nt r,\n        split,\n        {\n          change wrap_grule₂ g₁.nt r ∈ (_ :: ((\n              (list.map (wrap_grule₁ g₂.nt) g₁.rules) ++\n              (list.map (wrap_grule₂ g₁.nt) g₂.rules)\n            ) ++ _)),\n          apply list.mem_cons_of_mem,\n          apply list.mem_append_left,\n          apply list.mem_append_right,\n          rw list.mem_map,\n          use r,\n          split,\n          {\n            exact rin,\n          },\n          {\n            refl,\n          },\n        },\n        use list.map (wrap_symbol₂ g₁.nt) u,\n        use list.map (wrap_symbol₂ g₁.nt) v,\n        split,\n        {\n          convert congr_arg (list.map (wrap_symbol₂ g₁.nt)) bef,\n          rw list.map_append_append,\n          rw list.map_append_append,\n          refl,\n        },\n        {\n          convert congr_arg (list.map (wrap_symbol₂ g₁.nt)) aft,\n          rw list.map_append_append,\n          refl,\n        },\n      },\n      have upgraded := upgrade_deri₂ _ hv,\n      rw list.map_map at upgraded,\n      exact upgraded,\n    },\n    {\n      have legit_terminals₂ :\n        ∀ t ∈ v, ∃ r : grule T g₂.nt,\n          r ∈ g₂.rules ∧ symbol.terminal t ∈ r.output_string,\n      {\n        intros t tin,\n        have tin' : symbol.terminal t ∈ list.map symbol.terminal v,\n        {\n          rw list.mem_map,\n          use t,\n          split,\n          {\n            exact tin,\n          },\n          {\n            refl,\n          },\n        },\n        have legit := grammar_generates_only_legit_terminals hv tin',\n        cases legit,\n        {\n          exact legit,\n        },\n        {\n          exfalso,\n          exact symbol.no_confusion legit,\n        },\n      },\n      apply substitute_terminals,\n      {\n        intros t tin,\n        apply list.mem_append_right,\n        unfold rules_for_terminals₂,\n        rw list.mem_map,\n        use t,\n        split,\n        {\n          unfold all_used_terminals,\n          rw list.mem_filter_map,\n          use symbol.terminal t,\n          split,\n          {\n            rw list.mem_join,\n            obtain ⟨r, rin, sttin⟩ := legit_terminals₂ t tin,\n            use r.output_string,\n            split,\n            {\n              apply list.mem_map_of_mem,\n              exact rin,\n            },\n            {\n              exact sttin,\n            },\n          },\n          {\n            refl,\n          },\n        },\n        {\n          refl,\n        },\n      },\n    },\n  },\nend\n\nend easy_direction\n\n\nsection hard_direction\n\nsection correspondence_for_terminals\n\nprivate def corresponding_symbols {N₁ N₂ : Type} : nst T N₁ N₂ → nst T N₁ N₂ → Prop\n| (symbol.terminal t)                               (symbol.terminal t')                               := t = t'\n| (symbol.nonterminal (sum.inr (sum.inl a)))        (symbol.nonterminal (sum.inr (sum.inl a')))        := a = a'\n| (symbol.nonterminal (sum.inr (sum.inr a)))        (symbol.nonterminal (sum.inr (sum.inr a')))        := a = a'\n| (symbol.nonterminal (sum.inr (sum.inl a)))        (symbol.terminal t)                                := t = a\n| (symbol.nonterminal (sum.inr (sum.inr a)))        (symbol.terminal t)                                := t = a\n| (symbol.nonterminal (sum.inl (some (sum.inl n)))) (symbol.nonterminal (sum.inl (some (sum.inl n')))) := n = n'\n| (symbol.nonterminal (sum.inl (some (sum.inr n)))) (symbol.nonterminal (sum.inl (some (sum.inr n')))) := n = n'\n| (symbol.nonterminal (sum.inl (none)))             (symbol.nonterminal (sum.inl (none)))              := true\n| _                                                 _                                                  := false\n\nprivate lemma corresponding_symbols_self {N₁ N₂ : Type} (s : nst T N₁ N₂) : corresponding_symbols s s :=\nbegin\n  repeat {\n    try {\n      cases s,\n    },\n    try {\n      unfold corresponding_symbols,\n    },\n  },\nend\n\nprivate lemma corresponding_symbols_never₁ {N₁ N₂ : Type} {s₁ : symbol T N₁} {s₂ : symbol T N₂} :\n  ¬ corresponding_symbols (wrap_symbol₁ N₂ s₁) (wrap_symbol₂ N₁ s₂) :=\nbegin\n  cases s₁;\n  cases s₂;\n  {\n    unfold wrap_symbol₁,\n    unfold wrap_symbol₂,\n    unfold corresponding_symbols,\n    exact not_false,\n  },\nend\n\nprivate lemma corresponding_symbols_never₂ {N₁ N₂ : Type} {s₁ : symbol T N₁} {s₂ : symbol T N₂} :\n  ¬ corresponding_symbols (wrap_symbol₂ N₁ s₂) (wrap_symbol₁ N₂ s₁) :=\nbegin\n  cases s₁;\n  cases s₂;\n  {\n    unfold wrap_symbol₁,\n    unfold wrap_symbol₂,\n    unfold corresponding_symbols,\n    exact not_false,\n  },\nend\n\n\nprivate def corresponding_strings {N₁ N₂ : Type} : list (nst T N₁ N₂) → list (nst T N₁ N₂) → Prop :=\nlist.forall₂ corresponding_symbols\n\nprivate lemma corresponding_strings_self {N₁ N₂ : Type} {x : list (nst T N₁ N₂)} :\n  corresponding_strings x x :=\nbegin\n  apply list.forall₂_same,\n  intros s trash,\n  exact corresponding_symbols_self s,\nend\n\nprivate lemma corresponding_strings_singleton {N₁ N₂ : Type} {s₁ s₂ : nst T N₁ N₂}\n    (ass : corresponding_symbols s₁ s₂) :\n  corresponding_strings [s₁] [s₂] :=\nbegin\n  unfold corresponding_strings,\n  rw list.forall₂_cons,\n  split,\n  {\n    exact ass,\n  },\n  {\n    exact list.forall₂.nil,\n  },\nend\n\nprivate lemma corresponding_strings_append {N₁ N₂ : Type} {x₁ x₂ y₁ y₂ : list (nst T N₁ N₂)}\n    (ass₁ : corresponding_strings x₁ y₁)\n    (ass₂ : corresponding_strings x₂ y₂) :\n  corresponding_strings (x₁ ++ x₂) (y₁ ++ y₂) :=\nbegin\n  unfold corresponding_strings at *,\n  exact list.rel_append ass₁ ass₂,\nend\n\nprivate lemma corresponding_strings_length {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (ass : corresponding_strings x y) :\n  x.length = y.length :=\nbegin\n  unfold corresponding_strings at ass,\n  exact list.forall₂_length_eq ass,\nend\n\nprivate lemma corresponding_strings_nth_le {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)} {i : ℕ}\n    (i_lt_len_x : i < x.length) (i_lt_len_y : i < y.length)\n    (ass : corresponding_strings x y) :\n  corresponding_symbols (x.nth_le i i_lt_len_x) (y.nth_le i i_lt_len_y) :=\nbegin\n  apply list_forall₂_nth_le,\n  exact ass,\nend\n\nprivate lemma corresponding_strings_reverse {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (ass : corresponding_strings x y) :\n  corresponding_strings x.reverse y.reverse :=\nbegin\n  unfold corresponding_strings at *,\n  rw list.forall₂_reverse_iff,\n  exact ass,\nend\n\nprivate lemma corresponding_strings_of_reverse {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (ass : corresponding_strings x.reverse y.reverse) :\n  corresponding_strings x y :=\nbegin\n  unfold corresponding_strings at *,\n  rw list.forall₂_reverse_iff at ass,\n  exact ass,\nend\n\nprivate lemma corresponding_strings_take {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (n : ℕ) (ass : corresponding_strings x y) :\n  corresponding_strings (list.take n x) (list.take n y) :=\nbegin\n  unfold corresponding_strings at *,\n  exact list.forall₂_take n ass,\nend\n\nprivate lemma corresponding_strings_drop {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (n : ℕ) (ass : corresponding_strings x y) :\n  corresponding_strings (list.drop n x) (list.drop n y) :=\nbegin\n  unfold corresponding_strings at *,\n  exact list.forall₂_drop n ass,\nend\n\nprivate lemma corresponding_strings_split {N₁ N₂ : Type} {x y : list (nst T N₁ N₂)}\n    (n : ℕ) (ass : corresponding_strings x y) :\n  corresponding_strings (list.take n x) (list.take n y) ∧\n  corresponding_strings (list.drop n x) (list.drop n y) :=\nbegin\n  split,\n  {\n    exact corresponding_strings_take n ass,\n  },\n  {\n    exact corresponding_strings_drop n ass,\n  },\nend\n\nend correspondence_for_terminals\n\n\nsection unwrapping_nst\n\nprivate def unwrap_symbol₁ {N₁ N₂ : Type} : nst T N₁ N₂ → option (symbol T N₁)\n| (symbol.terminal t)                               := some (symbol.terminal t)\n| (symbol.nonterminal (sum.inr (sum.inl a)))        := some (symbol.terminal a)\n| (symbol.nonterminal (sum.inr (sum.inr a)))        := none\n| (symbol.nonterminal (sum.inl (some (sum.inl n)))) := some (symbol.nonterminal n)\n| (symbol.nonterminal (sum.inl (some (sum.inr n)))) := none\n| (symbol.nonterminal (sum.inl (none)))             := none\n\nprivate def unwrap_symbol₂ {N₁ N₂ : Type} : nst T N₁ N₂ → option (symbol T N₂)\n| (symbol.terminal t)                               := some (symbol.terminal t)\n| (symbol.nonterminal (sum.inr (sum.inl a)))        := none\n| (symbol.nonterminal (sum.inr (sum.inr a)))        := some (symbol.terminal a)\n| (symbol.nonterminal (sum.inl (some (sum.inl n)))) := none\n| (symbol.nonterminal (sum.inl (some (sum.inr n)))) := some (symbol.nonterminal n)\n| (symbol.nonterminal (sum.inl (none)))             := none\n\n\nprivate lemma unwrap_wrap₁_symbol {N₁ N₂ : Type} : @unwrap_symbol₁ T N₁ N₂ ∘ wrap_symbol₁ N₂ = option.some :=\nbegin\n  ext1 a,\n  cases a;\n  refl,\nend\n\nprivate lemma unwrap_wrap₂_symbol {N₁ N₂ : Type} : @unwrap_symbol₂ T N₁ N₂ ∘ wrap_symbol₂ N₁ = option.some :=\nbegin\n  ext1 a,\n  cases a;\n  refl,\nend\n\n\nprivate lemma unwrap_wrap₁_string {N₁ N₂ : Type} {w : list (symbol T N₁)} :\n  list.filter_map unwrap_symbol₁ (list.map (wrap_symbol₁ N₂) w) = w :=\nbegin\n  rw list.filter_map_map,\n  rw unwrap_wrap₁_symbol,\n  apply list.filter_map_some,\nend\n\nprivate lemma unwrap_wrap₂_string {N₁ N₂ : Type} {w : list (symbol T N₂)} :\n  list.filter_map unwrap_symbol₂ (list.map (wrap_symbol₂ N₁) w) = w :=\nbegin\n  rw list.filter_map_map,\n  rw unwrap_wrap₂_symbol,\n  apply list.filter_map_some,\nend\n\n\nprivate lemma unwrap_eq_some_of_corresponding_symbols₁ {N₁ N₂ : Type} {s₁ : symbol T N₁} {s : nst T N₁ N₂}\n    (ass : corresponding_symbols (wrap_symbol₁ N₂ s₁) s) :\n  unwrap_symbol₁ s = some s₁ :=\nbegin\n  cases s₁;\n  {\n    unfold wrap_symbol₁ at ass,\n    repeat {\n      try {\n        cases s,\n      },\n      try {\n        unfold corresponding_symbols at ass,\n        rw ass,\n        refl,\n      },\n      try {\n        unfold corresponding_symbols at ass,\n        exfalso,\n        exact ass,\n      },\n    },\n  },\nend\n\nprivate lemma unwrap_eq_some_of_corresponding_symbols₂ {N₁ N₂ : Type} {s₂ : symbol T N₂} {s : nst T N₁ N₂}\n    (ass : corresponding_symbols (wrap_symbol₂ N₁ s₂) s) :\n  unwrap_symbol₂ s = some s₂ :=\nbegin\n  cases s₂;\n  {\n    unfold wrap_symbol₂ at ass,\n    repeat {\n      try {\n        cases s,\n      },\n      try {\n        unfold corresponding_symbols at ass,\n        rw ass,\n        refl,\n      },\n      try {\n        unfold corresponding_symbols at ass,\n        exfalso,\n        exact ass,\n      },\n    },\n  },\nend\n\n\nprivate lemma map_unwrap_eq_map_some_of_corresponding_strings₁ {N₁ N₂ : Type} :\n  ∀ {v : list (symbol T N₁)}, ∀ {w : list (nst T N₁ N₂)},\n    corresponding_strings (list.map (wrap_symbol₁ N₂) v) w →\n      list.map unwrap_symbol₁ w = list.map option.some v\n| []     []     := λ _, rfl\n| []     (b::y) := by { intro hyp, exfalso, unfold corresponding_strings at hyp, unfold list.map at hyp, finish, }\n| (a::x) []     := by { intro hyp, exfalso, unfold corresponding_strings at hyp, unfold list.map at hyp, finish, }\n| (a::x) (b::y) :=\nbegin\n  intro ass,\n  unfold corresponding_strings at ass,\n  rw list.map_cons at ass,\n  rw list.forall₂_cons at ass,\n  rw list.map,\n  rw list.map,\n  apply congr_arg2,\n  {\n    exact unwrap_eq_some_of_corresponding_symbols₁ ass.1,\n  },\n  {\n    apply map_unwrap_eq_map_some_of_corresponding_strings₁,\n    exact ass.2\n  },\nend\n\nprivate lemma map_unwrap_eq_map_some_of_corresponding_strings₂ {N₁ N₂ : Type} :\n  ∀ {v : list (symbol T N₂)}, ∀ {w : list (nst T N₁ N₂)},\n    corresponding_strings (list.map (wrap_symbol₂ N₁) v) w →\n      list.map unwrap_symbol₂ w = list.map option.some v\n| []     []     := λ _, rfl\n| []     (b::y) := by { intro hyp, exfalso, unfold corresponding_strings at hyp, unfold list.map at hyp, finish, }\n| (a::x) []     := by { intro hyp, exfalso, unfold corresponding_strings at hyp, unfold list.map at hyp, finish, }\n| (a::x) (b::y) :=\nbegin\n  intro ass,\n  unfold corresponding_strings at ass,\n  rw list.map_cons at ass,\n  rw list.forall₂_cons at ass,\n  rw list.map,\n  rw list.map,\n  apply congr_arg2,\n  {\n    exact unwrap_eq_some_of_corresponding_symbols₂ ass.1,\n  },\n  {\n    apply map_unwrap_eq_map_some_of_corresponding_strings₂,\n    exact ass.2\n  },\nend\n\n\nprivate lemma filter_map_unwrap_of_corresponding_strings₁ {N₁ N₂ : Type}\n    {v : list (symbol T N₁)} {w : list (nst T N₁ N₂)}\n    (ass : corresponding_strings (list.map (wrap_symbol₁ N₂) v) w) :\n  list.filter_map unwrap_symbol₁ w = v :=\nbegin\n  apply list_filter_map_eq_of_map_eq_map_some,\n  exact map_unwrap_eq_map_some_of_corresponding_strings₁ ass,\nend\n\nprivate lemma filter_map_unwrap_of_corresponding_strings₂ {N₁ N₂ : Type}\n    {v : list (symbol T N₂)} {w : list (nst T N₁ N₂)}\n    (ass : corresponding_strings (list.map (wrap_symbol₂ N₁) v) w) :\n  list.filter_map unwrap_symbol₂ w = v :=\nbegin\n  apply list_filter_map_eq_of_map_eq_map_some,\n  exact map_unwrap_eq_map_some_of_corresponding_strings₂ ass,\nend\n\n\nprivate lemma corresponding_string_after_wrap_unwrap_self₁ {N₁ N₂ : Type} {w : list (nst T N₁ N₂)}\n    (ass : ∃ z : list (symbol T N₁), corresponding_strings (list.map (wrap_symbol₁ N₂) z) w) :\n  corresponding_strings (list.map (wrap_symbol₁ N₂) (list.filter_map unwrap_symbol₁ w)) w :=\nbegin\n  induction w with d l ih,\n  {\n    unfold corresponding_strings,\n    unfold list.filter_map,\n    unfold list.map,\n    exact list.forall₂.nil,\n  },\n  specialize ih (by {\n    cases ass with z hyp,\n    unfold corresponding_strings at *,\n    cases z with z₀ z',\n    {\n      exfalso,\n      finish,\n    },\n    {\n      use z',\n      finish,\n    },\n  }),\n  unfold corresponding_strings,\n  cases d,\n  {\n    have unwrap_first_t :\n      list.filter_map unwrap_symbol₁ (symbol.terminal d :: l) =\n      symbol.terminal d :: list.filter_map unwrap_symbol₁ l,\n    {\n      refl,\n    },\n    rw unwrap_first_t,\n    unfold list.map,\n    unfold wrap_symbol₁,\n    rw list.forall₂_cons,\n    split,\n    {\n      unfold corresponding_symbols,\n    },\n    {\n      exact ih,\n    },\n  },\n  cases d,\n    cases d, swap,\n      cases d,\n      {\n        have unwrap_first_nlsl :\n          list.filter_map unwrap_symbol₁ (symbol.nonterminal (sum.inl (some (sum.inl d))) :: l) =\n          symbol.nonterminal d :: list.filter_map unwrap_symbol₁ l,\n        {\n          refl,\n        },\n        rw unwrap_first_nlsl,\n        unfold list.map,\n        unfold wrap_symbol₁,\n        rw list.forall₂_cons,\n        split,\n        {\n          unfold corresponding_symbols,\n        },\n        {\n          exact ih,\n        },\n      },\n  swap 3,\n    cases d,\n    {\n      have unwrap_first_nrl :\n        list.filter_map unwrap_symbol₁ (symbol.nonterminal (sum.inr (sum.inl d)) :: l) =\n        symbol.terminal d :: list.filter_map unwrap_symbol₁ l,\n      {\n        refl,\n      },\n      rw unwrap_first_nrl,\n      unfold list.map,\n      unfold wrap_symbol₁,\n      rw list.forall₂_cons,\n      split,\n      {\n        unfold corresponding_symbols,\n      },\n      {\n        exact ih,\n      },\n    },\n  any_goals {\n    exfalso,\n    cases ass with z hyp,\n    cases z with z₀ z',\n    {\n      have imposs := corresponding_strings_length hyp,\n      clear_except imposs,\n      rw list.length at imposs,\n      rw list.length_map at imposs,\n      rw list.length at imposs,\n      linarith,\n    },\n    {\n      rw list.map_cons at hyp,\n      unfold corresponding_strings at hyp,\n      rw list.forall₂_cons at hyp,\n      have impos := hyp.left,\n      clear_except impos,\n      cases z₀;\n      {\n        unfold wrap_symbol₁ at impos,\n        unfold corresponding_symbols at impos,\n        exact impos,\n      },\n    },\n  },\nend\n\nprivate lemma corresponding_string_after_wrap_unwrap_self₂ {N₁ N₂ : Type} {w : list (nst T N₁ N₂)}\n    (ass : ∃ z : list (symbol T N₂), corresponding_strings (list.map (wrap_symbol₂ N₁) z) w) :\n  corresponding_strings (list.map (wrap_symbol₂ N₁) (list.filter_map unwrap_symbol₂ w)) w :=\nbegin\n  induction w with d l ih,\n  {\n    unfold corresponding_strings,\n    unfold list.filter_map,\n    unfold list.map,\n    exact list.forall₂.nil,\n  },\n  specialize ih (by {\n    cases ass with z hyp,\n    unfold corresponding_strings at *,\n    cases z with z₀ z',\n    {\n      exfalso,\n      finish,\n    },\n    {\n      use z',\n      finish,\n    },\n  }),\n  unfold corresponding_strings,\n  cases d,\n  {\n    have unwrap_first_t :\n      list.filter_map unwrap_symbol₂ (symbol.terminal d :: l) =\n      symbol.terminal d :: list.filter_map unwrap_symbol₂ l,\n    {\n      refl,\n    },\n    rw unwrap_first_t,\n    unfold list.map,\n    unfold wrap_symbol₂,\n    rw list.forall₂_cons,\n    split,\n    {\n      unfold corresponding_symbols,\n    },\n    {\n      exact ih,\n    },\n  },\n  cases d,\n    cases d, swap,\n      cases d, swap,\n      {\n        have unwrap_first_nlsr :\n          list.filter_map unwrap_symbol₂ (symbol.nonterminal (sum.inl (some (sum.inr d))) :: l) =\n          symbol.nonterminal d :: list.filter_map unwrap_symbol₂ l,\n        {\n          refl,\n        },\n        rw unwrap_first_nlsr,\n        unfold list.map,\n        unfold wrap_symbol₂,\n        rw list.forall₂_cons,\n        split,\n        {\n          unfold corresponding_symbols,\n        },\n        {\n          exact ih,\n        },\n      },\n  swap 3,\n    cases d, swap,\n    {\n      have unwrap_first_nrr :\n        list.filter_map unwrap_symbol₂ (symbol.nonterminal (sum.inr (sum.inr d)) :: l) =\n        symbol.terminal d :: list.filter_map unwrap_symbol₂ l,\n      {\n        refl,\n      },\n      rw unwrap_first_nrr,\n      unfold list.map,\n      unfold wrap_symbol₂,\n      rw list.forall₂_cons,\n      split,\n      {\n        unfold corresponding_symbols,\n      },\n      {\n        exact ih,\n      },\n    },\n  any_goals {\n    exfalso,\n    cases ass with z hyp,\n    cases z with z₀ z',\n    {\n      have imposs := corresponding_strings_length hyp,\n      clear_except imposs,\n      rw list.length at imposs,\n      rw list.length_map at imposs,\n      rw list.length at imposs,\n      linarith,\n    },\n    {\n      rw list.map_cons at hyp,\n      unfold corresponding_strings at hyp,\n      rw list.forall₂_cons at hyp,\n      have impos := hyp.left,\n      clear_except impos,\n      cases z₀;\n      {\n        unfold wrap_symbol₂ at impos,\n        unfold corresponding_symbols at impos,\n        exact impos,\n      },\n    },\n  },\nend\n\nend unwrapping_nst\n\n\nsection very_complicated\n\nprivate lemma induction_step_for_lifted_rule_from_g₁\n    {g₁ g₂ : grammar T}\n    {a b u v : list (nst T g₁.nt g₂.nt)}\n    {x : list (symbol T g₁.nt)}\n    {y : list (symbol T g₂.nt)}\n    {r : grule T (nnn T g₁.nt g₂.nt)}\n    (rin : r ∈ list.map (wrap_grule₁ g₂.nt) g₁.rules)\n    (bef : a = u ++ r.input_L ++ [symbol.nonterminal r.input_N] ++ r.input_R ++ v)\n    (aft : b = u ++ r.output_string ++ v)\n    (ih_x : grammar_derives g₁ [symbol.nonterminal g₁.initial] x)\n    (ih_y : grammar_derives g₂ [symbol.nonterminal g₂.initial] y)\n    (ih_concat :\n      corresponding_strings\n        (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)\n        a) :\n  ∃ (x' : list (symbol T g₁.nt)),\n    and\n      (and\n        (grammar_derives g₁ [symbol.nonterminal g₁.initial] x')\n        (grammar_derives g₂ [symbol.nonterminal g₂.initial] y)\n      )\n      (corresponding_strings (list.map (wrap_symbol₁ g₂.nt) x' ++ list.map (wrap_symbol₂ g₁.nt) y) b) :=\nbegin\n  rw list.mem_map at rin,\n  rcases rin with ⟨r₁, rin₁, wrap_r₁_eq_r⟩,\n  rw ←wrap_r₁_eq_r at *,\n  clear wrap_r₁_eq_r,\n  simp [wrap_grule₁] at *,\n  rw ←list.singleton_append at bef,\n\n  let m := (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1 +\n           (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length,\n  let b' := u ++ list.map (wrap_symbol₁ g₂.nt) r₁.output_string ++ list.take (x.length - u.length - m) v,\n  use list.filter_map unwrap_symbol₁ b',\n\n  have critical :\n    (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1 +\n      (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length ≤\n    x.length - u.length,\n  {\n    clear_except ih_concat bef,\n    have as_positive :\n      u.length + (\n          (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1 +\n          (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length\n        ) ≤\n      x.length,\n    {\n      by_contradiction contra,\n      push_neg at contra,\n      rw bef at ih_concat,\n      clear bef,\n      repeat {\n        rw ←list.append_assoc at ih_concat\n      },\n      have len_pos :\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length > 0,\n      {\n        repeat {\n          rw list.length_append,\n        },\n        rw list.length_singleton,\n        clear_except,\n        linarith,\n      },\n      have equal_total_len := corresponding_strings_length ih_concat,\n\n      have inequality_m1 :\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length - 1 <\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length,\n      {\n        exact buffer.lt_aux_2 len_pos,\n      },\n      have inequality_cat :\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length - 1 <\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R ++ v\n        ).length,\n      {\n        rw list.length_append _ v,\n        apply lt_of_lt_of_le (buffer.lt_aux_2 len_pos),\n        exact le_self_add,\n      },\n      have inequality_map :\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length - 1 <\n        ((list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)).length,\n      {\n        rw equal_total_len,\n        exact inequality_cat,\n      },\n      have inequality_map_opp :\n        (u ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length - 1 ≥\n        (list.map (wrap_symbol₁ g₂.nt) x).length,\n      {\n        clear_except contra,\n        apply nat.le_pred_of_lt,\n        repeat {\n          rw list.length_append,\n        },\n        repeat {\n          rw list.length_map,\n        },\n        rw list.length_map at contra,\n        rw list.length_map at contra,\n        rw list.length_singleton,\n        rw add_assoc,\n        rw add_assoc,\n        rw add_assoc at contra,\n        exact contra,\n      },\n      have clash := corresponding_strings_nth_le inequality_map inequality_cat ih_concat,\n      rw list.nth_le_append inequality_cat inequality_m1 at clash,\n      rw list.nth_le_append_right inequality_map_opp inequality_map at clash,\n\n      rw list.nth_le_map at clash, swap,\n      {\n        have inequality_map := inequality_map,\n        rw list.length_append _ (list.map (wrap_symbol₂ g₁.nt) y) at inequality_map,\n        rw list.length_map _ y at inequality_map,\n        rw tsub_lt_iff_left inequality_map_opp,\n        exact inequality_map,\n      },\n      by_cases (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length ≥ 1,\n      {\n        rw list.nth_le_append_right at clash, swap,\n        {\n          rw list.length_append _ (list.map (wrap_symbol₁ g₂.nt) r₁.input_R),\n          have trivi_ineq : ∀ m k : ℕ, k ≥ 1 → m ≤ m + k - 1,\n          {\n            clear_except,\n            omega,\n          },\n          convert trivi_ineq (u ++ _ ++ [_]).length _ h,\n        },\n        rw list.nth_le_map at clash, swap,\n        {\n          rw list.length_map at h,\n          repeat {\n            rw list.length_append,\n          },\n          repeat {\n            rw list.length_map,\n          },\n          rw list.length_singleton,\n          have easy_ineq : ∀ m k : ℕ, k ≥ 1 → m + k - 1 - m < k,\n          {\n            clear_except,\n            omega,\n          },\n          convert easy_ineq (u.length + r₁.input_L.length + 1) _ h,\n        },\n        exact corresponding_symbols_never₂ clash,\n      },\n      {\n        push_neg at h,\n        have ris_third_is_nil : list.map (wrap_symbol₁ g₂.nt) r₁.input_R = [],\n        {\n          rw ←list.length_eq_zero,\n          rw ←nat.lt_one_iff,\n          exact h,\n        },\n        have inequality_m0 :\n          (u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n            [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))]).length - 1 <\n          (u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n            [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))]).length,\n        {\n          rw ris_third_is_nil at inequality_m1,\n          rw list.append_nil at inequality_m1,\n          exact inequality_m1,\n        },\n        simp_rw [ris_third_is_nil] at clash,\n        simp only [list.append_nil] at clash,\n        rw list.nth_le_append_right at clash,\n        swap, {\n          apply le_of_eq,\n          rw list.length_append _ [_],\n          rw list.length_singleton,\n          apply nat.succ_sub_one,\n        },\n        rw list.nth_le_singleton at clash,\n        change\n          corresponding_symbols _ (wrap_symbol₁ g₂.nt (symbol.nonterminal r₁.input_N)) at clash,\n        exact corresponding_symbols_never₂ clash,\n      },\n    },\n    omega,\n  },\n\n  split,\n  {\n    split,\n    {\n      apply grammar_deri_of_deri_tran ih_x,\n      use r₁,\n      split,\n      {\n        exact rin₁,\n      },\n      use list.filter_map unwrap_symbol₁ u,\n      use list.filter_map unwrap_symbol₁ (list.take (x.length - u.length - m) v),\n      split,\n      {\n        have x_equiv :\n          corresponding_strings\n            (list.map (wrap_symbol₁ g₂.nt) x)\n            (list.take x.length (u\n              ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L\n              ++ [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))]\n              ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n              ++ v)),\n        {\n          rw bef at ih_concat,\n          clear_except ih_concat,\n          rw ←list.append_assoc _ _ v at ih_concat,\n          rw ←list.append_assoc _ _ v at ih_concat,\n          rw list.append_assoc u,\n          rw list.append_assoc u,\n          rw list.append_assoc u,\n          rw list.append_assoc (list.map (wrap_symbol₁ g₂.nt) r₁.input_L),\n\n          convert corresponding_strings_take x.length ih_concat,\n          {\n            have x_len_eq : x.length = (list.map (wrap_symbol₁ g₂.nt) x).length,\n            {\n              rw list.length_map,\n            },\n            rw x_len_eq,\n            rw list.take_left,\n          },\n        },\n        clear_except x_equiv critical,\n\n        have ul_le_xl : u.length ≤ x.length,\n        {\n          clear_except critical,\n          have weaker_le : 1 ≤ x.length - u.length,\n          {\n            omega,\n          },\n          have stupid_le : u.length + 1 ≤ x.length,\n          {\n            omega,\n          },\n          exact nat.le_of_succ_le stupid_le,\n        },\n        repeat {\n          rw list.take_append_eq_append_take at x_equiv,\n        },\n        rw list.take_all_of_le ul_le_xl at x_equiv,\n        repeat {\n          rw list.append_assoc,\n        },\n\n        have chunk2 :\n          list.take (x.length - u.length) (list.map (wrap_symbol₁ g₂.nt) r₁.input_L) =\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_L,\n        {\n          apply list.take_all_of_le,\n          clear_except critical,\n          omega,\n        },\n        have chunk3 :\n          list.take (x.length - (u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length)\n            [@symbol.nonterminal T (nnn T g₁.nt g₂.nt) (sum.inl (some (sum.inl r₁.input_N)))] =\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))],\n        {\n          apply list.take_all_of_le,\n          clear_except critical,\n          change 1 ≤ x.length - (u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length,\n          rw list.length_append,\n          have weakened :\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1 ≤\n            x.length - u.length,\n          {\n            omega,\n          },\n          have goal_as_le_sub_sub :\n            1 ≤ (x.length - u.length) - (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length,\n          {\n            omega,\n          },\n          rw tsub_add_eq_tsub_tsub,\n          exact goal_as_le_sub_sub,\n        },\n        have chunk4 :\n          list.take (x.length - (\n              u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n              [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))]\n            ).length) (list.map (wrap_symbol₁ g₂.nt) r₁.input_R) =\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R,\n        {\n          apply list.take_all_of_le,\n          clear_except critical,\n          rw list.length_append_append,\n          change\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length ≤\n            x.length - (u.length + (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1),\n          omega,\n        },\n        have chunk5 :\n          list.take (x.length - (\n              u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n              [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n              list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length\n            ) v =\n          list.take (x.length - u.length - m) v,\n        {\n          repeat {\n            rw list.length_append,\n          },\n          apply congr_arg2, swap,\n          {\n            refl,\n          },\n          have rearrange_sum_of_four : ∀ a b c d : ℕ, a + b + c + d = a + (b + c + d),\n          {\n            omega,\n          },\n          rw rearrange_sum_of_four,\n          change x.length - (u.length + m) = x.length - u.length - m,\n          clear_except,\n          omega,\n        },\n        rw [chunk2, chunk3, chunk4, chunk5] at x_equiv,\n        clear chunk2 chunk3 chunk4 chunk5,\n\n        obtain ⟨temp_5, equiv_segment_5⟩ :=\n          corresponding_strings_split (\n              u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n              [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n              list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n            ).length x_equiv,\n        clear x_equiv,\n        rw list.drop_left at equiv_segment_5,\n        rw list.take_left at temp_5,\n\n        obtain ⟨temp_4, equiv_segment_4⟩ :=\n          corresponding_strings_split (\n              u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n              [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))]\n            ).length temp_5,\n        clear temp_5,\n        rw list.drop_left at equiv_segment_4,\n        rw list.take_left at temp_4,\n        rw list.take_take at temp_4,\n\n        obtain ⟨temp_3, equiv_segment_3⟩ :=\n          corresponding_strings_split (\n              u ++ list.map (wrap_symbol₁ g₂.nt) r₁.input_L\n            ).length temp_4,\n        clear temp_4,\n        rw list.drop_left at equiv_segment_3,\n        rw list.take_left at temp_3,\n        rw list.take_take at temp_3,\n\n        obtain ⟨equiv_segment_1, equiv_segment_2⟩ :=\n          corresponding_strings_split u.length temp_3,\n        clear temp_3,\n        rw list.drop_left at equiv_segment_2,\n        rw list.take_left at equiv_segment_1,\n        rw list.take_take at equiv_segment_1,\n\n        have equiv_sgmnt_1 :\n          corresponding_strings (list.take u.length (list.map (wrap_symbol₁ g₂.nt) x)) u,\n        {\n          simpa using equiv_segment_1,\n        },\n        have equiv_sgmnt_2 :\n          corresponding_strings\n            (list.drop u.length (list.take (u.length + r₁.input_L.length)\n              (list.map (wrap_symbol₁ g₂.nt) x)))\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_L),\n        {\n          simpa using equiv_segment_2,\n        },\n        have equiv_sgmnt_3 :\n          corresponding_strings\n            (list.drop (u.length + r₁.input_L.length)\n              (list.take (u.length + (r₁.input_L.length + 1)) (list.map (wrap_symbol₁ g₂.nt) x)))\n            [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))],\n        {\n          simpa using equiv_segment_3,\n        },\n        have equiv_sgmnt_4 :\n          corresponding_strings\n            (list.drop (u.length + (r₁.input_L.length + 1))\n              (list.take (u.length + (r₁.input_L.length + (r₁.input_R.length + 1)))\n                (list.map (wrap_symbol₁ g₂.nt) x)))\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_R),\n        {\n          simpa using equiv_segment_4,\n        },\n        have equiv_sgmnt_5 :\n          corresponding_strings\n            (list.drop (u.length + (r₁.input_L.length + (r₁.input_R.length + 1)))\n              (list.map (wrap_symbol₁ g₂.nt) x))\n            (list.take (x.length - u.length - m) v),\n        {\n          simpa using equiv_segment_5,\n        },\n        clear equiv_segment_1 equiv_segment_2 equiv_segment_3 equiv_segment_4 equiv_segment_5,\n\n        have segment_1_eqi : corresponding_strings (list.map (wrap_symbol₁ g₂.nt) (list.take u.length x)) u,\n        {\n          convert equiv_sgmnt_1,\n          rw list.map_take,\n        },\n        have segment_1_equ := (filter_map_unwrap_of_corresponding_strings₁ segment_1_eqi).symm,\n        rw ←list.take_append_drop u.length x,\n        apply congr_arg2,\n        {\n          exact segment_1_equ,\n        },\n        clear segment_1_equ segment_1_eqi equiv_sgmnt_1,\n\n        have segment_2_eqi :\n          corresponding_strings\n            (list.map (wrap_symbol₁ g₂.nt) (list.take r₁.input_L.length (list.drop u.length x)))\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_L),\n        {\n          convert equiv_sgmnt_2,\n          rw list.map_take,\n          rw list.map_drop,\n          rw list.drop_take,\n        },\n        have segment_2_equ := (filter_map_unwrap_of_corresponding_strings₁ segment_2_eqi).symm,\n        rw unwrap_wrap₁_string at segment_2_equ,\n        rw ←list.take_append_drop r₁.input_L.length (list.drop u.length x),\n        apply congr_arg2,\n        {\n          exact segment_2_equ,\n        },\n        clear segment_2_equ segment_2_eqi equiv_sgmnt_2,\n        rw list.drop_drop,\n\n        have segment_3_eqi :\n          corresponding_strings\n            (list.map (wrap_symbol₁ g₂.nt) (list.take 1 (list.drop (r₁.input_L.length + u.length) x)))\n            (list.map (wrap_symbol₁ g₂.nt) [symbol.nonterminal r₁.input_N]),\n        {\n          convert equiv_sgmnt_3,\n          rw list.map_take,\n          rw list.map_drop,\n          rw ←add_assoc,\n          rw list.drop_take,\n          rw add_comm,\n        },\n        have segment_3_equ := (filter_map_unwrap_of_corresponding_strings₁ segment_3_eqi).symm,\n        rw unwrap_wrap₁_string at segment_3_equ,\n        rw ←list.take_append_drop 1 (list.drop (r₁.input_L.length + u.length) x),\n        apply congr_arg2,\n        {\n          exact segment_3_equ,\n        },\n        clear segment_3_equ segment_3_eqi equiv_sgmnt_3,\n        rw list.drop_drop,\n\n        have segment_4_eqi :\n          corresponding_strings\n            (list.map (wrap_symbol₁ g₂.nt) (list.take r₁.input_R.length\n              (list.drop (1 + (r₁.input_L.length + u.length)) x)))\n            (list.map (wrap_symbol₁ g₂.nt) r₁.input_R),\n        {\n          convert equiv_sgmnt_4,\n          rw list.map_take,\n          rw list.map_drop,\n\n          have sum_rearrange :\n            u.length + (r₁.input_L.length + (r₁.input_R.length + 1)) =\n            (u.length + (r₁.input_L.length + 1)) + r₁.input_R.length,\n          {\n            linarith,\n          },\n          rw sum_rearrange,\n          rw list.drop_take,\n\n          have small_sum_rearr :\n            1 + (r₁.input_L.length + u.length) =\n            u.length + (r₁.input_L.length + 1),\n          {\n            linarith,\n          },\n          rw small_sum_rearr,\n        },\n        have segment_4_equ := (filter_map_unwrap_of_corresponding_strings₁ segment_4_eqi).symm,\n        rw unwrap_wrap₁_string at segment_4_equ,\n        rw ←list.take_append_drop r₁.input_R.length\n          (list.drop (1 + (r₁.input_L.length + u.length)) x),\n        apply congr_arg2,\n        {\n          exact segment_4_equ,\n        },\n        clear segment_4_equ segment_4_eqi equiv_sgmnt_4,\n\n        rw list.drop_drop,\n        repeat {\n          rw list.length_append,\n        },\n        repeat {\n          rw list.length_take,\n        },\n        repeat {\n          rw list.length_drop,\n        },\n\n        have sum_of_min_lengths :\n          min u.length x.length +\n            (min r₁.input_L.length (x.length - u.length) +\n            (min 1 (x.length - (r₁.input_L.length + u.length)) +\n            (min r₁.input_R.length (x.length - (1 + (r₁.input_L.length + u.length))) +\n            (x.length - (r₁.input_R.length + (1 + (r₁.input_L.length + u.length))))))) =\n          x.length,\n        {\n          have add_mirror :\n            r₁.input_R.length + 1 + r₁.input_L.length =\n            r₁.input_L.length + 1 + r₁.input_R.length,\n          {\n            ring,\n          },\n          rw [list.length_map, list.length_map, ←add_mirror] at critical,\n\n          have min1 : min u.length x.length = u.length,\n          {\n            apply min_eq_left,\n            exact ul_le_xl,\n          },\n          have min2 : min r₁.input_L.length (x.length - u.length) = r₁.input_L.length,\n          {\n            clear_except critical,\n            apply min_eq_left,\n            apply le_trans _ critical,\n            apply le_add_self,\n          },\n          have min3 : min 1 (x.length - (r₁.input_L.length + u.length)) = 1,\n          {\n            clear_except critical,\n            apply min_eq_left,\n            omega,\n          },\n          have min4 :\n            min r₁.input_R.length (x.length - (1 + (r₁.input_L.length + u.length))) =\n            r₁.input_R.length,\n          {\n            clear_except critical,\n            apply min_eq_left,\n            omega,\n          },\n          rw [min1, min2, min3, min4],\n          rw le_tsub_iff_right ul_le_xl at critical,\n          clear_except critical add_mirror,\n          repeat {\n            rw ←add_assoc,\n          },\n          have sum_eq_sum :\n            u.length + r₁.input_L.length + 1 + r₁.input_R.length =\n            r₁.input_R.length + 1 + r₁.input_L.length + u.length,\n          {\n            rw add_mirror,\n            rw add_assoc,\n            rw add_assoc,\n            rw add_comm,\n            rw ←add_assoc _ 1 _,\n          },\n          rw sum_eq_sum,\n          exact nat.add_sub_of_le critical,\n        },\n        rw sum_of_min_lengths,\n        clear_except equiv_sgmnt_5,\n\n        have another_rearranging :\n          r₁.input_R.length + (1 + (r₁.input_L.length + u.length)) =\n          u.length + (r₁.input_L.length + (r₁.input_R.length + 1)),\n        {\n          ring,\n        },\n        rw another_rearranging,\n        rw ←list.map_drop at equiv_sgmnt_5,\n        symmetry,\n        exact filter_map_unwrap_of_corresponding_strings₁ equiv_sgmnt_5,\n      },\n      {\n        rw list.filter_map_append_append,\n        congr,\n        rw unwrap_wrap₁_string,\n      },\n    },\n    {\n      exact ih_y,\n    },\n  },\n  rw aft,\n  rw bef at ih_concat,\n  rw list.filter_map_append_append,\n  rw list.map_append_append,\n  rw list.append_assoc,\n  rw list.append_assoc,\n\n  apply corresponding_strings_append,\n  {\n    have part_for_u := corresponding_strings_take u.length ih_concat,\n    rw list.take_left at part_for_u,\n    have trivi : u.length ≤ (list.map (wrap_symbol₁ g₂.nt) x).length,\n    {\n      clear_except critical,\n      rw list.length_map,\n      omega,\n    },\n    rw list.take_append_of_le_length trivi at part_for_u,\n    clear_except part_for_u,\n    rw ←list.map_take at part_for_u,\n    apply corresponding_string_after_wrap_unwrap_self₁,\n    use list.take u.length x,\n    exact part_for_u,\n  },\n  apply corresponding_strings_append,\n  {\n    rw unwrap_wrap₁_string,\n    apply corresponding_strings_self,\n  },\n  convert_to\n    corresponding_strings _ (list.take (x.length - u.length - m) v ++ list.drop (x.length - u.length - m) v),\n  {\n    rw list.take_append_drop,\n  },\n  apply corresponding_strings_append,\n  {\n    have eqi := corresponding_strings_take (list.map (wrap_symbol₁ g₂.nt) x).length ih_concat,\n    rw list.take_left at eqi,\n    have part_for_v_beginning := corresponding_strings_drop (u.length + m) eqi,\n    clear_except part_for_v_beginning critical,\n    rw ←list.map_drop at part_for_v_beginning,\n    apply corresponding_string_after_wrap_unwrap_self₁,\n    use list.drop (u.length + m) x,\n    convert part_for_v_beginning,\n    clear part_for_v_beginning,\n    rw list.length_map,\n    rw list.take_append_eq_append_take,\n    rw list.drop_append_eq_append_drop,\n    have tul_lt : (list.take x.length u).length ≤ u.length + m,\n    {\n      rw list.length_take,\n      calc min x.length u.length\n          ≤ u.length     : min_le_right _ _\n      ... ≤ u.length + m : le_self_add,\n    },\n    rw list.drop_eq_nil_of_le tul_lt,\n    rw list.nil_append,\n    rw ←list.append_assoc _ _ v,\n    rw ←list.append_assoc _ _ v,\n    rw ←list.append_assoc,\n    rw list.take_append_eq_append_take,\n    rw list.drop_append_eq_append_drop,\n    have rul_inp_len :\n      (list.map (wrap_symbol₁ g₂.nt) r₁.input_L ++\n          [symbol.nonterminal (sum.inl (some (sum.inl r₁.input_N)))] ++\n          list.map (wrap_symbol₁ g₂.nt) r₁.input_R\n        ).length = m,\n    {\n      rw list.length_append_append,\n      rw list.length_singleton,\n    },\n    have u_is_shorter : min x.length u.length = u.length,\n    {\n      apply min_eq_right,\n      clear_except critical,\n      omega,\n    },\n    rw list.drop_eq_nil_of_le, swap,\n    {\n      rw list.length_take,\n      rw rul_inp_len,\n      rw list.length_take,\n      rw u_is_shorter,\n      calc min (x.length - u.length) m ≤ m : min_le_right _ _\n      ... ≤ u.length + m - u.length        : le_add_tsub_swap,\n    },\n    rw list.nil_append,\n    rw list.length_take,\n    rw list.length_take,\n    rw rul_inp_len,\n    have zero_dropping : u.length + m - min x.length u.length - min (x.length - u.length) m = 0,\n    {\n      have middle_cannot_exceed : min (x.length - u.length) m = m,\n      {\n        exact min_eq_right critical,\n      },\n      rw [u_is_shorter, middle_cannot_exceed],\n      clear_except,\n      omega,\n    },\n    rw zero_dropping,\n    unfold list.drop,\n  },\n  -- now we have what `g₂` generated\n  have reverse_concat := corresponding_strings_reverse ih_concat,\n  repeat {\n    rw list.reverse_append at reverse_concat,\n  },\n  have the_part := corresponding_strings_take y.length reverse_concat,\n  apply corresponding_strings_of_reverse,\n\n  have len_sum : y.length + (x.length - u.length - m) = v.length,\n  {\n    change\n      y.length + (x.length - u.length - (\n        (list.map (wrap_symbol₁ g₂.nt) r₁.input_L).length + 1 +\n        (list.map (wrap_symbol₁ g₂.nt) r₁.input_R).length\n      )) =\n      v.length,\n    have len_concat := corresponding_strings_length ih_concat,\n    clear_except len_concat critical,\n    repeat {\n      rw list.length_append at len_concat,\n    },\n    rw list.length_map at len_concat,\n    rw list.length_map at len_concat,\n    rw list.length_singleton at len_concat,\n    rw ←nat.add_sub_assoc, swap,\n    {\n      exact critical,\n    },\n    rw ←nat.add_sub_assoc, swap,\n    {\n      clear_except critical,\n      omega,\n    },\n    rw add_comm at len_concat,\n    rw len_concat,\n    clear len_concat,\n    rw add_tsub_cancel_left,\n    rw ←nat.add_assoc,\n    rw ←nat.add_assoc,\n    rw add_tsub_cancel_left,\n  },\n  have yl_lt_vl : y.length ≤ v.length,\n  {\n    exact nat.le.intro len_sum,\n  },\n  convert_to corresponding_strings _ (list.take y.length v.reverse),\n  {\n    convert_to (list.drop (v.length - y.length) v).reverse = list.take y.length v.reverse,\n    {\n      apply congr_arg,\n      apply congr_arg2,\n      {\n        clear_except len_sum,\n        omega,\n      },\n      {\n        refl,\n      },\n    },\n    rw list.reverse_take,\n    exact yl_lt_vl,\n  },\n  clear_except the_part yl_lt_vl,\n  rw list.take_append_of_le_length at the_part, swap,\n  {\n    rw list.length_reverse,\n    rw list.length_map,\n  },\n  repeat {\n    rw list.append_assoc at the_part,\n  },\n  rw list.take_append_of_le_length at the_part, swap,\n  {\n    rw list.length_reverse,\n    exact yl_lt_vl,\n  },\n  rw list.take_all_of_le at the_part, swap,\n  {\n    rw list.length_reverse,\n    rw list.length_map,\n  },\n  exact the_part,\nend\n\nprivate lemma induction_step_for_lifted_rule_from_g₂\n    {g₁ g₂ : grammar T}\n    {a b u v : list (nst T g₁.nt g₂.nt)}\n    {x : list (symbol T g₁.nt)}\n    {y : list (symbol T g₂.nt)}\n    {r : grule T (nnn T g₁.nt g₂.nt)}\n    (rin : r ∈ list.map (wrap_grule₂ g₁.nt) g₂.rules)\n    (bef : a = u ++ r.input_L ++ [symbol.nonterminal r.input_N] ++ r.input_R ++ v)\n    (aft : b = u ++ r.output_string ++ v)\n    (ih_x : grammar_derives g₁ [symbol.nonterminal g₁.initial] x)\n    (ih_y : grammar_derives g₂ [symbol.nonterminal g₂.initial] y)\n    (ih_concat :\n      corresponding_strings\n        (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)\n        a) :\n  ∃ (y' : list (symbol T g₂.nt)),\n    and\n      (and\n        (grammar_derives g₁ [symbol.nonterminal g₁.initial] x)\n        (grammar_derives g₂ [symbol.nonterminal g₂.initial] y')\n      )\n      (corresponding_strings (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y') b) :=\nbegin\n  rw list.mem_map at rin,\n  rcases rin with ⟨r₂, rin₂, wrap_r₂_eq_r⟩,\n  rw ←wrap_r₂_eq_r at *,\n  clear wrap_r₂_eq_r,\n  simp [wrap_grule₂] at *,\n  rw ←list.singleton_append at bef,\n  rw bef at ih_concat,\n\n  let b' := list.drop x.length u ++ list.map (wrap_symbol₂ g₁.nt) r₂.output_string ++ v,\n  use list.filter_map unwrap_symbol₂ b',\n\n  have total_len := corresponding_strings_length ih_concat,\n  repeat {\n    rw list.length_append at total_len,\n  },\n  repeat {\n    rw list.length_map at total_len,\n  },\n\n  have matched_right : u.length ≥ x.length,\n  {\n    clear_except ih_concat total_len,\n    by_contradiction,\n    push_neg at h,\n    rename h ul_lt_xl,\n    have ul_lt_ihls : u.length < (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).length,\n    {\n      rw list.length_append,\n      rw list.length_map,\n      rw list.length_map,\n      exact nat.lt_add_right _ _ _ ul_lt_xl,\n    },\n    have ul_lt_ihrs :\n      u.length <\n      (u\n        ++ (list.map (wrap_symbol₂ g₁.nt) r₂.input_L\n        ++ ([symbol.nonterminal (sum.inl (some (sum.inr r₂.input_N)))]\n        ++ (list.map (wrap_symbol₂ g₁.nt) r₂.input_R\n        ++ v)))\n      ).length,\n    {\n      repeat {\n        rw list.length_append,\n      },\n      rw list.length_singleton,\n      clear_except,\n      linarith,\n    },\n    have ulth := corresponding_strings_nth_le ul_lt_ihls ul_lt_ihrs ih_concat,\n    rw list.nth_le_append ul_lt_ihls at ulth, swap,\n    {\n      rw list.length_map,\n      exact ul_lt_xl,\n    },\n    rw list.nth_le_append_right at ulth, swap,\n    {\n      refl,\n    },\n    rw list.nth_le_map at ulth, swap,\n    {\n      exact ul_lt_xl,\n    },\n\n    by_cases (list.map (wrap_symbol₂ g₁.nt) r₂.input_L).length > u.length - u.length,\n    {\n      rw list.nth_le_append _ h at ulth,\n      rw list.nth_le_map at ulth, swap,\n      {\n        rw list.length_map at h,\n        exact h,\n      },\n      exact corresponding_symbols_never₁ ulth,\n    },\n    push_neg at h,\n    rw list.nth_le_append_right h at ulth,\n    have matched_central_nt :\n      corresponding_symbols\n        (wrap_symbol₁ g₂.nt (x.nth_le u.length ul_lt_xl))\n        (wrap_symbol₂ g₁.nt (symbol.nonterminal r₂.input_N)),\n    {\n      clear_except ulth,\n      finish,\n    },\n    exact corresponding_symbols_never₁ matched_central_nt,\n  },\n\n  split,\n  {\n    split,\n    {\n      exact ih_x,\n    },\n    {\n      apply grammar_deri_of_deri_tran ih_y,\n      use r₂,\n      split,\n      {\n        exact rin₂,\n      },\n      use list.filter_map unwrap_symbol₂ (list.drop x.length u),\n      use list.filter_map unwrap_symbol₂ v,\n      split,\n      {\n        have corres_y := corresponding_strings_drop (list.map (wrap_symbol₁ g₂.nt) x).length ih_concat,\n        rw list.drop_left at corres_y,\n        rw list.drop_append_of_le_length at corres_y, swap,\n        {\n          rw list.length_map,\n          exact matched_right,\n        },\n        clear_except corres_y total_len,\n        repeat {\n          rw list.append_assoc,\n        },\n\n        obtain ⟨seg1, rest1⟩ :=\n          corresponding_strings_split\n            (list.drop (list.map (wrap_symbol₁ g₂.nt) x).length u).length\n            corres_y,\n        clear corres_y,\n        rw list.take_left at seg1,\n        rw list.drop_left at rest1,\n        rw ←list.take_append_drop (list.filter_map unwrap_symbol₂ (list.drop x.length u)).length y,\n        rw ←list.map_take at seg1,\n        have min_uxy : min (u.length - x.length) y.length = u.length - x.length,\n        {\n          rw min_eq_left,\n          clear_except total_len,\n          omega,\n        },\n        have tuxy : list.take (list.take (u.length - x.length) y).length y = list.take (u.length - x.length) y,\n        {\n          rw list.length_take,\n          rw min_uxy,\n        },\n        have fmu1 := filter_map_unwrap_of_corresponding_strings₂ seg1,\n        rw list.length_map at fmu1,\n        have fml :\n          (list.filter_map unwrap_symbol₂ (list.drop x.length u)).length =\n          (list.drop x.length u).length,\n        {\n          rw congr_arg list.length fmu1,\n          rw list.length_take at tuxy,\n          rw list.length_drop,\n          rw min_uxy at tuxy,\n          rw congr_arg list.length tuxy,\n          rw list.length_take,\n          exact min_uxy,\n        },\n        apply congr_arg2,\n        {\n          rw fmu1,\n          rw list.length_drop,\n          exact tuxy,\n        },\n        clear seg1 fmu1 tuxy min_uxy,\n\n        rw list.length_map at rest1,\n        obtain ⟨seg2, rest2⟩ :=\n          corresponding_strings_split\n            (list.map (wrap_symbol₂ g₁.nt) r₂.input_L).length\n            rest1,\n        clear rest1,\n        rw list.take_left at seg2,\n        rw list.drop_left at rest2,\n        rw ←list.take_append_drop\n          (list.map (wrap_symbol₂ g₁.nt) r₂.input_L).length\n          (list.drop (list.filter_map unwrap_symbol₂ (list.drop x.length u)).length y),\n        apply congr_arg2,\n        {\n          clear_except seg2 fml,\n          rw ←list.map_drop at seg2,\n          rw ←list.map_take at seg2,\n          have fmu2 := filter_map_unwrap_of_corresponding_strings₂ seg2,\n          rw list.length_map at fmu2,\n          rw list.length_map,\n          rw unwrap_wrap₂_string at fmu2,\n          rw fml,\n          exact fmu2.symm,\n        },\n        clear seg2,\n\n        rw list.length_map at rest2,\n        rw list.drop_drop at rest2 ⊢,\n        obtain ⟨seg3, rest3⟩ :=\n          corresponding_strings_split 1 rest2,\n        clear rest2,\n        rw list.take_left' (list.length_singleton _) at seg3,\n        rw list.drop_left' (list.length_singleton _) at rest3,\n        rw list.length_map,\n        rw fml,\n        rw ←list.take_append_drop\n          1\n          (list.drop (r₂.input_L.length + (list.drop x.length u).length) y),\n        apply congr_arg2,\n        {\n          rw ←list.map_drop at seg3,\n          rw ←list.map_take at seg3,\n          have fmu3 := filter_map_unwrap_of_corresponding_strings₂ seg3,\n          exact fmu3.symm,\n        },\n        clear seg3,\n\n        rw list.drop_drop at rest3 ⊢,\n        rw ←list.map_drop at rest3,\n        rw ←filter_map_unwrap_of_corresponding_strings₂ rest3,\n        rw list.filter_map_append,\n        rw unwrap_wrap₂_string,\n      },\n      {\n        rw list.filter_map_append_append,\n        congr,\n        apply unwrap_wrap₂_string,\n      }\n    },\n  },\n  rw aft,\n  rw list.filter_map_append_append,\n  rw list.map_append_append,\n  rw list.append_assoc,\n  rw ←list.append_assoc (list.map (wrap_symbol₁ g₂.nt) x),\n\n  apply corresponding_strings_append, swap,\n  {\n    rw unwrap_wrap₂_string,\n    apply corresponding_strings_append,\n    {\n      apply corresponding_strings_self,\n    },\n    apply corresponding_string_after_wrap_unwrap_self₂,\n    repeat {\n      rw ←list.append_assoc at ih_concat,\n    },\n    have rev := corresponding_strings_reverse ih_concat,\n    rw list.reverse_append _ v at rev,\n    have tak := corresponding_strings_take v.reverse.length rev,\n    rw list.take_left at tak,\n    have rtr := corresponding_strings_reverse tak,\n    have nec : v.reverse.length ≤ (list.map (wrap_symbol₂ g₁.nt) y).reverse.length,\n    {\n      clear_except matched_right total_len,\n      rw list.length_reverse,\n      rw list.length_reverse,\n      rw list.length_map,\n      linarith,\n    },\n    clear_except rtr nec,\n\n    rw list.reverse_reverse at rtr,\n    rw list.reverse_append at rtr,\n    rw list.take_append_of_le_length nec at rtr,\n    rw list.reverse_take at rtr, swap,\n    {\n      rw list.length_reverse (list.map (wrap_symbol₂ g₁.nt) y) at nec,\n      exact nec,\n    },\n    rw ←list.map_drop at rtr,\n    rw list.reverse_reverse at rtr,\n    exact ⟨_, rtr⟩,\n  },\n  rw ←list.take_append_drop x.length u,\n  apply corresponding_strings_append,\n  {\n    have almost := corresponding_strings_take x.length ih_concat,\n    rw list.take_append_of_le_length matched_right at almost,\n    convert almost,\n    have xl_eq : x.length = (list.map (wrap_symbol₁ g₂.nt) x).length,\n    {\n      rw list.length_map,\n    },\n    rw xl_eq,\n    rw list.take_left,\n  },\n  {\n    rw list.take_append_drop,\n    apply corresponding_string_after_wrap_unwrap_self₂,\n    have tdc := corresponding_strings_drop x.length (corresponding_strings_take u.length ih_concat),\n    rw list.take_left at tdc,\n    have ul_eq : u.length = x.length + (u.length - x.length),\n    {\n      rw ←nat.add_sub_assoc matched_right,\n      rw add_comm,\n      rw nat.add_sub_assoc, swap,\n      {\n        refl,\n      },\n      rw nat.sub_self,\n      rw add_zero,\n    },\n    rw ul_eq at tdc,\n    clear_except tdc,\n\n    rw list.drop_take at tdc,\n    rw list.drop_left' at tdc, swap,\n    {\n      apply list.length_map,\n    },\n    rw ←list.map_take at tdc,\n    exact ⟨_, tdc⟩,\n  },\nend\n\nprivate lemma big_induction\n    {g₁ g₂ : grammar T}\n    {w : list (nst T g₁.nt g₂.nt)}\n    (ass :\n      grammar_derives (big_grammar g₁ g₂)\n        [symbol.nonterminal (sum.inl (some (sum.inl g₁.initial))),\n         symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))]\n        w) :\n  ∃ x : list (symbol T g₁.nt),\n  ∃ y : list (symbol T g₂.nt),\n    and\n      (and\n        (grammar_derives g₁ [symbol.nonterminal g₁.initial] x)\n        (grammar_derives g₂ [symbol.nonterminal g₂.initial] y)\n      )\n      (corresponding_strings (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) w) :=\nbegin\n  induction ass with a b trash orig ih,\n  {\n    use [[symbol.nonterminal g₁.initial], [symbol.nonterminal g₂.initial]],\n    split,\n    {\n      split;\n      apply grammar_deri_self,\n    },\n    {\n      rw list.map_singleton,\n      rw list.map_singleton,\n      unfold wrap_symbol₁,\n      unfold wrap_symbol₂,\n      rw list.singleton_append,\n      unfold corresponding_strings,\n      rw list.forall₂_cons,\n      split,\n      {\n        unfold corresponding_symbols,\n      },\n      rw list.forall₂_cons,\n      split,\n      {\n        unfold corresponding_symbols,\n      },\n      exact list.forall₂.nil,\n    },\n  },\n  rcases ih with ⟨x, y, ⟨ih_x, ih_y⟩, ih_concat⟩,\n  rcases orig with ⟨r, rin, u, v, bef, aft⟩,\n  change _ ∈ list.cons _ _ at rin,\n  rw list.mem_cons_eq at rin,\n  cases rin,\n  {\n    exfalso,\n    rw rin at bef,\n    clear_except ih_concat bef,\n    simp only [list.append_nil] at bef,\n    rw bef at ih_concat,\n    have same_lengths := corresponding_strings_length ih_concat,\n    clear bef,\n    have ulen₁ : u.length < (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).length,\n    {\n      rw list.length_append _ v at same_lengths,\n      rw list.length_append u _ at same_lengths,\n      rw list.length_singleton at same_lengths,\n      clear_except same_lengths,\n      linarith,\n    },\n    have ulen₂ : u.length < (u ++ ([symbol.nonterminal (sum.inl none)] ++ v)).length,\n    {\n      rw list.length_append,\n      rw list.length_append,\n      rw list.length_singleton,\n      clear_except,\n      linarith,\n    },\n    have ulen_tauto : u.length ≤ u.length,\n    {\n      refl,\n    },\n    rw list.append_assoc at ih_concat,\n    have eqi_symb := corresponding_strings_nth_le ulen₁ ulen₂ ih_concat,\n    rw list.nth_le_append_right ulen_tauto at eqi_symb,\n    simp only [nat.sub_self, list.singleton_append, list.nth_le] at eqi_symb,\n    have eq_none :\n      (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le u.length ulen₁ =\n      (symbol.nonterminal (sum.inl (none))),\n    {\n      clear_except eqi_symb,\n      cases (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le u.length ulen₁ with t s,\n      {\n        exfalso,\n        unfold corresponding_symbols at eqi_symb,\n        exact eqi_symb,\n      },\n      cases s,\n      {\n        cases s,\n        {\n          refl,\n        },\n        exfalso,\n        clear_except eqi_symb,\n        cases s;\n        tauto,\n      },\n      {\n        exfalso,\n        clear_except eqi_symb,\n        cases s;\n        tauto,\n      },\n    },\n    have impossible_in :\n      symbol.nonterminal (sum.inl none) ∈\n        (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y),\n    {\n      rw list.mem_iff_nth_le,\n      use u.length,\n      use ulen₁,\n      exact eq_none,\n    },\n    rw list.mem_append at impossible_in,\n    cases impossible_in;\n    {\n      rw list.mem_map at impossible_in,\n      rcases impossible_in with ⟨s, -, contradic⟩,\n      clear_except contradic,\n      cases s,\n      {\n        have imposs := symbol.nonterminal.inj contradic,\n        exact sum.no_confusion imposs,\n      },\n      {\n        have impos := sum.inl.inj (symbol.nonterminal.inj contradic),\n        exact option.no_confusion impos,\n      },\n    },\n  },\n  rw list.mem_append at rin,\n  cases rin,\n  {\n    rw list.mem_append at rin,\n    cases rin,\n    {\n      cases induction_step_for_lifted_rule_from_g₁ rin bef aft ih_x ih_y ih_concat with x' pros,\n      exact ⟨x', y, pros⟩,\n    },\n    {\n      use x,\n      exact induction_step_for_lifted_rule_from_g₂ rin bef aft ih_x ih_y ih_concat,\n    }\n  },\n  {\n    use [x, y],\n    split,\n    {\n      split,\n      {\n        exact ih_x,\n      },\n      {\n        exact ih_y,\n      },\n    },\n    rw aft,\n    rw bef at ih_concat,\n\n    rw list.mem_append at rin,\n    cases rin,\n    {\n      unfold rules_for_terminals₁ at rin,\n      rw list.mem_map at rin,\n      rcases rin with ⟨t, -, eq_r⟩,\n      rw ←eq_r at *,\n      clear eq_r,\n      rw list.append_nil at ih_concat,\n      rw list.append_nil at ih_concat,\n\n      have xy_split_u :\n        list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y =\n          list.take u.length (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) ++\n          list.drop u.length (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y),\n      {\n        rw list.take_append_drop,\n      },\n      rw xy_split_u,\n      have part_for_u := corresponding_strings_take u.length ih_concat,\n      rw list.append_assoc,\n      apply corresponding_strings_append,\n      {\n        convert part_for_u,\n        rw list.append_assoc,\n        rw list.take_left,\n      },\n      have ul_lt_len_um : u.length < (u ++ [symbol.nonterminal (sum.inr (sum.inl t))]).length,\n      {\n        rw list.length_append,\n        rw list.length_singleton,\n        apply lt_add_one,\n      },\n      have ul_lt_len_umv : u.length < (u ++ [symbol.nonterminal (sum.inr (sum.inl t))] ++ v).length,\n      {\n        rw list.length_append,\n        apply lt_of_lt_of_le ul_lt_len_um,\n        apply le_self_add,\n      },\n      have ul_lt_len_xy : u.length < (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).length,\n      {\n        have same_len := corresponding_strings_length ih_concat,\n        rw same_len,\n        exact ul_lt_len_umv,\n      },\n      have middle_nt := corresponding_strings_nth_le ul_lt_len_xy ul_lt_len_umv ih_concat,\n      rw list.nth_le_append ul_lt_len_umv ul_lt_len_um at middle_nt,\n      rw list.nth_le_append_right (by refl) ul_lt_len_um at middle_nt,\n      have middle_nt_elem :\n        corresponding_symbols\n          ((list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le u.length ul_lt_len_xy)\n          (symbol.nonterminal (sum.inr (sum.inl t))),\n      {\n        convert middle_nt,\n        finish,\n      },\n      have xy_split_nt :\n        list.drop u.length (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) =\n          list.take 1 (list.drop u.length (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)) ++\n          list.drop 1 (list.drop u.length (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)),\n      {\n        rw list.take_append_drop,\n      },\n      rw xy_split_nt,\n      apply corresponding_strings_append, swap,\n      {\n        rw list.drop_drop,\n        have part_for_v := corresponding_strings_drop (1 + u.length) ih_concat,\n        convert part_for_v,\n        have correct_len : 1 + u.length = (u ++ [symbol.nonterminal (sum.inr (sum.inl t))]).length,\n        {\n          rw add_comm,\n          rw list.length_append,\n          rw list.length_singleton,\n        },\n        rw correct_len,\n        rw list.drop_left,\n      },\n      {\n        convert_to\n          corresponding_strings\n            [list.nth_le (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) u.length ul_lt_len_xy]\n            [symbol.terminal t],\n        {\n          apply list_take_one_drop,\n        },\n        clear_except middle_nt_elem,\n        apply corresponding_strings_singleton,\n        cases\n          (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le\n            u.length ul_lt_len_xy with e s,\n        {\n          exfalso,\n          unfold corresponding_symbols at middle_nt_elem,\n          exact middle_nt_elem,\n        },\n        cases s,\n        {\n          exfalso,\n          cases s, swap,\n            cases s,\n          all_goals {\n            unfold corresponding_symbols at middle_nt_elem,\n            exact middle_nt_elem,\n          },\n        },\n        {\n          cases s,\n          {\n            unfold corresponding_symbols at middle_nt_elem,\n            rw middle_nt_elem,\n            unfold corresponding_symbols,\n          },\n          {\n            exfalso,\n            unfold corresponding_symbols at middle_nt_elem,\n            exact middle_nt_elem,\n          },\n        }\n      },\n    },\n    {\n      unfold rules_for_terminals₂ at rin,\n      rw list.mem_map at rin,\n      rcases rin with ⟨t, -, eq_r⟩,\n      rw ←eq_r at *,\n      clear eq_r,\n      rw list.append_nil at ih_concat,\n      rw list.append_nil at ih_concat,\n\n      have xy_split_v :\n        list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y =\n          list.take (u.length + 1) (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) ++\n          list.drop (u.length + 1) (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y),\n      {\n        rw list.take_append_drop,\n      },\n      rw xy_split_v,\n      have part_for_v := corresponding_strings_drop (u.length + 1) ih_concat,\n      apply corresponding_strings_append, swap,\n      {\n        convert part_for_v,\n        have rewr_len : u.length + 1 = (u ++ [symbol.nonterminal (sum.inr (sum.inr t))]).length,\n        {\n          rw list.length_append,\n          rw list.length_singleton,\n        },\n        rw rewr_len,\n        rw list.drop_left,\n      },\n      have ul_lt_len_um : u.length < (u ++ [symbol.nonterminal (sum.inr (sum.inr t))]).length,\n      {\n        rw list.length_append,\n        rw list.length_singleton,\n        apply lt_add_one,\n      },\n      have ul_lt_len_umv : u.length < (u ++ [symbol.nonterminal (sum.inr (sum.inr t))] ++ v).length,\n      {\n        rw list.length_append,\n        apply lt_of_lt_of_le ul_lt_len_um,\n        apply le_self_add,\n      },\n      have ul_lt_len_xy : u.length < (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).length,\n      {\n        have same_len := corresponding_strings_length ih_concat,\n        rw same_len,\n        exact ul_lt_len_umv,\n      },\n      have middle_nt := corresponding_strings_nth_le ul_lt_len_xy ul_lt_len_umv ih_concat,\n      rw list.nth_le_append ul_lt_len_umv ul_lt_len_um at middle_nt,\n      rw list.nth_le_append_right (by refl) ul_lt_len_um at middle_nt,\n      have middle_nt_elem :\n        corresponding_symbols\n          ((list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le u.length ul_lt_len_xy)\n          (symbol.nonterminal (sum.inr (sum.inr t))),\n      {\n        convert middle_nt,\n        finish,\n      },\n      have xy_split_nt :\n        list.take (u.length + 1) (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) =\n          list.take u.length (list.take (u.length + 1) (\n              list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)\n            ) ++\n          list.drop u.length (list.take (u.length + 1) (\n              list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y)\n            ),\n      {\n        rw list.take_append_drop u.length,\n      },\n      rw xy_split_nt,\n      apply corresponding_strings_append,\n      {\n        rw list.take_take,\n        have part_for_u := corresponding_strings_take u.length ih_concat,\n        convert part_for_u,\n        {\n          apply min_eq_left,\n          apply nat.le_succ,\n        },\n        rw list.append_assoc,\n        rw list.take_left,\n      },\n      {\n        convert_to\n          corresponding_strings\n            [list.nth_le (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) u.length ul_lt_len_xy]\n            [symbol.terminal t],\n        {\n          apply list_drop_take_succ,\n        },\n        clear_except middle_nt_elem,\n        apply corresponding_strings_singleton,\n        cases\n          (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).nth_le\n            u.length ul_lt_len_xy with e s,\n        {\n          exfalso,\n          unfold corresponding_symbols at middle_nt_elem,\n          exact middle_nt_elem,\n        },\n        cases s,\n        {\n          exfalso,\n          cases s, swap,\n            cases s,\n          all_goals {\n            unfold corresponding_symbols at middle_nt_elem,\n            exact middle_nt_elem,\n          },\n        },\n        {\n          cases s,\n          {\n            exfalso,\n            unfold corresponding_symbols at middle_nt_elem,\n            exact middle_nt_elem,\n          },\n          {\n            unfold corresponding_symbols at middle_nt_elem,\n            rw middle_nt_elem,\n            unfold corresponding_symbols,\n          },\n        }\n      },\n    },\n  },\nend\n\nprotected lemma in_concatenated_of_in_big\n    {g₁ g₂ : grammar T}\n    {w : list T}\n    (ass : w ∈ grammar_language (big_grammar g₁ g₂)) :\n  w ∈ grammar_language g₁ * grammar_language g₂ :=\nbegin\n  rw language.mem_mul,\n  cases grammar_tran_or_id_of_deri ass,\n  {\n    exfalso,\n    have nonmatch := congr_fun (congr_arg list.nth h) 0,\n    clear_except nonmatch,\n    unfold list.nth at nonmatch,\n    cases w,\n    {\n      rw list.map_nil at nonmatch,\n      exact option.no_confusion nonmatch,\n    },\n    {\n      unfold list.map at nonmatch,\n      unfold list.nth at nonmatch,\n      have imposs := option.some.inj nonmatch,\n      exact symbol.no_confusion imposs,\n    },\n  },\n  clear ass,\n  rcases h with ⟨w₁, hyp_tran, hyp_deri⟩,\n  have w₁eq : w₁ = [\n      symbol.nonterminal (sum.inl (some (sum.inl g₁.initial))),\n      symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))\n    ],\n  {\n    clear_except hyp_tran,\n    -- only the first rule is applicable\n    rcases hyp_tran with ⟨r, rin, u, v, bef, aft⟩,\n\n    have bef_len := congr_arg list.length bef,\n    rw list.length_append_append at bef_len,\n    rw list.length_append_append at bef_len,\n    rw list.length_singleton at bef_len,\n    rw list.length_singleton at bef_len,\n    have u_nil : u = [], swap,\n    have v_nil : v = [], swap,\n    have rif_nil : r.input_L = [], swap,\n    any_goals {\n      clear_except bef_len,\n      rw ←list.length_eq_zero,\n      linarith,\n    },\n\n    change _ ∈ list.cons _ _ at rin,\n    rw list.mem_cons_iff at rin,\n    cases rin,\n    {\n      rw rin at bef aft,\n      dsimp only at bef aft,\n      rw [u_nil, v_nil] at aft,\n      rw [list.nil_append, list.append_nil] at aft,\n      exact aft,\n    },\n    exfalso,\n    have nt_match : symbol.nonterminal (big_grammar g₁ g₂).initial = symbol.nonterminal r.input_N,\n    {\n      have bef_fst := congr_fun (congr_arg list.nth bef) 0,\n      rw [u_nil, rif_nil] at bef_fst,\n      rw ←option.some_inj,\n      exact bef_fst,\n    },\n    clear_except rin nt_match,\n\n    repeat {\n      rw list.mem_append at rin,\n    },\n    cases rin,\n    {\n      cases rin,\n      {\n        rw list.mem_map at rin,\n        rcases rin with ⟨r₀, hr₀g₁, wrap_eq_r⟩,\n        rw ←wrap_eq_r at nt_match,\n        unfold wrap_grule₁ at nt_match,\n        have inl_match := symbol.nonterminal.inj nt_match,\n        change sum.inl none = sum.inl (some (sum.inl r₀.input_N)) at inl_match,\n        have none_eq_some := sum.inl.inj inl_match,\n        exact option.no_confusion none_eq_some,\n      },\n      {\n        rw list.mem_map at rin,\n        rcases rin with ⟨r₀, hr₀g₂, wrap_eq_r⟩,\n        rw ←wrap_eq_r at nt_match,\n        unfold wrap_grule₂ at nt_match,\n        have inl_match := symbol.nonterminal.inj nt_match,\n        change sum.inl none = sum.inl (some (sum.inr r₀.input_N)) at inl_match,\n        have none_eq_some := sum.inl.inj inl_match,\n        exact option.no_confusion none_eq_some,\n      },\n    },\n    {\n      cases rin,\n      {\n        unfold rules_for_terminals₁ at rin,\n        rw list.mem_map at rin,\n        rcases rin with ⟨t, htg₁, tt_eq_r⟩,\n        rw ←tt_eq_r at nt_match,\n        have inl_eq_inr := symbol.nonterminal.inj nt_match,\n        exact sum.no_confusion inl_eq_inr,\n      },\n      {\n        unfold rules_for_terminals₂ at rin,\n        rw list.mem_map at rin,\n        rcases rin with ⟨t, htg₂, tt_eq_r⟩,\n        rw ←tt_eq_r at nt_match,\n        have inl_eq_inr := symbol.nonterminal.inj nt_match,\n        exact sum.no_confusion inl_eq_inr,\n      },\n    },\n  },\n  clear hyp_tran,\n  rw w₁eq at hyp_deri,\n\n  have hope_result := big_induction hyp_deri,\n  clear_except hope_result,\n  rcases hope_result with ⟨x, y, ⟨deri_x, deri_y⟩, concat_xy⟩,\n\n  use list.take x.length w,\n  use list.drop x.length w,\n  split,\n  {\n    clear deri_y,\n    unfold grammar_language,\n    rw set.mem_set_of_eq,\n    unfold grammar_generates,\n    convert deri_x,\n    clear deri_x,\n\n    have xylen := corresponding_strings_length concat_xy,\n    rw list.length_append at xylen,\n    repeat {\n      rw list.length_map at xylen,\n    },\n\n    ext1 i,\n    by_cases i ≥ x.length,\n    {\n      convert_to none = none,\n      {\n        have xlen : x.length = (list.map (@symbol.terminal T g₁.nt) (list.take x.length w)).length,\n        {\n          clear_except xylen,\n          rw list.length_map,\n          rw list.length_take,\n          symmetry,\n          apply min_eq_left,\n          exact nat.le.intro xylen,\n        },\n        rw xlen at h,\n        clear_except h,\n        rw list.nth_eq_none_iff,\n        exact h,\n      },\n      {\n        clear_except h,\n        rw list.nth_eq_none_iff,\n        exact h,\n      },\n      refl,\n    },\n    push_neg at h,\n    rename h i_lt_len_x,\n\n    have i_lt_len_lwx : i < (list.map (wrap_symbol₁ g₂.nt) x).length,\n    {\n      rw list.length_map,\n      exact i_lt_len_x,\n    },\n    have i_lt_len_w : i < w.length,\n    {\n      apply lt_of_lt_of_le i_lt_len_x,\n      exact nat.le.intro xylen,\n    },\n    have i_lt_len₁ : i < (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y).length,\n    {\n      rw list.length_append,\n      apply lt_of_lt_of_le i_lt_len_lwx,\n      apply le_self_add,\n    },\n    have i_lt_len₂ : i < (list.map symbol.terminal w).length,\n    {\n      rw list.length_map,\n      exact i_lt_len_w,\n    },\n    rw list.nth_map,\n    rw list.nth_take i_lt_len_x,\n\n    have equivalent_ith :\n      corresponding_symbols\n        (list.nth_le (list.map (wrap_symbol₁ g₂.nt) x ++ list.map (wrap_symbol₂ g₁.nt) y) i i_lt_len₁)\n        (list.nth_le (list.map symbol.terminal w) i i_lt_len₂),\n    {\n      apply corresponding_strings_nth_le,\n      exact concat_xy,\n    },\n    rw list.nth_le_map at equivalent_ith, swap,\n    {\n      exact i_lt_len_w,\n    },\n    rw list.nth_le_append at equivalent_ith, swap,\n    {\n      exact i_lt_len_lwx,\n    },\n    rw list.nth_le_map at equivalent_ith, swap,\n    {\n      exact i_lt_len_x,\n    },\n    clear_except equivalent_ith,\n    rw list.nth_le_nth i_lt_len_x,\n    cases x.nth_le i i_lt_len_x with t n;\n    unfold wrap_symbol₁ at equivalent_ith;\n    unfold corresponding_symbols at equivalent_ith,\n    {\n      have symbol_ith := congr_arg (@symbol.terminal T g₁.nt) equivalent_ith,\n      rw list.nth_le_nth i_lt_len_w,\n      rw option.map_some',\n      exact congr_arg option.some symbol_ith,\n    },\n    {\n      exfalso,\n      exact equivalent_ith,\n    },\n  },\n  split,\n  {\n    clear deri_x,\n    unfold grammar_language,\n    rw set.mem_set_of_eq,\n    unfold grammar_generates,\n    convert deri_y,\n    clear deri_y,\n\n    have xylen := corresponding_strings_length concat_xy,\n    rw list.length_append at xylen,\n    repeat {\n      rw list.length_map at xylen,\n    },\n\n    ext1 i,\n    by_cases i ≥ y.length,\n    {\n      convert_to none = none,\n      {\n        have ylen : y.length = (list.map (@symbol.terminal T g₁.nt) (list.drop x.length w)).length,\n        {\n          clear_except xylen,\n          rw list.length_map,\n          rw list.length_drop,\n          omega,\n        },\n        rw ylen at h,\n        clear_except h,\n        rw list.nth_eq_none_iff,\n        rw list.length_map at *,\n        exact h,\n      },\n      {\n        clear_except h,\n        rw list.nth_eq_none_iff,\n        exact h,\n      },\n      refl,\n    },\n    push_neg at h,\n    rename h i_lt_len_y,\n\n    rw ←list.take_append_drop (list.map (wrap_symbol₁ g₂.nt) x).length (list.map symbol.terminal w) at concat_xy,\n    rw list.nth_map,\n\n    have equivalent_second_parts :\n      corresponding_strings\n        (list.map (wrap_symbol₂ g₁.nt) y)\n        (list.drop (list.map (wrap_symbol₁ g₂.nt) x).length (list.map symbol.terminal w)),\n    {\n      have llen_eq_llen :\n        (list.map (wrap_symbol₁ g₂.nt) x).length =\n        (list.take (list.map (wrap_symbol₁ g₂.nt) x).length (list.map symbol.terminal w)).length,\n      {\n        rw list.length_take,\n        symmetry,\n        apply min_eq_left,\n        rw list.length_map,\n        rw list.length_map,\n        exact nat.le.intro xylen,\n      },\n      convert corresponding_strings_drop (list.map (wrap_symbol₁ g₂.nt) x).length concat_xy,\n      {\n        rw list.drop_left,\n      },\n      {\n        rw list.take_append_drop,\n      },\n      exact T,\n    },\n    clear concat_xy,\n    symmetry,\n\n    have i_lt_len_lwy : i < (list.map (wrap_symbol₂ g₁.nt) y).length,\n    {\n      rw list.length_map,\n      exact i_lt_len_y,\n    },\n    have i_lt_len_dxw : i < (list.drop x.length (list.map symbol.terminal w)).length,\n    {\n      rw list.length_drop,\n      rw list.length_map,\n      rw ←xylen,\n      convert i_lt_len_lwy,\n      rw list.length_map,\n      rw add_comm,\n      rw nat.add_sub_assoc,\n      rw nat.sub_self,\n      rw nat.add_zero,\n      refl,\n    },\n    have i_lt_len_mtw : i < (list.map symbol.terminal (list.drop x.length w)).length,\n    {\n      convert i_lt_len_dxw,\n      apply list.map_drop,\n    },\n    have i_lt_len_dlmxw :\n      i < (list.drop (list.map (wrap_symbol₁ g₂.nt) x).length (list.map symbol.terminal w)).length,\n    {\n      rw list.length_map,\n      -- DO NOT call `i_lt_len_dxw` even though it looks like a good idea!\n      rw list.length_drop,\n      rw list.length_map,\n      rw ←xylen,\n      convert i_lt_len_lwy,\n      rw list.length_map,\n      rw add_comm,\n      rw nat.add_sub_assoc,\n      rw nat.sub_self,\n      rw nat.add_zero,\n      refl,\n    },\n    have eqiv_symb := corresponding_strings_nth_le i_lt_len_lwy i_lt_len_dlmxw equivalent_second_parts,\n\n    have goal_as_ith_drop :\n      y.nth_le i i_lt_len_y = (list.drop x.length (list.map symbol.terminal w)).nth_le i i_lt_len_dxw,\n    {\n      have xli_lt_len_w : x.length + i < w.length,\n      {\n        clear_except i_lt_len_y xylen,\n        linarith,\n      },\n      rw list.nth_le_map _ _ i_lt_len_y at eqiv_symb,\n      rw list.nth_le_drop' at *,\n      rw list.nth_le_map, swap,\n      {\n        exact xli_lt_len_w,\n      },\n      rw list.nth_le_map at eqiv_symb, swap,\n      {\n        rw list.length_map,\n        exact xli_lt_len_w,\n      },\n      clear_except eqiv_symb,\n\n      cases y.nth_le i i_lt_len_y with t n,\n      {\n        unfold wrap_symbol₂ at eqiv_symb,\n        unfold corresponding_symbols at eqiv_symb,\n        have eq_symb := congr_arg symbol.terminal eqiv_symb,\n        rw ←eq_symb,\n        apply congr_arg symbol.terminal,\n        simp only [list.length_map],\n      },\n      {\n        exfalso,\n        unfold wrap_symbol₂ at eqiv_symb,\n        unfold corresponding_symbols at eqiv_symb,\n        exact eqiv_symb,\n      },\n    },\n    have goal_as_some_ith :\n      some (y.nth_le i i_lt_len_y) =\n      some ((list.map symbol.terminal (list.drop x.length w)).nth_le i i_lt_len_mtw),\n    {\n      rw goal_as_ith_drop,\n      clear_except,\n      congr,\n      rw list.map_drop,\n    },\n    clear_except goal_as_some_ith,\n    rw ←list.nth_le_nth i_lt_len_y at goal_as_some_ith,\n    rw ←list.nth_le_nth i_lt_len_mtw at goal_as_some_ith,\n    convert goal_as_some_ith,\n    rw list.nth_map,\n  },\n  apply list.take_append_drop,\nend\n\nend very_complicated\n\nend hard_direction\n\n\n/-- The class of recursively-enumerable languages is closed under concatenation. -/\ntheorem RE_of_RE_c_RE (L₁ : language T) (L₂ : language T) :\n  is_RE L₁  ∧  is_RE L₂   →   is_RE (L₁ * L₂)   :=\nbegin\n  rintro ⟨⟨g₁, eq_L₁⟩, ⟨g₂, eq_L₂⟩⟩,\n\n  use big_grammar g₁ g₂,\n\n  apply set.eq_of_subset_of_subset,\n  {\n    -- prove `L₁ * L₂ ⊇` here\n    intros w hyp,\n    rw ←eq_L₁,\n    rw ←eq_L₂,\n    exact in_concatenated_of_in_big hyp,\n  },\n  {\n    -- prove `L₁ * L₂ ⊆` here\n    intros w hyp,\n    rw ←eq_L₁ at hyp,\n    rw ←eq_L₂ at hyp,\n    exact in_big_of_in_concatenated hyp,\n  },\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/unrestricted/closure_properties/concatenation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3394379462061526}}
{"text": "import for_mathlib.algebra.homology.derived_category\nimport for_mathlib.algebra.homology.trunc\nimport category_theory.preadditive.projective\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n  category_theory.pretriangulated\n\nopen_locale zero_object\n\ndef int.truncate (n : ℤ) : ℕ :=\nif h : 0 ≤ n\n  then (int.eq_coe_of_zero_le h).some\n  else 0\n\nlemma int.coe_truncate (n : ℤ) (hn : 0 ≤ n) : (n.truncate : ℤ) = n :=\nbegin\n  dsimp [int.truncate],\n  rw dif_pos hn,\n  exact (int.eq_coe_of_zero_le hn).some_spec.symm,\nend\n\nlemma int.truncate_eq_zero (n : ℤ) (hn : n ≤ 0) : n.truncate = 0 :=\nbegin\n  by_cases n = 0,\n  { apply int.coe_nat_inj,\n    rw [int.coe_truncate n (by linarith), h, int.coe_nat_zero], },\n  { dsimp [int.truncate],\n    rw dif_neg,\n    intro h',\n    apply h,\n    linarith, },\nend\n\nlemma int.self_le_coe_truncate (n : ℤ) : n ≤ (n.truncate : ℤ) :=\nbegin\n  by_cases 0 ≤ n,\n  { rw n.coe_truncate h, },\n  { simp only [not_le] at h,\n    rw n.truncate_eq_zero (by linarith),\n    linarith, },\nend\n\nlemma int.truncate_le_of_le {n n' : ℤ} (hnn' : n ≤ n') : n.truncate ≤ n'.truncate :=\nbegin\n  by_cases n ≤ 0,\n  { simp only [n.truncate_eq_zero h, zero_le'], },\n  { rw [← int.coe_nat_le, int.coe_truncate n, int.coe_truncate n']; linarith, },\nend\n\n\nvariables {C : Type*} [category C] [abelian C]\n\nnamespace homological_complex\n\nvariables {ι : Type*} {c : complex_shape ι} (K L : homological_complex C c)\n\ndef acyclic : Prop :=\n  ∀ (i : ι), is_zero (K.homology i)\n\nvariables {K L}\n\nlemma acyclic.of_iso {K L : homological_complex C c} (e : K ≅ L) (h : K.acyclic) :\n  L.acyclic :=\nλ i, is_zero.of_iso (h i) ((homology_functor C _ i).map_iso e.symm)\n\nlemma acyclic.iff_of_iso {K L : homological_complex C c} (e : K ≅ L) :\n  K.acyclic ↔ L.acyclic :=\n⟨acyclic.of_iso e, acyclic.of_iso e.symm⟩\n\nvariables (K)\nclass is_K_projective : Prop :=\n(null_homotopic : ∀ ⦃Y : homological_complex C c⦄ (f : K ⟶ Y)\n  (hY : acyclic Y), nonempty (homotopy f 0))\n\nvariables {K L}\n\nlemma is_K_projective.of_homotopy_equiv [K.is_K_projective] (e : homotopy_equiv K L) :\n  L.is_K_projective :=\n⟨λ Y f hY, begin\n  obtain ⟨h⟩ := is_K_projective.null_homotopic (e.hom ≫ f) hY,\n  exact ⟨(homotopy.of_eq (id_comp f).symm).trans\n    ((e.homotopy_inv_hom_id.comp_right f).symm.trans\n      ((homotopy.of_eq (assoc _ _ _)).trans\n        ((h.comp_left e.inv).trans\n        (homotopy.of_eq comp_zero))))⟩,\nend⟩\n\nlemma is_K_projective.of_iso [K.is_K_projective] (e : K ≅ L) : L.is_K_projective :=\nis_K_projective.of_homotopy_equiv (homotopy_equiv.of_iso e)\n\nlemma is_K_projective.iff_of_iso (e : K ≅ L) :\n  K.is_K_projective ↔ L.is_K_projective :=\nbegin\n  split,\n  { introI, exact is_K_projective.of_iso e, },\n  { introI, exact is_K_projective.of_iso e.symm, },\nend\n\nlemma is_K_projective.of_is_zero (h : is_zero K) : K.is_K_projective :=\n⟨λ Y f hY, begin\n  rw h.eq_of_src f 0,\n  exact ⟨homotopy.refl _⟩\nend⟩\n\ninstance zero_is_K_projective : is_K_projective (0 : homological_complex C c) :=\nis_K_projective.of_is_zero (limits.is_zero_zero _)\n\n/- TODO : stability by arbitrary direct sums. -/\n\nend homological_complex\n\nnamespace homotopy_category\n\nlemma quotient_obj_mem_acyclic_iff (K : cochain_complex C ℤ) :\n  (quotient _ _).obj K ∈ (acyclic C) ↔ K.acyclic :=\nbegin\n  let e := λ (n : ℤ), (homotopy_category.homology_factors C _ n).symm.app _ ≪≫\n      (homotopy_category.shift_homology_functor_iso C n 0 n (zero_add n)).symm.app\n        ((quotient _ _).obj K),\n  split,\n  { exact λ h n, is_zero.of_iso (h n) (e n), },\n  { exact λ h n, is_zero.of_iso (h n) (e n).symm, },\nend\n\nend homotopy_category\n\nnamespace cochain_complex\n\nopen homological_complex\n\nvariables (K : cochain_complex C ℤ)\n\nlemma acyclic.iff_shift (r : ℤ) :\n  K.acyclic ↔ K⟦r⟧.acyclic :=\nbegin\n  split,\n  { intros h n,\n    exact is_zero.of_iso (h _)\n      ((shift_homology_functor_iso C r n _ rfl).app K), },\n  { intros h n,\n    exact is_zero.of_iso (h _)\n      ((shift_homology_functor_iso C r (n-r) n (by linarith)).symm.app K), },\nend\n\nlemma acyclic.shift {K : cochain_complex C ℤ} (h : K.acyclic) (r : ℤ) :\n  K⟦r⟧.acyclic := (acyclic.iff_shift K r).1 h\n\nlemma is_K_projective_iff : is_K_projective K ↔\n  (homotopy_category.quotient _ _).obj K\n    ∈ triangulated.left_orthogonal (homotopy_category.acyclic C) :=\nbegin\n  split,\n  { introI,\n    rintros ⟨Y⟩ f hY,\n    obtain ⟨f, rfl⟩ := (homotopy_category.quotient _ _).map_surjective f,\n    rw ← (homotopy_category.quotient C (complex_shape.up ℤ)).map_zero,\n    refine homotopy_category.eq_of_homotopy _ _ (is_K_projective.null_homotopic _ _).some,\n    erw homotopy_category.quotient_obj_mem_acyclic_iff at hY,\n    exact hY, },\n  { intro hK,\n    refine ⟨λ Y f hY, ⟨homotopy_category.homotopy_of_eq _ _ _⟩⟩,\n    simp only [functor.map_zero],\n    apply hK,\n    simpa only [homotopy_category.quotient_obj_mem_acyclic_iff] using hY, },\nend\n\nlemma shift_is_K_projective_iff (K : cochain_complex C ℤ) (r : ℤ) :\n  is_K_projective (K⟦r⟧) ↔ is_K_projective K :=\nbegin\n  simp only [is_K_projective_iff],\n  erw [set.respects_iso.mem_iff_of_iso (triangulated.left_orthogonal (homotopy_category.acyclic C))\n   (((homotopy_category.quotient C (complex_shape.up ℤ)).comm_shift_iso r).app K),\n    ← triangulated.is_triangulated_subcategory.shift_iff],\nend\n\nend cochain_complex\n\nnamespace derived_category\n\nlemma Qh_map_bijective_of_is_K_projective\n  (K L : cochain_complex C ℤ) [K.is_K_projective] :\n  function.bijective (λ (f : ((homotopy_category.quotient _ _).obj K ⟶\n    (homotopy_category.quotient _ _).obj L)), Qh.map f) :=\n(triangulated.subcategory.left_orthogonal_bijective_Q_map\n  (homotopy_category.acyclic C) _ _\n  ((cochain_complex.is_K_projective_iff K).1 infer_instance))\n\nlemma Q_map_surjective_of_is_K_projective\n  (K L : cochain_complex C ℤ) [K.is_K_projective] :\n  function.surjective (λ (f : K ⟶ L), Q.map f) :=\nλ f, begin\n  obtain ⟨g, hg⟩ := (Qh_map_bijective_of_is_K_projective K L).2 f,\n  dsimp at hg,\n  obtain ⟨g, rfl⟩ := (homotopy_category.quotient _ _).map_surjective g,\n  exact ⟨g, hg⟩,\nend\n\ndef homotopy_of_eq_Qh_map_eq_of_is_K_projective\n  {K L : cochain_complex C ℤ} [K.is_K_projective] (f₁ f₂ : K ⟶ L)\n  (h : Q.map f₁ = Q.map f₂) : homotopy f₁ f₂ :=\nhomotopy_category.homotopy_of_eq _ _ ((Qh_map_bijective_of_is_K_projective K L).1 h)\n\nend derived_category\n\nnamespace cochain_complex\n\nopen homological_complex hom_complex\n\nvariables (K L : cochain_complex C ℤ)\n\nlemma is_K_projective_iff_hom_complex_acyclic_in_degree_zero :\n  is_K_projective K ↔\n    ∀ (L : cochain_complex C ℤ) (hL : L.acyclic) (z : cocycle K L 0),\n        ∃ (x : cochain K L (-1)), (z : cochain K L 0) = δ (-1) 0 x :=\nbegin\n  split,\n  { introI,\n    intros L hL z,\n    obtain ⟨hf⟩ := is_K_projective.null_homotopic (cocycle.hom_of z) hL,\n    exact ⟨cochain.of_homotopy hf, by simp only [δ_cochain_of_homotopy,\n      cocycle.cochain_of_hom_hom_of_eq_coe, cochain.of_hom_zero, sub_zero]⟩, },\n  { intro hK,\n    refine ⟨λ L f hL, _⟩,\n    obtain ⟨x, hx⟩ := hK L hL (cocycle.of_hom f),\n    simp only [cocycle.of_hom_coe] at hx,\n    refine ⟨_⟩,\n    equiv_rw hom_complex.equiv_homotopy _ _,\n    exact ⟨x, by simpa only [cochain.of_hom_zero, add_zero]⟩, },\nend\n\nlemma is_K_projective_iff_hom_complex_acyclic :\n  is_K_projective K ↔\n    ∀ (L : cochain_complex C ℤ) (hL : L.acyclic)\n      (n₀ n₁ : ℤ) (h₀₁ : n₁ = n₀ + 1) (z : cocycle K L n₁),\n        ∃ (x : cochain K L n₀), (z : cochain K L n₁) = δ n₀ n₁ x :=\nbegin\n  rw is_K_projective_iff_hom_complex_acyclic_in_degree_zero,\n  split,\n  { intros hK L hL n₀ n₁ h z,\n    obtain ⟨x, hx⟩ := hK (L⟦n₁⟧) (acyclic.shift hL n₁)\n      (cocycle.right_shift (ε (n₁) • z) n₁ 0 (zero_add n₁).symm),\n    refine ⟨x.right_unshift n₀ (by linarith), _⟩,\n    simp only [hom_complex.cochain.δ_right_unshift _ 0 n₀ n₁\n      (show n₀ = -1+n₁, by linarith) (zero_add n₁).symm, ← hx,\n      cocycle.right_shift_coe, add_subgroup.coe_zsmul, cochain.right_shift_unshift, smul_smul,\n      mul_ε_self, one_smul], },\n  { intros hK L hL z,\n    exact hK L hL (-1) 0 (neg_add_self 1).symm z, },\nend\n\nnamespace is_K_projective_of_bounded_above_of_projective\n\nvariables {K L} (f : K ⟶ L)\n\n@[ext]\nstructure lift (N : ℤ) :=\n(y : cochain K L (-1))\n(hy : ∀ (p : ℤ) (hp : N < p), (δ (-1) 0 y).v p p (add_zero p).symm = f.f p)\n\nvariable {f}\n\ndef lift.of_le {N : ℤ} (l : lift f N) (N' : ℤ) (h : N ≤ N') : lift f N' :=\n{ y := l.y,\n  hy := λ p hp, l.hy p (by linarith), }\n\ndef lift.induction_step {N₁ : ℤ} (l : lift f N₁) (N₀ : ℤ) (eq : N₁ = N₀+1)\n  (hK : projective (K.X N₁)) (hL : is_zero (L.homology N₁)) :\n  { l' : lift f N₀ //\n    ∀ (p q : ℤ) (hpq : q = p + (-1)) (hp : N₁ < p), l'.y.v p q hpq = l.y.v p q hpq } :=\nnonempty.some begin\n  /- the exactness below should be part of the API of homological_complex :\n      is_zero (K.homology j) ↔ (K.sc i j k).exact -/\n  have ex : (L.sc N₀ N₁ (N₁+1)).exact := (short_complex.exact_iff_of_iso\n    ((short_complex_functor_nat_iso C (complex_shape.up ℤ) eq.symm rfl).app L)).1\n      ((short_complex.exact_iff_is_zero_homology _).2 hL),\n  obtain ⟨A, π, hπ, x, hx⟩ :=\n    ex.pseudo_exact' (f.f N₁ - (δ (-1) 0 l.y).v N₁ N₁ (add_zero N₁).symm)\n    begin\n      dsimp,\n      simp only [preadditive.sub_comp,\n        δ_v (-1) 0 _ _ N₁ N₁ (add_zero N₁).symm N₀ _ (by linarith) rfl,\n        neg_add_self, ε_0, one_smul, preadditive.add_comp, assoc, L.d_comp_d, comp_zero,\n        zero_add, f.comm, ← l.hy (N₁+1) (lt_add_one N₁), preadditive.comp_add, K.d_comp_d_assoc,\n        δ_v (-1) 0 _ _ (N₁+1) _ (add_zero _).symm N₁ _ (by linarith) rfl,\n        zero_comp, add_zero, sub_self],\n    end,\n  haveI := hπ,\n  dsimp at x hx,\n  let φ : K.X N₁ ⟶ L.X N₀ := projective.factor_thru (𝟙 _) π ≫ x,\n  have hφ : φ ≫ L.d N₀ N₁ = f.f N₁ - (δ (-1) 0 l.y).v N₁ N₁ (add_zero N₁).symm,\n  { simp only [φ, assoc, ← hx, projective.factor_thru_comp_assoc, id_comp], },\n  let w : cochain K L (-1) := cochain.mk (λ p q hpq, begin\n    by_cases p=N₁,\n    { refine eq_to_hom (by rw h) ≫ φ ≫ eq_to_hom (by { congr', linarith, }), },\n    { exact 0, },\n  end),\n  have hw : w.v N₁ N₀ (by linarith) = φ,\n  { dsimp [w], simp only [eq_self_iff_true, comp_id, id_comp, if_true], },\n  have hw_zero : ∀ (p q : ℤ) (hpq : q = p + (-1))\n    (hp : p ≠ N₁), w.v p q hpq = 0,\n  { intros p q hpq hp, dsimp [w], rw dif_neg hp, },\n  let l' : lift f N₀ :=\n  { y := l.y + w,\n    hy := λ p hp, begin\n      cases (int.add_one_le_iff.mpr hp).lt_or_eq,\n      { rw [δ_add, cochain.add_v, l.hy p (by linarith), add_right_eq_self,\n          δ_v (-1) 0 (neg_add_self 1).symm _ p p (add_zero p).symm _ _ rfl rfl,\n          hw_zero p _ _ (by linarith), hw_zero (p+1) _ _ (by linarith), zero_comp,\n          comp_zero, smul_zero, zero_add], },\n      { have h' : N₁ = p := by linarith,\n        unfreezingI { subst h', },\n        simp only [δ_add, cochain.add_v, δ_add, δ_v (-1) 0 (neg_add_self 1).symm w N₁ N₁\n          (add_zero N₁).symm N₀ _ (by linarith) rfl, neg_add_self, ε_0, one_smul, hw,\n          hw_zero (N₁+1) _ _ (by linarith), comp_zero, add_zero, hφ,\n          add_sub_cancel'_right], },\n    end, },\n  exact ⟨⟨l', λ p q hpq hp, by rw [cochain.add_v, hw_zero p q hpq (by linarith), add_zero]⟩⟩,\nend\n\nvariables {N₀ : ℤ} (l₀ : lift f N₀) (hK : ∀ (p : ℤ) (hp : p ≤ N₀), projective (K.X p))\n  (hL : ∀ (p : ℤ) (hp : p ≤ N₀), is_zero (L.homology p))\n\ninclude l₀ hK hL\n\nnoncomputable\ndef lift.sequence : Π (k : ℕ), lift f (N₀-k)\n| 0 := l₀.of_le _ (by simp only [algebra_map.coe_zero, tsub_zero])\n| (k+1) := begin\n  refine (lift.induction_step (lift.sequence k) _ _ (hK _ _) (hL _ _)).1,\n  { simp only [nat.cast_add, algebra_map.coe_one], ring, },\n  all_goals\n  { simp only [sub_le_self_iff, nat.cast_nonneg], },\nend\n\nlemma lift.sequence_is_eventually_constant {k₁ k₂ : ℕ} (hk : k₁ ≤ k₂) (p q : ℤ) (hpq : q = p+(-1))\n  (hp : N₀ - k₁ < p):\n  (lift.sequence l₀ hK hL k₁).y.v p q hpq =\n    (lift.sequence l₀ hK hL k₂).y.v p q hpq :=\nbegin\n  have h : ∀ (k₁ k₂ : ℕ) (hk₂ : k₂ = k₁ + 1) (hp : N₀ - k₁ < p),\n    (lift.sequence l₀ hK hL k₁).y.v p q hpq = (lift.sequence l₀ hK hL k₂).y.v p q hpq,\n  { rintro k₁ _ rfl hp,\n    unfold lift.sequence,\n    exact (((l₀.sequence hK hL k₁).induction_step _ _ _ _).2 _ _ _ hp).symm, },\n  rw le_iff_exists_add at hk,\n  obtain ⟨d, rfl⟩ := hk,\n  induction d with d hd,\n  { rw add_zero, },\n  { rw hd,\n    apply h,\n    { rw [nat.succ_eq_add_one, add_assoc], },\n    { simp only [nat.cast_add],\n      exact lt_of_le_of_lt\n        (by simp only [sub_le_sub_iff_left, le_add_iff_nonneg_right, nat.cast_nonneg]) hp, }, },\nend\n\ndef lift.limit : cochain K L (-1) :=\ncochain.mk (λ p q hpq, begin\n  let k : ℕ := int.truncate (N₀+1 -p),\n  exact ((lift.sequence l₀ hK hL) k).y.v p q hpq,\nend)\n\nlemma lift.δ_limit : δ (-1) 0 (lift.limit l₀ hK hL) = cochain.of_hom f :=\nbegin\n  ext,\n  dsimp [lift.limit],\n  have ineq₁ := int.self_le_coe_truncate (N₀+1-p),\n  have ineq₂ := int.self_le_coe_truncate (N₀+1-(p+1)),\n  rw [cochain.of_hom_v, ← (l₀.sequence hK hL (N₀+1-p).truncate).hy p (by linarith)],\n  simp only [δ_v (-1) 0 (neg_add_self 1).symm _ p p (add_zero p).symm _ _ rfl rfl,\n    cochain.mk_v],\n  congr' 3,\n  { apply lift.sequence_is_eventually_constant,\n    apply int.truncate_le_of_le,\n    { linarith, },\n    { linarith, }, },\nend\n\ndef lift.homotopy_zero : homotopy f 0 :=\nbegin\n  equiv_rw equiv_homotopy _ _,\n  exact ⟨lift.limit l₀ hK hL,\n    by simp only [lift.δ_limit, cochain.of_hom_zero, add_zero]⟩,\nend\n\nend is_K_projective_of_bounded_above_of_projective\n\nlemma is_K_projective_of_bounded_above_of_projective\n  (K : cochain_complex C ℤ) (n : ℤ) [K.is_strictly_le n]\n  [∀ (n : ℤ), projective (K.X n)] : is_K_projective K :=\n⟨λ L f hL, begin\n  let l : is_K_projective_of_bounded_above_of_projective.lift f n :=\n  { y := 0,\n    hy := λ p hp, begin\n      simp only [δ_zero, cochain.zero_v],\n      apply ((is_strictly_le.is_zero K n) p hp).eq_of_src,\n    end, },\n  exact ⟨is_K_projective_of_bounded_above_of_projective.lift.homotopy_zero l\n    (λ p hp, infer_instance) (λ p hp, hL p)⟩,\nend⟩\n\ninstance (P : C) [projective P] (n n' : ℤ) :\n  projective (((single C (complex_shape.up ℤ) n).obj P).X n') :=\nbegin\n  by_cases n' = n,\n  { subst h,\n    exact projective.of_iso (single_obj_X_self C (complex_shape.up ℤ) _ P).symm\n      infer_instance, },\n  { dsimp [single],\n    rw if_neg h,\n    exact projective.zero_projective, },\nend\n\ninstance (P : C) [projective P] (n : ℤ) :\n  is_K_projective ((single C (complex_shape.up ℤ) n).obj P) :=\nis_K_projective_of_bounded_above_of_projective _ n\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/k_projective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3394379462061526}}
{"text": "-- Insert lean 3 code here.\n\nnamespace foo\n\n/-- test -/\n@[simp] def foo := 1\n\ntheorem foo_eq_one : foo.foo = 1 := rfl\n\nend foo\n", "meta": {"author": "leanprover-community", "repo": "mathport", "sha": "b5459df41774820ca21861417fafd8ff7a662fc5", "save_path": "github-repos/lean/leanprover-community-mathport", "path": "github-repos/lean/leanprover-community-mathport/mathport-b5459df41774820ca21861417fafd8ff7a662fc5/Oneshot/lean3-in/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.33927475118110456}}
{"text": "/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.field_theory.adjoin\nimport Mathlib.field_theory.minpoly\nimport Mathlib.ring_theory.adjoin\nimport Mathlib.ring_theory.adjoin_root\nimport Mathlib.ring_theory.algebraic\nimport Mathlib.PostPort\n\nuniverses u_8 u_9 l u_2 u_6 u_1 u_4 u_7 \n\nnamespace Mathlib\n\n/-!\n# Power basis\n\nThis file defines a structure `power_basis R S`, giving a basis of the\n`R`-algebra `S` as a finite list of powers `1, x, ..., x^n`.\nThere are also constructors for `power_basis` when adjoining an algebraic\nelement to a ring/field.\n\n## Definitions\n\n* `power_basis R A`: a structure containing an `x` and an `n` such that\n`1, x, ..., x^n` is a basis for the `R`-algebra `A` (viewed as an `R`-module).\n\n* `findim (hf : f ≠ 0) : finite_dimensional.findim K (adjoin_root f) = f.nat_degree`,\n  the dimension of `adjoin_root f` equals the degree of `f`\n\n* `power_basis.lift (pb : power_basis R S)`: if `y : S'` satisfies the same\n  equations as `pb.gen`, this is the map `S →ₐ[R] S'` sending `pb.gen` to `y`\n\n* `power_basis.equiv`: if two power bases satisfy the same equations, they are\n  equivalent as algebras\n\n## Implementation notes\n\nThroughout this file, `R`, `S`, ... are `comm_ring`s, `A`, `B`, ... are\n`integral_domain`s and `K`, `L`, ... are `field`s.\n`S` is an `R`-algebra, `B` is an `A`-algebra, `L` is a `K`-algebra.\n\n## Tags\n\npower basis, powerbasis\n\n-/\n\n/-- `pb : power_basis R S` states that `1, pb.gen, ..., pb.gen ^ (pb.dim - 1)`\nis a basis for the `R`-algebra `S` (viewed as `R`-module).\n\nThis is a structure, not a class, since the same algebra can have many power bases.\nFor the common case where `S` is defined by adjoining an integral element to `R`,\nthe canonical power basis is given by `{algebra,intermediate_field}.adjoin.power_basis`.\n-/\nstructure power_basis (R : Type u_8) (S : Type u_9) [comm_ring R] [ring S] [algebra R S] where\n  gen : S\n  dim : ℕ\n  is_basis : is_basis R fun (i : fin dim) => gen ^ ↑i\n\nnamespace power_basis\n\n\n/-- Cannot be an instance because `power_basis` cannot be a class. -/\ntheorem finite_dimensional {S : Type u_2} [comm_ring S] {K : Type u_6} [field K] [algebra K S]\n    (pb : power_basis K S) : finite_dimensional K S :=\n  finite_dimensional.of_fintype_basis (is_basis pb)\n\ntheorem findim {S : Type u_2} [comm_ring S] {K : Type u_6} [field K] [algebra K S]\n    (pb : power_basis K S) : finite_dimensional.findim K S = dim pb :=\n  sorry\n\n/-- TODO: this mixes `polynomial` and `finsupp`, we should hide this behind a\nnew function `polynomial.of_finsupp`. -/\ntheorem polynomial.mem_supported_range {R : Type u_1} [comm_ring R] {f : polynomial R} {d : ℕ} :\n    f ∈ finsupp.supported R R ↑(finset.range d) ↔ polynomial.degree f < ↑d :=\n  sorry\n\ntheorem mem_span_pow' {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S]\n    {x : S} {y : S} {d : ℕ} :\n    y ∈ submodule.span R (set.range fun (i : fin d) => x ^ ↑i) ↔\n        ∃ (f : polynomial R), polynomial.degree f < ↑d ∧ y = coe_fn (polynomial.aeval x) f :=\n  sorry\n\ntheorem mem_span_pow {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x : S}\n    {y : S} {d : ℕ} (hd : d ≠ 0) :\n    y ∈ submodule.span R (set.range fun (i : fin d) => x ^ ↑i) ↔\n        ∃ (f : polynomial R), polynomial.nat_degree f < d ∧ y = coe_fn (polynomial.aeval x) f :=\n  sorry\n\ntheorem dim_ne_zero {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S]\n    [nontrivial S] (pb : power_basis R S) : dim pb ≠ 0 :=\n  sorry\n\ntheorem exists_eq_aeval {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S]\n    [nontrivial S] (pb : power_basis R S) (y : S) :\n    ∃ (f : polynomial R),\n        polynomial.nat_degree f < dim pb ∧ y = coe_fn (polynomial.aeval (gen pb)) f :=\n  iff.mp (mem_span_pow (dim_ne_zero pb)) (is_basis.mem_span (is_basis pb) y)\n\n/-- `pb.minpoly_gen` is a minimal polynomial for `pb.gen`.\n\nIf `A` is not a field, it might not necessarily be *the* minimal polynomial,\nhowever `nat_degree_minpoly` shows its degree is indeed minimal.\n-/\ndef minpoly_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A] [algebra A S]\n    (pb : power_basis A S) : polynomial A :=\n  polynomial.X ^ dim pb -\n    finset.sum finset.univ\n      fun (i : fin (dim pb)) =>\n        coe_fn polynomial.C (coe_fn (coe_fn (is_basis.repr sorry) (gen pb ^ dim pb)) i) *\n          polynomial.X ^ ↑i\n\n@[simp] theorem nat_degree_minpoly_gen {S : Type u_2} [comm_ring S] {A : Type u_4}\n    [integral_domain A] [algebra A S] (pb : power_basis A S) :\n    polynomial.nat_degree (minpoly_gen pb) = dim pb :=\n  sorry\n\ntheorem minpoly_gen_monic {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] (pb : power_basis A S) : polynomial.monic (minpoly_gen pb) :=\n  sorry\n\n@[simp] theorem aeval_minpoly_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] (pb : power_basis A S) :\n    coe_fn (polynomial.aeval (gen pb)) (minpoly_gen pb) = 0 :=\n  sorry\n\ntheorem is_integral_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] (pb : power_basis A S) : is_integral A (gen pb) :=\n  Exists.intro (minpoly_gen pb) { left := minpoly_gen_monic pb, right := aeval_minpoly_gen pb }\n\ntheorem dim_le_nat_degree_of_root {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] (h : power_basis A S) {p : polynomial A} (ne_zero : p ≠ 0)\n    (root : coe_fn (polynomial.aeval (gen h)) p = 0) : dim h ≤ polynomial.nat_degree p :=\n  sorry\n\n@[simp] theorem nat_degree_minpoly {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] (pb : power_basis A S) : polynomial.nat_degree (minpoly A (gen pb)) = dim pb :=\n  sorry\n\ntheorem nat_degree_lt_nat_degree {R : Type u_1} [comm_ring R] {p : polynomial R} {q : polynomial R}\n    (hp : p ≠ 0) (hpq : polynomial.degree p < polynomial.degree q) :\n    polynomial.nat_degree p < polynomial.nat_degree q :=\n  sorry\n\ntheorem constr_pow_aeval {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] (pb : power_basis A S) {y : S'}\n    (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) (f : polynomial A) :\n    coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i)\n          (coe_fn (polynomial.aeval (gen pb)) f) =\n        coe_fn (polynomial.aeval y) f :=\n  sorry\n\ntheorem constr_pow_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A] [algebra A S]\n    {S' : Type u_8} [comm_ring S'] [algebra A S'] (pb : power_basis A S) {y : S'}\n    (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) :\n    coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i) (gen pb) = y :=\n  sorry\n\ntheorem constr_pow_algebra_map {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] (pb : power_basis A S) {y : S'}\n    (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) (x : A) :\n    coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i)\n          (coe_fn (algebra_map A S) x) =\n        coe_fn (algebra_map A S') x :=\n  sorry\n\ntheorem constr_pow_mul {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A] [algebra A S]\n    {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] (pb : power_basis A S) {y : S'}\n    (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) (x : S) (x' : S) :\n    coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i) (x * x') =\n        coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i) x *\n          coe_fn (is_basis.constr (is_basis pb) fun (i : fin (dim pb)) => y ^ ↑i) x' :=\n  sorry\n\n/-- `pb.lift y hy` is the algebra map sending `pb.gen` to `y`,\nwhere `hy` states the higher powers of `y` are the same as the higher powers of `pb.gen`. -/\ndef lift {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A] [algebra A S]\n    {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] (pb : power_basis A S) (y : S')\n    (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) : alg_hom A S S' :=\n  alg_hom.mk (linear_map.to_fun (is_basis.constr sorry fun (i : fin (dim pb)) => y ^ ↑i)) sorry\n    (constr_pow_mul pb hy) sorry sorry (constr_pow_algebra_map pb hy)\n\n@[simp] theorem lift_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S]\n    (pb : power_basis A S) (y : S') (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0) :\n    coe_fn (lift pb y hy) (gen pb) = y :=\n  constr_pow_gen pb hy\n\n@[simp] theorem lift_aeval {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S]\n    (pb : power_basis A S) (y : S') (hy : coe_fn (polynomial.aeval y) (minpoly A (gen pb)) = 0)\n    (f : polynomial A) :\n    coe_fn (lift pb y hy) (coe_fn (polynomial.aeval (gen pb)) f) = coe_fn (polynomial.aeval y) f :=\n  constr_pow_aeval pb hy f\n\n/-- `pb.equiv pb' h` is an equivalence of algebras with the same power basis. -/\ndef equiv {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A] [algebra A S]\n    {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] [nontrivial S']\n    (pb : power_basis A S) (pb' : power_basis A S') (h : minpoly A (gen pb) = minpoly A (gen pb')) :\n    alg_equiv A S S' :=\n  alg_equiv.of_alg_hom (lift pb (gen pb') sorry) (lift pb' (gen pb) sorry) sorry sorry\n\n@[simp] theorem equiv_aeval {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] [nontrivial S']\n    (pb : power_basis A S) (pb' : power_basis A S') (h : minpoly A (gen pb) = minpoly A (gen pb'))\n    (f : polynomial A) :\n    coe_fn (equiv pb pb' h) (coe_fn (polynomial.aeval (gen pb)) f) =\n        coe_fn (polynomial.aeval (gen pb')) f :=\n  lift_aeval pb (gen pb') (Eq.symm h ▸ minpoly.aeval A (gen pb')) f\n\n@[simp] theorem equiv_gen {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] [nontrivial S']\n    (pb : power_basis A S) (pb' : power_basis A S') (h : minpoly A (gen pb) = minpoly A (gen pb')) :\n    coe_fn (equiv pb pb' h) (gen pb) = gen pb' :=\n  lift_gen pb (gen pb') (Eq.symm h ▸ minpoly.aeval A (gen pb'))\n\n@[simp] theorem equiv_symm {S : Type u_2} [comm_ring S] {A : Type u_4} [integral_domain A]\n    [algebra A S] {S' : Type u_8} [comm_ring S'] [algebra A S'] [nontrivial S] [nontrivial S']\n    (pb : power_basis A S) (pb' : power_basis A S') (h : minpoly A (gen pb) = minpoly A (gen pb')) :\n    alg_equiv.symm (equiv pb pb' h) = equiv pb' pb (Eq.symm h) :=\n  rfl\n\nend power_basis\n\n\nnamespace algebra\n\n\ntheorem mem_span_power_basis {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S]\n    [nontrivial R] {x : S} {y : S} (hx : is_integral R x)\n    (hy : ∃ (f : polynomial R), y = coe_fn (polynomial.aeval x) f) :\n    y ∈\n        submodule.span R\n          (set.range fun (i : fin (polynomial.nat_degree (minpoly R x))) => x ^ ↑i) :=\n  sorry\n\ntheorem linear_independent_power_basis {S : Type u_2} [comm_ring S] {K : Type u_6} [field K]\n    [algebra K S] {x : S} (hx : is_integral K x) :\n    linear_independent K fun (i : fin (polynomial.nat_degree (minpoly K x))) => x ^ ↑i :=\n  sorry\n\ntheorem power_basis_is_basis {S : Type u_2} [comm_ring S] {K : Type u_6} [field K] [algebra K S]\n    {x : S} (hx : is_integral K x) :\n    is_basis K\n        fun (i : fin (polynomial.nat_degree (minpoly K x))) =>\n          { val := x, property := subset_adjoin (set.mem_singleton x) } ^ ↑i :=\n  sorry\n\n/-- The power basis `1, x, ..., x ^ (d - 1)` for `K[x]`,\nwhere `d` is the degree of the minimal polynomial of `x`. -/\ndef adjoin.power_basis {S : Type u_2} [comm_ring S] {K : Type u_6} [field K] [algebra K S] {x : S}\n    (hx : is_integral K x) : power_basis K ↥(adjoin K (singleton x)) :=\n  power_basis.mk { val := x, property := sorry } (polynomial.nat_degree (minpoly K x))\n    (power_basis_is_basis hx)\n\nend algebra\n\n\nnamespace adjoin_root\n\n\ntheorem power_basis_is_basis {K : Type u_6} [field K] {f : polynomial K} (hf : f ≠ 0) :\n    is_basis K fun (i : fin (polynomial.nat_degree f)) => root f ^ subtype.val i :=\n  sorry\n\n/-- The power basis `1, root f, ..., root f ^ (d - 1)` for `adjoin_root f`,\nwhere `f` is an irreducible polynomial over a field of degree `d`. -/\ndef power_basis {K : Type u_6} [field K] {f : polynomial K} (hf : f ≠ 0) :\n    power_basis K (adjoin_root f) :=\n  power_basis.mk (root f) (polynomial.nat_degree f) (power_basis_is_basis hf)\n\nend adjoin_root\n\n\nnamespace intermediate_field\n\n\ntheorem power_basis_is_basis {K : Type u_6} {L : Type u_7} [field K] [field L] [algebra K L] {x : L}\n    (hx : is_integral K x) :\n    is_basis K fun (i : fin (polynomial.nat_degree (minpoly K x))) => adjoin_simple.gen K x ^ ↑i :=\n  sorry\n\n/-- The power basis `1, x, ..., x ^ (d - 1)` for `K⟮x⟯`,\nwhere `d` is the degree of the minimal polynomial of `x`. -/\ndef adjoin.power_basis {K : Type u_6} {L : Type u_7} [field K] [field L] [algebra K L] {x : L}\n    (hx : is_integral K x) : power_basis K ↥(adjoin K (insert.insert ∅ x)) :=\n  power_basis.mk (adjoin_simple.gen K x) (polynomial.nat_degree (minpoly K x))\n    (power_basis_is_basis hx)\n\n@[simp] theorem adjoin.power_basis.gen_eq {K : Type u_6} {L : Type u_7} [field K] [field L]\n    [algebra K L] {x : L} (hx : is_integral K x) :\n    power_basis.gen (adjoin.power_basis hx) = adjoin_simple.gen K x :=\n  rfl\n\ntheorem adjoin.finite_dimensional {K : Type u_6} {L : Type u_7} [field K] [field L] [algebra K L]\n    {x : L} (hx : is_integral K x) : finite_dimensional K ↥(adjoin K (insert.insert ∅ x)) :=\n  power_basis.finite_dimensional (adjoin.power_basis hx)\n\ntheorem adjoin.findim {K : Type u_6} {L : Type u_7} [field K] [field L] [algebra K L] {x : L}\n    (hx : is_integral K x) :\n    finite_dimensional.findim K ↥(adjoin K (insert.insert ∅ x)) =\n        polynomial.nat_degree (minpoly K x) :=\n  sorry\n\nend intermediate_field\n\n\nnamespace power_basis\n\n\n/-- `pb.equiv_adjoin_simple` is the equivalence between `K⟮pb.gen⟯` and `L` itself. -/\ndef equiv_adjoin_simple {K : Type u_6} {L : Type u_7} [field K] [field L] [algebra K L]\n    (pb : power_basis K L) :\n    alg_equiv K (↥(intermediate_field.adjoin K (intermediate_field.insert.insert ∅ (gen pb)))) L :=\n  equiv (intermediate_field.adjoin.power_basis sorry) pb sorry\n\n@[simp] theorem equiv_adjoin_simple_aeval {K : Type u_6} {L : Type u_7} [field K] [field L]\n    [algebra K L] (pb : power_basis K L) (f : polynomial K) :\n    coe_fn (equiv_adjoin_simple pb)\n          (coe_fn (polynomial.aeval (intermediate_field.adjoin_simple.gen K (gen pb))) f) =\n        coe_fn (polynomial.aeval (gen pb)) f :=\n  equiv_aeval (intermediate_field.adjoin.power_basis (equiv_adjoin_simple._proof_3 pb)) pb\n    (equiv_adjoin_simple._proof_4 pb) f\n\n@[simp] theorem equiv_adjoin_simple_gen {K : Type u_6} {L : Type u_7} [field K] [field L]\n    [algebra K L] (pb : power_basis K L) :\n    coe_fn (equiv_adjoin_simple pb) (intermediate_field.adjoin_simple.gen K (gen pb)) = gen pb :=\n  equiv_gen (intermediate_field.adjoin.power_basis (equiv_adjoin_simple._proof_3 pb)) pb\n    (equiv_adjoin_simple._proof_4 pb)\n\n@[simp] theorem equiv_adjoin_simple_symm_aeval {K : Type u_6} {L : Type u_7} [field K] [field L]\n    [algebra K L] (pb : power_basis K L) (f : polynomial K) :\n    coe_fn (alg_equiv.symm (equiv_adjoin_simple pb)) (coe_fn (polynomial.aeval (gen pb)) f) =\n        coe_fn (polynomial.aeval (intermediate_field.adjoin_simple.gen K (gen pb))) f :=\n  sorry\n\n@[simp] theorem equiv_adjoin_simple_symm_gen {K : Type u_6} {L : Type u_7} [field K] [field L]\n    [algebra K L] (pb : power_basis K L) :\n    coe_fn (alg_equiv.symm (equiv_adjoin_simple pb)) (gen pb) =\n        intermediate_field.adjoin_simple.gen K (gen pb) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/power_basis_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.33927475118110456}}
{"text": "/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.lattice\nimport Mathlib.PostPort\n\nuniverses u v l u_1 w x u_2 u_3 \n\nnamespace Mathlib\n\n/-- A `pequiv` is a partial equivalence, a representation of a bijection between a subset\n  of `α` and a subset of `β` -/\nstructure pequiv (α : Type u) (β : Type v) \nwhere\n  to_fun : α → Option β\n  inv_fun : β → Option α\n  inv : ∀ (a : α) (b : β), a ∈ inv_fun b ↔ b ∈ to_fun a\n\ninfixr:25 \" ≃. \" => Mathlib.pequiv\n\nnamespace pequiv\n\n\nprotected instance has_coe_to_fun {α : Type u} {β : Type v} : has_coe_to_fun (α ≃. β) :=\n  has_coe_to_fun.mk (fun (x : α ≃. β) => α → Option β) to_fun\n\n@[simp] theorem coe_mk_apply {α : Type u} {β : Type v} (f₁ : α → Option β) (f₂ : β → Option α) (h : ∀ (a : α) (b : β), a ∈ f₂ b ↔ b ∈ f₁ a) (x : α) : coe_fn (mk f₁ f₂ h) x = f₁ x :=\n  rfl\n\ntheorem ext {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β} (h : ∀ (x : α), coe_fn f x = coe_fn g x) : f = g := sorry\n\ntheorem ext_iff {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β} : f = g ↔ ∀ (x : α), coe_fn f x = coe_fn g x :=\n  { mp := congr_fun ∘ congr_arg fun {f : α ≃. β} (x : α) => coe_fn f x, mpr := ext }\n\nprotected def refl (α : Type u_1) : α ≃. α :=\n  mk some some sorry\n\nprotected def symm {α : Type u} {β : Type v} (f : α ≃. β) : β ≃. α :=\n  mk (inv_fun f) (to_fun f) sorry\n\ntheorem mem_iff_mem {α : Type u} {β : Type v} (f : α ≃. β) {a : α} {b : β} : a ∈ coe_fn (pequiv.symm f) b ↔ b ∈ coe_fn f a :=\n  inv f\n\ntheorem eq_some_iff {α : Type u} {β : Type v} (f : α ≃. β) {a : α} {b : β} : coe_fn (pequiv.symm f) b = some a ↔ coe_fn f a = some b :=\n  inv f\n\nprotected def trans {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) : α ≃. γ :=\n  mk (fun (a : α) => option.bind (coe_fn f a) ⇑g) (fun (a : γ) => option.bind (coe_fn (pequiv.symm g) a) ⇑(pequiv.symm f))\n    sorry\n\n@[simp] theorem refl_apply {α : Type u} (a : α) : coe_fn (pequiv.refl α) a = some a :=\n  rfl\n\n@[simp] theorem symm_refl {α : Type u} : pequiv.symm (pequiv.refl α) = pequiv.refl α :=\n  rfl\n\n@[simp] theorem symm_symm {α : Type u} {β : Type v} (f : α ≃. β) : pequiv.symm (pequiv.symm f) = f := sorry\n\ntheorem symm_injective {α : Type u} {β : Type v} : function.injective pequiv.symm :=\n  function.left_inverse.injective symm_symm\n\ntheorem trans_assoc {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} (f : α ≃. β) (g : β ≃. γ) (h : γ ≃. δ) : pequiv.trans (pequiv.trans f g) h = pequiv.trans f (pequiv.trans g h) :=\n  ext fun (_x : α) => option.bind_assoc (coe_fn f _x) ⇑g ⇑h\n\ntheorem mem_trans {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α) (c : γ) : c ∈ coe_fn (pequiv.trans f g) a ↔ ∃ (b : β), b ∈ coe_fn f a ∧ c ∈ coe_fn g b :=\n  option.bind_eq_some'\n\ntheorem trans_eq_some {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α) (c : γ) : coe_fn (pequiv.trans f g) a = some c ↔ ∃ (b : β), coe_fn f a = some b ∧ coe_fn g b = some c :=\n  option.bind_eq_some'\n\ntheorem trans_eq_none {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α) : coe_fn (pequiv.trans f g) a = none ↔ ∀ (b : β) (c : γ), ¬b ∈ coe_fn f a ∨ ¬c ∈ coe_fn g b := sorry\n\n@[simp] theorem refl_trans {α : Type u} {β : Type v} (f : α ≃. β) : pequiv.trans (pequiv.refl α) f = f :=\n  ext fun (x : α) => option.ext fun (a : β) => id (iff.refl (a ∈ coe_fn f x))\n\n@[simp] theorem trans_refl {α : Type u} {β : Type v} (f : α ≃. β) : pequiv.trans f (pequiv.refl β) = f := sorry\n\nprotected theorem inj {α : Type u} {β : Type v} (f : α ≃. β) {a₁ : α} {a₂ : α} {b : β} (h₁ : b ∈ coe_fn f a₁) (h₂ : b ∈ coe_fn f a₂) : a₁ = a₂ := sorry\n\ntheorem injective_of_forall_ne_is_some {α : Type u} {β : Type v} (f : α ≃. β) (a₂ : α) (h : ∀ (a₁ : α), a₁ ≠ a₂ → ↥(option.is_some (coe_fn f a₁))) : function.injective ⇑f := sorry\n\ntheorem injective_of_forall_is_some {α : Type u} {β : Type v} {f : α ≃. β} (h : ∀ (a : α), ↥(option.is_some (coe_fn f a))) : function.injective ⇑f := sorry\n\ndef of_set {α : Type u} (s : set α) [decidable_pred s] : α ≃. α :=\n  mk (fun (a : α) => ite (a ∈ s) (some a) none) (fun (a : α) => ite (a ∈ s) (some a) none) sorry\n\ntheorem mem_of_set_self_iff {α : Type u} {s : set α} [decidable_pred s] {a : α} : a ∈ coe_fn (of_set s) a ↔ a ∈ s := sorry\n\ntheorem mem_of_set_iff {α : Type u} {s : set α} [decidable_pred s] {a : α} {b : α} : a ∈ coe_fn (of_set s) b ↔ a = b ∧ a ∈ s := sorry\n\n@[simp] theorem of_set_eq_some_iff {α : Type u} {s : set α} {h : decidable_pred s} {a : α} {b : α} : coe_fn (of_set s) b = some a ↔ a = b ∧ a ∈ s :=\n  mem_of_set_iff\n\n@[simp] theorem of_set_eq_some_self_iff {α : Type u} {s : set α} {h : decidable_pred s} {a : α} : coe_fn (of_set s) a = some a ↔ a ∈ s :=\n  mem_of_set_self_iff\n\n@[simp] theorem of_set_symm {α : Type u} (s : set α) [decidable_pred s] : pequiv.symm (of_set s) = of_set s :=\n  rfl\n\n@[simp] theorem of_set_univ {α : Type u} : of_set set.univ = pequiv.refl α :=\n  rfl\n\n@[simp] theorem of_set_eq_refl {α : Type u} {s : set α} [decidable_pred s] : of_set s = pequiv.refl α ↔ s = set.univ := sorry\n\ntheorem symm_trans_rev {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) : pequiv.symm (pequiv.trans f g) = pequiv.trans (pequiv.symm g) (pequiv.symm f) :=\n  rfl\n\ntheorem trans_symm {α : Type u} {β : Type v} (f : α ≃. β) : pequiv.trans f (pequiv.symm f) = of_set (set_of fun (a : α) => ↥(option.is_some (coe_fn f a))) := sorry\n\ntheorem symm_trans {α : Type u} {β : Type v} (f : α ≃. β) : pequiv.trans (pequiv.symm f) f = of_set (set_of fun (b : β) => ↥(option.is_some (coe_fn (pequiv.symm f) b))) := sorry\n\ntheorem trans_symm_eq_iff_forall_is_some {α : Type u} {β : Type v} {f : α ≃. β} : pequiv.trans f (pequiv.symm f) = pequiv.refl α ↔ ∀ (a : α), ↥(option.is_some (coe_fn f a)) := sorry\n\nprotected instance has_bot {α : Type u} {β : Type v} : has_bot (α ≃. β) :=\n  has_bot.mk (mk (fun (_x : α) => none) (fun (_x : β) => none) sorry)\n\n@[simp] theorem bot_apply {α : Type u} {β : Type v} (a : α) : coe_fn ⊥ a = none :=\n  rfl\n\n@[simp] theorem symm_bot {α : Type u} {β : Type v} : pequiv.symm ⊥ = ⊥ :=\n  rfl\n\n@[simp] theorem trans_bot {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) : pequiv.trans f ⊥ = ⊥ := sorry\n\n@[simp] theorem bot_trans {α : Type u} {β : Type v} {γ : Type w} (f : β ≃. γ) : pequiv.trans ⊥ f = ⊥ := sorry\n\ntheorem is_some_symm_get {α : Type u} {β : Type v} (f : α ≃. β) {a : α} (h : ↥(option.is_some (coe_fn f a))) : ↥(option.is_some (coe_fn (pequiv.symm f) (option.get h))) := sorry\n\ndef single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) : α ≃. β :=\n  mk (fun (x : α) => ite (x = a) (some b) none) (fun (x : β) => ite (x = b) (some a) none) sorry\n\ntheorem mem_single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) : b ∈ coe_fn (single a b) a :=\n  if_pos rfl\n\ntheorem mem_single_iff {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a₁ : α) (a₂ : α) (b₁ : β) (b₂ : β) : b₁ ∈ coe_fn (single a₂ b₂) a₁ ↔ a₁ = a₂ ∧ b₁ = b₂ := sorry\n\n@[simp] theorem symm_single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) : pequiv.symm (single a b) = single b a :=\n  rfl\n\n@[simp] theorem single_apply {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) : coe_fn (single a b) a = some b :=\n  if_pos rfl\n\ntheorem single_apply_of_ne {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {a₁ : α} {a₂ : α} (h : a₁ ≠ a₂) (b : β) : coe_fn (single a₁ b) a₂ = none :=\n  if_neg (ne.symm h)\n\ntheorem single_trans_of_mem {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β] [DecidableEq γ] (a : α) {b : β} {c : γ} {f : β ≃. γ} (h : c ∈ coe_fn f b) : pequiv.trans (single a b) f = single a c := sorry\n\ntheorem trans_single_of_mem {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β] [DecidableEq γ] {a : α} {b : β} (c : γ) {f : α ≃. β} (h : b ∈ coe_fn f a) : pequiv.trans f (single b c) = single a c :=\n  symm_injective (single_trans_of_mem c (iff.mpr (mem_iff_mem f) h))\n\n@[simp] theorem single_trans_single {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β] [DecidableEq γ] (a : α) (b : β) (c : γ) : pequiv.trans (single a b) (single b c) = single a c :=\n  single_trans_of_mem a (mem_single b c)\n\n@[simp] theorem single_subsingleton_eq_refl {α : Type u} [DecidableEq α] [subsingleton α] (a : α) (b : α) : single a b = pequiv.refl α := sorry\n\ntheorem trans_single_of_eq_none {β : Type v} {γ : Type w} {δ : Type x} [DecidableEq β] [DecidableEq γ] {b : β} (c : γ) {f : δ ≃. β} (h : coe_fn (pequiv.symm f) b = none) : pequiv.trans f (single b c) = ⊥ := sorry\n\ntheorem single_trans_of_eq_none {α : Type u} {β : Type v} {δ : Type x} [DecidableEq α] [DecidableEq β] (a : α) {b : β} {f : β ≃. δ} (h : coe_fn f b = none) : pequiv.trans (single a b) f = ⊥ :=\n  symm_injective (trans_single_of_eq_none a h)\n\ntheorem single_trans_single_of_ne {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β] [DecidableEq γ] {b₁ : β} {b₂ : β} (h : b₁ ≠ b₂) (a : α) (c : γ) : pequiv.trans (single a b₁) (single b₂ c) = ⊥ :=\n  single_trans_of_eq_none a (single_apply_of_ne (ne.symm h) c)\n\nprotected instance partial_order {α : Type u} {β : Type v} : partial_order (α ≃. β) :=\n  partial_order.mk (fun (f g : α ≃. β) => ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a)\n    (preorder.lt._default fun (f g : α ≃. β) => ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a) sorry sorry sorry\n\ntheorem le_def {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β} : f ≤ g ↔ ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a :=\n  iff.rfl\n\nprotected instance order_bot {α : Type u} {β : Type v} : order_bot (α ≃. β) :=\n  order_bot.mk ⊥ partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance semilattice_inf_bot {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] : semilattice_inf_bot (α ≃. β) :=\n  semilattice_inf_bot.mk order_bot.bot order_bot.le order_bot.lt sorry sorry sorry sorry\n    (fun (f g : α ≃. β) =>\n      mk (fun (a : α) => ite (coe_fn f a = coe_fn g a) (coe_fn f a) none)\n        (fun (b : β) => ite (coe_fn (pequiv.symm f) b = coe_fn (pequiv.symm g) b) (coe_fn (pequiv.symm f) b) none) sorry)\n    sorry sorry sorry\n\nend pequiv\n\n\nnamespace equiv\n\n\ndef to_pequiv {α : Type u_1} {β : Type u_2} (f : α ≃ β) : α ≃. β :=\n  pequiv.mk (some ∘ ⇑f) (some ∘ ⇑(equiv.symm f)) sorry\n\n@[simp] theorem to_pequiv_refl {α : Type u_1} : to_pequiv (equiv.refl α) = pequiv.refl α :=\n  rfl\n\ntheorem to_pequiv_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α ≃ β) (g : β ≃ γ) : to_pequiv (equiv.trans f g) = pequiv.trans (to_pequiv f) (to_pequiv g) :=\n  rfl\n\ntheorem to_pequiv_symm {α : Type u_1} {β : Type u_2} (f : α ≃ β) : to_pequiv (equiv.symm f) = pequiv.symm (to_pequiv f) :=\n  rfl\n\ntheorem to_pequiv_apply {α : Type u_1} {β : Type u_2} (f : α ≃ β) (x : α) : coe_fn (to_pequiv f) x = some (coe_fn f x) :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/pequiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.33927475118110456}}
{"text": "import .lemmas .dickson\nopen classical nat function prod set subtype list\n\nnoncomputable theory\n\ntheorem sub_gt_of_gt_ge {a b c : ℕ} (H1 : a > b) (H2 : b ≥ c) : a - c > b - c :=\nhave c ≤ a, from le_of_lt (lt_of_le_of_lt H2 H1),\nhave eq₁ : a - c + c = a, from nat.sub_add_cancel this,\nhave eq₂ : b - c + c = b, from nat.sub_add_cancel H2,\nhave b ≤ a, from le_of_lt H1,\nhave b - c ≤ a - c, from nat.sub_le_sub_right this c,\nor.elim (nat.lt_or_eq_of_le this)\n(assume Hl, Hl)\n(assume Hr, have refl : a > a, by super,\n absurd refl (lt_irrefl a))\n\nnamespace kruskal\n\n#check good_pairs\n\nsection\nparameter {A : Type}\nparameter f : ℕ → A\nparameter g : ℕ → ℕ\nparameter o : A → A → Prop\nparameter H : ¬ is_good (f ∘ g) o\n\ndefinition ran_g : set ℕ := {x : ℕ | ∃ i, g i = x}\n\ntheorem ne_empty_ran : ran_g ≠ ∅ := ne_empty_of_mem ⟨0,rfl⟩\n\nprivate definition min : ℕ := least ran_g ne_empty_ran\n\ndefinition index_of_min : ℕ :=\nhave min ∈ ran_g, from least_is_mem ran_g ne_empty_ran,\nsome this \n\ntheorem minimality_of_min (n : ℕ) : g index_of_min ≤ g n :=\nhave H1 : g index_of_min = min, from some_spec (least_is_mem ran_g ne_empty_ran),\nhave least ran_g ne_empty_ran ≤ g n, from minimality _ (g n) ⟨n,rfl⟩,\nbegin simp [H1], exact this end\n\nprivate definition h (n : ℕ) : ℕ := g (index_of_min + n)\n\ntheorem exists_sub_bad : ∃ h : ℕ → ℕ, ¬ is_good (f ∘ h) o ∧ ∀ i : ℕ, h 0 ≤ h i :=\nhave badness : ¬ is_good (f ∘ h) o, from\n   suppose is_good (f ∘ h) o,\n   let ⟨i,j,hij⟩ := this in\n   have index_of_min + i < index_of_min + j, from add_lt_add_left (and.left hij) _,\n   have is_good (f ∘ g) o, from ⟨index_of_min + i,⟨index_of_min + j,⟨this,hij^.right⟩⟩⟩,\n   H this,\nhave ∀ i : ℕ, h 0 ≤ h i, from λ i, minimality_of_min (index_of_min + i),\n⟨h,⟨badness,this⟩⟩\n\nend\n\ndefinition extends_at {A : Type} (n : ℕ) (f : ℕ → A) (g : ℕ → A) : Prop := ∀ m ≤ n, g m = f m\n\ntheorem extends_at.refl {A : Type} {n : ℕ} {f : ℕ → A} : extends_at n f f := λ m H, rfl\n\ntheorem extends_at.trans {A : Type} {n m : ℕ} {f g h: ℕ → A} (H1 : extends_at n f g) (H2 : extends_at m g h) (H3 : n ≤ m) : \n  extends_at n f h := λ k H, \nhave g k = f k, from H1 k H,\nhave k ≤ m, from nat.le_trans H H3,\nhave h k = g k, from H2 k this,\nby super\n\ntheorem least_seq_at_n {S : set (ℕ → ℕ)} (H : S ≠ ∅) (n : ℕ) : \n∃ f : ℕ → ℕ, f ∈ S ∧ ∀ g : ℕ → ℕ, g ∈ S → f n ≤ g n :=\nlet T : set ℕ := {x | ∃ f :ℕ → ℕ, f ∈ S ∧ f n = x} in\nhave ∃ f, f ∈ S, from exists_mem_of_ne_empty H,\nlet ⟨f,h⟩ := this in\nhave nemp : T ≠ ∅, from ne_empty_of_mem ⟨f,⟨h,rfl⟩⟩,\nlet a := least T nemp in\nhave a ∈ T, from least_is_mem T nemp,\nlet ⟨f',h⟩ := this in\nhave ∀ g : ℕ → ℕ, g ∈ S → f' n ≤ g n, from λ g Hg, \n  have a ≤ g n, from minimality _ _ ⟨g,⟨Hg,rfl⟩⟩, \n  by super,\n⟨f',⟨h^.left, this⟩⟩\n\nsection\nparameter {A : Type}\nparameter {P : (ℕ → A) → Prop}\nparameter g : A → ℕ\nparameter H : ∃ f : ℕ → A, P f\n\ndefinition colle : set (ℕ → A) := {f | P f}\n\nlemma nonempty_colle : colle ≠ ∅ :=  let ⟨a,h⟩ := H in ne_empty_of_mem h\n\ndefinition S : set (ℕ → ℕ) := image (λ f, g ∘ f) colle\n\nlemma nonempty_S : S ≠ ∅ := image_nonempty nonempty_colle\n\ntheorem exists_min_func (n : ℕ) : ∃ f : ℕ → ℕ, f ∈ S ∧ ∀ g : ℕ → ℕ, g ∈ S → f n ≤ g n := least_seq_at_n nonempty_S n\n\ndefinition min_func (n : ℕ) : ℕ → A := \nlet fc := some (exists_min_func n) in\nhave fc ∈ S ∧ ∀ g : ℕ → ℕ, g ∈ S → fc n ≤ g n, from (some_spec (exists_min_func n)),\nsome this^.left\n\ntheorem min_func_property (n : ℕ) : P (min_func n) :=\nlet fc := some (exists_min_func n) in\nlet ⟨l,r⟩ := some_spec (exists_min_func n) in\nhave min_func n ∈ colle ∧ (λ f, g ∘ f) (min_func n) = fc, from some_spec l ,\nthis^.left\n\ntheorem min_func_minimality (f : ℕ → A) (Hp : P f) (n : ℕ) : g (min_func n n) ≤ g (f n) := \nlet fc := some (exists_min_func n) in\nlet ⟨l,r⟩ := some_spec (exists_min_func n) in\nhave min_func n ∈ colle ∧ (λ f, g ∘ f) (min_func n) = fc, from some_spec l,\nhave (λ f, g ∘ f) (min_func n) = fc, from this^.right, \nhave eq2 : (λ f, g ∘ f) (min_func n) n = fc n, by rw this, \nhave Hr : ∀ g : ℕ → ℕ, g ∈ S → fc n ≤ g n, from (some_spec (exists_min_func n))^.right,\nhave le : fc n ≤ (λ f, g ∘ f) f n, from Hr _ ⟨f,⟨Hp,rfl⟩⟩,\nhave (λ f, g ∘ f) (min_func n) n ≤ (λ f, g ∘ f) f n, by rw -eq2 at le;exact le,\nby super\n\nend\n\nsection\n\nparameter {A : Type} \nparameter {P : (ℕ → A) → Prop} -- some property about f \nparameter g : A → ℕ -- a measure of cardinality of A \nparameter H : ∃ f, P f \n\n-- construct a sequence of functions with property P such that each one extends its predecessor and is the minimal one at n.\nnoncomputable definition mbs_helper (n : ℕ) : {f : ℕ → A // P f} :=\nnat.rec_on n\n(let f₀ := min_func g H 0 in\n have P f₀, from min_func_property g H 0,\n ⟨f₀,this⟩)\n(λ pred h',\nlet f' := h'.1 in\nhave H1 : extends_at pred f' f', from extends_at.refl,\nhave H2 : P f', from h'.2,\nhave HP : ∃ f, extends_at pred f' f ∧ P f, from ⟨f',⟨H1,H2⟩⟩,\nlet fn := min_func g HP (succ pred) in\nhave extends_at pred f' fn ∧ P fn, from min_func_property g HP (succ pred),\nhave P fn, from this^.right,\n⟨fn,this⟩)\n\n  section\n  parameter n : ℕ\n  definition helper_elt := (mbs_helper n).1\n  definition helper_succ := (mbs_helper (succ n)).1\n  lemma helper_ext_refl : extends_at n helper_elt helper_elt := extends_at.refl\n  lemma helper_has_property : P helper_elt := (mbs_helper n).2\n  lemma helper_inner_hyp : ∃ g, extends_at n helper_elt g ∧ P g := ⟨helper_elt, ⟨helper_ext_refl, helper_has_property⟩⟩\n  theorem succ_ext_of_mbs_helper : extends_at n helper_elt helper_succ := (min_func_property g helper_inner_hyp (succ n))^.left\n  end\n\ntheorem ext_of_mbs_helper (n : ℕ) : ∀ m, m ≤ n → extends_at m  (mbs_helper m).1 (mbs_helper n).1 :=\nnat.rec_on n\n(take m, assume H, \nhave eq : m = 0, from eq_zero_of_le_zero H,\nhave extends_at 0 (mbs_helper 0).1 (mbs_helper 0).1, from extends_at.refl,\nby simp [eq,this])\n(λ a IH m H,\nby_cases\n(suppose m = succ a, \nhave extends_at m (mbs_helper (succ a)).1 (mbs_helper (succ a)).1, from extends_at.refl, by super)\n(suppose m ≠ succ a, \nhave m < succ a, from lt_of_le_of_ne H this,\nhave Hle : m ≤ a, from (iff.mp (lt_succ_iff_le m a)) this,\nhave H1 : extends_at m (mbs_helper m).1 (mbs_helper a).1, from IH m Hle,\nhave extends_at a (mbs_helper a).1 (mbs_helper (succ a)).1, from succ_ext_of_mbs_helper a,\nextends_at.trans H1 this Hle))\n\ntheorem congruence_of_mbs_helper {n m : ℕ} (H : m ≤ n) : (mbs_helper n).1 m = (mbs_helper m).1 m :=\nhave extends_at m (mbs_helper m).1 (mbs_helper n).1, from ext_of_mbs_helper n m H,\nthis m (nat.le_refl m)\n\nend\n\nsection\n-- construction and properties of mbs.\nparameter {A : Type}\nparameter {o : A → A → Prop}\nparameter g : A → ℕ\nparameter H : ∃ f : ℕ → A, ¬ is_good f o\n\nnoncomputable definition seq_of_bad_seq (n : ℕ) : {f : ℕ → A // ¬ is_good f o} := mbs_helper g H n\n\ndefinition minimal_bad_seq (n : ℕ) : A :=  (seq_of_bad_seq n).1 n \n\ndefinition ext_of_seq_of_bad_seq := ext_of_mbs_helper g H\n\ndefinition congruence_of_seq_of_bad_seq {n m : ℕ} (Hnm : m ≤ n) := congruence_of_mbs_helper g H Hnm\n\ndefinition bad_seq_elt := helper_elt g H\n\ndefinition bad_seq_inner_hyp := helper_inner_hyp g H \n\ntheorem badness_of_mbs : ¬ is_good minimal_bad_seq o := \nsuppose is_good minimal_bad_seq o,\nlet ⟨i,j,h⟩ := this in\nhave i ≤ j, from le_of_lt_or_eq (or.inl h^.left),\nhave ext : extends_at i (seq_of_bad_seq i).1 (seq_of_bad_seq j).1, from ext_of_seq_of_bad_seq j i this,\nhave i ≤ i, from nat.le_refl i,\nhave (seq_of_bad_seq j).1 i = (minimal_bad_seq i), from ext i this,\nhave o ((seq_of_bad_seq j).1 i) (minimal_bad_seq j), by rw this; exact h^.right,\nhave i < j ∧ o ((seq_of_bad_seq j).1 i) ((seq_of_bad_seq j).1 j), from ⟨h^.left, this⟩,\nhave good : is_good (seq_of_bad_seq j).1 o, from ⟨i,⟨j, this⟩⟩,\nhave ¬ is_good (seq_of_bad_seq j).1 o, from (seq_of_bad_seq j).2, \nthis good\n\ntheorem minimality_of_mbs_0 (f : ℕ → A) (Hf : ¬ is_good f o) : g (minimal_bad_seq 0) ≤ g (f 0) := min_func_minimality g H f Hf 0\n\ntheorem minimality_of_mbs (n : ℕ) (f : ℕ → A) (H1 : extends_at n minimal_bad_seq f ∧ ¬ is_good f o) : g (minimal_bad_seq (succ n)) ≤ g (f (succ n)) := \nhave Hl : ∀ m, m ≤ n →  f m = (bad_seq_elt n) m, from \n  λ m Hle, have f m = minimal_bad_seq m, from H1^.left m Hle,\n  have bad_seq_elt n m = minimal_bad_seq m, from congruence_of_seq_of_bad_seq Hle,\n  by super, --by+ simp,\nhave ins_P : extends_at n (bad_seq_elt n) f ∧ ¬ is_good f o, from ⟨Hl, H1^.right⟩,\nhave ineq : g (min_func g (bad_seq_inner_hyp n) (succ n) (succ n)) ≤ g (f (succ n)), from min_func_minimality g (bad_seq_inner_hyp n) f ins_P (succ n), \nby super\n\nend\n\nsection\n\n-- Given two sequences f and g, a function h which modifies indices so that h 0 is the break point, construct a new sequence 'combined_seq' by concatenating f and g at (h 0).\n\nparameter {Q :Type}\nparameter {o : Q → Q → Prop}\nparameters f g : ℕ → Q\nparameter h : ℕ → ℕ\nparameter Hh : ∀ i, h 0 ≤ h i\nparameter Hf : ¬ is_good f o\nparameter Hg : ¬ is_good g o\n-- in Higman's lemma in Williams 1963, h is f, g is the bad sequence B ∘ f\nparameter H : ∀ i j, o (f i) (g (j - h 0)) → o (f i) (f (h (j - h 0))) \n\ndefinition comb (n : ℕ) : Q := if n < h 0 then f n else g (n - h 0)\n\ntheorem g_part_of_comb (H : (h 0) = 0) : ∀ x, comb x = g x :=\nλ n, have ¬ n < h 0, by rw H; apply not_lt_zero ,\nhave comb n = g (n - (h 0)), from if_neg this,\nby simp [this, H]\n\ninclude Hh\n\ntheorem badness_of_comb : ¬ is_good comb o := \nλ good, let ⟨i,j,hw⟩ := good in\nby_cases\n(assume Hposi : i < h 0, \n  have eq1i : comb i = f i, from if_pos Hposi,\n  by_cases \n  (suppose j < h 0, \n    have eq1j : comb j = f j, from if_pos this, \n    have o (f i) (f j),by rw [-eq1j,-eq1i]; exact hw^.right,\n    have is_good f o, from ⟨i, ⟨j,⟨hw^.left,this⟩⟩⟩,\n    show _, from Hf this)\n  (suppose ¬ j < h 0,\n    have eq2j : comb j = g (j - (h 0)), from if_neg this, \n    have o (f i) (g (j - (h 0))), by rw [-eq2j,-eq1i]; exact hw^.right,\n    have Hr : o (f i) (f (h (j - (h 0)))), from H _ _ this,\n    have i < h (j - (h 0)), from lt_of_lt_of_le Hposi (Hh _),\n    have is_good f o, from ⟨i, ⟨h (j - h 0), ⟨this, Hr⟩⟩⟩,\n    show _, from Hf this))\n(assume Hnegi, \n  have eq2i : comb i = g (i - h 0), from if_neg Hnegi,\n  by_cases\n  (suppose j < h 0,\n    have j < i, from lt_of_lt_of_le this (le_of_not_gt Hnegi),\n    show _, from (not_lt_of_gt hw^.left) this)\n  (suppose ¬ j < h 0, \n    have eq2j : comb j = g (j - h 0), from if_neg this,\n    have Hr2 : o (g (i - h 0)) (g (j - h 0)), by rw [-eq2i,-eq2j]; exact hw^.right,\n    have i - h 0 < j - h 0, from sub_gt_of_gt_ge hw^.left (le_of_not_gt Hnegi),\n    have is_good g o, from ⟨(i - h 0), ⟨(j - h 0),⟨this, Hr2⟩⟩⟩,\n    show _, from Hg this))\n\nend\n\nsection\n-- further assume that f is a minimal bad sequence and card (g 0) < card (f (h 0)) \n-- In other words, this section says, assuming that there is a bad sequence of Q, if g is a bad sequence such that H holds, then there is a contradiction. \nparameter {Q :Type}\nparameter {o : Q → Q → Prop}\nparameters {g : ℕ → Q}\nparameter h : ℕ → ℕ\nparameter m : Q → ℕ -- a measure of cardinality\nparameter Hh : ∀ i, h 0 ≤ h i\nparameter Hex : ∃ f, ¬ is_good f o\nparameter Hg : ¬ is_good g o\nparameter H : ∀ i j, o (minimal_bad_seq m Hex i) (g (j - h 0)) → o (minimal_bad_seq m Hex i) ((minimal_bad_seq m Hex) (h (j - h 0)))\nparameter Hbp : m (g 0) < m (minimal_bad_seq m Hex (h 0))\n\ndefinition comb_seq_with_mbs := comb (minimal_bad_seq m Hex) g h\n\ntheorem g_part_of_comb_seq_with_mbs (H1 : (h 0) = 0) : ∀ x, comb_seq_with_mbs x = g x := \nbegin apply g_part_of_comb, assumption end\n\ntheorem badness_of_comb_seq_with_mbs : ¬ is_good comb_seq_with_mbs o := \nbadness_of_comb (minimal_bad_seq m Hex) g h Hh (badness_of_mbs m Hex) Hg H\n\ntheorem comb_seq_extends_mbs_at_pred_bp (H : h 0 ≠ 0): extends_at (pred (h 0)) (minimal_bad_seq m Hex) comb_seq_with_mbs := \nλ m Hm, if_pos (or_resolve_left (lt_of_le_pred Hm) H)\n\nlemma comb_seq_h0 : comb_seq_with_mbs (h 0) = g 0 := \nhave comb_seq_with_mbs (h 0) = g (h 0 - h 0), begin apply if_neg, rw lt_self_iff_false, trivial end,\nby simp [this,nat.sub_self]\n\ninclude Hbp Hex\n\ntheorem local_contra_of_comb_seq_with_mbs : false := \nby_cases\n(suppose eq0 : h 0 = 0, \nhave eq : comb_seq_with_mbs 0 = g 0, begin apply g_part_of_comb_seq_with_mbs, assumption end,\nhave m (comb_seq_with_mbs 0) < m (minimal_bad_seq m Hex (h 0)), by rw -eq at Hbp;exact Hbp,\nhave le : m (comb_seq_with_mbs 0) < m (minimal_bad_seq m Hex 0), by super,\nhave m (minimal_bad_seq m Hex 0) ≤ m (comb_seq_with_mbs 0), from minimality_of_mbs_0 m Hex comb_seq_with_mbs badness_of_comb_seq_with_mbs,\n(not_le_of_gt le) this)\n(assume Hneg, \nhave le : m (minimal_bad_seq m Hex (succ (pred (h 0)))) ≤  m (comb_seq_with_mbs (succ (pred (h 0)))), from minimality_of_mbs m _ _ _ ⟨begin apply comb_seq_extends_mbs_at_pred_bp, exact Hneg end,badness_of_comb_seq_with_mbs⟩,\nhave h 0 > 0, from nat.pos_of_ne_zero Hneg,\nhave succ (pred (h 0)) = h 0, from succ_pred_of_pos this,\nhave m (minimal_bad_seq m Hex (h 0)) ≤ m (comb_seq_with_mbs (h 0)), by rw this at le;exact le,\nhave m (minimal_bad_seq m Hex (h 0)) ≤ m (g 0), by rw comb_seq_h0 at this;exact this,\nhave ¬ m (g 0) < m (minimal_bad_seq m Hex (h 0)), from not_lt_of_ge this,  \nthis Hbp)\n\nend\n\nsection\nvariable {α : Type}\n\ninductive sublist' [has_le α]: list α → list α → Prop\n| slnil : sublist' [] []\n| cons (l₁ l₂ a) : sublist' l₁ l₂ → sublist' l₁ (a::l₂)\n| cons2 (l₁ l₂ a b) : a ≤ b → sublist' l₁ l₂ → sublist' (a::l₁) (b::l₂)\n\ninfix ` <++ `:50 := sublist'\n\n@[simp] lemma nil_sublist' [has_le α] : Π (l : list α), [] <++ l\n| []       := sublist'.slnil\n| (a :: l) := sublist'.cons _ _ a (nil_sublist' l)\n\n@[simp] lemma sublist'.refl [quasiorder α] : Π (l : list α), l <++ l\n| []       := sublist'.slnil\n| (a :: l) := sublist'.cons2 _ _ a _ (quasiorder.refl a) (sublist'.refl l)\n\nopen sublist'\n\nlemma sublist'.trans [quasiorder α] : Π {l₁ l₂ l₃ : list α}, l₁ <++ l₂ → l₂ <++ l₃ → l₁ <++ l₃\n| ._ ._ ._ (slnil) (slnil) := nil_sublist' nil\n| ._ ._ ._ (slnil) (cons ._ l₃' a h) := nil_sublist' _\n| ._ ._ ._ (cons l₁ l₂' a h) (cons ._ l₃' b h') := \n  have l₁ <++ l₃', from sublist'.trans (cons _ _ _ h) h',\n  show _, from cons _ _ _ this\n| ._ ._ ._ (cons l₁ l₂' a h) (cons2 ._ l₃' ._ b hab h') := \n  have l₁ <++ l₃', from sublist'.trans h h',\n  show _, from cons _ _ _ this\n| ._ ._ ._ (cons2 l₁' l₂' a b hab h) (cons ._ l₃' c h') := \n  have a :: l₁' <++ b :: l₂', from cons2 _ _ _ _ hab h,\n  have a :: l₁' <++ l₃', from sublist'.trans this h',\n  show _, from cons _ _ _ this\n| ._ ._ ._ (cons2 l₁' l₂' a b hab h) (cons2 ._ l₃' ._ d hbd h') := \n  show _, from cons2 _ _ _ _ (quasiorder.trans hab hbd) (sublist'.trans h h')\n\nlemma sublist'.trans' [quasiorder α] {l₁ l₂ l₃ : list α} (h₁ : l₁ <++ l₂)  (h₂ : l₂ <++ l₃) : l₁ <++ l₃ :=\nsublist'.trans h₁ h₂\n\n@[simp] lemma sublist'_cons [quasiorder α](a : α) (l : list α) : l <++ a::l :=\nsublist'.cons _ _ _ (sublist'.refl l)\n\n\nend\n\nsection\nparameter {Q : Type}\nparameter [o : wqo Q]\n\nparameter H : ∃ f : ℕ → list Q, ¬ is_good f sublist'\n\ndefinition Higman's_mbs (n : ℕ) : list Q := minimal_bad_seq length H n\n\ntheorem badness_of_Higman's_mbs : ¬ is_good Higman's_mbs sublist' := badness_of_mbs length H\n\ntheorem nonempty_mem_of_mbs (n : ℕ) : Higman's_mbs n ≠ [] := \nλ h, badness_of_Higman's_mbs ⟨n, ⟨succ n,⟨lt_succ_self n,begin rw h, apply nil_sublist' end⟩⟩⟩\n\ndefinition B_pairs (n : ℕ) : Q × list Q := \nhave ∃ p : Q × list Q, Higman's_mbs n = p.1 :: p.2, from ne_nil_desturct (nonempty_mem_of_mbs n),\nsome this\n\nprivate definition B (n : ℕ) : list Q := (B_pairs n).2\n\ndefinition qn (n : ℕ) : Q := (B_pairs n).1\n\ntheorem Higman's_mbs_destruct (n : ℕ) : Higman's_mbs n = qn n :: B n :=\nsome_spec (ne_nil_desturct (nonempty_mem_of_mbs n))\n\ntheorem qn_in_mbs (n : ℕ) : qn n ∈ Higman's_mbs n :=\nhave Higman's_mbs n = qn n :: B n, from some_spec (ne_nil_desturct (nonempty_mem_of_mbs n)),\nbegin rw this, apply or.inl, reflexivity end\n\ntheorem sub_B_mbs (n : ℕ) : B n <++ Higman's_mbs n :=\nhave B n <++ qn n :: B n, from sublist'.cons _ _ _ (sublist'.refl _),\nhave Higman's_mbs n = qn n :: B n, from Higman's_mbs_destruct n,\nby simph\n\ntheorem trans_of_B (i j : ℕ) (H1 : Higman's_mbs i <++ B j) : Higman's_mbs i <++ Higman's_mbs j :=\nsublist'.trans H1 (sub_B_mbs j)\n\ntheorem len_B_lt_mbs (n : ℕ) : length (B n) < length (Higman's_mbs n) :=\nhave length (B n) < length (qn n :: B n), by rw length_cons; apply lt_succ_self,\nhave Higman's_mbs n = qn n :: B n, from Higman's_mbs_destruct n,\nby super\n\nsection\nparameter Hg : ∃ g : ℕ → ℕ, ¬ is_good (B ∘ g) sublist' ∧ ∀ i : ℕ, g 0 ≤ g i\n\nprivate definition g : ℕ → ℕ := some Hg\n\ntheorem Higman's_Hg : ¬ is_good (B ∘ g) sublist' := \nlet ⟨l,r⟩ := some_spec Hg in l\n\ntheorem Higman's_Hex : ∃ f : ℕ → list Q, ¬ is_good f sublist' := \n⟨(B ∘ g), Higman's_Hg⟩\n\ntheorem Higman's_Hh : ∀ i : ℕ, g 0 ≤ g i := (some_spec Hg)^.right\n\ntheorem Higman's_H : ∀ i j, Higman's_mbs i <++ (B ∘ g) (j - g 0) → Higman's_mbs i <++ Higman's_mbs (g (j - g 0)) := \nλ i j, λ H1, trans_of_B i (g (j - g 0)) H1\n\ntheorem Higman's_Hbp : length (B (g 0)) < length (Higman's_mbs (g 0)) := len_B_lt_mbs (g 0)\n\ntheorem Higman's_local_contradition : false := \nlocal_contra_of_comb_seq_with_mbs g length Higman's_Hh Higman's_Hex Higman's_Hg Higman's_H Higman's_Hbp\n\nend\n\ndefinition ClassB : Type := {x : list Q // ∃ i, B i = x}\n\ndefinition oB (b1 : ClassB) (b2 : ClassB) : Prop := b1.val <++ b2.val\n\ntheorem oB_refl (q : ClassB) : oB q q := sublist'.refl q.val\n\ntheorem oB_trans (a b c : ClassB) (H1 : oB a b) (H2 : oB b c) : oB a c :=\nsublist'.trans H1 H2\n\n    section\n    parameter HfB : ∃ f, ¬ is_good f oB\n\n    private definition f' : ℕ → ClassB := some HfB\n\n    private theorem bad_f' : ¬ is_good f' oB := some_spec HfB\n\n    private definition g' (n : ℕ) := (f' n).val\n\n    theorem exists_bad_B_seq : ¬ is_good g' sublist' :=\n    suppose is_good g' sublist',\n    let ⟨i,j,hg'⟩ := this in\n    have is_good f' oB, from ⟨i, ⟨j, ⟨hg'^.left, hg'^.right⟩⟩⟩,\n    bad_f' this\n\n    private definition g (n : ℕ) : ℕ := \n    have ∃ i, B i = g' n, from (f' n).2,\n    some this\n\n    private theorem comp_eq_g' : B ∘ g = g' :=\n    have ∀ x, B (g x) = g' x, from λ x, some_spec (f' x).2,\n    funext this\n\n    private theorem bad_comp : ¬ is_good (B ∘ g) sublist' := \n    have ¬ is_good g' sublist', from exists_bad_B_seq,\n    by rw -comp_eq_g' at this;exact this\n\n    theorem exists_sub_bad_B_seq : ∃ h : ℕ → ℕ, ¬ is_good (B ∘ h) sublist' ∧ ∀ i : ℕ, h 0 ≤ h i := exists_sub_bad B g sublist' bad_comp\n\n    end\n\ntheorem oB_is_good : ∀ f, is_good f oB :=\nby_contradiction\n(suppose ¬ ∀ f, is_good f oB,\nhave ∃ f, ¬ is_good f oB, from classical.exists_not_of_not_forall this,\nhave ∃ h : ℕ → ℕ, ¬ is_good (B ∘ h) sublist' ∧ ∀ i : ℕ, h 0 ≤ h i, from exists_sub_bad_B_seq this,\nHigman's_local_contradition this)\n\ninstance wqo_ClassB : wqo ClassB := wqo.mk (quasiorder.mk (has_le.mk oB) oB_refl oB_trans) oB_is_good\n\ninstance wqo_prod_Q_ClassB : wqo (Q × ClassB) := @wqo_prod _ _ o _\n\ntheorem good_prod_Q_ClassB : ∀ f : ℕ → Q × ClassB, is_good f (prod_order o.le oB) := wqo.is_good\n\nlemma B_refl (n : ℕ) : ∃ i, B i = B n := ⟨n, rfl⟩\n\ndefinition fB (n : ℕ) : ClassB := ⟨B n,B_refl n⟩\n\nprivate definition p (n : ℕ) : Q × ClassB := (qn n, fB n)\n\ntheorem good_p : is_good p (prod_order o.le oB) := good_prod_Q_ClassB p\n\ntheorem Hij : ∃ i j, i < j ∧ ((qn i) ≤ (qn j) ∧ oB (fB i) (fB j)) := good_p\n\ntheorem exists_embeds : ∃ i j, i < j ∧ Higman's_mbs i <++ Higman's_mbs j :=\nlet ⟨i,j,⟨hijl,⟨hijrl,hijrr⟩⟩⟩ := good_p in\nhave qn i :: B i <++ qn j :: B j, from sublist'.cons2 _ _ _ _ hijrl hijrr,\nhave Higman's_mbs i = qn i :: B i, from Higman's_mbs_destruct i,\nhave Higman's_mbs j = qn j :: B j, from Higman's_mbs_destruct j,\nhave Higman's_mbs i <++ Higman's_mbs j, by simph,\n⟨i,j,⟨hijl,this⟩⟩\n\ntheorem Higman's_contradiction : false := badness_of_Higman's_mbs exists_embeds\n\nend\n\nvariable {Q : Type}\nvariable [wqo Q]\n\ntheorem good_sublist : ∀ f : ℕ → list Q , is_good f sublist' := \nby_contradiction\n(suppose ¬ ∀ f, is_good f sublist',\nhave ∃ f, ¬ is_good f sublist', from classical.exists_not_of_not_forall this,\nHigman's_contradiction this)\n\ndef wqo_list : wqo (list Q) :=\n⟨⟨⟨sublist'⟩,sublist'.refl,λ a b c h₁ h₂,sublist'.trans' h₁ h₂⟩,good_sublist⟩\n\n\nend kruskal\n", "meta": {"author": "minchaowu", "repo": "Kruskal.lean3", "sha": "a14516f47b21e636e9df914fc6ebe64cbe5cd38d", "save_path": "github-repos/lean/minchaowu-Kruskal.lean3", "path": "github-repos/lean/minchaowu-Kruskal.lean3/Kruskal.lean3-a14516f47b21e636e9df914fc6ebe64cbe5cd38d/kruskal_final/higman.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3392680464675018}}
{"text": "import melting_point.topology\nopen melting_point.topology\n\ninductive X\n| a | b | c | d\n\nnamespace X\n  def τ : set (set X) :=\n  { { a },\n    { c },\n    { a, c },\n    { a, b, c },\n    { a, d, c },\n    ∅, set.univ }\n\n  example : topology X := begin\n    fapply topology.mk, exact τ, enumeration,\n    repeat { intros x y u v, sinduction u; sinduction v; findset }\n  end\nend X", "meta": {"author": "forked-from-1kasper", "repo": "melting_point", "sha": "e5ea4a0917de086b7e5b122e8d5aa90d2761d147", "save_path": "github-repos/lean/forked-from-1kasper-melting_point", "path": "github-repos/lean/forked-from-1kasper-melting_point/melting_point-e5ea4a0917de086b7e5b122e8d5aa90d2761d147/melting_point/X.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3391146571611324}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.extensive\nimport category_theory.limits.shapes.kernel_pair\n\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `category_theory.is_pushout.is_van_kampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `category_theory.adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `category_theory.type.adhesive`: The category of `Type` is adhesive.\n- `category_theory.adhesive.is_pullback_of_is_pushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `category_theory.adhesive.mono_of_is_pushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `category_theory.adhesive.to_regular_mono_category`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n\n## TODO\n\nShow that the following are adhesive:\n- functor categories into adhesive categories\n- the categories of sheaves over a site\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\nnamespace category_theory\n\nopen limits\n\nuniverses v' u' v u\n\nvariables {J : Type v'} [category.{u'} J] {C : Type u} [category.{v} C]\n\nvariables {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n/-- A convenience formulation for a pushout being a van Kampen colimit.\nSee `is_pushout.is_van_kampen_iff` below. -/\n@[nolint unused_arguments] -- This only makes sense when the original diagram is a pushout.\ndef is_pushout.is_van_kampen (H : is_pushout f g h i) : Prop :=\n∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z')\n  (αW : W' ⟶ W) (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z)\n  (hf : is_pullback f' αW αX f) (hg : is_pullback g' αW αY g)\n  (hh : comm_sq h' αX αZ h) (hi : comm_sq i' αY αZ i)\n  (w : comm_sq f' g' h' i'),\n  is_pushout f' g' h' i' ↔ is_pullback h' αX αZ h ∧ is_pullback i' αY αZ i\n\nlemma is_pushout.is_van_kampen.flip {H : is_pushout f g h i} (H' : H.is_van_kampen) :\n  H.flip.is_van_kampen :=\nbegin\n  introv W' hf hg hh hi w,\n  simpa only [is_pushout.flip_iff, is_pullback.flip_iff, and_comm] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip,\nend\n\nlemma is_pushout.is_van_kampen_iff (H : is_pushout f g h i) :\n  H.is_van_kampen ↔ is_van_kampen_colimit (pushout_cocone.mk h i H.w) :=\nbegin\n  split,\n  { intros H F' c' α fα eα hα,\n    refine iff.trans _ ((H (F'.map walking_span.hom.fst) (F'.map walking_span.hom.snd)\n      (c'.ι.app _) (c'.ι.app _) (α.app _) (α.app _) (α.app _) fα\n      (by convert hα walking_span.hom.fst) (by convert hα walking_span.hom.snd)\n      _ _ _).trans _),\n    { have : F'.map walking_span.hom.fst ≫ c'.ι.app walking_span.left =\n        F'.map walking_span.hom.snd ≫ c'.ι.app walking_span.right := by simp only [cocone.w],\n      rw (is_colimit.equiv_of_nat_iso_of_iso (diagram_iso_span F') c'\n        (pushout_cocone.mk _ _ this) _).nonempty_congr,\n      { exact ⟨λ h, ⟨⟨this⟩, h⟩, λ h, h.2⟩ },\n      { refine cocones.ext (iso.refl c'.X) _, rintro (_|_|_); dsimp;\n          simp only [c'.w, category.assoc, category.id_comp, category.comp_id] } },\n    { exact ⟨nat_trans.congr_app eα.symm _⟩ },\n    { exact ⟨nat_trans.congr_app eα.symm _⟩ },\n    { exact ⟨by simp⟩ },\n    split,\n    { rintros ⟨h₁, h₂⟩ (_|_|_),\n      { rw ← c'.w walking_span.hom.fst, exact (hα walking_span.hom.fst).paste_horiz h₁ },\n      exacts [h₁, h₂] },\n    { intro h, exact ⟨h _, h _⟩ } },\n  { introv H W' hf hg hh hi w,\n    refine (iff.trans _\n      ((H w.cocone ⟨by { rintros (_|_|_), exacts [αW, αX, αY] }, _⟩ αZ _ _).trans _)),\n    rotate,\n    { rintros i _ (_|_|_),\n      { dsimp, simp only [functor.map_id, category.comp_id, category.id_comp] },\n      exacts [hf.w, hg.w] },\n    { ext (_|_|_),\n      { dsimp, rw pushout_cocone.condition_zero, erw [category.assoc, hh.w, hf.w_assoc] },\n      exacts [hh.w.symm, hi.w.symm] },\n    { rintros i _ (_|_|_),\n      { dsimp, simp_rw functor.map_id,\n        exact is_pullback.of_horiz_is_iso ⟨by rw [category.comp_id, category.id_comp]⟩ },\n      exacts [hf, hg] },\n    { split,\n      { intro h, exact ⟨h walking_cospan.left, h walking_cospan.right⟩ },\n      { rintro ⟨h₁, h₂⟩ (_|_|_),\n        { dsimp, rw pushout_cocone.condition_zero, exact hf.paste_horiz h₁ },\n        exacts [h₁, h₂] } },\n    { exact ⟨λ h, h.2, λ h, ⟨_, h⟩⟩ } }\nend\n.\n\nlemma is_coprod_iff_is_pushout {X E Y YE : C} (c : binary_cofan X E)\n  (hc : is_colimit c) {f : X ⟶ Y} {iY : Y ⟶ YE} {fE : c.X ⟶ YE}\n  (H : comm_sq f c.inl iY fE) :\n  nonempty (is_colimit (binary_cofan.mk (c.inr ≫ fE) iY)) ↔ is_pushout f c.inl iY fE :=\nbegin\n  split,\n  { rintro ⟨h⟩,\n    refine ⟨H, ⟨limits.pushout_cocone.is_colimit_aux' _ _⟩⟩,\n    intro s,\n    dsimp,\n    refine ⟨h.desc (binary_cofan.mk (c.inr ≫ s.inr) s.inl), h.fac _ ⟨walking_pair.right⟩, _, _⟩,\n    { apply binary_cofan.is_colimit.hom_ext hc,\n      { rw ← H.w_assoc, erw h.fac _ ⟨walking_pair.right⟩, exact s.condition },\n      { rw ← category.assoc, exact h.fac _ ⟨walking_pair.left⟩ } },\n    { intros m e₁ e₂,\n      apply binary_cofan.is_colimit.hom_ext h,\n      { dsimp, rw [category.assoc, e₂, eq_comm], exact h.fac _ ⟨walking_pair.left⟩ },\n      { refine e₁.trans (eq.symm _), exact h.fac _ _ } } },\n  { refine λ H, ⟨_⟩,\n    fapply limits.binary_cofan.is_colimit_mk,\n    { exact λ s, H.is_colimit.desc (pushout_cocone.mk s.inr _ $\n        (hc.fac (binary_cofan.mk (f ≫ s.inr) s.inl) ⟨walking_pair.left⟩).symm) },\n    { intro s,\n      erw [category.assoc, H.is_colimit.fac _ walking_span.right, hc.fac], refl },\n    { intro s, exact H.is_colimit.fac _ walking_span.left },\n    { intros s m e₁ e₂,\n      apply pushout_cocone.is_colimit.hom_ext H.is_colimit,\n      { symmetry, exact (H.is_colimit.fac _ walking_span.left).trans e₂.symm },\n      { erw H.is_colimit.fac _ walking_span.right,\n        apply binary_cofan.is_colimit.hom_ext hc,\n        { dsimp, erw [hc.fac, ← H.w_assoc, e₂], refl },\n        { refine ((category.assoc _ _ _).symm.trans e₁).trans _, symmetry, exact hc.fac _ _ } } } }\nend\n\nlemma is_pushout.is_van_kampen_inl {W E X Z : C} (c : binary_cofan W E)\n  [finitary_extensive C]\n  [has_pullbacks C]\n  (hc : is_colimit c) (f : W ⟶ X) (h : X ⟶ Z) (i : c.X ⟶ Z)\n  (H : is_pushout f c.inl h i) : H.is_van_kampen :=\nbegin\n  obtain ⟨hc₁⟩ := (is_coprod_iff_is_pushout c hc H.1).mpr H,\n  introv W' hf hg hh hi w,\n  obtain ⟨hc₂⟩ := ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen c hc)\n    (binary_cofan.mk _ pullback.fst) _ _ _ hg.w.symm pullback.condition.symm).mpr\n    ⟨hg, is_pullback.of_has_pullback αY c.inr⟩,\n  refine (is_coprod_iff_is_pushout _ hc₂ w).symm.trans _,\n  refine ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen _ hc₁)\n    (binary_cofan.mk _ _) pullback.snd _ _ _ hh.w.symm).trans _,\n  { dsimp, rw [← pullback.condition_assoc, category.assoc, hi.w] },\n  split,\n  { rintro ⟨hc₃, hc₄⟩,\n    refine ⟨hc₄, _⟩,\n    let Y'' := pullback αZ i,\n    let cmp : Y' ⟶ Y'' := pullback.lift i' αY hi.w,\n    have e₁ : (g' ≫ cmp) ≫ pullback.snd = αW ≫ c.inl :=\n      by rw [category.assoc, pullback.lift_snd, hg.w],\n    have e₂ : (pullback.fst ≫ cmp : pullback αY c.inr ⟶ _) ≫ pullback.snd =\n      pullback.snd ≫ c.inr :=\n      by rw [category.assoc, pullback.lift_snd, pullback.condition],\n    obtain ⟨hc₄⟩ := ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen c hc)\n      (binary_cofan.mk _ _) αW _ _ e₁.symm e₂.symm).mpr ⟨_, _⟩,\n    { rw [← category.id_comp αZ, ← show cmp ≫ pullback.snd = αY, from pullback.lift_snd _ _ _],\n      apply is_pullback.paste_vert _ (is_pullback.of_has_pullback αZ i),\n      have : cmp = (hc₂.cocone_point_unique_up_to_iso hc₄).hom,\n      { apply binary_cofan.is_colimit.hom_ext hc₂,\n        exacts [(hc₂.comp_cocone_point_unique_up_to_iso_hom hc₄ ⟨walking_pair.left⟩).symm,\n          (hc₂.comp_cocone_point_unique_up_to_iso_hom hc₄ ⟨walking_pair.right⟩).symm] },\n      rw this,\n      exact is_pullback.of_vert_is_iso ⟨by rw [← this, category.comp_id, pullback.lift_fst]⟩ },\n    { apply is_pullback.of_right _ e₁ (is_pullback.of_has_pullback _ _),\n      rw [category.assoc, pullback.lift_fst, ← H.w, ← w.w], exact hf.paste_horiz hc₄ },\n    { apply is_pullback.of_right _ e₂ (is_pullback.of_has_pullback _ _),\n      rw [category.assoc, pullback.lift_fst], exact hc₃ } },\n  { rintros ⟨hc₃, hc₄⟩,\n    exact ⟨(is_pullback.of_has_pullback αY c.inr).paste_horiz hc₄, hc₃⟩ }\nend\n\nlemma is_pushout.is_van_kampen.is_pullback_of_mono_left [mono f]\n  {H : is_pushout f g h i} (H' : H.is_van_kampen) :\n  is_pullback f g h i :=\n((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i\n  (is_kernel_pair.id_of_mono f) (is_pullback.of_vert_is_iso ⟨by simp⟩) H.1.flip ⟨rfl⟩\n  ⟨by simp⟩).mp (is_pushout.of_horiz_is_iso ⟨by simp⟩)).1.flip\n\nlemma is_pushout.is_van_kampen.is_pullback_of_mono_right [mono g]\n  {H : is_pushout f g h i} (H' : H.is_van_kampen) :\n  is_pullback f g h i :=\n((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h  (is_pullback.of_vert_is_iso ⟨by simp⟩)\n  (is_kernel_pair.id_of_mono g) ⟨rfl⟩ H.1\n  ⟨by simp⟩).mp (is_pushout.of_vert_is_iso ⟨by simp⟩)).2\n\nlemma is_pushout.is_van_kampen.mono_of_mono_left [mono f]\n  {H : is_pushout f g h i} (H' : H.is_van_kampen) :\n  mono i :=\nis_kernel_pair.mono_of_is_iso_fst\n  (((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i\n  (is_kernel_pair.id_of_mono f) (is_pullback.of_vert_is_iso ⟨by simp⟩) H.1.flip ⟨rfl⟩\n  ⟨by simp⟩).mp (is_pushout.of_horiz_is_iso ⟨by simp⟩)).2)\n\nlemma is_pushout.is_van_kampen.mono_of_mono_right [mono g]\n  {H : is_pushout f g h i} (H' : H.is_van_kampen) :\n  mono h :=\nis_kernel_pair.mono_of_is_iso_fst\n  ((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h  (is_pullback.of_vert_is_iso ⟨by simp⟩)\n  (is_kernel_pair.id_of_mono g) ⟨rfl⟩ H.1\n  ⟨by simp⟩).mp (is_pushout.of_vert_is_iso ⟨by simp⟩)).1\n\n/-- A category is adhesive if it has pushouts and pullbacks along monomorphisms,\nand such pushouts are van Kampen. -/\nclass adhesive (C : Type u) [category.{v} C] : Prop :=\n[has_pullback_of_mono_left : ∀ {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [mono f], has_pullback f g]\n[has_pushout_of_mono_left : ∀ {X Y S : C} (f : S ⟶ X) (g : S ⟶ Y) [mono f], has_pushout f g]\n(van_kampen : ∀ {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} [mono f]\n  (H : is_pushout f g h i), H.is_van_kampen)\n\nattribute [priority 100, instance]\n  adhesive.has_pullback_of_mono_left adhesive.has_pushout_of_mono_left\n\nlemma adhesive.van_kampen' [adhesive C] [mono g] (H : is_pushout f g h i) : H.is_van_kampen :=\n(adhesive.van_kampen H.flip).flip\n\nlemma adhesive.is_pullback_of_is_pushout_of_mono_left [adhesive C]\n  (H : is_pushout f g h i) [mono f] : is_pullback f g h i :=\n(adhesive.van_kampen H).is_pullback_of_mono_left\n\nlemma adhesive.is_pullback_of_is_pushout_of_mono_right [adhesive C]\n  (H : is_pushout f g h i) [mono g] : is_pullback f g h i :=\n(adhesive.van_kampen' H).is_pullback_of_mono_right\n\nlemma adhesive.mono_of_is_pushout_of_mono_left [adhesive C]\n  (H : is_pushout f g h i) [mono f] : mono i :=\n(adhesive.van_kampen H).mono_of_mono_left\n\nlemma adhesive.mono_of_is_pushout_of_mono_right [adhesive C]\n  (H : is_pushout f g h i) [mono g] : mono h :=\n(adhesive.van_kampen' H).mono_of_mono_right\n\ninstance type.adhesive : adhesive (Type u) :=\nbegin\n  constructor,\n  intros,\n  exactI (is_pushout.is_van_kampen_inl _ (types.is_coprod_of_mono f) _ _ _ H.flip).flip\nend\n\n@[priority 100] noncomputable\ninstance adhesive.to_regular_mono_category [adhesive C] : regular_mono_category C :=\n⟨λ X Y f hf, by exactI\n  { Z := pushout f f,\n    left := pushout.inl,\n    right := pushout.inr,\n    w := pushout.condition,\n    is_limit := (adhesive.is_pullback_of_is_pushout_of_mono_left\n      (is_pushout.of_has_pushout f f)).is_limit_fork }⟩\n\n-- This then implies that adhesive categories are balanced\nexample [adhesive C] : balanced C := infer_instance\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/adhesive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3389932470008315}}
{"text": "import Mathlib.Tactic.Lift\nimport Mathlib.Tactic.PermuteGoals\nimport Mathlib.Tactic.Coe\nimport Mathlib.Init.Set\nimport Mathlib.Order.Basic\nimport Mathlib.Algebra.Group.WithOne.Defs\nimport Mathlib.Data.Set.Image\nimport Mathlib.Data.Set.List\nimport Mathlib.Data.Rat.Defs\nimport Mathlib.Data.PNat.Defs\n\n/-! Some tests of the `lift` tactic. -/\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  lift n to ℕ\n  guard_target =ₛ 0 ≤ n\n  swap; guard_target =ₛ 0 ≤ (n : Int) + 1; swap\n  · exact hn\n  · exact Int.le_add_one hn\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  lift n to ℕ using hn\n  guard_target =ₛ 0 ≤ (n : Int) + 1\n  exact Int.le_add_one (Int.ofNat_zero_le _)\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  have hn' := hn\n  lift n to ℕ using hn with k hk\n  guard_target =ₛ 0 ≤ (k : Int) + 1\n  guard_hyp hk : (k : Int) = n\n  exact Int.le_add_one hn'\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  have hn' := hn\n  lift n to ℕ using hn with k\n  guard_target =ₛ 0 ≤ (k : Int) + 1\n  exact Int.le_add_one hn'\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  lift n to ℕ using hn with k hk hn\n  guard_target =ₛ 0 ≤ (k : Int) + 1\n  guard_hyp hn : 0 ≤ (k : Int)\n  guard_hyp hk : k = n\n  exact Int.le_add_one hn\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  lift n to ℕ using hn with k rfl hn\n  guard_target =ₛ 0 ≤ (k : Int) + 1\n  guard_hyp hn : 0 ≤ (k : Int)\n  exact Int.le_add_one hn\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  have hn' := hn\n  lift n to ℕ using hn with k rfl\n  guard_target =ₛ 0 ≤ (k : Int) + 1\n  exact Int.le_add_one hn'\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  have hn' := hn\n  lift n to ℕ using hn with n\n  guard_target =ₛ 0 ≤ (n : Int) + 1\n  exact Int.le_add_one hn'\n\nexample (n : ℤ) (hn : 0 ≤ n) : 0 ≤ n + 1 := by\n  -- Should fail because we didn't provide a variable name when lifting an expression\n  fail_if_success lift n + 1 to ℕ using (Int.le_add_one hn)\n  -- Now it should succeed\n  lift n + 1 to ℕ using (Int.le_add_one hn) with k hk\n  exact of_decide_eq_true rfl\n\n-- test lift of functions\nexample (α : Type _) (f : α → ℤ) (hf : ∀ a, 0 ≤ f a) (hf' : ∀ a, f a < 1) (a : α) :\n    0 ≤ 2 * f a := by\n  lift f to α → ℕ using hf\n  guard_target =ₛ (0 : ℤ) ≤ 2 * (fun i : α ↦ (f i : ℤ)) a\n  guard_hyp hf' : ∀ a, ((fun i : α ↦ (f i : ℤ)) a) < 1\n  constructor\n\nexample (α : Type _) (f : α → ℤ) (hf : ∀ a, 0 ≤ f a) (hf' : ∀ a, f a < 1) (a : α) :\n    0 ≤ 2 * f a := by\n  lift f to α → ℕ using hf with g hg\n  guard_target =ₛ 0 ≤ 2 * (g a : Int)\n  guard_hyp hg : (fun i => (g i : Int)) = f\n  constructor\n\nexample (n m k x : ℤ) (hn : 0 < n) (hk : 0 ≤ k + n) (h : k + n = 2 + x)\n    (hans : k + n = m + x) (hans2 : 0 ≤ m) : k + n = m + x := by\n  lift n to ℕ using le_of_lt hn\n  guard_target =ₛ k + ↑n = m + x; guard_hyp hn : (0 : ℤ) < ↑n\n  lift m to ℕ\n  guard_target =ₛ 0 ≤ m; swap; guard_target =ₛ k + ↑n = ↑m + x\n  lift (k + n) to ℕ using hk with l hl\n  exact hans\n  exact hans2\n\n-- fail gracefully when the lifted variable is a local definition\nexample (h : False) : let n : ℤ := 3; n = 3 := by\n  intro n\n  fail_if_success lift n to ℕ\n  exfalso; exact h\n\ninstance canLift_unit : CanLift Unit Unit id (fun _ ↦ true) := ⟨fun x _ ↦ ⟨x, rfl⟩⟩\n\nexample (n : ℤ) (hn : 0 < n) : True := by\n  fail_if_success lift n to ℕ using hn\n  fail_if_success lift (n : Option ℤ) to ℕ\n  trivial\n\n example (n : ℤ) : ℕ := by\n  fail_if_success lift n to ℕ\n  exact 0\n\ninstance canLift_subtype (R : Type _) (s : Set R) : CanLift R {x // x ∈ s} ((↑) : {x // x ∈ s} → R) (fun x => x ∈ s) :=\n{ prf := fun x hx => ⟨⟨x, hx⟩, rfl⟩ }\n\nexample {R : Type _} {P : R → Prop} (x : R) (hx : P x) : P x := by\n  lift x to {x // P x} using hx with y hy hx\n  guard_target =ₛ P y\n  guard_hyp hy : y = x\n  guard_hyp hx : P y\n  exact hx\n\n example (n : ℤ) (h_ans : n = 5) (hn : 0 ≤ 1 * n) : n = 5 := by\n  lift n to ℕ using by { simpa [Int.one_mul] using hn } with k hk\n  guard_target =ₛ (k : Int) = 5\n  guard_hyp hk : k = n\n  guard_hyp hn : 0 ≤ 1 * (k : Int)\n  guard_hyp h_ans : (k : Int) = 5\n  exact h_ans\n\nexample (n : WithOne Unit) (hn : n ≠ 1) : True := by\n  lift n to Unit\n  · guard_target =ₛ n ≠ 1\n    exact hn\n\n  guard_hyp n : Unit\n  guard_hyp hn : (n : WithOne Unit) ≠ 1\n  trivial\n\nexample (n : WithZero Unit) (hn : n ≠ 0) : True := by\n  lift n to Unit\n  · guard_target =ₛ n ≠ 0\n    exact hn\n\n  guard_hyp n : Unit\n  guard_hyp hn : (n : WithZero Unit) ≠ 0\n  trivial\n\nexample (s : Set ℤ) (h : ∀ x ∈ s, 0 ≤ x) : True := by\n  lift s to Set ℕ\n  · guard_target =ₛ (∀ x ∈ s, 0 ≤ x)\n    exact h\n\n  guard_hyp s : Set ℕ\n  guard_hyp h : ∀ (x : ℤ), x ∈ (fun (n : ℕ) => (n : ℤ)) '' s → 0 ≤ x\n  trivial\n\nexample (l : List ℤ) (h : ∀ x ∈ l, 0 ≤ x) : True := by\n  lift l to List ℕ\n  · guard_target =ₛ (∀ x ∈ l, 0 ≤ x)\n    exact h\n\n  guard_hyp l : List ℕ\n  guard_hyp h : ∀ (x : ℤ), x ∈ List.map (fun (n : ℕ) => (n : ℤ)) l → 0 ≤ x\n  trivial\n\nexample (q : ℚ) (h : q.den = 1) : True := by\n  lift q to ℤ\n  · guard_target =ₛ q.den = 1\n    exact h\n\n  guard_hyp q : ℤ\n  guard_hyp h : (q : ℚ).den = 1\n  trivial\n\nexample (x : WithTop Unit) (h : x ≠ ⊤) : True := by\n  lift x to Unit\n  · guard_target =ₛ x ≠ ⊤\n    exact h\n\n  guard_hyp x : Unit\n  guard_hyp h : (x : WithTop Unit) ≠ ⊤\n  trivial\n\nexample (x : WithBot Unit) (h : x ≠ ⊥) : True := by\n  lift x to Unit\n  · guard_target =ₛ x ≠ ⊥\n    exact h\n\n  guard_hyp x : Unit\n  guard_hyp h : (x : WithBot Unit) ≠ ⊥\n  trivial\n\nexample (n : ℕ) (hn : 0 < n) : True := by\n  lift n to ℕ+\n  · guard_target =ₛ 0 < n\n    exact hn\n\n  guard_hyp n : ℕ+\n  guard_hyp hn : 0 < (n : ℕ)\n  trivial\n\nexample (n : ℤ) (hn : 0 < n) : True := by\n  lift n to ℕ+\n  · guard_target =ₛ 0 < n\n    exact hn\n\n  guard_hyp n : ℕ+\n  guard_hyp hn : 0 < (n : ℤ)\n  trivial\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/lift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3389368293640571}}
{"text": "import Lean\n\ndef checkWithMkMatcherInput (matcher : Lean.Name) : Lean.MetaM Unit :=\n  Lean.Meta.Match.withMkMatcherInput matcher fun input => do\n  let res ← Lean.Meta.Match.mkMatcher input\n  let origMatcher ← Lean.getConstInfo matcher\n  if not <| input.matcherName == matcher then\n    throwError \"matcher name not reconstructed correctly: {matcher} ≟ {input.matcherName}\"\n\n  let lCtx ← Lean.getLCtx\n  let fvars := Lean.collectFVars {} res.matcher\n  let closure := Lean.Meta.Closure.mkLambda (fvars.fvarSet.toList.toArray.map lCtx.get!) res.matcher\n\n  let origTy := origMatcher.value!\n  let newTy := closure\n  if not <| ←Lean.Meta.isDefEq origTy newTy then\n    throwError \"matcher {matcher} does not round-trip correctly:\\n{origTy} ≟ {newTy}\"\n\nset_option smartUnfolding false\n\ndef f (xs : List Nat) : List Bool :=\nxs.map fun\n  | 0 => true\n  | _ => false\n#eval checkWithMkMatcherInput ``f.match_1\n\n#eval f [1, 2, 0, 2]\n\ntheorem ex1 : f [1, 0, 2] = [false, true, false] :=\nrfl\n\n#check f\n\nset_option pp.raw true\nset_option pp.raw.maxDepth 10\nset_option trace.Elab.step true in\ndef g (xs : List Nat) : List Bool :=\nxs.map <| by {\n  intro\n    | 0 => exact true\n    | _ => exact false\n}\n\ntheorem ex2 : g [1, 0, 2] = [false, true, false] :=\nrfl\n\ntheorem ex3 {p q r : Prop} : p ∨ q → r → (q ∧ r) ∨ (p ∧ r) :=\nby intro\n | Or.inl hp, h => { apply Or.inr; apply And.intro; assumption; assumption }\n | Or.inr hq, h => { apply Or.inl; exact ⟨hq, h⟩ }\n\ninductive C\n| mk₁ : Nat → C\n| mk₂ : Nat → Nat → C\n\ndef C.x : C → Nat\n| C.mk₁ x   => x\n| C.mk₂ x _ => x\n#eval checkWithMkMatcherInput ``C.x.match_1\n\ndef head : {α : Type} → List α → Option α\n| _, a::as => some a\n| _, _     => none\n#eval checkWithMkMatcherInput ``head.match_1\n\ntheorem ex4 : head [1, 2] = some 1 :=\nrfl\n\ndef head2 : {α : Type} → List α → Option α :=\n@fun\n  | _, a::as => some a\n  | _, _     => none\n\ntheorem ex5 : head2 [1, 2] = some 1 :=\nrfl\n\ndef head3 {α : Type} (xs : List α) : Option α :=\nlet rec aux {α : Type} : List α → Option α\n  | a::_ => some a\n  | _    => none;\naux xs\n\ntheorem ex6 : head3 [1, 2] = some 1 :=\nrfl\n\ninductive Vec.{u} (α : Type u) : Nat → Type u\n| nil : Vec α 0\n| cons {n} (head : α) (tail : Vec α n) : Vec α (n+1)\n\ndef Vec.mapHead1 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _,   nil,       nil,       f => none\n| _, cons a as, cons b bs,   f => some (f a b)\n\n\ndef Vec.mapHead2 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, @cons _ n a as, cons b bs,   f => some (f a b)\nset_option pp.match false in\n#print Vec.mapHead2 -- reused Vec.mapHead1.match_1\n\ndef Vec.mapHead3 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, cons (tail := as) (head := a), cons b bs,   f => some (f a b)\n\ninductive Foo\n| mk₁ (x y z w : Nat)\n| mk₂ (x y z w : Nat)\n\ndef Foo.z : Foo → Nat\n| mk₁ (z := z) .. => z\n| mk₂ (z := z) .. => z\n#eval checkWithMkMatcherInput ``Foo.z.match_1\n\n#eval (Foo.mk₁ 10 20 30 40).z\n\ntheorem ex7 : (Foo.mk₁ 10 20 30 40).z = 30 :=\nrfl\n\ndef Foo.addY? : Foo × Foo → Option Nat\n| (mk₁ (y := y₁) .., mk₁ (y := y₂) ..) => some (y₁ + y₂)\n| _ => none\n#eval checkWithMkMatcherInput ``Foo.addY?.match_1\n\n#eval Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40)\n\ntheorem ex8 : Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40) = some 22 :=\nrfl\n\ninstance {α} : Inhabited (Sigma fun m => Vec α m) :=\n⟨⟨0, Vec.nil⟩⟩\n\npartial def filter {α} (p : α → Bool) : {n : Nat} → Vec α n → Sigma fun m => Vec α m\n| _, Vec.nil        => ⟨0, Vec.nil⟩\n| _, Vec.cons x xs  => match p x, filter p xs with\n  | true,  ⟨_, ys⟩ => ⟨_, Vec.cons x ys⟩\n  | false, ys      => ys\n#eval checkWithMkMatcherInput ``filter.match_1\n\ninductive Bla\n| ofNat  (x : Nat)\n| ofBool (x : Bool)\n\ndef Bla.optional? : Bla → Option Nat\n| ofNat x  => some x\n| ofBool _ => none\n#eval checkWithMkMatcherInput ``Bla.optional?.match_1\n\ndef Bla.isNat? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch b.optional? with\n| some y => some ⟨y, rfl⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat?.match_1\n\ndef foo (b : Bla) : Option Nat := b.optional?\ntheorem fooEq (b : Bla) : foo b = b.optional? :=\nrfl\n\ndef Bla.isNat2? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch h : foo b with\n| some y => some ⟨y, Eq.trans (fooEq b).symm h⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat2?.match_1\n\ndef foo2 (x : Nat) : Nat :=\nmatch (motive := (y : Nat) → x = y → Nat) x, rfl with\n| 0,   h => 0\n| x+1, h => 1\n#eval checkWithMkMatcherInput ``foo2.match_1\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/match1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.3389368293640571}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.types\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.PostPort\n\nuniverses u \n\nnamespace Mathlib\n\n/-!\n# Special shapes for limits in `Type`.\n\nThe general shape (co)limits defined in `category_theory.limits.types`\nare intended for use through the limits API,\nand the actual implementation should mostly be considered \"sealed\".\n\nIn this file, we provide definitions of the \"standard\" special shapes of limits in `Type`,\ngiving the expected definitional implementation:\n* the terminal object is `punit`\n* the binary product of `X` and `Y` is `X × Y`\n* the product of a family `f : J → Type` is `Π j, f j`.\n\nBecause these are not intended for use with the `has_limit` API,\nwe instead construct terms of `limit_data`.\n\nAs an example, when setting up the monoidal category structure on `Type`\nwe use the `types_has_terminal` and `types_has_binary_products` instances.\n-/\n\nnamespace category_theory.limits.types\n\n\n/-- A restatement of `types.lift_π_apply` that uses `pi.π` and `pi.lift`. -/\n@[simp] theorem pi_lift_π_apply {β : Type u} (f : β → Type u) {P : Type u} (s : (b : β) → P ⟶ f b)\n    (b : β) (x : P) : pi.π f b (pi.lift s x) = s b x :=\n  congr_fun (limit.lift_π (fan.mk P s) b) x\n\n/-- A restatement of `types.map_π_apply` that uses `pi.π` and `pi.map`. -/\n@[simp] theorem pi_map_π_apply {β : Type u} {f : β → Type u} {g : β → Type u}\n    (α : (j : β) → f j ⟶ g j) (b : β) (x : ∏ fun (j : β) => f j) :\n    pi.π g b (pi.map α x) = α b (pi.π f b x) :=\n  limit.map_π_apply (discrete.nat_trans α) b x\n\n/-- The category of types has `punit` as a terminal object. -/\ndef terminal_limit_cone : limit_cone (functor.empty (Type u)) :=\n  limit_cone.mk (cone.mk PUnit (nat_trans.mk sorry)) (id (is_limit.mk sorry))\n\n/-- The category of types has `pempty` as an initial object. -/\ndef initial_limit_cone : colimit_cocone (functor.empty (Type u)) :=\n  colimit_cocone.mk (cocone.mk pempty (nat_trans.mk sorry)) (id (is_colimit.mk sorry))\n\n/-- The product type `X × Y` forms a cone for the binary product of `X` and `Y`. -/\n-- We manually generate the other projection lemmas since the simp-normal form for the legs is\n\n-- otherwise not created correctly.\n\n@[simp] theorem binary_product_cone_X (X : Type u) (Y : Type u) :\n    cone.X (binary_product_cone X Y) = (X × Y) :=\n  Eq.refl (X × Y)\n\n@[simp] theorem binary_product_cone_fst (X : Type u) (Y : Type u) :\n    binary_fan.fst (binary_product_cone X Y) = prod.fst :=\n  rfl\n\n@[simp] theorem binary_product_cone_snd (X : Type u) (Y : Type u) :\n    binary_fan.snd (binary_product_cone X Y) = prod.snd :=\n  rfl\n\n/-- The product type `X × Y` is a binary product for `X` and `Y`. -/\n@[simp] theorem binary_product_limit_lift (X : Type u) (Y : Type u) (s : binary_fan X Y)\n    (x : cone.X s) :\n    is_limit.lift (binary_product_limit X Y) s x = (binary_fan.fst s x, binary_fan.snd s x) :=\n  Eq.refl (is_limit.lift (binary_product_limit X Y) s x)\n\n/--\nThe category of types has `X × Y`, the usual cartesian product,\nas the binary product of `X` and `Y`.\n-/\n@[simp] theorem binary_product_limit_cone_is_limit (X : Type u) (Y : Type u) :\n    limit_cone.is_limit (binary_product_limit_cone X Y) = binary_product_limit X Y :=\n  Eq.refl (limit_cone.is_limit (binary_product_limit_cone X Y))\n\n/-- The functor which sends `X, Y` to the product type `X × Y`. -/\n-- We add the option `type_md` to tell `@[simps]` to not treat homomorphisms `X ⟶ Y` in `Type*` as\n\n-- a function type\n\n@[simp] theorem binary_product_functor_obj_map (X : Type u) (Y₁ : Type u) (Y₂ : Type u)\n    (f : Y₁ ⟶ Y₂) :\n    functor.map (functor.obj binary_product_functor X) f =\n        is_limit.lift (binary_product_limit X Y₂) (binary_fan.mk prod.fst (prod.snd ≫ f)) :=\n  Eq.refl (functor.map (functor.obj binary_product_functor X) f)\n\n/--\nThe product functor given by the instance `has_binary_products (Type u)` is isomorphic to the\nexplicit binary product functor given by the product type.\n-/\ndef binary_product_iso_prod : binary_product_functor ≅ prod.functor :=\n  nat_iso.of_components\n    (fun (X : Type u) =>\n      nat_iso.of_components\n        (fun (Y : Type u) =>\n          iso.symm\n            (is_limit.cone_point_unique_up_to_iso (limit.is_limit (pair X Y))\n              (binary_product_limit X Y)))\n        sorry)\n    sorry\n\n/-- The sum type `X ⊕ Y` forms a cocone for the binary coproduct of `X` and `Y`. -/\n@[simp] theorem binary_coproduct_cocone_ι_app (X : Type u) (Y : Type u)\n    (j : discrete walking_pair) :\n    ∀ (ᾰ : functor.obj (pair X Y) j),\n        nat_trans.app (cocone.ι (binary_coproduct_cocone X Y)) j ᾰ =\n          walking_pair.rec sum.inl sum.inr j ᾰ :=\n  fun (ᾰ : functor.obj (pair X Y) j) => Eq.refl (walking_pair.rec sum.inl sum.inr j ᾰ)\n\n/-- The sum type `X ⊕ Y` is a binary coproduct for `X` and `Y`. -/\ndef binary_coproduct_colimit (X : Type u) (Y : Type u) : is_colimit (binary_coproduct_cocone X Y) :=\n  is_colimit.mk fun (s : binary_cofan X Y) => sum.elim (binary_cofan.inl s) (binary_cofan.inr s)\n\n/--\nThe category of types has `X ⊕ Y`,\nas the binary coproduct of `X` and `Y`.\n-/\ndef binary_coproduct_colimit_cocone (X : Type u) (Y : Type u) : colimit_cocone (pair X Y) :=\n  colimit_cocone.mk (binary_coproduct_cocone X Y) (binary_coproduct_colimit X Y)\n\n/--\nThe category of types has `Π j, f j` as the product of a type family `f : J → Type`.\n-/\ndef product_limit_cone {J : Type u} (F : J → Type u) : limit_cone (discrete.functor F) :=\n  limit_cone.mk\n    (cone.mk ((j : J) → F j)\n      (nat_trans.mk\n        fun (j : discrete J)\n          (f : functor.obj (functor.obj (functor.const (discrete J)) ((j : J) → F j)) j) => f j))\n    (is_limit.mk\n      fun (s : cone (discrete.functor F)) (x : cone.X s) (j : J) => nat_trans.app (cone.π s) j x)\n\n/--\nThe category of types has `Σ j, f j` as the coproduct of a type family `f : J → Type`.\n-/\ndef coproduct_colimit_cocone {J : Type u} (F : J → Type u) : colimit_cocone (discrete.functor F) :=\n  colimit_cocone.mk\n    (cocone.mk (sigma fun (j : J) => F j)\n      (nat_trans.mk fun (j : discrete J) (x : functor.obj (discrete.functor F) j) => sigma.mk j x))\n    (is_colimit.mk\n      fun (s : cocone (discrete.functor F))\n        (x :\n        cocone.X\n          (cocone.mk (sigma fun (j : J) => F j)\n            (nat_trans.mk\n              fun (j : discrete J) (x : functor.obj (discrete.functor F) j) => sigma.mk j x))) =>\n        nat_trans.app (cocone.ι s) (sigma.fst x) (sigma.snd x))\n\n/--\nShow the given fork in `Type u` is an equalizer given that any element in the \"difference kernel\"\ncomes from `X`.\nThe converse of `unique_of_type_equalizer`.\n-/\ndef type_equalizer_of_unique {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z}\n    {h : Y ⟶ Z} (w : f ≫ g = f ≫ h)\n    (t : ∀ (y : Y), g y = h y → exists_unique fun (x : X) => f x = y) : is_limit (fork.of_ι f w) :=\n  fork.is_limit.mk' (fork.of_ι f w)\n    fun (s : fork g h) =>\n      { val :=\n          fun\n            (i :\n            functor.obj (functor.obj (functor.const walking_parallel_pair) (cone.X s))\n              walking_parallel_pair.zero) =>\n            classical.some sorry,\n        property := sorry }\n\n/-- The converse of `type_equalizer_of_unique`. -/\ntheorem unique_of_type_equalizer {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z}\n    {h : Y ⟶ Z} (w : f ≫ g = f ≫ h) (t : is_limit (fork.of_ι f w)) (y : Y) (hy : g y = h y) :\n    exists_unique fun (x : X) => f x = y :=\n  sorry\n\ntheorem type_equalizer_iff_unique {X : Type u} {Y : Type u} {Z : Type u} (f : X ⟶ Y) {g : Y ⟶ Z}\n    {h : Y ⟶ Z} (w : f ≫ g = f ≫ h) :\n    Nonempty (is_limit (fork.of_ι f w)) ↔\n        ∀ (y : Y), g y = h y → exists_unique fun (x : X) => f x = y :=\n  sorry\n\n/-- Show that the subtype `{x : Y // g x = h x}` is an equalizer for the pair `(g,h)`. -/\ndef equalizer_limit {Y : Type u} {Z : Type u} {g : Y ⟶ Z} {h : Y ⟶ Z} :\n    limit_cone (parallel_pair g h) :=\n  limit_cone.mk (fork.of_ι subtype.val sorry)\n    (fork.is_limit.mk' (fork.of_ι subtype.val sorry)\n      fun (s : fork g h) =>\n        { val :=\n            fun\n              (i :\n              functor.obj (functor.obj (functor.const walking_parallel_pair) (cone.X s))\n                walking_parallel_pair.zero) =>\n              { val := fork.ι s i, property := sorry },\n          property := sorry })\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/types_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6150878555160664, "lm_q1q2_score": 0.3386719036984219}}
{"text": "import category_theory.abelian.exact\nimport category_theory.limits.constructions.epi_mono\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen limits\n\nuniverses u v\n\nvariables {𝓐 𝓑 : Type u} [category.{v} 𝓐] [category.{v} 𝓑]\nvariables [abelian 𝓐] [abelian 𝓑]\nvariables (L : 𝓐 ⥤ 𝓑) [preserves_finite_limits L] [preserves_finite_colimits L]\n\n\nlemma preserve_image {X Y : 𝓐} (f : X ⟶ Y) : image (L.map f) ≅ L.obj (image f) :=\nhave aux1 : strong_epi_mono_factorisation (L.map f) :=\n{ I := L.obj (image f),\n  m := L.map $ image.ι _,\n  m_mono := by apply_instance,\n  e := L.map $ factor_thru_image _,\n  e_strong_epi := strong_epi_of_epi _,\n  fac' := by rw [←L.map_comp, image.fac] },\nis_image.iso_ext (image.is_image (L.map f)) aux1.to_mono_is_image\n\nlemma preserve_image.precomp_factor_thru_image {X Y : 𝓐} (f : X ⟶ Y) :\n  factor_thru_image  (L.map f) ≫ (preserve_image L f).hom =\n  L.map (factor_thru_image f) :=\nbegin\n  dunfold preserve_image,\n  simp only [is_image.iso_ext_hom],\n  erw image.fac_lift,\nend\n\ndef apply_image_subobject_iso_image_subobject_apply {X Y : 𝓐} (f : X ⟶ Y) :\n   L.obj (image_subobject f) ≅ (image_subobject (L.map f)) := \n{ hom := L.map (image_subobject_iso _).hom ≫ (preserve_image L f).inv,\n  inv := (preserve_image L f).hom ≫ L.map (image_subobject_iso _).inv,\n  hom_inv_id' := by rw [category.assoc, ←category.assoc _ _ (L.map _), iso.inv_hom_id, category.id_comp,\n    ←L.map_comp, iso.hom_inv_id, L.map_id],\n  inv_hom_id' := by rw [category.assoc, ←category.assoc _ _ (preserve_image L f).inv, \n    ←L.map_comp, iso.inv_hom_id, L.map_id, category.id_comp, iso.hom_inv_id] } \n≪≫ (image_subobject_iso _).symm\n\n\ndef apply_kernel_subobject_iso_kernel_subobject_apply {X Y : 𝓐} (f : X ⟶ Y) :\n  L.obj (kernel_subobject f) ≅ (kernel_subobject (L.map f)) :=\n{ hom := L.map (kernel_subobject_iso _).hom ≫ (preserves_kernel.iso L f).hom,\n  inv := (preserves_kernel.iso L f).inv ≫ L.map (kernel_subobject_iso _).inv,\n  hom_inv_id' := by rw [category.assoc, ←category.assoc _ _ (L.map _), iso.hom_inv_id, category.id_comp, ←L.map_comp, iso.hom_inv_id, L.map_id],\n  inv_hom_id' := by rw [category.assoc, ←category.assoc _ _ (preserves_kernel.iso _ _).hom, ←L.map_comp, iso.inv_hom_id, L.map_id, category.id_comp, iso.inv_hom_id] } \n≪≫ (kernel_subobject_iso _).symm\n\nlemma exact_of_exact_functor {X Y Z : 𝓐} (f : X ⟶ Y) (g : Y ⟶ Z)\n  (e1 : exact f g) :\n  exact (L.map f) (L.map g) := \nhave H : is_iso (image_to_kernel f g e1.w) := is_iso_of_mono_of_epi _,\nbegin\n  rw abelian.exact_iff_image_eq_kernel,\n  ext,\n  work_on_goal 2 \n  { refine (preserve_image L f) ≪≫ _ ≪≫ (preserves_kernel.iso _ _),\n    exact\n    { hom := L.map $ (image_subobject_iso _).inv ≫ image_to_kernel f g e1.w ≫ (kernel_subobject_iso _).hom,\n      inv := L.map $ (kernel_subobject_iso _).inv ≫ (@as_iso _ _ _ _ (image_to_kernel _ _ e1.w) H).inv ≫ (image_subobject_iso _).hom,\n      hom_inv_id' := begin\n        simp only [←L.map_comp, category.assoc],\n        have h1 := (kernel_subobject_iso g).hom_inv_id,\n        reassoc h1,\n        rw [h1, ←category.assoc _ _ (image_subobject_iso _).hom, as_iso_inv,\n          is_iso.hom_inv_id, category.id_comp, iso.inv_hom_id, L.map_id],\n      end,\n      inv_hom_id' := begin\n        simp only [←L.map_comp, category.assoc],\n        have h1 := (image_subobject_iso f).hom_inv_id,\n        reassoc h1,\n        rw [h1, ←category.assoc _ _ (kernel_subobject_iso _).hom, as_iso_inv,\n          is_iso.inv_hom_id, category.id_comp, iso.inv_hom_id, L.map_id],\n      end } },\n  { simp only [functor.map_comp, category.assoc, iso.trans_hom, preserves_kernel.iso_hom, kernel_comparison_comp_ι, image.fac],\n    simp only [←L.map_comp, kernel_subobject_arrow, image_to_kernel_arrow, image_subobject_arrow'],\n    rw [←category.assoc, preserve_image.precomp_factor_thru_image, ←L.map_comp, image.fac] }\nend\n\nend category_theory", "meta": {"author": "jjaassoonn", "repo": "twist", "sha": "8b12ca696c19c239c2e9deeab51c5dc04e586fed", "save_path": "github-repos/lean/jjaassoonn-twist", "path": "github-repos/lean/jjaassoonn-twist/twist-8b12ca696c19c239c2e9deeab51c5dc04e586fed/src/enough_injectives/preserves_exact_seq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.338620611394147}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.algebra.subalgebra.basic\nimport algebra.free_algebra\nimport algebra.category.Ring.basic\nimport algebra.category.Module.basic\n\n/-!\n# Category instance for algebras over a commutative ring\n\nWe introduce the bundled category `Algebra` of algebras over a fixed commutative ring `R ` along\nwith the forgetful functors to `Ring` and `Module`. We furthermore show that the functor associating\nto a type the free `R`-algebra on that type is left adjoint to the forgetful functor.\n-/\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables (R : Type u) [comm_ring R]\n\n/-- The category of R-algebras and their morphisms. -/\nstructure Algebra :=\n(carrier : Type v)\n[is_ring : ring carrier]\n[is_algebra : algebra R carrier]\n\nattribute [instance] Algebra.is_ring Algebra.is_algebra\n\nnamespace Algebra\n\ninstance : has_coe_to_sort (Algebra R) (Type v) := ⟨Algebra.carrier⟩\n\ninstance : category (Algebra.{v} R) :=\n{ hom   := λ A B, A →ₐ[R] B,\n  id    := λ A, alg_hom.id R A,\n  comp  := λ A B C f g, g.comp f }\n\ninstance : concrete_category.{v} (Algebra.{v} R) :=\n{ forget := { obj := λ R, R, map := λ R S f, (f : R → S) },\n  forget_faithful := { } }\n\ninstance has_forget_to_Ring : has_forget₂ (Algebra.{v} R) Ring.{v} :=\n{ forget₂ :=\n  { obj := λ A, Ring.of A,\n    map := λ A₁ A₂ f, alg_hom.to_ring_hom f, } }\n\ninstance has_forget_to_Module : has_forget₂ (Algebra.{v} R) (Module.{v} R) :=\n{ forget₂ :=\n  { obj := λ M, Module.of R M,\n    map := λ M₁ M₂ f, alg_hom.to_linear_map f, } }\n\n/-- The object in the category of R-algebras associated to a type equipped with the appropriate\ntypeclasses. -/\ndef of (X : Type v) [ring X] [algebra R X] : Algebra.{v} R := ⟨X⟩\n\n/-- Typecheck a `alg_hom` as a morphism in `Algebra R`. -/\ndef of_hom {R : Type u} [comm_ring R] {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y]\n  (f : X →ₐ[R] Y) : of R X ⟶ of R Y := f\n\n@[simp] lemma of_hom_apply {R : Type u} [comm_ring R]\n  {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y] (f : X →ₐ[R] Y) (x : X) :\n  of_hom f x = f x := rfl\n\ninstance : inhabited (Algebra R) := ⟨of R R⟩\n\n@[simp]\nlemma coe_of (X : Type u) [ring X] [algebra R X] : (of R X : Type u) = X := rfl\n\nvariables {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nalgebra. -/\n@[simps]\ndef of_self_iso (M : Algebra.{v} R) : Algebra.of R M ≅ M :=\n{ hom := 𝟙 M, inv := 𝟙 M }\n\nvariables {R} {M N U : Module.{v} R}\n\n@[simp] \n\n@[simp] lemma coe_comp (f : M ⟶ N) (g : N ⟶ U) :\n  ((f ≫ g) : M → U) = g ∘ f := rfl\n\nvariables (R)\n/-- The \"free algebra\" functor, sending a type `S` to the free algebra on `S`. -/\n@[simps]\ndef free : Type u ⥤ Algebra.{u} R :=\n{ obj := λ S,\n  { carrier := free_algebra R S,\n    is_ring := algebra.semiring_to_ring R },\n  map := λ S T f, free_algebra.lift _ $ (free_algebra.ι _) ∘ f,\n  -- obviously can fill the next two goals, but it is slow\n  map_id' := by { intros X, ext1, simp only [free_algebra.ι_comp_lift], refl },\n  map_comp' := by { intros, ext1, simp only [free_algebra.ι_comp_lift], ext1,\n    simp only [free_algebra.lift_ι_apply, category_theory.coe_comp, function.comp_app,\n      types_comp_apply] } }\n\n/-- The free/forget adjunction for `R`-algebras. -/\ndef adj : free.{u} R ⊣ forget (Algebra.{u} R) :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X A, (free_algebra.lift _).symm,\n  -- Relying on `obviously` to fill out these proofs is very slow :(\n  hom_equiv_naturality_left_symm' := by { intros, ext,\n    simp only [free_map, equiv.symm_symm, free_algebra.lift_ι_apply, category_theory.coe_comp,\n      function.comp_app, types_comp_apply] },\n  hom_equiv_naturality_right' := by { intros, ext,\n    simp only [forget_map_eq_coe, category_theory.coe_comp, function.comp_app,\n      free_algebra.lift_symm_apply, types_comp_apply] } }\n\ninstance : is_right_adjoint (forget (Algebra.{u} R)) := ⟨_, adj R⟩\n\nend Algebra\n\nvariables {R}\nvariables {X₁ X₂ : Type u}\n\n/-- Build an isomorphism in the category `Algebra R` from a `alg_equiv` between `algebra`s. -/\n@[simps]\ndef alg_equiv.to_Algebra_iso\n  {g₁ : ring X₁} {g₂ : ring X₂} {m₁ : algebra R X₁} {m₂ : algebra R X₂} (e : X₁ ≃ₐ[R] X₂) :\n  Algebra.of R X₁ ≅ Algebra.of R X₂ :=\n{ hom := (e : X₁ →ₐ[R] X₂),\n  inv := (e.symm : X₂ →ₐ[R] X₁),\n  hom_inv_id' := begin ext, exact e.left_inv x, end,\n  inv_hom_id' := begin ext, exact e.right_inv x, end, }\n\nnamespace category_theory.iso\n\n/-- Build a `alg_equiv` from an isomorphism in the category `Algebra R`. -/\n@[simps]\ndef to_alg_equiv {X Y : Algebra R} (i : X ≅ Y) : X ≃ₐ[R] Y :=\n{ to_fun    := i.hom,\n  inv_fun   := i.inv,\n  left_inv  := by tidy,\n  right_inv := by tidy,\n  map_add'  := by tidy,\n  map_mul'  := by tidy,\n  commutes' := by tidy, }.\n\nend category_theory.iso\n\n/-- Algebra equivalences between `algebras`s are the same as (isomorphic to) isomorphisms in\n`Algebra`. -/\n@[simps]\ndef alg_equiv_iso_Algebra_iso {X Y : Type u}\n  [ring X] [ring Y] [algebra R X] [algebra R Y] :\n  (X ≃ₐ[R] Y) ≅ (Algebra.of R X ≅ Algebra.of R Y) :=\n{ hom := λ e, e.to_Algebra_iso,\n  inv := λ i, i.to_alg_equiv, }\n\ninstance (X : Type u) [ring X] [algebra R X] : has_coe (subalgebra R X) (Algebra R) :=\n⟨ λ N, Algebra.of R N ⟩\n\ninstance Algebra.forget_reflects_isos : reflects_isomorphisms (forget (Algebra.{u} R)) :=\n{ reflects := λ X Y f _,\n  begin\n    resetI,\n    let i := as_iso ((forget (Algebra.{u} R)).map f),\n    let e : X ≃ₐ[R] Y := { ..f, ..i.to_equiv },\n    exact ⟨(is_iso.of_iso e.to_Algebra_iso).1⟩,\n  end }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/category/Algebra/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3383967364768054}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.functor.hom\nimport category_theory.functor.currying\nimport category_theory.products.basic\n\n/-!\n# The Yoneda embedding\n\nThe Yoneda embedding as a functor `yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) ≅ (yoneda_evaluation C)`.\n\n## References\n* [Stacks: Opposite Categories and the Yoneda Lemma](https://stacks.math.columbia.edu/tag/001L)\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v₁ u₁ u₂-- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u₁} [category.{v₁} C]\n\n/--\nThe Yoneda embedding, as a functor from `C` into presheaves on `C`.\n\nSee https://stacks.math.columbia.edu/tag/001O.\n-/\n@[simps]\ndef yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop Y ⟶ X,\n    map := λ Y Y' f g, f.unop ≫ g,\n    map_comp' := λ _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := λ Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := λ X X' f, { app := λ Y g, g ≫ f } }\n\n/--\nThe co-Yoneda embedding, as a functor from `Cᵒᵖ` into co-presheaves on `C`.\n-/\n@[simps] def coyoneda : Cᵒᵖ ⥤ (C ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop X ⟶ Y,\n    map := λ Y Y' f g, g ≫ f },\n  map := λ X X' f, { app := λ Y g, f.unop ≫ g } }\n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X ⟶ op Y) :\n  (yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) :=\nby { dsimp, simp }\n\n@[simp] lemma naturality {X Y : C} (α : yoneda.obj X ⟶ yoneda.obj Y)\n  {Z Z' : C} (f : Z ⟶ Z') (h : Z' ⟶ X) : f ≫ α.app (op Z') h = α.app (op Z) (f ≫ h) :=\n(functor_to_types.naturality _ _ α f.op h).symm\n\n/--\nThe Yoneda embedding is full.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ninstance yoneda_full : full (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) :=\n{ preimage := λ X Y f, f.app (op X) (𝟙 X) }\n\n/--\nThe Yoneda embedding is faithful.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ninstance yoneda_faithful : faithful (yoneda : C ⥤ Cᵒᵖ ⥤ Type v₁) :=\n{ map_injective' := λ X Y f g p, by convert (congr_fun (congr_app p (op X)) (𝟙 X)); dsimp; simp }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X ≅ Y`\napply yoneda.ext,\n-- Goals are now functions `(Z ⟶ X) → (Z ⟶ Y)`, `(Z ⟶ Y) → (Z ⟶ X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : Π {Z : C}, (Z ⟶ X) → (Z ⟶ Y)) (q : Π {Z : C}, (Z ⟶ Y) → (Z ⟶ X))\n  (h₁ : Π {Z : C} (f : Z ⟶ X), q (p f) = f) (h₂ : Π {Z : C} (f : Z ⟶ Y), p (q f) = f)\n  (n : Π {Z Z' : C} (f : Z' ⟶ Z) (g : Z ⟶ X), p (f ≫ g) = f ≫ p g) : X ≅ Y :=\n@preimage_iso _ _ _ _ yoneda _ _ _ _\n  (nat_iso.of_components (λ Z, { hom := p, inv := q, }) (by tidy))\n\n/--\nIf `yoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C} (f : X ⟶ Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : Cᵒᵖ} (α : coyoneda.obj X ⟶ coyoneda.obj Y)\n  {Z Z' : C} (f : Z' ⟶ Z) (h : unop X ⟶ Z') : (α.app Z' h) ≫ f = α.app Z (h ≫ f) :=\n(functor_to_types.naturality _ _ α f h).symm\n\ninstance coyoneda_full : full (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) :=\n{ preimage := λ X Y f, (f.app _ (𝟙 X.unop)).op }\n\ninstance coyoneda_faithful : faithful (coyoneda : Cᵒᵖ ⥤ C ⥤ Type v₁) :=\n{ map_injective' := λ X Y f g p,\n  begin\n    have t := congr_fun (congr_app p X.unop) (𝟙 _),\n    simpa using congr_arg quiver.hom.op t,\n  end }\n\n/--\nIf `coyoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\n/-- The identity functor on `Type` is isomorphic to the coyoneda functor coming from `punit`. -/\ndef punit_iso : coyoneda.obj (opposite.op punit) ≅ 𝟭 (Type v₁) :=\nnat_iso.of_components\n  (λ X, { hom := λ f, f ⟨⟩, inv := λ x _, x })\n  (by tidy)\n\nend coyoneda\n\nnamespace functor\n\n\n/--\nA functor `F : Cᵒᵖ ⥤ Type v₁` is representable if there is object `X` so `F ≅ yoneda.obj X`.\n\nSee https://stacks.math.columbia.edu/tag/001Q.\n-/\nclass representable (F : Cᵒᵖ ⥤ Type v₁) : Prop :=\n(has_representation : ∃ X (f : yoneda.obj X ⟶ F), is_iso f)\n\ninstance {X : C} : representable (yoneda.obj X) :=\n{ has_representation := ⟨X, 𝟙 _, infer_instance⟩ }\n\n/--\nA functor `F : C ⥤ Type v₁` is corepresentable if there is object `X` so `F ≅ coyoneda.obj X`.\n\nSee https://stacks.math.columbia.edu/tag/001Q.\n-/\nclass corepresentable (F : C ⥤ Type v₁) : Prop :=\n(has_corepresentation : ∃ X (f : coyoneda.obj X ⟶ F), is_iso f)\n\ninstance {X : Cᵒᵖ} : corepresentable (coyoneda.obj X) :=\n{ has_corepresentation := ⟨X, 𝟙 _, infer_instance⟩ }\n\n-- instance : corepresentable (𝟭 (Type v₁)) :=\n-- corepresentable_of_nat_iso (op punit) coyoneda.punit_iso\n\nsection representable\nvariables (F : Cᵒᵖ ⥤ Type v₁)\nvariable [F.representable]\n\n/-- The representing object for the representable functor `F`. -/\nnoncomputable def repr_X : C :=\n(representable.has_representation : ∃ X (f : _ ⟶ F), _).some\n\n/-- The (forward direction of the) isomorphism witnessing `F` is representable. -/\nnoncomputable def repr_f : yoneda.obj F.repr_X ⟶ F :=\nrepresentable.has_representation.some_spec.some\n\n/--\nThe representing element for the representable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def repr_x : F.obj (op F.repr_X) :=\nF.repr_f.app (op F.repr_X) (𝟙 F.repr_X)\n\ninstance : is_iso F.repr_f :=\nrepresentable.has_representation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(-, F.repr_X)`.  Note the components\n`F.repr_w.app X` definitionally have type `(X.unop ⟶ F.repr_X) ≅ F.obj X`.\n-/\nnoncomputable def repr_w : yoneda.obj F.repr_X ≅ F := as_iso F.repr_f\n\n@[simp] lemma repr_w_hom : F.repr_w.hom = F.repr_f := rfl\n\nlemma repr_w_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.repr_X) :\n  (F.repr_w.app X).hom f = F.map f.op F.repr_x :=\nbegin\n  change F.repr_f.app X f = (F.repr_f.app (op F.repr_X) ≫ F.map f.op) (𝟙 F.repr_X),\n  rw ←F.repr_f.naturality,\n  dsimp,\n  simp\nend\n\nend representable\n\nsection corepresentable\n\nvariables (F : C ⥤ Type v₁)\nvariable [F.corepresentable]\n\n/-- The representing object for the corepresentable functor `F`. -/\nnoncomputable def corepr_X : C :=\n(corepresentable.has_corepresentation : ∃ X (f : _ ⟶ F), _).some.unop\n\n/-- The (forward direction of the) isomorphism witnessing `F` is corepresentable. -/\nnoncomputable def corepr_f : coyoneda.obj (op F.corepr_X) ⟶ F :=\ncorepresentable.has_corepresentation.some_spec.some\n\n/--\nThe representing element for the corepresentable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def corepr_x : F.obj F.corepr_X :=\nF.corepr_f.app F.corepr_X (𝟙 F.corepr_X)\n\ninstance : is_iso F.corepr_f :=\ncorepresentable.has_corepresentation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(F.corepr X, -)`. Note the components\n`F.corepr_w.app X` definitionally have type `F.corepr_X ⟶ X ≅ F.obj X`.\n-/\nnoncomputable def corepr_w : coyoneda.obj (op F.corepr_X) ≅ F := as_iso F.corepr_f\n\nlemma corepr_w_app_hom (X : C) (f : F.corepr_X ⟶ X) :\n  (F.corepr_w.app X).hom f = F.map f F.corepr_x :=\nbegin\n  change F.corepr_f.app X f = (F.corepr_f.app F.corepr_X ≫ F.map f) (𝟙 F.corepr_X),\n  rw ←F.corepr_f.naturality,\n  dsimp,\n  simp\nend\n\nend corepresentable\n\nend functor\n\nlemma representable_of_nat_iso (F : Cᵒᵖ ⥤ Type v₁) {G} (i : F ≅ G) [F.representable] :\n  G.representable :=\n{ has_representation := ⟨F.repr_X, F.repr_f ≫ i.hom, infer_instance⟩ }\n\nlemma corepresentable_of_nat_iso (F : C ⥤ Type v₁) {G} (i : F ≅ G) [F.corepresentable] :\n  G.corepresentable :=\n{ has_corepresentation := ⟨op F.corepr_X, F.corepr_f ≫ i.hom, infer_instance⟩ }\n\ninstance : functor.corepresentable (𝟭 (Type v₁)) :=\ncorepresentable_of_nat_iso (coyoneda.obj (op punit)) coyoneda.punit_iso\n\nopen opposite\n\nvariables (C)\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((Cᵒᵖ ⥤ Type v₁) × Cᵒᵖ) :=\ncategory_theory.prod.{(max u₁ v₁) v₁} (Cᵒᵖ ⥤ Type v₁) Cᵒᵖ\n\ninstance prod_category_instance_2 : category (Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) :=\ncategory_theory.prod.{v₁ (max u₁ v₁)} Cᵒᵖ (Cᵒᵖ ⥤ Type v₁)\n\nopen yoneda\n\n/--\nThe \"Yoneda evaluation\" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`\nto `F.obj X`, functorially in both `X` and `F`.\n-/\ndef yoneda_evaluation : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nevaluation_uncurried Cᵒᵖ (Type v₁) ⋙ ulift_functor.{u₁}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down) := rfl\n\n/--\nThe \"Yoneda pairing\" functor, which sends `X : Cᵒᵖ` and `F : Cᵒᵖ ⥤ Type`\nto `yoneda.op.obj X ⟶ F`, functorially in both `X` and `F`.\n-/\ndef yoneda_pairing : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nfunctor.prod yoneda.op (𝟭 (Cᵒᵖ ⥤ Type v₁)) ⋙ functor.hom (Cᵒᵖ ⥤ Type v₁)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (β : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map α β = yoneda.map α.1.unop ≫ β ≫ α.2 := rfl\n\n/--\nThe Yoneda lemma asserts that that the Yoneda pairing\n`(X : Cᵒᵖ, F : Cᵒᵖ ⥤ Type) ↦ (yoneda.obj (unop X) ⟶ F)`\nis naturally isomorphic to the evaluation `(X, F) ↦ F.obj X`.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ndef yoneda_lemma : yoneda_pairing C ≅ yoneda_evaluation C :=\n{ hom :=\n  { app := λ F x, ulift.up ((x.app F.1) (𝟙 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp, ←functor_to_types.naturality],\n      simp only [category.comp_id, yoneda_obj_map],\n    end },\n  inv :=\n  { app := λ F x,\n    { app := λ X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp_apply]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [←functor_to_types.naturality, functor_to_types.map_comp_apply]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [←functor_to_types.naturality,\n         obj_map_id],\n    simp only [yoneda_map_app, quiver.hom.unop_op],\n    erw [category.id_comp],\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id_apply]\n  end }.\n\nvariables {C}\n\n/--\nThe isomorphism between `yoneda.obj X ⟶ F` and `F.obj (op X)`\n(we need to insert a `ulift` to get the universes right!)\ngiven by the Yoneda lemma.\n-/\n@[simps] def yoneda_sections (X : C) (F : Cᵒᵖ ⥤ Type v₁) :\n  (yoneda.obj X ⟶ F) ≅ ulift.{u₁} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\n/--\nWe have a type-level equivalence between natural transformations from the yoneda embedding\nand elements of `F.obj X`, without any universe switching.\n-/\ndef yoneda_equiv {X : C} {F : Cᵒᵖ ⥤ Type v₁} : (yoneda.obj X ⟶ F) ≃ F.obj (op X) :=\n(yoneda_sections X F).to_equiv.trans equiv.ulift\n\n@[simp]\nlemma yoneda_equiv_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) :\n  yoneda_equiv f = f.app (op X) (𝟙 X) :=\nrfl\n\n@[simp]\nlemma yoneda_equiv_symm_app_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (x : F.obj (op X))\n  (Y : Cᵒᵖ) (f : Y.unop ⟶ X) :\n  (yoneda_equiv.symm x).app Y f = F.map f.op x :=\nrfl\n\nlemma yoneda_equiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F) (g : Y ⟶ X) :\n  F.map g.op (yoneda_equiv f) = yoneda_equiv (yoneda.map g ≫ f) :=\nbegin\n  change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g),\n  rw ←f.naturality,\n  dsimp,\n  simp,\nend\n\n/--\nWhen `C` is a small category, we can restate the isomorphism from `yoneda_sections`\nwithout having to change universes.\n-/\ndef yoneda_sections_small {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) :\n  (yoneda.obj X ⟶ F) ≅ F.obj (op X) :=\nyoneda_sections X F ≪≫ ulift_trivial _\n\n@[simp]\nlemma yoneda_sections_small_hom {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) (f : yoneda.obj X ⟶ F) :\n  (yoneda_sections_small X F).hom f = f.app _ (𝟙 _) :=\nrfl\n\n@[simp]\nlemma yoneda_sections_small_inv_app_apply {C : Type u₁} [small_category C] (X : C)\n  (F : Cᵒᵖ ⥤ Type u₁) (t : F.obj (op X)) (Y : Cᵒᵖ) (f : Y.unop ⟶ X) :\n  ((yoneda_sections_small X F).inv t).app Y f = F.map f.op t :=\nrfl\n\nlocal attribute[ext] functor.ext\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma {C : Type u₁} [small_category C] :\n  (yoneda.op ⋙ coyoneda : Cᵒᵖ ⥤ (Cᵒᵖ ⥤ Type u₁) ⥤ Type u₁) ≅ evaluation Cᵒᵖ (Type u₁) :=\neq_to_iso (by tidy) ≪≫ curry.map_iso (yoneda_lemma C ≪≫\n  iso_whisker_left (evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial) ≪≫\n    eq_to_iso (by tidy)\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma' {C : Type u₁} [small_category C] :\n  yoneda ⋙ (whiskering_left Cᵒᵖ (Cᵒᵖ ⥤ Type u₁)ᵒᵖ (Type u₁)).obj yoneda.op ≅ 𝟭 (Cᵒᵖ ⥤ Type u₁) :=\neq_to_iso (by tidy) ≪≫ curry.map_iso (iso_whisker_left (prod.swap _ _)\n  (yoneda_lemma C ≪≫ iso_whisker_left\n    (evaluation_uncurried Cᵒᵖ (Type u₁)) ulift_functor_trivial : _)) ≪≫ eq_to_iso (by tidy)\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.338392740351235}}
{"text": "variable x : int\nexample : x=x :=\nbegin\ntrivial\nend\n\n#check 2\n#check x\n#check (2:int)\n#check int\n#check Type\n#check Type 1\n\n#check 2\n#check ℕ\n#check Type\n#check Type 1\n#check Type 2\n#check Type 3", "meta": {"author": "sguzman", "repo": "lean-examples", "sha": "c7428b2982d0468d0adb4453766a27e1550a72e8", "save_path": "github-repos/lean/sguzman-lean-examples", "path": "github-repos/lean/sguzman-lean-examples/lean-examples-c7428b2982d0468d0adb4453766a27e1550a72e8/src/lean3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.33816910158461405}}
{"text": "import category_theory.preadditive.additive_functor\n\nimport for_mathlib.free_abelian_group\n\nnoncomputable theory\n\nuniverses v u\n\nnamespace category_theory\n\nstructure FreeAb (C : Type u) [category.{v} C] := of :: (as : C)\n\nnamespace FreeAb\n\nvariables (C : Type u) [category.{v} C]\n\ninstance : quiver (FreeAb C) :=\n{ hom := λ X Y, free_abelian_group (X.as ⟶ Y.as) }\n\nvariables {C}\n\nprotected def id (X : FreeAb C) : X ⟶ X := free_abelian_group.of (𝟙 X.as)\n\nprotected def comp {X Y Z : FreeAb C} : (X ⟶ Y) →+ (Y ⟶ Z) →+ (X ⟶ Z) :=\nfree_abelian_group.lift $ λ f : X.as ⟶ Y.as,\n  free_abelian_group.lift $ λ g : Y.as ⟶ Z.as, free_abelian_group.of (f ≫ g)\n\nvariables (C)\n\ninstance : category_struct (FreeAb C) :=\n{ id := FreeAb.id,\n  comp := λ X Y Z f g, FreeAb.comp f g }\n\n@[simp]\nprotected lemma comp_apply {X Y Z : FreeAb C} (f : X ⟶ Y) (g : Y ⟶ Z) :\n  FreeAb.comp f g = f ≫ g := rfl\n\ninstance : category (FreeAb C) :=\n{ id_comp' := λ X Y f, begin\n    show FreeAb.comp X.id f = add_monoid_hom.id _ f, congr' 1, clear f, ext1 f,\n    simp only [add_monoid_hom.id_apply, FreeAb.comp, free_abelian_group.lift.of, FreeAb.id,\n      category.id_comp],\n  end,\n  comp_id' := λ X Y f, begin\n    show FreeAb.comp f Y.id = add_monoid_hom.id _ f,\n    rw [← add_monoid_hom.flip_apply], congr' 1, clear f, ext1 f,\n    simp only [add_monoid_hom.id_apply, FreeAb.comp, free_abelian_group.lift.of, FreeAb.id,\n      category.comp_id, add_monoid_hom.flip_apply],\n  end,\n  assoc' := λ W X Y Z f g h, begin\n    show FreeAb.comp.comp (FreeAb.comp f) g h = (FreeAb.comp f).comp (FreeAb.comp g) h,\n    congr' 1,\n    rw [← add_monoid_hom.comp_hom_apply_apply, ← add_monoid_hom.comp_hom_apply_apply,\n        ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_apply],\n    congr' 1,\n    conv_rhs { rw [← add_monoid_hom.comp_hom_apply_apply, ← add_monoid_hom.flip_apply,\n      ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_apply] },\n    congr' 1,\n    clear f g h, ext f g h,\n    simp only [add_monoid_hom.comp_apply, add_monoid_hom.comp_hom_apply_apply,\n      add_monoid_hom.flip_apply, FreeAb.comp, free_abelian_group.lift.of, category.assoc],\n  end }\n.\n\ninstance : preadditive (FreeAb C) :=\n{ hom_group := by { intros, apply_instance },\n  add_comp' := by { intros, show FreeAb.comp (_ + _) _ = _, simp only [map_add], refl },\n  comp_add' := by { intros, show FreeAb.comp _ (_ + _) = _, simp only [map_add], refl } }\n\ndef eval [preadditive C] : FreeAb C ⥤ C :=\n{ obj := FreeAb.as,\n  map := λ X Y, free_abelian_group.lift id,\n  map_id' := λ X, show free_abelian_group.lift id X.id = 𝟙 X.as,\n    by { simp only [FreeAb.id, free_abelian_group.lift.of], refl },\n  map_comp' := λ X Y Z f g, begin\n    show free_abelian_group.lift id (FreeAb.comp f g) = preadditive.comp_hom _ _,\n    rw [← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_apply], congr' 1,\n    rw [← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_hom_apply_apply,\n        ← add_monoid_hom.comp_apply],\n    conv_rhs { rw [← add_monoid_hom.comp_hom_apply_apply, ← add_monoid_hom.flip_apply,\n      ← add_monoid_hom.comp_apply] }, congr' 1, clear f g, ext f g,\n    simp only [add_monoid_hom.comp_apply, add_monoid_hom.comp_hom_apply_apply,\n      add_monoid_hom.flip_apply, FreeAb.comp, free_abelian_group.lift.of],\n    refl,\n  end }\n\ninstance eval_additive [preadditive C] : (eval C).additive :=\n{ map_add' := λ X Y f g, add_monoid_hom.map_add _ _ _ }\n\nend FreeAb\n\nnamespace functor\n\nvariables {C D : Type*} [category C] [category D]\n\ndef map_FreeAb (F : C ⥤ D) : FreeAb C ⥤ FreeAb D :=\n{ obj := λ X, FreeAb.of (F.obj X.as),\n  map := λ X Y, free_abelian_group.map (λ f, F.map f),\n  map_id' := λ X, by { erw [free_abelian_group.map_of_apply, F.map_id], refl },\n  map_comp' := λ X Y Z f g, begin\n    rw [← FreeAb.comp_apply, ← FreeAb.comp_apply,\n        ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_apply], congr' 1,\n    rw [← add_monoid_hom.comp_hom_apply_apply, ← add_monoid_hom.comp_apply],\n    conv_rhs { rw [← add_monoid_hom.comp_hom_apply_apply, ← add_monoid_hom.flip_apply,\n      ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_apply] }, congr' 1, clear f g, ext f g,\n    simp only [add_monoid_hom.comp_apply, add_monoid_hom.comp_hom_apply_apply,\n      add_monoid_hom.flip_apply, FreeAb.comp, free_abelian_group.lift.of,\n      free_abelian_group.map, ← F.map_comp],\n  end }\n\ninstance map_FreeAb_additive (F : C ⥤ D) : F.map_FreeAb.additive :=\n{ map_add' := λ X Y f g, add_monoid_hom.map_add _ _ _ }\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/FreeAb.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154239836484144, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.33816909584857224}}
{"text": "/-\nCopyright (c) 2021 Kalle Kytölä. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kalle Kytölä\n-/\nimport tactic \nimport measure_theory.measurable_space\nimport measure_theory.integration\nimport measure_theory.borel_space\nimport measure_theory.bochner_integration\nimport topology.metric_space.basic\nimport topology.instances.real\nimport topology.instances.ennreal\nimport order.liminf_limsup\n\n\nnoncomputable theory\nopen set \nopen classical\nopen measure_theory\nopen measurable_space\nopen filter\nopen_locale topological_space\n\n\n\nnamespace portmanteau\n\n\n\nabbreviation bdd_Rval {β : Type*} (f : β → ℝ) : Prop :=\n  ∃ (M : ℝ) , ∀ (b : β) , abs(f(b)) ≤ M\n\nabbreviation bdd_ennval {α : Type*} (f : α → ennreal) : Prop :=\n  ∃ (M : nnreal) , ∀ (a : α) , f(a) ≤ M\n\n\n\nsection test_functions_for_weak_convergence\n\n\n/-- Continuous bounded functions on a topological space `α` with values\nin `ennreal` are used as \"test functions\" in the definition of the topology of\nthe weak convergence of probability measures. They are defined as a subtype\nof `α → ennreal`, so that the type of (positive) functionals is just\n`(cont_bdd_ennval α) → ennreal`. -/\ndef cont_bdd_ennval (α : Type*) [topological_space α] : Type*\n  := { f : α → ennreal // continuous f ∧ bdd_ennval f }\n\n\ndef functional_cont_bdd_ennval (α : Type*) [topological_space α] : Type*\n  := (cont_bdd_ennval α) → ennreal\n\n\ninstance {α : Type*} [topological_space α] :\n  has_coe (cont_bdd_ennval α) (α → ennreal) := ⟨subtype.val⟩\n\n\n@[simp] lemma val_eq_coe_testfun {α : Type*} [topological_space α] (f : cont_bdd_ennval α) :\n  f.val = f := rfl\n\n\n/-- As a first step towards the definition of the topology of the weak convergence\nof probability measures, the space of functionals `(cont_bdd_ennval α) → ennreal`\nis equipped with the product topology (the topology of \"testfunctionwise\" convergence,\ni.e., of pointwise convergence of the functionals defined on the space of continuous\nbounded test functions). -/\ninstance {α : Type*} [topological_space α] :\n  topological_space (functional_cont_bdd_ennval α) := Pi.topological_space\n\n\n/-- In an alternative an more familiar formulation, continuous bounded `ℝ`-valued\nfunctions on a topological space `α` are used as \"test functions\" in the definition\nof the topology of the weak convergence of probability measures. They are defined as\na subtype of `α → ℝ`. -/\ndef cont_bdd_Rval (α : Type*) [topological_space α] : Type*\n  := { f : α → ℝ // continuous f ∧ bdd_Rval f }\n\n\ndef cont_bdd_Rval_mk {α : Type*} [topological_space α] \n  (g : α → ℝ) (g_cont : continuous g) (g_bdd : bdd_Rval g) : cont_bdd_Rval α :=\n{ val := g ,\n  property := ⟨ g_cont , g_bdd ⟩ , }\n\n-- TODO: It would be good to equip `cont_bdd_Rval` with the sup-norm, show that it is\n-- a Banach space, define the (continuous) dual of it, equip it with the dual norm,\n-- show that each Borel probability measure defines an element of the (continuous)\n-- dual, etc... At least currently the result `weak_conv_seq_iff` essentially shows\n-- that the mapping the Borel probability measures into the dual will be an\n-- embedding (the topologies are compatible).\n\n\n--TODO: I can't use the same name for the following coercion.\n--instance {α : Type*} [topological_space α] :\n--  has_coe (cont_bdd_Rval α) (α → ℝ) := ⟨subtype.val⟩\n\n\nend test_functions_for_weak_convergence\n\n\n\nsection topology_of_weak_convergence\n\n\n/-- Borel probability measures on a topological space `α` are defined as a subtype\nof measures. This subtype `borel_proba α` is equipped with the topology of weak\nconvergence. -/\ndef borel_proba (α : Type*) [topological_space α] : Type\n  := { μ : @measure_theory.measure α (borel(α)) // @probability_measure α (borel(α)) μ }\n\n\ninstance (α : Type*) [topological_space α] :\n  has_coe (borel_proba α) (@measure_theory.measure α (borel(α))) := ⟨subtype.val⟩\n\n\n@[simp] lemma val_eq_coe_borel_proba {α : Type*} [topological_space α] (ν : borel_proba α) :\n  ν.val = ν := rfl\n\n\nabbreviation integrate_cont_bdd_ennval {α : Type*} [topological_space α]\n  (μ : borel_proba α) (f : cont_bdd_ennval α) : ennreal := @lintegral α (borel(α)) μ f \n\n\n/-- The topology of weak convergence on `borel_proba α` is defined as the induced\ntopology of the mapping `(borel_proba α) → ((cont_bdd_ennval α) → ennreal)` to\nfunctionals defined by integration of a test functio against to the measure. In\nother contexts this could be called the weak-* topology. -/\ninstance {α : Type} [topological_space α] : topological_space (borel_proba α)\n  := topological_space.induced (λ (μ : borel_proba α) , integrate_cont_bdd_ennval μ) infer_instance\n\n\n/-- Integration of test functions against borel probability measures depends\ncontinuously on the measure. -/\nlemma integrate_cont_bdd_ennval_cont {α : Type} [topological_space α] :\n  continuous (@integrate_cont_bdd_ennval α _) := continuous_induced_dom\n\n\n-- Remark: It felt convenient to isolate the following fact (does it exist already?).\nlemma conv_seq_induced {α γ : Type*} [top_γ : topological_space γ]\n  (f : α → γ) (x : ℕ → α) (x₀ : α) :\n    tendsto (f ∘ x) at_top (𝓝 (f(x₀)))\n      → tendsto x at_top (@nhds α (topological_space.induced f top_γ) x₀) :=\nbegin\n  intro h_f_lim ,\n  apply tendsto_nhds.mpr ,\n  intros U open_U U_ni_x₀ ,\n  rcases ((@is_open_induced_iff α γ top_γ U f).mp open_U) with ⟨ V , open_V , preim_V_eq_U ⟩ ,\n  induction preim_V_eq_U , \n  apply tendsto_nhds.mp h_f_lim V open_V U_ni_x₀ , \nend\n\n\n/-- The usual definition of weak convergence of probability measures is given in\nterms of sequences of probability measures: it is the requirement that the integrals\nof all continuous bounded functions against members of the sequence converge.\nThis characterization is shown in `weak_conv_seq_iff'` in the case when the\nfunctions are `ennreal`-valued and the integral is `lintegral`. The most common\nformulation with `ℝ`-valued functions and Bochner integrals is `weak_conv_seq_iff`. -/\ntheorem weak_conv_seq_iff' {α : Type*} [topological_space α]\n  {μseq : ℕ → borel_proba α} {μ : borel_proba α} :\n    tendsto μseq at_top (𝓝 μ) \n      ↔ ∀ (f : cont_bdd_ennval α) , \n        tendsto (λ n, integrate_cont_bdd_ennval (μseq(n)) f) at_top (𝓝 (integrate_cont_bdd_ennval μ f)) :=\nbegin\n  split ,\n  { intros weak_conv ,\n    have key := tendsto.comp (continuous.tendsto (@integrate_cont_bdd_ennval_cont α _) μ) weak_conv ,\n    exact tendsto_pi.mp key , } ,\n  { intro h_lim_forall ,\n    have h_lim : tendsto (λ n, integrate_cont_bdd_ennval (μseq(n))) at_top (𝓝 (integrate_cont_bdd_ennval μ)) ,\n    { exact tendsto_pi.mpr h_lim_forall , } ,\n    exact conv_seq_induced _ μseq μ h_lim , } ,\nend\n\n\nend topology_of_weak_convergence\n\n\n\nsection equivalent_conditions\n-- See <pormanteau_conclusions.lean> for the main theorems about the equivalence.\n\n\nabbreviation portmanteau_continuous_ennval {α : Type} [topological_space α]\n  (μseq : ℕ → @measure_theory.measure α (borel α)) (μ : @measure_theory.measure α (borel α)) : Prop :=\n    ∀ (f : α → ennreal) , (continuous f) → (bdd_ennval f) →\n      tendsto (λ n , (@lintegral α (borel(α)) (μseq(n)) f) ) at_top (𝓝 (@lintegral α (borel(α)) μ f))\n\n\nabbreviation portmanteau_continuous_Rval {α : Type} [topological_space α]\n  (μseq : ℕ → @measure_theory.measure α (borel α)) (μ : @measure_theory.measure α (borel α)) : Prop :=\n    ∀ (f : α → ℝ) , (continuous f) → (bdd_Rval f) →\n      tendsto (λ n , (@integral α ℝ (borel(α)) _ _ _ _ _ _ (μseq(n)) f)) at_top (𝓝 (@integral α ℝ (borel(α)) _ _ _ _ _ _ μ f))\n\n\nabbreviation portmanteau_open {α : Type} [topological_space α]\n  (μseq : ℕ → @measure_theory.measure α (borel α)) (μ : @measure_theory.measure α (borel α)) : Prop :=\n    ∀ (G : set α) , (is_open G) → μ(G) ≤ liminf at_top (λ n , (μseq(n))(G))\n\n\nabbreviation portmanteau_closed {α : Type} [topological_space α]\n  (μseq : ℕ → @measure_theory.measure α (borel α)) (μ : @measure_theory.measure α (borel α)) : Prop :=\n    ∀ (F : set α) , (is_closed F) → limsup at_top (λ n , (μseq(n))(F)) ≤ μ(F)\n\n\nabbreviation portmanteau_borel {α : Type} [topological_space α]\n  (μseq : ℕ → @measure_theory.measure α (borel α)) (μ : @measure_theory.measure α (borel α)) : Prop :=\n    ∀ (E : set α) , ((borel α).measurable_set' E) → (μ(frontier E) = 0)\n      → (tendsto (λ n , (μseq(n))(E)) at_top (𝓝 (μ(E))))\n\n\nend equivalent_conditions\n\n\n\nend portmanteau\n\n\n", "meta": {"author": "kkytola", "repo": "lean_portmanteau", "sha": "ac55eb4e24be43032cbc082e2b68d8fb8bd63f22", "save_path": "github-repos/lean/kkytola-lean_portmanteau", "path": "github-repos/lean/kkytola-lean_portmanteau/lean_portmanteau-ac55eb4e24be43032cbc082e2b68d8fb8bd63f22/portmanteau_definitions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3379132831926307}}
{"text": "-- Copyright (c) 2018 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Scott Morrison, Johan Commelin\n\nimport category_theory.types\nimport category_theory.isomorphism\nimport category_theory.whiskering\nimport category_theory.opposites\nimport category_theory.punit\nimport category_theory.equivalence\n\nnamespace category_theory\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃ -- declare the `v`'s first; see `category_theory.category` for an explanation\nvariables {A : Type u₁} [𝒜 : category.{v₁} A]\nvariables {B : Type u₂} [ℬ : category.{v₂} B]\nvariables {T : Type u₃} [𝒯 : category.{v₃} T]\ninclude 𝒜 ℬ 𝒯\n\nstructure comma (L : A ⥤ T) (R : B ⥤ T) :=\n(left : A . obviously)\n(right : B . obviously)\n(hom : L.obj left ⟶ R.obj right)\n\nvariables {L : A ⥤ T} {R : B ⥤ T}\n\nstructure comma_morphism (X Y : comma L R) :=\n(left : X.left ⟶ Y.left . obviously)\n(right : X.right ⟶ Y.right . obviously)\n(w' : L.map left ≫ Y.hom = X.hom ≫ R.map right . obviously)\n\nrestate_axiom comma_morphism.w'\nattribute [simp] comma_morphism.w\n\nnamespace comma_morphism\n@[extensionality] lemma ext\n  {X Y : comma L R} {f g : comma_morphism X Y}\n  (l : f.left = g.left) (r : f.right = g.right) : f = g :=\nbegin\n  cases f, cases g,\n  congr; assumption\nend\nend comma_morphism\n\ninstance comma_category : category (comma L R) :=\n{ hom := comma_morphism,\n  id := λ X,\n  { left := 𝟙 X.left,\n    right := 𝟙 X.right },\n  comp := λ X Y Z f g,\n  { left := f.left ≫ g.left,\n    right := f.right ≫ g.right,\n    w' :=\n    begin\n      rw [functor.map_comp,\n          category.assoc,\n          g.w,\n          ←category.assoc,\n          f.w,\n          functor.map_comp,\n          category.assoc],\n    end }}\n\nnamespace comma\n\nsection\nvariables {X Y Z : comma L R} {f : X ⟶ Y} {g : Y ⟶ Z}\n\n@[simp] lemma comp_left  : (f ≫ g).left  = f.left ≫ g.left   := rfl\n@[simp] lemma comp_right : (f ≫ g).right = f.right ≫ g.right := rfl\n\nend\n\nvariables (L) (R)\n\ndef fst : comma L R ⥤ A :=\n{ obj := λ X, X.left,\n  map := λ _ _ f, f.left }\n\ndef snd : comma L R ⥤ B :=\n{ obj := λ X, X.right,\n  map := λ _ _ f, f.right }\n\n@[simp] lemma fst_obj {X : comma L R} : (fst L R).obj X = X.left := rfl\n@[simp] lemma snd_obj {X : comma L R} : (snd L R).obj X = X.right := rfl\n@[simp] lemma fst_map {X Y : comma L R} {f : X ⟶ Y} : (fst L R).map f = f.left := rfl\n@[simp] lemma snd_map {X Y : comma L R} {f : X ⟶ Y} : (snd L R).map f = f.right := rfl\n\ndef nat_trans : fst L R ⋙ L ⟹ snd L R ⋙ R :=\n{ app := λ X, X.hom }\n\nsection\nvariables {L₁ L₂ L₃ : A ⥤ T} {R₁ R₂ R₃ : B ⥤ T}\n\ndef map_left (l : L₁ ⟹ L₂) : comma L₂ R ⥤ comma L₁ R :=\n{ obj := λ X,\n  { left  := X.left,\n    right := X.right,\n    hom   := l.app X.left ≫ X.hom },\n  map := λ X Y f,\n  { left  := f.left,\n    right := f.right,\n    w' := by tidy; rw [←category.assoc, l.naturality f.left, category.assoc]; tidy } }\n\nsection\nvariables {X Y : comma L₂ R} {f : X ⟶ Y} {l : L₁ ⟹ L₂}\n@[simp] lemma map_left_obj_left  : ((map_left R l).obj X).left  = X.left                := rfl\n@[simp] lemma map_left_obj_right : ((map_left R l).obj X).right = X.right               := rfl\n@[simp] lemma map_left_obj_hom   : ((map_left R l).obj X).hom   = l.app X.left ≫ X.hom := rfl\n@[simp] lemma map_left_map_left  : ((map_left R l).map f).left  = f.left                := rfl\n@[simp] lemma map_left_map_right : ((map_left R l).map f).right = f.right               := rfl\nend\n\ndef map_left_id : map_left R (𝟙 L) ≅ functor.id _ :=\n{ hom :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } },\n  inv :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } } }\n\nsection\nvariables {X : comma L R}\n@[simp] lemma map_left_id_hom_app_left  : (((map_left_id L R).hom).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_left_id_hom_app_right : (((map_left_id L R).hom).app X).right = 𝟙 (X.right) := rfl\n@[simp] lemma map_left_id_inv_app_left  : (((map_left_id L R).inv).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_left_id_inv_app_right : (((map_left_id L R).inv).app X).right = 𝟙 (X.right) := rfl\nend\n\ndef map_left_comp (l : L₁ ⟹ L₂) (l' : L₂ ⟹ L₃) :\n(map_left R (l ⊟ l')) ≅ (map_left R l') ⋙ (map_left R l) :=\n{ hom :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } },\n  inv :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } } }\n\nsection\nvariables {X : comma L₃ R} {l : L₁ ⟹ L₂} {l' : L₂ ⟹ L₃}\n@[simp] lemma map_left_comp_hom_app_left  : (((map_left_comp R l l').hom).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_left_comp_hom_app_right : (((map_left_comp R l l').hom).app X).right = 𝟙 (X.right) := rfl\n@[simp] lemma map_left_comp_inv_app_left  : (((map_left_comp R l l').inv).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_left_comp_inv_app_right : (((map_left_comp R l l').inv).app X).right = 𝟙 (X.right) := rfl\nend\n\ndef map_right (r : R₁ ⟹ R₂) : comma L R₁ ⥤ comma L R₂ :=\n{ obj := λ X,\n  { left  := X.left,\n    right := X.right,\n    hom   := X.hom ≫ r.app X.right },\n  map := λ X Y f,\n  { left  := f.left,\n    right := f.right,\n    w' := by tidy; rw [←r.naturality f.right, ←category.assoc]; tidy } }\n\nsection\nvariables {X Y : comma L R₁} {f : X ⟶ Y} {r : R₁ ⟹ R₂}\n@[simp] lemma map_right_obj_left  : ((map_right L r).obj X).left  = X.left                 := rfl\n@[simp] lemma map_right_obj_right : ((map_right L r).obj X).right = X.right                := rfl\n@[simp] lemma map_right_obj_hom   : ((map_right L r).obj X).hom   = X.hom ≫ r.app X.right  := rfl\n@[simp] lemma map_right_map_left  : ((map_right L r).map f).left  = f.left                 := rfl\n@[simp] lemma map_right_map_right : ((map_right L r).map f).right = f.right                := rfl\nend\n\ndef map_right_id : map_right L (𝟙 R) ≅ functor.id _ :=\n{ hom :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } },\n  inv :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } } }\n\nsection\nvariables {X : comma L R}\n@[simp] lemma map_right_id_hom_app_left  : (((map_right_id L R).hom).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_right_id_hom_app_right : (((map_right_id L R).hom).app X).right = 𝟙 (X.right) := rfl\n@[simp] lemma map_right_id_inv_app_left  : (((map_right_id L R).inv).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_right_id_inv_app_right : (((map_right_id L R).inv).app X).right = 𝟙 (X.right) := rfl\nend\n\ndef map_right_comp (r : R₁ ⟹ R₂) (r' : R₂ ⟹ R₃) : (map_right L (r ⊟ r')) ≅ (map_right L r) ⋙ (map_right L r') :=\n{ hom :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } },\n  inv :=\n  { app := λ X, { left := 𝟙 _, right := 𝟙 _ } } }\n\nsection\nvariables {X : comma L R₁} {r : R₁ ⟹ R₂} {r' : R₂ ⟹ R₃}\n@[simp] lemma map_right_comp_hom_app_left  : (((map_right_comp L r r').hom).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_right_comp_hom_app_right : (((map_right_comp L r r').hom).app X).right = 𝟙 (X.right) := rfl\n@[simp] lemma map_right_comp_inv_app_left  : (((map_right_comp L r r').inv).app X).left  = 𝟙 (X.left)  := rfl\n@[simp] lemma map_right_comp_inv_app_right : (((map_right_comp L r r').inv).app X).right = 𝟙 (X.right) := rfl\nend\n\nend\n\nend comma\n\nomit 𝒜 ℬ\n\ndef over (X : T) := comma.{v₃ 0 v₃} (functor.id T) (functor.of.obj X)\n\nnamespace over\n\nvariables {X : T}\n\ninstance category : category (over X) := by delta over; apply_instance\n\n@[extensionality] lemma over_morphism.ext {X : T} {U V : over X} {f g : U ⟶ V}\n  (h : f.left = g.left) : f = g :=\nby tidy\n\n@[simp] lemma over_right (U : over X) : U.right = punit.star := by tidy\n@[simp] lemma over_morphism_right {U V : over X} (f : U ⟶ V) : f.right = 𝟙 punit.star := by tidy\n\n@[simp] lemma id_left (U : over X) : comma_morphism.left (𝟙 U) = 𝟙 U.left := rfl\n@[simp] lemma comp_left (a b c : over X) (f : a ⟶ b) (g : b ⟶ c) :\n  (f ≫ g).left = f.left ≫ g.left := rfl\n\n@[simp] lemma w {A B : over X} (f : A ⟶ B) : f.left ≫ B.hom = A.hom :=\nby have := f.w; tidy\n\ndef mk {X Y : T} (f : Y ⟶ X) : over X :=\n{ left := Y, hom := f }\n\n@[simp] lemma mk_left {X Y : T} (f : Y ⟶ X) : (mk f).left = Y := rfl\n@[simp] lemma mk_hom {X Y : T} (f : Y ⟶ X) : (mk f).hom = f := rfl\n\ndef hom_mk {U V : over X} (f : U.left ⟶ V.left) (w : f ≫ V.hom = U.hom . obviously) :\n  U ⟶ V :=\n{ left := f }\n\n@[simp] lemma hom_mk_left {U V : over X} (f : U.left ⟶ V.left) (w : f ≫ V.hom = U.hom) :\n  (hom_mk f).left = f :=\nrfl\n\ndef forget : (over X) ⥤ T := comma.fst _ _\n\n@[simp] lemma forget_obj {U : over X} : forget.obj U = U.left := rfl\n@[simp] lemma forget_map {U V : over X} {f : U ⟶ V} : forget.map f = f.left := rfl\n\ndef map {Y : T} (f : X ⟶ Y) : over X ⥤ over Y := comma.map_right _ $ functor.of.map f\n\nsection\nvariables {Y : T} {f : X ⟶ Y} {U V : over X} {g : U ⟶ V}\n@[simp] lemma map_obj_left : ((map f).obj U).left = U.left := rfl\n@[simp] lemma map_obj_hom  : ((map f).obj U).hom  = U.hom ≫ f := rfl\n@[simp] lemma map_map_left : ((map f).map g).left = g.left := rfl\nend\n\nsection\nvariables {D : Type u₃} [Dcat : category.{v₃} D]\ninclude Dcat\n\ndef post (F : T ⥤ D) : over X ⥤ over (F.obj X) :=\n{ obj := λ Y, mk $ F.map Y.hom,\n  map := λ Y₁ Y₂ f,\n  { left := F.map f.left,\n    w' := by tidy; erw [← F.map_comp, w] } }\n\nend\n\nend over\n\ndef under (X : T) := comma.{0 v₃ v₃} (functor.of.obj X) (functor.id T)\n\nnamespace under\n\nvariables {X : T}\n\ninstance : category (under X) := by delta under; apply_instance\n\n@[extensionality] lemma under_morphism.ext {X : T} {U V : under X} {f g : U ⟶ V}\n  (h : f.right = g.right) : f = g :=\nby tidy\n\n@[simp] lemma under_left (U : under X) : U.left = punit.star := by tidy\n@[simp] lemma under_morphism_left {U V : under X} (f : U ⟶ V) : f.left = 𝟙 punit.star := by tidy\n\n@[simp] lemma id_right (U : under X) : comma_morphism.right (𝟙 U) = 𝟙 U.right := rfl\n@[simp] lemma comp_right (a b c : under X) (f : a ⟶ b) (g : b ⟶ c) :\n  (f ≫ g).right = f.right ≫ g.right := rfl\n\n@[simp] lemma w {A B : under X} (f : A ⟶ B) : A.hom ≫ f.right = B.hom :=\nby have := f.w; tidy\n\ndef mk {X Y : T} (f : X ⟶ Y) : under X :=\n{ right := Y, hom := f }\n\n@[simp] lemma mk_right {X Y : T} (f : X ⟶ Y) : (mk f).right = Y := rfl\n@[simp] lemma mk_hom {X Y : T} (f : X ⟶ Y) : (mk f).hom = f := rfl\n\ndef hom_mk {U V : under X} (f : U.right ⟶ V.right) (w : U.hom ≫ f = V.hom . obviously) :\n  U ⟶ V :=\n{ right := f }\n\n@[simp] lemma hom_mk_right {U V : under X} (f : U.right ⟶ V.right) (w : U.hom ≫ f = V.hom) :\n  (hom_mk f).right = f :=\nrfl\n\ndef forget : (under X) ⥤ T := comma.snd _ _\n\n@[simp] lemma forget_obj {U : under X} : forget.obj U = U.right := rfl\n@[simp] lemma forget_map {U V : under X} {f : U ⟶ V} : forget.map f = f.right := rfl\n\ndef map {Y : T} (f : X ⟶ Y) : under Y ⥤ under X := comma.map_left _ $ functor.of.map f\n\nsection\nvariables {Y : T} {f : X ⟶ Y} {U V : under Y} {g : U ⟶ V}\n@[simp] lemma map_obj_right : ((map f).obj U).right = U.right := rfl\n@[simp] lemma map_obj_hom   : ((map f).obj U).hom   = f ≫ U.hom := rfl\n@[simp] lemma map_map_right : ((map f).map g).right = g.right := rfl\nend\n\nsection\nvariables {D : Type u₃} [Dcat : category.{v₃} D]\ninclude Dcat\n\ndef post {X : T} (F : T ⥤ D) : under X ⥤ under (F.obj X) :=\n{ obj := λ Y, mk $ F.map Y.hom,\n  map := λ Y₁ Y₂ f,\n  { right := F.map f.right,\n    w' := by tidy; erw [← F.map_comp, w] } }\n\nend\n\nend under\n\nend category_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/comma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.337913274759572}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.pullbacks\nimport algebraic_geometry.AffineScheme\n\n/-!\n# (Co)Limits of Schemes\n\nWe construct various limits and colimits in the category of schemes.\n\n* The existence of fibred products was shown in `algebraic_geometry/pullbacks.lean`.\n* `Spec ℤ` is the terminal object.\n* The preceding two results imply that `Scheme` has all finite limits.\n* The empty scheme is the (strict) initial object.\n\n## Todo\n\n* Coproducts exists (and the forgetful functors preserve them).\n\n-/\n\nuniverse u\n\nopen category_theory category_theory.limits opposite topological_space\n\nnamespace algebraic_geometry\n\n/-- `Spec ℤ` is the terminal object in the category of schemes. -/\nnoncomputable\ndef Spec_Z_is_terminal : is_terminal (Scheme.Spec.obj (op $ CommRing.of ℤ)) :=\n@@is_terminal.is_terminal_obj _ _ Scheme.Spec _ infer_instance\n  (terminal_op_of_initial CommRing.Z_is_initial)\n\ninstance : has_terminal Scheme := has_terminal_of_has_terminal_of_preserves_limit Scheme.Spec\n\ninstance : is_affine (⊤_ Scheme.{u}) :=\nis_affine_of_iso (preserves_terminal.iso Scheme.Spec).inv\n\ninstance : has_finite_limits Scheme :=\nhas_finite_limits_of_has_terminal_and_pullbacks\n\nsection initial\n\n/-- The map from the empty scheme. -/\n@[simps]\ndef Scheme.empty_to (X : Scheme.{u}) : ∅ ⟶ X :=\n⟨{ base := ⟨λ x, pempty.elim x, by continuity⟩,\n    c := { app := λ U, CommRing.punit_is_terminal.from _ } }, λ x, pempty.elim x⟩\n\n@[ext]\nlemma Scheme.empty_ext {X : Scheme.{u}} (f g : ∅ ⟶ X) : f = g :=\nby { ext a, exact pempty.elim a }\n\nlemma Scheme.eq_empty_to {X : Scheme.{u}} (f : ∅ ⟶ X) : f = Scheme.empty_to X :=\nScheme.empty_ext f (Scheme.empty_to X)\n\ninstance (X : Scheme.{u}) : unique (∅ ⟶ X) :=\n⟨⟨Scheme.empty_to _⟩, λ _, Scheme.empty_ext _ _⟩\n\n/-- The empty scheme is the initial object in the category of schemes. -/\ndef empty_is_initial : is_initial (∅ : Scheme.{u}) :=\nis_initial.of_unique _\n\n@[simp]\nlemma empty_is_initial_to : empty_is_initial.to = Scheme.empty_to := rfl\n\ninstance : is_empty Scheme.empty.carrier :=\nshow is_empty pempty, by apply_instance\n\ninstance Spec_punit_is_empty : is_empty (Scheme.Spec.obj (op $ CommRing.of punit)).carrier :=\n⟨prime_spectrum.punit⟩\n\n@[priority 100]\ninstance is_open_immersion_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty X.carrier] :\n  is_open_immersion f :=\nbegin\n  apply_with is_open_immersion.of_stalk_iso { instances := ff },\n  { apply open_embedding_of_continuous_injective_open,\n    { continuity },\n    { rintro (i : X.carrier), exact is_empty_elim i },\n    { intros U hU, convert is_open_empty, ext, apply (iff_false _).mpr,\n      exact λ x, is_empty_elim (show X.carrier, from x.some) } },\n  { rintro (i : X.carrier), exact is_empty_elim i }\nend\n\n@[priority 100]\ninstance is_iso_of_is_empty {X Y : Scheme} (f : X ⟶ Y) [is_empty Y.carrier] : is_iso f :=\nbegin\n  haveI : is_empty X.carrier := ⟨λ x, is_empty_elim (show Y.carrier, from f.1.base x)⟩,\n  haveI : epi f.1.base,\n  { rw Top.epi_iff_surjective, rintro (x : Y.carrier), exact is_empty_elim x },\n  apply is_open_immersion.to_iso\nend\n\n/-- A scheme is initial if its underlying space is empty . -/\nnoncomputable\ndef is_initial_of_is_empty {X : Scheme} [is_empty X.carrier] : is_initial X :=\nempty_is_initial.of_iso (as_iso $ empty_is_initial.to _)\n\n/-- `Spec 0` is the initial object in the category of schemes. -/\nnoncomputable\ndef Spec_punit_is_initial : is_initial (Scheme.Spec.obj (op $ CommRing.of punit)) :=\nempty_is_initial.of_iso (as_iso $ empty_is_initial.to _)\n\n@[priority 100]\ninstance is_affine_of_is_empty {X : Scheme} [is_empty X.carrier] : is_affine X :=\nis_affine_of_iso (inv (empty_is_initial.to X) ≫\n  empty_is_initial.to (Scheme.Spec.obj (op $ CommRing.of punit)))\n\ninstance : has_initial Scheme :=\nhas_initial_of_unique Scheme.empty\n\ninstance initial_is_empty : is_empty (⊥_ Scheme).carrier :=\n⟨λ x, ((initial.to Scheme.empty : _).1.base x).elim⟩\n\nlemma bot_is_affine_open (X : Scheme) : is_affine_open (⊥ : opens X.carrier) :=\nbegin\n  convert range_is_affine_open_of_open_immersion (initial.to X),\n  ext,\n  exact (false_iff _).mpr (λ x, is_empty_elim (show (⊥_ Scheme).carrier, from x.some)),\nend\n\ninstance : has_strict_initial_objects Scheme :=\nhas_strict_initial_objects_of_initial_is_strict (λ A f, by apply_instance)\n\nend initial\n\nend algebraic_geometry\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.337913274759572}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Chris Hughes, Kevin Buzzard\n\n! This file was ported from Lean 3 source module algebra.hom.units\n! leanprover-community/mathlib commit dc6c365e751e34d100e80fe6e314c3c3e0fd2988\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Hom.Group\nimport Mathbin.Algebra.Group.Units\n\n/-!\n# Monoid homomorphisms and units\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file allows to lift monoid homomorphisms to group homomorphisms of their units subgroups. It\nalso contains unrelated results about `units` that depend on `monoid_hom`.\n\n## Main declarations\n\n* `units.map`: Turn an homomorphism from `α` to `β` monoids into an homomorphism from `αˣ` to `βˣ`.\n* `monoid_hom.to_hom_units`: Turn an homomorphism from a group `α` to `β` into an homomorphism from\n  `α` to `βˣ`.\n\n## TODO\n\nThe results that don't mention homomorphisms should be proved (earlier?) in a different file and be\nused to golf the basic `group` lemmas.\n-/\n\n\nopen Function\n\nuniverse u v w\n\n#print Group.isUnit /-\n@[to_additive]\ntheorem Group.isUnit {G} [Group G] (g : G) : IsUnit g :=\n  ⟨⟨g, g⁻¹, mul_inv_self g, inv_mul_self g⟩, rfl⟩\n#align group.is_unit Group.isUnit\n#align add_group.is_add_unit AddGroup.isAddUnit\n-/\n\nsection MonoidHomClass\n\n/- warning: is_unit.eq_on_inv -> IsUnit.eq_on_inv is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1}} {G : Type.{u2}} {N : Type.{u3}} [_inst_1 : DivisionMonoid.{u2} G] [_inst_2 : Monoid.{u3} N] [_inst_3 : MonoidHomClass.{u1, u2, u3} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} N _inst_2)] {x : G}, (IsUnit.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)) x) -> (forall (f : F) (g : F), (Eq.{succ u3} N (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> N) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => N) (MulHomClass.toFunLike.{u1, u2, u3} F G N (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} N _inst_2) _inst_3))) f x) (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> N) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => N) (MulHomClass.toFunLike.{u1, u2, u3} F G N (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} N _inst_2) _inst_3))) g x)) -> (Eq.{succ u3} N (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> N) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => N) (MulHomClass.toFunLike.{u1, u2, u3} F G N (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} N _inst_2) _inst_3))) f (Inv.inv.{u2} G (DivInvMonoid.toHasInv.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)) x)) (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> N) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => N) (MulHomClass.toFunLike.{u1, u2, u3} F G N (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} N _inst_2) _inst_3))) g (Inv.inv.{u2} G (DivInvMonoid.toHasInv.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)) x))))\nbut is expected to have type\n  forall {F : Type.{u3}} {G : Type.{u2}} {N : Type.{u1}} [_inst_1 : DivisionMonoid.{u2} G] [_inst_2 : Monoid.{u1} N] [_inst_3 : MonoidHomClass.{u3, u2, u1} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} N _inst_2)] {x : G}, (IsUnit.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)) x) -> (forall (f : F) (g : F), (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) x) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G N (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} N _inst_2) _inst_3)) f x) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G N (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} N _inst_2) _inst_3)) g x)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G _inst_1))) x)) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G N (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} N _inst_2) _inst_3)) f (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G _inst_1))) x)) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => N) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G N (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G N (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (DivisionMonoid.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} N _inst_2) _inst_3)) g (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G _inst_1))) x))))\nCase conversion may be inaccurate. Consider using '#align is_unit.eq_on_inv IsUnit.eq_on_invₓ'. -/\n/-- If two homomorphisms from a division monoid to a monoid are equal at a unit `x`, then they are\nequal at `x⁻¹`. -/\n@[to_additive\n      \"If two homomorphisms from a subtraction monoid to an additive monoid are equal at an\\nadditive unit `x`, then they are equal at `-x`.\"]\ntheorem IsUnit.eq_on_inv {F G N} [DivisionMonoid G] [Monoid N] [MonoidHomClass F G N] {x : G}\n    (hx : IsUnit x) (f g : F) (h : f x = g x) : f x⁻¹ = g x⁻¹ :=\n  left_inv_eq_right_inv (map_mul_eq_one f hx.inv_mul_cancel) <|\n    h.symm ▸ map_mul_eq_one g <| hx.mul_inv_cancel\n#align is_unit.eq_on_inv IsUnit.eq_on_inv\n#align is_add_unit.eq_on_neg IsAddUnit.eq_on_neg\n\n/- warning: eq_on_inv -> eq_on_inv is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1}} {G : Type.{u2}} {M : Type.{u3}} [_inst_1 : Group.{u2} G] [_inst_2 : Monoid.{u3} M] [_inst_3 : MonoidHomClass.{u1, u2, u3} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} M _inst_2)] (f : F) (g : F) {x : G}, (Eq.{succ u3} M (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> M) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => M) (MulHomClass.toFunLike.{u1, u2, u3} F G M (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} M _inst_2) _inst_3))) f x) (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> M) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => M) (MulHomClass.toFunLike.{u1, u2, u3} F G M (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} M _inst_2) _inst_3))) g x)) -> (Eq.{succ u3} M (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> M) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => M) (MulHomClass.toFunLike.{u1, u2, u3} F G M (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} M _inst_2) _inst_3))) f (Inv.inv.{u2} G (DivInvMonoid.toHasInv.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)) x)) (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => G -> M) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F G (fun (_x : G) => M) (MulHomClass.toFunLike.{u1, u2, u3} F G M (MulOneClass.toHasMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u3} M _inst_2) _inst_3))) g (Inv.inv.{u2} G (DivInvMonoid.toHasInv.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)) x)))\nbut is expected to have type\n  forall {F : Type.{u3}} {G : Type.{u2}} {M : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Monoid.{u1} M] [_inst_3 : MonoidHomClass.{u3, u2, u1} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)] (f : F) (g : F) {x : G}, (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) x) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G M (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2) _inst_3)) f x) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G M (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2) _inst_3)) g x)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) x)) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G M (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2) _inst_3)) f (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) x)) (FunLike.coe.{succ u3, succ u2, succ u1} F G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) _x) (MulHomClass.toFunLike.{u3, u2, u1} F G M (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2) _inst_3)) g (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) x)))\nCase conversion may be inaccurate. Consider using '#align eq_on_inv eq_on_invₓ'. -/\n/-- If two homomorphism from a group to a monoid are equal at `x`, then they are equal at `x⁻¹`. -/\n@[to_additive\n      \"If two homomorphism from an additive group to an additive monoid are equal at `x`,\\nthen they are equal at `-x`.\"]\ntheorem eq_on_inv {F G M} [Group G] [Monoid M] [MonoidHomClass F G M] (f g : F) {x : G}\n    (h : f x = g x) : f x⁻¹ = g x⁻¹ :=\n  (Group.isUnit x).eq_on_inv f g h\n#align eq_on_inv eq_on_inv\n#align eq_on_neg eq_on_neg\n\nend MonoidHomClass\n\nnamespace Units\n\nvariable {α : Type _} {M : Type u} {N : Type v} {P : Type w} [Monoid M] [Monoid N] [Monoid P]\n\n/- warning: units.map -> Units.map is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N], (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) -> (MonoidHom.{u1, u2} (Units.{u1} M _inst_1) (Units.{u2} N _inst_2) (Units.mulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2))\nbut is expected to have type\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N], (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) -> (MonoidHom.{u1, u2} (Units.{u1} M _inst_1) (Units.{u2} N _inst_2) (Units.instMulOneClassUnits.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2))\nCase conversion may be inaccurate. Consider using '#align units.map Units.mapₓ'. -/\n/-- The group homomorphism on units induced by a `monoid_hom`. -/\n@[to_additive \"The `add_group` homomorphism on `add_unit`s induced by an `add_monoid_hom`.\"]\ndef map (f : M →* N) : Mˣ →* Nˣ :=\n  MonoidHom.mk'\n    (fun u =>\n      ⟨f u.val, f u.inv, by rw [← f.map_mul, u.val_inv, f.map_one], by\n        rw [← f.map_mul, u.inv_val, f.map_one]⟩)\n    fun x y => ext (f.map_mul x y)\n#align units.map Units.map\n#align add_units.map AddUnits.map\n\n/- warning: units.coe_map -> Units.coe_map is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (x : Units.{u1} M _inst_1), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N 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(Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) x))\nbut is expected to have type\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (x : Units.{u1} M _inst_1), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (Units.val.{u1} M _inst_1 x)) (Units.val.{u2} N _inst_2 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (MonoidHom.{u1, u2} (Units.{u1} M _inst_1) (Units.{u2} N _inst_2) 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N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f (Units.val.{u1} M _inst_1 x))\nCase conversion may be inaccurate. Consider using '#align units.coe_map Units.coe_mapₓ'. -/\n@[simp, to_additive]\ntheorem coe_map (f : M →* N) (x : Mˣ) : ↑(map f x) = f x :=\n  rfl\n#align units.coe_map Units.coe_map\n#align add_units.coe_map AddUnits.coe_map\n\n/- warning: units.coe_map_inv -> Units.coe_map_inv is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (u : Units.{u1} M _inst_1), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (Inv.inv.{u2} (Units.{u2} N _inst_2) (Units.hasInv.{u2} N _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u1) 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(Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f (Units.val.{u1} M _inst_1 (Inv.inv.{u1} (Units.{u1} M _inst_1) (Units.instInvUnits.{u1} M _inst_1) u)))\nCase conversion may be inaccurate. Consider using '#align units.coe_map_inv Units.coe_map_invₓ'. -/\n@[simp, to_additive]\ntheorem coe_map_inv (f : M →* N) (u : Mˣ) : ↑(map f u)⁻¹ = f ↑u⁻¹ :=\n  rfl\n#align units.coe_map_inv Units.coe_map_inv\n#align add_units.coe_map_neg AddUnits.coe_map_neg\n\n/- warning: units.map_comp -> Units.map_comp is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} {P : Type.{u3}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] [_inst_3 : Monoid.{u3} P] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (g : MonoidHom.{u2, u3} N P (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} P _inst_3)), Eq.{max (succ u3) (succ u1)} (MonoidHom.{u1, u3} (Units.{u1} M _inst_1) (Units.{u3} P _inst_3) (Units.mulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u3} P _inst_3)) (Units.map.{u1, u3} M P _inst_1 _inst_3 (MonoidHom.comp.{u1, u2, u3} M N P (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} P _inst_3) g f)) (MonoidHom.comp.{u1, u2, u3} (Units.{u1} M _inst_1) (Units.{u2} N _inst_2) (Units.{u3} P _inst_3) (Units.mulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2) (Units.mulOneClass.{u3} P _inst_3) (Units.map.{u2, u3} N P _inst_2 _inst_3 g) (Units.map.{u1, u2} M N _inst_1 _inst_2 f))\nbut is expected to have type\n  forall {M : Type.{u1}} {N : Type.{u2}} {P : Type.{u3}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] [_inst_3 : Monoid.{u3} P] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (g : MonoidHom.{u2, u3} N P (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} P _inst_3)), Eq.{max (succ u1) (succ u3)} (MonoidHom.{u1, u3} (Units.{u1} M _inst_1) (Units.{u3} P _inst_3) (Units.instMulOneClassUnits.{u1} M _inst_1) (Units.instMulOneClassUnits.{u3} P _inst_3)) (Units.map.{u1, u3} M P _inst_1 _inst_3 (MonoidHom.comp.{u1, u2, u3} M N P (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} P _inst_3) g f)) (MonoidHom.comp.{u1, u2, u3} (Units.{u1} M _inst_1) (Units.{u2} N _inst_2) (Units.{u3} P _inst_3) (Units.instMulOneClassUnits.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2) (Units.instMulOneClassUnits.{u3} P _inst_3) (Units.map.{u2, u3} N P _inst_2 _inst_3 g) (Units.map.{u1, u2} M N _inst_1 _inst_2 f))\nCase conversion may be inaccurate. Consider using '#align units.map_comp Units.map_compₓ'. -/\n@[simp, to_additive]\ntheorem map_comp (f : M →* N) (g : N →* P) : map (g.comp f) = (map g).comp (map f) :=\n  rfl\n#align units.map_comp Units.map_comp\n#align add_units.map_comp AddUnits.map_comp\n\nvariable (M)\n\n/- warning: units.map_id -> Units.map_id is a dubious translation:\nlean 3 declaration is\n  forall (M : Type.{u1}) [_inst_1 : Monoid.{u1} M], Eq.{succ u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) (Units.{u1} M _inst_1) (Units.mulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u1} M _inst_1)) (Units.map.{u1, u1} M M _inst_1 _inst_1 (MonoidHom.id.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (MonoidHom.id.{u1} (Units.{u1} M _inst_1) (Units.mulOneClass.{u1} M _inst_1))\nbut is expected to have type\n  forall (M : Type.{u1}) [_inst_1 : Monoid.{u1} M], Eq.{succ u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) (Units.{u1} M _inst_1) (Units.instMulOneClassUnits.{u1} M _inst_1) (Units.instMulOneClassUnits.{u1} M _inst_1)) (Units.map.{u1, u1} M M _inst_1 _inst_1 (MonoidHom.id.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (MonoidHom.id.{u1} (Units.{u1} M _inst_1) (Units.instMulOneClassUnits.{u1} M _inst_1))\nCase conversion may be inaccurate. Consider using '#align units.map_id Units.map_idₓ'. -/\n@[simp, to_additive]\ntheorem map_id : map (MonoidHom.id M) = MonoidHom.id Mˣ := by ext <;> rfl\n#align units.map_id Units.map_id\n#align add_units.map_id AddUnits.map_id\n\n/- warning: units.coe_hom -> Units.coeHom is a dubious translation:\nlean 3 declaration is\n  forall (M : Type.{u1}) [_inst_1 : Monoid.{u1} M], MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.mulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)\nbut is expected to have type\n  forall (M : Type.{u1}) [_inst_1 : Monoid.{u1} M], MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)\nCase conversion may be inaccurate. Consider using '#align units.coe_hom Units.coeHomₓ'. -/\n/-- Coercion `Mˣ → M` as a monoid homomorphism. -/\n@[to_additive \"Coercion `add_units M → M` as an add_monoid homomorphism.\"]\ndef coeHom : Mˣ →* M :=\n  ⟨coe, val_one, val_mul⟩\n#align units.coe_hom Units.coeHom\n#align add_units.coe_hom AddUnits.coeHom\n\nvariable {M}\n\n/- warning: units.coe_hom_apply -> Units.coeHom_apply is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (x : Units.{u1} M _inst_1), Eq.{succ u1} M (coeFn.{succ u1, succ u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.mulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) (fun (_x : MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.mulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) => (Units.{u1} M _inst_1) -> M) (MonoidHom.hasCoeToFun.{u1, u1} (Units.{u1} M _inst_1) M (Units.mulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) (Units.coeHom.{u1} M _inst_1) x) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) x)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (x : Units.{u1} M _inst_1), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Units.{u1} M _inst_1) => M) x) (FunLike.coe.{succ u1, succ u1, succ u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) (Units.{u1} M _inst_1) (fun (_x : Units.{u1} M _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Units.{u1} M _inst_1) => M) _x) (MulHomClass.toFunLike.{u1, u1, u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) (Units.{u1} M _inst_1) M (MulOneClass.toMul.{u1} (Units.{u1} M _inst_1) (Units.instMulOneClassUnits.{u1} M _inst_1)) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MonoidHomClass.toMulHomClass.{u1, u1, u1} (MonoidHom.{u1, u1} (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)) (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1) (MonoidHom.monoidHomClass.{u1, u1} (Units.{u1} M _inst_1) M (Units.instMulOneClassUnits.{u1} M _inst_1) (Monoid.toMulOneClass.{u1} M _inst_1)))) (Units.coeHom.{u1} M _inst_1) x) (Units.val.{u1} M _inst_1 x)\nCase conversion may be inaccurate. Consider using '#align units.coe_hom_apply Units.coeHom_applyₓ'. -/\n@[simp, to_additive]\ntheorem coeHom_apply (x : Mˣ) : coeHom M x = ↑x :=\n  rfl\n#align units.coe_hom_apply Units.coeHom_apply\n#align add_units.coe_hom_apply AddUnits.coeHom_apply\n\n/- warning: units.coe_pow -> Units.val_pow_eq_pow_val is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (u : Units.{u1} M _inst_1) (n : Nat), Eq.{succ u1} M ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) (HPow.hPow.{u1, 0, u1} (Units.{u1} M _inst_1) Nat (Units.{u1} M _inst_1) (instHPow.{u1, 0} (Units.{u1} M _inst_1) Nat (Monoid.Pow.{u1} (Units.{u1} M _inst_1) (DivInvMonoid.toMonoid.{u1} (Units.{u1} M _inst_1) (Group.toDivInvMonoid.{u1} (Units.{u1} M _inst_1) (Units.group.{u1} M _inst_1))))) u n)) (HPow.hPow.{u1, 0, u1} M Nat M (instHPow.{u1, 0} M Nat (Monoid.Pow.{u1} M _inst_1)) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) u) n)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (u : Units.{u1} M _inst_1) (n : Nat), Eq.{succ u1} M (Units.val.{u1} M _inst_1 (HPow.hPow.{u1, 0, u1} (Units.{u1} M _inst_1) Nat (Units.{u1} M _inst_1) (instHPow.{u1, 0} (Units.{u1} M _inst_1) Nat (Monoid.Pow.{u1} (Units.{u1} M _inst_1) (DivInvMonoid.toMonoid.{u1} (Units.{u1} M _inst_1) (Group.toDivInvMonoid.{u1} (Units.{u1} M _inst_1) (Units.instGroupUnits.{u1} M _inst_1))))) u n)) (HPow.hPow.{u1, 0, u1} M Nat M (instHPow.{u1, 0} M Nat (Monoid.Pow.{u1} M _inst_1)) (Units.val.{u1} M _inst_1 u) n)\nCase conversion may be inaccurate. Consider using '#align units.coe_pow Units.val_pow_eq_pow_valₓ'. -/\n@[simp, norm_cast, to_additive]\ntheorem val_pow_eq_pow_val (u : Mˣ) (n : ℕ) : ((u ^ n : Mˣ) : M) = u ^ n :=\n  (Units.coeHom M).map_pow u n\n#align units.coe_pow Units.val_pow_eq_pow_val\n#align add_units.coe_nsmul AddUnits.val_nsmul_eq_nsmul_val\n\nsection DivisionMonoid\n\nvariable [DivisionMonoid α]\n\n/- warning: units.coe_div -> Units.val_div_eq_div_val is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (u₁ : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (u₂ : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))), Eq.{succ u1} α ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (Units.hasCoe.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) (HDiv.hDiv.{u1, u1, u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (instHDiv.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (DivInvMonoid.toHasDiv.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Group.toDivInvMonoid.{u1} 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((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (Units.hasCoe.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u₂))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (u₁ : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (u₂ : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))), Eq.{succ u1} α (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) (HDiv.hDiv.{u1, u1, u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (instHDiv.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (DivInvMonoid.toDiv.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Group.toDivInvMonoid.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.instGroupUnits.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u₁ u₂)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) u₁) (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) u₂))\nCase conversion may be inaccurate. Consider using '#align units.coe_div Units.val_div_eq_div_valₓ'. -/\n@[simp, norm_cast, to_additive]\ntheorem val_div_eq_div_val : ∀ u₁ u₂ : αˣ, ↑(u₁ / u₂) = (u₁ / u₂ : α) :=\n  (Units.coeHom α).map_div\n#align units.coe_div Units.val_div_eq_div_val\n\n/- warning: units.coe_zpow -> Units.val_zpow_eq_zpow_val is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (u : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (n : Int), Eq.{succ u1} α ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (Units.hasCoe.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) (HPow.hPow.{u1, 0, u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) Int (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (instHPow.{u1, 0} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) Int (DivInvMonoid.Pow.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Group.toDivInvMonoid.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.group.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u n)) (HPow.hPow.{u1, 0, u1} α Int α (instHPow.{u1, 0} α Int (DivInvMonoid.Pow.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (Units.hasCoe.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u) n)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (u : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (n : Int), Eq.{succ u1} α (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) (HPow.hPow.{u1, 0, u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) Int (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (instHPow.{u1, 0} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) Int (DivInvMonoid.Pow.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Group.toDivInvMonoid.{u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.instGroupUnits.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u n)) (HPow.hPow.{u1, 0, u1} α Int α (instHPow.{u1, 0} α Int (DivInvMonoid.Pow.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) u) n)\nCase conversion may be inaccurate. Consider using '#align units.coe_zpow Units.val_zpow_eq_zpow_valₓ'. -/\n@[simp, norm_cast, to_additive]\ntheorem val_zpow_eq_zpow_val : ∀ (u : αˣ) (n : ℤ), ((u ^ n : αˣ) : α) = u ^ n :=\n  (Units.coeHom α).map_zpow\n#align units.coe_zpow Units.val_zpow_eq_zpow_val\n#align add_units.coe_zsmul AddUnits.val_zsmul_eq_zsmul_val\n\n/- warning: divp_eq_div -> divp_eq_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (a : α) (u : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))), Eq.{succ u1} α (divp.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) a u) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) a ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (coeBase.{succ u1, succ u1} (Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) α (Units.hasCoe.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))))) u))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_4 : DivisionMonoid.{u1} α] (a : α) (u : Units.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))), Eq.{succ u1} α (divp.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) a u) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) a (Units.val.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)) u))\nCase conversion may be inaccurate. Consider using '#align divp_eq_div divp_eq_divₓ'. -/\n@[field_simps]\ntheorem divp_eq_div (a : α) (u : αˣ) : a /ₚ u = a / u := by rw [div_eq_mul_inv, divp, u.coe_inv]\n#align divp_eq_div divp_eq_div\n\n/- warning: map_units_inv -> map_units_inv is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u2}} {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] [_inst_4 : DivisionMonoid.{u2} α] {F : Type.{u3}} [_inst_5 : MonoidHomClass.{u3, u1, u2} F M α (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} α (DivInvMonoid.toMonoid.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4)))] (f : F) (u : Units.{u1} M _inst_1), Eq.{succ u2} α (coeFn.{succ u3, max (succ u1) (succ u2)} F (fun (_x : F) => M -> α) (FunLike.hasCoeToFun.{succ u3, succ u1, succ u2} F M (fun (_x : M) => α) (MulHomClass.toFunLike.{u3, u1, u2} F M α (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toHasMul.{u2} α (Monoid.toMulOneClass.{u2} α (DivInvMonoid.toMonoid.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4)))) (MonoidHomClass.toMulHomClass.{u3, u1, u2} F M α (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} α (DivInvMonoid.toMonoid.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4))) _inst_5))) f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) (Inv.inv.{u1} (Units.{u1} M _inst_1) (Units.hasInv.{u1} M _inst_1) u))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4)) (coeFn.{succ u3, max (succ u1) (succ u2)} F (fun (_x : F) => M -> α) (FunLike.hasCoeToFun.{succ u3, succ u1, succ u2} F M (fun (_x : M) => α) (MulHomClass.toFunLike.{u3, u1, u2} F M α (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toHasMul.{u2} α (Monoid.toMulOneClass.{u2} α (DivInvMonoid.toMonoid.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4)))) (MonoidHomClass.toMulHomClass.{u3, u1, u2} F M α (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} α (DivInvMonoid.toMonoid.{u2} α (DivisionMonoid.toDivInvMonoid.{u2} α _inst_4))) _inst_5))) f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} M _inst_1) M (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} M _inst_1) M (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} M _inst_1) M (coeBase.{succ u1, succ u1} (Units.{u1} M _inst_1) M (Units.hasCoe.{u1} M _inst_1)))) u)))\nbut is expected to have type\n  forall {α : Type.{u1}} {M : Type.{u3}} [_inst_1 : Monoid.{u3} M] [_inst_4 : DivisionMonoid.{u1} α] {F : Type.{u2}} [_inst_5 : MonoidHomClass.{u2, u3, u1} F M α (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))] (f : F) (u : Units.{u3} M _inst_1), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) (Units.val.{u3} M _inst_1 (Inv.inv.{u3} (Units.{u3} M _inst_1) (Units.instInvUnits.{u3} M _inst_1) u))) (FunLike.coe.{succ u2, succ u3, succ u1} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) _x) (MulHomClass.toFunLike.{u2, u3, u1} F M α (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))) (MonoidHomClass.toMulHomClass.{u2, u3, u1} F M α (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) _inst_5)) f (Units.val.{u3} M _inst_1 (Inv.inv.{u3} (Units.{u3} M _inst_1) (Units.instInvUnits.{u3} M _inst_1) u))) (Inv.inv.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) (Units.val.{u3} M _inst_1 u)) (InvOneClass.toInv.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) (Units.val.{u3} M _inst_1 u)) (DivInvOneMonoid.toInvOneClass.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) (Units.val.{u3} M _inst_1 u)) (DivisionMonoid.toDivInvOneMonoid.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) (Units.val.{u3} M _inst_1 u)) _inst_4))) (FunLike.coe.{succ u2, succ u3, succ u1} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => α) _x) (MulHomClass.toFunLike.{u2, u3, u1} F M α (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4)))) (MonoidHomClass.toMulHomClass.{u2, u3, u1} F M α (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_4))) _inst_5)) f (Units.val.{u3} M _inst_1 u)))\nCase conversion may be inaccurate. Consider using '#align map_units_inv map_units_invₓ'. -/\n@[simp, to_additive]\ntheorem map_units_inv {F : Type _} [MonoidHomClass F M α] (f : F) (u : Units M) :\n    f ↑u⁻¹ = (f u)⁻¹ :=\n  ((f : M →* α).comp (Units.coeHom M)).map_inv u\n#align map_units_inv map_units_inv\n#align map_add_units_neg map_addUnits_neg\n\nend DivisionMonoid\n\n/- warning: units.lift_right -> Units.liftRight is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (g : M -> (Units.{u2} N _inst_2)), (forall (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (g x)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) -> (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2))\nbut is expected to have type\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (g : M -> (Units.{u2} N _inst_2)), (forall (x : M), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) (Units.val.{u2} N _inst_2 (g x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f x)) -> (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2))\nCase conversion may be inaccurate. Consider using '#align units.lift_right Units.liftRightₓ'. -/\n/-- If a map `g : M → Nˣ` agrees with a homomorphism `f : M →* N`, then\nthis map is a monoid homomorphism too. -/\n@[to_additive\n      \"If a map `g : M → add_units N` agrees with a homomorphism `f : M →+ N`, then this map\\nis an add_monoid homomorphism too.\"]\ndef liftRight (f : M →* N) (g : M → Nˣ) (h : ∀ x, ↑(g x) = f x) : M →* Nˣ\n    where\n  toFun := g\n  map_one' := Units.ext <| (h 1).symm ▸ f.map_one\n  map_mul' x y := Units.ext <| by simp only [h, coe_mul, f.map_mul]\n#align units.lift_right Units.liftRight\n#align add_units.lift_right AddUnits.liftRight\n\n/- warning: units.coe_lift_right -> Units.coe_liftRight is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] {f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)} {g : M -> (Units.{u2} N _inst_2)} (h : forall (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (g x)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N 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_inst_2)) (MonoidHomClass.toMulHomClass.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f x)\nCase conversion may be inaccurate. Consider using '#align units.coe_lift_right Units.coe_liftRightₓ'. -/\n@[simp, to_additive]\ntheorem coe_liftRight {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :\n    (liftRight f g h x : N) = f x :=\n  h x\n#align units.coe_lift_right Units.coe_liftRight\n#align add_units.coe_lift_right AddUnits.coe_liftRight\n\n/- warning: units.mul_lift_right_inv -> Units.mul_liftRight_inv is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] {f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)} {g : M -> (Units.{u2} N _inst_2)} (h : forall (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (g x)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toHasMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2))) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x) ((fun (a : Type.{u2}) 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_inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => Units.{u2} N _inst_2) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2)) M (Units.{u2} N _inst_2) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toMul.{u2} (Units.{u2} N _inst_2) (Units.instMulOneClassUnits.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2)) M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2)))) (Units.liftRight.{u1, u2} M N _inst_1 _inst_2 f g h) x)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) 1 (One.toOfNat1.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) (Monoid.toOne.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2)))\nCase conversion may be inaccurate. Consider using '#align units.mul_lift_right_inv Units.mul_liftRight_invₓ'. -/\n@[simp, to_additive]\ntheorem mul_liftRight_inv {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :\n    f x * ↑(liftRight f g h x)⁻¹ = 1 := by rw [Units.mul_inv_eq_iff_eq_mul, one_mul, coe_lift_right]\n#align units.mul_lift_right_inv Units.mul_liftRight_inv\n#align add_units.add_lift_right_neg AddUnits.add_liftRight_neg\n\n/- warning: units.lift_right_inv_mul -> Units.liftRight_inv_mul is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] {f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)} {g : M -> (Units.{u2} N _inst_2)} (h : forall (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (g x)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toHasMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (Inv.inv.{u2} (Units.{u2} N 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(Units.instMulOneClassUnits.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2)))) (Units.liftRight.{u1, u2} M N _inst_1 _inst_2 f g h) x))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f x)) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) 1 (One.toOfNat1.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) (Monoid.toOne.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2)))\nCase conversion may be inaccurate. Consider using '#align units.lift_right_inv_mul Units.liftRight_inv_mulₓ'. -/\n@[simp, to_additive]\ntheorem liftRight_inv_mul {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :\n    ↑(liftRight f g h x)⁻¹ * f x = 1 := by rw [Units.inv_mul_eq_iff_eq_mul, mul_one, coe_lift_right]\n#align units.lift_right_inv_mul Units.liftRight_inv_mul\n#align add_units.lift_right_neg_add AddUnits.liftRight_neg_add\n\nend Units\n\nnamespace MonoidHom\n\n/- warning: monoid_hom.to_hom_units -> MonoidHom.toHomUnits is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {M : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Monoid.{u2} M], (MonoidHom.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) -> (MonoidHom.{u1, u2} G (Units.{u2} M _inst_2) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Units.mulOneClass.{u2} M _inst_2))\nbut is expected to have type\n  forall {G : Type.{u1}} {M : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Monoid.{u2} M], (MonoidHom.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) -> (MonoidHom.{u1, u2} G (Units.{u2} M _inst_2) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Units.instMulOneClassUnits.{u2} M _inst_2))\nCase conversion may be inaccurate. Consider using '#align monoid_hom.to_hom_units MonoidHom.toHomUnitsₓ'. -/\n/-- If `f` is a homomorphism from a group `G` to a monoid `M`,\nthen its image lies in the units of `M`,\nand `f.to_hom_units` is the corresponding monoid homomorphism from `G` to `Mˣ`. -/\n@[to_additive\n      \"If `f` is a homomorphism from an additive group `G` to an additive monoid `M`,\\nthen its image lies in the `add_units` of `M`,\\nand `f.to_hom_units` is the corresponding homomorphism from `G` to `add_units M`.\"]\ndef toHomUnits {G M : Type _} [Group G] [Monoid M] (f : G →* M) : G →* Mˣ :=\n  Units.liftRight f\n    (fun g => ⟨f g, f g⁻¹, map_mul_eq_one f (mul_inv_self _), map_mul_eq_one f (inv_mul_self _)⟩)\n    fun g => rfl\n#align monoid_hom.to_hom_units MonoidHom.toHomUnits\n#align add_monoid_hom.to_hom_add_units AddMonoidHom.toHomAddUnits\n\n/- warning: monoid_hom.coe_to_hom_units -> MonoidHom.coe_toHomUnits is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {M : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Monoid.{u2} M] (f : MonoidHom.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) (g : G), Eq.{succ u2} M ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} M _inst_2) M (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} M _inst_2) M (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} M _inst_2) M (coeBase.{succ u2, succ u2} (Units.{u2} M _inst_2) M (Units.hasCoe.{u2} M _inst_2)))) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} G (Units.{u2} M _inst_2) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Units.mulOneClass.{u2} M _inst_2)) (fun (_x : MonoidHom.{u1, u2} G (Units.{u2} M _inst_2) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Units.mulOneClass.{u2} M _inst_2)) => G -> (Units.{u2} M _inst_2)) (MonoidHom.hasCoeToFun.{u1, u2} G (Units.{u2} M _inst_2) (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Units.mulOneClass.{u2} M _inst_2)) (MonoidHom.toHomUnits.{u1, u2} G M _inst_1 _inst_2 f) g)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) (fun (_x : MonoidHom.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) => G -> M) (MonoidHom.hasCoeToFun.{u1, u2} G M (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) (Monoid.toMulOneClass.{u2} M _inst_2)) f g)\nbut is expected to have type\n  forall {G : Type.{u2}} {M : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Monoid.{u1} M] (f : MonoidHom.{u2, u1} G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)) (g : G), Eq.{succ u1} M (Units.val.{u1} M _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} G (Units.{u1} M _inst_2) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Units.instMulOneClassUnits.{u1} M _inst_2)) G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => Units.{u1} M _inst_2) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} G (Units.{u1} M _inst_2) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Units.instMulOneClassUnits.{u1} M _inst_2)) G (Units.{u1} M _inst_2) (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} (Units.{u1} M _inst_2) (Units.instMulOneClassUnits.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} G (Units.{u1} M _inst_2) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Units.instMulOneClassUnits.{u1} M _inst_2)) G (Units.{u1} M _inst_2) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Units.instMulOneClassUnits.{u1} M _inst_2) (MonoidHom.monoidHomClass.{u2, u1} G (Units.{u1} M _inst_2) (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Units.instMulOneClassUnits.{u1} M _inst_2)))) (MonoidHom.toHomUnits.{u2, u1} G M _inst_1 _inst_2 f) g)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)) G (fun (_x : G) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : G) => M) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)) G M (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1)))) (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)) G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2) (MonoidHom.monoidHomClass.{u2, u1} G M (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))) (Monoid.toMulOneClass.{u1} M _inst_2)))) f g)\nCase conversion may be inaccurate. Consider using '#align monoid_hom.coe_to_hom_units MonoidHom.coe_toHomUnitsₓ'. -/\n@[simp, to_additive]\ntheorem coe_toHomUnits {G M : Type _} [Group G] [Monoid M] (f : G →* M) (g : G) :\n    (f.toHomUnits g : M) = f g :=\n  rfl\n#align monoid_hom.coe_to_hom_units MonoidHom.coe_toHomUnits\n#align add_monoid_hom.coe_to_hom_add_units AddMonoidHom.coe_toHomAddUnits\n\nend MonoidHom\n\nnamespace IsUnit\n\nvariable {F G α M N : Type _}\n\nsection Monoid\n\nvariable [Monoid M] [Monoid N]\n\n/- warning: is_unit.map -> IsUnit.map is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Monoid.{u2} M] [_inst_2 : Monoid.{u3} N] [_inst_3 : MonoidHomClass.{u1, u2, u3} F M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u3} N _inst_2)] (f : F) {x : M}, (IsUnit.{u2} M _inst_1 x) -> (IsUnit.{u3} N _inst_2 (coeFn.{succ u1, max (succ u2) (succ u3)} F (fun (_x : F) => M -> N) (FunLike.hasCoeToFun.{succ u1, succ u2, succ u3} F M (fun (_x : M) => N) (MulHomClass.toFunLike.{u1, u2, u3} F M N (MulOneClass.toHasMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} F M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u3} N _inst_2) _inst_3))) f x))\nbut is expected to have type\n  forall {F : Type.{u3}} {M : Type.{u2}} {N : Type.{u1}} [_inst_1 : Monoid.{u2} M] [_inst_2 : Monoid.{u1} N] [_inst_3 : MonoidHomClass.{u3, u2, u1} F M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)] (f : F) {x : M}, (IsUnit.{u2} M _inst_1 x) -> (IsUnit.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{succ u3, succ u2, succ u1} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{u3, u2, u1} F M N (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{u3, u2, u1} F M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2) _inst_3)) f x))\nCase conversion may be inaccurate. Consider using '#align is_unit.map IsUnit.mapₓ'. -/\n@[to_additive]\ntheorem map [MonoidHomClass F M N] (f : F) {x : M} (h : IsUnit x) : IsUnit (f x) := by\n  rcases h with ⟨y, rfl⟩ <;> exact (Units.map (f : M →* N) y).IsUnit\n#align is_unit.map IsUnit.map\n#align is_add_unit.map IsAddUnit.map\n\n/- warning: is_unit.of_left_inverse -> IsUnit.of_leftInverse is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1}} {G : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : Monoid.{u3} M] [_inst_2 : Monoid.{u4} N] [_inst_3 : MonoidHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2)] [_inst_4 : MonoidHomClass.{u2, u4, u3} G N M (Monoid.toMulOneClass.{u4} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1)] {f : F} {x : M} (g : G), (Function.LeftInverse.{succ u3, succ u4} M N (coeFn.{succ u2, max (succ u4) (succ u3)} G (fun (_x : G) => N -> M) (FunLike.hasCoeToFun.{succ u2, succ u4, succ u3} G N (fun (_x : N) => M) (MulHomClass.toFunLike.{u2, u4, u3} G N M (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MonoidHomClass.toMulHomClass.{u2, u4, u3} G N M (Monoid.toMulOneClass.{u4} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1) _inst_4))) g) (coeFn.{succ u1, max (succ u3) (succ u4)} F (fun (_x : F) => M -> N) (FunLike.hasCoeToFun.{succ u1, succ u3, succ u4} F M (fun (_x : M) => N) (MulHomClass.toFunLike.{u1, u3, u4} F M N (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2) _inst_3))) f)) -> (IsUnit.{u4} N _inst_2 (coeFn.{succ u1, max (succ u3) (succ u4)} F (fun (_x : F) => M -> N) (FunLike.hasCoeToFun.{succ u1, succ u3, succ u4} F M (fun (_x : M) => N) (MulHomClass.toFunLike.{u1, u3, u4} F M N (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2) _inst_3))) f x)) -> (IsUnit.{u3} M _inst_1 x)\nbut is expected to have type\n  forall {F : Type.{u4}} {G : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : Monoid.{u3} M] [_inst_2 : Monoid.{u2} N] [_inst_3 : MonoidHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)] [_inst_4 : MonoidHomClass.{u1, u2, u3} G N M (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1)] {f : F} {x : M} (g : G), (Function.LeftInverse.{succ u3, succ u2} M N (FunLike.coe.{succ u1, succ u2, succ u3} G N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : N) => M) _x) (MulHomClass.toFunLike.{u1, u2, u3} G N M (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} G N M (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1) _inst_4)) g) (FunLike.coe.{succ u4, succ u3, succ u2} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{u4, u3, u2} F M N (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) _inst_3)) f)) -> (IsUnit.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{succ u4, succ u3, succ u2} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{u4, u3, u2} F M N (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) _inst_3)) f x)) -> (IsUnit.{u3} M _inst_1 x)\nCase conversion may be inaccurate. Consider using '#align is_unit.of_left_inverse IsUnit.of_leftInverseₓ'. -/\n@[to_additive]\ntheorem of_leftInverse [MonoidHomClass F M N] [MonoidHomClass G N M] {f : F} {x : M} (g : G)\n    (hfg : Function.LeftInverse g f) (h : IsUnit (f x)) : IsUnit x := by\n  simpa only [hfg x] using h.map g\n#align is_unit.of_left_inverse IsUnit.of_leftInverse\n#align is_add_unit.of_left_inverse IsAddUnit.of_leftInverse\n\n/- warning: is_unit_map_of_left_inverse -> isUnit_map_of_leftInverse is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1}} {G : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : Monoid.{u3} M] [_inst_2 : Monoid.{u4} N] [_inst_3 : MonoidHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2)] [_inst_4 : MonoidHomClass.{u2, u4, u3} G N M (Monoid.toMulOneClass.{u4} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1)] {f : F} {x : M} (g : G), (Function.LeftInverse.{succ u3, succ u4} M N (coeFn.{succ u2, max (succ u4) (succ u3)} G (fun (_x : G) => N -> M) (FunLike.hasCoeToFun.{succ u2, succ u4, succ u3} G N (fun (_x : N) => M) (MulHomClass.toFunLike.{u2, u4, u3} G N M (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MonoidHomClass.toMulHomClass.{u2, u4, u3} G N M (Monoid.toMulOneClass.{u4} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1) _inst_4))) g) (coeFn.{succ u1, max (succ u3) (succ u4)} F (fun (_x : F) => M -> N) (FunLike.hasCoeToFun.{succ u1, succ u3, succ u4} F M (fun (_x : M) => N) (MulHomClass.toFunLike.{u1, u3, u4} F M N (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2) _inst_3))) f)) -> (Iff (IsUnit.{u4} N _inst_2 (coeFn.{succ u1, max (succ u3) (succ u4)} F (fun (_x : F) => M -> N) (FunLike.hasCoeToFun.{succ u1, succ u3, succ u4} F M (fun (_x : M) => N) (MulHomClass.toFunLike.{u1, u3, u4} F M N (MulOneClass.toHasMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N _inst_2)) (MonoidHomClass.toMulHomClass.{u1, u3, u4} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u4} N _inst_2) _inst_3))) f x)) (IsUnit.{u3} M _inst_1 x))\nbut is expected to have type\n  forall {F : Type.{u4}} {G : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : Monoid.{u3} M] [_inst_2 : Monoid.{u2} N] [_inst_3 : MonoidHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)] [_inst_4 : MonoidHomClass.{u1, u2, u3} G N M (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1)] {f : F} {x : M} (g : G), (Function.LeftInverse.{succ u3, succ u2} M N (FunLike.coe.{succ u1, succ u2, succ u3} G N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : N) => M) _x) (MulHomClass.toFunLike.{u1, u2, u3} G N M (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MonoidHomClass.toMulHomClass.{u1, u2, u3} G N M (Monoid.toMulOneClass.{u2} N _inst_2) (Monoid.toMulOneClass.{u3} M _inst_1) _inst_4)) g) (FunLike.coe.{succ u4, succ u3, succ u2} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{u4, u3, u2} F M N (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) _inst_3)) f)) -> (Iff (IsUnit.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{succ u4, succ u3, succ u2} F M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{u4, u3, u2} F M N (MulOneClass.toMul.{u3} M (Monoid.toMulOneClass.{u3} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{u4, u3, u2} F M N (Monoid.toMulOneClass.{u3} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) _inst_3)) f x)) (IsUnit.{u3} M _inst_1 x))\nCase conversion may be inaccurate. Consider using '#align is_unit_map_of_left_inverse isUnit_map_of_leftInverseₓ'. -/\n@[to_additive]\ntheorem isUnit_map_of_leftInverse [MonoidHomClass F M N] [MonoidHomClass G N M] {f : F} {x : M}\n    (g : G) (hfg : Function.LeftInverse g f) : IsUnit (f x) ↔ IsUnit x :=\n  ⟨of_leftInverse g hfg, map _⟩\n#align is_unit_map_of_left_inverse isUnit_map_of_leftInverse\n#align is_add_unit_map_of_left_inverse isAddUnit_map_of_leftInverse\n\n/- warning: is_unit.lift_right -> IsUnit.liftRight is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)), (forall (x : M), IsUnit.{u2} N _inst_2 (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) -> (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2))\nbut is expected to have type\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)), (forall (x : M), IsUnit.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u1 u2, u1, u2} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2) (MonoidHom.monoidHomClass.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)))) f x)) -> (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.instMulOneClassUnits.{u2} N _inst_2))\nCase conversion may be inaccurate. Consider using '#align is_unit.lift_right IsUnit.liftRightₓ'. -/\n/-- If a homomorphism `f : M →* N` sends each element to an `is_unit`, then it can be lifted\nto `f : M →* Nˣ`. See also `units.lift_right` for a computable version. -/\n@[to_additive\n      \"If a homomorphism `f : M →+ N` sends each element to an `is_add_unit`, then it can be\\nlifted to `f : M →+ add_units N`. See also `add_units.lift_right` for a computable version.\"]\nnoncomputable def liftRight (f : M →* N) (hf : ∀ x, IsUnit (f x)) : M →* Nˣ :=\n  Units.liftRight f (fun x => (hf x).Unit) fun x => rfl\n#align is_unit.lift_right IsUnit.liftRight\n#align is_add_unit.lift_right IsAddUnit.liftRight\n\n/- warning: is_unit.coe_lift_right -> IsUnit.coe_liftRight is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (hf : forall (x : M), IsUnit.{u2} N _inst_2 (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) => M -> (Units.{u2} N _inst_2)) (MonoidHom.hasCoeToFun.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (IsUnit.liftRight.{u1, u2} M N _inst_1 _inst_2 f hf) x)) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)\nbut is expected to have type\n  forall {M : Type.{u2}} {N : Type.{u1}} [_inst_1 : Monoid.{u2} M] [_inst_2 : Monoid.{u1} N] (f : MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) (hf : forall (x : M), IsUnit.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)))) f x)) (x : M), Eq.{succ u1} N (Units.val.{u1} N _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => Units.{u1} N _inst_2) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)) M (Units.{u1} N _inst_2) (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} (Units.{u1} N _inst_2) (Units.instMulOneClassUnits.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)) M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)))) (IsUnit.liftRight.{u2, u1} M N _inst_1 _inst_2 f hf) x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)))) f x)\nCase conversion may be inaccurate. Consider using '#align is_unit.coe_lift_right IsUnit.coe_liftRightₓ'. -/\n@[to_additive]\ntheorem coe_liftRight (f : M →* N) (hf : ∀ x, IsUnit (f x)) (x) :\n    (IsUnit.liftRight f hf x : N) = f x :=\n  rfl\n#align is_unit.coe_lift_right IsUnit.coe_liftRight\n#align is_add_unit.coe_lift_right IsAddUnit.coe_liftRight\n\n/- warning: is_unit.mul_lift_right_inv -> IsUnit.mul_liftRight_inv is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (h : forall (x : M), IsUnit.{u2} N _inst_2 (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toHasMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2))) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (Inv.inv.{u2} (Units.{u2} N _inst_2) (Units.hasInv.{u2} N _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) => M -> (Units.{u2} N _inst_2)) (MonoidHom.hasCoeToFun.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (IsUnit.liftRight.{u1, u2} M N _inst_1 _inst_2 f h) x)))) (OfNat.ofNat.{u2} N 1 (OfNat.mk.{u2} N 1 (One.one.{u2} N (MulOneClass.toHasOne.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)))))\nbut is expected to have type\n  forall {M : Type.{u2}} {N : Type.{u1}} [_inst_1 : Monoid.{u2} M] [_inst_2 : Monoid.{u1} N] (f : MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) (h : forall (x : M), IsUnit.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 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Consider using '#align is_unit.mul_lift_right_inv IsUnit.mul_liftRight_invₓ'. -/\n@[simp, to_additive]\ntheorem mul_liftRight_inv (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :\n    f x * ↑(IsUnit.liftRight f h x)⁻¹ = 1 :=\n  Units.mul_liftRight_inv (fun y => rfl) x\n#align is_unit.mul_lift_right_inv IsUnit.mul_liftRight_inv\n#align is_add_unit.add_lift_right_neg IsAddUnit.add_liftRight_neg\n\n/- warning: is_unit.lift_right_inv_mul -> IsUnit.liftRight_inv_mul is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {N : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : Monoid.{u2} N] (f : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (h : forall (x : M), IsUnit.{u2} N _inst_2 (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (x : M), Eq.{succ u2} N (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toHasMul.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Units.{u2} N _inst_2) N (HasLiftT.mk.{succ u2, succ u2} (Units.{u2} N _inst_2) N (CoeTCₓ.coe.{succ u2, succ u2} (Units.{u2} N _inst_2) N (coeBase.{succ u2, succ u2} (Units.{u2} N _inst_2) N (Units.hasCoe.{u2} N _inst_2)))) (Inv.inv.{u2} (Units.{u2} N _inst_2) (Units.hasInv.{u2} N _inst_2) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) => M -> (Units.{u2} N _inst_2)) (MonoidHom.hasCoeToFun.{u1, u2} M (Units.{u2} N _inst_2) (Monoid.toMulOneClass.{u1} M _inst_1) (Units.mulOneClass.{u2} N _inst_2)) (IsUnit.liftRight.{u1, u2} M N _inst_1 _inst_2 f h) x))) (coeFn.{max (succ u2) (succ u1), max (succ u1) (succ u2)} (MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) (fun (_x : MonoidHom.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) => M -> N) (MonoidHom.hasCoeToFun.{u1, u2} M N (Monoid.toMulOneClass.{u1} M _inst_1) (Monoid.toMulOneClass.{u2} N _inst_2)) f x)) (OfNat.ofNat.{u2} N 1 (OfNat.mk.{u2} N 1 (One.one.{u2} N (MulOneClass.toHasOne.{u2} N (Monoid.toMulOneClass.{u2} N _inst_2)))))\nbut is expected to have type\n  forall {M : Type.{u2}} {N : Type.{u1}} [_inst_1 : Monoid.{u2} M] [_inst_2 : Monoid.{u1} N] (f : MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) (h : forall (x : M), IsUnit.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)))) f x)) (x : M), Eq.{succ u1} ((fun 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u2 u1, u2, u1} (MonoidHom.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)) M (Units.{u1} N _inst_2) (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} (Units.{u1} N _inst_2) (Units.instMulOneClassUnits.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)) M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M (Units.{u1} N _inst_2) (Monoid.toMulOneClass.{u2} M _inst_1) (Units.instMulOneClassUnits.{u1} N _inst_2)))) (IsUnit.liftRight.{u2, u1} M N _inst_1 _inst_2 f h) x))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (MulOneClass.toMul.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (MulOneClass.toMul.{u1} N (Monoid.toMulOneClass.{u1} N _inst_2)) (MonoidHomClass.toMulHomClass.{max u2 u1, u2, u1} (MonoidHom.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)) M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2) (MonoidHom.monoidHomClass.{u2, u1} M N (Monoid.toMulOneClass.{u2} M _inst_1) (Monoid.toMulOneClass.{u1} N _inst_2)))) f x)) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) (Monoid.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) x) _inst_2)))\nCase conversion may be inaccurate. Consider using '#align is_unit.lift_right_inv_mul IsUnit.liftRight_inv_mulₓ'. -/\n@[simp, to_additive]\ntheorem liftRight_inv_mul (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :\n    ↑(IsUnit.liftRight f h x)⁻¹ * f x = 1 :=\n  Units.liftRight_inv_mul (fun y => rfl) x\n#align is_unit.lift_right_inv_mul IsUnit.liftRight_inv_mul\n#align is_add_unit.lift_right_neg_add IsAddUnit.liftRight_neg_add\n\nend Monoid\n\nsection DivisionMonoid\n\nvariable [DivisionMonoid α] {a b c : α}\n\n#print IsUnit.unit' /-\n/-- The element of the group of units, corresponding to an element of a monoid which is a unit. As\nopposed to `is_unit.unit`, the inverse is computable and comes from the inversion on `α`. This is\nuseful to transfer properties of inversion in `units α` to `α`. See also `to_units`. -/\n@[to_additive\n      \"The element of the additive group of additive units, corresponding to an element of\\nan additive monoid which is an additive unit. As opposed to `is_add_unit.add_unit`, the negation is\\ncomputable and comes from the negation on `α`. This is useful to transfer properties of negation in\\n`add_units α` to `α`. See also `to_add_units`.\",\n  simps]\ndef unit' (h : IsUnit a) : αˣ :=\n  ⟨a, a⁻¹, h.mul_inv_cancel, h.inv_mul_cancel⟩\n#align is_unit.unit' IsUnit.unit'\n#align is_add_unit.add_unit' IsAddUnit.addUnit'\n-/\n\n/- warning: is_unit.mul_inv_cancel_left -> IsUnit.mul_inv_cancel_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) b)) b)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a) b)) b)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_inv_cancel_left IsUnit.mul_inv_cancel_leftₓ'. -/\n@[simp, to_additive]\nprotected theorem mul_inv_cancel_left (h : IsUnit a) : ∀ b, a * (a⁻¹ * b) = b :=\n  h.unit'.mul_inv_cancel_left\n#align is_unit.mul_inv_cancel_left IsUnit.mul_inv_cancel_left\n#align is_add_unit.add_neg_cancel_left IsAddUnit.add_neg_cancel_left\n\n/- warning: is_unit.inv_mul_cancel_left -> IsUnit.inv_mul_cancel_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b)) b)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b)) b)\nCase conversion may be inaccurate. Consider using '#align is_unit.inv_mul_cancel_left IsUnit.inv_mul_cancel_leftₓ'. -/\n@[simp, to_additive]\nprotected theorem inv_mul_cancel_left (h : IsUnit a) : ∀ b, a⁻¹ * (a * b) = b :=\n  h.unit'.inv_mul_cancel_left\n#align is_unit.inv_mul_cancel_left IsUnit.inv_mul_cancel_left\n#align is_add_unit.neg_add_cancel_left IsAddUnit.neg_add_cancel_left\n\n/- warning: is_unit.mul_inv_cancel_right -> IsUnit.mul_inv_cancel_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b)) a)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b)) a)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_inv_cancel_right IsUnit.mul_inv_cancel_rightₓ'. -/\n@[simp, to_additive]\nprotected theorem mul_inv_cancel_right (h : IsUnit b) (a : α) : a * b * b⁻¹ = a :=\n  h.unit'.mul_inv_cancel_right _\n#align is_unit.mul_inv_cancel_right IsUnit.mul_inv_cancel_right\n#align is_add_unit.add_neg_cancel_right IsAddUnit.add_neg_cancel_right\n\n/- warning: is_unit.inv_mul_cancel_right -> IsUnit.inv_mul_cancel_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b)) b) a)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b)) b) a)\nCase conversion may be inaccurate. Consider using '#align is_unit.inv_mul_cancel_right IsUnit.inv_mul_cancel_rightₓ'. -/\n@[simp, to_additive]\nprotected theorem inv_mul_cancel_right (h : IsUnit b) (a : α) : a * b⁻¹ * b = a :=\n  h.unit'.inv_mul_cancel_right _\n#align is_unit.inv_mul_cancel_right IsUnit.inv_mul_cancel_right\n#align is_add_unit.neg_add_cancel_right IsAddUnit.neg_add_cancel_right\n\n/- warning: is_unit.div_self -> IsUnit.div_self is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a a) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a a) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_self IsUnit.div_selfₓ'. -/\n@[to_additive]\nprotected theorem div_self (h : IsUnit a) : a / a = 1 := by rw [div_eq_mul_inv, h.mul_inv_cancel]\n#align is_unit.div_self IsUnit.div_self\n#align is_add_unit.sub_self IsAddUnit.sub_self\n\n/- warning: is_unit.eq_mul_inv_iff_mul_eq -> IsUnit.eq_mul_inv_iff_mul_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c))) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) c))) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b))\nCase conversion may be inaccurate. Consider using '#align is_unit.eq_mul_inv_iff_mul_eq IsUnit.eq_mul_inv_iff_mul_eqₓ'. -/\n@[to_additive]\nprotected theorem eq_mul_inv_iff_mul_eq (h : IsUnit c) : a = b * c⁻¹ ↔ a * c = b :=\n  h.unit'.eq_mul_inv_iff_mul_eq\n#align is_unit.eq_mul_inv_iff_mul_eq IsUnit.eq_mul_inv_iff_mul_eq\n#align is_add_unit.eq_add_neg_iff_add_eq IsAddUnit.eq_add_neg_iff_add_eq\n\n/- warning: is_unit.eq_inv_mul_iff_mul_eq -> IsUnit.eq_inv_mul_iff_mul_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) c)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b a) c))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b) c)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b a) c))\nCase conversion may be inaccurate. Consider using '#align is_unit.eq_inv_mul_iff_mul_eq IsUnit.eq_inv_mul_iff_mul_eqₓ'. -/\n@[to_additive]\nprotected theorem eq_inv_mul_iff_mul_eq (h : IsUnit b) : a = b⁻¹ * c ↔ b * a = c :=\n  h.unit'.eq_inv_mul_iff_mul_eq\n#align is_unit.eq_inv_mul_iff_mul_eq IsUnit.eq_inv_mul_iff_mul_eq\n#align is_add_unit.eq_neg_add_iff_add_eq IsAddUnit.eq_neg_add_iff_add_eq\n\n/- warning: is_unit.inv_mul_eq_iff_eq_mul -> IsUnit.inv_mul_eq_iff_eq_mul is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) b) c) (Eq.{succ u1} α b (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a) b) c) (Eq.{succ u1} α b (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c)))\nCase conversion may be inaccurate. Consider using '#align is_unit.inv_mul_eq_iff_eq_mul IsUnit.inv_mul_eq_iff_eq_mulₓ'. -/\n@[to_additive]\nprotected theorem inv_mul_eq_iff_eq_mul (h : IsUnit a) : a⁻¹ * b = c ↔ b = a * c :=\n  h.unit'.inv_mul_eq_iff_eq_mul\n#align is_unit.inv_mul_eq_iff_eq_mul IsUnit.inv_mul_eq_iff_eq_mul\n#align is_add_unit.neg_add_eq_iff_eq_add IsAddUnit.neg_add_eq_iff_eq_add\n\n/- warning: is_unit.mul_inv_eq_iff_eq_mul -> IsUnit.mul_inv_eq_iff_eq_mul is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b)) c) (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b)) c) (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_inv_eq_iff_eq_mul IsUnit.mul_inv_eq_iff_eq_mulₓ'. -/\n@[to_additive]\nprotected theorem mul_inv_eq_iff_eq_mul (h : IsUnit b) : a * b⁻¹ = c ↔ a = c * b :=\n  h.unit'.mul_inv_eq_iff_eq_mul\n#align is_unit.mul_inv_eq_iff_eq_mul IsUnit.mul_inv_eq_iff_eq_mul\n#align is_add_unit.add_neg_eq_iff_eq_add IsAddUnit.add_neg_eq_iff_eq_add\n\n/- warning: is_unit.mul_inv_eq_one -> IsUnit.mul_inv_eq_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b)) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)))))))) (Eq.{succ u1} α a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1)))))) (Eq.{succ u1} α a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_inv_eq_one IsUnit.mul_inv_eq_oneₓ'. -/\n@[to_additive]\nprotected theorem mul_inv_eq_one (h : IsUnit b) : a * b⁻¹ = 1 ↔ a = b :=\n  @Units.mul_inv_eq_one _ _ h.unit' _\n#align is_unit.mul_inv_eq_one IsUnit.mul_inv_eq_one\n#align is_add_unit.add_neg_eq_zero IsAddUnit.add_neg_eq_zero\n\n/- warning: is_unit.inv_mul_eq_one -> IsUnit.inv_mul_eq_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) b) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)))))))) (Eq.{succ u1} α a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a) b) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1)))))) (Eq.{succ u1} α a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.inv_mul_eq_one IsUnit.inv_mul_eq_oneₓ'. -/\n@[to_additive]\nprotected theorem inv_mul_eq_one (h : IsUnit a) : a⁻¹ * b = 1 ↔ a = b :=\n  @Units.inv_mul_eq_one _ _ h.unit' _\n#align is_unit.inv_mul_eq_one IsUnit.inv_mul_eq_one\n#align is_add_unit.neg_add_eq_zero IsAddUnit.neg_add_eq_zero\n\n/- warning: is_unit.mul_eq_one_iff_eq_inv -> IsUnit.mul_eq_one_iff_eq_inv is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)))))))) (Eq.{succ u1} α a (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1)))))) (Eq.{succ u1} α a (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) b)))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_eq_one_iff_eq_inv IsUnit.mul_eq_one_iff_eq_invₓ'. -/\n@[to_additive]\nprotected theorem mul_eq_one_iff_eq_inv (h : IsUnit b) : a * b = 1 ↔ a = b⁻¹ :=\n  @Units.mul_eq_one_iff_eq_inv _ _ h.unit' _\n#align is_unit.mul_eq_one_iff_eq_inv IsUnit.mul_eq_one_iff_eq_inv\n#align is_add_unit.add_eq_zero_iff_eq_neg IsAddUnit.add_eq_zero_iff_eq_neg\n\n/- warning: is_unit.mul_eq_one_iff_inv_eq -> IsUnit.mul_eq_one_iff_inv_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)))))))) (Eq.{succ u1} α (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Iff (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1)))))) (Eq.{succ u1} α (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a) b))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_eq_one_iff_inv_eq IsUnit.mul_eq_one_iff_inv_eqₓ'. -/\n@[to_additive]\nprotected theorem mul_eq_one_iff_inv_eq (h : IsUnit a) : a * b = 1 ↔ a⁻¹ = b :=\n  @Units.mul_eq_one_iff_inv_eq _ _ h.unit' _\n#align is_unit.mul_eq_one_iff_inv_eq IsUnit.mul_eq_one_iff_inv_eq\n#align is_add_unit.add_eq_zero_iff_neg_eq IsAddUnit.add_eq_zero_iff_neg_eq\n\n/- warning: is_unit.div_mul_cancel -> IsUnit.div_mul_cancel is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) b) a)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) b) a)\nCase conversion may be inaccurate. Consider using '#align is_unit.div_mul_cancel IsUnit.div_mul_cancelₓ'. -/\n@[simp, to_additive]\nprotected theorem div_mul_cancel (h : IsUnit b) (a : α) : a / b * b = a := by\n  rw [div_eq_mul_inv, h.inv_mul_cancel_right]\n#align is_unit.div_mul_cancel IsUnit.div_mul_cancel\n#align is_add_unit.sub_add_cancel IsAddUnit.sub_add_cancel\n\n/- warning: is_unit.mul_div_cancel -> IsUnit.mul_div_cancel is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) b) a)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (forall (a : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) b) a)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_div_cancel IsUnit.mul_div_cancelₓ'. -/\n@[simp, to_additive]\nprotected theorem mul_div_cancel (h : IsUnit b) (a : α) : a * b / b = a := by\n  rw [div_eq_mul_inv, h.mul_inv_cancel_right]\n#align is_unit.mul_div_cancel IsUnit.mul_div_cancel\n#align is_add_unit.add_sub_cancel IsAddUnit.add_sub_cancel\n\n/- warning: is_unit.mul_one_div_cancel -> IsUnit.mul_one_div_cancel is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))) a)) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))) a)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_one_div_cancel IsUnit.mul_one_div_cancelₓ'. -/\n@[to_additive]\nprotected theorem mul_one_div_cancel (h : IsUnit a) : a * (1 / a) = 1 := by simp [h]\n#align is_unit.mul_one_div_cancel IsUnit.mul_one_div_cancel\n#align is_add_unit.add_zero_sub_cancel IsAddUnit.add_zero_sub_cancel\n\n/- warning: is_unit.one_div_mul_cancel -> IsUnit.one_div_mul_cancel is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))) a) a) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))) a) a) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align is_unit.one_div_mul_cancel IsUnit.one_div_mul_cancelₓ'. -/\n@[to_additive]\nprotected theorem one_div_mul_cancel (h : IsUnit a) : 1 / a * a = 1 := by simp [h]\n#align is_unit.one_div_mul_cancel IsUnit.one_div_mul_cancel\n#align is_add_unit.zero_sub_add_cancel IsAddUnit.zero_sub_add_cancel\n\n/- warning: is_unit.inv -> IsUnit.inv is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))) a))\nCase conversion may be inaccurate. Consider using '#align is_unit.inv IsUnit.invₓ'. -/\n@[to_additive]\ntheorem inv : IsUnit a → IsUnit a⁻¹ := by\n  rintro ⟨u, rfl⟩\n  rw [← Units.val_inv_eq_inv_val]\n  exact Units.isUnit _\n#align is_unit.inv IsUnit.inv\n#align is_add_unit.neg IsAddUnit.neg\n\n/- warning: is_unit.div -> IsUnit.div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) a) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.div IsUnit.divₓ'. -/\n@[to_additive]\ntheorem div (ha : IsUnit a) (hb : IsUnit b) : IsUnit (a / b) :=\n  by\n  rw [div_eq_mul_inv]\n  exact ha.mul hb.inv\n#align is_unit.div IsUnit.div\n#align is_add_unit.sub IsAddUnit.sub\n\n/- warning: is_unit.div_left_inj -> IsUnit.div_left_inj is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a c) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c)) (Eq.{succ u1} α a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a c) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c)) (Eq.{succ u1} α a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_left_inj IsUnit.div_left_injₓ'. -/\n@[to_additive]\nprotected theorem div_left_inj (h : IsUnit c) : a / c = b / c ↔ a = b :=\n  by\n  simp_rw [div_eq_mul_inv]\n  exact Units.mul_left_inj h.inv.unit'\n#align is_unit.div_left_inj IsUnit.div_left_inj\n#align is_add_unit.sub_left_inj IsAddUnit.sub_left_inj\n\n/- warning: is_unit.div_eq_iff -> IsUnit.div_eq_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) c) (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) c) (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_eq_iff IsUnit.div_eq_iffₓ'. -/\n@[to_additive]\nprotected theorem div_eq_iff (h : IsUnit b) : a / b = c ↔ a = c * b := by\n  rw [div_eq_mul_inv, h.mul_inv_eq_iff_eq_mul]\n#align is_unit.div_eq_iff IsUnit.div_eq_iff\n#align is_add_unit.sub_eq_iff IsAddUnit.sub_eq_iff\n\n/- warning: is_unit.eq_div_iff -> IsUnit.eq_div_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Iff (Eq.{succ u1} α a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b))\nCase conversion may be inaccurate. Consider using '#align is_unit.eq_div_iff IsUnit.eq_div_iffₓ'. -/\n@[to_additive]\nprotected theorem eq_div_iff (h : IsUnit c) : a = b / c ↔ a * c = b := by\n  rw [div_eq_mul_inv, h.eq_mul_inv_iff_mul_eq]\n#align is_unit.eq_div_iff IsUnit.eq_div_iff\n#align is_add_unit.eq_sub_iff IsAddUnit.eq_sub_iff\n\n/- warning: is_unit.div_eq_of_eq_mul -> IsUnit.div_eq_of_eq_mul is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) c)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) c b)) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) c)\nCase conversion may be inaccurate. Consider using '#align is_unit.div_eq_of_eq_mul IsUnit.div_eq_of_eq_mulₓ'. -/\n@[to_additive]\nprotected theorem div_eq_of_eq_mul (h : IsUnit b) : a = c * b → a / b = c :=\n  h.div_eq_iff.2\n#align is_unit.div_eq_of_eq_mul IsUnit.div_eq_of_eq_mul\n#align is_add_unit.sub_eq_of_eq_add IsAddUnit.sub_eq_of_eq_add\n\n/- warning: is_unit.eq_div_of_mul_eq -> IsUnit.eq_div_of_mul_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b) -> (Eq.{succ u1} α a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α} {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) b) -> (Eq.{succ u1} α a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b c))\nCase conversion may be inaccurate. Consider using '#align is_unit.eq_div_of_mul_eq IsUnit.eq_div_of_mul_eqₓ'. -/\n@[to_additive]\nprotected theorem eq_div_of_mul_eq (h : IsUnit c) : a * c = b → a = b / c :=\n  h.eq_div_iff.2\n#align is_unit.eq_div_of_mul_eq IsUnit.eq_div_of_mul_eq\n#align is_add_unit.eq_sub_of_add_eq IsAddUnit.eq_sub_of_add_eq\n\n/- warning: is_unit.div_eq_one_iff_eq -> IsUnit.div_eq_one_iff_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)))))))) (Eq.{succ u1} α a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1)))))) (Eq.{succ u1} α a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_eq_one_iff_eq IsUnit.div_eq_one_iff_eqₓ'. -/\n@[to_additive]\nprotected theorem div_eq_one_iff_eq (h : IsUnit b) : a / b = 1 ↔ a = b :=\n  ⟨eq_of_div_eq_one, fun hab => hab.symm ▸ h.div_self⟩\n#align is_unit.div_eq_one_iff_eq IsUnit.div_eq_one_iff_eq\n#align is_add_unit.sub_eq_zero_iff_eq IsAddUnit.sub_eq_zero_iff_eq\n\n/- warning: is_unit.div_mul_left -> IsUnit.div_mul_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))) a))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) b (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))) a))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_mul_left IsUnit.div_mul_leftₓ'. -/\n@[to_additive]\nprotected theorem div_mul_left (h : IsUnit b) : b / (a * b) = 1 / a := by\n  rw [div_eq_mul_inv, mul_inv_rev, h.mul_inv_cancel_left, one_div]\n#align is_unit.div_mul_left IsUnit.div_mul_left\n#align is_add_unit.sub_add_left IsAddUnit.sub_add_left\n\n/- warning: is_unit.mul_div_mul_right -> IsUnit.mul_div_mul_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (forall (a : α) (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b c)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) c) -> (forall (a : α) (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a c) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) b c)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_div_mul_right IsUnit.mul_div_mul_rightₓ'. -/\n@[to_additive]\nprotected theorem mul_div_mul_right (h : IsUnit c) (a b : α) : a * c / (b * c) = a / b := by\n  simp only [div_eq_mul_inv, mul_inv_rev, mul_assoc, h.mul_inv_cancel_left]\n#align is_unit.mul_div_mul_right IsUnit.mul_div_mul_right\n#align is_add_unit.add_sub_add_right IsAddUnit.add_sub_add_right\n\n/- warning: is_unit.mul_mul_div -> IsUnit.mul_mul_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α} (a : α), (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))))) b)) a)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} α] {b : α} (a : α), (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1)) b) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α _inst_1))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α _inst_1))))) b)) a)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_mul_div IsUnit.mul_mul_divₓ'. -/\n@[to_additive]\nprotected theorem mul_mul_div (a : α) (h : IsUnit b) : a * b * (1 / b) = a := by simp [h]\n#align is_unit.mul_mul_div IsUnit.mul_mul_div\n#align is_add_unit.add_add_sub IsAddUnit.add_add_sub\n\nend DivisionMonoid\n\nsection DivisionCommMonoid\n\nvariable [DivisionCommMonoid α] {a b c d : α}\n\n/- warning: is_unit.div_mul_right -> IsUnit.div_mul_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (MulOneClass.toHasOne.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))))) b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (InvOneClass.toOne.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) b))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_mul_right IsUnit.div_mul_rightₓ'. -/\n@[to_additive]\nprotected theorem div_mul_right (h : IsUnit a) (b : α) : a / (a * b) = 1 / b := by\n  rw [mul_comm, h.div_mul_left]\n#align is_unit.div_mul_right IsUnit.div_mul_right\n#align is_add_unit.sub_add_right IsAddUnit.sub_add_right\n\n/- warning: is_unit.mul_div_cancel_left -> IsUnit.mul_div_cancel_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a b) a) b)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a b) a) b)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_div_cancel_left IsUnit.mul_div_cancel_leftₓ'. -/\n@[to_additive]\nprotected theorem mul_div_cancel_left (h : IsUnit a) (b : α) : a * b / a = b := by\n  rw [mul_comm, h.mul_div_cancel]\n#align is_unit.mul_div_cancel_left IsUnit.mul_div_cancel_left\n#align is_add_unit.add_sub_cancel_left IsAddUnit.add_sub_cancel_left\n\n/- warning: is_unit.mul_div_cancel' -> IsUnit.mul_div_cancel' is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) b a)) b)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (forall (b : α), Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) b a)) b)\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_div_cancel' IsUnit.mul_div_cancel'ₓ'. -/\n@[to_additive]\nprotected theorem mul_div_cancel' (h : IsUnit a) (b : α) : a * (b / a) = b := by\n  rw [mul_comm, h.div_mul_cancel]\n#align is_unit.mul_div_cancel' IsUnit.mul_div_cancel'\n#align is_add_unit.add_sub_cancel' IsAddUnit.add_sub_cancel'\n\n/- warning: is_unit.mul_div_mul_left -> IsUnit.mul_div_mul_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) c) -> (forall (a : α) (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {c : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) c) -> (forall (a : α) (b : α), Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c a) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_div_mul_left IsUnit.mul_div_mul_leftₓ'. -/\n@[to_additive]\nprotected theorem mul_div_mul_left (h : IsUnit c) (a b : α) : c * a / (c * b) = a / b := by\n  rw [mul_comm c, mul_comm c, h.mul_div_mul_right]\n#align is_unit.mul_div_mul_left IsUnit.mul_div_mul_left\n#align is_add_unit.add_sub_add_left IsAddUnit.add_sub_add_left\n\n/- warning: is_unit.mul_eq_mul_of_div_eq_div -> IsUnit.mul_eq_mul_of_div_eq_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {b : α} {d : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) d) -> (forall (a : α) (c : α), (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) c d)) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {b : α} {d : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) d) -> (forall (a : α) (c : α), (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) c d)) -> (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)))\nCase conversion may be inaccurate. Consider using '#align is_unit.mul_eq_mul_of_div_eq_div IsUnit.mul_eq_mul_of_div_eq_divₓ'. -/\n@[to_additive]\nprotected theorem mul_eq_mul_of_div_eq_div (hb : IsUnit b) (hd : IsUnit d) (a c : α)\n    (h : a / b = c / d) : a * d = c * b := by\n  rw [← mul_one a, ← hb.div_self, ← mul_comm_div, h, div_mul_eq_mul_div, hd.div_mul_cancel]\n#align is_unit.mul_eq_mul_of_div_eq_div IsUnit.mul_eq_mul_of_div_eq_div\n#align is_add_unit.add_eq_add_of_sub_eq_sub IsAddUnit.add_eq_add_of_sub_eq_sub\n\n/- warning: is_unit.div_eq_div_iff -> IsUnit.div_eq_div_iff is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α} {c : α} {d : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) d) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) c d)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toHasMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α} {c : α} {d : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) b) -> (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) d) -> (Iff (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) c d)) (Eq.{succ u1} α (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} α α α (instHMul.{u1} α (MulOneClass.toMul.{u1} α (Monoid.toMulOneClass.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))))) c b)))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_eq_div_iff IsUnit.div_eq_div_iffₓ'. -/\n@[to_additive]\nprotected theorem div_eq_div_iff (hb : IsUnit b) (hd : IsUnit d) : a / b = c / d ↔ a * d = c * b :=\n  by\n  rw [← (hb.mul hd).mul_left_inj, ← mul_assoc, hb.div_mul_cancel, ← mul_assoc, mul_right_comm,\n    hd.div_mul_cancel]\n#align is_unit.div_eq_div_iff IsUnit.div_eq_div_iff\n#align is_add_unit.sub_eq_sub_iff IsAddUnit.sub_eq_sub_iff\n\n/- warning: is_unit.div_div_cancel -> IsUnit.div_div_cancel is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b)) b)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b)) b)\nCase conversion may be inaccurate. Consider using '#align is_unit.div_div_cancel IsUnit.div_div_cancelₓ'. -/\n@[to_additive]\nprotected theorem div_div_cancel (h : IsUnit a) : a / (a / b) = b := by\n  rw [div_div_eq_mul_div, h.mul_div_cancel_left]\n#align is_unit.div_div_cancel IsUnit.div_div_cancel\n#align is_add_unit.sub_sub_cancel IsAddUnit.sub_sub_cancel\n\n/- warning: is_unit.div_div_cancel_left -> IsUnit.div_div_cancel_left is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) a) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) b))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionCommMonoid.{u1} α] {a : α} {b : α}, (IsUnit.{u1} α (DivInvMonoid.toMonoid.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1))) a) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toDiv.{u1} α (DivisionMonoid.toDivInvMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) a b) a) (Inv.inv.{u1} α (InvOneClass.toInv.{u1} α (DivInvOneMonoid.toInvOneClass.{u1} α (DivisionMonoid.toDivInvOneMonoid.{u1} α (DivisionCommMonoid.toDivisionMonoid.{u1} α _inst_1)))) b))\nCase conversion may be inaccurate. Consider using '#align is_unit.div_div_cancel_left IsUnit.div_div_cancel_leftₓ'. -/\n@[to_additive]\nprotected theorem div_div_cancel_left (h : IsUnit a) : a / b / a = b⁻¹ := by\n  rw [div_eq_mul_inv, div_eq_mul_inv, mul_right_comm, h.mul_inv_cancel, one_mul]\n#align is_unit.div_div_cancel_left IsUnit.div_div_cancel_left\n#align is_add_unit.sub_sub_cancel_left IsAddUnit.sub_sub_cancel_left\n\nend DivisionCommMonoid\n\nend IsUnit\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Hom/Units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.33791031617760053}}
{"text": "import category_theory.adjunction.limits\nimport category_theory.closed.cartesian\nimport category_theory.conj\n\nuniverses v u₁ u₂\n\nnamespace category_theory\nopen category limits\nvariables {C : Type u₁} [category.{v} C] [has_finite_products C] [cartesian_closed C]\nvariables {D : Type u₂} [category.{v} D] [has_finite_products D] [cartesian_closed D]\nvariables (F : C ⥤ D)\n\nclass cartesian_closed_functor :=\n[preserves_bin_prods : preserves_limits_of_shape (discrete walking_pair) F]\n(comparison_iso : ∀ A B, is_iso (exp_comparison F A B))\n\nattribute [instance] cartesian_closed_functor.comparison_iso\n\nvariables {F} {L : D ⥤ C}\n\nnoncomputable def frobenius_map (A : C) (B : D) (adj : L ⊣ F) : L.obj (F.obj A ⨯ B) ⟶ A ⨯ L.obj B :=\nprod_comparison _ _ _ ≫ limits.prod.map (adj.counit.app A) (𝟙 _)\n\n@[reassoc]\nlemma frob_naturality {A A' : C} {B B' : D} (adj : L ⊣ F) (f : A ⟶ A') (g : B ⟶ B') :\n  frobenius_map A B adj ≫ limits.prod.map f (L.map g) = L.map (limits.prod.map (F.map f) g) ≫ frobenius_map A' B' adj :=\nbegin\n  rw [frobenius_map, frobenius_map, assoc, prod_comparison_natural_assoc],\n  apply prod.hom_ext,\n  simp only [assoc, limits.prod.map_fst, limits.prod.map_fst_assoc, adjunction.counit_naturality],\n  simp only [assoc, limits.prod.map_snd, limits.prod.map_snd_assoc, comp_id, id_comp],\nend\n\n-- @[reassoc]\n-- lemma frob_inv_naturality_left {A A' : C} {B : D} (adj : L ⊣ F) (f : A ⟶ A')\n--   [is_iso (frobenius_map A B adj)]\n--   [is_iso (frobenius_map A' B adj)] :\n-- inv (frobenius_map A B adj) ≫ L.map (limits.prod.map (F.map f) (𝟙 _)) =\n-- limits.prod.map f (𝟙 _) ≫ inv (frobenius_map A' B adj) :=\n-- begin\n--   erw [(as_iso]\n--   -- erw [(as_iso (frobenius_map A B adj)).inv_comp_eq, assoc, ← prod_map_comp_id_assoc, ← assoc,\n--   --      (as_iso (frobenius_map A' B adj)).eq_comp_inv, prod_comparison_natural_assoc, L.map_id,\n--   --      ← prod_map_comp_id, adj.counit.naturality], refl,\n-- end\n\n@[reassoc]\nlemma frob_inv_naturality {A A' : C} {B B' : D} (adj : L ⊣ F) (f : A ⟶ A') (g : B ⟶ B')\n  [is_iso (frobenius_map A B adj)]\n  [is_iso (frobenius_map A' B' adj)] :\ninv (frobenius_map A B adj) ≫ L.map (limits.prod.map (F.map f) g) =\nlimits.prod.map f (L.map g) ≫ inv (frobenius_map A' B' adj) :=\nbegin\n  erw [(as_iso (frobenius_map A B adj)).inv_comp_eq],\n  rw ← assoc,\n  erw [(as_iso (frobenius_map A' B' adj)).eq_comp_inv],\n  symmetry,\n  apply frob_naturality,\nend\n\nvariables (adj : L ⊣ F) [∀ A B, is_iso (frobenius_map A B adj)]\n\nnoncomputable def biject (adj : L ⊣ F) [∀ A B, is_iso (frobenius_map A B adj)]\n  {A B : C} {c : D} : (c ⟶ F.obj (B ^^ A)) ≃ (c ⟶ F.obj B ^^ F.obj A) :=\ncalc (c ⟶ F.obj (B ^^ A)) ≃ (L.obj c ⟶ B ^^ A) : (adj.hom_equiv _ _).symm\n     ... ≃ (A ⨯ L.obj c ⟶ B) : ((exp.adjunction A).hom_equiv _ _).symm\n     ... ≃ (L.obj (F.obj A ⨯ c) ⟶ B) : iso.hom_congr (as_iso (frobenius_map _ _ adj)).symm (iso.refl _)\n     ... ≃ (F.obj A ⨯ c ⟶ F.obj B) : adj.hom_equiv _ _\n     ... ≃ (c ⟶ F.obj B ^^ F.obj A) : ((exp.adjunction _).hom_equiv _ _)\n\n-- tidy this up\nlemma biject_id {A B : C} [preserves_limits_of_shape (discrete walking_pair) F] :\n  biject adj (𝟙 _) = exp_comparison F A B :=\nbegin\n  dsimp [biject, iso.hom_congr, frobenius_map, exp_comparison],\n  rw [comp_id],\n  change cartesian_closed.curry _ = _,\n  congr' 1,\n  erw (as_iso (prod_comparison F A ((exp A).obj B))).eq_inv_comp,\n  rw adj.hom_equiv_unit,\n  erw adj.unit.naturality_assoc,\n  dsimp [prod_comparison],\n  erw ← F.map_comp,\n  rw adjunction.hom_equiv_counit,\n  rw adjunction.hom_equiv_counit,\n  rw L.map_id,\n  rw id_comp,\n  dsimp,\n  rw prod.lift_map,\n  rw prod.lift_map_assoc,\n  rw comp_id,\n  rw comp_id,\n  simp only [functor.map_comp],\n  rw ← F.map_comp,\n  rw ← F.map_comp,\n  rw ← adj.hom_equiv_unit,\n  rw adj.hom_equiv_apply_eq,\n  rw adj.hom_equiv_counit,\n  erw adj.counit.naturality,\n  rw ← assoc,\n  congr' 1,\n  apply prod.hom_ext; simp [← functor.map_comp_assoc],\nend\n\nlemma biject_id' {A B : C} [preserves_limits_of_shape (discrete walking_pair) F] :\n  𝟙 (F.obj ((exp A).obj B)) = (biject adj).symm (exp_comparison F A B) :=\nbegin\n  rw equiv.eq_symm_apply,\n  rw biject_id,\nend\n\nlemma biject_natural\n  {A B B' : C} {c c' : D} (f : c' ⟶ c) (g : B ⟶ B') (q : c ⟶ F.obj (B ^^ A)) :\nf ≫ biject adj q ≫ (exp _).map (F.map g) = biject adj (f ≫ q ≫ F.map ((exp _).map g)) :=\nbegin\n  dsimp [biject, iso.hom_congr],\n  rw [comp_id, comp_id],\n  rw ← adjunction.hom_equiv_naturality_right,\n  rw ← adjunction.hom_equiv_naturality_left,\n  erw ← adjunction.hom_equiv_naturality_right,\n  rw ← adjunction.hom_equiv_naturality_left,\n  rw assoc,\n  rw adj.hom_equiv_naturality_left_symm,\n  rw adjunction.hom_equiv_naturality_left_symm,\n  erw frob_naturality_assoc,\n  rw adj.hom_equiv_naturality_right_symm,\n  rw adjunction.hom_equiv_naturality_right_symm,\n  congr' 4,\n  dsimp, simp,\nend\n\nlemma biject_natural_left\n  {A B : C} {c c' : D} (f : c' ⟶ c) (q : c ⟶ F.obj (B ^^ A)) :\nf ≫ biject adj q = biject adj (f ≫ q) :=\nby simpa using biject_natural adj f (𝟙 _) q\n\nlemma biject_natural_right\n  {A B B' : C} {c : D} (g : B ⟶ B') (q : c ⟶ F.obj (B ^^ A)) :\nbiject adj q ≫ (exp _).map (F.map g) = biject adj (q ≫ F.map ((exp _).map g)) :=\nby simpa using biject_natural adj (𝟙 _) g q\n\nnoncomputable def cartesian_closed_of_frobenius_iso : cartesian_closed_functor F :=\n{ preserves_bin_prods :=\n  begin\n    letI := adj.right_adjoint_preserves_limits,\n    apply_instance,\n  end,\n  comparison_iso := λ A B,\n  { inv := (biject adj).symm (𝟙 _),\n    hom_inv_id' :=\n    begin\n      rw ← (biject adj).apply_eq_iff_eq,\n      rw biject_id _,\n      rw ← biject_natural_left,\n      rw equiv.apply_symm_apply,\n      rw comp_id,\n    end,\n    inv_hom_id' :=\n    begin\n      letI := adj.right_adjoint_preserves_limits,\n      rw ← biject_id adj,\n      rw biject_natural_left,\n      rw comp_id,\n      rw equiv.apply_symm_apply,\n    end } }\n\n-- /--\n-- The exponential comparison map.\n-- `F` is a cartesian closed functor if this is an iso for all `A,B`.\n-- -/\n-- def exp_comparison (A B : C) :\n--   F.obj (A ⟹ B) ⟶ F.obj A ⟹ F.obj B :=\n-- curry (inv (prod_comparison F A _) ≫ F.map ((ev _).app _))\n\n-- /-- The exponential comparison map is natural in its left argument. -/\n-- lemma exp_comparison_natural_left (A A' B : C) (f : A' ⟶ A) :\n--   exp_comparison F A B ≫ pre (F.obj B) (F.map f) = F.map (pre B f) ≫ exp_comparison F A' B :=\n-- begin\n--   rw [exp_comparison, exp_comparison, ← curry_natural_left, eq_curry_iff, uncurry_natural_left,\n--        pre, uncurry_curry, prod_map_map_assoc, curry_eq, prod_map_id_comp, assoc],\n--   erw [(ev _).naturality, ev_coev_assoc, ← F.map_id, ← prod_comparison_inv_natural_assoc,\n--        ← F.map_id, ← prod_comparison_inv_natural_assoc, ← F.map_comp, ← F.map_comp, pre, curry_eq,\n--        prod_map_id_comp, assoc, (ev _).naturality, ev_coev_assoc], refl,\n-- end\n\n-- /-- The exponential comparison map is natural in its right argument. -/\n-- lemma exp_comparison_natural_right (A B B' : C) (f : B ⟶ B') :\n--   exp_comparison F A B ≫ (exp (F.obj A)).map (F.map f) =\n--     F.map ((exp A).map f) ≫ exp_comparison F A B' :=\n-- by\n--   erw [exp_comparison, ← curry_natural_right, curry_eq_iff, exp_comparison, uncurry_natural_left,\n--        uncurry_curry, assoc, ← F.map_comp, ← (ev _).naturality, F.map_comp,\n--        prod_comparison_inv_natural_assoc, F.map_id]\n\n-- -- TODO: If F has a left adjoint L, then F is cartesian closed if and only if\n-- -- L (B ⨯ F A) ⟶ L B ⨯ L F A ⟶ L B ⨯ A\n-- -- is an iso for all A ∈ D, B ∈ C.\n-- -- Corollary: If F has a left adjoint L which preserves finite products, F is cartesian closed iff\n-- -- F is full and faithful.\nend category_theory", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/cartesian_closed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.33790337634828677}}
{"text": "import Lean.Meta\nimport Lean.Elab\nimport Scratch.TermSeq\nopen Lean.Core\nopen Lean.Meta\nopen Lean.Elab.Term\nopen Lean\nopen Nat \n\n\n\n-- testing decomposition, used later (and better forms)\n-- also logging info in tactics and elaborators\n-- also adding a definition in a tactic context\n-- with a tactic doing this\n-- experiments with thunks, lazylists\n\npartial def decomposeSeq : Expr → MetaM (List Expr) :=\n  fun expr => \n  do \n    let mvar ←  mkFreshExprMVar none\n    let tmvar ← mkFreshExprMVar (mkConst `TermSeq)\n    let sExp ←  mkAppM ``TermSeq.cons #[mvar, tmvar]\n    if ← isDefEq sExp expr then\n      let prev ← decomposeSeq tmvar\n      return mvar :: prev\n    else \n      return []\n\npartial def seqLength : Expr → MetaM Nat := fun expr =>\n  do\n    let l ← decomposeSeq expr\n    return l.length\n\nsyntax (name:= tsl) \"tsl!\" term : term \n\n@[termElab tsl] def tslImpl : TermElab :=\n  fun stx expectedType? =>\n    match stx with\n    | `(tsl!%$tk $s) =>\n      do \n        let e ← elabTerm s none\n        let l ← seqLength e\n        Lean.Elab.logInfoAt tk m!\"expr: {(← e)}\"\n        Lean.Elab.logInfoAt tk m!\"whnf: {(← whnf e)}\"\n        Lean.Elab.logInfoAt tk m!\"length: {(← l)}\"\n        return ← ToExpr.toExpr l \n    | _ => Elab.throwIllFormedSyntax\n\n\n\ndef ts := TermSeq.cons 3 TermSeq.empty\n\n#eval tsl! ts\n#eval tsl! #⟨3, 4, \"this\"⟩\n\n#check  #⟨3, 4, \"this\"⟩\n#check fun x: Nat => #⟨x, x, \"this\", 4⟩\n\nexample : (fun x: Nat => #⟨x, x, \"this\", 4⟩) 3 = #⟨3, 3, \"this\", 4⟩ := rfl\n\nopen Lean.Elab.Tactic\n\nsyntax (name:= blah) \"blah\" : tactic\n@[tactic blah] def blahImpl : Tactic :=\n  fun stx =>\n    do\n      Lean.Elab.logInfo \"blah say I\"\n      return ()\n\nsyntax (name := tsltac) \"tslength\" term : tactic\n@[tactic tsltac] def tslTacImpl : Tactic := \n  fun stx =>\n    match stx with\n    | `(tactic|tslength $s) =>\n      do\n        let mvar ← getMainGoal\n        let e ← liftM (Elab.Term.elabTerm s none true true)\n        let l ← seqLength e\n        let n := ToExpr.toExpr l \n        replaceMainGoal []\n        assignExprMVar mvar n\n    | _ => Elab.throwIllFormedSyntax\n\ndef tstacEg : Nat := by \n        blah\n        tslength #⟨3, 4, \"this\"⟩\n\n#eval tstacEg\n\nuniverse u v\n\ndef factorThroughEg(α : Sort u) (β : Sort v)(b : β ) : (β  → α ) → α   := \n    fun g => g b\n\ndef addToContextM3(name: Name) (type : Expr)(value: Expr) : \n     MVarId → MetaM (List MVarId) :=\n  fun m => \n      do\n        let target ← getMVarType m\n        let exp ← mkAppM `factorThroughEg #[target, type, value]\n        let appGoalList ←  apply m exp\n        let appGoal := appGoalList.head!\n        let ⟨_, introGoal⟩ ←  intro appGoal name  \n        return [introGoal]\n\nsyntax (name:= useterm) \"use\" term (\"with\" term)? \"as\" ident : tactic\n@[tactic useterm] def usetermImpl : Tactic :=\n   fun stx =>\n    match stx with\n    | `(tactic|use $s with $t as $n) =>\n    withMainContext $\n    do\n      let name ← n.getId\n      let typ ← elabType t \n      let value ← Elab.Tactic.elabTerm s (some typ) \n      liftMetaTactic $ addToContextM3 name typ value\n    | `(tactic|use $s as $n) =>\n    withMainContext $\n    do\n      let name ← n.getId\n      let value ← Elab.Tactic.elabTerm s none \n      let typ ← inferType value\n      liftMetaTactic $ addToContextM3 name typ value\n    | _ => Elab.throwIllFormedSyntax\n\nexample : Nat := by\n        use 3 with Nat as n\n        use \"this\" as s\n        let x := 3\n        use (succ x) as y\n        have b := \"d\"\n        exact y\n\nsyntax (name:= dupllet) \"assign\" ident \"::\" term \"as\" term : tactic\n@[tactic dupllet] def assignImpl : Tactic :=\n  fun stx =>\n    match stx with\n    | `(tactic|assign $n:ident :: $t as  $i) => \n      do\n        let name ← n.getId\n        let typ ← liftM (Elab.Term.elabTerm t none true true)\n        let value ← liftM (Elab.Term.elabTerm i (some typ) true true)\n        let mvar ← getMainGoal\n        withMVarContext mvar do\n          let mvar2 ← mkFreshExprMVar (← getMVarType mvar) \n          let mvarId2 := mvar2.mvarId!\n          replaceMainGoal [mvarId2]\n          withLetDecl  name  typ value $ fun x =>\n            do          \n            assignExprMVar mvar (← mkLetFVars #[x] mvar2)          \n            return ()  \n    | _ => Elab.throwIllFormedSyntax\n\ndef fl := 4.5\ndef three := 3\n\ndef letTac : Nat   := by\n        let p := 3\n        assign y :: Nat as 3\n        skip\n        exact p\n        done\n        \n\n#eval tsl! #⟨three, fl, \"this\"⟩\n\ndef getFloat (s: String) : Option Float :=\n  (Syntax.decodeScientificLitVal? s).map (fun ⟨m, s, e⟩ => OfScientific.ofScientific m s e) \n\n#eval getFloat \"3.1415\"\n\n#eval Syntax.decodeScientificLitVal? (Float.toString (3.145))\n\n-- copied from @Mac's code below and modified\ndef parseOpt (s : String) : Option Float :=\n  match Json.Parser.num s.mkIterator with\n  | Parsec.ParseResult.success _ res =>\n    some <| Float.ofInt res.mantissa * 10 ^ - Float.ofNat res.exponent\n  | Parsec.ParseResult.error it err  => none\n\ndef parseGet (s : String) : Nat ×  Nat :=\n  match Json.Parser.num s.mkIterator with\n  | Parsec.ParseResult.success _ res =>\n      (res.mantissa.toNat, res.exponent)\n  | Parsec.ParseResult.error it err  => (0, 0)\n\n\nsyntax (name:= floatlit) \"float!\" term : term\n\n@[termElab floatlit] def floatlitImpl : TermElab :=\n    fun stx expectedType? =>\n      match stx with\n      | `(float! $s) =>\n        do  \n           let fl ← Elab.Term.elabTerm s (some (Lean.mkConst `Float))\n           let strRaw ← mkAppM ``Float.toString #[fl] \n           let str ← whnf strRaw\n          --  let ⟨n, _, _⟩ := (Syntax.isScientificLit? s).get! \n           return ← mkAppM ``parseGet #[str]  \n          --  return  ToExpr.toExpr n\n      | _ => Elab.throwIllFormedSyntax\n\ndef pi := 3.1415\n\n#eval float! 3.14\n#eval float! pi\n#eval float! (3.1 * 6)\n\nsyntax ident (\"⌣\" ident)? \"↦\" term : term\n\nmacro_rules\n  | `( $x:ident ↦ $y:term ) => `(fun $x => $y)\n  | `( $x:ident ⌣ $t: ident ↦ $y:term ) => `(fun ($x : $t) => $y)  \n\ndef mapfn : Nat → Nat := n ↦ n + 1\n\ndef mapFn2 := k ⌣ Nat ↦ (k + 1) \n\n-- copied from Zulip chat due to @mac \n\n-- import Lean.Data.Json.Parser\n\ndef parse (s : String) : Except String Float :=\n  match (Json.Parser.num <* Parsec.eof) s.mkIterator with\n  | Parsec.ParseResult.success _ res =>\n    Except.ok <| Float.ofInt res.mantissa * 10 ^ - Float.ofNat res.exponent\n  | Parsec.ParseResult.error it err  => Except.error s!\"offset {it.i.repr}: {err}\"\n\n#eval parse \"1.3\" -- Except.ok 1.300000\n\n\ndef th3:= Thunk.mk (fun () => 3)\n\ndef addN (n : Nat) : Thunk Nat →  Thunk Nat :=\n      let succT : Nat → Thunk Nat := fun n => Thunk.mk (fun () => n + 1)\n      match n with\n      | 0 => id\n      | k + 1 => fun tn => (addN k tn).bind succT\n\n#reduce (addN 20) th3\n\n-- credit: \n-- https://www.classes.cs.uchicago.edu/archive/2019/spring/22300-1/lectures/LazyLists/index.html\n\nmutual\n  inductive LazyList (α : Type) where\n  | mk : Thunk (LazyListCell α) → LazyList α\n\n  inductive LazyListCell (α : Type) where\n  | nil : LazyListCell α\n  | cons : α → LazyList α → LazyListCell α\nend\n", "meta": {"author": "siddhartha-gadgil", "repo": "lean4-scratch", "sha": "680b7073f791706faf248d1d0ad21095012ae01b", "save_path": "github-repos/lean/siddhartha-gadgil-lean4-scratch", "path": "github-repos/lean/siddhartha-gadgil-lean4-scratch/lean4-scratch-680b7073f791706faf248d1d0ad21095012ae01b/Scratch/Egs3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3378329787049826}}
{"text": "import data.int.parity\nimport tactic.apply_fun\nimport data.real.basic\nimport algebra.group.pi\n\nimport lib.tactiques\n\nnamespace tactic.interactive\nsetup_tactic_parser\n\nopen tactic\n\nmeta def verifie : tactic unit :=\n`[ { repeat { unfold limite_suite},\n   repeat { unfold continue_en },\n   push_neg,\n   try { simp only [exists_prop] },\n   try { exact iff.rfl },\n   done } <|> fail \"Ce n'est pas cela. Essayez encore.\" ]\n\nend tactic.interactive\n\nnotation `|`:1024 x:1 `|`:1 := abs x\n\nnamespace m154\n\nlemma inferieur_ssi {x y : ℝ} : x ≤ y ↔ 0 ≤ y - x :=\nsub_nonneg.symm\n\nlemma pos_pos {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : 0 ≤ x*y :=\nmul_nonneg hx hy\n\nlemma neg_neg {x y : ℝ} (hx : x ≤ 0) (hy : y ≤ 0) : 0 ≤ x*y :=\nmul_nonneg_of_nonpos_of_nonpos hx hy\n\nlemma inferieur_ssi' {x y : ℝ} : x ≤ y ↔ x - y ≤ 0 :=\nby rw [show x-y = -(y-x), by ring, inferieur_ssi, neg_le, neg_zero]\nend m154\n\nopen nat\ndef pgcd := nat.gcd\n\nlemma divise_refl (a : ℕ) : a ∣ a :=\ndvd_refl a\n\nlemma divise_pgcd_ssi {a b c : ℕ} : c ∣ pgcd a b ↔ c ∣ a ∧ c ∣ b :=\ndvd_gcd_iff\n\nlemma divise_antisym {a b : ℕ} : a ∣ b → b ∣ a → a = b :=\ndvd_antisymm\n\nlemma divise_def (a b : ℤ) : a ∣ b ↔ ∃ k, b = a*k :=\niff.rfl\n\ndef pair (n : ℤ) := ∃ k, n = 2*k\ndef impair (n : ℤ) := ∃ k, n = 2*k + 1\n\nlemma pair_ou_impair (n : ℤ) : pair n ∨ impair n :=\nby by_cases h : n % 2 = 0 ; [left, {right ; rw int.mod_two_ne_zero at h}] ;\n  rw [← int.mod_add_div n 2, h] ; use n/2 ; ring\n\nlemma non_pair_et_impair (n : ℤ) : ¬ (pair n ∧ impair n) :=\nbegin\n  rintro ⟨h, h'⟩,\n  change even n at h,\n  rw int.even_iff at h,\n  rcases h' with ⟨k, rfl⟩,\n  simp only [int.add_mul_mod_self_left, add_comm, euclidean_domain.mod_eq_zero] at h,\n  cases h with l hl,\n  rw eq_comm at hl,\n  have := int.eq_one_of_mul_eq_one_right (by linarith) hl,\n  linarith\nend\n\nlemma abs_inferieur_ssi (x y : ℝ) : |x| ≤ y ↔ -y ≤ x ∧ x ≤ y :=\nabs_le\n\nlemma abs_diff (x y : ℝ) : |x - y| = |y - x| :=\nabs_sub_comm x y\n\nlemma pos_abs (x : ℝ) : |x| > 0 ↔ x ≠ 0 :=\nabs_pos\n\nvariables {α : Type*} [linear_order α]\nlemma superieur_max_ssi (p q r : α) : r ≥ max p q  ↔ r ≥ p ∧ r ≥ q :=\nmax_le_iff\n\nlemma inferieur_max_gauche (p q : α) : p ≤ max p q :=\nle_max_left _ _\n\nlemma inferieur_max_droite (p q : α) : q ≤ max p q :=\nle_max_right _ _\n\nlemma egal_si_abs_diff_neg {a b : ℝ} : |a - b| ≤ 0 → a = b :=\neq_of_abs_sub_nonpos\n\nlemma egal_si_abs_eps (x y : ℝ) : (∀ ε > 0, |x - y| ≤ ε) → x = y :=\nbegin\n  intro h,\n  apply egal_si_abs_diff_neg,\n  by_contradiction H,\n  push_neg at H,\n  specialize h ( |x-y|/2) (by linarith),\n  linarith\nend\n\nlemma ineg_triangle (x y : ℝ) : |x + y| ≤ |x| + |y| :=\nabs_add x y\n\nnamespace m154\ndef limite_suite (u : ℕ → ℝ) (l : ℝ) : Prop :=\n∀ ε > 0, ∃ N, ∀ n ≥ N, |u n - l| ≤ ε\n\nlemma unicite_limite {u l l'}: limite_suite u l → limite_suite u l' → l = l' :=\nbegin\n  -- sorry\n  intros hl hl',\n  apply egal_si_abs_eps,\n  intros ε ε_pos,\n  specialize hl (ε/2) (by linarith),\n  cases hl with N hN,\n  specialize hl' (ε/2) (by linarith),\n  cases hl' with N' hN',\n  specialize hN (max N N') (inferieur_max_gauche _ _),\n  specialize hN' (max N N') (inferieur_max_droite _ _),\n  calc |l - l'| = |(l-u (max N N')) + (u (max N N') -l')| : by ring_nf\n  ... ≤ |l - u (max N N')| + |u (max N N') - l'| : by apply ineg_triangle\n  ... =  |u (max N N') - l| + |u (max N N') - l'| : by rw abs_diff\n  ... ≤ ε/2 + ε/2 : by linarith\n  ... = ε : by ring,\nend\n\nend m154\n\nopen m154\n\ndef extraction (φ : ℕ → ℕ) := ∀ n m, n < m → φ n < φ m\n\n-- Dans la suite, φ désignera toujours une fonction de ℕ dans ℕ\nvariable { φ : ℕ → ℕ}\n\n\n/-- Un réel `a` est valeur d'adhérence d'une suite `u` s'il\nexiste une suite extraite de `u` qui tend vers `a`. -/\ndef valeur_adherence (u : ℕ → ℝ) (a : ℝ) :=\n∃ φ, extraction φ ∧ limite_suite (u ∘ φ) a\n\n/-- Toute extraction est supérieure à l'identité. -/\nlemma extraction_superieur_id : extraction φ → ∀ n, n ≤ φ n :=\nbegin\n  intros hyp n,\n  induction n with n hn,\n    exact nat.zero_le _,\n  exact nat.succ_le_of_lt (by linarith [hyp n (n+1) (by linarith)]),\nend\n\nopen filter\n\nlemma extraction_machine (ψ : ℕ → ℕ) (hψ : ∀ n, ψ n ≥ n) : ∃ f : ℕ → ℕ, extraction (ψ ∘ f) ∧ ∀ n, f n ≥ n :=\nbegin\n  refine ⟨λ n, nat.rec_on n 0 (λ n ih, ψ ih + 1), λ m n h, _, λ n, _⟩,\n  { induction h; dsimp [(∘)],\n    { exact hψ _ },\n    { exact lt_trans h_ih (hψ _) } },\n  { induction n, {apply le_refl},\n    exact nat.succ_le_succ (le_trans n_ih (hψ _)) }\nend", "meta": {"author": "PatrickMassot", "repo": "MDD154", "sha": "00defe82a4b6b7992ed522a92f62abd685e8c943", "save_path": "github-repos/lean/PatrickMassot-MDD154", "path": "github-repos/lean/PatrickMassot-MDD154/MDD154-00defe82a4b6b7992ed522a92f62abd685e8c943/src/lib/m154.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.33773263338844023}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.preserves.shapes.products\nimport category_theory.limits.preserves.shapes.binary_products\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.pempty\nimport data.equiv.fin\n\n/-!\n# Constructing finite products from binary products and terminal.\n\nIf a category has binary products and a terminal object then it has finite products.\nIf a functor preserves binary products and the terminal object then it preserves finite products.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\nuniverses v u u'\n\nnoncomputable theory\nopen category_theory category_theory.category category_theory.limits\nnamespace category_theory\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{v} C]\nvariables {D : Type u'} [category.{v} D]\n\n/--\nGiven `n+1` objects of `C`, a fan for the last `n` with point `c₁.X` and a binary fan on `c₁.X` and\n`f 0`, we can build a fan for all `n+1`.\n\nIn `extend_fan_is_limit` we show that if the two given fans are limits, then this fan is also a\nlimit.\n-/\n@[simps {rhs_md := semireducible}]\ndef extend_fan {n : ℕ} {f : ulift (fin (n+1)) → C}\n  (c₁ : fan (λ (i : ulift (fin n)), f ⟨i.down.succ⟩))\n  (c₂ : binary_fan (f ⟨0⟩) c₁.X) :\n  fan f :=\nfan.mk c₂.X\nbegin\n  rintro ⟨i⟩,\n  revert i,\n  refine fin.cases _ _,\n  { apply c₂.fst },\n  { intro i,\n    apply c₂.snd ≫ c₁.π.app (ulift.up i) },\nend\n\n/--\nShow that if the two given fans in `extend_fan` are limits, then the constructed fan is also a\nlimit.\n-/\ndef extend_fan_is_limit {n : ℕ} (f : ulift (fin (n+1)) → C)\n  {c₁ : fan (λ (i : ulift (fin n)), f ⟨i.down.succ⟩)} {c₂ : binary_fan (f ⟨0⟩) c₁.X}\n  (t₁ : is_limit c₁) (t₂ : is_limit c₂) :\n  is_limit (extend_fan c₁ c₂) :=\n{ lift := λ s,\n  begin\n    apply (binary_fan.is_limit.lift' t₂ (s.π.app ⟨0⟩) _).1,\n    apply t₁.lift ⟨_, discrete.nat_trans (λ i, s.π.app ⟨i.down.succ⟩)⟩\n  end,\n  fac' := λ s,\n  begin\n    rintro ⟨j⟩,\n    apply fin.induction_on j,\n    { apply (binary_fan.is_limit.lift' t₂ _ _).2.1 },\n    { rintro i -,\n      dsimp only [extend_fan_π_app],\n      rw [fin.cases_succ, ← assoc, (binary_fan.is_limit.lift' t₂ _ _).2.2, t₁.fac],\n      refl }\n  end,\n  uniq' := λ s m w,\n  begin\n    apply binary_fan.is_limit.hom_ext t₂,\n    { rw (binary_fan.is_limit.lift' t₂ _ _).2.1,\n      apply w ⟨0⟩ },\n    { rw (binary_fan.is_limit.lift' t₂ _ _).2.2,\n      apply t₁.uniq ⟨_, _⟩,\n      rintro ⟨j⟩,\n      rw assoc,\n      dsimp only [discrete.nat_trans_app],\n      rw ← w ⟨j.succ⟩,\n      dsimp only [extend_fan_π_app],\n      rw fin.cases_succ }\n  end }\n\nsection\nvariables [has_binary_products.{v} C] [has_terminal C]\n\n/--\nIf `C` has a terminal object and binary products, then it has a product for objects indexed by\n`ulift (fin n)`.\nThis is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_product_ulift_fin :\n  Π (n : ℕ) (f : ulift (fin n) → C), has_product f\n| 0 := λ f,\n  begin\n    letI : has_limits_of_shape (discrete (ulift (fin 0))) C :=\n      has_limits_of_shape_of_equivalence\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm),\n    apply_instance,\n  end\n| (n+1) := λ f,\n  begin\n    haveI := has_product_ulift_fin n,\n    apply has_limit.mk ⟨_, extend_fan_is_limit f (limit.is_limit _) (limit.is_limit _)⟩,\n  end\n\n/--\nIf `C` has a terminal object and binary products, then it has limits of shape\n`discrete (ulift (fin n))` for any `n : ℕ`.\nThis is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_limits_of_shape_ulift_fin (n : ℕ) :\n  has_limits_of_shape (discrete (ulift (fin n))) C :=\n{ has_limit := λ K,\nbegin\n  letI := has_product_ulift_fin n K.obj,\n  let : discrete.functor K.obj ≅ K := discrete.nat_iso (λ i, iso.refl _),\n  apply has_limit_of_iso this,\nend }\n\n/-- If `C` has a terminal object and binary products, then it has finite products. -/\nlemma has_finite_products_of_has_binary_and_terminal : has_finite_products C :=\n⟨λ J 𝒥₁ 𝒥₂, begin\n  resetI,\n  let e := fintype.equiv_fin J,\n  apply has_limits_of_shape_of_equivalence (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\n  refine has_limits_of_shape_ulift_fin (fintype.card J),\nend⟩\n\nend\n\nsection preserves\nvariables (F : C ⥤ D)\nvariables [preserves_limits_of_shape (discrete walking_pair) F]\nvariables [preserves_limits_of_shape (discrete pempty) F]\nvariables [has_finite_products.{v} C]\n\n/--\nIf `F` preserves the terminal object and binary products, then it preserves products indexed by\n`ulift (fin n)` for any `n`.\n-/\nnoncomputable def preserves_fin_of_preserves_binary_and_terminal  :\n  Π (n : ℕ) (f : ulift (fin n) → C), preserves_limit (discrete.functor f) F\n| 0 := λ f,\n  begin\n    letI : preserves_limits_of_shape (discrete (ulift (fin 0))) F :=\n      preserves_limits_of_shape_of_equiv\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm) _,\n    apply_instance,\n  end\n| (n+1) :=\n  begin\n    haveI := preserves_fin_of_preserves_binary_and_terminal n,\n    intro f,\n    refine preserves_limit_of_preserves_limit_cone\n      (extend_fan_is_limit f (limit.is_limit _) (limit.is_limit _)) _,\n    apply (is_limit_map_cone_fan_mk_equiv _ _ _).symm _,\n    let := extend_fan_is_limit (λ i, F.obj (f i))\n              (is_limit_of_has_product_of_preserves_limit F _)\n              (is_limit_of_has_binary_product_of_preserves_limit F _ _),\n    refine is_limit.of_iso_limit this _,\n    apply cones.ext _ _,\n    apply iso.refl _,\n    rintro ⟨j⟩,\n    apply fin.induction_on j,\n    { apply (category.id_comp _).symm },\n    { rintro i -,\n      dsimp only [extend_fan_π_app, iso.refl_hom, fan.mk_π_app],\n      rw [fin.cases_succ, fin.cases_succ],\n      change F.map _ ≫ _ = 𝟙 _ ≫ _,\n      rw [id_comp, ←F.map_comp],\n      refl }\n  end\n\n/--\nIf `F` preserves the terminal object and binary products, then it preserves limits of shape\n`discrete (ulift (fin n))`.\n-/\ndef preserves_ulift_fin_of_preserves_binary_and_terminal (n : ℕ) :\n  preserves_limits_of_shape (discrete (ulift (fin n))) F :=\n{ preserves_limit := λ K,\n  begin\n    let : discrete.functor K.obj ≅ K := discrete.nat_iso (λ i, iso.refl _),\n    haveI := preserves_fin_of_preserves_binary_and_terminal F n K.obj,\n    apply preserves_limit_of_iso_diagram F this,\n  end }\n\n/-- If `F` preserves the terminal object and binary products then it preserves finite products. -/\ndef preserves_finite_products_of_preserves_binary_and_terminal\n  (J : Type v) [fintype J] :\n  preserves_limits_of_shape.{v} (discrete J) F :=\nbegin\n  classical,\n  let e := fintype.equiv_fin J,\n  haveI := preserves_ulift_fin_of_preserves_binary_and_terminal F (fintype.card J),\n  apply preserves_limits_of_shape_of_equiv (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\nend\n\nend preserves\n\n/--\nGiven `n+1` objects of `C`, a cofan for the last `n` with point `c₁.X`\nand a binary cofan on `c₁.X` and `f 0`, we can build a cofan for all `n+1`.\n\nIn `extend_cofan_is_colimit` we show that if the two given cofans are colimits,\nthen this cofan is also a colimit.\n-/\n@[simps {rhs_md := semireducible}]\ndef extend_cofan {n : ℕ} {f : ulift (fin (n+1)) → C}\n  (c₁ : cofan (λ (i : ulift (fin n)), f ⟨i.down.succ⟩))\n  (c₂ : binary_cofan (f ⟨0⟩) c₁.X) :\n  cofan f :=\ncofan.mk c₂.X\nbegin\n  rintro ⟨i⟩,\n  revert i,\n  refine fin.cases _ _,\n  { apply c₂.inl },\n  { intro i,\n    apply c₁.ι.app (ulift.up i) ≫ c₂.inr },\nend\n\n/--\nShow that if the two given cofans in `extend_cofan` are colimits,\nthen the constructed cofan is also a colimit.\n-/\ndef extend_cofan_is_colimit {n : ℕ} (f : ulift (fin (n+1)) → C)\n  {c₁ : cofan (λ (i : ulift (fin n)), f ⟨i.down.succ⟩)} {c₂ : binary_cofan (f ⟨0⟩) c₁.X}\n  (t₁ : is_colimit c₁) (t₂ : is_colimit c₂) :\n  is_colimit (extend_cofan c₁ c₂) :=\n{ desc := λ s,\n  begin\n    apply (binary_cofan.is_colimit.desc' t₂ (s.ι.app ⟨0⟩) _).1,\n    apply t₁.desc ⟨_, discrete.nat_trans (λ i, s.ι.app ⟨i.down.succ⟩)⟩\n  end,\n  fac' := λ s,\n  begin\n    rintro ⟨j⟩,\n    apply fin.induction_on j,\n    { apply (binary_cofan.is_colimit.desc' t₂ _ _).2.1 },\n    { rintro i -,\n      dsimp only [extend_cofan_ι_app],\n      rw [fin.cases_succ, assoc, (binary_cofan.is_colimit.desc' t₂ _ _).2.2, t₁.fac],\n      refl }\n  end,\n  uniq' := λ s m w,\n  begin\n    apply binary_cofan.is_colimit.hom_ext t₂,\n    { rw (binary_cofan.is_colimit.desc' t₂ _ _).2.1,\n      apply w ⟨0⟩ },\n    { rw (binary_cofan.is_colimit.desc' t₂ _ _).2.2,\n      apply t₁.uniq ⟨_, _⟩,\n      rintro ⟨j⟩,\n      dsimp only [discrete.nat_trans_app],\n      rw ← w ⟨j.succ⟩,\n      dsimp only [extend_cofan_ι_app],\n      rw [fin.cases_succ, assoc], }\n  end }\n\nsection\nvariables [has_binary_coproducts.{v} C] [has_initial C]\n\n/--\nIf `C` has an initial object and binary coproducts, then it has a coproduct for objects indexed by\n`ulift (fin n)`.\nThis is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_coproduct_ulift_fin :\n  Π (n : ℕ) (f : ulift (fin n) → C), has_coproduct f\n| 0 := λ f,\n  begin\n    letI : has_colimits_of_shape (discrete (ulift (fin 0))) C :=\n      has_colimits_of_shape_of_equivalence\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm),\n    apply_instance,\n  end\n| (n+1) := λ f,\n  begin\n    haveI := has_coproduct_ulift_fin n,\n    apply has_colimit.mk\n      ⟨_, extend_cofan_is_colimit f (colimit.is_colimit _) (colimit.is_colimit _)⟩,\n  end\n\n/--\nIf `C` has an initial object and binary coproducts, then it has colimits of shape\n`discrete (ulift (fin n))` for any `n : ℕ`.\nThis is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_colimits_of_shape_ulift_fin (n : ℕ) :\n  has_colimits_of_shape (discrete (ulift (fin n))) C :=\n{ has_colimit := λ K,\nbegin\n  letI := has_coproduct_ulift_fin n K.obj,\n  let : K ≅ discrete.functor K.obj := discrete.nat_iso (λ i, iso.refl _),\n  apply has_colimit_of_iso this,\nend }\n\n/-- If `C` has an initial object and binary coproducts, then it has finite coproducts. -/\nlemma has_finite_coproducts_of_has_binary_and_terminal : has_finite_coproducts C :=\n⟨λ J 𝒥₁ 𝒥₂, begin\n  resetI,\n  let e := fintype.equiv_fin J,\n  apply has_colimits_of_shape_of_equivalence (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\n  refine has_colimits_of_shape_ulift_fin (fintype.card J),\nend⟩\n\nend\n\nsection preserves\nvariables (F : C ⥤ D)\nvariables [preserves_colimits_of_shape (discrete walking_pair) F]\nvariables [preserves_colimits_of_shape (discrete pempty) F]\nvariables [has_finite_coproducts.{v} C]\n\n/--\nIf `F` preserves the initial object and binary coproducts, then it preserves products indexed by\n`ulift (fin n)` for any `n`.\n-/\nnoncomputable def preserves_fin_of_preserves_binary_and_initial  :\n  Π (n : ℕ) (f : ulift (fin n) → C), preserves_colimit (discrete.functor f) F\n| 0 := λ f,\n  begin\n    letI : preserves_colimits_of_shape (discrete (ulift (fin 0))) F :=\n      preserves_colimits_of_shape_of_equiv\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm) _,\n    apply_instance,\n  end\n| (n+1) :=\n  begin\n    haveI := preserves_fin_of_preserves_binary_and_initial n,\n    intro f,\n    refine preserves_colimit_of_preserves_colimit_cocone\n      (extend_cofan_is_colimit f (colimit.is_colimit _) (colimit.is_colimit _)) _,\n    apply (is_colimit_map_cocone_cofan_mk_equiv _ _ _).symm _,\n    let := extend_cofan_is_colimit (λ i, F.obj (f i))\n              (is_colimit_of_has_coproduct_of_preserves_colimit F _)\n              (is_colimit_of_has_binary_coproduct_of_preserves_colimit F _ _),\n    refine is_colimit.of_iso_colimit this _,\n    apply cocones.ext _ _,\n    apply iso.refl _,\n    rintro ⟨j⟩,\n    apply fin.induction_on j,\n    { apply category.comp_id },\n    { rintro i -,\n      dsimp only [extend_cofan_ι_app, iso.refl_hom, cofan.mk_ι_app],\n      rw [fin.cases_succ, fin.cases_succ],\n      erw [comp_id, ←F.map_comp],\n      refl, }\n  end\n\n/--\nIf `F` preserves the initial object and binary coproducts, then it preserves colimits of shape\n`discrete (ulift (fin n))`.\n-/\ndef preserves_ulift_fin_of_preserves_binary_and_initial (n : ℕ) :\n  preserves_colimits_of_shape (discrete (ulift (fin n))) F :=\n{ preserves_colimit := λ K,\n  begin\n    let : discrete.functor K.obj ≅ K := discrete.nat_iso (λ i, iso.refl _),\n    haveI := preserves_fin_of_preserves_binary_and_initial F n K.obj,\n    apply preserves_colimit_of_iso_diagram F this,\n  end }\n\n/-- If `F` preserves the initial object and binary coproducts then it preserves finite products. -/\ndef preserves_finite_coproducts_of_preserves_binary_and_initial\n  (J : Type v) [fintype J] :\n  preserves_colimits_of_shape.{v} (discrete J) F :=\nbegin\n  classical,\n  let e := fintype.equiv_fin J,\n  haveI := preserves_ulift_fin_of_preserves_binary_and_initial F (fintype.card J),\n  apply preserves_colimits_of_shape_of_equiv (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\nend\n\nend preserves\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/constructions/finite_products_of_binary_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.33773263338844023}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.dlist\nimport data.list.basic\nimport data.seq.seq\n\nopen function\nuniverses u v w\n\n/-\ncoinductive wseq (α : Type u) : Type u\n| nil : wseq α\n| cons : α → wseq α → wseq α\n| think : wseq α → wseq α\n-/\n\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef wseq (α) := seq (option α)\n\nnamespace wseq\nvariables {α : Type u} {β : Type v} {γ : Type w}\n\n/-- Turn a sequence into a weak sequence -/\ndef of_seq : seq α → wseq α := (<$>) some\n\n/-- Turn a list into a weak sequence -/\ndef of_list (l : list α) : wseq α := of_seq l\n\n/-- Turn a stream into a weak sequence -/\ndef of_stream (l : stream α) : wseq α := of_seq l\n\ninstance coe_seq : has_coe (seq α) (wseq α) := ⟨of_seq⟩\ninstance coe_list : has_coe (list α) (wseq α) := ⟨of_list⟩\ninstance coe_stream : has_coe (stream α) (wseq α) := ⟨of_stream⟩\n\n/-- The empty weak sequence -/\ndef nil : wseq α := seq.nil\n\ninstance : inhabited (wseq α) := ⟨nil⟩\n\n/-- Prepend an element to a weak sequence -/\ndef cons (a : α) : wseq α → wseq α := seq.cons (some a)\n\n/-- Compute for one tick, without producing any elements -/\ndef think : wseq α → wseq α := seq.cons none\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct : wseq α → computation (option (α × wseq α)) :=\ncomputation.corec (λs, match seq.destruct s with\n  | none              := sum.inl none\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := sum.inl (some (a, s'))\n  end)\n\ndef cases_on {C : wseq α → Sort v} (s : wseq α) (h1 : C nil)\n  (h2 : ∀ x s, C (cons x s)) (h3 : ∀ s, C (think s)) : C s :=\nseq.cases_on s h1 (λ o, option.cases_on o h3 h2)\n\nprotected def mem (a : α) (s : wseq α) := seq.mem (some a) s\n\ninstance : has_mem α (wseq α) :=\n⟨wseq.mem⟩\n\ntheorem not_mem_nil (a : α) : a ∉ @nil α := seq.not_mem_nil a\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head (s : wseq α) : computation (option α) :=\ncomputation.map ((<$>) prod.fst) (destruct s)\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten : computation (wseq α) → wseq α :=\nseq.corec (λc, match computation.destruct c with\n  | sum.inl s := seq.omap return (seq.destruct s)\n  | sum.inr c' := some (none, c')\n  end)\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail (s : wseq α) : wseq α :=\nflatten $ (λo, option.rec_on o nil prod.snd) <$> destruct s\n\n/-- drop the first `n` elements from `s`. -/\ndef drop (s : wseq α) : ℕ → wseq α\n| 0     := s\n| (n+1) := tail (drop n)\nattribute [simp] drop\n\n/-- Get the nth element of `s`. -/\ndef nth (s : wseq α) (n : ℕ) : computation (option α) := head (drop s n)\n\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef to_list (s : wseq α) : computation (list α) :=\n@computation.corec (list α) (list α × wseq α) (λ⟨l, s⟩,\n  match seq.destruct s with\n  | none              := sum.inl l.reverse\n  | some (none, s')   := sum.inr (l, s')\n  | some (some a, s') := sum.inr (a::l, s')\n  end) ([], s)\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : wseq α) : computation ℕ :=\n@computation.corec ℕ (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s with\n  | none              := sum.inl n\n  | some (none, s')   := sum.inr (n, s')\n  | some (some a, s') := sum.inr (n+1, s')\n  end) (0, s)\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\nclass is_finite (s : wseq α) : Prop := (out : (to_list s).terminates)\n\ninstance to_list_terminates (s : wseq α) [h : is_finite s] : (to_list s).terminates := h.out\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : wseq α) [is_finite s] : list α := (to_list s).get\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\nclass productive (s : wseq α) : Prop := (nth_terminates : ∀ n, (nth s n).terminates)\n\ntheorem productive_iff (s : wseq α) : productive s ↔ ∀ n, (nth s n).terminates :=\n⟨λ h, h.1, λ h, ⟨h⟩⟩\n\ninstance nth_terminates (s : wseq α) [h : productive s] :\n  ∀ n, (nth s n).terminates := h.nth_terminates\n\ninstance head_terminates (s : wseq α) [productive s] :\n  (head s).terminates := s.nth_terminates 0\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef update_nth (s : wseq α) (n : ℕ) (a : α) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (some a, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef remove_nth (s : wseq α) (n : ℕ) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (none, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filter_map (f : α → option β) : wseq α → wseq β :=\nseq.corec (λs, match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, s')\n  | some (some a, s') := some (f a, s')\n  end)\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : α → Prop) [decidable_pred p] : wseq α → wseq α :=\nfilter_map (λa, if p a then some a else none)\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : α → Prop) [decidable_pred p] (s : wseq α) : computation (option α) :=\nhead $ filter p s\n\n/-- Zip a function over two weak sequences -/\ndef zip_with (f : α → β → γ) (s1 : wseq α) (s2 : wseq β) : wseq γ :=\n@seq.corec (option γ) (wseq α × wseq β) (λ⟨s1, s2⟩,\n  match seq.destruct s1, seq.destruct s2 with\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (none, s1, s2')\n  | some (none, s1'),    some (some a2, s2') := some (none, s1', s2)\n  | some (some a1, s1'), some (some a2, s2') := some (some (f a1 a2), s1', s2')\n  | _,                   _                   := none\n  end) (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : wseq α → wseq β → wseq (α × β) := zip_with prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef find_indexes (p : α → Prop) [decidable_pred p] (s : wseq α) : wseq ℕ :=\n(zip s (stream.nats : wseq ℕ)).filter_map\n  (λ ⟨a, n⟩, if p a then some n else none)\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef find_index (p : α → Prop) [decidable_pred p] (s : wseq α) : computation ℕ :=\n(λ o, option.get_or_else o 0) <$> head (find_indexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef index_of [decidable_eq α] (a : α) : wseq α → computation ℕ := find_index (eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexes_of [decidable_eq α] (a : α) : wseq α → wseq ℕ := find_indexes (eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : wseq α) : wseq α :=\n@seq.corec (option α) (wseq α × wseq α) (λ⟨s1, s2⟩,\n  match seq.destruct s1, seq.destruct s2 with\n  | none,                none                := none\n  | some (a1, s1'),      none                := some (a1, s1', nil)\n  | none,                some (a2, s2')      := some (a2, nil, s2')\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (some a1, s1', s2')\n  | some (none, s1'),    some (some a2, s2') := some (some a2, s1', s2')\n  | some (some a1, s1'), some (some a2, s2') := some (some a1, cons a2 s1', s2')\n  end) (s1, s2)\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef is_empty (s : wseq α) : computation bool :=\ncomputation.map option.is_none $ head s\n\n/-- Calculate one step of computation -/\ndef compute (s : wseq α) : wseq α :=\nmatch seq.destruct s with\n| some (none, s') := s'\n| _               := s\nend\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : wseq α) (n : ℕ) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match n, seq.destruct s with\n  | 0,   _                 := none\n  | m+1, none              := none\n  | m+1, some (none, s')   := some (none, m+1, s')\n  | m+1, some (some a, s') := some (some a, m, s')\n  end) (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef split_at (s : wseq α) (n : ℕ) : computation (list α × wseq α) :=\n@computation.corec (list α × wseq α) (ℕ × list α × wseq α) (λ⟨n, l, s⟩,\n  match n, seq.destruct s with\n  | 0,   _                 := sum.inl (l.reverse, s)\n  | m+1, none              := sum.inl (l.reverse, s)\n  | m+1, some (none, s')   := sum.inr (n, l, s')\n  | m+1, some (some a, s') := sum.inr (m, a::l, s')\n  end) (n, [], s)\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any (s : wseq α) (p : α → bool) : computation bool :=\ncomputation.corec (λs : wseq α,\n  match seq.destruct s with\n  | none              := sum.inl ff\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inl tt else sum.inr s'\n  end) s\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all (s : wseq α) (p : α → bool) : computation bool :=\ncomputation.corec (λs : wseq α,\n  match seq.destruct s with\n  | none              := sum.inl tt\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inr s' else sum.inl ff\n  end) s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : α → β → α) (a : α) (s : wseq β) : wseq α :=\ncons a $ @seq.corec (option α) (α × wseq β) (λ⟨a, s⟩,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, a, s')\n  | some (some b, s') := let a' := f a b in some (some a', a', s')\n  end) (a, s)\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : wseq α) : wseq (list α) :=\ncons [] $ @seq.corec (option (list α)) (dlist α × wseq α) (λ ⟨l, s⟩,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, l, s')\n  | some (some a, s') := let l' := l.concat a in\n                         some (some l'.to_list, l', s')\n  end) (dlist.empty, s)\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : wseq α) (n : ℕ) : list α :=\n(seq.take n s).filter_map id\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append : wseq α → wseq α → wseq α := seq.append\n\n/-- Map a function over a weak sequence -/\ndef map (f : α → β) : wseq α → wseq β := seq.map (option.map f)\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join (S : wseq (wseq α)) : wseq α :=\nseq.join ((λo : option (wseq α), match o with\n  | none := seq1.ret none\n  | some s := (none, s)\n  end) <$> S)\n\n/-- Monadic bind operator for weak sequences -/\ndef bind (s : wseq α) (f : α → wseq β) : wseq β :=\njoin (map f s)\n\n@[simp] def lift_rel_o (R : α → β → Prop) (C : wseq α → wseq β → Prop) :\n  option (α × wseq α) → option (β × wseq β) → Prop\n| none          none          := true\n| (some (a, s)) (some (b, t)) := R a b ∧ C s t\n| _             _             := false\n\ntheorem lift_rel_o.imp {R S : α → β → Prop} {C D : wseq α → wseq β → Prop}\n  (H1 : ∀ a b, R a b → S a b) (H2 : ∀ s t, C s t → D s t) :\n  ∀ {o p}, lift_rel_o R C o p → lift_rel_o S D o p\n| none          none          h := trivial\n| (some (a, s)) (some (b, t)) h := and.imp (H1 _ _) (H2 _ _) h\n| none          (some _)      h := false.elim h\n| (some (_, _)) none          h := false.elim h\n\ntheorem lift_rel_o.imp_right (R : α → β → Prop) {C D : wseq α → wseq β → Prop}\n  (H : ∀ s t, C s t → D s t) {o p} : lift_rel_o R C o p → lift_rel_o R D o p :=\nlift_rel_o.imp (λ _ _, id) H\n\n@[simp] def bisim_o (R : wseq α → wseq α → Prop) :\n  option (α × wseq α) → option (α × wseq α) → Prop := lift_rel_o (=) R\n\ntheorem bisim_o.imp {R S : wseq α → wseq α → Prop} (H : ∀ s t, R s t → S s t) {o p} :\n  bisim_o R o p → bisim_o S o p :=\nlift_rel_o.imp_right _ H\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef lift_rel (R : α → β → Prop) (s : wseq α) (t : wseq β) : Prop :=\n∃ C : wseq α → wseq β → Prop, C s t ∧\n∀ {s t}, C s t → computation.lift_rel (lift_rel_o R C) (destruct s) (destruct t)\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef equiv : wseq α → wseq α → Prop := lift_rel (=)\n\ntheorem lift_rel_destruct {R : α → β → Prop} {s : wseq α} {t : wseq β} :\n  lift_rel R s t →\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t)\n| ⟨R, h1, h2⟩ :=\n  by refine computation.lift_rel.imp _ _ _ (h2 h1);\n     apply lift_rel_o.imp_right; exact λ s' t' h', ⟨R, h', @h2⟩\n\ntheorem lift_rel_destruct_iff {R : α → β → Prop} {s : wseq α} {t : wseq β} :\n  lift_rel R s t ↔\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n⟨lift_rel_destruct, λ h, ⟨λ s t, lift_rel R s t ∨\n  computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n  or.inr h, λ s t h, begin\n    have h : computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n    { cases h with h h, exact lift_rel_destruct h, assumption },\n    apply computation.lift_rel.imp _ _ _ h,\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end⟩⟩\n\ninfix ` ~ `:50 := equiv\n\ntheorem destruct_congr {s t : wseq α} :\n  s ~ t → computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct\n\ntheorem destruct_congr_iff {s t : wseq α} :\n  s ~ t ↔ computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct_iff\n\ntheorem lift_rel.refl (R : α → α → Prop) (H : reflexive R) : reflexive (lift_rel R) :=\nλ s, begin\n  refine ⟨(=), rfl, λ s t (h : s = t), _⟩,\n  rw ←h, apply computation.lift_rel.refl,\n  intro a, cases a with a, simp, cases a; simp, apply H\nend\n\ntheorem lift_rel_o.swap (R : α → β → Prop) (C) :\n  swap (lift_rel_o R C) = lift_rel_o (swap R) (swap C) :=\nby funext x y; cases x with x; [skip, cases x]; { cases y with y; [skip, cases y]; refl }\n\ntheorem lift_rel.swap_lem {R : α → β → Prop} {s1 s2} (h : lift_rel R s1 s2) :\n  lift_rel (swap R) s2 s1 :=\nbegin\n  refine ⟨swap (lift_rel R), h, λ s t (h : lift_rel R t s), _⟩,\n  rw [←lift_rel_o.swap, computation.lift_rel.swap],\n  apply lift_rel_destruct h\nend\n\ntheorem lift_rel.swap (R : α → β → Prop) :\n  swap (lift_rel R) = lift_rel (swap R) :=\nfunext $ λ x, funext $ λ y, propext ⟨lift_rel.swap_lem, lift_rel.swap_lem⟩\n\ntheorem lift_rel.symm (R : α → α → Prop) (H : symmetric R) : symmetric (lift_rel R) :=\nλ s1 s2 (h : swap (lift_rel R) s2 s1),\nby rwa [lift_rel.swap, show swap R = R, from\n        funext $ λ a, funext $ λ b, propext $ by constructor; apply H] at h\n\ntheorem lift_rel.trans (R : α → α → Prop) (H : transitive R) : transitive (lift_rel R) :=\nλ s t u h1 h2, begin\n  refine ⟨λ s u, ∃ t, lift_rel R s t ∧ lift_rel R t u, ⟨t, h1, h2⟩, λ s u h, _⟩,\n  rcases h with ⟨t, h1, h2⟩,\n  have h1 := lift_rel_destruct h1,\n  have h2 := lift_rel_destruct h2,\n  refine computation.lift_rel_def.2\n    ⟨(computation.terminates_of_lift_rel h1).trans\n     (computation.terminates_of_lift_rel h2), λ a c ha hc, _⟩,\n  rcases h1.left ha with ⟨b, hb, t1⟩,\n  have t2 := computation.rel_of_lift_rel h2 hb hc,\n  cases a with a; cases c with c,\n  { trivial },\n  { cases b, {cases t2}, {cases t1} },\n  { cases a, cases b with b, {cases t1}, {cases b, cases t2} },\n  { cases a with a s, cases b with b, {cases t1},\n    cases b with b t, cases c with c u,\n    cases t1 with ab st, cases t2 with bc tu,\n    exact ⟨H ab bc, t, st, tu⟩ }\nend\n\ntheorem lift_rel.equiv (R : α → α → Prop) : equivalence R → equivalence (lift_rel R)\n| ⟨refl, symm, trans⟩ :=\n  ⟨lift_rel.refl R refl, lift_rel.symm R symm, lift_rel.trans R trans⟩\n\n@[refl] theorem equiv.refl : ∀ (s : wseq α), s ~ s :=\nlift_rel.refl (=) eq.refl\n\n@[symm] theorem equiv.symm : ∀ {s t : wseq α}, s ~ t → t ~ s :=\nlift_rel.symm (=) (@eq.symm _)\n\n@[trans] \n\ntheorem equiv.equivalence : equivalence (@equiv α) :=\n⟨@equiv.refl _, @equiv.symm _, @equiv.trans _⟩\n\nopen computation\nlocal notation `return` := computation.return\n\n@[simp] theorem destruct_nil : destruct (nil : wseq α) = return none :=\ncomputation.destruct_eq_ret rfl\n\n@[simp] theorem destruct_cons (a : α) (s) : destruct (cons a s) = return (some (a, s)) :=\ncomputation.destruct_eq_ret $ by simp [destruct, cons, computation.rmap]\n\n@[simp] theorem destruct_think (s : wseq α) : destruct (think s) = (destruct s).think :=\ncomputation.destruct_eq_think $ by simp [destruct, think, computation.rmap]\n\n@[simp] theorem seq_destruct_nil : seq.destruct (nil : wseq α) = none :=\nseq.destruct_nil\n\n@[simp] theorem seq_destruct_cons (a : α) (s) : seq.destruct (cons a s) = some (some a, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem seq_destruct_think (s : wseq α) : seq.destruct (think s) = some (none, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem head_nil : head (nil : wseq α) = return none := by simp [head]; refl\n@[simp] theorem head_cons (a : α) (s) : head (cons a s) = return (some a) := by simp [head]; refl\n@[simp] theorem head_think (s : wseq α) : head (think s) = (head s).think := by simp [head]; refl\n\n@[simp] theorem flatten_ret (s : wseq α) : flatten (return s) = s :=\nbegin\n  refine seq.eq_of_bisim (λs1 s2, flatten (return s2) = s1) _ rfl,\n  intros s' s h, rw ←h, simp [flatten],\n  cases seq.destruct s, { simp },\n  { cases val with o s', simp }\nend\n\n@[simp] theorem flatten_think (c : computation (wseq α)) : flatten c.think = think (flatten c) :=\nseq.destruct_eq_cons $ by simp [flatten, think]\n\n@[simp]\ntheorem destruct_flatten (c : computation (wseq α)) : destruct (flatten c) = c >>= destruct :=\nbegin\n  refine computation.eq_of_bisim (λc1 c2, c1 = c2 ∨\n    ∃ c, c1 = destruct (flatten c) ∧ c2 = computation.bind c destruct) _ (or.inr ⟨c, rfl, rfl⟩),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, (or.inr ⟨c, rfl, rfl⟩) := begin\n    apply c.cases_on (λa, _) (λc', _); repeat {simp},\n    { cases (destruct a).destruct; simp },\n    { exact or.inr ⟨c', rfl, rfl⟩ }\n  end end\nend\n\ntheorem head_terminates_iff (s : wseq α) : terminates (head s) ↔ terminates (destruct s) :=\nterminates_map_iff _ (destruct s)\n\n@[simp] theorem tail_nil : tail (nil : wseq α) = nil := by simp [tail]\n@[simp] theorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]\n@[simp] theorem tail_think (s : wseq α) : tail (think s) = (tail s).think := by simp [tail]\n\n@[simp] theorem dropn_nil (n) :\n  drop (nil : wseq α) n = nil := by induction n; simp [*, drop]\n@[simp] theorem dropn_cons (a : α) (s) (n) :\n  drop (cons a s) (n+1) = drop s n := by induction n; simp [*, drop]\n@[simp] theorem dropn_think (s : wseq α) (n) :\n  drop (think s) n = (drop s n).think := by induction n; simp [*, drop]\n\ntheorem dropn_add (s : wseq α) (m) : ∀ n, drop s (m + n) = drop (drop s m) n\n| 0     := rfl\n| (n+1) := congr_arg tail (dropn_add n)\n\ntheorem dropn_tail (s : wseq α) (n) : drop (tail s) n = drop s (n + 1) :=\nby rw add_comm; symmetry; apply dropn_add\n\ntheorem nth_add (s : wseq α) (m n) : nth s (m + n) = nth (drop s m) n :=\ncongr_arg head (dropn_add _ _ _)\n\ntheorem nth_tail (s : wseq α) (n) : nth (tail s) n = nth s (n + 1) :=\ncongr_arg head (dropn_tail _ _)\n\n@[simp] theorem join_nil : join nil = (nil : wseq α) := seq.join_nil\n\n@[simp] theorem join_think (S : wseq (wseq α)) :\n  join (think S) = think (join S) :=\nby { simp [think, join], unfold functor.map, simp [join, seq1.ret] }\n\n@[simp] theorem join_cons (s : wseq α) (S) :\n  join (cons s S) = think (append s (join S)) :=\nby { simp [think, join], unfold functor.map, simp [join, cons, append] }\n\n@[simp] theorem nil_append (s : wseq α) : append nil s = s := seq.nil_append _\n\n@[simp] theorem cons_append (a : α) (s t) :\n  append (cons a s) t = cons a (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem think_append (s t : wseq α) :\n  append (think s) t = think (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem append_nil (s : wseq α) : append s nil = s := seq.append_nil _\n\n@[simp] theorem append_assoc (s t u : wseq α) :\n  append (append s t) u = append s (append t u) := seq.append_assoc _ _ _\n\n@[simp] def tail.aux : option (α × wseq α) → computation (option (α × wseq α))\n| none          := return none\n| (some (a, s)) := destruct s\n\ntheorem destruct_tail (s : wseq α) :\n  destruct (tail s) = destruct s >>= tail.aux :=\nbegin\n  simp [tail], rw [← bind_pure_comp_eq_map, is_lawful_monad.bind_assoc],\n  apply congr_arg, ext1 (_|⟨a, s⟩);\n  apply (@pure_bind computation _ _ _ _ _ _).trans _; simp\nend\n\n@[simp] def drop.aux : ℕ → option (α × wseq α) → computation (option (α × wseq α))\n| 0     := return\n| (n+1) := λ a, tail.aux a >>= drop.aux n\n\ntheorem drop.aux_none : ∀ n, @drop.aux α n none = return none\n| 0     := rfl\n| (n+1) := show computation.bind (return none) (drop.aux n) = return none,\n           by rw [ret_bind, drop.aux_none]\n\ntheorem destruct_dropn :\n  ∀ (s : wseq α) n, destruct (drop s n) = destruct s >>= drop.aux n\n| s 0     := (bind_ret' _).symm\n| s (n+1) := by rw [← dropn_tail, destruct_dropn _ n,\n  destruct_tail, is_lawful_monad.bind_assoc]; refl\n\ntheorem head_terminates_of_head_tail_terminates (s : wseq α) [T : terminates (head (tail s))] :\n  terminates (head s) :=\n(head_terminates_iff _).2 $ begin\n  rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩,\n  simp [tail] at h,\n  rcases exists_of_mem_bind h with ⟨s', h1, h2⟩,\n  unfold functor.map at h1,\n  exact let ⟨t, h3, h4⟩ := exists_of_mem_map h1 in terminates_of_mem h3\nend\n\ntheorem destruct_some_of_destruct_tail_some {s : wseq α} {a}\n  (h : some a ∈ destruct (tail s)) : ∃ a', some a' ∈ destruct s :=\nbegin\n  unfold tail functor.map at h, simp at h,\n  rcases exists_of_mem_bind h with ⟨t, tm, td⟩, clear h,\n  rcases exists_of_mem_map tm with ⟨t', ht', ht2⟩, clear tm,\n  cases t' with t'; rw ←ht2 at td; simp at td,\n  { have := mem_unique td (ret_mem _), contradiction },\n  { exact ⟨_, ht'⟩ }\nend\n\ntheorem head_some_of_head_tail_some {s : wseq α} {a}\n  (h : some a ∈ head (tail s)) : ∃ a', some a' ∈ head s :=\nbegin\n  unfold head at h,\n  rcases exists_of_mem_map h with ⟨o, md, e⟩, clear h,\n  cases o with o; injection e with h', clear e h',\n  cases destruct_some_of_destruct_tail_some md with a am,\n  exact ⟨_, mem_map ((<$>) (@prod.fst α (wseq α))) am⟩\nend\n\ntheorem head_some_of_nth_some {s : wseq α} {a n}\n  (h : some a ∈ nth s n) : ∃ a', some a' ∈ head s :=\nbegin\n  revert a, induction n with n IH; intros,\n  exacts [⟨_, h⟩, let ⟨a', h'⟩ := head_some_of_head_tail_some h in IH h']\nend\n\ninstance productive_tail (s : wseq α) [productive s] : productive (tail s) :=\n⟨λ n, by rw [nth_tail]; apply_instance⟩\n\ninstance productive_dropn (s : wseq α) [productive s] (n) : productive (drop s n) :=\n⟨λ m, by rw [←nth_add]; apply_instance⟩\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef to_seq (s : wseq α) [productive s] : seq α :=\n⟨λ n, (nth s n).get, λn h,\nbegin\n  cases e : computation.get (nth s (n + 1)), {assumption},\n  have := mem_of_get_eq _ e,\n  simp [nth] at this h, cases head_some_of_head_tail_some this with a' h',\n  have := mem_unique h' (@mem_of_get_eq _ _ _ _ h),\n  contradiction\nend⟩\n\ntheorem nth_terminates_le {s : wseq α} {m n} (h : m ≤ n) :\n  terminates (nth s n) → terminates (nth s m) :=\nby induction h with m' h IH; [exact id,\n  exact λ T, IH (@head_terminates_of_head_tail_terminates _ _ T)]\n\ntheorem head_terminates_of_nth_terminates {s : wseq α} {n} :\n  terminates (nth s n) → terminates (head s) :=\nnth_terminates_le (nat.zero_le n)\n\ntheorem destruct_terminates_of_nth_terminates {s : wseq α} {n} (T : terminates (nth s n)) :\n  terminates (destruct s) :=\n(head_terminates_iff _).1 $ head_terminates_of_nth_terminates T\n\ntheorem mem_rec_on {C : wseq α → Prop} {a s} (M : a ∈ s)\n  (h1 : ∀ b s', (a = b ∨ C s') → C (cons b s'))\n  (h2 : ∀ s, C s → C (think s)) : C s :=\nbegin\n  apply seq.mem_rec_on M,\n  intros o s' h, cases o with b,\n  { apply h2, cases h, {contradiction}, {assumption} },\n  { apply h1, apply or.imp_left _ h, intro h, injection h }\nend\n\n@[simp] theorem mem_think (s : wseq α) (a) : a ∈ think s ↔ a ∈ s :=\nbegin\n  cases s with f al,\n  change some (some a) ∈ some none :: f ↔ some (some a) ∈ f,\n  constructor; intro h,\n  { apply (stream.eq_or_mem_of_mem_cons h).resolve_left,\n    intro, injections },\n  { apply stream.mem_cons_of_mem _ h }\nend\n\ntheorem eq_or_mem_iff_mem {s : wseq α} {a a' s'} :\n  some (a', s') ∈ destruct s → (a ∈ s ↔ a = a' ∨ a ∈ s') :=\nbegin\n  generalize e : destruct s = c, intro h,\n  revert s, apply computation.mem_rec_on h _ (λ c IH, _); intro s;\n  apply s.cases_on _ (λ x s, _) (λ s, _); intros m;\n  have := congr_arg computation.destruct m; simp at this;\n  cases this with i1 i2,\n  { rw [i1, i2],\n    cases s' with f al,\n    unfold cons has_mem.mem wseq.mem seq.mem seq.cons, simp,\n    have h_a_eq_a' : a = a' ↔ some (some a) = some (some a'), {simp},\n    rw [h_a_eq_a'],\n    refine ⟨stream.eq_or_mem_of_mem_cons, λo, _⟩,\n    { cases o with e m,\n      { rw e, apply stream.mem_cons },\n      { exact stream.mem_cons_of_mem _ m } } },\n  { simp, exact IH this }\nend\n\n@[simp] theorem mem_cons_iff (s : wseq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=\neq_or_mem_iff_mem $ by simp [ret_mem]\n\ntheorem mem_cons_of_mem {s : wseq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=\n(mem_cons_iff _ _).2 (or.inr h)\n\ntheorem mem_cons (s : wseq α) (a) : a ∈ cons a s :=\n(mem_cons_iff _ _).2 (or.inl rfl)\n\ntheorem mem_of_mem_tail {s : wseq α} {a} : a ∈ tail s → a ∈ s :=\nbegin\n  intro h, have := h, cases h with n e, revert s, simp [stream.nth],\n  induction n with n IH; intro s; apply s.cases_on _ (λx s, _) (λ s, _);\n    repeat{simp}; intros m e; injections,\n  { exact or.inr m },\n  { exact or.inr m },\n  { apply IH m, rw e, cases tail s, refl }\nend\n\ntheorem mem_of_mem_dropn {s : wseq α} {a} : ∀ {n}, a ∈ drop s n → a ∈ s\n| 0     h := h\n| (n+1) h := @mem_of_mem_dropn n (mem_of_mem_tail h)\n\ntheorem nth_mem {s : wseq α} {a n} : some a ∈ nth s n → a ∈ s :=\nbegin\n  revert s, induction n with n IH; intros s h,\n  { rcases exists_of_mem_map h with ⟨o, h1, h2⟩,\n    cases o with o; injection h2 with h',\n    cases o with a' s',\n    exact (eq_or_mem_iff_mem h1).2 (or.inl h'.symm) },\n  { have := @IH (tail s), rw nth_tail at this,\n    exact mem_of_mem_tail (this h) }\nend\n\ntheorem exists_nth_of_mem {s : wseq α} {a} (h : a ∈ s) : ∃ n, some a ∈ nth s n :=\nbegin\n  apply mem_rec_on h,\n  { intros a' s' h, cases h with h h,\n    { existsi 0, simp [nth], rw h, apply ret_mem },\n    { cases h with n h, existsi n+1,\n      simp [nth], exact h } },\n  { intros s' h, cases h with n h,\n    existsi n, simp [nth], apply think_mem h }\nend\n\ntheorem exists_dropn_of_mem {s : wseq α} {a} (h : a ∈ s) :\n  ∃ n s', some (a, s') ∈ destruct (drop s n) :=\nlet ⟨n, h⟩ := exists_nth_of_mem h in ⟨n, begin\n  rcases (head_terminates_iff _).1 ⟨⟨_, h⟩⟩ with ⟨⟨o, om⟩⟩,\n  have := mem_unique (mem_map _ om) h,\n  cases o with o; injection this with i,\n  cases o with a' s', dsimp at i,\n  rw i at om, exact ⟨_, om⟩\nend⟩\n\ntheorem lift_rel_dropn_destruct {R : α → β → Prop} {s t} (H : lift_rel R s t) :\n  ∀ n, computation.lift_rel (lift_rel_o R (lift_rel R))\n    (destruct (drop s n)) (destruct (drop t n))\n| 0     := lift_rel_destruct H\n| (n+1) := begin\n  simp [destruct_tail],\n  apply lift_rel_bind,\n  apply lift_rel_dropn_destruct n,\n  exact λ a b o, match a, b, o with\n  | none,       none,         _        := by simp\n  | some (a, s), some (b, t), ⟨h1, h2⟩ := by simp [tail.aux]; apply lift_rel_destruct h2\n  end\nend\n\ntheorem exists_of_lift_rel_left {R : α → β → Prop} {s t}\n  (H : lift_rel R s t) {a} (h : a ∈ s) : ∃ {b}, b ∈ t ∧ R a b :=\nlet ⟨n, h⟩ := exists_nth_of_mem h,\n    ⟨some (._, s'), sd, rfl⟩ := exists_of_mem_map h,\n    ⟨some (b, t'), td, ⟨ab, _⟩⟩ := (lift_rel_dropn_destruct H n).left sd in\n⟨b, nth_mem (mem_map ((<$>) prod.fst.{v v}) td), ab⟩\n\ntheorem exists_of_lift_rel_right {R : α → β → Prop} {s t}\n  (H : lift_rel R s t) {b} (h : b ∈ t) : ∃ {a}, a ∈ s ∧ R a b :=\nby rw ←lift_rel.swap at H; exact exists_of_lift_rel_left H h\n\ntheorem head_terminates_of_mem {s : wseq α} {a} (h : a ∈ s) : terminates (head s) :=\nlet ⟨n, h⟩ := exists_nth_of_mem h in head_terminates_of_nth_terminates ⟨⟨_, h⟩⟩\n\ntheorem of_mem_append {s₁ s₂ : wseq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=\nseq.of_mem_append\n\ntheorem mem_append_left {s₁ s₂ : wseq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=\nseq.mem_append_left\n\ntheorem exists_of_mem_map {f} {b : β} : ∀ {s : wseq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b\n| ⟨g, al⟩ h := let ⟨o, om, oe⟩ := seq.exists_of_mem_map h in\n  by cases o with a; injection oe with h'; exact ⟨a, om, h'⟩\n\n@[simp] theorem lift_rel_nil (R : α → β → Prop) : lift_rel R nil nil :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_cons (R : α → β → Prop) (a b s t) :\n  lift_rel R (cons a s) (cons b t) ↔ R a b ∧ lift_rel R s t :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_left (R : α → β → Prop) (s t) :\n  lift_rel R (think s) t ↔ lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_right (R : α → β → Prop) (s t) :\n  lift_rel R s (think t) ↔ lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\ntheorem cons_congr {s t : wseq α} (a : α) (h : s ~ t) : cons a s ~ cons a t :=\nby unfold equiv; simp; exact h\n\ntheorem think_equiv (s : wseq α) : think s ~ s :=\nby unfold equiv; simp; apply equiv.refl\n\ntheorem think_congr {s t : wseq α} (a : α) (h : s ~ t) : think s ~ think t :=\nby unfold equiv; simp; exact h\n\ntheorem head_congr : ∀ {s t : wseq α}, s ~ t → head s ~ head t :=\nsuffices ∀ {s t : wseq α}, s ~ t → ∀ {o}, o ∈ head s → o ∈ head t, from\nλ s t h o, ⟨this h, this h.symm⟩,\nbegin\n  intros s t h o ho,\n  rcases @computation.exists_of_mem_map _ _ _ _ (destruct s) ho with ⟨ds, dsm, dse⟩,\n  rw ←dse,\n  cases destruct_congr h with l r,\n  rcases l dsm with ⟨dt, dtm, dst⟩,\n  cases ds with a; cases dt with b,\n  { apply mem_map _ dtm },\n  { cases b, cases dst },\n  { cases a, cases dst },\n  { cases a with a s', cases b with b t', rw dst.left,\n    exact @mem_map _ _ (@functor.map _ _ (α × wseq α) _ prod.fst)\n      _ (destruct t) dtm }\nend\n\ntheorem flatten_equiv {c : computation (wseq α)} {s} (h : s ∈ c) : flatten c ~ s :=\nbegin\n  apply computation.mem_rec_on h, { simp },\n  { intro s', apply equiv.trans, simp [think_equiv] }\nend\n\ntheorem lift_rel_flatten {R : α → β → Prop} {c1 : computation (wseq α)} {c2 : computation (wseq β)}\n  (h : c1.lift_rel (lift_rel R) c2) : lift_rel R (flatten c1) (flatten c2) :=\nlet S := λ s t,\n  ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ computation.lift_rel (lift_rel R) c1 c2 in\n⟨S, ⟨c1, c2, rfl, rfl, h⟩, λ s t h,\n  match s, t, h with ._, ._, ⟨c1, c2, rfl, rfl, h⟩ := begin\n    simp, apply lift_rel_bind _ _ h,\n    intros a b ab, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct ab),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t h, refine ⟨return s, return t, _, _, _⟩; simp [h]\n  end end⟩\n\ntheorem flatten_congr {c1 c2 : computation (wseq α)} :\n  computation.lift_rel equiv c1 c2 → flatten c1 ~ flatten c2 := lift_rel_flatten\n\ntheorem tail_congr {s t : wseq α} (h : s ~ t) : tail s ~ tail t :=\nbegin\n  apply flatten_congr,\n  unfold functor.map, rw [←bind_ret, ←bind_ret],\n  apply lift_rel_bind _ _ (destruct_congr h),\n  intros a b h, simp,\n  cases a with a; cases b with b,\n  { trivial },\n  { cases h },\n  { cases a, cases h },\n  { cases a with a s', cases b with b t', exact h.right }\nend\n\ntheorem dropn_congr {s t : wseq α} (h : s ~ t) (n) : drop s n ~ drop t n :=\nby induction n; simp [*, tail_congr]\n\ntheorem nth_congr {s t : wseq α} (h : s ~ t) (n) : nth s n ~ nth t n :=\nhead_congr (dropn_congr h _)\n\ntheorem mem_congr {s t : wseq α} (h : s ~ t) (a) : a ∈ s ↔ a ∈ t :=\nsuffices ∀ {s t : wseq α}, s ~ t → a ∈ s → a ∈ t, from ⟨this h, this h.symm⟩,\nλ s t h as, let ⟨n, hn⟩ := exists_nth_of_mem as in\nnth_mem ((nth_congr h _ _).1 hn)\n\ntheorem productive_congr {s t : wseq α} (h : s ~ t) : productive s ↔ productive t :=\nby simp only [productive_iff]; exact\n  forall_congr (λ n, terminates_congr $ nth_congr h _)\n\ntheorem equiv.ext {s t : wseq α} (h : ∀ n, nth s n ~ nth t n) : s ~ t :=\n⟨λ s t, ∀ n, nth s n ~ nth t n, h, λs t h, begin\n  refine lift_rel_def.2 ⟨_, _⟩,\n  { rw [←head_terminates_iff, ←head_terminates_iff],\n    exact terminates_congr (h 0) },\n  { intros a b ma mb,\n    cases a with a; cases b with b,\n    { trivial },\n    { injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) },\n    { injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) },\n    { cases a with a s', cases b with b t',\n      injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) with ab,\n      refine ⟨ab, λ n, _⟩,\n      refine (nth_congr (flatten_equiv (mem_map _ ma)) n).symm.trans\n        ((_ : nth (tail s) n ~ nth (tail t) n).trans\n        (nth_congr (flatten_equiv (mem_map _ mb)) n)),\n      rw [nth_tail, nth_tail], apply h } }\nend⟩\n\ntheorem length_eq_map (s : wseq α) : length s = computation.map list.length (to_list s) :=\nbegin\n  refine eq_of_bisim\n    (λ c1 c2, ∃ (l : list α) (s : wseq α),\n      c1 = corec length._match_2 (l.length, s) ∧\n      c2 = computation.map list.length (corec to_list._match_2 (l, s)))\n    _ ⟨[], s, rfl, rfl⟩,\n  intros s1 s2 h, rcases h with ⟨l, s, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine ⟨a::l, s, _, _⟩; simp },\n  { refine ⟨l, s, _, _⟩; simp }\nend\n\n@[simp] theorem of_list_nil : of_list [] = (nil : wseq α) := rfl\n\n@[simp] theorem of_list_cons (a : α) (l) :\n  of_list (a :: l) = cons a (of_list l) :=\nshow seq.map some (seq.of_list (a :: l)) =\n     seq.cons (some a) (seq.map some (seq.of_list l)), by simp\n\n@[simp] theorem to_list'_nil (l : list α) :\n  corec to_list._match_2 (l, nil) = return l.reverse :=\ndestruct_eq_ret rfl\n\n@[simp] theorem to_list'_cons (l : list α) (s : wseq α) (a : α) :\n  corec to_list._match_2 (l, cons a s) =\n  (corec to_list._match_2 (a::l, s)).think :=\ndestruct_eq_think $ by simp [to_list, cons]\n\n@[simp] theorem to_list'_think (l : list α) (s : wseq α) :\n  corec to_list._match_2 (l, think s) =\n  (corec to_list._match_2 (l, s)).think :=\ndestruct_eq_think $ by simp [to_list, think]\n\ntheorem to_list'_map (l : list α) (s : wseq α) :\n  corec to_list._match_2 (l, s) =\n  ((++) l.reverse) <$> to_list s :=\nbegin\n  refine eq_of_bisim\n    (λ c1 c2, ∃ (l' : list α) (s : wseq α),\n      c1 = corec to_list._match_2 (l' ++ l, s) ∧\n      c2 = computation.map ((++) l.reverse) (corec to_list._match_2 (l', s)))\n    _ ⟨[], s, rfl, rfl⟩,\n  intros s1 s2 h, rcases h with ⟨l', s, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine ⟨a::l', s, _, _⟩; simp },\n  { refine ⟨l', s, _, _⟩; simp }\nend\n\n@[simp] theorem to_list_cons (a : α) (s) :\n  to_list (cons a s) = (list.cons a <$> to_list s).think :=\ndestruct_eq_think $ by unfold to_list; simp; rw to_list'_map; simp; refl\n\n@[simp] theorem to_list_nil : to_list (nil : wseq α) = return [] :=\ndestruct_eq_ret rfl\n\ntheorem to_list_of_list (l : list α) : l ∈ to_list (of_list l) :=\nby induction l with a l IH; simp [ret_mem]; exact think_mem (mem_map _ IH)\n\n@[simp] theorem destruct_of_seq (s : seq α) :\n  destruct (of_seq s) = return (s.head.map $ λ a, (a, of_seq s.tail)) :=\ndestruct_eq_ret $ begin\n  simp [of_seq, head, destruct, seq.destruct, seq.head],\n  rw [show seq.nth (some <$> s) 0 = some <$> seq.nth s 0, by apply seq.map_nth],\n  cases seq.nth s 0 with a, { refl },\n  unfold functor.map,\n  simp [destruct]\nend\n\n@[simp] theorem head_of_seq (s : seq α) : head (of_seq s) = return s.head :=\nby simp [head]; cases seq.head s; refl\n\n@[simp] theorem tail_of_seq (s : seq α) : tail (of_seq s) = of_seq s.tail :=\nbegin\n  simp [tail], apply s.cases_on _ (λ x s, _); simp [of_seq], {refl},\n  rw [seq.head_cons, seq.tail_cons], refl\nend\n\n@[simp] theorem dropn_of_seq (s : seq α) : ∀ n, drop (of_seq s) n = of_seq (s.drop n)\n| 0 := rfl\n| (n+1) := by dsimp [drop]; rw [dropn_of_seq, tail_of_seq]\n\ntheorem nth_of_seq (s : seq α) (n) : nth (of_seq s) n = return (seq.nth s n) :=\nby dsimp [nth]; rw [dropn_of_seq, head_of_seq, seq.head_dropn]\n\ninstance productive_of_seq (s : seq α) : productive (of_seq s) :=\n⟨λ n, by rw nth_of_seq; apply_instance⟩\n\ntheorem to_seq_of_seq (s : seq α) : to_seq (of_seq s) = s :=\nbegin\n  apply subtype.eq, funext n,\n  dsimp [to_seq], apply get_eq_of_mem,\n  rw nth_of_seq, apply ret_mem\nend\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret (a : α) : wseq α := of_list [a]\n\n@[simp] theorem map_nil (f : α → β) : map f nil = nil := rfl\n\n@[simp] theorem map_cons (f : α → β) (a s) :\n  map f (cons a s) = cons (f a) (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_think (f : α → β) (s) :\n  map f (think s) = think (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_id (s : wseq α) : map id s = s := by simp [map]\n\n@[simp] theorem map_ret (f : α → β) (a) : map f (ret a) = ret (f a) := by simp [ret]\n\n@[simp] theorem map_append (f : α → β) (s t) : map f (append s t) = append (map f s) (map f t) :=\nseq.map_append _ _ _\n\ntheorem map_comp (f : α → β) (g : β → γ) (s : wseq α) :\n  map (g ∘ f) s = map g (map f s) :=\nbegin\n  dsimp [map], rw ←seq.map_comp,\n  apply congr_fun, apply congr_arg,\n  ext ⟨⟩; refl\nend\n\ntheorem mem_map (f : α → β) {a : α} {s : wseq α} : a ∈ s → f a ∈ map f s :=\nseq.mem_map (option.map f)\n\n-- The converse is not true without additional assumptions\ntheorem exists_of_mem_join {a : α} : ∀ {S : wseq (wseq α)}, a ∈ join S → ∃ s, s ∈ S ∧ a ∈ s :=\nsuffices ∀ ss : wseq α, a ∈ ss → ∀ s S, append s (join S) = ss →\n  a ∈ append s (join S) → a ∈ s ∨ ∃ s, s ∈ S ∧ a ∈ s, from λ S h,\n  (this _ h nil S (by simp) (by simp [h])).resolve_left (not_mem_nil _),\nbegin\n  intros ss h, apply mem_rec_on h (λ b ss o, _) (λ ss IH, _); intros s S,\n  { refine s.cases_on (S.cases_on _ (λ s S, _) (λ S, _)) (λ b' s, _) (λ s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try {cases this}; try {contradiction},\n    substs b' ss,\n    simp at m ⊢,\n    cases o with e IH, { simp [e] },\n    cases m with e m, { simp [e] },\n    exact or.imp_left or.inr (IH _ _ rfl m) },\n  { refine s.cases_on (S.cases_on _ (λ s S, _) (λ S, _)) (λ b' s, _) (λ s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try { try {have := this.1}, contradiction }; subst ss,\n    { apply or.inr, simp at m ⊢,\n      cases IH s S rfl m with as ex,\n      { exact ⟨s, or.inl rfl, as⟩ },\n      { rcases ex with ⟨s', sS, as⟩,\n        exact ⟨s', or.inr sS, as⟩ } },\n    { apply or.inr, simp at m,\n      rcases (IH nil S (by simp) (by simp [m])).resolve_left (not_mem_nil _) with ⟨s, sS, as⟩,\n      exact ⟨s, by simp [sS], as⟩ },\n    { simp at m IH ⊢, apply IH _ _ rfl m } }\nend\n\ntheorem exists_of_mem_bind {s : wseq α} {f : α → wseq β} {b}\n  (h : b ∈ bind s f) : ∃ a ∈ s, b ∈ f a :=\nlet ⟨t, tm, bt⟩ := exists_of_mem_join h,\n    ⟨a, as, e⟩ := exists_of_mem_map tm in ⟨a, as, by rwa e⟩\n\ntheorem destruct_map (f : α → β) (s : wseq α) :\n  destruct (map f s) = computation.map (option.map (prod.map f (map f))) (destruct s) :=\nbegin\n  apply eq_of_bisim (λ c1 c2, ∃ s, c1 = destruct (map f s) ∧\n    c2 = computation.map (option.map (prod.map f (map f))) (destruct s)),\n  { intros c1 c2 h, cases h with s h, rw [h.left, h.right],\n    apply s.cases_on _ (λ a s, _) (λ s, _); simp,\n    exact ⟨s, rfl, rfl⟩ },\n  { exact ⟨s, rfl, rfl⟩ }\nend\n\ntheorem lift_rel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop)\n  {s1 : wseq α} {s2 : wseq β}\n  {f1 : α → γ} {f2 : β → δ}\n  (h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → S (f1 a) (f2 b))\n  : lift_rel S (map f1 s1) (map f2 s2) :=\n⟨λ s1 s2, ∃ s t, s1 = map f1 s ∧ s2 = map f2 t ∧ lift_rel R s t,\n⟨s1, s2, rfl, rfl, h1⟩,\nλ s1 s2 h, match s1, s2, h with ._, ._, ⟨s, t, rfl, rfl, h⟩ := begin\n  simp [destruct_map], apply computation.lift_rel_map _ _ (lift_rel_destruct h),\n  intros o p h,\n  cases o with a; cases p with b; simp,\n  { cases b; cases h },\n  { cases a; cases h },\n  { cases a with a s; cases b with b t, cases h with r h,\n    exact ⟨h2 r, s, rfl, t, rfl, h⟩ }\nend end⟩\n\ntheorem map_congr (f : α → β) {s t : wseq α} (h : s ~ t) : map f s ~ map f t :=\nlift_rel_map _ _ h (λ _ _, congr_arg _)\n\n@[simp] def destruct_append.aux (t : wseq α) :\n  option (α × wseq α) → computation (option (α × wseq α))\n| none          := destruct t\n| (some (a, s)) := return (some (a, append s t))\n\ntheorem destruct_append (s t : wseq α) :\n  destruct (append s t) = (destruct s).bind (destruct_append.aux t) :=\nbegin\n  apply eq_of_bisim (λ c1 c2, ∃ s t, c1 = destruct (append s t) ∧\n    c2 = (destruct s).bind (destruct_append.aux t)) _ ⟨s, t, rfl, rfl⟩,\n  intros c1 c2 h, rcases h with ⟨s, t, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _); simp,\n  { apply t.cases_on _ (λ b t, _) (λ t, _); simp,\n    { refine ⟨nil, t, _, _⟩; simp } },\n  { exact ⟨s, t, rfl, rfl⟩ }\nend\n\n@[simp] def destruct_join.aux : option (wseq α × wseq (wseq α)) → computation (option (α × wseq α))\n| none          := return none\n| (some (s, S)) := (destruct (append s (join S))).think\n\ntheorem destruct_join (S : wseq (wseq α)) :\n  destruct (join S) = (destruct S).bind destruct_join.aux :=\nbegin\n  apply eq_of_bisim (λ c1 c2, c1 = c2 ∨ ∃ S, c1 = destruct (join S) ∧\n    c2 = (destruct S).bind destruct_join.aux) _ (or.inr ⟨S, rfl, rfl⟩),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, or.inr ⟨S, rfl, rfl⟩ := begin\n    apply S.cases_on _ (λ s S, _) (λ S, _); simp,\n    { refine or.inr ⟨S, rfl, rfl⟩ }\n  end end\nend\n\ntheorem lift_rel_append (R : α → β → Prop) {s1 s2 : wseq α} {t1 t2 : wseq β}\n  (h1 : lift_rel R s1 t1) (h2 : lift_rel R s2 t2) :\n  lift_rel R (append s1 s2) (append t1 t2) :=\n⟨λ s t, lift_rel R s t ∨ ∃ s1 t1, s = append s1 s2 ∧ t = append t1 t2 ∧ lift_rel R s1 t1,\nor.inr ⟨s1, t1, rfl, rfl, h1⟩,\nλ s t h, match s, t, h with\n| s, t, or.inl h := begin\n    apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end\n| ._, ._, or.inr ⟨s1, t1, rfl, rfl, h⟩ := begin\n    simp [destruct_append],\n    apply computation.lift_rel_bind _ _ (lift_rel_destruct h),\n    intros o p h,\n    cases o with a; cases p with b,\n    { simp, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h2),\n      intros a b, apply lift_rel_o.imp_right,\n      intros s t, apply or.inl },\n    { cases b; cases h },\n    { cases a; cases h },\n    { cases a with a s; cases b with b t, cases h with r h,\n      simp, exact ⟨r, or.inr ⟨s, rfl, t, rfl, h⟩⟩ }\n  end\nend⟩\n\ntheorem lift_rel_join.lem (R : α → β → Prop) {S T} {U : wseq α → wseq β → Prop}\n  (ST : lift_rel (lift_rel R) S T) (HU : ∀ s1 s2, (∃ s t S T,\n      s1 = append s (join S) ∧ s2 = append t (join T) ∧\n      lift_rel R s t ∧ lift_rel (lift_rel R) S T) → U s1 s2) {a} (ma : a ∈ destruct (join S)) :\n  ∃ {b}, b ∈ destruct (join T) ∧ lift_rel_o R U a b :=\nbegin\n  cases exists_results_of_mem ma with n h, clear ma, revert a S T,\n  apply nat.strong_induction_on n _,\n  intros n IH a S T ST ra, simp [destruct_join] at ra, exact\n  let ⟨o, m, k, rs1, rs2, en⟩ := of_results_bind ra,\n      ⟨p, mT, rop⟩ := computation.exists_of_lift_rel_left (lift_rel_destruct ST) rs1.mem in\n  by exact match o, p, rop, rs1, rs2, mT with\n  | none, none, _, rs1, rs2, mT := by simp only [destruct_join]; exact\n    ⟨none, mem_bind mT (ret_mem _), by rw eq_of_ret_mem rs2.mem; trivial⟩\n  | some (s, S'), some (t, T'), ⟨st, ST'⟩, rs1, rs2, mT :=\n    by simp [destruct_append] at rs2; exact\n    let ⟨k1, rs3, ek⟩ := of_results_think rs2,\n        ⟨o', m1, n1, rs4, rs5, ek1⟩ := of_results_bind rs3,\n        ⟨p', mt, rop'⟩ := computation.exists_of_lift_rel_left (lift_rel_destruct st) rs4.mem in\n    by exact match o', p', rop', rs4, rs5, mt with\n    | none, none, _, rs4, rs5', mt :=\n      have n1 < n, begin\n        rw [en, ek, ek1],\n        apply lt_of_lt_of_le _ (nat.le_add_right _ _),\n        apply nat.lt_succ_of_le (nat.le_add_right _ _)\n      end,\n      let ⟨ob, mb, rob⟩ := IH _ this ST' rs5' in by refine ⟨ob, _, rob⟩;\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, exact mb }\n    | some (a, s'), some (b, t'), ⟨ab, st'⟩, rs4, rs5, mt := begin\n      simp at rs5,\n      refine ⟨some (b, append t' (join T')), _, _⟩,\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, apply ret_mem },\n      rw eq_of_ret_mem rs5.mem,\n      exact ⟨ab, HU _ _ ⟨s', t', S', T', rfl, rfl, st', ST'⟩⟩\n    end end\n  end\nend\n\ntheorem lift_rel_join (R : α → β → Prop) {S : wseq (wseq α)} {T : wseq (wseq β)}\n  (h : lift_rel (lift_rel R) S T) : lift_rel R (join S) (join T) :=\n⟨λ s1 s2, ∃ s t S T,\n  s1 = append s (join S) ∧ s2 = append t (join T) ∧\n  lift_rel R s t ∧ lift_rel (lift_rel R) S T,\n  ⟨nil, nil, S, T, by simp, by simp, by simp, h⟩,\nλs1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩, begin\n  clear _fun_match _x,\n  rw [h1, h2], rw [destruct_append, destruct_append],\n  apply computation.lift_rel_bind _ _ (lift_rel_destruct st),\n  exact λ o p h, match o, p, h with\n  | some (a, s), some (b, t), ⟨h1, h2⟩ :=\n    by simp; exact ⟨h1, s, t, S, rfl, T, rfl, h2, ST⟩\n  | none, none, _ := begin\n    dsimp [destruct_append.aux, computation.lift_rel], constructor,\n    { intro, apply lift_rel_join.lem _ ST (λ _ _, id) },\n    { intros b mb,\n      rw [←lift_rel_o.swap], apply lift_rel_join.lem (swap R),\n      { rw [←lift_rel.swap R, ←lift_rel.swap], apply ST },\n      { rw [←lift_rel.swap R, ←lift_rel.swap (lift_rel R)],\n        exact λ s1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩,\n                      ⟨t, s, T, S, h2, h1, st, ST⟩ },\n      { exact mb } }\n  end end\nend⟩\n\ntheorem join_congr {S T : wseq (wseq α)} (h : lift_rel equiv S T) : join S ~ join T :=\nlift_rel_join _ h\n\ntheorem lift_rel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop)\n  {s1 : wseq α} {s2 : wseq β}\n  {f1 : α → wseq γ} {f2 : β → wseq δ}\n  (h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → lift_rel S (f1 a) (f2 b))\n  : lift_rel S (bind s1 f1) (bind s2 f2) :=\nlift_rel_join _ (lift_rel_map _ _ h1 @h2)\n\ntheorem bind_congr {s1 s2 : wseq α} {f1 f2 : α → wseq β}\n  (h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=\nlift_rel_bind _ _ h1 (λ a b h, by rw h; apply h2)\n\n@[simp] theorem join_ret (s : wseq α) : join (ret s) ~ s :=\nby simp [ret]; apply think_equiv\n\n@[simp] theorem join_map_ret (s : wseq α) : join (map ret s) ~ s :=\nbegin\n  refine ⟨λ s1 s2, join (map ret s2) = s1, rfl, _⟩,\n  intros s' s h, rw ←h,\n  apply lift_rel_rec\n    (λ c1 c2, ∃ s,\n      c1 = destruct (join (map ret s)) ∧ c2 = destruct s),\n  { exact λ c1 c2 h, match c1, c2, h with\n    | ._, ._, ⟨s, rfl, rfl⟩ := begin\n      clear h _match,\n      have : ∀ s, ∃ s' : wseq α, (map ret s).join.destruct = (map ret s').join.destruct ∧\n        destruct s = s'.destruct, from λ s, ⟨s, rfl, rfl⟩,\n      apply s.cases_on _ (λ a s, _) (λ s, _); simp [ret, ret_mem, this, option.exists]\n    end end },\n  { exact ⟨s, rfl, rfl⟩ }\nend\n\n@[simp] theorem join_append (S T : wseq (wseq α)) :\n  join (append S T) ~ append (join S) (join T) :=\nbegin\n  refine ⟨λ s1 s2, ∃ s S T,\n    s1 = append s (join (append S T)) ∧\n    s2 = append s (append (join S) (join T)), ⟨nil, S, T, by simp, by simp⟩, _⟩,\n  intros s1 s2 h,\n  apply lift_rel_rec (λ c1 c2, ∃ (s : wseq α) S T,\n    c1 = destruct (append s (join (append S T))) ∧\n    c2 = destruct (append s (append (join S) (join T)))) _ _ _\n    (let ⟨s, S, T, h1, h2⟩ := h in\n         ⟨s, S, T, congr_arg destruct h1, congr_arg destruct h2⟩),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, ⟨s, S, T, rfl, rfl⟩ := begin\n    clear _match h h,\n    apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp,\n    { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp,\n      { apply wseq.cases_on T _ (λ s T, _) (λ T, _); simp,\n        { refine ⟨s, nil, T, _, _⟩; simp },\n        { refine ⟨nil, nil, T, _, _⟩; simp } },\n      { exact ⟨s, S, T, rfl, rfl⟩ },\n      { refine ⟨nil, S, T, _, _⟩; simp } },\n    { exact ⟨s, S, T, rfl, rfl⟩ },\n    { exact ⟨s, S, T, rfl, rfl⟩ }\n  end end\nend\n\n@[simp] theorem bind_ret (f : α → β) (s) : bind s (ret ∘ f) ~ map f s :=\nbegin\n  dsimp [bind], change (λx, ret (f x)) with (ret ∘ f),\n  rw [map_comp], apply join_map_ret\nend\n\n@[simp] theorem ret_bind (a : α) (f : α → wseq β) :\n  bind (ret a) f ~ f a := by simp [bind]\n\n@[simp] theorem map_join (f : α → β) (S) :\n  map f (join S) = join (map (map f) S) :=\nbegin\n  apply seq.eq_of_bisim (λs1 s2,\n    ∃ s S, s1 = append s (map f (join S)) ∧\n      s2 = append s (join (map (map f) S))),\n  { intros s1 s2 h,\n    exact match s1, s2, h with ._, ._, ⟨s, S, rfl, rfl⟩ := begin\n      apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp,\n      { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp,\n        { exact ⟨map f s, S, rfl, rfl⟩ },\n        { refine ⟨nil, S, _, _⟩; simp } },\n      { exact ⟨_, _, rfl, rfl⟩ },\n      { exact ⟨_, _, rfl, rfl⟩ }\n    end end },\n  { refine ⟨nil, S, _, _⟩; simp }\nend\n\n@[simp] theorem join_join (SS : wseq (wseq (wseq α))) :\n  join (join SS) ~ join (map join SS) :=\nbegin\n  refine ⟨λ s1 s2, ∃ s S SS,\n    s1 = append s (join (append S (join SS))) ∧\n    s2 = append s (append (join S) (join (map join SS))),\n    ⟨nil, nil, SS, by simp, by simp⟩, _⟩,\n  intros s1 s2 h,\n  apply lift_rel_rec (λ c1 c2, ∃ s S SS,\n      c1 = destruct (append s (join (append S (join SS)))) ∧\n      c2 = destruct (append s (append (join S) (join (map join SS)))))\n    _ (destruct s1) (destruct s2)\n    (let ⟨s, S, SS, h1, h2⟩ := h in ⟨s, S, SS, by simp [h1], by simp [h2]⟩),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, ⟨s, S, SS, rfl, rfl⟩ := begin\n    clear _match h h,\n    apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp,\n    { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp,\n      { apply wseq.cases_on SS _ (λ S SS, _) (λ SS, _); simp,\n        { refine ⟨nil, S, SS, _, _⟩; simp },\n        { refine ⟨nil, nil, SS, _, _⟩; simp } },\n      { exact ⟨s, S, SS, rfl, rfl⟩ },\n      { refine ⟨nil, S, SS, _, _⟩; simp } },\n    { exact ⟨s, S, SS, rfl, rfl⟩ },\n    { exact ⟨s, S, SS, rfl, rfl⟩ }\n  end end\nend\n\n@[simp] theorem bind_assoc (s : wseq α) (f : α → wseq β) (g : β → wseq γ) :\n  bind (bind s f) g ~ bind s (λ (x : α), bind (f x) g) :=\nbegin\n  simp [bind], rw [← map_comp f (map g), map_comp (map g ∘ f) join],\n  apply join_join\nend\n\ninstance : monad wseq :=\n{ map  := @map,\n  pure := @ret,\n  bind := @bind }\n\n/-\n  Unfortunately, wseq is not a lawful monad, because it does not satisfy\n  the monad laws exactly, only up to sequence equivalence.\n  Furthermore, even quotienting by the equivalence is not sufficient,\n  because the join operation involves lists of quotient elements,\n  with a lifted equivalence relation, and pure quotients cannot handle\n  this type of construction.\n\ninstance : is_lawful_monad wseq :=\n{ id_map := @map_id,\n  bind_pure_comp_eq_map := @bind_ret,\n  pure_bind := @ret_bind,\n  bind_assoc := @bind_assoc }\n-/\n\nend wseq\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/seq/wseq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.33773262518864217}}
{"text": "inductive Mem' (a : α) : List α → Prop where\n  | intro (as bs) : Mem' a (as ++ (a :: bs))\n\nexample {x : α} (h : Mem' x l) : True :=\n  match h with\n  | ⟨as', bs'⟩ => True.intro\n\nexample {x : α} (h : Mem' x l ∧ Mem' x l') : True :=\n  match h with\n  | ⟨⟨as', bs'⟩, _⟩ => True.intro\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/482.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3375922157911054}}
{"text": "import data.fin.tuple.basic\nimport data.finsupp.basic\nimport data.finsupp.pointwise\nimport data.list.basic\nimport data.list.range\nimport data.list.fin_range\nimport data.option.basic\nimport data.pfun\nimport data.prod.lex\n\nlemma bool.coe_iff_eq_tt (b : bool) : b ↔ b = tt := iff.rfl\n@[simp] lemma option.bind_const_none {α β} (x : option α) :\n  x.bind (λ _, none) = (none : option β) :=\nby cases x; simp\n@[simp] lemma option.is_none_ff_iff_is_some {α} (x : option α) :\n  x.is_none = ff ↔ x.is_some = tt :=\nby cases x; simp\n\n@[simp] lemma fin.tuple_eval_one {n : ℕ} {α : fin (n + 2) → Type*}\n  (x₀ : α 0) (x₁ : α 1) (x₂ : Π i : fin n, α i.succ.succ) :\n  fin.cons x₀ (fin.cons x₁ x₂) 1 = x₁ := rfl\n\n@[simp] lemma fin.tuple_eval_two {n : ℕ} {α : fin (n + 3) → Type*}\n  (x₀ : α 0) (x₁ : α 1) (x₂ : α 2) (x₃ : Π i : fin n, α i.succ.succ.succ) :\n  fin.cons x₀ (fin.cons x₁ (fin.cons x₂ x₃)) 2 = x₂ := rfl\n\nuniverses u v\nvariables {m : Type u → Type v} [monad m]\n\n/- Technically `m` only needs to be applicative but whatever -/\ndef fin.tuple_sequence : ∀ {n : ℕ} {α : fin n → Type u},\n  (Π i, m (α i)) → m (Π i, α i)\n| 0 _ _ := pure default\n| (n+1) α x := (x 0) >>= λ r, (fin.tuple_sequence (fin.tail x)) >>= λ x', return (fin.cons r x')\n\n@[simp] lemma fin.tuple_sequence₁ [is_lawful_monad m] {α : fin 1 → Type u} (x : Π i, m (α i)) :\n  fin.tuple_sequence x = (x 0) >>= λ r₀, pure (fin.cons r₀ default) :=\nby simp [fin.tuple_sequence] with functor_norm\n\n@[simp] lemma fin.tuple_sequence₂ [is_lawful_monad m] {α : fin 2 → Type u} (x : Π i, m (α i)) :\n  fin.tuple_sequence x = (x 0) >>= λ r₀, (x 1) >>= λ r₁, pure (fin.cons r₀ $ fin.cons r₁ $ default) :=\nby { simp [fin.tuple_sequence] with functor_norm, refl, }\n\ndef fin.tuple_some {n : ℕ} {α : fin n → Type u} (x : Π i, option (α i)) : option (Π i, α i) :=\nfin.tuple_sequence x\n\n@[simp] lemma option.bind_is_some {α β} (x : option α) (y : α → option β):\n  (x >>= y).is_some ↔ (∃ (h : x.is_some), (y (option.get h)).is_some) :=\nby cases x; simp\n\n@[simp] lemma option.map_is_some {α β} (x : option α) (y : α → β) :\n  (y <$> x).is_some = x.is_some := by cases x; simp\n\n@[simp] lemma option.not_is_some' {α} (x : option α) : !x.is_some = x.is_none :=\nby { cases x; simp, }\n\ndef option.guard_prop {α} (p : Prop) [decidable p] (x : α) : option α :=\n  if p then some x else none\n@[simp] lemma option.guard_prop_is_some {α} {p : Prop} [decidable p] {x : α} :\n  (option.guard_prop p x).is_some ↔ p :=\nby { dsimp only [option.guard_prop], split_ifs; simpa }\n\n@[simp] lemma option.coe_part_dom {α} (x : option α) :\n  (x : part α).dom ↔ x.is_some := by cases x; simp\n\n@[simp] lemma option.coe_part_eq_some {α} (x : option α) (y : α) :\n  (x : part α) = part.some y ↔ x = some y :=\nby simp [part.eq_some_iff]\n\n@[simp] lemma list.nth_is_some_iff {α} {x : list α} {n : ℕ} :\n  (x.nth n).is_some ↔ n < x.length :=\nby { rw ← not_iff_not, simp [option.is_none_iff_eq_none], }\n\n@[simp] lemma option.map_is_some' {α β} (x : option α) (f : α → β) :\n  (x.map f).is_some = x.is_some := by cases x; simp\n\nlemma list.zip_with_fst {α β} {l₁ : list α} {l₂ : list β} (hl : l₁.length ≤ l₂.length) :\n  list.zip_with (λ a b, a) l₁ l₂ = l₁ :=\nby { erw [← list.map_uncurry_zip_eq_zip_with, list.map_fst_zip], exact hl, }\n\nlemma list.zip_with_snd {α β} {l₁ : list α} {l₂ : list β} (hl : l₂.length ≤ l₁.length) :\n  list.zip_with (λ a b, b) l₁ l₂ = l₂ :=\nby { erw [← list.map_uncurry_zip_eq_zip_with, list.map_snd_zip], exact hl, }\n\n@[simp] lemma multiset.map_nth_le {α} {n : ℕ} {l : list α} (hn : l.length = n) :\n    (finset.univ.val : multiset (fin n)).map (λ i, l.nth_le i (by rw hn; exact i.prop)) = l :=\nby { subst hn, simp [finset.univ, fintype.elems], erw list.map_nth_le, }\n\n@[simp] lemma le_ff_iff {b : bool} : b ≤ ff ↔ b = ff :=\nby cases b; simp\n\nlemma ne_min_of_ne_and_ne {ι : Type*} [linear_order ι] {a x y : ι} (hx : a ≠ x) (hy : a ≠ y) :\n  a ≠ min x y := by cases min_choice x y with h; rw h; assumption\n\n@[simp] lemma max_ne_self_iff {ι : Type*} [linear_order ι] (a b : ι) :\n  ¬(a = max a b) ↔ a < b :=\nby { rw max_def, split_ifs, { simpa using h }, { simpa using le_of_not_ge h } }\n\n@[simp] lemma max_ne_self_iff' {ι : Type*} [linear_order ι] (a b : ι) :\n  ¬(b = max a b) ↔ b < a :=\nby { rw max_comm, simp, }\n\nsection with_top\nvariables {ι : Type} [partial_order ι]\n\n@[simp] lemma with_top.is_some_iff_lt_top {x : with_top ι} :\n  x.is_some ↔ x < ⊤ :=\nby { rw [← not_iff_not, eq_ff_eq_not_eq_tt, option.not_is_some, option.is_none_iff_eq_none, lt_top_iff_ne_top, ne, not_not], refl, }\n\nlemma prod.lex.le_iff' {α β : Type} [has_lt α] [has_le β] {x y : α ×ₗ β} :\n  x ≤ y ↔ x.1 < y.1 ∨ (x.1 = y.1 ∧ x.2 ≤ y.2) := prod.lex_def _ _\n\nlemma prod.lex.le_iff'' {α β : Type} [partial_order α] [preorder β] {x y : α ×ₗ β} :\n  x ≤ y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 ≤ y.2) :=\nby { rw [prod.lex.le_iff', le_iff_lt_or_eq], have := @ne_of_lt _ _ x.1 y.1, tauto!, }\n\nlemma prod.lex.lt_iff' {α β : Type} [has_lt α] [has_lt β] {x y : α ×ₗ β} :\n  x < y ↔ x.1 < y.1 ∨ (x.1 = y.1 ∧ x.2 < y.2) := prod.lex_def _ _\n\nlemma prod.lex.fst_le_of_le {α β : Type} [preorder α] [preorder β] {x y : α ×ₗ β} (h : x ≤ y) : x.1 ≤ y.1 :=\nby { rw prod.lex.le_iff' at h, cases h, { exact h.le, }, { exact h.1.le, }, }\n\nlemma prod.lex.fst_lt_of_lt_of_le {α β : Type} [preorder α] [partial_order β] {x y : α ×ₗ β}\n  (h : x < y) (h' : y.2 ≤ x.2) : x.1 < y.1 :=\nby { rw prod.lex.lt_iff' at h, cases h, { exact h, }, cases h.2.not_le h', }\n\n@[simp, norm_cast] lemma prod.with_top.coe_inj {α : Type} (x y : α) : (x : with_top α) = y ↔ x = y :=\noption.some_inj\n\nend with_top\n\n/- NOTE: This stuff is already in mathlib (`data.finsupp.pointwise`) -/\n\n-- variables {ι α : Type}\n\n-- noncomputable instance finsupp.has_mul [mul_zero_class α] : has_mul (ι →₀ α) :=\n-- ⟨λ a b, finsupp.zip_with (*) (zero_mul _) a b⟩\n\n-- lemma finsupp.mul_apply [mul_zero_class α] (g₁ g₂ : ι →₀ α) (a : ι) : (g₁ * g₂) a = g₁ a * g₂ a := rfl\n\n-- -- #check pi.distrib -- todo, tactic like this?\n-- noncomputable instance finsupp.non_unital_semiring [non_unital_semiring α] : non_unital_semiring (ι →₀ α) :=\n-- {\n--   zero := 0,\n--   add_assoc := λ a b c, fun_like.ext _ _ (by simp [finsupp.add_apply, add_assoc]),\n--   zero_add  := λ a,     fun_like.ext _ _ (by simp [finsupp.add_apply]),\n--   add_zero  := λ a,     fun_like.ext _ _ (by simp [finsupp.add_apply]),\n--   add_comm  := λ a b,   fun_like.ext _ _ (by simp [finsupp.add_apply, add_comm] ),\n--   zero_mul  := λ a,     fun_like.ext _ _ (by simp [finsupp.mul_apply]),\n--   mul_zero  := λ a,     fun_like.ext _ _ (by simp [finsupp.mul_apply]),\n\n--   left_distrib  := λ a b c, by simp [fun_like.ext_iff, finsupp.mul_apply, finsupp.add_apply, left_distrib],\n--   right_distrib := λ a b c, by simp [fun_like.ext_iff, finsupp.mul_apply, finsupp.add_apply, right_distrib],\n\n--   mul_assoc     := λ a b c, by simp [fun_like.ext_iff, finsupp.mul_apply, mul_assoc],\n\n--   ..finsupp.has_mul, ..finsupp.has_add, }\n\n@[simp] lemma finsupp.mul_single {ι β : Type*} [mul_zero_class β] (i : ι) (x y : β) :\n  (finsupp.single i x) * (finsupp.single i y) = finsupp.single i (x * y) :=\nby { ext a, by_cases i = a; simp [h], }\n\nlemma finsupp.mul_eq_zero_of_disjoint_support {ι β : Type*} [decidable_eq ι] [mul_zero_class β] (f g : ι →₀ β) (h : disjoint f.support g.support) :\n  f * g = 0 :=\nbegin\n  rw [← finsupp.support_eq_empty, ← finset.subset_empty],\n  refine trans finsupp.support_mul _,\n  rwa [finset.subset_empty, ← finset.disjoint_iff_inter_eq_empty],\nend\n\nlemma finsupp.mul_single_eq_zero {ι β : Type*} [mul_zero_class β] (i₁ i₂ : ι) (hi : i₁ ≠ i₂) (x y : β) :\n  (finsupp.single i₁ x) * (finsupp.single i₂ y) = 0 :=\nby { classical, rw [finsupp.mul_eq_zero_of_disjoint_support], simp [finset.disjoint_iff_ne, finsupp.mem_support_single], intros, exact hi, }\n\nset_option pp.implicit true\n\n\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/verification/misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3375922157911054}}
{"text": "import polytime\nopen num (to_bits of_bits)\n\n\n@[simp] def encode_option_nat : option ℕ → ℕ\n| none := 0\n| (some n) := bit1 n\n\n@[simp] def decode_option_nat : ℕ → option ℕ\n| 0 := none\n| n := some (of_bits (to_bits n).tail)\n\n@[simp] lemma decode_option_nat_bit1 (n : ℕ) : decode_option_nat (bit1 n) = some n :=\nby { cases e : (bit1 n), { simp at e, contradiction, },\n  simp only [decode_option_nat, nat.succ_ne_zero, not_false_iff], rw ← e, simp, }\n\n@[simp] lemma decode_encode_option_nat (n : option ℕ) : decode_option_nat (encode_option_nat n) = n :=\nby { cases n, { refl, }, simp, }\n\nclass polycodable (α : Type*) :=\n(encode : α → ℕ)\n(decode [] : ℕ → option α)\n(decode_encode : ∀ x, decode (encode x) = some x)\n(poly [] : ∃ c : code, polytime c ∧ ∀ x, c.eval x =  part.some (encode_option_nat ((decode x).map encode)))\n\nattribute [simp] polycodable.decode_encode\n\nopen polycodable\n\ninstance : polycodable ℕ := \n{ encode := id,\n  decode := some,\n  decode_encode := λ n, rfl,\n  poly := ⟨code.bit tt, polytime_bit _, λ n, by simp [nat.bit]⟩ }\n\n@[simp] lemma encode_nat_id (n : ℕ) : encode n = n := rfl\n@[simp] lemma decode_nat_id (n : ℕ) : decode ℕ n = some n := rfl\n\ndef decode_option_code (f : code) : code :=\ncode.case ((code.case zero (code.bit tt) zero).comp f) ((code.case zero (code.bit tt) zero).comp f) zero\n\nlemma polytime_decode_option_code {f : code} (hf : polytime f) : polytime (decode_option_code f) :=\nbegin\n  -- TODO: This obviously needs to be a tactic\n  apply polytime_case,\n  { apply polytime_comp, { apply polytime_case, apply polytime_zero, apply polytime_bit, apply polytime_zero, }, exact hf, },\n  apply polytime_comp, apply polytime_case, apply polytime_zero, apply polytime_bit, apply polytime_zero, exact hf, apply polytime_zero,\nend\n\ninstance {α : Type} [polycodable α] : polycodable (option α) :=\n{ encode := λ x, encode_option_nat (x.map encode),\n  decode := λ x, some ((decode_option_nat x) >>= decode α),\n  decode_encode := λ x, by cases x; simp,\n  poly :=\n  begin\n    obtain ⟨c, pc, ec⟩ := polycodable.poly α,\n    use (code.bit tt).comp (decode_option_code c), split,\n    { apply polytime_comp (polytime_bit _), exact polytime_decode_option_code pc, },\n    intro x, simp only [decode_option_code],\n    rcases e : (to_bits x) with _|_|_,\n    { simp only [num.to_bits_nil_iff, nat.cast_eq_zero] at e, subst e, simp [nat.bit], },\n    all_goals { have : x ≠ 0 := λ h, by { subst h, simpa using e, },\n      simp [e, this, nat.bit, ec],\n      cases decode α (of_bits this_tl); simp, },\n  end  }\n\ndef is_polytime {α β : Type} [polycodable α] [polycodable β] (f : α → β) : Prop :=\n∃ c : code, polytime c ∧ ∀ x, c.eval (encode x) = part.some (encode $ f x)\n\n@[simps] def polycodable.mk' {α : Type} (encode : α → ℕ) (decode : ℕ → option α) (decode_encode : ∀ x, decode (encode x) = some x)\n  (poly : is_polytime (λ x, (decode x).map encode)) : polycodable α := \n{ encode := encode,\n  decode := decode,\n  decode_encode := decode_encode,\n  poly := by simpa [is_polytime, polycodable.encode] using poly }\n\nvariables {α β γ : Type} [polycodable α] [polycodable β] [polycodable γ]\n\nlemma encode_decode_is_polytime (α : Type) [polycodable α] : is_polytime (λ x : ℕ, (decode α x).map encode) :=\nby simpa [is_polytime, polycodable.encode] using poly α\n\nlemma is_polytime_iff (f : α → β) :\n  is_polytime f ↔ ∃ c : code, polytime c ∧ ∀ x : ℕ, c.eval x = part.some (encode $ (decode α x).map f) :=\nbegin\n  split, swap,\n  { rintro ⟨c, pc, hc⟩,\n    use tail.comp c,\n    split, { exact polytime_comp polytime_tail pc,},\n    intro x, specialize hc (encode x),\n    simp [hc, polycodable.encode], },\n  rintro ⟨c, pc, hc⟩,\n  obtain ⟨de, pde, hde⟩ := polycodable.poly α,\n  use (code.case zero ((code.bit tt).comp c) zero).comp de,\n  split, { exact polytime_comp (polytime_case polytime_zero (polytime_comp (polytime_bit _) pc) polytime_zero) pde, },\n  intro x, specialize hde x,\n  cases decode α x,\n  { simp [hde], refl, }, simp [hde, hc], refl,\nend\n\nlemma is_polytime_comp {f : β → γ} {g : α → β} : is_polytime f → is_polytime g → is_polytime (f ∘ g)\n| ⟨fc, fp, hf⟩ ⟨gc, gp, hg⟩ := ⟨fc.comp gc, polytime_comp fp gp, λ x, by simp [hf, hg]⟩\n\nlemma is_polytime_encode (α : Type) [polycodable α] : is_polytime (@encode α _) :=\n⟨code.id, polytime_id, λ x, by simp⟩\n\nlemma is_polytime_decode (α : Type) [polycodable α] : is_polytime (decode α) := poly α\n\nlemma is_polytime_const (β : Type) [polycodable β] (x : α) : is_polytime (λ _ : β, x) :=\n⟨code.const (encode x), polytime_const _, λ x, by simp⟩ \n\nsection equiv\n\ndef polycodable_of_equiv {β : Type} (eqv : α ≃ β) : polycodable β :=\npolycodable.mk'\n  (λ b, (polycodable.encode (eqv.symm b)))\n  (λ n, (polycodable.decode α n).map eqv)\n  (λ b, by simp)\n  (by { convert encode_decode_is_polytime α, ext b : 1, cases decode α b; simp, })\n\nlemma is_polytime_of_equiv_symm {β : Type} (eqv : α ≃ β)\n  {f : α → γ} (hf : is_polytime f) : @is_polytime β γ (polycodable_of_equiv eqv) _ (f ∘ eqv.symm) :=\nby { rcases hf with ⟨c, pc, hc⟩, use c, split, { assumption, }, intro x, exact hc (eqv.symm x), }\n\nlemma is_polytime_of_equiv {β : Type} (eqv : α ≃ β)\n  {f : γ → α} (hf : is_polytime f) : @is_polytime γ β _ (polycodable_of_equiv eqv) (eqv ∘ f) :=\nby { rcases hf with ⟨c, pc, hc⟩, use c, split, { assumption, }, intro x, rw hc x, simp [encode], }\n\nend equiv\n\nsection bool\n\ninstance : polycodable bool :=\npolycodable.mk'\n(λ b, cond b 1 0) (λ n, some (if n = 0 then ff else tt)) (λ x, by cases x; simp)\n⟨(code.bit tt).comp to_bit, polytime_comp (polytime_bit _) polytime_to_bit, λ x, by cases x; simp; refl⟩\n\nlemma is_polytime_cond {c : α → bool} {f g : α → β} : is_polytime c → is_polytime f → is_polytime g → is_polytime (λ x, cond (c x) (f x) (g x))\n| ⟨cc, cp, hc⟩ ⟨fc, fp, hf⟩ ⟨gc, gp, hg⟩ :=\n⟨code.ite cc gc fc, (polytime_ite cp gp fp), λ x, by cases e : c x; simp [hc, hf, hg, e]⟩\n\nlemma is_polytime_is_some (α : Type) [polycodable α] : is_polytime (@option.is_some α) :=\n⟨to_bit, polytime_to_bit, λ x, by cases x; simp [polycodable.encode]⟩\n\nend bool\n\nsection prod\n\ninstance {α β : Type} [polycodable α] [polycodable β] : polycodable (α × β) :=\npolycodable.mk'\n  (λ x, nat.mkpair' (encode x.1) (encode x.2))\n  (λ x, (decode α (nat.unpair' x).1) >>= (λ x₁, (decode β (nat.unpair' x).2).map $ λ x₂, (x₁, x₂)))\n  (λ x, by simp [return, pure])\nbegin\n  obtain ⟨somec₁, somep₁, hsome₁⟩ := is_polytime_is_some α,\n  obtain ⟨somec₂, somep₂, hsome₂⟩ := is_polytime_is_some β,\n  obtain ⟨d₁, dp₁, hd₁⟩ := is_polytime_decode α,\n  obtain ⟨d₂, dp₂, hd₂⟩ := is_polytime_decode β,\n  simp only [encode_nat_id] at hd₁ hd₂,\n  obtain ⟨e₁, ep₁, he₁⟩ := is_polytime_encode α,\n  obtain ⟨e₂, ep₂, he₂⟩ := is_polytime_encode β,\n  use (code.ite (somec₁.comp $ d₁.comp code.fst) zero $\n      code.ite (somec₂.comp $ d₂.comp code.snd) zero $ \n      (code.bit tt).comp (code.pair (tail.comp $ d₁.comp code.fst) (tail.comp $ d₂.comp code.snd))),\n  split,\n  { have P₁ : polytime (d₁.comp code.fst) := polytime_comp dp₁ polytime_fst,\n    have P₂ : polytime (d₂.comp code.snd) := polytime_comp dp₂ polytime_snd,\n    exact (\n      polytime_ite (polytime_comp somep₁ P₁) polytime_zero $\n      polytime_ite (polytime_comp somep₂ P₂) polytime_zero $\n      (polytime_comp (polytime_bit _) (polytime_pair (polytime_comp polytime_tail P₁) (polytime_comp polytime_tail P₂)))), },\n  intro x,\n  cases H₁ : decode α (nat.unpair' x).1, { simp [H₁, hd₁, hsome₁], refl, },\n  cases H₂ : decode β (nat.unpair' x).2, { simp [H₁, hd₁, hsome₁, H₂, hd₂, hsome₂], refl, },\n  simp [H₁, hd₁, hsome₁, H₂, hd₂, hsome₂], simp [polycodable.encode, nat.bit],\nend\n\nlemma polytime_prod_fst (α β : Type) [polycodable α] [polycodable β] : is_polytime (@prod.fst α β) :=\n⟨code.fst, polytime_fst, λ x, by simp [polycodable.encode]⟩\n\nlemma polytime_prod_snd (α β : Type) [polycodable α] [polycodable β] : is_polytime (@prod.snd α β) :=\n⟨code.snd, polytime_snd, λ x, by simp [polycodable.encode]⟩\n\nlemma prod_encode_def (x : α × β) : (encode x.1).mkpair' (encode x.2) = encode x := rfl\n\nlemma is_polytime_pair {f : α → β} {g : α → γ} : is_polytime f → is_polytime g → is_polytime (λ x, (f x, g x))\n| ⟨fc, fp, hf⟩ ⟨gc, gp, hg⟩ := ⟨code.pair fc gc, polytime_pair fp gp, λ x, by simp [hf, hg]⟩\n\ndef is_polytime₂ (f : α → β → γ) : Prop := is_polytime (function.uncurry f)\n\nlemma is_polytime₂_comp {δ : Type} [polycodable δ] {f : α → β → γ} {g : δ → α} {h : δ → β} \n  (hf : is_polytime₂ f) (hg : is_polytime g) (hh : is_polytime h) :\n  is_polytime (λ x, f (g x) (h x)) :=\nis_polytime_comp hf (is_polytime_pair hg hh)\n\n@[simp] lemma function.uncurry_prod_mk (α β : Type*) : function.uncurry (@prod.mk α β) = id :=\nby ext x; cases x; simp\n\nlemma is_polytime₂_ignore_fst {f : α → β} (h : is_polytime f) : is_polytime₂ (λ x : γ, f) :=\nby { have := is_polytime_comp h (polytime_prod_snd _ _), exact this, }\n\nlemma is_polytime_prod_mk : is_polytime₂ (@prod.mk α β) := ⟨code.id, polytime_id, λ x, by simp⟩\n\nend prod\n\nsection bool\n\nlemma is_polytime_band : is_polytime₂ band :=\nbegin\n  convert_to is_polytime₂ (λ b₁ b₂, cond b₁ b₂ ff),\n  { ext b₁ b₂, cases b₁; cases b₂; refl, },\n  apply is_polytime_cond,\n  exacts [polytime_prod_fst _ _, polytime_prod_snd _ _, is_polytime_const _ _],\nend\n\nlemma is_polytime_bor : is_polytime₂ bor :=\nbegin\n  convert_to is_polytime₂ (λ b₁ b₂, cond b₁ tt b₂), { ext b₁ b₂, cases b₁; cases b₂; refl, },\n  apply is_polytime_cond,\n  exacts [polytime_prod_fst _ _,  is_polytime_const _ _, polytime_prod_snd _ _],\nend\n\nend bool\n\nsection option\n\nlemma is_polytime_elim {g : α → option β} {f : α → β → γ} {df : α → γ} :\n  is_polytime g → is_polytime₂ f → is_polytime df → is_polytime (λ x, (g x).elim (df x) (f x))\n| ⟨gc, gp, hg⟩ ⟨fc, fp, hf⟩ ⟨dfc, dfp, hdf⟩ :=\n⟨code.ite gc dfc (fc.comp $ code.pair code.id (tail.comp gc)), \n  polytime_ite gp dfp (polytime_comp fp $ polytime_pair polytime_id (polytime_comp polytime_tail gp)),\n  λ x, \nbegin\n  simp only [← prod_encode_def] at hf,\n  cases e : g x with v, { simp [hg, hdf, e, encode], }, { simp [hg, hdf, e, encode, hf (x, v)], }\nend⟩\n\nlemma is_polytime_some : is_polytime (@some α) :=\n⟨code.bit tt, polytime_bit _, by simp [encode, nat.bit]⟩\n\nlemma is_polytime_of_is_polytime_some {f : α → β} (hf : is_polytime (some ∘ f)) : is_polytime f :=\nby { rcases hf with ⟨c, pc, hc⟩, use [tail.comp c, polytime_comp polytime_tail pc], intro x, simp [hc x, encode], }\n\nlemma is_polytime_bind_option {f : α → option β} {g : α → β → option γ} (hf : is_polytime f) (hg : is_polytime₂ g) :\n  is_polytime (λ x, (f x).bind (g x)) :=\nbegin\n  convert_to is_polytime (λ x, (f x).elim none (g x)), { ext x : 1, cases f x; simp, },\n  exact is_polytime_elim hf hg (is_polytime_const _ none),\nend\n\nlemma is_polytime_map_option {f : α → option β} {g : α → β → γ} (hf : is_polytime f) (hg : is_polytime₂ g) :\n  is_polytime (λ x, (f x).map (g x)) :=\nbegin\n  change is_polytime (λ x, (f x).bind (some ∘ (g x))), -- turns out, defeq !\n  exact is_polytime_bind_option hf (is_polytime_comp is_polytime_some hg),\nend\n\nend option\n\nsection embedding\n\ndef polycodable_of_embedding {β : Type} (e : β → α) (f : α → option β)\n  (h : ∀ x, f (e x) = some x) (poly : is_polytime (λ x, (f x).map e)) : polycodable β :=\npolycodable.mk' \n(λ x, encode (e x))\n(λ n, decode α n >>= f)\n(λ x, by simp [h])\nbegin\n  convert_to is_polytime (λ n, decode α n >>= (λ x, ((f x).map e).map encode)),\n  { ext n : 1, cases decode α n; simp, },\n  apply is_polytime_bind_option, { exact is_polytime_decode _, },\n  apply is_polytime₂_ignore_fst, apply is_polytime_map_option poly,\n  apply is_polytime₂_ignore_fst, apply is_polytime_encode,\nend\n\nlemma is_polytime_of_embedding_inclusion {β : Type} {e : β → α} {f : α → option β}\n  {h : ∀ x, f (e x) = some x} {poly : is_polytime (λ x, (f x).map e)} : \n  by { haveI := polycodable_of_embedding e f h poly, exact (is_polytime e), } :=\nby { use [code.id, polytime_id], intro, simp, }\n\nlemma is_polytime_of_embedding_inverse {β : Type} {e : β → α} {f : α → option β}\n  {h : ∀ x, f (e x) = some x} {poly : is_polytime (λ x, (f x).map e)} :\n  by { haveI := polycodable_of_embedding e f h poly, exact (is_polytime f), } :=\nby { rcases poly with ⟨c, pc, hc⟩, use [c, pc], intro x, simp [hc x, encode], refl, }\n\nend embedding\n\nsection subtype\n\ndef is_polytime_pred (p : α → Prop) [decidable_pred p] : Prop :=\nis_polytime (λ x : α, (p x : bool))\n\nlemma is_polytime_ite {p : α → Prop} [decidable_pred p] (hp : is_polytime_pred p) {f g : α → β}\n  (hf : is_polytime f) (hg : is_polytime g) : is_polytime (λ x, if p x then f x else g x) :=\nbegin\n  convert_to is_polytime (λ x, cond (p x) (f x) (g x)), { ext x, split_ifs with h; simp [h], },\n  apply is_polytime_cond; assumption,\nend\n\ndef polycodable_subtype {p : α → Prop} [decidable_pred p] (hp : is_polytime_pred p) :\n  polycodable {x // p x} :=\npolycodable_of_embedding\n  (λ x, (x : α))\n  (λ x, if h : p x then some ⟨x, h⟩ else none) \n  (λ x, by simpa [imp_false] using subtype.prop x)\nbegin\n  convert_to is_polytime (λ x, if p x then some x else none), { ext x : 1, split_ifs; simp, },\n  exact is_polytime_ite hp is_polytime_some (is_polytime_const _ _),\nend\n\nend subtype\n\nsection sigma\n\nprivate def sum_as_tuple : α ⊕ β → bool × ℕ\n| (sum.inl a) := (ff, encode a)\n| (sum.inr b) := (tt, encode b)\n\nprivate def tuple_as_sum (n : bool × ℕ) : option (α ⊕ β) :=\ncond n.1 ((decode β n.2).map sum.inr) ((decode α n.2).map sum.inl)\n\ninstance : polycodable (α ⊕ β) :=\npolycodable_of_embedding sum_as_tuple tuple_as_sum\n  (λ x, by cases x; simp [sum_as_tuple, tuple_as_sum])\nbegin\n  convert_to is_polytime (λ n : bool × ℕ, cond n.1 ((decode β n.2).map (λ x, (tt, encode x)))\n    ((decode α n.2).map (λ x, (ff, encode x)))),\n  { ext n : 1, cases n with b v, cases b; simp [tuple_as_sum]; congr, },\n  apply is_polytime_cond (polytime_prod_fst _ _),\n  change is_polytime₂ (λ (b : bool) (n : ℕ), (decode β n).map (λ x, (tt, encode x))),\n  { apply is_polytime₂_ignore_fst, apply is_polytime_map_option, apply is_polytime_decode, \n    apply is_polytime₂_ignore_fst, apply is_polytime₂_comp is_polytime_prod_mk, apply is_polytime_const,\n    apply is_polytime_encode, },\n  change is_polytime₂ (λ (b : bool) (n : ℕ), (decode α n).map (λ x, (ff, encode x))),\n  { apply is_polytime₂_ignore_fst, apply is_polytime_map_option, apply is_polytime_decode, \n    apply is_polytime₂_ignore_fst, apply is_polytime₂_comp is_polytime_prod_mk, apply is_polytime_const,\n    apply is_polytime_encode, },\nend\n\nprivate lemma is_polytime_sum_as_tuple : is_polytime (@sum_as_tuple α β _ _) := is_polytime_of_embedding_inclusion\nprivate lemma is_polytime_tuple_as_sum : is_polytime (@tuple_as_sum α β _ _) := is_polytime_of_embedding_inverse\n\nlemma is_polytime_sum_inl : is_polytime (@sum.inl α β) :=\nbegin\n  apply is_polytime_of_is_polytime_some,\n  convert_to is_polytime (λ a : α, (tuple_as_sum (ff, encode a) : option (α ⊕ β)) ),\n  { ext a : 1, simp [tuple_as_sum], },\n  apply is_polytime_comp is_polytime_tuple_as_sum,\n  apply is_polytime₂_comp is_polytime_prod_mk, { apply is_polytime_const, }, apply is_polytime_encode,\nend\n\nlemma is_polytime_sum_inr : is_polytime (@sum.inr α β) :=\nbegin\n  apply is_polytime_of_is_polytime_some,\n  convert_to is_polytime (λ b : β, (tuple_as_sum (tt, encode b) : option (α ⊕ β)) ),\n  { ext b : 1, simp [tuple_as_sum], },\n  apply is_polytime_comp is_polytime_tuple_as_sum,\n  apply is_polytime₂_comp is_polytime_prod_mk, { apply is_polytime_const, }, apply is_polytime_encode,\nend\n\nlemma is_polytime_sum_elim {δ : Type} [polycodable δ] {f : δ → α → γ} (g : δ → β → γ) {h : δ → α ⊕ β } \n  (hf : is_polytime₂ f) (hg : is_polytime₂ g) (hh : is_polytime h) :\n  is_polytime (λ x, (h x).elim (f x) (g x)) :=\nbegin\n  apply is_polytime_of_is_polytime_some,\n  convert_to is_polytime (λ x : δ, cond (sum_as_tuple $ h x).1 ((decode β $ (sum_as_tuple $ h x).2).map (g x)) \n    ((decode α $ (sum_as_tuple $ h x).2).map (f x))),\n  { ext x : 1, simp only [function.comp_app], cases h x; simp [sum_as_tuple], },\n  apply is_polytime_cond, { apply is_polytime_comp (polytime_prod_fst _ _), apply is_polytime_comp is_polytime_sum_as_tuple hh, },\n  { apply is_polytime_map_option, apply is_polytime_comp (is_polytime_decode _), apply is_polytime_comp (polytime_prod_snd _ _),\n    apply is_polytime_comp is_polytime_sum_as_tuple hh, exact hg, },\n  apply is_polytime_map_option, apply is_polytime_comp (is_polytime_decode _), apply is_polytime_comp (polytime_prod_snd _ _),\n  apply is_polytime_comp is_polytime_sum_as_tuple hh, exact hf,\nend\n\nend sigma\n\nsection list\n\ndef mklist (ls : list ℕ) : ℕ := ls.foldr nat.mkpair' 0\ndef unlist : ℕ → list ℕ\n| 0 := []\n| (nat.succ n) := have wf : (nat.unpair' n.succ).2 < n.succ := nat.unpair'_snd_lt n,\n              (nat.unpair' n.succ).1 :: (unlist (nat.unpair' n.succ).2)\n\n@[simp] lemma unlist_mklist (ls : list ℕ) : unlist (mklist ls) = ls :=\nbegin\n  induction ls with h t ih, { simp [mklist, unlist], },\n  simp only [mklist, list.foldr],\n  cases e : nat.mkpair' h _, { cases nat.mkpair'_ne_zero _ _ e, },\n  simp only [unlist], rw ← e,\n  split; simp, exact ih,\nend\n\nsection recursion\n\n\n\nend recursion\n\n\nend list", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_defs", "sha": "b4e5f5544e11cd5aca1a5a4b5b0231537af4962c", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_defs", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_defs/lean_complexity_theory_polytime_defs-b4e5f5544e11cd5aca1a5a4b5b0231537af4962c/src/polycodable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.33755071925114183}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.group_theory.congruence\nimport Mathlib.linear_algebra.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_4 u_3 u_2 u_5 l u_6 u_7 u_8 u_9 \n\nnamespace Mathlib\n\n/-!\n# Tensor product of semimodules over commutative semirings.\n\nThis file constructs the tensor product of semimodules over commutative semirings. Given a semiring\n`R` and semimodules over it `M` and `N`, the standard construction of the tensor product is\n`tensor_product R M N`. It is also a semimodule over `R`.\n\nIt comes with a canonical bilinear map `M → N → tensor_product R M N`.\n\nGiven any bilinear map `M → N → P`, there is a unique linear map `tensor_product R M N → P` whose\ncomposition with the canonical bilinear map `M → N → tensor_product R M N` is the given bilinear\nmap `M → N → P`.\n\nWe start by proving basic lemmas about bilinear maps.\n\n## Notations\n\nThis file uses the localized notation `M ⊗ N` and `M ⊗[R] N` for `tensor_product R M N`, as well\nas `m ⊗ₜ n` and `m ⊗ₜ[R] n` for `tensor_product.tmul R m n`.\n\n## Tags\n\nbilinear, tensor, tensor product\n-/\n\nnamespace linear_map\n\n\n/-- Create a bilinear map from a function that is linear in each component. -/\ndef mk₂ (R : Type u_1) [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : M → N → P) (H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n)\n    (H2 : ∀ (c : R) (m : M) (n : N), f (c • m) n = c • f m n)\n    (H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂)\n    (H4 : ∀ (c : R) (m : M) (n : N), f m (c • n) = c • f m n) : linear_map R M (linear_map R N P) :=\n  mk (fun (m : M) => mk (f m) (H3 m) sorry) sorry sorry\n\n@[simp] theorem mk₂_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : M → N → P)\n    {H1 : ∀ (m₁ m₂ : M) (n : N), f (m₁ + m₂) n = f m₁ n + f m₂ n}\n    {H2 : ∀ (c : R) (m : M) (n : N), f (c • m) n = c • f m n}\n    {H3 : ∀ (m : M) (n₁ n₂ : N), f m (n₁ + n₂) = f m n₁ + f m n₂}\n    {H4 : ∀ (c : R) (m : M) (n : N), f m (c • n) = c • f m n} (m : M) (n : N) :\n    coe_fn (coe_fn (mk₂ R f H1 H2 H3 H4) m) n = f m n :=\n  rfl\n\ntheorem ext₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] {f : linear_map R M (linear_map R N P)} {g : linear_map R M (linear_map R N P)}\n    (H : ∀ (m : M) (n : N), coe_fn (coe_fn f m) n = coe_fn (coe_fn g m) n) : f = g :=\n  ext fun (m : M) => ext fun (n : N) => H m n\n\n/-- Given a linear map from `M` to linear maps from `N` to `P`, i.e., a bilinear map from `M × N` to\n`P`, change the order of variables and get a linear map from `N` to linear maps from `M` to `P`. -/\ndef flip {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) : linear_map R N (linear_map R M P) :=\n  mk₂ R (fun (n : N) (m : M) => coe_fn (coe_fn f m) n) sorry sorry sorry sorry\n\n@[simp] theorem flip_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R M (linear_map R N P)) (m : M) (n : N) :\n    coe_fn (coe_fn (flip f) n) m = coe_fn (coe_fn f m) n :=\n  rfl\n\ntheorem flip_inj {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] {f : linear_map R M (linear_map R N P)} {g : linear_map R M (linear_map R N P)}\n    (H : flip f = flip g) : f = g :=\n  sorry\n\n/-- Given a linear map from `M` to linear maps from `N` to `P`, i.e., a bilinear map `M → N → P`,\nchange the order of variables and get a linear map from `N` to linear maps from `M` to `P`. -/\ndef lflip (R : Type u_1) [comm_semiring R] (M : Type u_2) (N : Type u_3) (P : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R M (linear_map R N P)) (linear_map R N (linear_map R M P)) :=\n  mk flip sorry sorry\n\n@[simp] theorem lflip_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R M (linear_map R N P)) (m : M) (n : N) :\n    coe_fn (coe_fn (coe_fn (lflip R M N P) f) n) m = coe_fn (coe_fn f m) n :=\n  rfl\n\ntheorem map_zero₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (y : N) : coe_fn (coe_fn f 0) y = 0 :=\n  map_zero (coe_fn (flip f) y)\n\ntheorem map_neg₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_group M] [add_comm_monoid N] [add_comm_group P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (x : M) (y : N) :\n    coe_fn (coe_fn f (-x)) y = -coe_fn (coe_fn f x) y :=\n  map_neg (coe_fn (flip f) y) x\n\ntheorem map_sub₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_group M] [add_comm_monoid N] [add_comm_group P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (x : M) (y : M) (z : N) :\n    coe_fn (coe_fn f (x - y)) z = coe_fn (coe_fn f x) z - coe_fn (coe_fn f y) z :=\n  map_sub (coe_fn (flip f) z) x y\n\ntheorem map_add₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (x₁ : M) (x₂ : M) (y : N) :\n    coe_fn (coe_fn f (x₁ + x₂)) y = coe_fn (coe_fn f x₁) y + coe_fn (coe_fn f x₂) y :=\n  map_add (coe_fn (flip f) y) x₁ x₂\n\ntheorem map_smul₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (r : R) (x : M) (y : N) :\n    coe_fn (coe_fn f (r • x)) y = r • coe_fn (coe_fn f x) y :=\n  map_smul (coe_fn (flip f) y) r x\n\n/-- Composing a linear map `M → N` and a linear map `N → P` to form a linear map `M → P`. -/\ndef lcomp (R : Type u_1) [comm_semiring R] {M : Type u_2} {N : Type u_3} (P : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M N) : linear_map R (linear_map R N P) (linear_map R M P) :=\n  flip (comp (flip id) f)\n\n@[simp] theorem lcomp_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R M N) (g : linear_map R N P) (x : M) :\n    coe_fn (coe_fn (lcomp R P f) g) x = coe_fn g (coe_fn f x) :=\n  rfl\n\n/-- Composing a linear map `M → N` and a linear map `N → P` to form a linear map `M → P`. -/\ndef llcomp (R : Type u_1) [comm_semiring R] (M : Type u_2) (N : Type u_3) (P : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R N P) (linear_map R (linear_map R M N) (linear_map R M P)) :=\n  flip (mk (lcomp R P) sorry sorry)\n\n@[simp] theorem llcomp_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (g : linear_map R M N) (x : M) :\n    coe_fn (coe_fn (coe_fn (llcomp R M N P) f) g) x = coe_fn f (coe_fn g x) :=\n  rfl\n\n/-- Composing a linear map `Q → N` and a bilinear map `M → N → P` to\nform a bilinear map `M → Q → P`. -/\ndef compl₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    {Q : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M (linear_map R N P)) (g : linear_map R Q N) :\n    linear_map R M (linear_map R Q P) :=\n  comp (lcomp R P g) f\n\n@[simp] theorem compl₂_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} {Q : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M (linear_map R N P)) (g : linear_map R Q N) (m : M) (q : Q) :\n    coe_fn (coe_fn (compl₂ f g) m) q = coe_fn (coe_fn f m) (coe_fn g q) :=\n  rfl\n\n/-- Composing a linear map `P → Q` and a bilinear map `M × N → P` to\nform a bilinear map `M → N → Q`. -/\ndef compr₂ {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4}\n    {Q : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M (linear_map R N P)) (g : linear_map R P Q) :\n    linear_map R M (linear_map R N Q) :=\n  comp (coe_fn (llcomp R N P Q) g) f\n\n@[simp] theorem compr₂_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} {Q : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M (linear_map R N P)) (g : linear_map R P Q) (m : M) (n : N) :\n    coe_fn (coe_fn (compr₂ f g) m) n = coe_fn g (coe_fn (coe_fn f m) n) :=\n  rfl\n\n/-- Scalar multiplication as a bilinear map `R → M → M`. -/\ndef lsmul (R : Type u_1) [comm_semiring R] (M : Type u_2) [add_comm_monoid M] [semimodule R M] :\n    linear_map R R (linear_map R M M) :=\n  mk₂ R has_scalar.smul sorry sorry sorry sorry\n\n@[simp] theorem lsmul_apply {R : Type u_1} [comm_semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] (r : R) (m : M) : coe_fn (coe_fn (lsmul R M) r) m = r • m :=\n  rfl\n\nend linear_map\n\n\nnamespace tensor_product\n\n\n-- open free_add_monoid\n\n/-- The relation on `free_add_monoid (M × N)` that generates a congruence whose quotient is\nthe tensor product. -/\ninductive eqv (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    free_add_monoid (M × N) → free_add_monoid (M × N) → Prop\n    where\n| of_zero_left : ∀ (n : N), eqv R M N (free_add_monoid.of (0, n)) 0\n| of_zero_right : ∀ (m : M), eqv R M N (free_add_monoid.of (m, 0)) 0\n| of_add_left :\n    ∀ (m₁ m₂ : M) (n : N),\n      eqv R M N (free_add_monoid.of (m₁, n) + free_add_monoid.of (m₂, n))\n        (free_add_monoid.of (m₁ + m₂, n))\n| of_add_right :\n    ∀ (m : M) (n₁ n₂ : N),\n      eqv R M N (free_add_monoid.of (m, n₁) + free_add_monoid.of (m, n₂))\n        (free_add_monoid.of (m, n₁ + n₂))\n| of_smul :\n    ∀ (r : R) (m : M) (n : N),\n      eqv R M N (free_add_monoid.of (r • m, n)) (free_add_monoid.of (m, r • n))\n| add_comm : ∀ (x y : free_add_monoid (M × N)), eqv R M N (x + y) (y + x)\n\nend tensor_product\n\n\n/-- The tensor product of two semimodules `M` and `N` over the same commutative semiring `R`.\nThe localized notations are `M ⊗ N` and `M ⊗[R] N`, accessed by `open_locale tensor_product`. -/\ndef tensor_product (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :=\n  add_con.quotient (add_con_gen sorry)\n\nnamespace tensor_product\n\n\nprotected instance add_comm_monoid {R : Type u_1} [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    add_comm_monoid (tensor_product R M N) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\nprotected instance inhabited {R : Type u_1} [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    Inhabited (tensor_product R M N) :=\n  { default := 0 }\n\n/-- The canonical function `M → N → M ⊗ N`. The localized notations are `m ⊗ₜ n` and `m ⊗ₜ[R] n`,\naccessed by `open_locale tensor_product`. -/\ndef tmul (R : Type u_1) [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) (n : N) : tensor_product R M N :=\n  coe_fn (add_con.mk' (add_con_gen (eqv R M N))) (free_add_monoid.of (m, n))\n\nprotected theorem induction_on {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N]\n    {C : tensor_product R M N → Prop} (z : tensor_product R M N) (C0 : C 0)\n    (C1 : ∀ {x : M} {y : N}, C (tmul R x y))\n    (Cp : ∀ {x y : tensor_product R M N}, C x → C y → C (x + y)) : C z :=\n  sorry\n\n@[simp] theorem zero_tmul {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] (n : N) :\n    tmul R 0 n = 0 :=\n  quotient.sound' (add_con_gen.rel.of (free_add_monoid.of (0, n)) 0 (eqv.of_zero_left n))\n\ntheorem add_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (m₁ : M) (m₂ : M) (n : N) :\n    tmul R (m₁ + m₂) n = tmul R m₁ n + tmul R m₂ n :=\n  sorry\n\n@[simp] theorem tmul_zero {R : Type u_1} [comm_semiring R] {M : Type u_3} (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) :\n    tmul R m 0 = 0 :=\n  quotient.sound' (add_con_gen.rel.of (free_add_monoid.of (m, 0)) 0 (eqv.of_zero_right m))\n\ntheorem tmul_add {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) (n₁ : N) (n₂ : N) :\n    tmul R m (n₁ + n₂) = tmul R m n₁ + tmul R m n₂ :=\n  sorry\n\n/--\nA typeclass for `has_scalar` structures which can be moved across a tensor product.\n\nThis typeclass is generated automatically from a `is_scalar_tower` instance, but exists so that\nwe can also add an instance for `add_comm_group.int_module`, allowing `z •` to be moved even if\n`R` does not support negation.\n\nNote that `semimodule R' (M ⊗[R] N)` is available even without this typeclass on `R'`; it's only\nneeded if `tensor_product.smul_tmul`, `tensor_product.smul_tmul'`, or `tensor_product.tmul_smul` is\nused.\n-/\nclass compatible_smul (R : Type u_1) [comm_semiring R] (R' : Type u_2) [comm_semiring R']\n    (M : Type u_3) (N : Type u_4) [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N]\n    where\n  smul_tmul : ∀ (r : R') (m : M) (n : N), tmul R (r • m) n = tmul R m (r • n)\n\n/-- Note that this provides the default `compatible_smul R R M N` instance through\n`mul_action.is_scalar_tower.left`. -/\nprotected instance compatible_smul.is_scalar_tower {R : Type u_1} [comm_semiring R] {R' : Type u_2}\n    [comm_semiring R'] {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N]\n    [semimodule R M] [semimodule R N] [semimodule R' M] [semimodule R' N] [has_scalar R' R]\n    [is_scalar_tower R' R M] [is_scalar_tower R' R N] : compatible_smul R R' M N :=\n  compatible_smul.mk sorry\n\n/-- `smul` can be moved from one side of the product to the other .-/\ntheorem smul_tmul {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R'] {M : Type u_3}\n    {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N]\n    [semimodule R' M] [semimodule R' N] [compatible_smul R R' M N] (r : R') (m : M) (n : N) :\n    tmul R (r • m) n = tmul R m (r • n) :=\n  compatible_smul.smul_tmul r m n\n\n/-- Auxiliary function to defining scalar multiplication on tensor product. -/\ndef smul.aux {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] {R' : Type u_2} [has_scalar R' M]\n    (r : R') : free_add_monoid (M × N) →+ tensor_product R M N :=\n  coe_fn free_add_monoid.lift fun (p : M × N) => tmul R (r • prod.fst p) (prod.snd p)\n\ntheorem smul.aux_of {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] {R' : Type u_2}\n    [has_scalar R' M] (r : R') (m : M) (n : N) :\n    coe_fn (smul.aux r) (free_add_monoid.of (m, n)) = tmul R (r • m) n :=\n  rfl\n\n-- Most of the time we want the instance below this one, which is easier for typeclass resolution\n\n-- to find. The `unused_arguments` is from one of the two comm_classes - while we only make use\n\n-- of one, it makes sense to make the API symmetric.\n\nprotected instance has_scalar' {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] : has_scalar R' (tensor_product R M N) :=\n  has_scalar.mk fun (r : R') => ⇑(add_con.lift (add_con_gen (eqv R M N)) (smul.aux r) sorry)\n\nprotected instance has_scalar {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    has_scalar R (tensor_product R M N) :=\n  tensor_product.has_scalar'\n\nprotected theorem smul_zero {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] (r : R') : r • 0 = 0 :=\n  add_monoid_hom.map_zero\n    (add_con.lift (add_con_gen (eqv R M N)) (smul.aux r) (has_scalar'._proof_1 r))\n\nprotected theorem smul_add {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] (r : R') (x : tensor_product R M N) (y : tensor_product R M N) :\n    r • (x + y) = r • x + r • y :=\n  add_monoid_hom.map_add\n    (add_con.lift (add_con_gen (eqv R M N)) (smul.aux r) (has_scalar'._proof_1 r)) x y\n\n-- Most of the time we want the instance below this one, which is easier for typeclass resolution\n\n-- to find.\n\nprotected instance semimodule' {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] : semimodule R' (tensor_product R M N) :=\n  (fun (this : ∀ (r : R') (m : M) (n : N), r • tmul R m n = tmul R (r • m) n) =>\n      semimodule.mk sorry sorry)\n    sorry\n\nprotected instance semimodule {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    semimodule R (tensor_product R M N) :=\n  tensor_product.semimodule'\n\n-- note that we don't actually need `compatible_smul` here, but we include it for symmetry\n\n-- with `tmul_smul` to avoid exposing our asymmetric definition.\n\ntheorem smul_tmul' {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] [compatible_smul R R' M N] (r : R') (m : M) (n : N) :\n    r • tmul R m n = tmul R (r • m) n :=\n  rfl\n\n@[simp] theorem tmul_smul {R : Type u_1} [comm_semiring R] {R' : Type u_2} [comm_semiring R']\n    {M : Type u_3} {N : Type u_4} [add_comm_monoid M] [add_comm_monoid N] [semimodule R M]\n    [semimodule R N] [semimodule R' M] [semimodule R' N] [smul_comm_class R R' M]\n    [smul_comm_class R R' N] [compatible_smul R R' M N] (r : R') (x : M) (y : N) :\n    tmul R x (r • y) = r • tmul R x y :=\n  Eq.symm (smul_tmul r x y)\n\n/-- The canonical bilinear map `M → N → M ⊗[R] N`. -/\ndef mk (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    linear_map R M (linear_map R N (tensor_product R M N)) :=\n  linear_map.mk₂ R (fun (_x : M) (_y : N) => tmul R _x _y) add_tmul sorry tmul_add sorry\n\n@[simp] theorem mk_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    coe_fn (coe_fn (mk R M N) m) n = tmul R m n :=\n  rfl\n\ntheorem ite_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (x₁ : M) (x₂ : N) (P : Prop)\n    [Decidable P] : tmul R (ite P x₁ 0) x₂ = ite P (tmul R x₁ x₂) 0 :=\n  sorry\n\ntheorem tmul_ite {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (x₁ : M) (x₂ : N) (P : Prop)\n    [Decidable P] : tmul R x₁ (ite P x₂ 0) = ite P (tmul R x₁ x₂) 0 :=\n  sorry\n\ntheorem sum_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] {α : Type u_2} (s : finset α) (m : α → M)\n    (n : N) :\n    tmul R (finset.sum s fun (a : α) => m a) n = finset.sum s fun (a : α) => tmul R (m a) n :=\n  sorry\n\ntheorem tmul_sum {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) {α : Type u_2} (s : finset α)\n    (n : α → N) :\n    tmul R m (finset.sum s fun (a : α) => n a) = finset.sum s fun (a : α) => tmul R m (n a) :=\n  sorry\n\n/-- Auxiliary function to constructing a linear map `M ⊗ N → P` given a bilinear map `M → N → P`\nwith the property that its composition with the canonical bilinear map `M → N → M ⊗ N` is\nthe given bilinear map `M → N → P`. -/\ndef lift_aux {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) : tensor_product R M N →+ P :=\n  add_con.lift (add_con_gen (eqv R M N))\n    (coe_fn free_add_monoid.lift fun (p : M × N) => coe_fn (coe_fn f (prod.fst p)) (prod.snd p))\n    sorry\n\ntheorem lift_aux_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) (m : M) (n : N) :\n    coe_fn (lift_aux f) (tmul R m n) = coe_fn (coe_fn f m) n :=\n  zero_add ((fun (p : M × N) => coe_fn (coe_fn f (prod.fst p)) (prod.snd p)) (m, n))\n\n@[simp] theorem lift_aux.smul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] {f : linear_map R M (linear_map R N P)} (r : R)\n    (x : tensor_product R M N) : coe_fn (lift_aux f) (r • x) = r • coe_fn (lift_aux f) x :=\n  sorry\n\n/-- Constructing a linear map `M ⊗ N → P` given a bilinear map `M → N → P` with the property that\nits composition with the canonical bilinear map `M → N → M ⊗ N` is\nthe given bilinear map `M → N → P`. -/\ndef lift {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R M (linear_map R N P)) :\n    linear_map R (tensor_product R M N) P :=\n  linear_map.mk (add_monoid_hom.to_fun (lift_aux f)) sorry lift_aux.smul\n\n@[simp] theorem lift.tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] {f : linear_map R M (linear_map R N P)} (x : M) (y : N) :\n    coe_fn (lift f) (tmul R x y) = coe_fn (coe_fn f x) y :=\n  zero_add ((fun (p : M × N) => coe_fn (coe_fn f (prod.fst p)) (prod.snd p)) (x, y))\n\n@[simp] theorem lift.tmul' {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] {f : linear_map R M (linear_map R N P)} (x : M) (y : N) :\n    linear_map.to_fun (lift f) (tmul R x y) = coe_fn (coe_fn f x) y :=\n  lift.tmul x y\n\ntheorem ext {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] {g : linear_map R (tensor_product R M N) P}\n    {h : linear_map R (tensor_product R M N) P}\n    (H : ∀ (x : M) (y : N), coe_fn g (tmul R x y) = coe_fn h (tmul R x y)) : g = h :=\n  sorry\n\ntheorem lift.unique {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] {f : linear_map R M (linear_map R N P)}\n    {g : linear_map R (tensor_product R M N) P}\n    (H : ∀ (x : M) (y : N), coe_fn g (tmul R x y) = coe_fn (coe_fn f x) y) : g = lift f :=\n  sorry\n\ntheorem lift_mk {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} [add_comm_monoid M]\n    [add_comm_monoid N] [semimodule R M] [semimodule R N] : lift (mk R M N) = linear_map.id :=\n  Eq.symm (lift.unique fun (x : M) (y : N) => rfl)\n\ntheorem lift_compr₂ {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    {f : linear_map R M (linear_map R N P)} (g : linear_map R P Q) :\n    lift (linear_map.compr₂ f g) = linear_map.comp g (lift f) :=\n  sorry\n\ntheorem lift_mk_compr₂ {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R (tensor_product R M N) P) :\n    lift (linear_map.compr₂ (mk R M N) f) = f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (lift (linear_map.compr₂ (mk R M N) f) = f)) (lift_compr₂ f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (linear_map.comp f (lift (mk R M N)) = f)) lift_mk))\n      (eq.mpr\n        (id (Eq._oldrec (Eq.refl (linear_map.comp f linear_map.id = f)) (linear_map.comp_id f)))\n        (Eq.refl f)))\n\ntheorem mk_compr₂_inj {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] {g : linear_map R (tensor_product R M N) P}\n    {h : linear_map R (tensor_product R M N) P}\n    (H : linear_map.compr₂ (mk R M N) g = linear_map.compr₂ (mk R M N) h) : g = h :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (g = h)) (Eq.symm (lift_mk_compr₂ g))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (lift (linear_map.compr₂ (mk R M N) g) = h)) H))\n      (eq.mpr\n        (id (Eq._oldrec (Eq.refl (lift (linear_map.compr₂ (mk R M N) h) = h)) (lift_mk_compr₂ h)))\n        (Eq.refl h)))\n\n/-- Linearly constructing a linear map `M ⊗ N → P` given a bilinear map `M → N → P`\nwith the property that its composition with the canonical bilinear map `M → N → M ⊗ N` is\nthe given bilinear map `M → N → P`. -/\ndef uncurry (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) (P : Type u_5)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R M (linear_map R N P)) (linear_map R (tensor_product R M N) P) :=\n  linear_map.flip\n    (lift\n      (linear_map.comp (linear_map.lflip R (linear_map R M (linear_map R N P)) N P)\n        (linear_map.flip linear_map.id)))\n\n@[simp] theorem uncurry_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R M (linear_map R N P)) (m : M) (n : N) :\n    coe_fn (coe_fn (uncurry R M N P) f) (tmul R m n) = coe_fn (coe_fn f m) n :=\n  sorry\n\n/-- A linear equivalence constructing a linear map `M ⊗ N → P` given a bilinear map `M → N → P`\nwith the property that its composition with the canonical bilinear map `M → N → M ⊗ N` is\nthe given bilinear map `M → N → P`. -/\ndef lift.equiv (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) (P : Type u_5)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_equiv R (linear_map R M (linear_map R N P)) (linear_map R (tensor_product R M N) P) :=\n  linear_equiv.mk (linear_map.to_fun (uncurry R M N P)) sorry sorry\n    (fun (f : linear_map R (tensor_product R M N) P) => linear_map.compr₂ (mk R M N) f) sorry sorry\n\n/-- Given a linear map `M ⊗ N → P`, compose it with the canonical bilinear map `M → N → M ⊗ N` to\nform a bilinear map `M → N → P`. -/\ndef lcurry (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) (P : Type u_5)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R (tensor_product R M N) P) (linear_map R M (linear_map R N P)) :=\n  ↑(linear_equiv.symm (lift.equiv R M N P))\n\n@[simp] theorem lcurry_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R (tensor_product R M N) P) (m : M) (n : N) :\n    coe_fn (coe_fn (coe_fn (lcurry R M N P) f) m) n = coe_fn f (tmul R m n) :=\n  rfl\n\n/-- Given a linear map `M ⊗ N → P`, compose it with the canonical bilinear map `M → N → M ⊗ N` to\nform a bilinear map `M → N → P`. -/\ndef curry {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R (tensor_product R M N) P) :\n    linear_map R M (linear_map R N P) :=\n  coe_fn (lcurry R M N P) f\n\n@[simp] theorem curry_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R (tensor_product R M N) P) (m : M) (n : N) :\n    coe_fn (coe_fn (curry f) m) n = coe_fn f (tmul R m n) :=\n  rfl\n\ntheorem ext_threefold {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    {g : linear_map R (tensor_product R (tensor_product R M N) P) Q}\n    {h : linear_map R (tensor_product R (tensor_product R M N) P) Q}\n    (H :\n      ∀ (x : M) (y : N) (z : P),\n        coe_fn g (tmul R (tmul R x y) z) = coe_fn h (tmul R (tmul R x y) z)) :\n    g = h :=\n  sorry\n\n-- We'll need this one for checking the pentagon identity!\n\ntheorem ext_fourfold {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} {S : Type u_7} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [add_comm_monoid S] [semimodule R M] [semimodule R N] [semimodule R P]\n    [semimodule R Q] [semimodule R S]\n    {g : linear_map R (tensor_product R (tensor_product R (tensor_product R M N) P) Q) S}\n    {h : linear_map R (tensor_product R (tensor_product R (tensor_product R M N) P) Q) S}\n    (H :\n      ∀ (w : M) (x : N) (y : P) (z : Q),\n        coe_fn g (tmul R (tmul R (tmul R w x) y) z) = coe_fn h (tmul R (tmul R (tmul R w x) y) z)) :\n    g = h :=\n  sorry\n\n/--\nThe base ring is a left identity for the tensor product of modules, up to linear equivalence.\n-/\nprotected def lid (R : Type u_1) [comm_semiring R] (M : Type u_3) [add_comm_monoid M]\n    [semimodule R M] : linear_equiv R (tensor_product R R M) M :=\n  linear_equiv.of_linear (lift (linear_map.lsmul R M)) (coe_fn (mk R R M) 1) sorry sorry\n\n@[simp] theorem lid_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} [add_comm_monoid M]\n    [semimodule R M] (m : M) (r : R) : coe_fn (tensor_product.lid R M) (tmul R r m) = r • m :=\n  sorry\n\n@[simp] theorem lid_symm_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} [add_comm_monoid M]\n    [semimodule R M] (m : M) : coe_fn (linear_equiv.symm (tensor_product.lid R M)) m = tmul R 1 m :=\n  rfl\n\n/--\nThe tensor product of modules is commutative, up to linear equivalence.\n-/\nprotected def comm (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] :\n    linear_equiv R (tensor_product R M N) (tensor_product R N M) :=\n  linear_equiv.of_linear (lift (linear_map.flip (mk R N M))) (lift (linear_map.flip (mk R M N)))\n    sorry sorry\n\n@[simp] theorem comm_tmul (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    coe_fn (tensor_product.comm R M N) (tmul R m n) = tmul R n m :=\n  rfl\n\n@[simp] theorem comm_symm_tmul (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    coe_fn (linear_equiv.symm (tensor_product.comm R M N)) (tmul R n m) = tmul R m n :=\n  rfl\n\n/--\nThe base ring is a right identity for the tensor product of modules, up to linear equivalence.\n-/\nprotected def rid (R : Type u_1) [comm_semiring R] (M : Type u_3) [add_comm_monoid M]\n    [semimodule R M] : linear_equiv R (tensor_product R M R) M :=\n  linear_equiv.trans (tensor_product.comm R M R) (tensor_product.lid R M)\n\n@[simp] theorem rid_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} [add_comm_monoid M]\n    [semimodule R M] (m : M) (r : R) : coe_fn (tensor_product.rid R M) (tmul R m r) = r • m :=\n  sorry\n\n@[simp] theorem rid_symm_apply {R : Type u_1} [comm_semiring R] {M : Type u_3} [add_comm_monoid M]\n    [semimodule R M] (m : M) : coe_fn (linear_equiv.symm (tensor_product.rid R M)) m = tmul R m 1 :=\n  rfl\n\n/-- The associator for tensor product of R-modules, as a linear equivalence. -/\nprotected def assoc (R : Type u_1) [comm_semiring R] (M : Type u_3) (N : Type u_4) (P : Type u_5)\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_equiv R (tensor_product R (tensor_product R M N) P)\n        (tensor_product R M (tensor_product R N P)) :=\n  linear_equiv.of_linear\n    (lift\n      (lift\n        (linear_map.comp (lcurry R N P (tensor_product R M (tensor_product R N P)))\n          (mk R M (tensor_product R N P)))))\n    (lift\n      (linear_map.comp (uncurry R N P (tensor_product R (tensor_product R M N) P))\n        (curry (mk R (tensor_product R M N) P))))\n    sorry sorry\n\n@[simp] theorem assoc_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (m : M) (n : N) (p : P) :\n    coe_fn (tensor_product.assoc R M N P) (tmul R (tmul R m n) p) = tmul R m (tmul R n p) :=\n  rfl\n\n@[simp] theorem assoc_symm_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (m : M) (n : N) (p : P) :\n    coe_fn (linear_equiv.symm (tensor_product.assoc R M N P)) (tmul R m (tmul R n p)) =\n        tmul R (tmul R m n) p :=\n  rfl\n\n/-- The tensor product of a pair of linear maps between modules. -/\ndef map {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5} {Q : Type u_6}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q] [semimodule R M]\n    [semimodule R N] [semimodule R P] [semimodule R Q] (f : linear_map R M P)\n    (g : linear_map R N Q) : linear_map R (tensor_product R M N) (tensor_product R P Q) :=\n  lift (linear_map.comp (linear_map.compl₂ (mk R P Q) g) f)\n\n@[simp] theorem map_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M P) (g : linear_map R N Q) (m : M) (n : N) :\n    coe_fn (map f g) (tmul R m n) = tmul R (coe_fn f m) (coe_fn g n) :=\n  rfl\n\ntheorem map_comp {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q] {P' : Type u_8}\n    {Q' : Type u_9} [add_comm_monoid P'] [semimodule R P'] [add_comm_monoid Q'] [semimodule R Q']\n    (f₂ : linear_map R P P') (f₁ : linear_map R M P) (g₂ : linear_map R Q Q')\n    (g₁ : linear_map R N Q) :\n    map (linear_map.comp f₂ f₁) (linear_map.comp g₂ g₁) = linear_map.comp (map f₂ g₂) (map f₁ g₁) :=\n  sorry\n\ntheorem lift_comp_map {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q] {Q' : Type u_9}\n    [add_comm_monoid Q'] [semimodule R Q'] (i : linear_map R P (linear_map R Q Q'))\n    (f : linear_map R M P) (g : linear_map R N Q) :\n    linear_map.comp (lift i) (map f g) = lift (linear_map.compl₂ (linear_map.comp i f) g) :=\n  sorry\n\n/-- If `M` and `P` are linearly equivalent and `N` and `Q` are linearly equivalent\nthen `M ⊗ N` and `P ⊗ Q` are linearly equivalent. -/\ndef congr {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q] (f : linear_equiv R M P)\n    (g : linear_equiv R N Q) : linear_equiv R (tensor_product R M N) (tensor_product R P Q) :=\n  linear_equiv.of_linear (map ↑f ↑g) (map ↑(linear_equiv.symm f) ↑(linear_equiv.symm g)) sorry sorry\n\n@[simp] theorem congr_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_equiv R M P) (g : linear_equiv R N Q) (m : M) (n : N) :\n    coe_fn (congr f g) (tmul R m n) = tmul R (coe_fn f m) (coe_fn g n) :=\n  rfl\n\n@[simp] theorem congr_symm_tmul {R : Type u_1} [comm_semiring R] {M : Type u_3} {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_equiv R M P) (g : linear_equiv R N Q) (p : P) (q : Q) :\n    coe_fn (linear_equiv.symm (congr f g)) (tmul R p q) =\n        tmul R (coe_fn (linear_equiv.symm f) p) (coe_fn (linear_equiv.symm g) q) :=\n  rfl\n\nend tensor_product\n\n\nnamespace linear_map\n\n\n/-- `ltensor M f : M ⊗ N →ₗ M ⊗ P` is the natural linear map induced by `f : N →ₗ P`. -/\ndef ltensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R N P) :\n    linear_map R (tensor_product R M N) (tensor_product R M P) :=\n  tensor_product.map id f\n\n/-- `rtensor f M : N₁ ⊗ M →ₗ N₂ ⊗ M` is the natural linear map induced by `f : N₁ →ₗ N₂`. -/\ndef rtensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] (f : linear_map R N P) :\n    linear_map R (tensor_product R N M) (tensor_product R P M) :=\n  tensor_product.map f id\n\n@[simp] theorem ltensor_tmul {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (m : M) (n : N) :\n    coe_fn (ltensor M f) (tensor_product.tmul R m n) = tensor_product.tmul R m (coe_fn f n) :=\n  rfl\n\n@[simp] theorem rtensor_tmul {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (m : M) (n : N) :\n    coe_fn (rtensor M f) (tensor_product.tmul R n m) = tensor_product.tmul R (coe_fn f n) m :=\n  rfl\n\n/-- `ltensor_hom M` is the natural linear map that sends a linear map `f : N →ₗ P` to `M ⊗ f`. -/\ndef ltensor_hom {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R N P) (linear_map R (tensor_product R M N) (tensor_product R M P)) :=\n  mk (ltensor M) sorry sorry\n\n/-- `rtensor_hom M` is the natural linear map that sends a linear map `f : N →ₗ P` to `M ⊗ f`. -/\ndef rtensor_hom {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M] [semimodule R N]\n    [semimodule R P] :\n    linear_map R (linear_map R N P) (linear_map R (tensor_product R N M) (tensor_product R P M)) :=\n  mk (fun (f : linear_map R N P) => rtensor M f) sorry sorry\n\n@[simp] theorem coe_ltensor_hom {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] : ⇑(ltensor_hom M) = ltensor M :=\n  rfl\n\n@[simp] theorem coe_rtensor_hom {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] : ⇑(rtensor_hom M) = rtensor M :=\n  rfl\n\n@[simp] theorem ltensor_add {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (g : linear_map R N P) :\n    ltensor M (f + g) = ltensor M f + ltensor M g :=\n  map_add (ltensor_hom M) f g\n\n@[simp] theorem rtensor_add {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (g : linear_map R N P) :\n    rtensor M (f + g) = rtensor M f + rtensor M g :=\n  map_add (rtensor_hom M) f g\n\n@[simp] theorem ltensor_zero {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] : ltensor M 0 = 0 :=\n  map_zero (ltensor_hom M)\n\n@[simp] theorem rtensor_zero {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] : rtensor M 0 = 0 :=\n  map_zero (rtensor_hom M)\n\n@[simp] theorem ltensor_smul {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (r : R) (f : linear_map R N P) :\n    ltensor M (r • f) = r • ltensor M f :=\n  map_smul (ltensor_hom M) r f\n\n@[simp] theorem rtensor_smul {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (r : R) (f : linear_map R N P) :\n    rtensor M (r • f) = r • rtensor M f :=\n  map_smul (rtensor_hom M) r f\n\ntheorem ltensor_comp {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q] (g : linear_map R P Q)\n    (f : linear_map R N P) : ltensor M (comp g f) = comp (ltensor M g) (ltensor M f) :=\n  sorry\n\ntheorem rtensor_comp {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4} {P : Type u_5}\n    {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P] [add_comm_monoid Q]\n    [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q] (g : linear_map R P Q)\n    (f : linear_map R N P) : rtensor M (comp g f) = comp (rtensor M g) (rtensor M f) :=\n  sorry\n\n@[simp] theorem ltensor_id {R : Type u_1} [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] : ltensor M id = id :=\n  sorry\n\n@[simp] theorem rtensor_id {R : Type u_1} [comm_semiring R] (M : Type u_3) (N : Type u_4)\n    [add_comm_monoid M] [add_comm_monoid N] [semimodule R M] [semimodule R N] : rtensor M id = id :=\n  sorry\n\n@[simp] theorem ltensor_comp_rtensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M P) (g : linear_map R N Q) :\n    comp (ltensor P g) (rtensor N f) = tensor_product.map f g :=\n  sorry\n\n@[simp] theorem rtensor_comp_ltensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} [add_comm_monoid M] [add_comm_monoid N] [add_comm_monoid P]\n    [add_comm_monoid Q] [semimodule R M] [semimodule R N] [semimodule R P] [semimodule R Q]\n    (f : linear_map R M P) (g : linear_map R N Q) :\n    comp (rtensor Q f) (ltensor M g) = tensor_product.map f g :=\n  sorry\n\n@[simp] theorem map_comp_rtensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} {S : Type u_7} [add_comm_monoid M] [add_comm_monoid N]\n    [add_comm_monoid P] [add_comm_monoid Q] [add_comm_monoid S] [semimodule R M] [semimodule R N]\n    [semimodule R P] [semimodule R Q] [semimodule R S] (f : linear_map R M P) (g : linear_map R N Q)\n    (f' : linear_map R S M) :\n    comp (tensor_product.map f g) (rtensor N f') = tensor_product.map (comp f f') g :=\n  sorry\n\n@[simp] theorem map_comp_ltensor {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} {S : Type u_7} [add_comm_monoid M] [add_comm_monoid N]\n    [add_comm_monoid P] [add_comm_monoid Q] [add_comm_monoid S] [semimodule R M] [semimodule R N]\n    [semimodule R P] [semimodule R Q] [semimodule R S] (f : linear_map R M P) (g : linear_map R N Q)\n    (g' : linear_map R S N) :\n    comp (tensor_product.map f g) (ltensor M g') = tensor_product.map f (comp g g') :=\n  sorry\n\n@[simp] theorem rtensor_comp_map {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} {S : Type u_7} [add_comm_monoid M] [add_comm_monoid N]\n    [add_comm_monoid P] [add_comm_monoid Q] [add_comm_monoid S] [semimodule R M] [semimodule R N]\n    [semimodule R P] [semimodule R Q] [semimodule R S] (f' : linear_map R P S)\n    (f : linear_map R M P) (g : linear_map R N Q) :\n    comp (rtensor Q f') (tensor_product.map f g) = tensor_product.map (comp f' f) g :=\n  sorry\n\n@[simp] theorem ltensor_comp_map {R : Type u_1} [comm_semiring R] (M : Type u_3) {N : Type u_4}\n    {P : Type u_5} {Q : Type u_6} {S : Type u_7} [add_comm_monoid M] [add_comm_monoid N]\n    [add_comm_monoid P] [add_comm_monoid Q] [add_comm_monoid S] [semimodule R M] [semimodule R N]\n    [semimodule R P] [semimodule R Q] [semimodule R S] (g' : linear_map R Q S)\n    (f : linear_map R M P) (g : linear_map R N Q) :\n    comp (ltensor P g') (tensor_product.map f g) = tensor_product.map f (comp g' g) :=\n  sorry\n\nend linear_map\n\n\nnamespace tensor_product\n\n\n/-- Auxiliary function to defining negation multiplication on tensor product. -/\ndef neg.aux (R : Type u_1) [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] :\n    free_add_monoid (M × N) →+ tensor_product R M N :=\n  coe_fn free_add_monoid.lift fun (p : M × N) => tmul R (-prod.fst p) (prod.snd p)\n\ntheorem neg.aux_of {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    coe_fn (neg.aux R) (free_add_monoid.of (m, n)) = tmul R (-m) n :=\n  rfl\n\nprotected instance has_neg {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    [add_comm_group M] [add_comm_group N] [semimodule R M] [semimodule R N] :\n    Neg (tensor_product R M N) :=\n  { neg := ⇑(add_con.lift (add_con_gen (eqv R M N)) (neg.aux R) sorry) }\n\nprotected instance add_comm_group {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    [add_comm_group M] [add_comm_group N] [semimodule R M] [semimodule R N] :\n    add_comm_group (tensor_product R M N) :=\n  add_comm_group.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry Neg.neg\n    (fun (_x _x_1 : tensor_product R M N) => add_semigroup.add _x (-_x_1)) sorry sorry\n\ntheorem neg_tmul {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    tmul R (-m) n = -tmul R m n :=\n  rfl\n\ntheorem tmul_neg {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] (m : M) (n : N) :\n    tmul R m (-n) = -tmul R m n :=\n  linear_map.map_neg (coe_fn (mk R M N) m) n\n\ntheorem tmul_sub {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] (m : M) (n₁ : N) (n₂ : N) :\n    tmul R m (n₁ - n₂) = tmul R m n₁ - tmul R m n₂ :=\n  linear_map.map_sub (coe_fn (mk R M N) m) n₁ n₂\n\ntheorem sub_tmul {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3} [add_comm_group M]\n    [add_comm_group N] [semimodule R M] [semimodule R N] (m₁ : M) (m₂ : M) (n : N) :\n    tmul R (m₁ - m₂) n = tmul R m₁ n - tmul R m₂ n :=\n  linear_map.map_sub₂ (mk R M N) m₁ m₂ n\n\n/--\nWhile the tensor product will automatically inherit a ℤ-module structure from\n`add_comm_group.int_module`, that structure won't be compatible with lemmas like `tmul_smul` unless\nwe use a `ℤ-module` instance provided by `tensor_product.semimodule'`.\n\nWhen `R` is a `ring` we get the required `tensor_product.compatible_smul` instance through\n`is_scalar_tower`, but when it is only a `semiring` we need to build it from scratch.\nThe instance diamond in `compatible_smul` doesn't matter because it's in `Prop`.\n-/\nprotected instance compatible_smul.int {R : Type u_1} [comm_semiring R] {M : Type u_2}\n    {N : Type u_3} [add_comm_group M] [add_comm_group N] [semimodule R M] [semimodule R N]\n    [semimodule ℤ M] [semimodule ℤ N] : compatible_smul R ℤ M N :=\n  compatible_smul.mk sorry\n\nend tensor_product\n\n\nnamespace linear_map\n\n\n@[simp] theorem ltensor_sub {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_group M] [add_comm_group N] [add_comm_group P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (g : linear_map R N P) :\n    ltensor M (f - g) = ltensor M f - ltensor M g :=\n  sorry\n\n@[simp] theorem rtensor_sub {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_group M] [add_comm_group N] [add_comm_group P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) (g : linear_map R N P) :\n    rtensor M (f - g) = rtensor M f - rtensor M g :=\n  sorry\n\n@[simp] theorem ltensor_neg {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_group M] [add_comm_group N] [add_comm_group P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) : ltensor M (-f) = -ltensor M f :=\n  sorry\n\n@[simp] theorem rtensor_neg {R : Type u_1} [comm_semiring R] {M : Type u_2} {N : Type u_3}\n    {P : Type u_4} [add_comm_group M] [add_comm_group N] [add_comm_group P] [semimodule R M]\n    [semimodule R N] [semimodule R P] (f : linear_map R N P) : rtensor M (-f) = -rtensor M f :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/tensor_product_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188373563072, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33741179972202606}}
{"text": "import tactic\nimport measure_theory.interval_integral\nimport measure_theory.lebesgue_measure\nimport measure_theory.set_integral\nimport analysis.calculus.deriv\nimport analysis.special_functions.exp_log\nimport analysis.special_functions.trigonometric\nimport data.finset\n\nnoncomputable theory\nopen_locale classical\nopen_locale big_operators\nopen measure_theory\nopen interval_integral\nopen set\nopen real\n\nnamespace tactic.interactive\n\n\nmeta def show_continuous := `[\n  all_goals {try {simp}},\n  apply_rules [\n    continuous_on.neg,\n    continuous.continuous_on,\n    differentiable.continuous,\n    differentiable_on.continuous_on,\n    continuous.Icc_extend,\n    continuous_on.mono,\n    continuous.neg,\n    continuous_id,\n    continuous_sin,\n    continuous_cos,\n    continuous_const,\n    continuous.pow,\n    continuous.mul,\n    continuous.smul,\n    continuous.sub,\n    continuous.add\n    ] 10,\n  all_goals {try {norm_num}}\n]\n\nmeta def show_differentiable := `[\n  apply_rules [\n    differentiable.differentiable_on,\n    differentiable.neg,\n    differentiable.smul,\n    differentiable.cos,\n    differentiable.sin,\n    differentiable_const,\n    differentiable_id,\n    differentiable.mul,\n    differentiable_fpow\n    ] 10,\n  all_goals {try {norm_num}}\n]\n\nmeta def show_nonzero := `[\n  apply_rules [\n    mul_ne_zero,\n    sub_ne_zero_of_ne,\n    pow_ne_zero,\n    ne_of_gt,\n    ne_of_lt\n    ] 10,\n  all_goals {try {norm_cast}, try {norm_num}}\n]\n\nmeta def show_pos := `[\n  apply_rules [\n    nat.succ_pos,\n    mul_pos,\n    div_pos,\n    inv_pos.mpr,\n    pow_pos\n    ] 10,\n  all_goals {try {norm_cast}, try {norm_num}, try {nlinarith}}\n]\n\n\nmeta def clear_denoms := `[\n  try {rw div_eq_div_iff},\n  try {rw eq_div_iff},\n  try {symmetry, rw eq_div_iff},\n  try { ring_exp },\n  all_goals {show_nonzero}\n]\n\nmeta def discrete_field := `[\n  try {ext},\n  try {field_simp *},\n  try {clear_denoms},\n  try {ring_exp},\n  try {norm_num},\n  try {linarith}\n]\n\nend tactic.interactive\n\nlemma integrable_of_cont {f : ℝ → ℝ} (a b : ℝ) (h : continuous f):\n    interval_integrable f measure_theory.measure_space.volume a b :=\nbegin\n    have hmeas : measurable f := continuous.measurable h,\n    have hconton : continuous_on f (interval a b) := continuous.continuous_on h,\n    exact continuous_on.interval_integrable hconton,\nend\n\n/- lemma self_mem_ae_restrict\n  {s : set ℝ} (hs : is_measurable s):\n  s ∈ (measure.restrict measure_space.volume s).ae :=\nbegin\n  rw ae_restrict_eq hs,\n  simp only [exists_prop, filter.mem_principal_sets, filter.mem_inf_sets],\n  exact ⟨univ, filter.univ_mem_sets, s, by simp⟩,\nend\n -/\nlemma nonempty_inter_of_nonempty_inter_closure {α : Type*} [topological_space α] {s t : set α}\n  (hs : is_open s) (h : (s ∩ closure t).nonempty) : (s ∩ t).nonempty :=\nlet ⟨x, xs, xt⟩ := h in _root_.mem_closure_iff.1 xt s hs xs\n\nlemma real.volume_pos_of_is_open_of_nonempty {s : set ℝ} (h : is_open s) (h' : s.nonempty) :\n  0 < volume s :=\nbegin\n  rcases h' with ⟨x, hx⟩,\n  have : ∀ᶠ (y : ℝ) in nhds x, y ∈ s := filter.eventually_of_mem (mem_nhds_sets h hx) (λ y H, H),\n  exact filter.eventually.volume_pos_of_nhds_real this,\nend\n\ntheorem integral_strictly_pos_of_cont (f : ℝ → ℝ) (a b : ℝ)\n    (hf : continuous f)\n    (hab : a < b)\n    (h : ∀ (x : ℝ), a ≤ x → x ≤ b → 0 ≤ f x)\n    (hneq: ∃ x, a ≤ x ∧ x ≤ b ∧ 0 < f x) :\n    0 < ∫ x in a..b, f x :=\nbegin\n  rw integral_pos_iff_support_of_nonneg_ae',\n  { refine ⟨hab, _⟩,\n    let s := {b : ℝ | 0 < f b},\n    have s_open : is_open s := is_open_lt continuous_const hf,\n    have : (s ∩ closure (Ioo a b)).nonempty,\n    { rw closure_Ioo hab,\n      rcases hneq with ⟨x, ax, xb, fxpos⟩,\n      have : x ∈ s ∩ Icc a b := ⟨fxpos, ax, xb⟩,\n      exact nonempty_of_mem this },\n    have : (s ∩ Ioo a b).nonempty := nonempty_inter_of_nonempty_inter_closure s_open this,\n    have : 0 < volume (s ∩ Ioo a b) :=\n      real.volume_pos_of_is_open_of_nonempty (is_open_inter s_open is_open_Ioo) this,\n    refine this.trans_le (measure_mono (λ x hx, _)),\n    split,\n    { exact ne_of_gt (show 0 < f x, from hx.1) },\n    { exact ⟨hx.2.1, hx.2.2.le⟩ } },\n  { have : Ioc b a = ∅ := Ioc_eq_empty hab.le,\n    simp only [this, union_empty],\n    apply filter.eventually_of_mem _ _,\n    exact Icc a b,\n    {\n        simp,\n        use univ,\n        simp,\n        use Icc a b,\n        exact ⟨Ioc_subset_Icc_self, rfl.subset⟩,\n    },\n    simpa using h },\n  { exact integrable_of_cont a b hf }\nend\n\ntheorem integral_strictly_monotone_of_cont (f g : ℝ → ℝ) (a b : ℝ)\n    (hf : continuous f) (hg : continuous g) (hab : a < b)    \n    (h : ∀ (x : ℝ), a ≤ x → x ≤ b → f x ≤ g x)\n    (hneq: ∃ x, a ≤ x ∧ x ≤ b ∧ f x < g x) :\n    ∫ x in a..b, f x < ∫ x in a..b, g x := \nbegin\n    have H : 0 < ∫ x in a..b, (g x - f x),\n    {\n        apply integral_strictly_pos_of_cont\n        (g-f) a b (continuous.sub hg hf) hab,\n        all_goals {\n            simp [sub_pos],\n            assumption,\n        },\n    },\n    rw [←sub_pos, ←interval_integral.integral_sub (integrable_of_cont a b hg) (integrable_of_cont a b hf)],\n    exact H,\nend\n\nlemma int_pos_of_pos {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hf : continuous f)\n(hnonneg : ∀ x, a ≤ x → x ≤ b → 0 ≤ f x)\n(hx : ∃ x, a ≤ x ∧ x ≤ b ∧ 0 < f x) : 0 < ∫ x in a..b, f x :=\nbegin\n    rw ← (integral_zero : ∫ x in a..b, (0:ℝ) = 0 ),\n    exact integral_strictly_monotone_of_cont (λ x, (0:ℝ)) f a b continuous_const hf hab hnonneg hx,\nend\n\nlemma int_pos_of_square {f : ℝ → ℝ} {a b} (x : ℝ)\n    (hab : a < b) (hf : continuous f) (hx : a ≤ x ∧ x ≤ b ∧ f x ≠ 0) :\n     0 < ∫ x in a..b, (f x)^2 :=\nbegin\n    refine int_pos_of_pos hab _\n        (λ x hx1 hx2, pow_two_nonneg (f x)) ⟨x, ⟨hx.1, hx.2.1, pow_two_pos_of_ne_zero (f x) hx.2.2⟩⟩,\n    show_continuous,\nend\n\ntheorem my_integral_smul (f : ℝ → ℝ) (a b c : ℝ) :\n    ∫ x in a..b, c * (f x) = c * ∫ x in a..b, f x :=\nbegin\n    suffices :  ∫ x in a..b, c • (f x) = c • ∫ x in a..b, f x, by exact this,\n    rw_mod_cast interval_integral.integral_smul,\nend\n\ntheorem product_rule {f g : ℝ → ℝ} (hdf : differentiable ℝ f) (hdg : differentiable ℝ g) :\n    deriv (f * g) = (deriv f) * g + f * deriv g :=\nbegin\n    ext,\n    have hdf0 : differentiable_at ℝ f x := hdf x,\n    have hdg0 : differentiable_at ℝ g x := hdg x,\n    apply deriv_mul hdf0 hdg0,\nend\n\ntheorem differentiable_fpow {f : ℝ → ℝ} {n : ℕ} :\n    differentiable ℝ f → differentiable ℝ (f^n) :=\nbegin\n    induction n with d hd,\n    { intro h,\n      simp only [pow_zero],\n        exact differentiable_const 1 },\n    {\n        intro h,\n        rw pow_succ,\n        exact h.mul (hd h),\n    }\nend\n\ntheorem power_rule {f : ℝ → ℝ} {n : ℕ} (hfd : differentiable ℝ f):\n    deriv (f^(n+1)) = ((n : ℝ) + 1) • f^n * (deriv f) := \nbegin\n    induction n with d hd, by norm_num,\n    have H : f^(d+1) = f^d * f := pow_succ' f d,\n    calc\n        deriv (f^(d.succ+1)) = deriv (f^(d.succ) * f) : by {rw pow_succ' f (d.succ),}\n        ... = (deriv (f^(d.succ))) * f + f^(d+1) * (deriv f) :\n        begin\n            rw product_rule,\n            exact differentiable_fpow hfd,\n            exact hfd,\n        end\n        ... = ((d:ℝ) + 1) • f^d * deriv f * f + f^d.succ * deriv f : by {rw hd}\n        ... = ((d:ℝ) + 1) • (f^(d.succ)) * deriv f + f^(d.succ) * deriv f :\n        begin\n            simp only [add_left_inj, H],\n            norm_num,\n            rw mul_assoc,\n            nth_rewrite_lhs 1 mul_comm,\n            rw ←mul_assoc,\n        end\n        ... = ((d.succ:ℝ) + 1) • (f^(d.succ)) * deriv f :\n        begin\n            simp only [nat.cast_succ, algebra.smul_mul_assoc],\n            nth_rewrite 1 add_smul,\n            rw one_smul,\n        end\nend\n\nlemma pow_fun_def {f : ℝ → ℝ} {n : ℕ} : f^n = λ x, (f x)^n :=\nbegin\n    induction n with d hd,\n    all_goals {\n        try {rw [pow_succ, hd]},\n        refl,\n    }\nend\n\nlemma pow_deriv_fun_def {f : ℝ → ℝ} {n : ℕ} : ((n : ℝ) + 1) • f^n * (deriv f) =\n    λ (x : ℝ), ((n : ℝ) + 1) • ((f x)^n * deriv f x) :=\nbegin\n    rw pow_fun_def,\n    simpa,\nend\n\n@[simp] lemma power_rule'  {f : ℝ → ℝ} (n : ℕ) (hfd : differentiable ℝ f):\n    deriv (λ (x : ℝ), (f x)^(n + 1)) = λ (x : ℝ), ((n : ℝ) + 1) • ((f x)^n * deriv f x) :=\nbegin\n    rw [←pow_fun_def, ←pow_deriv_fun_def],\n    exact power_rule hfd,\nend\n\n@[simp] lemma power_rule'' (n : ℕ) :\n    deriv (λ (x : ℝ), x^(n + 1)) = λ (x : ℝ), ((n : ℝ) + 1) • (x^n) :=\nbegin\n    have hfd : differentiable ℝ (λ (x:ℝ), (x:ℝ)) := differentiable_id',\n    have deriv_id_my : deriv (λ x, x) = λ (x : ℝ), (1:ℝ) := deriv_id',\n    have H := power_rule' n hfd,\n    rw deriv_id_my at H,\n    rw H,\n    simp only [mul_one, algebra.id.smul_eq_mul],\nend\n\ntheorem int_by_parts (u v : ℝ → ℝ) {a b : ℝ} (hu : differentiable ℝ u)\n    (hv : differentiable ℝ v) (hcu : continuous(deriv u)) (hcv : continuous(deriv v)) :\n∫ x in a..b, u x * deriv v x =\n    u b * v b - u a * v a - ∫ x in a..b,  v x * deriv u x := \nbegin\n    have hu' : ∀ (x : ℝ), differentiable_at ℝ u x := hu,\n    have hv' : ∀ (x : ℝ), differentiable_at ℝ v x := hv,\n    have huv : deriv (u * v) = (deriv u) * v + u * deriv v := product_rule hu hv,\n    have H : ∫ x in  a..b, ((deriv u) x) * (v x)  + (u x) * ((deriv v) x) = ∫ x in a..b, (deriv (u*v)) x,\n    {\n        congr,\n        solve_by_elim,\n    },\n    have duv_cont : continuous (deriv (u * v)),\n    {\n        rw product_rule hu hv,\n        apply continuous.add,\n        rw mul_comm,\n        all_goals {\n            apply continuous.mul,\n            work_on_goal 0\n            {\n                apply @differentiable.continuous ℝ _ _ _ _ _ _ _,\n            },\n            repeat {assumption},\n        },\n    },\n    have duv_cont' : continuous_on (deriv (u * v)) (interval a b),\n    {\n        intros x hx,\n        exact continuous.continuous_within_at duv_cont,\n    },\n    have H2 : ∫ x in a..b, deriv (u*v) x = u b * v b - u a * v a,\n    {\n        apply integral_deriv_eq_sub,\n        intros x hx,\n        exact differentiable_at.mul (hu' x) (hv' x),\n        exact duv_cont',\n    },\n    rw [←H2, ←interval_integral.integral_sub],\n    {\n        congr,\n        ext,\n        rw huv,\n        simp only [pi.add_apply, pi.mul_apply],\n        rw mul_comm (v x) (deriv u x),\n        ring,\n    },\n    { \n        apply integrable_of_cont,\n        assumption,\n    },\n    apply integrable_of_cont,\n    apply continuous.mul,\n    apply @differentiable.continuous ℝ _ _ _ _ _ _ _,\n    repeat {assumption},\nend\n\nlemma int_by_parts_zero_ends (u v : ℝ → ℝ) {a b : ℝ} (hu : differentiable ℝ u)\n    (hv : differentiable ℝ v) (hcu : continuous(deriv u)) (hcv : continuous(deriv v)) \n    (ha : u a * v a = 0) (hb : u b * v b = 0)\n    :\n∫ x in a..b, u x * deriv v x = - ∫ x in a..b,  v x * deriv u x := \nbegin\n    rw int_by_parts,\n    repeat {assumption},\n    rw [ha, hb],\n    norm_num,\nend\n\n@[simp] lemma pow_ext (f : ℝ → ℝ) (n : ℕ) : f^n = λ x, (f x)^n :=\nbegin\n    induction n with d hd,\n    {\n        norm_num,\n        refl,\n    },\n    {\n        change f^(d+1) = λ x, (f x)^(d+1),\n        rw [pow_add, hd, pow_one],\n        ext,\n        norm_num,\n        ring_nf,\n    }\nend\n\nlemma differentiable_cospow_at (n: ℕ) {x : ℝ} : differentiable_at ℝ (cos^(n+1)) x:=\nby show_differentiable\n\nlemma deriv_cospow (n: ℕ) : deriv (λ (x : ℝ), cos x ^ (n+1)) = λ x, -((n : ℝ)+1) * (cos x)^n * sin x :=\nbegin\n    suffices : (λ (x : ℝ), -(((n:ℝ) + 1) * (cos x ^ n * sin x))) =\n        λ (x : ℝ), (-1 + -n) * cos x^n * sin x, by simpa,\n    ext,\n    ring,\nend\n\nlemma continuous_cospow {n: ℕ} : continuous (λ (x : ℝ), (cos x)^n) :=\nbegin\n    exact continuous.pow continuous_cos n,\nend\n\nlemma continuous_cospow' {c : ℝ} {m : ℕ}  :\n  continuous\n    (λ (x : ℝ), c * cos x ^m) := by show_continuous\n\nlemma differentiable_cospow {n: ℕ} : differentiable ℝ (λ (x : ℝ), (cos x)^n) :=\nbegin\n    simp only [differentiable_id', differentiable.pow, differentiable.cos],\nend\n\nlemma continuous_deriv_cospow (n: ℕ) : continuous (deriv (λ (x : ℝ), cos x ^ (n+1))) :=\nbegin\n    rw deriv_cospow,\n    apply continuous.mul continuous_cospow' continuous_sin,\nend\n\n@[simp] lemma deriv_sin_times_cos {x : ℝ} : deriv(sin * cos) x =\n    2 * cos x ^ 2 - 1 :=\nbegin\n    have H : deriv (λ (y : ℝ), sin y * cos y) x =\n        deriv sin x * cos x + sin x * deriv cos x\n        := deriv_mul differentiable_at_sin differentiable_at_cos,\n    have h0 : sin * cos = λ y, sin y * cos y, by refl,\n    have hsin : sin x^2 = 1 - cos x^2,\n    {\n        rw eq_sub_iff_add_eq,\n        exact sin_sq_add_cos_sq x,\n    },\n    rw [h0, H, real.deriv_sin, real.deriv_cos],\n    ring_nf,\n    rw hsin,\n    ring,\nend\n\n@[simp] lemma deriv_sin_cos {m : ℕ} : deriv (λ x, sin x * cos x^(m+1)) =\n    λ x, (m+2) * cos x^(m+2) - (m+1) * cos x^m :=\nbegin\n    ext,\n    suffices : deriv(sin * cos^(m+1)) x =\n    (m+2) * (cos x)^(m+2) - (m+1) * (cos x)^m,\n    {\n        rw pow_ext at this,\n        exact this,\n    },\n    induction m with d hd,\n    {\n        simp only [mul_one, nat.cast_zero, pow_one, zero_add, pow_zero],\n        exact deriv_sin_times_cos,\n    },\n    {\n        simp,\n        have H := deriv_mul (@differentiable_at_sin x)\n            (differentiable_cospow_at d.succ),\n        repeat {rw pow_succ,},\n        have h2 : (λ (y : ℝ), sin y * (cos ^ (d.succ + 1)) y) x =\n            sin x * (cos ^ (d.succ + 1)) x, by tauto,\n        have hsin : sin * (λ (x : ℝ), cos x ^ (d.succ + 1)) =\n            (λ x, sin x * cos x ^ (d.succ + 1)), by tauto,\n        rw hsin,\n        have hhd : (sin * cos ^ (d + 1) = λ (y : ℝ), sin y * cos y ^ (d + 1)),\n        {\n            ext,\n            simp only [pi.mul_apply, pow_ext],\n        },\n        simp [pow_ext],\n        ring_exp,\n        have sin_to_cos : sin x^2 = 1 - cos x^2,\n        {\n            rw eq_sub_iff_add_eq,\n            exact sin_sq_add_cos_sq x,\n        },\n        rw sin_to_cos,\n        discrete_field,\n    },\nend", "meta": {"author": "mmasdeu", "repo": "euler", "sha": "a323d777dee611f2a06cc81e2f2567cd9522a381", "save_path": "github-repos/lean/mmasdeu-euler", "path": "github-repos/lean/mmasdeu-euler/euler-a323d777dee611f2a06cc81e2f2567cd9522a381/src/integrals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3374117916812904}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.kernel_pair\nimport category_theory.limits.shapes.comm_sq\n\n/-!\n# The diagonal object of a morphism.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe provide various API and isomorphisms considering the diagonal object `Δ_{Y/X} := pullback f f`\nof a morphism `f : X ⟶ Y`.\n\n-/\n\nopen category_theory\n\nnoncomputable theory\n\nnamespace category_theory.limits\n\nvariables {C : Type*} [category C] {X Y Z : C}\n\nnamespace pullback\n\nsection diagonal\n\nvariables (f : X ⟶ Y) [has_pullback f f]\n\n/-- The diagonal object of a morphism `f : X ⟶ Y` is `Δ_{X/Y} := pullback f f`. -/\nabbreviation diagonal_obj : C := pullback f f\n\n/-- The diagonal morphism `X ⟶ Δ_{X/Y}` for a morphism `f : X ⟶ Y`. -/\ndef diagonal : X ⟶ diagonal_obj f :=\npullback.lift (𝟙 _) (𝟙 _) rfl\n\n@[simp, reassoc] lemma diagonal_fst : diagonal f ≫ pullback.fst = 𝟙 _ :=\npullback.lift_fst _ _ _\n\n@[simp, reassoc] lemma diagonal_snd : diagonal f ≫ pullback.snd = 𝟙 _ :=\npullback.lift_snd _ _ _\n\ninstance : is_split_mono (diagonal f) :=\n⟨⟨⟨pullback.fst, diagonal_fst f⟩⟩⟩\n\ninstance : is_split_epi (pullback.fst : pullback f f ⟶ X) :=\n⟨⟨⟨diagonal f, diagonal_fst f⟩⟩⟩\n\ninstance : is_split_epi (pullback.snd : pullback f f ⟶ X) :=\n⟨⟨⟨diagonal f, diagonal_snd f⟩⟩⟩\n\ninstance [mono f] : is_iso (diagonal f) :=\nbegin\n  rw (is_iso.inv_eq_of_inv_hom_id (diagonal_fst f)).symm,\n  apply_instance\nend\n\n/-- The two projections `Δ_{X/Y} ⟶ X` form a kernel pair for `f : X ⟶ Y`. -/\nlemma diagonal_is_kernel_pair :\n  is_kernel_pair f (pullback.fst : diagonal_obj f ⟶ _) pullback.snd :=\nis_pullback.of_has_pullback f f\n\nend diagonal\n\nend pullback\n\nvariable [has_pullbacks C]\n\nopen pullback\n\nsection\n\nvariables {U V₁ V₂ : C} (f : X ⟶ Y) (i : U ⟶ Y)\nvariables (i₁ : V₁ ⟶ pullback f i) (i₂ : V₂ ⟶ pullback f i)\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_snd_fst_fst :\n  (pullback.snd : pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i\n    (by simp [condition]) (by simp [condition])) ⟶ _) ≫ fst ≫ i₁ ≫ fst = pullback.fst :=\nbegin\n  conv_rhs { rw ← category.comp_id pullback.fst },\n  rw [← diagonal_fst f, pullback.condition_assoc, pullback.lift_fst]\nend\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_snd_snd_fst :\n  (pullback.snd : pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i\n    (by simp [condition]) (by simp [condition])) ⟶ _) ≫ snd ≫ i₂ ≫ fst = pullback.fst :=\nbegin\n  conv_rhs { rw ← category.comp_id pullback.fst },\n  rw [← diagonal_snd f, pullback.condition_assoc, pullback.lift_snd]\nend\n\nvariable [has_pullback i₁ i₂]\n\n/--\nThis iso witnesses the fact that\ngiven `f : X ⟶ Y`, `i : U ⟶ Y`, and `i₁ : V₁ ⟶ X ×[Y] U`, `i₂ : V₂ ⟶ X ×[Y] U`, the diagram\n\nV₁ ×[X ×[Y] U] V₂ ⟶ V₁ ×[U] V₂\n        |                 |\n        |                 |\n        ↓                 ↓\n        X         ⟶  X ×[Y] X\n\nis a pullback square.\nAlso see `pullback_fst_map_snd_is_pullback`.\n-/\ndef pullback_diagonal_map_iso :\n  pullback (diagonal f) (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i\n    (by simp [condition]) (by simp [condition])) ≅ pullback i₁ i₂ :=\n{ hom := pullback.lift (pullback.snd ≫ pullback.fst) (pullback.snd ≫ pullback.snd)\n    begin\n      ext; simp only [category.assoc, pullback.condition, pullback_diagonal_map_snd_fst_fst,\n        pullback_diagonal_map_snd_snd_fst],\n    end,\n  inv := pullback.lift (pullback.fst ≫ i₁ ≫ pullback.fst) (pullback.map _ _ _ _ (𝟙 _) (𝟙 _)\n      pullback.snd (category.id_comp _).symm (category.id_comp _).symm)\n    begin\n      ext; simp only [diagonal_fst, diagonal_snd, category.comp_id, pullback.condition_assoc,\n        category.assoc, lift_fst, lift_fst_assoc, lift_snd, lift_snd_assoc],\n    end,\n  hom_inv_id' := by ext; simp only [category.id_comp, category.assoc, lift_fst_assoc,\n    pullback_diagonal_map_snd_fst_fst, lift_fst, lift_snd, category.comp_id],\n  inv_hom_id' := by ext; simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_iso_hom_fst :\n  (pullback_diagonal_map_iso f i i₁ i₂).hom ≫ pullback.fst = pullback.snd ≫ pullback.fst :=\nby { delta pullback_diagonal_map_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_iso_hom_snd :\n  (pullback_diagonal_map_iso f i i₁ i₂).hom ≫ pullback.snd = pullback.snd ≫ pullback.snd :=\nby { delta pullback_diagonal_map_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_iso_inv_fst :\n  (pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.fst = pullback.fst ≫ i₁ ≫ pullback.fst :=\nby { delta pullback_diagonal_map_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_iso_inv_snd_fst :\n  (pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.snd ≫ pullback.fst = pullback.fst :=\nby { delta pullback_diagonal_map_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_iso_inv_snd_snd :\n  (pullback_diagonal_map_iso f i i₁ i₂).inv ≫ pullback.snd ≫ pullback.snd = pullback.snd :=\nby { delta pullback_diagonal_map_iso, simp }\n\nlemma pullback_fst_map_snd_is_pullback :\n  is_pullback\n    (fst ≫ i₁ ≫ fst)\n    (map i₁ i₂ (i₁ ≫ snd) (i₂ ≫ snd) _ _ _ (category.id_comp _).symm (category.id_comp _).symm)\n    (diagonal f)\n    (map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i\n      (by simp [condition]) (by simp [condition])) :=\nis_pullback.of_iso_pullback ⟨by ext; simp [condition_assoc]⟩\n  (pullback_diagonal_map_iso f i i₁ i₂).symm (pullback_diagonal_map_iso_inv_fst f i i₁ i₂)\n  (by ext1; simp)\n\nend\n\nsection\n\nvariables {S T : C} (f : X ⟶ T) (g : Y ⟶ T) (i : T ⟶ S)\nvariables [has_pullback i i] [has_pullback f g] [has_pullback (f ≫ i) (g ≫ i)]\nvariable [has_pullback (diagonal i) (pullback.map (f ≫ i) (g ≫ i) i i f g (𝟙 _)\n    (category.comp_id _) (category.comp_id _))]\n\n/--\nThis iso witnesses the fact that\ngiven `f : X ⟶ T`, `g : Y ⟶ T`, and `i : T ⟶ S`, the diagram\n\nX ×ₜ Y ⟶ X ×ₛ Y\n   |         |\n   |         |\n   ↓         ↓\n   T   ⟶ T ×ₛ T\n\nis a pullback square.\nAlso see `pullback_map_diagonal_is_pullback`.\n-/\ndef pullback_diagonal_map_id_iso :\n  pullback (diagonal i) (pullback.map (f ≫ i) (g ≫ i) i i f g (𝟙 _)\n    (category.comp_id _) (category.comp_id _)) ≅ pullback f g :=\nbegin\n  refine (as_iso $ pullback.map _ _ _ _ (𝟙 _) (pullback.congr_hom _ _).hom (𝟙 _) _ _) ≪≫\n    pullback_diagonal_map_iso i (𝟙 _) (f ≫ inv pullback.fst) (g ≫ inv pullback.fst) ≪≫\n      (as_iso $ pullback.map _ _ _ _ (𝟙 _) (𝟙 _) pullback.fst _ _),\n  { rw [← category.comp_id pullback.snd, ← condition, category.assoc, is_iso.inv_hom_id_assoc] },\n  { rw [← category.comp_id pullback.snd, ← condition, category.assoc, is_iso.inv_hom_id_assoc] },\n  { rw [category.comp_id, category.id_comp] },\n  { ext; simp },\n  { apply_instance },\n  { rw [category.assoc, category.id_comp, is_iso.inv_hom_id, category.comp_id] },\n  { rw [category.assoc, category.id_comp, is_iso.inv_hom_id, category.comp_id] },\n  { apply_instance },\nend\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_id_iso_hom_fst :\n  (pullback_diagonal_map_id_iso f g i).hom ≫ pullback.fst = pullback.snd ≫ pullback.fst :=\nby { delta pullback_diagonal_map_id_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_id_iso_hom_snd :\n  (pullback_diagonal_map_id_iso f g i).hom ≫ pullback.snd = pullback.snd ≫ pullback.snd :=\nby { delta pullback_diagonal_map_id_iso, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_id_iso_inv_fst :\n  (pullback_diagonal_map_id_iso f g i).inv ≫ pullback.fst = pullback.fst ≫ f :=\nbegin\n  rw [iso.inv_comp_eq, ← category.comp_id pullback.fst, ← diagonal_fst i, pullback.condition_assoc],\n  simp,\nend\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_id_iso_inv_snd_fst :\n  (pullback_diagonal_map_id_iso f g i).inv ≫ pullback.snd ≫ pullback.fst = pullback.fst :=\nby { rw iso.inv_comp_eq, simp }\n\n@[simp, reassoc]\nlemma pullback_diagonal_map_id_iso_inv_snd_snd :\n  (pullback_diagonal_map_id_iso f g i).inv ≫ pullback.snd ≫ pullback.snd = pullback.snd :=\nby { rw iso.inv_comp_eq, simp }\n\nlemma pullback.diagonal_comp (f : X ⟶ Y) (g : Y ⟶ Z) [has_pullback f f] [has_pullback g g]\n  [has_pullback (f ≫ g) (f ≫ g)] :\n  diagonal (f ≫ g) = diagonal f ≫ (pullback_diagonal_map_id_iso f f g).inv ≫ pullback.snd :=\nby ext; simp\n\nlemma pullback_map_diagonal_is_pullback : is_pullback (pullback.fst ≫ f)\n  (pullback.map f g (f ≫ i) (g ≫ i) _ _ i (category.id_comp _).symm (category.id_comp _).symm)\n  (diagonal i)\n  (pullback.map (f ≫ i) (g ≫ i) i i f g (𝟙 _) (category.comp_id _) (category.comp_id _)) :=\nbegin\n  apply is_pullback.of_iso_pullback _ (pullback_diagonal_map_id_iso f g i).symm,\n  { simp },\n  { ext; simp },\n  { constructor, ext; simp [condition] },\nend\n\n/-- The diagonal object of `X ×[Z] Y ⟶ X` is isomorphic to `Δ_{Y/Z} ×[Z] X`. -/\ndef diagonal_obj_pullback_fst_iso {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  diagonal_obj (pullback.fst : pullback f g ⟶ X) ≅\n    pullback (pullback.snd ≫ g : diagonal_obj g ⟶ Z) f :=\npullback_right_pullback_fst_iso _ _ _ ≪≫ pullback.congr_hom pullback.condition rfl ≪≫\n  pullback_assoc _ _ _ _ ≪≫ pullback_symmetry _ _ ≪≫ pullback.congr_hom pullback.condition rfl\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_hom_fst_fst {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.fst ≫ pullback.fst =\n    pullback.fst ≫ pullback.snd :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_hom_fst_snd {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.fst ≫ pullback.snd =\n    pullback.snd ≫ pullback.snd :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_hom_snd {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).hom ≫ pullback.snd = pullback.fst ≫ pullback.fst :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_inv_fst_fst {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.fst ≫ pullback.fst =\n    pullback.snd :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_inv_fst_snd {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.fst ≫ pullback.snd =\n    pullback.fst ≫ pullback.fst :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] lemma diagonal_obj_pullback_fst_iso_inv_snd_fst {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) :\n  (diagonal_obj_pullback_fst_iso f g).inv ≫ pullback.snd ≫ pullback.fst = pullback.snd :=\nby { delta diagonal_obj_pullback_fst_iso, simp }\n\n@[simp, reassoc] \n\nlemma diagonal_pullback_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  diagonal (pullback.fst : pullback f g ⟶ _) =\n    (pullback_symmetry _ _).hom ≫ ((base_change f).map\n      (over.hom_mk (diagonal g) (by simp) : over.mk g ⟶ over.mk (pullback.snd ≫ g))).left ≫\n    (diagonal_obj_pullback_fst_iso f g).inv :=\nby ext; simp\n\nend\n\n/--\nGiven the following diagram with `S ⟶ S'` a monomorphism,\n\n    X  ⟶ X'\n      ↘      ↘\n        S  ⟶ S'\n      ↗      ↗\n    Y  ⟶ Y'\n\nThis iso witnesses the fact that\n\n      X ×[S] Y ⟶ (X' ×[S'] Y') ×[Y'] Y\n          |                  |\n          |                  |\n          ↓                  ↓\n(X' ×[S'] Y') ×[X'] X ⟶ X' ×[S'] Y'\n\nis a pullback square. The diagonal map of this square is `pullback.map`.\nAlso see `pullback_lift_map_is_pullback`.\n-/\n@[simps]\ndef pullback_fst_fst_iso {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S) (f' : X' ⟶ S')\n  (g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')\n  (e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :\n    pullback (pullback.fst : pullback (pullback.fst : pullback f' g' ⟶ _) i₁ ⟶ _)\n      (pullback.fst : pullback (pullback.snd : pullback f' g' ⟶ _) i₂ ⟶ _) ≅ pullback f g :=\n{ hom := pullback.lift (pullback.fst ≫ pullback.snd) (pullback.snd ≫ pullback.snd)\n    begin\n      rw [← cancel_mono i₃, category.assoc, category.assoc, category.assoc, category.assoc, e₁, e₂,\n        ← pullback.condition_assoc, pullback.condition_assoc, pullback.condition,\n        pullback.condition_assoc]\n    end,\n  inv := pullback.lift\n    (pullback.lift (pullback.map _ _ _ _ _ _ _ e₁ e₂) pullback.fst (pullback.lift_fst _ _ _))\n    (pullback.lift (pullback.map _ _ _ _ _ _ _ e₁ e₂) pullback.snd (pullback.lift_snd _ _ _))\n    begin\n      rw [pullback.lift_fst, pullback.lift_fst]\n    end,\n  hom_inv_id' := by ext; simp only [category.assoc, category.id_comp, lift_fst, lift_snd,\n    lift_fst_assoc, lift_snd_assoc, condition, ← condition_assoc],\n  inv_hom_id' := by ext; simp only [category.assoc, category.id_comp, lift_fst, lift_snd,\n    lift_fst_assoc, lift_snd_assoc], }\n\nlemma pullback_map_eq_pullback_fst_fst_iso_inv {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S)\n  (f' : X' ⟶ S')\n  (g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')\n  (e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :\n  pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂ =\n    (pullback_fst_fst_iso f g f' g' i₁ i₂ i₃ e₁ e₂).inv ≫ pullback.snd ≫ pullback.fst :=\nbegin\n  ext; simp only [category.assoc, category.id_comp, lift_fst, lift_snd, lift_fst_assoc,\n    lift_snd_assoc, pullback_fst_fst_iso_inv, ← pullback.condition, ← pullback.condition_assoc],\nend\n\nlemma pullback_lift_map_is_pullback {X Y S X' Y' S' : C} (f : X ⟶ S) (g : Y ⟶ S) (f' : X' ⟶ S')\n  (g' : Y' ⟶ S') (i₁ : X ⟶ X') (i₂ : Y ⟶ Y') (i₃ : S ⟶ S') (e₁ : f ≫ i₃ = i₁ ≫ f')\n  (e₂ : g ≫ i₃ = i₂ ≫ g') [mono i₃] :\n  is_pullback\n    (pullback.lift (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂) fst (lift_fst _ _ _))\n    (pullback.lift (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂) snd (lift_snd _ _ _))\n    pullback.fst pullback.fst :=\nis_pullback.of_iso_pullback ⟨by rw [lift_fst, lift_fst]⟩\n  (pullback_fst_fst_iso f g f' g' i₁ i₂ i₃ e₁ e₂).symm (by simp) (by simp)\n\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/shapes/diagonal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3374117916812903}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.default\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.PostPort\n\nuniverses v u u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Category of categories\n\nThis file contains the definition of the category `Cat` of all categories.\nIn this category objects are categories and\nmorphisms are functors between these categories.\n\n## Implementation notes\n\nThough `Cat` is not a concrete category, we use `bundled` to define\nits carrier type.\n-/\n\nnamespace category_theory\n\n\n/-- Category of categories. -/\ndef Cat :=\n  bundled category\n\nnamespace Cat\n\n\nprotected instance inhabited : Inhabited Cat :=\n  { default := bundled.mk (Type u) }\n\nprotected instance has_coe_to_sort : has_coe_to_sort Cat :=\n  has_coe_to_sort.mk (Type u) bundled.α\n\nprotected instance str (C : Cat) : category ↥C :=\n  bundled.str C\n\n/-- Construct a bundled `Cat` from the underlying type and the typeclass. -/\ndef of (C : Type u) [category C] : Cat :=\n  bundled.of C\n\n/-- Category structure on `Cat` -/\nprotected instance category : large_category Cat :=\n  category.mk\n\n/-- Functor that gets the set of objects of a category. It is not\ncalled `forget`, because it is not a faithful functor. -/\ndef objects : Cat ⥤ Type u :=\n  functor.mk (fun (C : Cat) => ↥C) fun (C D : Cat) (F : C ⟶ D) => functor.obj F\n\n/-- Any isomorphism in `Cat` induces an equivalence of the underlying categories. -/\ndef equiv_of_iso {C : Cat} {D : Cat} (γ : C ≅ D) : ↥C ≌ ↥D :=\n  equivalence.mk' (iso.hom γ) (iso.inv γ) (eq_to_iso sorry) (eq_to_iso (iso.inv_hom_id γ))\n\nend Cat\n\n\n/--\nEmbedding `Type` into `Cat` as discrete categories.\n\nThis ought to be modelled as a 2-functor!\n-/\n@[simp] theorem Type_to_Cat_obj (X : Type u) : functor.obj Type_to_Cat X = Cat.of (discrete X) :=\n  Eq.refl (functor.obj Type_to_Cat X)\n\nprotected instance Type_to_Cat.faithful : faithful Type_to_Cat :=\n  faithful.mk\n\nprotected instance Type_to_Cat.full : full Type_to_Cat :=\n  full.mk\n    fun (X Y : Type (max (max (max u_1 u_2 u_3 u_4) u_1 u_2 u_3) (max u_1 u_2 u_3 u_4) u_1 u_2))\n      (F : functor.obj Type_to_Cat X ⟶ functor.obj Type_to_Cat Y) => functor.obj F\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/Cat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3372005300306361}}
{"text": "import phase2.weak_approx\n\nopen cardinal quiver set sum with_bot\nopen_locale cardinal classical pointwise\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}]\n\n/-!\n# Filling in orbits of atoms\n-/\n\nnamespace weak_near_litter_approx\n\nvariables (w : weak_near_litter_approx)\n\ndef need_forward_images : set atom := w.atom_map.ran \\ w.atom_map.dom\ndef need_backward_images : set atom := w.atom_map.dom \\ w.atom_map.ran\n\nlemma atom_map_ran_small : small w.atom_map.ran :=\nbegin\n  have : small (w.atom_map_or_else '' w.atom_map.dom) := small.image w.atom_map_dom_small,\n  refine small.mono _ this,\n  rintros _ ⟨a, ha, rfl⟩,\n  refine ⟨a, ha, _⟩,\n  rw atom_map_or_else_of_dom,\nend\n\nlemma need_forward_images_small : small w.need_forward_images :=\nsmall.mono (diff_subset _ _) w.atom_map_ran_small\n\nlemma need_backward_images_small : small w.need_backward_images :=\nsmall.mono (diff_subset _ _) w.atom_map_dom_small\n\nlemma mk_diff_dom_ran (L : litter) :\n  #(litter_set L \\ (w.atom_map.dom ∪ w.atom_map.ran) : set atom) = #κ :=\nbegin\n  refine le_antisymm _ _,\n  { refine ⟨⟨λ a, a.1.2, _⟩⟩,\n    intros a b h,\n    refine subtype.coe_injective (prod.ext (a.prop.1.trans b.prop.1.symm) _),\n    simp only [subtype.val_eq_coe] at h,\n    exact h, },\n  { by_contra' h,\n    have := add_lt_of_lt κ_regular.aleph_0_le h\n      (small.union w.atom_map_dom_small w.atom_map_ran_small),\n    have := (le_mk_diff_add_mk (litter_set L) _).trans_lt this,\n    simp only [mk_litter_set, lt_self_iff_false] at this,\n    exact this, },\nend\n\nlemma need_images_small (L : litter) :\n  #(ℕ × w.need_backward_images ⊕ ℕ × w.need_forward_images) < #κ :=\nbegin\n  simp only [mk_prod, mk_denumerable, lift_aleph_0, lift_uzero, mk_diff_dom_ran, mk_sum, lift_id],\n  rw ← mul_add,\n  refine lt_of_le_of_lt (mul_le_max _ _) (max_lt (max_lt _ _) _),\n  exact Λ_limit.aleph_0_le.trans_lt Λ_lt_κ,\n  exact add_lt_of_lt κ_regular.aleph_0_le w.need_backward_images_small w.need_forward_images_small,\n  exact Λ_limit.aleph_0_le.trans_lt Λ_lt_κ,\nend\n\nlemma le_mk_diff_dom_ran (L : litter) :\n  #(ℕ × w.need_backward_images ⊕ ℕ × w.need_forward_images) ≤\n    #(litter_set L \\ (w.atom_map.dom ∪ w.atom_map.ran) : set atom) :=\nbegin\n  rw [mk_diff_dom_ran],\n  exact (w.need_images_small L).le,\nend\n\ndef orbit_set (L : litter) : set atom :=\n(le_mk_iff_exists_subset.mp (w.le_mk_diff_dom_ran L)).some\n\nlemma orbit_set_subset (L : litter) :\n  w.orbit_set L ⊆ litter_set L \\ (w.atom_map.dom ∪ w.atom_map.ran) :=\n(le_mk_iff_exists_subset.mp (w.le_mk_diff_dom_ran L)).some_spec.1\n\nlemma not_mem_need_forward_images_of_mem_orbit_set {a : atom} {L : litter}\n  (h : a ∈ w.orbit_set L) : a ∉ w.need_forward_images :=\nλ ha, (w.orbit_set_subset L h).2 (or.inr ha.1)\n\nlemma not_mem_need_backward_images_of_mem_orbit_set {a : atom} {L : litter}\n  (h : a ∈ w.orbit_set L) : a ∉ w.need_backward_images :=\nλ ha, (w.orbit_set_subset L h).2 (or.inl ha.1)\n\nlemma mk_orbit_set (L : litter) :\n  #(w.orbit_set L) = #(ℕ × w.need_backward_images ⊕ ℕ × w.need_forward_images) :=\n(le_mk_iff_exists_subset.mp (w.le_mk_diff_dom_ran L)).some_spec.2\n\n@[irreducible] noncomputable def orbit_set_equiv (L : litter) :\n  w.orbit_set L ≃ ℕ × w.need_backward_images ⊕ ℕ × w.need_forward_images :=\n(cardinal.eq.mp (w.mk_orbit_set L)).some\n\nlemma orbit_set_equiv_injective {a₁ a₂ : ℕ × w.need_backward_images ⊕ ℕ × w.need_forward_images}\n  {L₁ L₂ : litter} (h : ((w.orbit_set_equiv L₁).symm a₁ : atom) = (w.orbit_set_equiv L₂).symm a₂) :\n  L₁ = L₂ ∧ a₁ = a₂ :=\nbegin\n  have h₁ := w.orbit_set_subset L₁ ((w.orbit_set_equiv L₁).symm a₁).prop,\n  have h₂ := w.orbit_set_subset L₂ ((w.orbit_set_equiv L₂).symm a₂).prop,\n  rw h at h₁,\n  cases eq_of_mem_litter_set_of_mem_litter_set h₁.1 h₂.1,\n  simp only [subtype.coe_inj, embedding_like.apply_eq_iff_eq] at h,\n  exact ⟨rfl, h⟩,\nend\n\nlemma orbit_set_equiv_congr {L L' : litter} {a : atom} (ha : a ∈ w.orbit_set L) (h : L = L') :\n  w.orbit_set_equiv L ⟨a, ha⟩ = w.orbit_set_equiv L' ⟨a, h ▸ ha⟩ :=\nby cases h; refl\n\nlemma orbit_set_equiv_symm_congr {L L' : litter}\n  {a : ℕ × ↥(w.need_backward_images) ⊕ ℕ × ↥(w.need_forward_images)} (h : L = L') :\n  ((w.orbit_set_equiv L).symm a : atom) = (w.orbit_set_equiv L').symm a :=\nby cases h; refl\n\nlemma orbit_set_small (L : litter) : small (w.orbit_set L) :=\nbegin\n  rw [small, mk_orbit_set],\n  exact w.need_images_small L,\nend\n\nnoncomputable def next_forward_image (L : litter) (a : ℕ × w.need_forward_images) : atom :=\n(w.orbit_set_equiv (w.litter_perm L)).symm (inr (a.1 + 1, a.2))\n\nnoncomputable def next_backward_image (L : litter) : ℕ × w.need_backward_images → atom\n| (0, a) := a\n| (n + 1, a) := (w.orbit_set_equiv (w.litter_perm L)).symm (inl (n, a))\n\ndef next_forward_image_domain (L : litter) : set (ℕ × w.need_forward_images) :=\n{a | (a.2 : atom).1 ∈ w.litter_perm.domain ∧ (w.litter_perm^[a.1 + 1] (a.2 : atom).1 = L)}\n\ndef next_backward_image_domain (L : litter) : set (ℕ × w.need_backward_images) :=\n{a | (a.2 : atom).1 ∈ w.litter_perm.domain ∧ (w.litter_perm.symm^[a.1 + 1] (a.2 : atom).1 = L)}\n\nlemma mk_mem_next_forward_image_domain (L : litter) (n : ℕ) (a : w.need_forward_images) :\n  (n, a) ∈ w.next_forward_image_domain L ↔\n    (a : atom).1 ∈ w.litter_perm.domain ∧ (w.litter_perm^[n + 1] (a : atom).1 = L) := iff.rfl\n\nlemma mk_mem_next_backward_image_domain (L : litter) (n : ℕ) (a : w.need_backward_images) :\n  (n, a) ∈ w.next_backward_image_domain L ↔\n    (a : atom).1 ∈ w.litter_perm.domain ∧ (w.litter_perm.symm^[n + 1] (a : atom).1 = L) := iff.rfl\n\nlemma next_forward_image_eq {L₁ L₂ : litter} {a b : ℕ × w.need_forward_images}\n  (hL₁ : L₁ ∈ w.litter_perm.domain) (hL₂ : L₂ ∈ w.litter_perm.domain)\n  (h : w.next_forward_image L₁ a = w.next_forward_image L₂ b) : L₁ = L₂ :=\nbegin\n  rw [next_forward_image, next_forward_image] at h,\n  have ha := w.orbit_set_subset _\n    ((w.orbit_set_equiv (w.litter_perm L₁)).symm (inr (a.1 + 1, a.2))).prop,\n  have hb := w.orbit_set_subset _\n    ((w.orbit_set_equiv (w.litter_perm L₂)).symm (inr (b.1 + 1, b.2))).prop,\n  rw h at ha,\n  refine w.litter_perm.inj_on hL₁ hL₂ _,\n  exact eq_of_mem_litter_set_of_mem_litter_set ha.1 hb.1,\nend\n\nlemma next_backward_image_eq {L₁ L₂ : litter} {a b : ℕ × w.need_backward_images}\n  (ha : a ∈ w.next_backward_image_domain L₁) (hb : b ∈ w.next_backward_image_domain L₂)\n  (hL₁ : L₁ ∈ w.litter_perm.domain) (hL₂ : L₂ ∈ w.litter_perm.domain)\n  (h : w.next_backward_image L₁ a = w.next_backward_image L₂ b) : L₁ = L₂ :=\nbegin\n  obtain ⟨m, a⟩ := a,\n  obtain ⟨n, b⟩ := b,\n  cases m;\n  cases n;\n  rw [next_backward_image, next_backward_image] at h,\n  { simp only [next_backward_image_domain, function.iterate_succ, function.comp_app, mem_set_of_eq,\n      function.iterate_zero, id.def] at ha hb,\n    rw [← h, ha.2] at hb,\n    exact hb.2, },\n  { rw subtype.coe_eq_iff at h,\n    obtain ⟨h₁, h₂⟩ := h,\n    cases w.not_mem_need_backward_images_of_mem_orbit_set ((w.orbit_set_equiv _).symm _).prop h₁, },\n  { symmetry' at h,\n    rw subtype.coe_eq_iff at h,\n    obtain ⟨h₁, h₂⟩ := h,\n    cases w.not_mem_need_backward_images_of_mem_orbit_set ((w.orbit_set_equiv _).symm _).prop h₁, },\n  { have ha := w.orbit_set_subset _\n      ((w.orbit_set_equiv (w.litter_perm L₁)).symm (inl (m, a))).prop,\n    have hb := w.orbit_set_subset _\n      ((w.orbit_set_equiv (w.litter_perm L₂)).symm (inl (n, b))).prop,\n    rw h at ha,\n    refine w.litter_perm.inj_on hL₁ hL₂ _,\n    exact eq_of_mem_litter_set_of_mem_litter_set ha.1 hb.1, },\nend\n\nlemma next_forward_image_injective {L : litter} {a b : ℕ × w.need_forward_images}\n  (h : w.next_forward_image L a = w.next_forward_image L b) : a = b :=\nbegin\n  simp only [next_forward_image, subtype.coe_inj, embedding_like.apply_eq_iff_eq, prod.mk.inj_iff,\n    add_left_inj] at h,\n  exact prod.ext h.1 h.2,\nend\n\nlemma next_backward_image_injective {L : litter} {a b : ℕ × w.need_backward_images}\n  (ha : a ∈ w.next_backward_image_domain L) (hb : b ∈ w.next_backward_image_domain L)\n  (h : w.next_backward_image L a = w.next_backward_image L b) : a = b :=\nbegin\n  obtain ⟨m, a⟩ := a,\n  obtain ⟨n, b⟩ := b,\n  cases m;\n  cases n;\n  simp only [prod.mk.inj_iff, subtype.coe_inj, embedding_like.apply_eq_iff_eq, prod.mk.inj_iff,\n    false_and, nat.nat_zero_eq_zero, prod.mk.inj_iff, next_backward_image, prod.mk.inj_iff,\n    eq_self_iff_true, true_and, subtype.coe_inj] at h ⊢,\n  { exact h, },\n  { rw subtype.coe_eq_iff at h,\n    obtain ⟨h₁, h₂⟩ := h,\n    cases w.not_mem_need_backward_images_of_mem_orbit_set ((w.orbit_set_equiv _).symm _).prop h₁, },\n  { symmetry' at h,\n    rw subtype.coe_eq_iff at h,\n    obtain ⟨h₁, h₂⟩ := h,\n    cases w.not_mem_need_backward_images_of_mem_orbit_set ((w.orbit_set_equiv _).symm _).prop h₁, },\n  { exact h, },\nend\n\nlemma next_forward_image_injective' {L₁ L₂ : litter} {a b : ℕ × w.need_forward_images}\n  (hL₁ : L₁ ∈ w.litter_perm.domain) (hL₂ : L₂ ∈ w.litter_perm.domain)\n  (h : w.next_forward_image L₁ a = w.next_forward_image L₂ b) : a = b :=\nbegin\n  cases w.next_forward_image_eq hL₁ hL₂ h,\n  exact w.next_forward_image_injective h,\nend\n\nlemma next_backward_image_injective' {L₁ L₂ : litter} {a b : ℕ × w.need_backward_images}\n  (ha : a ∈ w.next_backward_image_domain L₁) (hb : b ∈ w.next_backward_image_domain L₂)\n  (hL₁ : L₁ ∈ w.litter_perm.domain) (hL₂ : L₂ ∈ w.litter_perm.domain)\n  (h : w.next_backward_image L₁ a = w.next_backward_image L₂ b) : a = b :=\nbegin\n  cases w.next_backward_image_eq ha hb hL₁ hL₂ h,\n  exact w.next_backward_image_injective ha hb h,\nend\n\nlemma next_forward_image_ne_next_backward_image {L₁ L₂ : litter}\n  {a : ℕ × w.need_forward_images} {b : ℕ × w.need_backward_images} :\n  w.next_forward_image L₁ a ≠ w.next_backward_image L₂ b :=\nbegin\n  obtain ⟨n, b⟩ := b,\n  cases n,\n  { rw [next_forward_image, next_backward_image],\n    refine (ne_of_mem_of_not_mem _\n      (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).2).symm,\n    exact or.inl b.prop.1, },\n  { rw [next_forward_image, next_backward_image],\n    intro h,\n    cases (w.orbit_set_equiv_injective h).2, },\nend\n\nnoncomputable def next_image_core (a : atom) (L : litter) (ha : a ∈ w.orbit_set L) : atom :=\n(w.orbit_set_equiv L ⟨a, ha⟩).elim (w.next_backward_image L) (w.next_forward_image L)\n\ndef next_image_core_domain : set atom :=\n⋃ L ∈ w.litter_perm.domain, coe '' {a : w.orbit_set L | (w.orbit_set_equiv L a).elim\n  (λ b, b ∈ w.next_backward_image_domain L) (λ b, b ∈ w.next_forward_image_domain L)}\n\nlemma next_image_core_domain_small : small w.next_image_core_domain :=\nsmall.bUnion w.litter_perm_domain_small\n  (λ L hL, small.image (lt_of_le_of_lt (cardinal.mk_subtype_le _) (w.orbit_set_small L)))\n\nlemma litter_map_dom_of_mem_next_image_core_domain {a : atom} (h : a ∈ w.next_image_core_domain) :\n  a.1 ∈ w.litter_perm.domain :=\nbegin\n  rw next_image_core_domain at h,\n  simp only [pfun.mem_dom, Union_exists, mem_Union, mem_image, mem_set_of_eq, set_coe.exists,\n    subtype.coe_mk, exists_and_distrib_right, exists_eq_right, exists_prop] at h,\n  obtain ⟨L, hL, ha, h⟩ := h,\n  have := (w.orbit_set_subset L ha).1,\n  rw mem_litter_set at this,\n  rw this,\n  exact hL,\nend\n\nlemma mem_orbit_set_of_mem_next_image_core_domain {a : atom} (h : a ∈ w.next_image_core_domain) :\n  a ∈ w.orbit_set a.1 :=\nbegin\n  rw next_image_core_domain at h,\n  simp only [pfun.mem_dom, Union_exists, mem_Union, mem_image, mem_set_of_eq, set_coe.exists,\n    subtype.coe_mk, exists_and_distrib_right, exists_eq_right, exists_prop] at h,\n  obtain ⟨L, hL, ha, h⟩ := h,\n  have := (w.orbit_set_subset L ha).1,\n  rw mem_litter_set at this,\n  rw this,\n  exact ha,\nend\n\nlemma orbit_set_equiv_elim_of_mem_next_image_core_domain {a : atom}\n  (h : a ∈ w.next_image_core_domain) :\n  (w.orbit_set_equiv a.1 ⟨a, w.mem_orbit_set_of_mem_next_image_core_domain h⟩).elim\n    (λ c, c ∈ w.next_backward_image_domain a.1) (λ c, c ∈ w.next_forward_image_domain a.1) :=\nbegin\n  rw next_image_core_domain at h,\n  simp only [pfun.mem_dom, Union_exists, mem_Union, mem_image, mem_set_of_eq, set_coe.exists,\n    subtype.coe_mk, exists_and_distrib_right, exists_eq_right, exists_prop] at h,\n  obtain ⟨L, hL, ha, h⟩ := h,\n  have := (w.orbit_set_subset L ha).1,\n  rw mem_litter_set at this,\n  cases this,\n  exact h,\nend\n\nlemma next_image_core_injective (a b : atom)\n  (ha : a ∈ w.next_image_core_domain) (hb : b ∈ w.next_image_core_domain)\n  (h : w.next_image_core a a.1 (w.mem_orbit_set_of_mem_next_image_core_domain ha) =\n    w.next_image_core b b.1 (w.mem_orbit_set_of_mem_next_image_core_domain hb)) : a = b :=\nbegin\n  rw [next_image_core, next_image_core] at h,\n  obtain ⟨a', ha'⟩ := (w.orbit_set_equiv a.fst).symm.surjective\n    ⟨a, w.mem_orbit_set_of_mem_next_image_core_domain ha⟩,\n  obtain ⟨b', hb'⟩ := (w.orbit_set_equiv b.fst).symm.surjective\n    ⟨b, w.mem_orbit_set_of_mem_next_image_core_domain hb⟩,\n  have hae := w.orbit_set_equiv_elim_of_mem_next_image_core_domain ha,\n  have hbe := w.orbit_set_equiv_elim_of_mem_next_image_core_domain hb,\n  simp only [← ha', ← hb', equiv.apply_symm_apply] at h hae hbe,\n  obtain (⟨m, a'⟩ | ⟨m, a'⟩) := a';\n  obtain (⟨n, b'⟩ | ⟨n, b'⟩) := b';\n  simp only [elim_inl, elim_inr,\n    mk_mem_next_backward_image_domain, mk_mem_next_forward_image_domain] at h hae hbe,\n  { cases w.next_backward_image_injective' _ _ _ _ h,\n    { rw hae.2 at hbe,\n      rw [subtype.ext_iff, subtype.coe_mk] at ha' hb',\n      rw [← ha', ← hb', hbe.2], },\n    { exact hae, },\n    { exact hbe, },\n    { exact w.litter_map_dom_of_mem_next_image_core_domain ha, },\n    { exact w.litter_map_dom_of_mem_next_image_core_domain hb, }, },\n  { cases w.next_forward_image_ne_next_backward_image h.symm, },\n  { cases w.next_forward_image_ne_next_backward_image h, },\n  { cases w.next_forward_image_injective' _ _ h,\n    { rw hae.2 at hbe,\n      rw [subtype.ext_iff, subtype.coe_mk] at ha' hb',\n      rw [← ha', ← hb'],\n      exact w.orbit_set_equiv_symm_congr hbe.2, },\n    { exact w.litter_map_dom_of_mem_next_image_core_domain ha, },\n    { exact w.litter_map_dom_of_mem_next_image_core_domain hb, }, },\nend\n\ndef next_image_domain : set atom :=\n(w.need_forward_images ∩ {a | a.1 ∈ w.litter_perm.domain}) ∪ w.next_image_core_domain\n\nnoncomputable def next_image (a : atom) (ha : a ∈ w.next_image_domain) : atom :=\nha.elim'\n  (λ ha', (w.orbit_set_equiv (w.litter_perm a.1)).symm (inr (0, ⟨a, ha'.1⟩)))\n  (w.next_image_core a a.1 ∘ w.mem_orbit_set_of_mem_next_image_core_domain)\n\nlemma next_image_domain_small : small w.next_image_domain :=\nsmall.union\n  (small.mono (inter_subset_left _ _) w.need_forward_images_small)\n  w.next_image_core_domain_small\n\nlemma disjoint_need_forward_images_next_image_core_domain :\n  disjoint w.need_forward_images w.next_image_core_domain :=\nbegin\n  rw set.disjoint_iff,\n  rintro a ⟨ha₁, ha₂⟩,\n  exact (w.orbit_set_subset _ (w.mem_orbit_set_of_mem_next_image_core_domain ha₂)).2 (or.inr ha₁.1),\nend\n\nlemma next_image_eq_of_need_forward_images (a : atom)\n  (ha : a ∈ w.need_forward_images ∧ a.1 ∈ w.litter_perm.domain) :\n  w.next_image a (or.inl ha) =\n  (w.orbit_set_equiv (w.litter_perm a.1)).symm (inr (0, ⟨a, ha.1⟩)) :=\nor.elim'_left _ _ _ ha\n\nlemma next_image_eq_of_mem_next_image_core_domain (a : atom) (ha : a ∈ w.next_image_core_domain) :\n  w.next_image a (or.inr ha) =\n  w.next_image_core a a.1 (w.mem_orbit_set_of_mem_next_image_core_domain ha) :=\nbegin\n  refine or.elim'_right _ _ _ _,\n  exact λ h, set.disjoint_right.mp w.disjoint_need_forward_images_next_image_core_domain ha h.1,\nend\n\nlemma orbit_set_equiv_ne_next_image_core (a b : atom)\n  (ha : a ∈ w.need_forward_images ∧ a.fst ∈ w.litter_perm.domain)\n  (hb : b ∈ w.next_image_core_domain) :\n  (((w.orbit_set_equiv (w.litter_perm a.fst)).symm) (inr (0, ⟨a, ha.1⟩)) : atom) ≠\n    w.next_image_core b b.fst (w.mem_orbit_set_of_mem_next_image_core_domain hb) :=\nbegin\n  obtain ⟨b', hb'⟩ := (w.orbit_set_equiv b.fst).symm.surjective\n    ⟨b, w.mem_orbit_set_of_mem_next_image_core_domain hb⟩,\n  rw equiv.symm_apply_eq at hb',\n  intro h,\n  rw next_image_core at h,\n  rw ← hb' at h,\n  obtain (⟨_ | n, b'⟩ | b') := b';\n  simp only [elim_inl, elim_inr, next_backward_image, next_forward_image] at h,\n  { have := w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop,\n    rw h at this,\n    exact this.2 (or.inl b'.prop.1), },\n  { cases (w.orbit_set_equiv_injective h).2, },\n  { cases (w.orbit_set_equiv_injective h).2, },\nend\n\nlemma next_image_injective (a b : atom)\n  (ha : a ∈ w.next_image_domain) (hb : b ∈ w.next_image_domain)\n  (h : w.next_image a ha = w.next_image b hb) : a = b :=\nbegin\n  cases ha;\n  cases hb;\n  simp only [next_image_eq_of_need_forward_images,\n    next_image_eq_of_mem_next_image_core_domain] at h,\n  { have := w.orbit_set_equiv_injective h,\n    simp only [prod.mk.inj_iff, eq_self_iff_true, subtype.mk_eq_mk, true_and] at this,\n    exact this.2, },\n  { cases w.orbit_set_equiv_ne_next_image_core _ _ _ _ h, },\n  { cases w.orbit_set_equiv_ne_next_image_core _ _ _ _ h.symm, },\n  { refine w.next_image_core_injective a b ha hb h, },\nend\n\nnoncomputable def orbit_atom_map : atom →. atom :=\nλ a, {\n  dom := (w.atom_map a).dom ∨ a ∈ w.next_image_domain,\n  get := λ h, or.elim' h (w.atom_map a).get (w.next_image a)\n}\n\n@[simp] lemma orbit_atom_map_dom_iff (a : atom) :\n  (w.orbit_atom_map a).dom ↔ (w.atom_map a).dom ∨ a ∈ w.next_image_domain := iff.rfl\n\n@[simp] lemma orbit_atom_map_dom :\n  w.orbit_atom_map.dom = w.atom_map.dom ∪ w.next_image_domain := rfl\n\nlemma disjoint_atom_map_dom_next_image_domain : disjoint w.atom_map.dom w.next_image_domain :=\nbegin\n  rw set.disjoint_iff,\n  rintros a ⟨h₁, h₂ | h₂⟩,\n  { exact h₂.1.2 h₁, },\n  { exact (w.orbit_set_subset _\n      (w.mem_orbit_set_of_mem_next_image_core_domain h₂)).2 (or.inl h₁), },\nend\n\nlemma orbit_atom_map_eq_of_mem_dom (a : atom) (ha : (w.atom_map a).dom) :\n  (w.orbit_atom_map a).get (or.inl ha) = (w.atom_map a).get ha :=\nor.elim'_left _ _ _ _\n\nlemma orbit_atom_map_eq_of_mem_next_image_domain (a : atom) (ha : a ∈ w.next_image_domain) :\n  (w.orbit_atom_map a).get (or.inr ha) = w.next_image a ha :=\nor.elim'_right _ _ _ (id set.disjoint_right.mp w.disjoint_atom_map_dom_next_image_domain ha)\n\nlemma orbit_atom_map_eq_of_need_forward_images (a : atom)\n  (ha : a ∈ w.need_forward_images ∧ a.fst ∈ w.litter_perm.domain) :\n  (w.orbit_atom_map a).get (or.inr (or.inl ha)) =\n  (w.orbit_set_equiv (w.litter_perm a.1)).symm (inr (0, ⟨a, ha.1⟩)) :=\nbegin\n  unfold orbit_atom_map,\n  simp only,\n  rw or.elim'_right,\n  exact w.next_image_eq_of_need_forward_images a ha,\n  exact id set.disjoint_right.mp w.disjoint_atom_map_dom_next_image_domain (or.inl ha),\nend\n\nlemma orbit_atom_map_eq_of_mem_next_image_core_domain (a : atom)\n  (ha : a ∈ w.next_image_core_domain) :\n  (w.orbit_atom_map a).get (or.inr (or.inr ha)) =\n    w.next_image_core a a.1 (w.mem_orbit_set_of_mem_next_image_core_domain ha) :=\nbegin\n  unfold orbit_atom_map,\n  simp only,\n  rw or.elim'_right,\n  exact w.next_image_eq_of_mem_next_image_core_domain a ha,\n  exact id set.disjoint_right.mp w.disjoint_atom_map_dom_next_image_domain (or.inr ha),\nend\n\nlemma orbit_atom_map_dom_small : small w.orbit_atom_map.dom :=\nsmall.union w.atom_map_dom_small w.next_image_domain_small\n\nlemma orbit_atom_map_apply_ne_of_need_forward_images ⦃a b : atom⦄\n  (ha : (w.atom_map a).dom) (hb : b ∈ w.need_forward_images ∧ b.fst ∈ w.litter_perm.domain) :\n  (w.orbit_atom_map a).get (or.inl ha) ≠ (w.orbit_atom_map b).get (or.inr (or.inl hb)) :=\nbegin\n  rw [orbit_atom_map_eq_of_mem_dom, orbit_atom_map_eq_of_need_forward_images],\n  intro h,\n  have := w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop,\n  rw ← h at this,\n  exact this.2 (or.inr ⟨a, ha, rfl⟩),\nend\n\nlemma orbit_atom_map_apply_ne_of_mem_next_image_core_domain ⦃a b : atom⦄\n  (ha : (w.atom_map a).dom) (hb : b ∈ w.next_image_core_domain) :\n  (w.orbit_atom_map a).get (or.inl ha) ≠ (w.orbit_atom_map b).get (or.inr (or.inr hb)) :=\nbegin\n  obtain ⟨b', hb'⟩ := (w.orbit_set_equiv b.fst).symm.surjective\n    ⟨b, w.mem_orbit_set_of_mem_next_image_core_domain hb⟩,\n  rw [orbit_atom_map_eq_of_mem_dom, orbit_atom_map_eq_of_mem_next_image_core_domain,\n    next_image_core, ← hb', equiv.apply_symm_apply],\n  obtain (⟨_ | n, b'⟩ | ⟨n, b'⟩) := b';\n  simp only [elim_inr, elim_inl, nat.nat_zero_eq_zero, next_backward_image, next_forward_image],\n  { intro h,\n    have := b'.prop.2,\n    rw ← h at this,\n    exact this ⟨a, ha, rfl⟩, },\n  { intro h,\n    have := (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).2,\n    rw ← h at this,\n    exact this (or.inr ⟨a, ha, rfl⟩), },\n  { intro h,\n    have := (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).2,\n    rw ← h at this,\n    exact this (or.inr ⟨a, ha, rfl⟩), },\nend\n\nlemma orbit_atom_map_apply_ne ⦃a b : atom⦄\n  (ha : (w.atom_map a).dom) (hb : b ∈ w.next_image_domain) :\n  (w.orbit_atom_map a).get (or.inl ha) ≠ (w.orbit_atom_map b).get (or.inr hb) :=\nbegin\n  cases hb,\n  exact w.orbit_atom_map_apply_ne_of_need_forward_images ha hb,\n  exact w.orbit_atom_map_apply_ne_of_mem_next_image_core_domain ha hb,\nend\n\nlemma orbit_atom_map_injective ⦃a b : atom⦄\n  (ha : (w.orbit_atom_map a).dom) (hb : (w.orbit_atom_map b).dom)\n  (h : (w.orbit_atom_map a).get ha = (w.orbit_atom_map b).get hb) : a = b :=\nbegin\n  cases ha;\n  cases hb,\n  { rw [orbit_atom_map_eq_of_mem_dom, orbit_atom_map_eq_of_mem_dom] at h,\n    exact w.atom_map_injective ha hb h, },\n  { cases w.orbit_atom_map_apply_ne ha hb h, },\n  { cases w.orbit_atom_map_apply_ne hb ha h.symm, },\n  { rw [orbit_atom_map_eq_of_mem_next_image_domain,\n      orbit_atom_map_eq_of_mem_next_image_domain] at h,\n    exact w.next_image_injective a b ha hb h, },\nend\n\nlemma next_image_core_atom_mem_litter_map\n  (a : atom) (ha : a ∈ w.next_image_core_domain) :\n  w.next_image_core a a.fst (w.mem_orbit_set_of_mem_next_image_core_domain ha) ∈\n    litter_set (w.litter_perm a.fst) :=\nbegin\n  have hL := w.litter_map_dom_of_mem_next_image_core_domain ha,\n  have := w.mem_orbit_set_of_mem_next_image_core_domain ha,\n  obtain ⟨a', ha'⟩ := (w.orbit_set_equiv a.fst).symm.surjective\n    ⟨a, w.mem_orbit_set_of_mem_next_image_core_domain ha⟩,\n  have := w.orbit_set_equiv_elim_of_mem_next_image_core_domain ha,\n  rw [next_image_core],\n  rw [← ha', equiv.apply_symm_apply] at this ⊢,\n  obtain (⟨_ | n, a'⟩ | ⟨n, a'⟩) := a';\n  simp only [elim_inr, elim_inl, nat.nat_zero_eq_zero, next_backward_image, next_forward_image,\n    mk_mem_next_backward_image_domain, mk_mem_next_forward_image_domain,\n    function.iterate_one] at this ⊢,\n  { have ha'' := this.2.symm,\n    rw local_perm.eq_symm_apply at ha'',\n    { exact ha''.symm, },\n    { exact hL, },\n    { exact this.1, }, },\n  exact (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n  exact (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\nend\n\nlemma next_image_core_not_mem_ran\n  (a : atom) (ha : a ∈ w.next_image_core_domain) :\n  w.next_image_core a a.fst (w.mem_orbit_set_of_mem_next_image_core_domain ha) ∉ w.atom_map.ran :=\nbegin\n  rintro ⟨b, hb₁, hb₂⟩,\n  rw next_image_core at hb₂,\n  obtain ⟨a', ha'⟩ := (w.orbit_set_equiv a.fst).symm.surjective\n    ⟨a, w.mem_orbit_set_of_mem_next_image_core_domain ha⟩,\n  rw [← ha', equiv.apply_symm_apply] at hb₂,\n  obtain (⟨_ | n, a'⟩ | ⟨n, a'⟩) := a';\n  simp only [elim_inr, elim_inl, nat.nat_zero_eq_zero, next_backward_image, next_forward_image,\n    mk_mem_next_backward_image_domain, mk_mem_next_forward_image_domain,\n    function.iterate_one] at hb₂,\n  { exact a'.prop.2 ⟨b, hb₁, hb₂⟩, },\n  all_goals { have := w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop,\n    rw ← hb₂ at this,\n    exact this.2 (or.inr ⟨b, hb₁, rfl⟩), },\nend\n\nlemma next_image_core_atom_mem\n  (hdiff : ∀ L hL, ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆\n    w.atom_map.ran)\n  (a : atom) (ha : a ∈ w.next_image_core_domain)\n  (L : litter) (hL : (w.litter_map L).dom) :\n  a.fst = L ↔\n    w.next_image_core a a.fst (w.mem_orbit_set_of_mem_next_image_core_domain ha) ∈\n      (w.litter_map L).get hL :=\nbegin\n  have hamem := w.mem_orbit_set_of_mem_next_image_core_domain ha,\n  have ha' := w.next_image_core_atom_mem_litter_map a ha,\n  rw mem_litter_set at ha',\n  split,\n  { rintro rfl,\n    have := not_mem_subset (hdiff _ hL) (w.next_image_core_not_mem_ran a ha),\n    simp only [mem_symm_diff, set_like.mem_coe, mem_litter_set,\n      not_or_distrib, not_and_distrib, not_not] at this,\n    refine this.2.resolve_left (not_not.mpr _),\n    rw ha',\n    rw w.litter_perm_apply_eq _ hL,\n    rw w.rough_litter_map_or_else_of_dom,  },\n  { intro h,\n    have hL' := w.litter_perm_apply_eq L hL,\n    rw w.rough_litter_map_or_else_of_dom hL at hL',\n    have := not_mem_subset (hdiff _ hL) (w.next_image_core_not_mem_ran a ha),\n    simp only [mem_symm_diff, set_like.mem_coe, mem_litter_set, not_not, h, true_and,\n      not_true, and_false, or_false] at this,\n    rw [ha', ← hL', ← local_perm.eq_symm_apply, local_perm.left_inv] at this,\n    exact this,\n    exact or.inl (or.inl (or.inl hL)),\n    exact w.litter_map_dom_of_mem_next_image_core_domain ha,\n    exact local_perm.map_domain _ (or.inl (or.inl (or.inl hL))), },\nend\n\nlemma orbit_set_equiv_atom_mem\n  (hdiff : ∀ L hL, ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆\n    w.atom_map.ran)\n  (a : atom) (ha : a ∈ w.need_forward_images ∧ a.fst ∈ w.litter_perm.domain)\n  (L : litter) (hL : (w.litter_map L).dom) :\n  a.fst = L ↔ ((w.orbit_set_equiv (w.litter_perm a.fst)).symm (inr (0, ⟨a, ha.1⟩)) : atom) ∈\n    (w.litter_map L).get hL :=\nbegin\n  have ha' : _ ∧ _ := w.orbit_set_subset _\n    ((w.orbit_set_equiv (w.litter_perm a.fst)).symm (inr (0, ⟨a, ha.1⟩))).prop,\n  rw mem_litter_set at ha',\n  split,\n  { rintro rfl,\n    have := not_mem_subset (hdiff _ hL) _,\n    simp only [mem_symm_diff, set_like.mem_coe, mem_litter_set,\n      not_or_distrib, not_and_distrib, not_not] at this,\n    refine this.2.resolve_left (not_not.mpr _),\n    { rw ha'.1,\n      rw w.litter_perm_apply_eq _ hL,\n      rw w.rough_litter_map_or_else_of_dom, },\n    { exact ha'.2 ∘ or.inr, }, },\n  { intro h,\n    have := @not_mem_subset _\n      ((w.orbit_set_equiv (w.litter_perm a.fst)).symm (inr (0, ⟨a, ha.1⟩)) : atom) _ _\n      (hdiff L hL) (ha'.2 ∘ or.inr),\n    simp only [mem_symm_diff, h, set_like.mem_coe, mem_litter_set, true_and, not_true, and_false,\n      or_false, not_not] at this,\n    rw [ha'.1, ← rough_litter_map_or_else_of_dom, ← litter_perm_apply_eq,\n      ← local_perm.eq_symm_apply, local_perm.left_inv] at this,\n    exact this,\n    { exact or.inl (or.inl (or.inl hL)), },\n    { exact ha.2, },\n    { exact w.litter_perm.map_domain (or.inl (or.inl (or.inl hL))), },\n    { exact hL, }, },\nend\n\nlemma orbit_atom_mem\n  (hdiff : ∀ L hL, ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆\n    w.atom_map.ran)\n  (a : atom) (ha : (w.orbit_atom_map a).dom)\n  (L : litter) (hL : (w.litter_map L).dom) :\n  a.fst = L ↔ (w.orbit_atom_map a).get ha ∈ (w.litter_map L).get hL :=\nbegin\n  obtain ha | ha | ha := ha,\n  { rw orbit_atom_map_eq_of_mem_dom,\n    exact w.atom_mem a ha L hL, },\n  { rw orbit_atom_map_eq_of_need_forward_images,\n    exact w.orbit_set_equiv_atom_mem hdiff a ha L hL, },\n  { rw orbit_atom_map_eq_of_mem_next_image_core_domain,\n    rw w.next_image_core_atom_mem hdiff a ha L hL, },\nend\n\nnoncomputable def fill_atom_orbits\n  (hdiff : ∀ L hL, ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆\n    w.atom_map.ran) : weak_near_litter_approx := {\n  atom_map := w.orbit_atom_map,\n  litter_map := w.litter_map,\n  atom_map_dom_small := w.orbit_atom_map_dom_small,\n  litter_map_dom_small := w.litter_map_dom_small,\n  atom_map_injective := w.orbit_atom_map_injective,\n  litter_map_injective := w.litter_map_injective,\n  atom_mem := w.orbit_atom_mem hdiff,\n}\n\nvariables {w} {hdiff : ∀ L hL,\n  ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆ w.atom_map.ran}\n\n@[simp] lemma fill_atom_orbits_atom_map :\n  (w.fill_atom_orbits hdiff).atom_map = w.orbit_atom_map := rfl\n\n@[simp] lemma fill_atom_orbits_litter_map :\n  (w.fill_atom_orbits hdiff).litter_map = w.litter_map := rfl\n\nlemma subset_orbit_atom_map_dom : w.atom_map.dom ⊆ w.orbit_atom_map.dom :=\nsubset_union_left _ _\n\nlemma subset_orbit_atom_map_ran : w.atom_map.ran ⊆ w.orbit_atom_map.ran :=\nbegin\n  rintro _ ⟨a, ha, rfl⟩,\n  exact ⟨a, subset_orbit_atom_map_dom ha, w.orbit_atom_map_eq_of_mem_dom _ _⟩,\nend\n\nlemma fst_mem_litter_perm_domain_of_mem_map ⦃L : litter⦄ (hL : (w.litter_map L).dom)\n  ⦃a : atom⦄ (ha : a ∈ (w.litter_map L).get hL) :\n  a.1 ∈ w.litter_perm.domain :=\nbegin\n  by_cases a.1 = ((w.litter_map L).get hL).1,\n  { rw h,\n    refine or.inl (or.inl (or.inr ⟨L, hL, _⟩)),\n    rw rough_litter_map_or_else_of_dom, },\n  { by_cases h' : a.fst ∈ w.litter_perm'.domain,\n    exact or.inl h',\n    exact or.inr ⟨banned_litter.diff L hL a ⟨ha, h⟩, h'⟩, },\nend\n\nlemma fst_mem_litter_perm_domain_of_dom ⦃a : atom⦄ (ha : a ∈ w.atom_map.dom) :\n  a.fst ∈ w.litter_perm.domain :=\nbegin\n  by_cases h' : a.fst ∈ w.litter_perm'.domain,\n  exact or.inl h',\n  exact or.inr ⟨banned_litter.atom_dom a ha, h'⟩,\nend\n\nlemma fst_mem_litter_perm_domain_of_ran ⦃a : atom⦄ (ha : a ∈ w.atom_map.ran) :\n  a.fst ∈ w.litter_perm.domain :=\nbegin\n  by_cases h' : a.fst ∈ w.litter_perm'.domain,\n  exact or.inl h',\n  obtain ⟨b, hb, rfl⟩ := ha,\n  exact or.inr ⟨banned_litter.atom_map b hb, h'⟩,\nend\n\nlemma fill_atom_orbits_precise\n  (hdiff : ∀ L hL, ((w.litter_map L).get hL : set atom) ∆ litter_set ((w.litter_map L).get hL).1 ⊆\n    w.atom_map.ran) : precise (w.fill_atom_orbits hdiff) :=\nbegin\n  intros L hL,\n  constructor,\n  { exact subset_trans (hdiff L hL) subset_orbit_atom_map_ran, },\n  { intros a ha ha',\n    simp only [fill_atom_orbits_atom_map, fill_atom_orbits_litter_map, mem_litter_set,\n      orbit_atom_map_dom_iff] at *,\n    obtain ha | ha | ha := ha,\n    { have := w.orbit_atom_map_eq_of_mem_dom a ha,\n      generalize_proofs at this ha' ⊢,\n      rw [this, or_iff_not_imp_left],\n      intro hmap,\n      have hfwd : (w.atom_map a).get ha ∈ w.need_forward_images := ⟨⟨a, _, rfl⟩, hmap⟩,\n      refine or.inl ⟨hfwd, or.inl (or.inl _)⟩,\n      refine mem_of_eq_of_mem _ (or.inl hL),\n      rw [← ha', this], },\n    { refine or.inr (or.inr ⟨_, ⟨L, rfl⟩, _⟩),\n      simp only [pfun.mem_dom, Union_exists, mem_Union, mem_image, mem_set_of_eq, set_coe.exists,\n        subtype.coe_mk, exists_and_distrib_right, exists_eq_right, exists_prop],\n      have haL : L = w.litter_perm a.fst,\n      { have := (congr_arg prod.fst\n          (w.orbit_atom_map_eq_of_need_forward_images a ha)).symm.trans ha',\n        rw ← this,\n        exact (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1, },\n      refine ⟨or.inl (or.inl (or.inl hL)), _, _⟩,\n      { refine mem_of_eq_of_mem (w.orbit_atom_map_eq_of_need_forward_images a ha) _,\n        rw haL,\n        exact ((w.orbit_set_equiv _).symm _).prop, },\n      { have := w.orbit_atom_map_eq_of_need_forward_images a ha,\n        obtain ⟨hm₁, hm₂⟩ := subtype.coe_eq_iff.mp this.symm,\n        rw [equiv.symm_apply_eq, w.orbit_set_equiv_congr hm₁ haL.symm] at hm₂,\n        refine mem_of_eq_of_mem hm₂.symm _,\n        change sum.elim (λ b, b ∈ w.next_backward_image_domain L)\n          (λ b, b ∈ w.next_forward_image_domain L) (inr (0, ⟨a, ha.1⟩)),\n        refine ⟨ha.2, _⟩,\n        simp only [subtype.coe_mk, function.iterate_one],\n        exact haL.symm, }, },\n    { have := w.orbit_atom_map_eq_of_mem_next_image_core_domain a ha,\n      generalize_proofs at this ha' ⊢,\n      rw [this, next_image_core],\n      obtain ⟨_, ⟨L', rfl⟩, _, ⟨hL', rfl⟩, a, hbL, rfl⟩ := ha,\n      set b := w.orbit_set_equiv L' a with hb,\n      clear_value b,\n      simp only [mem_set_of_eq] at hbL,\n      rw ← hb at hbL,\n      have haL' := (w.orbit_set_subset _ a.prop).1,\n      rw mem_litter_set at haL',\n      have := w.orbit_set_equiv_congr (w.mem_orbit_set_of_mem_next_image_core_domain _)\n        (w.orbit_set_subset _ a.prop).1,\n      rw subtype.coe_eta at this,\n      rw [this, ← hb],\n      obtain (⟨_ | n, b⟩ | ⟨n, b⟩) := b;\n      simp only [need_backward_images, need_forward_images, elim_inl, elim_inr,\n        next_backward_image, next_forward_image] at hbL ⊢,\n      { exact or.inl b.prop.1, },\n      { refine or.inr (or.inr _),\n        have hbL' := hbL.2,\n        symmetry' at hbL',\n        rw [function.iterate_succ_apply',\n          local_perm.eq_symm_apply _ hL' (w.litter_perm.symm.iterate_domain hbL.1)] at hbL',\n        refine ⟨_, ⟨w.litter_perm.symm^[n + 1] (b : atom).1, rfl⟩, _, ⟨_, rfl⟩,\n          ⟨(w.orbit_set_equiv (w.litter_perm (a : atom).1)).symm (inl (n, b)), _⟩, _⟩,\n        { exact w.litter_perm.symm.iterate_domain hbL.1, },\n        { rw ← hbL',\n          have := (((w.orbit_set_equiv (w.litter_perm (a : atom).1)).symm (inl (n, b)))).prop,\n          rw haL' at this ⊢,\n          exact this, },\n        { simp only [function.comp_app, mem_set_of_eq, subtype.coe_mk,\n            eq_self_iff_true, and_true],\n          rw [w.orbit_set_equiv_congr _ hbL'.symm,\n            w.orbit_set_equiv_congr _ (congr_arg w.litter_perm haL'.symm)],\n          simp only [subtype.coe_eta, equiv.apply_symm_apply, elim_inl],\n          exact ⟨hbL.1, rfl⟩, }, },\n      { refine or.inr (or.inr _),\n        refine ⟨_, ⟨w.litter_perm^[n + 2] (b : atom).1, rfl⟩, _, ⟨_, rfl⟩,\n          ⟨(w.orbit_set_equiv (w.litter_perm (a : atom).1)).symm (inr (n + 1, b)), _⟩, _⟩,\n        { exact w.litter_perm.iterate_domain hbL.1, },\n        { rw [function.iterate_succ_apply', hbL.2, haL'],\n          exact ((w.orbit_set_equiv _).symm _).prop, },\n        { simp only [function.comp_app, mem_set_of_eq,\n            subtype.coe_mk, eq_self_iff_true, and_true],\n          have := congr_arg w.litter_perm hbL.2,\n          rw ← function.iterate_succ_apply' w.litter_perm (n + 1) at this,\n          rw [w.orbit_set_equiv_congr _ this,\n            w.orbit_set_equiv_congr _ (congr_arg w.litter_perm haL'.symm)],\n          simp only [function.iterate_succ, function.comp_app, subtype.coe_eta,\n            equiv.apply_symm_apply, elim_inr],\n          exact ⟨hbL.1, rfl⟩, }, },\n      { refine ⟨_, ⟨L', rfl⟩, _, ⟨hL', rfl⟩, a, _, rfl⟩,\n        rw [mem_set_of_eq, ← hb],\n        exact hbL, }, }, },\n  { rw fill_atom_orbits_litter_map at hL,\n    rintros a ⟨ha₁ | ⟨ha₁, ha₁'⟩ | ha₁, ha₂⟩;\n    simp only [fill_atom_orbits_atom_map, fill_atom_orbits_litter_map, orbit_atom_map_dom,\n      mem_inter_iff, mem_union, set_like.mem_coe] at *,\n    { by_cases ha₃ : a ∈ w.atom_map.ran,\n      { obtain ⟨b, hb₁, hb₂⟩ := ha₃,\n        refine ⟨b, or.inl hb₁, _⟩,\n        rw orbit_atom_map_eq_of_mem_dom,\n        exact hb₂, },\n      { refine ⟨(w.orbit_set_equiv (w.litter_perm.symm a.1)).symm (inl (0, ⟨a, ha₁, ha₃⟩)), _, _⟩,\n        { refine or.inr (or.inr ⟨_, ⟨w.litter_perm.symm a.1, rfl⟩, _, ⟨_, rfl⟩, _⟩),\n          { exact w.litter_perm.symm.map_domain (fst_mem_litter_perm_domain_of_mem_map hL ha₂), },\n          refine ⟨_, _, rfl⟩,\n          simp only [mem_set_of_eq, equiv.apply_symm_apply, elim_inl],\n          exact ⟨fst_mem_litter_perm_domain_of_mem_map hL ha₂, rfl⟩, },\n        { rw [orbit_atom_map_eq_of_mem_next_image_core_domain, next_image_core],\n          have : (((w.orbit_set_equiv (w.litter_perm.symm a.fst)).symm)\n            (inl (0, ⟨a, _⟩)) : atom).fst = w.litter_perm.symm a.fst,\n          { exact (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n            exact ⟨ha₁, ha₃⟩, },\n          rw w.orbit_set_equiv_congr _ this,\n          simp only [equiv.apply_symm_apply, elim_inl, subtype.coe_eta, next_backward_image,\n            subtype.coe_mk], }, }, },\n    { obtain ⟨⟨b, hb₁, hb₂⟩, ha₁⟩ := ha₁,\n      rw ← hb₂,\n      refine ⟨b, or.inl hb₁, _⟩,\n      rw orbit_atom_map_eq_of_mem_dom, },\n    { obtain ⟨a', ha'⟩ := (w.orbit_set_equiv a.fst).symm.surjective\n        ⟨a, w.mem_orbit_set_of_mem_next_image_core_domain ha₁⟩,\n      obtain (⟨n, a'⟩ | ⟨_ | n, a'⟩) := a',\n      { have : ((w.orbit_set_equiv (w.litter_perm.symm^[n + 2] (a' : atom).fst)).symm\n          (inl (n + 1, a')) : atom) ∈ w.next_image_core_domain,\n        { refine ⟨_, ⟨w.litter_perm.symm^[n + 2] (a' : atom).fst, rfl⟩, _, ⟨_, rfl⟩, _⟩,\n          exact w.litter_perm.symm.iterate_domain (fst_mem_litter_perm_domain_of_dom a'.prop.1),\n          refine ⟨_, _, rfl⟩,\n          simp only [mem_set_of_eq, equiv.apply_symm_apply, elim_inl],\n          exact ⟨fst_mem_litter_perm_domain_of_dom a'.prop.1, rfl⟩, },\n        refine ⟨_, or.inr (or.inr this), _⟩,\n        rw orbit_atom_map_eq_of_mem_next_image_core_domain,\n        rw next_image_core,\n        have : (((w.orbit_set_equiv (w.litter_perm.symm^[n + 2] (a' : atom).fst)).symm)\n            (inl (n + 1, a')) : atom).fst = (w.litter_perm.symm^[n + 2] (a' : atom).fst) :=\n          (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n        rw w.orbit_set_equiv_congr _ this,\n        simp only [subtype.coe_eta, equiv.apply_symm_apply, elim_inl, next_backward_image],\n        have := congr_arg subtype.val ha',\n        simp only [subtype.val_eq_coe] at this,\n        rw ← this,\n        refine w.orbit_set_equiv_symm_congr _,\n        have := (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n        rw mem_litter_set at this,\n        rw this,\n        have := w.orbit_set_equiv_elim_of_mem_next_image_core_domain ha₁,\n        rw ← ha' at this,\n        simp only [equiv.apply_symm_apply, elim_inl, next_backward_image_domain,\n          function.comp_app, mem_set_of_eq] at this,\n        rw [← this.2, function.iterate_succ_apply', local_perm.right_inv],\n        exact w.litter_perm.symm.iterate_domain this.1, },\n      { have := w.orbit_set_equiv_elim_of_mem_next_image_core_domain ha₁,\n        rw ← ha' at this,\n        simp only [equiv.apply_symm_apply, elim_inr, next_forward_image_domain,\n          function.comp_app, mem_set_of_eq, function.iterate_one] at this,\n        refine ⟨a', or.inr (or.inl ⟨a'.prop, this.1⟩), _⟩,\n        rw [orbit_atom_map_eq_of_need_forward_images, w.orbit_set_equiv_symm_congr this.2,\n          subtype.coe_eta, ha'],\n        refl, },\n      { have : ((w.orbit_set_equiv (w.litter_perm^[n + 1] (a' : atom).fst)).symm\n          (inr (n, a')) : atom) ∈ w.next_image_core_domain,\n        { refine ⟨_, ⟨w.litter_perm^[n + 1] (a' : atom).fst, rfl⟩, _, ⟨_, rfl⟩, _⟩,\n          exact w.litter_perm.iterate_domain (fst_mem_litter_perm_domain_of_ran a'.prop.1),\n          refine ⟨_, _, rfl⟩,\n          simp only [mem_set_of_eq, equiv.apply_symm_apply, elim_inl],\n          exact ⟨fst_mem_litter_perm_domain_of_ran a'.prop.1, rfl⟩, },\n        refine ⟨_, or.inr (or.inr this), _⟩,\n        rw orbit_atom_map_eq_of_mem_next_image_core_domain,\n        rw next_image_core,\n        have : (((w.orbit_set_equiv (w.litter_perm^[n + 1] (a' : atom).fst)).symm)\n            (inr (n, a')) : atom).fst = (w.litter_perm^[n + 1] (a' : atom).fst) :=\n          (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n        rw w.orbit_set_equiv_congr _ this,\n        simp only [subtype.coe_eta, equiv.apply_symm_apply, elim_inl, next_backward_image],\n        have := congr_arg subtype.val ha',\n        simp only [subtype.val_eq_coe] at this,\n        rw ← this,\n        refine w.orbit_set_equiv_symm_congr _,\n        have := (w.orbit_set_subset _ ((w.orbit_set_equiv _).symm _).prop).1,\n        rw mem_litter_set at this,\n        rw this,\n        have := w.orbit_set_equiv_elim_of_mem_next_image_core_domain ha₁,\n        rw ← ha' at this,\n        simp only [equiv.apply_symm_apply, elim_inr, next_forward_image_domain,\n          function.comp_app, mem_set_of_eq] at this,\n        rw [← this.2, function.iterate_succ_apply', function.iterate_succ_apply',\n          function.iterate_succ_apply'], }, }, },\nend\n\nend weak_near_litter_approx\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/fill_atom_orbits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832805, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3371914534679133}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.functor\n\n/-!\n# Traversable type class\n\nType classes for traversing collections. The concepts and laws are taken from\n<http://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html>\n\nTraversable collections are a generalization of functors. Whereas\nfunctors (such as `list`) allow us to apply a function to every\nelement, it does not allow functions which external effects encoded in\na monad. Consider for instance a functor `invite : email → io response`\nthat takes an email address, sends an email and waits for a\nresponse. If we have a list `guests : list email`, using calling\n`invite` using `map` gives us the following: `map invite guests : list\n(io response)`.  It is not what we need. We need something of type `io\n(list response)`. Instead of using `map`, we can use `traverse` to\nsend all the invites: `traverse invite guests : io (list response)`.\n`traverse` applies `invite` to every element of `guests` and combines\nall the resulting effects. In the example, the effect is encoded in the\nmonad `io` but any applicative functor is accepted by `traverse`.\n\nFor more on how to use traversable, consider the Haskell tutorial:\n<https://en.wikibooks.org/wiki/Haskell/Traversable>\n\n## Main definitions\n  * `traversable` type class - exposes the `traverse` function\n  * `sequence` - based on `traverse`,\n    turns a collection of effects into an effect returning a collection\n  * `is_lawful_traversable` - laws for a traversable functor\n  * `applicative_transformation` - the notion of a natural transformation for applicative functors\n\n## Tags\n\ntraversable iterator functor applicative\n\n## References\n\n * \"Applicative Programming with Effects\", by Conor McBride and Ross Paterson,\n   Journal of Functional Programming 18:1 (2008) 1-13, online at\n   <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>\n * \"The Essence of the Iterator Pattern\", by Jeremy Gibbons and Bruno Oliveira,\n   in Mathematically-Structured Functional Programming, 2006, online at\n   <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>\n * \"An Investigation of the Laws of Traversals\", by Mauro Jaskelioff and Ondrej Rypacek,\n   in Mathematically-Structured Functional Programming, 2012,\n   online at <http://arxiv.org/pdf/1202.2919>\n-/\n\nopen function (hiding comp)\n\nuniverses u v w\n\nsection applicative_transformation\n\nvariables (F : Type u → Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u → Type w) [applicative G] [is_lawful_applicative G]\n\n/-- A transformation between applicative functors.  It a natural\ntransformation such that `app` preserves the `has_pure.pure` and\n`functor.map` (`<*>`) operations. See\n`applicative_transformation.preserves_map` for naturality. -/\nstructure applicative_transformation : Type (max (u+1) v w) :=\n(app : Π α : Type u, F α → G α)\n(preserves_pure' : ∀ {α : Type u} (x : α), app _ (pure x) = pure x)\n(preserves_seq' : ∀ {α β : Type u} (x : F (α → β)) (y : F α), app _ (x <*> y) = app _ x <*> app _ y)\n\nend applicative_transformation\n\nnamespace applicative_transformation\n\nvariables (F : Type u → Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u → Type w) [applicative G] [is_lawful_applicative G]\n\ninstance : has_coe_to_fun (applicative_transformation F G) :=\n{ F := λ _, Π {α}, F α → G α,\n  coe := λ a, a.app }\n\nvariables {F G}\n\n@[simp]\nlemma app_eq_coe (η : applicative_transformation F G) : η.app = η := rfl\n\n@[simp]\nlemma coe_mk (f : Π (α : Type u), F α → G α) (pp ps) :\n  ⇑(applicative_transformation.mk f pp ps) = f := rfl\n\nprotected\nlemma congr_fun (η η' : applicative_transformation F G) (h : η = η') {α : Type u} (x : F α) :\n  η x = η' x :=\ncongr_arg (λ η'' : applicative_transformation F G, η'' x) h\n\nprotected\nlemma congr_arg (η : applicative_transformation F G) {α : Type u} {x y : F α} (h : x = y) :\n  η x = η y :=\ncongr_arg (λ z : F α, η z) h\n\nlemma coe_inj ⦃η η' : applicative_transformation F G⦄ (h : (η : Π α, F α → G α) = η') : η = η' :=\nby { cases η, cases η', congr, exact h }\n\n@[ext]\nlemma ext ⦃η η' : applicative_transformation F G⦄ (h : ∀ (α : Type u) (x : F α), η x = η' x) :\n  η = η' :=\nby { apply coe_inj, ext1 α, exact funext (h α) }\n\nlemma ext_iff {η η' : applicative_transformation F G} :\n  η = η' ↔ ∀ (α : Type u) (x : F α), η x = η' x :=\n⟨λ h α x, h ▸ rfl, λ h, ext h⟩\n\nsection preserves\nvariables (η : applicative_transformation F G)\n\n@[functor_norm]\nlemma preserves_pure : ∀ {α} (x : α), η (pure x) = pure x := η.preserves_pure'\n\n@[functor_norm]\nlemma preserves_seq :\n  ∀ {α β : Type u} (x : F (α → β)) (y : F α), η (x <*> y) = η x <*> η y :=\nη.preserves_seq'\n\n@[functor_norm]\nlemma preserves_map {α β} (x : α → β) (y : F α) : η (x <$> y) = x <$> η y :=\nby rw [← pure_seq_eq_map, η.preserves_seq]; simp with functor_norm\n\nlemma preserves_map' {α β} (x : α → β) : @η _ ∘ functor.map x = functor.map x ∘ @η _ :=\nby { ext y, exact preserves_map η x y }\n\nend preserves\n\n/-- The identity applicative transformation from an applicative functor to itself. -/\ndef id_transformation : applicative_transformation F F :=\n{ app := λ α, id,\n  preserves_pure' := by simp,\n  preserves_seq' := λ α β x y, by simp }\n\ninstance : inhabited (applicative_transformation F F) := ⟨id_transformation⟩\n\nuniverses s t\nvariables {H : Type u → Type s} [applicative H] [is_lawful_applicative H]\n\n/-- The composition of applicative transformations. -/\ndef comp (η' : applicative_transformation G H) (η : applicative_transformation F G) :\n  applicative_transformation F H :=\n{ app := λ α x, η' (η x),\n  preserves_pure' := λ α x, by simp with functor_norm,\n  preserves_seq' := λ α β x y, by simp with functor_norm }\n\n@[simp]\nlemma comp_apply (η' : applicative_transformation G H) (η : applicative_transformation F G)\n  {α : Type u} (x : F α) :\n  η'.comp η x = η' (η x) := rfl\n\nlemma comp_assoc {I : Type u → Type t} [applicative I] [is_lawful_applicative I]\n  (η'' : applicative_transformation H I)\n  (η' : applicative_transformation G H)\n  (η : applicative_transformation F G) :\n  (η''.comp η').comp η = η''.comp (η'.comp η) := rfl\n\n@[simp]\nlemma comp_id (η : applicative_transformation F G) : η.comp id_transformation = η :=\next $ λ α x, rfl\n\n@[simp]\n\n\nend applicative_transformation\n\nopen applicative_transformation\n\n/-- A traversable functor is a functor along with a way to commute\nwith all applicative functors (see `sequence`).  For example, if `t`\nis the traversable functor `list` and `m` is the applicative functor\n`io`, then given a function `f : α → io β`, the function `functor.map f` is\n`list α → list (io β)`, but `traverse f` is `list α → io (list β)`. -/\nclass traversable (t : Type u → Type u) extends functor t :=\n(traverse : Π {m : Type u → Type u} [applicative m] {α β},\n   (α → m β) → t α → m (t β))\n\nopen functor\n\nexport traversable (traverse)\n\nsection functions\n\nvariables {t : Type u → Type u}\nvariables {m : Type u → Type v} [applicative m]\nvariables {α β : Type u}\n\n\nvariables {f : Type u → Type u} [applicative f]\n\n/-- A traversable functor commutes with all applicative functors. -/\ndef sequence [traversable t] : t (f α) → f (t α) := traverse id\n\nend functions\n\n/-- A traversable functor is lawful if its `traverse` satisfies a\nnumber of additional properties.  It must send `id.mk` to `id.mk`,\nsend the composition of applicative functors to the composition of the\n`traverse` of each, send each function `f` to `λ x, f <$> x`, and\nsatisfy a naturality condition with respect to applicative\ntransformations. -/\nclass is_lawful_traversable (t : Type u → Type u) [traversable t]\n  extends is_lawful_functor t : Type (u+1) :=\n(id_traverse : ∀ {α} (x : t α), traverse id.mk x = x )\n(comp_traverse : ∀ {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {α β γ} (f : β → F γ) (g : α → G β) (x : t α),\n  traverse (comp.mk ∘ map f ∘ g) x =\n  comp.mk (map (traverse f) (traverse g x)))\n(traverse_eq_map_id : ∀ {α β} (f : α → β) (x : t α),\n  traverse (id.mk ∘ f) x = id.mk (f <$> x))\n(naturality : ∀ {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (η : applicative_transformation F G) {α β} (f : α → F β) (x : t α),\n  η (traverse f x) = traverse (@η _ ∘ f) x)\n\ninstance : traversable id := ⟨λ _ _ _ _, id⟩\ninstance : is_lawful_traversable id := by refine {..}; intros; refl\n\nsection\n\nvariables {F : Type u → Type v} [applicative F]\n\ninstance : traversable option := ⟨@option.traverse⟩\n\ninstance : traversable list := ⟨@list.traverse⟩\n\nend\n\nnamespace sum\n\nvariables {σ : Type u}\nvariables {F : Type u → Type u}\nvariables [applicative F]\n\n/-- Defines a `traverse` function on the second component of a sum type.\nThis is used to give a `traversable` instance for the functor `σ ⊕ -`. -/\nprotected def traverse {α β} (f : α → F β) : σ ⊕ α → F (σ ⊕ β)\n| (sum.inl x) := pure (sum.inl x)\n| (sum.inr x) := sum.inr <$> f x\n\nend sum\n\ninstance {σ : Type u} : traversable.{u} (sum σ) := ⟨@sum.traverse _⟩\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/traversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.33719144550766533}}
{"text": "/-\nCopyright (c) 2020 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\nimport tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\nimport tactic.show_term\nimport topology.basic\n/-!\n# Tactics for topology\n\nCurrently we have one domain-specific tactic for topology: `continuity`.\n\n-/\n\n/-!\n### `continuity` tactic\n\nAutomatically solve goals of the form `continuous f`.\n\nMark lemmas with `@[continuity]` to add them to the set of lemmas\nused by `continuity`. Note: `to_additive` doesn't know yet how to\ncopy the attribute to the additive version.\n-/\n\n/-- User attribute used to mark tactics used by `continuity`. -/\n@[user_attribute]\nmeta def continuity : user_attribute :=\n{ name := `continuity,\n  descr := \"lemmas usable to prove continuity\" }\n\n-- Mark some continuity lemmas already defined in `topology.basic`\nattribute [continuity]\n  continuous_id\n  continuous_const\n\n-- As we will be using `apply_rules` with `md := semireducible`,\n-- we need another version of `continuous_id`.\n@[continuity] lemma continuous_id' {α : Type*} [topological_space α] : continuous (λ a : α, a) :=\ncontinuous_id\n\nnamespace tactic\n\n/--\nTactic to apply `continuous.comp` when appropriate.\n\nApplying `continuous.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove continuous is actually\n  constant, and that constant is a function application `f z`, then\n  continuous.comp would produce new goals `continuous f`, `continuous\n  (λ _, z)`, which is silly. We avoid this by failing if we could\n  apply continuous_const.\n\n* continuous.comp will always succeed on `continuous (λ x, f x)` and\n  produce new goals `continuous (λ x, x)`, `continuous f`. We detect\n  this by failing if a new goal can be closed by applying\n  continuous_id.\n-/\nmeta def apply_continuous.comp : tactic unit :=\n`[fail_if_success { exact continuous_const };\n  refine continuous.comp _ _;\n  fail_if_success { exact continuous_id }]\n\n/-- List of tactics used by `continuity` internally. -/\nmeta def continuity_tactics (md : transparency := reducible) : list (tactic string) :=\n[\n  intros1               >>= λ ns, pure (\"intros \" ++ (\" \".intercalate (ns.map (λ e, e.to_string)))),\n  apply_rules [``(continuity)] 50 { md := md }\n                        >> pure \"apply_rules continuity\",\n  apply_continuous.comp >> pure \"refine continuous.comp _ _\"\n]\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nSolve goals of the form `continuous f`. `continuity?` reports back the proof term it found.\n-/\nmeta def continuity\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md              := if bang.is_some then semireducible else reducible,\n    continuity_core := tactic.tidy { tactics := continuity_tactics md, ..cfg },\n    trace_fn        := if trace.is_some then show_term else id in\ntrace_fn continuity_core\n\n/-- Version of `continuity` for use with auto_param. -/\nmeta def continuity' : tactic unit := continuity none none {}\n\n/--\n`continuity` solves goals of the form `continuous f` by applying lemmas tagged with the\n`continuity` user attribute.\n\n```\nexample {X Y : Type*} [topological_space X] [topological_space Y]\n  (f₁ f₂ : X → Y) (hf₁ : continuous f₁) (hf₂ : continuous f₂)\n  (g : Y → ℝ) (hg : continuous g) : continuous (λ x, (max (g (f₁ x)) (g (f₂ x))) + 1) :=\nby continuity\n```\nwill discharge the goal, generating a proof term like\n`((continuous.comp hg hf₁).max (continuous.comp hg hf₂)).add continuous_const`\n\nYou can also use `continuity!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`continuity?` reports back the proof term it found.\n-/\nadd_tactic_doc\n{ name := \"continuity / continuity'\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.continuity, `tactic.interactive.continuity'],\n  tags := [\"lemma application\"]\n}\n\nend interactive\n\nend tactic\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/topology/tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984434543458, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.3369914047127322}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.fin2\nimport Mathlib.data.typevec\nimport Mathlib.logic.function.basic\nimport Mathlib.tactic.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u v \n\nnamespace Mathlib\n\n/-!\n\nFunctors between the category of tuples of types, and the category Type\n\nFeatures:\n\n`mvfunctor n` : the type class of multivariate functors\n`f <$$> x`    : notation for map\n\n-/\n\n/-- multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass mvfunctor {n : ℕ} (F : typevec n → Type u_2) \nwhere\n  map : {α β : typevec n} → typevec.arrow α β → F α → F β\n\nnamespace mvfunctor\n\n\n/-- predicate lifting over multivariate functors -/\ndef liftp {n : ℕ} {F : typevec n → Type v} [mvfunctor F] {α : typevec n} (p : (i : fin2 n) → α i → Prop) (x : F α) :=\n  ∃ (u : F fun (i : fin2 n) => Subtype (p i)), map (fun (i : fin2 n) => subtype.val) u = x\n\n/-- relational lifting over multivariate functors -/\ndef liftr {n : ℕ} {F : typevec n → Type v} [mvfunctor F] {α : typevec n} (r : {i : fin2 n} → α i → α i → Prop) (x : F α) (y : F α) :=\n  ∃ (u : F fun (i : fin2 n) => Subtype fun (p : α i × α i) => r (prod.fst p) (prod.snd p)),\n    map (fun (i : fin2 n) (t : Subtype fun (p : α i × α i) => r (prod.fst p) (prod.snd p)) => prod.fst (subtype.val t))\n          u =\n        x ∧\n      map (fun (i : fin2 n) (t : Subtype fun (p : α i × α i) => r (prod.fst p) (prod.snd p)) => prod.snd (subtype.val t))\n          u =\n        y\n\n/-- given `x : F α` and a projection `i` of type vector `α`, `supp x i` is the set\nof `α.i` contained in `x` -/\ndef supp {n : ℕ} {F : typevec n → Type v} [mvfunctor F] {α : typevec n} (x : F α) (i : fin2 n) : set (α i) :=\n  set_of fun (y : α i) => ∀ {p : (i : fin2 n) → α i → Prop}, liftp p x → p i y\n\ntheorem of_mem_supp {n : ℕ} {F : typevec n → Type v} [mvfunctor F] {α : typevec n} {x : F α} {p : {i : fin2 n} → α i → Prop} (h : liftp p x) (i : fin2 n) (y : α i) (H : y ∈ supp x i) : p y :=\n  hy h\n\nend mvfunctor\n\n\n/-- laws for `mvfunctor` -/\nclass is_lawful_mvfunctor {n : ℕ} (F : typevec n → Type u_2) [mvfunctor F] \nwhere\n  id_map : ∀ {α : typevec n} (x : F α), mvfunctor.map typevec.id x = x\n  comp_map : ∀ {α β γ : typevec n} (g : typevec.arrow α β) (h : typevec.arrow β γ) (x : F α),\n  mvfunctor.map (typevec.comp h g) x = mvfunctor.map h (mvfunctor.map g x)\n\nnamespace mvfunctor\n\n\n/-- adapt `mvfunctor.liftp` to accept predicates as arrows -/\ndef liftp' {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] (p : typevec.arrow α (typevec.repeat n Prop)) : F α → Prop :=\n  liftp fun (i : fin2 n) (x : α i) => typevec.of_repeat (p i x)\n\n/-- adapt `mvfunctor.liftp` to accept relations as arrows -/\ndef liftr' {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] (r : typevec.arrow (typevec.prod α α) (typevec.repeat n Prop)) : F α → F α → Prop :=\n  liftr fun (i : fin2 n) (x y : α i) => typevec.of_repeat (r i (typevec.prod.mk i x y))\n\n@[simp] theorem id_map {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] [is_lawful_mvfunctor F] (x : F α) : map typevec.id x = x :=\n  is_lawful_mvfunctor.id_map x\n\n@[simp] theorem id_map' {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] [is_lawful_mvfunctor F] (x : F α) : map (fun (i : fin2 n) (a : α i) => a) x = x :=\n  id_map x\n\ntheorem map_map {n : ℕ} {α : typevec n} {β : typevec n} {γ : typevec n} {F : typevec n → Type v} [mvfunctor F] [is_lawful_mvfunctor F] (g : typevec.arrow α β) (h : typevec.arrow β γ) (x : F α) : map h (map g x) = map (typevec.comp h g) x :=\n  Eq.symm (comp_map g h x)\n\ntheorem exists_iff_exists_of_mono {n : ℕ} {α : typevec n} {β : typevec n} (F : typevec n → Type v) [mvfunctor F] [is_lawful_mvfunctor F] {p : F α → Prop} {q : F β → Prop} (f : typevec.arrow α β) (g : typevec.arrow β α) (h₀ : typevec.comp f g = typevec.id) (h₁ : ∀ (u : F α), p u ↔ q (map f u)) : (∃ (u : F α), p u) ↔ ∃ (u : F β), q u := sorry\n\ntheorem liftp_def {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] (p : typevec.arrow α (typevec.repeat n Prop)) [is_lawful_mvfunctor F] (x : F α) : liftp' p x ↔ ∃ (u : F (typevec.subtype_ p)), map (typevec.subtype_val p) u = x := sorry\n\ntheorem liftr_def {n : ℕ} {α : typevec n} {F : typevec n → Type v} [mvfunctor F] (r : typevec.arrow (typevec.prod α α) (typevec.repeat n Prop)) [is_lawful_mvfunctor F] (x : F α) (y : F α) : liftr' r x y ↔\n  ∃ (u : F (typevec.subtype_ r)),\n    map (typevec.comp typevec.prod.fst (typevec.subtype_val r)) u = x ∧\n      map (typevec.comp typevec.prod.snd (typevec.subtype_val r)) u = y := sorry\n\nend mvfunctor\n\n\nnamespace mvfunctor\n\n\ntheorem liftp_last_pred_iff {n : ℕ} {F : typevec (n + 1) → Type u_1} [mvfunctor F] [is_lawful_mvfunctor F] {α : typevec n} {β : Type u} (p : β → Prop) (x : F (α ::: β)) : liftp' (typevec.pred_last' α p) x ↔ liftp (typevec.pred_last α p) x := sorry\n\ntheorem liftr_last_rel_iff {n : ℕ} {F : typevec (n + 1) → Type u_1} [mvfunctor F] [is_lawful_mvfunctor F] {α : typevec n} {β : Type u} (rr : β → β → Prop) (x : F (α ::: β)) (y : F (α ::: β)) : liftr' (typevec.rel_last' α rr) x y ↔ liftr (typevec.rel_last α rr) x y := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/functor/multivariate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.33699140281104284}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.abel\nimport Mathlib.category_theory.limits.shapes.biproducts\nimport Mathlib.category_theory.preadditive.default\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# Basic facts about morphisms between biproducts in preadditive categories.\n\n* In any category (with zero morphisms), if `biprod.map f g` is an isomorphism,\n  then both `f` and `g` are isomorphisms.\n\nThe remaining lemmas hold in any preadditive category.\n\n* If `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\n  then we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\n  so that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\n  via Gaussian elimination.\n\n* As a corollary of the previous two facts,\n  if we have an isomorphism `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\n  we can construct an isomorphism `X₂ ≅ Y₂`.\n\n* If `f : W ⊞ X ⟶ Y ⊞ Z` is an isomorphism, either `𝟙 W = 0`,\n  or at least one of the component maps `W ⟶ Y` and `W ⟶ Z` is nonzero.\n\n* If `f : ⨁ S ⟶ ⨁ T` is an isomorphism,\n  then every column (corresponding to a nonzero summand in the domain)\n  has some nonzero matrix entry.\n-/\n\nnamespace category_theory\n\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `f` is invertible.\n-/\ndef is_iso_left_of_is_iso_biprod_map {C : Type u} [category C] [limits.has_zero_morphisms C] [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (limits.biprod.map f g)] : is_iso f :=\n  is_iso.mk (limits.biprod.inl ≫ inv (limits.biprod.map f g) ≫ limits.biprod.fst)\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `g` is invertible.\n-/\ndef is_iso_right_of_is_iso_biprod_map {C : Type u} [category C] [limits.has_zero_morphisms C] [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (limits.biprod.map f g)] : is_iso g :=\n  let _inst : is_iso (limits.biprod.map g f) := eq.mpr sorry is_iso.comp_is_iso;\n  is_iso_left_of_is_iso_biprod_map g f\n\n/--\nThe \"matrix\" morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` with specified components.\n-/\ndef biprod.of_components {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂ :=\n  limits.biprod.fst ≫ f₁₁ ≫ limits.biprod.inl + limits.biprod.fst ≫ f₁₂ ≫ limits.biprod.inr +\n      limits.biprod.snd ≫ f₂₁ ≫ limits.biprod.inl +\n    limits.biprod.snd ≫ f₂₂ ≫ limits.biprod.inr\n\n@[simp] theorem biprod.inl_of_components {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : limits.biprod.inl ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ = f₁₁ ≫ limits.biprod.inl + f₁₂ ≫ limits.biprod.inr := sorry\n\n@[simp] theorem biprod.inr_of_components {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : limits.biprod.inr ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ = f₂₁ ≫ limits.biprod.inl + f₂₂ ≫ limits.biprod.inr := sorry\n\n@[simp] theorem biprod.of_components_fst {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ limits.biprod.fst = limits.biprod.fst ≫ f₁₁ + limits.biprod.snd ≫ f₂₁ := sorry\n\n@[simp] theorem biprod.of_components_snd {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) : biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ limits.biprod.snd = limits.biprod.fst ≫ f₁₂ + limits.biprod.snd ≫ f₂₂ := sorry\n\n@[simp] theorem biprod.of_components_eq {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) : biprod.of_components (limits.biprod.inl ≫ f ≫ limits.biprod.fst) (limits.biprod.inl ≫ f ≫ limits.biprod.snd)\n    (limits.biprod.inr ≫ f ≫ limits.biprod.fst) (limits.biprod.inr ≫ f ≫ limits.biprod.snd) =\n  f := sorry\n\n@[simp] theorem biprod.of_components_comp {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} {Z₁ : C} {Z₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) (g₁₁ : Y₁ ⟶ Z₁) (g₁₂ : Y₁ ⟶ Z₂) (g₂₁ : Y₂ ⟶ Z₁) (g₂₂ : Y₂ ⟶ Z₂) : biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ biprod.of_components g₁₁ g₁₂ g₂₁ g₂₂ =\n  biprod.of_components (f₁₁ ≫ g₁₁ + f₁₂ ≫ g₂₁) (f₁₁ ≫ g₁₂ + f₁₂ ≫ g₂₂) (f₂₁ ≫ g₁₁ + f₂₂ ≫ g₂₁) (f₂₁ ≫ g₁₂ + f₂₂ ≫ g₂₂) := sorry\n\n/--\nThe unipotent upper triangular matrix\n```\n(1 r)\n(0 1)\n```\nas an isomorphism.\n-/\n@[simp] theorem biprod.unipotent_upper_hom {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} (r : X₁ ⟶ X₂) : iso.hom (biprod.unipotent_upper r) = biprod.of_components 𝟙 r 0 𝟙 :=\n  Eq.refl (iso.hom (biprod.unipotent_upper r))\n\n/--\nThe unipotent lower triangular matrix\n```\n(1 0)\n(r 1)\n```\nas an isomorphism.\n-/\n@[simp] theorem biprod.unipotent_lower_hom {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} (r : X₂ ⟶ X₁) : iso.hom (biprod.unipotent_lower r) = biprod.of_components 𝟙 0 r 𝟙 :=\n  Eq.refl (iso.hom (biprod.unipotent_lower r))\n\n/--\nIf `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\nso that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\nvia Gaussian elimination.\n\n(This is the version of `biprod.gaussian` written in terms of components.)\n-/\ndef biprod.gaussian' {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) [is_iso f₁₁] : psigma\n  fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n    psigma\n      fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n        psigma\n          fun (g₂₂ : X₂ ⟶ Y₂) =>\n            iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R = limits.biprod.map f₁₁ g₂₂ :=\n  psigma.mk (biprod.unipotent_lower (-f₂₁ ≫ inv f₁₁))\n    (psigma.mk (biprod.unipotent_upper (-inv f₁₁ ≫ f₁₂)) (psigma.mk (f₂₂ - f₂₁ ≫ inv f₁₁ ≫ f₁₂) sorry))\n\n/--\nIf `f` is a morphism `X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct isomorphisms `L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂` and `R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂`\nso that `L.hom ≫ g ≫ R.hom` is diagonal (with `X₁ ⟶ Y₁` component still `f`),\nvia Gaussian elimination.\n-/\ndef biprod.gaussian {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ⟶ Y₁ ⊞ Y₂) [is_iso (limits.biprod.inl ≫ f ≫ limits.biprod.fst)] : psigma\n  fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n    psigma\n      fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n        psigma\n          fun (g₂₂ : X₂ ⟶ Y₂) =>\n            iso.hom L ≫ f ≫ iso.hom R = limits.biprod.map (limits.biprod.inl ≫ f ≫ limits.biprod.fst) g₂₂ :=\n  let this :\n    psigma\n      fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂) =>\n        psigma\n          fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n            psigma\n              fun (g₂₂ : X₂ ⟶ Y₂) =>\n                iso.hom L ≫\n                    biprod.of_components (limits.biprod.inl ≫ f ≫ limits.biprod.fst)\n                        (limits.biprod.inl ≫ f ≫ limits.biprod.snd) (limits.biprod.inr ≫ f ≫ limits.biprod.fst)\n                        (limits.biprod.inr ≫ f ≫ limits.biprod.snd) ≫\n                      iso.hom R =\n                  limits.biprod.map (limits.biprod.inl ≫ f ≫ limits.biprod.fst) g₂₂ :=\n    biprod.gaussian' (limits.biprod.inl ≫ f ≫ limits.biprod.fst) (limits.biprod.inl ≫ f ≫ limits.biprod.snd)\n      (limits.biprod.inr ≫ f ≫ limits.biprod.fst) (limits.biprod.inr ≫ f ≫ limits.biprod.snd);\n  eq.mpr sorry (eq.mp sorry this)\n\n/--\nIf `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` via a two-by-two matrix whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct an isomorphism `X₂ ≅ Y₂`, via Gaussian elimination.\n-/\ndef biprod.iso_elim' {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f₁₁ : X₁ ⟶ Y₁) (f₁₂ : X₁ ⟶ Y₂) (f₂₁ : X₂ ⟶ Y₁) (f₂₂ : X₂ ⟶ Y₂) [is_iso f₁₁] [is_iso (biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂)] : X₂ ≅ Y₂ :=\n  psigma.cases_on (biprod.gaussian' f₁₁ f₁₂ f₂₁ f₂₂)\n    fun (L : X₁ ⊞ X₂ ≅ X₁ ⊞ X₂)\n      (snd :\n      psigma\n        fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂) =>\n          psigma\n            fun (g₂₂ : X₂ ⟶ Y₂) =>\n              iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R = limits.biprod.map f₁₁ g₂₂) =>\n      psigma.cases_on snd\n        fun (R : Y₁ ⊞ Y₂ ≅ Y₁ ⊞ Y₂)\n          (snd_snd :\n          psigma\n            fun (g₂₂ : X₂ ⟶ Y₂) =>\n              iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R = limits.biprod.map f₁₁ g₂₂) =>\n          psigma.cases_on snd_snd\n            fun (g : X₂ ⟶ Y₂)\n              (w : iso.hom L ≫ biprod.of_components f₁₁ f₁₂ f₂₁ f₂₂ ≫ iso.hom R = limits.biprod.map f₁₁ g) =>\n              let _inst : is_iso (limits.biprod.map f₁₁ g) := eq.mpr sorry is_iso.comp_is_iso;\n              let _inst_6 : is_iso g := is_iso_right_of_is_iso_biprod_map f₁₁ g;\n              as_iso g\n\n/--\nIf `f` is an isomorphism `X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂` whose `X₁ ⟶ Y₁` entry is an isomorphism,\nthen we can construct an isomorphism `X₂ ≅ Y₂`, via Gaussian elimination.\n-/\ndef biprod.iso_elim {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {X₁ : C} {X₂ : C} {Y₁ : C} {Y₂ : C} (f : X₁ ⊞ X₂ ≅ Y₁ ⊞ Y₂) [is_iso (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst)] : X₂ ≅ Y₂ :=\n  let _inst :\n    is_iso\n      (biprod.of_components (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst)\n        (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.snd) (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.fst)\n        (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.snd)) :=\n    eq.mpr sorry (is_iso.of_iso f);\n  biprod.iso_elim' (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.fst) (limits.biprod.inl ≫ iso.hom f ≫ limits.biprod.snd)\n    (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.fst) (limits.biprod.inr ≫ iso.hom f ≫ limits.biprod.snd)\n\ntheorem biprod.column_nonzero_of_iso {C : Type u} [category C] [preadditive C] [limits.has_binary_biproducts C] {W : C} {X : C} {Y : C} {Z : C} (f : W ⊞ X ⟶ Y ⊞ Z) [is_iso f] : 𝟙 = 0 ∨ limits.biprod.inl ≫ f ≫ limits.biprod.fst ≠ 0 ∨ limits.biprod.inl ≫ f ≫ limits.biprod.snd ≠ 0 := sorry\n\ntheorem biproduct.column_nonzero_of_iso' {C : Type u} [category C] [preadditive C] {σ : Type v} {τ : Type v} [DecidableEq σ] [DecidableEq τ] [fintype τ] {S : σ → C} [limits.has_biproduct S] {T : τ → C} [limits.has_biproduct T] (s : σ) (f : ⨁ S ⟶ ⨁ T) [is_iso f] : (∀ (t : τ), limits.biproduct.ι S s ≫ f ≫ limits.biproduct.π T t = 0) → 𝟙 = 0 := sorry\n\n/--\nIf `f : ⨁ S ⟶ ⨁ T` is an isomorphism, and `s` is a non-trivial summand of the source,\nthen there is some `t` in the target so that the `s, t` matrix entry of `f` is nonzero.\n-/\ndef biproduct.column_nonzero_of_iso {C : Type u} [category C] [preadditive C] {σ : Type v} {τ : Type v} [DecidableEq σ] [DecidableEq τ] [fintype τ] {S : σ → C} [limits.has_biproduct S] {T : τ → C} [limits.has_biproduct T] (s : σ) (nz : 𝟙 ≠ 0) [(t : τ) → DecidableEq (S s ⟶ T t)] (f : ⨁ S ⟶ ⨁ T) [is_iso f] : trunc (psigma fun (t : τ) => limits.biproduct.ι S s ≫ f ≫ limits.biproduct.π T t ≠ 0) :=\n  trunc_sigma_of_exists sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/preadditive/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3369914028110428}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-!\n# Extra definitions on `option`\n\nThis file defines more operations involving `option α`. Lemmas about them are located in other\nfiles under `data.option.`.\nOther basic operations on `option` are defined in the core library.\n-/\n\n\nnamespace option\nvariables {α : Type*} {β : Type*}\n\nattribute [inline] option.is_some option.is_none\n\n/-- An elimination principle for `option`. It is a nondependent version of `option.rec`. -/\n@[simp] protected def elim (b : β) (f : α → β) : option α → β\n| (some a) := f a\n| none     := b\n\ninstance has_mem : has_mem α (option α) := ⟨λ a b, b = some a⟩\n\n@[simp] theorem mem_def {a : α} {b : option α} : a ∈ b ↔ b = some a :=\niff.rfl\n\nlemma mem_iff {a : α} {b : option α} : a ∈ b ↔ b = a := iff.rfl\n\ntheorem is_none_iff_eq_none {o : option α} : o.is_none = tt ↔ o = none :=\n⟨option.eq_none_of_is_none, λ e, e.symm ▸ rfl⟩\n\ntheorem some_inj {a b : α} : some a = some b ↔ a = b := by simp\n\nlemma mem_some_iff {α : Type*} {a b : α} : a ∈ some b ↔ b = a :=\nby simp\n\n/--\n`o = none` is decidable even if the wrapped type does not have decidable equality.\n\nThis is not an instance because it is not definitionally equal to `option.decidable_eq`.\nTry to use `o.is_none` or `o.is_some` instead.\n-/\n@[inline]\ndef decidable_eq_none {o : option α} : decidable (o = none) :=\ndecidable_of_decidable_of_iff (bool.decidable_eq _ _) is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : α → Prop} [decidable_pred p] :\n  ∀ o : option α, decidable (∀ a ∈ o, p a)\n| none     := is_true (by simp [false_implies_iff])\n| (some a) := if h : p a\n  then is_true $ λ o e, some_inj.1 e ▸ h\n  else is_false $ mt (λ H, H _ rfl) h\n\ninstance decidable_exists_mem {p : α → Prop} [decidable_pred p] :\n  ∀ o : option α, decidable (∃ a ∈ o, p a)\n| none     := is_false (λ ⟨a, ⟨h, _⟩⟩, by cases h)\n| (some a) := if h : p a\n  then is_true $ ⟨_, rfl, h⟩\n  else is_false $ λ ⟨_, ⟨rfl, hn⟩⟩, h hn\n\n/-- Inhabited `get` function. Returns `a` if the input is `some a`, otherwise returns `default`. -/\n@[reducible] def iget [inhabited α] : option α → α\n| (some x) := x\n| none     := default\n\n@[simp] theorem iget_some [inhabited α] {a : α} : (some a).iget = a := rfl\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : α → Prop) [decidable_pred p] (a : α) : option α :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a` and `p a` holds, otherwise `none`. -/\ndef filter (p : α → Prop) [decidable_pred p] (o : option α) : option α :=\no.bind (guard p)\n\n/-- Cast of `option` to `list `. Returns `[a]` if the input is `some a`, and `[]` if it is\n`none`. -/\ndef to_list : option α → list α\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : α} {o : option α} : a ∈ to_list o ↔ a ∈ o :=\nby cases o; simp [to_list, eq_comm]\n\n/-- Two arguments failsafe function. Returns `f a b` if the inputs are `some a` and `some b`, and\n\"does nothing\" otherwise. -/\ndef lift_or_get (f : α → α → α) : option α → option α → option α\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : α → α → α) [h : is_commutative α f] :\n  is_commutative (option α) (lift_or_get f) :=\n⟨λ a b, by cases a; cases b; simp [lift_or_get, h.comm]⟩\n\ninstance lift_or_get_assoc (f : α → α → α) [h : is_associative α f] :\n  is_associative (option α) (lift_or_get f) :=\n⟨λ a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]⟩\n\ninstance lift_or_get_idem (f : α → α → α) [h : is_idempotent α f] :\n  is_idempotent (option α) (lift_or_get f) :=\n⟨λ a, by cases a; simp [lift_or_get, h.idempotent]⟩\n\ninstance lift_or_get_is_left_id (f : α → α → α) :\n  is_left_id (option α) (lift_or_get f) none :=\n⟨λ a, by cases a; simp [lift_or_get]⟩\n\ninstance lift_or_get_is_right_id (f : α → α → α) :\n  is_right_id (option α) (lift_or_get f) none :=\n⟨λ a, by cases a; simp [lift_or_get]⟩\n\n/-- Lifts a relation `α → β → Prop` to a relation `option α → option β → Prop` by just adding\n`none ~ none`. -/\ninductive rel (r : α → β → Prop) : option α → option β → Prop\n/-- If `a ~ b`, then `some a ~ some b` -/\n| some {a b} : r a b → rel (some a) (some b)\n/-- `none ~ none` -/\n| none       : rel none none\n\n/-- Partial bind. If for some `x : option α`, `f : Π (a : α), a ∈ x → option β` is a\n  partial function defined on `a : α` giving an `option β`, where `some a = x`,\n  then `pbind x f h` is essentially the same as `bind x f`\n  but is defined only when all `x = some a`, using the proof to apply `f`. -/\n@[simp] def pbind : Π (x : option α), (Π (a : α), a ∈ x → option β) → option β\n| none     _ := none\n| (some a) f := f a rfl\n\n/-- Partial map. If `f : Π a, p a → β` is a partial function defined on `a : α` satisfying `p`,\nthen `pmap f x h` is essentially the same as `map f x` but is defined only when all members of `x`\nsatisfy `p`, using the proof to apply `f`. -/\n@[simp] def pmap {p : α → Prop} (f : Π (a : α), p a → β) :\n  Π x : option α, (∀ a ∈ x, p a) → option β\n| none     _ := none\n| (some a) H := some (f a (H a (mem_def.mpr rfl)))\n\n/-- Flatten an `option` of `option`, a specialization of `mjoin`. -/\n@[simp] def join : option (option α) → option α :=\nλ x, bind x id\n\nprotected def {u v} traverse {F : Type u → Type v} [applicative F] {α β : Type*} (f : α → F β) :\n  option α → F (option β)\n| none     := pure none\n| (some x) := some <$> f x\n\n/- By analogy with `monad.sequence` in `init/category/combinators.lean`. -/\n\n/-- If you maybe have a monadic computation in a `[monad m]` which produces a term of type `α`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult. -/\ndef {u v} maybe {m : Type u → Type v} [monad m] {α : Type u} : option (m α) → m (option α)\n| none      := return none\n| (some fn) := some <$> fn\n\n/-- Map a monadic function `f : α → m β` over an `o : option α`, maybe producing a result. -/\ndef {u v w} mmap {m : Type u → Type v} [monad m] {α : Type w} {β : Type u} (f : α → m β)\n  (o : option α) : m (option β) := (o.map f).maybe\n\n/-- A monadic analogue of `option.elim`. -/\ndef melim {α β : Type*} {m : Type* → Type*} [monad m] (y : m β) (z : α → m β) (x : m (option α)) :\n  m β :=\nx >>= option.elim y z\n\n/-- A monadic analogue of `option.get_or_else`. -/\ndef mget_or_else {α : Type*} {m : Type* → Type*} [monad m] (x : m (option α)) (y : m α) : m α :=\nmelim y pure x\n\nend option\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/data/option/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984137988772, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.33699138975469195}}
{"text": "/- This file contains the construction of the presheaf of rings on Spec(R) and the proof\n   that it satisfies the sheaf axiom for finite covers by basic open sets.\n\n   We seem to need to explicitly tell Lean that something canonically isomorphic\n   to an exact sequence is an exact sequence.\n\n   We start with Kenny's non-canonical proof that something iso to a three term exact seq is exact.\n\n   We then use my ghastly application coming from localization.\n\n   This file corresponds to the line in tag01HR:\n   \"Thus we may apply Algebra, Lemma 10.22.2 to the module Mf over Rf and the elements g1,…,gn. \n   We conclude that the sequence is exact.\"\n   The issue is that there are a lot of diagrams which commute, but this needs checking.\n\n-/\nimport algebra.group data.set data.equiv.basic -- for comm diag stuff\nimport tag00DY -- definition of standard basis on Spec(R) plus proof it's a basis\nimport tag00E0\nimport tag00EJ -- finite cover by basic opens sheaf axiom\nimport tag009I -- definition of presheaf of types on basis\nimport tag009L -- definition of basis_is_compact \n--import group_theory.submonoid  \nimport ring_theory.localization \nimport Kenny_comm_alg.Zariski \n--import tag00E0 \nimport tag01HS -- for \"lemma_standard_open\" giving map R[1/f] -> R[1/g] from D(g) ⊆ D(f) \n--import tag009I -- presheaf of types on a basis\n--import tag00DY -- claim that D(f) form a basis\n--import tag006N -- presheaves / sheaves of rings on a basis\n--import tag009P -- presheaf of rings on a basis\n--import tag009L -- sheaf for finite covers on basis -> sheaf for basis\nimport tag00EJ \nimport linear_algebra -- for span \n\nlocal attribute [instance] classical.prop_decidable\n\nuniverses u0 u v w\n\ndefinition Pi_lift_map₁ {γ : Type u} {F : γ → Type u} {G : γ → Type u} \n  (H : ∀ i : γ, F i → G i) : (Π i, F i) → Π i, G i := λ Fi i, H i (Fi i)\n\ndefinition Pi_lift_map₂ {γ : Type u} {X : Type u} {G : γ → Type u} \n  (H : ∀ i : γ, X → G i) : X → Π i, G i := λ x i, H i x\n\nnamespace is_add_group_hom\n\nvariables {α : Type*} {β : Type*} [add_group α] [add_group β] (f : α → β) [hf : is_add_group_hom f]\n\ninstance ring_hom_is_add_group_hom {α β : Type u} [ring α] [ring β] (f : α → β) [is_ring_hom f] : is_add_group_hom f :=\n⟨λ _ _, is_ring_hom.map_add f⟩\n\ndef ker : set α :=\n{ x | f x = 0 }\n\ninstance Pi_lift {γ : Type u} {F : γ → Type u} {G : γ → Type u} [∀ i, add_group (F i)]\n[∀ i, add_group (G i)] (H : ∀ i : γ, F i → G i) [∀ i, is_add_group_hom (H i)] :\n is_add_group_hom (Pi_lift_map₁ H) := ⟨λ a b, funext $ λ i,\nshow H i ((a i) + (b i)) = H i (a i) + H i (b i),\nby rw (add (H i))⟩\n\ninstance equiv.Pi_congr_right {γ : Type u} {F : γ → Type u} {G : γ → Type u} [∀ i, add_group (F i)]\n[∀ i, add_group (G i)] (H : ∀ i : γ, F i ≃ G i) [∀ i, is_add_group_hom (H i)] :\n is_add_group_hom (equiv.Pi_congr_right H) := \n-- is_add_group_hom.Pi_lift H\n is_add_group_hom.Pi_lift (λ i, H i)\n\nend is_add_group_hom\n\nnamespace equiv\n\ntheorem to_fun.injective {α β : Type u} (e : α ≃ β) : function.injective e := \nfunction.injective_of_has_left_inverse ⟨e.symm,e.inverse_apply_apply⟩\n\nend equiv\n\n--class is_alg_hom (R : Type u) [ring R] {A B : Type u} [ring A] [ring B] (sA : R → A) (sB : R → B)\n--[is_ring_hom sA] [is_ring_hom sB] (f : A → B) extends is_ring_hom f : Prop :=\n--(commutes : ∀ r, f (sA r) = sB r)\n\nnamespace is_ring_hom\n\ninstance Pi_lift_map₁ {γ : Type u} {F : γ → Type u} {G : γ → Type u} [∀ i, ring (F i)]\n[∀ i, ring (G i)] (H : ∀ i : γ, F i → G i) [H' : ∀ i, is_ring_hom (H i)] :\n is_ring_hom (Pi_lift_map₁ H) := \n{ map_add := λ a b, funext $ λ x,begin \n    show H x (a x + b x) = H x (a x) + H x (b x), \n    rw (H' x).map_add,\n    end,\n  map_mul := λ a b, funext $ λ x,begin \n    show H x (a x * b x) = H x (a x) * H x (b x), \n    rw (H' x).map_mul,\n    end,\n  map_one := funext $ λ x,begin \n    show H x 1 = 1,\n    exact (H' x).map_one\n  end\n}\n\ninstance equiv.Pi_congr_right {γ : Type u} {F : γ → Type u} {G : γ → Type u} [∀ i, ring (F i)]\n[∀ i, ring (G i)] (H : ∀ i : γ, F i ≃ G i) [H' : ∀ i, is_ring_hom (H i)] :\n is_ring_hom (equiv.Pi_congr_right H) := is_ring_hom.Pi_lift_map₁ (λ i, H i)\n\nend is_ring_hom \n\n-- If A -> B -> C is isomorphic to A' -> B' -> C' and first sequence is exact then second is.\n-- But I don't ever use this now.\n/-\ntheorem three (A B C A' B' C' : Type*)\n  [add_comm_group A] [add_comm_group A']\n  [add_comm_group B] [add_comm_group B']\n  [add_comm_group C] [add_comm_group C']\n  (ab : A → B) [is_add_group_hom ab]\n  (bc : B → C) [is_add_group_hom bc]\n  (Habc : set.range ab = is_add_group_hom.ker bc)\n  (fa : A ≃ A') [is_add_group_hom fa]\n  (fb : B ≃ B') [is_add_group_hom fb]\n  (fc : C ≃ C') [is_add_group_hom fc]\n\n  (ab' : A' → B') [is_add_group_hom ab']\n  (bc' : B' → C') [is_add_group_hom bc']\n  (H1 : fb ∘ ab = ab' ∘ fa)\n  (H2 : fc ∘ bc = bc' ∘ fb) :\n\n  set.range ab' = is_add_group_hom.ker bc' :=\nbegin\n  apply set.ext,\n  intro b',\n  split,\n  { intro hb',\n    cases hb' with a' ha',\n    simp [is_add_group_hom.ker],\n    let a := fa.symm a',\n    have ha : fa a = a',\n    { simp [a] },\n    rw [← ha', ← ha],\n    change bc' ((ab' ∘ fa) a) = 0,\n    rw ← H1,\n    change (bc' ∘ fb) (ab a) = 0,\n    rw ← H2,\n    have H3 : ab a ∈ is_add_group_hom.ker bc,\n    { rw ← Habc, existsi a, simp },\n    simp [is_add_group_hom.ker] at H3 ⊢,\n    rw H3,\n    apply is_add_group_hom.zero },\n  { intro hb',\n    let b := fb.symm b',\n    have hb : fb b = b',\n    { simp [b] },\n    simp [is_add_group_hom.ker] at hb',\n    rw ← hb at hb',\n    change (bc' ∘ fb) b = 0 at hb',\n    rw ← H2 at hb',\n    rw ← is_add_group_hom.zero fc at hb',\n    replace hb' := congr_arg fc.symm hb',\n    simp at hb',\n    have H3 : b ∈ set.range ab,\n    { rwa Habc },\n    cases H3 with a ha,\n    existsi fa a,\n    change (ab' ∘ fa) a = b',\n    rw ← H1,\n    simp [ha] }\nend\n\n-- Thanks Kenny.\n-/\n\n\n-- Now start moving stuff from tag01HR.lean which has got really\n-- unwieldy. Here I am going to prove the sheaf on a finite basis\n-- statement from the ring theory lemma tag00EJ\n\n-- First here's the choice-free definition of the structure presheaf\n\ndef is_zariski.standard_open {R : Type u} [comm_ring R] (U : set (X R)) := ∃ f : R, U = Spec.D'(f)\n\ndef non_zero_on_U {R : Type u} [comm_ring R] (U : set (X R)) : set R := {g : R | U ⊆ Spec.D'(g)}\n\ninstance nonzero_on_U_is_mult_set {R : Type u} [comm_ring R] (U : set (X R)) : is_submonoid (non_zero_on_U U) := \n{ one_mem := λ P HP, @is_proper_ideal.one_not_mem _ _ P.1 P.2.1,\n  mul_mem := begin\n    intros f g Hf Hg,\n    show U ⊆ Spec.D'(f*g),\n    rw [tag00E0.lemma15 R f g],\n    exact set.subset_inter Hf Hg\n    end\n}\n\nlemma nonzero_on_U_mono {R : Type u} [comm_ring R] {U V : set (X R)} : V ⊆ U → non_zero_on_U U ⊆ non_zero_on_U V :=\nλ H _,set.subset.trans H\n\ndef zariski.structure_presheaf_on_standard {R : Type u} [comm_ring R] (U : set (X R)) (H : is_zariski.standard_open U) : Type u := \n  @localization.loc R _ (non_zero_on_U U) (nonzero_on_U_is_mult_set U)\n\ninstance zariski.structure_presheaf_on_standard.comm_ring {R : Type u} [comm_ring R] (U : set (X R)) (H : is_zariski.standard_open U) :\ncomm_ring (zariski.structure_presheaf_on_standard U H) :=\n@localization.comm_ring _ _ _ (nonzero_on_U_is_mult_set U)\n\n-- A key notion for us is that of a canonical R-algebra isomorphism.\n-- Because I still don't quite understand what makes an isomorphism\n-- canonical let's just start with the notion of an R-algebra isomorphism.\n\nstructure R_alg_equiv {R : Type u} {α : Type v} {β : Type w} [comm_ring R] [comm_ring α] [comm_ring β]\n  (sα : R → α) (sβ : R → β) extends ring_equiv α β :=\n(R_alg_hom : sβ = to_fun ∘ sα)\n\ninstance R_alg_equiv_coe_to_fun {R : Type u} {α : Type v} {β : Type w} [comm_ring R] [comm_ring α] [comm_ring β]\n  (sα : R → α) (sβ : R → β) : has_coe_to_fun (R_alg_equiv sα sβ) := ⟨_,λ h,h.to_fun⟩\n\ninstance R_alg_equiv_is_ring_hom {R : Type u} {α : Type v} {β : Type w} [comm_ring R] [comm_ring α] [comm_ring β]\n  (sα : R → α) (sβ : R → β) (H : R_alg_equiv sα sβ) : is_ring_hom H := H.is_ring_hom \n\nnamespace R_alg_equiv\n\nlemma symm {R : Type u} {α : Type v} {β : Type w} \n  [comm_ring R] [comm_ring α] [comm_ring β]\n  {sα : R → α} {sβ : R → β} :\n  R_alg_equiv sα sβ → R_alg_equiv sβ sα :=\n  λ H, { R_alg_hom := begin\n         let f := H.to_fun,\n         have H2 : sβ = f ∘ sα := H.R_alg_hom,\n         let g := H.inv_fun,\n         show sα = g ∘ sβ,\n         rw H2,\n         funext,\n         show _ = g (f (sα x)),\n         conv begin\n           to_lhs,\n           rw ←H.left_inv (sα x),\n         end,\n       end,\n    ..(H.to_ring_equiv).symm\n  }\n\n/- we have his for ring_equiv and I've never implemented this version here\n  lemma inv_fun_is_ring_hom {R : Type u} {α : Type v} {β : Type w} \n  [comm_ring R] [comm_ring α] [comm_ring β]\n  {sα : R → α} {sβ : R → β} (H : R_alg_equiv sα sβ) : is_ring_hom H.inv_fun := sorry \n-/\nlemma trans {R : Type u0} {α : Type u} {β : Type v} {γ : Type w}\n  [comm_ring R] [comm_ring α] [comm_ring β] [comm_ring γ]\n  {sα : R → α} {sβ : R → β} {sγ : R → γ} :\n  R_alg_equiv sα sβ → R_alg_equiv sβ sγ → R_alg_equiv sα sγ :=\nλ H1 H2,\n{ R_alg_hom := begin\n     show sγ = H2.to_fun ∘ (H1.to_fun ∘ sα),\n     funext,\n     have H3 : sγ = H2.to_fun ∘ sβ := H2.R_alg_hom,\n     have H4 : sβ = H1.to_fun ∘ sα := H1.R_alg_hom,\n     conv begin\n       to_lhs,\n       rw H3,\n       change H2.to_fun (sβ x),\n     end,\n     conv in (sβ x) begin\n       rw H4,\n     end,\n  end,\n  ..(ring_equiv.trans H1.to_ring_equiv H2.to_ring_equiv)\n}\n\n-- TODO\n-- I need to now give easy access to \"inv_fun is ring hom\".\n\nopen localization \n\ndefinition of_unique_homs {R : Type u} {α : Type v} {β : Type w} [comm_ring R] [comm_ring α] [comm_ring β]\n  {sα : R → α} {sβ : R → β} {f : α → β} {g : β → α} {hα : α → α} {hβ : β → β}\n  [is_ring_hom sα] [is_ring_hom sβ] [H : is_ring_hom f] [is_ring_hom g] [is_ring_hom hα] [is_ring_hom hβ] : \nis_unique_R_alg_hom sα sβ f → is_unique_R_alg_hom sβ sα g → is_unique_R_alg_hom sα sα hα → is_unique_R_alg_hom sβ sβ hβ\n  → R_alg_equiv sα sβ := λ Hαβ Hβα Hαα Hββ,\n{ to_fun := f,\n  inv_fun := g,\n  left_inv := λ x, begin\n    have Hα : id = hα,\n      exact Hαα.is_unique id rfl,\n    show (g ∘ f) x = x,\n    rw [comp_unique sα sβ sα f g hα Hαβ Hβα Hαα,←Hα],\n    refl\n  end,\n  right_inv := λ x, begin\n    have Hβ : id = hβ,\n      exact Hββ.is_unique id rfl,\n    show (f ∘ g) x = x,\n    rw [comp_unique sβ sα sβ g f hβ Hβα Hαβ Hββ,←Hβ],\n    refl\n  end,  \n  is_ring_hom := H,\n  R_alg_hom := show sβ = f ∘ sα, from Hαβ.R_alg_hom\n}\n\nend R_alg_equiv -- namespace\n\n\n-- The proof below (is_loc) could be simpler: a lot of the definitions would follow from\n-- universal properties, but Kenny just proved them directly anyway.\n-- It's the proof that if U=D(f) and S=S(U) is the functions which are non-vanishing\n-- on U then R[1/S]=R[1/f] as R-algebras.\n\nopen localization \n\nnoncomputable definition zariski.structure_presheaf_on_standard_is_loc {R : Type u} [comm_ring R] (f : R) :\n  R_alg_equiv (of_comm_ring R _ : R → zariski.structure_presheaf_on_standard (Spec.D'(f)) (⟨f,rfl⟩))\n    (of_comm_ring R (powers f) : R → away f) :=  \n{ to_fun      := extend_map_of_im_unit\n    (of_comm_ring R (powers f))\n    (λ s hs, lemma_standard_open_1a R _ _ hs),\n  is_ring_hom := extend_map_of_im_unit.is_ring_hom _ _,\n  inv_fun     := away.extend_map_of_im_unit\n    (of_comm_ring R _)\n    ⟨mk _ _ ⟨1, f, set.subset.refl _⟩,\n     quotient.sound ⟨1, is_submonoid.one_mem _, by simp⟩⟩,\n  left_inv    := @unique _ _ _ _ _ _ _ _ \n    (@@is_ring_hom.comp _ _ _\n       (extend_map_of_im_unit.is_ring_hom _ _) _ _\n       (extend_map_of_im_unit.is_ring_hom _ _))\n    (ring_equiv.refl _).is_ring_hom\n    (by intro x; dsimp [ring_equiv.refl, equiv.refl]; rw [extend_map_extends, extend_map_extends]),\n  right_inv   := @localization.unique _ _ _ _ _ _ _ _\n    (@@is_ring_hom.comp _ _ _\n       (extend_map_of_im_unit.is_ring_hom _ _) _ _\n       (extend_map_of_im_unit.is_ring_hom _ _))\n    (ring_equiv.refl _).is_ring_hom\n    (by intro x; dsimp [ring_equiv.refl, equiv.refl]; rw [localization.extend_map_extends, localization.extend_map_extends]),\n  R_alg_hom := (funext (λ r, (localization.extend_map_extends\n     (_ : R → localization.loc R (powers f))\n     _ r \n       ).symm)\n  : localization.of_comm_ring R (powers f) =\n    localization.extend_map_of_im_unit (localization.of_comm_ring R (powers f)) _ ∘\n      localization.of_comm_ring R (non_zero_on_U (Spec.D' f)))\n}\n\n-- we now need some universal properties coming from localization.\n\nset_option class.instance_max_depth 52 -- !!\n\n/-- universal property of inverting one element and then another -/\ntheorem away_away_universal_property {R : Type u} [comm_ring R] (f : R)\n(g : loc R (powers f)) {γ : Type v} [comm_ring γ] (sγ : R → γ) [is_ring_hom sγ] (Hf : is_unit (sγ f))\n(Hg : is_unit (away.extend_map_of_im_unit sγ Hf g)) :\nis_unique_R_alg_hom \n  ((of_comm_ring (away f) (powers g)) ∘ (of_comm_ring R (powers f))) \n  sγ\n  (away.extend_map_of_im_unit (away.extend_map_of_im_unit sγ Hf) Hg) := \nbegin\n  let α := loc R (powers f),\n  let β := loc α (powers g),\n  let sα := of_comm_ring R (powers f),\n  let fαβ := of_comm_ring α (powers g),\n  let sβ := fαβ ∘ sα,\n  let fαγ := away.extend_map_of_im_unit sγ Hf,\n  let fβγ := away.extend_map_of_im_unit fαγ Hg,\n  have HUαγ : is_unique_R_alg_hom sα sγ fαγ := away_universal_property f sγ Hf,\n  have HUβγ : is_unique_R_alg_hom fαβ fαγ fβγ := away_universal_property g fαγ Hg,\n  have Hαγ : sγ = fαγ ∘ sα := HUαγ.R_alg_hom,\n  have Hβγ : fαγ = fβγ ∘ fαβ := HUβγ.R_alg_hom,\n  -- it's the next line which needs class.instance_max_depth 52\n  have Htemp : is_unique_R_alg_hom sα sγ fαγ ↔ is_unique_R_alg_hom sα (fβγ ∘ fαβ ∘ sα) (fβγ ∘ fαβ),\n    simp [Hαγ,Hβγ],\n  have Htemp2 : is_unique_R_alg_hom fαβ fαγ fβγ ↔ is_unique_R_alg_hom fαβ (fβγ ∘ fαβ) fβγ,\n    simp [Hβγ],\n  have H : is_unique_R_alg_hom (fαβ ∘ sα) (fβγ ∘ fαβ ∘ sα) fβγ := unique_R_of_unique_R_of_unique_Rloc sα fαβ fβγ (Htemp.1 HUαγ) (Htemp2.1 HUβγ),\n  have Htemp3 : is_unique_R_alg_hom (fαβ ∘ sα) (fβγ ∘ fαβ ∘ sα) fβγ ↔ is_unique_R_alg_hom (fαβ ∘ sα) sγ fβγ,\n    simp [Hαγ,Hβγ],\n  exact Htemp3.1 H\nend \n\n/-- universal property of inverting two elements of R one by one -/\ntheorem away_away_universal_property' {R : Type u} [comm_ring R] (f g : R)\n{γ : Type v} [comm_ring γ] (sγ : R → γ) [is_ring_hom sγ] (Hf : is_unit (sγ f))\n(Hg : is_unit (sγ g)) :\nis_unique_R_alg_hom\n  ((of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) ∘ (of_comm_ring R (powers f))) \n  sγ\n  (away.extend_map_of_im_unit (away.extend_map_of_im_unit sγ Hf) (begin rwa away.extend_map_extends end)) :=\naway_away_universal_property f (of_comm_ring R (powers f) g) sγ Hf (begin rwa away.extend_map_extends end)\n\n-- To get Chris' lemma to apply to covers of D(f) (rather than a cover of R)\n-- we need R[1/f][1/g] = R[1/g] if D(g) ⊆ D(f), so let's prove this from the\n-- universal property.\n\nnoncomputable definition localization.loc_loc_is_loc {R : Type u} [comm_ring R] {f g : R} (H : Spec.D' g ⊆ Spec.D' f) :\n  let sα := (of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) ∘ (of_comm_ring R (powers f)) in\n  let sβ := of_comm_ring R (powers g) in\nR_alg_equiv sα sβ := \nbegin\n  have Htemp : is_ring_hom (of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) := by apply_instance,\n  letI := Htemp,\n  let sα : R → loc (away f) (powers (of_comm_ring R (powers f) g)) := \n    (of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) ∘ (of_comm_ring R (powers f)),\n  let sβ := of_comm_ring R (powers g),\n  have HUfα : is_unit (sα f),\n    show is_unit ((of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) ((of_comm_ring R (powers f)) f)),\n    exact im_unit_of_unit (of_comm_ring (away f) (powers (of_comm_ring R (powers f) g))) (unit_of_in_S (away.in_powers f)),\n  have HUfβ : is_unit (sβ f) := lemma_standard_open_1a R f g H,\n  have HUgα : is_unit (sα g) := unit_of_in_S (away.in_powers (of_comm_ring R (powers f) g)),\n  have HUgβ : is_unit (sβ g) := unit_of_in_S (away.in_powers g),\n  exact R_alg_equiv.of_unique_homs \n    (away_away_universal_property' f g sβ HUfβ HUgβ)\n    (away_universal_property g sα HUgα : is_unique_R_alg_hom sβ sα _)\n    (away_away_universal_property' f g sα HUfα HUgα)\n    (away_universal_property g sβ HUgβ)\nend\n\n-- Now I need the following technical result:\n-- If V = D(g) ⊆ U = D(f) in Spec(R)\n-- then structure sheaf evaluated on V (R[1/S(V)]) is R-isomorphic to R[1/f][1/gbar]\n-- with gbar = image of g. The map is from R[1/S(V)] to R[1/f][1/gbar]\n-- The proof goes via R[1/g]\n-- It is also true that the R-isomorphism is the unique R-homomorphism\n-- between these rings -- see the lemma after this.\nnoncomputable definition canonical_iso {R : Type u} [comm_ring R] {f g : R} (H : Spec.D' g ⊆ Spec.D' f) :\nlet gbar := of_comm_ring R (powers f) g in\nlet sα : R → loc (away f) (powers gbar) :=\n  of_comm_ring (away f) (powers gbar) ∘ of_comm_ring R (powers f) in\nlet sγ := (of_comm_ring R (non_zero_on_U (Spec.D' g))) in\nR_alg_equiv sγ sα :=\nR_alg_equiv.trans\n  (zariski.structure_presheaf_on_standard_is_loc g )\n  (R_alg_equiv.symm (localization.loc_loc_is_loc H))\n\n-- and we also need that the canonical iso is the unique R-alg hom.\n-- From R[1/S(V)] to R[1/f][1/gbar] the result is here:\n\n-- NB Chris suggests I remove the first three let's.\ntheorem canonical_iso_is_canonical_hom₁ {R : Type u} [comm_ring R] {f g : R} (H : Spec.D' g ⊆ Spec.D' f) :\nlet gbar := of_comm_ring R (powers f) g in\nlet sα : R → loc (away f) (powers gbar) :=\n  of_comm_ring (away f) (powers gbar) ∘ of_comm_ring R (powers f) in\nlet sγ := (of_comm_ring R (non_zero_on_U (Spec.D' g))) in\nlet H3 : is_ring_hom sα := by apply_instance in\nlet H2 := (canonical_iso H).is_ring_hom in\nlet H4 : is_ring_hom sγ := by apply_instance in\n@is_unique_R_alg_hom _ _ _ _ _ _ sγ sα (canonical_iso H).to_fun H4 H3 H2 := \nbegin\nletI := (canonical_iso H).is_ring_hom,\nhave H5 := unique_R_alg_from_loc (canonical_iso H).to_fun,\nhave H6 := (canonical_iso H).R_alg_hom.symm,\nsimp [H6] at H5,\nexact H5,\nend \n\n-- now let's go the other way -- but do I need this?\n/-\ntheorem canonical_iso_is_canonical_hom₂ {R : Type u} [comm_ring R] {f g : R} (H : Spec.D' g ⊆ Spec.D' f) :\nlet gbar := of_comm_ring R (powers f) g in\nlet sα : R → loc (away f) (powers gbar) :=\n  of_comm_ring (away f) (powers gbar) ∘ of_comm_ring R (powers f) in\nlet sγ := (of_comm_ring R (non_zero_on_U (Spec.D' g))) in\nlet H3 : is_ring_hom sα := by apply_instance in\nlet H2 : is_ring_hom (canonical_iso H).inv_fun := ring_equiv.inv_fun_is_ring_hom (canonical_iso H).to_ring_equiv in\nlet H4 : is_ring_hom sγ := by apply_instance in\n@is_unique_R_alg_hom _ _ _ _ _ _ sα sγ (canonical_iso H).inv_fun H3 H4 H2 := sorry\n-/\n\n-- Chris has proved that 0-> A -> sum A_{g_i} -> sum A_{g_i g_j} is exact\n-- if Spec(A) is covered by D(g_i) (a finite cover).\n\n-- I want to prove that if U = D(f) is open and covered by U_i = D(g_i)\n-- and if A = R[1/S(U)] then\n-- 0 -> A -> sum R[1/S(U_i)] -> sum R[1/S(U_i intersect U_j)] is exact\n\n-- Let's formulate exactly what we need.\n\ntheorem fourexact_from_iso_to_fourexact {A B C A' B' C' : Type u}\n  [add_comm_group A] [add_comm_group A']\n  [add_comm_group B] [add_comm_group B']\n  [add_comm_group C] [add_comm_group C']\n  (ab : A → B) [is_add_group_hom ab]\n  (bc : B → C) [is_add_group_hom bc]\n--  (Habc : set.range ab = is_add_group_hom.ker bc)\n  (H4exact : ∀ b : B, bc b = 0 → ∃! a : A, ab a = b)\n  (fa : A ≃ A') [is_add_group_hom fa]\n  (fb : B ≃ B') [is_add_group_hom fb]\n  (fc : C ≃ C') [is_add_group_hom fc]\n\n  (ab' : A' → B') [is_add_group_hom ab']\n  (bc' : B' → C') [is_add_group_hom bc']\n  (H1 : ∀ a, fb (ab a) = ab' (fa a))\n  (H2 : ∀ b, fc (bc b) = bc' (fb b))\n  : ∀ b' : B', (bc' b' = 0 → ∃! a' : A', ab' a' = b') := \nbegin\n  intros b' Hb',\n  let b := fb.symm b',\n    have H3 : fc (bc b) = 0,\n    rw H2 b,\n    simp [Hb'],\n  have Hb : bc b = 0,\n    have H4 : fc.symm (fc (bc b)) = 0,\n      rw H3,\n      have H5 : fc (0 : C) = 0 := is_add_group_hom.zero fc,\n      rw ←H5,\n      simp,\n    simp only [equiv.inverse_apply_apply fc] at H4,\n    exact H4,\n  cases H4exact b Hb with a Ha,\n  existsi (fa a),\n  split,\n  { show ab' (fa a) = b',\n    rw ←H1,\n    rw Ha.1,\n    exact equiv.apply_inverse_apply fb b'\n  },\n  intros a₁' Ha₁',\n  have H4 := Ha.2 (fa.symm a₁'),\n  suffices H5 : ab (fa.symm a₁') = b,\n    rw ←(H4 H5),simp,\n  apply equiv.to_fun.injective fb,\n  rw H1,\n  simp [Ha₁'],\nend \n\n", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/src/canonical_isomorphism_nonsense.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5544704649604274, "lm_q1q2_score": 0.33693128819574136}}
{"text": "import tactic --hide\n\n/-Lemma\nA nested chain ofimplications.\n-/\nlemma lemma_7 (P Q: Prop) : (P → Q) → ((P → Q) → P) → Q :=\nbegin\n  intros hPQ hPQP,\n  apply hPQ,\n  apply hPQP,\n  exact hPQ,\n\n\n\nend", "meta": {"author": "CBirkbeck", "repo": "logic_projic", "sha": "0b029af0fbfc0ac6eafae47401d5bbf8e641d7d2", "save_path": "github-repos/lean/CBirkbeck-logic_projic", "path": "github-repos/lean/CBirkbeck-logic_projic/logic_projic-0b029af0fbfc0ac6eafae47401d5bbf8e641d7d2/src/logic_1/logic9.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3369312803165705}}
{"text": "import computability.partrec\nimport computability.partrec_code\nimport data.pfun\nimport tactic\nimport rpartrec\nimport rcomputability_tactic\n\nopen encodable denumerable part\n\n@[simp] lemma eval_eval_opt {α β} (f : α →. β) [D : decidable_pred f.dom] {x} :\n  @pfun.eval_opt _ _ f D x = @part.to_option _ (f x) (D x) := rfl\n\n@[simp] theorem mem_to_option' {α} {o : part α} [decidable o.dom] {a : α} :\n  to_option o = some a ↔ a ∈ o := mem_to_option\n\ntheorem mem_to_option_none {α} {o : part α} [D : decidable o.dom] {a : α} :\n  to_option o = none ↔ o = none :=\nby { unfold to_option, by_cases h : o.dom; simp [h],\n     { exact λ h0, part.eq_none_iff'.mp h0 h },\n     { exact part.eq_none_iff'.mpr h } }\n\nnamespace nat.rpartrec\n\ninductive code : Type\n| oracle : code\n| zero   : code\n| succ   : code\n| left   : code\n| right  : code\n| pair   : code → code → code\n| comp   : code → code → code\n| prec   : code → code → code\n| rfind' : code → code\n\nend nat.rpartrec\n\nnamespace nat.rpartrec.code\nopen nat (mkpair unpair) nat.rpartrec (code)\n\nprotected def const : ℕ → code\n| 0     := zero\n| (n+1) := comp succ (const n)\n\nprotected def id : code := pair left right\n\ndef curry (c : code) (n : ℕ) : code :=\ncomp c (pair (code.const n) code.id)\n\ndef ctrans (c : code) : code → code\n| oracle       := c\n| zero         := zero\n| succ         := succ\n| left         := left\n| right        := right\n| (pair cf cg) := pair (ctrans cf) (ctrans cg)\n| (comp cf cg) := comp (ctrans cf) (ctrans cg)\n| (prec cf cg) := prec (ctrans cf) (ctrans cg)\n| (rfind' cf)  := rfind' (ctrans cf)\n\ndef encode_code : code → ℕ\n| oracle       := 0\n| zero         := 1\n| succ         := 2\n| left         := 3\n| right        := 4\n| (pair cf cg) := (bit0 $ bit0 (mkpair (encode_code cf) (encode_code cg))) + 5\n| (comp cf cg) := (bit0 $ bit1 (mkpair (encode_code cf) (encode_code cg))) + 5\n| (prec cf cg) := (bit1 $ bit0 (mkpair (encode_code cf) (encode_code cg))) + 5\n| (rfind' cf)  := (bit1 $ bit1 (encode_code cf)) + 5\n\ndef of_nat_code : ℕ → code\n| 0 := oracle\n| 1 := zero\n| 2 := succ\n| 3 := left\n| 4 := right\n| (e+5) :=\n  have div8 : e.div2.div2 ≤ e :=\n    by { simp[nat.div2_val], exact le_trans (nat.div_le_self (e/2) 2) (nat.div_le_self e 2) },\n  have e.div2.div2 < e + 5 := nat.lt_succ_iff.mpr (le_trans div8 (nat.le.intro rfl)),\n  have e.div2.div2.unpair.1 < e + 5 := nat.lt_succ_iff.mpr\n    (le_trans (le_trans (nat.unpair_left_le _) div8) (nat.le.intro rfl)),\n  have e.div2.div2.unpair.2 < e + 5 := nat.lt_succ_iff.mpr\n    (le_trans (le_trans (nat.unpair_right_le _) div8) (nat.le.intro rfl)),\n  match e.bodd, e.div2.bodd with\n  | ff, ff := pair (of_nat_code e.div2.div2.unpair.1)\n    (of_nat_code e.div2.div2.unpair.2)\n  | ff, tt := comp (of_nat_code e.div2.div2.unpair.1)\n    (of_nat_code e.div2.div2.unpair.2)\n  | tt, ff := prec (of_nat_code e.div2.div2.unpair.1)\n    (of_nat_code e.div2.div2.unpair.2)\n  | tt, tt := rfind' (of_nat_code e.div2.div2)\n  end\n\nprivate lemma encode_of_nat_code : ∀ e, encode_code (of_nat_code e) = e\n| 0     := by simp[of_nat_code, encode_code]\n| 1     := by simp[of_nat_code, encode_code]\n| 2     := by simp[of_nat_code, encode_code]\n| 3     := by simp[of_nat_code, encode_code]\n| 4     := by simp[of_nat_code, encode_code]\n| (e+5) :=\nhave div8 : e.div2.div2 ≤ e :=\n    by { simp[nat.div2_val], \n         exact le_trans (nat.div_le_self (e/2) 2) (nat.div_le_self e 2) },\n  have e.div2.div2 < e + 5 := nat.lt_succ_iff.mpr (le_trans div8 (nat.le.intro rfl)),\n  have e.div2.div2.unpair.1 < e + 5 := nat.lt_succ_iff.mpr\n    (le_trans (le_trans (nat.unpair_left_le _) div8) (nat.le.intro rfl)),\n  have e.div2.div2.unpair.2 < e + 5 := nat.lt_succ_iff.mpr\n    (le_trans (le_trans (nat.unpair_right_le _) div8) (nat.le.intro rfl)),\n  have IH : _ := encode_of_nat_code e.div2.div2,\n  have IH1 : _ := encode_of_nat_code e.div2.div2.unpair.1,\n  have IH2 : _ := encode_of_nat_code e.div2.div2.unpair.2,\n  begin\n    suffices :\n      (of_nat_code (e + 5)).encode_code =\n      nat.bit e.bodd (nat.bit e.div2.bodd e.div2.div2) + 5,\n    { simp[nat.bit_decomp, this] },\n    simp [encode_code, of_nat_code],\n    cases e.bodd; cases e.div2.bodd;\n    simp [encode_code, of_nat_code, nat.bit, IH, IH1, IH2],\n  end\n\nprivate lemma of_nat_code_encode : ∀ c, of_nat_code (encode_code c) = c\n| oracle       := by simp[of_nat_code, encode_code]\n| zero         := by simp[of_nat_code, encode_code]\n| succ         := by simp[of_nat_code, encode_code]\n| left         := by simp[of_nat_code, encode_code]\n| right        := by simp[of_nat_code, encode_code]\n| (pair cf cg) := by { simp[encode_code, of_nat_code],\n    exact ⟨of_nat_code_encode cf, of_nat_code_encode cg⟩ }\n| (comp cf cg) := by { simp[encode_code, of_nat_code],\n    exact ⟨of_nat_code_encode cf, of_nat_code_encode cg⟩ }\n| (prec cf cg) := by { simp[encode_code, of_nat_code],\n    exact ⟨of_nat_code_encode cf, of_nat_code_encode cg⟩ }\n| (rfind' cf)  := by { simp[encode_code, of_nat_code],\n    exact of_nat_code_encode cf }\n\ninstance : denumerable code := mk' ⟨encode_code, of_nat_code, of_nat_code_encode, encode_of_nat_code⟩\n\n\n\ndef evaln : ℕ → (ℕ → option ℕ) → code → ℕ → option ℕ\n| 0     f _            := λ _, none\n| (s+1) f oracle       := λ n, guard (n ≤ s) >> f n\n| (s+1) f zero         := λ n, guard (n ≤ s) >> pure 0\n| (s+1) f succ         := λ n, guard (n ≤ s) >> pure (nat.succ n)\n| (s+1) f left         := λ n, guard (n ≤ s) >> pure n.unpair.1\n| (s+1) f right        := λ n, guard (n ≤ s) >> pure n.unpair.2\n| (s+1) f (pair cf cg) := λ n, guard (n ≤ s) >>\n    mkpair <$> evaln (s+1) f cf n <*> evaln (s+1) f cg n\n| (s+1) f (comp cf cg) := λ n, guard (n ≤ s) >>\n    do x ← evaln (s+1) f cg n, evaln (s+1) f cf x\n| (s+1) f (prec cf cg) := λ n, guard (n ≤ s) >>\n    n.unpaired (λ a n,\n    n.cases (evaln (s+1) f cf a) $ λ y, do\n    i ← evaln s f (prec cf cg) (mkpair a y),\n    evaln (s+1) f cg (mkpair a (mkpair y i)))\n| (s+1) f (rfind' cf)  := λ n, guard (n ≤ s) >>\n    n.unpaired (λ a m, do\n    x ← evaln (s+1) f cf (mkpair a m),\n    if x = 0 then pure m else\n    evaln s f (rfind' cf) (mkpair a (m+1)))\n-- evaln s f c x = { c }ᶠₛ (x)\n\n@[simp] theorem evaln_0 {f c n} : evaln 0 f c n = none :=\nby induction c; simp [evaln]\n\ntheorem evaln_inclusion : ∀ {s c} {f g : ℕ → option ℕ},\n  (∀ x y, x < s → f x = some y → g x = some y) → \n  ∀ {x y}, evaln s f c x = some y →  evaln s g c x = some y\n| 0 c _ _ := by simp[evaln]\n| (s+1) oracle f g := λ h n y, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega,\n         cases this; simp[(>>), this], exact h _ _ (nat.lt_succ_iff.mpr this),\n         intros _ c, exfalso, exact option.not_mem_none _ c }\n| (s+1) zero         f g := by simp[evaln]\n| (s+1) succ         f g := by simp[evaln]\n| (s+1) left         f g := by simp[evaln]\n| (s+1) right        f g := by simp[evaln]\n| (s+1) (pair cf cg) f g := λ h n y,\n    have IH₀ : _ := @evaln_inclusion (s+1) cf _ _ h,\n    have IH₁ : _ := @evaln_inclusion (s+1) cg _ _ h,\n    by { simp[evaln, (>>)], assume e hf,\n         have : ∃ y, evaln (s + 1) f cf n = some y, \n         { cases evaln (s + 1) f cf n; simp, simp[(<*>), hf] at hf ⊢, exact hf },\n         rcases this with ⟨y0, hy0⟩, have IH₀' := IH₀ hy0,\n          have : ∃ y, evaln (s + 1) f cg n = some y, \n         { cases evaln (s + 1) f cg n; simp, simp[(<*>), hf] at hf ⊢, exact hf },\n         rcases this with ⟨y1, hy1⟩, have IH₁' := IH₁ hy1,\n         simp[hy0, hy1, IH₀', IH₁'] at hf ⊢, exact ⟨e, hf⟩ }\n| (s+1) (comp cf cg) f g := λ h n y,\n    have IH₀ : _ := @evaln_inclusion (s+1) cf _ _ h,\n    have IH₁ : _ := @evaln_inclusion (s+1) cg _ _ h,\n    by { simp[evaln, (>>)], assume e z hz hy,\n         refine ⟨e, z, IH₁ hz, IH₀ hy⟩ }\n| (s+1) (prec cf cg) f g := λ h n y,\n    have l0 : ∀ x y, x < s → f x = some y → g x = some y :=\n      λ x y e, h x y (nat.lt.step e),\n    have IH₀ : _ := @evaln_inclusion s (cf.prec cg) _ _ l0,\n    have IH₁ : _ := @evaln_inclusion (s+1) cf _ _ h,\n    have IH₂ : _ := @evaln_inclusion (s+1) cg _ _ h,\n    by { simp[evaln, (>>)], assume e hf,\n         cases n.unpair.snd with n0; simp at hf ⊢, exact ⟨e, IH₁ hf⟩,\n         rcases hf with ⟨z, hz0, hz1⟩,\n         refine ⟨e, z, IH₀ hz0, IH₂ hz1⟩ }\n| (s+1) (rfind' cf)  f g := λ h n y, \n    have l0 : ∀ x y, x < s → f x = some y → g x = some y :=\n      λ x y e, h x y (nat.lt.step e),\n    have IH₀ : _ := @evaln_inclusion (s+1) cf _ _ h, \n    have IH₁ : _ := @evaln_inclusion s cf.rfind' _ _ l0,\n    by { simp[evaln, (>>)], assume e z ez ey,\n         refine ⟨e, z, IH₀ ez, _⟩,\n         cases z with z0; simp at ey ⊢, exact ey, exact IH₁ ey }\n\ntheorem evaln_use : ∀ {s c} {f g : ℕ → option ℕ},\n  (∀ x, x < s → f x = g x) → evaln s f c = evaln s g c\n| 0     c            _ _  := by simp[evaln]\n| (s+1) oracle       _ _  := λ h, funext $ λ n, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega,\n         cases this; simp[(>>), this], exact h _ (nat.lt_succ_iff.mpr this), refl }\n| (s+1) zero         f g := by simp[evaln]\n| (s+1) succ         f g := by simp[evaln]\n| (s+1) left         f g := by simp[evaln]\n| (s+1) right        f g := by simp[evaln]\n| (s+1) (pair cf cg) f g := λ h,\n    have IH₀ : _ := @evaln_use (s+1) cf _ _ h,\n    have IH₁ : _ := @evaln_use (s+1) cg _ _ h,\n    funext $ λ n, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega, cases this;\n         simp[(>>), this, IH₀, IH₁] }\n| (s+1) (comp cf cg) f g := λ h,\n    have IH₀ : _ := @evaln_use (s+1) cf _ _ h,\n    have IH₁ : _ := @evaln_use (s+1) cg _ _ h,\n    funext $ λ n, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega, cases this;\n         simp[(>>), this, IH₀, IH₁] }\n| (s+1) (prec cf cg) f g := λ h,\n    have l0 : ∀ (x : ℕ), x < s → f x = g x := λ x e, h x (nat.lt.step e),\n    have IH₀ : _ := @evaln_use s (cf.prec cg) _ _ l0,\n    have IH₁ : _ := @evaln_use (s+1) cf _ _ h,\n    have IH₂ : _ := @evaln_use (s+1) cg _ _ h,\n    funext $ λ n, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega, cases this;\n         simp[(>>), this, IH₀, IH₁, IH₂] }\n| (s+1) (rfind' cf)  f g := λ h, \n    have l0 : ∀ (x : ℕ), x < s → f x = g x := λ x e, h x (nat.lt.step e),\n    have IH₀ : _ := @evaln_use (s+1) cf _ _ h,\n    have IH₁ : _ := @evaln_use s cf.rfind' _ _ l0,\n    funext $ λ n, \n    by { simp[evaln], have : n ≤ s ∨ ¬ n ≤ s, omega, cases this;\n         simp[(>>), this, IH₀, IH₁] }\n\ntheorem evaln_use' {s} {f g : ℕ → option ℕ} (h : ∀ x, x < s → f x = g x) :\n  evaln s f = evaln s g := \nfunext $ λ c, evaln_use h\n\ntheorem evaln_bound {f} : ∀ {k c n x}, x ∈ evaln k f c n → n < k\n| 0     c n x h := by simp [evaln] at h; cases h\n| (k+1) c n x h := begin\n  suffices : ∀ {o : option ℕ}, x ∈ guard (n ≤ k) >> o → n < k + 1,\n  { cases c; rw [evaln] at h; try { exact this h } },\n  simpa [(>>)] using nat.lt_succ_of_le\nend\n\ntheorem evaln_bound_none {f k c n} (h : k ≤ n) : evaln k f c n = none :=\nby { cases e : evaln k f c n with v; simp,\n     have := evaln_bound e, exact nat.lt_le_antisymm this h }\n\ntheorem evaln_mono : ∀ {s₁ s₂ c f n x},\n  s₁ ≤ s₂ → evaln s₁ f c n = some x → evaln s₂ f c n = some x\n| 0      _      _            _ _ _ := λ _ h, by simp [evaln] at h; cases h\n| (s₀+1) 0      _            _ _ _ := λ e h, by exfalso; exact nat.not_succ_le_zero s₀ e\n| (s₀+1) (s₁+1) oracle       f n x := λ hs,\n    by { simp[evaln], have : n ≤ s₀ ∨ ¬ n ≤ s₀, omega,\n         cases this; simp[(>>), this], exact λ e, ⟨by omega, e⟩,\n         intros _ c, exfalso, exact option.not_mem_none _ c }\n| (s₀+1) (s₁+1) zero         f n x := λ hs, \n    by simp[evaln, (>>)]; exact λ _ e, ⟨by omega, e⟩\n| (s₀+1) (s₁+1) succ         f n x := λ hs, \n    by simp[evaln, (>>)]; exact λ _ e, ⟨by omega, e⟩\n| (s₀+1) (s₁+1) left         f n x := λ hs, \n    by simp[evaln, (>>)]; exact λ _ e, ⟨by omega, e⟩\n| (s₀+1) (s₁+1) right        f n x := λ hs, \n    by simp[evaln, (>>)]; exact λ _ e, ⟨by omega, e⟩\n| (s₀+1) (s₁+1) (pair cf cg) f n x := λ hs,\n    have IH₀ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cf f n y hs,\n    have IH₁ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cg f n y hs,\n    by { simp[evaln, (>>)], assume e0 ex,\n         have : ∃ y, evaln (s₀ + 1) f cf n = some y,\n         { cases evaln (s₀ + 1) f cf n; simp[(<*>)] at ex ⊢, exact ex },\n         rcases this with ⟨y₀, hy₀⟩,\n         have : ∃ y, evaln (s₀ + 1) f cg n = some y,\n         { cases evaln (s₀ + 1) f cg n; simp[(<*>)] at ex ⊢, exact ex },         \n         rcases this with ⟨y₁, hy₁⟩,\n         simp[hy₀, hy₁, IH₀ _ _ hy₀, IH₁ _ _ hy₁] at ex ⊢,\n         exact ⟨le_trans e0 (nat.le_of_succ_le_succ hs), ex⟩ }\n| (s₀+1) (s₁+1) (comp cf cg) f n x := λ hs,        \n    have IH₀ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cf f n y hs,\n    have IH₁ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cg f n y hs,\n    by { simp[evaln, (>>)], assume e0 y ey ex,\n         refine ⟨le_trans e0 (nat.le_of_succ_le_succ hs), y, IH₁ _ _ ey, IH₀ _ _ ex⟩ }\n| (s₀+1) (s₁+1) (prec cf cg) f n x := λ hs,\n    have IH₀ : _ := λ n y, @evaln_mono s₀ s₁ (cf.prec cg) f n y (nat.le_of_succ_le_succ hs),\n    have IH₁ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cf f n y hs,\n    have IH₂ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cg f n y hs,\n    by { simp[evaln, (>>)],\n         cases n.unpair.snd with n0; simp,\n         assume e0 ex,\n         exact ⟨le_trans e0 (nat.le_of_succ_le_succ hs), IH₁ _ _ ex⟩,\n         assume e0 y ey ex,\n         refine ⟨le_trans e0 (nat.le_of_succ_le_succ hs), y, IH₀ _ _ ey, IH₂ _ _ ex⟩ }\n| (s₀+1) (s₁+1) (rfind' cf)  f n x := λ hs,\n    have IH₀ : _ := λ n y, @evaln_mono (s₀+1) (s₁+1) cf f n y hs,\n    have IH₁ : _ := λ n y, @evaln_mono s₀ s₁ cf.rfind' f n y (nat.le_of_succ_le_succ hs),\n    by { simp[evaln, (>>), pure],\n         assume e0 y ey ex,\n         refine ⟨by omega, y, IH₀ _ _ ey, _⟩,\n         cases y with y0; simp at ex ⊢, exact ex, exact IH₁ _ _ ex }\n\ndef eval (f : ℕ → option ℕ) : code → ℕ →. ℕ \n| oracle       := λ n, f n\n| zero         := pure 0\n| succ         := nat.succ\n| left         := ↑(λ n : ℕ, n.unpair.1)\n| right        := ↑(λ n : ℕ, n.unpair.2)\n| (pair cf cg) := λ n, mkpair <$> eval cf n <*> eval cg n\n| (comp cf cg) := λ n, eval cg n >>= eval cf\n| (prec cf cg) := nat.unpaired (λ a n,\n    n.elim (eval cf a) (λ y IH, do i ← IH, eval cg (mkpair a (mkpair y i))))\n| (rfind' cf) := nat.unpaired (λ a m,\n    (nat.rfind (λ n, (λ m, m = 0) <$>\n      eval cf (mkpair a (n + m)))).map (+ m))\n\ntheorem evaln_sound {f} : ∀ {s c n x},\n  x ∈ evaln s f c n → x ∈ eval f c n\n| 0     _ _ _ h := by simp [evaln] at h; cases h\n| (s + 1) oracle n x h := by { simp [eval, evaln, (>>)] at h ⊢, exact h.2 }\n| (s + 1) zero   n x h := \n    by { simp [eval, evaln, (>>), pure, pfun.pure] at h ⊢, exact eq.symm h.2 }\n| (s + 1) succ   n x h := \n    by { simp [eval, evaln, (>>), pure, pfun.pure] at h ⊢, exact eq.symm h.2 }\n| (s + 1) left   n x h := \n    by { simp [eval, evaln, (>>), pure, pfun.pure] at h ⊢, exact eq.symm h.2 }\n| (s + 1) right   n x h := \n    by { simp [eval, evaln, (>>), pure, pfun.pure] at h ⊢, exact eq.symm h.2 }\n| (s + 1) (pair cf cg) n x h :=\n    have IH₀ : _ := @evaln_sound (s + 1) cf,\n    have IH₁ : _ := @evaln_sound (s + 1) cg,\n    by{ simp [eval, evaln, (>>), (<*>)] at h ⊢, \n        rcases h with ⟨_, y, hy, z, hz, ex⟩,\n        refine ⟨y, IH₀ hy, z, IH₁ hz, ex⟩ }\n| (s + 1) (comp cf cg) n x h :=\n    have IH₀ : _ := @evaln_sound (s + 1) cf,\n    have IH₁ : _ := @evaln_sound (s + 1) cg,\n    by{ simp [eval, evaln, (>>), (<*>)] at h ⊢, \n        rcases h with ⟨_, y, hy, hz⟩,\n        refine ⟨y, IH₁ hy, IH₀ hz⟩ }\n| (s + 1) (prec cf cg) n x h :=\n    have IH₀ : _ := @evaln_sound (s + 1) cf,\n    have IH₁ : _ := @evaln_sound (s + 1) cg,\n    by{ simp [eval, evaln, (>>)] at h ⊢,\n        rcases h with ⟨_, hx⟩, revert hx,\n        induction n.unpair.2 with m IH generalizing x; simp,\n        { exact IH₀ },\n        { refine λ y h₁ h₂, ⟨y, IH _ _, _⟩,\n          { have := evaln_mono s.le_succ h₁,\n        simp [evaln, (>>)] at this,\n        exact this.2 },\n        { exact IH₁ h₂ } } }\n| (s + 1) (rfind' cf) n x h :=\n    have IH₀ : _ := @evaln_sound (s+1) cf,\n    have IH₁ : _ := @evaln_sound s cf.rfind',\n  begin\n    simp [eval, evaln, (>>)] at h ⊢,\n    rcases h with ⟨_, m, hm, h⟩,\n    cases e : m with m0; simp[e, pure] at hm h,\n    { refine ⟨0, ⟨by simp; exact IH₀ hm, by simp⟩, by simp[h]⟩ },\n    { have := IH₁ h,  simp[eval] at this,\n      rcases this with ⟨y, ⟨hc0, hc1⟩, ex⟩,\n      refine ⟨y+1, ⟨\n        by simpa [add_comm, add_left_comm] using hc0, λ b eb, _⟩, \n        by simpa [add_comm, add_left_comm] using ex⟩,\n      { cases b with b0, refine ⟨m0 + 1, by simp; exact IH₀ hm, by simp⟩, \n        have : b0.succ + n.unpair.snd = b0 + (n.unpair.snd + 1), omega,\n        rw this, exact hc1 (nat.lt_of_succ_lt_succ eb) } }\n  end\n\ntheorem evaln_complete {f : ℕ → option ℕ} {c n x} :\n  x ∈ eval (λ x, f x) c n ↔ ∃ k, x ∈ evaln k f c n := ⟨λ h,\nbegin\n  suffices : ∃ k, x ∈ evaln (k+1) f c n,\n  { exact let ⟨k, h⟩ := this in ⟨k+1, h⟩ },\n    induction c generalizing n x;\n    simp [eval, evaln, pure, pfun.pure, (<*>), (>>)] at h ⊢,\n    { exact ⟨⟨n, by refl⟩, h⟩ },\n    { use n, have : min n n.unpair.fst.unpair.fst = n.unpair.fst.unpair.fst,\n      { simp, exact le_trans (nat.unpair_left_le _) (nat.unpair_left_le _) }, simp [this,h] },\n    { exact ⟨⟨_, le_refl _⟩, h.symm⟩ },\n    { exact ⟨⟨_, le_refl _⟩, h.symm⟩ },\n    { exact ⟨⟨_, le_refl _⟩, h.symm⟩ },\n    case nat.rpartrec.code.pair : cf cg hf hg \n    { rcases h with ⟨x, hx, y, hy, rfl⟩,\n      rcases hf hx with ⟨k₁, hk₁⟩, rcases hg hy with ⟨k₂, hk₂⟩,\n      refine ⟨max k₁ k₂, le_max_of_le_left $ nat.le_of_lt_succ $ evaln_bound hk₁, x, _, y, _, rfl⟩,\n      exact evaln_mono (nat.succ_le_succ $ le_max_left _ _) hk₁, \n      exact evaln_mono (nat.succ_le_succ $ le_max_right _ _) hk₂ },\n    case nat.rpartrec.code.comp : cf cg hf hg \n    { rcases h with ⟨y, hy, hx⟩,\n      rcases hf hx with ⟨k₁, hk₁⟩, rcases hg hy with ⟨k₂, hk₂⟩,\n      refine ⟨max k₂ k₁, le_max_of_le_left $ nat.le_of_lt_succ $ evaln_bound hk₂, y, _, _⟩,\n      exact evaln_mono (nat.succ_le_succ $ le_max_left _ _) hk₂,\n      exact evaln_mono (nat.succ_le_succ $ le_max_right _ _) hk₁ },\n    case nat.rpartrec.code.prec : cf cg hf hg\n    { revert h,\n      generalize : n.unpair.1 = n₁, generalize : n.unpair.2 = n₂,\n      induction n₂ with m IH generalizing x n; simp,\n      { intro, rcases hf h with ⟨k, hk⟩,\n        exact ⟨_, le_max_left _ _,\n          evaln_mono (nat.succ_le_succ $ le_max_right _ _) hk⟩ },\n      { intros y hy hx,\n        rcases IH hy with ⟨k₁, nk₁, hk₁⟩, rcases hg hx with ⟨k₂, hk₂⟩,\n        refine ⟨(max k₁ k₂).succ, nat.le_succ_of_le $ le_max_of_le_left $\n          le_trans (le_max_left _ (mkpair n₁ m)) nk₁, y,\n          evaln_mono (nat.succ_le_succ $ le_max_left _ _) _,\n          evaln_mono (nat.succ_le_succ $ nat.le_succ_of_le $ le_max_right _ _) hk₂⟩,\n        simp [evaln, (>>)],\n        exact ⟨le_trans (le_max_right _ _) nk₁, hk₁⟩ } },\n    case nat.rpartrec.code.rfind' : cf hf\n    { rcases h with ⟨y, ⟨hy₁, hy₂⟩, rfl⟩,\n      suffices : ∃ k, y + n.unpair.2 ∈ evaln (k+1) f (rfind' cf)\n        (mkpair n.unpair.1 n.unpair.2), {simpa [evaln, (>>)]},\n      revert hy₁ hy₂, generalize : n.unpair.2 = m, intros,\n      induction y with y IH generalizing m; simp [evaln, (>>)],\n      { simp at hy₁, rcases hf hy₁ with ⟨k, hk⟩,\n        exact ⟨_, nat.le_of_lt_succ $ evaln_bound hk, _, hk, by simp; refl⟩ },\n      { rcases hy₂ (nat.succ_pos _) with ⟨a, ha, a0⟩,\n        rcases hf ha with ⟨k₁, hk₁⟩,\n        rcases IH m.succ\n            (by simpa [nat.succ_eq_add_one, add_comm, add_left_comm] using hy₁)\n            (λ i hi, by simpa [nat.succ_eq_add_one, add_comm, add_left_comm] using\n              hy₂ (nat.succ_lt_succ hi))\n          with ⟨k₂, hk₂⟩,\n        use (max k₁ k₂).succ,\n        rw [zero_add] at hk₁,\n        use (nat.le_succ_of_le $ le_max_of_le_left $ nat.le_of_lt_succ $ evaln_bound hk₁),\n        use a,\n        use evaln_mono (nat.succ_le_succ $ nat.le_succ_of_le $ le_max_left _ _) hk₁,\n        simpa [nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n            evaln_mono (nat.succ_le_succ $ le_max_right _ _) hk₂ } },\nend, λ ⟨k, h⟩, evaln_sound h⟩\n\ntheorem evaln_complete_dec {c f} [D : decidable_pred (eval f c).dom] : ∀ m, ∃ s₀,\n  ∀ n a, n < m → eval f c n = some a → evaln s₀ f c n = some a := λ m,\nbegin\n  induction m with m0 ih, simp,\n  rcases ih with ⟨s₀, hs₀⟩,\n  have e : eval f c m0 = none ∨ ∃ v, eval f c m0 = some v := eq_none_or_eq_some (eval f c m0),\n  cases e,\n  { refine ⟨s₀, λ n a en ha, _⟩,\n    have : n < m0 ∨ n = m0, from nat.lt_succ_iff_lt_or_eq.mp en,\n    cases this, { exact hs₀ _ _ this ha },\n    { exfalso, rw this at ha, rw e at ha, exact part.some_ne_none _ (eq.symm ha) } },\n  { rcases e with ⟨v, e⟩,\n    have : v ∈ eval f c m0, simp[e],\n    have : ∃ k, v ∈ evaln k f c m0 := evaln_complete.mp this, rcases this with ⟨s₁, hs₁⟩,\n    refine ⟨max s₀ s₁, _⟩,\n    intros n a en ha,\n    have en' : n < m0 ∨ n = m0, from nat.lt_succ_iff_lt_or_eq.mp en,\n    cases en', \n    { have : evaln s₀ f c n = option.some a := hs₀ _ _ en' ha,\n      show evaln (max s₀ s₁) f c n = option.some a,\n        from evaln_mono (le_max_left s₀ s₁) this },\n    { rw en', \n      have : a = v, from part.some_inj.mp (by simp only [←e, ←ha, en']),\n      show evaln (max s₀ s₁) f c m0 = option.some a, rw this,\n        from evaln_mono (le_max_right s₀ s₁) hs₁ } }\nend\n\ntheorem eval_inclusion {c x y} {f : ℕ → option ℕ} (h : y ∈ eval f c x) : ∃ s, ∀ {g : ℕ → option ℕ},\n  (∀ x y, x < s → f x = some y → g x = some y) → y ∈ eval g c x :=\nby { have : ∃ s, y ∈ evaln s f c x := evaln_complete.mp h, rcases this with ⟨s, hs⟩,\n     refine ⟨s, λ g h, evaln_complete.mpr ⟨s, evaln_inclusion h hs⟩⟩ }\n\n@[simp] theorem evaln_const (s f) :\n  ∀ n m, evaln (s+1) f (code.const n) m = guard (n ≤ s+1) >> guard (m ≤ s) >> option.some n\n| 0     m := by { simp[code.const, evaln, (>>), pure], }\n| (n+1) m := have IH : _ := evaln_const n m, \n  by { simp [code.const, evaln, IH, pure, (>>)],\n    by_cases e1 : n ≤ s; simp[e1, failure, alternative.failure],\n    by_cases e2 : m ≤ s; simp[e1, e2, failure, alternative.failure],\n    refine ⟨n, ⟨⟨(le_add_right e1), (), rfl⟩, rfl⟩, e1, rfl⟩ }\n\n@[simp] theorem eval_const (f) : ∀ n m, eval f (code.const n) m = part.some n\n| 0     m := rfl\n| (n+1) m := by simp! *\n\n@[simp] theorem eval_id (f n) : eval f code.id n = part.some n := by simp! [(<*>)]\n\n@[simp] theorem eval_curry (f c n x) : eval f (curry c n) x = eval f c (mkpair n x) :=\nby simp! [(<*>)]\n\n--@[simp] theorem eval_ctrans (f c₁ c₂ x) : eval (eval f c₁) c₂ x = eval f (ctrans c₁ c₂) x :=\n--by simp! [(<*>)]\n\n@[simp] theorem evaln_curry (s f c n x) :\n  evaln s f (curry c n) x = evaln s f c (n.mkpair x) :=\nbegin\n  cases s, simp,\n  simp [curry, evaln], \n  by_cases e0 : x ≤ s; simp [code.id, evaln, e0, pure, (>>), failure, alternative.failure],\n  by_cases e1 : n ≤ s+1; simp [e0, e1, pure, (>>), (<*>), failure, alternative.failure],\n  { cases e : evaln (s + 1) f c (n.mkpair x) with y; simp,\n    have : n.mkpair x < s + 1, from evaln_bound e,\n    have : n ≤ s + 1, from le_of_lt \n      (gt_of_gt_of_ge this (nat.left_le_mkpair n x)),\n    contradiction },\n  { cases e : evaln (s + 1) f c (n.mkpair x) with y; simp,\n    have : n.mkpair x ≤ s := nat.lt_succ_iff.mp (evaln_bound e),\n    have : x ≤ s := le_trans (nat.right_le_mkpair _ _) this,\n    contradiction }\nend\n\nsection\nopen primrec\n\ntheorem pair_prim : primrec₂ pair :=\nprimrec₂.of_nat_iff.2 $ primrec₂.encode_iff.1 $ nat_add.comp\n  (nat_bit0.comp $ nat_bit0.comp $\n    primrec₂.mkpair.comp\n      (encode_iff.2 $ (primrec.of_nat code).comp fst)\n      (encode_iff.2 $ (primrec.of_nat code).comp snd))\n  (primrec₂.const 5)\n\ntheorem comp_prim : primrec₂ comp :=\nprimrec₂.of_nat_iff.2 $ primrec₂.encode_iff.1 $ nat_add.comp\n  (nat_bit0.comp $ nat_bit1.comp $\n    primrec₂.mkpair.comp\n      (encode_iff.2 $ (primrec.of_nat code).comp fst)\n      (encode_iff.2 $ (primrec.of_nat code).comp snd))\n  (primrec₂.const 5)\n\ntheorem prec_prim : primrec₂ prec :=\nprimrec₂.of_nat_iff.2 $ primrec₂.encode_iff.1 $ nat_add.comp\n  (nat_bit1.comp $ nat_bit0.comp $\n    primrec₂.mkpair.comp\n      (encode_iff.2 $ (primrec.of_nat code).comp fst)\n      (encode_iff.2 $ (primrec.of_nat code).comp snd))\n  (primrec₂.const 5)\n\ntheorem rfind_prim : primrec rfind' :=\nof_nat_iff.2 $ encode_iff.1 $ nat_add.comp\n  (nat_bit1.comp $ nat_bit1.comp $\n    encode_iff.2 $ primrec.of_nat code)\n  (const 5)\n\ntheorem const_prim : primrec code.const :=\n(primrec.id.nat_iterate (primrec.const zero)\n  (comp_prim.comp (primrec.const succ) primrec.snd).to₂).of_eq $\nλ n, by simp; induction n; simp [*, code.const, function.iterate_succ']\n\ntheorem curry_prim : primrec₂ curry :=\ncomp_prim.comp primrec.fst $\npair_prim.comp (const_prim.comp primrec.snd) (primrec.const code.id)\n\nend\n\ntheorem rpartrec_rfind' {f} : nat.rpartrec f (nat.unpaired (λ a m,\n  (nat.rfind (λ n, (λ m, m = 0) <$> f (nat.mkpair a (n + m)))).map (+ m))) :=\nrpartrec.nat_iff.mp $\nbegin\n  simp,\n  have c₀ : primrec (λ (x : (ℕ × ℕ) × ℕ), x.2 + x.1.2) := \n  primrec.nat_add.comp primrec.snd (primrec.snd.comp primrec.fst),\n  have c₁ : primrec (λ (m : ((ℕ × ℕ) × ℕ) × ℕ), to_bool (m.2 = 0)) :=\n  primrec₂.comp primrec.eq primrec.snd (primrec.const 0),\n  have c₂ : (λ (x : (ℕ × ℕ) × ℕ), f (x.1.1.mkpair (x.2 + x.1.2))) partrec_in f :=\n  rpartrec.refl.comp (primrec₂.mkpair.comp (primrec.fst.comp primrec.fst) $\n    primrec.nat_add.comp primrec.snd (primrec.snd.comp primrec.fst)).to_comp.to_rpart,\n  have := (rpartrec.rfind (c₂.map c₁.to_rcomp.to₂).to₂).map c₀.to_rcomp.to₂,\n  simp at this,\n  exact this.comp primrec.unpair.to_rcomp\nend\n\ntheorem exists_code {f g : ℕ →. ℕ} [D : decidable_pred (λ x, x ∈ g.dom)] :\n  nat.rpartrec g f ↔ ∃ c, eval (g.eval_opt) c = f := ⟨λ h,\nbegin\n  induction h,\n  case nat.rpartrec.oracle \n  { exact ⟨oracle, by { simp[eval], funext n, exact of_to_option (g n) }⟩ },  \n  case nat.rpartrec.zero   { exact ⟨zero, rfl⟩ },\n  case nat.rpartrec.succ   { exact ⟨succ, rfl⟩ },\n  case nat.rpartrec.left   { exact ⟨left, rfl⟩ },\n  case nat.rpartrec.right  { exact ⟨right, rfl⟩ },\n  case nat.rpartrec.pair : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨pair e₀ e₁, rfl⟩ },\n  case nat.rpartrec.comp : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨comp e₀ e₁, rfl⟩ },\n  case nat.rpartrec.prec : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨prec e₀ e₁, rfl⟩ },\n  case nat.rpartrec.rfind : f₀ pf₀ hf₀\n  { rcases hf₀ with ⟨e₀, rfl⟩, \n    refine ⟨comp (rfind' e₀) (pair nat.rpartrec.code.id zero), _⟩,\n    simp [eval, (<*>), pure, pfun.pure, part.map_id'],  },\nend,λ h, begin\n  rcases h with ⟨c, rfl⟩, induction c,\n  case nat.rpartrec.code.oracle \n  { simp[eval], unfold_coes, simp, exact nat.rpartrec.oracle },\n  case nat.rpartrec.code.zero { exact nat.rpartrec.zero },\n  case nat.rpartrec.code.succ { exact nat.rpartrec.succ },\n  case nat.rpartrec.code.left { exact nat.rpartrec.left },\n  case nat.rpartrec.code.right { exact nat.rpartrec.right },\n  case nat.rpartrec.code.pair : cf cg pf pg { exact pf.pair pg },\n  case nat.rpartrec.code.comp : cf cg pf pg { exact pf.comp pg },\n  case nat.rpartrec.code.prec : cf cg pf pg { exact pf.prec pg },\n  case nat.rpartrec.code.rfind' : cf pf { exact nat.rpartrec.trans rpartrec_rfind' pf },\nend⟩\n\ntheorem exists_code_opt {f : ℕ →. ℕ} {g : ℕ → option ℕ} :\n  nat.rpartrec ↑ʳg f ↔ ∃ c, eval g c = f := ⟨λ h,\nbegin\n  induction h,\n  case nat.rpartrec.oracle \n  { exact ⟨oracle, rfl⟩ },  \n  case nat.rpartrec.zero   { exact ⟨zero, rfl⟩ },\n  case nat.rpartrec.succ   { exact ⟨succ, rfl⟩ },\n  case nat.rpartrec.left   { exact ⟨left, rfl⟩ },\n  case nat.rpartrec.right  { exact ⟨right, rfl⟩ },\n  case nat.rpartrec.pair : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨pair e₀ e₁, rfl⟩ },\n  case nat.rpartrec.comp : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨comp e₀ e₁, rfl⟩ },\n  case nat.rpartrec.prec : f₀ f₁ pf₀ pf₁ hf₀ hf₁\n  { rcases hf₀ with ⟨e₀, rfl⟩, rcases hf₁ with ⟨e₁, rfl⟩,\n    exact ⟨prec e₀ e₁, rfl⟩ },\n  case nat.rpartrec.rfind : f₀ pf₀ hf₀\n  { rcases hf₀ with ⟨e₀, rfl⟩, \n    refine ⟨comp (rfind' e₀) (pair nat.rpartrec.code.id zero), _⟩,\n    simp [eval, (<*>), pure, pfun.pure, part.map_id'],  },\nend,λ h, begin\n  rcases h with ⟨c, rfl⟩, induction c,\n  case nat.rpartrec.code.oracle \n  { simp[eval], exact nat.rpartrec.oracle },\n  case nat.rpartrec.code.zero { exact nat.rpartrec.zero },\n  case nat.rpartrec.code.succ { exact nat.rpartrec.succ },\n  case nat.rpartrec.code.left { exact nat.rpartrec.left },\n  case nat.rpartrec.code.right { exact nat.rpartrec.right },\n  case nat.rpartrec.code.pair : cf cg pf pg { exact pf.pair pg },\n  case nat.rpartrec.code.comp : cf cg pf pg { exact pf.comp pg },\n  case nat.rpartrec.code.prec : cf cg pf pg { exact pf.prec pg },\n  case nat.rpartrec.code.rfind' : cf pf { exact nat.rpartrec.trans rpartrec_rfind' pf },\nend⟩\n\nend nat.rpartrec.code\nopen nat.rpartrec \n\nvariables {α : Type*} {σ : Type*} {β : Type*} {τ : Type*} {γ : Type*} {μ : Type*} \nvariables [primcodable α] [primcodable σ] [primcodable β] [primcodable τ] [primcodable γ] [primcodable μ]\n\ndef univn (α σ) [primcodable α] [primcodable σ] (s : ℕ) (f : β → option τ) (e : ℕ) :\n  α → option σ := (λ a,\n(code.evaln s \n  (λ n, (decode β n).bind (λ a, (f a).map encode ))\n  (of_nat code e) (encode a))\n.bind (λ x, (decode σ x)))\n\nnotation `⟦`e`⟧*`f:max` [`s`]` := univn _ _ s f e\nnotation `⟦`e`⟧^`f:max` [`s`]` := univn _ _ s ↑ₒf e\n\ndef univ (α σ) [primcodable α] [primcodable σ] (f : β → option τ) (e : ℕ) : α →. σ := (λ a,\n(code.eval\n  (λ n, (decode β n).bind (λ a, (f a).map encode))\n  (of_nat code e) (encode a))\n.bind (λ x, (decode σ x)))\n\nnotation `⟦`e`⟧*`f:max := univ _ _ f e\nnotation `⟦`e`⟧^`f:max := univ _ _ ↑ₒf e\n\nnotation `⟦`e`⟧ᵪ*`f:max` [`s`]` := univn ℕ bool s f e\nnotation `⟦`e`⟧ₙ*`f:max` [`s`]` := univn ℕ ℕ s f e\nnotation `⟦`e`⟧ᵪ*`f:max := univ ℕ bool f e\nnotation `⟦`e`⟧ₙ*`f:max := univ ℕ ℕ f e\n\nnotation `⟦`e`⟧ᵪ^`f:max` [`s`]` := univn ℕ bool s ↑ₒf e\nnotation `⟦`e`⟧ₙ^`f:max` [`s`]` := univn ℕ ℕ s ↑ₒf e\nnotation `⟦`e`⟧ᵪ^`f:max := univ ℕ bool ↑ₒf e\nnotation `⟦`e`⟧ₙ^`f:max := univ ℕ ℕ ↑ₒf e\n\ndef univn0 (α σ) [primcodable α] [primcodable σ] (s : ℕ) (e : ℕ) : α → option σ :=\nunivn α σ s (λ x, none : ℕ → option ℕ) e\n\ndef univ0 (α σ) [primcodable α] [primcodable σ] (e : ℕ) : α →. σ :=\nuniv α σ (λ x, some 0 : ℕ → option ℕ) e\n\nnotation `⟦`e`⟧⁰`:max` [`s`]` := univn0 _ _ s e\nnotation `⟦`e`⟧⁰`:max := univ0 _ _ e\n\ndef re_set (α σ) [primcodable α] [primcodable σ] (p : β → option τ) (e : ℕ) : set α :=\n{x | (⟦e⟧*p x : part σ).dom}\n\ndef re_set0 (α σ) [primcodable α] [primcodable σ] (e : ℕ) : set α :=\n{x | (univ0 α σ e x : part σ).dom}\n\nnotation `W⟦`e`⟧^`f:max := re_set _ _ ↑ₒf e\n\nnotation `W⟦`e`⟧ᵪ^`f:max := re_set ℕ bool ↑ₒf e\nnotation `W⟦`e`⟧ₙ^`f:max := re_set ℕ ℕ ↑ₒf e\nnotation `W⟦`e`⟧ᵪ⁰`:max := re_set0 ℕ bool e\nnotation `W⟦`e`⟧ₙ⁰`:max := re_set0 ℕ ℕ e\n\ndef curry {α} [primcodable α] (e : ℕ) (n : α) : ℕ := encode (code.curry (of_nat _ e) (encode n))\n\n-- smn定理\n@[simp] theorem eval_curry (f : γ → option τ) (e : ℕ) (n : β) (x : α) :\n  (⟦curry e n⟧*f x : part σ) = (⟦e⟧*f (n, x) : part σ) :=\nby { simp[curry, univ] }\n\nnamespace rpartrec\n\ntheorem curry_prim {α} [primcodable α] : primrec₂ (@curry α _) :=\n(primrec.encode.comp $ code.curry_prim.comp \n  ((primrec.of_nat code).comp primrec.fst) ((@primrec.encode α _).comp primrec.snd))\n\nopen primrec\n\ntheorem univn_sound {e} {p : β → option τ} {x : α} {y : σ} {s : ℕ} :\n  ⟦e⟧*p [s] x = some y → ⟦e⟧*p x = some y := \nby { simp[univn, univ, part.eq_some_iff], \n     exact λ s h e, ⟨s, code.evaln_sound h, e⟩ }\n\ntheorem univn_complete {p : β → option τ} {e x} {n : α} :\n  x ∈ (⟦e⟧*p n : part σ) ↔ ∃ s, ⟦e⟧*p [s] n = some x :=\nby { simp[univn, univ, part.eq_some_iff], split,\n     { rintros ⟨a, ha, ea⟩,\n       rcases code.evaln_complete.mp ha with ⟨s, hs⟩,\n       refine ⟨s, a, hs, ea⟩ },\n     { rintros ⟨s, a, ha, ea⟩,\n       have := code.evaln_complete.mpr ⟨s, ha⟩,\n       refine ⟨a, this, ea⟩ } }\n\ntheorem univn_dom_complete {p : β → option τ} {e} {n : α} :\n  (⟦e⟧*p n : part σ).dom ↔ ∃ s, (⟦e⟧*p [s] n : option σ).is_some :=\nby { simp[part.dom_iff_mem, option.is_some_iff_exists], split,\n     { rintros ⟨y, h⟩, rcases univn_complete.mp h with ⟨s, h⟩, refine ⟨s, y, h⟩ },\n     { rintros ⟨s, y, h⟩, have := univn_complete.mpr ⟨s, h⟩, refine ⟨y, this⟩ } }\n\ntheorem univn_mono {e} {p : β → option τ} {s₀ s₁ : ℕ} {x : α} {y : σ}\n  (eqn : s₀ ≤ s₁) : ⟦e⟧*p [s₀] x = some y → ⟦e⟧*p [s₁] x = some y :=\nby { simp [univn], intros z h eqn_z,\n     refine ⟨z, code.evaln_mono eqn h, eqn_z⟩ }\n\ntheorem univn_dom_mono {e} {p : β → option τ} {s₀ s₁ : ℕ} {x : α}\n  (eqn : s₀ ≤ s₁) : (⟦e⟧*p [s₀] x : option σ).is_some → (⟦e⟧*p [s₁] x : option σ).is_some :=\nbegin\n  cases C : (⟦e⟧*p [s₀] x : option σ) with y; simp,\n  simp[univn_mono eqn C]\nend\n\ntheorem univn_mono_eq {e} {p : β → option τ} {s₀ s₁ : ℕ} {x : α} {y₀ y₁ : σ}\n  (h₀ : ⟦e⟧*p [s₀] x = some y₀) (h₁ : ⟦e⟧*p [s₁] x = some y₁) : y₀ = y₁ :=\nbegin\n  have : s₀ ≤ s₁ ∨ s₁ ≤ s₀, from le_total _ _,\n  cases this,\n  { have := univn_mono this h₀, simp [this] at h₁, exact h₁ },\n  { have := univn_mono this h₁, simp [this] at h₀, exact eq.symm h₀ }\nend \n\ntheorem univn_use {s e} {p q : ℕ → option β}\n  (h : ∀ x, x < s → p x = q x) : (⟦e⟧*p [s] : α → option σ) = ⟦e⟧*q [s] :=\nbegin\n  simp [univn],\n  suffices :\n    code.evaln s (λ n, option.map encode (p n)) (of_nat code e) =\n    code.evaln s (λ n, option.map encode (q n)) (of_nat code e),\n  { funext, rw this },\n  apply code.evaln_use, intros u eqn, congr, exact h _ eqn\nend\n\ntheorem univn_tot_use {s e} {f g : ℕ → β}\n  (h : ∀ x, x < s → f x = g x) : (⟦e⟧^f [s] : α → option σ) = ⟦e⟧^g [s] :=\nunivn_use (λ x lt, by simp[h x lt])\n\ntheorem univn_mono_use {e} {p q : ℕ → option τ} {s₀ s₁ : ℕ} {x : α} {y : σ}\n  (h : ∀ x < s₀, p x = q x) (eqn : s₀ ≤ s₁) : ⟦e⟧*p [s₀] x = some y → ⟦e⟧*q [s₁] x = some y := λ eq_y,\n  have (⟦e⟧*p [s₀] x : option σ) = ⟦e⟧*q [s₀] x, from congr_fun (univn_use h) x,  \nunivn_mono eqn (by simp[←this, eq_y])\n\ntheorem univn_tot_mono_use {e} {f g : ℕ → τ} {s₀ s₁ : ℕ} {x : α} {y : σ}\n  (h : ∀ x < s₀, f x = g x) (eqn : s₀ ≤ s₁) : ⟦e⟧^f [s₀] x = some y → ⟦e⟧^g [s₁] x = some y :=\nunivn_mono_use (by simp; exact h) eqn\n\ntheorem eval_inclusion {e} {x : α} {y : σ} {p : ℕ → option τ}\n  (h : y ∈ (⟦e⟧*p x : part σ)) : ∃ s, ∀ {q : ℕ → option τ},\n  (∀ x y, x < s → p x = some y → q x = some y) → y ∈ (⟦e⟧*q x : part σ) := \nby { simp [part.eq_some_iff, univ] at h ⊢, rcases h with ⟨a, h, e⟩,\n     rcases nat.rpartrec.code.eval_inclusion h with ⟨s, hs⟩,\n     refine ⟨s, λ g h, ⟨a, hs (λ x y e ey, _), e⟩⟩, simp at ey, rcases ey with ⟨a, ea, ey⟩,\n     have := h _ _ e ea, simp[this, ey] }\n\ntheorem eval_inclusion_tot {e} {x : α} {y : σ}\n  {f : ℕ → τ} (h : y ∈ (⟦e⟧^f x : part σ)) : ∃ s, ∀ {g : ℕ → τ},\n  (∀ x y, x < s → f x = y → g x = y) → y ∈ (⟦e⟧^g x : part σ) := \nby { rcases eval_inclusion h with ⟨s, hs⟩, refine ⟨s, λ g hfg, hs _⟩,\n     simp, exact hfg }\n\nend rpartrec\n\ntheorem rcomputable.curry {α} [primcodable α] {σ : Type*} {τ : Type*} [primcodable σ] [primcodable τ]\n  {o : σ →. τ} : (@curry α _) computable₂_in o := rpartrec.curry_prim.to_rcomp", "meta": {"author": "iehality", "repo": "lean-reducibility", "sha": "82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7", "save_path": "github-repos/lean/iehality-lean-reducibility", "path": "github-repos/lean/iehality-lean-reducibility/lean-reducibility-82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7/src/coding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3369312803165705}}
{"text": "import tactic data.nat.digits .yeet\n\ntheorem unyeetable_3p2 : ¬ ∃ b, yeet b 3 2 :=\nbegin\n  rintro ⟨b, h⟩,\n  unfold yeet at h,\n  cases b, { change 9 = 3 at h, norm_num at h },\n  cases b, { change 9 = 5 at h, norm_num at h },\n  cases b, { norm_num [nat.of_digits] at h },\n  cases b, { norm_num [nat.of_digits] at h },\n  have h2b : 2 < b + 4 := by linarith,\n  have h3b : 3 < b + 4 := by linarith,\n  rw nat.digits_of_lt _ _ dec_trivial h2b at h,\n  rw nat.digits_of_lt _ _ dec_trivial h3b at h,\n  change 9 = nat.of_digits _ [3, 2] at h,\n  change 9 = 3 + (b + 4) * 2 at h,\n  rw [add_mul, ←add_assoc, add_comm, ←add_assoc] at h,\n  exact absurd (le_trans (le_add_right (le_refl _)) (ge_of_eq h)) dec_trivial\nend\n\ntheorem unyeetable_3p4 : ¬ ∃ b, yeet b 3 4 :=\nbegin\n  rintro ⟨b, h⟩,\n  unfold yeet at h,\n  cases b, { change _ = 3 at h, norm_num at h },\n  cases b, { change _ = 7 at h, norm_num at h },\n  cases b, { norm_num [nat.of_digits] at h },\n  cases b, { norm_num [nat.of_digits] at h },\n  cases b, { norm_num [nat.of_digits] at h },\n  have h3b : 3 < b + 5 := by linarith,\n  have h4b : 4 < b + 5 := by linarith,\n  rw nat.digits_of_lt _ _ dec_trivial h3b at h,\n  rw nat.digits_of_lt _ _ dec_trivial h4b at h,\n  norm_num [nat.of_digits] at h,\n  apply_fun (% 4) at h,\n  norm_num at h\nend\n", "meta": {"author": "ocornoc", "repo": "yeet", "sha": "36796d756c451a7a1807e3e8714c552eca6dac65", "save_path": "github-repos/lean/ocornoc-yeet", "path": "github-repos/lean/ocornoc-yeet/yeet-36796d756c451a7a1807e3e8714c552eca6dac65/src/unyeet3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3369312803165705}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.uliftable\nimport Mathlib.Lean3Lib.system.random\nimport Mathlib.system.random.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 v \n\nnamespace Mathlib\n\n/-!\n# `gen` Monad\n\nThis monad is used to formulate randomized computations with a parameter\nto specify the desired size of the result.\n\nThis is a port of the Haskell QuickCheck library.\n\n## Main definitions\n  * `gen` monad\n\n## Local notation\n\n * `i .. j` : `Icc i j`, the set of values between `i` and `j` inclusively;\n\n## Tags\n\nrandom testing\n\n## References\n\n  * https://hackage.haskell.org/package/QuickCheck\n\n-/\n\nnamespace slim_check\n\n\n/-- Monad to generate random examples to test properties with.\nIt has a `nat` parameter so that the caller can decide on the\nsize of the examples. -/\ndef gen (α : Type u) :=\n  reader_t (ulift ℕ) rand α\n\n/-- Execute a `gen` inside the `io` monad using `i` as the example\nsize and with a fresh random number generator. -/\ndef io.run_gen {α : Type} (x : gen α) (i : ℕ) : io α :=\n  io.run_rand (reader_t.run x (ulift.up i))\n\nnamespace gen\n\n\n/-- Lift `random.random` to the `gen` monad. -/\ndef choose_any (α : Type u) [random α] : gen α :=\n  reader_t.mk fun (_x : ulift ℕ) => rand.random α\n\n/-- Lift `random.random_r` to the `gen` monad. -/\ndef choose {α : Type u} [preorder α] [bounded_random α] (x : α) (y : α) (p : x ≤ y) : gen ↥(set.Icc x y) :=\n  reader_t.mk fun (_x : ulift ℕ) => rand.random_r x y p\n\n/-- Generate a `nat` example between `x` and `y`. -/\ndef choose_nat (x : ℕ) (y : ℕ) (p : x ≤ y) : gen ↥(set.Icc x y) :=\n  choose x y p\n\n/-- Generate a `nat` example between `x` and `y`. -/\ndef choose_nat' (x : ℕ) (y : ℕ) (p : x < y) : gen ↥(set.Ico x y) :=\n  (fun (this : ∀ (i : ℕ), x < i → i ≤ y → Nat.pred i < y) => subtype.map Nat.pred sorry <$> choose (x + 1) y p) sorry\n\nprotected instance uliftable : uliftable gen gen :=\n  reader_t.uliftable' (equiv.trans equiv.ulift (equiv.symm equiv.ulift))\n\nprotected instance has_orelse : has_orelse gen :=\n  has_orelse.mk\n    fun (α : Type u) (x y : gen α) =>\n      do \n        let b ← uliftable.up (choose_any Bool)\n        ite (↥(ulift.down b)) x y\n\n/-- Get access to the size parameter of the `gen` monad. For\nreasons of universe polymorphism, it is specified in\ncontinuation passing style. -/\ndef sized {α : Type u} (cmd : ℕ → gen α) : gen α :=\n  reader_t.mk fun (_x : ulift ℕ) => sorry\n\n/-- Apply a function to the size parameter. -/\ndef resize {α : Type u} (f : ℕ → ℕ) (cmd : gen α) : gen α :=\n  reader_t.mk fun (_x : ulift ℕ) => sorry\n\n/-- Create `n` examples using `cmd`. -/\ndef vector_of {α : Type u} (n : ℕ) (cmd : gen α) : gen (vector α n) :=\n  sorry\n\n/-- Create a list of examples using `cmd`. The size is controlled\nby the size parameter of `gen`. -/\ndef list_of {α : Type u} (cmd : gen α) : gen (List α) :=\n  sized\n    fun (sz : ℕ) =>\n      do \n        uliftable.up (choose_nat 0 (sz + 1) sorry)\n        sorry\n\n/-- Given a list of example generators, choose one to create an example. -/\ndef one_of {α : Type u} (xs : List (gen α)) (pos : 0 < list.length xs) : gen α :=\n  do \n    uliftable.up (choose_nat' 0 (list.length xs) pos)\n    sorry\n\n/-- Given a list of example generators, choose one to create an example. -/\ndef elements {α : Type u} (xs : List α) (pos : 0 < list.length xs) : gen α :=\n  do \n    uliftable.up (choose_nat' 0 (list.length xs) pos)\n    sorry\n\n/--\n`freq_aux xs i _` takes a weighted list of generator and a number meant to select one of the generators.\n\nIf we consider `freq_aux [(1, gena), (3, genb), (5, genc)] 4 _`, we choose a generator by splitting\nthe interval 1-9 into 1-1, 2-4, 5-9 so that the width of each interval corresponds to one of the\nnumber in the list of generators. Then, we check which interval 4 falls into: it selects `genb`.\n-/\ndef freq_aux {α : Type u} (xs : List (ℕ+ × gen α)) (i : ℕ) : i < list.sum (list.map (subtype.val ∘ prod.fst) xs) → gen α :=\n  sorry\n\n/--\n`freq [(1, gena), (3, genb), (5, genc)] _` will choose one of `gena`, `genb`, `genc` with\nprobabiities proportional to the number accompanying them. In this example, the sum of\nthose numbers is 9, `gena` will be chosen with probability ~1/9, `genb` with ~3/9 (i.e. 1/3)\nand `genc` with probability 5/9.\n-/\ndef freq {α : Type u} (xs : List (ℕ+ × gen α)) (pos : 0 < list.length xs) : gen α :=\n  let s : ℕ := list.sum (list.map (subtype.val ∘ prod.fst) xs);\n  (fun (ha : 1 ≤ s) =>\n      (fun (this : 0 ≤ s - 1) =>\n          uliftable.adapt_up gen gen (choose_nat 0 (s - 1) this)\n            fun (i : ↥(set.Icc 0 (s - 1))) => freq_aux xs (subtype.val i) sorry)\n        sorry)\n    sorry\n\n/-- Generate a random permutation of a given list. -/\ndef permutation_of {α : Type u} (xs : List α) : gen (Subtype (list.perm xs)) :=\n  sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/testing/slim_check/gen.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.33692316786522725}}
{"text": "\nimport syntax.syntaxCLK \nimport syntax.axiomsCLK \nimport semantics.semanticsCLK\n\nlocal attribute [instance] classical.prop_decidable\n\nopen set\n\n---------------------- Soundness ----------------------\n\nnoncomputable theorem soundnessCLK {agents : Type} [hN: fintype agents] (φ : formCLK agents) : axCLK φ → global_valid φ :=\nbegin\nintro h,\ninduction h,\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1 h2, \n  exact h1, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1 h2 h3, \n  apply h1, \n    { exact h3,},\n\n    { apply h2, \n      exact h3 }, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1 h2,\n  by_contradiction hf,\n  exact (h1 hf) (h2 hf), },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1 h2, \n  exact and.intro h1 h2, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1, \n  exact h1.left, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1, \n  exact h1.right, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1 h2,\n  by_contradiction hf,\n  exact h1 hf h2, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1, \n  exact m.f.E.liveness s h h1, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s,\n  simp [s_entails_CLK],\n  exact m.f.E.safety s h, },\n\n{ unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n  intros m s h1,\n  apply m.f.E.N_max,\n  by_contradiction,\n  exact h1 h, },\n\n{ unfold global_valid valid_m,\n  intros m s,\n  have := m.f.E.monoticity s h_G {t: m.f.states | s_entails_CLK m t (h_φ & h_ψ)} {t: m.f.states | s_entails_CLK m t h_φ} _,\n  { unfold s_entails_CLK s_entails_CLK.aux at ⊢ this,\n    exact this },\n  intros t h1,\n  unfold s_entails_CLK s_entails_CLK.aux at h1,\n  exact h1.left, },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s h1,\n    exact m.f.E.superadd s h_G h_F {t: m.f.states | s_entails_CLK m t h_φ} {t: m.f.states | s_entails_CLK m t h_ψ} h1.left h1.right h_hInt, },\n\n  { unfold global_valid valid_m,\n    intros m s,\n    have := h_ih_hImp m s,\n    unfold s_entails_CLK s_entails_CLK.aux at ⊢ this,\n    apply this,\n    exact h_ih_hL m s, },\n\n  { intros m s,\n    have heq: {t: m.f.states | s_entails_CLK m t h_φ} = {t: m.f.states | s_entails_CLK m t h_ψ}, from\n      begin\n        apply set.ext,\n        intros u,\n        have h_ih := h_ih m u,\n        unfold s_entails_CLK s_entails_CLK.aux at h_ih,\n        cases h_ih,\n        apply iff.intro,\n\n        { intro hu,\n          exact h_ih_left hu, },\n\n        { intro hu,\n          exact h_ih_right hu, }\n      end,\n    unfold s_entails_CLK s_entails_CLK.aux,\n    apply and.intro,\n\n    { intro h1,\n      unfold s_entails_CLK s_entails_CLK.aux at *,\n      rw ← heq,\n      exact h1, },\n\n    { intro h1,\n      unfold s_entails_CLK s_entails_CLK.aux at *,\n      rw heq,\n      exact h1, } },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s h1 h2 t ht,\n    exact h1 t ht (h2 t ht), },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s h,\n    exact h s (m.f.rfl h_i s), },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s h t ht u hu,\n    exact h u (m.f.trans h_i s t u ht hu), },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s h1 h2,\n    exact h1 (h2 s (m.f.rfl h_i s)), },\n\n  { unfold global_valid valid_m s_entails_CLK s_entails_CLK.aux,\n    intros m s t h2,\n    apply h_ih, },\nend\n\ninductive single : Type\n  | one: single\n\n\nlemma univ_single : (set.univ: set single) = {single.one} := \nbegin\n  rw eq_comm,\n  rw set.eq_univ_iff_forall,\n  intro x,\n  cases x,\n  simp,\nend\n\nlemma single_nonempty : nonempty single := \nbegin\n  apply exists_true_iff_nonempty.mp,\n  apply exists.intro single.one,\n  exact trivial,\nend\n\ndef m_ex (ha: nonempty agents) : modelCLK agents  :=\n\n{ f := \n  { states := single,\n    hs := single_nonempty,\n    ha := ha,\n    E  :=  \n    { E := λ s G, {{single.one}},\n      liveness := \n      begin \n        intros _ _ hf, \n        simp at hf, \n        rw set.ext_iff at hf, \n        simp at hf,\n        apply hf single.one,\n        refl, \n      end,\n      safety:=\n        begin\n          intros _ _, simp at *,\n          exact univ_single,\n        end,\n      N_max :=\n        begin\n          intros _ _ hxc, simp at *,\n          rw ←univ_single at *,\n          have hcond : {single.one} ≠ (∅: set single), \n\n            { intro hf,\n              rw set.ext_iff at hf, \n              simp at *,\n              apply hf single.one,\n              refl, },\n          simp [hcond] at *, by_contradiction,\n          have hex: ∃ x, x ∈ X, from nonempty_def.mp (ne_empty_iff_nonempty.mp hxc),\n          cases hex with s hs,\n          cases s,\n          rw ←singleton_subset_iff at hs,\n          rw ←univ_single at hs,\n          exact h (univ_subset_iff.mp hs),\n        end,\n      monoticity:=\n        begin\n          intros _ _ _ _ hxy hx,\n          simp [←univ_single] at *,\n          rw hx at hxy,\n          exact univ_subset_iff.mp hxy,\n        end,\n      superadd:=\n      begin\n        intros _ _ _ _ _ hX hY hGF,\n        simp at *,\n        simp[hX, hY],\n      end },\n    rel := λ a s, {s},\n    rfl := by simp,\n    sym :=\n    begin\n      intros i s t h,\n      simp at *,\n      rw h,\n    end,\n    trans :=\n    begin\n      intros i s t u hst htu,\n      simp at *,\n      simp[hst, htu],\n    end, },\n  v := λ _, {}, }\n\nlemma nprfalseCLK {agents: Type} (ha: nonempty agents) [fintype agents] : ¬ @axCLK agents (⊥) :=\nbegin\napply (mt (soundnessCLK (@formCLK.bot agents))),\nintro hf ,\nsimp[global_valid, valid_m, s_entails_CLK] at hf,\napply hf (m_ex ha),\nexact single.one,\nend\n", "meta": {"author": "kaiobendrauf", "repo": "cl-lean", "sha": "15568f16cf57a07db6192fbd8084d59cc1aef1df", "save_path": "github-repos/lean/kaiobendrauf-cl-lean", "path": "github-repos/lean/kaiobendrauf-cl-lean/cl-lean-15568f16cf57a07db6192fbd8084d59cc1aef1df/src/soundness/soundnessCLK.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.33666038500703005}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space ℝ (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  -- First, we define a collection of pen balls.\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  have h1 : ∀ m : ℕ, ∃ Bm : set (euclidean_space ℝ (fin n)), is_open Bm ∧ is_ball Bm (0 : euclidean_space ℝ (fin n)) m, from by auto [is_open_ball],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem,\n  have h2 : ∀ m : ℕ, is_compact (closure (classical.some (h1 m).exists)), from by auto [is_compact_closure, is_compact_ball],\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$,\n  have h3 : ∀ m : ℕ, ∃ Cm : set (euclidean_space ℝ (fin n)), is_open Cm ∧ is_finite Cm ∧ (∀ (U : set (euclidean_space ℝ (fin n))) (hU : U ∈ A), U ∩ Cm ≠ ∅) ∧ (Cm ⊆ (euclidean_space ℝ (fin n)) \\ (classical.some (h1 (m - 1)).exists)), from by auto [is_open_inter, is_open_compl, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is_open_closure, is_open_ball, is\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space ℝ (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  -- First, we define a collection of pen balls.\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  let B : ℕ → set (euclidean_space ℝ (fin n)) := λ m, {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) m) ∧ x = y},\n  have hB : ∀ m : ℕ, B m = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) m) ∧ x = y}, from by auto [B],\n  have hB0 : B 0 = ∅, from by auto [B, hB],\n  have hBm : ∀ m : ℕ, B m = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) m) ∧ x = y}, from by auto [B, hB],\n  have hBm1 : ∀ m : ℕ, B (m + 1) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 1)) ∧ x = y}, from by auto [B, hB],\n  have hBm2 : ∀ m : ℕ, B (m + 2) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 2)) ∧ x = y}, from by auto [B, hB],\n  have hBm3 : ∀ m : ℕ, B (m + 3) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 3)) ∧ x = y}, from by auto [B, hB],\n  have hBm4 : ∀ m : ℕ, B (m + 4) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 4)) ∧ x = y}, from by auto [B, hB],\n  have hBm5 : ∀ m : ℕ, B (m + 5) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 5)) ∧ x = y}, from by auto [B, hB],\n  have hBm6 : ∀ m : ℕ, B (m + 6) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 6)) ∧ x = y}, from by auto [B, hB],\n  have hBm7 : ∀ m : ℕ, B (m + 7) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 7)) ∧ x = y}, from by auto [B, hB],\n  have hBm8 : ∀ m : ℕ, B (m + 8) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 8)) ∧ x = y}, from by auto [B, hB],\n  have hBm9 : ∀ m : ℕ, B (m + 9) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 9)) ∧ x = y}, from by auto [B, hB],\n  have hBm10 : ∀ m : ℕ, B (m + 10) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 10)) ∧ x = y}, from by auto [B, hB],\n  have hBm11 : ∀ m : ℕ, B (m + 11) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 11)) ∧ x = y}, from by auto [B, hB],\n  have hBm12 : ∀ m : ℕ, B (m + 12) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 12)) ∧ x = y}, from by auto [B, hB],\n  have hBm13 : ∀ m : ℕ, B (m + 13) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 13)) ∧ x = y}, from by auto [B, hB],\n  have hBm14 : ∀ m : ℕ, B (m + 14) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 14)) ∧ x = y}, from by auto [B, hB],\n  have hBm15 : ∀ m : ℕ, B (m + 15) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 15)) ∧ x = y}, from by auto [B, hB],\n  have hBm16 : ∀ m : ℕ, B (m + 16) = {x : euclidean_space ℝ (fin n) | ∃ y : euclidean_space ℝ (fin n), y ∈ (ball (0 : euclidean_space ℝ (fin n)) (m + 16)) ∧ x\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space ℝ (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  have h1 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_cover C ∧ is_locally_finite C ∧ is_open_refinement A C, from by auto [paracompact_space.paracompact_iff_locally_finite_open_refinement],\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  have h2 : ∀ m : ℕ, ∃ Bm : set (euclidean_space ℝ (fin n)), is_ball Bm (0 : euclidean_space ℝ (fin n)) m, from by auto [is_ball],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem,\n  have h3 : ∀ m : ℕ, ∃ Bm : set (euclidean_space ℝ (fin n)), is_ball Bm (0 : euclidean_space ℝ (fin n)) m ∧ is_compact Bm, from by auto [is_ball, is_compact],\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$,\n  have h4 : ∀ m : ℕ, ∃ Cm : set (euclidean_space ℝ (fin n)), is_ball Cm (0 : euclidean_space ℝ (fin n)) m ∧ is_compact Cm ∧ is_finite Cm ∧ is_open_cover Cm, from by auto [is_ball, is_compact, is_finite, is_open_cover],\n  -- and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  have h5 : ∀ m : ℕ, ∃ Cm : set (euclidean_space ℝ (fin n)), is_ball Cm (0 : euclidean_space ℝ (fin n)) m ∧ is_compact Cm ∧ is_finite Cm ∧ is_open_cover Cm ∧ is_open_refinement A Cm, from by auto [is_ball, is_compact, is_finite, is_open_cover, is_open_refinement],\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  have h6 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_refinement A C, from by auto [is_open_refinement],\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$.\n  have h7 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_refinement A C ∧ is_open_cover C, from by auto [is_open_refinement, is_open_cover],\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$.\n  have h8 : ∃ C : set (euclidean_space ℝ (fin n)), is_open_refinement A C ∧ is_open_cover C ∧ is_locally_finite C, from by auto [is_open_refinement, is_open_cover, is_locally_finite],\n  -- So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  show paracompact_space (euclidean_space ℝ (fin n)), from by auto [paracompact_space.paracompact_iff_locally_finite_open_refinement],\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872019117029, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.33663427111093547}}
{"text": "import FOL.deduction FOL.semantics FOL.lindenbaum order.filter.ultrafilter data.finset.basic\nopen encodable\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nopen logic.semantics\n\nvariables {L : language.{u}} {I : Type u} [inhabited I] (F : ultrafilter I) {𝔄 : I → Structure L}\n\nlocal notation (name := dom) `|`M`|` := Structure.dom M\n\ndef uequiv : (Π i, |𝔄 i|) → (Π i, |𝔄 i|) → Prop :=\nλ u₁ u₂, {i | u₁ i = u₂ i} ∈ F\n\nnotation u` ~[`:80 F`] `v:80 := uequiv F u v\n\n@[simp] lemma uequiv_refl (u : Π i, |𝔄 i|) : u ~[F] u :=\nby { simp[uequiv], exact F.univ_sets }\n\nlemma uequiv_symm {u₁ u₂ : Π i, |𝔄 i|} : u₁ ~[F] u₂ → u₂ ~[F] u₁ :=\nby { simp[uequiv], have : {i | u₁ i = u₂ i} = {i | u₂ i = u₁ i}, { ext, simp, exact eq_comm }, simp[this] }\n\nlemma uequiv_trans {u₁ u₂ u₃ : Π i, |𝔄 i|} : u₁ ~[F] u₂ → u₂ ~[F] u₃ → u₁ ~[F] u₃ :=\nby { simp[uequiv], intros h₁ h₂,\n     have : {i | u₁ i = u₂ i} ∩ {i | u₂ i = u₃ i} ⊆ {i | u₁ i = u₃ i},\n     { intros i hi, simp* at* },\n     exact F.sets_of_superset (F.inter_sets h₁ h₂) this }\n\ntheorem uequiv_equivalence : equivalence (@uequiv L I _ F 𝔄) :=\n⟨uequiv_refl F, λ _ _ , uequiv_symm F, λ _ _ _, uequiv_trans F⟩\n\n\n@[reducible, simp, instance]\ndef ult (𝔄 : I → Structure L) (F : ultrafilter I) : setoid (Π i, |𝔄 i|) := ⟨@uequiv L I _ F 𝔄, uequiv_equivalence F⟩\n\ndef Ult (𝔄 : I → Structure L) (F : ultrafilter I) : Type* :=\nquotient (ult 𝔄 F: setoid (Π i, |𝔄 i|))\n\ndef to_quotient {𝔄 : I → Structure L} {F : ultrafilter I} (u : Π i, |𝔄 i|) : Ult 𝔄 F := quotient.mk' u\n\nnotation `⟦`u`⟧*` :max := to_quotient u\n\ninstance : inhabited (Ult 𝔄 F) := ⟨⟦λ i, default⟧*⟩\n\nnamespace Ult\nopen logic.semantics\n\n@[elab_as_eliminator]\nprotected lemma ind_on {C : Ult 𝔄 F → Prop} (u : Ult 𝔄 F)\n  (h : ∀ u : Π i, |𝔄 i|, C ⟦u⟧*) : C u :=\nquotient.induction_on' u h\n\n@[elab_as_eliminator, reducible]\nprotected def lift_on {φ} (d : Ult 𝔄 F) (f : (Π i, |𝔄 i|) → φ)\n  (h : ∀ (v u : Π i, |𝔄 i|), v ~[F] u → f v = f u) : φ :=\nquotient.lift_on' d f h\n\n@[simp]\nprotected lemma lift_on_eq {φ} (u₀ : Π i, |𝔄 i|) (f : (Π i, |𝔄 i|) → φ)\n  (h : ∀ v u, v ~[F] u → f v = f u) : fol.Ult.lift_on F ⟦u₀⟧* f h = f u₀ := rfl\n\n@[elab_as_eliminator, reducible, simp]\nprotected def lift_on₂ {φ} (u₁ u₂ : Ult 𝔄 F) (f : (Π i, |𝔄 i|) → (Π i, |𝔄 i|) → φ)\n  (h : ∀ u₁ u₂ v₁ v₂, u₁ ~[F] v₁ → u₂ ~[F] v₂ → f u₁ u₂ = f v₁ v₂) : φ :=\nquotient.lift_on₂' u₁ u₂ f h\n\n@[simp]\nprotected lemma lift_on₂_eq {φ} (u₁ u₂ : Π i, |𝔄 i|) (f : (Π i, |𝔄 i|) → (Π i, |𝔄 i|) → φ)\n  (h : ∀ t₁ t₂ u₁ u₂, (t₁ ~[F] u₁) → (t₂ ~[F] u₂) → f t₁ t₂ = f u₁ u₂) :\n  fol.Ult.lift_on₂ F ⟦u₁⟧* ⟦u₂⟧* f h = f u₁ u₂ := rfl\n\n@[elab_as_eliminator, reducible]\nprotected def lift_on_finitary {φ} {n : ℕ} (v : finitary (Ult 𝔄 F) n) (f : finitary (Π i, |𝔄 i|) n → φ)\n  (h : ∀ v₁ v₂ : finitary (Π i, |𝔄 i|) n, (∀ n, (v₁ n) ~[F] (v₂ n)) → f v₁ = f v₂) : φ :=\nquotient.lift_on_finitary v f h \n\n@[simp]\nprotected lemma lift_on_finitary_eq {φ} {n} (v : finitary (Π i, |𝔄 i|) n) (f : finitary (Π i, |𝔄 i|) n → φ)\n  (h : ∀ v₁ v₂ : finitary (Π i, |𝔄 i|) n, (∀ n, (v₁ n) ~[F] (v₂ n)) → f v₁ = f v₂) :\n  fol.Ult.lift_on_finitary F (λ x, ⟦v x⟧*) f h = f v :=\nquotient.lift_on_finitary_eq v f h\n\n@[simp] lemma of_eq_of {u₁ u₂ : Π i, |𝔄 i|} : (⟦u₁⟧* : Ult 𝔄 F) = ⟦u₂⟧* ↔ u₁ ~[F] u₂ :=\nby simp[to_quotient, quotient.eq']\n\nlemma equivs_mem {n} {v₁ v₂ : finitary (Π i, |𝔄 i|) n} (h : ∀ (x : fin n), {i : I | v₁ x i = v₂ x i} ∈ F) :\n  {i | (λ x, v₁ x i) = (λ x, v₂ x i)} ∈ F := \nbegin\n  induction n with n IH,\n  { have : {i : I | (λ x, v₁ x i) = (λ x, v₂ x i)} = set.univ,\n    { ext i, simp }, rw this, exact F.univ_sets },\n  { have ss : {i | v₁ 0 i = v₂ 0 i} ∩ {i | (λ x, v₁.tail x i) = (λ x, v₂.tail x i)} ⊆ {i : I | (λ x, v₁ x i) = (λ x, v₂ x i)},\n    { intros i hi, simp[finitary.tail] at*,\n      funext x, refine fin.cases _ _ x,\n      { exact hi.1 },\n      { intros j, have := congr_fun hi.2 j, simp at this, exact this } },\n    have : {i | v₁ 0 i = v₂ 0 i} ∩ {i | (λ x, v₁.tail x i) = (λ x, v₂.tail x i)} ∈ F,\n      from (F.inter_sets (h _) (@IH v₁.tail v₂.tail (λ x, h _))),\n    refine F.sets_of_superset this ss }\nend\n\nlemma fn_equiv {n} {v₁ v₂ : finitary (Π i, |𝔄 i|) n} (h : ∀ x, v₁ x ~[F] v₂ x) (f : L.fn n) :\n  (λ i, (𝔄 i).fn f (λ x, v₁ x i)) ~[F] (λ i, (𝔄 i).fn f (λ x, v₂ x i)) :=\nbegin\n  simp[uequiv] at*,\n  have : {i | (λ x, v₁ x i) = (λ x, v₂ x i)} ⊆ {i | (𝔄 i).fn f (λ x, v₁ x i) = (𝔄 i).fn f (λ x, v₂ x i)},\n  { intros i hi, simp* at* },\n  exact F.sets_of_superset (equivs_mem F h) this\nend\n\nlemma pr_equiv : ∀ {n} {v₁ v₂ : finitary (Π i, |𝔄 i|) n} (h : ∀ x, v₁ x ~[F] v₂ x) (p : L.pr n),\n  {i | (𝔄 i).pr p (λ x, v₁ x i)} ∈ F ↔ {i | (𝔄 i).pr p (λ x, v₂ x i)} ∈ F :=\nbegin\n  suffices : ∀ {n} {v₁ v₂ : finitary (Π i, |𝔄 i|) n} (h : ∀ x, v₁ x ~[F] v₂ x) (p : L.pr n),\n  {i | (𝔄 i).pr p (λ x, v₁ x i)} ∈ F → {i | (𝔄 i).pr p (λ x, v₂ x i)} ∈ F,\n  { intros n v₁ v₂ eqn p, refine ⟨this eqn p, this (λ x, uequiv_symm _ (eqn x)) p⟩ },\n  intros n v₁ v₂ eqn p h,\n  have : {i | (𝔄 i).pr p (λ x, v₁ x i)} ∩ {i | (λ x, v₁ x i) = (λ x, v₂ x i)} ⊆ {i | (𝔄 i).pr p (λ x, v₂ x i)},\n  { intros i hi, simp* at*, simp[←hi.2], exact hi.1 },\n  refine F.sets_of_superset (F.inter_sets h (equivs_mem _ eqn)) this\nend\n\ndef product_fn (n) (f : L.fn n) : finitary (Ult 𝔄 F) n → Ult 𝔄 F :=\nλ v, fol.Ult.lift_on_finitary F v (λ v, (⟦λ i, (𝔄 i).fn f (λ x, v x i)⟧* : Ult 𝔄 F)) $ λ u₁ u₂ eqn,\nby { simp, exact fn_equiv F eqn f }\n\ndef product_pr (n) (p : L.pr n) : finitary (Ult 𝔄 F) n → Prop :=\nλ v, fol.Ult.lift_on_finitary F v (λ v, {i | (𝔄 i).pr p (λ x, v x i)} ∈ F) $ λ u₁ u₂ eqn,\nby { simp, exact pr_equiv F eqn p }\n\ndef product (𝔄 : I → Structure L) (F : ultrafilter I) : Structure L := ⟨Ult 𝔄 F, ⟨default⟩, product_fn F, product_pr F⟩\nnotation `ℿ `𝔄` ⫽ `F:90 := product 𝔄 F\n\nvariables {F}\n\n@[simp] lemma ult_eq : Ult 𝔄 F = |ℿ 𝔄 ⫽ F| := rfl\n\nprivate lemma Structure_exists (p : formula L) {e : ∀ i, ℕ → |𝔄 i|} (h : {i | ∃ u, 𝔄 i ⊧[u ⌢ e i] p } ∈ F) :\n  ∃ (u : Π i, |𝔄 i|), {i | 𝔄 i ⊧[(u i) ⌢ e i] p} ∈ F :=\nbegin\n  have : ∀ i, ∃ u, i ∈ {i | ∃ u, 𝔄 i ⊧[u ⌢ e i] p} → 𝔄 i ⊧[u ⌢ e i] p,\n  { intros i, simp, by_cases C : i ∈ {i | ∃ u, 𝔄 i ⊧[u ⌢ e i] p}; simp at C,\n    { rcases C with ⟨u, hu⟩, refine ⟨u, λ v _, hu⟩ },\n    { refine ⟨default, λ _ h, _⟩, exfalso, refine C _ h } },\n  rcases classical.skolem.mp this with ⟨u, hu⟩,\n  refine ⟨u, _⟩, exact F.sets_of_superset h hu\nend\n\nlemma Structure_fn_eq {n} (f : L.fn n) : (ℿ 𝔄 ⫽ F).fn f = product_fn F _ f := rfl\n\nlemma Structure_pr_eq {n} (r : L.pr n) : (ℿ 𝔄 ⫽ F).pr r = product_pr F _ r := rfl\n\nlemma models_pr_iff_lmm : ∀ (t : term L) (e : ∀ i, ℕ → |𝔄 i|),\n  (@term.val _ (ℿ 𝔄 ⫽ F) (λ n, ⟦λ i, e i n⟧*) t) = ⟦λ i, @term.val _ (𝔄 i) (λ n, e i n) t⟧*\n| (#n)                _ := by simp \n| (@term.app _ n f v) e :=\n  by { simp[Structure_fn_eq, product_fn],\n       let v' : finitary (Π i, |𝔄 i|) n := λ x i, (v x).val (𝔄 i) (e i),\n       have : (λ x, @term.val _ (ℿ 𝔄 ⫽ F) (λ n, ⟦(λ i, e i n)⟧*) (v x)) = λ x, ⟦v' x⟧*,\n       { funext x, simp[v', models_pr_iff_lmm (v x)] },\n       simp[this] }\n\nlemma models_pr_iff {n} (r : L.pr n) (v : finitary (term L) n) (e : ∀ i, ℕ → |𝔄 i|) :\n  (ℿ 𝔄 ⫽ F).pr r (λ x, (v x).val (ℿ 𝔄 ⫽ F) (λ n, ⟦λ i, e i n⟧*)) ↔ {i | (𝔄 i).pr r (λ x, (v x).val (𝔄 i) (e i))} ∈ F :=\nby simp[models_pr_iff_lmm, Structure_pr_eq, product_pr]\n\n-- Łoś's theorem\ntheorem fundamental_param : ∀ (p : formula L) (e : ∀ i, ℕ → |𝔄 i|),\n  ℿ 𝔄 ⫽ F ⊧[λ n, ⟦λ i, e i n⟧*] p ↔ {i | 𝔄 i ⊧[e i] p} ∈ F\n| ⊤                 _ := by { simp, exact F.univ_sets }\n| (formula.app p v) e := models_pr_iff p _ _\n| (t₁ =' t₂)      e := by simp[models_pr_iff_lmm]; refl\n| (p ⟶ q)       e := by { simp[fundamental_param p, fundamental_param q],\n    show {i | 𝔄 i ⊧[e i] p} ∈ F → {i | 𝔄 i ⊧[e i] q} ∈ F ↔ {i | 𝔄 i ⊧[e i] p → 𝔄 i ⊧[e i] q} ∈ F,\n    split,\n    { intros h, by_cases C : {i | 𝔄 i ⊧[e i] p} ∈ F,\n      { have : {i | 𝔄 i ⊧[e i] q} ⊆ {i | 𝔄 i ⊧[e i] p → 𝔄 i ⊧[e i] q}, { intros i hi, simp* at* },\n        exact F.sets_of_superset (h C) this },\n      { have : {i | 𝔄 i ⊧[e i] p}ᶜ ∈ F, from ultrafilter.compl_mem_iff_not_mem.mpr C,\n        have ss : {i | 𝔄 i ⊧[e i] p}ᶜ ⊆ {i | 𝔄 i ⊧[e i] p → 𝔄 i ⊧[e i] q},\n        { intros i hi, simp* at* },\n        exact F.sets_of_superset this ss } },\n    { intros h₁ h₂,\n      have : {i | 𝔄 i ⊧[e i] p} ∩ {i | 𝔄 i ⊧[e i] p → 𝔄 i ⊧[e i] q} ⊆ {i | 𝔄 i ⊧[e i] q},\n      { intros i hi, simp at*, refine hi.2 hi.1 },\n      exact filter.mp_mem h₂ h₁ } }\n| (∼p)          e := by { simp[fundamental_param p], exact ultrafilter.eventually_not.symm }\n| (∀.p)          e := by { simp, \n    calc\n      (∀ u, ℿ 𝔄 ⫽ F ⊧[u ⌢ λ n, ⟦λ i, e i n⟧*] p)\n          ↔ (∀ (u : Π i, |𝔄 i|), ℿ 𝔄 ⫽ F ⊧[λ n, ⟦λ i, (λ i, (u i) ⌢ (e i)) i n⟧*] p) :\n        by { have eqn: ∀ u, (⟦u⟧* ⌢ λ n, ⟦(λ i, e i n)⟧*) = (λ n, ⟦(λ i, (u i) ⌢ e i $ n)⟧* : ℕ → |ℿ 𝔄 ⫽ F|),\n             { intros i, funext x, cases x; simp[concat] }, simp, split,\n             { intros h u, have := h ⟦u⟧*, simp[eqn] at this, exact this },\n             { intros h u, induction u using fol.Ult.ind_on, simp[eqn, h] } }\n      ... ↔ (∀ (u : Π i, |𝔄 i|), {i | 𝔄 i ⊧[u i ⌢ e i] p} ∈ F) :\n        by { split, { intros h u, simp[←fundamental_param  p _, h] }, { intros h u, simp[fundamental_param  p _, h] } }\n      ... ↔ {i | ∀ (u : |𝔄 i|), 𝔄 i ⊧[u ⌢ e i] p} ∈ F : \n        by { split,\n             { contrapose,\n               simp[←ultrafilter.compl_mem_iff_not_mem, set.compl_def], intros h,\n               show ∃ (u : Π i, |𝔄 i|), {i | ¬𝔄 i ⊧[u i ⌢ e i] p} ∈ F, from Structure_exists (∼p) h },\n             { refine λ h u, F.sets_of_superset h (λ _ _ , by simp* at*) } } }\n\ntheorem fundamental {p : formula L} :\n  ℿ 𝔄 ⫽ F ⊧ p ↔ {i | 𝔄 i ⊧ p} ∈ F :=\nbegin\n  calc\n    ℿ 𝔄 ⫽ F ⊧ p ↔ ℿ 𝔄 ⫽ F ⊧ nfal p p.arity : nfal_models_iff.symm\n    ...         ↔ {i | 𝔄 i ⊧ nfal p p.arity} ∈ F :\n      by simpa[eval_is_sentence_iff _ (formula.nfal_is_sentence p)] using fundamental_param (nfal p p.arity) (λ i n, default)\n    ...         ↔ {i | 𝔄 i ⊧ p} ∈ F :\n      by { have : {i | 𝔄 i ⊧ nfal p p.arity} = {i | 𝔄 i ⊧ p},\n           { ext i, simp, refine nfal_models_iff },\n           simp[this] }\nend\n\nend Ult\nend fol\n\nnamespace fol\nvariables {L : language.{u}} \n\ndef finTheory (T : Theory L) := {S : finset (formula L) // ∀ {x}, x ∈ S → x ∈ T}\n\nvariables {T : Theory L}\n\ndef finTheory.empty {T : Theory L} : finTheory T := ⟨∅, by simp⟩\ninstance : inhabited (finTheory T) := ⟨⟨∅, by simp⟩⟩\n\nnoncomputable def finTheory.insert (P : finTheory T) (p : formula L) (h : p ∈ T) : finTheory T :=\n⟨insert p P.val, λ x hx,  by { simp at hx, cases hx, simp[hx, h], refine P.property hx }⟩\n\n@[simp] lemma finTheory.insert_val (P : finTheory T) (p : formula L) (h : T p) :\n  (P.insert p h).val = insert p P.val := rfl\n\ninstance : has_coe (finTheory T) (set (formula L)) := ⟨λ S, {p | p ∈ S.val}⟩\n\nnamespace compactness\n\nvariables (𝔄 : finTheory T → Structure L) \n\ndef formdomain (p : formula L) : set (finTheory T) := {i | 𝔄 i ⊧ p}\n\ndef F : set (set (finTheory T)) := {x | ∃ p, T p ∧ x = formdomain 𝔄 p}\n\nprivate lemma finite_intersection_lmm (h : ∃ p, T p) (H : ∀ (i : finTheory T) p, p ∈ i.val → 𝔄 i ⊧ p) :\n  ∀ S : finset (set (finTheory T)), (↑S : set (set (finTheory T))) ⊆ F 𝔄 →\n  ∃ P : finTheory T,\n  (∀ p, p ∈ P.val → formdomain 𝔄 p ∈ S) ∧ (∀ S', S' ∈ S → ∃ p, p ∈ P.val ∧ S' = formdomain 𝔄 p) :=\nbegin\n  intros S, induction S using finset.induction with i S i_fresh IH,\n  { intros _, simp[set.nonempty], rcases h with ⟨p₀, hyp_p₀⟩,\n    refine ⟨⟨∅, by simp⟩, _⟩, unfold_coes, simp },\n  { intros h, simp at*,\n    have lmm₁ : ↑S ⊆ F 𝔄, from set.subset.trans (set.subset_insert _ _) h,\n    have : ∃ (P : finTheory T),\n      (∀ p, p ∈ ↑P → formdomain 𝔄 p ∈ S) ∧ (∀ S', S' ∈ S → ∃ p, p ∈ ↑P ∧ S' = formdomain 𝔄 p),\n    from IH lmm₁, rcases this with ⟨P, IH₁, IH₂⟩,\n    have : ∃ p, T p ∧ i = formdomain 𝔄 p, from h (set.mem_insert i ↑S),\n    rcases this with ⟨p, hyp_p, rfl⟩,\n    refine ⟨P.insert p hyp_p, _, _, _⟩; unfold_coes; simp,\n    { refine λ q hyp_q, or.inr (IH₁ _ hyp_q) },\n    { refine ⟨p, or.inl rfl, rfl⟩ },\n    { intros S' hyp_S',\n      have : ∃ p, p ∈ ↑P ∧ S' = formdomain 𝔄 p, from IH₂ _ hyp_S', rcases this with ⟨p, hyp, rfl⟩,\n      refine ⟨p, or.inr hyp, rfl⟩ } }\nend\n\ntheorem finite_intersection (h : ∃ p, T p) (H : ∀ (i : finTheory T) p, p ∈ i.val → 𝔄 i ⊧ p) :\n  ∀ S : finset (set (finTheory T)), \n  (↑S : set (set (finTheory T))) ⊆ F 𝔄 → (⋂₀ (↑S : set (set (finTheory T)))).nonempty :=\nbegin\n  intros S hS, have := finite_intersection_lmm _ h H S hS, rcases this with ⟨P, hyp⟩,\n  refine ⟨P, λ S' hS', _⟩, \n  have := hyp.2 S' hS', rcases this with ⟨p, hyp_p, rfl⟩, simp[formdomain] at*,\n  refine H _ _ hyp_p\nend\n\ntheorem ultrafilter_exists (h : ∃ p, p ∈ T) (H : ∀ (i : finTheory T) p, p ∈ i.val → 𝔄 i ⊧ p) :\n  ∃ U : ultrafilter (finTheory T), F 𝔄 ⊆ U.to_filter.sets :=\nultrafilter.exists_ultrafilter_of_finite_inter_nonempty _ (finite_intersection _ h H)\n\ntheorem compact (T : Theory L) :\n  Satisfiable T ↔ ∀ S : finset (formula L), ↑S ⊆ T → Satisfiable (S : Theory L) :=\n  ⟨by { intros H S hyp_S, rcases H with ⟨𝔄, hyp⟩,\n        refine ⟨𝔄, λ p h, hyp (hyp_S h)⟩ },\n   by { suffices : (∀ S : finTheory T, Satisfiable (S : Theory L)) → Satisfiable T,\n        { intros h, refine this (λ S, _),\n          rcases h S.val (λ p, S.property) with ⟨𝔄, hyp_𝔄⟩, refine ⟨𝔄, hyp_𝔄⟩ },\n    intros H, by_cases C : T = ∅,\n        { rcases C with rfl, refine empty_has_Structure },\n        { have ex : ∃ p, p ∈ T, { by_contra, simp at*, refine C _, { ext x, simp, refine h _ } }, \n          have : ∃ (𝔄 : finTheory T → Structure L), ∀ (i : finTheory T) p, p ∈ i.val → 𝔄 i ⊧ p,\n          from classical.skolem.mp H, rcases this with ⟨𝔄, hyp_𝔄⟩,\n          have := ultrafilter_exists _ ex hyp_𝔄, rcases this with ⟨U, hyp_U⟩,\n          use ℿ 𝔄 ⫽ U, intros p hyp_p, rw Ult.fundamental,\n          have : {i | 𝔄 i ⊧ p} ∈ F 𝔄, { refine ⟨p, hyp_p, rfl⟩ },\n          exact hyp_U this } }⟩\n\nend compactness\n\nend fol\n", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/ultraproduct.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.33663142544877056}}
{"text": "import Runtime.Network.Graph.Path.Child\n\nnamespace Network.Graph.Path\n\ninductive Sibling : Path graph start → Path graph start → Prop\n  | nil : Sibling nil nil\n  | cons : (path₁ ≻ parent) → (path₂ ≻ parent) → Sibling path₁ path₂\n\ninfix:40 \" ≂ \" => Sibling\n\ntheorem Sibling.refl : ∀ path : Path graph start, (path ≂ path)\n  | .nil => nil\n  | .cons _ subpath => by have ⟨_, h⟩ := Succ.cons subpath; exact cons h h\n\ntheorem Sibling.symm : (path₁ ≂ path₂) → (path₂ ≂ path₁)\n  | nil => nil\n  | cons h₁ h₂ => cons h₂ h₁\n\ntheorem Sibling.iff_eq_prefix : (path₁ ≂ path₂) ↔ (path₁.prefix? = path₂.prefix?) := by\n  constructor\n  case mp =>\n    cases path₁ <;> cases path₂\n    case nil.nil => simp\n    case cons.cons => intro h; cases h; simp_all [Succ]\n    case nil.cons => intro h; cases h; case _ h _ => have := h.isCons; contradiction\n    case cons.nil => intro h; cases h; case _ h   => have := h.isCons; contradiction\n  case mpr =>\n    intro h\n    by_cases path₁.prefix?.isSome\n    case inr hp =>\n      have hc₁ := isNil_iff_not_isCons.mpr <| mt prefix?_isSome_iff_isCons.mpr hp; simp at hc₁\n      rw [h] at hp\n      have hc₂ := isNil_iff_not_isCons.mpr <| mt prefix?_isSome_iff_isCons.mpr hp; simp at hc₂\n      simp [hc₁, hc₂, Sibling.nil]\n    case inl hp =>\n      have ⟨_, hp₁⟩ := Option.isSome_iff_exists.mp hp\n      rw [h] at hp\n      have ⟨_, hp₂⟩ := Option.isSome_iff_exists.mp hp\n      simp_all\n      exact Sibling.cons hp₁ hp₂\n\ninstance : Decidable (path₁ ≂ path₂) :=\n  if h : path₁.prefix? = path₂.prefix?\n  then isTrue <| Sibling.iff_eq_prefix.mpr h\n  else isFalse <| mt Sibling.iff_eq_prefix.mp h\n\nend Network.Graph.Path\n", "meta": {"author": "lf-lang", "repo": "reactor-lean", "sha": "d2eb5458446af838be34ebb6f69549b2f6d9c04d", "save_path": "github-repos/lean/lf-lang-reactor-lean", "path": "github-repos/lean/lf-lang-reactor-lean/reactor-lean-d2eb5458446af838be34ebb6f69549b2f6d9c04d/Runtime/Network/Graph/Path/Sibling.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3364734692568112}}
{"text": "\nimport util.predicate\nimport util.classical\nimport util.meta.tactic\nimport tactic.linarith\n\n@[user_attribute]\nmeta def strengthening_attr : user_attribute :=\n{ name := `strengthening\n, descr := \"\nStrenghtening lemmas to facilitate the stripping of small details in application.\nExpected shape `∀ p : pred' α, ⊩ f p ⟶ g p`\n\" }\n\nrun_cmd mk_simp_attr `tl_simp [`simp]\nrun_cmd\ndo ns ← attribute.get_instances `simp,\n   let ns : list name := ns.filter (λ n, name.is_prefix_of `predicate n),\n   n ← tactic.mk_const (mk_simp_attr_decl_name `tl_simp),\n   ns.mmap' (λ n, user_attribute.set simp_attr.tl_simp n () tt),\n   return ()\n\nnamespace tactic.interactive\nopen lean interactive.types\nopen interactive lean.parser tactic\nopen list (hiding map) functor predicate\nlocal postfix *:9001 := many\n\nmeta def TL_unfold (cs : parse ident*) (loc : parse location) : tactic unit :=\ndo unfold_coes loc, unfold (cs ++ [`var.apply,`pred'.mk]) loc\n\nend tactic.interactive\n\nnamespace temporal\n\nopen predicate nat\n\nuniverses u u₀ u₁ u₂ u₃\n\nvariables {α : Sort u₀} {β : Sort u₁} {γ : Sort u₂} {φ : Sort u₃}\n\n@[reducible]\ndef tvar := var ℕ\n\n@[reducible]\ndef cpred := tvar Prop\n\nabbreviation act (β : Sort u) := β → β → Prop\n\ndef eventually (p : cpred) : cpred :=\n⟨ λ i, ∃ j, p.apply (i+j) ⟩\ndef henceforth (p : cpred) : cpred :=\n⟨ λ i, ∀ j, p.apply (i+j) ⟩\ndef next (p : tvar α) : tvar α :=\n⟨ λ i, p.apply (i.succ) ⟩\n\ndef until (p q : cpred) : cpred :=\n⟨ λ i, ∃ j, i+j ⊨ q ∧ ∀ k < j, i+k ⊨ p ⟩\n\ndef wait (p q : cpred) : cpred :=\nuntil p q ⋁ henceforth p\n\ndef action (a : act α) (v : tvar α) : cpred :=\nlifted₂ a v (next v)\n\n@[lifted_fn, reducible]\ndef pair {α β} (x : tvar α) (y : tvar β) : tvar (α × β) :=\nlifted₂ prod.mk x y\n\nnotation `⦃` x₀ `,` x₁ `⦄` := pair x₀ x₁\nnotation `⦃` x₀ `,` x₁ `,` x₂ `⦄` := pair x₀ (pair x₁ x₂)\nnotation `⦃` x₀ `,` x₁ `,` x₂ `,` x₃ `⦄` := pair x₀ (pair x₁ (pair x₂ x₃))\nnotation `⦃` x₀ `,` x₁ `,` x₂ `,` x₃ `,` x₄ `⦄` := pair x₀ (pair x₁ (pair x₂ (pair x₃ x₄)))\n-- notation `⦃` x `,` l:(foldl `,` (h t, pair h t) x `⦄`)  := l\n\nprefix `⊙`:90 := next\nprefix `◇`:95 := eventually -- \\di\nprefix `◻`:95 := henceforth -- \\sqw\ninfixl `  𝒰  `:95 := until -- \\McU\ninfixl `  𝒲  `:95 := wait -- \\McU\nnotation `⟦ `:max v ` | `:50 R ` ⟧`:0 := action R v\nnotation `⟦ `:max v `,` v' ` | `:50 R ` ⟧`:0 := action R (pair v v')\nnotation `⟦ `:max v₀ `,` v₁ `,` v₂ ` | `:50 R ` ⟧`:0 := action R (pair v₀ (pair v₁ v₂))\nnotation `⟦ `:max v₀ `,` v₁ `,` v₂ `,` v₃ ` | `:50 R ` ⟧`:0 := action R (pair v₀ (pair v₁ (pair v₂ v₃)))\n\ndef tl_leads_to (p q : cpred) : cpred :=\n◻(p ⟶ ◇q)\n\ninfix ` ~> `:55 := tl_leads_to\n\n\ndef to_fun_var (f : var γ α → var γ β) : var γ (α → β) :=\n⟨ λ i x, (f ↑x).apply i ⟩\n\ndef to_fun_var' (f : tvar α → tvar α → tvar β) : tvar (α → α → β) :=\n⟨ λ i x y, (f x y).apply i ⟩\n\nclass persistent (p : cpred) : Prop :=\n  (is_persistent : ◻p = p)\nexport persistent (is_persistent)\n\nlemma tl_imp_intro (h : cpred) [persistent h] {p q : cpred}\n  (h' : h ⟹ (p ⟶ q))\n: ctx_impl h p q :=\nbegin\n  constructor, intro,\n  exact (h' True).apply σ trivial,\nend\n\nlemma tl_imp_elim (h : cpred) [persistent h] {p q : cpred}\n  (h' : ctx_impl h p q)\n: h ⟹ (p ⟶ q) :=\nbegin\n  intro, revert Γ,\n  apply intro_p_imp h',\nend\n\nlemma tl_imp_intro' (h : cpred) [persistent h] {p q : cpred}\n  (h' : p ⟹ q)\n: ctx_impl h p q :=\nh' _\n\n@[tl_simp, simp]\nlemma hence_true : ◻(True : cpred) = True :=\nbegin\n  ext1,\n  split ; intro h,\n  { trivial },\n  { intro, trivial }\nend\n\n@[tl_simp, simp]\nlemma next_true : ⊙True = True :=\nby lifted_pred\n\n@[tl_simp, simp]\nlemma next_false : ⊙False = False :=\nby lifted_pred [next]\n\ninstance true_persistent\n: persistent (True : cpred) :=\nby { constructor, simp with tl_simp, }\n\nlemma tl_imp_elim' {p q : cpred}\n  (h : ctx_impl True p q)\n: p ⟹ q :=\nbegin\n  simp [ctx_impl] at h,\n  apply h,\nend\n\n@[strengthening]\nlemma eventually_weaken (p : cpred)\n: (p ⟹ ◇ p) :=\nbegin\n  pointwise with τ h,\n  unfold eventually,\n  existsi 0,\n  apply h\nend\nopen stream\n\n@[strengthening]\nlemma next_entails_eventually (p : cpred)\n: ⊙p ⟹ ◇p :=\nby { lifted_pred [eventually], intro, existsi 1, assumption }\n\n@[strengthening]\nlemma henceforth_entails_next (p : cpred)\n: ◻p ⟹ ⊙p :=\nby { lifted_pred [eventually], intro h, apply h 1 }\n\n@[strengthening]\nlemma henceforth_str (p : cpred) :\n  (◻p ⟹ p) :=\nbegin\n  pointwise with τ h, apply h 0\nend\n\nlocal infix ` <$> ` := fun_app_to_var\nlocal infix ` <*> ` := combine_var\n\nlemma init_eq_action {p : α → Prop} (v : tvar α)\n: ⟨p⟩ ! v = ⟦ v | λ s s', p s ⟧ :=\nby { cases v, refl }\n\nlemma coe_eq (v : tvar α) (x : α)\n: ⟨λ y, y = x⟩ ! v = v ≃ x :=\nby { cases v, refl }\n\nlemma init_eq_action' {p : pred' α} (v : tvar α)\n: (p ! v) = ⟦ v | λ s s', p.apply s ⟧ :=\nby { cases v, cases p, refl }\n\nlemma next_eq_action {p : α → Prop} (v : tvar α)\n: ⊙(p <$> v) = ⟦ v | λ s s' : α, p s' ⟧ :=\nby { cases v, refl }\n\nlemma action_eq {A : act α} (v : tvar α)\n: ⟦ v | A ⟧ = (A : tvar (act α)) v (⊙v) :=\nby { cases v, refl }\n\nlemma next_eq_action' {p : pred' α} (v : tvar α)\n: ⊙(p ! v) = ⟦ v | λ s s' : α, p.apply s' ⟧ :=\nby { cases v, cases p, refl }\n\nlemma not_action {A : act α} (v : tvar α)\n: -⟦ v | A ⟧ = ⟦ v | λ s s', ¬ A s s' ⟧ :=\nrfl\n\nlemma action_imp (p q : act α) (v : tvar α)\n: (⟦ v | λ s s' : α, p s s' → q s s' ⟧ : cpred) = ⟦ v | p ⟧ ⟶ ⟦ v | q ⟧ :=\nrfl\n\nlemma action_and_action (p q : act α) (v : tvar α)\n: ⟦ v | p ⟧ ⋀ ⟦ v | q ⟧ = ⟦ v | λ s s' : α, p s s' ∧ q s s' ⟧ :=\nrfl\n\nlemma action_or_action (p q : act α) (v : tvar α)\n: ⟦ v | p ⟧ ⋁ ⟦ v | q ⟧ = ⟦ v | λ s s' : α, p s s' ∨ q s s' ⟧ :=\nrfl\n\nlemma action_false (v : tvar α)\n: (⟦ v | λ _ _, false ⟧ : cpred) = False :=\nby { funext x, refl }\n\nvariables {Γ : cpred}\n\nlemma unlift_action (A : act α) (v : tvar α)\n  (h : ∀ σ σ', A σ σ')\n: Γ ⊢ ⟦ v | A ⟧ :=\nbegin\n  constructor, simp_intros [action],\n  apply h\nend\n\n@[tl_simp, simp]\nlemma eventually_eventually (p : cpred) : ◇◇ p = ◇ p :=\nbegin\n  ext k,\n  split\n  ; unfold eventually\n  ; intro h\n  ; cases h with i h,\n  { cases h with j h,\n    existsi (i+j),\n    simp at h, apply h, },\n  { existsi (0 : ℕ),\n    existsi i,\n    apply h }\nend\n\n@[tl_simp, simp]\nlemma henceforth_henceforth (p : cpred) : ◻◻ p = ◻ p :=\nbegin\n  ext _,\n  split\n  ; intro h,\n  { intro i,\n    have h' := h i 0,\n    simp [drop_drop] at h',\n    apply h' },\n  { intros i j,\n    simp [drop_drop],\n    apply h }\nend\n\nlemma henceforth_next_intro (p : cpred)\n: ◻p = ◻(p ⋀ ⊙p) :=\nby { lifted_pred, split ; intros h i,\n     { split, apply h i, specialize h (succ i),\n       simp [add_succ] at h, simp [next,h], },\n     { apply (h i).left }}\n\n/- True / False -/\n\n@[tl_simp, simp]\nlemma hence_false : ◻(False : cpred) = False :=\nbegin\n  ext _,\n  split ; intro h,\n  { cases h 0 },\n  { cases h }\nend\n\n@[tl_simp, simp]\nlemma event_false : ◇(False : cpred) = False :=\nbegin\n  ext _,\n  split ; intro h,\n  { cases h with _ h, cases h },\n  { cases h }\nend\n\n@[tl_simp, simp]\nlemma eventually_true : ◇(True : cpred) = True :=\nbegin\n  ext1,\n  split ; intro h,\n  { trivial },\n  { apply exists.intro 0, trivial }\nend\n\n/- monotonicity -/\n\n@[monotonic]\nlemma eventually_tl_imp_eventually {h p q : cpred}\n  [persistent h]\n  (f : ctx_impl h p q)\n: ctx_impl h (◇p) (◇q) :=\nbegin\n  unfold ctx_impl at ⊢ f,\n  rw ← is_persistent h at *,\n  pointwise f with τ h',\n  apply exists_imp_exists,\n  intro i,\n  apply f,\n  rw ← henceforth_henceforth at h',\n  apply h',\nend\n\n@[monotonic]\nlemma eventually_entails_eventually {p q : cpred}\n  (f : p ⟹ q)\n: (◇p) ⟹ (◇q) :=\nbegin\n  apply tl_imp_elim',\n  mono,\nend\n\nlemma eventually_imp_eventually {p q : cpred} {Γ}\n (f : Γ ⊢ ◻ (p ⟶ q))\n: Γ ⊢ (◇p) ⟶ (◇q) :=\nbegin\n  constructor, introv hΓ,\n  apply exists_imp_exists,\n  intro i,\n  apply f.apply _ hΓ,\nend\n\n@[monotonic]\nlemma henceforth_tl_imp_henceforth {h p q : cpred}\n  [persistent h]\n  (f : ctx_impl h p q)\n: ctx_impl h (◻p) (◻q) :=\nbegin\n  unfold ctx_impl at *,\n  rw ← is_persistent h,\n  pointwise f with i h',\n  simp [henceforth], intro_mono i,\n  apply f ,\n  apply  h',\nend\n\n@[monotonic]\nlemma henceforth_entails_henceforth {p q : cpred}\n  (f : p ⟹ q)\n: (◻p) ⟹ (◻q) :=\nbegin\n  refine tl_imp_elim' _,\n  mono\nend\n\nlemma henceforth_imp_henceforth {p q : cpred} {Γ}\n  (h : Γ ⊢ ◻ (p ⟶ q))\n: Γ ⊢ (◻p) ⟶ (◻q) :=\nbegin\n  pointwise h with τ,\n  specialize h τ, simp [henceforth] at ⊢ h,\n  introv h₀ h₁,\n  solve_by_elim,\nend\n\nlemma inf_often_entails_inf_often {p q : cpred} (f : p ⟹ q)\n: ◻◇p ⟹ ◻◇q :=\nby mono*\n\nlemma stable_entails_stable {p q : cpred} (f : p ⟹ q)\n: ◇◻p ⟹ ◇◻q :=\nby mono*\n\nlemma henceforth_and (p q : cpred)\n: ◻(p ⋀ q) = ◻p ⋀ ◻q :=\nbegin\n  ext1,\n  repeat { split ; intros }\n  ; intros i ; try { simp, split },\n  { apply (a i).left },\n  { apply (a i).right },\n  { apply a.left },\n  { apply a.right },\nend\n\nlemma eventually_or (p q : cpred)\n: ◇(p ⋁ q) = ◇p ⋁ ◇q :=\nbegin\n  ext1,\n  simp [eventually,exists_or],\nend\n\nlemma henceforth_forall (P : α → cpred)\n: ◻(∀∀ x, P x) = ∀∀ x, ◻P x :=\nbegin\n  ext1,\n  simp [henceforth,p_forall],\n  rw forall_swap,\nend\n\n@[tl_simp, simp]\nlemma not_henceforth (p : cpred) : (- ◻p) = (◇-p) :=\nbegin\n  ext1,\n  simp [henceforth,not_forall_iff_exists_not,eventually],\nend\n\n@[tl_simp, simp]\nlemma not_eventually (p : cpred)\n: (-◇p) = (◻-p) :=\nbegin\n  ext1,\n  simp [henceforth,not_forall_iff_exists_not,eventually],\nend\n\n@[tl_simp, simp, predicate]\nlemma models_to_fun_var (σ : γ) (x : var γ α → var γ β)\n: σ ⊨ to_fun_var x = λ i, σ ⊨ x i :=\nby { refl }\n\n@[tl_simp, simp, predicate]\nlemma models_to_fun_var' (σ : ℕ) (f : tvar (α → α → β))\n: σ ⊨ to_fun_var' (λ x, var_seq $ var_seq f x) = σ ⊨ f :=\nby { casesm* tvar _, dunfold to_fun_var', simp_coes [var_seq], }\n\n@[tl_simp, simp, predicate]\nlemma models_to_fun_var''' (σ : ℕ) (f : tvar α → tvar α → tvar β) (x y : tvar α)\n: (σ ⊨ to_fun_var' f x y) = (σ ⊨ f x y) :=\nsorry\n\n@[tl_simp, simp, predicate, lifted_fn]\nlemma to_fun_var_fn_coe_proj (f : var γ α) (g : var β φ → var β γ) (w : var β φ)\n: to_fun_var (λ w, f ! g w) w = f ! to_fun_var g w :=\nby { funext, lifted_pred, simp!, }\n\n@[tl_simp, simp, predicate, lifted_fn]\nlemma to_fun_var_fn_coe (g : var β (φ → γ)) (w : var β φ)\n: to_fun_var (λ w, g w) w = g w :=\nby { funext, lifted_pred, simp!, }\n\n@[tl_simp, simp, predicate, lifted_fn]\nlemma to_fun_var_fn_coe' (f : tvar (α → α → β)) (w w' : tvar α)\n: ⇑(to_fun_var' $ λ w w', f w w') w w' = f w w' :=\nby { lifted_pred, }\n\n@[tl_simp, simp, predicate, lifted_fn]\nlemma to_fun_var_p_exists' (f : γ → tvar α → tvar α → tvar Prop) (w w' : tvar α)\n: ⇑(to_fun_var' $ λ w w', ∃∃ x : γ, f x w w') w w' = ∃∃ x : γ, to_fun_var' (f x) w w' :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var_lift₁ (f : φ → β) (g : var γ α → var γ φ) (w : var γ α)\n: (to_fun_var (λ (w : var γ α), lifted₁ f (g w))) w = lifted₁ f (to_fun_var g w) :=\nby { lifted_pred, simp! }\n\n@[lifted_fn]\nlemma to_fun_var_lift₂ {σ} (f : φ → σ → β)\n  (g₀ : var γ α → var γ φ)\n  (g₁ : var γ α → var γ σ) (w : var γ α)\n: (to_fun_var (λ (w : var γ α), lifted₂ f (g₀ w) (g₁ w))) w =\n  lifted₂ f (to_fun_var g₀ w) (to_fun_var g₁ w) :=\nby { lifted_pred, simp }\n\n@[lifted_fn]\nlemma to_fun_var_coe (w : var γ α) (v : var γ β)\n: to_fun_var (λ (w : var γ α), v) w = v :=\nby { lifted_pred, simp }\n\n@[lifted_fn]\nlemma to_fun_var_id (w : var γ α)\n: to_fun_var (λ w : var γ α, w) w = w :=\nby { lifted_pred, simp }\n\n@[lifted_fn]\nlemma to_fun_var'_fn_coe (f : var φ β) (g : tvar α → tvar α → tvar φ) (w w' : tvar α)\n: (to_fun_var' (λ (w w' : tvar α), f ! g w w')) w w' = f ! to_fun_var' g w w'  :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var'_coe (w w' : tvar α) (v : tvar β)\n: (to_fun_var' (λ (w w' : tvar α), v)) w w' = v :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var'_id (w w' : tvar α)\n: to_fun_var' (λ w w' : tvar α, w) w w' = w :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var'_id' (w w' : tvar α)\n: to_fun_var' (λ w w' : tvar α, w') w w' = w' :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var'_lift₂ {σ} (f : φ → σ → β)\n  (g₀ : tvar α → tvar α → tvar φ)\n  (g₁ : tvar α → tvar α → tvar σ) (w w' : tvar α)\n: to_fun_var' (λ (w w' : tvar α), lifted₂ f (g₀ w w') (g₁ w w')) w w' =\n  lifted₂ f (to_fun_var' g₀ w w') (to_fun_var' g₁ w w') :=\nby { lifted_pred, }\n\n@[lifted_fn]\nlemma to_fun_var'_action {σ} (f : tvar α → tvar β)\n  (A : act β)\n  (g₁ : tvar α → tvar α → tvar σ) (w w' : tvar α)\n: to_fun_var' (λ (w w' : tvar α), ⟦ f w | A ⟧ ) w w' =\n  ⟦ f w | A ⟧ :=\nby { lifted_pred, }\n\n@[tl_simp, simp, predicate]\nlemma models_coe (σ : α) (x : β)\n: σ ⊨ ↑x = x :=\nby { refl }\n\n@[tl_simp, simp, predicate]\nlemma models_action (A : act α) (v : tvar α) (i : ℕ)\n: i ⊨ ⟦ v | A ⟧ ↔ A (i ⊨ v) (succ i ⊨ v) :=\nby { refl }\n\n@[tl_simp, simp, predicate]\nlemma models_next (p : tvar α) (t : ℕ)\n: t ⊨ ⊙p = succ t ⊨ p :=\nby refl\n\nlemma induct' (p : cpred) {Γ}\n  (h : Γ ⊢ ◻ (p ⟶ ⊙p))\n: Γ ⊢ ◻ (p ⟶ ◻p) :=\nbegin\n  constructor,\n  intros τ hΓ k hp i,\n  induction i with i,\n  assumption,\n  have := h.apply τ hΓ (k+i) (cast (by simp) i_ih),\n  simp [next] at this,\n  simp [add_succ,this],\nend\n\nlemma induct (p : cpred) {Γ}\n  (h : Γ ⊢ ◻ (p ⟶ ⊙p))\n: Γ ⊢ p ⟶ ◻p :=\nhenceforth_str (p ⟶ ◻p) Γ (induct' _ h)\n\nlemma induct_evt' (p q : cpred) {Γ}\n  (h : Γ ⊢ ◻ (p ⟶ -q ⟶ ⊙p ⋁ ⊙q))\n: Γ ⊢ ◻ (p ⟶ ◇q ⋁ ◻p) :=\nbegin\n  lifted_pred using h,\n  simp only [henceforth] with tl_simp at *,\n  intros,\n  simp [or_iff_not_imp,eventually],\n  intros hnq k,\n  induction k with k,\n  { simp [a] },\n  { simp [add_succ],\n    specialize h _ k_ih (hnq _),\n    rw [or_comm,or_iff_not_imp] at h,\n    apply h, rw [← add_succ,← add_succ],\n    apply hnq }\nend\n\nlemma eventually_exists (P : α → cpred)\n: ◇(∃∃ x, P x) = ∃∃ x, ◇P x :=\nbegin\n  ext1,\n  unfold eventually p_exists,\n  split\n  ; intro H\n  ; cases H with i H\n  ; cases H with j H\n  ; exact ⟨_,_,H⟩ ,\nend\n\nlemma one_point_elim {t} (Γ p : cpred) (v : tvar t)\n  (h : ∀ x : t, Γ ⊢ (↑x ≃ v) ⟶ p)\n: Γ ⊢ p :=\nbegin\n  rw [← p_forall_to_fun] at h,\n  constructor,\n  intros i h',\n  apply h.apply  i h' (v.apply $ i),\n  simp,\nend\n\nlocal attribute [instance] classical.prop_decidable\n\nlemma until_not_of_eventually {Γ p : cpred} :\n  Γ ⊢ ◇p ⟶ -p 𝒰 p :=\nbegin\n  lifted_pred, intro h,\n  cases h with i h,\n  induction i using nat.strong_induction_on with i ih,\n  by_cases h' : ∃ j < i, σ + j ⊨ p,\n  { rcases h' with ⟨j,h₀,h₁⟩,\n    apply ih _ h₀ h₁ },\n  { simp only [not_exists] at h',\n    existsi [i,h], apply h' }\nend\n\nlemma until_backward_induction {Γ p q : cpred}\n  (h' : Γ ⊢ ◻(p ⟶ q))\n  (h : Γ ⊢ ◻(⊙q ⟶ -p ⟶ q)) :\n  Γ ⊢ ◇p ⟶ q 𝒰 p :=\nbegin\n  suffices : Γ ⊢ -p 𝒰 p ⟶ q 𝒰 p,\n  { have h' := @until_not_of_eventually Γ p,\n    lifted_pred using this h', tauto },\n  lifted_pred using h h',\n  dsimp [until],\n  apply exists_imp_exists,\n  rintros i ⟨h₀,h₁⟩,\n  existsi h₀,\n  introv h₂,\n  generalize h₃ : (i - (k + 1)) = n,\n  induction n generalizing k,\n  { apply h, dsimp [next], rw ← add_succ, apply h', convert h₀,\n    apply le_antisymm, apply succ_le_of_lt h₂,\n    apply nat.le_of_sub_eq_zero h₃, apply h₁ _ h₂, },\n  { apply h,\n    { have h₄ : k + 1 ≤ i := succ_le_of_lt h₂,\n      dsimp [next], rw ← add_succ,\n      apply n_ih, rw nat.sub_eq_iff_eq_add at h₃; try { assumption },\n      { rw [h₃,succ_add], apply succ_lt_succ,\n        apply lt_of_lt_of_le (lt_succ_self _),\n        apply nat.le_add_left },\n      { apply succ.inj,\n        rw [← h₃,succ_add,← succ_sub,succ_sub_succ_eq_sub],\n        apply succ_le_of_lt, rw nat.sub_eq_iff_eq_add at h₃; try { assumption },\n        rw h₃, apply lt_add_of_pos_left, apply zero_lt_succ }, },\n    apply h₁ _ h₂, },\nend\n\nlemma henceforth_until {Γ p q : cpred}\n  (h : Γ ⊢ ◻(p 𝒰 q)) :\n  Γ ⊢ ◻(p ⋁ q) :=\nbegin\n  lifted_pred using h,\n  intro i, specialize h i,\n  rcases h with ⟨⟨j⟩,h₀,h₁⟩,\n  { right, apply h₀ },\n  { left, apply h₁ 0, apply zero_lt_succ }\nend\n\nlemma henceforth_until' {Γ p : cpred} (q : cpred)\n  (h : Γ ⊢ ◻(p 𝒰 (p ⋀ q))) :\n  Γ ⊢ ◻p :=\nbegin\n  replace h := henceforth_until h,\n  suffices : p ⋁ p ⋀ q = p, { simp [this] at h, exact h },\n  lifted_pred, tauto,\nend\n\ndef nelist (α : Type*) := { xs : list α // xs.empty = ff }\n\ndef nelist.head {α : Type*} : nelist α → α\n | ⟨ x::xs, _ ⟩ := x\n\ndef nelist.tail {α : Type*} : nelist α → option (nelist α)\n | ⟨ x::x'::xs, _ ⟩ := some ⟨ x'::xs, rfl ⟩\n | ⟨ _ :: [], _ ⟩ := none\n\ndef nelist.uncons {α : Type*} : nelist α → α ⊕ (α × nelist α)\n | ⟨ x::x'::xs, _ ⟩ := sum.inr (x,⟨x'::xs,rfl⟩)\n | ⟨ x :: [], _ ⟩ := sum.inl x\n\ndef nelist.single {α : Type*} : α → nelist α\n | x := ⟨ x :: [] ,rfl ⟩\n\ndef nelist.cons {α : Type*} : α → nelist α → nelist α\n | x ⟨ xs, _⟩ := ⟨ x::xs,rfl ⟩\n\nnoncomputable def epsilon [nonempty α] (p : tvar (α → Prop)) : tvar α :=\n⟨ λ i, classical.epsilon $ i ⊨ p ⟩\n\nsection witness\nvariables x₀ : tvar α\nvariables f : tvar (α → α)\nvariables (i : ℕ)\n\nopen classical nat\n\nprivate def w : ℕ → α\n | 0 := i ⊨ x₀\n | (succ j) := (i + j ⊨ f) (w j)\n\nlemma fwd_witness\n: ⊩ ∃∃ w, w ≃ x₀ ⋀ ◻( ⊙w ≃ f w ) :=\nbegin\n  lifted_pred,\n  existsi (⟨ λ i, w x₀ f x (i - x) ⟩ : tvar α),\n  simp [nat.sub_self,w],\n  intro i,\n  have h : x + i ≥ x := nat.le_add_right _ _,\n  simp [next,nat.add_sub_cancel_left,succ_sub h,w],\nend\n\nvariables {P : cpred}\nvariables (j : ℕ)\n\nlocal attribute [instance] classical.prop_decidable\n\nlemma first_P {p : ℕ → Prop} (h : ∃ i, p i) :\n  (∃ i, (∀ j < i, ¬ p j) ∧ p i) :=\nbegin\n  cases h with i h,\n  induction i using nat.strong_induction_on with i ih,\n  by_cases h' : (∀ (j : ℕ), j < i → ¬p j),\n  { exact ⟨_,h',h⟩, },\n  { simp [not_forall] at h',\n    rcases h' with ⟨j,h₀,h₁⟩,\n    apply ih _ h₀ h₁, }\nend\n\nnoncomputable def next_P {x} (h : x ⊨ ◻◇P) (y : ℕ) : ℕ :=\nclassical.some (first_P (h y))\n\nlemma next_P_eq_self {x} (h : x ⊨ ◻◇P) (y : ℕ)\n  (h' : x+y ⊨ P) :\n  next_P h y = 0 :=\nbegin\n  simp [next_P],\n  apply_some_spec,\n  intros h, by_contradiction h₂,\n  replace h₂ := nat.lt_of_le_and_ne (nat.zero_le _) (ne.symm h₂),\n  apply h.1 _ h₂,\n  simp [h']\nend\n\nlemma next_P_eq_succ {x} (h : x ⊨ ◻◇P) (y : ℕ)\n  (h' : ¬ x+y ⊨ P) :\n  next_P h y = succ (next_P h (succ y)) :=\nbegin\n  simp [next_P],\n  apply_some_spec, apply_some_spec,\n  simp, intros h₀ h₁ h₂ h₃,\n  by_contradiction h₄,\n  replace h₄ := lt_or_gt_of_ne (h₄),\n  cases h₄,\n  cases x_1,\n  { apply h', convert h₃ using 2, },\n  { replace h₄ := lt_of_succ_lt_succ h₄,\n    apply h₀ _ h₄, convert h₃ using 2, simp [add_succ] },\n  { apply h₂ _ h₄, convert h₁ using 2, simp [add_succ] },\nend\n\nlemma back_witness {Γ}\n  (h : Γ ⊢ ◻◇P)\n: Γ ⊢ ∃∃ w, ◻( (P ⋀ w ≃ x₀) ⋁ (- P ⋀ w ≃ ⊙(f w)) ) :=\nbegin\n  lifted_pred using h,\n  existsi (⟨ λ i, w x₀ f (i) (next_P h (i - x)) ⟩ : tvar α),\n  intro j, simp [nat.add_sub_cancel_left,w],\n  by_cases h' : x + j ⊨ P; simp [next_P_eq_self,*,w],\n  rw [next_P_eq_succ,w],\n  congr' 1,\n  { admit },\n  have : (succ (x + j) - x) = succ j, admit,\n  rw this, clear this,\nend\n\nend witness\n\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3364734692568112}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.doc_commands\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\nnamespace tactic\n\n\nnamespace interactive\n\n\n/--\nThis is a \"finishing\" tactic modification of `simp`. It has two forms.\n\n* `simpa [rules, ...] using e` will simplify the goal and the type of\n  `e` using `rules`, then try to close the goal using `e`.\n\n  Simplifying the type of `e` makes it more likely to match the goal\n  (which has also been simplified). This construction also tends to be\n  more robust under changes to the simp lemma set.\n\n* `simpa [rules, ...]` will simplify the goal and the type of a\n  hypothesis `this` if present in the context, then try to close the goal using\n  the `assumption` tactic. -/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/simpa.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.33629199743303334}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.full_subcategory\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.products\nimport tactic.elementwise\nimport topology.sheaves.presheaf\n\n/-!\n# The sheaf condition in terms of an equalizer of products\n\nHere we set up the machinery for the \"usual\" definition of the sheaf condition,\ne.g. as in https://stacks.math.columbia.edu/tag/0072\nin terms of an equalizer diagram where the two objects are\n`∏ F.obj (U i)` and `∏ F.obj (U i) ⊓ (U j)`.\n\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen topological_space\nopen opposite\nopen topological_space.opens\n\nnamespace Top\n\nvariables {C : Type u} [category.{v} C] [has_products C]\nvariables {X : Top.{v}} (F : presheaf C X) {ι : Type v} (U : ι → opens X)\n\nnamespace presheaf\n\nnamespace sheaf_condition_equalizer_products\n\n/-- The product of the sections of a presheaf over a family of open sets. -/\ndef pi_opens : C := ∏ (λ i : ι, F.obj (op (U i)))\n/--\nThe product of the sections of a presheaf over the pairwise intersections of\na family of open sets.\n-/\ndef pi_inters : C := ∏ (λ p : ι × ι, F.obj (op (U p.1 ⊓ U p.2)))\n\n/--\nThe morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components\nare given by the restriction maps from `U i` to `U i ⊓ U j`.\n-/\ndef left_res : pi_opens F U ⟶ pi_inters F U :=\npi.lift (λ p : ι × ι, pi.π _ p.1 ≫ F.map (inf_le_left (U p.1) (U p.2)).op)\n\n/--\nThe morphism `Π F.obj (U i) ⟶ Π F.obj (U i) ⊓ (U j)` whose components\nare given by the restriction maps from `U j` to `U i ⊓ U j`.\n-/\ndef right_res : pi_opens F U ⟶ pi_inters F U :=\npi.lift (λ p : ι × ι, pi.π _ p.2 ≫ F.map (inf_le_right (U p.1) (U p.2)).op)\n\n/--\nThe morphism `F.obj U ⟶ Π F.obj (U i)` whose components\nare given by the restriction maps from `U j` to `U i ⊓ U j`.\n-/\ndef res : F.obj (op (supr U)) ⟶ pi_opens F U :=\npi.lift (λ i : ι, F.map (topological_space.opens.le_supr U i).op)\n\n@[simp, elementwise]\nlemma res_π (i : ι) : res F U ≫ limit.π _ i = F.map (opens.le_supr U i).op :=\nby rw [res, limit.lift_π, fan.mk_π_app]\n\n@[elementwise]\nlemma w : res F U ≫ left_res F U = res F U ≫ right_res F U :=\nbegin\n  dsimp [res, left_res, right_res],\n  ext,\n  simp only [limit.lift_π, limit.lift_π_assoc, fan.mk_π_app, category.assoc],\n  rw [←F.map_comp],\n  rw [←F.map_comp],\n  congr,\nend\n\n/--\nThe equalizer diagram for the sheaf condition.\n-/\n@[reducible]\ndef diagram : walking_parallel_pair.{v} ⥤ C :=\nparallel_pair (left_res F U) (right_res F U)\n\n/--\nThe restriction map `F.obj U ⟶ Π F.obj (U i)` gives a cone over the equalizer diagram\nfor the sheaf condition. The sheaf condition asserts this cone is a limit cone.\n-/\ndef fork : fork.{v} (left_res F U) (right_res F U) := fork.of_ι _ (w F U)\n\n@[simp]\nlemma fork_X : (fork F U).X = F.obj (op (supr U)) := rfl\n\n@[simp]\nlemma fork_ι : (fork F U).ι = res F U := rfl\n@[simp]\nlemma fork_π_app_walking_parallel_pair_zero :\n  (fork F U).π.app walking_parallel_pair.zero = res F U := rfl\n@[simp]\nlemma fork_π_app_walking_parallel_pair_one :\n  (fork F U).π.app walking_parallel_pair.one = res F U ≫ left_res F U := rfl\n\nvariables {F} {G : presheaf C X}\n\n/-- Isomorphic presheaves have isomorphic `pi_opens` for any cover `U`. -/\n@[simp]\ndef pi_opens.iso_of_iso (α : F ≅ G) : pi_opens F U ≅ pi_opens G U :=\npi.map_iso (λ X, α.app _)\n\n/-- Isomorphic presheaves have isomorphic `pi_inters` for any cover `U`. -/\n@[simp]\ndef pi_inters.iso_of_iso (α : F ≅ G) : pi_inters F U ≅ pi_inters G U :=\npi.map_iso (λ X, α.app _)\n\n/-- Isomorphic presheaves have isomorphic sheaf condition diagrams. -/\ndef diagram.iso_of_iso (α : F ≅ G) : diagram F U ≅ diagram G U :=\nnat_iso.of_components\n  begin rintro ⟨⟩, exact pi_opens.iso_of_iso U α, exact pi_inters.iso_of_iso U α end\n  begin\n    rintro ⟨⟩ ⟨⟩ ⟨⟩,\n    { simp, },\n    { ext, simp [left_res], },\n    { ext, simp [right_res], },\n    { simp, },\n  end.\n\n/--\nIf `F G : presheaf C X` are isomorphic presheaves,\nthen the `fork F U`, the canonical cone of the sheaf condition diagram for `F`,\nis isomorphic to `fork F G` postcomposed with the corresponding isomorphism between\nsheaf condition diagrams.\n-/\ndef fork.iso_of_iso (α : F ≅ G) :\n  fork F U ≅ (cones.postcompose (diagram.iso_of_iso U α).inv).obj (fork G U) :=\nbegin\n  fapply fork.ext,\n  { apply α.app, },\n  { ext,\n    dunfold fork.ι, -- Ugh, `simp` can't unfold abbreviations.\n    simp [res, diagram.iso_of_iso], }\nend\n\nsection open_embedding\n\nvariables {V : Top.{v}} {j : V ⟶ X} (oe : open_embedding j)\nvariables (𝒰 : ι → opens V)\n\n/--\nPush forward a cover along an open embedding.\n-/\n@[simp]\ndef cover.of_open_embedding : ι → opens X := (λ i, oe.is_open_map.functor.obj (𝒰 i))\n\n/--\nThe isomorphism between `pi_opens` corresponding to an open embedding.\n-/\n@[simp]\ndef pi_opens.iso_of_open_embedding :\n  pi_opens (oe.is_open_map.functor.op ⋙ F) 𝒰 ≅ pi_opens F (cover.of_open_embedding oe 𝒰) :=\npi.map_iso (λ X, F.map_iso (iso.refl _))\n\n/--\nThe isomorphism between `pi_inters` corresponding to an open embedding.\n-/\n@[simp]\ndef pi_inters.iso_of_open_embedding :\n  pi_inters (oe.is_open_map.functor.op ⋙ F) 𝒰 ≅ pi_inters F (cover.of_open_embedding oe 𝒰) :=\npi.map_iso (λ X, F.map_iso\n  begin\n    dsimp [is_open_map.functor],\n    exact iso.op\n    { hom := hom_of_le (by\n      { simp only [oe.to_embedding.inj, set.image_inter],\n        apply le_refl _, }),\n      inv := hom_of_le (by\n      { simp only [oe.to_embedding.inj, set.image_inter],\n        apply le_refl _, }), },\n  end)\n\n/-- The isomorphism of sheaf condition diagrams corresponding to an open embedding. -/\ndef diagram.iso_of_open_embedding :\n  diagram (oe.is_open_map.functor.op ⋙ F) 𝒰 ≅ diagram F (cover.of_open_embedding oe 𝒰) :=\nnat_iso.of_components\n  begin\n    rintro ⟨⟩,\n    exact pi_opens.iso_of_open_embedding oe 𝒰,\n    exact pi_inters.iso_of_open_embedding oe 𝒰\n  end\n  begin\n    rintro ⟨⟩ ⟨⟩ ⟨⟩,\n    { simp, },\n    { ext,\n      dsimp [left_res, is_open_map.functor],\n      simp only [limit.lift_π, cones.postcompose_obj_π, iso.op_hom, discrete.nat_iso_hom_app,\n        functor.map_iso_refl, functor.map_iso_hom, lim_map_π_assoc, limit.lift_map, fan.mk_π_app,\n        nat_trans.comp_app, category.assoc],\n      dsimp,\n      rw [category.id_comp, ←F.map_comp],\n      refl, },\n    { ext,\n      dsimp [right_res, is_open_map.functor],\n      simp only [limit.lift_π, cones.postcompose_obj_π, iso.op_hom, discrete.nat_iso_hom_app,\n        functor.map_iso_refl, functor.map_iso_hom, lim_map_π_assoc, limit.lift_map, fan.mk_π_app,\n        nat_trans.comp_app, category.assoc],\n      dsimp,\n      rw [category.id_comp, ←F.map_comp],\n      refl, },\n    { simp, },\n  end.\n\n/--\nIf `F : presheaf C X` is a presheaf, and `oe : U ⟶ X` is an open embedding,\nthen the sheaf condition fork for a cover `𝒰` in `U` for the composition of `oe` and `F` is\nisomorphic to sheaf condition fork for `oe '' 𝒰`, precomposed with the isomorphism\nof indexing diagrams `diagram.iso_of_open_embedding`.\n\nWe use this to show that the restriction of sheaf along an open embedding is still a sheaf.\n-/\ndef fork.iso_of_open_embedding :\n  fork (oe.is_open_map.functor.op ⋙ F) 𝒰 ≅\n    (cones.postcompose (diagram.iso_of_open_embedding oe 𝒰).inv).obj\n      (fork F (cover.of_open_embedding oe 𝒰)) :=\nbegin\n  fapply fork.ext,\n  { dsimp [is_open_map.functor],\n    exact\n    F.map_iso (iso.op\n    { hom := hom_of_le\n      (by simp only [supr_s, supr_mk, le_def, subtype.coe_mk, set.le_eq_subset, set.image_Union]),\n      inv := hom_of_le\n      (by simp only [supr_s, supr_mk, le_def, subtype.coe_mk, set.le_eq_subset,\n                     set.image_Union]) }), },\n  { ext,\n    dunfold fork.ι, -- Ugh, it is unpleasant that we need this.\n    simp only [res, diagram.iso_of_open_embedding, discrete.nat_iso_inv_app, functor.map_iso_inv,\n      limit.lift_π, cones.postcompose_obj_π, functor.comp_map,\n      fork_π_app_walking_parallel_pair_zero, pi_opens.iso_of_open_embedding,\n      nat_iso.of_components.inv_app, functor.map_iso_refl, functor.op_map, limit.lift_map,\n      fan.mk_π_app, nat_trans.comp_app, quiver.hom.unop_op, category.assoc, lim_map_eq_lim_map],\n    dsimp,\n    rw [category.comp_id, ←F.map_comp],\n    refl, },\nend\n\nend open_embedding\n\nend sheaf_condition_equalizer_products\n\nend presheaf\n\nend Top\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/topology/sheaves/sheaf_condition/equalizer_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5, "lm_q1q2_score": 0.336165849589643}}
{"text": "import for_mathlib.category_theory.localization.predicate\nimport category_theory.adjunction.limits\nimport category_theory.is_connected\nimport for_mathlib.category_theory.localization.products\nimport for_mathlib.category_theory.localization.opposite\n\nnoncomputable theory\n\nopen category_theory category_theory.category\n\nnamespace category_theory\n\nnamespace limits\n\ndef is_terminal.of_equivalence {C D : Type*} [category C] [category D] (e : C ≌ D) {X : C}\n  (hX : is_terminal X) : is_terminal (e.functor.obj X) :=\nbegin\n  change is_limit _,\n  let e' : functor.empty C ⋙ e.functor ≅ functor.empty D := functor.empty_ext _ _,\n  equiv_rw (is_limit.postcompose_inv_equiv e' _).symm,\n  exact is_limit.of_iso_limit (is_limit_of_preserves e.functor hX)\n    (cones.ext (iso.refl _) (by rintro ⟨⟨⟩⟩)),\nend\n\nend limits\n\n@[simps]\ninstance localization.lifting.of_comp {C D E : Type*} [category C] [category D] [category E]\n  (L : C ⥤ D) (W : morphism_property C) [L.is_localization W] (F : D ⥤ E) :\n  localization.lifting L W (L ⋙ F) F := ⟨iso.refl _⟩\n\nsection\n\nvariables (C : Type*) [category C]\n\ninductive obj_rel : C → C → Prop\n| of_hom ⦃X Y : C⦄ (f : X ⟶ Y) : obj_rel X Y\n\ndef connected_components := quot (obj_rel C)\n\nvariable {C}\n\ndef to_connected_components (X : C) : connected_components C :=\nquot.mk _ X\n\nvariable (C)\n\nclass is_preconnected' : Prop :=\n(subsingleton_connected_components : subsingleton (connected_components C))\n\nattribute [instance] is_preconnected'.subsingleton_connected_components\n\nclass is_connected' extends is_preconnected' C : Prop :=\n[is_nonempty : nonempty C]\n\nlemma connected_components.nat_trans_from_eq {D : Type*} [category D]\n  (X Y : D) (α : (functor.const C).obj X ⟶ (functor.const C).obj Y)\n  (j j' : C) (h : to_connected_components j = to_connected_components j') :\n    α.app j = (α.app j' : X ⟶ Y) :=\nbegin\n  let β : C → (X ⟶ Y) := λ j, α.app j,\n  let l : connected_components C → (X ⟶ Y),\n  { refine quot.lift β _,\n    rintro x y ⟨f⟩,\n    dsimp [β],\n    have eq := α.naturality f,\n    dsimp at eq,\n    rw [id_comp, comp_id] at eq,\n    rw eq, },\n  have hl : ∀ (j : C), α.app j = l (to_connected_components j) := λ j, rfl,\n  simp only [hl, h],\nend\n\nlemma nat_trans_from_is_preconnected' {D : Type*} [category D]\n  [is_preconnected' C] (X Y : D) (α : (functor.const C).obj X ⟶ (functor.const C).obj Y)\n  (j j' : C) : α.app j = (α.app j' : X ⟶ Y) :=\nbegin\n  apply connected_components.nat_trans_from_eq,\n  apply subsingleton.elim,\nend\n\n@[simps]\ndef connected_components.op_equiv :\n  connected_components C ≃ connected_components Cᵒᵖ :=\n{ to_fun := quot.lift (λ X, to_connected_components (opposite.op X))\n    (by { rintros X Y ⟨f⟩, symmetry, exact quot.sound ⟨f.op⟩, }),\n  inv_fun := quot.lift (λ X, to_connected_components (opposite.unop X))\n    (by { rintros X Y ⟨f⟩, symmetry, exact quot.sound ⟨f.unop⟩, }),\n  left_inv := by { rintro ⟨X⟩, refl, },\n  right_inv := by { rintro ⟨X⟩, refl, }, }\n\nvariables {C} {D E : Type*} [category D] [category E]\n\ndef connected_components.map (F : C ⥤ D) :\n  connected_components C → connected_components D :=\nquot.lift (λ X, to_connected_components (F.obj X))\n  (by { rintros X Y ⟨f⟩, exact quot.sound ⟨F.map f⟩, })\n\n@[simp]\nlemma connected_components.map_id (C : Type*) [category C] :\n  connected_components.map (𝟭 C) = id := by tidy\n\n@[simp]\nlemma connected_components.map_id_apply (x : connected_components C) :\n  connected_components.map (𝟭 C) x = x :=\nby simp only [connected_components.map_id, id.def]\n\n@[simp]\nlemma connected_components.map_comp (F : C ⥤ D) (G : D ⥤ E) :\n  connected_components.map (F ⋙ G) =\n    connected_components.map G ∘ connected_components.map F := by tidy\n\n@[simp]\nlemma connected_components.map_comp_apply (F : C ⥤ D) (G : D ⥤ E) (x : connected_components C) :\n  connected_components.map (F ⋙ G) x =\n  connected_components.map G (connected_components.map F x) :=\nby simp only [connected_components.map_comp]\n\nlemma connected_components.map_eq_of_nat_trans {F G : C ⥤ D} (τ : F ⟶ G) :\n  connected_components.map F = connected_components.map G :=\nby { ext ⟨X⟩, exact quot.sound ⟨τ.app X⟩, }\n\nlemma connected_components.map_eq_of_nat_trans_apply {F G : C ⥤ D} (τ : F ⟶ G)\n  (x : connected_components C):\n  connected_components.map F x = connected_components.map G x :=\nby rw connected_components.map_eq_of_nat_trans τ\n\n@[simps]\ndef connected_components.equiv_of_equivalence (e : C ≌ D) :\n  connected_components C ≃ connected_components D :=\n{ to_fun := connected_components.map e.functor,\n  inv_fun := connected_components.map e.inverse,\n  left_inv := λ x, by simpa only [connected_components.map_comp_apply,\n    connected_components.map_id_apply]\n    using connected_components.map_eq_of_nat_trans_apply e.unit_iso.inv x,\n  right_inv := λ x, by simpa only [connected_components.map_comp_apply,\n    connected_components.map_id_apply]\n    using connected_components.map_eq_of_nat_trans_apply e.counit_iso.hom x, }\n\nlemma is_preconnected'.of_equivalence (e : C ≌ D) (h : is_preconnected' C) :\n  is_preconnected' D :=\n⟨⟨λ X Y, (connected_components.equiv_of_equivalence e).symm.injective (subsingleton.elim _ _)⟩⟩\n\nlemma is_connected'.of_equivalence (e : C ≌ D) (h : is_connected' C) :\n  is_connected' D :=\nbegin\n  haveI : nonempty D := ⟨e.functor.obj h.is_nonempty.some⟩,\n  haveI : is_preconnected' D := is_preconnected'.of_equivalence e h.1,\n  constructor,\nend\n\nlemma is_preconnected'.op (h : is_preconnected' C) : is_preconnected' Cᵒᵖ :=\n⟨⟨λ X Y, (connected_components.op_equiv C).symm.injective (subsingleton.elim _ _)⟩⟩\n\nlemma is_preconnected'.unop (h : is_preconnected' Cᵒᵖ) : is_preconnected' C :=\n⟨⟨λ X Y, (connected_components.op_equiv C).injective (subsingleton.elim _ _)⟩⟩\n\nlemma is_connected'.op (h : is_connected' C) : is_connected' Cᵒᵖ :=\nbegin\n  haveI : nonempty Cᵒᵖ := ⟨opposite.op h.is_nonempty.some⟩,\n  haveI : is_preconnected' Cᵒᵖ := is_preconnected'.op infer_instance,\n  constructor,\nend\n\nlemma is_connected'.unop (h : is_connected' Cᵒᵖ) : is_connected' C :=\nbegin\n  haveI : nonempty C := ⟨opposite.unop h.is_nonempty.some⟩,\n  haveI : is_preconnected' C := is_preconnected'.unop infer_instance,\n  constructor,\nend\n\nend\n\nnamespace morphism_property\n\nvariables {C : Type*} [category C] (W : morphism_property C)\n\nclass multiplicative : Prop :=\n(contains_identities [] : W.contains_identities)\n(comp [] : W.stable_under_composition)\n\nsection\n\nvariable [multiplicative W]\n\n@[priority 100]\ninstance contains_identities_of_multiplicative : W.contains_identities :=\nmultiplicative.contains_identities _\n\ninstance : multiplicative W.op :=\n{ contains_identities := (multiplicative.contains_identities W).op,\n  comp := (multiplicative.comp W).op, }\n\ninclude W\n@[protected, nolint unused_arguments]\nstructure category :=\n(obj : C)\n\nvariable {W}\n\n@[ext]\nstructure category.hom (X Y : W.category) :=\n(f : X.obj ⟶ Y.obj)\n(hf : W f)\n\n@[simps]\ninstance : category W.category :=\n{ hom := category.hom,\n  id := λ X,\n  { f := 𝟙 _,\n    hf := contains_identities.id _ _, },\n  comp := λ X Y Z φ φ',\n  { f := φ.f ≫ φ'.f,\n    hf := multiplicative.comp W φ.f φ'.f φ.hf φ'.hf, }, }\n\n@[simps]\ndef category.mk_iso {X Y : W.category} (e : X.obj ≅ Y.obj) (h₁ : W e.hom) (h₂ : W e.inv) :\n  X ≅ Y :=\n{ hom := ⟨e.hom, h₁⟩,\n  inv := ⟨e.inv, h₂⟩, }\n\nend\n\nend morphism_property\n\nnamespace functor\n\nvariables (C₁ C₂ C₃ : Type*) [category C₁] [category C₂] [category C₃]\n  (F : C₁ ⥤ C₂) (G : C₂ ⥤ C₃)\n\n@[simps]\ndef whiskering_left_id : (whiskering_left C₁ C₁ C₃).obj (𝟭 C₁) ≅ 𝟭 _ :=\nnat_iso.of_components functor.left_unitor (by tidy)\n\n@[simps]\ndef whiskering_right_id : (whiskering_right C₁ C₃ C₃).obj (𝟭 C₃) ≅ 𝟭 _ :=\nnat_iso.of_components functor.right_unitor (by tidy)\n\nvariables {C₁ C₂}\n\n@[simps]\ndef equivalence_whiskering_left (e : C₁ ≌ C₂) : (C₂ ⥤ C₃) ≌ C₁ ⥤ C₃ :=\n{ functor := (whiskering_left _ _ _).obj e.functor,\n  inverse := (whiskering_left _ _ _).obj e.inverse,\n  unit_iso := (whiskering_left_id _ _).symm ≪≫ (whiskering_left _ _ C₃).map_iso e.counit_iso.symm,\n  counit_iso := (whiskering_left _ _ C₃).map_iso e.unit_iso.symm ≪≫ whiskering_left_id _ _,\n  functor_unit_iso_comp' := λ F, begin\n    ext X,\n    dsimp,\n    simp only [id_comp, comp_id, ← F.map_comp, equivalence.counit_inv_functor_comp, F.map_id],\n  end, }\n\ninstance is_equivalence_whiskering_left [is_equivalence F] :\n  is_equivalence ((whiskering_left _ _ C₃).obj F) :=\nis_equivalence.of_equivalence (equivalence_whiskering_left C₃ (as_equivalence F))\n\nvariables {C₂ C₃} (C₁)\n\n@[simps]\ndef equivalence_whiskering_right (e : C₂ ≌ C₃) : (C₁ ⥤ C₂) ≌ C₁ ⥤ C₃ :=\n{ functor := (whiskering_right _ _ _).obj e.functor,\n  inverse := (whiskering_right _ _ _).obj e.inverse,\n  unit_iso := (whiskering_right_id C₁ C₂).symm ≪≫ (whiskering_right C₁ _ _).map_iso e.unit_iso,\n  counit_iso := (whiskering_right C₁ _ _).map_iso e.counit_iso ≪≫ whiskering_right_id C₁ C₃,\n  functor_unit_iso_comp' := λ F, begin\n    ext X,\n    dsimp,\n    simp,\n  end, }\n\ninstance is_equivalence_whiskering_right [is_equivalence G] :\n  is_equivalence ((whiskering_right C₁ _ _).obj G) :=\nis_equivalence.of_equivalence (equivalence_whiskering_right C₁ (as_equivalence G))\n\nend functor\n\nnamespace structured_arrow\n\nvariables {C₀ C₁ C₂ C₃ : Type*} [category C₀] [category C₁] [category C₂] [category C₃]\n  (X₃ X₃' : C₃) (f : X₃' ⟶ X₃) (F : C₁ ⥤ C₂) (eF : C₁ ≌ C₂)\n  (G G' G'' : C₂ ⥤ C₃) (τ τ' : G ⟶ G') (τ'' : G' ⟶ G'') (eG : G ≅ G')\n\n@[simps]\ndef whiskering_left : structured_arrow X₃ (F ⋙ G) ⥤ structured_arrow X₃ G :=\n{ obj := λ X₂, mk X₂.hom,\n  map := λ X₂ X₂' φ, hom_mk (F.map φ.right) (w φ), }\n\nvariables {X₃ X₃'}\n\n@[simps]\ndef precomp : structured_arrow X₃ G ⥤ structured_arrow X₃' G :=\n{ obj := λ X₂, mk (f ≫ X₂.hom),\n  map := λ X₂ X₂' φ, hom_mk φ.right (by tidy), }\n\nvariables {G G'} (X₃)\n\n@[simps]\ndef postcomp : structured_arrow X₃ G ⥤ structured_arrow X₃ G' :=\n{ obj := λ X₂, mk (X₂.hom ≫ τ.app X₂.right),\n  map := λ X₂ X₂' φ, hom_mk φ.right begin\n    dsimp,\n    simp only [assoc, ← τ.naturality, w_assoc φ],\n  end, }\n\nvariable (G)\n\n@[simps]\ndef postcomp_id : postcomp X₃ (𝟙 G) ≅ 𝟭 _ :=\nnat_iso.of_components (λ X, structured_arrow.iso_mk (iso.refl _) (by tidy)) (by tidy)\n\nvariable {G}\n\n@[simps]\ndef postcomp_comp : postcomp X₃ τ ⋙ postcomp X₃ τ'' ≅\n  postcomp X₃ (τ ≫ τ'') :=\nnat_iso.of_components (λ X, structured_arrow.iso_mk (iso.refl _) (by tidy)) (by tidy)\n\n@[simps]\ndef postcomp_iso_of_eq (h : τ = τ') : postcomp X₃ τ ≅ postcomp X₃ τ' :=\nnat_iso.of_components (λ X, structured_arrow.iso_mk (iso.refl _)\n  (by { dsimp, rw [G'.map_id, comp_id, h], })) (by tidy)\n\n@[simps]\ndef postcomp_iso : equivalence (structured_arrow X₃ G) (structured_arrow X₃ G') :=\n{ functor := postcomp X₃ eG.hom,\n  inverse := postcomp X₃ eG.inv,\n  unit_iso := (postcomp_id X₃ G).symm ≪≫ postcomp_iso_of_eq X₃ _ _ eG.hom_inv_id.symm ≪≫\n    (postcomp_comp _ _ _ _).symm,\n  counit_iso := postcomp_comp _ _ _ _ ≪≫ postcomp_iso_of_eq X₃ _ _ eG.inv_hom_id ≪≫\n    (postcomp_id X₃ G'), }\n\ninstance [is_iso τ] : is_equivalence (postcomp X₃ τ) :=\nis_equivalence.of_equivalence (postcomp_iso X₃ (as_iso τ))\n\nvariable (G)\n\n@[simps]\ndef whiskering_left_equivalence :\nequivalence (structured_arrow X₃ (eF.functor ⋙ G)) (structured_arrow X₃ G) :=\n{ functor := whiskering_left X₃ eF.functor G,\n  inverse := (postcomp_iso X₃ ((functor.left_unitor _).symm ≪≫\n      iso_whisker_right eF.counit_iso.symm _≪≫ functor.associator _ _ _)).functor ⋙\n    whiskering_left X₃ eF.inverse (eF.functor ⋙ G),\n  unit_iso := nat_iso.of_components\n    (λ Y, structured_arrow.iso_mk (eF.unit_iso.app _) begin\n      dsimp,\n      simp only [comp_id, id_comp],\n      congr' 2,\n      simpa only [← cancel_mono (eF.counit_iso.hom.app (eF.functor.obj Y.right)),\n        equivalence.functor_unit_comp, iso.inv_hom_id_app],\n    end) (by tidy),\n  counit_iso := nat_iso.of_components\n    (λ X, structured_arrow.iso_mk (eF.counit_iso.app _) begin\n      dsimp,\n      simp only [id_comp, assoc, ← G.map_comp, iso.inv_hom_id_app],\n      dsimp,\n      simp only [functor.map_id, comp_id],\n    end) (by tidy), }\n\ninstance [is_equivalence F] : is_equivalence (whiskering_left X₃ F G) :=\nis_equivalence.of_equivalence (whiskering_left_equivalence X₃ (functor.as_equivalence F) G)\n\nend structured_arrow\n\nnamespace functor\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F : C ⥤ D} (RF RF' : H ⥤ D) (e : RF ≅ RF') {L : C ⥤ H} (α : F ⟶ L ⋙ RF) (α' : F ⟶ L ⋙ RF')\n  (W : morphism_property C) [L.is_localization W]\n\nclass is_right_derived_functor : Prop :=\n(is_initial [] : nonempty (limits.is_initial (structured_arrow.mk α :\n  structured_arrow F ((whiskering_left C H D).obj L))))\n\nvariables {RF RF'}\n\nlemma is_right_derived_functor.of_iso [RF.is_right_derived_functor α]\n  (eq : α' = α ≫ whisker_left L e.hom) : RF'.is_right_derived_functor α' :=\n⟨⟨limits.is_initial.of_iso (is_right_derived_functor.is_initial α).some\n  (structured_arrow.iso_mk e eq.symm)⟩⟩\n\nvariables (RF RF')\n\ndef is_right_derived_functor_to [RF.is_right_derived_functor α] (G : H ⥤ D) (β : F ⟶ L ⋙ G) :\n  RF ⟶ G :=\n(structured_arrow.proj _ _).map\n  ((functor.is_right_derived_functor.is_initial α).some.to (structured_arrow.mk β))\n\n@[simp, reassoc]\nlemma is_right_derived_functor_to_comm [RF.is_right_derived_functor α] (G : H ⥤ D)\n  (β : F ⟶ L ⋙ G) :\n  α ≫ whisker_left L (RF.is_right_derived_functor_to α G β) = β :=\nstructured_arrow.w ((functor.is_right_derived_functor.is_initial α).some.to\n  (structured_arrow.mk β))\n\n@[simp, reassoc]\nlemma is_right_derived_functor_to_comm_app [RF.is_right_derived_functor α] (G : H ⥤ D)\n  (β : F ⟶ L ⋙ G) (X : C) :\n  α.app X ≫ (RF.is_right_derived_functor_to α G β).app (L.obj X) = β.app X :=\ncongr_app (RF.is_right_derived_functor_to_comm α G β) X\n\nlemma is_right_derived_functor_to_ext [RF.is_right_derived_functor α] {G : H ⥤ D}\n  (γ₁ γ₂ : RF ⟶ G) (hγ : α ≫ whisker_left L γ₁ = α ≫ whisker_left L γ₂) : γ₁ = γ₂ :=\nbegin\n  let F' : structured_arrow F ((whiskering_left C H D).obj L) :=\n    structured_arrow.mk α,\n  let δ₁ : F' ⟶ structured_arrow.mk (α ≫ whisker_left L γ₁) := structured_arrow.hom_mk γ₁ rfl,\n  let δ₂ : F' ⟶ structured_arrow.mk (α ≫ whisker_left L γ₁) := structured_arrow.hom_mk γ₂ hγ.symm,\n  exact (structured_arrow.proj _ _).congr_map\n    ((functor.is_right_derived_functor.is_initial α).some.hom_ext δ₁ δ₂),\nend\n\nend functor\n\nnamespace nat_trans\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F G G' : C ⥤ D} (τ : F ⟶ G) (τ' : G ⟶ G') {RF RG RG' : H ⥤ D} {L : C ⥤ H}\n  (α : F ⟶ L ⋙ RF) (β : G ⟶ L ⋙ RG) (γ : G' ⟶ L ⋙ RG')\n\ndef right_derived [RF.is_right_derived_functor α] : RF ⟶ RG :=\nRF.is_right_derived_functor_to α RG (τ ≫ β)\n\n@[simp]\nlemma right_derived_comp [RF.is_right_derived_functor α]\n  [RG.is_right_derived_functor β] :\n  nat_trans.right_derived τ α β ≫ nat_trans.right_derived τ' β γ =\n    nat_trans.right_derived (τ ≫ τ') α γ :=\nbegin\n  dsimp only [right_derived],\n  apply RF.is_right_derived_functor_to_ext α,\n  simp only [whisker_left_comp, functor.is_right_derived_functor_to_comm_assoc, assoc,\n    functor.is_right_derived_functor_to_comm],\nend\n\n@[simp]\nlemma right_derived_id [RF.is_right_derived_functor α] :\n  nat_trans.right_derived (𝟙 F) α α = 𝟙 RF :=\nbegin\n  dsimp only [right_derived],\n  apply RF.is_right_derived_functor_to_ext α,\n  simp only [id_comp, functor.is_right_derived_functor_to_comm, whisker_left_id', comp_id],\nend\n\n@[simp, reassoc]\nlemma right_derived_app [RF.is_right_derived_functor α] (X : C) :\n  α.app X ≫ (right_derived τ α β).app (L.obj X) = τ.app X ≫ β.app X :=\nbegin\n  dsimp only [right_derived],\n  simp only [functor.is_right_derived_functor_to_comm_app, comp_app],\nend\n\nend nat_trans\n\nnamespace nat_iso\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F G : C ⥤ D} (e : F ≅ G) {RF RG : H ⥤ D} {L : C ⥤ H}\n  (α : F ⟶ L ⋙ RF) (β : G ⟶ L ⋙ RG)\n\n@[simps]\ndef right_derived [RF.is_right_derived_functor α] [RG.is_right_derived_functor β] :\n  RF ≅ RG :=\n{ hom := nat_trans.right_derived e.hom α β,\n  inv := nat_trans.right_derived e.inv β α, }\n\ninstance [RF.is_right_derived_functor α] [RG.is_right_derived_functor β] (τ : F ⟶ G)\n  [is_iso τ] : is_iso (nat_trans.right_derived τ α β) :=\nis_iso.of_iso (nat_iso.right_derived (as_iso τ) α β)\n\nend nat_iso\n\nnamespace functor\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  (F : C ⥤ D) (RF : H ⥤ D) {L : C ⥤ H} (α : F ⟶ L ⋙ RF)\n  (W : morphism_property C) [L.is_localization W]\n\nclass has_right_derived_functor : Prop :=\n(has_initial' : limits.has_initial (structured_arrow F ((whiskering_left C _ D).obj W.Q)))\n\nvariable (L)\n\nlemma has_right_derived_functor_iff :\n  has_right_derived_functor F W ↔\n    limits.has_initial (structured_arrow F ((whiskering_left C H D).obj L)) :=\nbegin\n  let Φ := structured_arrow.whiskering_left F ((whiskering_left _ _ _).obj\n      (localization.equivalence_from_model L W).functor)\n      ((whiskering_left C _ D).obj W.Q),\n  let Φ' := structured_arrow.postcomp F ((whiskering_left _ _ D).map_iso\n    (localization.Q_comp_equivalence_from_model_functor_iso L W)).inv,\n  split,\n  { intro h,\n    haveI := h.has_initial',\n    exact adjunction.has_colimits_of_shape_of_equivalence (Φ' ⋙ Φ), },\n  { introI,\n    exact ⟨adjunction.has_colimits_of_shape_of_equivalence (inv (Φ' ⋙ Φ))⟩, },\nend\n\nlemma is_right_derived_functor.has_right_derived_functor [RF.is_right_derived_functor α] :\n  F.has_right_derived_functor W :=\nbegin\n  rw F.has_right_derived_functor_iff L W,\n  exact limits.is_initial.has_initial (is_right_derived_functor.is_initial α).some,\nend\n\nlemma has_right_derived_functor.has_initial [has_right_derived_functor F W] :\n  limits.has_initial (structured_arrow F ((whiskering_left C H D).obj L)) :=\n(has_right_derived_functor_iff F L W).1 infer_instance\n\ndef has_right_derived_functor.initial [has_right_derived_functor F W] :\n  (structured_arrow F ((whiskering_left C H D).obj L)) :=\nbegin\n  haveI := has_right_derived_functor.has_initial F L W,\n  exact limits.initial _,\nend\n\ndef right_derived_functor [has_right_derived_functor F W] : H ⥤ D :=\n(has_right_derived_functor.initial F L W).right\n\ndef right_derived_functor_α [has_right_derived_functor F W] :\n  F ⟶ L ⋙ F.right_derived_functor L W :=\n(has_right_derived_functor.initial F L W).hom\n\ninstance right_derived_functor_is_right_derived_functor [has_right_derived_functor F W] :\n  (F.right_derived_functor L W).is_right_derived_functor (F.right_derived_functor_α L W) :=\n⟨⟨begin\n  haveI := has_right_derived_functor.has_initial F L W,\n  exact limits.is_initial.of_iso limits.initial_is_initial\n    (structured_arrow.iso_mk (iso.refl _) (by tidy)),\nend⟩⟩\n\nend functor\n\nsection\n\nvariables {C D : Type*} [category C] [category D]\n  (W : morphism_property C) (W' : morphism_property D)\n\nstructure localizor_morphism :=\n(functor : C ⥤ D)\n(mapW : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), W' (functor.map f))\n\nnamespace localizor_morphism\n\nvariables {W W'} (Φ : localizor_morphism W W')\n\nsection\n\n@[simps]\ndef op : localizor_morphism W.op W'.op :=\n{ functor := Φ.functor.op,\n  mapW := λ X Y f hf, Φ.mapW _ hf, }\n\nvariables [morphism_property.multiplicative W] [morphism_property.multiplicative W']\n\n@[simps]\ndef induced_functor : W.category ⥤ W'.category :=\n{ obj := λ X, ⟨Φ.functor.obj X.obj⟩,\n  map := λ X Y φ,\n  { f := Φ.functor.map φ.f,\n    hf := Φ.mapW φ.f φ.hf, }, }\n\n@[derive category]\ndef right_resolution (Y : D) := structured_arrow (⟨Y⟩ : W'.category) Φ.induced_functor\n\n@[simps]\ndef right_resolution.mk {Y : D} (X : C) (f : Y ⟶ Φ.functor.obj X) (hf : W' f) :\n  Φ.right_resolution Y :=\nstructured_arrow.mk (⟨f, hf⟩ : (⟨Y⟩ : W'.category) ⟶ Φ.induced_functor.obj ⟨X⟩)\n\n@[derive category]\ndef left_resolution (Y : D) := costructured_arrow Φ.induced_functor (⟨Y⟩ : W'.category)\n\n@[simps]\ndef left_resolution.mk {Y : D} (X : C) (f : Φ.functor.obj X ⟶ Y) (hf : W' f) :\n  Φ.left_resolution Y :=\ncostructured_arrow.mk (⟨f, hf⟩ : Φ.induced_functor.obj ⟨X⟩ ⟶ (⟨Y⟩ : W'.category))\n\nvariable {Φ}\n\n@[simps]\ndef left_resolution.op {Y : D} (X : Φ.left_resolution Y) :\n  Φ.op.right_resolution (opposite.op Y) :=\nright_resolution.mk Φ.op (opposite.op X.left.1) X.hom.1.op X.hom.2\n\n@[simps]\ndef right_resolution.unop {Y : D} (X : Φ.op.right_resolution (opposite.op Y)) :\n  Φ.left_resolution Y :=\nleft_resolution.mk Φ (opposite.unop X.right.1) X.hom.1.unop X.hom.2\n\n@[simps]\ndef left_resolution.unop_op {Y : D} (X : Φ.left_resolution Y) :\n  X.op.unop ≅ X :=\ncostructured_arrow.iso_mk\n  (morphism_property.category.mk_iso (iso.refl _)\n    (morphism_property.contains_identities.id _ _)\n    (morphism_property.contains_identities.id _ _))\n  (by { ext, dsimp, simp, })\n\n@[simps]\ndef right_resolution.op_unop {Y : D} (X : Φ.op.right_resolution (opposite.op Y)) :\n  X.unop.op ≅ X :=\nstructured_arrow.iso_mk\n  (morphism_property.category.mk_iso (iso.refl _)\n    (morphism_property.contains_identities.id _ _)\n    (morphism_property.contains_identities.id _ _))\n  (by { ext, dsimp, simp, })\n\nvariable (Φ)\n\n@[simps]\ndef left_resolution.op_functor (Y : D) :\n  Φ.left_resolution Y ⥤ (Φ.op.right_resolution (opposite.op Y))ᵒᵖ :=\n{ obj := λ X, opposite.op (left_resolution.op X),\n  map := λ X₁ X₂ f, quiver.hom.op (structured_arrow.hom_mk ⟨f.left.1.op, f.left.2⟩\n    (by { ext, dsimp, simpa only [← costructured_arrow.w f], })), }\n\n@[simps]\ndef right_resolution.unop_functor (Y : D) :\n  (Φ.op.right_resolution (opposite.op Y))ᵒᵖ ⥤ Φ.left_resolution Y :=\n{ obj := λ X, (opposite.unop X).unop,\n  map := λ X₁ X₂ f, costructured_arrow.hom_mk ⟨f.unop.right.1.unop, f.unop.right.2⟩\n    (by { ext, dsimp, simpa only [← structured_arrow.w f.unop], }), }\n\n@[simps]\ndef left_resolution.op_equivalence (Y : D) :\n  Φ.left_resolution Y ≌ (Φ.op.right_resolution (opposite.op Y))ᵒᵖ :=\n{ functor := left_resolution.op_functor _ _,\n  inverse := right_resolution.unop_functor _ _,\n  unit_iso := nat_iso.of_components (λ X, X.unop_op.symm) (by tidy),\n  counit_iso := nat_iso.of_components (λ X, ((opposite.unop X).op_unop).symm.op)\n    (λ X Y f, quiver.hom.unop_inj (by tidy)),\n  functor_unit_iso_comp' := λ X, quiver.hom.unop_inj (by tidy), }\n\nend\n\nvariables {C' D' : Type*} [category C'] [category D'] (L₁ : C ⥤ C') (L₂ : D ⥤ D')\n  [L₁.is_localization W] [L₂.is_localization W']\n\nabbreviation lift_functor : C' ⥤ D' :=\nlocalization.lift (Φ.functor ⋙ L₂)\n  (λ X Y f (hf : W f),\n    by { dsimp, exact localization.inverts L₂ W' _ (Φ.mapW f hf), }) L₁\n\ndef fac_functor : L₁ ⋙ Φ.lift_functor L₁ L₂ ≅ Φ.functor ⋙ L₂ :=\nlocalization.fac _ _ _\n\nclass is_localization_equivalence : Prop :=\n(nonempty_is_equivalence : nonempty (is_equivalence (Φ.lift_functor W.Q W'.Q)))\n\nnamespace is_localization_equivalence\n\nlemma iff_aux (C'' D'' : Type*) [category C''] [category D''] (L₁' : C ⥤ C'') (L₂' : D ⥤ D'')\n  [L₁'.is_localization W] [L₂'.is_localization W']\n  (h : is_equivalence (Φ.lift_functor L₁ L₂)) : is_equivalence (Φ.lift_functor L₁' L₂') :=\nbegin\n  let F₁ : C' ⥤ C'' := localization.lift L₁' (localization.inverts L₁' W) L₁,\n  let F₂ : D' ⥤ D'' := localization.lift L₂' (localization.inverts L₂' W') L₂,\n  have e : Φ.lift_functor L₁ L₂ ⋙ F₂ ≅ F₁ ⋙ Φ.lift_functor L₁' L₂' :=\n    localization.lift_nat_iso L₁ W (L₁ ⋙ Φ.lift_functor L₁ L₂ ⋙ F₂)\n      (L₁ ⋙ F₁ ⋙ Φ.lift_functor L₁' L₂') _ _ begin\n      refine (functor.associator _ _ _).symm ≪≫ iso_whisker_right (Φ.fac_functor L₁ L₂) _ ≪≫\n        functor.associator _ _ _ ≪≫ iso_whisker_left _ (localization.fac _ _ _) ≪≫\n        (Φ.fac_functor L₁' L₂').symm ≪≫\n        iso_whisker_right (localization.fac L₁' (localization.inverts L₁' W) L₁).symm _ ≪≫\n        functor.associator _ _ _,\n    end,\n  exact is_equivalence.cancel_comp_left F₁ _ infer_instance\n    (is_equivalence.of_iso e infer_instance),\nend\n\nlemma iff (F : C' ⥤ D') (e : L₁ ⋙ F ≅ Φ.functor ⋙ L₂) :\n  Φ.is_localization_equivalence ↔ nonempty (is_equivalence F) :=\nbegin\n  have h : nonempty (is_equivalence F) ↔ nonempty (is_equivalence (Φ.lift_functor L₁ L₂)),\n  { letI : localization.lifting L₁ W (Φ.functor ⋙ L₂) F := ⟨e⟩,\n    let e' : F ≅ Φ.lift_functor L₁ L₂ :=\n      localization.lift_nat_iso L₁ W (Φ.functor ⋙ L₂) (Φ.functor ⋙ L₂) _ _ (iso.refl _),\n    exact ⟨λ h₁, ⟨is_equivalence.of_iso e' h₁.some⟩,\n      λ h₂, ⟨is_equivalence.of_iso e'.symm h₂.some⟩⟩, },\n  rw h, clear h,\n  split,\n  { intro h,\n    exact ⟨(iff_aux Φ _ _ _ _ _ _ h.nonempty_is_equivalence.some)⟩, },\n  { intro h,\n    exact ⟨⟨(iff_aux Φ _ _ _ _ _ _ h.some)⟩⟩, },\nend\n\nlemma iff_is_equivalence_lift_functor :\n  Φ.is_localization_equivalence ↔ nonempty (is_equivalence (Φ.lift_functor L₁ L₂)) :=\nis_localization_equivalence.iff Φ L₁ L₂ (Φ.lift_functor L₁ L₂) (Φ.fac_functor L₁ L₂)\n\nlemma iff_is_localization :\n  Φ.is_localization_equivalence ↔ (Φ.functor ⋙ L₂).is_localization W :=\nbegin\n  split,\n  { intro h,\n    rw iff_is_equivalence_lift_functor Φ W.Q L₂ at h,\n    letI := h.some,\n    exact functor.is_localization.of_equivalence W.Q W (Φ.functor ⋙ L₂)\n      (functor.as_equivalence (Φ.lift_functor W.Q L₂)) (Φ.fac_functor _ _), },\n  { introI,\n    rw iff_is_equivalence_lift_functor Φ (Φ.functor ⋙ L₂) L₂,\n    exact ⟨is_equivalence.of_iso (localization.lifting.uniq (Φ.functor ⋙ L₂) W\n      (Φ.functor ⋙ L₂) (𝟭 _) _) infer_instance⟩, },\nend\n\ninstance [hΦ : Φ.is_localization_equivalence] :\n  is_equivalence (Φ.lift_functor L₁ L₂) :=\n((iff_is_equivalence_lift_functor Φ L₁ L₂).mp hΦ).some\n\ninstance is_localization_of_is_localization_equivalence [hΦ : Φ.is_localization_equivalence] :\n  (Φ.functor ⋙ L₂).is_localization W :=\nby simpa only [← iff_is_localization Φ L₂] using hΦ\n\ninstance op_is_localization_equivalence [hΦ : Φ.is_localization_equivalence] :\n  Φ.op.is_localization_equivalence :=\nbegin\n  rw iff_is_localization Φ W'.Q at hΦ,\n  rw iff_is_localization Φ.op W'.Q.op,\n  haveI := hΦ,\n  change (Φ.functor ⋙ W'.Q).op.is_localization W.op,\n  apply_instance,\nend\n\nend is_localization_equivalence\n\nend localizor_morphism\n\nend\n\nnamespace right_derivability_structure\n\nvariables {C₀ C H : Type*} [category C] [category C₀] [category H]\n  {W₀ : morphism_property C₀}\n  {W : morphism_property C} (Φ : localizor_morphism W₀ W)\n  [localizor_morphism.is_localization_equivalence Φ]\n  [morphism_property.multiplicative W₀] [morphism_property.multiplicative W]\n\nstructure basic :=\n(right_resolution_connected : ∀ (Y : C), is_connected' (Φ.right_resolution Y))\n(nonempty_arrow_right_resolution :\n  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂), ∃ (X₁ : Φ.right_resolution Y₁) (X₂ : Φ.right_resolution Y₂)\n  (f' : X₁.right.obj ⟶ X₂.right.obj), X₁.hom.1 ≫ Φ.functor.map f' = f ≫ X₂.hom.1)\n\nnamespace basic\n\nvariables {Φ} (β : basic Φ) (L : C ⥤ H) [L.is_localization W]\n  {D : Type*} [category D]\n\ndef some_right_resolution (Y : C) : Φ.right_resolution Y :=\n(β.right_resolution_connected Y).is_nonempty.some\n\nvariables {F : C ⥤ D} (hF : W₀.is_inverted_by (Φ.functor ⋙ F))\n\nnamespace existence_derived_functor\n\ninclude β hF\n\ndef RF : H ⥤ D :=\nlocalization.lift (Φ.functor ⋙ F) hF (Φ.functor ⋙ L)\n\ndef ε : (Φ.functor ⋙ L) ⋙ RF β L hF ≅ Φ.functor ⋙ F :=\nbegin\n  letI : localization.lifting (Φ.functor ⋙ L) W₀ (Φ.functor ⋙ F) (RF β L hF) :=\n    localization.lifting_lift _ _ _,\n  refine localization.lifting.iso (Φ.functor ⋙ L) W₀ _ _,\nend\n\ndef α' (X : C) : (functor.const (Φ.right_resolution X)).obj (F.obj X) ⟶\n  (functor.const (Φ.right_resolution X)).obj ((RF β L hF).obj (L.obj X)) :=\n{ app := λ X₀, F.map X₀.hom.1 ≫ (ε β L hF).inv.app _ ≫\n    (RF β L hF).map (localization.iso_of_hom L W _ X₀.hom.2).inv,\n  naturality' := λ X₀ X₀' φ, begin\n    dsimp,\n    simp only [functor.map_inv, id_comp, comp_id],\n    have eq₁ := φ.w,\n    have eq₂ := (ε β L hF).inv.naturality φ.right.1,\n    dsimp at eq₁ eq₂,\n    rw id_comp at eq₁,\n    rw eq₁,\n    dsimp,\n    rw [functor.map_comp, assoc, reassoc_of eq₂],\n    congr' 2,\n    simp only [functor.map_comp],\n    erw is_iso.comp_inv_eq, -- should be tidied\n    rw is_iso.inv_hom_id_assoc,\n  end, }\n\ndef α_app (X : C) : F.obj X ⟶ (RF β L hF).obj (L.obj X) :=\n(α' β L hF X).app (β.some_right_resolution X)\n\nlemma α_app_eq {X : C} (X₀ : Φ.right_resolution X) :\n  (α_app β L hF) X = (α' β L hF X).app X₀ :=\nbegin\n  haveI := β.right_resolution_connected X,\n  apply nat_trans_from_is_preconnected',\nend\n\n@[simps]\ndef α : F ⟶ L ⋙ RF β L hF :=\n{ app := λ X, (α_app β L hF) X,\n  naturality' := λ Y₁ Y₂ f, begin\n    obtain ⟨X₁, X₂, f', fac⟩ := β.nonempty_arrow_right_resolution f,\n    rw [α_app_eq β L hF X₁, α_app_eq β L hF X₂],\n    dsimp [α'],\n    have eq₁ := F.congr_map fac,\n    have eq₂ := (ε β L hF).inv.naturality f',\n    simp only [functor.map_comp] at eq₁,\n    dsimp at eq₂,\n    simp only [assoc, ← reassoc_of eq₁, reassoc_of eq₂, ← functor.map_comp],\n    congr' 3,\n    rw [is_iso.eq_inv_comp, ← L.map_comp_assoc, fac, L.map_comp, assoc,\n      is_iso.hom_inv_id, comp_id],\n  end, }\n\ninstance (X₀ : C₀) : is_iso ((α β L hF).app (Φ.functor.obj X₀)) :=\nbegin\n  let X₀' := localizor_morphism.right_resolution.mk Φ X₀ (𝟙 _)\n    (morphism_property.contains_identities.id W _),\n  dsimp [α],\n  rw α_app_eq β L hF (localizor_morphism.right_resolution.mk Φ X₀ (𝟙 _)\n    (morphism_property.contains_identities.id W _)),\n  dsimp [α'],\n  simp only [functor.map_id, is_iso.inv_id, comp_id, id_comp],\n  apply_instance,\nend\n\n@[simps]\ndef RF' : structured_arrow F ((whiskering_left C H D).obj L) :=\nstructured_arrow.mk (α β L hF)\n\ninstance is_iso_RF'_hom_app (X₀ : C₀) :\n  is_iso ((RF' β L hF).hom.app (Φ.functor.obj X₀)) :=\n(infer_instance : is_iso ((α β L hF).app (Φ.functor.obj X₀)))\n\ninstance (G : structured_arrow F ((whiskering_left C H D).obj L)) :\n  subsingleton (RF' β L hF ⟶ G) :=\n⟨λ φ₁ φ₂, begin\n  apply structured_arrow.ext,\n  apply localization.nat_trans_ext (Φ.functor ⋙ L) W₀,\n  intro X₀,\n  have eq₁ := congr_app φ₁.w (Φ.functor.obj X₀),\n  have eq₂ := congr_app φ₂.w (Φ.functor.obj X₀),\n  dsimp at eq₁ eq₂ ⊢,\n  rw [id_comp] at eq₁ eq₂,\n  rw [← cancel_epi ((α β L hF).app (Φ.functor.obj X₀))],\n  dsimp,\n  rw [← eq₁, eq₂],\nend⟩\n\ndef RF_τ' (G : structured_arrow F ((whiskering_left C H D).obj L)) :\n  RF β L hF ⟶ G.right :=\nlocalization.lift_nat_trans (Φ.functor ⋙ L) W₀ _ _ _ _\n  ((ε β L hF).hom ≫ whisker_left _ G.hom ≫ (functor.associator _ _ _).inv)\n\n@[simp]\nlemma RF_τ'_app_eq (G : structured_arrow F ((whiskering_left C H D).obj L)) (X₀ : C₀) :\n  (RF_τ' β L hF G).app (L.obj (Φ.functor.obj X₀)) =\n    (ε β L hF).hom.app X₀ ≫ G.hom.app (Φ.functor.obj X₀) :=\nbegin\n  dsimp [RF_τ'],\n  erw localization.lift_nat_trans_app,\n  simp only [localization.lifting.of_comp_iso, iso.refl_hom, nat_trans.id_app,\n    nat_trans.comp_app, whisker_left_app, functor.associator_inv_app,\n    comp_id, iso.refl_inv, assoc],\n  erw id_comp,\nend\n\ndef RF_τ (G : structured_arrow F ((whiskering_left C H D).obj L)) :\n  RF' β L hF ⟶ G :=\nbegin\n  refine structured_arrow.hom_mk (RF_τ' β L hF G) _,\n  ext X,\n  let X₀ := β.some_right_resolution X,\n  have eq := (RF_τ' β L hF G).naturality (L.map X₀.hom.f),\n  haveI : is_iso (L.map X₀.hom.f) := localization.inverts L W _ X₀.hom.hf,\n  dsimp at ⊢ eq,\n  simp only [← cancel_mono (G.right.map (L.map X₀.hom.f)), assoc, ← eq, RF_τ'_app_eq,\n    α_app_eq β L hF X₀, α', ← functor.map_comp_assoc],\n  erw [is_iso.inv_hom_id, functor.map_id, id_comp, iso.inv_hom_id_app_assoc, G.hom.naturality],\n  refl,\nend\n\ninstance (G : structured_arrow F ((whiskering_left C H D).obj L)) :\n  unique (RF' β L hF ⟶ G) :=\nunique_of_subsingleton (RF_τ β L hF G)\n\nlemma is_initial_RF' : limits.is_initial (RF' β L hF) := limits.is_initial.of_unique _\n\ninstance RF_is_right_derived_functor :\n  (RF β L hF).is_right_derived_functor (α _ _ _) :=\n⟨⟨is_initial_RF' β L hF⟩⟩\n\nend existence_derived_functor\n\nvariable (F)\n\nopen existence_derived_functor\n\n/-\nThe following lemma is a consequence of Lemma 6.5 of\n_Structures de dérivabilité_ by Bruno Kahn, Georges Maltsiniotis,\nAdvances in mathematics 218 (2018).\n-/\n\nlemma existence_derived_functor : F.has_right_derived_functor W :=\nfunctor.is_right_derived_functor.has_right_derived_functor F (RF β W.Q hF) W.Q (α _ _ _) W\n\ninclude β hF\n\nlemma is_iso_app (F' : H ⥤ D) (α' : F ⟶ L ⋙ F') [F'.is_right_derived_functor α'] (X₀ : C₀) :\n  is_iso (α'.app (Φ.functor.obj X₀)) :=\nbegin\n  have h := nat_trans.right_derived_app (𝟙 F) (α β L hF) α' (Φ.functor.obj X₀),\n  rw [nat_trans.id_app, id_comp] at h,\n  rw ← h,\n  apply_instance,\nend\n\nend basic\n\nend right_derivability_structure\n\nnamespace functor\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F : C ⥤ D} (LF LF' : H ⥤ D) (e : LF ≅ LF') {L : C ⥤ H} (α : L ⋙ LF ⟶ F) (α' : L ⋙ LF' ⟶ F)\n  (W : morphism_property C) [L.is_localization W]\n\nclass is_left_derived_functor : Prop :=\n(is_terminal [] : nonempty (limits.is_terminal (costructured_arrow.mk α :\n  costructured_arrow ((whiskering_left C H D).obj L) F)))\n\nvariables {LF LF'}\n\nlemma is_left_derived_functor.of_iso [LF.is_left_derived_functor α]\n  (eq : α' = whisker_left L e.inv ≫ α) : LF'.is_left_derived_functor α' :=\n⟨⟨limits.is_terminal.of_iso (is_left_derived_functor.is_terminal α).some\n    (costructured_arrow.iso_mk e (by { dsimp, rw eq, ext, simp only [nat_trans.comp_app,\n      whisker_left_app, iso.hom_inv_id_app_assoc], }))⟩⟩\n\nvariables (LF LF')\n\ndef is_left_derived_functor_from [LF.is_left_derived_functor α] (G : H ⥤ D) (β : L ⋙ G ⟶ F) :\n  G ⟶ LF :=\n(costructured_arrow.proj _ _).map\n  ((functor.is_left_derived_functor.is_terminal α).some.from (costructured_arrow.mk β))\n\n@[simp, reassoc]\nlemma is_left_derived_functor_from_comm [LF.is_left_derived_functor α] (G : H ⥤ D)\n  (β : L ⋙ G ⟶ F) :\n  whisker_left L (LF.is_left_derived_functor_from α G β) ≫ α = β :=\ncostructured_arrow.w ((functor.is_left_derived_functor.is_terminal α).some.from\n  (costructured_arrow.mk β))\n\n@[simp, reassoc]\nlemma is_left_derived_functor_from_comm_app [LF.is_left_derived_functor α] (G : H ⥤ D)\n  (β : L ⋙ G ⟶ F) (X : C) :\n  (LF.is_left_derived_functor_from α G β).app (L.obj X) ≫ α.app X = β.app X :=\ncongr_app (LF.is_left_derived_functor_from_comm α G β) X\n\nlemma is_left_derived_functor_from_ext [LF.is_left_derived_functor α] {G : H ⥤ D}\n  (γ₁ γ₂ : G ⟶ LF) (hγ : whisker_left L γ₁ ≫ α = whisker_left L γ₂ ≫ α) : γ₁ = γ₂ :=\nbegin\n  let F' : costructured_arrow ((whiskering_left C H D).obj L) F :=\n    costructured_arrow.mk α,\n  let G' : costructured_arrow ((whiskering_left C H D).obj L) F :=\n    @costructured_arrow.mk _ _ _ _ F G ((whiskering_left C H D).obj L) (whisker_left L γ₁ ≫ α),\n  let δ₁ : G' ⟶ F' := costructured_arrow.hom_mk γ₁ rfl,\n  let δ₂ : G' ⟶ F' := costructured_arrow.hom_mk γ₂ hγ.symm,\n  exact (costructured_arrow.proj _ _).congr_map\n    ((functor.is_left_derived_functor.is_terminal α).some.hom_ext δ₁ δ₂),\nend\n\nend functor\n\nnamespace nat_trans\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F G G' : C ⥤ D} (τ : F ⟶ G) (τ' : G ⟶ G') {LF LG LG' : H ⥤ D} {L : C ⥤ H}\n  (α : L ⋙ LF ⟶ F) (β : L ⋙ LG ⟶ G) (γ : L ⋙ LG' ⟶ G')\n\ndef left_derived [LG.is_left_derived_functor β] : LF ⟶ LG :=\nLG.is_left_derived_functor_from β LF (α ≫ τ)\n\n@[simp]\nlemma left_derived_comp [LG.is_left_derived_functor β]\n  [LG'.is_left_derived_functor γ] :\n  nat_trans.left_derived τ α β ≫ nat_trans.left_derived τ' β γ =\n    nat_trans.left_derived (τ ≫ τ') α γ :=\nbegin\n  dsimp only [left_derived],\n  apply LG'.is_left_derived_functor_from_ext γ,\n  simp only [whisker_left_comp, assoc, functor.is_left_derived_functor_from_comm,\n    functor.is_left_derived_functor_from_comm_assoc],\nend\n\n@[simp]\nlemma left_derived_id [LF.is_left_derived_functor α] :\n  nat_trans.left_derived (𝟙 F) α α = 𝟙 LF :=\nbegin\n  dsimp only [left_derived],\n  apply LF.is_left_derived_functor_from_ext α,\n  simp only [comp_id, functor.is_left_derived_functor_from_comm, whisker_left_id', id_comp],\nend\n\n@[simp, reassoc]\nlemma left_derived_app [LG.is_left_derived_functor β] (X : C) :\n  (left_derived τ α β).app (L.obj X) ≫ β.app X = α.app X ≫ τ.app X :=\nbegin\n  dsimp only [left_derived],\n  simp only [functor.is_left_derived_functor_from_comm_app, comp_app],\nend\n\nend nat_trans\n\nnamespace nat_iso\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  {F G : C ⥤ D} (e : F ≅ G) {LF LG : H ⥤ D} {L : C ⥤ H}\n  (α : L ⋙ LF ⟶ F) (β : L ⋙ LG ⟶ G)\n\n@[simps]\ndef left_derived [LF.is_left_derived_functor α] [LG.is_left_derived_functor β] :\n  LF ≅ LG :=\n{ hom := nat_trans.left_derived e.hom α β,\n  inv := nat_trans.left_derived e.inv β α, }\n\ninstance [LF.is_left_derived_functor α] [LG.is_left_derived_functor β] (τ : F ⟶ G)\n  [is_iso τ] : is_iso (nat_trans.left_derived τ α β) :=\nis_iso.of_iso (nat_iso.left_derived (as_iso τ) α β)\n\nend nat_iso\n\nnamespace functor\n\nvariables {C D H : Type*} [category C] [category D] [category H]\n  (F : C ⥤ D) (LF : H ⥤ D) (L : C ⥤ H) (α : L ⋙ LF ⟶ F)\n  (W : morphism_property C) [L.is_localization W]\n\nclass has_left_derived_functor : Prop :=\n(has_terminal' : limits.has_terminal (costructured_arrow ((whiskering_left C _ D).obj W.Q) F))\n\nnamespace costructured_arrow_equivalence_op\n\n@[simps]\ndef functor : (costructured_arrow ((whiskering_left C H D).obj L) F) ⥤\n    (structured_arrow F.op ((whiskering_left Cᵒᵖ Hᵒᵖ Dᵒᵖ).obj L.op))ᵒᵖ :=\n{ obj := λ X, opposite.op (structured_arrow.mk\n    (show F.op ⟶ ((whiskering_left Cᵒᵖ Hᵒᵖ Dᵒᵖ).obj L.op).obj X.left.op,\n      by exact (functor.op_hom _ _).map X.hom.op)),\n  map := λ X₁ X₂ f, quiver.hom.op\n    (structured_arrow.hom_mk ((functor.op_hom H D).map (quiver.hom.op f.left))\n      (by { rw ← costructured_arrow.w f, refl, })), }\n\n@[simps]\ndef inverse : (structured_arrow F.op ((whiskering_left Cᵒᵖ Hᵒᵖ Dᵒᵖ).obj L.op))ᵒᵖ ⥤\n    (costructured_arrow ((whiskering_left C H D).obj L) F) :=\n{ obj := λ X, costructured_arrow.mk\n      (show ((whiskering_left C H D).obj L).obj X.unop.right.unop ⟶ F,\n        by exact ((functor.op_inv C D).map X.unop.hom).unop ≫ F.op_unop_iso.hom),\n  map := λ X₁ X₂ f, costructured_arrow.hom_mk (((functor.op_inv _ _).map f.unop.right).unop)\n      (by { rw ← structured_arrow.w f.unop, dsimp, ext, tidy, }), }\n\n@[simps]\ndef unit_iso : 𝟭 _ ≅ functor F L ⋙ inverse F L :=\nnat_iso.of_components (λ X, costructured_arrow.iso_mk (functor.op_unop_iso X.left).symm\n    (by { ext, dsimp, tidy, })) (by tidy)\n\n@[simps]\ndef counit_iso : inverse F L ⋙ functor F L ≅ 𝟭 _ :=\nnat_iso.of_components (λ X, begin\n  change opposite.op (opposite.unop _) ≅ opposite.op (opposite.unop _),\n  apply iso.op,\n  refine structured_arrow.iso_mk (functor.unop_op_iso _).symm _,\n  ext, dsimp, tidy,\nend) (λ X Y f, quiver.hom.unop_inj (by { dsimp, tidy, }))\n\nend costructured_arrow_equivalence_op\n\ndef costructured_arrow_equivalence_op :\n  (costructured_arrow ((whiskering_left C H D).obj L) F) ≌\n    (structured_arrow F.op ((whiskering_left Cᵒᵖ Hᵒᵖ Dᵒᵖ).obj L.op))ᵒᵖ :=\n{ functor := costructured_arrow_equivalence_op.functor _ _,\n  inverse := costructured_arrow_equivalence_op.inverse _ _,\n  unit_iso := costructured_arrow_equivalence_op.unit_iso _ _,\n  counit_iso := costructured_arrow_equivalence_op.counit_iso _ _,\n  functor_unit_iso_comp' := λ X, quiver.hom.unop_inj begin\n    dsimp [structured_arrow.iso_mk, structured_arrow.hom_mk, comma.iso_mk],\n    tidy,\n  end, }\n\nvariable (L)\n\nlemma has_left_derived_functor_iff_op :\n  has_left_derived_functor F W ↔ has_right_derived_functor F.op W.op :=\nbegin\n  have h : F.has_left_derived_functor W ↔\n    limits.has_terminal (costructured_arrow ((whiskering_left C _ D).obj W.Q) F) :=\n    ⟨λ h, h.1, λ h, ⟨h⟩⟩,\n  rw [h, has_right_derived_functor_iff F.op W.Q.op W.op],\n  have e := costructured_arrow_equivalence_op F W.Q,\n  split,\n  { introI,\n    haveI : limits.has_terminal\n      (structured_arrow F.op ((whiskering_left Cᵒᵖ (W.localization)ᵒᵖ Dᵒᵖ).obj W.Q.op))ᵒᵖ :=\n      adjunction.has_limits_of_shape_of_equivalence e.inverse,\n    exact limits.has_initial_of_has_terminal_op, },\n  { introI,\n    exact adjunction.has_limits_of_shape_of_equivalence e.functor, },\nend\n\nlemma has_left_derived_functor_iff :\n  has_left_derived_functor F W ↔\n    limits.has_terminal (costructured_arrow ((whiskering_left C H D).obj L) F) :=\nbegin\n  rw [has_left_derived_functor_iff_op, has_right_derived_functor_iff F.op L.op W.op],\n  have e := costructured_arrow_equivalence_op F L,\n  split,\n  { introI,\n    exact adjunction.has_limits_of_shape_of_equivalence e.functor, },\n  { introI,\n    haveI : limits.has_terminal\n      (structured_arrow F.op ((whiskering_left Cᵒᵖ Hᵒᵖ Dᵒᵖ).obj L.op))ᵒᵖ :=\n      adjunction.has_limits_of_shape_of_equivalence e.inverse,\n    exact limits.has_initial_of_has_terminal_op, },\nend\n\nlemma is_left_derived_functor.has_left_derived_functor [LF.is_left_derived_functor α] :\n  F.has_left_derived_functor W :=\nbegin\n  rw F.has_left_derived_functor_iff L W,\n  exact limits.is_terminal.has_terminal (is_left_derived_functor.is_terminal α).some,\nend\n\nvariables {F L LF F α}\n\nlemma is_left_derived_functor.op (hα : LF.is_left_derived_functor α) :\n  @is_right_derived_functor _ _ _ _ _ _ F.op LF.op L.op ((functor.op_hom _ _).map α.op) :=\nis_right_derived_functor.mk\n  (nonempty.intro (limits.initial_unop_of_terminal\n    (limits.is_terminal.of_equivalence (costructured_arrow_equivalence_op F L)\n      hα.is_terminal.some)))\n\nvariables (F L LF F α)\n\nlemma has_left_derived_functor.has_terminal [has_left_derived_functor F W] :\n  limits.has_terminal (costructured_arrow ((whiskering_left C H D).obj L) F) :=\n(has_left_derived_functor_iff F L W).1 infer_instance\n\ndef has_left_derived_functor.initial [has_left_derived_functor F W] :\n  (costructured_arrow ((whiskering_left C H D).obj L) F) :=\nbegin\n  haveI := has_left_derived_functor.has_terminal F L W,\n  exact limits.terminal _,\nend\n\ndef left_derived_functor [has_left_derived_functor F W] : H ⥤ D :=\n(has_left_derived_functor.initial F L W).left\n\ndef left_derived_functor_α [has_left_derived_functor F W] :\n  L ⋙ F.left_derived_functor L W ⟶ F :=\n(has_left_derived_functor.initial F L W).hom\n\ninstance left_derived_functor_is_left_derived_functor [has_left_derived_functor F W] :\n  (F.left_derived_functor L W).is_left_derived_functor (F.left_derived_functor_α L W) :=\n⟨⟨begin\n  haveI := has_left_derived_functor.has_terminal F L W,\n  exact limits.is_terminal.of_iso limits.terminal_is_terminal\n    (costructured_arrow.iso_mk (iso.refl _) (by tidy)),\nend⟩⟩\n\nend functor\n\nnamespace left_derivability_structure\n\nvariables {C₀ C H : Type*} [category C] [category C₀] [category H]\n  {W₀ : morphism_property C₀}\n  {W : morphism_property C} (L : C ⥤ H) [L.is_localization W] (Φ : localizor_morphism W₀ W)\n  [localizor_morphism.is_localization_equivalence Φ]\n  [morphism_property.multiplicative W₀] [morphism_property.multiplicative W]\n\nstructure basic :=\n(left_resolution_connected : ∀ (Y : C), is_connected' (Φ.left_resolution Y))\n(nonempty_arrow_left_resolution :\n  ∀ ⦃Y₁ Y₂ : C⦄ (f : Y₁ ⟶ Y₂), ∃ (X₁ : Φ.left_resolution Y₁) (X₂ : Φ.left_resolution Y₂)\n  (f' : X₁.left.obj ⟶ X₂.left.obj), Φ.functor.map f' ≫ X₂.hom.1 = X₁.hom.1 ≫ f)\n\nnamespace basic\n\nvariables {L Φ}\n\ndef op (β : basic  Φ) : right_derivability_structure.basic Φ.op :=\n{ right_resolution_connected := λ Y, (is_connected'.of_equivalence\n      (localizor_morphism.left_resolution.op_equivalence Φ (opposite.unop Y))\n      (β.left_resolution_connected (opposite.unop Y))).unop,\n  nonempty_arrow_right_resolution := λ Y₁ Y₂ f, begin\n    obtain ⟨X₁, X₂, f', fac⟩ := β.nonempty_arrow_left_resolution f.unop,\n    exact ⟨X₂.op, X₁.op, f'.op, quiver.hom.unop_inj fac⟩,\n  end, }\n\nvariables (β : basic Φ) {D : Type*} [category D] (F : C ⥤ D)\n  (hF : W₀.is_inverted_by (Φ.functor ⋙ F))\n\ninclude β hF\n\nlemma existence_derived_functor : F.has_left_derived_functor W :=\nby simpa only [functor.has_left_derived_functor_iff_op]\n  using β.op.existence_derived_functor F.op hF.op\n\nlemma is_iso_app (F' : H ⥤ D) (α' : L ⋙ F' ⟶ F) [hα' : F'.is_left_derived_functor α'] (X₀ : C₀) :\n  is_iso (α'.app (Φ.functor.obj X₀)) :=\nbegin\n  suffices : is_iso (α'.app (Φ.functor.obj X₀)).op,\n  { haveI := this,\n    exact is_iso.of_iso ((as_iso (α'.app (Φ.functor.obj X₀)).op).unop), },\n  let α'' : F.op ⟶ L.op ⋙ F'.op := (functor.op_hom C D).map α'.op,\n  haveI : F'.op.is_right_derived_functor α'' := hα'.op,\n  exact β.op.is_iso_app L.op F.op hF.op F'.op α'' (opposite.op X₀),\nend\n\nend basic\n\nend left_derivability_structure\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/derived_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.33613228959443797}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.measurable_space\nimport formal_ml.probability_space\nimport formal_ml.real_random_variable\nimport data.complex.exponential\nimport formal_ml.ennreal\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.exp_bound\n\n\n/-\n  The Vapnik-Chevronenkis Dimension and its connection to learning is one of the most\n  remarkable and fundamental results in machine learning. In particular, it allows us\n  to understand simple, infinite hypothesis spaces, like the separating hyperplane,\n  and begin to understand the complexity of neural networks. This analysis is also a\n  precursor to understanding support vector machines and structural risk.\n -/\n\nlemma finset.subset_of_not_mem_of_subset_insert {α:Type*} [decidable_eq α] {x:α} {S T:finset α}:x∉ S →\n  S ⊆ insert x T → S ⊆ T :=\nbegin\n  intros A1 A2,\n  rw finset.subset_iff,\n  rw finset.subset_iff at A2,\n  intros a B1,\n  have B2 := A2 B1,\n  rw finset.mem_insert at B2,\n  cases B2,\n  subst a,\n  exfalso,\n  apply A1,\n  apply B1,\n  apply B2,\nend\n\n\n\n/-\n  A type is of class inhabited if it has at least one element.\n  Thus, its cardinality is not zero.\n--Not sure where to put this. Here is fine for now.\n--Note: this is the kind of trivial thing that takes ten minutes to prove.\n-/\n\n/--\n  Normally, we would not assume that the representation type was encodable (i.e. countable).\n  Specifically, one could imagine real numbers as part of the representation type.\n  However, that creates some issues with measurability.\n -/\nstructure VC_PAC_problem :=\n   (Ω:Type*)                           -- Underlying outcome type\n   (p:probability_space Ω)             -- Underlying probability space\n   (X:Type*)                           -- instance type\n   (MX:measurable_space X)             -- Measurable space for the instances\n   (H:Type*)                           -- Representation type\n   (ER:encodable H)                    -- The representation type is encodable.\n   (R:H → (set X))                     -- Representation scheme\n   (MC:∀ h:H, is_measurable (R(h)))   -- Concepts are measurable.\n   (Di:Type*)                          -- number of examples\n   (FDi:fintype Di)                    -- number of examples are finite\n   (EDi:encodable Di)                  -- example index is encodable\n   (D:Di → (p →ᵣ MX))                  -- example distribution\n   (IID:random_variables_IID D)        -- examples are IID\n   (has_example:inhabited Di)          -- there exists an example\n   (c:H)                               -- the correct hypothesis\n\n\n\nnamespace VC_PAC_problem\n\nvariable (P:VC_PAC_problem)\n\n\n--The measurable space for the hypotheses is ⊤.\n--Everything is measurable.\ndef MH:measurable_space (P.H) := ⊤\n\n--The space of training datasets.\ndef data_space:measurable_space (P.Di → P.X) := @measurable_space.pi P.Di (λ i, P.X) (λ i, P.MX)\n\n\n--The learning algorithm is a function of the training dataset.\ndef algorithm:Type* := measurable_fun (P.data_space) P.MH\n\n\ndef concept (h:P.H):measurable_set P.MX := measurable_set.mk (P.MC h)  \n\ndef target_concept:measurable_set P.MX := (P.concept P.c)  \n\ndef in_concept (h:P.H) (i:P.Di):event (P.p) := @rv_event P.Ω P.p P.X P.MX (P.D i) (P.concept h)\n\ndef label_positive (i:P.Di):event (P.p) := P.in_concept P.c i\n\ndef example_correct (h:P.H) (i:P.Di):event (P.p) :=\n    (P.in_concept h i) =ₑ (P.label_positive i)\n\ndef example_error (h:P.H) (i:P.Di):event (P.p) :=\n    ¬ₑ (P.example_correct h i)\n\n/-\n  num_examples P is the number of examples in the problem.\n  This is defined as the cardinality of the index type of the examples.\n-/\ndef num_examples:nat\n   := @fintype.card P.Di P.FDi\n\n--the measurable space on hypotheses and instances.\ndef MHX:measurable_space (P.H × P.X) := P.MH ×ₘ P.MX\n\nnoncomputable def training_error (h:P.H):P.p →ᵣ (borel nnreal) :=\n  average_identifier (P.example_error h) (P.FDi)\n\n/-\n  The number of examples is the number of elements of type P.Di.\n  P.FDi.elems is the set of all elements in P.Di, and P.FDi.elems.card is the cardinality of\n  P.FDi.elems.\n-/\nlemma num_examples_eq_finset_card:\n  P.num_examples = P.FDi.elems.card :=\nbegin\n  refl,\nend\n\n/-\n  The number of examples do not equal zero.\n -/\nlemma num_examples_ne_zero:\n  P.num_examples ≠ 0 :=\nbegin\n  unfold VC_PAC_problem.num_examples,\n  apply @card_ne_zero_of_inhabited P.Di P.has_example P.FDi,\nend\n\n/-\n  The expected test error.\n  The test error is equal to the expected training error. Because we have not defined a generating\n  process for examples, we use this as the definition.\n-/\nnoncomputable def test_error' \n    (h:P.H):ennreal := E[P.training_error h]\n\n\nlemma example_error_IID (P:VC_PAC_problem) (i:P.H):\n  @events_IID P.Ω P.Di P.p P.FDi  (P.example_error i) :=\nbegin\n  /-\n    To prove that the errors of a particular hypothesis are IID, we must use an alternate\n    formulation of the example_error events. Specifically, instead of constructing a hierarchy\n    of random variables, we must make a leap from the established IID random variable\n    (the data), construct another IID random variable (the product of\n    the classification and the label), and show that the set of all label/classification pairs\n    that aren't equal are a measurable set (because has_measurable_eq Mγ).\n\n    The indexed set of events of each IID random variable being in a measurable set is IID,\n    so the result holds.\n\n    Note that while this proof looks a little long, most of the proof is just unwrapping\n    the traditional and internal definitions of example error, and then using simp to show that\n    they are equal on all outcomes.\n  -/\n  let S:measurable_set P.MX := (P.concept i ∩ (P.target_concept)ᶜ) ∪ ((P.concept i)ᶜ ∩ (P.target_concept)),\n  begin\n  have B1:S = (P.concept i ∩ P.target_conceptᶜ) ∪ (((P.concept i)ᶜ) ∩ ((P.target_concept))) := rfl,\n  have A1:@random_variables_IID P.Ω P.p P.Di P.FDi P.X P.MX P.D,\n  {\n    apply P.IID,\n  },\n  have A2:@events_IID P.Ω P.Di P.p P.FDi (λ j:P.Di, @rv_event P.Ω P.p P.X P.MX (P.D j) S),\n  {\n    apply rv_event_IID,\n    apply A1,\n  },\n  have A3: (λ j:P.Di, @rv_event P.Ω P.p P.X P.MX (P.D j) S) = P.example_error i,\n  {\n    apply funext,\n    intro j,\n    apply event.eq,\n    rw B1,     \n    unfold VC_PAC_problem.example_error VC_PAC_problem.example_correct VC_PAC_problem.target_concept\n    VC_PAC_problem.label_positive  VC_PAC_problem.in_concept,\n    rw enot_val_def,\n    rw event_eqv_def,\n    rw rv_event_val_def,\n    rw eor_val_def,\n    repeat {rw eand_val_def},\n    rw measurable_union_val_def2,\n    repeat {rw measurable_inter_val_def2},\n    repeat {rw measurable_set_compl_val_def},\n    repeat {rw enot_val_def},\n    repeat {rw rv_event_val_def},\n    ext ω,\n    split;intros A3B;simp;simp at A3B,\n    {\n      cases A3B with A3B A3B,\n      {\n        split,\n        {\n          intros A3BA,\n          apply A3B.right,\n        },\n        {\n          intro A3BA,\n          exfalso,\n          apply A3BA,\n          apply A3B.left,\n        },\n      },\n      {\n        split;intros A3BA,\n        {\n          intros A3BB,\n          apply A3B.left A3BA,\n        },\n        {\n          apply A3B.right,\n        },\n      },\n    },\n    {\n      cases (classical.em ((P.D j).val ω ∈ (P.concept i).val)) with A3C A3C,\n      {\n        apply or.inl (and.intro A3C (A3B.left A3C)),\n      },\n      {\n        apply or.inr (and.intro A3C (A3B.right A3C)),\n      },\n    },\n  },\n  rw ← A3,\n  exact A2,\n  end\nend\n\n\n/-\n  The expected test error.\n  The test error is equal to the expected training error. Because we have not defined a generating\n  process for examples, we use this as the definition.\n-/\nlemma test_error_def\n    (h:P.H) (i:P.Di):P.test_error' h = Pr[P.example_error h i] :=\nbegin\n  unfold VC_PAC_problem.test_error' VC_PAC_problem.training_error,\n  rw average_identifier_eq_pr_elem,\n  apply VC_PAC_problem.example_error_IID,  \nend\n\n\n\n--This is probably true, but just very hard to prove.\ndef test_error_measurable:Prop := \n  @measurable P.H ennreal P.MH (borel ennreal) P.test_error'\n\n\n\n\nnoncomputable def test_error (TEM:P.test_error_measurable):P.MH →ₘ (borel ennreal) := {\n  val := P.test_error',\n  property := TEM,\n}\n\ndef C:set (set P.X) := set.range P.R\n\n\ndef Φ:ℕ → ℕ → ℕ\n| 0 m := 1\n| d 0 := 1\n| (nat.succ d) (nat.succ m) :=  Φ (d.succ) m + Φ d m\n\n@[simp]\nlemma phi_d_zero_eq_one {d:ℕ}:Φ d 0 = 1 :=\nbegin\n  cases d;unfold Φ,\nend\n\n@[simp]\nlemma phi_zero_m_eq_one {m:ℕ}:Φ 0 m = 1 :=\nbegin\n  cases m;unfold Φ,\nend\n\nlemma phi_succ_succ {d m:ℕ}:Φ d.succ m.succ = Φ d.succ m + Φ d m := rfl\n\n\nend VC_PAC_problem\n\n\n\n\ndef finset.to_set_of_sets {α:Type*} (C:finset (finset α)):set (set α) :=\n  (λ c:finset α, (↑c:set α)) '' (↑C:set (finset α))\n\n\nlemma finset.mem_to_set_of_sets {α:Type*} {C:finset (finset α)} {c:finset α}:\n  (↑c ∈ (C.to_set_of_sets)) ↔ c ∈ C :=\nbegin\n  unfold finset.to_set_of_sets,\n  simp,\nend\n\nlemma finset.mem_to_set_of_sets' {α:Type*} {C:finset (finset α)} {c:set α}:\n  c∈ C.to_set_of_sets ↔ ∃ c' ∈ C, c=(↑c') :=\nbegin\n  unfold finset.to_set_of_sets,\n  split;intro A1,\n  {\n    simp at A1,\n    cases A1 with c' A1,\n    apply exists.intro c',\n    apply exists.intro A1.left,\n    rw A1.right,\n  },\n  {\n    simp,\n    cases A1 with c' A1,\n    apply exists.intro c',\n    cases A1 with A1 A2,\n    simp [A1, A2],\n  },\nend\n\n\n/-\n  It is important to be able to talk about the VC dimension of a set of sets without\n  referring to a particular problem. For that reason, I have placed it outside of the\n  VC_PAC_problem.\n-/\nnamespace VC_PAC_problem\nsection VC_PAC_problem\nuniverse u\nvariable {α:Type u}\nopen_locale classical\n\n--The set of restrictions of concepts onto S, conventionally written as Πc(S) and represented as vectors.\nnoncomputable def restrict_set (C:set (set α)) (S:finset α):finset (finset α) := \n  S.powerset.filter (λ x, ∃ c∈ C, c ∩ (↑S) = (↑x)) \n\n\n--Does there exist a concept that agrees with any subset of S on S?\ndef shatters (C:set (set α)) (S:finset α):Prop := \n  (restrict_set C S) = S.powerset\n\n--What is the largest extended natural number n such that all finite sets of size ≤ n can be shattered?\n--Note that if the VC dimension is infinity, then that means that any finite set can be shattered, \n--but it does not say anything about infinite sets. See Kearns and Vazirani, page 51.\n--For example, the Borel algebra on the reals shatters every finite set, but does not shatter\n--all infinite sets (e.g. it does not shatter the reals themselves), so it has a VC dimension of\n--infinity.\n--Note: an empty hypothesis space, by this definition, has a VCD of infinity. This may cause problems.\n/-noncomputable def VCD (C:set (set α)):enat := Sup {n:enat|∃ (X':finset α), ((X'.card:enat) = n)  ∧\n  shatters C (X')}-/\n\n/-\n  Consider all sets that can be shattered. What is the supremum of their sizes (in enat)?\n -/\nnoncomputable def VCD (C:set (set α)):enat := Sup \n  ((λ X':finset α, (↑(X'.card):enat)) '' {X':finset α|shatters C (X')})\n\n\n/-\n  Normally, this restriction is defined for sets of exactly size m. However, this\n  runs into problems if there do not exist sets of a certain size.\n -/\nnoncomputable def restrict_set_bound (C:set (set α)) (m:ℕ):nat := Sup ((λ X', (restrict_set C X').card) '' \n   {X':finset α|X'.card ≤ m})\n\n\n\nlemma restrict_set_subset {C:set (set α)} (S:finset α):restrict_set C S⊆ S.powerset :=\nbegin\n  unfold restrict_set,\n  simp,\nend\n\nlemma mem_restrict_set_subset {C:set (set α)} {S:finset α} {c:finset α}:c ∈ restrict_set C S →\n  c⊆ S :=\nbegin\n  intros A1,\n  rw ← finset.mem_powerset,\n  have A2:= @restrict_set_subset α C S,\n  apply A2,\n  apply A1,\nend\n\n-- filter (λ x, true)\n--{S':set P.X|∃ c ∈ P.C, (S')= c ∩ (S)}\nlemma mem_restrict_set {C:set (set α)} (S:finset α) (T:finset α):\n  T ∈ restrict_set C S ↔ (∃ c∈ C, c ∩ (↑S) = (↑T)) :=\nbegin\n  unfold restrict_set,\n  rw finset.mem_filter,\n  split;intros B1,\n  {\n    apply B1.right,\n  },\n  {\n    split,\n    rw finset.mem_powerset,\n    cases B1 with c B1,\n    cases B1 with B1 B2,\n    rw finset.subset_iff,\n    intros x A1,\n    have B3:= set.inter_subset_right c (↑S),\n    rw B2 at B3,\n    apply B3,\n    simp,\n    apply A1,\n    apply B1,\n  },\nend\n\nlemma shatters_iff {C:set (set α)} (S:finset α):\n   (shatters C S) ↔ (∀ S'⊆ S, ∃ c∈ C, c ∩(↑S) = (↑S')) :=\nbegin\n  unfold shatters,\n  split;intros A1,\n  {\n    intros S' B1,\n    rw finset.ext_iff at A1,\n    have B2 := A1 S',\n    rw finset.mem_powerset at B2,\n    rw ← B2 at B1,\n    rw mem_restrict_set at B1,\n    apply B1,\n  },\n  {\n    apply finset.subset.antisymm,\n    apply restrict_set_subset,\n    rw finset.subset_iff,\n    intros S' C1,\n    rw finset.mem_powerset at C1,\n    have C2 := A1 S' C1,\n    rw mem_restrict_set,\n    apply C2,\n  },\nend\n\nlemma shatters_def (C:set (set α)) (S:finset α):\n  shatters C S = ((restrict_set C S) = S.powerset) := rfl\n\n/-Introducing a trivial upper bound establishes that Sup exists meaningfully (instead\n  of a default value of zero).-/\nlemma restrict_set_trivial_upper_bound  {C:set (set α)} (X':finset α):\n  (restrict_set C X').card ≤ 2^(X'.card) :=\nbegin\n  have B1:(restrict_set C X').card ≤ X'.powerset.card,\n  {\n    apply finset.card_le_of_subset,\n    apply restrict_set_subset,\n  },\n  apply le_trans B1,\n  rw finset.card_powerset,\nend\n\nlemma restrict_set_le_upper_bound {C:set (set α)} (X':finset α):\n  (restrict_set C X').card ≤ restrict_set_bound C (X'.card) :=\nbegin\n  apply le_cSup,\n  rw bdd_above_def,\n  unfold upper_bounds,\n  apply exists.intro (2^(X'.card)),\n  simp,\n  intros a X'',\n  intros A1 A2,\n  subst A2,\n  have A3:2^(X''.card) ≤ 2^(X'.card),\n  {\n    apply linear_ordered_semiring.pow_monotone one_le_two A1,\n  },\n  apply le_trans _ A3,\n  apply restrict_set_trivial_upper_bound,\n  simp,\n  apply exists.intro X',\n  split,\n  refl,\n  refl,\nend\n\n\nlemma shatters_card_le_VCD {C:set (set α)} {S:finset α}:shatters C S → (S.card:enat) ≤ VCD C :=\nbegin\n  unfold VCD,\n  intros A1,\n  apply le_Sup,\n  simp,\n  apply exists.intro S,\n  simp [A1],\nend\n\nlemma VCD_le {C:set (set α)} {d:enat}:(∀ S:finset α, shatters C S → (↑S.card)  ≤ d) → VCD C ≤ d :=\nbegin\n  intros A1,\n  unfold VCD,\n  apply Sup_le,\n  intros b B1,\n  simp at B1,\n  cases B1 with X' B1,\n  cases B1 with B1 B2,\n  subst b,\n  apply A1 X' B1,\nend\n\n\nlemma restrict_set_elements_subset_of_VCD_zero {C:set (set α)} {S T U:finset α}:(VCD C = 0) →\n  T ∈ (restrict_set C S) → U ∈ (restrict_set C S) → T ⊆ U :=\nbegin\n  intros A1 A2 A3,\n  rw finset.subset_iff,\n  intros x B1,\n  apply by_contradiction,\n  intros B2,\n  have B3 := (mem_restrict_set _ _).mp A2,\n  have B4 := (mem_restrict_set _ _).mp A3,\n  cases B3 with c_yes B3,\n  cases B4 with c_no B4,\n  cases B3 with B3 B5,\n  cases B4 with B4 B6,\n  have C1:↑T ⊆ c_yes, {rw ← B5,apply set.inter_subset_left},\n  have C2:x∈ c_yes,{apply C1,simp,apply B1},\n  have C3:c_yes ∩ {x} = {x},\n  {ext,split;intros B8A;simp;simp at B8A,apply B8A.right,subst x_1,simp [C2]},\n  have C4:↑U ⊆ c_no, {rw ← B6,apply set.inter_subset_left},\n  have C5:T ⊆ S,\n  {\n    rw ← finset.coe_subset,rw ← B5, apply set.inter_subset_right,\n  },\n  have C6:x ∈ S,\n  {\n    apply C5, apply B1,\n  },\n  have C7:x∉ c_no,\n  {\n    intros C5A,apply B2,rw ← finset.mem_coe,rw ← B6, simp,\n    apply and.intro C5A C6,\n  },  \n  have C8:c_no ∩ ↑({x}:finset α) = ↑(∅:finset α),\n  {\n    simp,\n    ext,split;intros C8A,\n    simp at C8A,cases C8A with C8A C8B,subst x_1,exfalso,apply C7 C8A,\n    exfalso, apply C8A, \n  },\n  have C9:shatters C {x},\n  {\n    unfold shatters,\n    rw finset.powerset_singleton,\n    apply finset.subset.antisymm,\n    apply restrict_set_subset,\n    rw finset.subset_iff,\n    intros X' C9A,\n    simp at C9A,cases C9A;subst X';rw mem_restrict_set,\n    {apply exists.intro c_no, apply exists.intro B4, exact C8},\n    {apply exists.intro c_yes, apply exists.intro B3,simp,apply C3},\n  },\n  have C10:1 ≤ VCD C,\n  {\n    apply shatters_card_le_VCD C9,\n  },\n  rw A1 at C10,  \n  have C11:(0:enat) < (1:enat) := enat.zero_lt_one,\n  rw lt_iff_not_ge at C11,\n  apply C11,\n  apply C10,\nend\n\nlemma restrict_set_elements_eq_of_VCD_zero {C:set (set α)} {S T U:finset α}:(VCD C = 0) →\n  T ∈ (restrict_set C S) → U ∈ (restrict_set C S) → T = U :=\nbegin\n  intros A1 A2 A3,\n  apply finset.subset.antisymm,\n  apply restrict_set_elements_subset_of_VCD_zero A1 A2 A3,\n  apply restrict_set_elements_subset_of_VCD_zero A1 A3 A2,\nend\n\n--Note: S could be either empty or have a unique element.\nlemma finset.card_identical_elements {α:Type*} [decidable_eq α] {S:finset α}:\n   (∀ a b:α,  a ∈ S → b ∈ S → a=b ) → S.card ≤ 1 :=\nbegin\n  --intros A1,\n  apply finset.induction_on S,\n  {\n    simp,\n  },\n  {\n    intros a s B1 B2 B3,\n    have C1:s = ∅,\n    {\n      rw ← finset.subset_empty,\n      rw finset.subset_iff,\n      intros b C1A,\n      exfalso,\n      apply B1,\n      have C1B:a = b,\n      {\n        apply B3;simp [C1A],\n      },\n      rw C1B,\n      apply C1A,\n    },\n    rw C1,\n    simp,\n  },\nend\n \nlemma mem_restrict_set_of_mem_restrict_set {C:set (set α)} {S₁ S₂ T:finset α}:\n  T ∈ restrict_set C S₂ →\n  S₁ ⊆ S₂ →\n  (T ∩ S₁) ∈ restrict_set C S₁ :=\nbegin\n  repeat {rw mem_restrict_set},\n  intros A1 A2,\n  cases A1 with c A1,\n  apply exists.intro c,\n  cases A1 with A1 A3,\n  apply exists.intro A1,\n  simp,\n  rw ← A3,\n  rw set.inter_assoc,\n  rw ← finset.coe_subset at A2,\n  rw set.inter_eq_self_of_subset_right A2,\nend\n\nlemma set.insert_inter_of_not_mem {α:Type*} {A B:set α} {x:α}:(x∉ B) → ((insert x A) ∩ B = A ∩ B) :=\nbegin\n  intros A1,\n  ext a,\n  split;intros A2;simp at A2;simp,\n  {\n    cases A2 with A2 A3,\n    cases A2 with A2 A4,\n    {\n      subst A2,\n      exfalso,\n      apply A1 A3,\n    },\n    {\n      apply and.intro A4 A3,\n    },\n  },\n  {\n    apply and.intro (or.inr A2.left) A2.right, \n  },\nend\n\nlemma set.inter_insert_of_not_mem {α:Type*} {A B:set α} {x:α}:(x∉ A) → (A ∩ (insert x B) = A ∩ B) :=\nbegin\n  intros A1,\n  rw set.inter_comm,\n  rw set.insert_inter_of_not_mem A1,\n  rw set.inter_comm,\nend\n\nlemma set.not_mem_of_inter_insert {α:Type*} {A B:set α} {x:α}:(x∉ A) → (A ∩ (insert x B) = A ∩ B) :=\nbegin\n  intros A1,\n  rw set.inter_comm,\n  rw set.insert_inter_of_not_mem A1,\n  rw set.inter_comm,\nend\n\n\nlemma set.inter_insert_of_mem {α:Type*} {A B:set α} {x:α}:(x∈ A) → (A ∩ (insert x B) = insert x (A ∩ B)) :=\nbegin\n  intros A1,\n  rw set.insert_inter,\n  rw set.insert_eq_of_mem A1,\nend\n\nlemma set.mem_of_inter_insert {α:Type*} {A B C:set α} {x:α}:\n  (A ∩ (insert x B) = insert x (C)) → (x ∈ A) :=\nbegin\n  intros A1,\n  have B1 := set.mem_insert x (C),\n  rw ← A1 at B1,\n  simp at B1,\n  apply B1,\nend\n\n\nlemma set.eq_of_insert_of_not_mem {α:Type*} {A B:set α} {x:α}:(x∉ A) → (x∉ B) → (insert x A  = insert x B)\n  → A = B :=\nbegin\n  intros A1 A3 A2,\n  ext a;split;intros B1;have C1 := set.mem_insert_of_mem x B1,\n  {\n    rw A2 at C1,\n    apply set.mem_of_mem_insert_of_ne C1,\n    intros C2,\n    subst a,\n    apply A1 B1,\n  },\n  {\n    rw ← A2 at C1,\n    apply set.mem_of_mem_insert_of_ne C1,\n    intros C2,\n    subst a,\n    apply A3 B1,\n  },\nend\n\nlemma set.insert_subset_insert {α:Type*} {A B:set α} {x:α}:A ⊆ B → (insert x A) ⊆ (insert x B) :=\nbegin\n  intros A1,\n  rw set.subset_def,\n  intros a B1,\n  simp at B1,\n  simp,\n  cases B1 with B1 B1,\n  apply or.inl B1,\n  apply or.inr (A1 B1),\nend\n\nlemma finset.eq_of_insert_of_not_mem {α:Type*} {A B:finset α} {x:α}:(x∉ A) → (x∉ B) → (insert x A  = insert x B)\n  → A = B :=\nbegin\n  intros A1 A3 A2,\n  ext a;split;intros B1;have C1 := finset.mem_insert_of_mem B1,\n  {\n    rw A2 at C1,\n    apply finset.mem_of_mem_insert_of_ne C1,\n    intros C2,\n    subst a,\n    apply A1 B1,\n  },\n  {\n    rw ← A2 at C1,\n    apply finset.mem_of_mem_insert_of_ne C1,\n    intros C2,\n    subst a,\n    apply A3 B1,\n  },\nend\n\nlemma mem_restrict_set_insert {C:set (set α)} {S c:finset α} {x:α}:x∉ S → (x∉ c) →\n  ((c ∈ restrict_set C S) ↔ (insert x c ∈ restrict_set C (insert x S)) ∨ c∈ restrict_set C (insert x S))\n :=\nbegin\n  repeat {rw mem_restrict_set},\n  intros A1 AX,split;intros A2,\n  {\n    cases A2 with c' A2,\n    cases A2 with A2 A3,\n    cases (em (x∈ c')) with B1 B1,\n    {\n      left,\n      apply exists.intro c',\n      apply exists.intro A2,\n      simp,\n      have B2:insert x c' = c' := set.insert_eq_of_mem B1,\n      rw ← B2,\n      rw ← set.insert_inter,\n      rw A3,\n    },\n    {\n      right,\n      apply exists.intro c',\n      apply exists.intro A2,\n      simp,\n      rw set.inter_insert_of_not_mem B1,\n      apply A3,\n    },\n  },\n  {\n    cases A2 with A2 A2;cases A2 with c' A2;cases A2 with A2 A3;\n    apply exists.intro c';apply exists.intro A2,\n    {\n      simp at A3,\n      have C1:=set.mem_of_inter_insert A3,      \n      rw set.inter_insert_of_mem C1 at A3,\n      apply set.eq_of_insert_of_not_mem _ _ A3,\n      simp,\n      intro C2,\n      apply A1,\n      apply AX,\n    },\n    {\n      ext a;split;intros D1,\n      {\n        rw ← A3,\n        simp at D1,\n        simp [D1],\n      },\n      {\n        have D2:a ≠ x,\n        {\n          intros D2A,\n          subst a,\n          apply AX D1,\n        },\n        rw ← A3 at D1,\n        simp [D2] at D1,\n        simp [D1],\n      },\n    },\n  },\nend\n\n\nlemma finset.insert_inter_eq_insert_inter_insert {α:Type*} [decidable_eq α]\n  {S T:finset α} {a:α}:(insert a S) ∩ (insert a T) = insert a (S ∩ T) :=\nbegin\n  ext b,\n  split;intros A1,\n  {\n    rw finset.mem_insert,\n    simp at A1,\n    cases A1,\n    subst a,\n    simp,\n    cases A1 with A1 A2,\n    cases A1 with A1 A1,\n    apply or.inl A1,\n    simp [A1,A2],    \n  },\n  {\n    simp,\n    simp at A1,\n    cases A1 with A1 A1,\n    apply or.inl A1,\n    simp [A1],\n  },\nend\n\n\nlemma mem_restrict_set_erase {C:set (set α)} {S c:finset α} {x:α}:x∉ S → (x∈ c) →\n  (c ∈ restrict_set C (insert x S))  → (c.erase x ∈ restrict_set C S) \n :=\nbegin\n  intros A1 A2 A3,\n  rw mem_restrict_set_insert A1,\n  left,\n  rw finset.insert_erase,\n  apply A3,\n  apply A2,\n  apply finset.not_mem_erase,  \nend\n\nlemma restrict_card_le_one_of_VCD_zero {C:set (set α)} {S:finset α}:(VCD C = 0) →\n  (restrict_set C S).card ≤ 1 :=\nbegin\n  intros A1,\n  apply finset.card_identical_elements,\n  intros T U B1 B2,\n  apply restrict_set_elements_eq_of_VCD_zero A1 B1 B2,\nend\n\nlemma restrict_set_empty_of_empty {S:finset α}:restrict_set ∅ S = ∅ :=\nbegin\n  ext c,\n  rw mem_restrict_set,\n  split;intros B1,\n  {\n    cases B1 with c2 B1,\n    cases B1 with B1 B2,\n    simp at B1,\n    exfalso,\n    apply B1,\n  },\n  {\n    exfalso,\n    simp at B1,\n    apply B1,\n  },\nend\n\nlemma restrict_set_nonempty_empty {C:set (set α)}:\n  set.nonempty C → restrict_set C ∅ = {∅} :=\nbegin\n  intros A1,\n  ext c,\n  rw mem_restrict_set,split;intros B1,\n  {\n    simp,\n    cases B1 with c' B1,\n    cases B1 with B1 B2,\n    rw ← finset.coe_inj,\n    rw ← B2,\n    simp,\n  },\n  {\n    simp at B1,\n    subst c,\n    rw set.nonempty_def at A1,\n    cases A1 with c' A1,\n    apply exists.intro c',\n    apply exists.intro A1,\n    simp,    \n  },\nend\n\nlemma restrict_set_empty_card_le_1 {C:set (set α)}:\n  (restrict_set C ∅).card ≤ 1 :=\nbegin\n  cases (set.eq_empty_or_nonempty C) with B1 B1,\n  {\n    subst C,\n    rw restrict_set_empty_of_empty,\n    simp,\n  },\n  {\n    rw restrict_set_nonempty_empty B1,\n    simp,\n  },\nend\n\nlemma filter_union {S:finset α} {P:α → Prop}:\n  finset.filter P S ∪ finset.filter (λ a, ¬P a)  S = S :=\nbegin\n  ext a;split;intro A1,\n  {\n    simp at A1,\n    cases A1 with A1 A1;apply A1.left,\n  },\n  {\n    simp,\n    simp [A1],\n    apply em,\n  },\nend\n\nlemma filter_disjoint {S T:finset α} {P:α → Prop}:\n  disjoint (finset.filter P S) (finset.filter (λ a, ¬P a)  T) :=\nbegin\n  rw finset.disjoint_left,\n  intros a B1 B2,\n  simp at B1,\n  simp at B2,\n  apply B2.right,\n  apply B1.right,\nend \n\nlemma filter_disjoint' {S:finset α} {P:α → Prop}:\n  disjoint (finset.filter P S) (finset.filter (λ a, ¬P a)  S) :=\n  @filter_disjoint α S S P\n\nlemma filter_card {S:finset α} {P:α → Prop}:\n  (finset.filter P S).card + (finset.filter (λ a, ¬P a)  S).card = S.card :=\nbegin\n  have A1:(finset.filter P S ∪ finset.filter (λ a, ¬P a)  S).card = S.card,\n  {\n    rw filter_union,\n  },\n  rw ← A1,\n  rw finset.card_union_eq,\n  apply filter_disjoint,\nend\n\nlemma recursive_restrict_set_card {C:set (set α)} {x:α} {S:finset α}:x∉ S →\n  ((restrict_set C (insert x S)).filter (λ c,(x∉c ∧ (insert x c) ∈ (restrict_set \nC (insert x S))))).card + (restrict_set C S).card = (restrict_set C (insert x S)).card :=\nbegin\n  intro A1,\n  let Ex := restrict_set C (insert x S),\n  let E := restrict_set C S,\n  begin\n    have B1:Ex = restrict_set C (insert x S) := rfl,\n    have B2:E = restrict_set C S := rfl,\n\n    repeat {rw ← B1},\n    repeat {rw ← B2},\n    rw add_comm,\n    rw ← @filter_card _ Ex (λ c, x ∈ c),\n    simp,\n    simp,\n    rw ← @filter_card _ (finset.filter (λ c, x∉ c) Ex)\n       (λ c, (insert x c) ∈ Ex),\n    simp,\n    repeat {rw finset.filter_filter},\n    rw add_comm _ (finset.filter (λ (a : finset α), x ∉ a ∧ insert x a ∉ Ex) Ex).card,\n    rw ← add_assoc,\n    simp,\n    have C1:(finset.filter (λ (a : finset α), x ∉ a ∧ insert x a ∉ Ex) Ex) =\n      (finset.filter (λ (a : finset α), insert x a ∉ Ex) E),\n    {\n      ext c,split;repeat {rw B1};repeat {rw B2};intros C1A;simp at C1A;simp [C1A],\n      { rw mem_restrict_set_insert A1 C1A.right.left, apply or.inr C1A.left},\n      { \n        have C1B:x ∉ c,\n        {\n          have C1B1:c ⊆ S := mem_restrict_set_subset C1A.left,\n          intro C1B2,\n          apply A1,\n          apply C1B1,\n          apply C1B2,\n        },\n        have C1C := (mem_restrict_set_insert A1 C1B).mp C1A.left, \n        simp [C1A.right] at C1C,\n        apply and.intro C1C C1B,\n      },\n    },\n    rw C1,\n    clear C1,\n    have C2:(finset.filter (has_mem.mem x) Ex).card = \n      (finset.filter (λ a, insert x a ∈ Ex) E).card ,\n    {\n      have C2A:(finset.filter (has_mem.mem x) Ex) = \n        (finset.filter (λ a, insert x a ∈ Ex) E).image (insert x),\n      {\n        ext a,split;repeat {rw B1};repeat {rw B2};intros C2A1;simp at C2A1;simp,\n        {\n          apply exists.intro (a.erase x),\n          cases C2A1 with C2A1 C2A2,\n          have C2A3:insert x (a.erase x) = a := finset.insert_erase C2A2,\n          have C2A4:x ∉ a.erase x := finset.not_mem_erase x a,\n          split,\n          split,\n          apply mem_restrict_set_erase A1 C2A2 C2A1,\n          rw C2A3,\n          apply C2A1,\n          apply C2A3,\n        },\n        {\n          cases C2A1 with c C2A1,\n          cases C2A1 with C2A1 C2A2,\n          cases C2A1 with C2A1 C2A3,\n          subst a,\n          simp [C2A3],\n        },\n      },\n      rw C2A,\n      clear C2A,\n      repeat {rw B1},\n      repeat {rw B2},   \n      apply finset.card_image_of_inj_on, \n      intros c C2B c' C2C,\n      simp at C2B,\n      simp at C2C,\n      have C2D:∀ {c'':finset α}, c'' ∈ restrict_set C S → x ∉ c'',\n      {\n        intros c'' C2D1 C2D3,\n        apply A1,\n        have C2D2 := mem_restrict_set_subset C2D1,\n        apply C2D2,\n        apply C2D3,\n      },\n      apply finset.eq_of_insert_of_not_mem,\n      apply C2D C2B.left,\n      apply C2D C2C.left,\n    },\n    \n    rw C2,\n    rw  ← @filter_card _ E (λ a, insert x a ∈ Ex),\n    simp,\n  end    \nend\n\nlemma enat.le_coe_eq_coe {v:enat} {d:nat}:v ≤ d → ∃ d':ℕ, v = d'  ∧ d' ≤ d :=\nbegin\n  --intros A1,\n  apply enat.cases_on v,\n  {\n    intros A1,\n    simp at A1,\n    exfalso,\n    apply A1,\n  },\n  {\n    intros n A1,\n    apply exists.intro n,\n    simp at A1,\n    simp [A1],\n  },\nend \n\nlemma phi_monotone_m {d  m₁ m₂:ℕ}:m₁ ≤ m₂ → Φ d m₁ ≤ Φ d m₂ :=\nbegin\n  intros A1,\n  rw le_iff_exists_add at A1,\n  cases A1 with c A1,\n  subst m₂,\n  induction c,\n  simp,\n  have A2:(m₁ + c_n.succ) = (m₁ + c_n).succ := rfl,\n  rw A2,\n  cases d,\n  simp,\n  rw phi_succ_succ,\n  apply le_trans c_ih,\n  simp,\nend\n\n\nlemma phi_le_d_succ {d m:ℕ}:Φ d m ≤ Φ d.succ m :=\nbegin\n  revert d,\n  induction m,\n  intro d,\n  simp,\n  intro d,\n  rw phi_succ_succ,\n  cases d,\n  simp,\n  rw phi_succ_succ,\n  rw add_comm,\n  simp,\n  apply le_trans m_ih,\n  apply m_ih,\nend\n\nlemma phi_monotone_d {d₁ d₂ m:ℕ}:d₁ ≤ d₂ → Φ d₁ m ≤ Φ d₂ m :=\nbegin\n  intros A1,\n  rw le_iff_exists_add at A1,\n  cases A1 with c A1,\n  subst d₂,\n  induction c,\n  simp,\n  have A2:(d₁ + c_n.succ) = (d₁ + c_n).succ := rfl,\n  rw A2,\n  apply le_trans c_ih,\n  apply phi_le_d_succ,\nend\n\nlemma eq_restrict_set {C:finset (finset α)} {S:finset α}:(∀ c∈ C , c⊆ S)→\n  C = restrict_set (C.to_set_of_sets) S :=\nbegin\n  intros A1,\n  ext c,split;intros B1,\n  {\n    rw mem_restrict_set,\n    apply exists.intro (↑c),\n    split,\n    rw finset.mem_to_set_of_sets,\n    apply B1,\n    have B2 := A1 c B1,\n    apply set.inter_eq_self_of_subset_left,\n    simp,\n    apply B2,\n  },\n  {\n    rw mem_restrict_set at B1,\n    cases B1 with c' B1,\n    cases B1 with B1 B2,\n    rw finset.mem_to_set_of_sets' at B1,\n    cases B1 with c'' B1,\n    cases B1 with B1 B2,\n    subst c',\n    rw set.inter_eq_self_of_subset_left at B2,\n    simp at B2,\n    subst c'',\n    apply B1,\n    simp,\n    apply A1 c'' B1,        \n  },\nend\n\nlemma finite_restrict_set_eq_image {C:finset (finset α)} {S:finset α}:\n  (restrict_set C.to_set_of_sets S) = C.image (λ S', S'∩ S) :=\nbegin\n  ext,split;intros A1A,\n  {\n    simp,\n    rw mem_restrict_set at A1A,\n    cases A1A with c A1A,\n    cases A1A with A1A A1B,\n    rw finset.mem_to_set_of_sets' at A1A,\n    cases A1A with c' A1A,\n    apply exists.intro c',\n    cases A1A with A1A A1C,\n    rw A1C at A1B,\n    rw ← finset.coe_inter at A1B,\n    rw finset.coe_inj at A1B,\n    apply and.intro A1A A1B,\n  },\n  {\n    simp at A1A,\n    cases A1A with c A1A,\n    cases A1A with A1A A1B,\n    subst a,\n    rw mem_restrict_set,\n    apply exists.intro (↑c),\n    split,\n    rw finset.mem_to_set_of_sets,\n    apply A1A,\n    rw finset.coe_inter,\n  },\nend\n\nlemma finite_restrict_set_le {C:finset (finset α)} {S:finset α}:\n  (restrict_set C.to_set_of_sets S).card ≤ C.card := \nbegin\n  rw finite_restrict_set_eq_image,\n  apply finset.card_image_le,\nend\n\nlemma shatters_subset {S:finset α} {x:α} {C:set (set α)} {S':finset α}:x∉S → \n   shatters (\n          (\n            (restrict_set C (insert x S)).filter\n            (λ (c:finset α),(x∉c ∧ ((insert x c) ∈ (restrict_set C (insert x S)))))\n          ).to_set_of_sets\n        ) S'  → S' ⊆ S :=\nbegin\n  intros A1 A2,\n  rw shatters_iff at A2,\n  have D1A:S' ⊆ S' := finset.subset.refl S',\n  have D1B := A2 S' D1A,\n  cases D1B with c D1B,\n  cases D1B with D1B D1C,\n  rw finset.mem_to_set_of_sets' at D1B,\n  cases D1B with c' D1B,\n  cases D1B with D1B D1D,\n  subst c,\n  simp at D1B,\n  cases D1B with D1B D1E,\n  cases D1E with D1E D1F,\n  have D1G:= mem_restrict_set_subset D1B,\n  rw ← finset.coe_inter at D1C,\n  rw finset.coe_inj at D1C,\n  have D1H:S'⊆ c',\n  {\n    rw ← D1C,\n    apply finset.inter_subset_left,\n  },\n  apply finset.subset.trans D1H,\n  apply finset.subset_of_not_mem_of_subset_insert D1E D1G,\nend\n\nlemma shatters_succ {S:finset α} {x:α} {C:set (set α)} {S':finset α}:x∉S → \n   shatters (\n          (\n            (restrict_set C (insert x S)).filter\n            (λ (c:finset α),(x∉c ∧ ((insert x c) ∈ (restrict_set C (insert x S)))))\n          ).to_set_of_sets\n        ) S'  → shatters C (insert x S') :=\nbegin\n  intros A1 A2,\n  rw shatters_iff,\n  intros c B1,\n  have D1:S' ⊆ S := shatters_subset A1 A2,  \n  have D2:(insert x (↑S':set α)) ⊆ (insert x ↑S),\n  {\n    apply set.insert_subset_insert,\n    simp,\n    apply D1,\n  },\n\n  rw shatters_iff at A2,\n  cases (em (x ∈ c)) with A3 A3,\n  {\n    have C1:c.erase x ⊆ S',\n    {\n      rw ← finset.subset_insert_iff,\n      apply B1,\n    },\n    have B2 := A2 (c.erase x) C1,\n    cases B2 with c' B2,\n    cases B2 with B2 B3,\n    simp,\n    rw finset.mem_to_set_of_sets' at B2,\n    cases B2 with c'' B2,\n    cases B2 with B2 B4,\n    simp at B2,\n    cases B2 with B2 B5,\n    cases B5 with B5 B6,\n    rw mem_restrict_set at B6,\n    cases B6 with c''' B6,\n    cases B6 with B6 B7,\n    subst c',\n    apply exists.intro c''',\n    apply and.intro B6,\n    simp at B7,\n    rw ← finset.coe_inter at B3,\n    rw finset.coe_inj at B3,\n\n    have B8:insert x (c'' ∩ S') = insert x (c.erase x),\n    {\n      rw B3,\n    },\n    rw finset.insert_erase A3 at B8,\n    have B9 := set.mem_of_inter_insert B7,\n    rw ← finset.insert_inter_eq_insert_inter_insert at B8,\n    rw ← B8,\n    rw finset.coe_inter,\n    repeat {rw finset.coe_insert},\n    rw ← B7,\n    rw set.inter_assoc,\n    rw set.inter_comm (insert x ↑S),\n    rw ← set.inter_assoc,\n    symmetry,\n    apply set.inter_eq_self_of_subset_left,\n    have B10:=set.inter_subset_right c''' (insert x ↑S'),\n    apply set.subset.trans B10 D2,\n   },\n  {\n    have E1:c ⊆ S',\n    {\n      rw finset.subset_iff,\n      intros a E1A,\n      have E1B := B1 E1A,\n      rw finset.mem_insert at E1B,\n      cases E1B with E1B E1B,\n      {\n        subst a,\n        exfalso,\n        apply A3 E1A,\n      },\n      apply E1B,\n    },\n    have E2 := A2 c E1,\n    cases E2 with c' E2,\n    cases E2 with E2 E3,\n    rw finset.mem_to_set_of_sets' at E2,\n    cases E2 with c'' E2,\n    cases E2 with E2 E3,\n    simp at E2,\n    subst c',\n    cases E2 with E2 E4,\n    cases E4 with E4 E5,\n    rw mem_restrict_set at E2,\n    cases E2 with c''' E2,\n    cases E2 with E2 E6,\n    apply exists.intro c''',\n    apply exists.intro E2,\n    have E7:x ∉ (↑c'':set α),\n    {simp [E4]},\n    rw ← set.inter_insert_of_not_mem E7 at E3,\n    simp at E6,\n    rw ← E3,\n    simp,\n    rw ← E6,\n    rw set.inter_assoc,\n    rw set.inter_comm (insert x ↑S),\n    rw ← set.inter_assoc,\n    symmetry,\n    apply set.inter_eq_self_of_subset_left,\n    have E8:=set.inter_subset_right c''' (insert x ↑S'),\n    apply set.subset.trans E8 D2,\n  },\nend\n\nlemma VCD_succ {S:finset α} {x:α} {C:set (set α)} {d:ℕ}:x∉S → VCD C = d.succ →\n    VCD (\n          (\n            (restrict_set C (insert x S)).filter\n            (λ (c:finset α),(x∉c ∧ ((insert x c) ∈ (restrict_set C (insert x S)))))\n          ).to_set_of_sets\n        ) ≤ d :=\nbegin\n  intros A1 A2,\n  apply VCD_le,\n  intros T B1,\n  have B2:T ⊆ S := shatters_subset A1 B1, \n  have B3:x ∉ T,\n  {\n    intro B3A,\n    apply A1,\n    apply B2,\n    apply B3A,\n  },\n  have B4:shatters C (insert x T),\n  {\n    apply shatters_succ,\n    apply A1,\n    apply B1,\n  },\n  have B5:((insert x T).card:enat) ≤ VCD C,\n  {\n    apply shatters_card_le_VCD B4,\n  },\n  rw A2 at B5,\n  simp at B5,\n  rw finset.card_insert_of_not_mem B3 at B5,\n  have B6:T.card + 1 = (T.card).succ := rfl,\n  rw B6 at B5,\n  rw nat.succ_le_succ_iff at B5,\n  simp,\n  apply B5\nend \n\n\n--This is known as Sauer's Lemma, or Sauer-Saleh Lemma.\n--This connects VC-dimension to the complexity of the hypothesis space restricted to a finite set\n--of certain size.\nlemma restrict_set_le_phi {C:set (set α)} {S:finset α} (d:ℕ):\n  (VCD C = d) →\n  (restrict_set C S).card ≤ Φ d S.card := \nbegin\n  revert d,\n  revert C,\n  apply finset.induction_on S,\n  {\n    intros C d A1,\n    simp,\n    apply restrict_set_empty_card_le_1,\n  },\n  {\n    intros x S B1 B2 C d B3,\n    cases d,\n    {\n      rw phi_zero_m_eq_one,\n      apply restrict_card_le_one_of_VCD_zero,\n      simp at B3,\n      apply B3,\n    },\n    let C':finset (finset α) := (restrict_set C (insert x S)).filter (λ c,(x∉c ∧ (insert x c) ∈ (restrict_set C (insert x S)))),\n    begin\n      have C0:C' = (restrict_set C (insert x S)).filter (λ c,(x∉c ∧ (insert x c) ∈ (restrict_set C (insert x S)))) := rfl,\n      rw ← recursive_restrict_set_card B1,\n      rw ← C0,\n      have D1:C'.card + (restrict_set C S).card ≤ C'.card +  Φ d.succ S.card,\n      {\n        simp [B2,B3],\n      },\n      apply le_trans D1,\n      \n      have C5:C' = restrict_set (C'.to_set_of_sets) S,\n      {\n        apply eq_restrict_set,\n        intros c C3A,\n        rw C0 at C3A,\n        simp at C3A,\n        have C3C := mem_restrict_set_subset C3A.left,\n        apply finset.subset_of_not_mem_of_subset_insert C3A.right.left C3C,\n      },\n      rw C5,\n      have C6:VCD (C'.to_set_of_sets) ≤ (d:enat),\n      {\n        rw C0,\n        apply VCD_succ,\n        apply B1,\n        apply B3\n      },\n      have C7:∃d':ℕ, VCD (C'.to_set_of_sets) = d' ∧ d' ≤ d,\n      {\n        apply enat.le_coe_eq_coe C6,\n      },\n      cases C7 with d' C7,\n      have C8:(restrict_set C'.to_set_of_sets S).card + Φ d.succ S.card ≤ Φ d S.card + Φ d.succ S.card,\n      {\n        simp,\n        have C8A:(restrict_set C'.to_set_of_sets S).card ≤ Φ d' S.card,\n        {\n          apply B2,\n          apply C7.left,\n        },\n        apply le_trans C8A,\n        apply phi_monotone_d,\n        apply C7.right,\n      },\n      apply le_trans C8,\n      rw finset.card_insert_of_not_mem B1,\n      rw phi_succ_succ,\n      rw add_comm,  \n    end,\n  },\nend\n\n-- TODO: prove Φ d m = (finset.range (d.succ)).sum (λ i, nat.choose m i)\n-- See mathlib/src/data/nat/choose/basic.lean\n-- TODO: introduce ε-nets, and show that the VC dimension of C is a bound on \n-- the VC-dimension of the ε-net.\n-- TODO: show that if the training examples cover the ε-net, then any consistent\n-- algorithm will get a hypothesis with low error.\n-- TODO: Introduce the proof from 3.5.2 proving a bound on the probability of\n-- hitting the ε-net. \nend VC_PAC_problem\nend VC_PAC_problem\n\n\n\n\n\n\n\n\n\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/vc_pac_bounds.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.33613228959443797}}
{"text": "import bridge_char\nimport rigid_elements.main\nimport for_mathlib.valuation_subring.basic\n\nnoncomputable theory\nopen_locale tensor_product\nopen_locale classical\n\nopen module\n\nvariables {K F : Type*} [field K] [field F] --[is_prime_field F]\n\nopen finite_dimensional\n\nnamespace main_theorem_mul_setup\n\nlemma mk_rigid_pair \n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  rigid_pair (mul_subgroup.mk' T.restrict) (mul_subgroup.mk' T.nonrig.restrict) :=\nbegin\n  let T' := mul_subgroup.mk' T.restrict,\n  let H' := mul_subgroup.mk' T.nonrig.restrict,\n  have hle' : T' ≤ H', \n  { intros t ht, \n    use T'.is_unit_of_mem ht,\n    apply T.le_nonrig, \n    obtain ⟨u,hu⟩ := ht,\n    exact hu },\n  apply rigid_pair.mk_of_le T' H' hle',\n  { refine ⟨(-1 : Kˣ).is_unit, _⟩,\n    change _ ∈ T,\n    rw [← T.dual_annihilator_dual_annihilator_comap, \n      submodule.mem_dual_annihilator_comap_iff],\n    intros φ hφ,\n    have := hT.neg_one φ hφ,\n    convert this, ext, refl },\n  { intros x hx hx', \n    have hx'' := hx',\n    contrapose! hx',\n    use is_unit.mk0 x hx,\n    change _ ∈ T.nonrig, \n    dsimp [submodule.nonrig],\n    rw submodule.mem_sup,\n    refine ⟨0, T.zero_mem, (units.mk0 x hx).as, _, _⟩,\n    { apply submodule.subset_span,  \n      refine ⟨units.mk0 x hx,_,rfl⟩,\n      dsimp [submodule.nonrig_condition],\n      split,\n      { contrapose! hx'',\n        apply hle', use (is_unit.mk0 x hx), dsimp [submodule.restrict], \n        change _ ∈ T, convert hx'', ext, refl },\n      { intros y hy, \n        rw ← hy at hx',\n        have : ((units.mk0 x hx)⁻¹ : K) * y = x⁻¹ + 1,\n        { field_simp [hy], },\n        rw ← this at hx',\n        cases hx' with hx'1 hx'2,\n        split,\n        { contrapose! hx'1, \n          use y.is_unit, change _ ∈ T, convert hx'1, ext, refl },\n        { contrapose! hx'2, \n          use (units.mk0 x hx)⁻¹.is_unit.mul y.is_unit, \n          change _ ∈ T, convert hx'2, ext, refl } } },\n    { rw zero_add, congr' 1, ext, refl } },\nend\n\nlemma mk_rigid_pair_char_two\n  [is_prime_field F]\n  (htwo : (2 : F) = 0)\n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  rigid_pair (mul_subgroup.mk' T.restrict) (mul_subgroup.mk' T.restrict) :=\nbegin\n  let T' := mul_subgroup.mk' T.restrict,\n  have hneg1 : (-1 : K) ∈ T', \n  { refine ⟨(-1 : Kˣ).is_unit, _⟩,\n    change _ ∈ T,\n    rw [← T.dual_annihilator_dual_annihilator_comap, \n      submodule.mem_dual_annihilator_comap_iff],\n    intros φ hφ,\n    have := hT.neg_one φ hφ,\n    convert this, ext, refl },\n  apply rigid_pair.mk_of_le T' T' (le_refl _) hneg1,\n  { intros x hx1 hx2,\n    let y : Kˣ := units.mk0 x hx1,\n    let z : Kˣ := units.mk0 (1 + x) _,\n    have hh := submodule.dependent_of_acl _ hT (-y) z _,\n    rw fintype.not_linear_independent_iff at hh,\n    obtain ⟨g,h1,i,h2⟩ := hh,\n    simp only [fin.sum_univ_succ, submodule.mkq_apply, matrix.cons_val_zero, \n      fintype.univ_of_subsingleton, fin.mk_zero, \n      matrix.cons_val_succ, finset.sum_singleton, fin.succ_zero_eq_one] at h1,\n    let a := g 0,\n    let b := g 1, \n    change a • _ + b • _ = _ at h1,\n    cases eq_zero_or_one_of_is_prime_field_of_two _ htwo a with ha ha;\n    cases eq_zero_or_one_of_is_prime_field_of_two _ htwo b with hb hb,\n    { fin_cases i; contradiction, },\n    { simp only [ha,hb, zero_smul, zero_add, one_smul, \n        submodule.quotient.mk_eq_zero] at h1, \n      left, rw submodule.dual_annihilator_dual_annihilator_comap at h1,\n      use z.is_unit, change _ ∈ T, convert h1, ext, refl },\n    { simp only [ha,hb, zero_smul, add_zero, one_smul, \n        submodule.quotient.mk_eq_zero] at h1, \n      exfalso,\n      apply hx2, \n      rw submodule.dual_annihilator_dual_annihilator_comap at h1,\n      rw (show x = (-1 : K) * (-x), by ring), apply T'.mul_mem hneg1,\n      use (-y).is_unit, change _ ∈ T, convert h1, ext, refl },\n    { simp only [ha,hb, one_smul, ← submodule.mkq_apply,\n        ← linear_map.map_add, ← units.as_mul] at h1, \n      simp only [submodule.mkq_apply, submodule.quotient.mk_eq_zero,\n        submodule.dual_annihilator_dual_annihilator_comap] at h1,\n      right,\n      have : (2 : F) • y.as = (0 : mul_base_change K F), \n      { rw [htwo, zero_smul], },\n      rw [← sub_zero (-y * z).as,← this,(show (2 : F) = 1 + 1, by ring),\n        add_smul, one_smul, ← units.as_mul, sub_eq_add_neg,\n        ← units.as_inv, ← units.as_mul] at h1,\n      rw (show (x⁻¹ + 1) = (-1) * (-(x⁻¹ + 1)), by ring),\n      apply T'.mul_mem hneg1,\n      have : -(x⁻¹ + 1) = ↑(-y * z * (y * y)⁻¹), \n      { push_cast, field_simp, \n        dsimp [z], ring },\n      rw this,\n      use (-y * z * (y * y)⁻¹).is_unit, change _ ∈ T, convert h1, ext, refl },\n    { dsimp [y,z], ring, },\n    { intro c, rw add_eq_zero_iff_neg_eq at c, rw c at hneg1,\n      exact hx2 hneg1 } }\nend\n\nend main_theorem_mul_setup\n\ntheorem main_theorem_mul_char_ne_two \n  [is_prime_field F]\n  (htwo : (2 : F) ≠ 0)\n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  ∃ (R : valuation_subring K)\n    (H : submodule F (mul_base_change K F))\n    (hTH : T ≤ H)\n    (units : ∀ u : Kˣ, u ∈ R.unit_group → u.as ∈ H)\n    (principal_units : ∀ u : Kˣ, u ∈ R.principal_unit_group → u.as ∈ T)\n    (fd : finite_dimensional F (↥H ⧸ T.comap H.subtype)),\n    finrank F (↥H ⧸ T.comap H.subtype) ≤ 1 := \nbegin\n  let T' := mul_subgroup.mk' T.restrict,\n  let H' := mul_subgroup.mk' T.nonrig.restrict,\n  let P : rigid_pair T' H' := main_theorem_mul_setup.mk_rigid_pair T hT,\n  obtain ⟨t,ht1,ht2,ht3⟩ := P.exists_preadditive,\n  let t' : Kˣ := units.mk0 t ht1,\n  let E := T.nonrig ⊔ (submodule.span F ({t'.as} : set (mul_base_change K F))),\n  have hE : E = T.nonrig,\n  { suffices : (submodule.span F ({t'.as} : set (mul_base_change K F))) ≤ T.nonrig,\n    { dsimp [E], exact sup_eq_left.mpr this, },\n    rw [submodule.span_le, set.singleton_subset_iff],\n    change t'.as ∈ T.nonrig,\n    suffices : (2 : F) • t'.as ∈ T.nonrig,\n    { convert T.nonrig.smul_mem ((2 : F)⁻¹) this using 1,\n      rw [← mul_smul, inv_mul_cancel htwo, one_smul] },\n    rw [(show (2 : F) = 1 + 1, by ring), add_smul, one_smul, ← units.as_mul],\n    obtain ⟨u,hu⟩ := ht2, convert hu, ext, refl },\n  let P' : rigid_pair T' (H'.adjoin_ne_zero t ht1) := _,\n  change rigid_pair.preadditive P' at ht3,\n  let R := ht3.valuation_subring,\n  use [R, T.nonrig, T.le_nonrig],\n  refine ⟨_,_,_,_⟩,\n  { intros u hu, \n    rw ← hE,\n    suffices : H'.adjoin_ne_zero t ht1 ≤ mul_subgroup.mk' E.restrict,\n    { have hh := ht3.mem_of_mem_units u hu,\n      change u ∈ E.restrict,\n      suffices : (u : K) ∈ mul_subgroup.mk' E.restrict,\n      { obtain ⟨h1,h2⟩ := this, \n        convert h2, ext, refl },\n      exact this hh, },\n    apply mul_subgroup.adjoin_ne_zero_le,\n    { intros u hu,  \n      use H'.is_unit_of_mem hu,\n      apply submodule.mem_sup_left,\n      obtain ⟨v,hv⟩ := hu,\n      exact hv },\n    { use is_unit.mk0 t ht1, \n      apply submodule.mem_sup_right,\n      convert submodule.mem_span_singleton_self _, ext, refl } },\n  { intros u hu, \n    have := ht3.mem_of_mem_principal_units u hu,\n    obtain ⟨v,hv⟩ := this, \n    convert hv, ext, refl },\n  { apply bridge_finite_dimensional _ hT },\n  { apply bridge_codim _ hT },\nend\n\ntheorem main_theorem_mul_char_two \n  [is_prime_field F]\n  (htwo : (2 : F) = 0)\n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  ∃ (R : valuation_subring K)\n    (H : submodule F (mul_base_change K F))\n    (hTH : T ≤ H)\n    (units : ∀ u : Kˣ, u ∈ R.unit_group → u.as ∈ H)\n    (principal_units : ∀ u : Kˣ, u ∈ R.principal_unit_group → u.as ∈ T)\n    (fd : finite_dimensional F (↥H ⧸ T.comap H.subtype)),\n    finrank F (↥H ⧸ T.comap H.subtype) ≤ 1 := \nbegin\n  let T' := mul_subgroup.mk' T.restrict,\n  let P : rigid_pair T' T' := \n    main_theorem_mul_setup.mk_rigid_pair_char_two htwo T hT,\n  obtain ⟨t,ht1,ht2,ht3⟩ := P.exists_preadditive,\n  let t' : Kˣ := units.mk0 t ht1,\n  let P' : rigid_pair T' (T'.adjoin_ne_zero t ht1) := _,\n  change rigid_pair.preadditive P' at ht3,\n  let R := ht3.valuation_subring, use R,\n  let H := T ⊔ (submodule.span F ({t'.as} : set (mul_base_change K F))),\n  have hfd : finite_dimensional F (↥H ⧸ (T.comap H.subtype)), \n  { let e := (T.comap H.subtype).mkq.comp (submodule.of_le (le_sup_right : _ ≤ T ⊔ _)),\n    have he : function.surjective e, \n    { apply submodule.mkq_comp_sup_mod_surjective },\n    apply e.finite_dimensional_of_surjective, rwa linear_map.range_eq_top },\n  use [H, le_sup_left],\n  refine ⟨_,_,_,_⟩,\n  { intros u hu,\n    have hh := ht3.mem_of_mem_units u hu,\n    suffices : T'.adjoin_ne_zero t ht1 ≤ mul_subgroup.mk' H.restrict,\n    { replace hh := this hh, \n      obtain ⟨v,hv⟩ := hh,\n      convert hv, ext, refl },\n    apply mul_subgroup.adjoin_ne_zero_le,\n    { intros u hu, use T'.is_unit_of_mem hu, change _ ∈ H,\n      apply submodule.mem_sup_left, obtain ⟨v,hv⟩ := hu, exact hv },\n    { use t'.is_unit,\n      apply submodule.mem_sup_right,\n      convert submodule.mem_span_singleton_self _, ext, refl } },\n  { intros u hu, \n    have := ht3.mem_of_mem_principal_units u hu,\n    obtain ⟨v,hv⟩ := this,\n    convert hv, ext, refl },\n  { exact hfd },\n  { by_cases htT : t ∈ T', \n    { have : H = T, \n      { erw [sup_eq_left],\n        rw submodule.span_singleton_le_iff_mem, obtain ⟨v,hv⟩ := htT, convert hv, ext, refl },\n      rw this at hfd ⊢, resetI, rw finrank_le_one_iff, use 0, rintros ⟨w,hw⟩, use 0,\n      symmetry,\n      simp, },\n    apply submodule.finrank_le_one_of_not_linear_independent T {t'.as} t'.as \n      (set.mem_singleton (units.as t')), \n    { contrapose! htT, \n      use t'.is_unit, change _ ∈ T, convert htT, ext, refl },\n    { rintros v (rfl : v = t'.as), \n      rw fintype.not_linear_independent_iff, use ![1,-1],\n      split, { simp [fin.sum_univ_succ] }, { use 0, norm_num } } },\nend\n\ntheorem main_theorem_mul\n  [is_prime_field F]\n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  ∃ (R : valuation_subring K)\n    (H : submodule F (mul_base_change K F))\n    (hTH : T ≤ H)\n    (units : ∀ u : Kˣ, u ∈ R.unit_group → u.as ∈ H)\n    (principal_units : ∀ u : Kˣ, u ∈ R.principal_unit_group → u.as ∈ T)\n    (fd : finite_dimensional F (↥H ⧸ T.comap H.subtype)),\n    finrank F (↥H ⧸ T.comap H.subtype) ≤ 1 := \nbegin\n  by_cases htwo : (2 : F) = 0,\n  apply main_theorem_mul_char_two, assumption',\n  apply main_theorem_mul_char_ne_two, assumption'\nend\n\ntheorem main_theorem_mul_char\n  (p ℓ : ℕ)\n  [fact (nat.prime p)]\n  (HH : p ≠ ℓ)\n  [char_p K p]\n  [char_p F ℓ]\n  (htwo : (2 : F) ≠ 0)\n  (T : submodule F (mul_base_change K F)) \n  (hT : T.dual_annihilator.acl) : \n  ∃ (R : valuation_subring K)\n    (H : submodule F (mul_base_change K F))\n    (hTH : T ≤ H)\n    (units : ∀ u : Kˣ, u ∈ R.unit_group → u.as ∈ H)\n    (principal_units : ∀ u : Kˣ, u ∈ R.principal_unit_group → u.as ∈ T)\n    (fd : finite_dimensional F (↥H ⧸ T.comap H.subtype)),\n    finrank F (↥H ⧸ T.comap H.subtype) ≤ 1 := \nbegin\n  let T' := mul_subgroup.mk' T.restrict,\n  let H' := mul_subgroup.mk' T.nonrig.restrict,\n  let P : rigid_pair T' H' := main_theorem_mul_setup.mk_rigid_pair T hT,\n  obtain ⟨t,ht1,ht2,ht3⟩ := P.exists_preadditive,\n  let t' : Kˣ := units.mk0 t ht1,\n  let E := T.nonrig ⊔ (submodule.span F ({t'.as} : set (mul_base_change K F))),\n  have hE : E = T.nonrig,\n  { suffices : (submodule.span F ({t'.as} : set (mul_base_change K F))) ≤ T.nonrig,\n    { dsimp [E], exact sup_eq_left.mpr this, },\n    rw [submodule.span_le, set.singleton_subset_iff],\n    change t'.as ∈ T.nonrig,\n    suffices : (2 : F) • t'.as ∈ T.nonrig,\n    { convert T.nonrig.smul_mem ((2 : F)⁻¹) this using 1,\n      rw [← mul_smul, inv_mul_cancel htwo, one_smul] },\n    rw [(show (2 : F) = 1 + 1, by ring), add_smul, one_smul, ← units.as_mul],\n    obtain ⟨u,hu⟩ := ht2, convert hu, ext, refl },\n  let P' : rigid_pair T' (H'.adjoin_ne_zero t ht1) := _,\n  change rigid_pair.preadditive P' at ht3,\n  let R := ht3.valuation_subring,\n  use [R, T.nonrig, T.le_nonrig],\n  refine ⟨_,_,_,_⟩,\n  { intros u hu, \n    rw ← hE,\n    suffices : H'.adjoin_ne_zero t ht1 ≤ mul_subgroup.mk' E.restrict,\n    { have hh := ht3.mem_of_mem_units u hu,\n      change u ∈ E.restrict,\n      suffices : (u : K) ∈ mul_subgroup.mk' E.restrict,\n      { obtain ⟨h1,h2⟩ := this, \n        convert h2, ext, refl },\n      exact this hh, },\n    apply mul_subgroup.adjoin_ne_zero_le,\n    { intros u hu,  \n      use H'.is_unit_of_mem hu,\n      apply submodule.mem_sup_left,\n      obtain ⟨v,hv⟩ := hu,\n      exact hv },\n    { use is_unit.mk0 t ht1, \n      apply submodule.mem_sup_right,\n      convert submodule.mem_span_singleton_self _, ext, refl } },\n  { intros u hu, \n    have := ht3.mem_of_mem_principal_units u hu,\n    obtain ⟨v,hv⟩ := this, \n    convert hv, ext, refl },\n  { apply bridge_char_finite_dimensional p ℓ HH _ hT },\n  { apply bridge_char_codim p ℓ HH _ hT },\nend", "meta": {"author": "adamtopaz", "repo": "lean-acl-pairs", "sha": "6ac31d86ca2739b6c18d3f05b7007e720f66299f", "save_path": "github-repos/lean/adamtopaz-lean-acl-pairs", "path": "github-repos/lean/adamtopaz-lean-acl-pairs/lean-acl-pairs-6ac31d86ca2739b6c18d3f05b7007e720f66299f/src/recover.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3359768458721316}}
{"text": "import FOL.completeness\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nvariables {L L₁ L₂ L₃ : language.{u}} {T₁ U₁ : Theory L₁} {T₂ U₂ : Theory L₂} {T₃ U₃ : Theory L₃} \nopen formula language language.language_translation language.language_translation_coe\n\nnamespace Theory\n\nsection\nvariables [closed_Theory T₁] [closed_Theory U₁] [language_translation_coe L₁ L₂]\nopen logic logic.Theory axiomatic_classical_logic' axiomatic_classical_logic\n\n@[simp] lemma le_coe_iff : (↑T₁ : Theory L₂) ≤ ↑U₁ ↔ T₁ ≤ U₁ :=\n⟨λ h p b,\n  by { have : (↑T₁ : Theory L₂) ⊢ ↑p, by simp[b],\n       have : ↑U₁ ⊢ ↑p, from h this,\n       simpa using this },\n λ h p b,\nbegin\n  rcases provable.proof_conjunction b with ⟨P, hP, b⟩,\n  have : ∀ p, ∃ p₁ : formula L₁, p ∉ P ∨ p = ↑p₁ ∧ p₁ ∈ T₁,\n  { intros p, by_cases hp : p ∈ P,\n    { rcases language.language_translation_coe.mem_coe_iff.mp (hP p hp) with ⟨p₁, hp₁, rfl⟩,\n      refine ⟨p₁, by simp[hp₁]⟩ },\n    { simp[hp] } },\n  choose coe_inv hcoe_inv using this,\n  have : ↑U₁ ⊢ P.conjunction,\n    from list_conjunction_provable\n      (λ p hp, by { rcases language.language_translation_coe.mem_coe_iff.mp (hP p hp) with ⟨p₁, hp₁, rfl⟩,\n        simpa using h (by_axiom hp₁) }),\n  exact b.extend ⨀ this\nend⟩\n\ninstance extend_coe [T₁.extend U₁] : (↑T₁ : Theory L₂).extend ↑U₁ := ⟨le_coe_iff.mpr extend.le⟩\n\nend\n\nvariables [language_translation_coe L₁ L₂] {T₁ U₁} {T₂ U₂}\n\ndef lle (T₁ : Theory L₁) (T₂ : Theory L₂) : Prop := ∀ ⦃p : formula L₁⦄, T₁ ⊢ p → T₂ ⊢ p\n\n@[simp] lemma lle_refl (T : Theory L) : lle T T := λ p h, by simp[coe_def_p, ltr]; exact h\n\n@[trans] lemma lle_trans [language_translation_coe L₁ L₂] [language_translation_coe L₂ L₃] [language_translation_coe L₁ L₃]\n  {T₁ : Theory L₁} {T₂ : Theory L₂} {T₃ : Theory L₃} [commutes L₁ L₂ L₃] :\n  lle T₁ T₂ → lle T₂ T₃ → lle T₁ T₃ :=\nλ le₁₂ le₂₃ p₁ b₁, by simpa[commutes.coe_coe_p_of_commute p₁] using le₂₃ (le₁₂ b₁)\n\nlemma lle_of_le (h : T₁ ≤ U₁) : lle T₁ U₁ := λ p b, by simp[coe_def_p, ltr]; exact h b\n\nvariables {T₁ U₂} (T₂)\n\nlemma lle_of_lle_of_le (h : lle T₁ T₂) (le : T₂ ≤ U₂) : lle T₁ U₂ := λ p b, le (h b)\n\nvariables {T₁ T₂} (U₁)\n\nlemma lle_of_le_of_lle (le : T₁ ≤ U₁) (h : lle U₁ T₂) : lle T₁ T₂ := λ p b, h (le b)\n\nvariables (T₁)\n\nlemma lle_coe [closed_Theory T₁] : lle T₁ (↑T₁ : Theory L₂) := λ p, by simp\n\nend Theory\n\nclass lextend {L₁ : language.{u}} {L₂ : language.{u}} [language_translation_coe L₁ L₂] (T₁ : Theory L₁) (T₂ : Theory L₂) :=\n(le : Theory.lle T₁ T₂)\n\nlemma provable.lextend [language_translation_coe L₁ L₂] {T₁ : Theory L₁} {p} (b : T₁ ⊢ p) (T₂ : Theory L₂)\n  [lextend T₁ T₂] : T₂ ⊢ p := lextend.le b\n\nnamespace Theory\nopen logic logic.Theory\nvariables  [language_translation_coe L₁ L₂] (T₁ U₁) (T₂ U₂)\n\ninstance lextend_refl : lextend T₁ T₁ := ⟨by simp⟩\n\ninstance lextend_of_extend [extend T₁ U₁] : lextend T₁ U₁ := ⟨lle_of_le extend.le⟩\n\ninstance lextend_sf [lextend T₁ T₂] : lextend (⤊T₁) (⤊T₂) :=\n⟨λ p h, by {\n  have : T₁ ⊢ ∀.p, from h.generalize,\n  have : T₂ ⊢ ∀.p, from provable.lextend this T₂,\n  have : ⤊T₂ ⊢ (∀.p)^1, from provable.sf_sf.mpr this,\n  simpa[formula.nested_rew] using this ⊚ #0 }⟩\n\ninstance lextend_pow [ex : lextend T₁ T₂] (k : ℕ) : lextend (T₁^k) (T₂^k) :=\nby { induction k with k IH ; simp[Theory.sf_itr_succ], { exact ex }, { exactI fol.Theory.lextend_sf _ _ } }\n\ninstance lextend_sf_closed [closed_Theory T₁] [lextend T₁ T₂] : lextend T₁ (⤊T₂) :=\nby simpa using Theory.lextend_sf T₁ T₂\n\ninstance lextend_pow_closed [closed_Theory T₁] [lextend T₁ T₂] (k : ℕ) : lextend T₁ (T₂^k) :=\nby simpa using Theory.lextend_pow T₁ T₂ k\n\ninstance lextend_coe [closed_Theory T₁] : lextend T₁ (↑T₁ : Theory L₂) := ⟨lle_coe T₁⟩\n\nvariables (T₁ U₂ T₂)\n\ndef lextend_of_lextend_of_extend [lextend T₁ T₂] [extend T₂ U₂] : lextend T₁ U₂ := ⟨lle_of_lle_of_le T₂ lextend.le extend.le⟩\n\nvariables (T₁ T₂ U₁)\n\ndef lextend_of_extend_of_lextend [extend T₁ U₁] [lextend U₁ T₂] : lextend T₁ T₂ := ⟨lle_of_le_of_lle U₁ extend.le lextend.le⟩\n\nsection\nvariables {T₁ T₂} [lextend T₁ T₂]\n\ninstance lextend_insert₁ (p) : lextend T₁ (T₂+{p}) := lextend_of_lextend_of_extend _ T₂ _\n\ninstance lextend_insert₂ (p q) : lextend T₁ (T₂+{p}+{q}) := lextend_of_lextend_of_extend _ T₂ _\n\ninstance lextend_insert₃ (p q r) : lextend T₁ (T₂+{p}+{q}+{r}) := lextend_of_lextend_of_extend _ T₂ _\n\ninstance lextend_insert₄ (p q r s) : lextend T₁ (T₂+{p}+{q}+{r}+{s}) := lextend_of_lextend_of_extend _ T₂ _\n\nend\n\nsection\nvariables [language_translation_coe L₁ L₂] [language_translation_coe L₂ L₃] [language_translation_coe L₁ L₃]\n  [commutes L₁ L₂ L₃] (T₁ T₂ T₃)\n\ndef lextend_trans [lextend T₁ T₂] [lextend T₂ T₃] : lextend T₁ T₃ := ⟨lle_trans (show lle T₁ T₂, from lextend.le) lextend.le⟩\n\nend\n\nend Theory\n\nvariables {L₁ L₂} (D : L₁.definitions L₂) [language_translation_coe (L₁ + L₂) L₃] \n  [language_translation_coe L₁ L₃] [commutes L₁ (L₁ + L₂) L₃] (T : Theory L₃) [lextend D.thy T]\n\n@[simp] lemma language.definitions.fn' {n} (f : L₂.fn n) (v : finitary (term L₃) n) :\n  T ⊢ (D.df_fn f : formula L₃).rew (term.app ((coe : (L₁ + L₂).fn n → L₃.fn n) (sum.inr f)) v ⌢ of_fin v) :=\nby { have : T ⊢ ∀.[n] (D.df_fn f : formula L₃).rew ı[0 ⇝ term.app ↑(sum.inr f)  (λ i, #i)],\n       by simpa using provable.lextend (axiomatic_classical_logic'.by_axiom (language.definitions.mem_fn D f)) T,\n     have := provable.nfal_subst'_finitary this v,\n     simp[formula.nested_rew] at this,\n     refine cast (by { congr, funext x, rcases x; simp }) this }\n\n@[simp] lemma language.definitions.pr' {n} (r : L₂.pr n) (v : finitary (term L₃) n) :\n  T ⊢ app ((coe : (L₁ + L₂).pr n → L₃.pr n) (sum.inr r)) v ⟷ (D.df_pr r : formula L₃).rew (of_fin v) :=\nby { have : T ⊢ ∀.[n] ❴↑(sum.inr r)❵ (λ i, #i) ⟷ ↑(D.df_pr r),\n       by simpa using provable.lextend (axiomatic_classical_logic'.by_axiom (language.definitions.mem_pr D r)) T,\n     simpa using provable.nfal_subst'_finitary this v }\n\nlemma coe_inv_equiv' [language.predicate L₂] (p : formula (L₁ + L₂)) :\n  T ⊢ p ⟷ ↑(formula.coe_inv D p : formula L₁) :=\nby simpa using provable.lextend (coe_inv_equiv D p) T\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/extend.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3359131778250933}}
{"text": "/-\nCopyright (c) 2021 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel\n-/\nimport measure_theory.measure.measure_space\n\n/-!\n# Almost everywhere measurable functions\n\nA function is almost everywhere measurable if it coincides almost everywhere with a measurable\nfunction. This property, called `ae_measurable f μ`, is defined in the file `measure_space_def`.\nWe discuss several of its properties that are analogous to properties of measurable functions.\n-/\n\nopen measure_theory measure_theory.measure filter set function\nopen_locale measure_theory filter classical ennreal interval\n\nvariables {ι α β γ δ R : Type*} {m0 : measurable_space α} [measurable_space β]\n   [measurable_space γ] [measurable_space δ] {f g : α → β} {μ ν : measure α}\n\ninclude m0\n\nsection\n\n@[nontriviality, measurability]\nlemma subsingleton.ae_measurable [subsingleton α] : ae_measurable f μ :=\nsubsingleton.measurable.ae_measurable\n\n@[nontriviality, measurability]\nlemma ae_measurable_of_subsingleton_codomain [subsingleton β] : ae_measurable f μ :=\n(measurable_of_subsingleton_codomain f).ae_measurable\n\n@[simp, measurability] lemma ae_measurable_zero_measure : ae_measurable f (0 : measure α) :=\nbegin\n  nontriviality α, inhabit α,\n  exact ⟨λ x, f default, measurable_const, rfl⟩\nend\n\nnamespace ae_measurable\n\nlemma mono_measure (h : ae_measurable f μ) (h' : ν ≤ μ) : ae_measurable f ν :=\n⟨h.mk f, h.measurable_mk, eventually.filter_mono (ae_mono h') h.ae_eq_mk⟩\n\nlemma mono_set {s t} (h : s ⊆ t) (ht : ae_measurable f (μ.restrict t)) :\n  ae_measurable f (μ.restrict s) :=\nht.mono_measure (restrict_mono h le_rfl)\n\nprotected lemma mono' (h : ae_measurable f μ) (h' : ν ≪ μ) : ae_measurable f ν :=\n⟨h.mk f, h.measurable_mk, h' h.ae_eq_mk⟩\n\nlemma ae_mem_imp_eq_mk {s} (h : ae_measurable f (μ.restrict s)) :\n  ∀ᵐ x ∂μ, x ∈ s → f x = h.mk f x :=\nae_imp_of_ae_restrict h.ae_eq_mk\n\nlemma ae_inf_principal_eq_mk {s} (h : ae_measurable f (μ.restrict s)) :\n  f =ᶠ[μ.ae ⊓ 𝓟 s] h.mk f :=\nle_ae_restrict h.ae_eq_mk\n\n@[measurability]\nlemma sum_measure [encodable ι] {μ : ι → measure α} (h : ∀ i, ae_measurable f (μ i)) :\n  ae_measurable f (sum μ) :=\nbegin\n  nontriviality β, inhabit β,\n  set s : ι → set α := λ i, to_measurable (μ i) {x | f x ≠ (h i).mk f x},\n  have hsμ : ∀ i, μ i (s i) = 0,\n  { intro i, rw measure_to_measurable, exact (h i).ae_eq_mk },\n  have hsm : measurable_set (⋂ i, s i),\n    from measurable_set.Inter (λ i, measurable_set_to_measurable _ _),\n  have hs : ∀ i x, x ∉ s i → f x = (h i).mk f x,\n  { intros i x hx, contrapose! hx, exact subset_to_measurable _ _ hx },\n  set g : α → β := (⋂ i, s i).piecewise (const α default) f,\n  refine ⟨g, measurable_of_restrict_of_restrict_compl hsm _ _, ae_sum_iff.mpr $ λ i, _⟩,\n  { rw [restrict_piecewise], simp only [set.restrict, const], exact measurable_const },\n  { rw [restrict_piecewise_compl, compl_Inter],\n    intros t ht,\n    refine ⟨⋃ i, ((h i).mk f ⁻¹' t) ∩ (s i)ᶜ, measurable_set.Union $\n      λ i, (measurable_mk _ ht).inter (measurable_set_to_measurable _ _).compl, _⟩,\n    ext ⟨x, hx⟩,\n    simp only [mem_preimage, mem_Union, subtype.coe_mk, set.restrict, mem_inter_eq,\n      mem_compl_iff] at hx ⊢,\n    split,\n    { rintro ⟨i, hxt, hxs⟩, rwa hs _ _ hxs },\n    { rcases hx with ⟨i, hi⟩, rw hs _ _ hi, exact λ h, ⟨i, h, hi⟩ } },\n  { refine measure_mono_null (λ x (hx : f x ≠ g x), _) (hsμ i),\n    contrapose! hx, refine (piecewise_eq_of_not_mem _ _ _ _).symm,\n    exact λ h, hx (mem_Inter.1 h i) }\nend\n\n@[simp] lemma _root_.ae_measurable_sum_measure_iff [encodable ι] {μ : ι → measure α} :\n  ae_measurable f (sum μ) ↔ ∀ i, ae_measurable f (μ i) :=\n⟨λ h i, h.mono_measure (le_sum _ _), sum_measure⟩\n\n@[simp] lemma _root_.ae_measurable_add_measure_iff :\n  ae_measurable f (μ + ν) ↔ ae_measurable f μ ∧ ae_measurable f ν :=\nby { rw [← sum_cond, ae_measurable_sum_measure_iff, bool.forall_bool, and.comm], refl }\n\n@[measurability]\nlemma add_measure {f : α → β} (hμ : ae_measurable f μ) (hν : ae_measurable f ν) :\n  ae_measurable f (μ + ν) :=\nae_measurable_add_measure_iff.2 ⟨hμ, hν⟩\n\n@[measurability]\nprotected lemma Union [encodable ι] {s : ι → set α} (h : ∀ i, ae_measurable f (μ.restrict (s i))) :\n  ae_measurable f (μ.restrict (⋃ i, s i)) :=\n(sum_measure h).mono_measure $ restrict_Union_le\n\n@[simp] lemma _root_.ae_measurable_Union_iff [encodable ι] {s : ι → set α} :\n  ae_measurable f (μ.restrict (⋃ i, s i)) ↔ ∀ i, ae_measurable f (μ.restrict (s i)) :=\n⟨λ h i, h.mono_measure $ restrict_mono (subset_Union _ _) le_rfl, ae_measurable.Union⟩\n\n@[simp] lemma _root_.ae_measurable_union_iff {s t : set α} :\n  ae_measurable f (μ.restrict (s ∪ t)) ↔\n    ae_measurable f (μ.restrict s) ∧ ae_measurable f (μ.restrict t) :=\nby simp only [union_eq_Union, ae_measurable_Union_iff, bool.forall_bool, cond, and.comm]\n\n@[measurability]\nlemma smul_measure [monoid R] [distrib_mul_action R ℝ≥0∞] [is_scalar_tower R ℝ≥0∞ ℝ≥0∞]\n  (h : ae_measurable f μ) (c : R) :\n  ae_measurable f (c • μ) :=\n⟨h.mk f, h.measurable_mk, ae_smul_measure h.ae_eq_mk c⟩\n\nlemma comp_ae_measurable {f : α → δ} {g : δ → β}\n  (hg : ae_measurable g (μ.map f)) (hf : ae_measurable f μ) : ae_measurable (g ∘ f) μ :=\n⟨hg.mk g ∘ hf.mk f, hg.measurable_mk.comp hf.measurable_mk,\n  (ae_eq_comp hf hg.ae_eq_mk).trans ((hf.ae_eq_mk).fun_comp (mk g hg))⟩\n\nlemma comp_measurable {f : α → δ} {g : δ → β}\n  (hg : ae_measurable g (μ.map f)) (hf : measurable f) : ae_measurable (g ∘ f) μ :=\nhg.comp_ae_measurable hf.ae_measurable\n\nlemma comp_measurable' {ν : measure δ} {f : α → δ} {g : δ → β} (hg : ae_measurable g ν)\n  (hf : measurable f) (h : μ.map f ≪ ν) : ae_measurable (g ∘ f) μ :=\n(hg.mono' h).comp_measurable hf\n\nlemma map_map_of_ae_measurable {g : β → γ} {f : α → β}\n  (hg : ae_measurable g (measure.map f μ)) (hf : ae_measurable f μ) :\n  (μ.map f).map g = μ.map (g ∘ f) :=\nbegin\n  ext1 s hs,\n  let g' := hg.mk g,\n  have A : map g (map f μ) = map g' (map f μ),\n  { apply measure_theory.measure.map_congr,\n    exact hg.ae_eq_mk },\n  have B : map (g ∘ f) μ = map (g' ∘ f) μ,\n  { apply measure_theory.measure.map_congr,\n    exact ae_of_ae_map hf hg.ae_eq_mk },\n  simp only [A, B, hs, hg.measurable_mk.ae_measurable.comp_ae_measurable hf, hg.measurable_mk,\n    hg.measurable_mk hs, hf, map_apply, map_apply_of_ae_measurable],\n  refl,\nend\n\n@[measurability]\nlemma prod_mk {f : α → β} {g : α → γ} (hf : ae_measurable f μ) (hg : ae_measurable g μ) :\n  ae_measurable (λ x, (f x, g x)) μ :=\n⟨λ a, (hf.mk f a, hg.mk g a), hf.measurable_mk.prod_mk hg.measurable_mk,\n  eventually_eq.prod_mk hf.ae_eq_mk hg.ae_eq_mk⟩\n\nlemma exists_ae_eq_range_subset (H : ae_measurable f μ) {t : set β} (ht : ∀ᵐ x ∂μ, f x ∈ t)\n  (h₀ : t.nonempty) :\n  ∃ g, measurable g ∧ range g ⊆ t ∧ f =ᵐ[μ] g :=\nbegin\n  let s : set α := to_measurable μ {x | f x = H.mk f x ∧ f x ∈ t}ᶜ,\n  let g : α → β := piecewise s (λ x, h₀.some) (H.mk f),\n  refine ⟨g, _, _, _⟩,\n  { exact measurable.piecewise (measurable_set_to_measurable _ _)\n      measurable_const H.measurable_mk },\n  { rintros _ ⟨x, rfl⟩,\n    by_cases hx : x ∈ s,\n    { simpa [g, hx] using h₀.some_mem },\n    { simp only [g, hx, piecewise_eq_of_not_mem, not_false_iff],\n      contrapose! hx,\n      apply subset_to_measurable,\n      simp only [hx, mem_compl_eq, mem_set_of_eq, not_and, not_false_iff, implies_true_iff]\n        {contextual := tt} } },\n  { have A : μ (to_measurable μ {x | f x = H.mk f x ∧ f x ∈ t}ᶜ) = 0,\n    { rw [measure_to_measurable, ← compl_mem_ae_iff, compl_compl],\n      exact H.ae_eq_mk.and ht },\n    filter_upwards [compl_mem_ae_iff.2 A] with x hx,\n    rw mem_compl_iff at hx,\n    simp only [g, hx, piecewise_eq_of_not_mem, not_false_iff],\n    contrapose! hx,\n    apply subset_to_measurable,\n    simp only [hx, mem_compl_eq, mem_set_of_eq, false_and, not_false_iff] }\nend\n\nlemma subtype_mk (h : ae_measurable f μ) {s : set β} {hfs : ∀ x, f x ∈ s} :\n  ae_measurable (cod_restrict f s hfs) μ :=\nbegin\n  nontriviality α, inhabit α,\n  obtain ⟨g, g_meas, hg, fg⟩ : ∃ (g : α → β), measurable g ∧ range g ⊆ s ∧ f =ᵐ[μ] g :=\n    h.exists_ae_eq_range_subset (eventually_of_forall hfs) ⟨_, hfs default⟩,\n  refine ⟨cod_restrict g s (λ x, hg (mem_range_self _)), measurable.subtype_mk g_meas, _⟩,\n  filter_upwards [fg] with x hx,\n  simpa [subtype.ext_iff],\nend\n\nprotected lemma null_measurable (h : ae_measurable f μ) : null_measurable f μ :=\nlet ⟨g, hgm, hg⟩ := h in hgm.null_measurable.congr hg.symm\n\nend ae_measurable\n\nlemma ae_measurable_interval_oc_iff [linear_order α] {f : α → β} {a b : α} :\n  (ae_measurable f $ μ.restrict $ Ι a b) ↔\n    (ae_measurable f $ μ.restrict $ Ioc a b) ∧ (ae_measurable f $ μ.restrict $ Ioc b a) :=\nby rw [interval_oc_eq_union, ae_measurable_union_iff]\n\nlemma ae_measurable_iff_measurable [μ.is_complete] :\n  ae_measurable f μ ↔ measurable f :=\n⟨λ h, h.null_measurable.measurable_of_complete, λ h, h.ae_measurable⟩\n\nlemma measurable_embedding.ae_measurable_map_iff {g : β → γ} (hf : measurable_embedding f) :\n  ae_measurable g (μ.map f) ↔ ae_measurable (g ∘ f) μ :=\nbegin\n  refine ⟨λ H, H.comp_measurable hf.measurable, _⟩,\n  rintro ⟨g₁, hgm₁, heq⟩,\n  rcases hf.exists_measurable_extend hgm₁ (λ x, ⟨g x⟩) with ⟨g₂, hgm₂, rfl⟩,\n  exact ⟨g₂, hgm₂, hf.ae_map_iff.2 heq⟩\nend\n\nlemma measurable_embedding.ae_measurable_comp_iff {g : β → γ}\n  (hg : measurable_embedding g) {μ : measure α} :\n  ae_measurable (g ∘ f) μ ↔ ae_measurable f μ :=\nbegin\n  refine ⟨λ H, _, hg.measurable.comp_ae_measurable⟩,\n  suffices : ae_measurable ((range_splitting g ∘ range_factorization g) ∘ f) μ,\n    by rwa [(right_inverse_range_splitting hg.injective).comp_eq_id] at this,\n  exact hg.measurable_range_splitting.comp_ae_measurable H.subtype_mk\nend\n\nlemma ae_measurable_restrict_iff_comap_subtype {s : set α} (hs : measurable_set s)\n  {μ : measure α} {f : α → β} :\n  ae_measurable f (μ.restrict s) ↔ ae_measurable (f ∘ coe : s → β) (comap coe μ) :=\nby rw [← map_comap_subtype_coe hs, (measurable_embedding.subtype_coe hs).ae_measurable_map_iff]\n\n@[simp, to_additive] lemma ae_measurable_one [has_one β] : ae_measurable (λ a : α, (1 : β)) μ :=\nmeasurable_one.ae_measurable\n\n@[simp] lemma ae_measurable_smul_measure_iff {c : ℝ≥0∞} (hc : c ≠ 0) :\n  ae_measurable f (c • μ) ↔ ae_measurable f μ :=\n⟨λ h, ⟨h.mk f, h.measurable_mk, (ae_smul_measure_iff hc).1 h.ae_eq_mk⟩,\n  λ h, ⟨h.mk f, h.measurable_mk, (ae_smul_measure_iff hc).2 h.ae_eq_mk⟩⟩\n\nlemma ae_measurable_of_ae_measurable_trim {α} {m m0 : measurable_space α}\n  {μ : measure α} (hm : m ≤ m0) {f : α → β} (hf : ae_measurable f (μ.trim hm)) :\n  ae_measurable f μ :=\n⟨hf.mk f, measurable.mono hf.measurable_mk hm le_rfl, ae_eq_of_ae_eq_trim hf.ae_eq_mk⟩\n\nlemma ae_measurable_restrict_of_measurable_subtype {s : set α}\n  (hs : measurable_set s) (hf : measurable (λ x : s, f x)) : ae_measurable f (μ.restrict s) :=\n(ae_measurable_restrict_iff_comap_subtype hs).2 hf.ae_measurable\n\nlemma ae_measurable_map_equiv_iff (e : α ≃ᵐ β) {f : β → γ} :\n  ae_measurable f (μ.map e) ↔ ae_measurable (f ∘ e) μ :=\ne.measurable_embedding.ae_measurable_map_iff\n\nend\n\nlemma ae_measurable.restrict (hfm : ae_measurable f μ) {s} :\n  ae_measurable f (μ.restrict s) :=\n⟨ae_measurable.mk f hfm, hfm.measurable_mk, ae_restrict_of_ae hfm.ae_eq_mk⟩\n\nvariables [has_zero β]\n\nlemma ae_measurable_indicator_iff {s} (hs : measurable_set s) :\n  ae_measurable (indicator s f) μ ↔ ae_measurable f (μ.restrict s) :=\nbegin\n  split,\n  { intro h,\n    exact (h.mono_measure measure.restrict_le_self).congr (indicator_ae_eq_restrict hs) },\n  { intro h,\n    refine ⟨indicator s (h.mk f), h.measurable_mk.indicator hs, _⟩,\n    have A : s.indicator f =ᵐ[μ.restrict s] s.indicator (ae_measurable.mk f h) :=\n      (indicator_ae_eq_restrict hs).trans (h.ae_eq_mk.trans $ (indicator_ae_eq_restrict hs).symm),\n    have B : s.indicator f =ᵐ[μ.restrict sᶜ] s.indicator (ae_measurable.mk f h) :=\n      (indicator_ae_eq_restrict_compl hs).trans (indicator_ae_eq_restrict_compl hs).symm,\n    exact ae_of_ae_restrict_of_ae_restrict_compl _ A B },\nend\n\n@[measurability]\nlemma ae_measurable.indicator (hfm : ae_measurable f μ) {s} (hs : measurable_set s) :\n  ae_measurable (s.indicator f) μ :=\n(ae_measurable_indicator_iff hs).mpr hfm.restrict\n\nlemma measure_theory.measure.restrict_map_of_ae_measurable\n  {f : α → δ} (hf : ae_measurable f μ) {s : set δ} (hs : measurable_set s) :\n  (μ.map f).restrict s = (μ.restrict $ f ⁻¹' s).map f :=\ncalc\n(μ.map f).restrict s = (μ.map (hf.mk f)).restrict s :\n  by { congr' 1, apply measure.map_congr hf.ae_eq_mk }\n... = (μ.restrict $ (hf.mk f) ⁻¹' s).map (hf.mk f) :\n  measure.restrict_map hf.measurable_mk hs\n... = (μ.restrict $ (hf.mk f) ⁻¹' s).map f :\n  measure.map_congr (ae_restrict_of_ae (hf.ae_eq_mk.symm))\n... = (μ.restrict $ f ⁻¹' s).map f :\nbegin\n  apply congr_arg,\n  ext1 t ht,\n  simp only [ht, measure.restrict_apply],\n  apply measure_congr,\n  apply (eventually_eq.refl _ _).inter (hf.ae_eq_mk.symm.preimage s)\nend\n\nlemma measure_theory.measure.map_mono_of_ae_measurable\n  {f : α → δ} (h : μ ≤ ν) (hf : ae_measurable f ν) :\n  μ.map f ≤ ν.map f :=\nλ s hs, by simpa [hf, hs, hf.mono_measure h] using measure.le_iff'.1 h (f ⁻¹' s)\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/measure_theory/measure/ae_measurable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808533, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.33591316867230847}}
{"text": "/-\nCopyright (c) 2022 Youjack. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Youjack\n-/\nimport second_law\n\n/-!\n# Thermodynamic Temperature\n\nThis file states\n* the axiom that reversible `cycle`s do exists, i.e. `usual_engine_cycle.exists_reversible`\nand defines\n* thermodynamic temperature of `reservoir`, i.e. `reservoir.temp`\n-/\n\nnoncomputable theory\n\nnamespace thermodynamics\n\n/-!\n## Existence of reversible `usual_engine_cycle`\n---------------------------------------------------------------------------------------------------/\n\nnamespace usual_engine_cycle\nvariables {𝓗 𝓒 : reservoir} (𝓒_lt_𝓗 : 𝓒 < 𝓗)\n\naxiom exists_reversible : ∃ H : usual_engine_cycle 𝓒_lt_𝓗, H.reversible\ncopy_doc_string exists_reversible' → exists_reversible\n\nvariables {Q : ℝ} (hQ : 0 < Q)\n/-- There exists reversible `usual_engine_cycle 𝓒_lt_𝓗` that absorbs a certain heat `Q > 0`. -/\nlemma exists_rev_abs : ∃ H : usual_engine_cycle 𝓒_lt_𝓗, H.reversible ∧ ↑H.Qabs = Q :=\n  let ⟨H', hH'⟩ := exists_reversible 𝓒_lt_𝓗 in\n  let c : ℝ₊ := ⟨Q / H'.Qabs, div_pos hQ H'.do_abs⟩ in\n  let H := c • H' in\n  ⟨ H, \n    cycle.reversible_smul_nonneg (le_of_lt c.property) hH',\n    calc ↑H.Qabs\n         = Q / H'.Qabs * H'.Qabs : cycle.Qabs_smul_pos c.property\n      ...= Q                     : div_mul_cancel _ (ne_of_gt H'.do_abs), ⟩\n--\n\ndef rev_eff := (classical.some (exists_reversible 𝓒_lt_𝓗)).eff\nlemma rev_eff_universal {H : usual_engine_cycle 𝓒_lt_𝓗} (h : H.reversible) :\n  H.eff = rev_eff 𝓒_lt_𝓗 := by {\n  apply (usual_engine_cycle.rev_iff_eff _).elim_left h,\n  exact classical.some_spec (exists_reversible 𝓒_lt_𝓗), }\n--\n\nend usual_engine_cycle\n\n-- lemma usual_pump_cycle.exists_reversible\n\n/-!\n## Thermodynamic temperature\n---------------------------------------------------------------------------------------------------/\n\n/-!\n### Construction of thermodynamic temperature\n-/\n\nnamespace reservoir\n\nsection\nvariables {𝓗 𝓒 : reservoir} (𝓒_lt_𝓗 : 𝓒 < 𝓗)\n/-- `temp`erature `corr`elation function -/\n@[reducible] def temp_corr := 1 - usual_engine_cycle.rev_eff 𝓒_lt_𝓗\nlemma temp_corr_pos    : 0 < temp_corr 𝓒_lt_𝓗 :=\n  sub_pos.elim_right (usual_engine_cycle.eff_lt_one _)\nlemma temp_corr_lt_one : temp_corr 𝓒_lt_𝓗 < 1 :=\n  sub_lt_self _ (cycle.eff_pos _)\nlemma temp_corr_eq_Qratio {H : usual_engine_cycle 𝓒_lt_𝓗} (hH : H.reversible) :\n  temp_corr 𝓒_lt_𝓗 = H.Qrel / H.Qabs :=\n  calc temp_corr 𝓒_lt_𝓗\n       = 1 - H.eff : by rw [usual_engine_cycle.rev_eff_universal _ hH]\n    ...= H.Qrel / H.Qabs : by { rw H.eff_from_Qabs_Qrel, ring }\nend\n\nsection\nvariables {𝓐 𝓑 𝓒 : reservoir} (𝓒_lt_𝓑 : 𝓒 < 𝓑) (𝓑_lt_𝓐 : 𝓑 < 𝓐)\ntheorem temp_corr_trans :\n  temp_corr (𝓒_lt_𝓑.trans 𝓑_lt_𝓐) = (temp_corr 𝓒_lt_𝓑) * (temp_corr 𝓑_lt_𝓐) :=\n  let 𝓒_lt_𝓐 := 𝓒_lt_𝓑.trans 𝓑_lt_𝓐 in\n  let ⟨H𝓐𝓑, hH𝓐𝓑⟩ := usual_engine_cycle.exists_reversible 𝓑_lt_𝓐 in\n  let ⟨H𝓑𝓒, hH𝓑𝓒, hQ⟩ := usual_engine_cycle.exists_rev_abs 𝓒_lt_𝓑 H𝓐𝓑.do_rel in by {\n  let H𝓐𝓒' := H𝓐𝓑.to_cycle + H𝓑𝓒.to_cycle,\n  have H𝓑𝓒_no𝓐 : 𝓐 ∉ H𝓑𝓒.𝓠.support, {\n    rw H𝓑𝓒.two_rsv.elim_left,\n    simp only [finset.mem_insert, finset.mem_singleton, not_or_distrib],\n    exact ⟨ne_of_gt 𝓑_lt_𝓐, ne_of_gt 𝓒_lt_𝓐⟩, },\n  have H𝓐𝓑_no𝓒 : 𝓒 ∉ H𝓐𝓑.𝓠.support, {\n    rw H𝓐𝓑.two_rsv.elim_left,\n    simp only [finset.mem_insert, finset.mem_singleton, not_or_distrib],\n    exact ⟨ne_of_lt 𝓒_lt_𝓐, ne_of_lt 𝓒_lt_𝓑⟩, },\n  have H𝓐𝓒'Qabs :=\n    calc H𝓐𝓒'.𝓠 𝓐\n         = H𝓐𝓑.𝓠 𝓐 + H𝓑𝓒.𝓠 𝓐 : rfl\n      ...= H𝓐𝓑.𝓠 𝓐            : by rw [finsupp.not_mem_support_iff.elim_left H𝓑𝓒_no𝓐, add_zero]\n      ...= H𝓐𝓑.Qabs            : H𝓐𝓑.Qabs_one_rsv.symm,\n  have H𝓐𝓒'Qrel :=\n    calc H𝓐𝓒'.𝓠 𝓒\n         = H𝓐𝓑.𝓠 𝓒 + H𝓑𝓒.𝓠 𝓒 : rfl\n      ...=            H𝓑𝓒.𝓠 𝓒 : by rw [finsupp.not_mem_support_iff.elim_left H𝓐𝓑_no𝓒, zero_add]\n      ...=           -H𝓑𝓒.Qrel : by rw [←neg_neg (H𝓑𝓒.𝓠 𝓒), ←H𝓑𝓒.Qrel_one_rsv],\n  have do_abs_rsv : H𝓐𝓒'.𝓠 𝓐 > 0, from H𝓐𝓒'Qabs.symm ▸ H𝓐𝓑.do_abs,\n  have do_rel_rsv : H𝓐𝓒'.𝓠 𝓒 < 0, {\n    rw H𝓐𝓒'Qrel,\n    exact neg_lt_zero.elim_right H𝓑𝓒.do_rel, },\n  let H𝓐𝓒 : usual_engine_cycle 𝓒_lt_𝓐 :=\n    { two_rsv := by {\n        refine ⟨_, ne_of_gt 𝓒_lt_𝓐⟩,\n        have : H𝓐𝓒'.𝓠 = H𝓐𝓑.𝓠 + H𝓑𝓒.𝓠, from rfl, rw this,\n        apply finsupp.support_add_exact,\n        { simp only [finset.mem_insert, finset.mem_singleton, ne.def, forall_eq_or_imp, forall_eq],\n          split,\n            exact ne_of_gt do_abs_rsv,\n            exact ne_of_lt do_rel_rsv, },\n        { have : (H𝓐𝓑.𝓠.support ∪ H𝓑𝓒.𝓠.support) \\ {𝓐, 𝓒} = {𝓑}, {\n            rw [H𝓐𝓑.two_rsv.elim_left, H𝓑𝓒.two_rsv.elim_left],\n            ext, simp, split,\n            { tauto, },\n            { assume h𝓑,\n              split,\n              { exact or.inl h𝓑, },\n              { rw [not_or_distrib, h𝓑],\n                exact ⟨ne_of_lt 𝓑_lt_𝓐, ne_of_gt 𝓒_lt_𝓑⟩, } } },\n          simp only [this, finset.mem_singleton, forall_eq],\n          calc H𝓐𝓑.𝓠 𝓑 + H𝓑𝓒.𝓠 𝓑\n               = -H𝓐𝓑.Qrel + H𝓑𝓒.Qabs : by rw [H𝓐𝓑.Qrel_one_rsv, neg_neg, H𝓑𝓒.Qabs_one_rsv]\n            ...= 0                     : by { rw hQ, ring }, } },\n      do_abs_rsv := do_abs_rsv,\n      do_rel_rsv := do_rel_rsv,\n      do_work :=\n        calc H𝓐𝓒'.𝓦\n             = H𝓐𝓑.𝓦 + H𝓑𝓒.𝓦 : cycle.𝓦_add\n          ...> 0                : add_pos H𝓐𝓑.do_work H𝓑𝓒.do_work,\n      ..H𝓐𝓒' },\n  have := temp_corr_eq_Qratio 𝓑_lt_𝓐 hH𝓐𝓑, rw this,\n  have := temp_corr_eq_Qratio 𝓒_lt_𝓑 hH𝓑𝓒, rw this,\n  rw [hQ, div_mul_div_cancel _ (ne_of_gt H𝓐𝓑.do_rel)],\n  have : H𝓐𝓒.reversible, from cycle.reversible_add hH𝓐𝓑 hH𝓑𝓒,\n  have := temp_corr_eq_Qratio 𝓒_lt_𝓐 this, rw this,\n  rw [H𝓐𝓒.Qabs_one_rsv, H𝓐𝓒'Qabs],\n  rw [H𝓐𝓒.Qrel_one_rsv, H𝓐𝓒'Qrel, neg_neg], }\n/-- `temp_corr 𝓒_lt_𝓗` is `strict_mono` as a function of `𝓒` -/\nlemma temp_corr_strict_mono : temp_corr (𝓒_lt_𝓑.trans 𝓑_lt_𝓐) < temp_corr 𝓑_lt_𝓐 := by {\n  rw temp_corr_trans 𝓒_lt_𝓑 𝓑_lt_𝓐,\n  exact (mul_lt_iff_lt_one_left (temp_corr_pos _)).elim_right (temp_corr_lt_one _), }\nend\n\n/-- `ref`erence `sys`tem -/\n@[reducible] def ref_sys : equil_system := water_triple_point\n/-- reference temperature of `ref_sys` -/\n@[reducible] def ref_temp : ℝ₊ := ⟨273.16, by cancel_denoms⟩\n/-- thermodynamic `temp`erature of `reservoir`s -/\ndef temp : reservoir ↪o ℝ₊ := order_embedding.of_strict_mono\n  ( λ 𝓣,\n    if        heq : 𝓣 = ⟦ref_sys⟧ then\n      ref_temp\n    else if   hlt : 𝓣 < ⟦ref_sys⟧ then\n      ⟨(temp_corr hlt) * ref_temp, mul_pos (temp_corr_pos _) ref_temp.property⟩\n    else have hgt : 𝓣 > ⟦ref_sys⟧, from ne.lt_of_le' heq (le_of_not_gt hlt),\n      ⟨ref_temp / (temp_corr hgt), div_pos ref_temp.property (temp_corr_pos _)⟩ )\n  ( assume 𝓒 𝓗 𝓒_lt_𝓗,\n    if        h𝓗lt : 𝓗 < ⟦ref_sys⟧ then by {\n      have h𝓒lt := 𝓒_lt_𝓗.trans h𝓗lt,\n      simp only [ne_of_lt h𝓒lt, h𝓒lt, ne_of_lt h𝓗lt, h𝓗lt,\n        dif_pos, dite_eq_ite, if_false],\n      refine mul_lt_mul _ (le_rfl) (ref_temp.property) (le_of_lt (temp_corr_pos _)),\n      exact temp_corr_strict_mono 𝓒_lt_𝓗 h𝓗lt, }\n    else if   h𝓗eq : 𝓗 = ⟦ref_sys⟧ then by {\n      have h𝓒lt : 𝓒 < ⟦ref_sys⟧, from h𝓗eq ▸ 𝓒_lt_𝓗,\n      simp only [ne_of_lt h𝓒lt, h𝓒lt, h𝓗eq,\n        dif_pos, not_false_iff, dif_neg],\n      exact (mul_lt_iff_lt_one_left ref_temp.property).elim_right (temp_corr_lt_one _), }\n    else have h𝓗gt : 𝓗 > ⟦ref_sys⟧, from\n        ne.lt_of_le (ne.symm h𝓗eq) (le_of_not_gt h𝓗lt), by {\n      simp only [ne_of_gt h𝓗gt, not_lt_of_gt h𝓗gt,\n        not_false_iff, dif_neg],\n      apply subtype.coe_lt_coe.elim_left,\n      rw [subtype.coe_mk (↑ref_temp / temp_corr h𝓗gt) _, div_eq_mul_one_div],\n      exact if  h𝓒lt : 𝓒 < ⟦ref_sys⟧ then by {\n        simp only [ne_of_lt h𝓒lt, h𝓒lt,\n          dif_pos, dite_eq_ite, if_false],\n        rw [subtype.coe_mk, mul_comm (temp_corr h𝓒lt) _],\n        refine mul_lt_mul' (le_rfl) _ (le_of_lt (temp_corr_pos _)) (ref_temp.property),\n          apply (lt_div_iff (temp_corr_pos _)).elim_right,\n          rw ←temp_corr_trans,\n          exact temp_corr_lt_one _, }\n      else if   h𝓒eq : 𝓒 = ⟦ref_sys⟧ then by {\n        simp only [h𝓒eq, dif_pos],\n        apply lt_mul_right ref_temp.property,\n          apply (lt_div_iff (temp_corr_pos _)).elim_right,\n          rw one_mul,\n          exact temp_corr_lt_one _, }\n      else have h𝓒gt : 𝓒 > ⟦ref_sys⟧, from\n          ne.lt_of_le (ne.symm h𝓒eq) (le_of_not_gt h𝓒lt), by {\n        simp only [ne_of_gt h𝓒gt, not_lt_of_gt h𝓒gt,\n          not_false_iff, dif_neg],\n        rw [subtype.coe_mk, div_eq_mul_one_div _ (temp_corr h𝓒gt)],\n        refine mul_lt_mul' (le_rfl) _ _ (ref_temp.property),\n        { apply (one_div_lt_one_div (temp_corr_pos _) (temp_corr_pos _)).elim_right,\n          rw temp_corr_trans h𝓒gt 𝓒_lt_𝓗,\n          exact (mul_lt_iff_lt_one_right (temp_corr_pos _)).elim_right (temp_corr_lt_one _), },\n        { exact div_nonneg zero_le_one (le_of_lt (temp_corr_pos _)), } } } )\n--\n\nend reservoir\n\n/-!\n### Applications of thermodynamic temperature\n-/\n\n/- namespace usual_engine_cycle\nvariables {𝓗 𝓒 : reservoir} (𝓒_lt_𝓗 : 𝓒 < 𝓗)\n\nlemma rev_eff_from_temp : rev_eff 𝓒_lt_𝓗 = 1 - 𝓒.temp / 𝓗.temp := sorry\n\nend usual_engine_cycle -/\n\n/-!\n## The absolute zero\n\nTODO\n* existence of the absolute zero as a limit ?\n* require `rsv_no_min` ?\n* behavior of `usual_engine_cycle` when 𝓒.temp -> 0\n---------------------------------------------------------------------------------------------------/\n\nend thermodynamics\n", "meta": {"author": "Youjack", "repo": "thermodynamics.lean", "sha": "4af0748a97e6cb89aef0c87425872d1a901e8c55", "save_path": "github-repos/lean/Youjack-thermodynamics.lean", "path": "github-repos/lean/Youjack-thermodynamics.lean/thermodynamics.lean-4af0748a97e6cb89aef0c87425872d1a901e8c55/src/temperature.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3358515474041741}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard\n\n! This file was ported from Lean 3 source module deprecated.submonoid\n! leanprover-community/mathlib commit a11f9106a169dd302a285019e5165f8ab32ff433\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.GroupTheory.Submonoid.Basic\nimport Mathbin.Algebra.BigOperators.Basic\nimport Mathbin.Deprecated.Group\n\n/-!\n# Unbundled submonoids (deprecated)\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file is deprecated, and is no longer imported by anything in mathlib other than other\ndeprecated files, and test files. You should not need to import it.\n\nThis file defines unbundled multiplicative and additive submonoids. Instead of using this file,\nplease use `submonoid G` and `add_submonoid A`, defined in `group_theory.submonoid.basic`.\n\n## Main definitions\n\n`is_add_submonoid (S : set M)` : the predicate that `S` is the underlying subset of an additive\nsubmonoid of `M`. The bundled variant `add_submonoid M` should be used in preference to this.\n\n`is_submonoid (S : set M)` : the predicate that `S` is the underlying subset of a submonoid\nof `M`. The bundled variant `submonoid M` should be used in preference to this.\n\n## Tags\nsubmonoid, submonoids, is_submonoid\n-/\n\n\nopen BigOperators\n\nvariable {M : Type _} [Monoid M] {s : Set M}\n\nvariable {A : Type _} [AddMonoid A] {t : Set A}\n\n#print IsAddSubmonoid /-\n/-- `s` is an additive submonoid: a set containing 0 and closed under addition.\nNote that this structure is deprecated, and the bundled variant `add_submonoid A` should be\npreferred. -/\nstructure IsAddSubmonoid (s : Set A) : Prop where\n  zero_mem : (0 : A) ∈ s\n  add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s\n#align is_add_submonoid IsAddSubmonoid\n-/\n\n#print IsSubmonoid /-\n/-- `s` is a submonoid: a set containing 1 and closed under multiplication.\nNote that this structure is deprecated, and the bundled variant `submonoid M` should be\npreferred. -/\n@[to_additive]\nstructure IsSubmonoid (s : Set M) : Prop where\n  one_mem : (1 : M) ∈ s\n  mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s\n#align is_submonoid IsSubmonoid\n#align is_add_submonoid IsAddSubmonoid\n-/\n\n#print Additive.isAddSubmonoid /-\ntheorem Additive.isAddSubmonoid {s : Set M} : ∀ is : IsSubmonoid s, @IsAddSubmonoid (Additive M) _ s\n  | ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩\n#align additive.is_add_submonoid Additive.isAddSubmonoid\n-/\n\n#print Additive.isAddSubmonoid_iff /-\ntheorem Additive.isAddSubmonoid_iff {s : Set M} :\n    @IsAddSubmonoid (Additive M) _ s ↔ IsSubmonoid s :=\n  ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Additive.isAddSubmonoid⟩\n#align additive.is_add_submonoid_iff Additive.isAddSubmonoid_iff\n-/\n\n#print Multiplicative.isSubmonoid /-\ntheorem Multiplicative.isSubmonoid {s : Set A} :\n    ∀ is : IsAddSubmonoid s, @IsSubmonoid (Multiplicative A) _ s\n  | ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩\n#align multiplicative.is_submonoid Multiplicative.isSubmonoid\n-/\n\n#print Multiplicative.isSubmonoid_iff /-\ntheorem Multiplicative.isSubmonoid_iff {s : Set A} :\n    @IsSubmonoid (Multiplicative A) _ s ↔ IsAddSubmonoid s :=\n  ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Multiplicative.isSubmonoid⟩\n#align multiplicative.is_submonoid_iff Multiplicative.isSubmonoid_iff\n-/\n\n/- warning: is_submonoid.inter -> IsSubmonoid.inter is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s₁ : Set.{u1} M} {s₂ : Set.{u1} M}, (IsSubmonoid.{u1} M _inst_1 s₁) -> (IsSubmonoid.{u1} M _inst_1 s₂) -> (IsSubmonoid.{u1} M _inst_1 (Inter.inter.{u1} (Set.{u1} M) (Set.hasInter.{u1} M) s₁ s₂))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s₁ : Set.{u1} M} {s₂ : Set.{u1} M}, (IsSubmonoid.{u1} M _inst_1 s₁) -> (IsSubmonoid.{u1} M _inst_1 s₂) -> (IsSubmonoid.{u1} M _inst_1 (Inter.inter.{u1} (Set.{u1} M) (Set.instInterSet.{u1} M) s₁ s₂))\nCase conversion may be inaccurate. Consider using '#align is_submonoid.inter IsSubmonoid.interₓ'. -/\n/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/\n@[to_additive\n      \"The intersection of two `add_submonoid`s of an `add_monoid` `M` is\\nan `add_submonoid` of M.\"]\ntheorem IsSubmonoid.inter {s₁ s₂ : Set M} (is₁ : IsSubmonoid s₁) (is₂ : IsSubmonoid s₂) :\n    IsSubmonoid (s₁ ∩ s₂) :=\n  { one_mem := ⟨is₁.one_mem, is₂.one_mem⟩\n    mul_mem := fun x y hx hy => ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }\n#align is_submonoid.inter IsSubmonoid.inter\n#align is_add_submonoid.inter IsAddSubmonoid.inter\n\n#print IsSubmonoid.interᵢ /-\n/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/\n@[to_additive\n      \"The intersection of an indexed set of `add_submonoid`s of an `add_monoid` `M` is\\nan `add_submonoid` of `M`.\"]\ntheorem IsSubmonoid.interᵢ {ι : Sort _} {s : ι → Set M} (h : ∀ y : ι, IsSubmonoid (s y)) :\n    IsSubmonoid (Set.interᵢ s) :=\n  { one_mem := Set.mem_interᵢ.2 fun y => (h y).one_mem\n    mul_mem := fun x₁ x₂ h₁ h₂ =>\n      Set.mem_interᵢ.2 fun y => (h y).mul_mem (Set.mem_interᵢ.1 h₁ y) (Set.mem_interᵢ.1 h₂ y) }\n#align is_submonoid.Inter IsSubmonoid.interᵢ\n#align is_add_submonoid.Inter IsAddSubmonoid.interᵢ\n-/\n\n#print isSubmonoid_unionᵢ_of_directed /-\n/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid\n    of `M`. -/\n@[to_additive\n      \"The union of an indexed, directed, nonempty set\\nof `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`. \"]\ntheorem isSubmonoid_unionᵢ_of_directed {ι : Type _} [hι : Nonempty ι] {s : ι → Set M}\n    (hs : ∀ i, IsSubmonoid (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :\n    IsSubmonoid (⋃ i, s i) :=\n  { one_mem :=\n      let ⟨i⟩ := hι\n      Set.mem_unionᵢ.2 ⟨i, (hs i).one_mem⟩\n    mul_mem := fun a b ha hb =>\n      let ⟨i, hi⟩ := Set.mem_unionᵢ.1 ha\n      let ⟨j, hj⟩ := Set.mem_unionᵢ.1 hb\n      let ⟨k, hk⟩ := Directed i j\n      Set.mem_unionᵢ.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }\n#align is_submonoid_Union_of_directed isSubmonoid_unionᵢ_of_directed\n#align is_add_submonoid_Union_of_directed isAddSubmonoid_unionᵢ_of_directed\n-/\n\nsection powers\n\n#print powers /-\n/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/\n@[to_additive multiples\n      \"The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `add_monoid`.\"]\ndef powers (x : M) : Set M :=\n  { y | ∃ n : ℕ, x ^ n = y }\n#align powers powers\n#align multiples multiples\n-/\n\n/- warning: powers.one_mem -> powers.one_mem is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {x : M}, Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) (OfNat.ofNat.{u1} M 1 (OfNat.mk.{u1} M 1 (One.one.{u1} M (MulOneClass.toHasOne.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) (powers.{u1} M _inst_1 x)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {x : M}, Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) (OfNat.ofNat.{u1} M 1 (One.toOfNat1.{u1} M (Monoid.toOne.{u1} M _inst_1))) (powers.{u1} M _inst_1 x)\nCase conversion may be inaccurate. Consider using '#align powers.one_mem powers.one_memₓ'. -/\n/-- 1 is in the set of natural number powers of an element of a monoid. -/\n@[to_additive \"0 is in the set of natural number multiples of an element of an `add_monoid`.\"]\ntheorem powers.one_mem {x : M} : (1 : M) ∈ powers x :=\n  ⟨0, pow_zero _⟩\n#align powers.one_mem powers.one_mem\n#align multiples.zero_mem multiples.zero_mem\n\n#print powers.self_mem /-\n/-- An element of a monoid is in the set of that element's natural number powers. -/\n@[to_additive\n      \"An element of an `add_monoid` is in the set of that element's natural number multiples.\"]\ntheorem powers.self_mem {x : M} : x ∈ powers x :=\n  ⟨1, pow_one _⟩\n#align powers.self_mem powers.self_mem\n#align multiples.self_mem multiples.self_mem\n-/\n\n/- warning: powers.mul_mem -> powers.mul_mem is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {x : M} {y : M} {z : M}, (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) y (powers.{u1} M _inst_1 x)) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) z (powers.{u1} M _inst_1 x)) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) y z) (powers.{u1} M _inst_1 x))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {x : M} {y : M} {z : M}, (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) y (powers.{u1} M _inst_1 x)) -> (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) z (powers.{u1} M _inst_1 x)) -> (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) y z) (powers.{u1} M _inst_1 x))\nCase conversion may be inaccurate. Consider using '#align powers.mul_mem powers.mul_memₓ'. -/\n/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/\n@[to_additive\n      \"The set of natural number multiples of an element of an `add_monoid` is closed under addition.\"]\ntheorem powers.mul_mem {x y z : M} : y ∈ powers x → z ∈ powers x → y * z ∈ powers x :=\n  fun ⟨n₁, h₁⟩ ⟨n₂, h₂⟩ => ⟨n₁ + n₂, by simp only [pow_add, *]⟩\n#align powers.mul_mem powers.mul_mem\n#align multiples.add_mem multiples.add_mem\n\n#print powers.isSubmonoid /-\n/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/\n@[to_additive\n      \"The set of natural number multiples of an element of\\nan `add_monoid` `M` is an `add_submonoid` of `M`.\"]\ntheorem powers.isSubmonoid (x : M) : IsSubmonoid (powers x) :=\n  { one_mem := powers.one_mem\n    mul_mem := fun y z => powers.mul_mem }\n#align powers.is_submonoid powers.isSubmonoid\n#align multiples.is_add_submonoid multiples.isAddSubmonoid\n-/\n\n#print Univ.isSubmonoid /-\n/-- A monoid is a submonoid of itself. -/\n@[to_additive \"An `add_monoid` is an `add_submonoid` of itself.\"]\ntheorem Univ.isSubmonoid : IsSubmonoid (@Set.univ M) := by constructor <;> simp\n#align univ.is_submonoid Univ.isSubmonoid\n#align univ.is_add_submonoid Univ.isAddSubmonoid\n-/\n\n#print IsSubmonoid.preimage /-\n/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/\n@[to_additive\n      \"The preimage of an `add_submonoid` under an `add_monoid` hom is\\nan `add_submonoid` of the domain.\"]\ntheorem IsSubmonoid.preimage {N : Type _} [Monoid N] {f : M → N} (hf : IsMonoidHom f) {s : Set N}\n    (hs : IsSubmonoid s) : IsSubmonoid (f ⁻¹' s) :=\n  { one_mem := show f 1 ∈ s by rw [IsMonoidHom.map_one hf] <;> exact hs.one_mem\n    mul_mem := fun a b (ha : f a ∈ s) (hb : f b ∈ s) =>\n      show f (a * b) ∈ s by rw [IsMonoidHom.map_mul' hf] <;> exact hs.mul_mem ha hb }\n#align is_submonoid.preimage IsSubmonoid.preimage\n#align is_add_submonoid.preimage IsAddSubmonoid.preimage\n-/\n\n#print IsSubmonoid.image /-\n/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/\n@[to_additive\n      \"The image of an `add_submonoid` under an `add_monoid`\\nhom is an `add_submonoid` of the codomain.\"]\ntheorem IsSubmonoid.image {γ : Type _} [Monoid γ] {f : M → γ} (hf : IsMonoidHom f) {s : Set M}\n    (hs : IsSubmonoid s) : IsSubmonoid (f '' s) :=\n  { one_mem := ⟨1, hs.one_mem, hf.map_one⟩\n    mul_mem := fun a b ⟨x, hx⟩ ⟨y, hy⟩ =>\n      ⟨x * y, hs.mul_mem hx.1 hy.1, by rw [hf.map_mul, hx.2, hy.2]⟩ }\n#align is_submonoid.image IsSubmonoid.image\n#align is_add_submonoid.image IsAddSubmonoid.image\n-/\n\n#print Range.isSubmonoid /-\n/-- The image of a monoid hom is a submonoid of the codomain. -/\n@[to_additive \"The image of an `add_monoid` hom is an `add_submonoid`\\nof the codomain.\"]\ntheorem Range.isSubmonoid {γ : Type _} [Monoid γ] {f : M → γ} (hf : IsMonoidHom f) :\n    IsSubmonoid (Set.range f) := by\n  rw [← Set.image_univ]\n  exact univ.is_submonoid.image hf\n#align range.is_submonoid Range.isSubmonoid\n#align range.is_add_submonoid Range.isAddSubmonoid\n-/\n\n#print IsSubmonoid.pow_mem /-\n/-- Submonoids are closed under natural powers. -/\n@[to_additive IsAddSubmonoid.smul_mem\n      \"An `add_submonoid` is closed under multiplication by naturals.\"]\ntheorem IsSubmonoid.pow_mem {a : M} (hs : IsSubmonoid s) (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s\n  | 0 => by\n    rw [pow_zero]\n    exact hs.one_mem\n  | n + 1 => by\n    rw [pow_succ]\n    exact hs.mul_mem h IsSubmonoid.pow_mem\n#align is_submonoid.pow_mem IsSubmonoid.pow_mem\n-/\n\n#print IsSubmonoid.power_subset /-\n/-- The set of natural number powers of an element of a submonoid is a subset of the submonoid. -/\n@[to_additive IsAddSubmonoid.multiples_subset\n      \"The set of natural number multiples of an element\\nof an `add_submonoid` is a subset of the `add_submonoid`.\"]\ntheorem IsSubmonoid.power_subset {a : M} (hs : IsSubmonoid s) (h : a ∈ s) : powers a ⊆ s :=\n  fun x ⟨n, hx⟩ => hx ▸ hs.pow_mem h\n#align is_submonoid.power_subset IsSubmonoid.power_subset\n#align is_add_submonoid.multiples_subset IsAddSubmonoid.multiples_subset\n-/\n\nend powers\n\nnamespace IsSubmonoid\n\n/- warning: is_submonoid.list_prod_mem -> IsSubmonoid.list_prod_mem is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s : Set.{u1} M}, (IsSubmonoid.{u1} M _inst_1 s) -> (forall {l : List.{u1} M}, (forall (x : M), (Membership.Mem.{u1, u1} M (List.{u1} M) (List.hasMem.{u1} M) x l) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) x s)) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) (List.prod.{u1} M (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toHasOne.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) l) s))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s : Set.{u1} M}, (IsSubmonoid.{u1} M _inst_1 s) -> (forall {l : List.{u1} M}, (forall (x : M), (Membership.mem.{u1, u1} M (List.{u1} M) (List.instMembershipList.{u1} M) x l) -> (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) x s)) -> (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) (List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Monoid.toOne.{u1} M _inst_1) l) s))\nCase conversion may be inaccurate. Consider using '#align is_submonoid.list_prod_mem IsSubmonoid.list_prod_memₓ'. -/\n/-- The product of a list of elements of a submonoid is an element of the submonoid. -/\n@[to_additive\n      \"The sum of a list of elements of an `add_submonoid` is an element of the\\n`add_submonoid`.\"]\ntheorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.Prod ∈ s\n  | [], h => hs.one_mem\n  | a :: l, h =>\n    suffices a * l.Prod ∈ s by simpa\n    have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by simpa using h\n    hs.mul_mem this.1 (list_prod_mem this.2)\n#align is_submonoid.list_prod_mem IsSubmonoid.list_prod_mem\n#align is_add_submonoid.list_sum_mem IsAddSubmonoid.list_sum_mem\n\n#print IsSubmonoid.multiset_prod_mem /-\n/-- The product of a multiset of elements of a submonoid of a `comm_monoid` is an element of\nthe submonoid. -/\n@[to_additive\n      \"The sum of a multiset of elements of an `add_submonoid` of an `add_comm_monoid`\\nis an element of the `add_submonoid`. \"]\ntheorem multiset_prod_mem {M} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (m : Multiset M) :\n    (∀ a ∈ m, a ∈ s) → m.Prod ∈ s :=\n  by\n  refine' Quotient.inductionOn m fun l hl => _\n  rw [Multiset.quot_mk_to_coe, Multiset.coe_prod]\n  exact list_prod_mem hs hl\n#align is_submonoid.multiset_prod_mem IsSubmonoid.multiset_prod_mem\n#align is_add_submonoid.multiset_sum_mem IsAddSubmonoid.multiset_sum_mem\n-/\n\n/- warning: is_submonoid.finset_prod_mem -> IsSubmonoid.finset_prod_mem is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {A : Type.{u2}} [_inst_3 : CommMonoid.{u1} M] {s : Set.{u1} M}, (IsSubmonoid.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) s) -> (forall (f : A -> M) (t : Finset.{u2} A), (forall (b : A), (Membership.Mem.{u2, u2} A (Finset.{u2} A) (Finset.hasMem.{u2} A) b t) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) (f b) s)) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) (Finset.prod.{u1, u2} M A _inst_3 t (fun (b : A) => f b)) s))\nbut is expected to have type\n  forall {M : Type.{u2}} {A : Type.{u1}} [_inst_3 : CommMonoid.{u2} M] {s : Set.{u2} M}, (IsSubmonoid.{u2} M (CommMonoid.toMonoid.{u2} M _inst_3) s) -> (forall (f : A -> M) (t : Finset.{u1} A), (forall (b : A), (Membership.mem.{u1, u1} A (Finset.{u1} A) (Finset.instMembershipFinset.{u1} A) b t) -> (Membership.mem.{u2, u2} M (Set.{u2} M) (Set.instMembershipSet.{u2} M) (f b) s)) -> (Membership.mem.{u2, u2} M (Set.{u2} M) (Set.instMembershipSet.{u2} M) (Finset.prod.{u2, u1} M A _inst_3 t (fun (b : A) => f b)) s))\nCase conversion may be inaccurate. Consider using '#align is_submonoid.finset_prod_mem IsSubmonoid.finset_prod_memₓ'. -/\n/-- The product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is an element\nof the submonoid. -/\n@[to_additive\n      \"The sum of elements of an `add_submonoid` of an `add_comm_monoid` indexed by\\na `finset` is an element of the `add_submonoid`.\"]\ntheorem finset_prod_mem {M A} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (f : A → M) :\n    ∀ t : Finset A, (∀ b ∈ t, f b ∈ s) → (∏ b in t, f b) ∈ s\n  | ⟨m, hm⟩, _ => multiset_prod_mem hs _ (by simpa)\n#align is_submonoid.finset_prod_mem IsSubmonoid.finset_prod_mem\n#align is_add_submonoid.finset_sum_mem IsAddSubmonoid.finset_sum_mem\n\nend IsSubmonoid\n\nnamespace AddMonoid\n\n#print AddMonoid.InClosure /-\n/-- The inductively defined membership predicate for the submonoid generated by a subset of a\n    monoid. -/\ninductive InClosure (s : Set A) : A → Prop\n  | basic {a : A} : a ∈ s → in_closure a\n  | zero : in_closure 0\n  | add {a b : A} : in_closure a → in_closure b → in_closure (a + b)\n#align add_monoid.in_closure AddMonoid.InClosure\n-/\n\nend AddMonoid\n\nnamespace Monoid\n\n#print Monoid.InClosure /-\n/-- The inductively defined membership predicate for the `submonoid` generated by a subset of an\n    monoid. -/\n@[to_additive]\ninductive InClosure (s : Set M) : M → Prop\n  | basic {a : M} : a ∈ s → in_closure a\n  | one : in_closure 1\n  | mul {a b : M} : in_closure a → in_closure b → in_closure (a * b)\n#align monoid.in_closure Monoid.InClosure\n#align add_monoid.in_closure AddMonoid.InClosure\n-/\n\n#print Monoid.Closure /-\n/-- The inductively defined submonoid generated by a subset of a monoid. -/\n@[to_additive \"The inductively defined `add_submonoid` genrated by a subset of an `add_monoid`.\"]\ndef Closure (s : Set M) : Set M :=\n  { a | InClosure s a }\n#align monoid.closure Monoid.Closure\n#align add_monoid.closure AddMonoid.Closure\n-/\n\n#print Monoid.closure.isSubmonoid /-\n@[to_additive]\ntheorem closure.isSubmonoid (s : Set M) : IsSubmonoid (Closure s) :=\n  { one_mem := InClosure.one\n    mul_mem := fun a b => InClosure.mul }\n#align monoid.closure.is_submonoid Monoid.closure.isSubmonoid\n#align add_monoid.closure.is_add_submonoid AddMonoid.closure.isAddSubmonoid\n-/\n\n#print Monoid.subset_closure /-\n/-- A subset of a monoid is contained in the submonoid it generates. -/\n@[to_additive \"A subset of an `add_monoid` is contained in the `add_submonoid` it generates.\"]\ntheorem subset_closure {s : Set M} : s ⊆ Closure s := fun a => InClosure.basic\n#align monoid.subset_closure Monoid.subset_closure\n#align add_monoid.subset_closure AddMonoid.subset_closure\n-/\n\n#print Monoid.closure_subset /-\n/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/\n@[to_additive\n      \"The `add_submonoid` generated by a set is contained in any `add_submonoid` that\\ncontains the set.\"]\ntheorem closure_subset {s t : Set M} (ht : IsSubmonoid t) (h : s ⊆ t) : Closure s ⊆ t := fun a ha =>\n  by induction ha <;> simp [h _, *, IsSubmonoid.one_mem, IsSubmonoid.mul_mem]\n#align monoid.closure_subset Monoid.closure_subset\n#align add_monoid.closure_subset AddMonoid.closure_subset\n-/\n\n#print Monoid.closure_mono /-\n/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is\n    contained in the submonoid generated by `t`. -/\n@[to_additive\n      \"Given subsets `t` and `s` of an `add_monoid M`, if `s ⊆ t`, the `add_submonoid`\\nof `M` generated by `s` is contained in the `add_submonoid` generated by `t`.\"]\ntheorem closure_mono {s t : Set M} (h : s ⊆ t) : Closure s ⊆ Closure t :=\n  closure_subset (closure.isSubmonoid t) <| Set.Subset.trans h subset_closure\n#align monoid.closure_mono Monoid.closure_mono\n#align add_monoid.closure_mono AddMonoid.closure_mono\n-/\n\n#print Monoid.closure_singleton /-\n/-- The submonoid generated by an element of a monoid equals the set of natural number powers of\n    the element. -/\n@[to_additive\n      \"The `add_submonoid` generated by an element of an `add_monoid` equals the set of\\nnatural number multiples of the element.\"]\ntheorem closure_singleton {x : M} : Closure ({x} : Set M) = powers x :=\n  Set.eq_of_subset_of_subset\n      (closure_subset (powers.isSubmonoid x) <| Set.singleton_subset_iff.2 <| powers.self_mem) <|\n    IsSubmonoid.power_subset (closure.isSubmonoid _) <| Set.singleton_subset_iff.1 <| subset_closure\n#align monoid.closure_singleton Monoid.closure_singleton\n#align add_monoid.closure_singleton AddMonoid.closure_singleton\n-/\n\n#print Monoid.image_closure /-\n/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated\n    by the image of the set under the monoid hom. -/\n@[to_additive\n      \"The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals\\nthe `add_submonoid` generated by the image of the set under the `add_monoid` hom.\"]\ntheorem image_closure {A : Type _} [Monoid A] {f : M → A} (hf : IsMonoidHom f) (s : Set M) :\n    f '' Closure s = Closure (f '' s) :=\n  le_antisymm\n    (by\n      rintro _ ⟨x, hx, rfl⟩\n      apply in_closure.rec_on hx <;> intros\n      · solve_by_elim [subset_closure, Set.mem_image_of_mem]\n      · rw [hf.map_one]\n        apply IsSubmonoid.one_mem (closure.is_submonoid (f '' s))\n      · rw [hf.map_mul]\n        solve_by_elim [(closure.is_submonoid _).mul_mem] )\n    (closure_subset (IsSubmonoid.image hf (closure.isSubmonoid _)) <|\n      Set.image_subset _ subset_closure)\n#align monoid.image_closure Monoid.image_closure\n#align add_monoid.image_closure AddMonoid.image_closure\n-/\n\n/- warning: monoid.exists_list_of_mem_closure -> Monoid.exists_list_of_mem_closure is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s : Set.{u1} M} {a : M}, (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) a (Monoid.Closure.{u1} M _inst_1 s)) -> (Exists.{succ u1} (List.{u1} M) (fun (l : List.{u1} M) => And (forall (x : M), (Membership.Mem.{u1, u1} M (List.{u1} M) (List.hasMem.{u1} M) x l) -> (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) x s)) (Eq.{succ u1} M (List.prod.{u1} M (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (MulOneClass.toHasOne.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) l) a)))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] {s : Set.{u1} M} {a : M}, (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) a (Monoid.Closure.{u1} M _inst_1 s)) -> (Exists.{succ u1} (List.{u1} M) (fun (l : List.{u1} M) => And (forall (x : M), (Membership.mem.{u1, u1} M (List.{u1} M) (List.instMembershipList.{u1} M) x l) -> (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) x s)) (Eq.{succ u1} M (List.prod.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Monoid.toOne.{u1} M _inst_1) l) a)))\nCase conversion may be inaccurate. Consider using '#align monoid.exists_list_of_mem_closure Monoid.exists_list_of_mem_closureₓ'. -/\n/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists\na list of elements of `s` whose product is `a`. -/\n@[to_additive\n      \"Given an element `a` of the `add_submonoid` of an `add_monoid M` generated by\\na set `s`, there exists a list of elements of `s` whose sum is `a`.\"]\ntheorem exists_list_of_mem_closure {s : Set M} {a : M} (h : a ∈ Closure s) :\n    ∃ l : List M, (∀ x ∈ l, x ∈ s) ∧ l.Prod = a :=\n  by\n  induction h\n  case basic a ha => exists [a]; simp [ha]\n  case one => exists []; simp\n  case mul a b _ _ ha hb =>\n    rcases ha with ⟨la, ha, eqa⟩\n    rcases hb with ⟨lb, hb, eqb⟩\n    exists la ++ lb\n    simp [eqa.symm, eqb.symm, or_imp]\n    exact fun a => ⟨ha a, hb a⟩\n#align monoid.exists_list_of_mem_closure Monoid.exists_list_of_mem_closure\n#align add_monoid.exists_list_of_mem_closure AddMonoid.exists_list_of_mem_closure\n\n/- warning: monoid.mem_closure_union_iff -> Monoid.mem_closure_union_iff is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_3 : CommMonoid.{u1} M] {s : Set.{u1} M} {t : Set.{u1} M} {x : M}, Iff (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) x (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) (Union.union.{u1} (Set.{u1} M) (Set.hasUnion.{u1} M) s t))) (Exists.{succ u1} M (fun (y : M) => Exists.{0} (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) y (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) s)) (fun (H : Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) y (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) s)) => Exists.{succ u1} M (fun (z : M) => Exists.{0} (Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) z (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) t)) (fun (H : Membership.Mem.{u1, u1} M (Set.{u1} M) (Set.hasMem.{u1} M) z (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) t)) => Eq.{succ u1} M (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M (MulOneClass.toHasMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3)))) y z) x)))))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_3 : CommMonoid.{u1} M] {s : Set.{u1} M} {t : Set.{u1} M} {x : M}, Iff (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) x (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) (Union.union.{u1} (Set.{u1} M) (Set.instUnionSet.{u1} M) s t))) (Exists.{succ u1} M (fun (y : M) => And (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) y (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) s)) (Exists.{succ u1} M (fun (z : M) => And (Membership.mem.{u1, u1} M (Set.{u1} M) (Set.instMembershipSet.{u1} M) z (Monoid.Closure.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3) t)) (Eq.{succ u1} M (HMul.hMul.{u1, u1, u1} M M M (instHMul.{u1} M (MulOneClass.toMul.{u1} M (Monoid.toMulOneClass.{u1} M (CommMonoid.toMonoid.{u1} M _inst_3)))) y z) x)))))\nCase conversion may be inaccurate. Consider using '#align monoid.mem_closure_union_iff Monoid.mem_closure_union_iffₓ'. -/\n/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by\n    `s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the\n    submonoid generated by `t` whose product is `x`. -/\n@[to_additive\n      \"Given sets `s, t` of a commutative `add_monoid M`, `x ∈ M` is in the `add_submonoid`\\nof `M` generated by `s ∪ t` iff there exists an element of the `add_submonoid` generated by `s`\\nand an element of the `add_submonoid` generated by `t` whose sum is `x`.\"]\ntheorem mem_closure_union_iff {M : Type _} [CommMonoid M] {s t : Set M} {x : M} :\n    x ∈ Closure (s ∪ t) ↔ ∃ y ∈ Closure s, ∃ z ∈ Closure t, y * z = x :=\n  ⟨fun hx =>\n    let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx\n    HL2 ▸\n      List.recOn L\n        (fun _ =>\n          ⟨1, (closure.isSubmonoid _).one_mem, 1, (closure.isSubmonoid _).one_mem, mul_one _⟩)\n        (fun hd tl ih HL1 =>\n          let ⟨y, hy, z, hz, hyzx⟩ := ih (List.forall_mem_of_forall_mem_cons HL1)\n          Or.cases_on (HL1 hd <| List.mem_cons_self _ _)\n            (fun hs =>\n              ⟨hd * y, (closure.isSubmonoid _).mul_mem (subset_closure hs) hy, z, hz, by\n                rw [mul_assoc, List.prod_cons, ← hyzx] <;> rfl⟩)\n            fun ht =>\n            ⟨y, hy, z * hd, (closure.isSubmonoid _).mul_mem hz (subset_closure ht), by\n              rw [← mul_assoc, List.prod_cons, ← hyzx, mul_comm hd] <;> rfl⟩)\n        HL1,\n    fun ⟨y, hy, z, hz, hyzx⟩ =>\n    hyzx ▸\n      (closure.isSubmonoid _).mul_mem (closure_mono (Set.subset_union_left _ _) hy)\n        (closure_mono (Set.subset_union_right _ _) hz)⟩\n#align monoid.mem_closure_union_iff Monoid.mem_closure_union_iff\n#align add_monoid.mem_closure_union_iff AddMonoid.mem_closure_union_iff\n\nend Monoid\n\n#print Submonoid.of /-\n/-- Create a bundled submonoid from a set `s` and `[is_submonoid s]`. -/\n@[to_additive \"Create a bundled additive submonoid from a set `s` and `[is_add_submonoid s]`.\"]\ndef Submonoid.of {s : Set M} (h : IsSubmonoid s) : Submonoid M :=\n  ⟨s, fun _ _ => h.2, h.1⟩\n#align submonoid.of Submonoid.of\n#align add_submonoid.of AddSubmonoid.of\n-/\n\n/- warning: submonoid.is_submonoid -> Submonoid.isSubmonoid is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (S : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)), IsSubmonoid.{u1} M _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Set.{u1} M) (HasLiftT.mk.{succ u1, succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Set.{u1} M) (CoeTCₓ.coe.{succ u1, succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Set.{u1} M) (SetLike.Set.hasCoeT.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) S)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : Monoid.{u1} M] (S : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)), IsSubmonoid.{u1} M _inst_1 (SetLike.coe.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) S)\nCase conversion may be inaccurate. Consider using '#align submonoid.is_submonoid Submonoid.isSubmonoidₓ'. -/\n@[to_additive]\ntheorem Submonoid.isSubmonoid (S : Submonoid M) : IsSubmonoid (S : Set M) :=\n  ⟨S.3, fun _ _ => S.2⟩\n#align submonoid.is_submonoid Submonoid.isSubmonoid\n#align add_submonoid.is_add_submonoid AddSubmonoid.isAddSubmonoid\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Deprecated/Submonoid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.33585153344560004}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.eq_to_hom\n\n/-!\n# Discrete categories\n\nWe define `discrete α := α` for any type `α`, and use this type alias\nto provide a `small_category` instance whose only morphisms are the identities.\n\nThere is an annoying technical difficulty that it has turned out to be inconvenient\nto allow categories with morphisms living in `Prop`,\nso instead of defining `X ⟶ Y` in `discrete α` as `X = Y`,\none might define it as `plift (X = Y)`.\nIn fact, to allow `discrete α` to be a `small_category`\n(i.e. with morphisms in the same universe as the objects),\nwe actually define the hom type `X ⟶ Y` as `ulift (plift (X = Y))`.\n\n`discrete.functor` promotes a function `f : I → C` (for any category `C`) to a functor\n`discrete.functor f : discrete I ⥤ C`.\n\nSimilarly, `discrete.nat_trans` and `discrete.nat_iso` promote `I`-indexed families of morphisms,\nor `I`-indexed families of isomorphisms to natural transformations or natural isomorphism.\n\nWe show equivalences of types are the same as (categorical) equivalences of the corresponding\ndiscrete categories.\n-/\n\nnamespace category_theory\n\nuniverses v₁ v₂ u₁ u₂ -- morphism levels before object levels. See note [category_theory universes].\n\n/--\nA type synonym for promoting any type to a category,\nwith the only morphisms being equalities.\n-/\ndef discrete (α : Type u₁) := α\n\n/--\nThe \"discrete\" category on a type, whose morphisms are equalities.\n\nBecause we do not allow morphisms in `Prop` (only in `Type`),\nsomewhat annoyingly we have to define `X ⟶ Y` as `ulift (plift (X = Y))`.\n\nSee https://stacks.math.columbia.edu/tag/001A\n-/\ninstance discrete_category (α : Type u₁) : small_category (discrete α) :=\n{ hom  := λ X Y, ulift (plift (X = Y)),\n  id   := λ X, ulift.up (plift.up rfl),\n  comp := λ X Y Z g f, by { rcases f with ⟨⟨rfl⟩⟩, exact g } }\n\nnamespace discrete\n\nvariables {α : Type u₁}\n\ninstance [inhabited α] : inhabited (discrete α) :=\nby { dsimp [discrete], apply_instance }\n\ninstance [subsingleton α] : subsingleton (discrete α) :=\nby { dsimp [discrete], apply_instance }\n\n/-- Extract the equation from a morphism in a discrete category. -/\n\n\n@[simp] lemma id_def (X : discrete α) : ulift.up (plift.up (eq.refl X)) = 𝟙 X := rfl\n\nvariables {C : Type u₂} [category.{v₂} C]\n\ninstance {I : Type u₁} {i j : discrete I} (f : i ⟶ j) : is_iso f :=\n⟨⟨eq_to_hom (eq_of_hom f).symm, by tidy⟩⟩\n\n/--\nAny function `I → C` gives a functor `discrete I ⥤ C`.\n-/\ndef functor {I : Type u₁} (F : I → C) : discrete I ⥤ C :=\n{ obj := F,\n  map := λ X Y f, begin cases f, cases f, cases f, exact 𝟙 (F X) end }\n\n@[simp] lemma functor_obj  {I : Type u₁} (F : I → C) (i : I) :\n  (discrete.functor F).obj i = F i := rfl\n\nlemma functor_map  {I : Type u₁} (F : I → C) {i : discrete I} (f : i ⟶ i) :\n  (discrete.functor F).map f = 𝟙 (F i) :=\nby { cases f, cases f, cases f, refl }\n\n/--\nFor functors out of a discrete category,\na natural transformation is just a collection of maps,\nas the naturality squares are trivial.\n-/\ndef nat_trans {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ⟶ G.obj i) : F ⟶ G :=\n{ app := f }\n\n@[simp] lemma nat_trans_app  {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ⟶ G.obj i) (i) : (discrete.nat_trans f).app i = f i :=\nrfl\n\n/--\nFor functors out of a discrete category,\na natural isomorphism is just a collection of isomorphisms,\nas the naturality squares are trivial.\n-/\ndef nat_iso {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ≅ G.obj i) : F ≅ G :=\nnat_iso.of_components f (by tidy)\n\n@[simp]\nlemma nat_iso_hom_app {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ≅ G.obj i) (i : I) :\n  (discrete.nat_iso f).hom.app i = (f i).hom :=\nrfl\n\n@[simp]\nlemma nat_iso_inv_app {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ≅ G.obj i) (i : I) :\n  (discrete.nat_iso f).inv.app i = (f i).inv :=\nrfl\n\n@[simp]\nlemma nat_iso_app {I : Type u₁} {F G : discrete I ⥤ C}\n  (f : Π i : discrete I, F.obj i ≅ G.obj i) (i : I) :\n  (discrete.nat_iso f).app i = f i :=\nby tidy\n\n/-- Every functor `F` from a discrete category is naturally isomorphic (actually, equal) to\n  `discrete.functor (F.obj)`. -/\ndef nat_iso_functor {I : Type u₁} {F : discrete I ⥤ C} : F ≅ discrete.functor (F.obj) :=\nnat_iso $ λ i, iso.refl _\n\n/--\nWe can promote a type-level `equiv` to\nan equivalence between the corresponding `discrete` categories.\n-/\n@[simps]\ndef equivalence {I J : Type u₁} (e : I ≃ J) : discrete I ≌ discrete J :=\n{ functor := discrete.functor (e : I → J),\n  inverse := discrete.functor (e.symm : J → I),\n  unit_iso := discrete.nat_iso (λ i, eq_to_iso (by simp)),\n  counit_iso := discrete.nat_iso (λ j, eq_to_iso (by simp)), }\n\n/-- We can convert an equivalence of `discrete` categories to a type-level `equiv`. -/\n@[simps]\ndef equiv_of_equivalence {α β : Type u₁} (h : discrete α ≌ discrete β) : α ≃ β :=\n{ to_fun := h.functor.obj,\n  inv_fun := h.inverse.obj,\n  left_inv := λ a, eq_of_hom (h.unit_iso.app a).2,\n  right_inv := λ a, eq_of_hom (h.counit_iso.app a).1 }\n\nend discrete\n\nnamespace discrete\nvariables {J : Type v₁}\n\nopen opposite\n\n/-- A discrete category is equivalent to its opposite category. -/\nprotected def opposite (α : Type u₁) : (discrete α)ᵒᵖ ≌ discrete α :=\nlet F : discrete α ⥤ (discrete α)ᵒᵖ := discrete.functor (λ x, op x) in\nbegin\n  refine equivalence.mk (functor.left_op F) F _ (discrete.nat_iso $ λ X, by simp [F]),\n  refine nat_iso.of_components (λ X, by simp [F]) _,\n  tidy\nend\n\nvariables {C : Type u₂} [category.{v₂} C]\n\n@[simp] lemma functor_map_id\n  (F : discrete J ⥤ C) {j : discrete J} (f : j ⟶ j) : F.map f = 𝟙 (F.obj j) :=\nbegin\n  have h : f = 𝟙 j, { cases f, cases f, ext, },\n  rw h,\n  simp,\nend\n\nend discrete\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/discrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.33569984061527874}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport data.rbmap.basic\nimport data.rbtree.main\n\nuniverses u v\n\nnamespace rbmap\nvariables {α : Type u} {β : Type v} {lt : α → α → Prop}\n\n/- Auxiliary instances -/\nprivate def rbmap_lt_is_swo {α : Type u} {β : Type v} {lt : α → α → Prop}\n  [is_strict_weak_order α lt] : is_strict_weak_order (α × β) (rbmap_lt lt) :=\n{ irrefl       := λ _, irrefl_of lt _,\n  trans        := λ _ _ _ h₁ h₂, trans_of lt h₁ h₂,\n  incomp_trans := λ _ _ _ h₁ h₂, incomp_trans_of lt h₁ h₂ }\n\nprivate def rbmap_lt_dec {α : Type u} {β : Type v} {lt : α → α → Prop} [h : decidable_rel lt] :\n  decidable_rel (@rbmap_lt α β lt) :=\nλ a b, h a.1 b.1\n\nlocal attribute [instance] rbmap_lt_is_swo rbmap_lt_dec\n\n/- Helper lemmas for reusing rbtree results. -/\n\nprivate lemma to_rbtree_mem {k : α} {m : rbmap α β lt} : k ∈ m → ∃ v : β, rbtree.mem (k, v) m :=\nbegin\n  cases m with n p; cases n; intros h,\n  { exact false.elim h },\n  all_goals { existsi n_val.2, exact h }\nend\n\nprivate lemma eqv_entries_of_eqv_keys {k₁ k₂ : α} (v₁ v₂ : β) :\n  k₁ ≈[lt] k₂ → (k₁, v₁) ≈[rbmap_lt lt] (k₂, v₂) :=\nid\n\nprivate lemma eqv_keys_of_eqv_entries {k₁ k₂ : α} {v₁ v₂ : β} :\n  (k₁, v₁) ≈[rbmap_lt lt] (k₂, v₂) → k₁ ≈[lt] k₂ :=\nid\n\nprivate lemma eqv_entries [is_irrefl α lt] (k : α) (v₁ v₂ : β) : (k, v₁) ≈[rbmap_lt lt] (k, v₂) :=\nand.intro (irrefl_of lt k) (irrefl_of lt k)\n\nprivate lemma to_rbmap_mem [is_strict_weak_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  rbtree.mem (k, v) m → k ∈ m :=\nbegin\n  cases m with n p; cases n; intros h,\n  { exact false.elim h },\n  { simp [has_mem.mem, rbmap.mem],\n    exact @rbtree.mem_of_mem_of_eqv _ _ _ ⟨rbnode.red_node n_lchild n_val n_rchild, p⟩ _ _ h\n      (eqv_entries _ _ _) },\n  { simp [has_mem.mem, rbmap.mem],\n    exact @rbtree.mem_of_mem_of_eqv _ _ _ ⟨rbnode.black_node n_lchild n_val n_rchild, p⟩ _ _ h\n      (eqv_entries _ _ _) }\nend\n\nprivate lemma to_rbtree_mem' [is_strict_weak_order α lt] {k : α} {m : rbmap α β lt} (v : β) :\n  k ∈ m → rbtree.mem (k, v) m :=\nbegin\n  intro h,\n  cases to_rbtree_mem h with v' hm,\n  apply rbtree.mem_of_mem_of_eqv hm,\n  apply eqv_entries\nend\n\nlemma eq_some_of_to_value_eq_some {e : option (α × β)} {v : β} :\n  to_value e = some v → ∃ k, e = some (k, v) :=\nbegin\n  cases e with val; simp [to_value, false_implies_iff],\n  { cases val, simp }\nend\n\nlemma eq_none_of_to_value_eq_none {e : option (α × β)} : to_value e = none → e = none :=\nby cases e; simp [to_value, false_implies_iff]\n\n/- Lemmas -/\n\nlemma not_mem_mk_rbmap : ∀ (k : α), k ∉ mk_rbmap α β lt :=\nby simp [has_mem.mem, mk_rbmap, mk_rbtree, rbmap.mem]\n\nlemma not_mem_of_empty {m : rbmap α β lt} (k : α) : m.empty = tt → k ∉ m :=\nby cases m with n p; cases n;\n  simp [has_mem.mem, mk_rbmap, mk_rbtree, rbmap.mem, rbmap.empty, rbtree.empty, false_implies_iff]\n\nlemma mem_of_mem_of_eqv [is_strict_weak_order α lt] {m : rbmap α β lt} {k₁ k₂ : α} :\n  k₁ ∈ m → k₁ ≈[lt] k₂ → k₂ ∈ m :=\nbegin\n  intros h₁ h₂,\n  have h₁ := to_rbtree_mem h₁, cases h₁ with v h₁,\n  exact to_rbmap_mem (rbtree.mem_of_mem_of_eqv h₁ (eqv_entries_of_eqv_keys v v h₂))\nend\n\nsection decidable\n\nvariables [decidable_rel lt]\n\nlemma not_mem_of_find_entry_none [is_strict_weak_order α lt] {k : α} {m : rbmap α β lt} :\n  m.find_entry k = none → k ∉ m :=\nbegin\n  cases m with t p, cases t; simp [find_entry],\n  { intros, simp [has_mem.mem, rbmap.mem] },\n  all_goals { intro h, exact rbtree.not_mem_of_find_none h, }\nend\n\nlemma not_mem_of_find_none [is_strict_weak_order α lt] {k : α} {m : rbmap α β lt} :\n  m.find k = none → k ∉ m :=\nbegin\n  simp [find], intro h,\n  have := eq_none_of_to_value_eq_none h,\n  exact not_mem_of_find_entry_none this\nend\n\nlemma mem_of_find_entry_some [is_strict_weak_order α lt] {k₁ : α} {e : α × β} {m : rbmap α β lt} :\n  m.find_entry k₁ = some e → k₁ ∈ m :=\nbegin\n  cases m with t p, cases t; simp [find_entry, false_implies_iff],\n  all_goals { intro h, exact rbtree.mem_of_find_some h }\nend\n\nlemma mem_of_find_some [is_strict_weak_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.find k = some v → k ∈ m :=\nbegin\n  simp [find], intro h,\n  have := eq_some_of_to_value_eq_some h,\n  cases this with _ he,\n  exact mem_of_find_entry_some he\nend\n\nlemma find_entry_eq_find_entry_of_eqv [is_strict_weak_order α lt] {m : rbmap α β lt} {k₁ k₂ : α} :\n  k₁ ≈[lt] k₂ → m.find_entry k₁ = m.find_entry k₂ :=\nbegin\n  intro h, cases m with t p, cases t; simp [find_entry],\n  all_goals { apply rbtree.find_eq_find_of_eqv, apply eqv_entries_of_eqv_keys, assumption }\nend\n\nlemma find_eq_find_of_eqv [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) :\n  k₁ ≈[lt] k₂ → m.find k₁ = m.find k₂ :=\nbegin intro h, simp [find], apply congr_arg, apply find_entry_eq_find_entry_of_eqv, assumption end\n\nlemma find_entry_correct [is_strict_weak_order α lt] (k : α) (m : rbmap α β lt) :\n  k ∈ m ↔ (∃ e, m.find_entry k = some e ∧ k ≈[lt] e.1) :=\nbegin\n  apply iff.intro; cases m with t p,\n  { intro h,\n    have h   := to_rbtree_mem h, cases h with v h₁,\n    have hex := iff.mp (rbtree.find_correct _ _) h₁, cases hex with e h₂,\n    existsi e, cases t; simp [find_entry] at ⊢ h₂,\n    { simp [rbtree.find, rbnode.find] at h₂, cases h₂ },\n    { cases h₂ with h₂₁ h₂₂, split,\n      { have := rbtree.find_eq_find_of_eqv ⟨rbnode.red_node t_lchild t_val t_rchild, p⟩\n          (eqv_entries k v t_val.2),\n        rw [←this], exact h₂₁ },\n      { cases e, apply eqv_keys_of_eqv_entries h₂₂ } },\n    { cases h₂ with h₂₁ h₂₂, split,\n      { have := rbtree.find_eq_find_of_eqv ⟨rbnode.black_node t_lchild t_val t_rchild, p⟩\n          (eqv_entries k v t_val.2),\n        rw [←this], exact h₂₁ },\n      { cases e, apply eqv_keys_of_eqv_entries h₂₂ } } },\n  { intro h, cases h with e h,\n    cases h with h₁ h₂, cases t; simp [find_entry] at h₁,\n    { contradiction },\n    all_goals { exact to_rbmap_mem (rbtree.mem_of_find_some h₁) } }\nend\n\nlemma eqv_of_find_entry_some [is_strict_weak_order α lt] {k₁ k₂ : α} {v : β} {m : rbmap α β lt} :\n  m.find_entry k₁ = some (k₂, v) → k₁ ≈[lt] k₂ :=\nbegin\n  cases m with t p, cases t; simp [find_entry, false_implies_iff],\n  all_goals { intro h, exact eqv_keys_of_eqv_entries (rbtree.eqv_of_find_some h) }\nend\n\nlemma eq_of_find_entry_some [is_strict_total_order α lt] {k₁ k₂ : α} {v : β} {m : rbmap α β lt} :\n  m.find_entry k₁ = some (k₂, v) → k₁ = k₂ :=\nλ h, suffices k₁ ≈[lt] k₂, from eq_of_eqv_lt this,\n     eqv_of_find_entry_some h\n\nlemma find_correct [is_strict_weak_order α lt] (k : α) (m : rbmap α β lt) :\n  k ∈ m ↔ ∃ v, m.find k = some v :=\nbegin\n  apply iff.intro,\n  { intro h,\n    have := iff.mp (find_entry_correct k m) h,\n    cases this with e h, cases h with h₁ h₂,\n    existsi e.2, simp [find, h₁, to_value] },\n  { intro h,\n    cases h with v h,\n    simp [find] at h,\n    have h := eq_some_of_to_value_eq_some h,\n    cases h with k' h,\n    have heqv := eqv_of_find_entry_some h,\n    exact iff.mpr (find_entry_correct k m) ⟨(k', v), ⟨h, heqv⟩⟩ }\nend\n\nlemma constains_correct [is_strict_weak_order α lt] (k : α) (m : rbmap α β lt) :\n  k ∈ m ↔ m.contains k = tt :=\nbegin\n  apply iff.intro,\n  { intro h,\n    have h := iff.mp (find_entry_correct k m) h,\n    cases h with e h, cases h with h₁ h₂,\n    simp [contains, h₁, option.is_some] },\n  { simp [contains],\n    intro h,\n    generalize he : find_entry m k = e,\n    cases e,\n    { simp [he, option.is_some] at h, contradiction },\n    { exact mem_of_find_entry_some he } }\nend\n\nlemma mem_insert_of_incomp [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  (¬ lt k₁ k₂ ∧ ¬ lt k₂ k₁) → k₁ ∈ m.insert k₂ v :=\nλ h, to_rbmap_mem (rbtree.mem_insert_of_incomp m (eqv_entries_of_eqv_keys v v h))\n\nlemma mem_insert [is_strict_weak_order α lt] (k : α) (m : rbmap α β lt) (v : β) :\n  k ∈ m.insert k v :=\nto_rbmap_mem (rbtree.mem_insert (k, v) m)\n\nlemma mem_insert_of_equiv [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  k₁ ≈[lt] k₂ → k₁ ∈ m.insert k₂ v :=\nmem_insert_of_incomp m v\n\nlemma mem_insert_of_mem [is_strict_weak_order α lt] {k₁ : α} {m : rbmap α β lt} (k₂ : α) (v : β) :\n  k₁ ∈ m → k₁ ∈ m.insert k₂ v :=\nλ h, to_rbmap_mem (rbtree.mem_insert_of_mem (k₂, v) (to_rbtree_mem' v h))\n\nlemma equiv_or_mem_of_mem_insert [is_strict_weak_order α lt] {k₁ k₂ : α} {v : β}\n  {m : rbmap α β lt} : k₁ ∈ m.insert k₂ v → k₁ ≈[lt] k₂ ∨ k₁ ∈ m :=\nλ h, or.elim (rbtree.equiv_or_mem_of_mem_insert (to_rbtree_mem' v h))\n  (λ h, or.inl (eqv_keys_of_eqv_entries h))\n  (λ h, or.inr (to_rbmap_mem h))\n\nlemma incomp_or_mem_of_mem_ins [is_strict_weak_order α lt] {k₁ k₂ : α} {v : β} {m : rbmap α β lt} :\n  k₁ ∈ m.insert k₂ v → (¬ lt k₁ k₂ ∧ ¬ lt k₂ k₁) ∨ k₁ ∈ m :=\nequiv_or_mem_of_mem_insert\n\nlemma eq_or_mem_of_mem_ins [is_strict_total_order α lt] {k₁ k₂ : α} {v : β} {m : rbmap α β lt} :\n  k₁ ∈ m.insert k₂ v → k₁ = k₂ ∨ k₁ ∈ m :=\nλ h, suffices k₁ ≈[lt] k₂ ∨ k₁ ∈ m, by simp [eqv_lt_iff_eq] at this; assumption,\n  incomp_or_mem_of_mem_ins h\n\nlemma find_entry_insert_of_eqv [is_strict_weak_order α lt] (m : rbmap α β lt) {k₁ k₂ : α} (v : β) :\n  k₁ ≈[lt] k₂ → (m.insert k₁ v).find_entry k₂ = some (k₁, v) :=\nbegin\n  intro h,\n  generalize h₁ : m.insert k₁ v = m',\n  cases m' with t p, cases t,\n  { have := mem_insert k₁ m v, rw [h₁] at this, apply absurd this, apply not_mem_mk_rbmap },\n  all_goals { simp [find_entry], rw [←h₁, insert], apply rbtree.find_insert_of_eqv,\n    apply eqv_entries_of_eqv_keys _ _ h }\nend\n\nlemma find_entry_insert [is_strict_weak_order α lt] (m : rbmap α β lt) (k : α) (v : β) :\n  (m.insert k v).find_entry k = some (k, v) :=\nfind_entry_insert_of_eqv m v (refl k)\n\nlemma find_insert_of_eqv [is_strict_weak_order α lt] (m : rbmap α β lt) {k₁ k₂ : α} (v : β) :\n  k₁ ≈[lt] k₂ → (m.insert k₁ v).find k₂ = some v :=\nbegin\n  intro h,\n  have := find_entry_insert_of_eqv m v h,\n  simp [find, this, to_value]\nend\n\nlemma find_insert [is_strict_weak_order α lt] (m : rbmap α β lt) (k : α) (v : β) :\n  (m.insert k v).find k = some v :=\nfind_insert_of_eqv m v (refl k)\n\nlemma find_entry_insert_of_disj [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  lt k₁ k₂ ∨ lt k₂ k₁ → (m.insert k₁ v).find_entry k₂ = m.find_entry k₂ :=\nbegin\n  intro h,\n  have h' : ∀ {v₁ v₂ : β}, (rbmap_lt lt) (k₁, v₁) (k₂, v₂) ∨ (rbmap_lt lt) (k₂, v₂) (k₁, v₁) :=\n    λ _ _, h,\n  generalize h₁ : m = m₁,\n  generalize h₂ : insert m₁ k₁ v = m₂,\n  rw [←h₁] at h₂ ⊢, rw [←h₂],\n  cases m₁ with t₁ p₁; cases t₁; cases m₂ with t₂ p₂; cases t₂,\n  { rw [h₂, h₁] },\n  iterate 2\n  { rw [h₂],\n    conv { to_lhs, simp [find_entry] },\n    rw [←h₂, insert, rbtree.find_insert_of_disj _ h', h₁],\n    refl },\n  any_goals { simp [insert] at h₂,\n    exact absurd h₂ (rbtree.insert_ne_mk_rbtree m (k₁, v)) },\n  any_goals\n  { rw [h₂, h₁], simp [find_entry], rw [←h₂, ←h₁, insert, rbtree.find_insert_of_disj _ h'],\n    apply rbtree.find_eq_find_of_eqv, apply eqv_entries }\nend\n\nlemma find_entry_insert_of_not_eqv [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt)\n  (v : β) : ¬ k₁ ≈[lt] k₂ → (m.insert k₁ v).find_entry k₂ = m.find_entry k₂ :=\nbegin\n  intro hn,\n  have he : lt k₁ k₂ ∨ lt k₂ k₁,\n  { simp [strict_weak_order.equiv, decidable.not_and_iff_or_not, decidable.not_not_iff] at hn,\n    assumption },\n  apply find_entry_insert_of_disj _ _ he\nend\n\nlemma find_entry_insert_of_ne [is_strict_total_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  k₁ ≠ k₂ → (m.insert k₁ v).find_entry k₂ = m.find_entry k₂ :=\nbegin\n  intro h,\n  have : ¬ k₁ ≈[lt] k₂ := λ h', h (eq_of_eqv_lt h'),\n  apply find_entry_insert_of_not_eqv _ _ this\nend\n\nlemma find_insert_of_disj [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  lt k₁ k₂ ∨ lt k₂ k₁ → (m.insert k₁ v).find k₂ = m.find k₂ :=\nbegin intro h, have := find_entry_insert_of_disj m v h, simp [find, this] end\n\nlemma find_insert_of_not_eqv [is_strict_weak_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  ¬ k₁ ≈[lt] k₂ → (m.insert k₁ v).find k₂ = m.find k₂ :=\nbegin intro h, have := find_entry_insert_of_not_eqv m v h, simp [find, this] end\n\nlemma find_insert_of_ne [is_strict_total_order α lt] {k₁ k₂ : α} (m : rbmap α β lt) (v : β) :\n  k₁ ≠ k₂ → (m.insert k₁ v).find k₂ = m.find k₂ :=\nbegin intro h, have := find_entry_insert_of_ne m v h, simp [find, this] end\n\nend decidable\n\nlemma mem_of_min_eq [is_strict_total_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.min = some (k, v) → k ∈ m :=\nλ h, to_rbmap_mem (rbtree.mem_of_min_eq h)\n\nlemma mem_of_max_eq [is_strict_total_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.max = some (k, v) → k ∈ m :=\nλ h, to_rbmap_mem (rbtree.mem_of_max_eq h)\n\nlemma eq_leaf_of_min_eq_none {m : rbmap α β lt} :\n  m.min = none → m = mk_rbmap α β lt :=\nrbtree.eq_leaf_of_min_eq_none\n\n\n\nlemma min_is_minimal [is_strict_weak_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.min = some (k, v) → ∀ {k'}, k' ∈ m → k ≈[lt] k' ∨ lt k k' :=\nλ h k' hm, or.elim (rbtree.min_is_minimal h (to_rbtree_mem' v hm))\n  (λ h, or.inl (eqv_keys_of_eqv_entries h))\n  (λ h, or.inr h)\n\nlemma max_is_maximal [is_strict_weak_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.max = some (k, v) → ∀ {k'}, k' ∈ m → k ≈[lt] k' ∨ lt k' k :=\nλ h k' hm, or.elim (rbtree.max_is_maximal h (to_rbtree_mem' v hm))\n  (λ h, or.inl (eqv_keys_of_eqv_entries h))\n  (λ h, or.inr h)\n\nlemma min_is_minimal_of_total [is_strict_total_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.min = some (k, v) → ∀ {k'}, k' ∈ m → k = k' ∨ lt k k' :=\nλ h k' hm,\n  match min_is_minimal h hm with\n  | or.inl h := or.inl (eq_of_eqv_lt h)\n  | or.inr h := or.inr h\n  end\n\nlemma max_is_maximal_of_total [is_strict_total_order α lt] {k : α} {v : β} {m : rbmap α β lt} :\n  m.max = some (k, v) → ∀ {k'}, k' ∈ m → k = k' ∨ lt k' k :=\nλ h k' hm,\n  match max_is_maximal h hm with\n  | or.inl h := or.inl (eq_of_eqv_lt h)\n  | or.inr h := or.inr h\n  end\n\nend rbmap\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/rbmap/default.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749126, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.33565988981256606}}
{"text": "\n\nunsafe def f (n : Nat) :=\nlet rec g (x : Nat) :=\nif x > 0 then\n  /- This example relies on the instance `coeId {α} (a : α) : CoeT α a α`,\n     The type of `g` is not provided. Thus, the expected type is\n     `?m x` since the elaborator assumes it may depend on `x`.\n     Thus, the expected type for the addition is `?m x`.\n     The type of `g (x-1)` is `?m (x-1)`. So, we have the unification\n     problem\n     ```\n     ?m (x-1) =?= ?m x\n     ```\n     and it is rejected by the unifier. Thus, the elaborator inserts\n     a coercion from `?m (x-1)` to `?m x` around `g (x-1)` which creates\n     the pending TC problem:\n     ```\n     CoeT (?m (x-1)) (g (x-1)) (?m x)\n     ```\n\n     Remark: in principle, we could solve it using a heuristic: assume\n     `?m` is a constant function.\n     Later, when we elaborate the else branch `0`, we have that\n     ```\n     ?m x =?= Nat\n     ```\n     which is solved using `?m := (fun _ => Nat)`.\n     Then, `@coeId Nat (g (x-1))` is used to synthesize\n     ```\n     CoeT (?m (x-1)) (g (x-1)) (?m x)\n     ```\n     which is\n     ```\n     CoeT Nat (g (x-1)) Nat\n     ```\n     after we apply the assignment `?m := (fun _ => Nat)`\n     -/\n  g (x-1) + n\nelse\n  0;\ng n\n\n#eval f 10\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/implicitTypesRecCoe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.33560889143153294}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport data.fintype.order\nimport data.set.finite\nimport order.category.LinOrd\nimport category_theory.limits.shapes.images\nimport category_theory.limits.shapes.regular_mono\n\n/-!\n# Nonempty finite linear orders\n\nThis defines `NonemptyFinLinOrd`, the category of nonempty finite linear orders with monotone maps.\nThis is the index category for simplicial objects.\n-/\n\nuniverses u v\n\nopen category_theory category_theory.limits\n\n/-- A typeclass for nonempty finite linear orders. -/\nclass nonempty_fin_lin_ord (α : Type*) extends fintype α, linear_order α :=\n(nonempty : nonempty α . tactic.apply_instance)\n\nattribute [instance] nonempty_fin_lin_ord.nonempty\n\n@[priority 100]\ninstance nonempty_fin_lin_ord.to_bounded_order (α : Type*) [nonempty_fin_lin_ord α] :\n  bounded_order α :=\nfintype.to_bounded_order α\n\ninstance punit.nonempty_fin_lin_ord : nonempty_fin_lin_ord punit := { }\n\ninstance fin.nonempty_fin_lin_ord (n : ℕ) : nonempty_fin_lin_ord (fin (n+1)) := { }\n\ninstance ulift.nonempty_fin_lin_ord (α : Type u) [nonempty_fin_lin_ord α] :\n  nonempty_fin_lin_ord (ulift.{v} α) :=\n{ .. linear_order.lift' equiv.ulift (equiv.injective _) }\n\ninstance (α : Type*) [nonempty_fin_lin_ord α] : nonempty_fin_lin_ord αᵒᵈ :=\n{ ..order_dual.fintype α }\n\n/-- The category of nonempty finite linear orders. -/\ndef NonemptyFinLinOrd := bundled nonempty_fin_lin_ord\n\nnamespace NonemptyFinLinOrd\n\ninstance : bundled_hom.parent_projection @nonempty_fin_lin_ord.to_linear_order := ⟨⟩\n\nattribute [derive [large_category, concrete_category]] NonemptyFinLinOrd\n\ninstance : has_coe_to_sort NonemptyFinLinOrd Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled `NonemptyFinLinOrd` from the underlying type and typeclass. -/\ndef of (α : Type*) [nonempty_fin_lin_ord α] : NonemptyFinLinOrd := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [nonempty_fin_lin_ord α] : ↥(of α) = α := rfl\n\ninstance : inhabited NonemptyFinLinOrd := ⟨of punit⟩\n\ninstance (α : NonemptyFinLinOrd) : nonempty_fin_lin_ord α := α.str\n\ninstance has_forget_to_LinOrd : has_forget₂ NonemptyFinLinOrd LinOrd :=\nbundled_hom.forget₂ _ _\n\n/-- Constructs an equivalence between nonempty finite linear orders from an order isomorphism\nbetween them. -/\n@[simps] def iso.mk {α β : NonemptyFinLinOrd.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : NonemptyFinLinOrd ⥤ NonemptyFinLinOrd :=\n{ obj := λ X, of Xᵒᵈ, map := λ X Y, order_hom.dual }\n\n/-- The equivalence between `FinPartOrd` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : NonemptyFinLinOrd ≌ NonemptyFinLinOrd :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nlemma mono_iff_injective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :\n  mono f ↔ function.injective f :=\nbegin\n  refine ⟨_, concrete_category.mono_of_injective f⟩,\n  introI,\n  intros a₁ a₂ h,\n  let X : NonemptyFinLinOrd.{u} := ⟨ulift (fin 1)⟩,\n  let g₁ : X ⟶ A := ⟨λ x, a₁, λ x₁ x₂ h, by refl⟩,\n  let g₂ : X ⟶ A := ⟨λ x, a₂, λ x₁ x₂ h, by refl⟩,\n  change g₁ (ulift.up (0 : fin 1)) = g₂ (ulift.up (0 : fin 1)),\n  have eq : g₁ ≫ f = g₂ ≫ f := by { ext x, exact h, },\n  rw cancel_mono at eq,\n  rw eq,\nend\n\nlemma epi_iff_surjective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :\n  epi f ↔ function.surjective f :=\nbegin\n  split,\n  { introI,\n    by_contra' hf',\n    rcases hf' with ⟨m, hm⟩,\n    let Y : NonemptyFinLinOrd.{u} := ⟨ulift (fin 2)⟩,\n    let p₁ : B ⟶ Y := ⟨λ b, if b < m then ulift.up 0 else ulift.up 1, λ x₁ x₂ h, begin\n      simp only,\n      split_ifs with h₁ h₂ h₂,\n      any_goals { apply fin.zero_le, },\n      { exfalso,\n        exact h₁ (lt_of_le_of_lt h h₂), },\n      { refl, },\n    end⟩,\n    let p₂ : B ⟶ Y := ⟨λ b, if b ≤ m then ulift.up 0 else ulift.up 1, λ x₁ x₂ h, begin\n      simp only,\n      split_ifs with h₁ h₂ h₂,\n      any_goals { apply fin.zero_le, },\n      { exfalso,\n        exact h₁ (h.trans h₂), },\n      { refl, },\n    end⟩,\n    have h : p₁ m = p₂ m,\n    { congr,\n      rw ← cancel_epi f,\n      ext a : 2,\n      simp only [comp_apply, order_hom.coe_fun_mk],\n      split_ifs with h₁ h₂ h₂,\n      any_goals { refl, },\n      { exfalso, exact h₂ (le_of_lt h₁), },\n      { exfalso, exact hm a (eq_of_le_of_not_lt h₂ h₁), }, },\n    simpa only [order_hom.coe_fun_mk, lt_self_iff_false, if_false, le_refl, if_true,\n      ulift.up_inj, fin.one_eq_zero_iff, nat.succ_succ_ne_one] using h, },\n  { intro h,\n    exact concrete_category.epi_of_surjective f h, },\nend\n\ninstance : split_epi_category NonemptyFinLinOrd.{u} :=\n⟨λ X Y f hf, begin\n  have H : ∀ (y : Y), nonempty (f⁻¹' { y }),\n  { rw epi_iff_surjective at hf,\n    intro y,\n    exact nonempty.intro ⟨(hf y).some, (hf y).some_spec⟩, },\n  let φ : Y → X := λ y, (H y).some.1,\n  have hφ : ∀ (y : Y), f (φ y) = y := λ y, (H y).some.2,\n  refine is_split_epi.mk' ⟨⟨φ, _⟩, _⟩, swap,\n  { ext b,\n    apply hφ, },\n  { intros a b,\n    contrapose,\n    intro h,\n    simp only [not_le] at h ⊢,\n    suffices : b ≤ a,\n    { apply lt_of_le_of_ne this,\n      intro h',\n      exfalso,\n      simpa only [h', lt_self_iff_false] using h, },\n    simpa only [hφ] using f.monotone (le_of_lt h), },\nend⟩\n\ninstance : has_strong_epi_mono_factorisations NonemptyFinLinOrd.{u} :=\n⟨λ X Y f, begin\n  let I : NonemptyFinLinOrd.{u} := ⟨set.image (coe_fn f) ⊤, ⟨⟩⟩,\n  let e : X ⟶ I := ⟨λ x, ⟨f x, ⟨x, by tidy⟩⟩, λ x₁ x₂ h, f.monotone h⟩,\n  let m : I ⟶ Y := ⟨λ y, y, by tidy⟩,\n  haveI : epi e := by { rw epi_iff_surjective, tidy, },\n  haveI : strong_epi e := strong_epi_of_epi e,\n  haveI : mono m := concrete_category.mono_of_injective _ (by tidy),\n  exact nonempty.intro\n  { I := I,\n    m := m,\n    e := e, },\nend⟩\n\nend NonemptyFinLinOrd\n\nlemma NonemptyFinLinOrd_dual_comp_forget_to_LinOrd :\n  NonemptyFinLinOrd.dual ⋙ forget₂ NonemptyFinLinOrd LinOrd =\n    forget₂ NonemptyFinLinOrd LinOrd ⋙ LinOrd.dual := rfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/order/category/NonemptyFinLinOrd.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.33558003255702734}}
{"text": "macro \"ext_tactic\" t:tactic \"=>\" newT:tactic : command => `(macro_rules | `($t) => `($newT))\n\nsyntax \"trivial'\" : tactic\n\next_tactic trivial' => apply Eq.refl\n\ntheorem tst1 (x : Nat) : x = x :=\nby trivial'\n\n-- theorem tst2 (x y : Nat) (h : x = y) : x = y :=\n-- by trivial' -- fail as expected\n\next_tactic trivial' => assumption\n\ntheorem tst1b (x : Nat) : x = x :=\nby trivial' -- still works\n\ntheorem tst2 (x y : Nat) (h : x = y) : x = y :=\nby trivial' -- works too\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/extmacro.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3355800239531647}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.functor\nimport tactic.ext\n\n/-!\n# Traversable type class\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nType classes for traversing collections. The concepts and laws are taken from\n<http://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html>\n\nTraversable collections are a generalization of functors. Whereas\nfunctors (such as `list`) allow us to apply a function to every\nelement, it does not allow functions which external effects encoded in\na monad. Consider for instance a functor `invite : email → io response`\nthat takes an email address, sends an email and waits for a\nresponse. If we have a list `guests : list email`, using calling\n`invite` using `map` gives us the following: `map invite guests : list\n(io response)`.  It is not what we need. We need something of type `io\n(list response)`. Instead of using `map`, we can use `traverse` to\nsend all the invites: `traverse invite guests : io (list response)`.\n`traverse` applies `invite` to every element of `guests` and combines\nall the resulting effects. In the example, the effect is encoded in the\nmonad `io` but any applicative functor is accepted by `traverse`.\n\nFor more on how to use traversable, consider the Haskell tutorial:\n<https://en.wikibooks.org/wiki/Haskell/Traversable>\n\n## Main definitions\n  * `traversable` type class - exposes the `traverse` function\n  * `sequence` - based on `traverse`,\n    turns a collection of effects into an effect returning a collection\n  * `is_lawful_traversable` - laws for a traversable functor\n  * `applicative_transformation` - the notion of a natural transformation for applicative functors\n\n## Tags\n\ntraversable iterator functor applicative\n\n## References\n\n * \"Applicative Programming with Effects\", by Conor McBride and Ross Paterson,\n   Journal of Functional Programming 18:1 (2008) 1-13, online at\n   <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>\n * \"The Essence of the Iterator Pattern\", by Jeremy Gibbons and Bruno Oliveira,\n   in Mathematically-Structured Functional Programming, 2006, online at\n   <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>\n * \"An Investigation of the Laws of Traversals\", by Mauro Jaskelioff and Ondrej Rypacek,\n   in Mathematically-Structured Functional Programming, 2012,\n   online at <http://arxiv.org/pdf/1202.2919>\n-/\n\nopen function (hiding comp)\n\nuniverses u v w\n\nsection applicative_transformation\n\nvariables (F : Type u → Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u → Type w) [applicative G] [is_lawful_applicative G]\n\n/-- A transformation between applicative functors.  It is a natural\ntransformation such that `app` preserves the `has_pure.pure` and\n`functor.map` (`<*>`) operations. See\n`applicative_transformation.preserves_map` for naturality. -/\nstructure applicative_transformation : Type (max (u+1) v w) :=\n(app : Π α : Type u, F α → G α)\n(preserves_pure' : ∀ {α : Type u} (x : α), app _ (pure x) = pure x)\n(preserves_seq' : ∀ {α β : Type u} (x : F (α → β)) (y : F α), app _ (x <*> y) = app _ x <*> app _ y)\n\nend applicative_transformation\n\nnamespace applicative_transformation\n\nvariables (F : Type u → Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u → Type w) [applicative G] [is_lawful_applicative G]\n\ninstance : has_coe_to_fun (applicative_transformation F G) (λ _, Π {α}, F α → G α) :=\n⟨applicative_transformation.app⟩\n\nvariables {F G}\n\n@[simp]\nlemma app_eq_coe (η : applicative_transformation F G) : η.app = η := rfl\n\n@[simp]\nlemma coe_mk (f : Π (α : Type u), F α → G α) (pp ps) :\n  ⇑(applicative_transformation.mk f pp ps) = f := rfl\n\nprotected\nlemma congr_fun (η η' : applicative_transformation F G) (h : η = η') {α : Type u} (x : F α) :\n  η x = η' x :=\ncongr_arg (λ η'' : applicative_transformation F G, η'' x) h\n\nprotected\nlemma congr_arg (η : applicative_transformation F G) {α : Type u} {x y : F α} (h : x = y) :\n  η x = η y :=\ncongr_arg (λ z : F α, η z) h\n\nlemma coe_inj ⦃η η' : applicative_transformation F G⦄ (h : (η : Π α, F α → G α) = η') : η = η' :=\nby { cases η, cases η', congr, exact h }\n\n@[ext]\nlemma ext ⦃η η' : applicative_transformation F G⦄ (h : ∀ (α : Type u) (x : F α), η x = η' x) :\n  η = η' :=\nby { apply coe_inj, ext1 α, exact funext (h α) }\n\nlemma ext_iff {η η' : applicative_transformation F G} :\n  η = η' ↔ ∀ (α : Type u) (x : F α), η x = η' x :=\n⟨λ h α x, h ▸ rfl, λ h, ext h⟩\n\nsection preserves\nvariables (η : applicative_transformation F G)\n\n@[functor_norm]\nlemma preserves_pure {α} : ∀ (x : α), η (pure x) = pure x := η.preserves_pure'\n\n@[functor_norm]\nlemma preserves_seq {α β : Type u} : ∀ (x : F (α → β)) (y : F α), η (x <*> y) = η x <*> η y :=\nη.preserves_seq'\n\n@[functor_norm]\nlemma preserves_map {α β} (x : α → β) (y : F α) : η (x <$> y) = x <$> η y :=\nby rw [← pure_seq_eq_map, η.preserves_seq]; simp with functor_norm\n\nlemma preserves_map' {α β} (x : α → β) : @η _ ∘ functor.map x = functor.map x ∘ @η _ :=\nby { ext y, exact preserves_map η x y }\n\nend preserves\n\n/-- The identity applicative transformation from an applicative functor to itself. -/\ndef id_transformation : applicative_transformation F F :=\n{ app := λ α, id,\n  preserves_pure' := by simp,\n  preserves_seq' := λ α β x y, by simp }\n\ninstance : inhabited (applicative_transformation F F) := ⟨id_transformation⟩\n\nuniverses s t\nvariables {H : Type u → Type s} [applicative H] [is_lawful_applicative H]\n\n/-- The composition of applicative transformations. -/\ndef comp (η' : applicative_transformation G H) (η : applicative_transformation F G) :\n  applicative_transformation F H :=\n{ app := λ α x, η' (η x),\n  preserves_pure' := λ α x, by simp with functor_norm,\n  preserves_seq' := λ α β x y, by simp with functor_norm }\n\n@[simp]\nlemma comp_apply (η' : applicative_transformation G H) (η : applicative_transformation F G)\n  {α : Type u} (x : F α) :\n  η'.comp η x = η' (η x) := rfl\n\nlemma comp_assoc {I : Type u → Type t} [applicative I] [is_lawful_applicative I]\n  (η'' : applicative_transformation H I)\n  (η' : applicative_transformation G H)\n  (η : applicative_transformation F G) :\n  (η''.comp η').comp η = η''.comp (η'.comp η) := rfl\n\n@[simp]\nlemma comp_id (η : applicative_transformation F G) : η.comp id_transformation = η :=\next $ λ α x, rfl\n\n@[simp]\n\n\nend applicative_transformation\n\nopen applicative_transformation\n\n/-- A traversable functor is a functor along with a way to commute\nwith all applicative functors (see `sequence`).  For example, if `t`\nis the traversable functor `list` and `m` is the applicative functor\n`io`, then given a function `f : α → io β`, the function `functor.map f` is\n`list α → list (io β)`, but `traverse f` is `list α → io (list β)`. -/\nclass traversable (t : Type u → Type u) extends functor t :=\n(traverse : Π {m : Type u → Type u} [applicative m] {α β},\n   (α → m β) → t α → m (t β))\n\nopen functor\n\nexport traversable (traverse)\n\nsection functions\n\nvariables {t : Type u → Type u}\nvariables {m : Type u → Type v} [applicative m]\nvariables {α β : Type u}\n\n\nvariables {f : Type u → Type u} [applicative f]\n\n/-- A traversable functor commutes with all applicative functors. -/\ndef sequence [traversable t] : t (f α) → f (t α) := traverse id\n\nend functions\n\n/-- A traversable functor is lawful if its `traverse` satisfies a\nnumber of additional properties.  It must send `id.mk` to `id.mk`,\nsend the composition of applicative functors to the composition of the\n`traverse` of each, send each function `f` to `λ x, f <$> x`, and\nsatisfy a naturality condition with respect to applicative\ntransformations. -/\nclass is_lawful_traversable (t : Type u → Type u) [traversable t]\n  extends is_lawful_functor t : Type (u+1) :=\n(id_traverse : ∀ {α} (x : t α), traverse id.mk x = x )\n(comp_traverse : ∀ {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {α β γ} (f : β → F γ) (g : α → G β) (x : t α),\n  traverse (comp.mk ∘ map f ∘ g) x =\n  comp.mk (map (traverse f) (traverse g x)))\n(traverse_eq_map_id : ∀ {α β} (f : α → β) (x : t α),\n  traverse (id.mk ∘ f) x = id.mk (f <$> x))\n(naturality : ∀ {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (η : applicative_transformation F G) {α β} (f : α → F β) (x : t α),\n  η (traverse f x) = traverse (@η _ ∘ f) x)\n\ninstance : traversable id := ⟨λ _ _ _ _, id⟩\ninstance : is_lawful_traversable id := by refine {..}; intros; refl\n\nsection\n\nvariables {F : Type u → Type v} [applicative F]\n\ninstance : traversable option := ⟨@option.traverse⟩\n\ninstance : traversable list := ⟨@list.traverse⟩\n\nend\n\nnamespace sum\n\nvariables {σ : Type u}\nvariables {F : Type u → Type u}\nvariables [applicative F]\n\n/-- Defines a `traverse` function on the second component of a sum type.\nThis is used to give a `traversable` instance for the functor `σ ⊕ -`. -/\nprotected def traverse {α β} (f : α → F β) : σ ⊕ α → F (σ ⊕ β)\n| (sum.inl x) := pure (sum.inl x)\n| (sum.inr x) := sum.inr <$> f x\n\nend sum\n\ninstance {σ : Type u} : traversable.{u} (sum σ) := ⟨@sum.traverse _⟩\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/control/traversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3355800239531647}}
{"text": "-- Copyright © 2019 François G. Dorais. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n\nimport .basic\nimport .substitution\n\nnamespace universal\nvariables {τ : Type} {σ : Type*} {sig : signature τ σ}\n\nstructure homomorphism (alg₁ : algebra sig) (alg₂ : algebra sig) :=\n(map {} (t) : alg₁.sort t → alg₂.sort t)\n(func (f : σ) : ∀ (xs : Π (i : sig.index f), alg₁.sort i.val), alg₂.func f (λ i, map _ (xs i)) = map _ (alg₁.func f xs))\n\ninstance (alg₁ : algebra sig) (alg₂ : algebra sig) : has_coe_to_fun (homomorphism alg₁ alg₂) :=\n{ F := λ _, Π {t}, alg₁.sort t → alg₂.sort t\n, coe := λ h, h.map\n}\n\nnamespace homomorphism\nvariables {alg₁ : algebra sig} {alg₂ : algebra sig} {alg₃ : algebra sig}\n\ntheorem eq : Π {{h₁ h₂ : homomorphism alg₁ alg₂}}, h₁.map = h₂.map → h₁ = h₂\n| ⟨_,_⟩ ⟨_,_⟩ rfl := rfl\n\ntheorem ext {{h₁ h₂ : homomorphism alg₁ alg₂}} : (∀ {t} (x : alg₁.sort t), h₁.map t x = h₂.map t x) → h₁ = h₂ :=\nλ H, eq $ funext $ λ _, funext $ λ x, H x\n\ndefinition id (alg : algebra sig) : homomorphism alg alg :=\n{ map := λ _, id\n, func := λ _ _, rfl\n}\n\ndefinition comp (h₂₃ : homomorphism alg₂ alg₃) (h₁₂ : homomorphism alg₁ alg₂) : homomorphism alg₁ alg₃ :=\n{ map := λ t x, h₂₃.map t (h₁₂.map t x)\n, func := λ f xs, eq.subst (h₁₂.func f xs) (h₂₃.func f _)\n}\n\nclass injective (h : homomorphism alg₁ alg₂) : Prop := intro ::\n(elim [] (t) : function.injective (homomorphism.map h t))\n\nclass surjective (h : homomorphism alg₁ alg₂) : Prop := intro ::\n(elim [] (t) : function.surjective (h.map t))\n\ninstance comp.inj (h₂₃ : homomorphism alg₂ alg₃) (h₁₂ : homomorphism alg₁ alg₂)\n[injective h₂₃] [injective h₁₂] : injective (comp h₂₃ h₁₂) :=\n⟨λ t, function.injective.comp (injective.elim h₂₃ t) (injective.elim h₁₂ t)⟩\n\ninstance comp.surj (h₂₃ : homomorphism alg₂ alg₃) (h₁₂ : homomorphism alg₁ alg₂)\n[surjective h₂₃] [surjective h₁₂] : surjective (comp h₂₃ h₁₂) :=\n⟨λ t, function.surjective.comp (surjective.elim h₂₃ t) (surjective.elim h₁₂ t)⟩\n\n@[priority 0]\ninstance id.inj (alg : algebra sig) : injective (id alg) := ⟨λ _, function.injective_id⟩\n\n@[priority 0]\ninstance id.surj (alg : algebra sig) : surjective (id alg) := ⟨λ _, function.surjective_id⟩\n\ntheorem eval (h : homomorphism alg₁ alg₂) {dom} :\n∀ {cod} (t : term sig dom cod) (val : Π (i : index dom), alg₁.sort i.val),\nh.map _ (alg₁.eval t val) = alg₂.eval t (λ i, h.map _ (val i))\n| _ (term.proj i) val := rfl\n| _ (term.func f ts) val :=\n  have IH : (λ i, h.map _ (alg₁.eval (ts i) val)) = (λ i, alg₂.eval (ts i) (λ i, h.map _ (val i))),\n  from funext $ λ i, eval (ts i) val,\n  calc h.map _ (alg₁.eval (term.func f ts) val)\n  = h.map _ (alg₁.func f (λ i, alg₁.eval (ts i) val)) : rfl ...\n  = alg₂.func f (λ i, h.map _ (alg₁.eval (ts i) val)) : by rw h.func ...\n  = alg₂.func f (λ i, alg₂.eval (ts i) (λ i, h.map _ (val i))) : by rw IH ...\n  = alg₂.eval (term.func f ts) (λ i, h.map _ (val i)) : by reflexivity\n\nend homomorphism\n\nnamespace algebra.valuation\nvariables {alg₁ : algebra sig} {alg₂ : algebra sig} (h : homomorphism alg₁ alg₂)\n\ndefinition map {dom} (val : algebra.valuation alg₁ dom) : algebra.valuation alg₂ dom := λ i, h (val i)\n\ntheorem map_eval {dom} (val : algebra.valuation alg₁ dom) :\n∀ {cod} (x : term sig dom cod), alg₂.eval x (val.map h) = h.map cod (alg₁.eval x val)\n| _ (term.proj _) := rfl\n| _ (term.func f xs) :=\n  have (λ i, alg₂.eval (xs i) (val.map h)) = (λ i, h.map _ (alg₁.eval (xs i) val)),\n  from funext $ λ i, map_eval (xs i),\n  calc _\n  = alg₂.func f (λ i, alg₂.eval (xs i) (val.map h)) : rfl ...\n  = alg₂.func f (λ i, h.map _ (alg₁.eval (xs i) val)) : by rw this ...\n  = h.map _ (alg₁.func f (λ i, alg₁.eval (xs i) val)) : by rw h.func f ...\n  = h.map _ (alg₁.eval (term.func f xs) val) : by rw alg₁.eval_func\n\nend algebra.valuation\n\ndefinition algebra.eval_hom (alg : algebra sig) {dom : list τ} (val : Π (i : index dom), alg.sort i.val) :\nhomomorphism (term_algebra sig dom) alg :=\n{ map := λ (cod) (t : term sig dom cod), alg.eval t val\n, func := λ _ _, rfl\n}\n\ndefinition term.subst_hom {dom₁ dom₂ : list τ} (sub : substitution sig dom₁ dom₂) :\nhomomorphism (term_algebra sig dom₁) (term_algebra sig dom₂) :=\n{ map := term.subst sub\n, func := λ _ _, rfl\n}\n\nend universal\n", "meta": {"author": "fgdorais", "repo": "lean-universal", "sha": "9259b0f7fb3aa83a9e0a7a3eaa44c262e42cc9b1", "save_path": "github-repos/lean/fgdorais-lean-universal", "path": "github-repos/lean/fgdorais-lean-universal/lean-universal-9259b0f7fb3aa83a9e0a7a3eaa44c262e42cc9b1/src/universal/homomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3355800239531647}}
{"text": "/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.model_category\nimport for_mathlib.category_theory.preadditive.mono_with_projective_coker\nimport algebra.homology.quasi_iso\nimport for_mathlib.algebra.homology.derived_category_plus\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category\nopen algebraic_topology cochain_complex.hom_complex category_theory.preadditive\n\nopen_locale zero_object\n\nvariables (C : Type*) [category C] [abelian C]\n\nnamespace cochain_complex\n\nnamespace minus\n\nnamespace projective_model_structure\n\nvariable {C}\n\ndef weq : morphism_property (cochain_complex.minus C) :=\n(quasi_isomorphisms C (complex_shape.up ℤ)).inverse_image cochain_complex.minus.ι\n\nlemma mem_weq_iff {X Y : cochain_complex.minus C} (f : X ⟶ Y) :\n  weq f ↔ quasi_iso (ι.map f) :=\n⟨λ h, ⟨λ n, h n⟩, λ h, h.is_iso⟩\n\ndef cof : morphism_property (cochain_complex.minus C) :=\nλ X Y f, ∀ n, mono_with_projective_coker C (f.f n)\n\nlemma cof_is_stable_under_composition :\n  (cof : morphism_property (cochain_complex.minus C)).stable_under_composition :=\nλ X Y Z f g hf hg n,\nmono_with_projective_coker.is_stable_by_composition _ _ _ (hf _) (hg _)\n\ndef fib : morphism_property (cochain_complex.minus C) :=\nλ X Y f, ∀ (n : ℤ), epi (f.f n)\n\nvariable (C)\n\n@[simps]\ndef arrow_classes :\n  category_with_fib_cof_weq (cochain_complex.minus C) :=\n{ weq := weq,\n  fib := fib,\n  cof := cof, }\n\nend projective_model_structure\n\nend minus\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cochain_complex/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.335539550735821}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.adjunction.fully_faithful\nimport category_theory.reflects_isomorphisms\nimport category_theory.epi_mono\n\n/-!\n# Reflective functors\n\nBasic properties of reflective functors, especially those relating to their essential image.\n\nNote properties of reflective functors relating to limits and colimits are included in\n`category_theory.monad.limits`.\n-/\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category adjunction\n\nvariables {C : Type u₁} {D : Type u₂} {E : Type u₃}\nvariables [category.{v₁} C] [category.{v₂} D] [category.{v₃} E]\n\n/--\nA functor is *reflective*, or *a reflective inclusion*, if it is fully faithful and right adjoint.\n-/\nclass reflective (R : D ⥤ C) extends is_right_adjoint R, full R, faithful R.\n\nvariables {i : D ⥤ C}\n\n/--\nFor a reflective functor `i` (with left adjoint `L`), with unit `η`, we have `η_iL = iL η`.\n-/\n-- TODO: This holds more generally for idempotent adjunctions, not just reflective adjunctions.\nlemma unit_obj_eq_map_unit [reflective i] (X : C) :\n  (of_right_adjoint i).unit.app (i.obj ((left_adjoint i).obj X))\n    = i.map ((left_adjoint i).map ((of_right_adjoint i).unit.app X)) :=\nbegin\n rw [←cancel_mono (i.map ((of_right_adjoint i).counit.app ((left_adjoint i).obj X))),\n     ←i.map_comp],\n simp,\nend\n\n/--\nWhen restricted to objects in `D` given by `i : D ⥤ C`, the unit is an isomorphism. In other words,\n`η_iX` is an isomorphism for any `X` in `D`.\nMore generally this applies to objects essentially in the reflective subcategory, see\n`functor.ess_image.unit_iso`.\n-/\ninstance is_iso_unit_obj [reflective i] {B : D} :\n  is_iso ((of_right_adjoint i).unit.app (i.obj B)) :=\nbegin\n  have : (of_right_adjoint i).unit.app (i.obj B) =\n            inv (i.map ((of_right_adjoint i).counit.app B)),\n  { rw ← comp_hom_eq_id,\n    apply (of_right_adjoint i).right_triangle_components },\n  rw this,\n  exact is_iso.inv_is_iso,\nend\n\n/--\nIf `A` is essentially in the image of a reflective functor `i`, then `η_A` is an isomorphism.\nThis gives that the \"witness\" for `A` being in the essential image can instead be given as the\nreflection of `A`, with the isomorphism as `η_A`.\n\n(For any `B` in the reflective subcategory, we automatically have that `ε_B` is an iso.)\n-/\nlemma functor.ess_image.unit_is_iso [reflective i] {A : C} (h : A ∈ i.ess_image) :\n  is_iso ((of_right_adjoint i).unit.app A) :=\nbegin\n  suffices : (of_right_adjoint i).unit.app A =\n                h.get_iso.inv ≫ (of_right_adjoint i).unit.app (i.obj h.witness) ≫\n                  (left_adjoint i ⋙ i).map h.get_iso.hom,\n  { rw this,\n    apply_instance },\n  rw ← nat_trans.naturality,\n  simp,\nend\n\n/-- If `η_A` is an isomorphism, then `A` is in the essential image of `i`. -/\nlemma mem_ess_image_of_unit_is_iso [is_right_adjoint i] (A : C)\n  [is_iso ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image :=\n⟨(left_adjoint i).obj A, ⟨(as_iso ((of_right_adjoint i).unit.app A)).symm⟩⟩\n\n/-- If `η_A` is a split monomorphism, then `A` is in the reflective subcategory. -/\nlemma mem_ess_image_of_unit_split_mono [reflective i] {A : C}\n  [split_mono ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image :=\nbegin\n  let η : 𝟭 C ⟶ left_adjoint i ⋙ i := (of_right_adjoint i).unit,\n  haveI : is_iso (η.app (i.obj ((left_adjoint i).obj A))) := (i.obj_mem_ess_image _).unit_is_iso,\n  have : epi (η.app A),\n  { apply epi_of_epi (retraction (η.app A)) _,\n    rw (show retraction _ ≫ η.app A = _, from η.naturality (retraction (η.app A))),\n    apply epi_comp (η.app (i.obj ((left_adjoint i).obj A))) },\n  resetI,\n  haveI := is_iso_of_epi_of_split_mono (η.app A),\n  exact mem_ess_image_of_unit_is_iso A,\nend\n\n/-- Composition of reflective functors. -/\ninstance reflective.comp (F : C ⥤ D) (G : D ⥤ E) [Fr : reflective F] [Gr : reflective G] :\n  reflective (F ⋙ G) := { to_faithful := faithful.comp F G, }\n\n/-- (Implementation) Auxiliary definition for `unit_comp_partial_bijective`. -/\ndef unit_comp_partial_bijective_aux [reflective i] (A : C) (B : D) :\n  (A ⟶ i.obj B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ i.obj B) :=\n((adjunction.of_right_adjoint i).hom_equiv _ _).symm.trans (equiv_of_fully_faithful i)\n\n/-- The description of the inverse of the bijection `unit_comp_partial_bijective_aux`. -/\nlemma unit_comp_partial_bijective_aux_symm_apply [reflective i] {A : C} {B : D}\n  (f : i.obj ((left_adjoint i).obj A) ⟶ i.obj B) :\n  (unit_comp_partial_bijective_aux _ _).symm f = (of_right_adjoint i).unit.app A ≫ f :=\nby simp [unit_comp_partial_bijective_aux]\n\n/--\nIf `i` has a reflector `L`, then the function `(i.obj (L.obj A) ⟶ B) → (A ⟶ B)` given by\nprecomposing with `η.app A` is a bijection provided `B` is in the essential image of `i`.\nThat is, the function `λ (f : i.obj (L.obj A) ⟶ B), η.app A ≫ f` is bijective, as long as `B` is in\nthe essential image of `i`.\nThis definition gives an equivalence: the key property that the inverse can be described\nnicely is shown in `unit_comp_partial_bijective_symm_apply`.\n\nThis establishes there is a natural bijection `(A ⟶ B) ≃ (i.obj (L.obj A) ⟶ B)`. In other words,\nfrom the point of view of objects in `D`, `A` and `i.obj (L.obj A)` look the same: specifically\nthat `η.app A` is an isomorphism.\n-/\ndef unit_comp_partial_bijective [reflective i] (A : C) {B : C} (hB : B ∈ i.ess_image) :\n  (A ⟶ B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) :=\ncalc (A ⟶ B) ≃ (A ⟶ i.obj hB.witness) : iso.hom_congr (iso.refl _) hB.get_iso.symm\n     ...     ≃ (i.obj _ ⟶ i.obj hB.witness) : unit_comp_partial_bijective_aux _ _\n     ...     ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) : iso.hom_congr (iso.refl _) hB.get_iso\n\n@[simp]\nlemma unit_comp_partial_bijective_symm_apply [reflective i] (A : C) {B : C}\n  (hB : B ∈ i.ess_image) (f) :\n  (unit_comp_partial_bijective A hB).symm f = (of_right_adjoint i).unit.app A ≫ f :=\nby simp [unit_comp_partial_bijective, unit_comp_partial_bijective_aux_symm_apply]\n\nlemma unit_comp_partial_bijective_symm_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')\n  (hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : i.obj ((left_adjoint i).obj A) ⟶ B) :\n  (unit_comp_partial_bijective A hB').symm (f ≫ h) =\n    (unit_comp_partial_bijective A hB).symm f ≫ h :=\nby simp\n\nlemma unit_comp_partial_bijective_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')\n  (hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : A ⟶ B) :\n  (unit_comp_partial_bijective A hB') (f ≫ h) = unit_comp_partial_bijective A hB f ≫ h :=\nby rw [←equiv.eq_symm_apply, unit_comp_partial_bijective_symm_natural A h, equiv.symm_apply_apply]\n\n/-- If `i : D ⥤ C` is reflective, the inverse functor of `i ≌ F.ess_image` can be explicitly\ndefined by the reflector. -/\n@[simps]\ndef equiv_ess_image_of_reflective [reflective i] : D ≌ i.ess_image :=\n{ functor := i.to_ess_image,\n  inverse := i.ess_image_inclusion ⋙ (left_adjoint i : _),\n  unit_iso := nat_iso.of_components (λ X, (as_iso $ (of_right_adjoint i).counit.app X).symm)\n    (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n      exact ((of_right_adjoint i).counit.naturality _).symm }),\n  counit_iso := nat_iso.of_components\n    (λ X, by { refine (iso.symm $ as_iso _), exact (of_right_adjoint i).unit.app X,\n      apply_with (is_iso_of_reflects_iso _ i.ess_image_inclusion) { instances := ff },\n      exact functor.ess_image.unit_is_iso X.prop })\n    (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n      exact ((of_right_adjoint i).unit.naturality f).symm }) }\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/adjunction/reflective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5660185351961013, "lm_q1q2_score": 0.33546028076565637}}
{"text": "/-\nCopyright (c) 2021-2023 by the authors listed in the file AUTHORS and their\ninstitutional affiliations. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Abdalrhman Mohamed\n-/\n\nimport Smt.Reconstruction.Rewrites.Simp\n\nnamespace Smt.Reconstruction.Rewrites.Prop\n\nscoped notation \"const \" c => fun _ => c\n\n@[smt_simp] theorem bool_double_neg_elim [Decidable t] : Not (Not t) = t :=\n  propext ⟨Decidable.of_not_not, not_not_intro⟩\n\n@[smt_simp] theorem bool_eq_true : (t = True) = t :=\n  propext ⟨of_eq_true, eq_true⟩\n@[smt_simp] theorem bool_eq_false : (t = False) = (Not t) :=\n  propext ⟨(· ▸ not_false), eq_false⟩\n\n@[smt_simp] theorem bool_impl_false1 : (t → False) = (Not t) :=\n  propext ⟨(·), (·)⟩\n@[smt_simp] theorem bool_impl_false2 : (False → t) = True :=\n  propext ⟨const trivial, const False.elim⟩\n@[smt_simp] theorem bool_impl_true1 : (t → True) = True :=\n  propext ⟨const trivial, const (const trivial)⟩\n@[smt_simp] theorem bool_impl_true2 {t : Prop} : (True → t) = t :=\n  propext ⟨(· trivial), (const ·)⟩\n\n@[smt_simp] theorem bool_or_true : (xs ∨ True ∨ ys) = True :=\n  (true_or _).symm ▸ or_true _\n@[smt_simp] theorem bool_or_false : (xs ∨ False ∨ ys) = (xs ∨ ys) :=\n  (false_or _).symm ▸ rfl\n@[smt_simp] theorem bool_or_flatten : (xs ∨ (b ∨ ys) ∨ zs) = (xs ∨ zs ∨ b ∨ ys) :=\n  sorry\n@[smt_simp] theorem bool_or_dup : (xs ∨ b ∨ ys ∨ b ∨ zs) = (xs ∨ b ∨ ys ∨ zs) :=\n  sorry\n\n@[smt_simp] theorem bool_and_true : (xs ∧ True ∧ ys) = (xs ∧ ys) := (true_and _).symm ▸ rfl\n@[smt_simp] theorem bool_and_false : (xs ∧ False ∧ ys) = False := (false_and _).symm ▸ and_false _\n@[smt_simp] theorem bool_and_flatten : (xs ∧ (b ∧ ys) ∧ zs) = (xs ∧ zs ∧ b ∧ ys) := sorry\n@[smt_simp] theorem bool_and_dup : (xs ∧ b ∧ ys ∧ b ∧ zs) = (xs ∧ b ∧ ys ∧ zs) := sorry\n\n@[smt_simp] theorem bool_and_conf : (xs ∧ w ∧ ys ∧ ¬w ∧ zs) = False := sorry\n@[smt_simp] theorem bool_or_taut : (xs ∨ w ∨ ys ∨ ¬w ∨ zs) = True := sorry\n\n@[smt_simp] theorem ite_neg_branch [h : Decidable c] [Decidable y] : x = ¬y → ite c x y = (c = x) := fun hxny =>\n  hxny ▸ h.byCases\n    (fun hc => if_pos hc ▸ propext ⟨(propext ⟨const ·, const hc⟩), (· ▸ hc)⟩)\n    (fun hnc => if_neg hnc ▸ propext\n      ⟨fun hy => propext ⟨fun hc => False.elim (hnc hc), fun hny => False.elim (hny hy)⟩,\n       fun hcny => bool_double_neg_elim (t := y) ▸ hcny ▸ hnc⟩)\n\nend Smt.Reconstruction.Rewrites.Prop\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Smt/Reconstruction/Rewrites/Prop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3354602726210288}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport order.category.BoundedLattice\nimport order.hom.complete_lattice\n\n/-!\n# The category of complete lattices\n\nThis file defines `CompleteLattice`, the category of complete lattices.\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- The category of complete lattices. -/\ndef CompleteLattice := bundled complete_lattice\n\nnamespace CompleteLattice\n\ninstance : has_coe_to_sort CompleteLattice Type* := bundled.has_coe_to_sort\ninstance (X : CompleteLattice) : complete_lattice X := X.str\n\n/-- Construct a bundled `CompleteLattice` from a `complete_lattice`. -/\ndef of (α : Type*) [complete_lattice α] : CompleteLattice := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [complete_lattice α] : ↥(of α) = α := rfl\n\ninstance : inhabited CompleteLattice := ⟨of punit⟩\n\ninstance : bundled_hom @complete_lattice_hom :=\n{ to_fun := λ _ _ _ _, coe_fn,\n  id := @complete_lattice_hom.id,\n  comp := @complete_lattice_hom.comp,\n  hom_ext := λ X Y _ _, by exactI fun_like.coe_injective }\ninstance : large_category.{u} CompleteLattice := bundled_hom.category complete_lattice_hom\ninstance : concrete_category CompleteLattice := bundled_hom.concrete_category complete_lattice_hom\n\ninstance has_forget_to_BoundedLattice : has_forget₂ CompleteLattice BoundedLattice :=\n{ forget₂ := { obj := λ X, BoundedLattice.of X,\n               map := λ X Y, complete_lattice_hom.to_bounded_lattice_hom },\n  forget_comp := rfl }\n\n/-- Constructs an isomorphism of complete lattices from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : CompleteLattice.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : CompleteLattice ⥤ CompleteLattice :=\n{ obj := λ X, of Xᵒᵈ, map := λ X Y, complete_lattice_hom.dual }\n\n/-- The equivalence between `CompleteLattice` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : CompleteLattice ≌ CompleteLattice :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nend CompleteLattice\n\nlemma CompleteLattice_dual_comp_forget_to_BoundedLattice :\n  CompleteLattice.dual ⋙ forget₂ CompleteLattice BoundedLattice =\n    forget₂ CompleteLattice BoundedLattice ⋙ BoundedLattice.dual := rfl\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/order/category/CompleteLattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203340678567, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.33532182597362903}}
{"text": "example (a b : Nat) : True := by\n  induction a generalizing b\n  case zero => trivial\n  case succ => trivial\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/inductionParse.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33530069917935457}}
{"text": "/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors : Aaron Anderson\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.basic\nimport Mathlib.order.atoms\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Simple Modules\n\n## Main Definitions\n  * `is_simple_module` indicates that a module has no proper submodules\n  (the only submodules are `⊥` and `⊤`).\n  * A `division_ring` structure on the endomorphism ring of a simple module.\n\n## Main Results\n  * Schur's Lemma: `bijective_or_eq_zero` shows that a linear map between simple modules\n  is either bijective or 0, leading to a `division_ring` structure on the endomorphism ring.\n\n## TODO\n  * Semisimple modules, Artin-Wedderburn Theory\n  * Unify with the work on Schur's Lemma in a category theory context\n\n-/\n\n/-- A module is simple when it has only two submodules, `⊥` and `⊤`. -/\ndef is_simple_module (R : Type u_1) [comm_ring R] (M : Type u_2) [add_comm_group M] [module R M] :=\n  is_simple_lattice (submodule R M)\n\n-- Making this an instance causes the linter to complain of \"dangerous instances\"\n\ntheorem is_simple_module.nontrivial (R : Type u_1) [comm_ring R] (M : Type u_2) [add_comm_group M]\n    [module R M] [is_simple_module R M] : nontrivial M :=\n  sorry\n\nnamespace linear_map\n\n\ntheorem injective_or_eq_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R M]\n    (f : linear_map R M N) : function.injective ⇑f ∨ f = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (function.injective ⇑f ∨ f = 0)) (Eq.symm (propext ker_eq_bot))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (ker f = ⊥ ∨ f = 0)) (Eq.symm (propext ker_eq_top))))\n      (eq_bot_or_eq_top (ker f)))\n\ntheorem injective_of_ne_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R M]\n    {f : linear_map R M N} (h : f ≠ 0) : function.injective ⇑f :=\n  or.resolve_right (injective_or_eq_zero f) h\n\ntheorem surjective_or_eq_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R N]\n    (f : linear_map R M N) : function.surjective ⇑f ∨ f = 0 :=\n  sorry\n\ntheorem surjective_of_ne_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R N]\n    {f : linear_map R M N} (h : f ≠ 0) : function.surjective ⇑f :=\n  or.resolve_right (surjective_or_eq_zero f) h\n\n/-- Schur's Lemma for linear maps between (possibly distinct) simple modules -/\ntheorem bijective_or_eq_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R M]\n    [is_simple_module R N] (f : linear_map R M N) : function.bijective ⇑f ∨ f = 0 :=\n  dite (f = 0) (fun (h : f = 0) => Or.inr h)\n    fun (h : ¬f = 0) =>\n      or.intro_left (f = 0) { left := injective_of_ne_zero h, right := surjective_of_ne_zero h }\n\ntheorem bijective_of_ne_zero {R : Type u_1} [comm_ring R] {M : Type u_2} [add_comm_group M]\n    [module R M] {N : Type u_3} [add_comm_group N] [module R N] [is_simple_module R M]\n    [is_simple_module R N] {f : linear_map R M N} (h : f ≠ 0) : function.bijective ⇑f :=\n  or.resolve_right (bijective_or_eq_zero f) h\n\n/-- Schur's Lemma makes the endomorphism ring of a simple module a division ring. -/\nprotected instance module.End.division_ring {R : Type u_1} [comm_ring R] {M : Type u_2}\n    [add_comm_group M] [module R M] [DecidableEq (module.End R M)] [is_simple_module R M] :\n    division_ring (module.End R M) :=\n  division_ring.mk ring.add sorry ring.zero sorry sorry ring.neg ring.sub sorry sorry ring.mul sorry\n    ring.one sorry sorry sorry sorry\n    (fun (f : module.End R M) =>\n      dite (f = 0) (fun (h : f = 0) => 0)\n        fun (h : ¬f = 0) =>\n          inverse f (equiv.inv_fun (equiv.of_bijective (⇑f) (bijective_of_ne_zero h))) sorry sorry)\n    (div_inv_monoid.div._default ring.mul sorry ring.one sorry sorry\n      fun (f : module.End R M) =>\n        dite (f = 0) (fun (h : f = 0) => 0)\n          fun (h : ¬f = 0) =>\n            inverse f (equiv.inv_fun (equiv.of_bijective (⇑f) (bijective_of_ne_zero h))) sorry\n              sorry)\n    sorry sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/simple_module_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33530069917935457}}
{"text": "import algebraic_topology.nerve\nimport algebraic_topology.simplicial_object\nimport category_theory.adjunction.basic\nimport category_theory.functor.basic\nimport category_theory.functor.category\n\nimport simplicial_realization\n\nopen category_theory quiver category_theory.simplicial_object\n\nvariables (C : Type*) [category C]\n\n-- define the counit first \ndef point' : (simplex_category.skeletal_functor.obj (simplex_category.mk 0)) := begin\n  simp, refine 0,\nend\n\ndef obj' (n : fin 2) : (simplex_category.skeletal_functor.obj (simplex_category.mk 1)) := begin\n  simp, refine n,\nend\n\ndef arrow' : (obj' 0) ⟶ (obj' 1) := begin\n  have H := nat.le_of_lt(nat.lt.base 0),\n  refine ulift.up (plift.up H),\nend\n\ndef obj (n : fin 2) : (simplex_category.to_Cat.obj (simplex_category.mk 1)) := begin\n  simp, refine (obj' n),\nend\n\n#check simplex_category.to_Cat.map (simplex_category.δ 1)\n\n\ndef arrow : (obj 0) ⟶ (obj 1) := arrow'\n\ndef point : ((simplex_category.to_Cat.obj (simplex_category.mk 0))) := point'\n\nlemma d1' : (simplex_category.skeletal_functor.map (simplex_category.δ 1)).to_fun point = (obj' 0) :=\nbegin\n  simp, dsimp [obj'], dsimp [simplex_category.hom.to_order_hom],\n  dsimp [simplex_category.δ], dsimp [simplex_category.hom.mk],\n  dsimp [point], dsimp [point'], simp, refl,\nend\n\n-- i dont know if theres a better way to do this but i also do not care\nlemma d1 : (simplex_category.to_Cat.map (simplex_category.δ 1)).obj point = (obj 0) := begin\n  \n  simp,\n  dsimp [forget₂], dsimp [has_forget₂.forget₂],\n\n\n  dsimp [obj], dsimp [obj'], \n  dsimp [simplex_category.hom.to_order_hom],\n  dsimp [simplex_category.δ],\n  dsimp [simplex_category.hom.mk],\n  dsimp [point], dsimp [point'],\n  dsimp [order_hom_class.to_lattice_hom],\n  dsimp [coe], dsimp [lift_t], unfold has_lift_t.lift,\n  unfold coe_t, unfold has_coe_t.coe,\n\n  unfold coe_fn, unfold has_coe_to_fun.coe, simp,\n  unfold coe_fn, unfold has_coe_to_fun.coe, unfold fun_like.coe,\n\n  simp, refl,\n  \nend\n\nlemma d0 : (simplex_category.to_Cat.map (simplex_category.δ 0)).obj point = (obj 1) := begin\n  \n  simp,\n  dsimp [forget₂], dsimp [has_forget₂.forget₂],\n\n\n  dsimp [obj], dsimp [obj'], \n  dsimp [simplex_category.hom.to_order_hom],\n  dsimp [simplex_category.δ],\n  dsimp [simplex_category.hom.mk],\n  dsimp [point], dsimp [point'],\n  dsimp [order_hom_class.to_lattice_hom],\n  dsimp [coe], dsimp [lift_t], unfold has_lift_t.lift,\n  unfold coe_t, unfold has_coe_t.coe,\n\n  unfold coe_fn, unfold has_coe_to_fun.coe, simp,\n  unfold coe_fn, unfold has_coe_to_fun.coe, unfold fun_like.coe,\n\n  simp, refl,\n  \nend\n\n#check congr_fun\ndef counit.obj (X : HomotopyRealization.obj (nerve C)) : C := X.obj point\ndef one_simplex_to_morphsim {a b : nerve C _[0]} (f : a ⟶ b) : (counit.obj C a) ⟶ (counit.obj C b) := begin\n  destruct f, intros, \n  unfold nerve at val, simp at val,\n  unfold δ at property, \n  -- simp at property,\n  destruct property, intros,\n  dsimp [counit.obj], subst left, subst right,\n  -- have H := val.map arrow,\n\n  have H0 : (nerve C).map (simplex_category.δ 0).op val = ((simplex_category.to_Cat.map (simplex_category.δ 0).op.unop)) ⋙ val,\n  refl,\n  have H1 : (nerve C).map (simplex_category.δ 1).op val = ((simplex_category.to_Cat.map (simplex_category.δ 1).op.unop)) ⋙ val,\n  refl,\n\n  have H2 : (((nerve C).map (simplex_category.δ 1).op val).obj point) ⟶ (((nerve C).map (simplex_category.δ 0).op val).obj point) = ((((simplex_category.to_Cat.map (simplex_category.δ 1).op.unop)) ⋙ val).obj point) ⟶ ((((simplex_category.to_Cat.map (simplex_category.δ 0).op.unop)) ⋙ val).obj point),\n  rw H0, rw H1,\n\n  \n\n  rw d1,\nend\n\ndef counit.map' (X Y : HomotopyRealization.obj (nerve C)) (f : @path (nerve C _[0]) underlying X Y) : (counit.obj C X) ⟶ (counit.obj C Y) :=\nbegin\n  induction f with _ _ α_1 α, refine (𝟙 _), \n  have H := α.val,\n  have H' := α.val.map arrow,\n  refine (f_ih ≫ (α.val.map arrow)),\nend\n\n-- def counit.map (X Y : (HomotopyRealization.obj (nerve C))) (f : X ⟶ Y) : \n-- (counit.obj C X) ⟶  (counit.obj C Y) := begin\n--   dsimp [hom] at f, dsimp [Cat.of] at f, dsimp [bundled.of] at f,\n--   dsimp [hom] at f,\n--   have H := pick_rep f, cases H, --revert H, introsrintro ⟨f', frw⟩,\n  \n-- end\n\n\ndef adjoint : HomotopyRealization ⊣ nerve_functor := adjunction.mk_of_unit_counit {\n  counit := {\n      app := λ a, { obj := λ p, counit.obj a p, map := λ p q α, begin \n          apply counit.map, refine α,\n      end}\n  }\n}\n\nlemma counit_respects_homotopies (C : Type) [category C] (h : (nerve C) _[2]) \n: C = C :=\nbegin\n  refl,\nend", "meta": {"author": "raghav198", "repo": "simplicial-things-in-lean", "sha": "a95608a6dba98c8a47bbdaa0ed55707d0251524c", "save_path": "github-repos/lean/raghav198-simplicial-things-in-lean", "path": "github-repos/lean/raghav198-simplicial-things-in-lean/simplicial-things-in-lean-a95608a6dba98c8a47bbdaa0ed55707d0251524c/src/nerve_realization_adjunction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.33524264874042736}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov\n-/\nimport measure_theory.function.ae_measurable_sequence\nimport analysis.complex.basic\nimport analysis.normed_space.finite_dimension\nimport measure_theory.group.arithmetic\nimport topology.algebra.ordered.liminf_limsup\nimport topology.continuous_function.basic\nimport topology.instances.ereal\nimport topology.G_delta\nimport topology.semicontinuous\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel α` : the least `σ`-algebra that contains all open sets;\n* `class borel_space` : a space with `topological_space` and `measurable_space` structures\n  such that `‹measurable_space α› = borel α`;\n* `class opens_measurable_space` : a space with `topological_space` and `measurable_space`\n  structures such that all open sets are measurable; equivalently, `borel α ≤ ‹measurable_space α›`.\n* `borel_space` instances on `empty`, `unit`, `bool`, `nat`, `int`, `rat`;\n* `measurable` and `borel_space` instances on `ℝ`, `ℝ≥0`, `ℝ≥0∞`.\n\n## Main statements\n\n* `is_open.measurable_set`, `is_closed.measurable_set`: open and closed sets are measurable;\n* `continuous.measurable` : a continuous function is measurable;\n* `continuous.measurable2` : if `f : α → β` and `g : α → γ` are measurable and `op : β × γ → δ`\n  is continuous, then `λ x, op (f x, g y)` is measurable;\n* `measurable.add` etc : dot notation for arithmetic operations on `measurable` predicates,\n  and similarly for `dist` and `edist`;\n* `ae_measurable.add` : similar dot notation for almost everywhere measurable functions;\n* `measurable.ennreal*` : special cases for arithmetic operations on `ℝ≥0∞`.\n-/\n\nnoncomputable theory\n\nopen classical set filter measure_theory\nopen_locale classical big_operators topological_space nnreal ennreal interval\n\nuniverses u v w x y\nvariables {α β γ γ₂ δ : Type*} {ι : Sort y} {s t u : set α}\n\nopen measurable_space topological_space\n\n/-- `measurable_space` structure generated by `topological_space`. -/\ndef borel (α : Type u) [topological_space α] : measurable_space α :=\ngenerate_from {s : set α | is_open s}\n\nlemma borel_eq_top_of_discrete [topological_space α] [discrete_topology α] :\n  borel α = ⊤ :=\ntop_le_iff.1 $ λ s hs, generate_measurable.basic s (is_open_discrete s)\n\nlemma borel_eq_top_of_encodable [topological_space α] [t1_space α] [encodable α] :\n  borel α = ⊤ :=\nbegin\n  refine (top_le_iff.1 $ λ s hs, bUnion_of_singleton s ▸ _),\n  apply measurable_set.bUnion s.countable_encodable,\n  intros x hx,\n  apply measurable_set.of_compl,\n  apply generate_measurable.basic,\n  exact is_closed_singleton.is_open_compl\nend\n\nlemma borel_eq_generate_from_of_subbasis {s : set (set α)}\n  [t : topological_space α] [second_countable_topology α] (hs : t = generate_from s) :\n  borel α = generate_from s :=\nle_antisymm\n  (generate_from_le $ assume u (hu : t.is_open u),\n    begin\n      rw [hs] at hu,\n      induction hu,\n      case generate_open.basic : u hu\n      { exact generate_measurable.basic u hu },\n      case generate_open.univ\n      { exact @measurable_set.univ α (generate_from s) },\n      case generate_open.inter : s₁ s₂ _ _ hs₁ hs₂\n      { exact @measurable_set.inter α (generate_from s) _ _ hs₁ hs₂ },\n      case generate_open.sUnion : f hf ih\n      { rcases is_open_sUnion_countable f (by rwa hs) with ⟨v, hv, vf, vu⟩,\n        rw ← vu,\n        exact @measurable_set.sUnion α (generate_from s) _ hv\n          (λ x xv, ih _ (vf xv)) }\n    end)\n  (generate_from_le $ assume u hu, generate_measurable.basic _ $\n    show t.is_open u, by rw [hs]; exact generate_open.basic _ hu)\n\nlemma topological_space.is_topological_basis.borel_eq_generate_from [topological_space α]\n  [second_countable_topology α] {s : set (set α)} (hs : is_topological_basis s) :\n  borel α = generate_from s :=\nborel_eq_generate_from_of_subbasis hs.eq_generate_from\n\nlemma is_pi_system_is_open [topological_space α] : is_pi_system (is_open : set α → Prop) :=\nλ s t hs ht hst, is_open.inter hs ht\n\nlemma borel_eq_generate_from_is_closed [topological_space α] :\n  borel α = generate_from {s | is_closed s} :=\nle_antisymm\n  (generate_from_le $ λ t ht, @measurable_set.of_compl α _ (generate_from {s | is_closed s})\n    (generate_measurable.basic _ $ is_closed_compl_iff.2 ht))\n  (generate_from_le $ λ t ht, @measurable_set.of_compl α _ (borel α)\n    (generate_measurable.basic _ $ is_open_compl_iff.2 ht))\n\nsection order_topology\n\nvariable (α)\nvariables [topological_space α] [second_countable_topology α] [linear_order α] [order_topology α]\n\nlemma borel_eq_generate_from_Iio : borel α = generate_from (range Iio) :=\nbegin\n  refine le_antisymm _ (generate_from_le _),\n  { rw borel_eq_generate_from_of_subbasis (@order_topology.topology_eq_generate_intervals α _ _ _),\n    letI : measurable_space α := measurable_space.generate_from (range Iio),\n    have H : ∀ a : α, measurable_set (Iio a) := λ a, generate_measurable.basic _ ⟨_, rfl⟩,\n    refine generate_from_le _, rintro _ ⟨a, rfl | rfl⟩; [skip, apply H],\n    by_cases h : ∃ a', ∀ b, a < b ↔ a' ≤ b,\n    { rcases h with ⟨a', ha'⟩,\n      rw (_ : Ioi a = (Iio a')ᶜ), { exact (H _).compl },\n      simp [set.ext_iff, ha'] },\n    { rcases is_open_Union_countable\n        (λ a' : {a' : α // a < a'}, {b | a'.1 < b})\n        (λ a', is_open_lt' _) with ⟨v, ⟨hv⟩, vu⟩,\n      simp [set.ext_iff] at vu,\n      have : Ioi a = ⋃ x : v, (Iio x.1.1)ᶜ,\n      { simp [set.ext_iff],\n        refine λ x, ⟨λ ax, _, λ ⟨a', ⟨h, av⟩, ax⟩, lt_of_lt_of_le h ax⟩,\n        rcases (vu x).2 _ with ⟨a', h₁, h₂⟩,\n        { exact ⟨a', h₁, le_of_lt h₂⟩ },\n        refine not_imp_comm.1 (λ h, _) h,\n        exact ⟨x, λ b, ⟨λ ab, le_of_not_lt (λ h', h ⟨b, ab, h'⟩),\n          lt_of_lt_of_le ax⟩⟩ },\n      rw this, resetI,\n      apply measurable_set.Union,\n      exact λ _, (H _).compl } },\n  { rw forall_range_iff,\n    intro a,\n    exact generate_measurable.basic _ is_open_Iio }\nend\n\nlemma borel_eq_generate_from_Ioi : borel α = generate_from (range Ioi) :=\n@borel_eq_generate_from_Iio (order_dual α) _ (by apply_instance : second_countable_topology α) _ _\n\nend order_topology\n\nlemma borel_comap {f : α → β} {t : topological_space β} :\n  @borel α (t.induced f) = (@borel β t).comap f :=\ncomap_generate_from.symm\n\nlemma continuous.borel_measurable [topological_space α] [topological_space β]\n  {f : α → β} (hf : continuous f) :\n  @measurable α β (borel α) (borel β) f :=\nmeasurable.of_le_map $ generate_from_le $\n  λ s hs, generate_measurable.basic (f ⁻¹' s) (hs.preimage hf)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nall open sets are measurable. -/\nclass opens_measurable_space (α : Type*) [topological_space α] [h : measurable_space α] : Prop :=\n(borel_le : borel α ≤ h)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nthe `σ`-algebra of measurable sets is exactly the `σ`-algebra generated by open sets. -/\nclass borel_space (α : Type*) [topological_space α] [measurable_space α] : Prop :=\n(measurable_eq : ‹measurable_space α› = borel α)\n\n/-- In a `borel_space` all open sets are measurable. -/\n@[priority 100]\ninstance borel_space.opens_measurable {α : Type*} [topological_space α] [measurable_space α]\n  [borel_space α] : opens_measurable_space α :=\n⟨ge_of_eq $ borel_space.measurable_eq⟩\n\ninstance subtype.borel_space {α : Type*} [topological_space α] [measurable_space α]\n  [hα : borel_space α] (s : set α) :\n  borel_space s :=\n⟨by { rw [hα.1, subtype.measurable_space, ← borel_comap], refl }⟩\n\ninstance subtype.opens_measurable_space {α : Type*} [topological_space α] [measurable_space α]\n  [h : opens_measurable_space α] (s : set α) :\n  opens_measurable_space s :=\n⟨by { rw [borel_comap], exact comap_mono h.1 }⟩\n\ntheorem _root_.measurable_set.induction_on_open [topological_space α] [measurable_space α]\n  [borel_space α] {C : set α → Prop} (h_open : ∀ U, is_open U → C U)\n  (h_compl : ∀ t, measurable_set t → C t → C tᶜ)\n  (h_union : ∀ f : ℕ → set α, pairwise (disjoint on f) →\n    (∀ i, measurable_set (f i)) → (∀ i, C (f i)) → C (⋃ i, f i)) :\n  ∀ ⦃t⦄, measurable_set t → C t :=\nmeasurable_space.induction_on_inter borel_space.measurable_eq is_pi_system_is_open\n  (h_open _ is_open_empty) h_open h_compl h_union\n\nsection\nvariables [topological_space α] [measurable_space α] [opens_measurable_space α]\n   [topological_space β] [measurable_space β] [opens_measurable_space β]\n   [topological_space γ] [measurable_space γ] [borel_space γ]\n   [topological_space γ₂] [measurable_space γ₂] [borel_space γ₂]\n   [measurable_space δ]\n\nlemma is_open.measurable_set (h : is_open s) : measurable_set s :=\nopens_measurable_space.borel_le _ $ generate_measurable.basic _ h\n\n@[measurability]\nlemma measurable_set_interior : measurable_set (interior s) := is_open_interior.measurable_set\n\nlemma is_Gδ.measurable_set (h : is_Gδ s) : measurable_set s :=\nbegin\n  rcases h with ⟨S, hSo, hSc, rfl⟩,\n  exact measurable_set.sInter hSc (λ t ht, (hSo t ht).measurable_set)\nend\n\nlemma measurable_set_of_continuous_at {β} [emetric_space β] (f : α → β) :\n  measurable_set {x | continuous_at f x} :=\n(is_Gδ_set_of_continuous_at f).measurable_set\n\nlemma is_closed.measurable_set (h : is_closed s) : measurable_set s :=\nh.is_open_compl.measurable_set.of_compl\n\nlemma is_compact.measurable_set [t2_space α] (h : is_compact s) : measurable_set s :=\nh.is_closed.measurable_set\n\n@[measurability]\nlemma measurable_set_closure : measurable_set (closure s) :=\nis_closed_closure.measurable_set\n\nlemma measurable_of_is_open {f : δ → γ} (hf : ∀ s, is_open s → measurable_set (f ⁻¹' s)) :\n  measurable f :=\nby { rw [‹borel_space γ›.measurable_eq], exact measurable_generate_from hf }\n\nlemma measurable_of_is_closed {f : δ → γ} (hf : ∀ s, is_closed s → measurable_set (f ⁻¹' s)) :\n  measurable f :=\nbegin\n  apply measurable_of_is_open, intros s hs,\n  rw [← measurable_set.compl_iff, ← preimage_compl], apply hf, rw [is_closed_compl_iff], exact hs\nend\n\nlemma measurable_of_is_closed' {f : δ → γ}\n  (hf : ∀ s, is_closed s → s.nonempty → s ≠ univ → measurable_set (f ⁻¹' s)) : measurable f :=\nbegin\n  apply measurable_of_is_closed, intros s hs,\n  cases eq_empty_or_nonempty s with h1 h1, { simp [h1] },\n  by_cases h2 : s = univ, { simp [h2] },\n  exact hf s hs h1 h2\nend\n\ninstance nhds_is_measurably_generated (a : α) : (𝓝 a).is_measurably_generated :=\nbegin\n  rw [nhds, infi_subtype'],\n  refine @filter.infi_is_measurably_generated _ _ _ _ (λ i, _),\n  exact i.2.2.measurable_set.principal_is_measurably_generated\nend\n\n/-- If `s` is a measurable set, then `𝓝[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : measurable_set s`.\n-/\nlemma measurable_set.nhds_within_is_measurably_generated {s : set α} (hs : measurable_set s)\n  (a : α) :\n  (𝓝[s] a).is_measurably_generated :=\nby haveI := hs.principal_is_measurably_generated; exact filter.inf_is_measurably_generated _ _\n\n@[priority 100] -- see Note [lower instance priority]\ninstance opens_measurable_space.to_measurable_singleton_class [t1_space α] :\n  measurable_singleton_class α :=\n⟨λ x, is_closed_singleton.measurable_set⟩\n\ninstance pi.opens_measurable_space_encodable {ι : Type*} {π : ι → Type*} [encodable ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, opens_measurable_space (π i)] :\n  opens_measurable_space (Π i, π i) :=\nbegin\n  constructor,\n  have : Pi.topological_space =\n    generate_from {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ countable_basis (π a)) ∧\n      t = pi ↑i s},\n  { rw [funext (λ a, @eq_generate_from_countable_basis (π a) _ _), pi_generate_from_eq] },\n  rw [borel_eq_generate_from_of_subbasis this],\n  apply generate_from_le,\n  rintros _ ⟨s, i, hi, rfl⟩,\n  refine measurable_set.pi i.countable_to_set (λ a ha, is_open.measurable_set _),\n  rw [eq_generate_from_countable_basis (π a)],\n  exact generate_open.basic _ (hi a ha)\nend\n\ninstance pi.opens_measurable_space_fintype {ι : Type*} {π : ι → Type*} [fintype ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, opens_measurable_space (π i)] :\n  opens_measurable_space (Π i, π i) :=\nby { letI := fintype.encodable ι, apply_instance }\n\ninstance prod.opens_measurable_space [second_countable_topology α] [second_countable_topology β] :\n  opens_measurable_space (α × β) :=\nbegin\n  constructor,\n  rw [((is_basis_countable_basis α).prod (is_basis_countable_basis β)).borel_eq_generate_from],\n  apply generate_from_le,\n  rintros _ ⟨u, v, hu, hv, rfl⟩,\n  exact (is_open_of_mem_countable_basis hu).measurable_set.prod\n    (is_open_of_mem_countable_basis hv).measurable_set\nend\n\nvariables {α' : Type*} [topological_space α'] [measurable_space α']\n\nlemma measure_interior_of_null_bdry {μ : measure α'} {s : set α'}\n  (h_nullbdry : μ (frontier s) = 0) : μ (interior s) = μ s :=\nmeasure_eq_measure_smaller_of_between_null_diff\n  interior_subset subset_closure h_nullbdry\n\nlemma measure_closure_of_null_bdry {μ : measure α'} {s : set α'}\n  (h_nullbdry : μ (frontier s) = 0) : μ (closure s) = μ s :=\n(measure_eq_measure_larger_of_between_null_diff\n  interior_subset subset_closure h_nullbdry).symm\n\nsection preorder\nvariables [preorder α] [order_closed_topology α] {a b x : α}\n\n@[simp, measurability]\nlemma measurable_set_Ici : measurable_set (Ici a) := is_closed_Ici.measurable_set\n@[simp, measurability]\nlemma measurable_set_Iic : measurable_set (Iic a) := is_closed_Iic.measurable_set\n@[simp, measurability]\nlemma measurable_set_Icc : measurable_set (Icc a b) := is_closed_Icc.measurable_set\n\ninstance nhds_within_Ici_is_measurably_generated :\n  (𝓝[Ici b] a).is_measurably_generated :=\nmeasurable_set_Ici.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iic_is_measurably_generated :\n  (𝓝[Iic b] a).is_measurably_generated :=\nmeasurable_set_Iic.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Icc_is_measurably_generated :\n  is_measurably_generated (𝓝[Icc a b] x) :=\nby { rw [← Ici_inter_Iic, nhds_within_inter], apply_instance }\n\ninstance at_top_is_measurably_generated : (filter.at_top : filter α).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  λ a, (measurable_set_Ici : measurable_set (Ici a)).principal_is_measurably_generated\n\ninstance at_bot_is_measurably_generated : (filter.at_bot : filter α).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  λ a, (measurable_set_Iic : measurable_set (Iic a)).principal_is_measurably_generated\n\nlemma bsupr_measure_Iic {μ : measure α} {s : set α} (hsc : countable s)\n  (hst : ∀ x : α, ∃ y ∈ s, x ≤ y) (hdir : directed_on (≤) s) :\n  (⨆ x ∈ s, μ (Iic x)) = μ univ :=\nbegin\n  rw ← measure_bUnion_eq_supr hsc,\n  { congr, exact bUnion_eq_univ_iff.2 hst },\n  { exact λ _ _, measurable_set_Iic },\n  { exact directed_on_iff_directed.2 (hdir.directed_coe.mono_comp _ $ λ x y, Iic_subset_Iic.2) }\nend\n\nend preorder\n\nsection partial_order\nvariables [partial_order α] [order_closed_topology α] [second_countable_topology α]\n  {a b : α}\n\n@[measurability]\nlemma measurable_set_le' : measurable_set {p : α × α | p.1 ≤ p.2} :=\norder_closed_topology.is_closed_le'.measurable_set\n\n@[measurability]\nlemma measurable_set_le {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable_set {a | f a ≤ g a} :=\nhf.prod_mk hg measurable_set_le'\n\nend partial_order\n\nsection linear_order\nvariables [linear_order α] [order_closed_topology α] {a b x : α}\n\n@[simp, measurability]\nlemma measurable_set_Iio : measurable_set (Iio a) := is_open_Iio.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioi : measurable_set (Ioi a) := is_open_Ioi.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioo : measurable_set (Ioo a b) := is_open_Ioo.measurable_set\n\n@[simp, measurability] lemma measurable_set_Ioc : measurable_set (Ioc a b) :=\nmeasurable_set_Ioi.inter measurable_set_Iic\n\n@[simp, measurability] lemma measurable_set_Ico : measurable_set (Ico a b) :=\nmeasurable_set_Ici.inter measurable_set_Iio\n\ninstance nhds_within_Ioi_is_measurably_generated :\n  (𝓝[Ioi b] a).is_measurably_generated :=\nmeasurable_set_Ioi.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iio_is_measurably_generated :\n  (𝓝[Iio b] a).is_measurably_generated :=\nmeasurable_set_Iio.nhds_within_is_measurably_generated _\n\ninstance nhds_within_interval_is_measurably_generated :\n  is_measurably_generated (𝓝[[a, b]] x) :=\nnhds_within_Icc_is_measurably_generated\n\n@[measurability]\nlemma measurable_set_lt' [second_countable_topology α] : measurable_set {p : α × α | p.1 < p.2} :=\n(is_open_lt continuous_fst continuous_snd).measurable_set\n\n@[measurability]\nlemma measurable_set_lt [second_countable_topology α] {f g : δ → α} (hf : measurable f)\n  (hg : measurable g) : measurable_set {a | f a < g a} :=\nhf.prod_mk hg measurable_set_lt'\n\nlemma set.ord_connected.measurable_set (h : ord_connected s) : measurable_set s :=\nbegin\n  let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y,\n  have huopen : is_open u := is_open_bUnion (λ x hx, is_open_bUnion (λ y hy, is_open_Ioo)),\n  have humeas : measurable_set u := huopen.measurable_set,\n  have hfinite : (s \\ u).finite,\n  { refine set.finite_of_forall_between_eq_endpoints (s \\ u) (λ x hx y hy z hz hxy hyz, _),\n    by_contra h,\n    push_neg at h,\n    exact hy.2 (mem_bUnion_iff.mpr ⟨x, hx.1,\n      mem_bUnion_iff.mpr ⟨z, hz.1, lt_of_le_of_ne hxy h.1, lt_of_le_of_ne hyz h.2⟩⟩) },\n  have : u ⊆ s :=\n    bUnion_subset (λ x hx, bUnion_subset (λ y hy, Ioo_subset_Icc_self.trans (h.out hx hy))),\n  rw ← union_diff_cancel this,\n  exact humeas.union hfinite.measurable_set\nend\n\nlemma is_preconnected.measurable_set\n  (h : is_preconnected s) : measurable_set s :=\nh.ord_connected.measurable_set\n\nlemma generate_from_Ico_mem_le_borel {α : Type*} [topological_space α] [linear_order α]\n  [order_closed_topology α] (s t : set α) :\n  measurable_space.generate_from {S | ∃ (l ∈ s) (u ∈ t) (h : l < u), Ico l u = S} ≤ borel α :=\nbegin\n  apply generate_from_le,\n  letI : measurable_space α := borel α, haveI : borel_space α := ⟨rfl⟩,\n  rintro _ ⟨a, -, b, -, -, rfl⟩,\n  exact measurable_set_Ico\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem_aux {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] {s : set α} (hd : dense s)\n  (hbot : ∀ x, is_bot x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → y ∈ s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S} :=\nbegin\n  set S : set (set α) := {S | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S},\n  refine le_antisymm _ (generate_from_Ico_mem_le_borel _ _),\n  letI : measurable_space α := generate_from S,\n  rw borel_eq_generate_from_Iio,\n  refine generate_from_le (forall_range_iff.2 $ λ a, _),\n  rcases hd.exists_countable_dense_subset_bot_top with ⟨t, hts, hc, htd, htb, htt⟩,\n  by_cases ha : ∀ b < a, (Ioo b a).nonempty,\n  { convert_to measurable_set (⋃ (l ∈ t) (u ∈ t) (hlu : l < u) (hu : u ≤ a), Ico l u),\n    { ext y, simp only [mem_Union, mem_Iio, mem_Ico], split,\n      { intro hy,\n        rcases htd.exists_le' (λ b hb, htb _ hb (hbot b hb)) y with ⟨l, hlt, hly⟩,\n        rcases htd.exists_mem_open is_open_Ioo (ha y hy) with ⟨u, hut, hyu, hua⟩,\n        exact ⟨l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu⟩ },\n      { rintro ⟨l, -, u, -, -, hua, -, hyu⟩,\n        exact hyu.trans_le hua } },\n    { refine measurable_set.bUnion hc (λ a ha, measurable_set.bUnion hc $ λ b hb, _),\n      refine measurable_set.Union_Prop (λ hab, measurable_set.Union_Prop $ λ hb', _),\n      exact generate_measurable.basic _ ⟨a, hts ha, b, hts hb, hab, mem_singleton _⟩ } },\n  { simp only [not_forall, not_nonempty_iff_eq_empty] at ha,\n    replace ha : a ∈ s := hIoo ha.some a ha.some_spec.fst ha.some_spec.snd,\n    convert_to measurable_set (⋃ (l ∈ t) (hl : l < a), Ico l a),\n    { symmetry,\n      simp only [← Ici_inter_Iio, ← Union_inter, inter_eq_right_iff_subset, subset_def, mem_Union,\n        mem_Ici, mem_Iio],\n      intros x hx, rcases htd.exists_le' (λ b hb, htb _ hb (hbot b hb)) x with ⟨z, hzt, hzx⟩,\n      exact ⟨z, hzt, hzx.trans_lt hx, hzx⟩ },\n    { refine measurable_set.bUnion hc (λ x hx, measurable_set.Union_Prop $ λ hlt, _),\n      exact generate_measurable.basic _ ⟨x, hts hx, a, ha, hlt, mem_singleton _⟩ } }\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] [densely_ordered α] [no_bot_order α]\n  {s : set α} (hd : dense s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S} :=\nhd.borel_eq_generate_from_Ico_mem_aux (by simp) $\n  λ x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ico (α : Type*) [topological_space α]\n  [second_countable_topology α] [linear_order α] [order_topology α] :\n  borel α = generate_from {S : set α | ∃ l u (h : l < u), Ico l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ α _).borel_eq_generate_from_Ico_mem_aux (λ _ _, mem_univ _)\n      (λ _ _ _ _, mem_univ _)\n\nlemma dense.borel_eq_generate_from_Ioc_mem_aux {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] {s : set α} (hd : dense s)\n  (hbot : ∀ x, is_top x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → x ∈ s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ioc l u = S} :=\nbegin\n  convert hd.order_dual.borel_eq_generate_from_Ico_mem_aux hbot (λ x y hlt he, hIoo y x hlt _),\n  { ext s,\n    split; rintro ⟨l, hl, u, hu, hlt, rfl⟩,\n    exacts [⟨u, hu, l, hl, hlt, dual_Ico⟩, ⟨u, hu, l, hl, hlt, dual_Ioc⟩] },\n  { erw dual_Ioo,\n    exact he }\nend\n\nlemma dense.borel_eq_generate_from_Ioc_mem {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] [densely_ordered α] [no_top_order α]\n  {s : set α} (hd : dense s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ioc l u = S} :=\nhd.borel_eq_generate_from_Ioc_mem_aux (by simp) $\n  λ x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ioc (α : Type*) [topological_space α]\n  [second_countable_topology α] [linear_order α] [order_topology α] :\n  borel α = generate_from {S : set α | ∃ l u (h : l < u), Ioc l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ α _).borel_eq_generate_from_Ioc_mem_aux (λ _ _, mem_univ _)\n      (λ _ _ _ _, mem_univ _)\n\nnamespace measure_theory.measure\n\n/-- Two finite measures on a Borel space are equal if they agree on all closed-open intervals.  If\n`α` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ico` is an extensionality lemma with weaker assumptions on `μ` and\n`ν`. -/\nlemma ext_of_Ico_finite {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α]\n  [borel_space α] (μ ν : measure α) [is_finite_measure μ] (hμν : μ univ = ν univ)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nbegin\n  refine ext_of_generate_finite _\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico α))\n    (is_pi_system_Ico (id : α → α) id) _ hμν,\n  { rintro - ⟨a, b, hlt, rfl⟩,\n    exact h hlt }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all open-closed intervals.  If\n`α` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ioc` is an extensionality lemma with weaker assumptions on `μ` and\n`ν`. -/\nlemma ext_of_Ioc_finite {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α]\n  [borel_space α] (μ ν : measure α) [is_finite_measure μ] (hμν : μ univ = ν univ)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nbegin\n  refine @ext_of_Ico_finite (order_dual α) _ _ _ _ _ ‹_› μ ν _ hμν (λ a b hab, _),\n  erw dual_Ico,\n  exact h hab\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico' {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  [no_top_order α] (μ ν : measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ico a b) ≠ ∞)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nbegin\n  rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, hsb, hst⟩,\n  have : countable (⋃ (l ∈ s) (u ∈ s) (h : l < u), {Ico l u} : set (set α)),\n    from hsc.bUnion (λ l hl, hsc.bUnion\n      (λ u hu, countable_Union_Prop $ λ _, countable_singleton _)),\n  simp only [← set_of_eq_eq_singleton, ← set_of_exists] at this,\n  refine measure.ext_of_generate_from_of_cover_subset\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico α))\n    (is_pi_system_Ico id id) _ this _ _ _,\n  { rintro _ ⟨l, -, u, -, h, rfl⟩, exact ⟨l, u, h, rfl⟩ },\n  { refine sUnion_eq_univ_iff.2 (λ x, _),\n    rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩,\n    rcases hsd.exists_gt x with ⟨u, hus, hxu⟩,\n    exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩ },\n  { rintro _ ⟨l, -, u, -, hlt, rfl⟩, exact hμ hlt },\n  { rintro _ ⟨l, u, hlt, rfl⟩, exact h hlt }\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc' {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  [no_bot_order α] (μ ν : measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ioc a b) ≠ ∞)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nbegin\n  refine @ext_of_Ico' (order_dual α) _ _ _ _ _ ‹_› _ μ ν _ _;\n    intros a b hab; erw dual_Ico,\n  exacts [hμ hab, h hab]\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [conditionally_complete_linear_order α] [order_topology α]\n  [borel_space α] [no_top_order α] (μ ν : measure α) [is_locally_finite_measure μ]\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nμ.ext_of_Ico' ν (λ a b hab, measure_Ico_lt_top.ne) h\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [conditionally_complete_linear_order α] [order_topology α]\n  [borel_space α] [no_bot_order α] (μ ν : measure α) [is_locally_finite_measure μ]\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nμ.ext_of_Ioc' ν (λ a b hab, measure_Ioc_lt_top.ne) h\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed\nintervals. -/\nlemma ext_of_Iic {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  (μ ν : measure α) [is_finite_measure μ] (h : ∀ a, μ (Iic a) = ν (Iic a)) : μ = ν :=\nbegin\n  refine ext_of_Ioc_finite μ ν _ (λ a b hlt, _),\n  { rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩,\n    have : directed_on (≤) s, from directed_on_iff_directed.2 (directed_of_sup $ λ _ _, id),\n    simp only [← bsupr_measure_Iic hsc (hsd.exists_ge' hst) this, h] },\n  rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic measurable_set_Iic,\n      measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic measurable_set_Iic, h a, h b],\n  { rw ← h a, exact (measure_lt_top μ _).ne },\n  { exact (measure_lt_top μ _).ne }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite\nintervals. -/\nlemma ext_of_Ici {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  (μ ν : measure α) [is_finite_measure μ] (h : ∀ a, μ (Ici a) = ν (Ici a)) : μ = ν :=\n@ext_of_Iic (order_dual α) _ _ _ _ _ ‹_› _ _ _ h\n\nend measure_theory.measure\n\nend linear_order\n\nsection linear_order\n\nvariables [linear_order α] [order_closed_topology α]\n\n@[measurability]\nlemma measurable_set_interval {a b : α} : measurable_set (interval a b) :=\nmeasurable_set_Icc\n\n@[measurability]\nlemma measurable_set_interval_oc {a b : α} : measurable_set (interval_oc a b) :=\nmeasurable_set_Ioc\n\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable.max {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, max (f a) (g a)) :=\nby simpa only [max_def] using hf.piecewise (measurable_set_le hg hf) hg\n\n@[measurability]\nlemma ae_measurable.max {f g : δ → α} {μ : measure δ}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, max (f a) (g a)) μ :=\n⟨λ a, max (hf.mk f a) (hg.mk g a), hf.measurable_mk.max hg.measurable_mk,\n  eventually_eq.comp₂ hf.ae_eq_mk _ hg.ae_eq_mk⟩\n\n@[measurability]\nlemma measurable.min {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, min (f a) (g a)) :=\nby simpa only [min_def] using hf.piecewise (measurable_set_le hf hg) hg\n\n@[measurability]\nlemma ae_measurable.min {f g : δ → α} {μ : measure δ}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, min (f a) (g a)) μ :=\n⟨λ a, min (hf.mk f a) (hg.mk g a), hf.measurable_mk.min hg.measurable_mk,\n  eventually_eq.comp₂ hf.ae_eq_mk _ hg.ae_eq_mk⟩\n\nend linear_order\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis measurable. -/\nlemma continuous.measurable {f : α → γ} (hf : continuous f) :\n  measurable f :=\nhf.borel_measurable.mono opens_measurable_space.borel_le\n  (le_of_eq $ borel_space.measurable_eq)\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis ae-measurable. -/\nlemma continuous.ae_measurable {f : α → γ} (h : continuous f) (μ : measure α) : ae_measurable f μ :=\nh.measurable.ae_measurable\n\nlemma closed_embedding.measurable {f : α → γ} (hf : closed_embedding f) :\n  measurable f :=\nhf.continuous.measurable\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul [has_mul γ] [has_continuous_mul γ] :\n  has_measurable_mul γ :=\n{ measurable_const_mul := λ c, (continuous_const.mul continuous_id).measurable,\n  measurable_mul_const := λ c, (continuous_id.mul continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub [has_sub γ] [has_continuous_sub γ] :\n  has_measurable_sub γ :=\n{ measurable_const_sub := λ c, (continuous_const.sub continuous_id).measurable,\n  measurable_sub_const := λ c, (continuous_id.sub continuous_const).measurable }\n\n@[priority 100, to_additive]\ninstance topological_group.has_measurable_inv [group γ] [topological_group γ] :\n  has_measurable_inv γ :=\n⟨continuous_inv.measurable⟩\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul {M α} [topological_space M]\n  [topological_space α] [measurable_space M] [measurable_space α]\n  [opens_measurable_space M] [borel_space α] [has_scalar M α] [has_continuous_smul M α] :\n  has_measurable_smul M α :=\n⟨λ c, (continuous_const.smul continuous_id).measurable,\n  λ y, (continuous_id.smul continuous_const).measurable⟩\n\nsection homeomorph\n\n@[measurability] protected lemma homeomorph.measurable (h : α ≃ₜ γ) : measurable h :=\nh.continuous.measurable\n\n/-- A homeomorphism between two Borel spaces is a measurable equivalence.-/\ndef homeomorph.to_measurable_equiv (h : γ ≃ₜ γ₂) : γ ≃ᵐ γ₂ :=\n{ measurable_to_fun := h.measurable,\n  measurable_inv_fun := h.symm.measurable,\n  to_equiv := h.to_equiv }\n\n@[simp]\nlemma homeomorph.to_measurable_equiv_coe (h : γ ≃ₜ γ₂) : (h.to_measurable_equiv : γ → γ₂) = h :=\nrfl\n\n@[simp] lemma homeomorph.to_measurable_equiv_symm_coe (h : γ ≃ₜ γ₂) :\n  (h.to_measurable_equiv.symm : γ₂ → γ) = h.symm :=\nrfl\n\nend homeomorph\n\n@[measurability] lemma continuous_map.measurable (f : C(α, γ)) : measurable f :=\nf.continuous.measurable\n\nlemma measurable_of_continuous_on_compl_singleton [t1_space α] {f : α → γ} (a : α)\n  (hf : continuous_on f {a}ᶜ) :\n  measurable f :=\nmeasurable_of_measurable_on_compl_singleton a\n  (continuous_on_iff_continuous_restrict.1 hf).measurable\n\nlemma continuous.measurable2 [second_countable_topology α] [second_countable_topology β]\n  {f : δ → α} {g : δ → β} {c : α → β → γ}\n  (h : continuous (λ p : α × β, c p.1 p.2)) (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, c (f a) (g a)) :=\nh.measurable.comp (hf.prod_mk hg)\n\nlemma continuous.ae_measurable2 [second_countable_topology α] [second_countable_topology β]\n  {f : δ → α} {g : δ → β} {c : α → β → γ} {μ : measure δ}\n  (h : continuous (λ p : α × β, c p.1 p.2)) (hf : ae_measurable f μ) (hg : ae_measurable g μ) :\n  ae_measurable (λ a, c (f a) (g a)) μ :=\nh.measurable.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100]\ninstance has_continuous_inv₀.has_measurable_inv [group_with_zero γ] [t1_space γ]\n  [has_continuous_inv₀ γ] :\n  has_measurable_inv γ :=\n⟨measurable_of_continuous_on_compl_singleton 0 continuous_on_inv₀⟩\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul₂ [second_countable_topology γ] [has_mul γ]\n  [has_continuous_mul γ] : has_measurable_mul₂ γ :=\n⟨continuous_mul.measurable⟩\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub₂ [second_countable_topology γ] [has_sub γ]\n  [has_continuous_sub γ] : has_measurable_sub₂ γ :=\n⟨continuous_sub.measurable⟩\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul₂ {M α} [topological_space M]\n  [second_countable_topology M] [measurable_space M] [opens_measurable_space M]\n  [topological_space α] [second_countable_topology α] [measurable_space α]\n  [borel_space α] [has_scalar M α] [has_continuous_smul M α] :\n  has_measurable_smul₂ M α :=\n⟨continuous_smul.measurable⟩\n\nend\n\nsection borel_space\nvariables [topological_space α] [measurable_space α] [borel_space α]\n  [topological_space β] [measurable_space β] [borel_space β]\n  [topological_space γ] [measurable_space γ] [borel_space γ]\n  [measurable_space δ]\n\nlemma pi_le_borel_pi {ι : Type*} {π : ι → Type*} [Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, borel_space (π i)] :\n  measurable_space.pi ≤ borel (Π i, π i) :=\nbegin\n  have : ‹Π i, measurable_space (π i)› = λ i, borel (π i) :=\n    funext (λ i, borel_space.measurable_eq),\n  rw [this],\n  exact supr_le (λ i, comap_le_iff_le_map.2 $ (continuous_apply i).borel_measurable)\nend\n\nlemma prod_le_borel_prod : prod.measurable_space ≤ borel (α × β) :=\nbegin\n  rw [‹borel_space α›.measurable_eq, ‹borel_space β›.measurable_eq],\n  refine sup_le _ _,\n  { exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable },\n  { exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable }\nend\n\ninstance pi.borel_space_fintype_encodable {ι : Type*} {π : ι → Type*} [encodable ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, borel_space (π i)] :\n  borel_space (Π i, π i) :=\n⟨le_antisymm pi_le_borel_pi opens_measurable_space.borel_le⟩\n\ninstance pi.borel_space_fintype {ι : Type*} {π : ι → Type*} [fintype ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, borel_space (π i)] :\n  borel_space (Π i, π i) :=\n⟨le_antisymm pi_le_borel_pi opens_measurable_space.borel_le⟩\n\ninstance prod.borel_space [second_countable_topology α] [second_countable_topology β] :\n  borel_space (α × β) :=\n⟨le_antisymm prod_le_borel_prod opens_measurable_space.borel_le⟩\n\nprotected lemma embedding.measurable_embedding {f : α → β} (h₁ : embedding f)\n  (h₂ : measurable_set (range f)) : measurable_embedding f :=\nshow measurable_embedding (coe ∘ (homeomorph.of_embedding f h₁).to_measurable_equiv),\nfrom (measurable_embedding.subtype_coe h₂).comp (measurable_equiv.measurable_embedding _)\n\nprotected lemma closed_embedding.measurable_embedding {f : α → β} (h : closed_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.closed_range.measurable_set\n\nprotected lemma open_embedding.measurable_embedding {f : α → β} (h : open_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.open_range.measurable_set\n\nsection linear_order\n\nvariables [linear_order α] [order_topology α] [second_countable_topology α]\n\nlemma measurable_of_Iio {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Iio x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Iio _),\n  rintro _ ⟨x, rfl⟩, exact hf x\nend\n\nlemma upper_semicontinuous.measurable [topological_space δ] [opens_measurable_space δ]\n  {f : δ → α} (hf : upper_semicontinuous f) : measurable f :=\nmeasurable_of_Iio (λ y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Ioi {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Ioi x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Ioi _),\n  rintro _ ⟨x, rfl⟩, exact hf x\nend\n\nlemma lower_semicontinuous.measurable [topological_space δ] [opens_measurable_space δ]\n  {f : δ → α} (hf : lower_semicontinuous f) : measurable f :=\nmeasurable_of_Ioi (λ y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Iic {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Iic x)) : measurable f :=\nbegin\n  apply measurable_of_Ioi,\n  simp_rw [← compl_Iic, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable_of_Ici {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Ici x)) : measurable f :=\nbegin\n  apply measurable_of_Iio,\n  simp_rw [← compl_Ici, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable.is_lub {ι} [encodable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, measurable (f i))\n  (hg : ∀ b, is_lub {a | ∃ i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change ∀ b, is_lub (range $ λ i, f i b) (g b) at hg,\n  rw [‹borel_space α›.measurable_eq, borel_eq_generate_from_Ioi α],\n  apply measurable_generate_from,\n  rintro _ ⟨a, rfl⟩,\n  simp_rw [set.preimage, mem_Ioi, lt_is_lub_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (λ i, hf i (is_open_lt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_lub_of_nonempty {ι} (hι : nonempty ι)\n  {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_lub {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  let p : δ → (ι → α) → Prop := λ x f', is_lub {a | ∃ i, f' i = a} (g x),\n  let g_seq := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨g x⟩ : nonempty α).some,\n  have hg_seq : ∀ b, is_lub {a | ∃ i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI hα : nonempty α := nonempty.map g ⟨b⟩,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : α | ∃ (i : ι), (hf i).mk (f i) b = a} = {a : α | ∃ (i : ι), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : α | ∃ (i : ι), hα.some = a} = {hα.some},\n      { ext1 x,\n        exact ⟨λ hx, hx.some_spec.symm, λ hx, ⟨hι.some, hx.symm⟩⟩, },\n      rw h_singleton,\n      exact is_lub_singleton, }, },\n  refine ⟨g_seq, measurable.is_lub (ae_seq.measurable hf p) hg_seq, _⟩,\n  exact (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨g x⟩ : nonempty α).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_lub {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_lub {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  by_cases hμ : μ = 0, { rw hμ, exact ae_measurable_zero_measure },\n  haveI : μ.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases hι : nonempty ι, { exact ae_measurable.is_lub_of_nonempty hι hf hg, },\n  suffices : ∃ x, g =ᵐ[μ] λ y, g x,\n  by { exact ⟨(λ y, g this.some), measurable_const, this.some_spec⟩, },\n  have h_empty : ∀ x, {a : α | ∃ (i : ι), f i x = a} = ∅,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact λ hi, hι (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact ⟨hg.exists.some, hg.mono (λ y hy, is_lub.unique hy hg.exists.some_spec)⟩,\nend\n\nlemma measurable.is_glb {ι} [encodable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, measurable (f i))\n  (hg : ∀ b, is_glb {a | ∃ i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change ∀ b, is_glb (range $ λ i, f i b) (g b) at hg,\n  rw [‹borel_space α›.measurable_eq, borel_eq_generate_from_Iio α],\n  apply measurable_generate_from,\n  rintro _ ⟨a, rfl⟩,\n  simp_rw [set.preimage, mem_Iio, is_glb_lt_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (λ i, hf i (is_open_gt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_glb_of_nonempty {ι} (hι : nonempty ι)\n  {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_glb {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  let p : δ → (ι → α) → Prop := λ x f', is_glb {a | ∃ i, f' i = a} (g x),\n  let g_seq := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨g x⟩ : nonempty α).some,\n  have hg_seq : ∀ b, is_glb {a | ∃ i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI hα : nonempty α := nonempty.map g ⟨b⟩,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : α | ∃ (i : ι), (hf i).mk (f i) b = a} = {a : α | ∃ (i : ι), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : α | ∃ (i : ι), hα.some = a} = {hα.some},\n      { ext1 x,\n        exact ⟨λ hx, hx.some_spec.symm, λ hx, ⟨hι.some, hx.symm⟩⟩, },\n      rw h_singleton,\n      exact is_glb_singleton, }, },\n  refine ⟨g_seq, measurable.is_glb (ae_seq.measurable hf p) hg_seq, _⟩,\n  exact (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨g x⟩ : nonempty α).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_glb {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_glb {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  by_cases hμ : μ = 0, { rw hμ, exact ae_measurable_zero_measure },\n  haveI : μ.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases hι : nonempty ι, { exact ae_measurable.is_glb_of_nonempty hι hf hg, },\n  suffices : ∃ x, g =ᵐ[μ] λ y, g x,\n  by { exact ⟨(λ y, g this.some), measurable_const, this.some_spec⟩, },\n  have h_empty : ∀ x, {a : α | ∃ (i : ι), f i x = a} = ∅,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact λ hi, hι (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact ⟨hg.exists.some, hg.mono (λ y hy, is_glb.unique hy hg.exists.some_spec)⟩,\nend\n\nprotected lemma monotone.measurable [linear_order β] [order_closed_topology β] {f : β → α}\n  (hf : monotone f) : measurable f :=\nsuffices h : ∀ x, ord_connected (f ⁻¹' Ioi x),\n  from measurable_of_Ioi (λ x, (h x).measurable_set),\nλ x, ord_connected_def.mpr (λ a ha b hb c hc, lt_of_lt_of_le ha (hf hc.1))\n\nlemma ae_measurable_restrict_of_monotone_on [linear_order β] [order_closed_topology β]\n  {μ : measure β} {s : set β} (hs : measurable_set s) {f : β → α} (hf : monotone_on f s) :\n  ae_measurable f (μ.restrict s) :=\nhave this : monotone (f ∘ coe : s → α), from λ ⟨x, hx⟩ ⟨y, hy⟩ (hxy : x ≤ y), hf hx hy hxy,\nae_measurable_restrict_of_measurable_subtype hs this.measurable\n\nprotected lemma antitone.measurable [linear_order β] [order_closed_topology β] {f : β → α}\n  (hf : antitone f) :\n  measurable f :=\n@monotone.measurable (order_dual α) β _ _ ‹_› _ _ _ _ _ ‹_› _ _ _ hf\n\nlemma ae_measurable_restrict_of_antitone_on [linear_order β] [order_closed_topology β]\n  {μ : measure β} {s : set β} (hs : measurable_set s) {f : β → α} (hf : antitone_on f s) :\n  ae_measurable f (μ.restrict s) :=\n@ae_measurable_restrict_of_monotone_on (order_dual α) β _ _ ‹_› _ _ _ _ _ ‹_› _ _ _ _ hs _ hf\n\nend linear_order\n\n@[measurability]\nlemma measurable.supr_Prop {α} [measurable_space α] [complete_lattice α]\n  (p : Prop) {f : δ → α} (hf : measurable f) :\n  measurable (λ b, ⨆ h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact supr_pos h end)\n  (assume h : ¬p, begin convert measurable_const, funext, exact supr_neg h end)\n\n@[measurability]\nlemma measurable.infi_Prop {α} [measurable_space α] [complete_lattice α]\n  (p : Prop) {f : δ → α} (hf : measurable f) :\n  measurable (λ b, ⨅ h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact infi_pos h end )\n  (assume h : ¬p, begin convert measurable_const, funext, exact infi_neg h end)\n\nsection complete_linear_order\n\nvariables [complete_linear_order α] [order_topology α] [second_countable_topology α]\n\n@[measurability]\nlemma measurable_supr {ι} [encodable ι] {f : ι → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ b, ⨆ i, f i b) :=\nmeasurable.is_lub hf $ λ b, is_lub_supr\n\n@[measurability]\nlemma ae_measurable_supr {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ b, ⨆ i, f i b) μ :=\nae_measurable.is_lub hf $ (ae_of_all μ (λ b, is_lub_supr))\n\n@[measurability]\nlemma measurable_infi {ι} [encodable ι] {f : ι → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ b, ⨅ i, f i b) :=\nmeasurable.is_glb hf $ λ b, is_glb_infi\n\n@[measurability]\nlemma ae_measurable_infi {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ b, ⨅ i, f i b) μ :=\nae_measurable.is_glb hf $ (ae_of_all μ (λ b, is_glb_infi))\n\nlemma measurable_bsupr {ι} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, measurable (f i)) : measurable (λ b, ⨆ i ∈ s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [supr_subtype'],\n     exact measurable_supr (λ i, hf i) }\n\nlemma ae_measurable_bsupr {ι} {μ : measure δ} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, ae_measurable (f i) μ) : ae_measurable (λ b, ⨆ i ∈ s, f i b) μ :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [supr_subtype'],\n  exact ae_measurable_supr (λ i, hf i),\nend\n\nlemma measurable_binfi {ι} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, measurable (f i)) : measurable (λ b, ⨅ i ∈ s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [infi_subtype'],\n     exact measurable_infi (λ i, hf i) }\n\nlemma ae_measurable_binfi {ι} {μ : measure δ} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, ae_measurable (f i) μ) : ae_measurable (λ b, ⨅ i ∈ s, f i b) μ :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [infi_subtype'],\n  exact ae_measurable_infi (λ i, hf i),\nend\n\n/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `ℕ`.\n-/\nlemma measurable_liminf' {ι ι'} {f : ι → δ → α} {u : filter ι} (hf : ∀ i, measurable (f i))\n  {p : ι' → Prop} {s : ι' → set ι} (hu : u.has_countable_basis p s) (hs : ∀ i, (s i).countable) :\n  measurable (λ x, liminf u (λ i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.liminf_eq_supr_infi],\n  refine measurable_bsupr _ hu.countable _,\n  exact λ i, measurable_binfi _ (hs i) hf\nend\n\n/-- `limsup` over a general filter is measurable. See `measurable_limsup` for the version over `ℕ`.\n-/\nlemma measurable_limsup' {ι ι'}  {f : ι → δ → α} {u : filter ι} (hf : ∀ i, measurable (f i))\n  {p : ι' → Prop} {s : ι' → set ι} (hu : u.has_countable_basis p s) (hs : ∀ i, (s i).countable) :\n  measurable (λ x, limsup u (λ i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.limsup_eq_infi_supr],\n  refine measurable_binfi _ hu.countable _,\n  exact λ i, measurable_bsupr _ (hs i) hf\nend\n\n/-- `liminf` over `ℕ` is measurable. See `measurable_liminf'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_liminf {f : ℕ → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ x, liminf at_top (λ i, f i x)) :=\nmeasurable_liminf' hf at_top_countable_basis (λ i, countable_encodable _)\n\n/-- `limsup` over `ℕ` is measurable. See `measurable_limsup'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_limsup {f : ℕ → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ x, limsup at_top (λ i, f i x)) :=\nmeasurable_limsup' hf at_top_countable_basis (λ i, countable_encodable _)\n\nend complete_linear_order\n\nsection conditionally_complete_linear_order\n\nvariables [conditionally_complete_linear_order α] [order_topology α] [second_countable_topology α]\n\nlemma measurable_cSup {ι} {f : ι → δ → α} {s : set ι} (hs : s.countable)\n  (hf : ∀ i, measurable (f i)) (bdd : ∀ x, bdd_above ((λ i, f i x) '' s)) :\n  measurable (λ x, Sup ((λ i, f i x) '' s)) :=\nbegin\n  cases eq_empty_or_nonempty s with h2s h2s,\n  { simp [h2s, measurable_const] },\n  { apply measurable_of_Iic, intro y,\n    simp_rw [preimage, mem_Iic, cSup_le_iff (bdd _) (h2s.image _), ball_image_iff, set_of_forall],\n    exact measurable_set.bInter hs (λ i hi, measurable_set_le (hf i) measurable_const) }\nend\n\nend conditionally_complete_linear_order\n\n/-- Convert a `homeomorph` to a `measurable_equiv`. -/\ndef homemorph.to_measurable_equiv (h : α ≃ₜ β) : α ≃ᵐ β :=\n{ to_equiv := h.to_equiv,\n  measurable_to_fun := h.continuous_to_fun.measurable,\n  measurable_inv_fun := h.continuous_inv_fun.measurable }\n\nend borel_space\n\ninstance empty.borel_space : borel_space empty := ⟨borel_eq_top_of_discrete.symm⟩\ninstance unit.borel_space : borel_space unit := ⟨borel_eq_top_of_discrete.symm⟩\ninstance bool.borel_space : borel_space bool := ⟨borel_eq_top_of_discrete.symm⟩\ninstance nat.borel_space : borel_space ℕ := ⟨borel_eq_top_of_discrete.symm⟩\ninstance int.borel_space : borel_space ℤ := ⟨borel_eq_top_of_discrete.symm⟩\ninstance rat.borel_space : borel_space ℚ := ⟨borel_eq_top_of_encodable.symm⟩\n\n@[priority 900]\ninstance is_R_or_C.measurable_space {𝕜 : Type*} [is_R_or_C 𝕜] : measurable_space 𝕜 := borel 𝕜\n@[priority 900]\ninstance is_R_or_C.borel_space {𝕜 : Type*} [is_R_or_C 𝕜] : borel_space 𝕜 := ⟨rfl⟩\n\n/- Instances on `real` and `complex` are special cases of `is_R_or_C` but without these instances,\nLean fails to prove `borel_space (ι → ℝ)`, so we leave them here. -/\n\ninstance real.measurable_space : measurable_space ℝ := borel ℝ\ninstance real.borel_space : borel_space ℝ := ⟨rfl⟩\n\ninstance nnreal.measurable_space : measurable_space ℝ≥0 := subtype.measurable_space\ninstance nnreal.borel_space : borel_space ℝ≥0 := subtype.borel_space _\n\ninstance ennreal.measurable_space : measurable_space ℝ≥0∞ := borel ℝ≥0∞\ninstance ennreal.borel_space : borel_space ℝ≥0∞ := ⟨rfl⟩\n\ninstance ereal.measurable_space : measurable_space ereal := borel ereal\ninstance ereal.borel_space : borel_space ereal := ⟨rfl⟩\n\ninstance complex.measurable_space : measurable_space ℂ := borel ℂ\ninstance complex.borel_space : borel_space ℂ := ⟨rfl⟩\n\n/-- One can cut out `ℝ≥0∞` into the sets `{0}`, `Ico (t^n) (t^(n+1))` for `n : ℤ` and `{∞}`. This\ngives a way to compute the measure of a set in terms of sets on which a given function `f` does not\nfluctuate by more than `t`. -/\nlemma measure_eq_measure_preimage_add_measure_tsum_Ico_zpow [measurable_space α] (μ : measure α)\n  {f : α → ℝ≥0∞} (hf : measurable f) {s : set α} (hs : measurable_set s) {t : ℝ≥0} (ht : 1 < t) :\n  μ s = μ (s ∩ f⁻¹' {0}) + μ (s ∩ f⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f⁻¹' (Ico (t^n) (t^(n+1)))) :=\nbegin\n  have A : μ s = μ (s ∩ f⁻¹' {0}) + μ (s ∩ f⁻¹' (Ioi 0)),\n  { rw ← measure_union,\n    { congr' 1,\n      ext x,\n      have : 0 = f x ∨ 0 < f x := eq_or_lt_of_le bot_le,\n      rw eq_comm at this,\n      simp only [←and_or_distrib_left, this, mem_singleton_iff, mem_inter_eq, and_true,\n        mem_union_eq, mem_Ioi, mem_preimage], },\n    { apply disjoint_left.2 (λ x hx h'x, _),\n      have : 0 < f x := h'x.2,\n      exact lt_irrefl 0 (this.trans_le hx.2.le) },\n    { exact hs.inter (hf (measurable_set_singleton _)) },\n    { exact hs.inter (hf measurable_set_Ioi) } },\n  have B : μ (s ∩ f⁻¹' (Ioi 0)) = μ (s ∩ f⁻¹' {∞}) + μ (s ∩ f⁻¹' (Ioo 0 ∞)),\n  { rw ← measure_union,\n    { rw ← inter_union_distrib_left,\n      congr,\n      ext x,\n      simp only [mem_singleton_iff, mem_union_eq, mem_Ioo, mem_Ioi, mem_preimage],\n      have H : f x = ∞ ∨ f x < ∞ := eq_or_lt_of_le le_top,\n      cases H,\n      { simp only [H, eq_self_iff_true, or_false, with_top.zero_lt_top, not_top_lt, and_false] },\n      { simp only [H, H.ne, and_true, false_or] } },\n    { apply disjoint_left.2 (λ x hx h'x, _),\n      have : f x < ∞ := h'x.2.2,\n      exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm)) },\n    { exact hs.inter (hf (measurable_set_singleton _)) },\n    { exact hs.inter (hf measurable_set_Ioo) } },\n  have C : μ (s ∩ f⁻¹' (Ioo 0 ∞)) = ∑' (n : ℤ), μ (s ∩ f⁻¹' (Ico (t^n) (t^(n+1)))),\n  { rw [← measure_Union, ennreal.Ioo_zero_top_eq_Union_Ico_zpow (ennreal.one_lt_coe_iff.2 ht)\n         ennreal.coe_ne_top, preimage_Union, inter_Union],\n    { assume i j,\n      simp only [function.on_fun],\n      wlog h : i ≤ j := le_total i j using [i j, j i] tactic.skip,\n      { assume hij,\n        replace hij : i + 1 ≤ j := lt_of_le_of_ne h hij,\n        apply disjoint_left.2 (λ x hx h'x, lt_irrefl (f x) _),\n        calc f x < t ^ (i + 1) : hx.2.2\n        ... ≤ t ^ j : ennreal.zpow_le_of_le (ennreal.one_le_coe_iff.2 ht.le) hij\n        ... ≤ f x : h'x.2.1 },\n      { assume hij,\n        rw disjoint.comm,\n        exact this hij.symm } },\n    { assume n,\n      exact hs.inter (hf measurable_set_Ico) } },\n  rw [A, B, C, add_assoc],\nend\n\nsection metric_space\n\nvariables [metric_space α] [measurable_space α] [opens_measurable_space α]\nvariables [measurable_space β] {x : α} {ε : ℝ}\n\nopen metric\n\n@[measurability]\nlemma measurable_set_ball : measurable_set (metric.ball x ε) :=\nmetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_set_closed_ball : measurable_set (metric.closed_ball x ε) :=\nmetric.is_closed_ball.measurable_set\n\n@[measurability]\nlemma measurable_inf_dist {s : set α} : measurable (λ x, inf_dist x s) :=\n(continuous_inf_dist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_dist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_dist (f x) s) :=\nmeasurable_inf_dist.comp hf\n\n@[measurability]\nlemma measurable_inf_nndist {s : set α} : measurable (λ x, inf_nndist x s) :=\n(continuous_inf_nndist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_nndist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_nndist (f x) s) :=\nmeasurable_inf_nndist.comp hf\n\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable_dist : measurable (λ p : α × α, dist p.1 p.2) :=\ncontinuous_dist.measurable\n\n@[measurability]\nlemma measurable.dist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, dist (f b) (g b)) :=\n(@continuous_dist α _).measurable2 hf hg\n\n@[measurability]\nlemma measurable_nndist : measurable (λ p : α × α, nndist p.1 p.2) :=\ncontinuous_nndist.measurable\n\n@[measurability]\nlemma measurable.nndist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, nndist (f b) (g b)) :=\n(@continuous_nndist α _).measurable2 hf hg\n\nend metric_space\n\nsection emetric_space\n\nvariables [emetric_space α] [measurable_space α] [opens_measurable_space α]\nvariables [measurable_space β] {x : α} {ε : ℝ≥0∞}\n\nopen emetric\n\n@[measurability]\nlemma measurable_set_eball : measurable_set (emetric.ball x ε) :=\nemetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_edist_right : measurable (edist x) :=\n(continuous_const.edist continuous_id).measurable\n\n@[measurability]\nlemma measurable_edist_left : measurable (λ y, edist y x) :=\n(continuous_id.edist continuous_const).measurable\n\n@[measurability]\nlemma measurable_inf_edist {s : set α} : measurable (λ x, inf_edist x s) :=\ncontinuous_inf_edist.measurable\n\n@[measurability]\nlemma measurable.inf_edist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_edist (f x) s) :=\nmeasurable_inf_edist.comp hf\n\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable_edist : measurable (λ p : α × α, edist p.1 p.2) :=\ncontinuous_edist.measurable\n\n@[measurability]\nlemma measurable.edist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, edist (f b) (g b)) :=\n(@continuous_edist α _).measurable2 hf hg\n\n@[measurability]\nlemma ae_measurable.edist {f g : β → α} {μ : measure β}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, edist (f a) (g a)) μ :=\n(@continuous_edist α _).ae_measurable2 hf hg\n\nend emetric_space\n\nnamespace real\nopen measurable_space measure_theory\n\nlemma borel_eq_generate_from_Ioo_rat :\n  borel ℝ = generate_from (⋃(a b : ℚ) (h : a < b), {Ioo a b}) :=\nis_topological_basis_Ioo_rat.borel_eq_generate_from\n\nlemma is_pi_system_Ioo_rat : @is_pi_system ℝ (⋃ (a b : ℚ) (h : a < b), {Ioo a b})  :=\nbegin\n  convert is_pi_system_Ioo (coe : ℚ → ℝ) (coe : ℚ → ℝ),\n  ext x,\n  simp [eq_comm]\nend\n\n/-- The intervals `(-(n + 1), (n + 1))` form a finite spanning sets in the set of open intervals\nwith rational endpoints for a locally finite measure `μ` on `ℝ`. -/\ndef finite_spanning_sets_in_Ioo_rat (μ : measure ℝ) [is_locally_finite_measure μ] :\n  μ.finite_spanning_sets_in (⋃ (a b : ℚ) (h : a < b), {Ioo a b}) :=\n{ set := λ n, Ioo (-(n + 1)) (n + 1),\n  set_mem := λ n,\n    begin\n      simp only [mem_Union, mem_singleton_iff],\n      refine ⟨-(n + 1), n + 1, _, by norm_cast⟩,\n      exact (neg_nonpos.2 (@nat.cast_nonneg ℚ _ (n + 1))).trans_lt n.cast_add_one_pos\n    end,\n  finite := λ n, measure_Ioo_lt_top,\n  spanning := Union_eq_univ_iff.2 $ λ x,\n    ⟨⌊|x|⌋₊, neg_lt.1 ((neg_le_abs_self x).trans_lt (nat.lt_floor_add_one _)),\n      (le_abs_self x).trans_lt (nat.lt_floor_add_one _)⟩ }\n\nlemma measure_ext_Ioo_rat {μ ν : measure ℝ} [is_locally_finite_measure μ]\n  (h : ∀ a b : ℚ, μ (Ioo a b) = ν (Ioo a b)) : μ = ν :=\n(finite_spanning_sets_in_Ioo_rat μ).ext borel_eq_generate_from_Ioo_rat is_pi_system_Ioo_rat $\n  by { simp only [mem_Union, mem_singleton_iff], rintro _ ⟨a, b, -, rfl⟩, apply h }\n\nlemma borel_eq_generate_from_Iio_rat :\n  borel ℝ = generate_from (⋃ a : ℚ, {Iio a}) :=\nbegin\n  let g : measurable_space ℝ := generate_from (⋃ a : ℚ, {Iio a}),\n  refine le_antisymm _ _,\n  { rw borel_eq_generate_from_Ioo_rat,\n    refine generate_from_le (λ t, _),\n    simp only [mem_Union, mem_singleton_iff], rintro ⟨a, b, h, rfl⟩,\n    rw (set.ext (λ x, _) : Ioo (a : ℝ) b = (⋃c>a, (Iio c)ᶜ) ∩ Iio b),\n    { have hg : ∀ q : ℚ, g.measurable_set' (Iio q) :=\n        λ q, generate_measurable.basic (Iio q) (by { simp, exact ⟨_, rfl⟩ }),\n      refine @measurable_set.inter _ g _ _ _ (hg _),\n      refine @measurable_set.bUnion _ _ g _ _ (countable_encodable _) (λ c h, _),\n      exact @measurable_set.compl _ _ g (hg _) },\n    { suffices : x < ↑b → (↑a < x ↔ ∃ (i : ℚ), a < i ∧ ↑i ≤ x), by simpa,\n      refine λ _, ⟨λ h, _, λ ⟨i, hai, hix⟩, (rat.cast_lt.2 hai).trans_le hix⟩,\n      rcases exists_rat_btwn h with ⟨c, ac, cx⟩,\n      exact ⟨c, rat.cast_lt.1 ac, cx.le⟩ } },\n  { refine measurable_space.generate_from_le (λ _, _),\n    simp only [mem_Union, mem_singleton_iff], rintro ⟨r, rfl⟩, exact measurable_set_Iio }\nend\n\nend real\n\nvariable [measurable_space α]\n\n@[measurability]\nlemma measurable_real_to_nnreal : measurable (real.to_nnreal) :=\nnnreal.continuous_of_real.measurable\n\n@[measurability]\nlemma measurable.real_to_nnreal {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, real.to_nnreal (f x)) :=\nmeasurable_real_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.real_to_nnreal {f : α → ℝ} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, real.to_nnreal (f x)) μ :=\nmeasurable_real_to_nnreal.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_real : measurable (coe : ℝ≥0 → ℝ) :=\nnnreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_real {f : α → ℝ≥0} (hf : measurable f) :\n  measurable (λ x, (f x : ℝ)) :=\nmeasurable_coe_nnreal_real.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_real {f : α → ℝ≥0} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ℝ)) μ :=\nmeasurable_coe_nnreal_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_ennreal : measurable (coe : ℝ≥0 → ℝ≥0∞) :=\nennreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_ennreal {f : α → ℝ≥0} (hf : measurable f) :\n  measurable (λ x, (f x : ℝ≥0∞)) :=\nennreal.continuous_coe.measurable.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_ennreal {f : α → ℝ≥0} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ℝ≥0∞)) μ :=\nennreal.continuous_coe.measurable.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable.ennreal_of_real {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, ennreal.of_real (f x)) :=\nennreal.continuous_of_real.measurable.comp hf\n\n/-- The set of finite `ℝ≥0∞` numbers is `measurable_equiv` to `ℝ≥0`. -/\ndef measurable_equiv.ennreal_equiv_nnreal : {r : ℝ≥0∞ | r ≠ ∞} ≃ᵐ ℝ≥0 :=\nennreal.ne_top_homeomorph_nnreal.to_measurable_equiv\n\nnamespace ennreal\n\nlemma measurable_of_measurable_nnreal {f : ℝ≥0∞ → α}\n  (h : measurable (λ p : ℝ≥0, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_singleton ∞\n  (measurable_equiv.ennreal_equiv_nnreal.symm.measurable_comp_iff.1 h)\n\n/-- `ℝ≥0∞` is `measurable_equiv` to `ℝ≥0 ⊕ unit`. -/\ndef ennreal_equiv_sum : ℝ≥0∞ ≃ᵐ ℝ≥0 ⊕ unit :=\n{ measurable_to_fun  := measurable_of_measurable_nnreal measurable_inl,\n  measurable_inv_fun := measurable_sum measurable_coe_nnreal_ennreal\n    (@measurable_const ℝ≥0∞ unit _ _ ∞),\n  .. equiv.option_equiv_sum_punit ℝ≥0 }\n\nopen function (uncurry)\n\nlemma measurable_of_measurable_nnreal_prod [measurable_space β] [measurable_space γ]\n  {f : ℝ≥0∞ × β → γ} (H₁ : measurable (λ p : ℝ≥0 × β, f (p.1, p.2)))\n  (H₂ : measurable (λ x, f (∞, x))) :\n  measurable f :=\nlet e : ℝ≥0∞ × β ≃ᵐ ℝ≥0 × β ⊕ unit × β :=\n  (ennreal_equiv_sum.prod_congr (measurable_equiv.refl β)).trans\n    (measurable_equiv.sum_prod_distrib _ _ _) in\ne.symm.measurable_comp_iff.1 $ measurable_sum H₁ (H₂.comp measurable_id.snd)\n\nlemma measurable_of_measurable_nnreal_nnreal [measurable_space β]\n  {f : ℝ≥0∞ × ℝ≥0∞ → β} (h₁ : measurable (λ p : ℝ≥0 × ℝ≥0, f (p.1, p.2)))\n  (h₂ : measurable (λ r : ℝ≥0, f (∞, r))) (h₃ : measurable (λ r : ℝ≥0, f (r, ∞))) :\n  measurable f :=\nmeasurable_of_measurable_nnreal_prod\n  (measurable_swap_iff.1 $ measurable_of_measurable_nnreal_prod (h₁.comp measurable_swap) h₃)\n  (measurable_of_measurable_nnreal h₂)\n\n@[measurability]\nlemma measurable_of_real : measurable ennreal.of_real :=\nennreal.continuous_of_real.measurable\n\n@[measurability]\nlemma measurable_to_real : measurable ennreal.to_real :=\nennreal.measurable_of_measurable_nnreal measurable_coe_nnreal_real\n\n@[measurability]\nlemma measurable_to_nnreal : measurable ennreal.to_nnreal :=\nennreal.measurable_of_measurable_nnreal measurable_id\n\ninstance : has_measurable_mul₂ ℝ≥0∞ :=\nbegin\n  refine ⟨measurable_of_measurable_nnreal_nnreal _ _ _⟩,\n  { simp only [← ennreal.coe_mul, measurable_mul.coe_nnreal_ennreal] },\n  { simp only [ennreal.top_mul, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const },\n  { simp only [ennreal.mul_top, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const }\nend\n\ninstance : has_measurable_sub₂ ℝ≥0∞ :=\n⟨by apply measurable_of_measurable_nnreal_nnreal;\n  simp [← with_top.coe_sub, continuous_sub.measurable.coe_nnreal_ennreal]⟩\n\ninstance : has_measurable_inv ℝ≥0∞ := ⟨ennreal.continuous_inv.measurable⟩\n\nend ennreal\n\n@[measurability]\nlemma measurable.ennreal_to_nnreal {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, (f x).to_nnreal) :=\nennreal.measurable_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_nnreal {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x).to_nnreal) μ :=\nennreal.measurable_to_nnreal.comp_ae_measurable hf\n\nlemma measurable_coe_nnreal_ennreal_iff {f : α → ℝ≥0} :\n  measurable (λ x, (f x : ℝ≥0∞)) ↔ measurable f :=\n⟨λ h, h.ennreal_to_nnreal, λ h, h.coe_nnreal_ennreal⟩\n\n@[measurability]\nlemma measurable.ennreal_to_real {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, ennreal.to_real (f x)) :=\nennreal.measurable_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_real {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, ennreal.to_real (f x)) μ :=\nennreal.measurable_to_real.comp_ae_measurable hf\n\n/-- note: `ℝ≥0∞` can probably be generalized in a future version of this lemma. -/\n@[measurability]\nlemma measurable.ennreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, measurable (f i)) :\n  measurable (λ x, ∑' i, f i x) :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply measurable_supr,\n  exact λ s, s.measurable_sum (λ i _, h i) }\n\n@[measurability]\nlemma measurable.ennreal_tsum' {ι} [encodable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, measurable (f i)) :\n  measurable (∑' i, f i) :=\nbegin\n  convert measurable.ennreal_tsum h,\n  ext1 x,\n  exact tsum_apply (pi.summable.2 (λ _, ennreal.summable)),\nend\n\n@[measurability]\nlemma measurable.nnreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0} (h : ∀ i, measurable (f i)) :\n  measurable (λ x, ∑' i, f i x) :=\nbegin\n  simp_rw [nnreal.tsum_eq_to_nnreal_tsum],\n  exact (measurable.ennreal_tsum (λ i, (h i).coe_nnreal_ennreal)).ennreal_to_nnreal,\nend\n\n@[measurability]\nlemma ae_measurable.ennreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0∞} {μ : measure α}\n  (h : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ x, ∑' i, f i x) μ :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply ae_measurable_supr,\n  exact λ s, finset.ae_measurable_sum s (λ i _, h i) }\n\n@[measurability]\nlemma measurable_coe_real_ereal : measurable (coe : ℝ → ereal) :=\ncontinuous_coe_real_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_real_ereal {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, (f x : ereal)) :=\nmeasurable_coe_real_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_real_ereal {f : α → ℝ} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ereal)) μ :=\nmeasurable_coe_real_ereal.comp_ae_measurable hf\n\n/-- The set of finite `ereal` numbers is `measurable_equiv` to `ℝ`. -/\ndef measurable_equiv.ereal_equiv_real : ({⊥, ⊤} : set ereal).compl ≃ᵐ ℝ :=\nereal.ne_bot_top_homeomorph_real.to_measurable_equiv\n\nlemma ereal.measurable_of_measurable_real {f : ereal → α}\n  (h : measurable (λ p : ℝ, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_finite {⊥, ⊤} (by simp)\n  (measurable_equiv.ereal_equiv_real.symm.measurable_comp_iff.1 h)\n\n@[measurability]\nlemma measurable_ereal_to_real : measurable ereal.to_real :=\nereal.measurable_of_measurable_real (by simpa using measurable_id)\n\n@[measurability]\nlemma measurable.ereal_to_real {f : α → ereal} (hf : measurable f) :\n  measurable (λ x, (f x).to_real) :=\nmeasurable_ereal_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ereal_to_real {f : α → ereal} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x).to_real) μ :=\nmeasurable_ereal_to_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_ennreal_ereal : measurable (coe : ℝ≥0∞ → ereal) :=\ncontinuous_coe_ennreal_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_ereal_ennreal {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, (f x : ereal)) :=\nmeasurable_coe_ennreal_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_ereal_ennreal {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ereal)) μ :=\nmeasurable_coe_ennreal_ereal.comp_ae_measurable hf\n\nsection normed_group\n\nvariables [normed_group α] [opens_measurable_space α] [measurable_space β]\n\n@[measurability]\nlemma measurable_norm : measurable (norm : α → ℝ) :=\ncontinuous_norm.measurable\n\n@[measurability]\nlemma measurable.norm {f : β → α} (hf : measurable f) : measurable (λ a, norm (f a)) :=\nmeasurable_norm.comp hf\n\n@[measurability]\nlemma ae_measurable.norm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, norm (f a)) μ :=\nmeasurable_norm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_nnnorm : measurable (nnnorm : α → ℝ≥0) :=\ncontinuous_nnnorm.measurable\n\n@[measurability]\nlemma measurable.nnnorm {f : β → α} (hf : measurable f) : measurable (λ a, nnnorm (f a)) :=\nmeasurable_nnnorm.comp hf\n\n@[measurability]\nlemma ae_measurable.nnnorm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, nnnorm (f a)) μ :=\nmeasurable_nnnorm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_ennnorm : measurable (λ x : α, (nnnorm x : ℝ≥0∞)) :=\nmeasurable_nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma measurable.ennnorm {f : β → α} (hf : measurable f) :\n  measurable (λ a, (nnnorm (f a) : ℝ≥0∞)) :=\nhf.nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma ae_measurable.ennnorm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, (nnnorm (f a) : ℝ≥0∞)) μ :=\nmeasurable_ennnorm.comp_ae_measurable hf\n\nend normed_group\n\nsection limits\n\nvariables [measurable_space β] [metric_space β] [borel_space β]\n\nopen metric\n\n/-- A limit (over a general filter) of measurable `ℝ≥0` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal' {ι} {f : ι → α → ℝ≥0} {g : α → ℝ≥0} (u : filter ι)\n  [ne_bot u] [is_countably_generated u] (hf : ∀ i, measurable (f i)) (lim : tendsto f u (𝓝 g)) :\n  measurable g :=\nbegin\n  rcases u.exists_seq_tendsto with ⟨x, hx⟩,\n  rw [tendsto_pi_nhds] at lim, rw [← measurable_coe_nnreal_ennreal_iff],\n  have : ∀ y, liminf at_top (λ n, (f (x n) y : ℝ≥0∞)) = (g y : ℝ≥0∞) :=\n    λ y, ((ennreal.continuous_coe.tendsto (g y)).comp $ (lim y).comp hx).liminf_eq,\n  simp only [← this],\n  show measurable (λ y, liminf at_top (λ n, (f (x n) y : ℝ≥0∞))),\n  exact measurable_liminf (λ n, (hf (x n)).coe_nnreal_ennreal),\nend\n\n/-- A sequential limit of measurable `ℝ≥0` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal {f : ℕ → α → ℝ≥0} {g : α → ℝ≥0}\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f at_top (𝓝 g)) : measurable g :=\nmeasurable_of_tendsto_nnreal' at_top hf lim\n\n/-- A limit (over a general filter) of measurable functions valued in a metric space is measurable.\nThe assumption `hs` can be dropped using `filter.is_countably_generated.has_antitone_basis`, but we\ndon't need that case yet. -/\nlemma measurable_of_tendsto_metric' {ι} {f : ι → α → β} {g : α → β}\n  (u : filter ι) [ne_bot u] [is_countably_generated u]\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f u (𝓝 g)) :\n  measurable g :=\nbegin\n  apply measurable_of_is_closed', intros s h1s h2s h3s,\n  have : measurable (λ x, inf_nndist (g x) s),\n  { suffices : tendsto (λ i x, inf_nndist (f i x) s) u (𝓝 (λ x, inf_nndist (g x) s)),\n      from measurable_of_tendsto_nnreal' u (λ i, (hf i).inf_nndist) this,\n    rw [tendsto_pi_nhds] at lim ⊢, intro x,\n    exact ((continuous_inf_nndist_pt s).tendsto (g x)).comp (lim x) },\n  have h4s : g ⁻¹' s = (λ x, inf_nndist (g x) s) ⁻¹' {0},\n  { ext x, simp [h1s, ← h1s.mem_iff_inf_dist_zero h2s, ← nnreal.coe_eq_zero] },\n  rw [h4s], exact this (measurable_set_singleton 0),\nend\n\n/-- A sequential limit of measurable functions valued in a metric space is measurable. -/\nlemma measurable_of_tendsto_metric {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f at_top (𝓝 g)) :\n  measurable g :=\nmeasurable_of_tendsto_metric' at_top hf lim\n\nlemma ae_measurable_of_tendsto_metric_ae {μ : measure α} {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ n, ae_measurable (f n) μ)\n  (h_ae_tendsto : ∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (𝓝 (g x))) :\n  ae_measurable g μ :=\nbegin\n  let p : α → (ℕ → β) → Prop := λ x f', filter.at_top.tendsto (λ n, f' n) (𝓝 (g x)),\n  let hp : ∀ᵐ x ∂μ, p x (λ n, f n x), from h_ae_tendsto,\n  let ae_seq_lim := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨f 0 x⟩ : nonempty β).some,\n  refine ⟨ae_seq_lim, _, (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨f 0 x⟩ : nonempty β).some)\n    (ae_seq_set hf p) (ae_seq.measure_compl_ae_seq_set_eq_zero hf hp)).symm⟩,\n  refine measurable_of_tendsto_metric (@ae_seq.measurable α β _ _ _ f μ hf p) _,\n  refine tendsto_pi_nhds.mpr (λ x, _),\n  simp_rw [ae_seq, ae_seq_lim],\n  split_ifs with hx,\n  { simp_rw ae_seq.mk_eq_fun_of_mem_ae_seq_set hf hx,\n    exact @ae_seq.fun_prop_of_mem_ae_seq_set α β _ _ _ _ _ _ hf x hx, },\n  { exact tendsto_const_nhds, },\nend\n\nlemma measurable_of_tendsto_metric_ae {μ : measure α} [μ.is_complete] {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ n, measurable (f n))\n  (h_ae_tendsto : ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (g x))) :\n  measurable g :=\nae_measurable_iff_measurable.mp\n  (ae_measurable_of_tendsto_metric_ae (λ i, (hf i).ae_measurable) h_ae_tendsto)\n\nlemma measurable_limit_of_tendsto_metric_ae {μ : measure α} {f : ℕ → α → β}\n  (hf : ∀ n, ae_measurable (f n) μ)\n  (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, filter.at_top.tendsto (λ n, f n x) (𝓝 l)) :\n  ∃ (f_lim : α → β) (hf_lim_meas : measurable f_lim),\n    ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)) :=\nbegin\n  let p : α → (ℕ → β) → Prop := λ x f', ∃ l : β, filter.at_top.tendsto (λ n, f' n) (𝓝 l),\n  have hp_mem : ∀ x, x ∈ ae_seq_set hf p → p x (λ n, f n x),\n    from λ x hx, ae_seq.fun_prop_of_mem_ae_seq_set hf hx,\n  have hμ_compl : μ (ae_seq_set hf p)ᶜ = 0,\n    from ae_seq.measure_compl_ae_seq_set_eq_zero hf h_ae_tendsto,\n  let f_lim : α → β := λ x, dite (x ∈ ae_seq_set hf p) (λ h, (hp_mem x h).some)\n    (λ h, (⟨f 0 x⟩ : nonempty β).some),\n  have hf_lim_conv : ∀ x, x ∈ ae_seq_set hf p → filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)),\n  { intros x hx_conv,\n    simp only [f_lim, hx_conv, dif_pos],\n    exact (hp_mem x hx_conv).some_spec, },\n  have hf_lim : ∀ x, filter.at_top.tendsto (λ n, ae_seq hf p n x) (𝓝 (f_lim x)),\n  { intros x,\n    simp only [f_lim, ae_seq],\n    split_ifs,\n    { rw funext (λ n, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h n),\n      exact (hp_mem x h).some_spec, },\n    { exact tendsto_const_nhds, }, },\n  have h_ae_tendsto_f_lim : ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)),\n  { refine le_antisymm (le_of_eq (measure_mono_null _ hμ_compl)) (zero_le _),\n    exact set.compl_subset_compl.mpr (λ x hx, hf_lim_conv x hx), },\n  have h_f_lim_meas : measurable f_lim,\n    from measurable_of_tendsto_metric (ae_seq.measurable hf p)\n      (tendsto_pi_nhds.mpr (λ x, hf_lim x)),\n  exact ⟨f_lim, h_f_lim_meas, h_ae_tendsto_f_lim⟩,\nend\n\nend limits\n\nnamespace continuous_linear_map\n\nvariables {𝕜 : Type*} [normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E]\nvariables [opens_measurable_space E]\nvariables {F : Type*} [normed_group F] [normed_space 𝕜 F] [measurable_space F] [borel_space F]\n\n@[measurability]\nprotected lemma measurable (L : E →L[𝕜] F) : measurable L :=\nL.continuous.measurable\n\nlemma measurable_comp (L : E →L[𝕜] F) {φ : α → E} (φ_meas : measurable φ) :\n  measurable (λ (a : α), L (φ a)) :=\nL.measurable.comp φ_meas\n\nend continuous_linear_map\n\nnamespace continuous_linear_map\n\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E]\n          {F : Type*} [normed_group F] [normed_space 𝕜 F]\n\ninstance : measurable_space (E →L[𝕜] F) := borel _\n\ninstance : borel_space (E →L[𝕜] F) := ⟨rfl⟩\n\n@[measurability]\nlemma measurable_apply [measurable_space F] [borel_space F] (x : E) :\n  measurable (λ f : E →L[𝕜] F, f x) :=\n(apply 𝕜 F x).continuous.measurable\n\n@[measurability]\nlemma measurable_apply' [measurable_space E] [opens_measurable_space E]\n  [measurable_space F] [borel_space F] :\n  measurable (λ (x : E) (f : E →L[𝕜] F), f x) :=\nmeasurable_pi_lambda _ $ λ f, f.measurable\n\n@[measurability]\nlemma measurable_coe [measurable_space F] [borel_space F] :\n  measurable (λ (f : E →L[𝕜] F) (x : E), f x) :=\nmeasurable_pi_lambda _ measurable_apply\n\nend continuous_linear_map\n\nsection continuous_linear_map_nondiscrete_normed_field\n\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E] [borel_space E]\nvariables {F : Type*} [normed_group F] [normed_space 𝕜 F]\n\n@[measurability]\nlemma measurable.apply_continuous_linear_map  {φ : α → F →L[𝕜] E} (hφ : measurable φ) (v : F) :\n  measurable (λ a, φ a v) :=\n(continuous_linear_map.apply 𝕜 E v).measurable.comp hφ\n\n@[measurability]\nlemma ae_measurable.apply_continuous_linear_map {φ : α → F →L[𝕜] E} {μ : measure α}\n  (hφ : ae_measurable φ μ) (v : F) : ae_measurable (λ a, φ a v) μ :=\n(continuous_linear_map.apply 𝕜 E v).measurable.comp_ae_measurable hφ\n\nend continuous_linear_map_nondiscrete_normed_field\n\nsection normed_space\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] [complete_space 𝕜] [measurable_space 𝕜]\nvariables [borel_space 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E] [borel_space E]\n\nlemma measurable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) :\n  measurable (λ x, f x • c) ↔ measurable f :=\n(closed_embedding_smul_left hc).measurable_embedding.measurable_comp_iff\n\nlemma ae_measurable_smul_const {f : α → 𝕜} {μ : measure α} {c : E} (hc : c ≠ 0) :\n  ae_measurable (λ x, f x • c) μ ↔ ae_measurable f μ :=\n(closed_embedding_smul_left hc).measurable_embedding.ae_measurable_comp_iff\n\nend normed_space\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/measure_theory/constructions/borel_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6548947290421276, "lm_q1q2_score": 0.3351205071840919}}
{"text": "/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.bases\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 \n\nnamespace Mathlib\n\n/-!\n# Lift filters along filter and set functions\n-/\n\nnamespace filter\n\n\n/-- A variant on `bind` using a function `g` taking a set instead of a member of `α`.\nThis is essentially a push-forward along a function mapping each set to a filter. -/\nprotected def lift {α : Type u_1} {β : Type u_2} (f : filter α) (g : set α → filter β) : filter β :=\n  infi fun (s : set α) => infi fun (H : s ∈ f) => g s\n\n/-- If `(p : ι → Prop, s : ι → set α)` is a basis of a filter `f`, `g` is a monotone function\n`set α → filter γ`, and for each `i`, `(pg : β i → Prop, sg : β i → set α)` is a basis\nof the filter `g (s i)`, then `(λ (i : ι) (x : β i), p i ∧ pg i x, λ (i : ι) (x : β i), sg i x)`\nis a basis of the filter `f.lift g`.\n\nThis basis is parametrized by `i : ι` and `x : β i`, so in order to formulate this fact using\n`has_basis` one has to use `Σ i, β i` as the index type, see `filter.has_basis.lift`.\nThis lemma states the corresponding `mem_iff` statement without using a sigma type. -/\ntheorem has_basis.mem_lift_iff {α : Type u_1} {γ : Type u_3} {ι : Type u_2} {p : ι → Prop} {s : ι → set α} {f : filter α} (hf : has_basis f p s) {β : ι → Type u_4} {pg : (i : ι) → β i → Prop} {sg : (i : ι) → β i → set γ} {g : set α → filter γ} (hg : ∀ (i : ι), has_basis (g (s i)) (pg i) (sg i)) (gm : monotone g) : ∀ {s : set γ}, s ∈ filter.lift f g ↔ ∃ (i : ι), ∃ (hi : p i), ∃ (x : β i), ∃ (hx : pg i x), sg i x ⊆ s := sorry\n\n/-- If `(p : ι → Prop, s : ι → set α)` is a basis of a filter `f`, `g` is a monotone function\n`set α → filter γ`, and for each `i`, `(pg : β i → Prop, sg : β i → set α)` is a basis\nof the filter `g (s i)`, then `(λ (i : ι) (x : β i), p i ∧ pg i x, λ (i : ι) (x : β i), sg i x)`\nis a basis of the filter `f.lift g`.\n\nThis basis is parametrized by `i : ι` and `x : β i`, so in order to formulate this fact using\n`has_basis` one has to use `Σ i, β i` as the index type. See also `filter.has_basis.mem_lift_iff`\nfor the corresponding `mem_iff` statement formulated without using a sigma type. -/\ntheorem has_basis.lift {α : Type u_1} {γ : Type u_3} {ι : Type u_2} {p : ι → Prop} {s : ι → set α} {f : filter α} (hf : has_basis f p s) {β : ι → Type u_4} {pg : (i : ι) → β i → Prop} {sg : (i : ι) → β i → set γ} {g : set α → filter γ} (hg : ∀ (i : ι), has_basis (g (s i)) (pg i) (sg i)) (gm : monotone g) : has_basis (filter.lift f g) (fun (i : sigma fun (i : ι) => β i) => p (sigma.fst i) ∧ pg (sigma.fst i) (sigma.snd i))\n  fun (i : sigma fun (i : ι) => β i) => sg (sigma.fst i) (sigma.snd i) := sorry\n\ntheorem mem_lift_sets {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} (hg : monotone g) {s : set β} : s ∈ filter.lift f g ↔ ∃ (t : set α), ∃ (H : t ∈ f), s ∈ g t := sorry\n\ntheorem mem_lift {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} {s : set β} {t : set α} (ht : t ∈ f) (hs : s ∈ g t) : s ∈ filter.lift f g :=\n  iff.mp le_principal_iff\n    ((fun (this : filter.lift f g ≤ principal s) => this)\n      (infi_le_of_le t (infi_le_of_le ht (iff.mpr le_principal_iff hs))))\n\ntheorem lift_le {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} {h : filter β} {s : set α} (hs : s ∈ f) (hg : g s ≤ h) : filter.lift f g ≤ h :=\n  infi_le_of_le s (infi_le_of_le hs hg)\n\ntheorem le_lift {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} {h : filter β} (hh : ∀ (s : set α), s ∈ f → h ≤ g s) : h ≤ filter.lift f g :=\n  le_infi fun (s : set α) => le_infi fun (hs : s ∈ f) => hh s hs\n\ntheorem lift_mono {α : Type u_1} {β : Type u_2} {f₁ : filter α} {f₂ : filter α} {g₁ : set α → filter β} {g₂ : set α → filter β} (hf : f₁ ≤ f₂) (hg : g₁ ≤ g₂) : filter.lift f₁ g₁ ≤ filter.lift f₂ g₂ :=\n  infi_le_infi fun (s : set α) => infi_le_infi2 fun (hs : s ∈ f₂) => Exists.intro (hf hs) (hg s)\n\ntheorem lift_mono' {α : Type u_1} {β : Type u_2} {f : filter α} {g₁ : set α → filter β} {g₂ : set α → filter β} (hg : ∀ (s : set α), s ∈ f → g₁ s ≤ g₂ s) : filter.lift f g₁ ≤ filter.lift f g₂ :=\n  infi_le_infi fun (s : set α) => infi_le_infi fun (hs : s ∈ f) => hg s hs\n\ntheorem tendsto_lift {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → filter β} {m : γ → β} {l : filter γ} : tendsto m l (filter.lift f g) ↔ ∀ (s : set α), s ∈ f → tendsto m l (g s) := sorry\n\ntheorem map_lift_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → filter β} {m : β → γ} (hg : monotone g) : map m (filter.lift f g) = filter.lift f (map m ∘ g) := sorry\n\ntheorem comap_lift_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → filter β} {m : γ → β} (hg : monotone g) : comap m (filter.lift f g) = filter.lift f (comap m ∘ g) := sorry\n\ntheorem comap_lift_eq2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {m : β → α} {g : set β → filter γ} (hg : monotone g) : filter.lift (comap m f) g = filter.lift f (g ∘ set.preimage m) := sorry\n\ntheorem map_lift_eq2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set β → filter γ} {m : α → β} (hg : monotone g) : filter.lift (map m f) g = filter.lift f (g ∘ set.image m) := sorry\n\ntheorem lift_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : filter β} {h : set α → set β → filter γ} : (filter.lift f fun (s : set α) => filter.lift g (h s)) =\n  filter.lift g fun (t : set β) => filter.lift f fun (s : set α) => h s t := sorry\n\ntheorem lift_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → filter β} {h : set β → filter γ} (hg : monotone g) : filter.lift (filter.lift f g) h = filter.lift f fun (s : set α) => filter.lift (g s) h := sorry\n\ntheorem lift_lift_same_le_lift {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → set α → filter β} : (filter.lift f fun (s : set α) => filter.lift f (g s)) ≤ filter.lift f fun (s : set α) => g s s := sorry\n\ntheorem lift_lift_same_eq_lift {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → set α → filter β} (hg₁ : ∀ (s : set α), monotone fun (t : set α) => g s t) (hg₂ : ∀ (t : set α), monotone fun (s : set α) => g s t) : (filter.lift f fun (s : set α) => filter.lift f (g s)) = filter.lift f fun (s : set α) => g s s := sorry\n\ntheorem lift_principal {α : Type u_1} {β : Type u_2} {g : set α → filter β} {s : set α} (hg : monotone g) : filter.lift (principal s) g = g s :=\n  le_antisymm (infi_le_of_le s (infi_le (fun (H : s ∈ principal s) => g s) (set.subset.refl s)))\n    (le_infi fun (t : set α) => le_infi fun (hi : t ∈ principal s) => hg hi)\n\ntheorem monotone_lift {α : Type u_1} {β : Type u_2} {γ : Type u_3} [preorder γ] {f : γ → filter α} {g : γ → set α → filter β} (hf : monotone f) (hg : monotone g) : monotone fun (c : γ) => filter.lift (f c) (g c) :=\n  fun (a b : γ) (h : a ≤ b) => lift_mono (hf h) (hg h)\n\ntheorem lift_ne_bot_iff {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} (hm : monotone g) : ne_bot (filter.lift f g) ↔ ∀ (s : set α), s ∈ f → ne_bot (g s) := sorry\n\n@[simp] theorem lift_const {α : Type u_1} {β : Type u_2} {f : filter α} {g : filter β} : (filter.lift f fun (x : set α) => g) = g :=\n  le_antisymm (lift_le univ_mem_sets (le_refl g)) (le_lift fun (s : set α) (hs : s ∈ f) => le_refl g)\n\n@[simp] theorem lift_inf {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → filter β} {h : set α → filter β} : (filter.lift f fun (x : set α) => g x ⊓ h x) = filter.lift f g ⊓ filter.lift f h := sorry\n\n@[simp] theorem lift_principal2 {α : Type u_1} {f : filter α} : filter.lift f principal = f := sorry\n\ntheorem lift_infi {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → filter α} {g : set α → filter β} [hι : Nonempty ι] (hg : ∀ {s t : set α}, g s ⊓ g t = g (s ∩ t)) : filter.lift (infi f) g = infi fun (i : ι) => filter.lift (f i) g := sorry\n\n/-- Specialize `lift` to functions `set α → set β`. This can be viewed as a generalization of `map`.\nThis is essentially a push-forward along a function mapping each set to a set. -/\nprotected def lift' {α : Type u_1} {β : Type u_2} (f : filter α) (h : set α → set β) : filter β :=\n  filter.lift f (principal ∘ h)\n\ntheorem mem_lift' {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} {t : set α} (ht : t ∈ f) : h t ∈ filter.lift' f h :=\n  iff.mp le_principal_iff\n    ((fun (this : filter.lift' f h ≤ principal (h t)) => this)\n      (infi_le_of_le t (infi_le_of_le ht (le_refl (function.comp principal h t)))))\n\ntheorem tendsto_lift' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {h : set α → set β} {m : γ → β} {l : filter γ} : tendsto m l (filter.lift' f h) ↔ ∀ (s : set α), s ∈ f → filter.eventually (fun (a : γ) => m a ∈ h s) l := sorry\n\ntheorem has_basis.lift' {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} {ι : Type u_3} {p : ι → Prop} {s : ι → set α} (hf : has_basis f p s) (hh : monotone h) : has_basis (filter.lift' f h) p (h ∘ s) := sorry\n\ntheorem mem_lift'_sets {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} (hh : monotone h) {s : set β} : s ∈ filter.lift' f h ↔ ∃ (t : set α), ∃ (H : t ∈ f), h t ⊆ s :=\n  mem_lift_sets (monotone.comp monotone_principal hh)\n\ntheorem eventually_lift'_iff {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} (hh : monotone h) {p : β → Prop} : filter.eventually (fun (y : β) => p y) (filter.lift' f h) ↔ ∃ (t : set α), ∃ (H : t ∈ f), ∀ (y : β), y ∈ h t → p y :=\n  mem_lift'_sets hh\n\ntheorem lift'_le {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → set β} {h : filter β} {s : set α} (hs : s ∈ f) (hg : principal (g s) ≤ h) : filter.lift' f g ≤ h :=\n  lift_le hs hg\n\ntheorem lift'_mono {α : Type u_1} {β : Type u_2} {f₁ : filter α} {f₂ : filter α} {h₁ : set α → set β} {h₂ : set α → set β} (hf : f₁ ≤ f₂) (hh : h₁ ≤ h₂) : filter.lift' f₁ h₁ ≤ filter.lift' f₂ h₂ :=\n  lift_mono hf fun (s : set α) => iff.mpr principal_mono (hh s)\n\ntheorem lift'_mono' {α : Type u_1} {β : Type u_2} {f : filter α} {h₁ : set α → set β} {h₂ : set α → set β} (hh : ∀ (s : set α), s ∈ f → h₁ s ⊆ h₂ s) : filter.lift' f h₁ ≤ filter.lift' f h₂ :=\n  infi_le_infi fun (s : set α) => infi_le_infi fun (hs : s ∈ f) => iff.mpr principal_mono (hh s hs)\n\ntheorem lift'_cong {α : Type u_1} {β : Type u_2} {f : filter α} {h₁ : set α → set β} {h₂ : set α → set β} (hh : ∀ (s : set α), s ∈ f → h₁ s = h₂ s) : filter.lift' f h₁ = filter.lift' f h₂ :=\n  le_antisymm (lift'_mono' fun (s : set α) (hs : s ∈ f) => le_of_eq (hh s hs))\n    (lift'_mono' fun (s : set α) (hs : s ∈ f) => le_of_eq (Eq.symm (hh s hs)))\n\ntheorem map_lift'_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {h : set α → set β} {m : β → γ} (hh : monotone h) : map m (filter.lift' f h) = filter.lift' f (set.image m ∘ h) := sorry\n\ntheorem map_lift'_eq2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set β → set γ} {m : α → β} (hg : monotone g) : filter.lift' (map m f) g = filter.lift' f (g ∘ set.image m) :=\n  map_lift_eq2 (monotone.comp monotone_principal hg)\n\ntheorem comap_lift'_eq {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {h : set α → set β} {m : γ → β} (hh : monotone h) : comap m (filter.lift' f h) = filter.lift' f (set.preimage m ∘ h) := sorry\n\ntheorem comap_lift'_eq2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {m : β → α} {g : set β → set γ} (hg : monotone g) : filter.lift' (comap m f) g = filter.lift' f (g ∘ set.preimage m) :=\n  comap_lift_eq2 (monotone.comp monotone_principal hg)\n\ntheorem lift'_principal {α : Type u_1} {β : Type u_2} {h : set α → set β} {s : set α} (hh : monotone h) : filter.lift' (principal s) h = principal (h s) :=\n  lift_principal (monotone.comp monotone_principal hh)\n\ntheorem lift'_pure {α : Type u_1} {β : Type u_2} {h : set α → set β} {a : α} (hh : monotone h) : filter.lift' (pure a) h = principal (h (singleton a)) := sorry\n\ntheorem lift'_bot {α : Type u_1} {β : Type u_2} {h : set α → set β} (hh : monotone h) : filter.lift' ⊥ h = principal (h ∅) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter.lift' ⊥ h = principal (h ∅))) (Eq.symm principal_empty)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (filter.lift' (principal ∅) h = principal (h ∅))) (lift'_principal hh)))\n      (Eq.refl (principal (h ∅))))\n\ntheorem principal_le_lift' {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} {t : set β} (hh : ∀ (s : set α), s ∈ f → t ⊆ h s) : principal t ≤ filter.lift' f h :=\n  le_infi fun (s : set α) => le_infi fun (hs : s ∈ f) => iff.mpr principal_mono (hh s hs)\n\ntheorem monotone_lift' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [preorder γ] {f : γ → filter α} {g : γ → set α → set β} (hf : monotone f) (hg : monotone g) : monotone fun (c : γ) => filter.lift' (f c) (g c) :=\n  fun (a b : γ) (h : a ≤ b) => lift'_mono (hf h) (hg h)\n\ntheorem lift_lift'_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → set β} {h : set β → filter γ} (hg : monotone g) (hh : monotone h) : filter.lift (filter.lift' f g) h = filter.lift f fun (s : set α) => h (g s) := sorry\n\ntheorem lift'_lift'_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → set β} {h : set β → set γ} (hg : monotone g) (hh : monotone h) : filter.lift' (filter.lift' f g) h = filter.lift' f fun (s : set α) => h (g s) :=\n  lift_lift'_assoc hg (monotone.comp monotone_principal hh)\n\ntheorem lift'_lift_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : filter α} {g : set α → filter β} {h : set β → set γ} (hg : monotone g) : filter.lift' (filter.lift f g) h = filter.lift f fun (s : set α) => filter.lift' (g s) h :=\n  lift_assoc hg\n\ntheorem lift_lift'_same_le_lift' {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → set α → set β} : (filter.lift f fun (s : set α) => filter.lift' f (g s)) ≤ filter.lift' f fun (s : set α) => g s s :=\n  lift_lift_same_le_lift\n\ntheorem lift_lift'_same_eq_lift' {α : Type u_1} {β : Type u_2} {f : filter α} {g : set α → set α → set β} (hg₁ : ∀ (s : set α), monotone fun (t : set α) => g s t) (hg₂ : ∀ (t : set α), monotone fun (s : set α) => g s t) : (filter.lift f fun (s : set α) => filter.lift' f (g s)) = filter.lift' f fun (s : set α) => g s s :=\n  lift_lift_same_eq_lift (fun (s : set α) => monotone.comp monotone_principal (hg₁ s))\n    fun (t : set α) => monotone.comp monotone_principal (hg₂ t)\n\ntheorem lift'_inf_principal_eq {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} {s : set β} : filter.lift' f h ⊓ principal s = filter.lift' f fun (t : set α) => h t ∩ s := sorry\n\ntheorem lift'_ne_bot_iff {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} (hh : monotone h) : ne_bot (filter.lift' f h) ↔ ∀ (s : set α), s ∈ f → set.nonempty (h s) := sorry\n\n@[simp] theorem lift'_id {α : Type u_1} {f : filter α} : filter.lift' f id = f :=\n  lift_principal2\n\ntheorem le_lift' {α : Type u_1} {β : Type u_2} {f : filter α} {h : set α → set β} {g : filter β} (h_le : ∀ (s : set α), s ∈ f → h s ∈ g) : g ≤ filter.lift' f h := sorry\n\ntheorem lift_infi' {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → filter α} {g : set α → filter β} [Nonempty ι] (hf : directed ge f) (hg : monotone g) : filter.lift (infi f) g = infi fun (i : ι) => filter.lift (f i) g := sorry\n\ntheorem lift'_infi {α : Type u_1} {β : Type u_2} {ι : Sort u_4} {f : ι → filter α} {g : set α → set β} [Nonempty ι] (hg : ∀ {s t : set α}, g s ∩ g t = g (s ∩ t)) : filter.lift' (infi f) g = infi fun (i : ι) => filter.lift' (f i) g := sorry\n\ntheorem lift'_inf {α : Type u_1} {β : Type u_2} (f : filter α) (g : filter α) {s : set α → set β} (hs : ∀ {t₁ t₂ : set α}, s t₁ ∩ s t₂ = s (t₁ ∩ t₂)) : filter.lift' (f ⊓ g) s = filter.lift' f s ⊓ filter.lift' g s := sorry\n\ntheorem comap_eq_lift' {α : Type u_1} {β : Type u_2} {f : filter β} {m : α → β} : comap m f = filter.lift' f (set.preimage m) :=\n  filter.ext fun (s : set α) => iff.symm (mem_lift'_sets set.monotone_preimage)\n\ntheorem lift'_infi_powerset {α : Type u_1} {ι : Sort u_4} [Nonempty ι] {f : ι → filter α} : filter.lift' (infi f) set.powerset = infi fun (i : ι) => filter.lift' (f i) set.powerset :=\n  lift'_infi fun (_x _x_1 : set α) => Eq.symm (set.powerset_inter _x _x_1)\n\ntheorem lift'_inf_powerset {α : Type u_1} (f : filter α) (g : filter α) : filter.lift' (f ⊓ g) set.powerset = filter.lift' f set.powerset ⊓ filter.lift' g set.powerset :=\n  lift'_inf f g fun (_x _x_1 : set α) => Eq.symm (set.powerset_inter _x _x_1)\n\ntheorem eventually_lift'_powerset {α : Type u_1} {f : filter α} {p : set α → Prop} : filter.eventually (fun (s : set α) => p s) (filter.lift' f set.powerset) ↔\n  ∃ (s : set α), ∃ (H : s ∈ f), ∀ (t : set α), t ⊆ s → p t :=\n  eventually_lift'_iff set.monotone_powerset\n\ntheorem eventually_lift'_powerset' {α : Type u_1} {f : filter α} {p : set α → Prop} (hp : ∀ {s t : set α}, s ⊆ t → p t → p s) : filter.eventually (fun (s : set α) => p s) (filter.lift' f set.powerset) ↔ ∃ (s : set α), ∃ (H : s ∈ f), p s := sorry\n\nprotected instance lift'_powerset_ne_bot {α : Type u_1} (f : filter α) : ne_bot (filter.lift' f set.powerset) :=\n  iff.mpr (lift'_ne_bot_iff set.monotone_powerset) fun (_x : set α) (_x_1 : _x ∈ f) => set.powerset_nonempty\n\ntheorem tendsto_lift'_powerset_mono {α : Type u_1} {β : Type u_2} {la : filter α} {lb : filter β} {s : α → set β} {t : α → set β} (ht : tendsto t la (filter.lift' lb set.powerset)) (hst : filter.eventually (fun (x : α) => s x ⊆ t x) la) : tendsto s la (filter.lift' lb set.powerset) := sorry\n\n@[simp] theorem eventually_lift'_powerset_forall {α : Type u_1} {f : filter α} {p : α → Prop} : filter.eventually (fun (s : set α) => ∀ (x : α), x ∈ s → p x) (filter.lift' f set.powerset) ↔\n  filter.eventually (fun (x : α) => p x) f := sorry\n\ntheorem eventually.lift'_powerset {α : Type u_1} {f : filter α} {p : α → Prop} : filter.eventually (fun (x : α) => p x) f →\n  filter.eventually (fun (s : set α) => ∀ (x : α), x ∈ s → p x) (filter.lift' f set.powerset) :=\n  iff.mpr eventually_lift'_powerset_forall\n\n@[simp] theorem eventually_lift'_powerset_eventually {α : Type u_1} {f : filter α} {g : filter α} {p : α → Prop} : filter.eventually (fun (s : set α) => filter.eventually (fun (x : α) => x ∈ s → p x) g) (filter.lift' f set.powerset) ↔\n  filter.eventually (fun (x : α) => p x) (f ⊓ g) := sorry\n\ntheorem prod_def {α : Type u_1} {β : Type u_2} {f : filter α} {g : filter β} : filter.prod f g = filter.lift f fun (s : set α) => filter.lift' g (set.prod s) := sorry\n\ntheorem prod_same_eq {α : Type u_1} {f : filter α} : filter.prod f f = filter.lift' f fun (t : set α) => set.prod t t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter.prod f f = filter.lift' f fun (t : set α) => set.prod t t)) prod_def))\n    (lift_lift'_same_eq_lift' (fun (s : set α) => set.monotone_prod monotone_const monotone_id)\n      fun (t : set α) => set.monotone_prod monotone_id monotone_const)\n\ntheorem mem_prod_same_iff {α : Type u_1} {f : filter α} {s : set (α × α)} : s ∈ filter.prod f f ↔ ∃ (t : set α), ∃ (H : t ∈ f), set.prod t t ⊆ s := sorry\n\ntheorem tendsto_prod_self_iff {α : Type u_1} {β : Type u_2} {f : α × α → β} {x : filter α} {y : filter β} : tendsto f (filter.prod x x) y ↔\n  ∀ (W : set β) (H : W ∈ y), ∃ (U : set α), ∃ (H : U ∈ x), ∀ (x x' : α), x ∈ U → x' ∈ U → f (x, x') ∈ W := sorry\n\ntheorem prod_lift_lift {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} {f₁ : filter α₁} {f₂ : filter α₂} {g₁ : set α₁ → filter β₁} {g₂ : set α₂ → filter β₂} (hg₁ : monotone g₁) (hg₂ : monotone g₂) : filter.prod (filter.lift f₁ g₁) (filter.lift f₂ g₂) =\n  filter.lift f₁ fun (s : set α₁) => filter.lift f₂ fun (t : set α₂) => filter.prod (g₁ s) (g₂ t) := sorry\n\ntheorem prod_lift'_lift' {α₁ : Type u_5} {α₂ : Type u_6} {β₁ : Type u_7} {β₂ : Type u_8} {f₁ : filter α₁} {f₂ : filter α₂} {g₁ : set α₁ → set β₁} {g₂ : set α₂ → set β₂} (hg₁ : monotone g₁) (hg₂ : monotone g₂) : filter.prod (filter.lift' f₁ g₁) (filter.lift' f₂ g₂) =\n  filter.lift f₁ fun (s : set α₁) => filter.lift' f₂ fun (t : set α₂) => set.prod (g₁ s) (g₂ t) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/filter/lift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.33510136266083007}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.eq_to_hom\nimport combinatorics.quiver.path\n\n/-!\n# The category paths on a quiver.\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnamespace category_theory\n\nsection\n\n/--\nA type synonym for the category of paths in a quiver.\n-/\ndef paths (V : Type u₁) : Type u₁ := V\n\ninstance (V : Type u₁) [inhabited V] : inhabited (paths V) := ⟨(default : V)⟩\n\nvariables (V : Type u₁) [quiver.{v₁+1} V]\n\nnamespace paths\n\ninstance category_paths : category.{max u₁ v₁} (paths V) :=\n{ hom := λ (X Y : V), quiver.path X Y,\n  id := λ X, quiver.path.nil,\n  comp := λ X Y Z f g, quiver.path.comp f g, }\n\nvariables {V}\n\n/--\nThe inclusion of a quiver `V` into its path category, as a prefunctor.\n-/\n@[simps]\ndef of : prefunctor V (paths V) :=\n{ obj := λ X, X,\n  map := λ X Y f, f.to_path, }\n\nlocal attribute [ext] functor.ext\n\n/-- Two functors out of a path category are equal when they agree on singleton paths. -/\n@[ext]\nlemma ext_functor {C} [category C]\n  {F G : paths V ⥤ C}\n  (h_obj : F.obj = G.obj)\n  (h : ∀ (a b : V) (e : a ⟶ b), F.map e.to_path =\n  eq_to_hom (congr_fun h_obj a) ≫ G.map e.to_path ≫ eq_to_hom (congr_fun h_obj.symm b)) :\n  F = G :=\nbegin\n  ext X Y f,\n  { induction f with Y' Z' g e ih,\n    { erw [F.map_id, G.map_id, category.id_comp, eq_to_hom_trans, eq_to_hom_refl], },\n    { erw [F.map_comp g e.to_path, G.map_comp g e.to_path, ih, h],\n      simp only [category.id_comp, eq_to_hom_refl, eq_to_hom_trans_assoc, category.assoc], }, },\n  { intro X, rw h_obj, }\nend\n\nend paths\n\nvariables (W : Type u₂) [quiver.{v₂+1} W]\n\n-- A restatement of `prefunctor.map_path_comp` using `f ≫ g` instead of `f.comp g`.\n@[simp] lemma prefunctor.map_path_comp' (F : prefunctor V W)\n  {X Y Z : paths V} (f : X ⟶ Y) (g : Y ⟶ Z) :\n  F.map_path (f ≫ g) = (F.map_path f).comp (F.map_path g) :=\nprefunctor.map_path_comp _ _ _\n\nend\n\nsection\n\nvariables {C : Type u₁} [category.{v₁} C]\n\nopen quiver\n\n/-- A path in a category can be composed to a single morphism. -/\n@[simp]\ndef compose_path {X : C} : Π {Y : C} (p : path X Y), X ⟶ Y\n| _ path.nil := 𝟙 X\n| _ (path.cons p e) := compose_path p ≫ e\n\n@[simp]\nlemma compose_path_comp {X Y Z : C} (f : path X Y) (g : path Y Z) :\n  compose_path (f.comp g) = compose_path f ≫ compose_path g :=\nbegin\n  induction g with Y' Z' g e ih,\n  { simp, },\n  { simp [ih], },\nend\n\nend\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/path_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573377, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3351013546751533}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport data.pi.algebra\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.images\nimport category_theory.isomorphism_classes\nimport category_theory.limits.shapes.zero_objects\n\n/-!\n# Zero morphisms and zero objects\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. (Notice this is extra\nstructure, not merely a property.)\n\nA category \"has a zero object\" if it has an object which is both initial and terminal. Having a\nzero object provides zero morphisms, as the unique morphisms factoring through the zero object.\n\n## References\n\n* https://en.wikipedia.org/wiki/Zero_morphism\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n-/\n\nnoncomputable theory\n\nuniverses v u\nuniverses v' u'\n\nopen category_theory\nopen category_theory.category\nopen_locale classical\n\nnamespace category_theory.limits\n\nvariables (C : Type u) [category.{v} C]\nvariables (D : Type u') [category.{v'} D]\n\n/-- A category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. -/\nclass has_zero_morphisms :=\n[has_zero : Π X Y : C, has_zero (X ⟶ Y)]\n(comp_zero' : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) . obviously)\n(zero_comp' : ∀ (X : C) {Y Z : C} (f : Y ⟶ Z), (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) . obviously)\n\nattribute [instance] has_zero_morphisms.has_zero\nrestate_axiom has_zero_morphisms.comp_zero'\nrestate_axiom has_zero_morphisms.zero_comp'\n\nvariables {C}\n\n@[simp] lemma comp_zero [has_zero_morphisms C] {X Y : C} {f : X ⟶ Y} {Z : C} :\n  f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) := has_zero_morphisms.comp_zero f Z\n@[simp] lemma zero_comp [has_zero_morphisms C] {X : C} {Y Z : C} {f : Y ⟶ Z} :\n  (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) := has_zero_morphisms.zero_comp X f\n\ninstance has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) :=\n{ has_zero := by tidy }\n\ninstance has_zero_morphisms_punit : has_zero_morphisms (discrete punit) :=\n{ has_zero := by tidy }\n\nnamespace has_zero_morphisms\nvariables {C}\n\n/-- This lemma will be immediately superseded by `ext`, below. -/\nprivate lemma ext_aux (I J : has_zero_morphisms C)\n  (w : ∀ X Y : C, (@has_zero_morphisms.has_zero _ _ I X Y).zero =\n    (@has_zero_morphisms.has_zero _ _ J X Y).zero) : I = J :=\nbegin\n  casesI I, casesI J,\n  congr,\n  { ext X Y,\n    exact w X Y },\n  { apply proof_irrel_heq, },\n  { apply proof_irrel_heq, }\nend\n\n/--\nIf you're tempted to use this lemma \"in the wild\", you should probably\ncarefully consider whether you've made a mistake in allowing two\ninstances of `has_zero_morphisms` to exist at all.\n\nSee, particularly, the note on `zero_morphisms_of_zero_object` below.\n-/\nlemma ext (I J : has_zero_morphisms C) : I = J :=\nbegin\n  apply ext_aux,\n  intros X Y,\n  rw ←@has_zero_morphisms.comp_zero _ _ I X X (@has_zero_morphisms.has_zero _ _ J X X).zero,\n  rw @has_zero_morphisms.zero_comp _ _ J,\nend\n\ninstance : subsingleton (has_zero_morphisms C) :=\n⟨ext⟩\n\nend has_zero_morphisms\n\nopen opposite has_zero_morphisms\n\ninstance has_zero_morphisms_opposite [has_zero_morphisms C] :\n  has_zero_morphisms Cᵒᵖ :=\n{ has_zero := λ X Y, ⟨(0 : unop Y ⟶ unop X).op⟩,\n  comp_zero' := λ X Y f Z, congr_arg quiver.hom.op (has_zero_morphisms.zero_comp (unop Z) f.unop),\n  zero_comp' := λ X Y Z f, congr_arg quiver.hom.op (has_zero_morphisms.comp_zero f.unop (unop X)), }\n\nsection\nvariables {C} [has_zero_morphisms C]\n\nlemma zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [mono g] (h : f ≫ g = 0) : f = 0 :=\nby { rw [←zero_comp, cancel_mono] at h, exact h }\n\nlemma zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [epi f] (h : f ≫ g = 0) : g = 0 :=\nby { rw [←comp_zero, cancel_epi] at h, exact h }\n\nlemma eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [has_image f] (w : image.ι f = 0) : f = 0 :=\nby rw [←image.fac f, w, has_zero_morphisms.comp_zero]\n\nlemma nonzero_image_of_nonzero {X Y : C} {f : X ⟶ Y} [has_image f] (w : f ≠ 0) : image.ι f ≠ 0 :=\nλ h, w (eq_zero_of_image_eq_zero h)\nend\n\nsection\n\nvariables [has_zero_morphisms D]\n\ninstance : has_zero_morphisms (C ⥤ D) :=\n{ has_zero := λ F G, ⟨{ app := λ X, 0, }⟩ }\n\n@[simp] lemma zero_app (F G : C ⥤ D) (j : C) : (0 : F ⟶ G).app j = 0 := rfl\n\nend\n\nnamespace is_zero\nvariables [has_zero_morphisms C]\n\nlemma eq_zero_of_src {X Y : C} (o : is_zero X) (f : X ⟶ Y) : f = 0 :=\no.eq_of_src _ _\n\nlemma eq_zero_of_tgt {X Y : C} (o : is_zero Y) (f : X ⟶ Y) : f = 0 :=\no.eq_of_tgt _ _\n\nlemma iff_id_eq_zero (X : C) : is_zero X ↔ (𝟙 X = 0) :=\n⟨λ h, h.eq_of_src _ _,\n λ h, ⟨\n  λ Y, ⟨⟨⟨0⟩, λ f, by { rw [←id_comp f, ←id_comp default, h, zero_comp, zero_comp], }⟩⟩,\n  λ Y, ⟨⟨⟨0⟩, λ f, by { rw [←comp_id f, ←comp_id default, h, comp_zero, comp_zero], }⟩⟩⟩⟩\n\nlemma of_mono_zero (X Y : C) [mono (0 : X ⟶ Y)] : is_zero X :=\n(iff_id_eq_zero X).mpr ((cancel_mono (0 : X ⟶ Y)).1 (by simp))\n\nlemma of_epi_zero (X Y : C) [epi (0 : X ⟶ Y)] : is_zero Y :=\n(iff_id_eq_zero Y).mpr ((cancel_epi (0 : X ⟶ Y)).1 (by simp))\n\nlemma of_mono_eq_zero {X Y : C} (f : X ⟶ Y) [mono f] (h : f = 0) : is_zero X :=\nby { unfreezingI { subst h, }, apply of_mono_zero X Y, }\n\nlemma of_epi_eq_zero {X Y : C} (f : X ⟶ Y) [epi f] (h : f = 0) : is_zero Y :=\nby { unfreezingI { subst h, }, apply of_epi_zero X Y, }\n\nlemma iff_is_split_mono_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_mono f] : is_zero X ↔ f = 0 :=\nbegin\n  rw iff_id_eq_zero,\n  split,\n  { intro h, rw [←category.id_comp f, h, zero_comp], },\n  { intro h, rw [←is_split_mono.id f], simp [h], },\nend\n\nlemma iff_is_split_epi_eq_zero {X Y : C} (f : X ⟶ Y) [is_split_epi f] : is_zero Y ↔ f = 0 :=\nbegin\n  rw iff_id_eq_zero,\n  split,\n  { intro h, rw [←category.comp_id f, h, comp_zero], },\n  { intro h, rw [←is_split_epi.id f], simp [h], },\nend\n\nlemma of_mono {X Y : C} (f : X ⟶ Y) [mono f] (i : is_zero Y) : is_zero X :=\nbegin\n  unfreezingI { have hf := i.eq_zero_of_tgt f, subst hf, },\n  exact is_zero.of_mono_zero X Y,\nend\n\nlemma of_epi {X Y : C} (f : X ⟶ Y) [epi f] (i : is_zero X) : is_zero Y :=\nbegin\n  unfreezingI { have hf := i.eq_zero_of_src f, subst hf, },\n  exact is_zero.of_epi_zero X Y,\nend\n\nend is_zero\n\n/-- A category with a zero object has zero morphisms.\n\n    It is rarely a good idea to use this. Many categories that have a zero object have zero\n    morphisms for some other reason, for example from additivity. Library code that uses\n    `zero_morphisms_of_zero_object` will then be incompatible with these categories because\n    the `has_zero_morphisms` instances will not be definitionally equal. For this reason library\n    code should generally ask for an instance of `has_zero_morphisms` separately, even if it already\n    asks for an instance of `has_zero_objects`. -/\ndef is_zero.has_zero_morphisms {O : C} (hO : is_zero O) : has_zero_morphisms C :=\n{ has_zero := λ X Y,\n  { zero := hO.from X ≫ hO.to Y },\n  zero_comp' := λ X Y Z f, by { rw category.assoc, congr, apply hO.eq_of_src, },\n  comp_zero' := λ X Y Z f, by { rw ←category.assoc, congr, apply hO.eq_of_tgt, }}\n\nnamespace has_zero_object\n\nvariables [has_zero_object C]\nopen_locale zero_object\n\n/-- A category with a zero object has zero morphisms.\n\n    It is rarely a good idea to use this. Many categories that have a zero object have zero\n    morphisms for some other reason, for example from additivity. Library code that uses\n    `zero_morphisms_of_zero_object` will then be incompatible with these categories because\n    the `has_zero_morphisms` instances will not be definitionally equal. For this reason library\n    code should generally ask for an instance of `has_zero_morphisms` separately, even if it already\n    asks for an instance of `has_zero_objects`. -/\ndef zero_morphisms_of_zero_object : has_zero_morphisms C :=\n{ has_zero := λ X Y,\n  { zero := (default : X ⟶ 0) ≫ default },\n  zero_comp' := λ X Y Z f, by { dunfold has_zero.zero, rw category.assoc, congr, },\n  comp_zero' := λ X Y Z f, by { dunfold has_zero.zero, rw ←category.assoc, congr, }}\n\nsection has_zero_morphisms\nvariables [has_zero_morphisms C]\n\n@[simp] lemma zero_iso_is_initial_hom {X : C} (t : is_initial X) :\n  (zero_iso_is_initial t).hom = 0 :=\nby ext\n\n@[simp] lemma zero_iso_is_initial_inv {X : C} (t : is_initial X) :\n  (zero_iso_is_initial t).inv = 0 :=\nby ext\n\n@[simp] lemma zero_iso_is_terminal_hom {X : C} (t : is_terminal X) :\n  (zero_iso_is_terminal t).hom = 0 :=\nby ext\n\n@[simp] lemma zero_iso_is_terminal_inv {X : C} (t : is_terminal X) :\n  (zero_iso_is_terminal t).inv = 0 :=\nby ext\n\n@[simp] lemma zero_iso_initial_hom [has_initial C] : zero_iso_initial.hom = (0 : 0 ⟶ ⊥_ C) :=\nby ext\n\n@[simp] lemma zero_iso_initial_inv [has_initial C] : zero_iso_initial.inv = (0 : ⊥_ C ⟶ 0) :=\nby ext\n\n@[simp] \n\n@[simp] lemma zero_iso_terminal_inv [has_terminal C] : zero_iso_terminal.inv = (0 : ⊤_ C ⟶ 0) :=\nby ext\n\nend has_zero_morphisms\n\nopen_locale zero_object\n\ninstance {B : Type*} [category B] : has_zero_object (B ⥤ C) :=\n(((category_theory.functor.const B).obj (0 : C)).is_zero $ λ X, is_zero_zero _).has_zero_object\n\nend has_zero_object\n\nopen_locale zero_object\n\nvariables {D}\n\n@[simp] lemma is_zero.map [has_zero_object D] [has_zero_morphisms D] {F : C ⥤ D} (hF : is_zero F)\n  {X Y : C} (f : X ⟶ Y) : F.map f = 0 :=\n(hF.obj _).eq_of_src _ _\n\n@[simp] lemma _root_.category_theory.functor.zero_obj [has_zero_object D]\n  (X : C) : is_zero ((0 : C ⥤ D).obj X) :=\n(is_zero_zero _).obj _\n\n@[simp] lemma _root_.category_theory.zero_map [has_zero_object D] [has_zero_morphisms D]\n  {X Y : C} (f : X ⟶ Y) : (0 : C ⥤ D).map f = 0 :=\n(is_zero_zero _).map _\n\nsection\nvariables [has_zero_object C] [has_zero_morphisms C]\nopen_locale zero_object\n\n@[simp]\nlemma id_zero : 𝟙 (0 : C) = (0 : 0 ⟶ 0) :=\nby ext\n\n/--  An arrow ending in the zero object is zero -/\n-- This can't be a `simp` lemma because the left hand side would be a metavariable.\nlemma zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 :=\nby ext\n\nlemma zero_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 :=\nbegin\n  have h : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv := by simp only [iso.hom_inv_id, id_comp, comp_id],\n  simpa using h,\nend\n\n/-- An arrow starting at the zero object is zero -/\nlemma zero_of_from_zero {X : C} (f : 0 ⟶ X) : f = 0 :=\nby ext\n\nlemma zero_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : f = 0 :=\nbegin\n  have h : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f := by simp only [iso.hom_inv_id_assoc, id_comp, comp_id],\n  simpa using h,\nend\n\nlemma zero_of_source_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic X 0) : f = 0 :=\nzero_of_source_iso_zero f (nonempty.some i)\nlemma zero_of_target_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic Y 0) : f = 0 :=\nzero_of_target_iso_zero f (nonempty.some i)\n\nlemma mono_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : mono f :=\n⟨λ Z g h w, by rw [zero_of_target_iso_zero g i, zero_of_target_iso_zero h i]⟩\n\nlemma epi_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : epi f :=\n⟨λ Z g h w, by rw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]⟩\n\n/--\nAn object `X` has `𝟙 X = 0` if and only if it is isomorphic to the zero object.\n\nBecause `X ≅ 0` contains data (even if a subsingleton), we express this `↔` as an `≃`.\n-/\ndef id_zero_equiv_iso_zero (X : C) : (𝟙 X = 0) ≃ (X ≅ 0) :=\n{ to_fun    := λ h, { hom := 0, inv := 0, },\n  inv_fun   := λ i, zero_of_target_iso_zero (𝟙 X) i,\n  left_inv  := by tidy,\n  right_inv := by tidy, }\n\n@[simp]\nlemma id_zero_equiv_iso_zero_apply_hom (X : C) (h : 𝟙 X = 0) :\n  ((id_zero_equiv_iso_zero X) h).hom = 0 := rfl\n\n@[simp]\nlemma id_zero_equiv_iso_zero_apply_inv (X : C) (h : 𝟙 X = 0) :\n  ((id_zero_equiv_iso_zero X) h).inv = 0 := rfl\n\n/-- If `0 : X ⟶ Y` is an monomorphism, then `X ≅ 0`. -/\n@[simps]\ndef iso_zero_of_mono_zero {X Y : C} (h : mono (0 : X ⟶ Y)) : X ≅ 0 :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' := (cancel_mono (0 : X ⟶ Y)).mp (by simp) }\n\n/-- If `0 : X ⟶ Y` is an epimorphism, then `Y ≅ 0`. -/\n@[simps]\ndef iso_zero_of_epi_zero {X Y : C} (h : epi (0 : X ⟶ Y)) : Y ≅ 0 :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' := (cancel_epi (0 : X ⟶ Y)).mp (by simp) }\n\n/-- If a monomorphism out of `X` is zero, then `X ≅ 0`. -/\ndef iso_zero_of_mono_eq_zero {X Y : C} {f : X ⟶ Y} [mono f] (h : f = 0) : X ≅ 0 :=\nby { unfreezingI { subst h, }, apply iso_zero_of_mono_zero ‹_›, }\n\n/-- If an epimorphism in to `Y` is zero, then `Y ≅ 0`. -/\ndef iso_zero_of_epi_eq_zero {X Y : C} {f : X ⟶ Y} [epi f] (h : f = 0) : Y ≅ 0 :=\nby { unfreezingI { subst h, }, apply iso_zero_of_epi_zero ‹_›, }\n\n/-- If an object `X` is isomorphic to 0, there's no need to use choice to construct\nan explicit isomorphism: the zero morphism suffices. -/\ndef iso_of_is_isomorphic_zero {X : C} (P : is_isomorphic X 0) : X ≅ 0 :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' :=\n  begin\n    casesI P,\n    rw ←P.hom_inv_id,\n    rw ←category.id_comp P.inv,\n    simp,\n  end,\n  inv_hom_id' := by simp, }\n\nend\n\nsection is_iso\nvariables [has_zero_morphisms C]\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\nthe identities on both `X` and `Y` are zero.\n-/\n@[simps]\ndef is_iso_zero_equiv (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (𝟙 X = 0 ∧ 𝟙 Y = 0) :=\n{ to_fun := by { introsI i, rw ←is_iso.hom_inv_id (0 : X ⟶ Y),\n    rw ←is_iso.inv_hom_id (0 : X ⟶ Y), simp },\n  inv_fun := λ h, ⟨⟨(0 : Y ⟶ X), by tidy⟩⟩,\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\nthe identity on `X` is zero.\n-/\ndef is_iso_zero_self_equiv (X : C) : is_iso (0 : X ⟶ X) ≃ (𝟙 X = 0) :=\nby simpa using is_iso_zero_equiv X X\n\nvariables [has_zero_object C]\nopen_locale zero_object\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\n`X` and `Y` are isomorphic to the zero object.\n-/\ndef is_iso_zero_equiv_iso_zero (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (X ≅ 0) × (Y ≅ 0) :=\nbegin\n  -- This is lame, because `prod` can't cope with `Prop`, so we can't use `equiv.prod_congr`.\n  refine (is_iso_zero_equiv X Y).trans _,\n  symmetry,\n  fsplit,\n  { rintros ⟨eX, eY⟩, fsplit,\n    exact (id_zero_equiv_iso_zero X).symm eX,\n    exact (id_zero_equiv_iso_zero Y).symm eY, },\n  { rintros ⟨hX, hY⟩, fsplit,\n    exact (id_zero_equiv_iso_zero X) hX,\n    exact (id_zero_equiv_iso_zero Y) hY, },\n  { tidy, },\n  { tidy, },\nend\n\nlemma is_iso_of_source_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) (j : Y ≅ 0) : is_iso f :=\nbegin\n  rw zero_of_source_iso_zero f i,\n  exact (is_iso_zero_equiv_iso_zero _ _).inv_fun ⟨i, j⟩,\nend\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\n`X` is isomorphic to the zero object.\n-/\ndef is_iso_zero_self_equiv_iso_zero (X : C) : is_iso (0 : X ⟶ X) ≃ (X ≅ 0) :=\n(is_iso_zero_equiv_iso_zero X X).trans subsingleton_prod_self_equiv\n\nend is_iso\n\n/-- If there are zero morphisms, any initial object is a zero object. -/\nlemma has_zero_object_of_has_initial_object\n  [has_zero_morphisms C] [has_initial C] : has_zero_object C :=\nbegin\n  refine ⟨⟨⊥_ C, λ X, ⟨⟨⟨0⟩, by tidy⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩⟩,\n  calc\n    f = f ≫ 𝟙 _ : (category.comp_id _).symm\n    ... = f ≫ 0 : by congr\n    ... = 0     : has_zero_morphisms.comp_zero _ _\nend\n\n/-- If there are zero morphisms, any terminal object is a zero object. -/\nlemma has_zero_object_of_has_terminal_object\n  [has_zero_morphisms C] [has_terminal C] : has_zero_object C :=\nbegin\n  refine ⟨⟨⊤_ C, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, by tidy⟩⟩⟩⟩,\n  calc\n    f = 𝟙 _ ≫ f : (category.id_comp _).symm\n    ... = 0 ≫ f : by congr\n    ... = 0     : zero_comp\nend\n\n\nsection image\nvariable [has_zero_morphisms C]\n\nlemma image_ι_comp_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image f]\n  [epi (factor_thru_image f)] (h : f ≫ g = 0) : image.ι f ≫ g = 0 :=\nzero_of_epi_comp (factor_thru_image f) $ by simp [h]\n\nlemma comp_factor_thru_image_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image g]\n  (h : f ≫ g = 0) : f ≫ factor_thru_image g = 0 :=\nzero_of_comp_mono (image.ι g) $ by simp [h]\n\nvariables [has_zero_object C]\nopen_locale zero_object\n\n/--\nThe zero morphism has a `mono_factorisation` through the zero object.\n-/\n@[simps]\ndef mono_factorisation_zero (X Y : C) : mono_factorisation (0 : X ⟶ Y) :=\n{ I := 0, m := 0, e := 0, }\n\n/--\nThe factorisation through the zero object is an image factorisation.\n-/\ndef image_factorisation_zero (X Y : C) : image_factorisation (0 : X ⟶ Y) :=\n{ F := mono_factorisation_zero X Y,\n  is_image := { lift := λ F', 0 } }\n\n\ninstance has_image_zero {X Y : C} : has_image (0 : X ⟶ Y) :=\nhas_image.mk $ image_factorisation_zero _ _\n\n/-- The image of a zero morphism is the zero object. -/\ndef image_zero {X Y : C} : image (0 : X ⟶ Y) ≅ 0 :=\nis_image.iso_ext (image.is_image (0 : X ⟶ Y)) (image_factorisation_zero X Y).is_image\n\n/-- The image of a morphism which is equal to zero is the zero object. -/\ndef image_zero' {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] : image f ≅ 0 :=\nimage.eq_to_iso h ≪≫ image_zero\n\n@[simp]\nlemma image.ι_zero {X Y : C} [has_image (0 : X ⟶ Y)] : image.ι (0 : X ⟶ Y) = 0 :=\nbegin\n  rw ←image.lift_fac (mono_factorisation_zero X Y),\n  simp,\nend\n\n/--\nIf we know `f = 0`,\nit requires a little work to conclude `image.ι f = 0`,\nbecause `f = g` only implies `image f ≅ image g`.\n-/\n@[simp]\nlemma image.ι_zero' [has_equalizers C] {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] :\n  image.ι f = 0 :=\nby { rw image.eq_fac h, simp }\n\nend image\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance is_split_mono_sigma_ι {β : Type u'} [has_zero_morphisms C] (f : β → C)\n  [has_colimit (discrete.functor f)] (b : β) : is_split_mono (sigma.ι f b) := is_split_mono.mk'\n{ retraction := sigma.desc $ pi.single b (𝟙 _) }\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance is_split_epi_pi_π {β : Type u'} [has_zero_morphisms C] (f : β → C)\n  [has_limit (discrete.functor f)] (b : β) : is_split_epi (pi.π f b) := is_split_epi.mk'\n{ section_ := pi.lift $ pi.single b (𝟙 _) }\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance is_split_mono_coprod_inl [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :\n  is_split_mono (coprod.inl : X ⟶ X ⨿ Y) := is_split_mono.mk'\n{ retraction := coprod.desc (𝟙 X) 0, }\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance is_split_mono_coprod_inr [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :\n  is_split_mono (coprod.inr : Y ⟶ X ⨿ Y) := is_split_mono.mk'\n{ retraction := coprod.desc 0 (𝟙 Y), }\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance is_split_epi_prod_fst [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :\n  is_split_epi (prod.fst : X ⨯ Y ⟶ X) := is_split_epi.mk'\n{ section_ := prod.lift (𝟙 X) 0, }\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance is_split_epi_prod_snd [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :\n  is_split_epi (prod.snd : X ⨯ Y ⟶ Y) := is_split_epi.mk'\n{ section_ := prod.lift 0 (𝟙 Y), }\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/shapes/zero_morphisms.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3351013538390312}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  apply iff.intro,\n  assume h1 : G.colorable 2,\n  cases h1 with f hf,\n  let A := f ⁻¹' {1},\n  let B := f ⁻¹' {2},\n  have h1 : (A ⊕ B) = V, from by auto [set.ext],\n\n  let f2 := cast (congr_arg _ h1) (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)),\n  have h2 : f2 : V → fin 2, from by auto [set.ext, set.image_preimage_eq_of_inverse (λ (x : ℕ), x ∈ ({1, 2} : set ℕ)) (λ (x : ℕ), x ∈ {1, 2}) (@fin.of_nat 2) (λ (x : ℕ), by auto [fin.of_nat_inj])],\n\n  have h3 : (∀ (v : A ⊕ B), (f2 v).val = (if v.is_inl then 1 else 2)), from by auto [fin.coe_val_of_ne_zero],\n \n  have h4 : (f2 = cast (congr_arg _ h1) (λ (t : A ⊕ B), (if t.is_inl then 1 else 2))), from rfl,\n  have h5 : f2 = f, from by auto [h1, h4, funext_iff, h3, cast_eq f],\n\n  have h6 : (∀ (v w : A ⊕ B), (f2 v).val ≠ (f2 w).val → (∃ (x : A) (y : B), v = x ⊕ y ∧ degree G x ≠ 0 ∧ degree G y ≠ 0)), from by auto using [hf, h5, fin.coe_val_of_ne_zero, fin.coe_val_of_ne_zero],\n\n  have h7 : (∀ (v w : A ⊕ B), ((v.is_inl) ∧ (w.is_inl)) ∨ ((v.is_inl) ∧ (w.is_inr)) ∨ ((v.is_inr) ∧ (w.is_inl)) ∨ ((v.is_inr) ∧ (w.is_inr))), from by auto [or.left_comm],\n\n  have h8 : (∀ (v w : A ⊕ B), (f2 v).val ≠ (f2 w).val → (((v.is_inl) ∧ (w.is_inl)) ∨ ((v.is_inl) ∧ (w.is_inr)) ∨ ((v.is_inr) ∧ (w.is_inl)) ∨ ((v.is_inr) ∧ (w.is_inr)))), from by auto [h7],\n\n  have h9 : (∀ (v w : V), w ≠ v → (∃ (x : A) (y : B), v = x ⊕ y ∧ degree G x ≠ 0 ∧ degree G y ≠ 0)), from by auto using [h1, h8, h6],\n\n  have h10 : (∀ (v : V), degree G v ≠ 0 → (∃ (p : A) (q : B), v = p ⊕ q ∧ degree G p ≠ 0 ∧ degree G q ≠ 0)), from by auto [h9],\n\n  have h11 : (∀ (v : V), v = v ∧ degree G v ≠ 0 → (∃ (p : A) (q : B), v = p ⊕ q ∧ degree G p ≠ 0 ∧ degree G q ≠ 0)), from by auto [h10],\n\n  have h12 : (∀ (v : V), (∃ (p : A) (q : B), v = p ⊕ q ∧ degree G p ≠ 0 ∧ degree G q ≠ 0)), from let v := v in by auto [h11],\n\n  have h13 : (∀ (v : V), (∃ (p : A) (q : B), v = p ⊕ q ∧ degree G p ≠ 0 ∧ degree G q ≠ 0) → (∃ (v : A ⊕ B), (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0)), from \n  begin\n    assume (v : V),\n    assume h13 : (∃ (p : A) (q : B), v = p ⊕ q ∧ degree G p ≠ 0 ∧ degree G q ≠ 0),\n    cases h13 with p h13,\n    cases h13 with q h13,\n    cases h13 with h13 h13,\n    cases h13 with h13 h13,\n    suffices : (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) (p ⊕ q) ≠ 0, from ⟨p ⊕ q, this⟩,\n    suffices : (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) (p ⊕ q) = 1, from by auto [congr_arg],\n  end,\n\n  have h14 : (∀ (v : A ⊕ B), (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0), from by auto [h12, h13],\n\n  have h15 : (∀ (v : A ⊕ B), (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0 → (∃ (x : A) (y : B), v = x ⊕ y ∧ degree G x ≠ 0 ∧ degree G y ≠ 0)), from \n  begin\n    assume (v : A ⊕ B),\n    assume h15 : (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0,\n    suffices : (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v = 1, from by auto [congr_arg],\n    cases v,\n    show (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) ⟨a, or.inl h⟩ = 1, from rfl,\n    assume h : b,\n    show (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) ⟨a, or.inr h⟩ = 2, from rfl,\n  end,\n\n  have h16 : (∃ (v : A ⊕ B), (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0), from classical.by_contradiction (λ h17, by auto [h14, not_exists.elim h17]),\n\n  have h17 : (∃ (v : A ⊕ B), (λ (t : A ⊕ B), (if t.is_inl then 1 else 2)) v ≠ 0 → (∃ (x : A) (y : B), v = x ⊕ y ∧ degree G x ≠ 0 ∧ degree G y ≠ 0)), from by auto [h15],\n\n  have h18 : (∃ (x : A) (y : B), (exists.elim h16 (λ (v : A ⊕ B), (exists.elim (h17 v (exists.elim h16 (λ (v' : A ⊕ B), (exists.elim h16 (λ (h18 : v' = v), h18.symm))))) (λ (v' : A ⊕ B), v')))) ∧ degree G x ≠ 0 ∧ degree G y ≠ 0), from let v := (exists.elim h16 (λ (v : A ⊕ B), v)) in let h19 := (exists.elim h16 (λ (v : A ⊕ B), exists.elim (h17 v (exists.elim h16 (λ (v' : A ⊕ B), (exists.elim h16 (λ (h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  have h1 : ∀ n, n ≥ 1 → n-1 < n, from by auto [nat.sub_one],\n  split,\n  {\n    assume h,\n    cases h,\n    cases h with f h1,\n    cases h1 with hf h1,\n    cases h1 with h2 h3,\n    have h4 : fintype (Σ (x : G.V), (f x) ≠ f x), from by auto [@fintype.of_equiv],\n    have h5 : fintype.card (Σ (x : G.V), (f x) ≠ f x) = 2, from by auto [fintype.card_congr, nat.sub_eq_of_eq_add, nat.add_sub_cancel (nat.pos_of_ne_zero h2)] using [@fintype.card_sigma, @fintype.card_sigma, @fintype.card_sigma, @fintype.card_sigma],\n    have h6 : fintype.card ((Σ (x : G.V), (f x) ≠ f x) : set (G.V)) < fintype.card G.V, from by auto [fintype.card_congr, nat.le_of_lt, fintype.card_lt_iff_not_eq_empty],\n    have h7 : (∀ (A B : fintype.card ((Σ (x : G.V), (f x) ≠ f x))), ∀ (f : (Σ (x : G.V), (f x) ≠ f x) → G.V), ∃ (H : G.V), ∀ (x : G.V), (x = H) ∨ (x ≠ H)), from by auto [fintype.card_lt_iff_not_eq_empty, fintype.card_congr, @fintype.card_sigma, @fintype.card_sigma, @fintype.card_sigma, @fintype.card_sigma, nat.le_of_lt],\n    have h8 : (∃ (H : G.V), ∀ (x : G.V), (x = H) ∨ (x ≠ H)), from by auto [h7],\n    cases h8 with H h9,\n    have h10 : (f H) ≠ f H, from by auto [h9],\n    have h11 : (f H) ≠ (f H), from by auto [h10],\n    have h12 : G.V ≃ (Σ (x : G.V), (f x) ≠ f x), from \n      begin\n        apply equiv.of_bijective _ _,\n        {\n          assume a,\n          cases a with x h13,\n          apply subtype.eq,\n          show (f x) ≠ f x, from by auto [h13]\n        },\n        {\n          assume a,\n          cases a with x h13,\n          have h14 : ∃ (a : G.V), a = x ∧ (f a) ≠ (f a), from by auto [set.mem_univ, @exists_prop, h13],\n          cases h14 with h14 h15,\n          cases h15 with h15 h16,\n          show ∃ (a : G.V), (f a) ≠ (f a), from by auto [h14]\n        },\n        {\n          assume a ha,\n          cases a with x h13,\n          have h14 : ∃ (a : G.V), a = x ∧ (f a) ≠ (f a), from by auto [set.mem_univ, @exists_prop, h13],\n          cases h14 with h14 h15,\n          cases h15 with h15 h16,\n          show (f h14) ≠ (f h14), from by auto [h14, h15, h16]\n        }\n      end,\n    have h13 : fintype.card G.V = fintype.card (Σ (x : G.V), (f x) ≠ f x), from by auto [fintype.card_congr, h12],\n    have h14 : (Σ (x : G.V), (f x) ≠ f x) ≃ (sigma G.V (λ x, x ≠ x)), from by auto [is_equiv.equiv_sigma_eq_sigma],\n    have h15 : sigma G.V (λ x, x ≠ x) ≃ (sigma G.V (λ x, x.1 ≠ x.2)), from by auto [equiv.trans, h14] using [@is_equiv.sigma_equiv_sigma_of_equiv],\n    have h16 : fintype.card (sigma G.V (λ x, x ≠ x)) = (fintype.card G.V)*(fintype.card G.V), from by auto [fintype.card_congr, h15],\n    have h17 : fintype.card G.V = 2, from by auto [eq_comm, eq.trans, h5, h13, h16, nat.mul_left_cancel, nat.pos_of_ne_zero h2],\n    have h18 : G.V ≃ (ulift bool), from by auto [function.funext_iff, @equiv_bool_ulift_def, h17],\n    have h19 : (ulift bool) ≃ (bool : Type*), from by auto [equiv.trans, h18, @equiv_ulift_def],\n    have h20 : G.V ≃ (bool : Type*), from by auto [h19, eq_comm],\n    let g : G.V → bool := by auto [h20],\n    use g,\n    split,\n    {\n      cases h3 with h3 h3a,\n      assume a b h21,\n      have h22 : (g a) ≠ (g b), from by auto [h20, h3, h21],\n      apply subtype.eq,\n      show (g a) ≠ (g b), from by auto [h22]\n    },\n    {\n      split,\n      {\n        assume a,\n        apply subtype.eq,\n        show f a = g a, from by auto [h20, h1, h2],\n      },\n      {\n        assume a,\n        apply subtype.eq,\n        show g a = f a, from by auto [h20, h1, h2],\n      }\n    }\n  },\n  {\n    assume h,\n    cases h with A B h1,\n    cases h1,\n    have h2 : fintype (A ⊕ B), from by auto [@fintype.of_equiv],\n    have h3 : ((equiv.ulift : bool ↝ ulift bool) : bool ↝ A ⊕ B) = (cast (congr_arg _ h1) (equiv.bool_bool : bool ↝ bool)), from by auto [funext],\n    let f : G.V → A ⊕ B := cast (congr_arg _ h1) (equiv.bool_bool),\n    have h6 : fintype G.V, from by auto [@fintype.of_equiv],\n    have h7 : ((equiv.ulift : bool ↝ ulift bool) : bool ↝ A ⊕ B) = (cast (congr_arg _ h1) (equiv.bool_bool : bool ↝ bool)) := begin auto [funext], end,\n    have h4 : fintype.card G.V = fintype.card A ⊕ B, from by auto [fintype.card_congr, h3],\n    have h5 : fintype.card A ⊕ B = 2, from by auto [@fintype.card_sigma, h2, @fintype.card_sigma, @fintype.card_sigma, nat.mul_left_cancel, nat.pos_of_ne_zero, mul_one],\n    have h8 : (fintype.card A = 1 ∧ fintype.card B = 1), from by auto [nat.le_add_right, nat.le_add_left, fintype.card_le_one_iff,  h4, h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    intros h1,\n    have h2 : ∃ R B : set V, ∀ v : V, (v ∈ R) ∨ (v ∈ B), from h1.to_has_decidable_eq, \n    let R := h2.1,\n    let B := h2.2,\n    have h3 : ∀ a b : V, a ≠ b → a ∈ R → b ∈ B, from\n      begin\n        assume (a b : V) (h4 : a ≠ b) (h5 : a ∈ R),\n        have h6 : (a ∈ B) ∨ (b ∈ R), from h2.3 a, \n        have h7 : (a ∈ B) → (b ∈ R), from\n          begin\n            intro h8,\n            have h9 : a ∈ B ∧ b ∈ R, from by auto [h8, h5],\n            have h10 : (a = b), from by auto,\n            show b ∈ R, from by auto [h10, h4],\n          end,\n        show b ∈ B, from by auto [h6, h7],\n      end,\n    have h4 : ∀ a b : V, a ∈ R → b ∈ B → a ≠ b, from\n      begin\n        assume (a b : V) (h5 : a ∈ R) (h6 : b ∈ B),\n        assume h7 : a = b,\n        have h8 : a ∈ R ∧ b ∈ R, from by auto [h5, h7],\n        show false, from h2.3 a h8,\n      end,\n    have h5 : ∀ (a b : V) (h6 : G.adj a b), a ∈ R → b ∈ B, from\n      begin\n        assume (a b : V) (h6 : G.adj a b),\n        show a ∈ R → b ∈ B, from\n          begin\n            intro h7,\n            have h8 : (a ∈ B) ∨ (b ∈ R), from h2.3 a,\n            have h9 : a ∈ B → b ∈ R, from\n              begin\n                intro h10,\n                have h11 : a ∈ B ∧ b ∈ R, from by auto [h10, h7],\n                have h12 : (a = b), from by auto,\n                show b ∈ R, from by auto [h12, h4],\n              end,\n            show b ∈ B, from by auto [h8, h9],\n          end,\n      end,\n    have h6 : ∀ (a b : V) (h7 : G.adj a b), a ∈ R → b ∈ B, from\n      begin\n        assume (a b : V) (h7 : G.adj a b),\n        show a ∈ R → b ∈ B, from\n          begin\n            intro h8,\n            have h9 : (a ∈ B) ∨ (b ∈ R), from h2.3 a,\n            have h10 : a ∈ B → b ∈ R, from\n              begin\n                intro h11,\n                have h12 : a ∈ B ∧ b ∈ R, from by auto [h11, h8],\n                have h13 : (a = b), from by auto,\n                show b ∈ R, from by auto [h13, h4],\n              end,\n            show b ∈ B, from by auto [h9, h10],\n          end\n      end,\n    have h7 : ∀ (a b : V) (h8 : G.adj a b), a ∈ B → b ∈ R, from\n      begin\n        assume (a b : V) (h8 : G.adj a b),\n        show a ∈ B → b ∈ R, from\n          begin\n            intro h9,\n            have h10 : (a ∈ R) ∨ (b ∈ B), from h2.3 a,\n            have h11 : a ∈ R → b ∈ B, from\n              begin\n                intro h12,\n                have h13 : a ∈ R ∧ b ∈ B, from by auto [h12, h9],\n                have h14 : (a = b), from by auto,\n                show b ∈ B, from by auto [h14, h4],\n              end,\n            show b ∈ R, from by auto [h10, h11],\n          end\n      end,\n    have h8 : ∀ (a b : V) (h9 : G.adj a b), a ∈ R → b ∈ B, from\n      begin\n        assume (a b : V) (h9 : G.adj a b),\n        show a ∈ R → b ∈ B, from\n          begin\n            intro h10,\n            have h11 : (a ∈ B) ∨ (b ∈ R), from h2.3 a,\n            have h12 : a ∈ B → b ∈ R, from\n              begin\n                intro h13,\n                have h14 : a ∈ B ∧ b ∈ R, from by auto [h13, h10],\n                have h15 : (a = b), from by auto,\n                show b ∈ R, from by auto [h15, h4],\n              end,\n            show b ∈ B, from by auto [h11, h12],\n          end\n      end,\n    have h9 : ∀ (a b : V) (h10 : G.adj a b), a ∈ B → b ∈ R, from\n      begin\n        assume (a b : V) (h10 : G.adj a b),\n        show a ∈ B → b ∈ R, from\n          begin\n            intro h11,\n            have h12 : (a ∈ R) ∨ (b ∈ B), from h2.3 a,\n            have h13 : a ∈ R → b ∈ B, from\n              begin\n                intro h14,\n                have h15 : a ∈ R ∧ b ∈ B, from by auto [h14, h11],\n                have h16 : (a = b), from by auto,\n                show b ∈ B, from by auto [h16, h4],\n              end,\n            show b ∈ R, from by auto [h12, h13],\n          end\n      end,\n    have h11 : ∀ (a b : V) (h12 : G.adj a b), a ∈ B → b ∈ R, from\n      begin\n        assume (a b : V) (h12 : G.adj a b),\n        show a ∈ B → b ∈ R, from\n          begin\n            intro h13,\n            have h14 : (a ∈ R) ∨ (b ∈ B), from h2.3 a,\n            have h15 : a ∈ R → b ∈ B, from\n              begin\n                intro h16,\n                have h17 : a ∈ R ∧ b ∈ B, from by auto [h16, h13],\n                have h18 : (a = b), from by auto,\n                show b ∈ B, from by auto [h18, h4],\n              end,\n            show b ∈ R, from by auto [h14, h15],\n          end\n      end,\n    have h12 : ∀ (a b : V) (h13 : G.adj a b), a ∈ R → b ∈ B, from\n      begin\n        assume (a b : V) (h13 : G.adj a b),\n        show a ∈ R → b ∈ B, from\n          begin\n            intro h14,\n            have h15 : (a ∈ B) ∨ (b ∈ R), from h2.3 a,\n            have h16 : a ∈ B → b ∈ R, from\n              begin\n                intro h17,\n                have h18 : a ∈ B ∧ b ∈ R, from by auto [h17, h14],\n                have h19 : (a = b), from by auto,\n                show b ∈ R, from by auto [h19, h4],\n              end,\n            show b ∈ B, from by auto [h15, h16],\n          end\n      end,\n    have h14 : ∀ (a b : V) (h15 : G.adj a b), a ∈ B → b ∈ R, from\n      begin\n        assume (a b :\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1 : G.colorable 2,\n    show ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from\n    begin\n      let h2 : V → fin 2 := (λ (v : V), classical.some (h1 v).is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some.is_some),\n      let A : Type* := (fin 2 ⟶ V),\n      let B : Type* := V,\n      let h3 : (A ⊕ B) = V := by tidy,\n      let h4 : finset A → finset B → finset ℕ, h5 : finset A → finset B → finset V := λ x y, ⟨x.1.to_fun ⁻¹' y.1, sorry⟩, h6 : finset A → finset B → finset V → Prop := λ x y z, ∃ x' ∈ x.1, ∃ y' ∈ y.1, z.1 = G.adj x' y',\n      let h7 : (finset A → finset B → finset V) = (finset A → finset B → finset V → Prop), h8 : finset A → finset B → finset V → Prop := by apply_instance, h9 : (finset A → finset B → finset V) = (finset A → finset B → finset V → Prop), h10 : finset A → finset B → finset V → Prop := by apply_instance,\n      let h11 : (finset A → finset B → finset ℕ) = (finset A → finset B → finset V), h12 : finset A → finset B → finset V := by apply_instance, h13 : (finset A → finset B → finset ℕ) = (finset A → finset B → finset V), h14 : finset A → finset B → finset V := by apply_instance,\n      let h15 : (finset A → finset B) = (finset A → finset B → finset ℕ), h16 : finset A → finset B → finset ℕ := by apply_instance, h17 : (finset A → finset B) = (finset A → finset B → finset ℕ), h18 : finset A → finset B → finset ℕ := by apply_instance,\n      let h19 : (finset A) = (finset A → finset B), h20 : finset A → finset B := by apply_instance, h21 : (finset A) = (finset A → finset B), h22 : finset A → finset B := by apply_instance,\n      let h23 : ∀ x, ∃! y, x = y, h24 : ∀ x, ∃! y, x = y := by apply_instance,\n      let h25 : (finset A) = finset A, h26 : finset A := by apply_instance,\n      let h27 : (finset B) = finset B, h28 : finset B := by apply_instance,\n      let h29 : (finset V) = finset V, h30 : finset V := by apply_instance,\n      let h31 : (finset ℕ) = finset ℕ, h32 : finset ℕ := by apply_instance,\n      have h33 : (finset A → finset B → finset V) = (finset A → finset B → finset ℕ), from by tidy,\n      have h34 : ∀ x, ∃! y, x = y, from by apply_instance,\n      show ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by tidy\n    end\n  },\n  {\n    assume h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B),\n    show G.colorable 2, from\n    begin\n      cases h1 with h2 h3,\n      cases h3 with h4 h5,\n      cases h4 with h6 h7,\n      \n      have h8 : ∀ x, ∃! y, x = y, from by apply_instance,\n      let h9 := by apply_instance,\n      let h10 := by apply_instance,\n      let h11 := by apply_instance,\n      let h12 := by apply_instance,\n      let h13 := by apply_instance,\n      let h14 := by apply_instance,\n      let h15 := by apply_instance,\n      let h16 := by apply_instance,\n      let h17 := by apply_instance,\n      let h18 := by apply_instance,\n      let h19 := by apply_instance,\n      let h20 := by apply_instance,\n      let h21 := by apply_instance,\n      let h22 := by apply_instance,\n      let h23 := by apply_instance,\n      let h24 := by apply_instance,\n      let h25 := by apply_instance,\n      let h26 := by apply_instance,\n      let h27 := by apply_instance,\n      let h28 := by apply_instance,\n      let h29 := by apply_instance,\n      let h30 := by apply_instance,\n      let h31 := by apply_instance,\n      let h32 := by apply_instance,\n      let h33 := by apply_instance,\n      let h34 := by apply_instance,\n      let h35 := by apply_instance,\n      let h36 := by apply_instance,\n      let h37 := by apply_instance,\n      let h38 := by apply_instance,\n      let h39 := by apply_instance,\n      let h40 := by apply_instance,\n      let h41 := by apply_instance,\n      let h42 := by apply_instance,\n      let h43 := by apply_instance,\n      let h44 := by apply_instance,\n      let h45 := by apply_instance,\n      let h46 := by apply_instance,\n      let h47 := by apply_instance,\n      let h48 := by apply_instance,\n      let h49 := by apply_instance,\n      let h50 := by apply_instance,\n      let h51 := by apply_instance,\n      let h52 := by apply_instance,\n      let h53 := by apply_instance,\n      let h54 := by apply_instance,\n      let h55 := by apply_instance,\n      let h56 := by apply_instance,\n      let h57 := by apply_instance,\n      let h58 := by apply_instance,\n      let h59 := by apply_instance,\n      let h60 := by apply_instance,\n      let h61 := by apply\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) := \nbegin\n  split,\n\n  assume h1 : G.colorable 2,\n  rcases h1 with ⟨f, h2, h3, h4⟩,\n    \n  have h5 : ∀ (v : V), (f v) = 1 ∨ (f v) = 2, from and.elim_right h3 v,\n\n  let A : set V := {v | (f v) = 1},\n  let B : set V := {v | (f v) = 2},\n\n  have h6 : ∀ (v : V), f v = 1 ↔ (v ∈ A), from by intros v; split; intro h; auto [mem_set_of_eq],\n  have h7 : ∀ (v : V), f v = 2 ↔ (v ∈ B), from by intros v; split; intro h; auto [mem_set_of_eq],\n\n  have h8 : ∀ (v : V), (v ∉ A) ↔ (v ∈ B), from by intros v; split; intro h; rcases h5 v with ⟨h8, h9⟩; auto [exfalso, h8]; auto [h9],\n\n  have h9 : ∀ (v : V), (v ∉ B) ↔ (v ∈ A), from by intros v; split; intro h; rcases h5 v with ⟨h8, h9⟩; auto [h8]; auto [exfalso, h9],\n\n  have h10 : disjoint A B, from by intros v hv; auto [exfalso, h4 v, hv],\n\n  have h11 : A ⊆ V, from by auto [subset_univ],\n  have h12 : B ⊆ V, from by auto [subset_univ],\n\n  have h13 : ∅ ∈ {x|x ⊆ V}, from by auto [subset_empty],\n  have h14 : A ∈ {x|x ⊆ V}, from by auto [h11],\n  have h15 : B ∈ {x|x ⊆ V}, from by auto [h12],\n\n  have h16 : fintype A, from by auto [fintype.of_subset h11],\n  have h17 : fintype B, from by auto [fintype.of_subset h12],\n\n  have h18 : fintype {x | x ⊆ V}, from by auto [set.finite_subset (fintype.powerset (fintype.univ V)), set.subset_univ],\n\n  have h19 : ∃ (y : {x | x ⊆ V}), (1 : ℕ) ≤ y.card ∧ (y.card + 1) ≤ A.card + B.card, from by auto [fintype.card_ne_zero, h16, h17, h13, h14, h15, card_union_of_disjoint, h10,\n    exists.intro _ (and.intro (le_of_succ_le_succ (nat.zero_le _) (nat.le_add_right _ _)) (le_add_of_nonneg_right _ (nat.zero_le _)))],\n\n  rcases h19 with ⟨y, h19t⟩,\n\n  have h20 : (1 : ℕ) ≤ y.card, from and.elim_left h19t,\n\n  have h21 : 2 ≤ A.card + B.card, from by auto [le_of_add_le_add_right, h19t],\n\n  have h22 : ∀ (x : {x | x ⊆ V}), (x ⊆ A ∨ x ⊆ B) → ∃! e : {x | x ⊆ V}, 1 ≤ e.card ∧ (e.card + 1) ≤ x.card,\n  {\n    assume x, intro h23,\n    have h24 : ∃ (e : {x | x ⊆ V}), 1 ≤ e.card ∧ (e.card + 1) ≤ x.card, from by auto [h18.card_le_card_of_subset h23],\n    show ∃! e : {x | x ⊆ V}, 1 ≤ e.card ∧ (e.card + 1) ≤ x.card, from by auto [exists_unique.intro _ h24],\n  },\n\n  have h25 : ∃! (e : {x | x ⊆ V}), 1 ≤ e.card ∧ (e.card + 1) ≤ A.card ∧ (e.card + 1) ≤ B.card, from\n    by auto [exists_unique.intro y, h20, h21, and.left, and.right],\n\n  rcases h25 with ⟨e, h26, h27⟩,\n\n  have h28 : ∀ (y : {x | x ⊆ V}), (1 ≤ y.card ∧ (y.card + 1) ≤ A.card ∧ (y.card + 1) ≤ B.card) → y = e, from and.elim_left h26,\n\n  have h29 : ∀ (v : V), (v ∈ A) → (v ∈ e), from by auto [le_of_add_le_add_right, h27, h6, h14],\n  have h30 : ∀ (v : V), (v ∈ B) → (v ∈ e), from by auto [le_of_add_le_add_right, h27, h7, h15],\n\n  have h31 : ∀ (v : V), (v ∈ e) → (v ∈ A ∨ v ∈ B), from by auto [h29, h30],\n\n  have h32 : ∀ (v : V), (v ∈ A) ≃ (v ∈ e), from by auto [h29, h8, h9],\n\n  have h33 : ∀ (v : V), (v ∈ B) ≃ (v ∈ e), from by auto [h30, h8, h9],\n\n  have h34 : ∀ (v : V), (v ∈ e) ↔ (v ∈ A ∨ v ∈ B), from by auto [h31, h29, h30],\n\n  have h35 : ∀ (v : V), (v ∈ A) ↔ (v ∈ e), from by auto [h7, h32, h34],\n  have h36 : ∀ (v : V), (v ∈ B) ↔ (v ∈ e), from by auto [h7, h33, h34],\n\n  have h37 : A = e, from ext_bijective.ext (λ v, v ∈ A) (λ v, v ∈ e) h35,\n  have h38 : B = e, from ext_bijective.ext (λ v, v ∈ B) (λ v, v ∈ e) h36,\n\n  have h39 : exist.intro (λ (v : V), v ∈ e) (begin intros v hv, have h : v ∈ e, from hv, rcases h with ⟨h⟩, show v ∈ A ∨ v ∈ B, from h, end) = exist.intro (λ (v : V), v ∈ A ∨ v ∈ B) (begin intros v hv, show v ∈ e, from or.elim hv (λ h, h) (λ h, h), end), from funext (λ v, propext (or_iff_not_and_not.elim_left v)),\n  have h40 : exist.intro (λ (v : V), v ∈ e) (begin intros v hv, have h : v ∈ e, from hv, rcases h with ⟨h⟩, show v ∈ A ∨ v ∈ B, from h, end) = exist.intro (λ (v : V), v ∈ B) (begin intros v hv, have h : v ∈ e, from hv, rcases h with ⟨h⟩, show v ∈ A ∨ v ∈ B, from h, end), from funext (λ v, propext (or_iff_not_and_not.elim_right v)),\n\n  have h41 : exist.intro (λ (v : V), v ∈ A) (begin intros v hv, show v ∈\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  sorry,\nend\n\n--OUTPUT 7\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume : ℕ,\n    intro H,\n    cases H with f H,\n    cases H with H1 H2,\n    cases H1 with hf hf,\n    have  H1 : (f '' G.vertices) ⊆ ({0, 1} : set ℕ), from set.image_subset_iff.mpr H2,\n    have H2 : f '' G.vertices = {0,1}, from set.eq_singleton_iff_eq_empty.mp (set.eq_empty_iff_forall_not_mem.mpr (λ a h, by {cases h with b hb, cases hb with hb1 hb2, rw hb1 at hb2,exact H1 hb2})),\n    have  H1 : (set.range f) ⊆ ({0, 1} : set ℕ), from set.range_subset_iff.mpr H2,\n    have H2 : set.range f = {0, 1}, from set.eq_singleton_iff_eq_empty.mp (set.eq_empty_iff_forall_not_mem.mpr (λ a h, by {cases h with b hb, cases hb with hb1 hb2, rw ← hb1 at hb2,exact H1 hb2})),\n    let α : Type* := by {cases H2, assumption},\n    let β : Type* := by {cases H2, assumption},\n    let h : (α ⊕ β) = V := by {exact congr_arg _ (set.range_eq_univ.mp (set.eq_univ_singleton.mp H2))},\n    let f₂ : (α ⊕ β) →  ℕ := λ x, f x,\n    have H1 : (f '' G.vertices) ⊆ ({0, 1} : set ℕ), from set.image_subset_iff.mpr H2,\n    have H2 : f₂ '' (α ⊕ β) = {0,1}, from set.eq_singleton_iff_eq_empty.mp (set.eq_empty_iff_forall_not_mem.mpr (λ a h, by {cases h with b hb, cases hb with hb1 hb2, rw hb1 at hb2,exact H1 hb2})),\n    have  H1 : (set.range f₂) ⊆ ({0, 1} : set ℕ), from set.range_subset_iff.mpr H2,\n    have H2 : set.range f₂ = {0, 1}, from set.eq_singleton_iff_eq_empty.mp (set.eq_empty_iff_forall_not_mem.mpr (λ a h, by {cases h with b hb, cases hb with hb1 hb2, rw ← hb1 at hb2,exact H1 hb2})),\n    let f₃ : V → ℕ := by {exact congr_arg _ (set.range_eq_univ.mp (set.eq_univ_singleton.mp H2))},\n    have H1 : (f₃ '' V) ⊆ ({0, 1} : set ℕ), from set.image_subset_iff.mpr H2,\n    have H2 : f₃ '' V = {0,1}, from set.eq_singleton_iff_eq_empty.mp (set.eq_empty_iff_forall_not_mem.mpr (λ a h, by {cases h with b hb, cases hb with hb1 hb2, rw hb1 at hb2,exact H1 hb2})),\n    have H3 : (f₂ '' (α ⊕ β)) = (f₃ '' V), from congr_arg _ (set.range_eq_univ.mp (set.eq_univ_singleton.mp H2)),\n      let f₄ : (α ⊕ β) → V := by {exact congr_arg _ (set.range_eq_univ.mp (set.eq_univ_singleton.mp H3))},\n      have H4 : set.range f₄ = V, from set.range_eq_univ.mpr (set.eq_univ_singleton.mpr H3),\n      have H5 : f₄ '' (α ⊕ β) = V := by {exact congr_arg _ H4},\n      let A : Type* := @classical.some (fintype (α ⊕ β)) _ (fintype.exists_fintype_iff.mp (fintype.fintype_image ((α ⊕ β)) V f₄)),\n      let B : Type* := @classical.some (fintype (α ⊕ β)) _ (fintype.exists_fintype_iff.mp (fintype.fintype_image ((α ⊕ β)) V f₄)),\n      let h1 : (α ⊕ β) = A ⊕ B := by {exact classical.some_spec (fintype.exists_fintype_iff.mp (fintype.fintype_image ((α ⊕ β)) V f₄))},\n      let h2 : A ∈ set.powerset (α ⊕ β), from by {rw ← h1 at *, exact simple_graph.fintype_edge_set G},\n      let h3 : B ∈ set.powerset (α ⊕ β), from by {rw ← h1 at *, exact simple_graph.fintype_edge_set G},\n      let h4 : A ∩ B = ∅ := (set.pairwise_disjoint_of_nonempty_inter_eq_empty (set.inter_subset_left _ _) (set.inter_subset_right _ _) (by {rw ← h1, exact simple_graph.fintype_edge_set G})),\n      let f₅ : (α ⊕ β) → α ⊕ β := λ x, by {have H6 : x ∈ (fintype.fintype_image (α ⊕ β) V f₄).fintype := by {rw ← h1 at *, exact (set.fintype_of_fintype_image_fintype f₄) x},rw ← h1 at *, exact (classical.some_spec (fintype.exists_fintype_iff.mp (fintype.fintype_image ((α ⊕ β)) V f₄))) x},\n      have H5 : (f₅ '' α ⊕ β) ⊆ V, from (set.image_subset_iff.mpr H4),\n      have H6 : f₅ '' α ⊕ β = V := set.eq_univ_of_univ_subset H5,\n      have H7 : set.range f₅ = V := set.range_eq_univ.mpr (set.eq_univ_singleton.mpr H6),\n      let f₆ : α ⊕ β → V := λ x, by {have H6 : x ∈ (fintype.fintype_image (α ⊕ β) V f₄).fintype := by {rw ← h1 at *, exact (set.fintype_of_fintype_image_fintype f₄) x},rw ← h1 at *, exact (classical.some_spec (fintype.exists_fintype_iff.mp (fintype.fintype_image ((α ⊕ β)) V f₄))) x},\n      have H5 : (f₆ '' α ⊕ β) ⊆ V, from (set.image_subset_iff.mpr H4),\n      have H6 : f₆ '' α ⊕ β = V := set.eq_univ_of_univ_subset H5,\n      have H7 : set.range f₆ = V := set.range_eq_univ.mpr (set.eq_univ_singleton.mpr\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  admit,\nend\n\n--OUTPUT 9\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n    split,\n    assume h,\n    have h1 := G.colorable_iff_edge_disjoint_complete_graph h,\n    have h2 := G.edge_disjoint_complete_graph_iff_complete_bipartite h1,\n    cases h2 with A B (hA : A = G.edge_disjoint_component 0) (hB : B = G.edge_disjoint_component 1),\n    rw ←hA, rw ←hB,\n    let tmp := disjoint_union_type.mk A B,\n    have h3 : (A ⊕ B) = G.V, from disjoint_union_type.mk_ne_empty A B,\n    let tmp2 := cast (congr_arg _ h3) (complete_bipartite_graph A B),\n    have h4 : G ≤ tmp2, by exact h2,\n    exact ⟨A, B, h3, h4⟩,\nend\n\n--OUTPUT 10\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume H : G.colorable 2,\n    have H1 : (powerset V).card = fintype.card (powerset V), from by { simp [powerset, card_powerset] },\n    have H2 : (powerset V).card = 2 ^ V.card, from by { simp },\n    have H3 : V.card = (finset.range 2).card, from congr_arg card H1.symm.trans H2.symm,\n    have H4 : V.card = 2, from by { simp [card_range_eq_succ_pred_of_nat, *] at H3 },\n    have H5 : (finset.range 2) ≃ V, from by { have H5 : fintype.card (finset.range 2) = 2, by { simp }, simp [fintype.equiv_fin, *] at H5, exact H5.trans H4 },\n    have H6 : V ≃ (finset.range 2), from equiv.symm H5,\n    have H7 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by { simp [*, @add_congr_arg] at H6, exact ⟨ (fin 2), (fin 2), H6, eq.trans (simple_graph.eq_complete_bipartite_graph_iff_is_bipartite G H6) (simple_graph.is_bipartite_iff_two_colors G H H6)⟩ },\n    exact H7,\n  },\n  {\n    assume H : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B),\n    let ⟨ A, B, H1, H2 ⟩ := H in\n    let h := by { convert H1, simp },\n    let f := by { simp [h, *] },\n    have H3 : G.colorable 2, from by exact ⟨ 2, f ⟩,\n    exact H3,\n  }\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.3350438123311148}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura, Jeremy Avigad, Floris van Doorn\n\n! This file was ported from Lean 3 source module init.data.sigma.basic\n! leanprover-community/mathlib commit 87932f1c56b7c05aefd2810d05e69da9500a0064\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Logic\nimport Leanbin.Init.Wf\n\nuniverse u v\n\n/- warning: ex_of_psig -> ex_of_psig is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {p : α -> Prop}, (PSigma.{succ u1, 0} α (fun (x : α) => p x)) -> (Exists.{succ u1} α (fun (x : α) => p x))\nbut is expected to have type\n  forall {α : Sort.{u1}} {p : α -> Prop}, (PSigma.{u1, 0} α (fun (x : α) => p x)) -> (Exists.{u1} α (fun (x : α) => p x))\nCase conversion may be inaccurate. Consider using '#align ex_of_psig ex_of_psigₓ'. -/\ntheorem ex_of_psig {α : Type u} {p : α → Prop} : (Σ'x, p x) → ∃ x, p x\n  | ⟨x, hx⟩ => ⟨x, hx⟩\n#align ex_of_psig ex_of_psig\n\nsection\n\nvariable {α : Type u} {β : α → Type v}\n\n#print Sigma.eq /-\nprotected theorem Sigma.eq :\n    ∀ {p₁ p₂ : Σa : α, β a} (h₁ : p₁.1 = p₂.1), (Eq.recOn h₁ p₁.2 : β p₂.1) = p₂.2 → p₁ = p₂\n  | ⟨a, b⟩, ⟨a, b⟩, rfl, rfl => rfl\n#align sigma.eq Sigma.eq\n-/\n\nend\n\nsection\n\nvariable {α : Sort u} {β : α → Sort v}\n\n#print PSigma.eq /-\nprotected theorem PSigma.eq :\n    ∀ {p₁ p₂ : PSigma β} (h₁ : p₁.1 = p₂.1), (Eq.recOn h₁ p₁.2 : β p₂.1) = p₂.2 → p₁ = p₂\n  | ⟨a, b⟩, ⟨a, b⟩, rfl, rfl => rfl\n#align psigma.eq PSigma.eq\n-/\n\nend\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Data/Sigma/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241911813151, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.33503807082622983}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.basic\n\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but aren't implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `lists ℕ` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `α` could be modelled as\n`α` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-steps definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `lists' α ff`: Atoms as ZFA prelists. Basically a copy of `α`.\n* `lists' α tt`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist (`lists'.nil`)\n  and from appending a ZFA prelist to a proper ZFA prelist (`lists'.cons a l`).\n* `lists α`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n\n## TODO\n\nThe next step is to define ZFA sets as lists quotiented by `lists.equiv`.\n(-/\n\nvariables {α : Type*}\n\n/-- Prelists, helper type to define `lists`. `lists' α ff` are the \"atoms\", a copy of `α`.\n`lists' α tt` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and from\nappending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything to an\natom while having only one appending function for appending both atoms and proper ZFC prelists to a\nproper ZFA prelist. -/\n@[derive decidable_eq]\ninductive {u} lists' (α : Type u) : bool → Type u\n| atom : α → lists' ff\n| nil : lists' tt\n| cons' {b} : lists' b → lists' tt → lists' tt\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = ff`), corresponding\nto an element of `α`, or a \"proper\" ZFA list, inductively defined from the empty ZFA list and from\nappending a ZFA list to a proper ZFA list. -/\ndef lists (α : Type*) := Σ b, lists' α b\n\nnamespace lists'\n\ninstance [inhabited α] : ∀ b, inhabited (lists' α b)\n| tt := ⟨nil⟩\n| ff := ⟨atom default⟩\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : lists α → lists' α tt → lists' α tt\n| ⟨b, a⟩ l := cons' a l\n\n/-- Converts a ZFA prelist to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : ∀ {b}, lists' α b → list (lists α)\n| _ (atom a)    := []\n| _ nil         := []\n| _ (cons' a l) := ⟨_, a⟩ :: l.to_list\n\n@[simp] theorem to_list_cons (a : lists α) (l) :\n  to_list (cons a l) = a :: l.to_list :=\nby cases a; simp [cons]\n\n/-- Converts a `list` of ZFA lists to a proper ZFA prelist. -/\n@[simp] def of_list : list (lists α) → lists' α tt\n| []       := nil\n| (a :: l) := cons a (of_list l)\n\n@[simp] theorem to_of_list (l : list (lists α)) : to_list (of_list l) = l :=\nby induction l; simp *\n\n@[simp] theorem of_to_list : ∀ (l : lists' α tt), of_list (to_list l) = l :=\nsuffices ∀ b (h : tt = b) (l : lists' α b),\n  let l' : lists' α tt := by rw h; exact l in\n  of_list (to_list l') = l', from this _ rfl,\nλ b h l, begin\n  induction l, {cases h}, {exact rfl},\n  case lists'.cons' : b a l IH₁ IH₂\n  { intro, change l' with cons' a l,\n    simpa [cons] using IH₂ rfl }\nend\n\nend lists'\n\nmutual inductive lists.equiv, lists'.subset\nwith lists.equiv : lists α → lists α → Prop\n| refl (l) : lists.equiv l l\n| antisymm {l₁ l₂ : lists' α tt} :\n  lists'.subset l₁ l₂ → lists'.subset l₂ l₁ → lists.equiv ⟨_, l₁⟩ ⟨_, l₂⟩\nwith lists'.subset : lists' α tt → lists' α tt → Prop\n| nil {l} : lists'.subset lists'.nil l\n| cons {a a' l l'} : lists.equiv a a' → a' ∈ lists'.to_list l' →\n  lists'.subset l l' → lists'.subset (lists'.cons a l) l'\nlocal infix ` ~ `:50 := lists.equiv\n\n/-- Equivalence of ZFA lists. Defined inductively. -/\nadd_decl_doc lists.equiv\n\n/-- Subset relation for ZFA lists. Defined inductively. -/\nadd_decl_doc lists'.subset\n\nnamespace lists'\n\ninstance : has_subset (lists' α tt) := ⟨lists'.subset⟩\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : has_mem (lists α) (lists' α b) :=\n⟨λ a l, ∃ a' ∈ l.to_list, a ~ a'⟩\n\ntheorem mem_def {b a} {l : lists' α b} :\n  a ∈ l ↔ ∃ a' ∈ l.to_list, a ~ a' := iff.rfl\n\n@[simp] theorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l :=\nby simp [mem_def, or_and_distrib_right, exists_or_distrib]\n\ntheorem cons_subset {a} {l₁ l₂ : lists' α tt} :\n  lists'.cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂ :=\nbegin\n  refine ⟨λ h, _, λ ⟨⟨a', m, e⟩, s⟩, subset.cons e m s⟩,\n  generalize_hyp h' : lists'.cons a l₁ = l₁' at h,\n  cases h with l a' a'' l l' e m s, {cases a, cases h'},\n  cases a, cases a', cases h', exact ⟨⟨_, m, e⟩, s⟩\nend\n\ntheorem of_list_subset {l₁ l₂ : list (lists α)} (h : l₁ ⊆ l₂) :\n  lists'.of_list l₁ ⊆ lists'.of_list l₂ :=\nbegin\n  induction l₁, {exact subset.nil},\n  refine subset.cons (lists.equiv.refl _) _ (l₁_ih (list.subset_of_cons_subset h)),\n  simp at h, simp [h]\nend\n\n@[refl] theorem subset.refl {l : lists' α tt} : l ⊆ l :=\nby rw ← lists'.of_to_list l; exact\n   of_list_subset (list.subset.refl _)\n\ntheorem subset_nil {l : lists' α tt} :\n  l ⊆ lists'.nil → l = lists'.nil :=\nbegin\n  rw ← of_to_list l,\n  induction to_list l; intro h, {refl},\n  rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩\nend\n\ntheorem mem_of_subset' {a} {l₁ l₂ : lists' α tt}\n  (s : l₁ ⊆ l₂) (h : a ∈ l₁.to_list) : a ∈ l₂ :=\nbegin\n  induction s with _ a a' l l' e m s IH, {cases h},\n  simp at h, rcases h with rfl|h,\n  exacts [⟨_, m, e⟩, IH h]\nend\n\ntheorem subset_def {l₁ l₂ : lists' α tt} :\n  l₁ ⊆ l₂ ↔ ∀ a ∈ l₁.to_list, a ∈ l₂ :=\n⟨λ H a, mem_of_subset' H, λ H, begin\n  rw ← of_to_list l₁,\n  revert H, induction to_list l₁; intro,\n  { exact subset.nil },\n  { simp at H, exact cons_subset.2 ⟨H.1, ih H.2⟩ }\nend⟩\n\nend lists'\n\nnamespace lists\n\n/-- Sends `a : α` to the corresponding atom in `lists α`. -/\n@[pattern] def atom (a : α) : lists α := ⟨_, lists'.atom a⟩\n\n/-- Converts a proper ZFA prelist to a ZFA list. -/\n@[pattern] def of' (l : lists' α tt) : lists α := ⟨_, l⟩\n\n/-- Converts a ZFA list to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : lists α → list (lists α)\n| ⟨b, l⟩ := l.to_list\n\n/-- Predicate stating that a ZFA list is proper. -/\ndef is_list (l : lists α) : Prop := l.1\n\n/-- Converts a `list` of ZFA lists to a ZFA list. -/\ndef of_list (l : list (lists α)) : lists α := of' (lists'.of_list l)\n\ntheorem is_list_to_list (l : list (lists α)) : is_list (of_list l) :=\neq.refl _\n\ntheorem to_of_list (l : list (lists α)) : to_list (of_list l) = l :=\nby simp [of_list, of']\n\ntheorem of_to_list : ∀ {l : lists α}, is_list l → of_list (to_list l) = l\n| ⟨tt, l⟩ _ := by simp [of_list, of']\n\ninstance : inhabited (lists α) :=\n⟨of' lists'.nil⟩\n\ninstance [decidable_eq α] : decidable_eq (lists α) :=\nby unfold lists; apply_instance\n\ninstance [has_sizeof α] : has_sizeof (lists α) :=\nby unfold lists; apply_instance\n\n/-- A recursion principle for pairs of ZFA lists and proper ZFA prelists. -/\ndef induction_mut (C : lists α → Sort*) (D : lists' α tt → Sort*)\n  (C0 : ∀ a, C (atom a)) (C1 : ∀ l, D l → C (of' l))\n  (D0 : D lists'.nil) (D1 : ∀ a l, C a → D l → D (lists'.cons a l)) :\n  pprod (∀ l, C l) (∀ l, D l) :=\nbegin\n  suffices : ∀ {b} (l : lists' α b),\n    pprod (C ⟨_, l⟩) (match b, l with\n    | tt, l := D l\n    | ff, l := punit\n    end),\n  { exact ⟨λ ⟨b, l⟩, (this _).1, λ l, (this l).2⟩ },\n  intros, induction l with a b a l IH₁ IH₂,\n  { exact ⟨C0 _, ⟨⟩⟩ },\n  { exact ⟨C1 _ D0, D0⟩ },\n  { suffices, {exact ⟨C1 _ this, this⟩},\n    exact D1 ⟨_, _⟩ _ IH₁.1 IH₂.2 }\nend\n\n/-- Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a\nproper ZFA prelist. An atom has no members. -/\ndef mem (a : lists α) : lists α → Prop\n| ⟨ff, l⟩ := false\n| ⟨tt, l⟩ := a ∈ l\n\ninstance : has_mem (lists α) (lists α) := ⟨mem⟩\n\ntheorem is_list_of_mem {a : lists α} : ∀ {l : lists α}, a ∈ l → is_list l\n| ⟨_, lists'.nil⟩       _ := rfl\n| ⟨_, lists'.cons' _ _⟩ _ := rfl\n\ntheorem equiv.antisymm_iff {l₁ l₂ : lists' α tt} :\n  of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁ :=\nbegin\n  refine ⟨λ h, _, λ ⟨h₁, h₂⟩, equiv.antisymm h₁ h₂⟩,\n  cases h with _ _ _ h₁ h₂,\n  { simp [lists'.subset.refl] }, { exact ⟨h₁, h₂⟩ }\nend\n\nattribute [refl] equiv.refl\n\ntheorem equiv_atom {a} {l : lists α} : atom a ~ l ↔ atom a = l :=\n⟨λ h, by cases h; refl, λ h, h ▸ equiv.refl _⟩\n\ntheorem equiv.symm {l₁ l₂ : lists α} (h : l₁ ~ l₂) : l₂ ~ l₁ :=\nby cases h with _ _ _ h₁ h₂; [refl, exact equiv.antisymm h₂ h₁]\n\ntheorem equiv.trans : ∀ {l₁ l₂ l₃ : lists α}, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=\nbegin\n  let trans := λ (l₁ : lists α), ∀ ⦃l₂ l₃⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃,\n  suffices : pprod (∀ l₁, trans l₁)\n    (∀ (l : lists' α tt) (l' ∈ l.to_list), trans l'), {exact this.1},\n  apply induction_mut,\n  { intros a l₂ l₃ h₁ h₂,\n    rwa ← equiv_atom.1 h₁ at h₂ },\n  { intros l₁ IH l₂ l₃ h₁ h₂,\n    cases h₁ with _ _ l₂, {exact h₂},\n    cases h₂ with _ _ l₃, {exact h₁},\n    cases equiv.antisymm_iff.1 h₁ with hl₁ hr₁,\n    cases equiv.antisymm_iff.1 h₂ with hl₂ hr₂,\n    apply equiv.antisymm_iff.2; split; apply lists'.subset_def.2,\n    { intros a₁ m₁,\n      rcases lists'.mem_of_subset' hl₁ m₁ with ⟨a₂, m₂, e₁₂⟩,\n      rcases lists'.mem_of_subset' hl₂ m₂ with ⟨a₃, m₃, e₂₃⟩,\n      exact ⟨a₃, m₃, IH _ m₁ e₁₂ e₂₃⟩ },\n    { intros a₃ m₃,\n      rcases lists'.mem_of_subset' hr₂ m₃ with ⟨a₂, m₂, e₃₂⟩,\n      rcases lists'.mem_of_subset' hr₁ m₂ with ⟨a₁, m₁, e₂₁⟩,\n      exact ⟨a₁, m₁, (IH _ m₁ e₂₁.symm e₃₂.symm).symm⟩ } },\n  { rintro _ ⟨⟩ },\n  { intros a l IH₁ IH₂, simpa [IH₁] using IH₂ }\nend\n\ninstance : setoid (lists α) :=\n⟨(~), equiv.refl, @equiv.symm _, @equiv.trans _⟩\n\nsection decidable\n\n@[simp] def equiv.decidable_meas :\n  (psum (Σ' (l₁ : lists α), lists α) $\n   psum (Σ' (l₁ : lists' α tt), lists' α tt)\n   Σ' (a : lists α), lists' α tt) → ℕ\n| (psum.inl ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂\n| (psum.inr $ psum.inl ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂\n| (psum.inr $ psum.inr ⟨l₁, l₂⟩) := sizeof l₁ + sizeof l₂\n\nopen well_founded_tactics\n\ntheorem sizeof_pos {b} (l : lists' α b) : 0 < sizeof l :=\nby cases l; unfold_sizeof; trivial_nat_lt\n\ntheorem lt_sizeof_cons' {b} (a : lists' α b) (l) :\n  sizeof (⟨b, a⟩ : lists α) < sizeof (lists'.cons' a l) :=\nby {unfold_sizeof, apply sizeof_pos}\n\n@[instance] mutual def equiv.decidable, subset.decidable, mem.decidable [decidable_eq α]\nwith equiv.decidable : ∀ l₁ l₂ : lists α, decidable (l₁ ~ l₂)\n| ⟨ff, l₁⟩ ⟨ff, l₂⟩ := decidable_of_iff' (l₁ = l₂) $\n  by cases l₁; refine equiv_atom.trans (by simp [atom])\n| ⟨ff, l₁⟩ ⟨tt, l₂⟩ := is_false $ by rintro ⟨⟩\n| ⟨tt, l₁⟩ ⟨ff, l₂⟩ := is_false $ by rintro ⟨⟩\n| ⟨tt, l₁⟩ ⟨tt, l₂⟩ := begin\n  haveI :=\n    have sizeof l₁ + sizeof l₂ <\n         sizeof (⟨tt, l₁⟩ : lists α) + sizeof (⟨tt, l₂⟩ : lists α),\n    by default_dec_tac,\n    subset.decidable l₁ l₂,\n  haveI :=\n    have sizeof l₂ + sizeof l₁ <\n         sizeof (⟨tt, l₁⟩ : lists α) + sizeof (⟨tt, l₂⟩ : lists α),\n    by default_dec_tac,\n    subset.decidable l₂ l₁,\n  exact decidable_of_iff' _ equiv.antisymm_iff,\nend\nwith subset.decidable : ∀ l₁ l₂ : lists' α tt, decidable (l₁ ⊆ l₂)\n| lists'.nil l₂ := is_true subset.nil\n| (@lists'.cons' _ b a l₁) l₂ := begin\n  haveI :=\n    have sizeof (⟨b, a⟩ : lists α) + sizeof l₂ <\n         sizeof (lists'.cons' a l₁) + sizeof l₂,\n    from add_lt_add_right (lt_sizeof_cons' _ _) _,\n    mem.decidable ⟨b, a⟩ l₂,\n  haveI :=\n    have sizeof l₁ + sizeof l₂ <\n         sizeof (lists'.cons' a l₁) + sizeof l₂,\n    by default_dec_tac,\n    subset.decidable l₁ l₂,\n  exact decidable_of_iff' _ (@lists'.cons_subset _ ⟨_, _⟩ _ _)\nend\nwith mem.decidable : ∀ (a : lists α) (l : lists' α tt), decidable (a ∈ l)\n| a lists'.nil := is_false $ by rintro ⟨_, ⟨⟩, _⟩\n| a (lists'.cons' b l₂) := begin\n  haveI :=\n    have sizeof a + sizeof (⟨_, b⟩ : lists α) <\n         sizeof a + sizeof (lists'.cons' b l₂),\n    from add_lt_add_left (lt_sizeof_cons' _ _) _,\n    equiv.decidable a ⟨_, b⟩,\n  haveI :=\n    have sizeof a + sizeof l₂ <\n         sizeof a + sizeof (lists'.cons' b l₂),\n    by default_dec_tac,\n    mem.decidable a l₂,\n  refine decidable_of_iff' (a ~ ⟨_, b⟩ ∨ a ∈ l₂) _,\n  rw ← lists'.mem_cons, refl\nend\nusing_well_founded\n{ rel_tac := λ _ _, `[exact ⟨_, measure_wf equiv.decidable_meas⟩],\n  dec_tac := `[assumption] }\n\nend decidable\n\nend lists\n\nnamespace lists'\n\ntheorem mem_equiv_left {l : lists' α tt} :\n  ∀ {a a'}, a ~ a' → (a ∈ l ↔ a' ∈ l) :=\nsuffices ∀ {a a'}, a ~ a' → a ∈ l → a' ∈ l,\n  from λ a a' e, ⟨this e, this e.symm⟩,\nλ a₁ a₂ e₁ ⟨a₃, m₃, e₂⟩, ⟨_, m₃, e₁.symm.trans e₂⟩\n\ntheorem mem_of_subset {a} {l₁ l₂ : lists' α tt}\n  (s : l₁ ⊆ l₂) : a ∈ l₁ → a ∈ l₂ | ⟨a', m, e⟩ :=\n(mem_equiv_left e).2 (mem_of_subset' s m)\n\ntheorem subset.trans {l₁ l₂ l₃ : lists' α tt}\n  (h₁ : l₁ ⊆ l₂) (h₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=\nsubset_def.2 $ λ a₁ m₁, mem_of_subset h₂ $ mem_of_subset' h₁ m₁\n\nend lists'\n\ndef finsets (α : Type*) := quotient (@lists.setoid α)\n\nnamespace finsets\n\ninstance : has_emptyc (finsets α) := ⟨⟦lists.of' lists'.nil⟧⟩\n\ninstance : inhabited (finsets α) := ⟨∅⟩\n\ninstance [decidable_eq α] : decidable_eq (finsets α) :=\nby unfold finsets; apply_instance\n\nend finsets\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/set_theory/lists.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.33486237255569956}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.limits.types\nimport category_theory.concrete_category\nimport category_theory.morphism_property\nimport category_theory.limits.mono_coprod\nimport data.set.finite\n\nuniverses v u\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {C : Type*} [category C] [has_finite_coproducts C]\n\nsection\n\nlemma congr_colimit_ι {J D : Type*} [category J] [category D] (F : J ⥤ D) [has_colimit F]\n  {j₁ j₂ : J} (h : j₁ = j₂) :\n  colimit.ι F j₁ = eq_to_hom (by rw h) ≫ colimit.ι F j₂ :=\nby { subst h, rw [eq_to_hom_refl, id_comp], }\n\nlemma congr_sigma_ι {J D : Type*} [category D] (X : J → D) [has_coproduct X]\n  {j₁ j₂ : J} (h : j₁ = j₂) :\n  sigma.ι X j₁ = eq_to_hom (by rw h) ≫ sigma.ι X j₂ :=\ncongr_colimit_ι (discrete.functor X) (congr_arg discrete.mk h)\n\n@[simp]\ndef coproduct_pullback {A B : Type*} (X : B → C) (f : A → B) [has_coproduct X]\n  [has_coproduct (X ∘ f)] : ∐ (X ∘ f) ⟶ ∐ X := sigma.desc (λ a, sigma.ι _ (f a))\n\n@[simps]\ndef coproduct_pullback_iso {A B : Type*} (X : B → C) (e : A ≃ B) [has_coproduct X]\n  [has_coproduct (X ∘ e)] : ∐ (X ∘ e) ≅ ∐ X :=\n{ hom := coproduct_pullback X e,\n  inv := sigma.desc (λ b, eq_to_hom (by simp) ≫ sigma.ι _ (e.symm b)),\n  hom_inv_id' := begin\n    ext a,\n    discrete_cases,\n    simp only [coproduct_pullback, colimit.ι_desc_assoc, cofan.mk_ι_app, colimit.ι_desc, comp_id],\n    exact (congr_sigma_ι (X ∘ e) (e.symm_apply_apply a).symm).symm,\n  end,\n  inv_hom_id' := begin\n    ext b,\n    discrete_cases,\n    simp only [coproduct_pullback, colimit.ι_desc_assoc, cofan.mk_ι_app, assoc, colimit.ι_desc,\n      comp_id],\n    exact (congr_sigma_ι X (e.apply_symm_apply b).symm).symm,\n  end, }\n\ninstance mono_coproduct_pullback_inl [mono_coprod C] {A B : Type*} (X : A ⊕ B → C)\n  [has_coproduct X] [has_coproduct (X ∘ sum.inl)] [has_coproduct (X ∘ sum.inr)] :\n  mono (coproduct_pullback X sum.inl) :=\nbegin\n  let c : binary_cofan (∐ (X ∘ sum.inl)) ((∐ (X ∘ sum.inr))) := binary_cofan.mk\n    (coproduct_pullback X sum.inl) (coproduct_pullback X sum.inr),\n  have hc : is_colimit c := begin\n    refine binary_cofan.is_colimit.mk c _ _ _ _,\n    { intros T f₁ f₂,\n      refine sigma.desc (λ x, _),\n      cases x,\n      { refine _ ≫ f₁, exact sigma.ι (X ∘ sum.inl) x, },\n      { refine _ ≫ f₂, exact sigma.ι (X ∘ sum.inr) x, }, },\n    { intros T f₁ f₂, ext, discrete_cases, simp, },\n    { intros T f₁ f₂, ext, discrete_cases, simp, },\n    { intros T f₁ f₂ m hm₁ hm₂,\n      ext x,\n      discrete_cases,\n      tidy, },\n  end,\n  exact mono_coprod.binary_cofan_inl c hc,\nend\n\nlemma mono_coproduct_pullback_of_injective [mono_coprod C] [has_finite_coproducts C]\n  {A B : Type*} [finite A] [finite B] (X : B → C) (f : A → B) (hf : function.injective f) :\n  mono (coproduct_pullback X f) :=\nbegin\n  let A' := (set.image f set.univ)ᶜ,\n  let g : A ⊕ A' → B := λ x, by { cases x, exacts [f x, x.1], },\n  have hg : function.bijective g,\n  { split,\n    { rintros (a₁|a₁') (a₂|a₂'),\n      { tidy, },\n      { intro h,\n        exfalso,\n        exact a₂'.2 ⟨a₁, ⟨set.mem_univ _, h⟩⟩, },\n      { intro h,\n        exfalso,\n        exact a₁'.2 ⟨a₂, ⟨set.mem_univ _, h.symm⟩⟩, },\n      { tidy, }, },\n    { intro b,\n      by_cases ∃ (a : A), b = f a,\n      { cases h with a ha,\n        exact ⟨sum.inl a, ha.symm⟩, },\n      { exact ⟨sum.inr ⟨b, λ h', h ⟨h'.some, h'.some_spec.2.symm⟩⟩, rfl⟩, }, }, },\n  let γ := equiv.of_bijective g hg,\n  haveI : finite A' := finite.of_injective (λ a', a'.1) (λ a₁' a₂' h, by { ext, exact h, }),\n  let E : arrow.mk (coproduct_pullback (X ∘ γ) sum.inl) ≅ arrow.mk (coproduct_pullback X f),\n  { refine arrow.iso_mk (iso.refl _) (coproduct_pullback_iso X γ) _,\n    ext a,\n    discrete_cases,\n    dsimp,\n    simp only [id_comp, colimit.ι_desc_assoc, cofan.mk_ι_app, colimit.ι_desc],\n    erw [colimit.ι_desc, cofan.mk_ι_app], },\n  exact ((morphism_property.respects_iso.monomorphisms C).arrow_mk_iso_iff E).mp\n    (morphism_property.monomorphisms.infer_property _),\nend\n\nend\n\nend limits\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/mono_coprod_in.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.33486236465043556}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport ring_theory.finite_type\nimport ring_theory.localization.at_prime\nimport ring_theory.localization.away\nimport ring_theory.localization.integer\nimport ring_theory.localization.submodule\nimport ring_theory.nilpotent\nimport ring_theory.ring_hom_properties\n\n/-!\n# Local properties of commutative rings\n\nIn this file, we provide the proofs of various local properties.\n\n## Naming Conventions\n\n* `localization_P` : `P` holds for `S⁻¹R` if `P` holds for `R`.\n* `P_of_localization_maximal` : `P` holds for `R` if `P` holds for `Rₘ` for all maximal `m`.\n* `P_of_localization_prime` : `P` holds for `R` if `P` holds for `Rₘ` for all prime `m`.\n* `P_of_localization_span` : `P` holds for `R` if given a spanning set `{fᵢ}`, `P` holds for all\n  `R_{fᵢ}`.\n\n## Main results\n\nThe following properties are covered:\n\n* The triviality of an ideal or an element:\n  `ideal_eq_bot_of_localization`, `eq_zero_of_localization`\n* `is_reduced` : `localization_is_reduced`, `is_reduced_of_localization_maximal`.\n* `finite`: `localization_finite`, `finite_of_localization_span`\n* `finite_type`: `localization_finite_type`, `finite_type_of_localization_span`\n\n-/\n\nopen_locale pointwise classical big_operators\n\nuniverse u\n\nvariables {R S : Type u} [comm_ring R] [comm_ring S] (M : submonoid R)\nvariables (N : submonoid S) (R' S' : Type u) [comm_ring R'] [comm_ring S'] (f : R →+* S)\nvariables [algebra R R'] [algebra S S']\n\nsection properties\n\nsection comm_ring\n\nvariable (P : ∀ (R : Type u) [comm_ring R], Prop)\n\ninclude P\n\n/-- A property `P` of comm rings is said to be preserved by localization\n  if `P` holds for `M⁻¹R` whenever `P` holds for `R`. -/\ndef localization_preserves : Prop :=\n  ∀ {R : Type u} [hR : comm_ring R] (M : by exactI submonoid R) (S : Type u) [hS : comm_ring S]\n    [by exactI algebra R S] [by exactI is_localization M S], @P R hR → @P S hS\n\n/-- A property `P` of comm rings satisfies `of_localization_maximal` if\n  if `P` holds for `R` whenever `P` holds for `Rₘ` for all maximal ideal `m`. -/\ndef of_localization_maximal : Prop :=\n  ∀ (R : Type u) [comm_ring R],\n    by exactI (∀ (J : ideal R) (hJ : J.is_maximal), by exactI P (localization.at_prime J)) → P R\n\nend comm_ring\n\nsection ring_hom\n\nvariable (P : ∀ {R S : Type u} [comm_ring R] [comm_ring S] (f : by exactI R →+* S), Prop)\n\ninclude P\n\n/-- A property `P` of ring homs is said to be preserved by localization\n if `P` holds for `M⁻¹R →+* M⁻¹S` whenever `P` holds for `R →+* S`. -/\ndef ring_hom.localization_preserves :=\n  ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S) (M : by exactI submonoid R)\n    (R' S' : Type u) [comm_ring R'] [comm_ring S'] [by exactI algebra R R']\n    [by exactI algebra S S'] [by exactI is_localization M R']\n    [by exactI is_localization (M.map f) S'],\n    by exactI (P f → P (is_localization.map S' f (submonoid.le_comap_map M) : R' →+* S'))\n\n/-- A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span`\nif `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `R` such that\n`P` holds for `Rᵣ →+* Sᵣ`.\n\nNote that this is equivalent to `ring_hom.of_localization_span` via\n`ring_hom.of_localization_span_iff_finite`, but this is easier to prove. -/\ndef ring_hom.of_localization_finite_span :=\n  ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)\n    (s : finset R) (hs : by exactI ideal.span (s : set R) = ⊤)\n    (H : by exactI (∀ (r : s), P (localization.away_map f r))), by exactI P f\n\n/-- A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span`\nif `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `R` such that\n`P` holds for `Rᵣ →+* Sᵣ`.\n\nNote that this is equivalent to `ring_hom.of_localization_finite_span` via\n`ring_hom.of_localization_span_iff_finite`, but this has less restrictions when applying. -/\ndef ring_hom.of_localization_span :=\n  ∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)\n    (s : set R) (hs : by exactI ideal.span s = ⊤)\n    (H : by exactI (∀ (r : s), P (localization.away_map f r))), by exactI P f\n\n/-- A property `P` of ring homs satisfies `ring_hom.holds_for_localization_away`\n if `P` holds for each localization map `R →+* Rᵣ`. -/\ndef ring_hom.holds_for_localization_away : Prop :=\n∀ ⦃R : Type u⦄ (S : Type u) [comm_ring R] [comm_ring S] [by exactI algebra R S] (r : R)\n  [by exactI is_localization.away r S], by exactI P (algebra_map R S)\n\n/-- A property `P` of ring homs satisfies `ring_hom.of_localization_finite_span_target`\nif `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `S` such that\n`P` holds for `R →+* Sᵣ`.\n\nNote that this is equivalent to `ring_hom.of_localization_span_target` via\n`ring_hom.of_localization_span_target_iff_finite`, but this is easier to prove. -/\ndef ring_hom.of_localization_finite_span_target : Prop :=\n∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)\n  (s : finset S) (hs : by exactI ideal.span (s : set S) = ⊤)\n  (H : by exactI (∀ (r : s), P ((algebra_map S (localization.away (r : S))).comp f))),\n  by exactI P f\n\n/-- A property `P` of ring homs satisfies `ring_hom.of_localization_span_target`\nif `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `S` such that\n`P` holds for `R →+* Sᵣ`.\n\nNote that this is equivalent to `ring_hom.of_localization_finite_span_target` via\n`ring_hom.of_localization_span_target_iff_finite`, but this has less restrictions when applying. -/\ndef ring_hom.of_localization_span_target : Prop :=\n∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S)\n  (s : set S) (hs : by exactI ideal.span s = ⊤)\n  (H : by exactI (∀ (r : s), P ((algebra_map S (localization.away (r : S))).comp f))),\n  by exactI P f\n\n/-- A property `P` of ring homs satisfies `of_localization_prime` if\n  if `P` holds for `R` whenever `P` holds for `Rₘ` for all prime ideals `p`. -/\ndef ring_hom.of_localization_prime : Prop :=\n∀ ⦃R S : Type u⦄ [comm_ring R] [comm_ring S] (f : by exactI R →+* S),\n  by exactI (∀ (J : ideal S) (hJ : J.is_prime),\n    by exactI P (localization.local_ring_hom _ J f rfl)) → P f\n\n/-- A property of ring homs is local if it is preserved by localizations and compositions, and for\neach `{ r }` that spans `S`, we have `P (R →+* S) ↔ ∀ r, P (R →+* Sᵣ)`. -/\nstructure ring_hom.property_is_local : Prop :=\n(localization_preserves : ring_hom.localization_preserves @P)\n(of_localization_span_target : ring_hom.of_localization_span_target @P)\n(stable_under_composition : ring_hom.stable_under_composition @P)\n(holds_for_localization_away : ring_hom.holds_for_localization_away @P)\n\nlemma ring_hom.of_localization_span_iff_finite :\n  ring_hom.of_localization_span @P ↔ ring_hom.of_localization_finite_span @P :=\nbegin\n  delta ring_hom.of_localization_span ring_hom.of_localization_finite_span,\n  apply forall₅_congr, -- TODO: Using `refine` here breaks `resetI`.\n  introsI,\n  split,\n  { intros h s, exact h s },\n  { intros h s hs hs',\n    obtain ⟨s', h₁, h₂⟩ := (ideal.span_eq_top_iff_finite s).mp hs,\n    exact h s' h₂ (λ x, hs' ⟨_, h₁ x.prop⟩) }\nend\n\nlemma ring_hom.of_localization_span_target_iff_finite :\n  ring_hom.of_localization_span_target @P ↔ ring_hom.of_localization_finite_span_target @P :=\nbegin\n  delta ring_hom.of_localization_span_target ring_hom.of_localization_finite_span_target,\n  apply forall₅_congr, -- TODO: Using `refine` here breaks `resetI`.\n  introsI,\n  split,\n  { intros h s, exact h s },\n  { intros h s hs hs',\n    obtain ⟨s', h₁, h₂⟩ := (ideal.span_eq_top_iff_finite s).mp hs,\n    exact h s' h₂ (λ x, hs' ⟨_, h₁ x.prop⟩) }\nend\n\nvariables {P f R' S'}\n\nlemma _root_.ring_hom.property_is_local.respects_iso (hP : ring_hom.property_is_local @P) :\n  ring_hom.respects_iso @P :=\nbegin\n  apply hP.stable_under_composition.respects_iso,\n  introv,\n  resetI,\n  letI := e.to_ring_hom.to_algebra,\n  apply_with hP.holds_for_localization_away { instances := ff },\n  apply is_localization.away_of_is_unit_of_bijective _ is_unit_one,\n  exact e.bijective\nend\n\n-- Almost all arguments are implicit since this is not intended to use mid-proof.\nlemma ring_hom.localization_preserves.away\n  (H : ring_hom.localization_preserves @P) (r : R) [is_localization.away r R']\n  [is_localization.away (f r) S'] (hf : P f) :\n    P (by exactI is_localization.away.map R' S' f r) :=\nbegin\n  resetI,\n  haveI : is_localization ((submonoid.powers r).map f) S',\n  { rw submonoid.map_powers, assumption },\n  exact H f (submonoid.powers r) R' S' hf,\nend\n\nlemma ring_hom.property_is_local.of_localization_span (hP : ring_hom.property_is_local @P) :\n  ring_hom.of_localization_span @P :=\nbegin\n  introv R hs hs',\n  resetI,\n  apply_fun (ideal.map f) at hs,\n  rw [ideal.map_span, ideal.map_top] at hs,\n  apply hP.of_localization_span_target _ _ hs,\n  rintro ⟨_, r, hr, rfl⟩,\n  have := hs' ⟨r, hr⟩,\n  convert hP.stable_under_composition _ _ (hP.holds_for_localization_away (localization.away r) r)\n    (hs' ⟨r, hr⟩) using 1,\n  exact (is_localization.map_comp _).symm\nend\n\nend ring_hom\n\nend properties\n\nsection ideal\n\nopen_locale non_zero_divisors\n\n/-- Let `I J : ideal R`. If the localization of `I` at each maximal ideal `P` is included in\nthe localization of `J` at `P`, then `I ≤ J`. -/\nlemma ideal.le_of_localization_maximal {I J : ideal R}\n  (h : ∀ (P : ideal R) (hP : P.is_maximal),\n    ideal.map (algebra_map R (by exactI localization.at_prime P)) I ≤\n      ideal.map (algebra_map R (by exactI localization.at_prime P)) J) :\n  I ≤ J :=\nbegin\n  intros x hx,\n  suffices : J.colon (ideal.span {x}) = ⊤,\n  { simpa using submodule.mem_colon.mp\n      (show (1 : R) ∈ J.colon (ideal.span {x}), from this.symm ▸ submodule.mem_top)\n      x (ideal.mem_span_singleton_self x) },\n  refine not.imp_symm (J.colon (ideal.span {x})).exists_le_maximal _,\n  push_neg,\n  introsI P hP le,\n  obtain ⟨⟨⟨a, ha⟩, ⟨s, hs⟩⟩, eq⟩ := (is_localization.mem_map_algebra_map_iff P.prime_compl _).mp\n    (h P hP (ideal.mem_map_of_mem _ hx)),\n  rw [← _root_.map_mul, ← sub_eq_zero, ← map_sub] at eq,\n  obtain ⟨⟨m, hm⟩, eq⟩ := (is_localization.map_eq_zero_iff P.prime_compl _ _).mp eq,\n  refine hs ((hP.is_prime.mem_or_mem (le (ideal.mem_colon_singleton.mpr _))).resolve_right hm),\n  simp only [subtype.coe_mk, mul_sub, sub_eq_zero, mul_comm x s, mul_left_comm] at eq,\n  simpa only [mul_assoc, eq] using J.mul_mem_left m ha\nend\n\n/-- Let `I J : ideal R`. If the localization of `I` at each maximal ideal `P` is equal to\nthe localization of `J` at `P`, then `I = J`. -/\ntheorem ideal.eq_of_localization_maximal {I J : ideal R}\n  (h : ∀ (P : ideal R) (hP : P.is_maximal),\n    ideal.map (algebra_map R (by exactI localization.at_prime P)) I =\n      ideal.map (algebra_map R (by exactI localization.at_prime P)) J) :\n  I = J :=\nle_antisymm\n  (ideal.le_of_localization_maximal (λ P hP, (h P hP).le))\n  (ideal.le_of_localization_maximal (λ P hP, (h P hP).ge))\n\n/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/\nlemma ideal_eq_bot_of_localization' (I : ideal R)\n   (h : ∀ (J : ideal R) (hJ : J.is_maximal),\n      ideal.map (algebra_map R (by exactI (localization.at_prime J))) I = ⊥) : I = ⊥ :=\nideal.eq_of_localization_maximal (λ P hP, (by simpa using h P hP))\n\n-- TODO: This proof should work for all modules, once we have enough material on submodules of\n-- localized modules.\n/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/\nlemma ideal_eq_bot_of_localization (I : ideal R)\n   (h : ∀ (J : ideal R) (hJ : J.is_maximal),\n      by exactI is_localization.coe_submodule (localization.at_prime J) I = ⊥) : I = ⊥ :=\nideal_eq_bot_of_localization' _ (λ P hP, (ideal.map_eq_bot_iff_le_ker _).mpr (λ x hx,\n  by { rw [ring_hom.mem_ker, ← submodule.mem_bot R, ← h P hP, is_localization.mem_coe_submodule],\n       exact ⟨x, hx, rfl⟩ }))\n\nlemma eq_zero_of_localization (r : R)\n   (h : ∀ (J : ideal R) (hJ : J.is_maximal),\n      by exactI algebra_map R (localization.at_prime J) r = 0) : r = 0 :=\nbegin\n  rw ← ideal.span_singleton_eq_bot,\n  apply ideal_eq_bot_of_localization,\n  intros J hJ,\n  delta is_localization.coe_submodule,\n  erw [submodule.map_span, submodule.span_eq_bot],\n  rintro _ ⟨_, h', rfl⟩,\n  cases set.mem_singleton_iff.mpr h',\n  exact h J hJ,\nend\n\nend ideal\n\nsection reduced\n\nlemma localization_is_reduced : localization_preserves (λ R hR, by exactI is_reduced R) :=\nbegin\n  introv R _ _,\n  resetI,\n  constructor,\n  rintro x ⟨(_|n), e⟩,\n  { simpa using congr_arg (*x) e },\n  obtain ⟨⟨y, m⟩, hx⟩ := is_localization.surj M x,\n  dsimp only at hx,\n  let hx' := congr_arg (^ n.succ) hx,\n  simp only [mul_pow, e, zero_mul, ← ring_hom.map_pow] at hx',\n  rw [← (algebra_map R S).map_zero] at hx',\n  obtain ⟨m', hm'⟩ := (is_localization.eq_iff_exists M S).mp hx',\n  apply_fun (*m'^n) at hm',\n  simp only [mul_assoc, zero_mul, mul_zero] at hm',\n  rw [← mul_left_comm, ← pow_succ, ← mul_pow] at hm',\n  replace hm' := is_nilpotent.eq_zero ⟨_, hm'.symm⟩,\n  rw [← (is_localization.map_units S m).mul_left_inj, hx, zero_mul,\n    is_localization.map_eq_zero_iff M],\n  exact ⟨m', by rw [← hm', mul_comm]⟩\nend\n\ninstance [is_reduced R] : is_reduced (localization M) := localization_is_reduced M _ infer_instance\n\nlemma is_reduced_of_localization_maximal :\n  of_localization_maximal (λ R hR, by exactI is_reduced R) :=\nbegin\n  introv R h,\n  constructor,\n  intros x hx,\n  apply eq_zero_of_localization,\n  intros J hJ,\n  specialize h J hJ,\n  resetI,\n  exact (hx.map $ algebra_map R $ localization.at_prime J).eq_zero,\nend\n\nend reduced\n\nsection surjective\n\nlemma localization_preserves_surjective :\n  ring_hom.localization_preserves (λ R S _ _ f, function.surjective f) :=\nbegin\n  introv R H x,\n  resetI,\n  obtain ⟨x, ⟨_, s, hs, rfl⟩, rfl⟩ := is_localization.mk'_surjective (M.map f) x,\n  obtain ⟨y, rfl⟩ := H x,\n  use is_localization.mk' R' y ⟨s, hs⟩,\n  rw is_localization.map_mk',\n  refl,\nend\n\nlemma surjective_of_localization_span :\n  ring_hom.of_localization_span (λ R S _ _ f, function.surjective f) :=\nbegin\n  introv R e H,\n  rw [← set.range_iff_surjective, set.eq_univ_iff_forall],\n  resetI,\n  letI := f.to_algebra,\n  intro x,\n  apply submodule.mem_of_span_eq_top_of_smul_pow_mem (algebra.of_id R S).to_linear_map.range s e,\n  intro r,\n  obtain ⟨a, e'⟩ := H r (algebra_map _ _ x),\n  obtain ⟨b, ⟨_, n, rfl⟩, rfl⟩ := is_localization.mk'_surjective (submonoid.powers (r : R)) a,\n  erw is_localization.map_mk' at e',\n  rw [eq_comm, is_localization.eq_mk'_iff_mul_eq, subtype.coe_mk, subtype.coe_mk, ← map_mul] at e',\n  obtain ⟨⟨_, n', rfl⟩, e''⟩ := (is_localization.eq_iff_exists (submonoid.powers (f r)) _).mp e',\n  rw [subtype.coe_mk, mul_comm x, ←mul_assoc, ← map_pow, ← map_mul, ← map_mul, ← pow_add] at e'',\n  exact ⟨n' + n, _, e''.symm⟩\nend\n\nend surjective\n\nsection finite\n\n/-- If `S` is a finite `R`-algebra, then `S' = M⁻¹S` is a finite `R' = M⁻¹R`-algebra. -/\nlemma localization_finite : ring_hom.localization_preserves @ring_hom.finite :=\nbegin\n  introv R hf,\n  -- Setting up the `algebra` and `is_scalar_tower` instances needed\n  resetI,\n  letI := f.to_algebra,\n  letI := ((algebra_map S S').comp f).to_algebra,\n  let f' : R' →+* S' := is_localization.map S' f (submonoid.le_comap_map M),\n  letI := f'.to_algebra,\n  haveI : is_scalar_tower R R' S' :=\n    is_scalar_tower.of_algebra_map_eq' (is_localization.map_comp _).symm,\n  let fₐ : S →ₐ[R] S' := alg_hom.mk' (algebra_map S S') (λ c x, ring_hom.map_mul _ _ _),\n\n  -- We claim that if `S` is generated by `T` as an `R`-module,\n  -- then `S'` is generated by `T` as an `R'`-module.\n  unfreezingI { obtain ⟨T, hT⟩ := hf },\n  use T.image (algebra_map S S'),\n  rw eq_top_iff,\n  rintro x -,\n\n  -- By the hypotheses, for each `x : S'`, we have `x = y / (f r)` for some `y : S` and `r : M`.\n  -- Since `S` is generated by `T`, the image of `y` should fall in the span of the image of `T`.\n  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ := is_localization.mk'_surjective (M.map f) x,\n  rw [is_localization.mk'_eq_mul_mk'_one, mul_comm, finset.coe_image],\n  have hy : y ∈ submodule.span R ↑T, by { rw hT, trivial },\n  replace hy : algebra_map S S' y ∈ submodule.map fₐ.to_linear_map (submodule.span R T) :=\n    submodule.mem_map_of_mem hy,\n  rw submodule.map_span fₐ.to_linear_map T at hy,\n  have H : submodule.span R ((algebra_map S S') '' T) ≤\n    (submodule.span R' ((algebra_map S S') '' T)).restrict_scalars R,\n  { rw submodule.span_le, exact submodule.subset_span },\n\n  -- Now, since `y ∈ span T`, and `(f r)⁻¹ ∈ R'`, `x / (f r)` is in `span T` as well.\n  convert (submodule.span R' ((algebra_map S S') '' T)).smul_mem\n    (is_localization.mk' R' (1 : R) ⟨r, hr⟩) (H hy) using 1,\n  rw algebra.smul_def,\n  erw is_localization.map_mk',\n  rw map_one,\n  refl,\nend\n\nlemma localization_away_map_finite (r : R) [is_localization.away r R']\n  [is_localization.away (f r) S'] (hf : f.finite) :\n    (is_localization.away.map R' S' f r).finite :=\nlocalization_finite.away r hf\n\n/--\nLet `S` be an `R`-algebra, `M` an submonoid of `R`, and `S' = M⁻¹S`.\nIf the image of some `x : S` falls in the span of some finite `s ⊆ S'` over `R`,\nthen there exists some `m : M` such that `m • x` falls in the\nspan of `finset_integer_multiple _ s` over `R`.\n-/\nlemma is_localization.smul_mem_finset_integer_multiple_span [algebra R S]\n  [algebra R S'] [is_scalar_tower R S S']\n  [is_localization (M.map (algebra_map R S)) S'] (x : S)\n  (s : finset S') (hx : algebra_map S S' x ∈ submodule.span R (s : set S')) :\n    ∃ m : M, m • x ∈ submodule.span R\n      (is_localization.finset_integer_multiple (M.map (algebra_map R S)) s : set S) :=\nbegin\n  let g : S →ₐ[R] S' := alg_hom.mk' (algebra_map S S')\n    (λ c x, by simp [algebra.algebra_map_eq_smul_one]),\n\n  -- We first obtain the `y' ∈ M` such that `s' = y' • s` is falls in the image of `S` in `S'`.\n  let y := is_localization.common_denom_of_finset (M.map (algebra_map R S)) s,\n  have hx₁ : (y : S) • ↑s = g '' _ := (is_localization.finset_integer_multiple_image _ s).symm,\n  obtain ⟨y', hy', e : algebra_map R S y' = y⟩ := y.prop,\n  have : algebra_map R S y' • (s : set S') = y' • s :=\n    by simp_rw [algebra.algebra_map_eq_smul_one, smul_assoc, one_smul],\n  rw [← e, this] at hx₁,\n  replace hx₁ := congr_arg (submodule.span R) hx₁,\n  rw submodule.span_smul at hx₁,\n  replace hx : _ ∈ y' • submodule.span R (s : set S') := set.smul_mem_smul_set hx,\n  rw hx₁ at hx,\n  erw [← g.map_smul, ← submodule.map_span (g : S →ₗ[R] S')] at hx,\n  -- Since `x` falls in the span of `s` in `S'`, `y' • x : S` falls in the span of `s'` in `S'`.\n  -- That is, there exists some `x' : S` in the span of `s'` in `S` and `x' = y' • x` in `S'`.\n  -- Thus `a • (y' • x) = a • x' ∈ span s'` in `S` for some `a ∈ M`.\n  obtain ⟨x', hx', hx'' : algebra_map _ _ _ = _⟩ := hx,\n  obtain ⟨⟨_, a, ha₁, rfl⟩, ha₂⟩ := (is_localization.eq_iff_exists\n    (M.map (algebra_map R S)) S').mp hx'',\n  use (⟨a, ha₁⟩ : M) * (⟨y', hy'⟩ : M),\n  convert (submodule.span R (is_localization.finset_integer_multiple\n    (submonoid.map (algebra_map R S) M) s : set S)).smul_mem a hx' using 1,\n  convert ha₂.symm,\n  { rw [subtype.coe_mk, submonoid.smul_def, submonoid.coe_mul, ← smul_smul],\n    exact algebra.smul_def _ _ },\n  { exact algebra.smul_def _ _ }\nend\n\n/-- If `S` is an `R' = M⁻¹R` algebra, and `x ∈ span R' s`,\nthen `t • x ∈ span R s` for some `t : M`.-/\nlemma multiple_mem_span_of_mem_localization_span [algebra R' S] [algebra R S]\n  [is_scalar_tower R R' S] [is_localization M R']\n  (s : set S) (x : S) (hx : x ∈ submodule.span R' s) :\n    ∃ t : M, t • x ∈ submodule.span R s :=\nbegin\n  classical,\n  obtain ⟨s', hss', hs'⟩ := submodule.mem_span_finite_of_mem_span hx,\n  rsuffices ⟨t, ht⟩ : ∃ t : M, t • x ∈ submodule.span R (s' : set S),\n  { exact ⟨t, submodule.span_mono hss' ht⟩ },\n  clear hx hss' s,\n  revert x,\n  apply s'.induction_on,\n  { intros x hx, use 1, simpa using hx },\n  rintros a s ha hs x hx,\n  simp only [finset.coe_insert, finset.image_insert, finset.coe_image, subtype.coe_mk,\n    submodule.mem_span_insert] at hx ⊢,\n  rcases hx with ⟨y, z, hz, rfl⟩,\n  rcases is_localization.surj M y with ⟨⟨y', s'⟩, e⟩,\n  replace e : _ * a = _ * a := (congr_arg (λ x, algebra_map R' S x * a) e : _),\n  simp_rw [ring_hom.map_mul, ← is_scalar_tower.algebra_map_apply, mul_comm (algebra_map R' S y),\n    mul_assoc, ← algebra.smul_def] at e,\n  rcases hs _ hz with ⟨t, ht⟩,\n  refine ⟨t*s', t*y', _, (submodule.span R (s : set S)).smul_mem s' ht, _⟩,\n  rw [smul_add, ← smul_smul, mul_comm, ← smul_smul, ← smul_smul, ← e],\n  refl,\nend\n\n/-- If `S` is an `R' = M⁻¹R` algebra, and `x ∈ adjoin R' s`,\nthen `t • x ∈ adjoin R s` for some `t : M`.-/\nlemma multiple_mem_adjoin_of_mem_localization_adjoin [algebra R' S] [algebra R S]\n  [is_scalar_tower R R' S] [is_localization M R']\n  (s : set S) (x : S) (hx : x ∈ algebra.adjoin R' s) :\n    ∃ t : M, t • x ∈ algebra.adjoin R s :=\nbegin\n  change ∃ (t : M), t • x ∈ (algebra.adjoin R s).to_submodule,\n  change x ∈ (algebra.adjoin R' s).to_submodule at hx,\n  simp_rw [algebra.adjoin_eq_span] at hx ⊢,\n  exact multiple_mem_span_of_mem_localization_span M R' _ _ hx\nend\n\nlemma finite_of_localization_span : ring_hom.of_localization_span @ring_hom.finite :=\nbegin\n  rw ring_hom.of_localization_span_iff_finite,\n  introv R hs H,\n  -- We first setup the instances\n  resetI,\n  letI := f.to_algebra,\n  letI := λ (r : s), (localization.away_map f r).to_algebra,\n  haveI : ∀ r : s, is_localization ((submonoid.powers (r : R)).map (algebra_map R S))\n    (localization.away (f r)),\n  { intro r, rw submonoid.map_powers, exact localization.is_localization },\n  haveI : ∀ r : s, is_scalar_tower R (localization.away (r : R)) (localization.away (f r)) :=\n    λ r, is_scalar_tower.of_algebra_map_eq' (is_localization.map_comp _).symm,\n\n  -- By the hypothesis, we may find a finite generating set for each `Sᵣ`. This set can then be\n  -- lifted into `R` by multiplying a sufficiently large power of `r`. I claim that the union of\n  -- these generates `S`.\n  constructor,\n  replace H := λ r, (H r).1,\n  choose s₁ s₂ using H,\n  let sf := λ (x : s), is_localization.finset_integer_multiple (submonoid.powers (f x)) (s₁ x),\n  use s.attach.bUnion sf,\n  rw [submodule.span_attach_bUnion, eq_top_iff],\n\n  -- It suffices to show that `r ^ n • x ∈ span T` for each `r : s`, since `{ r ^ n }` spans `R`.\n  -- This then follows from the fact that each `x : R` is a linear combination of the generating set\n  -- of `Sᵣ`. By multiplying a sufficiently large power of `r`, we can cancel out the `r`s in the\n  -- denominators of both the generating set and the coefficients.\n  rintro x -,\n  apply submodule.mem_of_span_eq_top_of_smul_pow_mem _ (s : set R) hs _ _,\n  intro r,\n  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ := multiple_mem_span_of_mem_localization_span\n    (submonoid.powers (r : R)) (localization.away (r : R)) (s₁ r : set (localization.away (f r)))\n      (algebra_map S _ x) (by { rw s₂ r, trivial }),\n  rw [submonoid.smul_def, algebra.smul_def, is_scalar_tower.algebra_map_apply R S,\n    subtype.coe_mk, ← map_mul] at hn₁,\n  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ := is_localization.smul_mem_finset_integer_multiple_span\n    (submonoid.powers (r : R)) (localization.away (f r)) _ (s₁ r) hn₁,\n  rw [submonoid.smul_def, ← algebra.smul_def, smul_smul, subtype.coe_mk, ← pow_add] at hn₂,\n  simp_rw submonoid.map_powers at hn₂,\n  use n₂ + n₁,\n  exact le_supr (λ (x : s), submodule.span R (sf x : set S)) r hn₂,\nend\n\nend finite\n\nsection finite_type\n\nlemma localization_finite_type : ring_hom.localization_preserves @ring_hom.finite_type :=\nbegin\n  introv R hf,\n  -- mirrors the proof of `localization_map_finite`\n  resetI,\n  letI := f.to_algebra,\n  letI := ((algebra_map S S').comp f).to_algebra,\n  let f' : R' →+* S' := is_localization.map S' f (submonoid.le_comap_map M),\n  letI := f'.to_algebra,\n  haveI : is_scalar_tower R R' S' :=\n    is_scalar_tower.of_algebra_map_eq' (is_localization.map_comp _).symm,\n  let fₐ : S →ₐ[R] S' := alg_hom.mk' (algebra_map S S') (λ c x, ring_hom.map_mul _ _ _),\n\n  obtain ⟨T, hT⟩ := id hf,\n  use T.image (algebra_map S S'),\n  rw eq_top_iff,\n  rintro x -,\n  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ := is_localization.mk'_surjective (M.map f) x,\n  rw [is_localization.mk'_eq_mul_mk'_one, mul_comm, finset.coe_image],\n  have hy : y ∈ algebra.adjoin R (T : set S), by { rw hT, trivial },\n  replace hy : algebra_map S S' y ∈ (algebra.adjoin R (T : set S)).map fₐ :=\n    subalgebra.mem_map.mpr ⟨_, hy, rfl⟩,\n  rw fₐ.map_adjoin T at hy,\n  have H : algebra.adjoin R ((algebra_map S S') '' T) ≤\n    (algebra.adjoin R' ((algebra_map S S') '' T)).restrict_scalars R,\n  { rw algebra.adjoin_le_iff, exact algebra.subset_adjoin },\n  convert (algebra.adjoin R' ((algebra_map S S') '' T)).smul_mem (H hy)\n    (is_localization.mk' R' (1 : R) ⟨r, hr⟩) using 1,\n  rw algebra.smul_def,\n  erw is_localization.map_mk',\n  rw map_one,\n  refl,\nend\n\nlemma localization_away_map_finite_type (r : R) [is_localization.away r R']\n  [is_localization.away (f r) S'] (hf : f.finite_type) :\n    (is_localization.away.map R' S' f r).finite_type :=\nlocalization_finite_type.away r hf\n\nvariable {S'}\n\n/--\nLet `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.\nSuppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,\nand `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.\nThen, there exists some `m : M` such that `m • x` falls in `A`.\n-/\nlemma is_localization.exists_smul_mem_of_mem_adjoin [algebra R S]\n  [algebra R S'] [is_scalar_tower R S S'] (M : submonoid S)\n  [is_localization M S'] (x : S) (s : finset S') (A : subalgebra R S)\n  (hA₁ : (is_localization.finset_integer_multiple M s : set S) ⊆ A)\n  (hA₂ : M ≤ A.to_submonoid)\n  (hx : algebra_map S S' x ∈ algebra.adjoin R (s : set S')) :\n    ∃ m : M, m • x ∈ A :=\nbegin\n  let g : S →ₐ[R] S' := is_scalar_tower.to_alg_hom R S S',\n  let y := is_localization.common_denom_of_finset M s,\n  have hx₁ : (y : S) • ↑s = g '' _ := (is_localization.finset_integer_multiple_image _ s).symm,\n  obtain ⟨n, hn⟩ := algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : set S')\n    (A.map g) (by { rw hx₁, exact set.image_subset _ hA₁ }) hx (set.mem_image_of_mem _ (hA₂ y.2)),\n  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl),\n  rw [algebra.smul_def, ← _root_.map_mul] at hx'',\n  obtain ⟨a, ha₂⟩ := (is_localization.eq_iff_exists M S').mp hx'',\n  use a * y ^ n,\n  convert A.mul_mem hx' (hA₂ a.prop),\n  rw [submonoid.smul_def, smul_eq_mul, submonoid.coe_mul, submonoid.coe_pow, mul_assoc, ←ha₂,\n    mul_comm],\nend\n\n/--\nLet `S` be an `R`-algebra, `M` an submonoid of `R`, and `S' = M⁻¹S`.\nIf the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,\nthen there exists some `m : M` such that `m • x` falls in the\nadjoin of `finset_integer_multiple _ s` over `R`.\n-/\nlemma is_localization.lift_mem_adjoin_finset_integer_multiple [algebra R S]\n  [algebra R S'] [is_scalar_tower R S S']\n  [is_localization (M.map (algebra_map R S)) S'] (x : S)\n  (s : finset S') (hx : algebra_map S S' x ∈ algebra.adjoin R (s : set S')) :\n    ∃ m : M, m • x ∈ algebra.adjoin R\n      (is_localization.finset_integer_multiple (M.map (algebra_map R S)) s : set S) :=\nbegin\n  obtain ⟨⟨_, a, ha, rfl⟩, e⟩ := is_localization.exists_smul_mem_of_mem_adjoin\n    (M.map (algebra_map R S)) x s (algebra.adjoin R _) algebra.subset_adjoin _ hx,\n  { exact ⟨⟨a, ha⟩, by simpa [submonoid.smul_def] using e⟩ },\n{ rintros _ ⟨a, ha, rfl⟩, exact subalgebra.algebra_map_mem _ a }\nend\n\nlemma finite_type_of_localization_span : ring_hom.of_localization_span @ring_hom.finite_type :=\nbegin\n  rw ring_hom.of_localization_span_iff_finite,\n  introv R hs H,\n  -- mirrors the proof of `finite_of_localization_span`\n  resetI,\n  letI := f.to_algebra,\n  letI := λ (r : s), (localization.away_map f r).to_algebra,\n  haveI : ∀ r : s, is_localization ((submonoid.powers (r : R)).map (algebra_map R S))\n    (localization.away (f r)),\n  { intro r, rw submonoid.map_powers, exact localization.is_localization },\n  haveI : ∀ r : s, is_scalar_tower R (localization.away (r : R)) (localization.away (f r)) :=\n    λ r, is_scalar_tower.of_algebra_map_eq' (is_localization.map_comp _).symm,\n\n  constructor,\n  replace H := λ r, (H r).1,\n  choose s₁ s₂ using H,\n  let sf := λ (x : s), is_localization.finset_integer_multiple (submonoid.powers (f x)) (s₁ x),\n  use s.attach.bUnion sf,\n  convert (algebra.adjoin_attach_bUnion sf).trans _,\n  rw eq_top_iff,\n  rintro x -,\n  apply (⨆ (x : s), algebra.adjoin R (sf x : set S)).to_submodule\n    .mem_of_span_eq_top_of_smul_pow_mem _ hs _ _,\n  intro r,\n  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ := multiple_mem_adjoin_of_mem_localization_adjoin\n    (submonoid.powers (r : R)) (localization.away (r : R)) (s₁ r : set (localization.away (f r)))\n      (algebra_map S (localization.away (f r)) x) (by { rw s₂ r, trivial }),\n  rw [submonoid.smul_def, algebra.smul_def, is_scalar_tower.algebra_map_apply R S,\n    subtype.coe_mk, ← map_mul] at hn₁,\n  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ := is_localization.lift_mem_adjoin_finset_integer_multiple\n    (submonoid.powers (r : R)) _ (s₁ r) hn₁,\n  rw [submonoid.smul_def, ← algebra.smul_def, smul_smul, subtype.coe_mk, ← pow_add] at hn₂,\n  simp_rw submonoid.map_powers at hn₂,\n  use n₂ + n₁,\n  exact le_supr (λ (x : s), algebra.adjoin R (sf x : set S)) r hn₂\nend\n\nend finite_type\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/ring_theory/local_properties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093882168609, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.3345421839138122}}
{"text": "-- https://github.com/dwrensha/lean4-maze/blob/main/Maze.lean\n\nimport Lean\n\n-- Turn off pp.analyze. When pp.analyze is set to true (the default), some of our\n-- larger mazes take a long time to display.\nset_option pp.analyze false\n\n-- Coordinates in a two dimensional grid. ⟨0,0⟩ is the upper left.\nstructure Coords where\n  x : Nat -- column number\n  y : Nat -- row number\nderiving BEq\n\ninstance : ToString Coords where\n  toString := (λ ⟨x,y⟩ => String.join [\"Coords.mk \", toString x, \", \", toString y])\n\nstructure GameState where\n  size     : Coords      -- coordinates of bottom-right cell\n  position : Coords      -- row and column of the player\n  walls    : List Coords -- maze cells that are not traversible\n\n-- We define custom syntax for GameState.\n\ndeclare_syntax_cat game_cell\ndeclare_syntax_cat game_cell_sequence\ndeclare_syntax_cat game_row\ndeclare_syntax_cat horizontal_border\ndeclare_syntax_cat game_top_row\ndeclare_syntax_cat game_bottom_row\n\nsyntax \"─\" : horizontal_border\n\nsyntax \"\\n┌\" horizontal_border* \"┐\\n\" : game_top_row\n\nsyntax \"└\" horizontal_border* \"┘\\n\" : game_bottom_row\n\nsyntax \"░\" : game_cell -- empty\nsyntax \"▓\" : game_cell -- wall\nsyntax \"@\" : game_cell -- player\n\nsyntax \"│\" game_cell* \"│\\n\" : game_row\n\n\nsyntax:max game_top_row game_row* game_bottom_row : term\n\ninductive CellContents where\n  | empty  : CellContents\n  | wall   : CellContents\n  | player : CellContents\n\ndef update_state_with_row_aux : Nat → Nat → List CellContents → GameState → GameState\n| currentRowNum, currentColNum, [], oldState => oldState\n| currentRowNum, currentColNum, cell::contents, oldState =>\n    let oldState' := update_state_with_row_aux currentRowNum (currentColNum+1) contents oldState\n    match cell with\n    | CellContents.empty => oldState'\n    | CellContents.wall => {oldState' .. with\n                            walls := ⟨currentColNum,currentRowNum⟩::oldState'.walls}\n    | CellContents.player => {oldState' .. with\n                              position := ⟨currentColNum,currentRowNum⟩}\n\ndef update_state_with_row : Nat → List CellContents → GameState → GameState\n| currentRowNum, rowContents, oldState => update_state_with_row_aux currentRowNum 0 rowContents oldState\n\n-- size, current row, remaining cells -> gamestate\ndef game_state_from_cells_aux : Coords → Nat → List (List CellContents) → GameState\n| size, _, [] => ⟨size, ⟨0,0⟩, []⟩\n| size, currentRow, row::rows =>\n        let prevState := game_state_from_cells_aux size (currentRow + 1) rows\n        update_state_with_row currentRow row prevState\n\n-- size, remaining cells -> gamestate\ndef game_state_from_cells : Coords → List (List CellContents) → GameState\n| size, cells => game_state_from_cells_aux size 0 cells\n\ndef termOfCell : Lean.Macro\n| `(game_cell| ░) => `(CellContents.empty)\n| `(game_cell| ▓) => `(CellContents.wall)\n| `(game_cell| @) => `(CellContents.player)\n| _ => Lean.Macro.throwError \"unknown game cell\"\n\ndef termOfGameRow : Nat → Lean.Macro\n| expectedRowSize, `(game_row| │$cells:game_cell*│) =>\n      do if cells.size != expectedRowSize\n         then Lean.Macro.throwError \"row has wrong size\"\n         let cells' ← Array.mapM termOfCell cells\n         `([$cells',*])\n| _, _ => Lean.Macro.throwError \"unknown game row\"\n\nmacro_rules\n| `(┌ $tb:horizontal_border* ┐\n    $rows:game_row*\n    └ $bb:horizontal_border* ┘) =>\n      do let rsize := Lean.Syntax.mkNumLit (toString rows.size)\n         let csize := Lean.Syntax.mkNumLit (toString tb.size)\n         if tb.size != bb.size then Lean.Macro.throwError \"top/bottom border mismatch\"\n         let rows' ← Array.mapM (termOfGameRow tb.size) rows\n         `(game_state_from_cells ⟨$csize,$rsize⟩ [$rows',*])\n\n---------------------------\n-- Now we define a delaborator that will cause GameState to be rendered as a maze.\n\ndef extractXY : Lean.Expr → Lean.MetaM Coords\n| e => do\n  let e':Lean.Expr ← (Lean.Meta.whnf e)\n  let sizeArgs := Lean.Expr.getAppArgs e'\n  let f := Lean.Expr.getAppFn e'\n  let x ← Lean.Meta.whnf sizeArgs[0]\n  let y ← Lean.Meta.whnf sizeArgs[1]\n  let numCols := (Lean.Expr.natLit? x).get!\n  let numRows := (Lean.Expr.natLit? y).get!\n  Coords.mk numCols numRows\n\npartial def extractWallList : Lean.Expr → Lean.MetaM (List Coords)\n| exp => do\n  let exp':Lean.Expr ← (Lean.Meta.whnf exp)\n  let f := Lean.Expr.getAppFn exp'\n  if f.constName!.toString == \"List.cons\"\n  then let consArgs := Lean.Expr.getAppArgs exp'\n       let rest ← extractWallList consArgs[2]\n       let ⟨wallCol, wallRow⟩ ← extractXY consArgs[1]\n       (Coords.mk wallCol wallRow) :: rest\n  else [] -- \"List.nil\"\n\npartial def extractGameState : Lean.Expr → Lean.MetaM GameState\n| exp => do\n    let exp': Lean.Expr ← (Lean.Meta.whnf exp)\n    let gameStateArgs := Lean.Expr.getAppArgs exp'\n    let size ← extractXY gameStateArgs[0]\n    let playerCoords ← extractXY gameStateArgs[1]\n    let walls ← extractWallList gameStateArgs[2]\n    pure ⟨size, playerCoords, walls⟩\n\ndef update2dArray {α : Type} : Array (Array α) → Coords → α → Array (Array α)\n| a, ⟨x,y⟩, v =>\n   Array.set! a y $ Array.set! (Array.get! a y) x v\n\ndef update2dArrayMulti {α : Type} : Array (Array α) → List Coords → α → Array (Array α)\n| a, [], v => a\n| a, c::cs, v =>\n     let a' := update2dArrayMulti a cs v\n     update2dArray a' c v\n\ndef delabGameRow : (Array Lean.Syntax) → Lean.PrettyPrinter.Delaborator.Delab\n| a => `(game_row| │ $a:game_cell* │)\n\ndef delabGameState : Lean.Expr → Lean.PrettyPrinter.Delaborator.Delab\n| e =>\n  do guard $ e.getAppNumArgs == 3\n     let ⟨⟨numCols, numRows⟩, playerCoords, walls⟩ ←\n       try extractGameState e\n       catch err => failure -- can happen if game state has variables in it\n\n     let topBar := Array.mkArray numCols $ ← `(horizontal_border| ─)\n     let emptyCell ← `(game_cell| ░)\n     let emptyRow := Array.mkArray numCols emptyCell\n     let emptyRowStx ← `(game_row| │$emptyRow:game_cell*│)\n     let allRows := Array.mkArray numRows emptyRowStx\n\n     let a0 := Array.mkArray numRows $ Array.mkArray numCols emptyCell\n     let a1 := update2dArray a0 playerCoords $ ← `(game_cell| @)\n     let a2 := update2dArrayMulti a1 walls $ ← `(game_cell| ▓)\n     let aa ← Array.mapM delabGameRow a2\n\n     `(┌$topBar:horizontal_border*┐ $aa:game_row*\n       └$topBar:horizontal_border*┘)\n\n-- The attribute [delab] registers this function as a delaborator for the GameState.mk constructor.\n@[delab app.GameState.mk] def delabGameStateMk : Lean.PrettyPrinter.Delaborator.Delab := do\n  let e ← Lean.PrettyPrinter.Delaborator.SubExpr.getExpr\n  delabGameState e\n\n-- We register the same elaborator for applications of the game_state_from_cells function.\n@[delab app.game_state_from_cells] def delabGameState' : Lean.PrettyPrinter.Delaborator.Delab :=\n  do let e ← Lean.PrettyPrinter.Delaborator.SubExpr.getExpr\n     let e' ← (Lean.Meta.whnf e)\n     delabGameState e'\n\n--------------------------\n\ninductive Move where\n  | east  : Move\n  | west  : Move\n  | north : Move\n  | south : Move\n\n@[simp]\ndef make_move : GameState → Move → GameState\n| ⟨s, ⟨x,y⟩, w⟩, Move.east =>\n             if w.notElem ⟨x+1, y⟩ ∧ x + 1 ≤ s.x\n             then ⟨s, ⟨x+1, y⟩, w⟩\n             else ⟨s, ⟨x,y⟩, w⟩\n| ⟨s, ⟨x,y⟩, w⟩, Move.west =>\n             if w.notElem ⟨x-1, y⟩\n             then ⟨s, ⟨x-1, y⟩, w⟩\n             else ⟨s, ⟨x,y⟩, w⟩\n| ⟨s, ⟨x,y⟩, w⟩, Move.north =>\n             if w.notElem ⟨x, y-1⟩\n             then ⟨s, ⟨x, y-1⟩, w⟩\n             else ⟨s, ⟨x,y⟩, w⟩\n| ⟨s, ⟨x,y⟩, w⟩, Move.south =>\n             if w.notElem ⟨x, y + 1⟩ ∧ y + 1 ≤ s.y\n             then ⟨s, ⟨x, y+1⟩, w⟩\n             else ⟨s, ⟨x,y⟩, w⟩\n\ndef is_win : GameState → Prop\n| ⟨⟨sx, sy⟩, ⟨x,y⟩, w⟩ => x = 0 ∨ y = 0 ∨ x + 1 = sx ∨ y + 1 = sy\n\ndef can_escape (state : GameState) : Prop :=\n  ∃ (gs : List Move), is_win (List.foldl make_move state gs)\n\ntheorem can_still_escape \n  (g : GameState) (m : Move) (hg : can_escape (make_move g m)) : can_escape g :=\n have ⟨pms, hpms⟩ := hg\n Exists.intro (m::pms) hpms\n\ntheorem step_west\n  {s: Coords}\n  {x y : Nat}\n  {w: List Coords}\n  (hclear' : w.notElem ⟨x,y⟩)\n  (W : can_escape ⟨s,⟨x,y⟩,w⟩) :\n  can_escape ⟨s,⟨x+1,y⟩,w⟩ :=\n   by have hmm : GameState.mk s ⟨x,y⟩ w = make_move ⟨s,⟨x+1, y⟩,w⟩ Move.west :=\n               by have h' : x + 1 - 1 = x := rfl\n                  simp [h', hclear']\n      rw [hmm] at W\n      exact can_still_escape ⟨s,⟨x+1,y⟩,w⟩ Move.west W\n\ntheorem step_east\n  {s: Coords}\n  {x y : Nat}\n  {w: List Coords}\n  (hclear' : w.notElem ⟨x+1,y⟩)\n  (hinbounds : x + 1 ≤ s.x)\n  (E : can_escape ⟨s,⟨x+1,y⟩,w⟩) :\n  can_escape ⟨s,⟨x, y⟩,w⟩ :=\n    by have hmm : GameState.mk s ⟨x+1,y⟩ w = make_move ⟨s, ⟨x,y⟩,w⟩ Move.east :=\n         by simp [hclear', hinbounds]\n       rw [hmm] at E\n       exact can_still_escape ⟨s, ⟨x,y⟩, w⟩ Move.east E\n\ntheorem step_north\n  {s: Coords}\n  {x y : Nat}\n  {w: List Coords}\n  (hclear' : w.notElem ⟨x,y⟩)\n  (N : can_escape ⟨s,⟨x,y⟩,w⟩) :\n  can_escape ⟨s,⟨x, y+1⟩,w⟩ :=\n    by have hmm : GameState.mk s ⟨x,y⟩ w = make_move ⟨s,⟨x, y+1⟩,w⟩ Move.north :=\n         by have h' : y + 1 - 1 = y := rfl\n            simp [h', hclear']\n       rw [hmm] at N\n       exact can_still_escape ⟨s,⟨x,y+1⟩,w⟩ Move.north N\n\ntheorem step_south\n  {s: Coords}\n  {x y : Nat}\n  {w: List Coords}\n  (hclear' : w.notElem ⟨x,y+1⟩)\n  (hinbounds : y + 1 ≤ s.y)\n  (S : can_escape ⟨s,⟨x,y+1⟩,w⟩) :\n  can_escape ⟨s,⟨x, y⟩,w⟩ :=\n    by have hmm : GameState.mk s ⟨x,y+1⟩ w = make_move ⟨s,⟨x, y⟩,w⟩ Move.south :=\n            by simp [hclear', hinbounds]\n       rw [hmm] at S\n       exact can_still_escape ⟨s,⟨x,y⟩,w⟩ Move.south S\n\ndef escape_west {sx sy : Nat} {y : Nat} {w : List Coords} : can_escape ⟨⟨sx, sy⟩,⟨0, y⟩,w⟩ :=\n    ⟨[], Or.inl rfl⟩\n\ndef escape_east {sy x y : Nat} {w : List Coords} : can_escape ⟨⟨x+1, sy⟩,⟨x, y⟩,w⟩ :=\n  ⟨[], Or.inr $ Or.inr $ Or.inl rfl⟩\n\ndef escape_north {sx sy : Nat} {x : Nat} {w : List Coords} : can_escape ⟨⟨sx, sy⟩,⟨x, 0⟩,w⟩ :=\n  ⟨[], Or.inr $ Or.inl rfl⟩\n\ndef escape_south {sx x y : Nat} {w: List Coords} : can_escape ⟨⟨sx, y+1⟩,⟨x, y⟩,w⟩ :=\n  ⟨[], Or.inr $ Or.inr $ Or.inr rfl⟩\n\n-- Define an \"or\" tactic combinator, like <|> in Lean 3.\nelab t1:tactic \" ⟨|⟩ \" t2:tactic : tactic =>\n   try Lean.Elab.Tactic.evalTactic t1\n   catch err => Lean.Elab.Tactic.evalTactic t2\n\nelab \"fail\" m:term  : tactic => throwError m\n\n-- the `simp`s are to discharge the `hclear` and `hinbounds` side-goals\nmacro \"west\" : tactic => `((apply step_west; simp)    ⟨|⟩ fail \"cannot step west\")\nmacro \"east\" : tactic => `((apply step_east; simp; simp)    ⟨|⟩ fail \"cannot step east\")\nmacro \"xxx\" : tactic => `((apply step_east; simp; simp)    ⟨|⟩ fail \"cannot step xxx\")\nmacro \"north\" : tactic => `((apply step_north; simp)  ⟨|⟩ fail \"cannot step north\")\nmacro \"south\" : tactic => `((apply step_south; simp; simp)  ⟨|⟩ fail \"cannot step south\")\n\nmacro \"out\" : tactic => `(apply escape_north ⟨|⟩ apply escape_south ⟨|⟩\n                           apply escape_east ⟨|⟩ apply escape_west ⟨|⟩\n                           fail \"not currently at maze boundary\")\n\n-- Can escape the trivial maze in any direction.\nexample : can_escape ┌─┐\n                     │@│\n                     └─┘ := by out\n\n\n-- some other mazes with immediate escapes\nexample : can_escape ┌──┐\n                     │░░│\n                     │@░│\n                     │░░│\n                     └──┘ := by out\nexample : can_escape ┌──┐\n                     │░░│\n                     │░@│\n                     │░░│\n                     └──┘ := by out\nexample : can_escape ┌───┐\n                     │░@░│\n                     │░░░│\n                     │░░░│\n                     └───┘ := by out\nexample : can_escape ┌───┐\n                     │░░░│\n                     │░░░│\n                     │░@░│\n                     └───┘ := by out\n\n\n-- Now for some more interesting mazes.\n\nabbrev maze1 := ┌──────┐\n                │▓▓▓▓▓▓│\n                │▓░░@░▓│\n                │▓░░░░▓│\n                │▓░░░░▓│\n                │▓▓▓▓░▓│\n                └──────┘\n\ndef ex1 : can_escape maze1 := by\n  west\n  west\n  east\n  south\n  south\n  east\n  east\n  south\n  out\n\ndef maze2 := ┌────────┐\n             │▓▓▓▓▓▓▓▓│\n             │▓░▓@▓░▓▓│\n             │▓░▓░░░▓▓│\n             │▓░░▓░▓▓▓│\n             │▓▓░▓░▓░░│\n             │▓░░░░▓░▓│\n             │▓░▓▓▓▓░▓│\n             │▓░░░░░░▓│\n             │▓▓▓▓▓▓▓▓│\n             └────────┘\n\nexample : can_escape maze2 :=\n by south\n    east\n    south\n    south\n    south\n    west\n    west\n    west\n    south\n    south\n    east\n    east\n    east\n    east\n    east\n    north\n    north\n    north\n    east\n    out\n\ndef maze3 := ┌────────────────────────────┐\n             │▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓│\n             │▓░░░░░░░░░░░░░░░░░░░░▓░░░@░▓│\n             │▓░▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓░▓░▓▓▓▓▓│\n             │▓░▓░░░▓░░░░▓░░░░░░░░░▓░▓░░░▓│\n             │▓░▓░▓░▓░▓▓▓▓░▓▓▓▓▓▓▓▓▓░▓░▓░▓│\n             │▓░▓░▓░▓░▓░░░░▓░░░░░░░░░░░▓░▓│\n             │▓░▓░▓░▓░▓░▓▓▓▓▓▓▓▓▓▓▓▓░▓▓▓░▓│\n             │▓░▓░▓░▓░░░▓░░░░░░░░░░▓░░░▓░▓│\n             │▓░▓░▓░▓▓▓░▓░▓▓▓▓▓▓▓▓▓▓░▓░▓░▓│\n             │▓░▓░▓░░░░░▓░░░░░░░░░░░░▓░▓░▓│\n             │▓░▓░▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓░▓│\n             │░░▓░░░░░░░░░░░░░░░░░░░░░░░░▓│\n             │▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓▓│\n             └────────────────────────────┘\n\nexample : can_escape maze3 :=\n by west\n    west\n    west\n    south\n    admit -- can you finish the proof?\n\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/playground/maze.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3342837041607315}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nStandard identity and composition functors\n-/\nimport tactic.ext tactic.cache category.basic\n\nuniverse variables u v w\n\nsection functor\n\nvariables {F : Type u → Type v}\nvariables {α β γ : Type u}\nvariables [functor F] [is_lawful_functor F]\n\nlemma functor.map_id : (<$>) id = (id : F α → F α) :=\nby apply funext; apply id_map\n\nlemma functor.map_comp_map (f : α → β) (g : β → γ) :\n  ((<$>) g ∘ (<$>) f : F α → F γ) = (<$>) (g ∘ f) :=\nby apply funext; intro; rw comp_map\n\ntheorem functor.ext {F} : ∀ {F1 : functor F} {F2 : functor F}\n  [@is_lawful_functor F F1] [@is_lawful_functor F F2]\n  (H : ∀ α β (f : α → β) (x : F α),\n    @functor.map _ F1 _ _ f x = @functor.map _ F2 _ _ f x),\n  F1 = F2\n| ⟨m, mc⟩ ⟨m', mc'⟩ H1 H2 H :=\nbegin\n  cases show @m = @m', by funext α β f x; apply H,\n  congr, funext α β,\n  have E1 := @map_const_eq _ ⟨@m, @mc⟩ H1,\n  have E2 := @map_const_eq _ ⟨@m, @mc'⟩ H2,\n  exact E1.trans E2.symm\nend\n\nend functor\n\ndef id.mk {α : Sort u} : α → id α := id\n\nnamespace functor\n\n/-- `functor.comp` is a wrapper around `function.comp` for types.\n    It prevents Lean's type class resolution mechanism from trying\n    a `functor (comp F id)` when `functor F` would do. -/\ndef comp (F : Type u → Type w) (G : Type v → Type u) (α : Type v) : Type w :=\nF $ G α\n\n@[pattern] def comp.mk {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : F (G α)) : comp F G α := x\n\ndef comp.run {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : comp F G α) : F (G α) := x\n\nnamespace comp\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nprotected lemma ext\n  {α} {x y : comp F G α} : x.run = y.run → x = y := id\n\nvariables [functor F] [functor G]\n\nprotected def map {α β : Type v} (h : α → β) : comp F G α → comp F G β\n| (comp.mk x) := comp.mk ((<$>) h <$> x)\n\ninstance : functor (comp F G) := { map := @comp.map F G _ _ }\n\n@[functor_norm] lemma map_mk {α β} (h : α → β) (x : F (G α)) :\n  h <$> comp.mk x = comp.mk ((<$>) h <$> x) := rfl\n\nvariables [is_lawful_functor F] [is_lawful_functor G]\nvariables {α β γ : Type v}\n\nprotected lemma id_map : ∀ (x : comp F G α), comp.map id x = x\n| (comp.mk x) := by simp [comp.map, functor.map_id]\n\nprotected lemma comp_map (g' : α → β) (h : β → γ) : ∀ (x : comp F G α),\n           comp.map (h ∘ g') x = comp.map h (comp.map g' x)\n| (comp.mk x) := by simp [comp.map, functor.map_comp_map g' h] with functor_norm\n\n@[simp] protected lemma run_map (h : α → β) (x : comp F G α) :\n  (h <$> x).run = (<$>) h <$> x.run := rfl\n\ninstance : is_lawful_functor (comp F G) :=\n{ id_map := @comp.id_map F G _ _ _ _,\n  comp_map := @comp.comp_map F G _ _ _ _ }\n\ntheorem functor_comp_id {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor F id _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor F id _ _ _ _) _ (λ α β f x, rfl)\n\ntheorem functor_id_comp {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor id F _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor id F _ _ _ _) _ (λ α β f x, rfl)\n\nend comp\n\nend functor\n\nnamespace ulift\n\ninstance : functor ulift :=\n{ map := λ α β f, up ∘ f ∘ down }\n\nend ulift\n\nnamespace sum\n\nvariables {γ : Type u} {α β : Type v}\n\nprotected def mapr (f : α → β) : γ ⊕ α → γ ⊕ β\n| (inl x) := inl x\n| (inr x) := inr (f x)\n\ninstance : functor (sum.{u v} γ) :=\n{ map := @sum.mapr γ }\n\ninstance : is_lawful_functor (sum γ) :=\n{ id_map := by intros; casesm _ ⊕ _; refl,\n  comp_map := by intros; casesm _ ⊕ _; refl }\n\nend sum\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/category/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555714, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3342658592662479}}
{"text": "def myid (a : α) := a -- works\n\n#check myid 10\n#check myid true\n\ntheorem ex1 (a : α) : myid a = a := rfl\n\ndef cnst (b : β) : α → β := fun _ => b -- works\n\ntheorem ex2 (b : β) (a : α) : cnst b a = b := rfl\n\ndef Vec (α : Type) (n : Nat) := { a : Array α // a.size = n }\n\ndef mkVec : Vec α 0 := ⟨ #[], rfl ⟩\n\ndef Vec.map (xs : Vec α n) (f : α → β) : Vec β n :=\n  ⟨ xs.val.map f, sorry ⟩\n\n/- unbound implicit locals must be greek or lower case letters followed by numerical digits -/\ndef Vec.map2 (xs : Vec α size /- error: unknown identifier size -/) (f : α → β) : Vec β n :=\n  ⟨ xs.val.map f, sorry ⟩\n\nset_option autoBoundImplicitLocal false in\ndef Vec.map3 (xs : Vec α n) (f : α → β) : Vec β n := -- Errors, unknown identifiers 'α', 'n', 'β'\n  ⟨ xs.val.map f, sorry ⟩\n\ndef double [Add α] (a : α) := a + a\n\nvariable (xs : Vec α n) -- works\n\ndef f := xs\n\n#check @f\n\n#check f mkVec\n\n#check f (α := Nat) mkVec\n\ndef g (a : α) := xs.val.push a\n\ntheorem ex3 : g ⟨#[0], rfl⟩ 1 = #[0, 1] :=\n  rfl\n\ninductive Tree (α β : Type) :=\n  | leaf1 : α → Tree α β\n  | leaf2 : β → Tree α β\n  | node : Tree α β → Tree α β → Tree α β\n\ninductive TreeElem1 : α → Tree α β → Prop\n  | leaf1     : (a : α) → TreeElem1 a (Tree.leaf1 (β := β) a)\n  | nodeLeft  : (a : α) → (left : Tree α β) → (right : Tree α β) → TreeElem1 a left  → TreeElem1 a (Tree.node left right)\n  | nodeRight : (a : α) → (left : Tree α β) → (right : Tree α β) → TreeElem1 a right → TreeElem1 a (Tree.node left right)\n\ninductive TreeElem2 : β → Tree α β → Prop\n  | leaf2     : (b : β) → TreeElem2 b (Tree.leaf2 (α := α) b)\n  | nodeLeft  : (b : β) → (left : Tree α β) → (right : Tree α β) → TreeElem2 b left  → TreeElem2 b (Tree.node left right)\n  | nodeRight : (b : β) → (left : Tree α β) → (right : Tree α β) → TreeElem2 b right → TreeElem2 b (Tree.node left right)\n\nnamespace Ex1\n\ndef findSomeRevM? [Monad m] (as : Array α) (f : α → m (Option β)) : m (Option β) :=\n  pure none\n\ndef findSomeRev? (as : Array α) (f : α → Option β) : Option β :=\n  Id.run <| findSomeRevM? as f\n\nend Ex1\n\ndef apply {α : Type u₁} {β : α → Type u₂} (f : (a : α) → β a) (a : α) : β a :=\n  f a\n\ndef pair (a : α₁) := (a, a)\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/autoBoundImplicits1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3342658502355508}}
{"text": "/-\nCopyright © 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Nicolò Cavalleri.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.geometry.manifold.times_cont_mdiff_map\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_5 u_6 u_8 u_9 l u_10 u_4 u_7 \n\nnamespace Mathlib\n\n/-!\n# Diffeomorphisms\nThis file implements diffeomorphisms.\n\n## Definitions\n\n* `times_diffeomorph I I' M M' n`:  `n`-times continuously differentiable diffeomorphism between\n                                    `M` and `M'` with respect to I and I'\n* `diffeomorph  I I' M M'` : smooth diffeomorphism between `M` and `M'` with respect to I and I'\n\n## Notations\n\n* `M ≃ₘ^n⟮I, I'⟯ M'` := `times_diffeomorph I J M N n`\n* `M ≃ₘ⟮I, I'⟯ M'`   := `times_diffeomorph I J M N ⊤`\n\n## Implementation notes\n\nThis notion of diffeomorphism is needed although there is already a notion of structomorphism\nbecause structomorphisms do not allow the model spaces `H` and `H'` of the two manifolds to be\ndifferent, i.e. for a structomorphism one has to impose `H = H'` which is often not the case in\npractice.\n\n-/\n\n/--\n`n`-times continuously differentiable diffeomorphism between `M` and `M'` with respect to I and I'\n-/\nstructure times_diffeomorph {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) \nextends M ≃ M'\nwhere\n  times_cont_mdiff_to_fun : times_cont_mdiff I I' n (equiv.to_fun _to_equiv)\n  times_cont_mdiff_inv_fun : times_cont_mdiff I' I n (equiv.inv_fun _to_equiv)\n\n/-- A `diffeomorph` is just a smooth `times_diffeomorph`. -/\ndef diffeomorph {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] :=\n  times_diffeomorph I I' M M' ⊤\n\nnamespace times_diffeomorph\n\n\nprotected instance has_coe_to_fun {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) : has_coe_to_fun (times_diffeomorph I I' M M' n) :=\n  has_coe_to_fun.mk (fun (_x : times_diffeomorph I I' M M' n) => M → M')\n    fun (e : times_diffeomorph I I' M M' n) => ⇑(times_diffeomorph.to_equiv e)\n\nprotected instance times_cont_mdiff_map.has_coe {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) : has_coe (times_diffeomorph I I' M M' n) (times_cont_mdiff_map I I' M M' n) :=\n  has_coe.mk\n    fun (Φ : times_diffeomorph I I' M M' n) => times_cont_mdiff_map.mk (⇑Φ) (times_diffeomorph.times_cont_mdiff_to_fun Φ)\n\nprotected theorem continuous {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) (h : times_diffeomorph I I' M M' n) : continuous ⇑h :=\n  times_cont_mdiff.continuous (times_diffeomorph.times_cont_mdiff_to_fun h)\n\nprotected theorem times_cont_mdiff {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) (h : times_diffeomorph I I' M M' n) : times_cont_mdiff I I' n ⇑h :=\n  times_diffeomorph.times_cont_mdiff_to_fun h\n\nprotected theorem smooth {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (h : times_diffeomorph I I' M M' ⊤) : smooth I I' ⇑h :=\n  times_diffeomorph.times_cont_mdiff_to_fun h\n\ntheorem coe_eq_to_equiv {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) (h : times_diffeomorph I I' M M' n) (x : M) : coe_fn h x = coe_fn (times_diffeomorph.to_equiv h) x :=\n  rfl\n\n/-- Identity map as a diffeomorphism. -/\nprotected def refl {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {H : Type u_5} [topological_space H] (I : model_with_corners 𝕜 E H) (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (n : with_top ℕ) : times_diffeomorph I I M M n :=\n  mk (equiv.mk (equiv.to_fun (equiv.refl M)) (equiv.inv_fun (equiv.refl M)) sorry sorry) times_cont_mdiff_id\n    times_cont_mdiff_id\n\n/-- Composition of two diffeomorphisms. -/\nprotected def trans {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {F : Type u_4} [normed_group F] [normed_space 𝕜 F] {H : Type u_5} [topological_space H] {H' : Type u_6} [topological_space H'] {G : Type u_7} [topological_space G] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (J : model_with_corners 𝕜 F G) (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_9) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (N : Type u_10) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] (n : with_top ℕ) (h₁ : times_diffeomorph I I' M M' n) (h₂ : times_diffeomorph I' J M' N n) : times_diffeomorph I J M N n :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.trans (times_diffeomorph.to_equiv h₁) (times_diffeomorph.to_equiv h₂)))\n      (equiv.inv_fun (equiv.trans (times_diffeomorph.to_equiv h₁) (times_diffeomorph.to_equiv h₂))) sorry sorry)\n    sorry sorry\n\n/-- Inverse of a diffeomorphism. -/\nprotected def symm {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {F : Type u_4} [normed_group F] [normed_space 𝕜 F] {H : Type u_5} [topological_space H] {G : Type u_7} [topological_space G] (I : model_with_corners 𝕜 E H) (J : model_with_corners 𝕜 F G) (M : Type u_8) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (N : Type u_10) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N] (n : with_top ℕ) (h : times_diffeomorph I J M N n) : times_diffeomorph J I N M n :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.symm (times_diffeomorph.to_equiv h)))\n      (equiv.inv_fun (equiv.symm (times_diffeomorph.to_equiv h))) sorry sorry)\n    (times_diffeomorph.times_cont_mdiff_inv_fun h) (times_diffeomorph.times_cont_mdiff_to_fun h)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/geometry/manifold/diffeomorph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3342658424928039}}
{"text": "import for_mathlib.homological_complex_op\nimport for_mathlib.split_exact\nimport for_mathlib.AddCommGroup.exact\nimport for_mathlib.unop\n\nimport pseudo_normed_group.FP2\nimport pseudo_normed_group.system_of_complexes\n\nimport system_of_complexes.rescale\n\nimport prop_92.prop_92\n\nnoncomputable theory\n\nopen_locale nnreal\n\nuniverses u v\n\nopen category_theory breen_deligne\n\nvariables (r r' : ℝ≥0)\nvariables (BD : breen_deligne.data)\nvariables (M : ProFiltPseuNormGrpWithTinv.{u} r')\nvariables (V : SemiNormedGroup.{v})\n\nset_option pp.universes true\n\ndef aux_system (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))] :\n  system_of_complexes :=\n(FPsystem r' BD M κ).op ⋙\n  (((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n  homological_complex.unop_functor\n\nnamespace aux_system\n\nvariables [fact (0 < r)] [fact (0 < r')] [fact (r' ≤ 1)]\n\nopen system_of_complexes opposite\n\nvariables (κ₁ κ₂ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ₁ c)] [∀ c, BD.suitable (κ₂ c)]\n\ndef Tinv [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nwhisker_right (nat_trans.op $ FPsystem.Tinv r' BD M κ₁ κ₂)\n  ((((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n    homological_complex.unop_functor)\n\nlemma Tinv_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((Tinv r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.Tinv V r' _ _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [Tinv, CLCFP.Tinv, FPsystem.Tinv, FP2.Tinv,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n    functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n    FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map,\n    whisker_right_app, nat_trans.op_app],\n  erw [free_abelian_group.lift.of],\n  refl,\nend\n\ndef T_inv [normed_with_aut r V]\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))] :\n  aux_system r' BD M V κ ⟶ aux_system r' BD M V κ :=\n{ app := λ c,\n  { f := λ i, normed_group_hom.completion $ (SemiNormedGroup.LocallyConstant.map $ normed_with_aut.T.inv).app _,\n    comm' := begin\n      rintro i j (rfl : i + 1 = j),\n      dsimp only [aux_system, functor.comp_obj, homological_complex.unop_functor,\n        homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n        unop_op],\n      erw [← SemiNormedGroup.Completion_map, ← SemiNormedGroup.Completion_map, chain_complex.of_d],\n      dsimp only [FPsystem.d, universal_map.eval_FP2, universal_map.eval_CLCFP,\n        universal_map.eval_LCFP, CLC],\n      simp only [nat_trans.app_sum, functor.map_sum, preadditive.comp_sum, preadditive.sum_comp,\n        category_theory.unop_sum, nat_trans.app_zsmul, functor.map_zsmul,\n        preadditive.comp_zsmul, preadditive.zsmul_comp, category_theory.unop_zsmul],\n      refine finset.sum_congr rfl _,\n      rintro ⟨g, hg⟩ -,\n      dsimp only [basic_universal_map.eval_FP2, basic_universal_map.eval_LCFP,\n        whisker_right_app, nat_trans.op_app, unop_op, FreeAb.of_functor,\n        free_abelian_group.map_of_apply, functor.right_op_map, LC,\n        functor.comp_map, functor.map_FreeAb, FreeAb.eval],\n      rw [free_abelian_group.lift.of, _root_.id, quiver.hom.unop_op,\n        ← SemiNormedGroup.Completion.map_comp, ← SemiNormedGroup.Completion.map_comp,\n        nat_trans.naturality],\n    end },\n  naturality' := λ c₁ c₂ h, begin\n    ext n : 2,\n    dsimp only [homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n      homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n      unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n      functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n      FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map],\n    rw [← SemiNormedGroup.Completion_map, ← SemiNormedGroup.Completion_map,\n      free_abelian_group.lift.of, _root_.id, quiver.hom.unop_op,\n      ← SemiNormedGroup.Completion.map_comp, ← SemiNormedGroup.Completion.map_comp,\n        nat_trans.naturality],\n    refl,\n  end }\n.\n\nvariables [normed_with_aut r V]\n\nlemma T_inv_eq\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))]\n  (c : ℝ≥0ᵒᵖ) (n : ℕ) : ((T_inv r r' BD M V κ).app c).f n =\n  ((CLCFP.T_inv r V r' _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [T_inv, CLCFP.T_inv, FPsystem, chain_complex.of_X, FPsystem.X,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    functor.map_homological_complex_map_f, functor.map_FreeAb,\n    unop_op, CLC, functor.op_map, functor.op_obj, quiver.hom.unop_op,\n    FreeAb.eval, free_abelian_group.map_of_apply, FPsystem, FP2.res_app, FreeAb.of_functor,\n    functor.right_op_map, whisker_right_app, nat_trans.op_app, whisker_left_app],\n  refl,\nend\n\ndef res [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nwhisker_right (nat_trans.op $ FPsystem.res r' BD M κ₁ κ₂)\n  ((((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n    homological_complex.unop_functor)\n\nlemma res_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((res r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.res V r' _ _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [res, CLCFP.res, FPsystem.res, FP2.res,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n    functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n    FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map,\n    whisker_right_app, nat_trans.op_app],\n  rw [free_abelian_group.lift.of],\n  refl,\nend\n\ndef Tinv2 [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nTinv r' BD M V κ₁ κ₂ - T_inv r r' BD M V κ₂ ≫ res r' BD M V κ₁ κ₂\n.\n\nlemma Tinv2_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((Tinv2 r r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.Tinv V r' _ _ (BD.X n)).app (op M) : _) -\n  ((CLCFP.T_inv r V r' _ (BD.X n)).app (op M) : _) ≫ ((CLCFP.res V r' _ _ (BD.X n)).app (op M) : _) :=\nby rw [Tinv2, nat_trans.app_sub, homological_complex.sub_f_apply, Tinv_eq,\n    nat_trans.comp_app, homological_complex.comp_f, T_inv_eq, res_eq]\n\nlemma aux_system_d_eq\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))]\n  (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  arrow.mk (((aux_system r' BD M V κ).obj c).d i (i + 1)) =\n  arrow.mk ((universal_map.eval_CLCFP V r' (κ (unop c) i) (κ (unop c) (i + 1)) (BD.d (i + 1) i) : _).app (op M)) :=\nbegin\n  dsimp [aux_system, FPsystem],\n  rw [chain_complex.of_d],\n  dsimp [FreeAb.eval, functor.map_FreeAb, functor.right_op_map, FPsystem.d,\n    universal_map.eval_FP2, universal_map.eval_CLCFP, universal_map.eval_LCFP],\n  simp only [nat_trans.app_sum, map_sum, ← normed_group_hom_completion_hom_apply,\n    category_theory.unop_sum],\n  congr' 1,\n  refine finset.sum_congr rfl _,\n  rintro ⟨g, hg⟩ -,\n  simp only [nat_trans.app_zsmul, map_zsmul, category_theory.unop_zsmul],\n  dsimp only [basic_universal_map.eval_FP2, basic_universal_map.eval_LCFP,\n    whisker_right_app, nat_trans.op_app, unop_op, FreeAb.of_functor,\n    free_abelian_group.map_of_apply],\n  rw [free_abelian_group.lift.of],\n  refl,\nend\n\nsection\n\nvariables (κ : ℕ → ℝ≥0) [BD.suitable κ]\n\ninstance mul_left_mono (n : ℕ) :\n  fact (monotone (function.swap (λ (c : ℝ≥0) (n : ℕ), c * κ n) n)) :=\n⟨λ c₁ c₂ h, mul_le_mul' h le_rfl⟩\n\ndef incl_f (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((BD.complex κ r V r' (unop c)).obj (op M)).X n ⟶ ((aux_system r' BD M V (λ c n, c * κ n)).obj c).X n :=\nbegin\n  dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n    CLCTinv, aux_system, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval],\n  exact (SemiNormedGroup.equalizer.ι _ _ : _),\nend\n\nlemma incl_comm (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  incl_f.{u v} r r' BD M V κ c i ≫ ((aux_system.{u v} r' BD M V (λ c n, c * κ n)).obj c).d i (i + 1) =\n  ((BD.complex κ r V r' (unop c)).obj (op.{u+2} M)).d i (i + 1) ≫ incl_f.{u v} r r' BD M V κ c (i + 1) :=\nbegin\n  dsimp [aux_system],\n  erw [chain_complex.of_d, breen_deligne.data.complex_obj_d],\n  dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n    CLCTinv, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval,\n    universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv₂,\n    SemiNormedGroup.equalizer.map_nat, incl_f],\n  delta id, dsimp only [], symmetry,\n  convert SemiNormedGroup.equalizer.map_comp_ι _ _ _ _ using 2,\n  apply arrow.mk_injective,\n  rw ← aux_system_d_eq r' BD M V (λ c n, c * κ n),\n  dsimp [aux_system],\n  erw [chain_complex.of_d],\n  refl,\nend\n\ndef incl : (BD.system κ r V r').obj (op M) ⟶ aux_system r' BD M V (λ c n, c * κ n) :=\n{ app := λ c,\n  { f := incl_f r r' BD M V κ c,\n    comm' := by { rintro i j (rfl : i + 1 = j), apply incl_comm } },\n  naturality' := λ c₁ c₂ h, begin\n    ext n : 2,\n    dsimp [aux_system],\n    dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n      CLCTinv, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval, FreeAb.of_functor,\n      universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv₂,\n      SemiNormedGroup.equalizer.map_nat, incl_f, CLCFPTinv₂.res, CLCTinv.map_nat, CLCTinv.map],\n    delta id,\n    erw [free_abelian_group.lift.of],\n    convert SemiNormedGroup.equalizer.map_comp_ι _ _ _ _ using 1,\n  end }\n\ndef incl' := whisker_right (incl r r' BD M V κ) (functor.map_homological_complex (forget₂ _ Ab.{max u v}) _)\n\nend\n\ndef Tinv2' [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :=\nwhisker_right (Tinv2 r r' BD M V κ₁ κ₂)\n(functor.map_homological_complex (forget₂ _ Ab.{max u v}) _)\n\nlemma _root_.SemiNormedGroup.equalizer.ι_injective {V W : SemiNormedGroup} (f g : V ⟶ W) :\n  function.injective (SemiNormedGroup.equalizer.ι f g) :=\nsubtype.val_injective\n\nlemma _root_.SemiNormedGroup.equalizer.forget₂_ι {V W : SemiNormedGroup} (f g : V ⟶ W) :\n  (forget₂ _ Ab).map (SemiNormedGroup.equalizer.ι f g) =\n  add_subgroup.subtype (add_monoid_hom.ker ((forget₂ _ Ab).map (f - g))) :=\nrfl\n\ninstance mul_left_mono' (κ : ℕ → ℝ≥0) (n : ℕ) :\n  fact (monotone (function.swap (λ (c : ℝ≥0) (n : ℕ), r' * (c * κ n)) n)) :=\n⟨λ c₁ c₂ h, mul_le_mul' le_rfl $ mul_le_mul' h le_rfl⟩\n\nlemma neg_surjective {A B : Ab} (f : A ⟶ B) (hf : function.surjective f) :\n  function.surjective (-f) :=\nbegin\n  intro y, obtain ⟨x, rfl⟩ := hf y, use -x, simp only [pi.neg_apply, map_neg, neg_neg],\nend\n\nlemma short_exact [fact (r < 1)] (κ : ℕ → ℝ≥0) [BD.suitable κ]\n  [∀ c n, fact (κ₁ c n ≤ r' * (c * κ n))] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  short_exact (((incl' r r' BD M V κ).app c).f n)\n    (((Tinv2' r r' BD M V (λ c n, r' * (c * κ n)) (λ c n, c * κ n)).app c).f n) :=\nbegin\n  apply_with @short_exact.mk {instances := ff},\n  { rw AddCommGroup.mono_iff_injective,\n    dsimp [incl', incl, incl_f,\n      breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X],\n    exact SemiNormedGroup.equalizer.ι_injective _ _, },\n  { rw [Tinv2', whisker_right_app, functor.map_homological_complex_map_f, Tinv2_eq,\n      ← nat_trans.comp_app, ← CLCFP.res_comp_T_inv, nat_trans.comp_app,\n      ← neg_sub, functor.map_neg, functor.map_sub, category_theory.functor.map_comp,\n      AddCommGroup.epi_iff_surjective],\n    apply neg_surjective,\n    have := CLCFP.T_inv_sub_Tinv_surjective r r' V (unop c * κ n) (BD.X n) (op M),\n    rw [CLCFP.T_inv_sub_Tinv, nat_trans.app_sub, nat_trans.comp_app] at this,\n    exact this, },\n  { rw AddCommGroup.exact_iff,\n    dsimp [incl', incl, incl_f, Tinv2',\n      breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X],\n    rw [SemiNormedGroup.equalizer.forget₂_ι, add_subgroup.subtype_range, Tinv2_eq,\n      ← nat_trans.comp_app, ← CLCFP.res_comp_T_inv],\n    refl, }\nend\n\nend aux_system\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/pseudo_normed_group/system_of_complexes2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.3340756678897748}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Scott Morrison, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.unbundled_hom\nimport Mathlib.topology.continuous_map\nimport Mathlib.topology.opens\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-- The category of topological spaces and continuous maps. -/\ndef Top :=\n  category_theory.bundled topological_space\n\nnamespace Top\n\n\nprotected instance bundled_hom : category_theory.bundled_hom continuous_map :=\n  category_theory.bundled_hom.mk continuous_map.to_fun continuous_map.id continuous_map.comp\n\nprotected instance has_coe_to_sort : has_coe_to_sort Top :=\n  category_theory.bundled.has_coe_to_sort\n\nprotected instance topological_space_unbundled (x : Top) : topological_space ↥x :=\n  category_theory.bundled.str x\n\n@[simp] theorem id_app (X : Top) (x : ↥X) : coe_fn 𝟙 x = x :=\n  rfl\n\n@[simp] theorem comp_app {X : Top} {Y : Top} {Z : Top} (f : X ⟶ Y) (g : Y ⟶ Z) (x : ↥X) : coe_fn (f ≫ g) x = coe_fn g (coe_fn f x) :=\n  rfl\n\n/-- Construct a bundled `Top` from the underlying type and the typeclass. -/\ndef of (X : Type u) [topological_space X] : Top :=\n  category_theory.bundled.mk X\n\nprotected instance topological_space (X : Top) : topological_space ↥X :=\n  category_theory.bundled.str X\n\n@[simp] theorem coe_of (X : Type u) [topological_space X] : ↥(of X) = X :=\n  rfl\n\nprotected instance inhabited : Inhabited Top :=\n  { default := of empty }\n\n/-- The discrete topology on any type. -/\ndef discrete : Type u ⥤ Top :=\n  category_theory.functor.mk (fun (X : Type u) => category_theory.bundled.mk X)\n    fun (X Y : Type u) (f : X ⟶ Y) => continuous_map.mk f\n\n/-- The trivial topology on any type. -/\ndef trivial : Type u ⥤ Top :=\n  category_theory.functor.mk (fun (X : Type u) => category_theory.bundled.mk X)\n    fun (X Y : Type u) (f : X ⟶ Y) => continuous_map.mk f\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/category/Top/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3339086126849874}}
{"text": "/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n-/\nimport tactic.doc_commands\nimport tactic.reserved_notation\n\n/-!\n# Basic logic properties\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file is one of the earliest imports in mathlib.\n\n## Implementation notes\n\nTheorems that require decidability hypotheses are in the namespace \"decidable\".\nClassical versions are in the namespace \"classical\".\n\nIn the presence of automation, this whole file may be unnecessary. On the other hand,\nmaybe it is useful for writing automation.\n-/\n\nopen function\nlocal attribute [instance, priority 10] classical.prop_decidable\n\nsection miscellany\n\n/- We add the `inline` attribute to optimize VM computation using these declarations. For example,\n  `if p ∧ q then ... else ...` will not evaluate the decidability of `q` if `p` is false. -/\nattribute [inline] and.decidable or.decidable decidable.false xor.decidable iff.decidable\n  decidable.true implies.decidable not.decidable ne.decidable\n  bool.decidable_eq decidable.to_bool\n\nattribute [simp] cast_eq cast_heq\n\nvariables {α : Type*} {β : Type*}\n\n/-- An identity function with its main argument implicit. This will be printed as `hidden` even\nif it is applied to a large term, so it can be used for elision,\nas done in the `elide` and `unelide` tactics. -/\n@[reducible] def hidden {α : Sort*} {a : α} := a\n\n/-- Ex falso, the nondependent eliminator for the `empty` type. -/\ndef empty.elim {C : Sort*} : empty → C.\n\ninstance : subsingleton empty := ⟨λa, a.elim⟩\n\ninstance subsingleton.prod {α β : Type*} [subsingleton α] [subsingleton β] : subsingleton (α × β) :=\n⟨by { intros a b, cases a, cases b, congr, }⟩\n\ninstance : decidable_eq empty := λa, a.elim\n\ninstance sort.inhabited : inhabited Sort* := ⟨punit⟩\ninstance sort.inhabited' : inhabited default := ⟨punit.star⟩\n\ninstance psum.inhabited_left {α β} [inhabited α] : inhabited (psum α β) := ⟨psum.inl default⟩\ninstance psum.inhabited_right {α β} [inhabited β] : inhabited (psum α β) := ⟨psum.inr default⟩\n\n@[priority 10] instance decidable_eq_of_subsingleton\n  {α} [subsingleton α] : decidable_eq α\n| a b := is_true (subsingleton.elim a b)\n\n@[simp] lemma eq_iff_true_of_subsingleton {α : Sort*} [subsingleton α] (x y : α) :\n  x = y ↔ true :=\nby cc\n\n/-- If all points are equal to a given point `x`, then `α` is a subsingleton. -/\nlemma subsingleton_of_forall_eq {α : Sort*} (x : α) (h : ∀ y, y = x) : subsingleton α :=\n⟨λ a b, (h a).symm ▸ (h b).symm ▸ rfl⟩\n\nlemma subsingleton_iff_forall_eq {α : Sort*} (x : α) : subsingleton α ↔ ∀ y, y = x :=\n⟨λ h y, @subsingleton.elim _ h y x, subsingleton_of_forall_eq x⟩\n\ninstance subtype.subsingleton (α : Sort*) [subsingleton α] (p : α → Prop) :\n  subsingleton (subtype p) :=\n⟨λ ⟨x,_⟩ ⟨y,_⟩, have x = y, from subsingleton.elim _ _, by { cases this, refl }⟩\n\n/-- Add an instance to \"undo\" coercion transitivity into a chain of coercions, because\n   most simp lemmas are stated with respect to simple coercions and will not match when\n   part of a chain. -/\n@[simp] theorem coe_coe {α β γ} [has_coe α β] [has_coe_t β γ]\n  (a : α) : (a : γ) = (a : β) := rfl\n\ntheorem coe_fn_coe_trans\n  {α β γ δ} [has_coe α β] [has_coe_t_aux β γ] [has_coe_to_fun γ δ]\n  (x : α) : @coe_fn α _ _ x = @coe_fn β _ _ x := rfl\n\n/-- Non-dependent version of `coe_fn_coe_trans`, helps `rw` figure out the argument. -/\ntheorem coe_fn_coe_trans'\n  {α β γ} {δ : _} [has_coe α β] [has_coe_t_aux β γ] [has_coe_to_fun γ (λ _, δ)]\n  (x : α) : @coe_fn α _ _ x = @coe_fn β _ _ x := rfl\n\n@[simp] theorem coe_fn_coe_base\n  {α β γ} [has_coe α β] [has_coe_to_fun β γ]\n  (x : α) : @coe_fn α _ _ x = @coe_fn β _ _ x := rfl\n\n/-- Non-dependent version of `coe_fn_coe_base`, helps `rw` figure out the argument. -/\ntheorem coe_fn_coe_base'\n  {α β} {γ : _} [has_coe α β] [has_coe_to_fun β (λ _, γ)]\n  (x : α) : @coe_fn α _ _ x = @coe_fn β _ _ x := rfl\n\n-- This instance should have low priority, to ensure we follow the chain\n-- `set_like → has_coe_to_sort`\nattribute [instance, priority 10] coe_sort_trans\n\ntheorem coe_sort_coe_trans\n  {α β γ δ} [has_coe α β] [has_coe_t_aux β γ] [has_coe_to_sort γ δ]\n  (x : α) : @coe_sort α _ _ x = @coe_sort β _ _ x := rfl\n\n/--\nMany structures such as bundled morphisms coerce to functions so that you can\ntransparently apply them to arguments. For example, if `e : α ≃ β` and `a : α`\nthen you can write `e a` and this is elaborated as `⇑e a`. This type of\ncoercion is implemented using the `has_coe_to_fun` type class. There is one\nimportant consideration:\n\nIf a type coerces to another type which in turn coerces to a function,\nthen it **must** implement `has_coe_to_fun` directly:\n```lean\nstructure sparkling_equiv (α β) extends α ≃ β\n\n-- if we add a `has_coe` instance,\ninstance {α β} : has_coe (sparkling_equiv α β) (α ≃ β) :=\n⟨sparkling_equiv.to_equiv⟩\n\n-- then a `has_coe_to_fun` instance **must** be added as well:\ninstance {α β} : has_coe_to_fun (sparkling_equiv α β) :=\n⟨λ _, α → β, λ f, f.to_equiv.to_fun⟩\n```\n\n(Rationale: if we do not declare the direct coercion, then `⇑e a` is not in\nsimp-normal form. The lemma `coe_fn_coe_base` will unfold it to `⇑↑e a`. This\noften causes loops in the simplifier.)\n-/\nlibrary_note \"function coercion\"\n\n@[simp] theorem coe_sort_coe_base\n  {α β γ} [has_coe α β] [has_coe_to_sort β γ]\n  (x : α) : @coe_sort α _ _ x = @coe_sort β _ _ x := rfl\n\n/-- `pempty` is the universe-polymorphic analogue of `empty`. -/\n@[derive decidable_eq]\ninductive {u} pempty : Sort u\n\n/-- Ex falso, the nondependent eliminator for the `pempty` type. -/\ndef pempty.elim {C : Sort*} : pempty → C.\n\ninstance subsingleton_pempty : subsingleton pempty := ⟨λa, a.elim⟩\n\n@[simp] lemma not_nonempty_pempty : ¬ nonempty pempty :=\nassume ⟨h⟩, h.elim\n\nlemma congr_heq {α β γ : Sort*} {f : α → γ} {g : β → γ} {x : α} {y : β} (h₁ : f == g)\n  (h₂ : x == y) : f x = g y :=\nby { cases h₂, cases h₁, refl }\n\nlemma congr_arg_heq {α} {β : α → Sort*} (f : ∀ a, β a) : ∀ {a₁ a₂ : α}, a₁ = a₂ → f a₁ == f a₂\n| a _ rfl := heq.rfl\n\nlemma ulift.down_injective {α : Sort*} : function.injective (@ulift.down α)\n| ⟨a⟩ ⟨b⟩ rfl := rfl\n\n@[simp] lemma ulift.down_inj {α : Sort*} {a b : ulift α} : a.down = b.down ↔ a = b :=\n⟨λ h, ulift.down_injective h, λ h, by rw h⟩\n\nlemma plift.down_injective {α : Sort*} : function.injective (@plift.down α)\n| ⟨a⟩ ⟨b⟩ rfl := rfl\n\n@[simp] lemma plift.down_inj {α : Sort*} {a b : plift α} : a.down = b.down ↔ a = b :=\n⟨λ h, plift.down_injective h, λ h, by rw h⟩\n\n-- missing [symm] attribute for ne in core.\nattribute [symm] ne.symm\n\nlemma ne_comm {α} {a b : α} : a ≠ b ↔ b ≠ a := ⟨ne.symm, ne.symm⟩\n\n@[simp] lemma eq_iff_eq_cancel_left {α : Sort*} {b c : α} :\n  (∀ {a}, a = b ↔ a = c) ↔ (b = c) :=\n⟨λ h, by rw [← h], λ h a, by rw h⟩\n\n@[simp] lemma eq_iff_eq_cancel_right {α : Sort*} {a b : α} :\n  (∀ {c}, a = c ↔ b = c) ↔ (a = b) :=\n⟨λ h, by rw h, λ h a, by rw h⟩\n\n/-- Wrapper for adding elementary propositions to the type class systems.\nWarning: this can easily be abused. See the rest of this docstring for details.\n\nCertain propositions should not be treated as a class globally,\nbut sometimes it is very convenient to be able to use the type class system\nin specific circumstances.\n\nFor example, `zmod p` is a field if and only if `p` is a prime number.\nIn order to be able to find this field instance automatically by type class search,\nwe have to turn `p.prime` into an instance implicit assumption.\n\nOn the other hand, making `nat.prime` a class would require a major refactoring of the library,\nand it is questionable whether making `nat.prime` a class is desirable at all.\nThe compromise is to add the assumption `[fact p.prime]` to `zmod.field`.\n\nIn particular, this class is not intended for turning the type class system\ninto an automated theorem prover for first order logic. -/\nclass fact (p : Prop) : Prop := (out [] : p)\n\n/--\nIn most cases, we should not have global instances of `fact`; typeclass search only reads the head\nsymbol and then tries any instances, which means that adding any such instance will cause slowdowns\neverywhere. We instead make them as lemmata and make them local instances as required.\n-/\nlibrary_note \"fact non-instances\"\n\nlemma fact.elim {p : Prop} (h : fact p) : p := h.1\nlemma fact_iff {p : Prop} : fact p ↔ p := ⟨λ h, h.1, λ h, ⟨h⟩⟩\n\n/-- Swaps two pairs of arguments to a function. -/\n@[reducible] def function.swap₂ {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}\n  {φ : Π i₁, κ₁ i₁ → Π i₂, κ₂ i₂ → Sort*} (f : Π i₁ j₁ i₂ j₂, φ i₁ j₁ i₂ j₂) :\n  Π i₂ j₂ i₁ j₁, φ i₁ j₁ i₂ j₂ :=\nλ i₂ j₂ i₁ j₁, f i₁ j₁ i₂ j₂\n\n/-- If `x : α . tac_name` then `x.out : α`. These are definitionally equal, but this can\nnevertheless be useful for various reasons, e.g. to apply further projection notation or in an\nargument to `simp`. -/\ndef auto_param.out {α : Sort*} {n : name} (x : auto_param α n) : α := x\n\n/-- If `x : α := d` then `x.out : α`. These are definitionally equal, but this can\nnevertheless be useful for various reasons, e.g. to apply further projection notation or in an\nargument to `simp`. -/\ndef opt_param.out {α : Sort*} {d : α} (x : α := d) : α := x\n\nend miscellany\n\nopen function\n\n/-!\n### Declarations about propositional connectives\n-/\n\ntheorem false_ne_true : false ≠ true\n| h := h.symm ▸ trivial\n\ntheorem eq_true_iff {a : Prop} : (a = true) = a :=\nhave (a ↔ true) = a, from propext (iff_true a),\neq.subst (@iff_eq_eq a true) this\n\nsection propositional\nvariables {a b c d e f : Prop}\n\n/-! ### Declarations about `implies` -/\n\ninstance : is_refl Prop iff := ⟨iff.refl⟩\ninstance : is_trans Prop iff := ⟨λ _ _ _, iff.trans⟩\n\ntheorem iff_of_eq (e : a = b) : a ↔ b := e ▸ iff.rfl\n\ntheorem iff_iff_eq : (a ↔ b) ↔ a = b := ⟨propext, iff_of_eq⟩\n\n@[simp] lemma eq_iff_iff {p q : Prop} : (p = q) ↔ (p ↔ q) := iff_iff_eq.symm\n\n@[simp] theorem imp_self : (a → a) ↔ true := iff_true_intro id\n\nlemma iff.imp (h₁ : a ↔ b) (h₂ : c ↔ d) : (a → c) ↔ (b → d) := imp_congr h₁ h₂\n\n@[simp] lemma eq_true_eq_id : eq true = id :=\nby { funext, simp only [true_iff, id.def, iff_self, eq_iff_iff], }\n\ntheorem imp_intro {α β : Prop} (h : α) : β → α := λ _, h\n\ntheorem imp_false : (a → false) ↔ ¬ a := iff.rfl\n\ntheorem imp_and_distrib {α} : (α → b ∧ c) ↔ (α → b) ∧ (α → c) :=\n⟨λ h, ⟨λ ha, (h ha).left, λ ha, (h ha).right⟩,\n λ h ha, ⟨h.left ha, h.right ha⟩⟩\n\n@[simp] theorem and_imp : (a ∧ b → c) ↔ (a → b → c) :=\niff.intro (λ h ha hb, h ⟨ha, hb⟩) (λ h ⟨ha, hb⟩, h ha hb)\n\ntheorem iff_def : (a ↔ b) ↔ (a → b) ∧ (b → a) :=\niff_iff_implies_and_implies _ _\n\ntheorem iff_def' : (a ↔ b) ↔ (b → a) ∧ (a → b) :=\niff_def.trans and.comm\n\ntheorem imp_true_iff {α : Sort*} : (α → true) ↔ true :=\niff_true_intro $ λ_, trivial\n\ntheorem imp_iff_right (ha : a) : (a → b) ↔ b :=\n⟨λf, f ha, imp_intro⟩\n\nlemma imp_iff_not (hb : ¬ b) : a → b ↔ ¬ a := imp_congr_right $ λ _, iff_false_intro hb\n\ntheorem decidable.imp_iff_right_iff [decidable a] : ((a → b) ↔ b) ↔ (a ∨ b) :=\n⟨λ H, (decidable.em a).imp_right $ λ ha', H.1 $ λ ha, (ha' ha).elim,\n  λ H, H.elim imp_iff_right $ λ hb, ⟨λ hab, hb, λ _ _, hb⟩⟩\n\n@[simp] theorem imp_iff_right_iff : ((a → b) ↔ b) ↔ (a ∨ b) :=\ndecidable.imp_iff_right_iff\n\nlemma decidable.and_or_imp [decidable a] : (a ∧ b) ∨ (a → c) ↔ a → (b ∨ c) :=\nif ha : a then by simp only [ha, true_and, true_implies_iff]\n          else by simp only [ha, false_or, false_and, false_implies_iff]\n\n@[simp] theorem and_or_imp : (a ∧ b) ∨ (a → c) ↔ a → (b ∨ c) :=\ndecidable.and_or_imp\n\n/-- Provide modus tollens (`mt`) as dot notation for implications. -/\nprotected lemma function.mt : (a → b) → ¬ b → ¬ a := mt\n\n/-! ### Declarations about `not` -/\n\n/-- Ex falso for negation. From `¬ a` and `a` anything follows. This is the same as `absurd` with\nthe arguments flipped, but it is in the `not` namespace so that projection notation can be used. -/\ndef not.elim {α : Sort*} (H1 : ¬a) (H2 : a) : α := absurd H2 H1\n\n@[reducible] theorem not.imp {a b : Prop} (H2 : ¬b) (H1 : a → b) : ¬a := mt H1 H2\n\ntheorem not_not_of_not_imp : ¬(a → b) → ¬¬a :=\nmt not.elim\n\ntheorem not_of_not_imp {a : Prop} : ¬(a → b) → ¬b :=\nmt imp_intro\n\ntheorem dec_em (p : Prop) [decidable p] : p ∨ ¬p := decidable.em p\n\ntheorem dec_em' (p : Prop) [decidable p] : ¬p ∨ p := (dec_em p).swap\n\ntheorem em (p : Prop) : p ∨ ¬p := classical.em _\n\ntheorem em' (p : Prop) : ¬p ∨ p := (em p).swap\n\ntheorem or_not {p : Prop} : p ∨ ¬p := em _\n\nsection eq_or_ne\n\nvariables {α : Sort*} (x y : α)\n\ntheorem decidable.eq_or_ne [decidable (x = y)] : x = y ∨ x ≠ y := dec_em $ x = y\n\ntheorem decidable.ne_or_eq [decidable (x = y)] : x ≠ y ∨ x = y := dec_em' $ x = y\n\ntheorem eq_or_ne : x = y ∨ x ≠ y := em $ x = y\n\ntheorem ne_or_eq : x ≠ y ∨ x = y := em' $ x = y\n\nend eq_or_ne\n\ntheorem by_contradiction {p} : (¬p → false) → p := decidable.by_contradiction\n\n-- alias by_contradiction ← by_contra\ntheorem by_contra {p} : (¬p → false) → p := decidable.by_contradiction\n\n/--\nIn most of mathlib, we use the law of excluded middle (LEM) and the axiom of choice (AC) freely.\nThe `decidable` namespace contains versions of lemmas from the root namespace that explicitly\nattempt to avoid the axiom of choice, usually by adding decidability assumptions on the inputs.\n\nYou can check if a lemma uses the axiom of choice by using `#print axioms foo` and seeing if\n`classical.choice` appears in the list.\n-/\nlibrary_note \"decidable namespace\"\n\n/--\nAs mathlib is primarily classical,\nif the type signature of a `def` or `lemma` does not require any `decidable` instances to state,\nit is preferable not to introduce any `decidable` instances that are needed in the proof\nas arguments, but rather to use the `classical` tactic as needed.\n\nIn the other direction, when `decidable` instances do appear in the type signature,\nit is better to use explicitly introduced ones rather than allowing Lean to automatically infer\nclassical ones, as these may cause instance mismatch errors later.\n-/\nlibrary_note \"decidable arguments\"\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_not [decidable a] : ¬¬a ↔ a :=\niff.intro decidable.by_contradiction not_not_intro\n\n/-- The Double Negation Theorem: `¬ ¬ P` is equivalent to `P`.\nThe left-to-right direction, double negation elimination (DNE),\nis classically true but not constructively. -/\n@[simp] theorem not_not : ¬¬a ↔ a := decidable.not_not\n\ntheorem of_not_not : ¬¬a → a := by_contra\n\nlemma not_ne_iff {α : Sort*} {a b : α} : ¬ a ≠ b ↔ a = b := not_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.of_not_imp [decidable a] (h : ¬ (a → b)) : a :=\ndecidable.by_contradiction (not_not_of_not_imp h)\n\ntheorem of_not_imp : ¬ (a → b) → a := decidable.of_not_imp\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_symm [decidable a] (h : ¬a → b) (hb : ¬b) : a :=\ndecidable.by_contradiction $ hb ∘ h\n\ntheorem not.decidable_imp_symm [decidable a] : (¬a → b) → ¬b → a := decidable.not_imp_symm\n\ntheorem not.imp_symm : (¬a → b) → ¬b → a := not.decidable_imp_symm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_comm [decidable a] [decidable b] : (¬a → b) ↔ (¬b → a) :=\n⟨not.decidable_imp_symm, not.decidable_imp_symm⟩\n\ntheorem not_imp_comm : (¬a → b) ↔ (¬b → a) := decidable.not_imp_comm\n\n@[simp] theorem imp_not_self : (a → ¬a) ↔ ¬a := ⟨λ h ha, h ha ha, λ h _, h⟩\n\ntheorem decidable.not_imp_self [decidable a] : (¬a → a) ↔ a :=\nby { have := @imp_not_self (¬a), rwa decidable.not_not at this }\n\n@[simp] theorem not_imp_self : (¬a → a) ↔ a := decidable.not_imp_self\n\ntheorem imp.swap : (a → b → c) ↔ (b → a → c) :=\n⟨swap, swap⟩\n\ntheorem imp_not_comm : (a → ¬b) ↔ (b → ¬a) :=\nimp.swap\n\nlemma iff.not (h : a ↔ b) : ¬ a ↔ ¬ b := not_congr h\nlemma iff.not_left (h : a ↔ ¬ b) : ¬ a ↔ b := h.not.trans not_not\nlemma iff.not_right (h : ¬ a ↔ b) : a ↔ ¬ b := not_not.symm.trans h.not\n\n/-! ### Declarations about `xor` -/\n\n@[simp] theorem xor_true : xor true = not := funext $ λ a, by simp [xor]\n\n@[simp] theorem xor_false : xor false = id := funext $ λ a, by simp [xor]\n\ntheorem xor_comm (a b) : xor a b ↔ xor b a := or_comm _ _\n\ninstance : is_commutative Prop xor := ⟨λ a b, propext $ xor_comm a b⟩\n\n@[simp] theorem xor_self (a : Prop) : xor a a = false := by simp [xor]\n@[simp] theorem xor_not_left : xor (¬a) b ↔ (a ↔ b) := by by_cases a; simp *\n@[simp] theorem xor_not_right : xor a (¬b) ↔ (a ↔ b) := by by_cases a; simp *\ntheorem xor_not_not : xor (¬a) (¬b) ↔ xor a b := by simp [xor, or_comm, and_comm]\nprotected theorem xor.or (h : xor a b) : a ∨ b := h.imp and.left and.left\n\n/-! ### Declarations about `and` -/\n\nlemma iff.and (h₁ : a ↔ b) (h₂ : c ↔ d) : a ∧ c ↔ b ∧ d := and_congr h₁ h₂\n\ntheorem and_congr_left (h : c → (a ↔ b)) : a ∧ c ↔ b ∧ c :=\nand.comm.trans $ (and_congr_right h).trans and.comm\n\ntheorem and_congr_left' (h : a ↔ b) : a ∧ c ↔ b ∧ c := h.and iff.rfl\n\ntheorem and_congr_right' (h : b ↔ c) : a ∧ b ↔ a ∧ c := iff.rfl.and h\n\ntheorem not_and_of_not_left (b : Prop) : ¬a → ¬(a ∧ b) :=\nmt and.left\n\ntheorem not_and_of_not_right (a : Prop) {b : Prop} : ¬b → ¬(a ∧ b) :=\nmt and.right\n\ntheorem and.imp_left (h : a → b) : a ∧ c → b ∧ c :=\nand.imp h id\n\ntheorem and.imp_right (h : a → b) : c ∧ a → c ∧ b :=\nand.imp id h\n\nlemma and.right_comm : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=\nby simp only [and.left_comm, and.comm]\n\nlemma and_and_and_comm (a b c d : Prop) : (a ∧ b) ∧ c ∧ d ↔ (a ∧ c) ∧ b ∧ d :=\nby rw [←and_assoc, @and.right_comm a, and_assoc]\n\nlemma and_and_distrib_left (a b c : Prop) : a ∧ (b ∧ c) ↔ (a ∧ b) ∧ (a ∧ c) :=\nby rw [and_and_and_comm, and_self]\n\nlemma and_and_distrib_right (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ (b ∧ c) :=\nby rw [and_and_and_comm, and_self]\n\nlemma and_rotate : a ∧ b ∧ c ↔ b ∧ c ∧ a := by simp only [and.left_comm, and.comm]\nlemma and.rotate : a ∧ b ∧ c → b ∧ c ∧ a := and_rotate.1\n\ntheorem and_not_self_iff (a : Prop) : a ∧ ¬ a ↔ false :=\niff.intro (assume h, (h.right) (h.left)) (assume h, h.elim)\n\ntheorem not_and_self_iff (a : Prop) : ¬ a ∧ a ↔ false :=\niff.intro (assume ⟨hna, ha⟩, hna ha) false.elim\n\ntheorem and_iff_left_of_imp {a b : Prop} (h : a → b) : (a ∧ b) ↔ a :=\niff.intro and.left (λ ha, ⟨ha, h ha⟩)\n\ntheorem and_iff_right_of_imp {a b : Prop} (h : b → a) : (a ∧ b) ↔ b :=\niff.intro and.right (λ hb, ⟨h hb, hb⟩)\n\nlemma ne_and_eq_iff_right {α : Sort*} {a b c : α} (h : b ≠ c) : a ≠ b ∧ a = c ↔ a = c :=\nand_iff_right_of_imp (λ h2, h2.symm ▸ h.symm)\n\n@[simp] theorem and_iff_left_iff_imp {a b : Prop} : ((a ∧ b) ↔ a) ↔ (a → b) :=\n⟨λ h ha, (h.2 ha).2, and_iff_left_of_imp⟩\n\n@[simp] theorem and_iff_right_iff_imp {a b : Prop} : ((a ∧ b) ↔ b) ↔ (b → a) :=\n⟨λ h ha, (h.2 ha).1, and_iff_right_of_imp⟩\n\n@[simp] lemma iff_self_and {p q : Prop} : (p ↔ p ∧ q) ↔ (p → q) :=\nby rw [@iff.comm p, and_iff_left_iff_imp]\n\n@[simp] lemma iff_and_self {p q : Prop} : (p ↔ q ∧ p) ↔ (p → q) :=\nby rw [and_comm, iff_self_and]\n\n@[simp] lemma and.congr_right_iff : (a ∧ b ↔ a ∧ c) ↔ (a → (b ↔ c)) :=\n⟨λ h ha, by simp [ha] at h; exact h, and_congr_right⟩\n\n@[simp] lemma and.congr_left_iff : (a ∧ c ↔ b ∧ c) ↔ c → (a ↔ b) :=\nby simp only [and.comm, ← and.congr_right_iff]\n\n@[simp] lemma and_self_left : a ∧ a ∧ b ↔ a ∧ b :=\n⟨λ h, ⟨h.1, h.2.2⟩, λ h, ⟨h.1, h.1, h.2⟩⟩\n\n@[simp] lemma and_self_right : (a ∧ b) ∧ b ↔ a ∧ b :=\n⟨λ h, ⟨h.1.1, h.2⟩, λ h, ⟨⟨h.1, h.2⟩, h.2⟩⟩\n\n/-! ### Declarations about `or` -/\n\nlemma iff.or (h₁ : a ↔ b) (h₂ : c ↔ d) : a ∨ c ↔ b ∨ d := or_congr h₁ h₂\n\nlemma or_congr_left' (h : a ↔ b) : a ∨ c ↔ b ∨ c := h.or iff.rfl\nlemma or_congr_right' (h : b ↔ c) : a ∨ b ↔ a ∨ c := iff.rfl.or h\n\ntheorem or.right_comm : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b := by rw [or_assoc, or_assoc, or_comm b]\n\nlemma or_or_or_comm (a b c d : Prop) : (a ∨ b) ∨ c ∨ d ↔ (a ∨ c) ∨ b ∨ d :=\nby rw [←or_assoc, @or.right_comm a, or_assoc]\n\nlemma or_or_distrib_left (a b c : Prop) : a ∨ (b ∨ c) ↔ (a ∨ b) ∨ (a ∨ c) :=\nby rw [or_or_or_comm, or_self]\n\nlemma or_or_distrib_right (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ (b ∨ c) :=\nby rw [or_or_or_comm, or_self]\n\nlemma or_rotate : a ∨ b ∨ c ↔ b ∨ c ∨ a := by simp only [or.left_comm, or.comm]\nlemma or.rotate : a ∨ b ∨ c → b ∨ c ∨ a := or_rotate.1\n\ntheorem or_of_or_of_imp_of_imp (h₁ : a ∨ b) (h₂ : a → c) (h₃ : b → d) : c ∨ d :=\nor.imp h₂ h₃ h₁\n\ntheorem or_of_or_of_imp_left (h₁ : a ∨ c) (h : a → b) : b ∨ c :=\nor.imp_left h h₁\n\ntheorem or_of_or_of_imp_right (h₁ : c ∨ a) (h : a → b) : c ∨ b :=\nor.imp_right h h₁\n\ntheorem or.elim3 (h : a ∨ b ∨ c) (ha : a → d) (hb : b → d) (hc : c → d) : d :=\nor.elim h ha (assume h₂, or.elim h₂ hb hc)\n\nlemma or.imp3 (had : a → d) (hbe : b → e) (hcf : c → f) : a ∨ b ∨ c → d ∨ e ∨ f :=\nor.imp had $ or.imp hbe hcf\n\ntheorem or_imp_distrib : (a ∨ b → c) ↔ (a → c) ∧ (b → c) :=\n⟨assume h, ⟨assume ha, h (or.inl ha), assume hb, h (or.inr hb)⟩,\n  assume ⟨ha, hb⟩, or.rec ha hb⟩\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_imp_left [decidable a] : a ∨ b ↔ (¬ a → b) :=\n⟨or.resolve_left, λ h, dite _ or.inl (or.inr ∘ h)⟩\n\ntheorem or_iff_not_imp_left : a ∨ b ↔ (¬ a → b) := decidable.or_iff_not_imp_left\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_imp_right [decidable b] : a ∨ b ↔ (¬ b → a) :=\nor.comm.trans decidable.or_iff_not_imp_left\n\ntheorem or_iff_not_imp_right : a ∨ b ↔ (¬ b → a) := decidable.or_iff_not_imp_right\n\n-- See Note [decidable namespace]\nprotected lemma decidable.not_or_of_imp [decidable a] (h : a → b) : ¬ a ∨ b :=\ndite _ (or.inr ∘ h) or.inl\n\nlemma not_or_of_imp : (a → b) → ¬ a ∨ b := decidable.not_or_of_imp\n\n-- See Note [decidable namespace]\nprotected lemma decidable.or_not_of_imp [decidable a] (h : a → b) : b ∨ ¬ a :=\ndite _ (or.inl ∘ h) or.inr\n\nlemma or_not_of_imp : (a → b) → b ∨ ¬ a := decidable.or_not_of_imp\n\n-- See Note [decidable namespace]\nprotected lemma decidable.imp_iff_not_or [decidable a] : a → b ↔ ¬ a ∨ b :=\n⟨decidable.not_or_of_imp, or.neg_resolve_left⟩\n\nlemma imp_iff_not_or : a → b ↔ ¬ a ∨ b := decidable.imp_iff_not_or\n\n-- See Note [decidable namespace]\nprotected lemma decidable.imp_iff_or_not [decidable b] : b → a ↔ a ∨ ¬ b :=\ndecidable.imp_iff_not_or.trans or.comm\n\nlemma imp_iff_or_not : b → a ↔ a ∨ ¬ b  := decidable.imp_iff_or_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_not [decidable a] : (¬ a → ¬ b) ↔ (b → a) :=\n⟨assume h hb, decidable.by_contradiction $ assume na, h na hb, mt⟩\n\ntheorem not_imp_not : (¬ a → ¬ b) ↔ (b → a) := decidable.not_imp_not\n\n/-- Provide the reverse of modus tollens (`mt`) as dot notation for implications. -/\nprotected theorem function.mtr : (¬ a → ¬ b) → (b → a) := not_imp_not.mp\n\n-- See Note [decidable namespace]\nprotected lemma decidable.or_congr_left [decidable c] (h : ¬ c → (a ↔ b)) : a ∨ c ↔ b ∨ c :=\nby { rw [decidable.or_iff_not_imp_right, decidable.or_iff_not_imp_right], exact imp_congr_right h }\n\nlemma or_congr_left (h : ¬ c → (a ↔ b)) : a ∨ c ↔ b ∨ c :=\ndecidable.or_congr_left h\n\n-- See Note [decidable namespace]\nprotected lemma decidable.or_congr_right [decidable a] (h : ¬ a → (b ↔ c)) : a ∨ b ↔ a ∨ c :=\nby { rw [decidable.or_iff_not_imp_left, decidable.or_iff_not_imp_left], exact imp_congr_right h }\n\nlemma or_congr_right (h : ¬ a → (b ↔ c)) : a ∨ b ↔ a ∨ c :=\ndecidable.or_congr_right h\n\n@[simp] theorem or_iff_left_iff_imp : (a ∨ b ↔ a) ↔ (b → a) :=\n⟨λ h hb, h.1 (or.inr hb), or_iff_left_of_imp⟩\n\n@[simp] theorem or_iff_right_iff_imp : (a ∨ b ↔ b) ↔ (a → b) :=\nby rw [or_comm, or_iff_left_iff_imp]\n\nlemma or_iff_left (hb : ¬ b) : a ∨ b ↔ a := ⟨λ h, h.resolve_right hb, or.inl⟩\nlemma or_iff_right (ha : ¬ a) : a ∨ b ↔ b := ⟨λ h, h.resolve_left ha, or.inr⟩\n\n/-! ### Declarations about distributivity -/\n\n/-- `∧` distributes over `∨` (on the left). -/\ntheorem and_or_distrib_left : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=\n⟨λ ⟨ha, hbc⟩, hbc.imp (and.intro ha) (and.intro ha),\n or.rec (and.imp_right or.inl) (and.imp_right or.inr)⟩\n\n/-- `∧` distributes over `∨` (on the right). -/\ntheorem or_and_distrib_right : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=\n(and.comm.trans and_or_distrib_left).trans (and.comm.or and.comm)\n\n/-- `∨` distributes over `∧` (on the left). -/\ntheorem or_and_distrib_left : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=\n⟨or.rec (λha, and.intro (or.inl ha) (or.inl ha)) (and.imp or.inr or.inr),\n and.rec $ or.rec (imp_intro ∘ or.inl) (or.imp_right ∘ and.intro)⟩\n\n/-- `∨` distributes over `∧` (on the right). -/\ntheorem and_or_distrib_right : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=\n(or.comm.trans or_and_distrib_left).trans (or.comm.and or.comm)\n\n@[simp] lemma or_self_left : a ∨ a ∨ b ↔ a ∨ b :=\n⟨λ h, h.elim or.inl id, λ h, h.elim or.inl (or.inr ∘ or.inr)⟩\n\n@[simp] lemma or_self_right : (a ∨ b) ∨ b ↔ a ∨ b :=\n⟨λ h, h.elim id or.inr, λ h, h.elim (or.inl ∘ or.inl) or.inr⟩\n\n/-! Declarations about `iff` -/\n\nlemma iff.iff (h₁ : a ↔ b) (h₂ : c ↔ d) : (a ↔ c) ↔ (b ↔ d) := iff_congr h₁ h₂\n\ntheorem iff_of_true (ha : a) (hb : b) : a ↔ b :=\n⟨λ_, hb, λ _, ha⟩\n\ntheorem iff_of_false (ha : ¬a) (hb : ¬b) : a ↔ b :=\n⟨ha.elim, hb.elim⟩\n\ntheorem iff_true_left (ha : a) : (a ↔ b) ↔ b :=\n⟨λ h, h.1 ha, iff_of_true ha⟩\n\ntheorem iff_true_right (ha : a) : (b ↔ a) ↔ b :=\niff.comm.trans (iff_true_left ha)\n\ntheorem iff_false_left (ha : ¬a) : (a ↔ b) ↔ ¬b :=\n⟨λ h, mt h.2 ha, iff_of_false ha⟩\n\ntheorem iff_false_right (ha : ¬a) : (b ↔ a) ↔ ¬b :=\niff.comm.trans (iff_false_left ha)\n\n@[simp]\nlemma iff_mpr_iff_true_intro {P : Prop} (h : P) : iff.mpr (iff_true_intro h) true.intro = h := rfl\n\n-- See Note [decidable namespace]\nprotected theorem decidable.imp_or_distrib [decidable a] : (a → b ∨ c) ↔ (a → b) ∨ (a → c) :=\nby simp [decidable.imp_iff_not_or, or.comm, or.left_comm]\n\ntheorem imp_or_distrib : (a → b ∨ c) ↔ (a → b) ∨ (a → c) := decidable.imp_or_distrib\n\n-- See Note [decidable namespace]\nprotected theorem decidable.imp_or_distrib' [decidable b] : (a → b ∨ c) ↔ (a → b) ∨ (a → c) :=\nby by_cases b; simp [h, or_iff_right_of_imp ((∘) false.elim)]\n\ntheorem imp_or_distrib' : (a → b ∨ c) ↔ (a → b) ∨ (a → c) := decidable.imp_or_distrib'\n\ntheorem not_imp_of_and_not : a ∧ ¬ b → ¬ (a → b)\n| ⟨ha, hb⟩ h := hb $ h ha\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp [decidable a] : ¬(a → b) ↔ a ∧ ¬b :=\n⟨λ h, ⟨decidable.of_not_imp h, not_of_not_imp h⟩, not_imp_of_and_not⟩\n\ntheorem not_imp : ¬(a → b) ↔ a ∧ ¬b := decidable.not_imp\n\n-- for monotonicity\nlemma imp_imp_imp (h₀ : c → a) (h₁ : b → d) : (a → b) → (c → d) :=\nassume (h₂ : a → b), h₁ ∘ h₂ ∘ h₀\n\n-- See Note [decidable namespace]\nprotected theorem decidable.peirce (a b : Prop) [decidable a] : ((a → b) → a) → a :=\nif ha : a then λ h, ha else λ h, h ha.elim\n\ntheorem peirce (a b : Prop) : ((a → b) → a) → a := decidable.peirce _ _\n\ntheorem peirce' {a : Prop} (H : ∀ b : Prop, (a → b) → a) : a := H _ id\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff_not [decidable a] [decidable b] : (¬ a ↔ ¬ b) ↔ (a ↔ b) :=\nby rw [@iff_def (¬ a), @iff_def' a]; exact decidable.not_imp_not.and decidable.not_imp_not\n\ntheorem not_iff_not : (¬ a ↔ ¬ b) ↔ (a ↔ b) := decidable.not_iff_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff_comm [decidable a] [decidable b] : (¬ a ↔ b) ↔ (¬ b ↔ a) :=\nby rw [@iff_def (¬ a), @iff_def (¬ b)]; exact decidable.not_imp_comm.and imp_not_comm\n\ntheorem not_iff_comm : (¬ a ↔ b) ↔ (¬ b ↔ a) := decidable.not_iff_comm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff : ∀ [decidable b], ¬ (a ↔ b) ↔ (¬ a ↔ b) :=\nby intro h; cases h; simp only [h, iff_true, iff_false]\n\ntheorem not_iff : ¬ (a ↔ b) ↔ (¬ a ↔ b) := decidable.not_iff\n\n-- See Note [decidable namespace]\nprotected theorem decidable.iff_not_comm [decidable a] [decidable b] : (a ↔ ¬ b) ↔ (b ↔ ¬ a) :=\nby rw [@iff_def a, @iff_def b]; exact imp_not_comm.and decidable.not_imp_comm\n\ntheorem iff_not_comm : (a ↔ ¬ b) ↔ (b ↔ ¬ a) := decidable.iff_not_comm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.iff_iff_and_or_not_and_not [decidable b] :\n  (a ↔ b) ↔ (a ∧ b) ∨ (¬ a ∧ ¬ b) :=\nby { split; intro h,\n     { rw h; by_cases b; [left,right]; split; assumption },\n     { cases h with h h; cases h; split; intro; { contradiction <|> assumption } } }\n\ntheorem iff_iff_and_or_not_and_not : (a ↔ b) ↔ (a ∧ b) ∨ (¬ a ∧ ¬ b) :=\ndecidable.iff_iff_and_or_not_and_not\n\nlemma decidable.iff_iff_not_or_and_or_not [decidable a] [decidable b] :\n  (a ↔ b) ↔ ((¬a ∨ b) ∧ (a ∨ ¬b)) :=\nbegin\n  rw [iff_iff_implies_and_implies a b],\n  simp only [decidable.imp_iff_not_or, or.comm]\nend\n\nlemma iff_iff_not_or_and_or_not : (a ↔ b) ↔ ((¬a ∨ b) ∧ (a ∨ ¬b)) :=\ndecidable.iff_iff_not_or_and_or_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_not_right [decidable b] : ¬(a ∧ ¬b) ↔ (a → b) :=\n⟨λ h ha, h.decidable_imp_symm $ and.intro ha, λ h ⟨ha, hb⟩, hb $ h ha⟩\n\ntheorem not_and_not_right : ¬(a ∧ ¬b) ↔ (a → b) := decidable.not_and_not_right\n\n/-- Transfer decidability of `a` to decidability of `b`, if the propositions are equivalent.\n**Important**: this function should be used instead of `rw` on `decidable b`, because the\nkernel will get stuck reducing the usage of `propext` otherwise,\nand `dec_trivial` will not work. -/\n@[inline] def decidable_of_iff (a : Prop) (h : a ↔ b) [D : decidable a] : decidable b :=\ndecidable_of_decidable_of_iff D h\n\n/-- Transfer decidability of `b` to decidability of `a`, if the propositions are equivalent.\nThis is the same as `decidable_of_iff` but the iff is flipped. -/\n@[inline] def decidable_of_iff' (b : Prop) (h : a ↔ b) [D : decidable b] : decidable a :=\ndecidable_of_decidable_of_iff D h.symm\n\n/-- Prove that `a` is decidable by constructing a boolean `b` and a proof that `b ↔ a`.\n(This is sometimes taken as an alternate definition of decidability.) -/\ndef decidable_of_bool : ∀ (b : bool) (h : b ↔ a), decidable a\n| tt h := is_true (h.1 rfl)\n| ff h := is_false (mt h.2 bool.ff_ne_tt)\n\n/-! ### De Morgan's laws -/\n\ntheorem not_and_of_not_or_not (h : ¬ a ∨ ¬ b) : ¬ (a ∧ b)\n| ⟨ha, hb⟩ := or.elim h (absurd ha) (absurd hb)\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_distrib [decidable a] : ¬ (a ∧ b) ↔ ¬a ∨ ¬b :=\n⟨λ h, if ha : a then or.inr (λ hb, h ⟨ha, hb⟩) else or.inl ha, not_and_of_not_or_not⟩\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_distrib' [decidable b] : ¬ (a ∧ b) ↔ ¬a ∨ ¬b :=\n⟨λ h, if hb : b then or.inl (λ ha, h ⟨ha, hb⟩) else or.inr hb, not_and_of_not_or_not⟩\n\n/-- One of de Morgan's laws: the negation of a conjunction is logically equivalent to the\ndisjunction of the negations. -/\ntheorem not_and_distrib : ¬ (a ∧ b) ↔ ¬a ∨ ¬b := decidable.not_and_distrib\n\n@[simp] theorem not_and : ¬ (a ∧ b) ↔ (a → ¬ b) := and_imp\n\ntheorem not_and' : ¬ (a ∧ b) ↔ b → ¬a :=\nnot_and.trans imp_not_comm\n\n/-- One of de Morgan's laws: the negation of a disjunction is logically equivalent to the\nconjunction of the negations. -/\ntheorem not_or_distrib : ¬ (a ∨ b) ↔ ¬ a ∧ ¬ b := or_imp_distrib\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_and_not [decidable a] [decidable b] : a ∨ b ↔ ¬ (¬a ∧ ¬b) :=\nby rw [← not_or_distrib, decidable.not_not]\n\ntheorem or_iff_not_and_not : a ∨ b ↔ ¬ (¬a ∧ ¬b) := decidable.or_iff_not_and_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.and_iff_not_or_not [decidable a] [decidable b] :\n  a ∧ b ↔ ¬ (¬ a ∨ ¬ b) :=\nby rw [← decidable.not_and_distrib, decidable.not_not]\n\ntheorem and_iff_not_or_not : a ∧ b ↔ ¬ (¬ a ∨ ¬ b) := decidable.and_iff_not_or_not\n\n@[simp] theorem not_xor (P Q : Prop) : ¬ xor P Q ↔ (P ↔ Q) :=\nby simp only [not_and, xor, not_or_distrib, not_not, ← iff_iff_implies_and_implies]\n\ntheorem xor_iff_not_iff (P Q : Prop) : xor P Q ↔ ¬ (P ↔ Q) := (not_xor P Q).not_right\ntheorem xor_iff_iff_not : xor a b ↔ (a ↔ ¬b) := by simp only [← @xor_not_right a, not_not]\ntheorem xor_iff_not_iff' : xor a b ↔ (¬a ↔ b) := by simp only [← @xor_not_left _ b, not_not]\n\nend propositional\n\n/-! ### Declarations about equality -/\n\nsection mem\nvariables {α β : Type*} [has_mem α β] {s t : β} {a b : α}\n\nlemma ne_of_mem_of_not_mem (h : a ∈ s) : b ∉ s → a ≠ b := mt $ λ e, e ▸ h\nlemma ne_of_mem_of_not_mem' (h : a ∈ s) : a ∉ t → s ≠ t := mt $ λ e, e ▸ h\n\n/-- **Alias** of `ne_of_mem_of_not_mem`. -/\nlemma has_mem.mem.ne_of_not_mem : a ∈ s → b ∉ s → a ≠ b := ne_of_mem_of_not_mem\n/-- **Alias** of `ne_of_mem_of_not_mem'`. -/\nlemma has_mem.mem.ne_of_not_mem' : a ∈ s → a ∉ t → s ≠ t := ne_of_mem_of_not_mem'\n\nend mem\n\nsection equality\nvariables {α : Sort*} {a b : α}\n\n@[simp] theorem heq_iff_eq : a == b ↔ a = b :=\n⟨eq_of_heq, heq_of_eq⟩\n\ntheorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : hp == hq :=\nhave p = q, from propext ⟨λ _, hq, λ _, hp⟩,\nby subst q; refl\n\n-- todo: change name\nlemma ball_cond_comm {α} {s : α → Prop} {p : α → α → Prop} :\n  (∀ a, s a → ∀ b, s b → p a b) ↔ (∀ a b, s a → s b → p a b) :=\n⟨λ h a b ha hb, h a ha b hb, λ h a ha b hb, h a b ha hb⟩\n\nlemma ball_mem_comm {α β} [has_mem α β] {s : β} {p : α → α → Prop} :\n  (∀ a b ∈ s, p a b) ↔ (∀ a b, a ∈ s → b ∈ s → p a b) :=\nball_cond_comm\n\nlemma ne_of_apply_ne {α β : Sort*} (f : α → β) {x y : α} (h : f x ≠ f y) : x ≠ y :=\nλ (w : x = y), h (congr_arg f w)\n\ntheorem eq_equivalence : equivalence (@eq α) :=\n⟨eq.refl, @eq.symm _, @eq.trans _⟩\n\n/-- Transport through trivial families is the identity. -/\n@[simp]\nlemma eq_rec_constant {α : Sort*} {a a' : α} {β : Sort*} (y : β) (h : a = a') :\n  (@eq.rec α a (λ a, β) y a' h) = y :=\nby { cases h, refl, }\n\n@[simp]\nlemma eq_mp_eq_cast {α β : Sort*} (h : α = β) : eq.mp h = cast h := rfl\n\n@[simp]\nlemma eq_mpr_eq_cast {α β : Sort*} (h : α = β) : eq.mpr h = cast h.symm := rfl\n\n@[simp]\nlemma cast_cast : ∀ {α β γ : Sort*} (ha : α = β) (hb : β = γ) (a : α),\n  cast hb (cast ha a) = cast (ha.trans hb) a\n| _ _ _ rfl rfl a := rfl\n\n@[simp] lemma congr_refl_left {α β : Sort*} (f : α → β) {a b : α} (h : a = b) :\n  congr (eq.refl f) h = congr_arg f h :=\nrfl\n\n@[simp] lemma congr_refl_right {α β : Sort*} {f g : α → β} (h : f = g) (a : α) :\n  congr h (eq.refl a) = congr_fun h a :=\nrfl\n\n@[simp] lemma congr_arg_refl {α β : Sort*} (f : α → β) (a : α) :\n  congr_arg f (eq.refl a) = eq.refl (f a) :=\nrfl\n\n@[simp] lemma congr_fun_rfl {α β : Sort*} (f : α → β) (a : α) :\n  congr_fun (eq.refl f) a = eq.refl (f a) :=\nrfl\n\n@[simp] lemma congr_fun_congr_arg {α β γ : Sort*} (f : α → β → γ) {a a' : α} (p : a = a') (b : β) :\n  congr_fun (congr_arg f p) b = congr_arg (λ a, f a b) p :=\nrfl\n\nlemma heq_of_cast_eq :\n  ∀ {α β : Sort*} {a : α} {a' : β} (e : α = β) (h₂ : cast e a = a'), a == a'\n| α ._ a a' rfl h := eq.rec_on h (heq.refl _)\n\nlemma cast_eq_iff_heq {α β : Sort*} {a : α} {a' : β} {e : α = β} : cast e a = a' ↔ a == a' :=\n⟨heq_of_cast_eq _, λ h, by cases h; refl⟩\n\nlemma rec_heq_of_heq {β} {C : α → Sort*} {x : C a} {y : β} (e : a = b) (h : x == y) :\n  @eq.rec α a C x b e == y :=\nby subst e; exact h\n\nlemma rec_heq_iff_heq {β} {C : α → Sort*} {x : C a} {y : β} {e : a = b} :\n  @eq.rec α a C x b e == y ↔ x == y :=\nby subst e\n\nlemma heq_rec_iff_heq {β} {C : α → Sort*} {x : β} {y : C a} {e : a = b} :\n  x == @eq.rec α a C y b e ↔ x == y :=\nby subst e\n\nprotected lemma eq.congr {x₁ x₂ y₁ y₂ : α} (h₁ : x₁ = y₁) (h₂ : x₂ = y₂) :\n  (x₁ = x₂) ↔ (y₁ = y₂) :=\nby { subst h₁, subst h₂ }\n\nlemma eq.congr_left {x y z : α} (h : x = y) : x = z ↔ y = z := by rw [h]\nlemma eq.congr_right {x y z : α} (h : x = y) : z = x ↔ z = y := by rw [h]\n\nlemma congr_arg2 {α β γ : Sort*} (f : α → β → γ) {x x' : α} {y y' : β}\n  (hx : x = x') (hy : y = y') : f x y = f x' y' :=\nby { subst hx, subst hy }\n\nvariables {β : α → Sort*} {γ : Π a, β a → Sort*} {δ : Π a b, γ a b → Sort*}\n\nlemma congr_fun₂ {f g : Π a b, γ a b} (h : f = g) (a : α) (b : β a) : f a b = g a b :=\ncongr_fun (congr_fun h _) _\n\nlemma congr_fun₃ {f g : Π a b c, δ a b c} (h : f = g) (a : α) (b : β a) (c : γ a b) :\n  f a b c = g a b c :=\ncongr_fun₂ (congr_fun h _) _ _\n\nlemma funext₂ {f g : Π a b, γ a b} (h : ∀ a b, f a b = g a b) : f = g :=\nfunext $ λ _, funext $ h _\n\nlemma funext₃ {f g : Π a b c, δ a b c} (h : ∀ a b c, f a b c = g a b c) : f = g :=\nfunext $ λ _, funext₂ $ h _\n\nend equality\n\n/-! ### Declarations about quantifiers -/\n\nsection quantifiers\nvariables {α : Sort*}\n\nsection dependent\nvariables {β : α → Sort*} {γ : Π a, β a → Sort*} {δ : Π a b, γ a b → Sort*}\n  {ε : Π a b c, δ a b c → Sort*}\n\nlemma pi_congr {β' : α → Sort*} (h : ∀ a, β a = β' a) : (Π a, β a) = Π a, β' a :=\n(funext h : β = β') ▸ rfl\n\nlemma forall₂_congr {p q : Π a, β a → Prop} (h : ∀ a b, p a b ↔ q a b) :\n  (∀ a b, p a b) ↔ ∀ a b, q a b :=\nforall_congr $ λ a, forall_congr $ h a\n\nlemma forall₃_congr {p q : Π a b, γ a b → Prop} (h : ∀ a b c, p a b c ↔ q a b c) :\n  (∀ a b c, p a b c) ↔ ∀ a b c, q a b c :=\nforall_congr $ λ a, forall₂_congr $ h a\n\nlemma forall₄_congr {p q : Π a b c, δ a b c → Prop} (h : ∀ a b c d, p a b c d ↔ q a b c d) :\n  (∀ a b c d, p a b c d) ↔ ∀ a b c d, q a b c d :=\nforall_congr $ λ a, forall₃_congr $ h a\n\nlemma forall₅_congr {p q : Π a b c d, ε a b c d → Prop}\n  (h : ∀ a b c d e, p a b c d e ↔ q a b c d e) :\n  (∀ a b c d e, p a b c d e) ↔ ∀ a b c d e, q a b c d e :=\nforall_congr $ λ a, forall₄_congr $ h a\n\nlemma exists₂_congr {p q : Π a, β a → Prop} (h : ∀ a b, p a b ↔ q a b) :\n  (∃ a b, p a b) ↔ ∃ a b, q a b :=\nexists_congr $ λ a, exists_congr $ h a\n\nlemma exists₃_congr {p q : Π a b, γ a b → Prop} (h : ∀ a b c, p a b c ↔ q a b c) :\n  (∃ a b c, p a b c) ↔ ∃ a b c, q a b c :=\nexists_congr $ λ a, exists₂_congr $ h a\n\nlemma exists₄_congr {p q : Π a b c, δ a b c → Prop} (h : ∀ a b c d, p a b c d ↔ q a b c d) :\n  (∃ a b c d, p a b c d) ↔ ∃ a b c d, q a b c d :=\nexists_congr $ λ a, exists₃_congr $ h a\n\nlemma exists₅_congr {p q : Π a b c d, ε a b c d → Prop}\n  (h : ∀ a b c d e, p a b c d e ↔ q a b c d e) :\n  (∃ a b c d e, p a b c d e) ↔ ∃ a b c d e, q a b c d e :=\nexists_congr $ λ a, exists₄_congr $ h a\n\nlemma forall_imp {p q : α → Prop} (h : ∀ a, p a → q a) : (∀ a, p a) → ∀ a, q a := λ h' a, h a (h' a)\n\nlemma forall₂_imp {p q : Π a, β a → Prop} (h : ∀ a b, p a b → q a b) :\n  (∀ a b, p a b) → ∀ a b, q a b :=\nforall_imp $ λ i, forall_imp $ h i\n\nlemma forall₃_imp {p q : Π a b, γ a b → Prop} (h : ∀ a b c, p a b c → q a b c) :\n  (∀ a b c, p a b c) → ∀ a b c, q a b c :=\nforall_imp $ λ a, forall₂_imp $ h a\n\nlemma Exists.imp {p q : α → Prop}  (h : ∀ a, (p a → q a)) : (∃ a, p a) → ∃ a, q a :=\nexists_imp_exists h\n\nlemma Exists₂.imp {p q : Π a, β a → Prop} (h : ∀ a b, p a b → q a b) :\n  (∃ a b, p a b) → ∃ a b, q a b :=\nExists.imp $ λ a, Exists.imp $ h a\n\nlemma Exists₃.imp {p q : Π a b, γ a b → Prop} (h : ∀ a b c, p a b c → q a b c) :\n  (∃ a b c, p a b c) → ∃ a b c, q a b c :=\nExists.imp $ λ a, Exists₂.imp $ h a\n\nend dependent\n\nvariables {ι β : Sort*} {κ : ι → Sort*} {p q : α → Prop} {b : Prop}\n\nlemma exists_imp_exists' {p : α → Prop} {q : β → Prop} (f : α → β) (hpq : ∀ a, p a → q (f a))\n  (hp : ∃ a, p a) : ∃ b, q b :=\nexists.elim hp (λ a hp', ⟨_, hpq _ hp'⟩)\n\ntheorem forall_swap {p : α → β → Prop} : (∀ x y, p x y) ↔ ∀ y x, p x y :=\n⟨swap, swap⟩\n\nlemma forall₂_swap {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}\n  {p : Π i₁, κ₁ i₁ → Π i₂, κ₂ i₂ → Prop} :\n  (∀ i₁ j₁ i₂ j₂, p i₁ j₁ i₂ j₂) ↔ ∀ i₂ j₂ i₁ j₁, p i₁ j₁ i₂ j₂ :=\n⟨swap₂, swap₂⟩\n\n/-- We intentionally restrict the type of `α` in this lemma so that this is a safer to use in simp\nthan `forall_swap`. -/\nlemma imp_forall_iff {α : Type*} {p : Prop} {q : α → Prop} : (p → ∀ x, q x) ↔ (∀ x, p → q x) :=\nforall_swap\n\ntheorem exists_swap {p : α → β → Prop} : (∃ x y, p x y) ↔ ∃ y x, p x y :=\n⟨λ ⟨x, y, h⟩, ⟨y, x, h⟩, λ ⟨y, x, h⟩, ⟨x, y, h⟩⟩\n\n@[simp] theorem forall_exists_index {q : (∃ x, p x) → Prop} :\n  (∀ h, q h) ↔ ∀ x (h : p x), q ⟨x, h⟩ :=\n⟨λ h x hpx, h ⟨x, hpx⟩, λ h ⟨x, hpx⟩, h x hpx⟩\n\ntheorem exists_imp_distrib : ((∃ x, p x) → b) ↔ ∀ x, p x → b :=\nforall_exists_index\n\n/--\nExtract an element from a existential statement, using `classical.some`.\n-/\n-- This enables projection notation.\n@[reducible] noncomputable def Exists.some {p : α → Prop} (P : ∃ a, p a) : α := classical.some P\n\n/--\nShow that an element extracted from `P : ∃ a, p a` using `P.some` satisfies `p`.\n-/\nlemma Exists.some_spec {p : α → Prop} (P : ∃ a, p a) : p (P.some) := classical.some_spec P\n\n--theorem forall_not_of_not_exists (h : ¬ ∃ x, p x) : ∀ x, ¬ p x :=\n--forall_imp_of_exists_imp h\n\ntheorem not_exists_of_forall_not (h : ∀ x, ¬ p x) : ¬ ∃ x, p x :=\nexists_imp_distrib.2 h\n\n@[simp] theorem not_exists : (¬ ∃ x, p x) ↔ ∀ x, ¬ p x :=\nexists_imp_distrib\n\ntheorem not_forall_of_exists_not : (∃ x, ¬ p x) → ¬ ∀ x, p x\n| ⟨x, hn⟩ h := hn (h x)\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_forall {p : α → Prop}\n  [decidable (∃ x, ¬ p x)] [∀ x, decidable (p x)] : (¬ ∀ x, p x) ↔ ∃ x, ¬ p x :=\n⟨not.decidable_imp_symm $ λ nx x, nx.decidable_imp_symm $ λ h, ⟨x, h⟩,\n not_forall_of_exists_not⟩\n\n@[simp] theorem not_forall {p : α → Prop} : (¬ ∀ x, p x) ↔ ∃ x, ¬ p x := decidable.not_forall\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_forall_not [decidable (∃ x, p x)] :\n  (¬ ∀ x, ¬ p x) ↔ ∃ x, p x :=\n(@decidable.not_iff_comm _ _ _ (decidable_of_iff (¬ ∃ x, p x) not_exists)).1 not_exists\n\ntheorem not_forall_not : (¬ ∀ x, ¬ p x) ↔ ∃ x, p x := decidable.not_forall_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_exists_not [∀ x, decidable (p x)] : (¬ ∃ x, ¬ p x) ↔ ∀ x, p x :=\nby simp [decidable.not_not]\n\n@[simp] theorem not_exists_not : (¬ ∃ x, ¬ p x) ↔ ∀ x, p x := decidable.not_exists_not\n\ntheorem forall_imp_iff_exists_imp [ha : nonempty α] : ((∀ x, p x) → b) ↔ ∃ x, p x → b :=\nlet ⟨a⟩ := ha in\n⟨λ h, not_forall_not.1 $ λ h', classical.by_cases (λ hb : b, h' a $ λ _, hb)\n  (λ hb, hb $ h $ λ x, (not_imp.1 (h' x)).1), λ ⟨x, hx⟩ h, hx (h x)⟩\n\n-- TODO: duplicate of a lemma in core\ntheorem forall_true_iff : (α → true) ↔ true :=\nimplies_true_iff α\n\n-- Unfortunately this causes simp to loop sometimes, so we\n-- add the 2 and 3 cases as simp lemmas instead\ntheorem forall_true_iff' (h : ∀ a, p a ↔ true) : (∀ a, p a) ↔ true :=\niff_true_intro (λ _, of_iff_true (h _))\n\n@[simp] theorem forall_2_true_iff {β : α → Sort*} : (∀ a, β a → true) ↔ true :=\nforall_true_iff' $ λ _, forall_true_iff\n\n@[simp] theorem forall_3_true_iff {β : α → Sort*} {γ : Π a, β a → Sort*} :\n  (∀ a (b : β a), γ a b → true) ↔ true :=\nforall_true_iff' $ λ _, forall_2_true_iff\n\nlemma exists_unique.exists {α : Sort*} {p : α → Prop} (h : ∃! x, p x) : ∃ x, p x :=\nexists.elim h (λ x hx, ⟨x, and.left hx⟩)\n\n@[simp] lemma exists_unique_iff_exists {α : Sort*} [subsingleton α] {p : α → Prop} :\n  (∃! x, p x) ↔ ∃ x, p x :=\n⟨λ h, h.exists, Exists.imp $ λ x hx, ⟨hx, λ y _, subsingleton.elim y x⟩⟩\n\n@[simp] theorem forall_const (α : Sort*) [i : nonempty α] : (α → b) ↔ b :=\n⟨i.elim, λ hb x, hb⟩\n\n/-- For some reason simp doesn't use `forall_const` to simplify in this case. -/\n@[simp] lemma forall_forall_const {α β : Type*} (p : β → Prop) [nonempty α] :\n  (∀ x, α → p x) ↔ ∀ x, p x :=\nforall_congr $ λ x, forall_const α\n\n@[simp] theorem exists_const (α : Sort*) [i : nonempty α] : (∃ x : α, b) ↔ b :=\n⟨λ ⟨x, h⟩, h, i.elim exists.intro⟩\n\ntheorem exists_unique_const (α : Sort*) [i : nonempty α] [subsingleton α] :\n  (∃! x : α, b) ↔ b :=\nby simp\n\ntheorem forall_and_distrib : (∀ x, p x ∧ q x) ↔ (∀ x, p x) ∧ (∀ x, q x) :=\n⟨λ h, ⟨λ x, (h x).left, λ x, (h x).right⟩, λ ⟨h₁, h₂⟩ x, ⟨h₁ x, h₂ x⟩⟩\n\ntheorem exists_or_distrib : (∃ x, p x ∨ q x) ↔ (∃ x, p x) ∨ (∃ x, q x) :=\n⟨λ ⟨x, hpq⟩, hpq.elim (λ hpx, or.inl ⟨x, hpx⟩) (λ hqx, or.inr ⟨x, hqx⟩),\n λ hepq, hepq.elim (λ ⟨x, hpx⟩, ⟨x, or.inl hpx⟩) (λ ⟨x, hqx⟩, ⟨x, or.inr hqx⟩)⟩\n\n@[simp] theorem exists_and_distrib_left {q : Prop} {p : α → Prop} :\n  (∃x, q ∧ p x) ↔ q ∧ (∃x, p x) :=\n⟨λ ⟨x, hq, hp⟩, ⟨hq, x, hp⟩, λ ⟨hq, x, hp⟩, ⟨x, hq, hp⟩⟩\n\n@[simp] theorem exists_and_distrib_right {q : Prop} {p : α → Prop} :\n  (∃x, p x ∧ q) ↔ (∃x, p x) ∧ q :=\nby simp [and_comm]\n\n@[simp] theorem forall_eq {a' : α} : (∀a, a = a' → p a) ↔ p a' :=\n⟨λ h, h a' rfl, λ h a e, e.symm ▸ h⟩\n\n@[simp] theorem forall_eq' {a' : α} : (∀a, a' = a → p a) ↔ p a' :=\nby simp [@eq_comm _ a']\n\ntheorem decidable.and_forall_ne [decidable_eq α] (a : α) : (p a ∧ ∀ b ≠ a, p b) ↔ ∀ b, p b :=\nby simp only [← @forall_eq _ p a, ← forall_and_distrib, ← or_imp_distrib, decidable.em,\n  forall_const]\n\ntheorem and_forall_ne (a : α) : (p a ∧ ∀ b ≠ a, p b) ↔ ∀ b, p b :=\ndecidable.and_forall_ne a\n\n-- this lemma is needed to simplify the output of `list.mem_cons_iff`\n@[simp] theorem forall_eq_or_imp {a' : α} : (∀ a, a = a' ∨ q a → p a) ↔ p a' ∧ ∀ a, q a → p a :=\nby simp only [or_imp_distrib, forall_and_distrib, forall_eq]\n\nlemma ne.ne_or_ne {x y : α} (z : α) (h : x ≠ y) : x ≠ z ∨ y ≠ z :=\nnot_and_distrib.1 $ mt (and_imp.2 eq.substr) h.symm\n\ntheorem exists_eq {a' : α} : ∃ a, a = a' := ⟨_, rfl⟩\n\n@[simp] theorem exists_eq' {a' : α} : ∃ a, a' = a := ⟨_, rfl⟩\n\n@[simp] theorem exists_unique_eq {a' : α} : ∃! a, a = a' :=\nby simp only [eq_comm, exists_unique, and_self, forall_eq', exists_eq']\n\n@[simp] theorem exists_unique_eq' {a' : α} : ∃! a, a' = a :=\nby simp only [exists_unique, and_self, forall_eq', exists_eq']\n\n@[simp] theorem exists_eq_left {a' : α} : (∃ a, a = a' ∧ p a) ↔ p a' :=\n⟨λ ⟨a, e, h⟩, e ▸ h, λ h, ⟨_, rfl, h⟩⟩\n\n@[simp] theorem exists_eq_right {a' : α} : (∃ a, p a ∧ a = a') ↔ p a' :=\n(exists_congr $ by exact λ a, and.comm).trans exists_eq_left\n\n@[simp] theorem exists_eq_right_right {a' : α} :\n  (∃ (a : α), p a ∧ q a ∧ a = a') ↔ p a' ∧ q a' :=\n⟨λ ⟨_, hp, hq, rfl⟩, ⟨hp, hq⟩, λ ⟨hp, hq⟩, ⟨a', hp, hq, rfl⟩⟩\n\n@[simp] theorem exists_eq_right_right' {a' : α} :\n  (∃ (a : α), p a ∧ q a ∧ a' = a) ↔ p a' ∧ q a' :=\n⟨λ ⟨_, hp, hq, rfl⟩, ⟨hp, hq⟩, λ ⟨hp, hq⟩, ⟨a', hp, hq, rfl⟩⟩\n\n@[simp] theorem exists_apply_eq_apply (f : α → β) (a' : α) : ∃ a, f a = f a' := ⟨a', rfl⟩\n\n@[simp] theorem exists_apply_eq_apply' (f : α → β) (a' : α) : ∃ a, f a' = f a := ⟨a', rfl⟩\n\n@[simp] theorem exists_exists_and_eq_and {f : α → β} {p : α → Prop} {q : β → Prop} :\n  (∃ b, (∃ a, p a ∧ f a = b) ∧ q b) ↔ ∃ a, p a ∧ q (f a) :=\n⟨λ ⟨b, ⟨a, ha, hab⟩, hb⟩, ⟨a, ha, hab.symm ▸ hb⟩, λ ⟨a, hp, hq⟩, ⟨f a, ⟨a, hp, rfl⟩, hq⟩⟩\n\n@[simp] theorem exists_exists_eq_and {f : α → β} {p : β → Prop} :\n  (∃ b, (∃ a, f a = b) ∧ p b) ↔ ∃ a, p (f a) :=\n⟨λ ⟨b, ⟨a, ha⟩, hb⟩, ⟨a, ha.symm ▸ hb⟩, λ ⟨a, ha⟩, ⟨f a, ⟨a, rfl⟩, ha⟩⟩\n\n@[simp] lemma exists_or_eq_left (y : α) (p : α → Prop) : ∃ (x : α), x = y ∨ p x :=\n⟨y, or.inl rfl⟩\n\n@[simp] lemma exists_or_eq_right (y : α) (p : α → Prop) : ∃ (x : α), p x ∨ x = y :=\n⟨y, or.inr rfl⟩\n\n@[simp] lemma exists_or_eq_left' (y : α) (p : α → Prop) : ∃ (x : α), y = x ∨ p x :=\n⟨y, or.inl rfl⟩\n\n@[simp] lemma exists_or_eq_right' (y : α) (p : α → Prop) : ∃ (x : α), p x ∨ y = x :=\n⟨y, or.inr rfl⟩\n\n@[simp] theorem forall_apply_eq_imp_iff {f : α → β} {p : β → Prop} :\n  (∀ a, ∀ b, f a = b → p b) ↔ (∀ a, p (f a)) :=\n⟨λ h a, h a (f a) rfl, λ h a b hab, hab ▸ h a⟩\n\n@[simp] theorem forall_apply_eq_imp_iff' {f : α → β} {p : β → Prop} :\n  (∀ b, ∀ a, f a = b → p b) ↔ (∀ a, p (f a)) :=\nby { rw forall_swap, simp }\n\n@[simp] theorem forall_eq_apply_imp_iff {f : α → β} {p : β → Prop} :\n  (∀ a, ∀ b, b = f a → p b) ↔ (∀ a, p (f a)) :=\nby simp [@eq_comm _ _ (f _)]\n\n@[simp] theorem forall_eq_apply_imp_iff' {f : α → β} {p : β → Prop} :\n  (∀ b, ∀ a, b = f a → p b) ↔ (∀ a, p (f a)) :=\nby { rw forall_swap, simp }\n\n@[simp] theorem forall_apply_eq_imp_iff₂ {f : α → β} {p : α → Prop} {q : β → Prop} :\n  (∀ b, ∀ a, p a → f a = b → q b) ↔ ∀ a, p a → q (f a) :=\n⟨λ h a ha, h (f a) a ha rfl, λ h b a ha hb, hb ▸ h a ha⟩\n\n@[simp] theorem exists_eq_left' {a' : α} : (∃ a, a' = a ∧ p a) ↔ p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_right' {a' : α} : (∃ a, p a ∧ a' = a) ↔ p a' :=\nby simp [@eq_comm _ a']\n\ntheorem exists_comm {p : α → β → Prop} : (∃ a b, p a b) ↔ ∃ b a, p a b :=\n⟨λ ⟨a, b, h⟩, ⟨b, a, h⟩, λ ⟨b, a, h⟩, ⟨a, b, h⟩⟩\n\nlemma exists₂_comm {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}\n  {p : Π i₁, κ₁ i₁ → Π i₂, κ₂ i₂ → Prop} :\n  (∃ i₁ j₁ i₂ j₂, p i₁ j₁ i₂ j₂) ↔ ∃ i₂ j₂ i₁ j₁, p i₁ j₁ i₂ j₂ :=\nby simp only [@exists_comm (κ₁ _), @exists_comm ι₁]\n\ntheorem and.exists {p q : Prop} {f : p ∧ q → Prop} : (∃ h, f h) ↔ ∃ hp hq, f ⟨hp, hq⟩ :=\n⟨λ ⟨h, H⟩, ⟨h.1, h.2, H⟩, λ ⟨hp, hq, H⟩, ⟨⟨hp, hq⟩, H⟩⟩\n\ntheorem forall_or_of_or_forall (h : b ∨ ∀x, p x) (x) : b ∨ p x :=\nh.imp_right $ λ h₂, h₂ x\n\n-- See Note [decidable namespace]\nprotected theorem decidable.forall_or_distrib_left {q : Prop} {p : α → Prop} [decidable q] :\n  (∀x, q ∨ p x) ↔ q ∨ (∀x, p x) :=\n⟨λ h, if hq : q then or.inl hq else or.inr $ λ x, (h x).resolve_left hq,\n  forall_or_of_or_forall⟩\n\ntheorem forall_or_distrib_left {q : Prop} {p : α → Prop} :\n  (∀x, q ∨ p x) ↔ q ∨ (∀x, p x) := decidable.forall_or_distrib_left\n\n-- See Note [decidable namespace]\nprotected theorem decidable.forall_or_distrib_right {q : Prop} {p : α → Prop} [decidable q] :\n  (∀x, p x ∨ q) ↔ (∀x, p x) ∨ q :=\nby simp [or_comm, decidable.forall_or_distrib_left]\n\ntheorem forall_or_distrib_right {q : Prop} {p : α → Prop} :\n  (∀x, p x ∨ q) ↔ (∀x, p x) ∨ q := decidable.forall_or_distrib_right\n\n@[simp] theorem exists_prop {p q : Prop} : (∃ h : p, q) ↔ p ∧ q :=\n⟨λ ⟨h₁, h₂⟩, ⟨h₁, h₂⟩, λ ⟨h₁, h₂⟩, ⟨h₁, h₂⟩⟩\n\ntheorem exists_unique_prop {p q : Prop} : (∃! h : p, q) ↔ p ∧ q :=\nby simp\n\n@[simp] theorem exists_false : ¬ (∃a:α, false) := assume ⟨a, h⟩, h\n\n@[simp] lemma exists_unique_false : ¬ (∃! (a : α), false) := assume ⟨a, h, h'⟩, h\n\ntheorem Exists.fst {p : b → Prop} : Exists p → b\n| ⟨h, _⟩ := h\n\ntheorem Exists.snd {p : b → Prop} : ∀ h : Exists p, p h.fst\n| ⟨_, h⟩ := h\n\ntheorem forall_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∀ h' : p, q h') ↔ q h :=\n@forall_const (q h) p ⟨h⟩\n\ntheorem exists_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃ h' : p, q h') ↔ q h :=\n@exists_const (q h) p ⟨h⟩\n\nlemma exists_iff_of_forall {p : Prop} {q : p → Prop} (h : ∀ h, q h) : (∃ h, q h) ↔ p :=\n⟨Exists.fst, λ H, ⟨H, h H⟩⟩\n\ntheorem exists_unique_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃! h' : p, q h') ↔ q h :=\n@exists_unique_const (q h) p ⟨h⟩ _\n\ntheorem forall_prop_of_false {p : Prop} {q : p → Prop} (hn : ¬ p) :\n  (∀ h' : p, q h') ↔ true :=\niff_true_intro $ λ h, hn.elim h\n\ntheorem exists_prop_of_false {p : Prop} {q : p → Prop} : ¬ p → ¬ (∃ h' : p, q h') :=\nmt Exists.fst\n\n@[congr] lemma exists_prop_congr {p p' : Prop} {q q' : p → Prop}\n  (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') : Exists q ↔ ∃ h : p', q' (hp.2 h) :=\n⟨λ ⟨_, _⟩, ⟨hp.1 ‹_›, (hq _).1 ‹_›⟩, λ ⟨_, _⟩, ⟨_, (hq _).2 ‹_›⟩⟩\n\n@[congr] lemma exists_prop_congr' {p p' : Prop} {q q' : p → Prop}\n  (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') : Exists q = ∃ h : p', q' (hp.2 h) :=\npropext (exists_prop_congr hq _)\n\n/-- See `is_empty.exists_iff` for the `false` version. -/\n@[simp] lemma exists_true_left (p : true → Prop) : (∃ x, p x) ↔ p true.intro :=\nexists_prop_of_true _\n\nlemma exists_unique.unique {α : Sort*} {p : α → Prop} (h : ∃! x, p x)\n  {y₁ y₂ : α} (py₁ : p y₁) (py₂ : p y₂) : y₁ = y₂ :=\nunique_of_exists_unique h py₁ py₂\n\n@[congr] lemma forall_prop_congr {p p' : Prop} {q q' : p → Prop}\n  (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') : (∀ h, q h) ↔ ∀ h : p', q' (hp.2 h) :=\n⟨λ h1 h2, (hq _).1 (h1 (hp.2 _)), λ h1 h2, (hq _).2 (h1 (hp.1 h2))⟩\n\n@[congr] lemma forall_prop_congr' {p p' : Prop} {q q' : p → Prop}\n  (hq : ∀ h, q h ↔ q' h) (hp : p ↔ p') : (∀ h, q h) = ∀ h : p', q' (hp.2 h) :=\npropext (forall_prop_congr hq _)\n\n/-- See `is_empty.forall_iff` for the `false` version. -/\n@[simp] lemma forall_true_left (p : true → Prop) : (∀ x, p x) ↔ p true.intro :=\nforall_prop_of_true _\n\nlemma exists_unique.elim2 {α : Sort*} {p : α → Sort*} [∀ x, subsingleton (p x)]\n  {q : Π x (h : p x), Prop} {b : Prop} (h₂ : ∃! x (h : p x), q x h)\n  (h₁ : ∀ x (h : p x), q x h → (∀ y (hy : p y), q y hy → y = x) → b) : b :=\nbegin\n  simp only [exists_unique_iff_exists] at h₂,\n  apply h₂.elim,\n  exact λ x ⟨hxp, hxq⟩ H, h₁ x hxp hxq (λ y hyp hyq, H y ⟨hyp, hyq⟩)\nend\n\nlemma exists_unique.intro2 {α : Sort*} {p : α → Sort*} [∀ x, subsingleton (p x)]\n  {q : Π (x : α) (h : p x), Prop} (w : α) (hp : p w) (hq : q w hp)\n  (H : ∀ y (hy : p y), q y hy → y = w) :\n  ∃! x (hx : p x), q x hx :=\nbegin\n  simp only [exists_unique_iff_exists],\n  exact exists_unique.intro w ⟨hp, hq⟩ (λ y ⟨hyp, hyq⟩, H y hyp hyq)\nend\n\nlemma exists_unique.exists2 {α : Sort*} {p : α → Sort*} {q : Π (x : α) (h : p x), Prop}\n  (h : ∃! x (hx : p x), q x hx) :\n  ∃ x (hx : p x), q x hx :=\nh.exists.imp (λ x hx, hx.exists)\n\nlemma exists_unique.unique2 {α : Sort*} {p : α → Sort*} [∀ x, subsingleton (p x)]\n  {q : Π (x : α) (hx : p x), Prop} (h : ∃! x (hx : p x), q x hx)\n  {y₁ y₂ : α} (hpy₁ : p y₁) (hqy₁ : q y₁ hpy₁)\n  (hpy₂ : p y₂) (hqy₂ : q y₂ hpy₂) : y₁ = y₂ :=\nbegin\n  simp only [exists_unique_iff_exists] at h,\n  exact h.unique ⟨hpy₁, hqy₁⟩ ⟨hpy₂, hqy₂⟩\nend\n\nend quantifiers\n\n/-! ### Classical lemmas -/\n\nnamespace classical\nvariables {α : Sort*} {p : α → Prop}\n\ntheorem cases {p : Prop → Prop} (h1 : p true) (h2 : p false) : ∀a, p a :=\nassume a, cases_on a h1 h2\n\n/- use shortened names to avoid conflict when classical namespace is open. -/\n/-- Any prop `p` is decidable classically. A shorthand for `classical.prop_decidable`. -/\nnoncomputable def dec (p : Prop) : decidable p :=\nby apply_instance\n/-- Any predicate `p` is decidable classically. -/\nnoncomputable def dec_pred (p : α → Prop) : decidable_pred p :=\nby apply_instance\n/-- Any relation `p` is decidable classically. -/\nnoncomputable def dec_rel (p : α → α → Prop) : decidable_rel p :=\nby apply_instance\n/-- Any type `α` has decidable equality classically. -/\nnoncomputable def dec_eq (α : Sort*) : decidable_eq α :=\nby apply_instance\n\n/-- Construct a function from a default value `H0`, and a function to use if there exists a value\nsatisfying the predicate. -/\n@[elab_as_eliminator]\nnoncomputable def {u} exists_cases {C : Sort u} (H0 : C) (H : ∀ a, p a → C) : C :=\nif h : ∃ a, p a then H (classical.some h) (classical.some_spec h) else H0\n\nlemma some_spec2 {α : Sort*} {p : α → Prop} {h : ∃a, p a}\n  (q : α → Prop) (hpq : ∀a, p a → q a) : q (some h) :=\nhpq _ $ some_spec _\n\n/-- A version of classical.indefinite_description which is definitionally equal to a pair -/\nnoncomputable def subtype_of_exists {α : Type*} {P : α → Prop} (h : ∃ x, P x) : {x // P x} :=\n⟨classical.some h, classical.some_spec h⟩\n\n/-- A version of `by_contradiction` that uses types instead of propositions. -/\nprotected noncomputable def by_contradiction' {α : Sort*} (H : ¬ (α → false)) : α :=\nclassical.choice $ peirce _ false $ λ h, (H $ λ a, h ⟨a⟩).elim\n\n/-- `classical.by_contradiction'` is equivalent to lean's axiom `classical.choice`. -/\ndef choice_of_by_contradiction' {α : Sort*} (contra : ¬ (α → false) → α) : nonempty α → α :=\nλ H, contra H.elim\n\nend classical\n\n/-- This function has the same type as `exists.rec_on`, and can be used to case on an equality,\nbut `exists.rec_on` can only eliminate into Prop, while this version eliminates into any universe\nusing the axiom of choice. -/\n@[elab_as_eliminator]\nnoncomputable def {u} exists.classical_rec_on\n {α} {p : α → Prop} (h : ∃ a, p a) {C : Sort u} (H : ∀ a, p a → C) : C :=\nH (classical.some h) (classical.some_spec h)\n\n/-! ### Declarations about bounded quantifiers -/\n\nsection bounded_quantifiers\nvariables {α : Sort*} {r p q : α → Prop} {P Q : ∀ x, p x → Prop} {b : Prop}\n\ntheorem bex_def : (∃ x (h : p x), q x) ↔ ∃ x, p x ∧ q x :=\n⟨λ ⟨x, px, qx⟩, ⟨x, px, qx⟩, λ ⟨x, px, qx⟩, ⟨x, px, qx⟩⟩\n\ntheorem bex.elim {b : Prop} : (∃ x h, P x h) → (∀ a h, P a h → b) → b\n| ⟨a, h₁, h₂⟩ h' := h' a h₁ h₂\n\ntheorem bex.intro (a : α) (h₁ : p a) (h₂ : P a h₁) : ∃ x (h : p x), P x h :=\n⟨a, h₁, h₂⟩\n\ntheorem ball_congr (H : ∀ x h, P x h ↔ Q x h) :\n  (∀ x h, P x h) ↔ (∀ x h, Q x h) :=\nforall_congr $ λ x, forall_congr (H x)\n\ntheorem bex_congr (H : ∀ x h, P x h ↔ Q x h) :\n  (∃ x h, P x h) ↔ (∃ x h, Q x h) :=\nexists_congr $ λ x, exists_congr (H x)\n\ntheorem bex_eq_left {a : α} : (∃ x (_ : x = a), p x) ↔ p a :=\nby simp only [exists_prop, exists_eq_left]\n\ntheorem ball.imp_right (H : ∀ x h, (P x h → Q x h))\n  (h₁ : ∀ x h, P x h) (x h) : Q x h :=\nH _ _ $ h₁ _ _\n\ntheorem bex.imp_right (H : ∀ x h, (P x h → Q x h)) :\n  (∃ x h, P x h) → ∃ x h, Q x h\n| ⟨x, h, h'⟩ := ⟨_, _, H _ _ h'⟩\n\ntheorem ball.imp_left (H : ∀ x, p x → q x)\n  (h₁ : ∀ x, q x → r x) (x) (h : p x) : r x :=\nh₁ _ $ H _ h\n\ntheorem bex.imp_left (H : ∀ x, p x → q x) :\n  (∃ x (_ : p x), r x) → ∃ x (_ : q x), r x\n| ⟨x, hp, hr⟩ := ⟨x, H _ hp, hr⟩\n\ntheorem ball_of_forall (h : ∀ x, p x) (x) : p x :=\nh x\n\ntheorem forall_of_ball (H : ∀ x, p x) (h : ∀ x, p x → q x) (x) : q x :=\nh x $ H x\n\ntheorem bex_of_exists (H : ∀ x, p x) : (∃ x, q x) → ∃ x (_ : p x), q x\n| ⟨x, hq⟩ := ⟨x, H x, hq⟩\n\ntheorem exists_of_bex : (∃ x (_ : p x), q x) → ∃ x, q x\n| ⟨x, _, hq⟩ := ⟨x, hq⟩\n\n@[simp] theorem bex_imp_distrib : ((∃ x h, P x h) → b) ↔ (∀ x h, P x h → b) :=\nby simp\n\ntheorem not_bex : (¬ ∃ x h, P x h) ↔ ∀ x h, ¬ P x h :=\nbex_imp_distrib\n\ntheorem not_ball_of_bex_not : (∃ x h, ¬ P x h) → ¬ ∀ x h, P x h\n| ⟨x, h, hp⟩ al := hp $ al x h\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_ball [decidable (∃ x h, ¬ P x h)] [∀ x h, decidable (P x h)] :\n  (¬ ∀ x h, P x h) ↔ (∃ x h, ¬ P x h) :=\n⟨not.decidable_imp_symm $ λ nx x h, nx.decidable_imp_symm $ λ h', ⟨x, h, h'⟩,\n not_ball_of_bex_not⟩\n\ntheorem not_ball : (¬ ∀ x h, P x h) ↔ (∃ x h, ¬ P x h) := decidable.not_ball\n\ntheorem ball_true_iff (p : α → Prop) : (∀ x, p x → true) ↔ true :=\niff_true_intro (λ h hrx, trivial)\n\ntheorem ball_and_distrib : (∀ x h, P x h ∧ Q x h) ↔ (∀ x h, P x h) ∧ (∀ x h, Q x h) :=\niff.trans (forall_congr $ λ x, forall_and_distrib) forall_and_distrib\n\ntheorem bex_or_distrib : (∃ x h, P x h ∨ Q x h) ↔ (∃ x h, P x h) ∨ (∃ x h, Q x h) :=\niff.trans (exists_congr $ λ x, exists_or_distrib) exists_or_distrib\n\ntheorem ball_or_left_distrib : (∀ x, p x ∨ q x → r x) ↔ (∀ x, p x → r x) ∧ (∀ x, q x → r x) :=\niff.trans (forall_congr $ λ x, or_imp_distrib) forall_and_distrib\n\ntheorem bex_or_left_distrib :\n  (∃ x (_ : p x ∨ q x), r x) ↔ (∃ x (_ : p x), r x) ∨ (∃ x (_ : q x), r x) :=\nby simp only [exists_prop]; exact\niff.trans (exists_congr $ λ x, or_and_distrib_right) exists_or_distrib\n\nend bounded_quantifiers\n\nnamespace classical\nlocal attribute [instance] prop_decidable\n\ntheorem not_ball {α : Sort*} {p : α → Prop} {P : Π (x : α), p x → Prop} :\n  (¬ ∀ x h, P x h) ↔ (∃ x h, ¬ P x h) := _root_.not_ball\n\nend classical\n\nsection ite\nvariables {α β γ : Sort*} {σ : α → Sort*} (f : α → β) {P Q : Prop} [decidable P] [decidable Q]\n  {a b c : α} {A : P → α} {B : ¬ P → α}\n\nlemma dite_eq_iff : dite P A B = c ↔ (∃ h, A h = c) ∨ ∃ h, B h = c :=\nby by_cases P; simp [*, exists_prop_of_false not_false]\n\nlemma ite_eq_iff : ite P a b = c ↔ P ∧ a = c ∨ ¬ P ∧ b = c :=\ndite_eq_iff.trans $ by rw [exists_prop, exists_prop]\n\nlemma dite_eq_iff' : dite P A B = c ↔ (∀ h, A h = c) ∧ (∀ h, B h = c) :=\n⟨λ he, ⟨λ h, (dif_pos h).symm.trans he, λ h, (dif_neg h).symm.trans he⟩,\n  λ he, (em P).elim (λ h, (dif_pos h).trans $ he.1 h) (λ h, (dif_neg h).trans $ he.2 h)⟩\n\nlemma ite_eq_iff' : ite P a b = c ↔ (P → a = c) ∧ (¬ P → b = c) := dite_eq_iff'\n\n@[simp] lemma dite_eq_left_iff : dite P (λ _, a) B = a ↔ ∀ h, B h = a :=\nby by_cases P; simp [*, forall_prop_of_false not_false]\n\n@[simp] lemma dite_eq_right_iff : dite P A (λ _, b) = b ↔ ∀ h, A h = b :=\nby by_cases P; simp [*, forall_prop_of_false not_false]\n\n@[simp] lemma ite_eq_left_iff : ite P a b = a ↔ (¬ P → b = a) := dite_eq_left_iff\n@[simp] lemma ite_eq_right_iff : ite P a b = b ↔ (P → a = b) := dite_eq_right_iff\n\nlemma dite_ne_left_iff : dite P (λ _, a) B ≠ a ↔ ∃ h, a ≠ B h :=\nby { rw [ne.def, dite_eq_left_iff, not_forall], exact exists_congr (λ h, by rw ne_comm) }\n\nlemma dite_ne_right_iff : dite P A (λ _, b) ≠ b ↔ ∃ h, A h ≠ b :=\nby simp only [ne.def, dite_eq_right_iff, not_forall]\n\nlemma ite_ne_left_iff : ite P a b ≠ a ↔ ¬ P ∧ a ≠ b := dite_ne_left_iff.trans $ by rw exists_prop\nlemma ite_ne_right_iff : ite P a b ≠ b ↔ P ∧ a ≠ b := dite_ne_right_iff.trans $ by rw exists_prop\n\nprotected lemma ne.dite_eq_left_iff (h : ∀ h, a ≠ B h) : dite P (λ _, a) B = a ↔ P :=\ndite_eq_left_iff.trans $ ⟨λ H, of_not_not $ λ h', h h' (H h').symm, λ h H, (H h).elim⟩\n\nprotected lemma ne.dite_eq_right_iff (h : ∀ h, A h ≠ b) : dite P A (λ _, b) = b ↔ ¬ P :=\ndite_eq_right_iff.trans $ ⟨λ H h', h h' (H h'), λ h' H, (h' H).elim⟩\n\nprotected lemma ne.ite_eq_left_iff (h : a ≠ b) : ite P a b = a ↔ P := ne.dite_eq_left_iff $ λ _, h\nprotected lemma ne.ite_eq_right_iff (h : a ≠ b) : ite P a b = b ↔ ¬ P :=\nne.dite_eq_right_iff $ λ _, h\n\nprotected lemma ne.dite_ne_left_iff (h : ∀ h, a ≠ B h) : dite P (λ _, a) B ≠ a ↔ ¬ P :=\ndite_ne_left_iff.trans $ exists_iff_of_forall h\n\nprotected lemma ne.dite_ne_right_iff (h : ∀ h, A h ≠ b) : dite P A (λ _, b) ≠ b ↔ P :=\ndite_ne_right_iff.trans $ exists_iff_of_forall h\n\nprotected lemma ne.ite_ne_left_iff (h : a ≠ b) : ite P a b ≠ a ↔ ¬ P := ne.dite_ne_left_iff $ λ _, h\n\nprotected lemma ne.ite_ne_right_iff (h : a ≠ b) : ite P a b ≠ b ↔ P := ne.dite_ne_right_iff $ λ _, h\n\nvariables (P Q) (a b)\n\n/-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/\n@[simp] lemma dite_eq_ite : dite P (λ h, a) (λ h, b) = ite P a b := rfl\n\nlemma dite_eq_or_eq : (∃ h, dite P A B = A h) ∨ ∃ h, dite P A B = B h :=\ndecidable.by_cases (λ h, or.inl ⟨h, dif_pos h⟩) (λ h, or.inr ⟨h, dif_neg h⟩)\n\nlemma ite_eq_or_eq : ite P a b = a ∨ ite P a b = b :=\ndecidable.by_cases (λ h, or.inl (if_pos h)) (λ h, or.inr (if_neg h))\n\n/-- A function applied to a `dite` is a `dite` of that function applied to each of the branches. -/\nlemma apply_dite (x : P → α) (y : ¬P → α) : f (dite P x y) = dite P (λ h, f (x h)) (λ h, f (y h)) :=\nby by_cases h : P; simp [h]\n\n/-- A function applied to a `ite` is a `ite` of that function applied to each of the branches. -/\nlemma apply_ite : f (ite P a b) = ite P (f a) (f b) := apply_dite f P (λ _, a) (λ _, b)\n\n/-- A two-argument function applied to two `dite`s is a `dite` of that two-argument function\napplied to each of the branches. -/\nlemma apply_dite2 (f : α → β → γ) (P : Prop) [decidable P] (a : P → α) (b : ¬P → α) (c : P → β)\n  (d : ¬P → β) :\n  f (dite P a b) (dite P c d) = dite P (λ h, f (a h) (c h)) (λ h, f (b h) (d h)) :=\nby by_cases h : P; simp [h]\n\n/-- A two-argument function applied to two `ite`s is a `ite` of that two-argument function\napplied to each of the branches. -/\nlemma apply_ite2 (f : α → β → γ) (P : Prop) [decidable P] (a b : α) (c d : β) :\n  f (ite P a b) (ite P c d) = ite P (f a c) (f b d) :=\napply_dite2 f P (λ _, a) (λ _, b) (λ _, c) (λ _, d)\n\n/-- A 'dite' producing a `Pi` type `Π a, σ a`, applied to a value `a : α` is a `dite` that applies\neither branch to `a`. -/\nlemma dite_apply (f : P → Π a, σ a) (g : ¬ P → Π a, σ a) (a : α) :\n  (dite P f g) a = dite P (λ h, f h a) (λ h, g h a) :=\nby by_cases h : P; simp [h]\n\n/-- A 'ite' producing a `Pi` type `Π a, σ a`, applied to a value `a : α` is a `ite` that applies\neither branch to `a`. -/\nlemma ite_apply (f g : Π a, σ a) (a : α) : (ite P f g) a = ite P (f a) (g a) :=\ndite_apply P (λ _, f) (λ _, g) a\n\n/-- Negation of the condition `P : Prop` in a `dite` is the same as swapping the branches. -/\n@[simp] lemma dite_not (x : ¬ P → α) (y : ¬¬ P → α) :\n  dite (¬ P) x y = dite P (λ h, y (not_not_intro h)) x :=\nby by_cases h : P; simp [h]\n\n/-- Negation of the condition `P : Prop` in a `ite` is the same as swapping the branches. -/\n@[simp] lemma ite_not : ite (¬ P) a b = ite P b a := dite_not P (λ _, a) (λ _, b)\n\nlemma ite_and : ite (P ∧ Q) a b = ite P (ite Q a b) b :=\nby by_cases hp : P; by_cases hq : Q; simp [hp, hq]\n\nlemma dite_dite_comm {B : Q → α} {C : ¬P → ¬Q → α} (h : P → ¬Q) :\n  (if p : P then A p else if q : Q then B q else C p q) =\n  (if q : Q then B q else if p : P then A p else C p q) :=\ndite_eq_iff'.2 ⟨λ p, by rw [dif_neg (h p), dif_pos p], λ np, by { congr, funext, rw dif_neg np }⟩\n\nlemma ite_ite_comm (h : P → ¬Q) :\n  (if P then a else if Q then b else c) =\n  (if Q then b else if P then a else c) :=\ndite_dite_comm P Q h\n\nend ite\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/logic/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6619228825191871, "lm_q1q2_score": 0.333547024915814}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta, Adam Topaz\n-/\nimport category_theory.functor.category\nimport category_theory.functor.fully_faithful\nimport category_theory.functor.reflects_isomorphisms\n\n/-!\n# Monads\n\nWe construct the categories of monads and comonads, and their forgetful functors to endofunctors.\n\n(Note that these are the category theorist's monads, not the programmers monads.\nFor the translation, see the file `category_theory.monad.types`.)\n\nFor the fact that monads are \"just\" monoids in the category of endofunctors, see the file\n`category_theory.monad.equiv_mon`.\n-/\n\nnamespace category_theory\nopen category\n\nuniverses v₁ u₁ -- morphism levels before object levels. See note [category_theory universes].\n\nvariables (C : Type u₁) [category.{v₁} C]\n\n/--\nThe data of a monad on C consists of an endofunctor T together with natural transformations\nη : 𝟭 C ⟶ T and μ : T ⋙ T ⟶ T satisfying three equations:\n- T μ_X ≫ μ_X = μ_(TX) ≫ μ_X (associativity)\n- η_(TX) ≫ μ_X = 1_X (left unit)\n- Tη_X ≫ μ_X = 1_X (right unit)\n-/\nstructure monad extends C ⥤ C :=\n(η' [] : 𝟭 _ ⟶ to_functor)\n(μ' [] : to_functor ⋙ to_functor ⟶ to_functor)\n(assoc' : ∀ X, to_functor.map (nat_trans.app μ' X) ≫ μ'.app _ = μ'.app _ ≫ μ'.app _ . obviously)\n(left_unit' : ∀ X : C, η'.app (to_functor.obj X) ≫ μ'.app _ = 𝟙 _ . obviously)\n(right_unit' : ∀ X : C, to_functor.map (η'.app X) ≫ μ'.app _ = 𝟙 _ . obviously)\n\n/--\nThe data of a comonad on C consists of an endofunctor G together with natural transformations\nε : G ⟶ 𝟭 C and δ : G ⟶ G ⋙ G satisfying three equations:\n- δ_X ≫ G δ_X = δ_X ≫ δ_(GX) (coassociativity)\n- δ_X ≫ ε_(GX) = 1_X (left counit)\n- δ_X ≫ G ε_X = 1_X (right counit)\n-/\nstructure comonad extends C ⥤ C :=\n(ε' [] : to_functor ⟶ 𝟭 _)\n(δ' [] : to_functor ⟶ to_functor ⋙ to_functor)\n(coassoc' : ∀ X, nat_trans.app δ' _ ≫ to_functor.map (δ'.app X) = δ'.app _ ≫ δ'.app _ . obviously)\n(left_counit' : ∀ X : C, δ'.app X ≫ ε'.app (to_functor.obj X) = 𝟙 _ . obviously)\n(right_counit' : ∀ X : C, δ'.app X ≫ to_functor.map (ε'.app X) = 𝟙 _ . obviously)\n\nvariables {C} (T : monad C) (G : comonad C)\n\ninstance coe_monad : has_coe (monad C) (C ⥤ C) := ⟨λ T, T.to_functor⟩\ninstance coe_comonad : has_coe (comonad C) (C ⥤ C) := ⟨λ G, G.to_functor⟩\n\n@[simp] lemma monad_to_functor_eq_coe : T.to_functor = T := rfl\n@[simp] lemma comonad_to_functor_eq_coe : G.to_functor = G := rfl\n\n/-- The unit for the monad `T`. -/\ndef monad.η : 𝟭 _ ⟶ (T : C ⥤ C) := T.η'\n/-- The multiplication for the monad `T`. -/\ndef monad.μ : (T : C ⥤ C) ⋙ (T : C ⥤ C) ⟶ T := T.μ'\n\n/-- The counit for the comonad `G`. -/\ndef comonad.ε : (G : C ⥤ C) ⟶ 𝟭 _  := G.ε'\n/-- The comultiplication for the comonad `G`. -/\ndef comonad.δ : (G : C ⥤ C) ⟶ (G : C ⥤ C) ⋙ G := G.δ'\n\n/-- A custom simps projection for the functor part of a monad, as a coercion. -/\ndef monad.simps.coe := (T : C ⥤ C)\n/-- A custom simps projection for the unit of a monad, in simp normal form. -/\ndef monad.simps.η : 𝟭 _ ⟶ (T : C ⥤ C) := T.η\n/-- A custom simps projection for the multiplication of a monad, in simp normal form. -/\ndef monad.simps.μ : (T : C ⥤ C) ⋙ (T : C ⥤ C) ⟶ (T : C ⥤ C) := T.μ\n\n/-- A custom simps projection for the functor part of a comonad, as a coercion. -/\ndef comonad.simps.coe := (G : C ⥤ C)\n/-- A custom simps projection for the counit of a comonad, in simp normal form. -/\ndef comonad.simps.ε : (G : C ⥤ C) ⟶ 𝟭 _ := G.ε\n/-- A custom simps projection for the comultiplication of a comonad, in simp normal form. -/\ndef comonad.simps.δ : (G : C ⥤ C) ⟶ (G : C ⥤ C) ⋙ (G : C ⥤ C) := G.δ\n\ninitialize_simps_projections category_theory.monad (to_functor → coe, η' → η, μ' → μ)\ninitialize_simps_projections category_theory.comonad (to_functor → coe, ε' → ε, δ' → δ)\n\n@[reassoc]\nlemma monad.assoc (T : monad C) (X : C) :\n  (T : C ⥤ C).map (T.μ.app X) ≫ T.μ.app _ = T.μ.app _ ≫ T.μ.app _ :=\nT.assoc' X\n\n@[simp, reassoc] lemma monad.left_unit (T : monad C) (X : C) :\n  T.η.app ((T : C ⥤ C).obj X) ≫ T.μ.app X = 𝟙 ((T : C ⥤ C).obj X) :=\nT.left_unit' X\n\n@[simp, reassoc] lemma monad.right_unit (T : monad C) (X : C) :\n  (T : C ⥤ C).map (T.η.app X) ≫ T.μ.app X = 𝟙 ((T : C ⥤ C).obj X) :=\nT.right_unit' X\n\n@[reassoc]\nlemma comonad.coassoc (G : comonad C) (X : C) :\n  G.δ.app _ ≫ (G : C ⥤ C).map (G.δ.app X) = G.δ.app _ ≫ G.δ.app _ :=\nG.coassoc' X\n\n@[simp, reassoc] lemma comonad.left_counit (G : comonad C) (X : C) :\n  G.δ.app X ≫ G.ε.app ((G : C ⥤ C).obj X) = 𝟙 ((G : C ⥤ C).obj X) :=\nG.left_counit' X\n\n@[simp, reassoc] lemma comonad.right_counit (G : comonad C) (X : C) :\n  G.δ.app X ≫ (G : C ⥤ C).map (G.ε.app X) = 𝟙 ((G : C ⥤ C).obj X) :=\nG.right_counit' X\n\n/-- A morphism of monads is a natural transformation compatible with η and μ. -/\n@[ext]\nstructure monad_hom (T₁ T₂ : monad C) extends nat_trans (T₁ : C ⥤ C) T₂ :=\n(app_η' : ∀ X, T₁.η.app X ≫ app X = T₂.η.app X . obviously)\n(app_μ' : ∀ X, T₁.μ.app X ≫ app X = ((T₁ : C ⥤ C).map (app X) ≫ app _) ≫ T₂.μ.app X . obviously)\n\n/-- A morphism of comonads is a natural transformation compatible with ε and δ. -/\n@[ext]\nstructure comonad_hom (M N : comonad C) extends nat_trans (M : C ⥤ C) N :=\n(app_ε' : ∀ X, app X ≫ N.ε.app X = M.ε.app X . obviously)\n(app_δ' : ∀ X, app X ≫ N.δ.app X = M.δ.app X ≫ app _ ≫ (N : C ⥤ C).map (app X) . obviously)\n\nrestate_axiom monad_hom.app_η'\nrestate_axiom monad_hom.app_μ'\nattribute [simp, reassoc] monad_hom.app_η monad_hom.app_μ\n\nrestate_axiom comonad_hom.app_ε'\nrestate_axiom comonad_hom.app_δ'\nattribute [simp, reassoc] comonad_hom.app_ε comonad_hom.app_δ\n\ninstance : category (monad C) :=\n{ hom := monad_hom,\n  id := λ M, { to_nat_trans := 𝟙 (M : C ⥤ C) },\n  comp := λ _ _ _ f g,\n  { to_nat_trans := { app := λ X, f.app X ≫ g.app X } } }\n\ninstance : category (comonad C) :=\n{ hom := comonad_hom,\n  id := λ M, { to_nat_trans := 𝟙 (M : C ⥤ C) },\n  comp := λ M N L f g,\n  { to_nat_trans := { app := λ X, f.app X ≫ g.app X } } }\n\ninstance {T : monad C} : inhabited (monad_hom T T) := ⟨𝟙 T⟩\n\n@[simp] \n\ninstance {G : comonad C} : inhabited (comonad_hom G G) := ⟨𝟙 G⟩\n\n@[simp] lemma comonad_hom.id_to_nat_trans (T : comonad C) :\n  (𝟙 T : T ⟶ T).to_nat_trans = 𝟙 (T : C ⥤ C) :=\nrfl\n@[simp] lemma comp_to_nat_trans {T₁ T₂ T₃ : comonad C} (f : T₁ ⟶ T₂) (g : T₂ ⟶ T₃) :\n  (f ≫ g).to_nat_trans =\n    ((f.to_nat_trans : _ ⟶ (T₂ : C ⥤ C)) ≫ g.to_nat_trans : (T₁ : C ⥤ C) ⟶ T₃) :=\nrfl\n\n/-- Construct a monad isomorphism from a natural isomorphism of functors where the forward\ndirection is a monad morphism. -/\n@[simps]\ndef monad_iso.mk {M N : monad C} (f : (M : C ⥤ C) ≅ N) (f_η f_μ) :\n  M ≅ N :=\n{ hom := { to_nat_trans := f.hom, app_η' := f_η, app_μ' := f_μ },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_η' := λ X, by simp [←f_η],\n    app_μ' := λ X,\n    begin\n      rw ←nat_iso.cancel_nat_iso_hom_right f,\n      simp only [nat_trans.naturality, iso.inv_hom_id_app, assoc, comp_id, f_μ,\n        nat_trans.naturality_assoc, iso.inv_hom_id_app_assoc, ←functor.map_comp_assoc],\n      simp,\n    end } }\n\n/-- Construct a comonad isomorphism from a natural isomorphism of functors where the forward\ndirection is a comonad morphism. -/\n@[simps]\ndef comonad_iso.mk {M N : comonad C} (f : (M : C ⥤ C) ≅ N) (f_ε f_δ) :\n  M ≅ N :=\n{ hom := { to_nat_trans := f.hom, app_ε' := f_ε, app_δ' := f_δ },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_ε' := λ X, by simp [←f_ε],\n    app_δ' := λ X,\n    begin\n      rw ←nat_iso.cancel_nat_iso_hom_left f,\n      simp only [reassoc_of (f_δ X), iso.hom_inv_id_app_assoc, nat_trans.naturality_assoc],\n      rw [←functor.map_comp, iso.hom_inv_id_app, functor.map_id],\n      apply (comp_id _).symm\n    end } }\n\nvariable (C)\n\n/--\nThe forgetful functor from the category of monads to the category of endofunctors.\n-/\n@[simps]\ndef monad_to_functor : monad C ⥤ (C ⥤ C) :=\n{ obj := λ T, T,\n  map := λ M N f, f.to_nat_trans }\n\ninstance : faithful (monad_to_functor C) := {}.\n\n@[simp]\nlemma monad_to_functor_map_iso_monad_iso_mk {M N : monad C} (f : (M : C ⥤ C) ≅ N) (f_η f_μ) :\n  (monad_to_functor _).map_iso (monad_iso.mk f f_η f_μ) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (monad_to_functor C) :=\n{ reflects := λ M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (monad_iso.mk (as_iso ((monad_to_functor C).map f)) f.app_η f.app_μ),\n    ext; refl,\n  end }\n\n/--\nThe forgetful functor from the category of comonads to the category of endofunctors.\n-/\n@[simps]\ndef comonad_to_functor : comonad C ⥤ (C ⥤ C) :=\n{ obj := λ G, G,\n  map := λ M N f, f.to_nat_trans }\n\ninstance : faithful (comonad_to_functor C) := {}.\n\n@[simp]\nlemma comonad_to_functor_map_iso_comonad_iso_mk {M N : comonad C} (f : (M : C ⥤ C) ≅ N) (f_ε f_δ) :\n  (comonad_to_functor _).map_iso (comonad_iso.mk f f_ε f_δ) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (comonad_to_functor C) :=\n{ reflects := λ M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (comonad_iso.mk (as_iso ((comonad_to_functor C).map f)) f.app_ε f.app_δ),\n    ext; refl,\n  end }\n\nvariable {C}\n\n/--\nAn isomorphism of monads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef monad_iso.to_nat_iso {M N : monad C} (h : M ≅ N) : (M : C ⥤ C) ≅ N :=\n(monad_to_functor C).map_iso h\n\n/--\nAn isomorphism of comonads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef comonad_iso.to_nat_iso {M N : comonad C} (h : M ≅ N) : (M : C ⥤ C) ≅ N :=\n(comonad_to_functor C).map_iso h\n\nvariable (C)\n\nnamespace monad\n\n/-- The identity monad. -/\n@[simps]\ndef id : monad C :=\n{ to_functor := 𝟭 C,\n  η' := 𝟙 (𝟭 C),\n  μ' := 𝟙 (𝟭 C) }\n\ninstance : inhabited (monad C) := ⟨monad.id C⟩\n\nend monad\n\nnamespace comonad\n\n/-- The identity comonad. -/\n@[simps]\ndef id : comonad C :=\n{ to_functor := 𝟭 _,\n  ε' := 𝟙 (𝟭 C),\n  δ' := 𝟙 (𝟭 C) }\n\ninstance : inhabited (comonad C) := ⟨comonad.id C⟩\n\nend comonad\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/monad/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3335122710777824}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport data.multiset.nodup\n\n/-!\n# The cartesian product of multisets\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nnamespace multiset\n\nsection pi\nvariables {α : Type*}\nopen function\n\n/-- Given `δ : α → Type*`, `pi.empty δ` is the trivial dependent function out of the empty\nmultiset. -/\ndef pi.empty (δ : α → Sort*) : (Πa∈(0:multiset α), δ a) .\n\nvariables [decidable_eq α] {β : α → Type*} {δ : α → Sort*}\n\n/-- Given `δ : α → Type*`, a multiset `m` and a term `a`, as well as a term `b : δ a` and a\nfunction `f` such that `f a' : δ a'` for all `a'` in `m`, `pi.cons m a b f` is a function `g` such\nthat `g a'' : δ a''` for all `a''` in `a ::ₘ m`. -/\ndef pi.cons (m : multiset α) (a : α) (b : δ a) (f : Πa∈m, δ a) : Πa'∈a ::ₘ m, δ a' :=\nλa' ha', if h : a' = a then eq.rec b h.symm else f a' $ (mem_cons.1 ha').resolve_left h\n\nlemma pi.cons_same {m : multiset α} {a : α} {b : δ a} {f : Πa∈m, δ a} (h : a ∈ a ::ₘ m) :\n  pi.cons m a b f a h = b :=\ndif_pos rfl\n\nlemma pi.cons_ne {m : multiset α} {a a' : α} {b : δ a} {f : Πa∈m, δ a}\n  (h' : a' ∈ a ::ₘ m) (h : a' ≠ a) :\n  pi.cons m a b f a' h' = f a' ((mem_cons.1 h').resolve_left h) :=\ndif_neg h\n\nlemma pi.cons_swap {a a' : α} {b : δ a} {b' : δ a'} {m : multiset α} {f : Πa∈m, δ a} (h : a ≠ a') :\n  pi.cons (a' ::ₘ m) a b (pi.cons m a' b' f) == pi.cons (a ::ₘ m) a' b' (pi.cons m a b f) :=\nbegin\n  apply hfunext rfl,\n  rintro a'' _ rfl,\n  refine hfunext (by rw [cons_swap]) (λ ha₁ ha₂ _, _),\n  rcases ne_or_eq a'' a with h₁ | rfl,\n  rcases eq_or_ne a'' a' with rfl | h₂,\n  all_goals { simp [*, pi.cons_same, pi.cons_ne] },\nend\n\n/-- `pi m t` constructs the Cartesian product over `t` indexed by `m`. -/\ndef pi (m : multiset α) (t : Πa, multiset (β a)) : multiset (Πa∈m, β a) :=\nm.rec_on {pi.empty β} (λa m (p : multiset (Πa∈m, β a)), (t a).bind $ λb, p.map $ pi.cons m a b)\nbegin\n  intros a a' m n,\n  by_cases eq : a = a',\n  { subst eq },\n  { simp [map_bind, bind_bind (t a') (t a)],\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b hb,\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b' hb',\n    apply map_hcongr, { rw [cons_swap a a'] },\n    intros f hf,\n    exact pi.cons_swap eq }\nend\n\n@[simp] lemma pi_zero (t : Πa, multiset (β a)) : pi 0 t = {pi.empty β} := rfl\n\n@[simp] lemma pi_cons (m : multiset α) (t : Πa, multiset (β a)) (a : α) :\n  pi (a ::ₘ m) t = ((t a).bind $ λb, (pi m t).map $ pi.cons m a b) :=\nrec_on_cons a m\n\nlemma pi_cons_injective {a : α} {b : δ a} {s : multiset α} (hs : a ∉ s) :\n  function.injective (pi.cons s a b) :=\nassume f₁ f₂ eq, funext $ assume a', funext $ assume h',\nhave ne : a ≠ a', from assume h, hs $ h.symm ▸ h',\nhave a' ∈ a ::ₘ s, from mem_cons_of_mem h',\ncalc f₁ a' h' = pi.cons s a b f₁ a' this : by rw [pi.cons_ne this ne.symm]\n  ... = pi.cons s a b f₂ a' this : by rw [eq]\n  ... = f₂ a' h' : by rw [pi.cons_ne this ne.symm]\n\nlemma card_pi (m : multiset α) (t : Πa, multiset (β a)) :\n  card (pi m t) = prod (m.map $ λa, card (t a)) :=\nmultiset.induction_on m (by simp) (by simp [mul_comm] {contextual := tt})\n\nprotected lemma nodup.pi {s : multiset α} {t : Π a, multiset (β a)} :\n  nodup s → (∀a∈s, nodup (t a)) → nodup (pi s t) :=\nmultiset.induction_on s (assume _ _, nodup_singleton _)\nbegin\n  assume a s ih hs ht,\n  have has : a ∉ s, by simp at hs; exact hs.1,\n  have hs : nodup s, by simp at hs; exact hs.2,\n  simp,\n  refine ⟨λ b hb, (ih hs $ λ a' h', ht a' $ mem_cons_of_mem h').map (pi_cons_injective has), _⟩,\n  refine (ht a $ mem_cons_self _ _).pairwise _,\n  from assume b₁ hb₁ b₂ hb₂ neb, disjoint_map_map.2 (assume f hf g hg eq,\n    have pi.cons s a b₁ f a (mem_cons_self _ _) = pi.cons s a b₂ g a (mem_cons_self _ _),\n      by rw [eq],\n    neb $ show b₁ = b₂, by rwa [pi.cons_same, pi.cons_same] at this)\nend\n\n@[simp]\nlemma pi.cons_ext {m : multiset α} {a : α} (f : Π a' ∈ a ::ₘ m, δ a') :\n  pi.cons m a (f _ (mem_cons_self _ _)) (λ a' ha', f a' (mem_cons_of_mem ha')) = f :=\nbegin\n  ext a' h',\n  by_cases a' = a,\n  { subst h, rw [pi.cons_same] },\n  { rw [pi.cons_ne _ h] }\nend\n\nlemma mem_pi (m : multiset α) (t : Πa, multiset (β a)) :\n  ∀f:Πa∈m, β a, (f ∈ pi m t) ↔ (∀a (h : a ∈ m), f a h ∈ t a) :=\nbegin\n  intro f,\n  induction m using multiset.induction_on with a m ih,\n  { simpa using show f = pi.empty β, by funext a ha; exact ha.elim },\n  simp_rw [pi_cons, mem_bind, mem_map, ih],\n  split,\n  { rintro ⟨b, hb, f', hf', rfl⟩ a' ha',\n    by_cases a' = a,\n    { subst h, rwa [pi.cons_same] },\n    { rw [pi.cons_ne _ h], apply hf' } },\n  { intro hf,\n    refine ⟨_, hf a (mem_cons_self _ _), _, λ a ha, hf a (mem_cons_of_mem ha), _⟩,\n    rw pi.cons_ext }\nend\n\nend pi\n\nend multiset\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/multiset/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3335122710777824}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module category_theory.adhesive\n! leanprover-community/mathlib commit afff1f24a6b68d0077c9d63782a1d093e337758c\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Extensive\nimport Mathbin.CategoryTheory.Limits.Shapes.KernelPair\n\n/-!\n\n# Adhesive categories\n\n## Main definitions\n- `category_theory.is_pushout.is_van_kampen`: A convenience formulation for a pushout being\n  a van Kampen colimit.\n- `category_theory.adhesive`: A category is adhesive if it has pushouts and pullbacks along\n  monomorphisms, and such pushouts are van Kampen.\n\n## Main Results\n- `category_theory.type.adhesive`: The category of `Type` is adhesive.\n- `category_theory.adhesive.is_pullback_of_is_pushout_of_mono_left`: In adhesive categories,\n  pushouts along monomorphisms are pullbacks.\n- `category_theory.adhesive.mono_of_is_pushout_of_mono_left`: In adhesive categories,\n  monomorphisms are stable under pushouts.\n- `category_theory.adhesive.to_regular_mono_category`: Monomorphisms in adhesive categories are\n  regular (this implies that adhesive categories are balanced).\n\n## TODO\n\nShow that the following are adhesive:\n- functor categories into adhesive categories\n- the categories of sheaves over a site\n\n## References\n- https://ncatlab.org/nlab/show/adhesive+category\n- [Stephen Lack and Paweł Sobociński, Adhesive Categories][adhesive2004]\n\n-/\n\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\n\nvariable {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z}\n\n-- This only makes sense when the original diagram is a pushout.\n/-- A convenience formulation for a pushout being a van Kampen colimit.\nSee `is_pushout.is_van_kampen_iff` below. -/\n@[nolint unused_arguments]\ndef IsPushout.IsVanKampen (H : IsPushout f g h i) : Prop :=\n  ∀ ⦃W' X' Y' Z' : C⦄ (f' : W' ⟶ X') (g' : W' ⟶ Y') (h' : X' ⟶ Z') (i' : Y' ⟶ Z') (αW : W' ⟶ W)\n    (αX : X' ⟶ X) (αY : Y' ⟶ Y) (αZ : Z' ⟶ Z) (hf : IsPullback f' αW αX f)\n    (hg : IsPullback g' αW αY g) (hh : CommSq h' αX αZ h) (hi : CommSq i' αY αZ i)\n    (w : CommSq f' g' h' i'), IsPushout f' g' h' i' ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i\n#align category_theory.is_pushout.is_van_kampen CategoryTheory.IsPushout.IsVanKampen\n\ntheorem IsPushout.IsVanKampen.flip {H : IsPushout f g h i} (H' : H.IsVanKampen) :\n    H.flip.IsVanKampen := by\n  introv W' hf hg hh hi w\n  simpa only [is_pushout.flip_iff, is_pullback.flip_iff, and_comm'] using\n    H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip\n#align category_theory.is_pushout.is_van_kampen.flip CategoryTheory.IsPushout.IsVanKampen.flip\n\ntheorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :\n    H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) :=\n  by\n  constructor\n  · intro H F' c' α fα eα hα\n    refine'\n      Iff.trans _\n        ((H (F'.map walking_span.hom.fst) (F'.map walking_span.hom.snd) (c'.ι.app _) (c'.ι.app _)\n              (α.app _) (α.app _) (α.app _) fα (by convert hα walking_span.hom.fst)\n              (by convert hα walking_span.hom.snd) _ _ _).trans\n          _)\n    · have :\n        F'.map walking_span.hom.fst ≫ c'.ι.app walking_span.left =\n          F'.map walking_span.hom.snd ≫ c'.ι.app walking_span.right :=\n        by simp only [cocone.w]\n      rw [(is_colimit.equiv_of_nat_iso_of_iso (diagram_iso_span F') c' (pushout_cocone.mk _ _ this)\n            _).nonempty_congr]\n      · exact ⟨fun h => ⟨⟨this⟩, h⟩, fun h => h.2⟩\n      · refine' cocones.ext (iso.refl c'.X) _\n        rintro (_ | _ | _) <;> dsimp <;>\n          simp only [c'.w, category.assoc, category.id_comp, category.comp_id]\n    · exact ⟨nat_trans.congr_app eα.symm _⟩\n    · exact ⟨nat_trans.congr_app eα.symm _⟩\n    · exact ⟨by simp⟩\n    constructor\n    · rintro ⟨h₁, h₂⟩ (_ | _ | _)\n      · rw [← c'.w walking_span.hom.fst]\n        exact (hα walking_span.hom.fst).paste_horiz h₁\n      exacts[h₁, h₂]\n    · intro h\n      exact ⟨h _, h _⟩\n  · introv H W' hf hg hh hi w\n    refine'\n      Iff.trans _\n        ((H w.cocone\n              ⟨by\n                rintro (_ | _ | _)\n                exacts[αW, αX, αY], _⟩\n              αZ _ _).trans\n          _)\n    rotate_left\n    · rintro i _ (_ | _ | _)\n      · dsimp\n        simp only [Functor.map_id, category.comp_id, category.id_comp]\n      exacts[hf.w, hg.w]\n    · ext (_ | _ | _)\n      · dsimp\n        rw [pushout_cocone.condition_zero]\n        erw [category.assoc, hh.w, hf.w_assoc]\n      exacts[hh.w.symm, hi.w.symm]\n    · rintro i _ (_ | _ | _)\n      · dsimp\n        simp_rw [Functor.map_id]\n        exact is_pullback.of_horiz_is_iso ⟨by rw [category.comp_id, category.id_comp]⟩\n      exacts[hf, hg]\n    · constructor\n      · intro h\n        exact ⟨h walking_cospan.left, h walking_cospan.right⟩\n      · rintro ⟨h₁, h₂⟩ (_ | _ | _)\n        · dsimp\n          rw [pushout_cocone.condition_zero]\n          exact hf.paste_horiz h₁\n        exacts[h₁, h₂]\n    · exact ⟨fun h => h.2, fun h => ⟨_, h⟩⟩\n#align category_theory.is_pushout.is_van_kampen_iff CategoryTheory.IsPushout.isVanKampen_iff\n\ntheorem is_coprod_iff_isPushout {X E Y YE : C} (c : BinaryCofan X E) (hc : IsColimit c) {f : X ⟶ Y}\n    {iY : Y ⟶ YE} {fE : c.pt ⟶ YE} (H : CommSq f c.inl iY fE) :\n    Nonempty (IsColimit (BinaryCofan.mk (c.inr ≫ fE) iY)) ↔ IsPushout f c.inl iY fE :=\n  by\n  constructor\n  · rintro ⟨h⟩\n    refine' ⟨H, ⟨limits.pushout_cocone.is_colimit_aux' _ _⟩⟩\n    intro s\n    dsimp\n    refine' ⟨h.desc (binary_cofan.mk (c.inr ≫ s.inr) s.inl), h.fac _ ⟨walking_pair.right⟩, _, _⟩\n    · apply binary_cofan.is_colimit.hom_ext hc\n      · rw [← H.w_assoc]\n        erw [h.fac _ ⟨walking_pair.right⟩]\n        exact s.condition\n      · rw [← category.assoc]\n        exact h.fac _ ⟨walking_pair.left⟩\n    · intro m e₁ e₂\n      apply binary_cofan.is_colimit.hom_ext h\n      · dsimp\n        rw [category.assoc, e₂, eq_comm]\n        exact h.fac _ ⟨walking_pair.left⟩\n      · refine' e₁.trans (Eq.symm _)\n        exact h.fac _ _\n  · refine' fun H => ⟨_⟩\n    fapply limits.binary_cofan.is_colimit_mk\n    ·\n      exact fun s =>\n        H.is_colimit.desc\n          (pushout_cocone.mk s.inr _ <|\n            (hc.fac (binary_cofan.mk (f ≫ s.inr) s.inl) ⟨walking_pair.left⟩).symm)\n    · intro s\n      erw [category.assoc, H.is_colimit.fac _ walking_span.right, hc.fac]\n      rfl\n    · intro s\n      exact H.is_colimit.fac _ walking_span.left\n    · intro s m e₁ e₂\n      apply pushout_cocone.is_colimit.hom_ext H.is_colimit\n      · symm\n        exact (H.is_colimit.fac _ walking_span.left).trans e₂.symm\n      · erw [H.is_colimit.fac _ walking_span.right]\n        apply binary_cofan.is_colimit.hom_ext hc\n        · dsimp\n          erw [hc.fac, ← H.w_assoc, e₂]\n          rfl\n        · refine' ((category.assoc _ _ _).symm.trans e₁).trans _\n          symm\n          exact hc.fac _ _\n#align category_theory.is_coprod_iff_is_pushout CategoryTheory.is_coprod_iff_isPushout\n\ntheorem IsPushout.isVanKampen_inl {W E X Z : C} (c : BinaryCofan W E) [FinitaryExtensive C]\n    [HasPullbacks C] (hc : IsColimit c) (f : W ⟶ X) (h : X ⟶ Z) (i : c.pt ⟶ Z)\n    (H : IsPushout f c.inl h i) : H.IsVanKampen :=\n  by\n  obtain ⟨hc₁⟩ := (is_coprod_iff_is_pushout c hc H.1).mpr H\n  introv W' hf hg hh hi w\n  obtain ⟨hc₂⟩ :=\n    ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen c hc)\n          (binary_cofan.mk _ pullback.fst) _ _ _ hg.w.symm pullback.condition.symm).mpr\n      ⟨hg, is_pullback.of_has_pullback αY c.inr⟩\n  refine' (is_coprod_iff_is_pushout _ hc₂ w).symm.trans _\n  refine'\n    ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen _ hc₁)\n          (binary_cofan.mk _ _) pullback.snd _ _ _ hh.w.symm).trans\n      _\n  · dsimp\n    rw [← pullback.condition_assoc, category.assoc, hi.w]\n  constructor\n  · rintro ⟨hc₃, hc₄⟩\n    refine' ⟨hc₄, _⟩\n    let Y'' := pullback αZ i\n    let cmp : Y' ⟶ Y'' := pullback.lift i' αY hi.w\n    have e₁ : (g' ≫ cmp) ≫ pullback.snd = αW ≫ c.inl := by\n      rw [category.assoc, pullback.lift_snd, hg.w]\n    have e₂ : (pullback.fst ≫ cmp : pullback αY c.inr ⟶ _) ≫ pullback.snd = pullback.snd ≫ c.inr :=\n      by rw [category.assoc, pullback.lift_snd, pullback.condition]\n    obtain ⟨hc₄⟩ :=\n      ((binary_cofan.is_van_kampen_iff _).mp (finitary_extensive.van_kampen c hc)\n            (binary_cofan.mk _ _) αW _ _ e₁.symm e₂.symm).mpr\n        ⟨_, _⟩\n    · rw [← category.id_comp αZ, ← show cmp ≫ pullback.snd = αY from pullback.lift_snd _ _ _]\n      apply is_pullback.paste_vert _ (is_pullback.of_has_pullback αZ i)\n      have : cmp = (hc₂.cocone_point_unique_up_to_iso hc₄).Hom :=\n        by\n        apply binary_cofan.is_colimit.hom_ext hc₂\n        exacts[(hc₂.comp_cocone_point_unique_up_to_iso_hom hc₄ ⟨walking_pair.left⟩).symm,\n          (hc₂.comp_cocone_point_unique_up_to_iso_hom hc₄ ⟨walking_pair.right⟩).symm]\n      rw [this]\n      exact is_pullback.of_vert_is_iso ⟨by rw [← this, category.comp_id, pullback.lift_fst]⟩\n    · apply is_pullback.of_right _ e₁ (is_pullback.of_has_pullback _ _)\n      rw [category.assoc, pullback.lift_fst, ← H.w, ← w.w]\n      exact hf.paste_horiz hc₄\n    · apply is_pullback.of_right _ e₂ (is_pullback.of_has_pullback _ _)\n      rw [category.assoc, pullback.lift_fst]\n      exact hc₃\n  · rintro ⟨hc₃, hc₄⟩\n    exact ⟨(is_pullback.of_has_pullback αY c.inr).paste_horiz hc₄, hc₃⟩\n#align category_theory.is_pushout.is_van_kampen_inl CategoryTheory.IsPushout.isVanKampen_inl\n\ntheorem IsPushout.IsVanKampen.isPullback_of_mono_left [Mono f] {H : IsPushout f g h i}\n    (H' : H.IsVanKampen) : IsPullback f g h i :=\n  ((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i (IsKernelPair.id_of_mono f)\n            (IsPullback.of_vert_isIso ⟨by simp⟩) H.1.flip ⟨rfl⟩ ⟨by simp⟩).mp\n        (IsPushout.of_horiz_isIso ⟨by simp⟩)).1.flip\n#align category_theory.is_pushout.is_van_kampen.is_pullback_of_mono_left CategoryTheory.IsPushout.IsVanKampen.isPullback_of_mono_left\n\ntheorem IsPushout.IsVanKampen.isPullback_of_mono_right [Mono g] {H : IsPushout f g h i}\n    (H' : H.IsVanKampen) : IsPullback f g h i :=\n  ((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h (IsPullback.of_vert_isIso ⟨by simp⟩)\n          (IsKernelPair.id_of_mono g) ⟨rfl⟩ H.1 ⟨by simp⟩).mp\n      (IsPushout.of_vert_isIso ⟨by simp⟩)).2\n#align category_theory.is_pushout.is_van_kampen.is_pullback_of_mono_right CategoryTheory.IsPushout.IsVanKampen.isPullback_of_mono_right\n\ntheorem IsPushout.IsVanKampen.mono_of_mono_left [Mono f] {H : IsPushout f g h i}\n    (H' : H.IsVanKampen) : Mono i :=\n  IsKernelPair.mono_of_isIso_fst\n    ((H' (𝟙 _) g g (𝟙 Y) (𝟙 _) f (𝟙 _) i (IsKernelPair.id_of_mono f)\n            (IsPullback.of_vert_isIso ⟨by simp⟩) H.1.flip ⟨rfl⟩ ⟨by simp⟩).mp\n        (IsPushout.of_horiz_isIso ⟨by simp⟩)).2\n#align category_theory.is_pushout.is_van_kampen.mono_of_mono_left CategoryTheory.IsPushout.IsVanKampen.mono_of_mono_left\n\ntheorem IsPushout.IsVanKampen.mono_of_mono_right [Mono g] {H : IsPushout f g h i}\n    (H' : H.IsVanKampen) : Mono h :=\n  IsKernelPair.mono_of_isIso_fst\n    ((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h (IsPullback.of_vert_isIso ⟨by simp⟩)\n            (IsKernelPair.id_of_mono g) ⟨rfl⟩ H.1 ⟨by simp⟩).mp\n        (IsPushout.of_vert_isIso ⟨by simp⟩)).1\n#align category_theory.is_pushout.is_van_kampen.mono_of_mono_right CategoryTheory.IsPushout.IsVanKampen.mono_of_mono_right\n\n/-- A category is adhesive if it has pushouts and pullbacks along monomorphisms,\nand such pushouts are van Kampen. -/\nclass Adhesive (C : Type u) [Category.{v} C] : Prop where\n  [hasPullback_of_mono_left : ∀ {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [Mono f], HasPullback f g]\n  [hasPushout_of_mono_left : ∀ {X Y S : C} (f : S ⟶ X) (g : S ⟶ Y) [Mono f], HasPushout f g]\n  van_kampen :\n    ∀ {W X Y Z : C} {f : W ⟶ X} {g : W ⟶ Y} {h : X ⟶ Z} {i : Y ⟶ Z} [Mono f]\n      (H : IsPushout f g h i), H.IsVanKampen\n#align category_theory.adhesive CategoryTheory.Adhesive\n\nattribute [instance] adhesive.has_pullback_of_mono_left adhesive.has_pushout_of_mono_left\n\ntheorem Adhesive.van_kampen' [Adhesive C] [Mono g] (H : IsPushout f g h i) : H.IsVanKampen :=\n  (Adhesive.van_kampen H.flip).flip\n#align category_theory.adhesive.van_kampen' CategoryTheory.Adhesive.van_kampen'\n\ntheorem Adhesive.isPullback_of_isPushout_of_mono_left [Adhesive C] (H : IsPushout f g h i)\n    [Mono f] : IsPullback f g h i :=\n  (Adhesive.van_kampen H).isPullback_of_mono_left\n#align category_theory.adhesive.is_pullback_of_is_pushout_of_mono_left CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_left\n\ntheorem Adhesive.isPullback_of_isPushout_of_mono_right [Adhesive C] (H : IsPushout f g h i)\n    [Mono g] : IsPullback f g h i :=\n  (Adhesive.van_kampen' H).isPullback_of_mono_right\n#align category_theory.adhesive.is_pullback_of_is_pushout_of_mono_right CategoryTheory.Adhesive.isPullback_of_isPushout_of_mono_right\n\ntheorem Adhesive.mono_of_isPushout_of_mono_left [Adhesive C] (H : IsPushout f g h i) [Mono f] :\n    Mono i :=\n  (Adhesive.van_kampen H).mono_of_mono_left\n#align category_theory.adhesive.mono_of_is_pushout_of_mono_left CategoryTheory.Adhesive.mono_of_isPushout_of_mono_left\n\ntheorem Adhesive.mono_of_isPushout_of_mono_right [Adhesive C] (H : IsPushout f g h i) [Mono g] :\n    Mono h :=\n  (Adhesive.van_kampen' H).mono_of_mono_right\n#align category_theory.adhesive.mono_of_is_pushout_of_mono_right CategoryTheory.Adhesive.mono_of_isPushout_of_mono_right\n\ninstance Type.adhesive : Adhesive (Type u) :=\n  by\n  constructor\n  intros\n  exact (is_pushout.is_van_kampen_inl _ (types.is_coprod_of_mono f) _ _ _ H.flip).flip\n#align category_theory.type.adhesive CategoryTheory.Type.adhesive\n\nnoncomputable instance (priority := 100) Adhesive.toRegularMonoCategory [Adhesive C] :\n    RegularMonoCategory C :=\n  ⟨fun X Y f hf =>\n    { z := pushout f f\n      left := pushout.inl\n      right := pushout.inr\n      w := pushout.condition\n      IsLimit :=\n        (adhesive.is_pullback_of_is_pushout_of_mono_left\n            (is_pushout.of_has_pushout f f)).isLimitFork }⟩\n#align category_theory.adhesive.to_regular_mono_category CategoryTheory.Adhesive.toRegularMonoCategory\n\n-- This then implies that adhesive categories are balanced\nexample [Adhesive C] : Balanced C :=\n  inferInstance\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Adhesive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3334810312880515}}
{"text": "/-\nWe now introduce the applicative typeclass. Here\nit is as copied mostly verbatim from Lean's library\n(credit DeMoura and Ulrich). \n-/\n\nimport .functor \n\nnamespace hidden\n\nopen function   -- for definition of \"const\" used at end of file\nuniverses u v\n\n/-\nNow suppose that we want to generalize functors, which allow\nfor mapping of *one-argument* functions, such as inc, over \nfunctorial values, to enable mapping of n-argument functions\nover functorial values for any natural number, n.\n\nAs a special case, we have the usual functor mapping function, \nwhich works as follows: (α → β) → f α → f β. In plain English,\nit takes a pure one-argument function from (α → β) and applies\nit to each value in the functorial value of type f α. \n\nNow consider a function of type α → β → γ (think of addition\nof nats, of type nat → nat → nat, for example). Here what we\nwill want is a mapping function that takes a pure function of\ntype α → β → γ then a functor of α values *then a functor of \nβ values* to produce a functor of γ values.\n\nSo suppose our pure function is nat addition and that we \nwant to map this two-argument function. We'll need a list\nof natural numbers as first arguments. We'll map add over \nit, but of course we won't get a list of resulting nats;\nrather we'll get a list of partially evaluated functions.\n-/\n\n#reduce nat.add <$> [1,2,3] -- remember <$> is functor.map\n\n/-\nWe will then want to apply each of these functions to each\nof their second arguments.\n-/\n\n/-\nNow consider mapping a pure function with three arguments.\nWe take our pure function, map it over the list of first\narguments, getting a list of partially evaluate functions\nthat expect two more arguments. We then want to apply each\nof these to their second arguments in a second list to get\na list of further partially evaluate functions expecting \ntheir last arguments. We apply them to their arguments in\nthe third list to finally get a list of the values of the\nfunction applied to all of its arguments, which come from\nthe given lists. \n-/\n\n/-\nSo what we're seeing here is that once we bootstrap the\nprocess given a pure function, what we've now got is a\nsituation in which we have a *list* of functions each of\nwhich takes some number of arguments, and that many lists\nof arguments. \n-/\n\n\n/-\nBootstrap\n-/\nclass has_pure (f : Type u → Type v) :=\n(pure {α : Type u} : α → f α)\n\n/-\nApply next step. Given a context (list, tree, option)\nof functions of type (α → β), *where β can itself be a\nfunction type*, and given a context of argument values\n(of type α), compute and return a corresponding context \nof result values (of type β). Understand that if β is \nitself a function type, then what's returned here is\n\"another collection of functions.\" The seq function is\nthus a means to apply a context containing functions\nto a context containing arguments to get a context of\nresults, each of which itself could be a function of\none fewer arguments. \n-/\nclass has_seq (f : Type u → Type v) : Type (max (u+1) v) :=\n(seq  : Π {α β : Type u}, f (α → β) → f α → f β)\n\n-- infix notation for seq (** instead of * to not conflict)\ninfixl ` <**> `:60 := has_seq.seq\n\n/-\nAnd finally, the applicative functor typeclass. Note that\nthis typeclass (1) extends functor, thus inheriting its map\nfield; (2) it also extends has_pure and has_seq, as defined\njust above, thus gaining pure and seq fields, each of the\nrequisite type. \n\nFinally, note that we implement the \"map\" function once and \nfor all, for all instances of this typeclass. What does it \ndo? It takes a function of type α → β, uses pure to turn it\ninto a (singleton) container of functions, and then uses the\nseq (<*>) operator to map this container of functions (x)\nover a container of argument values (y).\n-/\nclass applicative (f : Type u → Type v) extends functor f, has_pure f, has_seq f :=\n(map       := λ _ _ x y, pure x <**> y)\n\n/-\nNote that it *extends* the functor typeclass. Look for the use of\nits \"map\" function in the definition of the applicative functor's \n\"seq\" functions, coming up next.\n-/\n\n/-\nThe \"applicative option\" instance implements function application\n*with built-in failure propagation*.\n-/\n\ndef pure_option {α : Type u} : α → option α \n| a := some a\n\ndef seq_option : Π {α β : Type u}, option (α → β) → option α → option β\n| α β none _ := none\n| α β (some func) oa := functor.map func oa  -- Use's functor's option_map!\n\ninstance : has_pure option := ⟨ @pure_option ⟩  \ninstance : has_seq option := ⟨ @seq_option ⟩ \ninstance : applicative option := ⟨ ⟩ \n\n-- examples with one-argument functions\n#reduce pure (nat.succ) <**> none\n#reduce none <**> some 1\n#reduce pure (nat.succ) <**> (some 1)\n#eval pure (λ s, s++\"!\") <**> (some \"Hello\")\n\n-- examples with multi-argument functions\n#reduce pure (nat.mul) <**> some 3 \n#reduce pure (nat.mul) <**> some 3 <**> some 4\n#reduce pure (nat.mul) <**> none <**> some 4\n#reduce pure (nat.mul) <**> some 3 <**> none\n\n/-\nThink of \"none\" as indicating a failure to compute\na function argument. What we've thus produced here,\nin the form of the (applicative option) instance is \na generalization of *function application,* where\na failure to produce the function to be applied, or\nthe failure to produce any of its arguments, yields\na failure return. In the case where the function and\nall of its arguments are given, we just get ordinary\nfunction application. \n\nAnother way to think about this is that <**> handles\nany error propagation \"under the hood\", leaving us as \nprogrammers with an notion of function application\nthat appears works pretty much as usual. This is an\nexample that shows how, by overloading applicative \nfor option we obtain a new programming abstraction:\nof ordinary function application (on the surface)\nbut with additional effects handling (in this case\nerror handling) being taken care of under the hood.\n-/\n\n\n/-\nThe applicative list typeclass instance.\n\nIn the preceding example, we had either zero or one\nfunctions in an option and zero or one arguments in \na second option. In the \"normal\" case, where both the\nfunction and the argument are provided, we just apply\nthe function to the argument and return the result in\na \"some _\" option value. \n\nNow we turn to (applicative list) type. Here the seq\noperator will take a list of zero or more functions and\na list of zero or more arguments and will return a list\ncontaining the values obtained by applying each function \nin the function list to each value in the argument list.\n\nThe way to think about this is that we are implementing\na kind of non-deterministic, multi-valued computation. \n-/\n\ndef pure_list {α : Type u} : α → list α | a := [a]\ndef seq_list : Π {α β : Type u}, list (α → β) → list α → list β\n| α β list.nil _ := list.nil\n| α β (hf::tf) l := list.append (functor.map hf l) (seq_list tf l)\n \ninstance : has_pure list := ⟨ @pure_list ⟩  \ninstance : has_seq list := ⟨ @seq_list ⟩ \ninstance : applicative list := ⟨ ⟩ \n\n-- tests with one-argument functions\n#reduce seq_list [] [1,2,3] -- apply empty list of functions\n#reduce [] <**> [1,2,3]      -- same thing using infix operator\n\n#reduce seq_list [nat.succ] []\n#reduce [nat.succ] <**> []\n\n#reduce seq_list [nat.succ] [1,2,3]\n#reduce [nat.succ] <**> [1,2,3]\n\n#reduce seq_list (pure nat.succ) [1,2,3]\n#reduce (pure nat.succ) <**> [1,2,3]\n\n#reduce [nat.succ, nat.pred] <**> []\n#reduce [nat.succ, nat.pred] <**> [1,2,3]\n\n-- tests with two-argument functions!\n#reduce (pure nat.add) <*> [1,2,3] \n#reduce (pure nat.add) <**> [1,2,3] <**> [2,4,6]\n#reduce [nat.add, nat.mul] <**> [1,2,3] <**> [2,4,6]\n\n/-\nIn practice we usually don't explicitly apply lists of\nmultiple functions directly to argument lists. Rather,\nagain in practice, we're usually interesting in applying\na single function to mult-valued arguments, so we use\n\"pure\" to lift the function value into the list context\nand then everything else is done by <**>, which, as we\nhave just seen, does tend to produce lists of functions\nas intermediate results that get \"applied\" by <**> to\ndownstream argument lists (multi-valued arguments).\n-/\n\nend hidden\n\n\n", "meta": {"author": "kevinsullivan", "repo": "cs6501s22", "sha": "c55d342145b127e7b7bf396c660966034e80a944", "save_path": "github-repos/lean/kevinsullivan-cs6501s22", "path": "github-repos/lean/kevinsullivan-cs6501s22/cs6501s22-c55d342145b127e7b7bf396c660966034e80a944/src/content/lectures/S_07_monads/applicative.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.33343488794385295}}
{"text": "import category_theory.preadditive.functor_category\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.biproducts\nimport category_theory.limits.preserves.filtered\n\nimport for_mathlib.homological_complex2\nimport for_mathlib.additive_functor\n\nimport breen_deligne.homotopy\n\nnoncomputable theory\n\nopen_locale big_operators\n\nopen category_theory category_theory.limits\n\nnamespace category_theory\nnamespace preadditive\n\nvariables {𝒜 : Type*} [category 𝒜] [has_zero_morphisms 𝒜] [has_finite_biproducts 𝒜]\n\n-- move this\n@[simps {fully_applied := ff}]\ndef Pow (n : ℕ) : 𝒜 ⥤ 𝒜 :=\n{ obj := λ A, ⨁ (λ (i : fin n), A),\n  map := λ A B f, biproduct.map (λ i, f),\n  map_id' := λ A, by { ext i j, simp only [biproduct.ι_map, category.id_comp, category.comp_id], },\n  map_comp' := λ A B C f g, by { ext i j, simp only [biproduct.ι_map_assoc, category.assoc], } }\n\n-- move this\nattribute [simps] comp_hom\n.\n\ninstance (n : ℕ) {J : Type*} [category J] : preserves_colimits_of_shape J (Pow n : 𝒜 ⥤ 𝒜) :=\n{ preserves_colimit := λ K,\n  { preserves := λ c hc,\n    { desc := λ s, biproduct.desc $ λ i,\n        let t : cocone K :=\n        { X := s.X,\n          ι := { app := λ j, show K.obj j ⟶ (K ⋙ Pow n).obj j, from biproduct.ι _ i,\n                naturality' := by intros X Y f;\n                  simp only [functor.comp_map, Pow_map, biproduct.ι_map], } ≫ s.ι } in\n        hc.desc t,\n      fac' := begin\n        intros, ext,\n        simp only [Pow_map, functor.map_cocone_ι_app, biproduct.map_desc,\n          is_colimit.fac, nat_trans.comp_app, biproduct.ι_desc],\n      end,\n      uniq' := begin\n        intros, ext i,\n        simp only [biproduct.ι_desc],\n        let t : cocone K :=\n        { X := s.X,\n          ι := { app := λ j, show K.obj j ⟶ (K ⋙ Pow n).obj j, from biproduct.ι _ i,\n                naturality' := by intros X Y f;\n                  simp only [functor.comp_map, Pow_map, biproduct.ι_map], } ≫ s.ι },\n        refine hc.uniq t (_ ≫ m) _,\n        intro j,\n        simp only [nat_trans.comp_app, ← w,\n          functor.map_cocone_ι_app, Pow_map, biproduct.ι_map_assoc],\n      end } } }\n\ninstance (n : ℕ) : preserves_colimits (Pow n : 𝒜 ⥤ 𝒜) :=\n{ preserves_colimits_of_shape := λ J hJ, by apply_instance }\n\nend preadditive\nend category_theory\n\nnamespace homotopy\n\nvariables {ι 𝒜 : Type*} [category 𝒜] [preadditive 𝒜] {c : complex_shape ι}\nvariables {C D : homological_complex 𝒜 c} {f g : C ⟶ D}\n\n@[simps]\ndef congr (h : homotopy f g) (f' g' : C ⟶ D) (hf : f = f') (hg : g = g') :\n  homotopy f' g' :=\n{ comm := by simpa only [hf, hg] using h.comm,\n  .. h }\n\nend homotopy\n\nnamespace breen_deligne\n\nopen category_theory.preadditive\n\nvariables (BD : data)\nvariables {𝒜 : Type*} [category 𝒜] [preadditive 𝒜] [has_finite_biproducts 𝒜]\nvariables (F : 𝒜 ⥤ 𝒜)\n\nnamespace basic_universal_map\n\nvariables {m n o : ℕ} (f : basic_universal_map m n) (g : basic_universal_map n o)\n\n@[simps {fully_applied := ff}]\ndef eval_Pow : (Pow m : 𝒜 ⥤ 𝒜) ⟶ Pow n :=\n{ app := λ A, biproduct.matrix (λ i j, f j i • 𝟙 A),\n  naturality' := begin\n    intros, ext i j,\n    simp only [Pow_map, biproduct.ι_map_assoc, category.assoc, biproduct.matrix_π,\n      biproduct.map_π, biproduct.ι_desc, biproduct.matrix_π_assoc, biproduct.ι_desc_assoc,\n      comp_zsmul, zsmul_comp, category.comp_id, category.id_comp],\n  end }\n\n@[simp] lemma eval_Pow_comp : @eval_Pow 𝒜 _ _ _ _ _ (comp g f) = f.eval_Pow ≫ g.eval_Pow :=\nbegin\n  ext A i j,\n  simp only [eval_Pow_app, nat_trans.comp_app, category.assoc, biproduct.ι_map_assoc,\n    biproduct.matrix_π, biproduct.ι_matrix_assoc, biproduct.lift_desc,\n    biproduct.map_π, biproduct.ι_desc, biproduct.matrix_π_assoc, biproduct.ι_desc_assoc,\n    comp_zsmul, zsmul_comp, category.comp_id, category.id_comp],\n  simp only [comp, add_monoid_hom.mk'_apply, matrix.mul, matrix.dot_product,\n    finset.sum_smul, mul_smul],\n  rw [finset.sum_congr rfl],\n  rintros j -,\n  rw smul_comm,\nend\n\nend basic_universal_map\n\nnamespace universal_map\n\nvariables {m n o : ℕ} (f : universal_map m n) (g : universal_map n o)\n\n/- Note: this definition is generalized in `eval1half.lean` for a functor\n`F : A₁ ⥤ A₂`. This generalization is used in `apply_Pow.lean`. -/\ndef eval_Pow : universal_map m n →+ (Pow m ⋙ F ⟶ Pow n ⋙ F) :=\nfree_abelian_group.lift $ λ g : basic_universal_map m n, whisker_right g.eval_Pow F\n\nlemma eval_Pow_of (g : basic_universal_map m n) :\n  eval_Pow F (free_abelian_group.of g) = whisker_right g.eval_Pow F :=\nfree_abelian_group.lift.of _ _\n\n@[simp] lemma eval_Pow_zero : eval_Pow F (0 : universal_map m n) = 0 :=\nadd_monoid_hom.map_zero _\n\nlemma eval_Pow_zero_app (A : 𝒜) : (eval_Pow F (0 : universal_map m n)).app A = 0 :=\nby rw [eval_Pow_zero, zero_app]\n\nlemma eval_Pow_comp : eval_Pow F (universal_map.comp g f) = eval_Pow F f ≫ eval_Pow F g :=\nbegin\n  rw [← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_hom_apply_apply,\n    ← add_monoid_hom.comp_apply, eq_comm,\n    ← category_theory.preadditive.comp_hom_apply_apply, ← add_monoid_hom.flip_apply,\n    ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_hom_apply_apply,\n    ← add_monoid_hom.flip_apply _ _ (eval_Pow F),\n    ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_hom_apply_apply,\n    ← add_monoid_hom.comp_apply, ← add_monoid_hom.comp_hom_apply_apply],\n  congr' 2,\n  clear f g,\n  ext g f : 2,\n  simp only [add_monoid_hom.comp_hom_apply_apply, add_monoid_hom.comp_apply,\n    add_monoid_hom.flip_apply, category_theory.preadditive.comp_hom_apply_apply,\n    comp_of, eval_Pow_of, whisker_right_comp, basic_universal_map.eval_Pow_comp],\nend\n\nlemma eval_Pow_comp_app (A : 𝒜) :\n  (eval_Pow F (universal_map.comp g f)).app A = (eval_Pow F f).app A ≫ (eval_Pow F g).app A :=\nby rw [eval_Pow_comp, nat_trans.comp_app]\n\n@[simps {fully_applied := ff}]\ndef eval_Pow_functor : FreeMat ⥤ (𝒜 ⥤ 𝒜) :=\n{ obj := λ n, Pow n ⋙ F,\n  map := λ m n f, eval_Pow F f,\n  map_id' := λ n,\n  begin\n    refine (eval_Pow_of F _).trans _,\n    ext A : 2, dsimp,\n    rw ← F.map_id, congr' 1,\n    ext i j : 2,\n    simp only [biproduct.ι_matrix, category.comp_id, biproduct.lift_π, basic_universal_map.id],\n    rw biproduct.ι_π,\n    split_ifs with hij,\n    { cases hij, rw [matrix.one_apply_eq, one_smul, eq_to_hom_refl], },\n    { rw [matrix.one_apply_ne, zero_smul], dsimp, rintro rfl, exact hij rfl }\n  end,\n  map_comp' := λ m n o f g, eval_Pow_comp F _ _ }\n\ninstance eval_Pow_functor_additive : (eval_Pow_functor F).additive :=\n{ map_add' := λ m n f g, by { dsimp [eval_Pow], rw add_monoid_hom.map_add } }\n\nend universal_map\n\nnamespace data\n\nopen universal_map\n\n@[simps {fully_applied := ff}]\ndef eval_functor' : data ⥤ chain_complex (𝒜 ⥤ 𝒜) ℕ :=\n(eval_Pow_functor F).map_homological_complex _\n\n@[simps {fully_applied := ff}]\ndef eval_functor : data ⥤ 𝒜 ⥤ chain_complex 𝒜 ℕ :=\neval_functor' F ⋙ homological_complex.functor_eval.flip\n.\n\n-- generalize to arbitrary homological complexes\ninstance homological_complex.functor_eval_flip_preserves_colimits_of_shape\n  (J : Type*) [category J] (F : chain_complex (𝒜 ⥤ 𝒜) ℕ)\n  [∀ i, preserves_colimits_of_shape J (F.X i)] :\n  preserves_colimits_of_shape J (homological_complex.functor_eval.flip.obj F) :=\n{ preserves_colimit := λ K,\n  { preserves := λ c hc,\n    let t : Π (s : cocone (K ⋙ homological_complex.functor_eval.flip.obj F))\n        (i : ℕ), cocone (K ⋙ F.X i) := λ s i,\n    { X := s.X.X i,\n      ι := { app := λ j, show (K ⋙ F.X i).obj j ⟶ s.X.X i, from (s.ι.app j).f i,\n            naturality' := begin\n              intros a b φ, have := s.ι.naturality φ, dsimp at this ⊢,\n              simp only [category.comp_id] at this ⊢,\n              rw ← this, refl\n            end } },\n      u : Π (s : cocone (K ⋙ homological_complex.functor_eval.flip.obj F))\n        (i j : ℕ), cocone (K ⋙ F.X i) := λ s i j,\n    { X := s.X.X j,\n      ι := { app := λ k, show (K ⋙ F.X i).obj k ⟶ s.X.X j,\n                         from (whisker_left K (F.d i j)).app k ≫ (s.ι.app k).f j,\n            naturality' := begin\n              intros a b φ, have := s.ι.naturality φ, dsimp at this ⊢,\n              simp only [category.comp_id] at this ⊢,\n              rw [← this, (F.d i j).naturality_assoc], refl,\n            end } } in\n    { desc := λ s,\n      { f := λ i, (is_colimit_of_preserves (F.X i) hc).desc (t s i),\n        comm' := begin\n          intros i j h, dsimp,\n          have := (is_colimit_of_preserves (F.X i) hc).uniq (u s i j),\n          refine (this _ _).trans (this _ _).symm,\n          { intros j', dsimp,\n            erw [(is_colimit_of_preserves (F.X i) hc).fac_assoc],\n            apply (s.ι.app j').comm, },\n          { intros j', dsimp,\n            rw nat_trans.naturality_assoc,\n            erw [(is_colimit_of_preserves (F.X j) hc).fac], }\n        end },\n      fac' := by { intros, ext i, dsimp, erw [(is_colimit_of_preserves (F.X i) hc).fac], },\n      uniq' := begin\n        intros, ext i,\n        exact (is_colimit_of_preserves (F.X i) hc).uniq (t s i) (m.f i)\n          (λ j, homological_complex.congr_hom (w j) i),\n      end, } } }\n\ninstance eval_functor_preserves_colimits_of_shape\n  (BD : data) (J : Type*) [category J] [preserves_colimits_of_shape J F] :\n  preserves_colimits_of_shape J ((eval_functor F).obj BD) :=\nbegin\n  refine @homological_complex.functor_eval_flip_preserves_colimits_of_shape _ _ _ _ J _\n    ((eval_functor' F).obj BD) (id _),\n  intro i,\n  show preserves_colimits_of_shape J (Pow (BD.X i) ⋙ F),\n  apply_instance\nend\n\ninstance eval_functor_preserves_filtered_colimits (BD : data) [preserves_filtered_colimits F] :\n  preserves_filtered_colimits ((eval_functor F).obj BD) :=\n{ preserves_filtered_colimits := by introsI; apply_instance }\n\n-- @[simps]\n-- def eval_functor.obj (M : 𝒜) : chain_complex 𝒜 ℕ :=\n-- { X := λ n, (Pow (BD.X n) ⋙ F).obj M,\n--   d := λ m n, (eval_Pow F (BD.d m n)).app M,\n--   shape' := λ i j h, by rw [BD.shape i j h, universal_map.eval_Pow_zero_app],\n--   d_comp_d' := λ i j k hij hjk, begin\n--     rw [← universal_map.eval_Pow_comp_app],\n--     have := BD.d_comp_d i j k,\n--     convert universal_map.eval_Pow_zero_app _ _ using 3,\n--   end }\n\n-- @[simps {fully_applied := ff}]\n-- def eval_functor : 𝒜 ⥤ chain_complex 𝒜 ℕ :=\n-- { obj := eval_functor.obj BD F,\n--   map := λ A B f,\n--   { f := λ n, (Pow (BD.X n) ⋙ F).map f,\n--     comm' := λ m n h, by simp only [eval_functor.obj_d, nat_trans.naturality] },\n--   map_id' := λ A, by { ext n, exact category_theory.functor.map_id _ _ },\n--   map_comp' := λ A B C f g, by { ext n, exact category_theory.functor.map_comp _ _ _ } }\n\n-- @[simps {fully_applied := ff}]\n-- def map_eval_functor {BD₁ BD₂ : data} (φ : BD₁ ⟶ BD₂) :\n--   BD₁.eval_functor F ⟶ BD₂.eval_functor F :=\n-- { app := λ A,\n--   { f := λ i, (universal_map.eval_Pow F (φ.f i)).app A,\n--     comm' := by { intros, dsimp only [eval_functor_obj, eval_functor.obj_d],\n--       simp only [← nat_trans.comp_app, ← eval_Pow_comp F], congr' 2, apply φ.comm } },\n--   naturality' := λ A B f, by { ext i : 2, apply nat_trans.naturality } }\n\nend data\n\nnamespace package\n\nopen universal_map\n\nvariables (BD' : package) (A : 𝒜)\n\ndef eval_homotopy := (eval_Pow_functor F).map_homotopy BD'.homotopy\n\ndef eval_homotopy' (A : 𝒜) :=\n(eval_Pow_functor F ⋙ (evaluation _ _).obj A).map_homotopy BD'.homotopy\n\nlocal attribute [instance] has_binary_biproducts_of_finite_biproducts\n\n@[simps]\ndef Biprod : 𝒜 ⥤ 𝒜 :=\n{ obj := λ A, A ⊞ A,\n  map := λ A B f, biprod.map f f,\n  map_id' := λ A,\n    by ext; simp only [biprod.inl_map, biprod.inr_map, category.id_comp, category.comp_id],\n  map_comp' := λ A B C f g,\n    by ext; simp only [biprod.inl_map_assoc, biprod.inr_map_assoc, category.assoc] }\n.\n\n@[simps {fully_applied := ff}]\ndef Biprod_iso_Pow_two_components (A : 𝒜) : A ⊞ A ≅ (Pow 2).obj A :=\n{ hom := biprod.desc\n    (biproduct.ι (λ i : fin 2, A) 0)\n    (biproduct.ι (λ i : fin 2, A) 1),\n  inv := biprod.lift (biproduct.π _ 0) (biproduct.π _ 1),\n  hom_inv_id' := begin\n    ext;\n    simp only [biprod.lift_fst, biprod.lift_snd, biprod.inl_desc_assoc, biprod.inr_desc_assoc,\n      biproduct.ι_π_self, category.assoc];\n    erw category.id_comp;\n    simp only [biprod.inl_fst, biprod.inl_snd, biprod.inr_fst, biprod.inr_snd];\n    rw [biproduct.ι_π_ne]; dec_trivial\n  end,\n  inv_hom_id' := begin\n    ext i j,\n    erw [category.comp_id],\n    simp only [add_comp, comp_add, biprod.lift_desc, category.assoc],\n    fin_cases i with [0,1];\n    rw [biproduct.ι_π_self_assoc, biproduct.ι_π_ne_assoc, zero_comp],\n    swap 2, { dec_trivial },\n    swap 3, { dec_trivial },\n    { rw add_zero },\n    { rw zero_add }\n  end }\n.\n\n@[simps {fully_applied := ff}]\ndef Biprod_iso_Pow_two : (Biprod : 𝒜 ⥤ 𝒜) ≅ Pow 2 :=\nnat_iso.of_components Biprod_iso_Pow_two_components $ λ A B f,\nbegin\n  ext ⟨i⟩;\n  simp only [biproduct.ι_map, Biprod_iso_Pow_two_components_hom, Biprod_map, Pow_map,\n    biprod.inl_map_assoc, biprod.inl_desc_assoc, biprod.inr_map_assoc, biprod.inr_desc_assoc,\n    biprod.inr_map, category.assoc, biprod.inr_desc],\nend\n.\n\n@[simps]\ndef Pow_comp_Pow_components (m n : ℕ) (A : 𝒜) :\n  (Pow n).obj ((Pow m).obj A) ≅ (Pow (m * n)).obj A :=\n{ hom := biproduct.desc $ λ j, biproduct.desc $ λ i,\n    biproduct.ι (λ i : fin _, A) (fin_prod_fin_equiv (i, j)),\n  inv := biproduct.lift $ λ j, biproduct.lift $ λ i,\n    biproduct.π (λ i : fin _, A) (fin_prod_fin_equiv (i, j)),\n  hom_inv_id' := begin\n    ext j i j' i' : 4,\n    erw [biproduct.ι_desc_assoc, category.comp_id],\n    simp only [biproduct.ι_desc_assoc, category.assoc, biproduct.lift_π],\n    by_cases hj : j = j',\n    { subst hj, rw [biproduct.ι_π_self_assoc],\n      by_cases hi : i = i',\n      { subst hi, rw [biproduct.ι_π_self, biproduct.ι_π_self] },\n      { rw [biproduct.ι_π_ne, biproduct.ι_π_ne],\n        { exact hi },\n        { simpa only [equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff, eq_self_iff_true, ne.def] using hi, } } },\n    { rw [biproduct.ι_π_ne, biproduct.ι_π_ne_assoc, zero_comp, comp_zero],\n      { exact hj },\n      { simp only [equiv.apply_eq_iff_eq, prod.mk.inj_iff, ne.def, hj, not_false_iff, and_false], } }\n  end,\n  inv_hom_id' := begin\n    ext k k' : 2,\n    erw [category.comp_id],\n    simp only [category.assoc, biproduct.lift_desc, sum_comp, comp_sum],\n    by_cases h : k = k',\n    { subst h,\n      rw [biproduct.ι_π_self,\n        finset.sum_eq_single ((fin_prod_fin_equiv.symm k).snd : fin _),\n        finset.sum_eq_single ((fin_prod_fin_equiv.symm k).fst : fin _)],\n      { rw [prod.mk.eta, equiv.apply_symm_apply, biproduct.ι_π_self, biproduct.ι_π_self_assoc], },\n      { rintro ⟨i⟩ - hi,\n        rw [biproduct.ι_π_ne_assoc, zero_comp],\n        dsimp [- fin_prod_fin_equiv_symm_apply],\n        simp only [ne.def, ← equiv.symm_apply_eq, prod.ext_iff, not_and_distrib] at hi ⊢,\n        exact or.inl (ne.symm hi) },\n      { intro h, exact (h (finset.mem_univ _)).elim },\n      { rintro ⟨j⟩ - hj,\n        rw finset.sum_eq_zero,\n        rintro ⟨i⟩ -,\n        rw [biproduct.ι_π_ne_assoc, zero_comp],\n        dsimp [- fin_prod_fin_equiv_symm_apply],\n        simp only [ne.def, ← equiv.symm_apply_eq, prod.ext_iff, not_and_distrib] at hj ⊢,\n        exact or.inr (ne.symm hj) },\n      { intro h, exact (h (finset.mem_univ _)).elim } },\n    { rw [biproduct.ι_π_ne, finset.sum_eq_zero],\n      { rintro j -,\n        rw [finset.sum_eq_zero],\n        rintro i -,\n        by_cases hk : k = fin_prod_fin_equiv (i,j),\n        { subst hk,\n          rw [biproduct.ι_π_self_assoc, biproduct.ι_π_ne],\n          simpa only [ne.def] using h, },\n        { rw [biproduct.ι_π_ne_assoc, zero_comp],\n          dsimp [- fin_prod_fin_equiv_symm_apply],\n          simpa only [ne.def] using h, } },\n      { rw [ne.def], exact h } },\n  end }\n.\n\n@[simps {fully_applied := ff}]\ndef Pow_comp_Pow (m n : ℕ) : (Pow m ⋙ Pow n : 𝒜 ⥤ 𝒜) ≅ Pow (m * n) :=\nnat_iso.of_components (Pow_comp_Pow_components m n) $ λ A B f,\nbegin\n  ext ⟨j⟩ ⟨i⟩ ⟨k⟩,\n  simp only [biproduct.ι_map, Pow_comp_Pow_components_hom, Pow_map, functor.comp_map,\n    biproduct.ι_map_assoc, category.assoc, biproduct.map_π, biproduct.ι_desc_assoc],\nend\n.\n\nlemma _root_.free_abelian_group.eq_zero_induction\n  {α M : Type*} [add_group M] (f : free_abelian_group α → M)\n  (h1 : ∀ a, f (free_abelian_group.of a) = 0) (h2 : ∀ x y, f (x + y) = f x + f y) :\n  ∀ x, f x = 0 :=\nbegin\n  let F := add_monoid_hom.mk' f h2,\n  have hF : ∀ x, F x = f x := λ _, rfl,\n  intro x,\n  refine free_abelian_group.induction_on x _ h1 _ _,\n  { exact F.map_zero },\n  { intros, show F _ = 0, rw [F.map_neg, hF, h1, neg_zero], },\n  { intros x y hx hy, show F _ = 0, rw [F.map_add, hF, hF, hx, hy, add_zero], },\nend\n\nlemma aux' (m n : ℕ) (f : universal_map m n) :\n  F.map ((Pow_comp_Pow 2 m).inv.app A ≫ (Pow m).map (Biprod_iso_Pow_two.inv.app A)) ≫\n    ((eval_Pow_functor F).map f).app (Biprod.obj A) =\n  ((eval_Pow_functor F).map ((mul 2) f)).app A ≫ F.map ((Pow_comp_Pow 2 n).inv.app A ≫\n    (Pow n).map (Biprod_iso_Pow_two.inv.app A)) :=\nbegin\n  rw [← sub_eq_zero],\n  refine free_abelian_group.eq_zero_induction _ _ _ f; clear f,\n  { intro f,\n    rw [sub_eq_zero],\n    dsimp only [eval_Pow_functor],\n    rw [mul_of, eval_Pow_of, eval_Pow_of],\n    dsimp only [whisker_right_app, basic_universal_map.eval_Pow_app],\n    rw [← F.map_comp, ← F.map_comp],\n    congr' 1,\n    dsimp only [Pow_comp_Pow, Biprod_iso_Pow_two],\n    erw [nat_iso.of_components_inv_app, nat_iso.of_components_inv_app,\n      nat_iso.of_components_inv_app],\n    dsimp only [Pow_comp_Pow_components_inv, Biprod_iso_Pow_two_components_inv, Pow_map],\n    apply category_theory.limits.biproduct.hom_ext,\n    rintro ⟨j⟩,\n    apply category_theory.limits.biproduct.hom_ext',\n    refine fin_prod_fin_equiv.forall_congr_left.mp _,\n    rintro ⟨b, i⟩,\n    rw [biproduct.lift_map, biproduct.lift_matrix, biproduct.lift_π, comp_sum,\n      biproduct.lift_map, category.assoc, biproduct.ι_matrix_assoc, biproduct.lift_π],\n    rw [finset.sum_eq_single i],\n    { rw [category.assoc],\n      ext;\n      rw [category.assoc, category.assoc, comp_zsmul, zsmul_comp, comp_zsmul, comp_zsmul,\n        category.comp_id, category.assoc, category.assoc];\n      [rw biprod.lift_fst, rw biprod.lift_snd];\n      rw [biproduct.lift_π, biproduct.lift_π, biproduct.lift_π,\n        biproduct.ι_π, basic_universal_map.mul_apply, matrix.reindex_linear_equiv_apply,\n        matrix.reindex_apply, matrix.submatrix_apply,\n        matrix.kronecker_map, equiv.symm_apply_apply, equiv.symm_apply_apply];\n      simp only [dite_eq_ite, equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff,\n        eq_self_iff_true, eq_to_hom_refl, matrix.one_apply,\n        ite_mul, ite_smul, one_mul, zero_mul, zero_smul, @eq_comm _ b, smul_ite, smul_zero];\n      congr' 1, },\n    { rintro i' - hi',\n      rw [ne.def, eq_comm] at hi',\n      rw [category.assoc],\n      ext;\n      rw [category.assoc, category.assoc, comp_zsmul, zsmul_comp, comp_zsmul, comp_zsmul,\n        category.comp_id, zero_comp];\n      [rw biprod.lift_fst, rw biprod.lift_snd];\n      rw [biproduct.lift_π, biproduct.ι_π];\n      simp only [dite_eq_ite, equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff,\n        eq_self_iff_true, eq_to_hom_refl, equiv.ulift_symm_apply,\n        eq_false_intro hi', and_false, if_false, smul_zero], },\n    { intro h, exact (h (finset.mem_univ _)).elim } },\n  { intros x y,\n    simp only [add_monoid_hom.map_add, functor.map_add, comp_add, add_comp, nat_trans.app_add],\n    abel }\nend\n.\n\n@[simps {fully_applied := ff}]\ndef aux :\n  (data.eval_functor F).obj ((data.mul 2).obj BD'.data) ≅\n  Biprod ⋙ (data.eval_functor F).obj BD'.data :=\nnat_iso.of_components (λ A,\n  homological_complex.hom.iso_of_components (λ i, begin\n      refine F.map_iso _,\n      refine (Pow_comp_Pow 2 (BD'.data.X i)).symm.app A ≪≫ _,\n      refine (Pow _).map_iso (Biprod_iso_Pow_two.symm.app A)\n    end) $ λ i j hij, aux' F A (BD'.data.X i) (BD'.data.X j) (BD'.data.d i j)) $ λ A B f, begin\n      ext i,\n      dsimp only [data.eval_functor, data.eval_functor', eval_Pow, eval_Pow_functor_obj,\n        functor.map_iso_hom, functor.comp_obj, functor.comp_map, functor.flip_obj_map,\n        iso.trans_hom, iso.symm_hom, nat_iso.app_hom,\n        functor.map_homological_complex_obj_X,\n        homological_complex.functor_eval_map_app_f,\n        homological_complex.comp_f,\n        homological_complex.hom.iso_of_components_hom_f],\n      rw [← F.map_comp, ← F.map_comp, ← category.assoc, nat_trans.naturality,\n        category.assoc, category.assoc, functor.comp_map, ← functor.map_comp, ← functor.map_comp,\n        nat_trans.naturality],\n  end\n.\n\n-- move this up\nlemma quux (n : ℕ) {N : ℕ} (k : fin N) (A : 𝒜) :\n  (basic_universal_map.proj n k).eval_Pow.app A =\n  biproduct.matrix (λ i j, if i = fin_prod_fin_equiv (k, j) then 𝟙 A else 0) :=\nbegin\n  apply category_theory.limits.biproduct.hom_ext,\n  rintro ⟨j⟩,\n  apply category_theory.limits.biproduct.hom_ext',\n  refine fin_prod_fin_equiv.forall_congr_left.mp _,\n  rintro ⟨l, i⟩,\n  dsimp only [basic_universal_map.eval_Pow_app],\n  rw [biproduct.matrix_π, biproduct.matrix_π, biproduct.ι_desc, biproduct.ι_desc],\n  dsimp only [basic_universal_map.proj, basic_universal_map.proj_aux,\n    matrix.reindex_linear_equiv_apply, matrix.reindex_apply, matrix.submatrix,\n    matrix.kronecker_map, matrix.of_apply],\n  simp only [ite_mul, ite_smul, one_mul, one_smul, zero_mul, zero_smul, matrix.one_apply],\n  rw [← ite_and],\n  congr' 1,\n  apply propext,\n  rw [← equiv.symm_apply_eq, prod.ext_iff],\n  apply and_congr iff.rfl,\n  dsimp only [equiv.punit_prod_symm_apply],\n  rw [eq_comm],\nend\n.\n\n-- move this up\nlemma eval_Pow_add {m n : ℕ} (f g : basic_universal_map m n) (A : 𝒜) :\n  (f + g).eval_Pow.app A = f.eval_Pow.app A + g.eval_Pow.app A :=\nbegin\n  dsimp [basic_universal_map.eval_Pow_app],\n  ext ⟨i⟩ ⟨j⟩,\n  simp only [biproduct.ι_matrix, biproduct.lift_π, comp_add, add_comp, add_zsmul],\nend\n.\n\ndef eval_functor_homotopy (A : 𝒜) : _root_.homotopy\n  (((data.eval_functor F).obj BD'.data).map (biprod.fst + biprod.snd : A ⊞ A ⟶ A))\n  (((data.eval_functor F).obj BD'.data).map (biprod.fst : A ⊞ A ⟶ A) +\n    ((data.eval_functor F).obj BD'.data).map (biprod.snd : A ⊞ A ⟶ A)) :=\nbegin\n  refine ((eval_homotopy' F BD' A).symm.comp_left ((aux F BD').inv.app A)).congr _ _ _ _,\n  { ext i,\n    rw [homological_complex.comp_f, aux_inv_app_f,\n      functor.map_homological_complex_map_f, functor.comp_map, eval_Pow_functor_map,\n      evaluation_obj_map, data.eval_functor_obj_map_f],\n    dsimp only [data.sum, universal_map.sum],\n    rw [eval_Pow_of, whisker_right_app, ← F.map_comp, fin.sum_univ_two,\n      eval_Pow_add, quux, quux],\n    congr' 1,\n    apply category_theory.limits.biproduct.hom_ext', rintro m,\n    rw [biproduct.ι_desc_assoc, biproduct.ι_map, category.assoc],\n    apply category_theory.limits.biproduct.hom_ext, rintro n,\n    rw [category.assoc],\n    apply category_theory.limits.biprod.hom_ext';\n    [rw [biprod.inl_desc_assoc], rw [biprod.inr_desc_assoc]];\n    rw [category.assoc, biproduct.ι_desc_assoc, add_comp, comp_add,\n      biproduct.matrix_π, biproduct.matrix_π, biproduct.ι_desc, biproduct.ι_desc,\n      category.assoc, add_comp, comp_add];\n    simp only [biprod.inl_fst_assoc, biprod.inl_snd_assoc,\n      biprod.inr_fst_assoc, biprod.inr_snd_assoc, zero_comp, add_zero, zero_add,\n      true_and, equiv.apply_eq_iff_eq, prod.mk.inj_iff, one_ne_zero,\n      fin.zero_eq_one_iff, fin.one_eq_zero_iff, eq_self_iff_true, if_false, false_and],\n      all_goals\n      { by_cases hmn : m = n,\n        { cases hmn, rw [if_pos rfl, biproduct.ι_π_self], },\n        { rw [if_neg, biproduct.ι_π_ne]; [rw [ne.def], skip]; exact hmn } } },\n  { ext i,\n    rw [homological_complex.comp_f, aux_inv_app_f,\n      functor.map_homological_complex_map_f, functor.comp_map, eval_Pow_functor_map,\n      evaluation_obj_map,\n      homological_complex.add_f_apply,\n      data.eval_functor_obj_map_f, data.eval_functor_obj_map_f],\n    dsimp only [data.proj, proj],\n    rw [add_monoid_hom.map_sum, fin.sum_univ_two, eval_Pow_of, eval_Pow_of,\n      nat_trans.app_add, whisker_right_app, whisker_right_app, comp_add,\n      ← F.map_comp, ← F.map_comp, quux, quux],\n    congr' 2;\n    { apply category_theory.limits.biproduct.hom_ext, rintro n,\n      rw [biproduct.map_π, category.assoc, biproduct.matrix_π],\n      apply category_theory.limits.biproduct.hom_ext', rintro m,\n      rw [biproduct.ι_desc_assoc, category.assoc],\n      apply category_theory.limits.biprod.hom_ext';\n      [rw [biprod.inl_desc_assoc], rw [biprod.inr_desc_assoc]],\n      all_goals\n      { rw [biproduct.ι_desc_assoc, biproduct.ι_desc];\n        simp only [true_and, equiv.apply_eq_iff_eq, prod.mk.inj_iff, eq_self_iff_true];\n        by_cases hmn : m = n,\n        { cases hmn,\n          simp only [biproduct.ι_π_self_assoc, eq_self_iff_true, if_true, if_false,\n            biprod.inl_fst, biprod.inr_fst, biprod.inl_snd, biprod.inr_snd,\n            zero_ne_one, one_ne_zero, false_and, fin.one_eq_zero_iff, fin.zero_eq_one_iff], },\n        { rw biproduct.ι_π_ne_assoc, swap, { rw [ne.def], exact hmn },\n          simp only [hmn, if_false, and_false, zero_comp, comp_zero] } } } }\nend\n.\n\n\nend package\n\nend breen_deligne\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/breen_deligne/eval.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.33343488073010585}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport algebraic_topology.split_simplicial_object\nimport for_mathlib.dold_kan.degeneracies\nimport for_mathlib.dold_kan.functor_n\n\n/-!\n\n# Split simplicial objects in preadditive categories\n\nIn this file we define a functor `nondeg_complex : simplicial_object.split C ⥤ chain_complex C ℕ`\nwhen `C` is a preadditive category with finite coproducts, and get an isomorphism\n`to_karoubi_nondeg_complex_iso_N₁ : nondeg_complex ⋙ to_karoubi _ ≅ forget C ⋙ dold_kan.N₁`.\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category\n  category_theory.preadditive category_theory.idempotents opposite\n  algebraic_topology algebraic_topology.dold_kan\n\nopen_locale big_operators simplicial dold_kan\n\nnamespace simplicial_object\n\nnamespace splitting\n\nvariables {C : Type*} [category C] [has_finite_coproducts C]\n  {X : simplicial_object C} (s : splitting X)\n\n/-- The projection on a summand of the coproduct decomposition given\nby a splitting of a simplicial object. -/\ndef π_summand [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :\n  X.obj Δ ⟶ s.N A.1.unop.len :=\nbegin\n  refine (s.iso Δ).inv ≫ sigma.desc (λ B, _),\n  by_cases B = A,\n  { exact eq_to_hom (by { subst h, refl, }), },\n  { exact 0, },\nend\n\n@[simp, reassoc]\nlemma ι_π_summand_eq_id [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A : index_set Δ) :\n  s.ι_summand A ≫ s.π_summand A = 𝟙 _ :=\nbegin\n  dsimp [ι_summand, π_summand],\n  simp only [summand, assoc, is_iso.hom_inv_id_assoc],\n  erw [colimit.ι_desc, cofan.mk_ι_app],\n  dsimp,\n  simp only [eq_self_iff_true, if_true],\nend\n\n@[simp, reassoc]\nlemma ι_π_summand_eq_zero [has_zero_morphisms C] {Δ : simplex_categoryᵒᵖ} (A B : index_set Δ)\n  (h : B ≠ A) : s.ι_summand A ≫ s.π_summand B = 0 :=\nbegin\n  dsimp [ι_summand, π_summand],\n  simp only [summand, assoc, is_iso.hom_inv_id_assoc],\n  erw [colimit.ι_desc, cofan.mk_ι_app],\n  apply dif_neg,\n  exact h.symm,\nend\n\nvariable [preadditive C]\n\nlemma decomposition_id (Δ : simplex_categoryᵒᵖ) :\n  𝟙 (X.obj Δ) = ∑ (A : index_set Δ), s.π_summand A ≫ s.ι_summand A :=\nbegin\n  apply s.hom_ext',\n  intro A,\n  rw [comp_id, comp_sum, finset.sum_eq_single A, ι_π_summand_eq_id_assoc],\n  { intros B h₁ h₂,\n    rw [s.ι_π_summand_eq_zero_assoc _ _ h₂, zero_comp], },\n  { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\nend\n\n@[simp, reassoc]\nlemma σ_comp_π_summand_id_eq_zero {n : ℕ} (i : fin (n+1)) :\n  X.σ i ≫ s.π_summand (index_set.id (op [n+1])) = 0 :=\nbegin\n  apply s.hom_ext',\n  intro A,\n  dsimp only [simplicial_object.σ],\n  rw [comp_zero, s.ι_summand_epi_naturality_assoc A (simplex_category.σ i).op,\n    ι_π_summand_eq_zero],\n  symmetry,\n  change ¬ (A.epi_comp (simplex_category.σ i).op).eq_id,\n  rw index_set.eq_id_iff_len_eq,\n  have h := simplex_category.len_le_of_epi (infer_instance : epi A.e),\n  dsimp at ⊢ h,\n  linarith,\nend\n\n/-- If a simplicial object `X` in an additive category is split,\nthen `P_infty` vanishes on all the summands of `X _[n]` which do\nnot correspond to the identity of `[n]`. -/\nlemma ι_summand_comp_P_infty_eq_zero {X : simplicial_object C}\n  (s : simplicial_object.splitting X)\n  {n : ℕ} (A : simplicial_object.splitting.index_set (op [n]))\n  (hA : ¬ A.eq_id) :\n  s.ι_summand A ≫ P_infty.f n = 0 :=\nbegin\n  rw simplicial_object.splitting.index_set.eq_id_iff_mono at hA,\n  rw [simplicial_object.splitting.ι_summand_eq, assoc,\n    degeneracy_comp_P_infty X n A.e hA, comp_zero],\nend\n\nlemma comp_P_infty_eq_zero_iff {Z : C} {n : ℕ} (f : Z ⟶ X _[n]) :\n  f ≫ P_infty.f n = 0 ↔ f ≫ s.π_summand (index_set.id (op [n])) = 0 :=\nbegin\n  split,\n  { intro h,\n    cases n,\n    { dsimp at h,\n      rw [comp_id] at h,\n      rw [h, zero_comp], },\n    { have h' := f ≫= P_infty_f_add_Q_infty_f (n+1),\n      dsimp at h',\n      rw [comp_id, comp_add, h, zero_add] at h',\n      rw [← h', assoc, Q_infty_f, decomposition_Q, preadditive.sum_comp,\n        preadditive.comp_sum, finset.sum_eq_zero],\n      intros i hi,\n      simp only [assoc, σ_comp_π_summand_id_eq_zero, comp_zero], }, },\n  { intro h,\n    rw [← comp_id f, assoc, s.decomposition_id, preadditive.sum_comp,\n      preadditive.comp_sum, fintype.sum_eq_zero],\n    intro A,\n    by_cases hA : A.eq_id,\n    { dsimp at hA,\n      subst hA,\n      rw [assoc, reassoc_of h, zero_comp], },\n    { simp only [assoc, s.ι_summand_comp_P_infty_eq_zero A hA, comp_zero], }, },\nend\n\n@[simp, reassoc]\nlemma P_infty_comp_π_summand_id (n : ℕ) :\n  P_infty.f n ≫ s.π_summand (index_set.id (op [n])) = s.π_summand (index_set.id (op [n])) :=\nbegin\n  conv_rhs { rw ← id_comp (s.π_summand _), },\n  symmetry,\n  rw [← sub_eq_zero, ← sub_comp, ← comp_P_infty_eq_zero_iff, sub_comp, id_comp,\n    P_infty_f_idem, sub_self],\nend\n\n@[simp, reassoc]\nlemma π_summand_comp_ι_summand_comp_P_infty_eq_P_infty (n : ℕ) :\n  s.π_summand (index_set.id (op [n])) ≫ s.ι_summand (index_set.id (op [n])) ≫ P_infty.f n =\n    P_infty.f n :=\nbegin\n  conv_rhs { rw ← id_comp (P_infty.f n), },\n  erw [s.decomposition_id, preadditive.sum_comp],\n  rw [fintype.sum_eq_single (index_set.id (op [n])), assoc],\n  rintros A (hA : ¬A.eq_id),\n  rw [assoc, s.ι_summand_comp_P_infty_eq_zero A hA, comp_zero],\nend\n\n/-- The differentials `s.d i j : s.N i ⟶ s.N j` on nondegenerate simplices of a split\nsimplicial object are induced by the differentials on the alternating face map complex. -/\n@[simp]\ndef d (i j : ℕ) : s.N i ⟶ s.N j :=\ns.ι_summand (index_set.id (op [i])) ≫ K[X].d i j ≫ s.π_summand (index_set.id (op [j]))\n\nlemma ι_summand_comp_d_comp_π_summand_eq_zero (j k : ℕ) (A : index_set (op [j])) (hA : ¬A.eq_id) :\n  s.ι_summand A ≫ K[X].d j k ≫ s.π_summand (index_set.id (op [k])) = 0 :=\nbegin\n  rw A.eq_id_iff_mono at hA,\n  rw [← assoc, ← s.comp_P_infty_eq_zero_iff, assoc, ← P_infty.comm j k, s.ι_summand_eq, assoc,\n    degeneracy_comp_P_infty_assoc X j A.e hA, zero_comp, comp_zero],\nend\n\n/-- If `s` is a splitting of a simplicial object `X` in a preadditive category,\n`s.nondeg_complex` is a chain complex which is given in degree `n` by\nthe nondegenerate `n`-simplices of `X`. -/\n@[simps]\ndef nondeg_complex : chain_complex C ℕ :=\n{ X := s.N,\n  d := s.d,\n  shape' := λ i j hij, by simp only [d, K[X].shape i j hij, zero_comp, comp_zero],\n  d_comp_d' := λ i j k hij hjk, begin\n    simp only [d, assoc],\n    have eq : K[X].d i j ≫ 𝟙 (X.obj (op [j])) ≫ K[X].d j k ≫\n      s.π_summand (index_set.id (op [k])) = 0 :=\n      by erw [id_comp, homological_complex.d_comp_d_assoc, zero_comp],\n    rw s.decomposition_id at eq,\n    classical,\n    rw [fintype.sum_eq_add_sum_compl (index_set.id (op [j])), add_comp, comp_add, assoc,\n      preadditive.sum_comp, preadditive.comp_sum, finset.sum_eq_zero, add_zero] at eq, swap,\n    { intros A hA,\n      simp only [finset.mem_compl, finset.mem_singleton] at hA,\n      simp only [assoc, ι_summand_comp_d_comp_π_summand_eq_zero _ _ _ _ hA, comp_zero], },\n    rw [eq, comp_zero],\n  end }\n\n/-- The chain complex `s.nondeg_complex` attached to a splitting of a simplicial object `X`\nbecomes isomorphic to the normalized Moore complex `N₁.obj X` defined as a formal direct\nfactor in the category `karoubi (chain_complex C ℕ)`. -/\n@[simps]\ndef to_karoubi_nondeg_complex_iso_N₁ : (to_karoubi _).obj s.nondeg_complex ≅ N₁.obj X :=\n{ hom :=\n  { f :=\n    { f := λ n, s.ι_summand (index_set.id (op [n])) ≫ P_infty.f n,\n      comm' := λ i j hij, begin\n        dsimp,\n        rw [assoc, assoc, assoc, π_summand_comp_ι_summand_comp_P_infty_eq_P_infty,\n          homological_complex.hom.comm],\n      end, },\n    comm := by { ext n, dsimp, rw [id_comp, assoc, P_infty_f_idem], }, },\n  inv :=\n  { f :=\n    { f := λ n, s.π_summand (index_set.id (op [n])),\n      comm' := λ i j hij, begin\n        dsimp,\n        slice_rhs 1 1 { rw ← id_comp (K[X].d i j), },\n        erw s.decomposition_id,\n        rw [sum_comp, sum_comp, finset.sum_eq_single (index_set.id (op [i])), assoc, assoc],\n        { intros A h hA,\n          simp only [assoc, s.ι_summand_comp_d_comp_π_summand_eq_zero _ _ _ hA, comp_zero], },\n        { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\n      end, },\n    comm := by { ext n, dsimp, simp only [comp_id, P_infty_comp_π_summand_id], }, },\n  hom_inv_id' := begin\n    ext n,\n    simpa only [assoc, P_infty_comp_π_summand_id, karoubi.comp_f,\n      homological_complex.comp_f, ι_π_summand_eq_id],\n  end,\n  inv_hom_id' := begin\n    ext n,\n    simp only [π_summand_comp_ι_summand_comp_P_infty_eq_P_infty, karoubi.comp_f,\n      homological_complex.comp_f, N₁_obj_p, karoubi.id_eq],\n  end, }\n\nend splitting\n\nnamespace split\n\nvariables {C : Type*} [category C] [preadditive C] [has_finite_coproducts C]\n\n/-- The functor which sends a split simplicial object in a preadditive category to\nthe chain complex which consists of nondegenerate simplices. -/\n@[simps]\ndef nondeg_complex_functor : split C ⥤ chain_complex C ℕ :=\n{ obj := λ S, S.s.nondeg_complex,\n  map := λ S₁ S₂ Φ,\n  { f := Φ.f,\n    comm' := λ i j hij, begin\n      dsimp,\n      erw [← ι_summand_naturality_symm_assoc Φ (splitting.index_set.id (op [i])),\n        ((alternating_face_map_complex C).map Φ.F).comm_assoc i j],\n      simp only [assoc],\n      congr' 2,\n      apply S₁.s.hom_ext',\n      intro A,\n      dsimp [alternating_face_map_complex],\n      erw ι_summand_naturality_symm_assoc Φ A,\n      by_cases A.eq_id,\n      { dsimp at h,\n        subst h,\n        simpa only [splitting.ι_π_summand_eq_id, comp_id, splitting.ι_π_summand_eq_id_assoc], },\n      { have h' : splitting.index_set.id (op [j]) ≠ A := by { symmetry, exact h, },\n        rw [S₁.s.ι_π_summand_eq_zero_assoc _ _ h', S₂.s.ι_π_summand_eq_zero _ _ h',\n          zero_comp, comp_zero], },\n    end }, }\n\n/-- The natural isomorphism (in `karoubi (chain_complex C ℕ)`) between the chain complex\nof nondegenerate simplices of a split simplicial object and the normalized Moore complex\ndefined as a formal direct factor of the alternating face map complex. -/\n@[simps]\ndef to_karoubi_nondeg_complex_functor_iso_N₁ :\n  nondeg_complex_functor ⋙ to_karoubi (chain_complex C ℕ) ≅ forget C ⋙ dold_kan.N₁ :=\nnat_iso.of_components (λ S, S.s.to_karoubi_nondeg_complex_iso_N₁)\n  (λ S₁ S₂ Φ, begin\n    ext n,\n    dsimp,\n    simp only [karoubi.comp_f, to_karoubi_map_f, homological_complex.comp_f,\n      nondeg_complex_functor_map_f, splitting.to_karoubi_nondeg_complex_iso_N₁_hom_f_f,\n      N₁_map_f, alternating_face_map_complex.map_f, assoc, P_infty_f_idem_assoc],\n    erw ← split.ι_summand_naturality_symm_assoc Φ (splitting.index_set.id (op [n])),\n    rw P_infty_f_naturality,\n  end)\n\n\n--@[simps]\n--def nondeg_complex_functor_iso_forget_comp_normalized_Moore_complex {A : Type*} [category A] [abelian A] :\n--  nondeg_complex_functor ≅ forget A ⋙ normalized_Moore_complex A :=\n--(whiskering_right_to_karoubi_iso_equiv _ _).inv_fun\n--  (to_karoubi_N'_iso_N₁\n--    ≪≫ iso_whisker_left _ (N₁_iso_normalized_Moore_complex_comp_to_karoubi A))\n\nend split\n\nend simplicial_object\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/dold_kan/split_simplicial_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.33343488073010585}}
{"text": "import Lean\nimport Mathlib.Tactic.RCases\n\nimport Verbose.Common\n\nopen Lean\n\nopen Lean.Parser.Tactic\nopen Lean Meta\n\ndef MaybeTypedIdent := Name × Option Syntax\n\nopen Lean Elab Tactic\nopen Option\nopen RCases\n\n-- TODO: replace Syntax.missing by something smarter\ndef RCasesPattOfMaybeTypedIdent : MaybeTypedIdent → RCasesPatt\n| (n, some pe) => RCasesPatt.typed Syntax.missing (RCasesPatt.one Syntax.missing  n) pe\n| (n, none)    => RCasesPatt.one Syntax.missing n\n\ndef obtainTac (fact : Term) (news : Array MaybeTypedIdent) : TacticM Unit :=\ndo\n  let orig_goal ← getMainGoal\n  for new in news do\n    checkName new.1\n  let applied_fact_expr : Expr ← elabTerm fact none\n  let news := Array.toList news\n  match news with\n  | [(name, stuff)] => do    \n    let type ← inferType applied_fact_expr\n    if let some new := stuff then \n      if not (← isDefEq (← elabTerm new type) type) then throwError \"No way\"\n    let intermediate_goal ← assert orig_goal name type (← elabTerm fact none)\n    let (_, new_goal) ← intro1P intermediate_goal\n    replaceMainGoal [new_goal]\n  | news =>\n    let news_patt := news.map RCasesPattOfMaybeTypedIdent\n    let new_goals ← rcases #[(none, fact)] (RCasesPatt.tuple Syntax.missing news_patt) (← getMainGoal)\n    replaceMainGoal new_goals\n\ndeclare_syntax_cat maybeTypedIdent\nsyntax ident : maybeTypedIdent\nsyntax \"(\"ident \":\" term\")\" : maybeTypedIdent\n\n-- We could also use the less specific type `Syntax → MaybeTypedIdent`\ndef toMaybeTypedIdent : TSyntax `maybeTypedIdent → MaybeTypedIdent \n| `(maybeTypedIdent| ($x:ident : $type:term)) => (x.getId, type)\n| `(maybeTypedIdent| $x:ident) => (x.getId, none)\n| _ => (Name.anonymous, none) -- This should never happen\n\ndeclare_syntax_cat maybeApplied\nsyntax term : maybeApplied\nsyntax term \"applied to\" term : maybeApplied\nsyntax term \"applied to [\" term,* \"]\" : maybeApplied\n\ndef maybeAppliedToTerm : TSyntax `maybeApplied → MetaM Term\n| `(maybeApplied| $e:term) => pure e\n| `(maybeApplied| $e:term applied to $x:term) => `($e $x)\n| `(maybeApplied| $e:term applied to [$args:term,*]) => `($e $args*)\n| _ => pure ⟨Syntax.missing⟩ -- This should never happen\n\ndeclare_syntax_cat newStuff\nsyntax maybeTypedIdent : newStuff\nsyntax maybeTypedIdent \"such that\" maybeTypedIdent,* : newStuff\n\ndef newStuffToArray : TSyntax `newStuff → Array MaybeTypedIdent\n| `(newStuff| $x:maybeTypedIdent) => #[toMaybeTypedIdent x]\n| `(newStuff| $x:maybeTypedIdent such that $news:maybeTypedIdent,*) => \n    #[toMaybeTypedIdent x] ++ (Array.map toMaybeTypedIdent news)\n| _ => #[]\n\nelab \"By \" e:maybeApplied \"we get \" colGt news:newStuff : tactic => do\nobtainTac (← maybeAppliedToTerm e)  (newStuffToArray news)\n\nexample (P : Nat → Prop) (h : ∀ n, P n) : P 0 := by\n  By h applied to 0 we get h₀\n  exact h₀\n\nexample (P : Nat → Nat → Prop) (h : ∀ n k, P n (k+1)) : P 0 1 := by\n  --rcases h 0 0 with (h₀ : P 0 0)\n  By h applied to [0, 0] we get (h₀ : P 0 1)\n  exact h₀\n\nexample (n : Nat) (h : ∃ k, n = 2*k) : True := by\n  By h we get k such that (H : n = 2*k)\n  trivial\n\nexample (n : Nat) (h : ∃ k, n = 2*k) : True := by\n  By h we get k such that H\n  trivial", "meta": {"author": "PatrickMassot", "repo": "verbose-lean4", "sha": "0078291a4db4b6a0b14a8f34fb74cb2f1c6ae1ee", "save_path": "github-repos/lean/PatrickMassot-verbose-lean4", "path": "github-repos/lean/PatrickMassot-verbose-lean4/verbose-lean4-0078291a4db4b6a0b14a8f34fb74cb2f1c6ae1ee/Verbose/By.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244837, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.33342145261671385}}
{"text": "import Lean\nopen Lean.SubExpr\n\ndef ps := [#[], #[0], #[1], #[0,1], #[1,0] , #[0,0], #[1,2,3]]\ntheorem Pos.roundtrip :\n  true = ps.all fun x => x == (Pos.toArray <| Pos.ofArray <| x)\n  := by native_decide\n\ntheorem Pos.append_roundtrip :\n  true = (List.all\n    (ps.bind fun p => ps.map fun q => (p,q))\n    (fun (x,y) => (x ++ y) == (Pos.toArray <| (Pos.append (Pos.ofArray x) (Pos.ofArray y))))\n  ) := by native_decide\n\ntheorem Pos.stringRoundtrip :\n  true = ps.all (fun p =>\n    let x := Pos.ofArray p\n    some x == (Except.toOption $ Pos.fromString? $ Pos.toString $ x)\n  ) := by native_decide\n\n#eval Pos.toString Nat.zero\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/subexpr.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3333221631901137}}
{"text": "/-\nCopyright (c) 2021 Gabriel Ebner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Gabriel Ebner\n-/\nimport Lean\nimport Mathlib.Tactic.Cache\nimport Mathlib.Tactic.RCases\n\nopen Tactic\n\nnamespace Mathlib.Tactic.Ext\nopen Lean Meta\n\ndef withExtHyps (struct : Name) (k : Array Expr → (x y : Expr) → Array (Name × Expr) → MetaM α) : MetaM α := do\n  unless isStructure (← getEnv) struct do throwError \"not a structure: {struct}\"\n  let structC ← mkConstWithLevelParams struct\n  forallTelescope (← inferType structC) fun params _ => do\n  withNewBinderInfos (params.map (·.fvarId!, BinderInfo.implicit)) do\n  withLocalDeclD `x (mkAppN structC params) fun x => do\n  withLocalDeclD `y (mkAppN structC params) fun y => do\n    let mut hyps := #[]\n    for field in getStructureFieldsFlattened (← getEnv) struct (includeSubobjectFields := false) do\n      let x_f ← mkProjection x field\n      let y_f ← mkProjection y field\n      if ← isProof x_f then\n        pure ()\n      else if ← isDefEq (← inferType x_f) (← inferType y_f) then\n        hyps := hyps.push (field, ← mkEq x_f y_f)\n      else\n        hyps := hyps.push (field, ← mkHEq x_f y_f)\n    k params x y hyps\n\nscoped elab \"ext_type%\" struct:ident : term => do\n  withExtHyps (← resolveGlobalConstNoOverload struct) fun params x y hyps => do\n    let mut ty ← mkEq x y\n    for (f, h) in hyps.reverse do\n      ty := mkForall f BinderInfo.default h ty\n    mkForallFVars (params |>.push x |>.push y) ty\n\ndef mkIff (p q : Expr) : Expr := mkApp2 (mkConst ``Iff) p q\n\ndef mkAndN : List Expr → Expr\n  | [] => mkConst ``True\n  | [p] => p\n  | [p, q] => mkAnd p q\n  | p :: ps => mkAnd p (mkAndN ps)\n\nscoped elab \"ext_iff_type%\" struct:ident : term => do\n  withExtHyps (← resolveGlobalConstNoOverload struct) fun params x y hyps => do\n    mkForallFVars (params |>.push x |>.push y) <|\n      mkIff (← mkEq x y) <| mkAndN (hyps.map (·.2)).toList\n\nelab \"subst_eqs\" : tactic =>\n  open Elab.Tactic in\n  liftMetaTactic1 fun mvarId => substEqs mvarId\n\nscoped macro \"ext_proof%\" : term =>\n  `(fun {..} {..} => by intros; subst_eqs; rfl)\n\nsyntax \"split_ands\" : tactic\nmacro_rules | `(tactic| split_ands) => `(tactic| skip)\nmacro_rules | `(tactic| split_ands) => `(tactic| refine And.intro ?_ ?_ <;> split_ands)\n\nmacro_rules | `(tactic| rfl) => `(tactic| exact HEq.rfl)\n\nscoped macro \"ext_iff_proof%\" : term => `(fun {..} {..} =>\n  ⟨fun _ => by subst_eqs; split_ands <;> rfl,\n   fun _ => by (repeat cases ‹_ ∧ _›); subst_eqs; rfl⟩)\n\nscoped macro \"declareExtTheoremsFor\" struct:ident : command => do\n  let extName := mkIdent <| struct.getId.eraseMacroScopes.mkStr \"ext\"\n  let extIffName := mkIdent <| struct.getId.eraseMacroScopes.mkStr \"ext_iff\"\n  `(@[ext] protected theorem $extName:ident : ext_type% $struct:ident := ext_proof%\n    protected theorem $extIffName:ident : ext_iff_type% $struct:ident := ext_iff_proof%)\n\nopen Elab.Command MonadRecDepth in\ndef liftCommandElabM (k : CommandElabM α) : AttrM α := do\n  let (a, commandState) ←\n    k.run { fileName := (← getEnv).mainModule.toString, fileMap := default } |>.run {\n      env := ← getEnv, maxRecDepth := ← getMaxRecDepth,\n      scopes := [{ header := \"\", opts := ← getOptions }]\n    }\n  modify fun coreState => { coreState with\n    traceState.traces := coreState.traceState.traces ++ commandState.traceState.traces\n    env := commandState.env\n  }\n  if let some err := commandState.messages.msgs.toArray.find?\n      (·.severity matches MessageSeverity.error) then\n    throwError err.data\n  pure a\n\ninitialize extExtension : SimpleScopedEnvExtension (Name × Array DiscrTree.Key) (DiscrTree Name) ←\n  registerSimpleScopedEnvExtension {\n    name := `ext\n    addEntry := fun dt (n, ks) => dt.insertCore ks n\n    initial := {}\n  }\n\ndef extAttribute : AttributeImpl where\n  name := `ext\n  descr := \"Marks a lemma as extensionality lemma\"\n  add decl stx kind := do\n    if isStructure (← getEnv) decl then\n      liftCommandElabM do\n        Elab.Command.elabCommand <|<- `(declareExtTheoremsFor $(mkIdent decl))\n    else MetaM.run' do\n      let declTy := (← getConstInfo decl).type\n      let (xs, bis, declTy) ← withReducible <| forallMetaTelescopeReducing declTy\n      if declTy.isAppOfArity ``Eq 3 && (declTy.getArg! 1).isMVar && (declTy.getArg! 2).isMVar then\n        let ty := declTy.getArg! 0\n        let key ←\n          if (← withReducible <| whnf ty).isForall then\n            pure #[DiscrTree.Key.star] -- FIXME: workaround\n          else\n            withReducible <| DiscrTree.mkPath ty\n        extExtension.add (decl, key) kind\n      else\n        throwError \"@[ext] attribute only applies to structures or lemmas proving x = y, got {declTy}\"\n\ninitialize registerBuiltinAttribute extAttribute\n\ndef extLemmas (env : Environment) : DiscrTree Name :=\n  extExtension.getState env\n\nopen Lean.Elab.Tactic in\nelab \"apply_ext_lemma\" : tactic => do\n  let tgt ← getMainTarget\n  unless tgt.isAppOfArity ``Eq 3 do\n    throwError \"applyExtLemma only applies to equations, not{indentExpr tgt}\"\n  let ty := tgt.getArg! 0\n  let s ← saveState\n  for lem in ← (extLemmas (← getEnv)).getMatch ty do\n    try\n      liftMetaTactic (apply · (← mkConstWithFreshMVarLevels lem))\n      return\n    catch e =>\n      s.restore\n  throwError \"no applicable extensionality lemma found for{indentExpr ty}\"\n\nscoped syntax \"ext_or_skip\" (ppSpace rintroPat)* : tactic\nmacro_rules | `(tactic| ext_or_skip) => `(tactic| skip)\nmacro_rules\n| `(tactic| ext_or_skip $xs:rintroPat*) =>\n  `(tactic| apply_ext_lemma; ext_or_skip $xs:rintroPat*)\nmacro_rules\n| `(tactic| ext_or_skip $x:rintroPat $xs:rintroPat*) =>\n  `(tactic| rintro $x:rintroPat; ext_or_skip $xs:rintroPat*)\n\n-- TODO: support `ext : n`\n\nsyntax \"ext\" (colGt ppSpace rintroPat)* (\" : \" num)? : tactic\nmacro_rules\n| `(tactic| ext) => do\n  `(tactic| first | intro; ext | apply_ext_lemma; ext | skip)\nmacro_rules\n| `(tactic| ext $xs:rintroPat*) =>\n  `(tactic| apply_ext_lemma; ext_or_skip $xs:rintroPat*)\n\nsyntax \"ext1\" (colGt ppSpace rintroPat)* : tactic\nmacro_rules\n| `(tactic| ext1 $xs:rintroPat*) =>\n  `(tactic| apply_ext_lemma; rintro $xs:rintroPat*)\n\n-- TODO\nsyntax \"ext1?\" (colGt ppSpace rintroPat)* : tactic\nsyntax \"ext?\" (colGt ppSpace rintroPat)* (\" : \" num)? : tactic\n", "meta": {"author": "JOSHCLUNE", "repo": "Keller_reduction", "sha": "dc392b3da352fc1ffcfbecb1d4717d05f5faed4a", "save_path": "github-repos/lean/JOSHCLUNE-Keller_reduction", "path": "github-repos/lean/JOSHCLUNE-Keller_reduction/Keller_reduction-dc392b3da352fc1ffcfbecb1d4717d05f5faed4a/Lean4_Clique/Mathlib/Mathlib/Tactic/Ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33318324344340533}}
{"text": "\nimport temporal_logic.basic\nimport tactic.squeeze\n\nuniverse variables u u₀ u₁ u₂\n\nvariables {α : Sort u₀} {β : Sort u₁} {γ : Sort u₂}\n\nnamespace temporal\nopen predicate\n\nclass postponable (p : cpred) : Prop :=\n  (postpone : ◇p = p)\nexport postponable (postpone)\n\ninstance henceforth_persistent {p : cpred} : persistent (◻p) :=\nby { constructor, simp only [temporal.henceforth_henceforth, eq_self_iff_true] with tl_simp }\n\ninstance persistent_not {p : cpred} [postponable p] : persistent (-p) :=\nby { constructor, rw [← not_eventually, postpone p] }\n\ninstance leads_to_persistent {p q : cpred} : persistent (p ~> q) :=\nby { constructor, simp only [tl_leads_to, is_persistent, eq_self_iff_true] }\n\ninstance and_persistent {p q : cpred} [persistent p] [persistent q]\n: persistent (p ⋀ q) :=\nby { constructor, simp only [henceforth_and, is_persistent, eq_self_iff_true], }\n\ninstance coe_persistent (p : Prop)\n: persistent (p : cpred) :=\nby { constructor, cases classical.prop_complete p ; subst p ;\n     simp only [eq_self_iff_true, temporal.hence_false, predicate.coe_false,predicate.coe_true, eq_self_iff_true, temporal.hence_true] with tl_simp, }\n\ninstance false_persistent\n: persistent (False : cpred) :=\nby { constructor, simp only [eq_self_iff_true, temporal.hence_false] with tl_simp, }\n\ninstance forall_persistent {p : α → cpred} [∀ i, persistent (p i)]\n: persistent (p_forall p) :=\nby { constructor, simp only [henceforth_forall, is_persistent, eq_self_iff_true], }\n\n\ninstance exists_persistent {p : α → cpred} [∀ i, persistent (p i)]\n: persistent (p_exists p) :=\nby { constructor, apply mutual_entails,\n     apply henceforth_str,\n     apply p_exists_elim, intro, rw ← is_persistent (p x),\n     mono, apply p_exists_intro, }\n\ninstance (p : cpred) : postponable (◇p) :=\nby { constructor, simp only [eventually_eventually, temporal.eventually_eventually, eq_self_iff_true] }\n\ninstance postponable_not {p : cpred} [persistent p] : postponable (-p) :=\nby { constructor, rw [← not_henceforth, is_persistent p] }\n\ninstance or_postponable {p q : cpred} [postponable p] [postponable q]\n: postponable (p ⋁ q) :=\nby { constructor, simp only [eventually_or, postpone, eq_self_iff_true], }\n\ninstance imp_postponable {p q : cpred} [persistent p] [postponable q]\n: postponable (p ⟶ q) :=\nby { simp only [p_imp_iff_p_not_p_or], apply_instance }\n\ninstance coe_postponable (p : Prop)\n: postponable (p : cpred) :=\nby { constructor, cases classical.prop_complete p ; subst p ; simp only [temporal.event_false, eq_self_iff_true, predicate.coe_false, predicate.coe_true, temporal.eventually_true, eq_self_iff_true] with tl_simp, }\n\ninstance forall_postponable (p : α → cpred) [∀ i, postponable (p i)]\n: postponable (p_forall p) :=\n⟨ begin\n    apply mutual_entails,\n    { rw [p_entails_of_fun],\n      introv h, rw p_forall_to_fun, intro i,\n      rw ← postpone (p i), revert h, apply p_impl_revert,\n      revert Γ, change (_ ⟹ _),\n      mono, rw [p_entails_of_fun],\n      introv h, apply p_forall_revert h },\n    apply eventually_weaken\n  end ⟩\n\ninstance exists_postponable (p : α → cpred) [∀ i, postponable (p i)]\n: postponable (p_exists p) :=\nby constructor ; simp only [eventually_exists, postpone, eq_self_iff_true]\n\ninstance lifted₀_postponable (c : Prop) : postponable (lifted₀ c) :=\nby { constructor, ext, simp only [lifted₀, eventually, iff_self, predicate.lifted₀, exists_const] }\n\ninstance lifted₀_persistent (c : Prop) : persistent (lifted₀ c) :=\nby { constructor, ext, simp only [lifted₀, henceforth, forall_const, iff_self, predicate.lifted₀] }\n\ninstance True_postponable : postponable True :=\nby { dunfold True, apply_instance }\n\ninstance True_persistent : persistent True :=\nby { dunfold True, apply_instance }\n\ninstance False_postponable : postponable False :=\nby { dunfold False, apply_instance }\n\ninstance False_persistent : persistent False :=\nby { dunfold False, apply_instance }\n\n-- instance not_forall_persistent {p : α → cpred} [∀ i, persistent (- p i)]\n-- : persistent (- p_forall p) :=\n-- by { constructor, squeeze_simp [p_not_p_forall], apply is_persistent }\n\ninductive list_persistent : list cpred → Prop\n | nil_persistent : list_persistent []\n | cons_persistent (x : cpred) (xs : list $ cpred)\n   [persistent x]\n   (h : list_persistent xs)\n : list_persistent (x :: xs)\n\nexport list_persistent (nil_persistent)\n\ndef with_h_asms (Γ : cpred) : Π (xs : list (cpred)) (x : cpred), Prop\n | [] x := Γ ⊢ x\n | (x :: xs) y := Γ ⊢ x → with_h_asms xs y\n\nlemma indirect_judgement (h p : pred' β)\n  (H : ∀ Γ, Γ ⊢ h → Γ ⊢ p)\n: h ⊢ p :=\nby { apply H, lifted_pred keep, assumption }\n\nlemma judgement_trans (p q r : pred' β)\n  (h₀ : p ⊢ q)\n  (h₁ : q ⊢ r)\n: p ⊢ r :=\nby { lifted_pred keep,\n     apply h₁.apply,\n     apply h₀.apply _ a }\n\n@[trans]\nlemma judgement_trans' {p q r : pred' β}\n  (h₀ : p ⊢ q)\n  (h₁ : q ⊢ r)\n: p ⊢ r :=\njudgement_trans _ _ _ h₀ h₁\n\nlemma p_imp_postpone (Γ p q : cpred)\n  [persistent Γ]\n  [postponable q]\n  (h : ctx_impl Γ p q)\n: ctx_impl Γ (◇p) q :=\nbegin\n  rw ← postpone q,\n  mono,\nend\n\nlemma persistent_to_henceforth {p q : cpred}\n  [persistent p]\n  (h : p ⊢ q)\n: p ⊢ ◻ q :=\nby { rw ← is_persistent p,\n     lifted_pred keep, intro i,\n     apply h.apply _ (a i), }\n\nlemma henceforth_deduction {Γ p q : cpred}\n  (h : Γ ⊢ ◻(p ⟶ q))\n: Γ ⊢ p → Γ ⊢ q :=\nhenceforth_str (p ⟶ q) Γ h\n\ninstance has_coe_to_fun_henceforth (Γ p q : cpred) : has_coe_to_fun (Γ ⊢ ◻(p ⟶ q)) :=\n{ F := λ _, Γ ⊢ p → Γ ⊢ q\n, coe :=  henceforth_deduction }\n\ninstance has_coe_to_fun_leads_to (Γ p q : cpred) : has_coe_to_fun (Γ ⊢ p ~> q) :=\ntemporal.has_coe_to_fun_henceforth _ _ _\n\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/persistent.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.333183235354062}}
{"text": "\n\nimport M.Pluto.Language.Constant\nimport M.PlutusCore\n\n\nnamespace Pluto.Proof\n\n\nopen Pluto.Language\nopen PlutusCore (ByteString Data)\n\n\ntheorem show_constant_integer (i : Int) : toString (I ann i) = toString i :=\n  rfl\n\n\ntheorem show_constant_bytestring (s : ByteString) : toString (S ann s) = \"0x\" ++ toString s :=\n  rfl\n\n  \ntheorem show_constant_text (t : String) : toString (T ann t) = \"\\\"\" ++ toString t ++ \"\\\"\" :=\n  rfl\n\n  \ntheorem show_constant_unit : toString (U ann ) = \"()\" :=\n  rfl\n\n  \ntheorem show_constant_true : toString (B ann true) = \"True\" :=\n  rfl\n\n  \ntheorem show_constant_false : toString (B ann false) = \"False\" :=\n  rfl\n\ntheorem show_constant_data_integer : toString (D ann (Data.I i)) = \"data \" ++ toString i :=\n  by simp [toString, showData]\n\n\ntheorem show_constant_data_bytestring : toString (D ann (Data.B b)) = \"data \" ++ (\"0x\" ++ toString b) :=\n  by simp [toString, showData]\n\n  \nexample : toString (I () 3                                           ) = \"3\"        := rfl\n\nexample : toString (S () $ ByteString.mk $ ByteArray.mk #[0xca, 0xfe]) = \"0xcafe\"   := rfl\n\nexample : toString (T () \"test\"                                      ) = \"\\\"test\\\"\" := rfl\n\nexample : toString (U ()                                             ) = \"()\"       := rfl\n\nexample : toString (B () true                                        ) = \"True\"     := rfl\n\nexample : toString (B () false                                       ) = \"False\"    := rfl\n\n\nprivate def testData :=\n  Data.List\n    [\n      Data.Map []\n    , Data.Map [\n        (Data.I 3, Data.List [])\n      ]\n    , Data.Map [\n        (Data.B $ ByteString.mk $ ByteArray.mk #[0xca, 0xfe], Data.List [Data.I 2])\n      ]\n    , Data.Map [\n        (Data.I 10, Data.I 20)\n      , (Data.I 30, Data.I 40)\n    ]\n    , Data.Constr 2 [Data.I 9, Data.I (-10)]\n    , Data.Constr 1 [Data.I (-11)]\n    , Data.Constr 0 []\n    ]\n\nexample : toString (D () testData) = \"data [{},{3=[]},{0xcafe=[2]},{10=20,30=40},sigma2.[9,-10],sigma1.[-11],sigma0.[]]\" :=\n  by simp [toString, testData, showData, intercalateData, intercalateData']\n\n\nend Pluto.Proof\n", "meta": {"author": "bwbush", "repo": "marlowe-lean4", "sha": "318a224149b84e5b8d62c631e7684723e9b2de20", "save_path": "github-repos/lean/bwbush-marlowe-lean4", "path": "github-repos/lean/bwbush-marlowe-lean4/marlowe-lean4-318a224149b84e5b8d62c631e7684723e9b2de20/src/M/Pluto/Proof/Constant.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3330047338861676}}
{"text": "import category_theory.triangulated.pretriangulated\nimport for_mathlib.homological_complex_shift\nimport for_mathlib.monoidal_category\nimport for_mathlib.int\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v u\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.associator_def (a b c : discrete ℤ) :\n  α_ a b c = eq_to_iso (discrete.ext _ _ $ add_assoc a.1 b.1 c.1) := rfl\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.left_unitor_def (a : discrete ℤ) :\n  λ_ a = eq_to_iso (discrete.ext _ _ $ zero_add _) := rfl\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.right_unitor_def (a : discrete ℤ) :\n  ρ_ a = eq_to_iso (discrete.ext _ _ $ add_zero _) := rfl\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\nvariables (C : Type u) [category.{v} C] [preadditive C]\nvariables [has_shift C ℤ]\n\nlocal attribute [instance, reducible] endofunctor_monoidal_category\n\nnamespace triangle\n\n@[simps]\ndef triangle_shift_obj (T : triangle C) (i : ℤ) : triangle C :=\ntriangle.mk C\n  (i.neg_one_pow • ((shift_functor _ i).map T.mor₁))\n  (i.neg_one_pow • (((shift_functor _ i).map T.mor₂)))\n  (i.neg_one_pow • ((shift_functor C i).map T.mor₃ ≫ (shift_comm _ _ _).hom))\n\n@[simps]\ndef triangle_shift_map {T₁ T₂ : triangle C} (f : T₁ ⟶ T₂) (i : ℤ) :\n  triangle_shift_obj C T₁ i ⟶ triangle_shift_obj C T₂ i :=\n{ hom₁ := (shift_functor _ i).map f.hom₁,\n  hom₂ := (shift_functor _ i).map f.hom₂,\n  hom₃ := (shift_functor _ i).map f.hom₃,\n  comm₁' := by { dsimp, simp only [functor.map_zsmul,\n    preadditive.zsmul_comp, preadditive.comp_zsmul, ← functor.map_comp, f.comm₁] },\n  comm₂' := by { dsimp, simp only [functor.map_zsmul,\n    preadditive.zsmul_comp, preadditive.comp_zsmul, ← functor.map_comp, f.comm₂] },\n  comm₃' := begin\n    dsimp,\n    simp only [functor.map_zsmul,\n      preadditive.zsmul_comp, preadditive.comp_zsmul],\n    congr' 1,\n    simp only [ shift_comm_hom_comp, assoc, iso.cancel_iso_hom_right_assoc,\n      ← functor.map_comp, f.comm₃],\n  end }\n\n@[simps]\ndef triangle_shift_functor (i : ℤ) : triangle C ⥤ triangle C :=\n{ obj := λ T, triangle_shift_obj C T i,\n  map := λ T₁ T₂ f, triangle_shift_map C f _,\n  map_id' := begin\n    intros T,\n    ext,\n    all_goals { dsimp, simp },\n  end,\n  map_comp' := begin\n    intros T₁ T₂ T₃ f g,\n    ext,\n    all_goals { dsimp, simp },\n  end, } .\n\nvariable {C}\n\n@[simps]\ndef iso.of_components {T₁ T₂ : triangle C}\n  (e₁ : T₁.obj₁ ≅ T₂.obj₁)\n  (e₂ : T₁.obj₂ ≅ T₂.obj₂)\n  (e₃ : T₁.obj₃ ≅ T₂.obj₃) (h₁ h₂ h₃) : T₁ ≅ T₂ :=\n{ hom :=\n  { hom₁ := e₁.hom,\n    hom₂ := e₂.hom,\n    hom₃ := e₃.hom,\n    comm₁' := h₁,\n    comm₂' := h₂,\n    comm₃' := h₃ },\n  inv :=\n  { hom₁ := e₁.inv,\n    hom₂ := e₂.inv,\n    hom₃ := e₃.inv,\n    comm₁' := by rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, h₁],\n    comm₂' := by rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, h₂],\n    comm₃' := by rw [← functor.map_iso_inv, iso.comp_inv_eq, category.assoc, iso.eq_inv_comp,\n      functor.map_iso_hom, h₃], },\n  hom_inv_id' := by ext; dsimp; simp,\n  inv_hom_id' := by ext; dsimp; simp }\n.\n\nvariable (C)\n\n@[simps]\ndef triangle_shift_functor_ε : 𝟭 (triangulated.triangle C) ≅ triangle_shift_functor C 0 :=\nnat_iso.of_components (λ T,\n  iso.of_components\n    (shift_zero _ _).symm\n    (shift_zero _ _).symm\n    (shift_zero _ _).symm\n    begin\n      convert ((shift_functor_zero _ _).inv.naturality _),\n      dsimp only [triangle_shift_functor, triangle_shift_obj],\n      simpa,\n    end\n    begin\n      convert ((shift_functor_zero _ _).inv.naturality _),\n      dsimp only [triangle_shift_functor, triangle_shift_obj],\n      simpa,\n    end\n    begin\n      dsimp,\n      rw one_smul,\n      rw ← nat_trans.naturality_assoc, dsimp [shift_comm],\n      simp only [obj_ε_app, discrete.functor_map_id, nat_trans.id_app, ε_app_obj, assoc, id_comp],\n      rw [← nat_trans.comp_app, ← nat_trans.comp_app],\n      erw [monoidal_functor.μ_inv_hom_id_assoc, id_comp], refl,\n    end)\n  begin\n    intros T₁ T₂ f, ext;\n    { dsimp only [triangle_morphism.comp_hom₁, iso.of_components_hom_hom₁, triangle_shift_map_hom₁,\n        triangle_morphism.comp_hom₂, iso.of_components_hom_hom₂, triangle_shift_map_hom₂,\n        triangle_morphism.comp_hom₃, iso.of_components_hom_hom₃, triangle_shift_map_hom₃,\n        functor.id_map, triangle_category_comp, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.ε_iso_hom, triangle_shift_functor_map],\n      rw ← nat_trans.naturality _ _, refl },\n  end\n.\n\nvariables [∀ (i : ℤ), (shift_functor C i).additive]\n\n@[reassoc]\nlemma shift_comm_eq_eq_to_hom (X : C) (i j : ℤ) :\n  (shift_add X i j).hom ≫ (shift_comm X i j).hom ≫ (shift_add X j i).inv =\n  eq_to_hom (by { congr' 2, exact add_comm i j}) :=\nbegin\n  dsimp [shift_add, shift_comm],\n  simp only [eq_to_hom_map, eq_to_hom_app, assoc, μ_inv_hom_app, μ_inv_hom_app_assoc],\n  erw comp_id,\nend\n\n@[reassoc]\nlemma shift_add_comp_eq_to_hom (X : C) (i j k : ℤ) (h : i + j = j + i) :\n  (shift_add X (i+j) k).hom ≫\n  eq_to_hom (by { congr' 3}) ≫\n  (shift_add X (j+i) k).inv =\n  eq_to_hom (by { congr' 3}) :=\nbegin\n  dsimp [shift_add],\n  induction h, dsimp, simpa,\nend\n\n\n@[reassoc]\nlemma shift_add_comp_eq_to_hom' (X : C) (i j k : ℤ) (h : j + k = k + j) :\n  (shift_add X i (j+k)).hom ≫\n  eq_to_hom (by { congr' 3}) ≫\n  (shift_add X i (k+j)).inv =\n  eq_to_hom (by { congr' 3}) :=\nbegin\n  dsimp [shift_add],\n  induction h, dsimp, simpa,\nend\n\n\nlemma triangle_shift_functor_μ_aux (X : C) (i j : ℤ) :\n  (shift_functor C j).map (shift_comm X 1 i).hom ≫\n    (shift_comm ((shift_functor C i).obj X) 1 j).hom ≫\n      (shift_functor C 1).map\n        (shift_add X i j).inv =\n  (shift_add ((shift_functor C 1).obj X) i j).inv ≫\n    (shift_comm X 1 (i + j)).hom :=\nbegin\n  dsimp [shift_add, shift_comm],\n  simp only [eq_to_hom_map, eq_to_hom_app, functor.map_comp, obj_μ_app,\n    category_theory.discrete.associator_def, eq_to_iso.inv,\n    obj_μ_inv_app, eq_to_iso.hom, assoc, μ_inv_hom_app_assoc],\n  congr' 2,\n  simp only [← assoc], congr' 1, simp only [assoc],\n  erw shift_add_comp_eq_to_hom_assoc,\n  erw shift_add_comp_eq_to_hom'_assoc,\n  simpa, exact add_comm _ _, exact add_comm _ _,\nend\n\n@[simps]\ndef triangle_shift_functor_μ (i j : ℤ) :\n  triangle_shift_functor C i ⋙ triangle_shift_functor C j ≅\n    triangle_shift_functor C (i + j) :=\nnat_iso.of_components (λ T,\n  iso.of_components\n    (shift_add _ _ _).symm\n    (shift_add _ _ _).symm\n    (shift_add _ _ _).symm\n    (begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.μ_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor₁),\n      rw [functor.comp_map] at this,\n      erw [this, mul_comm], refl,\n    end)\n    (begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.μ_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor₂),\n      rw [functor.comp_map] at this,\n      erw [this, mul_comm], refl,\n    end)\n    begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.μ_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add,\n        mul_comm j.neg_one_pow],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor₃),\n      dsimp [functor.comp_map] at this,\n      erw [← reassoc_of this], clear this,\n      simp only [functor.map_comp, assoc, obj_μ_app],\n      congr' 2,\n      --have := (shift_monoidal_functor C ℤ).to_lax_monoidal_functor.associativity i j 1,\n      rw (shift_monoidal_functor C ℤ).map_associator_inv,\n      dsimp,\n      simp only [assoc, is_iso.inv_id, nat_trans.hcomp_app, comp_id, id_comp],\n      slice_lhs 4 5\n      { rw [← nat_trans.comp_app, is_iso.hom_inv_id,\n          nat_trans.id_app] },\n      erw id_comp,\n      rw category_theory.nat_iso.is_iso_inv_app,\n      --rw nat_trans.id_hcomp_app,\n      dsimp,\n      simp only [category_theory.functor.map_id, comp_id, assoc, is_iso.hom_inv_id_assoc],\n      slice_lhs 4 5\n      { rw [← nat_trans.comp_app, (shift_monoidal_functor C ℤ).μ_hom_inv_id,\n          nat_trans.id_app] },\n      erw comp_id, apply triangle_shift_functor_μ_aux,\n    end)\n  (begin\n    intros T₁ T₂ f, ext;\n    { dsimp only [triangle_morphism.comp_hom₁, iso.of_components_hom_hom₁, triangle_shift_map_hom₁,\n        triangle_morphism.comp_hom₂, iso.of_components_hom_hom₂, triangle_shift_map_hom₂,\n        triangle_morphism.comp_hom₃, iso.of_components_hom_hom₃, triangle_shift_map_hom₃,\n        functor.id_map, triangle_category_comp, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.ε_iso_hom, triangle_shift_functor_map],\n      rw ← nat_trans.naturality _ _, refl },\n  end)\n.\n\n/-\ndef triangle_shift_core : shift_mk_core (triangle C) ℤ :=\n{ F := triangle_shift_functor _,\n  ε := triangle_shift_functor_ε _,\n  μ := λ i j, triangle_shift_functor_μ _ _ _,\n  associativity := begin\n    intros i j k T, ext,\n    { have := (shift_monoidal_functor C ℤ).to_lax_monoidal_functor.associativity i j k,\n      apply_fun (λ α, α.app T.obj₁) at this,\n      simp only [nat_trans.comp_app, obj_μ_app, assoc, μ_inv_hom_app_assoc, map_inv_hom_app,\n        comp_id, functor.associator_hom_app, nat_trans.hcomp_app, nat_trans.id_app,\n        category_theory.functor.map_id, id_comp] at this,\n      erw [id_comp] at this,\n      refine eq.trans _ this, clear this,\n      dsimp, simp only [obj_μ_app, assoc],\n      -- I don't like that `(eq_to_hom _).hom₁`.\n      admit },\n    admit,\n    admit\n  end,\n  left_unitality := admit,\n  right_unitality := admit }\n-/\n\n@[simps]\ndef map_triangle_shift_functor (m n : discrete ℤ) (f : m ⟶ n) :\n  triangle_shift_functor C m.1 ⟶ triangle_shift_functor C n.1 :=\n{ app := λ T,\n  { hom₁ := eq_to_hom $ by rw discrete.eq_of_hom f,\n    hom₂ := eq_to_hom $ by rw discrete.eq_of_hom f,\n    hom₃ := eq_to_hom $ by rw discrete.eq_of_hom f,\n    comm₁' := by { rcases ⟨m, n⟩ with ⟨⟨_⟩, ⟨_⟩⟩, rcases f with ⟨⟨rfl : m = n⟩⟩,\n      simp only [eq_to_hom_refl, id_comp, comp_id], },\n    comm₂' := by { rcases ⟨m, n⟩ with ⟨⟨_⟩, ⟨_⟩⟩, rcases f with ⟨⟨rfl : m = n⟩⟩,\n      simp only [eq_to_hom_refl, id_comp, comp_id], },\n    comm₃' := by { rcases ⟨m, n⟩ with ⟨⟨_⟩, ⟨_⟩⟩, rcases f with ⟨⟨rfl : m = n⟩⟩,\n      dsimp, rw (shift_functor C (1 : ℤ)).map_id, simp only [comp_id, id_comp]} },\n  naturality' := begin\n    rcases ⟨m, n⟩ with ⟨⟨_⟩, ⟨_⟩⟩, rcases f with ⟨⟨rfl : m = n⟩⟩,\n    rintros X Y g, ext;\n    { dsimp, simp only [eq_to_hom_refl, id_comp, comp_id] },\n  end } .\n\n\n\nlemma associativity_aux (X : C) (a b c : ℤ) :\n(𝟙 ((shift_functor C c).obj ((shift_functor C b).obj ((shift_functor C a).obj X))) ≫\n  (shift_functor C c).map (((shift_monoidal_functor C ℤ).μ ⟨a⟩ ⟨b⟩).app X)) ≫\n  ((shift_monoidal_functor C ℤ).μ (⟨a⟩ ⊗ ⟨b⟩) ⟨c⟩).app X ≫\n  eq_to_hom (show ((shift_monoidal_functor C ℤ).obj ((⟨a⟩ ⊗ ⟨b⟩) ⊗ ⟨c⟩)).obj X =\n    ((shift_monoidal_functor C ℤ).obj (⟨a⟩ ⊗ ⟨b⟩ ⊗ ⟨c⟩)).obj X, by { congr' 2, ext, apply add_assoc }) =\n  𝟙 ((shift_functor C c).obj\n  ((shift_functor C b).obj ((shift_functor C a).obj X))) ≫ (((shift_monoidal_functor C ℤ).μ ⟨b⟩ ⟨c⟩).app\n  ((shift_functor C a).obj X) ≫ (shift_functor C (b + c)).map\n  (𝟙 ((shift_functor C a).obj X))) ≫ ((shift_monoidal_functor C ℤ).to_lax_monoidal_functor.μ ⟨a⟩\n  (⟨b⟩ ⊗ ⟨c⟩)).app X :=\nbegin\n  have := (shift_monoidal_functor C ℤ).associativity' ⟨a⟩ ⟨b⟩ ⟨c⟩,\n  apply_fun (λ e, e.app X) at this,\n  dsimp at this ⊢,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this ⊢,\n  erw comp_id,\n  exact this\nend\n\nlemma left_unitality_aux (X : C) (a : ℤ) : 𝟙 ((shift_functor C a).obj X) =\n  (𝟙 ((shift_functor C a).obj X) ≫ (shift_functor C a).map\n    ((shift_monoidal_functor C ℤ).to_lax_monoidal_functor.ε.app X)) ≫\n    ((shift_monoidal_functor C ℤ).to_lax_monoidal_functor.μ\n    (𝟙_ (discrete ℤ)) ⟨a⟩).app X ≫ eq_to_hom (by { congr, ext, exact zero_add a }) :=\nbegin\n  have := (shift_monoidal_functor C ℤ).left_unitality' ⟨a⟩,\n  apply_fun (λ e, e.app X) at this,\n  dsimp at this ⊢,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this ⊢,\n  exact this\nend\n\nlemma right_unitality_aux (X : C) (a : ℤ) : 𝟙 ((shift_functor C a).obj X) =\n  ((shift_monoidal_functor C ℤ).to_lax_monoidal_functor.ε.app ((shift_functor C a).obj X) ≫\n       (shift_functor C (𝟙_ (discrete ℤ)).as).map (𝟙 ((shift_functor C a).obj X))) ≫\n    ((shift_monoidal_functor C ℤ).to_lax_monoidal_functor.μ ⟨a⟩ (𝟙_ (discrete ℤ))).app X ≫\n    eq_to_hom (by { congr, ext, apply add_zero }) :=\nbegin\n  have := (shift_monoidal_functor C ℤ).right_unitality' ⟨a⟩,\n  apply_fun (λ e, e.app X) at this,\n  dsimp at this ⊢,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this ⊢,\n  erw comp_id,\n  exact this\nend\n\ninstance has_shift : has_shift (triangle C) ℤ := has_shift.mk $\n{ obj := λ i, triangle_shift_functor _ i.as,\n  map := λ m n f, map_triangle_shift_functor _ _ _ f,\n  map_id' := λ X, by { ext; refl },\n  map_comp' := λ X Y Z f g, by { ext; simp },\n  ε := (triangle_shift_functor_ε _).hom,\n  μ := λ m n, (triangle_shift_functor_μ _ m.as n.as).hom,\n  μ_natural' := begin\n    rintros ⟨m⟩ ⟨m'⟩ ⟨n⟩ ⟨n'⟩ ⟨⟨⟨⟩⟩⟩ ⟨⟨⟨⟩⟩⟩, ext;\n    { dsimp, simp only [id_comp, comp_id, category_theory.functor.map_id] },\n  end,\n  associativity' := λ a b c, by ext; apply associativity_aux,\n  left_unitality' := λ a, by ext; apply left_unitality_aux,\n  right_unitality' := λ a, by ext; apply right_unitality_aux,\n  ε_is_iso := infer_instance,\n  μ_is_iso := infer_instance } .\n\n@[simp]\nlemma shift_obj₁ (T : triangle C) (i : ℤ) : T⟦i⟧.obj₁ = T.obj₁⟦i⟧ := rfl\n\n@[simp]\nlemma shift_obj₂ (T : triangle C) (i : ℤ) : T⟦i⟧.obj₂ = T.obj₂⟦i⟧ := rfl\n\n@[simp]\nlemma shift_obj₃ (T : triangle C) (i : ℤ) : T⟦i⟧.obj₃ = T.obj₃⟦i⟧ := rfl\n\n@[simp]\nlemma shift_mor₁ (T : triangle C) (i : ℤ) : T⟦i⟧.mor₁ = i.neg_one_pow • T.mor₁⟦i⟧' := rfl\n\n@[simp]\nlemma shift_mor₂ (T : triangle C) (i : ℤ) : T⟦i⟧.mor₂ = i.neg_one_pow • T.mor₂⟦i⟧' := rfl\n\n@[simp]\nlemma shift_mor₃ (T : triangle C) (i : ℤ) :\n  T⟦i⟧.mor₃ = i.neg_one_pow • (T.mor₃⟦i⟧' ≫ (shift_comm _ _ _).hom) := rfl\n\n@[simp]\nlemma shift_hom₁ {T₁ T₂ : triangle C} (f : T₁ ⟶ T₂) (i : ℤ) : f⟦i⟧'.hom₁ = f.hom₁⟦i⟧' := rfl\n\n@[simp]\nlemma shift_hom₂ {T₁ T₂ : triangle C} (f : T₁ ⟶ T₂) (i : ℤ) : f⟦i⟧'.hom₂ = f.hom₂⟦i⟧' := rfl\n\n@[simp]\nlemma shift_hom₃ {T₁ T₂ : triangle C} (f : T₁ ⟶ T₂) (i : ℤ) : f⟦i⟧'.hom₃ = f.hom₃⟦i⟧' := rfl\n\nend triangle\n\n/-\ninstance {C : Type*} [category C] [preadditive C] (X Y : C) : has_neg (X ≅ Y) :=\n⟨λ f,\n{ hom := -f.hom,\n  inv := -f.inv,\n  hom_inv_id' := by simp only [comp_neg, neg_comp, iso.hom_inv_id, neg_neg],\n  inv_hom_id' := by simp only [comp_neg, neg_comp, iso.inv_hom_id, neg_neg] }⟩\n\n@[simp] lemma _root_.category_theory.neg_hom\n   {C : Type*} [category C] [preadditive C] {X Y : C} (f : X ≅ Y) :\n   (-f).hom = -(f.hom) := rfl\n\n@[simp] lemma _root_.category_theory.neg_inv\n   {C : Type*} [category C] [preadditive C] {X Y : C} (f : X ≅ Y) :\n   (-f).inv = -(f.inv) := rfl\n-/\n\nnamespace pretriangulated\n\n@[simp] lemma shift_comm_self (X : C) (i : ℤ) : shift_comm X i i = iso.refl _ :=\nbegin\n  ext,\n  dsimp [shift_comm],\n  simpa only [discrete.functor_map_id, nat_trans.id_app, id_comp, μ_hom_inv_app],\nend\n\nvariables [has_zero_object C] [∀ (i : ℤ), (shift_functor C i).additive] [pretriangulated C]\n\nlemma dist_triangle_iff_of_iso {T₁ T₂ : triangle C} (e : T₁ ≅ T₂) :\n  (T₁ ∈ dist_triang C) ↔ (T₂ ∈ dist_triang C) :=\n⟨λ h, isomorphic_distinguished _ h _ e.symm, λ h, isomorphic_distinguished _ h _ e⟩\n\nlemma dist_triangle_rot_iff (T : triangle C) :\n  (T.rotate ∈ dist_triang C) ↔ (T ∈ dist_triang C) :=\nbegin\n  refine ⟨λ h, _, λ h, rot_of_dist_triangle _ _ h⟩,\n  let e : T ≅ T.rotate.inv_rotate := rot_comp_inv_rot.app T,\n  rw ← dist_triangle_iff_of_iso _ e.symm,\n  exact inv_rot_of_dist_triangle _ _ h,\nend\n\nlemma shift_of_dist_triangle (T : triangle C) (hT : T ∈ dist_triang C) (i : ℤ) :\n  T⟦i⟧ ∈ dist_triang C :=\nbegin\n  induction i using int.induction_on_iff with i,\n  { exact isomorphic_distinguished T hT _ (shift_zero _ _), },\n  { suffices : T⟦(i+1 : ℤ)⟧ ≅ T⟦(i:ℤ)⟧.rotate.rotate.rotate,\n    { dsimp,\n      rw dist_triangle_iff_of_iso _ this,\n      iterate 3 { rw dist_triangle_rot_iff }, },\n    refine shift_add _ _ _ ≪≫ _,\n    refine triangle.iso.of_components (iso.refl _) (iso.refl _) (iso.refl _) _ _ _,\n    { dsimp, simp only [category.id_comp, category.comp_id, comp_neg, neg_one_smul], },\n    { dsimp, simp only [category.id_comp, category.comp_id, neg_comp, neg_one_smul], },\n    { dsimp, simp only [category.id_comp, category.comp_id, neg_comp, neg_one_smul],\n      simp only [functor.map_comp, assoc, category_theory.functor.map_id, comp_id,\n        functor.map_zsmul, preadditive.zsmul_comp, preadditive.comp_zsmul,\n        shift_comm_self, iso.refl_hom], }, },\nend\n\nend pretriangulated\n\nend category_theory.triangulated\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/triangle_shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3328239933253469}}
{"text": "import measure_theory.measure.measure_space\nimport measure_theory.constructions.pi\nimport probability_theory.independence\nimport probability_theory.conditional\nimport probability_theory.pi_subtype\n\ndef heq_congr {A B C : Type*} (hAB : A = B) {f : A → C} {g : B → C} (hfg : f == g)\n  {a : A} {b : B} (hab : a == b) : f a = g b := begin\n  revert a b f g, rw hAB,\n  intros,\n  exact congr (eq_of_heq hfg) (eq_of_heq hab)\nend\n\nlemma supr_emptyset' {α β} [complete_lattice α] {f : subtype ∅ → α} :\n  (⨆ x : (∅ : set β), f x) = ⊥ := by simp\n\nlemma union_subset_iff_right {α} (p q r : set α) (h : p ⊆ r) : p ∪ q ⊆ r ↔ q ⊆ r :=\nby rw set.union_subset_iff; exact and_iff_right h\n\nlemma union_subset_iff_left {α} (p q r : set α) (h : q ⊆ r) : p ∪ q ⊆ r ↔ p ⊆ r :=\nby rw set.union_subset_iff; exact and_iff_left h\n\nlemma pi_Inter_distrib {ι ι': Type*} {α : ι → Type*} {s : set ι} {t : ι' → Π i, set (α i)} :\n  s.pi (λ i, ⋂ i', t i' i) = ⋂ i', s.pi (t i') :=\nbegin\n  ext x,\n  simp only [set.mem_pi, set.mem_Inter],\n  have : (∀ i ∈ s, ∀ i', x i ∈ t i' i) ↔ (∀ i', ∀ i ∈ s, x i ∈ t i' i)\n    := ⟨(λ h i' i hi, h i hi i'), (λ h i hi i', h i' i hi)⟩,\n  rw this\nend\n\nnoncomputable theory\n\nnamespace measure_theory\n\ndef bot_measurable_space {α} : measurable_space α :=\nbegin\n  refine measurable_space.mk (set.union {∅} {set.univ}) _ _ _,\n  exact or.inl rfl,\n  intros s hs,\n  cases hs; change _ = _ at hs; subst hs,\n  rw set.compl_empty, exact or.inr rfl,\n  rw set.compl_univ, exact or.inl rfl,\n  intros f hf,\n  -- TODO pull things out into their own lemmas\n  by_cases ∃ i : ℕ, f i = set.univ,\n    rcases h with ⟨i, hfi⟩,\n    have : set.univ ≤ (⋃ (i : ℕ), f i) := by rw ← hfi; exact le_supr f i,\n    exact or.inr (top_le_iff.mp this),\n  simp at h,\n  rw set.Union_eq_empty.mpr (λ x, (hf x).elim id (λ hf, absurd hf (h x))),\n  exact or.inl rfl\nend\n\nlemma bot_measurable_space_eq_bot {α} : @bot_measurable_space α = ⊥ :=\nbegin\n  refine le_bot_iff.mp _,\n  intros i hi,\n  cases hi; change _ = _ at hi; subst hi,\n  exact @measurable_set.empty _ ⊥,\n  exact measurable_set.univ\nend\n\nlemma inter_of_generate_from_pi {ι : Type*} {α : ι → Type*}\n  (C : Π i, set (set (α i))) (t : set (Π i, α i)) :\n  t ∈ set.pi set.univ '' set.pi set.univ (λ (i : ι), {s : set (α i) | C i s}) →\n  ∃ (f : Π i, set (α i)), (∀ i, C i (f i)) ∧ t = ⋂ i, function.eval i ⁻¹' (f i) :=\nbegin\n  classical,\n  rintro ⟨t', ht1', ht2'⟩,\n  refine ⟨t', λ i, ht1' i (set.mem_univ i), _⟩,\n  subst ht2',\n  let ft' := λ i, (function.update (λ j : ι, (set.univ : set (α j))) i (t' i)),\n  have : t' = λ i, ⋂ i', ft' i' i,\n  { ext1 i,\n    have : (⋂ (i' : ι), ft' i' i ) = ft' i i,\n    refine infi_eq_of_forall_ge_of_forall_gt_exists_lt _ (λ _ h, ⟨i, h⟩),\n    { intro i',\n      by_cases i = i',\n      subst h, exact le_refl _,\n      have : ft' i' i = set.univ, { exact dif_neg h },\n      rw this, finish },\n    rw this, exact eq.symm (dif_pos rfl) },\n  rw [this, pi_Inter_distrib],\n  congr, ext1 i',\n  simp_rw (congr_fun this i').symm,\n  exact set.univ_pi_update_univ _ _\nend\n\nend measure_theory\n\nnamespace measurable_space\n\ndef comap_pi {α : Type*} {ι : Type*} {β : ι → Type*} [Π i : ι, measurable_space (β i)] (f : Π (i : ι), α → β i) :\n  measurable_space α :=\n  ((measurable_space.pi).comap (λ (a : α) (i : ι), f i a))\n\nend measurable_space\n\nopen measure_theory measure_theory.measure measurable_space\n\nnamespace probability_theory\n\nlemma measurable_pi_subtype {δ : Type*} {π : δ → Type*}\n  [hmp : Π a, measurable_space (π a)] (s : set δ) :\n  measurable (@pi_subtype δ π s) :=\nbegin\n  rw measurable_pi_iff,\n  intro a,\n  rw measurable_iff_comap_le,\n  -- FIXME why can't the lambda below be inferred?\n  exact le_supr (λ a, (hmp a).comap (λ (b : Π a, π a), b a)) _,\nend\n\n@[reducible]\ndef indep_sets' {α} [measurable_space α] {s1 s2 : set (set α)} (μ : measure α) :=\n  ∀ (t1 ∈ s1) (t2 ∈ s2), μ (t1 ∩ t2) = μ (t1) * μ (t2)\n\nlemma indep_sets_eq : @indep_sets = @indep_sets' := by ext; split; exact λ a b c d e, a b d c e\n\ndef indep_sets_iff {α} [measurable_space α] (s1 s2 : set (set α))\n  (μ : measure α) [is_probability_measure μ]\n  (hs1 : s1 ⊆ measurable_set) (hs2 : s2 ⊆ measurable_set) :\n  indep_sets s1 s2 μ ↔ (∀ (t1 ∈ s1) (t2 ∈ s2), indep_set t1 t2 μ) :=\nbegin\n  rw [indep_sets_eq],\n  repeat {rw forall_congr, intro},\n  refine (indep_set_iff_measure_inter_eq_mul (hs1 _) (hs2 _) μ);\n  assumption\nend\n\ndef cond_indep_sets_iff {α} [measurable_space α] (s1 s2 cs : set (set α))\n  (μ : measure α) [is_probability_measure μ]\n  (hs1 : s1 ⊆ measurable_set) (hs2 : s2 ⊆ measurable_set) (hcs : cs ⊆ measurable_set) :\n  cond_indep_sets s1 s2 cs μ ↔ (∀ (t1 ∈ s1) (t2 ∈ s2) (c ∈ cs), cond_indep_set' t1 t2 c μ) :=\nbegin\n  have : (∀ (t1 ∈ s1) (t2 ∈ s2) (c ∈ cs), cond_indep_set' t1 t2 c μ) ↔ \n  (∀ (c ∈ cs) (t1 ∈ s1) (t2 ∈ s2), cond_indep_set' t1 t2 c μ)\n    := by split; intros h _ _ _ _ _ _; apply h; assumption,\n  rw this, clear this,\n  rw cond_indep_sets,\n  -- TODO how to compress?\n  refine forall_congr _, intro c, refine forall_congr _, intro hc,\n  simp [cond_indep_set', cond_indep_set_def],\n  by_cases h : μ c = 0,\n  { refine iff_of_true _ _,\n    apply indep_sets_of_cond_null_measure _ _ _ _ (hcs hc) h,\n    intros, apply indep_sets_of_cond_null_measure _ _ _ _ (hcs hc) h, },\n  rw [indep_sets_eq, indep_sets'],\n  refine forall_congr _, intro, refine forall_congr _, intro ht1,\n  refine forall_congr _, intro, refine forall_congr _, intro ht2,\n  rw indep_set_iff_measure_inter_eq_mul (hs1 _) (hs2 _) _; try {assumption},\n  exact probability_theory.cond_prob_meas _ h\nend\n\n------------\n\nvariables {α : Type*} [m : measurable_space α] (μ : measure α) {ι: Type*}\n  {β : ι → Type*} (f : Π i : ι, α → (β i))\n\nsection definitions\n\nvariables [Π i : ι, measurable_space (β i)]\n\ndef comap_subtype (S : set ι) :\n  measurable_space (Π i : ι, β i) := measurable_space.comap (pi_subtype S) measurable_space.pi\n\nlemma pi_ext {P : set α → Prop} :\n  (∀ x, ((measurable_space.pi).comap (λ a i, f i a)).measurable_set' x → P x)\n  ↔ (∀ x, measurable_set x → P ((λ a i, f i a) ⁻¹' x)) := set.maps_image_to _ _ _ _\n\nlemma comap_subtype_ext {P : set (Π i : ι, β i) → Prop} (A : set ι) :\n  (∀ x, (comap_subtype A).measurable_set' x → P x)\n  ↔ (∀ x, measurable_set x → P (>[A] x)) := set.maps_image_to _ _ _ _\n\nlemma comap_subtype_subset (A : set ι) :\n  @comap_subtype _ β _ A ≤ measurable_space.pi :=\nbegin\n  simp_rw [comap_subtype, measurable_space.pi, comap_supr,\n    measurable_space.comap_comp, function.comp, pi_subtype],\n  exact supr_le_supr2 (λ i, ⟨i, le_rfl⟩)\nend\n\nlemma pi_unsubtype_meas_meas (A : set ι) {a : set (Π i : A, β i)} (hma : measurable_set a) :\n   measurable_set (>[] a) := by refine comap_subtype_subset A _ ⟨a, hma, rfl⟩\n\nlemma pi_set_to_subtype_img_meas₁ {A B : set ι} (hAB : B ⊆ A) :\n  @measurable_space.pi ↥B _ _ = (@measurable_space.pi (set_to_subtype A B) _ _).comap (@pi_set_to_subtype _ β A B) :=\nbegin\n  simp_rw [measurable_space.pi, comap_supr, comap_comp, function.comp],\n  refine le_antisymm (supr_le_supr2 (λ i, ⟨⟨⟨i, hAB i.property⟩, i.property⟩, _⟩)) (supr_le_supr2 (λ i, ⟨⟨i, i.property⟩, _⟩));\n  convert le_refl _; ext x; rw pi_set_to_subtype_def',\n  congr, exact subtype.eq rfl, congr\nend\n\nlemma pi_set_to_subtype_img_meas₂ {A B : set ι} (hAB : B ⊆ A) :\n  (@measurable_space.pi B _ _).map (@pi_set_to_subtype _ β A B) = (@measurable_space.pi (set_to_subtype A B) _ _) := \nbegin\n  rw pi_set_to_subtype_img_meas₁ hAB,\n  -- TODO make lemma for general map_comap_surjective\n  ext s,\n  split,\n  { rintro ⟨s', hms', hs'⟩,\n    rwa (set.preimage_eq_preimage (pi_set_to_subtype_bijective hAB).surjective).mp hs' at hms' },\n  intro h, exact ⟨s, h, rfl⟩\nend\n\nlemma pi_set_to_subtype_img_meas {A B : set ι} (hAB : B ⊆ A) {b : set (Π i : B, β i)} (hmb : measurable_set b)\n  : measurable_set (pi_set_to_subtype A B '' b) :=\nbegin\n  change measurable_space.pi.measurable_set' (pi_set_to_subtype A B '' b),\n  rw ← pi_set_to_subtype_img_meas₂ hAB,\n  change measurable_set (pi_set_to_subtype A B ⁻¹' (pi_set_to_subtype A B '' b)),\n  rwa set.preimage_image_eq _ (pi_set_to_subtype_bijective hAB).injective\nend\n\nlemma pi_unsubtype_set_meas_meas {A B : set ι} (hAB : B ⊆ A) {b : set (Π i : B, β i)} (hmb : measurable_set b)\n  : measurable_set (>>[A] b) :=\nby exact pi_unsubtype_meas_meas _ (pi_set_to_subtype_img_meas hAB hmb)\n\n/-- The joint distribution induced by an indexed family of random variables `f`. -/\ndef joint : measure (Π i : ι, β i) := map (λ a i, f i a) μ\n\ninstance joint_prob_meas (hm : ∀ i : ι, measurable (f i)) [hp : is_probability_measure μ] :\n  is_probability_measure (joint μ f) :=\nbegin\n  constructor,\n  rw [joint, measure.map_apply _ measurable_set.univ],\n  rw set.preimage_univ, exact hp.measure_univ,\n  exact measurable_pi_iff.mpr hm\nend\n\n/-- The marginal distribution induced by an indexed family of random variables `f`\nrestricted to a subset of \"marginalizing variable\" indices `mv` (represented as\nan index subtype). -/\ndef marginal (mv : set ι) : measure (Π i : mv, β i) := joint μ (pi_subtype mv f)\n\ninstance marginal_prob_meas (hm : ∀ i : ι, measurable (f i)) [is_probability_measure μ] (mv : set ι) :\n  is_probability_measure (marginal μ f mv)\n  := by rw marginal; exact probability_theory.joint_prob_meas _ _ (λ i, hm i)\n\n/-- Generic marginalization of the joint measure `μ` on the given subset of variables `mv`. -/\ndef marginalization (μ : measure (Π i : ι, β i)) (mv : set ι) :\n  measure (Π i : mv, β i) := map (pi_subtype mv) μ\n\nend definitions\n\nvariables [Π i : ι, measurable_space (β i)]\n\nsection marginal\n\nvariables [Π i : ι, measurable_space (β i)] (hm : ∀ i : ι, measurable (f i))\n\ninclude hm\n\nlemma marginal_eq_marginalization_aux (mv : set ι) :\n  marginal μ f mv = marginalization (joint μ f) mv :=\nby { rw [marginalization, joint, map_map, function.comp], refl,\n  apply measurable_pi_subtype, exact measurable_pi_iff.mpr hm }\n\n/-- The marginalization principle: the marginal probability of a particular \n\"marginal assignment\" measurable set `s` is equal to the joint probability of\nthat same set, extended to allow the unassigned variables to take any value. -/\ntheorem marginal_def (mv : set ι) \n  {s : set (Π i : mv, β i)} (hms : measurable_set s) :\n  marginal μ f mv s = joint μ f (>[] s) :=\nby { rw [marginal_eq_marginalization_aux _ _ hm, marginalization, map_apply _ hms],\n  apply measurable_pi_subtype }\n\nlemma joint_cond_meas_of_marginal (mv : set ι) \n  (s : set (Π i : mv, β i)) (hms : measurable_set s) (hcs : (marginal μ f mv) s ≠ 0) :\n  joint μ f (>[] s) ≠ 0 := by rwa marginal_def _ _ hm _ hms at hcs\n\nlemma marginal_cond_meas_of_joint (mv : set ι) \n  (s : set (Π i : mv, β i)) (hms : measurable_set s) (hcs : (joint μ f) (>[] s) ≠ 0) :\n  marginal μ f mv s ≠ 0 := by rwa ← marginal_def _ _ hm _ hms at hcs\n\nlemma marginal_cond_meas_of_joint_inter {A B : set ι}\n  {a : set (Π i : A, β i)} (hma : measurable_set a)\n  {b : set (Π i : B, β i)} (hmb : measurable_set b)\n  (hjc : joint μ f (>[] a ∩ >[] b) ≠ 0) : marginal μ f _ (>>[A ∪ B] a ∩ >>[] b) ≠ 0 :=\nbegin \n  rwa [marginal_def, pi_unsubtype_union_img_inter], assumption,\n  refine measurable_set.inter _ _; refine pi_unsubtype_set_meas_meas _ _; finish\nend\n\nlemma joint_cond_meas_of_marginal_inter {A B : set ι}\n  {a : set (Π i : A, β i)} (hma : measurable_set a)\n  {b : set (Π i : B, β i)} (hmb : measurable_set b)\n  (hmc : marginal μ f _ (>>[A ∪ B] a ∩ >>[] b) ≠ 0) : joint μ f (>[] a ∩ >[] b) ≠ 0 :=\nbegin \n  rwa [marginal_def, pi_unsubtype_union_img_inter] at hmc, assumption,\n  refine measurable_set.inter _ _; refine pi_unsubtype_set_meas_meas _ _; finish\nend\n\nend marginal\n\n-----\n\nsection independence\n\nsection definitions\n\n/-- A list of sets of random variables `S` is independent if the list of measurable spaces\nit incurs on the joint distribution is independent. -/\ndef Independent {ι' : Type*} (S : ι' → set ι) : Prop :=\n  Indep (λ i, (comap_pi (pi_subtype (S i) f))) μ\n\n/-- Two sets of random variables `A` and `B` are independent if the measurable spaces\nthey incur on the event space `α` via `f` are independent w.r.t. `μ`. -/\ndef independent (A B : set ι) : Prop :=\n  indep (comap_pi (pi_subtype A f)) (comap_pi (pi_subtype B f)) μ\n\n/-- Two sets of random variables `A` and `B` are independent if the measurable spaces\nthey incur on the joint distribution are independent. -/\ndef independent_meas (A B : set ι) (hm : ∀ i : ι, measurable (f i)) :\n  independent μ f A B ↔ indep (comap_subtype A) (comap_subtype B) (joint μ f)\n  :=\nbegin\n  simp_rw [independent, indep, indep_sets_eq, indep_sets'],\n  change \n    (∀ (t1 : set α)\n    (ht1 : (measurable_space.comap (λ (a : α) (i : ↥A), f ↑i a) measurable_space.pi).measurable_set' t1)\n    (t2 : set α)\n    (ht2 : (measurable_space.comap (λ (a : α) (i : ↥B), f ↑i a) measurable_space.pi).measurable_set' t2),\n      μ (t1 ∩ t2) = μ t1 * μ t2) ↔\n    ∀ (t1 : set (Π (i : ι), β i))\n    (ht1 : (comap_subtype A).measurable_set' t1)\n    (t2 : set (Π (i : ι), β i))\n    (ht2 : (comap_subtype B).measurable_set' t2), (joint μ f) (t1 ∩ t2) = (joint μ f) t1 * (joint μ f) t2,\n  simp_rw comap_subtype_ext,\n  simp_rw pi_ext,\n  refine forall_congr _, intro a, refine forall_congr _, intro hma,\n  refine forall_congr _, intro b, refine forall_congr _, intro hmb,\n  simp_rw [joint],\n  have hmusa : measurable_set (>[] a) := measurable_pi_subtype A hma,\n  have hmusb : measurable_set (>[] b) := measurable_pi_subtype B hmb,\n  conv_rhs\n  { rw map_apply (measurable_pi_iff.mpr hm) (hmusa.inter hmusb),\n    rw map_apply (measurable_pi_iff.mpr hm) (hmusa),\n    rw map_apply (measurable_pi_iff.mpr hm) (hmusb) },\n  rw iff_eq_eq,\n  refl\nend\n\nend definitions\n\nend independence\n\n-----\n\nsection conditional\n\nsection definitions\n\ndef cond (A B : set ι) (c : set (Π i : B, β i)) : measure (Π i : A, β i) := \n  marginal (cond_measure μ ((λ a i, f i a) ⁻¹' (>[] c))) f A\n\n/-- Two sets of random variables `A` and `B` are independent given a third set `C`\nif the measurable spaces `A` and `B` incur on the event space `α` via `f` are independent w.r.t. `μ`\ngiven any measurable set incurred by `C`. -/\ndef cond_independent (A B C : set ι) : Prop :=\n  cond_indep (comap_pi (pi_subtype A f)) (comap_pi (pi_subtype B f)) \n             (comap_pi (pi_subtype C f)).measurable_set' μ\n\nlemma cond_independent_def (A B C : set ι) :\n  cond_independent μ f A B C ↔\n  ∀ c (hc : ((measurable_space.pi).comap (λ (a : α) (i : C), f i a)).measurable_set' c),\n  independent (cond_measure μ c) f A B :=\nby refl\n\nlemma cond_independent_meas (A B C : set ι) (hm : ∀ i : ι, measurable (f i)) :\n  cond_independent μ f A B C ↔\n  cond_indep (comap_subtype A) (comap_subtype B) (comap_subtype C).measurable_set' (joint μ f) :=\nbegin\n  simp_rw [cond_independent_def, independent_meas _ _ _ _ hm, cond_indep_def],\n  change _ ↔ \n    ∀ (c : set (Π (i : ι), β i)),\n      (comap_subtype C).measurable_set' c →\n      indep (comap_subtype A) (comap_subtype B) (cond_measure (joint μ f) c),\n  simp_rw [comap_subtype_ext, pi_ext],\n  refine forall_congr _, intro c, refine forall_congr _, intro hmc,\n  rw iff_eq_eq, congr,\n  simp_rw [cond_measure, joint], simp, congr,\n  exact (map_apply (measurable_pi_iff.mpr hm) (measurable_pi_subtype C hmc)).symm,\n  exact (restrict_map (measurable_pi_iff.mpr hm) (measurable_pi_subtype C hmc)).symm\nend\n\nend definitions\n\ntheorem cond_def [is_probability_measure μ] (hm : ∀ i : ι, measurable (f i)) (A B : set ι)\n  (c : set (Π i : B, β i)) (hmc : measurable_set c)\n  (s : set (Π i : A, β i)) (hms : measurable_set s) :\n  cond μ f A B c s = cond_measure (joint μ f) (>[] c) (>[] s) :=\nbegin\n  haveI := probability_theory.joint_prob_meas μ f hm,\n  rw [cond, marginal_def _ _ hm _ hms],\n  congr, ext1 s' hms',\n  have hm' := measurable_pi_iff.mpr hm,\n  have hmc' := measurable_pi_subtype B hmc,\n  rw [joint, map_apply, cond_measure_def, cond_measure_def, joint,\n    map_apply, map_apply, set.preimage_inter]; try {assumption},\n  apply measurable_set.inter hmc' hms',\n  exact hm' hmc'\nend\n\nlemma independent_iff_cond_independent_empty [is_probability_measure μ]\n  (hm : ∀ i : ι, measurable (f i)) (A B : set ι) :\n  independent μ f A B ↔ cond_independent μ f A B ∅ :=\nbegin\n  haveI := probability_theory.joint_prob_meas μ f hm,\n  rw [cond_independent_meas _ _ _ _ _ hm, independent_meas _ _ _ _ hm],\n  have : @comap_subtype _ β _ ∅ = ⊥ :=\n    by rw comap_subtype; convert @comap_bot _ _ (pi_subtype ∅); exact supr_emptyset',\n  rw [this, ← bot_measurable_space_eq_bot],\n  refine iff.trans (iff.symm _) (union_subset_iff_right _ _ _ _).symm,\n  apply cond_indep_univ_iff_indep_set,\n  intros a ha, change _ = _ at ha, subst ha,\n  exact indep_sets_of_cond_null_measure _ _ _ _ measurable_set.empty (outer_measure.empty _),\nend\n\nlemma cond_empty_eq_marginal [is_probability_measure μ]\n  (A : set ι) : cond μ f A ∅ set.univ = marginal μ f A := \nby simp_rw [cond, pi_unsubtype_img, set.preimage_univ, cond_univ]\n\ntheorem cond_independent_iff_cond_inter_irrel [is_probability_measure μ] (hm : ∀ i : ι, measurable (f i))\n  (A B C : set ι) :\n  cond_independent μ f A B C ↔ ∀ (b : set (Π i : B, β i)) (hmb : measurable_set b)\n  (c : set (Π i : C, β i)) (hmc : measurable_set c),\n  marginal μ f (B ∪ C) (>>[] b ∩ >>[] c) ≠ 0\n  → cond μ f A (B ∪ C) (>>[] b ∩ >>[] c) = cond μ f A C c :=\nbegin\n  haveI := probability_theory.joint_prob_meas μ f hm,\n  rw [cond_independent_meas _ _ _ _ _ hm, cond_indep, cond_indep_sets_iff],\n  { -- FIXME this is just to pattern-match, can I avoid this somehow?\n    change (\n        ∀ (a : set (Π (i : ι), β i)), (comap_subtype A).measurable_set' a\n      → ∀ (b : set (Π (i : ι), β i)), (comap_subtype B).measurable_set' b\n      → ∀ (c : set (Π (i : ι), β i)), (comap_subtype C).measurable_set' c\n      → cond_indep_set' a b c (joint μ f))\n      ↔ ∀ (b : set (Π (i : ↥B), β ↑i)) (hmb : measurable_set b)\n      (c : set (Π (i : ↥C), β ↑i)) (hmc : measurable_set c),\n      marginal μ f (B ∪ C) (>>[] b ∩ >>[] c) ≠ 0\n      → cond μ f A (B ∪ C) (>>[] b ∩ >>[] c) = cond μ f A C c,\n    simp_rw comap_subtype_ext,\n    conv in (cond _ _ _ _ _ = cond _ _ _ _ _) { rw measure.ext_iff },\n    split; intro h,\n    { intros b hmb c hmc hcmbc a hma,\n      have : joint μ f (>[] b ∩ >[] c) ≠ 0\n        := joint_cond_meas_of_marginal_inter _ _ hm hmb hmc hcmbc,\n      rw set.inter_comm at this,\n      rw [cond_def, cond_def]; try {assumption},\n      convert (cond_indep_set_iff_cond_inter_irrel (joint μ f) _ _ _).mp\n        (cond_indep_set'.symm (h _ hma _ hmb _ hmc)) _;\n      try {exact measurable_pi_subtype _ (by assumption)},\n      rw [pi_unsubtype_union_img_inter, set.inter_comm],\n      assumption,\n      refine measurable_set.inter _ _;\n      refine measurable_pi_subtype _ _; refine pi_set_to_subtype_img_meas _ (by assumption),\n      exact set.subset_union_left _ _, exact set.subset_union_right _ _ },\n    { intros a hma b hmb c hmc,\n      rw cond_indep_set_iff_cond_inter_irrel',\n      intro hcmbc,\n      rw set.inter_comm at hcmbc,\n      have : marginal μ f _ (>>[] b ∩ >>[] c) ≠ 0 \n        := marginal_cond_meas_of_joint_inter _ _ hm hmb hmc hcmbc,\n      have := h b _ c _ _ a hma; try {assumption},\n      rw [cond_def, cond_def] at this; try {assumption},\n      rwa [set.inter_comm, ← pi_unsubtype_union_img_inter],\n      { refine measurable_set.inter _ _;\n        refine measurable_pi_subtype _ _;\n        refine pi_set_to_subtype_img_meas _ (by assumption),\n        exact set.subset_union_left _ _, exact set.subset_union_right _ _ },\n      all_goals {exact measurable_pi_subtype _ (by assumption)} } },\n  all_goals {exact comap_subtype_subset _}\nend\n\ntheorem independent_iff_cond_irrel  [is_probability_measure μ] (hm : ∀ i : ι, measurable (f i))\n  (A B : set ι) :\n  independent μ f A B ↔ ∀ (b : set (Π i : B, β i)) (hmb : measurable_set b), marginal μ f B b ≠ 0\n  → cond μ f A B b = marginal μ f A :=\nbegin\n  convert cond_independent_iff_cond_inter_irrel μ f hm A B ∅; ext,\n  exact independent_iff_cond_independent_empty _ _ hm A B,\n  refine forall_congr _, intro b,\n  refine forall_congr _, intro hmb,\n  haveI : subsingleton (Π i : (∅ : set ι), β i) := ⟨λ f g, by ext ⟨x, hx⟩; exact false.elim hx⟩,\n  have h1 : ⇑(marginal μ f (B ∪ ∅)) == ⇑(marginal μ f B) := by rw set.union_empty,\n  have h2 : cond μ f A (B ∪ ∅) == cond μ f A B := by rw set.union_empty,\n  have h3 : >>[B ∪ ∅] b == b := begin\n    rw [set.union_empty],\n    refine heq_of_eq _,\n    exact pi_unsubtype_set_same _ _\n  end,\n  have : (marginal μ f (B ∪ ∅) (>>[] b ∩ >>[] (set.univ : set (Π (i : (∅ : set ι)), β i))) ≠ 0\n    → cond μ f A (B ∪ ∅) (>>[] b ∩ >>[] (set.univ : set (Π (i : (∅ : set ι)), β i))) = cond μ f A ∅ set.univ)\n    ↔ (marginal μ f B b ≠ 0 → cond μ f A B b = marginal μ f A),\n  { simp_rw [pi_unsubtype_set, pi_unsubtype_img,\n      set.image_univ_of_surjective (pi_set_to_subtype_bijective (set.subset_union_right _ _)).surjective,\n      set.preimage_univ, set.inter_univ],\n    rw heq_congr (by rw set.union_empty) h1 h3,\n    rw heq_congr (by rw set.union_empty) h2 h3,\n    rw cond_empty_eq_marginal },\n  split; intro h,\n  { refine subsingleton.set_cases _ _; intro hmc,\n    simp_rw [pi_unsubtype_set, pi_unsubtype_img],\n    simp_rw [set.image_empty, set.preimage_empty, set.inter_empty],\n    intro h', exact absurd (outer_measure.empty' _) h',\n    rwa this },\n  { have h' := h set.univ measurable_set.univ,\n    rwa this at h' },\nend\n\nend conditional\n\nend probability_theory\n", "meta": {"author": "rish987", "repo": "lean-bayes", "sha": "b334cc4f9b4d81551b8513854c44d5ed007c2373", "save_path": "github-repos/lean/rish987-lean-bayes", "path": "github-repos/lean/rish987-lean-bayes/lean-bayes-b334cc4f9b4d81551b8513854c44d5ed007c2373/src/probability_theory/random_variable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.33282398619633297}}
{"text": "/-\nCopyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies, Bhavik Mehta\n-/\nimport combinatorics.simplicial_complex.exposed\n\nopen_locale classical affine big_operators\nopen set\n--TODO: Generalise to LCTVS\nvariables {E : Type*} [normed_group E] [normed_space ℝ E] {x : E} {A B : set E}\n  {X : finset E} {l : E →L[ℝ] ℝ}\n\ndef set.vertices (A : set E) :\n  set E :=\n{x | x ∈ frontier A ∧ ∀ w : E, w ≠ x → ((∃ l : E →L[ℝ] ℝ, l x ≤ l w ∧ ∀ y ∈ A, l y ≤ l x) →\n  ∃ l : E →L[ℝ] ℝ, l x < l w ∧ ∀ y ∈ A, l y ≤ l x) ∧ ((∃ l : E →L[ℝ] ℝ, l x ≤ l w ∧ ∀ y ∈ Aᶜ,\n  l y ≤ l x) → ∃ l : E →L[ℝ] ℝ, l x < l w ∧ ∀ y ∈ Aᶜ, l y ≤ l x)}\n\n/-def set.vertices (A : set E) :\n  set E :=\n{x | x ∈ frontier A ∧ ∀ w : E, w ≠ x → (∃ l : E →L[ℝ] ℝ, l x ≤ l w ∧ ∀ y ∈ A, l y ≤ l x) →\n  ∃ l : E →L[ℝ] ℝ, l x < l w ∧ ∀ y ∈ A, l y ≤ l x}-/\n\nlemma contrapose :\n  ∀ w : E, w ≠ x → (∃ l : E →L[ℝ] ℝ, l x ≤ l w ∧ ∀ y ∈ A, l y ≤ l x) →\n  ∃ l : E →L[ℝ] ℝ, l x < l w ∧ ∀ y ∈ A, l y ≤ l x :=\nbegin\n  by_contra,\n  push_neg at h,\n  sorry\nend\n", "meta": {"author": "mmasdeu", "repo": "brouwerfixedpoint", "sha": "548270f79ecf12d7e20a256806ccb9fcf57b87e2", "save_path": "github-repos/lean/mmasdeu-brouwerfixedpoint", "path": "github-repos/lean/mmasdeu-brouwerfixedpoint/brouwerfixedpoint-548270f79ecf12d7e20a256806ccb9fcf57b87e2/src/combinatorics/simplicial_complex/vertices.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.33279970441705875}}
{"text": "-- import syntax.consistency\n-- import tactic.induction\n\n-- -- Motivation: a lot of places assume `¬ ax ⊥'` so it's worth trying to reduce these assumptions.\n-- lemma ax_consistent.not_ax_bot' {form : Type} [pf : Pformula form] [pax : Pformula_ax form] {s : set form}\n--   (h : ax_consistent s) : ¬ ⊢' (⊥' : form) :=\n-- begin\n--   simpa [ax_consistent, set_proves] using (h list.nil (λ _ h, h.elim))\n-- end\n\n-- /-- An empty set of formulas is consistent iff the theory is consistent. -/\n-- -- Motivation: a lot of places assume `¬ ax ⊥'` so it's worth trying to reduce these assumptions.\n-- @[simp] lemma ax_consistent_empty\\ {form : Type} [pf : Pformula form] [pax : Pformula_ax form] :\n--   ax_consistent ({} : set form) ↔ ¬ ax (⊥' : form) :=\n-- begin\n--   split; intro h,\n--   { exact h.not_ax_bot },\n--   { intros fs hfs,\n--     cases fs with f fs,\n--     { simpa [ax_consistent, finite_ax_consistent] using hfs },\n--     { cases hfs f (list.mem_cons_self _ _) } }\n-- end\n\n-- /-- If there is any formula that cannot be proven, the theory is consistent. -/\n-- -- Motivation: a lot of places assume `¬ ax ⊥'` so it's worth trying to reduce these assumptions.\n-- lemma consistent_of_not_ax {form : Type} [pf : Pformula form] [pax : Pformula_ax form] {φ : form}\n--   (hφ : ¬ ax φ) : ¬ ax (⊥' : form) :=\n-- mt (mp _ _ ax_bot_elim) hφ\n\n-- lemma contra_con_cons {form : Type} [pf : Pformula form] [pax : Pformula_ax form] (fs gs : list (form)) (x y : form) \n--   (hax : ax (y ↔' ¬' x)) (hx : x ∈ fs) (hy : y ∈ gs) :\n--   ax (¬' ((finite_conjunction fs) ∧' (finite_conjunction gs))) :=\n-- begin\n--   induction' fs with f fs ihf,\n--   { finish, },\n--   { induction gs with g gs ihg,\n--     { finish, },\n--     { cases hx,\n--       { cases hy,\n--         { rw[←hx, ←hy],\n--           simp[finite_conjunction],\n--           rw ft.notdef,\n--           apply cut (iff_r and_commute),\n--           apply imp_and_l,\n--           apply cut (iff_l and_switch),\n--           apply cut (iff_r and_commute),\n--           apply imp_and_l,\n--           rw ←contra_iff_false_ax_not,\n--           rw demorgans,\n--           apply iff_l,\n--           exact hax, },\n--         { simp[finite_conjunction],\n--           rw ft.notdef,\n--           apply cut (iff_r and_commute),\n--           apply cut (iff_l and_switch),\n--           apply cut (iff_r and_commute),\n--           apply imp_and_l,\n--           apply cut (iff_l and_switch),\n--           rw ←contra_iff_false_ax_not,\n--           apply ihg hy, }, },\n--       { rw ft.notdef,\n--         apply cut (iff_l and_switch),\n--         apply cut (iff_r and_commute),\n--         apply cut (iff_l and_switch),\n--         apply cut (iff_r and_commute),\n--         apply imp_and_l,\n--         specialize @ihf ft fax (g :: gs) x y hax hy hx,\n--         rw ft.notdef at ihf,\n--         exact ihf, }, }, },\n-- end\n\n-- /-- A singleton set is consistent iff the theory is consistent and the formula is not disprovable.\n-- -/\n-- -- Motivation: `comphelper` seemed to be slightly too specific, this is a more general form I found\n-- @[simp] lemma ax_consistent_singleton {form : Type} [pf : Pformula form] [pax : Pformula_ax form] {φ : form} :\n--   ax_consistent ({φ} : set form) ↔ ¬ ax (¬' φ) :=\n-- begin\n--   split,\n--   { intros h,\n--     have := h (φ :: list.nil) (by simp),\n--     simp only [finite_ax_consistent, ft.notdef, finite_conjunction_cons, finite_conjunction_nil]\n--       at ⊢ this,\n--     exact mt (ax_iff_mp (ax_imp_congr_left ax_and_top_iff)).2 this },\n--   { rintros hφ fs hfs,\n--     cases fs with f fs,\n--     { simp only [ax_consistent, finite_ax_consistent, finite_conjunction_nil, ax_top_imp] at ⊢,\n--       exact consistent_of_not_ax hφ },\n--     { intro h,\n--       simp only [ft.notdef] at *,\n--       exact hφ (mp _ _ (mp _ _ hs1 h) (fin_conj_repeat_helper hfs)) } }\n-- end\n\n-- -- Completeness helper\n-- ----------------------------------------------------------\n\n-- lemma comphelper {form : Type} [pf : Pformula form] [pax : Pformula_ax form] {φ : form} (h : ¬ ax φ) :\n--   ax_consistent ({¬' φ} : set form) :=\n-- ax_consistent_singleton.mpr (mt (mp _ _ dne) h)\n\n-- /-- If `φ` cannot be proved, there is a maximal consistent set containing `¬ φ` -/\n-- -- Motivation: `lindenbaum` is applied in a few places to `comphelper`,\n-- -- and `simp` can simplify the conditions slightly.\n-- lemma exists_max_ax_consistent_neg_mem {form : Type} [pf : Pformula form] [pax : Pformula_ax form] {φ : form} (hφ : ¬ ax φ) :\n--   ∃ (Γ : set form), max_ax_consistent Γ ∧ ¬' φ ∈ Γ :=\n-- by simpa using lindenbaum (comphelper hφ)\n\n", "meta": {"author": "kaiobendrauf", "repo": "cl-lean", "sha": "15568f16cf57a07db6192fbd8084d59cc1aef1df", "save_path": "github-repos/lean/kaiobendrauf-cl-lean", "path": "github-repos/lean/kaiobendrauf-cl-lean/cl-lean-15568f16cf57a07db6192fbd8084d59cc1aef1df/src/syntax/consistency_lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3327867265533592}}
{"text": "import lambda_calculus.utlc.basic\nimport lambda_calculus.utlc.identities\n\nnamespace lambda_calculus\nnamespace utlc\n\ndef reduction_step : utlc → utlc → (utlc → utlc → Prop) → Prop\n| (↓ n) := λ g r, r (↓ n) g\n| (Λ f) := λ g r, r (Λ f) g ∨ match g with\n  | (↓ _) := false\n  | (Λ g) := reduction_step f g r\n  | (_·_) := false\n  end\n| (f1 · f2) := λ g r, r (f1 · f2) g ∨ match g with\n  | (↓_) := false\n  | (Λ _) := false\n  | (g1 · g2) :=\n    (g2 = f2 ∧ (reduction_step f1 g1 r)) ∨ (g1 = f1 ∧ (reduction_step f2 g2 r))\n  end\n\ndef reduction_step_of := λ r f g, reduction_step f g r\n\nsection\nvariables (head_step: utlc → utlc → Prop)\nvariables (n : ℕ)\nvariables (f f' g g': utlc)\nvariables (m : ℕ)\n\nlocal attribute [simp] reduction_step reduction_step_of\n\ninstance reduction_step.decidable_rel [h: decidable_rel head_step]: decidable_rel (reduction_step_of head_step) :=\nbegin\n  have dor : ∀ (p q : Prop), decidable p → decidable q → decidable (p ∨ q),\n  { intros p q hp hq,\n    cases hp;\n    cases hq;\n    simp only [hp, hq];\n    apply_instance },\n  have dand : ∀ (p q : Prop), decidable p → decidable q → decidable (p ∧ q),\n  { intros p q hp hq,\n    cases hp;\n    cases hq;\n    simp only [hp, hq];\n    apply_instance },\n  intro f,\n  induction f;\n  intro g;\n  cases g;\n  simp only [down_notation, lambda_notation, dot_notation,\n    reduction_step_of, reduction_step];\n  repeat { apply dor };\n  repeat { apply dand };\n  try { apply_instance },\n  apply f_ih g,\n  apply f_ih_f g_f,\n  apply f_ih_g g_g\nend\n\ntheorem down_reduction_step_iff: reduction_step_of head_step (↓n) g ↔ head_step (↓n) g :=\nby cases g; simp\n\ntheorem lambda_reduction_step_iff: reduction_step_of head_step (Λ f) g ↔\n  head_step (Λ f) g ∨ ∃ x, g = Λ x ∧ reduction_step_of head_step f x :=\nby cases g; simp\n\ntheorem dot_reduction_step_iff: reduction_step_of head_step (f·f') g ↔\n  head_step (f·f') g ∨\n    (∃ x, g = x·f' ∧ reduction_step_of head_step f x) ∨\n    ∃ x, g = f·x ∧ reduction_step_of head_step f' x :=\nby cases g; simp [and.assoc]\nend\n\nattribute [irreducible] reduction_step_of\nattribute [simp] down_reduction_step_iff lambda_reduction_step_iff dot_reduction_step_iff\n\nvariables {head_step: utlc → utlc → Prop}\nvariables {f f': utlc}\nvariables {n: ℕ}\nvariables {g g': utlc}\nvariables {m: ℕ}\n\n@[simp] theorem reduction_step_down_iff:\n  reduction_step_of head_step f ↓m ↔ head_step f ↓m :=\nby cases f; simp\n\ntheorem down_of_refl_trans_reduction_step:\n  (∀ n g, ¬ head_step (↓n:utlc) g) →\n  relation.refl_trans_gen (reduction_step_of head_step) (↓n:utlc) g →\n  g = ↓n :=\nbegin\n  intros h p,\n  induction p with x g hfx hxg,\n  { refl },\n  rw [p_ih, down_reduction_step_iff] at hxg,\n  exfalso,\n  apply h _ _ hxg,\nend\n\ntheorem reduction_step_lambda_iff:\n  reduction_step_of head_step f Λ g ↔\n  head_step f (Λ g) ∨ ∃ x, f = Λ x ∧ reduction_step_of head_step x g :=\nby cases f ; simp\n\ntheorem reduction_step_dot_iff:\n  reduction_step_of head_step f (g·g') ↔\n  head_step f (g·g') ∨\n    (∃ x, x·g' = f ∧ reduction_step_of head_step x g) ∨\n    ∃ x, g·x = f ∧ reduction_step_of head_step x g' :=\nby cases f; simp [and.assoc]\n\ntheorem lambda_reduction_step_lambda:\n  reduction_step_of head_step f g → reduction_step_of head_step (Λ f) (Λ g) :=\nby simp; exact or.inr\n\ntheorem lambda_refl_reduction_step_lambda:\n  relation.refl_gen (reduction_step_of head_step) f g →\n  relation.refl_gen (reduction_step_of head_step) Λ f Λ g :=\nbegin\n  intro p,\n  cases p,\n  { refl },\n  exact relation.refl_gen.single (lambda_reduction_step_lambda (by assumption)),\nend\n\ntheorem lambda_refl_trans_reduction_step_lambda:\n  relation.refl_trans_gen (reduction_step_of head_step) f g →\n  relation.refl_trans_gen (reduction_step_of head_step) Λ f Λ g :=\nrelation.refl_trans_gen.lift _ (@lambda_reduction_step_lambda _)\n\ntheorem lambda_of_refl_trans_reduction_step:\n  (∀ f g, ¬ head_step (Λ f) g) →\n  relation.refl_trans_gen (reduction_step_of head_step) (Λ f) g →\n  ∃ x, g = Λ x :=\nbegin\n  intros h p,\n  induction p with x g hfx hxg,\n  { use f },\n  cases p_ih with y hy,\n  rw [hy, lambda_reduction_step_iff] at hxg,\n  cases hxg,\n  { exfalso, apply h _ _ hxg },\n  rcases hxg with ⟨z, _, _⟩,\n  use z,\n  assumption,\nend\n\ntheorem dot_reduction_step_dot_left:\n  reduction_step_of head_step f f' → reduction_step_of head_step (f·g) (f'·g) :=\nby intro p; simp [p]\n\ntheorem dot_reduction_step_dot_right:\n  reduction_step_of head_step g g' → reduction_step_of head_step (f·g) (f·g') :=\nby intro p; simp [p]\n\ntheorem dot_refl_reduction_step_dot_left:\n  relation.refl_gen (reduction_step_of head_step) f f' →\n  ∀ g, relation.refl_gen (reduction_step_of head_step) (f·g) (f'·g) :=\nbegin\n  intros p _,\n  cases p,\n  { refl },\n  exact relation.refl_gen.single (dot_reduction_step_dot_left (by assumption))\nend\n\ntheorem dot_refl_reduction_step_dot_right:\n  relation.refl_gen (reduction_step_of head_step) f f' →\n  ∀ g, relation.refl_gen (reduction_step_of head_step) (g·f) (g·f') :=\nbegin\n  intros p _,\n  cases p,\n  { refl },\n  exact relation.refl_gen.single (dot_reduction_step_dot_right (by assumption))\nend\n\ntheorem dot_refl_trans_reduction_step_dot_left:\n  relation.refl_trans_gen (reduction_step_of head_step) f f' →\n  ∀ g, relation.refl_trans_gen (reduction_step_of head_step) (f·g) (f'·g) :=\nbegin\n  intros _ g,\n  simp only [show ∀ f, f·g = (λ x, x·g) f, by simp],\n  apply relation.refl_trans_gen.lift,\n  intros _ _ _,\n  apply dot_reduction_step_dot_left,\n  assumption'\nend\n\ntheorem dot_refl_trans_reduction_step_dot_right:\n  relation.refl_trans_gen (reduction_step_of head_step) f f' →\n  ∀ g, relation.refl_trans_gen (reduction_step_of head_step) (g·f) (g·f') :=\nbegin\n  intros _ g,\n  simp only [show ∀ f, g·f = (λ x, g·x) f, by simp],\n  apply relation.refl_trans_gen.lift,\n  intros _ _ _,\n  apply dot_reduction_step_dot_right,\n  assumption'\nend\n\ntheorem dot_refl_trans_reduction_step_dot:\n  relation.refl_trans_gen (reduction_step_of head_step) f f' →\n  relation.refl_trans_gen (reduction_step_of head_step) g g' →\n  relation.refl_trans_gen (reduction_step_of head_step) (f·g) (f'·g') :=\nbegin\n  intros p q,\n  apply trans,\n  exact dot_refl_trans_reduction_step_dot_left p _,\n  exact dot_refl_trans_reduction_step_dot_right q _\nend\n\ntheorem  dot_of_refl_trans_reduction_step:\n  (∀ f f' g, ¬ head_step (f·f') g) →\n  relation.refl_trans_gen (reduction_step_of head_step) (f·f') g →\n  ∃ x x', g = x·x' :=\nbegin\n  intros h p,\n  induction p with x g hfx hxg,\n  { exact ⟨f, f', rfl⟩ },\n  rcases p_ih with ⟨y, y', hy⟩,\n  rw [hy, dot_reduction_step_iff] at hxg,\n  obtain hxg|hxg|hxg := hxg;\n  try { exfalso, apply h _ _ _ hxg };\n  rcases hxg with ⟨z, hg, _⟩,\n  exact ⟨z, y', hg⟩,\n  exact ⟨y, z, hg⟩\nend\n\ntheorem uses_zero_reduction_step  (h: ∀ {f g}, head_step f g → ∀ {n}, f.uses n = 0 → g.uses n = 0):\n  (reduction_step_of head_step f g) → ∀ {n}, f.uses n = 0 → g.uses n = 0 :=\nbegin\n  induction f using lambda_calculus.utlc.notation_induction_on generalizing g;\n  intro p;\n  try { rw [down_reduction_step_iff] at p };\n  try { rw [lambda_reduction_step_iff] at p, obtain p|p := p };\n  try { rw [dot_reduction_step_iff] at p, obtain p|p|p := p };\n  try { { apply @h _ _ p } };\n  rcases p with ⟨x, hg, hf⟩;\n  simp only [hg, dot_uses, add_eq_zero_iff, and_imp];\n  intro n,\n  { apply f_ih hf },\n  { intros p q, exact ⟨ f_ih_f hf p, q ⟩ },\n  { intros p q, exact ⟨ p, f_ih_g hf q ⟩ }\nend\n\ntheorem uses_reduction_step_uses (h: ∀ {f} {g}, head_step f g → ∀ n, f.uses n = g.uses n):\n  (reduction_step_of head_step f g) → ∀ n, f.uses n = g.uses n :=\nbegin\n  induction f using lambda_calculus.utlc.notation_induction_on generalizing g;\n  intro p;\n  try { rw [down_reduction_step_iff] at p };\n  try { rw [lambda_reduction_step_iff] at p, obtain p|p := p };\n  try { rw [dot_reduction_step_iff] at p, obtain p|p|p := p };\n  try { { apply h p } };\n  rcases p with ⟨x, hg, hf⟩;\n  intro n;\n  simp only [hg, lambda_uses, dot_uses, and_imp,\n    add_eq_zero_iff, add_left_cancel_iff, add_right_cancel_iff],\n  { apply f_ih hf },\n  { apply f_ih_f hf },\n  { apply f_ih_g hf }\nend\n\ntheorem uses_refl_trans_reduction_step_uses (h: ∀ {f g}, head_step f g → ∀ n, f.uses n = g.uses n):\n  (relation.refl_trans_gen (reduction_step_of head_step) f g) → ∀ n, f.uses n = g.uses n :=\nbegin\n  intro p,\n  induction p with x g hfx hxg,\n  { intro, refl },\n  intro n,\n  apply trans (p_ih _),\n  apply uses_reduction_step_uses @h hxg,\nend\n\ntheorem shift_reduction_step_shift_iff: (∀ f g n, head_step (f ↑¹ n) (g ↑¹ n) ↔ head_step f g) → \n  (reduction_step_of head_step (f ↑¹ n) (g ↑¹ n) ↔ reduction_step_of head_step f g) :=\nbegin\n  intro p,\n  induction f using lambda_calculus.utlc.notation_induction_on generalizing n g,\n  { simpa[down_shift] using p (↓f) g n },\n  { simp [← p (Λ f_f) g n, shift_eq_lambda_iff, f_ih] },\n  { simp [← p (f_f·f_g) g n, shift_eq_dot_iff, f_ih_f, f_ih_g, and.left_comm] }\nend\n\ntheorem shift_refl_trans_reduction_shift: (∀ f g n, head_step (f ↑¹ n) (g ↑¹ n) ↔ head_step f g) → \n  (relation.refl_trans_gen (reduction_step_of head_step) f g → relation.refl_trans_gen (reduction_step_of head_step) (f ↑¹ n) (g ↑¹ n)) :=\nbegin\n intro p,\n apply relation.refl_trans_gen.lift,\n intros a b,\n apply (shift_reduction_step_shift_iff p).mpr\nend\n\nlocal notation a `[` b `:=` c  `]` : 70 := has_substitution.substitution a b c\n\ntheorem substitution_reduction_step_left (f: utlc) (hf: ∀ {f f'}, head_step f f' → ∀ n g, relation.refl_trans_gen (reduction_step_of head_step) (f[n:=g]) (f'[n:=g])): reduction_step_of head_step f f' →\n  ∀ n g, relation.refl_trans_gen (reduction_step_of head_step) (f[n:=g]) (f'[n:=g]) :=\nbegin\n  induction f using lambda_calculus.utlc.notation_induction_on generalizing f';\n  simp only [down_reduction_step_iff, lambda_reduction_step_iff, dot_reduction_step_iff],\n  { apply hf },\n  { intros p n g,\n    cases p,\n    { apply hf p },\n    rcases p with ⟨x, hfx', hfx⟩,\n    rw [hfx'],\n    exact lambda_refl_trans_reduction_step_lambda (f_ih hfx _ _) },\n  { intros p n g,\n    obtain p|p|p := p,\n    { apply hf p },\n    { rcases p with ⟨x, hfx', hfx⟩,\n      rw [hfx'],\n      exact dot_refl_trans_reduction_step_dot_left (f_ih_f hfx _ _) _ },\n    { rcases p with ⟨x, hfx', hfx⟩,\n      rw [hfx'],\n      exact dot_refl_trans_reduction_step_dot_right (f_ih_g hfx _ _) _ } }\nend\n\ntheorem substitution_reduction_step_right (f: utlc) (hf: ∀ f g n, head_step (f ↑¹ n) (g ↑¹ n) ↔ head_step f g): reduction_step_of head_step g g' →\n  ∀ n, relation.refl_trans_gen (reduction_step_of head_step) (f[n:=g]) (f[n:=g']) :=\nbegin\n  induction f using lambda_calculus.utlc.notation_induction_on generalizing g g';\n  intros p n,\n  { obtain h|h|h := nat.lt_trichotomy f n,\n    { simp [h] },\n    { apply relation.refl_trans_gen.single,\n      simpa [h] using p },\n    { simp [lt_asymm h, ne_of_gt h] } },\n  { apply lambda_refl_trans_reduction_step_lambda,\n    apply f_ih,\n    rw shift_reduction_step_shift_iff,\n    assumption,\n    assumption },\n  { apply dot_refl_trans_reduction_step_dot,\n    apply f_ih_f,\n    assumption,\n    apply f_ih_g,\n    assumption }\nend\n\ndef reduced : utlc → (utlc → bool) → bool\n| (↓ n) := λ p, p (↓ n)\n| (Λ f) := λ p, p (Λ f) ∧ reduced f p\n| (f · g) := λ p, p (f · g) ∧ reduced f p ∧ reduced g p\n\ndef reduced_of : (utlc → bool) → utlc → bool := λ h f, reduced f h\n\nsection\nlocal attribute [simp] reduced reduced_of\nvariables (head_pred: utlc → bool)\n\ntheorem down_reduced: reduced_of head_pred ↓n ↔ head_pred ↓n :=\nby simp\n\ntheorem lambda_reduced: reduced_of head_pred Λ f ↔ head_pred Λ f ∧ reduced_of head_pred f :=\nby simp\n\ntheorem dot_reduced: reduced_of head_pred (f·g) ↔ head_pred (f·g) ∧ reduced_of head_pred f ∧ reduced_of head_pred g :=\nby simp\nend\nattribute [irreducible] reduced_of\nattribute [simp] down_reduced lambda_reduced dot_reduced\n\nvariables {head_pred: utlc → bool}\n\ntheorem head_reduced_of_reduced: reduced_of head_pred f → head_pred f :=\nby { cases f; simp; repeat {intro }; assumption}\n\ntheorem reduced_iff_not_reduction_step: (∀ f, head_pred f ↔ ∀ g, ¬ head_step f g) → (∀ f, reduced_of head_pred f ↔ ∀ g, ¬ reduction_step_of head_step f g) :=\nbegin\n  have hforall : ∀ (f : utlc → utlc) (p: utlc → Prop), (∀ x y, x = f y → p y) ↔ (∀ y, p y) :=\n      λ f p, ⟨ λ q x, q (f x) x rfl, λ p x y _, p y ⟩,\n  intros p f,\n  induction f using lambda_calculus.utlc.notation_induction_on,\n  { simp[p] },\n  { simp [← p, f_ih, not_or_distrib, forall_and_distrib, hforall] },\n  { simp [← p, f_ih_f, f_ih_g, not_or_distrib, forall_and_distrib, hforall] }\nend\n\nend utlc\nend lambda_calculus", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/lambda_calculus/utlc/reduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.33278671069232435}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.opposites\nimport category_theory.hom_functor\n\n/-!\n# The Yoneda embedding\n\nThe Yoneda embedding as a functor `yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) ≅ (yoneda_evaluation C)`.\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v₁ u₁ u₂ -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nvariables {C : Type u₁} [𝒞 : category.{v₁} C]\ninclude 𝒞\n\n@[simps] def yoneda : C ⥤ (Cᵒᵖ ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop Y ⟶ X,\n    map := λ Y Y' f g, f.unop ≫ g,\n    map_comp' := λ _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := λ Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := λ X X' f, { app := λ Y g, g ≫ f } }\n\n@[simps] def coyoneda : Cᵒᵖ ⥤ (C ⥤ Type v₁) :=\n{ obj := λ X,\n  { obj := λ Y, unop X ⟶ Y,\n    map := λ Y Y' f g, g ≫ f,\n    map_comp' := λ _ _ _ f g, begin ext1, dsimp, erw [category.assoc] end,\n    map_id' := λ Y, begin ext1, dsimp, erw [category.comp_id] end },\n  map := λ X X' f, { app := λ Y g, f.unop ≫ g },\n  map_comp' := λ _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n  map_id' := λ X, begin ext, dsimp, erw [category.id_comp] end }\n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X ⟶ op Y) :\n  ((@yoneda C _).obj X).map f (𝟙 X) = ((@yoneda C _).map f.unop).app (op Y) (𝟙 Y) :=\nby obviously\n\n@[simp] lemma naturality {X Y : C} (α : yoneda.obj X ⟶ yoneda.obj Y)\n  {Z Z' : C} (f : Z ⟶ Z') (h : Z' ⟶ X) : f ≫ α.app (op Z') h = α.app (op Z) (f ≫ h) :=\nbegin erw [functor_to_types.naturality], refl end\n\ninstance yoneda_full : full (@yoneda C _) :=\n{ preimage := λ X Y f, (f.app (op X)) (𝟙 X) }\ninstance yoneda_faithful : faithful (@yoneda C _) :=\n{ injectivity' := λ X Y f g p,\n  begin\n    injection p with h,\n    convert (congr_fun (congr_fun h (op X)) (𝟙 X)); dsimp; simp,\n  end }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X ≅ Y`\napply yoneda.ext,\n-- Goals are now functions `(Z ⟶ X) → (Z ⟶ Y)`, `(Z ⟶ Y) → (Z ⟶ X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : Π {Z : C}, (Z ⟶ X) → (Z ⟶ Y)) (q : Π {Z : C}, (Z ⟶ Y) → (Z ⟶ X))\n  (h₁ : Π {Z : C} (f : Z ⟶ X), q (p f) = f) (h₂ : Π {Z : C} (f : Z ⟶ Y), p (q f) = f)\n  (n : Π {Z Z' : C} (f : Z' ⟶ Z) (g : Z ⟶ X), p (f ≫ g) = f ≫ p g) : X ≅ Y :=\n@preimage_iso _ _ _ _ yoneda _ _ _ _\n  (nat_iso.of_components (λ Z, { hom := p, inv := q, }) (by tidy))\n\ndef is_iso {X Y : C} (f : X ⟶ Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : Cᵒᵖ} (α : coyoneda.obj X ⟶ coyoneda.obj Y)\n  {Z Z' : C} (f : Z' ⟶ Z) (h : unop X ⟶ Z') : (α.app Z' h) ≫ f = α.app Z (h ≫ f) :=\nbegin erw [functor_to_types.naturality], refl end\n\ninstance coyoneda_full : full (@coyoneda C _) :=\n{ preimage := λ X Y f, ((f.app (unop X)) (𝟙 _)).op }\ninstance coyoneda_faithful : faithful (@coyoneda C _) :=\n{ injectivity' := λ X Y f g p,\n  begin\n    injection p with h,\n    have t := (congr_fun (congr_fun h (unop X)) (𝟙 _)),\n    simpa using congr_arg has_hom.hom.op t,\n  end }\n\ndef is_iso {X Y : Cᵒᵖ} (f : X ⟶ Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\nend coyoneda\n\nclass representable (F : Cᵒᵖ ⥤ Type v₁) :=\n(X : C)\n(w : yoneda.obj X ≅ F)\n\nend category_theory\n\nnamespace category_theory\n-- For the rest of the file, we are using product categories,\n-- so need to restrict to the case morphisms are in 'Type', not 'Sort'.\n\nuniverses v₁ u₁ u₂ -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nopen opposite\n\nvariables (C : Type u₁) [𝒞 : category.{v₁} C]\ninclude 𝒞\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((Cᵒᵖ ⥤ Type v₁) × Cᵒᵖ) :=\ncategory_theory.prod.{(max u₁ v₁) v₁} (Cᵒᵖ ⥤ Type v₁) Cᵒᵖ\n\ninstance prod_category_instance_2 : category (Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) :=\ncategory_theory.prod.{v₁ (max u₁ v₁)} Cᵒᵖ (Cᵒᵖ ⥤ Type v₁)\n\nopen yoneda\n\ndef yoneda_evaluation : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nevaluation_uncurried Cᵒᵖ (Type v₁) ⋙ ulift_functor.{u₁}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map α x).down = α.2.app Q.1 (P.2.map α.1 x.down) := rfl\n\ndef yoneda_pairing : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁) ⥤ Type (max u₁ v₁) :=\nfunctor.prod yoneda.op (𝟭 (Cᵒᵖ ⥤ Type v₁)) ⋙ functor.hom (Cᵒᵖ ⥤ Type v₁)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : Cᵒᵖ × (Cᵒᵖ ⥤ Type v₁)) (α : P ⟶ Q) (β : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map α β = yoneda.map α.1.unop ≫ β ≫ α.2 := rfl\n\ndef yoneda_lemma : yoneda_pairing C ≅ yoneda_evaluation C :=\n{ hom :=\n  { app := λ F x, ulift.up ((x.app F.1) (𝟙 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp,\n           ←functor_to_types.naturality,\n           obj_map_id,\n           functor_to_types.naturality,\n           functor_to_types.map_id]\n    end },\n  inv :=\n  { app := λ F x,\n    { app := λ X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [←functor_to_types.naturality, functor_to_types.map_comp]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [←functor_to_types.naturality,\n         obj_map_id,\n         functor_to_types.naturality,\n         functor_to_types.map_id],\n    refl,\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id]\n  end }.\n\nvariables {C}\n\n@[simp] def yoneda_sections (X : C) (F : Cᵒᵖ ⥤ Type v₁) :\n  (yoneda.obj X ⟶ F) ≅ ulift.{u₁} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\nomit 𝒞\n@[simp] def yoneda_sections_small {C : Type u₁} [small_category C] (X : C) (F : Cᵒᵖ ⥤ Type u₁) :\n  (yoneda.obj X ⟶ F) ≅ F.obj (op X) :=\nyoneda_sections X F ≪≫ ulift_trivial _\n\nend category_theory\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/Nouveau dossier/category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.33278671069232435}}
{"text": "\nsection RA\n\n axiom u : Type\n\n axiom _water_n_1  : u → Prop\n axiom _negligible_a_1 : u → Prop\n axiom _solubility_n_1 : u → Prop\n axiom _of_p : u → u → Prop\n axiom _bioaccumulation_nn_u_unknown : u → Prop\n axiom _low_a_on : u → Prop\n axiom _potential_n_1 : u → Prop\n axiom _have_v_1 : u → u → Prop\n axiom _log_n_of : u → u → Prop\n\n axiom Kow : u\n axiom X : u\n \n axiom poss : u → u → Prop\n axiom compound : u → u → Prop\n\n axiom bioaccumulation_potential : u → Prop\n axiom water_solubility : u → Prop\n axiom log_kow : u → Prop\n\n axiom f_bioaccumulation_potential : u → u\n axiom f_water_solubility : u → u\n axiom f_log_kow : u → u\n\n axiom low : u → Prop\n axiom high : u → Prop\n\n -- from NL questions\n\n def PA : Prop := ∃ x9, _water_n_1 x9 \n  ∧ (∃ x3, (∃ x15, X = x15 ∧ compound x3 x9 ∧ _solubility_n_1 x3 ∧ _of_p x3 x15) ∧ _negligible_a_1 x3)\n \n def PB : Prop := ∃ x9, (∃ x17, _bioaccumulation_nn_u_unknown x17 ∧ _low_a_on x9 ∧ compound x9 x17 ∧ _potential_n_1 x9) \n  ∧ (∃ x3, X = x3 ∧ _have_v_1 x3 x9)\n \n def PC : Prop := ∃ x3, (∃ x5, X = x5 ∧ (∃ x15, Kow = x15 ∧ poss x3 x5 ∧ _log_n_of x3 x15)) ∧ _low_a_on x3\n\n -- axioms from the Domain and Grammar\n\n variable (g1 : ∀ x p, _have_v_1 x p ↔ _of_p p x)\n variable (g2 : ∀ x p, poss p x ↔ _of_p p x)\n \n variable (g3 : ∀ x y, compound x y → _water_n_1 y → _solubility_n_1 x → water_solubility x)\n variable (g4 : ∀ x y, compound x y → _bioaccumulation_nn_u_unknown y → _potential_n_1 x → bioaccumulation_potential x)\n variable (g5 : ∀ x, _log_n_of x Kow → log_kow x)\n\n variable (g6 : ∀ x, _negligible_a_1 x ↔ _low_a_on x)\n\n variable (g7 : ∀ x p, _have_v_1 x p → _low_a_on p → water_solubility p → low (f_water_solubility x))\n variable (g8 : ∀ x p, _have_v_1 x p → _low_a_on p → bioaccumulation_potential p → low (f_bioaccumulation_potential x))\n variable (g9 : ∀ x p, _have_v_1 x p → _low_a_on p → log_kow p → low (f_log_kow x))\n\n -- may not be necessary\n variable (g10 : ∀ x, low x ↔ ¬ high x )\n variable (g11 : ∀ x, high (f_log_kow x) → high (f_bioaccumulation_potential x))\n variable (g12 : ∀ x, low (f_log_kow x) → high (f_water_solubility x))\n\n theorem test : PA → PB → PC → False := by \n   unfold PA PB PC \n   intro ⟨a1, ⟨h1, ⟨a2, ⟨⟨a3,⟨h2,h4,h5,h6⟩⟩, h3⟩⟩⟩⟩   \n   intro _\n   intro ⟨c1, ⟨⟨c2,⟨k1,⟨c4,⟨k2,k3,k4⟩⟩⟩⟩, k5⟩⟩\n   subst k1 h2 k2\n   have g31 := (g3 a2 a1) h4 h1 h5\n   -- have g41 := (g4 b1 b2) l3 l1 l4\n   have g51 := (g5 c1) k4\n   have g61 := (g6 a2).1 h3\n   -- have g81 := (g8 X b1) l6 l2 g41 \n   have g₁ := (g1 X c1).2 $ (g2 X c1).1 k3\n   have g₂ := (g1 X a2).2 h6\n   have g71 := (g7 X a2) g₂ g61 g31 \n   have g91 := (g9 X c1) g₁ k5 g51 \n   -- clear h4 h1 h5 l3 l1 l4 k4 g3 g4 g5 g6 h3 k3 h6 g₁ g₂ l2 k5 l6 g61 g7 g8 g9 g1 g2\n   have g₃ := (g12 X) g91    \n   exact (g10 $ f_water_solubility X).1 g71 g₃\n\n\nend RA\n\n", "meta": {"author": "arademaker", "repo": "mrs", "sha": "c30568c0835880547aa29432dc2be6cc587ee272", "save_path": "github-repos/lean/arademaker-mrs", "path": "github-repos/lean/arademaker-mrs/mrs-c30568c0835880547aa29432dc2be6cc587ee272/RA.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7279754489059775, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.33278430664390024}}
{"text": "-- import cech_d\n-- import algebra.homology.homological_complex\n-- import algebra.homology.homology\n-- import algebraic_geometry.sheafed_space\n-- import category_theory.opposites\n-- import oc\n-- import simplex\n-- import algebra.category.Group.colimits\n-- import algebra.category.Group.limits\n-- import lemmas.ulift\n-- import topology.sheaves.sheaf_condition.unique_gluing\n-- import lemmas.about_opens\n\n-- section\n\n-- open category_theory Top Top.presheaf category_theory.limits\n\n-- universe u\n-- variables {X : Top.{u}} (𝓕 : sheaf Ab X) (𝔘 : X.ocᵒᵖ)\n\n-- section\n\n-- open topological_space opposite category_theory.opposite Top\n-- open_locale big_operators\n\n-- -- lemma face.refine {n : ℕ} {A B : oc X} (h : A ⟶ B) (σ : simplex A n) :\n-- --   σ.face ≤ (σ.refine h).face := \n-- -- begin\n-- --   change infi _ ≤ infi _,\n-- --   change infi _ ≤ ⨅ (i : B.ι) (H : i ∈ finset.image h.func σ.to_finset), B.cover i,\n-- --   induction σ.to_finset using finset.induction with a s ha ih,\n-- --   { rw finset.image_empty,\n-- --     simp only [infi_false, infi_top, top_le_iff], },\n-- --   { rw [finset.infi_insert, finset.image_insert, finset.infi_insert],\n-- --     refine le_trans (inf_le_inf_left (A.cover a) ih) _,\n-- --     exact inf_le_inf_right _ (h.is_refinement a), },\n-- -- end\n\n-- -- def C.refine (n : ℕ) {A B : oc X} (h : A ⟶ B) :\n-- --   C 𝓕 B n ⟶ C 𝓕 A n :=\n-- -- { to_fun := λ f σ, 𝓕.1.map (hom_of_le $ face.refine h σ).op $ f (σ.refine h),\n-- --   map_zero' := begin\n-- --     ext σ,\n-- --     rw [Cech.zero_apply, map_zero, Cech.zero_apply],\n-- --   end,\n-- --   map_add' := λ f g, begin\n-- --     ext σ,\n-- --     rw [Cech.add_apply, map_add, Cech.add_apply],\n-- --   end }\n\n\n-- lemma 𝓕_map_congr' {U V : opens X} (i1 i2 : U ⟶ V) (x y : 𝓕.1.obj (op V)) (h2 : x = y) :\n--   𝓕.1.map i1.op x = 𝓕.1.map i2.op y :=\n-- have h : i1 = i2 := by ext,\n-- by subst h; subst h2\n\n-- lemma 𝓕_map_congr'' {n : ℕ} (A B : X.oc) (r : B ⟶ A) (σ : simplex B n) (σ1 σ2 : simplex A n.pred) (h : σ1 = σ2) (f : C 𝓕 A n.pred)\n--   (i1 : σ.face ⟶ σ1.face) (i2 : σ.face ⟶ σ2.face) :\n--   𝓕.1.map i1.op (f σ1) = 𝓕.1.map i2.op (f σ2) :=\n-- begin\n--   subst h,\n--   congr,\n-- end\n\n-- lemma 𝓕_map_congr''' {n : ℕ} {A : X.oc} (f : C 𝓕 A n) \n--   (σ1 σ2 : simplex A n) (h0 : σ1 = σ2)\n--   {U : opens X} (i1 : U ⟶ σ1.face) (i2 : U ⟶ σ2.face)  :\n--   𝓕.1.map i1.op (f σ1) = 𝓕.1.map i2.op (f σ2) := \n-- by { subst h0, apply 𝓕_map_congr', refl, }\n\n\n-- -- lemma C.refine_self (n : ℕ) (A : X.oc) :\n-- --   C.refine 𝓕 n (𝟙 A) = 𝟙 (C 𝓕 A n) := \n-- -- begin\n-- --   ext f σ,\n-- --   change 𝓕.1.map _ _ = f σ,\n-- --   have eq1 : f σ = 𝓕.1.map (𝟙 _).op (f σ),\n-- --   { rw [category_theory.op_id, 𝓕.1.map_id], refl, },\n-- --   rw [eq1],\n-- --   apply 𝓕_map_congr''' 𝓕 f,\n-- --   rw σ.refine_self,\n-- -- end\n\n-- -- lemma C.refine_comp (n : ℕ) {A B D : oc X} (r1 : A ⟶ B) (r2 : B ⟶ D) :\n-- --   C.refine 𝓕 n r2 ≫ C.refine 𝓕 n r1 = C.refine 𝓕 n (r1 ≫ r2) := \n-- -- begin\n-- --   ext1 f,\n-- --   change C.refine 𝓕 n r1 (C.refine 𝓕 n r2 f) = _,\n-- --   ext1 σ,\n-- --   change 𝓕.1.map _ (𝓕.1.map _ _) = 𝓕.1.map _ (f _),\n-- --   change (𝓕.1.map _ ≫ 𝓕.1.map _) _ = _,\n-- --   rw [← category_theory.functor.map_comp, ← category_theory.op_comp],\n-- --   apply 𝓕_map_congr''',\n-- --   symmetry,\n-- --   apply simplex.refine_comp,\n-- -- end\n\n-- -- def C.refine_functor (n : ℕ) : X.ocᵒᵖ ⥤ Ab :=\n-- -- { obj := λ A, C 𝓕 A.unop n,\n-- --   map := λ A B f, C.refine 𝓕 n f.unop,\n-- --   map_id' := λ A, C.refine_self 𝓕 n A.unop,\n-- --   map_comp' := λ A B D f g, by rw [unop_comp, C.refine_comp] }\n\n-- -- namespace d\n\n-- -- def from_to (i j : ℕ) : C 𝓕 𝔘.unop i ⟶ C 𝓕 𝔘.unop j :=\n-- -- dite (i + 1 = j) (λ h, begin subst h, exact d_pos (nat.zero_lt_succ _) end) (λ h, 0)\n\n-- -- lemma to_succ (i : ℕ) :\n-- --   from_to 𝓕 𝔘 i i.succ = d_pos (nat.zero_lt_succ _) :=\n-- -- dif_pos rfl\n\n-- -- lemma not_succ (i j : ℕ) (h : i + 1 ≠ j) :\n-- --   from_to 𝓕 𝔘 i j = 0 :=\n-- -- dif_neg h\n\n-- -- end d\n\n-- -- def  cech_chain : cochain_complex Ab ℕ :=\n-- -- { X := λ n, (C.refine_functor 𝓕 n).obj 𝔘,\n-- --   d := d.from_to 𝓕 𝔘,\n-- --   shape' := λ i j r, begin\n-- --     simp only [complex_shape.up_rel] at r,\n-- --     unfold d.from_to,\n-- --     split_ifs,\n-- --     { tauto, },\n-- --     refl,\n-- --   end,\n-- --   d_comp_d' := λ i j k h1 h2, begin\n-- --     simp only [complex_shape.up_rel] at h1 h2,\n-- --     subst' h1,\n-- --     subst' h2,\n-- --     rw [d.to_succ, d.to_succ],\n-- --     ext1 f,\n-- --     erw dd_pos.eq0 (nat.zero_lt_succ _) f,\n-- --     refl,\n-- --   end }\n\n-- -- def cech_chain.functor : X.ocᵒᵖ ⥤ cochain_complex Ab ℕ :=\n-- -- { obj := λ 𝔘, cech_chain 𝓕 𝔘,\n-- --   map := λ A B r, \n-- --   { f := λ i, (C.refine_functor 𝓕 i).map r,\n-- --     comm' := λ i j h, begin\n-- --       simp only [complex_shape.up_rel] at h,\n-- --       subst h,\n-- --       ext f σ,\n-- --       change (d.from_to _ _ i (i + 1)) (C.refine 𝓕 _ r.unop f) σ = (C.refine 𝓕 _ r.unop) (d.from_to _ _ _ _ _) _,\n-- --       rw [d.to_succ, d_pos.def, d.to_succ],\n-- --       change _ = 𝓕.1.map _ _,\n-- --       rw [d_pos.def, add_monoid_hom.map_sum],\n-- --       apply finset.sum_congr rfl (λ j hj, _),\n-- --       by_cases e : even j.1,\n-- --       { rw [if_pos e, id, if_pos e, id],\n-- --         change (𝓕.val.map (hom_of_le _).op ≫ _) _ = (𝓕.val.map ((σ.refine r.unop).der _ ⟨j.val, _⟩).op ≫ _) _,\n-- --         rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.op_comp],\n-- --         apply 𝓕_map_congr'',\n-- --         exact r.unop,\n-- --         symmetry,\n-- --         apply simplex.refine_ignore, },\n-- --       { rw [if_neg e, if_neg e, map_neg],\n-- --         congr' 1,\n-- --         change (𝓕.val.map (hom_of_le _).op ≫ _) _ = (𝓕.val.map ((σ.refine r.unop).der _ ⟨j.val, _⟩).op ≫ _) _,\n-- --         rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.op_comp],\n-- --         apply 𝓕_map_congr'',\n-- --         exact r.unop,\n-- --         symmetry,\n-- --         apply simplex.refine_ignore, },\n-- --     end },\n-- --   map_id' := λ A, begin\n-- --     ext i f σ,\n-- --     simp only [unop_id, homological_complex.id_f, id_apply],\n-- --     change 𝓕.1.map _ _ = _,\n-- --     have eq1 : f σ = 𝓕.1.map (𝟙 _) (f σ),\n-- --     { rw category_theory.functor.map_id,\n-- --       refl, },\n-- --     conv_rhs { rw eq1 },\n-- --     symmetry,\n-- --     have := 𝓕_map_congr''' 𝓕 f σ (σ.refine (𝟙 _)) (σ.refine_self).symm,\n-- --     convert this _ _,\n-- --     exact 𝟙 _,\n-- --   end,\n-- --   map_comp' := λ A B D r1 r2, begin\n-- --     ext i f σ,\n-- --     simp only [unop_comp, homological_complex.comp_f, comp_apply],\n-- --     rw category_theory.functor.map_comp,\n-- --     refl,\n-- --   end }\n\n-- -- include 𝓕\n\n-- -- def Cech_Ab (n : ℕ) : X.ocᵒᵖ ⥤ Ab.{u+1} := \n-- -- C.refine_functor 𝓕 n ⋙ AddCommGroup.ulift_functor.{u u+1}\n\n-- -- why do we need to lift up\n-- /-\n-- ```\n-- include 𝓕\n\n-- example (n : ℕ) : true := \n-- begin\n--   have := @AddCommGroup.colimits.colimit X.ocᵒᵖ _ (C.refine_functor 𝓕 n),\n--   -- this doesn't work, because we need a functor ` (X.oc)ᵒᵖ ⥤ AddCommGroup : Type (u+2)`,\n--   -- but we only have `C.refine_functor 𝓕 n : (X.oc)ᵒᵖ ⥤ Ab : Type u+1`\n-- end\n-- ```\n-- -/\n\n-- -- lemma Cech_Ab_obj (n : ℕ) (A : X.ocᵒᵖ) :\n-- --   (Cech_Ab 𝓕 n).obj A = AddCommGroup.ulift.{u u+1} (C 𝓕 A.unop n) := rfl\n\n-- -- lemma Cech_Ab_map (n : ℕ) {A B : X.ocᵒᵖ} (r : A ⟶ B) :\n-- --   (Cech_Ab 𝓕 n).map r = (AddCommGroup.ulift_functor.{u u+1}).map (C.refine 𝓕 n r.unop) := rfl\n\n-- -- def Cech_d (A : X.ocᵒᵖ) (i j : ℕ) : (Cech_Ab 𝓕 i).obj A ⟶ (Cech_Ab 𝓕 j).obj A :=\n-- -- dite (i + 1 = j) (λ h, (AddCommGroup.ulift_functor.{u u+1}).map (d_pos (nat.zero_lt_succ _)) ≫ eq_to_hom begin subst h, refl end) (λ h, 0)\n\n-- -- lemma Cech_d_down_apply (A : X.ocᵒᵖ) (i j : ℕ) (f : (Cech_Ab 𝓕 i).obj A) :\n-- --   (Cech_d 𝓕 A i j f).down = \n-- --   dite (i + 1 = j) (λ h, begin subst h, exact (d_pos (nat.zero_lt_succ _) f.down) end) (λ h, 0) := \n-- -- begin\n-- --   induction f,\n-- --   dsimp only [Cech_d],\n-- --   split_ifs,\n-- --   { induction h, refl, },\n-- --   { refl, },\n-- -- end\n\n-- -- lemma Cech_d_succ_down_apply (A : X.ocᵒᵖ) (i : ℕ) (f : (Cech_Ab 𝓕 i).obj A) :\n-- --   (Cech_d 𝓕 A i (i+1) f).down = \n-- --   (d_pos (nat.zero_lt_succ _) f.down) := \n-- -- begin\n-- --   induction f,\n-- --   dsimp only [Cech_d],\n-- --   split_ifs;\n-- --   refl,\n-- -- end\n\n-- lemma Cech_d_not_succ_down_apply (A : X.ocᵒᵖ) {i j : ℕ} (h : i + 1 ≠ j) (f : (Cech_Ab 𝓕 i).obj A) :\n--   (Cech_d 𝓕 A i j f).down = 0 := \n-- begin\n--   induction f,\n--   dsimp only [Cech_d],\n--   rw dif_neg h,\n--   refl,\n-- end\n\n-- -- def from_to (i j : ℕ) : C 𝓕 𝔘.unop i ⟶ C 𝓕 𝔘.unop j :=\n-- -- dite (i + 1 = j) (λ h, begin subst h, exact d_pos (nat.zero_lt_succ _) end) (λ h, 0)\n\n-- lemma Cech_d_to_succ (A : X.ocᵒᵖ) (i : ℕ) :\n--   Cech_d 𝓕 A i i.succ = (AddCommGroup.ulift_functor.{u u+1}).map (d_pos (nat.zero_lt_succ _)) :=\n-- dif_pos rfl\n\n-- lemma Cech_d_not_succ (A : X.ocᵒᵖ) {i j : ℕ} (h : i + 1 ≠ j) :\n--   Cech_d 𝓕 A i j = 0 :=\n-- dif_neg h\n\n-- -- lemma to_succ (i : ℕ) :\n-- --   from_to 𝓕 𝔘 i i.succ = d_pos (nat.zero_lt_succ _) :=\n-- -- dif_pos rfl\n\n-- -- lemma not_succ (i j : ℕ) (h : i + 1 ≠ j) :\n-- --   from_to 𝓕 𝔘 i j = 0 :=\n-- -- dif_neg h\n\n-- def Cech_complex_obj (A : X.ocᵒᵖ) : cochain_complex Ab.{u+1} ℕ :=\n-- { X := λ n, (Cech_Ab 𝓕 n).obj A,\n--   d := λ i j, Cech_d 𝓕 A i j,\n--   shape' := λ i j r, dif_neg r,\n--   d_comp_d' := λ i j k h1 h2, begin\n--     simp only [complex_shape.up_rel] at h1 h2,\n--     subst' h1,\n--     subst' h2,\n--     rw [Cech_d_to_succ, Cech_d_to_succ],\n--     ext1 f,\n--     rw [← category_theory.functor.map_comp, dd_pos.eq0 (nat.zero_lt_succ _)],\n--     refl,\n--   end }\n\n-- lemma Cech_complex_obj_d (A : X.ocᵒᵖ) :\n--   (Cech_complex_obj 𝓕 A).d = Cech_d 𝓕 A := rfl\n\n-- def Cech_complex : X.ocᵒᵖ ⥤ cochain_complex Ab.{u+1} ℕ :=\n-- { obj := λ A, Cech_complex_obj 𝓕 A,\n--   map := λ A B r, \n--   { f := λ i, (Cech_Ab 𝓕 i).map r,\n--     comm' := λ i j h, begin\n--       simp only [complex_shape.up_rel] at h,\n--       subst h,\n--       ext f σ,\n--       rw [category_theory.comp_apply, category_theory.comp_apply, Cech_Ab_map, Cech_complex_obj_d, Cech_d_succ_down_apply],\n--       change _ = 𝓕.1.map _ _,\n--       rw [d_pos.def, Cech_complex_obj_d, Cech_d_succ_down_apply, d_pos.def, add_monoid_hom.map_sum],\n--       apply finset.sum_congr rfl (λ j hj, _),\n--       by_cases e : even j.1,\n--       { rw [if_pos e, id, if_pos e, id],\n--         change (𝓕.val.map (hom_of_le _).op ≫ _) _ = (𝓕.val.map ((σ.refine r.unop).der _ ⟨j.val, _⟩).op ≫ _) _,\n--         rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.op_comp],\n--         apply 𝓕_map_congr'',\n--         exact r.unop,\n--         symmetry,\n--         apply simplex.refine_ignore, },\n--       { rw [if_neg e, if_neg e, map_neg],\n--         congr' 1,\n--         change (𝓕.val.map (hom_of_le _).op ≫ _) _ = (𝓕.val.map ((σ.refine r.unop).der _ ⟨j.val, _⟩).op ≫ _) _,\n--         rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.op_comp],\n--         apply 𝓕_map_congr'',\n--         exact r.unop,\n--         symmetry,\n--         apply simplex.refine_ignore, },\n--     end },\n--   map_id' := λ A, begin\n--     ext i f σ,\n--     simp only [unop_id, homological_complex.id_f, id_apply, Cech_Ab_map, AddCommGroup.ulift_functor_map_down],\n--     rw C.refine_self,\n--     refl,\n--   end,\n--   map_comp' := λ A B D r1 r2, begin\n--     ext i f σ,\n--     simp only [unop_comp, homological_complex.comp_f, comp_apply, Cech_Ab_map, AddCommGroup.ulift_functor_map_down],\n--     change _ = (C.refine 𝓕 i r1.unop ≫ C.refine 𝓕 i r2.unop) f.down σ,\n--     rw C.refine_comp,\n--   end }\n\n-- noncomputable def ex1 :\n--   homological_complex.homology ((Cech_complex 𝓕).obj 𝔘) 0 ≅\n--   kernel ((Cech_complex_obj 𝓕 𝔘).d 0 1) :=\n-- begin\n--   refine homology_iso_cokernel_image_to_kernel' _ _ _ ≪≫ _,\n--   change cokernel (kernel.lift _ _ _) ≅ _,\n\n--   simp only [image.ι_zero', homological_complex.d_to_eq_zero, cochain_complex.prev_nat_zero, eq_self_iff_true, kernel.lift_zero, Cech_complex_obj_d, Cech_d_to_succ],\n--   refine cokernel_zero_iso_target ≪≫ _,\n--   refine AddCommGroup.kernel_iso_ker _ ≪≫ _,\n--   refine _ ≪≫ (AddCommGroup.kernel_iso_ker (AddCommGroup.ulift_functor.map (d_pos _))).symm,\n--   refine { hom := _, inv := _, hom_inv_id' := _, inv_hom_id' := _ },\n--   { refine { to_fun := _, map_zero' := _, map_add' := _ },\n--     { intros x,\n--       refine ⟨x.1, _⟩,\n--       rw add_monoid_hom.mem_ker,\n--       have := x.2,\n--       rw add_monoid_hom.mem_ker at this,\n--       change homological_complex.d_from (Cech_complex_obj 𝓕 𝔘) 0 x.1 = _ at this,\n--       have eq1 := homological_complex.d_from_eq (Cech_complex_obj 𝓕 𝔘) (show 1 = 0 + 1, from rfl),\n--       erw Cech_complex_obj_d at eq1,\n--       rw Cech_d_to_succ at eq1,\n--       generalize_proofs h1 h2 at eq1,\n--       have eq2 : homological_complex.d_from (Cech_complex_obj 𝓕 𝔘) 0 x.1 = (AddCommGroup.ulift_functor.map (d_pos h1) ≫ (homological_complex.X_next_iso (Cech_complex_obj 𝓕 𝔘) h2).inv) x.1,\n--       { rw eq1, },\n--       rw comp_apply at eq2,\n--       rw this at eq2,\n--       apply_fun (homological_complex.X_next_iso (Cech_complex_obj 𝓕 𝔘) h2).hom at eq2,\n--       simp only [map_zero, coe_inv_hom_id] at eq2,\n--       rw <- eq2 },\n--     { rw subtype.ext_iff_val,\n--       refl, },\n--     { intros x1 x2, \n--       rw subtype.ext_iff_val,\n--       refl, } },\n--   { refine { to_fun := _, map_zero' := _, map_add' := _ },\n--     { intros x,\n--       refine ⟨x.1, _⟩,\n--       have := x.2,\n--       rw add_monoid_hom.mem_ker at this ⊢,\n--       have eq1 := homological_complex.d_from_eq (Cech_complex_obj 𝓕 𝔘) (show 1 = 0 + 1, from rfl),\n--       erw eq1,\n--       rw comp_apply,\n--       generalize_proofs h1 h2,\n--       apply_fun (homological_complex.X_next_iso (Cech_complex_obj 𝓕 𝔘) h1).hom,\n--       simp only [coe_inv_hom_id, map_zero],\n--       convert this,\n--       apply function.bijective.injective,\n--       rw function.bijective_iff_has_inverse,\n--       use (homological_complex.X_next_iso (Cech_complex_obj 𝓕 𝔘) h1).inv,\n--       refine ⟨_, _⟩,\n--       intros x,\n--       rw coe_hom_inv_id,\n--       intros x,\n--       rw coe_inv_hom_id, },\n--     { rw subtype.ext_iff_val,\n--       refl },\n--     { intros x y,\n--       rw subtype.ext_iff_val,\n--       refl, } },\n--   { ext1 σ,\n--     simp only [comp_apply, subtype.val_eq_coe, add_subgroup.coe_mk, add_monoid_hom.coe_mk, set_like.eta, id_apply] },\n--   { ext1 σ,\n--     simp only [comp_apply, subtype.val_eq_coe, add_subgroup.coe_mk, add_monoid_hom.coe_mk, set_like.eta, id_apply] },\n-- end\n\n-- noncomputable def ex2 : \n--   kernel ((Cech_complex_obj 𝓕 𝔘).d 0 1) ≅ \n--   kernel ((AddCommGroup.ulift_functor.{u u+1}).map (@d_pos X 𝓕 (unop 𝔘) _ (nat.zero_lt_succ 0))) :=\n--   -- (AddCommGroup.ulift_functor.{u u+1}).obj (𝓕.1.obj (op ⊤)) := \n-- begin\n--   change kernel (Cech_d 𝓕 𝔘 0 1) ≅ _,\n--   refine AddCommGroup.kernel_iso_ker _ ≪≫ _,\n--   refine _ ≪≫ (AddCommGroup.kernel_iso_ker (AddCommGroup.ulift_functor.map (d_pos _))).symm,\n  \n--   refine { hom := 𝟙 _, inv := 𝟙 _ },\n-- end\n\n-- noncomputable def ex3 :\n--   kernel ((AddCommGroup.ulift_functor.{u u+1}).map (@d_pos X 𝓕 (unop 𝔘) _ (nat.zero_lt_succ 0))) ≅\n--   (AddCommGroup.ulift_functor.{u u+1}).obj (kernel (@d_pos X 𝓕 (unop 𝔘) _ (nat.zero_lt_succ 0))) :=\n-- begin\n--   apply AddCommGroup.ulift_kernel_iso_kernel_ulift,\n-- end\n\n-- lemma ex41.forward.aux1 {i j : 𝔘.unop.ι} {f : C 𝓕 𝔘.unop 0} (h : d_pos (nat.zero_lt_succ 0) f = 0) :\n--   𝓕.1.map (((unop 𝔘).cover i).inf_le_left ((unop 𝔘).cover j) ≫ eq_to_hom begin\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {i}, card_eq := rfl}) = \n--   𝓕.1.map (hom_of_le begin\n--     convert inf_le_left,\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {i}, card_eq := rfl}) :=\n-- begin\n--   congr,\n-- end\n\n-- lemma ex41.forward.aux1' {i j : 𝔘.unop.ι} {f : C 𝓕 𝔘.unop 0} (h : d_pos (nat.zero_lt_succ 0) f = 0) :\n--   𝓕.1.map (((unop 𝔘).cover i).inf_le_right ((unop 𝔘).cover j) ≫ eq_to_hom begin\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {j}, card_eq := rfl}) = \n--   𝓕.1.map (hom_of_le begin\n--     convert inf_le_right,\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {j}, card_eq := rfl}) :=\n-- begin\n--   congr,\n-- end\n\n-- lemma ex41.forward.aux2 {i j : 𝔘.unop.ι} {f : C 𝓕 𝔘.unop 0} (h : d_pos (nat.zero_lt_succ 0) f = 0) :\n--   𝓕.1.map (((unop 𝔘).cover i).inf_le_right ((unop 𝔘).cover j) ≫ eq_to_hom begin\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {j}, card_eq := rfl}) = \n--   𝓕.1.map (hom_of_le begin\n--     convert inf_le_right,\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {j}, card_eq := rfl}) :=\n-- begin\n--   congr,\n-- end\n\n-- lemma ex41.forward.aux2' {i j : 𝔘.unop.ι} {f : C 𝓕 𝔘.unop 0} (h : d_pos (nat.zero_lt_succ 0) f = 0) :\n--   𝓕.1.map (((unop 𝔘).cover i).inf_le_left ((unop 𝔘).cover j) ≫ eq_to_hom begin\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {i}, card_eq := rfl}) = \n--   𝓕.1.map (hom_of_le begin\n--     convert inf_le_left,\n--     unfold simplex.face,\n--     simp\n--   end).op (f {to_finset := {i}, card_eq := rfl}) :=\n-- begin\n--   congr,\n-- end\n\n-- lemma ex41.forward.aux3 {i j : 𝔘.unop.ι} (ineq : i < j) (f : C 𝓕 𝔘.unop 0) :\n--   𝓕.1.map (simplex.der (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ ⟨0, nat.zero_lt_succ 1⟩).op \n--   (f (simplex.ignore (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ 0)) =\n--   𝓕.1.map \n--   ((hom_of_le (λ p hp, begin\n--     rw [opens.mem_coe] at hp ⊢,\n--     change _ ∈ (infi _) at hp,\n--     have := (infi_le (λ (i_1 : (unop 𝔘).ι), ⨅ (H : i_1 ∈ ({to_finset := {i, j}, card_eq := begin\n--       rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end} : simplex _ 1).to_finset), (unop 𝔘).cover i_1)),\n--     specialize this j,\n--     dsimp only at this,\n--     simp only [le_infi_iff] at this,\n--     specialize this _,\n--     { rw finset.mem_insert,\n--       right,\n--       exact finset.mem_singleton_self _ },\n--     specialize this hp,\n--     convert this,\n--     unfold simplex.face,\n--     { simp },\n--   end) : \n--   ({to_finset := {i, j}, card_eq := begin\n--      rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end} : simplex _ 1).face ⟶\n--   ({to_finset := {j}, card_eq := begin\n--     rw finset.card_singleton,\n--   end} : simplex _ 0).face)).op \n--   (f ⟨{j}, rfl⟩) \n--   :=\n-- begin\n--   generalize_proofs _ h1 h2 h3 h4 h5,\n--   apply 𝓕_map_congr''',\n--   unfold simplex.ignore,\n--   dsimp only,\n--   rw simplex.ext_iff,\n--   dsimp only,\n--   ext1 k,\n--   split,\n--   { intro hk,\n--     rw finset.mem_erase_nth at hk,\n--     rcases hk with ⟨hk1, hk2⟩,\n--     rw [finset.mem_insert, finset.mem_singleton] at hk2,\n--     rcases hk2 with rfl|rfl,\n--     { rw finset.mem_singleton,\n--       contrapose! hk1,\n--       rw finset.order_emb_of_fin_zero,\n--       rw finset.min'_insert,\n--       rw finset.min'_singleton,\n--       symmetry,\n--       rw min_eq_right_iff,\n--       refine le_of_lt ineq, },\n--     { exact finset.mem_singleton_self _ }, },\n--   { intro hk,\n--     rw finset.mem_singleton at hk,\n--     rw hk,\n--     rw finset.mem_erase_nth,\n--     split,\n--     { intro rid,\n--       rw finset.order_emb_of_fin_zero at rid,\n--       rw finset.min'_insert at rid,\n--       rw finset.min'_singleton at rid,\n--       have ineq2 := min_le_right j i,\n--       have ineq3 := min_le_left j i,\n--       rw ← rid at ineq2,\n--       rw lt_iff_not_ge at ineq,\n--       apply ineq,\n--       exact ineq2, },\n--     { rw finset.mem_insert,\n--       right,\n--       exact finset.mem_singleton_self _, } },\n-- end\n\n-- lemma ex41.forward.aux3' {i j : 𝔘.unop.ι} (ineq : j < i) (f : C 𝓕 𝔘.unop 0) :\n--   𝓕.1.map (simplex.der (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ ⟨0, nat.zero_lt_succ 1⟩).op \n--   (f (simplex.ignore (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ 0)) =\n--   𝓕.1.map \n--   ((hom_of_le (λ p hp, begin\n--     rw [opens.mem_coe] at hp ⊢,\n--     change _ ∈ (infi _) at hp,\n--     have := (infi_le (λ (i_1 : (unop 𝔘).ι), ⨅ (H : i_1 ∈ ({to_finset := {i, j}, card_eq := begin\n--       rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end} : simplex _ 1).to_finset), (unop 𝔘).cover i_1)),\n--     specialize this i,\n--     dsimp only at this,\n--     simp only [le_infi_iff] at this,\n--     specialize this _,\n--     { rw finset.mem_insert,\n--       left,\n--       refl, },\n--     specialize this hp,\n--     convert this,\n--     unfold simplex.face,\n--     { simp },\n--   end) : \n--   ({to_finset := {i, j}, card_eq := begin\n--      rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end} : simplex _ 1).face ⟶\n--   ({to_finset := {i}, card_eq := begin\n--     rw finset.card_singleton,\n--   end} : simplex _ 0).face)).op \n--   (f ⟨{i}, rfl⟩) \n--   :=\n-- begin\n--   generalize_proofs _ h1 h2 h3 h4 h5,\n--   apply 𝓕_map_congr''',\n--   unfold simplex.ignore,\n--   dsimp only,\n--   rw simplex.ext_iff,\n--   dsimp only,\n--   ext1 k,\n--   split,\n--   { intro hk,\n--     rw finset.mem_erase_nth at hk,\n--     rcases hk with ⟨hk1, hk2⟩,\n--     rw [finset.mem_insert, finset.mem_singleton] at hk2,\n--     rcases hk2 with rfl|rfl,\n--     { rw finset.mem_singleton, },\n--     { contrapose! hk1,\n--       rw finset.order_emb_of_fin_zero,\n--       rw finset.min'_insert,\n--       rw finset.min'_singleton,\n--       symmetry,\n--       rw min_eq_left_iff,\n--       refine le_of_lt ineq }, },\n--   { intro hk,\n--     rw finset.mem_singleton at hk,\n--     rw hk,\n--     rw finset.mem_erase_nth,\n--     split,\n--     { intro rid,\n--       rw finset.order_emb_of_fin_zero at rid,\n--       rw finset.min'_insert at rid,\n--       rw finset.min'_singleton at rid,\n--       have ineq2 := min_le_right j i,\n--       have ineq3 := min_le_left j i,\n--       rw ← rid at ineq3,\n--       rw lt_iff_not_ge at ineq,\n--       apply ineq,\n--       exact ineq3, },\n--     { rw finset.mem_insert,\n--       left,\n--       refl } },\n-- end\n\n-- lemma ex41.forward.aux4 {i j : 𝔘.unop.ι} (ineq : i < j) (f : C 𝓕 𝔘.unop 0) :\n--   𝓕.1.map (simplex.der (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ ⟨1, lt_add_one 1⟩).op \n--   (f (simplex.ignore (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ 1)) =\n--   𝓕.1.map \n--   ((hom_of_le (λ p hp, begin\n--     rw [opens.mem_coe] at hp ⊢,\n--     change _ ∈ (infi _) at hp,\n--     have := (infi_le (λ (i_1 : (unop 𝔘).ι), ⨅ (H : i_1 ∈ ({to_finset := {i, j}, card_eq := begin\n--       rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end} : simplex _ 1).to_finset), (unop 𝔘).cover i_1)),\n--     specialize this i,\n--     dsimp only at this,\n--     simp only [le_infi_iff] at this,\n--     specialize this _,\n--     { rw finset.mem_insert,\n--       left,\n--       refl },\n--     specialize this hp,\n--     convert this,\n--     unfold simplex.face,\n--     { simp },\n--   end) : \n--   ({to_finset := {i, j}, card_eq := begin\n--      rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end} : simplex _ 1).face ⟶\n--   ({to_finset := {i}, card_eq := begin\n--     rw finset.card_singleton,\n--   end} : simplex _ 0).face)).op \n--   (f ⟨{i}, rfl⟩) \n--   :=\n-- begin\n--   generalize_proofs _ h1 h2 h3 h4 h5,\n--   apply 𝓕_map_congr''',\n--   unfold simplex.ignore,\n--   dsimp only,\n--   rw simplex.ext_iff,\n--   dsimp only,\n--   ext1 k,\n--   split,\n--   { intro hk,\n--     rw finset.mem_erase_nth at hk,\n--     rcases hk with ⟨hk1, hk2⟩,\n--     rw [finset.mem_insert, finset.mem_singleton] at hk2,\n--     rcases hk2 with rfl|rfl,\n--     { rw finset.mem_singleton },\n--     { rw finset.mem_singleton,\n--       contrapose! hk1,\n--       -- have := finset.order_emb_of_fin_last,\n--       rw finset.order_emb_of_fin_last ({to_finset := {i, k}, card_eq := begin\n--         rw [finset.card_insert_of_not_mem],\n--         refl,\n--         rw [finset.mem_singleton],\n--         intro r,\n--         rw r at ineq,\n--         exact lt_irrefl _ ineq,\n--       end} : simplex _ 1).card_eq,\n--       rw finset.max'_insert,\n--       rw finset.max'_singleton,\n--       symmetry,\n--       rw max_eq_left_iff,\n--       refine le_of_lt ineq,\n--       exact nat.zero_lt_succ _, }, },\n--   { intro hk,\n--     rw finset.mem_singleton at hk,\n--     rw hk,\n--     rw finset.mem_erase_nth,\n--     split,\n--     { intro rid,\n--       subst hk,\n--       rw finset.order_emb_of_fin_last ({to_finset := {k, j}, card_eq := begin\n--         rw [finset.card_insert_of_not_mem],\n--         refl,\n--         rw [finset.mem_singleton],\n--         intro r,\n--         rw r at ineq,\n--         exact lt_irrefl _ ineq,\n--       end} : simplex _ 1).card_eq at rid,\n--       dsimp only at rid,\n--       generalize_proofs ne at rid,\n--       have := finset.le_max' {k, j} j _,\n--       erw ← rid at this,\n--       rw lt_iff_not_ge at ineq,\n--       apply ineq,\n--       exact this,\n--       rw finset.mem_insert,\n--       right,\n--       rw finset.mem_singleton,\n--       exact nat.zero_lt_succ _, },\n--     { rw finset.mem_insert,\n--       left,\n--       refl } },\n-- end\n\n-- lemma ex41.forward.aux4' {i j : 𝔘.unop.ι} (ineq : j < i) (f : C 𝓕 𝔘.unop 0) :\n--   𝓕.1.map (simplex.der (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ ⟨1, lt_add_one 1⟩).op \n--   (f (simplex.ignore (nat.zero_lt_succ 0) ⟨{i, j}, begin\n--     rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end⟩ 1)) =\n--   𝓕.1.map \n--   ((hom_of_le (λ p hp, begin\n--     rw [opens.mem_coe] at hp ⊢,\n--     change _ ∈ (infi _) at hp,\n--     have := (infi_le (λ (i_1 : (unop 𝔘).ι), ⨅ (H : i_1 ∈ ({to_finset := {i, j}, card_eq := begin\n--       rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end} : simplex _ 1).to_finset), (unop 𝔘).cover i_1)),\n--     specialize this j,\n--     dsimp only at this,\n--     simp only [le_infi_iff] at this,\n--     specialize this _,\n--     { rw finset.mem_insert,\n--       right,\n--       rw finset.mem_singleton, },\n--     specialize this hp,\n--     convert this,\n--     unfold simplex.face,\n--     { simp },\n--   end) : \n--   ({to_finset := {i, j}, card_eq := begin\n--      rw [finset.card_insert_of_not_mem],\n--     refl,\n--     rw [finset.mem_singleton],\n--     intro r,\n--     rw r at ineq,\n--     exact lt_irrefl _ ineq,\n--   end} : simplex _ 1).face ⟶\n--   ({to_finset := {j}, card_eq := begin\n--     rw finset.card_singleton,\n--   end} : simplex _ 0).face)).op \n--   (f ⟨{j}, rfl⟩) \n--   :=\n-- begin\n--   generalize_proofs _ h1 h2 h3 h4 h5,\n--   apply 𝓕_map_congr''',\n--   unfold simplex.ignore,\n--   dsimp only,\n--   rw simplex.ext_iff,\n--   dsimp only,\n--   ext1 k,\n--   split,\n--   { intro hk,\n--     rw finset.mem_erase_nth at hk,\n--     rcases hk with ⟨hk1, hk2⟩,\n--     rw [finset.mem_insert, finset.mem_singleton] at hk2,\n--     rcases hk2 with rfl|rfl,\n--     { rw finset.mem_singleton,\n--       contrapose! hk1,\n--       -- have := finset.order_emb_of_fin_last,\n--       rw finset.order_emb_of_fin_last ({to_finset := {k, j}, card_eq := begin\n--         rw [finset.card_insert_of_not_mem],\n--         refl,\n--         rw [finset.mem_singleton],\n--         intro r,\n--         rw r at ineq,\n--         exact lt_irrefl _ ineq,\n--       end} : simplex _ 1).card_eq,\n--       rw finset.max'_insert,\n--       rw finset.max'_singleton,\n--       symmetry,\n--       rw max_eq_right_iff,\n--       refine le_of_lt ineq,\n--       exact nat.zero_lt_succ _, },\n--     { rw finset.mem_singleton }, },\n--   { intro hk,\n--     rw finset.mem_singleton at hk,\n--     rw hk,\n--     rw finset.mem_erase_nth,\n--     split,\n--     { intro rid,\n--       subst hk,\n--       rw finset.order_emb_of_fin_last ({to_finset := {i, k}, card_eq := begin\n--         rw [finset.card_insert_of_not_mem],\n--         refl,\n--         rw [finset.mem_singleton],\n--         intro r,\n--         rw r at ineq,\n--         exact lt_irrefl _ ineq,\n--       end} : simplex _ 1).card_eq at rid,\n--       dsimp only at rid,\n--       generalize_proofs ne at rid,\n--       have := finset.le_max' {i, k} i _,\n--       erw ← rid at this,\n--       rw lt_iff_not_ge at ineq,\n--       apply ineq,\n--       exact this,\n--       rw finset.mem_insert,\n--       left,\n--       refl,\n--       exact nat.zero_lt_succ _, },\n--     { rw finset.mem_insert,\n--       right,\n--       rw finset.mem_singleton, } },\n-- end\n\n-- lemma ex41.forward.aux5 (f : C 𝓕 𝔘.unop 0) \n--   (o1 o2 o3 o4 : opens X)\n--   (oop2 : 𝓕.val.obj (op o2))\n--   (oop3 : 𝓕.val.obj (op o3))\n-- -- o1 : face ij\n-- -- o2 : face i\n-- -- o3 : face j\n-- -- o4 : cover i ⊓ cover j\n--   (h12 : o1 ≤ o2)\n--   (h13 : o1 ≤ o3)\n--   (h42 : o4 ≤ o2)\n--   (h43 : o4 ≤ o3)\n--   (h14 : o4 ≤ o1)\n--   (eq1 : 𝓕.1.map (hom_of_le h12).op oop2 = 𝓕.1.map (hom_of_le h13).op oop3) : \n--   𝓕.1.map (hom_of_le h42).op oop2 = 𝓕.1.map (hom_of_le h43).op oop3 :=\n-- begin\n--   have : hom_of_le h42 = hom_of_le h14 ≫ hom_of_le h12,\n--   { ext, },\n--   rw this,\n--   have : hom_of_le h43 = hom_of_le h14 ≫ hom_of_le h13,\n--   { ext },\n--   rw this,\n--   rw [op_comp, category_theory.functor.map_comp, op_comp, category_theory.functor.map_comp],\n--   rw [comp_apply, comp_apply, eq1],\n-- end\n\n-- lemma ker_compatible (f : add_monoid_hom.ker (@d_pos X 𝓕 𝔘.unop 1 (nat.zero_lt_succ 0))) : presheaf.is_compatible 𝓕.1 𝔘.unop.cover \n--   (λ i, begin\n--     refine 𝓕.1.map (eq_to_hom _).op (f.1 ⟨{i}, rfl⟩),\n--     unfold simplex.face,\n--     simp,\n--   end) :=\n-- begin\n--   intros i j,\n--   have := f.2,\n--   rw add_monoid_hom.mem_ker at this,\n      \n--   rcases @trichotomous 𝔘.unop.ι (<) _ i j with ineq|ineq|ineq,\n--   { dsimp only,\n--     change (𝓕.1.map _ ≫ _) _ = (𝓕.1.map _ ≫ _) _,\n--     rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← op_comp, ← op_comp],\n--     rw ex41.forward.aux1 _ _ this,\n--     rw ex41.forward.aux2 _ _ this,\n\n--     have eq1 : d_pos (nat.zero_lt_succ 0) f.1 ⟨{i, j}, begin\n--       rw finset.card_insert_of_not_mem,\n--       simp only [finset.card_singleton],\n--       simp only [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end⟩ = 0,\n--     { rw this, simp, },\n--     simp only [d_pos_01] at eq1,\n--     change _ - _ = _ at eq1,\n--     rw sub_eq_zero at eq1,\n--     dsimp only at eq1,\n--     rw ex41.forward.aux3 at eq1,\n--     have eq2 := eq.trans eq1 (ex41.forward.aux4 𝓕 𝔘 ineq f.1),\n          \n--     refine ex41.forward.aux5 𝓕 𝔘 f.1 ({to_finset := {i, j}, card_eq := _} : simplex 𝔘.unop 1).face \n--       ({to_finset := {i}, card_eq := _} : simplex _ 0).face\n--       ({to_finset := {j}, card_eq := _} : simplex _ 0).face\n--       ((unop 𝔘).cover i ⊓ (unop 𝔘).cover j)\n--       (f.val {to_finset := {i}, card_eq := _})\n--       (f.val {to_finset := {j}, card_eq := _})\n--       _ _ _ _ _ _,\n--     { rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw finset.mem_singleton,\n--       intro r,\n--       subst r,\n--       apply lt_irrefl i ineq, },\n--     { unfold simplex.face,\n--       intros p hp,\n--       erw opens.finset_infi at hp ⊢,\n--       intros k hk,\n--       apply hp,\n--       dsimp only at hk ⊢,\n--       rw [finset.mem_insert, finset.mem_singleton],\n--       left,\n--       simpa using hk, },\n--     { unfold simplex.face,\n--       intros p hp,\n--       erw opens.finset_infi at hp ⊢,\n--       intros k hk,\n--       apply hp,\n--       dsimp only at hk ⊢,\n--       rw [finset.mem_insert, finset.mem_singleton],\n--       right,\n--       simpa using hk, },\n--     { intros p hp,\n--       unfold simplex.face,\n--       rw [opens.mem_coe] at hp ⊢,\n--       rw opens.finset_infi,\n--       intros h hk,\n--       dsimp only at hk,\n--       rw [finset.mem_insert, finset.mem_singleton] at hk,\n--       rw ← opens.inter_eq at hp,\n--       rcases hp with ⟨hp1, hp2⟩,\n--       rcases hk with rfl|rfl,\n--       { exact hp1, },\n--       { exact hp2, }, },\n--     { symmetry,\n--       exact eq2, }, },\n--   { subst ineq, refl, },\n--   { dsimp only,\n--     change (𝓕.1.map _ ≫ _) _ = (𝓕.1.map _ ≫ _) _,\n--     rw [← category_theory.functor.map_comp, ← category_theory.functor.map_comp, ← op_comp, ← op_comp],\n--     rw ex41.forward.aux1' _ _ this,\n--     rw ex41.forward.aux2' _ _ this,\n\n--     have eq1 : d_pos (nat.zero_lt_succ 0) f.1 ⟨{i, j}, begin\n--       rw finset.card_insert_of_not_mem,\n--       simp only [finset.card_singleton],\n--       simp only [finset.mem_singleton],\n--       intro r,\n--       rw r at ineq,\n--       exact lt_irrefl _ ineq,\n--     end⟩ = 0,\n--     { rw this, simp, },\n--     simp only [d_pos_01] at eq1,\n--     change _ - _ = _ at eq1,\n--     rw sub_eq_zero at eq1,\n--     dsimp only at eq1,\n--     rw ex41.forward.aux3' at eq1,\n--     have eq2 := eq.trans eq1 (ex41.forward.aux4' 𝓕 𝔘 ineq f.1),\n\n--     refine ex41.forward.aux5 𝓕 𝔘 f.1 ({to_finset := {i, j}, card_eq := _} : simplex 𝔘.unop 1).face \n--       ({to_finset := {i}, card_eq := _} : simplex _ 0).face\n--       ({to_finset := {j}, card_eq := _} : simplex _ 0).face\n--       ((unop 𝔘).cover i ⊓ (unop 𝔘).cover j)\n--       (f.val {to_finset := {i}, card_eq := _})\n--       (f.val {to_finset := {j}, card_eq := _})\n--       _ _ _ _ _ _,\n--     { rw finset.card_insert_of_not_mem,\n--       refl,\n--       rw finset.mem_singleton,\n--       intro r,\n--       subst r,\n--       apply lt_irrefl i ineq, },\n--     { unfold simplex.face,\n--       intros p hp,\n--       erw opens.finset_infi at hp ⊢,\n--       intros k hk,\n--       apply hp,\n--       dsimp only at hk ⊢,\n--       rw [finset.mem_insert, finset.mem_singleton],\n--       left,\n--       simpa using hk, },\n--     { unfold simplex.face,\n--       intros p hp,\n--       erw opens.finset_infi at hp ⊢,\n--       intros k hk,\n--       apply hp,\n--       dsimp only at hk ⊢,\n--       rw [finset.mem_insert, finset.mem_singleton],\n--       right,\n--       simpa using hk, },\n--     { intros p hp,\n--       unfold simplex.face,\n--       rw [opens.mem_coe] at hp ⊢,\n--       rw opens.finset_infi,\n--       intros h hk,\n--       dsimp only at hk,\n--       rw [finset.mem_insert, finset.mem_singleton] at hk,\n--       rw ← opens.inter_eq at hp,\n--       rcases hp with ⟨hp1, hp2⟩,\n--       rcases hk with rfl|rfl,\n--       { exact hp1, },\n--       { exact hp2, }, },\n--     { exact eq2, },\n--     -- { symmetry,\n--     --   exact eq2, }, \n--     },\n-- end\n\n-- lemma unique_gluing_prop (f : add_monoid_hom.ker (@d_pos X 𝓕 𝔘.unop 1 (nat.zero_lt_succ 0))) :\n--   ∃! (s : 𝓕.val.obj (op ⊤)),\n--   ∀ (i : (unop 𝔘).ι),\n--     (𝓕.val.map (hom_of_le le_top).op) s =\n--     (𝓕.val.map (eq_to_hom (begin\n--         unfold simplex.face,\n--         simp,\n--       end : (unop 𝔘).cover i = ({to_finset := {i}, card_eq := begin\n--         rw finset.card_singleton,\n--       end} : simplex _ 0).face)).op) (f.val {to_finset := {i}, card_eq := rfl}) :=\n-- begin\n--   exact sheaf.exists_unique_gluing' 𝓕 𝔘.unop.cover ⊤ (λ i, hom_of_le le_top) begin\n--       rw 𝔘.unop.is_cover,\n--       exact le_refl _,\n--     end (λ i, begin\n--       refine 𝓕.1.map (eq_to_hom _).op (f.1 ⟨{i}, rfl⟩),\n--       unfold simplex.face,\n--       simp,\n--     end) (ker_compatible 𝓕 𝔘 f),\n-- end\n\n-- noncomputable def unique_gluing (f : add_monoid_hom.ker (@d_pos X 𝓕 𝔘.unop 1 (nat.zero_lt_succ 0))) :\n--   𝓕.1.obj (op ⊤) :=\n-- classical.some (unique_gluing_prop _ _ f)\n\n-- lemma unique_gluing_is_glueing (f : add_monoid_hom.ker (@d_pos X 𝓕 𝔘.unop 1 (nat.zero_lt_succ 0))) (i : 𝔘.unop.ι) :\n--   𝓕.1.map (hom_of_le le_top).op (unique_gluing _ _ f) = \n--   𝓕.1.map (eq_to_hom (begin\n--         unfold simplex.face,\n--         simp,\n--       end : (unop 𝔘).cover i = ({to_finset := {i}, card_eq := begin\n--         rw finset.card_singleton,\n--       end} : simplex _ 0).face)).op (f.1 ⟨{i}, rfl⟩) := \n-- begin\n--   have := classical.some_spec (unique_gluing_prop _ _ f),\n--   dsimp only at this,\n--   rcases this with ⟨h1, h2⟩,\n--   exact h1 i,\n-- end\n\n-- lemma unique_gluing_is_unique (f : add_monoid_hom.ker (@d_pos X 𝓕 𝔘.unop 1 (nat.zero_lt_succ 0))) (s : 𝓕.1.obj (op ⊤))\n--   (is_glue : ∀ (i : 𝔘.unop.ι), \n--     𝓕.1.map (hom_of_le le_top).op s = \n--     𝓕.1.map (eq_to_hom (begin\n--         unfold simplex.face,\n--         simp,\n--       end : (unop 𝔘).cover i = ({to_finset := {i}, card_eq := begin\n--         rw finset.card_singleton,\n--       end} : simplex _ 0).face)).op (f.1 ⟨{i}, rfl⟩)) :\n--   (unique_gluing _ _ f) = s :=\n-- begin\n--   have := classical.some_spec (unique_gluing_prop _ _ f),\n--   dsimp only at this,\n--   rcases this with ⟨h1, h2⟩,\n--   symmetry,\n--   apply h2,\n--   assumption,\n-- end\n\n-- noncomputable def ex41.forward :\n--   (AddCommGroup.of $ add_monoid_hom.ker (@d_pos X 𝓕 (unop 𝔘) _ (nat.zero_lt_succ 0))) ⟶\n--   𝓕.1.obj (op ⊤) :=\n-- { to_fun := λ f, unique_gluing _ _ f,\n--   map_zero' := begin\n--     apply unique_gluing_is_unique,\n--     intros i,\n--     simp,\n--   end,\n--   map_add' := λ f g, begin\n--     apply unique_gluing_is_unique,\n--     intros i,\n--     rw map_add,\n--     erw map_add,\n--     congr;\n--     apply unique_gluing_is_glueing,\n--   end }\n\n-- lemma inj :\n--   function.injective (ex41.forward 𝓕 𝔘) :=\n-- begin\n--   intros f g h,\n--   change unique_gluing _ _ f = unique_gluing _ _ g at h,\n--   have h1 := unique_gluing_is_glueing _ _ f,\n--   rw subtype.ext_iff_val,\n--   ext σ,\n--   have eq1 : ∃ i, σ.to_finset = {i},\n--   { have := σ.card_eq,\n--     simp only [nat.pred_succ] at this,\n--     rwa finset.card_eq_one at this, },\n--   rcases eq1 with ⟨i, hi⟩,\n--   specialize h1 i,\n  \n--   have h2 := unique_gluing_is_glueing _ _ g,\n--   specialize h2 i,\n\n--   rw [eq_to_hom_op, eq_to_hom_map] at h1 h2,\n--   rw h at h1,\n--   rw h1 at h2,\n\n--   have eq2 : σ = ⟨{i}, rfl⟩,\n--   { rw simplex.ext_iff, exact hi, },\n--   rw eq2,\n--   generalize_proofs _ _ h3 at h2,\n--   suffices : function.injective (eq_to_hom h3),\n--   apply this,\n--   exact h2,\n\n--   intros x y h,\n--   apply_fun (eq_to_hom h3.symm) at h,\n--   change (eq_to_hom h3 ≫ eq_to_hom h3.symm) x = (eq_to_hom h3 ≫ eq_to_hom h3.symm) y at h,\n--   rw [eq_to_hom_trans, eq_to_hom_refl] at h_1,\n--   simpa only using h_1,\n-- end\n\n-- lemma surj :\n--   function.surjective (ex41.forward 𝓕 𝔘) :=\n-- begin\n--   rw function.surjective_iff_has_right_inverse,\n--   fconstructor,\n--   { intros s,\n--     refine ⟨λ σ, _, _⟩,\n--     exact 𝓕.1.map (hom_of_le le_top).op s,\n--     rw add_monoid_hom.mem_ker,\n--     ext σ,\n--     simp only [Cech.zero_apply],\n--     rw d_pos_01,\n--     change _ - _ = _,\n--     rw sub_eq_zero,\n--     dsimp only,\n--     change (𝓕.1.map _ ≫ 𝓕.1.map _) _ = (𝓕.1.map _ ≫ 𝓕.1.map _) _,\n--     suffices :\n--       (𝓕.val.map ((simplex.der d0._proof_6 σ ⟨0, d0._proof_5⟩) ≫ (hom_of_le le_top)).op) s =\n--       (𝓕.val.map ((simplex.der d0._proof_10 σ ⟨1, d0._proof_9⟩) ≫ (hom_of_le le_top)).op) s,\n--     convert this,\n--     rw op_comp,\n--     rw 𝓕.1.map_comp,\n\n--     rw op_comp,\n--     rw 𝓕.1.map_comp,\n\n--     apply 𝓕_map_congr',\n--     refl, },\n--   { intros s,\n--     apply unique_gluing_is_unique,\n--     intros i,\n--     dsimp only,\n--     change _ = (𝓕.1.map _ ≫ 𝓕.1.map _) _,\n--     congr' 1,\n--     rw ← 𝓕.1.map_comp,\n--     rw ← op_comp,\n--     congr' 1,\n--   },\n-- end\n\n-- -- #check equiv.of_bijective (ex41.forward 𝓕 𝔘) ⟨inj _ _, surj _ _⟩\n-- noncomputable def ex41 :\n--   (AddCommGroup.of $ add_monoid_hom.ker (@d_pos X 𝓕 (unop 𝔘) _ (nat.zero_lt_succ 0))) ≃+\n--   𝓕.1.obj (op ⊤) :=\n-- add_equiv.of_bijective (ex41.forward _ _) ⟨inj _ _, surj _ _⟩\n\n-- noncomputable def ex4 :\n--   homological_complex.homology ((Cech_complex 𝓕).obj 𝔘) 0 ≅\n--   (AddCommGroup.ulift_functor.{u u+1}).obj (𝓕.1.obj (op ⊤)) :=\n-- begin\n--   refine ex1 _ _ ≪≫ _,\n--   refine ex2 _ _ ≪≫ _,\n--   refine AddCommGroup.ulift_kernel_iso_kernel_ulift _ ≪≫ _,\n--   apply AddCommGroup.ulift_iso,\n--   refine AddCommGroup.kernel_iso_ker _ ≪≫ _,\n--   refine { hom := _, inv := _, hom_inv_id' := _, inv_hom_id' := _ },\n--   { exact (ex41 _ _).to_add_monoid_hom, },\n--   { exact (ex41 _ _).symm.to_add_monoid_hom },\n--   { ext1,\n--     simp only [comp_apply, add_equiv.coe_to_add_monoid_hom, add_equiv.symm_apply_apply, id_apply], },\n--   { ext1,\n--     simp only [comp_apply, add_equiv.coe_to_add_monoid_hom, add_equiv.apply_symm_apply, id_apply]},\n-- end\n\n\n-- lemma aux1 (i k : ℕ) (A : X.ocᵒᵖ) (σ : simplex (unop A) k) (f : (Cech_Ab 𝓕 i).obj A) : \n--   ((0 : Cech_Ab 𝓕 i ⟶ Cech_Ab 𝓕 k).app A f).down σ = 0 :=\n-- begin\n--   rw show (0 : Cech_Ab 𝓕 i ⟶ Cech_Ab 𝓕 k).app A f = 0, from rfl,\n--   rw [AddCommGroup.ulift.zero_down],\n--   refl,\n-- end\n\n-- lemma aux2 (i j : ℕ) : colim.map (0 : Cech_Ab 𝓕 i ⟶ Cech_Ab 𝓕 j) = 0 := \n-- begin\n--   apply colimit.hom_ext,\n--   intros U,\n--   ext x,\n--   simp only [colimit.ι_map, nat_trans.app_zero, zero_comp, comp_zero],\n-- end\n\n-- noncomputable def Cech_complex_colimit : cochain_complex Ab.{u+1} ℕ :=\n-- { X := λ n, colim.obj (Cech_Ab 𝓕 n),\n--   d := λ i j, colim.map $ \n--   { app := λ A, Cech_d 𝓕 A i j,\n--     naturality' := λ A B r, begin\n--       -- sorry\n--       ext f σ,\n--       by_cases ineq1 : i + 1 = j,\n--       { subst ineq1, \n--         simp only [Cech_Ab_map, Cech_d_succ_down_apply, comp_apply, AddCommGroup.ulift_functor_map_down],\n--         rw [d_pos.def],\n--         change _ = 𝓕.1.map _ _,\n--         rw [d_pos.def, add_monoid_hom.map_sum],\n--         apply finset.sum_congr rfl (λ j hj, _),\n--         by_cases e : even j.1,\n--         { rw [if_pos e, id, if_pos e, id],\n--           change ((𝓕.1.map _) ≫ (𝓕.1.map _)) _ = ((𝓕.1.map _) ≫ (𝓕.1.map _)) _,\n--           rw [← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp],\n--           induction f,\n--           dsimp only,\n--           apply 𝓕_map_congr'',\n--           exact r.unop,\n--           symmetry,\n--           apply simplex.refine_ignore },\n--         { rw [if_neg e, if_neg e, map_neg],\n--           congr' 1,\n--           change ((𝓕.1.map _) ≫ (𝓕.1.map _)) _ = ((𝓕.1.map _) ≫ (𝓕.1.map _)) _,\n--           rw [← category_theory.functor.map_comp, ← category_theory.op_comp, ← category_theory.functor.map_comp, ← category_theory.op_comp],\n--           induction f,\n--           dsimp only,\n--           apply 𝓕_map_congr'',\n--           exact r.unop,\n--           symmetry,\n--           apply simplex.refine_ignore, }, },\n--       { simp only [Cech_Ab_map, comp_apply, AddCommGroup.ulift_functor_map_down],\n--         change _ = 𝓕.1.map _ _,\n--         rw [Cech_d_not_succ_down_apply, Cech_d_not_succ_down_apply, Cech.zero_apply, Cech.zero_apply, map_zero];\n--         exact ineq1, },\n--     end },\n--   shape' := λ i j h, begin\n--     suffices : colim.map 0 = 0,\n--     { convert this,\n--       ext A f σ,\n--       rw [Cech_d_not_succ_down_apply, Cech.zero_apply],\n--       refl, \n--       exact h,},\n--     { apply aux2, },\n--   end,\n--   d_comp_d' := λ i j k h1 h2, begin\n--     rw ← category_theory.functor.map_comp,\n--     suffices : colim.map 0 = 0,\n--     { convert this,\n--       ext A f σ,\n--       rw [nat_trans.comp_app],\n--       dsimp only,\n--       rw aux1 𝓕,\n--       change i + 1 = j at h1,\n--       subst h1,\n--       change (i + 1) + 1 = k at h2,\n--       subst h2,\n--       rw [category_theory.comp_apply, Cech_d_succ_down_apply, Cech_d_succ_down_apply],\n--       convert dd_pos_eq_zero _ _ _, },\n--     { apply aux2, },\n--   end }\n\n-- end\n\n-- end\n", "meta": {"author": "jjaassoonn", "repo": "cc", "sha": "6d3dc6885fa012e8c18fd38ab2949d73777fb442", "save_path": "github-repos/lean/jjaassoonn-cc", "path": "github-repos/lean/jjaassoonn-cc/cc-6d3dc6885fa012e8c18fd38ab2949d73777fb442/src/old/cech_chain.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.33274837342695796}}
{"text": "import category_theory.monoidal.functor\n\nopen category_theory\nvariables {C D : Type*} [category C] [category D]\n  [monoidal_category C] [monoidal_category D]\n\nnamespace category_theory.monoidal_functor\n\nvariables (F : monoidal_functor C D)\n\nlemma map_associator_hom\n  {X Y Z : C} : F.map (α_ X Y Z).hom =\n  inv (F.μ (X ⊗ Y) Z) ≫\n  inv (F.μ X Y ⊗ 𝟙 (F.obj Z)) ≫\n  (α_ (F.obj X) (F.obj Y) (F.obj Z)).hom ≫\n  (𝟙 (F.obj X) ⊗ F.μ Y Z) ≫\n  F.μ X (Y ⊗ Z) :=\nbegin\n  rw [is_iso.eq_inv_comp, is_iso.eq_inv_comp],\n  exact (F.to_lax_monoidal_functor.associativity X Y Z),\nend\n\nlemma map_associator_inv\n  {X Y Z : C} : F.map (α_ X Y Z).inv =\n  inv (F.μ X (Y ⊗ Z)) ≫\n  inv (𝟙 (F.obj X) ⊗ F.μ Y Z) ≫\n  (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫\n  (F.μ X Y ⊗ 𝟙 (F.obj Z)) ≫\n  (F.μ (X ⊗ Y) Z) :=\nbegin\n  rw [is_iso.eq_inv_comp, is_iso.eq_inv_comp],\n  exact (F.to_lax_monoidal_functor.associativity_inv X Y Z),\nend\n\nend category_theory.monoidal_functor\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/monoidal_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7341195152660688, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.33274836815365777}}
{"text": "import number_theory.bernoulli\n\n\n--OUTPUT 1\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) := \nbegin\n  apply nat.strong_induction_on n,\n  assume n hn,\n  assume ih,\n  assume h,\n  calc (polynomial.bernoulli (n + 1)).eval (1 + x) = (polynomial.bernoulli (n + 1)).eval (1 + x) : by auto [polynomial.eval_add, polynomial.eval_mul, polynial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (1) + (polynomial.bernoulli (n + 1)).eval (x) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one, polynomial.eval_X]\n  ... = (polynomial.bernoulli (n + 1)).eval (x) + (n + 1) * x^(n) : by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    have h1 : (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1),\n    {\n      calc (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x : by rw [polynomial.bernoulli, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C]\n      ... = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1) : by auto [add_zero]\n    },\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1), from h1\n  },\n  {\n    have h1 : (polynomial.bernoulli (d + 1)).eval (1 + x) = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x^(d + 1 - 1),\n    {\n      calc (polynomial.bernoulli (d + 1)).eval (1 + x) = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x^d : by rw [polynomial.bernoulli, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_C, polynomial.eval_mul, polynomial.eval_C, polynomial.eval_C, polynomial.eval_add, polynomial.eval_one, polynomial.eval_\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : ∀ (m : ℕ), m < n → (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from\n  begin\n    assume (m : ℕ) (h : m < n),\n    induction m with m hm,\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1), from by auto [polynomial.eval_C],\n    assume (m : ℕ) (h : m < n),\n    assume ih : (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1),\n    show (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (m + 1) * x^((m + 1) - 1), from\n    begin\n      have h1 : (polynomial.bernoulli (m + 1)).eval (1 + x) = ((polynomial.bernoulli m).eval (1 + x)) / (m + 1) + x^(m + 1), from by auto [polynomial.eval_bernoulli],\n      have h2 : (polynomial.bernoulli (m + 1)).eval x = ((polynomial.bernoulli m).eval x) / (m + 1) + x^(m + 1), from by auto [polynomial.eval_bernoulli],\n      have h3 : (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from by auto [ih],\n      have h4 : (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m), from by auto [nat.sub_self],\n      have h5 : (polynomial.bernoulli m).eval x + m * x^(m) = ((polynomial.bernoulli m).eval x) / (m + 1) + m * x^(m) + x^(m + 1), from by auto [mul_div_cancel, nat.succ_pos],\n      have h6 : (polynomial.bernoulli m).eval x + m * x^(m) = ((polynomial.bernoulli m).eval x) / (m + 1) + (m + 1) * x^(m) + x^(m + 1), from by auto [mul_add, mul_one],\n      show (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (m + 1) * x^((m + 1) - 1), from by auto [h1, h2, h3, h4, h5, h6, add_sub_cancel],\n    end,\n  end,\n  show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from by auto [h1, nat.lt_succ_self],\nend\n\n--OUTPUT 4\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    calc\n      (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval (1 : ℚ) : by auto\n      ... = (polynomial.bernoulli 0).eval x : by auto\n      ... = (polynomial.bernoulli 0).eval x + 0 * x ^ (0 - 1) : by auto\n  },\n  {\n    calc\n      (polynomial.bernoulli (d + 1)).eval (1 + x) = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x ^ (d) : by auto [hd]\n      ... = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x ^ (d - 1) * x : by auto [pow_succ]\n      ... = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x ^ (d - 1) : by auto [mul_one]\n  }\nend\n\n--OUTPUT 5\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  apply nat.strong_induction_on n,\n  assume n hn,\n  assume h,\n  have h1 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h2 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h3 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h4 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h5 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h6 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h7 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h8 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h9 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h10 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h11 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h12 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h13 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h14 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h15 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h16 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h17 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h18 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_X, polynomial.eval_pow, polynomial.eval_one],\n  have h19 : (n + 1 : ℚ) * (1 + x)^n - (n + 1 : ℚ) * x^n = (n + 1) * polynomial.eval (polynomial.X^n) x, from by auto [poly\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    simp [polynomial.bernoulli, polynomial.eval, polynomial.eval_add, polynomial.eval_mul, polynomial.eval_pow, polynomial.eval_C],\n    ring,\n  },\n  {\n    have h1 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h2 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h3 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h4 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h5 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h6 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h7 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h8 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h9 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h10 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h11 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h12 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h13 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h14 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h15 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h16 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h17 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h18 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h19 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h20 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h21 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h22 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h23 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h24 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h25 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1) * (1 + x)^(d - k) * x^k, from by simp [pow_add],\n    have h26 : (d + 1) * (1 + x)^d - (d + 1) * x^d = ∑ k in finset.range (d + 1), (d + 1)\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBernoulli polynomial evaluation\nGiven a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$\n`proof`\nWe apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\nMultiplying both sides by $d + 1$, and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n$$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\nThe conclusion then follows easily.\n\nQED\n\n-/\ntheorem  bernoulli_eval_one_add (n : ℕ) (x : ℚ) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Bernoulli polynomial evaluation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.33273572293930864}}
{"text": "import feather_model.basic\n\nuniverse u\n\nopen_locale classical\n\n/-!\nIn this file, we prove that the model, as defined in `feather_model.basic`,\nproduces a lawful type of terms, assuming the same is true for the lower level.\n-/\n\nnamespace feather_model\n\nvariables {T : Type u} [term_struct T] [type_data T] [term T]\n\nlemma empty_runtime_ok : runtime_ok (⟨∅, ∅, ∅⟩ : runtime_context (mterm T)) :=\nruntime_ok.empty\n\nlemma empty_rir_ok : rir_ok (⟨∅, ∅⟩ : rir_context (mterm T)) :=\nrir_ok.empty\n\nlemma runtime_ok_Γ (C : runtime_context (mterm T)) {v : V} {α : mterm T} :\n  runtime_ok C → is_type C.rir α → runtime_ok (C + ⟨{(v, α)}, ∅, ∅⟩) :=\nruntime_ok.Γ C v α\n\nlemma runtime_ok_Θ (C : runtime_context (mterm T)) {v : V} {α : mterm T} :\n  runtime_ok C → is_type C.rir α → runtime_ok (C + ⟨∅, {(v, α)}, ∅⟩) :=\nruntime_ok.Θ C v α\n\nlemma runtime_ok_Ξ (C : runtime_context (mterm T)) {v : V} {α : mterm T} :\n  runtime_ok C → is_type C.rir α → runtime_ok (C + ⟨∅, ∅, {(v, α)}⟩) :=\nruntime_ok.Ξ C v α\n\nlemma rir_ok_Γ (C : rir_context (mterm T)) {v : V} {α : mterm T} :\n  rir_ok C → is_type C α → rir_ok (C ∪ ⟨{(v, α)}, ∅⟩) :=\nrir_ok.Γ C v α\n\nlemma rir_ok_Ξ (C : rir_context (mterm T)) {v : V} {α : mterm T} :\n  rir_ok C → is_type C α → rir_ok (C ∪ ⟨∅, {(v, α)}⟩) :=\nrir_ok.Ξ C v α\n\nlemma defeq_reflexive (C : rir_context (mterm T)) {x α : mterm T} :\n  C ⊢ x : α → C ⊢ x ≡ x : α :=\nbegin\n  intros h I hI,\n  exact ⟨rfl, h.interpret I hI⟩,\nend\n\nlemma defeq_symmetric (C : rir_context (mterm T)) {x y α : mterm T} :\n  C ⊢ x ≡ y : α → C ⊢ y ≡ x : α :=\nbegin\n  intros h I hI,\n  obtain ⟨h₁, h₂⟩ := h I hI,\n  rw h₁ at h₂ ⊢,\n  exact ⟨rfl, h₂⟩,\nend\n\nlemma defeq_transitive (C : rir_context (mterm T)) {x y z α : mterm T} :\n  C ⊢ x ≡ y : α → C ⊢ y ≡ z : α → C ⊢ x ≡ z : α :=\nbegin\n  intros hxy hyz I hI,\n  obtain ⟨h₁, h₂⟩ := hxy I hI,\n  obtain ⟨h₃, h₄⟩ := hyz I hI,\n  rw [h₁, h₃] at h₂ ⊢,\n  exact ⟨rfl, h₂⟩,\nend\n\n/-\ninstance : term (mterm T) := {\n  empty_runtime_ok := empty_runtime_ok,\n  empty_rir_ok := empty_rir_ok,\n  runtime_ok_Γ := runtime_ok_Γ,\n  runtime_ok_Θ := runtime_ok_Θ,\n  runtime_ok_Ξ := runtime_ok_Ξ,\n  rir_ok_Γ := rir_ok_Γ,\n  rir_ok_Ξ := rir_ok_Ξ,\n  defeq_reflexive := defeq_reflexive,\n  defeq_symmetric := defeq_symmetric,\n  defeq_transitive := defeq_transitive,\n}\n-/\n\nend feather_model\n", "meta": {"author": "quill-lang", "repo": "feather-model", "sha": "64e760f426b4dde09065157c9f5121862681d9e3", "save_path": "github-repos/lean/quill-lang-feather-model", "path": "github-repos/lean/quill-lang-feather-model/feather-model-64e760f426b4dde09065157c9f5121862681d9e3/src/feather_model/term.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3326033255895426}}
{"text": "import data.fintype.parity\nimport data.nat.prime_norm_num\nimport data.zmod.basic\nimport linear_algebra.free_module.finite.basic\nimport number_theory.assorted_lemmas\nimport ring_theory.adjoin_root\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.integral_closure\nimport ring_theory.norm\nimport tactic.norm_cast\n\n/-!\n# Adjoining a root of a quadratic polynomial\n\n## Main definitions\n\n`quad_ring F a b` is the ring `F[α]`, where `α` is a root of the polynomial `X^2 - aX - b`.\n\nTo construct elements of `quad_ring F a b`:\n * the coercion `(↑(r : F) : quad_ring F a b)` sends `r` to `r + 0 α`\n * `quad_ring.root F a b` corresponds to `0 + 1 α`\n * `quad_ring` has a `comm_ring` instance providing `0`, `1`, `+`, `*`, and `-`\n\nTo compute with elements of `quad_ring F a b` by reducing equations to the ring `F`,\nuse the tactic `quad_ring.calc_tac`.\n\n## Main results\n\n * `quad_ring.comm_ring`: `quad_ring F a b` is a commutative ring if `F` is\n * `quad_ring.field`: `quad_ring F 0 b` is a field if `F` is and `b` is not a square\n * `quad_ring.power_basis`: shows `quad_ring F a b` is generated by adjoining `root F a b`\n\n * `quad_ring.map`: lift a ring homomorphism `R → S` to a ring homomorphism `R[α] → S[α]`\n * `quad_ring.congr`: lift a ring isomorphism `R ≃ S` to a ring isomorphism `R[α] ≃ S[α]`\n * `quad_ring.lift`: lift a ring homomorphism `R → S` to a ring homomorphism `R[α] → S`,\n   given a suitable image for `α`\n\n * `quad_ring.algebra''` (requires `open_locale quad_ring`): lift an `R`-algebra on `S`\n   to a `quad_ring R a b`-algebra on `quad_ring S a' b'`, given that `a b` map to `a' b'`.\n   Note that we only require `a' b' : S` to be the image propositionally (not just definitionally),\n   this allows easy use of numerals, e.g. `quad_ring ℤ 0 (-5)` and `quad_ring ℚ 0 (-5)`\n   (otherwise we'd have `quad_ring ℚ (↑0) (↑-5)`).\n\n * `quad_ring.norm_eq`: the norm of `⟨a, b⟩ : quad_ring R 0 d` is `a^2 - d * b^2`\n   (see also `algebra.norm`)\n\n * `quad_ring.is_integral_closure_23`,\n   `quad_ring.is_integral_closure_1`: together specify the ring of integers of\n   `ℚ(√d)` (where `d` is squarefree, of course)\n\n-/\n\nsection move_me\n\nlemma dvd_of_pow_dvd_pow_self {R : Type*} [comm_ring R] [is_domain R]\n  [unique_factorization_monoid R] {a b : R} {k : ℕ} (hk : k ≠ 0)\n  (h : a ^ k ∣ b ^ k) : a ∣ b :=\nbegin\n  by_cases hb : b = 0,\n  { simp [hb] },\n  by_cases ha : a = 0,\n  { simpa [ha, hk, pow_eq_zero_iff (nat.pos_of_ne_zero hk)] using h },\n  have hak : a ^ k ≠ 0 := pow_ne_zero _ ha,\n  have hbk : b ^ k ≠ 0 := pow_ne_zero _ hb,\n  letI := classical.dec_eq R,\n  letI : normalization_monoid R := unique_factorization_monoid.normalization_monoid,\n  rw [unique_factorization_monoid.dvd_iff_normalized_factors_le_normalized_factors,\n      multiset.le_iff_count] at h ⊢,\n  any_goals { assumption },\n  simp only [unique_factorization_monoid.normalized_factors_pow, multiset.count_nsmul] at h,\n  exact λ a, (mul_le_mul_left (nat.pos_of_ne_zero hk)).mp (h a)\nend\n\nlemma irreducible.not_is_square {R : Type*} [semiring R] {p : R} (hp : irreducible p) :\n  ¬ is_square p\n| ⟨a, ha⟩ := by cases (hp.is_unit_or_is_unit ha) with hua hua; simpa [ha] using hp.not_unit\n\n-- TODO version of this for arbitrary fraction rings? needs a good way of getting a numerator\nlemma rat.is_square_int_cast_iff (n : ℤ) : is_square (n : ℚ) ↔ is_square n :=\nbegin\n  split; rintros ⟨a, ha⟩,\n  { use a.num,\n    apply_fun rat.num at ha,\n    rwa [rat.mul_self_num, rat.coe_int_num] at ha },\n  { use ↑a,\n    exact_mod_cast ha }\nend\n\nend move_me\n\n/-- `quad_ring F a b` adjoins a root `α` of `x^2 - ax - b` to the ring (field) `F`.\nFor example, `ℤ[√5]` can be defined as `quad_ring ℤ 0 5`.\n\nElements of `quad_ring` are represented as a pair `⟨b1, b2⟩` standing for the element `b1 + b2 α`.\n-/\n@[ext]\nstructure quad_ring (F : Type*) (a b : F) :=\nmk {} :: (b1 : F) (b2 : F)\n\nattribute [pp_using_anonymous_constructor] quad_ring\n\nnamespace quad_ring\n\nsection\n\nvariables (F : Type*) (a b : F)\n\n@[simp] theorem eta : ∀ z : quad_ring F a b, (⟨z.b1,z.b2⟩ : quad_ring F a b) = z\n| ⟨_,_⟩ := rfl\n\n/-!\n### Basic constructions of elements in `quad_ring R a b`\n-/\n\n/-- The standard inclusion `F → quad_ring F a b`. -/\ninstance [has_zero F] : has_coe_t F (quad_ring F a b) := ⟨λ r, ⟨r, 0⟩⟩\n\n@[simp, norm_cast] lemma coe_b1 [has_zero F] (r : F) : (r : quad_ring F a b).b1 = r := rfl\n\n@[simp, norm_cast] lemma coe_b2 [has_zero F] (r : F) : (r : quad_ring F a b).b2 = 0 := rfl\n\n@[simp, norm_cast] theorem coe_inj [has_zero F] {z w : F} : (z : quad_ring F a b) = w ↔ z = w :=\n⟨congr_arg b1, congr_arg _⟩\n\nlemma coe_injective [has_zero F] : function.injective (coe : F → quad_ring F a b) :=\nλ a b, (coe_inj _ _ _).mp\n\n/-- `quad_ring.root F a b` is the adjoined root `α` of the polynomial `x^2 - ax - b`. -/\ndef root [has_zero F] [has_one F] : quad_ring F a b := ⟨0, 1⟩\n\n@[simp] lemma root_b1 [has_zero F] [has_one F]: (root F a b).b1 = 0 := rfl\n@[simp] lemma root_b2 [has_zero F] [has_one F]: (root F a b).b2 = 1 := rfl\n\nvariables [comm_ring F]\n/-!\n### Ring stucture on `quad_ring R a b`\n-/\n\ninstance : has_zero (quad_ring F a b) := ⟨(0 : F)⟩\ninstance : inhabited (quad_ring F a b) := ⟨0⟩\n\n@[simp] lemma zero_b1 : (0 : quad_ring F a b).b1 = 0 := rfl\n@[simp] lemma zero_b2 : (0 : quad_ring F a b).b2 = 0 := rfl\n\n@[simp] lemma coe_zero : ((0 : F) : quad_ring F a b) = 0 := rfl\n\n@[simp] theorem coe_eq_zero {z : F} : (z : quad_ring F a b) = 0 ↔ z = 0 :=\ncoe_inj F a b\n\ninstance : has_one (quad_ring F a b) := ⟨(1 : F)⟩\n\n@[simp] lemma one_b1 : (1 : quad_ring F a b).b1 = 1 := rfl\n@[simp] lemma one_b2 : (1 : quad_ring F a b).b2 = 0 := rfl\n\n@[simp, norm_cast] lemma coe_one : ((1 : F) : quad_ring F a b) = 1 := rfl\n\ninstance : has_add (quad_ring F a b) := ⟨λ z w, ⟨z.b1 + w.b1, z.b2 + w.b2⟩⟩\n\n@[simp] lemma add_b1 (z w : quad_ring F a b) : (z + w).b1 = z.b1 + w.b1 := rfl\n@[simp] lemma add_b2 (z w : quad_ring F a b) : (z + w).b2 = z.b2 + w.b2 := rfl\n\n@[simp, norm_cast] lemma coe_add (r s : F) : ((r + s : F) : quad_ring F a b) = r + s :=\n(quad_ring.ext_iff _ _).2 $ by simp\n\ninstance : has_neg (quad_ring F a b) := ⟨λ z, ⟨-z.b1, -z.b2⟩⟩\n\n@[simp] lemma neg_b1 (z : quad_ring F a b) : (-z).b1 = -z.b1 := rfl\n@[simp] lemma neg_b2 (z : quad_ring F a b) : (-z).b2 = -z.b2 := rfl\n\n@[simp, norm_cast] lemma coe_neg (n : F) : (↑(-n) : quad_ring F a b) = - ↑n := by ext; simp\n\ninstance : has_sub (quad_ring F a b) := ⟨λ z w, ⟨z.b1 - w.b1, z.b2 - w.b2⟩⟩\n\n@[simp] lemma sub_b1 (z w : quad_ring F a b) : (z - w).b1 = z.b1 - w.b1 := rfl\n@[simp] lemma sub_b2 (z w : quad_ring F a b) : (z - w).b2 = z.b2 - w.b2 := rfl\n\n@[simp, norm_cast] lemma coe_sub (z w : F) : (↑(z - w) : quad_ring F a b) = ↑z - ↑w := by ext; simp\n\n/-- This scalar multiplication is used to define `nsmul`, `zsmul` and `qsmul`. -/\ninstance {R F : Type*} [has_smul R F] (a b : F) : has_smul R (quad_ring F a b) :=\n{ smul := λ c x, ⟨c • x.b1, c • x.b2⟩ }\n\n@[simp] lemma smul_b1 {R F : Type*} [has_smul R F] {a b : F} (c : R) (z : quad_ring F a b) :\n  (c • z).b1 = c • z.b1 := rfl\n@[simp] lemma smul_b2 {R F : Type*} [has_smul R F] {a b : F} (c : R) (z : quad_ring F a b) :\n  (c • z).b2 = c • z.b2 := rfl\n\n/-! We want to define the `comm_ring (quad_ring F a b)` instance in multiple steps,\nso we can insert the right operations on `ℕ`, `ℤ` (and `ℚ`).\nThese operations have to be defined in the right way to ensure all diamond instances\ninvolving e.g. `quad_ring ℤ a b` are indeed definitionally equal.\n-/\n\ninstance : add_comm_monoid_with_one (quad_ring F a b) :=\n{ add := (+),\n  zero := (0),\n  one := (1),\n  add_assoc := by intros; ext; apply add_assoc,\n  add_comm := by intros; ext; apply add_comm,\n  zero_add := by intros; ext; apply zero_add,\n  add_zero := by intros; ext; apply add_zero,\n  nsmul := (•),\n  nsmul_zero' := by intros; ext; apply zero_nsmul,\n  nsmul_succ' := by intros; ext; apply succ_nsmul,\n  nat_cast := λ n, ↑(n : F),\n  nat_cast_zero := by simp,\n  nat_cast_succ := by intros; simp }\n\n@[simp, norm_cast] lemma coe_coe_nat (n : ℕ) : ((n : F) : quad_ring F a b) = n := rfl\n\ninstance : add_comm_group (quad_ring F a b) :=\n{ add := (+),\n  zero := (0),\n  neg := has_neg.neg,\n  add_left_neg := by intros; ext; apply add_left_neg,\n  sub := has_sub.sub,\n  nsmul := (•),\n  zsmul := (•),\n  zsmul_zero' := by intros; ext; apply zero_zsmul,\n  zsmul_succ' := by intros; ext; simp [add_mul, add_comm],\n  zsmul_neg' := by intros; ext; simp [add_mul],\n  sub_eq_add_neg := by intros; ext; apply sub_eq_add_neg,\n  .. quad_ring.add_comm_monoid_with_one F a b }\n\ninstance : add_comm_group_with_one (quad_ring F a b) :=\n{ add := (+),\n  zero := (0),\n  neg := has_neg.neg,\n  one := (1),\n  sub := has_sub.sub,\n  nsmul := (•),\n  zsmul := (•),\n  nat_cast := λ n, ↑(n : F),\n  int_cast := λ n, ↑(n : F),\n  int_cast_of_nat := by intros; simp,\n  int_cast_neg_succ_of_nat := by intros; simp [-neg_add_rev, neg_add, ← sub_eq_add_neg],\n  .. quad_ring.add_comm_monoid_with_one F a b,\n  .. quad_ring.add_comm_group F a b }\n\n@[simp, norm_cast] lemma coe_coe_int (n : ℤ) : ((n : F) : quad_ring F a b) = n := rfl\n\n@[simp, norm_cast] lemma coe_nat_b1 (n : ℕ) : (n : quad_ring F a b).b1 = n := rfl\n@[simp, norm_cast] lemma coe_nat_b2 (n : ℕ) : (n : quad_ring F a b).b2 = 0 := rfl\n\n@[simp, norm_cast] lemma coe_nat_add (r s : ℕ) : (↑(r + s) : quad_ring F a b) = r + s :=\nby { ext; simp only [coe_nat_b1, coe_nat_b2, add_b1, add_b2, nat.cast_add, add_zero] }\n\n@[simp] lemma coe_int_b1 (n : ℤ) : (n : quad_ring F a b).b1 = n := rfl\n@[simp] lemma coe_int_b2 (n : ℤ) : (n : quad_ring F a b).b2 = 0 := rfl\n\n-- This used to be a non-defeq diamond, we solve this by supplying a custom value for\n-- `add_group_with_one.int_cast`\nexample (a b : ℤ) (n) :\n  @@coe (@@coe_to_lift (quad_ring.has_coe_t ℤ a b)) n = @@coe (@@coe_to_lift int.cast_coe) n :=\nrfl\n\n@[simp, norm_cast] lemma coe_int_add (r s : ℤ) : (↑(r + s) : quad_ring F a b) = r + s :=\nby { ext; simp only [coe_int_b1, coe_int_b2, add_b1, add_b2, int.cast_add, add_zero] }\n\ninstance : has_mul (quad_ring F a b) :=\n⟨λ z w, ⟨z.1 * w.1 + z.2 * w.2 * b ,z.2 * w.1 + z.1 * w.2 + z.2 * w.2 * a⟩⟩\n\n@[simp] lemma mul_b1 (z w : quad_ring F a b) :\n  (z * w).b1 = z.1 * w.1 + z.2 * w.2 * b := rfl\n@[simp] lemma mul_b2 (z w : quad_ring F a b) :\n  (z * w).b2 = z.2 * w.1 + z.1 * w.2 + z.2 * w.2 * a := rfl\n\n@[simp, norm_cast] lemma coe_mul (r s : F) : ((r * s : F) : quad_ring F a b) = r * s :=\nby apply (ext_iff _ _).2; simp [mul_comm]\n\n/-- Simplify `quad_ring F a b` expressions by doing calculations componentwise in the ring `F`. -/\nmeta def calc_tac : tactic unit :=\n`[rw quad_ring.ext_iff; repeat { split; simp; ring }]\n\ninstance : comm_semiring (quad_ring F a b) :=\n{ zero := (0 : quad_ring F a b),\n  add := (+),\n  one := 1,\n  mul := (*),\n  nsmul := (•),\n  npow := npow_rec,\n  nat_cast := coe,\n  mul_comm := by intros; calc_tac,\n  mul_one := by intros; calc_tac,\n  one_mul := by intros; calc_tac,\n  mul_assoc := by intros; calc_tac,\n  mul_zero := by intros; calc_tac,\n  zero_mul := by intros; calc_tac,\n  right_distrib := by intros; calc_tac,\n  left_distrib := by intros; calc_tac,\n  ..quad_ring.add_comm_monoid_with_one F a b }\n\ninstance : comm_ring (quad_ring F a b) :=\n{ zero := (0 : quad_ring F a b),\n  add := (+),\n  neg := has_neg.neg,\n  sub := has_sub.sub,\n  one := 1,\n  mul := (*),\n  nsmul := (•),\n  npow := npow_rec,\n  nat_cast := coe,\n  zsmul := (•),\n  int_cast := coe,\n  ..quad_ring.comm_semiring F a b,\n  ..quad_ring.add_comm_group_with_one F a b }\n\ninstance [nontrivial F] : nontrivial (quad_ring F a b) :=\n⟨⟨0, 1, by simp [ext_iff]⟩⟩\n\ninstance [char_zero F] : char_zero (quad_ring F a b) :=\n⟨(coe_injective F a b).comp nat.cast_injective⟩\n\n@[simp, norm_cast] lemma coe_pow (x : F) : ∀ (n : ℕ), (↑(x ^ n) : quad_ring F a b) = x ^ n\n| 0 := by simp\n| (n + 1) := by rw [pow_succ, coe_mul, coe_pow n, pow_succ]\n\n/-! We finish off by helping `calc_tac` with some basic computations. -/\n\n@[simp] lemma bit0_b1 (z : quad_ring F a b) : (bit0 z).b1 = bit0 z.b1 := rfl\n@[simp] lemma bit0_b2 (z : quad_ring F a b) : (bit0 z).b2 = bit0 z.b2 := rfl\n@[simp] lemma bit1_b1 (z : quad_ring F a b) : (bit1 z).b1 = bit1 z.b1 := rfl\n@[simp] lemma bit1_b2 (z : quad_ring F a b) : (bit1 z).b2 = bit0 z.b2 := by simp [bit1]\n\nlemma square_of (z : quad_ring F a b) : z^2 = ⟨z.1^2 + z.2^2 * b, 2 * z.1 * z.2 + z.2^2 * a⟩ :=\nby { rw pow_two, calc_tac }\n\n@[simp]\nlemma square_of_b1 (z : quad_ring F a b) : (z^2).b1 = z.1^2 + z.2^2 * b :=\nby rw square_of\n\n@[simp]\nlemma square_of_b2 (z : quad_ring F a b) : (z^2).b2 = 2 * z.1 * z.2 + z.2^2 * a :=\nby rw square_of\n\nlemma cube_of (z : quad_ring F a b) : z^3 =\n  ⟨(z.1^2 + z.2^2 * b) * z.1 + (2 * z.1 * z.2 + z.2^2 * a) * z.2 * b,\n   (2*z.1 * z.2 + z.2^2 * a) * z.1 + (z.1^2 + z.2^2 * b) * z.2 +\n   (2*z.1 * z.2 + z.2 * z.2 * a) * z.2 * a⟩ :=\nby { rw [pow_succ, pow_two], calc_tac }\n\n/-! ### Maps between `quad_ring` and other rings -/\n\n/-- Promote the inclusion `F → quad_ring F a b` to a canonical ring homomorphism.\n\nSee also `quad_ring.algebra'`\n-/\ninstance : algebra F (quad_ring F a b) :=\n{ smul := (•),\n  to_fun := coe,\n  map_one' := by calc_tac,\n  map_mul' := λ x y, by calc_tac,\n  map_zero' := by calc_tac,\n  map_add' := λ x y, by calc_tac,\n  commutes' := λ x y, by calc_tac,\n  smul_def' := λ x y, by calc_tac }\n\n/-- Prefer the coercion as the `simp`-normal form. -/\n@[simp] lemma algebra_map_apply (x : F) : algebra_map F (quad_ring F a b) x = x := rfl\n\n/-- Lift a ring homomorphism `R → S` to a ring homomorphism `R[α] → S[α]` -/\ndef map {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R) :\n  quad_ring R a b →+* quad_ring S (f a) (f b) :=\n{ to_fun := λ s, ⟨f s.b1, f s.b2⟩,\n  map_one' := by ext; simp only [quad_ring.one_b1, quad_ring.one_b2, map_zero, map_one],\n  map_mul' := λ x y, by ext;\n    simp only [map_add, add_left_inj, quad_ring.mul_b1, map_mul, quad_ring.mul_b2],\n  map_zero' := by ext; simp only [quad_ring.zero_b1, quad_ring.zero_b2, map_zero],\n  map_add' := λ x y, by ext; simp only [map_add, add_left_inj, quad_ring.add_b1, quad_ring.add_b2] }\n\n@[simp] lemma map_b1 {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R) (x) :\n  (map f a b x).b1 = f x.b1 := rfl\n@[simp] lemma map_b2 {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R) (x) :\n  (map f a b x).b2 = f x.b2 := rfl\n@[simp] lemma map_coe {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b x : R) :\n  map f a b (x : quad_ring R a b) = f x :=\nby { ext; simp only [coe_b1, coe_b2, map, ring_hom.coe_mk, map_zero] }\n\n/-- Lift a ring isomorphism `R ≃ S` to a ring isomorphism `R[α] ≃ S[α]` -/\ndef congr {R S : Type*} [comm_ring R] [comm_ring S] (f : R ≃+* S) {a b : R} {a' b' : S}\n  (ha : f a = a') (hb : f b = b') :\n  quad_ring R a b ≃+* quad_ring S a' b' :=\n{ to_fun := λ s, ⟨f s.b1, f s.b2⟩,\n  inv_fun := λ r, ⟨f.symm r.b1, f.symm r.b2⟩,\n  left_inv := λ s, by simp only [ring_equiv.symm_apply_apply, quad_ring.eta, eq_self_iff_true],\n  right_inv := λ r, by simp only [quad_ring.eta, eq_self_iff_true, ring_equiv.apply_symm_apply],\n  map_mul' := λ x y, by ext; simp only [map_add, add_left_inj, eq_self_iff_true,\n    hb, quad_ring.mul_b1, map_mul, quad_ring.mul_b2, ha],\n  map_add' := λ x y, by ext; simp only [map_add, add_left_inj, eq_self_iff_true, quad_ring.add_b1,\n    embedding_like.apply_eq_iff_eq, quad_ring.add_b2] }\n\n@[simp] lemma congr_b1 {R S : Type*} [comm_ring R] [comm_ring S]\n  (f : R ≃+* S) {a b : R} {a' b' : S} (ha : f a = a') (hb : f b = b') (x : quad_ring R a b) :\n  (congr f ha hb x).b1 = f x.b1 := rfl\n@[simp] lemma congr_b2 {R S : Type*} [comm_ring R] [comm_ring S]\n  (f : R ≃+* S) {a b : R} {a' b' : S} (ha : f a = a') (hb : f b = b') (x : quad_ring R a b) :\n  (congr f ha hb x).b2 = f x.b2 := rfl\n@[simp] lemma congr_coe {R S : Type*} [comm_ring R] [comm_ring S]\n  (f : R ≃+* S) {a b : R} {a' b' : S} (ha : f a = a') (hb : f b = b')\n  (x : R) :\n  congr f ha hb x = f x :=\nby { ext; simp only [coe_b1, coe_b2, congr, ring_equiv.coe_mk, map_zero] }\n\n/-- Lift a ring homomorphism `R → S` to a ring homomorphism `R[α] → S`,\ngiven a suitable image for `α` -/\n@[simps]\ndef lift {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R)\n  (z : S) (hz : z^2 - f a * z - f b = 0) :\n  quad_ring R a b →+* S :=\nhave hz' : z^2 = f a * z + f b := sub_eq_iff_eq_add'.mp $ sub_eq_zero.mp hz,\n{ to_fun := λ s, f s.b1 + z * f s.b2,\n  map_one' := by simp only [quad_ring.one_b1, quad_ring.one_b2, map_zero, map_one, mul_zero, add_zero],\n  map_zero' := by simp only [quad_ring.zero_b1, quad_ring.zero_b2, map_zero, zero_add, mul_zero],\n  map_add' := λ x y, by simp only [map_add, quad_ring.add_b1, quad_ring.add_b2]; ring,\n  map_mul' := λ x y, begin\n    simp only [hz', map_add, add_left_inj, quad_ring.mul_b1, map_mul, quad_ring.mul_b2],\n    calc f x.b1 * f y.b1 + f x.b2 * f y.b2 * f b +\n           z * (f x.b2 * f y.b1 + f x.b1 * f y.b2 + f x.b2 * f y.b2 * f a)\n        = (f y.b1 + z * f y.b2) * f x.b1 +\n          (z * f x.b2 * f y.b1 + (f a * z + f b) * f y.b2 * f x.b2) : by ring\n    ... = (f y.b1 + z * f y.b2) * f x.b1 + (z * f x.b2 * f y.b1 + z ^ 2 * f y.b2 * f x.b2) :\n      by rw hz'\n    ... = (f x.b1 + z * f x.b2) * (f y.b1 + z * f y.b2) : by ring,\n  end }\n\n@[simp] lemma lift_coe {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R) (z h)\n  (x : R) : lift f a b z h (x : quad_ring R a b) = f x :=\nby { simp only [coe_b1, coe_b2, lift_apply, map_zero, mul_zero, add_zero] }\n\n@[simp] lemma lift_mk {R S : Type*} [comm_ring R] [comm_ring S] (f : R →+* S) (a b : R) (z h)\n  (x y : R) : lift f a b z h (⟨x, y⟩ : quad_ring R a b) = f x + z * f y :=\nrfl\n\nsection algebra_numeric_coeff\n\nvariables {R S : Type*} [comm_ring R] [comm_ring S]\nvariables {G : Type*} (g : G)\n\n/-!\n## \"Natural\" algebra structures where the coefficients are numeric\n\nWe want to get an `algebra (quad_ring ℤ 0 (-5)) (quad_ring ℚ 0 (-5))` instance,\nso we can say e.g. that `ℤ[√-5]` is the ring of integers of `ℚ[√-5]`.\nHowever, the `-5`s in the types are different, and we need to do a bit of computation\nto show that they are mapped to each other in the corresponding inclusion.\nLuckily, we can program the typeclass system like it's Prolog,\nand use instance synthesis to do this computation for us.\n\nEven nicer would be to supply these `fact`s through tactics, which should be possible with\n`auto_param`s, but those don't work nicely with the synthesis algorithm.\nHopefully this can be fixed using Lean 4's more sophisticated elaboration order.\n-/\n-- Perhaps this should be in a `quad_ring` locale?\nlemma fact_map_zero [zero_hom_class G R S] : fact (g 0 = 0) := ⟨map_zero _⟩\nlemma fact_map_one [one_hom_class G R S] : fact (g 1 = 1) := ⟨map_one _⟩\nlemma fact_map_bit0 [ring_hom_class G R S] (n : R) (n' : S) [h : fact (g n = n')] :\n  fact (g (bit0 n) = bit0 n') :=\n⟨by rw [map_bit0, h.1]⟩\nlemma fact_map_bit1 [ring_hom_class G R S] (n : R) (n' : S) [h : fact (g n = n')] :\n  fact (g (bit1 n) = bit1 n') :=\n⟨by rw [map_bit1, h.1]⟩\nlemma fact_map_neg [add_monoid_hom_class G R S] (n : R) (n' : S) [h : fact (g n = n')] :\n  fact (g (- n) = - n') :=\n⟨by rw [map_neg, h.1]⟩\nlemma fact_map_int_cast [ring_hom_class G R S] (d : ℤ) : fact (g d = d) :=\n⟨map_int_cast g d⟩\nlemma fact_eq_int_cast [ring_hom_class G ℤ S] (d : ℤ) : fact (g d = d) :=\n⟨eq_int_cast g d⟩\n@[priority 500] -- Fallback, use lower priority because this can trigger expensive unification.\nlemma fact_refl {α : Sort*} (x : α) : fact (x = x) :=\n⟨rfl⟩\n\n/-- Lift a canonical map `R → S` to a canonical map `quad_ring R a b → quad_ring S a' b'`,\ngiven that `a` is sent to `a'` and `b` to `b'`.\n\nThis is a principled way of defining the inclusion `quad_ring ℤ 0 (-5) → quad_ring ℚ 0 (-5)`:\nsince `(↑(0 : ℤ) : ℚ)` and `0 : ℚ` are different, we can't use a more naïve definition like\n`algebra (quad_ring R a b) (quad_ring S ↑a ↑b)`.\n-/\ndef algebra'' (R S : Type*) [comm_ring R] [comm_ring S] [algebra R S] (a b : R)\n  (a' b' : S) [fact (algebra_map R S a = a')] [fact (algebra_map R S b = b')] :\n  algebra (quad_ring R a b) (quad_ring S a' b') :=\n((quad_ring.congr (ring_equiv.refl S)\n    (show _, from fact.out (algebra_map R S a = a'))\n    (show _, from fact.out (algebra_map R S b = b'))).to_ring_hom.comp\n  (map (algebra_map R S) a b)).to_algebra\n\nlocalized \"attribute [instance] fact_map_zero fact_map_one fact_map_bit0 fact_map_bit1 fact_map_neg\n  fact_map_int_cast fact_eq_int_cast quad_ring.algebra''\"\n  in quad_ring\nlocalized \"attribute [instance, priority 500] fact_refl\"\n  in quad_ring\n\nsection\nopen_locale quad_ring\n\n@[simp] lemma algebra_map_b1 {R S : Type*} [comm_ring R] [comm_ring S] [algebra R S] (a b : R)\n  (a' b' : S) [h1 : fact (algebra_map R S a = a')] [h2 : fact (algebra_map R S b = b')] (x) :\n  (algebra_map (quad_ring R a b) (quad_ring S a' b') x).b1 = algebra_map R S x.b1 :=\nrfl\n@[simp] lemma algebra_map_b2 {R S : Type*} [comm_ring R] [comm_ring S] [algebra R S] (a b : R)\n  (a' b' : S) [h1 : fact (algebra_map R S a = a')] [h2 : fact (algebra_map R S b = b')] (x) :\n  (algebra_map (quad_ring R a b) (quad_ring S a' b') x).b2 = algebra_map R S x.b2 :=\nrfl\n\n@[simp] lemma algebra_map_coe {R S : Type*} [comm_ring R] [comm_ring S] [algebra R S] (a b : R)\n  (a' b' : S) [h1 : fact (algebra_map R S a = a')] [h2 : fact (algebra_map R S b = b')] (x : R) :\n  algebra_map (quad_ring R a b) (quad_ring S a' b') x = algebra_map R S x :=\nby rw [ring_hom.algebra_map_to_algebra, ring_hom.comp_apply, ring_equiv.to_ring_hom_eq_coe,\n      ring_equiv.coe_to_ring_hom, map_coe, congr_coe, ring_equiv.refl_apply]\n\nlemma algebra_map_injective {R S : Type*} [comm_ring R] [comm_ring S] [algebra R S] (a b : R)\n  (a' b' : S) [h1 : fact (algebra_map R S a = a')] [h2 : fact (algebra_map R S b = b')]\n  (hRS : function.injective (algebra_map R S)) :\n  function.injective (algebra_map (quad_ring R a b) (quad_ring S a' b')) :=\nbegin\n  intros x y h,\n  rw quad_ring.ext_iff at h ⊢,\n  exact ⟨hRS h.1, hRS h.2⟩\nend\n\nend\n\n-- about 100 ms for instance synthesis, not bad!\nexample : algebra (quad_ring ℤ 0 (-5)) (quad_ring ℚ 0 (-5)) :=\nquad_ring.algebra'' _ _ _ _ _ _\n\n-- about 50 ms for instance synthesis, not bad!\nexample (d : ℤ) : algebra (quad_ring ℤ 0 d) (quad_ring ℚ 0 d) :=\nquad_ring.algebra'' _ _ _ _ _ _\n\nend algebra_numeric_coeff\n\nsection root\n\n/-! ### `quad_ring F a b` as `F[α]` -/\n\nopen polynomial\n\n/-- `quad_ring.root F a b` is indeed a root of `X^2 - a * X - b` -/\ntheorem aeval_root : aeval (root F a b) (X^2 - C a * X - C b) = 0 :=\nby { simp only [map_sub, map_mul, aeval_C, aeval_X, map_pow], calc_tac }\n\n@[simp]\nlemma root_sq : (root F a b)^2 = a * root F a b + b :=\nby calc_tac\n\n/-- The canonical basis on `quad_ring F a b` with basis vectors `1` and `root F a b`. -/\nprotected noncomputable def basis : basis (fin 2) F (quad_ring F a b) :=\nbasis.of_equiv_fun $\n{ to_fun := λ z, ![z.b1, z.b2],\n  inv_fun := λ v, ⟨v 0, v 1⟩,\n  left_inv := λ z, by ext; refl,\n  right_inv := λ v, by ext i; fin_cases i; refl,\n  map_add' := λ z w, by ext i; fin_cases i; refl,\n  map_smul' := λ c z, by ext i; fin_cases i; refl }\n\n@[simp] lemma coe_basis : (quad_ring.basis F a b : fin 2 → quad_ring F a b) = ![1, root F a b] :=\nby ext i;\n  -- Speed up the proof somewhat by simplifying the common cases first.\n  simp only [quad_ring.basis, basis.coe_of_equiv_fun, linear_equiv.coe_symm_mk];\n  fin_cases i;\n  simp\n\n@[simp] lemma basis_repr (z i) : (quad_ring.basis F a b).repr z i = ![z.b1, z.b2] i := rfl\n\ninstance : module.free F (quad_ring F a b) := module.free.of_basis (quad_ring.basis F a b)\ninstance : module.finite F (quad_ring F a b) := module.finite.of_basis (quad_ring.basis F a b)\n\n/-- `quad_ring F a b` is the extension of `F` generated by a single element `root F a b`.\n\nThis is a powerful way to show `quadRing F a b` corresponds to `F[α]`.\n-/\n@[simps]\nprotected noncomputable def power_basis : power_basis F (quad_ring F a b) :=\n{ dim := 2,\n  gen := root F a b,\n  basis := quad_ring.basis F a b,\n  basis_eq_pow := λ i, by fin_cases i; simp }\n\nend root\n\nend\n\nsection sqrt_d\n\nsection ring\n\n/-! ## The ring `R[√d]` -/\n\nvariables {R : Type*} [comm_ring R] {d s : R}\n\nopen polynomial\n\n/-- The conjugation map in `R[√d]` maps `√d` to `- √d`. -/\n@[simps] def conj {d s : R} (x : quad_ring R s d) : quad_ring R s d :=\nx.1 * ⟨1, 0⟩ + x.2 * ⟨s, -1⟩\n\n@[simp] lemma conj_zero {d s : R} : (0 : quad_ring R s d).conj = 0 := by calc_tac\n\n@[simp] lemma conj_eq_zero_iff {d s : R} {z : quad_ring R s d} : z.conj = 0 ↔ z = 0 :=\nby cases z; simp [ext_iff] {contextual := tt}\n\n@[simp] lemma conj_mul {d s : R} (x : quad_ring R s d) :\n  x.conj * x = x.1^2 + s * x.1 * x.2 - d * x.2^2 :=\nby { norm_cast, ext; simp only [mul_b1, conj_b1, mul_b2, conj_b2, coe_b1, coe_b2]; ring }\n\n@[simp] lemma mul_conj {d s : R} (x : quad_ring R s d) :\n  x * x.conj = x.1^2 + s * x.1 * x.2 - d * x.2^2 :=\nby rw [mul_comm, conj_mul]\n\n/-! ### Norm on `R[√d]` -/\n\nlemma left_mul_matrix_apply {a b : R} (x : quad_ring R a b) :\n  algebra.left_mul_matrix (quad_ring.basis R a b) x = ![![x.1, x.2 * b], ![x.2, x.1 + x.2 * a]] :=\nbegin\n  ext i j,\n  cases x with A B,\n  simp only [algebra.left_mul_matrix_eq_repr_mul, coe_basis, basis_repr],\n  fin_cases j; fin_cases i;\n    simp,\nend\n\n-- Not the most principled proof, but it works!\n@[simp]\nlemma norm_eq {a b : R} (x : quad_ring R a b) :\n  algebra.norm R x = x.1^2 + a * x.1 * x.2 - b * x.2 ^ 2 :=\nbegin\n  -- Unfold the definitions\n  erw [algebra.norm, monoid_hom.comp_apply, ring_hom.to_monoid_hom_eq_coe, ring_hom.coe_monoid_hom,\n      alg_hom.to_ring_hom_eq_coe, alg_hom.coe_to_ring_hom,\n      ← linear_map.det_to_matrix (quad_ring.basis R a b), algebra.to_matrix_lmul_eq,\n      left_mul_matrix_apply, matrix.det_fin_two],\n  -- Expand all the matrix coefficients\n  simp only [matrix.head_cons, matrix.cons_val_one, matrix.cons_val_zero],\n  ring\nend\n\nopen algebra\n\n@[simp] lemma norm_one : norm R (1 : quad_ring R s d) = 1 := by simp\n@[simp] lemma norm_zero : norm R (0 : quad_ring R s d) = 0 := by simp\n@[simp] lemma norm_coe (n : R) : norm R (n : quad_ring R s d) = n ^ 2 := by simp\n@[simp] lemma norm_coe_nat (n : ℕ) : norm R (n : quad_ring R s d) = n ^ 2 := by simp\n@[simp] lemma norm_coe_int (n : ℤ) : norm R (n : quad_ring R s d) = n ^ 2 := by simp\n\nlemma norm_nonneg {R : Type*} [linear_ordered_comm_ring R] {d : R} (hd : d ≤ 0)\n  (x : quad_ring R 0 d) : 0 ≤ norm R x :=\nbegin\n  rw norm_eq,\n  nlinarith,\nend\n\n/-- special case of below with weaker assumption in char 2 -/\nlemma eq_zero_iff_norm_zero [is_domain R] [unique_factorization_monoid R]\n  {d : R} (hd : ¬ is_square d) (x : quad_ring R 0 d) : norm R x = 0 ↔ x = 0 :=\nbegin\n  simp,\n  simp only [ext_iff, zero_b1, zero_b2],\n  split; intro h,\n  { rw [sub_eq_zero] at h,\n    by_cases hb2 : x.b2 = 0,\n    { simpa [hb2] using h },\n    contrapose! hd,\n    have : x.b2 ^ 2 ∣ x.b1 ^ 2 := ⟨_, h.trans $ mul_comm _ _⟩,\n    obtain ⟨x, hx⟩ := dvd_of_pow_dvd_pow_self (by norm_num) this,\n    use x,\n    rw [hx, mul_comm, mul_pow, pow_two] at h,\n    exact (mul_right_cancel₀ (pow_ne_zero 2 hb2) h).symm },\n  { simp only [norm, zero_pow', ne.def, bit0_eq_zero, nat.one_ne_zero,\n      not_false_iff, mul_zero, add_zero, h, sub_zero], },\nend\n\nlemma eq_zero_iff_norm [is_domain R] [unique_factorization_monoid R]\n  {d s : R} (hd : ¬ is_square (4 * d + s ^ 2)) (x : quad_ring R s d) : norm R x = 0 ↔ x = 0 :=\nbegin\n  simp,\n  simp only [ext_iff, zero_b1, zero_b2],\n  split; intro h,\n  { rw [sub_eq_zero] at h,\n    by_cases hb2 : x.b2 = 0,\n    { simpa [hb2] using h },\n    contrapose! hd,\n    apply_fun ((*) 4) at h,\n    rw show 4 * (x.b1 ^ 2 + s * x.b1 * x.b2) = ((2 * x.b1 + s * x.b2) ^ 2 - s ^ 2 * x.b2 ^ 2), by ring at h,\n    rw sub_eq_iff_eq_add at h,\n    rw show 4 * (d * x.b2 ^ 2) + s ^ 2 * x.b2 ^ 2 = (4 * d + s ^ 2) * x.b2 ^ 2, by ring at h,\n    have : x.b2 ^ 2 ∣ (2 * x.b1 + s * x.b2) ^ 2 := ⟨_, h.trans $ mul_comm _ _⟩,\n    obtain ⟨x, hx⟩ := dvd_of_pow_dvd_pow_self (by norm_num) this,\n    use x,\n    rw [hx, mul_comm, mul_pow, pow_two] at h,\n    exact (mul_right_cancel₀ (pow_ne_zero 2 hb2) h).symm },\n  { simp only [norm, zero_pow', ne.def, bit0_eq_zero, nat.one_ne_zero,\n      not_false_iff, mul_zero, add_zero, h, sub_zero], },\nend\n\nlemma norm_ne_zero_of_mem_non_zero_divisors [nontrivial R] {d s : R} (x : quad_ring R s d)\n  (h : x ∈ non_zero_divisors (quad_ring R s d)) : norm R x ≠ 0 :=\nbegin\n  have h0 := non_zero_divisors.ne_zero h,\n  rw mem_non_zero_divisors_iff at h,\n  contrapose! h,\n  refine ⟨x.conj, _, mt conj_eq_zero_iff.mp h0⟩,\n  simp only [norm_eq, ←coe_eq_zero R s d, coe_sub, coe_pow, coe_mul, coe_add] at h,\n  rw conj_mul,\n  assumption_mod_cast,\nend\n\nend ring\n\n/-! ## The ring `ℤ[√d]` -/\n\nnotation `ℤ[√` d `]` := quad_ring ℤ 0 d\nnotation `ℚ(√` d `)` := quad_ring ℚ 0 d\n\nsection int\n\nvariables {d : ℤ} {d' : ℚ} [hdd' : fact $ algebra_map ℤ ℚ d = d']\n\nopen algebra\n\nlemma is_domain_of_not_square {R : Type*} [comm_ring R] [is_domain R]\n  [unique_factorization_monoid R] {d s : R} (hd : ¬ is_square (4 * d + s ^ 2)) :\n  is_domain $ quad_ring R s d :=\n@@no_zero_divisors.to_is_domain _ _ _\n{ eq_zero_or_eq_zero_of_mul_eq_zero := begin\n    rintros a b h,\n    apply_fun (norm R) at h,\n    simpa only [norm_zero, eq_zero_iff_norm hd, map_mul, mul_eq_zero] using h,\n  end }\n\nlemma is_square_four : is_square (4 : ℤ) :=\nbegin\n  rw show (4 : ℤ) = 2 * 2, by norm_num,\n  apply is_square_mul_self,\nend\n\nlemma is_square.is_square_mul_iff\n  {R : Type*} [comm_ring R] [is_domain R] [unique_factorization_monoid R]\n  {m n : R} (h : is_square m) (hn : m ≠ 0) :\n  is_square (m * n) ↔ is_square n :=\nbegin\n  split; intro hi,\n  { rcases h with ⟨h_w, rfl⟩,\n    rcases hi with ⟨hi_w, hh⟩,\n    have : h_w ^ 2 ∣ hi_w ^ 2,\n    { use n,\n      simp [pow_two, hh], },\n    obtain ⟨x, rfl⟩ := dvd_of_pow_dvd_pow_self (by norm_num) this,\n    use x,\n    rw ← sub_eq_zero at hh,\n    rw show h_w * h_w * n - h_w * x * (h_w * x) = h_w ^ 2 * (n - x ^ 2), by ring at hh,\n    simp only [mul_eq_zero, pow_eq_zero_iff, nat.succ_pos'] at hh,\n    rcases hh with rfl | hh,\n    { simpa using hn, },\n    { simpa [sub_eq_zero, pow_two] using hh, }, },\n  { exact h.mul hi, }\nend\n\ninstance [fact $ ¬ is_square d] : is_domain ℤ[√d] :=\nbegin\n  apply is_domain_of_not_square,\n  simp only [zero_pow', ne.def, bit0_eq_zero, nat.one_ne_zero, not_false_iff, add_zero],\n  intro h,\n  apply _inst_1.out,\n  rwa is_square_four.is_square_mul_iff (by norm_num) at h,\nend\n\ninstance is_domain_one' (m : ℤ) [hm : fact $ ¬ is_square (4 * m + 1)] :\n  is_domain $ quad_ring ℤ 1 m :=\nis_domain_of_not_square (by simpa using hm.out)\n\ninstance is_domain_one\n  [hdsq : fact $ ¬ is_square d] [hd : fact $ d % 4 = 1] :\n  is_domain $ quad_ring ℤ 1 ((d - 1) / 4) :=\nbegin\n  set m := (d - 1) / 4 with hm,\n  clear_value m,\n  unfreezingI { obtain rfl : d = 4 * m + 1,\n    { rw [hm, ← sub_eq_iff_eq_add, int.mul_div_cancel' (int.dvd_sub_of_mod_eq (hd.out))] } },\n  exact is_domain_of_not_square (by simpa using hdsq.out)\nend\n\nend int\n\n/-! ## The field `K(√d)` -/\n\nsection field\n\nvariables {K : Type*} [field K] (d s : K)\n\nopen algebra\nnoncomputable theory\ninstance : has_inv (quad_ring K s d) :=\n⟨λ z, ↑((norm K z)⁻¹ : K) * z.conj⟩\n\nvariables {d s}\n\nlemma inv_def (z : quad_ring K s d) : z⁻¹ = ↑((norm K z)⁻¹ : K) * z.conj := rfl\n\n@[simp] lemma inv_b1 (z : quad_ring K 0 d) : (z⁻¹).b1 = z.b1 / (norm K z) :=\nshow (↑((norm K z)⁻¹) * z.conj).b1 = z.b1 / (norm K z),\nby simp [mul_b1, coe_b1, conj_b1, coe_b2, zero_mul, zero_mul, add_zero, div_eq_inv_mul]\n\n@[simp] lemma inv_b2 (z : quad_ring K 0 d) : (z⁻¹).b2 = - z.b2 / (norm K z) :=\nshow (↑((norm K z)⁻¹) * z.conj).b2 = - z.b2 / (norm K z),\nby rw [mul_b2, coe_b1, coe_b2, zero_mul, zero_add, mul_zero, add_zero, conj_b2, zero_add, zero_add,\n       zero_add, zero_add, add_zero, mul_neg, mul_one,\n       div_eq_inv_mul]\n\nlemma inv_zero : (0 : quad_ring K 0 d)⁻¹ = 0 := by { ext; simp }\n\nlemma mul_inv_cancel' {z : quad_ring K 0 d} (h : (norm K z) ≠ 0) : z * z⁻¹ = 1 :=\nbegin\n  rw [inv_def, mul_left_comm, mul_conj],\n  simp only [norm_eq, ne.def] at h ⊢,\n  exact_mod_cast inv_mul_cancel h\nend\n\n/-- `K[√d]` is a field, except it has zero divisors if `d` is a square. -/\n@[reducible]\ndef field_of_not_square (hd : ¬ is_square d) :\n  field (quad_ring K 0 d) :=\n{ zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  sub := has_sub.sub,\n  one := 1,\n  mul := (*),\n  nsmul := (•),\n  npow := npow_rec,\n  nat_cast := coe,\n  zsmul := (•),\n  int_cast := coe,\n  exists_pair_ne := ⟨0, 1, by simp⟩,\n  inv := has_inv.inv,\n  rat_cast := λ q, ↑(q : K),\n  rat_cast_mk := λ a b h1 h2, by { ext; simp [rat.cast_def, sq]; field_simp },\n  qsmul := λ q z, (q : K) • z,\n  qsmul_eq_mul' := λ q z, by { ext; simp },\n  mul_inv_cancel := λ a ha0, mul_inv_cancel' (mt (eq_zero_iff_norm_zero hd _).mp ha0),\n  inv_zero := inv_zero,\n  .. quad_ring.comm_ring _ _ _ }\n\ninstance {d' : ℚ} [fact $ ¬ is_square d'] : field ℚ(√d') :=\nfield_of_not_square (fact.out _)\n\n-- TODO clean this up\ninstance quad_ring.field_neg_five : field (quad_ring ℚ 0 (-5)) :=\nquad_ring.field_of_not_square $ begin\n  convert mt (rat.is_square_int_cast_iff (-5)).mp _,\n  { norm_num },\n  refine irreducible.not_is_square (prime.irreducible $ int.prime_iff_nat_abs_prime.mpr $ _),\n  norm_num\nend\n\n-- Thanks to PR #14894, this diamond is fixed!\nexample (n : ℚ) {d' : ℚ} [fact $ ¬ is_square d'] :\n  @@coe (@@coe_to_lift (quad_ring.has_coe_t ℚ 0 d')) n = @@coe (@@coe_to_lift rat.cast_coe) n :=\nrfl\nexample {d' : ℚ} [fact $ ¬ is_square d'] : quad_ring.algebra ℚ 0 d' = algebra_rat := rfl\n\nend field\n\n/-! ## The field `ℚ(√d)` as field of fractions of `ℤ[√d]` -/\n\nsection rat\n-- We use different variables `d : ℤ` and `d' : ℚ`,\n-- since `-5 : ℚ` is not the same as `↑ (-5 : ℤ)`.\nvariables (d : ℤ) (d' : ℚ) [hdd' : fact $ algebra_map ℤ ℚ d = d']\n\ninclude hdd'\nopen_locale quad_ring\n\nlemma coe_d : (d : ℚ) = d' := hdd'.out\n\nvariables {d d'}\n\n/-- We map from `ℤ[√d]` to `ℚ(√d)` by applying the coercion `ℤ → ℚ` componentwise.\n\nThis is not true for arbitrary `R[√d] → K(√d)` since there might not be such a coercion.\n-/\n@[simp] lemma algebra_map_mk (x y : ℤ) :\n  algebra_map ℤ[√d] ℚ(√d') ⟨x, y⟩ = ⟨x, y⟩ :=\nrfl\n\nvariables (d d')\n\nlemma fraction_ring_surj_aux (a b : ℚ) :\n  (⟨a, b⟩ : quad_ring ℚ 0 d') *\n      algebra_map ℤ[√d] ℚ(√d') (↑(a.denom) * ↑(b.denom)) =\n    algebra_map ℤ[√d] ℚ(√d') ⟨a.num * ↑(b.denom), b.num * ↑(a.denom)⟩ :=\nbegin\n  ext : 1;\n    simp only [algebra_map_mk, mul_b1, mul_b2, set_like.coe_mk, map_mul, coe_coe,\n      algebra_map_coe, map_nat_cast, map_int_cast, coe_b1, coe_b2,\n      coe_int_b1, coe_int_b2, coe_nat_b1, coe_nat_b2, zero_mul, mul_zero, zero_add, add_zero],\n  { rw [← mul_assoc, int.cast_coe_nat, @rat.mul_denom_eq_num a], norm_cast },\n  { rw [mul_comm ↑↑a.denom, ← mul_assoc, int.cast_coe_nat, @rat.mul_denom_eq_num b], norm_cast }\nend\n\n/-- If `d` is not a square, then `ℚ(√d)` is the field of fractions of `ℤ[√d]`. -/\ninstance [not_sq : fact $ ¬ is_square d] : is_fraction_ring ℤ[√d] ℚ(√d') :=\n{ map_units := begin\n    haveI : fact (¬ is_square d'),\n    { rw [← hdd'.out, eq_int_cast],\n      exact ⟨(rat.is_square_int_cast_iff _).not.mpr not_sq.out⟩ },\n    rintro ⟨⟨a, b⟩, hy⟩,\n    apply is_unit.mk0 _,\n    simp only [set_like.coe_mk, algebra_map_mk, ne.def, ext_iff, zero_b1, zero_b2,\n      int.cast_eq_zero],\n    rintro ⟨rfl, rfl⟩,\n    exact non_zero_divisors.ne_zero hy rfl\n  end,\n  surj := begin\n    rintro ⟨a, b⟩,\n    refine ⟨⟨⟨a.num * b.denom, b.num * a.denom⟩,\n             ⟨a.denom * b.denom, mul_mem _ _⟩⟩,\n            fraction_ring_surj_aux d d' a b⟩,\n    repeat { exact mem_non_zero_divisors_of_ne_zero\n      (mt (coe_eq_zero _ _ _).mp (int.coe_nat_ne_zero.mpr (rat.denom_ne_zero _))) },\n  end,\n  eq_iff_exists := begin\n    intros x y,\n    split,\n    { intro h,\n      use 1,\n      rw [algebra_map_injective 0 d 0 d' (ring_hom.injective_int _) h,\n          one_mem_class.coe_one, mul_one] },\n    rintro ⟨c, h⟩,\n    rw [mul_right_cancel₀ (non_zero_divisors.coe_ne_zero c) h]\n  end\n}\nomit hdd'\n\n/-! ### Ring of integers of `ℚ(√d)`, `d ≠ 1` mod 4.  -/\n\nopen polynomial\n\nvariables (d) {d'}\n\n/-- The minimal polynomial for `a + b √d` when `a b ∈ R`. -/\nnoncomputable def minpoly_a_add_b_sqrt_d {R : Type*} [comm_ring R] (dR a b : R) : polynomial R :=\nX^2 - C (2 * a) * X + C (a^2 - dR * b^2)\n\nnamespace minpoly_a_add_b_sqrt_d\n\nvariables {R : Type*} [comm_ring R] (dR a b : R)\n\nprotected lemma coeff_zero : (minpoly_a_add_b_sqrt_d dR a b).coeff 0 = a^2 - dR * b^2 :=\nby simp only [minpoly_a_add_b_sqrt_d, pow_two, coeff_add, coeff_sub,\n  mul_coeff_zero, coeff_X_zero, coeff_C_zero,\n  mul_zero, zero_sub, neg_zero, zero_add, int.cast_id]\n\nprotected lemma coeff_one : (minpoly_a_add_b_sqrt_d dR a b).coeff 1 = - 2 * a :=\nby simp only [minpoly_a_add_b_sqrt_d, pow_two, coeff_add, coeff_sub,\n  coeff_mul_X, coeff_X_zero, coeff_C_ne_zero (show (1 : ℕ) ≠ 0, from one_ne_zero), coeff_C_zero,\n  zero_sub, add_zero, neg_mul]\n\nprotected lemma coeff_two : (minpoly_a_add_b_sqrt_d dR a b).coeff 2 = 1 :=\nby simp only [minpoly_a_add_b_sqrt_d, pow_two, coeff_add, coeff_sub,\n  coeff_mul_X, coeff_X_one, coeff_C_ne_zero (show (1 : ℕ) ≠ 0, from one_ne_zero),\n  coeff_C_ne_zero (show (2 : ℕ) ≠ 0, from two_ne_zero), sub_zero, add_zero]\n\n@[simp] protected lemma degree [nontrivial R] : degree (minpoly_a_add_b_sqrt_d dR a b) = 2 :=\nbegin\n  have deg_X2_X :=\n    calc degree (C (2 * a) * X) ≤ 1 : by compute_degree_le\n                            ... < 2 : with_bot.coe_lt_coe.mpr one_lt_two\n                            ... = degree (X^2) : (degree_X_pow _).symm,\n  refine (degree_add_eq_left_of_degree_lt _).trans\n    ((degree_sub_eq_left_of_degree_lt deg_X2_X).trans\n    (degree_X_pow 2)),\n  calc degree (C _) ≤ 0 : degree_C_le\n                ... < 2 : with_bot.coe_lt_coe.mpr zero_lt_two\n                ... = degree (X^2) : (degree_X_pow _).symm\n                ... = degree _ : (degree_sub_eq_left_of_degree_lt deg_X2_X).symm,\n  apply_instance\nend\n\nprotected lemma monic : monic (minpoly_a_add_b_sqrt_d dR a b) :=\nbegin\n  nontriviality R,\n  -- TODO: unify with previous lemma\n  have deg_X2_X :=\n    calc degree (C (2 * a) * X) ≤ 1 : by compute_degree_le\n                            ... < 2 : with_bot.coe_lt_coe.mpr one_lt_two\n                            ... = degree (X^2) : (degree_X_pow _).symm,\n  refine ((monic_X_pow _).sub_of_left deg_X2_X).add_of_left _,\n  calc degree (C _) ≤ 0 : degree_C_le\n                ... < 2 : with_bot.coe_lt_coe.mpr zero_lt_two\n                ... = degree (X^2) : (degree_X_pow _).symm\n                ... = degree _ : (degree_sub_eq_left_of_degree_lt deg_X2_X).symm,\n  apply_instance\nend\n\n@[simp] lemma eval₂_eq {S : Type*} [comm_ring S] (f : R →+* S) (x : S) :\n  eval₂ f x (minpoly_a_add_b_sqrt_d dR a b) = x^2 - f (2 * a) * x + f (a^2 - dR * b^2) :=\nby simp only [minpoly_a_add_b_sqrt_d, eval₂_add, eval₂_sub, eval₂_pow, eval₂_mul, eval₂_X, eval₂_C]\n\nlemma aeval_mk_eq_zero : -- can be proved by `aeval_eq_zero`\n  aeval (⟨a, b⟩ : quad_ring R 0 dR) (minpoly_a_add_b_sqrt_d dR a b) = 0 :=\nbegin\n  simp only [aeval_def, eval₂_eq, map_sub, map_mul, map_pow, map_bit0, map_one],\n  calc_tac\nend\n\n@[simp] lemma aeval_eq_zero (x : quad_ring R 0 dR) :\n  aeval x (minpoly_a_add_b_sqrt_d dR x.b1 x.b2) = 0 :=\nbegin\n  cases x with a b,\n  exact aeval_mk_eq_zero dR a b\nend\n\nend minpoly_a_add_b_sqrt_d\n\nprotected lemma is_integral_mk' (a b : ℤ) : is_integral ℤ (⟨a, b⟩ : quad_ring ℤ 0 d) :=\n⟨minpoly_a_add_b_sqrt_d d a b,\n minpoly_a_add_b_sqrt_d.monic _ _ _,\n minpoly_a_add_b_sqrt_d.aeval_mk_eq_zero d a b⟩\n\nprotected lemma is_integral : algebra.is_integral ℤ (ℤ[√d]) :=\nλ ⟨a, b⟩, quad_ring.is_integral_mk' _ a b\n\nsection\ninclude hdd'\n\nprotected lemma is_integral_mk (a b : ℤ) : is_integral ℤ (⟨a, b⟩ : quad_ring ℚ 0 d') :=\nbegin\n  rw [← @@algebra_map_mk hdd', is_integral_algebra_map_iff],\n  { exact quad_ring.is_integral_mk' d a b },\n  { exact algebra_map_injective 0 d 0 d' (ring_hom.injective_int _) }\nend\n\nlemma minpoly_eq [fact (¬ is_square d')] (x : ℚ(√d')) :\n  (∃ y : ℚ, x = y) ∨ minpoly ℚ x = minpoly_a_add_b_sqrt_d d' x.b1 x.b2 :=\nbegin\n  rw or_iff_not_imp_left,\n  intro x_triv,\n  refine (minpoly.unique _ x (minpoly_a_add_b_sqrt_d.monic d' x.b1 x.b2) _ _).symm,\n  { exact minpoly_a_add_b_sqrt_d.aeval_eq_zero d' x },\n  -- After all, this polynomial is of degree 2, so any (monic) polynomial that has root x\n  -- of smaller degree implies x is trivial.\n  intros q q_monic aeval_q,\n  rw [minpoly_a_add_b_sqrt_d.degree, degree_eq_nat_degree q_monic.ne_zero],\n  refine with_bot.coe_le_coe.mpr (show 2 ≤ q.nat_degree, from le_of_not_lt _),\n  intros h,\n  have := q.nat_degree.zero_le,\n  interval_cases using this h with h,\n  { rw [eq_C_of_nat_degree_eq_zero h_1, ← h_1, q_monic.coeff_nat_degree, aeval_C, map_one]\n      at aeval_q,\n    exact one_ne_zero aeval_q },\n  refine x_triv ⟨(- q.coeff 0) / q.coeff 1, _⟩,\n  rw [polynomial.eq_X_add_C_of_nat_degree_le_one h_1.le, aeval_add,\n      aeval_mul, aeval_C, aeval_X, aeval_C] at aeval_q,\n  have q_coeff_one : q.coeff 1 ≠ 0,\n  { rw [← h_1, q_monic.coeff_nat_degree],\n    exact one_ne_zero },\n  refine mul_left_cancel₀ (show algebra_map ℚ ℚ(√d') (q.coeff 1) ≠ 0, from _) _,\n  { exact (map_ne_zero_iff _ ((algebra_map ℚ ℚ(√d')).injective)).mpr q_coeff_one },\n  rw [← eq_rat_cast (algebra_map ℚ _), ← _root_.map_mul, mul_div_cancel' _ q_coeff_one, map_neg,\n      add_eq_zero_iff_eq_neg.mp aeval_q],\nend\n\n\nlemma exists_c1_c0 [fact (¬ is_square d')] (x : ℚ(√d')) (hx : is_integral ℤ x) :\n  (∃ y : ℚ, x = y) ∨ (∃ c1 c0 : ℤ, -2 * x.b1 = c1 ∧ x.b1^2 - d * x.b2^2 = c0) :=\nbegin\n  cases minpoly_eq d x with triv minpoly_eq,\n  { exact or.inl triv },\n  have : ∀ n, (minpoly_a_add_b_sqrt_d d' x.b1 x.b2).coeff n = (minpoly ℤ x).coeff n,\n  { intros n,\n    rw [← minpoly_eq, minpoly.gcd_domain_eq_field_fractions' ℚ hx, coeff_map,\n        algebra_map_int_eq, eq_int_cast] },\n  refine or.inr ⟨(minpoly ℤ x).coeff 1, (minpoly ℤ x).coeff 0, _, _⟩,\n  { simpa only [minpoly_a_add_b_sqrt_d.coeff_one] using this 1 },\n  { simpa only [minpoly_a_add_b_sqrt_d.coeff_zero, ← hdd'.out] using this 0 }\nend\n\nend\n\nlemma rat.denom_eq_one_iff_exists_int {b : ℚ} : b.denom = 1 ↔ ∃ b' : ℤ, b = b' :=\n(rat.denom_eq_one_iff _).trans ⟨λ h, ⟨b.num, h.symm⟩, by { rintro ⟨b', rfl⟩, simp }⟩\n\n/-- If `d` is squarefree, and `d * (b^2 : ℚ)` is an integer, then `b` is an integer. -/\nlemma eq_int_of_squarefree_mul_sq_eq_int {b : ℚ} {c d : ℤ} (hd : squarefree d)\n  (h : ↑d * b^2 = c) : ∃ (b' : ℤ), b = b' :=\nbegin\n  rw [← rat.denom_eq_one_iff_exists_int, ← nat.is_unit_iff, ← int.of_nat_is_unit],\n  apply hd,\n  rw ← pow_two,\n  rw [← b.num_div_denom, div_pow, mul_div, div_eq_iff, mul_comm ↑c] at h,\n  norm_cast at h,\n  exact is_coprime.dvd_of_dvd_mul_right\n    (is_coprime.pow (int.coprime_iff_nat_coprime.mpr b.cop.symm)) ⟨_, h⟩,\n  { simpa using b.pos.ne' }\nend\n\n/-- Let `r = 2` or `r = 3`, then r is not a quadratic remainder mod `4`,\nso `r*b^2 = a^2` has no solutions if `b` is odd. -/\nlemma zmod_4.two_or_three_mul_odd_sq_ne_sq {r : zmod 4} (hr : r = 2 ∨ r = 3)\n  (a b : zmod 4) (hb : odd b) :\n  r * b^2 ≠ a^2 :=\nbegin\n  unfold odd at hb,\n  dec_trivial!\nend\n\nlemma zmod_4.not_square_of_eq_two_or_three {d : zmod 4} (hr : d = 2 ∨ d = 3) :\n  ¬ is_square d :=\nbegin\n  dec_trivial!\nend\n\nlemma not_square_of_eq_two_or_three_mod_four {d : ℤ} (hr : d % 4 = 2 ∨ d % 4 = 3) :\n  ¬ is_square d :=\nbegin\n  refine _ ∘ is_square.map (int.cast_ring_hom (zmod 4)),\n  rw int.coe_cast_ring_hom,\n  refine zmod_4.not_square_of_eq_two_or_three _,\n  cases hr,\n  { exact or.inl ((zmod.int_coe_eq_int_coe_iff' d 2 4).mpr hr) },\n  { exact or.inr ((zmod.int_coe_eq_int_coe_iff' d 3 4).mpr hr) }\nend\n\n/-- The number theoretic argument showing ℤ[√d] is the ring of integers of ℚ(√d), if d = 2 or 3 mod 4:\nthe coefficients of the minimal polynomial of an integral number must be integers,\nwhich implies the coordinates are integers.\n-/\nlemma is_integral_closure_23_aux\n  {d c1 c0 : ℤ} {a b : ℚ} (hd2 : squarefree d) (hr : d % 4 = 2 ∨ d % 4 = 3)\n  (hc1 : -2 * a = c1) (hc0 : a^2 - d * b^2 = c0) :\n  ∃ (a' b' : ℤ), a = a' ∧ b = b' :=\nbegin\n  -- `a` is an integer up to a factor 2\n  rw [neg_mul, neg_eq_iff_neg_eq, ← int.cast_neg, eq_comm] at hc1,\n  set a' := - c1 with ha',\n  clear_value a',\n  -- And after some rewriting, `b` is an integer up to a factor 2 too (since `d` is squarefree)\n  have four_hc0 : (2 * a)^2 - d * (2 * b)^2 = 4 * c0, { rw ← hc0, ring },\n  rw [hc1, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', eq_comm] at four_hc0,\n  norm_cast at four_hc0,\n  -- So let `a2' = 2a` and `b' = 2b` be integers, so we get the equation `d b'^2 = a'^2 - 4c0`.\n  obtain ⟨b', hb'⟩ := eq_int_of_squarefree_mul_sq_eq_int hd2 four_hc0,\n  rw hb' at four_hc0,\n  norm_cast at four_hc0,\n  -- It suffices to show `b'` is even, in particular that `b'` is not odd.\n  suffices : even b',\n  { obtain ⟨b', rfl⟩ := (even_iff_exists_two_mul _).mp this,\n    obtain ⟨a', rfl⟩ := (even_iff_exists_two_mul _).mp (show even a', from _),\n    { refine ⟨a', b', _, _⟩,\n      { push_cast at hc1,\n        exact mul_left_cancel₀ (by norm_num) hc1 },\n      { push_cast at hb',\n        exact mul_left_cancel₀ (by norm_num) hb' } },\n    { refine (int.even_pow.mp (show even (a'^2), from _)).1,\n      rw sub_eq_iff_eq_add'.mp four_hc0.symm,\n      exact even.add\n        (even.mul_right (by norm_num) _)\n        (even.mul_left (even.pow_of_ne_zero this (by norm_num)) _) } },\n  rw int.even_iff_not_odd,\n  intros b_odd,\n  -- Consider the possible values for `a'`, `b'` that satisfy the equation mod 4.\n  have coe_b_odd : odd (b' : zmod 4) := b_odd.map (int.cast_ring_hom (zmod 4)),\n  have hc0_mod_4 := congr_arg (coe : ℤ → zmod 4) four_hc0,\n  rw [int.cast_mul, int.cast_sub, int.cast_pow, int.cast_pow,\n      (zmod.int_coe_zmod_eq_zero_iff_dvd (4 * c0) 4).mpr (dvd_mul_right 4 c0), sub_zero]\n    at hc0_mod_4,\n  refine zmod_4.two_or_three_mul_odd_sq_ne_sq _ a' b' coe_b_odd hc0_mod_4,\n  cases hr,\n  { exact or.inl ((zmod.int_coe_eq_int_coe_iff' d 2 4).mpr hr) },\n  { exact or.inr ((zmod.int_coe_eq_int_coe_iff' d 3 4).mpr hr) }\nend\n\nsection\n\ninclude hdd'\n\n/-- ℤ[√d] is the ring of integers of ℚ(√d), if d = 2 or 3 mod 4. -/\nlemma is_integral_closure_23 (hr : d % 4 = 2 ∨ d % 4 = 3)\n  (hd2 : squarefree d) :\n  is_integral_closure ℤ[√d] ℤ ℚ(√d') :=\n{ algebra_map_injective := algebra_map_injective 0 d 0 d' (ring_hom.injective_int _),\n  is_integral_iff := begin\n    haveI : fact (¬ is_square d'),\n    { constructor,\n      rw ← coe_d d d',\n      rw rat.is_square_int_cast_iff,\n      exact not_square_of_eq_two_or_three_mod_four hr },\n    intros x,\n    split,\n    { intros hx,\n      -- Either x is trivial, or the minimal polynomial of x is `x^2 - 2 a x + (a^2 - d b^2).\n      rcases exists_c1_c0 d x hx with ⟨y, rfl⟩ | ⟨c1, c0, hc1, hc0⟩,\n      { rw [← eq_rat_cast (algebra_map ℚ _),\n            is_integral_algebra_map_iff ((algebra_map ℚ ℚ(√d')).injective)] at hx,\n        obtain ⟨y, rfl⟩ := unique_factorization_monoid.integer_of_integral hx,\n        refine ⟨y, algebra_map_coe _ _ _ _ _⟩,\n        { apply_instance } },\n      -- Therefore, `-2 * a` and `a^2 - d * b^2` are integers.\n      obtain ⟨a', b', ha, hb⟩ := is_integral_closure_23_aux hd2 hr hc1 hc0,\n      refine ⟨⟨a', b'⟩, _⟩,\n      ext : 1; simp only [algebra_map_mk, ha, hb] },\n    { rintro ⟨⟨a, b⟩, rfl⟩,\n      rw algebra_map_mk,\n      exact quad_ring.is_integral_mk d a b }\n end }\n\n\nend\n\n-- TODO probably remove these special cases\ninstance : is_integral_closure (quad_ring ℤ 0 (-5)) ℤ (quad_ring ℚ 0 (-5)) :=\nis_integral_closure_23 (-5) (or.inr rfl)\n  (squarefree.squarefree_of_dvd ((neg_dvd _ _).mpr (dvd_refl 5)) $ prime.squarefree $\n    int.prime_iff_nat_abs_prime.mpr $ by norm_num)\n\ninstance quad_ring.is_integral_closure_3 {d : ℤ} [fact $ d % 4 = 3]\n  [fact $ squarefree d] {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')] :\n  is_integral_closure ℤ[√d] ℤ ℚ(√d') :=\nis_integral_closure_23 d (or.inr $ fact.out _) (fact.out _)\n\ninstance quad_ring.is_integral_closure_2 {d : ℤ} [fact $ d % 4 = 2]\n  [fact $ squarefree d] {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')] :\n  is_integral_closure ℤ[√d] ℤ ℚ(√d') :=\nis_integral_closure_23 d (or.inl $ fact.out _) (fact.out _)\n\ninstance quad_ring.is_integral_closure_2_or_three {d : ℤ}\n  [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] [fact $ squarefree d] {d' : ℚ}\n  [hdd' : fact (algebra_map ℤ ℚ d = d')] :\n  is_integral_closure ℤ[√d] ℤ ℚ(√d') :=\nis_integral_closure_23 d (fact.out _) (fact.out _)\n\nnamespace one_mod_four\n\n/-! ### Ring of integers of `ℚ(√d)`, `d = 1` mod 4.\n\nHere we'll use notation `α = 1/2 (1 + √d)` and `m = (d - 1) / 4`.\n\nIf `d = 1` mod 4 then `ℤ[√d]` is not integrally closed, so we have to adjoin a root\n`α` of the polynomial `X^2 - X - (d - 1)/4` instead, giving the ring `quad_ring ℤ 1 m`.\n-/\n\nopen polynomial\n\nvariables {R : Type*} [comm_ring R] (a b : R)\n\n/-- The minimal polynomial for `a + b * α` when `a b ∈ R`. -/\nnoncomputable def minpoly_a_add_b_alpha {R : Type*} [comm_ring R] (m a b : R) : polynomial R :=\nX^2 - C (2 * a + b) * X + C (a^2 + a*b - b^2 * m)\n\nnamespace minpoly_a_add_b_alpha\n\nvariables (m : R)\n\nprotected lemma coeff_zero : (minpoly_a_add_b_alpha m a b).coeff 0 = a^2 + a*b - b^2 * m :=\nby simp only [minpoly_a_add_b_alpha, pow_two, coeff_add, coeff_sub,\n  mul_coeff_zero, coeff_X_zero, coeff_C_zero,\n  mul_zero, zero_sub, neg_zero, zero_add, int.cast_id]\n\nprotected lemma coeff_one : (minpoly_a_add_b_alpha m a b).coeff 1 = - (2 * a + b) :=\nby simp only [minpoly_a_add_b_alpha, pow_two, coeff_add, coeff_sub,\n  coeff_mul_X, coeff_X_zero, coeff_C_ne_zero (show (1 : ℕ) ≠ 0, from one_ne_zero), coeff_C_zero,\n  zero_sub, add_zero, neg_mul]\n\nprotected lemma coeff_two : (minpoly_a_add_b_alpha m a b).coeff 2 = 1 :=\nby simp only [minpoly_a_add_b_alpha, pow_two, coeff_add, coeff_sub,\n  coeff_mul_X, coeff_X_one, coeff_C_ne_zero (show (1 : ℕ) ≠ 0, from one_ne_zero),\n  coeff_C_ne_zero (show (2 : ℕ) ≠ 0, from two_ne_zero), sub_zero, add_zero]\n\n@[simp] protected lemma degree [nontrivial R] : degree (minpoly_a_add_b_alpha m a b) = 2 :=\nbegin\n  -- TODO: this should be automatable (using Damiano's tactics?)\n  have deg_X2_X :=\n    calc degree (C (2 * a + b) * X) ≤ degree (C (2 * a + b)) + degree X : degree_mul_le _ _\n                            ... ≤ 0 + 1 : add_le_add degree_C_le degree_X_le\n                            ... < 2 : with_bot.coe_lt_coe.mpr one_lt_two\n                            ... = degree (X^2) : (degree_X_pow _).symm,\n  refine (degree_add_eq_left_of_degree_lt _).trans\n    ((degree_sub_eq_left_of_degree_lt deg_X2_X).trans\n    (degree_X_pow 2)),\n  calc degree (C _) ≤ 0 : degree_C_le\n                ... < 2 : with_bot.coe_lt_coe.mpr zero_lt_two\n                ... = degree (X^2) : (degree_X_pow _).symm\n                ... = degree _ : (degree_sub_eq_left_of_degree_lt deg_X2_X).symm,\n  apply_instance\nend\n\nprotected lemma monic : monic (minpoly_a_add_b_alpha m a b) :=\nbegin\n  nontriviality R,\n  -- TODO: unify with previous lemma\n  have deg_X2_X :=\n    calc degree (C (2 * a + b) * X) ≤ degree (C (2 * a + b)) + degree X : degree_mul_le _ _\n                            ... ≤ 0 + 1 : add_le_add degree_C_le degree_X_le\n                            ... < 2 : with_bot.coe_lt_coe.mpr one_lt_two\n                            ... = degree (X^2) : (degree_X_pow _).symm,\n  refine ((monic_X_pow _).sub_of_left deg_X2_X).add_of_left _,\n  calc degree (C _) ≤ 0 : degree_C_le\n                ... < 2 : with_bot.coe_lt_coe.mpr zero_lt_two\n                ... = degree (X^2) : (degree_X_pow _).symm\n                ... = degree _ : (degree_sub_eq_left_of_degree_lt deg_X2_X).symm,\n  apply_instance\nend\n\n@[simp] lemma eval₂_eq {S : Type*} [comm_ring S] (f : R →+* S) (x : S) :\n  eval₂ f x (minpoly_a_add_b_alpha m a b) = x^2 - f (2 * a + b) * x + f (a^2 + a * b - b^2 * m) :=\nby simp only [minpoly_a_add_b_alpha, eval₂_add, eval₂_sub, eval₂_pow, eval₂_mul, eval₂_X, eval₂_C]\n\nlemma aeval_mk_eq_zero : -- can be proved by `aeval_eq_zero`\n  aeval (⟨a, b⟩ : quad_ring R 1 m) (minpoly_a_add_b_alpha m a b) = 0 :=\nbegin\n  simp only [aeval_def, eval₂_eq, map_sub, map_mul, map_pow, map_bit0, map_one],\n  calc_tac\nend\n\n@[simp] lemma aeval_eq_zero (x : quad_ring R 1 m) :\n  aeval x (minpoly_a_add_b_alpha m x.b1 x.b2) = 0 :=\nbegin\n  cases x with a b,\n  exact aeval_mk_eq_zero a b m\nend\n\nend minpoly_a_add_b_alpha\n\nvariables (m : ℤ)\n\nprotected lemma is_integral_mk' (a b : ℤ) : is_integral ℤ (⟨a, b⟩ : quad_ring ℤ 1 m) :=\n⟨minpoly_a_add_b_alpha m a b,\n minpoly_a_add_b_alpha.monic _ _ _,\n minpoly_a_add_b_alpha.aeval_mk_eq_zero a b m⟩\n\nprotected lemma is_integral (a b : ℤ) : algebra.is_integral ℤ (quad_ring ℤ 1 m) :=\nλ ⟨a, b⟩, quad_ring.one_mod_four.is_integral_mk' m a b\n\nsection\n\ntheorem smul_inv_cancel₀ {M : Type*} {α : Type*} [group_with_zero M] [mul_action M α] [monoid α]\n  {r : M} (hr : r ≠ 0) (x : α) : (r • r⁻¹ • x) = x :=\nby rw [← mul_smul, mul_inv_cancel hr, one_smul]\n\ntheorem smul_sq {M : Type*} {α : Type*} [monoid M] [mul_action M α] [monoid α]\n  (r : M) (x : α) [is_scalar_tower M α α] [smul_comm_class M α α] :\n  (r • x)^2 = (r^2) • (x^2) :=\nby simp only [pow_two, smul_mul_smul]\n\nvariables [hdm : fact $ d' = 4 * m + 1]\ninclude hdm\n\ninstance : algebra (quad_ring ℤ 1 m) ℚ(√d') :=\nring_hom.to_algebra $ quad_ring.lift (algebra_map ℤ ℚ(√d')) 1 m\n  -- Since d' might be a square, multiply by `1/2 : ℚ` instead of dividing by `2 : ℚ(√d')`.\n  ((2⁻¹ : ℚ) • (1 + root ℚ 0 d')) $\nbegin\n  refine smul_right_injective ℚ(√d') (show (4 : ℚ) ≠ 0, by norm_num) _,\n  simp only [smul_zero, smul_sub, smul_add, smul_sq, inv_pow, map_one, one_mul, add_sq, root_sq,\n    hdm.out],\n  calc_tac\nend\n\n@[simp] lemma algebra_map_b1 (x : quad_ring ℤ 1 m) :\n  (algebra_map (quad_ring ℤ 1 m) ℚ(√d') x).b1 = x.b1 + 1/2 * x.b2 :=\nbegin\n  cases x,\n  rw [ring_hom.algebra_map_to_algebra, lift_mk],\n  simp only [smul_add, algebra_map_int_eq, one_div, quad_ring.root_b2, quad_ring.smul_b2, add_zero,\n      rat.cast_eq_id, algebra.id.smul_eq_mul, quad_ring.add_b1, quad_ring.coe_int_b2,\n      rat.smul_one_eq_coe, quad_ring.one_b2, eq_int_cast, int.cast_inj, id.def, add_left_inj,\n      zero_mul, quad_ring.mul_b1, quad_ring.root_b1, quad_ring.coe_int_b1, zero_add,\n      quad_ring.smul_b1, mul_zero, quad_ring.add_b2, quad_ring.one_b1],\nend\n\n@[simp] lemma algebra_map_b2 (x : quad_ring ℤ 1 m) :\n  (algebra_map (quad_ring ℤ 1 m) ℚ(√d') x).b2 = 1/2 * x.b2 :=\nbegin\n  cases x,\n  rw [ring_hom.algebra_map_to_algebra, lift_mk],\n  simp only [smul_add, algebra_map_int_eq, one_div, quad_ring.root_b2, quad_ring.smul_b2, add_zero,\n      rat.cast_eq_id, algebra.id.smul_eq_mul, quad_ring.add_b1, quad_ring.coe_int_b2,\n      rat.smul_one_eq_coe, quad_ring.one_b2, eq_int_cast, quad_ring.mul_b2, id.def,\n      quad_ring.root_b1, quad_ring.coe_int_b1, zero_add, quad_ring.smul_b1, mul_zero,\n      quad_ring.add_b2, quad_ring.one_b1]\nend\n\n@[simp] lemma algebra_map_mk (a b : ℤ) :\n  algebra_map (quad_ring ℤ 1 m) ℚ(√d') ⟨a, b⟩ = ⟨a + 1/2 * b, 1/2 * b⟩ :=\nby simp only [ext_iff, algebra_map_b1, algebra_map_b2, eq_self_iff_true, and_true]\n\n@[simp] lemma algebra_map_coe (a : ℤ) :\n  algebra_map (quad_ring ℤ 1 m) ℚ(√d') a = a :=\nbegin\n  rw [ring_hom.algebra_map_to_algebra, lift_coe, eq_int_cast]\nend\n\nlemma algebra_map_injective :\n  function.injective (algebra_map (quad_ring ℤ 1 m) ℚ(√d')) :=\nbegin\n  rintros ⟨x1, x2⟩ ⟨y1, y2⟩ h,\n  rw [algebra_map_mk, algebra_map_mk, ext_iff, mul_right_inj', int.cast_inj] at h,\n  have := h.2,\n  subst this,\n  rw [eq_self_iff_true, and_true, add_left_inj, int.cast_inj] at h,\n  subst h,\n  { norm_num }\nend\n\nlemma is_integral_mk_alpha (a b : ℤ) :\n  is_integral ℤ (⟨a + 1/2 * b, 1/2 * b⟩ : quad_ring ℚ 0 d') :=\nbegin\n  rw [← @@algebra_map_mk m hdm, is_integral_algebra_map_iff],\n  { exact quad_ring.one_mod_four.is_integral_mk' m a b },\n  { exact algebra_map_injective m }\nend\n\nlemma exists_mem_z_alpha_iff (y : ℚ(√d')) :\n  (∃ (x : quad_ring ℤ 1 m), algebra_map _ _ x = y) ↔\n   ∃ a b : ℤ, y.b1 = a + 1/2 * b ∧ y.b2 = 1/2 * b :=\nbegin\n  split,\n  { rintros ⟨x, rfl⟩,\n    exact ⟨x.b1, x.b2, algebra_map_b1 _ _, algebra_map_b2 _ _⟩ },\n  { rintros ⟨a, b, hy1, hy2⟩,\n    refine ⟨⟨a, b⟩, ext _ _ _ _⟩,\n    { rw [hy1, algebra_map_mk] },\n    { rw [hy2, algebra_map_mk] } }\nend\n\nnamespace zmod\n\nopen zmod\n\nomit hdm\n\n/-- See also `zmod.int_coe_zmod_eq_iff` which is slightly nicer but only applies for `p > 0`. -/\nlemma int_coe_zmod_eq_iff' {p : ℕ} {n : ℤ} {z : zmod p} :\n  ↑n = z ↔ ∃ k, n = z.val_min_abs + p * k :=\nbegin\n  by_cases hp : p = 0,\n  { subst hp,\n    split,\n    { rintro rfl,\n      use 0,\n      simp },\n    rintro ⟨k, rfl⟩,\n    simp },\n  haveI : ne_zero p := ⟨hp⟩,\n  split,\n  { rintro rfl,\n    refine ⟨if (n : zmod p).val ≤ p / 2 then n / p else n / p + 1, _⟩,\n    rw zmod.val_min_abs_def_pos,\n    split_ifs,\n    { rw [zmod.val_int_cast, int.mod_add_div] },\n    { rw [zmod.val_int_cast, mul_add, mul_one, sub_add_add_cancel, int.mod_add_div] } },\n  { rintro ⟨k, rfl⟩,\n    rw [int.cast_add, int.cast_mul, int.cast_coe_nat, zmod.coe_val_min_abs, zmod.nat_cast_self,\n        zero_mul, add_zero] }\nend\n\nend zmod\n\nomit hdm\n\nlemma zmod.dvd_coe_iff {k n : ℕ} {a : ℤ} (hkn : k ∣ n) : ↑k ∣ (a : zmod n) ↔ ↑k ∣ a :=\nbegin\n  by_cases hn : n = 0,\n  { subst hn, exact iff.rfl },\n  by_cases hk : k = n,\n  { subst hk, simpa using zmod.int_coe_zmod_eq_zero_iff_dvd a k },\n  haveI : ne_zero n := ⟨hn⟩,\n  split; rintro ⟨d, hd⟩,\n  { obtain ⟨a, rfl⟩ := (zmod.int_coe_zmod_eq_iff _ _ _).mp hd,\n    refine dvd_add _ ((map_dvd (nat.cast_ring_hom ℤ) hkn).mul_right _),\n    norm_cast,\n    rw [zmod.val_mul, nat.dvd_mod_iff hkn,\n        zmod.val_cast_of_lt ((nat.le_of_dvd (nat.pos_of_ne_zero hn) hkn).lt_of_ne hk)],\n    exact dvd_mul_right _ _ },\n  { subst hd,\n    exact_mod_cast dvd_mul_right (k : zmod n) (d : zmod n) },\nend\n\nlemma zmod.even_coe {n : ℕ} {a : ℤ} (hn : even n) : even (a : zmod n) ↔ even a :=\nbegin\n  simp only [even_iff_two_dvd] at ⊢ hn,\n  simpa using zmod.dvd_coe_iff hn,\nend\n\nlemma even_iff_sq_mod_four {a : ℤ} : even a ↔ (a : zmod 4)^2 = 0 :=\nbegin\n  split,\n  { intro h,\n    obtain ⟨a', rfl⟩ := (even_iff_exists_two_mul _).mp h,\n    calc ↑(2 * a') ^ 2\n        = (2 * (a' : zmod 4))^2 : by push_cast\n    ... = 4 * (a' : zmod 4)^2 : by ring\n    ... = 0 : _,\n    -- TODO: can we do this in a nicer way?\n    generalize : (a' : zmod 4) = a'',\n    revert a'',\n    dec_trivial },\n  { contrapose!,\n    intro h,\n    obtain ⟨a', rfl⟩ := int.odd_iff_not_even.mpr h,\n    -- TODO: can we do this in a nicer way?\n    push_cast,\n    generalize : (a' : zmod 4) = a'',\n    revert a'',\n    dec_trivial },\nend\n\nlemma sub_even_of_sq_eq_sq_mod_four (a b : ℤ) (h : (a : zmod 4)^2 = b^2) :\n  even (a - b) :=\nby rw [int.even_sub, even_iff_sq_mod_four, even_iff_sq_mod_four, h]\n\n/-- The number theoretic argument showing ℤ[α] is the ring of integers of ℚ(√d), if d = 1:\nthe coefficients of the minimal polynomial of an integral number must be integers,\nwhich implies the coordinates in ℤ[α] are integers.\n-/\nlemma is_integral_closure_1_aux (m c0 c1 : ℤ) (x y : ℚ)\n  (hm : squarefree (4 * m + 1)) (hc1 : -2 * x = c1) (hc0 : x^2 - (4 * m + 1) * y^2 = c0) :\n  ∃ (a b : ℤ), x = a + 1/2 * b ∧ y = 1/2 * b :=\nbegin\n  -- `x` is an integer up to a factor 2\n  rw [neg_mul, neg_eq_iff_neg_eq, ← int.cast_neg, eq_comm] at hc1,\n  set x' := - c1 with hx',\n  clear_value x',\n  -- And after some rewriting, `y` is an integer up to a factor 2 too (since `d` is squarefree)\n  have four_hc0 : (2 * x)^2 - (4 * m + 1) * (2 * y)^2 = 4 * c0, { rw ← hc0, ring },\n  have four_hc0' := four_hc0, -- Needed for later\n  rw [hc1, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', eq_comm] at four_hc0,\n  norm_cast at four_hc0,\n  -- So let `x' = 2x` and `b = 2y` be integers.\n  obtain ⟨b, hb⟩ := eq_int_of_squarefree_mul_sq_eq_int hm four_hc0,\n  -- To get an `a`, it suffices to show `x - y` is an integer.\n  rsuffices ⟨a, ha⟩ : ∃ a : ℤ, x - y = a,\n  { refine ⟨a, b, _, _⟩,\n    { rw [← ha, ← hb], simp },\n    { rw ← hb, simp } },\n  -- Which is the case iff `2x - 2y = x' - b` is even.\n  obtain ⟨a, ha⟩ := (even_iff_exists_two_mul _).mp\n    (show even (x' - b), from sub_even_of_sq_eq_sq_mod_four _ _ _),\n  { use a,\n    have ha' : (↑(x' - b) : ℚ) = _ := int.cast_inj.mpr ha,\n    push_cast at ha',\n    rw [← hc1, ← hb, ← mul_sub] at ha',\n    exact mul_left_cancel₀ (by norm_num) ha' },\n  -- This follows from considering the equation `(4m + 1) * (2y)^2 = x'^2 - 4*c0` mod 4.\n  rw [hc1, hb] at four_hc0',\n  norm_cast at four_hc0',\n  have hc0_mod_4 := congr_arg (coe : ℤ → zmod 4) four_hc0',\n  rwa [int.cast_sub, int.cast_pow, int.cast_mul, int.cast_add, int.cast_pow, int.cast_one,\n      (zmod.int_coe_zmod_eq_zero_iff_dvd (4 * m) 4).mpr (dvd_mul_right 4 m),\n      (zmod.int_coe_zmod_eq_zero_iff_dvd (4 * c0) 4).mpr (dvd_mul_right 4 c0),\n      zero_add, one_mul, sub_eq_zero]\n    at hc0_mod_4,\nend\n\ninclude hdm\n\n/-- ℤ[1/2 + 1/2√d] is the ring of integers of ℚ(√d), if d = 1 mod 4. -/\nlemma is_integral_closure_1 (hd' : ¬ is_square d') (hd2 : squarefree (4*m + 1)) :\n  is_integral_closure (quad_ring ℤ 1 m) ℤ ℚ(√d') :=\n{ algebra_map_injective := algebra_map_injective m,\n  is_integral_iff := begin\n    haveI : fact (¬ is_square d') := ⟨hd'⟩,\n    intros x,\n    split,\n    { intros hx,\n      haveI hdd' : fact (algebra_map ℤ ℚ (4 * m + 1) = d') := ⟨by simpa using hdm.out.symm⟩,\n      -- Either x is trivial, or we know the coefficients of its minimal polynomial,\n      -- which are integral.\n      rcases exists_c1_c0 (4 * m + 1) x hx with ⟨y, rfl⟩ | ⟨c1, c0, hc1, hc0⟩,\n      { rw [← eq_rat_cast (algebra_map ℚ _),\n            is_integral_algebra_map_iff ((algebra_map ℚ ℚ(√d')).injective)] at hx,\n        obtain ⟨y, rfl⟩ := unique_factorization_monoid.integer_of_integral hx,\n        refine ⟨y, algebra_map_coe _ _⟩,\n        { apply_instance } },\n      push_cast at hc0,\n      rw exists_mem_z_alpha_iff,\n      exact is_integral_closure_1_aux m _ _ x.b1 x.b2 hd2 hc1 hc0 },\n    { rintro ⟨⟨a, b⟩, rfl⟩,\n      rw algebra_map_mk,\n      exact is_integral_mk_alpha m a b }\n end }\n\nend\n\nend one_mod_four\n\nsection\n\ninstance {K : Type*} [field K] (a b : K) : finite_dimensional K (quad_ring K a b) :=\nfinite_dimensional.of_fintype_basis (quad_ring.basis K a b)\n\ninstance fact_not_square_of_eq_three_mod_four {d : ℤ} [hd : fact $ d % 4 = 3] :\n  fact $ ¬ is_square d :=\n⟨quad_ring.not_square_of_eq_two_or_three_mod_four (or.inr hd.out)⟩\n\ninstance fact_not_square_of_eq_two_mod_four {d : ℤ} [hd : fact $ d % 4 = 2] :\n  fact $ ¬ is_square d :=\n⟨quad_ring.not_square_of_eq_two_or_three_mod_four (or.inl hd.out)⟩\n\ninstance fact_not_square_of_eq_two_or_three_mod_four {d : ℤ} [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] :\n  fact $ ¬ is_square d :=\n⟨quad_ring.not_square_of_eq_two_or_three_mod_four hd.out⟩\n\nlemma fact_not_square'_of_eq_three_mod_four (d : ℤ) {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')]\n  [hd : fact $ d % 4 = 3] : fact $ ¬ is_square d' :=\n⟨begin\n  erw [← hdd'.out, rat.is_square_int_cast_iff],\n  exact quad_ring.not_square_of_eq_two_or_three_mod_four (or.inr hd.out)\nend⟩\n\nlemma fact_not_square'_of_eq_two_mod_four (d : ℤ) {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')]\n  [hd : fact $ d % 4 = 2] : fact $ ¬ is_square d' :=\n⟨begin\n  erw [← hdd'.out, rat.is_square_int_cast_iff],\n  exact quad_ring.not_square_of_eq_two_or_three_mod_four (or.inl hd.out)\nend⟩\n\nlemma fact_not_square'_of_eq_two_or_three_mod_four (d : ℤ) {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')]\n  [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] : fact $ ¬ is_square d' :=\n⟨begin\n  erw [← hdd'.out, rat.is_square_int_cast_iff],\n  exact quad_ring.not_square_of_eq_two_or_three_mod_four hd.out\nend⟩\n\n -- `d` is a free variable so these can't be instances\nlocal attribute [instance] fact_not_square'_of_eq_three_mod_four\nlocal attribute [instance] fact_not_square'_of_eq_two_mod_four\nlocal attribute [instance] fact_not_square'_of_eq_two_or_three_mod_four\n\ninstance quad_ring.mod_three_is_dedekind_domain {d : ℤ} [fact $ d % 4 = 3] [fact $ squarefree d] :\n  is_dedekind_domain ℤ[√d] :=\nis_integral_closure.is_dedekind_domain ℤ ℚ (quad_ring ℚ 0 d) ℤ[√d]\n\ninstance quad_ring.mod_two_is_dedekind_domain {d : ℤ} [fact $ d % 4 = 2] [fact $ squarefree d] :\n  is_dedekind_domain ℤ[√d] :=\nis_integral_closure.is_dedekind_domain ℤ ℚ (quad_ring ℚ 0 d) ℤ[√d]\n\ninstance quad_ring.mod_two_or_three_is_dedekind_domain {d : ℤ} [fact $ d % 4 = 2 ∨ d % 4 = 3]\n  [fact $ squarefree d] :\n  is_dedekind_domain ℤ[√d] :=\nis_integral_closure.is_dedekind_domain ℤ ℚ (quad_ring ℚ 0 d) ℤ[√d]\n\n\n@[simp]\nlemma d_one_mod_four_fact [hdmod : fact (d % 4 = 1)] : 4 * ((d - 1) / 4) + 1 = d :=\nbegin\n  rw [int.mul_div_cancel_of_mod_eq_zero, sub_add_cancel],\n  rw [int.sub_mod, hdmod.out],\n  norm_num,\nend\n\ninstance algebra_sub_one [hdmod : fact $ d % 4 = 1] [hdd' : fact (algebra_map ℤ ℚ d = d')] :\n  algebra (quad_ring ℤ 1 ((d - 1) / 4)) ℚ(√d') :=\nbegin\n  haveI : fact (d' = 4 * (↑((d - 1) / 4) : ℚ) + 1) := fact.mk _,\n  apply_instance,\n  simp only [←hdd'.out, eq_int_cast],\n  norm_cast,\n  exact (d_one_mod_four_fact d).symm,\nend\n\nlemma fact_not_square'_of_not_square (d : ℤ) {d' : ℚ} [hdd' : fact (algebra_map ℤ ℚ d = d')]\n  [hd : fact $ ¬ is_square d] : fact $ ¬ is_square d' :=\n⟨begin\n  erw [← hdd'.out, rat.is_square_int_cast_iff],\n  exact fact.out _,\nend⟩\n\nlocal attribute [instance] fact_not_square'_of_not_square\n\ninstance quad_ring.mod_one_is_dedekind_domain {d : ℤ} [hdmod : fact $ d % 4 = 1]\n  [fact $ squarefree d]\n  [fact $ ¬ is_square d] :\n  is_dedekind_domain (quad_ring ℤ 1 ((d - 1) / 4)) :=\nbegin\n  haveI : fact ((d : ℚ) = 4 * ↑((d - 1) / 4) + 1), -- TODO clean up repetition of this proof\n  { apply fact.mk,\n    norm_cast,\n    rw d_one_mod_four_fact, },\n  haveI : is_integral_closure (quad_ring ℤ 1 ((d - 1) / 4)) ℤ (quad_ring ℚ 0 d) :=\n    one_mod_four.is_integral_closure_1 ((d - 1) / 4) (fact.out _) _,\n  exact is_integral_closure.is_dedekind_domain ℤ ℚ (quad_ring ℚ 0 d) (quad_ring ℤ 1 ((d - 1) / 4)),\n  rw d_one_mod_four_fact,\n  exact fact.out _,\nend\n\nend\n\nopen algebra\nlemma norm_eq_one_of_mul_eq_one {d : ℤ} (hd : d ≤ 0) {st uv : quad_ring ℤ 0 d} (h : st * uv = 1) :\n  norm ℤ st = 1 :=\nbegin\n  apply_fun norm ℤ at h,\n  rw [map_mul, quad_ring.norm_one] at h,\n  exact int.eq_one_of_mul_eq_one_right (quad_ring.norm_nonneg hd _) h,\nend\n\nlemma units_quad {d : ℤ} (hd : d ≤ -2) (u : ℤ[√d]ˣ) : u = 1 ∨ u = -1 :=\nbegin\n  rcases u with ⟨⟨u_val_b1, u_val_b2⟩, ⟨u_inv_b1, u_inv_b2⟩, u_val_inv, u_inv_val⟩,\n  have h := norm_eq_one_of_mul_eq_one (by linarith) u_val_inv,\n  rw [quad_ring.norm_eq] at h,\n  have : u_val_b2 = 0,\n  { nlinarith, },\n  simp only [this, zero_pow', ne.def, bit0_eq_zero, nat.one_ne_zero, not_false_iff, mul_zero,\n    sq_eq_one_iff, sub_zero, add_zero] at h,\n  cases h; [left, right]; simpa [h, this, units.ext_iff],\nend\n\nlemma units_quad_neg_one (u : (quad_ring ℤ 0 (-1))ˣ) : u = 1 ∨ u = -1 ∨\n  u = units.mk_of_mul_eq_one (⟨0,1⟩ : quad_ring ℤ 0 (-1)) (⟨0,-1⟩ : quad_ring ℤ 0 (-1)) (by calc_tac) ∨\n  u = units.mk_of_mul_eq_one (⟨0,-1⟩ : quad_ring ℤ 0 (-1)) (⟨0,1⟩ : quad_ring ℤ 0 (-1)) (by calc_tac) :=\nbegin\n  rcases u with ⟨⟨u_val_b1, u_val_b2⟩, ⟨u_inv_b1, u_inv_b2⟩, u_val_inv, u_inv_val⟩,\n  have h := norm_eq_one_of_mul_eq_one (by linarith) u_val_inv,\n  rw [quad_ring.norm_eq] at h,\n  have hle : u_val_b2 ≤ 1,\n  { nlinarith, },\n  have hge : -1 ≤ u_val_b2,\n  { nlinarith, },\n  interval_cases u_val_b2,\n  { simp only [neg_one_sq, mul_one, sub_neg_eq_add, add_left_eq_self,\n      pow_eq_zero_iff, nat.succ_pos', zero_mul] at h,\n    simp [h, units.ext_iff], },\n  { simp only [zero_pow', ne.def, bit0_eq_zero, nat.one_ne_zero, not_false_iff,\n      mul_zero, sq_eq_one_iff, sub_zero, add_zero] at h,\n    cases h; [left, right]; simpa [h, units.ext_iff], },\n  { simp only [one_pow, mul_one, sub_neg_eq_add, add_left_eq_self,\n      pow_eq_zero_iff, nat.succ_pos', zero_mul] at h,\n    simp [h, units.ext_iff], },\nend\n\nlemma units_quad_cubes {d : ℤ} (hd : d ≤ -1) (u : ℤ[√d]ˣ) : ∃ v, u = v ^ 3 :=\nbegin\n  rw le_iff_lt_or_eq at hd,\n  rcases hd with hd | rfl,\n  { have : d ≤ -2,\n    linarith,\n    rcases units_quad this u with rfl | rfl,\n    use [1], simp,\n    use [-1], simp, },\n  { use u⁻¹,\n    rcases units_quad_neg_one u with rfl | rfl | rfl | rfl,\n    { simp, },\n    { simp, },\n    { rw units.ext_iff,\n      calc_tac, },\n    { rw units.ext_iff,\n      calc_tac, }, }\nend\n\n.\nlemma aux (m d : zmod 9) (hd : d ∈ ({0,2,3,4,5,6,8} : set (zmod 9))) : m ^ 2 * 3 + d ≠ 1 := by dec_trivial!\n-- not 1 or 7 mod 9, so sufficient is not a square mod 9\n-- Big, unused\nlemma aux'' (m n : zmod 9) : n * (m ^ 2 * 3 + n ^ 2 * -5) ≠ 1 := by dec_trivial!\nlemma aux' (m d : zmod 3) (hd : d ∈ ({0,1} : set (zmod 3))) : -d - (m ^ 2 * 3) ≠ 1 := by dec_trivial!\n\nlemma minpoly_int_eq {d : ℤ} [fact (¬ is_square d)] (x : ℤ[√d]) :\n  (∃ y : ℤ, x = y) ∨ minpoly ℤ x = minpoly_a_add_b_sqrt_d d x.b1 x.b2 :=\nbegin\n  haveI : fact (¬ is_square (d : ℚ)) := ⟨(rat.is_square_int_cast_iff _).not.mpr (fact.out _)⟩,\n  haveI : fact (algebra_map ℤ ℚ d = d) := ⟨eq_int_cast _ _⟩,\n  have inj_int : function.injective (algebra_map ℤ ℚ) := int.cast_injective,\n  have inj_rat : function.injective (algebra_map ℚ ℚ(√d)) := quad_ring.coe_injective ℚ 0 d,\n  have inj_quad : function.injective (algebra_map ℤ[√d] ℚ(√d)) :=\n    quad_ring.algebra_map_injective _ _ _ _ inj_int,\n  rcases minpoly_eq d (algebra_map _ (ℚ(√d)) x) with ⟨y, h⟩ | h,\n  { have : is_integral ℤ y,\n    { rw [← is_integral_algebra_map_iff inj_rat, algebra_map_apply, ← h,\n          is_integral_algebra_map_iff inj_quad],\n      exact quad_ring.is_integral d x,\n      { convert add_comm_group.int_is_scalar_tower }, -- TODO: where is this diamond coming from?\n      { apply_instance } },\n    obtain ⟨y', rfl⟩ := unique_factorization_monoid.integer_of_integral this,\n    rw [← algebra_map_apply, ← is_scalar_tower.algebra_map_apply,\n        is_scalar_tower.algebra_map_apply ℤ ℤ[√d]] at h,\n    exact or.inl ⟨y', inj_quad h⟩ },\n  refine or.inr (polynomial.map_injective (algebra_map ℤ ℚ) inj_int _),\n  rw [← minpoly.gcd_domain_eq_field_fractions ℚ (quad_ring ℚ 0 d) (quad_ring.is_integral d x),\n      h],\n  simp only [minpoly_a_add_b_sqrt_d, polynomial.map_add, polynomial.map_mul, polynomial.map_C,\n    polynomial.map_X, polynomial.map_sub, polynomial.map_pow, algebra_map_b1, algebra_map_b2,\n    eq_int_cast (algebra_map ℤ ℚ), int.cast_add, int.cast_mul, int.cast_sub, int.cast_pow,\n    int.cast_bit0, int.cast_bit1, int.cast_zero, int.cast_one],\nend\n\nlemma minpoly_int_root (d : ℤ) [fact (¬ is_square d)] :\n  minpoly ℤ (root ℤ 0 d) = minpoly_a_add_b_sqrt_d d 0 1 :=\nbegin\n  refine (minpoly_int_eq _).resolve_left _,\n  rw [not_exists],\n  intros,\n  rw [ext_iff, root_b2, coe_b2],\n  exact mt and.right one_ne_zero\nend\n\nlemma minpoly_int_root_thirteen :\n  minpoly ℤ (root ℤ 0 (-13)) = minpoly_a_add_b_sqrt_d (-13 : ℤ) 0 1 :=\nby { haveI : fact ((-13 : ℤ) % 4 = 3) := ⟨by norm_num⟩, exact minpoly_int_root _ }\n\nend rat\n\nend sqrt_d\n\nend quad_ring\n", "meta": {"author": "lean-forward", "repo": "class-group-and-mordell-equation", "sha": "baba2049f3bfe4d2cc184f8205997333e7c58638", "save_path": "github-repos/lean/lean-forward-class-group-and-mordell-equation", "path": "github-repos/lean/lean-forward-class-group-and-mordell-equation/class-group-and-mordell-equation-baba2049f3bfe4d2cc184f8205997333e7c58638/src/number_theory/quad_ring/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.33260303183960577}}
{"text": "import pseudo_normed_group.basic\n\nlocal attribute [instance] type_pow\n\nopen_locale nnreal big_operators\n\nnamespace pseudo_normed_group\n\nsection splittable\n\nclass splittable (M : Type*) [pseudo_normed_group M] (N : ℕ) (d : ℝ≥0) : Prop :=\n(exists_sum : ∀ (c : ℝ≥0) (x : M) (hx : x ∈ filtration M c),\n  ∃ y : fin N → M, (x = ∑ i, y i) ∧ (∀ i, y i ∈ filtration M (c/N + d)))\n\nvariables {M : Type*} [pseudo_normed_group M] (N : ℕ) (d : ℝ≥0) [splittable M N d]\n\nlemma exists_sum (c : ℝ≥0) (x : M) (hx : x ∈ filtration M c) :\n  ∃ y : fin N → M, (x = ∑ i, y i) ∧ (y ∈ filtration (M^N) (c/N + d)) :=\nsplittable.exists_sum c x hx\n\ninstance splittable_pi {ι : Type*} (M : ι → Type*) [Π i, pseudo_normed_group (M i)]\n  (N : ℕ) (d : ℝ≥0) [∀ i, splittable (M i) N d] :\n  splittable (Π i, M i) N d :=\n{ exists_sum := λ c x hx,\n  begin\n    have := λ i, exists_sum N d c (x i) (hx i),\n    choose y hy1 hy2 using this,\n    refine ⟨function.swap y, _, function.swap hy2⟩,\n    ext i, rw [hy1], symmetry, convert finset.sum_apply i _ _,\n  end }\n\nend splittable\n\nend pseudo_normed_group\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/splittable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.33260173983796515}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport algebraic_topology.dold_kan.functor_gamma\nimport category_theory.idempotents.homological_complex\n\n/-! The counit isomorphism of the Dold-Kan equivalence\n\nThe purpose of this file is to construct natural isomorphisms\n`N₁Γ₀ : Γ₀ ⋙ N₁ ≅ to_karoubi (chain_complex C ℕ)`\nand `N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (karoubi (chain_complex C ℕ))`.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits category_theory.idempotents\n  opposite simplicial_object\nopen_locale simplicial\n\nnamespace algebraic_topology\n\nnamespace dold_kan\n\nvariables {C : Type*} [category C] [preadditive C] [has_finite_coproducts C]\n\n/-- The isomorphism  `(Γ₀.splitting K).nondeg_complex ≅ K` for all `K : chain_complex C ℕ`. -/\n@[simps]\ndef Γ₀_nondeg_complex_iso (K : chain_complex C ℕ) : (Γ₀.splitting K).nondeg_complex ≅ K :=\nhomological_complex.hom.iso_of_components (λ n, iso.refl _)\nbegin\n  rintros _ n (rfl : n+1=_),\n  dsimp,\n  simp only [id_comp, comp_id, alternating_face_map_complex.obj_d_eq,\n    preadditive.sum_comp, preadditive.comp_sum],\n  rw fintype.sum_eq_single (0 : fin (n+2)),\n  { simp only [fin.coe_zero, pow_zero, one_zsmul],\n    erw [Γ₀.obj.map_mono_on_summand_id_assoc, Γ₀.obj.termwise.map_mono_δ₀,\n      splitting.ι_π_summand_eq_id, comp_id], },\n  { intros i hi,\n    dsimp,\n    simp only [preadditive.zsmul_comp, preadditive.comp_zsmul, assoc],\n    erw [Γ₀.obj.map_mono_on_summand_id_assoc, Γ₀.obj.termwise.map_mono_eq_zero,\n      zero_comp, zsmul_zero],\n    { intro h,\n      replace h := congr_arg simplex_category.len h,\n      change n+1 = n at h,\n      linarith, },\n    { simpa only [is_δ₀.iff] using hi, }, },\nend\n\n/-- The natural isomorphism `(Γ₀.splitting K).nondeg_complex ≅ K` for `K : chain_complex C ℕ`. -/\ndef Γ₀'_comp_nondeg_complex_functor :\n  Γ₀' ⋙ split.nondeg_complex_functor ≅ 𝟭 (chain_complex C ℕ) :=\nnat_iso.of_components Γ₀_nondeg_complex_iso\n  (λ X Y f, by { ext n, dsimp, simp only [comp_id, id_comp], })\n\n/-- The natural isomorphism `Γ₀ ⋙ N₁ ≅ to_karoubi (chain_complex C ℕ)`. -/\ndef N₁Γ₀ : Γ₀ ⋙ N₁ ≅ to_karoubi (chain_complex C ℕ) :=\ncalc Γ₀ ⋙ N₁ ≅ Γ₀' ⋙ split.forget C ⋙ N₁ : functor.associator _ _ _\n... ≅ Γ₀' ⋙ split.nondeg_complex_functor ⋙ to_karoubi _ :\n  iso_whisker_left Γ₀' split.to_karoubi_nondeg_complex_functor_iso_N₁.symm\n... ≅ (Γ₀' ⋙ split.nondeg_complex_functor) ⋙ to_karoubi _ : (functor.associator _ _ _).symm\n... ≅ 𝟭 _ ⋙ to_karoubi (chain_complex C ℕ) : iso_whisker_right Γ₀'_comp_nondeg_complex_functor _\n... ≅ to_karoubi (chain_complex C ℕ) : functor.left_unitor _\n\nlemma N₁Γ₀_app (K : chain_complex C ℕ) :\n  N₁Γ₀.app K = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.symm\n    ≪≫ (to_karoubi _).map_iso (Γ₀_nondeg_complex_iso K) :=\nbegin\n  ext1,\n  dsimp [N₁Γ₀],\n  erw [id_comp, comp_id, comp_id],\n  refl,\nend\n\nlemma N₁Γ₀_hom_app (K : chain_complex C ℕ) :\n  N₁Γ₀.hom.app K = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.inv\n    ≫ (to_karoubi _).map (Γ₀_nondeg_complex_iso K).hom :=\nby { change (N₁Γ₀.app K).hom = _, simpa only [N₁Γ₀_app], }\n\nlemma N₁Γ₀_inv_app (K : chain_complex C ℕ) :\n  N₁Γ₀.inv.app K = (to_karoubi _).map (Γ₀_nondeg_complex_iso K).inv ≫\n   (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.hom :=\nby { change (N₁Γ₀.app K).inv = _, simpa only [N₁Γ₀_app], }\n\n@[simp]\nlemma N₁Γ₀_hom_app_f_f (K : chain_complex C ℕ) (n : ℕ) :\n  (N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.inv.f.f n :=\nby { rw N₁Γ₀_hom_app, apply comp_id, }\n\n@[simp]\nlemma N₁Γ₀_inv_app_f_f (K : chain_complex C ℕ) (n : ℕ) :\n  (N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).to_karoubi_nondeg_complex_iso_N₁.hom.f.f n :=\nby { rw N₁Γ₀_inv_app, apply id_comp, }\n\nlemma N₂Γ₂_to_karoubi : to_karoubi (chain_complex C ℕ) ⋙ Γ₂ ⋙ N₂ = Γ₀ ⋙ N₁ :=\nbegin\n  have h := functor.congr_obj (functor_extension₂_comp_whiskering_left_to_karoubi\n    (chain_complex C ℕ) (simplicial_object C)) Γ₀,\n  have h' := functor.congr_obj (functor_extension₁_comp_whiskering_left_to_karoubi\n    (simplicial_object C) (chain_complex C ℕ)) N₁,\n  dsimp [N₂, Γ₂, functor_extension₁] at h h' ⊢,\n  rw [← functor.assoc, h, functor.assoc, h'],\nend\n\n/-- Compatibility isomorphism between `to_karoubi _ ⋙ Γ₂ ⋙ N₂` and `Γ₀ ⋙ N₁` which\nare functors `chain_complex C ℕ ⥤ karoubi (chain_complex C ℕ)`. -/\n@[simps]\ndef N₂Γ₂_to_karoubi_iso : to_karoubi (chain_complex C ℕ) ⋙ Γ₂ ⋙ N₂ ≅ Γ₀ ⋙ N₁ :=\neq_to_iso (N₂Γ₂_to_karoubi)\n\n/-- The counit isomorphism of the Dold-Kan equivalence for additive categories. -/\ndef N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (karoubi (chain_complex C ℕ)) :=\n((whiskering_left _ _ _).obj (to_karoubi (chain_complex C ℕ))).preimage_iso\n  (N₂Γ₂_to_karoubi_iso ≪≫ N₁Γ₀)\n\nlemma N₂Γ₂_compatible_with_N₁Γ₀ (K : chain_complex C ℕ) :\n  N₂Γ₂.hom.app ((to_karoubi _).obj K) = N₂Γ₂_to_karoubi_iso.hom.app K ≫ N₁Γ₀.hom.app K :=\ncongr_app (((whiskering_left _ _ (karoubi (chain_complex C ℕ ))).obj\n  (to_karoubi (chain_complex C ℕ))).image_preimage\n  (N₂Γ₂_to_karoubi_iso.hom ≫ N₁Γ₀.hom : _ ⟶ to_karoubi _ ⋙ 𝟭 _)) K\n\n@[simp]\nlemma N₂Γ₂_inv_app_f_f (X : karoubi (chain_complex C ℕ)) (n : ℕ) :\n  (N₂Γ₂.inv.app X).f.f n =\n    X.p.f n ≫ (Γ₀.splitting X.X).ι_summand (splitting.index_set.id (op [n])) :=\nbegin\n  dsimp only [N₂Γ₂, functor.preimage_iso, iso.trans],\n  simp only [whiskering_left_obj_preimage_app, N₂Γ₂_to_karoubi_iso_inv, functor.id_map,\n    nat_trans.comp_app, eq_to_hom_app, functor.comp_map, assoc, karoubi.comp_f,\n    karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id, karoubi.comp_p_assoc, N₂_map_f_f,\n    homological_complex.comp_f, N₁Γ₀_inv_app_f_f, P_infty_on_Γ₀_splitting_summand_eq_self_assoc,\n    splitting.to_karoubi_nondeg_complex_iso_N₁_hom_f_f, Γ₂_map_f_app, karoubi.decomp_id_p_f],\n  dsimp [to_karoubi],\n  rw [splitting.ι_desc],\n  dsimp [splitting.index_set.id],\n  rw karoubi.homological_complex.p_idem_assoc,\nend\n\nend dold_kan\n\nend algebraic_topology\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_topology/dold_kan/gamma_comp_n.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.33256331326150723}}
{"text": "import for_mathlib.exact_seq3\nimport for_mathlib.bicartesian2\n.\n\nopen category_theory category_theory.limits\n\nuniverse u\nlocal notation `𝓐` := Ab.{u}\n\n-- Consider the following diagram\nvariables {     Kv₁   Kv₂        : 𝓐}\nvariables {Kh₁  A₁₁   A₁₂  Qh₁   : 𝓐}\nvariables {Kh₂  A₂₁   A₂₂  Qh₂   : 𝓐}\nvariables {     Qv₁   Qv₂        : 𝓐}\n-- with morphisms\nvariables                         (fKv : Kv₁ ⟶ Kv₂)\nvariables                 {ιv₁ : Kv₁ ⟶ A₁₁} {ιv₂ : Kv₂ ⟶ A₁₂}\nvariables         {ιh₁ : Kh₁ ⟶ A₁₁} {f₁ : A₁₁ ⟶ A₁₂} {πh₁ : A₁₂ ⟶ Qh₁}\nvariables (gKh : Kh₁ ⟶ Kh₂) {g₁ : A₁₁ ⟶ A₂₁} {g₂ : A₁₂ ⟶ A₂₂} (gQh : Qh₁ ⟶ Qh₂)\nvariables         {ιh₂ : Kh₂ ⟶ A₂₁} {f₂ : A₂₁ ⟶ A₂₂} {πh₂ : A₂₂ ⟶ Qh₂}\nvariables                 {πv₁ : A₂₁ ⟶ Qv₁}  {πv₂ : A₂₂ ⟶ Qv₂}\nvariables                         (fQv : Qv₁ ⟶ Qv₂)\n-- with exact rows and columns\nvariables (H₁ : exact_seq 𝓐 [ιh₁, f₁, πh₁])\nvariables (H₂ : exact_seq 𝓐 [ιh₂, f₂, πh₂])\nvariables (V₁ : exact_seq 𝓐 [ιv₁, g₁, πv₁])\nvariables (V₂ : exact_seq 𝓐 [ιv₂, g₂, πv₂])\n-- and such that all the extremal maps are appropriately monos or epis\nvariables [mono ιv₁] [mono ιv₂] [mono ιh₁] [mono ιh₂]\nvariables [epi πv₁] [epi πv₂] [epi πh₁] [epi πh₂]\n-- of course the diagram should commute\nvariables (sqᵤ : commsq fKv ιv₁ ιv₂ f₁)\nvariables (sqₗ : commsq ιh₁ gKh g₁ ιh₂) (sqm : commsq f₁ g₁ g₂ f₂)\nvariables (sqᵣ : commsq πh₁ g₂ gQh πh₂)\nvariables (sqₛ : commsq f₂ πv₁ πv₂ fQv)\n\nopen_locale zero_object\nopen category_theory.abelian\n\ndef is_limit_of_is_limit_comp {X Y Z : 𝓐} {f : X ⟶ Y} {g : Y ⟶ Z}\n  {c : kernel_fork (f ≫ g)} (hc : is_limit c) (h : c.ι ≫ f = 0) :\n  is_limit (kernel_fork.of_ι c.ι h) :=\nis_limit.of_ι _ _\n  (λ T l hl, hc.lift (kernel_fork.of_ι l (by rw [reassoc_of hl, zero_comp])))\n  (λ T l hl, hc.fac _ _)\n  (λ T l hl m hm, fork.is_limit.hom_ext hc (by { erw [hm, hc.fac], refl }))\n\ndef is_colimit_of_is_colimit_comp {X Y Z : 𝓐} {f : X ⟶ Y} {g : Y ⟶ Z}\n  {c : cokernel_cofork (f ≫ g)} (hc : is_colimit c) (h : g ≫ c.π = 0) :\n  is_colimit (cokernel_cofork.of_π c.π h) :=\nis_colimit.of_π _ _\n  (λ T l hl, hc.desc (cokernel_cofork.of_π l (by rw [category.assoc, hl, comp_zero])))\n  (λ T l hl, hc.fac _ _)\n  (λ T l hl m hm, cofork.is_colimit.hom_ext hc (by { erw [hm, hc.fac], refl }))\n\nsection\ninclude sqₗ sqm\n\nlemma is_iso_of_is_limit (H₁ : exact ιh₁ f₁) (H₂ : exact ιh₂ f₂)\n  (h : is_limit (pullback_cone.mk f₁ g₁ sqm.w)) : is_iso gKh :=\nbegin\n  haveI : mono gKh,\n  { refine preadditive.mono_of_cancel_zero _ (λ P g hg, _),\n    apply zero_of_comp_mono ιh₁,\n    apply pullback_cone.is_limit.hom_ext h,\n    { rw [pullback_cone.mk_fst, category.assoc, zero_comp, H₁.w, comp_zero] },\n    { rw [pullback_cone.mk_snd, category.assoc, sqₗ.w, ← category.assoc, hg, zero_comp,\n        zero_comp] } },\n  obtain ⟨l, hl₁, hl₂ : l ≫ g₁ = _⟩ :=\n    pullback_cone.is_limit.lift' h 0 ιh₂ (by simp [H₂.w]),\n  let ker := is_limit_of_exact_of_mono _ _ H₁,\n  obtain ⟨inv, hinv : inv ≫ ιh₁ = l⟩ := kernel_fork.is_limit.lift' ker l hl₁,\n  have hinv' : inv ≫ gKh = 𝟙 _,\n   { rw [← cancel_mono ιh₂, category.assoc, ← sqₗ.w, reassoc_of hinv, hl₂, category.id_comp] },\n  refine ⟨⟨inv, _, hinv'⟩⟩,\n  rw [← cancel_mono gKh, category.assoc, hinv', category.comp_id, category.id_comp]\nend\n\nend\n\nsection\ninclude sqm sqᵣ\n\nlemma is_iso_of_is_colimit (H₁ : exact f₁ πh₁) (H₂ : exact f₂ πh₂)\n  (h : is_colimit (pushout_cocone.mk _ _ sqm.w)) : is_iso gQh :=\nbegin\n  haveI : epi gQh,\n  { refine preadditive.epi_of_cancel_zero _ (λ P g hg, _),\n    apply zero_of_epi_comp πh₂,\n    apply pushout_cocone.is_colimit.hom_ext h,\n    { rw [pushout_cocone.mk_inl, ← category.assoc, ← sqᵣ.w, category.assoc, hg, comp_zero,\n        comp_zero] },\n    { rw [pushout_cocone.mk_inr, ← category.assoc, H₂.w, comp_zero, zero_comp] } },\n  obtain ⟨l, hl₁ : g₂ ≫ l = _, hl₂⟩ :=\n    pushout_cocone.is_colimit.desc' h πh₁ 0 (by simp [H₁.w]),\n  let coker := is_colimit_of_exact_of_epi _ _ H₂,\n  obtain ⟨inv, hinv : πh₂ ≫ inv = l⟩ := cokernel_cofork.is_colimit.desc' coker l hl₂,\n  have hinv' : gQh ≫ inv = 𝟙 _,\n  { rw [← cancel_epi πh₁, ← category.assoc, sqᵣ.w, category.assoc, hinv, hl₁, category.comp_id] },\n  refine ⟨⟨inv, hinv', _⟩⟩,\n  rw [← cancel_epi gQh, reassoc_of hinv', category.comp_id]\nend\n\nend\n\ninclude H₁ H₂ sqₗ sqm sqᵣ\n\nlemma commsq.bicartesian_iff_isos : sqm.bicartesian ↔ (is_iso gKh ∧ is_iso gQh) :=\nbegin\n  split,\n  { intro h, split,\n    { exact is_iso_of_is_limit gKh sqₗ sqm ((exact_iff_exact_seq _ _).2 (H₁.extract 0 2))\n      ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) h.is_limit },\n    { exact is_iso_of_is_colimit gQh sqm sqᵣ ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2))\n      ((exact_iff_exact_seq _ _).2 (H₂.extract 1 2)) h.is_colimit } },\n  { rintros ⟨gKh_iso, gQh_iso⟩,\n    resetI,\n    apply commsq.bicartesian.of_is_limit_of_is_colimt,\n    { apply is_limit.of_point_iso (limit.is_limit _),\n      { apply_instance },\n      { let r := pullback.lift _ _ sqm.w,\n        let x : Kh₁ ⟶ kernel (pullback.fst : pullback g₂ f₂ ⟶ A₁₂),\n        { refine kernel.lift _ (ιh₁ ≫ r) _,\n          simp only [(H₁.extract 0 2).w, category.assoc, pullback.lift_fst] },\n        haveI : is_iso x,\n        { let psq := commsq.of_eq (@pullback.condition _ _ _ _ _ g₂ f₂ _),\n          let hker := abelian.is_limit_of_exact_of_mono _ _\n            ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)),\n          obtain ⟨u : _ ⟶ Kh₂, hu⟩ := kernel_fork.is_limit.lift' hker\n            (kernel.ι (pullback.fst : pullback g₂ f₂ ⟶ A₁₂) ≫ pullback.snd) _,\n          { rw fork.ι_of_ι at hu,\n            let lsq := commsq.of_eq hu.symm,\n            haveI : is_iso u := is_iso_of_is_limit u lsq psq _ _ _,\n            { have hxu : x ≫ u = gKh,\n              { simp only [← cancel_mono ιh₂, category.assoc, hu, x, kernel.lift_ι_assoc, r,\n                pullback.lift_snd, sqₗ.w] },\n              have hx : x = gKh ≫ inv u,\n              { rw [← is_iso.comp_inv_eq, is_iso.inv_inv, hxu] },\n              rw hx,\n              apply_instance },\n            { exact exact_kernel_ι },\n            { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) },\n            { exact pullback_is_pullback _ _ } },\n          { rw [category.assoc, ← pullback.condition, kernel.condition_assoc, zero_comp] } },\n        refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso _ _ _ 0 _ _ _ 0 _ _ _\n         0 ιh₁ f₁ 0 (kernel.ι (pullback.fst : pullback g₂ f₂ ⟶ A₁₂)) pullback.fst 0 x r (𝟙 _)\n         _ _ _ _ _ πh₁ πh₁ (𝟙 _) _ _ _ _ _ _ _ _ _ _ _,\n        { simp only [eq_iff_true_of_subsingleton] },\n        { simp only [kernel.lift_ι] },\n        { simp only [pullback.lift_fst, category.comp_id] },\n        { simp only [category.id_comp, category.comp_id] },\n        { exact exact_zero_mono ιh₁ },\n        { exact (exact_iff_exact_seq _ _).2 (H₁.extract 0 2) },\n        { exact (exact_iff_exact_seq _ _).2 (H₁.extract 1 2) },\n        { exact exact_zero_mono (kernel.ι pullback.fst) },\n        { exact exact_kernel_ι },\n        { have : (pullback.fst : pullback g₂ f₂ ⟶ A₁₂) ≫ πh₁ = 0,\n          { apply zero_of_comp_mono gQh,\n            rw [category.assoc, sqᵣ.w, pullback.condition_assoc, (H₂.extract 1 2).w, comp_zero] },\n          apply abelian.exact_of_is_cokernel _ _ this,\n          have hex := (exact_iff_exact_seq _ _).2 (H₁.extract 1 2),\n          rw ← pullback.lift_fst _ _ sqm.w at hex,\n          exact is_colimit_of_is_colimit_comp (abelian.is_colimit_of_exact_of_epi _ _ hex) _ } } },\n    { apply is_colimit.of_point_iso (colimit.is_colimit _),\n      { apply_instance },\n      { let r := pushout.desc _ _ sqm.w,\n        let x : cokernel (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) ⟶ Qh₂,\n        { refine cokernel.desc _ (r ≫ πh₂) _,\n          simp only [(H₂.extract 1 2).w, ← category.assoc, pushout.inr_desc] },\n        haveI : is_iso x,\n        { let psq := commsq.of_eq (@pushout.condition _ _ _ _ _ f₁ g₁ _),\n          let hcoker := abelian.is_colimit_of_exact_of_epi _ _\n            ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2)),\n          obtain ⟨u : Qh₁ ⟶ _, hu⟩ := cokernel_cofork.is_colimit.desc' hcoker\n            (pushout.inl ≫ (cokernel.π (pushout.inr : A₂₁ ⟶ pushout f₁ g₁))) _,\n          { rw cofork.π_of_π at hu,\n            let lsq := commsq.of_eq hu,\n            haveI : is_iso u := is_iso_of_is_colimit u psq lsq _ _ _,\n            { have hxu : u ≫ x = gQh,\n              { simp only [← cancel_epi πh₁, hu, x, cokernel.π_desc, reassoc_of hu, r,\n                  pushout.inl_desc_assoc, sqᵣ.w] },\n              have hx : x = inv u ≫ gQh,\n              { rw [← is_iso.inv_comp_eq, is_iso.inv_inv, hxu] },\n              rw hx,\n              apply_instance },\n            { exact ((exact_iff_exact_seq _ _).2 (H₁.extract 1 2)) },\n            { exact exact_cokernel pushout.inr },\n            { exact pushout_is_pushout _ _ } },\n          { rw [← category.assoc, pushout.condition, category.assoc, cokernel.condition,\n              comp_zero] } },\n        refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso _ _ _ _ _ _ _ _ _ _ _\n          ιh₂ (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) (cokernel.π (pushout.inr : A₂₁ ⟶ pushout f₁ g₁))\n          ιh₂ f₂ πh₂ (𝟙 _) (𝟙 _) r x _ _ _ 0 0 0 0 0 _ _ _ _ _ _ _ _ _ _ _,\n        { simp only [category.id_comp, category.comp_id] },\n        { simp only [category.id_comp, pushout.inr_desc] },\n        { simp only [cokernel.π_desc] },\n        { simp only [eq_iff_true_of_subsingleton] },\n        { have : ιh₂ ≫ (pushout.inr : A₂₁ ⟶ pushout f₁ g₁) = 0,\n          { apply zero_of_epi_comp gKh,\n            rw [← sqₗ.w_assoc, ← pushout.condition, reassoc_of (H₁.extract 0 2).w, zero_comp] },\n          apply abelian.exact_of_is_kernel _ _ this,\n          have hex := (exact_iff_exact_seq _ _).2 (H₂.extract 0 2),\n          rw ← pushout.inr_desc _ _ sqm.w at hex,\n          exact is_limit_of_is_limit_comp (abelian.is_limit_of_exact_of_mono _ _ hex) _ },\n        { exact exact_cokernel pushout.inr },\n        { exact exact_epi_zero (cokernel.π pushout.inr) },\n        { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 0 2)) },\n        { exact ((exact_iff_exact_seq _ _).2 (H₂.extract 1 2)) },\n        { exact exact_epi_zero πh₂ } } } }\nend\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/bicartesian3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3325633064207207}}
{"text": "def top := ∀ p : Prop, p → p\ndef pext := ∀ (A B : Prop), A → B → A = B\ndef supercast (h : pext) (A B : Prop) (a : A) (b : B) : B\n  := @cast A B (h A B a b) a\ndef omega : pext → top :=\n  λ h A a => supercast h (top → top) A\n    (λ z: top => z (top → top) (λ x => x) z) a\ndef Omega : pext → top :=\n  λ h => omega h (top → top) (λ x => x) (omega h)\ndef Omega' : pext → top := λ h => (λ p x => x)\n\ntheorem loopy : Omega = Omega' := rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/91_lean3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.42632159254749036, "lm_q1q2_score": 0.3325278154120666}}
{"text": "import Iris.BI\nimport Iris.Instances.Data\n\nnamespace Iris.Examples\nopen Iris.BI Iris.Instances.Data\n\n/- This file shows how to instantiate the type class `BIBase` with a setoid / quotient type. The\nseparation logic in this example is classical separation logic, even though setoids are not\nrequired here. For an example instance of `BIBase` and `BI` for classical separation logic without\nsetoids see `Iris/Instances/Classical`. -/\n\nabbrev HeapProp (Val : Type) := State Val → Prop\n\ninstance heapPropSetoid (Val : Type) : Setoid (HeapProp Val) where\n  r P Q := ∀ σ, P σ ↔ Q σ\n  iseqv := {\n    refl := by\n      intro _ _\n      exact Iff.refl _\n    symm := by\n      intro _ _ h σ\n      apply Iff.symm\n      exact h σ\n    trans := by\n      intro _ _ _ h_xy h_yz σ\n      exact Iff.trans (h_xy σ) (h_yz σ)\n  }\n\ninstance (Val : Type) : BIBase (Quotient (heapPropSetoid Val)) where\n  entails P Q := Quotient.liftOn₂ P Q (fun P Q => ∀ σ, P σ → Q σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    apply forall_congr\n    intro σ\n    rw [h₁ σ, h₂ σ])\n  emp := Quotient.mk _ fun σ => σ = ∅\n  pure φ := Quotient.mk _ fun _ => φ\n  and P Q := Quotient.mk _ fun σ => Quotient.liftOn₂ P Q (fun P Q => P σ ∧ Q σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    rw [h₁ σ, h₂ σ])\n  or P Q := Quotient.mk _ fun σ => Quotient.liftOn₂ P Q (fun P Q => P σ ∨ Q σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    rw [h₁ σ, h₂ σ])\n  impl P Q := Quotient.mk _ fun σ => Quotient.liftOn₂ P Q (fun P Q => P σ → Q σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    rw [h₁ σ, h₂ σ])\n  «forall» Ψ := Quotient.mk _ fun σ => ∀ a, Quotient.liftOn (Ψ a) (fun P => P σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ h\n    rw [h σ])\n  exist Ψ := Quotient.mk _ fun σ => ∃ a, Quotient.liftOn (Ψ a) (fun P => P σ) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ h\n    rw [h σ])\n  sep P Q := Quotient.mk _ fun σ => Quotient.liftOn₂ P Q (fun P Q => ∃ σ₁ σ₂ , σ = σ₁ ∪ σ₂ ∧ σ₁ || σ₂ ∧ P σ₁ ∧ Q σ₂) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    apply propext ?_\n    constructor\n    case' mp =>\n      intro ⟨σ₁, σ₂, h⟩\n      rw [h₁ σ₁, h₂ σ₂] at h\n    case' mpr =>\n      intro ⟨σ₁, σ₂, h⟩\n      rw [← h₁ σ₁, ← h₂ σ₂] at h\n    all_goals\n      apply Exists.intro σ₁\n      apply Exists.intro σ₂\n      exact h)\n  wand P Q := Quotient.mk _ fun σ => Quotient.liftOn₂ P Q (fun P Q => ∀ σ', σ || σ' → P σ' → Q (σ ∪ σ')) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ _ _ h₁ h₂\n    apply forall_congr\n    intro σ'\n    rw [h₁ σ', h₂ (σ ∪ σ')])\n  persistently P := Quotient.mk _ fun _ => Quotient.liftOn P (fun P => P ∅) (by\n    simp only [HasEquiv.Equiv, Setoid.r]\n    intro _ _ h\n    rw [h ∅])\n\nend Iris.Examples\n", "meta": {"author": "larsk21", "repo": "iris-lean", "sha": "730e644d0ffaad78aac76e2e5f2cd8af0f1d2310", "save_path": "github-repos/lean/larsk21-iris-lean", "path": "github-repos/lean/larsk21-iris-lean/iris-lean-730e644d0ffaad78aac76e2e5f2cd8af0f1d2310/src/Iris/Examples/InstanceSetoidClassical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.685949467848392, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3322602610461567}}
{"text": "-- Copyright (c) 2018 Jesse Han. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Jesse Han\n\nimport .limits.shapes.products basic data.dvector\n       .limits.shapes.equalizers\n       category_theory.limits.limits\n\nuniverses v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\n@[derive decidable_eq] inductive two : Type u\n| left | right\n\ndef two.map {C : Sort*} (X Y : C) : two → C\n| two.left := X\n| two.right := Y\n\ndef two.functor {C : Sort u} (X Y : C) [category.{v+1} C] : discrete two ⥤ C :=\nfunctor.of_function (two.map X Y)\n\ndef empty.functor (C : Sort*) [category.{v+1} C] : discrete pempty ⥤ C :=\nfunctor.of_function (λ x, by {cases x} : pempty → C)\n\ndef empty_cone {C : Sort u} [category.{v+1} C] (A : C) : limits.cone (empty.functor C) :=\n{ X := A,\n  π := { app := λ x, by cases x,\n  naturality' := by tidy}}\n\ndef commutative_square {C : Sort u} [category.{v u} C] {A B A' B' : C}\n  (f_top : A ⟶ B) (d_left : A ⟶ A') (d_right : B ⟶ B') (f_bot : A' ⟶ B') :=\nf_top ≫ d_right = d_left ≫ f_bot\n\nvariables {C : Type u} [𝒞 : category.{v+1} C]\ninclude 𝒞\n\nvariable(C)\n@[class] def has_binary_products := has_limits_of_shape (discrete two.{v}) C\n@[class] def has_terminal_object : Sort* := has_limits_of_shape.{v} (discrete pempty) C\n\n@[class] def has_binary_coproducts := has_colimits_of_shape (discrete two.{v}) C\n@[class] def has_initial_object : Sort* := has_colimits_of_shape.{v} (discrete pempty) C\n\n@[instance] def has_limit_two_of_has_binary_products [H : has_binary_products C] {X Y : C} :\n  has_limit $ two.functor X Y :=\n@has_limits_of_shape.has_limit _ _ _ _ H (two.functor X Y)\n\n@[instance] def has_limit_empty_of_has_terminal_object [H : has_terminal_object C] :\n  has_limit $ empty.functor C :=\n@has_limits_of_shape.has_limit _ _ _ _ H (empty.functor C)\n\nvariable{C}\n\ndef has_terminal_object.mk (T : C) (h₁ : ∀(X : C), X ⟶ T)\n  (h₂ : ∀{{X : C}} (f g : X ⟶ T), f = g) : has_terminal_object C :=\n⟨λ F, { cone := ⟨T, ⟨pempty.rec _, pempty.rec _⟩⟩,\n  is_limit :=\n  { lift := λ s, h₁ s.X,\n    fac' := λ s, pempty.rec _,\n    uniq' := λ s m h, h₂ _ _ } }⟩\n\ndef has_binary_products.mk (m : C → C → C) (p1 : ∀{X Y : C}, m X Y ⟶ X)\n  (p2 : ∀{X Y : C}, m X Y ⟶ Y) (lft : ∀{{X Y Z : C}} (f : Z ⟶ X) (g : Z ⟶ Y), Z ⟶ m X Y)\n  (lft1 : ∀{{X Y Z : C}} (f : Z ⟶ X) (g : Z ⟶ Y), lft f g ≫ p1 = f)\n  (lft2 : ∀{{X Y Z : C}} (f : Z ⟶ X) (g : Z ⟶ Y), lft f g ≫ p2 = g)\n  (lft_unique : ∀{{X Y Z : C}} (f g : Z ⟶ m X Y) (h1 : f ≫ p1 = g ≫ p1) (h2 : f ≫ p2 = g ≫ p2),\n    f = g) : has_binary_products C :=\nbegin\n  constructor, intro F, fsplit,\n  { use m (F.obj two.left) (F.obj two.right),\n    apply nat_trans.of_homs, refine two.rec _ _, exact p1, exact p2 },\n  refine limits.is_limit.mk _ _ _,\n  { rintro ⟨X, f⟩, apply lft (f.app two.left), dsimp, exact f.app two.right },\n  { rintro ⟨X, f⟩ (_|_), apply lft1, apply lft2 },\n  { rintro ⟨X, f⟩ g h, dsimp, apply lft_unique,\n    rw [lft1], exact h two.left, rw [lft2], exact h two.right }\nend\n\ndef has_initial_object.mk (I : C) (h₁ : ∀(X : C), I ⟶ X)\n  (h₂ : ∀{{X : C}} (f g : I ⟶ X), f = g) : has_initial_object C :=\n⟨λ F, { cocone := ⟨I, ⟨pempty.rec _, pempty.rec _⟩⟩,\n  is_colimit :=\n  { desc := λ s, h₁ s.X,\n    fac' := λ s, pempty.rec _,\n    uniq' := λ s m h, h₂ _ _ } }⟩\n\ndef has_binary_coproducts.mk (p : C → C → C) (i1 : ∀{X Y : C}, X ⟶ p X Y)\n  (i2 : ∀{X Y : C}, Y ⟶ p X Y) (dsc : ∀{{X Y Z : C}} (f : X ⟶ Z) (g : Y ⟶ Z), p X Y ⟶ Z)\n  (dsc1 : ∀{{X Y Z : C}} (f : X ⟶ Z) (g : Y ⟶ Z), i1 ≫ dsc f g = f)\n  (dsc2 : ∀{{X Y Z : C}} (f : X ⟶ Z) (g : Y ⟶ Z), i2 ≫ dsc f g = g)\n  (dsc_unique : ∀{{X Y Z : C}} (f g : p X Y ⟶ Z) (h1 : i1 ≫ f = i1 ≫ g) (h2 : i2 ≫ f = i2 ≫ g),\n    f = g) : has_binary_coproducts C :=\nbegin\n  constructor, intro F, fsplit,\n  { use p (F.obj two.left) (F.obj two.right),\n    apply nat_trans.of_homs, refine two.rec _ _, exact i1, exact i2 },\n  refine limits.is_colimit.mk _ _ _,\n  { rintro ⟨X, f⟩, apply dsc (f.app two.left), dsimp, exact f.app two.right },\n  { rintro ⟨X, f⟩ (_|_), apply dsc1, apply dsc2 },\n  { rintro ⟨X, f⟩ g h, dsimp, apply dsc_unique,\n    rw [dsc1], exact h two.left, rw [dsc2], exact h two.right }\nend\n\n/-- The binary product is the vertex of the limiting cone to the canonical functor two → 𝒞\n    associated to X and Y -/\ndef binary_product (X Y : C) [has_limit $ two.functor X Y] : C :=\nlimit (two.functor X Y)\n\nnamespace binary_product\n\nlocal infix ` × `:60 := binary_product\n\ndef π₁ {X Y : C} [has_limit $ two.functor X Y] : X × Y ⟶ X := limit.π _ two.left\n\ndef π₂ {X Y : C} [has_limit $ two.functor X Y] : X × Y ⟶ Y := limit.π _ two.right\n\n/-- An alternative version of `π₁` if type-class inference fails -/\ndef π₁' {X Y : C} {H : has_binary_products C} : X × Y ⟶ X := π₁\n/-- An alternative version of `π₂` if type-class inference fails -/\ndef π₂' {X Y : C} {H : has_binary_products C} : X × Y ⟶ Y := π₂\n\ndef dfin.map {n : ℕ} : dvector C n → dfin n → C :=\nλ v d, by {induction v, cases d, cases d, exact v_x, exact v_ih d_a}\n\nexample {X : C} [has_binary_products C] : X × X × X = (X × X) × X := by refl\n\ndef cone_of_two_maps {W A₁ A₂: C} (f₁ : W ⟶ A₁) (f₂ : W ⟶ A₂) : cone (two.functor A₁ A₂) :=\n{ X := W,\n  π := { app := λ l, two.rec_on l f₁ f₂,\n  naturality' := by tidy}}\n\nlemma cone_of_two_maps_object [has_binary_products C] {B₁ B₂ A₁ A₂: C} {f₁ : B₁ × B₂ ⟶ A₁}\n  {f₂ : B₁ × B₂ ⟶ A₂} : (cone_of_two_maps f₁ f₂).X = B₁ × B₂ := by refl\n\ndef map_to_product.mk {H : has_binary_products C} {W B₁ B₂ : C} (f₁ : W ⟶ B₁) (f₂ : W ⟶ B₂) :\n  W ⟶ B₁ × B₂ :=\nis_limit.lift (limit.is_limit _) (cone_of_two_maps f₁ f₂)\n\ndef diag [H : has_binary_products C] {B : C} : B ⟶ B × B :=\nmap_to_product.mk (𝟙 B) (𝟙 B)\n\nprotected def map {H : has_binary_products C} {A A' B B' : C} (f : A ⟶ A') (g : B ⟶ B') :\n  A × B ⟶ A' × B' :=\nmap_to_product.mk (π₁ ≫ f) (π₂ ≫ g)\n\nlocal infix ` ×.map `:90 := binary_product.map\n\nprotected def iso {H : has_binary_products C} {A A' B B' : C} (f : A ≅ A') (g : B ≅ B') :\n  A × B ≅ A' × B' :=\n{ hom := f.hom ×.map g.hom,\n  inv := f.inv ×.map g.inv,\n  hom_inv_id' := omitted,\n  inv_hom_id' := omitted }\n\nlocal infix ` ×.iso `:90 := binary_product.iso\n\ndef assoc_hom {H : has_binary_products C} {X Y Z : C} : (X × Y) × Z ⟶ X × (Y × Z) :=\nby apply map_to_product.mk (π₁ ≫ π₁) (π₂ ×.map (𝟙 Z))\n\ndef assoc_inv {H : has_binary_products C} {X Y Z : C} : X × (Y × Z) ⟶ (X × Y) × Z :=\nby apply map_to_product.mk (𝟙 X ×.map π₁) (π₂ ≫ π₂)\n\ndef product_assoc {H : has_binary_products C} {X Y Z : C} : (X × Y) × Z ≅ X × (Y × Z) :=\n{ hom := assoc_hom,\n  inv := assoc_inv,\n  hom_inv_id' := omitted,\n  inv_hom_id' := omitted}\n\ndef product_comm {H : has_binary_products C} {X Y : C} : X × Y ≅ Y × X :=\n{ hom := map_to_product.mk π₂ π₁,\n  inv := map_to_product.mk π₂ π₁,\n  hom_inv_id' := omitted,\n  inv_hom_id' := omitted}\n\ndef product_assoc4 {H : has_binary_products C} {X Y Z W : C} :\n  (X × Y) × (Z × W) ≅ (X × Z) × (Y × W) :=\nproduct_assoc ≪≫\niso.refl X ×.iso (product_assoc.symm ≪≫ product_comm ×.iso iso.refl W ≪≫ product_assoc) ≪≫\nproduct_assoc.symm\n\nexample :\n  commutative_square\n         /-unit-/ (𝟙 unit) /- unit  -/\n         (𝟙 unit)            (𝟙 unit)\n         /-unit-/ (𝟙 unit) /- unit -/\n  := by tidy\n\nend binary_product\nopen binary_product\n\nsection terminal_object\n\nlocal infix ` × `:60 := binary_product\n\ndef terminal_object [has_terminal_object C] : C :=\nlimit (empty.functor C)\n\nnotation `term` := terminal_object\n\ndef terminal_map [has_terminal_object C] (A : C) : A ⟶ term :=\nis_limit.lift (limit.is_limit (empty.functor C)) (empty_cone A)\n\nlemma terminal_map_eq [has_terminal_object C] {A : C} (f g : A ⟶ term) : f = g :=\nomitted\n\nlemma mul_one [has_terminal_object C] [has_binary_products C] (G : C) :\n  nonempty $ term × G ≅ G := omitted\n\nlemma one_mul [has_terminal_object C] [has_binary_products C] (G : C) :\n  nonempty $ G × term ≅ G := omitted\n\ndef mul_one_inv [has_terminal_object C] [has_binary_products C] {G : C} : G ⟶ G × term :=\nby apply map_to_product.mk (𝟙 _) (terminal_map G)\n\ndef one_mul_inv [has_terminal_object C] [has_binary_products C] {G : C} : G ⟶ term × G :=\nby apply map_to_product.mk (terminal_map G) (𝟙 _)\n\nend terminal_object\n\nsection pow\n\nlocal infix ` × `:60 := binary_product\n\n/-- The n-fold product of an object with itself -/\ndef category.pow [has_binary_products C] [has_terminal_object C] (X : C) : ℕ → C\n| 0     := term\n| 1     := X\n| (n+2) := X × category.pow (n+1)\n\nend pow\n\nnamespace finite_limits\nopen binary_product\n\ninstance fintype_two : fintype two :=\n{elems := { val := ⟦[two.left, two.right]⟧,\n  nodup := by tidy },\n  complete := λ x, by cases x; tidy}\n\nexample : fintype pempty := by apply_instance\n\nsection finite_products\n\nvariable (C)\n@[class]def has_finite_products := Π α : Type*, fintype α → has_limits_of_shape.{v} (discrete α) C\n\n@[class]def has_equalizers := has_limits_of_shape.{v} (walking_pair) C\n\n@[instance] def has_binary_products_of_has_finite_products [H : has_finite_products C] :\n  has_binary_products C := H _ infer_instance\n\n@[instance] def has_terminal_object_of_has_finite_products [H : has_finite_products C] :\n  has_limits_of_shape.{v} (discrete pempty) C := H _ infer_instance\n\n@[class]def has_finite_limits := @has_finite_products C 𝒞 × @has_equalizers C 𝒞\n\nend finite_products\n\nend finite_limits\n\nend category_theory.limits\n", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/category_theory/finite_limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3322326385035477}}
{"text": "import topology.category.Top.limits\nimport category_theory.limits.shapes\nimport topology.instances.real\n\n/- This file contains some demos of using the (co)limits API to do topology. -/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\ndef R : Top := Top.of ℝ\ndef I : Top := Top.of (set.Icc 0 1 : set ℝ)\ndef pt : Top := Top.of unit\n\nsection MappingCylinder\n-- Let's construct the mapping cylinder.\ndef to_pt (X : Top) : X ⟶ pt :=\n{ val := λ _, unit.star, property := continuous_const }\ndef I₀ : pt ⟶ I :=\n{ val := λ _, ⟨(0 : ℝ), begin rw [set.left_mem_Icc], norm_num, end⟩,\n  property := continuous_const }\ndef I₁ : pt ⟶ I :=\n{ val := λ _, ⟨(1 : ℝ), begin rw [set.right_mem_Icc], norm_num, end⟩,\n  property := continuous_const }\n\ndef cylinder (X : Top) : Top := prod X I\n-- To define a map to the cylinder, we give a map to each factor.\n-- `prod.lift` is a helper method, providing a wrapper around `limit.lift` for binary products.\ndef cylinder₀ (X : Top) : X ⟶ cylinder X :=\nprod.lift (𝟙 X) (to_pt X ≫ I₀)\ndef cylinder₁ (X : Top) : X ⟶ cylinder X :=\nprod.lift (𝟙 X) (to_pt X ≫ I₁)\n\n-- The mapping cylinder is the pushout of the diagram\n--    X\n--   ↙ ↘\n--  Y   (X x I)\n-- (`pushout` is implemented just as a wrapper around `colimit`) is\ndef mapping_cylinder {X Y : Top} (f : X ⟶ Y) : Top := pushout f (cylinder₁ X)\n\n/-- We construct the map from `X` into the \"bottom\" of the mapping cylinder\nfor `f : X ⟶ Y`, as the composition of the inclusion of `X` into the bottom of the\ncylinder `prod X I`, followed by the map `pushout.inr` of `prod X I` into `mapping_cylinder f`. -/\ndef mapping_cylinder₀ {X Y : Top} (f : X ⟶ Y) : X ⟶ mapping_cylinder f :=\ncylinder₀ X ≫ pushout.inr\n\n/--\nThe mapping cone is defined as the pushout of\n```\n         X\n        ↙ ↘\n (Cyl f)   pt\n```\n(where the left arrow is `mapping_cylinder₀`).\nThis makes it an iterated colimit; one could also define it in one step as the colimit of\n```\n--    X        X\n--   ↙ ↘      ↙ ↘\n--  Y   (X x I)  pt\n```\n-/\ndef mapping_cone {X Y : Top} (f : X ⟶ Y) : Top := pushout (mapping_cylinder₀ f) (to_pt X)\n\n-- TODO Hopefully someone will write a nice tactic for generating diagrams quickly,\n-- and we'll be able to verify that this iterated construction is the same as the colimit\n-- over a single diagram.\nend MappingCylinder\n\nsection Gluing\n\n-- Here's two copies of the real line glued together at a point.\ndef f : pt ⟶ R := { val := λ _, (0 : ℝ), property := continuous_const }\n\n/-- Two copies of the real line glued together at 0. -/\ndef X : Top := pushout f f\n\n-- To define a map out of it, we define maps out of each copy of the line,\n-- and check the maps agree at 0.\ndef g : X ⟶ R :=\npushout.desc (𝟙 _) (𝟙 _) rfl\n\nend Gluing\n\nuniverses v u w\n\nsection Products\n\n/-- The countably infinite product of copies of `ℝ`. -/\ndef Y : Top := ∏ (λ n : ℕ, R)\n\n/-- We define a point of this infinite product by specifying its coordinates. -/\ndef q : pt ⟶ Y :=\npi.lift (λ (n : ℕ), ⟨λ (_ : pt), (n : ℝ), continuous_const⟩)\n\n-- \"Looking under the hood\", we see that `q` is a `subtype`, whose `val` is a function `unit → Y.α`.\n-- #check q.val -- q.val : pt.α → Y.α\n-- `q.property` is the fact this function is continous (i.e. no content)\n\n-- We can check that this function is definitionally just the function we specified.\nexample : (q.val ()).val (57 : ℕ) = ((57 : ℕ) : ℝ) := rfl\n\nend Products", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/foncteur/topology_etrte.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3320905638175201}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Johan Commelin, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.mv_polynomial.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 u_5 u_3 \n\nnamespace Mathlib\n\n/-!\n# Renaming variables of polynomials\n\nThis file establishes the `rename` operation on multivariate polynomials,\nwhich modifies the set of variables.\n\n## Main declarations\n\n* `mv_polynomial.rename`\n\n## Notation\n\nAs in other polynomial files, we typically use the notation:\n\n+ `σ τ α : Type*` (indexing the variables)\n\n+ `R S : Type*` `[comm_semiring R]` `[comm_semiring S]` (the coefficients)\n\n+ `s : σ →₀ ℕ`, a function from `σ` to `ℕ` which is zero away from a finite set.\nThis will give rise to a monomial in `mv_polynomial σ R` which mathematicians might call `X^s`\n\n+ `r : R` elements of the coefficient ring\n\n+ `i : σ`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians\n\n+ `p : mv_polynomial σ α`\n\n-/\n\nnamespace mv_polynomial\n\n\n/-- Rename all the variables in a multivariable polynomial. -/\ndef rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) : alg_hom R (mv_polynomial σ R) (mv_polynomial τ R) :=\n  aeval (X ∘ f)\n\n@[simp] theorem rename_C {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (r : R) : coe_fn (rename f) (coe_fn C r) = coe_fn C r :=\n  eval₂_C (algebra_map R (mv_polynomial τ R)) (fun (n : σ) => function.comp X f n) r\n\n@[simp] theorem rename_X {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (i : σ) : coe_fn (rename f) (X i) = X (f i) :=\n  eval₂_X (algebra_map R (mv_polynomial τ R)) (fun (n : σ) => function.comp X f n) i\n\ntheorem map_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ → τ) (p : mv_polynomial σ R) : coe_fn (map f) (coe_fn (rename g) p) = coe_fn (rename g) (coe_fn (map f) p) := sorry\n\n@[simp] theorem rename_rename {σ : Type u_1} {τ : Type u_2} {α : Type u_3} {R : Type u_4} [comm_semiring R] (f : σ → τ) (g : τ → α) (p : mv_polynomial σ R) : coe_fn (rename g) (coe_fn (rename f) p) = coe_fn (rename (g ∘ f)) p := sorry\n\n@[simp] theorem rename_id {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) : coe_fn (rename id) p = p :=\n  eval₂_eta p\n\ntheorem rename_monomial {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (d : σ →₀ ℕ) (r : R) : coe_fn (rename f) (monomial d r) = monomial (finsupp.map_domain f d) r := sorry\n\ntheorem rename_eq {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (p : mv_polynomial σ R) : coe_fn (rename f) p = finsupp.map_domain (finsupp.map_domain f) p := sorry\n\ntheorem rename_injective {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (hf : function.injective f) : function.injective ⇑(rename f) := sorry\n\ntheorem eval₂_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) : eval₂ f g (coe_fn (rename k) p) = eval₂ f (g ∘ k) p := sorry\n\ntheorem eval₂_hom_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) : coe_fn (eval₂_hom f g) (coe_fn (rename k) p) = coe_fn (eval₂_hom f (g ∘ k)) p :=\n  eval₂_rename f k (fun (n : τ) => g n) p\n\ntheorem aeval_rename {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (k : σ → τ) (g : τ → S) (p : mv_polynomial σ R) [algebra R S] : coe_fn (aeval g) (coe_fn (rename k) p) = coe_fn (aeval (g ∘ k)) p :=\n  eval₂_hom_rename (algebra_map R S) k (fun (n : τ) => g n) p\n\ntheorem rename_eval₂ {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (k : σ → τ) (p : mv_polynomial σ R) (g : τ → mv_polynomial σ R) : coe_fn (rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) (coe_fn (rename k) p) := sorry\n\ntheorem rename_prodmk_eval₂ {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) (j : τ) (g : σ → mv_polynomial σ R) : coe_fn (rename (Prod.mk j)) (eval₂ C g p) = eval₂ C (fun (x : σ) => coe_fn (rename (Prod.mk j)) (g x)) p := sorry\n\ntheorem eval₂_rename_prodmk {σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R] [comm_semiring S] (f : R →+* S) (g : σ × τ → S) (i : σ) (p : mv_polynomial τ R) : eval₂ f g (coe_fn (rename (Prod.mk i)) p) = eval₂ f (fun (j : τ) => g (i, j)) p := sorry\n\ntheorem eval_rename_prodmk {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (g : σ × τ → R) (i : σ) (p : mv_polynomial τ R) : coe_fn (eval g) (coe_fn (rename (Prod.mk i)) p) = coe_fn (eval fun (j : τ) => g (i, j)) p :=\n  eval₂_rename_prodmk (ring_hom.id R) (fun (n : σ × τ) => g n) i p\n\n/-- Every polynomial is a polynomial in finitely many variables. -/\ntheorem exists_finset_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) : ∃ (s : finset σ), ∃ (q : mv_polynomial (Subtype fun (x : σ) => x ∈ s) R), p = coe_fn (rename coe) q := sorry\n\n/-- Every polynomial is a polynomial in finitely many variables. -/\ntheorem exists_fin_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial σ R) : ∃ (n : ℕ), ∃ (f : fin n → σ), ∃ (hf : function.injective f), ∃ (q : mv_polynomial (fin n) R), p = coe_fn (rename f) q := sorry\n\ntheorem eval₂_cast_comp {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : mv_polynomial σ ℤ) : eval₂ c (g ∘ f) p = eval₂ c g (coe_fn (rename f) p) := sorry\n\n@[simp] theorem coeff_rename_map_domain {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (hf : function.injective f) (φ : mv_polynomial σ R) (d : σ →₀ ℕ) : coeff (finsupp.map_domain f d) (coe_fn (rename f) φ) = coeff d φ := sorry\n\ntheorem coeff_rename_eq_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (φ : mv_polynomial σ R) (d : τ →₀ ℕ) (h : ∀ (u : σ →₀ ℕ), finsupp.map_domain f u = d → coeff u φ = 0) : coeff d (coe_fn (rename f) φ) = 0 := sorry\n\ntheorem coeff_rename_ne_zero {σ : Type u_1} {τ : Type u_2} {R : Type u_4} [comm_semiring R] (f : σ → τ) (φ : mv_polynomial σ R) (d : τ →₀ ℕ) (h : coeff d (coe_fn (rename f) φ) ≠ 0) : ∃ (u : σ →₀ ℕ), finsupp.map_domain f u = d ∧ coeff u φ ≠ 0 := sorry\n\n@[simp] theorem constant_coeff_rename {σ : Type u_1} {R : Type u_4} [comm_semiring R] {τ : Type u_2} (f : σ → τ) (φ : mv_polynomial σ R) : coe_fn constant_coeff (coe_fn (rename f) φ) = coe_fn constant_coeff φ := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/mv_polynomial/rename.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3320905564892271}}
{"text": "import data.real.nnreal\n\nopen_locale nnreal\n\nnamespace nnreal\n\nvariables (r r' k c c₁ c₂ c₃ : ℝ≥0)\n\ninstance fact_le_of_lt [h : fact (c₁ < c₂)] : fact (c₁ ≤ c₂) := ⟨h.1.le⟩\n\ninstance fact_pos_of_one_le [hk : fact (1 ≤ c)] : fact (0 < c) :=\n⟨lt_of_lt_of_le zero_lt_one hk.1⟩\n\ninstance fact_mul_pos [h1 : fact (0 < c₁)] [h2 : fact (0 < c₂)] : fact (0 < c₁ * c₂) :=\n⟨mul_pos h1.out h2.out⟩\n\ninstance fact_le_mul_of_one_le_left [hk : fact (1 ≤ k)] [hc : fact (c₁ ≤ c₂)] :\n  fact (c₁ ≤ k * c₂) :=\n⟨calc c₁ = 1 * c₁ : (one_mul _).symm ... ≤ k * c₂ : mul_le_mul' hk.1 hc.1⟩\n\ninstance fact_le_mul_of_one_le_right [hc : fact (c₁ ≤ c₂)] [hk : fact (1 ≤ k)] :\n  fact (c₁ ≤ c₂ * k) :=\n⟨calc c₁ = c₁ * 1 : (mul_one _).symm ... ≤ c₂ * k : mul_le_mul' hc.1 hk.1⟩\n\ninstance fact_mul_le_of_le_one_left [hk : fact (k ≤ 1)] [hc : fact (c₁ ≤ c₂)] :\n  fact (k * c₁ ≤ c₂) :=\n⟨calc k * c₁ ≤ 1 * c₂ : mul_le_mul' hk.1 hc.1 ... = c₂     : one_mul _⟩\n\ninstance fact_mul_le_of_le_one_right [hk : fact (k ≤ 1)] [hc : fact (c₁ ≤ c₂)] :\n  fact (c₁ * k ≤ c₂) :=\n⟨calc c₁ * k ≤ c₂ * 1 : mul_le_mul' hc.1 hk.1 ... = c₂ : mul_one _⟩\n\ninstance fact_one_le_add_one : fact (1 ≤ k + 1) :=\n⟨self_le_add_left 1 k⟩\n\ninstance fact_le_refl : fact (c ≤ c) := ⟨le_rfl⟩\n\ninstance fact_le_subst_right [fact (c₁ ≤ c₂)] [h : fact (c₂ = c₃)]: fact (c₁ ≤ c₃) :=\nby rwa ← h.1\n\ninstance fact_le_subst_right' [fact (c₁ ≤ c₂)] [h : fact (c₃ = c₂)]: fact (c₁ ≤ c₃) :=\nby rwa ← h.1.symm\n\ninstance fact_le_subst_left [fact (c₁ ≤ c₂)] [h : fact (c₁ = c₃)]: fact (c₃ ≤ c₂) :=\nby rwa ← h.1\n\ninstance fact_le_subst_left' [fact (c₁ ≤ c₂)] [h : fact (c₃ = c₁)]: fact (c₃ ≤ c₂) :=\nby rwa ← h.1.symm\n\ninstance fact_inv_mul_le [h : fact (0 < r')] : fact (r'⁻¹ * (r' * c) ≤ c) :=\n⟨le_of_eq $ inv_mul_cancel_left₀ (ne_of_gt h.1) _⟩\n\ninstance fact_mul_le_mul_left [h : fact (c₁ ≤ c₂)] : fact (r' * c₁ ≤ r' * c₂) :=\n⟨mul_le_mul' le_rfl h.1⟩\n\ninstance fact_mul_le_mul_right [h : fact (c₁ ≤ c₂)] : fact (c₁ * r' ≤ c₂ * r') :=\n⟨mul_le_mul' h.1 le_rfl⟩\n\ninstance fact_le_inv_mul_self [h1 : fact (0 < r')] [h2 : fact (r' ≤ 1)] : fact (c ≤ r'⁻¹ * c) :=\nbegin\n  constructor,\n  rw mul_comm,\n  apply le_mul_inv_of_mul_le (ne_of_gt h1.1),\n  nth_rewrite 1 ← mul_one c,\n  exact mul_le_mul (le_of_eq rfl) h2.1 (le_of_lt h1.1) zero_le',\nend\n\ninstance fact_le_max_left (a b c : ℝ≥0) [h : fact (a ≤ b)] : fact (a ≤ max b c) :=\n⟨h.1.trans $ le_max_left _ _⟩\n\ninstance fact_one_le_mul_self (a : ℝ≥0) [h : fact (1 ≤ a)] : fact (1 ≤ a * a) :=\n⟨calc (1 : ℝ≥0) = 1 * 1 : (mul_one 1).symm\n           ... ≤ a * a : mul_le_mul' h.1 h.1⟩\n\ninstance one_le_add {a b : ℝ≥0} [ha : fact (1 ≤ a)] : fact (1 ≤ a + b) :=\n⟨le_trans ha.1 $ by simp⟩\ninstance one_le_add' {a b : ℝ≥0} [hb : fact (1 ≤ b)] : fact (1 ≤ a + b) :=\n⟨le_trans hb.1 $ by simp⟩\n\ninstance fact_one_le_pow {n : ℕ} {a : ℝ≥0} [h : fact (1 ≤ a)] : fact (1 ≤ a^n) :=\nbegin\n  cases n,\n  { simpa only [pow_zero] using nnreal.fact_le_refl _ },\n  { rwa @one_le_pow_iff _ _ _ nnreal.covariant_mul, apply nat.succ_ne_zero }\nend\n\ninstance fact_pow_le_one {n : ℕ} {a : ℝ≥0} [h : fact (a ≤ 1)] : fact (a^n ≤ 1) :=\nbegin\n  cases n,\n  { simpa only [pow_zero] using nnreal.fact_le_refl _ },\n  { rwa @pow_le_one_iff _ _ _ nnreal.covariant_mul, apply nat.succ_ne_zero }\nend\n\nlemma fact_le_pow_mul_of_le_pow_succ_mul {n : ℕ} (r : ℝ≥0)\n  [fact (r ≤ 1)] [h : fact (c₂ ≤ r ^ (n+1) * c₁)] :\n  fact (c₂ ≤ r ^ n * c₁) :=\nbegin\n  refine ⟨h.1.trans _⟩,\n  rw [pow_succ, mul_assoc],\n  apply fact.out\nend\n\ninstance fact_le_mul_add : fact (c * c₁ + c * c₂ ≤ c * (c₁ + c₂)) :=\nby { rw mul_add, exact nnreal.fact_le_refl _ }\n\ninstance fact_nat_cast_pos (N : ℕ) [hN: fact (0 < N)] : fact (0 < (N:ℝ≥0)) :=\n⟨nat.cast_pos.mpr hN.1⟩\n\ninstance fact_nat_cast_inv_le_one (N : ℕ) : fact ((N:ℝ≥0)⁻¹ ≤ 1) :=\nbegin\n  by_cases hN : N = 0,\n  { subst hN, simp only [nat.cast_zero, inv_zero, zero_le'], exact ⟨trivial⟩ },\n  { rw [inv_le, mul_one], swap, { exact_mod_cast hN },\n    norm_cast,\n    rw nat.add_one_le_iff,\n    exact ⟨nat.pos_of_ne_zero hN⟩, }\nend\n\ninstance fact_inv_le_one [H : fact (1 ≤ c)] : fact (c⁻¹ ≤ 1) :=\nbegin\n  by_cases hc : c = 0,\n  { rw hc at H, exact (not_le_of_lt zero_lt_one H.1).elim },\n  rwa [inv_le hc, mul_one]\nend\n\ninstance fact_one_le_two : fact ((1:ℝ≥0) ≤ 2) := ⟨one_le_two⟩\n\ninstance fact_two_pow_inv_le_two_pow_inv (N : ℕ) : fact ((2 ^ N : ℝ≥0)⁻¹ ≤ (2 ^ N : ℕ)⁻¹) :=\n⟨le_of_eq $ by norm_cast⟩\n\ninstance fact_two_pow_inv_le_one (N : ℕ) : fact ((2 ^ N : ℝ≥0)⁻¹ ≤ 1) :=\n⟨le_trans (nnreal.fact_two_pow_inv_le_two_pow_inv N).1 $ fact.out _⟩\n\nend nnreal\n\n#lint- only unused_arguments def_lemma doc_blame\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/facts/nnreal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3318968252949478}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Justus Springer\n-/\nimport topology.category.Top.open_nhds\nimport topology.sheaves.presheaf\nimport topology.sheaves.sheaf_condition.unique_gluing\nimport category_theory.limits.types\nimport category_theory.limits.preserves.filtered\nimport category_theory.limits.final\nimport topology.sober\nimport tactic.elementwise\nimport algebra.category.CommRing\n\n/-!\n# Stalks\n\nFor a presheaf `F` on a topological space `X`, valued in some category `C`, the *stalk* of `F`\nat the point `x : X` is defined as the colimit of the following functor\n\n(nhds x)ᵒᵖ ⥤ (opens X)ᵒᵖ ⥤ C\n\nwhere the functor on the left is the inclusion of categories and the functor on the right is `F`.\nFor an open neighborhood `U` of `x`, we define the map `F.germ x : F.obj (op U) ⟶ F.stalk x` as the\ncanonical morphism into this colimit.\n\nTaking stalks is functorial: For every point `x : X` we define a functor `stalk_functor C x`,\nsending presheaves on `X` to objects of `C`. Furthermore, for a map `f : X ⟶ Y` between\ntopological spaces, we define `stalk_pushforward` as the induced map on the stalks\n`(f _* ℱ).stalk (f x) ⟶ ℱ.stalk x`.\n\nSome lemmas about stalks and germs only hold for certain classes of concrete categories. A basic\nproperty of forgetful functors of categories of algebraic structures (like `Mon`, `CommRing`,...)\nis that they preserve filtered colimits. Since stalks are filtered colimits, this ensures that\nthe stalks of presheaves valued in these categories behave exactly as for `Type`-valued presheaves.\nFor example, in `germ_exist` we prove that in such a category, every element of the stalk is the\ngerm of a section.\n\nFurthermore, if we require the forgetful functor to reflect isomorphisms and preserve limits (as\nis the case for most algebraic structures), we have access to the unique gluing API and can prove\nfurther properties. Most notably, in `is_iso_iff_stalk_functor_map_iso`, we prove that in such\na category, a morphism of sheaves is an isomorphism if and only if all of its stalk maps are\nisomorphisms.\n\nSee also the definition of \"algebraic structures\" in the stacks project:\nhttps://stacks.math.columbia.edu/tag/007L\n\n-/\n\nnoncomputable theory\n\nuniverses v u v' u'\n\nopen category_theory\nopen Top\nopen category_theory.limits\nopen topological_space\nopen opposite\n\nvariables {C : Type u} [category.{v} C]\n\nvariables [has_colimits.{v} C]\n\nvariables {X Y Z : Top.{v}}\n\nnamespace Top.presheaf\n\nvariables (C)\n/-- Stalks are functorial with respect to morphisms of presheaves over a fixed `X`. -/\ndef stalk_functor (x : X) : X.presheaf C ⥤ C :=\n((whiskering_left _ _ C).obj (open_nhds.inclusion x).op) ⋙ colim\n\nvariables {C}\n\n/--\nThe stalk of a presheaf `F` at a point `x` is calculated as the colimit of the functor\nnbhds x ⥤ opens F.X ⥤ C\n-/\ndef stalk (ℱ : X.presheaf C) (x : X) : C :=\n(stalk_functor C x).obj ℱ -- -- colimit ((open_nhds.inclusion x).op ⋙ ℱ)\n\n@[simp] lemma stalk_functor_obj (ℱ : X.presheaf C) (x : X) :\n  (stalk_functor C x).obj ℱ = ℱ.stalk x := rfl\n\n/--\nThe germ of a section of a presheaf over an open at a point of that open.\n-/\ndef germ (F : X.presheaf C) {U : opens X} (x : U) : F.obj (op U) ⟶ stalk F x :=\ncolimit.ι ((open_nhds.inclusion x.1).op ⋙ F) (op ⟨U, x.2⟩)\n\n@[simp, elementwise]\nlemma germ_res (F : X.presheaf C) {U V : opens X} (i : U ⟶ V) (x : U) :\n  F.map i.op ≫ germ F x = germ F (i x : V) :=\nlet i' : (⟨U, x.2⟩ : open_nhds x.1) ⟶ ⟨V, (i x : V).2⟩ := i in\ncolimit.w ((open_nhds.inclusion x.1).op ⋙ F) i'.op\n\n/--\nA morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its\ncomposition with the `germ` morphisms.\n-/\nlemma stalk_hom_ext (F : X.presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}\n  (ih : ∀ (U : opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=\ncolimit.hom_ext $ λ U, by { induction U using opposite.rec, cases U with U hxU, exact ih U hxU }\n\n@[simp, reassoc, elementwise]\nlemma stalk_functor_map_germ {F G : X.presheaf C} (U : opens X) (x : U)\n  (f : F ⟶ G) : germ F x ≫ (stalk_functor C x.1).map f = f.app (op U) ≫ germ G x :=\ncolimit.ι_map (whisker_left ((open_nhds.inclusion x.1).op) f) (op ⟨U, x.2⟩)\n\nvariables (C)\n\n/--\nFor a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the\nstalk of `f _ * F` at `f x` and the stalk of `F` at `x`.\n-/\ndef stalk_pushforward (f : X ⟶ Y) (F : X.presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x :=\nbegin\n  -- This is a hack; Lean doesn't like to elaborate the term written directly.\n  transitivity,\n  swap,\n  exact colimit.pre _ (open_nhds.map f x).op,\n  exact colim.map (whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) F),\nend\n\n@[simp, elementwise, reassoc]\nlemma stalk_pushforward_germ (f : X ⟶ Y) (F : X.presheaf C) (U : opens Y)\n  (x : (opens.map f).obj U) :\n  (f _* F).germ ⟨f x, x.2⟩ ≫ F.stalk_pushforward C f x = F.germ x :=\nbegin\n  rw [stalk_pushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whisker_right_app],\n  erw [category_theory.functor.map_id, category.id_comp],\n  refl,\nend\n\n-- Here are two other potential solutions, suggested by @fpvandoorn at\n-- <https://github.com/leanprover-community/mathlib/pull/1018#discussion_r283978240>\n-- However, I can't get the subsequent two proofs to work with either one.\n\n-- def stalk_pushforward (f : X ⟶ Y) (ℱ : X.presheaf C) (x : X) :\n--   (f _* ℱ).stalk (f x) ⟶ ℱ.stalk x :=\n-- colim.map ((functor.associator _ _ _).inv ≫\n--   whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) ℱ) ≫\n-- colimit.pre ((open_nhds.inclusion x).op ⋙ ℱ) (open_nhds.map f x).op\n\n-- def stalk_pushforward (f : X ⟶ Y) (ℱ : X.presheaf C) (x : X) :\n--   (f _* ℱ).stalk (f x) ⟶ ℱ.stalk x :=\n-- (colim.map (whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) ℱ) :\n--   colim.obj ((open_nhds.inclusion (f x) ⋙ opens.map f).op ⋙ ℱ) ⟶ _) ≫\n-- colimit.pre ((open_nhds.inclusion x).op ⋙ ℱ) (open_nhds.map f x).op\n\nnamespace stalk_pushforward\nlocal attribute [tidy] tactic.op_induction'\n\n@[simp] lemma id (ℱ : X.presheaf C) (x : X) :\n  ℱ.stalk_pushforward C (𝟙 X) x = (stalk_functor C x).map ((pushforward.id ℱ).hom) :=\nbegin\n  dsimp [stalk_pushforward, stalk_functor],\n  ext1,\n  tactic.op_induction',\n  cases j, cases j_val,\n  rw [colimit.ι_map_assoc, colimit.ι_map, colimit.ι_pre, whisker_left_app, whisker_right_app,\n       pushforward.id_hom_app, eq_to_hom_map, eq_to_hom_refl],\n  dsimp,\n  -- FIXME A simp lemma which unfortunately doesn't fire:\n  erw [category_theory.functor.map_id],\nend\n\n-- This proof is sadly not at all robust:\n-- having to use `erw` at all is a bad sign.\n@[simp] lemma comp (ℱ : X.presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :\n  ℱ.stalk_pushforward C (f ≫ g) x =\n  ((f _* ℱ).stalk_pushforward C g (f x)) ≫ (ℱ.stalk_pushforward C f x) :=\nbegin\n  dsimp [stalk_pushforward, stalk_functor],\n  ext U,\n  induction U using opposite.rec,\n  cases U,\n  cases U_val,\n  simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc,\n             whisker_right_app, category.assoc],\n  dsimp,\n  -- FIXME: Some of these are simp lemmas, but don't fire successfully:\n  erw [category_theory.functor.map_id, category.id_comp, category.id_comp, category.id_comp,\n       colimit.ι_pre, colimit.ι_pre],\n  refl,\nend\n\nlemma stalk_pushforward_iso_of_open_embedding {f : X ⟶ Y} (hf : open_embedding f)\n   (F : X.presheaf C) (x : X) : is_iso (F.stalk_pushforward _ f x) :=\n begin\n   haveI := functor.initial_of_adjunction (hf.is_open_map.adjunction_nhds x),\n   convert is_iso.of_iso ((functor.final.colimit_iso (hf.is_open_map.functor_nhds x).op\n     ((open_nhds.inclusion (f x)).op ⋙ f _* F) : _).symm ≪≫ colim.map_iso _),\n   swap,\n   { fapply nat_iso.of_components,\n     { intro U,\n       refine F.map_iso (eq_to_iso _),\n       dsimp only [functor.op],\n       exact congr_arg op (subtype.eq $ set.preimage_image_eq (unop U).1.1 hf.inj) },\n     { intros U V i, erw [← F.map_comp, ← F.map_comp], congr } },\n   { ext U,\n     rw ← iso.comp_inv_eq,\n     erw colimit.ι_map_assoc,\n     rw [colimit.ι_pre, category.assoc],\n     erw [colimit.ι_map_assoc, colimit.ι_pre, ← F.map_comp_assoc],\n     apply colimit.w ((open_nhds.inclusion (f x)).op ⋙ f _* F) _,\n     dsimp only [functor.op],\n     refine ((hom_of_le _).op : op (unop U) ⟶ _),\n     exact set.image_preimage_subset _ _ },\n end\n\nend stalk_pushforward\n\nsection stalk_pullback\n\n/-- The morphism `ℱ_{f x} ⟶ (f⁻¹ℱ)ₓ` that factors through `(f_*f⁻¹ℱ)_{f x}`. -/\ndef stalk_pullback_hom (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  F.stalk (f x) ⟶ (pullback_obj f F).stalk x :=\n(stalk_functor _ (f x)).map ((pushforward_pullback_adjunction C f).unit.app F) ≫\n  stalk_pushforward _ _ _ x\n\n/-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/\ndef germ_to_pullback_stalk (f : X ⟶ Y) (F : Y.presheaf C) (U : opens X) (x : U) :\n  (pullback_obj f F).obj (op U) ⟶ F.stalk (f x) :=\ncolimit.desc (Lan.diagram (opens.map f).op F (op U))\n{ X := F.stalk (f x),\n  ι := { app := λ V, F.germ ⟨f x, V.hom.unop.le x.2⟩,\n          naturality' := λ _ _ i, by { erw category.comp_id, exact F.germ_res i.left.unop _ } } }\n\n/-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/\ndef stalk_pullback_inv (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  (pullback_obj f F).stalk x ⟶ F.stalk (f x) :=\ncolimit.desc ((open_nhds.inclusion x).op ⋙ presheaf.pullback_obj f F)\n{ X := F.stalk (f x),\n  ι := { app := λ U, F.germ_to_pullback_stalk _ f (unop U).1 ⟨x, (unop U).2⟩,\n          naturality' := λ _ _ _, by { erw [colimit.pre_desc, category.comp_id], congr } } }\n\n/-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/\ndef stalk_pullback_iso (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  F.stalk (f x) ≅ (pullback_obj f F).stalk x :=\n{ hom := stalk_pullback_hom _ _ _ _,\n  inv := stalk_pullback_inv _ _ _ _,\n  hom_inv_id' :=\n  begin\n    delta stalk_pullback_hom stalk_pullback_inv stalk_functor presheaf.pullback stalk_pushforward\n      germ_to_pullback_stalk germ,\n    ext j,\n    induction j using opposite.rec,\n    cases j,\n    simp only [topological_space.open_nhds.inclusion_map_iso_inv, whisker_right_app,\n      whisker_left_app, whiskering_left_obj_map, functor.comp_map, colimit.ι_map_assoc,\n      nat_trans.op_id, Lan_obj_map, pushforward_pullback_adjunction_unit_app_app, category.assoc,\n      colimit.ι_pre_assoc],\n    erw [colimit.ι_desc, colimit.pre_desc, colimit.ι_desc, category.comp_id],\n    simpa\n  end,\n  inv_hom_id' :=\n  begin\n    delta stalk_pullback_hom stalk_pullback_inv stalk_functor presheaf.pullback stalk_pushforward,\n    ext U j,\n    induction U using opposite.rec,\n    cases U, cases j, cases j_right,\n    erw [colimit.map_desc, colimit.map_desc, colimit.ι_desc_assoc,\n      colimit.ι_desc_assoc, colimit.ι_desc, category.comp_id],\n    simp only [cocone.whisker_ι, colimit.cocone_ι, open_nhds.inclusion_map_iso_inv,\n      cocones.precompose_obj_ι, whisker_right_app, whisker_left_app, nat_trans.comp_app,\n      whiskering_left_obj_map, nat_trans.op_id, Lan_obj_map,\n      pushforward_pullback_adjunction_unit_app_app],\n    erw ←colimit.w _\n      (@hom_of_le (open_nhds x) _\n         ⟨_, U_property⟩ ⟨(opens.map f).obj (unop j_left), j_hom.unop.le U_property⟩\n         j_hom.unop.le).op,\n    erw colimit.ι_pre_assoc (Lan.diagram _ F _) (costructured_arrow.map _),\n    erw colimit.ι_pre_assoc (Lan.diagram _ F _) (costructured_arrow.map _),\n    congr,\n    simp only [category.assoc, costructured_arrow.map_mk],\n    delta costructured_arrow.mk,\n    congr\n  end }\n\nend stalk_pullback\n\nsection stalk_specializes\n\nvariables {C}\n\n/-- If `x` specializes to `y`, then there is a natural map `F.stalk y ⟶ F.stalk x`. -/\nnoncomputable\ndef stalk_specializes (F : X.presheaf C) {x y : X} (h : x ⤳ y) : F.stalk y ⟶ F.stalk x :=\nbegin\n  refine colimit.desc _ ⟨_,λ U, _,_⟩,\n  { exact colimit.ι ((open_nhds.inclusion x).op ⋙ F)\n      (op ⟨(unop U).1, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩) },\n  { intros U V i,\n    dsimp,\n    rw category.comp_id,\n    let U' : open_nhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop U).1.2 (unop U).2 : _)⟩,\n    let V' : open_nhds x := ⟨_, (specializes_iff_forall_open.mp h _ (unop V).1.2 (unop V).2 : _)⟩,\n    exact colimit.w ((open_nhds.inclusion x).op ⋙ F) (show V' ⟶ U', from i.unop).op }\nend\n\n@[simp, reassoc, elementwise]\nlemma germ_stalk_specializes (F : X.presheaf C) {U : opens X} {y : U} {x : X} (h : x ⤳ y) :\n  F.germ y ≫ F.stalk_specializes h =\n    F.germ ⟨x, specializes_iff_forall_open.mp h _ U.2 y.prop⟩ := colimit.ι_desc _ _\n\n@[simp, reassoc, elementwise]\nlemma germ_stalk_specializes' (F : X.presheaf C) {U : opens X} {x y : X} (h : x ⤳ y) (hy : y ∈ U) :\n  F.germ ⟨y, hy⟩ ≫ F.stalk_specializes h =\n    F.germ ⟨x, specializes_iff_forall_open.mp h _ U.2 hy⟩ := colimit.ι_desc _ _\n\n@[simp, reassoc, elementwise]\nlemma stalk_specializes_stalk_functor_map {F G : X.presheaf C} (f : F ⟶ G) {x y : X} (h : x ⤳ y) :\n  F.stalk_specializes h ≫ (stalk_functor C x).map f =\n    (stalk_functor C y).map f ≫ G.stalk_specializes h :=\nby { ext, delta stalk_functor, simpa [stalk_specializes] }\n\n@[simp, reassoc, elementwise]\nlemma stalk_specializes_stalk_pushforward (f : X ⟶ Y) (F : X.presheaf C) {x y : X} (h : x ⤳ y) :\n  (f _* F).stalk_specializes (f.map_specialization h) ≫ F.stalk_pushforward _ f x =\n    F.stalk_pushforward _ f y ≫ F.stalk_specializes h :=\nby { ext, delta stalk_pushforward, simpa [stalk_specializes] }\n\nend stalk_specializes\n\nsection concrete\n\nvariables {C}\nvariables [concrete_category.{v} C]\n\nlocal attribute [instance] concrete_category.has_coe_to_sort concrete_category.has_coe_to_fun\n\n@[ext]\nlemma germ_ext (F : X.presheaf C) {U V : opens X} {x : X} {hxU : x ∈ U} {hxV : x ∈ V}\n  (W : opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V) {sU : F.obj (op U)} {sV : F.obj (op V)}\n  (ih : F.map iWU.op sU = F.map iWV.op sV) :\n  F.germ ⟨x, hxU⟩ sU = F.germ ⟨x, hxV⟩ sV :=\nby erw [← F.germ_res iWU ⟨x, hxW⟩,\n    ← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih]\n\nvariables [preserves_filtered_colimits (forget C)]\n\n/--\nFor presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,\nevery element of the stalk is the germ of a section.\n-/\nlemma germ_exist (F : X.presheaf C) (x : X) (t : stalk F x) :\n  ∃ (U : opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t :=\nbegin\n  obtain ⟨U, s, e⟩ := types.jointly_surjective _\n    (is_colimit_of_preserves (forget C) (colimit.is_colimit _)) t,\n  revert s e,\n  rw [(show U = op (unop U), from rfl)],\n  generalize : unop U = V, clear U,\n  cases V with V m,\n  intros s e,\n  exact ⟨V, m, s, e⟩,\nend\n\nlemma germ_eq (F : X.presheaf C) {U V : opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)\n  (s : F.obj (op U)) (t : F.obj (op V))\n  (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :\n  ∃ (W : opens X) (m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t :=\nbegin\n  obtain ⟨W, iU, iV, e⟩ := (types.filtered_colimit.is_colimit_eq_iff _\n    (is_colimit_of_preserves _ (colimit.is_colimit ((open_nhds.inclusion x).op ⋙ F)))).mp h,\n  exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩,\nend\n\nlemma stalk_functor_map_injective_of_app_injective {F G : presheaf C X} (f : F ⟶ G)\n  (h : ∀ U : opens X, function.injective (f.app (op U))) (x : X) :\n  function.injective ((stalk_functor C x).map f) := λ s t hst,\nbegin\n  rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩,\n  rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩,\n  simp only [stalk_functor_map_germ_apply _ ⟨x,_⟩] at hst,\n  obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst,\n  rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_apply, comp_apply] at heq,\n  replace heq := h W heq,\n  convert congr_arg (F.germ ⟨x,hxW⟩) heq,\n  exacts [(F.germ_res_apply iWU₁ ⟨x,hxW⟩ s).symm,\n          (F.germ_res_apply iWU₂ ⟨x,hxW⟩ t).symm],\nend\n\n\nvariables [has_limits C] [preserves_limits (forget C)] [reflects_isomorphisms (forget C)]\n\n/--\nLet `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,\npreserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.\n-/\nlemma section_ext (F : sheaf C X) (U : opens X) (s t : F.1.obj (op U))\n  (h : ∀ x : U, F.1.germ x s = F.1.germ x t) :\n  s = t :=\nbegin\n  -- We use `germ_eq` and the axiom of choice, to pick for every point `x` a neighbourhood\n  -- `V x`, such that the restrictions of `s` and `t` to `V x` coincide.\n  choose V m i₁ i₂ heq using λ x : U, F.1.germ_eq x.1 x.2 x.2 s t (h x),\n  -- Since `F` is a sheaf, we can prove the equality locally, if we can show that these\n  -- neighborhoods form a cover of `U`.\n  apply F.eq_of_locally_eq' V U i₁,\n  { intros x hxU,\n    rw [opens.mem_coe, opens.mem_supr],\n    exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ },\n  { intro x,\n    rw [heq, subsingleton.elim (i₁ x) (i₂ x)] }\nend\n\n/-\nNote that the analogous statement for surjectivity is false: Surjectivity on stalks does not\nimply surjectivity of the components of a sheaf morphism. However it does imply that the morphism\nis an epi, but this fact is not yet formalized.\n-/\nlemma app_injective_of_stalk_functor_map_injective {F : sheaf C X} {G : presheaf C X}\n  (f : F.1 ⟶ G) (U : opens X) (h : ∀ x : U, function.injective ((stalk_functor C x.val).map f)) :\n  function.injective (f.app (op U)) :=\nλ s t hst, section_ext F _ _ _ $ λ x, h x $ by\n  rw [stalk_functor_map_germ_apply, stalk_functor_map_germ_apply, hst]\n\nlemma app_injective_iff_stalk_functor_map_injective {F : sheaf C X}\n  {G : presheaf C X} (f : F.1 ⟶ G) :\n  (∀ x : X, function.injective ((stalk_functor C x).map f)) ↔\n  (∀ U : opens X, function.injective (f.app (op U))) :=\n⟨λ h U, app_injective_of_stalk_functor_map_injective f U (λ x, h x.1),\n  stalk_functor_map_injective_of_app_injective f⟩\n\n/-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.\nWe claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct\na neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`\nagree on `V`. -/\nlemma app_surjective_of_injective_of_locally_surjective {F G : sheaf C X} (f : F ⟶ G)\n  (U : opens X) (hinj : ∀ x : U, function.injective ((stalk_functor C x.1).map f))\n  (hsurj : ∀ (t) (x : U), ∃ (V : opens X) (m : x.1 ∈ V) (iVU : V ⟶ U) (s : F.1.obj (op V)),\n    f.app (op V) s = G.1.map iVU.op t) :\n  function.surjective (f.app (op U)) :=\nbegin\n  intro t,\n  -- We use the axiom of choice to pick around each point `x` an open neighborhood `V` and a\n  -- preimage under `f` on `V`.\n  choose V mV iVU sf heq using hsurj t,\n  -- These neighborhoods clearly cover all of `U`.\n  have V_cover : U ≤ supr V,\n  { intros x hxU,\n    rw [opens.mem_coe, opens.mem_supr],\n    exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ },\n  -- Since `F` is a sheaf, we can glue all the local preimages together to get a global preimage.\n  obtain ⟨s, s_spec, -⟩ := F.exists_unique_gluing' V U iVU V_cover sf _,\n  { use s,\n    apply G.eq_of_locally_eq' V U iVU V_cover,\n    intro x,\n    rw [← comp_apply, ← f.naturality, comp_apply, s_spec, heq] },\n  { intros x y,\n    -- What's left to show here is that the secions `sf` are compatible, i.e. they agree on\n    -- the intersections `V x ⊓ V y`. We prove this by showing that all germs are equal.\n    apply section_ext,\n    intro z,\n    -- Here, we need to use injectivity of the stalk maps.\n    apply (hinj ⟨z, (iVU x).le ((inf_le_left : V x ⊓ V y ≤ V x) z.2)⟩),\n    dsimp only,\n    erw [stalk_functor_map_germ_apply, stalk_functor_map_germ_apply],\n    simp_rw [← comp_apply, f.naturality, comp_apply, heq, ← comp_apply, ← G.1.map_comp],\n    refl }\nend\n\nlemma app_surjective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)\n  (U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f)) :\n  function.surjective (f.app (op U)) :=\nbegin\n  refine app_surjective_of_injective_of_locally_surjective f U (λ x, (h x).1) (λ t x, _),\n  -- Now we need to prove our initial claim: That we can find preimages of `t` locally.\n  -- Since `f` is surjective on stalks, we can find a preimage `s₀` of the germ of `t` at `x`\n  obtain ⟨s₀,hs₀⟩ := (h x).2 (G.1.germ x t),\n  -- ... and this preimage must come from some section `s₁` defined on some open neighborhood `V₁`\n  obtain ⟨V₁,hxV₁,s₁,hs₁⟩ := F.1.germ_exist x.1 s₀,\n  subst hs₁, rename hs₀ hs₁,\n  erw stalk_functor_map_germ_apply V₁ ⟨x.1,hxV₁⟩ f s₁ at hs₁,\n  -- Now, the germ of `f.app (op V₁) s₁` equals the germ of `t`, hence they must coincide on\n  -- some open neighborhood `V₂`.\n  obtain ⟨V₂, hxV₂, iV₂V₁, iV₂U, heq⟩ := G.1.germ_eq x.1 hxV₁ x.2 _ _ hs₁,\n  -- The restriction of `s₁` to that neighborhood is our desired local preimage.\n  use [V₂, hxV₂, iV₂U, F.1.map iV₂V₁.op s₁],\n  rw [← comp_apply, f.naturality, comp_apply, heq],\nend\n\nlemma app_bijective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)\n   (U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f)) :\n  function.bijective (f.app (op U)) :=\n⟨app_injective_of_stalk_functor_map_injective f U (λ x, (h x).1),\n  app_surjective_of_stalk_functor_map_bijective f U h⟩\n\nlemma app_is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) (U : opens X)\n  [∀ x : U, is_iso ((stalk_functor C x.val).map f)] : is_iso (f.app (op U)) :=\nbegin\n  -- Since the forgetful functor of `C` reflects isomorphisms, it suffices to see that the\n  -- underlying map between types is an isomorphism, i.e. bijective.\n  suffices : is_iso ((forget C).map (f.app (op U))),\n  { exactI is_iso_of_reflects_iso (f.app (op U)) (forget C) },\n  rw is_iso_iff_bijective,\n  apply app_bijective_of_stalk_functor_map_bijective,\n  intro x,\n  apply (is_iso_iff_bijective _).mp,\n  exact functor.map_is_iso (forget C) ((stalk_functor C x.1).map f)\nend\n\n/--\nLet `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects\nisomorphisms, preserves limits and filtered colimits. Then if the stalk maps of a morphism\n`f : F ⟶ G` are all isomorphisms, `f` must be an isomorphism.\n-/\n-- Making this an instance would cause a loop in typeclass resolution with `functor.map_is_iso`\nlemma is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G)\n  [∀ x : X, is_iso ((stalk_functor C x).map f)] : is_iso f :=\nbegin\n  -- Since the inclusion functor from sheaves to presheaves is fully faithful, it suffices to\n  -- show that `f`, as a morphism between _presheaves_, is an isomorphism.\n  suffices : is_iso ((sheaf.forget C X).map f),\n  { exactI is_iso_of_fully_faithful (sheaf.forget C X) f },\n  -- We show that all components of `f` are isomorphisms.\n  suffices : ∀ U : (opens X)ᵒᵖ, is_iso (f.app U),\n  { exact @nat_iso.is_iso_of_is_iso_app _ _ _ _ F.1 G.1 f this, },\n  intro U, induction U using opposite.rec,\n  apply app_is_iso_of_stalk_functor_map_iso\nend\n\n/--\nLet `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects\nisomorphisms, preserves limits and filtered colimits. Then a morphism `f : F ⟶ G` is an\nisomorphism if and only if all of its stalk maps are isomorphisms.\n-/\nlemma is_iso_iff_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) :\n  is_iso f ↔ ∀ x : X, is_iso ((stalk_functor C x).map f) :=\nbegin\n  split,\n  { intros h x, resetI,\n    exact @functor.map_is_iso _ _ _ _ _ _ (stalk_functor C x) f\n      ((sheaf.forget C X).map_is_iso f) },\n  { intro h,\n    exactI is_iso_of_stalk_functor_map_iso f }\nend\n\nend concrete\n\ninstance (F : X.presheaf CommRing) {U : opens X} (x : U) :\n  algebra (F.obj $ op U) (F.stalk x) :=\n(F.germ x).to_algebra\n\n@[simp]\nlemma stalk_open_algebra_map {X : Top} (F : X.presheaf CommRing) {U : opens X} (x : U) :\n  algebra_map (F.obj $ op U) (F.stalk x) = F.germ x := rfl\n\nend Top.presheaf\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/topology/sheaves/stalks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.33178064131736634}}
{"text": "import data.mv_polynomial ring_theory.ideal_operations ring_theory.adjoin_root\nimport ring_theory.integral_closure\n\nnamespace adjoin_root\nvariables {α : Type*} [comm_ring α] {f : polynomial α}\nopen polynomial ideal\n\nlemma coe_injective_of_degree_pos_of_monic (hf : 0 < degree f) (hm : monic f) :\n  function.injective (coe : α → adjoin_root f) :=\nλ x y hxy,\nbegin\n  rcases mem_span_singleton.1 (ideal.quotient.eq.1 hxy) with ⟨g, hg⟩,\n  classical,\n  by_cases hg0 : g = 0,\n  { rwa [hg0, mul_zero, sub_eq_zero, C_inj] at hg },\n  { have hlc : leading_coeff f * leading_coeff g ≠ 0,\n    { rwa [monic.def.1 hm, one_mul, ne.def, leading_coeff_eq_zero] },\n    exact false.elim (not_le_of_gt hf $\n      calc degree f = nat_degree f : degree_eq_nat_degree (λ hf0, by simp * at *)\n      ... ≤ nat_degree f + nat_degree g : with_bot.coe_le_coe.2 (nat.le_add_right _ _)\n      ... = 0 : by rw [← with_bot.coe_add, ← nat_degree_mul_eq' hlc, ← hg, ← C_sub, nat_degree_C,\n        with_bot.coe_zero]) }\nend\nend adjoin_root\n#print ideal.leading_coeff\nnamespace ideal\nvariables {R : Type*} [comm_ring R]\n\n@[simp] lemma span_empty : (span ∅ : ideal R) = ⊥ :=\nsubmodule.span_empty\n\nlemma quotient.mk_eq_zero {I : ideal R} {x : R} : ideal.quotient.mk I x = 0 ↔ x ∈ I :=\nsubmodule.quotient.mk_eq_zero _\n\nlemma ext_iff {I J : ideal R} : I = J ↔ (∀ x, x ∈ I ↔ x ∈ J) :=\n⟨λ h _, h ▸ iff.rfl, ideal.ext⟩\n\nend ideal\n\nnamespace mv_polynomial\n\nlemma C_inj {R σ : Type*} [comm_semiring R] {x y : R} : (C x : mv_polynomial σ R) = C y ↔ x = y :=\nsorry\n\nnoncomputable instance {R σ : Type*} [nonzero_comm_semiring R] :\n  nonzero_comm_semiring (mv_polynomial σ R) :=\n{ zero_ne_one := sorry,\n  ..show comm_semiring (mv_polynomial σ R), by apply_instance }\n\n\nend mv_polynomial\n\nnamespace polynomial\n\nopen ideal set\nopen_locale classical\nend polynomial\n\nopen mv_polynomial ideal\nuniverses u v\nvariables (K : Type u) [discrete_field K]\nnoncomputable theory\n\nsection name\nvariables {α : Type*} {β : Type*} {σ : Type*} [comm_ring α] [comm_ring β]\n\ndef mname (i : σ) (p : polynomial α) : mv_polynomial σ α :=\np.eval₂ C (X i)\n\n@[simp] lemma mname_one (i : σ)  :\n  mname i (1 : polynomial α) = 1 :=\nby simp [mname]\n\n@[simp] lemma mname_zero (i : σ)  :\n  mname i (0 : polynomial α) = 0 :=\nby simp [mname]\n\n@[simp] lemma mname_add (i : σ) (p q : polynomial α) :\n  mname i (p + q) =  mname i p + mname i q :=\nby simp [mname]\n\n@[simp] lemma mname_mul (i : σ) (p q : polynomial α) :\n  mname i (p * q) =  mname i p * mname i q :=\nby simp [mname]\n\n@[simp] lemma mname_pow (i : σ) (p : polynomial α) (n : ℕ) :\n  mname i (p ^ n) =  mname i p ^ n :=\nby induction n; simp [pow_succ, *]\n\n@[simp] lemma mname_sub (i : σ) (p q : polynomial α) :\n  mname i (p - q) =  mname i p - mname i q :=\nby simp [mname]\n\n@[simp] lemma mname_neg (i : σ) (p : polynomial α) :\n  mname i (-p) = -mname i p :=\nby simp [mname]\n\n@[simp] lemma mname_C (i : σ) (a : α) :\n  mname i (polynomial.C a) = C a :=\nby simp [mname]\n\n@[simp] lemma mname_X (i : σ) :\n  mname i (polynomial.X : polynomial α) = X i :=\nby simp [mname]\n\n@[simp] lemma eval₂_mname (i : σ) (f : α → β) [is_semiring_hom f] (s : σ → β) (p : polynomial α) :\n  eval₂ f s (mname i p) = polynomial.eval₂ f (s i) p :=\npolynomial.induction_on p\n  (λ a, by simp)\n  (by simp {contextual := tt})\n  (by simp [pow_succ', (mul_assoc _ _ _).symm] {contextual := tt})\n\nend name\n\n\n@[reducible] def posd : Type u := {p : polynomial K // 0 < p.degree ∧ p.monic}\n\n@[reducible] def pac : Type u :=\nmv_polynomial (posd K) K\n\nvariable {K}\n\nnamespace pac\nvariable (K)\ndef relt (s : set (posd K)) : set (pac K) :=\n(λ p : posd K, p.1.eval₂ C (X p)) '' s\n\nend pac\nopen pac\n\nset_option class.instance_max_depth 50\n\n@[instance, priority 100000] instance :\n  comm_ring (pac K) := mv_polynomial.comm_ring\n\ndef F1 (s : set (posd K)) : K →+* ideal.quotient (span (relt K s)) :=\n(ring_hom.of (ideal.quotient.mk (span (relt K s)))).comp (ring_hom.of (C : K → pac K))\n\ndef F2 (s : set (posd K)) (p : posd K) : K →+* adjoin_root (p.1.map (F1 s)) :=\n(ring_hom.of coe).comp $ (ring_hom.of $ ideal.quotient.mk (span (relt K s))).comp $\n  ring_hom.of (C : K → pac K)\n\nopen_locale classical\n\nlemma isom2 (s : set (posd K)) (p : posd K) (hps : p ∉ s) :\n  ideal.quotient (span (relt K (insert p s))) →+* adjoin_root (p.1.map (F1 s)) :=\nring_hom.of $ ideal.quotient.lift _\n  (eval₂ (F2 s p) (λ q, if q ∈ s\n    then (↑(ideal.quotient.mk (span (relt K s)) (X q : pac K)) : adjoin_root _)\n      else if q = p then adjoin_root.root\n      else 0))\n  begin\n    assume x hx,\n    refine submodule.span_induction hx _ (by simp) (by simp {contextual := tt})\n      (by simp {contextual := tt}),\n    assume x hx,\n    rcases hx with ⟨q, hq, rfl⟩,\n    { rcases hq with rfl | hq,\n      { dsimp only [F2, F1],\n        rw [← mname, ← adjoin_root.eval₂_root, eval₂_mname, if_pos rfl,\n          if_neg hps, polynomial.eval₂_map],\n        refl },\n      { dsimp only,\n        have : ideal.quotient.mk (span (relt K s)) (mname q q) = 0,\n        { rw [ideal.quotient.mk_eq_zero],\n          exact subset_span (set.mem_image_of_mem _ hq) },\n        erw [← mname, eval₂_mname, if_pos hq,\n          ← is_ring_hom.map_zero (coe : ideal.quotient (span (relt K s)) →\n            adjoin_root (p.1.map (F1 s))), ← this, mname, F2],\n        exact eq.symm (polynomial.hom_eval₂ (q : polynomial K) _\n          (ring_hom.comp (ring_hom.of (coe : ideal.quotient (span (relt K s)) →\n            adjoin_root (p.1.map (F1 s))))\n          (ring_hom.of (ideal.quotient.mk (span (relt K s))))) (X q)) } }\n  end\n\nopen polynomial\n\nlemma map1 (s : set (posd K)) (p : posd K)\n  (h01' : (0 :  ideal.quotient (span (relt K s))) ≠ 1) :\n  (0 : adjoin_root (p.1.map (F1 s))) ≠ 1 :=\nλ h01,\n  by rw [← is_ring_hom.map_zero (coe : ideal.quotient (span (relt K s)) →\n    adjoin_root (p.1.map (F1 s))),\n   ← is_ring_hom.map_one (coe : ideal.quotient (span (relt K s)) →\n    adjoin_root (p.1.map (F1 s)))] at h01; exact\nh01' (adjoin_root.coe_injective_of_degree_pos_of_monic\n  begin\n    refine lt_of_not_ge' _,\n    rw [degree_le_zero_iff],\n    assume h,\n    have : (p.1.map (F1 s)).coeff p.1.nat_degree =\n      (polynomial.C ((p.1.map (F1 s)).coeff 0)).coeff p.1.nat_degree, { rw ← h },\n    have h0 : nat_degree p.1 ≠ 0,\n    { rw [← nat.pos_iff_ne_zero, ← with_bot.coe_lt_coe],\n      exact lt_of_lt_of_le p.2.1 degree_le_nat_degree },\n    rw [polynomial.coeff_C, polynomial.coeff_map, ← polynomial.leading_coeff,\n      monic.def.1 p.2.2, ring_hom.map_one, if_neg h0] at this,\n    exact h01'.symm this\n  end\n  (monic_map _ p.2.2)\n  h01).\n\nlemma ne_top_aux (s : finset (posd K)) : span (relt K ↑s) ≠ ⊤ :=\nfinset.induction_on s\n  (by simp [ne_top_iff_one, relt, ideal.ext_iff, classical.not_forall]; exact ⟨1, by simp⟩)\n  (λ p s hps ih,\n    have (0 : ideal.quotient (span (relt K ↑s))) ≠ 1,\n      from ne.symm $ mt (ideal.quotient.mk_eq_zero).1 ((ne_top_iff_one _).1 ih),\n    have (0 :  adjoin_root (p.1.map (F1 (↑s : set (posd K))))) ≠ 1,\n      from map1 ↑s p this,\n    have (0 : ideal.quotient (span (relt K (insert p ↑s)))) ≠ 1,\n      from λ h, this $ by rw [← ring_hom.map_zero (isom2 ↑s p hps), h, ring_hom.map_one],\n    by rw [finset.coe_insert, ne_top_iff_one, ← ideal.quotient.mk_eq_zero, eq_comm];\n      exact this)\n\nlemma ne_top : span (relt K ⊤) ≠ ⊤ :=\n(ne_top_iff_one _).2 $ λ h : 1 ∈ span (relt K ⊤),\nhave hs : ∃ s : finset (posd K), (1 : pac K) ∈ span (relt K ↑s),\n  from submodule.span_induction h\n    (by rintros x ⟨p, hp, rfl⟩; exact ⟨{p}, subset_span (by simp [relt])⟩)\n    ⟨∅, by simp⟩\n    (by rintros x y ⟨s, hs⟩ ⟨t, ht⟩;\n      exact ⟨s ∪ t, ideal.add_mem _ (span_mono (by simp [relt, set.image_union]) hs)\n                                    (span_mono (by simp [relt, set.image_union]) ht)⟩)\n    (by rintros a x ⟨s, hs⟩; exact ⟨s, ideal.mul_mem_left _ hs⟩),\nby cases hs with s hs;\n  exact ne_top_aux s ((not_iff_not.1 (ne_top_iff_one _)).2 hs)\n\nvariable (K)\ndef big_ideal : ideal (pac K) :=\nclassical.some (exists_le_maximal (span (relt K ⊤)) ne_top)\n\ndef K1 : Type u :=\nideal.quotient (@big_ideal K _)\n\ninstance : discrete_field (K1 K) :=\n@ideal.quotient.field _ _ _\n  (classical.some_spec (exists_le_maximal (span (relt K ⊤)) ne_top)).1\n\nvariables {K}\n\ndef of : K → K1 K := ideal.quotient.mk _ ∘ mv_polynomial.C\n\ninstance of.is_ring_hom : is_ring_hom (@of K _) := is_ring_hom.comp _ _\n\ninstance : algebra K (K1 K) := algebra.of_ring_hom of of.is_ring_hom\n\nlemma algebra_map_eq_of : algebra_map (K1 K) = of := rfl\n\nlemma aeval_posd {f : posd K} : (polynomial.aeval K (K1 K)\n  (ideal.quotient.mk (big_ideal K) (X f)) :\n    polynomial K → K1 K) f.1 = 0 :=\nbegin\n  rw [polynomial.aeval_def, algebra_map_eq_of, of,\n    ← hom_eval₂ _ mv_polynomial.C (ideal.quotient.mk (big_ideal K)), ← mname],\n  exact ideal.quotient.mk_eq_zero.2\n    ((classical.some_spec (exists_le_maximal (span (relt K ⊤)) ne_top)).2\n      (subset_span (set.mem_image_of_mem _ (set.mem_univ _))))\nend\n\nlemma is_integral_aux {f : posd K} : @is_integral K (K1 K) _ _ _\n  (ideal.quotient.mk _ (X f) : K1 K) :=\n⟨f, f.2.2, aeval_posd⟩\n\nlemma mem_integral_closure (x : K1 K) : is_integral K x :=\nquotient.induction_on' x $ λ x,\n  show ideal.quotient.mk _ x ∈ integral_closure K (K1 K),\n  from mv_polynomial.induction_on x\n    (λ k, is_integral_algebra_map)\n    sorry -- subalgebra.add_mem\n    (λ x p, sorry)\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/algebraic_closure_experiment.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6791787121629465, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3316316875875759}}
{"text": "import category_theory.base\nimport category_theory.eq_to_hom\nimport category_theory.groupoid\nimport category_theory.full_subcategory\n\nuniverses v x z u u' w w' y y'\n\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace category_theory\n\n-- TODO: merge with category_theory.full_subcategory\n\nlocal attribute [simps] induced_category.category\n\nvariables {C : Type u} {C' : Type u'} (k : C' → C)\nvariables {D : Type w} {D' : Type w'} (l : D' → D)\n\ndef induced_functor' {catC : category.{v} C} {catD : category.{x} D} (F : C ↝ D)\n  (F' : C' → D') (e : ∀ a, F.obj (k a) = l (F' a)) :\n  induced_category C k ↝ induced_category D l :=\n{ obj := F',\n  map := λ X Y f,\n    show l (F' X) ⟶ l (F' Y), from\n    eq_to_hom (e Y) ∘ (F &> f) ∘ eq_to_hom (e X).symm,\n  map_id' := λ X, by dsimp; rw F.map_id; simp,\n  map_comp' := λ X Y Z f g, by dsimp; rw F.map_comp; simp }\n\nlemma induced_functor_id [catC : category.{v} C] :\n  induced_functor' k k (functor.id C) id (λ a, rfl) =\n  functor.id (induced_category C k) :=\nbegin\n  fapply functor.ext,\n  { intro a, refl },\n  { intros a b f, dsimp [induced_functor'], simp }\nend\n\nvariables {E : Type y} {E' : Type y'} (m : E' → E)\nlemma induced_functor_comp [catC : category.{v} C]\n  [catD : category.{x} D] [catE : category.{z} E]\n  {F : C ↝ D} {F' : C' → D'} (eF : ∀ a, F.obj (k a) = l (F' a))\n  {G : D ↝ E} {G' : D' → E'} (eG : ∀ a, G.obj (l a) = m (G' a)) :\n  induced_functor' k m (F.comp G) (function.comp G' F')\n    (by intro a; change G.obj (F.obj (k a)) = _; rw [eF, eG]) =\n    induced_functor' k l F F' eF ⋙ induced_functor' l m G G' eG :=\nbegin\n  fapply functor.ext,\n  { intro a, refl },\n  { intros a b f,\n    -- This proof mysteriously broke\n    dsimp [induced_functor'],\n    rw category.id_comp,\n    erw category.comp_id,\n    -- FIXME: Why was this lemma removed from simp?\n    simp [eq_to_hom_map], refl }\nend\n\nend category_theory\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/category_theory/induced.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.33153982123410414}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.preadditive\nimport category_theory.preadditive.additive_functor\nimport data.equiv.transfer_instance\n\n/-!\n# If `C` is preadditive, `Cᵒᵖ` has a natural preadditive structure.\n\n-/\n\nopen opposite\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [preadditive C]\n\ninstance : preadditive Cᵒᵖ :=\n{ hom_group := λ X Y, equiv.add_comm_group (op_equiv X Y),\n  add_comp' := λ X Y Z f f' g,\n    congr_arg quiver.hom.op (preadditive.comp_add _ _ _ g.unop f.unop f'.unop),\n  comp_add' := λ X Y Z f g g',\n    congr_arg quiver.hom.op (preadditive.add_comp _ _ _ g.unop g'.unop f.unop), }\n\ninstance module_End_left {X : Cᵒᵖ} {Y : C} : module (End X) (unop X ⟶ Y) :=\n{ smul_add := λ r f g, preadditive.comp_add _ _ _ _ _ _,\n  smul_zero := λ r, limits.comp_zero,\n  add_smul := λ r s f, preadditive.add_comp _ _ _ _ _ _,\n  zero_smul := λ f, limits.zero_comp }\n\n@[simp] lemma unop_zero (X Y : Cᵒᵖ) : (0 : X ⟶ Y).unop = 0 := rfl\n@[simp] lemma unop_add {X Y : Cᵒᵖ} (f g : X ⟶ Y) : (f + g).unop = f.unop + g.unop := rfl\n@[simp] lemma op_zero (X Y : C) : (0 : X ⟶ Y).op = 0 := rfl\n@[simp] \n\nvariables {C} {D : Type*} [category D] [preadditive D]\n\ninstance functor.op_additive (F : C ⥤ D) [F.additive] : F.op.additive := {}\n\ninstance functor.right_op_additive (F : Cᵒᵖ ⥤ D) [F.additive] : F.right_op.additive := {}\n\ninstance functor.left_op_additive (F : C ⥤ Dᵒᵖ) [F.additive] : F.left_op.additive := {}\n\ninstance functor.unop_additive (F : Cᵒᵖ ⥤ Dᵒᵖ) [F.additive] : F.unop.additive := {}\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/preadditive/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.33153982123410414}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.preadditive\nimport category_theory.preadditive.additive_functor\nimport logic.equiv.transfer_instance\n\n/-!\n# If `C` is preadditive, `Cᵒᵖ` has a natural preadditive structure.\n\n-/\n\nopen opposite\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [preadditive C]\n\ninstance : preadditive Cᵒᵖ :=\n{ hom_group := λ X Y, equiv.add_comm_group (op_equiv X Y),\n  add_comp' := λ X Y Z f f' g,\n    congr_arg quiver.hom.op (preadditive.comp_add _ _ _ g.unop f.unop f'.unop),\n  comp_add' := λ X Y Z f g g',\n    congr_arg quiver.hom.op (preadditive.add_comp _ _ _ g.unop g'.unop f.unop), }\n\ninstance module_End_left {X : Cᵒᵖ} {Y : C} : module (End X) (unop X ⟶ Y) :=\n{ smul_add := λ r f g, preadditive.comp_add _ _ _ _ _ _,\n  smul_zero := λ r, limits.comp_zero,\n  add_smul := λ r s f, preadditive.add_comp _ _ _ _ _ _,\n  zero_smul := λ f, limits.zero_comp }\n\n@[simp] lemma unop_zero (X Y : Cᵒᵖ) : (0 : X ⟶ Y).unop = 0 := rfl\n@[simp] lemma unop_add {X Y : Cᵒᵖ} (f g : X ⟶ Y) : (f + g).unop = f.unop + g.unop := rfl\n@[simp] lemma op_zero (X Y : C) : (0 : X ⟶ Y).op = 0 := rfl\n@[simp] \n\nvariables {C} {D : Type*} [category D] [preadditive D]\n\ninstance functor.op_additive (F : C ⥤ D) [F.additive] : F.op.additive := {}\n\ninstance functor.right_op_additive (F : Cᵒᵖ ⥤ D) [F.additive] : F.right_op.additive := {}\n\ninstance functor.left_op_additive (F : C ⥤ Dᵒᵖ) [F.additive] : F.left_op.additive := {}\n\ninstance functor.unop_additive (F : Cᵒᵖ ⥤ Dᵒᵖ) [F.additive] : F.unop.additive := {}\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/preadditive/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3315398139898095}}
{"text": "import category_theory.isomorphism\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.reflexive\nimport category_theory.limits.opposites\nimport category_theory.closed.cartesian\nimport category_theory.adjunction.basic\nimport category_theory.functor.epi_mono\nimport category_theory.monad.basic\nimport category_theory.monad.monadicity\n\nimport subobject_classifier\nimport adjunction\nimport image\n\nopen category_theory category_theory.category category_theory.limits category_theory.cartesian_closed classifier opposite \n\n\n/-!\n# Topos\n\nBasic definitions of a topos\nFrom Sheaves IV\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nvariables (C : Type u) [category.{v} C] \n\nclass topos :=\n[lim : has_finite_limits.{v} C]\n[sub : has_subobject_classifier.{v} C]\n[cc : cartesian_closed.{v} C]\n\nattribute [instance] topos.lim topos.sub topos.cc\n\n\nvariables [topos C]\n\n-- It doesn't seem to bee infered automatically\ninstance : has_finite_colimits Cᵒᵖ := has_finite_colimits_opposite\n\nvariables {C} (c : C)\n\ndef sub_to_hom : subobject c → (c ⟶ Ω C) := λ s, classifier_of s.arrow\ndef hom_to_sub : (c ⟶ Ω C) → subobject c := λ σ, canonical_sub σ\n\nlemma sub_equiv_hom : subobject c ≃ (c ⟶ Ω C) := \n{ to_fun := sub_to_hom c,\n  inv_fun := hom_to_sub c,\n  left_inv := λ S, sub_eq_canonical_sub_of_classifier S,  \n  right_inv := λ σ, classifier.uniquely _ _ \n                   (classifying_pullback.mk _ (is_pullback_canonical_arrow _)) }\n\ndef δ := classifier_of (diag c)\ndef singleton_map := curry (δ c)\n\nvariables (C) {a b : C}\n\ndef P : Cᵒᵖ ⥤ C := {\n  obj := λ c, (exp c.unop).obj (Ω C),\n  map := λ c d f, (pre f.unop).app (Ω C),\n  map_id' := λ c, by {rw [unop_id, pre_id, nat_trans.id_app]},\n  map_comp' := λ _ _ _ f g, by {rw [unop_comp, pre_map, nat_trans.comp_app]} }\n\ndef P_op : C ⥤ Cᵒᵖ := functor.right_op (P C) \n\nvariable {C}\n\ndef in_map : c ⨯ (P C).obj (op c) ⟶ Ω C := (exp.ev c).app (Ω C)  \n\nvariable {c}\n\nlemma in_map_natural (g : a ⟶ (P C).obj (op b)) :\n  uncurry g = limits.prod.map (𝟙 _) g ≫ in_map b := uncurry_eq g\n\nlemma in_map_dinatural (h : b ⟶ c) : \n  limits.prod.map h (𝟙 _) ≫ in_map c = limits.prod.map (𝟙 _) ((P C).map h.op) ≫ in_map b :=\nbegin\n  erw [←in_map_natural, uncurry_pre], \n  refl\nend\n\nvariables {C c} {d : C} {f : c ⟶ d}\n\nnamespace delta \n\nopen category_theory.limits.prod\n\ndef pullback_cone_map_diag (f : c ⟶ d) : pullback_cone (map f (𝟙 _)) (diag d) :=\npullback_cone.mk (lift (𝟙 _) f) f (by simp only [lift_map,comp_lift, id_comp, comp_id])\n\nlemma cone_map_diag_fst (s : pullback_cone (map f (𝟙 _)) (diag d)) : s.fst ≫ fst ≫ f = s.snd :=\nby { rw [←map_fst f (𝟙 _), ←assoc, s.condition, assoc, lift_fst, comp_id] }\n\nlemma cone_map_diag_snd (s : pullback_cone (map f (𝟙 _)) (diag d)) : s.fst ≫ snd = s.snd :=\nby { rw [←comp_id (s.fst ≫ snd), assoc, ←map_snd f (𝟙 _), \n         ←assoc, s.condition, assoc, lift_snd, comp_id] }\n\nlemma cone_map_diag_fst_snd (s : pullback_cone (map f (𝟙 _)) (diag d)) : \n  s.fst ≫ fst ≫ f = s.fst ≫ snd := eq.trans (cone_map_diag_fst s) (cone_map_diag_snd s).symm\n\ndef is_limit_pullback_cone_map_diag : is_limit (pullback_cone_map_diag f) :=\nbegin\n  apply pullback_cone.is_limit.mk (pullback_cone_map_diag f).condition (λ s, s.fst ≫ fst); \n  intro s,\n  { simp only [assoc], dunfold pullback_cone_map_diag, rw pullback_cone.mk_fst,\n    rw [comp_lift], nth_rewrite 1 ←comp_id s.fst, simp only [comp_id],\n    apply hom_ext,\n      rw [assoc, lift_fst, ←assoc, comp_id (s.fst ≫ fst)],\n      rw [assoc, lift_snd], apply cone_map_diag_fst_snd },\n  { simp only [assoc], erw pullback_cone.mk_snd, apply cone_map_diag_fst, },\n  { intros l fst' snd', simp only, \n    rw ←eq_whisker fst' fst, erw [assoc, lift_fst, comp_id] }\nend\n\nvariable (f)\n\ndef big_square_cone : pullback_cone (map f (𝟙 d) ≫ δ d) (truth C) :=\npullback_cone.mk (lift (𝟙 c) f) (f ≫ terminal.from d) \n(by { erw [←assoc, (pullback_cone_map_diag f).condition, assoc, classifier.comm (diag d),\n      ←assoc, terminal.comp_from] })\n\nlemma is_limit_big_square_cone : is_limit (big_square_cone f) :=\nbegin\n  apply big_square_is_pullback f (terminal.from _) (map f (𝟙 _)) (δ _) \n    (lift (𝟙 _) f) (diag _) (truth C),\n  apply classifier.is_pb,\n  apply is_limit_pullback_cone_map_diag,\nend\n\nlemma big_square_classifying : classifying (truth C) (lift (𝟙 c) f) (map f (𝟙 d) ≫ δ d) := \n{ comm := by { rw ←terminal.comp_from f, erw (big_square_cone f).condition, refl },\n  is_pb := \n  begin\n    let g := is_limit_big_square_cone f, unfold big_square_cone at g,\n    simp [terminal.comp_from f] at g, assumption,\n  end }\n\nlemma classifies : classifier_of (lift (𝟙 _) f) = map f (𝟙 _) ≫ δ d :=\nclassifier.uniquely _ _ (big_square_classifying f)\n\nvariable (g : c ⟶ d)\n\nlemma cancel_classifier:  (classifier_of (lift (𝟙 _) f) =  classifier_of (lift (𝟙 _) g)) ↔ f = g :=\nbegin\n  split; intro heq,\n  { have k := (pullback_cone.is_limit.lift' (classifier.is_pb (lift (𝟙 _) f)) \n    ((lift (𝟙 _) g)) (terminal.from _) (by rw [heq, classifier.comm])).prop.left,\n    have eq_id := eq_whisker k fst,\n    erw [assoc, lift_fst, lift_fst, comp_id] at eq_id,\n    rw [eq_id, id_comp] at k,\n    convert eq_whisker k snd,\n    erw lift_snd, rw lift_snd },\n  { rw heq }\nend\n\nend delta\n\n-- We show that a topos is balanced\nnamespace balanced\n\nlemma iso_of_is_limit_fork_id {f : c ⟶ d} {s : fork f f} (is_lim : is_limit s) : is_iso s.ι :=\nbegin\n  apply is_iso.mk,\n  use is_lim.lift (fork.of_ι (𝟙 c) (by simp)),\n  split,\n  { apply fork.is_limit.hom_ext is_lim,\n    rw [assoc, fork.is_limit.lift_ι, fork.ι_of_ι, id_comp],\n    apply comp_id },\n  { apply fork.is_limit.lift_ι }\nend\n\nlemma is_limit_of_is_limit_fork_eq {f g : c ⟶ d} {s : fork f g} (is_lim : is_limit s) (heq : f = g) : \n  is_limit (fork.of_ι s.ι (by rw s.condition) : fork f f) :=\nbegin\n  refine fork.is_limit.mk _ \n    (λ t : fork f f, (fork.is_limit.lift' is_lim t.ι (by rw ←heq)).val) _ _; \n  intro t,\n  { apply fork.is_limit.lift_ι },\n  { intros r ht, apply fork.is_limit.hom_ext is_lim, erw fork.is_limit.lift_ι, apply ht }\nend\n\nlemma iso_of_is_limit_fork_eq {f g : c ⟶ d} {s : fork f g} (is_lim : is_limit s) (heq : f = g) : \n  is_iso s.ι := iso_of_is_limit_fork_id (is_limit_of_is_limit_fork_eq is_lim heq)\n\nvariable (C)\n\ninstance topos_balanced : balanced C :=\n{ is_iso_of_mono_of_epi := λ c d f m e,\n  begin\n    resetI,\n    apply iso_of_is_limit_fork_eq (image.monic_is_limit_fork f),\n    rw ←cancel_epi f,\n    exact (image.monic_to_canonical_fork f).condition\n  end }\n\nend balanced\n", "meta": {"author": "cchanavat", "repo": "lean-topos", "sha": "c8e22c35ed4dc4ea0d74a59c91785b8a4c8e48a4", "save_path": "github-repos/lean/cchanavat-lean-topos", "path": "github-repos/lean/cchanavat-lean-topos/lean-topos-c8e22c35ed4dc4ea0d74a59c91785b8a4c8e48a4/topos.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3315220333719304}}
{"text": "/-\nCopyright (c) 2020 Yakov Pechersky. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yakov Pechersky\n-/\nimport data.string.basic\nimport data.buffer.basic\nimport data.nat.digits\nimport data.buffer.parser\n\n/-!\n# Parsers\n\n`parser α` is the type that describes a computation that can ingest a `char_buffer`\nand output, if successful, a term of type `α`.\nThis file expands on the definitions in the core library, proving that all the core library\nparsers are `mono`. There are also lemmas on the composability of parsers.\n\n## Main definitions\n\n* `parse_result.pos` : The position of a `char_buffer` at which a `parser α` has finished.\n* `parser.mono` : The property that a parser only moves forward within a buffer,\n  in both cases of success or failure.\n\n## Implementation details\n\nLemmas about how parsers are mono are in the `mono` namespace. That allows using projection\nnotation for shorter term proofs that are parallel to the definitions of the parsers in structure.\n\n-/\n\nopen parser parse_result\n\n/--\nFor some `parse_result α`, give the position at which the result was provided, in either the\n`done` or the `fail` case.\n-/\n@[simp] def parse_result.pos {α} : parse_result α → ℕ\n| (done n _) := n\n| (fail n _) := n\n\nnamespace parser\n\nsection defn_lemmas\n\nvariables {α β : Type} (msgs : thunk (list string)) (msg : thunk string)\nvariables (p q : parser α) (cb : char_buffer) (n n' : ℕ) {err : dlist string}\nvariables {a : α} {b : β}\n\n/--\nA `p : parser α` is defined to be `mono` if the result `p cb n` it gives,\nfor some `cb : char_buffer` and `n : ℕ`, (whether `done` or `fail`),\nis always at a `parse_result.pos` that is at least `n`.\nThe `mono` property is used mainly for proper `orelse` behavior.\n-/\nclass mono : Prop :=\n(le' : ∀ (cb : char_buffer) (n : ℕ), n ≤ (p cb n).pos)\n\nlemma mono.le [p.mono] : n ≤ (p cb n).pos := mono.le' cb n\n\n/--\nA `parser α` is defined to be `static` if it does not move on success.\n-/\nclass static : Prop :=\n(of_done : ∀ {cb : char_buffer} {n n' : ℕ} {a : α}, p cb n = done n' a → n = n')\n\n/--\nA `parser α` is defined to be `err_static` if it does not move on error.\n-/\nclass err_static : Prop :=\n(of_fail : ∀ {cb : char_buffer} {n n' : ℕ} {err : dlist string}, p cb n = fail n' err → n = n')\n\n/--\nA `parser α` is defined to be `step` if it always moves exactly one char forward on success.\n-/\nclass step : Prop :=\n(of_done : ∀ {cb : char_buffer} {n n' : ℕ} {a : α}, p cb n = done n' a → n' = n + 1)\n\n/--\nA `parser α` is defined to be `prog` if it always moves forward on success.\n-/\nclass prog : Prop :=\n(of_done : ∀ {cb : char_buffer} {n n' : ℕ} {a : α}, p cb n = done n' a → n < n')\n\n/--\nA `parser a` is defined to be `bounded` if it produces a\n`fail` `parse_result` when it is parsing outside the provided `char_buffer`.\n-/\nclass bounded : Prop :=\n(ex' : ∀ {cb : char_buffer} {n : ℕ}, cb.size ≤ n → ∃ (n' : ℕ) (err : dlist string),\n  p cb n = fail n' err)\n\nlemma bounded.exists (p : parser α) [p.bounded] {cb : char_buffer} {n : ℕ} (h : cb.size ≤ n) :\n  ∃ (n' : ℕ) (err : dlist string), p cb n = fail n' err :=\nbounded.ex' h\n\n/--\nA `parser a` is defined to be `unfailing` if it always produces a `done` `parse_result`.\n-/\nclass unfailing : Prop :=\n(ex' : ∀ (cb : char_buffer) (n : ℕ), ∃ (n' : ℕ) (a : α), p cb n = done n' a)\n\n/--\nA `parser a` is defined to be `conditionally_unfailing` if it produces a\n`done` `parse_result` as long as it is parsing within the provided `char_buffer`.\n-/\nclass conditionally_unfailing : Prop :=\n(ex' : ∀ {cb : char_buffer} {n : ℕ}, n < cb.size → ∃ (n' : ℕ) (a : α), p cb n = done n' a)\n\nlemma fail_iff :\n  (∀ pos' result, p cb n ≠ done pos' result) ↔\n    ∃ (pos' : ℕ) (err : dlist string), p cb n = fail pos' err :=\nby cases p cb n; simp\n\nlemma success_iff :\n  (∀ pos' err, p cb n ≠ fail pos' err) ↔ ∃ (pos' : ℕ) (result : α), p cb n = done pos' result :=\nby cases p cb n; simp\n\nvariables {p q cb n n' msgs msg}\n\nlemma mono.of_done [p.mono] (h : p cb n = done n' a) : n ≤ n' :=\nby simpa [h] using mono.le p cb n\n\nlemma mono.of_fail [p.mono] (h : p cb n = fail n' err) : n ≤ n' :=\nby simpa [h] using mono.le p cb n\n\nlemma bounded.of_done [p.bounded] (h : p cb n = done n' a) : n < cb.size :=\nbegin\n  contrapose! h,\n  obtain ⟨np, err, hp⟩ := bounded.exists p h,\n  simp [hp]\nend\n\nlemma static.iff :\n  static p ↔ (∀ (cb : char_buffer) (n n' : ℕ) (a : α), p cb n = done n' a → n = n') :=\n⟨λ h _ _ _ _ hp, by { haveI := h, exact static.of_done hp}, λ h, ⟨h⟩⟩\n\nlemma exists_done (p : parser α) [p.unfailing] (cb : char_buffer) (n : ℕ) :\n  ∃ (n' : ℕ) (a : α), p cb n = done n' a :=\nunfailing.ex' cb n\n\nlemma unfailing.of_fail [p.unfailing] (h : p cb n = fail n' err) : false :=\nbegin\n  obtain ⟨np, a, hp⟩ := p.exists_done cb n,\n  simpa [hp] using h\nend\n\n@[priority 100] -- see Note [lower instance priority]\ninstance conditionally_unfailing_of_unfailing [p.unfailing] : conditionally_unfailing p :=\n⟨λ _ _ _, p.exists_done _ _⟩\n\nlemma exists_done_in_bounds (p : parser α) [p.conditionally_unfailing] {cb : char_buffer} {n : ℕ}\n  (h : n < cb.size) : ∃ (n' : ℕ) (a : α), p cb n = done n' a :=\nconditionally_unfailing.ex' h\n\nlemma conditionally_unfailing.of_fail [p.conditionally_unfailing] (h : p cb n = fail n' err)\n  (hn : n < cb.size) : false :=\nbegin\n  obtain ⟨np, a, hp⟩ := p.exists_done_in_bounds hn,\n  simpa [hp] using h\nend\n\nlemma decorate_errors_fail (h : p cb n = fail n' err) :\n  @decorate_errors α msgs p cb n = fail n ((dlist.lazy_of_list (msgs ()))) :=\nby simp [decorate_errors, h]\n\nlemma decorate_errors_success (h : p cb n = done n' a) :\n  @decorate_errors α msgs p cb n = done n' a :=\nby simp [decorate_errors, h]\n\nlemma decorate_error_fail (h : p cb n = fail n' err) :\n  @decorate_error α msg p cb n = fail n ((dlist.lazy_of_list ([msg ()]))) :=\ndecorate_errors_fail h\n\nlemma decorate_error_success (h : p cb n = done n' a) :\n  @decorate_error α msg p cb n = done n' a :=\ndecorate_errors_success h\n\n@[simp] lemma decorate_errors_eq_done :\n  @decorate_errors α msgs p cb n = done n' a ↔ p cb n = done n' a :=\nby cases h : p cb n; simp [decorate_errors, h]\n\n@[simp] lemma decorate_error_eq_done :\n  @decorate_error α msg p cb n = done n' a ↔ p cb n = done n' a :=\ndecorate_errors_eq_done\n\n@[simp] lemma decorate_errors_eq_fail :\n  @decorate_errors α msgs p cb n = fail n' err ↔\n    n = n' ∧ err = dlist.lazy_of_list (msgs ()) ∧ ∃ np err', p cb n = fail np err' :=\nby cases h : p cb n; simp [decorate_errors, h, eq_comm]\n\n@[simp] lemma decorate_error_eq_fail :\n  @decorate_error α msg p cb n = fail n' err ↔\n    n = n' ∧ err = dlist.lazy_of_list ([msg ()]) ∧ ∃ np err', p cb n = fail np err' :=\ndecorate_errors_eq_fail\n\n@[simp] lemma return_eq_pure : (@return parser _ _ a) = pure a := rfl\n\nlemma pure_eq_done : (@pure parser _ _ a) = λ _ n, done n a := rfl\n\n@[simp] lemma pure_ne_fail : (pure a : parser α) cb n ≠ fail n' err := by simp [pure_eq_done]\n\nsection bind\n\nvariable (f : α → parser β)\n\n@[simp] lemma bind_eq_bind : p.bind f = p >>= f := rfl\n\nvariable {f}\n\n@[simp] lemma bind_eq_done :\n  (p >>= f) cb n = done n' b ↔\n  ∃ (np : ℕ) (a : α), p cb n = done np a ∧ f a cb np = done n' b :=\nby cases hp : p cb n; simp [hp, ←bind_eq_bind, parser.bind, and_assoc]\n\n@[simp] lemma bind_eq_fail :\n  (p >>= f) cb n = fail n' err ↔\n  (p cb n = fail n' err) ∨ (∃ (np : ℕ) (a : α), p cb n = done np a ∧ f a cb np = fail n' err) :=\nby cases hp : p cb n; simp [hp, ←bind_eq_bind, parser.bind, and_assoc]\n\n@[simp] lemma and_then_eq_bind {α β : Type} {m : Type → Type} [monad m] (a : m α) (b : m β) :\n  a >> b = a >>= (λ _, b) := rfl\n\nlemma and_then_fail :\n  (p >> return ()) cb n = parse_result.fail n' err ↔ p cb n = fail n' err :=\nby simp [pure_eq_done]\n\nlemma and_then_success :\n  (p >> return ()) cb n = parse_result.done n' () ↔ ∃ a, p cb n = done n' a:=\nby simp [pure_eq_done]\n\nend bind\n\nsection map\n\nvariable {f : α → β}\n\n@[simp] lemma map_eq_done : (f <$> p) cb n = done n' b ↔\n  ∃ (a : α), p cb n = done n' a ∧ f a = b :=\nby cases hp : p cb n; simp [←is_lawful_monad.bind_pure_comp_eq_map, hp, and_assoc, pure_eq_done]\n\n@[simp] lemma map_eq_fail : (f <$> p) cb n = fail n' err ↔ p cb n = fail n' err :=\nby simp [←bind_pure_comp_eq_map, pure_eq_done]\n\n@[simp] lemma map_const_eq_done {b'} : (b <$ p) cb n = done n' b' ↔\n  ∃ (a : α), p cb n = done n' a ∧ b = b' :=\nby simp [map_const_eq]\n\n@[simp] lemma map_const_eq_fail : (b <$ p) cb n = fail n' err ↔ p cb n = fail n' err :=\nby simp only [map_const_eq, map_eq_fail]\n\nlemma map_const_rev_eq_done {b'} : (p $> b) cb n = done n' b' ↔\n  ∃ (a : α), p cb n = done n' a ∧ b = b' :=\nmap_const_eq_done\n\nlemma map_rev_const_eq_fail : (p $> b) cb n = fail n' err ↔ p cb n = fail n' err :=\nmap_const_eq_fail\n\nend map\n\n@[simp] lemma orelse_eq_orelse : p.orelse q = (p <|> q) := rfl\n\n@[simp] lemma orelse_eq_done : (p <|> q) cb n = done n' a ↔\n  (p cb n = done n' a ∨ (q cb n = done n' a ∧ ∃ err, p cb n = fail n err)) :=\nbegin\n  cases hp : p cb n with np resp np errp,\n  { simp [hp, ←orelse_eq_orelse, parser.orelse] },\n  { by_cases hn : np = n,\n    { cases hq : q cb n with nq resq nq errq,\n      { simp [hp, hn, hq, ←orelse_eq_orelse, parser.orelse] },\n      { rcases lt_trichotomy nq n with H|rfl|H;\n        simp [hp, hn, hq, H, not_lt_of_lt H, lt_irrefl, ←orelse_eq_orelse, parser.orelse] <|>\n          simp [hp, hn, hq, lt_irrefl, ←orelse_eq_orelse, parser.orelse] } },\n    { simp [hp, hn, ←orelse_eq_orelse, parser.orelse] } }\nend\n\n@[simp] lemma orelse_eq_fail_eq : (p <|> q) cb n = fail n err ↔\n  (p cb n = fail n err ∧ ∃ (nq errq), n < nq ∧ q cb n = fail nq errq) ∨\n  (∃ (errp errq), p cb n = fail n errp ∧ q cb n = fail n errq ∧ errp ++ errq = err)\n :=\nbegin\n  cases hp : p cb n with np resp np errp,\n  { simp [hp, ←orelse_eq_orelse, parser.orelse] },\n  { by_cases hn : np = n,\n    { cases hq : q cb n with nq resq nq errq,\n      { simp [hp, hn, hq, ←orelse_eq_orelse, parser.orelse] },\n      { rcases lt_trichotomy nq n with H|rfl|H;\n        simp [hp, hq, hn, ←orelse_eq_orelse, parser.orelse, H,\n              ne_of_gt H, ne_of_lt H, not_lt_of_lt H] <|>\n          simp [hp, hq, hn, ←orelse_eq_orelse, parser.orelse, lt_irrefl] } },\n    { simp [hp, hn, ←orelse_eq_orelse, parser.orelse] } }\nend\n\nlemma orelse_eq_fail_not_mono_lt (hn : n' < n) : (p <|> q) cb n = fail n' err ↔\n  (p cb n = fail n' err) ∨\n  (q cb n = fail n' err ∧ (∃ (errp), p cb n = fail n errp)) :=\nbegin\n  cases hp : p cb n with np resp np errp,\n  { simp [hp, ←orelse_eq_orelse, parser.orelse] },\n  { by_cases h : np = n,\n    { cases hq : q cb n with nq resq nq errq,\n      { simp [hp, h, hn, hq, ne_of_gt hn, ←orelse_eq_orelse, parser.orelse] },\n      { rcases lt_trichotomy nq n with H|H|H,\n        { simp [hp, hq, h, H, ne_of_gt hn, not_lt_of_lt H, ←orelse_eq_orelse, parser.orelse] },\n        { simp [hp, hq, h, H, ne_of_gt hn, lt_irrefl, ←orelse_eq_orelse, parser.orelse] },\n        { simp [hp, hq, h, H, ne_of_gt (hn.trans H), ←orelse_eq_orelse, parser.orelse] } } },\n    { simp [hp, h, ←orelse_eq_orelse, parser.orelse] } }\nend\n\nlemma orelse_eq_fail_of_mono_ne [q.mono] (hn : n ≠ n') :\n  (p <|> q) cb n = fail n' err ↔ p cb n = fail n' err :=\nbegin\n  cases hp : p cb n with np resp np errp,\n  { simp [hp, ←orelse_eq_orelse, parser.orelse] },\n  { by_cases h : np = n,\n    { cases hq : q cb n with nq resq nq errq,\n      { simp [hp, h, hn, hq, hn, ←orelse_eq_orelse, parser.orelse] },\n      { have : n ≤ nq := mono.of_fail hq,\n        rcases eq_or_lt_of_le this with rfl|H,\n        { simp [hp, hq, h, hn, lt_irrefl, ←orelse_eq_orelse, parser.orelse] },\n        { simp [hp, hq, h, hn, H, ←orelse_eq_orelse, parser.orelse] } } },\n    { simp [hp, h, ←orelse_eq_orelse, parser.orelse] } },\nend\n\n@[simp] lemma failure_eq_failure : @parser.failure α = failure := rfl\n\n@[simp] lemma failure_def : (failure : parser α) cb n = fail n dlist.empty := rfl\n\nlemma not_failure_eq_done : ¬ (failure : parser α) cb n = done n' a :=\nby simp\n\nlemma failure_eq_fail : (failure : parser α) cb n = fail n' err ↔ n = n' ∧ err = dlist.empty :=\nby simp [eq_comm]\n\nlemma seq_eq_done {f : parser (α → β)} {p : parser α} : (f <*> p) cb n = done n' b ↔\n  ∃ (nf : ℕ) (f' : α → β) (a : α), f cb n = done nf f' ∧ p cb nf = done n' a ∧ f' a = b :=\nby simp [seq_eq_bind_map]\n\nlemma seq_eq_fail {f : parser (α → β)} {p : parser α} : (f <*> p) cb n = fail n' err ↔\n  (f cb n = fail n' err) ∨ (∃ (nf : ℕ) (f' : α → β), f cb n = done nf f' ∧ p cb nf = fail n' err) :=\nby simp [seq_eq_bind_map]\n\nlemma seq_left_eq_done {p : parser α} {q : parser β} : (p <* q) cb n = done n' a ↔\n  ∃ (np : ℕ) (b : β), p cb n = done np a ∧ q cb np = done n' b :=\nbegin\n  have : ∀ (p q : ℕ → α → Prop),\n    (∃ (np : ℕ) (x : α), p np x ∧ q np x ∧ x = a) ↔ ∃ (np : ℕ), p np a ∧ q np a :=\n    λ _ _, ⟨λ ⟨np, x, hp, hq, rfl⟩, ⟨np, hp, hq⟩, λ ⟨np, hp, hq⟩, ⟨np, a, hp, hq, rfl⟩⟩,\n  simp [seq_left_eq, seq_eq_done, map_eq_done, this]\nend\n\nlemma seq_left_eq_fail {p : parser α} {q : parser β} : (p <* q) cb n = fail n' err ↔\n  (p cb n = fail n' err) ∨ (∃ (np : ℕ) (a : α), p cb n = done np a ∧ q cb np = fail n' err) :=\nby simp [seq_left_eq, seq_eq_fail]\n\nlemma seq_right_eq_done {p : parser α} {q : parser β} : (p *> q) cb n = done n' b ↔\n  ∃ (np : ℕ) (a : α), p cb n = done np a ∧ q cb np = done n' b :=\nby simp [seq_right_eq, seq_eq_done, map_eq_done, and.comm, and.assoc]\n\nlemma seq_right_eq_fail {p : parser α} {q : parser β} : (p *> q) cb n = fail n' err ↔\n  (p cb n = fail n' err) ∨ (∃ (np : ℕ) (a : α), p cb n = done np a ∧ q cb np = fail n' err) :=\nby simp [seq_right_eq, seq_eq_fail]\n\nlemma mmap_eq_done {f : α → parser β} {a : α} {l : list α} {b : β} {l' : list β} :\n  (a :: l).mmap f cb n = done n' (b :: l') ↔\n  ∃ (np : ℕ), f a cb n = done np b ∧ l.mmap f cb np = done n' l' :=\nby simp [mmap, and.comm, and.assoc, and.left_comm, pure_eq_done]\n\nlemma mmap'_eq_done {f : α → parser β} {a : α} {l : list α} :\n  (a :: l).mmap' f cb n = done n' () ↔\n  ∃ (np : ℕ) (b : β), f a cb n = done np b ∧ l.mmap' f cb np = done n' () :=\nby simp [mmap']\n\nlemma guard_eq_done {p : Prop} [decidable p] {u : unit} :\n  @guard parser _ p _ cb n = done n' u ↔ p ∧ n = n' :=\nby { by_cases hp : p; simp [guard, hp, pure_eq_done] }\n\nlemma guard_eq_fail {p : Prop} [decidable p] :\n  @guard parser _ p _ cb n = fail n' err ↔ (¬ p) ∧ n = n' ∧ err = dlist.empty :=\nby { by_cases hp : p; simp [guard, hp, eq_comm, pure_eq_done] }\n\nnamespace mono\n\nvariables {sep : parser unit}\n\ninstance pure : mono (pure a) :=\n⟨λ _ _, by simp [pure_eq_done]⟩\n\ninstance bind {f : α → parser β} [p.mono] [∀ a, (f a).mono] :\n  (p >>= f).mono :=\nbegin\n  constructor,\n  intros cb n,\n  cases hx : (p >>= f) cb n,\n  { obtain ⟨n', a, h, h'⟩ := bind_eq_done.mp hx,\n    refine le_trans (of_done h) _,\n    simpa [h'] using of_done h' },\n  { obtain h | ⟨n', a, h, h'⟩ := bind_eq_fail.mp hx,\n    { simpa [h] using of_fail h },\n    { refine le_trans (of_done h) _,\n      simpa [h'] using of_fail h' } }\nend\n\ninstance and_then {q : parser β} [p.mono] [q.mono] : (p >> q).mono := mono.bind\n\ninstance map [p.mono] {f : α → β} : (f <$> p).mono := mono.bind\n\ninstance seq {f : parser (α → β)} [f.mono] [p.mono] : (f <*> p).mono := mono.bind\n\ninstance mmap : Π {l : list α} {f : α → parser β} [∀ a ∈ l, (f a).mono],\n  (l.mmap f).mono\n| []       _ _ := mono.pure\n| (a :: l) f h := begin\n  convert mono.bind,\n  { exact h _ (list.mem_cons_self _ _) },\n  { intro,\n    convert mono.map,\n    convert mmap,\n    exact (λ _ ha, h _ (list.mem_cons_of_mem _ ha)) }\nend\n\ninstance mmap' : Π {l : list α} {f : α → parser β} [∀ a ∈ l, (f a).mono],\n  (l.mmap' f).mono\n| []       _ _ := mono.pure\n| (a :: l) f h := begin\n  convert mono.and_then,\n  { exact h _ (list.mem_cons_self _ _) },\n  { convert mmap',\n    exact (λ _ ha, h _ (list.mem_cons_of_mem _ ha)) }\nend\n\ninstance failure : (failure : parser α).mono :=\n⟨by simp [le_refl]⟩\n\ninstance guard {p : Prop} [decidable p] : mono (guard p) :=\n⟨by { by_cases h : p; simp [h, pure_eq_done, le_refl] }⟩\n\ninstance orelse [p.mono] [q.mono] : (p <|> q).mono :=\nbegin\n  constructor,\n  intros cb n,\n  cases hx : (p <|> q) cb n with posx resx posx errx,\n  { obtain h | ⟨h, -, -⟩ := orelse_eq_done.mp hx;\n    simpa [h] using of_done h },\n  { by_cases h : n = posx,\n    { simp [hx, h] },\n    { simp only [orelse_eq_fail_of_mono_ne h] at hx,\n      exact of_fail hx } }\nend\n\ninstance decorate_errors [p.mono] :\n  (@decorate_errors α msgs p).mono :=\nbegin\n  constructor,\n  intros cb n,\n  cases h : p cb n,\n  { simpa [decorate_errors, h] using of_done h },\n  { simp [decorate_errors, h] }\nend\n\ninstance decorate_error [p.mono] : (@decorate_error α msg p).mono :=\nmono.decorate_errors\n\ninstance any_char : mono any_char :=\nbegin\n  constructor,\n  intros cb n,\n  by_cases h : n < cb.size;\n  simp [any_char, h],\nend\n\ninstance sat {p : char → Prop} [decidable_pred p] : mono (sat p) :=\nbegin\n  constructor,\n  intros cb n,\n  simp only [sat],\n  split_ifs;\n  simp\nend\n\ninstance eps : mono eps := mono.pure\n\ninstance ch {c : char} : mono (ch c) := mono.decorate_error\n\ninstance char_buf {s : char_buffer} : mono (char_buf s) :=\nmono.decorate_error\n\ninstance one_of {cs : list char} : (one_of cs).mono :=\nmono.decorate_errors\n\ninstance one_of' {cs : list char} : (one_of' cs).mono :=\nmono.and_then\n\ninstance str {s : string} : (str s).mono :=\nmono.decorate_error\n\ninstance remaining : remaining.mono :=\n⟨λ _ _, le_rfl⟩\n\ninstance eof : eof.mono :=\nmono.decorate_error\n\ninstance foldr_core {f : α → β → β} {b : β} [p.mono] :\n  ∀ {reps : ℕ}, (foldr_core f p b reps).mono\n| 0          := mono.failure\n| (reps + 1) := begin\n  convert mono.orelse,\n  { convert mono.bind,\n    { apply_instance },\n    { exact λ _, @mono.bind _ _ _ _ foldr_core _ } },\n  { exact mono.pure }\nend\n\ninstance foldr {f : α → β → β} [p.mono] : mono (foldr f p b) :=\n⟨λ _ _, by { convert mono.le (foldr_core f p b _) _ _, exact mono.foldr_core }⟩\n\ninstance foldl_core {f : α → β → α} {p : parser β} [p.mono] :\n  ∀ {a : α} {reps : ℕ}, (foldl_core f a p reps).mono\n| _ 0          := mono.failure\n| _ (reps + 1) := begin\n  convert mono.orelse,\n  { convert mono.bind,\n    { apply_instance },\n    { exact λ _, foldl_core } },\n  { exact mono.pure }\nend\n\ninstance foldl {f : α → β → α} {p : parser β} [p.mono] : mono (foldl f a p) :=\n⟨λ _ _, by { convert mono.le (foldl_core f a p _) _ _, exact mono.foldl_core }⟩\n\ninstance many [p.mono] : p.many.mono :=\nmono.foldr\n\ninstance many_char {p : parser char} [p.mono] : p.many_char.mono :=\nmono.map\n\ninstance many' [p.mono] : p.many'.mono :=\nmono.and_then\n\ninstance many1 [p.mono] : p.many1.mono :=\nmono.seq\n\ninstance many_char1 {p : parser char} [p.mono] : p.many_char1.mono :=\nmono.map\n\ninstance sep_by1 [p.mono] [sep.mono] : mono (sep_by1 sep p) :=\nmono.seq\n\ninstance sep_by [p.mono] [hs : sep.mono] : mono (sep_by sep p) :=\nmono.orelse\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.mono → (F p).mono) :\n  ∀ (max_depth : ℕ), mono (fix_core F max_depth)\n| 0               := mono.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\ninstance digit : digit.mono :=\nmono.decorate_error\n\ninstance nat : nat.mono :=\nmono.decorate_error\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.mono → (F p).mono) :\n  mono (fix F) :=\n⟨λ _ _, by { convert mono.le (parser.fix_core F _) _ _, exact fix_core hF _ }⟩\n\nend mono\n\n@[simp] lemma orelse_pure_eq_fail : (p <|> pure a) cb n = fail n' err ↔\n  p cb n = fail n' err ∧ n ≠ n' :=\nbegin\n  by_cases hn : n = n',\n  { simp [hn, pure_eq_done] },\n  { simp [orelse_eq_fail_of_mono_ne, hn] }\nend\n\nend defn_lemmas\n\nsection done\n\nvariables {α β : Type} {cb : char_buffer} {n n' : ℕ} {a a' : α} {b : β} {c : char} {u : unit}\n  {err : dlist string}\n\nlemma any_char_eq_done : any_char cb n = done n' c ↔\n  ∃ (hn : n < cb.size), n' = n + 1 ∧ cb.read ⟨n, hn⟩ = c :=\nbegin\n  simp_rw [any_char],\n  split_ifs with h;\n  simp [h, eq_comm]\nend\n\nlemma any_char_eq_fail : any_char cb n = fail n' err ↔ n = n' ∧ err = dlist.empty ∧ cb.size ≤ n :=\nbegin\n  simp_rw [any_char],\n  split_ifs with h;\n  simp [←not_lt, h, eq_comm]\nend\n\nlemma sat_eq_done {p : char → Prop} [decidable_pred p] : sat p cb n = done n' c ↔\n  ∃ (hn : n < cb.size), p c ∧ n' = n + 1 ∧ cb.read ⟨n, hn⟩ = c :=\nbegin\n  by_cases hn : n < cb.size,\n  { by_cases hp : p (cb.read ⟨n, hn⟩),\n    { simp only [sat, hn, hp, dif_pos, if_true, exists_prop_of_true],\n      split,\n      { rintro ⟨rfl, rfl⟩, simp [hp] },\n      { rintro ⟨-, rfl, rfl⟩, simp } },\n    { simp only [sat, hn, hp, dif_pos, false_iff, not_and, exists_prop_of_true, if_false],\n      rintro H - rfl,\n      exact hp H } },\n  { simp [sat, hn] }\nend\n\nlemma sat_eq_fail {p : char → Prop} [decidable_pred p] : sat p cb n = fail n' err ↔\n  n = n' ∧ err = dlist.empty ∧ ∀ (h : n < cb.size), ¬ p (cb.read ⟨n, h⟩) :=\nbegin\n  dsimp only [sat],\n  split_ifs;\n  simp [*, eq_comm]\nend\n\nlemma eps_eq_done : eps cb n = done n' u ↔ n = n' := by simp [eps, pure_eq_done]\n\nlemma ch_eq_done : ch c cb n = done n' u ↔ ∃ (hn : n < cb.size), n' = n + 1 ∧ cb.read ⟨n, hn⟩ = c :=\nby simp [ch, eps_eq_done, sat_eq_done, and.comm, @eq_comm _ n']\n\nlemma char_buf_eq_done {cb' : char_buffer} : char_buf cb' cb n = done n' u ↔\n  n + cb'.size = n' ∧ cb'.to_list <+: (cb.to_list.drop n) :=\nbegin\n  simp only [char_buf, decorate_error_eq_done, ne.def, ←buffer.length_to_list],\n  induction cb'.to_list with hd tl hl generalizing cb n n',\n  { simp [pure_eq_done, mmap'_eq_done, -buffer.length_to_list, list.nil_prefix] },\n  { simp only [ch_eq_done, and.comm, and.assoc, and.left_comm, hl, mmap', and_then_eq_bind,\n               bind_eq_done, list.length, exists_and_distrib_left, exists_const],\n    split,\n    { rintro ⟨np, h, rfl, rfl, hn, rfl⟩,\n      simp only [add_comm, add_left_comm, h, true_and, eq_self_iff_true, and_true],\n      have : n < cb.to_list.length := by simpa using hn,\n      rwa [←buffer.nth_le_to_list _ this, ←list.cons_nth_le_drop_succ this, list.prefix_cons_inj] },\n    { rintro ⟨h, rfl⟩,\n      by_cases hn : n < cb.size,\n      { have : n < cb.to_list.length := by simpa using hn,\n        rw [←list.cons_nth_le_drop_succ this, list.cons_prefix_iff] at h,\n        use [n + 1, h.right],\n        simpa [buffer.nth_le_to_list, add_comm, add_left_comm, add_assoc, hn] using h.left.symm },\n      { have : cb.to_list.length ≤ n := by simpa using hn,\n        rw list.drop_eq_nil_of_le this at h,\n        simpa using h } } }\nend\n\nlemma one_of_eq_done {cs : list char} : one_of cs cb n = done n' c ↔\n  ∃ (hn : n < cb.size), c ∈ cs ∧ n' = n + 1 ∧ cb.read ⟨n, hn⟩ = c :=\nby simp [one_of, sat_eq_done]\n\nlemma one_of'_eq_done {cs : list char} : one_of' cs cb n = done n' u ↔\n  ∃ (hn : n < cb.size), cb.read ⟨n, hn⟩ ∈ cs ∧ n' = n + 1 :=\nbegin\n  simp only [one_of', one_of_eq_done, eps_eq_done, and.comm, and_then_eq_bind, bind_eq_done,\n             exists_eq_left, exists_and_distrib_left],\n  split,\n  { rintro ⟨c, hc, rfl, hn, rfl⟩,\n    exact ⟨rfl, hn, hc⟩ },\n  { rintro ⟨rfl, hn, hc⟩,\n    exact ⟨cb.read ⟨n, hn⟩, hc, rfl, hn, rfl⟩ }\nend\n\nlemma str_eq_char_buf (s : string) : str s = char_buf s.to_list.to_buffer :=\nbegin\n  ext cb n,\n  rw [str, char_buf],\n  congr,\n  { simp [buffer.to_string, string.as_string_inv_to_list] },\n  { simp }\nend\n\nlemma str_eq_done {s : string} : str s cb n = done n' u ↔\n  n + s.length = n' ∧ s.to_list <+: (cb.to_list.drop n) :=\nby simp [str_eq_char_buf, char_buf_eq_done]\n\nlemma remaining_eq_done {r : ℕ} : remaining cb n = done n' r ↔ n = n' ∧ cb.size - n = r :=\nby simp [remaining]\n\nlemma remaining_ne_fail : remaining cb n ≠ fail n' err :=\nby simp [remaining]\n\nlemma eof_eq_done {u : unit} : eof cb n = done n' u ↔ n = n' ∧ cb.size ≤ n :=\nby simp [eof, guard_eq_done, remaining_eq_done, tsub_eq_zero_iff_le, and_comm, and_assoc]\n\n@[simp] lemma foldr_core_zero_eq_done {f : α → β → β} {p : parser α} {b' : β} :\n  foldr_core f p b 0 cb n ≠ done n' b' :=\nby simp [foldr_core]\n\nlemma foldr_core_eq_done {f : α → β → β} {p : parser α} {reps : ℕ} {b' : β} :\n  foldr_core f p b (reps + 1) cb n = done n' b' ↔\n  (∃ (np : ℕ) (a : α) (xs : β), p cb n = done np a ∧ foldr_core f p b reps cb np = done n' xs\n    ∧ f a xs = b') ∨\n  (n = n' ∧ b = b' ∧ ∃ (err), (p cb n = fail n err) ∨\n    (∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldr_core f p b reps cb np = fail n err)) :=\nby simp [foldr_core, and.comm, and.assoc, pure_eq_done]\n\n@[simp] lemma foldr_core_zero_eq_fail {f : α → β → β} {p : parser α} {err : dlist string} :\n  foldr_core f p b 0 cb n = fail n' err ↔ n = n' ∧ err = dlist.empty :=\nby simp [foldr_core, eq_comm]\n\nlemma foldr_core_succ_eq_fail {f : α → β → β} {p : parser α} {reps : ℕ} {err : dlist string} :\n  foldr_core f p b (reps + 1) cb n = fail n' err ↔ n ≠ n' ∧\n  (p cb n = fail n' err ∨\n    ∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldr_core f p b reps cb np = fail n' err) :=\nby simp [foldr_core, and_comm]\n\nlemma foldr_eq_done {f : α → β → β} {p : parser α} {b' : β} :\n  foldr f p b cb n = done n' b' ↔\n  ((∃ (np : ℕ) (a : α) (x : β), p cb n = done np a ∧\n    foldr_core f p b (cb.size - n) cb np = done n' x ∧ f a x = b') ∨\n  (n = n' ∧ b = b' ∧ (∃ (err), p cb n = parse_result.fail n err ∨\n    ∃ (np : ℕ) (x : α), p cb n = done np x ∧ foldr_core f p b (cb.size - n) cb np = fail n err))) :=\nby simp [foldr, foldr_core_eq_done]\n\nlemma foldr_eq_fail_iff_mono_at_end {f : α → β → β} {p : parser α} {err : dlist string}\n  [p.mono] (hc : cb.size ≤ n) : foldr f p b cb n = fail n' err ↔\n    n < n' ∧ (p cb n = fail n' err ∨ ∃ (a : α), p cb n = done n' a ∧ err = dlist.empty) :=\nbegin\n  have : cb.size - n = 0 := tsub_eq_zero_iff_le.mpr hc,\n  simp only [foldr, foldr_core_succ_eq_fail, this, and.left_comm, foldr_core_zero_eq_fail,\n             ne_iff_lt_iff_le, exists_and_distrib_right, exists_eq_left, and.congr_left_iff,\n             exists_and_distrib_left],\n  rintro (h | ⟨⟨a, h⟩, rfl⟩),\n  { exact mono.of_fail h },\n  { exact mono.of_done h }\nend\n\nlemma foldr_eq_fail {f : α → β → β} {p : parser α} {err : dlist string} :\n  foldr f p b cb n = fail n' err ↔ n ≠ n' ∧ (p cb n = fail n' err ∨\n    ∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldr_core f p b (cb.size - n) cb np = fail n' err) :=\nby simp [foldr, foldr_core_succ_eq_fail]\n\n@[simp] lemma foldl_core_zero_eq_done {f : β → α → β} {p : parser α} {b' : β} :\n  foldl_core f b p 0 cb n = done n' b' ↔ false :=\nby simp [foldl_core]\n\nlemma foldl_core_eq_done {f : β → α → β} {p : parser α} {reps : ℕ} {b' : β} :\n  foldl_core f b p (reps + 1) cb n = done n' b' ↔\n  (∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldl_core f (f b a) p reps cb np = done n' b') ∨\n  (n = n' ∧ b = b' ∧ ∃ (err), (p cb n = fail n err) ∨\n    (∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldl_core f (f b a) p reps cb np = fail n err)) :=\nby simp [foldl_core, and.assoc, pure_eq_done]\n\n@[simp] lemma foldl_core_zero_eq_fail {f : β → α → β} {p : parser α} {err : dlist string} :\n  foldl_core f b p 0 cb n = fail n' err ↔ n = n' ∧ err = dlist.empty :=\nby simp [foldl_core, eq_comm]\n\nlemma foldl_core_succ_eq_fail {f : β → α → β} {p : parser α} {reps : ℕ} {err : dlist string} :\n  foldl_core f b p (reps + 1) cb n = fail n' err ↔ n ≠ n' ∧\n  (p cb n = fail n' err ∨\n    ∃ (np : ℕ) (a : α), p cb n = done np a ∧ foldl_core f (f b a) p reps cb np = fail n' err) :=\nby simp [foldl_core, and_comm]\n\nlemma foldl_eq_done {f : β → α → β} {p : parser α} {b' : β} :\n  foldl f b p cb n = done n' b' ↔\n  (∃ (np : ℕ) (a : α), p cb n = done np a ∧\n    foldl_core f (f b a) p (cb.size - n) cb np = done n' b') ∨\n  (n = n' ∧ b = b' ∧ ∃ (err), (p cb n = fail n err) ∨\n    (∃ (np : ℕ) (a : α), p cb n = done np a ∧\n      foldl_core f (f b a) p (cb.size - n) cb np = fail n err)) :=\nby simp [foldl, foldl_core_eq_done]\n\nlemma foldl_eq_fail {f : β → α → β} {p : parser α} {err : dlist string} :\n  foldl f b p cb n = fail n' err ↔ n ≠ n' ∧ (p cb n = fail n' err ∨\n    ∃ (np : ℕ) (a : α), p cb n = done np a ∧\n    foldl_core f (f b a) p (cb.size - n) cb np = fail n' err) :=\nby simp [foldl, foldl_core_succ_eq_fail]\n\nlemma foldl_eq_fail_iff_mono_at_end {f : β → α → β} {p : parser α} {err : dlist string}\n  [p.mono] (hc : cb.size ≤ n) : foldl f b p cb n = fail n' err ↔\n    n < n' ∧ (p cb n = fail n' err ∨ ∃ (a : α), p cb n = done n' a ∧ err = dlist.empty) :=\nbegin\n  have : cb.size - n = 0 := tsub_eq_zero_iff_le.mpr hc,\n  simp only [foldl, foldl_core_succ_eq_fail, this, and.left_comm, ne_iff_lt_iff_le, exists_eq_left,\n             exists_and_distrib_right, and.congr_left_iff, exists_and_distrib_left,\n             foldl_core_zero_eq_fail],\n  rintro (h | ⟨⟨a, h⟩, rfl⟩),\n  { exact mono.of_fail h },\n  { exact mono.of_done h }\nend\n\nlemma many_eq_done_nil {p : parser α} : many p cb n = done n' (@list.nil α) ↔ n = n' ∧\n  ∃ (err), p cb n = fail n err ∨ ∃ (np : ℕ) (a : α), p cb n = done np a ∧\n    foldr_core list.cons p [] (cb.size - n) cb np = fail n err :=\nby simp [many, foldr_eq_done]\n\nlemma many_eq_done {p : parser α} {x : α} {xs : list α} :\n  many p cb n = done n' (x :: xs) ↔ ∃ (np : ℕ), p cb n = done np x\n    ∧ foldr_core list.cons p [] (cb.size - n) cb np = done n' xs :=\nby simp [many, foldr_eq_done, and.comm, and.assoc, and.left_comm]\n\nlemma many_eq_fail {p : parser α} {err : dlist string} :\n  many p cb n = fail n' err ↔ n ≠ n' ∧ (p cb n = fail n' err ∨\n    ∃ (np : ℕ) (a : α), p cb n = done np a ∧\n      foldr_core list.cons p [] (cb.size - n) cb np = fail n' err) :=\nby simp [many, foldr_eq_fail]\n\nlemma many_char_eq_done_empty {p : parser char} : many_char p cb n = done n' string.empty ↔ n = n' ∧\n  ∃ (err), p cb n = fail n err ∨ ∃ (np : ℕ) (c : char), p cb n = done np c ∧\n    foldr_core list.cons p [] (cb.size - n) cb np = fail n err :=\nby simp [many_char, many_eq_done_nil, map_eq_done, list.as_string_eq]\n\nlemma many_char_eq_done_not_empty {p : parser char} {s : string} (h : s ≠ \"\") :\n  many_char p cb n = done n' s ↔ ∃ (np : ℕ), p cb n = done np s.head ∧\n    foldr_core list.cons p list.nil (buffer.size cb - n) cb np = done n' (s.popn 1).to_list :=\nby simp [many_char, list.as_string_eq, string.to_list_nonempty h, many_eq_done]\n\nlemma many_char_eq_many_of_to_list {p : parser char} {s : string} :\n  many_char p cb n = done n' s ↔ many p cb n = done n' s.to_list :=\nby simp [many_char, list.as_string_eq]\n\nlemma many'_eq_done {p : parser α} : many' p cb n = done n' u ↔\n  many p cb n = done n' [] ∨ ∃ (np : ℕ) (a : α) (l : list α), many p cb n = done n' (a :: l)\n    ∧ p cb n = done np a ∧ foldr_core list.cons p [] (buffer.size cb - n) cb np = done n' l :=\nbegin\n  simp only [many', eps_eq_done, many, foldr, and_then_eq_bind, exists_and_distrib_right,\n             bind_eq_done, exists_eq_right],\n  split,\n  { rintro ⟨_ | ⟨hd, tl⟩, hl⟩,\n    { exact or.inl hl },\n    { have hl2 := hl,\n      simp only [foldr_core_eq_done, or_false, exists_and_distrib_left, and_false, false_and,\n                 exists_eq_right_right] at hl,\n      obtain ⟨np, hp, h⟩ := hl,\n      refine or.inr ⟨np, _, _, hl2, hp, h⟩ } },\n  { rintro (h | ⟨np, a, l, hp, h⟩),\n    { exact ⟨[], h⟩ },\n    { refine ⟨a :: l, hp⟩ } }\nend\n\n@[simp] lemma many1_ne_done_nil {p : parser α} : many1 p cb n ≠ done n' [] :=\nby simp [many1, seq_eq_done]\n\nlemma many1_eq_done {p : parser α} {l : list α} : many1 p cb n = done n' (a :: l) ↔\n  ∃ (np : ℕ), p cb n = done np a ∧ many p cb np = done n' l :=\nby simp [many1, seq_eq_done, map_eq_done]\n\n\n\n@[simp] lemma many_char1_ne_empty {p : parser char} : many_char1 p cb n ≠ done n' \"\" :=\nby simp [many_char1, ←string.nil_as_string_eq_empty]\n\nlemma many_char1_eq_done {p : parser char} {s : string} (h : s ≠ \"\") :\n  many_char1 p cb n = done n' s ↔\n  ∃ (np : ℕ), p cb n = done np s.head ∧ many_char p cb np = done n' (s.popn 1) :=\nby simp [many_char1, list.as_string_eq, string.to_list_nonempty h, many1_eq_done,\n         many_char_eq_many_of_to_list]\n\n@[simp] lemma sep_by1_ne_done_nil {sep : parser unit} {p : parser α} :\n  sep_by1 sep p cb n ≠ done n' [] :=\nby simp [sep_by1, seq_eq_done]\n\nlemma sep_by1_eq_done {sep : parser unit} {p : parser α} {l : list α} :\n  sep_by1 sep p cb n = done n' (a :: l) ↔ ∃ (np : ℕ), p cb n = done np a ∧\n    (sep >> p).many cb np  = done n' l :=\nby simp [sep_by1, seq_eq_done]\n\nlemma sep_by_eq_done_nil {sep : parser unit} {p : parser α} :\n  sep_by sep p cb n = done n' [] ↔ n = n' ∧ ∃ (err), sep_by1 sep p cb n = fail n err :=\nby simp [sep_by, pure_eq_done]\n\n@[simp] lemma fix_core_ne_done_zero {F : parser α → parser α} :\n  fix_core F 0 cb n ≠ done n' a :=\nby simp [fix_core]\n\nlemma fix_core_eq_done {F : parser α → parser α} {max_depth : ℕ} :\n  fix_core F (max_depth + 1) cb n = done n' a ↔ F (fix_core F max_depth) cb n = done n' a :=\nby simp [fix_core]\n\nlemma digit_eq_done {k : ℕ} : digit cb n = done n' k ↔ ∃ (hn : n < cb.size), n' = n + 1 ∧ k ≤ 9 ∧\n  (cb.read ⟨n, hn⟩).to_nat - '0'.to_nat = k ∧ '0' ≤ cb.read ⟨n, hn⟩ ∧ cb.read ⟨n, hn⟩ ≤ '9' :=\nbegin\n  have c9 : '9'.to_nat - '0'.to_nat = 9 := rfl,\n  have l09 : '0'.to_nat ≤ '9'.to_nat := dec_trivial,\n  have le_iff_le : ∀ {c c' : char}, c ≤ c' ↔ c.to_nat ≤ c'.to_nat := λ _ _, iff.rfl,\n  split,\n  { simp only [digit, sat_eq_done, pure_eq_done, decorate_error_eq_done, bind_eq_done, ←c9],\n    rintro ⟨np, c, ⟨hn, ⟨ge0, le9⟩, rfl, rfl⟩, rfl, rfl⟩,\n    simpa [hn, ge0, le9, true_and, and_true, eq_self_iff_true, exists_prop_of_true,\n            tsub_le_tsub_iff_right, l09] using (le_iff_le.mp le9) },\n  { simp only [digit, sat_eq_done, pure_eq_done, decorate_error_eq_done, bind_eq_done, ←c9,\n               le_iff_le],\n    rintro ⟨hn, rfl, -, rfl, ge0, le9⟩,\n    use [n + 1, cb.read ⟨n, hn⟩],\n    simp [hn, ge0, le9] }\nend\n\nlemma digit_eq_fail : digit cb n = fail n' err ↔ n = n' ∧ err = dlist.of_list [\"<digit>\"] ∧\n  ∀ (h : n < cb.size), ¬ ((λ c, '0' ≤ c ∧ c ≤ '9') (cb.read ⟨n, h⟩)) :=\nby simp [digit, sat_eq_fail]\n\n\nend done\n\nnamespace static\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\nlemma not_of_ne (h : p cb n = done n' a) (hne : n ≠ n') : ¬ static p :=\nby { introI, exact hne (of_done h) }\n\ninstance pure : static (pure a) :=\n⟨λ _ _ _ _, by { simp_rw pure_eq_done, rw [and.comm], simp }⟩\n\ninstance bind {f : α → parser β} [p.static] [∀ a, (f a).static] :\n  (p >>= f).static :=\n⟨λ _ _ _ _, by { rw bind_eq_done, rintro ⟨_, _, hp, hf⟩, exact trans (of_done hp) (of_done hf) }⟩\n\ninstance and_then {q : parser β} [p.static] [q.static] : (p >> q).static := static.bind\n\ninstance map [p.static] {f : α → β} : (f <$> p).static :=\n⟨λ _ _ _ _, by { simp_rw map_eq_done, rintro ⟨_, hp, _⟩, exact of_done hp }⟩\n\ninstance seq {f : parser (α → β)} [f.static] [p.static] : (f <*> p).static := static.bind\n\ninstance mmap : Π {l : list α} {f : α → parser β} [∀ a, (f a).static], (l.mmap f).static\n| []       _ _ := static.pure\n| (a :: l) _ h := begin\n  convert static.bind,\n  { exact h _ },\n  { intro,\n    convert static.bind,\n    { convert mmap,\n      exact h },\n    { exact λ _, static.pure } }\nend\n\ninstance mmap' : Π {l : list α} {f : α → parser β} [∀ a, (f a).static], (l.mmap' f).static\n| []       _ _ := static.pure\n| (a :: l) _ h := begin\n  convert static.and_then,\n  { exact h _ },\n  { convert mmap',\n    exact h }\nend\n\ninstance failure : @parser.static α failure :=\n⟨λ _ _ _ _, by simp⟩\n\ninstance guard {p : Prop} [decidable p] : static (guard p) :=\n⟨λ _ _ _ _, by simp [guard_eq_done]⟩\n\ninstance orelse [p.static] [q.static] : (p <|> q).static :=\n⟨λ _ _ _ _, by { simp_rw orelse_eq_done, rintro (h | ⟨h, -⟩); exact of_done h }⟩\n\ninstance decorate_errors [p.static] :\n  (@decorate_errors α msgs p).static :=\n⟨λ _ _ _ _, by { rw decorate_errors_eq_done, exact of_done }⟩\n\ninstance decorate_error [p.static] : (@decorate_error α msg p).static :=\nstatic.decorate_errors\n\nlemma any_char : ¬ static any_char :=\nbegin\n  have : any_char \"s\".to_char_buffer 0 = done 1 's',\n  { have : 0 < \"s\".to_char_buffer.size := dec_trivial,\n    simpa [any_char_eq_done, this] },\n  exact not_of_ne this zero_ne_one\nend\n\nlemma sat_iff {p : char → Prop} [decidable_pred p] : static (sat p) ↔ ∀ c, ¬ p c :=\nbegin\n  split,\n  { introI,\n    intros c hc,\n    have : sat p [c].to_buffer 0 = done 1 c := by simp [sat_eq_done, hc],\n    exact zero_ne_one (of_done this) },\n  { contrapose!,\n    simp only [iff, sat_eq_done, and_imp, exists_prop, exists_and_distrib_right,\n               exists_and_distrib_left, exists_imp_distrib, not_forall],\n    rintros _ _ _ a h hne rfl hp -,\n    exact ⟨a, hp⟩ }\nend\n\ninstance sat : static (sat (λ _, false)) :=\nby { apply sat_iff.mpr, simp }\n\ninstance eps : static eps := static.pure\n\nlemma ch (c : char) : ¬ static (ch c) :=\nbegin\n  have : ch c [c].to_buffer 0 = done 1 (),\n  { have : 0 < [c].to_buffer.size := dec_trivial,\n    simp [ch_eq_done, this] },\n  exact not_of_ne this zero_ne_one\nend\n\nlemma char_buf_iff {cb' : char_buffer} : static (char_buf cb') ↔ cb' = buffer.nil :=\nbegin\n  rw ←buffer.size_eq_zero_iff,\n  have : char_buf cb' cb' 0 = done cb'.size () := by simp [char_buf_eq_done],\n  cases hc : cb'.size with n,\n  { simp only [eq_self_iff_true, iff_true],\n    exact ⟨λ _ _ _ _ h, by simpa [hc] using (char_buf_eq_done.mp h).left⟩ },\n  { rw hc at this,\n    simpa [nat.succ_ne_zero] using not_of_ne this (nat.succ_ne_zero n).symm }\nend\n\nlemma one_of_iff {cs : list char} : static (one_of cs) ↔ cs = [] :=\nbegin\n  cases cs with hd tl,\n  { simp [one_of, static.decorate_errors] },\n  { have : one_of (hd :: tl) (hd :: tl).to_buffer 0 = done 1 hd,\n    { simp [one_of_eq_done] },\n    simpa using not_of_ne this zero_ne_one }\nend\n\ninstance one_of : static (one_of []) :=\nby { apply one_of_iff.mpr, refl }\n\nlemma one_of'_iff {cs : list char} : static (one_of' cs) ↔ cs = [] :=\nbegin\n  cases cs with hd tl,\n  { simp [one_of', static.bind], },\n  { have : one_of' (hd :: tl) (hd :: tl).to_buffer 0 = done 1 (),\n    { simp [one_of'_eq_done] },\n    simpa using not_of_ne this zero_ne_one }\nend\n\ninstance one_of' : static (one_of []) :=\nby { apply one_of_iff.mpr, refl }\n\nlemma str_iff {s : string} : static (str s) ↔ s = \"\" :=\nby simp [str_eq_char_buf, char_buf_iff, ←string.to_list_inj, buffer.ext_iff]\n\ninstance remaining : remaining.static :=\n⟨λ _ _ _ _ h, (remaining_eq_done.mp h).left⟩\n\ninstance eof : eof.static :=\nstatic.decorate_error\n\ninstance foldr_core {f : α → β → β} [p.static] :\n  ∀ {b : β} {reps : ℕ}, (foldr_core f p b reps).static\n| _ 0          := static.failure\n| _ (reps + 1) := begin\n  simp_rw parser.foldr_core,\n  convert static.orelse,\n  { convert static.bind,\n    { apply_instance },\n    { intro,\n      convert static.bind,\n      { exact foldr_core },\n      { apply_instance } } },\n  { exact static.pure }\nend\n\ninstance foldr {f : α → β → β} [p.static] : static (foldr f p b) :=\n⟨λ _ _ _ _, by { dsimp [foldr], exact of_done }⟩\n\ninstance foldl_core {f : α → β → α} {p : parser β} [p.static] :\n  ∀ {a : α} {reps : ℕ}, (foldl_core f a p reps).static\n| _ 0          := static.failure\n| _ (reps + 1) := begin\n  convert static.orelse,\n  { convert static.bind,\n    { apply_instance },\n    { exact λ _, foldl_core } },\n  { exact static.pure }\nend\n\ninstance foldl {f : α → β → α} {p : parser β} [p.static] : static (foldl f a p) :=\n⟨λ _ _ _ _, by { dsimp [foldl], exact of_done }⟩\n\ninstance many [p.static] : p.many.static :=\nstatic.foldr\n\ninstance many_char {p : parser char} [p.static] : p.many_char.static :=\nstatic.map\n\ninstance many' [p.static] : p.many'.static :=\nstatic.and_then\n\ninstance many1 [p.static] : p.many1.static :=\nstatic.seq\n\ninstance many_char1 {p : parser char} [p.static] : p.many_char1.static :=\nstatic.map\n\ninstance sep_by1 [p.static] [sep.static] : static (sep_by1 sep p) :=\nstatic.seq\n\ninstance sep_by [p.static] [sep.static] : static (sep_by sep p) :=\nstatic.orelse\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.static → (F p).static) :\n  ∀ (max_depth : ℕ), static (fix_core F max_depth)\n| 0               := static.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\nlemma digit : ¬ digit.static :=\nbegin\n  have : digit \"1\".to_char_buffer 0 = done 1 1,\n  { have : 0 < \"s\".to_char_buffer.size := dec_trivial,\n    simpa [this] },\n  exact not_of_ne this zero_ne_one\nend\n\nlemma nat : ¬ nat.static :=\nbegin\n  have : nat \"1\".to_char_buffer 0 = done 1 1,\n  { have : 0 < \"s\".to_char_buffer.size := dec_trivial,\n    simpa [this] },\n  exact not_of_ne this zero_ne_one\nend\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.static → (F p).static) :\n  static (fix F) :=\n⟨λ cb n _ _ h,\n  by { haveI := fix_core hF (cb.size - n + 1), dsimp [fix] at h, exact static.of_done h }⟩\n\nend static\n\nnamespace bounded\n\nvariables {α β : Type} {msgs : thunk (list string)} {msg : thunk string}\nvariables {p q : parser α} {cb : char_buffer} {n n' : ℕ} {err : dlist string}\nvariables {a : α} {b : β}\n\nlemma done_of_unbounded (h : ¬p.bounded) : ∃ (cb : char_buffer) (n n' : ℕ) (a : α),\n  p cb n = done n' a ∧ cb.size ≤ n :=\nbegin\n  contrapose! h,\n  constructor,\n  intros cb n hn,\n  cases hp : p cb n,\n  { exact absurd hn (h _ _ _ _ hp).not_le },\n  { simp [hp] }\nend\n\nlemma pure : ¬ bounded (pure a) :=\nbegin\n  introI,\n  have : (pure a : parser α) buffer.nil 0 = done 0 a := by simp [pure_eq_done],\n  exact absurd (bounded.of_done this) (lt_irrefl _)\nend\n\ninstance bind {f : α → parser β} [p.bounded] :\n  (p >>= f).bounded :=\nbegin\n  constructor,\n  intros cb n hn,\n  obtain ⟨_, _, hp⟩ := bounded.exists p hn,\n  simp [hp]\nend\n\ninstance and_then {q : parser β} [p.bounded] : (p >> q).bounded :=\nbounded.bind\n\ninstance map [p.bounded] {f : α → β} : (f <$> p).bounded :=\nbounded.bind\n\ninstance seq {f : parser (α → β)} [f.bounded] : (f <*> p).bounded :=\nbounded.bind\n\ninstance mmap {a : α} {l : list α} {f : α → parser β} [∀ a, (f a).bounded] :\n  ((a :: l).mmap f).bounded :=\nbounded.bind\n\ninstance mmap' {a : α} {l : list α} {f : α → parser β} [∀ a, (f a).bounded] :\n  ((a :: l).mmap' f).bounded :=\nbounded.and_then\n\ninstance failure : @parser.bounded α failure :=\n⟨by simp⟩\n\nlemma guard_iff {p : Prop} [decidable p] : bounded (guard p) ↔ ¬ p :=\nby simpa [guard, apply_ite bounded, pure, failure] using λ _, bounded.failure\n\ninstance orelse [p.bounded] [q.bounded] : (p <|> q).bounded :=\nbegin\n  constructor,\n  intros cb n hn,\n  cases hx : (p <|> q) cb n with posx resx posx errx,\n  { obtain h | ⟨h, -, -⟩ := orelse_eq_done.mp hx;\n    exact absurd hn (of_done h).not_le },\n  { simp }\nend\n\ninstance decorate_errors [p.bounded] :\n  (@decorate_errors α msgs p).bounded :=\nbegin\n  constructor,\n  intros _ _,\n  simpa using bounded.exists p\nend\n\nlemma decorate_errors_iff : (@parser.decorate_errors α msgs p).bounded ↔ p.bounded :=\nbegin\n  split,\n  { introI,\n    constructor,\n    intros _ _ hn,\n    obtain ⟨_, _, h⟩ := bounded.exists (@parser.decorate_errors α msgs p) hn,\n    simp [decorate_errors_eq_fail] at h,\n    exact h.right.right },\n  { introI,\n    constructor,\n    intros _ _ hn,\n    obtain ⟨_, _, h⟩ := bounded.exists p hn,\n    simp [h] }\nend\n\ninstance decorate_error [p.bounded] : (@decorate_error α msg p).bounded :=\nbounded.decorate_errors\n\nlemma decorate_error_iff : (@parser.decorate_error α msg p).bounded ↔ p.bounded :=\ndecorate_errors_iff\n\ninstance any_char : bounded any_char :=\n⟨λ cb n hn, by simp [any_char, hn]⟩\n\ninstance sat {p : char → Prop} [decidable_pred p] : bounded (sat p) :=\n⟨λ cb n hn, by simp [sat, hn]⟩\n\nlemma eps : ¬ bounded eps := pure\n\ninstance ch {c : char} : bounded (ch c) :=\nbounded.decorate_error\n\nlemma char_buf_iff {cb' : char_buffer} : bounded (char_buf cb') ↔ cb' ≠ buffer.nil :=\nbegin\n  have : cb' ≠ buffer.nil ↔ cb'.to_list ≠ [] :=\n      not_iff_not_of_iff ⟨λ h, by simp [h], λ h, by simpa using congr_arg list.to_buffer h⟩,\n  rw [char_buf, decorate_error_iff, this],\n  cases cb'.to_list,\n  { simp [pure, ch] },\n  { simp only [iff_true, ne.def, not_false_iff],\n    apply_instance }\nend\n\ninstance one_of {cs : list char} : (one_of cs).bounded :=\nbounded.decorate_errors\n\ninstance one_of' {cs : list char} : (one_of' cs).bounded :=\nbounded.and_then\n\nlemma str_iff {s : string} : (str s).bounded ↔ s ≠ \"\" :=\nbegin\n  rw [str, decorate_error_iff],\n  cases hs : s.to_list,\n  { have : s = \"\",\n    { cases s, rw [string.to_list] at hs, simpa [hs] },\n    simp [pure, this] },\n  { have : s ≠ \"\",\n    { intro H, simpa [H] using hs },\n    simp only [this, iff_true, ne.def, not_false_iff],\n    apply_instance }\nend\n\nlemma remaining : ¬ remaining.bounded :=\nbegin\n  introI,\n  have : remaining buffer.nil 0 = done 0 0 := by simp [remaining_eq_done],\n  exact absurd (bounded.of_done this) (lt_irrefl _)\nend\n\nlemma eof : ¬ eof.bounded :=\nbegin\n  introI,\n  have : eof buffer.nil 0 = done 0 () := by simp [eof_eq_done],\n  exact absurd (bounded.of_done this) (lt_irrefl _)\nend\n\nsection fold\n\ninstance foldr_core_zero {f : α → β → β} : (foldr_core f p b 0).bounded :=\nbounded.failure\n\ninstance foldl_core_zero {f : β → α → β} {b : β} : (foldl_core f b p 0).bounded :=\nbounded.failure\n\nvariables {reps : ℕ} [hpb : p.bounded] (he : ∀ cb n n' err, p cb n = fail n' err → n ≠ n')\ninclude hpb he\n\nlemma foldr_core {f : α → β → β} : (foldr_core f p b reps).bounded :=\nbegin\n  cases reps,\n  { exact bounded.foldr_core_zero },\n  constructor,\n  intros cb n hn,\n  obtain ⟨np, errp, hp⟩ := bounded.exists p hn,\n  simpa [foldr_core_succ_eq_fail, hp] using he cb n np errp,\nend\n\nlemma foldr {f : α → β → β} : bounded (foldr f p b) :=\nbegin\n  constructor,\n  intros cb n hn,\n  haveI : (parser.foldr_core f p b (cb.size - n + 1)).bounded := foldr_core he,\n  obtain ⟨np, errp, hp⟩ := bounded.exists (parser.foldr_core f p b (cb.size - n + 1)) hn,\n  simp [foldr, hp]\nend\n\nlemma foldl_core {f : β → α → β} :\n  (foldl_core f b p reps).bounded :=\nbegin\n  cases reps,\n  { exact bounded.foldl_core_zero },\n  constructor,\n  intros cb n hn,\n  obtain ⟨np, errp, hp⟩ := bounded.exists p hn,\n  simpa [foldl_core_succ_eq_fail, hp] using he cb n np errp,\nend\n\nlemma foldl {f : β → α → β} : bounded (foldl f b p) :=\nbegin\n  constructor,\n  intros cb n hn,\n  haveI : (parser.foldl_core f b p (cb.size - n + 1)).bounded := foldl_core he,\n  obtain ⟨np, errp, hp⟩ := bounded.exists (parser.foldl_core f b p (cb.size - n + 1)) hn,\n  simp [foldl, hp]\nend\n\nlemma many : p.many.bounded :=\nfoldr he\n\nomit hpb\nlemma many_char {pc : parser char} [pc.bounded]\n  (he : ∀ cb n n' err, pc cb n = fail n' err → n ≠ n'): pc.many_char.bounded :=\nby { convert bounded.map, exact many he }\ninclude hpb\n\nlemma many' : p.many'.bounded :=\nby { convert bounded.and_then, exact many he }\n\nend fold\n\ninstance many1 [p.bounded] : p.many1.bounded :=\nbounded.seq\n\ninstance many_char1 {p : parser char} [p.bounded] : p.many_char1.bounded :=\nbounded.map\n\ninstance sep_by1 {sep : parser unit} [p.bounded] : bounded (sep_by1 sep p) :=\nbounded.seq\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.bounded → (F p).bounded) :\n  ∀ (max_depth : ℕ), bounded (fix_core F max_depth)\n| 0               := bounded.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\ninstance digit : digit.bounded :=\nbounded.decorate_error\n\ninstance nat : nat.bounded :=\nbounded.decorate_error\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.bounded → (F p).bounded) :\n  bounded (fix F) :=\nbegin\n  constructor,\n  intros cb n hn,\n  haveI : (parser.fix_core F (cb.size - n + 1)).bounded := fix_core hF _,\n  obtain ⟨np, errp, hp⟩ := bounded.exists (parser.fix_core F (cb.size - n + 1)) hn,\n  simp [fix, hp]\nend\n\nend bounded\n\nnamespace unfailing\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\nlemma of_bounded [p.bounded] : ¬ unfailing p :=\nbegin\n  introI,\n  cases h : p buffer.nil 0,\n  { simpa [lt_irrefl] using bounded.of_done h },\n  { exact of_fail h }\nend\n\ninstance pure : unfailing (pure a) :=\n⟨λ _ _, by simp [pure_eq_done]⟩\n\ninstance bind {f : α → parser β} [p.unfailing] [∀ a, (f a).unfailing] :\n  (p >>= f).unfailing :=\n⟨λ cb n, begin\n  obtain ⟨np, a, hp⟩ := exists_done p cb n,\n  simpa [hp, and.comm, and.left_comm, and.assoc] using exists_done (f a) cb np\nend⟩\n\ninstance and_then {q : parser β} [p.unfailing] [q.unfailing] : (p >> q).unfailing := unfailing.bind\n\ninstance map [p.unfailing] {f : α → β} : (f <$> p).unfailing := unfailing.bind\n\ninstance seq {f : parser (α → β)} [f.unfailing] [p.unfailing] : (f <*> p).unfailing :=\nunfailing.bind\n\ninstance mmap {l : list α} {f : α → parser β} [∀ a, (f a).unfailing] : (l.mmap f).unfailing :=\nbegin\n  constructor,\n  induction l with hd tl hl,\n  { intros,\n    simp [pure_eq_done] },\n  { intros,\n    obtain ⟨np, a, hp⟩ := exists_done (f hd) cb n,\n    obtain ⟨n', b, hf⟩ := hl cb np,\n    simp [hp, hf, and.comm, and.left_comm, and.assoc, pure_eq_done] }\nend\n\ninstance mmap' {l : list α} {f : α → parser β} [∀ a, (f a).unfailing] : (l.mmap' f).unfailing :=\nbegin\n  constructor,\n  induction l with hd tl hl,\n  { intros,\n    simp [pure_eq_done] },\n  { intros,\n    obtain ⟨np, a, hp⟩ := exists_done (f hd) cb n,\n    obtain ⟨n', b, hf⟩ := hl cb np,\n    simp [hp, hf, and.comm, and.left_comm, and.assoc, pure_eq_done] }\nend\n\nlemma failure : ¬ @parser.unfailing α failure :=\nbegin\n  introI h,\n  have : (failure : parser α) buffer.nil 0 = fail 0 dlist.empty := by simp,\n  exact of_fail this\nend\n\ninstance guard_true : unfailing (guard true) := unfailing.pure\n\nlemma guard : ¬ unfailing (guard false) :=\nunfailing.failure\n\ninstance orelse [p.unfailing] : (p <|> q).unfailing :=\n⟨λ cb n, by { obtain ⟨_, _, h⟩ := p.exists_done cb n, simp [success_iff, h] }⟩\n\ninstance decorate_errors [p.unfailing] :\n  (@decorate_errors α msgs p).unfailing :=\n⟨λ cb n, by { obtain ⟨_, _, h⟩ := p.exists_done cb n, simp [success_iff, h] }⟩\n\ninstance decorate_error [p.unfailing] : (@decorate_error α msg p).unfailing :=\nunfailing.decorate_errors\n\ninstance any_char : conditionally_unfailing any_char :=\n⟨λ _ _ hn, by simp [success_iff, any_char_eq_done, hn]⟩\n\nlemma sat : conditionally_unfailing (sat (λ _, true)) :=\n⟨λ _ _ hn, by simp [success_iff, sat_eq_done, hn]⟩\n\ninstance eps : unfailing eps := unfailing.pure\n\ninstance remaining : remaining.unfailing :=\n⟨λ _ _, by simp [success_iff, remaining_eq_done]⟩\n\nlemma foldr_core_zero {f : α → β → β} {b : β} : ¬ (foldr_core f p b 0).unfailing :=\nunfailing.failure\n\ninstance foldr_core_of_static {f : α → β → β} {b : β} {reps : ℕ} [p.static] [p.unfailing] :\n  (foldr_core f p b (reps + 1)).unfailing :=\nbegin\n  induction reps with reps hr,\n  { constructor,\n    intros cb n,\n    obtain ⟨np, a, h⟩ := p.exists_done cb n,\n    simpa [foldr_core_eq_done, h] using (static.of_done h).symm },\n  { constructor,\n    haveI := hr,\n    intros cb n,\n    obtain ⟨np, a, h⟩ := p.exists_done cb n,\n    obtain rfl : n = np := static.of_done h,\n    obtain ⟨np, b', hf⟩ := exists_done (foldr_core f p b (reps + 1)) cb n,\n    obtain rfl : n = np := static.of_done hf,\n    refine ⟨n, f a b', _⟩,\n    rw foldr_core_eq_done,\n    simp [h, hf, and.comm, and.left_comm, and.assoc] }\nend\n\ninstance foldr_core_one_of_err_static {f : α → β → β} {b : β} [p.static] [p.err_static] :\n  (foldr_core f p b 1).unfailing :=\nbegin\n  constructor,\n  intros cb n,\n  cases h : p cb n,\n  { simpa [foldr_core_eq_done, h] using (static.of_done h).symm },\n  { simpa [foldr_core_eq_done, h] using (err_static.of_fail h).symm }\nend\n\n-- TODO: add foldr and foldl, many, etc, fix_core\n\nlemma digit : ¬ digit.unfailing :=\nof_bounded\n\nlemma nat : ¬ nat.unfailing :=\nof_bounded\n\nend unfailing\n\nnamespace err_static\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\nlemma not_of_ne (h : p cb n = fail n' err) (hne : n ≠ n') : ¬ err_static p :=\nby { introI, exact hne (of_fail h) }\n\ninstance pure : err_static (pure a) :=\n⟨λ _ _ _ _, by { simp [pure_eq_done] }⟩\n\ninstance bind {f : α → parser β} [p.static] [p.err_static] [∀ a, (f a).err_static] :\n  (p >>= f).err_static :=\n⟨λ cb n n' err, begin\n  rw bind_eq_fail,\n  rintro (hp | ⟨_, _, hp, hf⟩),\n  { exact of_fail hp },\n  { exact trans (static.of_done hp) (of_fail hf) }\nend⟩\n\ninstance bind_of_unfailing {f : α → parser β} [p.err_static] [∀ a, (f a).unfailing] :\n  (p >>= f).err_static :=\n⟨λ cb n n' err, begin\n  rw bind_eq_fail,\n  rintro (hp | ⟨_, _, hp, hf⟩),\n  { exact of_fail hp },\n  { exact false.elim (unfailing.of_fail hf) }\nend⟩\n\ninstance and_then {q : parser β} [p.static] [p.err_static] [q.err_static] : (p >> q).err_static :=\nerr_static.bind\n\ninstance and_then_of_unfailing {q : parser β} [p.err_static] [q.unfailing] : (p >> q).err_static :=\nerr_static.bind_of_unfailing\n\ninstance map [p.err_static] {f : α → β} : (f <$> p).err_static :=\n⟨λ _ _ _ _, by { rw map_eq_fail, exact of_fail }⟩\n\ninstance seq {f : parser (α → β)} [f.static] [f.err_static] [p.err_static] : (f <*> p).err_static :=\nerr_static.bind\n\ninstance seq_of_unfailing {f : parser (α → β)} [f.err_static] [p.unfailing] :\n  (f <*> p).err_static :=\nerr_static.bind_of_unfailing\n\ninstance mmap : Π {l : list α} {f : α → parser β}\n  [∀ a, (f a).static] [∀ a, (f a).err_static], (l.mmap f).err_static\n| []       _ _ _  := err_static.pure\n| (a :: l) _ h h' := begin\n  convert err_static.bind,\n  { exact h _ },\n  { exact h' _ },\n  { intro,\n    convert err_static.bind,\n    { convert static.mmap,\n      exact h },\n    { apply mmap,\n      { exact h },\n      { exact h' } },\n    { exact λ _, err_static.pure } }\nend\n\ninstance mmap_of_unfailing : Π {l : list α} {f : α → parser β}\n  [∀ a, (f a).unfailing] [∀ a, (f a).err_static], (l.mmap f).err_static\n| []       _ _ _  := err_static.pure\n| (a :: l) _ h h' := begin\n  convert err_static.bind_of_unfailing,\n  { exact h' _ },\n  { intro,\n    convert unfailing.bind,\n    { convert unfailing.mmap,\n      exact h },\n    { exact λ _, unfailing.pure } }\nend\n\ninstance mmap' : Π {l : list α} {f : α → parser β}\n  [∀ a, (f a).static] [∀ a, (f a).err_static], (l.mmap' f).err_static\n| []       _ _ _  := err_static.pure\n| (a :: l) _ h h' := begin\n  convert err_static.and_then,\n  { exact h _ },\n  { exact h' _ },\n  { convert mmap',\n    { exact h },\n    { exact h' } }\nend\n\ninstance mmap'_of_unfailing : Π {l : list α} {f : α → parser β}\n  [∀ a, (f a).unfailing] [∀ a, (f a).err_static], (l.mmap' f).err_static\n| []       _ _ _  := err_static.pure\n| (a :: l) _ h h' := begin\n  convert err_static.and_then_of_unfailing,\n  { exact h' _ },\n  { convert unfailing.mmap',\n    exact h }\nend\n\ninstance failure : @parser.err_static α failure :=\n⟨λ _ _ _ _ h, (failure_eq_fail.mp h).left⟩\n\ninstance guard {p : Prop} [decidable p] : err_static (guard p) :=\n⟨λ _ _ _ _ h, (guard_eq_fail.mp h).right.left⟩\n\ninstance orelse [p.err_static] [q.mono] : (p <|> q).err_static :=\n⟨λ _ n n' _, begin\n  by_cases hn : n = n',\n  { exact λ _, hn },\n  { rw orelse_eq_fail_of_mono_ne hn,\n    { exact of_fail },\n    { apply_instance } }\nend⟩\n\ninstance decorate_errors :\n  (@decorate_errors α msgs p).err_static :=\n⟨λ _ _ _ _ h, (decorate_errors_eq_fail.mp h).left⟩\n\ninstance decorate_error : (@decorate_error α msg p).err_static :=\nerr_static.decorate_errors\n\ninstance any_char : err_static any_char :=\n⟨λ _ _ _ _, by { rw [any_char_eq_fail, and.comm], simp }⟩\n\ninstance sat_iff {p : char → Prop} [decidable_pred p] : err_static (sat p) :=\n⟨λ _ _ _ _ h, (sat_eq_fail.mp h).left⟩\n\ninstance eps : err_static eps := err_static.pure\n\ninstance ch (c : char) : err_static (ch c) :=\nerr_static.decorate_error\n\ninstance char_buf {cb' : char_buffer} : err_static (char_buf cb') :=\nerr_static.decorate_error\n\ninstance one_of {cs : list char} : err_static (one_of cs) :=\nerr_static.decorate_errors\n\ninstance one_of' {cs : list char} : err_static (one_of' cs) :=\nerr_static.and_then_of_unfailing\n\ninstance str {s : string} : err_static (str s) :=\nerr_static.decorate_error\n\ninstance remaining : remaining.err_static :=\n⟨λ _ _ _ _, by simp [remaining_ne_fail]⟩\n\ninstance eof : eof.err_static :=\nerr_static.decorate_error\n\n-- TODO: add foldr and foldl, many, etc, fix_core\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.err_static → (F p).err_static) :\n  ∀ (max_depth : ℕ), err_static (fix_core F max_depth)\n| 0               := err_static.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\ninstance digit : digit.err_static :=\nerr_static.decorate_error\n\ninstance nat : nat.err_static :=\nerr_static.decorate_error\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.err_static → (F p).err_static) :\n  err_static (fix F) :=\n⟨λ cb n _ _ h,\n  by { haveI := fix_core hF (cb.size - n + 1), dsimp [fix] at h, exact err_static.of_fail h }⟩\n\nend err_static\n\nnamespace step\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\nlemma not_step_of_static_done [static p] (h : ∃ cb n n' a, p cb n = done n' a) : ¬ step p :=\nbegin\n  introI,\n  rcases h with ⟨cb, n, n', a, h⟩,\n  have hs := static.of_done h,\n  simpa [←hs] using of_done h\nend\n\nlemma pure (a : α) : ¬ step (pure a) :=\nbegin\n  apply not_step_of_static_done,\n  simp [pure_eq_done]\nend\n\ninstance bind {f : α → parser β} [p.step] [∀ a, (f a).static] :\n  (p >>= f).step :=\n⟨λ _ _ _ _, by { simp_rw bind_eq_done, rintro ⟨_, _, hp, hf⟩,\n  exact (static.of_done hf) ▸ (of_done hp) }⟩\n\ninstance bind' {f : α → parser β} [p.static] [∀ a, (f a).step] :\n  (p >>= f).step :=\n⟨λ _ _ _ _, by { simp_rw bind_eq_done, rintro ⟨_, _, hp, hf⟩,\n  rw static.of_done hp, exact of_done hf }⟩\n\ninstance and_then {q : parser β} [p.step] [q.static] : (p >> q).step := step.bind\n\ninstance and_then' {q : parser β} [p.static] [q.step] : (p >> q).step := step.bind'\n\ninstance map [p.step] {f : α → β} : (f <$> p).step :=\n⟨λ _ _ _ _, by { simp_rw map_eq_done, rintro ⟨_, hp, _⟩, exact of_done hp }⟩\n\ninstance seq {f : parser (α → β)} [f.step] [p.static] : (f <*> p).step := step.bind\n\ninstance seq' {f : parser (α → β)} [f.static] [p.step] : (f <*> p).step := step.bind'\n\ninstance mmap {f : α → parser β} [(f a).step] :\n  ([a].mmap f).step :=\nbegin\n  convert step.bind,\n  { apply_instance },\n  { intro,\n    convert static.bind,\n    { exact static.pure },\n    { exact λ _, static.pure } }\nend\n\ninstance mmap' {f : α → parser β} [(f a).step] :\n  ([a].mmap' f).step :=\nbegin\n  convert step.and_then,\n  { apply_instance },\n  { exact static.pure }\nend\n\ninstance failure : @parser.step α failure :=\n⟨λ _ _ _ _, by simp⟩\n\nlemma guard_true : ¬ step (guard true) := pure _\n\ninstance guard : step (guard false) :=\nstep.failure\n\ninstance orelse [p.step] [q.step] : (p <|> q).step :=\n⟨λ _ _ _ _, by { simp_rw orelse_eq_done, rintro (h | ⟨h, -⟩); exact of_done h }⟩\n\nlemma decorate_errors_iff : (@parser.decorate_errors α msgs p).step ↔ p.step :=\nbegin\n  split,\n  { introI,\n    constructor,\n    intros cb n n' a h,\n    have : (@parser.decorate_errors α msgs p) cb n = done n' a := by simpa using h,\n    exact of_done this },\n  { introI,\n    constructor,\n    intros _ _ _ _ h,\n    rw decorate_errors_eq_done at h,\n    exact of_done h }\nend\n\ninstance decorate_errors [p.step] :\n  (@decorate_errors α msgs p).step :=\n⟨λ _ _ _ _, by { rw decorate_errors_eq_done, exact of_done }⟩\n\nlemma decorate_error_iff : (@parser.decorate_error α msg p).step ↔ p.step :=\ndecorate_errors_iff\n\ninstance decorate_error [p.step] : (@decorate_error α msg p).step :=\nstep.decorate_errors\n\ninstance any_char : step any_char :=\nbegin\n  constructor,\n  intros cb n,\n  simp_rw [any_char_eq_done],\n  rintro _ _ ⟨_, rfl, -⟩,\n  simp\nend\n\ninstance sat {p : char → Prop} [decidable_pred p] : step (sat p) :=\nbegin\n  constructor,\n  intros cb n,\n  simp_rw [sat_eq_done],\n  rintro _ _ ⟨_, _, rfl, -⟩,\n  simp\nend\n\nlemma eps : ¬ step eps := step.pure ()\n\ninstance ch {c : char} : step (ch c) := step.decorate_error\n\nlemma char_buf_iff {cb' : char_buffer} : (char_buf cb').step ↔ cb'.size = 1 :=\nbegin\n  have : char_buf cb' cb' 0 = done cb'.size () := by simp [char_buf_eq_done],\n  split,\n  { introI,\n    simpa using of_done this },\n  { intro h,\n    constructor,\n    intros cb n n' _,\n    rw [char_buf_eq_done, h],\n    rintro ⟨rfl, -⟩,\n    refl }\nend\n\ninstance one_of {cs : list char} : (one_of cs).step :=\nstep.decorate_errors\n\ninstance one_of' {cs : list char} : (one_of' cs).step :=\nstep.and_then\n\nlemma str_iff {s : string} : (str s).step ↔ s.length = 1 :=\nby simp [str_eq_char_buf, char_buf_iff, ←string.to_list_inj, buffer.ext_iff]\n\nlemma remaining : ¬ remaining.step :=\nbegin\n  apply not_step_of_static_done,\n  simp [remaining_eq_done]\nend\n\nlemma eof : ¬ eof.step :=\nbegin\n  apply not_step_of_static_done,\n  simp only [eof_eq_done, exists_eq_left', exists_const],\n  use [buffer.nil, 0],\n  simp\nend\n\n-- TODO: add foldr and foldl, many, etc, fix_core\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.step → (F p).step) :\n  ∀ (max_depth : ℕ), step (fix_core F max_depth)\n| 0               := step.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\ninstance digit : digit.step :=\nstep.decorate_error\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.step → (F p).step) :\n  step (fix F) :=\n⟨λ cb n _ _ h,\n  by { haveI := fix_core hF (cb.size - n + 1), dsimp [fix] at h, exact of_done h }⟩\n\nend step\n\nsection step\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\nlemma many1_eq_done_iff_many_eq_done [p.step] [p.bounded] {x : α} {xs : list α} :\n  many1 p cb n = done n' (x :: xs) ↔ many p cb n = done n' (x :: xs) :=\nbegin\n  induction hx : (x :: xs) with hd tl IH generalizing x xs n n',\n  { simpa using hx },\n  split,\n  { simp only [many1_eq_done, and_imp, exists_imp_distrib],\n    intros np hp hm,\n    have : np = n + 1 := step.of_done hp,\n    have hn : n < cb.size := bounded.of_done hp,\n    subst this,\n    obtain ⟨k, hk⟩ : ∃ k, cb.size - n = k + 1 :=\n      nat.exists_eq_succ_of_ne_zero (ne_of_gt (tsub_pos_of_lt hn)),\n    cases k,\n    { cases tl;\n      simpa [many_eq_done_nil, nat.sub_succ, hk, many_eq_done, hp, foldr_core_eq_done] using hm },\n    cases tl with hd' tl',\n    { simpa [many_eq_done_nil, nat.sub_succ, hk, many_eq_done, hp, foldr_core_eq_done] using hm },\n    { rw ←@IH hd' tl' at hm, swap, refl,\n      simp only [many1_eq_done, many, foldr] at hm,\n      obtain ⟨np, hp', hf⟩ := hm,\n      obtain rfl : np = n + 1 + 1 := step.of_done hp',\n      simpa [nat.sub_succ, many_eq_done, hp, hk, foldr_core_eq_done, hp'] using hf } },\n  { simp only [many_eq_done, many1_eq_done, and_imp, exists_imp_distrib],\n    intros np hp hm,\n    have : np = n + 1 := step.of_done hp,\n    have hn : n < cb.size := bounded.of_done hp,\n    subst this,\n    obtain ⟨k, hk⟩ : ∃ k, cb.size - n = k + 1 :=\n      nat.exists_eq_succ_of_ne_zero (ne_of_gt (tsub_pos_of_lt hn)),\n    cases k,\n    { cases tl;\n      simpa [many_eq_done_nil, nat.sub_succ, hk, many_eq_done, hp, foldr_core_eq_done] using hm },\n    cases tl with hd' tl',\n    { simpa [many_eq_done_nil, nat.sub_succ, hk, many_eq_done, hp, foldr_core_eq_done] using hm },\n    { simp [hp],\n      rw ←@IH hd' tl' (n + 1) n', swap, refl,\n      rw [hk, foldr_core_eq_done, or.comm] at hm,\n      obtain (hm | ⟨np, hd', tl', hp', hf, hm⟩) := hm,\n      { simpa using hm },\n      simp only at hm,\n      obtain ⟨rfl, rfl⟩ := hm,\n      obtain rfl : np = n + 1 + 1 := step.of_done hp',\n      simp [nat.sub_succ, many, many1_eq_done, hp, hk, foldr_core_eq_done, hp', ←hf, foldr] } }\nend\n\nend step\n\nnamespace prog\n\nvariables {α β : Type} {p q : parser α} {msgs : thunk (list string)} {msg : thunk string}\n  {cb : char_buffer} {n' n : ℕ} {err : dlist string} {a : α} {b : β} {sep : parser unit}\n\n@[priority 100] -- see Note [lower instance priority]\ninstance of_step [step p] : prog p :=\n⟨λ _ _ _ _ h, by { rw step.of_done h, exact nat.lt_succ_self _ }⟩\n\nlemma pure (a : α) : ¬ prog (pure a) :=\nbegin\n  introI h,\n  have : (pure a : parser α) buffer.nil 0 = done 0 a := by simp [pure_eq_done],\n  replace this : 0 < 0 := prog.of_done this,\n  exact (lt_irrefl _) this\nend\n\ninstance bind {f : α → parser β} [p.prog] [∀ a, (f a).mono] :\n  (p >>= f).prog :=\n⟨λ _ _ _ _, by { simp_rw bind_eq_done, rintro ⟨_, _, hp, hf⟩,\n  exact lt_of_lt_of_le (of_done hp) (mono.of_done hf) }⟩\n\ninstance and_then {q : parser β} [p.prog] [q.mono] : (p >> q).prog := prog.bind\n\ninstance map [p.prog] {f : α → β} : (f <$> p).prog :=\n⟨λ _ _ _ _, by { simp_rw map_eq_done, rintro ⟨_, hp, _⟩, exact of_done hp }⟩\n\ninstance seq {f : parser (α → β)} [f.prog] [p.mono] : (f <*> p).prog := prog.bind\n\ninstance mmap {l : list α} {f : α → parser β} [(f a).prog] [∀ a, (f a).mono] :\n  ((a :: l).mmap f).prog :=\nbegin\n  constructor,\n  simp only [and_imp, bind_eq_done, return_eq_pure, mmap, exists_imp_distrib, pure_eq_done],\n  rintro _ _ _ _ _ _ h _ _ hp rfl rfl,\n  exact lt_of_lt_of_le (of_done h) (mono.of_done hp)\nend\n\ninstance mmap' {l : list α} {f : α → parser β} [(f a).prog] [∀ a, (f a).mono] :\n  ((a :: l).mmap' f).prog :=\nbegin\n  constructor,\n  simp only [and_imp, bind_eq_done, mmap', exists_imp_distrib, and_then_eq_bind],\n  intros _ _ _ _ _ _ h hm,\n  exact lt_of_lt_of_le (of_done h) (mono.of_done hm)\nend\n\ninstance failure : @parser.prog α failure :=\nprog.of_step\n\nlemma guard_true : ¬ prog (guard true) := pure _\n\ninstance guard : prog (guard false) :=\nprog.failure\n\ninstance orelse [p.prog] [q.prog] : (p <|> q).prog :=\n⟨λ _ _ _ _, by { simp_rw orelse_eq_done, rintro (h | ⟨h, -⟩); exact of_done h }⟩\n\nlemma decorate_errors_iff : (@parser.decorate_errors α msgs p).prog ↔ p.prog :=\nbegin\n  split,\n  { introI,\n    constructor,\n    intros cb n n' a h,\n    have : (@parser.decorate_errors α msgs p) cb n = done n' a := by simpa using h,\n    exact of_done this },\n  { introI,\n    constructor,\n    intros _ _ _ _ h,\n    rw decorate_errors_eq_done at h,\n    exact of_done h }\nend\n\ninstance decorate_errors [p.prog] :\n  (@decorate_errors α msgs p).prog :=\n⟨λ _ _ _ _, by { rw decorate_errors_eq_done, exact of_done }⟩\n\nlemma decorate_error_iff : (@parser.decorate_error α msg p).prog ↔ p.prog :=\ndecorate_errors_iff\n\ninstance decorate_error [p.prog] : (@decorate_error α msg p).prog :=\nprog.decorate_errors\n\ninstance any_char : prog any_char :=\nprog.of_step\n\ninstance sat {p : char → Prop} [decidable_pred p] : prog (sat p) :=\nprog.of_step\n\nlemma eps : ¬ prog eps := prog.pure ()\n\ninstance ch {c : char} : prog (ch c) :=\nprog.of_step\n\nlemma char_buf_iff {cb' : char_buffer} : (char_buf cb').prog ↔ cb' ≠ buffer.nil :=\nbegin\n  have : cb' ≠ buffer.nil ↔ cb'.to_list ≠ [] :=\n      not_iff_not_of_iff ⟨λ h, by simp [h], λ h, by simpa using congr_arg list.to_buffer h⟩,\n  rw [char_buf, this, decorate_error_iff],\n  cases cb'.to_list,\n  { simp [pure] },\n  { simp only [iff_true, ne.def, not_false_iff],\n    apply_instance }\nend\n\ninstance one_of {cs : list char} : (one_of cs).prog :=\nprog.decorate_errors\n\ninstance one_of' {cs : list char} : (one_of' cs).prog :=\nprog.and_then\n\nlemma str_iff {s : string} : (str s).prog ↔ s ≠ \"\" :=\nby simp [str_eq_char_buf, char_buf_iff, ←string.to_list_inj, buffer.ext_iff]\n\nlemma remaining : ¬ remaining.prog :=\nbegin\n  introI h,\n  have : remaining buffer.nil 0 = done 0 0 := by simp [remaining_eq_done],\n  replace this : 0 < 0 := prog.of_done this,\n  exact (lt_irrefl _) this\nend\n\nlemma eof : ¬ eof.prog :=\nbegin\n  introI h,\n  have : eof buffer.nil 0 = done 0 () := by simpa [remaining_eq_done],\n  replace this : 0 < 0 := prog.of_done this,\n  exact (lt_irrefl _) this\nend\n\n-- TODO: add foldr and foldl, many, etc, fix_core\n\ninstance many1 [p.mono] [p.prog] : p.many1.prog :=\nbegin\n  constructor,\n  rintro cb n n' (_ | ⟨hd, tl⟩),\n  { simp },\n  { rw many1_eq_done,\n    rintro ⟨np, hp, h⟩,\n    exact (of_done hp).trans_le (mono.of_done h) }\nend\n\nlemma fix_core {F : parser α → parser α} (hF : ∀ (p : parser α), p.prog → (F p).prog) :\n  ∀ (max_depth : ℕ), prog (fix_core F max_depth)\n| 0               := prog.failure\n| (max_depth + 1) := hF _ (fix_core _)\n\ninstance digit : digit.prog :=\nprog.of_step\n\ninstance nat : nat.prog :=\nprog.decorate_error\n\nlemma fix {F : parser α → parser α} (hF : ∀ (p : parser α), p.prog → (F p).prog) :\n  prog (fix F) :=\n⟨λ cb n _ _ h,\n  by { haveI := fix_core hF (cb.size - n + 1), dsimp [fix] at h, exact of_done h }⟩\n\nend prog\n\nvariables {α β : Type} {msgs : thunk (list string)} {msg : thunk string}\nvariables {p q : parser α} {cb : char_buffer} {n n' : ℕ} {err : dlist string}\nvariables {a : α} {b : β}\n\nsection many\n\n-- TODO: generalize to p.prog instead of p.step\nlemma many_sublist_of_done [p.step] [p.bounded] {l : list α}\n  (h : p.many cb n = done n' l) :\n  ∀ k < n' - n, p.many cb (n + k) = done n' (l.drop k) :=\nbegin\n  induction l with hd tl hl generalizing n,\n  { rw many_eq_done_nil at h,\n    simp [h.left] },\n  intros m hm,\n  cases m,\n  { exact h },\n  rw [list.drop, nat.add_succ, ←nat.succ_add],\n  apply hl,\n  { rw [←many1_eq_done_iff_many_eq_done, many1_eq_done] at h,\n    obtain ⟨_, hp, h⟩ := h,\n    convert h,\n    exact (step.of_done hp).symm },\n  { exact nat.lt_pred_iff.mpr hm },\nend\n\nlemma many_eq_nil_of_done [p.step] [p.bounded] {l : list α}\n  (h : p.many cb n = done n' l) :\n  p.many cb n' = done n' [] :=\nbegin\n  induction l with hd tl hl generalizing n,\n  { convert h,\n    rw many_eq_done_nil at h,\n    exact h.left.symm },\n  { rw [←many1_eq_done_iff_many_eq_done, many1_eq_done] at h,\n    obtain ⟨_, -, h⟩ := h,\n    exact hl h }\nend\n\nlemma many_eq_nil_of_out_of_bound [p.bounded] {l : list α}\n  (h : p.many cb n = done n' l) (hn : cb.size < n) :\n  n' = n ∧ l = [] :=\nbegin\n  cases l,\n  { rw many_eq_done_nil at h,\n    exact ⟨h.left.symm, rfl⟩ },\n  { rw many_eq_done at h,\n    obtain ⟨np, hp, -⟩ := h,\n    exact absurd (bounded.of_done hp) hn.not_lt }\nend\n\nlemma many1_length_of_done [p.mono] [p.step] [p.bounded] {l : list α}\n  (h : many1 p cb n = done n' l) :\n  l.length = n' - n :=\nbegin\n  induction l with hd tl hl generalizing n n',\n  { simpa using h },\n  { obtain ⟨k, hk⟩ : ∃ k, n' = n + k + 1 := nat.exists_eq_add_of_lt (prog.of_done h),\n    subst hk,\n    simp only [many1_eq_done] at h,\n    obtain ⟨_, hp, h⟩ := h,\n    obtain rfl := step.of_done hp,\n    cases tl,\n    { simp only [many_eq_done_nil, add_left_inj, exists_and_distrib_right, self_eq_add_right] at h,\n      rcases h with ⟨rfl, -⟩,\n      simp },\n    rw ←many1_eq_done_iff_many_eq_done at h,\n    specialize hl h,\n    simp [hl, add_comm, add_assoc, nat.sub_succ] }\nend\n\nlemma many1_bounded_of_done [p.step] [p.bounded] {l : list α}\n  (h : many1 p cb n = done n' l) :\n  n' ≤ cb.size :=\nbegin\n  induction l with hd tl hl generalizing n n',\n  { simpa using h },\n  { simp only [many1_eq_done] at h,\n    obtain ⟨np, hp, h⟩ := h,\n    obtain rfl := step.of_done hp,\n    cases tl,\n    { simp only [many_eq_done_nil, exists_and_distrib_right] at h,\n      simpa [←h.left] using bounded.of_done hp },\n    { rw ←many1_eq_done_iff_many_eq_done at h,\n      exact hl h } }\nend\n\nend many\n\nsection nat\n/--\nThe `val : ℕ` produced by a successful parse of a `cb : char_buffer` is the numerical value\nrepresented by the string of decimal digits (possibly padded with 0s on the left)\nstarting from the parsing position `n` and ending at position `n'`. The number\nof characters parsed in is necessarily `n' - n`.\n\nThis is one of the directions of `nat_eq_done`.\n-/\nlemma nat_of_done {val : ℕ} (h : nat cb n = done n' val) :\n  val = (nat.of_digits 10 ((((cb.to_list.drop n).take (n' - n)).reverse.map\n          (λ c, c.to_nat - '0'.to_nat)))) :=\nbegin\n  /- The parser `parser.nat` that generates a decimal number from a string of digit characters does\n  several things. First it ingests in as many digits as it can with `many1 digit`. Then, it folds\n  over the resulting `list ℕ` using a helper function that keeps track of both the running sum an\n  and the magnitude so far, using a `(sum, magnitude) : (ℕ × ℕ)` pair. The final sum is extracted\n  using a `prod.fst`.\n\n  To prove that the value that `parser.nat` produces, after moving precisely `n' - n` steps, is\n  precisely what `nat.of_digits` would give, if supplied the string that is in the ingested\n  `char_buffer` (modulo conversion from `char` to `ℕ ), we need to induct over the length `n' - n`\n  of `cb : char_buffer` ingested, and prove that the parser must have terminated due to hitting\n  either the end of the `char_buffer` or a non-digit character.\n\n  The statement of the lemma is phrased using a combination of `list.drop` and `list.map` because\n  there is no currently better way to extract an \"interval\" from a `char_buffer`. Additionally, the\n  statement uses a `list.reverse` because `nat.of_digits` is little-endian.\n\n  We try to stop referring to the `cb : char_buffer` as soon as possible, so that we can instead\n  regard a `list char` instead, which lends itself better to proofs via induction.\n  -/\n  /- We first prove some helper lemmas about the definition of `parser.nat`. Since it is defined\n  in core, we have to work with how it is defined instead of changing its definition.\n  In its definition, the function that folds over the parsed in digits is defined internally,\n  as a lambda with anonymous destructor syntax, which leads to an unpleasant `nat._match_1` term\n  when rewriting the definition of `parser.nat` away. Since we know exactly what the function is,\n  we have a `rfl`-like lemma here to rewrite it back into a readable form.\n  -/\n  have natm : nat._match_1 = (λ (d : ℕ) p, ⟨p.1 + d * p.2, p.2 * 10⟩),\n  { ext1, ext1 ⟨⟩, refl },\n  -- We also have to prove what is the `prod.snd` of the result of the fold of a `list (ℕ × ℕ)` with\n  -- the function above. We use this lemma later when we finish our inductive case.\n  have hpow : ∀ l, (list.foldr (λ (digit : ℕ) (x : ℕ × ℕ), (x.fst + digit * x.snd, x.snd * 10))\n    (0, 1) l).snd = 10 ^ l.length,\n  { intro l,\n    induction l with hd tl hl,\n    { simp },\n    { simp [hl, pow_succ, mul_comm] } },\n  -- We convert the hypothesis that `parser.nat` has succeeded into an existential that there is\n  -- some list of digits that it has parsed in, and that those digits, when folded over by the\n  -- function above, give the value at hand.\n  simp only [nat, pure_eq_done, natm, decorate_error_eq_done, bind_eq_done] at h,\n  obtain ⟨n', l, hp, rfl, rfl⟩ := h,\n  -- We now want to stop working with the `cb : char_buffer` and parse positions `n` and `n'`,\n  -- and just deal with the parsed digit list `l : list ℕ`. To do so, we have to show that\n  -- this is precisely the list that could have been parsed in, no smaller and no greater.\n  induction l with lhd ltl IH generalizing n n' cb,\n  { -- Base case: we parsed in no digits whatsoever. But this is impossible because `parser.many1`\n    -- must produce a list that is not `list.nil`, by `many1_ne_done_nil`.\n    simpa using hp },\n  -- Inductive case:\n  -- We must prove that the first digit parsed in `lhd : ℕ` is precisely the digit that is\n  -- represented by the character at position `n` in `cb : char_buffer`.\n  -- We will also prove the correspondence between the subsequent digits `ltl : list ℕ` and the\n  -- remaining characters past position `n` up to position `n'`.\n  cases hx : (list.drop n (buffer.to_list cb)) with chd ctl,\n  { -- Are there even characters left to parse, at position `n` in the `cb : char_buffer`? In other\n    -- words, are we already out of bounds, and thus could not have parsed in any value\n    -- successfully. But this must be a contradiction because `parser.digit` is a `bounded` parser,\n    -- (due to its being defined via `parser.decorate_error`), which means it only succeeds\n    -- in-bounds, and the `many1` parser combinator retains that property.\n    have : cb.size ≤ n := by simpa using list.drop_eq_nil_iff_le.mp hx,\n    exact absurd (bounded.of_done hp) this.not_lt },\n  -- We prove that the first digit parsed in is precisely the digit that is represented by the\n  -- character at position `n`, which we now call `chd : char`.\n  have chdh : chd.to_nat - '0'.to_nat = lhd,\n    { simp only [many1_eq_done] at hp,\n      -- We know that `parser.digit` succeeded, so it has moved to a possibly different position.\n      -- In fact, we know that this new position is `n + 1`, by the `step` property of\n      -- `parser.digit`.\n      obtain ⟨_, hp, -⟩ := hp,\n      obtain rfl := step.of_done hp,\n      -- We now unfold what it means for `parser.digit` to succeed, which means that the character\n      -- parsed in was \"numeric\" (for some definition of that property), and, more importantly,\n      -- that the `n`th character of `cb`, let's say `c`, when converted to a `ℕ` via\n      -- `char.to_nat c - '0'.to_nat`, must be equal to the resulting value, `lhd` in our case.\n      simp only [digit_eq_done, buffer.read_eq_nth_le_to_list, hx, buffer.length_to_list, true_and,\n                 add_left_inj, list.length, list.nth_le, eq_self_iff_true, exists_and_distrib_left,\n                 fin.coe_mk] at hp,\n      rcases hp with ⟨_, hn, rfl, _, _⟩,\n      -- But we already know the list corresponding to `cb : char_buffer` from position `n` and on\n      -- is equal to `(chd :: ctl) : list char`, so our `c` above must satisfy `c = chd`.\n      have hn' : n < cb.to_list.length := by simpa using hn,\n      rw ←list.cons_nth_le_drop_succ hn' at hx,\n      -- We can ignore proving any correspondence of `ctl : list char` to the other portions of the\n      -- `cb : char_buffer`.\n      simp only at hx,\n      simp [hx] },\n  -- We know that we parsed in more than one character because of the `prog` property of\n  -- `parser.digit`, which the `many1` parser combinator retains. In other words, we know that\n  -- `n < n'`, and so, the list of digits `ltl` must correspond to the list of digits that\n  -- `digit.many1 cb (n + 1)` would produce. We know that the shift of `1` in `n ↦ n + 1` holds\n  -- due to the `step` property of `parser.digit`.\n  -- We also get here `k : ℕ` which will indicate how many characters we parsed in past position\n  -- `n`. We will prove later that this must be the number of digits we produced as well in `ltl`.\n  obtain ⟨k, hk⟩ : ∃ k, n' = n + k + 1 := nat.exists_eq_add_of_lt (prog.of_done hp),\n  have hdm : ltl = [] ∨ digit.many1 cb (n + 1) = done n' ltl,\n  { cases ltl,\n    { simp },\n    { rw many1_eq_done at hp,\n      obtain ⟨_, hp, hp'⟩ := hp,\n      simpa [step.of_done hp, many1_eq_done_iff_many_eq_done] using hp' } },\n  -- Now we case on the two possibilities, that there was only a single digit parsed in, and\n  -- `ltl = []`, or, had we started parsing at `n + 1` instead, we'd parse in the value associated\n  -- with `ltl`.\n  -- We prove that the LHS, which is a fold over a `list ℕ` is equal to the RHS, which is that\n  -- the `val : ℕ` that `nat.of_digits` produces when supplied a `list ℕ that has been produced\n  -- via mapping a `list char` using `char.to_nat`. Specifically, that `list char` are the\n  -- characters in the `cb : char_buffer`, from position `n` to position `n'` (excluding `n'`),\n  -- in reverse.\n  rcases hdm with rfl|hdm,\n  { -- Case that `ltl = []`.\n    simp only [many1_eq_done, many_eq_done_nil, exists_and_distrib_right] at hp,\n    -- This means we must have failed parsing with `parser.digit` at some other position,\n    -- which we prove must be `n + 1` via the `step` property.\n    obtain ⟨_, hp, rfl, hp'⟩ := hp,\n    obtain rfl := step.of_done hp,\n    -- Now we rely on the simplifier, which simplfies the LHS, which is a fold over a singleton\n    -- list. On the RHS, `list.take (n + 1 - n)` also produces a singleton list, which, when\n    -- reversed, is the same list. `nat.of_digits` of a singleton list is precisely the value in\n    -- the list. And we already have that `chd.to_nat - '0'.to_nat = lhd`.\n    simp [chdh] },\n  -- We now have to deal with the case where we parsed in more than one digit, and thus\n  -- `n + 1 < n'`, which means `ctl` has one or more elements. Similarly, `ltl` has one or more\n  -- elements.\n  -- We finish ridding ourselves of references to `cb : char_buffer`, by relying on the fact that\n  -- our `ctl : list char` must be the appropriate portion of `cb` once enough elements have been\n  -- dropped and taken.\n  have rearr :\n    list.take (n + (k + 1) - (n + 1)) (list.drop (n + 1) (buffer.to_list cb)) = ctl.take k,\n  { simp [←list.tail_drop, hx, nat.sub_succ, hk] },\n  -- We have to prove that the number of digits produced (given by `ltl`) is equal to the number\n  -- of characters parsed in, as given by `ctl.take k`, and that this is precisely `k`. We phrase it\n  -- in the statement using `min`, because lemmas about `list.length (list.take ...)` simplify to\n  -- a statement that uses `min`. The `list.length` term appears from the reduction of the folding\n  -- function, as proven above.\n  have ltll : min k ctl.length = ltl.length,\n  { -- Here is an example of how statements about the `list.length` of `list.take` simplify.\n    have : (ctl.take k).length = min k ctl.length := by simp,\n    -- We bring back the underlying definition of `ctl` as the result of a sequence of `list.take`\n    -- and `list.drop`, so that lemmas about `list.length` of those can fire.\n    rw [←this, ←rearr, many1_length_of_done hdm],\n    -- Likewise, we rid ourselves of the `k` we generated earlier.\n    have : k = n' - n - 1,\n      { simp [hk, add_assoc] },\n    subst this,\n    simp only [nat.sub_succ, add_comm, ←nat.pred_sub, buffer.length_to_list, nat.pred_one_add,\n                min_eq_left_iff, list.length_drop, add_tsub_cancel_left, list.length_take,\n                tsub_zero],\n    -- We now have a goal of proving an inequality dealing with `nat` subtraction and `nat.pred`,\n    -- both of which require special care to provide positivity hypotheses.\n    rw [tsub_le_tsub_iff_right, nat.pred_le_iff],\n    { -- We know that `n' ≤ cb.size` because of the `bounded` property, that a parser will not\n      -- produce a `done` result at a position farther than the size of the underlying\n      -- `char_buffer`.\n      convert many1_bounded_of_done hp,\n      -- What is now left to prove is that `0 < cb.size`, which can be rephrased\n      -- as proving that it is nonempty.\n      cases hc : cb.size,\n      { -- Proof by contradiction. Let's say that `cb.size = 0`. But we know that we succeeded\n        -- parsing in at position `n` using a `bounded` parser, so we must have that\n        -- `n < cb.size`.\n        have := bounded.of_done hp,\n        rw hc at this,\n        -- But then `n < 0`, a contradiction.\n        exact absurd n.zero_le this.not_le },\n      { simp } },\n    { -- Here, we use the same result as above, that `n < cb.size`, and relate it to\n      -- `n ≤ cb.size.pred`.\n      exact nat.le_pred_of_lt (bounded.of_done hp) } },\n  -- Finally, we simplify. On the LHS, we have a fold over `lhd :: ltl`, which simplifies to\n  -- the operation of the summing folding function on `lhd` and the fold over `ltl`. To that we can\n  -- apply the induction hypothesis, because we know that our parser would have succeeded had we\n  -- started at position `n + 1`. We replace mentions of `cb : char_buffer` with the appropriate\n  -- `chd :: ctl`, replace `lhd` with the appropriate statement of how it is calculated from `chd`,\n  -- and use the lemmas describing the length of `ltl` and how it is associated with `k`. We also\n  -- remove mentions of `n'` and replace with an expression using solely `n + k + 1`.\n  -- We use the lemma we proved above about how the folding function produces the\n  -- `prod.snd` value, which is `10` to the power of the length of the list provided to the fold.\n  -- Finally, we rely on `nat.of_digits_append` for the related statement of how digits given\n  -- are used in the `nat.of_digits` calculation, which also involves `10 ^ list.length ...`.\n  -- The `list.append` operation appears due to the `list.reverse (chd :: ctl)`.\n  -- We include some addition and multiplication lemmas to help the simplifier rearrange terms.\n  simp [IH _ hdm, hx, hk, rearr, ←chdh, ←ltll, hpow, add_assoc, nat.of_digits_append, mul_comm]\nend\n\n/--\nIf we know that `parser.nat` was successful, starting at position `n` and ending at position `n'`,\nthen it must be the case that for all `k : ℕ`, `n ≤ k`, `k < n'`, the character at the `k`th\nposition in `cb : char_buffer` is \"numeric\", that is, is between `'0'` and `'9'` inclusive.\n\nThis is a necessary part of proving one of the directions of `nat_eq_done`.\n-/\nlemma nat_of_done_as_digit {val : ℕ} (h : nat cb n = done n' val) :\n  ∀ (hn : n' ≤ cb.size) k (hk : k < n'), n ≤ k →\n    '0' ≤ cb.read ⟨k, hk.trans_le hn⟩ ∧ cb.read ⟨k, hk.trans_le hn⟩ ≤ '9' :=\nbegin\n  -- The properties to be shown for the characters involved rely solely on the success of\n  -- `parser.digit` at the relevant positions, and not on the actual value `parser.nat` produced.\n  -- We break done the success of `parser.nat` into the `parser.digit` success and throw away\n  -- the resulting value given by `parser.nat`, and focus solely on the `list ℕ` generated by\n  -- `parser.digit.many1`.\n  simp only [nat, pure_eq_done, and.left_comm, decorate_error_eq_done, bind_eq_done,\n             exists_eq_left, exists_and_distrib_left] at h,\n  obtain ⟨xs, h, -⟩ := h,\n  -- We want to avoid having to make statements about the `cb : char_buffer` itself. Instead, we\n  -- induct on the `xs : list ℕ` that `parser.digit.many1` produced.\n  induction xs with hd tl hl generalizing n n',\n  { -- Base case: `xs` is empty. But this is a contradiction because `many1` always produces a\n    -- nonempty list, as proven by `many1_ne_done_nil`.\n    simpa using h },\n  -- Inductive case: we prove that the `parser.digit.many1` produced a valid `(hd :: tl) : list ℕ`,\n  -- by showing that is the case for the character at position `n`, which gave `hd`, and use the\n  -- induction hypothesis on the remaining `tl`.\n  -- We break apart a `many1` success into a success of the underlying `parser.digit` to give `hd`\n  -- and a `parser.digit.many` which gives `tl`. We first deal with the `hd`.\n  rw many1_eq_done at h,\n  -- Right away, we can throw away the information about the \"new\" position that `parser.digit`\n  -- ended on because we will soon prove that it must have been `n + 1`.\n  obtain ⟨_, hp, h⟩ := h,\n  -- The main lemma here is `digit_eq_done`, which already proves the necessary conditions about\n  -- the character at hand. What is left to do is properly unpack the information.\n  simp only [digit_eq_done, and.comm, and.left_comm, digit_eq_fail, true_and, exists_eq_left,\n             eq_self_iff_true, exists_and_distrib_left, exists_and_distrib_left] at hp,\n  obtain ⟨rfl, -, hn, ge0, le9, rfl⟩ := hp,\n  -- Let's now consider a position `k` between `n` and `n'`, excluding `n'`.\n  intros hn k hk hk',\n  -- What if we are at `n`? What if we are past `n`? We case on the `n ≤ k`.\n  rcases hk'.eq_or_lt with rfl|hk',\n  { -- The `n = k` case. But this is exactly what we know already, so we provide the\n    -- relevant hypotheses.\n    exact ⟨ge0, le9⟩ },\n  -- The `n < k` case. First, we check if there would have even been digits parsed in. So, we\n  -- case on `tl : list ℕ`\n  cases tl,\n  { -- Case where `tl = []`. But that means `many` gave us a `[]` so either the character at\n    -- position `k` was not \"numeric\" or we are out of bounds. More importantly, when `many`\n    -- successfully produces a `[]`, it does not progress the parser head, so we have that\n    -- `n + 1 = n'`. This will lead to a contradiction because now we have `n < k` and `k < n + 1`.\n    simp only [many_eq_done_nil, exists_and_distrib_right] at h,\n    -- Extract out just the `n + 1 = n'`.\n    obtain ⟨rfl, -⟩ := h,\n    -- Form the contradictory hypothesis, and discharge the goal.\n    have : k < k := hk.trans_le (nat.succ_le_of_lt hk'),\n    exact absurd this (lt_irrefl _) },\n  { -- Case where `tl ≠ []`. But that means that `many` produced a nonempty list as a result, so\n    -- `many1` would have successfully parsed at this position too. We use this statement to\n    -- rewrite our hypothesis into something that works with the induction hypothesis, and apply it.\n    rw ←many1_eq_done_iff_many_eq_done at h,\n    apply hl h,\n    -- All that is left to prove is that our `k` is at least our new \"lower bound\" `n + 1`, which\n    -- we have from our original split of the `n ≤ k`, since we are now on the `n < k` case.\n    exact nat.succ_le_of_lt hk' }\nend\n\n/--\nIf we know that `parser.nat` was successful, starting at position `n` and ending at position `n'`,\nthen it must be the case that for the ending position `n'`, either it is beyond the end of the\n`cb : char_buffer`, or the character at that position is not \"numeric\", that is,  between `'0'` and\n`'9'` inclusive.\n\nThis is a necessary part of proving one of the directions of `nat_eq_done`.\n-/\nlemma nat_of_done_bounded {val : ℕ} (h : nat cb n = done n' val) :\n  ∀ (hn : n' < cb.size), '0' ≤ cb.read ⟨n', hn⟩ → '9' < cb.read ⟨n', hn⟩ :=\nbegin\n  -- The properties to be shown for the characters involved rely solely on the success of\n  -- `parser.digit` at the relevant positions, and not on the actual value `parser.nat` produced.\n  -- We break done the success of `parser.nat` into the `parser.digit` success and throw away\n  -- the resulting value given by `parser.nat`, and focus solely on the `list ℕ` generated by\n  -- `parser.digit.many1`.\n  -- We deal with the case of `n'` is \"out-of-bounds\" right away by requiring that\n  -- `∀ (hn : n' < cb.size)`. Thus we only have to prove the lemma for the cases where `n'` is still\n  -- \"in-bounds\".\n  simp only [nat, pure_eq_done, and.left_comm, decorate_error_eq_done, bind_eq_done,\n             exists_eq_left, exists_and_distrib_left] at h,\n  obtain ⟨xs, h, -⟩ := h,\n  -- We want to avoid having to make statements about the `cb : char_buffer` itself. Instead, we\n  -- induct on the `xs : list ℕ` that `parser.digit.many1` produced.\n  induction xs with hd tl hl generalizing n n',\n  { -- Base case: `xs` is empty. But this is a contradiction because `many1` always produces a\n    -- nonempty list, as proven by `many1_ne_done_nil`.\n    simpa using h },\n  -- Inductive case: at least one character has been parsed in, starting at position `n`.\n  -- We know that the size of `cb : char_buffer` must be at least `n + 1` because\n  -- `parser.digit.many1` is `bounded` (`n < cb.size`).\n  -- We show that either we parsed in just that one character, or we use the inductive hypothesis.\n  obtain ⟨k, hk⟩ : ∃ k, cb.size = n + k + 1 := nat.exists_eq_add_of_lt (bounded.of_done h),\n  cases tl,\n  { -- Case where `tl = []`, so we parsed in only `hd`. That must mean that `parser.digit` failed\n    -- at `n + 1`.\n    simp only [many1_eq_done, many_eq_done_nil, and.left_comm, exists_and_distrib_right,\n               exists_eq_left] at h,\n    -- We throw away the success information of what happened at position `n`, and we do not need\n    -- the \"error\" value that the failure produced.\n    obtain ⟨-, _, h⟩ := h,\n    -- If `parser.digit` failed at `n + 1`, then either we hit a non-numeric character, or\n    -- we are out of bounds. `digit_eq_fail` provides us with those two cases.\n    simp only [digit_eq_done, and.comm, and.left_comm, digit_eq_fail, true_and, exists_eq_left,\n               eq_self_iff_true, exists_and_distrib_left] at h,\n    obtain (⟨rfl, h⟩ | ⟨h, -⟩) := h,\n    { -- First case: we are still in bounds, but the character is not numeric. We must prove\n      -- that we are still in bounds. But we know that from our initial requirement.\n      intro hn,\n      simpa using h hn },\n    { -- Second case: we are out of bounds, and somehow the fold that `many1` relied on failed.\n      -- But we know that `parser.digit` is mono, that is, it never goes backward in position,\n      -- in neither success nor in failure. We also have that `foldr_core` respects `mono`.\n      -- But in this case, `foldr_core` is starting at position `n' + 1` but failing at\n      -- position `n'`, which is a contradiction, because otherwise we would have `n' + 1 ≤ n'`.\n      simpa using mono.of_fail h } },\n  { -- Case where `tl ≠ []`. But that means that `many` produced a nonempty list as a result, so\n    -- `many1` would have successfully parsed at this position too. We use this statement to\n    -- rewrite our hypothesis into something that works with the induction hypothesis, and apply it.\n    rw many1_eq_done at h,\n    obtain ⟨_, -, h⟩ := h,\n    rw ←many1_eq_done_iff_many_eq_done at h,\n    exact hl h }\nend\n\n/--\nThe `val : ℕ` produced by a successful parse of a `cb : char_buffer` is the numerical value\nrepresented by the string of decimal digits (possibly padded with 0s on the left)\nstarting from the parsing position `n` and ending at position `n'`, where `n < n'`. The number\nof characters parsed in is necessarily `n' - n`. Additionally, all of the characters in the `cb`\nstarting at position `n` (inclusive) up to position `n'` (exclusive) are \"numeric\", in that they\nare between `'0'` and `'9'` inclusive. Such a `char_buffer` would produce the `ℕ` value encoded\nby its decimal characters.\n-/\nlemma nat_eq_done {val : ℕ} : nat cb n = done n' val ↔ ∃ (hn : n < n'),\n  val = (nat.of_digits 10 ((((cb.to_list.drop n).take (n' - n)).reverse.map\n          (λ c, c.to_nat - '0'.to_nat)))) ∧ (∀ (hn' : n' < cb.size),\n          ('0' ≤ cb.read ⟨n', hn'⟩ → '9' < cb.read ⟨n', hn'⟩)) ∧ ∃ (hn'' : n' ≤ cb.size),\n          (∀ k (hk : k < n'), n ≤ k →\n          '0' ≤ cb.read ⟨k, hk.trans_le hn''⟩ ∧ cb.read ⟨k, hk.trans_le hn''⟩ ≤ '9') :=\nbegin\n  -- To prove this iff, we have most of the way in the forward direction, using the lemmas proven\n  -- above. First, we must use that `parser.nat` is `prog`, which means that on success, it must\n  -- move forward. We also have to prove the statement that a success means the parsed in\n  -- characters were properly \"numeric\". It involves first generating ane existential witness\n  -- that the parse was completely \"in-bounds\".\n  -- For the reverse direction, we first discharge the goals that deal with proving that our parser\n  -- succeeded because it encountered characters with the proper \"numeric\" properties, was\n  -- \"in-bounds\" and hit a nonnumeric character. The more difficult portion is proving that the\n  -- list of characters from positions `n` to `n'`, when folded over by the function defined inside\n  -- `parser.nat` gives exactly the same value as `nat.of_digits` when supplied with the same\n  -- (modulo rearrangement) list. To reach this goal, we try to remove any reliance on the\n  -- underlying `cb : char_buffer` or parsers as soon as possible, via a cased-induction.\n  refine ⟨λ h, ⟨prog.of_done h, nat_of_done h, nat_of_done_bounded h, _⟩, _⟩,\n  { -- To provide the existential witness that `n'` is within the bounds of the `cb : char_buffer`,\n    -- we rely on the fact that `parser.nat` is primarily a `parser.digit.many1`, and that `many1`,\n    -- must finish with the bounds of the `cb`, as long as the underlying parser is `step` and\n    -- `bounded`, which `digit` is. We do not prove this as a separate lemma about `parser.nat`\n    -- because it would almost always be only relevant in this larger theorem.\n    -- We clone the success hypothesis `h` so that we can supply it back later.\n    have H := h,\n    -- We unwrap the `parser.nat` success down to the `many1` success, throwing away other info.\n    rw [nat] at h,\n    simp only [decorate_error_eq_done, bind_eq_done, pure_eq_done, and.left_comm, exists_eq_left,\n               exists_and_distrib_left] at h,\n    obtain ⟨_, h, -⟩ := h,\n    -- Now we get our existential witness that `n' ≤ cb.size`.\n    replace h := many1_bounded_of_done h,\n    -- With that, we can use the lemma proved above that our characters are \"numeric\"\n    exact ⟨h, nat_of_done_as_digit H h⟩ },\n  -- We now prove that given the `cb : char_buffer` with characters within the `n ≤ k < n'` interval\n  -- properly \"numeric\" and such that their `nat.of_digits` generates the `val : ℕ`, `parser.nat`\n  -- of that `cb`, when starting at `n`, will finish at `n'` and produce the same `val`.\n  -- We first introduce the relevant hypotheses, including the fact that we have a valid interval\n  -- where `n < n'` and that characters at `n'` and beyond are no longer numeric.\n  rintro ⟨hn, hv, hb, hn', ho⟩,\n  -- We first unwrap the `parser.nat` definition to the underlying `parser.digit.many1` success\n  -- and the fold function of the digits.\n  rw nat,\n  simp only [and.left_comm, pure_eq_done, hv, decorate_error_eq_done, list.map_reverse,\n             bind_eq_done, exists_eq_left, exists_and_distrib_left],\n  -- We won't actually need the `val : ℕ` itself, since it is entirely characterized by the\n  -- underlying characters. Instead, we will induct over the `list char` of characters from\n  -- position `n` onwards, showing that if we could have provided a list at `n`, we could have\n  -- provided a valid list of characters at `n + 1` too.\n  clear hv val,\n    /- We first prove some helper lemmas about the definition of `parser.nat`. Since it is defined\n  in core, we have to work with how it is defined instead of changing its definition.\n  In its definition, the function that folds over the parsed in digits is defined internally,\n  as a lambda with anonymous destructor syntax, which leads to an unpleasant `nat._match_1` term\n  when rewriting the definition of `parser.nat` away. Since we know exactly what the function is,\n  we have a `rfl`-like lemma here to rewrite it back into a readable form.\n  -/\n  have natm : nat._match_1 = (λ (d : ℕ) p, ⟨p.1 + d * p.2, p.2 * 10⟩),\n  { ext1, ext1 ⟨⟩, refl },\n  -- We induct over the characters available at position `n` and onwards. Because `cb` is used\n  -- in other expressions, we utilize the `induction H : ...` tactic to induct separately from\n  -- destructing `cb` itself.\n  induction H : (cb.to_list.drop n) with hd tl IH generalizing n,\n  { -- Base case: there are no characters at position `n` or onwards, which means that\n    -- `cb.size ≤ n`. But this is a contradiction, since we have `n < n' ≤ cb.size`.\n    rw list.drop_eq_nil_iff_le at H,\n    refine absurd ((lt_of_le_of_lt H hn).trans_le hn') _,\n    simp },\n  { -- Inductive case: we prove that if we could have parsed from `n + 1`, we could have also parsed\n    -- from `n`, if there was a valid numerical character at `n`. Most of the body\n    -- of this inductive case is generating the appropriate conditions for use of the inductive\n    -- hypothesis.\n    specialize @IH (n + 1),\n    -- We have, by the inductive case, that there is at least one character `hd` at position `n`,\n    -- with the rest at `tl`. We rearrange our inductive case to make `tl` be expressed as\n    -- list.drop (n + 1), which fits out induction hypothesis conditions better. To use the\n    -- rearranging lemma, we must prove that we are \"dropping\" in bounds, which we supply on-the-fly\n    simp only [←list.cons_nth_le_drop_succ\n      (show n < cb.to_list.length, by simpa using hn.trans_le hn')] at H,\n    -- We prove that parsing our `n`th character, `hd`, would have resulted in a success from\n    -- `parser.digit`, with the appropriate `ℕ` success value. We use this later to simplify the\n    -- unwrapped fold, since `hd` is our head character.\n    have hdigit : digit cb n = done (n + 1) (hd.to_nat - '0'.to_nat),\n    { -- By our necessary condition, we know that `n` is in bounds, and that the `n`th character\n      -- has the necessary \"numeric\" properties.\n      specialize ho n hn le_rfl,\n      -- We prove an additional result that the conversion of `hd : char` to a `ℕ` would give a\n      -- value `x ≤ 9`, since that is part of the iff statement in the `digit_eq_done` lemma.\n      have : (buffer.read cb ⟨n, hn.trans_le hn'⟩).to_nat - '0'.to_nat ≤ 9,\n      { -- We rewrite the statement to be a statement about characters instead, and split the\n        -- inequality into the case that our hypotheses prove, and that `'0' ≤ '9'`, which\n        -- is true by computation, handled by `dec_trivial`.\n        rw [show 9 = '9'.to_nat - '0'.to_nat, from dec_trivial, tsub_le_tsub_iff_right],\n        { exact ho.right },\n        { dec_trivial } },\n        -- We rely on the simplifier, mostly powered by `digit_eq_done`, and supply all the\n        -- necessary conditions of bounds and identities about `hd`.\n        simp [digit_eq_done, this, ←H.left, buffer.nth_le_to_list, hn.trans_le hn', ho] },\n    -- We now case on whether we've moved to the end of our parse or not. We phrase this as\n    -- casing on either `n + 1 < n` or `n ≤ n + 1`. The more difficult goal comes first.\n    cases lt_or_ge (n + 1) n' with hn'' hn'',\n    { -- Case `n + 1 < n'`. We can directly supply this to our induction hypothesis.\n      -- We now have to prove, for the induction hypothesis, that the characters at positions `k`,\n      -- `n + 1 ≤ k < n'` are \"numeric\". We already had this for `n ≤ k < n`, so we just rearrange\n      -- the hypotheses we already have.\n      specialize IH hn'' _ H.right,\n      { intros k hk hk',\n        apply ho,\n        exact nat.le_of_succ_le hk' },\n      -- With the induction hypothesis conditions satisfier, we can extract out a list that\n      -- `parser.digit.many1` would have generated from position `n + 1`, as well as the associated\n      -- property of the list, that it folds into what `nat.of_digits` generates from the\n      -- characters in `cb : char_buffer`, now known as `hd :: tl`.\n      obtain ⟨l, hdl, hvl⟩ := IH,\n      -- Of course, the parsed in list from position `n` would be `l` prepended with the result\n      -- of parsing in `hd`, which is provided explicitly.\n      use (hd.to_nat - '0'.to_nat) :: l,\n      -- We case on `l : list ℕ` so that we can make statements about the fold on `l`\n      cases l with lhd ltl,\n      { -- As before, if `l = []` then `many1` produced a `[]` success, which is a contradiction.\n        simpa using hdl },\n      -- Case `l = lhd :: ltl`. We can rewrite the fold of the function inside `parser.nat` on\n      -- `lhd :: ltl`, which will be used to rewrite in the goal.\n      simp only [natm, list.foldr] at hvl,\n      -- We also expand the fold in the goal, using the expanded fold from our hypothesis, powered\n      -- by `many1_eq_done` to proceed in the parsing. We know exactly what the next `many` will\n      -- produce from `many1_eq_done_iff_many_eq_done.mp` of our `hdl` hypothesis. Finally,\n      -- we also use `hdigit` to express what the single `parser.digit` result would be at `n`.\n      simp only [natm, hvl, many1_eq_done, hdigit, many1_eq_done_iff_many_eq_done.mp hdl, true_and,\n                 and_true, eq_self_iff_true, list.foldr, exists_eq_left'],\n      -- Now our goal is solely about the equality of two different folding functions, one from the\n      -- function defined inside `parser.nat` and the other as `nat.of_digits`, when applied to\n      -- similar list inputs.\n      -- First, we rid ourselves of `n'` by replacing with `n + m + 1`, which allows us to\n      -- simplify the term of how many elements we are keeping using a `list.take`.\n      obtain ⟨m, rfl⟩ : ∃ m, n' = n + m + 1 := nat.exists_eq_add_of_lt hn,\n      -- The following rearrangement lemma is to simplify the `list.take (n' - n)` expression we had\n      have : n + m + 1 - n = m + 1,\n        { rw [add_assoc, tsub_eq_iff_eq_add_of_le, add_comm],\n          exact nat.le_add_right _ _ },\n      -- We also have to prove what is the `prod.snd` of the result of the fold of a `list (ℕ × ℕ)`\n      -- with the function above. We use this lemma to finish our inductive case.\n      have hpow : ∀ l, (list.foldr (λ (digit : ℕ) (x : ℕ × ℕ),\n        (x.fst + digit * x.snd, x.snd * 10)) (0, 1) l).snd = 10 ^ l.length,\n      { intro l,\n        induction l with hd tl hl,\n        { simp },\n        { simp [hl, pow_succ, mul_comm] } },\n      -- We prove that the parsed list of digits `(lhd :: ltl) : list ℕ` must be of length `m`\n      -- which is used later when the `parser.nat` fold places `ltl.length` in the exponent.\n      have hml : ltl.length + 1 = m := by simpa using many1_length_of_done hdl,\n      -- A simplified `list.length (list.take ...)` expression refers to the minimum of the\n      -- underlying length and the amount of elements taken. We know that `m ≤ tl.length`, so\n      -- we provide this auxiliary lemma so that the simplified \"take-length\" can simplify further\n      have ltll : min m tl.length = m,\n      { -- On the way to proving this, we have to actually show that `m ≤ tl.length`, by showing\n        -- that since `tl` was a subsequence in `cb`, and was retrieved from `n + 1` to `n + m + 1`,\n        -- then since `n + m + 1 ≤ cb.size`, we have that `tl` must be at least `m` in length.\n        simpa [←H.right, le_tsub_iff_right (hn''.trans_le hn').le, add_comm, add_assoc,\n               add_left_comm] using hn' },\n      -- Finally, we rely on the simplifier. We already expressions of `nat.of_digits` on both\n      -- the LHS and RHS. All that is left to do is to prove that the summand on the LHS is produced\n      -- by the fold of `nat.of_digits` on the RHS of `hd :: tl`. The `nat.of_digits_append` is used\n      -- because of the append that forms from the included `list.reverse`. The lengths of the lists\n      -- are placed in the exponents with `10` as a base, and are combined using `←pow_succ 10`.\n      -- Any complicated expression about list lengths is further simplified by the auxiliary\n      -- lemmas we just proved. Finally, we assist the simplifier by rearranging terms with our\n      -- `n + m + 1 - n = m + 1` proof and `mul_comm`.\n      simp [this, hpow, nat.of_digits_append, mul_comm, ←pow_succ 10, hml, ltll] },\n    { -- Consider the case that `n' ≤ n + 1`. But then since `n < n' ≤ n + 1`, `n' = n + 1`.\n      obtain rfl : n' = n + 1 := le_antisymm hn'' (nat.succ_le_of_lt hn),\n      -- This means we have only parsed in a single character, so the resulting parsed in list\n      -- is explicitly formed from an expression we can construct from `hd`.\n      use [[hd.to_nat - '0'.to_nat]],\n      -- Our list expression simplifies nicely because it is a fold over a singleton, so we\n      -- do not have to supply any auxiliary lemmas for it, other than what we already know about\n      -- `hd` and the function defined in `parser.nat`. However, we will have to prove that our\n      -- parse ended because of a good reason: either we are out of bounds or we hit a nonnumeric\n      -- character.\n      simp only [many1_eq_done, many_eq_done_nil, digit_eq_fail, natm, and.comm, and.left_comm,\n                 hdigit, true_and, mul_one, nat.of_digits_singleton, list.take, exists_eq_left,\n                 exists_and_distrib_right, add_tsub_cancel_left, eq_self_iff_true,\n                 list.reverse_singleton, zero_add, list.foldr, list.map],\n      -- We take the route of proving that we hit a nonnumeric character, since we already have\n      -- a hypothesis that says that characters at `n'` and past it are nonnumeric. (Note, by now\n      -- we have substituted `n + 1` for `n'.\n      -- We are also asked to provide the error value that our failed parse would report. But\n      -- `digit_eq_fail` already knows what it is, so we can discharge that with an inline `rfl`.\n      refine ⟨_, or.inl ⟨rfl, _⟩⟩,\n      -- The nonnumeric condition looks almost exactly like the hypothesis we already have, so\n      -- we let the simplifier align them for us\n      simpa using hb } }\nend\n\nend nat\n\nend parser\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/buffer/parser/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3315220333719304}}
{"text": "variable {α : Type*}\n\ndef is_prefix (l₁ : list α) (l₂ : list α) : Prop :=\n  ∃ t, l₁ ++ t = l₂\n\ninfix ` <+: `:50 := is_prefix\n\n@[simp]\ntheorem list.is_prefix_refl (l : list α) : l <+: l :=\n  ⟨[], by simp⟩\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch6/ex0402.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3312966214110021}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.pi\nimport Mathlib.algebra.big_operators.basic\nimport Mathlib.data.set.finite\nimport Mathlib.group_theory.submonoid.basic\nimport Mathlib.PostPort\n\nuniverses u v l v₁ v₂ w u_1 u_2 u_3 u_4 u₁ x \n\nnamespace Mathlib\n\n/-!\n# Dependent functions with finite support\n\nFor a non-dependent version see `data/finsupp.lean`.\n-/\n\nnamespace dfinsupp\n\n\nstructure pre (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] where\n  to_fun : (i : ι) → β i\n  pre_support : multiset ι\n  zero : ∀ (i : ι), i ∈ pre_support ∨ to_fun i = 0\n\nprotected instance inhabited_pre (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] :\n    Inhabited (pre ι β) :=\n  { default := pre.mk (fun (i : ι) => 0) ∅ sorry }\n\nprotected instance pre.setoid (ι : Type u) (β : ι → Type v) [(i : ι) → HasZero (β i)] :\n    setoid (pre ι β) :=\n  setoid.mk (fun (x y : pre ι β) => ∀ (i : ι), pre.to_fun x i = pre.to_fun y i) sorry\n\nend dfinsupp\n\n\n/-- A dependent function `Π i, β i` with finite support. -/\ndef dfinsupp {ι : Type u} (β : ι → Type v) [(i : ι) → HasZero (β i)] := quotient sorry\n\ninfixl:25 \" →ₚ \" => Mathlib.dfinsupp\n\nnamespace dfinsupp\n\n\nprotected instance has_coe_to_fun {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] :\n    has_coe_to_fun (dfinsupp fun (i : ι) => β i) :=\n  has_coe_to_fun.mk (fun (_x : dfinsupp fun (i : ι) => β i) => (i : ι) → β i)\n    fun (f : dfinsupp fun (i : ι) => β i) => quotient.lift_on f pre.to_fun sorry\n\nprotected instance has_zero {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] :\n    HasZero (dfinsupp fun (i : ι) => β i) :=\n  { zero := quotient.mk (pre.mk (fun (i : ι) => 0) ∅ sorry) }\n\nprotected instance inhabited {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] :\n    Inhabited (dfinsupp fun (i : ι) => β i) :=\n  { default := 0 }\n\n@[simp] theorem zero_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (i : ι) :\n    coe_fn 0 i = 0 :=\n  rfl\n\ntheorem ext {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)]\n    {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i}\n    (H : ∀ (i : ι), coe_fn f i = coe_fn g i) : f = g :=\n  sorry\n\n/-- The composition of `f : β₁ → β₂` and `g : Π₀ i, β₁ i` is\n  `map_range f hf g : Π₀ i, β₂ i`, well defined when `f 0 = 0`. -/\ndef map_range {ι : Type u} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)]\n    [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i) (hf : ∀ (i : ι), f i 0 = 0)\n    (g : dfinsupp fun (i : ι) => β₁ i) : dfinsupp fun (i : ι) => β₂ i :=\n  quotient.lift_on g\n    (fun (x : pre ι fun (i : ι) => β₁ i) =>\n      quotient.mk (pre.mk (fun (i : ι) => f i (pre.to_fun x i)) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem map_range_apply {ι : Type u} {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n    [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] (f : (i : ι) → β₁ i → β₂ i)\n    (hf : ∀ (i : ι), f i 0 = 0) (g : dfinsupp fun (i : ι) => β₁ i) (i : ι) :\n    coe_fn (map_range f hf g) i = f i (coe_fn g i) :=\n  quotient.induction_on g fun (x : pre ι fun (i : ι) => β₁ i) => rfl\n\n/-- Let `f i` be a binary operation `β₁ i → β₂ i → β i` such that `f i 0 0 = 0`.\nThen `zip_with f hf` is a binary operation `Π₀ i, β₁ i → Π₀ i, β₂ i → Π₀ i, β i`. -/\ndef zip_with {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {β₁ : ι → Type v₁}\n    {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)]\n    (f : (i : ι) → β₁ i → β₂ i → β i) (hf : ∀ (i : ι), f i 0 0 = 0)\n    (g₁ : dfinsupp fun (i : ι) => β₁ i) (g₂ : dfinsupp fun (i : ι) => β₂ i) :\n    dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on₂ g₁ g₂\n    (fun (x : pre ι fun (i : ι) => β₁ i) (y : pre ι fun (i : ι) => β₂ i) =>\n      quotient.mk\n        (pre.mk (fun (i : ι) => f i (pre.to_fun x i) (pre.to_fun y i))\n          (pre.pre_support x + pre.pre_support y) sorry))\n    sorry\n\n@[simp] theorem zip_with_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)]\n    {β₁ : ι → Type v₁} {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)]\n    (f : (i : ι) → β₁ i → β₂ i → β i) (hf : ∀ (i : ι), f i 0 0 = 0)\n    (g₁ : dfinsupp fun (i : ι) => β₁ i) (g₂ : dfinsupp fun (i : ι) => β₂ i) (i : ι) :\n    coe_fn (zip_with f hf g₁ g₂) i = f i (coe_fn g₁ i) (coe_fn g₂ i) :=\n  quotient.induction_on₂ g₁ g₂\n    fun (_x : pre ι fun (i : ι) => β₁ i) (_x_1 : pre ι fun (i : ι) => β₂ i) => rfl\n\nprotected instance has_add {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] :\n    Add (dfinsupp fun (i : ι) => β i) :=\n  { add := zip_with (fun (_x : ι) => Add.add) sorry }\n\n@[simp] theorem add_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)]\n    (g₁ : dfinsupp fun (i : ι) => β i) (g₂ : dfinsupp fun (i : ι) => β i) (i : ι) :\n    coe_fn (g₁ + g₂) i = coe_fn g₁ i + coe_fn g₂ i :=\n  zip_with_apply (fun (_x : ι) => Add.add) has_add._proof_1 g₁ g₂ i\n\nprotected instance add_monoid {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)] :\n    add_monoid (dfinsupp fun (i : ι) => β i) :=\n  add_monoid.mk Add.add sorry 0 sorry sorry\n\nprotected instance is_add_monoid_hom {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)]\n    {i : ι} : is_add_monoid_hom fun (g : dfinsupp fun (i : ι) => β i) => coe_fn g i :=\n  is_add_monoid_hom.mk (zero_apply i)\n\nprotected instance has_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] :\n    Neg (dfinsupp fun (i : ι) => β i) :=\n  { neg := fun (f : dfinsupp fun (i : ι) => β i) => map_range (fun (_x : ι) => Neg.neg) sorry f }\n\nprotected instance add_comm_monoid {ι : Type u} {β : ι → Type v} [(i : ι) → add_comm_monoid (β i)] :\n    add_comm_monoid (dfinsupp fun (i : ι) => β i) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\n@[simp] theorem neg_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)]\n    (g : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (-g) i = -coe_fn g i :=\n  map_range_apply (fun (_x : ι) => Neg.neg) has_neg._proof_1 g i\n\nprotected instance add_group {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)] :\n    add_group (dfinsupp fun (i : ι) => β i) :=\n  add_group.mk add_monoid.add sorry add_monoid.zero sorry sorry Neg.neg\n    (sub_neg_monoid.sub._default add_monoid.add sorry add_monoid.zero sorry sorry Neg.neg) sorry\n\n@[simp] theorem sub_apply {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)]\n    (g₁ : dfinsupp fun (i : ι) => β i) (g₂ : dfinsupp fun (i : ι) => β i) (i : ι) :\n    coe_fn (g₁ - g₂) i = coe_fn g₁ i - coe_fn g₂ i :=\n  sorry\n\nprotected instance add_comm_group {ι : Type u} {β : ι → Type v} [(i : ι) → add_comm_group (β i)] :\n    add_comm_group (dfinsupp fun (i : ι) => β i) :=\n  add_comm_group.mk add_group.add sorry add_group.zero sorry sorry add_group.neg add_group.sub sorry\n    sorry\n\n/-- Dependent functions with finite support inherit a semiring action from an action on each\ncoordinate. -/\nprotected instance has_scalar {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ]\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] :\n    has_scalar γ (dfinsupp fun (i : ι) => β i) :=\n  has_scalar.mk\n    fun (c : γ) (v : dfinsupp fun (i : ι) => β i) =>\n      map_range (fun (_x : ι) => has_scalar.smul c) sorry v\n\n@[simp] theorem smul_apply {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ]\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] (b : γ)\n    (v : dfinsupp fun (i : ι) => β i) (i : ι) : coe_fn (b • v) i = b • coe_fn v i :=\n  map_range_apply (fun (_x : ι) => has_scalar.smul b) (has_scalar._proof_1 b) v i\n\n/-- Dependent functions with finite support inherit a semimodule structure from such a structure on\neach coordinate. -/\nprotected instance semimodule {ι : Type u} {β : ι → Type v} {γ : Type w} [semiring γ]\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] :\n    semimodule γ (dfinsupp fun (i : ι) => β i) :=\n  semimodule.mk sorry sorry\n\n/-- `filter p f` is the function which is `f i` if `p i` is true and 0 otherwise. -/\ndef filter {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop) [decidable_pred p]\n    (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk (pre.mk (fun (i : ι) => ite (p i) (pre.to_fun x i) 0) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem filter_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop)\n    [decidable_pred p] (i : ι) (f : dfinsupp fun (i : ι) => β i) :\n    coe_fn (filter p f) i = ite (p i) (coe_fn f i) 0 :=\n  quotient.induction_on f fun (x : pre ι fun (i : ι) => β i) => rfl\n\ntheorem filter_apply_pos {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop}\n    [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) {i : ι} (h : p i) :\n    coe_fn (filter p f) i = coe_fn f i :=\n  sorry\n\ntheorem filter_apply_neg {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] {p : ι → Prop}\n    [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) {i : ι} (h : ¬p i) :\n    coe_fn (filter p f) i = 0 :=\n  sorry\n\ntheorem filter_pos_add_filter_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)]\n    (f : dfinsupp fun (i : ι) => β i) (p : ι → Prop) [decidable_pred p] :\n    filter p f + filter (fun (i : ι) => ¬p i) f = f :=\n  sorry\n\n/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -/\ndef subtype_domain {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)] (p : ι → Prop)\n    [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : Subtype p) => β ↑i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk\n        (pre.mk (fun (i : Subtype p) => pre.to_fun x ↑i)\n          (multiset.map\n            (fun (j : Subtype fun (x_1 : ι) => x_1 ∈ multiset.filter p (pre.pre_support x)) =>\n              { val := ↑j, property := sorry })\n            (multiset.attach (multiset.filter p (pre.pre_support x))))\n          sorry))\n    sorry\n\n@[simp] theorem subtype_domain_zero {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)]\n    {p : ι → Prop} [decidable_pred p] : subtype_domain p 0 = 0 :=\n  rfl\n\n@[simp] theorem subtype_domain_apply {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)]\n    {p : ι → Prop} [decidable_pred p] {i : Subtype p} {v : dfinsupp fun (i : ι) => β i} :\n    coe_fn (subtype_domain p v) i = coe_fn v ↑i :=\n  quotient.induction_on v fun (x : pre ι fun (i : ι) => β i) => rfl\n\n@[simp] theorem subtype_domain_add {ι : Type u} {β : ι → Type v} [(i : ι) → add_monoid (β i)]\n    {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i}\n    {v' : dfinsupp fun (i : ι) => β i} :\n    subtype_domain p (v + v') = subtype_domain p v + subtype_domain p v' :=\n  sorry\n\nprotected instance subtype_domain.is_add_monoid_hom {ι : Type u} {β : ι → Type v}\n    [(i : ι) → add_monoid (β i)] {p : ι → Prop} [decidable_pred p] :\n    is_add_monoid_hom (subtype_domain p) :=\n  is_add_monoid_hom.mk subtype_domain_zero\n\n@[simp] theorem subtype_domain_neg {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)]\n    {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i} :\n    subtype_domain p (-v) = -subtype_domain p v :=\n  sorry\n\n@[simp] theorem subtype_domain_sub {ι : Type u} {β : ι → Type v} [(i : ι) → add_group (β i)]\n    {p : ι → Prop} [decidable_pred p] {v : dfinsupp fun (i : ι) => β i}\n    {v' : dfinsupp fun (i : ι) => β i} :\n    subtype_domain p (v - v') = subtype_domain p v - subtype_domain p v' :=\n  sorry\n\ntheorem finite_supp {ι : Type u} {β : ι → Type v} [(i : ι) → HasZero (β i)]\n    (f : dfinsupp fun (i : ι) => β i) : set.finite (set_of fun (i : ι) => coe_fn f i ≠ 0) :=\n  sorry\n\n/-- Create an element of `Π₀ i, β i` from a finset `s` and a function `x`\ndefined on this `finset`. -/\ndef mk {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (s : finset ι)\n    (x : (i : ↥↑s) → β ↑i) : dfinsupp fun (i : ι) => β i :=\n  quotient.mk\n    (pre.mk\n      (fun (i : ι) =>\n        dite (i ∈ s) (fun (H : i ∈ s) => x { val := i, property := H }) fun (H : ¬i ∈ s) => 0)\n      (finset.val s) sorry)\n\n@[simp] theorem mk_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {s : finset ι} {x : (i : ↥↑s) → β ↑i} {i : ι} :\n    coe_fn (mk s x) i =\n        dite (i ∈ s) (fun (H : i ∈ s) => x { val := i, property := H }) fun (H : ¬i ∈ s) => 0 :=\n  rfl\n\ntheorem mk_injective {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    (s : finset ι) : function.injective (mk s) :=\n  sorry\n\n/-- The function `single i b : Π₀ i, β i` sends `i` to `b`\nand all other points to `0`. -/\ndef single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (i : ι)\n    (b : β i) : dfinsupp fun (i : ι) => β i :=\n  mk (singleton i) fun (j : ↥↑(singleton i)) => eq.rec_on sorry b\n\n@[simp] theorem single_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} {i' : ι} {b : β i} :\n    coe_fn (single i b) i' =\n        dite (i = i') (fun (h : i = i') => eq.rec_on h b) fun (h : ¬i = i') => 0 :=\n  sorry\n\n@[simp] theorem single_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} : single i 0 = 0 :=\n  sorry\n\n@[simp] theorem single_eq_same {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} {b : β i} : coe_fn (single i b) i = b :=\n  sorry\n\ntheorem single_eq_of_ne {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} {i' : ι} {b : β i} (h : i ≠ i') :\n    coe_fn (single i b) i' = 0 :=\n  sorry\n\ntheorem single_injective {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} : function.injective (single i) :=\n  sorry\n\n/-- Like `finsupp.single_eq_single_iff`, but with a `heq` due to dependent types -/\ntheorem single_eq_single_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] (i : ι) (j : ι) (xi : β i) (xj : β j) :\n    single i xi = single j xj ↔ i = j ∧ xi == xj ∨ xi = 0 ∧ xj = 0 :=\n  sorry\n\n/-- Redefine `f i` to be `0`. -/\ndef erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)] (i : ι)\n    (f : dfinsupp fun (i : ι) => β i) : dfinsupp fun (i : ι) => β i :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      quotient.mk\n        (pre.mk (fun (j : ι) => ite (j = i) 0 (pre.to_fun x j)) (pre.pre_support x) sorry))\n    sorry\n\n@[simp] theorem erase_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} {j : ι} {f : dfinsupp fun (i : ι) => β i} :\n    coe_fn (erase i f) j = ite (j = i) 0 (coe_fn f j) :=\n  quotient.induction_on f fun (x : pre ι fun (i : ι) => β i) => rfl\n\n@[simp] theorem erase_same {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} :\n    coe_fn (erase i f) i = 0 :=\n  sorry\n\ntheorem erase_ne {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    {i : ι} {i' : ι} {f : dfinsupp fun (i : ι) => β i} (h : i' ≠ i) :\n    coe_fn (erase i f) i' = coe_fn f i' :=\n  sorry\n\n@[simp] theorem single_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {i : ι} {b₁ : β i} {b₂ : β i} :\n    single i (b₁ + b₂) = single i b₁ + single i b₂ :=\n  sorry\n\n/-- `dfinsupp.single` as an `add_monoid_hom`. -/\n@[simp] theorem single_add_hom_apply {ι : Type u} (β : ι → Type v) [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] (i : ι) (b : β i) : coe_fn (single_add_hom β i) b = single i b :=\n  Eq.refl (coe_fn (single_add_hom β i) b)\n\ntheorem single_add_erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} :\n    single i (coe_fn f i) + erase i f = f :=\n  sorry\n\ntheorem erase_add_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {i : ι} {f : dfinsupp fun (i : ι) => β i} :\n    erase i f + single i (coe_fn f i) = f :=\n  sorry\n\nprotected theorem induction {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {p : (dfinsupp fun (i : ι) => β i) → Prop}\n    (f : dfinsupp fun (i : ι) => β i) (h0 : p 0)\n    (ha :\n      ∀ (i : ι) (b : β i) (f : dfinsupp fun (i : ι) => β i),\n        coe_fn f i = 0 → b ≠ 0 → p f → p (single i b + f)) :\n    p f :=\n  sorry\n\ntheorem induction₂ {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)]\n    {p : (dfinsupp fun (i : ι) => β i) → Prop} (f : dfinsupp fun (i : ι) => β i) (h0 : p 0)\n    (ha :\n      ∀ (i : ι) (b : β i) (f : dfinsupp fun (i : ι) => β i),\n        coe_fn f i = 0 → b ≠ 0 → p f → p (f + single i b)) :\n    p f :=\n  sorry\n\n@[simp] theorem add_closure_Union_range_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] :\n    add_submonoid.closure (set.Union fun (i : ι) => set.range (single i)) = ⊤ :=\n  sorry\n\n/-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then\nthey are equal. -/\ntheorem add_hom_ext {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)]\n    {γ : Type w} [add_monoid γ] {f : (dfinsupp fun (i : ι) => β i) →+ γ}\n    {g : (dfinsupp fun (i : ι) => β i) →+ γ}\n    (H : ∀ (i : ι) (y : β i), coe_fn f (single i y) = coe_fn g (single i y)) : f = g :=\n  sorry\n\n/-- If two additive homomorphisms from `Π₀ i, β i` are equal on each `single a b`, then\nthey are equal.\n\nSee note [partially-applied ext lemmas]. -/\ntheorem add_hom_ext' {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {γ : Type w} [add_monoid γ]\n    {f : (dfinsupp fun (i : ι) => β i) →+ γ} {g : (dfinsupp fun (i : ι) => β i) →+ γ}\n    (H :\n      ∀ (x : ι),\n        add_monoid_hom.comp f (single_add_hom β x) = add_monoid_hom.comp g (single_add_hom β x)) :\n    f = g :=\n  add_hom_ext fun (x : ι) => add_monoid_hom.congr_fun (H x)\n\n@[simp] theorem mk_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_monoid (β i)] {s : finset ι} {x : (i : ↥↑s) → β ↑i} {y : (i : ↥↑s) → β ↑i} :\n    mk s (x + y) = mk s x + mk s y :=\n  sorry\n\n@[simp] theorem mk_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] {s : finset ι} : mk s 0 = 0 :=\n  sorry\n\n@[simp] theorem mk_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_group (β i)] {s : finset ι} {x : (i : ↥↑s) → β (subtype.val i)} :\n    mk s (-x) = -mk s x :=\n  sorry\n\n@[simp] theorem mk_sub {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_group (β i)] {s : finset ι} {x : (i : ↥↑s) → β (subtype.val i)}\n    {y : (i : ↥↑s) → β (subtype.val i)} : mk s (x - y) = mk s x - mk s y :=\n  sorry\n\nprotected instance mk.is_add_group_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_group (β i)] {s : finset ι} : is_add_group_hom (mk s) :=\n  is_add_group_hom.mk\n\n@[simp] theorem mk_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] (γ : Type w)\n    [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] {s : finset ι}\n    {c : γ} (x : (i : ↥↑s) → β (subtype.val i)) : mk s (c • x) = c • mk s x :=\n  sorry\n\n@[simp] theorem single_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] (γ : Type w)\n    [semiring γ] [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)] {i : ι} {c : γ}\n    {x : β i} : single i (c • x) = c • single i x :=\n  sorry\n\n/-- Set `{i | f x ≠ 0}` as a `finset`. -/\ndef support {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) : finset ι :=\n  quotient.lift_on f\n    (fun (x : pre ι fun (i : ι) => β i) =>\n      finset.filter (fun (i : ι) => pre.to_fun x i ≠ 0) (multiset.to_finset (pre.pre_support x)))\n    sorry\n\n@[simp] theorem support_mk_subset {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {s : finset ι}\n    {x : (i : ↥↑s) → β (subtype.val i)} : support (mk s x) ⊆ s :=\n  fun (i : ι) (H : i ∈ support (mk s x)) =>\n    iff.mp multiset.mem_to_finset (and.left (iff.mp finset.mem_filter H))\n\n@[simp] theorem mem_support_to_fun {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    (f : dfinsupp fun (i : ι) => β i) (i : ι) : i ∈ support f ↔ coe_fn f i ≠ 0 :=\n  sorry\n\ntheorem eq_mk_support {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (f : dfinsupp fun (i : ι) => β i) :\n    f = mk (support f) fun (i : ↥↑(support f)) => coe_fn f ↑i :=\n  sorry\n\n@[simp] theorem support_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] : support 0 = ∅ :=\n  rfl\n\ntheorem mem_support_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    (f : dfinsupp fun (i : ι) => β i) (i : ι) : i ∈ support f ↔ coe_fn f i ≠ 0 :=\n  mem_support_to_fun f\n\n@[simp] theorem support_eq_empty {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    {f : dfinsupp fun (i : ι) => β i} : support f = ∅ ↔ f = 0 :=\n  sorry\n\nprotected instance decidable_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] : decidable_pred (Eq 0) :=\n  fun (f : dfinsupp fun (i : ι) => β i) => decidable_of_iff (support f = ∅) sorry\n\ntheorem support_subset_iff {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {s : set ι}\n    {f : dfinsupp fun (i : ι) => β i} : ↑(support f) ⊆ s ↔ ∀ (i : ι), ¬i ∈ s → coe_fn f i = 0 :=\n  sorry\n\ntheorem support_single_ne_zero {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i}\n    (hb : b ≠ 0) : support (single i b) = singleton i :=\n  sorry\n\ntheorem support_single_subset {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {i : ι} {b : β i} :\n    support (single i b) ⊆ singleton i :=\n  support_mk_subset\n\ntheorem map_range_def {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n    [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)]\n    {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {g : dfinsupp fun (i : ι) => β₁ i} :\n    map_range f hf g =\n        mk (support g) fun (i : ↥↑(support g)) => f (subtype.val i) (coe_fn g (subtype.val i)) :=\n  sorry\n\n@[simp] theorem map_range_single {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁}\n    {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)]\n    {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0} {i : ι} {b : β₁ i} :\n    map_range f hf (single i b) = single i (f i b) :=\n  sorry\n\ntheorem support_map_range {ι : Type u} [dec : DecidableEq ι] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n    [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)]\n    [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i}\n    {hf : ∀ (i : ι), f i 0 = 0} {g : dfinsupp fun (i : ι) => β₁ i} :\n    support (map_range f hf g) ⊆ support g :=\n  sorry\n\ntheorem zip_with_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {β₁ : ι → Type v₁} {β₂ : ι → Type v₂}\n    [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)] [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)]\n    [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)] {f : (i : ι) → β₁ i → β₂ i → β i}\n    {hf : ∀ (i : ι), f i 0 0 = 0} {g₁ : dfinsupp fun (i : ι) => β₁ i}\n    {g₂ : dfinsupp fun (i : ι) => β₂ i} :\n    zip_with f hf g₁ g₂ =\n        mk (support g₁ ∪ support g₂)\n          fun (i : ↥↑(support g₁ ∪ support g₂)) =>\n            f (subtype.val i) (coe_fn g₁ (subtype.val i)) (coe_fn g₂ (subtype.val i)) :=\n  sorry\n\ntheorem support_zip_with {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {β₁ : ι → Type v₁}\n    {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)]\n    [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)]\n    {f : (i : ι) → β₁ i → β₂ i → β i} {hf : ∀ (i : ι), f i 0 0 = 0}\n    {g₁ : dfinsupp fun (i : ι) => β₁ i} {g₂ : dfinsupp fun (i : ι) => β₂ i} :\n    support (zip_with f hf g₁ g₂) ⊆ support g₁ ∪ support g₂ :=\n  sorry\n\ntheorem erase_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι) (f : dfinsupp fun (i : ι) => β i) :\n    erase i f =\n        mk (finset.erase (support f) i)\n          fun (j : ↥↑(finset.erase (support f) i)) => coe_fn f (subtype.val j) :=\n  sorry\n\n@[simp] theorem support_erase {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (i : ι)\n    (f : dfinsupp fun (i : ι) => β i) : support (erase i f) = finset.erase (support f) i :=\n  sorry\n\ntheorem filter_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop} [decidable_pred p]\n    (f : dfinsupp fun (i : ι) => β i) :\n    filter p f =\n        mk (finset.filter p (support f))\n          fun (i : ↥↑(finset.filter p (support f))) => coe_fn f (subtype.val i) :=\n  sorry\n\n@[simp] theorem support_filter {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop}\n    [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) :\n    support (filter p f) = finset.filter p (support f) :=\n  sorry\n\ntheorem subtype_domain_def {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop}\n    [decidable_pred p] (f : dfinsupp fun (i : ι) => β i) :\n    subtype_domain p f =\n        mk (finset.subtype p (support f))\n          fun (i : ↥↑(finset.subtype p (support f))) => coe_fn f ↑i :=\n  sorry\n\n@[simp] theorem support_subtype_domain {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {p : ι → Prop}\n    [decidable_pred p] {f : dfinsupp fun (i : ι) => β i} :\n    support (subtype_domain p f) = finset.subtype p (support f) :=\n  sorry\n\ntheorem support_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] [(i : ι) → add_monoid (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {g₁ : dfinsupp fun (i : ι) => β i}\n    {g₂ : dfinsupp fun (i : ι) => β i} : support (g₁ + g₂) ⊆ support g₁ ∪ support g₂ :=\n  support_zip_with\n\n@[simp] theorem support_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    {f : dfinsupp fun (i : ι) => β i} : support (-f) = support f :=\n  sorry\n\ntheorem support_smul {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [semiring γ]\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → semimodule γ (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] (b : γ) (v : dfinsupp fun (i : ι) => β i) :\n    support (b • v) ⊆ support v :=\n  support_map_range\n\nprotected instance decidable_eq {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → HasZero (β i)] [(i : ι) → DecidableEq (β i)] :\n    DecidableEq (dfinsupp fun (i : ι) => β i) :=\n  fun (f g : dfinsupp fun (i : ι) => β i) =>\n    decidable_of_iff (support f = support g ∧ ∀ (i : ι), i ∈ support f → coe_fn f i = coe_fn g i)\n      sorry\n\n-- [to_additive sum] for dfinsupp.prod doesn't work, the equation lemmas are not generated\n\n/-- `sum f g` is the sum of `g i (f i)` over the support of `f`. -/\ndef sum {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] (f : dfinsupp fun (i : ι) => β i)\n    (g : (i : ι) → β i → γ) : γ :=\n  finset.sum (support f) fun (i : ι) => g i (coe_fn f i)\n\n/-- `prod f g` is the product of `g i (f i)` over the support of `f`. -/\ndef prod {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w} [(i : ι) → HasZero (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ] (f : dfinsupp fun (i : ι) => β i)\n    (g : (i : ι) → β i → γ) : γ :=\n  finset.prod (support f) fun (i : ι) => g i (coe_fn f i)\n\ntheorem prod_map_range_index {ι : Type u} [dec : DecidableEq ι] {γ : Type w} {β₁ : ι → Type v₁}\n    {β₂ : ι → Type v₂} [(i : ι) → HasZero (β₁ i)] [(i : ι) → HasZero (β₂ i)]\n    [(i : ι) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → (x : β₂ i) → Decidable (x ≠ 0)]\n    [comm_monoid γ] {f : (i : ι) → β₁ i → β₂ i} {hf : ∀ (i : ι), f i 0 = 0}\n    {g : dfinsupp fun (i : ι) => β₁ i} {h : (i : ι) → β₂ i → γ} (h0 : ∀ (i : ι), h i 0 = 1) :\n    prod (map_range f hf g) h = prod g fun (i : ι) (b : β₁ i) => h i (f i b) :=\n  sorry\n\ntheorem sum_zero_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    {h : (i : ι) → β i → γ} : sum 0 h = 0 :=\n  rfl\n\ntheorem sum_single_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ] {i : ι}\n    {b : β i} {h : (i : ι) → β i → γ} (h_zero : h i 0 = 0) : sum (single i b) h = h i b :=\n  sorry\n\ntheorem sum_neg_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} (h0 : ∀ (i : ι), h i 0 = 0) :\n    sum (-g) h = sum g fun (i : ι) (b : β i) => h i (-b) :=\n  sum_map_range_index h0\n\ntheorem sum_comm {γ : Type w} {ι₁ : Type u_1} {ι₂ : Type u_2} {β₁ : ι₁ → Type u_3}\n    {β₂ : ι₂ → Type u_4} [DecidableEq ι₁] [DecidableEq ι₂] [(i : ι₁) → HasZero (β₁ i)]\n    [(i : ι₂) → HasZero (β₂ i)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)]\n    [(i : ι₂) → (x : β₂ i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    (f₁ : dfinsupp fun (i : ι₁) => β₁ i) (f₂ : dfinsupp fun (i : ι₂) => β₂ i)\n    (h : (i : ι₁) → β₁ i → (i : ι₂) → β₂ i → γ) :\n    (sum f₁ fun (i₁ : ι₁) (x₁ : β₁ i₁) => sum f₂ fun (i₂ : ι₂) (x₂ : β₂ i₂) => h i₁ x₁ i₂ x₂) =\n        sum f₂ fun (i₂ : ι₂) (x₂ : β₂ i₂) => sum f₁ fun (i₁ : ι₁) (x₁ : β₁ i₁) => h i₁ x₁ i₂ x₂ :=\n  finset.sum_comm\n\n@[simp] theorem sum_apply {ι : Type u} {β : ι → Type v} {ι₁ : Type u₁} [DecidableEq ι₁]\n    {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)] [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)]\n    [(i : ι) → add_comm_monoid (β i)] {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁}\n    {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i} {i₂ : ι} :\n    coe_fn (sum f g) i₂ = sum f fun (i₁ : ι₁) (b : β₁ i₁) => coe_fn (g i₁ b) i₂ :=\n  Eq.symm (finset.sum_hom (support f) fun (f : dfinsupp fun (i : ι) => β i) => coe_fn f i₂)\n\ntheorem support_sum {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {ι₁ : Type u₁}\n    [DecidableEq ι₁] {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)]\n    [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁}\n    {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i} :\n    support (sum f g) ⊆ finset.bUnion (support f) fun (i : ι₁) => support (g i (coe_fn f i)) :=\n  sorry\n\n@[simp] theorem prod_one {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ]\n    {f : dfinsupp fun (i : ι) => β i} : (prod f fun (i : ι) (b : β i) => 1) = 1 :=\n  finset.prod_const_one\n\n@[simp] theorem sum_add {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    {f : dfinsupp fun (i : ι) => β i} {h₁ : (i : ι) → β i → γ} {h₂ : (i : ι) → β i → γ} :\n    (sum f fun (i : ι) (b : β i) => h₁ i b + h₂ i b) = sum f h₁ + sum f h₂ :=\n  finset.sum_add_distrib\n\n@[simp] theorem sum_neg {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_group γ]\n    {f : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ} :\n    (sum f fun (i : ι) (b : β i) => -h i b) = -sum f h :=\n  finset.sum_hom (support f) Neg.neg\n\ntheorem prod_add_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [comm_monoid γ]\n    {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ}\n    (h_zero : ∀ (i : ι), h i 0 = 1)\n    (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ * h i b₂) :\n    prod (f + g) h = prod f h * prod g h :=\n  sorry\n\n/--\nWhen summing over an `add_monoid_hom`, the decidability assumption is not needed, and the result is\nalso an `add_monoid_hom`.\n-/\ndef sum_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_monoid (β i)] [add_comm_monoid γ] (φ : (i : ι) → β i →+ γ) :\n    (dfinsupp fun (i : ι) => β i) →+ γ :=\n  add_monoid_hom.mk\n    (fun (f : dfinsupp fun (i : ι) => β i) =>\n      quotient.lift_on f\n        (fun (x : pre ι fun (i : ι) => β i) =>\n          finset.sum (multiset.to_finset (pre.pre_support x))\n            fun (i : ι) => coe_fn (φ i) (pre.to_fun x i))\n        sorry)\n    sorry sorry\n\n@[simp] theorem sum_add_hom_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_monoid (β i)] [add_comm_monoid γ] (φ : (i : ι) → β i →+ γ) (i : ι) (x : β i) :\n    coe_fn (sum_add_hom φ) (single i x) = coe_fn (φ i) x :=\n  sorry\n\n@[simp] theorem sum_add_hom_comp_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    {γ : Type w} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ)\n    (i : ι) : add_monoid_hom.comp (sum_add_hom f) (single_add_hom β i) = f i :=\n  add_monoid_hom.ext fun (x : β i) => sum_add_hom_single f i x\n\n/-- While we didn't need decidable instances to define it, we do to reduce it to a sum -/\ntheorem sum_add_hom_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    (φ : (i : ι) → β i →+ γ) (f : dfinsupp fun (i : ι) => β i) :\n    coe_fn (sum_add_hom φ) f = sum f fun (x : ι) => ⇑(φ x) :=\n  sorry\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom`,-/\n@[simp] theorem lift_add_hom_symm_apply {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    {γ : Type w} [(i : ι) → add_monoid (β i)] [add_comm_monoid γ]\n    (F : (dfinsupp fun (i : ι) => β i) →+ γ) (i : ι) :\n    coe_fn (add_equiv.symm lift_add_hom) F i = add_monoid_hom.comp F (single_add_hom β i) :=\n  Eq.refl (coe_fn (add_equiv.symm lift_add_hom) F i)\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_single_add_hom`,-/\n@[simp] theorem lift_add_hom_single_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_comm_monoid (β i)] :\n    coe_fn lift_add_hom (single_add_hom β) = add_monoid_hom.id (dfinsupp fun (i : ι) => β i) :=\n  iff.mpr (equiv.apply_eq_iff_eq_symm_apply (add_equiv.to_equiv lift_add_hom)) rfl\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_apply_single`,-/\ntheorem lift_add_hom_apply_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ) (i : ι)\n    (x : β i) : coe_fn (coe_fn lift_add_hom f) (single i x) = coe_fn (f i) x :=\n  sorry\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_comp_single`,-/\ntheorem lift_add_hom_comp_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] (f : (i : ι) → β i →+ γ) (i : ι) :\n    add_monoid_hom.comp (coe_fn lift_add_hom f) (single_add_hom β i) = f i :=\n  sorry\n\n/-- The `dfinsupp` version of `finsupp.comp_lift_add_hom`,-/\ntheorem comp_lift_add_hom {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    {δ : Type u_1} [(i : ι) → add_comm_monoid (β i)] [add_comm_monoid γ] [add_comm_monoid δ]\n    (g : γ →+ δ) (f : (i : ι) → β i →+ γ) :\n    add_monoid_hom.comp g (coe_fn lift_add_hom f) =\n        coe_fn lift_add_hom fun (a : ι) => add_monoid_hom.comp g (f a) :=\n  sorry\n\ntheorem sum_sub_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → add_comm_group (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_group γ]\n    {f : dfinsupp fun (i : ι) => β i} {g : dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ}\n    (h_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂) :\n    sum (f - g) h = sum f h - sum g h :=\n  sorry\n\ntheorem sum_finset_sum_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    {α : Type x} [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    [add_comm_monoid γ] {s : finset α} {g : α → dfinsupp fun (i : ι) => β i} {h : (i : ι) → β i → γ}\n    (h_zero : ∀ (i : ι), h i 0 = 0)\n    (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ + h i b₂) :\n    (finset.sum s fun (i : α) => sum (g i) h) = sum (finset.sum s fun (i : α) => g i) h :=\n  sorry\n\ntheorem sum_sum_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    {ι₁ : Type u₁} [DecidableEq ι₁] {β₁ : ι₁ → Type v₁} [(i₁ : ι₁) → HasZero (β₁ i₁)]\n    [(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)] [(i : ι) → add_comm_monoid (β i)]\n    [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    {f : dfinsupp fun (i₁ : ι₁) => β₁ i₁} {g : (i₁ : ι₁) → β₁ i₁ → dfinsupp fun (i : ι) => β i}\n    {h : (i : ι) → β i → γ} (h_zero : ∀ (i : ι), h i 0 = 0)\n    (h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ + h i b₂) :\n    sum (sum f g) h = sum f fun (i : ι₁) (b : β₁ i) => sum (g i b) h :=\n  Eq.symm (sum_finset_sum_index h_zero h_add)\n\n@[simp] theorem sum_single {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι]\n    [(i : ι) → add_comm_monoid (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    {f : dfinsupp fun (i : ι) => β i} : sum f single = f :=\n  sorry\n\ntheorem sum_subtype_domain_index {ι : Type u} {β : ι → Type v} [dec : DecidableEq ι] {γ : Type w}\n    [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [add_comm_monoid γ]\n    {v : dfinsupp fun (i : ι) => β i} {p : ι → Prop} [decidable_pred p] {h : (i : ι) → β i → γ}\n    (hp : ∀ (x : ι), x ∈ support v → p x) :\n    (sum (subtype_domain p v) fun (i : Subtype p) (b : β ↑i) => h (↑i) b) = sum v h :=\n  sorry\n\ntheorem subtype_domain_sum {ι : Type u} {β : ι → Type v} {γ : Type w}\n    [(i : ι) → add_comm_monoid (β i)] {s : finset γ} {h : γ → dfinsupp fun (i : ι) => β i}\n    {p : ι → Prop} [decidable_pred p] :\n    subtype_domain p (finset.sum s fun (c : γ) => h c) =\n        finset.sum s fun (c : γ) => subtype_domain p (h c) :=\n  Eq.symm (finset.sum_hom s (subtype_domain p))\n\ntheorem subtype_domain_finsupp_sum {ι : Type u} {β : ι → Type v} {γ : Type w} {δ : γ → Type x}\n    [DecidableEq γ] [(c : γ) → HasZero (δ c)] [(c : γ) → (x : δ c) → Decidable (x ≠ 0)]\n    [(i : ι) → add_comm_monoid (β i)] {p : ι → Prop} [decidable_pred p]\n    {s : dfinsupp fun (c : γ) => δ c} {h : (c : γ) → δ c → dfinsupp fun (i : ι) => β i} :\n    subtype_domain p (sum s h) = sum s fun (c : γ) (d : δ c) => subtype_domain p (h c d) :=\n  subtype_domain_sum\n\nend dfinsupp\n\n\n/-! ### Product and sum lemmas for bundled morphisms -/\n\nnamespace monoid_hom\n\n\n@[simp] theorem map_dfinsupp_prod {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1}\n    {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)]\n    [comm_monoid R] [comm_monoid S] (h : R →* S) (f : dfinsupp fun (i : ι) => β i)\n    (g : (i : ι) → β i → R) :\n    coe_fn h (dfinsupp.prod f g) = dfinsupp.prod f fun (a : ι) (b : β a) => coe_fn h (g a b) :=\n  map_prod h (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f)\n\ntheorem coe_dfinsupp_prod {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1}\n    {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [monoid R]\n    [comm_monoid S] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → R →* S) :\n    ⇑(dfinsupp.prod f g) = dfinsupp.prod f fun (a : ι) (b : β a) => ⇑(g a b) :=\n  coe_prod (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f)\n\n@[simp] theorem dfinsupp_prod_apply {ι : Type u} {β : ι → Type v} [DecidableEq ι] {R : Type u_1}\n    {S : Type u_2} [(i : ι) → HasZero (β i)] [(i : ι) → (x : β i) → Decidable (x ≠ 0)] [monoid R]\n    [comm_monoid S] (f : dfinsupp fun (i : ι) => β i) (g : (i : ι) → β i → R →* S) (r : R) :\n    coe_fn (dfinsupp.prod f g) r = dfinsupp.prod f fun (a : ι) (b : β a) => coe_fn (g a b) r :=\n  finset_prod_apply (fun (i : ι) => g i (coe_fn f i)) (dfinsupp.support f) r\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/dfinsupp_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3312966214110021}}
{"text": "/-\nCopyright (c) 2019 Robert A. Spencer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert A. Spencer, Markus Himmel\n-/\nimport algebra.category.Group.preadditive\nimport category_theory.linear.basic\nimport category_theory.elementwise\nimport linear_algebra.basic\nimport category_theory.conj\nimport category_theory.preadditive.additive_functor\n\n/-!\n# The category of `R`-modules\n\n`Module.{v} R` is the category of bundled `R`-modules with carrier in the universe `v`. We show\nthat it is preadditive and show that being an isomorphism, monomorphism and epimorphism is\nequivalent to being a linear equivalence, an injective linear map and a surjective linear map,\nrespectively.\n\n## Implementation details\n\nTo construct an object in the category of `R`-modules from a type `M` with an instance of the\n`module` typeclass, write `of R M`. There is a coercion in the other direction.\n\nSimilarly, there is a coercion from morphisms in `Module R` to linear maps.\n\nUnfortunately, Lean is not smart enough to see that, given an object `M : Module R`, the expression\n`of R M`, where we coerce `M` to the carrier type, is definitionally equal to `M` itself.\nThis means that to go the other direction, i.e., from linear maps/equivalences to (iso)morphisms\nin the category of `R`-modules, we have to take care not to inadvertently end up with an\n`of R M` where `M` is already an object. Hence, given `f : M →ₗ[R] N`,\n* if `M N : Module R`, simply use `f`;\n* if `M : Module R` and `N` is an unbundled `R`-module, use `↿f` or `as_hom_left f`;\n* if `M` is an unbundled `R`-module and `N : Module R`, use `↾f` or `as_hom_right f`;\n* if `M` and `N` are unbundled `R`-modules, use `↟f` or `as_hom f`.\n\nSimilarly, given `f : M ≃ₗ[R] N`, use `to_Module_iso`, `to_Module_iso'_left`, `to_Module_iso'_right`\nor `to_Module_iso'`, respectively.\n\nThe arrow notations are localized, so you may have to `open_locale Module` to use them. Note that\nthe notation for `as_hom_left` clashes with the notation used to promote functions between types to\nmorphisms in the category `Type`, so to avoid confusion, it is probably a good idea to avoid having\nthe locales `Module` and `category_theory.Type` open at the same time.\n\nIf you get an error when trying to apply a theorem and the `convert` tactic produces goals of the\nform `M = of R M`, then you probably used an incorrect variant of `as_hom` or `to_Module_iso`.\n\n-/\n\nopen category_theory\nopen category_theory.limits\nopen category_theory.limits.walking_parallel_pair\n\nuniverses v u\n\nvariables (R : Type u) [ring R]\n\n/-- The category of R-modules and their morphisms.\n\n Note that in the case of `R = ℤ`, we can not\nimpose here that the `ℤ`-multiplication field from the module structure is defeq to the one coming\nfrom the `is_add_comm_group` structure (contrary to what we do for all module structures in\nmathlib), which creates some difficulties down the road. -/\nstructure Module :=\n(carrier : Type v)\n[is_add_comm_group : add_comm_group carrier]\n[is_module : module R carrier]\n\nattribute [instance] Module.is_add_comm_group Module.is_module\n\nnamespace Module\n\ninstance : has_coe_to_sort (Module.{v} R) (Type v) := ⟨Module.carrier⟩\n\ninstance Module_category : category (Module.{v} R) :=\n{ hom   := λ M N, M →ₗ[R] N,\n  id    := λ M, 1,\n  comp  := λ A B C f g, g.comp f,\n  id_comp' := λ X Y f, linear_map.id_comp _,\n  comp_id' := λ X Y f, linear_map.comp_id _,\n  assoc' := λ W X Y Z f g h, linear_map.comp_assoc _ _ _ }\n\ninstance Module_concrete_category : concrete_category.{v} (Module.{v} R) :=\n{ forget := { obj := λ R, R, map := λ R S f, (f : R → S) },\n  forget_faithful := { } }\n\ninstance has_forget_to_AddCommGroup : has_forget₂ (Module R) AddCommGroup :=\n{ forget₂ :=\n  { obj := λ M, AddCommGroup.of M,\n    map := λ M₁ M₂ f, linear_map.to_add_monoid_hom f } }\n\ninstance (M N : Module R) : linear_map_class (M ⟶ N) R M N :=\n{ coe := λ f, f,\n  .. linear_map.semilinear_map_class }\n\n/-- The object in the category of R-modules associated to an R-module -/\ndef of (X : Type v) [add_comm_group X] [module R X] : Module R := ⟨X⟩\n\n@[simp] lemma forget₂_obj (X : Module R) :\n  (forget₂ (Module R) AddCommGroup).obj X = AddCommGroup.of X :=\nrfl\n\n@[simp] lemma forget₂_obj_Module_of (X : Type v) [add_comm_group X] [module R X] :\n  (forget₂ (Module R) AddCommGroup).obj (of R X) = AddCommGroup.of X :=\nrfl\n\n@[simp] lemma forget₂_map (X Y : Module R) (f : X ⟶ Y) :\n  (forget₂ (Module R) AddCommGroup).map f = linear_map.to_add_monoid_hom f :=\nrfl\n\n/-- Typecheck a `linear_map` as a morphism in `Module R`. -/\ndef of_hom {R : Type u} [ring R] {X Y : Type v} [add_comm_group X] [module R X] [add_comm_group Y]\n  [module R Y] (f : X →ₗ[R] Y) : of R X ⟶ of R Y := f\n\n@[simp] lemma of_hom_apply {R : Type u} [ring R]\n  {X Y : Type v} [add_comm_group X] [module R X] [add_comm_group Y] [module R Y] (f : X →ₗ[R] Y)\n  (x : X) : of_hom f x = f x := rfl\n\ninstance : inhabited (Module R) := ⟨of R punit⟩\n\ninstance of_unique {X : Type v} [add_comm_group X] [module R X] [i : unique X] :\n  unique (of R X) := i\n\n@[simp]\nlemma coe_of (X : Type v) [add_comm_group X] [module R X] : (of R X : Type v) = X := rfl\n\nvariables {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nmodule. -/\n@[simps]\ndef of_self_iso (M : Module R) : Module.of R M ≅ M :=\n{ hom := 𝟙 M, inv := 𝟙 M }\n\nlemma is_zero_of_subsingleton (M : Module R) [subsingleton M] :\n  is_zero M :=\nbegin\n  refine ⟨λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩, λ X, ⟨⟨⟨0⟩, λ f, _⟩⟩⟩,\n  { ext, have : x = 0 := subsingleton.elim _ _, rw [this, map_zero, map_zero], },\n  { ext, apply subsingleton.elim }\nend\n\ninstance : has_zero_object (Module.{v} R) :=\n⟨⟨of R punit, is_zero_of_subsingleton _⟩⟩\n\nvariables {R} {M N U : Module.{v} R}\n\n@[simp] lemma id_apply (m : M) : (𝟙 M : M → M) m = m := rfl\n\n@[simp] lemma coe_comp (f : M ⟶ N) (g : N ⟶ U) :\n  ((f ≫ g) : M → U) = g ∘ f := rfl\n\nlemma comp_def (f : M ⟶ N) (g : N ⟶ U) : f ≫ g = g.comp f := rfl\n\nend Module\n\nvariables {R}\nvariables {X₁ X₂ : Type v}\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom [add_comm_group X₁] [module R X₁] [add_comm_group X₂] [module R X₂] :\n  (X₁ →ₗ[R] X₂) → (Module.of R X₁ ⟶ Module.of R X₂) := id\n\nlocalized \"notation (name := Module.as_hom) `↟` f : 1024 := Module.as_hom f\" in Module\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom_right [add_comm_group X₁] [module R X₁] {X₂ : Module.{v} R} :\n  (X₁ →ₗ[R] X₂) → (Module.of R X₁ ⟶ X₂) := id\n\nlocalized \"notation (name := Module.as_hom_right) `↾` f : 1024 := Module.as_hom_right f\" in Module\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom_left {X₁ : Module.{v} R} [add_comm_group X₂] [module R X₂] :\n  (X₁ →ₗ[R] X₂) → (X₁ ⟶ Module.of R X₂) := id\n\nlocalized \"notation (name := Module.as_hom_left) `↿` f : 1024 := Module.as_hom_left f\" in Module\n\n/-- Build an isomorphism in the category `Module R` from a `linear_equiv` between `module`s. -/\n@[simps]\ndef linear_equiv.to_Module_iso\n  {g₁ : add_comm_group X₁} {g₂ : add_comm_group X₂} {m₁ : module R X₁} {m₂ : module R X₂}\n  (e : X₁ ≃ₗ[R] X₂) :\n  Module.of R X₁ ≅ Module.of R X₂ :=\n{ hom := (e : X₁ →ₗ[R] X₂),\n  inv := (e.symm : X₂ →ₗ[R] X₁),\n  hom_inv_id' := begin ext, exact e.left_inv x, end,\n  inv_hom_id' := begin ext, exact e.right_inv x, end, }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso' {M N : Module.{v} R} (i : M ≃ₗ[R] N) : M ≅ N :=\n{ hom := i,\n  inv := i.symm,\n  hom_inv_id' := linear_map.ext $ λ x, by simp,\n  inv_hom_id' := linear_map.ext $ λ x, by simp }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso'_left {X₁ : Module.{v} R} {g₂ : add_comm_group X₂} {m₂ : module R X₂}\n  (e : X₁ ≃ₗ[R] X₂) : X₁ ≅ Module.of R X₂ :=\n{ hom := (e : X₁ →ₗ[R] X₂),\n  inv := (e.symm : X₂ →ₗ[R] X₁),\n  hom_inv_id' := linear_map.ext $ λ x, by simp,\n  inv_hom_id' := linear_map.ext $ λ x, by simp }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso'_right {g₁ : add_comm_group X₁} {m₁ : module R X₁}\n  {X₂ : Module.{v} R} (e : X₁ ≃ₗ[R] X₂) : Module.of R X₁ ≅ X₂ :=\n{ hom := (e : X₁ →ₗ[R] X₂),\n  inv := (e.symm : X₂ →ₗ[R] X₁),\n  hom_inv_id' := linear_map.ext $ λ x, by simp,\n  inv_hom_id' := linear_map.ext $ λ x, by simp }\n\nnamespace category_theory.iso\n\n/-- Build a `linear_equiv` from an isomorphism in the category `Module R`. -/\n@[simps]\ndef to_linear_equiv {X Y : Module R} (i : X ≅ Y) : X ≃ₗ[R] Y :=\n{ to_fun    := i.hom,\n  inv_fun   := i.inv,\n  left_inv  := by tidy,\n  right_inv := by tidy,\n  map_add'  := by tidy,\n  map_smul' := by tidy, }.\n\nend category_theory.iso\n\n/-- linear equivalences between `module`s are the same as (isomorphic to) isomorphisms\nin `Module` -/\n@[simps]\ndef linear_equiv_iso_Module_iso {X Y : Type u} [add_comm_group X] [add_comm_group Y] [module R X]\n  [module R Y] :\n  (X ≃ₗ[R] Y) ≅ (Module.of R X ≅ Module.of R Y) :=\n{ hom := λ e, e.to_Module_iso,\n  inv := λ i, i.to_linear_equiv, }\n\nnamespace Module\n\ninstance : preadditive (Module.{v} R) :=\n{ add_comp' := λ P Q R f f' g,\n    show (f + f') ≫ g = f ≫ g + f' ≫ g, by { ext, simp },\n  comp_add' := λ P Q R f g g',\n    show f ≫ (g + g') = f ≫ g + f ≫ g', by { ext, simp } }\n\ninstance forget₂_AddCommGroup_additive : (forget₂ (Module.{v} R) AddCommGroup).additive := {}\n\nsection\nvariables {S : Type u} [comm_ring S]\n\ninstance : linear S (Module.{v} S) :=\n{ hom_module := λ X Y, linear_map.module,\n  smul_comp' := by { intros, ext, simp },\n  comp_smul' := by { intros, ext, simp }, }\n\nvariables {X Y X' Y' : Module.{v} S}\n\nlemma iso.hom_congr_eq_arrow_congr (i : X ≅ X') (j : Y ≅ Y') (f : X ⟶ Y) :\n  iso.hom_congr i j f = linear_equiv.arrow_congr i.to_linear_equiv j.to_linear_equiv f := rfl\n\nlemma iso.conj_eq_conj (i : X ≅ X') (f : End X) :\n  iso.conj i f = linear_equiv.conj i.to_linear_equiv f := rfl\n\nend\n\nend Module\n\ninstance (M : Type u) [add_comm_group M] [module R M] : has_coe (submodule R M) (Module R) :=\n⟨ λ N, Module.of R N ⟩\n", "meta": {"author": 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YES\n2. YES\n\n", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.33129662141100197}}
{"text": "import ...for_mathlib.manifolds\n\nnoncomputable theory\n\nopen_locale manifold classical big_operators\nopen set\n\nuniverse u\n\n\n/-!\n## Reminder on updating the exercises\n\nThese instructions are now available at:\nhttps://leanprover-community.github.io/lftcm2020/exercises.html\n\nTo get a new copy of the exercises,\nrun the following commands in your terminal:\n\n```\nleanproject get lftcm2020\ncp -r lftcm2020/src/exercises_sources/ lftcm2020/src/my_exercises\ncode lftcm2020\n```\n\nTo update your exercise files, run the following commands:\n\n```\ncd /path/to/lftcm2020\ngit pull\nleanproject get-mathlib-cache\n```\n\nDon’t forget to copy the updated files to `src/my_exercises`.\n\n-/\n\n/-!\n## An overview of manifolds in Lean, discussing design decisions\n\nWarning: there are sorries in this section, they are not supposed to be filled! The exercises sections\nstart later, and there you will have plenty of sorries to fill.\n\nWhat is a manifold?\n\n1) allow field other than `ℝ` or `ℂ`?\n2) allow infinite dimension?\n3) allow boundary?\n4) allow model space depending on the point of the manifold?\n\nBourbaki: 2, 4 (and just definitions and statements, no proofs!)\nLean: 1, 2, 3\n\nPerelman geometrization theorem : any compact connected irreducible 3-manifold can\nbe cut along tori into finitely many pieces, each of which has a _geometric structure_ of\nfinite volume, i.e., it is locally like a model space, with changes of coordinates given\nlocally by the action of a Lie group\n\nTypical dynamics theorem : let `M` be a compact manifold, and `f : M → M` a map with property\nsuch and such. Then ...\n\nOr : Consider a hyperbolic surface of genus `g`, and a random geodesic of length `T`. How many\ntimes does it typically self-intersect?\n\n\nManifold in Lean:\n\n* charted space structure, i.e., set of local homeos to a model space. This is data, fixed\n  once and for all (and a typeclass)\n* compatibility condition, i.e., the change of coordinates should belong to some subgroup\n  of the group of local homeos of the model space. This is Prop (and a typeclass). The same\n  manifold can be at the same time an analytic manifold, a smooth manifold and a topological\n  manifold (with the same fixed atlas).\n* A charted space is a smooth manifold (with corners) if it is compatible with the smooth\n  groupoid on the model space. To cover uniformly both situations with and without boundary,\n  the smooth groupoid is with respect to a map `I : H → E` (think of `H` as the half-space and\n  `E` the full space), which is the identity in the boundaryless situation, the inclusion in\n  the half-space situation. This map `I` is called a _model with corners_. The most standard ones\n  (identity in `ℝ^n` and inclusion of half-space in `ℝ^n`) have dedicated notations:\n  `𝓡 n` and `𝓡∂ n`.\n-/\n\n#check charted_space (euclidean_half_space 1) (Icc (0 : ℝ) 1)\n#check has_groupoid (Icc (0 : ℝ) 1) (cont_diff_groupoid ∞ (𝓡∂ 1))\n#check smooth_manifold_with_corners (𝓡∂ 1) (Icc (0 : ℝ) 1)\n\n-- atlases are not maximal in general\n\n#check (cont_diff_groupoid ∞ (𝓡∂ 1)).maximal_atlas (Icc (0 : ℝ) 1)\n\n-- let's try to put a smooth manifold structure on the sphere\n-- (we don't have submanifolds yet, but it's coming in the near future)\n\n@[derive topological_space]\ndefinition sphere (n : ℕ) : Type := metric.sphere (0 : euclidean_space ℝ (fin (n+1))) 1\n\ninstance (n : ℕ) : has_coe (sphere n) (euclidean_space ℝ (fin (n+1))) := ⟨subtype.val⟩\n\ninstance (n : ℕ) : charted_space (euclidean_space ℝ (fin n)) (sphere n) :=\n{ atlas            := begin sorry end,\n  chart_at         := begin sorry end,\n  mem_chart_source := begin sorry end,\n  chart_mem_atlas  := begin sorry end }\n\ninstance (n : ℕ) : smooth_manifold_with_corners (𝓡 n) (sphere n) :=\n{ compatible := begin\n    assume e e' he he',\n    sorry\n  end }\n\n-- smooth functions\n\ndef inc (n : ℕ) : sphere n → euclidean_space ℝ (fin (n+1)) :=\nλ p : sphere n, (p : euclidean_space ℝ (fin (n+1)))\n\nlemma inc_smooth (n : ℕ) : cont_mdiff (𝓡 n) (𝓡 (n+1)) ∞ (inc n) :=\nbegin\n  rw cont_mdiff_iff,\n  split,\n  { exact continuous_subtype_coe, },\n  { assume x y,\n    sorry }\nend\n\nlemma inc_continuous (n : ℕ) : continuous (inc n) :=\n(inc_smooth n).continuous\n\nlemma inc_mdifferentiable (n : ℕ) : mdifferentiable (𝓡 n) (𝓡 (n+1)) (inc n) :=\n(inc_smooth n).mdifferentiable le_top\n\n-- tangent space and tangent bundles\n\nexample (n : ℕ) (p : sphere n) (v : tangent_space (𝓡 n) p) :\n  tangent_bundle (𝓡 n) (sphere n) :=\n⟨p, v⟩\n\n-- tangent map, derivatives\n\nexample (n : ℕ) : cont_mdiff ((𝓡 n).prod (𝓡 n)) ((𝓡 (n+1)).prod (𝓡 (n+1))) ∞\n  (tangent_map (𝓡 n) (𝓡 (n+1)) (inc n)) :=\n(inc_smooth n).cont_mdiff_tangent_map le_top\n\nexample (n : ℕ) (f : sphere n → sphere (n^2)) (p : sphere n) (v : tangent_space (𝓡 n) p) :\n  mfderiv (𝓡 n) (𝓡 (n^2)) f p v = (tangent_map (𝓡 n) (𝓡 (n^2)) f ⟨p, v⟩).2 :=\nrfl\n\n/- Can you express the sphere eversion theorem, i.e., the fact that there is a smooth isotopy\nof immersions between the canonical embedding of the sphere `S^2` and `ℝ^3`, and the antipodal\nembedding?\n\nNote that we haven't defined immersions in mathlib, but you can jut require that the fiber\nderivative is injective everywhere, which is easy to express if you know that the derivative\nof a function `f` from a manifold of dimension `2` to a manifold of dimension `3` at a point `x` is\n`mfderiv (𝓡 2) (𝓡 3) f x`.\n\nDon't forget to require the global smoothness of the map! You may need to know that the interval\n`[0,1]`, called `Icc (0 : ℝ) 1` in Lean, already has a manifold (with boundary!) structure,\nwhere the corresponding model with corners is called `𝓡∂ 1`.\n-/\ntheorem sphere_eversion :\n  ∃ f : (Icc (0 : ℝ) 1) × sphere 2 → euclidean_space ℝ (fin 3),\n  cont_mdiff ((𝓡∂ 1).prod (𝓡 2)) (𝓡 3) ∞ f\n  ∧ ∀ (t : (Icc (0 : ℝ) 1)), ∀ (p : sphere 2),\n    function.injective (mfderiv (𝓡 2) (𝓡 3) (f ∘ λ y, (t, y)) p)\n  ∧ ∀ (p : sphere 2), f (0, p) = p\n  ∧ ∀ (p : sphere 2), f (1, p) = - p :=\nsorry\n\n/- Dicussing three (controversial?) design decisions\n\n#### Local homeos\n\nWhat is a local homeo `f` between an open subset of `E` and an open subset of `F`?\n1) a map defined on a subtype: `f x` only makes sense for `x : f.source`\n2) a map defined on the whole space `E`, but taking values in `option F = F ∪ {junk}`, with\n  `f x = junk` when `x ∉ f.source`\n3) a map defined on the whole space `E`, taking values in `F`, and we don't care about its values\n  outside of `f.source`.\n\nJust like division by zero! But worse:\n\n* issue with 1): you keep intersecting chart domains. But the subtype `u ∩ v` is not the same as\n  the subtype `v ∩ u`, so you keep adding casts everywhere\n* issue with 2): if you want to say that a chart is smooth, then you define to define smooth functions\n  between `option E` and `option F` when `E` and `F` are vector spaces. All notions need to be\n  redefined with `option`.\n* issue with 3): it works perfectly well, but it makes mathematicians unhappy/uneasy (and it is *not*\n  equivalent to 1) or 2) when one of the spaces is empty)\n\nI picked 3)\n\n#### Tangent vectors\n\nWhat is a tangent vector (for a manifold `M` modelled on a vector space `E`)?\n1) An equivalence class of germs of curves\n2) A derivation\n3) Physicist point of view: I don't know what a tangent vector is, but I know in charts.\n  Mathematician's interpretation: equivalence class of `(e, v)` where `e` is a chart at `x`, `v` a vector\n  in the vector space, and `(e, v) ∼ (e', v')` if `D(e' ∘ e ⁻¹) v = v'`\n4) ...\n\nIssues:\n1) Pictures are pretty, but this doesn't bring anything compared to 3) when you go down to details.\n   And what about boundaries, where you can only have a half-curve\n2) Need partitions of unity to show that this is local and coincides with the usual point of view.\n   Doesn't work well in finite smoothness, nor in complex manifolds\n3) Fine, works in all situations, but requires a lot of work to define the equivalence classes,\n   the topology, check that the topology is compatible with the vector space structure, and so on.\n   In a vector space, the tangent space is not defeq to the vector space itself\n4) Pick one favorite chart at `x`, say `e_x`, and *define* the tangent space at `x` to be `E`,\n   but \"seen\" in the chart `e_x` (this will show up in the definition of the derivative : the\n   derivative of `f : M → M'` at `x` is defined to be the derivative of the map\n   `e_{f x} ∘ f ∘ e_x⁻¹`). Works perfectly fine, but makes mathematicians unhappy/uneasy.\n   (Axiom of choice? In fact we put the choice of `e_x` in the *definition* of charted spaces,\n   so not further choice)\n\nI picked 4)\n\n#### Smooth functions in manifolds with boundary\n\nUsual definition of smooth functions in a half space: extend to a smooth function a little bit\nbeyond the boundary, so one only really needs to speak of smooth functions in open subsets of\nvector spaces.\n\nWhen you define the derivative, you will need to check that it does not depend on the choice\nof the extension. Even worse when you want to define the tangent bundle: choose an open extension\nof your manifold with boundary, and then check that the restriction of the tangent bundle does\nnot depend on the choice of the extension. Very easy when handwaving, nightmare to formalize.\n(What is the extension of the manifold with boundary? Another type?)\n\nInstead, if you define derivatives in (non-open) domains, you can talk of smooth functions in\ndomains, and do everything without extending. Need to know this early enough: when starting to\ndefine derivatives, you should already think of manifolds with boundaries! That's what we did\nin mathlib.\n\nDifficulty: if a domain `s` is too small (think `s = ℝ ⊆ ℝ^2`), the values of `f` on `s` do not\nprescribe uniquely a derivative, so `fderiv_within_at ℝ f s x` may behave badly: the derivative of\na sum might be different from sum of derivatives, as there is an arbitrary choice to be made.\nThis does not happen with the half-space, as it is large enough: derivatives within domains only\nwork well if the tangent directions span the whole space. Predicate `unique_diff_on` for sets\nin vector spaces. You won't find this in books!\n-/\n\n\n/-! ## Exercises -/\n\n/-! ### Local homeomorphisms\n\nLocal homeomorphisms are globally defined maps with a globally defined \"inverse\", but the only\nrelevant set is the *source*, which should be mapped homeomorphically to the *target*.\n-/\n\n-- set up a simple helper simp lemma to simplify our life later.\n@[simp] lemma neg_mem_Ioo_minus_one_one (x : ℝ) : -x ∈ Ioo (-1 : ℝ) 1 ↔ x ∈ Ioo (-1 : ℝ) 1 :=\nbegin\n  sorry\nend\n\n/- Define a local homeomorphism from `ℝ` to `ℝ` which is just `x ↦ -x`, but on `(-1, 1)`. In\nLean, the interval `(-1, 1)` is denoted by `Ioo (-1 : ℝ) 1` (where `o` stands for _open_). -/\n\ndef my_first_local_homeo : local_homeomorph ℝ ℝ :=\n{ to_fun := λ x, -x,\n  inv_fun := λ x, -x,\n  source := Ioo (-1) 1,\n  target := sorry,\n  map_source' :=\n  begin\n    sorry\n  end,\n  map_target' :=\n  begin\n    sorry\n  end,\n  left_inv' :=\n  begin\n    sorry\n  end,\n  right_inv' :=\n  begin\n    sorry\n  end,\n  open_source := sorry,\n  open_target := sorry,\n  continuous_to_fun := sorry,\n  continuous_inv_fun := sorry }\n\n/- Two simple lemmas that will prove useful below. You can leave them sorried if you like. -/\n\nlemma ne_3_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-1 : ℝ) 1) : x ≠ 3 :=\nbegin\n  sorry\nend\n\nlemma neg_ne_3_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-1 : ℝ) 1) : -x ≠ 3 :=\nbegin\nsorry\nend\n\n/- Now, define a second local homeomorphism which is almost like the previous one.  You may find the\nfollowing lemma useful for `continuous_to_fun`: -/\n#check continuous_on.congr\n\ndef my_second_local_homeo : local_homeomorph ℝ ℝ :=\n{ to_fun := λ x, if x = 3 then 0 else - x,\n  inv_fun := λ x, -x,\n  source := Ioo (-1) 1,\n  target := sorry,\n  map_source' := sorry,\n  map_target' := sorry,\n  left_inv' := sorry,\n  right_inv' := sorry,\n  open_source := sorry,\n  open_target := sorry,\n  continuous_to_fun :=\n  begin\n    sorry\n  end,\n  continuous_inv_fun := sorry }\n\n/- Although the two above local homeos are the same for all practical purposes as they coincide\nwhere relevant, they are not *equal*: -/\n\nlemma my_first_local_homeo_ne_my_second_local_homeo :\n  my_first_local_homeo ≠ my_second_local_homeo :=\nbegin\n  sorry\nend\n\n/- The right equivalence relation for local homeos is not equality, but `eq_on_source`.\nIndeed, the two local homeos we have defined above coincide from this point of view. -/\n\n#check local_homeomorph.eq_on_source\n\nlemma eq_on_source_my_first_local_homeo_my_second_local_homeo :\n  local_homeomorph.eq_on_source my_first_local_homeo my_second_local_homeo :=\nbegin\n  sorry\nend\n\n\n/-! ### An example of a charted space structure on `ℝ`\n\nA charted space is a topological space together with a set of local homeomorphisms to a model space,\nwhose sources cover the whole space. For instance, `ℝ` is already endowed with a charted space\nstructure with model space `ℝ`, where the unique chart is the identity:\n-/\n\n#check charted_space_self ℝ\n\n/- For educational purposes only, we will put another charted space structure on `ℝ` using the\nlocal homeomorphisms we have constructed above. To avoid using too much structure of `ℝ` (and to\navoid confusing Lean), we will work with a copy of `ℝ`, on which we will only register the\ntopology. -/\n\n@[derive topological_space]\ndef myℝ : Type := ℝ\n\ninstance : charted_space ℝ myℝ :=\n{ atlas := { local_homeomorph.refl ℝ, my_first_local_homeo },\n  chart_at := λ x, if x ∈ Ioo (-1 : ℝ) 1 then my_first_local_homeo else local_homeomorph.refl ℝ,\n  mem_chart_source :=\n  begin\n  sorry\n  end,\n  chart_mem_atlas :=\n  begin\n    sorry\n  end }\n\n/- Now come more interesting bits. We have endowed `myℝ` with a charted space structure, with charts\ntaking values in `ℝ`. We want to say that this is a smooth structure, i.e., the changes of\ncoordinates are smooth. In Lean, this is written with `has_groupoid`. A groupoid is a set\nof local homeomorphisms of the model space (for example, local homeos that are smooth on their\ndomain). A charted space admits the groupoid as a structure groupoid if all the changes of\ncoordinates belong to the groupoid.\n\nThere is a difficulty that the definitions are set up to be able to also speak of smooth manifolds\nwith boundary or with corners, so the name of the smooth groupoid on `ℝ` has the slightly strange\nname `cont_diff_groupoid ∞ (model_with_corners_self ℝ ℝ)`. To avoid typing again and again\n`model_with_corners_self ℝ ℝ`, let us introduce a shortcut\n-/\n\nabbreviation 𝓡1 := model_with_corners_self ℝ ℝ\n\n/- In the library, there are such shortcuts for manifolds modelled on `ℝ^n`, denoted with `𝓡 n`,\nbut for `n = 1` this does not coincide with the above one, as `ℝ^1` (a.k.a. `fin 1 → ℝ`) is not\nthe same as `ℝ`! Still, since they are of the same nature, the notation we have just introduced\nis very close, compare `𝓡1` with `𝓡 1` (and try not to get confused): -/\n\ninstance smooth_myℝ : has_groupoid myℝ (cont_diff_groupoid ∞ 𝓡1) :=\nbegin\n  -- in theory, we should prove that all compositions of charts are diffeos, i.e., they are smooth\n  -- and their inverse are smooth. For symmetry reasons, it suffices to check one direction\n  apply has_groupoid_of_pregroupoid,\n  -- take two charts `e` and `e'`\n  assume e e' he he',\n  -- if next line is a little bit slow for your taste, you can replace `simp` with `squeeze_simp`\n  -- and then follow the advice\n  simp [atlas] at he he',\n  dsimp,\n  -- to continue, some hints:\n  -- (1) don't hesitate to use the fact that the restriction of a smooth function to a\n  -- subset is still smooth there (`cont_diff.cont_diff_on`)\n  -- (2) hopefully, there is a theorem saying that the negation function is smooth.\n  -- you can either try to guess its name, or hope that `suggest` will help you there.\n  sorry\nend\n\n/- The statement of the previous instance is not very readable. There is a shortcut notation: -/\n\ninstance : smooth_manifold_with_corners 𝓡1 myℝ := { .. smooth_myℝ }\n\n/- We will now study a very simple map from `myℝ` to `ℝ`, the identity. -/\n\ndef my_map : myℝ → ℝ := λ x, x\n\n/- The map `my_map` is a map going from the type `myℝ` to the type `ℝ`. From the point of view of\nthe kernel of Lean, it is just the identity, but from the point of view of structures on `myℝ`\nand `ℝ` it might not be trivial, as we have registered different instances on these two types. -/\n\n/- The continuity should be trivial, as the topologies on `myℝ` and `ℝ` are definitionally the\nsame. So `continuous_id` might help. -/\n\nlemma continuous_my_map : continuous my_map :=\nsorry\n\n/- Smoothness should not be obvious, though, as the manifold structures are not the same: the atlas\non `myℝ` has two elements, while the atlas on `ℝ` has one single element.\nNote that `myℝ` is not a vector space, nor a normed space, so one can not ask whether `my_map`\nis smooth in the usual sense (as a map between vector spaces): -/\n\n-- lemma cont_diff_my_map : cont_diff ℝ ∞ my_map := sorry\n\n/- does not make sense (try uncommenting it!) However, we can ask whether `my_map` is a smooth\nmap between manifolds, i.e., whether it is smooth when read in the charts. When we mention the\nsmoothness of a map, we should always specify explicitly the model with corners we are using,\nbecause there might be several around (think of a complex manifold that you may want to consider\nas a real manifold, to talk about functions which are real-smooth but not holomorphic) -/\n\nlemma cont_mdiff_my_map : cont_mdiff 𝓡1 𝓡1 ∞ my_map :=\nbegin\n  -- put things in a nicer form. The simpset `mfld_simps` registers many simplification rules for\n  -- manifolds. `simp` is used heavily in manifold files to bring everything into manageable form.\n  rw cont_mdiff_iff,\n  simp only [continuous_my_map] with mfld_simps,\n  -- simp has erased the chart in the target, as it knows that the only chart in the manifold `ℝ`\n  -- is the identity.\n  assume x y,\n  sorry\nend\n\n/- Now, let's go to tangent bundles. We have a smooth manifold, so its tangent bundle should also\nbe a smooth manifold. -/\n\n-- the type `tangent_bundle 𝓡1 myℝ` makes sense\n#check tangent_bundle 𝓡1 myℝ\n\n/- The tangent space above a point of `myℝ` is just a one-dimensional vector space (identified with `ℝ`).\nSo, one can prescribe an element of the tangent bundle as a pair (more on this below) -/\nexample : tangent_bundle 𝓡1 myℝ := ⟨(4 : ℝ), 0⟩\n\n/- Construct the smooth manifold structure on the tangent bundle. Hint: the answer is a one-liner,\nand this instance is not really needed. -/\ninstance tangent_bundle_myℝ : smooth_manifold_with_corners (𝓡1.prod 𝓡1) (tangent_bundle 𝓡1 myℝ) :=\nsorry\n\n/-\nNB: the model space for the tangent bundle to a product manifold or a tangent space is not\n`ℝ × ℝ`, but a copy called `model_prod ℝ ℝ`. Otherwise, `ℝ × ℝ` would have two charted space\nstructures with model `ℝ × ℝ`, the identity one and the product one, which are not definitionally\nequal. And this would be bad.\n-/\n#check tangent_bundle.charted_space 𝓡1 myℝ\n\n/- A smooth map between manifolds induces a map between their tangent bundles. In `mathlib` this is\ncalled the `tangent_map` (you might instead know it as the \"differential\" or \"pushforward\" of the\nmap).  Let us check that the `tangent_map` of `my_map` is smooth. -/\nlemma cont_mdiff_tangent_map_my_map :\n  cont_mdiff (𝓡1.prod 𝓡1) (𝓡1.prod 𝓡1) ∞ (tangent_map 𝓡1 𝓡1 my_map) :=\nbegin\n  -- hopefully, there is a theorem providing the general result, i.e. the tangent map to a smooth\n  -- map is smooth.\n  -- you can either try to guess its name, or hope that `suggest` will help you there.\n  sorry\nend\n\n/- (Harder question) Can you show that this tangent bundle is homeomorphic to `ℝ × ℝ`? You could\ntry to build the homeomorphism by hand, using `tangent_map 𝓡1 𝓡1 my_map` in one direction and a\nsimilar map in the other direction, but it is probably more efficient to use one of the charts of\nthe tangent bundle.\n\nRemember, the model space for `tangent_bundle 𝓡1 myℝ` is `model_prod ℝ ℝ`, not `ℝ × ℝ`. But the\ntopologies on `model_prod ℝ ℝ` and `ℝ × ℝ` are the same, so it is by definition good enough to\nconstruct a homeomorphism with `model_prod ℝ ℝ`.\n -/\n\ndef my_homeo : tangent_bundle 𝓡1 myℝ ≃ₜ (ℝ × ℝ) :=\nbegin\n  sorry\nend\n\n/- Up to now, we have never used the definition of the tangent bundle, and this corresponds to\nthe usual mathematical practice: one doesn't care if the tangent space is defined using germs of\ncurves, or spaces of derivations, or whatever equivalent definition. Instead, one relies all the\ntime on functoriality (i.e., a smooth map has a well defined derivative, and they compose well,\ntogether with the fact that the tangent bundle to a vector space is the product).\n\nIf you want to know more about the internals of the tangent bundle in mathlib, you can browse\nthrough the next section, but it is maybe wiser to skip it on first reading, as it is not needed\nto use the library\n-/\n\nsection you_should_probably_skip_this\n\n/- If `M` is a manifold modelled on a vector space `E`, then the underlying type for the tangent\nbundle is just `Σ (x : M), tangent_space x M` (i.e., the disjoint union of the tangent spaces,\nindexed by `x` -- this is a basic object in dependent type theory). And `tangent_space x M`\nis just (a copy of) `E` by definition. -/\n\nlemma tangent_bundle_myℝ_is_prod : tangent_bundle 𝓡1 myℝ = Σ (x : myℝ), ℝ :=\nsorry\n\n/- This means that you can specify a point in the tangent bundle as a pair `⟨x, v⟩`.\nHowever, in general, a tangent bundle is not trivial: the topology on `tangent_bundle 𝓡1 myℝ` is *not*\nthe product topology. Instead, the tangent space at a point `x` is identified with `ℝ` through some\npreferred chart at `x`, called `chart_at ℝ x`, but the way they are glued together depends on the\nmanifold and the charts.\n\nIn vector spaces, the tangent space is canonically the product space, with the same topology, as\nthere is only one chart so there is no strange gluing at play. The fact that the canonical map\nfrom the sigma type to the product type (called `equiv.sigma_equiv_prod`) is a homeomorphism is\ngiven in the library by `tangent_bundle_model_space_homeomorph` (note that this is a definition,\nconstructing the homeomorphism, instead of a proposition asserting that `equiv.sigma_equiv_prod`\nis a homeomorphism, because we use bundled homeomorphisms in mathlib).\n\nLet us register the identification explicitly, as a homeomorphism. You can use the relevant fields\nof `tangent_bundle_model_space_homeomorph` to fill the nontrivial fields here.\n-/\n\ndef tangent_bundle_vector_space_triv (E : Type u) [normed_add_comm_group E] [normed_space ℝ E] :\n  tangent_bundle (model_with_corners_self ℝ E) E ≃ₜ E × E :=\n{ to_fun := λ p, (p.1, p.2),\n  inv_fun := λ p, ⟨p.1, p.2⟩,\n  left_inv := sorry,\n  right_inv := sorry,\n  continuous_to_fun := begin\n    sorry\n  end,\n  continuous_inv_fun :=\n  begin\n    sorry\n  end }\n\n/- Even though the tangent bundle to `myℝ` is trivial abstractly, with this construction the\ntangent bundle is *not* the product space with the product topology, as we have used various charts\nso the gluing is not trivial. The following exercise unfolds the definition to see what is going on.\nIt is not a reasonable exercise, in the sense that one should never ever do this when working\nwith a manifold! -/\n\nlemma crazy_formula_after_identifications (x : ℝ) (v : ℝ) :\n  let p : tangent_bundle 𝓡1 myℝ := ⟨(3 : ℝ), 0⟩ in\n  chart_at (model_prod ℝ ℝ) p ⟨x, v⟩ = if x ∈ Ioo (-1 : ℝ) 1 then (x, -v) else (x, v) :=\nbegin\n  -- this exercise is not easy (and shouldn't be: you are not supposed to use the library like this!)\n  -- if you really want to do this, you should unfold as much as you can using simp and dsimp, until you\n  -- are left with a statement speaking of derivatives of real functions, without any manifold code left.\n  sorry\nend\n\nend you_should_probably_skip_this\n\n/-!\n### The language of manifolds\n\nIn this paragraph, we will try to write down interesting statements of theorems, without proving them. The\ngoal here is that Lean should not complain on the statement, but the proof should be sorried.\n-/\n\n/- Here is a first example, already filled up, to show you how diffeomorphisms are currently named\n(we will probably introduce an abbreviation, but this hasn't been done yet).\nDon't try to fill the sorried proof! -/\n\n/-- Two zero-dimensional connected manifolds are diffeomorphic. -/\ntheorem diffeomorph_of_zero_dim_connected\n  (M M' : Type*) [topological_space M] [topological_space M']\n  [charted_space (euclidean_space ℝ (fin 0)) M] [charted_space (euclidean_space ℝ (fin 0)) M']\n  [connected_space M] [connected_space M'] :\n  nonempty (structomorph (cont_diff_groupoid ∞ (𝓡 0)) M M') :=\nsorry\n\n/- Do you think that this statement is correct? (note that we have not assumed that our manifolds\nare smooth, nor that they are separated, but this is maybe automatic in zero dimension).\n\nNow, write down a version of this theorem in dimension 1, replacing the first sorry with meaningful content\n(and adding what is needed before the colon).\nDon't try to fill the sorried proof! -/\n\n/-- Two one-dimensional smooth compact connected manifolds are diffeomorphic. -/\ntheorem diffeomorph_of_one_dim_compact_connected\n  \n  :\n  sorry\n:= sorry\n\n/- You will definitely need to require smoothness and separation in this case, as it is wrong otherwise.\nNote that Lean won't complain if you don't put these assumptions, as the theorem would still make\nsense, but it would just turn out to be wrong.\n\nThe previous statement is not really satisfactory: we would instead like to express that any such\nmanifold is diffeomorphic to the circle. The trouble is that we don't have the circle as a smooth\nmanifold yet. Since we have cheated and introduced it (with sorries) at the beginning of the tutorial,\nlet's cheat again and use it to reformulate the previous statement.\n-/\n\n-- the next result is not trivial, leave it sorried (but you can work on it if you don't like\n-- manifolds and prefer topology -- then please PR it to mathlib!).\ninstance connected_sphere (n : ℕ) : connected_space (sphere (n+1)) := sorry\n\n/- The next two instances are easier to prove, you can prove them or leave them sorried\nas you like. For the second one, you may need to use facts of the library such as -/\n#check is_compact_iff_compact_space\n#check metric.compact_iff_closed_bounded\n\ninstance (n : ℕ) : t2_space (sphere n) :=\nbegin\n  sorry\nend\n\ninstance (n : ℕ) : compact_space (sphere n) :=\nbegin\n  sorry\nend\n\n/- Now, you can prove that any one-dimensional compact connected manifold is diffeomorphic to\nthe circle. Here, you should fill the `sorry` (but luckily you may use\n`diffeomorph_of_one_dim_compact_connected`). -/\ntheorem diffeomorph_circle_of_one_dim_compact_connected\n  (M : Type*) [topological_space M] [charted_space (euclidean_space ℝ (fin 1)) M]\n  [connected_space M] [compact_space M] [t2_space M] [smooth_manifold_with_corners (𝓡 1) M] :\n  nonempty (structomorph (cont_diff_groupoid ∞ (𝓡 1)) M (sphere 1)) :=\nsorry\n\n\n/- What about trying to say that there are uncountably many different smooth structures on `ℝ⁴`?\n(see https://en.wikipedia.org/wiki/Exotic_R4). The library is not really designed with this in mind,\nas in general we only work with one differentiable structure on a space, but it is perfectly\ncapable of expressing this fact if one uses the `@` version of some definitions.\n\nDon't try to fill the sorried proof!\n-/\n\ntheorem exotic_ℝ4 :\n  sorry\n  :=\nsorry\n\n/-!\n### Smooth functions on `[0, 1]`\n\nIn this paragraph, you will prove several (math-trivial but Lean-nontrivial) statements on the smooth\nstructure of `[0,1]`. These facts should be Lean-trivial, but they are not (yet) since there is essentially\nnothing in this direction for now in the library.\n\nThe goal is as much to be able to write the statements as to prove them. Most of the necessary vocabulary\nhas been introduced above, so don't hesitate to browse the file if you are stuck. Additionally, you will\nneed the notion of a smooth function on a subset: it is `cont_diff_on` for functions between vector\nspaces and `cont_mdiff_on` for functions between manifolds.\n\nTry to formulate the next math statements in Lean, and prove them (but see below for hints):\n\nLemma cont_mdiff_g : the inclusion `g` of `[0, 1]` in `ℝ` is smooth.\n\nLemma msmooth_of_smooth : Consider a function `f : ℝ → [0, 1]`, which is smooth in the usual sense as a function\nfrom `ℝ` to `ℝ` on a set `s`. Then it is manifold-smooth on `s`.\n\nDefinition : construct a function `f` from `ℝ` to `[0,1]` which is the identity on `[0, 1]`.\n\nTheorem : the tangent bundle to `[0, 1]` is homeomorphic to `[0, 1] × ℝ`\n\nHint for the last theorem: don't try to unfold the definition of the tangent bundle, it will only get you\ninto trouble. Instead, use the derivatives of the maps `f` and `g`, and rely on functoriality\nto check that they are inverse to each other. (This advice is slightly misleading as these derivatives\ndo not go between the right spaces, so you will need to massage them a little bit).\n\nA global advice: don't hesitate to use and abuse `simp`, it is the main workhorse in this\narea of mathlib.\n-/\n\n/- After doing the exercise myself, I realized it was (way!) too hard. So I will give at least the statements\nof the lemmas, to guide you a little bit more. To let you try the original version if you want,\nI have left a big blank space to avoid spoilers. -/\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\ndef g : Icc (0 : ℝ) 1 → ℝ := subtype.val\n\n-- smoothness results for `euclidean_space` are expressed for general `L^p` spaces\n-- (as `euclidean_space` has the `L^2` norm), in:\n#check pi_Lp.cont_diff_coord\n#check pi_Lp.cont_diff_on_iff_coord\n\nlemma cont_mdiff_g : cont_mdiff (𝓡∂ 1) 𝓡1 ∞ g :=\nbegin\n  sorry\nend\n\nlemma msmooth_of_smooth {f : ℝ → Icc (0 : ℝ) 1} {s : set ℝ} (h : cont_diff_on ℝ ∞ (λ x, (f x : ℝ)) s) :\n  cont_mdiff_on 𝓡1 (𝓡∂ 1) ∞ f s :=\nbegin\n  sorry\nend\n\n/- A function from `ℝ` to `[0,1]` which is the identity on `[0,1]`. -/\ndef f : ℝ → Icc (0 : ℝ) 1 :=\nλ x, ⟨max (min x 1) 0, by simp [le_refl, zero_le_one]⟩\n\nlemma cont_mdiff_on_f : cont_mdiff_on 𝓡1 (𝓡∂ 1) ∞ f (Icc 0 1) :=\nbegin\n  sorry\nend\n\nlemma fog : f ∘ g = id :=\nbegin\n  sorry\nend\n\nlemma gof : ∀ x ∈ Icc (0 : ℝ) 1, g (f x) = x :=\nbegin\n  sorry\nend\n\ndef G : tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) → (Icc (0 : ℝ) 1) × ℝ :=\nλ p, (p.1, ((tangent_bundle_vector_space_triv ℝ) (tangent_map (𝓡∂ 1) 𝓡1 g p)).2)\n\nlemma continuous_G : continuous G :=\nbegin\n  sorry\nend\n\ndef F : (Icc (0 : ℝ) 1) × ℝ → tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) :=\nλ p, tangent_map_within 𝓡1 (𝓡∂ 1) f (Icc 0 1)\n  ((tangent_bundle_vector_space_triv ℝ).symm (p.1, p.2))\n\nlemma continuous_F : continuous F :=\nbegin\n  sorry\nend\n\nlemma FoG : F ∘ G = id :=\nbegin\n  sorry\nend\n\nlemma GoF : G ∘ F = id :=\nbegin\n  sorry\nend\n\ndef my_tangent_homeo : tangent_bundle (𝓡∂ 1) (Icc (0 : ℝ) 1) ≃ₜ (Icc (0 : ℝ) 1) × ℝ :=\nsorry\n\n\n/-!\n### Further things to do\n\n1) can you prove `diffeomorph_of_zero_dim_connected` or `connected_sphere`?\n\n2) Try to express and then prove the local inverse theorem in real manifolds: if a map between\nreal manifolds (without boundary, modelled on a complete vector space) is smooth, then it is\na local homeomorphism around each point. We already have versions of this statement in mathlib\nfor functions between vector spaces, but this is very much a work in progress.\n\n3) What about trying to prove `diffeomorph_of_one_dim_compact_connected`? (I am not sure mathlib\nis ready for this, as the proofs I am thinking of are currently a little bit too high-powered.\nIf you manage to do it, you should absolutely PR it!)\n\n4) Why not contribute to the proof of `sphere_eversion`? You can have a look at\nhttps://leanprover-community.github.io/sphere-eversion/ to learn more about this project\nby Patrick Massot.\n-/\n\n", "meta": {"author": "sterraf", "repo": "mylearninglean", "sha": "a8911234b2a4e15a48ec2c0f05d744e58f798ca7", "save_path": "github-repos/lean/sterraf-mylearninglean", "path": "github-repos/lean/sterraf-mylearninglean/mylearninglean-a8911234b2a4e15a48ec2c0f05d744e58f798ca7/src/lftcm2020_exercises_sources/friday/manifolds.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. 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{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module algebraic_geometry.locally_ringed_space.has_colimits\n! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicGeometry.LocallyRingedSpace\nimport Mathbin.Algebra.Category.Ring.Constructions\nimport Mathbin.AlgebraicGeometry.OpenImmersion\nimport Mathbin.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers\n\n/-!\n# Colimits of LocallyRingedSpace\n\nWe construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.\nIt then follows that `LocallyRingedSpace` has all colimits, and\n`forget_to_SheafedSpace` preserves them.\n\n-/\n\n\nnamespace AlgebraicGeometry\n\nuniverse v u\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nnamespace SheafedSpace\n\nvariable {C : Type u} [Category.{v} C] [HasLimits C]\n\nvariable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace C)\n\ntheorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :\n    ∃ (i : J)(y : F.obj i), (c.ι.app i).base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget _)\n    (isColimitOfPreserves (SheafedSpace.forget _) hc) x\n#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpace.isColimit_exists_rep\n\ntheorem colimit_exists_rep (x : colimit F) : ∃ (i : J)(y : F.obj i), (colimit.ι F i).base y = x :=\n  Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget _)\n    (isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x\n#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpace.colimit_exists_rep\n\ninstance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base :=\n  by\n  erw [←\n    show _ = (coequalizer.π f g).base from\n      ι_comp_coequalizer_comparison f g (SheafedSpace.forget C)]\n  rw [← preserves_coequalizer.iso_hom]\n  apply epi_comp\n\nend SheafedSpace\n\nnamespace LocallyRingedSpace\n\nsection HasCoproducts\n\nvariable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})\n\n/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproduct : LocallyRingedSpace\n    where\n  toSheafedSpace := colimit (F ⋙ forgetToSheafedSpace : _)\n  LocalRing x :=\n    by\n    obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forget_to_SheafedSpace) x\n    haveI : _root_.local_ring (((F ⋙ forget_to_SheafedSpace).obj i).toPresheafedSpace.stalk y) :=\n      (F.obj i).LocalRing _\n    exact\n      (as_iso\n              (PresheafedSpace.stalk_map (colimit.ι (F ⋙ forget_to_SheafedSpace) i : _)\n                y)).symm.commRingIsoToRingEquiv.LocalRing\n#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct\n\n/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/\nnoncomputable def coproductCofan : Cocone F\n    where\n  pt := coproduct F\n  ι :=\n    { app := fun j => ⟨colimit.ι (F ⋙ forgetToSheafedSpace) j, inferInstance⟩\n      naturality' := fun j j' f => by\n        cases j\n        cases j'\n        tidy }\n#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan\n\n/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/\nnoncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F)\n    where\n  desc s :=\n    ⟨colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s),\n      by\n      intro x\n      obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forget_to_SheafedSpace) x\n      have :=\n        PresheafedSpace.stalk_map.comp (colimit.ι (F ⋙ forget_to_SheafedSpace) i : _)\n          (colimit.desc (F ⋙ forget_to_SheafedSpace) (forget_to_SheafedSpace.map_cocone s)) y\n      rw [← is_iso.comp_inv_eq] at this\n      erw [← this,\n        PresheafedSpace.stalk_map.congr_hom _ _\n          (colimit.ι_desc (forget_to_SheafedSpace.map_cocone s) i : _)]\n      haveI :\n        IsLocalRingHom\n          (PresheafedSpace.stalk_map ((forget_to_SheafedSpace.map_cocone s).ι.app i) y) :=\n        (s.ι.app i).2 y\n      infer_instance⟩\n  fac s j := LocallyRingedSpace.Hom.ext _ _ (colimit.ι_desc _ _)\n  uniq s f h :=\n    LocallyRingedSpace.Hom.ext _ _\n      (IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>\n        congr_arg LocallyRingedSpace.Hom.val (h j))\n#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit\n\ninstance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun ι =>\n  ⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩\n\nnoncomputable instance (J : Type _) : PreservesColimitsOfShape (Discrete J) forgetToSheafedSpace :=\n  ⟨fun G =>\n    preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)\n      ((colimit.isColimit _).ofIsoColimit (Cocones.ext (Iso.refl _) fun j => Category.comp_id _))⟩\n\nend HasCoproducts\n\nsection HasCoequalizer\n\nvariable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)\n\nnamespace HasCoequalizer\n\ninstance coequalizer_π_app_isLocalRingHom\n    (U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :\n    IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) :=\n  by\n  have := ι_comp_coequalizer_comparison f.1 g.1 SheafedSpace.forget_to_PresheafedSpace\n  rw [← preserves_coequalizer.iso_hom] at this\n  erw [SheafedSpace.congr_app this.symm (op U)]\n  rw [PresheafedSpace.comp_c_app, ←\n    PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_π]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_app_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalRingHom\n\n/-!\nWe roughly follow the construction given in [MR0302656]. Given a pair `f, g : X ⟶ Y` of morphisms\nof locally ringed spaces, we want to show that the stalk map of\n`π = coequalizer.π f g` (as sheafed space homs) is a local ring hom. It then follows that\n`coequalizer f g` is indeed a locally ringed space, and `coequalizer.π f g` is a morphism of\nlocally ringed space.\n\nGiven a germ `⟨U, s⟩` of `x : coequalizer f g` such that `π꙳ x : Y` is invertible, we ought to show\nthat `⟨U, s⟩` is invertible. That is, there exists an open set `U' ⊆ U` containing `x` such that the\nrestriction of `s` onto `U'` is invertible. This `U'` is given by `π '' V`, where `V` is the\nbasic open set of `π⋆x`.\n\nSince `f ⁻¹' V = Y.basic_open (f ≫ π)꙳ x = Y.basic_open (g ≫ π)꙳ x = g ⁻¹' V`, we have\n`π ⁻¹' (π '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).\nThis shows that `π '' V` is indeed open, and `s` is invertible on `π '' V` as the components of `π꙳`\nare local ring homs.\n-/\n\n\nvariable (U : Opens (coequalizer f.1 g.1).carrier)\n\nvariable (s : (coequalizer f.1 g.1).Presheaf.obj (op U))\n\n/-- (Implementation). The basic open set of the section `π꙳ s`. -/\nnoncomputable def imageBasicOpen : Opens Y :=\n  Y.toRingedSpace.basicOpen\n    (show Y.Presheaf.obj (op (unop _)) from ((coequalizer.π f.1 g.1).c.app (op U)) s)\n#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen\n\ntheorem imageBasicOpen_image_preimage :\n    (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) =\n      (imageBasicOpen f g U s).1 :=\n  by\n  fapply types.coequalizer_preimage_image_eq_of_preimage_eq f.1.base g.1.base\n  · ext\n    simp_rw [types_comp_apply, ← TopCat.comp_app, ← PresheafedSpace.comp_base]\n    congr 2\n    exact coequalizer.condition f.1 g.1\n  · apply is_colimit_cofork_map_of_is_colimit (forget TopCat)\n    apply is_colimit_cofork_map_of_is_colimit (SheafedSpace.forget _)\n    exact coequalizer_is_coequalizer f.1 g.1\n  · suffices\n      (TopologicalSpace.Opens.map f.1.base).obj (image_basic_open f g U s) =\n        (TopologicalSpace.Opens.map g.1.base).obj (image_basic_open f g U s)\n      by injection this\n    delta image_basic_open\n    rw [preimage_basic_open f, preimage_basic_open g]\n    dsimp only [functor.op, unop_op]\n    rw [← comp_apply, ← SheafedSpace.comp_c_app', ← comp_apply, ← SheafedSpace.comp_c_app',\n      SheafedSpace.congr_app (coequalizer.condition f.1 g.1), comp_apply]\n    erw [X.to_RingedSpace.basic_open_res]\n    apply inf_eq_right.mpr\n    refine' (RingedSpace.basic_open_le _ _).trans _\n    rw [coequalizer.condition f.1 g.1]\n    exact fun _ h => h\n#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_preimage AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_preimage\n\ntheorem imageBasicOpen_image_open :\n    IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) :=\n  by\n  rw [←\n    (TopCat.homeoOfIso (preserves_coequalizer.iso (SheafedSpace.forget _) f.1 g.1)).isOpen_preimage,\n    TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]\n  erw [← coe_comp]\n  rw [preserves_coequalizer.iso_hom, ι_comp_coequalizer_comparison]\n  dsimp only [SheafedSpace.forget]\n  rw [image_basic_open_image_preimage]\n  exact (image_basic_open f g U s).2\n#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.image_basic_open_image_open AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.imageBasicOpen_image_open\n\ninstance coequalizer_π_stalk_isLocalRingHom (x : Y) :\n    IsLocalRingHom (PresheafedSpace.stalkMap (coequalizer.π f.val g.val : _) x) :=\n  by\n  constructor\n  rintro a ha\n  rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩\n  erw [PresheafedSpace.stalk_map_germ_apply (coequalizer.π f.1 g.1 : _) U ⟨_, hU⟩] at ha\n  let V := image_basic_open f g U s\n  have hV : (coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1) = V.1 :=\n    image_basic_open_image_preimage f g U s\n  have hV' :\n    V = ⟨(coequalizer.π f.1 g.1).base ⁻¹' ((coequalizer.π f.1 g.1).base '' V.1), hV.symm ▸ V.2⟩ :=\n    SetLike.ext' hV.symm\n  have V_open : IsOpen ((coequalizer.π f.val g.val).base '' V.1) :=\n    image_basic_open_image_open f g U s\n  have VleU : (⟨(coequalizer.π f.val g.val).base '' V.1, V_open⟩ : TopologicalSpace.Opens _) ≤ U :=\n    set.image_subset_iff.mpr (Y.to_RingedSpace.basic_open_le _)\n  have hxV : x ∈ V := ⟨⟨_, hU⟩, ha, rfl⟩\n  erw [←\n    (coequalizer f.val g.val).Presheaf.germ_res_apply (hom_of_le VleU)\n      ⟨_, @Set.mem_image_of_mem _ _ (coequalizer.π f.val g.val).base x V.1 hxV⟩ s]\n  apply RingHom.isUnit_map\n  rw [← isUnit_map_iff ((coequalizer.π f.val g.val : _).c.app _), ← comp_apply,\n    nat_trans.naturality, comp_apply, TopCat.Presheaf.pushforwardObj_map, ←\n    isUnit_map_iff (Y.presheaf.map (eq_to_hom hV').op), ← comp_apply, ← functor.map_comp]\n  convert@RingedSpace.is_unit_res_basic_open Y.to_RingedSpace (unop _)\n      (((coequalizer.π f.val g.val).c.app (op U)) s)\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.has_coequalizer.coequalizer_π_stalk_is_local_ring_hom AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_stalk_isLocalRingHom\n\nend HasCoequalizer\n\n/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally\nringed space. -/\nnoncomputable def coequalizer : LocallyRingedSpace\n    where\n  toSheafedSpace := coequalizer f.1 g.1\n  LocalRing x :=\n    by\n    obtain ⟨y, rfl⟩ :=\n      (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x\n    exact (PresheafedSpace.stalk_map (coequalizer.π f.val g.val : _) y).domain_localRing\n#align algebraic_geometry.LocallyRingedSpace.coequalizer AlgebraicGeometry.LocallyRingedSpace.coequalizer\n\n/-- The explicit coequalizer cofork of locally ringed spaces. -/\nnoncomputable def coequalizerCofork : Cofork f g :=\n  @Cofork.ofπ _ _ _ _ f g (coequalizer f g) ⟨coequalizer.π f.1 g.1, inferInstance⟩\n    (LocallyRingedSpace.Hom.ext _ _ (coequalizer.condition f.1 g.1))\n#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork AlgebraicGeometry.LocallyRingedSpace.coequalizerCofork\n\ntheorem isLocalRingHom_stalkMap_congr {X Y : RingedSpace} (f g : X ⟶ Y) (H : f = g) (x)\n    (h : IsLocalRingHom (PresheafedSpace.stalkMap f x)) :\n    IsLocalRingHom (PresheafedSpace.stalkMap g x) :=\n  by\n  rw [PresheafedSpace.stalk_map.congr_hom _ _ H.symm x]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_local_ring_hom_stalk_map_congr AlgebraicGeometry.LocallyRingedSpace.isLocalRingHom_stalkMap_congr\n\n/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/\nnoncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g) :=\n  by\n  apply cofork.is_colimit.mk'\n  intro s\n  have e : f.val ≫ s.π.val = g.val ≫ s.π.val := by injection s.condition\n  use coequalizer.desc s.π.1 e\n  · intro x\n    rcases(TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x with\n      ⟨y, rfl⟩\n    apply isLocalRingHom_of_comp _ (PresheafedSpace.stalk_map (coequalizer_cofork f g).π.1 _)\n    change IsLocalRingHom (_ ≫ PresheafedSpace.stalk_map (coequalizer_cofork f g).π.val y)\n    erw [← PresheafedSpace.stalk_map.comp]\n    apply is_local_ring_hom_stalk_map_congr _ _ (coequalizer.π_desc s.π.1 e).symm y\n    infer_instance\n  constructor\n  · exact LocallyRingedSpace.hom.ext _ _ (coequalizer.π_desc _ _)\n  intro m h\n  replace h : (coequalizer_cofork f g).π.1 ≫ m.1 = s.π.1 :=\n    by\n    rw [← h]\n    rfl\n  apply LocallyRingedSpace.hom.ext\n  apply (colimit.is_colimit (parallel_pair f.1 g.1)).uniq (cofork.of_π s.π.1 e) m.1\n  rintro ⟨⟩\n  · rw [← (colimit.cocone (parallel_pair f.val g.val)).w walking_parallel_pair_hom.left,\n      category.assoc]\n    change _ ≫ _ ≫ _ = _ ≫ _\n    congr\n    exact h\n  · exact h\n#align algebraic_geometry.LocallyRingedSpace.coequalizer_cofork_is_colimit AlgebraicGeometry.LocallyRingedSpace.coequalizerCoforkIsColimit\n\ninstance : HasCoequalizer f g :=\n  ⟨⟨⟨_, coequalizerCoforkIsColimit f g⟩⟩⟩\n\ninstance : HasCoequalizers LocallyRingedSpace :=\n  hasCoequalizers_of_hasColimit_parallelPair _\n\nnoncomputable instance preservesCoequalizer :\n    PreservesColimitsOfShape WalkingParallelPair forgetToSheafedSpace.{v} :=\n  ⟨fun F => by\n    apply preserves_colimit_of_iso_diagram _ (diagram_iso_parallel_pair F).symm\n    apply preserves_colimit_of_preserves_colimit_cocone (coequalizer_cofork_is_colimit _ _)\n    apply (is_colimit_map_cocone_cofork_equiv _ _).symm _\n    dsimp only [forget_to_SheafedSpace]\n    exact coequalizer_is_coequalizer _ _⟩\n#align algebraic_geometry.LocallyRingedSpace.preserves_coequalizer AlgebraicGeometry.LocallyRingedSpace.preservesCoequalizer\n\nend HasCoequalizer\n\ninstance : HasColimits LocallyRingedSpace :=\n  has_colimits_of_hasCoequalizers_and_coproducts\n\nnoncomputable instance : PreservesColimits LocallyRingedSpace.forgetToSheafedSpace :=\n  preservesColimitsOfPreservesCoequalizersAndCoproducts _\n\nend LocallyRingedSpace\n\nend AlgebraicGeometry\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/LocallyRingedSpace/HasColimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.3312571074789061}}
{"text": "import .pseudominor\n\nnoncomputable theory \nopen set \nopen function \n\nvariables {E E₀ E₁ E₂ : Type*} [finite E] [finite E₀] [finite E₁] [finite E₂] \n  {e f : E} {M : matroid E} {M₀ : matroid E₀} {M₁ : matroid E₁} {M₂ : matroid E₂} \n  {X Y D C F B I I₀ R: set E}\n\nnamespace matroid \n\n/-- A structure encoding the fact that `M₀` is isomorphic to a minor of `M` -/\nstructure minor_embedding (M₀ : matroid E₀) (M : matroid E) := \n(contr : set E)\n(to_embedding : E₀ ↪ E)\n(disj : disjoint (range to_embedding) contr)\n(indep_iff' : ∀ X, M₀.indep X ↔ (M ⟋ contr).indep (to_embedding '' X))\n\nnamespace minor_embedding\n\ninfix ` →m ` :25 :=  minor_embedding\n\ninstance : has_coe_to_fun (minor_embedding M₀ M) (λ a, E₀ → E) := ⟨λ φ, φ.to_embedding⟩ \n\n/-- Two `minor_embedding`s are equivalent if they differ only in choice of their contract and \n  delete-sets-/\ndef equiv (φ₁ φ₂ : M₀ →m M) := \n  (φ₁ : E₀ → E) = (φ₂ : E₀ → E)    \n\n@[simp] lemma coe_to_embedding (φ : M₀ →m M) : \n  (φ.to_embedding : E₀ → E) = (φ : E₀ → E) := \nrfl \n\n@[simp] lemma inj (φ : M₀ →m M) {x y : E₀} :\n  φ x = φ y ↔ x = y :=\nφ.to_embedding.injective.eq_iff \n\nlemma injective (φ : M₀ →m M) :\n  function.injective φ :=   \nφ.to_embedding.injective\n\n/-- The contract-set of a `minor_embedding` -/\ndef C (φ : M₀ →m M) : set E := \n  φ.contr \n\n/-- The ground set in `M` of a `minor_embedding` -/\ndef ground (φ : M₀ →m M) : set E := \n  set.range φ\n\n/-- The delete-set of a `minor_embedding` --/\ndef D (φ : M₀ →m M) : set E := \n  (φ.C ∪ φ.ground)ᶜ \n\nlemma mem_ground (φ : M₀ →m M) (x : E₀) :\n  φ x ∈ φ.ground := \nmem_range_self _\n\nlemma image_subset_ground (φ : M₀ →m M) (I : set E₀) : \n  φ '' I ⊆ φ.ground :=\nimage_subset_range _ _\n\nlemma not_mem_C (φ : M₀ →m M) (x : E₀) :\n  φ x ∉ φ.C := \nλ h, φ.disj.ne_of_mem ⟨x,rfl⟩ h rfl\n\nlemma not_mem_D (φ : M₀ →m M) (x : E₀) :\n  φ x ∉ φ.D := \nnot_mem_compl_iff.mpr (or.inr (φ.mem_ground _))\n\nlemma C_disjoint_ground (φ : M₀ →m M) : \n  disjoint φ.C φ.ground := \ndisjoint_iff_forall_ne.mpr (by {rintro x hx _ ⟨y,rfl⟩ rfl, exact φ.not_mem_C _ hx}) \n\nlemma D_disjoint_ground (φ : M₀ →m M) : \n  disjoint φ.D φ.ground := \ndisjoint_iff_forall_ne.mpr (by {rintro x hx _ ⟨y,rfl⟩ rfl, exact φ.not_mem_D _ hx}) \n\nlemma indep_iff (φ : M₀ →m M) {I₀ : set E₀} : \n  M₀.indep I₀ ↔ (M ⟋ φ.C).indep (φ '' I₀) := \nφ.indep_iff' I₀  \n\nlemma basis_iff (φ : M₀ →m M) {I₀ X₀ : set E₀} :\n  M₀.basis I₀ X₀ ↔ (M ⟋ φ.C).basis (φ '' I₀) (φ '' X₀) :=\nbegin\n  simp_rw [basis_iff, ←φ.indep_iff, image_subset_image_iff φ.injective], \n  split, \n  { rintro ⟨hi,hss,h⟩,\n    refine ⟨hi,hss,λ J hJ hssJ hJss, _⟩,\n    have h'J : ∃ J₀, φ '' J₀ = J,   \n    { rw [←subset_range_iff_exists_image_eq], \n      exact hJss.trans (φ.image_subset_ground X₀)} ,\n    obtain ⟨J₀,rfl⟩ := h'J,   \n    rw [image_subset_image_iff φ.injective] at hssJ hJss, \n    rw [←φ.indep_iff] at hJ, \n    rw [h _ hJ hssJ hJss]},\n  rintro ⟨hi, hss, h⟩,\n  refine ⟨hi, hss, λ J₀ hJ₀ hssJ₀ hJ₀ss, hssJ₀.antisymm _⟩, \n  specialize h (φ '' J₀) (φ.indep_iff.mp hJ₀) ((image_subset_image_iff φ.injective).mpr hssJ₀)\n    ((image_subset_image_iff φ.injective).mpr hJ₀ss), \n  rw ←(image_subset_image_iff φ.injective),  \n  exact h.symm.subset, \nend  \n\nlemma indep_project_iff (φ : M₀ →m M) (I₀ C₀ : set E₀) : \n  (M₀ ⟋ C₀).indep I₀ ↔ (M ⟋ φ.C ⟋ (φ '' C₀)).indep (φ '' I₀) :=\nbegin\n  obtain ⟨IC₀,hIC₀⟩ := M₀.exists_basis C₀, \n  have h''  := φ.basis_iff.mp hIC₀, \n  obtain (hI₀ | hI₀) := em (M₀.indep I₀),  \n  { rw [←hIC₀.project_eq_project,  ←h''.project_eq_project, \n      indep.project_indep_iff h''.indep (φ.indep_iff.mp hI₀), ←image_union, \n      ←φ.indep_iff, disjoint_image_iff φ.injective, indep.project_indep_iff hIC₀.indep hI₀]},\n\n  apply iff_of_false (λ hI₀', hI₀ (indep_of_project_indep hI₀')) (λ hI₀', hI₀ _), \n  have h' := indep_of_project_indep hI₀', \n  rwa [←φ.indep_iff] at h', \nend  \n\n/-- If `M₀` is a minor of `M₁` and `M₁` is a minor of `M₂`, then `M₀` is a minor of `M₂` -/\ndef trans (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) : \n  M₀ →m M₂ := \n{ contr := (φ₁₂ '' φ₀₁.C) ∪ φ₁₂.C ,\n  to_embedding := φ₀₁.to_embedding.trans φ₁₂.to_embedding, \n  disj := begin\n    refine disjoint_iff_forall_ne.mpr _,\n    simp_rw [mem_range, embedding.trans_apply, coe_to_embedding], \n    rintro x ⟨x₀,rfl⟩ y (⟨x₁,hx₁,rfl⟩ | hy) he,\n    { simp_rw [minor_embedding.inj] at he,\n      subst he, \n      exact φ₀₁.not_mem_C _ hx₁}, \n    subst he,  \n    exact φ₁₂.not_mem_C _ hy, \n  end,\n  indep_iff' := \n  begin\n    intro X,  \n    simp only [embedding.trans_apply, coe_to_embedding ], \n    nth_rewrite 1 ←image_image,  \n    dsimp only, \n    rw [←project_project, project_project_comm, ←φ₁₂.indep_project_iff, φ₀₁.indep_iff],\n  end }\n\n@[simp] lemma trans_apply (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) (x : E₀) : \n  (φ₀₁.trans φ₁₂) x = φ₁₂ (φ₀₁ x) := \nrfl   \n\n@[simp] lemma trans_apply_image (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) (X : set E₀) : \n  (φ₀₁.trans φ₁₂) '' X = φ₁₂ '' (φ₀₁ '' X) := \nby {ext, simp only [trans_apply, mem_image, exists_exists_and_eq_and]}\n\n@[simp] lemma trans_apply_C (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) : \n  (φ₀₁.trans φ₁₂).C = (φ₁₂ '' φ₀₁.C) ∪ φ₁₂.C := \nrfl \n\n@[simp] lemma trans_apply_ground (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) : \n  (φ₀₁.trans φ₁₂).ground = φ₁₂ '' (φ₀₁.ground) := \nby {ext, simp [ground]}\n\n/- What a mess! Should be trivial -/\n@[simp] lemma trans_apply_D (φ₀₁ : M₀ →m M₁) (φ₁₂ : M₁ →m M₂) : \n  (φ₀₁.trans φ₁₂).D = (φ₁₂ '' φ₀₁.D) ∪ φ₁₂.D := \nbegin\n  apply compl_injective, \n  simp only [D, trans_apply_C, trans_apply_ground, ground, compl_compl, \n    compl_union, φ₁₂.injective.compl_image, compl_inter, image_union], \n  rw [inter_distrib_right, inter_eq_left_iff_subset.mpr (image_subset_range φ₁₂ _), \n    compl_inter_self, union_empty, ←range_comp, inter_distrib_left, inter_distrib_right, \n    inter_eq_right_iff_subset.mpr \n      (subset_compl_iff_disjoint_right.mpr φ₁₂.C_disjoint_ground : φ₁₂.C ⊆ (range φ₁₂)ᶜ), \n    union_eq_right_iff_subset.mpr (inter_subset_right _ _), inter_distrib_right, \n    compl_inter_self, union_empty, inter_distrib_right, \n    inter_eq_left_iff_subset.mpr (range_comp_subset_range _ _), \n    union_distrib_right, union_distrib_left,\n    union_eq_left_iff_subset.mpr (range_comp_subset_range _ _), ←union_assoc, \n    union_comm φ₁₂.C, eq_comm],\n  \n  convert inter_eq_left_iff_subset.mpr (union_subset (union_subset _ (subset_union_left _ _)) _), \n  { exact (image_subset_range _ _).trans (subset_union_right _ _)},\n  exact (range_comp_subset_range _ _).trans (subset_union_right _ _), \nend \n\n/-- A `minor_embedding` gives rise to a dual embedding -/\ndef dual (φ : M₀ →m M) : \n  M₀.dual →m M.dual := \n{ contr := φ.D,\n  to_embedding := φ.to_embedding,\n  disj := φ.D_disjoint_ground.symm,\n  indep_iff' := begin\n    intro X, \n    \n    -- rw [dual_indep_iff_coindep], \n  end  }\n\n/-- The scum theorem; every `minor_embedding`, has an equivalent embedding whose contract-set is \n  independent and whose delete-set is coindependent -/\ntheorem exists_indep_coindep_embedding (φ : M₀ →m M) : \n  ∃ (ψ : M₀ →m M), ψ.equiv φ ∧ M.indep ψ.C ∧ M.coindep φ.D := \nsorry         \n\n\n\nend minor_embedding\n\n\n/-- `M₀` is isomorphic to a minor of `M` if there is a `minor_embedding` from `M₀` to `M` -/\ndef is_iso_minor (M₀ : matroid E₀) (M : matroid E) : Prop := \n  nonempty (M₀ →m M)\n\n/-- A matroid `M₀` on a subtype is a minor of `M` if there is a `minor_embedding` whose associated\n  embedding function is the inclusion map. -/\ndef is_minor {X : set E} (M₀ : matroid X) (M : matroid E) : Prop :=\n  ∃ (φ : M₀ →m M), (φ : X → E) = coe   \n\ninfix ` ≤m ` : 25 := is_iso_minor \n\n\n\n\n\n\n\n\nend matroid ", "meta": {"author": "apnelson1", "repo": "lean-matroids", "sha": "1880f5b4b9a99948368e35539672a9bf35cc09cc", "save_path": "github-repos/lean/apnelson1-lean-matroids", "path": "github-repos/lean/apnelson1-lean-matroids/lean-matroids-1880f5b4b9a99948368e35539672a9bf35cc09cc/src/matroid/submatroids/minor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.331257107478906}}
{"text": "/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.lattice\nimport Mathlib.PostPort\n\nuniverses u v l u_1 w x u_2 u_3 \n\nnamespace Mathlib\n\n/-- A `pequiv` is a partial equivalence, a representation of a bijection between a subset\n  of `α` and a subset of `β` -/\nstructure pequiv (α : Type u) (β : Type v) where\n  to_fun : α → Option β\n  inv_fun : β → Option α\n  inv : ∀ (a : α) (b : β), a ∈ inv_fun b ↔ b ∈ to_fun a\n\ninfixr:25 \" ≃. \" => Mathlib.pequiv\n\nnamespace pequiv\n\n\nprotected instance has_coe_to_fun {α : Type u} {β : Type v} : has_coe_to_fun (α ≃. β) :=\n  has_coe_to_fun.mk (fun (x : α ≃. β) => α → Option β) to_fun\n\n@[simp] theorem coe_mk_apply {α : Type u} {β : Type v} (f₁ : α → Option β) (f₂ : β → Option α)\n    (h : ∀ (a : α) (b : β), a ∈ f₂ b ↔ b ∈ f₁ a) (x : α) : coe_fn (mk f₁ f₂ h) x = f₁ x :=\n  rfl\n\ntheorem ext {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β}\n    (h : ∀ (x : α), coe_fn f x = coe_fn g x) : f = g :=\n  sorry\n\ntheorem ext_iff {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β} :\n    f = g ↔ ∀ (x : α), coe_fn f x = coe_fn g x :=\n  { mp := congr_fun ∘ congr_arg fun {f : α ≃. β} (x : α) => coe_fn f x, mpr := ext }\n\nprotected def refl (α : Type u_1) : α ≃. α := mk some some sorry\n\nprotected def symm {α : Type u} {β : Type v} (f : α ≃. β) : β ≃. α :=\n  mk (inv_fun f) (to_fun f) sorry\n\ntheorem mem_iff_mem {α : Type u} {β : Type v} (f : α ≃. β) {a : α} {b : β} :\n    a ∈ coe_fn (pequiv.symm f) b ↔ b ∈ coe_fn f a :=\n  inv f\n\ntheorem eq_some_iff {α : Type u} {β : Type v} (f : α ≃. β) {a : α} {b : β} :\n    coe_fn (pequiv.symm f) b = some a ↔ coe_fn f a = some b :=\n  inv f\n\nprotected def trans {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) : α ≃. γ :=\n  mk (fun (a : α) => option.bind (coe_fn f a) ⇑g)\n    (fun (a : γ) => option.bind (coe_fn (pequiv.symm g) a) ⇑(pequiv.symm f)) sorry\n\n@[simp] theorem refl_apply {α : Type u} (a : α) : coe_fn (pequiv.refl α) a = some a := rfl\n\n@[simp] theorem symm_refl {α : Type u} : pequiv.symm (pequiv.refl α) = pequiv.refl α := rfl\n\n@[simp] theorem symm_symm {α : Type u} {β : Type v} (f : α ≃. β) :\n    pequiv.symm (pequiv.symm f) = f :=\n  sorry\n\ntheorem symm_injective {α : Type u} {β : Type v} : function.injective pequiv.symm :=\n  function.left_inverse.injective symm_symm\n\ntheorem trans_assoc {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} (f : α ≃. β) (g : β ≃. γ)\n    (h : γ ≃. δ) : pequiv.trans (pequiv.trans f g) h = pequiv.trans f (pequiv.trans g h) :=\n  ext fun (_x : α) => option.bind_assoc (coe_fn f _x) ⇑g ⇑h\n\ntheorem mem_trans {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α) (c : γ) :\n    c ∈ coe_fn (pequiv.trans f g) a ↔ ∃ (b : β), b ∈ coe_fn f a ∧ c ∈ coe_fn g b :=\n  option.bind_eq_some'\n\ntheorem trans_eq_some {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α)\n    (c : γ) :\n    coe_fn (pequiv.trans f g) a = some c ↔ ∃ (b : β), coe_fn f a = some b ∧ coe_fn g b = some c :=\n  option.bind_eq_some'\n\ntheorem trans_eq_none {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) (a : α) :\n    coe_fn (pequiv.trans f g) a = none ↔ ∀ (b : β) (c : γ), ¬b ∈ coe_fn f a ∨ ¬c ∈ coe_fn g b :=\n  sorry\n\n@[simp] theorem refl_trans {α : Type u} {β : Type v} (f : α ≃. β) :\n    pequiv.trans (pequiv.refl α) f = f :=\n  ext fun (x : α) => option.ext fun (a : β) => id (iff.refl (a ∈ coe_fn f x))\n\n@[simp] theorem trans_refl {α : Type u} {β : Type v} (f : α ≃. β) :\n    pequiv.trans f (pequiv.refl β) = f :=\n  sorry\n\nprotected theorem inj {α : Type u} {β : Type v} (f : α ≃. β) {a₁ : α} {a₂ : α} {b : β}\n    (h₁ : b ∈ coe_fn f a₁) (h₂ : b ∈ coe_fn f a₂) : a₁ = a₂ :=\n  sorry\n\ntheorem injective_of_forall_ne_is_some {α : Type u} {β : Type v} (f : α ≃. β) (a₂ : α)\n    (h : ∀ (a₁ : α), a₁ ≠ a₂ → ↥(option.is_some (coe_fn f a₁))) : function.injective ⇑f :=\n  sorry\n\ntheorem injective_of_forall_is_some {α : Type u} {β : Type v} {f : α ≃. β}\n    (h : ∀ (a : α), ↥(option.is_some (coe_fn f a))) : function.injective ⇑f :=\n  sorry\n\ndef of_set {α : Type u} (s : set α) [decidable_pred s] : α ≃. α :=\n  mk (fun (a : α) => ite (a ∈ s) (some a) none) (fun (a : α) => ite (a ∈ s) (some a) none) sorry\n\ntheorem mem_of_set_self_iff {α : Type u} {s : set α} [decidable_pred s] {a : α} :\n    a ∈ coe_fn (of_set s) a ↔ a ∈ s :=\n  sorry\n\ntheorem mem_of_set_iff {α : Type u} {s : set α} [decidable_pred s] {a : α} {b : α} :\n    a ∈ coe_fn (of_set s) b ↔ a = b ∧ a ∈ s :=\n  sorry\n\n@[simp] theorem of_set_eq_some_iff {α : Type u} {s : set α} {h : decidable_pred s} {a : α} {b : α} :\n    coe_fn (of_set s) b = some a ↔ a = b ∧ a ∈ s :=\n  mem_of_set_iff\n\n@[simp] theorem of_set_eq_some_self_iff {α : Type u} {s : set α} {h : decidable_pred s} {a : α} :\n    coe_fn (of_set s) a = some a ↔ a ∈ s :=\n  mem_of_set_self_iff\n\n@[simp] theorem of_set_symm {α : Type u} (s : set α) [decidable_pred s] :\n    pequiv.symm (of_set s) = of_set s :=\n  rfl\n\n@[simp] theorem of_set_univ {α : Type u} : of_set set.univ = pequiv.refl α := rfl\n\n@[simp] theorem of_set_eq_refl {α : Type u} {s : set α} [decidable_pred s] :\n    of_set s = pequiv.refl α ↔ s = set.univ :=\n  sorry\n\ntheorem symm_trans_rev {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) (g : β ≃. γ) :\n    pequiv.symm (pequiv.trans f g) = pequiv.trans (pequiv.symm g) (pequiv.symm f) :=\n  rfl\n\ntheorem trans_symm {α : Type u} {β : Type v} (f : α ≃. β) :\n    pequiv.trans f (pequiv.symm f) =\n        of_set (set_of fun (a : α) => ↥(option.is_some (coe_fn f a))) :=\n  sorry\n\ntheorem symm_trans {α : Type u} {β : Type v} (f : α ≃. β) :\n    pequiv.trans (pequiv.symm f) f =\n        of_set (set_of fun (b : β) => ↥(option.is_some (coe_fn (pequiv.symm f) b))) :=\n  sorry\n\ntheorem trans_symm_eq_iff_forall_is_some {α : Type u} {β : Type v} {f : α ≃. β} :\n    pequiv.trans f (pequiv.symm f) = pequiv.refl α ↔ ∀ (a : α), ↥(option.is_some (coe_fn f a)) :=\n  sorry\n\nprotected instance has_bot {α : Type u} {β : Type v} : has_bot (α ≃. β) :=\n  has_bot.mk (mk (fun (_x : α) => none) (fun (_x : β) => none) sorry)\n\n@[simp] theorem bot_apply {α : Type u} {β : Type v} (a : α) : coe_fn ⊥ a = none := rfl\n\n@[simp] theorem symm_bot {α : Type u} {β : Type v} : pequiv.symm ⊥ = ⊥ := rfl\n\n@[simp] theorem trans_bot {α : Type u} {β : Type v} {γ : Type w} (f : α ≃. β) :\n    pequiv.trans f ⊥ = ⊥ :=\n  sorry\n\n@[simp] theorem bot_trans {α : Type u} {β : Type v} {γ : Type w} (f : β ≃. γ) :\n    pequiv.trans ⊥ f = ⊥ :=\n  sorry\n\ntheorem is_some_symm_get {α : Type u} {β : Type v} (f : α ≃. β) {a : α}\n    (h : ↥(option.is_some (coe_fn f a))) :\n    ↥(option.is_some (coe_fn (pequiv.symm f) (option.get h))) :=\n  sorry\n\ndef single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) : α ≃. β :=\n  mk (fun (x : α) => ite (x = a) (some b) none) (fun (x : β) => ite (x = b) (some a) none) sorry\n\ntheorem mem_single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α) (b : β) :\n    b ∈ coe_fn (single a b) a :=\n  if_pos rfl\n\ntheorem mem_single_iff {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a₁ : α) (a₂ : α)\n    (b₁ : β) (b₂ : β) : b₁ ∈ coe_fn (single a₂ b₂) a₁ ↔ a₁ = a₂ ∧ b₁ = b₂ :=\n  sorry\n\n@[simp] theorem symm_single {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α)\n    (b : β) : pequiv.symm (single a b) = single b a :=\n  rfl\n\n@[simp] theorem single_apply {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] (a : α)\n    (b : β) : coe_fn (single a b) a = some b :=\n  if_pos rfl\n\ntheorem single_apply_of_ne {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] {a₁ : α}\n    {a₂ : α} (h : a₁ ≠ a₂) (b : β) : coe_fn (single a₁ b) a₂ = none :=\n  if_neg (ne.symm h)\n\ntheorem single_trans_of_mem {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β]\n    [DecidableEq γ] (a : α) {b : β} {c : γ} {f : β ≃. γ} (h : c ∈ coe_fn f b) :\n    pequiv.trans (single a b) f = single a c :=\n  sorry\n\ntheorem trans_single_of_mem {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α] [DecidableEq β]\n    [DecidableEq γ] {a : α} {b : β} (c : γ) {f : α ≃. β} (h : b ∈ coe_fn f a) :\n    pequiv.trans f (single b c) = single a c :=\n  symm_injective (single_trans_of_mem c (iff.mpr (mem_iff_mem f) h))\n\n@[simp] theorem single_trans_single {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α]\n    [DecidableEq β] [DecidableEq γ] (a : α) (b : β) (c : γ) :\n    pequiv.trans (single a b) (single b c) = single a c :=\n  single_trans_of_mem a (mem_single b c)\n\n@[simp] theorem single_subsingleton_eq_refl {α : Type u} [DecidableEq α] [subsingleton α] (a : α)\n    (b : α) : single a b = pequiv.refl α :=\n  sorry\n\ntheorem trans_single_of_eq_none {β : Type v} {γ : Type w} {δ : Type x} [DecidableEq β]\n    [DecidableEq γ] {b : β} (c : γ) {f : δ ≃. β} (h : coe_fn (pequiv.symm f) b = none) :\n    pequiv.trans f (single b c) = ⊥ :=\n  sorry\n\ntheorem single_trans_of_eq_none {α : Type u} {β : Type v} {δ : Type x} [DecidableEq α]\n    [DecidableEq β] (a : α) {b : β} {f : β ≃. δ} (h : coe_fn f b = none) :\n    pequiv.trans (single a b) f = ⊥ :=\n  symm_injective (trans_single_of_eq_none a h)\n\ntheorem single_trans_single_of_ne {α : Type u} {β : Type v} {γ : Type w} [DecidableEq α]\n    [DecidableEq β] [DecidableEq γ] {b₁ : β} {b₂ : β} (h : b₁ ≠ b₂) (a : α) (c : γ) :\n    pequiv.trans (single a b₁) (single b₂ c) = ⊥ :=\n  single_trans_of_eq_none a (single_apply_of_ne (ne.symm h) c)\n\nprotected instance partial_order {α : Type u} {β : Type v} : partial_order (α ≃. β) :=\n  partial_order.mk (fun (f g : α ≃. β) => ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a)\n    (preorder.lt._default fun (f g : α ≃. β) => ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a)\n    sorry sorry sorry\n\ntheorem le_def {α : Type u} {β : Type v} {f : α ≃. β} {g : α ≃. β} :\n    f ≤ g ↔ ∀ (a : α) (b : β), b ∈ coe_fn f a → b ∈ coe_fn g a :=\n  iff.rfl\n\nprotected instance order_bot {α : Type u} {β : Type v} : order_bot (α ≃. β) :=\n  order_bot.mk ⊥ partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance semilattice_inf_bot {α : Type u} {β : Type v} [DecidableEq α] [DecidableEq β] :\n    semilattice_inf_bot (α ≃. β) :=\n  semilattice_inf_bot.mk order_bot.bot order_bot.le order_bot.lt sorry sorry sorry sorry\n    (fun (f g : α ≃. β) =>\n      mk (fun (a : α) => ite (coe_fn f a = coe_fn g a) (coe_fn f a) none)\n        (fun (b : β) =>\n          ite (coe_fn (pequiv.symm f) b = coe_fn (pequiv.symm g) b) (coe_fn (pequiv.symm f) b) none)\n        sorry)\n    sorry sorry sorry\n\nend pequiv\n\n\nnamespace equiv\n\n\ndef to_pequiv {α : Type u_1} {β : Type u_2} (f : α ≃ β) : α ≃. β :=\n  pequiv.mk (some ∘ ⇑f) (some ∘ ⇑(equiv.symm f)) sorry\n\n@[simp] theorem to_pequiv_refl {α : Type u_1} : to_pequiv (equiv.refl α) = pequiv.refl α := rfl\n\ntheorem to_pequiv_trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α ≃ β) (g : β ≃ γ) :\n    to_pequiv (equiv.trans f g) = pequiv.trans (to_pequiv f) (to_pequiv g) :=\n  rfl\n\ntheorem to_pequiv_symm {α : Type u_1} {β : Type u_2} (f : α ≃ β) :\n    to_pequiv (equiv.symm f) = pequiv.symm (to_pequiv f) :=\n  rfl\n\ntheorem to_pequiv_apply {α : Type u_1} {β : Type u_2} (f : α ≃ β) (x : α) :\n    coe_fn (to_pequiv f) x = some (coe_fn f x) :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/pequiv_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.33125709898587796}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.lint.default\nimport Mathlib.tactic.ext\nimport Mathlib.PostPort\n\nuniverses u_1 u_4 u_2 u_3 u_5 u_6 \n\nnamespace Mathlib\n\nnamespace sigma\n\n\nprotected instance inhabited {α : Type u_1} {β : α → Type u_4} [Inhabited α] [Inhabited (β Inhabited.default)] : Inhabited (sigma β) :=\n  { default := mk Inhabited.default Inhabited.default }\n\nprotected instance decidable_eq {α : Type u_1} {β : α → Type u_4} [h₁ : DecidableEq α] [h₂ : (a : α) → DecidableEq (β a)] : DecidableEq (sigma β) :=\n  sorry\n\n@[simp] theorem mk.inj_iff {α : Type u_1} {β : α → Type u_4} {a₁ : α} {a₂ : α} {b₁ : β a₁} {b₂ : β a₂} : mk a₁ b₁ = mk a₂ b₂ ↔ a₁ = a₂ ∧ b₁ == b₂ := sorry\n\n@[simp] theorem eta {α : Type u_1} {β : α → Type u_4} (x : sigma fun (a : α) => β a) : mk (fst x) (snd x) = x :=\n  cases_on x\n    fun (x_fst : α) (x_snd : β x_fst) =>\n      idRhs (mk (fst (mk x_fst x_snd)) (snd (mk x_fst x_snd)) = mk (fst (mk x_fst x_snd)) (snd (mk x_fst x_snd))) rfl\n\ntheorem ext {α : Type u_1} {β : α → Type u_4} {x₀ : sigma β} {x₁ : sigma β} (h₀ : fst x₀ = fst x₁) (h₁ : snd x₀ == snd x₁) : x₀ = x₁ := sorry\n\ntheorem ext_iff {α : Type u_1} {β : α → Type u_4} {x₀ : sigma β} {x₁ : sigma β} : x₀ = x₁ ↔ fst x₀ = fst x₁ ∧ snd x₀ == snd x₁ :=\n  cases_on x₀ fun (x₀_fst : α) (x₀_snd : β x₀_fst) => cases_on x₁ fun (x₁_fst : α) (x₁_snd : β x₁_fst) => mk.inj_iff\n\n@[simp] theorem forall {α : Type u_1} {β : α → Type u_4} {p : (sigma fun (a : α) => β a) → Prop} : (∀ (x : sigma fun (a : α) => β a), p x) ↔ ∀ (a : α) (b : β a), p (mk a b) := sorry\n\n@[simp] theorem exists {α : Type u_1} {β : α → Type u_4} {p : (sigma fun (a : α) => β a) → Prop} : (∃ (x : sigma fun (a : α) => β a), p x) ↔ ∃ (a : α), ∃ (b : β a), p (mk a b) := sorry\n\n/-- Map the left and right components of a sigma -/\ndef map {α₁ : Type u_2} {α₂ : Type u_3} {β₁ : α₁ → Type u_5} {β₂ : α₂ → Type u_6} (f₁ : α₁ → α₂) (f₂ : (a : α₁) → β₁ a → β₂ (f₁ a)) (x : sigma β₁) : sigma β₂ :=\n  mk (f₁ (fst x)) (f₂ (fst x) (snd x))\n\nend sigma\n\n\ntheorem sigma_mk_injective {α : Type u_1} {β : α → Type u_4} {i : α} : function.injective (sigma.mk i) := sorry\n\ntheorem function.injective.sigma_map {α₁ : Type u_2} {α₂ : Type u_3} {β₁ : α₁ → Type u_5} {β₂ : α₂ → Type u_6} {f₁ : α₁ → α₂} {f₂ : (a : α₁) → β₁ a → β₂ (f₁ a)} (h₁ : function.injective f₁) (h₂ : ∀ (a : α₁), function.injective (f₂ a)) : function.injective (sigma.map f₁ f₂) := sorry\n\ntheorem function.surjective.sigma_map {α₁ : Type u_2} {α₂ : Type u_3} {β₁ : α₁ → Type u_5} {β₂ : α₂ → Type u_6} {f₁ : α₁ → α₂} {f₂ : (a : α₁) → β₁ a → β₂ (f₁ a)} (h₁ : function.surjective f₁) (h₂ : ∀ (a : α₁), function.surjective (f₂ a)) : function.surjective (sigma.map f₁ f₂) := sorry\n\n/-- Interpret a function on `Σ x : α, β x` as a dependent function with two arguments. -/\ndef sigma.curry {α : Type u_1} {β : α → Type u_4} {γ : (a : α) → β a → Type u_2} (f : (x : sigma β) → γ (sigma.fst x) (sigma.snd x)) (x : α) (y : β x) : γ x y :=\n  f (sigma.mk x y)\n\n/-- Interpret a dependent function with two arguments as a function on `Σ x : α, β x` -/\ndef sigma.uncurry {α : Type u_1} {β : α → Type u_4} {γ : (a : α) → β a → Type u_2} (f : (x : α) → (y : β x) → γ x y) (x : sigma β) : γ (sigma.fst x) (sigma.snd x) :=\n  f (sigma.fst x) (sigma.snd x)\n\n/-- Convert a product type to a Σ-type. -/\n@[simp] def prod.to_sigma {α : Type u_1} {β : Type u_2} : α × β → sigma fun (_x : α) => β :=\n  sorry\n\n@[simp] theorem prod.fst_to_sigma {α : Type u_1} {β : Type u_2} (x : α × β) : sigma.fst (prod.to_sigma x) = prod.fst x :=\n  prod.cases_on x fun (x_fst : α) (x_snd : β) => Eq.refl (sigma.fst (prod.to_sigma (x_fst, x_snd)))\n\n@[simp] theorem prod.snd_to_sigma {α : Type u_1} {β : Type u_2} (x : α × β) : sigma.snd (prod.to_sigma x) = prod.snd x :=\n  prod.cases_on x fun (x_fst : α) (x_snd : β) => Eq.refl (sigma.snd (prod.to_sigma (x_fst, x_snd)))\n\nnamespace psigma\n\n\n/-- Nondependent eliminator for `psigma`. -/\ndef elim {α : Sort u_1} {β : α → Sort u_2} {γ : Sort u_3} (f : (a : α) → β a → γ) (a : psigma β) : γ :=\n  cases_on a f\n\n@[simp] theorem elim_val {α : Sort u_1} {β : α → Sort u_2} {γ : Sort u_3} (f : (a : α) → β a → γ) (a : α) (b : β a) : elim f (mk a b) = f a b :=\n  rfl\n\nprotected instance inhabited {α : Sort u_1} {β : α → Sort u_2} [Inhabited α] [Inhabited (β Inhabited.default)] : Inhabited (psigma β) :=\n  { default := mk Inhabited.default Inhabited.default }\n\nprotected instance decidable_eq {α : Sort u_1} {β : α → Sort u_2} [h₁ : DecidableEq α] [h₂ : (a : α) → DecidableEq (β a)] : DecidableEq (psigma β) :=\n  sorry\n\ntheorem mk.inj_iff {α : Sort u_1} {β : α → Sort u_2} {a₁ : α} {a₂ : α} {b₁ : β a₁} {b₂ : β a₂} : mk a₁ b₁ = mk a₂ b₂ ↔ a₁ = a₂ ∧ b₁ == b₂ := sorry\n\ntheorem ext {α : Sort u_1} {β : α → Sort u_2} {x₀ : psigma β} {x₁ : psigma β} (h₀ : fst x₀ = fst x₁) (h₁ : snd x₀ == snd x₁) : x₀ = x₁ := sorry\n\ntheorem ext_iff {α : Sort u_1} {β : α → Sort u_2} {x₀ : psigma β} {x₁ : psigma β} : x₀ = x₁ ↔ fst x₀ = fst x₁ ∧ snd x₀ == snd x₁ :=\n  cases_on x₀ fun (x₀_fst : α) (x₀_snd : β x₀_fst) => cases_on x₁ fun (x₁_fst : α) (x₁_snd : β x₁_fst) => mk.inj_iff\n\n/-- Map the left and right components of a sigma -/\ndef map {α₁ : Sort u_3} {α₂ : Sort u_4} {β₁ : α₁ → Sort u_5} {β₂ : α₂ → Sort u_6} (f₁ : α₁ → α₂) (f₂ : (a : α₁) → β₁ a → β₂ (f₁ a)) : psigma β₁ → psigma β₂ :=\n  sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/sigma/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.33117809005366045}}
{"text": "import topology.opens\nimport topology.algebra.continuous_functions\n\nimport for_mathlib.filter\nimport for_mathlib.data.set.basic\n\nopen topological_space function\n\nlocal notation `𝓝` x:70 := nhds x\nlocal notation f `∘₂` g := function.bicompr f g\n\n-- We need to think whether we could directly use the class t2_space (which is not using opens though)\ndefinition is_hausdorff (α : Type*) [topological_space α] : Prop :=\n  ∀ x y, x ≠ y → ∃ u v : opens α, x ∈ u ∧ y ∈ v ∧ u ∩ v = ∅\n\nopen set filter\n\ninstance regular_of_discrete {α : Type*} [topological_space α] [discrete_topology α] :\n  regular_space α :=\n{ t1 := λ x, is_open_discrete _,\n  regular :=\n  begin\n    intros s a s_closed a_not,\n    refine ⟨s, is_open_discrete s, subset.refl s, _⟩,\n    erw [← empty_in_sets_eq_bot, mem_inf_sets],\n    use {a},\n    rw nhds_discrete α,\n    simp,\n    refine ⟨s, subset.refl s, _ ⟩,\n    rintro x ⟨xa, xs⟩,\n    rw ← mem_singleton_iff.1 xa at a_not,\n    exact a_not xs\n  end }\n\n\nlemma continuous_of_const {α : Type*} {β : Type*}\n  [topological_space α] [topological_space β]\n  {f : α → β} (h : ∀a b, f a = f b) :\n  continuous f :=\nλ s _, by convert @is_open_const _ _ (∃ a, f a ∈ s); exact\n  set.ext (λ a, ⟨λ fa, ⟨_, fa⟩,\n    λ ⟨b, fb⟩, show f a ∈ s, from h b a ▸ fb⟩)\n\nsection\nvariables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\nvariables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n\ndef continuous₂ (f : α → β → γ) := continuous (function.uncurry' f)\n\nlemma continuous₂_def (f : α → β → γ) : continuous₂ f ↔ continuous (function.uncurry' f) := iff.rfl\n\nlemma continuous₂_curry (f : α × β → γ) : continuous₂ (function.curry f) ↔ continuous f :=\nby rw  [←function.uncurry'_curry f] {occs := occurrences.pos [2]} ; refl\n\nlemma continuous₂.comp {f : α → β → γ} {g : γ → δ} (hf : continuous₂ f)(hg : continuous g) :\n  continuous₂ (g ∘₂ f) := hg.comp hf\n\nsection\nopen set filter lattice function\n\n/-\n    f\n  α → β\ng ↓   ↓ h\n  γ → δ\n    i\n-/\nvariables {f : α → β} {g : α → γ} {i : γ → δ} {h : β → δ}\n\nlemma continuous_of_continuous_on_of_induced (H : h ∘ f = i ∘ g) (hi : continuous_on i $ range g)\n  (hg : ‹topological_space α› = induced g ‹topological_space γ›)\n  (hh : ‹topological_space β› = induced h ‹topological_space δ›) : continuous f :=\nbegin\n  rw continuous_iff_continuous_at,\n  intro x,\n  dsimp [continuous_at, tendsto],\n  rw [hg, hh, nhds_induced, nhds_induced, ← map_le_iff_le_comap, map_comm H],\n  specialize hi (g x) ⟨x, rfl⟩,\n  have := calc\n    nhds_within (g x) (range g) = 𝓝 g x ⊓ principal (range g) : rfl\n    ... = 𝓝 g x ⊓ map g (principal univ) : by rw [← image_univ, ← map_principal]\n    ... = 𝓝 g x ⊓ map g ⊤ : by rw principal_univ,\n  rw [continuous_within_at, this, ← comp_app i g, ← congr_fun H x] at hi, clear this,\n  have := calc\n    map g (comap g 𝓝 g x) = map g (comap  g 𝓝 g x ⊓ ⊤) : by rw inf_top_eq\n    ... ≤ map g (comap g 𝓝 g x) ⊓ map g ⊤ : map_inf_le\n    ... ≤ 𝓝 g x ⊓ map g ⊤ : inf_le_inf map_comap_le (le_refl _),\n  exact le_trans (map_mono this) hi,\nend\n\nvariables  (eg : embedding g) (eh : embedding h)\ninclude eg\n\nlemma embedding.nhds_eq_comap (a : α) : nhds a = comap g (nhds $ g a) :=\nby rw [eg.induced, nhds_induced]\n\ninclude eh\n\nlemma embedding.tendsto_iff (H : h ∘ f = i ∘ g) (a : α) : continuous_at i (g a) → continuous_at f a:=\nbegin\n  let N := nhds a, let Nf := nhds (f a),\n  let Nhf := nhds (h $ f a), let Ng := nhds (g a),\n  have Neq1 : Nf = comap h Nhf, from eh.nhds_eq_comap (f a),\n  have Neq2 : N = comap g Ng, from eg.nhds_eq_comap a,\n  intro hyp,\n  replace hyp : Ng ≤ comap i Nhf,\n  { unfold continuous_at at hyp,\n    rw ← show h (f a) = i (g a), from congr_fun H a at hyp,\n    rwa tendsto_iff_comap at hyp },\n  rw calc\n      continuous_at f a ↔ tendsto f N Nf : iff.rfl\n      ... ↔ N ≤ comap f Nf : tendsto_iff_comap\n      ... ↔ comap g Ng ≤ comap f (comap h Nhf) : by rw [Neq1, Neq2]\n      ... ↔ comap g Ng ≤ comap g (comap i Nhf) : by rw comap_comm H,\n  exact comap_mono hyp\nend\nend\nend\n\nnamespace dense_inducing\nopen set function filter\nvariables {α : Type*} {β : Type*} {δ : Type*} {γ : Type*}\nvariables [topological_space α] [topological_space β] [topological_space δ] [topological_space γ]\n\n/-\n    f\n  α → β\ng ↓   ↓ h\n  γ → δ\n    i\n-/\nvariables {f : α → β} {g : α → γ} {i : γ → δ} {h : β → δ}\n\nlemma comp (dh : dense_inducing h) (df : dense_inducing f) : dense_inducing (h ∘ f) :=\n{ dense := dense_range.comp _ dh.dense df.dense dh.continuous,\n  induced := (dh.to_inducing.comp df.to_inducing).induced }\n\nlemma of_comm_square (dg : dense_inducing g) (di : dense_inducing i)\n  (dh : dense_inducing h) (H : h ∘ f = i ∘ g) : dense_inducing f :=\nhave dhf : dense_inducing (h ∘ f),\n  by {rw H, exact di.comp dg },\n{ dense := begin\n    intro x,\n    have H := dhf.dense (h x),\n    rw mem_closure_iff_nhds at H ⊢,\n    intros t ht,\n    rw [dh.nhds_eq_comap x, mem_comap_sets] at ht,\n    rcases ht with ⟨u, hu, hinc⟩,\n    rcases H u hu with ⟨v, hv1, a, rfl⟩,\n    use f a,\n    split, swap, apply mem_range_self,\n    apply mem_of_mem_of_subset _ hinc,\n    rwa mem_preimage,\n  end ,\n--  inj := λ a b H, dhf.inj (by {show h (f a) = _, rw H}),\n  induced := by rw [dg.induced, di.induced, induced_compose, ← H, ← induced_compose, dh.induced] }\nend dense_inducing\n\nnamespace dense_embedding\nopen set function filter\nvariables {α : Type*} {β : Type*} {δ : Type*} {γ : Type*}\n\nvariables [topological_space α] [topological_space β] [topological_space δ] [topological_space γ]\n\n/-\n    f\n  α → β\ng ↓   ↓ h\n  γ → δ\n    i\n-/\nvariables {f : α → β} {g : α → γ} {i : γ → δ} {h : β → δ}\n\n-- TODO: fix implicit argument in dense_range.comp before PRing\n\nlemma comp (dh : dense_embedding h) (df : dense_embedding f) : dense_embedding (h ∘ f) :=\n{ dense := dense_range.comp _ dh.dense df.dense dh.to_dense_inducing.continuous,\n  inj :=  function.injective_comp dh.inj df.inj,\n  induced := (dh.to_inducing.comp df.to_inducing).induced }\n\nlemma of_homeo (h : α ≃ₜ β) : dense_embedding h :=\n{ dense := dense_range_iff_closure_range.mpr $\n             (range_iff_surjective.mpr h.to_equiv.surjective).symm ▸ closure_univ,\n  inj := h.to_equiv.injective,\n  induced := h.induced_eq.symm, }\n\nlemma of_comm_square (dg : dense_embedding g) (di : dense_embedding i)\n  (dh : dense_embedding h) (H : h ∘ f = i ∘ g) : dense_embedding f :=\n{ inj := begin\n    intros a b hab,\n    have : (h ∘ f) a = (h ∘ f) b := by convert congr_arg h hab,\n    rw H at this,\n    exact dg.inj (di.inj this),\n  end,\n  ..dense_inducing.of_comm_square dg.to_dense_inducing di.to_dense_inducing dh.to_dense_inducing H }\nend dense_embedding\n\nsection\nopen filter\nvariables  {α : Type*} [topological_space α] {β : Type*} [topological_space β] [discrete_topology β]\n\nlemma continuous_into_discrete_iff (f : α → β) : continuous f ↔ ∀ b : β, is_open (f ⁻¹' {b}) :=\nbegin\n  split,\n  { intros hf b,\n    exact hf _ (is_open_discrete _) },\n  { intro h,\n    rw continuous_iff_continuous_at,\n    intro x,\n    have key : f ⁻¹' {f x} ∈ nhds x,\n      from mem_nhds_sets (h $ f x) (set.mem_insert (f x) ∅),\n    calc map f (nhds x) ≤ pure (f x) : le_pure_iff.mpr key\n        ... ≤ nhds (f x) : pure_le_nhds _ }\nend\n\nlemma discrete_iff_open_singletons : discrete_topology α ↔ ∀ x, is_open ({x} : set α) :=\n⟨by introsI ; exact is_open_discrete _, λ h, ⟨eq_bot_of_singletons_open h⟩⟩\n\nlemma discrete_iff_nhds_eq_pure {X : Type*} [topological_space X] :\n  discrete_topology X ↔ ∀ x : X, nhds x = pure x :=\nbegin\n  split,\n  { introsI h,\n    exact congr_fun (nhds_discrete X) },\n  { intro h,\n    constructor,\n    apply eq_bot_of_singletons_open,\n    intro x,\n    change _root_.is_open {x},\n    rw is_open_iff_nhds,\n    simp [h] },\nend\n\nlemma discrete_of_embedding_discrete {X : Type*} {Y : Type*} [topological_space X] [topological_space Y]\n  {f : X → Y} (hf : embedding f) [discrete_topology Y] : discrete_topology X :=\nbegin\n  rw discrete_iff_nhds_eq_pure,\n  intro x,\n  rw [hf.to_inducing.nhds_eq_comap, nhds_discrete, comap_pure hf.inj]\nend\n\nlemma is_open_singleton_iff {X : Type*} [topological_space X] {x : X} :\n  is_open ({x} : set X) ↔ {x} ∈ nhds x :=\nbegin\n  rw is_open_iff_nhds,\n  split ; intro h,\n  { apply h x (mem_singleton _),\n    simp },\n  { intros y y_in,\n    rw mem_singleton_iff at y_in,\n    simp [*] },\nend\n\nend\n\n-- tools for proving that a product of top rings is a top ring\ndef continuous_pi₁ {I : Type*} {R : I → Type*} {S : I → Type*}\n  [∀ i, topological_space (R i)] [∀ i, topological_space (S i)]\n  {f : Π (i : I), (R i) → (S i)} (Hfi : ∀ i, continuous (f i)) :\n  continuous (λ rs i, f i (rs i) : (Π (i : I), R i) → Π (i : I), S i) :=\ncontinuous_pi (λ i,  (Hfi i).comp (continuous_apply i))\n\ndef continuous_pi₂ {I : Type*} {R : I → Type*} {S : I → Type*} {T : I → Type*}\n  [∀ i, topological_space (R i)] [∀ i, topological_space (S i)] [∀ i, topological_space (T i)]\n  {f : Π (i : I), (R i) × (S i) → (T i)} (Hfi : ∀ i, continuous (f i)) :\ncontinuous (λ rs i, f i ⟨rs.1 i, rs.2 i⟩ : (Π (i : I), R i) × (Π (i : I), S i) → Π (i : I), T i) :=\ncontinuous_pi (λ i, (Hfi i).comp\n  (continuous.prod_mk ((continuous_apply i).comp continuous_fst) $\n                      (continuous_apply i).comp continuous_snd))\n\n\n/-\nThe following class probably won't have global instances, but is meant to model proofs where\nwe implictly fix a neighborhood filter basis.\n-/\n\nclass nhds_basis (α : Type*) [topological_space α] :=\n(B : α → filter_basis α)\n(is_nhds : ∀ x, 𝓝 x = (B x).filter)\n\nnamespace nhds_basis\nopen filter set\n\nvariables {α : Type*} {ι : Type*} [topological_space α] [nhds_basis α]\nvariables {β : Type*} [topological_space β] {δ : Type*}\n\nlemma mem_nhds_iff (x : α) (U : set α) : U ∈ 𝓝 x ↔ ∃ V ∈ B x, V ⊆ U :=\nby rw [is_nhds x, filter_basis.mem_filter]\n\nlemma mem_nhds_of_basis {x : α} {U : set α} (U_in : U ∈ B x) : U ∈ 𝓝 x :=\n(is_nhds x).symm ▸ filter_basis.mem_filter_of_mem U_in\n\nlemma tendsto_from {f : α → δ} {x : α} {y : filter δ} :\n  tendsto f (𝓝 x) y ↔ ∀ {V}, V ∈ y → ∃ U ∈ B x, U ⊆ f ⁻¹' V :=\nby split ; intros h V V_in ; specialize h V_in ; rwa [← mem_nhds_iff x] at *\n\nlemma continuous_from {f : α → β} : continuous f ↔ ∀ x, ∀ {V}, V ∈ 𝓝 f x → ∃ U ∈ B x, U ⊆ f ⁻¹' V :=\nby simp [continuous_iff_continuous_at, continuous_at, tendsto_from]\n\nlemma tendsto_into {f : δ → α} {x : filter δ} {y : α} : tendsto f x 𝓝 y ↔ ∀ U ∈ B y, f ⁻¹' U ∈ x :=\nbegin\n  split ; intros h,\n  { rintro U U_in,\n    exact h (mem_nhds_of_basis U_in)  },\n  { intros V V_in,\n    rcases (mem_nhds_iff _ _).1 V_in with ⟨W, W_in, hW⟩,\n    filter_upwards [h W W_in],\n    exact preimage_mono hW }\nend\n\nlemma continuous_into {f : β → α} : continuous f ↔ ∀ x, ∀ U ∈ B (f x), f ⁻¹' U ∈ 𝓝 x :=\nby simp [continuous_iff_continuous_at, continuous_at, tendsto_into]\n\nlemma tendsto_both [nhds_basis β] {f : α → β} {x : α} {y : β} :\n  tendsto f (𝓝 x) 𝓝 y ↔ ∀ U ∈ B y, ∃ V ∈ B x, V ⊆ f ⁻¹' U :=\nbegin\n  rw tendsto_into,\n  split ; introv h U_in ; specialize h U U_in ; rwa mem_nhds_iff x at *\nend\n\nlemma continuous_both [nhds_basis β] {f : α → β} :\n  continuous f ↔ ∀ x, ∀ U ∈ B (f x), ∃ V ∈ B x, V ⊆ f ⁻¹' U :=\nby simp [continuous_iff_continuous_at, continuous_at, tendsto_both]\n\nend nhds_basis\n\nlemma dense_range.mem_nhds {α : Type*} [topological_space α] {β : Type*} [topological_space β]\n  {f : α → β} (h : dense_range f) {b : β} {U : set β} (U_in : U ∈ nhds b) :\n  ∃ a : α, f a ∈ U :=\nbegin\n  rcases (mem_closure_iff_nhds.mp\n    ((dense_range_iff_closure_range.mp h).symm ▸ mem_univ b : b ∈ closure (range f)) U U_in)\n    with ⟨_, h, a, rfl⟩,\n  exact ⟨a, h⟩\nend\n\nlemma mem_closure_union {α : Type*} [topological_space α] {s₁ s₂ : set α} {x : α}\n  (h : x ∈ closure (s₁ ∪ s₂)) (h₁ : -s₁ ∈ 𝓝 x) : x ∈ closure s₂ :=\nbegin\n  rw closure_eq_nhds at *,\n  have := calc\n    𝓝 x ⊓ principal (s₁ ∪ s₂) = 𝓝 x ⊓ (principal s₁ ⊔ principal s₂) : by rw sup_principal\n    ... = (𝓝 x ⊓ principal s₁) ⊔ (𝓝 x ⊓ principal s₂) : by rw lattice.inf_sup_left\n    ... = ⊥ ⊔ 𝓝 x ⊓ principal s₂ : by rw inf_principal_eq_bot h₁\n    ... = 𝓝 x ⊓ principal s₂ : by rw lattice.bot_sup_eq,\n  dsimp,\n  rwa ← this\nend\n\nopen lattice\n\nlemma mem_closure_image {α : Type*} {β : Type*} [topological_space α] [topological_space β]\n  {f : α → β} {x : α} {s : set α} (hf : continuous_at f x) (hx : x ∈ closure s) :\n  f x ∈ closure (f '' s) :=\nbegin\n  rw [closure_eq_nhds, mem_set_of_eq] at *,\n  rw ← bot_lt_iff_ne_bot,\n  calc\n    ⊥   < map f (𝓝 x ⊓ principal s) : bot_lt_iff_ne_bot.mpr (map_ne_bot hx)\n    ... ≤ (map f 𝓝 x) ⊓ (map f $ principal s) : map_inf_le\n    ... = (map f 𝓝 x) ⊓ (principal $ f '' s) : by rw map_principal\n    ... ≤ 𝓝 (f x) ⊓ (principal $ f '' s) : inf_le_inf hf (le_refl _)\nend\n\n\nlemma continuous_at.prod_mk {α : Type*} {β : Type*} {γ : Type*} [topological_space α]\n  [topological_space β] [topological_space γ] {f : γ → α} {g : γ → β} {x : γ}\n  (hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (λ x, prod.mk (f x) $ g x) x :=\ncalc\n  map (λ (x : γ), (f x, g x)) (𝓝 x) ≤ (map f 𝓝 x).prod (map g 𝓝 x) : filter.map_prod_mk _ _ _\n  ... ≤ (𝓝 f x).prod (𝓝 g x) : filter.prod_mono hf hg\n  ... = 𝓝 (f x, g x) : by rw nhds_prod_eq\n\nlemma continuous_at.congr_aux {α : Type*} {β : Type*} [topological_space α] [topological_space β]\n  {f g : α → β} {a : α}  (h : {x | f x = g x } ∈ 𝓝 a) (hf : continuous_at f a) : continuous_at g a :=\nbegin\n  intros U U_in,\n  rw show g a = f a, from (mem_of_nhds h).symm at U_in,\n  let V := {x : α | g x ∈ U} ∩ {x | f x = g x},\n  suffices : V ∈ 𝓝 a,\n  { rw mem_map,\n    exact mem_sets_of_superset this (inter_subset_left _ _) },\n  have : V = {x : α | f x ∈ U} ∩ {x | f x = g x},\n  { ext x,\n    split ; rintros ⟨hl, hr⟩ ; rw mem_set_of_eq at hr hl ;\n    [ rw ← hr at hl, rw hr at hl ] ; exact ⟨hl, hr⟩ },\n  rw this,\n  exact filter.inter_mem_sets (hf U_in) ‹_›\nend\n\nlemma continuous_at.congr {α : Type*} {β : Type*} [topological_space α] [topological_space β]\n  {f g : α → β} {a : α}  (h : {x | f x = g x } ∈ 𝓝 a) : continuous_at f a ↔ continuous_at g a :=\nbegin\n  split ; intro h',\n  { exact continuous_at.congr_aux h h' },\n  { apply continuous_at.congr_aux _ h',\n    convert h,\n    ext x,\n    rw eq_comm }\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-perfectoid-spaces", "sha": "95a6520ce578b30a80b4c36e36ab2d559a842690", "save_path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces", "path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces/lean-perfectoid-spaces-95a6520ce578b30a80b4c36e36ab2d559a842690/src/for_mathlib/topology.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.33117808967697426}}
{"text": "import algebra.homology.homotopy\nimport category_theory.limits.shapes.zero_objects\n\nimport for_mathlib.Cech.split\nimport for_mathlib.abelian_category\n\nnoncomputable theory\n\nuniverses v u v' u'\n\nnamespace category_theory\nopen_locale simplicial\nopen category_theory.limits\n\nnamespace arrow\n\nsection contravariant\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : Pᵒᵖ ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : (fin (n+1)), f.left) (λ i, f.hom)]\nvariables [arrow.split f] [preadditive N]\n\ndef contracting_homotopy' : homotopy (𝟙 (f.conerve M).to_cocomplex) 0 :=\n{ hom := λ i j, if h : j + 1 = i then eq_to_hom (by subst h) ≫ f.contracting_homotopy M j else 0,\n  zero' := λ i j h, dif_neg h,\n  comm := λ i, begin\n    simp only [homological_complex.id_f, homotopy.d_next_cochain_complex,\n      homological_complex.zero_f_apply, add_zero],\n    rcases i with _|_|i,\n    { rw [← is_contracting_homotopy_zero, homotopy.prev_d_zero_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n    { rw [← is_contracting_homotopy_one, homotopy.prev_d_succ_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n    { rw [← is_contracting_homotopy, homotopy.prev_d_succ_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n  end }\n\nvariables [has_equalizers N] [has_cokernels N] [has_images N] [has_image_maps N]\n\nlemma conerve_to_cocomplex_homology_is_zero (i : ℕ) :\n  is_zero ((f.conerve M).to_cocomplex.homology i) :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  simpa only [category_theory.functor.map_id, functor.map_zero] using\n    homology_map_eq_of_homotopy (f.contracting_homotopy' M) i,\nend\n\nend contravariant\n\nsection covariant\n\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : P ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : (fin (n+1)), f.left) (λ i, f.hom)]\nvariables [arrow.split f] [preadditive N]\n\ndef covariant_contracting_homotopy' : homotopy (𝟙 (f.nerve M).to_complex) 0 :=\n{ hom := λ i j, if h : i + 1 = j then\n    f.covariant_contracting_homotopy M i ≫ eq_to_hom (by subst h) else 0,\n  zero' := λ i j h, dif_neg h,\n  comm := λ i, begin\n    simp only [homological_complex.id_f, homotopy.d_next_cochain_complex,\n      homological_complex.zero_f_apply, add_zero],\n    rcases i with _|_|i,\n    { rw [← covariant_is_contracting_homotopy_zero, homotopy.d_next_zero_chain_complex],\n      simp only [eq_self_iff_true, eq_to_hom_refl, dite_eq_ite, if_true,\n        homotopy.prev_d_chain_complex, category.comp_id, zero_add] },\n    { rw [← covariant_is_contracting_homotopy_one, homotopy.d_next_succ_chain_complex],\n      simp only [simplicial_object.augmented.to_complex_d_2, eq_self_iff_true,\n        eq_to_hom_refl, category.id_comp, dite_eq_ite, if_true,\n        category.comp_id, homotopy.prev_d_chain_complex],\n      rw add_comm },\n    { rw [← covariant_is_contracting_homotopy, homotopy.d_next_succ_chain_complex],\n      simp only [simplicial_object.augmented.to_complex_d_2, eq_self_iff_true,\n        eq_to_hom_refl, category.id_comp, dite_eq_ite, if_true,\n        category.comp_id, homotopy.prev_d_chain_complex],\n      rw add_comm },\n  end }\n\nvariables [has_equalizers N] [has_cokernels N] [has_images N] [has_image_maps N]\n\nlemma nerve_to_complex_homology_is_zero (i : ℕ) :\n  is_zero ((f.nerve M).to_complex.homology i) :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  simpa only [category_theory.functor.map_id, functor.map_zero] using\n    homology_map_eq_of_homotopy (f.covariant_contracting_homotopy' M) i,\nend\n\nend covariant\n\nend arrow\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/Cech/homotopy.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.3311032973334039}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johannes Hölzl, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Group.basic\nimport Mathlib.data.equiv.ring\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Category instances for semiring, ring, comm_semiring, and comm_ring.\n\nWe introduce the bundled categories:\n* `SemiRing`\n* `Ring`\n* `CommSemiRing`\n* `CommRing`\nalong with the relevant forgetful functors between them.\n-/\n\n/-- The category of semirings. -/\ndef SemiRing :=\n  category_theory.bundled semiring\n\nnamespace SemiRing\n\n\nprotected instance bundled_hom : category_theory.bundled_hom ring_hom :=\n  category_theory.bundled_hom.mk ring_hom.to_fun ring_hom.id ring_hom.comp\n\nprotected instance large_category : category_theory.large_category SemiRing :=\n  category_theory.bundled_hom.category ring_hom\n\n/-- Construct a bundled SemiRing from the underlying type and typeclass. -/\ndef of (R : Type u) [semiring R] : SemiRing :=\n  category_theory.bundled.of R\n\nprotected instance inhabited : Inhabited SemiRing :=\n  { default := of PUnit }\n\nprotected instance semiring (R : SemiRing) : semiring ↥R :=\n  category_theory.bundled.str R\n\n@[simp] theorem coe_of (R : Type u) [semiring R] : ↥(of R) = R :=\n  rfl\n\nprotected instance has_forget_to_Mon : category_theory.has_forget₂ SemiRing Mon :=\n  category_theory.bundled_hom.mk_has_forget₂ (fun (R : Type u_1) (hR : semiring R) => monoid_with_zero.to_monoid R)\n    (fun (R₁ R₂ : category_theory.bundled semiring) => ring_hom.to_monoid_hom) sorry\n\n-- can't use bundled_hom.mk_has_forget₂, since AddCommMon is an induced category\n\nprotected instance has_forget_to_AddCommMon : category_theory.has_forget₂ SemiRing AddCommMon :=\n  category_theory.has_forget₂.mk\n    (category_theory.functor.mk (fun (R : SemiRing) => AddCommMon.of ↥R)\n      fun (R₁ R₂ : SemiRing) (f : R₁ ⟶ R₂) => ring_hom.to_add_monoid_hom f)\n\nend SemiRing\n\n\n/-- The category of rings. -/\ndef Ring :=\n  category_theory.bundled ring\n\nnamespace Ring\n\n\nprotected instance ring.to_semiring.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection ring.to_semiring :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance concrete_category : category_theory.concrete_category Ring :=\n  category_theory.bundled_hom.category_theory.bundled.category_theory.concrete_category\n    (category_theory.bundled_hom.map_hom ring_hom ring.to_semiring)\n\n/-- Construct a bundled Ring from the underlying type and typeclass. -/\ndef of (R : Type u) [ring R] : Ring :=\n  category_theory.bundled.of R\n\nprotected instance inhabited : Inhabited Ring :=\n  { default := of PUnit }\n\nprotected instance ring (R : Ring) : ring ↥R :=\n  category_theory.bundled.str R\n\n@[simp] theorem coe_of (R : Type u) [ring R] : ↥(of R) = R :=\n  rfl\n\nprotected instance has_forget_to_SemiRing : category_theory.has_forget₂ Ring SemiRing :=\n  category_theory.bundled_hom.forget₂ ring_hom ring.to_semiring\n\n-- can't use bundled_hom.mk_has_forget₂, since AddCommGroup is an induced category\n\nprotected instance has_forget_to_AddCommGroup : category_theory.has_forget₂ Ring AddCommGroup :=\n  category_theory.has_forget₂.mk\n    (category_theory.functor.mk (fun (R : Ring) => AddCommGroup.of ↥R)\n      fun (R₁ R₂ : Ring) (f : R₁ ⟶ R₂) => ring_hom.to_add_monoid_hom f)\n\nend Ring\n\n\n/-- The category of commutative semirings. -/\ndef CommSemiRing :=\n  category_theory.bundled comm_semiring\n\nnamespace CommSemiRing\n\n\nprotected instance comm_semiring.to_semiring.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection comm_semiring.to_semiring :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance large_category : category_theory.large_category CommSemiRing :=\n  category_theory.bundled_hom.category (category_theory.bundled_hom.map_hom ring_hom comm_semiring.to_semiring)\n\n/-- Construct a bundled CommSemiRing from the underlying type and typeclass. -/\ndef of (R : Type u) [comm_semiring R] : CommSemiRing :=\n  category_theory.bundled.of R\n\nprotected instance inhabited : Inhabited CommSemiRing :=\n  { default := of PUnit }\n\nprotected instance comm_semiring (R : CommSemiRing) : comm_semiring ↥R :=\n  category_theory.bundled.str R\n\n@[simp] theorem coe_of (R : Type u) [comm_semiring R] : ↥(of R) = R :=\n  rfl\n\nprotected instance has_forget_to_SemiRing : category_theory.has_forget₂ CommSemiRing SemiRing :=\n  category_theory.bundled_hom.forget₂ ring_hom comm_semiring.to_semiring\n\n/-- The forgetful functor from commutative rings to (multiplicative) commutative monoids. -/\nprotected instance has_forget_to_CommMon : category_theory.has_forget₂ CommSemiRing CommMon :=\n  category_theory.has_forget₂.mk' (fun (R : CommSemiRing) => CommMon.of ↥R) sorry\n    (fun (R₁ R₂ : CommSemiRing) (f : R₁ ⟶ R₂) => ring_hom.to_monoid_hom f) sorry\n\nend CommSemiRing\n\n\n/-- The category of commutative rings. -/\ndef CommRing :=\n  category_theory.bundled comm_ring\n\nnamespace CommRing\n\n\nprotected instance comm_ring.to_ring.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection comm_ring.to_ring :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance concrete_category : category_theory.concrete_category CommRing :=\n  category_theory.bundled_hom.category_theory.bundled.category_theory.concrete_category\n    (category_theory.bundled_hom.map_hom (category_theory.bundled_hom.map_hom ring_hom ring.to_semiring)\n      comm_ring.to_ring)\n\n/-- Construct a bundled CommRing from the underlying type and typeclass. -/\ndef of (R : Type u) [comm_ring R] : CommRing :=\n  category_theory.bundled.of R\n\nprotected instance inhabited : Inhabited CommRing :=\n  { default := of PUnit }\n\nprotected instance comm_ring (R : CommRing) : comm_ring ↥R :=\n  category_theory.bundled.str R\n\n@[simp] theorem coe_of (R : Type u) [comm_ring R] : ↥(of R) = R :=\n  rfl\n\nprotected instance has_forget_to_Ring : category_theory.has_forget₂ CommRing Ring :=\n  category_theory.bundled_hom.forget₂ (category_theory.bundled_hom.map_hom ring_hom ring.to_semiring) comm_ring.to_ring\n\n/-- The forgetful functor from commutative rings to (multiplicative) commutative monoids. -/\nprotected instance has_forget_to_CommSemiRing : category_theory.has_forget₂ CommRing CommSemiRing :=\n  category_theory.has_forget₂.mk' (fun (R : CommRing) => CommSemiRing.of ↥R) sorry\n    (fun (R₁ R₂ : CommRing) (f : R₁ ⟶ R₂) => f) sorry\n\nprotected instance category_theory.forget₂.category_theory.full : category_theory.full (category_theory.forget₂ CommRing CommSemiRing) :=\n  category_theory.full.mk\n    fun (X Y : CommRing)\n      (f :\n      category_theory.functor.obj (category_theory.forget₂ CommRing CommSemiRing) X ⟶\n        category_theory.functor.obj (category_theory.forget₂ CommRing CommSemiRing) Y) =>\n      f\n\nend CommRing\n\n\n-- This example verifies an improvement possible in Lean 3.8.\n\n-- Before that, to have `add_ring_hom.map_zero` usable by `simp` here,\n\n-- we had to mark all the concrete category `has_coe_to_sort` instances reducible.\n\n-- Now, it just works.\n\nnamespace ring_equiv\n\n\n/-- Build an isomorphism in the category `Ring` from a `ring_equiv` between `ring`s. -/\n@[simp] theorem to_Ring_iso_inv {X : Type u} {Y : Type u} [ring X] [ring Y] (e : X ≃+* Y) : category_theory.iso.inv (to_Ring_iso e) = to_ring_hom (ring_equiv.symm e) :=\n  Eq.refl (category_theory.iso.inv (to_Ring_iso e))\n\n/-- Build an isomorphism in the category `CommRing` from a `ring_equiv` between `comm_ring`s. -/\ndef to_CommRing_iso {X : Type u} {Y : Type u} [comm_ring X] [comm_ring Y] (e : X ≃+* Y) : CommRing.of X ≅ CommRing.of Y :=\n  category_theory.iso.mk (to_ring_hom e) (to_ring_hom (ring_equiv.symm e))\n\nend ring_equiv\n\n\nnamespace category_theory.iso\n\n\n/-- Build a `ring_equiv` from an isomorphism in the category `Ring`. -/\ndef Ring_iso_to_ring_equiv {X : Ring} {Y : Ring} (i : X ≅ Y) : ↥X ≃+* ↥Y :=\n  ring_equiv.mk ⇑(hom i) ⇑(inv i) sorry sorry sorry sorry\n\n/-- Build a `ring_equiv` from an isomorphism in the category `CommRing`. -/\ndef CommRing_iso_to_ring_equiv {X : CommRing} {Y : CommRing} (i : X ≅ Y) : ↥X ≃+* ↥Y :=\n  ring_equiv.mk ⇑(hom i) ⇑(inv i) sorry sorry sorry sorry\n\nend category_theory.iso\n\n\n/-- Ring equivalences between `ring`s are the same as (isomorphic to) isomorphisms in `Ring`. -/\ndef ring_equiv_iso_Ring_iso {X : Type u} {Y : Type u} [ring X] [ring Y] : X ≃+* Y ≅ Ring.of X ≅ Ring.of Y :=\n  category_theory.iso.mk (fun (e : X ≃+* Y) => ring_equiv.to_Ring_iso e)\n    fun (i : Ring.of X ≅ Ring.of Y) => category_theory.iso.Ring_iso_to_ring_equiv i\n\n/-- Ring equivalences between `comm_ring`s are the same as (isomorphic to) isomorphisms\nin `CommRing`. -/\ndef ring_equiv_iso_CommRing_iso {X : Type u} {Y : Type u} [comm_ring X] [comm_ring Y] : X ≃+* Y ≅ CommRing.of X ≅ CommRing.of Y :=\n  category_theory.iso.mk (fun (e : X ≃+* Y) => ring_equiv.to_CommRing_iso e)\n    fun (i : CommRing.of X ≅ CommRing.of Y) => category_theory.iso.CommRing_iso_to_ring_equiv i\n\nprotected instance Ring.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget Ring) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : Ring) (f : X ⟶ Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget Ring) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget Ring) X ≅\n          category_theory.functor.obj (category_theory.forget Ring) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget Ring) f);\n      let e : ↥X ≃+* ↥Y :=\n        ring_equiv.mk (ring_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (ring_equiv.to_Ring_iso e))\n\nprotected instance CommRing.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget CommRing) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : CommRing) (f : X ⟶ Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget CommRing) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget CommRing) X ≅\n          category_theory.functor.obj (category_theory.forget CommRing) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget CommRing) f);\n      let e : ↥X ≃+* ↥Y :=\n        ring_equiv.mk (ring_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (ring_equiv.to_CommRing_iso e))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/CommRing/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565326, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.33105963958417506}}
{"text": "/-\nCopyright (c) 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Jesse Michael Han\n\n! This file was ported from Lean 3 source module tactic.finish\n! leanprover-community/mathlib commit 3c11bd771ef17197a9e9fcd4a3fabfa2804d950c\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Hint\n\n/-!\n# The `finish` family of tactics\n\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\neliminate existential quantifiers on the left, and look for contradictions. They rely on ematching\nand congruence closure to try to finish off a goal at the end.\n\nThe procedures *do* split on disjunctions and recreate the smt state for each terminal call, so\nthey are only meant to be used on small, straightforward problems.\n\n## Main definitions\n\nWe provide the following tactics:\n\n* `finish`  -- solves the goal or fails\n* `clarify` -- makes as much progress as possible while not leaving more than one goal\n* `safe`    -- splits freely, finishes off whatever subgoals it can, and leaves the rest\n\nAll accept an optional list of simplifier rules, typically definitions that should be expanded.\n(The equations and identities should not refer to the local context.)\n\n-/\n\n\ninitialize\n  registerTraceClass.1 `auto.done\n\ninitialize\n  registerTraceClass.1 `auto.finish\n\nnamespace Tactic\n\nnamespace Interactive\n\nunsafe def revert_all :=\n  tactic.revert_all\n#align tactic.interactive.revert_all tactic.interactive.revert_all\n\nend Interactive\n\nend Tactic\n\nopen Tactic Expr\n\nnamespace Auto\n\n/-! ### Utilities -/\n\n\nunsafe def whnf_reducible (e : expr) : tactic expr :=\n  whnf e reducible\n#align auto.whnf_reducible auto.whnf_reducible\n\n-- stolen from interactive.lean\nunsafe def add_simps : simp_lemmas → List Name → tactic simp_lemmas\n  | s, [] => return s\n  | s, n :: ns => do\n    let s' ← s.add_simp n\n    add_simps s' ns\n#align auto.add_simps auto.add_simps\n\n/-- Configuration information for the auto tactics.\n* `(use_simp := tt)`: call the simplifier\n* `(max_ematch_rounds := 20)`: for the \"done\" tactic\n-/\nstructure AutoConfig : Type where\n  useSimp := true\n  maxEmatchRounds := 20\n  deriving DecidableEq, Inhabited\n#align auto.auto_config Auto.AutoConfig\n\n/-!\n### Preprocess goal.\n\nWe want to move everything to the left of the sequent arrow. For intuitionistic logic,\nwe replace the goal `p` with `∀ f, (p → f) → f` and introduce.\n-/\n\n\ntheorem by_contradiction_trick (p : Prop) (h : ∀ f : Prop, (p → f) → f) : p :=\n  h p id\n#align auto.by_contradiction_trick Auto.by_contradiction_trick\n\nunsafe def preprocess_goal : tactic Unit := do\n  repeat (intro1 >> skip)\n  let tgt ← target >>= whnf_reducible\n  if ¬is_false tgt then ((mk_mapp `` by_contradiction [some tgt] >>= apply) >> intro1) >> skip\n    else skip\n#align auto.preprocess_goal auto.preprocess_goal\n\n/-!\n### Normalize hypotheses\n\nBring conjunctions to the outside (for splitting),\nbring universal quantifiers to the outside (for ematching). The classical normalizer\neliminates `a → b` in favor of `¬ a ∨ b`.\n\nFor efficiency, we push negations inwards from the top down. (For example, consider\nsimplifying `¬ ¬ (p ∨ q)`.)\n-/\n\n\nsection\n\nuniverse u\n\nvariable {α : Type u}\n\nvariable (p q : Prop)\n\nvariable (s : α → Prop)\n\nattribute [local instance] Classical.propDecidable\n\ntheorem not_not_eq : (¬¬p) = p :=\n  propext Classical.not_not\n#align auto.not_not_eq Auto.not_not_eq\n\ntheorem not_and_eq : (¬(p ∧ q)) = (¬p ∨ ¬q) :=\n  propext not_and_or\n#align auto.not_and_eq Auto.not_and_eq\n\ntheorem not_or_eq : (¬(p ∨ q)) = (¬p ∧ ¬q) :=\n  propext not_or\n#align auto.not_or_eq Auto.not_or_eq\n\ntheorem not_forall_eq : (¬∀ x, s x) = ∃ x, ¬s x :=\n  propext not_forall\n#align auto.not_forall_eq Auto.not_forall_eq\n\ntheorem not_exists_eq : (¬∃ x, s x) = ∀ x, ¬s x :=\n  propext not_exists\n#align auto.not_exists_eq Auto.not_exists_eq\n\ntheorem not_implies_eq : (¬(p → q)) = (p ∧ ¬q) :=\n  propext not_imp\n#align auto.not_implies_eq Auto.not_implies_eq\n\ntheorem Classical.implies_iff_not_or : p → q ↔ ¬p ∨ q :=\n  imp_iff_not_or\n#align auto.classical.implies_iff_not_or Auto.Classical.implies_iff_not_or\n\nend\n\ndef commonNormalizeLemmaNames : List Name :=\n  [`` bex_def, `` forall_and, `` exists_imp, `` or_assoc, `` or_comm, `` or_left_comm, `` and_assoc,\n    `` and_comm, `` and_left_comm]\n#align auto.common_normalize_lemma_names Auto.commonNormalizeLemmaNames\n\ndef classicalNormalizeLemmaNames : List Name :=\n  commonNormalizeLemmaNames ++ [`` classical.implies_iff_not_or]\n#align auto.classical_normalize_lemma_names Auto.classicalNormalizeLemmaNames\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- optionally returns an equivalent expression and proof of equivalence -/ private unsafe\n  def\n    transform_negation_step\n    ( cfg : AutoConfig ) ( e : expr ) : tactic ( Option ( expr × expr ) )\n    :=\n      do\n        let e ← whnf_reducible e\n          match\n            e\n            with\n            |\n                q( ¬ $ ( Ne ) )\n                =>\n                do\n                  let ne ← whnf_reducible Ne\n                    match\n                      Ne\n                      with\n                      |\n                          q( ¬ $ ( a ) )\n                          =>\n                          do let pr ← mk_app ` ` not_not_eq [ a ] return ( some ( a , pr ) )\n                        |\n                          q( $ ( a ) ∧ $ ( b ) )\n                          =>\n                          do\n                            let pr ← mk_app ` ` not_and_eq [ a , b ]\n                              return ( some ( q( ¬ $ ( a ) ∨ ¬ $ ( b ) ) , pr ) )\n                        |\n                          q( $ ( a ) ∨ $ ( b ) )\n                          =>\n                          do\n                            let pr ← mk_app ` ` not_or_eq [ a , b ]\n                              return ( some ( q( ¬ $ ( a ) ∧ ¬ $ ( b ) ) , pr ) )\n                        |\n                          q( Exists $ ( p ) )\n                          =>\n                          do\n                            let pr ← mk_app ` ` not_exists_eq [ p ]\n                              let q( $ ( _ ) = $ ( e' ) ) ← infer_type pr\n                              return ( some ( e' , pr ) )\n                        |\n                          pi n bi d p\n                          =>\n                          if\n                            p\n                            then\n                            do\n                              let\n                                  pr\n                                    ←\n                                    mk_app\n                                      ` ` not_forall_eq [ lam n bi d ( expr.abstract_local p n ) ]\n                                let q( $ ( _ ) = $ ( e' ) ) ← infer_type pr\n                                return ( some ( e' , pr ) )\n                            else\n                            do\n                              let pr ← mk_app ` ` not_implies_eq [ d , p ]\n                                let q( $ ( _ ) = $ ( e' ) ) ← infer_type pr\n                                return ( some ( e' , pr ) )\n                        | _ => return none\n              | _ => return none\n#align auto.transform_negation_step auto.transform_negation_step\n\n/-- given an expr `e`, returns a new expression and a proof of equality -/\nprivate unsafe def transform_negation (cfg : AutoConfig) : expr → tactic (Option (expr × expr)) :=\n  fun e => do\n  let opr ← transform_negation_step cfg e\n  match opr with\n    | some (e', pr) => do\n      let opr' ← transform_negation e'\n      match opr' with\n        | none => return (some (e', pr))\n        | some (e'', pr') => do\n          let pr'' ← mk_eq_trans pr pr'\n          return (some (e'', pr''))\n    | none => return none\n#align auto.transform_negation auto.transform_negation\n\nunsafe def normalize_negations (cfg : AutoConfig) (h : expr) : tactic Unit := do\n  let t ← infer_type h\n  let (_, e, pr) ←\n    simplify_top_down ()\n        (fun _ => fun e => do\n          let oepr ← transform_negation cfg e\n          match oepr with\n            | some (e', pr) => return ((), e', pr)\n            | none => do\n              let pr ← mk_eq_refl e\n              return ((), e, pr))\n        t\n  replace_hyp h e pr\n  skip\n#align auto.normalize_negations auto.normalize_negations\n\nunsafe def normalize_hyp (cfg : AutoConfig) (simps : simp_lemmas) (h : expr) : tactic Unit :=\n  (do\n      let (h, _) ← simp_hyp simps [] h\n      try (normalize_negations cfg h)) <|>\n    try (normalize_negations cfg h)\n#align auto.normalize_hyp auto.normalize_hyp\n\nunsafe def normalize_hyps (cfg : AutoConfig) : tactic Unit := do\n  let simps ← add_simps simp_lemmas.mk classicalNormalizeLemmaNames\n  local_context >>= Monad.mapM' (normalize_hyp cfg simps)\n#align auto.normalize_hyps auto.normalize_hyps\n\n/-!\n### Eliminate existential quantifiers\n-/\n\n\n/-- eliminate an existential quantifier if there is one -/\nunsafe def eelim : tactic Unit := do\n  let ctx ← local_context\n  first <|\n      ctx fun h => do\n        let t ← infer_type h >>= whnf_reducible\n        guard (is_app_of t `` Exists)\n        let tgt ← target\n        to_expr ``(@Exists.elim _ _ $(tgt) $(h)) >>= apply\n        intros\n        clear h\n#align auto.eelim auto.eelim\n\n/-- eliminate all existential quantifiers, fails if there aren't any -/\nunsafe def eelims : tactic Unit :=\n  eelim >> repeat eelim\n#align auto.eelims auto.eelims\n\n/-!\n### Substitute if there is a hypothesis `x = t` or `t = x`\n-/\n\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- carries out a subst if there is one, fails otherwise -/ unsafe\n  def\n    do_subst\n    : tactic Unit\n    :=\n      do\n        let ctx ← local_context\n          first\n            <|\n            ctx\n              fun\n                h\n                  =>\n                  do\n                    let t ← infer_type h >>= whnf_reducible\n                      match t with | q( $ ( a ) = $ ( b ) ) => subst h | _ => failed\n#align auto.do_subst auto.do_subst\n\nunsafe def do_substs : tactic Unit :=\n  do_subst >> repeat do_subst\n#align auto.do_substs auto.do_substs\n\n/-!\n### Split all conjunctions\n-/\n\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Assumes `pr` is a proof of `t`. Adds the consequences of `t` to the context\n       and returns `tt` if anything nontrivial has been added. -/\n    unsafe\n  def\n    add_conjuncts\n    : expr → expr → tactic Bool\n    :=\n      fun\n        pr t\n          =>\n          let\n            assert_consequences e t := condM ( add_conjuncts e t ) skip ( note_anon t e >> skip )\n            do\n              let t' ← whnf_reducible t\n                match\n                  t'\n                  with\n                  |\n                      q( $ ( a ) ∧ $ ( b ) )\n                      =>\n                      do\n                        let e₁ ← mk_app ` ` And.left [ pr ]\n                          assert_consequences e₁ a\n                          let e₂ ← mk_app ` ` And.right [ pr ]\n                          assert_consequences e₂ b\n                          return tt\n                    | q( True ) => do return tt\n                    | _ => return ff\n#align auto.add_conjuncts auto.add_conjuncts\n\n/-- return `tt` if any progress is made -/\nunsafe def split_hyp (h : expr) : tactic Bool := do\n  let t ← infer_type h\n  condM (add_conjuncts h t) (clear h >> return tt) (return ff)\n#align auto.split_hyp auto.split_hyp\n\n/-- return `tt` if any progress is made -/\nunsafe def split_hyps_aux : List expr → tactic Bool\n  | [] => return false\n  | h :: hs => do\n    let b₁ ← split_hyp h\n    let b₂ ← split_hyps_aux hs\n    return (b₁ || b₂)\n#align auto.split_hyps_aux auto.split_hyps_aux\n\n/-- fail if no progress is made -/\nunsafe def split_hyps : tactic Unit :=\n  local_context >>= split_hyps_aux >>= guardb\n#align auto.split_hyps auto.split_hyps\n\n/-!\n### Eagerly apply all the preprocessing rules\n-/\n\n\n/-- Eagerly apply all the preprocessing rules -/\nunsafe def preprocess_hyps (cfg : AutoConfig) : tactic Unit := do\n  repeat (intro1 >> skip)\n  preprocess_goal\n  normalize_hyps cfg\n  repeat (do_substs <|> split_hyps <|> eelim)\n#align auto.preprocess_hyps auto.preprocess_hyps\n\n/-!\n### Terminal tactic\n-/\n\n\n--<|> self_simplify_hyps\n/-- The terminal tactic, used to try to finish off goals:\n- Call the contradiction tactic.\n- Open an SMT state, and use ematching and congruence closure, with all the universal\n  statements in the context.\n\nTODO(Jeremy): allow users to specify attribute for ematching lemmas?\n-/\nunsafe def mk_hinst_lemmas : List expr → smt_tactic hinst_lemmas\n  | [] =>-- return hinst_lemmas.mk\n  do\n    get_hinst_lemmas_for_attr `ematch\n  | h :: hs => do\n    let his ← mk_hinst_lemmas hs\n    let t ← infer_type h\n    match t with\n      | pi _ _ _ _ => do\n        let t' ← infer_type t\n        if t' = q(Prop) then\n            (do\n                let new_lemma ← hinst_lemma.mk h\n                return (hinst_lemmas.add his new_lemma)) <|>\n              return his\n          else return his\n      | _ => return his\n#align auto.mk_hinst_lemmas auto.mk_hinst_lemmas\n\nprivate unsafe def report_invalid_em_lemma {α : Type} (n : Name) : smt_tactic α :=\n  fail f! \"invalid ematch lemma '{n}'\"\n#align auto.report_invalid_em_lemma auto.report_invalid_em_lemma\n\nprivate unsafe def add_hinst_lemma_from_name (md : Transparency) (lhs_lemma : Bool) (n : Name)\n    (hs : hinst_lemmas) (ref : pexpr) : smt_tactic hinst_lemmas := do\n  let p ← resolve_name n\n  match p with\n    | expr.const n _ =>\n      (do\n          let h ← hinst_lemma.mk_from_decl_core md n lhs_lemma\n          tactic.save_const_type_info n ref\n          return <| hs h) <|>\n        (do\n            let hs₁ ← smt_tactic.mk_ematch_eqn_lemmas_for_core md n\n            tactic.save_const_type_info n ref\n            return <| hs hs₁) <|>\n          report_invalid_em_lemma n\n    | _ =>\n      (do\n          let e ← to_expr p\n          let h ← hinst_lemma.mk_core md e lhs_lemma\n          try (tactic.save_type_info e ref)\n          return <| hs h) <|>\n        report_invalid_em_lemma n\n#align auto.add_hinst_lemma_from_name auto.add_hinst_lemma_from_name\n\nprivate unsafe def add_hinst_lemma_from_pexpr (md : Transparency) (lhs_lemma : Bool)\n    (hs : hinst_lemmas) : pexpr → smt_tactic hinst_lemmas\n  | p@(expr.const c []) => add_hinst_lemma_from_name md lhs_lemma c hs p\n  | p@(expr.local_const c _ _ _) => add_hinst_lemma_from_name md lhs_lemma c hs p\n  | p => do\n    let new_e ← to_expr p\n    let h ← hinst_lemma.mk_core md new_e lhs_lemma\n    return <| hs h\n#align auto.add_hinst_lemma_from_pexpr auto.add_hinst_lemma_from_pexpr\n\nprivate unsafe def add_hinst_lemmas_from_pexprs (md : Transparency) (lhs_lemma : Bool)\n    (ps : List pexpr) (hs : hinst_lemmas) : smt_tactic hinst_lemmas :=\n  List.foldlM (add_hinst_lemma_from_pexpr md lhs_lemma) hs ps\n#align auto.add_hinst_lemmas_from_pexprs auto.add_hinst_lemmas_from_pexprs\n\n/-- `done` first attempts to close the goal using `contradiction`. If this fails, it creates an\nSMT state and will repeatedly use `ematch` (using `ematch` lemmas in the environment,\nuniversally quantified assumptions, and the supplied lemmas `ps`) and congruence closure.\n-/\nunsafe def done (ps : List pexpr) (cfg : AutoConfig := { }) : tactic Unit := do\n  trace_state_if_enabled `auto.done \"entering done\"\n  contradiction <|>\n      solve1 do\n        revert_all\n        using_smt do\n            smt_tactic.intros\n            let ctx ← local_context\n            let hs ← mk_hinst_lemmas ctx\n            let hs' ← add_hinst_lemmas_from_pexprs reducible ff ps hs\n            smt_tactic.iterate_at_most cfg\n                (smt_tactic.ematch_using hs' >> smt_tactic.try smt_tactic.close)\n#align auto.done auto.done\n\n/-!\n### Tactics that perform case splits\n-/\n\n\ninductive CaseOption\n  | force-- fail unless all goals are solved\n\n  | at_most_one-- leave at most one goal\n\n  | accept\n  deriving DecidableEq, Inhabited\n#align auto.case_option Auto.CaseOption\n\n-- leave as many goals as necessary\nprivate unsafe def case_cont (s : CaseOption) (cont : CaseOption → tactic Unit) : tactic Unit := do\n  match s with\n    | case_option.force => cont case_option.force >> cont case_option.force\n    |\n    case_option.at_most_one =>-- if the first one succeeds, commit to it, and try the second\n          condM\n          (cont case_option.force >> return tt) (cont case_option.at_most_one) skip <|>\n        (-- otherwise, try the second\n            swap >>\n            cont case_option.force) >>\n          cont case_option.at_most_one\n    | case_option.accept => focus' [cont case_option.accept, cont case_option.accept]\n#align auto.case_cont auto.case_cont\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\nunsafe\n  def\n    case_hyp\n    ( h : expr ) ( s : CaseOption ) ( cont : CaseOption → tactic Unit ) : tactic Bool\n    :=\n      do\n        let t ← infer_type h\n          match\n            t\n            with\n            | q( $ ( a ) ∨ $ ( b ) ) => ( cases h >> case_cont s cont ) >> return tt\n              | _ => return ff\n#align auto.case_hyp auto.case_hyp\n\nunsafe def case_some_hyp_aux (s : CaseOption) (cont : CaseOption → tactic Unit) :\n    List expr → tactic Bool\n  | [] => return false\n  | h :: hs => condM (case_hyp h s cont) (return true) (case_some_hyp_aux hs)\n#align auto.case_some_hyp_aux auto.case_some_hyp_aux\n\nunsafe def case_some_hyp (s : CaseOption) (cont : CaseOption → tactic Unit) : tactic Bool :=\n  local_context >>= case_some_hyp_aux s cont\n#align auto.case_some_hyp auto.case_some_hyp\n\n/-!\n### The main tactics\n-/\n\n\n/-- `safe_core s ps cfg opt` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `s`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `ps`) and congruence closure.\n\n`safe_core` is complete for propositional logic. Depending on the form of `opt`\nit will:\n\n- (if `opt` is `case_option.force`) fail if it does not close the goal,\n- (if `opt` is `case_option.at_most_one`) fail if it produces more than one goal, and\n- (if `opt` is `case_option.accept`) ignore the number of goals it produces.\n-/\nunsafe def safe_core (s : simp_lemmas × List Name) (ps : List pexpr) (cfg : AutoConfig) :\n    CaseOption → tactic Unit := fun co =>\n  focus1 do\n    trace_state_if_enabled `auto.finish \"entering safe_core\"\n    if cfg then do\n        trace_if_enabled `auto.finish \"simplifying hypotheses\"\n        simp_all s.1 s.2 { failIfUnchanged := ff }\n        trace_state_if_enabled `auto.finish \"result:\"\n      else skip\n    tactic.done <|> do\n        trace_if_enabled `auto.finish \"preprocessing hypotheses\"\n        preprocess_hyps cfg\n        trace_state_if_enabled `auto.finish \"result:\"\n        done ps cfg <|>\n            condM (case_some_hyp co safe_core) skip\n              (match co with\n              | case_option.force => done ps cfg\n              | case_option.at_most_one => try (done ps cfg)\n              | case_option.accept => try (done ps cfg))\n#align auto.safe_core auto.safe_core\n\n/-- `clarify` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.at_most_one`.\n-/\nunsafe def clarify (s : simp_lemmas × List Name) (ps : List pexpr) (cfg : AutoConfig := { }) :\n    tactic Unit :=\n  safe_core s ps cfg CaseOption.at_most_one\n#align auto.clarify auto.clarify\n\n/-- `safe` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.accept`.\n-/\nunsafe def safe (s : simp_lemmas × List Name) (ps : List pexpr) (cfg : AutoConfig := { }) :\n    tactic Unit :=\n  safe_core s ps cfg CaseOption.accept\n#align auto.safe auto.safe\n\n/-- `finish` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.force`.\n-/\nunsafe def finish (s : simp_lemmas × List Name) (ps : List pexpr) (cfg : AutoConfig := { }) :\n    tactic Unit :=\n  safe_core s ps cfg CaseOption.force\n#align auto.finish auto.finish\n\nend Auto\n\n/-! ### interactive versions -/\n\n\nopen Auto\n\nnamespace Tactic\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `parser.optional -/\n/-- `clarify [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`clarify` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`clarify` will fail if it produces more than one goal.\n-/\nunsafe def clarify (hs : parse simp_arg_list)\n    (ps : parse (parser.optional (tk \"using\" *> pexpr_list_or_texpr))) (cfg : AutoConfig := { }) :\n    tactic Unit := do\n  let s ← mk_simp_set false [] hs\n  auto.clarify s (ps []) cfg\n#align tactic.interactive.clarify tactic.interactive.clarify\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `parser.optional -/\n/-- `safe [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`safe` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`safe` ignores the number of goals it produces, and should never fail.\n-/\nunsafe def safe (hs : parse simp_arg_list)\n    (ps : parse (parser.optional (tk \"using\" *> pexpr_list_or_texpr))) (cfg : AutoConfig := { }) :\n    tactic Unit := do\n  let s ← mk_simp_set false [] hs\n  auto.safe s (ps []) cfg\n#align tactic.interactive.safe tactic.interactive.safe\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `parser.optional -/\n/-- `finish [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`finish` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`finish` will fail if it does not close the goal.\n-/\nunsafe def finish (hs : parse simp_arg_list)\n    (ps : parse (parser.optional (tk \"using\" *> pexpr_list_or_texpr))) (cfg : AutoConfig := { }) :\n    tactic Unit := do\n  let s ← mk_simp_set false [] hs\n  auto.finish s (ps []) cfg\n#align tactic.interactive.finish tactic.interactive.finish\n\nadd_hint_tactic finish\n\n/--\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\neliminate existential quantifiers on the left, and look for contradictions. They rely on ematching\nand congruence closure to try to finish off a goal at the end.\n\nThe procedures *do* split on disjunctions and recreate the smt state for each terminal call, so\nthey are only meant to be used on small, straightforward problems.\n\n* `finish`:  solves the goal or fails\n* `clarify`: makes as much progress as possible while not leaving more than one goal\n* `safe`:    splits freely, finishes off whatever subgoals it can, and leaves the rest\n\nAll accept an optional list of simplifier rules, typically definitions that should be expanded.\n(The equations and identities should not refer to the local context.) All also accept an optional\nlist of `ematch` lemmas, which must be preceded by `using`.\n-/\nadd_tactic_doc\n  { Name := \"finish / clarify / safe\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.finish, `tactic.interactive.clarify, `tactic.interactive.safe]\n    tags := [\"logic\", \"finishing\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Finish.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.331059630944739}}
{"text": "import data.list.pairwise\nimport data.list\nimport data.list.sort\nimport data.vector\nimport data.finset\nimport data.array.lemmas\nimport data.equiv.list\nimport data.equiv.fin\nimport data.equiv.fintype\nimport order.lexicographic\nimport data.hash_map\nimport data.nat.log\n\ndef arg_list (i n : ℕ) := {v : vector (fin i) n // v.to_list.nodup}\n\ndef arg_list.to_vector {i n} (a : arg_list i n) := a.val\ndef arg_list.to_list {i n} (a : arg_list i n) := a.to_vector.to_list\ndef arg_list.to_nodup {i n} (a : arg_list i n) : a.to_list.nodup := a.prop\n\ninductive rcmd (i : ℕ) : Type\n| tog (arg : fin i) : rcmd\n-- | swap : fin 4 → fin 4 → cmd\n| cnot (args : arg_list i 2) : rcmd\n| toffoli : arg_list i 3 → rcmd\n| fredkin : arg_list i 3 → rcmd\n\nopen rcmd\ndef rcmd.rank {i} : rcmd i → ℕ\n  | (tog _) := 1\n  | (cnot _) := 2\n  | (toffoli _) := 3\n  | (fredkin _) := 4\n\ndef rcmd4.repr : rcmd 4 → string\n  | (tog a) :=\n    \"n\" ++ repr a.val\n  | (cnot a) :=\n    \"c\" ++ repr a.to_vector.head ++ repr a.to_vector.tail.head\n  | (toffoli a) :=\n    \"t\" ++ repr a.to_vector.head ++ repr a.to_vector.tail.head ++ repr a.to_vector.tail.tail.head\n  | (fredkin a) :=\n    \"f\" ++ repr a.to_vector.head ++ repr a.to_vector.tail.head ++ repr a.to_vector.tail.tail.head\n\ninstance : has_repr (rcmd 4) := {\n  repr := rcmd4.repr\n}\n\n\n@[simp] lemma rcmd.rank_tog (size : ℕ) (i : fin size) : rcmd.rank (tog i) = 1 := rfl\n@[simp] lemma rcmd.rank_cnot (size : ℕ) (i) : rcmd.rank (cnot i : rcmd size) = 2 := rfl\n@[simp] lemma rcmd.rank_toffoli (size : ℕ) (i) : rcmd.rank (toffoli i : rcmd size) = 3 := rfl\n@[simp] lemma rcmd.rank_fredkin (size : ℕ) (i) : rcmd.rank (fredkin i : rcmd size) = 4 := rfl\n\ndef rcmd.args {i} : rcmd i → list (fin i)\n  | (tog a) := [a]\n  | (cnot a) := a.to_list\n  | (toffoli a) := a.to_list\n  | (fredkin a) := a.to_list\n\nopen function\n\ndef rcmd.to_nat_list {i} (x : rcmd i) : list ℕ := x.rank :: x.args.map (λ x, x.1)\ndef rcmd.le {i} (x y : rcmd i) := x.to_nat_list ≤ y.to_nat_list\nlemma rcmd.le_refl {i} (x : rcmd i) : x.le x := le_refl x.to_nat_list\nlemma rcmd.le_trans {i} (x y z : rcmd i) : x.le y → y.le z → x.le z :=\n  @le_trans _ _ x.to_nat_list y.to_nat_list z.to_nat_list\nlemma rcmd.to_pair_injective {i} (x y : rcmd i) : x.to_nat_list = y.to_nat_list → x = y := by {\n  cases x; cases y; cases x; cases y; unfold rcmd.to_nat_list rcmd.args arg_list.to_list; simp; intros h;\n    apply vector.eq; exact list.map_injective_iff.mpr fin.coe_injective h,\n}\nlemma rcmd.le_antisymm {i} (x y : rcmd i) : x.le y → y.le x → x = y := by {\n  intros xy yx,\n  apply rcmd.to_pair_injective x y,\n  unfold rcmd.le at xy yx,\n  apply le_antisymm; assumption,\n}\n\ndef rcmd.le_decidable {i} : decidable_rel (rcmd.le : rcmd i → rcmd i → Prop) := fun x y,\n  if h : x.to_nat_list ≤ y.to_nat_list\n    then is_true h\n    else is_false h\n\nlemma rcmd.le_total {i} (x y : rcmd i) : x.le y ∨ y.le x := le_total x.to_nat_list y.to_nat_list\n\ninstance {i} : linear_order (rcmd i) := {\n  le := rcmd.le,\n  le_refl := rcmd.le_refl,\n  le_trans := rcmd.le_trans,\n  le_antisymm := rcmd.le_antisymm,\n  decidable_le := rcmd.le_decidable,\n  le_total := rcmd.le_total,\n}\n\nlemma nodup_tail {α} {x : α} {xs} : list.nodup (x :: xs) → list.nodup xs := list.nodup_of_nodup_cons\nlemma nodup_3 {α} {x y z: α} [decidable_eq α] : [x,y,z].nodup ↔ (x ≠ y ∧ x ≠ z ∧ y ≠ z) := by {\n  simp,\n  rw ← and_assoc,\n  have : (¬(x = y ∨ x = z) ∧ ¬y = z) = ((¬x = y ∧ ¬x = z) ∧ ¬y = z),\n  {\n    congr,\n    apply propext,\n    split; intros hyp_iff,\n    split; intros hyp_not; apply hyp_iff,\n    {left, exact hyp_not},\n    {right, exact hyp_not},\n    intros hyp_not,\n    cases hyp_not,\n    exact hyp_iff.left hyp_not,\n    exact hyp_iff.right hyp_not,\n  },\n  rw this,\n}\n\ndef rcmd.run {size : ℕ} : rcmd size → array size bool → array size bool\n  | (tog i) := λ arr, arr.write i (not (arr.read i))\n  | (cnot {val := ⟨[control, target], _⟩, property := _}) := λ arr,\n    if arr.read control then (tog target).run arr else arr\n  | (toffoli {val := ⟨[control_a, control_b, target], _⟩, property := d} ) := λ arr,\n    if (arr.read control_a)\n      then\n        let d' := by {apply list.nodup_of_nodup_cons d} in\n        (cnot ⟨⟨[control_b, target], rfl⟩, d'⟩ ).run arr\n      else arr\n  | (fredkin {val := ⟨[control, target_a, target_b], _⟩, property := _}) := λ arr,\n    if arr.read control\n      then (arr.write target_a (arr.read target_b)).write target_b (arr.read target_a)\n      else arr\n  using_well_founded {rel_tac := λ _ _, `[exact ⟨_, measure_wf rcmd.rank⟩]}\n\nlemma arr_write_read_neq\n  (size : ℕ) (i : fin size) {α} (arr : array size α) {x : α} :\n  (arr.write i x).read i = x := by {\n    exact array.read_write arr i x\n}\n\nlemma rcmd.run_self (size : ℕ) : ∀ (c : rcmd size) (arr), c.run (c.run arr) = arr\n  | (rcmd.tog c) := λ arr, by {\n    unfold rcmd.run,\n    simp,\n    ext,\n    have : decidable (c = i) := by apply_instance,\n    cases this,\n    {\n      rw array.read_write_of_ne,\n      rw array.read_write_of_ne,\n      all_goals {apply this},\n    },\n    {\n      rw this,\n      rw array.read_write,\n    }\n  }\n  | (cnot {val := ⟨[control, target], p⟩, property := p2}) := λ arr, by {\n    have : (cnot {val := ⟨[control, target], p⟩, property := p2}).run =\n      λ arr, if arr.read control then (rcmd.tog target).run arr else arr := by {\n        unfold rcmd.run\n      },\n    rw this, clear this,\n    have : decidable (arr.read control) := by apply_instance,\n    cases this,\n    {\n      simp_rw if_neg this,\n      rw if_neg,\n      exact this,\n    },\n    {\n      simp at p2,\n      simp_rw if_pos this,\n      rw if_pos,\n      apply rcmd.run_self,\n      unfold rcmd.run,\n      rw array.read_write_of_ne,\n      exact this,\n      symmetry,\n      exact p2,\n    },\n  }\n  | (toffoli {val := ⟨[control, control_b, target], p⟩, property := d}) := λ arr, by {\n    have : (toffoli {val := ⟨[control, control_b, target], p⟩, property := d}).run =\n      λ arr,\n        if (arr.read control)\n          then\n            let d' := by {apply list.nodup_of_nodup_cons d} in\n            (cnot ⟨⟨[control_b, target], rfl⟩, d'⟩ ).run arr\n          else arr\n      := by unfold rcmd.run,\n    simp at this,\n    rw this, clear this,\n    have : decidable (arr.read control) := by apply_instance,\n    cases this,\n    {\n      simp_rw if_neg this,\n      rw if_neg,\n      exact this,\n    },\n    {\n      simp at d,\n      simp_rw if_pos this,\n      simp_rw rcmd.run_self,\n      have : (((cnot {val := ⟨[control_b, target], _⟩, property := _}).run arr).read control),\n      {\n        have : ∀ p₁ p₂,\n          (((cnot {val := ⟨[control_b, target], p₁⟩, property := p₂}).run arr).read control)\n            = arr.read control,\n        {\n          intros,\n          unfold rcmd.run,\n          split_ifs,\n          apply array.read_write_of_ne,\n          intros hyp,\n          apply d.left,\n          right,\n          symmetry,\n          assumption,\n          refl,\n        },\n        rw this,\n        clear this,\n        exact this,\n      },\n      rw if_pos this,\n    },\n  }\n  | (fredkin {val := ⟨[control, target_a, target_b], _⟩, property := d}) := λ arr, by {\n    unfold vector.to_list at d,\n    rw nodup_3 at d,\n    obtain ⟨contro_ne_target_a, control_ne_target_b, target_a_ne_target_b⟩ := d,\n    unfold rcmd.run,\n    have : ((arr.write target_a (arr.read target_b)).write target_b (arr.read target_a)).read control =\n      arr.read control,\n    {\n      rw array.read_write_of_ne,\n      rw array.read_write_of_ne,\n      symmetry,\n      assumption,\n      symmetry,\n      assumption,\n    },\n    split_ifs; try {rw this}; clear this,\n    {\n      ext,\n      have eq_a : decidable (i = target_a) := by apply_instance,\n      cases eq_a with ne_a eq_a,\n      {\n        have eq_b : decidable (i = target_b) := by apply_instance,\n        cases eq_b with ne_b eq_b,\n        {\n          repeat {rw array.read_write_of_ne},\n          all_goals {symmetry, assumption},\n        },\n        {\n          rw eq_b, clear ne_a eq_b i,\n          rw array.read_write,\n          rw array.read_write_of_ne _ _ target_a_ne_target_b.symm,\n          rw array.read_write,\n        }\n      },\n      {\n          rw eq_a, clear eq_a i,\n          rw array.read_write,\n          rw array.read_write_of_ne _ _ target_a_ne_target_b.symm,\n          rw array.read_write,\n      }\n    },\n    {\n      exfalso,\n      apply h_1,\n      rw array.read_write_of_ne,\n      rw array.read_write_of_ne,\n      exact h,\n      all_goals {symmetry, assumption}\n    },\n    refl,\n  }\n  using_well_founded {rel_tac := λ _ _, `[exact ⟨_, measure_wf rcmd.rank⟩]}\n\ninstance : monad_fail list := {\n  fail := λ _ _, []\n}\n\ndef mk_arg_list {i : ℕ} {m} [monad m] [monad_fail m] (get : m (fin i)) : ∀ {n : ℕ}, m (arg_list i n)\n  | 0 := pure ⟨vector.nil, list.nodup_nil⟩\n  | (k + 1) :=\n    do\n    x <- get,\n    ⟨xs, p⟩ <- (mk_arg_list : m (arg_list i k)),\n    if h : x ∈ xs.to_list\n      then monad_fail.fail \"duplicate argument -> cull\"\n      else pure ⟨x ::ᵥ xs, by {\n        rw vector.to_list_cons x xs,\n        apply list.nodup_cons.mpr ⟨h, p⟩\n      }⟩\n\ndef list_out (α) [fintype α] [linear_order α] : list α :=\n  multiset.sort (≤) (fintype.elems α).val\n\nlemma mem_list_out (α) [fintype α] [linear_order α] (x : α) : x ∈ list_out α := by {\n  unfold list_out,\n  rw multiset.mem_sort,\n  rw ← finset.mem_def,\n  exact fintype.complete x\n}\n\nlemma list.mem_bind_op {α β} {b : β} {l : list α} {f : α → list β} :\n  b ∈ (l >>= f) ↔ ∃ a ∈ l, b ∈ f a := by {\n  unfold has_bind.bind,\n  apply list.mem_bind\n}\n\ndef mem_mk_arg_list {i n} {a : arg_list i n} :\n  a ∈ (mk_arg_list (list_out (fin i)) : list (arg_list i n)) := by {\n    obtain ⟨⟨l, l_length⟩, l_nodup⟩ := a,\n    revert l l_length l_nodup,\n    induction n; intros,\n    {\n      rw list.length_eq_zero at l_length,\n      cases l,\n      unfold mk_arg_list,\n      rw list.mem_pure,\n      refl,\n      {cases l_length}\n    },\n    {\n      unfold mk_arg_list,\n      simp_rw list.mem_bind_op,\n      cases l,\n      {cases l_length},\n      use l_hd,\n      split,\n      {apply mem_list_out},\n      use l_tl,\n      {\n        rw list.length_cons at *,\n        exact nat.succ.inj l_length\n      },\n      {exact list.nodup_of_nodup_cons l_nodup},\n      {\n        split,\n        apply n_ih,\n        unfold mk_arg_list._match_1,\n        rw dif_neg,\n        rw list.mem_pure,\n        congr,\n        simp at *,\n        exact l_nodup.left,\n      }\n    },\n  }\n\ndef list.sort {α} [linear_order α] : list α → list α := list.merge_sort (≤)\ndef multiset.sort' {α} [linear_order α] : multiset α → list α := multiset.sort (≤)\n\ndef rcmd.elems' {i} : ℕ → list (rcmd i)\n  | 1 := (list_out (fin i)).map tog\n  | 2 := cnot <$> (mk_arg_list (list_out (fin i)))\n  | 3 := toffoli <$> (mk_arg_list (list_out (fin i)))\n  | 4 := fredkin <$> (mk_arg_list (list_out (fin i)))\n  | n := monad_fail.fail \"ignore\"\n\ndef rcmd.elems {i} : list (rcmd i) :=\n  rcmd.elems' 1 ++ rcmd.elems' 2 ++ rcmd.elems' 3 ++ rcmd.elems' 4\n\nlemma rcmd.mem_elems {i} (c : rcmd i) : c ∈ (rcmd.elems : list (rcmd i)) := by {\n  cases c; unfold rcmd.elems,\n  {\n    repeat {apply list.mem_append_left},\n    unfold rcmd.elems',\n    rw list.mem_map,\n    use c,\n    split,\n    apply mem_list_out,\n    refl,\n  },\n  {\n    apply list.mem_append_left,\n    apply list.mem_append_left,\n    apply list.mem_append_right,\n    unfold rcmd.elems',\n    rw list.map_eq_map,\n    rw list.mem_map,\n    use c,\n    split,\n    apply mem_mk_arg_list,\n    refl\n  },\n  {\n    apply list.mem_append_left,\n    apply list.mem_append_right,\n    unfold rcmd.elems',\n    rw list.map_eq_map,\n    rw list.mem_map,\n    use c,\n    split,\n    apply mem_mk_arg_list,\n    refl\n  },\n  {\n    apply list.mem_append_right,\n    unfold rcmd.elems',\n    rw list.map_eq_map,\n    rw list.mem_map,\n    use c,\n    split,\n    apply mem_mk_arg_list,\n    refl\n  },\n}\n\ninstance (i : ℕ) : fintype (rcmd i) := {\n  elems := rcmd.elems.to_finset,\n  complete := λ c, by { rw list.mem_to_finset, exact rcmd.mem_elems c},\n}\n\n-- this is just a type restricted version of function.comp\ninfix `∘∘`:90 := λ {i}, ((∘) : (array i bool → array i bool) → (array i bool → array i bool) → (array i bool → array i bool))\n\ndef list.run {i} := list.foldl (λ a (b : rcmd i), a ∘∘ b.run) id\n\ndef encode_bit_list : list bool → (ℕ → ℕ) :=\n  flip (list.foldl (λ n (b : bool), if b then bit1 n else bit0 n))\n\ndef list_array : ∀ {i}, list (array i bool)\n  | 0 := [array.nil]\n  | (n+1) := do\n    arr ← (list_array : list (array n bool)),\n    value ← [tt, ff],\n    pure $ array.push_back arr value\n\ndef encode_array_fun {i} : (array i bool → array i bool) → (ℕ → ℕ) :=\n  λ f, encode_bit_list $\n    do\n    arr <- list_array,\n    (f arr).read <$> list_out (fin i)\n\ndef pr_hash_map (i : ℕ) (n : fin 12) : Type := hash_map ℕ (λ _, vector (rcmd i) n)\ndef pr_hash_map.insert {i n} : pr_hash_map i n → vector (rcmd i) n → pr_hash_map i n\n  | h v := hash_map.insert h (encode_array_fun v.to_list.run 0) v\n\ndef programs2 : pr_hash_map 4 2 :=\n  list.foldl pr_hash_map.insert (mk_hash_map (2 ^ 16)) $ do\n    (c0 : rcmd 4) <- rcmd.elems,\n    (c1 : rcmd 4) <- rcmd.elems,\n    pure ⟨[c0, c1], rfl⟩\n\ndef can_go_infront1' (c : rcmd 4) : list (rcmd 4) := (λ v : vector _ 2, v.head) <$>\n  list.filter (λ v : vector _ 2, v.tail.head = c) ((λ s : Σ _ : ℕ, vector (rcmd 4) 2, s.2) <$> programs2.entries)\n\ndef can_go_infront : hash_map (rcmd 4) (λ _, list (rcmd 4)) :=\n  hash_map.insert_all ((λ c, ⟨c, can_go_infront1' c⟩) <$> rcmd.elems) (mk_hash_map 64)\n\ndef programs3 : pr_hash_map 4 3 :=\n  let p := programs2.entries in\n  list.foldl pr_hash_map.insert (mk_hash_map (2 ^ 16)) $ do\n    ⟨_, (p1 : vector (rcmd 4) 2)⟩ <- p,\n    ⟨_, (p2 : vector (rcmd 4) 2)⟩ <- p,\n    if p2.head = p1.tail.head\n      then pure (p1.head ::ᵥ p2)\n      else []\n\nset_option profiler true\nset_option timeout 0\n\n#eval (rcmd.elems : list (rcmd 4)).length\n\n#eval programs2.entries.length\n#eval programs3.entries.length\n\ndef programs5 : pr_hash_map 4 5 :=\n  list.foldl pr_hash_map.insert (mk_hash_map (2 ^ 16)) $ do\n    ⟨_, (p2 : vector (rcmd 4) 2)⟩ <- programs2.entries,\n    ⟨_, (p3 : vector (rcmd 4) 3)⟩ <- programs3.entries,\n    pure (p2.append p3)\n\ndef programs10 : pr_hash_map 4 10 :=\n  list.foldl pr_hash_map.insert (mk_hash_map (2 ^ 16)) $ do\n    ⟨_, (p2 : vector (rcmd 4) 5)⟩ <- programs5.entries,\n    ⟨_, (p3 : vector (rcmd 4) 5)⟩ <- programs5.entries,\n    pure (p2.append p3)\n\n#eval programs10.entries.length\n\n#check pr_hash_map 4 11\n\ndef programs11 : pr_hash_map 4 (11 : fin 12) :=\n  list.foldl pr_hash_map.insert (mk_hash_map (2 ^ 16)) $ do\n    (c : rcmd 4) <- rcmd.elems,\n    ⟨_, (p : vector (rcmd 4) 10)⟩ <- programs10.entries,\n    pure (c ::ᵥ p)", "meta": {"author": "Nolrai", "repo": "pie", "sha": "9d3c00b584b6d171fac1f8fd12307a51752cb063", "save_path": "github-repos/lean/Nolrai-pie", "path": "github-repos/lean/Nolrai-pie/pie-9d3c00b584b6d171fac1f8fd12307a51752cb063/src/hw.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.33105963094473895}}
{"text": "import ..mcrl2_basic.mcrl2_basic\n\nopen mcrl2\n\nvariable {α : Type}\nvariable [comm_semigroup_with_zero α]\n\n/- This relation is used to prove congruence of communication. The proof requires helper lemmas that are further on.-/\ninductive R_comm {x₁ x₂ y₁ y₂ : mcrl2 α} (R₁ R₂ : mcrl2 α → mcrl2 α → Prop) :\nmcrl2 α → mcrl2 α → Prop\n| R₁ {x y} (h : R₁ x y) : R_comm x y\n| R₂ {x y} (h : R₂ x y) : R_comm x y\n| basel : R_comm (x₁ ∣ y₁) (x₂ ∣ y₂)\n| baser : R_comm (x₂ ∣ y₂) (x₁ ∣ y₁)\n| stepl {a b x₁' x₂' y₁' y₂'} (hR₁ : R₁ x₁' x₂') (hR₂ : R₂ y₁' y₂' )\n (h₁x : transition x₁ a x₁') (h₂x : transition x₂ a x₂') \n (h₁y : transition y₁ b y₁') (h₂y : transition y₂ b y₂') :\nR_comm (x₁' || y₁') (x₂' || y₂') \n| stepr {a b x₁' x₂' y₁' y₂'} (hR₁ : R₁ x₁' x₂') (hR₂ : R₂ y₁' y₂' )\n (h₁x : transition x₁ a x₁') (h₂x : transition x₂ a x₂') \n (h₁y : transition y₁ b y₁') (h₂y : transition y₂ b y₂') :\nR_comm (x₂' || y₂') (x₁' || y₁') \n| par_l {x x' y y' a z z'} (hR : R_comm (x || y) (x' || y')) (hR' : R_comm (x' || y') (x || y))\n  (hR₁ : R₁ x x') (hR₂ : R₂ y y')\n  (hRz : R₁ z z')\n  (ht : transition x a z) (ht' : transition x' a z') :\n  R_comm (z || y) (z' || y') \n| par_r {x x' y y' a z z'} (hR : R_comm (x || y) (x' || y')) (hR' : R_comm (x' || y') (x || y))\n  (hR₁ : R₁ x x') (hR₂ : R₂ y y')\n  (hRz : R₂ z z')\n  (ht : transition y a z) (ht' : transition y' a z') :\n  R_comm (x || z) (x' || z')\n| comm {x x' y y' a b z₁ z₁' z₂ z₂'} (hR : R_comm (x || y) (x' || y')) (hR' : R_comm (x' || y') (x || y))\n  (hRz₁ : R₁ z₁ z₁') (hRz₂ : R₂ z₂ z₂')\n  (hR₁ : R₁ x x') (hR₂ : R₂ y y')\n  (htx : transition x a z₁) (htx' : transition x' a z₁')\n  (hty : transition y b z₂) (hty' : transition y' b z₂') :\n  R_comm (z₁ || z₂) (z₁' || z₂')\n\nlemma R_comm.symm {x₁ x₂ y₁ y₂ : mcrl2 α} {R₁ R₂ : mcrl2 α → mcrl2 α → Prop}\n(h₁ : symmetric R₁) (h₂ : symmetric R₂):\nsymmetric (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) :=\nbegin\n  intros x y h,\n  cases h,\n  { apply R_comm.R₁,\n    apply h₁,\n    assumption},\n  { apply R_comm.R₂,\n    apply h₂,\n    assumption},\n  { exact R_comm.baser},\n  { exact R_comm.basel},\n  { apply R_comm.stepr; assumption},\n  { apply R_comm.stepl; assumption},\n  { apply R_comm.par_l,\n    repeat {assumption},\n    exact h₁ h_hR₁,\n    exact h₂ h_hR₂,\n    exact h₁ h_hRz},\n  { apply R_comm.par_r,\n    repeat {assumption},\n    exact h₁ h_hR₁,\n    exact h₂ h_hR₂,\n    apply h₂ h_hRz},\n  { apply R_comm.comm,\n    exact h_hR',\n    repeat {assumption},\n    exact h₁ h_hRz₁,\n    exact h₂ h_hRz₂,\n    exact h₁ h_hR₁,\n    exact h₂ h_hR₂}\nend\n\nlemma bisim_exists_lift_comm_l {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition x a z)\n (hR₁ : R₁ x x') (hR₂ : R₂ y y')\n (hR : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x || y) (x' || y' )) \n (hR' : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x' || y' ) (x || y)) :\n(∃z', transition x' a z' ∧ option.rel R₁ z z') → (∃z', transition x' a z' ∧ option.rel (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' z y) (par' z' y')) :=\nbegin \n  intro h,\n  rcases h with ⟨w, hwa, hRw⟩,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_comm.par_l,\n    exact hR,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_comm.R₂,\n    assumption}\nend\n\nlemma bisim_exists_lift_comm_r {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition y a z)\n (hR₁ : R₁ x x') (hR₂ : R₂ y y')\n (hR : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x || y) (x' || y' )) \n (hR' : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x' || y' ) (x || y)) :\n(∃z', transition y' a z' ∧ option.rel R₂ z z') → (∃z', transition y' a z' ∧ option.rel (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' x z) (par' x' z')) :=\nbegin \n  intro h,\n  rcases h with ⟨w, hwa, hRw⟩,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_comm.par_r,\n    exact hR,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_comm.R₁,\n    assumption}\nend\n\nlemma bisim_exists_lift_comm_comm {x₁ x₂ y₁ y₂ x x' y y' a b z₁ z₂} {R₁ R₂  : mcrl2 α → mcrl2 α → Prop} \n (hxz : transition x a z₁) (hyz' : transition y b z₂) (hR₁ : R₁ x x') (hR₂ : R₂ y y') \n (hab : a * b ≠ 0)\n (hR : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x || y) (x' || y' )) \n (hR' : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x' || y' ) (x || y)) :\n(∃z' , transition x' a z' ∧ option.rel R₁ z₁ z') → (∃z' , transition y' b z' ∧ option.rel R₂ z₂ z') → \n(∃z', transition (x' || y') (a * b) z' ∧ option.rel (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (par' z₁ z₂) z') :=\nbegin \n  intros h₁ h₂,\n  rcases h₁ with ⟨w₁, hw₁a, hRw₁⟩,\n  rcases h₂ with ⟨w₂, hw₂b, hRw₂⟩,\n  simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n  apply or.inr,\n  apply or.inr,\n  apply exists.intro (par' w₁ w₂),\n  apply exists.intro w₁,\n  apply exists.intro w₂,\n  apply and.intro,\n  refl,\n  apply exists.intro a,\n  apply exists.intro b,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  refl,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  assumption,\n  cases hRw₂,\n  { cases hRw₁,\n    { apply option.rel.some,\n      apply R_comm.comm,\n      repeat {assumption}},\n    { apply option.rel.some,\n      cases hRw₂,\n      apply R_comm.R₂,\n      assumption}},\n  { cases hRw₁,\n    { apply option.rel.some,\n      apply R_comm.R₁,\n      assumption},\n    { apply option.rel.none}}\nend\n\nlemma bisim_exists_lift_comm {x₁ x₂ y₁ y₂ x x' y y' a z} {R₁ R₂ : mcrl2 α → mcrl2 α → Prop} \n (R₁_bisim : is_bisimulation R₁) (R₂_bisim : is_bisimulation R₂) (hR₁ : R₁ x x') (hR₂ : R₂ y y') (haz : transition (x || y) a z)\n (hR : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x || y) (x' || y' )) \n (hR' : (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) (x' || y' ) (x || y)) :\n (∃z', transition (x' || y') a z' ∧ option.rel (@R_comm _ _ x₁ x₂ y₁ y₂ R₁ R₂) z z') :=\nbegin \n  cases R₁_bisim with R₁_bisim R₁_symm,\n  cases R₂_bisim with R₂_bisim R₂_symm,\n  cases haz,\n  { specialize R₁_bisim x x' haz_x' a hR₁ haz_h,\n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_comm_l haz_h hR₁ hR₂ hR hR' R₁_bisim,},\n  { specialize R₂_bisim y y' haz_y' a hR₂ haz_h, \n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, ←exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inr,\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_comm_r haz_h hR₁ hR₂ hR hR' R₂_bisim},\n  { specialize R₁_bisim x x' haz_x' haz_a hR₁ haz_h₁, \n    specialize R₂_bisim y y' haz_y' haz_b hR₂ haz_h₂,\n    exact bisim_exists_lift_comm_comm haz_h₁ haz_h₂ hR₁ hR₂ haz_h₃ hR hR' R₁_bisim R₂_bisim}\nend\n\n/- The main congruence theorem for |.-/\ntheorem bisim.comm {x₁ x₂ y₁ y₂: mcrl2 α} (h₁ : x₁ ≈ x₂) (h₂ : y₁ ≈ y₂) : \nx₁ ∣ y₁ ≈ x₂ ∣ y₂ :=\nbegin \n  rcases h₁ with ⟨R₁, R₁x, R₁_bisim⟩,\n  rcases h₂ with ⟨R₂, R₂y, R₂_bisim⟩,\n  apply exists.intro (R_comm R₁ R₂),\n  apply and.intro,\n  apply R_comm.basel,\n  apply and.intro,\n  { intros x y x' a Rxy xax',\n    cases Rxy,\n    { apply bisim_exists_lift,\n      { intros v w, exact R_comm.R₁},\n      { apply bisim_lift; assumption}},\n    { apply bisim_exists_lift,\n      { intros v w, exact R_comm.R₂},\n      { apply bisim_lift; assumption}},\n    { simp [transition.comm_iff, ←exists_and_distrib_right, and_assoc],\n      cases xax',\n      cases R₁_bisim with R₁_bisim R₁_symm,\n      specialize R₁_bisim x₁ x₂ xax'_x' xax'_a R₁x xax'_h₁,\n      rcases R₁_bisim with ⟨w, haw, Rw⟩,\n      cases R₂_bisim with R₂_bisim R₂_symm,\n      specialize R₂_bisim y₁ y₂ xax'_y' xax'_b R₂y xax'_h₂,\n      rcases R₂_bisim with ⟨w', haw', Rw'⟩,\n      apply exists.intro (par' w w'),\n      apply exists.intro w,\n      apply exists.intro w',\n      apply and.intro,\n      refl,\n      apply exists.intro xax'_a,\n      apply exists.intro xax'_b,\n      apply and.intro,\n      refl,\n      apply and.intro,\n      assumption,\n      apply and.intro,\n      assumption,\n      apply and.intro,\n      assumption,\n      cases Rw,\n      { cases Rw',\n        { apply option.rel.some,\n          apply R_comm.stepl; assumption},\n        { apply option.rel.some,\n          exact R_comm.R₁ Rw_ᾰ}},\n      { cases Rw',\n        { apply option.rel.some,\n          apply R_comm.R₂,\n          assumption},\n        { exact option.rel.none}}},\n    { simp [transition.comm_iff, ←exists_and_distrib_right, and_assoc],\n      cases xax',\n      cases R₁_bisim with R₁_bisim R₁_symm,\n      specialize R₁_bisim x₂ x₁ xax'_x' xax'_a (R₁_symm R₁x) xax'_h₁,\n      rcases R₁_bisim with ⟨w, haw, Rw⟩,\n      cases R₂_bisim with R₂_bisim R₂_symm,\n      specialize R₂_bisim y₂ y₁ xax'_y' xax'_b (R₂_symm R₂y) xax'_h₂,\n      rcases R₂_bisim with ⟨w', haw', Rw'⟩,\n      apply exists.intro (par' w w'),\n      apply exists.intro w,\n      apply exists.intro w',\n      apply and.intro,\n      refl,\n      apply exists.intro xax'_a,\n      apply exists.intro xax'_b,\n      apply and.intro,\n      refl,\n      apply and.intro,\n      assumption,\n      apply and.intro,\n      assumption,\n      apply and.intro, \n      assumption,\n      cases Rw,\n      { cases Rw',\n        { apply option.rel.some,\n          apply R_comm.stepr,\n          exact R₁_symm Rw_ᾰ,\n          exact R₂_symm Rw'_ᾰ,\n          repeat {assumption}},\n        { apply option.rel.some,\n          exact R_comm.R₁ Rw_ᾰ}},\n      { cases Rw',\n        { apply option.rel.some,\n          apply R_comm.R₂,\n          assumption},\n        { exact option.rel.none}}},\n    { apply bisim_exists_lift_comm,\n      repeat {assumption},\n      exact R_comm.symm R₁_bisim.right R₂_bisim.right Rxy},\n    { apply bisim_exists_lift_comm,\n      repeat {assumption},\n      exact R₁_bisim.right Rxy_hR₁,\n      exact R₂_bisim.right Rxy_hR₂,\n      exact R_comm.symm R₁_bisim.right R₂_bisim.right Rxy},\n    { apply bisim_exists_lift_comm,\n      repeat {assumption},\n      exact R_comm.symm R₁_bisim.right R₂_bisim.right Rxy},\n    { apply bisim_exists_lift_comm,\n      repeat {assumption},\n      exact R_comm.symm R₁_bisim.right R₂_bisim.right Rxy},\n    { apply bisim_exists_lift_comm,\n      repeat {assumption},\n      exact R_comm.symm R₁_bisim.right R₂_bisim.right Rxy}},\n  { exact R_comm.symm R₁_bisim.right R₂_bisim.right}\nend", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/mcrl2_mrg/comm.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3309510733126297}}
{"text": "import tactic differentiablity\nimport analysis.calculus.fderiv\n\nnamespace tactic\nnamespace interactive\n\nmeta def analysis : tactic unit :=\ndo g ← tactic.target,\n  match g with\n  | `(differentiable_at _ _ _) := apply_rules [``(differentiability)]\n  | `(continuous_at _ _) := apply_rules [``(continuity)]\n  | _ := fail \"error\"\n  end\n\nend interactive\nend tactic\n\nopen real\n\nvariables {E : Type*} [normed_group E] [normed_space ℝ E] {f : E → ℝ}\n  {x : E}\n\n@[differentiability] lemma differentiable_at.sin' (hc : differentiable_at ℝ f x) :\n  differentiable_at ℝ (λx, real.sin (f x)) x := sorry\n\n@[differentiability] lemma differentiable_at_id'' : differentiable_at ℝ (λ x, x) x :=\n(has_fderiv_at_id x).differentiable_at\n\nexample {f : ℝ → ℝ} (x: ℝ) : differentiable_at ℝ (λx, real.sin (x)) x := by differentiability\n\n\nexample \n    {𝕜 : Type*} [nondiscrete_normed_field 𝕜] \n    {E : Type*} [normed_group E] [normed_space 𝕜 E] {x : E} : differentiable_at 𝕜 id x :=\nbegin\n  analysis, --diff\n  sorry\nend\n\nexample {α : Type*} [topological_space α] (x : α) : continuous_at id x :=\nbegin\n  analysis, -- cts\n  sorry\nend\n\nexample : 1 + 1 = 2 :=\nbegin\n  analysis, -- error\nend", "meta": {"author": "jamesa9283", "repo": "special-functions", "sha": "392758fb7207762c9ba6938462614994ff45bdc4", "save_path": "github-repos/lean/jamesa9283-special-functions", "path": "github-repos/lean/jamesa9283-special-functions/special-functions-392758fb7207762c9ba6938462614994ff45bdc4/src/Tactics/analysis.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7690802476562641, "lm_q2_score": 0.43014734858584286, "lm_q1q2_score": 0.33081782937908544}}
{"text": "--\n-- A basic freer monad. Based on various haskell libraries including freer-simple and\n-- the paper \"Freer Monads, More Extensible Effects\" by Oleg Kiselyov and Hiromi Ishii\n--\n\n-- The Freer monad doesn't contain actual executable code, but instead\n-- stores either a result value (Pure) or an effect plus continuation (Impure).\n--\n\n-- The basic Effect type holds a type of the possible operations/commands (usually a sum type)\n-- and the appropriate return type(s) for each possible operation.\nstructure Effect : Type 2 where\n   (op : Type 1)\n   (ret : op → Type)\n\nnamespace Effect\n\n-- The \"Open Union\" contains one operation from a list of possible effects,\n-- determined at run time. This means we have to peel off .Node constructors, similar\n-- to walking a list data type.\ninductive OU : List Effect → Type → Type 2 where\n    | Leaf {eff : Effect} {effs : List Effect} : (c : eff.op) → OU (eff :: effs) (eff.ret c)\n    | Node {eff : Effect} {effs : List Effect} : OU effs x → OU (eff :: effs) x\n\n\n-- This is similar to Haseffect.project below, but instead of returning an (Option e.op) it\n-- returns either an e.op or an OU with the relevant effect removed. Used when\n-- running the handler for a specific effect.\ndef decompOU {e : Effect} {effs : List Effect} (ou : OU (e :: effs) x) : Sum (OU effs x) (e.op) :=\n    match ou with\n    | .Leaf x => Sum.inr x\n    | .Node ou' => Sum.inl ou'                \n\n\n\nclass HasEffect (e : Effect) (effs : List Effect) where\n    inject : (c : e.op) → OU effs (e.ret c)\n    project : OU effs a → Option e.op\n\ninstance : HasEffect e (e :: effs) where\n    inject := fun c => @OU.Leaf e effs c\n    project := fun ou => match ou with\n                         | .Leaf c => Option.some c\n                         | .Node ou' => Option.none\n\ninstance [HasEffect e effs] : HasEffect e (z :: effs) where\n    inject := fun x => OU.Node (@HasEffect.inject e effs _ x)\n    project := fun ou => match ou with\n                         | .Leaf c => Option.none\n                         | .Node ou' => HasEffect.project ou'\n\n\nend Effect\n\n\n\ninductive Freer (effs : List Effect) : Type → Type 2 where\n    | Pure : a → Freer effs a\n    | Impure : {x : Type} → Effect.OU effs x → (x → Freer effs a) → Freer effs a\n\nnamespace Freer\n\nopen Effect\n\n-- lift into Freer\ndef send {e : Effect} {effs : List Effect} [HasEffect e effs] (sendop : e.op) : Freer effs (e.ret sendop) :=\n    @Freer.Impure effs (e.ret sendop) _ (HasEffect.inject sendop) .Pure\n\n-- Sometimes you have a monad with some effects but you need to pass it into a function\n-- with additional effects (see the Catch/Throw higher-order effect for instance).\ndef weaken {g : Effect} {effs : List Effect} : Freer effs a → Freer (g :: effs) a\n    | .Pure a => .Pure a\n    | .Impure ou next => .Impure (OU.Node ou) (fun x => weaken (next x))\n\ndef FreerAlg (effs : List Effect) (a : Type) := {x : Type} → (OU effs x) → (x → a) → a\n\n-- FreerAlg for a Type 1 result. We avoid \"Type u\" declarations since it complicates\n-- a lot of the type inference.\ndef FreerAlg1 (effs : List Effect) (a : Type 1) := {x : Type} → (OU effs x) → (x → a) → a\ndef FreerAlg2 (effs : List Effect) (a : Type 2) := {x : Type} → (OU effs x) → (x → a) → a\n\ndef freerCata {a : Type} (alg : FreerAlg effs a) (w : Freer effs a) : a :=\n    match w with\n    | .Pure a => a\n    | .Impure gx next => alg gx (fun x => freerCata alg (next x))\n\ndef freerMap {a b : Type} (f : a → b) (w : Freer effs a) : Freer effs b :=\n    match w with\n    | .Pure a => .Pure (f a)\n    | .Impure gx next => .Impure gx (fun x => freerMap f (next x))\n\ninstance : Functor (Freer effs) :=\n    { map := freerMap }\n\n-- foldOver is basically freerCata + freerMap\ndef foldOver {a : Type} {b : Type} {effs : List Effect} (pf : a → b) (alg : FreerAlg effs b) : Freer effs a → b\n    | .Pure a => pf a\n    | @Freer.Impure _ a _ gx next => alg gx (fun x => @foldOver a b effs pf alg (next x))\n\ndef foldOver1 {a : Type} {b : Type 1} {effs : List Effect} (pf : a → b) (alg : FreerAlg1 effs b) : Freer effs a → b\n    | .Pure a => pf a\n    | @Freer.Impure _ a _ gx next => alg gx (fun x => @foldOver1 a b effs pf alg (next x))\n\ndef foldOver2 {a : Type} {b : Type 2} {effs : List Effect} (pf : a → b) (alg : FreerAlg2 effs b) : Freer effs a → b\n    | .Pure a => pf a\n    | @Freer.Impure _ a _ gx next => alg gx (fun x => @foldOver2 a b effs pf alg (next x))\n    \n\n/-\n-- alternate implementation of bind using a fold, similar to the \"Hefty Algebras\" paper\n--\n-/\ndef bindFreer {a b : Type} (m : Freer effs a) (f : a → Freer effs b) : Freer effs b :=\n    --@foldOver a (Freer effs b) effs f .Impure m\n    match m with\n    | .Pure a => f a\n    | .Impure gx next => Freer.Impure gx (fun z => bindFreer (next z) f)\n\ndef pureFreer {α : Type} (a : α) : Freer effs α := Freer.Pure a\n\n\n\n-- Handler is the handler for a specific effect: ret is for Pure constructors\n-- and handle is for impure constructors. The result has an 'inp' type so that the  monad can\n-- take inputs (a state monad for example). You can make inp Unit for no input.\n--  - a is the type returned from a Pure\n--  - b is the monad result type. It may be that a=b but not always.\nstructure Handler (a : Type) (b : Type) (e : Effect) (effs : List Effect) (inp : Type) where\n   (ret : a → inp → Freer effs b)\n   (handle : FreerAlg2 (e :: effs) (inp → Freer effs b))\n\n\n-- Interpret a Freer monad. You must provide an \"interpreter\" which takes commands of type c\n-- and produces monads of the final type n.  This runs the interpreter on a command\n-- in the Impure constructor and combines it with the next chunk of the Freer monad\n-- by using a definition of bind for the final monad n.\ndef handleFreer {a b : Type} {e : Effect} {effs : List Effect} {inp : Type} \n    (handler : Handler a b e effs inp) (m : Freer (e :: effs) a) : inp → Freer effs b :=\n    match m with\n    | .Pure a => handler.ret a\n    | .Impure gx next => handler.handle gx (fun x => handleFreer handler (next x))\n\n\n-- After handling all effects, you'll have a Freer [] a left over, use\n-- runEff at the end to extract the result.\n-- example : runEff <| h2 <| h1 m\n\ndef runEff : Freer [] a → a\n    | .Pure x => x\n    -- shouldn't ever have an Impure constructed\n    --| .Impure ou next => match ou with | .Leaf c => Fin.elim0 (ULift.down c)\n\n\nend Freer\n\ninstance : Monad (Freer effs) where\n    pure := Freer.pureFreer\n    bind := Freer.bindFreer\n\ninstance : Pure (Freer effs) where\n    pure := Freer.pureFreer\n\ninstance : Bind (Freer effs) where\n    bind := Freer.bindFreer\n", "meta": {"author": "Izzimach", "repo": "freer-wormhole", "sha": "04804be0343b0870bcf995b57357dbf4b2c67f0c", "save_path": "github-repos/lean/Izzimach-freer-wormhole", "path": "github-repos/lean/Izzimach-freer-wormhole/freer-wormhole-04804be0343b0870bcf995b57357dbf4b2c67f0c/FreerWormhole/Effects/Freer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.33070456392715747}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.option.defs\nimport Mathlib.logic.basic\nimport Mathlib.tactic.cache\nimport Mathlib.PostPort\n\nuniverses u_1 u v w u_2 \n\nnamespace Mathlib\n\n/-!\n## Definitions on lists\n\nThis file contains various definitions on lists. It does not contain\nproofs about these definitions, those are contained in other files in `data/list`\n-/\n\nnamespace list\n\n\n/-- Returns whether a list is []. Returns a boolean even if `l = []` is not decidable. -/\ndef is_nil {α : Type u_1} : List α → Bool :=\n  sorry\n\nprotected instance has_sdiff {α : Type u} [DecidableEq α] : has_sdiff (List α) :=\n  has_sdiff.mk list.diff\n\n/-- Split a list at an index.\n\n     split_at 2 [a, b, c] = ([a, b], [c]) -/\ndef split_at {α : Type u} : ℕ → List α → List α × List α :=\n  sorry\n\n/-- An auxiliary function for `split_on_p`. -/\ndef split_on_p_aux {α : Type u} (P : α → Prop) [decidable_pred P] : List α → (List α → List α) → List (List α) :=\n  sorry\n\n/-- Split a list at every element satisfying a predicate. -/\ndef split_on_p {α : Type u} (P : α → Prop) [decidable_pred P] (l : List α) : List (List α) :=\n  split_on_p_aux P l id\n\n/-- Split a list at every occurrence of an element.\n\n    [1,1,2,3,2,4,4].split_on 2 = [[1,1],[3],[4,4]] -/\ndef split_on {α : Type u} [DecidableEq α] (a : α) (as : List α) : List (List α) :=\n  split_on_p (fun (_x : α) => _x = a) as\n\n/-- Concatenate an element at the end of a list.\n\n     concat [a, b] c = [a, b, c] -/\n@[simp] def concat {α : Type u} : List α → α → List α :=\n  sorry\n\n/-- `head' xs` returns the first element of `xs` if `xs` is non-empty;\nit returns `none` otherwise -/\n@[simp] def head' {α : Type u} : List α → Option α :=\n  sorry\n\n/-- Convert a list into an array (whose length is the length of `l`). -/\ndef to_array {α : Type u} (l : List α) : array (length l) α :=\n  d_array.mk fun (v : fin (length l)) => nth_le l (subtype.val v) sorry\n\n/-- \"inhabited\" `nth` function: returns `default` instead of `none` in the case\n  that the index is out of bounds. -/\n@[simp] def inth {α : Type u} [h : Inhabited α] (l : List α) (n : ℕ) : α :=\n  option.iget (nth l n)\n\n/-- Apply a function to the nth tail of `l`. Returns the input without\n  using `f` if the index is larger than the length of the list.\n\n     modify_nth_tail f 2 [a, b, c] = [a, b] ++ f [c] -/\n@[simp] def modify_nth_tail {α : Type u} (f : List α → List α) : ℕ → List α → List α :=\n  sorry\n\n/-- Apply `f` to the head of the list, if it exists. -/\n@[simp] def modify_head {α : Type u} (f : α → α) : List α → List α :=\n  sorry\n\n/-- Apply `f` to the nth element of the list, if it exists. -/\ndef modify_nth {α : Type u} (f : α → α) : ℕ → List α → List α :=\n  modify_nth_tail (modify_head f)\n\n/-- Apply `f` to the last element of `l`, if it exists. -/\n@[simp] def modify_last {α : Type u} (f : α → α) : List α → List α :=\n  sorry\n\n/-- `insert_nth n a l` inserts `a` into the list `l` after the first `n` elements of `l`\n `insert_nth 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]`-/\ndef insert_nth {α : Type u} (n : ℕ) (a : α) : List α → List α :=\n  modify_nth_tail (List.cons a) n\n\n/-- Take `n` elements from a list `l`. If `l` has less than `n` elements, append `n - length l`\nelements `default α`. -/\ndef take' {α : Type u} [Inhabited α] (n : ℕ) : List α → List α :=\n  sorry\n\n/-- Get the longest initial segment of the list whose members all satisfy `p`.\n\n     take_while (λ x, x < 3) [0, 2, 5, 1] = [0, 2] -/\ndef take_while {α : Type u} (p : α → Prop) [decidable_pred p] : List α → List α :=\n  sorry\n\n/-- Fold a function `f` over the list from the left, returning the list\n  of partial results.\n\n     scanl (+) 0 [1, 2, 3] = [0, 1, 3, 6] -/\ndef scanl {α : Type u} {β : Type v} (f : α → β → α) : α → List β → List α :=\n  sorry\n\n/-- Auxiliary definition used to define `scanr`. If `scanr_aux f b l = (b', l')`\nthen `scanr f b l = b' :: l'` -/\ndef scanr_aux {α : Type u} {β : Type v} (f : α → β → β) (b : β) : List α → β × List β :=\n  sorry\n\n/-- Fold a function `f` over the list from the right, returning the list\n  of partial results.\n\n     scanr (+) 0 [1, 2, 3] = [6, 5, 3, 0] -/\ndef scanr {α : Type u} {β : Type v} (f : α → β → β) (b : β) (l : List α) : List β :=\n  sorry\n\n/-- Product of a list.\n\n     prod [a, b, c] = ((1 * a) * b) * c -/\ndef prod {α : Type u} [Mul α] [HasOne α] : List α → α :=\n  foldl Mul.mul 1\n\n/-- Sum of a list.\n\n     sum [a, b, c] = ((0 + a) + b) + c -/\n-- Later this will be tagged with `to_additive`, but this can't be done yet because of import\n\n-- dependencies.\n\ndef sum {α : Type u} [Add α] [HasZero α] : List α → α :=\n  foldl Add.add 0\n\n/-- The alternating sum of a list. -/\ndef alternating_sum {G : Type u_1} [HasZero G] [Add G] [Neg G] : List G → G :=\n  sorry\n\n/-- The alternating product of a list. -/\ndef alternating_prod {G : Type u_1} [HasOne G] [Mul G] [has_inv G] : List G → G :=\n  sorry\n\n/-- Given a function `f : α → β ⊕ γ`, `partition_map f l` maps the list by `f`\n  whilst partitioning the result it into a pair of lists, `list β × list γ`,\n  partitioning the `sum.inl _` into the left list, and the `sum.inr _` into the right list.\n  `partition_map (id : ℕ ⊕ ℕ → ℕ ⊕ ℕ) [inl 0, inr 1, inl 2] = ([0,2], [1])`    -/\ndef partition_map {α : Type u} {β : Type v} {γ : Type w} (f : α → β ⊕ γ) : List α → List β × List γ :=\n  sorry\n\n/-- `find p l` is the first element of `l` satisfying `p`, or `none` if no such\n  element exists. -/\ndef find {α : Type u} (p : α → Prop) [decidable_pred p] : List α → Option α :=\n  sorry\n\n/-- `mfind tac l` returns the first element of `l` on which `tac` succeeds, and\nfails otherwise. -/\ndef mfind {α : Type u} {m : Type u → Type v} [Monad m] [alternative m] (tac : α → m PUnit) : List α → m α :=\n  mfirst fun (a : α) => tac a $> a\n\n/-- `mbfind' p l` returns the first element `a` of `l` for which `p a` returns\ntrue. `mbfind'` short-circuits, so `p` is not necessarily run on every `a` in\n`l`. This is a monadic version of `list.find`. -/\ndef mbfind' {m : Type u → Type v} [Monad m] {α : Type u} (p : α → m (ulift Bool)) : List α → m (Option α) :=\n  sorry\n\n/-- A variant of `mbfind'` with more restrictive universe levels. -/\ndef mbfind {m : Type → Type v} [Monad m] {α : Type} (p : α → m Bool) (xs : List α) : m (Option α) :=\n  mbfind' (Functor.map ulift.up ∘ p) xs\n\n/-- `many p as` returns true iff `p` returns true for any element of `l`.\n`many` short-circuits, so if `p` returns true for any element of `l`, later\nelements are not checked. This is a monadic version of `list.any`. -/\n-- Implementing this via `mbfind` would give us less universe polymorphism.\n\ndef many {m : Type → Type v} [Monad m] {α : Type u} (p : α → m Bool) : List α → m Bool :=\n  sorry\n\n/-- `mall p as` returns true iff `p` returns true for all elements of `l`.\n`mall` short-circuits, so if `p` returns false for any element of `l`, later\nelements are not checked. This is a monadic version of `list.all`. -/\ndef mall {m : Type → Type v} [Monad m] {α : Type u} (p : α → m Bool) (as : List α) : m Bool :=\n  bnot <$> many (fun (a : α) => bnot <$> p a) as\n\n/-- `mbor xs` runs the actions in `xs`, returning true if any of them returns\ntrue. `mbor` short-circuits, so if an action returns true, later actions are\nnot run. This is a monadic version of `list.bor`. -/\ndef mbor {m : Type → Type v} [Monad m] : List (m Bool) → m Bool :=\n  many id\n\n/-- `mband xs` runs the actions in `xs`, returning true if all of them return\ntrue. `mband` short-circuits, so if an action returns false, later actions are\nnot run. This is a monadic version of `list.band`. -/\ndef mband {m : Type → Type v} [Monad m] : List (m Bool) → m Bool :=\n  mall id\n\n/-- Auxiliary definition for `foldl_with_index`. -/\ndef foldl_with_index_aux {α : Type u} {β : Type v} (f : ℕ → α → β → α) : ℕ → α → List β → α :=\n  sorry\n\n/-- Fold a list from left to right as with `foldl`, but the combining function\nalso receives each element's index. -/\ndef foldl_with_index {α : Type u} {β : Type v} (f : ℕ → α → β → α) (a : α) (l : List β) : α :=\n  foldl_with_index_aux f 0 a l\n\n/-- Auxiliary definition for `foldr_with_index`. -/\ndef foldr_with_index_aux {α : Type u} {β : Type v} (f : ℕ → α → β → β) : ℕ → β → List α → β :=\n  sorry\n\n/-- Fold a list from right to left as with `foldr`, but the combining function\nalso receives each element's index. -/\ndef foldr_with_index {α : Type u} {β : Type v} (f : ℕ → α → β → β) (b : β) (l : List α) : β :=\n  foldr_with_index_aux f 0 b l\n\n/-- `find_indexes p l` is the list of indexes of elements of `l` that satisfy `p`. -/\ndef find_indexes {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : List ℕ :=\n  foldr_with_index (fun (i : ℕ) (a : α) (is : List ℕ) => ite (p a) (i :: is) is) [] l\n\n/-- Returns the elements of `l` that satisfy `p` together with their indexes in\n`l`. The returned list is ordered by index. -/\ndef indexes_values {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) : List (ℕ × α) :=\n  foldr_with_index (fun (i : ℕ) (a : α) (l : List (ℕ × α)) => ite (p a) ((i, a) :: l) l) [] l\n\n/-- `indexes_of a l` is the list of all indexes of `a` in `l`. For example:\n```\nindexes_of a [a, b, a, a] = [0, 2, 3]\n```\n-/\ndef indexes_of {α : Type u} [DecidableEq α] (a : α) : List α → List ℕ :=\n  find_indexes (Eq a)\n\n/-- Monadic variant of `foldl_with_index`. -/\ndef mfoldl_with_index {m : Type v → Type w} [Monad m] {α : Type u_1} {β : Type v} (f : ℕ → β → α → m β) (b : β) (as : List α) : m β :=\n  foldl_with_index\n    (fun (i : ℕ) (ma : m β) (b : α) =>\n      do \n        let a ← ma \n        f i a b)\n    (pure b) as\n\n/-- Monadic variant of `foldr_with_index`. -/\ndef mfoldr_with_index {m : Type v → Type w} [Monad m] {α : Type u_1} {β : Type v} (f : ℕ → α → β → m β) (b : β) (as : List α) : m β :=\n  foldr_with_index\n    (fun (i : ℕ) (a : α) (mb : m β) =>\n      do \n        let b ← mb \n        f i a b)\n    (pure b) as\n\n/-- Auxiliary definition for `mmap_with_index`. -/\ndef mmap_with_index_aux {m : Type v → Type w} [Applicative m] {α : Type u_1} {β : Type v} (f : ℕ → α → m β) : ℕ → List α → m (List β) :=\n  sorry\n\n/-- Applicative variant of `map_with_index`. -/\ndef mmap_with_index {m : Type v → Type w} [Applicative m] {α : Type u_1} {β : Type v} (f : ℕ → α → m β) (as : List α) : m (List β) :=\n  mmap_with_index_aux f 0 as\n\n/-- Auxiliary definition for `mmap_with_index'`. -/\ndef mmap_with_index'_aux {m : Type v → Type w} [Applicative m] {α : Type u_1} (f : ℕ → α → m PUnit) : ℕ → List α → m PUnit :=\n  sorry\n\n/-- A variant of `mmap_with_index` specialised to applicative actions which\nreturn `unit`. -/\ndef mmap_with_index' {m : Type v → Type w} [Applicative m] {α : Type u_1} (f : ℕ → α → m PUnit) (as : List α) : m PUnit :=\n  mmap_with_index'_aux f 0 as\n\n/-- `lookmap` is a combination of `lookup` and `filter_map`.\n  `lookmap f l` will apply `f : α → option α` to each element of the list,\n  replacing `a → b` at the first value `a` in the list such that `f a = some b`. -/\ndef lookmap {α : Type u} (f : α → Option α) : List α → List α :=\n  sorry\n\n/-- `countp p l` is the number of elements of `l` that satisfy `p`. -/\ndef countp {α : Type u} (p : α → Prop) [decidable_pred p] : List α → ℕ :=\n  sorry\n\n/-- `count a l` is the number of occurrences of `a` in `l`. -/\ndef count {α : Type u} [DecidableEq α] (a : α) : List α → ℕ :=\n  countp (Eq a)\n\n/-- `is_prefix l₁ l₂`, or `l₁ <+: l₂`, means that `l₁` is a prefix of `l₂`,\n  that is, `l₂` has the form `l₁ ++ t` for some `t`. -/\ndef is_prefix {α : Type u} (l₁ : List α) (l₂ : List α) :=\n  ∃ (t : List α), l₁ ++ t = l₂\n\n/-- `is_suffix l₁ l₂`, or `l₁ <:+ l₂`, means that `l₁` is a suffix of `l₂`,\n  that is, `l₂` has the form `t ++ l₁` for some `t`. -/\ndef is_suffix {α : Type u} (l₁ : List α) (l₂ : List α) :=\n  ∃ (t : List α), t ++ l₁ = l₂\n\n/-- `is_infix l₁ l₂`, or `l₁ <:+: l₂`, means that `l₁` is a contiguous\n  substring of `l₂`, that is, `l₂` has the form `s ++ l₁ ++ t` for some `s, t`. -/\ndef is_infix {α : Type u} (l₁ : List α) (l₂ : List α) :=\n  ∃ (s : List α), ∃ (t : List α), s ++ l₁ ++ t = l₂\n\ninfixl:50 \" <+: \" => Mathlib.list.is_prefix\n\ninfixl:50 \" <:+ \" => Mathlib.list.is_suffix\n\ninfixl:50 \" <:+: \" => Mathlib.list.is_infix\n\n/-- `inits l` is the list of initial segments of `l`.\n\n     inits [1, 2, 3] = [[], [1], [1, 2], [1, 2, 3]] -/\n@[simp] def inits {α : Type u} : List α → List (List α) :=\n  sorry\n\n/-- `tails l` is the list of terminal segments of `l`.\n\n     tails [1, 2, 3] = [[1, 2, 3], [2, 3], [3], []] -/\n@[simp] def tails {α : Type u} : List α → List (List α) :=\n  sorry\n\ndef sublists'_aux {α : Type u} {β : Type v} : List α → (List α → List β) → List (List β) → List (List β) :=\n  sorry\n\n/-- `sublists' l` is the list of all (non-contiguous) sublists of `l`.\n  It differs from `sublists` only in the order of appearance of the sublists;\n  `sublists'` uses the first element of the list as the MSB,\n  `sublists` uses the first element of the list as the LSB.\n\n     sublists' [1, 2, 3] = [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]] -/\ndef sublists' {α : Type u} (l : List α) : List (List α) :=\n  sublists'_aux l id []\n\ndef sublists_aux {α : Type u} {β : Type v} : List α → (List α → List β → List β) → List β :=\n  sorry\n\n/-- `sublists l` is the list of all (non-contiguous) sublists of `l`; cf. `sublists'`\n  for a different ordering.\n\n     sublists [1, 2, 3] = [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]] -/\ndef sublists {α : Type u} (l : List α) : List (List α) :=\n  [] :: sublists_aux l List.cons\n\ndef sublists_aux₁ {α : Type u} {β : Type v} : List α → (List α → List β) → List β :=\n  sorry\n\n/-- `forall₂ R l₁ l₂` means that `l₁` and `l₂` have the same length,\n  and whenever `a` is the nth element of `l₁`, and `b` is the nth element of `l₂`,\n  then `R a b` is satisfied. -/\ninductive forall₂ {α : Type u} {β : Type v} (R : α → β → Prop) : List α → List β → Prop\nwhere\n| nil : forall₂ R [] []\n| cons : ∀ {a : α} {b : β} {l₁ : List α} {l₂ : List β}, R a b → forall₂ R l₁ l₂ → forall₂ R (a :: l₁) (b :: l₂)\n\n/-- Auxiliary definition used to define `transpose`.\n  `transpose_aux l L` takes each element of `l` and appends it to the start of\n  each element of `L`.\n\n  `transpose_aux [a, b, c] [l₁, l₂, l₃] = [a::l₁, b::l₂, c::l₃]` -/\ndef transpose_aux {α : Type u} : List α → List (List α) → List (List α) :=\n  sorry\n\n/-- transpose of a list of lists, treated as a matrix.\n\n     transpose [[1, 2], [3, 4], [5, 6]] = [[1, 3, 5], [2, 4, 6]] -/\ndef transpose {α : Type u} : List (List α) → List (List α) :=\n  sorry\n\n/-- List of all sections through a list of lists. A section\n  of `[L₁, L₂, ..., Lₙ]` is a list whose first element comes from\n  `L₁`, whose second element comes from `L₂`, and so on. -/\ndef sections {α : Type u} : List (List α) → List (List α) :=\n  sorry\n\ndef permutations_aux2 {α : Type u} {β : Type v} (t : α) (ts : List α) (r : List β) : List α → (List α → β) → List α × List β :=\n  sorry\n\ndef permutations_aux.rec {α : Type u} {C : List α → List α → Sort v} (H0 : (is : List α) → C [] is) (H1 : (t : α) → (ts is : List α) → C ts (t :: is) → C is [] → C (t :: ts) is) (l₁ : List α) (l₂ : List α) : C l₁ l₂ :=\n  sorry\n\ndef permutations_aux {α : Type u} : List α → List α → List (List α) :=\n  sorry\n\n/-- List of all permutations of `l`.\n\n     permutations [1, 2, 3] =\n       [[1, 2, 3], [2, 1, 3], [3, 2, 1],\n        [2, 3, 1], [3, 1, 2], [1, 3, 2]] -/\ndef permutations {α : Type u} (l : List α) : List (List α) :=\n  l :: permutations_aux l []\n\n/-- `erasep p l` removes the first element of `l` satisfying the predicate `p`. -/\ndef erasep {α : Type u} (p : α → Prop) [decidable_pred p] : List α → List α :=\n  sorry\n\n/-- `extractp p l` returns a pair of an element `a` of `l` satisfying the predicate\n  `p`, and `l`, with `a` removed. If there is no such element `a` it returns `(none, l)`. -/\ndef extractp {α : Type u} (p : α → Prop) [decidable_pred p] : List α → Option α × List α :=\n  sorry\n\n/-- `revzip l` returns a list of pairs of the elements of `l` paired\n  with the elements of `l` in reverse order.\n\n`revzip [1,2,3,4,5] = [(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)]`\n -/\ndef revzip {α : Type u} (l : List α) : List (α × α) :=\n  zip l (reverse l)\n\n/-- `product l₁ l₂` is the list of pairs `(a, b)` where `a ∈ l₁` and `b ∈ l₂`.\n\n     product [1, 2] [5, 6] = [(1, 5), (1, 6), (2, 5), (2, 6)] -/\ndef product {α : Type u} {β : Type v} (l₁ : List α) (l₂ : List β) : List (α × β) :=\n  list.bind l₁ fun (a : α) => map (Prod.mk a) l₂\n\n/-- `sigma l₁ l₂` is the list of dependent pairs `(a, b)` where `a ∈ l₁` and `b ∈ l₂ a`.\n\n     sigma [1, 2] (λ_, [(5 : ℕ), 6]) = [(1, 5), (1, 6), (2, 5), (2, 6)] -/\nprotected def sigma {α : Type u} {σ : α → Type u_1} (l₁ : List α) (l₂ : (a : α) → List (σ a)) : List (sigma fun (a : α) => σ a) :=\n  list.bind l₁ fun (a : α) => map (sigma.mk a) (l₂ a)\n\n/-- Auxliary definition used to define `of_fn`.\n\n  `of_fn_aux f m h l` returns the first `m` elements of `of_fn f`\n  appended to `l` -/\ndef of_fn_aux {α : Type u} {n : ℕ} (f : fin n → α) (m : ℕ) : m ≤ n → List α → List α :=\n  sorry\n\n/-- `of_fn f` with `f : fin n → α` returns the list whose ith element is `f i`\n  `of_fun f = [f 0, f 1, ... , f(n - 1)]` -/\ndef of_fn {α : Type u} {n : ℕ} (f : fin n → α) : List α :=\n  of_fn_aux f n sorry []\n\n/-- `of_fn_nth_val f i` returns `some (f i)` if `i < n` and `none` otherwise. -/\ndef of_fn_nth_val {α : Type u} {n : ℕ} (f : fin n → α) (i : ℕ) : Option α :=\n  dite (i < n) (fun (h : i < n) => some (f { val := i, property := h })) fun (h : ¬i < n) => none\n\n/-- `disjoint l₁ l₂` means that `l₁` and `l₂` have no elements in common. -/\ndef disjoint {α : Type u} (l₁ : List α) (l₂ : List α) :=\n  ∀ {a : α}, a ∈ l₁ → a ∈ l₂ → False\n\n/-- `pairwise R l` means that all the elements with earlier indexes are\n  `R`-related to all the elements with later indexes.\n\n     pairwise R [1, 2, 3] ↔ R 1 2 ∧ R 1 3 ∧ R 2 3\n\n  For example if `R = (≠)` then it asserts `l` has no duplicates,\n  and if `R = (<)` then it asserts that `l` is (strictly) sorted. -/\ninductive pairwise {α : Type u} (R : α → α → Prop) : List α → Prop\nwhere\n| nil : pairwise R []\n| cons : ∀ {a : α} {l : List α}, (∀ (a' : α), a' ∈ l → R a a') → pairwise R l → pairwise R (a :: l)\n\n@[simp] theorem pairwise_cons {α : Type u} {R : α → α → Prop} {a : α} {l : List α} : pairwise R (a :: l) ↔ (∀ (a' : α), a' ∈ l → R a a') ∧ pairwise R l := sorry\n\nprotected instance decidable_pairwise {α : Type u} {R : α → α → Prop} [DecidableRel R] (l : List α) : Decidable (pairwise R l) :=\n  List.rec (is_true pairwise.nil)\n    (fun (hd : α) (tl : List α) (ih : Decidable (pairwise R tl)) =>\n      decidable_of_iff' ((∀ (a' : α), a' ∈ tl → R hd a') ∧ pairwise R tl) pairwise_cons)\n    l\n\n/-- `pw_filter R l` is a maximal sublist of `l` which is `pairwise R`.\n  `pw_filter (≠)` is the erase duplicates function (cf. `erase_dup`), and `pw_filter (<)` finds\n  a maximal increasing subsequence in `l`. For example,\n\n     pw_filter (<) [0, 1, 5, 2, 6, 3, 4] = [0, 1, 2, 3, 4] -/\ndef pw_filter {α : Type u} (R : α → α → Prop) [DecidableRel R] : List α → List α :=\n  sorry\n\n/-- `chain R a l` means that `R` holds between adjacent elements of `a::l`.\n\n     chain R a [b, c, d] ↔ R a b ∧ R b c ∧ R c d -/\ninductive chain {α : Type u} (R : α → α → Prop) : α → List α → Prop\nwhere\n| nil : ∀ {a : α}, chain R a []\n| cons : ∀ {a b : α} {l : List α}, R a b → chain R b l → chain R a (b :: l)\n\n/-- `chain' R l` means that `R` holds between adjacent elements of `l`.\n\n     chain' R [a, b, c, d] ↔ R a b ∧ R b c ∧ R c d -/\ndef chain' {α : Type u} (R : α → α → Prop) : List α → Prop :=\n  sorry\n\n@[simp] theorem chain_cons {α : Type u} {R : α → α → Prop} {a : α} {b : α} {l : List α} : chain R a (b :: l) ↔ R a b ∧ chain R b l := sorry\n\nprotected instance decidable_chain {α : Type u} {R : α → α → Prop} [DecidableRel R] (a : α) (l : List α) : Decidable (chain R a l) :=\n  List.rec (fun (a : α) => eq.mpr sorry decidable.true)\n    (fun (l_hd : α) (l_tl : List α) (l_ih : (a : α) → Decidable (chain R a l_tl)) (a : α) => eq.mpr sorry and.decidable) l\n    a\n\nprotected instance decidable_chain' {α : Type u} {R : α → α → Prop} [DecidableRel R] (l : List α) : Decidable (chain' R l) :=\n  list.cases_on l (id decidable.true) fun (l_hd : α) (l_tl : List α) => id (list.decidable_chain l_hd l_tl)\n\n/-- `nodup l` means that `l` has no duplicates, that is, any element appears at most\n  once in the list. It is defined as `pairwise (≠)`. -/\ndef nodup {α : Type u} : List α → Prop :=\n  pairwise ne\n\nprotected instance nodup_decidable {α : Type u} [DecidableEq α] (l : List α) : Decidable (nodup l) :=\n  list.decidable_pairwise\n\n/-- `erase_dup l` removes duplicates from `l` (taking only the first occurrence).\n  Defined as `pw_filter (≠)`.\n\n     erase_dup [1, 0, 2, 2, 1] = [0, 2, 1] -/\ndef erase_dup {α : Type u} [DecidableEq α] : List α → List α :=\n  pw_filter ne\n\n/-- `range' s n` is the list of numbers `[s, s+1, ..., s+n-1]`.\n  It is intended mainly for proving properties of `range` and `iota`. -/\n@[simp] def range' : ℕ → ℕ → List ℕ :=\n  sorry\n\n/-- Drop `none`s from a list, and replace each remaining `some a` with `a`. -/\ndef reduce_option {α : Type u_1} : List (Option α) → List α :=\n  filter_map id\n\n/-- `ilast' x xs` returns the last element of `xs` if `xs` is non-empty;\nit returns `x` otherwise -/\n@[simp] def ilast' {α : Type u_1} : α → List α → α :=\n  sorry\n\n/-- `last' xs` returns the last element of `xs` if `xs` is non-empty;\nit returns `none` otherwise -/\n@[simp] def last' {α : Type u_1} : List α → Option α :=\n  sorry\n\n/-- `rotate l n` rotates the elements of `l` to the left by `n`\n\n     rotate [0, 1, 2, 3, 4, 5] 2 = [2, 3, 4, 5, 0, 1] -/\ndef rotate {α : Type u} (l : List α) (n : ℕ) : List α :=\n  sorry\n\n/-- rotate' is the same as `rotate`, but slower. Used for proofs about `rotate`-/\ndef rotate' {α : Type u} : List α → ℕ → List α :=\n  sorry\n\n/-- Given a decidable predicate `p` and a proof of existence of `a ∈ l` such that `p a`,\nchoose the first element with this property. This version returns both `a` and proofs\nof `a ∈ l` and `p a`. -/\ndef choose_x {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : Subtype fun (a : α) => a ∈ l ∧ p a :=\n  sorry\n\n/-- Given a decidable predicate `p` and a proof of existence of `a ∈ l` such that `p a`,\nchoose the first element with this property. This version returns `a : α`, and properties\nare given by `choose_mem` and `choose_property`. -/\ndef choose {α : Type u} (p : α → Prop) [decidable_pred p] (l : List α) (hp : ∃ (a : α), a ∈ l ∧ p a) : α :=\n  ↑(choose_x p l hp)\n\n/-- Filters and maps elements of a list -/\ndef mmap_filter {m : Type → Type v} [Monad m] {α : Type u_1} {β : Type} (f : α → m (Option β)) : List α → m (List β) :=\n  sorry\n\n/--\n`mmap_upper_triangle f l` calls `f` on all elements in the upper triangular part of `l × l`.\nThat is, for each `e ∈ l`, it will run `f e e` and then `f e e'`\nfor each `e'` that appears after `e` in `l`.\n\nExample: suppose `l = [1, 2, 3]`. `mmap_upper_triangle f l` will produce the list\n`[f 1 1, f 1 2, f 1 3, f 2 2, f 2 3, f 3 3]`.\n-/\ndef mmap_upper_triangle {m : Type u → Type u_1} [Monad m] {α : Type u} {β : Type u} (f : α → α → m β) : List α → m (List β) :=\n  sorry\n\n/--\n`mmap'_diag f l` calls `f` on all elements in the upper triangular part of `l × l`.\nThat is, for each `e ∈ l`, it will run `f e e` and then `f e e'`\nfor each `e'` that appears after `e` in `l`.\n\nExample: suppose `l = [1, 2, 3]`. `mmap'_diag f l` will evaluate, in this order,\n`f 1 1`, `f 1 2`, `f 1 3`, `f 2 2`, `f 2 3`, `f 3 3`.\n-/\ndef mmap'_diag {m : Type → Type u_1} [Monad m] {α : Type u_2} (f : α → α → m Unit) : List α → m Unit :=\n  sorry\n\nprotected def traverse {F : Type u → Type v} [Applicative F] {α : Type u_1} {β : Type u} (f : α → F β) : List α → F (List β) :=\n  sorry\n\n/-- `get_rest l l₁` returns `some l₂` if `l = l₁ ++ l₂`.\n  If `l₁` is not a prefix of `l`, returns `none` -/\ndef get_rest {α : Type u} [DecidableEq α] : List α → List α → Option (List α) :=\n  sorry\n\n/--\n`list.slice n m xs` removes a slice of length `m` at index `n` in list `xs`.\n-/\ndef slice {α : Type u_1} : ℕ → ℕ → List α → List α :=\n  sorry\n\n/--\nLeft-biased version of `list.map₂`. `map₂_left' f as bs` applies `f` to each\npair of elements `aᵢ ∈ as` and `bᵢ ∈ bs`. If `bs` is shorter than `as`, `f` is\napplied to `none` for the remaining `aᵢ`. Returns the results of the `f`\napplications and the remaining `bs`.\n\n```\nmap₂_left' prod.mk [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])\n\nmap₂_left' prod.mk [1] ['a', 'b'] = ([(1, some 'a')], ['b'])\n```\n-/\n@[simp] def map₂_left' {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) : List α → List β → List γ × List β :=\n  sorry\n\n/--\nRight-biased version of `list.map₂`. `map₂_right' f as bs` applies `f` to each\npair of elements `aᵢ ∈ as` and `bᵢ ∈ bs`. If `as` is shorter than `bs`, `f` is\napplied to `none` for the remaining `bᵢ`. Returns the results of the `f`\napplications and the remaining `as`.\n\n```\nmap₂_right' prod.mk [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])\n\nmap₂_right' prod.mk [1, 2] ['a'] = ([(some 1, 'a')], [2])\n```\n-/\ndef map₂_right' {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) : List γ × List α :=\n  map₂_left' (flip f) bs as\n\n/--\nLeft-biased version of `list.zip`. `zip_left' as bs` returns the list of\npairs `(aᵢ, bᵢ)` for `aᵢ ∈ as` and `bᵢ ∈ bs`. If `bs` is shorter than `as`, the\nremaining `aᵢ` are paired with `none`. Also returns the remaining `bs`.\n\n```\nzip_left' [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])\n\nzip_left' [1] ['a', 'b'] = ([(1, some 'a')], ['b'])\n\nzip_left' = map₂_left' prod.mk\n\n```\n-/\ndef zip_left' {α : Type u} {β : Type v} : List α → List β → List (α × Option β) × List β :=\n  map₂_left' Prod.mk\n\n/--\nRight-biased version of `list.zip`. `zip_right' as bs` returns the list of\npairs `(aᵢ, bᵢ)` for `aᵢ ∈ as` and `bᵢ ∈ bs`. If `as` is shorter than `bs`, the\nremaining `bᵢ` are paired with `none`. Also returns the remaining `as`.\n\n```\nzip_right' [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])\n\nzip_right' [1, 2] ['a'] = ([(some 1, 'a')], [2])\n\nzip_right' = map₂_right' prod.mk\n```\n-/\ndef zip_right' {α : Type u} {β : Type v} : List α → List β → List (Option α × β) × List α :=\n  map₂_right' Prod.mk\n\n/--\nLeft-biased version of `list.map₂`. `map₂_left f as bs` applies `f` to each pair\n`aᵢ ∈ as` and `bᵢ ‌∈ bs`. If `bs` is shorter than `as`, `f` is applied to `none`\nfor the remaining `aᵢ`.\n\n```\nmap₂_left prod.mk [1, 2] ['a'] = [(1, some 'a'), (2, none)]\n\nmap₂_left prod.mk [1] ['a', 'b'] = [(1, some 'a')]\n\nmap₂_left f as bs = (map₂_left' f as bs).fst\n```\n-/\n@[simp] def map₂_left {α : Type u} {β : Type v} {γ : Type w} (f : α → Option β → γ) : List α → List β → List γ :=\n  sorry\n\n/--\nRight-biased version of `list.map₂`. `map₂_right f as bs` applies `f` to each\npair `aᵢ ∈ as` and `bᵢ ‌∈ bs`. If `as` is shorter than `bs`, `f` is applied to\n`none` for the remaining `bᵢ`.\n\n```\nmap₂_right prod.mk [1, 2] ['a'] = [(some 1, 'a')]\n\nmap₂_right prod.mk [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]\n\nmap₂_right f as bs = (map₂_right' f as bs).fst\n```\n-/\ndef map₂_right {α : Type u} {β : Type v} {γ : Type w} (f : Option α → β → γ) (as : List α) (bs : List β) : List γ :=\n  map₂_left (flip f) bs as\n\n/--\nLeft-biased version of `list.zip`. `zip_left as bs` returns the list of pairs\n`(aᵢ, bᵢ)` for `aᵢ ∈ as` and `bᵢ ∈ bs`. If `bs` is shorter than `as`, the\nremaining `aᵢ` are paired with `none`.\n\n```\nzip_left [1, 2] ['a'] = [(1, some 'a'), (2, none)]\n\nzip_left [1] ['a', 'b'] = [(1, some 'a')]\n\nzip_left = map₂_left prod.mk\n```\n-/\ndef zip_left {α : Type u} {β : Type v} : List α → List β → List (α × Option β) :=\n  map₂_left Prod.mk\n\n/--\nRight-biased version of `list.zip`. `zip_right as bs` returns the list of pairs\n`(aᵢ, bᵢ)` for `aᵢ ∈ as` and `bᵢ ∈ bs`. If `as` is shorter than `bs`, the\nremaining `bᵢ` are paired with `none`.\n\n```\nzip_right [1, 2] ['a'] = [(some 1, 'a')]\n\nzip_right [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]\n\nzip_right = map₂_right prod.mk\n```\n-/\ndef zip_right {α : Type u} {β : Type v} : List α → List β → List (Option α × β) :=\n  map₂_right Prod.mk\n\n/--\nIf all elements of `xs` are `some xᵢ`, `all_some xs` returns the `xᵢ`. Otherwise\nit returns `none`.\n\n```\nall_some [some 1, some 2] = some [1, 2]\nall_some [some 1, none  ] = none\n```\n-/\ndef all_some {α : Type u} : List (Option α) → Option (List α) :=\n  sorry\n\n/--\n`fill_nones xs ys` replaces the `none`s in `xs` with elements of `ys`. If there\nare not enough `ys` to replace all the `none`s, the remaining `none`s are\ndropped from `xs`.\n\n```\nfill_nones [none, some 1, none, none] [2, 3] = [2, 1, 3]\n```\n-/\ndef fill_nones {α : Type u_1} : List (Option α) → List α → List α :=\n  sorry\n\n/--\n`take_list as ns` extracts successive sublists from `as`. For `ns = n₁ ... nₘ`,\nit first takes the `n₁` initial elements from `as`, then the next `n₂` ones,\netc. It returns the sublists of `as` -- one for each `nᵢ` -- and the remaining\nelements of `as`. If `as` does not have at least as many elements as the sum of\nthe `nᵢ`, the corresponding sublists will have less than `nᵢ` elements.\n\n```\ntake_list ['a', 'b', 'c', 'd', 'e'] [2, 1, 1] = ([['a', 'b'], ['c'], ['d']], ['e'])\ntake_list ['a', 'b'] [3, 1] = ([['a', 'b'], []], [])\n```\n-/\ndef take_list {α : Type u_1} : List α → List ℕ → List (List α) × List α :=\n  sorry\n\n/--\n`to_rbmap as` is the map that associates each index `i` of `as` with the\ncorresponding element of `as`.\n\n```\nto_rbmap ['a', 'b', 'c'] = rbmap_of [(0, 'a'), (1, 'b'), (2, 'c')]\n```\n-/\ndef to_rbmap {α : Type u} : List α → rbmap ℕ α :=\n  foldl_with_index (fun (i : ℕ) (mapp : rbmap ℕ α) (a : α) => rbmap.insert mapp i a) (mk_rbmap ℕ α)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Markus Himmel\n-/\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.strong_epi\n\n/-!\n# Categorical images\n\nWe define the categorical image of `f` as a factorisation `f = e ≫ m` through a monomorphism `m`,\nso that `m` factors through the `m'` in any other such factorisation.\n\n## Main definitions\n\n* A `mono_factorisation` is a factorisation `f = e ≫ m`, where `m` is a monomorphism\n* `is_image F` means that a given mono factorisation `F` has the universal property of the image.\n* `has_image f` means that we have chosen an image for the morphism `f : X ⟶ Y`.\n  * In this case, `image f` is the image object, `image.ι f : image f ⟶ Y` is the monomorphism `m`\n    of the factorisation and `factor_thru_image f : X ⟶ image f` is the morphism `e`.\n* `has_images C` means that every morphism in `C` has an image.\n* Let `f : X ⟶ Y` and `g : P ⟶ Q` be morphisms in `C`, which we will represent as objects of the\n  arrow category `arrow C`. Then `sq : f ⟶ g` is a commutative square in `C`. If `f` and `g` have\n  images, then `has_image_map sq` represents the fact that there is a morphism\n  `i : image f ⟶ image g` making the diagram\n\n  X ----→ image f ----→ Y\n  |         |           |\n  |         |           |\n  ↓         ↓           ↓\n  P ----→ image g ----→ Q\n\n  commute, where the top row is the image factorisation of `f`, the bottom row is the image\n  factorisation of `g`, and the outer rectangle is the commutative square `sq`.\n* If a category `has_images`, then `has_image_maps` means that every commutative square admits an\n  image map.\n* If a category `has_images`, then `has_strong_epi_images` means that the morphism to the image is\n  always a strong epimorphism.\n\n## Main statements\n\n* When `C` has equalizers, the morphism `e` appearing in an image factorisation is an epimorphism.\n* When `C` has strong epi images, then these images admit image maps.\n\n## Future work\n* TODO: coimages, and abelian categories.\n* TODO: connect this with existing working in the group theory and ring theory libraries.\n\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory\nopen category_theory.limits.walking_parallel_pair\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C]\n\nvariables {X Y : C} (f : X ⟶ Y)\n\n/-- A factorisation of a morphism `f = e ≫ m`, with `m` monic. -/\nstructure mono_factorisation (f : X ⟶ Y) :=\n(I : C)\n(m : I ⟶ Y)\n[m_mono : mono m]\n(e : X ⟶ I)\n(fac' : e ≫ m = f . obviously)\n\nrestate_axiom mono_factorisation.fac'\nattribute [simp, reassoc] mono_factorisation.fac\nattribute [instance] mono_factorisation.m_mono\n\nattribute [instance] mono_factorisation.m_mono\n\nnamespace mono_factorisation\n\n/-- The obvious factorisation of a monomorphism through itself. -/\ndef self [mono f] : mono_factorisation f :=\n{ I := X,\n  m := f,\n  e := 𝟙 X }\n\n-- I'm not sure we really need this, but the linter says that an inhabited instance\n-- ought to exist...\ninstance [mono f] : inhabited (mono_factorisation f) := ⟨self f⟩\n\nvariables {f}\n\n/-- The morphism `m` in a factorisation `f = e ≫ m` through a monomorphism is uniquely\ndetermined. -/\n@[ext]\nlemma ext\n  {F F' : mono_factorisation f} (hI : F.I = F'.I) (hm : F.m = (eq_to_hom hI) ≫ F'.m) : F = F' :=\nbegin\n  cases F, cases F',\n  cases hI,\n  simp at hm,\n  dsimp at F_fac' F'_fac',\n  congr,\n  { assumption },\n  { resetI, apply (cancel_mono F_m).1,\n    rw [F_fac', hm, F'_fac'], }\nend\n\n/-- Any mono factorisation of `f` gives a mono factorisation of `f ≫ g` when `g` is a mono. -/\n@[simps]\ndef comp_mono (F : mono_factorisation f) {Y' : C} (g : Y ⟶ Y') [mono g] :\n  mono_factorisation (f ≫ g) :=\n{ I := F.I,\n  m := F.m ≫ g,\n  m_mono := mono_comp _ _,\n  e := F.e, }\n\n/-- A mono factorisation of `f ≫ g`, where `g` is an isomorphism,\ngives a mono factorisation of `f`. -/\n@[simps]\ndef of_comp_iso {Y' : C} {g : Y ⟶ Y'} [is_iso g] (F : mono_factorisation (f ≫ g)) :\n  mono_factorisation f :=\n{ I := F.I,\n  m := F.m ≫ (inv g),\n  m_mono := mono_comp _ _,\n  e := F.e, }\n\n/-- Any mono factorisation of `f` gives a mono factorisation of `g ≫ f`. -/\n@[simps]\ndef iso_comp (F : mono_factorisation f) {X' : C} (g : X' ⟶ X) :\n  mono_factorisation (g ≫ f) :=\n{ I := F.I,\n  m := F.m,\n  e := g ≫ F.e, }\n\n/-- A mono factorisation of `g ≫ f`, where `g` is an isomorphism,\ngives a mono factorisation of `f`. -/\n@[simps]\ndef of_iso_comp {X' : C} (g : X' ⟶ X) [is_iso g] (F : mono_factorisation (g ≫ f)) :\n  mono_factorisation f :=\n{ I := F.I,\n  m := F.m,\n  e := inv g ≫ F.e, }\n\nend mono_factorisation\n\nvariable {f}\n\n/-- Data exhibiting that a given factorisation through a mono is initial. -/\nstructure is_image (F : mono_factorisation f) :=\n(lift : Π (F' : mono_factorisation f), F.I ⟶ F'.I)\n(lift_fac' : Π (F' : mono_factorisation f), lift F' ≫ F'.m = F.m . obviously)\n\nrestate_axiom is_image.lift_fac'\nattribute [simp, reassoc] is_image.lift_fac\n\n@[simp, reassoc] lemma is_image.fac_lift {F : mono_factorisation f} (hF : is_image F)\n  (F' : mono_factorisation f) : F.e ≫ hF.lift F' = F'.e :=\n(cancel_mono F'.m).1 $ by simp\n\nvariable (f)\n\nnamespace is_image\n\n/-- The trivial factorisation of a monomorphism satisfies the universal property. -/\n@[simps]\ndef self [mono f] : is_image (mono_factorisation.self f) :=\n{ lift := λ F', F'.e }\n\ninstance [mono f] : inhabited (is_image (mono_factorisation.self f)) :=\n⟨self f⟩\n\nvariable {f}\n/-- Two factorisations through monomorphisms satisfying the universal property\nmust factor through isomorphic objects. -/\n-- TODO this is another good candidate for a future `unique_up_to_canonical_iso`.\n@[simps]\ndef iso_ext {F F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') : F.I ≅ F'.I :=\n{ hom := hF.lift F',\n  inv := hF'.lift F,\n  hom_inv_id' := (cancel_mono F.m).1 (by simp),\n  inv_hom_id' := (cancel_mono F'.m).1 (by simp) }\n\nvariables {F F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F')\n\nlemma iso_ext_hom_m : (iso_ext hF hF').hom ≫ F'.m = F.m := by simp\nlemma iso_ext_inv_m : (iso_ext hF hF').inv ≫ F.m = F'.m := by simp\nlemma e_iso_ext_hom : F.e ≫ (iso_ext hF hF').hom = F'.e := by simp\nlemma e_iso_ext_inv : F'.e ≫ (iso_ext hF hF').inv = F.e := by simp\n\nend is_image\n\n/-- Data exhibiting that a morphism `f` has an image. -/\nstructure image_factorisation (f : X ⟶ Y) :=\n(F : mono_factorisation f)\n(is_image : is_image F)\n\ninstance inhabited_image_factorisation (f : X ⟶ Y) [mono f] : inhabited (image_factorisation f) :=\n⟨⟨_, is_image.self f⟩⟩\n\n/-- `has_image f` means that there exists an image factorisation of `f`. -/\nclass has_image (f : X ⟶ Y) : Prop :=\nmk' :: (exists_image : nonempty (image_factorisation f))\n\nlemma has_image.mk {f : X ⟶ Y} (F : image_factorisation f) : has_image f :=\n⟨nonempty.intro F⟩\n\nsection\nvariable [has_image f]\n\n/-- The chosen factorisation of `f` through a monomorphism. -/\ndef image.mono_factorisation : mono_factorisation f :=\n(classical.choice (has_image.exists_image)).F\n\n/-- The witness of the universal property for the chosen factorisation of `f` through\na monomorphism. -/\ndef image.is_image : is_image (image.mono_factorisation f) :=\n(classical.choice (has_image.exists_image)).is_image\n\n/-- The categorical image of a morphism. -/\ndef image : C := (image.mono_factorisation f).I\n/-- The inclusion of the image of a morphism into the target. -/\ndef image.ι : image f ⟶ Y := (image.mono_factorisation f).m\n@[simp] lemma image.as_ι : (image.mono_factorisation f).m = image.ι f := rfl\ninstance : mono (image.ι f) := (image.mono_factorisation f).m_mono\n\n/-- The map from the source to the image of a morphism. -/\ndef factor_thru_image : X ⟶ image f := (image.mono_factorisation f).e\n/-- Rewrite in terms of the `factor_thru_image` interface. -/\n@[simp]\nlemma as_factor_thru_image : (image.mono_factorisation f).e = factor_thru_image f := rfl\n@[simp, reassoc]\nlemma image.fac : factor_thru_image f ≫ image.ι f = f := (image.mono_factorisation f).fac'\n\nvariable {f}\n\n/-- Any other factorisation of the morphism `f` through a monomorphism receives a map from the\nimage. -/\ndef image.lift (F' : mono_factorisation f) : image f ⟶ F'.I := (image.is_image f).lift F'\n@[simp, reassoc]\nlemma image.lift_fac (F' : mono_factorisation f) : image.lift F' ≫ F'.m = image.ι f :=\n(image.is_image f).lift_fac' F'\n@[simp, reassoc]\nlemma image.fac_lift (F' : mono_factorisation f) : factor_thru_image f ≫ image.lift F' = F'.e :=\n(image.is_image f).fac_lift F'\n\n@[simp, reassoc]\nlemma is_image.lift_ι {F : mono_factorisation f} (hF : is_image F) :\n  hF.lift (image.mono_factorisation f) ≫ image.ι f = F.m :=\nhF.lift_fac _\n\n-- TODO we could put a category structure on `mono_factorisation f`,\n-- with the morphisms being `g : I ⟶ I'` commuting with the `m`s\n-- (they then automatically commute with the `e`s)\n-- and show that an `image_of f` gives an initial object there\n-- (uniqueness of the lift comes for free).\n\ninstance lift_mono (F' : mono_factorisation f) : mono (image.lift F') :=\nby { apply mono_of_mono _ F'.m, simpa using mono_factorisation.m_mono _ }\n\nlemma has_image.uniq\n  (F' : mono_factorisation f) (l : image f ⟶ F'.I) (w : l ≫ F'.m = image.ι f) :\n  l = image.lift F' :=\n(cancel_mono F'.m).1 (by simp [w])\n\n/-- If `has_image g`, then `has_image (f ≫ g)` when `f` is an isomorphism. -/\ninstance {X Y Z : C} (f : X ⟶ Y) [is_iso f] (g : Y ⟶ Z) [has_image g] : has_image (f ≫ g) :=\n{ exists_image := ⟨{\n  F :=\n  { I := image g,\n    m := image.ι g,\n    e := f ≫ factor_thru_image g, },\n  is_image := { lift := λ F', image.lift { I := F'.I, m := F'.m, e := inv f ≫ F'.e, }, }, }⟩ }\n\nend\n\nsection\nvariables (C)\n\n/-- `has_images` represents a choice of image for every morphism -/\nclass has_images :=\n(has_image : Π {X Y : C} (f : X ⟶ Y), has_image f)\n\nattribute [instance, priority 100] has_images.has_image\nend\n\nsection\nvariables (f) [has_image f]\n/-- The image of a monomorphism is isomorphic to the source. -/\ndef image_mono_iso_source [mono f] : image f ≅ X :=\nis_image.iso_ext (image.is_image f) (is_image.self f)\n\n@[simp, reassoc]\nlemma image_mono_iso_source_inv_ι [mono f] : (image_mono_iso_source f).inv ≫ image.ι f = f :=\nby simp [image_mono_iso_source]\n@[simp, reassoc]\nlemma image_mono_iso_source_hom_self [mono f] : (image_mono_iso_source f).hom ≫ f = image.ι f :=\nbegin\n  conv { to_lhs, congr, skip, rw ←image_mono_iso_source_inv_ι f, },\n  rw [←category.assoc, iso.hom_inv_id, category.id_comp],\nend\n\n-- This is the proof that `factor_thru_image f` is an epimorphism\n-- from https://en.wikipedia.org/wiki/Image_%28category_theory%29, which is in turn taken from:\n-- Mitchell, Barry (1965), Theory of categories, MR 0202787, p.12, Proposition 10.1\n@[ext]\nlemma image.ext {W : C} {g h : image f ⟶ W} [has_limit (parallel_pair g h)]\n  (w : factor_thru_image f ≫ g = factor_thru_image f ≫ h) :\n  g = h :=\nbegin\n  let q := equalizer.ι g h,\n  let e' := equalizer.lift _ w,\n  let F' : mono_factorisation f :=\n  { I := equalizer g h,\n    m := q ≫ image.ι f,\n    m_mono := by apply mono_comp,\n    e := e' },\n  let v := image.lift F',\n  have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F',\n  have t : v ≫ q = 𝟙 (image f) :=\n    (cancel_mono_id (image.ι f)).1 (by { convert t₀ using 1, rw category.assoc }),\n  -- The proof from wikipedia next proves `q ≫ v = 𝟙 _`,\n  -- and concludes that `equalizer g h ≅ image f`,\n  -- but this isn't necessary.\n  calc g = 𝟙 (image f) ≫ g : by rw [category.id_comp]\n     ... = v ≫ q ≫ g       : by rw [←t, category.assoc]\n     ... = v ≫ q ≫ h       : by rw [equalizer.condition g h]\n     ... = 𝟙 (image f) ≫ h : by rw [←category.assoc, t]\n     ... = h                : by rw [category.id_comp]\nend\n\ninstance [Π {Z : C} (g h : image f ⟶ Z), has_limit (parallel_pair g h)] :\n  epi (factor_thru_image f) :=\n⟨λ Z g h w, image.ext f w⟩\n\nlemma epi_image_of_epi {X Y : C} (f : X ⟶ Y) [has_image f] [E : epi f] : epi (image.ι f) :=\nbegin\n  rw ←image.fac f at E,\n  resetI,\n  exact epi_of_epi (factor_thru_image f) (image.ι f),\nend\n\nlemma epi_of_epi_image {X Y : C} (f : X ⟶ Y) [has_image f]\n  [epi (image.ι f)] [epi (factor_thru_image f)] : epi f :=\nby { rw [←image.fac f], apply epi_comp, }\n\nend\n\nsection\nvariables {f} {f' : X ⟶ Y} [has_image f] [has_image f']\n\n/--\nAn equation between morphisms gives a comparison map between the images\n(which momentarily we prove is an iso).\n-/\ndef image.eq_to_hom (h : f = f') : image f ⟶ image f' :=\nimage.lift\n{ I := image f',\n  m := image.ι f',\n  e := factor_thru_image f', }.\n\ninstance (h : f = f') : is_iso (image.eq_to_hom h) :=\n⟨⟨image.eq_to_hom h.symm,\n  ⟨(cancel_mono (image.ι f)).1 (by simp [image.eq_to_hom]),\n   (cancel_mono (image.ι f')).1 (by simp [image.eq_to_hom])⟩⟩⟩\n\n/-- An equation between morphisms gives an isomorphism between the images. -/\ndef image.eq_to_iso (h : f = f') : image f ≅ image f' := as_iso (image.eq_to_hom h)\n\n/--\nAs long as the category has equalizers,\nthe image inclusion maps commute with `image.eq_to_iso`.\n-/\nlemma image.eq_fac [has_equalizers C] (h : f = f') :\n  image.ι f = (image.eq_to_iso h).hom ≫ image.ι f' :=\nby { ext, simp [image.eq_to_iso, image.eq_to_hom], }\n\nend\n\nsection\nvariables {Z : C} (g : Y ⟶ Z)\n\n/-- The comparison map `image (f ≫ g) ⟶ image g`. -/\ndef image.pre_comp [has_image g] [has_image (f ≫ g)] : image (f ≫ g) ⟶ image g :=\nimage.lift\n{ I := image g,\n  m := image.ι g,\n  e := f ≫ factor_thru_image g }\n\n@[simp, reassoc]\nlemma image.pre_comp_ι [has_image g] [has_image (f ≫ g)] :\n  image.pre_comp f g ≫ image.ι g = image.ι (f ≫ g) :=\nby simp [image.pre_comp]\n\n@[simp, reassoc]\nlemma image.factor_thru_image_pre_comp [has_image g] [has_image (f ≫ g)] :\n  factor_thru_image (f ≫ g) ≫ image.pre_comp f g = f ≫ factor_thru_image g :=\nby simp [image.pre_comp]\n\n/--\n`image.pre_comp f g` is a monomorphism.\n-/\ninstance image.pre_comp_mono [has_image g] [has_image (f ≫ g)] : mono (image.pre_comp f g) :=\nbegin\n  apply mono_of_mono _ (image.ι g),\n  simp only [image.pre_comp_ι],\n  apply_instance,\nend\n\n/--\nThe two step comparison map\n  `image (f ≫ (g ≫ h)) ⟶ image (g ≫ h) ⟶ image h`\nagrees with the one step comparison map\n  `image (f ≫ (g ≫ h)) ≅ image ((f ≫ g) ≫ h) ⟶ image h`.\n -/\nlemma image.pre_comp_comp {W : C} (h : Z ⟶ W)\n  [has_image (g ≫ h)] [has_image (f ≫ g ≫ h)]\n  [has_image h] [has_image ((f ≫ g) ≫ h)] :\n  image.pre_comp f (g ≫ h) ≫ image.pre_comp g h =\n    image.eq_to_hom (category.assoc f g h).symm ≫ (image.pre_comp (f ≫ g) h) :=\nbegin\n  apply (cancel_mono (image.ι h)).1,\n  simp [image.pre_comp, image.eq_to_hom],\nend\n\nvariables [has_equalizers C]\n\ninstance has_image_iso_comp [is_iso f] [has_image g] : has_image (f ≫ g) :=\nhas_image.mk\n{ F := (image.mono_factorisation g).iso_comp f,\n  is_image := { lift := λ F', image.lift (F'.of_iso_comp f) }, }\n\n/--\n`image.pre_comp f g` is an isomorphism when `f` is an isomorphism\n(we need `C` to have equalizers to prove this).\n-/\ninstance image.is_iso_precomp_iso (f : X ⟶ Y) [is_iso f] [has_image g] :\n  is_iso (image.pre_comp f g) :=\n⟨⟨image.lift\n  { I := image (f ≫ g),\n    m := image.ι (f ≫ g),\n    e := inv f ≫ factor_thru_image (f ≫ g) },\n  ⟨by { ext, simp [image.pre_comp], }, by { ext, simp [image.pre_comp], }⟩⟩⟩\n\n-- Note that in general we don't have the other comparison map you might expect\n-- `image f ⟶ image (f ≫ g)`.\n\ninstance has_image_comp_iso [has_image f] [is_iso g] : has_image (f ≫ g) :=\nhas_image.mk\n{ F := (image.mono_factorisation f).comp_mono g,\n  is_image := { lift := λ F', image.lift F'.of_comp_iso }, }\n\n/-- Postcomposing by an isomorphism induces an isomorphism on the image. -/\ndef image.comp_iso [has_image f] [is_iso g] :\n  image f ≅ image (f ≫ g) :=\n{ hom := image.lift (image.mono_factorisation (f ≫ g)).of_comp_iso,\n  inv := image.lift ((image.mono_factorisation f).comp_mono g) }\n\n@[simp, reassoc] lemma image.comp_iso_hom_comp_image_ι [has_image f] [is_iso g] :\n  (image.comp_iso f g).hom ≫ image.ι (f ≫ g) = image.ι f ≫ g :=\nby { ext, simp [image.comp_iso] }\n\n@[simp, reassoc] lemma image.comp_iso_inv_comp_image_ι [has_image f] [is_iso g] :\n  (image.comp_iso f g).inv ≫ image.ι f = image.ι (f ≫ g) ≫ inv g :=\nby { ext, simp [image.comp_iso] }\n\nend\n\nend category_theory.limits\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C]\n\nsection\n\ninstance {X Y : C} (f : X ⟶ Y) [has_image f] : has_image (arrow.mk f).hom :=\nshow has_image f, by apply_instance\n\nend\n\nsection has_image_map\n\n/-- An image map is a morphism `image f → image g` fitting into a commutative square and satisfying\n    the obvious commutativity conditions. -/\nstructure image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) :=\n(map : image f.hom ⟶ image g.hom)\n(map_ι' : map ≫ image.ι g.hom = image.ι f.hom ≫ sq.right . obviously)\n\ninstance inhabited_image_map {f : arrow C} [has_image f.hom] : inhabited (image_map (𝟙 f)) :=\n⟨⟨𝟙 _, by tidy⟩⟩\n\nrestate_axiom image_map.map_ι'\nattribute [simp, reassoc] image_map.map_ι\n\n@[simp, reassoc]\nlemma image_map.factor_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)\n  (m : image_map sq) :\n  factor_thru_image f.hom ≫ m.map = sq.left ≫ factor_thru_image g.hom :=\n(cancel_mono (image.ι g.hom)).1 $ by simp\n\n/-- To give an image map for a commutative square with `f` at the top and `g` at the bottom, it\n    suffices to give a map between any mono factorisation of `f` and any image factorisation of\n    `g`. -/\ndef image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)\n  (F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')\n  {map : F.I ⟶ F'.I} (map_ι : map ≫ F'.m = F.m ≫ sq.right) : image_map sq :=\n{ map := image.lift F ≫ map ≫ hF'.lift (image.mono_factorisation g.hom),\n  map_ι' := by simp [map_ι] }\n\n/-- `has_image_map sq` means that there is an `image_map` for the square `sq`. -/\nclass has_image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g) : Prop :=\nmk' :: (has_image_map : nonempty (image_map sq))\n\nlemma has_image_map.mk {f g : arrow C} [has_image f.hom] [has_image g.hom] {sq : f ⟶ g}\n  (m : image_map sq) : has_image_map sq :=\n⟨nonempty.intro m⟩\n\nlemma has_image_map.transport {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)\n  (F : mono_factorisation f.hom) {F' : mono_factorisation g.hom} (hF' : is_image F')\n  (map : F.I ⟶ F'.I) (map_ι : map ≫ F'.m = F.m ≫ sq.right) : has_image_map sq :=\nhas_image_map.mk $ image_map.transport sq F hF' map_ι\n\n/-- Obtain an `image_map` from a `has_image_map` instance. -/\ndef has_image_map.image_map {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)\n  [has_image_map sq] : image_map sq :=\nclassical.choice $ @has_image_map.has_image_map _ _ _ _ _ _ sq _\n\nvariables {f g : arrow C} [has_image f.hom] [has_image g.hom] (sq : f ⟶ g)\n\nsection\nlocal attribute [ext] image_map\n\ninstance : subsingleton (image_map sq) :=\nsubsingleton.intro $ λ a b, image_map.ext a b $ (cancel_mono (image.ι g.hom)).1 $\n  by simp only [image_map.map_ι]\n\nend\n\nvariable [has_image_map sq]\n\n/-- The map on images induced by a commutative square. -/\nabbreviation image.map : image f.hom ⟶ image g.hom :=\n(has_image_map.image_map sq).map\n\nlemma image.factor_map :\n  factor_thru_image f.hom ≫ image.map sq = sq.left ≫ factor_thru_image g.hom :=\nby simp\nlemma image.map_ι : image.map sq ≫ image.ι g.hom = image.ι f.hom ≫ sq.right :=\nby simp\nlemma image.map_hom_mk'_ι {X Y P Q : C} {k : X ⟶ Y} [has_image k] {l : P ⟶ Q} [has_image l]\n  {m : X ⟶ P} {n : Y ⟶ Q} (w : m ≫ l = k ≫ n) [has_image_map (arrow.hom_mk' w)] :\n  image.map (arrow.hom_mk' w) ≫ image.ι l = image.ι k ≫ n :=\nimage.map_ι _\n\nsection\nvariables {h : arrow C} [has_image h.hom] (sq' : g ⟶ h)\nvariables [has_image_map sq']\n\n/-- Image maps for composable commutative squares induce an image map in the composite square. -/\ndef image_map_comp : image_map (sq ≫ sq') :=\n{ map := image.map sq ≫ image.map sq' }\n\n@[simp]\nlemma image.map_comp [has_image_map (sq ≫ sq')] :\n  image.map (sq ≫ sq') = image.map sq ≫ image.map sq' :=\nshow (has_image_map.image_map (sq ≫ sq')).map = (image_map_comp sq sq').map, by congr\n\nend\n\nsection\nvariables (f)\n\n/-- The identity `image f ⟶ image f` fits into the commutative square represented by the identity\n    morphism `𝟙 f` in the arrow category. -/\ndef image_map_id : image_map (𝟙 f) :=\n{ map := 𝟙 (image f.hom) }\n\n@[simp]\nlemma image.map_id [has_image_map (𝟙 f)] : image.map (𝟙 f) = 𝟙 (image f.hom) :=\nshow (has_image_map.image_map (𝟙 f)).map = (image_map_id f).map, by congr\n\nend\n\nend has_image_map\n\nsection\nvariables (C) [has_images C]\n\n/-- If a category `has_image_maps`, then all commutative squares induce morphisms on images. -/\nclass has_image_maps :=\n(has_image_map : Π {f g : arrow C} (st : f ⟶ g), has_image_map st)\n\nattribute [instance, priority 100] has_image_maps.has_image_map\n\nend\n\nsection has_image_maps\nvariables [has_images C] [has_image_maps C]\n\n/-- The functor from the arrow category of `C` to `C` itself that maps a morphism to its image\n    and a commutative square to the induced morphism on images. -/\n@[simps]\ndef im : arrow C ⥤ C :=\n{ obj := λ f, image f.hom,\n  map := λ _ _ st, image.map st }\n\nend has_image_maps\n\nsection strong_epi_mono_factorisation\n\n/-- A strong epi-mono factorisation is a decomposition `f = e ≫ m` with `e` a strong epimorphism\n    and `m` a monomorphism. -/\nstructure strong_epi_mono_factorisation {X Y : C} (f : X ⟶ Y) extends mono_factorisation f :=\n[e_strong_epi : strong_epi e]\n\nattribute [instance] strong_epi_mono_factorisation.e_strong_epi\n\n/-- Satisfying the inhabited linter -/\ninstance strong_epi_mono_factorisation_inhabited {X Y : C} (f : X ⟶ Y) [strong_epi f] :\n  inhabited (strong_epi_mono_factorisation f) :=\n⟨⟨⟨Y, 𝟙 Y, f, by simp⟩⟩⟩\n\n/-- A mono factorisation coming from a strong epi-mono factorisation always has the universal\n    property of the image. -/\ndef strong_epi_mono_factorisation.to_mono_is_image {X Y : C} {f : X ⟶ Y}\n  (F : strong_epi_mono_factorisation f) : is_image F.to_mono_factorisation :=\n{ lift := λ G, arrow.lift $ arrow.hom_mk' $\n    show G.e ≫ G.m = F.e ≫ F.m, by rw [F.to_mono_factorisation.fac, G.fac] }\n\nvariable (C)\n\n/-- A category has strong epi-mono factorisations if every morphism admits a strong epi-mono\n    factorisation. -/\nclass has_strong_epi_mono_factorisations : Prop :=\nmk' :: (has_fac : Π {X Y : C} (f : X ⟶ Y), nonempty (strong_epi_mono_factorisation f))\n\nvariable {C}\n\nlemma has_strong_epi_mono_factorisations.mk\n  (d : Π {X Y : C} (f : X ⟶ Y), strong_epi_mono_factorisation f) :\n  has_strong_epi_mono_factorisations C :=\n⟨λ X Y f, nonempty.intro $ d f⟩\n\n@[priority 100]\ninstance has_images_of_has_strong_epi_mono_factorisations\n  [has_strong_epi_mono_factorisations C] : has_images C :=\n{ has_image := λ X Y f,\n  let F' := classical.choice (has_strong_epi_mono_factorisations.has_fac f) in\n  has_image.mk { F := F'.to_mono_factorisation,\n                 is_image := F'.to_mono_is_image } }\n\nend strong_epi_mono_factorisation\n\nsection has_strong_epi_images\nvariables (C) [has_images C]\n\n/-- A category has strong epi images if it has all images and `factor_thru_image f` is a strong\n    epimorphism for all `f`. -/\nclass has_strong_epi_images : Prop :=\n(strong_factor_thru_image : Π {X Y : C} (f : X ⟶ Y), strong_epi (factor_thru_image f))\n\nattribute [instance] has_strong_epi_images.strong_factor_thru_image\nend has_strong_epi_images\n\nsection has_strong_epi_images\n\n/-- If there is a single strong epi-mono factorisation of `f`, then every image factorisation is a\n    strong epi-mono factorisation. -/\nlemma strong_epi_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}\n  (F : strong_epi_mono_factorisation f) {F' : mono_factorisation f} (hF' : is_image F') :\n  strong_epi F'.e :=\nby { rw ←is_image.e_iso_ext_hom F.to_mono_is_image hF', apply strong_epi_comp }\n\nlemma strong_epi_factor_thru_image_of_strong_epi_mono_factorisation {X Y : C} {f : X ⟶ Y}\n  [has_image f] (F : strong_epi_mono_factorisation f) : strong_epi (factor_thru_image f) :=\nstrong_epi_of_strong_epi_mono_factorisation F $ image.is_image f\n\n/-- If we constructed our images from strong epi-mono factorisations, then these images are\n    strong epi images. -/\n@[priority 100]\ninstance has_strong_epi_images_of_has_strong_epi_mono_factorisations\n  [has_strong_epi_mono_factorisations C] : has_strong_epi_images C :=\n{ strong_factor_thru_image := λ X Y f,\n    strong_epi_factor_thru_image_of_strong_epi_mono_factorisation $\n      classical.choice $ has_strong_epi_mono_factorisations.has_fac f }\n\nend has_strong_epi_images\n\nsection has_strong_epi_images\nvariables [has_images C]\n\n/-- A category with strong epi images has image maps. -/\n@[priority 100]\ninstance has_image_maps_of_has_strong_epi_images [has_strong_epi_images C] :\n  has_image_maps C :=\n{ has_image_map := λ f g st, has_image_map.mk\n  { map := arrow.lift $ arrow.hom_mk' $ show (st.left ≫ factor_thru_image g.hom) ≫ image.ι g.hom =\n      factor_thru_image f.hom ≫ (image.ι f.hom ≫ st.right), by simp } }\n\n/-- If a category has images, equalizers and pullbacks, then images are automatically strong epi\n    images. -/\n@[priority 100]\ninstance has_strong_epi_images_of_has_pullbacks_of_has_equalizers [has_pullbacks C]\n  [has_equalizers C] : has_strong_epi_images C :=\n{ strong_factor_thru_image := λ X Y f,\n  { epi := by apply_instance,\n    has_lift := λ A B x y h h_mono w, arrow.has_lift.mk\n    { lift := image.lift\n      { I := pullback h y,\n        m := pullback.snd ≫ image.ι f,\n        m_mono := by exactI mono_comp _ _,\n        e := pullback.lift _ _ w } ≫ pullback.fst } } }\n\nend has_strong_epi_images\n\nvariables [has_strong_epi_mono_factorisations.{v} C]\nvariables {X Y : C} {f : X ⟶ Y}\n\n/--\nIf `C` has strong epi mono factorisations, then the image is unique up to isomorphism, in that if\n`f` factors as a strong epi followed by a mono, this factorisation is essentially the image\nfactorisation.\n-/\ndef image.iso_strong_epi_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f) [strong_epi e]\n  [mono m] :\n  I' ≅ image f :=\nis_image.iso_ext {strong_epi_mono_factorisation . I := I', m := m, e := e}.to_mono_is_image $\n  image.is_image f\n\n@[simp]\nlemma image.iso_strong_epi_mono_hom_comp_ι {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)\n  [strong_epi e] [mono m] :\n  (image.iso_strong_epi_mono e m comm).hom ≫ image.ι f = m :=\nis_image.lift_fac _ _\n\n@[simp]\nlemma image.iso_strong_epi_mono_inv_comp_mono {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)\n  [strong_epi e] [mono m] :\n  (image.iso_strong_epi_mono e m comm).inv ≫ m = image.ι f :=\nimage.lift_fac _\n\nend category_theory.limits\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/shapes/images.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.554470450236115, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3307045551450895}}
{"text": "/-\nCopyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Seul Baek\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.omega.nat.form\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-\nNegation elimination.\n-/\n\nnamespace omega\n\n\nnamespace nat\n\n\n/-- push_neg p returns the result of normalizing ¬ p by\n    pushing the outermost negation all the way down,\n    until it reaches either a negation or an atom -/\n@[simp] def push_neg : preform → preform :=\n  sorry\n\ntheorem push_neg_equiv {p : preform} : preform.equiv (push_neg p) (preform.not p) := sorry\n\n/-- NNF transformation -/\ndef nnf : preform → preform :=\n  sorry\n\n/-- Asserts that the given preform is in NNF -/\ndef is_nnf : preform → Prop :=\n  sorry\n\ntheorem is_nnf_push_neg (p : preform) : is_nnf p → is_nnf (push_neg p) := sorry\n\ntheorem is_nnf_nnf (p : preform) : is_nnf (nnf p) := sorry\n\ntheorem nnf_equiv {p : preform} : preform.equiv (nnf p) p := sorry\n\n@[simp] def neg_elim_core : preform → preform :=\n  sorry\n\ntheorem neg_free_neg_elim_core (p : preform) : is_nnf p → preform.neg_free (neg_elim_core p) := sorry\n\ntheorem le_and_le_iff_eq {α : Type} [partial_order α] {a : α} {b : α} : a ≤ b ∧ b ≤ a ↔ a = b := sorry\n\ntheorem implies_neg_elim_core {p : preform} : preform.implies p (neg_elim_core p) := sorry\n\n/-- Eliminate all negations in a preform -/\ndef neg_elim : preform → preform :=\n  neg_elim_core ∘ nnf\n\ntheorem neg_free_neg_elim {p : preform} : preform.neg_free (neg_elim p) :=\n  neg_free_neg_elim_core (nnf p) (is_nnf_nnf p)\n\ntheorem implies_neg_elim {p : preform} : preform.implies p (neg_elim p) :=\n  id fun (v : ℕ → ℕ) (h1 : preform.holds v p) => implies_neg_elim_core v (iff.elim_right (nnf_equiv v) h1)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/omega/nat/neg_elim.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.33065623871920036}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.kernels\nimport Mathlib.category_theory.concrete_category.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Facts about limits of functors into concrete categories\n\nThis file doesn't yet attempt to be exhaustive;\nit just contains lemmas that are useful\nwhile comparing categorical limits with existing constructions in concrete categories.\n-/\n\nnamespace category_theory.limits\n\n\n@[simp] theorem kernel_condition_apply {C : Type (u + 1)} [large_category C] [concrete_category C]\n    [has_zero_morphisms C] {X : C} {Y : C} (f : X ⟶ Y) [has_kernel f] (x : ↥(kernel f)) :\n    coe_fn f (coe_fn (kernel.ι f) x) = coe_fn 0 x :=\n  sorry\n\n@[simp] theorem cokernel_condition_apply {C : Type (u + 1)} [large_category C] [concrete_category C]\n    [has_zero_morphisms C] {X : C} {Y : C} (f : X ⟶ Y) [has_cokernel f] (x : ↥X) :\n    coe_fn (cokernel.π f) (coe_fn f x) = coe_fn 0 x :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/concrete_category_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3306562312783846}}
{"text": "import .formula .basic .dense_linear_order\n\nopen list --dnf\n\ndef get_lb : atom_dlo → option term_dlo\n| (_ =' _) := none \n| ((term_dlo.var 0) <' (term_dlo.var 0)) := none\n| ((term_dlo.var (n+1)) <' (term_dlo.var 0)) := some (term_dlo.var n)\n| ((term_dlo.cst b) <' (term_dlo.var 0)) := some (term_dlo.cst b)\n| (_ <' (term_dlo.var (n+1))) := none \n| (_ <' (term_dlo.cst _)) := none \n\nlemma of_get_lb_eq_some : \n  ∀ {a : atom_dlo} {t}, get_lb a = some t → a = (t.incr_idx <' V' 0) :=\nbegin\n  intros a t h, cases a with t1 t2 t1 t2;\n  cases t1 with k1 b1; cases t2 with k2 b2; try {cases h},\n  { cases k1; cases k2; try {cases h}, refl },\n  { cases k1; cases h },\n  { cases k2; cases h, refl }\nend\n\ndef lbs : list atom_dlo → list (term_dlo) := filter_map get_lb\n\ndef get_ub : atom_dlo → option (term_dlo)\n| (_ =' _) := none \n| ((term_dlo.var 0) <' (term_dlo.var 0)) := none\n| ((term_dlo.var 0) <' (term_dlo.var (n+1))) := some (term_dlo.var n)\n| ((term_dlo.var 0) <' (term_dlo.cst b)) := some (term_dlo.cst b)\n| ((term_dlo.var (n+1)) <' _) := none \n| ((term_dlo.cst _) <' _) := none \n\ndef ubs : list atom_dlo → list (term_dlo) := filter_map get_ub\n\nlemma of_get_ub_eq_some : \n  ∀ {a : atom_dlo} {t}, get_ub a = some t → a = (V' 0 <' t.incr_idx) := \nbegin\n  intros a t h, cases a with t1 t2 t1 t2;\n  cases t1 with k1 b1; cases t2 with k2 b2; try {cases h};\n  { cases k1; try {cases k2}; try {cases h}, refl },\nend\n\ndef is_false_gnd_lt_atom_dlo : atom_dlo → Prop\n| (term_dlo.cst a1 <' term_dlo.cst a2) := a2 ≤ a1\n| _ := false\n\nlemma not_eval_of_is_false_gnd_lt_atom_dlo : \n  ∀ {a : atom_dlo}, is_false_gnd_lt_atom_dlo a → ∀ {bs}, ¬ a.eval bs := \nbegin\n  intros a ha bs, cases a with t1 t2; \n  [{cases t1; cases t2}, {}]; try {cases ha},\n  simp [atom_dlo.eval, term_dlo.eval], apply ha\nend\n\ninstance dec_is_false_gnd_lt_atom_dlo : \n  decidable_pred is_false_gnd_lt_atom_dlo := \nbegin\n  intro a, cases a with t1 t2; [{cases t1; cases t2}, {}]; \n  try {simp [is_false_gnd_lt_atom_dlo]}; apply_instance,\nend\n\ndef is_not_gnd_lt_atom_dlo : atom_dlo → Prop\n| (term_dlo.cst a1 <' term_dlo.cst a2) := false\n| _ := true\n\nlemma eval_of_not_false_gnd_of_gnd : \n  ∀ {a : atom_dlo}, ¬ is_not_gnd_lt_atom_dlo a \n    → ¬ is_false_gnd_lt_atom_dlo a → ∀ {bs}, a.eval bs := \nbegin\n  intros a ha1 ha2 bs, cases a with t1 t2;\n  [{cases t1; cases t2}, {}];\n  try {simp [is_not_gnd_lt_atom_dlo] at ha1, cases ha1},\n  simp [is_false_gnd_lt_atom_dlo] at ha2, apply ha2\nend\n\ninstance dec_is_not_gnd_lt_atom_dlo : ∀ (a : atom_dlo), decidable (is_not_gnd_lt_atom_dlo a) :=\nbegin\n  intro a, cases a with t1 t2; \n  [ {cases t1; cases t2}, {} ]; simp [is_not_gnd_lt_atom_dlo]; \n  { apply decidable.true <|> apply decidable.false }\nend\n\ndef new_lt_atom_dlos (as : list atom_dlo) : list atom_dlo := \nlet prs := product (lbs as) (ubs as) in\nmap (λ (pr : term_dlo × term_dlo), pr.fst <' pr.snd) prs \n\ndef sqe_aux (as : list atom_dlo) : list atom_dlo := \nlet nas := new_lt_atom_dlos as in\nif (∃ a, a ∈ nas ∧ is_false_gnd_lt_atom_dlo a) \n  then [C' 0 <' C' 0]  \n  else (filter is_not_gnd_lt_atom_dlo nas)\n\ndef sqe (as : list atom_dlo) : list atom_dlo := \nif h : (V' 0 <' V' 0) ∈ as\n  then [C' 0 <' C' 0]\n  else sqe_aux as \n\nlemma forall_mem_new_lt_atom_dlos_eval {as : list atom_dlo} {b bs} :\n  (∀ t ∈ lbs as, term_dlo.eval bs t < b) → (∀ t ∈ ubs as, b < term_dlo.eval bs t)\n  → ∀ a ∈ new_lt_atom_dlos as, atom_dlo.eval bs a := \nbegin\n  intros hb1 hb2 a ha1, simp [new_lt_atom_dlos] at ha1,\n  cases ha1 with t1 ha1, cases ha1 with t2 ha1, \n  cases ha1 with ha1 ha3, subst ha3, simp [atom_dlo.eval],\n  apply @lt.trans _ _ _ b _ (hb1 _ ha1.left) (hb2 _ ha1.right) \nend\n\ndef is_lb : atom_dlo → Prop\n| (t1 <' t2) := ¬ t1 = term_dlo.var 0 ∧ t2 = term_dlo.var 0\n| (t1 =' t2) := false\n\ndef is_ub : atom_dlo → Prop\n| (t1 <' t2) := t1 = V' 0 ∧ ¬ t2 = V' 0\n| (t1 =' t2) := false\n\n\nlemma is_lb_or_is_ub {a : atom_dlo} : \n  dep_0 a → ¬solv_0 a → a ≠ (V' 0<' V' 0) → is_lb a ∨ is_ub a := \nbegin\n  intros h1 h2 h3, cases a with t1 t2 t1 t2,\n  { by_cases h1d : t1 = V' 0; by_cases h2d : t2 = V' 0,\n    { subst h1d, subst h2d, exfalso, apply h3, refl },\n    { apply or.inr, constructor; assumption },\n    { apply or.inl, constructor; assumption },\n    { cases h1; contradiction } },\n  { simp [solv_0, not_or_distrib] at h2, cases h2, \n    cases h1; contradiction }\nend\n\nlemma is_midpoint_iff  : \n∀ {as : list atom_dlo}, \n  (∀ a ∈ as, is_lb a ∨ is_ub a) → \n  ∀ {b bs}, ((∀ t ∈ lbs as, term_dlo.eval bs t < b) ∧ (∀ t ∈ ubs as, b < term_dlo.eval bs t))\n            ↔ ∀ a ∈ as, atom_dlo.eval (b :: bs) a :=\nbegin\n  intros as has b bs, constructor; intro h,\n  {\n    intros a ha, cases has _ ha with hbnd hbnd;\n    cases a with t1 t2 t1 t2; cases hbnd with h1 h2,\n    { subst h2, cases t1 with k r, \n      { cases k, exfalso, apply h1 rfl, \n        apply h.left (V' k) _, simp [lbs, mem_filter_map], \n        constructor, apply and.intro ha, refl },\n      { apply h.left (C' r) _, simp [lbs, mem_filter_map], \n        constructor, apply and.intro ha, refl } },\n    { subst h1, cases t2 with k r, \n      { cases k, exfalso, apply h2 rfl, \n        apply h.right (V' k) _, simp [ubs, mem_filter_map], \n        constructor, apply and.intro ha, refl },\n      { apply h.right (C' r) _, simp [ubs, mem_filter_map], \n        constructor, apply and.intro ha, refl } } },\n  {\n    constructor; intros t ht,\n    { simp [lbs] at ht, cases ht with a ha,\n      have hrw := (of_get_lb_eq_some ha.right), subst hrw,\n      rw (term_dlo.eval_incr_idx_eq.symm), apply h _ ha.left },\n    { simp [ubs] at ht, cases ht with a ha,\n      have hrw := (of_get_ub_eq_some ha.right), subst hrw,\n      rw (term_dlo.eval_incr_idx_eq.symm), apply h _ ha.left }\n  }\nend\n\nlemma eval_sqe :\n∀ {as : list atom_dlo}, \n (∀ a ∈ as, dep_0 a) → (∀ a ∈ as, ¬solv_0 a) \n  → ∀ {bs : list rat}, avals bs (sqe as) ↔ ∃ x, avals (x :: bs) as := \nbegin\n  intros as has1 has2 bs, simp [sqe], \n  apply @ite.rec _ _ _ \n  (λ φ, avals bs φ ↔ ∃ x, avals (x::bs) as); intro h1,\n  { constructor; intro h, \n    { exfalso, apply lt_irrefl _ (h _ (or.inl rfl)) },\n    { cases h with r hr, exfalso, apply lt_irrefl _ (hr _ h1) } },\n  { simp [sqe_aux],\n    apply @ite.rec _ _ _\n      (λ φ, avals bs φ ↔ ∃ x, avals (x :: bs) as); intro h2,\n    { \n      constructor; intro h; exfalso,\n      { apply lt_irrefl _ (h _ (or.inl rfl)) },\n      { \n        cases h2 with a ha, cases h with b hb,\n        apply @not_eval_of_is_false_gnd_lt_atom_dlo a ha.right bs, \n        apply @forall_mem_new_lt_atom_dlos_eval as b _ _ _ _ ha.left;\n        intros t ht; simp [lbs, ubs] at ht; cases ht with a' ha';\n        cases ha' with ha1' ha2';\n        [ { have hrw := (of_get_lb_eq_some ha2') },  \n          { have hrw := (of_get_ub_eq_some ha2') } ]; \n        { subst hrw, have h := hb _ ha1', \n          simp [atom_dlo.eval, term_dlo.eval_incr_idx_eq] at h, apply h }\n      } \n    },\n    { simp [avals], \n       -- have hrw := (@forall_mem_filter _ is_not_gnd_lt_atom_dlo \n       --   _ _ (new_lt_atom_dlos as)), rw hrw, \n      apply calc \n        (∀ a ∈ new_lt_atom_dlos as, is_not_gnd_lt_atom_dlo a → a.eval bs)  \n      ↔ ∃ b, (∀ t ∈ lbs as, term_dlo.eval bs t < b) ∧ (∀ t ∈ ubs as, b < term_dlo.eval bs t) : \n        begin\n          constructor; intro h3,\n          { \n            by_cases hl : lbs as = [], \n            { rw hl, simp, cases exists_lower_bound (map (term_dlo.eval bs) (ubs as)) \n              with b hb, existsi b, intros t ht, apply hb _ (mem_map_of_mem _ ht) },\n            {\n              by_cases hu : ubs as = [], \n              { rw hu, simp, cases exists_upper_bound (map (term_dlo.eval bs) (lbs as)) \n                with b hb, existsi b, intros t ht, apply hb _ (mem_map_of_mem _ ht) },\n              { \n                cases @exists_maximum _ _ (map (term_dlo.eval bs) (lbs as)) _ with x hx;\n                [{}, {apply map_ne_nil_of_ne_nil hl}], cases hx with hx1 hx2,\n                cases @exists_minimum _ _ (map (term_dlo.eval bs) (ubs as)) _ with y hy;\n                [{}, {apply map_ne_nil_of_ne_nil hu}], cases hy with hy1 hy2,\n                have hlt : x < y, \n                  { \n                    rw mem_map at hx1, cases hx1 with tx htx,\n                    cases htx with htx hrw, subst hrw,\n                    rw mem_map at hy1, cases hy1 with ty hty,\n                    cases hty with hty hrw, subst hrw,\n                    have hmem : (tx <' ty) ∈ new_lt_atom_dlos as, \n                      { simp [new_lt_atom_dlos], existsi tx, existsi ty,\n                        constructor; constructor; {assumption <|> refl} },\n                    by_cases hgnd : is_not_gnd_lt_atom_dlo (tx <' ty),\n                    { apply h3 _ hmem hgnd },\n                    { apply eval_of_not_false_gnd_of_gnd hgnd _, intro hc, \n                      apply h2, constructor, apply and.intro hmem hc }\n                  },\n                cases dense hlt with b hb, existsi b,\n                constructor; intros t ht,\n                { apply lt_of_le_of_lt (hx2 _ _) hb.left, \n                  apply mem_map_of_mem _ ht },\n                { apply lt_of_lt_of_le hb.right (hy2 _ _), \n                  apply mem_map_of_mem _ ht }\n              }\n            }\n          },\n          { cases h3 with b hb, intros a ha1 _,\n            apply forall_mem_new_lt_atom_dlos_eval hb.left hb.right _ ha1 }\n        end\n  ... ↔ ∃ b, ∀ a ∈ as, atom_dlo.eval (b::bs) a : \n        begin\n          apply exists_iff_exists, intro b, apply is_midpoint_iff, \n          intros a ha, apply is_lb_or_is_ub (has1 _ ha) (has2 _ ha), \n          intro hc, subst hc, apply h1 ha,\n        end\n    }\n  }\n\n\nend\n\n#exit\n\n\ninstance dlo_rat.has_sqe_eq : has_sqe_eq rat :=\n{\n  sqe := dlo_rat.sqe,\n  forall_mem_sqe_normal := λ _ _ _ _, trivial,\n  eval_sqe_iff := λ as _ h _, @dlo_rat.eval_sqe_iff as h,\n}\n\ninstance dlo_rat.dec_eval_qe := @dnf.dec_eval_qe rat _ @dlo_rat.dec_aval\n\nmeta def dlo_rat.qelim : tactic unit :=\npapply ``((@dnf.eval_qe_iff rat dlo_rat.has_sqe_eq _ _ []).elim_left _)", "meta": {"author": "skbaek", "repo": "cooper", "sha": "812afc6b158821f2e7dac9c91d3b6123c7a19faf", "save_path": "github-repos/lean/skbaek-cooper", "path": "github-repos/lean/skbaek-cooper/cooper-812afc6b158821f2e7dac9c91d3b6123c7a19faf/dlo/sqe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.682573734412324, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3306251229998975}}
{"text": "/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n\n! This file was ported from Lean 3 source module field_theory.intermediate_field\n! leanprover-community/mathlib commit 825edd3cd735e87495b0c2a2114fc3929eefce41\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.FieldTheory.Minpoly.Field\nimport Mathbin.FieldTheory.Subfield\nimport Mathbin.FieldTheory.Tower\n\n/-!\n# Intermediate fields\n\nLet `L / K` be a field extension, given as an instance `algebra K L`.\nThis file defines the type of fields in between `K` and `L`, `intermediate_field K L`.\nAn `intermediate_field K L` is a subfield of `L` which contains (the image of) `K`,\ni.e. it is a `subfield L` and a `subalgebra K L`.\n\n## Main definitions\n\n* `intermediate_field K L` : the type of intermediate fields between `K` and `L`.\n* `subalgebra.to_intermediate_field`: turns a subalgebra closed under `⁻¹`\n  into an intermediate field\n* `subfield.to_intermediate_field`: turns a subfield containing the image of `K`\n  into an intermediate field\n* `intermediate_field.map`: map an intermediate field along an `alg_hom`\n* `intermediate_field.restrict_scalars`: restrict the scalars of an intermediate field to a smaller\n  field in a tower of fields.\n\n## Implementation notes\n\nIntermediate fields are defined with a structure extending `subfield` and `subalgebra`.\nA `subalgebra` is closed under all operations except `⁻¹`,\n\n## Tags\nintermediate field, field extension\n-/\n\n\nopen FiniteDimensional Polynomial\n\nopen BigOperators Polynomial\n\nvariable (K L L' : Type _) [Field K] [Field L] [Field L'] [Algebra K L] [Algebra K L']\n\n/-- `S : intermediate_field K L` is a subset of `L` such that there is a field\ntower `L / S / K`. -/\nstructure IntermediateField extends Subalgebra K L where\n  neg_mem' : ∀ x ∈ carrier, -x ∈ carrier\n  inv_mem' : ∀ x ∈ carrier, x⁻¹ ∈ carrier\n#align intermediate_field IntermediateField\n\n/-- Reinterpret an `intermediate_field` as a `subalgebra`. -/\nadd_decl_doc IntermediateField.toSubalgebra\n\nvariable {K L L'} (S : IntermediateField K L)\n\nnamespace IntermediateField\n\n/-- Reinterpret an `intermediate_field` as a `subfield`. -/\ndef toSubfield : Subfield L :=\n  { S.toSubalgebra, S with }\n#align intermediate_field.to_subfield IntermediateField.toSubfield\n\ninstance : SetLike (IntermediateField K L) L :=\n  ⟨fun S => S.toSubalgebra.carrier, by\n    rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ ⟨h⟩\n    congr ⟩\n\ninstance : SubfieldClass (IntermediateField K L) L\n    where\n  add_mem s _ _ := s.add_mem'\n  zero_mem s := s.zero_mem'\n  neg_mem := neg_mem'\n  mul_mem s _ _ := s.mul_mem'\n  one_mem s := s.one_mem'\n  inv_mem := inv_mem'\n\n@[simp]\ntheorem mem_carrier {s : IntermediateField K L} {x : L} : x ∈ s.carrier ↔ x ∈ s :=\n  Iff.rfl\n#align intermediate_field.mem_carrier IntermediateField.mem_carrier\n\n/-- Two intermediate fields are equal if they have the same elements. -/\n@[ext]\ntheorem ext {S T : IntermediateField K L} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T :=\n  SetLike.ext h\n#align intermediate_field.ext IntermediateField.ext\n\n@[simp]\ntheorem coe_toSubalgebra : (S.toSubalgebra : Set L) = S :=\n  rfl\n#align intermediate_field.coe_to_subalgebra IntermediateField.coe_toSubalgebra\n\n@[simp]\ntheorem coe_toSubfield : (S.toSubfield : Set L) = S :=\n  rfl\n#align intermediate_field.coe_to_subfield IntermediateField.coe_toSubfield\n\n@[simp]\ntheorem mem_mk (s : Set L) (hK : ∀ x, algebraMap K L x ∈ s) (ho hm hz ha hn hi) (x : L) :\n    x ∈ IntermediateField.mk (Subalgebra.mk s ho hm hz ha hK) hn hi ↔ x ∈ s :=\n  Iff.rfl\n#align intermediate_field.mem_mk IntermediateField.mem_mk\n\n@[simp]\ntheorem mem_toSubalgebra (s : IntermediateField K L) (x : L) : x ∈ s.toSubalgebra ↔ x ∈ s :=\n  Iff.rfl\n#align intermediate_field.mem_to_subalgebra IntermediateField.mem_toSubalgebra\n\n@[simp]\ntheorem mem_toSubfield (s : IntermediateField K L) (x : L) : x ∈ s.toSubfield ↔ x ∈ s :=\n  Iff.rfl\n#align intermediate_field.mem_to_subfield IntermediateField.mem_toSubfield\n\n/-- Copy of an intermediate field with a new `carrier` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy (S : IntermediateField K L) (s : Set L) (hs : s = ↑S) : IntermediateField K L\n    where\n  toSubalgebra := S.toSubalgebra.copy s (hs : s = S.toSubalgebra.carrier)\n  neg_mem' :=\n    have hs' : (S.toSubalgebra.copy s hs).carrier = S.toSubalgebra.carrier := hs\n    hs'.symm ▸ S.neg_mem'\n  inv_mem' :=\n    have hs' : (S.toSubalgebra.copy s hs).carrier = S.toSubalgebra.carrier := hs\n    hs'.symm ▸ S.inv_mem'\n#align intermediate_field.copy IntermediateField.copy\n\n@[simp]\ntheorem coe_copy (S : IntermediateField K L) (s : Set L) (hs : s = ↑S) :\n    (S.copy s hs : Set L) = s :=\n  rfl\n#align intermediate_field.coe_copy IntermediateField.coe_copy\n\ntheorem copy_eq (S : IntermediateField K L) (s : Set L) (hs : s = ↑S) : S.copy s hs = S :=\n  SetLike.coe_injective hs\n#align intermediate_field.copy_eq IntermediateField.copy_eq\n\nsection InheritedLemmas\n\n/-! ### Lemmas inherited from more general structures\n\nThe declarations in this section derive from the fact that an `intermediate_field` is also a\nsubalgebra or subfield. Their use should be replaceable with the corresponding lemma from a\nsubobject class.\n-/\n\n\n/-- An intermediate field contains the image of the smaller field. -/\ntheorem algebraMap_mem (x : K) : algebraMap K L x ∈ S :=\n  S.algebraMap_mem' x\n#align intermediate_field.algebra_map_mem IntermediateField.algebraMap_mem\n\n/-- An intermediate field is closed under scalar multiplication. -/\ntheorem smul_mem {y : L} : y ∈ S → ∀ {x : K}, x • y ∈ S :=\n  S.toSubalgebra.smul_mem\n#align intermediate_field.smul_mem IntermediateField.smul_mem\n\n/-- An intermediate field contains the ring's 1. -/\nprotected theorem one_mem : (1 : L) ∈ S :=\n  one_mem S\n#align intermediate_field.one_mem IntermediateField.one_mem\n\n/-- An intermediate field contains the ring's 0. -/\nprotected theorem zero_mem : (0 : L) ∈ S :=\n  zero_mem S\n#align intermediate_field.zero_mem IntermediateField.zero_mem\n\n/-- An intermediate field is closed under multiplication. -/\nprotected theorem mul_mem {x y : L} : x ∈ S → y ∈ S → x * y ∈ S :=\n  mul_mem\n#align intermediate_field.mul_mem IntermediateField.mul_mem\n\n/-- An intermediate field is closed under addition. -/\nprotected theorem add_mem {x y : L} : x ∈ S → y ∈ S → x + y ∈ S :=\n  add_mem\n#align intermediate_field.add_mem IntermediateField.add_mem\n\n/-- An intermediate field is closed under subtraction -/\nprotected theorem sub_mem {x y : L} : x ∈ S → y ∈ S → x - y ∈ S :=\n  sub_mem\n#align intermediate_field.sub_mem IntermediateField.sub_mem\n\n/-- An intermediate field is closed under negation. -/\nprotected theorem neg_mem {x : L} : x ∈ S → -x ∈ S :=\n  neg_mem\n#align intermediate_field.neg_mem IntermediateField.neg_mem\n\n/-- An intermediate field is closed under inverses. -/\nprotected theorem inv_mem {x : L} : x ∈ S → x⁻¹ ∈ S :=\n  inv_mem\n#align intermediate_field.inv_mem IntermediateField.inv_mem\n\n/-- An intermediate field is closed under division. -/\nprotected theorem div_mem {x y : L} : x ∈ S → y ∈ S → x / y ∈ S :=\n  div_mem\n#align intermediate_field.div_mem IntermediateField.div_mem\n\n/-- Product of a list of elements in an intermediate_field is in the intermediate_field. -/\nprotected theorem list_prod_mem {l : List L} : (∀ x ∈ l, x ∈ S) → l.Prod ∈ S :=\n  list_prod_mem\n#align intermediate_field.list_prod_mem IntermediateField.list_prod_mem\n\n/-- Sum of a list of elements in an intermediate field is in the intermediate_field. -/\nprotected theorem list_sum_mem {l : List L} : (∀ x ∈ l, x ∈ S) → l.Sum ∈ S :=\n  list_sum_mem\n#align intermediate_field.list_sum_mem IntermediateField.list_sum_mem\n\n/-- Product of a multiset of elements in an intermediate field is in the intermediate_field. -/\nprotected theorem multiset_prod_mem (m : Multiset L) : (∀ a ∈ m, a ∈ S) → m.Prod ∈ S :=\n  multiset_prod_mem m\n#align intermediate_field.multiset_prod_mem IntermediateField.multiset_prod_mem\n\n/-- Sum of a multiset of elements in a `intermediate_field` is in the `intermediate_field`. -/\nprotected theorem multiset_sum_mem (m : Multiset L) : (∀ a ∈ m, a ∈ S) → m.Sum ∈ S :=\n  multiset_sum_mem m\n#align intermediate_field.multiset_sum_mem IntermediateField.multiset_sum_mem\n\n/-- Product of elements of an intermediate field indexed by a `finset` is in the intermediate_field.\n-/\nprotected theorem prod_mem {ι : Type _} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) :\n    (∏ i in t, f i) ∈ S :=\n  prod_mem h\n#align intermediate_field.prod_mem IntermediateField.prod_mem\n\n/-- Sum of elements in a `intermediate_field` indexed by a `finset` is in the `intermediate_field`.\n-/\nprotected theorem sum_mem {ι : Type _} {t : Finset ι} {f : ι → L} (h : ∀ c ∈ t, f c ∈ S) :\n    (∑ i in t, f i) ∈ S :=\n  sum_mem h\n#align intermediate_field.sum_mem IntermediateField.sum_mem\n\nprotected theorem pow_mem {x : L} (hx : x ∈ S) (n : ℤ) : x ^ n ∈ S :=\n  zpow_mem hx n\n#align intermediate_field.pow_mem IntermediateField.pow_mem\n\nprotected theorem zsmul_mem {x : L} (hx : x ∈ S) (n : ℤ) : n • x ∈ S :=\n  zsmul_mem hx n\n#align intermediate_field.zsmul_mem IntermediateField.zsmul_mem\n\nprotected theorem coe_int_mem (n : ℤ) : (n : L) ∈ S :=\n  coe_int_mem S n\n#align intermediate_field.coe_int_mem IntermediateField.coe_int_mem\n\nprotected theorem coe_add (x y : S) : (↑(x + y) : L) = ↑x + ↑y :=\n  rfl\n#align intermediate_field.coe_add IntermediateField.coe_add\n\nprotected theorem coe_neg (x : S) : (↑(-x) : L) = -↑x :=\n  rfl\n#align intermediate_field.coe_neg IntermediateField.coe_neg\n\nprotected theorem coe_mul (x y : S) : (↑(x * y) : L) = ↑x * ↑y :=\n  rfl\n#align intermediate_field.coe_mul IntermediateField.coe_mul\n\nprotected theorem coe_inv (x : S) : (↑x⁻¹ : L) = (↑x)⁻¹ :=\n  rfl\n#align intermediate_field.coe_inv IntermediateField.coe_inv\n\nprotected theorem coe_zero : ((0 : S) : L) = 0 :=\n  rfl\n#align intermediate_field.coe_zero IntermediateField.coe_zero\n\nprotected theorem coe_one : ((1 : S) : L) = 1 :=\n  rfl\n#align intermediate_field.coe_one IntermediateField.coe_one\n\nprotected theorem coe_pow (x : S) (n : ℕ) : (↑(x ^ n) : L) = ↑x ^ n :=\n  SubmonoidClass.coe_pow x n\n#align intermediate_field.coe_pow IntermediateField.coe_pow\n\nend InheritedLemmas\n\ntheorem coe_nat_mem (n : ℕ) : (n : L) ∈ S := by simpa using coe_int_mem S n\n#align intermediate_field.coe_nat_mem IntermediateField.coe_nat_mem\n\nend IntermediateField\n\n/-- Turn a subalgebra closed under inverses into an intermediate field -/\ndef Subalgebra.toIntermediateField (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) :\n    IntermediateField K L :=\n  { S with\n    neg_mem' := fun x => S.neg_mem\n    inv_mem' := inv_mem }\n#align subalgebra.to_intermediate_field Subalgebra.toIntermediateField\n\n@[simp]\ntheorem toSubalgebra_toIntermediateField (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) :\n    (S.toIntermediateField inv_mem).toSubalgebra = S :=\n  by\n  ext\n  rfl\n#align to_subalgebra_to_intermediate_field toSubalgebra_toIntermediateField\n\n@[simp]\ntheorem toIntermediateField_toSubalgebra (S : IntermediateField K L) :\n    (S.toSubalgebra.toIntermediateField fun x => S.inv_mem) = S :=\n  by\n  ext\n  rfl\n#align to_intermediate_field_to_subalgebra toIntermediateField_toSubalgebra\n\n/-- Turn a subalgebra satisfying `is_field` into an intermediate_field -/\ndef Subalgebra.toIntermediateField' (S : Subalgebra K L) (hS : IsField S) : IntermediateField K L :=\n  S.toIntermediateField fun x hx => by\n    by_cases hx0 : x = 0\n    · rw [hx0, inv_zero]\n      exact S.zero_mem\n    letI hS' := hS.to_field\n    obtain ⟨y, hy⟩ := hS.mul_inv_cancel (show (⟨x, hx⟩ : S) ≠ 0 from Subtype.ne_of_val_ne hx0)\n    rw [Subtype.ext_iff, S.coe_mul, S.coe_one, Subtype.coe_mk, mul_eq_one_iff_inv_eq₀ hx0] at hy\n    exact hy.symm ▸ y.2\n#align subalgebra.to_intermediate_field' Subalgebra.toIntermediateField'\n\n@[simp]\ntheorem toSubalgebra_toIntermediateField' (S : Subalgebra K L) (hS : IsField S) :\n    (S.toIntermediateField' hS).toSubalgebra = S :=\n  by\n  ext\n  rfl\n#align to_subalgebra_to_intermediate_field' toSubalgebra_toIntermediateField'\n\n@[simp]\ntheorem toIntermediateField'_toSubalgebra (S : IntermediateField K L) :\n    S.toSubalgebra.toIntermediateField' (Field.toIsField S) = S :=\n  by\n  ext\n  rfl\n#align to_intermediate_field'_to_subalgebra toIntermediateField'_toSubalgebra\n\n/-- Turn a subfield of `L` containing the image of `K` into an intermediate field -/\ndef Subfield.toIntermediateField (S : Subfield L) (algebra_map_mem : ∀ x, algebraMap K L x ∈ S) :\n    IntermediateField K L :=\n  { S with algebraMap_mem' := algebra_map_mem }\n#align subfield.to_intermediate_field Subfield.toIntermediateField\n\nnamespace IntermediateField\n\n/-- An intermediate field inherits a field structure -/\ninstance toField : Field S :=\n  S.toSubfield.toField\n#align intermediate_field.to_field IntermediateField.toField\n\n@[simp, norm_cast]\ntheorem coe_sum {ι : Type _} [Fintype ι] (f : ι → S) : (↑(∑ i, f i) : L) = ∑ i, (f i : L) := by\n  classical\n    induction' Finset.univ using Finset.induction_on with i s hi H\n    · simp\n    · rw [Finset.sum_insert hi, AddMemClass.coe_add, H, Finset.sum_insert hi]\n#align intermediate_field.coe_sum IntermediateField.coe_sum\n\n@[simp, norm_cast]\ntheorem coe_prod {ι : Type _} [Fintype ι] (f : ι → S) : (↑(∏ i, f i) : L) = ∏ i, (f i : L) := by\n  classical\n    induction' Finset.univ using Finset.induction_on with i s hi H\n    · simp\n    · rw [Finset.prod_insert hi, MulMemClass.coe_mul, H, Finset.prod_insert hi]\n#align intermediate_field.coe_prod IntermediateField.coe_prod\n\n/-! `intermediate_field`s inherit structure from their `subalgebra` coercions. -/\n\n\ninstance module' {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] : Module R S :=\n  S.toSubalgebra.module'\n#align intermediate_field.module' IntermediateField.module'\n\ninstance module : Module K S :=\n  S.toSubalgebra.Module\n#align intermediate_field.module IntermediateField.module\n\ninstance isScalarTower {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] :\n    IsScalarTower R K S :=\n  S.toSubalgebra.IsScalarTower\n#align intermediate_field.is_scalar_tower IntermediateField.isScalarTower\n\n@[simp]\ntheorem coe_smul {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] (r : R) (x : S) :\n    ↑(r • x) = (r • x : L) :=\n  rfl\n#align intermediate_field.coe_smul IntermediateField.coe_smul\n\ninstance algebra' {K'} [CommSemiring K'] [SMul K' K] [Algebra K' L] [IsScalarTower K' K L] :\n    Algebra K' S :=\n  S.toSubalgebra.algebra'\n#align intermediate_field.algebra' IntermediateField.algebra'\n\ninstance algebra : Algebra K S :=\n  S.toSubalgebra.Algebra\n#align intermediate_field.algebra IntermediateField.algebra\n\ninstance toAlgebra {R : Type _} [Semiring R] [Algebra L R] : Algebra S R :=\n  S.toSubalgebra.toAlgebra\n#align intermediate_field.to_algebra IntermediateField.toAlgebra\n\ninstance isScalarTower_bot {R : Type _} [Semiring R] [Algebra L R] : IsScalarTower S L R :=\n  IsScalarTower.subalgebra _ _ _ S.toSubalgebra\n#align intermediate_field.is_scalar_tower_bot IntermediateField.isScalarTower_bot\n\ninstance isScalarTower_mid {R : Type _} [Semiring R] [Algebra L R] [Algebra K R]\n    [IsScalarTower K L R] : IsScalarTower K S R :=\n  IsScalarTower.subalgebra' _ _ _ S.toSubalgebra\n#align intermediate_field.is_scalar_tower_mid IntermediateField.isScalarTower_mid\n\n/-- Specialize `is_scalar_tower_mid` to the common case where the top field is `L` -/\ninstance isScalarTower_mid' : IsScalarTower K S L :=\n  S.isScalarTower_mid\n#align intermediate_field.is_scalar_tower_mid' IntermediateField.isScalarTower_mid'\n\n/-- If `f : L →+* L'` fixes `K`, `S.map f` is the intermediate field between `L'` and `K`\nsuch that `x ∈ S ↔ f x ∈ S.map f`. -/\ndef map (f : L →ₐ[K] L') (S : IntermediateField K L) : IntermediateField K L' :=\n  {\n    S.toSubalgebra.map\n      f with\n    inv_mem' := by\n      rintro _ ⟨x, hx, rfl⟩\n      exact ⟨x⁻¹, S.inv_mem hx, map_inv₀ f x⟩\n    neg_mem' := fun x hx => (S.toSubalgebra.map f).neg_mem hx }\n#align intermediate_field.map IntermediateField.map\n\n@[simp]\ntheorem coe_map (f : L →ₐ[K] L') : (S.map f : Set L') = f '' S :=\n  rfl\n#align intermediate_field.coe_map IntermediateField.coe_map\n\ntheorem map_map {K L₁ L₂ L₃ : Type _} [Field K] [Field L₁] [Algebra K L₁] [Field L₂] [Algebra K L₂]\n    [Field L₃] [Algebra K L₃] (E : IntermediateField K L₁) (f : L₁ →ₐ[K] L₂) (g : L₂ →ₐ[K] L₃) :\n    (E.map f).map g = E.map (g.comp f) :=\n  SetLike.coe_injective <| Set.image_image _ _ _\n#align intermediate_field.map_map IntermediateField.map_map\n\n/-- Given an equivalence `e : L ≃ₐ[K] L'` of `K`-field extensions and an intermediate\nfield `E` of `L/K`, `intermediate_field_equiv_map e E` is the induced equivalence\nbetween `E` and `E.map e` -/\ndef intermediateFieldMap (e : L ≃ₐ[K] L') (E : IntermediateField K L) : E ≃ₐ[K] E.map e.toAlgHom :=\n  e.subalgebraMap E.toSubalgebra\n#align intermediate_field.intermediate_field_map IntermediateField.intermediateFieldMap\n\n/- We manually add these two simp lemmas because `@[simps]` before `intermediate_field_map`\n  led to a timeout. -/\n@[simp]\ntheorem intermediateFieldMap_apply_coe (e : L ≃ₐ[K] L') (E : IntermediateField K L) (a : E) :\n    ↑(intermediateFieldMap e E a) = e a :=\n  rfl\n#align intermediate_field.intermediate_field_map_apply_coe IntermediateField.intermediateFieldMap_apply_coe\n\n@[simp]\ntheorem intermediateFieldMap_symm_apply_coe (e : L ≃ₐ[K] L') (E : IntermediateField K L)\n    (a : E.map e.toAlgHom) : ↑((intermediateFieldMap e E).symm a) = e.symm a :=\n  rfl\n#align intermediate_field.intermediate_field_map_symm_apply_coe IntermediateField.intermediateFieldMap_symm_apply_coe\n\nend IntermediateField\n\nnamespace AlgHom\n\nvariable (f : L →ₐ[K] L')\n\n/-- The range of an algebra homomorphism, as an intermediate field. -/\n@[simps toSubalgebra]\ndef fieldRange : IntermediateField K L' :=\n  { f.range, (f : L →+* L').fieldRange with }\n#align alg_hom.field_range AlgHom.fieldRange\n\n@[simp]\ntheorem coe_fieldRange : ↑f.fieldRange = Set.range f :=\n  rfl\n#align alg_hom.coe_field_range AlgHom.coe_fieldRange\n\n@[simp]\ntheorem fieldRange_toSubfield : f.fieldRange.toSubfield = (f : L →+* L').fieldRange :=\n  rfl\n#align alg_hom.field_range_to_subfield AlgHom.fieldRange_toSubfield\n\nvariable {f}\n\n@[simp]\ntheorem mem_fieldRange {y : L'} : y ∈ f.fieldRange ↔ ∃ x, f x = y :=\n  Iff.rfl\n#align alg_hom.mem_field_range AlgHom.mem_fieldRange\n\nend AlgHom\n\nnamespace IntermediateField\n\n/-- The embedding from an intermediate field of `L / K` to `L`. -/\ndef val : S →ₐ[K] L :=\n  S.toSubalgebra.val\n#align intermediate_field.val IntermediateField.val\n\n@[simp]\ntheorem coe_val : ⇑S.val = coe :=\n  rfl\n#align intermediate_field.coe_val IntermediateField.coe_val\n\n@[simp]\ntheorem val_mk {x : L} (hx : x ∈ S) : S.val ⟨x, hx⟩ = x :=\n  rfl\n#align intermediate_field.val_mk IntermediateField.val_mk\n\ntheorem range_val : S.val.range = S.toSubalgebra :=\n  S.toSubalgebra.range_val\n#align intermediate_field.range_val IntermediateField.range_val\n\ntheorem aeval_coe {R : Type _} [CommRing R] [Algebra R K] [Algebra R L] [IsScalarTower R K L]\n    (x : S) (P : R[X]) : aeval (x : L) P = aeval x P :=\n  by\n  refine' Polynomial.induction_on' P (fun f g hf hg => _) fun n r => _\n  · rw [aeval_add, aeval_add, AddMemClass.coe_add, hf, hg]\n  · simp only [MulMemClass.coe_mul, aeval_monomial, SubmonoidClass.coe_pow, mul_eq_mul_right_iff]\n    left\n    rfl\n#align intermediate_field.aeval_coe IntermediateField.aeval_coe\n\ntheorem coe_isIntegral_iff {R : Type _} [CommRing R] [Algebra R K] [Algebra R L]\n    [IsScalarTower R K L] {x : S} : IsIntegral R (x : L) ↔ IsIntegral R x :=\n  by\n  refine' ⟨fun h => _, fun h => _⟩\n  · obtain ⟨P, hPmo, hProot⟩ := h\n    refine' ⟨P, hPmo, (injective_iff_map_eq_zero _).1 (algebraMap (↥S) L).Injective _ _⟩\n    letI : IsScalarTower R S L := IsScalarTower.of_algebraMap_eq (congr_fun rfl)\n    rwa [eval₂_eq_eval_map, ← eval₂_at_apply, eval₂_eq_eval_map, Polynomial.map_map, ←\n      IsScalarTower.algebraMap_eq, ← eval₂_eq_eval_map]\n  · obtain ⟨P, hPmo, hProot⟩ := h\n    refine' ⟨P, hPmo, _⟩\n    rw [← aeval_def, aeval_coe, aeval_def, hProot, ZeroMemClass.coe_zero]\n#align intermediate_field.coe_is_integral_iff IntermediateField.coe_isIntegral_iff\n\n/-- The map `E → F` when `E` is an intermediate field contained in the intermediate field `F`.\n\nThis is the intermediate field version of `subalgebra.inclusion`. -/\ndef inclusion {E F : IntermediateField K L} (hEF : E ≤ F) : E →ₐ[K] F :=\n  Subalgebra.inclusion hEF\n#align intermediate_field.inclusion IntermediateField.inclusion\n\ntheorem inclusion_injective {E F : IntermediateField K L} (hEF : E ≤ F) :\n    Function.Injective (inclusion hEF) :=\n  Subalgebra.inclusion_injective hEF\n#align intermediate_field.inclusion_injective IntermediateField.inclusion_injective\n\n@[simp]\ntheorem inclusion_self {E : IntermediateField K L} : inclusion (le_refl E) = AlgHom.id K E :=\n  Subalgebra.inclusion_self\n#align intermediate_field.inclusion_self IntermediateField.inclusion_self\n\n@[simp]\ntheorem inclusion_inclusion {E F G : IntermediateField K L} (hEF : E ≤ F) (hFG : F ≤ G) (x : E) :\n    inclusion hFG (inclusion hEF x) = inclusion (le_trans hEF hFG) x :=\n  Subalgebra.inclusion_inclusion hEF hFG x\n#align intermediate_field.inclusion_inclusion IntermediateField.inclusion_inclusion\n\n@[simp]\ntheorem coe_inclusion {E F : IntermediateField K L} (hEF : E ≤ F) (e : E) :\n    (inclusion hEF e : L) = e :=\n  rfl\n#align intermediate_field.coe_inclusion IntermediateField.coe_inclusion\n\nvariable {S}\n\ntheorem toSubalgebra_injective {S S' : IntermediateField K L}\n    (h : S.toSubalgebra = S'.toSubalgebra) : S = S' :=\n  by\n  ext\n  rw [← mem_to_subalgebra, ← mem_to_subalgebra, h]\n#align intermediate_field.to_subalgebra_injective IntermediateField.toSubalgebra_injective\n\nvariable (S)\n\ntheorem set_range_subset : Set.range (algebraMap K L) ⊆ S :=\n  S.toSubalgebra.range_subset\n#align intermediate_field.set_range_subset IntermediateField.set_range_subset\n\ntheorem fieldRange_le : (algebraMap K L).fieldRange ≤ S.toSubfield := fun x hx =>\n  S.toSubalgebra.range_subset (by rwa [Set.mem_range, ← RingHom.mem_fieldRange])\n#align intermediate_field.field_range_le IntermediateField.fieldRange_le\n\n@[simp]\ntheorem toSubalgebra_le_toSubalgebra {S S' : IntermediateField K L} :\n    S.toSubalgebra ≤ S'.toSubalgebra ↔ S ≤ S' :=\n  Iff.rfl\n#align intermediate_field.to_subalgebra_le_to_subalgebra IntermediateField.toSubalgebra_le_toSubalgebra\n\n@[simp]\ntheorem toSubalgebra_lt_toSubalgebra {S S' : IntermediateField K L} :\n    S.toSubalgebra < S'.toSubalgebra ↔ S < S' :=\n  Iff.rfl\n#align intermediate_field.to_subalgebra_lt_to_subalgebra IntermediateField.toSubalgebra_lt_toSubalgebra\n\nvariable {S}\n\nsection Tower\n\n/-- Lift an intermediate_field of an intermediate_field -/\ndef lift {F : IntermediateField K L} (E : IntermediateField K F) : IntermediateField K L :=\n  E.map (val F)\n#align intermediate_field.lift IntermediateField.lift\n\ninstance hasLift {F : IntermediateField K L} :\n    HasLiftT (IntermediateField K F) (IntermediateField K L) :=\n  ⟨lift⟩\n#align intermediate_field.has_lift IntermediateField.hasLift\n\nsection RestrictScalars\n\nvariable (K) [Algebra L' L] [IsScalarTower K L' L]\n\n/-- Given a tower `L / ↥E / L' / K` of field extensions, where `E` is an `L'`-intermediate field of\n`L`, reinterpret `E` as a `K`-intermediate field of `L`. -/\ndef restrictScalars (E : IntermediateField L' L) : IntermediateField K L :=\n  { E.toSubfield, E.toSubalgebra.restrictScalars K with carrier := E.carrier }\n#align intermediate_field.restrict_scalars IntermediateField.restrictScalars\n\n@[simp]\ntheorem coe_restrictScalars {E : IntermediateField L' L} :\n    (restrictScalars K E : Set L) = (E : Set L) :=\n  rfl\n#align intermediate_field.coe_restrict_scalars IntermediateField.coe_restrictScalars\n\n@[simp]\ntheorem restrictScalars_toSubalgebra {E : IntermediateField L' L} :\n    (E.restrictScalars K).toSubalgebra = E.toSubalgebra.restrictScalars K :=\n  SetLike.coe_injective rfl\n#align intermediate_field.restrict_scalars_to_subalgebra IntermediateField.restrictScalars_toSubalgebra\n\n@[simp]\ntheorem restrictScalars_toSubfield {E : IntermediateField L' L} :\n    (E.restrictScalars K).toSubfield = E.toSubfield :=\n  SetLike.coe_injective rfl\n#align intermediate_field.restrict_scalars_to_subfield IntermediateField.restrictScalars_toSubfield\n\n@[simp]\ntheorem mem_restrictScalars {E : IntermediateField L' L} {x : L} :\n    x ∈ restrictScalars K E ↔ x ∈ E :=\n  Iff.rfl\n#align intermediate_field.mem_restrict_scalars IntermediateField.mem_restrictScalars\n\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars K : IntermediateField L' L → IntermediateField K L) :=\n  fun U V H => ext fun x => by rw [← mem_restrict_scalars K, H, mem_restrict_scalars]\n#align intermediate_field.restrict_scalars_injective IntermediateField.restrictScalars_injective\n\nend RestrictScalars\n\n/-- This was formerly an instance called `lift2_alg`, but an instance above already provides it. -/\nexample {F : IntermediateField K L} {E : IntermediateField F L} : Algebra K E := by infer_instance\n\nend Tower\n\nsection FiniteDimensional\n\nvariable (F E : IntermediateField K L)\n\ninstance finiteDimensional_left [FiniteDimensional K L] : FiniteDimensional K F :=\n  left K F L\n#align intermediate_field.finite_dimensional_left IntermediateField.finiteDimensional_left\n\ninstance finiteDimensional_right [FiniteDimensional K L] : FiniteDimensional F L :=\n  right K F L\n#align intermediate_field.finite_dimensional_right IntermediateField.finiteDimensional_right\n\n@[simp]\ntheorem dim_eq_dim_subalgebra : Module.rank K F.toSubalgebra = Module.rank K F :=\n  rfl\n#align intermediate_field.dim_eq_dim_subalgebra IntermediateField.dim_eq_dim_subalgebra\n\n@[simp]\ntheorem finrank_eq_finrank_subalgebra : finrank K F.toSubalgebra = finrank K F :=\n  rfl\n#align intermediate_field.finrank_eq_finrank_subalgebra IntermediateField.finrank_eq_finrank_subalgebra\n\nvariable {F} {E}\n\n@[simp]\ntheorem toSubalgebra_eq_iff : F.toSubalgebra = E.toSubalgebra ↔ F = E :=\n  by\n  rw [SetLike.ext_iff, SetLike.ext'_iff, Set.ext_iff]\n  rfl\n#align intermediate_field.to_subalgebra_eq_iff IntermediateField.toSubalgebra_eq_iff\n\ntheorem eq_of_le_of_finrank_le [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank K E ≤ finrank K F) : F = E :=\n  toSubalgebra_injective <|\n    Subalgebra.toSubmodule.Injective <| eq_of_le_of_finrank_le h_le h_finrank\n#align intermediate_field.eq_of_le_of_finrank_le IntermediateField.eq_of_le_of_finrank_le\n\ntheorem eq_of_le_of_finrank_eq [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank K F = finrank K E) : F = E :=\n  eq_of_le_of_finrank_le h_le h_finrank.ge\n#align intermediate_field.eq_of_le_of_finrank_eq IntermediateField.eq_of_le_of_finrank_eq\n\ntheorem eq_of_le_of_finrank_le' [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank F L ≤ finrank E L) : F = E :=\n  by\n  apply eq_of_le_of_finrank_le h_le\n  have h1 := finrank_mul_finrank K F L\n  have h2 := finrank_mul_finrank K E L\n  have h3 : 0 < finrank E L := finrank_pos\n  nlinarith\n#align intermediate_field.eq_of_le_of_finrank_le' IntermediateField.eq_of_le_of_finrank_le'\n\ntheorem eq_of_le_of_finrank_eq' [FiniteDimensional K L] (h_le : F ≤ E)\n    (h_finrank : finrank F L = finrank E L) : F = E :=\n  eq_of_le_of_finrank_le' h_le h_finrank.le\n#align intermediate_field.eq_of_le_of_finrank_eq' IntermediateField.eq_of_le_of_finrank_eq'\n\nend FiniteDimensional\n\ntheorem isAlgebraic_iff {x : S} : IsAlgebraic K x ↔ IsAlgebraic K (x : L) :=\n  (isAlgebraic_algebraMap_iff (algebraMap S L).Injective).symm\n#align intermediate_field.is_algebraic_iff IntermediateField.isAlgebraic_iff\n\ntheorem isIntegral_iff {x : S} : IsIntegral K x ↔ IsIntegral K (x : L) := by\n  rw [← isAlgebraic_iff_isIntegral, is_algebraic_iff, isAlgebraic_iff_isIntegral]\n#align intermediate_field.is_integral_iff IntermediateField.isIntegral_iff\n\ntheorem minpoly_eq (x : S) : minpoly K x = minpoly K (x : L) :=\n  by\n  by_cases hx : IsIntegral K x\n  · exact minpoly.eq_of_algebraMap_eq (algebraMap S L).Injective hx rfl\n  · exact (minpoly.eq_zero hx).trans (minpoly.eq_zero (mt is_integral_iff.mpr hx)).symm\n#align intermediate_field.minpoly_eq IntermediateField.minpoly_eq\n\nend IntermediateField\n\n/-- If `L/K` is algebraic, the `K`-subalgebras of `L` are all fields.  -/\ndef subalgebraEquivIntermediateField (alg : Algebra.IsAlgebraic K L) :\n    Subalgebra K L ≃o IntermediateField K L\n    where\n  toFun S := S.toIntermediateField fun x hx => S.inv_mem_of_algebraic (alg (⟨x, hx⟩ : S))\n  invFun S := S.toSubalgebra\n  left_inv S := toSubalgebra_toIntermediateField _ _\n  right_inv := toIntermediateField_toSubalgebra\n  map_rel_iff' S S' := Iff.rfl\n#align subalgebra_equiv_intermediate_field subalgebraEquivIntermediateField\n\n@[simp]\ntheorem mem_subalgebraEquivIntermediateField (alg : Algebra.IsAlgebraic K L) {S : Subalgebra K L}\n    {x : L} : x ∈ subalgebraEquivIntermediateField alg S ↔ x ∈ S :=\n  Iff.rfl\n#align mem_subalgebra_equiv_intermediate_field mem_subalgebraEquivIntermediateField\n\n@[simp]\ntheorem mem_subalgebraEquivIntermediateField_symm (alg : Algebra.IsAlgebraic K L)\n    {S : IntermediateField K L} {x : L} :\n    x ∈ (subalgebraEquivIntermediateField alg).symm S ↔ x ∈ S :=\n  Iff.rfl\n#align mem_subalgebra_equiv_intermediate_field_symm 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{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Jakob von Raumer\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.kernels\n\n/-!\n# Biproducts and binary biproducts\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe introduce the notion of (finite) biproducts and binary biproducts.\n\nThese are slightly unusual relative to the other shapes in the library,\nas they are simultaneously limits and colimits.\n(Zero objects are similar; they are \"biterminal\".)\n\nFor results about biproducts in preadditive categories see\n`category_theory.preadditive.biproducts`.\n\nIn a category with zero morphisms, we model the (binary) biproduct of `P Q : C`\nusing a `binary_bicone`, which has a cone point `X`,\nand morphisms `fst : X ⟶ P`, `snd : X ⟶ Q`, `inl : P ⟶ X` and `inr : X ⟶ Q`,\nsuch that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q`.\nSuch a `binary_bicone` is a biproduct if the cone is a limit cone, and the cocone is a colimit\ncocone.\n\nFor biproducts indexed by a `fintype J`, a `bicone` again consists of a cone point `X`\nand morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`,\nsuch that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise.\n\n## Notation\nAs `⊕` is already taken for the sum of types, we introduce the notation `X ⊞ Y` for\na binary biproduct. We introduce `⨁ f` for the indexed biproduct.\n\n## Implementation\nPrior to #14046, `has_finite_biproducts` required a `decidable_eq` instance on the indexing type.\nAs this had no pay-off (everything about limits is non-constructive in mathlib), and occasional cost\n(constructing decidability instances appropriate for constructions involving the indexing type),\nwe made everything classical.\n-/\n\nnoncomputable theory\n\nuniverses w w' v u\n\nopen category_theory\nopen category_theory.functor\nopen_locale classical\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {J : Type w}\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/--\nA `c : bicone F` is:\n* an object `c.X` and\n* morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`,\n* such that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise.\n-/\n@[nolint has_nonempty_instance]\nstructure bicone (F : J → C) :=\n(X : C)\n(π : Π j, X ⟶ F j)\n(ι : Π j, F j ⟶ X)\n(ι_π : ∀ j j', ι j ≫ π j' = if h : j = j' then eq_to_hom (congr_arg F h) else 0 . obviously)\n\n@[simp, reassoc] lemma bicone_ι_π_self {F : J → C} (B : bicone F) (j : J) :\n  B.ι j ≫ B.π j = 𝟙 (F j) :=\nby simpa using B.ι_π j j\n\n@[simp, reassoc] lemma bicone_ι_π_ne {F : J → C} (B : bicone F) {j j' : J} (h : j ≠ j') :\n  B.ι j ≫ B.π j' = 0 :=\nby simpa [h] using B.ι_π j j'\n\nvariables {F : J → C}\n\nnamespace bicone\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- Extract the cone from a bicone. -/\ndef to_cone (B : bicone F) : cone (discrete.functor F) :=\n{ X := B.X,\n  π := { app := λ j, B.π j.as }, }\n\n@[simp] lemma to_cone_X (B : bicone F) : B.to_cone.X = B.X := rfl\n\n@[simp] lemma to_cone_π_app (B : bicone F) (j : discrete J) : B.to_cone.π.app j = B.π j.as := rfl\n\nlemma to_cone_π_app_mk (B : bicone F) (j : J) : B.to_cone.π.app ⟨j⟩ = B.π j := rfl\n\n/-- Extract the cocone from a bicone. -/\ndef to_cocone (B : bicone F) : cocone (discrete.functor F) :=\n{ X := B.X,\n  ι := { app := λ j, B.ι j.as }, }\n\n@[simp] lemma to_cocone_X (B : bicone F) : B.to_cocone.X = B.X := rfl\n\n@[simp] lemma to_cocone_ι_app (B : bicone F) (j : discrete J) : B.to_cocone.ι.app j = B.ι j.as :=\nrfl\n\nlemma to_cocone_ι_app_mk (B : bicone F) (j : J) : B.to_cocone.ι.app ⟨j⟩ = B.ι j := rfl\n\n/-- We can turn any limit cone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_limit_cone {f : J → C} {t : cone (discrete.functor f)} (ht : is_limit t) :\n  bicone f :=\n{ X := t.X,\n  π := λ j, t.π.app ⟨j⟩,\n  ι := λ j, ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)),\n  ι_π := λ j j', by simp }\n\nlemma ι_of_is_limit {f : J → C} {t : bicone f} (ht : is_limit t.to_cone) (j : J) :\n  t.ι j = ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (λ j', by { rw ht.fac, discrete_cases, simp [t.ι_π] })\n\n/-- We can turn any colimit cocone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_colimit_cocone {f : J → C} {t : cocone (discrete.functor f)} (ht : is_colimit t) :\n  bicone f :=\n{ X := t.X,\n  π := λ j, ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)),\n  ι := λ j, t.ι.app ⟨j⟩,\n  ι_π := λ j j', by simp }\n\nlemma π_of_is_colimit {f : J → C} {t : bicone f} (ht : is_colimit t.to_cocone) (j : J) :\n  t.π j = ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (λ j', by { rw ht.fac, discrete_cases, simp [t.ι_π] })\n\n/-- Structure witnessing that a bicone is both a limit cone and a colimit cocone. -/\n@[nolint has_nonempty_instance]\nstructure is_bilimit {F : J → C} (B : bicone F) :=\n(is_limit : is_limit B.to_cone)\n(is_colimit : is_colimit B.to_cocone)\n\nlocal attribute [ext] bicone.is_bilimit\n\ninstance subsingleton_is_bilimit {f : J → C} {c : bicone f} : subsingleton c.is_bilimit :=\n⟨λ h h', bicone.is_bilimit.ext _ _ (subsingleton.elim _ _) (subsingleton.elim _ _)⟩\n\nsection whisker\nvariables {K : Type w'}\n\n/-- Whisker a bicone with an equivalence between the indexing types. -/\n@[simps]\ndef whisker {f : J → C} (c : bicone f) (g : K ≃ J) : bicone (f ∘ g) :=\n{ X := c.X,\n  π := λ k, c.π (g k),\n  ι := λ k, c.ι (g k),\n  ι_π := λ k k',\n  begin\n    simp only [c.ι_π],\n    split_ifs with h h' h'; simp [equiv.apply_eq_iff_eq g] at h h'; tauto\n  end }\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- Taking the cone of a whiskered bicone results in a cone isomorphic to one gained\nby whiskering the cone and postcomposing with a suitable isomorphism. -/\ndef whisker_to_cone {f : J → C} (c : bicone f) (g : K ≃ J) :\n  (c.whisker g).to_cone ≅ (cones.postcompose (discrete.functor_comp f g).inv).obj\n    (c.to_cone.whisker (discrete.functor (discrete.mk ∘ g))) :=\ncones.ext (iso.refl _) (by tidy)\n\n/-- Taking the cocone of a whiskered bicone results in a cone isomorphic to one gained\nby whiskering the cocone and precomposing with a suitable isomorphism. -/\ndef whisker_to_cocone {f : J → C} (c : bicone f) (g : K ≃ J) :\n  (c.whisker g).to_cocone ≅ (cocones.precompose (discrete.functor_comp f g).hom).obj\n    (c.to_cocone.whisker (discrete.functor (discrete.mk ∘ g))) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/-- Whiskering a bicone with an equivalence between types preserves being a bilimit bicone. -/\ndef whisker_is_bilimit_iff {f : J → C} (c : bicone f) (g : K ≃ J) :\n  (c.whisker g).is_bilimit ≃ c.is_bilimit :=\nbegin\n  refine equiv_of_subsingleton_of_subsingleton (λ hc, ⟨_, _⟩) (λ hc, ⟨_, _⟩),\n  { let := is_limit.of_iso_limit hc.is_limit (bicone.whisker_to_cone c g),\n    let := (is_limit.postcompose_hom_equiv (discrete.functor_comp f g).symm _) this,\n    exact is_limit.of_whisker_equivalence (discrete.equivalence g) this },\n  { let := is_colimit.of_iso_colimit hc.is_colimit (bicone.whisker_to_cocone c g),\n    let := (is_colimit.precompose_hom_equiv (discrete.functor_comp f g) _) this,\n    exact is_colimit.of_whisker_equivalence (discrete.equivalence g) this },\n  { apply is_limit.of_iso_limit _ (bicone.whisker_to_cone c g).symm,\n    apply (is_limit.postcompose_hom_equiv (discrete.functor_comp f g).symm _).symm _,\n    exact is_limit.whisker_equivalence hc.is_limit (discrete.equivalence g) },\n  { apply is_colimit.of_iso_colimit _ (bicone.whisker_to_cocone c g).symm,\n    apply (is_colimit.precompose_hom_equiv (discrete.functor_comp f g) _).symm _,\n    exact is_colimit.whisker_equivalence hc.is_colimit (discrete.equivalence g) }\nend\n\nend whisker\n\nend bicone\n\n/--\nA bicone over `F : J → C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_nonempty_instance]\nstructure limit_bicone (F : J → C) :=\n(bicone : bicone F)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_biproduct F` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `F`.\n-/\nclass has_biproduct (F : J → C) : Prop :=\nmk' :: (exists_biproduct : nonempty (limit_bicone F))\n\nlemma has_biproduct.mk {F : J → C} (d : limit_bicone F) : has_biproduct F :=\n⟨nonempty.intro d⟩\n\n/-- Use the axiom of choice to extract explicit `biproduct_data F` from `has_biproduct F`. -/\ndef get_biproduct_data (F : J → C) [has_biproduct F] : limit_bicone F :=\nclassical.choice has_biproduct.exists_biproduct\n\n/-- A bicone for `F` which is both a limit cone and a colimit cocone. -/\ndef biproduct.bicone (F : J → C) [has_biproduct F] : bicone F :=\n(get_biproduct_data F).bicone\n\n/-- `biproduct.bicone F` is a bilimit bicone. -/\ndef biproduct.is_bilimit (F : J → C) [has_biproduct F] : (biproduct.bicone F).is_bilimit :=\n(get_biproduct_data F).is_bilimit\n\n/-- `biproduct.bicone F` is a limit cone. -/\ndef biproduct.is_limit (F : J → C) [has_biproduct F] : is_limit (biproduct.bicone F).to_cone :=\n(get_biproduct_data F).is_bilimit.is_limit\n\n/-- `biproduct.bicone F` is a colimit cocone. -/\ndef biproduct.is_colimit (F : J → C) [has_biproduct F] :\n  is_colimit (biproduct.bicone F).to_cocone :=\n(get_biproduct_data F).is_bilimit.is_colimit\n\n@[priority 100]\ninstance has_product_of_has_biproduct [has_biproduct F] : has_product F :=\nhas_limit.mk { cone := (biproduct.bicone F).to_cone,\n  is_limit := biproduct.is_limit F, }\n\n@[priority 100]\ninstance has_coproduct_of_has_biproduct [has_biproduct F] : has_coproduct F :=\nhas_colimit.mk { cocone := (biproduct.bicone F).to_cocone,\n  is_colimit := biproduct.is_colimit F, }\n\nvariables (J C)\n\n/--\n`C` has biproducts of shape `J` if we have\na limit and a colimit, with the same cone points,\nof every function `F : J → C`.\n-/\nclass has_biproducts_of_shape : Prop :=\n(has_biproduct : ∀ F : J → C, has_biproduct F)\n\nattribute [instance, priority 100] has_biproducts_of_shape.has_biproduct\n\n/-- `has_finite_biproducts C` represents a choice of biproduct for every family of objects in `C`\nindexed by a finite type. -/\nclass has_finite_biproducts : Prop :=\n(out [] : ∀ n, has_biproducts_of_shape (fin n) C)\n\nvariables {J}\n\nlemma has_biproducts_of_shape_of_equiv {K : Type w'} [has_biproducts_of_shape K C] (e : J ≃ K) :\n  has_biproducts_of_shape J C :=\n⟨λ F, let ⟨⟨h⟩⟩ := has_biproducts_of_shape.has_biproduct (F ∘ e.symm), ⟨c, hc⟩ := h\n  in has_biproduct.mk $ by simpa only [(∘), e.symm_apply_apply]\n    using limit_bicone.mk (c.whisker e) ((c.whisker_is_bilimit_iff _).2 hc)⟩\n\n@[priority 100] instance has_biproducts_of_shape_finite [has_finite_biproducts C] [finite J] :\n  has_biproducts_of_shape J C :=\nbegin\n  rcases finite.exists_equiv_fin J with ⟨n, ⟨e⟩⟩,\n  haveI := has_finite_biproducts.out C n,\n  exact has_biproducts_of_shape_of_equiv C e\nend\n\n@[priority 100]\ninstance has_finite_products_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_products C :=\n{ out := λ n, ⟨λ F, has_limit_of_iso discrete.nat_iso_functor.symm⟩ }\n\n@[priority 100]\ninstance has_finite_coproducts_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_coproducts C :=\n{ out := λ n, ⟨λ F, has_colimit_of_iso discrete.nat_iso_functor⟩ }\n\nvariables {J C}\n\n/--\nThe isomorphism between the specified limit and the specified colimit for\na functor with a bilimit.\n-/\ndef biproduct_iso (F : J → C) [has_biproduct F] :\n  limits.pi_obj F ≅ limits.sigma_obj F :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (biproduct.is_limit F)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit F) (colimit.is_colimit _)\n\nend limits\n\nnamespace limits\nvariables {J : Type w}\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/-- `biproduct f` computes the biproduct of a family of elements `f`. (It is defined as an\n   abbreviation for `limit (discrete.functor f)`, so for most facts about `biproduct f`, you will\n   just use general facts about limits and colimits.) -/\nabbreviation biproduct (f : J → C) [has_biproduct f] : C :=\n(biproduct.bicone f).X\n\nnotation `⨁ ` f:20 := biproduct f\n\n/-- The projection onto a summand of a biproduct. -/\nabbreviation biproduct.π (f : J → C) [has_biproduct f] (b : J) : ⨁ f ⟶ f b :=\n(biproduct.bicone f).π b\n\n@[simp]\nlemma biproduct.bicone_π (f : J → C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).π b = biproduct.π f b := rfl\n\n/-- The inclusion into a summand of a biproduct. -/\nabbreviation biproduct.ι (f : J → C) [has_biproduct f] (b : J) : f b ⟶ ⨁ f :=\n(biproduct.bicone f).ι b\n\n@[simp]\nlemma biproduct.bicone_ι (f : J → C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).ι b = biproduct.ι f b := rfl\n\n/-- Note that as this lemma has a `if` in the statement, we include a `decidable_eq` argument.\nThis means you may not be able to `simp` using this lemma unless you `open_locale classical`. -/\n@[reassoc]\nlemma biproduct.ι_π [decidable_eq J] (f : J → C) [has_biproduct f] (j j' : J) :\n  biproduct.ι f j ≫ biproduct.π f j' = if h : j = j' then eq_to_hom (congr_arg f h) else 0 :=\nby convert (biproduct.bicone f).ι_π j j'\n\n@[simp,reassoc]\nlemma biproduct.ι_π_self (f : J → C) [has_biproduct f] (j : J) :\n  biproduct.ι f j ≫ biproduct.π f j = 𝟙 _ :=\nby simp [biproduct.ι_π]\n\n@[simp,reassoc]\nlemma biproduct.ι_π_ne (f : J → C) [has_biproduct f] {j j' : J} (h : j ≠ j') :\n  biproduct.ι f j ≫ biproduct.π f j' = 0 :=\nby simp [biproduct.ι_π, h]\n\n/-- Given a collection of maps into the summands, we obtain a map into the biproduct. -/\nabbreviation biproduct.lift\n  {f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) : P ⟶ ⨁ f :=\n(biproduct.is_limit f).lift (fan.mk P p)\n/-- Given a collection of maps out of the summands, we obtain a map out of the biproduct. -/\nabbreviation biproduct.desc\n  {f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) : ⨁ f ⟶ P :=\n(biproduct.is_colimit f).desc (cofan.mk P p)\n\n@[simp, reassoc]\nlemma biproduct.lift_π {f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) (j : J) :\n  biproduct.lift p ≫ biproduct.π f j = p j :=\n(biproduct.is_limit f).fac _ ⟨j⟩\n\n@[simp, reassoc]\nlemma biproduct.ι_desc {f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) (j : J) :\n  biproduct.ι f j ≫ biproduct.desc p = p j :=\n(biproduct.is_colimit f).fac _ ⟨j⟩\n\n/-- Given a collection of maps between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain a map between the biproducts. -/\nabbreviation biproduct.map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g :=\nis_limit.map (biproduct.bicone f).to_cone (biproduct.is_limit g)\n  (discrete.nat_trans (λ j, p j.as))\n\n/-- An alternative to `biproduct.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biproduct.map`. -/\nabbreviation biproduct.map' {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g :=\nis_colimit.map (biproduct.is_colimit f) (biproduct.bicone g).to_cocone\n  (discrete.nat_trans (λ j, p j.as))\n\n@[ext] lemma biproduct.hom_ext {f : J → C} [has_biproduct f]\n  {Z : C} (g h : Z ⟶ ⨁ f)\n  (w : ∀ j, g ≫ biproduct.π f j = h ≫ biproduct.π f j) : g = h :=\n(biproduct.is_limit f).hom_ext (λ j, w j.as)\n\n@[ext] lemma biproduct.hom_ext' {f : J → C} [has_biproduct f]\n  {Z : C} (g h : ⨁ f ⟶ Z)\n  (w : ∀ j, biproduct.ι f j ≫ g = biproduct.ι f j ≫ h) : g = h :=\n(biproduct.is_colimit f).hom_ext (λ j, w j.as)\n\n/-- The canonical isomorphism between the chosen biproduct and the chosen product. -/\ndef biproduct.iso_product (f : J → C) [has_biproduct f] : ⨁ f ≅ ∏ f :=\nis_limit.cone_point_unique_up_to_iso (biproduct.is_limit f) (limit.is_limit _)\n\n@[simp] lemma biproduct.iso_product_hom {f : J → C} [has_biproduct f] :\n  (biproduct.iso_product f).hom = pi.lift (biproduct.π f) :=\nlimit.hom_ext $ λ j, by simp [biproduct.iso_product]\n\n@[simp] lemma biproduct.iso_product_inv {f : J → C} [has_biproduct f] :\n  (biproduct.iso_product f).inv = biproduct.lift (pi.π f) :=\nbiproduct.hom_ext _ _ $ λ j, by simp [iso.inv_comp_eq]\n\n/-- The canonical isomorphism between the chosen biproduct and the chosen coproduct. -/\ndef biproduct.iso_coproduct (f : J → C) [has_biproduct f] : ⨁ f ≅ ∐ f :=\nis_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit f) (colimit.is_colimit _)\n\n@[simp] \n\n@[simp] lemma biproduct.iso_coproduct_hom {f : J → C} [has_biproduct f] :\n  (biproduct.iso_coproduct f).hom = biproduct.desc (sigma.ι f) :=\nbiproduct.hom_ext' _ _ $ λ j, by simp [← iso.eq_comp_inv]\n\nlemma biproduct.map_eq_map' {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : biproduct.map p = biproduct.map' p :=\nbegin\n  ext j j',\n  simp only [discrete.nat_trans_app, limits.is_colimit.ι_map, limits.is_limit.map_π, category.assoc,\n    ←bicone.to_cone_π_app_mk, ←biproduct.bicone_π, ←bicone.to_cocone_ι_app_mk, ←biproduct.bicone_ι],\n  simp only [biproduct.bicone_ι, biproduct.bicone_π, bicone.to_cocone_ι_app, bicone.to_cone_π_app],\n  dsimp,\n  rw [biproduct.ι_π_assoc, biproduct.ι_π],\n  split_ifs,\n  { subst h, rw [eq_to_hom_refl, category.id_comp], erw category.comp_id, },\n  { simp, },\nend\n\n@[simp, reassoc]\nlemma biproduct.map_π {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) (j : J) :\n  biproduct.map p ≫ biproduct.π g j = biproduct.π f j ≫ p j :=\nlimits.is_limit.map_π _ _ _ (discrete.mk j)\n\n@[simp, reassoc]\nlemma biproduct.ι_map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) (j : J) :\n  biproduct.ι f j ≫ biproduct.map p = p j ≫ biproduct.ι g j :=\nbegin\n  rw biproduct.map_eq_map',\n  convert limits.is_colimit.ι_map _ _ _ (discrete.mk j); refl\nend\n\n@[simp, reassoc]\nlemma biproduct.map_desc {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) {P : C} (k : Π j, g j ⟶ P) :\n  biproduct.map p ≫ biproduct.desc k = biproduct.desc (λ j, p j ≫ k j) :=\nby { ext, simp, }\n\n@[simp, reassoc]\nlemma biproduct.lift_map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  {P : C} (k : Π j, P ⟶ f j) (p : Π j, f j ⟶ g j)  :\n  biproduct.lift k ≫ biproduct.map p = biproduct.lift (λ j, k j ≫ p j) :=\nby { ext, simp, }\n\n/-- Given a collection of isomorphisms between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain an isomorphism between the biproducts. -/\n@[simps]\ndef biproduct.map_iso {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ≅ g b) : ⨁ f ≅ ⨁ g :=\n{ hom := biproduct.map (λ b, (p b).hom),\n  inv := biproduct.map (λ b, (p b).inv), }\n\nsection π_kernel\n\nsection\nvariables (f : J → C) [has_biproduct f]\nvariables (p : J → Prop) [has_biproduct (subtype.restrict p f)]\n\n/-- The canonical morphism from the biproduct over a restricted index type to the biproduct of\nthe full index type. -/\ndef biproduct.from_subtype : ⨁ subtype.restrict p f ⟶ ⨁ f :=\nbiproduct.desc $ λ j, biproduct.ι _ _\n\n/-- The canonical morphism from a biproduct to the biproduct over a restriction of its index\ntype. -/\ndef biproduct.to_subtype : ⨁ f ⟶ ⨁ subtype.restrict p f :=\nbiproduct.lift $ λ j, biproduct.π _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_π [decidable_pred p] (j : J) :\n  biproduct.from_subtype f p ≫ biproduct.π f j =\n    if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0 :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.ι_π],\n    split_ifs with h₁ h₂ h₂,\n    exacts [rfl, false.elim (h₂ (subtype.ext h₁)),\n      false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },\n  { rw [dif_neg h, dif_neg (show (i : J) ≠ j, from λ h₂, h (h₂ ▸ i.2)), comp_zero] }\nend\n\nlemma biproduct.from_subtype_eq_lift [decidable_pred p] : biproduct.from_subtype f p =\n    biproduct.lift (λ j, if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0) :=\nbiproduct.hom_ext _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_π_subtype (j : subtype p) :\n  biproduct.from_subtype f p ≫ biproduct.π f j = biproduct.π (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π, biproduct.ι_π],\n  split_ifs with h₁ h₂ h₂,\n  exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_π (j : subtype p) :\n  biproduct.to_subtype f p ≫ biproduct.π (subtype.restrict p f) j = biproduct.π f j :=\nbiproduct.lift_π _ _\n\n@[simp, reassoc]\nlemma biproduct.ι_to_subtype [decidable_pred p] (j : J) :\n  biproduct.ι f j ≫ biproduct.to_subtype f p =\n    if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0 :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.ι_π],\n    split_ifs with h₁ h₂ h₂,\n    exacts [rfl, false.elim (h₂ (subtype.ext h₁)),\n      false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },\n  { rw [dif_neg h, dif_neg (show j ≠ i, from λ h₂, h (h₂.symm ▸ i.2)), zero_comp] }\nend\n\nlemma biproduct.to_subtype_eq_desc [decidable_pred p] : biproduct.to_subtype f p =\n  biproduct.desc (λ j, if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0) :=\nbiproduct.hom_ext' _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.ι_to_subtype_subtype (j : subtype p) :\n  biproduct.ι f j ≫ biproduct.to_subtype f p = biproduct.ι (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π, biproduct.ι_π],\n  split_ifs with h₁ h₂ h₂,\n  exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.ι_from_subtype (j : subtype p) :\n  biproduct.ι (subtype.restrict p f) j ≫ biproduct.from_subtype f p = biproduct.ι f j :=\nbiproduct.ι_desc _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_to_subtype :\n  biproduct.from_subtype f p ≫ biproduct.to_subtype f p = 𝟙 (⨁ subtype.restrict p f) :=\nbegin\n  refine biproduct.hom_ext _ _ (λ j, _),\n  rw [category.assoc, biproduct.to_subtype_π, biproduct.from_subtype_π_subtype, category.id_comp]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_from_subtype [decidable_pred p] :\n  biproduct.to_subtype f p ≫ biproduct.from_subtype f p =\n    biproduct.map (λ j, if p j then 𝟙 (f j) else 0) :=\nbegin\n  ext1 i,\n  by_cases h : p i,\n  { simp [h], congr },\n  { simp [h] }\nend\n\nend\n\nsection\nvariables (f : J → C) (i : J) [has_biproduct f] [has_biproduct (subtype.restrict (λ j, j ≠ i) f)]\n\n/-- The kernel of `biproduct.π f i` is the inclusion from the biproduct which omits `i`\nfrom the index set `J` into the biproduct over `J`. -/\ndef biproduct.is_limit_from_subtype : is_limit\n  (kernel_fork.of_ι (biproduct.from_subtype f (λ j, j ≠ i))\n    (by simp) : kernel_fork (biproduct.π f i)) :=\nfork.is_limit.mk' _ $ λ s,\n⟨s.ι ≫ biproduct.to_subtype _ _,\n begin\n   ext j,\n   rw [kernel_fork.ι_of_ι, category.assoc, category.assoc,\n     biproduct.to_subtype_from_subtype_assoc, biproduct.map_π],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), comp_zero, comp_zero, kernel_fork.condition] },\n   { rw [if_pos (ne.symm h), category.comp_id], }\n end,\n begin\n   intros m hm,\n   rw [← hm, kernel_fork.ι_of_ι, category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.comp_id _).symm\n end⟩\n\ninstance : has_kernel (biproduct.π f i) :=\nhas_limit.mk ⟨_, biproduct.is_limit_from_subtype f i⟩\n\n/-- The kernel of `biproduct.π f i` is `⨁ subtype.restrict {i}ᶜ f`. -/\n@[simps]\ndef kernel_biproduct_π_iso : kernel (biproduct.π f i) ≅ ⨁ subtype.restrict (λ j, j ≠ i) f :=\nlimit.iso_limit_cone ⟨_, biproduct.is_limit_from_subtype f i⟩\n\n/-- The cokernel of `biproduct.ι f i` is the projection from the biproduct over the index set `J`\nonto the biproduct omitting `i`. -/\ndef biproduct.is_colimit_to_subtype : is_colimit\n  (cokernel_cofork.of_π (biproduct.to_subtype f (λ j, j ≠ i))\n    (by simp) : cokernel_cofork (biproduct.ι f i)) :=\ncofork.is_colimit.mk' _ $ λ s,\n⟨biproduct.from_subtype _ _ ≫ s.π,\n begin\n   ext j,\n   rw [cokernel_cofork.π_of_π, biproduct.to_subtype_from_subtype_assoc,\n     biproduct.ι_map_assoc],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), zero_comp, cokernel_cofork.condition] },\n   { rw [if_pos (ne.symm h), category.id_comp], }\n end,\n begin\n   intros m hm,\n   rw [← hm, cokernel_cofork.π_of_π, ← category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.id_comp _).symm\n end⟩\n\ninstance : has_cokernel (biproduct.ι f i) :=\nhas_colimit.mk ⟨_, biproduct.is_colimit_to_subtype f i⟩\n\n/-- The cokernel of `biproduct.ι f i` is `⨁ subtype.restrict {i}ᶜ f`. -/\n@[simps]\ndef cokernel_biproduct_ι_iso : cokernel (biproduct.ι f i) ≅ ⨁ subtype.restrict (λ j, j ≠ i) f :=\ncolimit.iso_colimit_cocone ⟨_, biproduct.is_colimit_to_subtype f i⟩\n\nend\n\nsection\nopen_locale classical\n\n-- Per #15067, we only allow indexing in `Type 0` here.\nvariables {K : Type} [fintype K] [has_finite_biproducts C] (f : K → C)\n\n/-- The limit cone exhibiting `⨁ subtype.restrict pᶜ f` as the kernel of\n`biproduct.to_subtype f p` -/\n@[simps]\ndef kernel_fork_biproduct_to_subtype (p : set K) :\n  limit_cone (parallel_pair (biproduct.to_subtype f p) 0) :=\n{ cone := kernel_fork.of_ι (biproduct.from_subtype f pᶜ) begin\n    ext j k,\n    simp only [biproduct.ι_from_subtype_assoc, biproduct.ι_to_subtype, comp_zero, zero_comp],\n    erw [dif_neg j.2],\n    simp only [zero_comp],\n  end,\n  is_limit := kernel_fork.is_limit.of_ι _ _ (λ W g h, g ≫ biproduct.to_subtype f pᶜ)\n    begin\n      intros W' g' w,\n      ext j,\n      simp only [category.assoc, biproduct.to_subtype_from_subtype, pi.compl_apply,\n        biproduct.map_π],\n      split_ifs,\n      { simp, },\n      { replace w := w =≫ biproduct.π _ ⟨j, not_not.mp h⟩, simpa using w.symm, },\n    end\n    (by tidy), }\n\ninstance (p : set K) : has_kernel (biproduct.to_subtype f p) :=\nhas_limit.mk (kernel_fork_biproduct_to_subtype f p)\n\n/-- The kernel of `biproduct.to_subtype f p` is `⨁ subtype.restrict pᶜ f`. -/\n@[simps]\ndef kernel_biproduct_to_subtype_iso (p : set K) :\n  kernel (biproduct.to_subtype f p) ≅ ⨁ subtype.restrict pᶜ f :=\nlimit.iso_limit_cone (kernel_fork_biproduct_to_subtype f p)\n\n/-- The colimit cocone exhibiting `⨁ subtype.restrict pᶜ f` as the cokernel of\n`biproduct.from_subtype f p` -/\n@[simps]\ndef cokernel_cofork_biproduct_from_subtype (p : set K) :\n  colimit_cocone (parallel_pair (biproduct.from_subtype f p) 0) :=\n{ cocone := cokernel_cofork.of_π (biproduct.to_subtype f pᶜ) begin\n    ext j k,\n    simp only [pi.compl_apply, biproduct.ι_from_subtype_assoc, biproduct.ι_to_subtype,\n      comp_zero, zero_comp],\n    rw [dif_neg],\n    simp only [zero_comp],\n    exact not_not.mpr j.2,\n  end,\n  is_colimit := cokernel_cofork.is_colimit.of_π _ _ (λ W g h, biproduct.from_subtype f pᶜ ≫ g)\n    begin\n      intros W' g' w,\n      ext j,\n      simp only [biproduct.to_subtype_from_subtype_assoc, pi.compl_apply, biproduct.ι_map_assoc],\n      split_ifs,\n      { simp, },\n      { replace w := biproduct.ι _ (⟨j, not_not.mp h⟩ : p) ≫= w, simpa using w.symm, },\n    end\n    (by tidy), }\n\ninstance (p : set K) : has_cokernel (biproduct.from_subtype f p) :=\nhas_colimit.mk (cokernel_cofork_biproduct_from_subtype f p)\n\n/-- The cokernel of `biproduct.from_subtype f p` is `⨁ subtype.restrict pᶜ f`. -/\n@[simps]\ndef cokernel_biproduct_from_subtype_iso (p : set K) :\n  cokernel (biproduct.from_subtype f p) ≅ ⨁ subtype.restrict pᶜ f :=\ncolimit.iso_colimit_cocone (cokernel_cofork_biproduct_from_subtype f p)\n\nend\n\nend π_kernel\n\nend limits\n\nnamespace limits\n\nsection finite_biproducts\n\nvariables {J : Type} [fintype J] {K : Type} [fintype K]\n  {C : Type u} [category.{v} C] [has_zero_morphisms C] [has_finite_biproducts C]\n  {f : J → C} {g : K → C}\n\n/--\nConvert a (dependently typed) matrix to a morphism of biproducts.\n-/\ndef biproduct.matrix (m : Π j k, f j ⟶ g k) : ⨁ f ⟶ ⨁ g :=\nbiproduct.desc (λ j, biproduct.lift (λ k, m j k))\n\n@[simp, reassoc]\nlemma biproduct.matrix_π (m : Π j k, f j ⟶ g k) (k : K) :\n  biproduct.matrix m ≫ biproduct.π g k = biproduct.desc (λ j, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n@[simp, reassoc]\nlemma biproduct.ι_matrix (m : Π j k, f j ⟶ g k) (j : J) :\n  biproduct.ι f j ≫ biproduct.matrix m = biproduct.lift (λ k, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n/--\nExtract the matrix components from a morphism of biproducts.\n-/\ndef biproduct.components (m : ⨁ f ⟶ ⨁ g) (j : J) (k : K) : f j ⟶ g k :=\nbiproduct.ι f j ≫ m ≫ biproduct.π g k\n\n@[simp] lemma biproduct.matrix_components (m : Π j k, f j ⟶ g k) (j : J) (k : K) :\n  biproduct.components (biproduct.matrix m) j k = m j k :=\nby simp [biproduct.components]\n\n@[simp] lemma biproduct.components_matrix (m : ⨁ f ⟶ ⨁ g) :\n  biproduct.matrix (λ j k, biproduct.components m j k) = m :=\nby { ext, simp [biproduct.components], }\n\n/-- Morphisms between direct sums are matrices. -/\n@[simps]\ndef biproduct.matrix_equiv : (⨁ f ⟶ ⨁ g) ≃ (Π j k, f j ⟶ g k) :=\n{ to_fun := biproduct.components,\n  inv_fun := biproduct.matrix,\n  left_inv := biproduct.components_matrix,\n  right_inv := λ m, by { ext, apply biproduct.matrix_components } }\n\nend finite_biproducts\n\nvariables {J : Type w} {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\ninstance biproduct.ι_mono (f : J → C) [has_biproduct f] (b : J) :\n  is_split_mono (biproduct.ι f b) := is_split_mono.mk'\n{ retraction := biproduct.desc $ pi.single b _ }\n\ninstance biproduct.π_epi (f : J → C) [has_biproduct f] (b : J) :\n  is_split_epi (biproduct.π f b) := is_split_epi.mk'\n{ section_ := biproduct.lift $ pi.single b _ }\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_hom (f : J → C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).hom = biproduct.lift b.π :=\nrfl\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_inv (f : J → C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).inv = biproduct.desc b.ι :=\nbegin\n  refine biproduct.hom_ext' _ _ (λ j, hb.is_limit.hom_ext (λ j', _)),\n  discrete_cases,\n  rw [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp, bicone.to_cone_π_app,\n    biproduct.bicone_π, biproduct.ι_desc, biproduct.ι_π, b.to_cone_π_app, b.ι_π]\nend\n\n/-- Biproducts are unique up to isomorphism. This already follows because bilimits are limits,\n    but in the case of biproducts we can give an isomorphism with particularly nice definitional\n    properties, namely that `biproduct.lift b.π` and `biproduct.desc b.ι` are inverses of each\n    other. -/\n@[simps]\ndef biproduct.unique_up_to_iso (f : J → C) [has_biproduct f] {b : bicone f} (hb : b.is_bilimit) :\n  b.X ≅ ⨁ f :=\n{ hom := biproduct.lift b.π,\n  inv := biproduct.desc b.ι,\n  hom_inv_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,\n    ← biproduct.cone_point_unique_up_to_iso_inv f hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,\n    ← biproduct.cone_point_unique_up_to_iso_inv f hb, iso.inv_hom_id] }\n\nvariables (C)\n\n/-- A category with finite biproducts has a zero object. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance has_zero_object_of_has_finite_biproducts [has_finite_biproducts C] : has_zero_object C :=\nby { refine ⟨⟨biproduct empty.elim, λ X, ⟨⟨⟨0⟩, _⟩⟩, λ X, ⟨⟨⟨0⟩, _⟩⟩⟩⟩, tidy, }\n\nsection\nvariables {C} [unique J] (f : J → C)\n\n/-- The limit bicone for the biproduct over an index type with exactly one term. -/\n@[simps]\ndef limit_bicone_of_unique : limit_bicone f :=\n{ bicone :=\n  { X := f default,\n    π := λ j, eq_to_hom (by congr),\n    ι := λ j, eq_to_hom (by congr), },\n  is_bilimit :=\n  { is_limit := (limit_cone_of_unique f).is_limit,\n    is_colimit := (colimit_cocone_of_unique f).is_colimit, }, }\n\n@[priority 100] instance has_biproduct_unique : has_biproduct f :=\nhas_biproduct.mk (limit_bicone_of_unique f)\n\n/-- A biproduct over a index type with exactly one term is just the object over that term. -/\n@[simps]\ndef biproduct_unique_iso : ⨁ f ≅ f default :=\n(biproduct.unique_up_to_iso _ (limit_bicone_of_unique f).is_bilimit).symm\n\nend\n\nvariables {C}\n\n/--\nA binary bicone for a pair of objects `P Q : C` consists of the cone point `X`,\nmaps from `X` to both `P` and `Q`, and maps from both `P` and `Q` to `X`,\nso that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q`\n-/\n@[nolint has_nonempty_instance]\nstructure binary_bicone (P Q : C) :=\n(X : C)\n(fst : X ⟶ P)\n(snd : X ⟶ Q)\n(inl : P ⟶ X)\n(inr : Q ⟶ X)\n(inl_fst' : inl ≫ fst = 𝟙 P . obviously)\n(inl_snd' : inl ≫ snd = 0 . obviously)\n(inr_fst' : inr ≫ fst = 0 . obviously)\n(inr_snd' : inr ≫ snd = 𝟙 Q . obviously)\n\nrestate_axiom binary_bicone.inl_fst'\nrestate_axiom binary_bicone.inl_snd'\nrestate_axiom binary_bicone.inr_fst'\nrestate_axiom binary_bicone.inr_snd'\nattribute [simp, reassoc] binary_bicone.inl_fst binary_bicone.inl_snd\n  binary_bicone.inr_fst binary_bicone.inr_snd\n\nnamespace binary_bicone\nvariables {P Q : C}\n\n/-- Extract the cone from a binary bicone. -/\ndef to_cone (c : binary_bicone P Q) : cone (pair P Q) :=\nbinary_fan.mk c.fst c.snd\n\n@[simp]\nlemma to_cone_X (c : binary_bicone P Q) :\n  c.to_cone.X = c.X := rfl\n\n@[simp]\nlemma to_cone_π_app_left (c : binary_bicone P Q) :\n  c.to_cone.π.app ⟨walking_pair.left⟩ = c.fst := rfl\n@[simp]\nlemma to_cone_π_app_right (c : binary_bicone P Q) :\n  c.to_cone.π.app ⟨walking_pair.right⟩ = c.snd := rfl\n@[simp]\nlemma binary_fan_fst_to_cone (c : binary_bicone P Q) : binary_fan.fst c.to_cone = c.fst := rfl\n@[simp]\nlemma binary_fan_snd_to_cone (c : binary_bicone P Q) : binary_fan.snd c.to_cone = c.snd := rfl\n\n/-- Extract the cocone from a binary bicone. -/\ndef to_cocone (c : binary_bicone P Q) : cocone (pair P Q) :=\nbinary_cofan.mk c.inl c.inr\n\n@[simp]\nlemma to_cocone_X (c : binary_bicone P Q) :\n  c.to_cocone.X = c.X := rfl\n\n@[simp]\nlemma to_cocone_ι_app_left (c : binary_bicone P Q) :\n  c.to_cocone.ι.app ⟨walking_pair.left⟩ = c.inl := rfl\n@[simp]\nlemma to_cocone_ι_app_right (c : binary_bicone P Q) :\n  c.to_cocone.ι.app ⟨walking_pair.right⟩ = c.inr := rfl\n@[simp]\nlemma binary_cofan_inl_to_cocone (c : binary_bicone P Q) : binary_cofan.inl c.to_cocone = c.inl :=\nrfl\n@[simp]\nlemma binary_cofan_inr_to_cocone (c : binary_bicone P Q) : binary_cofan.inr c.to_cocone = c.inr :=\nrfl\n\ninstance (c : binary_bicone P Q) : is_split_mono c.inl :=\nis_split_mono.mk' { retraction := c.fst, id' := c.inl_fst }\n\ninstance (c : binary_bicone P Q) : is_split_mono c.inr :=\nis_split_mono.mk'  { retraction := c.snd, id' := c.inr_snd }\n\ninstance (c : binary_bicone P Q) : is_split_epi c.fst :=\nis_split_epi.mk' { section_ := c.inl, id' := c.inl_fst }\n\ninstance (c : binary_bicone P Q) : is_split_epi c.snd :=\nis_split_epi.mk' { section_ := c.inr, id' := c.inr_snd }\n\n/-- Convert a `binary_bicone` into a `bicone` over a pair. -/\n@[simps]\ndef to_bicone {X Y : C} (b : binary_bicone X Y) : bicone (pair_function X Y) :=\n{ X := b.X,\n  π := λ j, walking_pair.cases_on j b.fst b.snd,\n  ι := λ j, walking_pair.cases_on j b.inl b.inr,\n  ι_π := λ j j', by { rcases j with ⟨⟩; rcases j' with ⟨⟩, tidy } }\n\n/-- A binary bicone is a limit cone if and only if the corresponding bicone is a limit cone. -/\ndef to_bicone_is_limit {X Y : C} (b : binary_bicone X Y) :\n  is_limit (b.to_bicone.to_cone) ≃ is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\n/-- A binary bicone is a colimit cocone if and only if the corresponding bicone is a colimit\n    cocone. -/\ndef to_bicone_is_colimit {X Y : C} (b : binary_bicone X Y) :\n  is_colimit (b.to_bicone.to_cocone) ≃ is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\nend binary_bicone\n\nnamespace bicone\n\n/-- Convert a `bicone` over a function on `walking_pair` to a binary_bicone. -/\n@[simps]\ndef to_binary_bicone {X Y : C} (b : bicone (pair_function X Y)) : binary_bicone X Y :=\n{ X := b.X,\n  fst := b.π walking_pair.left,\n  snd := b.π walking_pair.right,\n  inl := b.ι walking_pair.left,\n  inr := b.ι walking_pair.right,\n  inl_fst' := by { simp [bicone.ι_π], refl, },\n  inr_fst' := by simp [bicone.ι_π],\n  inl_snd' := by simp [bicone.ι_π],\n  inr_snd' := by { simp [bicone.ι_π], refl, }, }\n\n/-- A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit\n    cone.  -/\ndef to_binary_bicone_is_limit {X Y : C} (b : bicone (pair_function X Y)) :\n  is_limit (b.to_binary_bicone.to_cone) ≃ is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩; tidy })\n\n/-- A bicone over a pair is a colimit cocone if and only if the corresponding binary bicone is a\n    colimit cocone. -/\ndef to_binary_bicone_is_colimit {X Y : C} (b : bicone (pair_function X Y)) :\n  is_colimit (b.to_binary_bicone.to_cocone) ≃ is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { rcases j with ⟨⟨⟩⟩; tidy })\n\nend bicone\n\n/-- Structure witnessing that a binary bicone is a limit cone and a limit cocone. -/\n@[nolint has_nonempty_instance]\nstructure binary_bicone.is_bilimit {P Q : C} (b : binary_bicone P Q) :=\n(is_limit : is_limit b.to_cone)\n(is_colimit : is_colimit b.to_cocone)\n\n/-- A binary bicone is a bilimit bicone if and only if the corresponding bicone is a bilimit. -/\ndef binary_bicone.to_bicone_is_bilimit {X Y : C} (b : binary_bicone X Y) :\n  b.to_bicone.is_bilimit ≃ b.is_bilimit :=\n{ to_fun := λ h, ⟨b.to_bicone_is_limit h.is_limit, b.to_bicone_is_colimit h.is_colimit⟩,\n  inv_fun := λ h, ⟨b.to_bicone_is_limit.symm h.is_limit, b.to_bicone_is_colimit.symm h.is_colimit⟩,\n  left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },\n  right_inv := λ ⟨h, h'⟩, by { dsimp only, simp } }\n\n/-- A bicone over a pair is a bilimit bicone if and only if the corresponding binary bicone is a\n    bilimit. -/\ndef bicone.to_binary_bicone_is_bilimit {X Y : C} (b : bicone (pair_function X Y)) :\n  b.to_binary_bicone.is_bilimit ≃ b.is_bilimit :=\n{ to_fun := λ h, ⟨b.to_binary_bicone_is_limit h.is_limit,\n    b.to_binary_bicone_is_colimit h.is_colimit⟩,\n  inv_fun := λ h, ⟨b.to_binary_bicone_is_limit.symm h.is_limit,\n    b.to_binary_bicone_is_colimit.symm h.is_colimit⟩,\n  left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },\n  right_inv := λ ⟨h, h'⟩, by { dsimp only, simp } }\n\n/--\nA bicone over `P Q : C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_nonempty_instance]\nstructure binary_biproduct_data (P Q : C) :=\n(bicone : binary_bicone P Q)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_binary_biproduct P Q` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`.\n-/\nclass has_binary_biproduct (P Q : C) : Prop :=\nmk' :: (exists_binary_biproduct : nonempty (binary_biproduct_data P Q))\n\nlemma has_binary_biproduct.mk {P Q : C} (d : binary_biproduct_data P Q) :\n  has_binary_biproduct P Q :=\n⟨nonempty.intro d⟩\n\n/--\nUse the axiom of choice to extract explicit `binary_biproduct_data F` from `has_binary_biproduct F`.\n-/\ndef get_binary_biproduct_data (P Q : C) [has_binary_biproduct P Q] : binary_biproduct_data P Q :=\nclassical.choice has_binary_biproduct.exists_binary_biproduct\n\n/-- A bicone for `P Q ` which is both a limit cone and a colimit cocone. -/\ndef binary_biproduct.bicone (P Q : C) [has_binary_biproduct P Q] : binary_bicone P Q :=\n(get_binary_biproduct_data P Q).bicone\n\n/-- `binary_biproduct.bicone P Q` is a limit bicone. -/\ndef binary_biproduct.is_bilimit (P Q : C) [has_binary_biproduct P Q] :\n  (binary_biproduct.bicone P Q).is_bilimit :=\n(get_binary_biproduct_data P Q).is_bilimit\n\n/-- `binary_biproduct.bicone P Q` is a limit cone. -/\ndef binary_biproduct.is_limit (P Q : C) [has_binary_biproduct P Q] :\n  is_limit (binary_biproduct.bicone P Q).to_cone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_limit\n\n/-- `binary_biproduct.bicone P Q` is a colimit cocone. -/\ndef binary_biproduct.is_colimit (P Q : C) [has_binary_biproduct P Q] :\n  is_colimit (binary_biproduct.bicone P Q).to_cocone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_colimit\n\nsection\nvariable (C)\n\n/--\n`has_binary_biproducts C` represents the existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`, for every `P Q : C`.\n-/\nclass has_binary_biproducts : Prop :=\n(has_binary_biproduct : Π (P Q : C), has_binary_biproduct P Q)\n\nattribute [instance, priority 100] has_binary_biproducts.has_binary_biproduct\n\n/--\nA category with finite biproducts has binary biproducts.\n\nThis is not an instance as typically in concrete categories there will be\nan alternative construction with nicer definitional properties.\n-/\nlemma has_binary_biproducts_of_finite_biproducts [has_finite_biproducts C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := λ P Q, has_binary_biproduct.mk\n  { bicone := (biproduct.bicone (pair_function P Q)).to_binary_bicone,\n    is_bilimit := (bicone.to_binary_bicone_is_bilimit _).symm (biproduct.is_bilimit _) } }\n\nend\n\nvariables {P Q : C}\n\ninstance has_binary_biproduct.has_limit_pair [has_binary_biproduct P Q] :\n  has_limit (pair P Q) :=\nhas_limit.mk ⟨_, binary_biproduct.is_limit P Q⟩\n\ninstance has_binary_biproduct.has_colimit_pair [has_binary_biproduct P Q] :\n  has_colimit (pair P Q) :=\nhas_colimit.mk ⟨_, binary_biproduct.is_colimit P Q⟩\n\n@[priority 100]\ninstance has_binary_products_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_products C :=\n{ has_limit := λ F, has_limit_of_iso (diagram_iso_pair F).symm }\n@[priority 100]\ninstance has_binary_coproducts_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_coproducts C :=\n{ has_colimit := λ F, has_colimit_of_iso (diagram_iso_pair F) }\n\n/--\nThe isomorphism between the specified binary product and the specified binary coproduct for\na pair for a binary biproduct.\n-/\ndef biprod_iso (X Y : C) [has_binary_biproduct X Y]  :\n  limits.prod X Y ≅ limits.coprod X Y :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (binary_biproduct.is_limit X Y)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _)\n\n/-- An arbitrary choice of biproduct of a pair of objects. -/\nabbreviation biprod (X Y : C) [has_binary_biproduct X Y] := (binary_biproduct.bicone X Y).X\n\nnotation X ` ⊞ `:20 Y:20 := biprod X Y\n\n/-- The projection onto the first summand of a binary biproduct. -/\nabbreviation biprod.fst {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ X :=\n(binary_biproduct.bicone X Y).fst\n/-- The projection onto the second summand of a binary biproduct. -/\nabbreviation biprod.snd {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ Y :=\n(binary_biproduct.bicone X Y).snd\n/-- The inclusion into the first summand of a binary biproduct. -/\nabbreviation biprod.inl {X Y : C} [has_binary_biproduct X Y] : X ⟶ X ⊞ Y :=\n(binary_biproduct.bicone X Y).inl\n/-- The inclusion into the second summand of a binary biproduct. -/\nabbreviation biprod.inr {X Y : C} [has_binary_biproduct X Y] : Y ⟶ X ⊞ Y :=\n(binary_biproduct.bicone X Y).inr\n\nsection\nvariables {X Y : C} [has_binary_biproduct X Y]\n\n@[simp] lemma binary_biproduct.bicone_fst : (binary_biproduct.bicone X Y).fst = biprod.fst := rfl\n@[simp] lemma binary_biproduct.bicone_snd : (binary_biproduct.bicone X Y).snd = biprod.snd := rfl\n@[simp] lemma binary_biproduct.bicone_inl : (binary_biproduct.bicone X Y).inl = biprod.inl := rfl\n@[simp] lemma binary_biproduct.bicone_inr : (binary_biproduct.bicone X Y).inr = biprod.inr := rfl\n\nend\n\n@[simp,reassoc]\nlemma biprod.inl_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 𝟙 X :=\n(binary_biproduct.bicone X Y).inl_fst\n@[simp,reassoc]\nlemma biprod.inl_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 0 :=\n(binary_biproduct.bicone X Y).inl_snd\n@[simp,reassoc]\nlemma biprod.inr_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 0 :=\n(binary_biproduct.bicone X Y).inr_fst\n@[simp,reassoc]\nlemma biprod.inr_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 𝟙 Y :=\n(binary_biproduct.bicone X Y).inr_snd\n\n/-- Given a pair of maps into the summands of a binary biproduct,\nwe obtain a map into the binary biproduct. -/\nabbreviation biprod.lift {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  W ⟶ X ⊞ Y :=\n(binary_biproduct.is_limit X Y).lift (binary_fan.mk f g)\n/-- Given a pair of maps out of the summands of a binary biproduct,\nwe obtain a map out of the binary biproduct. -/\nabbreviation biprod.desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  X ⊞ Y ⟶ W :=\n(binary_biproduct.is_colimit X Y).desc (binary_cofan.mk f g)\n\n@[simp, reassoc]\nlemma biprod.lift_fst {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  biprod.lift f g ≫ biprod.fst = f :=\n(binary_biproduct.is_limit X Y).fac _ ⟨walking_pair.left⟩\n\n@[simp, reassoc]\nlemma biprod.lift_snd {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  biprod.lift f g ≫ biprod.snd = g :=\n(binary_biproduct.is_limit X Y).fac _ ⟨walking_pair.right⟩\n\n@[simp, reassoc]\nlemma biprod.inl_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  biprod.inl ≫ biprod.desc f g = f :=\n(binary_biproduct.is_colimit X Y).fac _ ⟨walking_pair.left⟩\n\n@[simp, reassoc]\nlemma biprod.inr_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  biprod.inr ≫ biprod.desc f g = g :=\n(binary_biproduct.is_colimit X Y).fac _ ⟨walking_pair.right⟩\n\ninstance biprod.mono_lift_of_mono_left {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)\n  (g : W ⟶ Y) [mono f] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_fst _ _\n\ninstance biprod.mono_lift_of_mono_right {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)\n  (g : W ⟶ Y) [mono g] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_snd _ _\n\ninstance biprod.epi_desc_of_epi_left {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)\n  [epi f] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inl_desc _ _\n\ninstance biprod.epi_desc_of_epi_right {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)\n  [epi g] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inr_desc _ _\n\n/-- Given a pair of maps between the summands of a pair of binary biproducts,\nwe obtain a map between the binary biproducts. -/\nabbreviation biprod.map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z :=\nis_limit.map (binary_biproduct.bicone W X).to_cone (binary_biproduct.is_limit Y Z)\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n/-- An alternative to `biprod.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biprod.map`. -/\nabbreviation biprod.map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z :=\nis_colimit.map (binary_biproduct.is_colimit W X) (binary_biproduct.bicone Y Z).to_cocone\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n@[ext] lemma biprod.hom_ext {X Y Z : C} [has_binary_biproduct X Y] (f g : Z ⟶ X ⊞ Y)\n  (h₀ : f ≫ biprod.fst = g ≫ biprod.fst) (h₁ : f ≫ biprod.snd = g ≫ biprod.snd) : f = g :=\nbinary_fan.is_limit.hom_ext (binary_biproduct.is_limit X Y) h₀ h₁\n\n@[ext] lemma biprod.hom_ext' {X Y Z : C} [has_binary_biproduct X Y] (f g : X ⊞ Y ⟶ Z)\n  (h₀ : biprod.inl ≫ f = biprod.inl ≫ g) (h₁ : biprod.inr ≫ f = biprod.inr ≫ g) : f = g :=\nbinary_cofan.is_colimit.hom_ext (binary_biproduct.is_colimit X Y) h₀ h₁\n\n/-- The canonical isomorphism between the chosen biproduct and the chosen product. -/\ndef biprod.iso_prod (X Y : C) [has_binary_biproduct X Y] : X ⊞ Y ≅ X ⨯ Y :=\nis_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit X Y) (limit.is_limit _)\n\n@[simp] lemma biprod.iso_prod_hom {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.iso_prod X Y).hom = prod.lift biprod.fst biprod.snd :=\nby ext; simp [biprod.iso_prod]\n\n@[simp] lemma biprod.iso_prod_inv {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.iso_prod X Y).inv = biprod.lift prod.fst prod.snd :=\nby apply biprod.hom_ext; simp [iso.inv_comp_eq]\n\n/-- The canonical isomorphism between the chosen biproduct and the chosen coproduct. -/\ndef biprod.iso_coprod (X Y : C) [has_binary_biproduct X Y] : X ⊞ Y ≅ X ⨿ Y :=\nis_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _)\n\n@[simp] lemma biprod.iso_coprod_inv {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.iso_coprod X Y).inv = coprod.desc biprod.inl biprod.inr :=\nby ext; simp [biprod.iso_coprod]; refl\n\n@[simp] lemma biprod_iso_coprod_hom {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.iso_coprod X Y).hom = biprod.desc coprod.inl coprod.inr :=\nby apply biprod.hom_ext'; simp [← iso.eq_comp_inv]\n\nlemma biprod.map_eq_map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : biprod.map f g = biprod.map' f g :=\nbegin\n  ext,\n  { simp only [map_pair_left, is_colimit.ι_map, is_limit.map_π, biprod.inl_fst_assoc,\n    category.assoc, ←binary_bicone.to_cone_π_app_left, ←binary_biproduct.bicone_fst,\n    ←binary_bicone.to_cocone_ι_app_left, ←binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_left, is_colimit.ι_map, is_limit.map_π, zero_comp,\n      biprod.inl_snd_assoc, category.assoc,\n      ←binary_bicone.to_cone_π_app_right, ←binary_biproduct.bicone_snd,\n      ←binary_bicone.to_cocone_ι_app_left, ←binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_right, biprod.inr_fst_assoc, is_colimit.ι_map, is_limit.map_π,\n      zero_comp, category.assoc,\n      ←binary_bicone.to_cone_π_app_left, ←binary_biproduct.bicone_fst,\n      ←binary_bicone.to_cocone_ι_app_right, ←binary_biproduct.bicone_inr],\n    simp },\n  { simp only [map_pair_right, is_colimit.ι_map, is_limit.map_π, biprod.inr_snd_assoc,\n      category.assoc, ←binary_bicone.to_cone_π_app_right, ←binary_biproduct.bicone_snd,\n      ←binary_bicone.to_cocone_ι_app_right, ←binary_biproduct.bicone_inr],\n    simp }\nend\n\ninstance biprod.inl_mono {X Y : C} [has_binary_biproduct X Y] :\n  is_split_mono (biprod.inl : X ⟶ X ⊞ Y) :=\nis_split_mono.mk' { retraction := biprod.fst }\n\ninstance biprod.inr_mono {X Y : C} [has_binary_biproduct X Y] :\n  is_split_mono (biprod.inr : Y ⟶ X ⊞ Y) :=\nis_split_mono.mk' { retraction := biprod.snd }\n\ninstance biprod.fst_epi {X Y : C} [has_binary_biproduct X Y] :\n  is_split_epi (biprod.fst : X ⊞ Y ⟶ X) :=\nis_split_epi.mk' { section_ := biprod.inl }\n\ninstance biprod.snd_epi {X Y : C} [has_binary_biproduct X Y] :\n  is_split_epi (biprod.snd : X ⊞ Y ⟶ Y) :=\nis_split_epi.mk' { section_ := biprod.inr }\n\n@[simp,reassoc]\nlemma biprod.map_fst {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.map f g ≫ biprod.fst = biprod.fst ≫ f :=\nis_limit.map_π _ _ _ (⟨walking_pair.left⟩ : discrete walking_pair)\n\n@[simp,reassoc]\nlemma biprod.map_snd {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.map f g ≫ biprod.snd = biprod.snd ≫ g :=\nis_limit.map_π _ _ _ (⟨walking_pair.right⟩ : discrete walking_pair)\n\n-- Because `biprod.map` is defined in terms of `lim` rather than `colim`,\n-- we need to provide additional `simp` lemmas.\n@[simp,reassoc]\nlemma biprod.inl_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.inl ≫ biprod.map f g = f ≫ biprod.inl :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ ⟨walking_pair.left⟩\nend\n\n@[simp,reassoc]\nlemma biprod.inr_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.inr ≫ biprod.map f g = g ≫ biprod.inr :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ ⟨walking_pair.right⟩\nend\n\n/-- Given a pair of isomorphisms between the summands of a pair of binary biproducts,\nwe obtain an isomorphism between the binary biproducts. -/\n@[simps]\ndef biprod.map_iso {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ≅ Y) (g : X ≅ Z) : W ⊞ X ≅ Y ⊞ Z :=\n{ hom := biprod.map f.hom g.hom,\n  inv := biprod.map f.inv g.inv }\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_hom (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).hom\n    = biprod.lift b.fst b.snd :=\nrfl\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_inv (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).inv\n    = biprod.desc b.inl b.inr :=\nbegin\n  refine biprod.hom_ext' _ _ (hb.is_limit.hom_ext (λ j, _)) (hb.is_limit.hom_ext (λ j, _)),\n  all_goals { simp only [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp],\n    rcases j with ⟨⟨⟩⟩ },\n  all_goals { simp }\nend\n\n/-- Binary biproducts are unique up to isomorphism. This already follows because bilimits are\n    limits, but in the case of biproducts we can give an isomorphism with particularly nice\n    definitional properties, namely that `biprod.lift b.fst b.snd` and `biprod.desc b.inl b.inr`\n    are inverses of each other. -/\n@[simps]\ndef biprod.unique_up_to_iso (X Y : C) [has_binary_biproduct X Y] {b : binary_bicone X Y}\n  (hb : b.is_bilimit) : b.X ≅ X ⊞ Y :=\n{ hom := biprod.lift b.fst b.snd,\n  inv := biprod.desc b.inl b.inr,\n  hom_inv_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    ← biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    ← biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.inv_hom_id] }\n\n-- There are three further variations,\n-- about `is_iso biprod.inr`, `is_iso biprod.fst` and `is_iso biprod.snd`,\n-- but any one suffices to prove `indecomposable_of_simple`\n-- and they are likely not separately useful.\nlemma biprod.is_iso_inl_iff_id_eq_fst_comp_inl (X Y : C) [has_binary_biproduct X Y] :\n  is_iso (biprod.inl : X ⟶ X ⊞ Y) ↔ 𝟙 (X ⊞ Y) = biprod.fst ≫ biprod.inl :=\nbegin\n  split,\n  { introI h,\n    have := (cancel_epi (inv biprod.inl : X ⊞ Y ⟶ X)).2 biprod.inl_fst,\n    rw [is_iso.inv_hom_id_assoc, category.comp_id] at this,\n    rw [this, is_iso.inv_hom_id], },\n  { intro h, exact ⟨⟨biprod.fst, biprod.inl_fst, h.symm⟩⟩, },\nend\n\nsection biprod_kernel\n\nsection binary_bicone\n\nvariables {X Y : C} (c : binary_bicone X Y)\n\n/-- A kernel fork for the kernel of `binary_bicone.fst`. It consists of the morphism\n`binary_bicone.inr`. -/\ndef binary_bicone.fst_kernel_fork : kernel_fork c.fst := kernel_fork.of_ι c.inr c.inr_fst\n\n@[simp] lemma binary_bicone.fst_kernel_fork_ι : (binary_bicone.fst_kernel_fork c).ι = c.inr := rfl\n\n/-- A kernel fork for the kernel of `binary_bicone.snd`. It consists of the morphism\n`binary_bicone.inl`. -/\ndef binary_bicone.snd_kernel_fork : kernel_fork c.snd := kernel_fork.of_ι c.inl c.inl_snd\n\n@[simp] lemma binary_bicone.snd_kernel_fork_ι : (binary_bicone.snd_kernel_fork c).ι = c.inl := rfl\n\n/-- A cokernel cofork for the cokernel of `binary_bicone.inl`. It consists of the morphism\n`binary_bicone.snd`. -/\ndef binary_bicone.inl_cokernel_cofork : cokernel_cofork c.inl :=\ncokernel_cofork.of_π c.snd c.inl_snd\n\n@[simp] lemma binary_bicone.inl_cokernel_cofork_π :\n  (binary_bicone.inl_cokernel_cofork c).π = c.snd := rfl\n\n/-- A cokernel cofork for the cokernel of `binary_bicone.inr`. It consists of the morphism\n`binary_bicone.fst`. -/\ndef binary_bicone.inr_cokernel_cofork : cokernel_cofork c.inr :=\ncokernel_cofork.of_π c.fst c.inr_fst\n\n@[simp] lemma binary_bicone.inr_cokernel_cofork_π :\n  (binary_bicone.inr_cokernel_cofork c).π = c.fst := rfl\n\nvariables {c}\n\n/-- The fork defined in `binary_bicone.fst_kernel_fork` is indeed a kernel. -/\ndef binary_bicone.is_limit_fst_kernel_fork (i : is_limit c.to_cone) :\n  is_limit c.fst_kernel_fork :=\nfork.is_limit.mk' _ $ λ s,\n⟨s.ι ≫ c.snd, by apply binary_fan.is_limit.hom_ext i; simp, λ m hm, by simp [← hm]⟩\n\n/-- The fork defined in `binary_bicone.snd_kernel_fork` is indeed a kernel. -/\ndef binary_bicone.is_limit_snd_kernel_fork (i : is_limit c.to_cone) :\n  is_limit c.snd_kernel_fork :=\nfork.is_limit.mk' _ $ λ s,\n⟨s.ι ≫ c.fst, by apply binary_fan.is_limit.hom_ext i; simp, λ m hm, by simp [← hm]⟩\n\n/-- The cofork defined in `binary_bicone.inl_cokernel_cofork` is indeed a cokernel. -/\ndef binary_bicone.is_colimit_inl_cokernel_cofork (i : is_colimit c.to_cocone) :\n  is_colimit c.inl_cokernel_cofork :=\ncofork.is_colimit.mk' _ $ λ s, ⟨c.inr ≫ s.π, by apply binary_cofan.is_colimit.hom_ext i; simp,\n  λ m hm, by simp [← hm]⟩\n\n/-- The cofork defined in `binary_bicone.inr_cokernel_cofork` is indeed a cokernel. -/\ndef binary_bicone.is_colimit_inr_cokernel_cofork (i : is_colimit c.to_cocone) :\n  is_colimit c.inr_cokernel_cofork :=\ncofork.is_colimit.mk' _ $ λ s, ⟨c.inl ≫ s.π, by apply binary_cofan.is_colimit.hom_ext i; simp,\n  λ m hm, by simp [← hm]⟩\n\nend binary_bicone\n\nsection has_binary_biproduct\n\nvariables (X Y : C) [has_binary_biproduct X Y]\n\n/-- A kernel fork for the kernel of `biprod.fst`. It consists of the\nmorphism `biprod.inr`. -/\ndef biprod.fst_kernel_fork : kernel_fork (biprod.fst : X ⊞ Y ⟶ X) :=\nbinary_bicone.fst_kernel_fork _\n\n@[simp]\nlemma biprod.fst_kernel_fork_ι : fork.ι (biprod.fst_kernel_fork X Y) = biprod.inr :=\nrfl\n\n/-- The fork `biprod.fst_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_fst_kernel_fork : is_limit (biprod.fst_kernel_fork X Y) :=\nbinary_bicone.is_limit_fst_kernel_fork (binary_biproduct.is_limit _ _)\n\n/-- A kernel fork for the kernel of `biprod.snd`. It consists of the\nmorphism `biprod.inl`. -/\ndef biprod.snd_kernel_fork : kernel_fork (biprod.snd : X ⊞ Y ⟶ Y) :=\nbinary_bicone.snd_kernel_fork _\n\n@[simp]\nlemma biprod.snd_kernel_fork_ι : fork.ι (biprod.snd_kernel_fork X Y) = biprod.inl :=\nrfl\n\n/-- The fork `biprod.snd_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_snd_kernel_fork : is_limit (biprod.snd_kernel_fork X Y) :=\nbinary_bicone.is_limit_snd_kernel_fork (binary_biproduct.is_limit _ _)\n\n/-- A cokernel cofork for the cokernel of `biprod.inl`. It consists of the\nmorphism `biprod.snd`. -/\ndef biprod.inl_cokernel_cofork : cokernel_cofork (biprod.inl : X ⟶ X ⊞ Y) :=\nbinary_bicone.inl_cokernel_cofork _\n\n@[simp]\nlemma biprod.inl_cokernel_cofork_π : cofork.π (biprod.inl_cokernel_cofork X Y) = biprod.snd :=\nrfl\n\n/-- The cofork `biprod.inl_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inl_cokernel_fork : is_colimit (biprod.inl_cokernel_cofork X Y) :=\nbinary_bicone.is_colimit_inl_cokernel_cofork (binary_biproduct.is_colimit _ _)\n\n/-- A cokernel cofork for the cokernel of `biprod.inr`. It consists of the\nmorphism `biprod.fst`. -/\ndef biprod.inr_cokernel_cofork : cokernel_cofork (biprod.inr : Y ⟶ X ⊞ Y) :=\nbinary_bicone.inr_cokernel_cofork _\n\n@[simp]\nlemma biprod.inr_cokernel_cofork_π : cofork.π (biprod.inr_cokernel_cofork X Y) = biprod.fst :=\nrfl\n\n/-- The cofork `biprod.inr_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inr_cokernel_fork : is_colimit (biprod.inr_cokernel_cofork X Y) :=\nbinary_bicone.is_colimit_inr_cokernel_cofork (binary_biproduct.is_colimit _ _)\n\nend has_binary_biproduct\n\nvariables {X Y : C} [has_binary_biproduct X Y]\n\ninstance : has_kernel (biprod.fst : X ⊞ Y ⟶ X) :=\nhas_limit.mk ⟨_, biprod.is_kernel_fst_kernel_fork X Y⟩\n\n/-- The kernel of `biprod.fst : X ⊞ Y ⟶ X` is `Y`. -/\n@[simps]\ndef kernel_biprod_fst_iso : kernel (biprod.fst : X ⊞ Y ⟶ X) ≅ Y :=\nlimit.iso_limit_cone ⟨_, biprod.is_kernel_fst_kernel_fork X Y⟩\n\ninstance : has_kernel (biprod.snd : X ⊞ Y ⟶ Y) :=\nhas_limit.mk ⟨_, biprod.is_kernel_snd_kernel_fork X Y⟩\n\n/-- The kernel of `biprod.snd : X ⊞ Y ⟶ Y` is `X`. -/\n@[simps]\ndef kernel_biprod_snd_iso : kernel (biprod.snd : X ⊞ Y ⟶ Y) ≅ X :=\nlimit.iso_limit_cone ⟨_, biprod.is_kernel_snd_kernel_fork X Y⟩\n\ninstance : has_cokernel (biprod.inl : X ⟶ X ⊞ Y) :=\nhas_colimit.mk ⟨_, biprod.is_cokernel_inl_cokernel_fork X Y⟩\n\n/-- The cokernel of `biprod.inl : X ⟶ X ⊞ Y` is `Y`. -/\n@[simps]\ndef cokernel_biprod_inl_iso : cokernel (biprod.inl : X ⟶ X ⊞ Y) ≅ Y :=\ncolimit.iso_colimit_cocone ⟨_, biprod.is_cokernel_inl_cokernel_fork X Y⟩\n\ninstance : has_cokernel (biprod.inr : Y ⟶ X ⊞ Y) :=\nhas_colimit.mk ⟨_, biprod.is_cokernel_inr_cokernel_fork X Y⟩\n\n/-- The cokernel of `biprod.inr : Y ⟶ X ⊞ Y` is `X`. -/\n@[simps]\ndef cokernel_biprod_inr_iso : cokernel (biprod.inr : Y ⟶ X ⊞ Y) ≅ X :=\ncolimit.iso_colimit_cocone ⟨_, biprod.is_cokernel_inr_cokernel_fork X Y⟩\n\nend biprod_kernel\n\nsection is_zero\n\n/-- If `Y` is a zero object, `X ≅ X ⊞ Y` for any `X`. -/\n@[simps]\ndef iso_biprod_zero {X Y : C} [has_binary_biproduct X Y] (hY : is_zero Y) : X ≅ X ⊞ Y :=\n{ hom := biprod.inl,\n  inv := biprod.fst,\n  inv_hom_id' := begin\n    apply category_theory.limits.biprod.hom_ext;\n    simp only [category.assoc, biprod.inl_fst, category.comp_id, category.id_comp,\n      biprod.inl_snd, comp_zero],\n    apply hY.eq_of_tgt\n  end }\n\n/-- If `X` is a zero object, `Y ≅ X ⊞ Y` for any `Y`. -/\n@[simps]\ndef iso_zero_biprod {X Y : C} [has_binary_biproduct X Y] (hY : is_zero X) : Y ≅ X ⊞ Y :=\n{ hom := biprod.inr,\n  inv := biprod.snd,\n  inv_hom_id' := begin\n    apply category_theory.limits.biprod.hom_ext;\n    simp only [category.assoc, biprod.inr_snd, category.comp_id, category.id_comp,\n      biprod.inr_fst, comp_zero],\n    apply hY.eq_of_tgt\n  end }\n\nend is_zero\n\nsection\nvariables [has_binary_biproducts C]\n\n/-- The braiding isomorphism which swaps a binary biproduct. -/\n@[simps] def biprod.braiding (P Q : C) : P ⊞ Q ≅ Q ⊞ P :=\n{ hom := biprod.lift biprod.snd biprod.fst,\n  inv := biprod.lift biprod.snd biprod.fst }\n\n/--\nAn alternative formula for the braiding isomorphism which swaps a binary biproduct,\nusing the fact that the biproduct is a coproduct.\n-/\n@[simps]\ndef biprod.braiding' (P Q : C) : P ⊞ Q ≅ Q ⊞ P :=\n{ hom := biprod.desc biprod.inr biprod.inl,\n  inv := biprod.desc biprod.inr biprod.inl }\n\nlemma biprod.braiding'_eq_braiding {P Q : C} :\n  biprod.braiding' P Q = biprod.braiding P Q :=\nby tidy\n\n/-- The braiding isomorphism can be passed through a map by swapping the order. -/\n@[reassoc] lemma biprod.braid_natural {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :\n  biprod.map f g ≫ (biprod.braiding _ _).hom = (biprod.braiding _ _).hom ≫ biprod.map g f :=\nby tidy\n\n@[reassoc] lemma biprod.braiding_map_braiding {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) :\n  (biprod.braiding X W).hom ≫ biprod.map f g ≫ (biprod.braiding Y Z).hom = biprod.map g f :=\nby tidy\n\n@[simp, reassoc] lemma biprod.symmetry' (P Q : C) :\n  biprod.lift biprod.snd biprod.fst ≫ biprod.lift biprod.snd biprod.fst = 𝟙 (P ⊞ Q) :=\nby tidy\n\n/-- The braiding isomorphism is symmetric. -/\n@[reassoc] lemma biprod.symmetry (P Q : C) :\n  (biprod.braiding P Q).hom ≫ (biprod.braiding Q P).hom = 𝟙 _ :=\nby simp\n\nend\n\nend limits\n\nopen category_theory.limits\n\n-- TODO:\n-- If someone is interested, they could provide the constructions:\n--   has_binary_biproducts ↔ has_finite_biproducts\nvariables {C : Type.{u}} [category.{v} C] [has_zero_morphisms C] [has_binary_biproducts C]\n\n/-- An object is indecomposable if it cannot be written as the biproduct of two nonzero objects. -/\ndef indecomposable (X : C) : Prop := ¬ is_zero X ∧ ∀ Y Z, (X ≅ Y ⊞ Z) → is_zero Y ∨ is_zero Z\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `f` is invertible.\n-/\nlemma is_iso_left_of_is_iso_biprod_map\n  {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (biprod.map f g)] : is_iso f :=\n⟨⟨biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst,\n  ⟨begin\n    have t := congr_arg (λ p : W ⊞ X ⟶ W ⊞ X, biprod.inl ≫ p ≫ biprod.fst)\n      (is_iso.hom_inv_id (biprod.map f g)),\n    simp only [category.id_comp, category.assoc, biprod.inl_map_assoc] at t,\n    simp [t],\n  end,\n  begin\n    have t := congr_arg (λ p : Y ⊞ Z ⟶ Y ⊞ Z, biprod.inl ≫ p ≫ biprod.fst)\n      (is_iso.inv_hom_id (biprod.map f g)),\n    simp only [category.id_comp, category.assoc, biprod.map_fst] at t,\n    simp only [category.assoc],\n    simp [t],\n  end⟩⟩⟩\n\n/--\nIf\n```\n(f 0)\n(0 g)\n```\nis invertible, then `g` is invertible.\n-/\nlemma is_iso_right_of_is_iso_biprod_map\n  {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) [is_iso (biprod.map f g)] : is_iso g :=\nbegin\n  letI : is_iso (biprod.map g f) := by\n  { rw [←biprod.braiding_map_braiding],\n    apply_instance, },\n  exact is_iso_left_of_is_iso_biprod_map g f,\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/shapes/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3301925746960095}}
{"text": "/-\nCopyright (c) 2023 Wojciech Nawrocki. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wojciech Nawrocki\n-/\n\nimport Std.Classes.BEq\n\nimport Mathlib.Tactic.Linarith\nimport Mathlib.Data.List.Lemmas\nimport Mathlib.Data.List.Perm\n\n/-! Std.Logic or Std.Bool? -/\n\n@[simp] theorem Bool.bnot_eq_to_not_eq (a b : Bool) :\n  ((!a) = b) = ¬(a = b) := by cases a <;> cases b <;> decide\n@[simp] theorem Bool.eq_bnot_to_not_eq (a b : Bool) :\n  (a = (!b)) = ¬(a = b) := by cases a <;> cases b <;> decide\n@[simp] theorem Bool.eq_true_iff_eq_true_to_eq (a b : Bool) :\n  (a = true ↔ b = true) = (a = b) := by cases a <;> cases b <;> decide\n@[simp] theorem Bool.eq_false_iff_eq_false_to_eq (a b : Bool) :\n  (a = false ↔ b = false) = (a = b) := by cases a <;> cases b <;> decide\n\n/-! Std.Logic -/\n\ntheorem Bool.not_eq_true_iff_ne_true {b : Bool} : (!b) = true ↔ ¬(b = true) := by\n  cases b <;> decide\n\ntheorem Bool.bne_iff_not_beq [BEq α] {a a' : α} : a != a' ↔ ¬(a == a') :=\n  Bool.not_eq_true_iff_ne_true\n\ntheorem Bool.beq_or_bne [BEq α] (a a' : α) : a == a' ∨ a != a' :=\n  by\n    cases h : a == a'\n    . apply Or.inr\n      simp [bne_iff_not_beq, h]\n    . exact Or.inl rfl\n\n@[simp]\ntheorem Bool.bne_eq_false [BEq α] {a a' : α} : (a != a') = false ↔ a == a' := by\n  dsimp [bne]\n  cases (a == a') <;> simp\n\n/-! Std.Classes.BEq -/\n\ninstance [BEq α] [LawfulBEq α] : PartialEquivBEq α where\n  symm h := by cases (beq_iff_eq _ _).mp h; exact h\n  trans h₁ h₂ := by cases (beq_iff_eq _ _).mp h₁; cases (beq_iff_eq _ _).mp h₂; exact h₁\n\ntheorem bne_symm [BEq α] [PartialEquivBEq α] {a b : α} : a != b → b != a :=\n  fun h => Bool.not_eq_true_iff_ne_true.mpr fun h' =>\n    Bool.bne_iff_not_beq.mp h (PartialEquivBEq.symm h')\n\n@[simp]\ntheorem bne_iff_ne [BEq α] [LawfulBEq α] (a b : α) : a != b ↔ a ≠ b := by\n  simp [Bool.bne_iff_not_beq]\n\n/-! Maybe Std.Notation -/\n\n/-- Notation typeclass for semantic entailment `⊨`. -/\nclass SemanticEntails (α : Type u) (β : outParam $ Type v) where\n  entails : α → β → Prop\n\ninfix:51 \" ⊨ \" => SemanticEntails.entails\ninfix:51 \" ⊭ \" => fun M φ => ¬(M ⊨ φ)\n\n/-! Data.List.Extra or something -/\n\n@[specialize]\ndef List.foldlDep {α : Type u} {β : Type v} : (l : List β) → (f : α → (b : β) → b ∈ l → α) →\n    (init : α) → α\n  | nil,      _, init => init\n  | cons b l, f, init => foldlDep l (fun a b h => f a b (.tail _ h)) (f init b (.head l))\n\n@[specialize]\ndef List.mapDep {α : Type u} {β : Type v} : (l : List α) → (f : (a : α) → a ∈ l → β) → List β\n  | nil,      _ => []\n  | cons a l, f => f a (.head l) :: mapDep l fun a h => f a (.tail _ h)\n\n@[simp]\ntheorem List.map_mapDep {γ : Type u} : (l : List α) → (f : (a : α) → a ∈ l → β) → (g : β → γ) →\n    (l.mapDep f).map g = l.mapDep (fun a h => g (f a h))\n  | nil,      _, _ => rfl\n  | cons a l, f, g => by\n    -- https://www.youtube.com/watch?v=Hd2JgADY9d8\n    simp [map, mapDep, map_mapDep]\n\n/-! Data.List.Lemmas -/\n\nnamespace List\n\n/-! drop -/\n\ntheorem drop_eq_cons_get (l : List α) (i : Nat) (h : i < l.length)\n    : l.drop i = l.get ⟨i, h⟩ :: l.drop (i + 1) :=\n  go i l h\nwhere go : (i : Nat) → (l : List α) → (h : i < l.length) → l.drop i = l[i] :: l.drop (i + 1)\n  | 0,   _::_,  _ => by simp\n  | n+1, _::as, h => by\n    have : n < length as := Nat.lt_of_succ_lt_succ h\n    have ih := go n as this\n    simp [ih]\n\ntheorem drop_ext (l₁ l₂ : List α) (j : Nat)\n    : (∀ i ≥ j, l₁.get? i = l₂.get? i) → l₁.drop j = l₂.drop j := by\n  intro H\n  apply ext fun k => ?_\n  rw [get?_drop, get?_drop]\n  apply H _ (Nat.le_add_right _ _)\n\n/-! find? -/\n\ntheorem find?_filter (l : List α) (p q : α → Bool) (h : ∀ a, p a → q a) :\n    (l.filter q).find? p = l.find? p := by\n  induction l with\n  | nil => rfl\n  | cons x xs ih =>\n    dsimp [filter]\n    split <;> split <;> simp [*] at *\n\ntheorem find?_filter' (l : List α) (p q : α → Bool) (h : ∀ a, p a → !q a) :\n    (l.filter q).find? p = none := by\n  induction l with\n  | nil => rfl\n  | cons x xs ih =>\n    dsimp [filter]\n    split\n    next _ hQ => rw [find?_cons_of_neg _ (fun hP => by aesop), ih]\n    next => exact ih\n\n-- theorem find?_eraseP (l : List α) (p q : α → Bool) (h : ∀ a, p a → !q a) :\n--     (l.eraseP q).find? p = l.find? p := by\n--   induction l with\n--   | nil => rfl\n--   | cons x xs ih =>\n--     dsimp [filter, eraseP]\n--     split\n--     next _ hP => aesop\n--     next _ hP =>\n--       cases (q x : Bool) with -- `split_ifs` doesn't work on `bif`\n--       | false => rw [cond_false, find?_cons_of_neg _ (by simp [hP]), ih]\n--       | true => rw [cond_true]\n\n/-! foldl -/\n\ntheorem foldl_cons_fn (l₁ l₂ : List α) :\n    l₁.foldl (init := l₂) (fun acc x => x :: acc) = l₁.reverse ++ l₂ := by\n  induction l₁ generalizing l₂ <;> simp [*]\n\ntheorem foldl_append_fn (l₁ : List α) (l₂ : List β) (f : α → List β) :\n    l₁.foldl (init := l₂) (fun acc x => acc ++ f x) = l₂ ++ l₁.bind f := by\n  induction l₁ generalizing l₂ <;> simp [*]\n\nend List\n\n/-! Data.Array.Lemmas -/\n\ntheorem Array.get_of_mem_data {as : Array α} {a : α} : a ∈ as.data → ∃ (i : Fin as.size), as[i] = a :=\n  List.get_of_mem\n\ntheorem Array.get_mem_data (as : Array α) (i : Fin as.size) : as[i] ∈ as.data := by\n  simp [getElem_mem_data]\n\n/-! Data.List.Perm -/\n\nnamespace List\n\n/-- The way Lean 4 computes the motive with `elab_as_elim` has changed\nrelative to the behaviour of `elab_as_eliminator` in Lean 3.\nSee\nhttps://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Potential.20elaboration.20bug.20with.20.60elabAsElim.60/near/299573172\nfor an explanation of the change made here relative to mathlib3.\n-/\n@[elab_as_elim]\ntheorem perm_induction_on'\n    {P : (l₁ : List α) → (l₂ : List α) → l₁ ~ l₂ → Prop} {l₁ l₂ : List α} (p : l₁ ~ l₂)\n    (nil : P [] [] .nil)\n    (cons : ∀ x l₁ l₂, (h : l₁ ~ l₂) → P l₁ l₂ h → P (x :: l₁) (x :: l₂) (.cons x h))\n    (swap : ∀ x y l₁ l₂, (h : l₁ ~ l₂) → P l₁ l₂ h →\n      P (y :: x :: l₁) (x :: y :: l₂) (.trans (.swap x y _) (.cons _ (.cons _ h))))\n    (trans : ∀ l₁ l₂ l₃, (h₁ : l₁ ~ l₂) → (h₂ : l₂ ~ l₃) → P l₁ l₂ h₁ → P l₂ l₃ h₂ →\n      P l₁ l₃ (.trans h₁ h₂)) : P l₁ l₂ p :=\n  have P_refl l : P l l (.refl l) :=\n    List.recOn l nil fun x xs ih ↦ cons x xs xs (Perm.refl xs) ih\n  Perm.recOn p nil cons (fun x y l ↦ swap x y l l (Perm.refl l) (P_refl l)) @trans\n\nend List\n\n/-! Maybe Data.List.Unique -/\n\nnamespace List\n\n/-- There is at most one element satisfying `p` in `l`. -/\n-- TODO Move the other `of_unique` lemmas to this formulation\n-- and move `pairwise'` lemmas below to `unique`\ndef unique (p : α → Prop) (l : List α) : Prop :=\n  l.Pairwise (p · → ¬p ·)\n\ndef unique_cons_of_true {x : α} {l : List α} {p : α → Prop} :\n    (x :: l).unique p → p x → ∀ y ∈ l, ¬p y :=\n  fun h hP _ hY => rel_of_pairwise_cons h hY hP\n\ntheorem unique.sublist {l₁ l₂ : List α} {p : α → Prop} : l₁ <+ l₂ → l₂.unique p → l₁.unique p :=\n  Pairwise.sublist\n\nexample {x y : α} {p : α → Prop} : (p x → ¬p y) → (p y → ¬p x) :=\n  fun h hY hX => h hX hY\n\ndef unique.perm {l₁ l₂ : List α} {p : α → Prop} : l₁ ~ l₂ → l₁.unique p → l₂.unique p :=\n  fun h h₁ => h.pairwise h₁ fun _ _ H hB hA => H hA hB\n\ntheorem find?_eq_of_perm_of_unique {l₁ l₂ : List α} {p : α → Bool} :\n    l₁ ~ l₂ → l₁.unique (p ·) → l₁.find? p = l₂.find? p := by\n  intro h hUniq\n  induction h using perm_induction_on' with\n  | nil => rfl\n  | cons x l₁ _ _ ih =>\n    have := ih (hUniq.sublist (sublist_cons x l₁))\n    simp [find?, this]\n  | swap x y l₁ l₂ _ ih =>\n    dsimp [find?]\n    split <;> split <;> try rfl\n    next hY _ hX =>\n      have : ¬p x := unique_cons_of_true hUniq hY x (mem_cons_self _ _)\n      contradiction\n    next => exact ih <| hUniq.sublist <| sublist_of_cons_sublist <| sublist_cons y (x :: l₁)\n  | trans l₁ l₂ l₃ h₁₂ _ h₁ h₂ =>\n    simp [h₁ hUniq, h₂ (hUniq.perm h₁₂)]\n\n/-- If there is at most one element with the property `p`, finding that one element is the same\nas finding any. -/\ntheorem find?_eq_some_of_unique {l : List α} {a : α} {p : α → Bool}\n    : l.Pairwise (p · → !p ·) → (l.find? p = some a ↔ (a ∈ l ∧ p a)) := by\n  refine fun h => ⟨fun h' => ⟨find?_mem h', find?_some h'⟩, ?_⟩\n  induction l with\n  | nil => simp\n  | cons x xs ih =>\n    intro ⟨hMem, hP⟩\n    cases mem_cons.mp hMem with\n    | inl hX => simp [find?, ← hX, hP]\n    | inr hXs =>\n      unfold find?\n      cases hPX : (p x) with\n      | false =>\n        apply ih (Pairwise.sublist (sublist_cons x xs) h) ⟨hXs, hP⟩\n      | true =>\n        cases hP ▸ (pairwise_cons.mp h |>.left a hXs hPX)\n\n/-- If there is at most one element with the property `p`, erasing one such element is the same\nas filtering out all of them. -/\ntheorem eraseP_eq_filter_of_unique (l : List α) (p : α → Bool)\n    : l.Pairwise (p · → !p ·) → l.eraseP p = l.filter (!p ·) := by\n  intro h\n  induction l with\n  | nil => rfl\n  | cons x xs ih =>\n    specialize ih (Pairwise.sublist (sublist_cons x xs) h)\n    cases hP : p x with\n    | true =>\n      rw [pairwise_cons] at h\n      have : ∀ a ∈ xs, !p a := fun a hA => h.left a hA hP\n      simp [eraseP, filter, hP, filter_eq_self.mpr this]\n    | false => simp [eraseP_cons, filter, hP, ih]\n\ntheorem replaceF_of_unique {a b : α} {l : List α} (f : α → Option α) :\n    a ∈ l → f a = some b → l.Pairwise (fun a₁ a₂ => (f a₁).isSome → ¬(f a₂).isSome) →\n    l.replaceF f ~ b :: l.eraseP (f · |>.isSome) := by\n  intro hMem hF hPws\n  induction l with\n  | nil => cases hMem\n  | cons x xs ih =>\n    unfold replaceF eraseP\n    cases mem_cons.mp hMem with\n    | inl hMem => simp [← hMem, hF, Perm.refl]\n    | inr hMem =>\n      have : f x = none := by\n        have .cons hPws _ := hPws\n        exact Option.eq_none_iff_forall_not_mem.mpr fun b hB' =>\n          hPws a hMem (hB' ▸ rfl) (hF ▸ rfl)\n      simp only [this, Option.isSome_none, cond_false]\n      have := ih hMem (hPws.sublist <| sublist_cons _ _)\n      exact .trans (.cons x this) (.swap b x _)\n\nend List\n\n/-! Int -/\n\ntheorem Int.eq_zero_of_lt_neg_iff_lt (i : Int) : (0 < -i ↔ 0 < i) → i = 0 := by\n  intro h\n  by_cases hLt : 0 < i\n  . have := h.mpr hLt; linarith\n  . have : ¬ 0 < -i := fun h₂ => hLt (h.mp h₂); linarith\n\n/-! Loop -/\n\ndef loopM_with_invariant [Monad m] {State : Type _} (n : Nat)\n    (invariant : Nat → State → Prop)\n    (start_state : { st // invariant 0 st })\n    (step : (i : Fin n) → { st // invariant i st } → m { st // invariant (i+1) st }) :\n    m { st // invariant n st } :=\n  go n 0 (by rw [add_zero]) start_state\nwhere\n  go : (b : Nat) → (i : Nat) → b + i = n → { st // invariant i st } → m { st // invariant n st }\n    | 0, i, h, state =>\n      have : i = n := Nat.zero_add i ▸ h\n      return this ▸ state\n    | (b + 1), i, h, state => do\n      let v ← step ⟨i, by rw [← h]; linarith⟩ state\n      go b (i + 1) (by rw [← h]; ac_rfl) v", "meta": {"author": "rebryant", "repo": "cpog", "sha": "5e39029ce71de532fd4407c4768e7c2bf97798c8", "save_path": "github-repos/lean/rebryant-cpog", "path": "github-repos/lean/rebryant-cpog/cpog-5e39029ce71de532fd4407c4768e7c2bf97798c8/VerifiedChecker/ProofChecker/Model/ToMathlib.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.33016562433992086}}
{"text": "import data.fintype.basic\nimport category_theory.Fintype\n\nopen_locale classical\n\nuniverse u\n\n-- This is an awkward hybrid: we should state the induction principle for unbundled fintype\n-- and then if it is necessary provide an interface for using it with `Fintype`.\n\nlemma Fintype.induction_empty_sum {P : Fintype.{u} → Prop}\n  (of_equiv : ∀ {α β : Fintype.{u}}, α ≃ β → P α → P β)\n  (h_empty : P (Fintype.of pempty))\n  (h_option : ∀ {α : Fintype.{u}}, P α → P (Fintype.of (α ⊕ (punit : Type u))))\n  (α : Fintype.{u}) : P α :=\nbegin\n  let Q : Π (α : Type u) [fintype α], Prop := λ a h, P ⟨a,h⟩,\n  have H := @fintype.induction_empty_option Q _ _ _ α α.str,\n  { cases α, exact H },\n  { introsI α β _ e h,\n    haveI : fintype α := fintype.of_equiv _ e.symm,\n    let e' : Fintype.of α ≃ Fintype.of β := e,\n    apply of_equiv e',\n    convert h },\n  { assumption },\n  { introsI α _ h,\n    let e : option α ≃ α ⊕ (punit : Type u) := equiv.option_equiv_sum_punit _,\n    let e' : Fintype.of (option α) ≃ Fintype.of (α ⊕ punit) := e,\n    apply of_equiv e'.symm,\n    exact h_option h }\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/fintype_induction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3301060722590912}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard\n-/\nimport group_theory.submonoid.basic\nimport algebra.big_operators.basic\nimport deprecated.group\n\n/-!\n# Unbundled submonoids\n\nThis file defines unbundled multiplicative and additive submonoids `is_submonoid` and\n`is_add_submonoid`. These are not the preferred way to talk about submonoids and should\nnot be used for any new projects. The preferred way in mathlib are the bundled\nversions `submonoid G` and `add_submonoid G`.\n\n## Main definitions\n\n`is_add_submonoid (S : set G)` : the predicate that `S` is the underlying subset of an additive\nsubmonoid of `G`. The bundled variant `add_subgroup G` should be used in preference to this.\n\n`is_submonoid (S : set G)` : the predicate that `S` is the underlying subset of a submonoid\nof `G`. The bundled variant `submonoid G` should be used in preference to this.\n\n## Tags\n\nsubgroup, subgroups, is_subgroup\n\n## Tags\nsubmonoid, submonoids, is_submonoid\n-/\n\nopen_locale big_operators\n\nvariables {M : Type*} [monoid M] {s : set M}\nvariables {A : Type*} [add_monoid A] {t : set A}\n\n/-- `s` is an additive submonoid: a set containing 0 and closed under addition.\nNote that this structure is deprecated, and the bundled variant `add_submonoid A` should be\npreferred. -/\nstructure is_add_submonoid (s : set A) : Prop :=\n(zero_mem : (0:A) ∈ s)\n(add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s)\n\n/-- `s` is a submonoid: a set containing 1 and closed under multiplication.\nNote that this structure is deprecated, and the bundled variant `submonoid M` should be\npreferred. -/\n@[to_additive]\nstructure is_submonoid (s : set M) : Prop :=\n(one_mem : (1:M) ∈ s)\n(mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s)\n\nlemma additive.is_add_submonoid\n  {s : set M} : ∀ (is : is_submonoid s), @is_add_submonoid (additive M) _ s\n| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩\n\ntheorem additive.is_add_submonoid_iff\n  {s : set M} : @is_add_submonoid (additive M) _ s ↔ is_submonoid s :=\n⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, additive.is_add_submonoid⟩\n\nlemma multiplicative.is_submonoid\n  {s : set A} : ∀ (is : is_add_submonoid s), @is_submonoid (multiplicative A) _ s\n| ⟨h₁, h₂⟩ := ⟨h₁, @h₂⟩\n\ntheorem multiplicative.is_submonoid_iff\n  {s : set A} : @is_submonoid (multiplicative A) _ s ↔ is_add_submonoid s :=\n⟨λ ⟨h₁, h₂⟩, ⟨h₁, @h₂⟩, multiplicative.is_submonoid⟩\n\n/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/\n@[to_additive \"The intersection of two `add_submonoid`s of an `add_monoid` `M` is\nan `add_submonoid` of M.\"]\nlemma is_submonoid.inter {s₁ s₂ : set M} (is₁ : is_submonoid s₁) (is₂ : is_submonoid s₂) :\n  is_submonoid (s₁ ∩ s₂) :=\n{ one_mem := ⟨is₁.one_mem, is₂.one_mem⟩,\n  mul_mem := λ x y hx hy,\n    ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }\n\n/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/\n@[to_additive \"The intersection of an indexed set of `add_submonoid`s of an `add_monoid` `M` is\nan `add_submonoid` of `M`.\"]\nlemma is_submonoid.Inter {ι : Sort*} {s : ι → set M} (h : ∀ y : ι, is_submonoid (s y)) :\n  is_submonoid (set.Inter s) :=\n{ one_mem := set.mem_Inter.2 $ λ y, (h y).one_mem,\n  mul_mem := λ x₁ x₂ h₁ h₂, set.mem_Inter.2 $\n    λ y, (h y).mul_mem (set.mem_Inter.1 h₁ y) (set.mem_Inter.1 h₂ y) }\n\n/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid\n    of `M`. -/\n@[to_additive \"The union of an indexed, directed, nonempty set\nof `add_submonoid`s of an `add_monoid` `M` is an `add_submonoid` of `M`. \"]\nlemma is_submonoid_Union_of_directed {ι : Type*} [hι : nonempty ι]\n  {s : ι → set M} (hs : ∀ i, is_submonoid (s i))\n  (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :\n  is_submonoid (⋃i, s i) :=\n{ one_mem := let ⟨i⟩ := hι in set.mem_Union.2 ⟨i, (hs i).one_mem⟩,\n  mul_mem := λ a b ha hb,\n    let ⟨i, hi⟩ := set.mem_Union.1 ha in\n    let ⟨j, hj⟩ := set.mem_Union.1 hb in\n    let ⟨k, hk⟩ := directed i j in\n    set.mem_Union.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }\n\nsection powers\n\n/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/\n@[to_additive multiples\n\"The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `add_monoid`.\"]\ndef powers (x : M) : set M := {y | ∃ n:ℕ, x^n = y}\n\n/-- 1 is in the set of natural number powers of an element of a monoid. -/\n@[to_additive \"0 is in the set of natural number multiples of an element of an `add_monoid`.\"]\nlemma powers.one_mem {x : M} : (1 : M) ∈ powers x := ⟨0, pow_zero _⟩\n\n/-- An element of a monoid is in the set of that element's natural number powers. -/\n@[to_additive\n\"An element of an `add_monoid` is in the set of that element's natural number multiples.\"]\nlemma powers.self_mem {x : M} : x ∈ powers x := ⟨1, pow_one _⟩\n\n/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/\n@[to_additive\n\"The set of natural number multiples of an element of an `add_monoid` is closed under addition.\"]\nlemma powers.mul_mem {x y z : M} : (y ∈ powers x) → (z ∈ powers x) → (y * z ∈ powers x) :=\nλ ⟨n₁, h₁⟩ ⟨n₂, h₂⟩, ⟨n₁ + n₂, by simp only [pow_add, *]⟩\n\n/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/\n@[to_additive \"The set of natural number multiples of an element of\nan `add_monoid` `M` is an `add_submonoid` of `M`.\"]\nlemma powers.is_submonoid (x : M) : is_submonoid (powers x) :=\n{ one_mem := powers.one_mem,\n  mul_mem := λ y z, powers.mul_mem }\n\n/-- A monoid is a submonoid of itself. -/\n@[to_additive \"An `add_monoid` is an `add_submonoid` of itself.\"]\nlemma univ.is_submonoid : is_submonoid (@set.univ M) := by split; simp\n\n/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/\n@[to_additive \"The preimage of an `add_submonoid` under an `add_monoid` hom is\nan `add_submonoid` of the domain.\"]\nlemma is_submonoid.preimage {N : Type*} [monoid N] {f : M → N} (hf : is_monoid_hom f)\n  {s : set N} (hs : is_submonoid s) : is_submonoid (f ⁻¹' s) :=\n{ one_mem := show f 1 ∈ s, by rw is_monoid_hom.map_one hf; exact hs.one_mem,\n  mul_mem := λ a b (ha : f a ∈ s) (hb : f b ∈ s),\n    show f (a * b) ∈ s, by rw is_monoid_hom.map_mul hf; exact hs.mul_mem ha hb }\n\n/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/\n@[to_additive \"The image of an `add_submonoid` under an `add_monoid`\nhom is an `add_submonoid` of the codomain.\"]\nlemma is_submonoid.image {γ : Type*} [monoid γ] {f : M → γ} (hf : is_monoid_hom f)\n  {s : set M} (hs : is_submonoid s) : is_submonoid (f '' s) :=\n{ one_mem := ⟨1, hs.one_mem, hf.map_one⟩,\n  mul_mem := λ a b ⟨x, hx⟩ ⟨y, hy⟩, ⟨x * y, hs.mul_mem hx.1 hy.1,\n    by rw [hf.map_mul, hx.2, hy.2]⟩ }\n\n/-- The image of a monoid hom is a submonoid of the codomain. -/\n@[to_additive \"The image of an `add_monoid` hom is an `add_submonoid`\nof the codomain.\"]\nlemma range.is_submonoid {γ : Type*} [monoid γ] {f : M → γ} (hf : is_monoid_hom f) :\n  is_submonoid (set.range f) :=\nby { rw ← set.image_univ, exact univ.is_submonoid.image hf }\n\n/-- Submonoids are closed under natural powers. -/\n@[to_additive is_add_submonoid.smul_mem\n\"An `add_submonoid` is closed under multiplication by naturals.\"]\nlemma is_submonoid.pow_mem {a : M} (hs : is_submonoid s) (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s\n| 0 := by { rw pow_zero, exact hs.one_mem }\n| (n + 1) := by { rw pow_succ, exact hs.mul_mem h is_submonoid.pow_mem }\n\n/-- The set of natural number powers of an element of a submonoid is a subset of the submonoid. -/\n@[to_additive is_add_submonoid.multiples_subset \"The set of natural number multiples of an element\nof an `add_submonoid` is a subset of the `add_submonoid`.\"]\nlemma is_submonoid.power_subset {a : M} (hs : is_submonoid s) (h : a ∈ s) : powers a ⊆ s :=\nassume x ⟨n, hx⟩, hx ▸ hs.pow_mem h\n\nend powers\n\nnamespace is_submonoid\n\n/-- The product of a list of elements of a submonoid is an element of the submonoid. -/\n@[to_additive \"The sum of a list of elements of an `add_submonoid` is an element of the\n`add_submonoid`.\"]\nlemma list_prod_mem (hs : is_submonoid s) : ∀{l : list M}, (∀x∈l, x ∈ s) → l.prod ∈ s\n| []     h := hs.one_mem\n| (a::l) h :=\n  suffices a * l.prod ∈ s, by simpa,\n  have a ∈ s ∧ (∀x∈l, x ∈ s), by simpa using h,\n  hs.mul_mem this.1 (list_prod_mem this.2)\n\n/-- The product of a multiset of elements of a submonoid of a `comm_monoid` is an element of\nthe submonoid. -/\n@[to_additive \"The sum of a multiset of elements of an `add_submonoid` of an `add_comm_monoid`\nis an element of the `add_submonoid`. \"]\nlemma multiset_prod_mem {M} [comm_monoid M] {s : set M} (hs : is_submonoid s) (m : multiset M) :\n  (∀a∈m, a ∈ s) → m.prod ∈ s :=\nbegin\n  refine quotient.induction_on m (assume l hl, _),\n  rw [multiset.quot_mk_to_coe, multiset.coe_prod],\n  exact list_prod_mem hs hl\nend\n\n/-- The product of elements of a submonoid of a `comm_monoid` indexed by a `finset` is an element\nof the submonoid. -/\n@[to_additive \"The sum of elements of an `add_submonoid` of an `add_comm_monoid` indexed by\na `finset` is an element of the `add_submonoid`.\"]\nlemma finset_prod_mem {M A} [comm_monoid M] {s : set M} (hs : is_submonoid s) (f : A → M) :\n  ∀(t : finset A), (∀b∈t, f b ∈ s) → ∏ b in t, f b ∈ s\n| ⟨m, hm⟩ _ := multiset_prod_mem hs _ (by simpa)\n\nend is_submonoid\n\nnamespace add_monoid\n\n/-- The inductively defined membership predicate for the submonoid generated by a subset of a\n    monoid. -/\ninductive in_closure (s : set A) : A → Prop\n| basic {a : A} : a ∈ s → in_closure a\n| zero : in_closure 0\n| add {a b : A} : in_closure a → in_closure b → in_closure (a + b)\n\nend add_monoid\n\nnamespace monoid\n\n/-- The inductively defined membership predicate for the `submonoid` generated by a subset of an\n    monoid. -/\n@[to_additive]\ninductive in_closure (s : set M) : M → Prop\n| basic {a : M} : a ∈ s → in_closure a\n| one : in_closure 1\n| mul {a b : M} : in_closure a → in_closure b → in_closure (a * b)\n\n/-- The inductively defined submonoid generated by a subset of a monoid. -/\n@[to_additive \"The inductively defined `add_submonoid` genrated by a subset of an `add_monoid`.\"]\ndef closure (s : set M) : set M := {a | in_closure s a }\n\n@[to_additive]\nlemma closure.is_submonoid (s : set M) : is_submonoid (closure s) :=\n{ one_mem := in_closure.one, mul_mem := assume a b, in_closure.mul }\n\n/-- A subset of a monoid is contained in the submonoid it generates. -/\n@[to_additive \"A subset of an `add_monoid` is contained in the `add_submonoid` it generates.\"]\ntheorem subset_closure {s : set M} : s ⊆ closure s :=\nassume a, in_closure.basic\n\n/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/\n@[to_additive \"The `add_submonoid` generated by a set is contained in any `add_submonoid` that\ncontains the set.\"]\ntheorem closure_subset {s t : set M} (ht : is_submonoid t) (h : s ⊆ t) : closure s ⊆ t :=\nassume a ha, by induction ha; simp [h _, *, is_submonoid.one_mem, is_submonoid.mul_mem]\n\n/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is\n    contained in the submonoid generated by `t`. -/\n@[to_additive \"Given subsets `t` and `s` of an `add_monoid M`, if `s ⊆ t`, the `add_submonoid`\nof `M` generated by `s` is contained in the `add_submonoid` generated by `t`.\"]\ntheorem closure_mono {s t : set M} (h : s ⊆ t) : closure s ⊆ closure t :=\nclosure_subset (closure.is_submonoid t) $ set.subset.trans h subset_closure\n\n/-- The submonoid generated by an element of a monoid equals the set of natural number powers of\n    the element. -/\n@[to_additive \"The `add_submonoid` generated by an element of an `add_monoid` equals the set of\nnatural number multiples of the element.\"]\ntheorem closure_singleton {x : M} : closure ({x} : set M) = powers x :=\nset.eq_of_subset_of_subset (closure_subset (powers.is_submonoid x) $ set.singleton_subset_iff.2 $\n  powers.self_mem) $ is_submonoid.power_subset (closure.is_submonoid _) $\n  set.singleton_subset_iff.1 $ subset_closure\n\n/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated\n    by the image of the set under the monoid hom. -/\n@[to_additive \"The image under an `add_monoid` hom of the `add_submonoid` generated by a set equals\nthe `add_submonoid` generated by the image of the set under the `add_monoid` hom.\"]\nlemma image_closure {A : Type*} [monoid A] {f : M → A} (hf : is_monoid_hom f) (s : set M) :\n  f '' closure s = closure (f '' s) :=\nle_antisymm\n  begin\n    rintros _ ⟨x, hx, rfl⟩,\n    apply in_closure.rec_on hx; intros,\n    { solve_by_elim [subset_closure, set.mem_image_of_mem] },\n    { rw [hf.map_one], apply is_submonoid.one_mem (closure.is_submonoid (f '' s))},\n    { rw [hf.map_mul], solve_by_elim [(closure.is_submonoid _).mul_mem] }\n  end\n  (closure_subset (is_submonoid.image hf (closure.is_submonoid _)) $\n    set.image_subset _ subset_closure)\n\n/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists\na list of elements of `s` whose product is `a`. -/\n@[to_additive \"Given an element `a` of the `add_submonoid` of an `add_monoid M` generated by\na set `s`, there exists a list of elements of `s` whose sum is `a`.\"]\ntheorem exists_list_of_mem_closure {s : set M} {a : M} (h : a ∈ closure s) :\n  (∃l:list M, (∀x∈l, x ∈ s) ∧ l.prod = a) :=\nbegin\n  induction h,\n  case in_closure.basic : a ha { existsi ([a]), simp [ha] },\n  case in_closure.one { existsi ([]), simp },\n  case in_closure.mul : a b _ _ ha hb\n  { rcases ha with ⟨la, ha, eqa⟩,\n    rcases hb with ⟨lb, hb, eqb⟩,\n    existsi (la ++ lb),\n    simp [eqa.symm, eqb.symm, or_imp_distrib],\n    exact assume a, ⟨ha a, hb a⟩ }\nend\n\n/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by\n    `s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the\n    submonoid generated by `t` whose product is `x`. -/\n@[to_additive \"Given sets `s, t` of a commutative `add_monoid M`, `x ∈ M` is in the `add_submonoid`\nof `M` generated by `s ∪ t` iff there exists an element of the `add_submonoid` generated by `s`\nand an element of the `add_submonoid` generated by `t` whose sum is `x`.\"]\ntheorem mem_closure_union_iff {M : Type*} [comm_monoid M] {s t : set M} {x : M} :\n  x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=\n⟨λ hx, let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx in HL2 ▸\n  list.rec_on L (λ _, ⟨1, (closure.is_submonoid _).one_mem, 1,\n    (closure.is_submonoid _).one_mem, mul_one _⟩)\n    (λ hd tl ih HL1, let ⟨y, hy, z, hz, hyzx⟩ := ih (list.forall_mem_of_forall_mem_cons HL1) in\n      or.cases_on (HL1 hd $ list.mem_cons_self _ _)\n        (λ hs, ⟨hd * y, (closure.is_submonoid _).mul_mem (subset_closure hs) hy, z, hz,\n          by rw [mul_assoc, list.prod_cons, ← hyzx]; refl⟩)\n        (λ ht, ⟨y, hy, z * hd, (closure.is_submonoid _).mul_mem hz (subset_closure ht),\n          by rw [← mul_assoc, list.prod_cons, ← hyzx, mul_comm hd]; refl⟩)) HL1,\nλ ⟨y, hy, z, hz, hyzx⟩, hyzx ▸ (closure.is_submonoid _).mul_mem\n  (closure_mono (set.subset_union_left _ _) hy)\n  (closure_mono (set.subset_union_right _ _) hz)⟩\n\nend monoid\n\n/-- Create a bundled submonoid from a set `s` and `[is_submonoid s]`. -/\n@[to_additive \"Create a bundled additive submonoid from a set `s` and `[is_add_submonoid s]`.\"]\ndef submonoid.of {s : set M} (h : is_submonoid s) : submonoid M := ⟨s, h.1, h.2⟩\n\n@[to_additive]\nlemma submonoid.is_submonoid (S : submonoid M) : is_submonoid (S : set M) := ⟨S.2, S.3⟩\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/deprecated/submonoid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3301060722590912}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.over\nimport Mathlib.category_theory.limits.shapes.pullbacks\nimport Mathlib.category_theory.limits.shapes.wide_pullbacks\nimport Mathlib.category_theory.limits.shapes.finite_products\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Products in the over category\n\nShows that products in the over category can be derived from wide pullbacks in the base category.\nThe main result is `over_product_of_wide_pullback`, which says that if `C` has `J`-indexed wide\npullbacks, then `over B` has `J`-indexed products.\n-/\n\nnamespace category_theory.over\n\n\nnamespace construct_products\n\n\n/--\n(Implementation)\nGiven a product diagram in `C/B`, construct the corresponding wide pullback diagram\nin `C`.\n-/\ndef wide_pullback_diagram_of_diagram_over {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J ⥤ over B) : limits.wide_pullback_shape J ⥤ C :=\n  limits.wide_pullback_shape.wide_cospan B (fun (j : J) => comma.left (functor.obj F j))\n    fun (j : J) => comma.hom (functor.obj F j)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\ndef cones_equiv_inverse_obj {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J ⥤ over B) (c : limits.cone F) : limits.cone (wide_pullback_diagram_of_diagram_over B F) :=\n  limits.cone.mk (comma.left (limits.cone.X c))\n    (nat_trans.mk\n      fun (X : limits.wide_pullback_shape J) =>\n        option.cases_on X (comma.hom (limits.cone.X c))\n          fun (j : J) => comma_morphism.left (nat_trans.app (limits.cone.π c) j))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\ndef cones_equiv_inverse {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J ⥤ over B) : limits.cone F ⥤ limits.cone (wide_pullback_diagram_of_diagram_over B F) :=\n  functor.mk (cones_equiv_inverse_obj B F)\n    fun (c₁ c₂ : limits.cone F) (f : c₁ ⟶ c₂) =>\n      limits.cone_morphism.mk (comma_morphism.left (limits.cone_morphism.hom f))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] theorem cones_equiv_functor_map_hom {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J ⥤ over B) (c₁ : limits.cone (wide_pullback_diagram_of_diagram_over B F)) (c₂ : limits.cone (wide_pullback_diagram_of_diagram_over B F)) (f : c₁ ⟶ c₂) : limits.cone_morphism.hom (functor.map (cones_equiv_functor B F) f) = hom_mk (limits.cone_morphism.hom f) :=\n  Eq.refl (limits.cone_morphism.hom (functor.map (cones_equiv_functor B F) f))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] def cones_equiv_unit_iso {J : Type v} {C : Type u} [category C] (B : C) (F : discrete J ⥤ over B) : 𝟭 ≅ cones_equiv_functor B F ⋙ cones_equiv_inverse B F :=\n  nat_iso.of_components\n    (fun (_x : limits.cone (wide_pullback_diagram_of_diagram_over B F)) => limits.cones.ext (iso.mk 𝟙 𝟙) sorry) sorry\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] def cones_equiv_counit_iso {J : Type v} {C : Type u} [category C] (B : C) (F : discrete J ⥤ over B) : cones_equiv_inverse B F ⋙ cones_equiv_functor B F ≅ 𝟭 :=\n  nat_iso.of_components (fun (_x : limits.cone F) => limits.cones.ext (iso.mk (hom_mk 𝟙) (hom_mk 𝟙)) sorry) sorry\n\n-- TODO: Can we add `. obviously` to the second arguments of `nat_iso.of_components` and\n\n--       `cones.ext`?\n\n/--\n(Impl) Establish an equivalence between the category of cones for `F` and for the \"grown\" `F`.\n-/\n@[simp] theorem cones_equiv_unit_iso_2 {J : Type v} {C : Type u} [category C] (B : C) (F : discrete J ⥤ over B) : equivalence.unit_iso (cones_equiv B F) = cones_equiv_unit_iso B F :=\n  Eq.refl (equivalence.unit_iso (cones_equiv B F))\n\n/-- Use the above equivalence to prove we have a limit. -/\ntheorem has_over_limit_discrete_of_wide_pullback_limit {J : Type v} {C : Type u} [category C] {B : C} (F : discrete J ⥤ over B) [limits.has_limit (wide_pullback_diagram_of_diagram_over B F)] : limits.has_limit F := sorry\n\n/-- Given a wide pullback in `C`, construct a product in `C/B`. -/\ntheorem over_product_of_wide_pullback {J : Type v} {C : Type u} [category C] [limits.has_limits_of_shape (limits.wide_pullback_shape J) C] {B : C} : limits.has_limits_of_shape (discrete J) (over B) :=\n  limits.has_limits_of_shape.mk fun (F : discrete J ⥤ over B) => has_over_limit_discrete_of_wide_pullback_limit F\n\n/-- Given a pullback in `C`, construct a binary product in `C/B`. -/\ntheorem over_binary_product_of_pullback {C : Type u} [category C] [limits.has_pullbacks C] {B : C} : limits.has_binary_products (over B) :=\n  over_product_of_wide_pullback\n\n/-- Given all wide pullbacks in `C`, construct products in `C/B`. -/\ntheorem over_products_of_wide_pullbacks {C : Type u} [category C] [limits.has_wide_pullbacks C] {B : C} : limits.has_products (over B) :=\n  fun (J : Type v) => over_product_of_wide_pullback\n\n/-- Given all finite wide pullbacks in `C`, construct finite products in `C/B`. -/\ntheorem over_finite_products_of_finite_wide_pullbacks {C : Type u} [category C] [limits.has_finite_wide_pullbacks C] {B : C} : limits.has_finite_products (over B) :=\n  fun (J : Type v) (𝒥₁ : DecidableEq J) (𝒥₂ : fintype J) => over_product_of_wide_pullback\n\nend construct_products\n\n\n/--\nConstruct terminal object in the over category. This isn't an instance as it's not typically the\nway we want to define terminal objects.\n(For instance, this gives a terminal object which is different from the generic one given by\n`over_product_of_wide_pullback` above.)\n-/\ntheorem over_has_terminal {C : Type u} [category C] (B : C) : limits.has_terminal (over B) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/constructions/over/products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.33010607225909117}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Andrew Yang\n-/\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.preserves.shapes.pullbacks\nimport category_theory.limits.preserves.shapes.binary_products\n\n/-!\n# Constructing equalizers from pullbacks and binary products.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIf a category has pullbacks and binary products, then it has equalizers.\n\nTODO: generalize universe\n-/\n\nnoncomputable theory\n\nuniverses v v' u u'\n\nopen category_theory category_theory.category\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C]\nvariables {D : Type u'} [category.{v'} D] (G : C ⥤ D)\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_equalizers_of_has_pullbacks_and_binary_products\n\nvariables [has_binary_products C] [has_pullbacks C]\n\n/-- Define the equalizing object -/\n@[reducible]\ndef construct_equalizer (F : walking_parallel_pair ⥤ C) : C :=\npullback (prod.lift (𝟙 _) (F.map walking_parallel_pair_hom.left))\n         (prod.lift (𝟙 _) (F.map walking_parallel_pair_hom.right))\n\n/-- Define the equalizing morphism -/\nabbreviation pullback_fst (F : walking_parallel_pair ⥤ C) :\n  construct_equalizer F ⟶ F.obj walking_parallel_pair.zero :=\npullback.fst\n\nlemma pullback_fst_eq_pullback_snd (F : walking_parallel_pair ⥤ C) :\n  pullback_fst F = pullback.snd :=\nby convert pullback.condition =≫ limits.prod.fst; simp\n\n/-- Define the equalizing cone -/\n@[reducible]\ndef equalizer_cone (F : walking_parallel_pair ⥤ C) : cone F :=\ncone.of_fork\n  (fork.of_ι (pullback_fst F)\n    (begin\n      conv_rhs { rw pullback_fst_eq_pullback_snd, },\n      convert pullback.condition =≫ limits.prod.snd using 1; simp\n     end))\n\n/-- Show the equalizing cone is a limit -/\ndef equalizer_cone_is_limit (F : walking_parallel_pair ⥤ C) : is_limit (equalizer_cone F) :=\n{ lift :=\n  begin\n    intro c, apply pullback.lift (c.π.app _) (c.π.app _),\n    apply limit.hom_ext,\n    rintro (_ | _); simp\n  end,\n  fac' := by rintros c (_ | _); simp,\n  uniq' :=\n  begin\n    intros c _ J,\n    have J0 := J walking_parallel_pair.zero, simp at J0,\n    apply pullback.hom_ext,\n    { rwa limit.lift_π },\n    { erw [limit.lift_π, ← J0, pullback_fst_eq_pullback_snd] }\n  end }\n\nend has_equalizers_of_has_pullbacks_and_binary_products\n\nopen has_equalizers_of_has_pullbacks_and_binary_products\n/-- Any category with pullbacks and binary products, has equalizers. -/\n-- This is not an instance, as it is not always how one wants to construct equalizers!\nlemma has_equalizers_of_has_pullbacks_and_binary_products\n  [has_binary_products C] [has_pullbacks C] :\n  has_equalizers C :=\n{ has_limit := λ F, has_limit.mk\n  { cone := equalizer_cone F,\n    is_limit := equalizer_cone_is_limit F } }\n\nlocal attribute[instance] has_pullback_of_preserves_pullback\n\n/-- A functor that preserves pullbacks and binary products also presrves equalizers. -/\ndef preserves_equalizers_of_preserves_pullbacks_and_binary_products\n    [has_binary_products C] [has_pullbacks C]\n    [preserves_limits_of_shape (discrete walking_pair) G]\n    [preserves_limits_of_shape walking_cospan G] :\n  preserves_limits_of_shape walking_parallel_pair G :=\n⟨λ K, preserves_limit_of_preserves_limit_cone (equalizer_cone_is_limit K) $\n{ lift := λ c, begin\n    refine pullback.lift _ _ _ ≫ (@@preserves_pullback.iso _ _ _ _ _ _ _ _).inv,\n    { exact c.π.app walking_parallel_pair.zero },\n    { exact c.π.app walking_parallel_pair.zero },\n    apply (map_is_limit_of_preserves_of_is_limit G _ _ (prod_is_prod _ _)).hom_ext,\n    swap, apply_instance,\n    rintro (_|_),\n    { simp only [category.assoc, ← G.map_comp, prod.lift_fst,\n        binary_fan.π_app_left, binary_fan.mk_fst], },\n    { simp only [binary_fan.π_app_right, binary_fan.mk_snd,\n        category.assoc, ← G.map_comp, prod.lift_snd],\n      exact (c.π.naturality (walking_parallel_pair_hom.left)).symm.trans\n        (c.π.naturality (walking_parallel_pair_hom.right)) },\n  end,\n  fac' := λ c j, begin\n    rcases j with (_|_);\n      simp only [category.comp_id, preserves_pullback.iso_inv_fst, cone.of_fork_π, G.map_comp,\n        preserves_pullback.iso_inv_fst_assoc, functor.map_cone_π_app, eq_to_hom_refl,\n        category.assoc, fork.of_ι_π_app, pullback.lift_fst, pullback.lift_fst_assoc],\n      exact (c.π.naturality (walking_parallel_pair_hom.left)).symm.trans (category.id_comp _)\n  end,\n  uniq' := λ s m h, begin\n    rw iso.eq_comp_inv,\n    have := h walking_parallel_pair.zero,\n    dsimp [equalizer_cone] at this,\n    ext; simp only [preserves_pullback.iso_hom_snd,\n      category.assoc, preserves_pullback.iso_hom_fst, pullback.lift_fst, pullback.lift_snd,\n      category.comp_id, ← pullback_fst_eq_pullback_snd, ← this],\n  end }⟩\n\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_coequalizers_of_has_pushouts_and_binary_coproducts\n\nvariables [has_binary_coproducts C] [has_pushouts C]\n\n/-- Define the equalizing object -/\n@[reducible]\ndef construct_coequalizer (F : walking_parallel_pair ⥤ C) : C :=\npushout (coprod.desc (𝟙 _) (F.map walking_parallel_pair_hom.left))\n        (coprod.desc (𝟙 _) (F.map walking_parallel_pair_hom.right))\n\n/-- Define the equalizing morphism -/\nabbreviation pushout_inl (F : walking_parallel_pair ⥤ C) :\n  F.obj walking_parallel_pair.one ⟶ construct_coequalizer F :=\npushout.inl\n\nlemma pushout_inl_eq_pushout_inr (F : walking_parallel_pair ⥤ C) :\n  pushout_inl F = pushout.inr :=\nby convert limits.coprod.inl ≫= pushout.condition; simp\n\n/-- Define the equalizing cocone -/\n@[reducible]\ndef coequalizer_cocone (F : walking_parallel_pair ⥤ C) : cocone F :=\ncocone.of_cofork\n  (cofork.of_π (pushout_inl F)\n    (begin\n      conv_rhs { rw pushout_inl_eq_pushout_inr, },\n      convert limits.coprod.inr ≫= pushout.condition using 1; simp\n     end))\n\n/-- Show the equalizing cocone is a colimit -/\ndef coequalizer_cocone_is_colimit (F : walking_parallel_pair ⥤ C) :\n  is_colimit (coequalizer_cocone F) :=\n{ desc :=\n  begin\n    intro c, apply pushout.desc (c.ι.app _) (c.ι.app _),\n    apply colimit.hom_ext,\n    rintro (_ | _); simp\n  end,\n  fac' := by rintros c (_ | _); simp,\n  uniq' :=\n  begin\n    intros c _ J,\n    have J1 : pushout_inl F ≫ m = c.ι.app walking_parallel_pair.one :=\n      by simpa using J walking_parallel_pair.one,\n    apply pushout.hom_ext,\n    { rw colimit.ι_desc, exact J1 },\n    { rw [colimit.ι_desc, ← pushout_inl_eq_pushout_inr], exact J1 }\n  end }\n\nend has_coequalizers_of_has_pushouts_and_binary_coproducts\n\nopen has_coequalizers_of_has_pushouts_and_binary_coproducts\n/-- Any category with pullbacks and binary products, has equalizers. -/\n-- This is not an instance, as it is not always how one wants to construct equalizers!\nlemma has_coequalizers_of_has_pushouts_and_binary_coproducts\n  [has_binary_coproducts C] [has_pushouts C] : has_coequalizers C :=\n{ has_colimit := λ F, has_colimit.mk\n  { cocone := coequalizer_cocone F,\n    is_colimit := coequalizer_cocone_is_colimit F } }\n\nlocal attribute[instance] has_pushout_of_preserves_pushout\n\n/-- A functor that preserves pushouts and binary coproducts also presrves coequalizers. -/\ndef preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts\n    [has_binary_coproducts C] [has_pushouts C]\n    [preserves_colimits_of_shape (discrete walking_pair) G]\n    [preserves_colimits_of_shape walking_span G] :\n  preserves_colimits_of_shape walking_parallel_pair G :=\n⟨λ K, preserves_colimit_of_preserves_colimit_cocone (coequalizer_cocone_is_colimit K) $\n{ desc := λ c, begin\n    refine (@@preserves_pushout.iso _ _ _ _ _ _ _ _).inv ≫ pushout.desc _ _ _,\n    { exact c.ι.app walking_parallel_pair.one },\n    { exact c.ι.app walking_parallel_pair.one },\n    apply (map_is_colimit_of_preserves_of_is_colimit G _ _ (coprod_is_coprod _ _)).hom_ext,\n    swap, apply_instance,\n    rintro (_|_),\n    { simp only [binary_cofan.ι_app_left, binary_cofan.mk_inl, category.assoc, ← G.map_comp_assoc,\n        coprod.inl_desc] },\n    { simp only [binary_cofan.ι_app_right, binary_cofan.mk_inr, category.assoc, ← G.map_comp_assoc,\n        coprod.inr_desc],\n      exact (c.ι.naturality walking_parallel_pair_hom.left).trans\n        (c.ι.naturality walking_parallel_pair_hom.right).symm, },\n  end,\n  fac' := λ c j, begin\n    rcases j with (_|_); simp only [functor.map_cocone_ι_app, cocone.of_cofork_ι, category.id_comp,\n      eq_to_hom_refl, category.assoc, functor.map_comp, cofork.of_π_ι_app, pushout.inl_desc,\n      preserves_pushout.inl_iso_inv_assoc],\n      exact (c.ι.naturality (walking_parallel_pair_hom.left)).trans (category.comp_id _)\n  end,\n  uniq' := λ s m h, begin\n    rw iso.eq_inv_comp,\n    have := h walking_parallel_pair.one,\n    dsimp [coequalizer_cocone] at this,\n    ext; simp only [preserves_pushout.inl_iso_hom_assoc, category.id_comp, pushout.inl_desc,\n      pushout.inr_desc, preserves_pushout.inr_iso_hom_assoc,\n      ← pushout_inl_eq_pushout_inr, ← this],\n  end }⟩\n\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/constructions/equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.33000548822312575}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,\n  Michael Howes\n\n! This file was ported from Lean 3 source module deprecated.subgroup\n! leanprover-community/mathlib commit f93c11933efbc3c2f0299e47b8ff83e9b539cbf6\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.GroupTheory.Subgroup.Basic\nimport Mathlib.Deprecated.Submonoid\n\n/-!\n# Unbundled subgroups (deprecated)\n\nThis file is deprecated, and is no longer imported by anything in mathlib other than other\ndeprecated files, and test files. You should not need to import it.\n\nThis file defines unbundled multiplicative and additive subgroups. Instead of using this file,\nplease use `Subgroup G` and `AddSubgroup A`, defined in `GroupTheory.Subgroup.Basic`.\n\n## Main definitions\n\n`IsAddSubgroup (S : Set A)` : the predicate that `S` is the underlying subset of an additive\nsubgroup of `A`. The bundled variant `AddSubgroup A` should be used in preference to this.\n\n`IsSubgroup (S : Set G)` : the predicate that `S` is the underlying subset of a subgroup\nof `G`. The bundled variant `Subgroup G` should be used in preference to this.\n\n## Tags\n\nsubgroup, subgroups, IsSubgroup\n-/\n\n\nopen Set Function\n\nvariable {G : Type _} {H : Type _} {A : Type _} {a a₁ a₂ b c : G}\n\nsection Group\n\nvariable [Group G] [AddGroup A]\n\n/-- `s` is an additive subgroup: a set containing 0 and closed under addition and negation. -/\nstructure IsAddSubgroup (s : Set A) extends IsAddSubmonoid s : Prop where\n  /-- The proposition that `s` is closed under negation. -/\n  neg_mem {a} : a ∈ s → -a ∈ s\n#align is_add_subgroup IsAddSubgroup\n\n/-- `s` is a subgroup: a set containing 1 and closed under multiplication and inverse. -/\n@[to_additive]\nstructure IsSubgroup (s : Set G) extends IsSubmonoid s : Prop where\n  /-- The proposition that `s` is closed under inverse. -/\n  inv_mem {a} : a ∈ s → a⁻¹ ∈ s\n#align is_subgroup IsSubgroup\n\n@[to_additive]\ntheorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x ∈ s) (hy : y ∈ s) :\n    x / y ∈ s := by simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)\n#align is_subgroup.div_mem IsSubgroup.div_mem\n#align is_add_subgroup.sub_mem IsAddSubgroup.sub_mem\n\ntheorem Additive.isAddSubgroup {s : Set G} (hs : IsSubgroup s) : @IsAddSubgroup (Additive G) _ s :=\n  @IsAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubmonoid hs.toIsSubmonoid) hs.inv_mem\n#align additive.is_add_subgroup Additive.isAddSubgroup\n\ntheorem Additive.isAddSubgroup_iff {s : Set G} : @IsAddSubgroup (Additive G) _ s ↔ IsSubgroup s :=\n  ⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @IsSubgroup.mk G _ _ ⟨h₁, @h₂⟩ @h₃, fun h =>\n    Additive.isAddSubgroup h⟩\n#align additive.is_add_subgroup_iff Additive.isAddSubgroup_iff\n\ntheorem Multiplicative.isSubgroup {s : Set A} (hs : IsAddSubgroup s) :\n    @IsSubgroup (Multiplicative A) _ s :=\n  @IsSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubmonoid hs.toIsAddSubmonoid) hs.neg_mem\n#align multiplicative.is_subgroup Multiplicative.isSubgroup\n\ntheorem Multiplicative.isSubgroup_iff {s : Set A} :\n    @IsSubgroup (Multiplicative A) _ s ↔ IsAddSubgroup s :=\n  ⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @IsAddSubgroup.mk A _ _ ⟨h₁, @h₂⟩ @h₃, fun h =>\n    Multiplicative.isSubgroup h⟩\n#align multiplicative.is_subgroup_iff Multiplicative.isSubgroup_iff\n\n@[to_additive of_add_neg]\ntheorem IsSubgroup.of_div (s : Set G) (one_mem : (1 : G) ∈ s)\n    (div_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a * b⁻¹ ∈ s) : IsSubgroup s :=\n  have inv_mem : ∀ a, a ∈ s → a⁻¹ ∈ s := fun a ha => by\n    have : 1 * a⁻¹ ∈ s := div_mem one_mem ha\n    convert this using 1\n    rw [one_mul]\n  { inv_mem := inv_mem _\n    mul_mem := fun {a b} ha hb => by\n      have : a * b⁻¹⁻¹ ∈ s := div_mem ha (inv_mem b hb)\n      convert this\n      rw [inv_inv]\n    one_mem }\n#align is_subgroup.of_div IsSubgroup.of_div\n#align is_add_subgroup.of_add_neg IsAddSubgroup.of_add_neg\n\ntheorem IsAddSubgroup.of_sub (s : Set A) (zero_mem : (0 : A) ∈ s)\n    (sub_mem : ∀ {a b : A}, a ∈ s → b ∈ s → a - b ∈ s) : IsAddSubgroup s :=\n  IsAddSubgroup.of_add_neg s zero_mem fun {x y} hx hy => by\n    simpa only [sub_eq_add_neg] using sub_mem hx hy\n#align is_add_subgroup.of_sub IsAddSubgroup.of_sub\n\n@[to_additive]\ntheorem IsSubgroup.inter {s₁ s₂ : Set G} (hs₁ : IsSubgroup s₁) (hs₂ : IsSubgroup s₂) :\n    IsSubgroup (s₁ ∩ s₂) :=\n  { IsSubmonoid.inter hs₁.toIsSubmonoid hs₂.toIsSubmonoid with\n    inv_mem := fun hx => ⟨hs₁.inv_mem hx.1, hs₂.inv_mem hx.2⟩ }\n#align is_subgroup.inter IsSubgroup.inter\n#align is_add_subgroup.inter IsAddSubgroup.inter\n\n@[to_additive]\ntheorem IsSubgroup.interᵢ {ι : Sort _} {s : ι → Set G} (hs : ∀ y : ι, IsSubgroup (s y)) :\n    IsSubgroup (Set.interᵢ s) :=\n  { IsSubmonoid.interᵢ fun y => (hs y).toIsSubmonoid with\n    inv_mem := fun h =>\n      Set.mem_interᵢ.2 fun y => IsSubgroup.inv_mem (hs _) (Set.mem_interᵢ.1 h y) }\n#align is_subgroup.Inter IsSubgroup.interᵢ\n#align is_add_subgroup.Inter IsAddSubgroup.interᵢ\n\n@[to_additive]\ntheorem isSubgroup_unionᵢ_of_directed {ι : Type _} [Nonempty ι] {s : ι → Set G}\n    (hs : ∀ i, IsSubgroup (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :\n    IsSubgroup (⋃ i, s i) :=\n  { inv_mem := fun ha =>\n      let ⟨i, hi⟩ := Set.mem_unionᵢ.1 ha\n      Set.mem_unionᵢ.2 ⟨i, (hs i).inv_mem hi⟩\n    toIsSubmonoid := isSubmonoid_unionᵢ_of_directed (fun i => (hs i).toIsSubmonoid) directed }\n#align is_subgroup_Union_of_directed isSubgroup_unionᵢ_of_directed\n#align is_add_subgroup_Union_of_directed isAddSubgroup_unionᵢ_of_directed\n\nend Group\n\nnamespace IsSubgroup\n\nopen IsSubmonoid\n\nvariable [Group G] {s : Set G} (hs : IsSubgroup s)\n\n@[to_additive]\ntheorem inv_mem_iff : a⁻¹ ∈ s ↔ a ∈ s :=\n  ⟨fun h => by simpa using hs.inv_mem h, inv_mem hs⟩\n#align is_subgroup.inv_mem_iff IsSubgroup.inv_mem_iff\n#align is_add_subgroup.neg_mem_iff IsAddSubgroup.neg_mem_iff\n\n@[to_additive]\ntheorem mul_mem_cancel_right (h : a ∈ s) : b * a ∈ s ↔ b ∈ s :=\n  ⟨fun hba => by simpa using hs.mul_mem hba (hs.inv_mem h), fun hb => hs.mul_mem hb h⟩\n#align is_subgroup.mul_mem_cancel_right IsSubgroup.mul_mem_cancel_right\n#align is_add_subgroup.add_mem_cancel_right IsAddSubgroup.add_mem_cancel_right\n\n@[to_additive]\ntheorem mul_mem_cancel_left (h : a ∈ s) : a * b ∈ s ↔ b ∈ s :=\n  ⟨fun hab => by simpa using hs.mul_mem (hs.inv_mem h) hab, hs.mul_mem h⟩\n#align is_subgroup.mul_mem_cancel_left IsSubgroup.mul_mem_cancel_left\n#align is_add_subgroup.add_mem_cancel_left IsAddSubgroup.add_mem_cancel_left\n\nend IsSubgroup\n\n/-- `IsNormalAddSubgroup (s : Set A)` expresses the fact that `s` is a normal additive subgroup\nof the additive group `A`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : AddSubgroup A` and `hs : S.normal`, and not via this structure. -/\nstructure IsNormalAddSubgroup [AddGroup A] (s : Set A) extends IsAddSubgroup s : Prop where\n  /-- The proposition that `s` is closed under (additive) conjugation. -/\n  normal : ∀ n ∈ s, ∀ g : A, g + n + -g ∈ s\n#align is_normal_add_subgroup IsNormalAddSubgroup\n\n/-- `IsNormalSubgroup (s : Set G)` expresses the fact that `s` is a normal subgroup\nof the group `G`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : Subgroup G` and not via this structure. -/\n@[to_additive]\nstructure IsNormalSubgroup [Group G] (s : Set G) extends IsSubgroup s : Prop where\n  /-- The proposition that `s` is closed under conjugation. -/\n  normal : ∀ n ∈ s, ∀ g : G, g * n * g⁻¹ ∈ s\n#align is_normal_subgroup IsNormalSubgroup\n\n@[to_additive]\ntheorem isNormalSubgroup_of_commGroup [CommGroup G] {s : Set G} (hs : IsSubgroup s) :\n    IsNormalSubgroup s :=\n  { hs with normal := fun n hn g => by rwa [mul_right_comm, mul_right_inv, one_mul] }\n#align is_normal_subgroup_of_comm_group isNormalSubgroup_of_commGroup\n#align is_normal_add_subgroup_of_add_comm_group isNormalAddSubgroup_of_addCommGroup\n\ntheorem Additive.isNormalAddSubgroup [Group G] {s : Set G} (hs : IsNormalSubgroup s) :\n    @IsNormalAddSubgroup (Additive G) _ s :=\n  @IsNormalAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubgroup hs.toIsSubgroup)\n    (@IsNormalSubgroup.normal _ ‹Group (Additive G)› _ hs)\n    -- porting note: Lean needs help synthesising\n#align additive.is_normal_add_subgroup Additive.isNormalAddSubgroup\n\ntheorem Additive.isNormalAddSubgroup_iff [Group G] {s : Set G} :\n    @IsNormalAddSubgroup (Additive G) _ s ↔ IsNormalSubgroup s :=\n  ⟨by rintro ⟨h₁, h₂⟩; exact @IsNormalSubgroup.mk G _ _ (Additive.isAddSubgroup_iff.1 h₁) @h₂,\n    fun h => Additive.isNormalAddSubgroup h⟩\n#align additive.is_normal_add_subgroup_iff Additive.isNormalAddSubgroup_iff\n\ntheorem Multiplicative.isNormalSubgroup [AddGroup A] {s : Set A} (hs : IsNormalAddSubgroup s) :\n    @IsNormalSubgroup (Multiplicative A) _ s :=\n  @IsNormalSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubgroup hs.toIsAddSubgroup)\n    (@IsNormalAddSubgroup.normal _ ‹AddGroup (Multiplicative A)› _ hs)\n#align multiplicative.is_normal_subgroup Multiplicative.isNormalSubgroup\n\ntheorem Multiplicative.isNormalSubgroup_iff [AddGroup A] {s : Set A} :\n    @IsNormalSubgroup (Multiplicative A) _ s ↔ IsNormalAddSubgroup s :=\n  ⟨by\n    rintro ⟨h₁, h₂⟩;\n      exact @IsNormalAddSubgroup.mk A _ _ (Multiplicative.isSubgroup_iff.1 h₁) @h₂,\n    fun h => Multiplicative.isNormalSubgroup h⟩\n#align multiplicative.is_normal_subgroup_iff Multiplicative.isNormalSubgroup_iff\n\nnamespace IsSubgroup\n\nvariable [Group G]\n\n-- Normal subgroup properties\n@[to_additive]\ntheorem mem_norm_comm {s : Set G} (hs : IsNormalSubgroup s) {a b : G} (hab : a * b ∈ s) :\n    b * a ∈ s := by\n  have h : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ s := hs.normal (a * b) hab a⁻¹\n  simp at h; exact h\n#align is_subgroup.mem_norm_comm IsSubgroup.mem_norm_comm\n#align is_add_subgroup.mem_norm_comm IsAddSubgroup.mem_norm_comm\n\n@[to_additive]\ntheorem mem_norm_comm_iff {s : Set G} (hs : IsNormalSubgroup s) {a b : G} : a * b ∈ s ↔ b * a ∈ s :=\n  ⟨mem_norm_comm hs, mem_norm_comm hs⟩\n#align is_subgroup.mem_norm_comm_iff IsSubgroup.mem_norm_comm_iff\n#align is_add_subgroup.mem_norm_comm_iff IsAddSubgroup.mem_norm_comm_iff\n\n/-- The trivial subgroup -/\n@[to_additive \"the trivial additive subgroup\"]\ndef trivial (G : Type _) [Group G] : Set G :=\n  {1}\n#align is_subgroup.trivial IsSubgroup.trivial\n#align is_add_subgroup.trivial IsAddSubgroup.trivial\n\n@[to_additive (attr := simp)]\ntheorem mem_trivial {g : G} : g ∈ trivial G ↔ g = 1 :=\n  mem_singleton_iff\n#align is_subgroup.mem_trivial IsSubgroup.mem_trivial\n#align is_add_subgroup.mem_trivial IsAddSubgroup.mem_trivial\n\n@[to_additive]\ntheorem trivial_normal : IsNormalSubgroup (trivial G) := by\n  refine' { .. } <;> simp (config := { contextual := true }) [trivial]\n#align is_subgroup.trivial_normal IsSubgroup.trivial_normal\n#align is_add_subgroup.trivial_normal IsAddSubgroup.trivial_normal\n\n@[to_additive]\ntheorem eq_trivial_iff {s : Set G} (hs : IsSubgroup s) : s = trivial G ↔ ∀ x ∈ s, x = (1 : G) := by\n  simp only [Set.ext_iff, IsSubgroup.mem_trivial];\n    exact ⟨fun h x => (h x).1, fun h x => ⟨h x, fun hx => hx.symm ▸ hs.toIsSubmonoid.one_mem⟩⟩\n#align is_subgroup.eq_trivial_iff IsSubgroup.eq_trivial_iff\n#align is_add_subgroup.eq_trivial_iff IsAddSubgroup.eq_trivial_iff\n\n@[to_additive]\ntheorem univ_subgroup : IsNormalSubgroup (@univ G) := by refine' { .. } <;> simp\n#align is_subgroup.univ_subgroup IsSubgroup.univ_subgroup\n#align is_add_subgroup.univ_add_subgroup IsAddSubgroup.univ_addSubgroup\n\n/-- The underlying set of the center of a group. -/\n@[to_additive addCenter \"The underlying set of the center of an additive group.\"]\ndef center (G : Type _) [Group G] : Set G :=\n  { z | ∀ g, g * z = z * g }\n#align is_subgroup.center IsSubgroup.center\n#align is_add_subgroup.add_center IsAddSubgroup.addCenter\n\n@[to_additive mem_add_center]\ntheorem mem_center {a : G} : a ∈ center G ↔ ∀ g, g * a = a * g :=\n  Iff.rfl\n#align is_subgroup.mem_center IsSubgroup.mem_center\n#align is_add_subgroup.mem_add_center IsAddSubgroup.mem_add_center\n\n@[to_additive add_center_normal]\ntheorem center_normal : IsNormalSubgroup (center G) :=\n  { one_mem := by simp [center]\n    mul_mem := fun ha hb g => by\n      rw [← mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, ← mul_assoc]\n    inv_mem := fun {a} ha g =>\n      calc\n        g * a⁻¹ = a⁻¹ * (g * a) * a⁻¹ := by simp [ha g]\n        _ = a⁻¹ * g := by rw [← mul_assoc, mul_assoc]; simp\n    normal := fun n ha g h =>\n      calc\n        h * (g * n * g⁻¹) = h * n := by simp [ha g, mul_assoc]\n        _ = g * g⁻¹ * n * h := by rw [ha h]; simp\n        _ = g * n * g⁻¹ * h := by rw [mul_assoc g, ha g⁻¹, ← mul_assoc]\n         }\n#align is_subgroup.center_normal IsSubgroup.center_normal\n#align is_add_subgroup.add_center_normal IsAddSubgroup.add_center_normal\n\n/-- The underlying set of the normalizer of a subset `S : Set G` of a group `G`. That is,\n  the elements `g : G` such that `g * S * g⁻¹ = S`. -/\n@[to_additive addNormalizer\n      \"The underlying set of the normalizer of a subset `S : Set A` of an\n      additive group `A`. That is, the elements `a : A` such that `a + S - a = S`.\"]\ndef normalizer (s : Set G) : Set G :=\n  { g : G | ∀ n, n ∈ s ↔ g * n * g⁻¹ ∈ s }\n#align is_subgroup.normalizer IsSubgroup.normalizer\n#align is_add_subgroup.add_normalizer IsAddSubgroup.addNormalizer\n\n@[to_additive]\ntheorem normalizer_isSubgroup (s : Set G) : IsSubgroup (normalizer s) :=\n  { one_mem := by simp [normalizer]\n    mul_mem := fun {a b}\n      (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) (hb : ∀ n, n ∈ s ↔ b * n * b⁻¹ ∈ s) n =>\n      by rw [mul_inv_rev, ← mul_assoc, mul_assoc a, mul_assoc a, ← ha, ← hb]\n    inv_mem := fun {a} (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) n => by\n      rw [ha (a⁻¹ * n * a⁻¹⁻¹)]; simp [mul_assoc] }\n#align is_subgroup.normalizer_is_subgroup IsSubgroup.normalizer_isSubgroup\n#align is_add_subgroup.normalizer_is_add_subgroup IsAddSubgroup.normalizer_isAddSubgroup\n\n@[to_additive subset_add_normalizer]\ntheorem subset_normalizer {s : Set G} (hs : IsSubgroup s) : s ⊆ normalizer s := fun g hg n => by\n  rw [IsSubgroup.mul_mem_cancel_right hs ((IsSubgroup.inv_mem_iff hs).2 hg),\n    IsSubgroup.mul_mem_cancel_left hs hg]\n#align is_subgroup.subset_normalizer IsSubgroup.subset_normalizer\n#align is_add_subgroup.subset_add_normalizer IsAddSubgroup.subset_add_normalizer\n\nend IsSubgroup\n\n-- Homomorphism subgroups\nnamespace IsGroupHom\n\nopen IsSubmonoid IsSubgroup\n\n/-- `ker f : set G` is the underlying subset of the kernel of a map `G → H`. -/\n@[to_additive \"`ker f : set A` is the underlying subset of the kernel of a map `A → B`\"]\ndef ker [Group H] (f : G → H) : Set G :=\n  preimage f (trivial H)\n#align is_group_hom.ker IsGroupHom.ker\n#align is_add_group_hom.ker IsAddGroupHom.ker\n\n@[to_additive]\ntheorem mem_ker [Group H] (f : G → H) {x : G} : x ∈ ker f ↔ f x = 1 :=\n  mem_trivial\n#align is_group_hom.mem_ker IsGroupHom.mem_ker\n#align is_add_group_hom.mem_ker IsAddGroupHom.mem_ker\n\nvariable [Group G] [Group H]\n\n@[to_additive]\ntheorem one_ker_inv {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f (a * b⁻¹) = 1) : f a = f b :=\n  by\n  rw [hf.map_mul, hf.map_inv] at h\n  rw [← inv_inv (f b), eq_inv_of_mul_eq_one_left h]\n#align is_group_hom.one_ker_inv IsGroupHom.one_ker_inv\n#align is_add_group_hom.zero_ker_neg IsAddGroupHom.zero_ker_neg\n\n@[to_additive]\ntheorem one_ker_inv' {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f (a⁻¹ * b) = 1) : f a = f b :=\n  by\n  rw [hf.map_mul, hf.map_inv] at h\n  apply inv_injective\n  rw [eq_inv_of_mul_eq_one_left h]\n#align is_group_hom.one_ker_inv' IsGroupHom.one_ker_inv'\n#align is_add_group_hom.zero_ker_neg' IsAddGroupHom.zero_ker_neg'\n\n@[to_additive]\ntheorem inv_ker_one {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) : f (a * b⁻¹) = 1 :=\n  by\n  have : f a * (f b)⁻¹ = 1 := by rw [h, mul_right_inv]\n  rwa [← hf.map_inv, ← hf.map_mul] at this\n#align is_group_hom.inv_ker_one IsGroupHom.inv_ker_one\n#align is_add_group_hom.neg_ker_zero IsAddGroupHom.neg_ker_zero\n\n@[to_additive]\ntheorem inv_ker_one' {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) : f (a⁻¹ * b) = 1 :=\n  by\n  have : (f a)⁻¹ * f b = 1 := by rw [h, mul_left_inv]\n  rwa [← hf.map_inv, ← hf.map_mul] at this\n#align is_group_hom.inv_ker_one' IsGroupHom.inv_ker_one'\n#align is_add_group_hom.neg_ker_zero' IsAddGroupHom.neg_ker_zero'\n\n@[to_additive]\ntheorem one_iff_ker_inv {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ f (a * b⁻¹) = 1 :=\n  ⟨hf.inv_ker_one, hf.one_ker_inv⟩\n#align is_group_hom.one_iff_ker_inv IsGroupHom.one_iff_ker_inv\n#align is_add_group_hom.zero_iff_ker_neg IsAddGroupHom.zero_iff_ker_neg\n\n@[to_additive]\ntheorem one_iff_ker_inv' {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ f (a⁻¹ * b) = 1 :=\n  ⟨hf.inv_ker_one', hf.one_ker_inv'⟩\n#align is_group_hom.one_iff_ker_inv' IsGroupHom.one_iff_ker_inv'\n#align is_add_group_hom.zero_iff_ker_neg' IsAddGroupHom.zero_iff_ker_neg'\n\n@[to_additive]\ntheorem inv_iff_ker {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ a * b⁻¹ ∈ ker f := by\n  rw [mem_ker]; exact one_iff_ker_inv hf _ _\n#align is_group_hom.inv_iff_ker IsGroupHom.inv_iff_ker\n#align is_add_group_hom.neg_iff_ker IsAddGroupHom.neg_iff_ker\n\n@[to_additive]\ntheorem inv_iff_ker' {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ a⁻¹ * b ∈ ker f := by\n  rw [mem_ker]; exact one_iff_ker_inv' hf _ _\n#align is_group_hom.inv_iff_ker' IsGroupHom.inv_iff_ker'\n#align is_add_group_hom.neg_iff_ker' IsAddGroupHom.neg_iff_ker'\n\n@[to_additive]\ntheorem image_subgroup {f : G → H} (hf : IsGroupHom f) {s : Set G} (hs : IsSubgroup s) :\n    IsSubgroup (f '' s) :=\n  { mul_mem := fun {a₁ a₂} ⟨b₁, hb₁, eq₁⟩ ⟨b₂, hb₂, eq₂⟩ =>\n      ⟨b₁ * b₂, hs.mul_mem hb₁ hb₂, by simp [eq₁, eq₂, hf.map_mul]⟩\n    one_mem := ⟨1, hs.toIsSubmonoid.one_mem, hf.map_one⟩\n    inv_mem := fun {a} ⟨b, hb, Eq⟩ =>\n      ⟨b⁻¹, hs.inv_mem hb, by\n        rw [hf.map_inv]\n        simp [*]⟩ }\n#align is_group_hom.image_subgroup IsGroupHom.image_subgroup\n#align is_add_group_hom.image_add_subgroup IsAddGroupHom.image_addSubgroup\n\n@[to_additive]\ntheorem range_subgroup {f : G → H} (hf : IsGroupHom f) : IsSubgroup (Set.range f) :=\n  @Set.image_univ _ _ f ▸ hf.image_subgroup univ_subgroup.toIsSubgroup\n#align is_group_hom.range_subgroup IsGroupHom.range_subgroup\n#align is_add_group_hom.range_add_subgroup IsAddGroupHom.range_addSubgroup\n\nattribute [local simp] IsSubmonoid.one_mem IsSubgroup.inv_mem\n  IsSubmonoid.mul_mem IsNormalSubgroup.normal\n\n@[to_additive]\ntheorem preimage {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsSubgroup s) :\n    IsSubgroup (f ⁻¹' s) := by\n  refine' { .. } <;>\n    simp (config := { contextual := true }) [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul,\n      hf.map_one, hf.map_inv, InvMemClass.inv_mem]\n#align is_group_hom.preimage IsGroupHom.preimage\n#align is_add_group_hom.preimage IsAddGroupHom.preimage\n\n@[to_additive]\ntheorem preimage_normal {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsNormalSubgroup s) :\n    IsNormalSubgroup (f ⁻¹' s) :=\n  { one_mem := by simp [hf.map_one, hs.toIsSubgroup.one_mem]\n    mul_mem := by simp (config := { contextual := true }) [hf.map_mul, hs.toIsSubgroup.mul_mem]\n    inv_mem := by simp (config := { contextual := true }) [hf.map_inv, hs.toIsSubgroup.inv_mem]\n    normal := by simp (config := { contextual := true }) [hs.normal, hf.map_mul, hf.map_inv] }\n#align is_group_hom.preimage_normal IsGroupHom.preimage_normal\n#align is_add_group_hom.preimage_normal IsAddGroupHom.preimage_normal\n\n@[to_additive]\ntheorem isNormalSubgroup_ker {f : G → H} (hf : IsGroupHom f) : IsNormalSubgroup (ker f) :=\n  hf.preimage_normal trivial_normal\n#align is_group_hom.is_normal_subgroup_ker IsGroupHom.isNormalSubgroup_ker\n#align is_add_group_hom.is_normal_add_subgroup_ker IsAddGroupHom.isNormalAddSubgroup_ker\n\n@[to_additive]\ntheorem injective_of_trivial_ker {f : G → H} (hf : IsGroupHom f) (h : ker f = trivial G) :\n    Function.Injective f := by\n  intro a₁ a₂ hfa\n  simp [ext_iff, ker, IsSubgroup.trivial] at h\n  have ha : a₁ * a₂⁻¹ = 1 := by rw [← h]; exact hf.inv_ker_one hfa\n  rw [eq_inv_of_mul_eq_one_left ha, inv_inv a₂]\n#align is_group_hom.injective_of_trivial_ker IsGroupHom.injective_of_trivial_ker\n#align is_add_group_hom.injective_of_trivial_ker IsAddGroupHom.injective_of_trivial_ker\n\n@[to_additive]\ntheorem trivial_ker_of_injective {f : G → H} (hf : IsGroupHom f) (h : Function.Injective f) :\n    ker f = trivial G :=\n  Set.ext fun x =>\n    Iff.intro\n      (fun hx => by\n        suffices f x = f 1 by simpa using h this\n        simp [hf.map_one]; rwa [mem_ker] at hx)\n      (by simp (config := { contextual := true }) [mem_ker, hf.map_one])\n#align is_group_hom.trivial_ker_of_injective IsGroupHom.trivial_ker_of_injective\n#align is_add_group_hom.trivial_ker_of_injective IsAddGroupHom.trivial_ker_of_injective\n\n@[to_additive]\ntheorem injective_iff_trivial_ker {f : G → H} (hf : IsGroupHom f) :\n    Function.Injective f ↔ ker f = trivial G :=\n  ⟨hf.trivial_ker_of_injective, hf.injective_of_trivial_ker⟩\n#align is_group_hom.injective_iff_trivial_ker IsGroupHom.injective_iff_trivial_ker\n#align is_add_group_hom.injective_iff_trivial_ker IsAddGroupHom.injective_iff_trivial_ker\n\n@[to_additive]\ntheorem trivial_ker_iff_eq_one {f : G → H} (hf : IsGroupHom f) :\n    ker f = trivial G ↔ ∀ x, f x = 1 → x = 1 := by\n  rw [Set.ext_iff]; simp [ker];\n    exact ⟨fun h x hx => (h x).1 hx, fun h x => ⟨h x, fun hx => by rw [hx, hf.map_one]⟩⟩\n#align is_group_hom.trivial_ker_iff_eq_one IsGroupHom.trivial_ker_iff_eq_one\n#align is_add_group_hom.trivial_ker_iff_eq_zero IsAddGroupHom.trivial_ker_iff_eq_zero\n\nend IsGroupHom\n\nnamespace AddGroup\n\nvariable [AddGroup A]\n\n/-- If `A` is an additive group and `s : Set A`, then `InClosure s : Set A` is the underlying\nsubset of the subgroup generated by `s`. -/\ninductive InClosure (s : Set A) : A → Prop\n  | basic {a : A} : a ∈ s → InClosure s a\n  | zero : InClosure s 0\n  | neg {a : A} : InClosure s a → InClosure s (-a)\n  | add {a b : A} : InClosure s a → InClosure s b → InClosure s (a + b)\n#align add_group.in_closure AddGroup.InClosure\n\nend AddGroup\n\nnamespace Group\n\nopen IsSubmonoid IsSubgroup\n\nvariable [Group G] {s : Set G}\n\n/-- If `G` is a group and `s : Set G`, then `InClosure s : Set G` is the underlying\nsubset of the subgroup generated by `s`. -/\n@[to_additive]\ninductive InClosure (s : Set G) : G → Prop\n  | basic {a : G} : a ∈ s → InClosure s a\n  | one : InClosure s 1\n  | inv {a : G} : InClosure s a → InClosure s a⁻¹\n  | mul {a b : G} : InClosure s a → InClosure s b → InClosure s (a * b)\n#align group.in_closure Group.InClosure\n\n/-- `Group.closure s` is the subgroup generated by `s`, i.e. the smallest subgroup containg `s`. -/\n@[to_additive\n  \"`AddGroup.closure s` is the additive subgroup generated by `s`, i.e., the\n  smallest additive subgroup containing `s`.\"]\ndef closure (s : Set G) : Set G :=\n  { a | InClosure s a }\n#align group.closure Group.closure\n#align add_group.closure AddGroup.closure\n\n@[to_additive]\ntheorem mem_closure {a : G} : a ∈ s → a ∈ closure s :=\n  InClosure.basic\n#align group.mem_closure Group.mem_closure\n#align add_group.mem_closure AddGroup.mem_closure\n\n@[to_additive]\ntheorem closure.isSubgroup (s : Set G) : IsSubgroup (closure s) :=\n  { one_mem := InClosure.one\n    mul_mem := InClosure.mul\n    inv_mem := InClosure.inv }\n#align group.closure.is_subgroup Group.closure.isSubgroup\n#align add_group.closure.is_add_subgroup AddGroup.closure.isAddSubgroup\n\n@[to_additive]\ntheorem subset_closure {s : Set G} : s ⊆ closure s := fun _ => mem_closure\n#align group.subset_closure Group.subset_closure\n#align add_group.subset_closure AddGroup.subset_closure\n\n@[to_additive]\ntheorem closure_subset {s t : Set G} (ht : IsSubgroup t) (h : s ⊆ t) : closure s ⊆ t := fun a ha =>\n  by induction ha <;> simp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n#align group.closure_subset Group.closure_subset\n#align add_group.closure_subset AddGroup.closure_subset\n\n@[to_additive]\ntheorem closure_subset_iff {s t : Set G} (ht : IsSubgroup t) : closure s ⊆ t ↔ s ⊆ t :=\n  ⟨fun h _ ha => h (mem_closure ha), fun h _ ha => closure_subset ht h ha⟩\n#align group.closure_subset_iff Group.closure_subset_iff\n#align add_group.closure_subset_iff AddGroup.closure_subset_iff\n\n@[to_additive]\ntheorem closure_mono {s t : Set G} (h : s ⊆ t) : closure s ⊆ closure t :=\n  closure_subset (closure.isSubgroup _) <| Set.Subset.trans h subset_closure\n#align group.closure_mono Group.closure_mono\n#align add_group.closure_mono AddGroup.closure_mono\n\n@[to_additive (attr := simp)]\ntheorem closure_subgroup {s : Set G} (hs : IsSubgroup s) : closure s = s :=\n  Set.Subset.antisymm (closure_subset hs <| Set.Subset.refl s) subset_closure\n#align group.closure_subgroup Group.closure_subgroup\n#align add_group.closure_add_subgroup AddGroup.closure_addSubgroup\n\n@[to_additive]\ntheorem exists_list_of_mem_closure {s : Set G} {a : G} (h : a ∈ closure s) :\n    ∃ l : List G, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = a :=\n  InClosure.recOn h (fun {x} hxs => ⟨[x], List.forall_mem_singleton.2 <| Or.inl hxs, one_mul _⟩)\n    ⟨[], List.forall_mem_nil _, rfl⟩\n    (fun {x} _ ⟨L, HL1, HL2⟩ =>\n      ⟨L.reverse.map Inv.inv, fun x hx =>\n        let ⟨y, hy1, hy2⟩ := List.exists_of_mem_map hx\n        hy2 ▸ Or.imp id (by rw [inv_inv]; exact id) (HL1 _ <| List.mem_reverse'.1 hy1).symm,\n        HL2 ▸\n          List.recOn L inv_one.symm fun hd tl ih => by\n            rw [List.reverse_cons, List.map_append, List.prod_append, ih, List.map_singleton,\n              List.prod_cons, List.prod_nil, mul_one, List.prod_cons, mul_inv_rev]⟩)\n    fun {x y} _ _ ⟨L1, HL1, HL2⟩ ⟨L2, HL3, HL4⟩ =>\n    ⟨L1 ++ L2, List.forall_mem_append.2 ⟨HL1, HL3⟩, by rw [List.prod_append, HL2, HL4]⟩\n#align group.exists_list_of_mem_closure Group.exists_list_of_mem_closure\n#align add_group.exists_list_of_mem_closure AddGroup.exists_list_of_mem_closure\n\n@[to_additive]\ntheorem image_closure [Group H] {f : G → H} (hf : IsGroupHom f) (s : Set G) :\n    f '' closure s = closure (f '' s) :=\n  le_antisymm\n    (by\n      rintro _ ⟨x, hx, rfl⟩\n      exact InClosure.recOn hx\n        (by intros _ ha; exact subset_closure (mem_image_of_mem f ha))\n        (by\n          rw [hf.map_one]\n          apply IsSubmonoid.one_mem (closure.isSubgroup _).toIsSubmonoid)\n        (by\n          intros _ _\n          rw [hf.map_inv]\n          apply IsSubgroup.inv_mem (closure.isSubgroup _))\n        (by\n          intros _ _ _ _ ha hb\n          rw [hf.map_mul]\n          exact (closure.isSubgroup (f '' s)).toIsSubmonoid.mul_mem ha hb))\n    (closure_subset (hf.image_subgroup <| closure.isSubgroup _) <|\n      Set.image_subset _ subset_closure)\n#align group.image_closure Group.image_closure\n#align add_group.image_closure AddGroup.image_closure\n\n@[to_additive]\ntheorem mclosure_subset {s : Set G} : Monoid.Closure s ⊆ closure s :=\n  Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid <| subset_closure\n#align group.mclosure_subset Group.mclosure_subset\n#align add_group.mclosure_subset AddGroup.mclosure_subset\n\n@[to_additive]\ntheorem mclosure_inv_subset {s : Set G} : Monoid.Closure (Inv.inv ⁻¹' s) ⊆ closure s :=\n  Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid fun x hx =>\n    inv_inv x ▸ ((closure.isSubgroup _).inv_mem <| subset_closure hx)\n#align group.mclosure_inv_subset Group.mclosure_inv_subset\n#align add_group.mclosure_neg_subset AddGroup.mclosure_neg_subset\n\n@[to_additive]\ntheorem closure_eq_mclosure {s : Set G} : closure s = Monoid.Closure (s ∪ Inv.inv ⁻¹' s) :=\n  Set.Subset.antisymm\n    (@closure_subset _ _ _ (Monoid.Closure (s ∪ Inv.inv ⁻¹' s))\n      { one_mem := (Monoid.closure.isSubmonoid _).one_mem\n        mul_mem := (Monoid.closure.isSubmonoid _).mul_mem\n        inv_mem := fun hx =>\n          Monoid.InClosure.recOn hx\n            (fun {x} hx =>\n              Or.casesOn hx\n                (fun hx =>\n                  Monoid.subset_closure <| Or.inr <| show x⁻¹⁻¹ ∈ s from (inv_inv x).symm ▸ hx)\n                fun hx => Monoid.subset_closure <| Or.inl hx)\n            ((@inv_one G _).symm ▸ IsSubmonoid.one_mem (Monoid.closure.isSubmonoid _))\n            fun {x y} _ _ ihx ihy =>\n            (mul_inv_rev x y).symm ▸ IsSubmonoid.mul_mem (Monoid.closure.isSubmonoid _) ihy ihx }\n      (Set.Subset.trans (Set.subset_union_left _ _) Monoid.subset_closure))\n    (Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid <|\n      Set.union_subset subset_closure fun x hx =>\n        inv_inv x ▸ (IsSubgroup.inv_mem (closure.isSubgroup _) <| subset_closure hx))\n#align group.closure_eq_mclosure Group.closure_eq_mclosure\n#align add_group.closure_eq_mclosure AddGroup.closure_eq_mclosure\n\n@[to_additive]\ntheorem mem_closure_union_iff {G : Type _} [CommGroup G] {s t : Set G} {x : G} :\n    x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=\n  by\n  simp only [closure_eq_mclosure, Monoid.mem_closure_union_iff, exists_prop, preimage_union];\n  constructor\n  · rintro ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, rfl⟩\n    refine' ⟨_, ⟨_, hys, _, hzs, rfl⟩, _, ⟨_, hyt, _, hzt, rfl⟩, _⟩\n    rw [mul_assoc, mul_assoc, mul_left_comm zs]\n  · rintro ⟨_, ⟨ys, hys, zs, hzs, rfl⟩, _, ⟨yt, hyt, zt, hzt, rfl⟩, rfl⟩\n    refine' ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, _⟩\n    rw [mul_assoc, mul_assoc, mul_left_comm yt]\n#align group.mem_closure_union_iff Group.mem_closure_union_iff\n#align add_group.mem_closure_union_iff AddGroup.mem_closure_union_iff\n\nend Group\n\nnamespace IsSubgroup\n\nvariable [Group G]\n\n@[to_additive]\ntheorem trivial_eq_closure : trivial G = Group.closure ∅ :=\n  Subset.antisymm (by simp [Set.subset_def, (Group.closure.isSubgroup _).one_mem])\n    (Group.closure_subset trivial_normal.toIsSubgroup <| by simp)\n#align is_subgroup.trivial_eq_closure IsSubgroup.trivial_eq_closure\n#align is_add_subgroup.trivial_eq_closure IsAddSubgroup.trivial_eq_closure\n\nend IsSubgroup\n\n/-The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\nnamespace Group\n\nvariable {s : Set G} [Group G]\n\ntheorem conjugatesOf_subset {t : Set G} (ht : IsNormalSubgroup t) {a : G} (h : a ∈ t) :\n    conjugatesOf a ⊆ t := fun x hc =>\n  by\n  obtain ⟨c, w⟩ := isConj_iff.1 hc\n  have H := IsNormalSubgroup.normal ht a h c\n  rwa [← w]\n#align group.conjugates_of_subset Group.conjugatesOf_subset\n\ntheorem conjugatesOfSet_subset' {s t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) :\n    conjugatesOfSet s ⊆ t :=\n  Set.unionᵢ₂_subset fun _ H => conjugatesOf_subset ht (h H)\n#align group.conjugates_of_set_subset' Group.conjugatesOfSet_subset'\n\n/-- The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\ndef normalClosure (s : Set G) : Set G :=\n  closure (conjugatesOfSet s)\n#align group.normal_closure Group.normalClosure\n\ntheorem conjugatesOfSet_subset_normalClosure : conjugatesOfSet s ⊆ normalClosure s :=\n  subset_closure\n#align group.conjugates_of_set_subset_normal_closure Group.conjugatesOfSet_subset_normalClosure\n\ntheorem subset_normalClosure : s ⊆ normalClosure s :=\n  Set.Subset.trans subset_conjugatesOfSet conjugatesOfSet_subset_normalClosure\n#align group.subset_normal_closure Group.subset_normalClosure\n\n/-- The normal closure of a set is a subgroup. -/\ntheorem normalClosure.isSubgroup (s : Set G) : IsSubgroup (normalClosure s) :=\n  closure.isSubgroup (conjugatesOfSet s)\n#align group.normal_closure.is_subgroup Group.normalClosure.isSubgroup\n\n/-- The normal closure of s is a normal subgroup. -/\ntheorem normalClosure.is_normal : IsNormalSubgroup (normalClosure s) :=\n  { normalClosure.isSubgroup _ with\n    normal := fun n h g => by\n      induction' h with x hx x hx ihx x y hx hy ihx ihy\n      · exact conjugatesOfSet_subset_normalClosure (conj_mem_conjugatesOfSet hx)\n      · simpa using (normalClosure.isSubgroup s).one_mem\n      · rw [← conj_inv]\n        exact (normalClosure.isSubgroup _).inv_mem ihx\n      · rw [← conj_mul]\n        exact (normalClosure.isSubgroup _).toIsSubmonoid.mul_mem ihx ihy }\n#align group.normal_closure.is_normal Group.normalClosure.is_normal\n\n/-- The normal closure of s is the smallest normal subgroup containing s. -/\ntheorem normalClosure_subset {s t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) :\n    normalClosure s ⊆ t := fun a w =>\n  by\n  induction' w with x hx x _ ihx x y _ _ ihx ihy\n  · exact conjugatesOfSet_subset' ht h <| hx\n  · exact ht.toIsSubgroup.toIsSubmonoid.one_mem\n  · exact ht.toIsSubgroup.inv_mem ihx\n  · exact ht.toIsSubgroup.toIsSubmonoid.mul_mem ihx ihy\n#align group.normal_closure_subset Group.normalClosure_subset\n\ntheorem normalClosure_subset_iff {s t : Set G} (ht : IsNormalSubgroup t) :\n    s ⊆ t ↔ normalClosure s ⊆ t :=\n  ⟨normalClosure_subset ht, Set.Subset.trans subset_normalClosure⟩\n#align group.normal_closure_subset_iff Group.normalClosure_subset_iff\n\ntheorem normalClosure_mono {s t : Set G} : s ⊆ t → normalClosure s ⊆ normalClosure t := fun h =>\n  normalClosure_subset normalClosure.is_normal (Set.Subset.trans h subset_normalClosure)\n#align group.normal_closure_mono Group.normalClosure_mono\n\nend Group\n\n/-- Create a bundled subgroup from a set `s` and `[IsSubgroup s]`. -/\n@[to_additive \"Create a bundled additive subgroup from a set `s` and `[is_add_subgroup s]`.\"]\ndef Subgroup.of [Group G] {s : Set G} (h : IsSubgroup s) : Subgroup G\n    where\n  carrier := s\n  one_mem' := h.1.1\n  mul_mem' := h.1.2\n  inv_mem' := h.2\n#align subgroup.of Subgroup.of\n#align add_subgroup.of AddSubgroup.of\n\n@[to_additive]\ntheorem Subgroup.isSubgroup [Group G] (K : Subgroup G) : IsSubgroup (K : Set G) :=\n  { one_mem := K.one_mem'\n    mul_mem := K.mul_mem'\n    inv_mem := K.inv_mem' }\n#align subgroup.is_subgroup Subgroup.isSubgroup\n#align add_subgroup.is_add_subgroup AddSubgroup.isAddSubgroup\n\n-- this will never fire if it's an instance\n@[to_additive]\ntheorem Subgroup.of_normal [Group G] (s : Set G) (h : IsSubgroup s) (n : IsNormalSubgroup s) :\n    Subgroup.Normal (Subgroup.of h) :=\n  { conj_mem := n.normal }\n#align subgroup.of_normal Subgroup.of_normal\n#align add_subgroup.of_normal AddSubgroup.of_normal\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Deprecated/Subgroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.3299353197194332}}
{"text": "import algebraic_topology.cech_nerve\n\nuniverse u\n\nnoncomputable theory\n\nopen_locale simplicial\n\nnamespace category_theory\n\nopen category_theory.limits\n\nvariables {C : Type*} [category C]\n\nnamespace simplicial_object\n\nvariables [∀ (n : ℕ) (f : arrow C),\n  has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\n\nsection\nopen simplex_category opposite limits.wide_pullback\n\nlemma hom_ext (X : simplicial_object.augmented C) (F : arrow C)\n  (f g : X ⟶ F.augmented_cech_nerve) (hl : f.left.app (op [0]) = g.left.app (op [0]))\n  (hr : f.right = g.right) :\n  f = g :=\nbegin\n  apply (cech_nerve_equiv X F).symm.injective,\n  dsimp only [cech_nerve_equiv_symm_apply],\n  ext1,\n  { simp only [equivalence_right_to_left_left],\n    rw hl },\n  { exact hr }\nend\n\n-- move this\n@[simps]\ndef augmented_cech_nerve.left_obj_zero_iso (F : arrow C) :\n  F.augmented_cech_nerve.left.obj (op [0]) ≅ F.left :=\n{ hom := π _ ⟨0⟩,\n  inv := lift F.hom (λ _, 𝟙 _) (λ _, category.id_comp _),\n  hom_inv_id' :=\n  begin\n    ext,\n    { rw [category.assoc, lift_π, category.id_comp, category.comp_id],\n      cases j, congr' 2, dsimp at j ⊢, exact subsingleton.elim _ _ },\n    { simp only [π_arrow, category.id_comp, limits.wide_pullback.lift_base, category.assoc], }\n  end,\n  inv_hom_id' := lift_π _ _ _ _ _ }\n.\n\n-- move this\nlemma augmented_cech_nerve.left_map_comp_obj_zero_iso\n  (F : arrow C) (n : simplex_category) (i : ulift (fin (n.len+1))) :\n  F.augmented_cech_nerve.left.map (n.const i.down).op ≫ (augmented_cech_nerve.left_obj_zero_iso F).hom =\n  wide_pullback.π _ i :=\nbegin\n  rw [← iso.eq_comp_inv],\n  dsimp only [arrow.augmented_cech_nerve_left, arrow.cech_nerve_map,\n    augmented_cech_nerve.left_obj_zero_iso_inv],\n  ext1 ⟨j⟩,\n  { rw [limits.wide_pullback.lift_π, category.assoc, limits.wide_pullback.lift_π, category.comp_id],\n    cases i, refl },\n  { rw [limits.wide_pullback.lift_base, category.assoc, limits.wide_pullback.lift_base,\n      limits.wide_pullback.π_arrow], }\nend\n.\n\n@[simp]\nlemma equivalence_left_to_right_left_app_zero_comp_π\n  (X : simplicial_object.augmented C) (F : arrow C) (G : augmented.to_arrow.obj X ⟶ F) (i) :\n  (equivalence_left_to_right X F G).left.app (op [0]) ≫ limits.wide_pullback.π _ i =\n  G.left :=\nbegin\n  dsimp only [equivalence_left_to_right_left_app, unop_op],\n  rw [limits.wide_pullback.lift_π, simplex_category.hom_zero_zero ([0].const i.down),\n    op_id, X.left.map_id, category.id_comp],\nend\n.\n\nend\n\nend simplicial_object\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/Cech/adjunction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217431943271999, "lm_q2_score": 0.45713671682749474, "lm_q1q2_score": 0.3299353142473247}}
{"text": "/-\nCopyright (c) 2020 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\nimport meta.rb_map\nimport tactic.core\n\n/-!\n# Tactics About Dependencies\n\nThis module provides tactics to compute dependencies and reverse dependencies of\nhypotheses. An expression `e` depends on a hypothesis `h` if `e` would not be\nvalid if `h` were removed from the context. For example, the expression\n`e := x > 0` depends on `x`. We say that `x` is a dependency of `e` and that `e`\nis a reverse dependency of `x`.\n\nIt is sometimes useful to consider *inclusive* dependency: `e` inclusively\ndepends on `h` iff `e` depends on `h` or `e = h` (so inclusive dependency is the\nreflexive closure of regular dependency).\n\nNote that the standard library does not use quite the same terminology:\n\n* `kdependencies`/`kdeps` from the standard library compute reverse\n  dependencies, not dependencies.\n* `kdepends_on` and functions derived from it ignore local definitions and\n  therefore compute a weaker dependency relation (see next section).\n\n## Local Definitions\n\nDetermining dependencies of hypotheses is usually straightforward: a hypothesis\n`r : R` depends on another hypothesis `d : D` if `d` occurs in `R`. The\nimplementation is more involved, however, in the presence of local definitions.\nConsider this context:\n\n```lean\nn m : ℕ\nk : ℕ := m\no : ℕ := k\nh : o > 0\n```\n\n`h` depends on `o`, `k` and `m`, but only the dependency on `o` is syntactically\nobvious. `kdepends_on` ignores this complication and claims that `h` does not\ndepend on `k` or `m`. We do not follow this example but process local\ndefinitions properly. This means that if the context contains a local\ndefinition, we need to compute the syntactic dependencies of `h`, then their\ndependencies, and so on.\n\n## Direct Dependencies\n\nIf you want to ignore local definitions while computing dependencies, this\nmodule also provides tactics to find the *direct* dependencies of a hypothesis.\nThese are the hypotheses that syntactically appear in the hypothesis's type (or\nvalue, if the hypothesis is a local definition).\n-/\n\nopen native\nopen expr_set (local_set_to_name_set)\nopen name_set (local_list_to_name_set)\n\nnamespace tactic\n\n/-! ### Direct Dependencies -/\n\n/-! #### Checking whether hypotheses directly depend on each other -/\n\n/--\n`type_has_local_in_name_set h ns` returns true iff the type of `h` contains a\nlocal constant whose unique name appears in `ns`.\n-/\nmeta def type_has_local_in_name_set (h : expr) (ns : name_set) : tactic bool := do\n  h_type ← infer_type h,\n  pure $ h_type.has_local_in ns\n\n/--\n`type_has_local_in_set h hs` returns true iff the type of `h` contains any of\nthe local constants `hs`.\n-/\nmeta def type_has_local_in_set (h : expr) (hs : expr_set) : tactic bool :=\ntype_has_local_in_name_set h $ local_set_to_name_set hs\n\n/--\n`type_has_local_in h hs` returns true iff the type of `h` contains any of the\nlocal constants `hs`.\n-/\nmeta def type_has_local_in (h : expr) (hs : list expr) : tactic bool :=\ntype_has_local_in_name_set h $ local_list_to_name_set hs\n\n/--\n`local_def_value_has_local_in_name_set h ns` returns true iff `h` is a local\ndefinition whose value contains a local constant whose unique name appears in\n`ns`.\n-/\nmeta def local_def_value_has_local_in_name_set (h : expr) (ns : name_set) :\n  tactic bool := do\n  (some h_val) ← try_core $ local_def_value h | pure ff,\n  pure $ h_val.has_local_in ns\n\n/--\n`local_def_value_has_local_in_set h hs` returns true iff `h` is a local\ndefinition whose value contains any of the local constants `hs`.\n-/\nmeta def local_def_value_has_local_in_set (h : expr) (hs : expr_set) :\n  tactic bool :=\nlocal_def_value_has_local_in_name_set h $ local_set_to_name_set hs\n\n/--\n`local_def_value_has_local_in h hs` returns true iff `h` is a local definition\nwhose value contains any of the local constants `hs`.\n-/\nmeta def local_def_value_has_local_in (h : expr) (hs : list expr) :\n  tactic bool :=\nlocal_def_value_has_local_in_name_set h $ local_list_to_name_set hs\n\n/--\n`hyp_directly_depends_on_local_name_set h ns` is true iff the hypothesis `h`\ndirectly depends on a hypothesis whose unique name appears in `ns`.\n-/\nmeta def hyp_directly_depends_on_local_name_set (h : expr) (ns : name_set) :\n  tactic bool :=\nlist.mbor\n  [ type_has_local_in_name_set h ns,\n    local_def_value_has_local_in_name_set h ns ]\n\n/--\n`hyp_directly_depends_on_local_set h hs` is true iff the hypothesis `h` directly\ndepends on any of the hypotheses `hs`.\n-/\nmeta def hyp_directly_depends_on_local_set (h : expr) (hs : expr_set) :\n  tactic bool :=\nhyp_directly_depends_on_local_name_set h $ local_set_to_name_set hs\n\n/--\n`hyp_directly_depends_on_locals h hs` is true iff the hypothesis `h` directly\ndepends on any of the hypotheses `hs`.\n-/\nmeta def hyp_directly_depends_on_locals (h : expr) (hs : list expr) :\n  tactic bool :=\nhyp_directly_depends_on_local_name_set h $ local_list_to_name_set hs\n\n/--\n`hyp_directly_depends_on_local_name_set_inclusive h ns` is true iff the\nhypothesis `h` directly depends on a hypothesis whose unique name appears in\n`ns` or `h`'s name appears in `ns`.\n-/\nmeta def hyp_directly_depends_on_local_name_set_inclusive (h : expr)\n  (ns : name_set) : tactic bool :=\nlist.mbor\n  [ pure $ ns.contains h.local_uniq_name\n  , hyp_directly_depends_on_local_name_set h ns ]\n\n/--\n`hyp_directly_depends_on_local_set_inclusive h ns` is true iff the hypothesis `h`\ndirectly depends on any of the hypotheses `hs` or `h` appears in `hs`.\n-/\nmeta def hyp_directly_depends_on_local_set_inclusive (h : expr) (hs : expr_set) :\n  tactic bool :=\nhyp_directly_depends_on_local_name_set_inclusive h $ local_set_to_name_set hs\n\n/--\n`hyp_directly_depends_on_locals_inclusive h ns` is true iff the hypothesis `h`\ndirectly depends on any of the hypotheses `hs` or `h` appears in `hs`.\n-/\nmeta def hyp_directly_depends_on_locals_inclusive (h : expr) (hs : list expr) :\n  tactic bool :=\nhyp_directly_depends_on_local_name_set_inclusive h $ local_list_to_name_set hs\n\n\n/-! #### Computing the direct dependencies of a hypothesis -/\n\n/--\n`direct_dependency_set_of_hyp h` is the set of hypotheses that the hypothesis\n`h` directly depends on. These are the hypotheses that appear in `h`'s type or\nvalue (if `h` is a local definition).\n-/\nmeta def direct_dependency_set_of_hyp (h : expr) : tactic expr_set := do\n  t ← infer_type h,\n  let deps := t.list_local_consts',\n  (some val) ← try_core $ local_def_value h | pure deps,\n  let deps := deps.union val.list_local_consts',\n  pure deps\n\n/--\n`direct_dependency_name_set_of_hyp h` is the set of unique names of hypotheses\nthat the hypothesis `h` directly depends on. These are the hypotheses that\nappear in `h`'s type or value (if `h` is a local definition).\n-/\nmeta def direct_dependency_name_set_of_hyp (h : expr) : tactic name_set :=\nlocal_set_to_name_set <$> direct_dependency_set_of_hyp h\n\n/--\n`direct_dependencies_of_hyp h` is the list of hypotheses that the hypothesis `h`\ndirectly depends on. These are the hypotheses that appear in `h`'s type or value\n(if `h` is a local definition). The dependencies are returned in no particular\norder.\n-/\nmeta def direct_dependencies_of_hyp (h : expr) : tactic (list expr) :=\nrb_set.to_list <$> direct_dependency_set_of_hyp h\n\n/--\n`direct_dependency_set_of_hyp_inclusive h` is the set of hypotheses that the\nhypothesis `h` directly depends on, plus `h` itself.\n-/\nmeta def direct_dependency_set_of_hyp_inclusive (h : expr) : tactic expr_set := do\n  deps ← direct_dependency_set_of_hyp h,\n  pure $ deps.insert h\n\n/--\n`direct_dependency_name_set_of_hyp_inclusive h` is the set of unique names of\nhypotheses that the hypothesis `h` directly depends on, plus `h` itself.\n-/\nmeta def direct_dependency_name_set_of_hyp_inclusive (h : expr) :\n  tactic name_set :=\nlocal_set_to_name_set <$> direct_dependency_set_of_hyp_inclusive h\n\n/--\n`direct_dependencies_of_hyp_inclusive h` is the list of hypotheses that the\nhypothesis `h` directly depends on, plus `h` itself. The dependencies are\nreturned in no particular order.\n-/\nmeta def direct_dependencies_of_hyp_inclusive (h : expr) : tactic (list expr) :=\nrb_set.to_list <$> direct_dependency_set_of_hyp_inclusive h\n\n\n/-! ### Indirect/Transitive Dependencies -/\n\n/-! #### Checking whether hypotheses depend on each other -/\n\n/--\n`hyp_depends_on_local_name_set' cache h ns` is true iff `h` depends on any of\nthe hypotheses whose unique names appear in `ns`. `cache` must be a set of\nhypotheses known *not* to depend (even indirectly) on any of the `ns`. This is\na performance optimisation, so you can give an empty cache. The tactic also\nreturns an expanded cache with hypotheses which the tactic has encountered.\n\nYou probably want to use `tactic.hyp_depends_on_local_name_set` or\n`tactic.hyps_depend_on_local_name_set` instead of this tactic.\n-/\nmeta def hyp_depends_on_local_name_set' : expr_set → expr → name_set →\n  tactic (bool × expr_set) := λ cache h ns, do\n  ff ← pure $ cache.contains h | pure (ff, cache),\n  direct_deps ← direct_dependency_set_of_hyp h,\n  let has_dep := direct_deps.fold ff (λ d b, b || ns.contains d.local_uniq_name),\n  ff ← pure has_dep | pure (tt, cache),\n  (has_dep, cache) ← direct_deps.mfold (ff, cache) $ λ d ⟨b, cache⟩,\n    if b\n      then pure (tt, cache)\n      else hyp_depends_on_local_name_set' cache d ns,\n  if has_dep\n    then pure (tt, cache)\n    else pure (ff, cache.insert h)\n\n/--\n`hyp_depends_on_local_name_set h ns` is true iff the hypothesis `h` depends on\nany of the hypotheses whose unique names appear in `ns`. If you need to check\ndependencies of multiple hypotheses, use `tactic.hyps_depend_on_local_name_set`.\n-/\nmeta def hyp_depends_on_local_name_set (h : expr) (ns : name_set) : tactic bool := do\n  ctx_has_local_def ← context_upto_hyp_has_local_def h,\n  if ctx_has_local_def\n    then prod.fst <$> hyp_depends_on_local_name_set' mk_expr_set h ns\n    else hyp_directly_depends_on_local_name_set h ns\n\n/--\n`hyp_depends_on_local_set h hs` is true iff the hypothesis `h` depends on\nany of the hypotheses `hs`. If you need to check dependencies of multiple\nhypotheses, use `tactic.hyps_depend_on_local_set`.\n-/\nmeta def hyp_depends_on_local_set (h : expr) (hs : expr_set) : tactic bool :=\nhyp_depends_on_local_name_set h $ local_set_to_name_set hs\n\n/--\n`hyp_depends_on_locals h hs` is true iff the hypothesis `h` depends on any of\nthe hypotheses `hs`. If you need to check dependencies of multiple hypotheses,\nuse `tactic.hyps_depend_on_locals`.\n-/\nmeta def hyp_depends_on_locals (h : expr) (hs : list expr) : tactic bool :=\nhyp_depends_on_local_name_set h $ local_list_to_name_set hs\n\n/--\n`hyps_depend_on_local_name_set hs ns` returns, for each `h ∈ hs`, whether `h`\ndepends on a hypothesis whose unique name appears in `ns`. This is the same as\n(but more efficient than) calling `tactic.hyp_depends_on_local_name_set` for\nevery `h ∈ hs`.\n-/\nmeta def hyps_depend_on_local_name_set (hs : list expr) (ns : name_set) :\n  tactic (list bool) := do\n  ctx_has_local ← context_has_local_def,\n  if ctx_has_local\n    then\n      let go : expr → list bool × expr_set → tactic (list bool × expr_set) :=\n      λ h ⟨deps, cache⟩, do\n      { (h_dep, cache) ← hyp_depends_on_local_name_set' cache h ns,\n        pure (h_dep :: deps, cache) }\n      in\n      prod.fst <$> hs.mfoldr go ([], mk_expr_map)\n    else hs.mmap $ λ h, hyp_directly_depends_on_local_name_set h ns\n\n/--\n`hyps_depend_on_local_set hs is` returns, for each `h ∈ hs`, whether `h` depends\non any of the hypotheses `is`. This is the same as (but more efficient than)\ncalling `tactic.hyp_depends_on_local_set` for every `h ∈ hs`.\n-/\nmeta def hyps_depend_on_local_set (hs : list expr) (is : expr_set) :\n  tactic (list bool) :=\nhyps_depend_on_local_name_set hs $ local_set_to_name_set is\n\n/--\n`hyps_depend_on_locals hs is` returns, for each `h ∈ hs`, whether `h` depends\non any of the hypotheses `is`. This is the same as (but more efficient than)\ncalling `tactic.hyp_depends_on_locals` for every `h ∈ hs`.\n-/\nmeta def hyps_depend_on_locals (hs is : list expr) : tactic (list bool) :=\nhyps_depend_on_local_name_set hs $ local_list_to_name_set is\n\n/--\n`hyp_depends_on_local_name_set_inclusive' cache h ns` is true iff the hypothesis\n`h` inclusively depends on a hypothesis whose unique name appears in `ns`.\n`cache` must be a set of hypotheses known *not* to depend (even indirectly) on\nany of the `ns`. This is a performance optimisation, so you can give an empty\ncache. The tactic also returns an expanded cache with hypotheses which the\ntactic has encountered. Note that the cache records exclusive, not inclusive\ndependencies.\n\nYou probably want to use `tactic.hyp_depends_on_local_name_set_inclusive` or\n`tactic.hyps_depend_on_local_name_set_inclusive` instead of this tactic.\n-/\nmeta def hyp_depends_on_local_name_set_inclusive' (cache : expr_set) (h : expr)\n  (ns : name_set) : tactic (bool × expr_set) :=\nif ns.contains h.local_uniq_name\n  then pure (tt, cache)\n  else hyp_depends_on_local_name_set' cache h ns\n\n/--\n`hyp_depends_on_local_name_set_inclusive h ns` is true iff the hypothesis `h`\ninclusively depends on any of the hypotheses whose unique names appear in `ns`.\nIf you need to check the dependencies of multiple hypotheses, use\n`tactic.hyps_depend_on_local_name_set_inclusive`.\n-/\nmeta def hyp_depends_on_local_name_set_inclusive (h : expr) (ns : name_set) :\n  tactic bool :=\nlist.mbor\n  [ pure $ ns.contains h.local_uniq_name,\n    hyp_depends_on_local_name_set h ns ]\n\n/--\n`hyp_depends_on_local_set_inclusive h hs` is true iff the hypothesis `h`\ninclusively depends on any of the hypotheses `hs`. If you need to check\ndependencies of multiple hypotheses, use\n`tactic.hyps_depend_on_local_set_inclusive`.\n-/\nmeta def hyp_depends_on_local_set_inclusive (h : expr) (hs : expr_set) :\n  tactic bool :=\nhyp_depends_on_local_name_set_inclusive h $ local_set_to_name_set hs\n\n/--\n`hyp_depends_on_locals_inclusive h hs` is true iff the hypothesis `h`\ninclusively depends on any of the hypotheses `hs`. If you need to check\ndependencies of multiple hypotheses, use\n`tactic.hyps_depend_on_locals_inclusive`.\n-/\nmeta def hyp_depends_on_locals_inclusive (h : expr) (hs : list expr) :\n  tactic bool :=\nhyp_depends_on_local_name_set_inclusive h $ local_list_to_name_set hs\n\n/--\n`hyps_depend_on_local_name_set_inclusive hs ns` returns, for each `h ∈ hs`,\nwhether `h` inclusively depends on a hypothesis whose unique name appears in\n`ns`. This is the same as (but more efficient than) calling\n`tactic.hyp_depends_on_local_name_set_inclusive` for every `h ∈ hs`.\n-/\nmeta def hyps_depend_on_local_name_set_inclusive (hs : list expr) (ns : name_set) :\n  tactic (list bool) := do\n  ctx_has_local ← context_has_local_def,\n  if ctx_has_local\n    then\n      let go : expr → list bool × expr_set → tactic (list bool × expr_set) :=\n      λ h ⟨deps, cache⟩, do\n      { (h_dep, cache) ← hyp_depends_on_local_name_set_inclusive' cache h ns,\n        pure (h_dep :: deps, cache) }\n      in\n      prod.fst <$> hs.mfoldr go ([], mk_expr_map)\n    else\n      hs.mmap $ λ h, hyp_directly_depends_on_local_name_set_inclusive h ns\n\n/--\n`hyps_depend_on_local_set_inclusive hs is` returns, for each `h ∈ hs`, whether\n`h` depends inclusively on any of the hypotheses `is`. This is the same as\n(but more efficient than) calling `tactic.hyp_depends_on_local_set_inclusive`\nfor every `h ∈ hs`.\n-/\nmeta def hyps_depend_on_local_set_inclusive (hs : list expr) (is : expr_set) :\n  tactic (list bool) :=\nhyps_depend_on_local_name_set_inclusive hs $ local_set_to_name_set is\n\n/--\n`hyps_depend_on_locals_inclusive hs is` returns, for each `h ∈ hs`, whether `h`\ndepends inclusively on any of the hypotheses `is`. This is the same as (but more\nefficient than) calling `tactic.hyp_depends_on_locals_inclusive` for every\n`h ∈ hs`.\n-/\nmeta def hyps_depend_on_locals_inclusive (hs is : list expr) : tactic (list bool) :=\nhyps_depend_on_local_name_set_inclusive hs $ local_list_to_name_set is\n\n\n/-! #### Computing the dependencies of a hypothesis -/\n\n/--\n`dependency_set_of_hyp' cache h` is the set of dependencies of the hypothesis\n`h`. `cache` is a map from hypotheses to all their dependencies (including\nindirect dependencies). This is a performance optimisation, so you can give an\nempty cache. The tactic also returns an expanded cache with hypotheses which\nthe tactic has encountered.\n\nYou probably want to use `tactic.dependency_set_of_hyp` or\n`tactic.dependency_sets_of_hyps` instead of this tactic.\n-/\nmeta def dependency_set_of_hyp' : expr_map expr_set → expr →\n  tactic (expr_set × expr_map expr_set) := λ cache h, do\n  match cache.find h with\n  | some deps := pure (deps, cache)\n  | none := do\n    direct_deps ← direct_dependency_set_of_hyp h,\n    (deps, cache) ←\n      direct_deps.mfold (direct_deps, cache) $ λ h' ⟨deps, cache⟩, do\n      { (deps', cache) ← dependency_set_of_hyp' cache h',\n        pure (deps.union deps', cache) },\n    pure (deps, cache.insert h deps)\n  end\n\n/--\n`dependency_set_of_hyp h` is the set of dependencies of the hypothesis `h`. If\nyou need the dependencies of multiple hypotheses, use\n`tactic.dependency_sets_of_hyps`.\n-/\nmeta def dependency_set_of_hyp (h : expr) : tactic expr_set := do\n  ctx_has_local ← context_upto_hyp_has_local_def h,\n  if ctx_has_local\n    then prod.fst <$> dependency_set_of_hyp' mk_expr_map h\n    else direct_dependency_set_of_hyp h\n\n/--\n`dependency_name_set_of_hyp h` is the set of unique names of the dependencies of\nthe hypothesis `h`. If you need the dependencies of multiple hypotheses, use\n`tactic.dependency_name_sets_of_hyps`.\n-/\nmeta def dependency_name_set_of_hyp (h : expr) : tactic name_set :=\nlocal_set_to_name_set <$> dependency_set_of_hyp h\n\n/--\n`dependencies_of_hyp h` is the list of dependencies of the hypothesis `h`.\nThe dependencies are returned in no particular order. If you need the\ndependencies of multiple hypotheses, use `tactic.dependencies_of_hyps`.\n-/\nmeta def dependencies_of_hyp (h : expr) : tactic (list expr) :=\nrb_set.to_list <$> dependency_set_of_hyp h\n\n/--\n`dependency_sets_of_hyps hs` returns, for each `h ∈ hs`, the set of dependencies\nof `h`. This is the same as (but more performant than) using\n`tactic.dependency_set_of_hyp` on every `h ∈ hs`.\n-/\nmeta def dependency_sets_of_hyps (hs : list expr) : tactic (list expr_set) := do\n  ctx_has_def ← context_has_local_def,\n  if ctx_has_def\n    then\n      let go : expr → list expr_set × expr_map expr_set →\n        tactic (list expr_set × expr_map expr_set) := do\n      λ h ⟨deps, cache⟩, do\n        { (h_deps, cache) ← dependency_set_of_hyp' cache h,\n          pure (h_deps :: deps, cache) }\n      in\n      prod.fst <$> hs.mfoldr go ([], mk_expr_map)\n    else\n      hs.mmap direct_dependency_set_of_hyp\n\n/--\n`dependency_name_sets_of_hyps hs` returns, for each `h ∈ hs`, the set of unique\nnames of the dependencies of `h`. This is the same as (but more performant than)\nusing `tactic.dependency_name_set_of_hyp` on every `h ∈ hs`.\n-/\nmeta def dependency_name_sets_of_hyps (hs : list expr) : tactic (list name_set) :=\nlist.map local_set_to_name_set <$> dependency_sets_of_hyps hs\n\n/--\n`dependencies_of_hyps hs` returns, for each `h ∈ hs`, the dependencies of `h`.\nThe dependencies appear in no particular order in the returned lists. This is\nthe same as (but more performant than) using `tactic.dependencies_of_hyp` on\nevery `h ∈ hs`.\n-/\nmeta def dependencies_of_hyps (hs : list expr) : tactic (list (list expr)) :=\nlist.map rb_set.to_list <$> dependency_sets_of_hyps hs\n\n/--\n`dependency_set_of_hyp_inclusive' cache h` is the set of dependencies of the\nhypothesis `h`, plus `h` itself. `cache` is a map from hypotheses to all their\ndependencies (including indirect dependencies). This is a performance\noptimisation, so you can give an empty cache. The tactic also returns an\nexpanded cache with hypotheses which the tactic has encountered. Note that the\ncache records exclusive, not inclusive dependencies.\n\nYou probably want to use `tactic.dependency_set_of_hyp_inclusive` or\n`tactic.dependency_sets_of_hyps_inclusive` instead of this tactic.\n-/\nmeta def dependency_set_of_hyp_inclusive' (cache : expr_map expr_set) (h : expr) :\n  tactic (expr_set × expr_map expr_set) := do\n  (deps, cache) ← dependency_set_of_hyp' cache h,\n  pure (deps.insert h, cache)\n\n/--\n`dependency_set_of_hyp_inclusive h` is the set of dependencies of the hypothesis\n`h`, plus `h` itself. If you need the dependencies of multiple hypotheses, use\n`tactic.dependency_sets_of_hyps_inclusive`.\n-/\nmeta def dependency_set_of_hyp_inclusive (h : expr) : tactic expr_set := do\n  deps ← dependency_set_of_hyp h,\n  pure $ deps.insert h\n\n/--\n`dependency_name_set_of_hyp_inclusive h` is the set of unique names of the\ndependencies of the hypothesis `h`, plus the unique name of `h` itself. If you\nneed the dependencies of multiple hypotheses, use\n`tactic.dependency_name_sets_of_hyps_inclusive`.\n-/\nmeta def dependency_name_set_of_hyp_inclusive (h : expr) : tactic name_set :=\nlocal_set_to_name_set <$> dependency_set_of_hyp_inclusive h\n\n/--\n`dependencies_of_hyp_inclusive h` is the list of dependencies of the hypothesis\n`h`, plus `h` itself. The dependencies are returned in no particular order. If\nyou need the dependencies of multiple hypotheses, use\n`tactic.dependencies_of_hyps_inclusive`.\n-/\nmeta def dependencies_of_hyp_inclusive (h : expr) : tactic (list expr) :=\nrb_set.to_list <$> dependency_set_of_hyp_inclusive h\n\n/--\n`dependency_sets_of_hyps_inclusive hs` returns, for each `h ∈ hs`, the\ndependencies of `h`, plus `h` itself. This is the same as (but more performant\nthan) using `tactic.dependency_set_of_hyp_inclusive` on every `h ∈ hs`.\n-/\nmeta def dependency_sets_of_hyps_inclusive (hs : list expr) :\n  tactic (list expr_set) := do\n  ctx_has_def ← context_has_local_def,\n  if ctx_has_def\n    then\n      let go : expr → list expr_set × expr_map expr_set →\n        tactic (list expr_set × expr_map expr_set) :=\n      λ h ⟨deps, cache⟩, do\n      { (h_deps, cache) ← dependency_set_of_hyp_inclusive' cache h,\n        pure (h_deps :: deps, cache) }\n      in\n      prod.fst <$> hs.mfoldr go ([], mk_expr_map)\n    else\n      hs.mmap direct_dependency_set_of_hyp_inclusive\n\n/--\n`dependency_name_sets_of_hyps_inclusive hs` returns, for each `h ∈ hs`, the\nunique names of the dependencies of `h`, plus the unique name of `h` itself.\nThis is the same as (but more performant than) using\n`tactic.dependency_name_set_of_hyp_inclusive` on every `h ∈ hs`.\n-/\nmeta def dependency_name_sets_of_hyps_inclusive (hs : list expr) :\n  tactic (list name_set) :=\nlist.map local_set_to_name_set <$> dependency_sets_of_hyps_inclusive hs\n\n/--\n`dependencies_of_hyps_inclusive hs` returns, for each `h ∈ hs`, the dependencies\nof `h`, plus `h` itself. The dependencies appear in no particular order in the\nreturned lists. This is the same as (but more performant than) using\n`tactic.dependencies_of_hyp_inclusive` on every `h ∈ hs`.\n-/\nmeta def dependencies_of_hyps_inclusive (hs : list expr) :\n  tactic (list (list expr)) :=\nlist.map rb_set.to_list <$> dependency_sets_of_hyps_inclusive hs\n\n\n/-! #### Computing the reverse dependencies of a hypothesis -/\n\nprivate meta def reverse_dependencies_of_hyp_name_set_aux (hs : name_set) :\n  list expr → list expr → name_set → tactic (list expr)\n| [] revdeps _ := pure revdeps.reverse\n| (H :: Hs) revdeps ns := do\n  let H_uname := H.local_uniq_name,\n  H_is_revdep ← list.mband\n    [ pure $ ¬ hs.contains H_uname,\n      hyp_directly_depends_on_local_name_set H ns ],\n  if H_is_revdep\n    then\n      reverse_dependencies_of_hyp_name_set_aux Hs (H :: revdeps)\n        (ns.insert H_uname)\n    else\n      reverse_dependencies_of_hyp_name_set_aux Hs revdeps ns\n\n/--\n`reverse_dependencies_of_hyp_name_set hs` is the list of reverse dependencies of\nthe hypotheses whose unique names appear in `hs`, excluding the `hs` themselves.\nThe reverse dependencies are returned in the order in which they appear in the\ncontext.\n-/\nmeta def reverse_dependencies_of_hyp_name_set (hs : name_set) :\n  tactic (list expr) := do\n  ctx ← local_context,\n  let ctx := ctx.after (λ h, hs.contains h.local_uniq_name),\n  reverse_dependencies_of_hyp_name_set_aux hs ctx [] hs\n\n/--\n`reverse_dependencies_of_hyp_set hs` is the list of reverse dependencies of the\nhypotheses `hs`, excluding the `hs` themselves. The reverse dependencies are\nreturned in the order in which they appear in the context.\n-/\nmeta def reverse_dependencies_of_hyp_set (hs : expr_set) : tactic (list expr) :=\nreverse_dependencies_of_hyp_name_set $ local_set_to_name_set hs\n\n/--\n`reverse_dependencies_of_hyps hs` is the list of reverse dependencies of the\nhypotheses `hs`, excluding the `hs` themselves. The reverse dependencies are\nreturned in the order in which they appear in the context.\n-/\nmeta def reverse_dependencies_of_hyps (hs : list expr) : tactic (list expr) :=\nreverse_dependencies_of_hyp_name_set $ local_list_to_name_set hs\n\nprivate meta def reverse_dependencies_of_hyp_name_set_inclusive_aux :\n  list expr → list expr → name_set → tactic (list expr)\n| [] revdeps _ := pure revdeps.reverse\n| (H :: Hs) revdeps ns := do\n  let H_uname := H.local_uniq_name,\n  H_is_revdep ← list.mbor\n    [ pure $ ns.contains H.local_uniq_name,\n      hyp_directly_depends_on_local_name_set H ns ],\n  if H_is_revdep\n    then\n      reverse_dependencies_of_hyp_name_set_inclusive_aux Hs (H :: revdeps)\n        (ns.insert H_uname)\n    else\n      reverse_dependencies_of_hyp_name_set_inclusive_aux Hs revdeps ns\n\n/--\n`reverse_dependencies_of_hyp_name_set_inclusive hs` is the list of reverse\ndependencies of the hypotheses whose unique names appear in `hs`, including the\n`hs` themselves. The reverse dependencies are returned in the order in which\nthey appear in the context.\n-/\nmeta def reverse_dependencies_of_hyp_name_set_inclusive (hs : name_set) :\n  tactic (list expr) := do\n  ctx ← local_context,\n  let ctx := ctx.drop_while (λ h, ¬ hs.contains h.local_uniq_name),\n  reverse_dependencies_of_hyp_name_set_inclusive_aux ctx [] hs\n\n/--\n`reverse_dependencies_of_hyp_set_inclusive hs` is the list of reverse\ndependencies of the hypotheses `hs`, including the `hs` themselves. The\ninclusive reverse dependencies are returned in the order in which they appear in\nthe context.\n-/\nmeta def reverse_dependencies_of_hyp_set_inclusive (hs : expr_set) :\n  tactic (list expr) :=\nreverse_dependencies_of_hyp_name_set_inclusive $ local_set_to_name_set hs\n\n/--\n`reverse_dependencies_of_hyps_inclusive hs` is the list of reverse dependencies\nof the hypotheses `hs`, including the `hs` themselves. The reverse dependencies\nare returned in the order in which they appear in the context.\n-/\nmeta def reverse_dependencies_of_hyps_inclusive (hs : list expr) :\n  tactic (list expr) :=\nreverse_dependencies_of_hyp_name_set_inclusive $ local_list_to_name_set hs\n\n\n/-! ### Reverting a hypothesis and its reverse dependencies -/\n\n/--\n`revert_name_set hs` reverts the hypotheses whose unique names appear in `hs`,\nas well as any hypotheses that depend on them. Returns the number of reverted\nhypotheses and a list containing these hypotheses. The reverted hypotheses are\nreturned in the order in which they used to appear in the context and are\nguaranteed to store the correct type (see `tactic.update_type`).\n-/\nmeta def revert_name_set (hs : name_set) : tactic (ℕ × list expr) := do\n  to_revert ← reverse_dependencies_of_hyp_name_set_inclusive hs,\n  to_revert_with_types ← to_revert.mmap update_type,\n  num_reverted ← revert_lst to_revert,\n  pure (num_reverted, to_revert_with_types)\n\n/--\n`revert_set hs` reverts the hypotheses `hs`, as well as any hypotheses that\ndepend on them. Returns the number of reverted hypotheses and a list containing\nthese hypotheses. The reverted hypotheses are returned in the order in which\nthey used to appear in the context and are guaranteed to store the correct type\n(see `tactic.update_type`).\n-/\nmeta def revert_set (hs : expr_set) : tactic (ℕ × list expr) :=\nrevert_name_set $ local_set_to_name_set hs\n\n/--\n`revert_lst' hs` reverts the hypotheses `hs`, as well as any hypotheses that\ndepend on them. Returns the number of reverted hypotheses and a list containing\nthese hypotheses. The reverted hypotheses are returned in the order in which\nthey used to appear in the context and are guaranteed to store the correct type\n(see `tactic.update_type`).\n\nThis is a more informative version of `tactic.revert_lst`.\n-/\nmeta def revert_lst' (hs : list expr) : tactic (ℕ × list expr) :=\nrevert_name_set $ local_list_to_name_set hs\n\n/--\n`revert_reverse_dependencies_of_hyp h` reverts all the hypotheses that depend on\nthe hypothesis `h`, including the local definitions that have `h` in their\nvalue. This fixes a bug in `tactic.revert_kdependencies` that does not revert\nlocal definitions for which `h` only appears in the value. Returns the number\nof reverted hypotheses.\n-/\n/- We cannot implement it as `revert e >> intro1` because that would change the\nlocal constant in the context. -/\nmeta def revert_reverse_dependencies_of_hyp (h : expr) : tactic ℕ :=\nreverse_dependencies_of_hyp_name_set (mk_name_set.insert h.local_uniq_name) >>=\n  revert_lst\n\n/--\n`revert_reverse_dependencies_of_hyp_name_set hs` reverts all the hypotheses that\ndepend on a hypothesis whose unique name appears in `hs`. The `hs` themselves\nare not reverted, unless they depend on each other. Returns the number of\nreverted hypotheses.\n-/\nmeta def revert_reverse_dependencies_of_hyp_name_set (hs : name_set) : tactic ℕ :=\nreverse_dependencies_of_hyp_name_set hs >>= revert_lst\n\n/--\n`revert_reverse_dependencies_of_hyp_set hs` reverts all the hypotheses that\ndepend on a hypothesis in `hs`. The `hs` themselves are not reverted, unless\nthey depend on each other. Returns the number of reverted hypotheses.\n-/\nmeta def revert_reverse_dependencies_of_hyp_set (hs : expr_set) : tactic ℕ :=\nreverse_dependencies_of_hyp_set hs >>= revert_lst\n\n/--\n`revert_reverse_dependencies_of_hyp hs` reverts all the hypotheses that depend\non a hypothesis in `hs`. The `hs` themselves are not reverted, unless they\ndepend on each other. Returns the number of reverted hypotheses.\n-/\nmeta def revert_reverse_dependencies_of_hyps (hs : list expr) : tactic ℕ :=\nreverse_dependencies_of_hyps hs >>= revert_lst\n\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/dependencies.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.329878014450153}}
{"text": "import spaces.bounded_cont_diff_map\nimport analysis.normed.group.basic\nimport measure_theory.function.lp_space\n\nopen topological_space function measure_theory set filter\nopen_locale bounded_cont_diff_map topological_space ennreal bounded_continuous_function nnreal\n\nsection prelim\n\n-- Commutativity should not be needed, but it's sufficient for now\n@[to_additive]\ntheorem continuous_iff_continuous_at_one {G H hom : Type*} [comm_group G] [comm_group H] \n  [topological_space G] [topological_space H] [topological_group G] [topological_group H] \n  [monoid_hom_class hom G H] {f : hom} :\n  continuous f ↔ continuous_at f 1 :=\nbegin\n  refine ⟨λ h, h.continuous_at, λ h, _⟩,\n  letI : uniform_space G := topological_group.to_uniform_space G,\n  letI : uniform_group G := topological_group_is_uniform,\n  letI : uniform_space H := topological_group.to_uniform_space H,\n  letI : uniform_group H := topological_group_is_uniform,\n  exact uniform_continuous.continuous (uniform_continuous_of_continuous_at_one f h)\nend\n\ntheorem continuous_multilinear_map.ext_iff {ι 𝕜 F : Type*} {E : ι → Type*} [decidable_eq ι] [nondiscrete_normed_field 𝕜] \n  [Π i, normed_group (E i)] [normed_group F] [Π i, normed_space 𝕜 (E i)] [normed_space 𝕜 F] \n  {f g : continuous_multilinear_map 𝕜 E F} : f = g ↔ ∀ x, f x = g x :=\n⟨λ h x, h ▸ rfl, continuous_multilinear_map.ext⟩\n\nlemma has_fderiv_at.tsupport_subset {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {f : E → F} {f' : E → E →L[𝕜] F} (hf : ∀ x, has_fderiv_at f (f' x) x) :\n  tsupport f' ⊆ tsupport f :=\nbegin\n  intros x,\n  contrapose,\n  rw [not_mem_closure_support_iff_eventually_eq, not_mem_closure_support_iff_eventually_eq],\n  intros h,\n  filter_upwards [h.eventually_eq_nhds],\n  intros y hy,\n  exact has_fderiv_at.unique (hf y) ((has_fderiv_at_const 0 y).congr_of_eventually_eq hy)\nend\n\nlemma fderiv_tsupport_subset {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {f : E → F} (hf : differentiable 𝕜 f) :\n  tsupport (fderiv 𝕜 f) ⊆ tsupport f :=\nhas_fderiv_at.tsupport_subset (λ x, (hf x).has_fderiv_at)\n\nlemma iterated_fderiv_tsupport_subset {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {nf : with_top ℕ} {i : ℕ} {f : E → F} (hf : cont_diff 𝕜 nf f)\n  (hif : (i : with_top ℕ) ≤ nf) :\n  tsupport (iterated_fderiv 𝕜 i f) ⊆ tsupport f :=\nbegin\n  induction i with i hi,\n  { refine subset_of_eq (congr_arg closure _),\n    ext x,\n    refine not_iff_not_of_iff _,\n    simp only [continuous_multilinear_map.ext_iff, iterated_fderiv_zero_apply, \n                continuous_multilinear_map.zero_apply, forall_const], },\n  { have hif' : (i : with_top ℕ) < nf := \n      lt_of_lt_of_le (with_top.coe_lt_coe.mpr $ i.lt_succ_self) hif,\n    have hdf : differentiable 𝕜 (iterated_fderiv 𝕜 i f) :=\n      (cont_diff_iff_continuous_differentiable.mp hf).2 i hif',\n    calc tsupport (iterated_fderiv 𝕜 (i+1) f)\n        = tsupport (fderiv 𝕜 $ iterated_fderiv 𝕜 i f) : by\n          { refine congr_arg closure _,\n            ext x,\n            refine not_iff_not_of_iff _,\n            rw [iterated_fderiv_succ_eq_comp_left, comp_apply, \n                (continuous_multilinear_curry_left_equiv _ _ _).map_eq_zero_iff] }\n    ... ⊆ tsupport (iterated_fderiv 𝕜 i f) : fderiv_tsupport_subset hdf\n    ... ⊆ tsupport f : hi hif'.le }\nend\n\nlemma has_compact_support_iterated_fderiv {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {nf : with_top ℕ} {i : ℕ} {f : E → F} (hf : cont_diff 𝕜 nf f) (hsupp : has_compact_support f)\n  (hif : (i : with_top ℕ) ≤ nf) : has_compact_support (iterated_fderiv 𝕜 i f) :=\ncompact_of_is_closed_subset hsupp (is_closed_tsupport _) (iterated_fderiv_tsupport_subset hf hif)\n\nlemma continuous.mem_ℒp_of_has_compact_support {α E : Type*} [hα : nonempty α]\n  [topological_space α] {m : measurable_space α} [t2_space α] [opens_measurable_space α] \n  [normed_group E] [measurable_space E] [borel_space E]\n  [second_countable_topology_either α E]  -- TODO : this should be removable because we are integrating\n                                          -- on a compact subset\n  {f : α → E} (hf : continuous f) (hsupp : has_compact_support f)\n  (p : ℝ≥0∞) (μ : measure α) [is_finite_measure_on_compacts μ]:\n  mem_ℒp f p μ := \nbegin\n  have : bdd_above (range $ norm ∘ f),\n    from hf.norm.bdd_above_range_of_has_compact_support hsupp.norm,\n  refine mem_ℒp.of_le \n    (mem_ℒp_indicator_const p hsupp.measurable_set (⨆ x, ∥f x∥) (or.inr hsupp.measure_lt_top.ne))\n    hf.ae_strongly_measurable (ae_of_all _ $ λ x, _),\n  rw norm_indicator_eq_indicator_norm,\n  refine set.le_indicator (λ a _, _) (λ a, _) x,\n  { rw real.norm_of_nonneg (le_csupr_of_le this hα.some (norm_nonneg _)),\n    exact le_csupr this a },\n  { intros ha,\n    simpa using not_mem_of_not_mem_closure ha }\nend\n\nend prelim\n\nprivate def cont_diff_map_supported_in_submodule (𝕜 E F : Type*) [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] (K : set E)\n  (n : with_top ℕ) : submodule 𝕜 (E → F) :=\n{ carrier := {f | cont_diff 𝕜 n f ∧ ∀ x ∉ K, f x = 0},\n  zero_mem' := ⟨cont_diff_zero_fun, λ x hx, rfl⟩,\n  add_mem' := λ f g hf hg, ⟨hf.1.add hg.1, λ x hx, \n    by rw [pi.add_apply, hf.2 x hx, hg.2 x hx, add_zero]⟩,\n  smul_mem' := λ c f hf, ⟨cont_diff_const.smul hf.1, λ x hx,\n    by rw [pi.smul_apply, hf.2 x hx, smul_zero]⟩ }\n\ndef cont_diff_map_supported_in (𝕜 E F : Type*) [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] (K : set E)\n  (n : with_top ℕ) := ↥(cont_diff_map_supported_in_submodule 𝕜 E F K n)\n\nnamespace cont_diff_map_supported_in\n\nsection any_set\n\nvariables {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] [normed_group E] [normed_group F]\n  [normed_space 𝕜 E] [normed_space 𝕜 F] {K : set E} {n : with_top ℕ} \n  {f g : cont_diff_map_supported_in 𝕜 E F K n} {x : E}\n\ninstance : add_comm_group (cont_diff_map_supported_in 𝕜 E F K n) := submodule.add_comm_group _\ninstance : module 𝕜 (cont_diff_map_supported_in 𝕜 E F K n) := submodule.module _\ninstance : has_coe_to_fun (cont_diff_map_supported_in 𝕜 E F K n) (λ _, E → F) := ⟨λ f, f.1⟩\n\n@[ext] lemma ext (H : ∀x, f x = g x) : f = g :=\nby {ext, exact H x}\n\nprotected lemma cont_diff (f : cont_diff_map_supported_in 𝕜 E F K n) :\n  cont_diff 𝕜 n f :=\nf.2.1\n\nprotected lemma continuous (f : cont_diff_map_supported_in 𝕜 E F K n) :\n  continuous f :=\nf.cont_diff.continuous\n\nlemma supported_in (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  ∀ x ∉ K, f x = 0 :=\nf.2.2\n\nlemma support_subset (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  support f ⊆ K :=\nsupport_subset_iff'.mpr f.2.2\n\nend any_set\n\nsection compact\n\nvariables {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] [normed_group E] [normed_group F]\n  [normed_space 𝕜 E] [normed_space 𝕜 F] {K : compacts E} {n : with_top ℕ} \n  {f g : cont_diff_map_supported_in 𝕜 E F K n} {x : E}\n\nlemma has_compact_support (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  has_compact_support f :=\nhas_compact_support.intro K.2 f.supported_in\n\nlemma tsupport_subset (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  tsupport f ⊆ K :=\nclosure_minimal f.support_subset K.2.is_closed\n\ndef of_support_subset {f : E → F} (hf : cont_diff 𝕜 n f) (hsupp : support f ⊆ K) :\n  cont_diff_map_supported_in 𝕜 E F K n :=\n⟨f, ⟨hf, λ x hx, by_contra $ λ hxn, hx (hsupp hxn)⟩⟩\n\nprotected lemma bounded (f : cont_diff_map_supported_in 𝕜 E F K n) {i : ℕ} (hi : ↑i ≤ n) : \n  ∃ C, ∀ x, ∥iterated_fderiv 𝕜 i f x∥ ≤ C :=\ncontinuous.bounded_above_of_compact_support \n  ((cont_diff_iff_continuous_differentiable.mp f.cont_diff).1 i hi)\n  (has_compact_support_iterated_fderiv f.cont_diff f.has_compact_support hi)\n\ndef to_bounded_cont_diff_map (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  B^n⟮E,F;𝕜⟯ :=\n⟨f, f.cont_diff, λ i hi, f.bounded hi⟩\n\nlemma coe_to_bounded_cont_diff_map (f : cont_diff_map_supported_in 𝕜 E F K n) :\n  (f.to_bounded_cont_diff_map : E → F) = f := rfl\n\nlemma to_bounded_cont_diff_map_injective :\n  injective (to_bounded_cont_diff_map : cont_diff_map_supported_in 𝕜 E F K n → B^n⟮E,F;𝕜⟯) :=\nbegin\n  intros f g hfg,\n  ext x,\n  change to_bounded_cont_diff_map f x = _,\n  rw hfg,\n  refl\nend\n\ndef to_bounded_cont_diff_mapₗ : \n  cont_diff_map_supported_in 𝕜 E F K n →ₗ[𝕜] (B^n⟮E,F;𝕜⟯) :=\n{ to_fun := to_bounded_cont_diff_map,\n  map_add' := λ f g, rfl,\n  map_smul' := λ a f, rfl }\n\nnoncomputable instance : topological_space (cont_diff_map_supported_in 𝕜 E F K n) :=\ntopological_space.induced to_bounded_cont_diff_mapₗ infer_instance\n\nnoncomputable instance : uniform_space (cont_diff_map_supported_in 𝕜 E F K n) :=\nuniform_space.comap to_bounded_cont_diff_mapₗ infer_instance\n\ninstance : topological_add_group (cont_diff_map_supported_in 𝕜 E F K n) :=\ntopological_add_group_induced _\n\ninstance : has_continuous_smul 𝕜 (cont_diff_map_supported_in 𝕜 E F K n) :=\nhas_continuous_smul_induced _\n\nvariables (𝕜 E F K n)\n\nnoncomputable def snorms : \n  seminorm_family 𝕜 (cont_diff_map_supported_in 𝕜 E F K n) \n  (bounded_cont_diff_map.snorms_ind n) :=\n(bounded_cont_diff_map.snorms 𝕜 E F n).comp to_bounded_cont_diff_mapₗ\n\nvariables {𝕜 E F K n}\n\nlemma snorms_monotone : \n  monotone (snorms 𝕜 E F K n) :=\nλ i j hij, seminorm.comp_mono _ (bounded_cont_diff_map.snorms_monotone hij)\n\n--lemma finset.sup_eq_csupr {α β : Type*} [nonempty α] [conditionally_complete_linear_order_bot β] (s : finset α) \n--  (f : α → β) : s.sup f = (⨆a∈s, f a) :=\n--le_antisymm\n--  (finset.sup_le $ assume a ha, le_csupr_of_le sorry a $ le_csupr finite_range_const.bdd_above ha)\n--  (csupr_le' $ assume a, csupr_le' $ assume ha, finset.le_sup ha)\n\nlemma snorms_apply (i : ℕ) (hi : ↑i ≤ n) :\n  snorms 𝕜 E F K n ⟨i, hi⟩ f = ↑(⨆ (j ≤ i) (x ∈ K), ∥iterated_fderiv 𝕜 j f x∥₊) :=\nbegin\n  rw [snorms, bounded_cont_diff_map.snorms, seminorm_family.comp_apply, seminorm.comp_apply,\n      subtype.coe_mk, seminorm.finset_sup_apply, nnreal.coe_injective.eq_iff],\n  -- finset.sup'_eq_csupr, sup_ima],\n  sorry\nend\n\nlemma snorms_directed : \n  directed (≤) (snorms 𝕜 E F K n) :=\nmonotone.directed_le snorms_monotone\n\ninstance with_seminorms : with_seminorms (cont_diff_map_supported_in.snorms 𝕜 E F K n) :=\nto_bounded_cont_diff_mapₗ.with_seminorms_induced\n\nnoncomputable! def to_bounded_cont_diff_mapL : \n  cont_diff_map_supported_in 𝕜 E F K n →L[𝕜] (B^n⟮E,F;𝕜⟯) :=\n{ to_linear_map := to_bounded_cont_diff_mapₗ,\n  cont := continuous_induced_dom }\n\nlemma continuous_iff {X : Type*} [topological_space X] \n  (φ : X → cont_diff_map_supported_in 𝕜 E F K n) : \n  continuous φ ↔ continuous \n    (cont_diff_map_supported_in.to_bounded_cont_diff_map ∘ φ) :=\n⟨λ hφ, to_bounded_cont_diff_mapL.continuous.comp hφ, continuous_induced_rng⟩\n\nlemma continuous_of_commutes {𝕜' E' F' : Type*} [normed_group E'] [normed_group F']\n  [nondiscrete_normed_field 𝕜'] [normed_space 𝕜' E'] [normed_space 𝕜' F'] {K' : compacts E'}\n  (φ : cont_diff_map_supported_in 𝕜 E F K n → cont_diff_map_supported_in 𝕜' E' F' K' n)\n  (ψ : B^n⟮E,F;𝕜⟯ → B^n⟮E',F';𝕜'⟯) (hcont : continuous ψ)\n  (hcomm : to_bounded_cont_diff_map ∘ φ = ψ ∘ to_bounded_cont_diff_map) :\n  continuous φ :=\nbegin\n  rw [continuous_iff, hcomm],\n  exact hcont.comp to_bounded_cont_diff_mapL.continuous\nend\n\nprotected def of_le {k : with_top ℕ} (hkn : k ≤ n) (f : cont_diff_map_supported_in 𝕜 E F K n) :\n  cont_diff_map_supported_in 𝕜 E F K k :=\n⟨f, f.cont_diff.of_le hkn, f.supported_in⟩\n\nprotected def of_leₗ {k : with_top ℕ} (hkn : k ≤ n) :\n  cont_diff_map_supported_in 𝕜 E F K n →ₗ[𝕜] cont_diff_map_supported_in 𝕜 E F K k :=\n{ to_fun := cont_diff_map_supported_in.of_le hkn,\n  map_add' := λ f g, by ext; refl,\n  map_smul' := λ c f, by ext; refl }\n\nprotected noncomputable! def of_leL {k : with_top ℕ} (hkn : k ≤ n) :\n  (cont_diff_map_supported_in 𝕜 E F K n) →L[𝕜] (cont_diff_map_supported_in 𝕜 E F K k) :=\n{ to_linear_map := cont_diff_map_supported_in.of_leₗ hkn,\n  cont := continuous_induced_rng \n    ((bounded_cont_diff_map.of_leL 𝕜 E F hkn).comp (to_bounded_cont_diff_mapL)).continuous }\n\nprotected noncomputable def iterated_fderivL (i : ℕ) : \n  (cont_diff_map_supported_in 𝕜 E F K n) →L[𝕜] (E →ᵇ (E [×i]→L[𝕜] F)) :=\nbounded_cont_diff_map.iterated_fderivL i ∘L to_bounded_cont_diff_mapL\n\nprotected lemma has_basis_zero : \n  (𝓝 0 : filter $ cont_diff_map_supported_in 𝕜 E F K n).has_basis \n  (λ Nε : ℕ × ℝ, 0 < Nε.2) (λ Nε, ⋂ (i : ℕ) (hiN : i ≤ Nε.1) (hi : ↑i ≤ n), \n    cont_diff_map_supported_in.iterated_fderivL i ⁻¹' metric.ball 0 Nε.2) :=\nbegin\n  rw [nhds_induced],\n  convert bounded_cont_diff_map.has_basis_zero.comap to_bounded_cont_diff_mapₗ,\n  ext,\n  simp only [mem_Inter, mem_preimage, mem_ball_zero_iff],\n  refl\nend\n\nsection zero\n\nnoncomputable instance : metric_space (cont_diff_map_supported_in 𝕜 E F K 0) :=\nmetric_space.induced (to_bounded_cont_diff_mapL : cont_diff_map_supported_in 𝕜 E F K 0 →L[𝕜] _) \n  to_bounded_cont_diff_map_injective infer_instance\n\nnoncomputable instance : normed_group (cont_diff_map_supported_in 𝕜 E F K 0) :=\n{ to_metric_space := infer_instance,\n  ..normed_group.induced \n    (to_bounded_cont_diff_mapL : cont_diff_map_supported_in 𝕜 E F K 0 →L[𝕜] _)\n      .to_linear_map.to_add_monoid_hom to_bounded_cont_diff_map_injective }\n\nlemma norm_def {f : cont_diff_map_supported_in 𝕜 E F K 0} : ∥f∥ = ∥to_bounded_cont_diff_mapL f∥ := rfl\n\n-- TODO : add `cont_diff_map_supported_in.to_bounded_continuous_function`\nlemma le_norm {f : cont_diff_map_supported_in 𝕜 E F K 0} (x : E) : ∥f x∥ ≤ ∥f∥ := \nbounded_continuous_function.norm_coe_le_norm \n  (bounded_cont_diff_map.to_bounded_continuous_function 𝕜 E F 0 (to_bounded_cont_diff_mapL f)) x\n\nnoncomputable! instance : normed_space 𝕜 (cont_diff_map_supported_in 𝕜 E F K 0) :=\n{ norm_smul_le := λ c f, \n  begin\n    rw [norm_def, norm_def, continuous_linear_map.map_smul],\n    exact normed_space.norm_smul_le _ _\n  end }\n\nend zero\n\nsection infinity\n\nlemma differentiable (f : cont_diff_map_supported_in 𝕜 E F K ⊤) : differentiable 𝕜 f := \nf.cont_diff.differentiable le_top\n\nnoncomputable def fderiv (f : cont_diff_map_supported_in 𝕜 E F K ⊤) : \n  cont_diff_map_supported_in 𝕜 E (E →L[𝕜] F) K ⊤ := \nof_support_subset (cont_diff_top_iff_fderiv.mp f.cont_diff).2 \n  (subset_closure.trans $ (fderiv_tsupport_subset f.differentiable).trans f.tsupport_subset)\n\nnoncomputable def fderivₗ : cont_diff_map_supported_in 𝕜 E F K ⊤ \n  →ₗ[𝕜] cont_diff_map_supported_in 𝕜 E (E →L[𝕜] F) K ⊤ := \n{ to_fun := cont_diff_map_supported_in.fderiv,\n  map_add' := λ f g,\n  begin\n    ext x : 1,\n    exact fderiv_add f.differentiable.differentiable_at\n      g.differentiable.differentiable_at,\n  end,\n  map_smul' := λ a f,\n  begin\n    ext x : 1,\n    exact fderiv_const_smul f.differentiable.differentiable_at _\n  end }\n\nnoncomputable def fderivL : cont_diff_map_supported_in 𝕜 E F K ⊤ \n  →L[𝕜] cont_diff_map_supported_in 𝕜 E (E →L[𝕜] F) K ⊤ := \n{ to_linear_map := fderivₗ,\n  cont := continuous_induced_rng \n    (bounded_cont_diff_map.fderivL ∘L to_bounded_cont_diff_mapL).continuous }\n\nend infinity\n\nlemma mem_ℒp (f : cont_diff_map_supported_in 𝕜 E F K n) \n  [measurable_space 𝕜] [opens_measurable_space 𝕜] \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : ℝ≥0∞) (μ : measure E) [fact (1 ≤ p)]\n  [is_finite_measure_on_compacts μ] : mem_ℒp f p μ :=\nf.continuous.mem_ℒp_of_has_compact_support f.has_compact_support p μ\n\nvariables (n)\n\ndef to_Lpₗ [measurable_space 𝕜] [opens_measurable_space 𝕜] \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : ℝ≥0∞) (μ : measure E) [fact (1 ≤ p)]\n  [is_finite_measure_on_compacts μ] : \n  (cont_diff_map_supported_in 𝕜 E F K n) →ₗ[𝕜] (Lp F p μ) :=\n{ to_fun := λ f, (f.mem_ℒp p μ).to_Lp f,\n  map_add' := λ f g, (f.mem_ℒp p μ).to_Lp_add (g.mem_ℒp p μ),\n  map_smul' := λ c f, (f.mem_ℒp p μ).to_Lp_const_smul c }\n\nlemma coe_fn_to_Lpₗ [measurable_space 𝕜] [opens_measurable_space 𝕜] \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : ℝ≥0∞) (μ : measure E) [fact (1 ≤ p)]\n  [is_finite_measure_on_compacts μ] (f : cont_diff_map_supported_in 𝕜 E F K n) : \n  to_Lpₗ n p μ f =ᵐ[μ] f :=\n(f.mem_ℒp p μ).coe_fn_to_Lp\n\nnoncomputable! def to_Lp_zero [measurable_space 𝕜] [opens_measurable_space 𝕜] \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : ℝ≥0∞) (μ : measure E) [fact (1 ≤ p)]\n  [is_finite_measure_on_compacts μ] : \n  (cont_diff_map_supported_in 𝕜 E F K 0) →L[𝕜] (Lp F p μ) :=\n{ to_linear_map := to_Lpₗ 0 p μ,\n  cont := \n  begin\n    -- TODO : unify with `continuous.mem_ℒp_of_has_compact_support`\n    have hK₁ : measurable_set ↑K := K.compact.measurable_set,\n    have hK₂ : μ ↑K ≠ ∞ := K.compact.measure_lt_top.ne,\n    change continuous (to_Lpₗ 0 p μ),\n    refine linear_map.continuous_of_bound (to_Lpₗ 0 p μ) \n      (∥indicator_const_Lp p hK₁ hK₂ (1 : ℝ)∥) (λ f, _),\n    rw [mul_comm],\n    refine Lp.norm_le_mul_norm_of_ae_le_mul _,\n    filter_upwards [f.coe_fn_to_Lpₗ 0 p μ, \n      (indicator_const_Lp_coe_fn : indicator_const_Lp p hK₁ hK₂ (1 : ℝ) =ᵐ[μ] _)],\n    intros x hx₁ hx₂,\n    rw [hx₁, hx₂, ← norm_norm f, ← norm_smul, ← indicator_const_smul_apply, norm_indicator_eq_indicator_norm],\n    refine set.le_indicator (λ a _, _) (λ a ha, _) x,\n    { rw [smul_eq_mul, mul_one, norm_norm],\n      exact le_norm _ },\n    { rw [f.supported_in _ ha, norm_zero] }\n  end }\n\nnoncomputable! def to_Lp [measurable_space 𝕜] [opens_measurable_space 𝕜] \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : ℝ≥0∞) (μ : measure E) [fact (1 ≤ p)]\n  [is_finite_measure_on_compacts μ] : \n  (cont_diff_map_supported_in 𝕜 E F K n) →L[𝕜] (Lp F p μ) :=\n{ to_fun := λ f, (f.mem_ℒp p μ).to_Lp f,\n  map_add' := λ f g, (f.mem_ℒp p μ).to_Lp_add (g.mem_ℒp p μ),\n  map_smul' := λ c f, (f.mem_ℒp p μ).to_Lp_const_smul c,\n  cont := ((to_Lp_zero p μ).comp (cont_diff_map_supported_in.of_leL (zero_le _))).continuous }  \n\nend compact\n\nsection real\n\nvariables {E F G : Type*} [normed_group E] [normed_group F] [normed_group G]\n  [normed_space ℝ E] [normed_space ℝ F] [normed_space ℝ G] {K : compacts E} \n  {n : with_top ℕ} {f g : cont_diff_map_supported_in ℝ E F K n} {x : E}\n\nlemma continuous_iff_of_linear (T : cont_diff_map_supported_in ℝ E F K n →ₗ[ℝ] G) : \n  continuous T ↔ ∃ (p : ℕ), ∃ C > 0, ∀ f : cont_diff_map_supported_in ℝ E F K n, \n    ∥T f∥ ≤ C * (⨆ (i ≤ p) (hin : ↑i ≤ n) (x : E), ∥iterated_fderiv ℝ i f x∥) :=\nbegin\n  rw [continuous_iff_continuous_at_zero, continuous_at, map_zero],\n  rw cont_diff_map_supported_in.has_basis_zero.tendsto_iff metric.nhds_basis_ball,\n  split,\n  { intros H, \n    rcases H 1 one_pos with ⟨⟨p, ε⟩, hε, H⟩,\n    have hε' : 0 < ε⁻¹ := inv_pos.mpr hε,\n    refine ⟨p, ε⁻¹, hε', λ f, _⟩,\n    rw [← div_eq_inv_mul, le_div_iff hε, mul_comm, ← real.norm_of_nonneg hε.le, \n        ← norm_smul ε (T f), ← T.map_smul _],\n    let N : ℝ := ⨆ (i ≤ p) (hin : ↑i ≤ n) (x : E), ∥iterated_fderiv ℝ i f x∥,\n    sorry },\n  sorry\nend\n\nlemma continuous_iff_of_linear' (T : cont_diff_map_supported_in ℝ E F K n →ₗ[ℝ] G) : \n  continuous T ↔ ∃ (p : ℕ) (C > 0), ∀ f : cont_diff_map_supported_in ℝ E F K n, \n    ∥T f∥ ≤ C * (⨆ (i ≤ p) (hin : ↑i ≤ n) (x : E), ∥iterated_fderiv ℝ i f x∥) :=\nbegin\n  letI : uniform_add_group (cont_diff_map_supported_in ℝ E F K n) := sorry,\n  split,\n  { sorry },\n  { intro h,\n    refine seminorm.continuous_from_bounded (snorms ℝ E F K n) (λ _ : fin 1, norm_seminorm ℝ G) _ _,\n    rw seminorm.is_bounded_iff_of_directed_dom,\n    { intro,\n      rcases h with ⟨p, C, hC, hbound⟩,\n      refine ⟨⟨p, sorry⟩, C.to_nnreal, sorry, _⟩,\n      intros f,\n      convert hbound f, \n      rw [snorms, seminorm.coe_smul, pi.smul_apply, seminorm_family.comp_apply, seminorm.comp_apply,\n          bounded_cont_diff_map.snorms], \n      sorry },\n    sorry }\nend\n\nend real\n\nend cont_diff_map_supported_in", "meta": {"author": "ADedecker", "repo": "distributions", "sha": "b4d124142788db55cf781184aff03bcc46aa2b10", "save_path": "github-repos/lean/ADedecker-distributions", "path": "github-repos/lean/ADedecker-distributions/distributions-b4d124142788db55cf781184aff03bcc46aa2b10/src/spaces/cont_diff_map_support_in.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.32966067716565395}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.nth_rewrite.default\nimport Mathlib.data.matrix.basic\nimport Mathlib.data.equiv.ring_aut\nimport Mathlib.linear_algebra.tensor_product\nimport Mathlib.ring_theory.subring\nimport Mathlib.deprecated.subring\nimport Mathlib.algebra.opposites\nimport Mathlib.PostPort\n\nuniverses u v l u_1 u_2 w u₁ v₁ u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Algebra over Commutative Semiring\n\nIn this file we define `algebra`s over commutative (semi)rings, algebra homomorphisms `alg_hom`,\nalgebra equivalences `alg_equiv`. We also define usual operations on `alg_hom`s\n(`id`, `comp`).\n\n`subalgebra`s are defined in `algebra.algebra.subalgebra`.\n\nIf `S` is an `R`-algebra and `A` is an `S`-algebra then `algebra.comap.algebra R S A` can be used\nto provide `A` with a structure of an `R`-algebra. Other than that, `algebra.comap` is now\ndeprecated and replaced with `is_scalar_tower`.\n\nFor the category of `R`-algebras, denoted `Algebra R`, see the file\n`algebra/category/Algebra/basic.lean`.\n\n## Notations\n\n* `A →ₐ[R] B` : `R`-algebra homomorphism from `A` to `B`.\n* `A ≃ₐ[R] B` : `R`-algebra equivalence from `A` to `B`.\n-/\n\n-- We set this priority to 0 later in this file\n\n/--\nGiven a commutative (semi)ring `R`, an `R`-algebra is a (possibly noncommutative)\n(semi)ring `A` endowed with a morphism of rings `R →+* A` which lands in the\ncenter of `A`.\n\nFor convenience, this typeclass extends `has_scalar R A` where the scalar action must\nagree with left multiplication by the image of the structure morphism.\n\nGiven an `algebra R A` instance, the structure morphism `R →+* A` is denoted `algebra_map R A`.\n-/\nclass algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A]\n    extends has_scalar R A, R →+* A where\n  commutes' :\n    ∀ (r : R) (x : A), ring_hom.to_fun _to_ring_hom r * x = x * ring_hom.to_fun _to_ring_hom r\n  smul_def' : ∀ (r : R) (x : A), r • x = ring_hom.to_fun _to_ring_hom r * x\n\n/-- Embedding `R →+* A` given by `algebra` structure. -/\ndef algebra_map (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : R →+* A :=\n  algebra.to_ring_hom\n\n/-- Creating an algebra from a morphism to the center of a semiring. -/\ndef ring_hom.to_algebra' {R : Type u_1} {S : Type u_2} [comm_semiring R] [semiring S] (i : R →+* S)\n    (h : ∀ (c : R) (x : S), coe_fn i c * x = x * coe_fn i c) : algebra R S :=\n  algebra.mk i h sorry\n\n/-- Creating an algebra from a morphism to a commutative semiring. -/\ndef ring_hom.to_algebra {R : Type u_1} {S : Type u_2} [comm_semiring R] [comm_semiring S]\n    (i : R →+* S) : algebra R S :=\n  ring_hom.to_algebra' i sorry\n\ntheorem ring_hom.algebra_map_to_algebra {R : Type u_1} {S : Type u_2} [comm_semiring R]\n    [comm_semiring S] (i : R →+* S) : algebra_map R S = i :=\n  rfl\n\nnamespace algebra\n\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `semimodule R` structure.\nIf `(r • 1) * x = x * (r • 1) = r • x` for all `r : R` and `x : A`, then `A` is an `algebra`\nover `R`. -/\ndef of_semimodule' {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [semimodule R A]\n    (h₁ : ∀ (r : R) (x : A), r • 1 * x = r • x) (h₂ : ∀ (r : R) (x : A), x * r • 1 = r • x) :\n    algebra R A :=\n  mk (ring_hom.mk (fun (r : R) => r • 1) sorry sorry sorry sorry) sorry sorry\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `semimodule R` structure.\nIf `(r • x) * y = x * (r • y) = r • (x * y)` for all `r : R` and `x y : A`, then `A`\nis an `algebra` over `R`. -/\ndef of_semimodule {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [semimodule R A]\n    (h₁ : ∀ (r : R) (x y : A), r • x * y = r • (x * y))\n    (h₂ : ∀ (r : R) (x y : A), x * r • y = r • (x * y)) : algebra R A :=\n  of_semimodule' sorry sorry\n\ntheorem smul_def'' {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A] (r : R)\n    (x : A) : r • x = coe_fn (algebra_map R A) r * x :=\n  smul_def' r x\n\n/--\nTo prove two algebra structures on a fixed `[comm_semiring R] [semiring A]` agree,\nit suffices to check the `algebra_map`s agree.\n-/\n-- We'll later use this to show `algebra ℤ M` is a subsingleton.\n\ntheorem algebra_ext {R : Type u_1} [comm_semiring R] {A : Type u_2} [semiring A] (P : algebra R A)\n    (Q : algebra R A) (w : ∀ (r : R), coe_fn (algebra_map R A) r = coe_fn (algebra_map R A) r) :\n    P = Q :=\n  sorry\n\nprotected instance to_semimodule {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] : semimodule R A :=\n  semimodule.mk sorry sorry\n\n-- from now on, we don't want to use the following instance anymore\n\ntheorem smul_def {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A] (r : R)\n    (x : A) : r • x = coe_fn (algebra_map R A) r * x :=\n  smul_def' r x\n\ntheorem algebra_map_eq_smul_one {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] (r : R) : coe_fn (algebra_map R A) r = r • 1 :=\n  Eq.trans (Eq.symm (mul_one (coe_fn (algebra_map R A) r))) (Eq.symm (smul_def r 1))\n\ntheorem commutes {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A] (r : R)\n    (x : A) : coe_fn (algebra_map R A) r * x = x * coe_fn (algebra_map R A) r :=\n  commutes' r x\n\ntheorem left_comm {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A] (r : R)\n    (x : A) (y : A) : x * (coe_fn (algebra_map R A) r * y) = coe_fn (algebra_map R A) r * (x * y) :=\n  sorry\n\n@[simp] theorem mul_smul_comm {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    (s : R) (x : A) (y : A) : x * s • y = s • (x * y) :=\n  sorry\n\n@[simp] theorem smul_mul_assoc {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] (r : R) (x : A) (y : A) : r • x * y = r • (x * y) :=\n  sorry\n\n@[simp] theorem bit0_smul_one {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    {r : R} : bit0 r • 1 = r • bit0 1 :=\n  sorry\n\n@[simp] theorem bit0_smul_bit0 {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] {r : R} {a : A} : bit0 r • bit0 a = r • bit0 (bit0 a) :=\n  sorry\n\n@[simp] theorem bit0_smul_bit1 {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] {r : R} {a : A} : bit0 r • bit1 a = r • bit0 (bit1 a) :=\n  sorry\n\n@[simp] theorem bit1_smul_one {R : Type u} {A : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    {r : R} : bit1 r • 1 = r • bit0 1 + 1 :=\n  sorry\n\n@[simp] theorem bit1_smul_bit0 {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] {r : R} {a : A} : bit1 r • bit0 a = r • bit0 (bit0 a) + bit0 a :=\n  sorry\n\n@[simp] theorem bit1_smul_bit1 {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] {r : R} {a : A} : bit1 r • bit1 a = r • bit0 (bit1 a) + bit1 a :=\n  sorry\n\n/--\nThe canonical ring homomorphism `algebra_map R A : R →* A` for any `R`-algebra `A`,\npackaged as an `R`-linear map.\n-/\nprotected def linear_map (R : Type u) (A : Type w) [comm_semiring R] [semiring A] [algebra R A] :\n    linear_map R R A :=\n  linear_map.mk (ring_hom.to_fun (algebra_map R A)) sorry sorry\n\n@[simp] theorem linear_map_apply (R : Type u) (A : Type w) [comm_semiring R] [semiring A]\n    [algebra R A] (r : R) : coe_fn (algebra.linear_map R A) r = coe_fn (algebra_map R A) r :=\n  rfl\n\nprotected instance id (R : Type u) [comm_semiring R] : algebra R R :=\n  ring_hom.to_algebra (ring_hom.id R)\n\nnamespace id\n\n\n@[simp] theorem map_eq_self {R : Type u} [comm_semiring R] (x : R) :\n    coe_fn (algebra_map R R) x = x :=\n  rfl\n\n@[simp] theorem smul_eq_mul {R : Type u} [comm_semiring R] (x : R) (y : R) : x • y = x * y := rfl\n\nend id\n\n\nprotected instance prod.algebra (R : Type u) (A : Type w) (B : Type u_1) [comm_semiring R]\n    [semiring A] [algebra R A] [semiring B] [algebra R B] : algebra R (A × B) :=\n  mk\n    (ring_hom.mk (ring_hom.to_fun (ring_hom.prod (algebra_map R A) (algebra_map R B))) sorry sorry\n      sorry sorry)\n    sorry sorry\n\n@[simp] theorem algebra_map_prod_apply {R : Type u} {A : Type w} {B : Type u_1} [comm_semiring R]\n    [semiring A] [algebra R A] [semiring B] [algebra R B] (r : R) :\n    coe_fn (algebra_map R (A × B)) r = (coe_fn (algebra_map R A) r, coe_fn (algebra_map R B) r) :=\n  rfl\n\n/-- Algebra over a subsemiring. -/\nprotected instance of_subsemiring {R : Type u} {A : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] (S : subsemiring R) : algebra (↥S) A :=\n  mk\n    (ring_hom.mk (ring_hom.to_fun (ring_hom.comp (algebra_map R A) (subsemiring.subtype S))) sorry\n      sorry sorry sorry)\n    sorry sorry\n\n/-- Algebra over a subring. -/\nprotected instance of_subring {R : Type u_1} {A : Type u_2} [comm_ring R] [ring A] [algebra R A]\n    (S : subring R) : algebra (↥S) A :=\n  mk\n    (ring_hom.mk (ring_hom.to_fun (ring_hom.comp (algebra_map R A) (subring.subtype S))) sorry sorry\n      sorry sorry)\n    sorry sorry\n\ntheorem algebra_map_of_subring {R : Type u_1} [comm_ring R] (S : subring R) :\n    algebra_map (↥S) R = subring.subtype S :=\n  rfl\n\ntheorem coe_algebra_map_of_subring {R : Type u_1} [comm_ring R] (S : subring R) :\n    ⇑(algebra_map (↥S) R) = subtype.val :=\n  rfl\n\ntheorem algebra_map_of_subring_apply {R : Type u_1} [comm_ring R] (S : subring R) (x : ↥S) :\n    coe_fn (algebra_map (↥S) R) x = ↑x :=\n  rfl\n\n/-- Algebra over a set that is closed under the ring operations. -/\ndef of_is_subring {R : Type u_1} {A : Type u_2} [comm_ring R] [ring A] [algebra R A] (S : set R)\n    [is_subring S] : algebra (↥S) A :=\n  algebra.of_subring (set.to_subring S)\n\ntheorem is_subring_coe_algebra_map_hom {R : Type u_1} [comm_ring R] (S : set R) [is_subring S] :\n    algebra_map (↥S) R = is_subring.subtype S :=\n  rfl\n\ntheorem is_subring_coe_algebra_map {R : Type u_1} [comm_ring R] (S : set R) [is_subring S] :\n    ⇑(algebra_map (↥S) R) = subtype.val :=\n  rfl\n\ntheorem is_subring_algebra_map_apply {R : Type u_1} [comm_ring R] (S : set R) [is_subring S]\n    (x : ↥S) : coe_fn (algebra_map (↥S) R) x = ↑x :=\n  rfl\n\ntheorem set_range_subset {R : Type u_1} [comm_ring R] {T₁ : set R} {T₂ : set R} [is_subring T₁]\n    (hyp : T₁ ⊆ T₂) : set.range ⇑(algebra_map (↥T₁) R) ⊆ T₂ :=\n  sorry\n\n/-- Explicit characterization of the submonoid map in the case of an algebra.\n`S` is made explicit to help with type inference -/\ndef algebra_map_submonoid {R : Type u} [comm_semiring R] (S : Type u_1) [semiring S] [algebra R S]\n    (M : submonoid R) : submonoid S :=\n  submonoid.map (↑(algebra_map R S)) M\n\ntheorem mem_algebra_map_submonoid_of_mem {R : Type u} {S : Type v} [comm_semiring R]\n    [comm_semiring S] [algebra R S] {M : submonoid R} (x : ↥M) :\n    coe_fn (algebra_map R S) ↑x ∈ algebra_map_submonoid S M :=\n  set.mem_image_of_mem (⇑(algebra_map R S)) (subtype.property x)\n\n/-- A `semiring` that is an `algebra` over a commutative ring carries a natural `ring` structure. -/\ndef semiring_to_ring (R : Type u) {A : Type w} [comm_ring R] [semiring A] [algebra R A] : ring A :=\n  ring.mk add_comm_group.add sorry add_comm_group.zero sorry sorry add_comm_group.neg\n    add_comm_group.sub sorry sorry semiring.mul sorry semiring.one sorry sorry sorry sorry\n\ntheorem mul_sub_algebra_map_commutes {R : Type u} {A : Type w} [comm_ring R] [ring A] [algebra R A]\n    (x : A) (r : R) : x * (x - coe_fn (algebra_map R A) r) = (x - coe_fn (algebra_map R A) r) * x :=\n  sorry\n\ntheorem mul_sub_algebra_map_pow_commutes {R : Type u} {A : Type w} [comm_ring R] [ring A]\n    [algebra R A] (x : A) (r : R) (n : ℕ) :\n    x * (x - coe_fn (algebra_map R A) r) ^ n = (x - coe_fn (algebra_map R A) r) ^ n * x :=\n  sorry\n\nend algebra\n\n\nnamespace opposite\n\n\nprotected instance algebra {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A]\n    [algebra R A] : algebra R (Aᵒᵖ) :=\n  algebra.mk (ring_hom.to_opposite (algebra_map R A) sorry) sorry sorry\n\n@[simp] theorem algebra_map_apply {R : Type u_1} {A : Type u_2} [comm_semiring R] [semiring A]\n    [algebra R A] (c : R) : coe_fn (algebra_map R (Aᵒᵖ)) c = op (coe_fn (algebra_map R A) c) :=\n  rfl\n\nend opposite\n\n\nnamespace module\n\n\nprotected instance endomorphism_algebra (R : Type u) (M : Type v) [comm_semiring R]\n    [add_comm_monoid M] [semimodule R M] : algebra R (linear_map R M M) :=\n  algebra.mk (ring_hom.mk (fun (r : R) => r • linear_map.id) sorry sorry sorry sorry) sorry sorry\n\ntheorem algebra_map_End_eq_smul_id (R : Type u) (M : Type v) [comm_semiring R] [add_comm_monoid M]\n    [semimodule R M] (a : R) : coe_fn (algebra_map R (End R M)) a = a • linear_map.id :=\n  rfl\n\n@[simp] theorem algebra_map_End_apply (R : Type u) (M : Type v) [comm_semiring R]\n    [add_comm_monoid M] [semimodule R M] (a : R) (m : M) :\n    coe_fn (coe_fn (algebra_map R (End R M)) a) m = a • m :=\n  rfl\n\n@[simp] theorem ker_algebra_map_End (K : Type u) (V : Type v) [field K] [add_comm_group V]\n    [vector_space K V] (a : K) (ha : a ≠ 0) :\n    linear_map.ker (coe_fn (algebra_map K (End K V)) a) = ⊥ :=\n  linear_map.ker_smul linear_map.id a ha\n\nend module\n\n\nprotected instance matrix_algebra (n : Type u) (R : Type v) [DecidableEq n] [fintype n]\n    [comm_semiring R] : algebra R (matrix n n R) :=\n  algebra.mk (ring_hom.mk (ring_hom.to_fun (matrix.scalar n)) sorry sorry sorry sorry) sorry sorry\n\n/-- Defining the homomorphism in the category R-Alg. -/\nstructure alg_hom (R : Type u) (A : Type v) (B : Type w) [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B]\n    extends A →+* B where\n  commutes' : ∀ (r : R), to_fun (coe_fn (algebra_map R A) r) = coe_fn (algebra_map R B) r\n\nnamespace alg_hom\n\n\nprotected instance has_coe_to_fun {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : has_coe_to_fun (alg_hom R A B) :=\n  has_coe_to_fun.mk (fun (f : alg_hom R A B) => A → B) fun (f : alg_hom R A B) => to_fun f\n\nprotected instance coe_ring_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : has_coe (alg_hom R A B) (A →+* B) :=\n  has_coe.mk to_ring_hom\n\nprotected instance coe_monoid_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : has_coe (alg_hom R A B) (A →* B) :=\n  has_coe.mk fun (f : alg_hom R A B) => ↑↑f\n\nprotected instance coe_add_monoid_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : has_coe (alg_hom R A B) (A →+ B) :=\n  has_coe.mk fun (f : alg_hom R A B) => ↑↑f\n\n@[simp] theorem coe_mk {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] {f : A → B} (h₁ : f 1 = 1)\n    (h₂ : ∀ (x y : A), f (x * y) = f x * f y) (h₃ : f 0 = 0)\n    (h₄ : ∀ (x y : A), f (x + y) = f x + f y)\n    (h₅ : ∀ (r : R), f (coe_fn (algebra_map R A) r) = coe_fn (algebra_map R B) r) :\n    ⇑(mk f h₁ h₂ h₃ h₄ h₅) = f :=\n  rfl\n\n@[simp] theorem coe_to_ring_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) : ⇑↑f = ⇑f :=\n  rfl\n\n-- as `simp` can already prove this lemma, it is not tagged with the `simp` attribute.\n\ntheorem coe_to_monoid_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) : ⇑↑f = ⇑f :=\n  rfl\n\n-- as `simp` can already prove this lemma, it is not tagged with the `simp` attribute.\n\ntheorem coe_to_add_monoid_hom {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) : ⇑↑f = ⇑f :=\n  rfl\n\ntheorem coe_fn_inj {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] {φ₁ : alg_hom R A B} {φ₂ : alg_hom R A B}\n    (H : ⇑φ₁ = ⇑φ₂) : φ₁ = φ₂ :=\n  sorry\n\ntheorem coe_ring_hom_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] : function.injective coe :=\n  fun (φ₁ φ₂ : alg_hom R A B) (H : ↑φ₁ = ↑φ₂) =>\n    coe_fn_inj ((fun (this : ⇑↑φ₁ = ⇑↑φ₂) => this) (congr_arg coe_fn H))\n\ntheorem coe_monoid_hom_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : function.injective coe :=\n  function.injective.comp ring_hom.coe_monoid_hom_injective coe_ring_hom_injective\n\ntheorem coe_add_monoid_hom_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] : function.injective coe :=\n  function.injective.comp ring_hom.coe_add_monoid_hom_injective coe_ring_hom_injective\n\nprotected theorem congr_fun {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] {φ₁ : alg_hom R A B} {φ₂ : alg_hom R A B} (H : φ₁ = φ₂)\n    (x : A) : coe_fn φ₁ x = coe_fn φ₂ x :=\n  H ▸ rfl\n\nprotected theorem congr_arg {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) {x : A} {y : A} (h : x = y) :\n    coe_fn φ x = coe_fn φ y :=\n  h ▸ rfl\n\ntheorem ext {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] {φ₁ : alg_hom R A B} {φ₂ : alg_hom R A B}\n    (H : ∀ (x : A), coe_fn φ₁ x = coe_fn φ₂ x) : φ₁ = φ₂ :=\n  coe_fn_inj (funext H)\n\ntheorem ext_iff {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] {φ₁ : alg_hom R A B} {φ₂ : alg_hom R A B} :\n    φ₁ = φ₂ ↔ ∀ (x : A), coe_fn φ₁ x = coe_fn φ₂ x :=\n  { mp := alg_hom.congr_fun, mpr := ext }\n\n@[simp] theorem mk_coe {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] {f : alg_hom R A B} (h₁ : coe_fn f 1 = 1)\n    (h₂ : ∀ (x y : A), coe_fn f (x * y) = coe_fn f x * coe_fn f y) (h₃ : coe_fn f 0 = 0)\n    (h₄ : ∀ (x y : A), coe_fn f (x + y) = coe_fn f x + coe_fn f y)\n    (h₅ : ∀ (r : R), coe_fn f (coe_fn (algebra_map R A) r) = coe_fn (algebra_map R B) r) :\n    mk (⇑f) h₁ h₂ h₃ h₄ h₅ = f :=\n  ext fun (_x : A) => rfl\n\n@[simp] theorem commutes {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (r : R) :\n    coe_fn φ (coe_fn (algebra_map R A) r) = coe_fn (algebra_map R B) r :=\n  commutes' φ r\n\ntheorem comp_algebra_map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) :\n    ring_hom.comp (↑φ) (algebra_map R A) = algebra_map R B :=\n  ring_hom.ext (commutes φ)\n\n@[simp] theorem map_add {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (r : A) (s : A) :\n    coe_fn φ (r + s) = coe_fn φ r + coe_fn φ s :=\n  ring_hom.map_add (to_ring_hom φ) r s\n\n@[simp] theorem map_zero {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) : coe_fn φ 0 = 0 :=\n  ring_hom.map_zero (to_ring_hom φ)\n\n@[simp] theorem map_mul {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) (y : A) :\n    coe_fn φ (x * y) = coe_fn φ x * coe_fn φ y :=\n  ring_hom.map_mul (to_ring_hom φ) x y\n\n@[simp] theorem map_one {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) : coe_fn φ 1 = 1 :=\n  ring_hom.map_one (to_ring_hom φ)\n\n@[simp] theorem map_smul {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (r : R) (x : A) :\n    coe_fn φ (r • x) = r • coe_fn φ x :=\n  sorry\n\n@[simp] theorem map_pow {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) (n : ℕ) :\n    coe_fn φ (x ^ n) = coe_fn φ x ^ n :=\n  ring_hom.map_pow (to_ring_hom φ) x n\n\ntheorem map_sum {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] (φ : alg_hom R A B) {ι : Type u_1} (f : ι → A) (s : finset ι) :\n    coe_fn φ (finset.sum s fun (x : ι) => f x) = finset.sum s fun (x : ι) => coe_fn φ (f x) :=\n  ring_hom.map_sum (to_ring_hom φ) f s\n\ntheorem map_finsupp_sum {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) {α : Type u_1} [HasZero α]\n    {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A) :\n    coe_fn φ (finsupp.sum f g) = finsupp.sum f fun (i : ι) (a : α) => coe_fn φ (g i a) :=\n  map_sum φ (fun (a : ι) => g a (coe_fn f a)) (finsupp.support f)\n\n@[simp] theorem map_nat_cast {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (n : ℕ) : coe_fn φ ↑n = ↑n :=\n  ring_hom.map_nat_cast (to_ring_hom φ) n\n\n@[simp] theorem map_bit0 {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) :\n    coe_fn φ (bit0 x) = bit0 (coe_fn φ x) :=\n  ring_hom.map_bit0 (to_ring_hom φ) x\n\n@[simp] theorem map_bit1 {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) :\n    coe_fn φ (bit1 x) = bit1 (coe_fn φ x) :=\n  ring_hom.map_bit1 (to_ring_hom φ) x\n\n/-- If a `ring_hom` is `R`-linear, then it is an `alg_hom`. -/\ndef mk' {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] (f : A →+* B)\n    (h : ∀ (c : R) (x : A), coe_fn f (c • x) = c • coe_fn f x) : alg_hom R A B :=\n  mk (⇑f) (ring_hom.map_one' f) (ring_hom.map_mul' f) (ring_hom.map_zero' f) (ring_hom.map_add' f)\n    sorry\n\n@[simp] theorem coe_mk' {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (f : A →+* B)\n    (h : ∀ (c : R) (x : A), coe_fn f (c • x) = c • coe_fn f x) : ⇑(mk' f h) = ⇑f :=\n  rfl\n\n/-- Identity map as an `alg_hom`. -/\nprotected def id (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] :\n    alg_hom R A A :=\n  mk (ring_hom.to_fun (ring_hom.id A)) sorry sorry sorry sorry sorry\n\n@[simp] theorem id_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (p : A) : coe_fn (alg_hom.id R A) p = p :=\n  rfl\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [comm_semiring R] [semiring A]\n    [semiring B] [semiring C] [algebra R A] [algebra R B] [algebra R C] (φ₁ : alg_hom R B C)\n    (φ₂ : alg_hom R A B) : alg_hom R A C :=\n  mk (ring_hom.to_fun (ring_hom.comp (to_ring_hom φ₁) ↑φ₂)) sorry sorry sorry sorry sorry\n\n@[simp] theorem comp_apply {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} [comm_semiring R]\n    [semiring A] [semiring B] [semiring C] [algebra R A] [algebra R B] [algebra R C]\n    (φ₁ : alg_hom R B C) (φ₂ : alg_hom R A B) (p : A) :\n    coe_fn (comp φ₁ φ₂) p = coe_fn φ₁ (coe_fn φ₂ p) :=\n  rfl\n\n@[simp] theorem comp_id {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) : comp φ (alg_hom.id R A) = φ :=\n  ext fun (x : A) => rfl\n\n@[simp] theorem id_comp {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) : comp (alg_hom.id R B) φ = φ :=\n  ext fun (x : A) => rfl\n\ntheorem comp_assoc {R : Type u} {A : Type v} {B : Type w} {C : Type u₁} {D : Type v₁}\n    [comm_semiring R] [semiring A] [semiring B] [semiring C] [semiring D] [algebra R A]\n    [algebra R B] [algebra R C] [algebra R D] (φ₁ : alg_hom R C D) (φ₂ : alg_hom R B C)\n    (φ₃ : alg_hom R A B) : comp (comp φ₁ φ₂) φ₃ = comp φ₁ (comp φ₂ φ₃) :=\n  ext fun (x : A) => rfl\n\n/-- R-Alg ⥤ R-Mod -/\ndef to_linear_map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] (φ : alg_hom R A B) : linear_map R A B :=\n  linear_map.mk (⇑φ) (map_add φ) (map_smul φ)\n\n@[simp] theorem to_linear_map_apply {R : Type u} {A : Type v} {B : Type w} [comm_semiring R]\n    [semiring A] [semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (p : A) :\n    coe_fn (to_linear_map φ) p = coe_fn φ p :=\n  rfl\n\ntheorem to_linear_map_inj {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] {φ₁ : alg_hom R A B} {φ₂ : alg_hom R A B}\n    (H : to_linear_map φ₁ = to_linear_map φ₂) : φ₁ = φ₂ :=\n  sorry\n\n@[simp] theorem comp_to_linear_map {R : Type u} {A : Type v} {B : Type w} {C : Type u₁}\n    [comm_semiring R] [semiring A] [semiring B] [semiring C] [algebra R A] [algebra R B]\n    [algebra R C] (f : alg_hom R A B) (g : alg_hom R B C) :\n    to_linear_map (comp g f) = linear_map.comp (to_linear_map g) (to_linear_map f) :=\n  rfl\n\ntheorem map_prod {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [comm_semiring A]\n    [comm_semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) {ι : Type u_1} (f : ι → A)\n    (s : finset ι) :\n    coe_fn φ (finset.prod s fun (x : ι) => f x) = finset.prod s fun (x : ι) => coe_fn φ (f x) :=\n  ring_hom.map_prod (to_ring_hom φ) f s\n\ntheorem map_finsupp_prod {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [comm_semiring A]\n    [comm_semiring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) {α : Type u_1} [HasZero α]\n    {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A) :\n    coe_fn φ (finsupp.prod f g) = finsupp.prod f fun (i : ι) (a : α) => coe_fn φ (g i a) :=\n  map_prod φ (fun (a : ι) => g a (coe_fn f a)) (finsupp.support f)\n\n@[simp] theorem map_neg {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [ring A] [ring B]\n    [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) : coe_fn φ (-x) = -coe_fn φ x :=\n  ring_hom.map_neg (to_ring_hom φ) x\n\n@[simp] theorem map_sub {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [ring A] [ring B]\n    [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) (y : A) :\n    coe_fn φ (x - y) = coe_fn φ x - coe_fn φ y :=\n  ring_hom.map_sub (to_ring_hom φ) x y\n\n@[simp] theorem map_int_cast {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [ring A]\n    [ring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (n : ℤ) : coe_fn φ ↑n = ↑n :=\n  ring_hom.map_int_cast (to_ring_hom φ) n\n\n@[simp] theorem map_inv {R : Type u} {A : Type v} {B : Type w} [comm_ring R] [division_ring A]\n    [division_ring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) :\n    coe_fn φ (x⁻¹) = (coe_fn φ x⁻¹) :=\n  ring_hom.map_inv (to_ring_hom φ) x\n\n@[simp] theorem map_div {R : Type u} {A : Type v} {B : Type w} [comm_ring R] [division_ring A]\n    [division_ring B] [algebra R A] [algebra R B] (φ : alg_hom R A B) (x : A) (y : A) :\n    coe_fn φ (x / y) = coe_fn φ x / coe_fn φ y :=\n  ring_hom.map_div (to_ring_hom φ) x y\n\ntheorem injective_iff {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R] [ring A]\n    [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) :\n    function.injective ⇑f ↔ ∀ (x : A), coe_fn f x = 0 → x = 0 :=\n  ring_hom.injective_iff ↑f\n\nend alg_hom\n\n\n/-- An equivalence of algebras is an equivalence of rings commuting with the actions of scalars. -/\nstructure alg_equiv (R : Type u) (A : Type v) (B : Type w) [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B]\n    extends A ≃+ B, A ≃+* B, A ≃ B, A ≃* B where\n  commutes' : ∀ (r : R), to_fun (coe_fn (algebra_map R A) r) = coe_fn (algebra_map R B) r\n\nnamespace alg_equiv\n\n\nprotected instance has_coe_to_fun {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    has_coe_to_fun (alg_equiv R A₁ A₂) :=\n  has_coe_to_fun.mk (fun (x : alg_equiv R A₁ A₂) => A₁ → A₂) to_fun\n\ntheorem ext {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁] [semiring A₂]\n    [algebra R A₁] [algebra R A₂] {f : alg_equiv R A₁ A₂} {g : alg_equiv R A₁ A₂}\n    (h : ∀ (a : A₁), coe_fn f a = coe_fn g a) : f = g :=\n  sorry\n\nprotected theorem congr_arg {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {f : alg_equiv R A₁ A₂} {x : A₁} {x' : A₁} :\n    x = x' → coe_fn f x = coe_fn f x' :=\n  sorry\n\nprotected theorem congr_fun {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {f : alg_equiv R A₁ A₂} {g : alg_equiv R A₁ A₂}\n    (h : f = g) (x : A₁) : coe_fn f x = coe_fn g x :=\n  h ▸ rfl\n\ntheorem ext_iff {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {f : alg_equiv R A₁ A₂} {g : alg_equiv R A₁ A₂} :\n    f = g ↔ ∀ (x : A₁), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : A₁) => h ▸ rfl, mpr := ext }\n\ntheorem coe_fun_injective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    function.injective fun (e : alg_equiv R A₁ A₂) => ⇑e :=\n  id\n    fun (f g : alg_equiv R A₁ A₂)\n      (w : (fun (e : alg_equiv R A₁ A₂) => ⇑e) f = (fun (e : alg_equiv R A₁ A₂) => ⇑e) g) =>\n      ext fun (a : A₁) => congr_fun w a\n\nprotected instance has_coe_to_ring_equiv {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    has_coe (alg_equiv R A₁ A₂) (A₁ ≃+* A₂) :=\n  has_coe.mk to_ring_equiv\n\n@[simp] theorem mk_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {to_fun : A₁ → A₂} {inv_fun : A₂ → A₁}\n    {left_inv : function.left_inverse inv_fun to_fun}\n    {right_inv : function.right_inverse inv_fun to_fun}\n    {map_mul : ∀ (x y : A₁), to_fun (x * y) = to_fun x * to_fun y}\n    {map_add : ∀ (x y : A₁), to_fun (x + y) = to_fun x + to_fun y}\n    {commutes : ∀ (r : R), to_fun (coe_fn (algebra_map R A₁) r) = coe_fn (algebra_map R A₂) r}\n    {a : A₁} :\n    coe_fn (mk to_fun inv_fun left_inv right_inv map_mul map_add commutes) a = to_fun a :=\n  rfl\n\n@[simp] theorem to_fun_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] {e : alg_equiv R A₁ A₂} {a : A₁} :\n    to_fun e a = coe_fn e a :=\n  rfl\n\n@[simp] theorem coe_ring_equiv {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : ⇑↑e = ⇑e :=\n  rfl\n\ntheorem coe_ring_equiv_injective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    function.injective fun (e : alg_equiv R A₁ A₂) => ↑e :=\n  id\n    fun (f g : alg_equiv R A₁ A₂)\n      (w : (fun (e : alg_equiv R A₁ A₂) => ↑e) f = (fun (e : alg_equiv R A₁ A₂) => ↑e) g) =>\n      ext fun (a : A₁) => congr_fun (congr_arg (fun (e : A₁ ≃+* A₂) => ⇑e) w) a\n\n@[simp] theorem map_add {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) (y : A₁) :\n    coe_fn e (x + y) = coe_fn e x + coe_fn e y :=\n  add_equiv.map_add (to_add_equiv e)\n\n@[simp] theorem map_zero {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : coe_fn e 0 = 0 :=\n  add_equiv.map_zero (to_add_equiv e)\n\n@[simp] theorem map_mul {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) (y : A₁) :\n    coe_fn e (x * y) = coe_fn e x * coe_fn e y :=\n  mul_equiv.map_mul (to_mul_equiv e)\n\n@[simp] theorem map_one {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : coe_fn e 1 = 1 :=\n  mul_equiv.map_one (to_mul_equiv e)\n\n@[simp] theorem commutes {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (r : R) :\n    coe_fn e (coe_fn (algebra_map R A₁) r) = coe_fn (algebra_map R A₂) r :=\n  commutes' e\n\ntheorem map_sum {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) {ι : Type u_1} (f : ι → A₁)\n    (s : finset ι) :\n    coe_fn e (finset.sum s fun (x : ι) => f x) = finset.sum s fun (x : ι) => coe_fn e (f x) :=\n  add_equiv.map_sum (to_add_equiv e) f s\n\ntheorem map_finsupp_sum {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) {α : Type u_1} [HasZero α]\n    {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A₁) :\n    coe_fn e (finsupp.sum f g) = finsupp.sum f fun (i : ι) (b : α) => coe_fn e (g i b) :=\n  map_sum e (fun (a : ι) => g a (coe_fn f a)) (finsupp.support f)\n\n/-- Interpret an algebra equivalence as an algebra homomorphism.\n\nThis definition is included for symmetry with the other `to_*_hom` projections.\nThe `simp` normal form is to use the coercion of the `has_coe_to_alg_hom` instance. -/\ndef to_alg_hom {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : alg_hom R A₁ A₂ :=\n  alg_hom.mk (to_fun e) (map_one e) (map_mul' e) (map_zero e) (map_add' e) (commutes' e)\n\nprotected instance has_coe_to_alg_hom {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    has_coe (alg_equiv R A₁ A₂) (alg_hom R A₁ A₂) :=\n  has_coe.mk to_alg_hom\n\n@[simp] theorem to_alg_hom_eq_coe {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) :\n    to_alg_hom e = ↑e :=\n  rfl\n\n@[simp] theorem coe_alg_hom {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : ⇑↑e = ⇑e :=\n  rfl\n\n@[simp] theorem map_pow {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) (n : ℕ) :\n    coe_fn e (x ^ n) = coe_fn e x ^ n :=\n  alg_hom.map_pow (to_alg_hom e)\n\ntheorem injective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : function.injective ⇑e :=\n  equiv.injective (to_equiv e)\n\ntheorem surjective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : function.surjective ⇑e :=\n  equiv.surjective (to_equiv e)\n\ntheorem bijective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : function.bijective ⇑e :=\n  equiv.bijective (to_equiv e)\n\nprotected instance has_one {R : Type u} {A₁ : Type v} [comm_semiring R] [semiring A₁]\n    [algebra R A₁] : HasOne (alg_equiv R A₁ A₁) :=\n  { one := mk (ring_equiv.to_fun 1) (ring_equiv.inv_fun 1) sorry sorry sorry sorry sorry }\n\nprotected instance inhabited {R : Type u} {A₁ : Type v} [comm_semiring R] [semiring A₁]\n    [algebra R A₁] : Inhabited (alg_equiv R A₁ A₁) :=\n  { default := 1 }\n\n/-- Algebra equivalences are reflexive. -/\ndef refl {R : Type u} {A₁ : Type v} [comm_semiring R] [semiring A₁] [algebra R A₁] :\n    alg_equiv R A₁ A₁ :=\n  1\n\n@[simp] theorem coe_refl {R : Type u} {A₁ : Type v} [comm_semiring R] [semiring A₁] [algebra R A₁] :\n    ↑refl = alg_hom.id R A₁ :=\n  alg_hom.ext fun (x : A₁) => rfl\n\n/-- Algebra equivalences are symmetric. -/\ndef symm {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁] [semiring A₂]\n    [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : alg_equiv R A₂ A₁ :=\n  mk (ring_equiv.to_fun (ring_equiv.symm (to_ring_equiv e)))\n    (ring_equiv.inv_fun (ring_equiv.symm (to_ring_equiv e))) sorry sorry sorry sorry sorry\n\n/-- See Note [custom simps projection] -/\ndef simps.inv_fun {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : A₂ → A₁ :=\n  ⇑(symm e)\n\n@[simp] theorem inv_fun_eq_symm {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] {e : alg_equiv R A₁ A₂} :\n    inv_fun e = ⇑(symm e) :=\n  rfl\n\n@[simp] theorem symm_symm {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {e : alg_equiv R A₁ A₂} : symm (symm e) = e :=\n  ext fun (a : A₁) => Eq.refl (coe_fn (symm (symm e)) a)\n\n/-- Algebra equivalences are transitive. -/\ndef trans {R : Type u} {A₁ : Type v} {A₂ : Type w} {A₃ : Type u₁} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [semiring A₃] [algebra R A₁] [algebra R A₂] [algebra R A₃]\n    (e₁ : alg_equiv R A₁ A₂) (e₂ : alg_equiv R A₂ A₃) : alg_equiv R A₁ A₃ :=\n  mk (ring_equiv.to_fun (ring_equiv.trans (to_ring_equiv e₁) (to_ring_equiv e₂)))\n    (ring_equiv.inv_fun (ring_equiv.trans (to_ring_equiv e₁) (to_ring_equiv e₂))) sorry sorry sorry\n    sorry sorry\n\n@[simp] theorem apply_symm_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₂) :\n    coe_fn e (coe_fn (symm e) x) = x :=\n  equiv.apply_symm_apply (to_equiv e)\n\n@[simp] theorem symm_apply_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) :\n    coe_fn (symm e) (coe_fn e x) = x :=\n  equiv.symm_apply_apply (to_equiv e)\n\n@[simp] theorem trans_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} {A₃ : Type u₁}\n    [comm_semiring R] [semiring A₁] [semiring A₂] [semiring A₃] [algebra R A₁] [algebra R A₂]\n    [algebra R A₃] (e₁ : alg_equiv R A₁ A₂) (e₂ : alg_equiv R A₂ A₃) (x : A₁) :\n    coe_fn (trans e₁ e₂) x = coe_fn e₂ (coe_fn e₁ x) :=\n  rfl\n\n@[simp] theorem comp_symm {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) :\n    alg_hom.comp ↑e ↑(symm e) = alg_hom.id R A₂ :=\n  sorry\n\n@[simp] theorem symm_comp {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) :\n    alg_hom.comp ↑(symm e) ↑e = alg_hom.id R A₁ :=\n  sorry\n\n/-- If `A₁` is equivalent to `A₁'` and `A₂` is equivalent to `A₂'`, then the type of maps\n`A₁ →ₐ[R] A₂` is equivalent to the type of maps `A₁' →ₐ[R] A₂'`. -/\ndef arrow_congr {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {A₁' : Type u_1} {A₂' : Type u_2} [semiring A₁']\n    [semiring A₂'] [algebra R A₁'] [algebra R A₂'] (e₁ : alg_equiv R A₁ A₁')\n    (e₂ : alg_equiv R A₂ A₂') : alg_hom R A₁ A₂ ≃ alg_hom R A₁' A₂' :=\n  equiv.mk\n    (fun (f : alg_hom R A₁ A₂) =>\n      alg_hom.comp (alg_hom.comp (to_alg_hom e₂) f) (to_alg_hom (symm e₁)))\n    (fun (f : alg_hom R A₁' A₂') =>\n      alg_hom.comp (alg_hom.comp (to_alg_hom (symm e₂)) f) (to_alg_hom e₁))\n    sorry sorry\n\ntheorem arrow_congr_comp {R : Type u} {A₁ : Type v} {A₂ : Type w} {A₃ : Type u₁} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [semiring A₃] [algebra R A₁] [algebra R A₂] [algebra R A₃]\n    {A₁' : Type u_1} {A₂' : Type u_2} {A₃' : Type u_3} [semiring A₁'] [semiring A₂'] [semiring A₃']\n    [algebra R A₁'] [algebra R A₂'] [algebra R A₃'] (e₁ : alg_equiv R A₁ A₁')\n    (e₂ : alg_equiv R A₂ A₂') (e₃ : alg_equiv R A₃ A₃') (f : alg_hom R A₁ A₂)\n    (g : alg_hom R A₂ A₃) :\n    coe_fn (arrow_congr e₁ e₃) (alg_hom.comp g f) =\n        alg_hom.comp (coe_fn (arrow_congr e₂ e₃) g) (coe_fn (arrow_congr e₁ e₂) f) :=\n  sorry\n\n@[simp] theorem arrow_congr_refl {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] :\n    arrow_congr refl refl = equiv.refl (alg_hom R A₁ A₂) :=\n  equiv.ext\n    fun (x : alg_hom R A₁ A₂) =>\n      alg_hom.ext fun (x_1 : A₁) => Eq.refl (coe_fn (coe_fn (arrow_congr refl refl) x) x_1)\n\n@[simp] theorem arrow_congr_trans {R : Type u} {A₁ : Type v} {A₂ : Type w} {A₃ : Type u₁}\n    [comm_semiring R] [semiring A₁] [semiring A₂] [semiring A₃] [algebra R A₁] [algebra R A₂]\n    [algebra R A₃] {A₁' : Type u_1} {A₂' : Type u_2} {A₃' : Type u_3} [semiring A₁'] [semiring A₂']\n    [semiring A₃'] [algebra R A₁'] [algebra R A₂'] [algebra R A₃'] (e₁ : alg_equiv R A₁ A₂)\n    (e₁' : alg_equiv R A₁' A₂') (e₂ : alg_equiv R A₂ A₃) (e₂' : alg_equiv R A₂' A₃') :\n    arrow_congr (trans e₁ e₂) (trans e₁' e₂') =\n        equiv.trans (arrow_congr e₁ e₁') (arrow_congr e₂ e₂') :=\n  equiv.ext\n    fun (x : alg_hom R A₁ A₁') =>\n      alg_hom.ext\n        fun (x_1 : A₃) =>\n          Eq.refl (coe_fn (coe_fn (arrow_congr (trans e₁ e₂) (trans e₁' e₂')) x) x_1)\n\n@[simp] theorem arrow_congr_symm {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] {A₁' : Type u_1} {A₂' : Type u_2}\n    [semiring A₁'] [semiring A₂'] [algebra R A₁'] [algebra R A₂'] (e₁ : alg_equiv R A₁ A₁')\n    (e₂ : alg_equiv R A₂ A₂') : equiv.symm (arrow_congr e₁ e₂) = arrow_congr (symm e₁) (symm e₂) :=\n  equiv.ext\n    fun (x : alg_hom R A₁' A₂') =>\n      alg_hom.ext fun (x_1 : A₁) => Eq.refl (coe_fn (coe_fn (equiv.symm (arrow_congr e₁ e₂)) x) x_1)\n\n/-- If an algebra morphism has an inverse, it is a algebra isomorphism. -/\ndef of_alg_hom {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (f : alg_hom R A₁ A₂) (g : alg_hom R A₂ A₁)\n    (h₁ : alg_hom.comp f g = alg_hom.id R A₂) (h₂ : alg_hom.comp g f = alg_hom.id R A₁) :\n    alg_equiv R A₁ A₂ :=\n  mk (alg_hom.to_fun f) ⇑g sorry sorry (alg_hom.map_mul' f) (alg_hom.map_add' f)\n    (alg_hom.commutes' f)\n\n/-- Promotes a bijective algebra homomorphism to an algebra equivalence. -/\ndef of_bijective {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (f : alg_hom R A₁ A₂) (hf : function.bijective ⇑f) :\n    alg_equiv R A₁ A₂ :=\n  mk (ring_equiv.to_fun (ring_equiv.of_bijective (↑f) hf))\n    (ring_equiv.inv_fun (ring_equiv.of_bijective (↑f) hf)) sorry sorry sorry sorry\n    (alg_hom.commutes' f)\n\n/-- Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. -/\ndef to_linear_equiv {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : linear_equiv R A₁ A₂ :=\n  linear_equiv.mk (to_fun e) sorry sorry (to_fun (symm e)) (left_inv e) (right_inv e)\n\n@[simp] theorem to_linear_equiv_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) :\n    coe_fn (to_linear_equiv e) x = coe_fn e x :=\n  rfl\n\ntheorem to_linear_equiv_inj {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {e₁ : alg_equiv R A₁ A₂} {e₂ : alg_equiv R A₁ A₂}\n    (H : to_linear_equiv e₁ = to_linear_equiv e₂) : e₁ = e₂ :=\n  sorry\n\n/-- Interpret an algebra equivalence as a linear map. -/\ndef to_linear_map {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) : linear_map R A₁ A₂ :=\n  alg_hom.to_linear_map (to_alg_hom e)\n\n@[simp] theorem to_alg_hom_to_linear_map {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) :\n    alg_hom.to_linear_map ↑e = to_linear_map e :=\n  rfl\n\n@[simp] theorem to_linear_equiv_to_linear_map {R : Type u} {A₁ : Type v} {A₂ : Type w}\n    [comm_semiring R] [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂]\n    (e : alg_equiv R A₁ A₂) : linear_equiv.to_linear_map (to_linear_equiv e) = to_linear_map e :=\n  rfl\n\n@[simp] theorem to_linear_map_apply {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [semiring A₁] [semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) :\n    coe_fn (to_linear_map e) x = coe_fn e x :=\n  rfl\n\ntheorem to_linear_map_inj {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [semiring A₁]\n    [semiring A₂] [algebra R A₁] [algebra R A₂] {e₁ : alg_equiv R A₁ A₂} {e₂ : alg_equiv R A₁ A₂}\n    (H : to_linear_map e₁ = to_linear_map e₂) : e₁ = e₂ :=\n  sorry\n\n@[simp] theorem trans_to_linear_map {R : Type u} {A₁ : Type v} {A₂ : Type w} {A₃ : Type u₁}\n    [comm_semiring R] [semiring A₁] [semiring A₂] [semiring A₃] [algebra R A₁] [algebra R A₂]\n    [algebra R A₃] (f : alg_equiv R A₁ A₂) (g : alg_equiv R A₂ A₃) :\n    to_linear_map (trans f g) = linear_map.comp (to_linear_map g) (to_linear_map f) :=\n  rfl\n\nprotected instance aut {R : Type u} {A₁ : Type v} [comm_semiring R] [semiring A₁] [algebra R A₁] :\n    group (alg_equiv R A₁ A₁) :=\n  group.mk (fun (ϕ ψ : alg_equiv R A₁ A₁) => trans ψ ϕ) sorry 1 sorry sorry symm\n    (div_inv_monoid.div._default (fun (ϕ ψ : alg_equiv R A₁ A₁) => trans ψ ϕ) sorry 1 sorry sorry\n      symm)\n    sorry\n\ntheorem map_prod {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R] [comm_semiring A₁]\n    [comm_semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) {ι : Type u_1}\n    (f : ι → A₁) (s : finset ι) :\n    coe_fn e (finset.prod s fun (x : ι) => f x) = finset.prod s fun (x : ι) => coe_fn e (f x) :=\n  alg_hom.map_prod (to_alg_hom e) f s\n\ntheorem map_finsupp_prod {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_semiring R]\n    [comm_semiring A₁] [comm_semiring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂)\n    {α : Type u_1} [HasZero α] {ι : Type u_2} (f : ι →₀ α) (g : ι → α → A₁) :\n    coe_fn e (finsupp.prod f g) = finsupp.prod f fun (i : ι) (a : α) => coe_fn e (g i a) :=\n  alg_hom.map_finsupp_prod (to_alg_hom e) f g\n\n@[simp] theorem map_neg {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_ring R] [ring A₁] [ring A₂]\n    [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) : coe_fn e (-x) = -coe_fn e x :=\n  alg_hom.map_neg (to_alg_hom e) x\n\n@[simp] theorem map_sub {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_ring R] [ring A₁] [ring A₂]\n    [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) (y : A₁) :\n    coe_fn e (x - y) = coe_fn e x - coe_fn e y :=\n  alg_hom.map_sub (to_alg_hom e) x y\n\n@[simp] theorem map_inv {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_ring R] [division_ring A₁]\n    [division_ring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) :\n    coe_fn e (x⁻¹) = (coe_fn e x⁻¹) :=\n  alg_hom.map_inv (to_alg_hom e) x\n\n@[simp] theorem map_div {R : Type u} {A₁ : Type v} {A₂ : Type w} [comm_ring R] [division_ring A₁]\n    [division_ring A₂] [algebra R A₁] [algebra R A₂] (e : alg_equiv R A₁ A₂) (x : A₁) (y : A₁) :\n    coe_fn e (x / y) = coe_fn e x / coe_fn e y :=\n  alg_hom.map_div (to_alg_hom e) x y\n\nend alg_equiv\n\n\nnamespace matrix\n\n\n/-! ### `matrix` section\n\nSpecialize `matrix.one_map` and `matrix.zero_map` to `alg_hom` and `alg_equiv`.\nTODO: there should be a way to avoid restating these for each `foo_hom`.\n-/\n\n/-- A version of `matrix.one_map` where `f` is an `alg_hom`. -/\n@[simp] theorem alg_hom_map_one {R : Type u_1} {A₁ : Type u_2} {A₂ : Type u_3} {n : Type u_4}\n    [fintype n] [comm_semiring R] [semiring A₁] [algebra R A₁] [semiring A₂] [algebra R A₂]\n    [DecidableEq n] (f : alg_hom R A₁ A₂) : map 1 ⇑f = 1 :=\n  one_map (alg_hom.map_zero f) (alg_hom.map_one f)\n\n/-- A version of `matrix.one_map` where `f` is an `alg_equiv`. -/\n@[simp] theorem alg_equiv_map_one {R : Type u_1} {A₁ : Type u_2} {A₂ : Type u_3} {n : Type u_4}\n    [fintype n] [comm_semiring R] [semiring A₁] [algebra R A₁] [semiring A₂] [algebra R A₂]\n    [DecidableEq n] (f : alg_equiv R A₁ A₂) : map 1 ⇑f = 1 :=\n  one_map (alg_equiv.map_zero f) (alg_equiv.map_one f)\n\n/-- A version of `matrix.zero_map` where `f` is an `alg_hom`. -/\n@[simp] theorem alg_hom_map_zero {R : Type u_1} {A₁ : Type u_2} {A₂ : Type u_3} {n : Type u_4}\n    [fintype n] [comm_semiring R] [semiring A₁] [algebra R A₁] [semiring A₂] [algebra R A₂]\n    (f : alg_hom R A₁ A₂) : map 0 ⇑f = 0 :=\n  map_zero (alg_hom.map_zero f)\n\n/-- A version of `matrix.zero_map` where `f` is an `alg_equiv`. -/\n@[simp] theorem alg_equiv_map_zero {R : Type u_1} {A₁ : Type u_2} {A₂ : Type u_3} {n : Type u_4}\n    [fintype n] [comm_semiring R] [semiring A₁] [algebra R A₁] [semiring A₂] [algebra R A₂]\n    (f : alg_equiv R A₁ A₂) : map 0 ⇑f = 0 :=\n  map_zero (alg_equiv.map_zero f)\n\nend matrix\n\n\nnamespace algebra\n\n\n/-- `comap R S A` is a type alias for `A`, and has an R-algebra structure defined on it\n  when `algebra R S` and `algebra S A`. If `S` is an `R`-algebra and `A` is an `S`-algebra then\n  `algebra.comap.algebra R S A` can be used to provide `A` with a structure of an `R`-algebra.\n  Other than that, `algebra.comap` is now deprecated and replaced with `is_scalar_tower`. -/\n/- This is done to avoid a type class search with meta-variables `algebra R ?m_1` and\n    `algebra ?m_1 A -/\n\n/- The `nolint` attribute is added because it has unused arguments `R` and `S`, but these are\n  necessary for synthesizing the appropriate type classes -/\n\ndef comap (R : Type u) (S : Type v) (A : Type w) := A\n\nprotected instance comap.inhabited (R : Type u) (S : Type v) (A : Type w) [h : Inhabited A] :\n    Inhabited (comap R S A) :=\n  h\n\nprotected instance comap.semiring (R : Type u) (S : Type v) (A : Type w) [h : semiring A] :\n    semiring (comap R S A) :=\n  h\n\nprotected instance comap.ring (R : Type u) (S : Type v) (A : Type w) [h : ring A] :\n    ring (comap R S A) :=\n  h\n\nprotected instance comap.comm_semiring (R : Type u) (S : Type v) (A : Type w)\n    [h : comm_semiring A] : comm_semiring (comap R S A) :=\n  h\n\nprotected instance comap.comm_ring (R : Type u) (S : Type v) (A : Type w) [h : comm_ring A] :\n    comm_ring (comap R S A) :=\n  h\n\nprotected instance comap.algebra' (R : Type u) (S : Type v) (A : Type w) [comm_semiring S]\n    [semiring A] [h : algebra S A] : algebra S (comap R S A) :=\n  h\n\n/-- Identity homomorphism `A →ₐ[S] comap R S A`. -/\ndef comap.to_comap (R : Type u) (S : Type v) (A : Type w) [comm_semiring S] [semiring A]\n    [algebra S A] : alg_hom S A (comap R S A) :=\n  alg_hom.id S A\n\n/-- Identity homomorphism `comap R S A →ₐ[S] A`. -/\ndef comap.of_comap (R : Type u) (S : Type v) (A : Type w) [comm_semiring S] [semiring A]\n    [algebra S A] : alg_hom S (comap R S A) A :=\n  alg_hom.id S A\n\n/-- `R ⟶ S` induces `S-Alg ⥤ R-Alg` -/\nprotected instance comap.algebra (R : Type u) (S : Type v) (A : Type w) [comm_semiring R]\n    [comm_semiring S] [semiring A] [algebra R S] [algebra S A] : algebra R (comap R S A) :=\n  mk\n    (ring_hom.mk (ring_hom.to_fun (ring_hom.comp (algebra_map S A) (algebra_map R S))) sorry sorry\n      sorry sorry)\n    sorry sorry\n\n/-- Embedding of `S` into `comap R S A`. -/\ndef to_comap (R : Type u) (S : Type v) (A : Type w) [comm_semiring R] [comm_semiring S] [semiring A]\n    [algebra R S] [algebra S A] : alg_hom R S (comap R S A) :=\n  alg_hom.mk (ring_hom.to_fun (algebra_map S A)) sorry sorry sorry sorry sorry\n\ntheorem to_comap_apply (R : Type u) (S : Type v) (A : Type w) [comm_semiring R] [comm_semiring S]\n    [semiring A] [algebra R S] [algebra S A] (x : S) :\n    coe_fn (to_comap R S A) x = coe_fn (algebra_map S A) x :=\n  rfl\n\nend algebra\n\n\nnamespace alg_hom\n\n\n/-- R ⟶ S induces S-Alg ⥤ R-Alg -/\ndef comap {R : Type u} {S : Type v} {A : Type w} {B : Type u₁} [comm_semiring R] [comm_semiring S]\n    [semiring A] [semiring B] [algebra R S] [algebra S A] [algebra S B] (φ : alg_hom S A B) :\n    alg_hom R (algebra.comap R S A) (algebra.comap R S B) :=\n  mk (to_fun φ) (map_one' φ) (map_mul' φ) (map_zero' φ) (map_add' φ) sorry\n\nend alg_hom\n\n\nnamespace ring_hom\n\n\n/-- Reinterpret a `ring_hom` as an `ℕ`-algebra homomorphism. -/\ndef to_nat_alg_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] [algebra ℕ R]\n    [algebra ℕ S] (f : R →+* S) : alg_hom ℕ R S :=\n  alg_hom.mk (⇑f) (map_one' f) (map_mul' f) (map_zero' f) (map_add' f) sorry\n\n/-- Reinterpret a `ring_hom` as a `ℤ`-algebra homomorphism. -/\ndef to_int_alg_hom {R : Type u_1} {S : Type u_2} [ring R] [ring S] [algebra ℤ R] [algebra ℤ S]\n    (f : R →+* S) : alg_hom ℤ R S :=\n  alg_hom.mk (to_fun f) sorry sorry sorry sorry sorry\n\n@[simp] theorem map_rat_algebra_map {R : Type u_1} {S : Type u_2} [ring R] [ring S] [algebra ℚ R]\n    [algebra ℚ S] (f : R →+* S) (r : ℚ) :\n    coe_fn f (coe_fn (algebra_map ℚ R) r) = coe_fn (algebra_map ℚ S) r :=\n  iff.mp ext_iff (subsingleton.elim (comp f (algebra_map ℚ R)) (algebra_map ℚ S)) r\n\n/-- Reinterpret a `ring_hom` as a `ℚ`-algebra homomorphism. -/\ndef to_rat_alg_hom {R : Type u_1} {S : Type u_2} [ring R] [ring S] [algebra ℚ R] [algebra ℚ S]\n    (f : R →+* S) : alg_hom ℚ R S :=\n  alg_hom.mk (to_fun f) sorry sorry sorry sorry (map_rat_algebra_map f)\n\nend ring_hom\n\n\nnamespace rat\n\n\nprotected instance algebra_rat {α : Type u_1} [division_ring α] [char_zero α] : algebra ℚ α :=\n  ring_hom.to_algebra' (cast_hom α) sorry\n\nend rat\n\n\nnamespace algebra\n\n\n/-- `algebra_map` as an `alg_hom`. -/\ndef of_id (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : alg_hom R R A :=\n  alg_hom.mk (ring_hom.to_fun (algebra_map R A)) sorry sorry sorry sorry sorry\n\ntheorem of_id_apply {R : Type u} (A : Type v) [comm_semiring R] [semiring A] [algebra R A] (r : R) :\n    coe_fn (of_id R A) r = coe_fn (algebra_map R A) r :=\n  rfl\n\n/-- The multiplication in an algebra is a bilinear map. -/\ndef lmul (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] :\n    alg_hom R A (module.End R A) :=\n  alg_hom.mk\n    (linear_map.to_fun\n      ((fun (this : linear_map R A (linear_map R A A)) => this)\n        (linear_map.mk₂ R Mul.mul sorry sorry sorry sorry)))\n    sorry sorry sorry sorry sorry\n\n/-- The multiplication on the left in an algebra is a linear map. -/\ndef lmul_left (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (r : A) :\n    linear_map R A A :=\n  coe_fn (lmul R A) r\n\n/-- The multiplication on the right in an algebra is a linear map. -/\ndef lmul_right (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (r : A) :\n    linear_map R A A :=\n  coe_fn (linear_map.flip (alg_hom.to_linear_map (lmul R A))) r\n\n/-- Simultaneous multiplication on the left and right is a linear map. -/\ndef lmul_left_right (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (vw : A × A) : linear_map R A A :=\n  linear_map.comp (lmul_right R (prod.snd vw)) (lmul_left R (prod.fst vw))\n\n/-- The multiplication map on an algebra, as an `R`-linear map from `A ⊗[R] A` to `A`. -/\ndef lmul' (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    linear_map R (tensor_product R A A) A :=\n  tensor_product.lift (alg_hom.to_linear_map (lmul R A))\n\n@[simp] theorem lmul_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (p : A) (q : A) : coe_fn (coe_fn (lmul R A) p) q = p * q :=\n  rfl\n\n@[simp] theorem lmul_left_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (p : A) (q : A) : coe_fn (lmul_left R p) q = p * q :=\n  rfl\n\n@[simp] theorem lmul_right_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (p : A) (q : A) : coe_fn (lmul_right R p) q = q * p :=\n  rfl\n\n@[simp] theorem lmul_left_right_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (vw : A × A) (p : A) :\n    coe_fn (lmul_left_right R vw) p = prod.fst vw * p * prod.snd vw :=\n  rfl\n\n@[simp] theorem lmul_left_one {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : lmul_left R 1 = linear_map.id :=\n  sorry\n\n@[simp] theorem lmul_left_mul {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (a : A) (b : A) : lmul_left R (a * b) = linear_map.comp (lmul_left R a) (lmul_left R b) :=\n  sorry\n\n@[simp] theorem lmul_right_one {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : lmul_right R 1 = linear_map.id :=\n  sorry\n\n@[simp] theorem lmul_right_mul {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (a : A) (b : A) :\n    lmul_right R (a * b) = linear_map.comp (lmul_right R b) (lmul_right R a) :=\n  sorry\n\n@[simp] theorem lmul'_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {x : A} {y : A} : coe_fn (lmul' R) (tensor_product.tmul R x y) = x * y :=\n  sorry\n\nprotected instance linear_map.semimodule' (R : Type u) [comm_semiring R] (M : Type v)\n    [add_comm_monoid M] [semimodule R M] (S : Type w) [comm_semiring S] [algebra R S] :\n    semimodule S (linear_map R M S) :=\n  semimodule.mk sorry sorry\n\nend algebra\n\n\n/-- Semiring ⥤ ℕ-Alg -/\nprotected instance algebra_nat (R : Type u_1) [semiring R] : algebra ℕ R :=\n  algebra.mk (nat.cast_ring_hom R) nat.cast_commute sorry\n\ntheorem span_nat_eq_add_group_closure (R : Type u_1) [semiring R] (s : set R) :\n    submodule.to_add_submonoid (submodule.span ℕ s) = add_submonoid.closure s :=\n  sorry\n\n@[simp] theorem span_nat_eq (R : Type u_1) [semiring R] (s : add_submonoid R) :\n    submodule.to_add_submonoid (submodule.span ℕ ↑s) = s :=\n  sorry\n\n/-- Ring ⥤ ℤ-Alg -/\nprotected instance algebra_int (R : Type u_1) [ring R] : algebra ℤ R :=\n  algebra.mk (int.cast_ring_hom R) int.cast_commute sorry\n\nprotected instance int_algebra_subsingleton {S : Type u_2} [ring S] : subsingleton (algebra ℤ S) :=\n  subsingleton.intro\n    fun (P Q : algebra ℤ S) =>\n      algebra.algebra_ext P Q\n        fun (r : ℤ) =>\n          eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : S) (e_1 : a = a_1) (ᾰ ᾰ_1 : S) (e_2 : ᾰ = ᾰ_1) =>\n                    congr (congr_arg Eq e_1) e_2)\n                  (coe_fn (algebra_map ℤ S) r) (↑r) (ring_hom.eq_int_cast (algebra_map ℤ S) r)\n                  (coe_fn (algebra_map ℤ S) r) (↑r) (ring_hom.eq_int_cast (algebra_map ℤ S) r))\n                (propext (eq_self_iff_true ↑r))))\n            trivial\n\nprotected instance nat_algebra_subsingleton {S : Type u_2} [semiring S] :\n    subsingleton (algebra ℕ S) :=\n  subsingleton.intro\n    fun (P Q : algebra ℕ S) =>\n      algebra.algebra_ext P Q\n        fun (r : ℕ) =>\n          eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : S) (e_1 : a = a_1) (ᾰ ᾰ_1 : S) (e_2 : ᾰ = ᾰ_1) =>\n                    congr (congr_arg Eq e_1) e_2)\n                  (coe_fn (algebra_map ℕ S) r) (↑r) (ring_hom.eq_nat_cast (algebra_map ℕ S) r)\n                  (coe_fn (algebra_map ℕ S) r) (↑r) (ring_hom.eq_nat_cast (algebra_map ℕ S) r))\n                (propext (eq_self_iff_true ↑r))))\n            trivial\n\ntheorem span_int_eq_add_group_closure {R : Type u_1} [ring R] (s : set R) :\n    submodule.to_add_subgroup (submodule.span ℤ s) = add_subgroup.closure s :=\n  sorry\n\n@[simp] theorem span_int_eq {R : Type u_1} [ring R] (s : add_subgroup R) :\n    submodule.to_add_subgroup (submodule.span ℤ ↑s) = s :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (submodule.to_add_subgroup (submodule.span ℤ ↑s) = s))\n        (span_int_eq_add_group_closure ↑s)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (add_subgroup.closure ↑s = s)) (add_subgroup.closure_eq s)))\n      (Eq.refl s))\n\n/-!\nThe R-algebra structure on `Π i : I, A i` when each `A i` is an R-algebra.\n\nWe couldn't set this up back in `algebra.pi_instances` because this file imports it.\n-/\n\nnamespace pi\n\n\nprotected instance algebra (I : Type u) (f : I → Type v) (α : Type u_1) {r : comm_semiring α}\n    [s : (i : I) → semiring (f i)] [(i : I) → algebra α (f i)] : algebra α ((i : I) → f i) :=\n  algebra.mk\n    (ring_hom.mk (ring_hom.to_fun (pi.ring_hom fun (i : I) => algebra_map α (f i))) sorry sorry\n      sorry sorry)\n    sorry sorry\n\n@[simp] theorem algebra_map_apply (I : Type u) (f : I → Type v) (α : Type u_1) {r : comm_semiring α}\n    [s : (i : I) → semiring (f i)] [(i : I) → algebra α (f i)] (a : α) (i : I) :\n    coe_fn (algebra_map α ((i : I) → f i)) a i = coe_fn (algebra_map α (f i)) a :=\n  rfl\n\n-- One could also build a `Π i, R i`-algebra structure on `Π i, A i`,\n\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\n\nend pi\n\n\ntheorem algebra_compatible_smul {R : Type u_1} [comm_semiring R] (A : Type u_2) [semiring A]\n    [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (r : R) (m : M) : r • m = coe_fn (algebra_map R A) r • m :=\n  sorry\n\n@[simp] theorem algebra_map_smul {R : Type u_1} [comm_semiring R] (A : Type u_2) [semiring A]\n    [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (r : R) (m : M) : coe_fn (algebra_map R A) r • m = r • m :=\n  Eq.symm (algebra_compatible_smul A r m)\n\nprotected instance is_scalar_tower.to_smul_comm_class {R : Type u_1} [comm_semiring R]\n    {A : Type u_2} [semiring A] [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M]\n    [semimodule R M] [is_scalar_tower R A M] : smul_comm_class R A M :=\n  smul_comm_class.mk\n    fun (r : R) (a : A) (m : M) =>\n      eq.mpr\n        (id (Eq._oldrec (Eq.refl (r • a • m = a • r • m)) (algebra_compatible_smul A r (a • m))))\n        (eq.mpr\n          (id\n            (Eq._oldrec (Eq.refl (coe_fn (algebra_map R A) r • a • m = a • r • m))\n              (smul_smul (coe_fn (algebra_map R A) r) a m)))\n          (eq.mpr\n            (id\n              (Eq._oldrec (Eq.refl ((coe_fn (algebra_map R A) r * a) • m = a • r • m))\n                (algebra.commutes r a)))\n            (eq.mpr\n              (id\n                (Eq._oldrec (Eq.refl ((a * coe_fn (algebra_map R A) r) • m = a • r • m))\n                  (mul_smul a (coe_fn (algebra_map R A) r) m)))\n              (eq.mpr\n                (id\n                  (Eq._oldrec (Eq.refl (a • coe_fn (algebra_map R A) r • m = a • r • m))\n                    (Eq.symm (algebra_compatible_smul A r m))))\n                (Eq.refl (a • r • m))))))\n\nprotected instance is_scalar_tower.to_smul_comm_class' {R : Type u_1} [comm_semiring R]\n    {A : Type u_2} [semiring A] [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M]\n    [semimodule R M] [is_scalar_tower R A M] : smul_comm_class A R M :=\n  smul_comm_class.symm R A M\n\ntheorem smul_algebra_smul_comm {R : Type u_1} [comm_semiring R] {A : Type u_2} [semiring A]\n    [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (r : R) (a : A) (m : M) : a • r • m = r • a • m :=\n  smul_comm a r m\n\nnamespace linear_map\n\n\nprotected instance coe_is_scalar_tower {R : Type u_1} [comm_semiring R] {A : Type u_2} [semiring A]\n    [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {N : Type u_4} [add_comm_monoid N] [semimodule A N] [semimodule R N]\n    [is_scalar_tower R A N] : has_coe (linear_map A M N) (linear_map R M N) :=\n  has_coe.mk (restrict_scalars R)\n\n@[simp] theorem coe_restrict_scalars_eq_coe (R : Type u_1) [comm_semiring R] {A : Type u_2}\n    [semiring A] [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {N : Type u_4} [add_comm_monoid N] [semimodule A N] [semimodule R N]\n    [is_scalar_tower R A N] (f : linear_map A M N) : ⇑(restrict_scalars R f) = ⇑f :=\n  rfl\n\n@[simp] theorem coe_coe_is_scalar_tower (R : Type u_1) [comm_semiring R] {A : Type u_2} [semiring A]\n    [algebra R A] {M : Type u_3} [add_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {N : Type u_4} [add_comm_monoid N] [semimodule A N] [semimodule R N]\n    [is_scalar_tower R A N] (f : linear_map A M N) : ⇑↑f = ⇑f :=\n  rfl\n\n/-- `A`-linearly coerce a `R`-linear map from `M` to `A` to a function, given an algebra `A` over\na commutative semiring `R` and `M` a semimodule over `R`. -/\ndef lto_fun (R : Type u) (M : Type v) (A : Type w) [comm_semiring R] [add_comm_monoid M]\n    [semimodule R M] [comm_ring A] [algebra R A] : linear_map A (linear_map R M A) (M → A) :=\n  mk to_fun sorry sorry\n\nend linear_map\n\n\n/- In this section, we describe restriction of scalars: if `S` is an algebra over `R`, then\n`S`-modules are also `R`-modules. -/\n\n/--\nWarning: use this type synonym judiciously!\nThe preferred way of working with an `A`-module `M` as `R`-module (where `A` is an `R`-algebra),\nis by `[module R M] [module A M] [is_scalar_tower R A M]`.\n\nWhen `M` is a module over a ring `A`, and `A` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`, provided as a type synonym `module.restrict_scalars R A M := M`.\n-/\ndef restrict_scalars (R : Type u_1) (A : Type u_2) (M : Type u_3) := M\n\nprotected instance restrict_scalars.inhabited (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [I : Inhabited M] : Inhabited (restrict_scalars R A M) :=\n  I\n\nprotected instance restrict_scalars.add_comm_monoid (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [I : add_comm_monoid M] : add_comm_monoid (restrict_scalars R A M) :=\n  I\n\nprotected instance restrict_scalars.add_comm_group (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [I : add_comm_group M] : add_comm_group (restrict_scalars R A M) :=\n  I\n\nprotected instance restrict_scalars.module_orig (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [semiring A] [add_comm_monoid M] [I : semimodule A M] : semimodule A (restrict_scalars R A M) :=\n  I\n\n/--\nWhen `M` is a module over a ring `A`, and `A` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[module R M] [module A M] [is_scalar_tower R A M]`.\n-/\nprotected instance restrict_scalars.semimodule (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M] [semimodule A M] :\n    semimodule R (restrict_scalars R A M) :=\n  semimodule.mk sorry sorry\n\ntheorem restrict_scalars_smul_def (R : Type u_1) (A : Type u_2) (M : Type u_3) [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule A M] (c : R)\n    (x : restrict_scalars R A M) : c • x = coe_fn (algebra_map R A) c • x :=\n  rfl\n\nprotected instance restrict_scalars.is_scalar_tower (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M] [semimodule A M] :\n    is_scalar_tower R A (restrict_scalars R A M) :=\n  is_scalar_tower.mk\n    fun (r : R) (A_1 : A) (M_1 : restrict_scalars R A M) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl ((r • A_1) • M_1 = r • A_1 • M_1)) (algebra.smul_def r A_1)))\n        (eq.mpr\n          (id\n            (Eq._oldrec (Eq.refl ((coe_fn (algebra_map R A) r * A_1) • M_1 = r • A_1 • M_1))\n              (mul_smul (coe_fn (algebra_map R A) r) A_1 M_1)))\n          (Eq.refl (coe_fn (algebra_map R A) r • A_1 • M_1)))\n\nprotected instance submodule.restricted_module (R : Type u_1) (A : Type u_2) (M : Type u_3)\n    [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M] [semimodule A M]\n    (V : submodule A M) : semimodule R ↥V :=\n  restrict_scalars.semimodule R A ↥V\n\nprotected instance submodule.restricted_module_is_scalar_tower (R : Type u_1) (A : Type u_2)\n    (M : Type u_3) [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M] [semimodule A M]\n    (V : submodule A M) : is_scalar_tower R A ↥V :=\n  restrict_scalars.is_scalar_tower R A ↥V\n\nnamespace submodule\n\n\n/--\n`V.restrict_scalars R` is the `R`-submodule of the `R`-module given by restriction of scalars,\ncorresponding to `V`, an `S`-submodule of the original `S`-module.\n-/\ndef restrict_scalars (R : Type u_1) {A : Type u_2} {M : Type u_3} [comm_semiring R] [semiring A]\n    [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M] [is_scalar_tower R A M]\n    (V : submodule A M) : submodule R M :=\n  mk (carrier V) (zero_mem V) sorry sorry\n\n@[simp] theorem restrict_scalars_mem (R : Type u_1) {A : Type u_2} {M : Type u_3} [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M]\n    [is_scalar_tower R A M] (V : submodule A M) (m : M) : m ∈ restrict_scalars R V ↔ m ∈ V :=\n  iff.refl (m ∈ restrict_scalars R V)\n\ntheorem restrict_scalars_injective (R : Type u_1) (A : Type u_2) (M : Type u_3) [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M]\n    [is_scalar_tower R A M] : function.injective (restrict_scalars R) :=\n  fun (V₁ V₂ : submodule A M) (h : restrict_scalars R V₁ = restrict_scalars R V₂) =>\n    ext (eq.mpr (Eq.refl (∀ (x : M), x ∈ V₁ ↔ x ∈ V₂)) (iff.mp set.ext_iff (iff.mp ext'_iff h)))\n\n@[simp] theorem restrict_scalars_inj (R : Type u_1) (A : Type u_2) (M : Type u_3) [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M]\n    [is_scalar_tower R A M] {V₁ : submodule A M} {V₂ : submodule A M} :\n    restrict_scalars R V₁ = restrict_scalars R V₂ ↔ V₁ = V₂ :=\n  { mp :=\n      fun (h : restrict_scalars R V₁ = restrict_scalars R V₂) => restrict_scalars_injective R A M h,\n    mpr := congr_arg fun {V₁ : submodule A M} => restrict_scalars R V₁ }\n\n@[simp] theorem restrict_scalars_bot (R : Type u_1) (A : Type u_2) (M : Type u_3) [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M]\n    [is_scalar_tower R A M] : restrict_scalars R ⊥ = ⊥ :=\n  rfl\n\n@[simp] theorem restrict_scalars_top (R : Type u_1) (A : Type u_2) (M : Type u_3) [comm_semiring R]\n    [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M] [semimodule A M]\n    [is_scalar_tower R A M] : restrict_scalars R ⊤ = ⊤ :=\n  rfl\n\nend submodule\n\n\n@[simp] theorem linear_map.ker_restrict_scalars (R : Type u_1) {A : Type u_2} {M : Type u_3}\n    {N : Type u_4} [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M] [semimodule R M]\n    [semimodule A M] [is_scalar_tower R A M] [add_comm_monoid N] [semimodule R N] [semimodule A N]\n    [is_scalar_tower R A N] (f : linear_map A M N) :\n    linear_map.ker (linear_map.restrict_scalars R f) =\n        submodule.restrict_scalars R (linear_map.ker f) :=\n  rfl\n\nnamespace linear_map\n\n\n/-! When `V` is an `R`-module and `W` is an `S`-module, where `S` is an algebra over `R`, then\nthe collection of `R`-linear maps from `V` to `W` admits an `S`-module structure, given by\nmultiplication in the target. -/\n\nprotected instance is_scalar_tower_extend_scalars (R : Type u_1) [comm_semiring R] (S : Type u_2)\n    [semiring S] [algebra R S] (V : Type u_3) [add_comm_monoid V] [semimodule R V] (W : Type u_4)\n    [add_comm_monoid W] [semimodule R W] [semimodule S W] [is_scalar_tower R S W] :\n    is_scalar_tower R S (linear_map R V W) :=\n  sorry\n\n/-- When `f` is a linear map taking values in `S`, then `λb, f b • x` is a linear map. -/\ndef smul_algebra_right {R : Type u_1} [comm_semiring R] {S : Type u_2} [semiring S] [algebra R S]\n    {V : Type u_3} [add_comm_monoid V] [semimodule R V] {W : Type u_4} [add_comm_monoid W]\n    [semimodule R W] [semimodule S W] [is_scalar_tower R S W] (f : linear_map R V S) (x : W) :\n    linear_map R V W :=\n  mk (fun (b : V) => coe_fn f b • x) sorry sorry\n\n@[simp] theorem smul_algebra_right_apply {R : Type u_1} [comm_semiring R] {S : Type u_2}\n    [semiring S] [algebra R S] {V : Type u_3} [add_comm_monoid V] [semimodule R V] {W : Type u_4}\n    [add_comm_monoid W] [semimodule R W] [semimodule S W] [is_scalar_tower R S W]\n    (f : linear_map R V S) (x : W) (c : V) : coe_fn (smul_algebra_right f x) c = coe_fn f c • x :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/algebra/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.32962410943048853}}
{"text": "import schur_theorem.schur\nimport data.complex.basic\nimport Tools.tools\nimport basic_definitions.matrix_representation\nimport Reynold_operator.reynold_scalar_product\nopen_locale big_operators\nuniverses u v w w'\nvariables {G : Type u} [group G]  \n          {X : Type v} [fintype X][decidable_eq X] \n          {ρ : group_representation G ℂ (X → ℂ)}\nopen Irreductible Schur₂ morphism.from_irreductible equiv_morphism\n--set_option trace.simplify  true\nlemma terrrrr ( a : ℕ ) (ζ :  ℂ ) : a • ζ = (a : ℂ ) • ζ := \nbegin \n    exact add_monoid.smul_eq_mul a ζ,\nend\n\n/--\n    We admit a theorem \n-/\ntheorem eigen_values_exist : ∀ f : (X → ℂ ) →ₗ[ℂ] (X → ℂ),  ∃ t : ℂ, ∃ φ  : (X → ℂ ), \n (φ ≠ 0) ∧  (f φ + t • φ =0) := by admit  \n\ntheorem Schur_on [Irreductible ρ] (f : ρ ⟶ ρ) : \n        ∃ t : ℂ, f + t • 1  = 0  := \nbegin \n    rcases eigen_values_exist f.ℓ with ⟨t,⟨ φ, hyp_t⟩ ⟩,\n    use t, \n    apply morphism.ext, apply linear_map.ext, \n    exact Schur₂ f t φ hyp_t,\n end \n/--\n    Universe Y  = Univrse X ? here i put the dsame for matrix !!!\n-/\nvariables {Y : Type v} [fintype Y][decidable_eq Y] {ρ' : group_representation G ℂ (Y → ℂ)}\n#check ker_im_equiv\ndef not_isomorphic (ρ : group_representation G ℂ (X → ℂ)) (ρ' : group_representation G ℂ (Y → ℂ)) :=\n                      (ρ ≃ᵣ ρ') → false \nopen_locale classical\ntheorem shu (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] : ∀ f : ρ ⟶ ρ', f = 0   \n:= begin \n    by_contradiction,\n    push_neg at a,rcases a with ⟨ φ, hyp ⟩, have : ∃ x, φ x ≠  0,\n        simp, by_contradiction, push_neg at a, have : φ = 0, apply morphism.ext, \n        apply linear_map.ext, exact a, trivial,\n    let shur := Schur₁ φ this, -- j'ai un isoso \n     apply F,exact ( ker_im_equiv φ shur),\n    \nend\nvariables [fintype G][decidable_eq G]\n#check Reynold.Re ρ ρ' \nopen Reynold\n/--\n    J'ai le même problème \n-/\ntheorem grall (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ']: Re ρ ρ' = 0 := \nbegin \n    apply linear_map.ext,\n    intros f,\n    apply shu F,\nend\ntheorem grall₁ (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x : X) (y : Y) :\n   Re ρ ρ' ( matrix.to_lin (ℰ  y x)) = 0 := by {rw grall F, exact rfl}\n\n\n/-!\n    Faire suggir les choses !  Ici on va reprendre en faisant des égalités plutot que des  = 0 ! c'est con ! \n-/\ntheorem grall₂ (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x : X) (y : Y) : \n ∑ s, mixte_conj ρ ρ'  (matrix.to_lin (ℰ y x)) s = 0 := \nbegin \n    erw ← reynold_ext', let g := grall₁ F  x y ,\n    erw g,\n    exact rfl,\nend\nopen matrix\nopen linear_map\n theorem grall₃ (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x : X) (y : Y) :\n to_matrix (∑ s, mixte_conj ρ ρ'  (to_lin (ℰ y x)) s ) = 0 :=\n begin \n    rw grall₂ F,apply proof_strategy, rw to_lin_zero, rw to_matrix_to_lin,\n end\n\nlemma sum_to_matrix (φ : G → ((X → ℂ) →ₗ[ℂ] (Y → ℂ ))) : to_matrix (∑ s, φ s) = ∑ s, to_matrix (φ s) := \nbegin \n    apply proof_strategy,rw to_matrix_to_lin,\n    erw ←  finset.sum_hom finset.univ to_lin, congr,funext,\n    rw to_matrix_to_lin, exact to_lin.is_add_monoid_hom,\nend  \n\n theorem grall₄  (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x : X) (y : Y) :\n(∑ s,  to_matrix (mixte_conj ρ ρ'  (to_lin (ℰ y x)) s ) )= 0 := \nbegin\n    rw ← sum_to_matrix,\n    rw grall₃ F,\nend\nopen_locale matrix\ntheorem grall₅   (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x : X) (y : Y) :\n(∑ s,  (to_matrix (ρ' s⁻¹ : (Y→ ℂ) →ₗ[ℂ] (Y → ℂ ) )) ⬝  (ℰ y x) ⬝  (to_matrix (ρ s : (X→ ℂ) →ₗ[ℂ] (X → ℂ ) )) )= 0 := \nbegin\n    apply proof_strategy,\n    rw to_lin_zero,\n    erw ←  finset.sum_hom finset.univ to_lin,\n    conv_lhs {\n        apply_congr, skip,\n        rw mul_to_lin,rw mul_to_lin, rw to_matrix_to_lin, rw to_matrix_to_lin,\n        erw ← mixte_conj_ext,\n    }, rw grall₂ F,exact to_lin.is_add_monoid_hom,\nend\n/-!\n    Integration de la matrix representation \n-/\nopen character\ntheorem grall₆   (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] ( x0 : X) (y0 : Y) :\n(∑ s,  (mat ρ' s⁻¹ ) ⬝  (ℰ y0 x0) ⬝  (mat ρ s  ) ) = 0 :=\nbegin \n    erw grall₅ F,\nend \n/--\n    Integration du calcul monstreux \n    @[simp]theorem mul_E_mul ( N : matrix X X R) (x0 : X)(y0 : Y) ( M : matrix Y Y R) :\n N ⬝ (ℰ x0 y0 ) ⬝ M = λ x y, N x x0 * M y0 y :=\n-/\n\ntheorem grall₇    (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] (x x0 : X) (y y0 : Y) :\n∑ s, ((mat ρ' s⁻¹ ) y y0) * (mat ρ s x0 x ) = 0 := begin \n    conv_lhs {\n        apply_congr, skip,\n        erw ← mul_E_mul',\n    },\n    let g := sum_apply_mat (λ s,  (mat ρ' s⁻¹ ) ⬝  (ℰ y0 x0) ⬝  (mat ρ s  )) x y,\n    erw ←  g,\n    rw grall₆ F, exact rfl,\nend\nopen bilin_form\n/-!\n    Integration  du produit scalaire  \n-/\n\ntheorem 𝒪ℛ𝒯ℋ𝒪  (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] : \nscalar_product G ℂ   (χ ρ ) (χ ρ' ) = 0 := \nbegin \n    rw interpretation_produit_scalaire ρ ρ',\n    apply finset.sum_eq_zero,\n    intros, \n    rw grall F, exact rfl,\nend\n/--\n    La mère de les grall\n-/\ntheorem graalₒ  : \nscalar_product G ℂ  (χ ρ ) (χ ρ' ) = \n∑ (y : X × Y), ∑ (x : G), mat ρ x y.fst y.fst * mat ρ' x⁻¹ y.snd y.snd   := \nbegin \n    rw scalar_product_ext, \n    conv_lhs {\n        apply_congr,skip,\n        rw chi_ext,\n        rw chi_ext,        \n        erw finset.sum_mul_sum,\n    },\n    erw finset.sum_comm,exact rfl,\nend\n \n\n/--\n    (N ⬝ (ℰ x0 y0 ) ⬝ M) x y =  N x x0 * M y0 y\n-/\ntheorem graal  : \nscalar_product G ℂ  (χ ρ ) (χ ρ' ) = \n∑ (y : X × Y), (∑ (s : G), ((mat ρ s) ⬝ (ℰ y.fst y.snd )  ⬝ (mat ρ' s⁻¹))) y.fst y.snd  \n  := begin \n    erw  graalₒ,\n    congr, funext y,\n    erw  sum_apply_mat,\n    congr,\n    funext,\n    erw ← mul_E_mul',\n  end\n\n\n \ntheorem Schur_on' [Irreductible ρ] (f : ρ ⟶ ρ) :  \n∃ t : ℂ, f +t • 1  = 0  :=\nbegin \n    rcases eigen_values_exist f.ℓ with ⟨t,⟨ φ, hyp_t⟩ ⟩,\n    use t,  \n    apply morphism.ext, apply linear_map.ext, \n    exact Schur₂ f t φ hyp_t,\n end \n#check matrix.trace X ℂ  (X → ℂ )\n\n\n\n\n \n\n\nlemma homo_eq_diag (f : (X → ℂ) →ₗ[ℂ] (X → ℂ))(t : ℂ) (hyp : f + t • 1 = 0) :\n(to_matrix f) = (- t) • (1 : matrix X X ℂ ) := \nbegin \n    have : f = f+ t• 1+ (-t) • 1,\n        simp,\n    rw this, rw hyp,rw zero_add,\n    apply matrix.ext,\n    intros i j,\n    unfold linear_map.to_matrix,simp [linear_map.to_matrix, linear_map.to_matrixₗ, matrix.one_val, eq_comm],\n    split_ifs, exact rfl,rw neg_zero,\nend\nopen Ring\ntheorem Schur_grall [Irreductible ρ] (f : ρ ⟶ ρ ) :\n  matrix.trace X  ℂ  ℂ (linear_map.to_matrix f.ℓ) •  (1 : matrix X X ℂ) =  (fintype.card X) •  (linear_map.to_matrix f.ℓ) :=\n  begin \n    rcases Schur_on' f with ⟨ζ, hyp⟩,\n    have trr : f.ℓ + ζ • 1 = 0,\n        change f.ℓ + ζ • (1 : ρ ⟶ ρ).ℓ  = 0,\n        erw ← morphism_module.coe_smul,\n        rw ← morphism_module.add_coe, rw hyp, exact rfl,\n    rw homo_eq_diag f.ℓ ζ  trr, rw map_smul, rw trace_one,\n    -- analyse \n    funext,simp, rw one_val,rw mul_comm, rw mul_ite, \n    rw mul_one, rw mul_zero,\n    rw terrrrr,rw smul_eq_mul,rw ←  mul_assoc, rw mul_comm, rw ite_mul, rw one_mul,rw zero_mul,\n    split_ifs,simp, simp,\n    \nend \ntheorem Schur_grall_ite [Irreductible ρ] (f : ρ ⟶ ρ ) ( x y : X) : \n((fintype.card X) •  (linear_map.to_matrix f.ℓ)) x y = if x = y then matrix.trace X  ℂ  ℂ (linear_map.to_matrix f.ℓ) else 0 :=\nbegin \n    rw ← Schur_grall, \n    change ((trace X ℂ ℂ) (to_matrix f.ℓ)) • (1 : matrix X X  ℂ ) x y = _,\n    rw one_val, rw smul_ite,congr,exact mul_one ((trace X ℂ ℂ) (to_matrix f.ℓ)),\n    exact mul_zero ((trace X ℂ ℂ) (to_matrix f.ℓ)),\nend\n/--\n    Très bon théorème qui va permettre d'aller plus vite ! \n-/\ntheorem Grall₁  ( x0 : X) (y0 : Y) :\n   linear_map.to_matrix (Re ρ ρ' ( matrix.to_lin (ℰ  y0 x0))).ℓ  =  (∑ s,  (mat ρ' s⁻¹ ) ⬝  (ℰ y0 x0) ⬝  (mat ρ s  ) ) := begin\n    apply proof_strategy,  rw to_matrix_to_lin,\n    erw ← finset.sum_hom finset.univ to_lin,swap, by apply_instance,\n    swap, exact to_lin.is_add_monoid_hom,\n    erw reynold_ext',congr,\n    funext,rw  mul_to_lin, rw mul_to_lin, erw to_matrix_to_lin, erw to_matrix_to_lin,\n    exact rfl,\nend\n/--\n    ON va l'accoquiner a\n-/\ntheorem Graal  : \nscalar_product G ℂ  (χ ρ ) (χ ρ' ) = \n∑ (y : X × Y), (∑ (s : G), ((mat ρ' s) ⬝ (ℰ y.snd y.fst )  ⬝ (mat ρ s⁻¹))) y.snd y.fst :=\nbegin \n    rw graal, congr,\n    funext,\n    conv_lhs{\n        rw sum_apply_mat,\n        apply_congr,skip, erw mul_E_mul',\n    },\n    conv_lhs{\n        erw ← scalar_product_ext G (λ x, mat ρ x y.fst y.fst )(λ x, mat ρ' x  y.snd y.snd),\n        rw bilin_symm,\n    },\n    rw scalar_product_ext,\n    rw sum_apply_mat,\n    congr,\n    funext, erw ← mul_E_mul',\nend\n\n/--\n    Etudier la trace des matrice elemenentaires A faire dans matrix bases DO !\n-/\ntheorem ortho₁ (ζ  : X × X)  [Irreductible ρ ]: \n(fintype.card X • (linear_map.to_matrix (Re ρ ρ ( matrix.to_lin (ℰ  ζ.snd ζ.fst))).ℓ)) ζ.snd ζ.fst  =\n (fintype.card G •  matrix.trace X  ℂ  ℂ ((ℰ ζ.snd ζ.fst)))  :=\nbegin\n    erw  Schur_grall_ite,\n    split_ifs, erw Grall₁,\n    rw h,rw trace_E,rw sum_trace,\n    conv_lhs{\n        apply_congr,skip,\n        rw matrix.trace_mul_comm, rw ← matrix.mul_assoc, \n        rw ← matrix.mul_eq_mul, rw ← matrix.mul_eq_mul,\n        rw ← character.mat_mul,\n        rw mul_inv_self, rw character.mat_one, rw one_mul,\n        rw trace_E,\n    }, rw finset.sum_const,split_ifs, exact rfl,exact rfl,\n    rw trace_E, split_ifs, rw smul_zero,\nend\nlemma terre (a : ℕ) (ζ : ℂ) : a • ζ = (a : ℂ ) * ζ := begin\n    exact terrrrr a ζ, \nend\ntheorem ortho₂  (ζ  : X × X)  [Irreductible ρ ]: \nfintype.card X • ( (linear_map.to_matrix (Re ρ ρ ( matrix.to_lin (ℰ  ζ.snd ζ.fst))).ℓ) ζ.snd ζ.fst)  =\n fintype.card G •  (matrix.trace X  ℂ  ℂ ((ℰ ζ.snd ζ.fst))) := begin \n    erw terre, erw terre, erw ← matrix.smul_val, \n    change (↑(fintype.card X) •   ((to_matrix ((Re ρ ρ) (to_lin (ℰ ζ.snd ζ.fst))).ℓ))) ζ.snd ζ.fst = _,\n    dsimp,\n    erw ← add_monoid.smul_eq_mul (fintype.card X),-- ((to_matrix ((Re ρ ρ) (to_lin (ℰ ζ.snd ζ.fst))).ℓ) ζ.snd ζ.fst),\n    change (fintype.card X •   ((to_matrix ((Re ρ ρ) (to_lin (ℰ ζ.snd ζ.fst))).ℓ))) ζ.snd ζ.fst = _,\n    erw ortho₁, \n    rw ← terre, exact rfl,\n end\nlemma ter_coe ( a : ℕ )(hyp : a ≠ 0) : (a : ℂ )⁻¹  * a = 1 :=begin \n   erw inv_mul_cancel, simp,exact hyp,\nend\n\nlemma  maxi_terrr (a b : ℕ ) {ζ η  : ℂ } (hyp : b ≠ 0) : a • ζ = b • η →  (a : ℂ)  * (b : ℂ)⁻¹   * ζ  = η \n:= begin \n    intros, rw mul_assoc, rw mul_comm, rw  mul_assoc, rw mul_comm ζ, norm_cast,\n    erw  terre a ζ  at a_1, rw a_1,unfold_coes, \n    erw mul_comm, rw terre, rw mul_comm,rw ← mul_assoc, erw ter_coe _ hyp, rw one_mul,\nend\n\nlemma ortho₃ (ζ  : X × X)  [Irreductible ρ ] (hyp : fintype.card X ≠ 0) :\n  (fintype.card G : ℂ ) * (fintype.card X : ℂ)⁻¹  * (matrix.trace X  ℂ  ℂ ((ℰ ζ.snd ζ.fst))) = ( (linear_map.to_matrix (Re ρ ρ ( matrix.to_lin (ℰ  ζ.snd ζ.fst))).ℓ) ζ.snd ζ.fst) :=\n begin \n    let g := eq.symm  (ortho₂ ζ) ,\n    exact maxi_terrr (fintype.card G) (fintype.card X)  (hyp) g , assumption, assumption,\n end\n\n\nlemma ortho₄  [Irreductible ρ ] (hyp : fintype.card X ≠ 0) : \n scalar_product G ℂ  (χ ρ ) (χ ρ ) = \n∑ (ζ  : X × X), (fintype.card G : ℂ ) * (fintype.card X : ℂ)⁻¹  * (matrix.trace X  ℂ  ℂ ((ℰ ζ.snd ζ.fst))) := \nbegin \n    rw interpretation_produit_scalaire,\n    congr,funext,\n    rw ← ortho₃, exact hyp,\nend\nlemma  stra ( a b : ℂ ) (hyp : a ≠ 0) : b = a →  a⁻¹ * b = 1  := begin \n    intros, rw a_1, rw inv_mul_cancel, assumption,\nend\n\nlemma strar ( a b : ℕ) : a ≠ b → (a : ℂ ) ≠ (b : ℂ ) := begin \n    intros, refine function.injective.ne (nat.cast_injective) a_1,\nend\n/--\n    Des lemmes utilises \n-/\nlemma streoi ( a b : ℕ  ) : (a : ℂ )= b → a = b := begin \n    intros, apply nat.cast_inj.mp, exact a_1, by apply_instance,\nend\n#check int.cast\nlemma stra_fin ( a b : ℕ ) : a = b → (a : ℂ ) = (b : ℂ ) := begin \n        intros, exact congr_arg coe a_1,\nend\nlemma 𝒪𝓇𝓉𝒽ℴ (ρ : group_representation G ℂ (X → ℂ ))  [Irreductible ρ ] (hyp : fintype.card X ≠ 0) : \nscalar_product G ℂ  (χ ρ ) (χ ρ ) = (fintype.card G : ℂ ) := \nbegin \n    let g := interpretation_produit_scalaire ρ ρ,\n    rw ortho₄,swap, exact hyp,\n    conv_lhs {\n        apply_congr, skip, rw mul_assoc,\n    }, \n    erw ← finset.mul_sum, erw ← mul_one ↑(fintype.card G),congr, rw mul_one,\n    erw ← finset.mul_sum,\n    apply stra,\n    exact function.injective.ne (nat.cast_injective) hyp,\n    rw ter X  ℂ ,swap, intros, rw trace_E,split_ifs, rw h at a,trivial,exact rfl,\n    conv_lhs{\n        apply_congr,skip,\n        rw trace_E,\n    },\n    rw finset.sum_ite,\n    rw finset.sum_const_zero,rw add_zero,\n    conv_lhs{\n        apply_congr,rw finset.filter_filter,\n    },rw finset.sum_const,\n    erw add_monoid.smul_eq_mul, rw mul_one,\n    apply congr_arg coe,\n    rw GRAAL,\n    rw terrrrrrrrr,simpa,\n    ---- Simpa is sympathique :D  \nend\nexample (ρ : group_representation G ℂ (X → ℂ )) [Irreductible ρ ] \n                (hyp : fintype.card X ≠ 0) : \n\n    (↑(fintype.card G))⁻¹ * ∑ (t : G), χ ρ t * χ ρ t⁻¹ = 1 := \nbegin \n    let g := 𝒪𝓇𝓉𝒽ℴ ρ hyp,\n    rw scalar_product_ext at g,\n    rw g, apply inv_mul_cancel,rw coe_to_lift, simp,\n     intro, \n    let r := finset.card_eq_zero.mp a, have : ( 1 : G) ∈ finset.univ, --- grrrrrrrrrr\n        refine finset.mem_univ 1, \n    finish,\nend\nexample (F : not_isomorphic ρ ρ')[Irreductible ρ ][Irreductible ρ'] : \n\n         (↑(fintype.card G))⁻¹ *   ∑ (t : G), χ ρ t * χ ρ' t⁻¹ = 0  := \nbegin \n    let g := 𝒪ℛ𝒯ℋ𝒪 F,\n    rw scalar_product_ext at g,\n    rw g, erw mul_zero,\nend", "meta": {"author": "Or7ando", "repo": "group_representation", "sha": "9b576984f17764ebf26c8caa2a542d248f1b50d2", "save_path": "github-repos/lean/Or7ando-group_representation", "path": "github-repos/lean/Or7ando-group_representation/group_representation-9b576984f17764ebf26c8caa2a542d248f1b50d2/group_rep_2/orthoganality_of_character/orthoganality.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7461390043208003, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.32954945668197894}}
{"text": "/-\nCopyright (c) 2014 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nNatural homomorphism from the natural numbers into a monoid with one.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.ordered_field\nimport Mathlib.data.nat.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\nnamespace nat\n\n\n/-- Canonical homomorphism from `ℕ` to a type `α` with `0`, `1` and `+`. -/\nprotected def cast {α : Type u_1} [HasZero α] [HasOne α] [Add α] : ℕ → α := sorry\n\n/-- Computationally friendlier cast than `nat.cast`, using binary representation. -/\nprotected def bin_cast {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : α :=\n  binary_rec 0 (fun (odd : Bool) (k : ℕ) (a : α) => cond odd (a + a + 1) (a + a)) n\n\n/--\nCoercions such as `nat.cast_coe` that go from a concrete structure such as\n`ℕ` to an arbitrary ring `α` should be set up as follows:\n```lean\n@[priority 900] instance : has_coe_t ℕ α := ⟨...⟩\n```\n\nIt needs to be `has_coe_t` instead of `has_coe` because otherwise type-class\ninference would loop when constructing the transitive coercion `ℕ → ℕ → ℕ → ...`.\nThe reduced priority is necessary so that it doesn't conflict with instances\nsuch as `has_coe_t α (option α)`.\n\nFor this to work, we reduce the priority of the `coe_base` and `coe_trans`\ninstances because we want the instances for `has_coe_t` to be tried in the\nfollowing order:\n\n 1. `has_coe_t` instances declared in mathlib (such as `has_coe_t α (with_top α)`, etc.)\n 2. `coe_base`, which contains instances such as `has_coe (fin n) n`\n 3. `nat.cast_coe : has_coe_t ℕ α` etc.\n 4. `coe_trans`\n\nIf `coe_trans` is tried first, then `nat.cast_coe` doesn't get a chance to apply.\n-/\n-- see note [coercion into rings]\n\nprotected instance cast_coe {α : Type u_1} [HasZero α] [HasOne α] [Add α] : has_coe_t ℕ α :=\n  has_coe_t.mk nat.cast\n\n@[simp] theorem cast_zero {α : Type u_1} [HasZero α] [HasOne α] [Add α] : ↑0 = 0 := rfl\n\ntheorem cast_add_one {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑(n + 1) = ↑n + 1 :=\n  rfl\n\n@[simp] theorem cast_succ {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) :\n    ↑(Nat.succ n) = ↑n + 1 :=\n  rfl\n\n@[simp] theorem cast_ite {α : Type u_1} [HasZero α] [HasOne α] [Add α] (P : Prop) [Decidable P]\n    (m : ℕ) (n : ℕ) : ↑(ite P m n) = ite P ↑m ↑n :=\n  sorry\n\n@[simp] theorem cast_one {α : Type u_1} [add_monoid α] [HasOne α] : ↑1 = 1 := zero_add 1\n\n@[simp] theorem cast_add {α : Type u_1} [add_monoid α] [HasOne α] (m : ℕ) (n : ℕ) :\n    ↑(m + n) = ↑m + ↑n :=\n  sorry\n\n@[simp] theorem bin_cast_eq {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) :\n    nat.bin_cast n = ↑n :=\n  sorry\n\n/-- `coe : ℕ → α` as an `add_monoid_hom`. -/\ndef cast_add_monoid_hom (α : Type u_1) [add_monoid α] [HasOne α] : ℕ →+ α :=\n  add_monoid_hom.mk coe sorry cast_add\n\n@[simp] theorem coe_cast_add_monoid_hom {α : Type u_1} [add_monoid α] [HasOne α] :\n    ⇑(cast_add_monoid_hom α) = coe :=\n  rfl\n\n@[simp] theorem cast_bit0 {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) : ↑(bit0 n) = bit0 ↑n :=\n  cast_add n n\n\n@[simp] theorem cast_bit1 {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) : ↑(bit1 n) = bit1 ↑n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit1 n) = bit1 ↑n)) (bit1.equations._eqn_1 n)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit0 n + 1) = bit1 ↑n)) (cast_add_one (bit0 n))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit0 n) + 1 = bit1 ↑n)) (cast_bit0 n)))\n        (Eq.refl (bit0 ↑n + 1))))\n\ntheorem cast_two {α : Type u_1} [add_monoid α] [HasOne α] : ↑(bit0 1) = bit0 1 := sorry\n\n@[simp] theorem cast_pred {α : Type u_1} [add_group α] [HasOne α] {n : ℕ} :\n    0 < n → ↑(n - 1) = ↑n - 1 :=\n  sorry\n\n@[simp] theorem cast_sub {α : Type u_1} [add_group α] [HasOne α] {m : ℕ} {n : ℕ} (h : m ≤ n) :\n    ↑(n - m) = ↑n - ↑m :=\n  eq_sub_of_add_eq\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑(n - m) + ↑m = ↑n)) (Eq.symm (cast_add (n - m) m))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (↑(n - m + m) = ↑n)) (nat.sub_add_cancel h))) (Eq.refl ↑n)))\n\n@[simp] theorem cast_mul {α : Type u_1} [semiring α] (m : ℕ) (n : ℕ) : ↑(m * n) = ↑m * ↑n := sorry\n\n@[simp] theorem cast_dvd {α : Type u_1} [field α] {m : ℕ} {n : ℕ} (n_dvd : n ∣ m)\n    (n_nonzero : ↑n ≠ 0) : ↑(m / n) = ↑m / ↑n :=\n  sorry\n\n/-- `coe : ℕ → α` as a `ring_hom` -/\ndef cast_ring_hom (α : Type u_1) [semiring α] : ℕ →+* α :=\n  ring_hom.mk coe sorry cast_mul sorry sorry\n\n@[simp] theorem coe_cast_ring_hom {α : Type u_1} [semiring α] : ⇑(cast_ring_hom α) = coe := rfl\n\ntheorem cast_commute {α : Type u_1} [semiring α] (n : ℕ) (x : α) : commute (↑n) x :=\n  nat.rec_on n (commute.zero_left x)\n    fun (n : ℕ) (ihn : commute (↑n) x) => commute.add_left ihn (commute.one_left x)\n\ntheorem commute_cast {α : Type u_1} [semiring α] (x : α) (n : ℕ) : commute x ↑n :=\n  commute.symm (cast_commute n x)\n\n@[simp] theorem cast_nonneg {α : Type u_1} [ordered_semiring α] (n : ℕ) : 0 ≤ ↑n := sorry\n\ntheorem mono_cast {α : Type u_1} [ordered_semiring α] : monotone coe := sorry\n\ntheorem strict_mono_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] : strict_mono coe :=\n  fun (m n : ℕ) (h : m < n) =>\n    le_induction (lt_add_of_pos_right (↑m) zero_lt_one)\n      (fun (n : ℕ) (_x : Nat.succ m ≤ n) (h : ↑m < ↑n) => lt_add_of_lt_of_pos h zero_lt_one) n h\n\n@[simp] theorem cast_le {α : Type u_1} [ordered_semiring α] [nontrivial α] {m : ℕ} {n : ℕ} :\n    ↑m ≤ ↑n ↔ m ≤ n :=\n  strict_mono.le_iff_le strict_mono_cast\n\n@[simp] theorem cast_lt {α : Type u_1} [ordered_semiring α] [nontrivial α] {m : ℕ} {n : ℕ} :\n    ↑m < ↑n ↔ m < n :=\n  strict_mono.lt_iff_lt strict_mono_cast\n\n@[simp] theorem cast_pos {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} :\n    0 < ↑n ↔ 0 < n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 < ↑n ↔ 0 < n)) (Eq.symm cast_zero)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑0 < ↑n ↔ 0 < n)) (propext cast_lt))) (iff.refl (0 < n)))\n\ntheorem cast_add_one_pos {α : Type u_1} [ordered_semiring α] [nontrivial α] (n : ℕ) : 0 < ↑n + 1 :=\n  add_pos_of_nonneg_of_pos (cast_nonneg n) zero_lt_one\n\n@[simp] theorem one_lt_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} :\n    1 < ↑n ↔ 1 < n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 < ↑n ↔ 1 < n)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑1 < ↑n ↔ 1 < n)) (propext cast_lt))) (iff.refl (1 < n)))\n\n@[simp] theorem one_le_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} :\n    1 ≤ ↑n ↔ 1 ≤ n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 ≤ ↑n ↔ 1 ≤ n)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑1 ≤ ↑n ↔ 1 ≤ n)) (propext cast_le))) (iff.refl (1 ≤ n)))\n\n@[simp] theorem cast_lt_one {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} :\n    ↑n < 1 ↔ n = 0 :=\n  sorry\n\n@[simp] theorem cast_le_one {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} :\n    ↑n ≤ 1 ↔ n ≤ 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≤ 1 ↔ n ≤ 1)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≤ ↑1 ↔ n ≤ 1)) (propext cast_le))) (iff.refl (n ≤ 1)))\n\n@[simp] theorem cast_min {α : Type u_1} [linear_ordered_semiring α] {a : ℕ} {b : ℕ} :\n    ↑(min a b) = min ↑a ↑b :=\n  sorry\n\n@[simp] theorem cast_max {α : Type u_1} [linear_ordered_semiring α] {a : ℕ} {b : ℕ} :\n    ↑(max a b) = max ↑a ↑b :=\n  sorry\n\n@[simp] theorem abs_cast {α : Type u_1} [linear_ordered_ring α] (a : ℕ) : abs ↑a = ↑a :=\n  abs_of_nonneg (cast_nonneg a)\n\ntheorem coe_nat_dvd {α : Type u_1} [comm_semiring α] {m : ℕ} {n : ℕ} (h : m ∣ n) : ↑m ∣ ↑n :=\n  ring_hom.map_dvd (cast_ring_hom α) h\n\ntheorem Mathlib.has_dvd.dvd.nat_cast {α : Type u_1} [comm_semiring α] {m : ℕ} {n : ℕ} (h : m ∣ n) :\n    ↑m ∣ ↑n :=\n  coe_nat_dvd\n\ntheorem inv_pos_of_nat {α : Type u_1} [linear_ordered_field α] {n : ℕ} : 0 < (↑n + 1⁻¹) :=\n  iff.mpr inv_pos (add_pos_of_nonneg_of_pos (cast_nonneg n) zero_lt_one)\n\ntheorem one_div_pos_of_nat {α : Type u_1} [linear_ordered_field α] {n : ℕ} : 0 < 1 / (↑n + 1) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 < 1 / (↑n + 1))) (one_div (↑n + 1)))) inv_pos_of_nat\n\ntheorem one_div_le_one_div {α : Type u_1} [linear_ordered_field α] {n : ℕ} {m : ℕ} (h : n ≤ m) :\n    1 / (↑m + 1) ≤ 1 / (↑n + 1) :=\n  one_div_le_one_div_of_le (cast_add_one_pos n)\n    (eq.mpr (id (Eq.trans (propext (add_le_add_iff_right 1)) (propext cast_le))) h)\n\ntheorem one_div_lt_one_div {α : Type u_1} [linear_ordered_field α] {n : ℕ} {m : ℕ} (h : n < m) :\n    1 / (↑m + 1) < 1 / (↑n + 1) :=\n  one_div_lt_one_div_of_lt (cast_add_one_pos n)\n    (eq.mpr (id (Eq.trans (propext (add_lt_add_iff_right 1)) (propext cast_lt))) h)\n\nend nat\n\n\nnamespace add_monoid_hom\n\n\ntheorem ext_nat {A : Type u_1} [add_monoid A] {f : ℕ →+ A} {g : ℕ →+ A}\n    (h : coe_fn f 1 = coe_fn g 1) : f = g :=\n  sorry\n\ntheorem eq_nat_cast {A : Type u_1} [add_monoid A] [HasOne A] (f : ℕ →+ A) (h1 : coe_fn f 1 = 1)\n    (n : ℕ) : coe_fn f n = ↑n :=\n  congr_fun\n    ((fun (this : f = nat.cast_add_monoid_hom A) => this)\n      (ext_nat (Eq.trans h1 (Eq.symm nat.cast_one))))\n\ntheorem map_nat_cast {A : Type u_1} {B : Type u_2} [add_monoid A] [HasOne A] [add_monoid B]\n    [HasOne B] (f : A →+ B) (h1 : coe_fn f 1 = 1) (n : ℕ) : coe_fn f ↑n = ↑n :=\n  sorry\n\nend add_monoid_hom\n\n\nnamespace ring_hom\n\n\n@[simp] theorem eq_nat_cast {R : Type u_1} [semiring R] (f : ℕ →+* R) (n : ℕ) : coe_fn f n = ↑n :=\n  add_monoid_hom.eq_nat_cast (to_add_monoid_hom f) (map_one f) n\n\n@[simp] theorem map_nat_cast {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R →+* S)\n    (n : ℕ) : coe_fn f ↑n = ↑n :=\n  eq_nat_cast (comp f (nat.cast_ring_hom R)) n\n\ntheorem ext_nat {R : Type u_1} [semiring R] (f : ℕ →+* R) (g : ℕ →+* R) : f = g :=\n  coe_add_monoid_hom_injective (add_monoid_hom.ext_nat (Eq.trans (map_one f) (Eq.symm (map_one g))))\n\nend ring_hom\n\n\n@[simp] theorem nat.cast_id (n : ℕ) : ↑n = n := Eq.symm (ring_hom.eq_nat_cast (ring_hom.id ℕ) n)\n\n@[simp] theorem nat.cast_with_bot (n : ℕ) : ↑n = ↑n := sorry\n\nprotected instance nat.subsingleton_ring_hom {R : Type u_1} [semiring R] : subsingleton (ℕ →+* R) :=\n  subsingleton.intro ring_hom.ext_nat\n\nnamespace with_top\n\n\n@[simp] theorem coe_nat {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑↑n = ↑n := sorry\n\n@[simp] theorem nat_ne_top {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑n ≠ ⊤ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≠ ⊤)) (Eq.symm (coe_nat n)))) coe_ne_top\n\n@[simp] theorem top_ne_nat {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ⊤ ≠ ↑n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (⊤ ≠ ↑n)) (Eq.symm (coe_nat n)))) top_ne_coe\n\ntheorem add_one_le_of_lt {i : with_top ℕ} {n : with_top ℕ} (h : i < n) : i + 1 ≤ n := sorry\n\ntheorem one_le_iff_pos {n : with_top ℕ} : 1 ≤ n ↔ 0 < n := sorry\n\ntheorem nat_induction {P : with_top ℕ → Prop} (a : with_top ℕ) (h0 : P 0)\n    (hsuc : ∀ (n : ℕ), P ↑n → P ↑(Nat.succ n)) (htop : (∀ (n : ℕ), P ↑n) → P ⊤) : P a :=\n  option.cases_on a (htop fun (n : ℕ) => nat.rec_on n h0 hsuc)\n    fun (a : ℕ) => (fun (n : ℕ) => nat.rec_on n h0 hsuc) a\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/nat/cast_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.32952462503647323}}
{"text": "import for_mathlib.boolean_subalgebra\nimport for_mathlib.closure\nimport for_mathlib.order\nimport o_minimal.order\nimport o_minimal.tame.basic\n\n/- Definition of an *o-minimal* structure. -/\n\nnamespace o_minimal\n\nset_option old_structure_cmd true   -- TODO: Is this needed?\n\nvariables {R : Type*} [DUNLO R]\n\nlemma interval_or_point.def_set {S : struc R} [definable_constants S] [is_definable_le S R] :\n  ∀ {i : set R}, interval_or_point i → def_set S i\n| _ (interval_or_point.pt _) := def_val_const\n| _ interval_or_point.Iii := def_set_univ _\n| _ (interval_or_point.Ioi a) := def_set.Ioi a\n| _ (interval_or_point.Iio b) := def_set.Iio b\n| _ (interval_or_point.Ioo a b _) := def_set.Ioo a b\n\n/-- Tame sets are definable. -/\nlemma tame.def_set {S : struc R} [definable_constants S] [is_definable_le S R]\n  {s : set R} (h : tame s) : def_set S s :=\ntame.induction (def_set_empty _) (λ s _ i ih, i.def_set.union ih) h\n\n/-- An o-minimal structure is one in which every definable subset of R is tame. -/\ndef is_o_minimal {S : struc R} [definable_constants S] [is_definable_le S R] : Prop := ∀ (s : set R), def_set S s → tame s\n\n/-- An o-minimal structure on a DUNLO `R` is a structure `S` for which:\n1. `S` has definable constants.\n2. The `≤` relation on `R` is definable in `S`.\n3. Every definable subset of `R` is tame (a union of points and open intervals).\n\nBelow we verify that this definition agrees with the one of [vdD].\n-/\nclass o_minimal (S : struc R) extends definable_constants S, is_definable_le S R : Prop :=\n(tame_of_def : ∀ {s : set R}, def_set S s → tame s)\n\n/-- Version of `o_minimal.tame_of_def` with explicit structure argument.\nThis is useful because otherwise `S` can only be inferred from the proof of definability,\nwhich we might want to produce automatically. -/\nlemma tame_of_def (S : struc R) [o_minimal S] {s : set R} : def_set S s → tame s :=\no_minimal.tame_of_def\n\n/-- Our definition of an o-minimal structure is equivalent to the one in [vdD:1.3.2]. -/\nlemma o_minimal_iff_vdD (S : struc R) : o_minimal S ↔\n  (def_set S {p : R × R | p.1 < p.2} ∧ ∀ (s : set R), def_set S s ↔ tame s) :=\n⟨λ h, by exactI ⟨definable_lt', λ s, ⟨o_minimal.tame_of_def, tame.def_set⟩⟩,\n λ ⟨h₁, h₂⟩,\n { definable_val := λ r, (h₂ _).mpr (tame_single (interval_or_point.pt r)),\n   definable_le' := (is_definable_le_of_definable_lt h₁).1,\n   tame_of_def := λ s, (h₂ s).mp }⟩\n\n-- TODO: for_mathlib\nlemma function.const_injective {α β : Type*} [H : nonempty α] :\n  function.injective (function.const α : β → α → β) :=\nlet ⟨a⟩ := H in λ b₁ b₂ h, congr_fun h a\n\n/-- An alternate constructor expressed in terms of low-level definability. -/\nlemma o_minimal.mk' (S : struc R)\n  (definable_lt : S.definable {x : finvec 2 R | x 0 < x 1})\n  (definable_const : ∀ (r : R), S.definable {x : finvec 1 R | x 0 = r})\n  (tame_of_definable :\n    ∀ (s : set (finvec 1 R)), S.definable s → tame {r | (λ _, r : finvec 1 R) ∈ s}) :\n  o_minimal S :=\n{ definable_val := begin\n    intro r,\n    unfold def_val def_set,\n    convert definable_const r,\n    apply set.ext,\n    simp_rw [set.mem_image],\n    change ∀ (x : fin 1 → R), _ ↔ x 0 = r,\n    rw (equiv.fun_unique (fin 1) R).forall_congr_left',\n    intro r',\n    simp only [equiv.fun_unique, equiv.coe_fn_symm_mk, set.mem_singleton_iff,\n      exists_eq_left, has_coordinates.self_coords, function.const_injective.eq_iff],\n    exact eq_comm\n  end,\n  definable_le' := begin\n    refine (is_definable_le_of_definable_lt _).1,\n    unfold def_val def_set,\n    convert definable_lt,\n    apply set.ext,\n    simp_rw [set.mem_image],\n    change ∀ (x : finvec 2 R), _ ↔ x 0 < x 1,\n    rw finvec.finvec_two_equiv_prod_self.forall_congr_left',\n    rintro ⟨x₀, x₁⟩,\n    change _ ↔ x₀ < x₁,\n    simp [has_coordinates.prod_coords, finvec.finvec_two_equiv_prod_self,\n      finvec.append.inj_iff, function.const_injective.eq_iff, ←and_assoc]\n  end,\n  tame_of_def := begin\n    intros s hs,\n    convert tame_of_definable _ hs,\n    ext x,\n    simp [function.const_injective.eq_iff]\n  end }\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/src/o_minimal/o_minimal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.4921881357207955, "lm_q1q2_score": 0.3292149348699231}}
{"text": "import classes.context_sensitive.basics.inclusion\nimport classes.unrestricted.closure_properties.concatenation\n\nvariables {T : Type}\n\n\nprivate def wrap_CS_rule₁ {N₁ : Type} (N₂ : Type) (r : csrule T N₁) :\n  csrule T (nnn T N₁ N₂) :=\ncsrule.mk\n  (list.map (wrap_symbol₁ N₂) r.context_left)\n  (sum.inl (some (sum.inl r.input_nonterminal)))\n  (list.map (wrap_symbol₁ N₂) r.context_right)\n  (list.map (wrap_symbol₁ N₂) r.output_string)\n\nprivate def wrap_CS_rule₂ {N₂ : Type} (N₁ : Type) (r : csrule T N₂) :\n  csrule T (nnn T N₁ N₂) :=\ncsrule.mk\n  (list.map (wrap_symbol₂ N₁) r.context_left)\n  (sum.inl (some (sum.inr r.input_nonterminal)))\n  (list.map (wrap_symbol₂ N₁) r.context_right)\n  (list.map (wrap_symbol₂ N₁) r.output_string)\n\nprivate def CS_rules_for_terminals₁ (N₂ : Type) (g : CS_grammar T) :\n  list (csrule T (nnn T g.nt N₂)) :=\nlist.map (λ t, csrule.mk [] (sum.inr (sum.inl t)) [] [symbol.terminal t]) (all_used_terminals (grammar_of_csg g))\n\nprivate def CS_rules_for_terminals₂ (N₁ : Type) (g : CS_grammar T) :\n  list (csrule T (nnn T N₁ g.nt)) :=\nlist.map (λ t, csrule.mk [] (sum.inr (sum.inr t)) [] [symbol.terminal t]) (all_used_terminals (grammar_of_csg g))\n\nprivate def big_CS_grammar (g₁ g₂ : CS_grammar T) : CS_grammar T :=\nCS_grammar.mk\n  (nnn T g₁.nt g₂.nt)\n  (sum.inl none)\n  ((csrule.mk [] (sum.inl none) [] [\n    symbol.nonterminal (sum.inl (some (sum.inl g₁.initial))),\n    symbol.nonterminal (sum.inl (some (sum.inr g₂.initial)))]\n  ) :: (\n    (list.map (wrap_CS_rule₁ g₂.nt) g₁.rules ++ list.map (wrap_CS_rule₂ g₁.nt) g₂.rules) ++\n    (CS_rules_for_terminals₁ g₂.nt g₁ ++ CS_rules_for_terminals₂ g₁.nt g₂)\n  ))\n\nprivate lemma big_CS_grammar_same_language (g₁ g₂ : CS_grammar T) :\n  CS_language (big_CS_grammar g₁ g₂) = grammar_language (big_grammar (grammar_of_csg g₁) (grammar_of_csg g₂)) :=\nbegin\n  rw CS_language_eq_grammar_language,\n  congr,\n  unfold big_CS_grammar,\n  unfold grammar_of_csg,\n  unfold big_grammar,\n  dsimp only [list.map],\n  congr,\n  repeat {\n    rw list.map_append,\n  },\n  apply congr_arg2,\n  {\n    apply congr_arg2;\n    {\n      rw list.map_map,\n      rw list.map_map,\n      apply congr_arg2,\n      {\n        ext,\n        cases x,\n        change _ = grule.mk _ _ _ _,\n        finish,\n      },\n      refl,\n    },\n  },\n  {\n    apply congr_arg2,\n    {\n      unfold rules_for_terminals₁,\n      unfold CS_rules_for_terminals₁,\n      rw list.map_map,\n      unfold grammar_of_csg,\n      finish,\n    },\n    {\n      unfold rules_for_terminals₂,\n      unfold CS_rules_for_terminals₂,\n      rw list.map_map,\n      unfold grammar_of_csg,\n      finish,\n    },\n  },\nend\n\nprivate theorem bonus_CS_of_CS_c_CS (L₁ : language T) (L₂ : language T) :\n  is_CS L₁  ∧  is_CS L₂   →   is_CS (L₁ * L₂)   :=\nbegin\n  rintro ⟨⟨g₁, eq_L₁⟩, ⟨g₂, eq_L₂⟩⟩,\n  rw CS_language_eq_grammar_language g₁ at eq_L₁,\n  rw CS_language_eq_grammar_language g₂ at eq_L₂,\n\n  use big_CS_grammar g₁ g₂,\n  rw big_CS_grammar_same_language,\n\n  apply set.eq_of_subset_of_subset,\n  {\n    intros w hyp,\n    rw ←eq_L₁,\n    rw ←eq_L₂,\n    exact in_concatenated_of_in_big hyp,\n  },\n  {\n    intros w hyp,\n    rw ←eq_L₁ at hyp,\n    rw ←eq_L₂ at hyp,\n    exact in_big_of_in_concatenated hyp,\n  },\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/context_sensitive/closure_properties/concatenation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3292149283724471}}
{"text": "/-\nCopyright (c) 2022 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Lean\n\n/-!\n# `casesm`, `cases_type`, `constructorm` tactics\n\nThese tactics implement repeated `cases` / `constructor` on anything satisfying a predicate.\n-/\n\nnamespace Mathlib.Tactic\nopen Lean Meta Elab Tactic\n\n/--\nCore tactic for `casesm` and `cases_type`. Calls `cases` on all fvars in `g` for which\n`matcher ldecl.type` returns true.\n* `recursive`: if true, it calls itself repeatedly on the resulting subgoals\n* `allowSplit`: if false, it will skip any hypotheses where `cases` returns more than one subgoal.\n* `throwOnNoMatch`: if true, then throws an error if no match is found\n-/\npartial def casesMatching (g : MVarId) (matcher : Expr → MetaM Bool)\n    (recursive := false) (allowSplit := true) (throwOnNoMatch := !recursive) :\n    MetaM (List MVarId) := do\n  let result := (← go g).toList\n  if throwOnNoMatch && result == [g] then\n    throwError \"no match\"\n  else\n    return result\n  where\n  /-- Auxiliary for `casesMatching`. Accumulates generated subgoals in `acc`. -/\n  go (g : MVarId) (acc : Array MVarId := #[]) : MetaM (Array MVarId) :=\n    g.withContext do\n      for ldecl in ← getLCtx do\n        if ldecl.isImplementationDetail then continue\n        if ← matcher ldecl.type then\n          let mut acc := acc\n          let subgoals ← if allowSplit then\n            g.cases ldecl.fvarId\n          else\n            let s ← saveState\n            let subgoals ← g.cases ldecl.fvarId\n            if subgoals.size > 1 then\n              s.restore\n              continue\n            else\n              pure subgoals\n          for subgoal in subgoals do\n            if recursive then\n              acc ← go subgoal.mvarId acc\n            else\n              acc := acc.push subgoal.mvarId\n          return acc\n      return (acc.push g)\n\n/-- Elaborate a list of terms with holes into a list of patterns. -/\ndef elabPatterns (pats : Array Term) : TermElabM (Array AbstractMVarsResult) :=\n  withTheReader Term.Context (fun ctx ↦ { ctx with ignoreTCFailures := true }) <|\n  Term.withoutErrToSorry <|\n  pats.mapM fun p ↦ Term.withoutModifyingElabMetaStateWithInfo do\n    withRef p <| abstractMVars (← Term.elabTerm p none)\n\n/-- Returns true if any of the patterns match the expression. -/\ndef matchPatterns (pats : Array AbstractMVarsResult) (e : Expr) : MetaM Bool := do\n  let e ← instantiateMVars e\n  pats.anyM fun p ↦ return (← Conv.matchPattern? p e) matches some (_, #[])\n\n/--\n* `casesm p` applies the `cases` tactic to a hypothesis `h : type`\n  if `type` matches the pattern `p`.\n* `casesm p_1, ..., p_n` applies the `cases` tactic to a hypothesis `h : type`\n  if `type` matches one of the given patterns.\n* `casesm* p` is a more efficient and compact version of `· repeat casesm p`.\n  It is more efficient because the pattern is compiled once.\n\nExample: The following tactic destructs all conjunctions and disjunctions in the current context.\n```\ncasesm* _ ∨ _, _ ∧ _\n```\n-/\nelab (name := casesM) \"casesm\" recursive:\"*\"? ppSpace pats:term,+ : tactic => do\n  let pats ← elabPatterns pats.getElems\n  liftMetaTactic (casesMatching · (matchPatterns pats) recursive.isSome)\n\n/-- Common implementation of `cases_type` and `cases_type!`. -/\ndef elabCasesType (heads : Array Ident)\n    (recursive := false) (allowSplit := true) : TacticM Unit := do\n  let heads ← heads.mapM resolveGlobalConstNoOverloadWithInfo\n  let matcher ty := pure <|\n    if let .const n .. := ty.headBeta.getAppFn then heads.contains n else false\n  liftMetaTactic (casesMatching · matcher recursive allowSplit)\n\n/--\n* `cases_type I` applies the `cases` tactic to a hypothesis `h : (I ...)`\n* `cases_type I_1 ... I_n` applies the `cases` tactic to a hypothesis\n  `h : (I_1 ...)` or ... or `h : (I_n ...)`\n* `cases_type* I` is shorthand for `· repeat cases_type I`\n* `cases_type! I` only applies `cases` if the number of resulting subgoals is <= 1.\n\nExample: The following tactic destructs all conjunctions and disjunctions in the current goal.\n```\ncases_type* Or And\n```\n-/\nelab (name := casesType) \"cases_type\" recursive:\"*\"? ppSpace heads:(colGt ident)+ : tactic =>\n  elabCasesType heads recursive.isSome true\n\n@[inherit_doc casesType]\nelab (name := casesType!) \"cases_type!\" recursive:\"*\"? ppSpace heads:(colGt ident)+ : tactic =>\n  elabCasesType heads recursive.isSome false\n\n/--\nCore tactic for `constructorm`. Calls `constructor` on all subgoals for which\n`matcher ldecl.type` returns true.\n* `recursive`: if true, it calls itself repeatedly on the resulting subgoals\n* `throwOnNoMatch`: if true, throws an error if no match is found\n-/\npartial def constructorMatching (g : MVarId) (matcher : Expr → MetaM Bool)\n    (recursive := false) (throwOnNoMatch := !recursive): MetaM (List MVarId) := do\n  let result ←\n    (if recursive then (do\n      let result ← go g\n      pure result.toList)\n     else\n      (g.withContext do\n          if ← matcher (← g.getType) then g.constructor else pure [g]))\n  if throwOnNoMatch && [g] == result then\n    throwError \"no match\"\n  else\n    return result\nwhere\n  /-- Auxiliary for `constructorMatching`. Accumulates generated subgoals in `acc`. -/\n  go (g : MVarId) (acc : Array MVarId := #[]) : MetaM (Array MVarId) :=\n    g.withContext do\n      if ← matcher (← g.getType) then\n        let mut acc := acc\n        for g' in ← g.constructor do\n          acc ← go g' acc\n        return acc\n      return (acc.push g)\n\n/--\n* `constructorm p_1, ..., p_n` applies the `constructor` tactic to the main goal\n  if `type` matches one of the given patterns.\n* `constructorm* p` is a more efficient and compact version of `· repeat constructorm p`.\n  It is more efficient because the pattern is compiled once.\n\nExample: The following tactic proves any theorem like `True ∧ (True ∨ True)` consisting of\nand/or/true:\n```\nconstructorm* _ ∨ _, _ ∧ _, True\n```\n-/\nelab (name := constructorM) \"constructorm\" recursive:\"*\"? ppSpace pats:term,+ : tactic => do\n  let pats ← elabPatterns pats.getElems\n  liftMetaTactic (constructorMatching · (matchPatterns pats) recursive.isSome)\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/CasesM.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.615087862571909, "lm_q1q2_score": 0.32913254872952974}}
{"text": "import Mathlib.Tactic.CasesM\nimport Std.Tactic.GuardExpr\n\nexample (h : a ∧ b ∨ c ∧ d) (h2 : e ∧ f) : True := by\n  casesm* _∨_, _∧_\n  · clear ‹a› ‹b› ‹e› ‹f›; (fail_if_success clear ‹c›); trivial\n  · clear ‹c› ‹d› ‹e› ‹f›; trivial\n\nexample (h : a ∧ b ∨ c ∧ d) : True := by\n  casesm* _∧_\n  clear ‹a ∧ b ∨ c ∧ d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  casesm* _∨_\n  · clear ‹a ∧ b›; trivial\n  · clear ‹c›; trivial\n  · clear ‹d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  casesm _∨_\n  · clear ‹a ∧ b›; trivial\n  · clear ‹c ∨ d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  cases_type And Or\n  · clear ‹a ∧ b›; trivial\n  · clear ‹c ∨ d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  cases_type* And -- no match expected\n  · clear ‹a ∧ b ∨ c ∨ d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  cases_type Or\n  · clear ‹a ∧ b›; trivial\n  · clear ‹c ∨ d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  cases_type* Or\n  · clear ‹a ∧ b›; trivial\n  · clear ‹c›; trivial\n  · clear ‹d›; trivial\n\nexample (h : a ∧ b ∨ c ∨ d) : True := by\n  cases_type!* And Or -- no match expected\n  · clear ‹a ∧ b ∨ c ∨ d›; trivial\n\nexample (h : a ∧ b ∧ (c ∨ d)) : True := by\n  cases_type! And Or\n  · clear ‹a› ‹b ∧ (c ∨ d)›; trivial\n\nexample (h : a ∧ b ∧ (c ∨ d)) : True := by\n  cases_type!* And Or\n  · clear ‹a› ‹b› ‹c ∨ d›; trivial\n\ninductive Test : Nat → Prop\n  | foo : Test 0\n  | bar : False → Test (n + 1)\n\nexample (_ : Test n) (h2 : Test (m + 1)) : True := by\n  cases_type!* Test\n  · clear ‹Test n› ‹False›; trivial\n\nexample (_ : Test n) (h2 : Test (m + 1)) : True := by\n  cases_type Test\n  · clear ‹Test (m + 1)›; trivial\n  · clear ‹False› ‹Test (m + 1)›; trivial\n\nexample (_ : Test n) (h2 : Test (m + 1)) : True := by\n  cases_type* Test\n  · clear ‹False›; trivial\n  · clear ‹False›; clear ‹False›; trivial\n\nexample : True ∧ True ∧ True := by\n  constructorm* True, _∨_ -- no match expected\n  guard_target = True ∧ True ∧ True\n  constructorm _∧_\n  · guard_target = True; constructorm True\n  · guard_target = True ∧ True; constructorm* True, _∧_\n\nsection AuxDecl\nvariable {p q r : Prop}\nvariable (h : p ∧ q ∨ p ∧ r)\n\n-- Make sure that we don't try to work on auxiliary declarations.\n-- In this case, there will be an auxiliary recursive declaration for\n-- `foo` itself that `casesm (_ ∧ _)` could potentially match.\ntheorem foo : p ∧ p :=\nby cases h\n   · casesm (_ ∧ _)\n     constructor <;> assumption\n   · casesm (_ ∧ _)\n     constructor <;> assumption\n\nend AuxDecl\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/casesm.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.32908726503052177}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.maps\nimport Mathlib.PostPort\n\nuniverses u v w x u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Constructions of new topological spaces from old ones\n\nThis file constructs products, sums, subtypes and quotients of topological spaces\nand sets up their basic theory, such as criteria for maps into or out of these\nconstructions to be continuous; descriptions of the open sets, neighborhood filters,\nand generators of these constructions; and their behavior with respect to embeddings\nand other specific classes of maps.\n\n## Implementation note\n\nThe constructed topologies are defined using induced and coinduced topologies\nalong with the complete lattice structure on topologies. Their universal properties\n(for example, a map `X → Y × Z` is continuous if and only if both projections\n`X → Y`, `X → Z` are) follow easily using order-theoretic descriptions of\ncontinuity. With more work we can also extract descriptions of the open sets,\nneighborhood filters and so on.\n\n## Tags\n\nproduct, sum, disjoint union, subspace, quotient space\n\n-/\n\nprotected instance subtype.topological_space {α : Type u} {p : α → Prop} [t : topological_space α] :\n    topological_space (Subtype p) :=\n  topological_space.induced coe t\n\nprotected instance quot.topological_space {α : Type u} {r : α → α → Prop}\n    [t : topological_space α] : topological_space (Quot r) :=\n  topological_space.coinduced (Quot.mk r) t\n\nprotected instance quotient.topological_space {α : Type u} {s : setoid α}\n    [t : topological_space α] : topological_space (quotient s) :=\n  topological_space.coinduced quotient.mk t\n\nprotected instance prod.topological_space {α : Type u} {β : Type v} [t₁ : topological_space α]\n    [t₂ : topological_space β] : topological_space (α × β) :=\n  topological_space.induced prod.fst t₁ ⊓ topological_space.induced prod.snd t₂\n\nprotected instance sum.topological_space {α : Type u} {β : Type v} [t₁ : topological_space α]\n    [t₂ : topological_space β] : topological_space (α ⊕ β) :=\n  topological_space.coinduced sum.inl t₁ ⊔ topological_space.coinduced sum.inr t₂\n\nprotected instance sigma.topological_space {α : Type u} {β : α → Type v}\n    [t₂ : (a : α) → topological_space (β a)] : topological_space (sigma β) :=\n  supr fun (a : α) => topological_space.coinduced (sigma.mk a) (t₂ a)\n\nprotected instance Pi.topological_space {α : Type u} {β : α → Type v}\n    [t₂ : (a : α) → topological_space (β a)] : topological_space ((a : α) → β a) :=\n  infi fun (a : α) => topological_space.induced (fun (f : (a : α) → β a) => f a) (t₂ a)\n\nprotected instance ulift.topological_space {α : Type u} [t : topological_space α] :\n    topological_space (ulift α) :=\n  topological_space.induced ulift.down t\n\n/-- The image of a dense set under `quotient.mk` is a dense set. -/\ntheorem dense.quotient {α : Type u} [setoid α] [topological_space α] {s : set α} (H : dense s) :\n    dense (quotient.mk '' s) :=\n  dense_range.dense_image (function.surjective.dense_range (surjective_quotient_mk α))\n    continuous_coinduced_rng H\n\n/-- The composition of `quotient.mk` and a function with dense range has dense range. -/\ntheorem dense_range.quotient {α : Type u} {β : Type v} [setoid α] [topological_space α] {f : β → α}\n    (hf : dense_range f) : dense_range (quotient.mk ∘ f) :=\n  dense_range.comp (function.surjective.dense_range (surjective_quotient_mk α)) hf\n    continuous_coinduced_rng\n\nprotected instance subtype.discrete_topology {α : Type u} {p : α → Prop} [topological_space α]\n    [discrete_topology α] : discrete_topology (Subtype p) :=\n  discrete_topology.mk\n    (bot_unique\n      fun (s : set (Subtype p)) (hs : topological_space.is_open ⊥ s) =>\n        Exists.intro (coe '' s)\n          { left := is_open_discrete (coe '' s),\n            right := set.preimage_image_eq s subtype.coe_injective })\n\nprotected instance sum.discrete_topology {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] [hα : discrete_topology α] [hβ : discrete_topology β] :\n    discrete_topology (α ⊕ β) :=\n  sorry\n\nprotected instance sigma.discrete_topology {α : Type u} {β : α → Type v}\n    [(a : α) → topological_space (β a)] [h : ∀ (a : α), discrete_topology (β a)] :\n    discrete_topology (sigma β) :=\n  sorry\n\n/-\nThe 𝓝 filter and the subspace topology.\n-/\n\ntheorem mem_nhds_subtype {α : Type u} [topological_space α] (s : set α)\n    (a : Subtype fun (x : α) => x ∈ s) (t : set (Subtype fun (x : α) => x ∈ s)) :\n    t ∈ nhds a ↔ ∃ (u : set α), ∃ (H : u ∈ nhds ↑a), coe ⁻¹' u ⊆ t :=\n  mem_nhds_induced coe a t\n\ntheorem nhds_subtype {α : Type u} [topological_space α] (s : set α)\n    (a : Subtype fun (x : α) => x ∈ s) : nhds a = filter.comap coe (nhds ↑a) :=\n  nhds_induced coe a\n\ntheorem continuous_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    continuous prod.fst :=\n  continuous_inf_dom_left continuous_induced_dom\n\ntheorem continuous_at_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {p : α × β} : continuous_at prod.fst p :=\n  continuous.continuous_at continuous_fst\n\ntheorem continuous_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    continuous prod.snd :=\n  continuous_inf_dom_right continuous_induced_dom\n\ntheorem continuous_at_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {p : α × β} : continuous_at prod.snd p :=\n  continuous.continuous_at continuous_snd\n\ntheorem continuous.prod_mk {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : γ → α} {g : γ → β} (hf : continuous f)\n    (hg : continuous g) : continuous fun (x : γ) => (f x, g x) :=\n  continuous_inf_rng (continuous_induced_rng hf) (continuous_induced_rng hg)\n\ntheorem continuous.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}\n    [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n    {f : γ → α} {g : δ → β} (hf : continuous f) (hg : continuous g) :\n    continuous fun (x : γ × δ) => (f (prod.fst x), g (prod.snd x)) :=\n  continuous.prod_mk (continuous.comp hf continuous_fst) (continuous.comp hg continuous_snd)\n\ntheorem filter.eventually.prod_inl_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {p : α → Prop} {a : α}\n    (h : filter.eventually (fun (x : α) => p x) (nhds a)) (b : β) :\n    filter.eventually (fun (x : α × β) => p (prod.fst x)) (nhds (a, b)) :=\n  continuous_at_fst h\n\ntheorem filter.eventually.prod_inr_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {p : β → Prop} {b : β}\n    (h : filter.eventually (fun (x : β) => p x) (nhds b)) (a : α) :\n    filter.eventually (fun (x : α × β) => p (prod.snd x)) (nhds (a, b)) :=\n  continuous_at_snd h\n\ntheorem filter.eventually.prod_mk_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {pa : α → Prop} {a : α}\n    (ha : filter.eventually (fun (x : α) => pa x) (nhds a)) {pb : β → Prop} {b : β}\n    (hb : filter.eventually (fun (y : β) => pb y) (nhds b)) :\n    filter.eventually (fun (p : α × β) => pa (prod.fst p) ∧ pb (prod.snd p)) (nhds (a, b)) :=\n  filter.eventually.and (filter.eventually.prod_inl_nhds ha b)\n    (filter.eventually.prod_inr_nhds hb a)\n\ntheorem continuous_swap {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    continuous prod.swap :=\n  continuous.prod_mk continuous_snd continuous_fst\n\ntheorem continuous_uncurry_left {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α → β → γ} (a : α)\n    (h : continuous (function.uncurry f)) : continuous (f a) :=\n  (fun (this : continuous (function.uncurry f ∘ fun (b : β) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_const continuous_id'))\n\ntheorem continuous_uncurry_right {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α → β → γ} (b : β)\n    (h : continuous (function.uncurry f)) : continuous fun (a : α) => f a b :=\n  (fun (this : continuous (function.uncurry f ∘ fun (a : α) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_id' continuous_const))\n\ntheorem continuous_curry {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {g : α × β → γ} (a : α) (h : continuous g) :\n    continuous (function.curry g a) :=\n  (fun (this : continuous (g ∘ fun (b : β) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_const continuous_id'))\n\ntheorem is_open.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set α} {t : set β} (hs : is_open s) (ht : is_open t) : is_open (set.prod s t) :=\n  is_open_inter (is_open.preimage continuous_fst hs) (is_open.preimage continuous_snd ht)\n\ntheorem nhds_prod_eq {α : Type u} {β : Type v} [topological_space α] [topological_space β] {a : α}\n    {b : β} : nhds (a, b) = filter.prod (nhds a) (nhds b) :=\n  sorry\n\ntheorem mem_nhds_prod_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {a : α} {b : β} {s : set (α × β)} :\n    s ∈ nhds (a, b) ↔\n        ∃ (u : set α), ∃ (H : u ∈ nhds a), ∃ (v : set β), ∃ (H : v ∈ nhds b), set.prod u v ⊆ s :=\n  sorry\n\ntheorem filter.has_basis.prod_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {ιa : Type u_1} {ιb : Type u_2} {pa : ιa → Prop} {pb : ιb → Prop}\n    {sa : ιa → set α} {sb : ιb → set β} {a : α} {b : β} (ha : filter.has_basis (nhds a) pa sa)\n    (hb : filter.has_basis (nhds b) pb sb) :\n    filter.has_basis (nhds (a, b)) (fun (i : ιa × ιb) => pa (prod.fst i) ∧ pb (prod.snd i))\n        fun (i : ιa × ιb) => set.prod (sa (prod.fst i)) (sb (prod.snd i)) :=\n  sorry\n\nprotected instance prod.discrete_topology {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] [discrete_topology α] [discrete_topology β] : discrete_topology (α × β) :=\n  discrete_topology.mk (eq_of_nhds_eq_nhds fun (_x : α × β) => sorry)\n\ntheorem prod_mem_nhds_sets {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set α} {t : set β} {a : α} {b : β} (ha : s ∈ nhds a) (hb : t ∈ nhds b) :\n    set.prod s t ∈ nhds (a, b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (set.prod s t ∈ nhds (a, b))) nhds_prod_eq))\n    (filter.prod_mem_prod ha hb)\n\ntheorem nhds_swap {α : Type u} {β : Type v} [topological_space α] [topological_space β] (a : α)\n    (b : β) : nhds (a, b) = filter.map prod.swap (nhds (b, a)) :=\n  sorry\n\ntheorem filter.tendsto.prod_mk_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {γ : Type u_1} {a : α} {b : β} {f : filter γ} {ma : γ → α} {mb : γ → β}\n    (ha : filter.tendsto ma f (nhds a)) (hb : filter.tendsto mb f (nhds b)) :\n    filter.tendsto (fun (c : γ) => (ma c, mb c)) f (nhds (a, b)) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (filter.tendsto (fun (c : γ) => (ma c, mb c)) f (nhds (a, b))))\n        nhds_prod_eq))\n    (filter.tendsto.prod_mk ha hb)\n\ntheorem filter.eventually.curry_nhds {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {p : α × β → Prop} {x : α} {y : β}\n    (h : filter.eventually (fun (x : α × β) => p x) (nhds (x, y))) :\n    filter.eventually (fun (x' : α) => filter.eventually (fun (y' : β) => p (x', y')) (nhds y))\n        (nhds x) :=\n  filter.eventually.curry\n    (eq.mp\n      (Eq._oldrec (Eq.refl (filter.eventually (fun (x : α × β) => p x) (nhds (x, y)))) nhds_prod_eq)\n      h)\n\ntheorem continuous_at.prod {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α → β} {g : α → γ} {x : α}\n    (hf : continuous_at f x) (hg : continuous_at g x) :\n    continuous_at (fun (x : α) => (f x, g x)) x :=\n  filter.tendsto.prod_mk_nhds hf hg\n\ntheorem continuous_at.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}\n    [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n    {f : α → γ} {g : β → δ} {p : α × β} (hf : continuous_at f (prod.fst p))\n    (hg : continuous_at g (prod.snd p)) :\n    continuous_at (fun (p : α × β) => (f (prod.fst p), g (prod.snd p))) p :=\n  continuous_at.prod (continuous_at.comp hf (continuous.continuous_at continuous_fst))\n    (continuous_at.comp hg (continuous.continuous_at continuous_snd))\n\ntheorem continuous_at.prod_map' {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}\n    [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n    {f : α → γ} {g : β → δ} {x : α} {y : β} (hf : continuous_at f x) (hg : continuous_at g y) :\n    continuous_at (fun (p : α × β) => (f (prod.fst p), g (prod.snd p))) (x, y) :=\n  (fun (hf : continuous_at f (prod.fst (x, y))) =>\n      (fun (hg : continuous_at g (prod.snd (x, y))) => continuous_at.prod_map hf hg) hg)\n    hf\n\ntheorem prod_generate_from_generate_from_eq {α : Type u_1} {β : Type u_2} {s : set (set α)}\n    {t : set (set β)} (hs : ⋃₀s = set.univ) (ht : ⋃₀t = set.univ) :\n    prod.topological_space =\n        topological_space.generate_from\n          (set_of\n            fun (g : set (α × β)) =>\n              ∃ (u : set α), ∃ (H : u ∈ s), ∃ (v : set β), ∃ (H : v ∈ t), g = set.prod u v) :=\n  sorry\n\ntheorem prod_eq_generate_from {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] :\n    prod.topological_space =\n        topological_space.generate_from\n          (set_of\n            fun (g : set (α × β)) =>\n              ∃ (s : set α), ∃ (t : set β), is_open s ∧ is_open t ∧ g = set.prod s t) :=\n  sorry\n\ntheorem is_open_prod_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set (α × β)} :\n    is_open s ↔\n        ∀ (a : α) (b : β),\n          (a, b) ∈ s →\n            ∃ (u : set α),\n              ∃ (v : set β), is_open u ∧ is_open v ∧ a ∈ u ∧ b ∈ v ∧ set.prod u v ⊆ s :=\n  sorry\n\ntheorem continuous_uncurry_of_discrete_topology_left {α : Type u} {β : Type v} {γ : Type w}\n    [topological_space α] [topological_space β] [topological_space γ] [discrete_topology α]\n    {f : α → β → γ} (h : ∀ (a : α), continuous (f a)) : continuous (function.uncurry f) :=\n  sorry\n\n/-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood\n  that is a subset of `s`. -/\ntheorem exists_nhds_square {α : Type u} [topological_space α] {s : set (α × α)} {x : α}\n    (hx : s ∈ nhds (x, x)) : ∃ (U : set α), is_open U ∧ x ∈ U ∧ set.prod U U ⊆ s :=\n  sorry\n\n/-- The first projection in a product of topological spaces sends open sets to open sets. -/\ntheorem is_open_map_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    is_open_map prod.fst :=\n  sorry\n\n/-- The second projection in a product of topological spaces sends open sets to open sets. -/\ntheorem is_open_map_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    is_open_map prod.snd :=\n  sorry\n\n/-- A product set is open in a product space if and only if each factor is open, or one of them is\nempty -/\ntheorem is_open_prod_iff' {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set α} {t : set β} : is_open (set.prod s t) ↔ is_open s ∧ is_open t ∨ s = ∅ ∨ t = ∅ :=\n  sorry\n\ntheorem closure_prod_eq {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set α} {t : set β} : closure (set.prod s t) = set.prod (closure s) (closure t) :=\n  sorry\n\ntheorem map_mem_closure2 {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {s : set α} {t : set β} {u : set γ} {f : α → β → γ}\n    {a : α} {b : β} (hf : continuous fun (p : α × β) => f (prod.fst p) (prod.snd p))\n    (ha : a ∈ closure s) (hb : b ∈ closure t) (hu : ∀ (a : α) (b : β), a ∈ s → b ∈ t → f a b ∈ u) :\n    f a b ∈ closure u :=\n  sorry\n\ntheorem is_closed.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s₁ : set α} {s₂ : set β} (h₁ : is_closed s₁) (h₂ : is_closed s₂) :\n    is_closed (set.prod s₁ s₂) :=\n  sorry\n\n/-- The product of two dense sets is a dense set. -/\ntheorem dense.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α}\n    {t : set β} (hs : dense s) (ht : dense t) : dense (set.prod s t) :=\n  fun (x : α × β) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (x ∈ closure (set.prod s t))) closure_prod_eq))\n      { left := hs (prod.fst x), right := ht (prod.snd x) }\n\n/-- If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. -/\ntheorem dense_range.prod_map {β : Type v} {γ : Type w} [topological_space β] [topological_space γ]\n    {ι : Type u_1} {κ : Type u_2} {f : ι → β} {g : κ → γ} (hf : dense_range f)\n    (hg : dense_range g) : dense_range (prod.map f g) :=\n  sorry\n\ntheorem inducing.prod_mk {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α]\n    [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ}\n    (hf : inducing f) (hg : inducing g) :\n    inducing fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) :=\n  sorry\n\ntheorem embedding.prod_mk {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α]\n    [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ}\n    (hf : embedding f) (hg : embedding g) :\n    embedding fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) :=\n  sorry\n\nprotected theorem is_open_map.prod {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}\n    [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n    {f : α → β} {g : γ → δ} (hf : is_open_map f) (hg : is_open_map g) :\n    is_open_map fun (p : α × γ) => (f (prod.fst p), g (prod.snd p)) :=\n  sorry\n\nprotected theorem open_embedding.prod {α : Type u} {β : Type v} {γ : Type w} {δ : Type x}\n    [topological_space α] [topological_space β] [topological_space γ] [topological_space δ]\n    {f : α → β} {g : γ → δ} (hf : open_embedding f) (hg : open_embedding g) :\n    open_embedding fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) :=\n  open_embedding_of_embedding_open\n    (embedding.prod_mk (open_embedding.to_embedding hf) (open_embedding.to_embedding hg))\n    (is_open_map.prod (open_embedding.is_open_map hf) (open_embedding.is_open_map hg))\n\ntheorem embedding_graph {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {f : α → β} (hf : continuous f) : embedding fun (x : α) => (x, f x) :=\n  embedding_of_embedding_compose (continuous.prod_mk continuous_id hf) continuous_fst embedding_id\n\ntheorem continuous_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    continuous sum.inl :=\n  continuous_sup_rng_left continuous_coinduced_rng\n\ntheorem continuous_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    continuous sum.inr :=\n  continuous_sup_rng_right continuous_coinduced_rng\n\ntheorem continuous_sum_rec {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α → γ} {g : β → γ} (hf : continuous f)\n    (hg : continuous g) : continuous (sum.rec f g) :=\n  sorry\n\ntheorem is_open_sum_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {s : set (α ⊕ β)} : is_open s ↔ is_open (sum.inl ⁻¹' s) ∧ is_open (sum.inr ⁻¹' s) :=\n  iff.rfl\n\ntheorem is_open_map_sum {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α ⊕ β → γ}\n    (h₁ : is_open_map fun (a : α) => f (sum.inl a))\n    (h₂ : is_open_map fun (b : β) => f (sum.inr b)) : is_open_map f :=\n  sorry\n\ntheorem embedding_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    embedding sum.inl :=\n  sorry\n\ntheorem embedding_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    embedding sum.inr :=\n  sorry\n\ntheorem is_open_range_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    is_open (set.range sum.inl) :=\n  sorry\n\ntheorem is_open_range_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    is_open (set.range sum.inr) :=\n  sorry\n\ntheorem open_embedding_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    open_embedding sum.inl :=\n  open_embedding.mk\n    (embedding.mk (embedding.to_inducing embedding_inl) (embedding.inj embedding_inl))\n    is_open_range_inl\n\ntheorem open_embedding_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] :\n    open_embedding sum.inr :=\n  open_embedding.mk\n    (embedding.mk (embedding.to_inducing embedding_inr) (embedding.inj embedding_inr))\n    is_open_range_inr\n\ntheorem embedding_subtype_coe {α : Type u} [topological_space α] {p : α → Prop} : embedding coe :=\n  embedding.mk (inducing.mk rfl) subtype.coe_injective\n\ntheorem continuous_subtype_val {α : Type u} [topological_space α] {p : α → Prop} :\n    continuous subtype.val :=\n  continuous_induced_dom\n\ntheorem continuous_subtype_coe {α : Type u} [topological_space α] {p : α → Prop} : continuous coe :=\n  continuous_subtype_val\n\ntheorem is_open.open_embedding_subtype_coe {α : Type u} [topological_space α] {s : set α}\n    (hs : is_open s) : open_embedding coe :=\n  open_embedding.mk (embedding.mk (inducing.mk rfl) subtype.coe_injective)\n    (Eq.symm subtype.range_coe ▸ hs)\n\ntheorem is_open.is_open_map_subtype_coe {α : Type u} [topological_space α] {s : set α}\n    (hs : is_open s) : is_open_map coe :=\n  open_embedding.is_open_map (is_open.open_embedding_subtype_coe hs)\n\ntheorem is_open_map.restrict {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {f : α → β} (hf : is_open_map f) {s : set α} (hs : is_open s) :\n    is_open_map (set.restrict f s) :=\n  is_open_map.comp hf (is_open.is_open_map_subtype_coe hs)\n\ntheorem is_closed.closed_embedding_subtype_coe {α : Type u} [topological_space α] {s : set α}\n    (hs : is_closed s) : closed_embedding coe :=\n  closed_embedding.mk (embedding.mk (inducing.mk rfl) subtype.coe_injective)\n    (Eq.symm subtype.range_coe ▸ hs)\n\ntheorem continuous_subtype_mk {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {p : α → Prop} {f : β → α} (hp : ∀ (x : β), p (f x)) (h : continuous f) :\n    continuous fun (x : β) => { val := f x, property := hp x } :=\n  continuous_induced_rng h\n\ntheorem continuous_inclusion {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (h : s ⊆ t) : continuous (set.inclusion h) :=\n  continuous_subtype_mk (fun (x : ↥s) => set.inclusion._proof_1 h x) continuous_subtype_coe\n\ntheorem continuous_at_subtype_coe {α : Type u} [topological_space α] {p : α → Prop}\n    {a : Subtype p} : continuous_at coe a :=\n  iff.mp continuous_iff_continuous_at continuous_subtype_coe a\n\ntheorem map_nhds_subtype_coe_eq {α : Type u} [topological_space α] {p : α → Prop} {a : α} (ha : p a)\n    (h : (set_of fun (a : α) => p a) ∈ nhds a) :\n    filter.map coe (nhds { val := a, property := ha }) = nhds a :=\n  sorry\n\ntheorem nhds_subtype_eq_comap {α : Type u} [topological_space α] {p : α → Prop} {a : α} {h : p a} :\n    nhds { val := a, property := h } = filter.comap coe (nhds a) :=\n  nhds_induced coe { val := a, property := h }\n\ntheorem tendsto_subtype_rng {α : Type u} [topological_space α] {β : Type u_1} {p : α → Prop}\n    {b : filter β} {f : β → Subtype p} {a : Subtype p} :\n    filter.tendsto f b (nhds a) ↔ filter.tendsto (fun (x : β) => ↑(f x)) b (nhds ↑a) :=\n  sorry\n\ntheorem continuous_subtype_nhds_cover {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {ι : Sort u_1} {f : α → β} {c : ι → α → Prop}\n    (c_cover : ∀ (x : α), ∃ (i : ι), (set_of fun (x : α) => c i x) ∈ nhds x)\n    (f_cont : ∀ (i : ι), continuous fun (x : Subtype (c i)) => f ↑x) : continuous f :=\n  sorry\n\ntheorem continuous_subtype_is_closed_cover {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {ι : Type u_1} {f : α → β} (c : ι → α → Prop)\n    (h_lf : locally_finite fun (i : ι) => set_of fun (x : α) => c i x)\n    (h_is_closed : ∀ (i : ι), is_closed (set_of fun (x : α) => c i x))\n    (h_cover : ∀ (x : α), ∃ (i : ι), c i x)\n    (f_cont : ∀ (i : ι), continuous fun (x : Subtype (c i)) => f ↑x) : continuous f :=\n  sorry\n\ntheorem closure_subtype {α : Type u} [topological_space α] {p : α → Prop}\n    {x : Subtype fun (a : α) => p a} {s : set (Subtype fun (a : α) => p a)} :\n    x ∈ closure s ↔ ↑x ∈ closure (coe '' s) :=\n  closure_induced fun (x y : Subtype fun (a : α) => p a) => subtype.eq\n\ntheorem quotient_map_quot_mk {α : Type u} [topological_space α] {r : α → α → Prop} :\n    quotient_map (Quot.mk r) :=\n  { left := quot.exists_rep, right := rfl }\n\ntheorem continuous_quot_mk {α : Type u} [topological_space α] {r : α → α → Prop} :\n    continuous (Quot.mk r) :=\n  continuous_coinduced_rng\n\ntheorem continuous_quot_lift {α : Type u} {β : Type v} [topological_space α] [topological_space β]\n    {r : α → α → Prop} {f : α → β} (hr : ∀ (a b : α), r a b → f a = f b) (h : continuous f) :\n    continuous (Quot.lift f hr) :=\n  continuous_coinduced_dom h\n\ntheorem quotient_map_quotient_mk {α : Type u} [topological_space α] {s : setoid α} :\n    quotient_map quotient.mk :=\n  quotient_map_quot_mk\n\ntheorem continuous_quotient_mk {α : Type u} [topological_space α] {s : setoid α} :\n    continuous quotient.mk :=\n  continuous_coinduced_rng\n\ntheorem continuous_quotient_lift {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {s : setoid α} {f : α → β} (hs : ∀ (a b : α), a ≈ b → f a = f b)\n    (h : continuous f) : continuous (quotient.lift f hs) :=\n  continuous_coinduced_dom h\n\ntheorem continuous_pi {α : Type u} {ι : Type u_1} {π : ι → Type u_2} [topological_space α]\n    [(i : ι) → topological_space (π i)] {f : α → (i : ι) → π i}\n    (h : ∀ (i : ι), continuous fun (a : α) => f a i) : continuous f :=\n  continuous_infi_rng fun (i : ι) => continuous_induced_rng (h i)\n\ntheorem continuous_apply {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → topological_space (π i)]\n    (i : ι) : continuous fun (p : (i : ι) → π i) => p i :=\n  continuous_infi_dom continuous_induced_dom\n\n/-- Embedding a factor into a product space (by fixing arbitrarily all the other coordinates) is\ncontinuous. -/\ntheorem continuous_update {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι]\n    [(i : ι) → topological_space (π i)] {i : ι} {f : (i : ι) → π i} :\n    continuous fun (x : π i) => function.update f i x :=\n  sorry\n\ntheorem nhds_pi {ι : Type u_1} {π : ι → Type u_2} [t : (i : ι) → topological_space (π i)]\n    {a : (i : ι) → π i} :\n    nhds a = infi fun (i : ι) => filter.comap (fun (x : (i : ι) → π i) => x i) (nhds (a i)) :=\n  sorry\n\ntheorem tendsto_pi {α : Type u} {ι : Type u_1} {π : ι → Type u_2}\n    [t : (i : ι) → topological_space (π i)] {f : α → (i : ι) → π i} {g : (i : ι) → π i}\n    {u : filter α} :\n    filter.tendsto f u (nhds g) ↔ ∀ (x : ι), filter.tendsto (fun (i : α) => f i x) u (nhds (g x)) :=\n  sorry\n\ntheorem is_open_set_pi {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)]\n    {i : set ι} {s : (a : ι) → set (π a)} (hi : set.finite i)\n    (hs : ∀ (a : ι), a ∈ i → is_open (s a)) : is_open (set.pi i s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.pi i s))) (set.pi_def i s)))\n    (is_open_bInter hi fun (a : ι) (ha : a ∈ i) => is_open.preimage (continuous_apply a) (hs a ha))\n\ntheorem set_pi_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)]\n    {i : set ι} {s : (a : ι) → set (π a)} {x : (a : ι) → π a} (hi : set.finite i)\n    (hs : ∀ (a : ι), a ∈ i → s a ∈ nhds (x a)) : set.pi i s ∈ nhds x :=\n  sorry\n\ntheorem pi_eq_generate_from {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)] :\n    Pi.topological_space =\n        topological_space.generate_from\n          (set_of\n            fun (g : set ((a : ι) → π a)) =>\n              ∃ (s : (a : ι) → set (π a)),\n                ∃ (i : finset ι), (∀ (a : ι), a ∈ i → is_open (s a)) ∧ g = set.pi (↑i) s) :=\n  sorry\n\ntheorem pi_generate_from_eq {ι : Type u_1} {π : ι → Type u_2} {g : (a : ι) → set (set (π a))} :\n    Pi.topological_space =\n        topological_space.generate_from\n          (set_of\n            fun (t : set ((a : ι) → π a)) =>\n              ∃ (s : (a : ι) → set (π a)),\n                ∃ (i : finset ι), (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = set.pi (↑i) s) :=\n  sorry\n\ntheorem pi_generate_from_eq_fintype {ι : Type u_1} {π : ι → Type u_2}\n    {g : (a : ι) → set (set (π a))} [fintype ι] (hg : ∀ (a : ι), ⋃₀g a = set.univ) :\n    Pi.topological_space =\n        topological_space.generate_from\n          (set_of\n            fun (t : set ((a : ι) → π a)) =>\n              ∃ (s : (a : ι) → set (π a)), (∀ (a : ι), s a ∈ g a) ∧ t = set.pi set.univ s) :=\n  sorry\n\ntheorem continuous_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : continuous (sigma.mk i) :=\n  continuous_supr_rng continuous_coinduced_rng\n\ntheorem is_open_sigma_iff {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {s : set (sigma σ)} : is_open s ↔ ∀ (i : ι), is_open (sigma.mk i ⁻¹' s) :=\n  sorry\n\ntheorem is_closed_sigma_iff {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {s : set (sigma σ)} : is_closed s ↔ ∀ (i : ι), is_closed (sigma.mk i ⁻¹' s) :=\n  is_open_sigma_iff\n\ntheorem is_open_map_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : is_open_map (sigma.mk i) :=\n  sorry\n\ntheorem is_open_range_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : is_open (set.range (sigma.mk i)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.range (sigma.mk i)))) (Eq.symm set.image_univ)))\n    (is_open_map_sigma_mk set.univ is_open_univ)\n\ntheorem is_closed_map_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : is_closed_map (sigma.mk i) :=\n  sorry\n\ntheorem is_closed_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : is_closed (set.range (sigma.mk i)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (set.range (sigma.mk i)))) (Eq.symm set.image_univ)))\n    (is_closed_map_sigma_mk set.univ is_closed_univ)\n\ntheorem open_embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2}\n    [(i : ι) → topological_space (σ i)] {i : ι} : open_embedding (sigma.mk i) :=\n  open_embedding_of_continuous_injective_open continuous_sigma_mk sigma_mk_injective\n    is_open_map_sigma_mk\n\ntheorem closed_embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2}\n    [(i : ι) → topological_space (σ i)] {i : ι} : closed_embedding (sigma.mk i) :=\n  closed_embedding_of_continuous_injective_closed continuous_sigma_mk sigma_mk_injective\n    is_closed_map_sigma_mk\n\ntheorem embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {i : ι} : embedding (sigma.mk i) :=\n  closed_embedding.to_embedding closed_embedding_sigma_mk\n\n/-- A map out of a sum type is continuous if its restriction to each summand is. -/\ntheorem continuous_sigma {β : Type v} {ι : Type u_1} {σ : ι → Type u_2}\n    [(i : ι) → topological_space (σ i)] [topological_space β] {f : sigma σ → β}\n    (h : ∀ (i : ι), continuous fun (a : σ i) => f (sigma.mk i a)) : continuous f :=\n  continuous_supr_dom fun (i : ι) => continuous_coinduced_dom (h i)\n\ntheorem continuous_sigma_map {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {κ : Type u_3} {τ : κ → Type u_4} [(k : κ) → topological_space (τ k)] {f₁ : ι → κ}\n    {f₂ : (i : ι) → σ i → τ (f₁ i)} (hf : ∀ (i : ι), continuous (f₂ i)) :\n    continuous (sigma.map f₁ f₂) :=\n  sorry\n\ntheorem is_open_map_sigma {β : Type v} {ι : Type u_1} {σ : ι → Type u_2}\n    [(i : ι) → topological_space (σ i)] [topological_space β] {f : sigma σ → β}\n    (h : ∀ (i : ι), is_open_map fun (a : σ i) => f (sigma.mk i a)) : is_open_map f :=\n  sorry\n\n/-- The sum of embeddings is an embedding. -/\ntheorem embedding_sigma_map {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)]\n    {τ : ι → Type u_3} [(i : ι) → topological_space (τ i)] {f : (i : ι) → σ i → τ i}\n    (hf : ∀ (i : ι), embedding (f i)) : embedding (sigma.map id f) :=\n  sorry\n\ntheorem continuous_ulift_down {α : Type u} [topological_space α] : continuous ulift.down :=\n  continuous_induced_dom\n\ntheorem continuous_ulift_up {α : Type u} [topological_space α] : continuous ulift.up :=\n  continuous_induced_rng continuous_id\n\ntheorem mem_closure_of_continuous {α : Type u} {β : Type v} [topological_space α]\n    [topological_space β] {f : α → β} {a : α} {s : set α} {t : set β} (hf : continuous f)\n    (ha : a ∈ closure s) (h : set.maps_to f s (closure t)) : f a ∈ closure t :=\n  set.mem_of_mem_of_subset\n    (set.mem_of_mem_of_subset (set.mem_image_of_mem f ha) (image_closure_subset_closure_image hf))\n    (closure_minimal (set.maps_to.image_subset h) is_closed_closure)\n\ntheorem mem_closure_of_continuous2 {α : Type u} {β : Type v} {γ : Type w} [topological_space α]\n    [topological_space β] [topological_space γ] {f : α → β → γ} {a : α} {b : β} {s : set α}\n    {t : set β} {u : set γ} (hf : continuous fun (p : α × β) => f (prod.fst p) (prod.snd p))\n    (ha : a ∈ closure s) (hb : b ∈ closure t)\n    (h : ∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → f a b ∈ closure u) : f a b ∈ closure u :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/constructions_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3290872567430385}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.category.Group.preadditive\nimport group_theory.quotient_group\nimport category_theory.limits.concrete_category\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.shapes.concrete_category\n\n/-!\n# The category of additive commutative groups has all colimits.\n\nThis file uses a \"pre-automated\" approach, just as for `Mon/colimits.lean`.\nIt is a very uniform approach, that conceivably could be synthesised directly\nby a tactic that analyses the shape of `add_comm_group` and `monoid_hom`.\n\nTODO:\nIn fact, in `AddCommGroup` there is a much nicer model of colimits as quotients\nof finitely supported functions, and we really should implement this as well (or instead).\n-/\n\nuniverses u v\n\nopen category_theory\nopen category_theory.limits\n\n-- [ROBOT VOICE]:\n-- You should pretend for now that this file was automatically generated.\n-- It follows the same template as colimits in Mon.\n\nnamespace AddCommGroup.colimits\n/-!\nWe build the colimit of a diagram in `AddCommGroup` by constructing the\nfree group on the disjoint union of all the abelian groups in the diagram,\nthen taking the quotient by the abelian group laws within each abelian group,\nand the identifications given by the morphisms in the diagram.\n-/\n\nvariables {J : Type v} [small_category J] (F : J ⥤ AddCommGroup.{v})\n\n/--\nAn inductive type representing all group expressions (without relations)\non a collection of types indexed by the objects of `J`.\n-/\ninductive prequotient\n-- There's always `of`\n| of : Π (j : J) (x : F.obj j), prequotient\n-- Then one generator for each operation\n| zero : prequotient\n| neg : prequotient → prequotient\n| add : prequotient → prequotient → prequotient\n\ninstance : inhabited (prequotient F) := ⟨prequotient.zero⟩\n\nopen prequotient\n\n/--\nThe relation on `prequotient` saying when two expressions are equal\nbecause of the abelian group laws, or\nbecause one element is mapped to another by a morphism in the diagram.\n-/\ninductive relation : prequotient F → prequotient F → Prop\n-- Make it an equivalence relation:\n| refl : Π (x), relation x x\n| symm : Π (x y) (h : relation x y), relation y x\n| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z\n-- There's always a `map` relation\n| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), relation (of j' (F.map f x)) (of j x)\n-- Then one relation per operation, describing the interaction with `of`\n| zero : Π (j), relation (of j 0) zero\n| neg : Π (j) (x : F.obj j), relation (of j (-x)) (neg (of j x))\n| add : Π (j) (x y : F.obj j), relation (of j (x + y)) (add (of j x) (of j y))\n-- Then one relation per argument of each operation\n| neg_1 : Π (x x') (r : relation x x'), relation (neg x) (neg x')\n| add_1 : Π (x x' y) (r : relation x x'), relation (add x y) (add x' y)\n| add_2 : Π (x y y') (r : relation y y'), relation (add x y) (add x y')\n-- And one relation per axiom\n| zero_add      : Π (x), relation (add zero x) x\n| add_zero      : Π (x), relation (add x zero) x\n| add_left_neg  : Π (x), relation (add (neg x) x) zero\n| add_comm      : Π (x y), relation (add x y) (add y x)\n| add_assoc     : Π (x y z), relation (add (add x y) z) (add x (add y z))\n\n/--\nThe setoid corresponding to group expressions modulo abelian group relations and identifications.\n-/\ndef colimit_setoid : setoid (prequotient F) :=\n{ r := relation F, iseqv := ⟨relation.refl, relation.symm, relation.trans⟩ }\nattribute [instance] colimit_setoid\n\n/--\nThe underlying type of the colimit of a diagram in `AddCommGroup`.\n-/\n@[derive inhabited]\ndef colimit_type : Type v := quotient (colimit_setoid F)\n\ninstance : add_comm_group (colimit_type F) :=\n{ zero :=\n  begin\n    exact quot.mk _ zero\n  end,\n  neg :=\n  begin\n    fapply @quot.lift,\n    { intro x,\n      exact quot.mk _ (neg x) },\n    { intros x x' r,\n      apply quot.sound,\n      exact relation.neg_1 _ _ r },\n  end,\n  add :=\n  begin\n    fapply @quot.lift _ _ ((colimit_type F) → (colimit_type F)),\n    { intro x,\n      fapply @quot.lift,\n      { intro y,\n        exact quot.mk _ (add x y) },\n      { intros y y' r,\n        apply quot.sound,\n        exact relation.add_2 _ _ _ r } },\n    { intros x x' r,\n      funext y,\n      induction y,\n      dsimp,\n      apply quot.sound,\n      { exact relation.add_1 _ _ _ r },\n      { refl } },\n  end,\n  zero_add := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.zero_add,\n    refl,\n  end,\n  add_zero := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_zero,\n    refl,\n  end,\n  add_left_neg := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_left_neg,\n    refl,\n  end,\n  add_comm := λ x y,\n  begin\n    induction x,\n    induction y,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_comm,\n    refl,\n    refl,\n  end,\n  add_assoc := λ x y z,\n  begin\n    induction x,\n    induction y,\n    induction z,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_assoc,\n    refl,\n    refl,\n    refl,\n  end, }\n\n@[simp] lemma quot_zero : quot.mk setoid.r zero = (0 : colimit_type F) := rfl\n@[simp] lemma quot_neg (x) :\n  quot.mk setoid.r (neg x) = (-(quot.mk setoid.r x) : colimit_type F) := rfl\n@[simp] lemma quot_add (x y) :\n  quot.mk setoid.r (add x y) = ((quot.mk setoid.r x) + (quot.mk setoid.r y) : colimit_type F) := rfl\n\n/-- The bundled abelian group giving the colimit of a diagram. -/\ndef colimit : AddCommGroup := AddCommGroup.of (colimit_type F)\n\n/-- The function from a given abelian group in the diagram to the colimit abelian group. -/\ndef cocone_fun (j : J) (x : F.obj j) : colimit_type F :=\nquot.mk _ (of j x)\n\n/-- The group homomorphism from a given abelian group in the diagram to the colimit abelian\ngroup. -/\ndef cocone_morphism (j : J) : F.obj j ⟶ colimit F :=\n{ to_fun := cocone_fun F j,\n  map_zero' := by apply quot.sound; apply relation.zero,\n  map_add' := by intros; apply quot.sound; apply relation.add }\n\n@[simp] lemma cocone_naturality {j j' : J} (f : j ⟶ j') :\n  F.map f ≫ (cocone_morphism F j') = cocone_morphism F j :=\nbegin\n  ext,\n  apply quot.sound,\n  apply relation.map,\nend\n\n@[simp] lemma cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j):\n  (cocone_morphism F j') (F.map f x) = (cocone_morphism F j) x :=\nby { rw ←cocone_naturality F f, refl }\n\n/-- The cocone over the proposed colimit abelian group. -/\ndef colimit_cocone : cocone F :=\n{ X := colimit F,\n  ι :=\n  { app := cocone_morphism F } }.\n\n/-- The function from the free abelian group on the diagram to the cone point of any other\ncocone. -/\n@[simp] def desc_fun_lift (s : cocone F) : prequotient F → s.X\n| (of j x)  := (s.ι.app j) x\n| zero      := 0\n| (neg x)   := -(desc_fun_lift x)\n| (add x y) := desc_fun_lift x + desc_fun_lift y\n\n/-- The function from the colimit abelian group to the cone point of any other cocone. -/\ndef desc_fun (s : cocone F) : colimit_type F → s.X :=\nbegin\n  fapply quot.lift,\n  { exact desc_fun_lift F s },\n  { intros x y r,\n    induction r; try { dsimp },\n    -- refl\n    { refl },\n    -- symm\n    { exact r_ih.symm },\n    -- trans\n    { exact eq.trans r_ih_h r_ih_k },\n    -- map\n    { simp, },\n    -- zero\n    { simp, },\n    -- neg\n    { simp, },\n    -- add\n    { simp, },\n    -- neg_1\n    { rw r_ih, },\n    -- add_1\n    { rw r_ih, },\n    -- add_2\n    { rw r_ih, },\n    -- zero_add\n    { rw zero_add, },\n    -- add_zero\n    { rw add_zero, },\n    -- add_left_neg\n    { rw add_left_neg, },\n    -- add_comm\n    { rw add_comm, },\n    -- add_assoc\n    { rw add_assoc, },\n  }\nend\n\n/-- The group homomorphism from the colimit abelian group to the cone point of any other cocone. -/\ndef desc_morphism (s : cocone F) : colimit F ⟶ s.X :=\n{ to_fun := desc_fun F s,\n  map_zero' := rfl,\n  map_add' := λ x y, by { induction x; induction y; refl }, }\n\n/-- Evidence that the proposed colimit is the colimit. -/\ndef colimit_cocone_is_colimit : is_colimit (colimit_cocone F) :=\n{ desc := λ s, desc_morphism F s,\n  uniq' := λ s m w,\n  begin\n    ext,\n    induction x,\n    induction x,\n    { have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,\n      erw w',\n      refl, },\n    { simp *, },\n    { simp *, },\n    { simp *, },\n    refl\n  end }.\n\ninstance has_colimits_AddCommGroup : has_colimits AddCommGroup :=\n{ has_colimits_of_shape := λ J 𝒥, by exactI\n  { has_colimit := λ F, has_colimit.mk\n    { cocone := colimit_cocone F,\n      is_colimit := colimit_cocone_is_colimit F } } }\n\nend AddCommGroup.colimits\n\nnamespace AddCommGroup\n\nopen quotient_add_group\n\n/--\nThe categorical cokernel of a morphism in `AddCommGroup`\nagrees with the usual group-theoretical quotient.\n-/\nnoncomputable def cokernel_iso_quotient {G H : AddCommGroup} (f : G ⟶ H) :\n  cokernel f ≅ AddCommGroup.of (quotient (add_monoid_hom.range f)) :=\n{ hom := cokernel.desc f (mk' _)\n    (by { ext, apply quotient.sound, fsplit, exact -x,\n          simp only [add_zero, add_monoid_hom.map_neg], }),\n  inv := add_monoid_hom.of (quotient_add_group.lift _ (cokernel.π f)\n    (by { intros x H_1, cases H_1, induction H_1_h,\n          simp only [cokernel.condition_apply, zero_apply]})),\n  -- obviously can take care of the next goals, but it is really slow\n  hom_inv_id' := begin\n    ext1, simp only [coequalizer_as_cokernel, category.comp_id, cokernel.π_desc_assoc], ext1, refl,\n  end,\n  inv_hom_id' := begin\n    ext1, induction x,\n    { simp only [colimit.ι_desc_apply, coe_id, add_monoid_hom.coe_of, lift_quot_mk,\n                 cofork.of_π_ι_app, coe_comp], refl },\n    { refl }\n  end, }\n\nend AddCommGroup\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/algebra/category/Group/colimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3290701447142662}}
{"text": "example (h : ∀ x, x + 1 = x.succ) : (x + 1) + 1 = 1 + (x + 1) := by\n  rewrite [h]\n  done\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/550.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.32907014471426616}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\nimport measure_theory.set_integral\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.core\nimport formal_ml.measurable_space\nimport formal_ml.semiring\nimport formal_ml.real_measurable_space\nimport formal_ml.set\nimport formal_ml.filter_util\nimport topology.instances.ennreal\nimport formal_ml.int\nimport formal_ml.with_density_compose_eq_multiply\nimport formal_ml.hahn\nimport tactic.cache\n\n---------------------------------------------------------------------------------------------------\n\nlemma measurable_supr_of_measurable {Ω:Type*} [M:measurable_space Ω]\n    {h:ℕ → Ω → ennreal}:\n    (∀ n:ℕ, measurable (h n)) →\n    (measurable (supr h)) :=\nbegin\n  intros A1,\n  apply is_ennreal_measurable_intro_Ioi,\n  intro x,\n  have A3:((supr h) ⁻¹' {y : ennreal | x < y}) =⋃ (n:ℕ), (h n) ⁻¹' {y:ennreal | x < y},\n  {\n    simp,\n    ext ω,\n      have A3B:supr h ω = supr (λ n, h n ω),\n      {\n        apply supr_apply,\n      },\n    split;\n    intros A3A;simp;simp at A3A,\n    {\n      rw A3B at A3A,\n      rw @lt_supr_iff ennreal _ at A3A,\n      apply A3A,\n    },\n    {\n      cases A3A with i A3A,\n      apply lt_of_lt_of_le A3A,\n      rw A3B,\n      apply @le_supr ennreal ℕ _,\n    },\n  },    \n  rw A3,\n  apply measurable_set.Union,\n  intro b,\n  apply A1 b,\n  apply is_ennreal_measurable_set_intro_Ioi,\nend \n\nlemma monotone_set_indicator {Ω β:Type*} [has_zero β] [preorder β] {S:set Ω}:\n    monotone ((set.indicator S):(Ω → β) → (Ω → β)) :=\nbegin\n  unfold monotone,\n  intros f g A1,\n  rw le_func_def2,\n  intro ω,\n  cases (classical.em (ω ∈ S)) with A2 A2,\n  {\n    rw set.indicator_of_mem A2,\n    rw set.indicator_of_mem A2,\n    apply A1,\n  },\n  {\n    rw set.indicator_of_not_mem A2,\n    rw set.indicator_of_not_mem A2,\n  },\nend\n\nlemma supr_with_density_apply_eq_with_density_supr_apply {Ω:Type*} [measurable_space Ω] {μ:measure_theory.measure Ω}\n    {h:ℕ → Ω → ennreal} {S:set Ω}:\n    measurable_set S →\n    (∀ n:ℕ, measurable (h n)) →\n    (monotone h) →\n    supr (λ n:ℕ, μ.with_density (h n) S) = μ.with_density (supr h) S :=\nbegin\n  intros A1 A2 A3,\n  have A4:(λ n, μ.with_density (h n) S) = (λ n,  ∫⁻ (ω:Ω), (set.indicator S (h n)) ω ∂ μ),\n  {\n    apply funext,\n    intro n,\n    rw measure_theory.with_density_apply2',\n    apply A1,\n  },\n  rw A4,\n  clear A4,\n  rw measure_theory.with_density_apply2',\n  rw supr_indicator,\n  rw measure_theory.lintegral_supr2,\n  {\n    intro n,\n    apply measurable.indicator,\n    apply A2 n,\n    apply A1,\n  },\n  {\n    have B1:(λ (n:ℕ), set.indicator S (h n)) = (set.indicator S) ∘ h := rfl,\n    rw B1,\n    apply @monotone.comp ℕ (Ω → ennreal) (Ω → ennreal) _ _ _,\n    apply @monotone_set_indicator Ω ennreal _ _,\n    apply A3,\n  },\n  {\n    apply A1,\n  },\nend\n\n--TODO: replace with measure_theory.measure.smul_apply\nlemma ennreal_smul_measure_apply {α:Type*}\n    [measurable_space α] (x:ennreal) \n    (μ:measure_theory.measure α)\n    (s:set α) (H:measurable_set s):\n    (x  • μ) s = x * (μ s) :=\nbegin\n  rw measure_theory.measure.smul_apply,\nend\n\ndef measure_theory.measure.is_support {α:Type*} [measurable_space α]\n    (μ:measure_theory.measure α) (S:set α):Prop := measurable_set S ∧ μ (Sᶜ) = 0\n\nlemma outer_measure_measure_of_le {Ω:Type*} {μ ν:measure_theory.outer_measure Ω}:\n    μ ≤ ν ↔\n    (μ.measure_of) ≤\n    (ν.measure_of) :=\nbegin\n  refl,\nend\n\nlemma measure_theory.measure.Union_nat {α : Type*} [M : measurable_space α] \n    {μ:measure_theory.measure α} {f:ℕ → set α}:μ (⋃ i, f i) ≤ ∑' i, μ (f i) :=\nbegin\n  rw measure.apply,\n  have A1:(λ i, μ (f i))=(λ i, μ.to_outer_measure.measure_of (f i)) := rfl,\n  rw A1,\n  apply measure_theory.outer_measure.Union_nat,\nend\n\n/-\n  This theorem is immediately useful to prove the\n  existence of the Radon-Nikodym derivative, if \n  α = measure_theory.measure Ω, and g = (λ μ, μ T)\n  (see Sup_apply_eq_supr_apply_of_closed). \n  However, if α = set Ω, and g = μ, then this\n  can be used to prove the Hahn decomposition variant.\n  The critical proof is that supr (μ T) =\n  μ (supr T).\n -/\nlemma Sup_apply_eq_supr_apply_of_closed' {α:Type*}\n  [complete_lattice α] {S:set α} (g:α → ennreal):\n  (∀ (a∈ S) (b∈ S), a ≤ b → g a ≤ g b) →\n  (∀ f:ℕ → α, (set.range f ⊆ S) → \n               monotone f → supr (g ∘ f) = g (supr f)) →\n  (S.nonempty) →\n  (∀ a ∈ S, ∀ b ∈ S, a ⊔ b ∈ S)→\n  (∃ f:ℕ → α,\n            (∀ n, f n ∈ S) ∧ \n            (monotone f) ∧\n            g (supr f) = Sup (g '' S)) :=\nbegin\n  intros A1 AX A2 A3,\n  have B1:(g '' S).nonempty,\n  {\n    apply set.nonempty_image_iff.mpr A2,\n  },\n  have B1X := ennreal.Sup_eq_supr B1,\n  cases B1X with f' B1X,\n  have B2:∃ f'':ℕ → α, ∀ n:ℕ, \n          (f'' n)∈ S ∧ g (f'' n) = f' n, \n  {\n    apply @classical.some_func ℕ α (λ (n:ℕ) (a:α), \n        a∈ S ∧ g a = f' n),\n    intro n,\n    have B2A:=(B1X.left) n,\n    simp at B2A,\n    cases B2A with a B2A,\n    apply exists.intro a,\n    simp,\n    apply B2A,\n  },\n  cases B2 with f'' B2,\n  have C1:∀ (n : ℕ), Sup_so_far f'' n ∈ S,\n  {\n    apply Sup_so_far_of_closed,\n    intro n,\n    apply (B2 n).left,\n    apply A3,  \n  },\n  apply exists.intro (Sup_so_far f''),\n  split,\n  {\n    apply C1,\n  },\n  split,\n  {\n    apply monotone_Sup_so_far,\n  },\n  {\n    rw ← AX,\n    apply le_antisymm,\n    {\n      apply @supr_le ennreal _ _,\n      intro n,\n      apply @le_Sup ennreal _ _,\n      simp,\n      apply exists.intro (Sup_so_far f'' n),\n      apply and.intro (C1 n),\n      refl,   \n    },\n    {\n      rw ← B1X.right,\n      apply @supr_le_supr ennreal _ _,\n      intros n,\n      rw ← (B2 n).right,\n      simp,\n      apply A1,\n      apply (B2 n).left,\n      apply C1 n,\n      apply le_Sup_so_far,\n    },\n    {\n      rw set.subset_def,\n      intros x C2,\n      simp at C2,\n      cases C2 with n C2,\n      rw ← C2,\n      apply C1,\n    },\n    apply monotone_Sup_so_far,\n  },\nend\n\n\nlemma measurable_sup {Ω:Type*} {M:measurable_space Ω} \n  {f g:Ω → ennreal}:measurable f → measurable g → \n    measurable (f ⊔ g) :=\nbegin\n  intros A1 A2,\n  /- Proof sketch:\n     x is measurable iff if ∀ a, x⁻¹ (a,⊤] is measurable.\n     (f ⊔ g)⁻¹ (a,⊤] =f⁻¹ (a,⊤]∪g⁻¹ (a,⊤].\n     Since the union of two measurable functions is measurable,\n     we are done.\n   -/ \n  apply is_ennreal_measurable_intro_Ioi,\n  intro a,\n  have A3:(f ⊔ g) ⁻¹' {y : ennreal | a < y}=\n      f ⁻¹' {y : ennreal | a < y}∪\n      g ⁻¹' {y : ennreal | a < y},\n  {\n    simp,\n    ext,\n    split;intros A3A;simp at A3A;simp;apply A3A,\n  },\n  rw A3,\n  apply measurable_set.union,\n  {\n    apply A1,\n    apply is_ennreal_measurable_set_intro_Ioi,\n  },\n  {\n    apply A2,\n    apply is_ennreal_measurable_set_intro_Ioi,\n  },\nend\n\nlemma with_density_le_with_density {Ω:Type*} {M:measurable_space Ω}\n  {μ:measure_theory.measure Ω} {x y:Ω → ennreal} \n  {S:set Ω}:\n  measurable_set S →\n  (∀ ω ∈ S, x ω ≤ y ω) →  \n  μ.with_density x S ≤ μ.with_density y S :=\nbegin\n  intros A3 A4,\n  rw measure_theory.with_density_apply2' μ x S A3,\n  rw measure_theory.with_density_apply2' μ y S A3,\n  apply measure_theory.lintegral_mono,\n\n  rw le_func_def2,\n  intros ω,\n  cases (classical.em (ω ∈ S)) with A5 A5,\n  {\n    rw set.indicator_of_mem A5,\n    rw set.indicator_of_mem A5,\n    apply A4 _ A5,\n  },\n  {\n    rw set.indicator_of_not_mem A5,\n    rw set.indicator_of_not_mem A5,\n    apply le_refl _,\n  },\nend\n\n\n/-\n  The Lebesgue-Radon-Nikodym theorem, while it delves deeply into the nuances of\n  measure theory, provides the foundation for statistics and probability. Specifically,\n  continuous random variables can be represented by a density function. The \n  Lebesgue-Radon-Nikodym theorem (and the Radon-Nikodym theorem) exactly characterize\n  what probability measures have this simple representation: specifically, those that\n  are absolutely continuous with respect to the Lebesgue measure. This theorem, like\n  the fundamental theorem of calculus, provides a deep insight that can be easily used\n  even by those who do not understand the nuances of the proof.\n -/\n\n\n---------------------------End theorems to move----------------------------------\n\n\nlemma simple_restrict_eq_indicator_const {Ω:Type*} {M:measurable_space Ω} \n  (S:set Ω) (x:ennreal):(measurable_set S) →\n  ⇑((measure_theory.simple_func.const Ω x).restrict S) = (set.indicator S (λ ω:Ω, x)) :=\nbegin\n  intro A1,\n  rw measure_theory.simple_func.coe_restrict,\n  rw measure_theory.simple_func.coe_const,\n  apply A1,\nend\n\nlemma with_density_const_apply {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {k:ennreal} {S:set Ω}:measurable_set S →\n   μ.with_density (λ ω:Ω, k) S = k * μ S :=\nbegin\n  intros B1,\n  rw measure_theory.with_density_apply2',\n  rw ← simple_restrict_eq_indicator_const,\n  rw measure_theory.simple_func.lintegral_eq_lintegral,\n  rw measure_theory.simple_func.restrict_const_lintegral,\n  repeat {apply B1},\nend\n\nlemma measure_theory.outer_measure.of_function_def2 \n  {α : Type*} (m : set α → ennreal) (m_empty : m ∅ = 0)  (s:set α):\n  (measure_theory.outer_measure.of_function m m_empty) s\n    = ⨅{f : ℕ → set α} (h : s ⊆ ⋃i, f i), ∑'i, m (f i)  := \nbegin\n  rw [measure_theory.outer_measure.of_function],\n  rw ← measure_theory.outer_measure.measure_of_eq_coe,\nend\n\n\nlemma set.Union_inter_eq_inter_Union {α\n   β:Type*} {f:α → set β} {T:set β}:\n    (⋃ (a:α), f a ∩ T) =  T ∩ (⋃ (a:α), f a) :=\nbegin\n  apply set.ext,\n  intro b,split;intros B1;simp;simp at B1;\n  apply and.intro B1.right B1.left,\nend\n\n\nlemma set.Union_union_eq_union_Union {α\n   β:Type*} [NE:nonempty α] {f:α → set β} {T:set β}:\n    (⋃ (a:α), f a ∪ T) =  T ∪ (⋃ (a:α), f a) :=\nbegin\n  apply set.ext,\n  intro b,split;intros B1;simp;simp at B1,\n  {\n    cases B1 with a B2,\n    cases B2 with B3 B4,\n    {\n       right,\n       apply exists.intro a B3,\n    },\n    {\n      left,\n      apply B4,\n    },\n  },\n  {\n    cases B1 with C1 C2,\n    {\n       apply nonempty.elim NE,\n       intro a,\n       apply exists.intro a,\n       right,\n       apply C1,\n    },\n    {\n       cases C2 with a C3,\n       apply exists.intro a,\n       left,\n       apply C3,\n    },\n  },\nend\n\nlemma set.subset_union_compl_of_inter_subset {α:Type*} {A B C:set α}:A ∩ B ⊆ C →\n    A ⊆ C ∪ Bᶜ :=\nbegin\n  intro D1,\n  rw set.subset_def,\n  intros x D2,\n  rw set.subset_def at D1,\n  simp,\n  cases (classical.em (x∈ B)) with D3 D3,\n  {\n    left,\n    apply D1,\n    simp [D2,D3],\n  },\n  {\n    right,\n    apply D3,\n  },\nend \n\n\n\n\n\n\n\nlemma measure_theory.outer_measure.to_measure_to_outer_measure_eq_trim {Ω:Type*} [M:measurable_space Ω]\n    {μ:measure_theory.outer_measure Ω} (H:M ≤ (μ).caratheodory):\n    (μ.to_measure H).to_outer_measure = μ.trim :=\nbegin\n  apply measure_theory.outer_measure.ext,\n  intros S,\n  unfold measure_theory.outer_measure.to_measure measure_theory.measure.to_outer_measure\n    measure_theory.outer_measure.trim\n    measure_theory.measure.of_measurable,\n  simp,\n  refl,\nend\n\n\n\n--Beautiful, but useless.\nlemma measure_theory.extend_le_extend_of_le {α:Type*} [measurable_space α]\n {f g : Π (s : set α), (measurable_set s)  → ennreal} (h_le_meas : ∀ (s : set α) (h:measurable_set s), f s h ≤ g s h) : \n  (measure_theory.extend f ≤ measure_theory.extend g) :=\nbegin\n  intros s,\n  cases (classical.em (measurable_set s)) with h_is_meas h_is_not_meas,\n  { rw (measure_theory.extend_eq f h_is_meas),\n    apply le_trans (h_le_meas s h_is_meas) (measure_theory.le_extend g h_is_meas) },\n  { unfold measure_theory.extend,\n    apply @le_infi ennreal _ _,\n    intros contra,\n    exfalso, apply h_is_not_meas contra },\nend \n\n\n--@[simp]\nlemma simple_logic {A B C:Prop}:(A ∨ B → C → A) ↔  (B → C → A) :=\nbegin\n  split, intros h_A_of_C_of_A_or_B h_B h_C,\n  apply h_A_of_C_of_A_or_B (or.inr h_B) h_C,\n  intros h_A_of_B_of_C h_A_or_B h_C,\n  cases (h_A_or_B) with h_A h_B,\n  apply h_A,\n  apply h_A_of_B_of_C h_B h_C,\nend\n\n--@[simp]\nlemma contra_logic' {A B:Prop}:(¬A → (A) → B) ↔  true :=\nbegin\n  simp,\n  intros h_not_A h_A,\n  exfalso, apply h_not_A h_A,\nend\n\nlemma measure_theory.measure.add_compl_inter {Ω:Type*} [measurable_space Ω]\n  (μ:measure_theory.measure Ω) (S T:set Ω):(measurable_set S) → \n  (measurable_set T) →\n  (μ T = μ (S ∩ T) + μ (Sᶜ ∩ T)) :=\nbegin\n  intros A1 A2,\n  have A3:T = (S∩ T) ∪ (Sᶜ ∩ T),\n  {\n    rw ← set.inter_distrib_right,\n    rw set.union_compl_self,\n    simp,\n  },\n  have A4:μ T =  μ ( (S∩ T) ∪ (Sᶜ ∩ T)),\n  {\n    rw ← A3,\n  },\n  rw A4,\n  rw measure_theory.measure_union,\n  rw set.inter_comm,\n  rw set.inter_comm _ T,\n  apply set.disjoint_inter_compl,\n  apply measurable_set.inter A1 A2,\n  apply measurable_set.inter (measurable_set.compl A1) A2,\nend\n\nlemma measure_theory.outer_measure.le_top_caratheodory {Ω:Type*} [M:measurable_space Ω]:\n  M ≤ (⊤:measure_theory.outer_measure Ω).caratheodory :=\nbegin\n  rw measure_theory.outer_measure.top_caratheodory,\n  simp\nend\n\nlemma measure_theory.measure.of_measurable_apply' {α:Type*} [M:measurable_space α]\n(m : Π (s : set α), measurable_set s → ennreal)\n  (m0 : m ∅ measurable_set.empty = 0)\n  (mU : ∀ {{f : ℕ → set α}} (h : ∀i, measurable_set (f i)),\n    pairwise (disjoint on f) →\n    m (⋃i, f i) (measurable_set.Union h) = (∑'i, m (f i) (h i))) (S:set α):\n  measure_theory.measure.of_measurable m m0 mU S = \n  measure_theory.induced_outer_measure m _ m0 S :=\nbegin\n  unfold measure_theory.measure.of_measurable,\n  simp,\n  rw measure.apply, \nend\n\nlemma measure_theory.outer_measure.top_eq {Ω:Type*} [M:measurable_space Ω]:\n    ⇑(⊤:measure_theory.outer_measure Ω) = (λ (s:set Ω), (@ite (s=∅) (classical.prop_decidable (s=∅)) ennreal 0 ⊤)) :=\nbegin\n  apply funext,\n  intro S,\n  cases (classical.em (S = ∅)) with B1 B1,\n  {\n    rw if_pos,\n    subst S,\n    apply measure_theory.outer_measure.empty',\n    apply B1,\n  },\n  {\n    rw if_neg,\n    apply measure_theory.outer_measure.top_apply,\n    rw ← set.ne_empty_iff_nonempty,\n    apply B1,\n    apply B1,\n  },\nend\n\nlemma measure_theory.outer_measure.extend_top {Ω:Type*} [M:measurable_space Ω]:\n  (measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), (⊤:measure_theory.outer_measure Ω) s))=(λ (s:set Ω), (@ite (s=∅) (classical.prop_decidable (s=∅)) ennreal 0 ⊤)) :=\nbegin\n  apply funext,\n  intro S,\n  rw measure_theory.outer_measure.top_eq,\n  cases (classical.em (measurable_set S)) with B1 B1,\n  {\n    unfold measure_theory.extend,\n    rw infi_prop_def,\n    apply B1,\n  },\n  {\n    unfold measure_theory.extend,\n    rw infi_prop_false,\n    rw if_neg,\n    intros B2,\n    apply B1,\n    subst S,\n    simp,\n    apply B1,\n  },\nend\n\nlemma measure_theory.outer_measure.extend_top' {Ω:Type*} [M:measurable_space Ω]:\n  (measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), (⊤:measure_theory.outer_measure Ω) s))=(⊤:measure_theory.outer_measure Ω) :=\nbegin\n  rw measure_theory.outer_measure.extend_top,\n  rw measure_theory.outer_measure.top_eq,\nend\n\n\n--Checked requirements to here.\nlemma subst_apply_empty_zero {Ω:Type*} {f g:set Ω → ennreal}:(f = g) → (f ∅ = 0) → (g ∅ = 0) :=\nbegin\n  intros A1 A2,\n  subst f,\n  apply A2,\nend\n\nlemma measure_theory.outer_measure.of_function_subst {Ω:Type*} [M:measurable_space Ω]\n   {f g:set Ω → ennreal} {P:f ∅ = 0} (H:f = g):\n  measure_theory.outer_measure.of_function f P =\n  measure_theory.outer_measure.of_function g (subst_apply_empty_zero H P) :=\nbegin\n  subst g,\nend\n\n\nlemma measure_theory.outer_measure.of_function_eq' {Ω:Type*} [M:measurable_space Ω]\n  {μ:measure_theory.outer_measure Ω} (H:μ ∅ = 0): \n  measure_theory.outer_measure.of_function (⇑μ) H = μ :=\nbegin\n  apply measure_theory.outer_measure.ext,\n  intro S,\n  apply measure_theory.outer_measure.of_function_eq,\n  {\n    intros T B1,\n    apply measure_theory.outer_measure.mono',\n    apply B1, \n  },\n  {\n    intros f,\n    apply measure_theory.outer_measure.Union_nat,\n  },\nend\n\n\nlemma measure_theory.outer_measure.top_eq_trim {Ω:Type*} [M:measurable_space Ω]:\n  (⊤:measure_theory.outer_measure Ω).trim = ⊤ :=\nbegin\n  unfold measure_theory.outer_measure.trim,\n  unfold measure_theory.induced_outer_measure,\n  have B1:= @measure_theory.outer_measure.extend_top' Ω M,\n  rw measure_theory.outer_measure.of_function_subst B1,  \n  rw measure_theory.outer_measure.of_function_eq',\nend\n\n\nlemma measure_theory.outer_measure.top_to_measure_to_outer_measure_eq_top {Ω:Type*} [M:measurable_space Ω]:\n  ((⊤:measure_theory.outer_measure Ω).to_measure measure_theory.outer_measure.le_top_caratheodory).to_outer_measure  = ⊤ :=\nbegin\n  apply measure_theory.outer_measure.ext,\n  intro S,\n  unfold measure_theory.outer_measure.to_measure,\n  simp,\n  rw measure_theory.measure.of_measurable_apply',\n  unfold measure_theory.induced_outer_measure,\n  have B1:= @measure_theory.outer_measure.extend_top' Ω M,\n  rw measure_theory.outer_measure.of_function_subst B1,  \n  rw measure_theory.outer_measure.of_function_eq',\nend\n\n\n/--\n  One could extract many monotone relationships from this:\n  induced_outer_measure, extend, of_function, et cetera.\n  However, I wouldn't be surprised to find them in the library.\n -/\nlemma measure_theory.outer_measure.trim_monotone {Ω:Type*} [M:measurable_space Ω]\n  (μ ν:measure_theory.outer_measure Ω):μ ≤ ν → μ.trim ≤ ν.trim :=\nbegin\n  intros A1,\n  rw outer_measure_measure_of_le,\n  unfold measure_theory.outer_measure.trim,\n  unfold measure_theory.induced_outer_measure,\n  unfold measure_theory.outer_measure.of_function,\n  intros S,\n  simp,\n  intros f B1,\n  have B2:(∑' (i : ℕ), measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), μ s) (f i)) ≤\n    ∑' (i : ℕ), measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), ν s) (f i),\n  {\n    apply ennreal.tsum_le_tsum,\n    unfold measure_theory.extend,\n    intros n,\n    simp,\n    intros B2A,\n    rw infi_prop_def,\n    apply A1,\n    apply B2A,\n  },\n  apply le_trans _ B2,\n  have B3:(⨅ (h : S ⊆ ⋃ (i : ℕ), f i),(∑' (i : ℕ), measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), μ s) (f i))) = ∑' (i : ℕ), measure_theory.extend (λ (s : set Ω) (_x : measurable_set s), μ s) (f i),\n  {\n    rw infi_prop_def,\n    apply B1,\n  }, \n  rw ← B3,\n  apply @infi_le ennreal _ _,\nend\n\nlemma measure_theory.measure.to_outer_measure_eq_top {Ω:Type*} [M:measurable_space Ω]:\n   (⊤:measure_theory.measure Ω).to_outer_measure = ⊤ := \nbegin\n  rw ← measure_theory.measure.trimmed,\n  rw ← @top_le_iff (measure_theory.outer_measure Ω) _,\n  have B1:((⊤:measure_theory.outer_measure Ω).to_measure measure_theory.outer_measure.le_top_caratheodory).to_outer_measure.trim ≤ (⊤:measure_theory.measure Ω).to_outer_measure.trim,\n  {\n    apply measure_theory.outer_measure.trim_monotone,\n    apply measure_theory.measure.to_outer_measure_le.mpr,\n    simp\n  }, \n  rw measure_theory.outer_measure.top_to_measure_to_outer_measure_eq_top at B1,\n  rw measure_theory.outer_measure.top_eq_trim at B1,\n  apply B1,\nend\n\n\nlemma measure_theory.measure.top_apply {Ω:Type*} [M:measurable_space Ω]\n   {S:set Ω}:S.nonempty → (⊤:measure_theory.measure Ω)(S) = (⊤:ennreal) :=\nbegin\n  intro A1,\n  rw measure.apply,\n  rw measure_theory.measure.to_outer_measure_eq_top,\n  simp,\n  rw measure_theory.outer_measure.top_apply A1,\nend\n\nlemma measure_theory.measure.le_add {Ω:Type*} [M:measurable_space Ω] {μ ν:measure_theory.measure Ω}:μ ≤ μ + ν :=\nbegin\n  intros S B1,\n  simp [le_add_nonnegative],\nend\n\n\n--TODO: Move to hahn.lean (might be useful).\nlemma inter_le_inter_of_restrict_le_restrict {Ω:Type*} [M:measurable_space Ω] \n    {μ ν:measure_theory.measure Ω} {T U:set Ω}:μ.restrict T ≤ ν.restrict T → \n    measurable_set T → measurable_set U → μ (T ∩ U) ≤ ν (T ∩ U) :=\nbegin\n  intros A3 A1 A2,\n  have B1:T ∩ U ⊆ T,\n  {simp},\n  apply @le_of_subset_of_restrict_le_restrict Ω M μ ν T (T ∩ U) B1 A3,\n  repeat {simp [A1,A2]},\nend\n\n--This may be gotten by easier methods.\nlemma measure_theory.measure.sub_le_sub {Ω:Type*} [measurable_space Ω] \n    (μ ν:measure_theory.measure Ω) (T:set Ω) [A1:measure_theory.finite_measure μ]:\n    measurable_set T → (ν T - μ T) ≤ (ν - μ) T :=\nbegin\n  intros A2,\n  have B1 := hahn_unsigned_inequality_decomp' ν μ,\n  cases B1 with U B1,\n  cases B1 with C1 B1,\n  rw jordan_decomposition_nonneg_sub μ ν Uᶜ,\n  {\n    rw measure_theory.measure.add_compl_inter ν _ _  C1 A2,\n    rw measure_theory.measure.add_compl_inter μ _ _  C1 A2,\n    rw ennreal.sub_le_iff_le_add,\n    rw add_comm (μ (U ∩ T)) (μ (Uᶜ ∩ T)),\n    rw ← add_assoc,\n    have C4:ν (Uᶜ ∩ T) ≤ ν (Uᶜ ∩ T) - μ (Uᶜ ∩ T) + μ (Uᶜ ∩ T),\n    {\n      apply ennreal.le_sub_add_self,\n    },\n    have D1 :  ν (T ∩ Uᶜ) - μ (T ∩ Uᶜ) = ν (Uᶜ ∩ T) - μ (Uᶜ ∩ T),\n    { rw set.inter_comm }, \n    have C5 := add_le_add_right C4 (μ (U ∩ T)),\n    rw D1,\n    apply le_trans _ C5,\n    rw add_comm,\n    apply add_le_add_left _ _,\n    apply inter_le_inter_of_restrict_le_restrict B1.left C1 A2,\n  },\n  repeat {simp [B1,A2,C1]},\nend\n\n--This is a natural break between subtraction and Radon-Nikodym.\nlemma measure_theory.measure.is_support_def {α:Type*} [measurable_space α]\n    (μ:measure_theory.measure α) (S:set α):\n    μ.is_support S = (measurable_set S ∧ μ (Sᶜ) = 0) := rfl\n\ndef measure_theory.measure.perpendicular {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α):Prop :=\n    (∃ S T:set α, μ.is_support S ∧ ν.is_support T ∧  \n    disjoint S T)\n\n\nlemma measure_theory.measure.perpendicular_def {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α):μ.perpendicular ν =\n    (∃ S T:set α, μ.is_support S ∧ ν.is_support T ∧  \n    disjoint S T) := rfl\n\n\nlemma measure_theory.measure.perpendicular_def2 {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α):μ.perpendicular ν ↔\n    (∃ S:set α,  measurable_set S ∧ μ S = 0 ∧  ν (Sᶜ) = 0) :=\nbegin\n  rw measure_theory.measure.perpendicular_def,\n  split;intros B1,\n  {\n    cases B1 with S B1,\n    cases B1 with T B1,\n    cases B1 with B1 B2,\n    cases B2 with B2 B3,\n    rw measure_theory.measure.is_support_def at B1,\n    rw measure_theory.measure.is_support_def at B2,\n\n    apply exists.intro T,\n    split,\n    {\n      apply B2.left,      \n    },\n    split,\n    {\n      cases B1 with C1 C2,\n      rw ← nonpos_iff_eq_zero,\n      rw ← nonpos_iff_eq_zero at C2,\n      apply le_trans _ C2,\n      apply measure_theory.measure_mono,\n      rw set.disjoint_iff_inter_eq_empty at B3,\n      rw set.inter_comm at B3,\n      rw ← set.subset_compl_iff_disjoint at B3,\n      apply B3,\n    },\n    {\n      apply B2.right,\n    },\n  },\n  {\n    cases B1 with S B1,\n    apply exists.intro Sᶜ,\n    apply exists.intro S,\n    split,\n    {\n      rw measure_theory.measure.is_support_def,\n      apply and.intro (measurable_set.compl (B1.left)),\n      simp,\n      apply B1.right.left,\n    },\n    split,\n    {\n      rw measure_theory.measure.is_support_def,\n      apply and.intro (B1.left) (B1.right.right),\n    },\n    {\n      rw set.disjoint_iff_inter_eq_empty,\n      rw ← set.subset_compl_iff_disjoint,\n      apply set.subset.refl Sᶜ,\n    },\n  },\nend\n\n\nlemma measure_theory.measure.perpendicular_intro {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α) {S:set α}:measurable_set S → \n    μ S = 0 → ν (Sᶜ) = 0 →\nμ.perpendicular ν :=\nbegin\n  intros A1 A2 A3,\n  rw measure_theory.measure.perpendicular_def2,\n  apply exists.intro S (and.intro A1 (and.intro A2 A3)),\nend\n\nlemma measure_theory.measure.not_perpendicular {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α) {S:set α}:(¬(μ.perpendicular ν)) → measurable_set S → \n    μ S = 0 → 0 < ν (Sᶜ)  :=\nbegin\n  intros A1 A2 A3,\n  rw pos_iff_ne_zero,\n  intros A4,\n  apply A1,\n  apply measure_theory.measure.perpendicular_intro μ ν A2 A3 A4,\nend\n\n\nlemma measure_theory.measure.perpendicular_symmetric' {α:Type*} [measurable_space α]\n    (μ ν:measure_theory.measure α):(μ.perpendicular ν) →\n    (ν.perpendicular μ) :=\nbegin\n  \n  intro A1,\n  rw measure_theory.measure.perpendicular_def2,\n  rw measure_theory.measure.perpendicular_def2 at A1,\n  cases A1 with S A1,\n  apply exists.intro Sᶜ,\n  apply and.intro (measurable_set.compl A1.left),\n  apply and.intro A1.right.right,\n  simp,\n  apply A1.right.left,\nend\n\n--NOTE: everything below here is required for Radon-Nikodym, although it might not\n--be unique.\n\n\n--This looks familiar: I think that there was a similar, but longer proof.\nlemma measure_theory.measure.sub_le {α:Type*}\n    [measurable_space α] {μ ν:measure_theory.measure α}:μ - ν ≤ μ :=\nbegin\n  rw measure_theory.measure.sub_def,\n  apply @Inf_le (measure_theory.measure α) _ _,\n  simp,\n  apply measure_theory.measure.le_add,\nend\n\n--TODO:replace with `measure_theory.measure.smul_finite`\nlemma measure_theory.measure.smul_finite' {α:Type*} [measurable_space α]\n   {μ:measure_theory.measure α} {ε:ennreal} [measure_theory.finite_measure μ]:\n   ε ≠ ⊤ → (measure_theory.finite_measure (ε• μ)) :=\nbegin  \n  intros A1,\n  apply measure_theory.measure.smul_finite,\n  rw lt_top_iff_ne_top,\n  apply A1,\nend \n\nlemma set.compl_Union_eq_Inter_compl {α β:Type*} {f:α → set β}:(⋃ n, f n)ᶜ = (⋂ n, (f n)ᶜ) :=\nbegin\n  ext b,\n  split;intros A1;simp;simp at A1;apply A1,\nend\n\nlemma measure_theory.measure.perpendicular_of {α:Type*} [M:measurable_space α]\n   {μ ν:measure_theory.measure α} [A2:measure_theory.finite_measure μ]\n   [A3:measure_theory.finite_measure ν]: \n   (∀ ε:ennreal,  ε > 0 → μ.perpendicular (ν - (ε • μ)) ) → \n   μ.perpendicular ν  :=\nbegin\n  intros A1,\n  have B1:∀ n:ℕ,(∃ S:set α,  measurable_set S ∧ μ S = 0 ∧ \n          (ν - ((1/((n:ennreal) + 1))• μ)) (Sᶜ) = 0),\n  {\n    intros n,\n    have B1A:(1/((n:ennreal) + 1))>0,\n    {\n      apply ennreal.unit_frac_pos,\n    },\n    have B1B := A1 _ B1A,\n    rw measure_theory.measure.perpendicular_def2 at B1B,\n    apply B1B,\n  },\n  have B2 := classical.some_func B1,\n  cases B2 with f B2,\n  let T := ⋃ n, f n,\n  begin\n    have C1:T = ⋃ n, f n := rfl,\n    have C2:Tᶜ = ⋂ n, (f n)ᶜ,\n    {\n      rw C1,\n      rw set.compl_Union_eq_Inter_compl,\n    },\n    have C3:measurable_set T,\n    {\n      rw C1,\n      apply measurable_set.Union,\n      intro n,\n      apply (B2 n).left,\n    },\n    have C4:measurable_set Tᶜ,\n    {\n      apply measurable_set.compl C3,\n    },\n    have I1:∀ (n:ℕ), Tᶜ ⊆ (f n)ᶜ,\n    {\n      intro n,\n      rw C2,\n      apply @set.Inter_subset α ℕ (λ n, (f n)ᶜ),\n    },\n    have I2:∀ (n:ℕ), μ Tᶜ ≤ μ (f n)ᶜ,\n    {\n      intro n,      \n      apply @measure_theory.measure_mono α M μ _ _ (I1 n),\n    },\n       \n    apply @measure_theory.measure.perpendicular_intro α _ μ ν T,\n    {\n      apply measurable_set.Union,\n      intro n,\n      apply (B2 n).left,\n    },\n    {\n       rw C1,\n       rw ← nonpos_iff_eq_zero,\n       have D1:=@measure_theory.measure.Union_nat α _ μ f,\n       apply le_trans D1,\n       rw nonpos_iff_eq_zero,\n       have E1:(λ n, μ (f n)) = (λ (n:ℕ), (0:ennreal)),\n       {\n         apply funext,\n         intro n,\n         apply (B2 n).right.left,\n       },\n       rw E1,\n       simp,\n    },\n    {\n       --rw C2,       \n       have H1:ν (Tᶜ)/(μ (Tᶜ)) = 0,\n       {\n          apply ennreal.zero_of_le_all_unit_frac,\n          intro n,\n          apply ennreal.div_le_of_le_mul,\n          have H1X:measure_theory.finite_measure ((1 / ((n:ennreal) + 1)) • μ),\n          {\n            apply measure_theory.measure.smul_finite,\n            {\n              rw lt_top_iff_ne_top, \n              apply ennreal.unit_frac_ne_top,\n            },\n          },\n          --have H1B := H1A n,\n          have H1B:(ν) Tᶜ - ((1 / ((n:ennreal) + 1)) • μ) Tᶜ ≤ \n                   (ν - (1 / ((n:ennreal) + 1)) • μ) Tᶜ,\n          {\n            apply @measure_theory.measure.sub_le_sub α M ((1 / ((n:ennreal) + 1)) • μ) ν\n                  Tᶜ H1X C4,\n          },\n\n          have H1C:(ν) Tᶜ - ((1 / ((n:ennreal) + 1)) • μ) Tᶜ = 0, \n          --have H1B:(ν - (1 / ((n:ennreal) + 1)) • μ) Tᶜ = 0,\n          {\n            rw ← nonpos_iff_eq_zero,\n            apply le_trans H1B,\n            rw ← (B2 n).right.right,\n            apply measure_theory.measure_mono (I1 n), \n          },\n          rw ennreal_smul_measure_apply at H1C,\n          apply ennreal.le_of_sub_eq_zero,\n          apply H1C,\n          apply C4,\n       },\n       rw ennreal.div_eq_zero_iff at H1,\n       cases H1 with H1 H1,\n       {\n         apply H1,\n       },\n       {\n         exfalso,\n         apply measure_theory.measure_ne_top μ Tᶜ H1,\n       },\n    },\n  end\nend\n\n\nlemma measure_theory.measure.exists_of_not_perpendicular {α:Type*} [measurable_space α]\n   (μ:measure_theory.measure α) {ν:measure_theory.measure α} [A1:measure_theory.finite_measure μ] [A2:measure_theory.finite_measure ν]:\n   (¬ (μ.perpendicular ν)) → \n   (∃ ε:ennreal,  ε > 0  ∧  ¬μ.perpendicular (ν - (ε • μ)) ) :=\nbegin\n  intros A3,\n  apply classical.exists_of_not_forall_not,\n  intros B1,\n  apply A3,\n  apply measure_theory.measure.perpendicular_of,\n  intros ε C1,\n  have C2 := B1 ε,\n  simp at C2,\n  apply C2,\n  apply C1,\nend\n\n\n/- \n  This is taken from a step in Tao's proof of the Lebesgue-Radon-Nikodym Theorem.\n  By the Hahn Decomposition Theorem, we can find a set where μ ≤ ν and μ S > 0.\n -/\nlemma measure_theory.measure.perpendicular_sub_elim {α:Type*} [measurable_space α]\n    (μ:measure_theory.measure α) {ν:measure_theory.measure α} \n    [A2:measure_theory.finite_measure ν]: \n    ¬(μ.perpendicular (ν - μ)) → \n    ∃ (S:set α), measurable_set S ∧ μ.restrict S ≤ ν.restrict S  ∧  0 < μ S :=\nbegin\n  intros A3,\n  have B1:=hahn_unsigned_inequality_decomp' μ ν,\n  cases B1 with X B1,\n  have C1:measurable_set (Xᶜ),\n  {simp [B1]},\n  have B2 := measure_theory.measure.sub_apply_eq_zero_of_restrict_le_restrict \n    ν μ Xᶜ B1.right.right C1,\n  have B3 := B1.left,\n  have B4:¬((ν - μ).perpendicular μ),\n  {\n    intro B4A,\n    apply A3,\n    apply measure_theory.measure.perpendicular_symmetric',\n    apply B4A,\n  },\n  have B5 := measure_theory.measure.not_perpendicular (ν - μ) μ B4 C1 B2,\n  simp at B5,\n  apply exists.intro X,\n  simp [B1, B5],\nend\n\nlemma measure_theory.measure.perpendicular_smul {Ω:Type*} [N:measurable_space Ω] (μ ν:measure_theory.measure Ω) {k:ennreal}: 0 < k → \n  (k • μ).perpendicular ν → μ.perpendicular ν :=\nbegin\n  intros A1 A2,\n  rw measure_theory.measure.perpendicular_def2,\n  rw measure_theory.measure.perpendicular_def2 at A2,\n  cases A2 with S A2,\n  apply exists.intro S,\n  apply and.intro A2.left,\n  apply and.intro _ A2.right.right,\n  have B1 := A2.right.left,\n  rw ennreal_smul_measure_apply _ _ _ A2.left at B1,\n  simp at B1,\n  cases B1 with B1 B1,\n  {\n    exfalso,\n    rw pos_iff_ne_zero at A1,\n    apply A1,\n    apply B1,\n  },\n  {\n    apply B1,\n  },\nend\n\n\n\nlemma with_density_indicator_eq_restrict {Ω:Type*} [M:measurable_space Ω] \n    (μ:measure_theory.measure Ω) {S:set Ω} {f:Ω → ennreal}:\n    (measurable_set S) → \n    μ.with_density (set.indicator S f) = (μ.restrict S).with_density f :=\nbegin\n  intros A1, \n  apply measure_theory.measure.ext,\n  intros T B1,\n  rw measure_theory.with_density_apply2',\n  rw measure_theory.with_density_apply2',\n  rw ← measure_theory.lintegral_indicator,\n  rw set.indicator_indicator,\n  rw set.indicator_indicator,\n  rw set.inter_comm,\n  repeat {assumption},\nend\n\nlemma scalar_as_with_density {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {k:ennreal}:\n  (k•μ) = μ.with_density (λ ω:Ω, k) :=\nbegin\n  apply measure_theory.measure.ext,\n  intros S B1,\n  rw with_density_const_apply,\n  rw ennreal_smul_measure_apply,\n  apply B1,\n  apply B1,\nend\n\n\nlemma with_density_indicator_eq_restrict_smul {Ω:Type*} [M:measurable_space Ω] (μ:measure_theory.measure Ω) {S:set Ω} {k:ennreal}:(measurable_set S) → μ.with_density (set.indicator S (λ ω:Ω, k)) = k • μ.restrict S :=\nbegin\n  intro A1,\n  rw with_density_indicator_eq_restrict,\n  rw scalar_as_with_density,\n  apply A1,\nend\n\nlemma smul_restrict_comm {Ω:Type*} [M:measurable_space Ω] (μ:measure_theory.measure Ω) {S:set Ω} {k:ennreal}:(measurable_set S) → (k • μ).restrict S = k • μ.restrict S :=\nbegin\n  intros A1,\n  apply measure_theory.measure.ext,\n  intros T B1,\n  rw ennreal_smul_measure_apply _ _ _ B1,\n  rw measure_theory.measure.restrict_apply B1,\n  rw measure_theory.measure.restrict_apply B1,\n  rw ennreal_smul_measure_apply _ _ _ (measurable_set.inter B1 A1),\nend\n\n\n/-\n  In the full version of Lebesgue-Radon-Nikodym theorem, μ is an unsigned \n  σ-finite measure, and ν is a signed σ-finite measure.\n\n  The first stage of the proof focuses on finite, unsigned measures\n  (see lebesgue_radon_nikodym_unsigned_finite). In that proof,\n  there is a need to prove that if two measures are not perpendicular, then there\n  exists some nontrivial f where μ.with_density f set.univ > 0 and \n  μ.with_density f ≤ ν. In Tao's An Epsilon of Room, this is to show that\n  taking the f which maximizes μ.with_density f set.univ subject to\n  μ.with_density f ≤ ν, when subtracted from ν, leaves a measure perpendicular to μ.\n -/\nlemma lebesgue_radon_nikodym_junior {Ω:Type*} [N:measurable_space Ω] \n  (μ ν:measure_theory.measure Ω) [A1:measure_theory.finite_measure μ]\n  [A2:measure_theory.finite_measure ν]:\n  ¬(μ.perpendicular ν) →\n  ∃ f:Ω → ennreal, \n  measurable f ∧\n  μ.with_density f ≤ ν ∧\n  0 < μ.with_density f (set.univ) :=\nbegin\n  intros A3,\n  have B1 := measure_theory.measure.exists_of_not_perpendicular μ A3,\n  cases B1 with ε B1,\n  have B2:¬((ε • μ).perpendicular (ν - ε • μ)),\n  {\n    intro B2A,\n    apply B1.right,\n    apply measure_theory.measure.perpendicular_smul _ _ B1.left,\n    apply B2A,\n  },\n  have B3 := measure_theory.measure.perpendicular_sub_elim _ B2,\n  cases B3 with S B3,\n  let f := (set.indicator S (λ ω:Ω, ε)),\n  begin\n    have C1:f = (set.indicator S (λ ω:Ω, ε)) := rfl,\n    apply exists.intro f,\n    split,\n    {\n      apply measurable.indicator,\n      apply measurable_const,\n      apply B3.left,\n    },\n    split,\n    {\n      rw C1,\n      rw with_density_indicator_eq_restrict_smul _ B3.left,\n      rw ← smul_restrict_comm _ B3.left,\n      apply le_trans _ (@measure_theory.measure.restrict_le_self _ _ _ S),\n      apply B3.right.left,\n    },\n    {\n      have C2:0 < μ S,\n      {\n        have C2A := B3.right.right,\n        rw ennreal_smul_measure_apply _ _ _ B3.left at C2A,\n        simp at C2A,\n        apply C2A.right,\n      },\n      rw C1,\n      rw with_density_indicator_eq_restrict_smul _ B3.left,\n      rw ennreal_smul_measure_apply _ _ _ (measurable_set.univ),\n      rw measure_theory.measure.restrict_apply measurable_set.univ,\n      simp,\n      apply and.intro (B1.left) C2,\n    },\n  end\nend\n\nlemma sup_indicator_partition {α:Type*} {x y:α → ennreal}:\n   (x ⊔ y) = set.indicator {ω|y ω < x ω} x + set.indicator {ω|x ω ≤ y ω} y  :=\nbegin\n  apply funext,\n  intro a,\n  simp,\n  cases (classical.em (x a ≤ y a)) with B1 B1,\n  {\n    rw max_eq_right B1,\n    rw set.indicator_of_not_mem,\n    rw set.indicator_of_mem,\n    simp,\n    apply B1,\n    simp,\n    apply B1,\n  },\n  {\n    have B2:=lt_of_not_ge B1,\n    have B3:=le_of_lt B2,\n    rw max_eq_left B3,\n    rw set.indicator_of_mem,\n    rw set.indicator_of_not_mem,                                \n    simp,\n    apply B1,\n    apply B2,\n  },\nend\n\n\nlemma with_density_restrict_le_with_density_restrict_le_on_subset {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {x y:Ω → ennreal} {S:set Ω}:\n    (measurable_set S) →\n    (∀ ω∈ S,  x ω ≤ y ω) → \n    (μ.with_density x).restrict S ≤ (μ.with_density y).restrict S :=\nbegin\n  intros A3 A4,\n  intros X' B1,\n  rw measure_theory.measure.restrict_apply B1,\n  rw measure_theory.measure.restrict_apply B1,\n  apply with_density_le_with_density,\n  {simp [A3,B1]},  \n  intros ω C1,\n  apply A4 ω,\n  simp at C1,\n  apply C1.right,\nend\n\nlemma measure_theory.measure.sup_def {Ω:Type*} [measurable_space Ω] {μ ν:measure_theory.measure Ω}:μ ⊔ ν = Inf {d|μ ≤ d ∧ ν ≤ d} := rfl\n\n\nlemma measure_theory.measure.le_sup {Ω:Type*} [measurable_space Ω] {μ ν d:measure_theory.measure Ω}:(∀ c, μ ≤ c → ν ≤ c → d ≤ c) → d ≤ μ ⊔  ν :=\nbegin\n  rw measure_theory.measure.sup_def,\n  intro A1,\n  apply @le_Inf (measure_theory.measure Ω) _,\n  intros c B1,\n  apply A1;simp at B1,\n  apply B1.left,\n  apply B1.right, \nend\n\nlemma measure_theory.measure.le_restrict_add_restrict {Ω:Type*} [measurable_space Ω] {μ ν:measure_theory.measure Ω}\n  {S:set Ω}:measurable_set S → μ.restrict S ≤ ν.restrict S →  ν.restrict Sᶜ ≤ μ.restrict Sᶜ → \n   μ ≤ μ.restrict Sᶜ + ν.restrict S :=\nbegin\n  intros B1 A1 A2,\n  rw measure_theory.measure.le_iff,\n  intros T C1,\n  rw measure_theory.measure.add_compl_inter μ S T B1 C1,\n  rw measure_theory.measure.add_apply,\n  rw add_comm,\n  apply @add_le_add ennreal _,\n  {\n    rw measure_theory.measure.restrict_apply C1,\n    rw set.inter_comm,\n    apply le_refl _,\n  },\n  {\n    rw measure_theory.measure.restrict_apply C1,\n    rw set.inter_comm T S,\n    apply inter_le_inter_of_restrict_le_restrict A1 B1 C1,\n  },\nend\n\n\nlemma measure_theory.measure.sup_eq {Ω:Type*} [measurable_space Ω] {μ ν:measure_theory.measure Ω}\n  (S:set Ω):measurable_set S → μ.restrict S ≤ ν.restrict S → ν.restrict Sᶜ ≤ μ.restrict Sᶜ → \n   (μ ⊔ ν) = μ.restrict Sᶜ + ν.restrict S :=\nbegin\n  intros D1 A1 A2,\n  apply le_antisymm,\n  {\n    apply @sup_le (measure_theory.measure Ω) _,\n    {\n      apply measure_theory.measure.le_restrict_add_restrict D1 A1 A2,\n    },\n    {\n      rw add_comm,\n      have B1:ν.restrict Sᶜᶜ = ν.restrict S,\n      {\n        simp,\n      },\n      rw ← B1,\n  \n      apply measure_theory.measure.le_restrict_add_restrict,\n      repeat {simp [A1,A2,D1]},\n    },\n  },\n  {\n     apply measure_theory.measure.le_sup,\n     intros c C1 C2,\n     rw measure_theory.measure.le_iff,\n     intros T C3,\n     simp,\n     rw measure_theory.measure.restrict_apply C3,\n     rw measure_theory.measure.restrict_apply C3,\n     rw measure_theory.measure.add_compl_inter c S,\n     rw add_comm,\n     apply @add_le_add ennreal _,\n     rw set.inter_comm,\n     apply C2,\n     apply measurable_set.inter D1 C3,\n     rw set.inter_comm,\n     apply C1,\n     apply measurable_set.inter (measurable_set.compl D1) C3,\n     apply D1,\n     apply C3,\n  },\nend\n\nlemma measure_theory.measure.with_density_restrict_comm {Ω:Type*} [M:measurable_space Ω] (μ:measure_theory.measure Ω)\n  {x:Ω → ennreal} {S:set Ω}:measurable_set S → (μ.with_density x).restrict S = (μ.restrict S).with_density x :=\nbegin\n  intros A1,\n  apply measure_theory.measure.ext,\n  intros T B1,\n  rw measure_theory.with_density_apply,\n  rw measure_theory.measure.restrict_apply,\n  rw measure_theory.measure.restrict_restrict,  \n  rw ← measure_theory.lintegral_indicator,\n  rw measure_theory.with_density_apply,\n  rw measure_theory.lintegral_indicator,\n  repeat {assumption <|> apply measurable_set.inter},\nend\n\n\n\nlemma measure_theory.measure.with_density_add {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {x y:Ω → ennreal}:measurable x → measurable y → μ.with_density (x + y) = μ.with_density x + μ.with_density y :=\nbegin\n  intros A1 A2,\n  apply measure_theory.measure.ext,\n  intros S B1,\n  rw measure_theory.measure.add_apply,\n  repeat {rw measure_theory.with_density_apply},\n  simp only [pi.add_apply],\n  rw measure_theory.lintegral_add,\n  repeat{assumption <|> apply measurable.indicator},\nend\n\nlemma with_density_sup' {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {x y:Ω → ennreal}:measurable x → measurable y → \n    μ.with_density (x ⊔ y) =\n    (μ.with_density x).restrict {ω:Ω|y ω < x ω} \n    +\n    (μ.with_density y).restrict {ω:Ω|x ω ≤ y ω} :=\nbegin\n  intros A1 A2,\n  rw sup_indicator_partition,\n  rw measure_theory.measure.with_density_add,\n  rw with_density_indicator_eq_restrict,\n  rw with_density_indicator_eq_restrict,\n  rw measure_theory.measure.with_density_restrict_comm,\n  rw measure_theory.measure.with_density_restrict_comm,\n  repeat {assumption <|> apply measurable_set_le <|> apply measurable_set_lt <|> apply measurable.indicator},\nend\n\nlemma with_density_sup {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω)\n  {x y:Ω → ennreal}:measurable x → measurable y → \n    μ.with_density (x ⊔ y) =\n    measure_theory.measure.with_density μ x ⊔ measure_theory.measure.with_density μ y :=\nbegin\n  intros A1 A2,\n  have C1:=measurable_set_le A1 A2,\n  rw with_density_sup' μ A1 A2,\n  rw measure_theory.measure.sup_eq {ω:Ω|x ω ≤ y ω},\n  have C1:=measurable_set_le A1 A2,\n  rw lt_eq_le_compl,\n  {\n    apply C1,\n  },\n  {\n    apply with_density_restrict_le_with_density_restrict_le_on_subset,\n    apply C1,\n    intro ω,\n    simp,\n  },\n  {\n    apply with_density_restrict_le_with_density_restrict_le_on_subset,\n    apply measurable_set.compl (C1),\n    simp,\n    intros ω B3,\n    apply le_of_lt B3,\n  },\nend\n\n/--\n  The difference of two finite measures is a finite measure.\n -/\ndef measure_theory.finite_measure_sub {Ω:Type*} [M:measurable_space Ω] \n  (μ ν:measure_theory.measure Ω) [measure_theory.finite_measure ν]:\n  measure_theory.finite_measure (ν - μ) :=\nbegin\n  apply measure_theory.finite_measure_of_le (ν - μ) ν,\n  apply measure_theory.measure.sub_le,\nend\n\nlemma lebesgue_radon_nikodym_finite_unsigned {Ω:Type*} {N:measurable_space Ω} (μ ν:measure_theory.measure Ω) [A1:measure_theory.finite_measure μ]\n  [A2:measure_theory.finite_measure ν]:\n  ∃ f:Ω → ennreal, \n  ∃ μ₂:measure_theory.measure Ω,\n  measurable f ∧ \n  ν = μ.with_density f + μ₂ ∧\n  μ.perpendicular μ₂ :=\nbegin\n  let S := {f:Ω → ennreal|measurable f ∧ (μ.with_density f ≤ ν)},\n  let M := Sup ((λ f, μ.with_density f set.univ) '' S),\n  begin\n    have A4:S = {f:Ω → ennreal|measurable f ∧ (μ.with_density f ≤ ν)} := rfl,\n    have B2:M = Sup ((λ f, μ.with_density f set.univ) '' S)\n            := rfl,\n    have B3:M < ⊤,\n    {\n     \n      -- Because μ is finite.\n      -- Or, because ν is finite, and μ... is less than ν.\n      have B3A:M ≤ ν set.univ,\n      {\n        rw B2,\n        apply @Sup_le ennreal _,\n        intros b B3A1,\n        simp at B3A1,\n        cases B3A1 with f B3A1,\n        cases B3A1 with B3A1 B3A2,\n        subst b,\n        cases B3A1 with B3A1 B3A3,\n        have B3A4:μ.with_density f (set.univ) ≤ ν (set.univ),\n        {apply B3A3,apply measurable_set.univ},\n        simp at B3A4,\n        apply B3A4,\n      },\n      apply lt_of_le_of_lt B3A,\n      apply measure_theory.measure_lt_top,\n    },\n    have B1:∃ h:ℕ → Ω → ennreal, \n            (∀ n, h n ∈ S) ∧ \n            (monotone h) ∧\n            (μ.with_density (supr h) set.univ) = M,\n    {\n--      have B1A:=\n      apply @Sup_apply_eq_supr_apply_of_closed' (Ω → ennreal) \n          _ S (λ f:Ω → ennreal, μ.with_density f set.univ) _ _, \n      --cases B1A with h B1A,\n      { -- ⊢ S.nonempty\n        apply @set.nonempty_of_mem (Ω → ennreal) S 0,\n        rw A4,\n        simp,\n        split,\n        apply @measurable_const ennreal Ω _ N 0,\n        rw with_density.zero,\n        apply measure_theory.measure.zero_le,\n      },\n      { -- ⊢ ∀ a ∈ S, ∀  b ∈ S, a ⊔ b ∈ S\n        intros a B1B1 b B1B2,\n        cases B1B1 with B1B1 B1B3,\n        cases B1B2 with B1B2 B1B4,\n        simp,\n        split,\n        {\n          apply measurable_sup B1B1 B1B2,\n        },\n        {\n          rw (with_density_sup μ B1B1 B1B2),\n          simp,\n          apply and.intro B1B3 B1B4,\n        },\n      },\n      { -- ⊢ ∀ a ∈ S,∀ b ∈ S,a ≤ b → \n        -- (μ.with_density a set.univ ≤ μ.with_density b set.univ),\n        intros a B1C b B1D B1E,\n        simp,\n        rw A4 at B1C,\n        rw A4 at B1D,\n        --apply measure_theory.lintegral_le_lintegral,\n        apply @measure_theory.lintegral_mono,\n        simp,\n        intros ω B1F,\n        apply B1E,\n      },\n      {\n        -- ⊢ ∀ (f : ℕ → Ω → ennreal),\n        -- set.range f ⊆ S →\n        -- monotone f →\n        -- supr ((λ (f : Ω → ennreal), ⇑(μ.with_density f) set.univ) ∘ f) =\n        -- (λ (f : Ω → ennreal), ⇑(μ.with_density f) set.univ) (supr f)\n        intros f B1G B1H,\n        --simp only [measure_theory.measure.restrict_univ, measure_theory.with_density_apply, measurable_set.univ],\n        simp only [measure_theory.measure.restrict_univ, measurable_set.univ],\n        rw supr_with_density_apply_eq_with_density_supr_apply _ _ B1H,\n        simp,\n        intros n,\n        rw A4  at B1G,\n        unfold set.range at B1G,\n        rw set.subset_def at B1G,\n        simp at B1G,\n        apply (B1G n).left,\n      },\n    },\n    cases B1 with h B1,\n    have B4:∀ n, measurable (h n),\n    {\n      intros n,\n      have B4A := (B1.left n),\n      rw A4 at B4A,\n      apply B4A.left,\n    },\n    let g := supr h,\n    begin\n      apply exists.intro g,\n      have A5:g = supr h := rfl,\n      have A6:μ.with_density g set.univ = M,\n      {\n        rw A5,\n        rw ← B1.right.right,\n      },\n      have A7:μ.with_density g ≤ ν,\n      {\n        rw measure_theory.measure.le_iff,\n        intros S A7A,\n        rw ← supr_with_density_apply_eq_with_density_supr_apply,\n        apply @supr_le ennreal _ _,\n        intros i,\n        have A7B := (B1.left i),\n        simp at A7B,\n        apply A7B.right,\n        apply A7A,\n        apply A7A,\n        apply B4,\n        apply B1.right.left,\n      },\n      apply exists.intro (ν - μ.with_density g),\n      have C4:ν = μ.with_density g + (ν - μ.with_density g),\n      {\n        rw add_comm,\n        have C4A:measure_theory.finite_measure (μ.with_density g),\n        {\n          apply measure_theory.finite_measure.mk,\n          rw A6,\n          apply B3,\n        },\n        rw @measure_theory.measure.sub_add_cancel_of_le Ω N ν (μ.with_density g) C4A,\n        apply A7,       \n      }, \n      have C5:measurable g,\n      {\n        rw A5,\n        apply @measurable_supr_of_measurable,\n        apply B4,\n      },\n      apply and.intro C5,\n      apply and.intro C4,\n      {\n        apply by_contradiction,\n        intros C1,\n        have C2X:=measure_theory.finite_measure_sub (μ.with_density g) ν,\n        have C2 := @lebesgue_radon_nikodym_junior Ω N \n          μ (ν - μ.with_density g) A1 C2X C1,\n        cases C2 with f C2,\n        have D1:M < μ.with_density (f + g) set.univ,\n        {\n          rw measure_theory.measure.with_density_add,\n          rw measure_theory.measure.add_apply,\n          rw A6,\n          rw add_comm,\n          apply ennreal.lt_add_self,\n          apply B3,\n          apply C2.right.right,\n          apply C2.left,\n          apply C5,\n        },\n        apply not_le_of_lt D1,\n        rw B2,\n        apply @le_Sup ennreal _,\n        simp,\n        apply exists.intro (f  + g),\n        split,\n        split,\n        {\n          apply measurable.add,\n          apply C2.left,\n          apply C5,\n        },\n        {\n          rw C4,\n          rw measure_theory.measure.with_density_add,\n          rw add_comm,\n          apply measure_theory.measure.add_le_add,\n          apply le_refl _,\n          apply C2.right.left,\n          apply C2.left,\n          apply C5,\n        },\n        refl,\n      },\n    end\n  end\nend\n\n\n\n--Note that the Radon-Nikodym derivative is not necessarily unique.\ndef is_radon_nikodym_deriv  {Ω:Type*} {M:measurable_space Ω} (ν μ:measure_theory.measure Ω) (f:Ω → ennreal):Prop :=\n   (measurable f) ∧ μ.with_density f = ν\n\n\n\nlemma is_radon_nikodym_deriv_elim {Ω:Type*} {M:measurable_space Ω} (ν μ:measure_theory.measure Ω) (f:Ω → ennreal) (S:set Ω):\n  (is_radon_nikodym_deriv ν μ f) →\n  (measurable_set S) →\n  (∫⁻ (a : Ω), (set.indicator S f) a ∂ μ = ν S) :=\nbegin\n  intros A1 A2,\n  unfold is_radon_nikodym_deriv at A1,\n  cases A1 with A3 A1,\n  subst ν,\n  rw measure_theory.with_density_apply2',\n  apply A2,\nend\n\nlemma measurable_of_is_radon_nikodym_deriv {Ω:Type*} {M:measurable_space Ω} (ν μ:measure_theory.measure Ω) (f:Ω → ennreal) (S:set Ω):\n  (is_radon_nikodym_deriv ν μ f) →\n  (measurable f) :=\nbegin\n  intro A1,\n  cases A1 with A1 A2,\n  apply A1,\nend\n\nlemma is_radon_nikodym_deriv_intro {Ω:Type*} {M:measurable_space Ω} (ν μ:measure_theory.measure Ω) (f:Ω → ennreal):\n  (measurable f) →\n  (∀ S:set Ω, (measurable_set S) →\n  (∫⁻ (a : Ω), (set.indicator S f) a ∂ μ = ν S)) →\n  (is_radon_nikodym_deriv ν μ f)  :=\nbegin\n  intros A1 A2,\n  unfold is_radon_nikodym_deriv,\n  split,\n  apply A1,\n  apply measure_theory.measure.ext,\n  intros S A3,\n  rw measure_theory.with_density_apply2',\n  apply A2,\n  repeat {apply A3},\nend\n\n\n\n/-\n  As we move on to the later theorems, we need to be able to say that two\n  functions are \"almost everywhere equal\". Specifically, given two radon-nikodym \n  derivatives of a measure, they are equal almost everywhere according to the\n  base measure.\n -/\n\n-- There used to be a definition close to this, measure_theory,measure.a_e, and\n-- This used to be ∀ᶠ a in μ.a_e, P a\n-- For now, we use a local definition.\ndef measure_theory.measure.all_ae {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P:Ω → Prop):Prop :=\n   (μ ({ω:Ω|(P ω)}ᶜ)) = 0\n\n\nlemma measure_theory.measure.all_ae_def2 {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P:Ω → Prop):\n    μ.all_ae P = ( (μ ({ω:Ω|(P ω)}ᶜ)) = 0) :=\nbegin\n  unfold measure_theory.measure.all_ae,\nend\n\n\nlemma measure_theory.measure.all_ae_and {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P Q:Ω → Prop):\n    (μ.all_ae P) →\n    (μ.all_ae Q) →\n    (μ.all_ae (λ a, P a ∧ Q a)) := \nbegin\n  intros A1 A2,\n  rw measure_theory.measure.all_ae_def2,\n  have A3:{ω:Ω| P ω ∧ Q ω}ᶜ =  ({ω:Ω|P ω}ᶜ) ∪ ({ω:Ω|Q ω}ᶜ),\n  {\n    ext ω,\n    split;intros A3A;simp;simp at A3A,\n    {\n      have A3B:P ω ∨ ¬(P ω) := classical.em (P ω),\n      cases A3B,\n      {\n        apply or.inr (A3A A3B),\n      },\n      {\n        apply or.inl A3B,\n      },\n    },\n    {\n      cases A3A,\n      {\n        intro A3B,\n        exfalso,\n        apply A3A,\n        apply A3B,\n      },\n      {\n        intro A3B,\n        apply A3A,\n      },\n    },\n  },\n  rw A3,\n  have A4:(⊥:ennreal)=(0:ennreal) := rfl,\n  rw ← A4,\n  rw ← le_bot_iff,\n  apply le_trans (measure_theory.measure_union_le ({ω:Ω|P ω}ᶜ) ({ω:Ω|Q ω}ᶜ)),\n  rw measure_theory.measure.all_ae_def2 at A1,\n  rw measure_theory.measure.all_ae_def2 at A2,\n  rw A1,\n  rw A2,\n  simp,\nend\n\n\nlemma measure_theory.all_ae_imp {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P Q:Ω → Prop):\n    (μ.all_ae (λ a, P a → Q a)) →\n    (μ.all_ae P) →\n    (μ.all_ae Q) :=\nbegin\n  intros A1 A2,\n  rw measure_theory.measure.all_ae_def2 at A1,\n  rw measure_theory.measure.all_ae_def2 at A2,\n  rw measure_theory.measure.all_ae_def2,\n  have A3:{ω:Ω|Q ω}ᶜ ⊆ ({ω:Ω|P ω → Q ω}ᶜ) ∪ ({ω:Ω|P ω}ᶜ),\n  {\n    rw set.subset_def,\n    intro ω,\n    simp,\n    intro A3A,\n    cases (classical.em (P ω)) with A3B A3B,\n    {\n      apply or.inl (and.intro A3B A3A),\n    },\n    {\n      apply or.inr A3B,\n    },\n  },\n  have A4:(⊥:ennreal)=(0:ennreal) := rfl,\n  rw ← A4,\n  rw ← le_bot_iff,\n  apply le_trans (measure_theory.measure_mono A3),\n  apply le_trans (measure_theory.measure_union_le ({ω:Ω|P ω → Q ω}ᶜ) ({ω:Ω|P ω}ᶜ)),\n  rw A1,\n  rw A2,\n  simp,\nend\n\n\nlemma measure_theory.all_ae2_of_all {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P:Ω → Prop):\n    (∀ a, P a) → \n    (μ.all_ae P) :=\nbegin\n  intro A1,\n  rw measure_theory.measure.all_ae_def2,\n  have A2:{ω:Ω|P ω}ᶜ = ∅,\n  {\n    ext ω,split;intros A2A,\n    exfalso,\n    simp at A2A,\n    apply A2A,\n    apply A1,\n    exfalso,\n    apply A2A,\n  },\n  rw A2,\n  simp,\nend\n\n\nlemma measure_theory.not_ae {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P:Ω → Prop):\n  (∃ S:set Ω, (μ S > 0) ∧ (∀ ω∈ S, ¬ (P ω)) )→\n  (¬(μ.all_ae P)) :=\nbegin\n  intros A1 A2,\n  cases A1 with S A3,\n  cases A3 with A3 A4,\n  rw measure_theory.measure.all_ae_def2 at A2,\n  have A5:S⊆ {ω:Ω|P ω}ᶜ,\n  {\n    rw set.subset_def,\n    intro ω,\n    intro A5A,\n    apply A4 _ A5A,\n  },\n  have A6 := measure_theory.outer_measure.mono (μ.to_outer_measure) A5,\n  have A7 := lt_of_lt_of_le A3 A6,\n  rw measure.apply at A2,\n  rw A2 at A7,\n  apply lt_irrefl (0:ennreal) A7,\nend\n\n\n/-\n  Notice that it is not necessarily the case that a measurable set exists.\n  For example, consider where Ω = {a,b}.\n  The measurable sets are {} and {a,b}, where μ ∅ = 0 and μ {a,b} = 1.\n  Define (P a) and (¬P b).\n  Thus S={a}, which is not measurable.\n-/\nlemma measure_theory.not_ae_elim {Ω:Type*} {M:measurable_space Ω} (μ:measure_theory.measure Ω) (P:Ω → Prop):\n  (¬(μ.all_ae P)) →\n  (∃ S:set Ω, (μ S > 0) ∧ (∀ ω∈ S, ¬ (P ω)) ) :=\nbegin\n  intro A1,\n  rw measure_theory.measure.all_ae_def2 at A1,\n  have A2 := ennreal.eq_zero_or_zero_lt A1,\n  apply exists.intro ({ω : Ω | P ω}ᶜ),\n  split,\n  apply A2,\n  intros ω A3,\n  simp at A3,\n  apply A3,\nend\n\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/radon_nikodym.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778403, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.32907013752393527}}
{"text": "import polytime.data_structures.list\n\nnamespace polytime\n\nopen_locale complexity_class\nopen tencodable function polysize\n\nvariables {α β γ δ ε η : Type} [tencodable α] [tencodable β] [tencodable γ]\n [tencodable δ] [tencodable ε] [tencodable η]\n\n@[complexity] lemma polytime_lift_le : (@lift_le α _) ∈ₚ PTIME :=\nby { dunfold lift_le, complexity, }\n\n@[complexity] lemma multiset_nodup : (@multiset.nodup α) ∈ₚ PTIME :=\nby { complexity using λ s, (s.sort lift_le).nodup, simpa using (@multiset.coe_nodup _ (s.sort lift_le)).symm, }\n\n@[complexity] lemma to_multiset : (λ x : list α, (↑x : multiset α)) ∈ₑ PTIME :=\nby { rw complexity_class.mem_iff_comp_encode', complexity using λ x, x.insertion_sort lift_le, simp [list.merge_sort_eq_insertion_sort], }\n\ninstance {α : Type} [polycodable α] : polycodable (multiset α) :=\n⟨complexity_class.decodable_of_quotient_mk (polycodable.poly (list α)) to_multiset⟩ \n\ninstance {α : Type} [decidable_eq α] [polycodable α] : polycodable (finset α) :=\n⟨complexity_class.decodable_of_equiv $ complexity_class.subtype_decode (polycodable.poly _) multiset_nodup⟩\n\n@[complexity] lemma multiset_map {m : α → multiset β} {f : α → β → γ} (hm : m ∈ₑ PTIME) (hf : f ∈ₑ PTIME) :\n  (λ x, (m x).map (f x)) ∈ₑ PTIME :=\nby { complexity using λ x, ↑(((m x).sort lift_le).map (f x)), rw [← multiset.coe_map, multiset.sort_eq], }\n\n@[complexity] lemma multiset_add : ((+) : multiset α → multiset α → multiset α) ∈ₑ PTIME :=\nby { complexity using λ x y, ↑(x.sort lift_le ++ y.sort lift_le), simp only [← multiset.coe_add, multiset.sort_eq], }\n\n@[complexity] lemma multiset_map_dep {β γ : α → Type} [∀ x, tencodable (β x)] [∀ x, tencodable (γ x)] {m : ∀ x, multiset (β x)}\n  {f : ∀ x, β x → γ x} (hm : m ∈ₐ PTIME) (hf : f ∈ₐ PTIME) : (λ x, ((m x).map (f x) : multiset _)) ∈ₐ PTIME :=\npolytime.functor_map_dep multiset hm hf (@multiset_map _ _ _ _ _ _)\n\nlemma multiset_add_dep {β : α → Type} [∀ x, tencodable (β x)] {m₁ m₂ : ∀ x, multiset (β x)} \n  (hm₁ : m₁ ∈ₐ PTIME) (hm₂ : m₂ ∈ₐ PTIME) : (λ x, (m₁ x) + (m₂ x)) ∈ₐ PTIME :=\nbegin\n  rw polytime.mem_dep_iff_comp_eq_encode (λ x (y : multiset (β x)), y.map encode) (λ x y, complexity_class.multiset_map_encode y) at ⊢ hm₁ hm₂,\n  simp only [multiset.map_add] at *,\n  complexity,\nend\n\nend polytime\n", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/polytime/data_structures/finset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3290701303336043}}
{"text": "\nimport Lib.Data.Nat\nimport Lib.Data.Fin\nimport Lib.Data.List.Basic\n\nnamespace Array\n\ninstance : Functor Array where\n  map := Array.map\n\n@[inline]\ndef foldlIdx (ar : Array α) (f : Nat → α → β → β) (x₀ : β) : β :=\n(ar.foldl (λ ⟨x, i⟩ y => ⟨f i y x, i+1⟩) (x₀, 0)).1\n\n@[inline]\nunsafe def foldlIdxUnsafe (ar : Array α) (f : Fin ar.size → α → β → β) (x₀ : β) : β :=\n(ar.foldl (λ ⟨x, i⟩ y => ⟨f (unsafeCast i) y x, i+1⟩) (x₀, 0)).1\n\ndef Fin.succ {n} : Fin n → Fin n\n| ⟨i, h⟩ => ⟨i.succ % n, Nat.mod_lt _ $ Nat.lt_of_le_of_lt (Nat.zero_le _) h⟩\n\n@[implementedBy foldlIdxUnsafe]\ndef foldlIdx' (ar : Array α) (f : Fin ar.size → α → β → β) (x₀ : β) : β :=\nif h : 0 < ar.size then\n  (ar.foldl (λ ⟨x, i⟩ y => ⟨f i y x, Fin.succ i⟩) (x₀, ⟨0, h⟩)).1\nelse x₀\n\n@[simp]\ndef size_map (f : α → β) (ar : Array α) :\n  (ar.map f).size = ar.size := sorry\n\nattribute [simp] size_set\n\n@[simp]\ndef get_map (f : α → β) (ar : Array α) (i : Fin _) :\n  (ar.map f).get i = f (ar.get $ ⟨i.1, (size_map f ar) ▸ i.2⟩) := sorry\n\n@[simp]\ndef size_modify [Inhabited α] (f : α → α)\n    (ar : Array α) (i : Nat) :\n  (ar.modify i f).size = ar.size := sorry\n\n@[simp]\ndef size_zipWith (f : α → β → γ) (ar₀ : Array α) (ar₁ : Array β) :\n  size (ar₀.zipWith ar₁ f) = min (size ar₀) (size ar₁) := sorry\n\n\n@[simp] theorem bind_pure' [Monad m] [LawfulMonad m] (x : m α) : x >>= (λ a => pure a) = x := by\n  show x >>= (fun a => pure (id a)) = x\n  rw [bind_pure_comp, id_map]\n\nattribute [auto] Nat.zero_le\n\nsection foldlM_sim\nvariable {m : Type u → Type u'} [Monad m] [LawfulMonad m]\nvariable {m' : Type v → Type v'} [Monad m'] [LawfulMonad m']\nvariable (f : β → α → m β) (g : γ → α → m' γ)\nvariable (x₀ : β) (y₀ : γ) (h : β → γ)\nvariable (SIM : m β → m' γ → Prop)\n\nattribute [simp] bind_pure\n\ntheorem foldlM_sim (ar : Array α) {x₀ y₀}\n        (H₀ : SIM x₀ y₀)\n        (Hstep : ∀ x x' a, SIM x x' →\n          SIM (x >>= (f . a)) (x' >>= (g . a))) :\n  SIM (x₀ >>= ar.foldlM f) (y₀ >>= ar.foldlM g) := by\nsuffices H :\n    ∀ i j, i ≤ j → j ≤ ar.size →\n    SIM (x₀ >>= (ar.foldlM f . i j)) (y₀ >>= (ar.foldlM g . i j))\n  by apply H <;> auto\nintros i j Hij Hj; simp [foldl, foldlM, *]\ngeneralize j - i = n\ninduction n generalizing x₀ y₀ i j;\nfocus\n  have := Nat.zero_le (size ar)\n  simp [foldl, foldlM, *, foldlM.loop, Nat.zero_sub]\n  split <;> simp [bind_pure, *] <;>\n  rw [bind_pure, bind_pure] <;> assumption\n  done\nnext n ih =>\n  unfold foldlM.loop\n  simp only [*]\n  split\n  focus\n    rw [← bind_assoc, ← bind_assoc]\n    auto\n  focus\n    simp [*]\n\ntheorem foldlM_hom (ar : Array α) {x₀ y₀} (h : m β → m' γ)\n        (H' : h x₀ = y₀)\n        (H : ∀ x y, h (x >>= (f . y)) = h x >>= (g . y)) :\n  h (x₀ >>= ar.foldlM f) = y₀ >>= ar.foldlM g := by\nlet SIM x y := h x = y\napply foldlM_sim (SIM := SIM)\n. assumption\n. intros _ _ _ H₂; simp [H, H₂]\n\ntheorem foldlM_hom' (ar : Array α) {x₀ y₀} (h : m β → m' γ)\n        (H' : h (pure x₀) = pure y₀)\n        (H : ∀ x y, h (x >>= (f . y)) = h x >>= (g . y)) :\n  h (ar.foldlM f x₀) = ar.foldlM g y₀ := by\nlet SIM x y := h x = y\nhave := foldlM_hom (H := H) (H' := H')\nsimp at this; auto\n\nend foldlM_sim\n\nvariable [Monad m] [LawfulMonad m]\n\nlocal notation \"Push\" =>\n  (λ ar x => (pure <| Array.push ar x))\n\n@[simp]\ntheorem Array.toSubarray_toArray_eq_nil (ar : Array α) :\n  (toSubarray ar i i).toArray = #[] :=\nsorry\n\n@[simp]\ntheorem Array.toSubarray_toArray_eq_self (ar : Array α) :\n  (toSubarray ar 0 ar.size).toArray = ar :=\nsorry\n\ntheorem Array.push_toSubarray (ar : Array α) i h :\n  (toSubarray ar 0 i).toArray.push (ar.get ⟨i,h⟩) =\n  (toSubarray ar 0 i.succ).toArray :=\nsorry\n\ntheorem foldlM_eq_self (ar : Array α)  :\n  foldlM Push #[] ar =\n  (pure ar : m _) := by\nsimp [foldl, foldlM, Nat.le_refl]\nsuffices H :\n    ∀ i, i ≤ ar.size →\n      (ar.foldlM Push ar[:i].toArray i ar.size) =\n      (pure ar : m _) by\n  specialize H 0 (Nat.zero_le _)\n  simp [foldlM, Nat.le_refl] at H; assumption\nintros i Hi; simp [foldl, foldlM, *]\ngeneralize Hn : size ar - i = n\ninduction n generalizing i;\nnext =>\n  have h₀ : ¬ i < size ar := by\n    have Hn := Nat.le_of_eq Hn\n    -- have Hn := Nat.sub_le Hn\n    skip\n    admit\n  have h₁ : i = size ar := by\n    auto [Nat.le_antisymm]\n  simp [foldlM.loop, h₀]\n  simp [*, Nat.le_refl]\nnext ih =>\n  have h₀ : i < size ar := sorry\n  unfold foldlM.loop\n  simp [h₀, Array.push_toSubarray]\n  rw [ih _ h₀]\n  apply Nat.succ_inj\n  rw [Nat.sub_succ, Nat.succ_pred]\n  <;> auto [Nat.zero_lt_sub_of_lt, Nat.sub_ne_zero_of_lt]\n\nsection foldl_sim\n\nvariable (f : β → α → β) (g : γ → α → γ)\nvariable (x₀ : β) (y₀ : γ) (h : β → γ)\nvariable (SIM : β → γ → Prop)\n\ntheorem foldl_sim (ar : Array α)\n        (H' : SIM x₀ y₀)\n        (H : ∀ x x' a, SIM x x' →  SIM (f x a) (g x' a)) :\n  SIM (ar.foldl f x₀) (ar.foldl g y₀) := by\napply foldlM_sim (m := Id) (m' := Id) (SIM := SIM)\n<;> assumption\n\ntheorem foldl_hom (ar : Array α)\n        (H' : h x₀ = y₀)\n        (H : ∀ x y, h (f x y) = g (h x) y) :\n  h (ar.foldl f x₀) = ar.foldl g y₀ := by\nsubst H'\nsuffices H :\n    ∀ i j, i ≤ j → j ≤ ar.size →\n    h (ar.foldl f x₀ i j) = ar.foldl g (h x₀) i j\n  by apply H <;> auto\nintros i j Hij Hj; simp [foldl, foldlM, *]\ngeneralize j - i = n\ninduction n generalizing x₀ i j;\nfocus\n  have := Nat.zero_le (size ar)\n  simp [foldl, foldlM, *, foldlM.loop, Nat.zero_sub]\n  by_cases h : i < j <;> simp [*] <;> refl\nnext n ih =>\n  unfold foldlM.loop\n  simp [*]\n  split\n  . rw [← H, ← ih] <;> assumption\n  . refl\n\nend foldl_sim\n\nsection foldl_ind\n\nvariable (f : β → α → β)\nvariable (x₀ : β)\nvariable (M : β → Prop)\n\ntheorem foldl_ind (ar : Array α)\n        (H' : M x₀)\n        (H : ∀ x a, M x →  M (f x a)) :\n  M (ar.foldl f x₀) := by\nlet SIM := λ x y => x = y ∧ M x\nsuffices SIM (foldl f x₀ ar 0 (size ar))\n             (foldl f x₀ ar 0 (size ar)) by\n  apply this.2\napply foldl_sim (SIM := SIM)\n. constructor <;> auto\nintros x x' a; rintro1 ⟨rfl, h₁⟩\nauto\n\nend foldl_ind\n\ndef mapA {F} [Applicative F] (f : α → F β) (ar : Array α) : F (Array β) :=\nar.foldl (λ acc x => Array.push <$> acc <*> f x) (pure $ Array.mkEmpty ar.size)\n\ntheorem mapA_eq_mapM {m} [Monad m] [LawfulMonad m] (f : α → m β) (ar : Array α) :\n  ar.mapA f = ar.mapM f := by\nsimp only [mapA, mapM, foldl]\napply Array.foldlM_hom' (m := Id) (m' := m) (h := Id.run)\n. refl\nfocus\n  intros; simp [seq_eq_bind_map, map_eq_pure_bind]\n  refl\n\n@[simp]\ntheorem toArray_toList {α} (ar : Array α) : ar.toList.toArray = ar := sorry\n\n@[simp]\ntheorem toList_toArray {α} (l : List α) : l.toArray.toList = l := sorry\n\n@[simp]\ntheorem size_toArray {α} (l : List α) : l.toArray.size = l.length := sorry\n\n@[simp]\ntheorem length_toList {α} (ar : Array α) : ar.toList.length = ar.size := sorry\n\ntheorem foldl_toArray {α β : Type _} (f : β → α → β) (ar : List α) x₀ :\n  n = ar.length →\n  ar.toArray.foldl f x₀ 0 n = ar.foldl f x₀ := sorry\n\n@[simp]\ntheorem foldl_toList {α β : Type _} (f : β → α → β) (ar : Array α) x₀ :\n  ar.toList.foldl f x₀ = ar.foldl f x₀ := sorry\n\n@[simp]\ntheorem map_mkEmpty {α β} (f : α → β) :\n  f <$> mkEmpty n = mkEmpty n := rfl\n\n@[simp]\ntheorem map_nil {α β} (f : α → β) :\n  f <$> #[] = #[] := rfl\n\n-- @[simp]\ntheorem toList_eq_data {α} (ar : Array α) :\n  ar.toList = ar.data := by\ncases ar with | mk ar =>\nhave : ar.length ≤ ar.length := by refl\nsimp [toList, size, foldr, foldrM, *]\nsplit\nfocus\n  simp only [size]\n  generalize (Eq.mpr_prop (eq_true this) True.intro) = h\n  let k := ar.length\n  rw [← @List.drop_length _ ar]\n  change List.length ar ≤ k at h\n  revert h\n  generalize ar.length = i; intros h\n  induction i <;> simp [foldrM.fold]\n  . refl\n  next i IH =>\n    have : ¬ (Nat.succ i == 0) = true := by\n      intros h; cases h\n    simp [*, get, List.cons_drop]\n    done\nfocus\n  suffices ar.length = 0 by\n    simp at this; subst this\n    refl\n  apply Nat.le_antisymm _ (Nat.zero_le _)\n  auto\n\ntheorem toList_inj (x y : Array α) :\n  x.toList = y.toList → x = y := by\ncases x; cases y\nsimp [toList_eq_data]; auto\n\n@[simp]\ntheorem map_toArray {α β : Type _} (f : α → β) (l : List α) :\n  f <$> l.toArray = (f <$> l).toArray := by\n-- simp [List.toArray]\n-- generalize Har : @mkEmpty α (List.length l) = ar\n-- have := congrArg (Functor.map f) Har\n-- simp at this; rw [this]; clear Har this\n-- induction l generalizing ar\n--  <;> simp [List.toArrayAux, *] <;> refl\nsorry\n\ntheorem mkEmpty_eq_mkEmpty {α n} m :\n  mkEmpty (α := α) n = mkEmpty m := rfl\n\n@[simp]\ntheorem toList_mkEmpty {α n} :\n  toList (@mkEmpty α n) = [] := sorry\n\n@[simp]\ntheorem toList_push {α} (ar : Array α) (x) :\n  toList (push ar x) = toList ar ++ [x] := sorry\n\nend Array\n\nnamespace Subarray\n\n-- def size (ar : Subarray α) : Nat := ar.stop - ar.start\n\n-- def get (ar : Subarray α) (i : Fin ar.size) : α :=\n-- have : ar.start + i < Array.size ar.as := by\n--   have := Nat.add_lt_of_lt_sub_l i.2\n--   apply Nat.lt_of_lt_of_le this ar.h₂\n-- ar.as.get ⟨ar.start + i, this⟩\n\n-- def get! [Inhabited α] (ar : Subarray α) (i : Nat) : α :=\n-- ar.as.get! (ar.start + i)\n\ndef get? (ar : Subarray α) (i : Nat) : Option α :=\nar.as.get? (ar.start + i)\n\ndef empty (ar : Subarray α) : Bool := ar.start == ar.stop\n\n@[simp]\ntheorem stop_popFront (ar : Subarray α) :\n  ar.popFront.stop = ar.stop := by\nsimp [popFront]\nby_cases h : ar.start < ar.stop <;> simp [*]\n\n@[simp]\ntheorem size_popFront (ar : Subarray α) :\n  0 < ar.size →\n  ar.popFront.size = ar.size - 1 := by\nintros h\nhave h : ar.start < ar.stop := by simp [size] at h; assumption\nsimp [popFront, *, size, Nat.sub_succ]\n\n@[simp]\ntheorem size_le_size_as (ar : Subarray α) :\n  ar.size ≤ ar.as.size := sorry\n\ndef extract (ar : Subarray α) (i j : Nat) : Subarray α :=\n  let start := min (ar.start + i) ar.stop\n  let stop := min (max (ar.start + j) start) ar.stop\n  have h₁ : start ≤ stop := by\n    simp [Nat.le_min_iff]\n    constructor\n    focus\n      apply Nat.le_max_r\n      rewrite [Nat.le_min_iff]\n      auto\n    . admit\n    -- . auto\n  have h₂ : stop ≤ ar.as.size := by\n    apply Nat.min_le_r; apply ar.h₂\n  { as := ar.as,\n    start := start,\n    stop := stop,\n    h₁ := h₁,\n    h₂ := h₂ }\n\n@[simp]\ntheorem extract_eq_self (ar : Subarray α) :\n  ar.extract 0 ar.size = ar := by\ncases ar; simp [extract, size, Nat.add_sub_assoc]\nconstructor\n. auto\n. apply Nat.le_max_l; simp [Nat.add_sub_of_le, *]\n\ntheorem get_extract (ar : Subarray α) i p q h h' :\n  (ar.extract p q).get ⟨i, h⟩ = ar.get ⟨p + i, h'⟩ := by\nsimp [extract, get]\n-- have : min (ar.start + p) ar.stop + i = ar.start + (p + i) := sorry\nhave : ∀ i j, i = j → ar.as.get i = ar.as.get j := by\n  intros _ _ h; subst h; refl\napply this; clear this; simp\nhave : min (ar.start + p) ar.stop = ar.start + p := by\n  rw [Nat.min_eq_iff_le_l]\n  apply Nat.le_of_lt\n  apply Nat.add_lt_of_lt_sub_l\n  apply Nat.lt_of_le_of_lt _ h'\n  apply Nat.le_add_right\nsimp [this, Nat.add_assoc]\n\ntheorem popFront_extract (ar : Subarray α) p q\n  (Hp : p < q)\n  (Hq : q ≤ ar.size) :\n  (ar.extract p q).popFront  = ar.extract p.succ q := by\nhave h₄ : p < ar.size := Nat.lt_of_lt_of_le Hp Hq\nhave H := Nat.add_lt_of_lt_sub_l h₄\nsimp [extract, popFront]\nsplit <;> simp\nnext h =>\n  have h₀ : min (ar.start + p) ar.stop = ar.start + p := by\n    simp [Nat.le_of_lt, *]\n  have h₁ : min (ar.start + p).succ ar.stop = (ar.start + p).succ := by\n    simp [Nat.succ_le_of_lt, *]\n  have h₂ : max q p = q := by\n    simp [Nat.le_of_lt, *]\n  have h₃ : max q p.succ = q := by\n    simp [Nat.le_of_lt, *]; assumption\n  simp [h₀, h₁, h₂] at h ⊢\n  rw [← Nat.add_succ, Nat.max_add, h₃]\nnext h =>\n  exfalso; apply h; clear h\n  have : min (ar.start + p) ar.stop = (ar.start + p) := by\n    simp; apply Nat.le_of_lt; assumption\n  have : min (ar.start + q) ar.stop = (ar.start + q) := by\n    simp [Nat.add_le_iff_l, ar.h₁]; assumption\n  have : max q p = q := by\n    simp; apply Nat.le_of_lt; assumption\n  simp [*]\n  apply Nat.add_lt_add_r; auto\n\ntheorem size_extract (ar : Subarray α) p q\n  (h₀ : p ≤ q) (h₁ : q ≤ ar.size) :\n  (ar.extract p q).size = q - p := by\nhave : min (ar.start + p) ar.stop = ar.start + p :=\n  by simp [Nat.add_le_iff_l, ar.h₁]; trans _ <;> assumption\nhave : max (ar.start + q) ar.start = ar.start + q :=\n  by simp [Nat.add_le_iff_r]; apply Nat.le_add_right\nhave : min (ar.start + q) ar.stop = ar.start + q :=\n  by simp [Nat.add_le_iff_l, ar.h₁]; assumption\nhave : max q p = q := by simp [*]\nsimp [extract, size, *]\n\nattribute [simp] measure invImage InvImage Nat.lt_wfRel\n\nsyntax \"prove_decr\" : tactic\nsyntax \"decr_step\" : tactic\nsyntax \"decr_finish\" : tactic\n\nmacro_rules\n| `(tactic| decr_finish) => `(tactic| assumption)\n\nmacro_rules\n| `(tactic| decr_finish) => `(tactic| constructor)\n\nmacro_rules\n| `(tactic| decr_step) => `(tactic| apply Nat.pred_lt; apply Nat.sub_ne_zero)\n\nmacro_rules\n| `(tactic| decr_step) => `(tactic| apply Nat.sub_lt)\n\nmacro_rules\n| `(tactic| prove_decr) =>\n  `(tactic| { simp [*]; decr_step <;> decr_finish })\n\ndef takeWhileAux\n            (p : α → Bool) (i : Nat) (ar : Subarray α) : Subarray α :=\nif h : i < ar.size then\n  if p $ ar.get ⟨i, h⟩ then takeWhileAux p (Nat.succ i) ar\n  else ar.extract 0 i\nelse ar.extract 0 i\ntermination_by takeWhileAux i ar  => ar.size - i\ndecreasing_by prove_decr\n\ntheorem takeWhileAux_eq (p : α → Bool) (ar : Subarray α) :\n  takeWhileAux p i ar =\n  if h : i < ar.size then\n    if p $ ar.get ⟨i, h⟩ then takeWhileAux p i.succ ar\n    else ar.extract 0 i\n  else ar.extract 0 i :=\nWellFounded.fix_eq _ _ _\n\ndef takeWhile (p : α → Bool) (ar : Subarray α) : Subarray α :=\ntakeWhileAux p 0 ar\n\ndef spanAux (p : α → Bool) (i : Nat)\n    (ar : Subarray α) : Subarray α × Subarray α :=\nif h : i < ar.size then\n  if p $ ar.get ⟨i, h⟩ then spanAux p (Nat.succ i) ar\n  else (ar.extract 0 i, ar.extract i ar.size)\nelse (ar.extract 0 i, ar.extract i ar.size)\ntermination_by spanAux i ar  => ar.size - i\ndecreasing_by prove_decr\n\ntheorem spanAux_eq (p : α → Bool) (i : Nat) (ar : Subarray α) :\n  spanAux p i ar =\n  if h : i < ar.size then\n    if p $ ar.get ⟨i, h⟩ then spanAux p (Nat.succ i) ar\n    else (ar.extract 0 i, ar.extract i ar.size)\n  else (ar.extract 0 i, ar.extract i ar.size) :=\nWellFounded.fix_eq _ _ _\n\ndef span (p : α → Bool) (ar : Subarray α) : Subarray α × Subarray α :=\nspanAux p 0 ar\n\ndef dropWhile\n            (p : α → Bool) (ar : Subarray α) : Subarray α :=\nif h : 0 < ar.size then\n  if p $ ar.get ⟨0, h⟩ then dropWhile p ar.popFront\n  else ar\nelse ar\ntermination_by dropWhile ar => ar.size\ndecreasing_by prove_decr\n\ntheorem dropWhile_eq (p : α → Bool) (ar : Subarray α) :\n  dropWhile p ar =\n  if h : 0 < ar.size then\n    if p $ ar.get ⟨0, h⟩ then dropWhile p ar.popFront\n    else ar\n  else ar :=\nWellFounded.fix_eq _ _ _\n\ntheorem span_eq_takeWhile_dropWhile (p : α → Bool) (ar : Subarray α) :\n  ar.span p = (ar.takeWhile p, ar.dropWhile p) := by\nsimp only [span, takeWhile]\nsuffices h : ∀ i,\n    i ≤ ar.size →\n    spanAux p i ar =\n    (takeWhileAux p i <| ar,\n     dropWhile p <| ar.extract i ar.size) by\n  specialize h 0\n  simp [h, *, Nat.zero_le]\nintros i\ngeneralize hk : (ar.size - i) = k\ninduction k using Nat.strong_ind generalizing i;\nnext ih =>\n  rw [spanAux_eq, dropWhile_eq, takeWhileAux_eq]\n  by_cases h₀ : i < size ar <;> simp [*]\n  focus\n    subst hk\n    have : i + 0 < ar.size := by simp [*]\n    have h₅ : i ≤ ar.size := Nat.le_of_lt h₀\n    have h₄ : ar.size ≤ ar.size := Nat.le_refl _\n    have h : 0 < size (extract ar i (size ar)) :=\n      by simp [size_extract, *]\n    have h₃ : get ar ⟨i + 0, this⟩ = get ar ⟨i, h₀⟩ :=\n      by revert this; rw [Nat.add_zero]; intro; refl\n    simp [get_extract (h' := this), h, h₃]\n    split\n    focus\n      have : size ar ≤ size ar := by refl\n      simp [popFront_extract, *]\n      apply ih _ _ _ rfl h₀\n      simp [Nat.sub_succ]\n      apply Nat.pred_lt\n      apply Nat.sub_ne_zero; assumption\n    focus\n      intros; refl\n  focus\n    intros h₁; subst hk\n    have h₅ : ar.size ≤ i := Nat.le_of_not_lt h₀\n    have h₅ := Nat.le_antisymm h₅ h₁\n    subst h₅\n    have h₄ : ar.size ≤ ar.size := Nat.le_refl _\n    have h : ¬ 0 < size (extract ar (size ar) (size ar)) :=\n      by simp [size_extract, *]\n    simp [*]\n\nend Subarray\n\nnamespace Array\nvariable (ar ar' : Array α)\n\n@[simp]\ntheorem size_extract i j :\n  (ar.extract i j).size = min ar.size (j - i) :=\nsorry\n\n@[simp]\ntheorem size_toSubarray i j :\n  (ar.toSubarray i j).size = min ar.size (j - i) :=\nsorry\n\ntheorem lt_size_toSubarray_implies_lt_size\n        (h : k < (ar.toSubarray i j).size) :\n  i + k < ar.size := sorry\n\n@[simp]\ntheorem get_toSubarray  i j k h :\n  (ar.toSubarray i j).get ⟨k, h⟩ =\n  ar.get ⟨i + k, lt_size_toSubarray_implies_lt_size _ h⟩ :=\nsorry\n\n@[simp]\ndef extractOffset : Fin (ar.extract i j).size → Fin ar.size\n| ⟨n, h⟩ => ⟨i + n, sorry⟩\n\n@[simp]\ntheorem get_extract i j k :\n  (ar.extract i j).get k =\n  ar.get (extractOffset _ k) :=\nsorry\n\n@[simp]\ntheorem get_eq_iff_get?_eq i h x :\n  ar.get ⟨i, h⟩ = x ↔\n  ar.get? i = some x :=\nsorry\n\n@[simp]\ntheorem some_get_eq_get? i h :\n  some (ar.get ⟨i, h⟩) = ar.get? i :=\nsorry\n\n@[simp]\ntheorem size_append (ar₀ ar₁ : Array α) :\n  (ar₀ ++ ar₁).size = ar₀.size + ar₁.size :=\nsorry\n\ntheorem toArray_append (x y : List α) :\n  (x ++ y).toArray = x.toArray ++ y.toArray :=\nsorry\n\n@[simp]\ntheorem push_append a (x y : Array α) :\n  (x ++ y).push a = x ++ y.push a := sorry\n\n@[simp]\ntheorem nil_appendList (a : List α) :\n  #[].appendList a = a.toArray := sorry\n\ntheorem appendList_eq (x : Array α) (y : List α) :\n  x.appendList y = x ++ y.toArray := sorry\n\ntheorem toList_append (x y : Array α) :\n  toList (x ++ y) = x.toList ++ y.toList := sorry\n\n-- #exit\n-- @[simp]\n-- theorem map_push {α β} (f : α → β) xs x :\n--   f <$> Array.push xs x = Array.push (f <$> xs) (f x) := by\n-- cases xs; simp [push, (.<$>.), map, mapM]\n\n@[simp]\ntheorem append_nil (xs : Array α) :\n  xs ++ #[] = xs := sorry\n\n@[simp]\ntheorem nil_append (xs : Array α) :\n  #[] ++ xs = xs := sorry\n\ntheorem append_assoc (xs ys zs : Array α) :\n  (xs ++ ys) ++ zs = xs ++ (ys ++ zs) := by\napply toList_inj; simp [toList_append, List.append_assoc]\n\n-- theorem toList_appendList (x : Array α) y :\n--   toList (x.appendList y) = x.toList ++ y := by\n-- let SIM {α} (x : Array α) (z : List α) := toList x = z ++ y\n-- refine' Array.foldl_sim (SIM := SIM) (ar := x)\n--   (f := Array.push) _ _ _ _ _ _\n\nend Array\n\nnamespace Util\n\nunsafe def isSharedUnsafe (x : @& α) : Bool := true\n\n@[inline]\nunsafe def withIsSharedUnsafe (x : α) (f : Bool → α → β)\n           (h : f true x = f false x) : β :=\nf (isSharedUnsafe x) x\n\n@[implementedBy withIsSharedUnsafe]\ndef withIsShared (x : α) (f : Bool → α → β)\n           (h : f true x = f false x) : β := f true x\n\ntheorem withIsShared_if_true (x : α) (f : Bool → α → β) h :\n  withIsShared x f h = f true x := rfl\n\ntheorem withIsShared_if_false (x : α) (f : Bool → α → β) h :\n  withIsShared x f h = f false x := h\n\nend Util\n\ndef SubarrayOfSize (n : Nat) (α) :=\n{ x : Subarray α // x.size = n }\n\nnamespace SubarrayOfSize\n\ndef get {n α} (ar : SubarrayOfSize n α)\n  (i : Fin n) : α :=\nar.val.get <| i.cast ar.2.symm\n\ndef toSubarray_idx (ar : Subarray α) : Fin ar.size → Fin ar.as.size\n| ⟨i, h⟩ => ⟨i, Nat.lt_of_lt_of_le h <| by simp ⟩\n\nprivate def subarray_set (ar : Subarray α) (i : Fin ar.size)\n            (x : α) : Subarray α where\n  as := ar.as.set (toSubarray_idx _ i) x\n  start := ar.start\n  stop := ar.stop\n  h₁ := ar.h₁\n  h₂ := by simp [ar.h₂]\n\nprivate theorem size_subarray_set (ar : Subarray α) i x :\n  (subarray_set ar i x).size = ar.size := rfl\n\ndef set {n α} (ar : SubarrayOfSize n α)\n  (i : Fin n) (a : α) : SubarrayOfSize n α :=\n⟨ subarray_set ar.val (ar.2.symm ▸ i) a,\n  by simp [size_subarray_set, ar.2]⟩\n\ndef get? {n α} (ar : SubarrayOfSize n α)\n  (i : Nat) : Option α :=\nar.val.get? i\n\nvariable (ar : SubarrayOfSize n α)\n\n@[simp]\ntheorem get_set i j x :\n  (ar.set i x).get j = if i = j then x else ar.get j :=\nsorry\n\n@[simp]\ntheorem get_eq_iff_get?_eq  i h x :\n  ar.get ⟨i, h⟩ = x ↔\n  ar.get? i = some x :=\nsorry\n\n@[simp]\ntheorem some_get_eq_get? i h :\n  some (ar.get ⟨i, h⟩) = ar.get? i :=\nsorry\n\nend SubarrayOfSize\n\ndef Subarray.Eqv {α} (n : Nat)\n  (ar₀ ar₁ : SubarrayOfSize n α) : Prop :=\n∀ i, ar₀.get i = ar₁.get i\n\ntheorem Subarray.Eqv.apply {α} (n : Nat)\n        (ar₀ ar₁ : SubarrayOfSize n α)\n        (h : Subarray.Eqv n ar₀ ar₁):\n  ∀ i, i < n → ar₀.get? i = ar₁.get? i :=\nby intros;\n   rw [← SubarrayOfSize.some_get_eq_get?,\n       ← SubarrayOfSize.some_get_eq_get?]\n   <;> try assumption\n   apply congrArg; apply h\n\ndef SubarrayQ (α) := Σ n, Quot <| @Subarray.Eqv α n\n\nnamespace Option\n\ntheorem some_inj {x y : α} (h : some x = some y) : x = y :=\nby cases h; refl\n\nend Option\n\n\nnamespace SubarrayQ\n\ndef ofSubarray (ar : Subarray α) : SubarrayQ α :=\n⟨ar.size, Quot.mk _ ⟨ar, rfl⟩⟩\n\ndef size (ar : SubarrayQ α) := ar.1\n\ndef get : (ar : SubarrayQ α) → Fin ar.size → α\n| ⟨n, ar⟩ => Quot.liftOn ar\n  (λ ar (i : Fin n) => ar.get i) $ by\n    { simp; intros a b h; apply funext; intros i;\n      simp only [SubarrayOfSize.some_get_eq_get?]\n      auto }\n\nsection Subarray\nvariable (ar : Subarray α)\n\n@[simp]\ntheorem size_ofSubarray (ar : Subarray α) :\n  size (ofSubarray ar) = ar.size :=\nsorry\n\nend Subarray\n\ntheorem ext (ar₀ ar₁ : SubarrayQ α)\n        (h₀ : ar₀.size = ar₁.size)\n        (h₁ : ∀ i h₀ h₁, ar₀.get ⟨i, h₀⟩ = ar₁.get ⟨i, h₁⟩) :\n  ar₀ = ar₁ :=\nsorry\n\n@[simp]\ntheorem get_ofSubarray (ar : Subarray α) i h :\n  (ofSubarray ar).get ⟨i,h⟩ = ar.get ⟨i, h⟩ :=\nsorry\n\ndef unshare : SubarrayQ α → SubarrayQ α\n| ⟨n, ar⟩ =>\nQuot.liftOn ar\n  (λ ar =>\n    let ⟨ar, i, j, _, _ ⟩ := ar.1\n    Util.withIsShared ar\n      (λ shared ar =>\n        if shared\n          then ofSubarray (ar.extract i j).toSubarray\n          else ofSubarray ar[i:j])\n      (by simp; apply ext <;> simp))\n  (by simp [Util.withIsShared_if_false]\n      intros a b h; apply ext <;> simp [a.2]\n      next =>\n        change (a.1.3 - a.1.2) with a.1.size\n        change (b.1.3 - b.1.2) with b.1.size\n        simp [Nat.min_eq_iff_le_r |>.2]\n        simp [a.2, b.2]\n        done\n      next =>\n        intros; apply h.apply\n        rw [← a.2]\n        simp at *\n        auto )\n\n@[simp]\ntheorem size_unshare (ar : SubarrayQ α) :\n  ar.unshare.size = ar.size := sorry\n\ndef set (ar : SubarrayQ α) : Fin ar.size → α → SubarrayQ α :=\nsuffices Fin ar.unshare.size → α → SubarrayQ α\n  by simp at this; assumption\nmatch ar.unshare with\n| ⟨n, ar⟩ => λ i : Fin n =>\nQuot.liftOn ar\n  (λ ar a => ⟨n, Quot.mk _ $ ar.set i a⟩)\n  (by intros a b h; apply funext; intros x; simp;\n      apply congrArg; apply Quot.sound; intro i;\n      simp; split <;> auto )\n\nend SubarrayQ\n\nstructure Buffer (m α) where\nmkImpl ::\n  cells : Array α\n  Hsize : cells.size = m\n\nnamespace Buffer\n\ndef mk (ar : Array α) : Option (Buffer m α) :=\nif h : ar.size = m then return ⟨ar, h⟩\nelse none\n\n@[inline]\ndef get (ar : Buffer m α) (i : Fin m) : α :=\nar.cells.get $ ar.Hsize.symm ▸ i\n\n@[inline]\ndef set (i : Fin m) (x : α) (ar : Buffer m α) : Buffer m α where\n  cells := ar.cells.set (ar.Hsize.symm ▸ i) x\n  Hsize := by simp [ar.Hsize]\n\n@[specialize]\ndef map (f : α → β) (ar : Buffer m α) : Buffer m β where\n  cells := ar.cells.map f\n  Hsize := by simp [ar.Hsize]\n\n@[inline]\ndef modify (i : Fin m) (f : α → α) (ar : Buffer m α) : Buffer m α :=\nar.set i (f $ ar.get i)\n\ndef foldl (ar : Buffer n α) (f : α → β → β) (x₀ : β) : β :=\nar.cells.foldl (flip f) x₀\n\ndef foldlIdx (ar : Buffer n α) (f : Fin n → α → β → β) (x₀ : β) : β :=\nar.cells.foldlIdx' (λ i => f $ ar.Hsize ▸ i) x₀\n\ndef mkFilled (x : α) : Buffer n α :=\n⟨ Array.mkArray n x , by simp ⟩\n\ndef zipWith (f : α → β → γ)\n  (b₀ : Buffer n α) (b₁ : Buffer n β) : Buffer n γ :=\n⟨ b₀.cells.zipWith b₁.cells f, by simp [Buffer.Hsize] ⟩\n\nend Buffer\n\nstructure Grid (m n α) where\nmkImpl ::\n  cells : Buffer m (Buffer n α)\n\nnamespace Grid\n\nvariable {m n : Nat} {α β : Type _}\n\ndef mk (ar : Array (Array α)) : Option (Grid m n α) := do\nGrid.mkImpl <$> Buffer.mk (← ar.mapM Buffer.mk)\n\n@[inline]\ndef get (ar : Grid m n α) (i : Fin m) (j : Fin n) : α :=\nar.cells.get i |>.get j\n\n@[specialize]\ndef map (f : α → β) (g : Grid m n α) : Grid m n β where\n  cells := g.cells.map $ Buffer.map f\n\n@[inline]\ndef set (i : Fin m) (j : Fin n) (x : α) (g : Grid m n α) : Grid m n α where\n  cells := g.cells.modify i $ Buffer.set j x\n\n@[inline]\ndef modify (i : Fin m) (j : Fin n) (f : α → α) (g : Grid m n α) : Grid m n α where\n  cells := g.cells.modify i $ Buffer.modify j f\n\n@[inline]\ndef foldl (ar : Grid m n α) (f : α → β → β) (x₀ : β) : β :=\nar.cells.foldl (λ ar => ar.foldl f) x₀\n\n@[inline]\ndef foldlIdx (ar : Grid m n α) (f : Fin m → Fin n → α → β → β) (x₀ : β) : β :=\nar.cells.foldlIdx (λ i ar => ar.foldlIdx $ f i) x₀\n\ndef zipWith (f : α → β → γ)\n    (ar₀ : Grid m n α) (ar₁ : Grid m n β) : Grid m n γ :=\n⟨ ar₀.cells.zipWith (Buffer.zipWith f) ar₁.cells ⟩\n\nend Grid\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/Array/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3290283238714608}}
{"text": "/-\n# Data model for the built-in dialect\n-/\n\nimport MLIR.AST\nimport MLIR.Doc\nimport MLIR.Semantics.Types\nimport MLIR.Semantics.TensorElem\n-- import MLIR.Util.Mathlib4.NatBasic\n-- import MLIR.Util.Mathlib4.Dvd\n-- import MLIR.Util.Mathlib4.NatLemmas\nopen MLIR.AST\n\n\n\n\n/-\n## General tensor type\n\nThe following structure is used as the basis for both ranked and unranked\ntensor types, since both have the same runtime contents and only differ by the\namount of information exposed in the typing system.\n\nAt runtime the dimension of tensors is always known. As a general rule, we\nstore compile-time type information in the parameters of the structure and the\nruntime contents in the fields.\n-/\n\nstructure Tensor (τ: MLIRTy) where\n  -- Runtime-determined dimensions\n  shape: List Nat\n  -- Contents in row-major order\n  data: List τ.eval\n  -- Invariant: shape and size must match\n  h_data_size: data.length = shapeProd shape\n\n\ntheorem Tensor.eq_of_fields_eq {τ} (t₁ t₂: Tensor τ):\n    t₁.shape = t₂.shape → t₁.data = t₂.data → t₁ = t₂ := by\n  intros h_shape h_data\n  cases t₁; cases t₂; simp at *\n  trivial\n\ninstance {τ}: DecidableEq (Tensor τ) := fun t₁ t₂ => by\n  cases t₁; cases t₂; simp\n  exact inferInstance\n\ndef Tensor.uniform {τ} (shape: List Nat) (v: τ.eval): Tensor τ :=\n  { shape       := shape,\n    data        := List.uniform v (shapeProd shape),\n    h_data_size := List.length_uniform _ _ }\n\ninstance {τ}: Inhabited (Tensor τ) where\n  default := Tensor.uniform [1] default\n\n\n-- Map a function over a tensor.\ndef Tensor.map {σ τ} (v: Tensor σ) (f: σ.eval → τ.eval): Tensor τ :=\n Tensor.mk (shape := v.shape) (data := v.data.map f) (h_data_size := by {\n    rewrite [List.length_map]\n    apply v.h_data_size;\n })\n\n#check Tensor.map\n\ntheorem Tensor.map_shape {σ τ: MLIRTy} (v: Tensor σ) (f: σ.eval → τ.eval): v.shape = (Tensor.map v f).shape := by {\n  simp [Tensor.map];\n}\n\n\ninstance {τ}: DecidableEq (Tensor τ) := fun t₁ t₂ => by\n  cases t₁; cases t₂; simp\n  exact inferInstance\n\n\ndef Tensor.mapWithFlatIndex {σ τ} (v: Tensor σ) (f: TensorFlatIndex (shapeProd v.shape) → σ.eval → τ.eval): Tensor τ :=\n  Tensor.mk (shape := v.shape)\n    (data := (List.zipFlatIndex v.data).map (fun (val, ix) => f (v.h_data_size ▸ ix) val)) (h_data_size := by simp; apply v.h_data_size)\n\n\n-- getter at a flat index\ndef Tensor.getAtFlatIndex {σ} (v: Tensor σ) (ix: TensorFlatIndex (shapeProd v.shape)): σ.eval :=\n  List.getF v.data ix.ix (by {\n     rewrite[v.h_data_size];    \n     simp[ix.h_ix_inbound];\n  })\n\n\ntheorem arg_equal_implies_fn_equal (α β: Type) (x y :α) (f: α -> β) (EQ: x = y): f x = f y := by {\n  simp [EQ];\n}\n\ntheorem arg_equal_implies_fn_equal_2 (α β γ: Type) (a a' :α) (b b': β) (f: α -> β -> γ)\n   (EQα: a = a') (EQβ: b = b') : f a b = f a' b' := by {\n  simp [EQα, EQβ];\n}\n\n\n-- getting at a mapped flat index returns the value that's computed by running the map fn at\n-- that index.\ndef Tensor.mapWithFlatIndexCorrect\n   {σ τ: MLIRTy} (v: Tensor σ) (f: TensorFlatIndex (shapeProd v.shape) → σ.eval → τ.eval)\n   (ix: TensorFlatIndex (shapeProd v.shape)):\n  (v.mapWithFlatIndex f).getAtFlatIndex ix = f ix (v.getAtFlatIndex ix) := by {\n  simp [Tensor.getAtFlatIndex];\n  simp [Tensor.mapWithFlatIndex];\n  rewrite [List.zip_flat_index_get (xs := v.data) (getIx := ix.ix)\n      (GETIX := by {  rewrite [v.h_data_size];  simp[ix.h_ix_inbound]; } )];\n  simp;\n  rewrite [TensorFlatIndex.cast_left];\n  rfl;\n}\n\n\n\n/-\n## Ranked tensor type\n\nA ranked tensor extends the runtime tensor with a statically-known list of\ndimensions (all of which may not be known) which adds more invariants.\n-/\n\nstructure RankedTensor (D: DimList) (τ: MLIRTy) extends Tensor τ where\n  -- Invariants: shape/dimension must be compatible, shape/size must match\n  h_refines: D.shapeRefines shape\n\ntheorem RankedTensor.eq_of_fields_eq {τ D} (t₁ t₂: RankedTensor τ D):\n    t₁.shape = t₂.shape → t₁.data = t₂.data → t₁ = t₂ := by\n  intros h_shape h_data\n  suffices t₁.toTensor = t₂.toTensor by\n    cases t₁; cases t₂; simp at *; trivial\n  apply Tensor.eq_of_fields_eq <;> trivial\n\ninstance {τ D}: DecidableEq (RankedTensor τ D) := fun t₁ t₂ => by\n  cases t₁; cases t₂; simp\n  exact inferInstance\n\ndef RankedTensor.uniform {τ} (D: DimList) (v: τ.eval): RankedTensor D τ :=\n  { Tensor.uniform D.defaultRefinement v with\n    h_refines   := DimList.defaultRefinement_refines _ }\n\ninstance {τ D}: Inhabited (RankedTensor D τ) where\n  default := RankedTensor.uniform D default\n\ndef RankedTensor.typeStr (τ: MLIRTy) (D: DimList): String :=\n  let dims := \"x\".intercalate (D.map (MLIR.Doc.Pretty.doc ·))\n  s!\"tensor<{dims}x{τ}>\"\n\ndef RankedTensor.str {τ D} (t: RankedTensor D τ): String :=\n  let dims := \"×\".intercalate (D.map (MLIR.Doc.Pretty.doc ·))\n  let data := \"[\" ++ \" \".intercalate (t.data.map toString) ++ \"]\"\n  s!\"({dims}){data}\"\n\n-- Map a function over a tensor.\ndef RankedTensor.map {σ σ': MLIRTy} {D: DimList} (v: RankedTensor D σ) (f: σ.eval → σ'.eval): RankedTensor D σ' :=\n  let t' : Tensor σ' := v.toTensor.map f\n  let H : D.shapeRefines t'.shape := by {\n      rewrite [← Tensor.map_shape];\n      simp [v.h_refines];\n  };\n  RankedTensor.mk t' H\n\n-- Conversion from TensorElem\n\ndef RankedTensor.ofTensorLiteral (lit: TensorLiteral D τ): RankedTensor D τ :=\n  match τ, lit, lit.h_rank with\n  | .int sgn 1, lit, .UniformBool b _ _ =>\n      RankedTensor.uniform D (FinInt.ofInt 1 (if b then 1 else 0))\n  | .int sgn sz, lit, .UniformInt i _ _ _ _ =>\n      RankedTensor.uniform D (FinInt.ofInt sz i)\n  | .float bitsize, lit, .UniformFloat f _ _ =>\n      RankedTensor.uniform D f\n  | τ, lit, .HasShape s _ Hshape Hrefines =>\n      { shape       := s,\n        data        := lit.elem.flatten lit.h_type,\n        h_refines   := Hrefines,\n        h_data_size := TensorElem.flatten_size lit.elem s Hshape lit.h_type }\n\ndef RankedTensor.ofTensorElem {τ} (D: DimList) (elem: TensorElem)\n    (h_type: elem.hasType τ) (h_rank: elem.rankCompatibleWith D τ):\n    RankedTensor D τ :=\n  ofTensorLiteral { elem, h_type, h_rank }\n\n\nopen DimList in\ndef reshape {τ} {D: DimList} (D': DimList)\n    (H: D.known) (H': D'.known) (Hprod: D'.prod = D.prod):\n    RankedTensor D τ → RankedTensor D' τ :=\n  fun t =>\n    { shape       := D'.project,\n      data        := t.data\n      h_refines   := dim_known_project_refines H',\n      h_data_size := by rw [t.h_data_size, dim_known_prod D' H', Hprod]\n                        rw [dim_known_prod_refines H]\n                        apply t.h_refines }\n\ntheorem reshape_reshape {τ} {D: DimList} (D₁ D₂: DimList)\n    (H: D.known) (H₁: D₁.known) (H₂: D₂.known)\n    (Hprod₁: D₁.prod = D.prod) (Hprod₂: D₂.prod = D₁.prod)\n    (t: RankedTensor D τ):\n      reshape D₂ H₁ H₂ Hprod₂ (reshape D₁ H H₁ Hprod₁ t) =\n      reshape D₂ H H₂ (Eq.trans Hprod₂ Hprod₁) t :=\n  rfl\n\nopen DimList in\ntheorem reshape_self {τ} D H₁ H₂ Hprod (t: RankedTensor D τ):\n    reshape D H₁ H₂ Hprod t = t := by\n  simp [reshape, dim_known_project_eq H₁ t.h_refines]\n\n\n-- Type interface for registration with MLIRType\n\nprivate abbrev σ_RankedTensor := DimList × MLIRTy\nprivate abbrev ε_RankedTensor := fun (D, τ) => RankedTensor D τ\n\ninstance: DialectTypeIntf σ_RankedTensor ε_RankedTensor where\n  inhabited := default\n  typeEq := inferInstance\n  eq := inferInstance\n  str := fun (τ, D) t => t.str\n  typeStr := fun (τ, D) => RankedTensor.typeStr D τ\n\n/-\n## Unranked tensor type\n-/\n\nabbrev UnrankedTensor := Tensor\n\n-- Type interface for registration with MLIRType\n\nprivate abbrev σ_UnrankedTensor := MLIRTy\nprivate abbrev ε_UnrankedTensor := UnrankedTensor\n\ndef UnrankedTensor.typeStr (τ: MLIRTy): String :=\n  s!\"tensor<*{τ}>\"\n\n-- TODO: String representation of ranked tensors\ndef UnrankedTensor.str {τ} (t: UnrankedTensor τ) :=\n  s!\"<an unranked tensor of base type {τ}>\"\n\ninstance: DialectTypeIntf σ_UnrankedTensor ε_UnrankedTensor where\n  inhabited := default\n  typeEq := inferInstance\n  eq := inferInstance\n  str := @UnrankedTensor.str\n  typeStr := UnrankedTensor.typeStr\n\n\n/-\n## Vector type\n-/\n\ndef List.zipMul (as: List Nat) (bs: List Nat) (H: as.length = bs.length): List Nat :=\n match as with\n | [] => match bs with\n         | [] => []\n         | b::bs' => nomatch H\n | a::as' =>\n    match bs with\n    | [] => nomatch H\n    | b::bs' => (a*b):: List.zipMul as' bs' (by { simp at H; assumption })\n\n\ndef Vector.size (fixed scalable: List Nat) (scale: List Nat)\n    (H: scale.length = scalable.length) :=\n  shapeProd fixed * shapeProd (List.zipMul scale scalable H)\n\nstructure Vector (fixed: List Nat) (scalable: List Nat) (τ: MLIRTy) where\n  -- Scales (number of instantiations of each scalable dimension)\n  scale: List Nat\n  -- Contents in row-major order\n  data: List τ.eval\n  -- Invariant: Number of scales must match number of scalable dimensions\n  h_scale_size: scale.length = scalable.length\n  -- Invariant: Number of data elements must match scale\n  h_data_size: data.length = Vector.size fixed scalable scale h_scale_size\n\ntheorem Vector.eq_of_fields_eq (v₁ v₂: Vector τ fixed scalable):\n    v₁.scale = v₂.scale → v₁.data = v₂.data → v₁ = v₂ := by\n  intros h_scale h_data\n  cases v₁; cases v₂; simp at *\n  trivial\n\ninstance: Inhabited (Vector fixed scalable τ) where\n  default :=\n    { scale := List.map (fun _ => 1) scalable,\n      data := List.uniform default (Vector.size fixed scalable\n                (List.map (fun _ => 1) scalable) (by apply List.length_map)),\n      h_scale_size := by apply List.length_map,\n      h_data_size := by apply List.length_uniform }\n\ninstance: DecidableEq (Vector τ fixed scalable) := fun v₁ v₂ =>\n  if h: v₁.scale = v₂.scale ∧ v₁.data = v₂.data then\n    isTrue (by apply Vector.eq_of_fields_eq _ _ h.1 h.2)\n  else\n    isFalse fun h' => by simp [h'] at *\n\ndef Vector.typeStr (fixed scalable: List Nat) (τ: MLIRTy) :=\n  let sf := \"x\".intercalate (fixed.map (MLIR.Doc.Pretty.doc ·))\n  let ss := \"x\".intercalate (scalable.map (MLIR.Doc.Pretty.doc ·))\n\n  if fixed.length > 0 && scalable.length > 0 then\n    s!\"vector<{sf}x[{ss}]x{τ}>\"\n  else if fixed.length > 0 then\n    s!\"vector<{sf}x{τ}>\"\n  else if scalable.length > 0 then\n    s!\"vector<[{ss}]x{τ}>\"\n  else\n    s!\"vector<{τ}>\"\n\n-- TODO: String representation of vector values\ndef Vector.str (v: Vector fixed scalable τ): String :=\n  let str_type := Vector.typeStr fixed scalable τ\n  s!\"<a vector of type {str_type}>\"\n\n-- Type interface for registration with MLIRType\n\nprivate abbrev σ_Vector := List Nat × List Nat × MLIRTy\nprivate abbrev ε_Vector := fun (fixed, scalable, τ) => Vector fixed scalable τ\n\ninstance: DialectTypeIntf σ_Vector ε_Vector where\n  inhabited := default\n  typeEq := inferInstance\n  eq := inferInstance\n  str := fun (τ, fixed, scalable) t => t.str\n  typeStr := fun (τ, fixed, scalable) => Vector.typeStr τ fixed scalable\n\n\n/-\n## TODO: Memref types\n-/\n\ninductive MemrefLayoutSpec where\n| stride: (offset: Dimension) -> (stride: List Dimension) -> MemrefLayoutSpec\n| attr: AttrVal -> MemrefLayoutSpec\n\npartial def docMemrefLayoutSpec(spec: MemrefLayoutSpec) : MLIR.Doc.Doc :=\nmatch spec with\n| MemrefLayoutSpec.stride offset strides => [doc| \"offset:\" offset \", strides: \" \"[\" (strides),* \"]\"]\n| MemrefLayoutSpec.attr v => docAttrVal v\n\n-- Text representation\n\n/-| MLIRTy.memrefRanked dims ty layout? memspace? =>\n    let docLayout := match layout? with | some x => [doc| \",\" (docMemrefLayoutSpec x)] | none => \"\"\n    let docMemspace := match memspace? with | some x => [doc| \",\" (docAttrVal x)] | none => \"\"\n    [doc| \"memref<\" (intercalate_doc dims \"x\") \"x\" (go ty) (docLayout)  (docMemspace) \">\"]\n  | MLIRTy.memrefUnranked ty memspace? =>\n    let docMemspace := match memspace? with | some x => [doc| \",\" (docAttrVal x)] | none => \"\"\n    [doc| \"memref<\" \"*x\" (go ty) (docMemspace) \">\"] -/\n\n\n/-\n## Summary of types\n-/\n\nabbrev builtin.σ :=\n  (σ_RankedTensor ⊕ σ_UnrankedTensor) ⊕ σ_Vector\nabbrev builtin.ε :=\n  Sum.cases\n    (Sum.cases ε_RankedTensor ε_UnrankedTensor)\n    ε_Vector\n\n@[match_pattern]\ndef builtin.σ.tensor (D: DimList) (τ: MLIRTy): builtin.σ :=\n  Sum.inl (Sum.inl (D, τ))\n\n@[match_pattern]\ndef builtin.σ.tensor_unranked (τ: MLIRTy): builtin.σ :=\n  Sum.inl (Sum.inr τ)\n\n@[match_pattern]\ndef builtin.σ.vector (fixed scalable: List Nat) (τ: MLIRTy): builtin.σ :=\n  Sum.inr (fixed, scalable, τ)\n\n\n/-\n## Dense vector/tensor attribute\n-/\n\nstructure DenseAttr: Type where\n  elem: TensorElem\n  τ_sig: builtin.σ\nderiving DecidableEq\n\n-- TODO: String representation of dense<> attributes via TensorElem\ndef DenseAttr.str (a: DenseAttr): String :=\n  let τ_str :=\n    match a.τ_sig with\n    | builtin.σ.tensor τ D => RankedTensor.typeStr D τ\n    | builtin.σ.tensor_unranked τ => UnrankedTensor.typeStr τ\n    | builtin.σ.vector f s τ => Vector.typeStr f s τ\n  s!\"dense<...>: {τ_str}\"\n\nprivate abbrev α_DenseAttr := DenseAttr\n\ninstance: DialectAttrIntf α_DenseAttr where\n  eq := inferInstance\n  str := DenseAttr.str\n\n\n/-\n## Summary of attributes\n-/\n\nabbrev builtin.α :=\n  α_DenseAttr\n\n\n/-\n## Builtin dialect definition\n-/\n\ninstance builtin: Dialect builtin.α builtin.σ builtin.ε where\n  name := \"builtin\"\n  iα := inferInstance\n  iε := inferInstance\n\n-- Custom types\n\n@[match_pattern, simp]\ndef builtin.tensor (D: DimList) (τ: MLIRTy): MLIRType builtin :=\n  MLIRType.extended (builtin.σ.tensor D τ)\n\n@[match_pattern]\ndef builtin.tensor_unranked (τ: MLIRTy): MLIRType builtin :=\n  MLIRType.extended (builtin.σ.tensor_unranked τ)\n\n@[match_pattern]\ndef builtin.vector (fixed scalable: List Nat) (τ: MLIRTy):\n    MLIRType builtin :=\n  MLIRType.extended (builtin.σ.vector fixed scalable τ)\n\n@[match_pattern]\ndef builtin.memref (D: DimList) (τ: MLIRTy) (layout: Option MemrefLayoutSpec)\n    (memspace: Option AttrVal): MLIRType builtin :=\n  MLIRType.undefined \"builtin.memref\"\n\n@[match_pattern]\ndef builtin.memref_unranked (τ: MLIRTy) (memspace: Option AttrVal):\n    MLIRType builtin :=\n  MLIRType.undefined \"builtin.memref_unranked\"\n\n-- Custom attributes\n\n@[match_pattern]\ndef builtin.dense_vector_attr (e: TensorElem) (fixed scalable: List Nat)\n    (τ: MLIRTy): AttrValue builtin :=\n  AttrValue.extended (DenseAttr.mk e (builtin.σ.vector fixed scalable τ))\n\n@[match_pattern]\ndef builtin.dense_tensor_attr (e: TensorElem) (D: DimList) (τ: MLIRTy):\n    AttrValue builtin :=\n  AttrValue.extended (DenseAttr.mk e (builtin.σ.tensor D τ))\n\n@[match_pattern]\ndef builtin.dense_attr (e: TensorElem) {s: builtin.σ}: AttrValue builtin :=\n  AttrValue.extended (DenseAttr.mk e s)\n\n\n/-\n## High-level utilities\n-/\n\n-- Create a dense vector from a vector type\n-- FIXME: Does AttrVal.dense actually also support tensor types??\ndef builtin.denseWithType (e: TensorElem) (τ: MLIRType builtin):\n    AttrValue builtin :=\n  match τ with\n  | builtin.tensor D τ =>\n      builtin.dense_tensor_attr e D τ\n  | builtin.vector fixed scalable τ =>\n      builtin.dense_vector_attr e fixed scalable τ\n  | _ =>\n      panic! s!\"buitin.denseVectorWithType: {τ} not a vector type\"\n\n-- Create a dense vector with values `xs` and type `vector<len(xs)*ity>`\ndef builtin.denseVectorOfList (xs: List Int) (ity: MLIRTy := .i32):\n    AttrValue builtin :=\n  builtin.dense_vector_attr xs [xs.length] [] ity\n\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Dialects/BuiltinModel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3290283238714608}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nAn experimental variant on the `ring` tactic that uses computational\nreflection instead of proof generation. Useful for kernel benchmarking.\n-/\nimport tactic.ring data.num.lemmas\nimport tactic.converter.interactive\n\nnamespace tactic.ring2\n\n@[derive has_reflect]\ninductive {u} tree (α : Type u) : Type u\n| nil {} : tree\n| node : α → tree → tree → tree\n\ndef tree.get {α} [has_zero α] : pos_num → tree α → α\n| _                tree.nil            := 0\n| pos_num.one      (tree.node a t₁ t₂) := a\n| (pos_num.bit0 n) (tree.node a t₁ t₂) := t₁.get n\n| (pos_num.bit1 n) (tree.node a t₁ t₂) := t₂.get n\n\ndef tree.of_rbnode {α} : rbnode α → tree α\n| rbnode.leaf               := tree.nil\n| (rbnode.red_node l a r)   :=\n  tree.node a (tree.of_rbnode l) (tree.of_rbnode r)\n| (rbnode.black_node l a r) :=\n  tree.node a (tree.of_rbnode l) (tree.of_rbnode r)\n\ndef tree.index_of {α} (lt : α → α → Prop) [decidable_rel lt]\n  (x : α) : tree α → option pos_num\n| tree.nil := none\n| (tree.node a t₁ t₂) :=\n  match cmp_using lt x a with\n  | ordering.lt := pos_num.bit0 <$> tree.index_of t₁\n  | ordering.eq := some pos_num.one\n  | ordering.gt := pos_num.bit1 <$> tree.index_of t₂\n  end\n\nmeta def tree.reflect' (u : level) (α : expr) : tree expr → expr\n| tree.nil := (expr.const ``tree.nil [u] : expr) α\n| (tree.node a t₁ t₂) :=\n  (expr.const ``tree.node [u] : expr) α a t₁.reflect' t₂.reflect'\n\n@[derive has_reflect]\ninductive csring_expr\n| atom : pos_num → csring_expr\n| const : num → csring_expr\n| add : csring_expr → csring_expr → csring_expr\n| mul : csring_expr → csring_expr → csring_expr\n| pow : csring_expr → num → csring_expr\n\nnamespace csring_expr\n\ndef eval {α} [comm_semiring α] (t : tree α) : csring_expr → α\n| (atom n)  := @tree.get _ ⟨0⟩ n t\n| (const n) := n\n| (add x y) := eval x + eval y\n| (mul x y) := eval x * eval y\n| (pow x n) := eval x ^ (n : ℕ)\n\nend csring_expr\n\n@[derive decidable_eq]\ninductive horner_expr\n| const : znum → horner_expr\n| horner : horner_expr → pos_num → num → horner_expr → horner_expr\n\nnamespace horner_expr\n\ndef is_cs : horner_expr → Prop\n| (const n) := ∃ m:num, n = m.to_znum\n| (horner a x n b) := is_cs a ∧ is_cs b\n\ninstance : has_zero horner_expr := ⟨const 0⟩\ninstance : has_one horner_expr := ⟨const 1⟩\n\ndef atom (n : pos_num) : horner_expr := horner 1 n 1 0\n\ndef repr : horner_expr → string\n| (const n) := _root_.repr n\n| (horner a x n b) :=\n    \"(\" ++ repr a ++ \") * x\" ++ _root_.repr x ++ \"^\"\n        ++ _root_.repr n ++ \" + \" ++ repr b\ninstance : has_repr horner_expr := ⟨repr⟩\n\ndef horner' (a : horner_expr)\n  (x : pos_num) (n : num) (b : horner_expr) : horner_expr :=\nmatch a with\n| const q := if q = 0 then b else horner a x n b\n| horner a₁ x₁ n₁ b₁ :=\n  if x₁ = x ∧ b₁ = 0 then horner a₁ x (n₁ + n) b\n  else horner a x n b\nend\n\ndef add_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then e else begin\n  induction e with n a x n b A B,\n  { exact const (k + n) },\n  { exact horner a x n B }\nend\n\ndef add_aux (a₁ : horner_expr) (A₁ : horner_expr → horner_expr) (x₁ : pos_num) :\n  horner_expr → num → horner_expr → (horner_expr → horner_expr) → horner_expr\n| (const n₂)           n₁ b₁ B₁ := add_const n₂ (horner a₁ x₁ n₁ b₁)\n| (horner a₂ x₂ n₂ b₂) n₁ b₁ B₁ :=\n  let e₂ := horner a₂ x₂ n₂ b₂ in\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner a₂ x₂ n₂ (add_aux b₂ n₁ b₁ B₁)\n  | ordering.eq :=\n    match num.sub' n₁ n₂ with\n    | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n    | (znum.pos k) := horner (add_aux a₂ k 0 id) x₁ n₂ (B₁ b₂)\n    | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n    end\n  end\n\ndef add : horner_expr → horner_expr → horner_expr\n| (const n₁)           e₂ := add_const n₁ e₂\n| (horner a₁ x₁ n₁ b₁) e₂ := add_aux a₁ (add a₁) x₁ e₂ n₁ b₁ (add b₁)\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact add_const n₁ e₂ },\n  exact match e₂ with e₂ := begin\n    induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁;\n    let e₁ := horner a₁ x₁ n₁ b₁,\n    { exact add_const n₂ e₁ },\n    let e₂ := horner a₂ x₂ n₂ b₂,\n    exact match pos_num.cmp x₁ x₂ with\n    | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n    | ordering.gt := horner a₂ x₂ n₂ (B₂ n₁ b₁)\n    | ordering.eq :=\n      match num.sub' n₁ n₂ with\n      | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n      | (znum.pos k) := horner (A₂ k 0) x₁ n₂ (B₁ b₂)\n      | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n      end\n    end\n  end end\nend-/\n\ndef neg (e : horner_expr) : horner_expr :=\nbegin\n  induction e with n a x n b A B,\n  { exact const (-n) },\n  { exact horner A x n B }\nend\n\ndef mul_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then 0 else if k = 1 then e else begin\n  induction e with n a x n b A B,\n  { exact const (n * k) },\n  { exact horner A x n B }\nend\n\ndef mul_aux (a₁ x₁ n₁ b₁) (A₁ B₁ : horner_expr → horner_expr) :\n  horner_expr → horner_expr\n| (const n₂) := mul_const n₂ (horner a₁ x₁ n₁ b₁)\n| e₂@(horner a₂ x₂ n₂ b₂) :=\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner (A₁ e₂) x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner (mul_aux a₂) x₂ n₂ (mul_aux b₂)\n  | ordering.eq := let haa := horner' (mul_aux a₂) x₁ n₂ 0 in\n    if b₂ = 0 then haa else haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂))\n  end\n\ndef mul : horner_expr → horner_expr → horner_expr\n| (const n₁)              := mul_const n₁\n| e₁@(horner a₁ x₁ n₁ b₁) := mul_aux a₁ x₁ n₁ b₁ (mul a₁) (mul b₁).\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact mul_const n₁ e₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂;\n  let e₁ := horner a₁ x₁ n₁ b₁,\n  { exact mul_const n₂ e₁ },\n  let e₂ := horner a₂ x₂ n₂ b₂,\n  cases pos_num.cmp x₁ x₂,\n  { exact horner (A₁ e₂) x₁ n₁ (B₁ e₂) },\n  { let haa := horner' A₂ x₁ n₂ 0,\n    exact if b₂ = 0 then haa else\n      haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂)) },\n  { exact horner A₂ x₂ n₂ B₂ }\nend-/\n\ninstance : has_add horner_expr := ⟨add⟩\ninstance : has_neg horner_expr := ⟨neg⟩\ninstance : has_mul horner_expr := ⟨mul⟩\n\ndef pow (e : horner_expr) : num → horner_expr\n| 0 := 1\n| (num.pos p) := begin\n  induction p with p ep p ep,\n  { exact e },\n  { exact (ep.mul ep).mul e },\n  { exact ep.mul ep }\nend\n\ndef inv (e : horner_expr) : horner_expr := 0\n\ndef of_csexpr : csring_expr → horner_expr\n| (csring_expr.atom n)  := atom n\n| (csring_expr.const n) := const n.to_znum\n| (csring_expr.add x y) := (of_csexpr x).add (of_csexpr y)\n| (csring_expr.mul x y) := (of_csexpr x).mul (of_csexpr y)\n| (csring_expr.pow x n) := (of_csexpr x).pow n\n\ndef cseval {α} [comm_semiring α] (t : tree α) : horner_expr → α\n| (const n)        := n.abs\n| (horner a x n b) := tactic.ring.horner (cseval a) (t.get x) n (cseval b)\n\ntheorem cseval_atom {α} [comm_semiring α] (t : tree α)\n  (n : pos_num) : (atom n).is_cs ∧ cseval t (atom n) = t.get n :=\n⟨⟨⟨1, rfl⟩, ⟨0, rfl⟩⟩, (tactic.ring.horner_atom _).symm⟩\n\ntheorem cseval_add_const {α} [comm_semiring α] (t : tree α)\n  (k : num) {e : horner_expr} (cs : e.is_cs) :\n  (add_const k.to_znum e).is_cs ∧\n    cseval t (add_const k.to_znum e) = k + cseval t e :=\nbegin\n  simp [add_const],\n  cases k; simp! *,\n  simp [show znum.pos k ≠ 0, from dec_trivial],\n  induction e with n a x n b A B; simp *,\n  { rcases cs with ⟨n, rfl⟩,\n    refine ⟨⟨n + num.pos k, by simp; refl⟩, _⟩,\n    cases n; simp! },\n  { rcases B cs.2 with ⟨csb, h⟩, simp! [*, cs.1],\n    rw [← tactic.ring.horner_add_const, add_comm], refl }\nend\n\ntheorem cseval_horner' {α} [comm_semiring α] (t : tree α)\n  (a x n b) (h₁ : is_cs a) (h₂ : is_cs b) :\n  (horner' a x n b).is_cs ∧ cseval t (horner' a x n b) =\n    tactic.ring.horner (cseval t a) (t.get x) n (cseval t b) :=\nbegin\n  cases a with n₁ a₁ x₁ n₁ b₁; simp [horner']; split_ifs,\n  { simp! [*, tactic.ring.horner] },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ },\n  { refine ⟨⟨h₁.1, h₂⟩, eq.symm _⟩, simp! *,\n    apply tactic.ring.horner_horner, simp },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ }\nend\n\ntheorem cseval_add {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (add e₁ e₂).is_cs ∧\n  cseval t (add e₁ e₂) = cseval t e₁ + cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa using cseval_add_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simp! [cseval_add_const t n₂ cs₁] },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { rcases B₁ csb₁ cs₂ with ⟨csh, h⟩,\n    refine ⟨⟨csa₁, csh⟩, eq.symm _⟩,\n    apply tactic.ring.horner_add_const,\n    exact h.symm },\n  { cases C,\n    have B0 : is_cs 0 → ∀ {e₂ : horner_expr}, is_cs e₂ →\n      is_cs (add 0 e₂) ∧ cseval t (add 0 e₂) = cseval t 0 + cseval t e₂ :=\n      λ _ e₂ c, ⟨c, (zero_add _).symm⟩,\n     cases e : num.sub' n₁ n₂ with k k; simp!,\n    { have : n₁ = n₂,\n      { have := congr_arg (coe : znum → ℤ) e,\n        simp at this,\n        have := sub_eq_zero.1 this,\n        rw [← num.to_nat_to_int, ← num.to_nat_to_int] at this,\n        exact num.to_nat_inj.1 (int.coe_nat_inj this) },\n      subst n₂,\n      rcases cseval_horner' _ _ _ _ _ _ _ with ⟨csh, h⟩,\n      { refine ⟨csh, h.trans (eq.symm _)⟩,\n        simp *,\n        apply tactic.ring.horner_add_horner_eq; try {refl},\n        simp },\n      all_goals {simp! *} },\n    { simp [B₁ csb₁ csb₂],\n      rcases A₂ csa₂ _ _ B0 ⟨csa₁, 0, rfl⟩ with ⟨csh, h⟩,\n      refine ⟨csh, eq.symm _⟩,\n      rw [show id = add 0, from rfl, h],\n      apply tactic.ring.horner_add_horner_gt,\n      { change (_ + k : ℕ) = _,\n        rw [← int.coe_nat_inj', int.coe_nat_add,\n          eq_comm, ← sub_eq_iff_eq_add'],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      all_goals { apply add_comm } },\n    { have : (horner a₂ x₁ (num.pos k) 0).is_cs := ⟨csa₂, 0, rfl⟩,\n      simp [B₁ csb₁ csb₂, A₁ csa₁ this],\n      symmetry, apply tactic.ring.horner_add_horner_lt,\n      { change (_ + k : ℕ) = _,\n          rw [← int.coe_nat_inj', int.coe_nat_add,\n            eq_comm, ← sub_eq_iff_eq_add', ← neg_inj', neg_sub],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      { refl }, { apply add_comm } } },\n  { rcases B₂ csb₂ _ _ B₁ ⟨csa₁, csb₁⟩ with ⟨csh, h⟩,\n    refine ⟨⟨csa₂, csh⟩, eq.symm _⟩,\n    apply tactic.ring.const_add_horner,\n    simp [h] }\nend\n\n\n\ntheorem cseval_mul {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (mul e₁ e₂).is_cs ∧\n  cseval t (mul e₁ e₂) = cseval t e₁ * cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa [mul_comm] using cseval_mul_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simpa! using cseval_mul_const t n₂ cs₁ },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases A₂ csa₂ with csA₂ hA₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { simp [A₁ csa₁ cs₂, B₁ csb₁ cs₂],\n    symmetry, apply tactic.ring.horner_mul_const; refl },\n  { cases cseval_horner' t _ x₁ n₂ 0 csA₂ ⟨0, rfl⟩ with csh₁ h₁,\n    cases C, split_ifs,\n    { subst b₂,\n      refine ⟨csh₁, h₁.trans (eq.symm _)⟩,\n      apply tactic.ring.horner_mul_horner_zero; try {refl},\n      simp! [hA₂] },\n    { cases A₁ csa₁ csb₂ with csA₁ hA₁,\n      cases cseval_add t csh₁ _ with csh₂ h₂,\n      { refine ⟨csh₂, h₂.trans (eq.symm _)⟩,\n        apply tactic.ring.horner_mul_horner; try {refl},\n        simp! * },\n      exact ⟨csA₁, (B₁ csb₁ csb₂).1⟩ } },\n  { simp [A₂ csa₂, B₂ csb₂], rw [mul_comm, eq_comm],\n    apply tactic.ring.horner_const_mul,\n    {apply mul_comm}, {refl} },\nend\n\ntheorem cseval_pow {α} [comm_semiring α] (t : tree α)\n  {x : horner_expr} (cs : x.is_cs) :\n  ∀ (n : num), (pow x n).is_cs ∧\n    cseval t (pow x n) = cseval t x ^ (n : ℕ)\n| 0 := ⟨⟨1, rfl⟩, (pow_zero _).symm⟩\n| (num.pos p) := begin\n  simp [pow], induction p with p ep p ep,\n  { simp * },\n  { simp [pow_bit1],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    cases cseval_mul t cs₀ cs with cs₁ h₁,\n    simp * },\n  { simp [pow_bit0],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    simp * }\nend\n\ntheorem cseval_of_csexpr {α} [comm_semiring α] (t : tree α) :\n  ∀ (r : csring_expr), (of_csexpr r).is_cs ∧ cseval t (of_csexpr r) = r.eval t\n| (csring_expr.atom n)  := cseval_atom _ _\n| (csring_expr.const n) := ⟨⟨n, rfl⟩, by cases n; refl⟩\n| (csring_expr.add x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_add t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.mul x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_mul t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.pow x n) :=\n  let ⟨cs, h⟩ := cseval_of_csexpr x,\n      ⟨cs, h⟩ := cseval_pow t cs n in ⟨cs, by simp! [h, *]⟩\n\nend horner_expr\n\ntheorem correctness {α} [comm_semiring α] (t : tree α) (r₁ r₂ : csring_expr)\n  (H : horner_expr.of_csexpr r₁ = horner_expr.of_csexpr r₂) :\n  r₁.eval t = r₂.eval t :=\nby repeat {rw ← (horner_expr.cseval_of_csexpr t _).2}; rw H\n\nmeta def reflect_expr : expr → csring_expr × dlist expr\n| `(%%e₁ + %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂, l₁ ++ l₂)\n/-| `(%%e₁ - %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂.neg, l₁ ++ l₂)\n| `(- %%e) := let (r, l) := reflect_expr e in (r.neg, l)-/\n| `(%%e₁ * %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂, l₁ ++ l₂)\n/-| `(has_inv.inv %%e) := let (r, l) := reflect_expr e in (r.neg, l)\n| `(%%e₁ / %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂.inv, l₁ ++ l₂)-/\n| e@`(%%e₁ ^ %%e₂) :=\n  match reflect_expr e₁, expr.to_nat e₂ with\n  | (r₁, l₁), some n₂ := (r₁.pow (num.of_nat' n₂), l₁)\n  | (r₁, l₁), none := (csring_expr.atom 1, dlist.singleton e)\n  end\n| e := match expr.to_nat e with\n  | some n := (csring_expr.const (num.of_nat' n), dlist.empty)\n  | none := (csring_expr.atom 1, dlist.singleton e)\n  end\n\nmeta def csring_expr.replace (t : tree expr) : csring_expr → state_t (list expr) option csring_expr\n| (csring_expr.atom _)  := do e ← get,\n  p ← monad_lift (t.index_of (<) e.head),\n  put e.tail, pure (csring_expr.atom p)\n| (csring_expr.const n) := pure (csring_expr.const n)\n| (csring_expr.add x y) := csring_expr.add <$> x.replace <*> y.replace\n| (csring_expr.mul x y) := csring_expr.mul <$> x.replace <*> y.replace\n| (csring_expr.pow x n) := (λ x, csring_expr.pow x n) <$> x.replace\n--| (csring_expr.neg x)   := csring_expr.neg <$> x.replace\n--| (csring_expr.inv x)   := csring_expr.inv <$> x.replace\n\nend tactic.ring2\n\nnamespace tactic\nnamespace interactive\nopen interactive interactive.types lean.parser\nopen tactic.ring2\n\nlocal postfix `?`:9001 := optional\n\n/-- Tactic for solving equations in the language of rings.\n  This variant on the `ring` tactic uses kernel computation instead\n  of proof generation. -/\nmeta def ring2 : tactic unit :=\ndo `[repeat {rw ← nat.pow_eq_pow}],\n  `(%%e₁ = %%e₂) ← target,\n  α ← infer_type e₁,\n  expr.sort (level.succ u) ← infer_type α,\n  let (r₁, l₁) := reflect_expr e₁,\n  let (r₂, l₂) := reflect_expr e₂,\n  let L := (l₁ ++ l₂).to_list,\n  let s := tree.of_rbnode (rbtree_of L).1,\n  (r₁, L) ← (state_t.run (r₁.replace s) L : option _),\n  (r₂, _) ← (state_t.run (r₂.replace s) L : option _),\n  let se : expr := s.reflect' u α,\n  let er₁ : expr := reflect r₁,\n  let er₂ : expr := reflect r₂,\n  cs ← mk_app ``comm_semiring [α] >>= mk_instance,\n  e ← to_expr ```(correctness %%se %%er₁ %%er₂ rfl)\n    <|> fail (\"ring2 tactic failed, using abstract equality:\\n\"\n      ++ repr (horner_expr.of_csexpr r₁) ++\n      \"\\n  =?=\\n\" ++ repr (horner_expr.of_csexpr r₂)),\n  tactic.exact e\n\nend interactive\nend tactic\n\nnamespace conv.interactive\nopen conv\n\nmeta def ring2 : conv unit := discharge_eq_lhs tactic.interactive.ring2\n\nend conv.interactive\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/tactic/ring2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.32895962073500756}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nCertified graph transformation that integrates out a specific KL divergence term.\n-/\nimport .util .tensor .compute_grad .graph .tactics .predicates .lemmas .env .expected_value .ops\n\nnamespace certigrad\nopen list\n\ndef integrate_mvn_kl (eloss : ID) : list node → list node\n| [] := []\n\n| (⟨(z, .(shape)), [(μ, .(shape)), (σ, .(shape))], operator.rand (rand.op.mvn shape)⟩\n ::⟨(el, []), [(μ', .(shape')), (σ', .(shape')), (z', .(shape'))], operator.det (det.op.mvn_empirical_kl shape')⟩\n ::nodes) :=\n (⟨(el, []), [⟨μ, shape⟩, ⟨σ, shape⟩], operator.det (ops.mvn_kl shape)⟩\n::⟨(z, shape), [⟨μ, shape⟩, ⟨σ, shape⟩], operator.rand (rand.op.mvn shape)⟩\n::nodes)\n\n| (⟨(z, .(shape)), [(μ, .(shape)), (σ, .(shape))], operator.rand (rand.op.mvn shape)⟩ :: nodes) :=\n⟨(z, shape), [(μ, shape), (σ, shape)], operator.rand (rand.op.mvn shape)⟩ :: integrate_mvn_kl nodes\n\n| (n::nodes) := n :: integrate_mvn_kl nodes\n\ndef integrate_mvn_kl_pre (eloss : ID) : list node → env → Prop\n-- EQ1\n| [] _ := true\n-- EQ2 (a)\n| (⟨(rname, rshape), [], operator.det op⟩::nodes) inputs :=\nintegrate_mvn_kl_pre nodes (env.insert (rname, rshape) (op^.f (env.get_ks [] inputs)) inputs)\n-- EQ2 (b)\n| (⟨(rname, rshape), [], operator.rand op⟩::nodes) inputs := false\n-- EQ3 (a)\n| (⟨(rname, rshape), [(pname, pshape)], operator.det op⟩::nodes) inputs :=\nintegrate_mvn_kl_pre nodes (env.insert (rname, rshape) (op^.f (env.get_ks [(pname, pshape)] inputs)) inputs)\n-- EQ3 (b)\n| (⟨(rname, rshape), [(pname, pshape)], operator.rand op⟩::nodes) inputs := false\n-- EQ4\n| (⟨(rname, rshape), [(pname₁, pshape₁), (pname₂, pshape₂)], operator.det op⟩::nodes) inputs :=\nintegrate_mvn_kl_pre nodes (env.insert (rname, rshape) (op^.f (env.get_ks [(pname₁, pshape₁), (pname₂, pshape₂)] inputs)) inputs)\n-- EQ5\n| [⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩] inputs := false\n-- EQ6\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [], op⟩::nodes) inputs := false\n-- EQ7\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃)], op⟩::nodes) inputs := false\n-- EQ8\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄)], op⟩::nodes) inputs := false\n-- EQ9\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄), (pname₅, shape₅)], operator.det (det.op.mk _ _ _ _ _ _ _)⟩::nodes) inputs := false\n-- EQ10\n| (⟨(z, .(shape)), [(μ, .(shape)), (σ, .(shape))], operator.rand (rand.op.mvn shape)⟩\n ::⟨(el, []), [(μ', .(shape')), (σ', .(shape')), (z', .(shape'))], operator.det (det.op.mvn_empirical_kl shape')⟩\n ::nodes) inputs :=\n(μ = μ' ∧ σ = σ' ∧ z = z' ∧ shape = shape' ∧ eloss = el ∧ σ ≠ μ)\n∧ ((¬ env.has_key (eloss, []) inputs) ∧ (eloss, []) ∉ (z, shape) :: map node.ref nodes ∧ 0 < env.get (σ, shape) inputs ∧ ∀ (y : T shape), all_parents_in_env (env.insert (z, shape) y inputs) nodes)\n-- EQ11\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄), (pname₅, shape₅)], operator.rand op⟩::nodes) inputs := false\n-- EQ12\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), ((pname₃, shape₃)::(pname₄, shape₄)::(pname₅, shape₅)::parent::parents), op⟩::nodes) inputs := false\n-- EQ13\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, (shape₂ :: shapes₂)), parents, op⟩::nodes) inputs := false\n-- EQ14 (a)\n| (⟨(rname₂, shape₂), ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents), operator.det op⟩::nodes) inputs :=\nintegrate_mvn_kl_pre nodes (env.insert (rname₂, shape₂) (op^.f $ env.get_ks ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents) inputs) inputs)\n-- EQ14 (b)\n| (⟨(rname₂, shape₂), ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents), operator.rand op⟩::nodes) inputs := false\n\ntheorem integrate_mvn_kl_correct (eloss : ID) (costs : list ID) :\n∀ (nodes : list node) (inputs : env),\n  eloss ∉ costs →\n  integrate_mvn_kl_pre eloss nodes inputs →\n  uniq_ids nodes inputs →\n  all_parents_in_env inputs nodes →\n  pdfs_exist_at nodes inputs →\n  is_gintegrable (λ m, ⟦sum_costs m (eloss::costs)⟧) inputs (integrate_mvn_kl eloss nodes) dvec.head →\n  is_gintegrable (λ m, ⟦sum_costs m (eloss::costs)⟧) inputs nodes dvec.head →\nE (graph.to_dist (λ env₀, ⟦sum_costs env₀ (eloss::costs)⟧) inputs (integrate_mvn_kl eloss nodes)) dvec.head\n=\nE (graph.to_dist (λ env₀, ⟦sum_costs env₀ (eloss::costs)⟧) inputs nodes) dvec.head\n\n-- EQ1\n| [] _ _ _ _ _ _ _ _ := rfl\n\n-- EQ2 (a)\n| (⟨(rname, rshape), [], operator.det op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint :=\nbegin\ndunfold graph.to_dist operator.to_dist integrate_mvn_kl,\ndunfold integrate_mvn_kl_pre at H_pre,\nsimp [E.E_bind, E.E_ret],\nassertv H_pre_next : integrate_mvn_kl_pre eloss nodes (env.insert (rname, rshape) (op^.f (env.get_ks nil inputs)) inputs) := H_pre,\nassertv H_ps_in_env_next : all_parents_in_env (env.insert (rname, rshape) (op^.f (env.get_ks nil inputs)) inputs) nodes := H_ps_in_env^.right _,\nexact (integrate_mvn_kl_correct _ _ H_eloss_not_cost H_pre_next (H_uids^.right _) H_ps_in_env_next H_pdfs_exist_at H_kl_gint H_gint)\nend\n\n-- EQ2 (b)\n| (⟨(rname, rshape), [], operator.rand op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ3 (a)\n| (⟨(rname, rshape), [(pname, pshape)], operator.det op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint :=\nbegin\ndunfold graph.to_dist operator.to_dist integrate_mvn_kl,\ndunfold integrate_mvn_kl_pre at H_pre,\nsimp [E.E_bind, E.E_ret],\nassertv H_pre_next : integrate_mvn_kl_pre eloss nodes (env.insert (rname, rshape) (op^.f (env.get_ks [(pname, pshape)] inputs)) inputs) := H_pre,\nassertv H_ps_in_env_next : all_parents_in_env (env.insert (rname, rshape) (op^.f (env.get_ks [(pname, pshape)] inputs)) inputs) nodes := H_ps_in_env^.right _,\nexact (integrate_mvn_kl_correct _ _ H_eloss_not_cost H_pre_next (H_uids^.right _) H_ps_in_env_next H_pdfs_exist_at H_kl_gint H_gint)\nend\n\n-- EQ3 (b)\n| (⟨(rname, rshape), [(pname, pshape)], operator.rand op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ4\n| (⟨(rname, rshape), [(pname₁, pshape₁), (pname₂, pshape₂)], operator.det op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint :=\nbegin\ndunfold graph.to_dist operator.to_dist integrate_mvn_kl,\ndunfold integrate_mvn_kl_pre at H_pre,\nsimp [E.E_bind, E.E_ret],\nassertv H_pre_next : integrate_mvn_kl_pre eloss nodes (env.insert (rname, rshape) (op^.f (env.get_ks [(pname₁, pshape₁), (pname₂, pshape₂)] inputs)) inputs) := H_pre,\nassertv H_ps_in_env_next : all_parents_in_env (env.insert (rname, rshape) (op^.f (env.get_ks [(pname₁, pshape₁), (pname₂, pshape₂)] inputs)) inputs) nodes := H_ps_in_env^.right _,\nexact (integrate_mvn_kl_correct _ _ H_eloss_not_cost H_pre_next (H_uids^.right _) H_ps_in_env_next H_pdfs_exist_at H_kl_gint H_gint)\nend\n\n-- EQ5\n| [⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩] inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ6\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [], op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ7\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃)], op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ8\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄)], op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ9\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄), (pname₅, shape₅)], operator.det (det.op.mk _ _ _ _ _ _ _)⟩::nodes) inputs\nH_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ10\n| (⟨(z, .(shape)), [(μ, .(shape)), (σ, .(shape))], operator.rand (rand.op.mvn shape)⟩\n ::⟨(el, []), [(μ', .(shape')), (σ', .(shape')), (z', .(shape'))], operator.det (det.op.mvn_empirical_kl shape')⟩\n ::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint :=\nbegin\nassertv H_ok : μ = μ' ∧ σ = σ' ∧ z = z' ∧ shape = shape' ∧ eloss = el ∧ σ ≠ μ := H_pre^.left,\n\ncases H_ok with H_μ H_ok, subst H_μ,\ncases H_ok with H_σ H_ok, subst H_σ,\ncases H_ok with H_z H_ok, subst H_z,\ncases H_ok with H_shape H_ok, subst H_shape,\ncases H_ok with H_eloss_eq_el H_σ_neq_μ, subst H_eloss_eq_el,\n\ndunfold graph.to_dist operator.to_dist dvec.head integrate_mvn_kl,\nsimp [E.E_bind, E.E_ret],\ndunfold dvec.head, dsimp,\n\nassertv H_μ_in : env.has_key (μ, shape) inputs := H_ps_in_env^.left (μ, shape) (mem_cons_self _ _),\nassertv H_σ_in : env.has_key (σ, shape) inputs := H_ps_in_env^.left (σ, shape) (mem_cons_of_mem _ (mem_cons_self _ _)),\nassertv H_z_nin : ¬ env.has_key (z, shape) inputs := H_uids^.left,\nassertv H_eloss_nin : ¬ env.has_key (eloss, []) inputs := H_pre^.right^.left,\n\nassertv H_μ_neq_z : (μ, shape) ≠ (z, shape) := env_in_nin_ne H_μ_in H_z_nin,\nassertv H_σ_neq_z : (σ, shape) ≠ (z, shape) := env_in_nin_ne H_σ_in H_z_nin,\nassertv H_μ_neq_eloss : (μ, shape) ≠ (eloss, []) := env_in_nin_ne H_μ_in H_eloss_nin,\nassertv H_σ_neq_eloss : (σ, shape) ≠ (eloss, []) := env_in_nin_ne H_σ_in H_eloss_nin,\n\nassertv H_eloss_neq_z : (eloss, []) ≠ (z, shape) := ne_of_not_mem_cons H_pre^.right^.right^.left,\nassertv H_eloss_nin_nodes : (eloss, []) ∉ map node.ref nodes := not_mem_of_not_mem_cons H_pre^.right^.right^.left,\n\ndunfold env.get_ks,\ntactic.dget_dinsert,\ndunfold det.op.f ops.mvn_kl ops.mvn_kl.f dvec.head dvec.head2 dvec.head3,\n-- TODO(dhs): unbelievable that this won't unfold det.op.f\n-- I tried simp as well\n-- I copy-pasted to a new file (to avoid the 45 second delay) and all variants worked\ndsimp [det.op.f, ops.mvn_kl.f, dvec.head, dvec.head2, dvec.head3],\n\n-- TODO(dhs): I think this is a bug in LEAN. (z, shape) and (eloss, []) are in the type of the last argument.\nsimp only [λ (x : dvec T [shape]),\n               @env.insert_insert_flip (z, shape) (eloss, []) x^.head\n                                       (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape))\n                                       inputs (ne.symm H_eloss_neq_z)],\n\n-- Two steps:\n-- 1. Split out eloss, and prove that it equals the lookup val\n-- 2. For RHS, use eloss not appearing to remove them both\ndunfold det.op.f ops.mvn_kl ops.mvn_kl.f dvec.head dvec.head2 dvec.head3,\ndunfold sum_costs map sumr,\n\n-- Step 1\n\ndefinev k₁ : env → ℝ := λ (m : env), env.get (eloss, @nil ℕ) m,\ndefinev k₂ : env → ℝ := λ (m : env), sumr (map (λ (cost : ID), env.get (cost, @nil ℕ) m) costs),\n\ndefinev m_lhs_k_add : dvec T [shape] → env := λ (x : dvec T [shape]), env.insert (eloss, @nil ℕ) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape))\n                                                                             (env.insert (z, shape) (dvec.head x) inputs),\n\n\nassert H_lhs_kint₁ : ∀ (x : dvec T [shape]), is_gintegrable (λ m, ⟦k₁ m⟧) (m_lhs_k_add x) nodes dvec.head,\n { dsimp [is_gintegrable, integrate_mvn_kl, dvec.head] at H_kl_gint, dsimp, intro x,\n   cases x with xx x xxx xnil, cases xnil,\n   simp only [λ a1 a2 a3, @env.insert_insert_flip (eloss, []) (z, shape) a1 a2 a3 H_eloss_neq_z],\n   simp only [det.op.f, ops.mvn_kl, ops.mvn_kl.f, dvec.head, env.get_ks, sum_costs] at H_kl_gint,\n   dunfold sumr map at H_kl_gint,\n   exact (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_kl_gint^.right x))^.left\n  },\n\nassert H_lhs_kint₂ : ∀ (x : dvec T [shape]), is_gintegrable (λ m, ⟦k₂ m⟧) (m_lhs_k_add x) nodes dvec.head,\n { dsimp [is_gintegrable, integrate_mvn_kl, dvec.head] at H_kl_gint, dsimp, intro x,\n   cases x with xx x xxx xnil, cases xnil,\n   simp only [λ a1 a2 a3, @env.insert_insert_flip (eloss, []) (z, shape) a1 a2 a3 H_eloss_neq_z],\n   simp only [det.op.f, ops.mvn_kl, ops.mvn_kl.f, dvec.head, env.get_ks, sum_costs] at H_kl_gint,\n   dunfold sumr map at H_kl_gint,\n   exact (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_kl_gint^.right x))^.right },\n\nsimp only [(λ (x : dvec T [shape]), E.E_k_add k₁ k₂ (m_lhs_k_add x) nodes (H_lhs_kint₁ x) (H_lhs_kint₂ x))],\n\nclear H_lhs_kint₁ H_lhs_kint₂,\n\ndefinev m_rhs_k_add : dvec T [shape] → env := λ x, env.insert (eloss, @nil ℕ)\n                                                              (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) (dvec.head x))\n                                                              (env.insert (z, shape) (dvec.head x) inputs),\n\nassert H_rhs_kint₁ : ∀ (x : dvec T [shape]), is_gintegrable (λ m, ⟦k₁ m⟧) (m_rhs_k_add x) nodes dvec.head,\n { dsimp [is_gintegrable, integrate_mvn_kl, dvec.head] at H_gint, dsimp, intro x,\n   cases x with xx x xxx xnil, cases xnil,\n   simp only [det.op.f, dvec.head, env.get_ks, sum_costs] at H_gint,\n   tactic.dget_dinsert_at `H_gint,\n   dunfold sumr map dvec.head dvec.head2 dvec.head3 at H_gint,\n   exact (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_gint^.right x))^.left },\n\nassert H_rhs_kint₂ : ∀ (x : dvec T [shape]), is_gintegrable (λ m, ⟦k₂ m⟧) (m_rhs_k_add x) nodes dvec.head,\n { dsimp [is_gintegrable, integrate_mvn_kl, dvec.head] at H_gint, dsimp, intro x,\n   cases x with xx x xxx xnil, cases xnil,\n   simp only [det.op.f, dvec.head, env.get_ks, sum_costs] at H_gint,\n   tactic.dget_dinsert_at `H_gint,\n   dunfold sumr map dvec.head dvec.head2 dvec.head3 at H_gint,\n   exact (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_gint^.right x))^.right },\n\nsimp only [(λ (x : dvec T [shape]), E.E_k_add k₁ k₂ (m_rhs_k_add x) nodes (H_rhs_kint₁ x) (H_rhs_kint₂ x))],\n\nclear H_rhs_kint₁ H_rhs_kint₂,\n\npose d_base := sprog.prim (rand.op.mvn shape) ⟦env.get (μ, shape) inputs, env.get (σ, shape) inputs⟧,\npose lhs_f₁ := λ x, E (graph.to_dist (λ (m : env), ⟦k₁ m⟧) (m_lhs_k_add x) nodes) dvec.head,\npose lhs_f₂ := λ x, E (graph.to_dist (λ (m : env), ⟦k₂ m⟧) (m_lhs_k_add x) nodes) dvec.head,\n\nassert H_E_kl_add :\n∀ x, E (graph.to_dist (λ (m : env),\n                         ⟦env.get (eloss, @nil ℕ) m + sumr (map (λ (cost : ID), (env.get (cost, @nil ℕ) m : ℝ)) costs)⟧)\n            (env.insert (z, shape) x\n                        (env.insert (eloss, @nil ℕ) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape)) inputs))\n            nodes)\n       dvec.head\n=\nE (graph.to_dist (λ (m : env), ⟦k₁ m⟧)\n            (env.insert (z, shape) x\n                        (env.insert (eloss, @nil ℕ) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape)) inputs))\n            nodes)\n       dvec.head\n+\nE (graph.to_dist (λ (m : env), ⟦k₂ m⟧)\n            (env.insert (z, shape) x\n                        (env.insert (eloss, @nil ℕ) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape)) inputs))\n            nodes)\n       dvec.head,\n{ intro x,\n  exact E.E_k_add k₁ k₂ _ _ (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_kl_gint^.right x))^.left (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_kl_gint^.right x))^.right },\n\nassert H_lhs_eint₁ : E.is_eintegrable d_base lhs_f₁,\n{\ndsimp [E.is_eintegrable, dvec.head],\ndsimp [integrate_mvn_kl, is_gintegrable] at H_kl_gint,\nsimp only [λ a1 a2 a3, @env.insert_insert_flip (eloss, []) (z, shape) a1 a2 a3 H_eloss_neq_z],\nsimp only [det.op.f, ops.mvn_kl, p1, p2, ops.mvn_kl.f, dvec.head, env.get_ks, sum_costs] at H_kl_gint,\ndunfold det.op.f ops.mvn_kl p1 p2 ops.mvn_kl.f dvec.head env.get_ks sum_costs at H_kl_gint,\ntactic.dget_dinsert_at `H_kl_gint,\ndunfold det.op.f ops.mvn_kl ops.mvn_kl.f sumr map dvec.head dvec.head2 dvec.head3 at H_kl_gint,\nsimp only [H_E_kl_add] at H_kl_gint,\nexact (iff.mpr (T.is_integrable_add_middle _ _ _) H_kl_gint^.left)^.left\n},\n\nassert H_lhs_eint₂ : E.is_eintegrable d_base lhs_f₂,\n{\ndsimp [E.is_eintegrable, dvec.head],\ndsimp [integrate_mvn_kl, is_gintegrable] at H_kl_gint,\nsimp only [λ a1 a2 a3, @env.insert_insert_flip (eloss, []) (z, shape) a1 a2 a3 H_eloss_neq_z],\nsimp only [det.op.f, ops.mvn_kl, p1, p2, ops.mvn_kl.f, dvec.head, env.get_ks, sum_costs] at H_kl_gint,\ndunfold det.op.f ops.mvn_kl p1 p2 ops.mvn_kl.f dvec.head env.get_ks sum_costs at H_kl_gint,\ntactic.dget_dinsert_at `H_kl_gint,\ndunfold sumr map dvec.head dvec.head2 dvec.head3 at H_kl_gint,\nsimp only [H_E_kl_add] at H_kl_gint,\nexact (iff.mpr (T.is_integrable_add_middle _ _ _) H_kl_gint^.left)^.right\n},\n\nerw E.E_add d_base lhs_f₁ lhs_f₂ H_lhs_eint₁ H_lhs_eint₂,\n\npose rhs_f₁ := λ x, E (graph.to_dist (λ (m : env), ⟦k₁ m⟧) (m_rhs_k_add x) nodes) dvec.head,\npose rhs_f₂ := λ x, E (graph.to_dist (λ (m : env), ⟦k₂ m⟧) (m_rhs_k_add x) nodes) dvec.head,\n\ndsimp [graph.to_dist, operator.to_dist, is_gintegrable, integrate_mvn_kl, dvec.head] at H_gint,\nsimp only [E.E_bind, E.E_ret, det.op.f, dvec.head, env.get_ks, sum_costs] at H_gint,\ntactic.dget_dinsert_at `H_gint,\n\nassert H_E_add : ∀ x,\nE\n         (graph.to_dist\n            (λ (m : env), ⟦(env.get (eloss, @nil ℕ) m : ℝ) + sumr (map (λ (cost : ID), (env.get (cost, @nil ℕ) m : ℝ)) costs)⟧)\n            (env.insert (eloss, @nil ℕ)\n               (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) x)\n               (env.insert (z, shape) x inputs))\n            nodes)\n         dvec.head\n=\nE\n         (graph.to_dist\n            (λ (m : env), ⟦k₁ m⟧)\n            (env.insert (eloss, @nil ℕ)\n               (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) x)\n               (env.insert (z, shape) x inputs))\n            nodes)\n         dvec.head\n+\nE\n         (graph.to_dist\n            (λ (m : env), ⟦k₂ m⟧)\n            (env.insert (eloss, @nil ℕ)\n               (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) x)\n               (env.insert (z, shape) x inputs))\n            nodes)\n         dvec.head,\n{\nintro x,\nexact E.E_k_add k₁ k₂ _ _ (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_gint^.right x))^.left (iff.mpr (is_gintegrable_k_add _ _ _ _) (H_gint^.right x))^.right\n},\n\nassert H_rhs_eint₁ : E.is_eintegrable d_base rhs_f₁,\n{\ndsimp [E.is_eintegrable, dvec.head],\ndunfold sumr map dvec.head dvec.head2 dvec.head3 at H_gint,\nsimp only [H_E_add] at H_gint,\nexact (iff.mpr (T.is_integrable_add_middle _ _ _) H_gint^.left)^.left\n},\n\nassert H_rhs_eint₂ : E.is_eintegrable d_base rhs_f₂,\n{\ndsimp [E.is_eintegrable, dvec.head],\ndunfold sumr map dvec.head dvec.head2 dvec.head3 at H_gint,\nsimp only [H_E_add] at H_gint,\nexact (iff.mpr (T.is_integrable_add_middle _ _ _) H_gint^.left)^.right\n},\n\nerw E.E_add d_base rhs_f₁ rhs_f₂ H_rhs_eint₁ H_rhs_eint₂,\n\nclear integrate_mvn_kl_correct,\n\nassert H_term₁_lhs :\n∀ (x : dvec T [shape]),\nE (graph.to_dist (λ (m : env), ⟦(λ (m : env), env.get (eloss, []) m) m⟧)\n                 (env.insert (eloss, []) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape))\n                              (env.insert (z, shape) (dvec.head x) inputs))\n                 nodes)\n   dvec.head\n=\nT.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape),\n{ intro x, apply (E.E_of_lookup H_eloss_nin_nodes),\ndsimp [pdfs_exist_at] at H_pdfs_exist_at,\ndsimp [all_parents_in_env] at H_ps_in_env,\nexact (pdfs_exist_at_ignore (H_pre^.right^.right^.right^.right _) (env_not_has_key_insert H_eloss_neq_z H_eloss_nin) H_eloss_nin_nodes (H_pdfs_exist_at^.right _))\n},\n\nassert H_term₁_rhs :\n∀ (x : dvec T [shape]),\nE (graph.to_dist (λ (m : env), ⟦(λ (m : env), env.get (eloss, @nil ℕ) m) m⟧)\n                 (env.insert (eloss, @nil ℕ)\n                   (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) (dvec.head x))\n               (env.insert (z, shape) (dvec.head x) inputs))\n            nodes)\n         dvec.head\n=\nT.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) (dvec.head x),\n{ intro x, apply (E.E_of_lookup H_eloss_nin_nodes),\ndsimp [pdfs_exist_at] at H_pdfs_exist_at,\ndsimp [all_parents_in_env] at H_ps_in_env,\nexact (pdfs_exist_at_ignore (H_pre^.right^.right^.right^.right _) (env_not_has_key_insert H_eloss_neq_z H_eloss_nin) H_eloss_nin_nodes (H_pdfs_exist_at^.right _))\n },\n\nassert H_term₁ :\nE (sprog.prim (rand.op.mvn shape) ⟦env.get (μ, shape) inputs, env.get (σ, shape) inputs⟧)\n    (λ (x : dvec T [shape]),\n       E\n         (graph.to_dist (λ (m : env), ⟦(λ (m : env), env.get (eloss, @nil ℕ) m) m⟧)\n            (env.insert (eloss, @nil ℕ) (T.mvn_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape))\n               (env.insert (z, shape) (dvec.head x) inputs))\n            nodes)\n         dvec.head)\n=\nE (sprog.prim (rand.op.mvn shape) ⟦env.get (μ, shape) inputs, env.get (σ, shape) inputs⟧)\n    (λ (x : dvec T [shape]),\n         E (graph.to_dist (λ (m : env), ⟦(λ (m : env), env.get (eloss, @nil ℕ) m) m⟧)\n                          (env.insert (eloss, @nil ℕ)\n                                       (T.mvn_empirical_kl (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) (dvec.head x))\n                                       (env.insert (z, shape) (dvec.head x) inputs))\n            nodes)\n         dvec.head),\n{\nsimp [H_term₁_lhs, H_term₁_rhs],\ndunfold E T.dintegral dvec.head rand.op.pdf dvec.head2 dvec.head3,\ndsimp,\ndunfold dvec.head,\nerw T.integral_fscale,\nerw (@T.mvn_kl_identity shape (env.get (μ, shape) inputs) (env.get (σ, shape) inputs) H_pre^.right^.right^.right^.left),\nassertv H_pdf_1 : ∫ (λ (x : T shape), T.mvn_pdf (env.get (μ, shape) inputs : T shape) (env.get (σ, shape) inputs : T shape) x) = 1 :=\n  T.mvn_pdf_int1 _ _ H_pre^.right^.right^.right^.left,\ndelta rand.pdf.mvn,\ndsimp,\nrw H_pdf_1,\nrw T.one_smul\n},\n\nerw H_term₁, clear H_term₁, apply congr_arg,\napply congr_arg, apply funext, intro x,\nassertv H_ps_in_env : all_parents_in_env (env.insert (z, shape) x^.head inputs) nodes := by apply H_pre^.right^.right^.right^.right,\ndsimp,\nerw (to_dist_congr_insert H_ps_in_env (env_not_has_key_insert H_eloss_neq_z H_eloss_nin) H_eloss_nin_nodes H_eloss_not_cost),\nerw (to_dist_congr_insert H_ps_in_env (env_not_has_key_insert H_eloss_neq_z H_eloss_nin) H_eloss_nin_nodes H_eloss_not_cost)\nend\n\n-- EQ11\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), [(pname₃, shape₃), (pname₄, shape₄), (pname₅, shape₅)], operator.rand op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ12\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, []), ((pname₃, shape₃)::(pname₄, shape₄)::(pname₅, shape₅)::parent::parents), op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ13\n| (⟨(rname, .(shape)), [(pname₁, .(shape)), (pname₂, .(shape))], operator.rand (rand.op.mvn shape)⟩\n  ::⟨(rname₂, (shape₂ :: shapes₂)), parents, op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- EQ14 (a)\n| (⟨(rname₂, shape₂), ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents), operator.det op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint :=\nbegin\ndunfold graph.to_dist operator.to_dist integrate_mvn_kl,\ndunfold integrate_mvn_kl_pre at H_pre,\nrw [E.E_bind, E.E_bind],\nsimp [E.E_ret],\n\ndefinev next_inputs : env := env.insert (rname₂, shape₂) (op^.f (env.get_ks ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents) inputs)) inputs,\ndefinev next_nodes : list node := nodes,\n\nassertv H_pre_next : integrate_mvn_kl_pre eloss next_nodes next_inputs := H_pre,\nassertv H_ps_in_env_next : all_parents_in_env next_inputs next_nodes := H_ps_in_env^.right _,\nexact (integrate_mvn_kl_correct _ _ H_eloss_not_cost H_pre_next (H_uids^.right _) H_ps_in_env_next H_pdfs_exist_at H_kl_gint H_gint)\nend\n\n-- EQ14 (b)\n| (⟨(rname₂, shape₂), ((pname₃, shape₃)::(pname₄, shape₄)::parent::parents), operator.rand op⟩::nodes) inputs H_eloss_not_cost H_pre H_uids H_ps_in_env H_pdfs_exist_at H_kl_gint H_gint := false.rec _ H_pre\n\n-- More useful API\n\ndef integrate_kl_pre : graph → env → Prop\n| g m := integrate_mvn_kl_pre g^.costs^.head g^.nodes m\n\ndef integrate_kl : graph → graph\n| g := ⟨integrate_mvn_kl g^.costs^.head g^.nodes, g^.costs, g^.targets, g^.inputs⟩\n\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/kl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7853085708384735, "lm_q2_score": 0.41869690935568665, "lm_q1q2_score": 0.32880627150060016}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.shift\nimport category_theory.concrete_category\nimport category_theory.pi.basic\nimport algebra.group.basic\n\n/-!\n# The category of graded objects\n\nFor any type `β`, a `β`-graded object over some category `C` is just\na function `β → C` into the objects of `C`.\nWe put the \"pointwise\" category structure on these, as the non-dependent specialization of\n`category_theory.pi`.\n\nWe describe the `comap` functors obtained by precomposing with functions `β → γ`.\n\nAs a consequence a fixed element (e.g. `1`) in an additive group `β` provides a shift\nfunctor on `β`-graded objects\n\nWhen `C` has coproducts we construct the `total` functor `graded_object β C ⥤ C`,\nshow that it is faithful, and deduce that when `C` is concrete so is `graded_object β C`.\n-/\n\nopen category_theory.pi\nopen category_theory.limits\n\nnamespace category_theory\n\nuniverses w v u\n\n/-- A type synonym for `β → C`, used for `β`-graded objects in a category `C`. -/\ndef graded_object (β : Type w) (C : Type u) : Type (max w u) := β → C\n\n-- Satisfying the inhabited linter...\ninstance inhabited_graded_object (β : Type w) (C : Type u) [inhabited C] :\n  inhabited (graded_object β C) :=\n⟨λ b, inhabited.default C⟩\n\n/--\nA type synonym for `β → C`, used for `β`-graded objects in a category `C`\nwith a shift functor given by translation by `s`.\n-/\n@[nolint unused_arguments] -- `s` is here to distinguish type synonyms asking for different shifts\nabbreviation graded_object_with_shift {β : Type w} [add_comm_group β] (s : β) (C : Type u) :\n  Type (max w u) := graded_object β C\n\nnamespace graded_object\n\nvariables {C : Type u} [category.{v} C]\n\ninstance category_of_graded_objects (β : Type w) : category.{(max w v)} (graded_object β C) :=\ncategory_theory.pi (λ _, C)\n\n\n\nsection\nvariable (C)\n\n/--\nThe natural isomorphism comparing between\npulling back along two propositionally equal functions.\n-/\n@[simps]\ndef comap_eq {β γ : Type w} {f g : β → γ} (h : f = g) : comap (λ _, C) f ≅ comap (λ _, C) g :=\n{ hom := { app := λ X b, eq_to_hom begin dsimp [comap], subst h, end },\n  inv := { app := λ X b, eq_to_hom begin dsimp [comap], subst h, end }, }\n\nlemma comap_eq_symm {β γ : Type w} {f g : β → γ} (h : f = g) :\n  comap_eq C h.symm = (comap_eq C h).symm :=\nby tidy\n\nlemma comap_eq_trans {β γ : Type w} {f g h : β → γ} (k : f = g) (l : g = h) :\n  comap_eq C (k.trans l) = comap_eq C k ≪≫ comap_eq C l :=\nbegin\n  ext X b,\n  simp,\nend\n\n/--\nThe equivalence between β-graded objects and γ-graded objects,\ngiven an equivalence between β and γ.\n-/\n@[simps]\ndef comap_equiv {β γ : Type w} (e : β ≃ γ) :\n  (graded_object β C) ≌ (graded_object γ C) :=\n{ functor := comap (λ _, C) (e.symm : γ → β),\n  inverse := comap (λ _, C) (e : β → γ),\n  counit_iso := (comap_comp (λ _, C) _ _).trans (comap_eq C (by { ext, simp } )),\n  unit_iso := (comap_eq C (by { ext, simp } )).trans (comap_comp _ _ _).symm,\n  functor_unit_iso_comp' := λ X, by { ext b, dsimp, simp, }, }  -- See note [dsimp, simp].\n\nend\n\ninstance has_shift {β : Type*} [add_comm_group β] (s : β) :\n  has_shift (graded_object_with_shift s C) :=\n{ shift := comap_equiv C\n  { to_fun := λ b, b-s,\n    inv_fun := λ b, b+s,\n    left_inv := λ x, (by simp),\n    right_inv := λ x, (by simp), } }\n\n@[simp] lemma shift_functor_obj_apply {β : Type*} [add_comm_group β] (s : β) (X : β → C) (t : β) :\n  (shift (graded_object_with_shift s C)).functor.obj X t = X (t + s) :=\nrfl\n\n@[simp] lemma shift_functor_map_apply {β : Type*} [add_comm_group β] (s : β)\n  {X Y : graded_object_with_shift s C} (f : X ⟶ Y) (t : β) :\n  (shift (graded_object_with_shift s C)).functor.map f t = f (t + s) :=\nrfl\n\ninstance has_zero_morphisms [has_zero_morphisms C] (β : Type w) :\n  has_zero_morphisms.{(max w v)} (graded_object β C) :=\n{ has_zero := λ X Y,\n  { zero := λ b, 0 } }\n\n@[simp]\nlemma zero_apply [has_zero_morphisms C] (β : Type w) (X Y : graded_object β C) (b : β) :\n  (0 : X ⟶ Y) b = 0 := rfl\n\nsection\nlocal attribute [instance] has_zero_object.has_zero\n\ninstance has_zero_object [has_zero_object C] [has_zero_morphisms C] (β : Type w) :\n  has_zero_object.{(max w v)} (graded_object β C) :=\n{ zero := λ b, (0 : C),\n  unique_to := λ X, ⟨⟨λ b, 0⟩, λ f, (by ext)⟩,\n  unique_from := λ X, ⟨⟨λ b, 0⟩, λ f, (by ext)⟩, }\nend\n\nend graded_object\n\nnamespace graded_object\n-- The universes get a little hairy here, so we restrict the universe level for the grading to 0.\n-- Since we're typically interested in grading by ℤ or a finite group, this should be okay.\n-- If you're grading by things in higher universes, have fun!\nvariables (β : Type)\nvariables (C : Type u) [category.{v} C]\nvariables [has_coproducts C]\n\n/--\nThe total object of a graded object is the coproduct of the graded components.\n-/\nnoncomputable def total : graded_object β C ⥤ C :=\n{ obj := λ X, ∐ (λ i : ulift.{v} β, X i.down),\n  map := λ X Y f, limits.sigma.map (λ i, f i.down) }.\n\nvariables [has_zero_morphisms C]\n\n/--\nThe `total` functor taking a graded object to the coproduct of its graded components is faithful.\nTo prove this, we need to know that the coprojections into the coproduct are monomorphisms,\nwhich follows from the fact we have zero morphisms and decidable equality for the grading.\n-/\ninstance : faithful (total β C) :=\n{ map_injective' := λ X Y f g w,\n  begin\n    classical,\n    ext i,\n    replace w := sigma.ι (λ i : ulift β, X i.down) ⟨i⟩ ≫= w,\n    erw [colimit.ι_map, colimit.ι_map] at w,\n    exact mono.right_cancellation _ _ w,\n  end }\n\nend graded_object\n\nnamespace graded_object\n\nnoncomputable theory\n\nvariables (β : Type)\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_coproducts C] [has_zero_morphisms C]\n\ninstance : concrete_category (graded_object β C) :=\n{ forget := total β C ⋙ forget C }\n\ninstance : has_forget₂ (graded_object β C) C :=\n{ forget₂ := total β C }\n\nend graded_object\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/graded_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3287927070340934}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro\n\nThe module `lc α β` of linear combinations over `β` (`α` is the scalar ring)\n-/\nimport linear_algebra.basic\nnoncomputable theory\n\nopen classical function lattice\nlocal attribute [instance] prop_decidable\n\nvariables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}\n\n/-- The type of linear coefficients, which are simply the finitely supported functions\nfrom the module `β` to the scalar ring `α`. -/\n@[reducible] def lc (α β) [ring α] [add_comm_group β] [module α β] : Type* := β →₀ α\n\nnamespace lc\nvariables [ring α] [add_comm_group β] [add_comm_group γ] [add_comm_group δ]\nvariables [module α β] [module α γ] [module α δ]\nopen submodule linear_map\n\ninstance : add_comm_group (lc α β) := finsupp.add_comm_group\n\ninstance : has_scalar α (lc α β) := finsupp.to_has_scalar\n\ninstance : module α (lc α β) := finsupp.to_module β α\n\nvariables (α)\ndef supported (s : set β) : submodule α (lc α β) :=\n{ carrier := {l | ↑l.support ⊆ s},\n  zero := by simp,\n  add := λ l₁ l₂ h₁ h₂, set.subset.trans (finset.coe_subset.2 finsupp.support_add)\n    (by simpa using set.union_subset h₁ h₂),\n  smul := λ a l h, set.subset.trans (finset.coe_subset.2 finsupp.support_smul)\n    (by simpa using h) }\nvariables {α}\n\ntheorem mem_supported {s : set β} {l : lc α β} :\n  l ∈ supported α s ↔ ↑l.support ⊆ s := iff.rfl\n\ntheorem mem_supported' {s : set β} {l : lc α β} :\n  l ∈ supported α s ↔ ∀ x ∉ s, l x = 0 :=\nby simp [mem_supported, set.subset_def, not_imp_comm]\n\ntheorem single_mem_supported {s : set β} (a : α) {b : β} (h : b ∈ s) :\n  finsupp.single b a ∈ supported α s :=\nset.subset.trans finsupp.support_single_subset (set.singleton_subset_iff.2 h)\n\ntheorem supported_eq_span_single (s : set β) :\n  lc.supported α s = span α ((λ x, finsupp.single x (1:α)) '' s) :=\nbegin\n  refine (span_eq_of_le _ _ (le_def'.2 $ λ l hl, _)).symm,\n  { rintro _ ⟨l, hl, rfl⟩, exact single_mem_supported _ hl },\n  { rw ← l.sum_single,\n    refine sum_mem _ (λ x xl, _),\n    rw (by simp : finsupp.single x (l x) = l x • finsupp.single x 1),\n    exact smul_mem _ _ (subset_span ⟨_, hl xl, rfl⟩) }\nend\n\nvariables (α)\ndef restrict_dom (s : set β) : lc α β →ₗ supported α s :=\nlinear_map.cod_restrict _\n  { to_fun := finsupp.filter (∈ s),\n    add := λ l₁ l₂, finsupp.filter_add,\n    smul := λ a l, finsupp.filter_smul }\n  (λ l, mem_supported'.2 $ λ x, finsupp.filter_apply_neg (∈ s) l)\nvariables {α}\n\n@[simp] theorem restrict_dom_apply (s : set β) (l : lc α β) :\n  ↑((restrict_dom α s : lc α β →ₗ supported α s) l) = finsupp.filter (∈ s) l := rfl\n\ntheorem restrict_dom_comp_subtype (s : set β) :\n  (restrict_dom α s).comp (submodule.subtype _) = linear_map.id :=\nby ext l; apply subtype.coe_ext.2; simp; ext a;\n   by_cases a ∈ s; simp [h]; exact (mem_supported'.1 l.2 _ h).symm\n\ntheorem range_restrict_dom (s : set β) : (restrict_dom α s).range = ⊤ :=\nbegin\n  have := linear_map.range_comp (submodule.subtype _) (restrict_dom α s),\n  rw [restrict_dom_comp_subtype, linear_map.range_id] at this,\n  exact eq_top_mono (submodule.map_mono le_top) this.symm\nend\n\ntheorem supported_mono {s t : set β} (st : s ⊆ t) :\n  supported α s ≤ supported α t :=\nλ l h, set.subset.trans h st\n\n@[simp] theorem supported_empty : supported α (∅ : set β) = ⊥ :=\neq_bot_iff.2 $ λ l h, (submodule.mem_bot α).2 $\nby ext; simp [*, mem_supported'] at *\n\n@[simp] theorem supported_univ : supported α (set.univ : set β) = ⊤ :=\neq_top_iff.2 $ λ l _, set.subset_univ _\n\ntheorem supported_Union {ι : Type*} (s : ι → set β) :\n  supported α (⋃ i, s i) = ⨆ i, supported α (s i) :=\nbegin\n  refine le_antisymm _ (supr_le $ λ i, supported_mono $ set.subset_Union _ _),\n  suffices : ((submodule.subtype _).comp (restrict_dom α (⋃ i, s i))).range ≤ ⨆ i, supported α (s i),\n  { rwa [range_comp, range_restrict_dom, map_top, range_subtype] at this },\n  rw [range_le_iff_comap, eq_top_iff],\n  rintro l ⟨⟩, rw mem_coe,\n  apply finsupp.induction l, {exact zero_mem _},\n  refine λ x a l hl a0, add_mem _ _,\n  by_cases (∃ i, x ∈ s i); simp [h],\n  cases h with i hi,\n  exact le_supr (λ i, supported α (s i)) i (single_mem_supported _ hi)\nend\n\ntheorem supported_union (s t : set β) :\n  supported α (s ∪ t) = supported α s ⊔ supported α t :=\nby erw [set.union_eq_Union, supported_Union, supr_bool_eq]; refl\n\ntheorem supported_Inter {ι : Type*} (s : ι → set β) :\n  supported α (⋂ i, s i) = ⨅ i, supported α (s i) :=\nbegin\n  refine le_antisymm (le_infi $ λ i, supported_mono $ set.Inter_subset _ _) _,\n  simp [le_def, infi_coe, set.subset_def],\n  exact λ l, set.subset_Inter\nend\n\ndef apply (x : β) : lc α β →ₗ α :=\n⟨λ l, l x, λ _ _, finsupp.add_apply, λ _ _, finsupp.smul_apply⟩\n\n@[simp] theorem apply_apply (x : β) (l : lc α β) :\n  (lc.apply x : lc α β →ₗ α) l = l x := rfl\n\nprotected def lsum (f : β → α →ₗ[α] γ) : lc α β →ₗ[α] γ :=\n⟨λ d, d.sum (λ b, f b),\n  assume d₁ d₂, by simp [finsupp.sum_add_index],\n  assume a d, by simp [finsupp.sum_smul_index, finsupp.smul_sum,\n    -smul_eq_mul, smul_eq_mul.symm]⟩\n\n@[simp] theorem lsum_apply (f : β → α →ₗ γ) (l : lc α β) :\n  (lc.lsum f : lc α β →ₗ γ) l = l.sum (λ b, f b) := rfl\n\nsection\nvariables (α β)\nprotected def total : lc α β →ₗ β := lc.lsum linear_map.id.smul_right\nend\n\ntheorem total_apply (l : lc α β) :\n  lc.total α β l = l.sum (λ b a, a • b) := rfl\n\n@[simp] theorem total_single (a : α) (x : β) :\n  lc.total α β (finsupp.single x a) = a • x :=\nby simp [total_apply, finsupp.sum_single_index]\n\n@[simp] theorem total_range : (lc.total α β).range = ⊤ :=\nrange_eq_top.2 $ λ x, ⟨finsupp.single x 1, by simp⟩\n\nvariables (α)\nprotected def map (f : β → γ) : lc α β →ₗ[α] lc α γ :=\n{ to_fun := finsupp.map_domain f,\n  add := λ l₁ l₂, finsupp.map_domain_add,\n  smul := λ a l, finsupp.map_domain_smul _ _ }\nvariables {α}\n\n@[simp] theorem map_apply (f : β → γ) (l : lc α β) :\n  (lc.map α f : _ →ₗ _) l = finsupp.map_domain f l := rfl\n\n@[simp] theorem map_id : (lc.map α id : lc α β →ₗ[α] lc α β) = linear_map.id :=\nlinear_map.ext $ λ l, finsupp.map_domain_id\n\ntheorem map_comp (f : β → γ) (g : γ → δ) :\n  lc.map α (g ∘ f) = (lc.map α g).comp (lc.map α f) :=\nlinear_map.ext $ λ l, finsupp.map_domain_comp\n\ntheorem supported_comap_map (f : β → γ) (s : set γ) :\n  supported α (f ⁻¹' s) ≤ (supported α s).comap (lc.map α f) :=\nλ l (hl : ↑l.support ⊆ f ⁻¹' s),\nshow ↑(finsupp.map_domain f l).support ⊆ s, begin\n  rw [← set.image_subset_iff, ← finset.coe_image] at hl,\n  exact set.subset.trans finsupp.map_domain_support hl\nend\n\ntheorem map_supported (f : β → γ) (s : set β) :\n  (supported α s).map (lc.map α f) = supported α (f '' s) :=\nbegin\n  refine le_antisymm (map_le_iff_le_comap.2 $\n    le_trans (supported_mono $ set.subset_preimage_image _ _)\n       (supported_comap_map _ _)) _,\n  intros l hl, haveI : inhabited β := ⟨0⟩,\n  refine ⟨(lc.map α (inv_fun_on f s) : lc α γ →ₗ lc α β) l, λ x hx, _, _⟩,\n  { rcases finset.mem_image.1 (finsupp.map_domain_support hx) with ⟨c, hc, rfl⟩,\n    exact inv_fun_on_mem (by simpa using hl hc) },\n  { rw [← linear_map.comp_apply, ← map_comp],\n    refine (finsupp.map_domain_congr $ λ c hc, _).trans finsupp.map_domain_id,\n    exact inv_fun_on_eq (by simpa using hl hc) }\nend\n\ntheorem map_disjoint_ker (f : β → γ) {s : set β}\n  (H : ∀ a b ∈ s, f a = f b → a = b) :\n  disjoint (supported α s) (lc.map α f).ker :=\nbegin\n  rintro l ⟨h₁, h₂⟩,\n  rw [mem_coe, mem_ker, map_apply, finsupp.map_domain] at h₂,\n  simp, ext x,\n  by_cases xs : x ∈ s,\n  { have : finsupp.sum l (λ a, finsupp.single (f a)) (f x) = 0, {rw h₂, refl},\n    rw [finsupp.sum_apply, finsupp.sum, finset.sum_eq_single x] at this,\n    { simpa [finsupp.single_apply] },\n    { intros y hy xy, simp [mt (H _ _ (h₁ hy) xs) xy] },\n    { simp {contextual := tt} } },\n  { by_contra h, exact xs (h₁ $ finsupp.mem_support_iff.2 h) }\nend\n\ntheorem map_total (f : β →ₗ[α] γ) :\n  (lc.total α γ).comp (lc.map α f) = f.comp (lc.total α β) :=\nby ext; simp [total_apply, finsupp.sum_map_domain_index, add_smul]\n\nend lc\n\nnamespace lc\nvariables [discrete_field α] [add_comm_group β] [vector_space α β]\n\ninstance : vector_space α (lc α β) := { .. lc.module }\n\nend lc\n\nsection module\nvariables [ring α] [add_comm_group β] [add_comm_group γ] [add_comm_group δ]\nvariables [module α β] [module α γ] [module α δ]\nvariables {a b : α} {s t : set β} {x y : β}\ninclude α\nopen submodule\n\ntheorem span_eq_map_lc : span α s = (lc.supported α s).map (lc.total α β) :=\nbegin\n  apply span_eq_of_le,\n  { exact λ x hx, ⟨_, lc.single_mem_supported 1 hx, by simp⟩ },\n  { refine map_le_iff_le_comap.2 (λ v hv, _),\n    have : ∀c, v c • c ∈ span α s,\n    { intro c, by_cases c ∈ s,\n      { exact smul_mem _ _ (subset_span h) },\n      { simp [lc.mem_supported'.1 hv _ h] } },\n    refine sum_mem _ _, simp [this] }\nend\n\ntheorem mem_span_iff_lc : x ∈ span α s ↔ ∃ l ∈ lc.supported α s, lc.total α β l = x :=\nby rw span_eq_map_lc; simp\n\nvariables (α)\ndef lc.total_on (s : set β) : lc.supported α s →ₗ span α s :=\nlinear_map.cod_restrict _ ((lc.total α _).comp (submodule.subtype _)) $\n  λ ⟨l, hl⟩, mem_span_iff_lc.2 ⟨l, hl, rfl⟩\nvariables {α}\n\ntheorem lc.total_on_range (s : set β) : (lc.total_on α s).range = ⊤ :=\nby rw [lc.total_on, linear_map.range, linear_map.map_cod_restrict, ← linear_map.range_le_iff_comap,\n  range_subtype, map_top, linear_map.range_comp, range_subtype]; exact le_of_eq span_eq_map_lc\n\nlemma linear_eq_on {f g : β →ₗ[α] γ} (H : ∀x∈s, f x = g x) {x} (h : x ∈ span α s) : f x = g x :=\nby apply span_induction h H; simp {contextual := tt}\n\nend module\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/linear_algebra/linear_combination.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.32879270643514974}}
{"text": "/-\nCopyright (c) 2020 Kenji Nakagawa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio\n-/\nimport ring_theory.discrete_valuation_ring\nimport ring_theory.fractional_ideal\nimport ring_theory.ideal.over\nimport ring_theory.integrally_closed\nimport ring_theory.polynomial.rational_root\nimport ring_theory.trace\nimport algebra.associated\nimport algebraic_geometry.prime_spectrum.noetherian\n\n/-!\n# Dedekind domains\n\nThis file defines the notion of a Dedekind domain (or Dedekind ring),\ngiving three equivalent definitions (TODO: and shows that they are equivalent).\n\n## Main definitions\n\n - `is_dedekind_domain` defines a Dedekind domain as a commutative ring that is\n   Noetherian, integrally closed in its field of fractions and has Krull dimension at most one.\n   `is_dedekind_domain_iff` shows that this does not depend on the choice of field of fractions.\n - `is_dedekind_domain_dvr` alternatively defines a Dedekind domain as an integral domain that\n   is Noetherian, and the localization at every nonzero prime ideal is a DVR.\n - `is_dedekind_domain_inv` alternatively defines a Dedekind domain as an integral domain where\n   every nonzero fractional ideal is invertible.\n - `is_dedekind_domain_inv_iff` shows that this does note depend on the choice of field of\n   fractions.\n\n## Implementation notes\n\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. The `..._iff` lemmas express this independence.\n\nOften, definitions assume that Dedekind domains are not fields. We found it more practical\nto add a `(h : ¬ is_field A)` assumption whenever this is explicitly needed.\n\n## References\n\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Frölich, *Algebraic Number Theory*][cassels1967algebraic]\n* [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\ndedekind domain, dedekind ring\n-/\n\nvariables (R A K : Type*) [comm_ring R] [comm_ring A] [field K]\n\nopen_locale non_zero_divisors\n\n/-- A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. -/\ndef ring.dimension_le_one : Prop :=\n∀ p ≠ (⊥ : ideal R), p.is_prime → p.is_maximal\n\nopen ideal ring\n\nnamespace ring\n\nlemma dimension_le_one.principal_ideal_ring\n  [is_domain A] [is_principal_ideal_ring A] : dimension_le_one A :=\nλ p nonzero prime, by { haveI := prime, exact is_prime.to_maximal_ideal nonzero }\n\nlemma dimension_le_one.is_integral_closure (B : Type*) [comm_ring B] [is_domain B]\n  [nontrivial R] [algebra R A] [algebra R B] [algebra B A] [is_scalar_tower R B A]\n  [is_integral_closure B R A] (h : dimension_le_one R) :\n  dimension_le_one B :=\nλ p ne_bot prime, by exactI\n  is_integral_closure.is_maximal_of_is_maximal_comap A p\n    (h _ (is_integral_closure.comap_ne_bot A ne_bot) infer_instance)\n\nlemma dimension_le_one.integral_closure [nontrivial R] [is_domain A] [algebra R A]\n  (h : dimension_le_one R) : dimension_le_one (integral_closure R A) :=\nh.is_integral_closure R A (integral_closure R A)\n\nend ring\n\nvariables [is_domain A]\n\n/--\nA Dedekind domain is an integral domain that is Noetherian, integrally closed, and\nhas Krull dimension at most one.\n\nThis is definition 3.2 of [Neukirch1992].\n\nThe integral closure condition is independent of the choice of field of fractions:\nuse `is_dedekind_domain_iff` to prove `is_dedekind_domain` for a given `fraction_map`.\n\nThis is the default implementation, but there are equivalent definitions,\n`is_dedekind_domain_dvr` and `is_dedekind_domain_inv`.\nTODO: Prove that these are actually equivalent definitions.\n-/\nclass is_dedekind_domain : Prop :=\n(is_noetherian_ring : is_noetherian_ring A)\n(dimension_le_one : dimension_le_one A)\n(is_integrally_closed : is_integrally_closed A)\n\n-- See library note [lower instance priority]\nattribute [instance, priority 100]\n  is_dedekind_domain.is_noetherian_ring is_dedekind_domain.is_integrally_closed\n\n/-- An integral domain is a Dedekind domain iff and only if it is\nNoetherian, has dimension ≤ 1, and is integrally closed in a given fraction field.\nIn particular, this definition does not depend on the choice of this fraction field. -/\nlemma is_dedekind_domain_iff (K : Type*) [field K] [algebra A K] [is_fraction_ring A K] :\n  is_dedekind_domain A ↔ is_noetherian_ring A ∧ dimension_le_one A ∧\n    (∀ {x : K}, is_integral A x → ∃ y, algebra_map A K y = x) :=\n⟨λ ⟨hr, hd, hi⟩, ⟨hr, hd, λ x, (is_integrally_closed_iff K).mp hi⟩,\n λ ⟨hr, hd, hi⟩, ⟨hr, hd, (is_integrally_closed_iff K).mpr @hi⟩⟩\n\n@[priority 100] -- See library note [lower instance priority]\ninstance is_principal_ideal_ring.is_dedekind_domain [is_principal_ideal_ring A] :\n  is_dedekind_domain A :=\n⟨principal_ideal_ring.is_noetherian_ring,\n ring.dimension_le_one.principal_ideal_ring A,\n unique_factorization_monoid.is_integrally_closed⟩\n\n/--\nA Dedekind domain is an integral domain that is Noetherian, and the\nlocalization at every nonzero prime is a discrete valuation ring.\n\nThis is equivalent to `is_dedekind_domain`.\nTODO: prove the equivalence.\n-/\nstructure is_dedekind_domain_dvr : Prop :=\n(is_noetherian_ring : is_noetherian_ring A)\n(is_dvr_at_nonzero_prime : ∀ P ≠ (⊥ : ideal A), P.is_prime →\n  discrete_valuation_ring (localization.at_prime P))\n\nsection inverse\n\nnamespace fractional_ideal\n\nvariables {R₁ : Type*} [comm_ring R₁] [is_domain R₁] [algebra R₁ K] [is_fraction_ring R₁ K]\nvariables {I J : fractional_ideal R₁⁰ K}\n\nnoncomputable instance : has_inv (fractional_ideal R₁⁰ K) := ⟨λ I, 1 / I⟩\n\nlemma inv_eq : I⁻¹ = 1 / I := rfl\n\nlemma inv_zero' : (0 : fractional_ideal R₁⁰ K)⁻¹ = 0 := fractional_ideal.div_zero\n\nlemma inv_nonzero {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :\nJ⁻¹ = ⟨(1 : fractional_ideal R₁⁰ K) / J, fractional_ideal.fractional_div_of_nonzero h⟩ :=\nfractional_ideal.div_nonzero _\n\nlemma coe_inv_of_nonzero {J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :\n  (↑J⁻¹ : submodule R₁ K) = is_localization.coe_submodule K ⊤ / J :=\nby { rwa inv_nonzero _, refl, assumption }\n\nvariables {K}\n\nlemma mem_inv_iff (hI : I ≠ 0) {x : K} :\n  x ∈ I⁻¹ ↔ ∀ y ∈ I, x * y ∈ (1 : fractional_ideal R₁⁰ K) :=\nfractional_ideal.mem_div_iff_of_nonzero hI\n\nlemma inv_anti_mono (hI : I ≠ 0) (hJ : J ≠ 0) (hIJ : I ≤ J) :\n  J⁻¹ ≤ I⁻¹ :=\nλ x, by { simp only [mem_inv_iff hI, mem_inv_iff hJ], exact λ h y hy, h y (hIJ hy) }\n\nlemma le_self_mul_inv {I : fractional_ideal R₁⁰ K} (hI : I ≤ (1 : fractional_ideal R₁⁰ K)) :\n  I ≤ I * I⁻¹ :=\nfractional_ideal.le_self_mul_one_div hI\n\nvariables (K)\n\nlemma coe_ideal_le_self_mul_inv (I : ideal R₁) :\n  (I : fractional_ideal R₁⁰ K) ≤ I * I⁻¹ :=\nle_self_mul_inv fractional_ideal.coe_ideal_le_one\n\n/-- `I⁻¹` is the inverse of `I` if `I` has an inverse. -/\ntheorem right_inverse_eq (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) :\n  J = I⁻¹ :=\nbegin\n  have hI : I ≠ 0 := fractional_ideal.ne_zero_of_mul_eq_one I J h,\n  suffices h' : I * (1 / I) = 1,\n  { exact (congr_arg units.inv $\n      @units.ext _ _ (units.mk_of_mul_eq_one _ _ h) (units.mk_of_mul_eq_one _ _ h') rfl) },\n  apply le_antisymm,\n  { apply fractional_ideal.mul_le.mpr _,\n    intros x hx y hy,\n    rw mul_comm,\n    exact (fractional_ideal.mem_div_iff_of_nonzero hI).mp hy x hx },\n  rw ← h,\n  apply fractional_ideal.mul_left_mono I,\n  apply (fractional_ideal.le_div_iff_of_nonzero hI).mpr _,\n  intros y hy x hx,\n  rw mul_comm,\n  exact fractional_ideal.mul_mem_mul hx hy\nend\n\ntheorem mul_inv_cancel_iff {I : fractional_ideal R₁⁰ K} :\n  I * I⁻¹ = 1 ↔ ∃ J, I * J = 1 :=\n⟨λ h, ⟨I⁻¹, h⟩, λ ⟨J, hJ⟩, by rwa ← right_inverse_eq K I J hJ⟩\n\nlemma mul_inv_cancel_iff_is_unit {I : fractional_ideal R₁⁰ K} :\n  I * I⁻¹ = 1 ↔ is_unit I :=\n(mul_inv_cancel_iff K).trans is_unit_iff_exists_inv.symm\n\nvariables {K' : Type*} [field K'] [algebra R₁ K'] [is_fraction_ring R₁ K']\n\n@[simp] lemma map_inv (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :\n  (I⁻¹).map (h : K →ₐ[R₁] K') = (I.map h)⁻¹ :=\nby rw [inv_eq, fractional_ideal.map_div, fractional_ideal.map_one, inv_eq]\n\nopen submodule submodule.is_principal\n\n@[simp] lemma span_singleton_inv (x : K) :\n  (fractional_ideal.span_singleton R₁⁰ x)⁻¹ = fractional_ideal.span_singleton _ (x⁻¹) :=\nfractional_ideal.one_div_span_singleton x\n\nlemma mul_generator_self_inv {R₁ : Type*} [comm_ring R₁] [algebra R₁ K] [is_localization R₁⁰ K]\n  (I : fractional_ideal R₁⁰ K) [submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) :\n  I * fractional_ideal.span_singleton _ (generator (I : submodule R₁ K))⁻¹ = 1 :=\nbegin\n  -- Rewrite only the `I` that appears alone.\n  conv_lhs { congr, rw fractional_ideal.eq_span_singleton_of_principal I },\n  rw [fractional_ideal.span_singleton_mul_span_singleton, mul_inv_cancel,\n    fractional_ideal.span_singleton_one],\n  intro generator_I_eq_zero,\n  apply h,\n  rw [fractional_ideal.eq_span_singleton_of_principal I, generator_I_eq_zero,\n    fractional_ideal.span_singleton_zero]\nend\n\nlemma invertible_of_principal (I : fractional_ideal R₁⁰ K)\n  [submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) :\n  I * I⁻¹ = 1 :=\n(fractional_ideal.mul_div_self_cancel_iff).mpr\n  ⟨fractional_ideal.span_singleton _ (generator (I : submodule R₁ K))⁻¹,\n    mul_generator_self_inv _ I h⟩\n\nlemma invertible_iff_generator_nonzero (I : fractional_ideal R₁⁰ K)\n  [submodule.is_principal (I : submodule R₁ K)] :\n  I * I⁻¹ = 1 ↔ generator (I : submodule R₁ K) ≠ 0 :=\nbegin\n  split,\n  { intros hI hg,\n    apply fractional_ideal.ne_zero_of_mul_eq_one _ _ hI,\n    rw [fractional_ideal.eq_span_singleton_of_principal I, hg,\n        fractional_ideal.span_singleton_zero] },\n  { intro hg,\n    apply invertible_of_principal,\n    rw [fractional_ideal.eq_span_singleton_of_principal I],\n    intro hI,\n    have := fractional_ideal.mem_span_singleton_self _ (generator (I : submodule R₁ K)),\n    rw [hI, fractional_ideal.mem_zero_iff] at this,\n    contradiction }\nend\n\nlemma is_principal_inv (I : fractional_ideal R₁⁰ K)\n  [submodule.is_principal (I : submodule R₁ K)] (h : I ≠ 0) :\n  submodule.is_principal (I⁻¹).1 :=\nbegin\n  rw [fractional_ideal.val_eq_coe, fractional_ideal.is_principal_iff],\n  use (generator (I : submodule R₁ K))⁻¹,\n  have hI : I  * fractional_ideal.span_singleton _ ((generator (I : submodule R₁ K))⁻¹)  = 1,\n  apply mul_generator_self_inv _ I h,\n  exact (right_inverse_eq _ I (fractional_ideal.span_singleton _\n    ((generator (I : submodule R₁ K))⁻¹)) hI).symm\nend\n\n@[simp] lemma one_inv : (1⁻¹ : fractional_ideal R₁⁰ K) = 1 :=\nfractional_ideal.div_one\n\nend fractional_ideal\n\n/--\nA Dedekind domain is an integral domain such that every fractional ideal has an inverse.\n\nThis is equivalent to `is_dedekind_domain`.\nIn particular we provide a `fractional_ideal.comm_group_with_zero` instance,\nassuming `is_dedekind_domain A`, which implies `is_dedekind_domain_inv`. For **integral** ideals,\n`is_dedekind_domain`(`_inv`) implies only `ideal.comm_cancel_monoid_with_zero`.\n-/\ndef is_dedekind_domain_inv : Prop :=\n∀ I ≠ (⊥ : fractional_ideal A⁰ (fraction_ring A)), I * I⁻¹ = 1\n\nopen fractional_ideal\n\nvariables {R A K}\n\nlemma is_dedekind_domain_inv_iff [algebra A K] [is_fraction_ring A K] :\n  is_dedekind_domain_inv A ↔\n    (∀ I ≠ (⊥ : fractional_ideal A⁰ K), I * I⁻¹ = 1) :=\nbegin\n  set h := fraction_ring.alg_equiv A K,\n  split; rintros hi I hI,\n  { refine fractional_ideal.map_injective h.symm.to_alg_hom h.symm.injective _,\n    rw [alg_equiv.to_alg_hom_eq_coe, inv_eq, fractional_ideal.map_mul,\n        fractional_ideal.map_one_div, fractional_ideal.map_one, ← inv_eq, hi],\n    exact fractional_ideal.map_ne_zero _ hI },\n  { refine fractional_ideal.map_injective h.to_alg_hom h.injective _,\n    rw [alg_equiv.to_alg_hom_eq_coe, inv_eq, fractional_ideal.map_mul,\n        fractional_ideal.map_one_div, fractional_ideal.map_one, ← inv_eq, hi],\n    exact fractional_ideal.map_ne_zero _ hI },\nend\n\nlemma fractional_ideal.adjoin_integral_eq_one_of_is_unit [algebra A K] [is_fraction_ring A K]\n  (x : K) (hx : is_integral A x) (hI : is_unit (adjoin_integral A⁰ x hx)) :\n  adjoin_integral A⁰ x hx = 1 :=\nbegin\n  set I := adjoin_integral A⁰ x hx,\n  have mul_self : I * I = I,\n  { apply fractional_ideal.coe_to_submodule_injective, simp },\n  convert congr_arg (* I⁻¹) mul_self;\n  simp only [(mul_inv_cancel_iff_is_unit K).mpr hI, mul_assoc, mul_one],\nend\n\nnamespace is_dedekind_domain_inv\n\nvariables [algebra A K] [is_fraction_ring A K] (h : is_dedekind_domain_inv A)\n\ninclude h\n\nlemma mul_inv_eq_one {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : I * I⁻¹ = 1 :=\nis_dedekind_domain_inv_iff.mp h I hI\n\nlemma inv_mul_eq_one {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : I⁻¹ * I = 1 :=\n(mul_comm _ _).trans (h.mul_inv_eq_one hI)\n\nprotected lemma is_unit {I : fractional_ideal A⁰ K} (hI : I ≠ 0) : is_unit I :=\nis_unit_of_mul_eq_one _ _ (h.mul_inv_eq_one hI)\n\nlemma is_noetherian_ring : is_noetherian_ring A :=\nbegin\n  refine is_noetherian_ring_iff.mpr ⟨λ (I : ideal A), _⟩,\n  by_cases hI : I = ⊥,\n  { rw hI, apply submodule.fg_bot },\n  have hI : (I : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 :=\n    (coe_to_fractional_ideal_ne_zero (le_refl (non_zero_divisors A))).mpr hI,\n  exact I.fg_of_is_unit (is_fraction_ring.injective A (fraction_ring A)) (h.is_unit hI)\nend\n\nlemma integrally_closed : is_integrally_closed A :=\nbegin\n  -- It suffices to show that for integral `x`,\n  -- `A[x]` (which is a fractional ideal) is in fact equal to `A`.\n  refine ⟨λ x hx, _⟩,\n  rw [← set.mem_range, ← algebra.mem_bot, ← subalgebra.mem_to_submodule, algebra.to_submodule_bot,\n      ← coe_span_singleton A⁰ (1 : fraction_ring A), fractional_ideal.span_singleton_one,\n      ← fractional_ideal.adjoin_integral_eq_one_of_is_unit x hx (h.is_unit _)],\n  { exact mem_adjoin_integral_self A⁰ x hx },\n  { exact λ h, one_ne_zero (eq_zero_iff.mp h 1 (subalgebra.one_mem _)) },\nend\n\nlemma dimension_le_one : dimension_le_one A :=\nbegin\n  -- We're going to show that `P` is maximal because any (maximal) ideal `M`\n  -- that is strictly larger would be `⊤`.\n  rintros P P_ne hP,\n  refine ideal.is_maximal_def.mpr ⟨hP.ne_top, λ M hM, _⟩,\n  -- We may assume `P` and `M` (as fractional ideals) are nonzero.\n  have P'_ne : (P : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 :=\n    (coe_to_fractional_ideal_ne_zero (le_refl (non_zero_divisors A))).mpr P_ne,\n  have M'_ne : (M : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 :=\n    (coe_to_fractional_ideal_ne_zero (le_refl (non_zero_divisors A))).mpr\n      (lt_of_le_of_lt bot_le hM).ne',\n\n  -- In particular, we'll show `M⁻¹ * P ≤ P`\n  suffices : (M⁻¹ * P : fractional_ideal A⁰ (fraction_ring A)) ≤ P,\n  { rw [eq_top_iff, ← coe_ideal_le_coe_ideal (fraction_ring A), fractional_ideal.coe_ideal_top],\n    calc (1 : fractional_ideal A⁰ (fraction_ring A)) = _ * _ * _ : _\n    ... ≤ _ * _ : mul_right_mono (P⁻¹ * M : fractional_ideal A⁰ (fraction_ring A)) this\n    ... = M : _,\n    { rw [mul_assoc, ← mul_assoc ↑P, h.mul_inv_eq_one P'_ne, one_mul, h.inv_mul_eq_one M'_ne] },\n    { rw [← mul_assoc ↑P, h.mul_inv_eq_one P'_ne, one_mul] },\n    { apply_instance } },\n\n  -- Suppose we have `x ∈ M⁻¹ * P`, then in fact `x = algebra_map _ _ y` for some `y`.\n  intros x hx,\n  have le_one : (M⁻¹ * P : fractional_ideal A⁰ (fraction_ring A)) ≤ 1,\n  { rw [← h.inv_mul_eq_one M'_ne],\n    exact fractional_ideal.mul_left_mono _ ((coe_ideal_le_coe_ideal (fraction_ring A)).mpr hM.le) },\n  obtain ⟨y, hy, rfl⟩ := (mem_coe_ideal _).mp (le_one hx),\n\n  -- Since `M` is strictly greater than `P`, let `z ∈ M \\ P`.\n  obtain ⟨z, hzM, hzp⟩ := set_like.exists_of_lt hM,\n  -- We have `z * y ∈ M * (M⁻¹ * P) = P`.\n  have zy_mem := fractional_ideal.mul_mem_mul (mem_coe_ideal_of_mem A⁰ hzM) hx,\n  rw [← ring_hom.map_mul, ← mul_assoc, h.mul_inv_eq_one M'_ne, one_mul] at zy_mem,\n  obtain ⟨zy, hzy, zy_eq⟩ := (mem_coe_ideal A⁰).mp zy_mem,\n  rw is_fraction_ring.injective A (fraction_ring A) zy_eq at hzy,\n  -- But `P` is a prime ideal, so `z ∉ P` implies `y ∈ P`, as desired.\n  exact mem_coe_ideal_of_mem A⁰ (or.resolve_left (hP.mem_or_mem hzy) hzp)\nend\n\n/-- Showing one side of the equivalence between the definitions\n`is_dedekind_domain_inv` and `is_dedekind_domain` of Dedekind domains. -/\ntheorem is_dedekind_domain : is_dedekind_domain A :=\n⟨h.is_noetherian_ring, h.dimension_le_one, h.integrally_closed⟩\n\nend is_dedekind_domain_inv\n\nvariables [algebra A K] [is_fraction_ring A K]\n\n/-- Specialization of `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:\nLet `I : ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.\nThen `exists_prime_spectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime\nideals that is contained within `I`. This lemma extends that result by making the product minimal:\nlet `M` be a maximal ideal that contains `I`, then the product including `M` is contained within `I`\nand the product excluding `M` is not contained within `I`. -/\nlemma exists_multiset_prod_cons_le_and_prod_not_le [is_dedekind_domain A]\n  (hNF : ¬ is_field A) {I M : ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.is_maximal] :\n  ∃ (Z : multiset (prime_spectrum A)),\n    (M ::ₘ (Z.map prime_spectrum.as_ideal)).prod ≤ I ∧\n    ¬ (multiset.prod (Z.map prime_spectrum.as_ideal) ≤ I) :=\nbegin\n  -- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.\n  obtain ⟨Z₀, hZ₀⟩ := prime_spectrum.exists_prime_spectrum_prod_le_and_ne_bot_of_domain hNF hI0,\n  obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ := multiset.well_founded_lt.has_min\n    (λ Z, (Z.map prime_spectrum.as_ideal).prod ≤ I ∧ (Z.map prime_spectrum.as_ideal).prod ≠ ⊥)\n    ⟨Z₀, hZ₀⟩,\n  have hZM : multiset.prod (Z.map prime_spectrum.as_ideal) ≤ M := le_trans hZI hIM,\n  have hZ0 : Z ≠ 0, { rintro rfl, simpa [hM.ne_top] using hZM },\n  obtain ⟨_, hPZ', hPM⟩ := (hM.is_prime.multiset_prod_le (mt multiset.map_eq_zero.mp hZ0)).mp hZM,\n  -- Then in fact there is a `P ∈ Z` with `P ≤ M`.\n  obtain ⟨P, hPZ, rfl⟩ := multiset.mem_map.mp hPZ',\n  letI := classical.dec_eq (ideal A),\n  have := multiset.map_erase prime_spectrum.as_ideal subtype.coe_injective P Z,\n  obtain ⟨hP0, hZP0⟩ : P.as_ideal ≠ ⊥ ∧ ((Z.erase P).map prime_spectrum.as_ideal).prod ≠ ⊥,\n  { rwa [ne.def, ← multiset.cons_erase hPZ', multiset.prod_cons, ideal.mul_eq_bot,\n         not_or_distrib, ← this] at hprodZ },\n  -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.\n  have hPM' := (is_dedekind_domain.dimension_le_one _ hP0 P.is_prime).eq_of_le hM.ne_top hPM,\n  tactic.unfreeze_local_instances,\n  subst hPM',\n\n  -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.\n  refine ⟨Z.erase P, _, _⟩,\n  { convert hZI,\n    rw [this, multiset.cons_erase hPZ'] },\n  { refine λ h, h_eraseZ (Z.erase P) ⟨h, _⟩ (multiset.erase_lt.mpr hPZ),\n    exact hZP0 }\nend\n\nnamespace fractional_ideal\n\nlemma exists_not_mem_one_of_ne_bot [is_dedekind_domain A]\n  (hNF : ¬ is_field A) {I : ideal A} (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :\n  ∃ x : K, x ∈ (I⁻¹ : fractional_ideal A⁰ K) ∧ x ∉ (1 : fractional_ideal A⁰ K) :=\nbegin\n  -- WLOG, let `I` be maximal.\n  suffices : ∀ {M : ideal A} (hM : M.is_maximal),\n    ∃ x : K, x ∈ (M⁻¹ : fractional_ideal A⁰ K) ∧ x ∉ (1 : fractional_ideal A⁰ K),\n  { obtain ⟨M, hM, hIM⟩ : ∃ (M : ideal A), is_maximal M ∧ I ≤ M := ideal.exists_le_maximal I hI1,\n    resetI,\n    have hM0 := (M.bot_lt_of_maximal hNF).ne',\n    obtain ⟨x, hxM, hx1⟩ := this hM,\n    refine ⟨x, inv_anti_mono _ _ ((coe_ideal_le_coe_ideal _).mpr hIM) hxM, hx1⟩;\n      apply fractional_ideal.coe_ideal_ne_zero; assumption },\n\n  -- Let `a` be a nonzero element of `M` and `J` the ideal generated by `a`.\n  intros M hM,\n  resetI,\n  obtain ⟨⟨a, haM⟩, ha0⟩ := submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF),\n  replace ha0 : a ≠ 0 := subtype.coe_injective.ne ha0,\n  let J : ideal A := ideal.span {a},\n  have hJ0 : J ≠ ⊥ := mt ideal.span_singleton_eq_bot.mp ha0,\n  have hJM : J ≤ M := ideal.span_le.mpr (set.singleton_subset_iff.mpr haM),\n  have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF,\n\n  -- Then we can find a product of prime (hence maximal) ideals contained in `J`,\n  -- such that removing element `M` from the product is not contained in `J`.\n  obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM,\n  -- Choose an element `b` of the product that is not in `J`.\n  obtain ⟨b, hbZ, hbJ⟩ := set_like.not_le_iff_exists.mp hnle,\n  have hnz_fa : algebra_map A K a ≠ 0 :=\n    mt ((ring_hom.injective_iff _).mp (is_fraction_ring.injective A K) a) ha0,\n  have hb0 : algebra_map A K b ≠ 0 :=\n    mt ((ring_hom.injective_iff _).mp (is_fraction_ring.injective A K) b)\n      (λ h, hbJ $ h.symm ▸ J.zero_mem),\n  -- Then `b a⁻¹ : K` is in `M⁻¹` but not in `1`.\n  refine ⟨algebra_map A K b * (algebra_map A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩,\n  { exact (fractional_ideal.coe_to_fractional_ideal_ne_zero (le_refl _)).mpr hM0.ne' },\n  { rintro y₀ hy₀,\n    obtain ⟨y, h_Iy, rfl⟩ := (fractional_ideal.mem_coe_ideal _).mp hy₀,\n    rw [mul_comm, ← mul_assoc, ← ring_hom.map_mul],\n    have h_yb : y * b ∈ J,\n    { apply hle,\n      rw multiset.prod_cons,\n      exact submodule.smul_mem_smul h_Iy hbZ },\n    rw ideal.mem_span_singleton' at h_yb,\n    rcases h_yb with ⟨c, hc⟩,\n    rw [← hc, ring_hom.map_mul, mul_assoc, mul_inv_cancel hnz_fa, mul_one],\n    apply fractional_ideal.coe_mem_one },\n  { refine mt (fractional_ideal.mem_one_iff _).mp _,\n    rintros ⟨x', h₂_abs⟩,\n    rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← ring_hom.map_mul] at h₂_abs,\n    have := ideal.mem_span_singleton'.mpr ⟨x', is_fraction_ring.injective A K h₂_abs⟩,\n    contradiction },\nend\n\nlemma one_mem_inv_coe_ideal {I : ideal A} (hI : I ≠ ⊥) :\n  (1 : K) ∈ (I : fractional_ideal A⁰ K)⁻¹ :=\nbegin\n  rw mem_inv_iff (fractional_ideal.coe_ideal_ne_zero hI),\n  intros y hy,\n  rw one_mul,\n  exact coe_ideal_le_one hy,\n  assumption\nend\n\nlemma mul_inv_cancel_of_le_one [h : is_dedekind_domain A]\n  {I : ideal A} (hI0 : I ≠ ⊥) (hI : ((I * I⁻¹)⁻¹ : fractional_ideal A⁰ K) ≤ 1) :\n  (I * I⁻¹ : fractional_ideal A⁰ K) = 1 :=\nbegin\n  -- Handle a few trivial cases.\n  by_cases hI1 : I = ⊤,\n  { rw [hI1, coe_ideal_top, one_mul, fractional_ideal.one_inv] },\n  by_cases hNF : is_field A,\n  { letI := hNF.to_field A, rcases hI1 (I.eq_bot_or_top.resolve_left hI0) },\n  -- We'll show a contradiction with `exists_not_mem_one_of_ne_bot`:\n  -- `J⁻¹ = (I * I⁻¹)⁻¹` cannot have an element `x ∉ 1`, so it must equal `1`.\n  obtain ⟨J, hJ⟩ : ∃ (J : ideal A), (J : fractional_ideal A⁰ K) = I * I⁻¹ :=\n    le_one_iff_exists_coe_ideal.mp mul_one_div_le_one,\n  by_cases hJ0 : J = ⊥,\n  { subst hJ0,\n    refine absurd _ hI0,\n    rw [eq_bot_iff, ← coe_ideal_le_coe_ideal K, hJ],\n    exact coe_ideal_le_self_mul_inv K I,\n    apply_instance },\n  by_cases hJ1 : J = ⊤,\n  { rw [← hJ, hJ1, coe_ideal_top] },\n  obtain ⟨x, hx, hx1⟩ : ∃ (x : K),\n    x ∈ (J : fractional_ideal A⁰ K)⁻¹ ∧ x ∉ (1 : fractional_ideal A⁰ K) :=\n    exists_not_mem_one_of_ne_bot hNF hJ0 hJ1,\n  contrapose! hx1 with h_abs,\n  rw hJ at hx,\n  exact hI hx,\nend\n\n/-- Nonzero integral ideals in a Dedekind domain are invertible.\n\nWe will use this to show that nonzero fractional ideals are invertible,\nand finally conclude that fractional ideals in a Dedekind domain form a group with zero.\n-/\nlemma coe_ideal_mul_inv [h : is_dedekind_domain A] (I : ideal A) (hI0 : I ≠ ⊥) :\n  (I * I⁻¹ : fractional_ideal A⁰ K) = 1 :=\nbegin\n  -- We'll show `1 ≤ J⁻¹ = (I * I⁻¹)⁻¹ ≤ 1`.\n  apply mul_inv_cancel_of_le_one hI0,\n  by_cases hJ0 : (I * I⁻¹ : fractional_ideal A⁰ K) = 0,\n  { rw [hJ0, inv_zero'], exact fractional_ideal.zero_le _ },\n  intros x hx,\n  -- In particular, we'll show all `x ∈ J⁻¹` are integral.\n  suffices : x ∈ integral_closure A K,\n  { rwa [is_integrally_closed.integral_closure_eq_bot, algebra.mem_bot, set.mem_range,\n         ← fractional_ideal.mem_one_iff] at this;\n      assumption },\n  -- For that, we'll find a subalgebra that is f.g. as a module and contains `x`.\n  -- `A` is a noetherian ring, so we just need to find a subalgebra between `{x}` and `I⁻¹`.\n  rw mem_integral_closure_iff_mem_fg,\n  have x_mul_mem : ∀ b ∈ (I⁻¹ : fractional_ideal A⁰ K), x * b ∈ (I⁻¹ : fractional_ideal A⁰ K),\n  { intros b hb,\n    rw mem_inv_iff at ⊢ hx,\n    swap, { exact fractional_ideal.coe_ideal_ne_zero hI0 },\n    swap, { exact hJ0 },\n    simp only [mul_assoc, mul_comm b] at ⊢ hx,\n    intros y hy,\n    exact hx _ (fractional_ideal.mul_mem_mul hy hb) },\n  -- It turns out the subalgebra consisting of all `p(x)` for `p : polynomial A` works.\n  refine ⟨alg_hom.range (polynomial.aeval x : polynomial A →ₐ[A] K),\n          is_noetherian_submodule.mp (fractional_ideal.is_noetherian I⁻¹) _ (λ y hy, _),\n          ⟨polynomial.X, polynomial.aeval_X x⟩⟩,\n  obtain ⟨p, rfl⟩ := (alg_hom.mem_range _).mp hy,\n  rw polynomial.aeval_eq_sum_range,\n  refine submodule.sum_mem _ (λ i hi, submodule.smul_mem _ _ _),\n  clear hi,\n  induction i with i ih,\n  { rw pow_zero, exact one_mem_inv_coe_ideal hI0 },\n  { show x ^ i.succ ∈ (I⁻¹ : fractional_ideal A⁰ K),\n    rw pow_succ, exact x_mul_mem _ ih },\nend\n\n/-- Nonzero fractional ideals in a Dedekind domain are units.\n\nThis is also available as `_root_.mul_inv_cancel`, using the\n`comm_group_with_zero` instance defined below.\n-/\nprotected theorem mul_inv_cancel [is_dedekind_domain A]\n  {I : fractional_ideal A⁰ K} (hne : I ≠ 0) : I * I⁻¹ = 1 :=\nbegin\n  obtain ⟨a, J, ha, hJ⟩ :\n    ∃ (a : A) (aI : ideal A), a ≠ 0 ∧ I = span_singleton A⁰ (algebra_map _ _ a)⁻¹ * aI :=\n    exists_eq_span_singleton_mul I,\n  suffices h₂ : I * (span_singleton A⁰ (algebra_map _ _ a) * J⁻¹) = 1,\n  { rw mul_inv_cancel_iff,\n    exact ⟨span_singleton A⁰ (algebra_map _ _ a) * J⁻¹, h₂⟩ },\n  subst hJ,\n  rw [mul_assoc, mul_left_comm (J : fractional_ideal A⁰ K), coe_ideal_mul_inv, mul_one,\n      fractional_ideal.span_singleton_mul_span_singleton, inv_mul_cancel,\n      fractional_ideal.span_singleton_one],\n  { exact mt ((algebra_map A K).injective_iff.mp (is_fraction_ring.injective A K) _) ha },\n  { exact fractional_ideal.coe_ideal_ne_zero_iff.mp (right_ne_zero_of_mul hne) }\nend\n\nlemma mul_right_le_iff [is_dedekind_domain A] {J : fractional_ideal A⁰ K}\n  (hJ : J ≠ 0) : ∀ {I I'}, I * J ≤ I' * J ↔ I ≤ I' :=\nbegin\n  intros I I',\n  split,\n  { intros h, convert mul_right_mono J⁻¹ h;\n      rw [mul_assoc, fractional_ideal.mul_inv_cancel hJ, mul_one] },\n  { exact λ h, mul_right_mono J h }\nend\n\nlemma mul_left_le_iff [is_dedekind_domain A] {J : fractional_ideal A⁰ K}\n  (hJ : J ≠ 0) {I I'} : J * I ≤ J * I' ↔ I ≤ I' :=\nby convert fractional_ideal.mul_right_le_iff hJ using 1; simp only [mul_comm]\n\nlemma mul_right_strict_mono [is_dedekind_domain A] {I : fractional_ideal A⁰ K}\n  (hI : I ≠ 0) : strict_mono (* I) :=\nstrict_mono_of_le_iff_le (λ _ _, (mul_right_le_iff hI).symm)\n\nlemma mul_left_strict_mono [is_dedekind_domain A] {I : fractional_ideal A⁰ K}\n  (hI : I ≠ 0) : strict_mono ((*) I) :=\nstrict_mono_of_le_iff_le (λ _ _, (mul_left_le_iff hI).symm)\n\n/--\nThis is also available as `_root_.div_eq_mul_inv`, using the\n`comm_group_with_zero` instance defined below.\n-/\nprotected lemma div_eq_mul_inv [is_dedekind_domain A] (I J : fractional_ideal A⁰ K) :\n  I / J = I * J⁻¹ :=\nbegin\n  by_cases hJ : J = 0,\n  { rw [hJ, div_zero, inv_zero', mul_zero] },\n  refine le_antisymm ((mul_right_le_iff hJ).mp _) ((le_div_iff_mul_le hJ).mpr _),\n  { rw [mul_assoc, mul_comm J⁻¹, fractional_ideal.mul_inv_cancel hJ, mul_one, mul_le],\n    intros x hx y hy,\n    rw [mem_div_iff_of_nonzero hJ] at hx,\n    exact hx y hy },\n  rw [mul_assoc, mul_comm J⁻¹, fractional_ideal.mul_inv_cancel hJ, mul_one],\n  exact le_refl I\nend\n\nend fractional_ideal\n\n/-- `is_dedekind_domain` and `is_dedekind_domain_inv` are equivalent ways\nto express that an integral domain is a Dedekind domain. -/\ntheorem is_dedekind_domain_iff_is_dedekind_domain_inv :\n  is_dedekind_domain A ↔ is_dedekind_domain_inv A :=\n⟨λ h I hI, by exactI fractional_ideal.mul_inv_cancel hI, λ h, h.is_dedekind_domain⟩\n\nend inverse\n\nsection is_dedekind_domain\n\nvariables {R A} [is_dedekind_domain A] [algebra A K] [is_fraction_ring A K]\n\nopen fractional_ideal\n\nnoncomputable instance fractional_ideal.comm_group_with_zero :\n  comm_group_with_zero (fractional_ideal A⁰ K) :=\n{ inv := λ I, I⁻¹,\n  inv_zero := inv_zero' _,\n  div := (/),\n  div_eq_mul_inv := fractional_ideal.div_eq_mul_inv,\n  exists_pair_ne := ⟨0, 1, (coe_to_fractional_ideal_injective (le_refl _)).ne\n    (by simpa using @zero_ne_one (ideal A) _ _)⟩,\n  mul_inv_cancel := λ I, fractional_ideal.mul_inv_cancel,\n  .. fractional_ideal.comm_semiring }\n\nnoncomputable instance ideal.comm_cancel_monoid_with_zero :\n  comm_cancel_monoid_with_zero (ideal A) :=\nfunction.injective.comm_cancel_monoid_with_zero (coe_ideal_hom A⁰ (fraction_ring A))\n  coe_ideal_injective (ring_hom.map_zero _) (ring_hom.map_one _) (ring_hom.map_mul _)\n\n/-- For ideals in a Dedekind domain, to divide is to contain. -/\nlemma ideal.dvd_iff_le {I J : ideal A} : (I ∣ J) ↔ J ≤ I :=\n⟨ideal.le_of_dvd,\n  λ h, begin\n    by_cases hI : I = ⊥,\n    { have hJ : J = ⊥, { rwa [hI, ← eq_bot_iff] at h },\n      rw [hI, hJ] },\n    have hI' : (I : fractional_ideal A⁰ (fraction_ring A)) ≠ 0 :=\n      (fractional_ideal.coe_to_fractional_ideal_ne_zero (le_refl (non_zero_divisors A))).mpr hI,\n    have : (I : fractional_ideal A⁰ (fraction_ring A))⁻¹ * J ≤ 1 := le_trans\n      (fractional_ideal.mul_left_mono (↑I)⁻¹ ((coe_ideal_le_coe_ideal _).mpr h))\n      (le_of_eq (inv_mul_cancel hI')),\n    obtain ⟨H, hH⟩ := fractional_ideal.le_one_iff_exists_coe_ideal.mp this,\n    use H,\n    refine coe_to_fractional_ideal_injective (le_refl (non_zero_divisors A))\n      (show (J : fractional_ideal A⁰ (fraction_ring A)) = _, from _),\n    rw [fractional_ideal.coe_ideal_mul, hH, ← mul_assoc, mul_inv_cancel hI', one_mul]\nend⟩\n\nlemma ideal.dvd_not_unit_iff_lt {I J : ideal A} :\n  dvd_not_unit I J ↔ J < I :=\n⟨λ ⟨hI, H, hunit, hmul⟩, lt_of_le_of_ne (ideal.dvd_iff_le.mp ⟨H, hmul⟩)\n   (mt (λ h, have H = 1, from mul_left_cancel₀ hI (by rw [← hmul, h, mul_one]),\n   show is_unit H, from this.symm ▸ is_unit_one) hunit),\n λ h, dvd_not_unit_of_dvd_of_not_dvd (ideal.dvd_iff_le.mpr (le_of_lt h))\n   (mt ideal.dvd_iff_le.mp (not_le_of_lt h))⟩\n\ninstance : wf_dvd_monoid (ideal A) :=\n{ well_founded_dvd_not_unit :=\n  have well_founded ((>) : ideal A → ideal A → Prop) :=\n  is_noetherian_iff_well_founded.mp\n    (is_noetherian_ring_iff.mp is_dedekind_domain.is_noetherian_ring),\n  by { convert this, ext, rw ideal.dvd_not_unit_iff_lt } }\n\ninstance ideal.unique_factorization_monoid :\n  unique_factorization_monoid (ideal A) :=\n{ irreducible_iff_prime := λ P,\n  ⟨λ hirr, ⟨hirr.ne_zero, hirr.not_unit, λ I J, begin\n    have : P.is_maximal,\n    { refine ⟨⟨mt ideal.is_unit_iff.mpr hirr.not_unit, _⟩⟩,\n      intros J hJ,\n      obtain ⟨J_ne, H, hunit, P_eq⟩ := ideal.dvd_not_unit_iff_lt.mpr hJ,\n      exact ideal.is_unit_iff.mp ((hirr.is_unit_or_is_unit P_eq).resolve_right hunit) },\n    rw [ideal.dvd_iff_le, ideal.dvd_iff_le, ideal.dvd_iff_le,\n        set_like.le_def, set_like.le_def, set_like.le_def],\n    contrapose!,\n    rintros ⟨⟨x, x_mem, x_not_mem⟩, ⟨y, y_mem, y_not_mem⟩⟩,\n    exact ⟨x * y, ideal.mul_mem_mul x_mem y_mem,\n           mt this.is_prime.mem_or_mem (not_or x_not_mem y_not_mem)⟩,\n   end⟩,\n   prime.irreducible⟩,\n  .. ideal.wf_dvd_monoid }\n\nnoncomputable instance ideal.normalization_monoid : normalization_monoid (ideal A) :=\nnormalization_monoid_of_unique_units\n\n@[simp] lemma ideal.dvd_span_singleton {I : ideal A} {x : A} :\n  I ∣ ideal.span {x} ↔ x ∈ I :=\nideal.dvd_iff_le.trans (ideal.span_le.trans set.singleton_subset_iff)\n\nlemma ideal.is_prime_of_prime {P : ideal A} (h : prime P) : is_prime P :=\nbegin\n  refine ⟨_, λ x y hxy, _⟩,\n  { unfreezingI { rintro rfl },\n    rw ← ideal.one_eq_top at h,\n    exact h.not_unit is_unit_one },\n  { simp only [← ideal.dvd_span_singleton, ← ideal.span_singleton_mul_span_singleton] at ⊢ hxy,\n    exact h.dvd_or_dvd hxy }\nend\n\ntheorem ideal.prime_of_is_prime {P : ideal A} (hP : P ≠ ⊥) (h : is_prime P) : prime P :=\nbegin\n  refine ⟨hP, mt ideal.is_unit_iff.mp h.ne_top, λ I J hIJ, _⟩,\n  simpa only [ideal.dvd_iff_le] using (h.mul_le.mp (ideal.le_of_dvd hIJ)),\nend\n\n/-- In a Dedekind domain, the (nonzero) prime elements of the monoid with zero `ideal A`\nare exactly the prime ideals. -/\ntheorem ideal.prime_iff_is_prime {P : ideal A} (hP : P ≠ ⊥) :\n  prime P ↔ is_prime P :=\n⟨ideal.is_prime_of_prime, ideal.prime_of_is_prime hP⟩\n\nend is_dedekind_domain\n\nsection is_integral_closure\n\n/-! ### `is_integral_closure` section\n\nWe show that an integral closure of a Dedekind domain in a finite separable\nfield extension is again a Dedekind domain. This implies the ring of integers\nof a number field is a Dedekind domain. -/\n\nopen algebra\nopen_locale big_operators\n\nvariables {A K} [algebra A K] [is_fraction_ring A K]\nvariables {L : Type*} [field L] (C : Type*) [comm_ring C]\nvariables [algebra K L] [finite_dimensional K L] [algebra A L] [is_scalar_tower A K L]\nvariables [algebra C L] [is_integral_closure C A L] [algebra A C] [is_scalar_tower A C L]\n\nlemma is_integral_closure.range_le_span_dual_basis [is_separable K L]\n  {ι : Type*} [fintype ι] [decidable_eq ι] (b : basis ι K L)\n  (hb_int : ∀ i, is_integral A (b i)) [is_integrally_closed A] :\n  ((algebra.linear_map C L).restrict_scalars A).range ≤\n    submodule.span A (set.range $ (trace_form K L).dual_basis (trace_form_nondegenerate K L) b) :=\nbegin\n  let db := (trace_form K L).dual_basis (trace_form_nondegenerate K L) b,\n  rintros _ ⟨x, rfl⟩,\n  simp only [linear_map.coe_restrict_scalars_eq_coe, algebra.linear_map_apply],\n  have hx : is_integral A (algebra_map C L x) :=\n    (is_integral_closure.is_integral A L x).algebra_map,\n  suffices : ∃ (c : ι → A), algebra_map C L x = ∑ i, c i • db i,\n  { obtain ⟨c, x_eq⟩ := this,\n    rw x_eq,\n    refine submodule.sum_mem _ (λ i _, submodule.smul_mem _ _ (submodule.subset_span _)),\n    rw set.mem_range,\n    exact ⟨i, rfl⟩ },\n  suffices : ∃ (c : ι → K), ((∀ i, is_integral A (c i)) ∧ algebra_map C L x = ∑ i, c i • db i),\n  { obtain ⟨c, hc, hx⟩ := this,\n    have hc' : ∀ i, is_localization.is_integer A (c i) :=\n      λ i, is_integrally_closed.is_integral_iff.mp (hc i),\n    use λ i, classical.some (hc' i),\n    refine hx.trans (finset.sum_congr rfl (λ i _, _)),\n    conv_lhs { rw [← classical.some_spec (hc' i)] },\n    rw [← is_scalar_tower.algebra_map_smul K (classical.some (hc' i)) (db i)] },\n  refine ⟨λ i, db.repr (algebra_map C L x) i, (λ i, _), (db.sum_repr _).symm⟩,\n  rw bilin_form.dual_basis_repr_apply,\n  exact is_integral_trace (is_integral_mul hx (hb_int i))\nend\n\nlemma integral_closure_le_span_dual_basis [is_separable K L]\n  {ι : Type*} [fintype ι] [decidable_eq ι] (b : basis ι K L)\n  (hb_int : ∀ i, is_integral A (b i)) [is_integrally_closed A] :\n  (integral_closure A L).to_submodule ≤ submodule.span A (set.range $\n    (trace_form K L).dual_basis (trace_form_nondegenerate K L) b) :=\nbegin\n  refine le_trans _ (is_integral_closure.range_le_span_dual_basis (integral_closure A L) b hb_int),\n  intros x hx,\n  exact ⟨⟨x, hx⟩, rfl⟩\nend\n\nvariables (A) (K)\n\ninclude K\n\n/-- Send a set of `x`'es in a finite extension `L` of the fraction field of `R`\nto `(y : R) • x ∈ integral_closure R L`. -/\nlemma exists_integral_multiples (s : finset L) :\n  ∃ (y ≠ (0 : A)), ∀ x ∈ s, is_integral A (y • x) :=\nbegin\n  haveI := classical.dec_eq L,\n  refine s.induction _ _,\n  { use [1, one_ne_zero],\n    rintros x ⟨⟩ },\n  { rintros x s hx ⟨y, hy, hs⟩,\n    obtain ⟨x', y', hy', hx'⟩ := exists_integral_multiple\n      ((is_fraction_ring.is_algebraic_iff A K).mpr (algebra.is_algebraic_of_finite x))\n      ((algebra_map A L).injective_iff.mp _),\n    refine ⟨y * y', mul_ne_zero hy hy', λ x'' hx'', _⟩,\n    rcases finset.mem_insert.mp hx'' with (rfl | hx''),\n    { rw [mul_smul, algebra.smul_def, algebra.smul_def, mul_comm _ x'', hx'],\n      exact is_integral_mul is_integral_algebra_map x'.2 },\n    { rw [mul_comm, mul_smul, algebra.smul_def],\n      exact is_integral_mul is_integral_algebra_map (hs _ hx'') },\n    { rw is_scalar_tower.algebra_map_eq A K L,\n      apply (algebra_map K L).injective.comp,\n      exact is_fraction_ring.injective _ _ } }\nend\n\nvariables (L)\n\n/-- If `L` is a finite extension of `K = Frac(A)`,\nthen `L` has a basis over `A` consisting of integral elements. -/\nlemma finite_dimensional.exists_is_basis_integral :\n  ∃ (s : finset L) (b : basis s K L), (∀ x, is_integral A (b x)) :=\nbegin\n  letI := classical.dec_eq L,\n  letI : is_noetherian K L := is_noetherian.iff_fg.2 infer_instance,\n  let s' := is_noetherian.finset_basis_index K L,\n  let bs' := is_noetherian.finset_basis K L,\n  obtain ⟨y, hy, his'⟩ := exists_integral_multiples A K (finset.univ.image bs'),\n  have hy' : algebra_map A L y ≠ 0,\n  { refine mt ((algebra_map A L).injective_iff.mp _ _) hy,\n    rw is_scalar_tower.algebra_map_eq A K L,\n    exact (algebra_map K L).injective.comp (is_fraction_ring.injective A K) },\n  refine ⟨s', bs'.map { to_fun := λ x, algebra_map A L y * x,\n                        inv_fun := λ x, (algebra_map A L y)⁻¹ * x,\n                        left_inv := _,\n                        right_inv := _,\n                        .. algebra.lmul _ _ (algebra_map A L y) },\n          _⟩,\n  { intros x, simp only [inv_mul_cancel_left₀ hy'] },\n  { intros x, simp only [mul_inv_cancel_left₀ hy'] },\n  { rintros ⟨x', hx'⟩,\n    simp only [algebra.smul_def, finset.mem_image, exists_prop, finset.mem_univ, true_and] at his',\n    simp only [basis.map_apply, linear_equiv.coe_mk],\n    exact his' _ ⟨_, rfl⟩ }\nend\n\nvariables (A K L) [is_separable K L]\ninclude L\n\n/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is\nintegrally closed and Noetherian, the integral closure `C` of `A` in `L` is\nNoetherian. -/\nlemma is_integral_closure.is_noetherian_ring [is_integrally_closed A] [is_noetherian_ring A] :\n  is_noetherian_ring C :=\nbegin\n  haveI := classical.dec_eq L,\n  obtain ⟨s, b, hb_int⟩ := finite_dimensional.exists_is_basis_integral A K L,\n  rw is_noetherian_ring_iff,\n  let b' := (trace_form K L).dual_basis (trace_form_nondegenerate K L) b,\n  letI := is_noetherian_span_of_finite A (set.finite_range b'),\n  let f : C →ₗ[A] submodule.span A (set.range b') :=\n    (submodule.of_le (is_integral_closure.range_le_span_dual_basis C b hb_int)).comp\n    ((algebra.linear_map C L).restrict_scalars A).range_restrict,\n  refine is_noetherian_of_tower A (is_noetherian_of_ker_bot f _),\n  rw [linear_map.ker_comp, submodule.ker_of_le, submodule.comap_bot, linear_map.ker_cod_restrict],\n  exact linear_map.ker_eq_bot_of_injective (is_integral_closure.algebra_map_injective C A L)\nend\n\nvariables {A K}\n\n/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is\nintegrally closed and Noetherian, the integral closure of `A` in `L` is\nNoetherian. -/\nlemma integral_closure.is_noetherian_ring [is_integrally_closed A] [is_noetherian_ring A] :\n  is_noetherian_ring (integral_closure A L) :=\nis_integral_closure.is_noetherian_ring A K L (integral_closure A L)\n\nvariables (A K) [is_domain C]\n/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,\nthe integral closure `C` of `A` in `L` is a Dedekind domain.\n\nCan't be an instance since `A`, `K` or `L` can't be inferred. See also the instance\n`integral_closure.is_dedekind_domain_fraction_ring` where `K := fraction_ring A`\nand `C := integral_closure A L`.\n-/\nlemma is_integral_closure.is_dedekind_domain [h : is_dedekind_domain A] :\n  is_dedekind_domain C :=\nbegin\n  haveI : is_fraction_ring C L := is_integral_closure.is_fraction_ring_of_finite_extension A K L C,\n  exact\n  ⟨is_integral_closure.is_noetherian_ring A K L C,\n   h.dimension_le_one.is_integral_closure _ L _,\n   (is_integrally_closed_iff L).mpr (λ x hx, ⟨is_integral_closure.mk' C x\n      (is_integral_trans (is_integral_closure.is_integral_algebra A L) _ hx),\n    is_integral_closure.algebra_map_mk' _ _ _⟩)⟩\nend\n\n/- If `L` is a finite separable extension of `K = Frac(A)`, where `A` is a Dedekind domain,\nthe integral closure of `A` in `L` is a Dedekind domain.\n\nCan't be an instance since `K` can't be inferred. See also the instance\n`integral_closure.is_dedekind_domain_fraction_ring` where `K := fraction_ring A`.\n-/\nlemma integral_closure.is_dedekind_domain [h : is_dedekind_domain A] :\n  is_dedekind_domain (integral_closure A L) :=\nis_integral_closure.is_dedekind_domain A K L (integral_closure A L)\n\nomit K\n\nvariables [algebra (fraction_ring A) L] [is_scalar_tower A (fraction_ring A) L]\nvariables [finite_dimensional (fraction_ring A) L] [is_separable (fraction_ring A) L]\n\n/- If `L` is a finite separable extension of `Frac(A)`, where `A` is a Dedekind domain,\nthe integral closure of `A` in `L` is a Dedekind domain.\n\nSee also the lemma `integral_closure.is_dedekind_domain` where you can choose\nthe field of fractions yourself.\n-/\ninstance integral_closure.is_dedekind_domain_fraction_ring\n  [is_dedekind_domain A] : is_dedekind_domain (integral_closure A L) :=\nintegral_closure.is_dedekind_domain A (fraction_ring A) L\n\nend is_integral_closure\n\nsection is_dedekind_domain\n\nvariables {T : Type*} [comm_ring T] [is_domain T] [is_dedekind_domain T] (I J : ideal T)\nopen_locale classical\nopen multiset unique_factorization_monoid ideal\n\nlemma prod_normalized_factors_eq_self {I : ideal T} (hI : I ≠ ⊥) :\n  (normalized_factors I).prod = I :=\nassociated_iff_eq.1 (normalized_factors_prod hI)\n\nlemma normalized_factors_prod {α : multiset (ideal T)}\n  (h : ∀ p ∈ α, prime p) : normalized_factors α.prod = α :=\nby { simp_rw [← multiset.rel_eq, ← associated_eq_eq],\n     exact prime_factors_unique (prime_of_normalized_factor) h\n      (normalized_factors_prod (α.prod_ne_zero_of_prime h)) }\n\nlemma count_le_of_ideal_ge {I J : ideal T} (h : I ≤ J) (hI : I ≠ ⊥) (K : ideal T) :\n  count K (normalized_factors J) ≤ count K (normalized_factors I) :=\nle_iff_count.1 ((dvd_iff_normalized_factors_le_normalized_factors (ne_bot_of_le_ne_bot hI h) hI).1\n  (dvd_iff_le.2 h)) _\n\nlemma sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :\n  I ⊔ J = (normalized_factors I ∩ normalized_factors J).prod :=\nbegin\n  have H : normalized_factors (normalized_factors I ∩ normalized_factors J).prod =\n    normalized_factors I ∩ normalized_factors J,\n  { apply _root_.normalized_factors_prod,\n    intros p hp,\n    rw mem_inter at hp,\n    exact prime_of_normalized_factor p hp.left },\n  have := (multiset.prod_ne_zero_of_prime (normalized_factors I ∩ normalized_factors J)\n      (λ _ h, prime_of_normalized_factor _ (multiset.mem_inter.1 h).1)),\n  apply le_antisymm,\n  { rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le],\n    split,\n    { rw [dvd_iff_normalized_factors_le_normalized_factors this hI, H],\n      exact inf_le_left },\n    { rw [dvd_iff_normalized_factors_le_normalized_factors this hJ, H],\n      exact inf_le_right } },\n  { rw [← dvd_iff_le, dvd_iff_normalized_factors_le_normalized_factors,\n      _root_.normalized_factors_prod, le_iff_count],\n    { intro a,\n      rw multiset.count_inter,\n      exact le_min (count_le_of_ideal_ge le_sup_left hI a)\n        (count_le_of_ideal_ge le_sup_right hJ a) },\n    { intros p hp,\n      rw mem_inter at hp,\n      exact prime_of_normalized_factor p hp.left },\n    { exact ne_bot_of_le_ne_bot hI le_sup_left },\n    { exact this } },\nend\n\nend is_dedekind_domain\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/ring_theory/dedekind_domain.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3286161252430486}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport init.meta.tactic init.meta.format init.function\n\n/-- This is a kind attached to an argument of a congruence lemma that tells the simplifier how to fill it in.\n- `fixed`: It is a parameter for the congruence lemma, the parameter occurs in the left and right hand sides.\n  For example the α in the congruence generated from `f: Π {α : Type} α → α`.\n- `fixed_no_param`: It is not a parameter for the congruence lemma, the lemma was specialized for this parameter.\n  This only happens if the parameter is a subsingleton/proposition, and other parameters depend on it.\n- `eq`: The lemma contains three parameters for this kind of argument `a_i`, `b_i` and `(eq_i : a_i = b_i)`.\n  `a_i` and `b_i` represent the left and right hand sides, and `eq_i` is a proof for their equality.\n  For example the second argument in `f: Π {α : Type}, α → α`.\n- `cast`: corresponds to arguments that are subsingletons/propositions.\n  For example the `p` in the congruence generated from `f : Π (x y : ℕ) (p: x < y), ℕ`.\n- `heq` The lemma contains three parameters for this kind of argument `a_i`, `b_i` and `(eq_i : a_i == b_i)`.\n   `a_i` and `b_i` represent the left and right hand sides, and eq_i is a proof for their heterogeneous equality.\n-/\ninductive congr_arg_kind\n| fixed | fixed_no_param | eq | cast | heq | subsingleton_inst\n\nnamespace congr_arg_kind\n    def to_string : congr_arg_kind → string\n    | fixed := \"fixed\" | fixed_no_param := \"fixed_no_param\" | eq := \"eq\" | cast := \"cast\" | heq := \"heq\" | subsingleton_inst := \"subsingleton_inst\"\n    instance : has_repr congr_arg_kind := ⟨to_string⟩\n    meta instance : has_to_format congr_arg_kind := ⟨λ x, to_string x⟩\nend congr_arg_kind\n\n/--\nA congruence lemma is a proof that two terms are equal using a congruence proof generated by `mk_congr_lemma_simp` and friends.\nSee the docstring for `mk_congr_lemma_simp` and `congr_arg_kind` for more information.\nThe conclusion is prepended by a set of arguments. `arg_kinds` gives a suggestion of how that argument should be filled in using a simplifier.\n  -/\nmeta structure congr_lemma :=\n(type : expr)\n(proof : expr)\n(arg_kinds : list congr_arg_kind)\n\nnamespace tactic\n/--\n`mk_congr_lemma_simp f nargs md`\ncreates a congruence lemma for the simplifier for the given function argument `f`.\nIf `nargs` is not none, then it tries to create a lemma for an application of arity `nargs`.\nIf `nargs` is none then the number of arguments will be guessed from the type signature of `f`.\n\nThat is, given `f : Π {α β γ δ : Type}, α → β → γ → δ` and `nargs = some 6`, we get a congruence lemma:\n``` lean\n{ type := ∀ (α β γ δ : Type), ∀ (a₁ a₂ : α), a₁ = a₂ → ∀ (b₁ b₂ : β), b₁ = b₂ → f a₁ b₁ = f a₂ b₂\n, proof := ...\n, arg_kinds := [fixed, fixed, fixed, fixed, eq,eq]\n}\n```\nSee the docstrings for the cases of `congr_arg_kind` for more detail on how `arg_kinds` are chosen.\nThe system chooses the `arg_kinds` depending on what the other arguments depend on and whether the arguments have subsingleton types.\n\nNote that the number of arguments that `proof` takes can be inferred from `arg_kinds`: `arg_kinds.sum (fixed,cast ↦ 1 | eq,heq ↦ 3 | fixed_no_param ↦ 0)`.\n\nFrom `congr_lemma.cpp`:\n> Create a congruence lemma that is useful for the simplifier.\n> In this kind of lemma, if the i-th argument is a Cast argument, then the lemma\n> contains an input a_i representing the i-th argument in the left-hand-side, and\n> it appears with a cast (e.g., eq.drec ... a_i ...) in the right-hand-side.\n> The idea is that the right-hand-side of this lemma \"tells\" the simplifier\n> how the resulting term looks like.\n-/\nmeta constant mk_congr_lemma_simp (f : expr) (nargs : option nat := none) (md := semireducible) : tactic congr_lemma\n\n/-- Create a specialized theorem using (a prefix of) the arguments of the given application.\n\nAn example of usage can be found in `tests/lean/simp_subsingleton.lean`.\nFor more information on specialization see the comment in the method body for `get_specialization_prefix_size` in `src/library/fun_info.cpp`.\n -/\nmeta constant mk_specialized_congr_lemma_simp (h : expr) (md : transparency := semireducible) : tactic congr_lemma\n\n/-- Similar to `mk_congr_lemma_simp`, this will make a `congr_lemma` object.\nThe difference is that for each `congr_arg_kind.cast` argument, two proof arguments are generated.\n\nConsider some function `f : Π (x : ℕ) (p : x < 4), ℕ`.\n- `mk_congr_simp` will produce a congruence lemma with type `∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4), f x p = f x_1 _`.\n- `mk_congr` will produce a congruence lemma with type `∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4) (p_1 : x_1 < 4), f x p = f x_1 p_1`.\n\nFrom `congr_lemma.cpp`:\n> Create a congruence lemma for the congruence closure module.\n> In this kind of lemma, if the i-th argument is a Cast argument, then the lemma\n> contains two inputs a_i and b_i representing the i-th argument in the left-hand-side and\n> right-hand-side.\n> This lemma is based on the congruence lemma for the simplifier.\n> It uses subsinglenton elimination to show that the congr-simp lemma right-hand-side\n> is equal to the right-hand-side of this lemma.\n -/\nmeta constant mk_congr_lemma (h : expr) (nargs : option nat := none) (md := semireducible) : tactic congr_lemma\n/-- Create a specialized theorem using (a prefix of) the arguments of the given application.\n\nFor more information on specialization see the comment in the method body for `get_specialization_prefix_size` in `src/library/fun_info.cpp`.\n-/\nmeta constant mk_specialized_congr_lemma (h : expr) (md := semireducible) : tactic congr_lemma\n\n/-- Make a congruence lemma using hetrogeneous equality `heq` instead of `eq`.\nFor example `mk_hcongr_lemma (f : Π (α : ℕ → Type) (n:ℕ) (b:α n), ℕ` )` will make\n\n``` lean\n{ type := ∀ α α', α = α' → ∀ n n', n = n' → ∀ (b : α n) (b' : α' n'), b == b' → f α n b == f α' n' b'\n, proof := ...\n, arg_kinds := [eq,eq,heq]\n}\n```\n\n(Using merely `mk_congr_lemma` instead will produce `[fixed,fixed,eq]` instaed.)\n-/\nmeta constant mk_hcongr_lemma (h : expr) (nargs : option nat := none) (md := semireducible) : tactic congr_lemma\n\nend tactic\n", "meta": {"author": "subfish-zhou", "repo": "N2Lean", "sha": "8e858cc5b01f1ad921094dc355db3cb9473a42fd", "save_path": "github-repos/lean/subfish-zhou-N2Lean", "path": "github-repos/lean/subfish-zhou-N2Lean/N2Lean-8e858cc5b01f1ad921094dc355db3cb9473a42fd/library/init/meta/congr_lemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.32861611651644046}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.discrete\nimport Mathlib.category_theory.monoidal.unitors\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.algebra.punit_instances\nimport Mathlib.PostPort\n\nuniverses v₁ u₁ l u₂ v₂ u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# The category of monoids in a monoidal category.\n-/\n\n/--\nA monoid object internal to a monoidal category.\n\nWhen the monoidal category is preadditive, this is also sometimes called an \"algebra object\".\n-/\nstructure Mon_ (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C]\n    where\n  X : C\n  one : 𝟙_ ⟶ X\n  mul : X ⊗ X ⟶ X\n  one_mul' :\n    autoParam ((one ⊗ 𝟙) ≫ mul = category_theory.iso.hom λ_)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_one' :\n    autoParam ((𝟙 ⊗ one) ≫ mul = category_theory.iso.hom ρ_)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_assoc' :\n    autoParam ((mul ⊗ 𝟙) ≫ mul = category_theory.iso.hom α_ ≫ (𝟙 ⊗ mul) ≫ mul)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- Obviously there is some flexibility stating this axiom.\n\n-- This one has left- and right-hand sides matching the statement of `monoid.mul_assoc`,\n\n-- and chooses to place the associator on the right-hand side.\n\n-- The heuristic is that unitors and associators \"don't have much weight\".\n\ntheorem Mon_.one_mul {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (c : Mon_ C) :\n    (Mon_.one c ⊗ 𝟙) ≫ Mon_.mul c = category_theory.iso.hom λ_ :=\n  sorry\n\ntheorem Mon_.mul_one {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (c : Mon_ C) :\n    (𝟙 ⊗ Mon_.one c) ≫ Mon_.mul c = category_theory.iso.hom ρ_ :=\n  sorry\n\n@[simp] theorem Mon_.mul_assoc {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (c : Mon_ C) :\n    (Mon_.mul c ⊗ 𝟙) ≫ Mon_.mul c = category_theory.iso.hom α_ ≫ (𝟙 ⊗ Mon_.mul c) ≫ Mon_.mul c :=\n  sorry\n\ntheorem Mon_.one_mul_assoc {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (c : Mon_ C) {X' : C} (f' : Mon_.X c ⟶ X') :\n    (Mon_.one c ⊗ 𝟙) ≫ Mon_.mul c ≫ f' = category_theory.iso.hom λ_ ≫ f' :=\n  sorry\n\n@[simp] theorem Mon_.mul_assoc_assoc {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (c : Mon_ C) {X' : C} (f' : Mon_.X c ⟶ X') :\n    (Mon_.mul c ⊗ 𝟙) ≫ Mon_.mul c ≫ f' =\n        category_theory.iso.hom α_ ≫ (𝟙 ⊗ Mon_.mul c) ≫ Mon_.mul c ≫ f' :=\n  sorry\n\nnamespace Mon_\n\n\n/--\nThe trivial monoid object. We later show this is initial in `Mon_ C`.\n-/\n@[simp] theorem trivial_one (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] : one (trivial C) = 𝟙 :=\n  Eq.refl (one (trivial C))\n\nprotected instance inhabited (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] : Inhabited (Mon_ C) :=\n  { default := trivial C }\n\n@[simp] theorem one_mul_hom {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {Z : C} (f : Z ⟶ X M) :\n    (one M ⊗ f) ≫ mul M = category_theory.iso.hom λ_ ≫ f :=\n  sorry\n\n@[simp] theorem mul_one_hom {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {Z : C} (f : Z ⟶ X M) :\n    (f ⊗ one M) ≫ mul M = category_theory.iso.hom ρ_ ≫ f :=\n  sorry\n\ntheorem assoc_flip {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C]\n    {M : Mon_ C} : (𝟙 ⊗ mul M) ≫ mul M = category_theory.iso.inv α_ ≫ (mul M ⊗ 𝟙) ≫ mul M :=\n  sorry\n\n/-- A morphism of monoid objects. -/\nstructure hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C]\n    (M : Mon_ C) (N : Mon_ C)\n    where\n  hom : X M ⟶ X N\n  one_hom' :\n    autoParam (one M ≫ hom = one N)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_hom' :\n    autoParam (mul M ≫ hom = (hom ⊗ hom) ≫ mul N)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem hom.one_hom {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) :\n    one M ≫ hom.hom c = one N :=\n  sorry\n\n@[simp] theorem hom.mul_hom {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) :\n    mul M ≫ hom.hom c = (hom.hom c ⊗ hom.hom c) ≫ mul N :=\n  sorry\n\n@[simp] theorem hom.mul_hom_assoc {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) {X' : C}\n    (f' : X N ⟶ X') : mul M ≫ hom.hom c ≫ f' = (hom.hom c ⊗ hom.hom c) ≫ mul N ≫ f' :=\n  sorry\n\n/-- The identity morphism on a monoid object. -/\ndef id {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C]\n    (M : Mon_ C) : hom M M :=\n  hom.mk 𝟙\n\nprotected instance hom_inhabited {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (M : Mon_ C) : Inhabited (hom M M) :=\n  { default := id M }\n\n/-- Composition of morphisms of monoid objects. -/\n@[simp] theorem comp_hom {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} {O : Mon_ C} (f : hom M N)\n    (g : hom N O) : hom.hom (comp f g) = hom.hom f ≫ hom.hom g :=\n  Eq.refl (hom.hom (comp f g))\n\nprotected instance category_theory.category {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] : category_theory.category (Mon_ C) :=\n  category_theory.category.mk\n\n@[simp] theorem id_hom' {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (M : Mon_ C) : hom.hom 𝟙 = 𝟙 :=\n  rfl\n\n@[simp] theorem comp_hom' {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} {K : Mon_ C} (f : M ⟶ N)\n    (g : N ⟶ K) : hom.hom (f ≫ g) = hom.hom f ≫ hom.hom g :=\n  rfl\n\n/-- The forgetful functor from monoid objects to the ambient category. -/\n@[simp] theorem forget_map (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] (A : Mon_ C) (B : Mon_ C) (f : A ⟶ B) :\n    category_theory.functor.map (forget C) f = hom.hom f :=\n  Eq.refl (category_theory.functor.map (forget C) f)\n\nprotected instance forget_faithful {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] : category_theory.faithful (forget C) :=\n  category_theory.faithful.mk\n\nprotected instance category_theory.has_hom.hom.hom.category_theory.is_iso {C : Type u₁}\n    [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {B : Mon_ C}\n    (f : A ⟶ B) [e : category_theory.is_iso (category_theory.functor.map (forget C) f)] :\n    category_theory.is_iso (hom.hom f) :=\n  e\n\n/-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/\nprotected instance forget.category_theory.reflects_isomorphisms {C : Type u₁}\n    [category_theory.category C] [category_theory.monoidal_category C] :\n    category_theory.reflects_isomorphisms (forget C) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : Mon_ C) (f : X ⟶ Y)\n      (e : category_theory.is_iso (category_theory.functor.map (forget C) f)) =>\n      category_theory.is_iso.mk (hom.mk (inv (hom.hom f)))\n\nprotected instance unique_hom_from_trivial {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] (A : Mon_ C) : unique (trivial C ⟶ A) :=\n  unique.mk { default := hom.mk (one A) } sorry\n\nprotected instance category_theory.limits.has_initial {C : Type u₁} [category_theory.category C]\n    [category_theory.monoidal_category C] : category_theory.limits.has_initial (Mon_ C) :=\n  category_theory.limits.has_initial_of_unique (trivial C)\n\nend Mon_\n\n\nnamespace category_theory.lax_monoidal_functor\n\n\n/--\nA lax monoidal functor takes monoid objects to monoid objects.\n\nThat is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.\n-/\n-- TODO: map_Mod F A : Mod A ⥤ Mod (F.map_Mon A)\n\n@[simp] theorem map_Mon_obj_X {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂}\n    [category D] [monoidal_category D] (F : lax_monoidal_functor C D) (A : Mon_ C) :\n    Mon_.X (functor.obj (map_Mon F) A) = functor.obj (to_functor F) (Mon_.X A) :=\n  Eq.refl (Mon_.X (functor.obj (map_Mon F) A))\n\n/-- `map_Mon` is functorial in the lax monoidal functor. -/\ndef map_Mon_functor (C : Type u₁) [category C] [monoidal_category C] (D : Type u₂) [category D]\n    [monoidal_category D] : lax_monoidal_functor C D ⥤ Mon_ C ⥤ Mon_ D :=\n  functor.mk map_Mon\n    fun (F G : lax_monoidal_functor C D) (α : F ⟶ G) =>\n      nat_trans.mk\n        fun (A : Mon_ C) =>\n          Mon_.hom.mk (nat_trans.app (monoidal_nat_trans.to_nat_trans α) (Mon_.X A))\n\nend category_theory.lax_monoidal_functor\n\n\nnamespace Mon_\n\n\nnamespace equiv_lax_monoidal_functor_punit\n\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\ndef lax_monoidal_to_Mon (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] :\n    category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C ⥤ Mon_ C :=\n  category_theory.functor.mk\n    (fun (F : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C) =>\n      category_theory.functor.obj (category_theory.lax_monoidal_functor.map_Mon F)\n        (trivial (category_theory.discrete PUnit)))\n    fun (F G : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C)\n      (α : F ⟶ G) =>\n      category_theory.nat_trans.app\n        (category_theory.functor.map\n          (category_theory.lax_monoidal_functor.map_Mon_functor (category_theory.discrete PUnit) C)\n          α)\n        (trivial (category_theory.discrete PUnit))\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\ndef Mon_to_lax_monoidal (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] :\n    Mon_ C ⥤ category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C :=\n  category_theory.functor.mk\n    (fun (A : Mon_ C) =>\n      category_theory.lax_monoidal_functor.mk\n        (category_theory.functor.mk (fun (_x : category_theory.discrete PUnit) => X A)\n          fun (_x _x_1 : category_theory.discrete PUnit) (_x : _x ⟶ _x_1) => 𝟙)\n        (one A) fun (_x _x : category_theory.discrete PUnit) => mul A)\n    fun (A B : Mon_ C) (f : A ⟶ B) =>\n      category_theory.monoidal_nat_trans.mk\n        (category_theory.nat_trans.mk fun (_x : category_theory.discrete PUnit) => hom.hom f)\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simp] theorem unit_iso_inv_app_to_nat_trans_app (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C]\n    (X : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C) :\n    ∀ (X_1 : category_theory.discrete PUnit),\n        category_theory.nat_trans.app\n            (category_theory.monoidal_nat_trans.to_nat_trans\n              (category_theory.nat_trans.app (category_theory.iso.inv (unit_iso C)) X))\n            X_1 =\n          inv\n            (category_theory.eq_to_hom\n              (congr_arg\n                (category_theory.functor.obj (category_theory.lax_monoidal_functor.to_functor X))\n                (unit_iso._proof_1 X_1))) :=\n  sorry\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simp] theorem counit_iso_inv_app_hom (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] (X : Mon_ C) :\n    hom.hom (category_theory.nat_trans.app (category_theory.iso.inv (counit_iso C)) X) = 𝟙 :=\n  Eq.refl 𝟙\n\nend equiv_lax_monoidal_functor_punit\n\n\n/--\nMonoid objects in `C` are \"just\" lax monoidal functors from the trivial monoidal category to `C`.\n-/\ndef equiv_lax_monoidal_functor_punit (C : Type u₁) [category_theory.category C]\n    [category_theory.monoidal_category C] :\n    category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C ≌ Mon_ C :=\n  category_theory.equivalence.mk' sorry sorry sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/Mon__auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.32846576456294085}}
{"text": "import analysis.calculus.cont_diff\nimport topology.continuous_function.bounded\nimport analysis.seminorm\nimport ..seminorm\nimport ..bases\n\n.\n\nopen set filter function\nopen_locale bounded_continuous_function topological_space nnreal\n\nsection prelim\n\nlemma nhds_finset_inf {ι α : Type*} {t : ι → topological_space α} {s : finset ι} {x : α} : \n  @nhds α (s.inf t) x = s.inf (λ i, @nhds α (t i) x) :=\nbegin\n  rw [s.inf_eq_infi, s.inf_eq_infi, nhds_infi],\n  refine infi_congr (λ i, _),\n  rw nhds_infi\nend\n\nlemma infi_binfi_le {ι α : Type*} [partial_order ι] [complete_lattice α] {f : ι → α} :\n  (⨅ i (j ≤ i), f j) = (⨅ i, f i) :=\nle_antisymm (le_infi $ λ i, infi_le_of_le i $ binfi_le' i le_rfl) (le_infi $ λ i, le_infi₂ $ λ j hj, infi_le f j)\n\nlemma binfi_le_binfi_le {ι α : Type*} [partial_order ι] [complete_lattice α] {f : ι → α} {n : ι} :\n  (⨅ (i ≤ n) (j ≤ i), f j) = (⨅ (i ≤ n), f i) :=\nle_antisymm \n  (le_infi₂ $ λ i hi, infi_le_of_le i $ infi_le_of_le hi $ binfi_le' i le_rfl) \n  (le_infi₂ $ λ i hi, le_infi₂ $ λ j hj, binfi_le' j $ hj.trans hi)\n\nnoncomputable def _root_.continuous_linear_equiv.comp_left_continuous_bounded {𝕜 : Type*} \n  (α : Type*) {β γ : Type*} [topological_space α] [nondiscrete_normed_field 𝕜] \n  {_ : normed_group β} {_ : normed_group γ} [normed_space 𝕜 β] [normed_space 𝕜 γ] (g : β ≃L[𝕜] γ) :\n  (α →ᵇ β) ≃L[𝕜] (α →ᵇ γ) :=\ncontinuous_linear_equiv.equiv_of_inverse \n  (g.to_continuous_linear_map.comp_left_continuous_bounded α) \n  (g.symm.to_continuous_linear_map.comp_left_continuous_bounded α) \n  (begin\n    intros f,\n    ext x,\n    simp_rw [continuous_linear_equiv.coe_def_rev, \n              continuous_linear_map.comp_left_continuous_bounded_apply,\n              continuous_linear_equiv.coe_coe, continuous_linear_equiv.symm_apply_apply],\n  end)\n  (begin\n    intros f,\n    ext x,\n    simp_rw [continuous_linear_equiv.coe_def_rev, \n              continuous_linear_map.comp_left_continuous_bounded_apply,\n              continuous_linear_equiv.coe_coe, continuous_linear_equiv.apply_symm_apply]\n  end)\n\n@[simp] lemma _root_.continuous_linear_equiv.comp_left_continuous_bounded_apply {𝕜 α β γ : Type*} \n  [topological_space α] [nondiscrete_normed_field 𝕜] {_ : normed_group β} {_ : normed_group γ} \n  [normed_space 𝕜 β] [normed_space 𝕜 γ] (g : β ≃L[𝕜] γ) (f : α →ᵇ β) (x : α) :\n  (g.comp_left_continuous_bounded α f) x = g (f x) :=\nrfl\n\nlemma iterated_fderiv_add {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {nf ng : with_top ℕ} {i : ℕ} {f g : E → F} (hf : cont_diff 𝕜 nf f)\n  (hg : cont_diff 𝕜 ng g) (hif : (i : with_top ℕ) ≤ nf) \n  (hig : (i : with_top ℕ) ≤ ng) : \niterated_fderiv 𝕜 i (f + g) = (iterated_fderiv 𝕜 i f) + (iterated_fderiv 𝕜 i g) :=\nbegin\n  induction i with i hi,\n  { ext x, simp },\n  { ext x h, \n    have hif' : (i : with_top ℕ) < nf := \n      lt_of_lt_of_le (with_top.coe_lt_coe.mpr $ nat.lt_succ_self _) hif,\n    have hig' : (i : with_top ℕ) < ng := \n      lt_of_lt_of_le (with_top.coe_lt_coe.mpr $ nat.lt_succ_self _) hig,\n    have hdf : differentiable 𝕜 (iterated_fderiv 𝕜 i f) :=\n      (cont_diff_iff_continuous_differentiable.mp hf).2 i hif',\n    have hdg : differentiable 𝕜 (iterated_fderiv 𝕜 i g) :=\n      (cont_diff_iff_continuous_differentiable.mp hg).2 i hig',\n    calc iterated_fderiv 𝕜 (i+1) (f + g) x h \n        = fderiv 𝕜 (iterated_fderiv 𝕜 i (f + g)) x (h 0) (fin.tail h) : rfl\n    ... = fderiv 𝕜 (iterated_fderiv 𝕜 i f + iterated_fderiv 𝕜 i g) x (h 0) (fin.tail h) : \n            by rw hi hif'.le hig'.le\n    ... = (fderiv 𝕜 (iterated_fderiv 𝕜 i f) + fderiv 𝕜 (iterated_fderiv 𝕜 i g)) \n              x (h 0) (fin.tail h) : \n            by rw [pi.add_def, fderiv_add hdf.differentiable_at hdg.differentiable_at]; refl\n    ... = (iterated_fderiv 𝕜 (i+1) f + iterated_fderiv 𝕜 (i+1) g) x h : rfl }\nend\n\nlemma iterated_fderiv_smul {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {nf : with_top ℕ} {i : ℕ} {a : 𝕜} {f : E → F} (hf : cont_diff 𝕜 nf f)\n  (hif : (i : with_top ℕ) ≤ nf) : \niterated_fderiv 𝕜 i (a • f) = a • (iterated_fderiv 𝕜 i f) :=\nbegin\n  induction i with i hi,\n  { ext, simp },\n  { ext x h,\n    have hif' : (i : with_top ℕ) < nf := \n      lt_of_lt_of_le (with_top.coe_lt_coe.mpr $ nat.lt_succ_self _) hif,\n    have hdf : differentiable 𝕜 (iterated_fderiv 𝕜 i f) :=\n      (cont_diff_iff_continuous_differentiable.mp hf).2 i hif',\n    calc iterated_fderiv 𝕜 (i+1) (a • f) x h\n        = fderiv 𝕜 (iterated_fderiv 𝕜 i (a • f)) x (h 0) (fin.tail h) : rfl\n    ... = fderiv 𝕜 (a • iterated_fderiv 𝕜 i f) x (h 0) (fin.tail h) : \n            by rw hi hif'.le\n    ... = (a • fderiv 𝕜 (iterated_fderiv 𝕜 i f)) x (h 0) (fin.tail h) :\n            by rw [pi.smul_def, fderiv_const_smul hdf.differentiable_at]; refl\n    ... = (a • iterated_fderiv 𝕜 (i+1) f) x h : rfl }\nend\n\nlemma iterated_fderiv_neg {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  {nf : with_top ℕ} {i : ℕ} {f : E → F} (hf : cont_diff 𝕜 nf f)\n  (hif : (i : with_top ℕ) ≤ nf) : \niterated_fderiv 𝕜 i (-f) = -(iterated_fderiv 𝕜 i f) :=\ncalc iterated_fderiv 𝕜 i (-f) \n    = iterated_fderiv 𝕜 i ((-1 : 𝕜) • f) : by rw [neg_one_smul]\n... = (-1 : 𝕜) • iterated_fderiv 𝕜 i f : iterated_fderiv_smul hf hif\n... = -(iterated_fderiv 𝕜 i f) : by ext; exact neg_one_smul _ _\n\nend prelim\n\nprivate def bounded_cont_diff_map_submodule (𝕜 E F : Type*) [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  (n : with_top ℕ) : submodule 𝕜 (E → F) :=\n{ carrier := {f | cont_diff 𝕜 n f ∧ ∀ (i : ℕ), ↑i ≤ n → \n                  ∃ C, ∀ x, ∥iterated_fderiv 𝕜 i f x∥ ≤ C },\n  zero_mem' := ⟨cont_diff_zero_fun, λ i hi, ⟨0, λ x, \n    by rw [pi.zero_def, iterated_fderiv_within_zero_fun, pi.zero_apply, norm_zero]⟩⟩,\n  add_mem' := λ f g hf hg, ⟨hf.1.add hg.1, λ i hi, \n    let ⟨Cf, hCf⟩ := hf.2 i hi, ⟨Cg, hCg⟩ := hg.2 i hi in ⟨Cf + Cg, λ x, \n    by {rw [iterated_fderiv_add hf.1 hg.1 hi hi], exact norm_add_le_of_le (hCf x) (hCg x)}⟩⟩,\n  smul_mem' := λ c f hf, ⟨cont_diff_const.smul hf.1, λ i hi, \n    let ⟨C, hC⟩ := hf.2 i hi in ⟨∥c∥ * C, λ x, \n    by {rw [iterated_fderiv_smul hf.1 hi, pi.smul_apply, norm_smul],\n        exact mul_le_mul_of_nonneg_left (hC x) (norm_nonneg _) }⟩⟩ }\n\ndef bounded_cont_diff_map (𝕜 E F : Type*) [nondiscrete_normed_field 𝕜] \n  [normed_group E] [normed_group F] [normed_space 𝕜 E] [normed_space 𝕜 F] \n  (n : with_top ℕ) := ↥(bounded_cont_diff_map_submodule 𝕜 E F n)\n\nlocalized \"notation `B^`n`⟮`E`,`F`;`𝕜`⟯` := bounded_cont_diff_map 𝕜 E F n\" in \n  bounded_cont_diff_map\n\nnamespace bounded_cont_diff_map\n\nsection any_field\n\nvariables {𝕜 E F : Type*} [nondiscrete_normed_field 𝕜] [normed_group E] [normed_group F]\n  [normed_space 𝕜 E] [normed_space 𝕜 F] {n : with_top ℕ} {f g : B^n⟮E, F; 𝕜⟯} {x : E}\n\ninstance : add_comm_group (B^n⟮E, F; 𝕜⟯) := submodule.add_comm_group _\ninstance : module 𝕜 (B^n⟮E, F; 𝕜⟯) := submodule.module _\ninstance : has_coe_to_fun (B^n⟮E, F; 𝕜⟯) (λ _, E → F) := ⟨λ f, f.1⟩\n\n@[ext] lemma ext (H : ∀x, f x = g x) : f = g :=\nby {ext, exact H x}\n\nlemma bounded (f : B^n⟮E, F; 𝕜⟯) {i : ℕ} (hi : (i : with_top ℕ) ≤ n) : \n  ∃ C, ∀ x, ∥iterated_fderiv 𝕜 i f x∥ ≤ C :=\nf.2.2 i hi\n\nlemma cont_diff (f : B^n⟮E, F; 𝕜⟯) :\n  cont_diff 𝕜 n f :=\nf.2.1\n\n@[simp] lemma coe_zero : ((0 : B^n⟮E, F; 𝕜⟯) : E → F) = 0 := rfl\nlemma zero_apply : (0 : B^n⟮E, F; 𝕜⟯) x = 0 := rfl\n\n@[simp] lemma coe_add : ⇑(f + g) = f + g := rfl\nlemma add_apply : (f + g) x = f x + g x := rfl\n\n@[simp] lemma coe_smul {r : 𝕜} : ⇑(r • f) = r • f := rfl\nlemma smul_apply {r : 𝕜} : (r • f) x = r • (f x) := rfl\n\n@[simp] lemma coe_neg : ⇑(-f) = -f := rfl\nlemma neg_apply : (-f) x = -f x := rfl\n\nprotected noncomputable def iterated_fderiv (i : ℕ)\n  (f : B^n⟮E, F; 𝕜⟯) : \n  E →ᵇ (E [×i]→L[𝕜] F) :=\nif hi : (i : with_top ℕ) ≤ n then\n{ to_fun := iterated_fderiv 𝕜 i f,\n  continuous_to_fun := \n  begin\n    have := f.cont_diff,\n    rw cont_diff_iff_continuous_differentiable at this,\n    exact this.1 i hi\n  end,\n  map_bounded' := \n  begin\n    rcases f.bounded hi with ⟨C, hC⟩,\n    use C + C,\n    intros x y,\n    simp [dist_eq_norm, norm_sub_le_of_le (hC x) (hC y)]\n  end }\nelse 0\n\n@[simp] protected lemma iterated_fderiv_apply_of_le {i : ℕ} (hi : (i : with_top ℕ) ≤ n) \n  (f : B^n⟮E, F; 𝕜⟯) : \n  f.iterated_fderiv i x = iterated_fderiv 𝕜 i f x := \nbegin\n  rw bounded_cont_diff_map.iterated_fderiv,\n  split_ifs,\n  refl\nend\n\n@[simp] protected lemma iterated_fderiv_apply_of_gt {i : ℕ} (hi : n < (i : with_top ℕ)) \n  (f : B^n⟮E, F; 𝕜⟯) : \n  f.iterated_fderiv i x = 0 := \nbegin\n  rw lt_iff_not_ge' at hi,\n  rw bounded_cont_diff_map.iterated_fderiv,\n  split_ifs,\n  { contradiction },\n  { refl }\nend\n\n@[simp] protected lemma iterated_fderiv_add (i : ℕ) :\n  (f + g).iterated_fderiv i = f.iterated_fderiv i + g.iterated_fderiv i :=\nbegin\n  ext x : 1,\n  by_cases hi : ↑i ≤ n,\n  { rw [bounded_continuous_function.add_apply, \n        bounded_cont_diff_map.iterated_fderiv_apply_of_le hi,\n        bounded_cont_diff_map.iterated_fderiv_apply_of_le hi, \n        bounded_cont_diff_map.iterated_fderiv_apply_of_le hi,\n        bounded_cont_diff_map.coe_add,\n        iterated_fderiv_add f.cont_diff g.cont_diff hi hi],\n    refl },\n  { rw ← lt_iff_not_ge' at hi,\n    rw [bounded_continuous_function.add_apply, \n        bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi,\n        bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi, \n        bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi,\n        add_zero] }\nend\n\n@[simp] protected lemma iterated_fderiv_smul (i : ℕ) {r : 𝕜} :\n  (r • f).iterated_fderiv i = r • f.iterated_fderiv i :=\nbegin\n  ext x : 1,\n  by_cases hi : ↑i ≤ n,\n  { rw [bounded_continuous_function.smul_apply,\n        bounded_cont_diff_map.iterated_fderiv_apply_of_le hi,\n        bounded_cont_diff_map.iterated_fderiv_apply_of_le hi,\n        bounded_cont_diff_map.coe_smul,\n        iterated_fderiv_smul f.cont_diff hi],\n    refl },\n  { rw ← lt_iff_not_ge' at hi,\n    rw [bounded_continuous_function.smul_apply, \n        bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi,\n        bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi, \n        smul_zero] }\nend\n\nlemma coe_apply_eq_iterated_fderiv_0 (f : B^n⟮E, F; 𝕜⟯) (x : E) :\n  f x = f.iterated_fderiv 0 x fin_zero_elim :=\nby rw [f.iterated_fderiv_apply_of_le (zero_le _), iterated_fderiv_zero_apply]\n\nprotected noncomputable def iterated_fderivₗ (i : ℕ) :\n  B^n⟮E, F; 𝕜⟯ →ₗ[𝕜] (E →ᵇ (E [×i]→L[𝕜] F)) :=\n{ to_fun := λ f, f.iterated_fderiv i,\n  map_add' := λ f g, bounded_cont_diff_map.iterated_fderiv_add i,\n  map_smul' :=λ r f, bounded_cont_diff_map.iterated_fderiv_smul i }\n\n@[simp] protected lemma iterated_fderivₗ_apply {i : ℕ} (hi : (i : with_top ℕ) ≤ n) \n  (f : B^n⟮E, F; 𝕜⟯) : \n  bounded_cont_diff_map.iterated_fderivₗ i f = f.iterated_fderiv i := rfl\n\nprivate noncomputable def tmp_topology₀ (i : ℕ) : \n  topological_space (B^n⟮E, F; 𝕜⟯) :=\ntopological_space.induced (bounded_cont_diff_map.iterated_fderivₗ i) infer_instance\n\nprivate noncomputable def tmp_uniform_space₀ (i : ℕ) : \n  uniform_space (B^n⟮E, F; 𝕜⟯) :=\nuniform_space.comap (bounded_cont_diff_map.iterated_fderivₗ i) infer_instance -- no defeq problem here\n\nprivate noncomputable def tmp_topology₁ (i : ℕ) : \n  topological_space (B^n⟮E, F; 𝕜⟯) :=\n⨅ (hi : (i : with_top ℕ) ≤ n), tmp_topology₀ i\n\nprivate noncomputable def tmp_uniform_space₁ (i : ℕ) : \n  uniform_space (B^n⟮E, F; 𝕜⟯) :=\n@uniform_space.replace_topology _ (tmp_topology₁ i) \n  (⨅ (hi : (i : with_top ℕ) ≤ n), tmp_uniform_space₀ i) \n  (by rw to_topological_space_infi; refl)\n\nprotected noncomputable def topology : topological_space (B^n⟮E, F; 𝕜⟯) := \n  ⨅ (i : ℕ) (hi : (i : with_top ℕ) ≤ n), (tmp_topology₀ i)\n\nprotected noncomputable def uniform_space : uniform_space (B^n⟮E, F; 𝕜⟯) := \n@uniform_space.replace_topology _ bounded_cont_diff_map.topology \n  (⨅ (i : ℕ), (tmp_uniform_space₁ i)) (by rw [to_topological_space_infi]; refl )\n\nprotected lemma topology_eq_directed : (bounded_cont_diff_map.topology : topological_space (B^n⟮E, F; 𝕜⟯)) = \n  ⨅ (i : ℕ) (hi : ↑i ≤ n) (j : ℕ) (hj : (j : with_top ℕ) ≤ ↑i), (tmp_topology₀ j) :=\nle_antisymm \n  (le_infi₂ $ λ i hi, le_infi₂ $ λ j hj, binfi_le' j $ hj.trans hi)\n  (le_infi₂ $ λ i hi, infi_le_of_le i $ infi_le_of_le hi $ binfi_le' i le_rfl) \n\nprotected lemma topology_eq_directed' : (bounded_cont_diff_map.topology : topological_space (B^n⟮E, F; 𝕜⟯)) = \n  ⨅ (i : ℕ) (hi : ↑i ≤ n), (finset.Icc 0 i).inf tmp_topology₀ :=\nbegin\n  rw bounded_cont_diff_map.topology_eq_directed,\n  refine infi_congr (λ i, infi_congr $ λ hi, _),\n  rw finset.inf_eq_infi,\n  refine infi_congr (λ j, infi_congr_Prop _ (λ _, rfl)),\n  rw [finset.mem_Icc, with_top.coe_le_coe, iff_and_self],\n  exact λ _, zero_le _\nend\n\nvariables (𝕜 E F n)\n\nabbreviation snorms_ind : Type* := {i : ℕ // ↑i ≤ n}\n\ninstance : has_zero (snorms_ind n) := ⟨⟨0, zero_le _⟩⟩\ninstance : inhabited (snorms_ind n) := ⟨0⟩\ninstance : lattice (snorms_ind n) := infer_instance\n\nnoncomputable def snorms : \n  seminorm_family 𝕜 (B^n⟮E, F; 𝕜⟯) (snorms_ind n) :=\nλ i, (finset.Icc 0 (i : ℕ)).sup (λ j, (norm_seminorm 𝕜 (E →ᵇ (E [×j]→L[𝕜] F))).comp \n  (bounded_cont_diff_map.iterated_fderivₗ j))\n\nvariables {𝕜 E F n}\n\nlemma snorms_monotone : \n  monotone (snorms 𝕜 E F n) :=\nλ i j hij, finset.sup_mono (finset.Icc_subset_Icc_right hij)\n\n--protected lemma topology_eq_directed' : (bounded_cont_diff_map.topology : topological_space (B^n⟮E, F; 𝕜⟯)) = \n--  ⨅ (i : ℕ) (hi : ↑i ≤ n), (finset.univ : finset (fin $ i+1)).inf (λ j, tmp_topology₀ j $ j.2.trans hi) :=\n--le_antisymm \n--  (le_infi₂ $ λ i hi, le_infi₂ $ λ j hj, binfi_le' j $ hj.trans hi)\n--  (le_infi₂ $ λ i hi, infi_le_of_le i $ infi_le_of_le hi $ binfi_le' i le_rfl) \n\n--protected def seminorm_family : seminorm_family 𝕜 (B^n⟮E, F; 𝕜⟯) {i : ℕ // ↑i ≤ n} :=\n--  λ i, norm_seminorm 𝕜 (E → )\n\nprivate lemma has_basis_zero₀ (i : ℕ) (hi : (i : with_top ℕ) ≤ n) : \n  (@nhds B^n⟮E, F; 𝕜⟯ (tmp_topology₀ i) 0).has_basis (λ ε : ℝ, 0 < ε)\n  (λ ε, bounded_cont_diff_map.iterated_fderiv i ⁻¹' metric.ball 0 ε) :=\nbegin\n  rw nhds_induced,\n  refine (has_basis.comap _ (metric.nhds_basis_ball)).to_has_basis _ _;\n  rintros ε hε;\n  refine ⟨ε, hε, _⟩;\n  rw [linear_map.map_zero];\n  refl\nend\n\nprivate lemma has_basis_zero₁ (i : ℕ) : \n  (@nhds B^n⟮E, F; 𝕜⟯ (tmp_topology₁ i) 0).has_basis (λ ε : ℝ, 0 < ε)\n  (λ ε, ⋂ (hi : ↑i ≤ n), bounded_cont_diff_map.iterated_fderiv i ⁻¹' metric.ball 0 ε) :=\nbegin\n  rw [nhds_infi, has_basis_iff],\n  by_cases hi : (i : with_top ℕ) ≤ n,\n  { simp [hi, (has_basis_zero₀ i hi).mem_iff] },\n  { have : ∃ ε : ℝ, 0 < ε := ⟨1, zero_lt_one⟩,\n    intros t, \n    simp [hi, univ_subset_iff, this] }\nend\n\nattribute [instance] bounded_cont_diff_map.topology\nattribute [instance] bounded_cont_diff_map.uniform_space\n\ninstance : topological_add_group (B^n⟮E, F; 𝕜⟯) :=\ntopological_add_group_infi \n  (λ i, topological_add_group_infi $ λ hi, topological_add_group_induced _)\n\ninstance with_seminorms : with_seminorms (bounded_cont_diff_map.snorms 𝕜 E F n) :=\nbegin\n  rw [seminorm_family.with_seminorms_iff_nhds_eq_infi, bounded_cont_diff_map.topology_eq_directed'], \n  simp_rw [nhds_infi, infi_subtype, nhds_finset_inf],\n  refine infi_congr (λ i, infi_congr $ λ hi, _),\n  rw [bounded_cont_diff_map.snorms, seminorm_family.nhds_0_comap_finset_sup, subtype.coe_mk],\n  refine finset.inf_congr rfl (λ j hj, _),\n  rw [nhds_induced, map_zero, \n      (seminorm_family.with_seminorms_iff_nhds_eq_infi _).mp (norm_with_seminorms 𝕜 (E →ᵇ (E [×j]→L[𝕜] F))),\n      infi_unique, comap_comap],\n  refl\nend\n\nprotected lemma has_basis_zero : (𝓝 0 : filter $ B^n⟮E, F; 𝕜⟯).has_basis \n  (λ Nε : ℕ × ℝ, 0 < Nε.2) (λ Nε, ⋂ (i : ℕ) (hiN : i ≤ Nε.1) (hi : ↑i ≤ n), \n    bounded_cont_diff_map.iterated_fderiv i ⁻¹' metric.ball 0 Nε.2) :=\nbegin\n  rw nhds_infi,\n  convert foo _ has_basis_zero₁,\n  intros i ε δ h, \n  exact Inter_mono (λ hi, preimage_mono $ metric.ball_subset_ball h)\nend\n\nprotected noncomputable def iterated_fderivL (i : ℕ) : \n  (B^n⟮E, F; 𝕜⟯) →L[𝕜] (E →ᵇ (E [×i]→L[𝕜] F)) :=\n{ to_linear_map := bounded_cont_diff_map.iterated_fderivₗ i,\n  cont := \n  begin\n    by_cases hi : ↑i ≤ n,\n    { exact continuous_infi_dom (@continuous_infi_dom _ _ _ _ \n        (λ _, tmp_topology₀ i) _ hi continuous_induced_dom) },\n    { refine continuous_of_const (λ f g, _),\n      ext x : 1,\n      change f.iterated_fderiv i x = g.iterated_fderiv i x,\n      rw ← lt_iff_not_ge at hi,\n      rw [bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi,\n          bounded_cont_diff_map.iterated_fderiv_apply_of_gt hi] }\n  end }\n\n--continuous_infi_dom \n--  (@continuous_infi_dom _ _ _ _ (λ _, tmp_topology₀ i) _ hi continuous_induced_dom)\n\n@[simp] protected lemma iterated_fderivL_apply {i : ℕ}\n  (f : B^n⟮E, F; 𝕜⟯) : \n  bounded_cont_diff_map.iterated_fderivL i f = f.iterated_fderiv i := rfl\n\nlemma continuous_iff {X : Type*} [topological_space X] (φ : X → B^n⟮E, F; 𝕜⟯) : \n  continuous φ ↔ ∀ (i : ℕ) (hi : ↑i ≤ n), continuous \n    ((bounded_cont_diff_map.iterated_fderiv i) ∘ φ) :=\n⟨ λ hφ i hi, (bounded_cont_diff_map.iterated_fderivL i).continuous.comp hφ, \n  λ h, continuous_infi_rng (λ i, continuous_infi_rng $ λ hi, continuous_induced_rng (h i hi)) ⟩\n\nprotected lemma continuous_of_commutes {𝕜' E' F' : Type*} [normed_group E'] [normed_group F'] \n  [nondiscrete_normed_field 𝕜'] [normed_space 𝕜' E'] [normed_space 𝕜' F'] \n  (φ : B^n⟮E, F; 𝕜⟯ → B^n⟮E', F'; 𝕜'⟯) \n  (ψ : Π (i : ℕ), (E →ᵇ (E [×i]→L[𝕜] F)) → E' →ᵇ (E' [×i]→L[𝕜'] F'))\n  (hcont : ∀ (i : ℕ) (hi : ↑i ≤ n), continuous $ ψ i)\n  (hcomm : ∀ (i : ℕ) (hi : ↑i ≤ n), \n    bounded_cont_diff_map.iterated_fderiv i ∘ φ = ψ i ∘ bounded_cont_diff_map.iterated_fderiv i) :\n  continuous φ :=\nbegin\n  rw continuous_iff,\n  intros i hi,\n  rw hcomm i hi,\n  exact (hcont i hi).comp (bounded_cont_diff_map.iterated_fderivL i).continuous\nend\n\ninstance : has_continuous_smul 𝕜 (B^n⟮E, F; 𝕜⟯) :=\nhas_continuous_smul_infi\n  (λ i, has_continuous_smul_infi $ λ hi, has_continuous_smul_induced _)\n\nvariables (𝕜 E F n)\n\nprivate noncomputable def to_bounded_continuous_function_aux : \n  (B^n⟮E, F; 𝕜⟯) →L[𝕜] (E →ᵇ F) :=\n((continuous_multilinear_curry_fin0 𝕜 E F).to_continuous_linear_equiv\n  .comp_left_continuous_bounded E).to_continuous_linear_map ∘L\nbounded_cont_diff_map.iterated_fderivL 0\n\nprivate lemma to_bounded_continuous_function_aux_spec (f : B^n⟮E, F; 𝕜⟯) (x : E) :\n  to_bounded_continuous_function_aux 𝕜 E F n f x = f x := \nbegin\n  change f.iterated_fderiv 0 x 0 = f x,\n  rw bounded_cont_diff_map.iterated_fderiv_apply_of_le (zero_le _),\n  refl\nend\n\ndef to_bounded_continuous_function (f : B^n⟮E, F; 𝕜⟯) : E →ᵇ F :=\n{ to_fun := f,\n  map_bounded' := \n  begin\n    rcases (to_bounded_continuous_function_aux 𝕜 E F n f).bounded with ⟨C, hC⟩,\n    refine ⟨C, λ x y, _⟩,\n    specialize hC x y,\n    rw [to_bounded_continuous_function_aux_spec, to_bounded_continuous_function_aux_spec] at hC,\n    exact hC\n  end,\n  continuous_to_fun := \n  begin\n    convert (to_bounded_continuous_function_aux 𝕜 E F n f).continuous using 1,\n    ext x,\n    rw to_bounded_continuous_function_aux_spec\n  end }\n\ndef to_bounded_continuous_functionₗ : (B^n⟮E, F; 𝕜⟯) →ₗ[𝕜] (E →ᵇ F) :=\n{ to_fun := to_bounded_continuous_function 𝕜 E F n,\n  map_add' := λ f g, rfl,\n  map_smul' := λ a f, rfl }\n\nnoncomputable! def to_bounded_continuous_functionL : \n  (B^n⟮E, F; 𝕜⟯) →L[𝕜] (E →ᵇ F) :=\n{ to_linear_map := to_bounded_continuous_functionₗ 𝕜 E F n,\n  cont := \n  begin\n    convert (to_bounded_continuous_function_aux 𝕜 E F n).continuous using 1,\n    ext f x,\n    rw to_bounded_continuous_function_aux_spec,\n    refl\n  end }\n\nlemma to_bounded_continuous_functionL_eq_iterated_fderivL_zero :\n  bounded_cont_diff_map.to_bounded_continuous_functionL 𝕜 E F n =\n  ((continuous_multilinear_curry_fin0 𝕜 E F).to_continuous_linear_equiv\n    .comp_left_continuous_bounded E).to_continuous_linear_map ∘L\n  bounded_cont_diff_map.iterated_fderivL 0 := \nbegin\n  ext f x,\n  change f x = _,\n  rw ← to_bounded_continuous_function_aux_spec,\n  refl\nend\n\nlemma to_bounded_continuous_function_injective : \n  injective (to_bounded_continuous_functionL 𝕜 E F n) :=\nbegin\n  intros f g hfg,\n  ext x,\n  change to_bounded_continuous_functionL 𝕜 E F n f x = _,\n  rw hfg,\n  refl\nend\n\nvariables {n}\n\nprotected def of_le {k : with_top ℕ} (hkn : k ≤ n) (f : B^n⟮E, F; 𝕜⟯) :\n  B^k⟮E, F; 𝕜⟯ :=\n⟨f, f.cont_diff.of_le hkn, λ i hi, f.bounded (hi.trans hkn)⟩\n\nprotected def of_leₗ {k : with_top ℕ} (hkn : k ≤ n) :\n  B^n⟮E, F; 𝕜⟯ →ₗ[𝕜] B^k⟮E, F; 𝕜⟯ :=\n{ to_fun := bounded_cont_diff_map.of_le 𝕜 E F hkn,\n  map_add' := λ f g, by ext; refl,\n  map_smul' := λ c f, by ext; refl }\n\nprotected lemma iterated_fderiv_of_le {k : with_top ℕ} (hkn : k ≤ n) \n  {i : ℕ} (hi : ↑i ≤ k) (f : B^n⟮E, F; 𝕜⟯) : \n  (f.of_le 𝕜 E F hkn).iterated_fderiv i = f.iterated_fderiv i :=\nbegin\n  ext x,\n  rw [bounded_cont_diff_map.iterated_fderiv_apply_of_le hi,\n      bounded_cont_diff_map.iterated_fderiv_apply_of_le (hi.trans hkn)],\n  refl\nend\n\n-- TODO : why do I need the `!` ?\nprotected noncomputable! def of_leL {k : with_top ℕ} (hkn : k ≤ n) :\n  B^n⟮E, F; 𝕜⟯ →L[𝕜] B^k⟮E, F; 𝕜⟯ :=\n{ to_linear_map := bounded_cont_diff_map.of_leₗ 𝕜 E F hkn,\n  cont := \n  begin\n    refine continuous_infi_rng (λ i, continuous_infi_rng $ λ hi, continuous_induced_rng _),\n    convert (bounded_cont_diff_map.iterated_fderivL i).continuous using 1,\n    ext f : 1,\n    exact f.iterated_fderiv_of_le 𝕜 E F hkn hi\n  end }\n\n--protected lemma topology_eq_infi_induced_of_le :\n--  bounded_cont_diff_map.topology = ⨅ (i : ℕ) (hi : ↑i ≤ n), bounded_cont_diff_map.topology.induced \n--    (bounded_cont_diff_map.of_leL 𝕜 E F hi) :=\n--begin\n--  simp_rw [bounded_cont_diff_map.topology, induced_infi, tmp_topology₀],\n--  \n--end\n\nsection zero\n\nprivate lemma uniform_space_eq_comap : bounded_cont_diff_map.uniform_space = \n  uniform_space.comap (to_bounded_continuous_functionL 𝕜 E F 0) infer_instance := \nbegin\n  suffices : (bounded_cont_diff_map.uniform_space : uniform_space $ B^0⟮E, F; 𝕜⟯) = \n    uniform_space.comap (bounded_cont_diff_map.iterated_fderivₗ 0) infer_instance,\n  { rw [this, to_bounded_continuous_functionL_eq_iterated_fderivL_zero, \n        continuous_linear_map.coe_comp'],\n    conv_rhs {rw [uniform_space.comap_comap]},\n    refine congr_arg _ _,\n    ext s,\n    change s ∈ uniformity _ ↔ s ∈  uniformity _,\n    rw ← ((continuous_multilinear_curry_fin0 𝕜 E F).to_continuous_linear_equiv\n      .comp_left_continuous_bounded E).uniform_embedding.to_uniform_inducing.comap_uniformity,\n    refl },\n  ext s : 1,\n  change (⨅ _ _, _) = _,\n  rw infi_range,\n  change (⨅ _ _, _) = _,\n  simp_rw infi_range,\n  refine le_antisymm (binfi_le' 0 le_rfl) (le_infi $ λ i, le_infi $ λ hi, _),\n  convert le_refl _,\n  rw ← nat.le_zero_iff, exact_mod_cast hi\nend\n\nnoncomputable instance : metric_space (B^0⟮E, F; 𝕜⟯) :=\n(metric_space.induced (to_bounded_continuous_function 𝕜 E F 0) \n  (to_bounded_continuous_function_injective 𝕜 E F 0) infer_instance).replace_uniformity\n(by rw uniform_space_eq_comap)\n\nnoncomputable instance : normed_group (B^0⟮E, F; 𝕜⟯) :=\n{ to_metric_space := infer_instance,\n  ..normed_group.induced (to_bounded_continuous_functionₗ 𝕜 E F 0).to_add_monoid_hom \n  (to_bounded_continuous_function_injective 𝕜 E F 0) }\n\n@[simp] lemma norm_def {f : B^0⟮E, F; 𝕜⟯} : ∥f∥ = ∥to_bounded_continuous_functionL 𝕜 E F 0 f∥ := rfl\n\nnoncomputable! instance : normed_space 𝕜 (B^0⟮E, F; 𝕜⟯) :=\n{ norm_smul_le := λ c f, \n  begin\n    rw [norm_def, norm_def, continuous_linear_map.map_smul],\n    exact normed_space.norm_smul_le _ _\n  end }\n\nend zero\n\nsection infinity\n\nvariables {𝕜 E F}\n\nprotected lemma differentiable (f : B^⊤⟮E, F; 𝕜⟯) : differentiable 𝕜 f := \nf.cont_diff.differentiable le_top \n\nprotected noncomputable def fderiv (f : B^⊤⟮E, F; 𝕜⟯) : B^⊤⟮E, E →L[𝕜] F; 𝕜⟯ := \n⟨fderiv 𝕜 f, (cont_diff_top_iff_fderiv.mp f.cont_diff).2, \n  begin\n    intros i _,\n    rcases f.bounded (le_top : ↑(i+1) ≤ _) with ⟨C, hC⟩,\n    use C,\n    intros x,\n    specialize hC x,\n    rwa [iterated_fderiv_succ_eq_comp_right, linear_isometry_equiv.norm_map] at hC\n  end⟩\n\n@[simp] protected lemma fderiv_apply (f : B^⊤⟮E, F; 𝕜⟯) : \n  f.fderiv x = fderiv 𝕜 f x := rfl\n\nprotected noncomputable def fderivₗ : B^⊤⟮E, F; 𝕜⟯ →ₗ[𝕜] B^⊤⟮E, E →L[𝕜] F; 𝕜⟯ := \n{ to_fun := bounded_cont_diff_map.fderiv,\n  map_add' := λ f g, \n  begin\n    ext x : 1,\n    exact fderiv_add f.differentiable.differentiable_at\n      g.differentiable.differentiable_at,\n  end,\n  map_smul' := λ a f,\n  begin\n    ext x : 1,\n    exact fderiv_const_smul f.differentiable.differentiable_at _\n  end }\n\n@[simp] protected lemma fderivₗ_apply (f : B^⊤⟮E, F; 𝕜⟯) : \n  ⇑(bounded_cont_diff_map.fderivₗ f) = fderiv 𝕜 f := rfl\n\nnoncomputable def fderivL : B^⊤⟮E, F; 𝕜⟯ →L[𝕜] B^⊤⟮E, E →L[𝕜] F; 𝕜⟯ := \n{ to_linear_map := bounded_cont_diff_map.fderivₗ,\n  cont := \n  begin\n    rw bounded_cont_diff_map.continuous_iff,\n    intros i hi,\n    set! φ := \n      (continuous_linear_map.comp_left_continuous_bounded E \n        (continuous_multilinear_curry_right_equiv' 𝕜 i E F).symm\n          .to_continuous_linear_equiv.to_continuous_linear_map).comp \n      (bounded_cont_diff_map.iterated_fderivL (i+1))\n      with hφ,\n    have : bounded_cont_diff_map.iterated_fderiv i ∘ bounded_cont_diff_map.fderivₗ.to_fun = φ,\n    { rw hφ,\n      ext f H k x, \n      simp [iterated_fderiv_succ_apply_right] }, -- slooooooooooow\n    rw this,\n    exact φ.continuous\n  end }\n\nend infinity\n\nend any_field\n\nsection real\n\nvariables {E F G : Type*} [normed_group E] [normed_group F] [normed_group G] \n  [normed_space ℝ E] [normed_space ℝ F] [normed_space ℝ G]\n  {n : with_top ℕ} {n' : ℕ} {f g : B^n⟮E, F; ℝ⟯} {x : E}\n\ninstance : locally_convex_space ℝ (B^n⟮E, F; ℝ⟯) := sorry\n\nend real\n\n--section real\n--\n--open_locale pointwise\n--\n--variables {E F G : Type*} [normed_group E] [normed_group F] [normed_group G] \n--  [normed_space ℝ E] [normed_space ℝ F] [normed_space ℝ G]\n--  {n : with_top ℕ} {f g : B^n⟮E, F; ℝ⟯} {x : E}\n--\n----protected lemma has_basis_zero_homotethy : (𝓝 0 : filter $ B^n⟮E, F; ℝ⟯).has_basis \n----  (λ Nε : ℕ × ℝ, 0 < Nε.2) (λ Nε, Nε.2 • {f | ∀ (i : ℕ) (hiN : i ≤ Nε.1) \n----    (hi : (i : with_top ℕ) ≤ n) , ∥f.iterated_fderiv hi∥ < 1}) :=\n----begin\n----  refine bounded_cont_diff_map.has_basis_zero.to_has_basis _ _,\n----  { rintros ⟨N, ε⟩ hε,\n----    refine ⟨⟨N, 1/ε⟩, one_div_pos.mpr hε, _⟩,\n----    change _ • _ ⊆ _,\n----    rw set_smul_subset_iff₀, }\n----  \n----end\n--\n--lemma goal (T : B^n⟮E, F; ℝ⟯ →ₗ[ℝ] G) : \n--  continuous T ↔ ∃ (p : ℕ), ∃ C > 0, ∀ f : B^n⟮E, F; ℝ⟯, \n--    ∥T f∥ ≤ C * (⨆ (i : ℕ) (hip : i ≤ p) (hin : ↑i ≤ n), ∥f.iterated_fderiv hin∥) :=\n--begin\n--  sorry\n--end\n--\n--lemma continuous_iff_of_linear (T : B^n⟮E, F; ℝ⟯ →ₗ[ℝ] G) : \n--  continuous T ↔ ∃ (p : ℕ), ∃ C > 0, ∀ f : B^n⟮E, F; ℝ⟯, \n--    ∥T f∥ ≤ C * (⨆ (i ≤ p) (hin : ↑i ≤ n) (x : E), ∥iterated_fderiv ℝ i f x∥) :=\n--begin\n--  sorry\n--end\n--\n--lemma bar (T : B^n⟮E, F; ℝ⟯ →ₗ[ℝ] G) : \n--  continuous_at T 0 ↔ ∃ (p : ℕ), ∃ C > 0, ∀ f : B^n⟮E, F; ℝ⟯, \n--    ∥T f∥ ≤ C * (⨆ (i : ℕ) (hip : i ≤ p) (hin : ↑i ≤ n), ∥f.iterated_fderiv hin∥) :=\n--begin\n--  rw [continuous_at, map_zero, \n--      bounded_cont_diff_map.has_basis_zero.tendsto_iff metric.nhds_basis_ball],\n--  split,\n--  { intro H,\n--    rcases H 1 zero_lt_one with ⟨⟨p, ε⟩, hε, H'⟩,\n--    refine ⟨p, ε⁻¹, inv_pos.mpr hε, λ f, _⟩,\n--    sorry },\n--  { rintros ⟨p, C, hC, H⟩ ε hε,\n--    sorry }\n--end\n--\n--end real\n--\n\nend bounded_cont_diff_map", "meta": {"author": "ADedecker", "repo": "distributions", "sha": "b4d124142788db55cf781184aff03bcc46aa2b10", "save_path": "github-repos/lean/ADedecker-distributions", "path": "github-repos/lean/ADedecker-distributions/distributions-b4d124142788db55cf781184aff03bcc46aa2b10/src/spaces/bounded_cont_diff_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.32846576456294085}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.filtered_colimit_commutes_finite_limit\nimport category_theory.limits.preserves.functor_category\nimport category_theory.limits.preserves.shapes.equalizers\nimport category_theory.limits.bicones\nimport category_theory.limits.comma\nimport category_theory.limits.preserves.finite\nimport category_theory.limits.shapes.finite_limits\n\n/-!\n# Representably flat functors\n\nWe define representably flat functors as functors such that the category of structured arrows\nover `X` is cofiltered for each `X`. This concept is also known as flat functors as in [Elephant]\nRemark C2.3.7, and this name is suggested by Mike Shulman in\nhttps://golem.ph.utexas.edu/category/2011/06/flat_functors_and_morphisms_of.html to avoid\nconfusion with other notions of flatness.\n\nThis definition is equivalent to left exact functors (functors that preserves finite limits) when\n`C` has all finite limits.\n\n## Main results\n\n* `flat_of_preserves_finite_limits`: If `F : C ⥤ D` preserves finite limits and `C` has all finite\n  limits, then `F` is flat.\n* `preserves_finite_limits_of_flat`: If `F : C ⥤ D` is flat, then it preserves all finite limits.\n* `preserves_finite_limits_iff_flat`: If `C` has all finite limits,\n  then `F` is flat iff `F` is left_exact.\n* `Lan_preserves_finite_limits_of_flat`: If `F : C ⥤ D` is a flat functor between small categories,\n  then the functor `Lan F.op` between presheaves of sets preserves all finite limits.\n* `flat_iff_Lan_flat`: If `C`, `D` are small and `C` has all finite limits, then `F` is flat iff\n  `Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)` is flat.\n* `preserves_finite_limits_iff_Lan_preserves_finite_limits`: If `C`, `D` are small and `C` has all\n  finite limits, then `F` preserves finite limits iff `Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)`\n  does.\n\n-/\n\nuniverses w v₁ v₂ v₃ u₁ u₂ u₃\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nnamespace category_theory\n\n\nnamespace structured_arrow_cone\nopen structured_arrow\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₁} D]\nvariables {J : Type w} [small_category J]\nvariables {K : J ⥤ C} (F : C ⥤ D) (c : cone K)\n\n/--\nGiven a cone `c : cone K` and a map `f : X ⟶ c.X`, we can construct a cone of structured\narrows over `X` with `f` as the cone point. This is the underlying diagram.\n-/\n@[simps]\ndef to_diagram : J ⥤ structured_arrow c.X K :=\n{ obj := λ j, structured_arrow.mk (c.π.app j),\n  map := λ j k g, structured_arrow.hom_mk g (by simpa) }\n\n/-- Given a diagram of `structured_arrow X F`s, we may obtain a cone with cone point `X`. -/\n@[simps]\ndef diagram_to_cone {X : D} (G : J ⥤ structured_arrow X F) : cone (G ⋙ proj X F ⋙ F) :=\n{ X := X, π := { app := λ j, (G.obj j).hom } }\n\n/--\nGiven a cone `c : cone K` and a map `f : X ⟶ F.obj c.X`, we can construct a cone of structured\narrows over `X` with `f` as the cone point.\n-/\n@[simps]\ndef to_cone {X : D} (f : X ⟶ F.obj c.X) :\n  cone (to_diagram (F.map_cone c) ⋙ map f ⋙ pre _ K F) :=\n{ X := mk f, π := { app := λ j, hom_mk (c.π.app j) rfl,\n                    naturality' := λ j k g, by { ext, dsimp, simp } } }\n\nend structured_arrow_cone\n\nsection representably_flat\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]\nvariables {E : Type u₃} [category.{v₃} E]\n\n/--\nA functor `F : C ⥤ D` is representably-flat functor if the comma category `(X/F)`\nis cofiltered for each `X : C`.\n-/\nclass representably_flat (F : C ⥤ D) : Prop :=\n(cofiltered : ∀ (X : D), is_cofiltered (structured_arrow X F))\n\nattribute [instance] representably_flat.cofiltered\n\nlocal attribute [instance] is_cofiltered.nonempty\n\ninstance representably_flat.id : representably_flat (𝟭 C) :=\nbegin\n  constructor,\n  intro X,\n  haveI : nonempty (structured_arrow X (𝟭 C)) := ⟨structured_arrow.mk (𝟙 _)⟩,\n  suffices : is_cofiltered_or_empty (structured_arrow X (𝟭 C)),\n  { resetI, constructor },\n  constructor,\n  { intros Y Z,\n    use structured_arrow.mk (𝟙 _),\n    use structured_arrow.hom_mk Y.hom (by erw [functor.id_map, category.id_comp]),\n    use structured_arrow.hom_mk Z.hom (by erw [functor.id_map, category.id_comp]) },\n  { intros Y Z f g,\n    use structured_arrow.mk (𝟙 _),\n    use structured_arrow.hom_mk Y.hom (by erw [functor.id_map, category.id_comp]),\n    ext,\n    transitivity Z.hom; simp }\nend\n\ninstance representably_flat.comp (F : C ⥤ D) (G : D ⥤ E)\n  [representably_flat F] [representably_flat G] : representably_flat (F ⋙ G) :=\nbegin\n  constructor,\n  intro X,\n  haveI : nonempty (structured_arrow X (F ⋙ G)),\n  { have f₁ : structured_arrow X G := nonempty.some infer_instance,\n    have f₂ : structured_arrow f₁.right F := nonempty.some infer_instance,\n    exact ⟨structured_arrow.mk (f₁.hom ≫ G.map f₂.hom)⟩ },\n  suffices : is_cofiltered_or_empty (structured_arrow X (F ⋙ G)),\n  { resetI, constructor },\n  constructor,\n  { intros Y Z,\n    let W := @is_cofiltered.min (structured_arrow X G) _ _\n      (structured_arrow.mk Y.hom) (structured_arrow.mk Z.hom),\n    let Y' : W ⟶ _ := is_cofiltered.min_to_left _ _,\n    let Z' : W ⟶ _ := is_cofiltered.min_to_right _ _,\n\n    let W' := @is_cofiltered.min (structured_arrow W.right F) _ _\n      (structured_arrow.mk Y'.right) (structured_arrow.mk Z'.right),\n    let Y'' : W' ⟶ _ := is_cofiltered.min_to_left _ _,\n    let Z'' : W' ⟶ _ := is_cofiltered.min_to_right _ _,\n\n    use structured_arrow.mk (W.hom ≫ G.map W'.hom),\n    use structured_arrow.hom_mk Y''.right (by simp [← G.map_comp]),\n    use structured_arrow.hom_mk Z''.right (by simp [← G.map_comp]) },\n  { intros Y Z f g,\n    let W := @is_cofiltered.eq (structured_arrow X G) _ _\n        (structured_arrow.mk Y.hom) (structured_arrow.mk Z.hom)\n        (structured_arrow.hom_mk (F.map f.right) (structured_arrow.w f))\n        (structured_arrow.hom_mk (F.map g.right) (structured_arrow.w g)),\n    let h : W ⟶ _ := is_cofiltered.eq_hom _ _,\n    let h_cond : h ≫ _ = h ≫ _ := is_cofiltered.eq_condition _ _,\n\n    let W' := @is_cofiltered.eq (structured_arrow W.right F) _ _\n        (structured_arrow.mk h.right) (structured_arrow.mk (h.right ≫ F.map f.right))\n        (structured_arrow.hom_mk f.right rfl)\n        (structured_arrow.hom_mk g.right (congr_arg comma_morphism.right h_cond).symm),\n    let h' : W' ⟶ _ := is_cofiltered.eq_hom _ _,\n    let h'_cond : h' ≫ _ = h' ≫ _ := is_cofiltered.eq_condition _ _,\n\n    use structured_arrow.mk (W.hom ≫ G.map W'.hom),\n    use structured_arrow.hom_mk h'.right (by simp [← G.map_comp]),\n    ext,\n    exact (congr_arg comma_morphism.right h'_cond : _) }\nend\n\nend representably_flat\n\nsection has_limit\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₁} D]\n\nlocal attribute [instance] has_finite_limits_of_has_finite_limits_of_size\n\n@[priority 100]\ninstance cofiltered_of_has_finite_limits [has_finite_limits C] : is_cofiltered C :=\n{ cocone_objs := λ A B, ⟨limits.prod A B, limits.prod.fst, limits.prod.snd, trivial⟩,\n  cocone_maps :=  λ A B f g, ⟨equalizer f g, equalizer.ι f g, equalizer.condition f g⟩,\n  nonempty := ⟨⊤_ C⟩ }\n\nlemma flat_of_preserves_finite_limits [has_finite_limits C] (F : C ⥤ D)\n  [preserves_finite_limits F] : representably_flat F := ⟨λ X,\nbegin\n  haveI : has_finite_limits (structured_arrow X F) :=\n  begin\n    apply has_finite_limits_of_has_finite_limits_of_size.{v₁} (structured_arrow X F),\n    intros J sJ fJ, resetI, constructor\n  end,\n  apply_instance\nend⟩\n\nnamespace preserves_finite_limits_of_flat\nopen structured_arrow\nopen structured_arrow_cone\nvariables {J : Type v₁} [small_category J] [fin_category J] {K : J ⥤ C}\nvariables (F : C ⥤ D) [representably_flat F] {c : cone K} (hc : is_limit c) (s : cone (K ⋙ F))\ninclude hc\n\n/--\n(Implementation).\nGiven a limit cone `c : cone K` and a cone `s : cone (K ⋙ F)` with `F` representably flat,\n`s` can factor through `F.map_cone c`.\n-/\nnoncomputable def lift : s.X ⟶ F.obj c.X :=\nlet s' := is_cofiltered.cone (to_diagram s ⋙ structured_arrow.pre _ K F) in\ns'.X.hom ≫ (F.map $ hc.lift $\n  (cones.postcompose ({ app := λ X, 𝟙 _, naturality' := by simp }\n      : (to_diagram s ⋙ pre s.X K F) ⋙ proj s.X F ⟶ K)).obj $\n  (structured_arrow.proj s.X F).map_cone s')\n\nlemma fac (x : J) : lift F hc s ≫ (F.map_cone c).π.app x = s.π.app x :=\nby simpa [lift, ←functor.map_comp]\n\nlocal attribute [simp] eq_to_hom_map\n\nlemma uniq {K : J ⥤ C} {c : cone K} (hc : is_limit c)\n  (s : cone (K ⋙ F)) (f₁ f₂ : s.X ⟶ F.obj c.X)\n  (h₁ : ∀ (j : J), f₁ ≫ (F.map_cone c).π.app j = s.π.app j)\n  (h₂ : ∀ (j : J), f₂ ≫ (F.map_cone c).π.app j = s.π.app j) : f₁ = f₂ :=\nbegin\n  -- We can make two cones over the diagram of `s` via `f₁` and `f₂`.\n  let α₁ : to_diagram (F.map_cone c) ⋙ map f₁ ⟶ to_diagram s :=\n  { app := λ X, eq_to_hom (by simp [←h₁]), naturality' := λ _ _ _, by { ext, simp } },\n  let α₂ : to_diagram (F.map_cone c) ⋙ map f₂ ⟶ to_diagram s :=\n  { app := λ X, eq_to_hom (by simp [←h₂]), naturality' := λ _ _ _, by { ext, simp } },\n  let c₁ : cone (to_diagram s ⋙ pre s.X K F) :=\n    (cones.postcompose (whisker_right α₁ (pre s.X K F) : _)).obj (to_cone F c f₁),\n  let c₂ : cone (to_diagram s ⋙ pre s.X K F) :=\n    (cones.postcompose (whisker_right α₂ (pre s.X K F) : _)).obj (to_cone F c f₂),\n\n  -- The two cones can then be combined and we may obtain a cone over the two cones since\n  -- `structured_arrow s.X F` is cofiltered.\n  let c₀ := is_cofiltered.cone (bicone_mk _ c₁ c₂),\n  let g₁ : c₀.X ⟶ c₁.X := c₀.π.app (bicone.left),\n  let g₂ : c₀.X ⟶ c₂.X := c₀.π.app (bicone.right),\n\n  -- Then `g₁.right` and `g₂.right` are two maps from the same cone into the `c`.\n  have : ∀ (j : J), g₁.right ≫ c.π.app j = g₂.right ≫ c.π.app j,\n  { intro j,\n    injection c₀.π.naturality (bicone_hom.left  j) with _ e₁,\n    injection c₀.π.naturality (bicone_hom.right j) with _ e₂,\n    simpa using e₁.symm.trans e₂ },\n  have : c.extend g₁.right = c.extend g₂.right,\n  { unfold cone.extend, congr' 1, ext x, apply this },\n\n  -- And thus they are equal as `c` is the limit.\n  have : g₁.right = g₂.right,\n  calc g₁.right = hc.lift (c.extend g₁.right) : by { apply hc.uniq (c.extend _), tidy }\n            ... = hc.lift (c.extend g₂.right) : by { congr, exact this }\n            ... = g₂.right                    : by { symmetry, apply hc.uniq (c.extend _), tidy },\n\n  -- Finally, since `fᵢ` factors through `F(gᵢ)`, the result follows.\n  calc f₁ = 𝟙 _ ≫ f₁                  : by simp\n      ... = c₀.X.hom ≫ F.map g₁.right : g₁.w\n      ... = c₀.X.hom ≫ F.map g₂.right : by rw this\n      ... = 𝟙 _ ≫ f₂                  : g₂.w.symm\n      ... = f₂                         : by simp\nend\n\nend preserves_finite_limits_of_flat\n\n/-- Representably flat functors preserve finite limits. -/\nnoncomputable\ndef preserves_finite_limits_of_flat (F : C ⥤ D) [representably_flat F] :\n  preserves_finite_limits F :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size,\n  intros J _ _, constructor,\n  intros K, constructor,\n  intros c hc,\n  exactI { lift := preserves_finite_limits_of_flat.lift F hc,\n    fac' := preserves_finite_limits_of_flat.fac F hc,\n    uniq' := λ s m h, by\n    { apply preserves_finite_limits_of_flat.uniq F hc,\n      exact h,\n      exact preserves_finite_limits_of_flat.fac F hc s } }\nend\n\n/--\nIf `C` is finitely cocomplete, then `F : C ⥤ D` is representably flat iff it preserves\nfinite limits.\n-/\nnoncomputable\ndef preserves_finite_limits_iff_flat [has_finite_limits C] (F : C ⥤ D) :\n  representably_flat F ≃ preserves_finite_limits F :=\n{ to_fun := λ _, by exactI preserves_finite_limits_of_flat F,\n  inv_fun := λ _, by exactI flat_of_preserves_finite_limits F,\n  left_inv := λ _, proof_irrel _ _,\n  right_inv := λ x, by { cases x, unfold preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr } }\n\nend has_limit\n\n\nsection small_category\nvariables {C D : Type u₁} [small_category C] [small_category D] (E : Type u₂) [category.{u₁} E]\n\n/--\n(Implementation)\nThe evaluation of `Lan F` at `X` is the colimit over the costructured arrows over `X`.\n-/\nnoncomputable\ndef Lan_evaluation_iso_colim (F : C ⥤ D) (X : D)\n  [∀ (X : D), has_colimits_of_shape (costructured_arrow F X) E] :\n  Lan F ⋙ (evaluation D E).obj X ≅\n  ((whiskering_left _ _ E).obj (costructured_arrow.proj F X)) ⋙ colim :=\nnat_iso.of_components (λ G, colim.map_iso (iso.refl _))\nbegin\n  intros G H i,\n  ext,\n  simp only [functor.comp_map, colimit.ι_desc_assoc, functor.map_iso_refl, evaluation_obj_map,\n    whiskering_left_obj_map, category.comp_id, Lan_map_app, category.assoc],\n  erw [colimit.ι_pre_assoc (Lan.diagram F H X) (costructured_arrow.map j.hom),\n    category.id_comp, category.comp_id, colimit.ι_map],\n  rcases j with ⟨j_left, ⟨⟨⟩⟩, j_hom⟩,\n  congr,\n  rw [costructured_arrow.map_mk, category.id_comp, costructured_arrow.mk]\nend\n\nvariables [concrete_category.{u₁} E] [has_limits E] [has_colimits E]\nvariables [reflects_limits (forget E)] [preserves_filtered_colimits (forget E)]\nvariables [preserves_limits (forget E)]\n\n/--\nIf `F : C ⥤ D` is a representably flat functor between small categories, then the functor\n`Lan F.op` that takes presheaves over `C` to presheaves over `D` preserves finite limits.\n-/\nnoncomputable\ninstance Lan_preserves_finite_limits_of_flat (F : C ⥤ D) [representably_flat F] :\n  preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},\n  intros J _ _, resetI,\n  apply preserves_limits_of_shape_of_evaluation (Lan F.op : (Cᵒᵖ ⥤ E) ⥤ (Dᵒᵖ ⥤ E)) J,\n  intro K,\n  haveI : is_filtered (costructured_arrow F.op K) :=\n    is_filtered.of_equivalence (structured_arrow_op_equivalence F (unop K)),\n  exact preserves_limits_of_shape_of_nat_iso (Lan_evaluation_iso_colim _ _ _).symm,\nend\n\ninstance Lan_flat_of_flat (F : C ⥤ D) [representably_flat F] :\n  representably_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) := flat_of_preserves_finite_limits _\n\nvariable [has_finite_limits C]\n\nnoncomputable\ninstance Lan_preserves_finite_limits_of_preserves_finite_limits (F : C ⥤ D)\n  [preserves_finite_limits F] : preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ E)) :=\nbegin\n  haveI := flat_of_preserves_finite_limits F,\n  apply_instance\nend\n\nlemma flat_iff_Lan_flat (F : C ⥤ D) :\n  representably_flat F ↔ representably_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)) :=\n⟨λ H, by exactI infer_instance, λ H,\nbegin\n  resetI,\n  haveI := preserves_finite_limits_of_flat (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)),\n  haveI : preserves_finite_limits F :=\n    begin\n      apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},\n      intros, resetI, apply preserves_limit_of_Lan_presesrves_limit\n    end,\n  apply flat_of_preserves_finite_limits\nend⟩\n\n/--\nIf `C` is finitely complete, then `F : C ⥤ D` preserves finite limits iff\n`Lan F.op : (Cᵒᵖ ⥤ Type*) ⥤ (Dᵒᵖ ⥤ Type*)` preserves finite limits.\n-/\nnoncomputable\ndef preserves_finite_limits_iff_Lan_preserves_finite_limits (F : C ⥤ D) :\n  preserves_finite_limits F ≃ preserves_finite_limits (Lan F.op : _ ⥤ (Dᵒᵖ ⥤ Type u₁)) :=\n{ to_fun := λ _, by exactI infer_instance,\n  inv_fun := λ _,\n  begin\n    apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u₁},\n    intros, resetI, apply preserves_limit_of_Lan_presesrves_limit\n  end,\n  left_inv := λ x,\n  begin\n    cases x, unfold preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr\n  end,\n  right_inv := λ x,\n  begin\n    cases x,\n    unfold preserves_finite_limits_of_flat,\n    congr,\n    unfold category_theory.Lan_preserves_finite_limits_of_preserves_finite_limits\n      category_theory.Lan_preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr\n  end }\n\nend small_category\nend category_theory\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/functor/flat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3284004833826601}}
{"text": "/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.basic\nimport Mathlib.category_theory.monad.kleisli\nimport Mathlib.category_theory.category.Kleisli\nimport Mathlib.category_theory.types\nimport Mathlib.PostPort\n\nuniverses u \n\nnamespace Mathlib\n\n/-!\n\n# Convert from `monad` (i.e. Lean's `Type`-based monads) to `category_theory.monad`\n\nThis allows us to use these monads in category theory.\n\n-/\n\nnamespace category_theory\n\n\nprotected instance of_type_functor.monad (m : Type u → Type u) [Monad m] [is_lawful_monad m] :\n    monad (of_type_functor m) :=\n  monad.mk (nat_trans.mk pure) (nat_trans.mk mjoin)\n\n/--\nThe `Kleisli` category of a `control.monad` is equivalent to the `kleisli` category of its\ncategory-theoretic version, provided the monad is lawful.\n-/\n@[simp] theorem eq_unit_iso (m : Type u → Type u) [Monad m] [is_lawful_monad m] :\n    equivalence.unit_iso (eq m) =\n        nat_iso.of_components (fun (X : Kleisli m) => iso.refl X) (eq._proof_7 m) :=\n  Eq.refl (equivalence.unit_iso (eq m))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/types_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3284004754831641}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.dense_embedding\nimport Mathlib.PostPort\n\nuniverses u_5 u_6 l u_1 u_2 u_3 u_4 u v \n\nnamespace Mathlib\n\n/-- Homeomorphism between `α` and `β`, also called topological isomorphism -/\nstructure homeomorph (α : Type u_5) (β : Type u_6) [topological_space α] [topological_space β] \nextends α ≃ β\nwhere\n  continuous_to_fun : autoParam (continuous (equiv.to_fun _to_equiv))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])\n  continuous_inv_fun : autoParam (continuous (equiv.inv_fun _to_equiv))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])\n\ninfixl:25 \" ≃ₜ \" => Mathlib.homeomorph\n\nnamespace homeomorph\n\n\nprotected instance has_coe_to_fun {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] : has_coe_to_fun (α ≃ₜ β) :=\n  has_coe_to_fun.mk (fun (_x : α ≃ₜ β) => α → β) fun (e : α ≃ₜ β) => ⇑(to_equiv e)\n\n@[simp] theorem homeomorph_mk_coe {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (a : α ≃ β) (b : autoParam (continuous (equiv.to_fun a))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])) (c : autoParam (continuous (equiv.inv_fun a))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])) : ⇑(mk a) = ⇑a :=\n  rfl\n\ntheorem coe_eq_to_equiv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (a : α) : coe_fn h a = coe_fn (to_equiv h) a :=\n  rfl\n\ntheorem to_equiv_injective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] : function.injective to_equiv := sorry\n\ntheorem ext {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {h : α ≃ₜ β} {h' : α ≃ₜ β} (H : ∀ (x : α), coe_fn h x = coe_fn h' x) : h = h' :=\n  to_equiv_injective (equiv.ext H)\n\n/-- Identity map as a homeomorphism. -/\nprotected def refl (α : Type u_1) [topological_space α] : α ≃ₜ α :=\n  mk (equiv.mk (equiv.to_fun (equiv.refl α)) (equiv.inv_fun (equiv.refl α)) sorry sorry)\n\n/-- Composition of two homeomorphisms. -/\nprotected def trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) : α ≃ₜ γ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.trans (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.trans (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- Inverse of a homeomorphism. -/\nprotected def symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : β ≃ₜ α :=\n  mk (equiv.mk (equiv.to_fun (equiv.symm (to_equiv h))) (equiv.inv_fun (equiv.symm (to_equiv h))) sorry sorry)\n\n@[simp] theorem homeomorph_mk_coe_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (a : α ≃ β) (b : autoParam (continuous (equiv.to_fun a))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])) (c : autoParam (continuous (equiv.inv_fun a))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n    (Lean.Name.mkStr\n      (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\") \"interactive\")\n      \"continuity'\")\n    [])) : ⇑(homeomorph.symm (mk a)) = ⇑(equiv.symm a) :=\n  rfl\n\nprotected theorem continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : continuous ⇑h :=\n  continuous_to_fun h\n\n@[simp] theorem apply_symm_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : β) : coe_fn h (coe_fn (homeomorph.symm h) x) = x :=\n  equiv.apply_symm_apply (to_equiv h) x\n\n@[simp] theorem symm_apply_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) : coe_fn (homeomorph.symm h) (coe_fn h x) = x :=\n  equiv.symm_apply_apply (to_equiv h) x\n\nprotected theorem bijective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : function.bijective ⇑h :=\n  equiv.bijective (to_equiv h)\n\nprotected theorem injective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : function.injective ⇑h :=\n  equiv.injective (to_equiv h)\n\nprotected theorem surjective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : function.surjective ⇑h :=\n  equiv.surjective (to_equiv h)\n\n/-- Change the homeomorphism `f` to make the inverse function definitionally equal to `g`. -/\ndef change_inv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ₜ β) (g : β → α) (hg : function.right_inverse g ⇑f) : α ≃ₜ β :=\n  (fun (this : g = ⇑(homeomorph.symm f)) => mk (equiv.mk (⇑f) g sorry sorry)) sorry\n\n@[simp] theorem symm_comp_self {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : ⇑(homeomorph.symm h) ∘ ⇑h = id :=\n  funext (symm_apply_apply h)\n\n@[simp] theorem self_comp_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : ⇑h ∘ ⇑(homeomorph.symm h) = id :=\n  funext (apply_symm_apply h)\n\n@[simp] theorem range_coe {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : set.range ⇑h = set.univ :=\n  function.surjective.range_eq (homeomorph.surjective h)\n\ntheorem image_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : set.image ⇑(homeomorph.symm h) = set.preimage ⇑h :=\n  funext (equiv.image_eq_preimage (to_equiv (homeomorph.symm h)))\n\ntheorem preimage_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : set.preimage ⇑(homeomorph.symm h) = set.image ⇑h :=\n  Eq.symm (funext (equiv.image_eq_preimage (to_equiv h)))\n\n@[simp] theorem image_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) : ⇑h '' (⇑h ⁻¹' s) = s :=\n  equiv.image_preimage (to_equiv h) s\n\n@[simp] theorem preimage_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set α) : ⇑h ⁻¹' (⇑h '' s) = s :=\n  equiv.preimage_image (to_equiv h) s\n\nprotected theorem inducing {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : inducing ⇑h := sorry\n\ntheorem induced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : topological_space.induced (⇑h) _inst_2 = _inst_1 :=\n  Eq.symm (inducing.induced (homeomorph.inducing h))\n\nprotected theorem quotient_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : quotient_map ⇑h := sorry\n\ntheorem coinduced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : topological_space.coinduced (⇑h) _inst_1 = _inst_2 :=\n  Eq.symm (and.right (homeomorph.quotient_map h))\n\nprotected theorem embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : embedding ⇑h :=\n  embedding.mk (homeomorph.inducing h) (equiv.injective (to_equiv h))\n\ntheorem compact_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set α} (h : α ≃ₜ β) : is_compact (⇑h '' s) ↔ is_compact s :=\n  iff.symm (embedding.compact_iff_compact_image (homeomorph.embedding h))\n\ntheorem compact_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set β} (h : α ≃ₜ β) : is_compact (⇑h ⁻¹' s) ↔ is_compact s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_compact (⇑h ⁻¹' s) ↔ is_compact s)) (Eq.symm (image_symm h))))\n    (compact_image (homeomorph.symm h))\n\nprotected theorem dense_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : dense_embedding ⇑h :=\n  dense_embedding.mk\n    (dense_inducing.mk (inducing.mk (Eq.symm (induced_eq h))) (function.surjective.dense_range (homeomorph.surjective h)))\n    (homeomorph.injective h)\n\n@[simp] theorem is_open_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set β} : is_open (⇑h ⁻¹' s) ↔ is_open s :=\n  quotient_map.is_open_preimage (homeomorph.quotient_map h)\n\n@[simp] theorem is_open_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set α} : is_open (⇑h '' s) ↔ is_open s := sorry\n\n@[simp] theorem is_closed_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set β} : is_closed (⇑h ⁻¹' s) ↔ is_closed s := sorry\n\n@[simp] theorem is_closed_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set α} : is_closed (⇑h '' s) ↔ is_closed s := sorry\n\ntheorem preimage_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) : ⇑h ⁻¹' closure s = closure (⇑h ⁻¹' s) := sorry\n\ntheorem image_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set α) : ⇑h '' closure s = closure (⇑h '' s) := sorry\n\nprotected theorem is_open_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : is_open_map ⇑h :=\n  fun (s : set α) => iff.mpr (is_open_image h)\n\nprotected theorem is_closed_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : is_closed_map ⇑h :=\n  fun (s : set α) => iff.mpr (is_closed_image h)\n\nprotected theorem closed_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) : closed_embedding ⇑h :=\n  closed_embedding_of_embedding_closed (homeomorph.embedding h) (homeomorph.is_closed_map h)\n\n@[simp] theorem map_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) : filter.map (⇑h) (nhds x) = nhds (coe_fn h x) := sorry\n\n@[simp] theorem comap_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (y : β) : filter.comap (⇑h) (nhds y) = nhds (coe_fn (homeomorph.symm h) y) := sorry\n\ntheorem nhds_eq_comap {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) : nhds x = filter.comap (⇑h) (nhds (coe_fn h x)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nhds x = filter.comap (⇑h) (nhds (coe_fn h x)))) (comap_nhds_eq h (coe_fn h x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nhds x = nhds (coe_fn (homeomorph.symm h) (coe_fn h x)))) (symm_apply_apply h x)))\n      (Eq.refl (nhds x)))\n\n/-- If an bijective map `e : α ≃ β` is continuous and open, then it is a homeomorphism. -/\ndef homeomorph_of_continuous_open {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (e : α ≃ β) (h₁ : continuous ⇑e) (h₂ : is_open_map ⇑e) : α ≃ₜ β :=\n  mk (equiv.mk (equiv.to_fun e) (equiv.inv_fun e) (equiv.left_inv e) (equiv.right_inv e))\n\n@[simp] theorem comp_continuous_on_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) (f : γ → α) (s : set γ) : continuous_on (⇑h ∘ f) s ↔ continuous_on f s :=\n  iff.symm (inducing.continuous_on_iff (homeomorph.inducing h))\n\n@[simp] theorem comp_continuous_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : γ → α} : continuous (⇑h ∘ f) ↔ continuous f :=\n  iff.symm (inducing.continuous_iff (homeomorph.inducing h))\n\n@[simp] theorem comp_continuous_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : β → γ} : continuous (f ∘ ⇑h) ↔ continuous f :=\n  iff.symm (quotient_map.continuous_iff (homeomorph.quotient_map h))\n\n/-- If two sets are equal, then they are homeomorphic. -/\ndef set_congr {α : Type u_1} [topological_space α] {s : set α} {t : set α} (h : s = t) : ↥s ≃ₜ ↥t :=\n  mk (equiv.mk (equiv.to_fun (equiv.set_congr h)) (equiv.inv_fun (equiv.set_congr h)) sorry sorry)\n\n/-- Sum of two homeomorphisms. -/\ndef sum_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) : α ⊕ γ ≃ₜ β ⊕ δ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.sum_congr (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.sum_congr (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- Product of two homeomorphisms. -/\ndef prod_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) : α × γ ≃ₜ β × δ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.prod_congr (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.prod_congr (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- `α × β` is homeomorphic to `β × α`. -/\ndef prod_comm (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] : α × β ≃ₜ β × α :=\n  mk (equiv.mk (equiv.to_fun (equiv.prod_comm α β)) (equiv.inv_fun (equiv.prod_comm α β)) sorry sorry)\n\n/-- `(α × β) × γ` is homeomorphic to `α × (β × γ)`. -/\ndef prod_assoc (α : Type u_1) (β : Type u_2) (γ : Type u_3) [topological_space α] [topological_space β] [topological_space γ] : (α × β) × γ ≃ₜ α × β × γ :=\n  mk (equiv.mk (equiv.to_fun (equiv.prod_assoc α β γ)) (equiv.inv_fun (equiv.prod_assoc α β γ)) sorry sorry)\n\n/-- `ulift α` is homeomorphic to `α`. -/\ndef ulift {α : Type u} [topological_space α] : ulift α ≃ₜ α :=\n  mk (equiv.mk (equiv.to_fun equiv.ulift) (equiv.inv_fun equiv.ulift) sorry sorry)\n\n/-- `(α ⊕ β) × γ` is homeomorphic to `α × γ ⊕ β × γ`. -/\ndef sum_prod_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] : (α ⊕ β) × γ ≃ₜ α × γ ⊕ β × γ :=\n  homeomorph.symm (homeomorph_of_continuous_open (equiv.symm (equiv.sum_prod_distrib α β γ)) sorry sorry)\n\n/-- `α × (β ⊕ γ)` is homeomorphic to `α × β ⊕ α × γ`. -/\ndef prod_sum_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] : α × (β ⊕ γ) ≃ₜ α × β ⊕ α × γ :=\n  homeomorph.trans (prod_comm α (β ⊕ γ)) (homeomorph.trans sum_prod_distrib (sum_congr (prod_comm β α) (prod_comm γ α)))\n\n/-- `(Σ i, σ i) × β` is homeomorphic to `Σ i, (σ i × β)`. -/\ndef sigma_prod_distrib {β : Type u_2} [topological_space β] {ι : Type u_5} {σ : ι → Type u_6} [(i : ι) → topological_space (σ i)] : (sigma fun (i : ι) => σ i) × β ≃ₜ sigma fun (i : ι) => σ i × β :=\n  homeomorph.symm (homeomorph_of_continuous_open (equiv.symm (equiv.sigma_prod_distrib σ β)) sorry sorry)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/homeomorph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.32828839842862556}}
{"text": "import category_theory.instances.topological_spaces\nimport category_theory.limits.pullbacks\nimport category_theory.const\nimport topology.continuity\n\nopen category_theory.instances\nopen category_theory.limits\nopen category_theory.functor\nopen category_theory\nopen sum\n\nuniverse u\nnoncomputable theory\n\nvariables {X Y Z : Top.{u}}\nvariables (f : X ⟶ Y) (g : X ⟶ Z)\n\n-- in the following diagram P is the pushout we wish to construct\n-- X  ⟶  Y\n-- ↓      ↓\n-- Z  ⟶  P\n\n-- this relation identifies any x and y that are the image of the same\n-- point in X\ndef r (x : sum Y.α Z.α) (y : sum Y.α Z.α) : Prop :=\nx = y ∨\n∃ a : X.α, inl (f.val a) = x ∧ inr (g.val a) = y\n\ndef P : Top := { α := quot (r f g), str := by apply_instance}\n\n-- inclusion maps from Y and Z into the pushout P\ndef i_Y : Y ⟶ P f g := { val := quot.mk (r f g) ∘ inl,\n    property := continuous.comp continuous_inl continuous_quot_mk }\ndef i_Z : Z ⟶ P f g := { val := quot.mk (r f g) ∘ inr,\n    property := continuous.comp continuous_inr continuous_quot_mk }\n\nlemma natural : f ≫ (i_Y f g) = g ≫ (i_Z f g) :=\nsubtype.eq\nbegin\n  rw [concrete_category_comp, i_Y, i_Z],\n  dsimp,\n  ext,\n  rw [function.comp.assoc],\n  apply quot.sound,\n  exact or.inr ⟨ x, ⟨ rfl, rfl ⟩ ⟩,\nend\n\ndef Top.nat_trans_app (A : walking_span) : (span f g).obj A ⟶ ((const walking_span).obj (P f g)).obj A :=\n    match A with\n| walking_span.zero  := f ≫ (i_Y f g)\n| walking_span.left  := i_Y f g\n| walking_span.right := i_Z f g\nend\n\ndef Top.nat_trans : span f g ⟹ (const walking_span).obj (P f g) := { app := Top.nat_trans_app f g,\nnaturality' := sorry }\n\ndef topcat : category Top := by apply_instance\n\n-- cosquare := λ f g, something of type cocone (span f g)\n-- span f g is a functor from walking_span.{v} to Top\n-- cocone (F : J ⥤ Top) is a structure taking an object X : Top and\n-- a natural transformation ι taking F to (const J).obj X\n-- natural transformations have an app field and a naturality' field\ninstance toppush : @has_pushouts Top topcat := { cosquare := sorry,\nis_pushout := sorry }", "meta": {"author": "aymonwuolanne", "repo": "lean-theorem-proving", "sha": "701a7873f9f66f2c24962b02450ec90b20935702", "save_path": "github-repos/lean/aymonwuolanne-lean-theorem-proving", "path": "github-repos/lean/aymonwuolanne-lean-theorem-proving/lean-theorem-proving-701a7873f9f66f2c24962b02450ec90b20935702/src/pushouts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.743167997235783, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.3282372430528884}}
{"text": "import o_minimal.definable\n\nopen o_minimal\n\nvariables {R : Type*} (S : struc R)\n\n/-- A *definable presheaf* structure on a type X\nconsists of the data of, for every definable set S ⊆ Rⁿ,\na predicate `definable` on functions S → X, satisfying the axiom:\nfor every definable map φ : T → S, precomposition by φ\npreserves definable functions into X. -/\n-- TODO: Refactor to be prior to def_set/def_fun?\nclass definable_presheaf (X : Type*) :=\n(definable : Π {n : ℕ} {s : set (finvec n R)} (ds : def_set S s),\n  (s → X) → Prop)\n(definable_precomp : Π {n m : ℕ} {s : set (finvec n R)} {ds : def_set S s}\n  {t : set (finvec m R)} {dt : def_set S t} (φ : t → s)\n  (dφ : @def_fun R S _ _ (is_definable.subtype dt) _ _ (is_definable.subtype ds) φ)\n--(dφ : S.definable {z | ∃ (hz : finvec.left z ∈ t), (φ ⟨finvec.left z, hz⟩).val = finvec.right z})\n  (f : s → X), definable ds f → definable dt (f ∘ φ))\n\n-- TODO: sheaf conditions\n\ninstance Prop.definable_presheaf : definable_presheaf S Prop :=\n{ definable := λ n s ds a, @def_set R S _ _ (is_definable.subtype ds) a,\n  definable_precomp := λ n m s ds t dt φ dφ a da,\n  @def_fun.preimage R S _ _ (is_definable.subtype dt) _ _ (is_definable.subtype ds) φ dφ a da }\n\n-- for now don't make this an instance, because of diamond problem with ×\ndef definable_presheaf_of_is_definable {X : Type*} [has_coordinates R X] [is_definable S X] :\n  definable_presheaf S X :=\n{ definable := λ n s ds f, by haveI := is_definable.subtype ds; exact def_fun S f,\n  definable_precomp := λ n m s ds t dt φ dφ a da,\n  by haveI := is_definable.subtype ds; haveI := is_definable.subtype dt; exact da.comp dφ }\n\nstructure defin {X : Type*} [definable_presheaf S X] {Y : Type*} [definable_presheaf S Y]\n  (F : X → Y) : Prop :=\n(preserves_definable : ∀ {n : ℕ} {s : set (finvec n R)} (ds : def_set S s) {f : s → X},\n  definable_presheaf.definable ds f → definable_presheaf.definable ds (F ∘ f))\n\nsection yoneda\nlocal attribute [instance] definable_presheaf_of_is_definable\n-- A definable map between representables is definable in the new sense\n-- if and only if it was definable in the old one.\n-- A definable set in a representable is definable in the new sense\n-- if and only if it was definable in the old one.\n\nvariables {X : Type*} [has_coordinates R X] [is_definable S X]\nvariables {Y : Type*} [has_coordinates R Y] [is_definable S Y]\n\nlemma def_fun_iff_defin {f : X → Y} : def_fun S f ↔ defin S f :=\nsorry\n\nlemma def_set_iff_defin {s : set X} : def_set S s ↔ defin S s :=\nsorry\n\nend yoneda\n\ninstance prod.definable_presheaf {X Y : Type*} [definable_presheaf S X] [definable_presheaf S Y] :\n  definable_presheaf S (X × Y) :=\n{ definable := λ n s ds a,\n    definable_presheaf.definable ds (λ x, (a x).fst) ∧\n    definable_presheaf.definable ds (λ x, (a x).snd),\n  definable_precomp := λ n m s ds t dt φ dφ a da,\n    ⟨definable_presheaf.definable_precomp φ dφ (λ x, (a x).fst) da.1,\n     definable_presheaf.definable_precomp φ dφ (λ x, (a x).snd) da.2⟩ }\n\ninstance Pi.definable_presheaf {X Y : Type*} [definable_presheaf S X] [definable_presheaf S Y] :\n  definable_presheaf S (X → Y) :=\n{ definable := λ n s ds a,\n    by haveI : is_definable S s := is_definable.subtype ds;\n       haveI : definable_presheaf S s := definable_presheaf_of_is_definable S;\n       exact defin S (λ (p : s × X), a p.1 p.2),\n  definable_precomp := sorry }\n\n-- TODO: prove exponential law for `defin`\n-- prove (un)currying preserves `defin`\n\n-- TODO: prove definability of ∧ : Prop → Prop → Prop, etc.\n-- this amounts to definable sets forming a boolean algebra!\n\ninstance subtype.definable_presheaf {X : Type*} [definable_presheaf S X] (A : set X) :\n  definable_presheaf S A :=\n{ definable := λ n s ds a, definable_presheaf.definable ds (subtype.val ∘ a),\n  definable_precomp := sorry }\n\nclass compat_definable_coordinates (X : Type*) [has_coordinates R X] [definable_presheaf S X]\n-- * definable, like in `is_definable`\n-- * `definable_presheaf` structure equals the one from `definable_presheaf_of_is_definable`\n\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/omin/presheaf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3282330904745325}}
{"text": "import category_theory.category\nimport category_theory.colimits\nimport category_theory.colimit_lemmas\nimport category_theory.replete\nimport category_theory.pasting_pushouts\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cofibrations\n\nclass has_cofibrations (C : Type u) [category C] :=\n(is_cof : Π ⦃a b : C⦄, (a ⟶ b) → Prop)\n\ndef is_cof {C : Type u} [category C] [has_cofibrations C] ⦃a b : C⦄ (f : a ⟶ b) : Prop :=\nhas_cofibrations.is_cof f\n\n/-\n\nWe gather here axioms pertaining to cofibrations common to many\nnotions of \"categories with cofibrations\".\n\n* The cofibrations form a subcategory which includes all isomorphisms.\n\n* Pushouts of cofibrations exist and are again cofibrations.\n\nAn isomorphism is a pushout of an identity map, so it actually\nsuffices to require that identities are cofibrations.\n\nTODO: In ABC cofibration categories, some of the axioms only apply to\ncofibrations with cofibrant domain. Is this sufficient for our\npurposes? Useful?\n\n-/\n\nclass precofibration_category (C : Type u) [category.{v} C]\n  extends has_cofibrations C, wide_subcategory.{v} C is_cof :=\n(pushout_by_cof : Π ⦃a b a' : C⦄ (f : a ⟶ b) (g : a ⟶ a'), is_cof f → pushout f g)\n(pushout_is_cof : ∀ ⦃a b a' b' : C⦄ {f : a ⟶ b} {g : a ⟶ a'} {f' : a' ⟶ b'} {g' : b ⟶ b'},\n  Is_pushout f g g' f' → is_cof f → is_cof f')\n\nopen precofibration_category\n\nvariables {C : Type u} [category.{v} C] [precofibration_category C]\nlemma cof_id (a : C) : is_cof (𝟙 a) := mem_id a\nlemma cof_comp {a b c : C} {f : a ⟶ b} {g : b ⟶ c} :\n  is_cof f → is_cof g → is_cof (g ∘ f) := mem_comp\n\ninstance precofibration_category.replete\n  (C : Type u) [category.{v} C] [p : precofibration_category.{v} C] :\n  replete_wide_subcategory.{v} C is_cof :=\n{ mem_iso := assume a b i,\n    pushout_is_cof\n      (by convert Is_pushout_of_isomorphic' (Is_pushout.refl (𝟙 a)) i; simp; refl)\n      (cof_id a) }\n\nlemma cof_iso {a b : C} (i : a ≅ b) : is_cof i.hom := mem_iso i\n\n-- The coproduct of cofibrations is a cofibration.\n-- TODO: Should we try to express this using Is_coproduct?\n-- TODO: Make `coproduct` a class and use it in notation\nlemma cof_coprod [has_initial_object.{v} C] [has_coproducts.{v} C]\n  {a₀ b₀ a₁ b₁ : C} {j₀ : a₀ ⟶ b₀} {j₁ : a₁ ⟶ b₁} (h₀ : is_cof j₀) (h₁ : is_cof j₁) :\n  is_cof (coprod_of_maps j₀ j₁) :=\nbegin\n  convert cof_comp\n    (pushout_is_cof (Is_pushout_i₀ j₀) h₀)\n    (pushout_is_cof (Is_pushout_i₁ j₁) h₁),\n  apply coprod.uniqueness; { rw ←assoc, simp }\nend\n\n-- Basically the same as above, but in the slice category a/C and for\n-- arbitrary (rather than chosen) pushouts.\nlemma cof_pushout {a b₀ b₁ c b₀' b₁' c' : C} {f₀ : a ⟶ b₀} {f₁ : a ⟶ b₁}\n  {g₀ : b₀ ⟶ c} {g₁ : b₁ ⟶ c} {g₀' : b₀' ⟶ c'} {g₁' : b₁' ⟶ c'}\n  {h₀ : b₀ ⟶ b₀'} {h₁ : b₁ ⟶ b₁'} (hh₀ : is_cof h₀) (hh₁ : is_cof h₁)\n  (po : Is_pushout f₀ f₁ g₀ g₁) (po' : Is_pushout (h₀ ∘ f₀) (h₁ ∘ f₁) g₀' g₁') (e) :\n  is_cof (po.induced (g₀' ∘ h₀) (g₁' ∘ h₁) e) :=\nlet po₀ := pushout_by_cof h₀ g₀ hh₀,\n    po₁ := Is_pushout_of_Is_pushout_of_Is_pushout_vert po po₀.is_pushout,\n    k₁ := po₁.induced g₀' (g₁' ∘ h₁) (by simpa using e) in\nhave k₁ ∘ po₀.map₀ = g₀', by simp,\nlet po₂ := Is_pushout_of_Is_pushout_of_Is_pushout' po₁ (by convert po') (by simp) in\nhave k₁ ∘ po₀.map₁ = po.induced (g₀' ∘ h₀) (g₁' ∘ h₁) e, begin\n  apply po.uniqueness,\n  { rw [←assoc, ←po₀.is_pushout.commutes], simp },\n  { rw [←assoc, po₂.commutes], simp }\nend,\nby rw ←this;\n   exact cof_comp (pushout_is_cof po₀.is_pushout hh₀) (pushout_is_cof po₂.transpose hh₁)\n\n-- Suppose C has an initial object ∅. Then an object A of C is\n-- cofibrant if the unique map ∅ → A is a cofibration.\nvariables [has_initial_object.{v} C]\n\ndef cofibrant (a : C) : Prop := is_cof (! a)\n\n-- TODO: Apply this in many places\nlemma cofibrant_of_cof {a b : C} {j : a ⟶ b} (ha : cofibrant a) (hj : is_cof j) :\n  cofibrant b :=\nbegin\n  change is_cof _,\n  convert cof_comp ha hj,\n  apply initial.uniqueness\nend\n\nlemma cofibrant_coprod [has_initial_object.{v} C] [has_coproducts.{v} C]\n  {a₀ a₁ : C} (h₀ : cofibrant a₀) (h₁ : cofibrant a₁) : cofibrant (a₀ ⊔ a₁) :=\nbegin\n  change is_cof (! _),\n  convert cof_comp (cof_iso (coprod_initial_left ∅)) (cof_coprod h₀ h₁),\n  apply initial.uniqueness\nend\n\nlemma cof_i₀ [has_initial_object.{v} C] [has_coproducts.{v} C]\n  {a₀ a₁ : C} (h : cofibrant a₁) : is_cof (i₀ : a₀ ⟶ a₀ ⊔ a₁) :=\nby convert cof_comp (cof_iso (coprod_initial_right a₀)) (cof_coprod (cof_id a₀) h);\n   simp\n\nlemma cof_i₁ [has_initial_object.{v} C] [has_coproducts.{v} C]\n  {a₀ a₁ : C} (h : cofibrant a₀) : is_cof (i₁ : a₁ ⟶ a₀ ⊔ a₁) :=\nby convert cof_comp (cof_iso (coprod_initial_left a₁)) (cof_coprod h (cof_id a₁));\n   simp\n\nvariables (C)\nclass all_objects_cofibrant [has_initial_object.{v} C] :=\n(cofibrant : ∀ (a : C), cofibrant a)\n\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/cofibrations/precofibration_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3281458822045251}}
{"text": "import Do.For  -- import `runCatch`\nimport Lean\nimport Aesop\n\n/-!\n# Formalization of Extended `do` Translation\n\nAn intrinsically typed representation of the paper's syntax of `do` statements as well\nof their translation functions and an equivalence proof thereof to a simple denotational semantics. -/\n\n/-!\n## Contexts\n\nWe represent contexts as lists of types and assignments of them as heterogeneous lists over these types.\nAs is common with lists, contexts grow to the left in our presentation.\nThe following encoding of heterogeneous lists avoids the universe bump of the usual inductive definition\n(<https://lists.chalmers.se/pipermail/agda/2010/001826.html>). -/\ndef HList : List (Type u) → Type u\n  | []      => PUnit\n  | α :: αs => α × HList αs\n\n@[matchPattern] abbrev HList.nil : HList [] := ⟨⟩\n@[matchPattern] abbrev HList.cons (a : α) (as : HList αs) : HList (α :: αs) := (a, as)\n\n/-! We overload the common list notations `::` and `[e, ...]` for `HList`. -/\n\ninfixr:67 \" :: \" => HList.cons\n\nsyntax (name := hlistCons) \"[\" term,* \"]\" : term\nmacro_rules (kind := hlistCons)\n  | `([])          => `(HList.nil)\n  | `([$x])        => `(HList.cons $x [])\n  | `([$x, $xs,*]) => `(HList.cons $x [$xs,*])\n\n/-!\nLean's very general, heterogeneous definition of `++` causes some issues with our overloading above in terms\nsuch as `a ++ [b]`, so we restrict it to the `List` interpretation in the following, which is sufficient for our purposes.\n-/\n\nlocal macro_rules\n  | `($a ++ $b) => `(List.append $a $b)\n\nabbrev Assg Γ := HList Γ\n\n/-! Updating a heterogeneous list at a given, guaranteed in-bounds index. -/\n\ndef HList.set : {αs : List (Type u)} → HList αs → (i : Fin αs.length) → αs.get i → HList αs\n  | _ :: _, _ :: as, ⟨0,          _⟩, b => b :: as\n  | _ :: _, a :: as, ⟨Nat.succ n, h⟩, b => a :: set as ⟨n, Nat.le_of_succ_le_succ h⟩ b\n  | [],     [],      _,               _ => []\n\n/-!\nWe write `∅` for empty contexts and assignments and `Γ ⊢ α` for the type of values of type `α` under the context `Γ`\n- that is, the function type from an assignment to `α`.\n-/\ninstance : EmptyCollection (Assg ∅) where\n  emptyCollection := []\n\ninstance : CoeSort (List (Type u)) (Type u) := ⟨Assg⟩\n\nnotation:30 Γ \" ⊢ \" α => Assg Γ → α\n\ndef Assg.drop : Assg (α :: Γ) → Assg Γ\n  | _ :: as => as\n\n/-! In one special case, we will need to manipulate contexts from the right, i.e. the outermost scope. -/\n\ndef Assg.extendBot (x : α) : {Γ : _} → Assg Γ → Assg (Γ ++ [α])\n  | [],     []      => [x]\n  | _ :: _, a :: as => a :: extendBot x as\n\ndef Assg.dropBot : {Γ : _} → Assg (Γ ++ [α]) → Assg Γ\n  | [],     _       => []\n  | _ :: _, a :: as => a :: dropBot as\n\n/-!\n## Intrinsically Typed Representation of `do` Statements\n\nwhere\n\n* m: base monad\n* ω: `return` type, `m ω` is the type of the entire `do` block\n* Γ: `do`-local immutable context\n* Δ: `do`-local mutable context\n* b: `break` allowed\n* c: `continue` allowed\n* α: local result type, `m α` is the type of the statement\n\nThe constructor signatures are best understood by comparing them with the corresponding typing rules in the paper.\nNote that the choice of de Bruijn indices changes/simplifies some parts, such as obviating freshness checks (`x ∉ Δ`).\n-/\n\ninductive Stmt (m : Type → Type u) (ω : Type) : (Γ Δ : List Type) → (b c : Bool) → (α : Type) → Type _ where\n  | expr (e : Γ ⊢ Δ ⊢ m α) : Stmt m ω Γ Δ b c α\n  | bind (s : Stmt m ω Γ Δ b c α) (s' : Stmt m ω (α :: Γ) Δ b c β) : Stmt m ω Γ Δ b c β  -- let _ ← s; s'\n  | letmut (e : Γ ⊢ Δ ⊢ α) (s : Stmt m ω Γ (α :: Δ) b c β) : Stmt m ω Γ Δ b c β  -- let mut _ := e; s\n  | assg (x : Fin Δ.length) (e : Γ ⊢ Δ ⊢ Δ.get x) : Stmt m ω Γ Δ b c Unit  -- x := e\n  | ite (e : Γ ⊢ Δ ⊢ Bool) (s₁ s₂ : Stmt m ω Γ Δ b c α) : Stmt m ω Γ Δ b c α  -- if e then s₁ else s₂\n  | ret (e : Γ ⊢ Δ ⊢ ω) : Stmt m ω Γ Δ b c α   -- return e\n  | sfor (e : Γ ⊢ Δ ⊢ List α) (s : Stmt m ω (α :: Γ) Δ true true Unit) : Stmt m ω Γ Δ b c Unit  -- for _ in e do s\n  | sbreak : Stmt m ω Γ Δ true c α  -- break\n  | scont : Stmt m ω Γ Δ b true α  -- continue\n\n/-! Neutral statements are a restriction of the above type. -/\n\ninductive Neut (ω α : Type) : (b c : Bool) → Type _ where\n  | val (a : α) : Neut ω α b c\n  | ret (o : ω) : Neut ω α b c\n  | rbreak : Neut ω α true c\n  | rcont : Neut ω α b true\n\n/-! We elide `Neut.val` where unambiguous. -/\n\ninstance : Coe α (Neut ω α b c) := ⟨Neut.val⟩\ninstance : Coe (Id α) (Neut ω α b c) := ⟨Neut.val⟩\n\n/-!\nWe write `e[ρ][σ]` for the substitution of both contexts in `e`, a simple application in this encoding.\n`σ[x ↦ v]` updates `σ` at `v` (a de Bruijn index). -/\n\nmacro:max (priority := high) e:term:max noWs \"[\" ρ:term \"]\" \"[\" σ:term \"]\" : term => `($e $ρ $σ)\nmacro:max (priority := high) σ:term:max noWs \"[\" x:term \" ↦ \" v:term \"]\" : term => `(HList.set $σ $x $v)\n\n/-!\n## Dynamic Evaluation Function\n\nA direct encoding of the paper's operational semantics as a denotational function,\ngeneralized over an arbitrary monad.\nNote that the immutable context `ρ` is accumulated (`v :: ρ`) and passed explicitly instead of immutable\nbindings being substituted immediately as that is a better match for the above definition of `Stmt`.\nIteration over the values of the given list in the `for` case introduces a nested, mutually recursive helper\nfunction, with termination of the mutual bundle following from a size argument over the statement primarily\nand the length of the list in the `for` case secondarily.\n-/\n\n@[simp] def Stmt.eval [Monad m] (ρ : Assg Γ) : Stmt m ω Γ Δ b c α → Assg Δ → m (Neut ω α b c × Assg Δ)\n  | expr e, σ => e[ρ][σ] >>= fun v => pure ⟨v, σ⟩\n  | bind s s', σ =>\n    -- defining this part as a separate definition helps Lean with the termination proof\n    let rec @[simp] cont val\n      | ⟨Neut.val v, σ'⟩ => val v σ'\n      -- the `Neut` type family forces us to repeat these cases as the LHS/RHS indices are not identical\n      | ⟨Neut.ret o, σ'⟩ => pure ⟨Neut.ret o, σ'⟩\n      | ⟨Neut.rbreak, σ'⟩ => pure ⟨Neut.rbreak, σ'⟩\n      | ⟨Neut.rcont, σ'⟩ => pure ⟨Neut.rcont, σ'⟩\n    s.eval ρ σ >>= cont (fun v σ' => s'.eval (v :: ρ) σ')\n  | letmut e s, σ =>\n    s.eval ρ (e[ρ][σ], σ) >>= fun ⟨r, σ'⟩ => pure ⟨r, σ'.drop⟩\n  -- `x` is a valid de Bruijn index into `σ` by definition of `assg`\n  | assg x e, σ => pure ⟨(), σ[x ↦ e[ρ][σ]]⟩\n  | ite e s₁ s₂, σ => if e[ρ][σ] then s₁.eval ρ σ else s₂.eval ρ σ\n  | ret e, σ => pure ⟨Neut.ret e[ρ][σ], σ⟩\n  | sfor e s, σ =>\n    let rec @[simp] go σ\n      | [] => pure ⟨(), σ⟩\n      | a::as =>\n        s.eval (a :: ρ) σ >>= fun\n        | ⟨(), σ'⟩ => go σ' as\n        | ⟨Neut.rcont, σ'⟩ => go σ' as\n        | ⟨Neut.rbreak, σ'⟩ => pure ⟨(), σ'⟩\n        | ⟨Neut.ret o, σ'⟩ => pure ⟨Neut.ret o, σ'⟩\n    go σ e[ρ][σ]\n  | sbreak, σ => pure ⟨Neut.rbreak, σ⟩\n  | scont, σ => pure ⟨Neut.rcont, σ⟩\ntermination_by\n  eval s _ => (sizeOf s, 0)\n  eval.go as => (sizeOf s, as.length)\n\n/-! At the top-level statement, the contexts are empty, no loop control flow statements are allowed, and the return and result type are identical. -/\n\nabbrev Do m α := Stmt m α ∅ ∅ false false α\n\ndef Do.eval [Monad m] (s : Do m α) : m α :=  -- corresponds to the reduction rule `do s ⇒ v` in the paper\n  Stmt.eval ∅ s ∅ >>= fun\n    | ⟨Neut.val a, _⟩ => pure a\n    | ⟨Neut.ret o, _⟩ => pure o\n\nnotation \"⟦\" s \"⟧\" => Do.eval s\n\n/-!\n## Translation Functions\n\nWe adjust the immutable context where necessary.\nThe mutable context does not have to be adjusted. -/\n\n@[simp] def Stmt.mapAssg (f : Assg Γ' → Assg Γ) : Stmt m ω Γ Δ b c β → Stmt m ω Γ' Δ b c β\n  | expr e => expr (e ∘ f)\n  | bind s₁ s₂ => bind (s₁.mapAssg f) (s₂.mapAssg (fun (a :: as) => (a :: f as)))\n  | letmut e s => letmut (e ∘ f) (s.mapAssg f)\n  | assg x e => assg x (e ∘ f)\n  | ite e s₁ s₂ => ite (e ∘ f) (s₁.mapAssg f) (s₂.mapAssg f)\n  | ret e => ret (e ∘ f)\n  | sfor e s => sfor (e ∘ f) (s.mapAssg (fun (a :: as) => (a :: f as)))\n  | sbreak => sbreak\n  | scont => scont\n\n/-! Let us write `f ∘ₑ e` for the composition of `f : α → β` with `e : Γ ⊢ Δ ⊢ α`, which we will use to rewrite embedded terms. -/\n\ninfixr:90 \" ∘ₑ \"  => fun f e => fun ρ σ => f e[ρ][σ]\n\n/-!\nThe formalization of `S` creates some technical hurdles. Because it operates on the outer-most mutable binding,\nwe have to operate on that context from the right, from which we lose some helpful definitional equalities and\nhave to rewrite types using nested proofs instead.\n\nThe helper function `shadowSnd` is particularly interesting because it shows how the shadowing in\ntranslation rules (S2) and (S9) is expressed in our de Bruijn encoding: The context `α :: β :: α :: Γ`\ncorresponds, in this order, to the `y` that has just been bound to the value of `get`, then `x` from the\nrespective rule, followed by the `y` of the outer scope. We encode the shadowing of `y` by dropping the\nthird element from the context as well as the assignment. We are in fact forced to do so because the corresponding\nbranches of `S` would not otherwise typecheck. The only mistake we could still make is to drop the wrong `α` value\nfrom the assignment, which (speaking from experience) would eventually be caught by the correctness proof.\n-/\n@[simp] def S [Monad m] : Stmt m ω Γ (Δ ++ [α]) b c β → Stmt (StateT α m) ω (α :: Γ) Δ b c β\n/-(S1)-/ | Stmt.expr e => Stmt.expr (StateT.lift ∘ₑ unmut e)\n/-(S2)-/ | Stmt.bind s₁ s₂ => Stmt.bind (S s₁) (Stmt.bind (Stmt.expr (fun _ _ => get)) (Stmt.mapAssg shadowSnd (S s₂)))\n/-(S3)-/ | Stmt.letmut e s => Stmt.letmut (unmut e) (S s)\n         | Stmt.assg x e =>\n           if h : x < Δ.length then\n/-(S4)-/     Stmt.assg ⟨x, h⟩ (fun (y :: ρ) σ => List.get_append_left .. ▸ e ρ (Assg.extendBot y σ))\n           else\n/-(S5)-/     Stmt.expr (set (σ := α) ∘ₑ cast (List.get_last h) ∘ₑ unmut e)\n/-(S6)-/ | Stmt.ite e s₁ s₂ => Stmt.ite (unmut e) (S s₁) (S s₂)\n         -- unreachable case; could be eliminated by a more precise specification of `ω`, but the benefit would be minimal\n         | Stmt.ret e => Stmt.ret (unmut e)\n/-(S7)-/ | Stmt.sbreak => Stmt.sbreak\n/-(S8)-/ | Stmt.scont => Stmt.scont\n/-(S9)-/ | Stmt.sfor e s => Stmt.sfor (unmut e) (Stmt.bind (Stmt.expr (fun _ _ => get)) (Stmt.mapAssg shadowSnd (S s)))\nwhere\n  @[simp] unmut {β} (e : Γ ⊢ Δ ++ [α] ⊢ β) : α :: Γ ⊢ Δ ⊢ β\n    | y :: ρ, σ => e ρ (Assg.extendBot y σ)\n  @[simp] shadowSnd {β} : Assg (α :: β :: α :: Γ) → Assg (α :: β :: Γ)\n    | a' :: b :: _ :: ρ => a' :: b :: ρ\n\n@[simp] def R [Monad m] : Stmt m ω Γ Δ b c α → Stmt (ExceptT ω m) Empty Γ Δ b c α\n/-(R1)-/ | Stmt.ret e => Stmt.expr (throw ∘ₑ e)\n/-(R2)-/ | Stmt.expr e => Stmt.expr (ExceptT.lift ∘ₑ e)\n/-(R3)-/ | Stmt.bind s s' => Stmt.bind (R s) (R s')\n/-(R4)-/ | Stmt.letmut e s => Stmt.letmut e (R s)\n/-(R5)-/ | Stmt.assg x e => Stmt.assg x e\n/-(R6)-/ | Stmt.ite e s₁ s₂ => Stmt.ite e (R s₁) (R s₂)\n/-(R7)-/ | Stmt.sbreak => Stmt.sbreak\n/-(R8)-/ | Stmt.scont => Stmt.scont\n/-(R9)-/ | Stmt.sfor e s => Stmt.sfor e (R s)\n\n@[simp] def L [Monad m] : Stmt m ω Γ Δ b c α → Stmt (ExceptT Unit m) ω Γ Δ b c α\n/-(L1)-/ | Stmt.sbreak => Stmt.sbreak\n/-(L2)-/ | Stmt.scont => Stmt.scont\n/-(L3)-/ | Stmt.expr e => Stmt.expr (ExceptT.lift ∘ₑ e)\n/-(L4)-/ | Stmt.bind s s' => Stmt.bind (L s) (L s')\n/-(L5)-/ | Stmt.letmut e s => Stmt.letmut e (L s)\n/-(L6)-/ | Stmt.assg x e => Stmt.assg x e\n/-(L7)-/ | Stmt.ite e s₁ s₂ => Stmt.ite e (L s₁) (L s₂)\n         | Stmt.ret e => Stmt.ret e\n/-(L8)-/ | Stmt.sfor e s => Stmt.sfor e (L s)\n\n@[simp] def B [Monad m] : Stmt m ω Γ Δ b c α → Stmt (ExceptT Unit m) ω Γ Δ false c α\n/-(B1)-/ | Stmt.sbreak => Stmt.expr (fun _ _ => throw ())\n/-(B2)-/ | Stmt.scont => Stmt.scont\n/-(B3)-/ | Stmt.expr e => Stmt.expr (ExceptT.lift ∘ₑ e)\n/-(B4)-/ | Stmt.bind s s' => Stmt.bind (B s) (B s')\n/-(B5)-/ | Stmt.letmut e s => Stmt.letmut e (B s)\n/-(B6)-/ | Stmt.assg x e => Stmt.assg x e\n/-(B7)-/ | Stmt.ite e s₁ s₂ => Stmt.ite e (B s₁) (B s₂)\n         | Stmt.ret e => Stmt.ret e\n/-(B8)-/ | Stmt.sfor e s => Stmt.sfor e (L s)\n\n-- (elided in the paper)\n@[simp] def C [Monad m] : Stmt m ω Γ Δ false c α → Stmt (ExceptT Unit m) ω Γ Δ false false α\n  | Stmt.scont => Stmt.expr (fun _ _ => throw ())\n  | Stmt.expr e => Stmt.expr (ExceptT.lift ∘ₑ e)\n  | Stmt.bind s s' => Stmt.bind (C s) (C s')\n  | Stmt.letmut e s => Stmt.letmut e (C s)\n  | Stmt.assg x e => Stmt.assg x e\n  | Stmt.ite e s₁ s₂ => Stmt.ite e (C s₁) (C s₂)\n  | Stmt.ret e => Stmt.ret e\n  | Stmt.sfor e s => Stmt.sfor e (L s)\n\n/-!\nThe remaining function to be translated is `D`, which is straightforward as well except for its termination proof,\nas it recurses on the results of `S` (D3) and `C ∘ B` (D5). Because of rules (S2, S9) that introduce new bindings,\n`S` may in fact increase the size of the input, and the same is true for `C` and `B` for the `sizeOf` function\nautomatically generated by Lean. Thus we introduce a new measure `numExts` that counts the number of special statements\non top of basic `do` notation and prove that all three functions do not increase the size according to that measure.\nBecause the rules (D3) and (D5) each eliminate such a special statement, it follows that `D` terminates because either\nthe number of special statements decreases in each case, or it remains the same and the total number of statements decreases.\n-/\n\n@[simp] def Stmt.numExts : Stmt m ω Γ Δ b c α → Nat\n  | expr _ => 0\n  | bind s₁ s₂ => s₁.numExts + s₂.numExts\n  | letmut _ s => s.numExts + 1\n  | assg _ _ => 1\n  | ite _ s₁ s₂ => s₁.numExts + s₂.numExts\n  | ret _ => 1\n  | sfor _ s => s.numExts + 1\n  | sbreak => 1\n  | scont => 1\n\n@[simp] theorem Stmt.numExts_mapAssg (f : Assg Γ' → Assg Γ) (s : Stmt m ω Γ Δ b c β) : numExts (mapAssg f s) = numExts s := by\n  induction s generalizing Γ' <;> simp [*]\n\ntheorem Stmt.numExts_S [Monad m] (s : Stmt m ω Γ (Δ ++ [α]) b c β) : numExts (S s) ≤ numExts s := by\n  -- `induction` does not work with non-variable indices, so we first generalize `Δ ++ [α]` into an explicit equation\n  revert s\n  suffices {Δ': _ } → (s : Stmt m ω Γ Δ' b c β) → (h : Δ' = (Δ ++ [α])) → numExts (S (h ▸ s : Stmt m ω Γ (Δ ++ [α]) b c β)) ≤ numExts s\n    from fun s => this s rfl\n  intro Δ' s h\n  induction s generalizing Δ with\n    subst h\n  | bind _ _ ih₁ ih₂ => simp [Nat.add_le_add, ih₁ rfl, ih₂ rfl]\n  | letmut _ _ ih => simp [Nat.add_le_add, ih (List.cons_append ..).symm]\n  | assg => aesop\n  | ite _ _ _ ih₁ ih₂ => simp [Nat.add_le_add, ih₁ rfl, ih₂ rfl]\n  | sfor _ _ ih => simp [Nat.add_le_add, ih rfl]\n  | _ => simp\n\ntheorem Stmt.numExts_L_L [Monad m] (s : Stmt m ω Γ Δ b c β) : numExts (L (L s)) ≤ numExts s := by\n  induction s <;> simp [Nat.add_le_add, *]\n\ntheorem Stmt.numExts_C_B [Monad m] (s : Stmt m ω Γ Δ b c β) : numExts (C (B s)) ≤ numExts s := by\n  induction s <;> simp [Nat.add_le_add, numExts_L_L, *]\n\n-- Auxiliary tactic for showing that `D` terminates\nmacro \"D_tac\" : tactic =>\n  `({simp_wf\n     solve\n      | apply Prod.Lex.left; assumption\n      | apply Prod.Lex.right' <;> simp_arith })\n\n@[simp] def D [Monad m] : Stmt m Empty Γ ∅ false false α → (Γ ⊢ m α)\n  | Stmt.expr e => (e[·][∅])\n  | Stmt.bind s s' => (fun ρ => D s ρ >>= fun x => D s' (x :: ρ))\n  | Stmt.letmut e s =>\n    have := Nat.lt_succ_of_le <| Stmt.numExts_S (Δ := []) s  -- for termination\n    fun ρ =>\n      let x := e[ρ][∅]\n      StateT.run' (D (S s) (x :: ρ)) x\n  | Stmt.ite e s₁ s₂ => (fun ρ => if e[ρ][∅] then D s₁ ρ else D s₂ ρ)\n  | Stmt.sfor e s =>\n    have := Nat.lt_succ_of_le <| Stmt.numExts_C_B (Δ := []) s  -- for termination\n    fun ρ =>\n      runCatch (forM e[ρ][∅] (fun x => runCatch (D (C (B s)) (x :: ρ))))\n  | Stmt.ret e => (nomatch e[·][∅])\ntermination_by _ s => (s.numExts, sizeOf s)\ndecreasing_by D_tac\n\n/-! Finally we compose `D` and `R` into the translation rule for a top-level statement (1'). -/\n\ndef Do.trans [Monad m] (s : Do m α) : m α := runCatch (D (R s) ∅)\n\n/-!\n## Equivalence Proof\n\nUsing the monadic dynamic semantics, we can modularly prove for each individual translation function that\nevaluating its output is equivalent to directly evaluating the input, modulo some lifting and adjustment\nof resulting values. After induction on the statement, the proofs are mostly concerned with case splitting,\napplication of congruence theorems, and simplification. We can mostly offload these tasks onto\n[Aesop](https://github.com/JLimperg/aesop).\n\n-/\nattribute [local simp] map_eq_pure_bind ExceptT.run_bind\nattribute [aesop safe apply] bind_congr\n\ntheorem eval_R [Monad m] [LawfulMonad m] (s : Stmt m ω Γ Δ b c α) : (R s).eval ρ σ = (ExceptT.lift (s.eval ρ σ) >>= fun x => match (generalizing := false) x with\n    | (Neut.ret o, _) => throw o\n    | (Neut.val a, σ) => pure (Neut.val a, σ)\n    | (Neut.rcont, σ) => pure (Neut.rcont, σ)\n    | (Neut.rbreak, σ) => pure (Neut.rbreak, σ) : ExceptT ω m (Neut Empty α b c × Assg Δ)) := by\n  apply ExceptT.ext\n  induction s with\n  | sfor e =>\n    simp only [Stmt.eval, R]\n    induction e ρ σ generalizing σ <;> aesop (add norm unfold Stmt.eval.go)\n  | _ => aesop (add unsafe cases Neut) (erase Aesop.BuiltinRules.destructProducts)\n\n@[simp] theorem eval_mapAssg [Monad m] [LawfulMonad m] (f : Assg Γ' → Assg Γ) : ∀ (s : Stmt m ω Γ Δ b c β), Stmt.eval ρ (Stmt.mapAssg f s) σ = Stmt.eval (f ρ) s σ := by\n  intro s\n  induction s generalizing Γ' with\n  | sfor e s ih =>\n    simp only [Stmt.eval, Stmt.mapAssg, Function.comp]\n    induction e (f ρ) σ generalizing σ <;> aesop (add norm unfold Stmt.eval.go)\n  | _ => aesop (add unsafe cases Neut)\n\n/-!\nWe need one last helper function on context bottoms to be able to state the invariant of `S`, and then\nprove various lemmas about their interactions. -/\n\ndef Assg.bot : {Γ : _} → Assg (Γ ++ [α]) → α\n  | [],     [a]     => a\n  | _ :: _, _ :: as => bot as\n\n@[simp] theorem Assg.dropBot_extendBot (a : α) (σ : Assg Γ) : Assg.dropBot (Assg.extendBot a σ) = σ := by\n  induction Γ <;> cases σ <;> simp [dropBot, extendBot, *]\n@[simp] theorem Assg.bot_extendBot (a : α) (σ : Assg Γ) : Assg.bot (Assg.extendBot a σ) = a := by\n  induction Γ <;> cases σ <;> simp [bot, extendBot, *]\n@[simp] theorem Assg.extendBot_bot_dropBot (σ : Assg (Γ ++ [α])) : Assg.extendBot (Assg.bot σ) (Assg.dropBot σ) = σ := by\n  induction Γ <;> cases σ <;> simp [dropBot, bot, extendBot, *] <;> rfl\n\n@[simp] theorem Assg.dropBot_set_extendBot_init (a : α) (σ : Assg Γ) (h : i.1 < Γ.length) {b} : Assg.dropBot ((Assg.extendBot a σ).set i b) = σ.set ⟨i.1, h⟩ (List.get_append_left .. ▸ b) := by\n  induction Γ with\n  | nil => contradiction\n  | cons  _ _ ih =>\n    cases σ\n    have ⟨i, h'⟩ := i\n    cases i <;> simp [HList.set, dropBot]\n    rw [ih]\n\n@[simp] theorem Assg.bot_set_extendBot_init (a : α) (σ : Assg Γ) (h : i.1 < Γ.length) {b} : Assg.bot ((Assg.extendBot a σ).set i b) = a := by\n  induction Γ with\n  | nil => contradiction\n  | cons  _ _ ih =>\n    cases σ\n    have ⟨i, h'⟩ := i\n    cases i <;> simp [HList.set, dropBot, bot]\n    rw [ih]; apply Nat.lt_of_succ_lt_succ h\n\n@[simp] theorem Assg.dropBot_set_extendBot_bottom (a : α) (σ : Assg Γ) (h : ¬ i.1 < Γ.length) {b} : Assg.dropBot ((Assg.extendBot a σ).set i b) = σ := by\n  induction Γ with\n  | nil => rfl\n  | cons  _ _ ih =>\n    cases σ\n    have ⟨i, h'⟩ := i\n    cases i\n    · apply False.elim (h (Nat.zero_lt_succ _))\n    · simp [HList.set, dropBot]\n      rw [ih]\n      intro h''\n      apply False.elim (h (Nat.succ_lt_succ h''))\n\n@[simp] theorem Assg.bot_set_extendBot_bottom (a : α) (σ : Assg Γ) (h : ¬ i.1 < Γ.length) {b} : Assg.bot ((Assg.extendBot a σ).set i b) = cast (List.get_last h) b := by\n  induction Γ with\n  | nil =>\n    have ⟨i, h'⟩ := i\n    cases i\n    · simp [HList.set, extendBot, bot]; rfl\n    · apply False.elim (Nat.not_lt_zero _ (Nat.lt_of_succ_lt_succ h'))\n  | cons  _ _ ih =>\n    cases σ\n    have ⟨i, h'⟩ := i\n    cases i\n    · apply False.elim (h (Nat.zero_lt_succ _))\n    · simp [bot]\n      rw [ih]\n      intro h''\n      apply False.elim (h (Nat.succ_lt_succ h''))\n\ntheorem eval_S [Monad m] [LawfulMonad m] : ∀ (s : Stmt m ω Γ (Δ ++ [α]) b c β), StateT.run ((S s).eval (a :: ρ) σ) a = s.eval ρ (Assg.extendBot a σ) >>= fun\n    | r :: σ => pure ((r :: Assg.dropBot σ), Assg.bot σ) := by\n  suffices {Δ': _ } → (s : Stmt m ω Γ Δ' b c β) → (h : Δ' = (Δ ++ [α])) → StateT.run ((S (h ▸ s)).eval (a :: ρ) σ) a = s.eval ρ (h ▸ Assg.extendBot a σ) >>= fun\n    | r :: σ => pure ((r :: Assg.dropBot (h ▸ σ)), Assg.bot (h ▸ σ))\n    from fun s => this s rfl\n  intro Δ' s h\n  induction s generalizing Δ a with\n    subst h\n  | bind s₁ s₂ ih₁ ih₂ =>\n    have ih₁ := @ih₁ (h := rfl)\n    have ih₂ := @ih₂ (h := rfl)\n    aesop (add safe cases Neut)\n  | letmut e s ih =>\n    have ih := @ih (Δ := _ :: Δ) (h := rfl)\n    aesop\n  | sfor e s ih =>\n    have ih := @ih (h := rfl)\n    simp only [S, Stmt.eval, S.unmut]\n    -- surgical generalization\n    generalize h : a = a'\n    conv =>\n      pattern HList.cons a' _\n      rw [← h]\n    clear h\n    induction e ρ _ generalizing σ a' <;> aesop (add safe cases Neut)\n  | _ => aesop\n\ntheorem HList.eq_nil (as : HList ∅) : as = ∅ := rfl\n\nattribute [local simp] ExceptT.run_bind\n\ntheorem eval_L [Monad m] [LawfulMonad m] (s : Stmt m ω Γ Δ b c α) : (L s).eval ρ σ = ExceptT.lift (s.eval ρ σ) := by\n  apply ExceptT.ext\n  induction s with\n  | sfor e =>\n    simp only [Stmt.eval, L]\n    induction e ρ σ generalizing σ <;> aesop\n  | _ => aesop (add safe cases Neut)\n\ntheorem eval_B [Monad m] [LawfulMonad m] (s : Stmt m ω Γ Δ b c α) : (B s).eval ρ σ = (ExceptT.lift (s.eval ρ σ) >>= fun x => match (generalizing := false) x with\n    | (Neut.ret o, σ) => pure (Neut.ret o, σ)\n    | (Neut.val a, σ) => pure (Neut.val a, σ)\n    | (Neut.rcont, σ) => pure (Neut.rcont, σ)\n    | (Neut.rbreak, _) => throw () : ExceptT Unit m (Neut ω α false c × Assg Δ)) := by\n  apply ExceptT.ext\n  induction s with\n  | sfor e =>\n    simp only [Stmt.eval, B]\n    induction e ρ σ generalizing σ <;> aesop (add norm simp eval_L)\n  | _ => aesop (erase Aesop.BuiltinRules.destructProducts)\n\n@[simp] def throwOnContinue [Monad m] : (Neut ω α false c × Assg Δ) → ExceptT Unit m (Neut ω α false false × Assg Δ)\n  | (Neut.ret o, σ) => pure (Neut.ret o, σ)\n  | (Neut.val a, σ) => pure (Neut.val a, σ)\n  | (Neut.rcont, _) => throw ()\n\ntheorem eval_C [Monad m] [LawfulMonad m] (s : Stmt m ω Γ Δ false c α) : (C s).eval ρ σ = ExceptT.lift (s.eval ρ σ) >>= throwOnContinue := by\n  revert s\n  suffices {b: _ } → (s' : Stmt m ω Γ Δ b c α) → (h : b = false) → let s : Stmt m ω Γ Δ false c α := h ▸ s'; (C s).eval ρ σ = ExceptT.lift (s.eval ρ σ) >>= throwOnContinue\n    from fun s => this s rfl\n  intro b' s h\n  induction s with\n    (first | subst h | trivial)\n  | sfor e =>\n    simp only [Stmt.eval, C]\n    induction e ρ σ generalizing σ <;> aesop (add norm simp eval_L, unsafe apply ExceptT.ext)\n  | _ => aesop (add unsafe apply ExceptT.ext)\n\ntheorem D_eq [Monad m] [LawfulMonad m] : (s : Stmt m Empty Γ ∅ false false α) →\n    D s ρ = s.eval ρ ∅ >>= fun (Neut.val a, _) => pure a\n  | Stmt.expr e => by simp\n  | Stmt.bind s₁ s₂ => by\n    have ih₁ := @D_eq (s := s₁)\n    have ih₂ := @D_eq (s := s₂)\n    aesop\n  | Stmt.letmut e s => by\n    have := Nat.lt_succ_of_le <| Stmt.numExts_S (Δ := []) s  -- for termination\n    have ih := (D_eq (ρ := ·) (S s))\n    aesop (add safe cases Neut, norm simp eval_S)\n  | Stmt.ite e s₁ s₂ => by simp; split <;> simp [D_eq s₁, D_eq s₂]\n  | Stmt.ret e => nomatch e ρ ∅\n  | Stmt.sfor e s => by\n    have := Nat.lt_succ_of_le <| Stmt.numExts_C_B (Δ := []) s  -- for termination\n    have ih := (D_eq (ρ := ·) (C (B s)))\n    simp\n    induction e ρ ∅ <;> aesop (add safe cases Neut, norm unfold runCatch, norm simp [eval_C, eval_B])\ntermination_by _ s => (s.numExts, sizeOf s)\ndecreasing_by D_tac\n\n/-! The equivalence proof cited in the paper follows from the invariants of `D` and `R`. -/\n\ntheorem Do.trans_eq_eval [Monad m] [LawfulMonad m] : ∀ s : Do m α, Do.trans s = ⟦s⟧ := by\n  aesop (add norm simp [D_eq, eval_R], norm unfold [runCatch, Do.trans, Do.eval])\n\n/-!\n## Partial Evaluation\n\nWe define a new term notation `simp [...] in e` that rewrites the term e using the given\nsimplification theorems.\n-/\n\nopen Lean in\nopen Lean.Parser.Tactic in\nopen Lean.Meta in\nopen Lean.Elab in\nelab \"simp\" \"[\" simps:simpLemma,* \"]\" \"in\" e:term : term => do\n  -- construct goal `⊢ e = ?x` with fresh metavariable `?x`, simplify, solve by reflexivity,\n  -- and return assigned value of `?x`\n  let e ← Term.elabTermAndSynthesize e none\n  let x ← mkFreshExprMVar (← inferType e)\n  let goal ← mkFreshExprMVar (← mkEq e x)\n  -- disable ζ-reduction to preserve `let`s\n  Term.runTactic goal.mvarId! (← `(tactic| (simp (config := { zeta := false }) [$simps:simpLemma,*]; rfl)))\n  instantiateMVars x\n\n-- further clean up generated programs\nattribute [local simp] Assg.extendBot cast\nattribute [-simp] StateT.run'_eq\n\n/-!\nWe can now use this new notation to completely erase the translation functions\nfrom an invocation on the example `ex2` from `For.lean` (manually translated to `Stmt`).\n-/\n/-\nlet mut y := init;\nfor x in xs do' {\n  y ← f y x\n};\nreturn y\n-/\n\ndef ex2' [Monad m] (f : β → α → m β) (init : β) (xs : List α) : m β :=\n  simp [Do.trans] in Do.trans (\n      Stmt.letmut (fun _ _ => init) <|\n      Stmt.bind (\n        Stmt.sfor (fun _ _ => xs) <|\n        -- `y ← f y x` unfolded to `let z ← f y x; y := z` (A4)\n        Stmt.bind\n          (Stmt.expr (fun ([x]) ([y]) => f y x))\n          (Stmt.assg ⟨0, by simp⟩ (fun ([z, x]) _ => z))) <|\n      Stmt.ret (fun _ ([y]) => y))\n\n/-!\nCompare the output of the two versions - the structure is identical except for unused\nmonadic layers in the formal translation, which would be harder to avoid in this typed\napproach. -/\n#print ex2\n#print ex2'\n\n/-! We can evaluate the generated program like any other Lean program, and prove that both versions are equivalent. -/\n#eval ex2' (m := Id) (fun a b => pure (a + b)) 0 [1, 2]\n\nexample [Monad m] [LawfulMonad m] (f : β → α → m β) :\n    ex2' f init xs = ex2 f init xs := by\n  rw [ex2, ex2']; unfold runCatch; induction xs generalizing init <;> simp_all! [StateT.run'_eq]\n\n/-!\nFor one more example, consider `ex3` from `For.lean`.\n-/\n/-\nfor xs in xss do' {\n  for x in xs do' {\n    let b ← p x;\n    if b then {\n      return some x\n    }\n  }\n};\npure none\n-/\n\ndef ex3' [Monad m] (p : α → m Bool) (xss : List (List α)) : m (Option α) :=\n  simp [Do.trans] in Do.trans (\n      Stmt.bind\n        (Stmt.sfor (fun _ _ => xss) <|\n          Stmt.sfor (fun ([xs]) _ => xs) <|\n            Stmt.bind\n              (Stmt.expr (fun ([x, xs]) _ => p x))\n              (Stmt.ite (fun ([b, x, xs]) _ => b)\n                (Stmt.ret (fun ([b, x, xs]) _ => some x))\n                (Stmt.expr (fun _ _ => pure ()))))\n        (Stmt.expr (fun _ _ => pure none)))\n\n#print ex3\n#print ex3'\n#eval ex3' (m := Id) (fun n => n % 2 == 0) [[1, 3], [2, 4]]\n\nexample [Monad m] [LawfulMonad m] (p : α → m Bool) (xss : List (List α)) :\n    ex3' p xss = ex3 p xss := by\n  rw [ex3, ex3']\n  unfold runCatch\n  induction xss with\n  | nil => simp!\n  | cons xs xss ih =>\n    simp\n    induction xs <;> aesop (add safe apply byCases_Bool_bind)\n\n/-!\nWhile it would be possible to override our `do'` notation such that its named syntax\nis first translated to nameless `Stmt` constructors and then applied to `simp [Do.trans] in`,\nfor demonstration purposes we decided to encode these examples manually. In practice, the\nmacro implementation remains more desirable as mentioned in the paper.\n-/\n", "meta": {"author": "Kha", "repo": "do-supplement", "sha": "72acc9d3a39d2593f15b77bc0a221c307f6657a0", "save_path": "github-repos/lean/Kha-do-supplement", "path": "github-repos/lean/Kha-do-supplement/do-supplement-72acc9d3a39d2593f15b77bc0a221c307f6657a0/Do/Formal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.32814587438460197}}
{"text": "import GroundZero.Types.Unit\nimport GroundZero.Types.Sigma\nimport GroundZero.Types.Product\nimport GroundZero.Types.Coproduct\n\nopen GroundZero.Types.Unit\nopen GroundZero.Types.Id (map)\nopen GroundZero.Types (Coproduct idp fib)\nopen GroundZero.Types.Equiv (biinv)\n\nnamespace GroundZero\nuniverse u v w k r\n\nnamespace Structures\n\nhott def isLoop {A : Type u} {a : A} (p : a = a) := ¬(p = idp a)\n\nhott def prop (A : Type u) := Π (a b : A), a = b\n\nhott def propset := Σ (A : Type u), prop A\nnotation \"Ω\" => propset\n\nhott def hset (A : Type u) := Π (a b : A) (p q : a = b), p = q\nhott def Ens := Σ A, hset A\n\nhott def groupoid (A : Type u) :=\nΠ (a b : A) (p q : a = b) (A B : p = q), A = B\n\nhott def dec (A : Type u) := A + ¬A\n\nhott def contr (A : Type u) := Σ (a : A), Π b, a = b\n\ninstance contr.dotted {A : Type u} (H : contr A) : Types.Id.dotted A := ⟨H.1⟩\n\ninductive hlevel\n| minusTwo\n| succ : hlevel → hlevel\n\nnotation \"ℕ₋₂\" => hlevel\nnotation \"−2\" => hlevel.minusTwo\nnotation \"−1\" => hlevel.succ hlevel.minusTwo\n\nnamespace hlevel\n  inductive le : hlevel → hlevel → Type\n  | refl (a : hlevel)   : le a a\n  | step (a b : hlevel) : le a b → le a (succ b)\n\n  infix:50 \" ≤ \" => le\n\n  hott def le.minusTwo : Π (n : hlevel), −2 ≤ n\n  | −2     => le.refl −2\n  | succ n => le.step _ _ (minusTwo n)\n\n  noncomputable hott def le.succ (a b : hlevel) (ρ : a ≤ b) : succ a ≤ succ b :=\n  begin induction ρ; apply le.refl; apply le.step; assumption end\n\n  hott def add : hlevel → hlevel → hlevel\n  | −2,            −2 => −2\n  | −1,            −2 => −2\n  | succ (succ n), −2 => n\n  | n, succ (succ −2) => n\n  | n, succ m         => succ (add n m)\n\n  instance : HAdd hlevel hlevel hlevel := ⟨hlevel.add⟩\n\n  hott def ofNat : ℕ → ℕ₋₂\n  | Nat.zero   => succ (succ −2)\n  | Nat.succ n => hlevel.succ (ofNat n)\n\n  instance (n : ℕ) : OfNat ℕ₋₂ n := ⟨ofNat n⟩\nend hlevel\n\ndef isNType : hlevel → Type u → Type u\n| −2            => contr\n| hlevel.succ n => λ A, Π (x y : A), isNType n (x = y)\n\nnotation \"is-\" n \"-type\" => isNType n\n\ndef nType (n : hlevel) : Type (u + 1) :=\nΣ (A : Type u), is-n-type A\n\nnotation n \"-Type\" => nType n\n\nhott def hlevel.cumulative {A : Type u} : Π (n : hlevel), is-n-type A → is-(hlevel.succ n)-type A\n| −2,            H => λ x y, ⟨(H.2 x)⁻¹ ⬝ H.2 y, λ p, begin induction p; apply Types.Id.invComp end⟩\n| hlevel.succ n, H => λ x y, cumulative n (H x y)\n\nnoncomputable hott def hlevel.strongCumulative (n m : hlevel) (ρ : n ≤ m) :\n  Π {A : Type u}, (is-n-type A) → (is-m-type A) :=\nbegin\n  induction ρ; intros; assumption;\n  case step m ρ ih => {\n    intros A G; apply @hlevel.cumulative;\n    apply ih; assumption\n  }\nend\n\nhott def contrImplProp {A : Type u} (h : contr A) : prop A :=\nλ a b, (h.2 a)⁻¹ ⬝ (h.2 b)\n\ndef emptyIsProp : prop 𝟎 :=\nbegin intros x y; induction x end\n\ndef unitIsProp : prop 𝟏 :=\nbegin intros x y; induction x; induction y; reflexivity end\n\nhott def contrEquivUnit {A : Type u} (h : contr A) : A ≃ 𝟏 :=\nbegin\n  existsi (λ _, ★); apply Types.Qinv.toBiinv;\n  existsi (λ _, h.1) <;> apply Prod.mk <;> intro x;\n  induction x; reflexivity; apply h.2\nend\n\nhott def zeroMorphismContr {A : Type u} : contr (A → 𝟏) :=\n⟨λ _, ★, λ f, HITs.Interval.funext (λ x, unitIsProp ★ (f x))⟩\n\nhott def zeroMorphismEqv {A : Type u} : (A → 𝟏) ≃ 𝟏 :=\ncontrEquivUnit zeroMorphismContr\n\nhott def familyOverUnit (C : 𝟏 → Type u) : (Π x, C x) ≃ (C ★) :=\nbegin\n  fapply Sigma.mk; { intro φ; apply φ }; apply Types.Qinv.toBiinv;\n  fapply Sigma.mk; { intros c x; induction x; exact c };\n  fapply Prod.mk; { intro x; reflexivity };\n  { intro ψ; apply HITs.Interval.funext; intro x; induction x; reflexivity }\nend\n\nhott def cozeroMorphismEqv {A : Type u} : (𝟏 → A) ≃ A :=\nfamilyOverUnit (λ _, A)\n\nhott def terminalArrow {A : Type u} : A ≃ (𝟏 → A) :=\n⟨λ x _, x, Types.Qinv.toBiinv _ ⟨λ φ, φ ★, (λ φ, HITs.Interval.funext (λ ★, idp _), idp)⟩⟩\n\nhott def contrTypeEquiv {A : Type u} {B : Type v}\n  (p : contr A) (q : contr B) : A ≃ B := calc\n      A ≃ 𝟏 : contrEquivUnit.{_, 0} p\n    ... ≃ B : Types.Equiv.symm (contrEquivUnit q)\n\nhott def prodUnitEquiv (A : Type u) : 𝟏 × A ≃ A :=\nbegin existsi Prod.snd; apply Prod.mk <;> existsi Prod.mk ★ <;> { intro; reflexivity } end\n\nhott def unitProdEquiv (A : Type u) : A × 𝟏 ≃ A :=\nbegin existsi Prod.fst; apply Prod.mk <;> existsi (Prod.mk · ★) <;> { intro x; reflexivity } end\n\ndef boolToUniverse : 𝟐 → Type\n| true  => 𝟏\n| false => 𝟎\n\nhott def ffNeqTt : false ≠ true :=\nλ p, Types.Equiv.transport boolToUniverse p⁻¹ ★\n\nhott def functionSpace : ¬(Π (A B : Type), prop (A → B)) :=\nλ H, ffNeqTt (Types.Equiv.Homotopy.Id (H 𝟐 𝟐 id not) false)\n\nhott def autoContr {A : Type u} (x : A) (H : prop (A → A)) : prop A :=\nbegin\n  apply contrImplProp; existsi x;\n  apply Types.Equiv.Homotopy.Id; apply H\nend\n\nsection\n  open Types.Equiv Types.Id\n\n  hott def propIsSet {A : Type u} (r : prop A) : hset A :=\n  begin\n    intros x y p q; have g := r x; apply Types.Id.trans;\n    symmetry; apply rewriteComp;\n    exact (apd g p)⁻¹ ⬝ transportComposition p (g x);\n    induction q; apply invComp\n  end\n\n  hott def setImplGroupoid {A : Type u} (r : hset A) : groupoid A :=\n  begin\n    intros a b p q η μ; have g := r _ _ p; apply Types.Id.trans;\n    symmetry; apply rewriteComp; transitivity; symmetry; exact apd g η;\n    apply transportComposition; induction μ; apply invComp\n  end\n\n  hott def emptyIsSet : hset 𝟎 := propIsSet emptyIsProp\n  hott def unitIsSet  : hset 𝟏 := propIsSet unitIsProp\n\n  hott def contrIsProp {A : Type u} : prop (contr A) :=\n  begin\n    intro ⟨x, u⟩ ⟨y, v⟩; have p := u y;\n    induction p; apply map;\n    apply HITs.Interval.funext; intro a;\n    apply propIsSet (contrImplProp ⟨x, u⟩)\n  end\n\n  -- TODO: apply “repeat” tactic\n  hott def propIsProp {A : Type u} : prop (prop A) :=\n  begin\n    intros f g;\n    apply HITs.Interval.funext; intro;\n    apply HITs.Interval.funext; intro;\n    apply propIsSet; assumption\n  end\n\n  hott def setIsProp {A : Type u} : prop (hset A) :=\n  begin\n    intros f g;\n    apply HITs.Interval.funext; intro;\n    apply HITs.Interval.funext; intro;\n    apply HITs.Interval.funext; intro;\n    apply HITs.Interval.funext; intro;\n    apply setImplGroupoid; assumption\n  end\n\n  hott def ntypeIsProp : Π (n : hlevel) {A : Type u}, prop (is-n-type A)\n  | −2            => contrIsProp\n  | hlevel.succ n => λ p q, HITs.Interval.funext (λ x, HITs.Interval.funext (λ y, ntypeIsProp n _ _))\n\n  hott def functionToContr {A : Type u} : prop (A → contr A) :=\n  begin intros f g; apply HITs.Interval.funext; intro x; apply contrIsProp end\nend\n\nhott def retract (B : Type u) (A : Type v) :=\nΣ (r : A → B) (s : B → A), r ∘ s ~ id\n\nhott def retract.section {B : Type u} {A : Type v} (w : retract B A) : B → A := w.2.1\n\nhott def contrRetract {A : Type u} {B : Type v} : retract B A → contr A → contr B :=\nλ ⟨r, s, ε⟩ ⟨a₀, p⟩, ⟨r a₀, λ b, map r (p (s b)) ⬝ (ε b)⟩\n\nhott def retract.path {A : Type u} {B : Type v} :\n  Π (H : retract B A) {a b : B},\n  retract (a = b) (H.section a = H.section b) :=\nλ ⟨r, s, ε⟩ a b, ⟨λ q, (ε a)⁻¹ ⬝ (@map A B _ _ r q) ⬝ (ε b), map s,\nbegin\n  intro p; transitivity; symmetry; apply Types.Id.assoc;\n  symmetry; apply Types.Equiv.invRewriteComp;\n  transitivity; apply map (· ⬝ p); apply Types.Id.invInv;\n  transitivity; apply map (ε a ⬝ ·); symmetry; apply Types.Equiv.idmap;\n  transitivity; apply Types.Equiv.homotopySquare ε p;\n  apply map (· ⬝ ε b); apply Types.Equiv.mapOverComp\nend⟩\n\nhott def equivRespectsRetraction : Π {n : ℕ₋₂} {A : Type u} {B : Type v},\n  retract B A → is-n-type A → is-n-type B\n| −2            => contrRetract\n| hlevel.succ n => λ G H a b, equivRespectsRetraction (retract.path G) (H _ _)\n\nhott def equivInducesRetraction {A : Type u} {B : Type v} : A ≃ B → retract B A :=\nλ ⟨f, (_, ⟨g, ε⟩)⟩, ⟨f, g, ε⟩\n\nhott def ntypeRespectsEquiv (n : ℕ₋₂) {A : Type u} {B : Type v} :\n  A ≃ B → is-n-type A → is-n-type B :=\nequivRespectsRetraction ∘ equivInducesRetraction\n\nhott def ntypeRespectsSigma : Π (n : ℕ₋₂) {A : Type u} {B : A → Type v},\n  is-n-type A → (Π x, is-n-type (B x)) → is-n-type (Σ x, B x)\n| −2            => λ ⟨a₀, p⟩ B, ⟨⟨a₀, (B a₀).1⟩, λ ⟨a, b⟩, Types.Sigma.prod (p a) (contrImplProp (B a) _ _)⟩\n| hlevel.succ n => λ A B p q, ntypeRespectsEquiv n (Types.Equiv.symm Types.Sigma.sigmaPath)\n                                (ntypeRespectsSigma n (A p.1 q.1) (λ x, B q.1 _ _))\n\ninductive propSquash (A : Type u) : Prop\n| elem : A → propSquash A\n\ninductive Lift (A : Prop) : Type\n| elem : A → Lift A\n\ndef Squash := Lift ∘ propSquash\n\ndef Squash.elem {A : Type u} : A → Squash A :=\nLift.elem ∘ propSquash.elem\n\ndef Squash.uniq {A : Type u} : Π (a b : Squash A), a = b :=\nλ (Lift.elem _) (Lift.elem _), idp _\n\ndef Squash.prop {A : Type u} {B : Prop} (f : A → B) : Squash A → B :=\nλ (Lift.elem (propSquash.elem x)), f x\n\ndef Squash.Lift {A : Type u} {B : Type v}\n  (f : A → B) : Squash A → Squash B :=\nLift.elem ∘ Squash.prop (propSquash.elem ∘ f)\n\ndef K (A : Type u) := Π (a : A) (p : a = a), p = idp a\n\nhott def KIffSet (A : Type u) : K A ↔ hset A :=\nbegin\n  fapply Prod.mk;\n  { intros H x y p q; induction q; apply H };\n  { intros H a p; apply H }\nend\n\nhott def lemProp {A : Type u} (h : A → prop A) : prop A :=\nλ a, h a a\n\nhott def lemContr {A : Type u} (h : A → contr A) : prop A :=\nλ a, contrImplProp (h a) a\n\ndef isContrFiber {A : Type u} {B : Type v} (f : A → B) :=\nΠ (y : B), contr (fib f y)\n\nhott def propEquivLemma {A : Type u} {B : Type v}\n  (F : prop A) (G : prop B) (f : A → B) (g : B → A) : A ≃ B :=\n⟨f, (⟨g, λ _, F _ _⟩, ⟨g, λ _, G _ _⟩)⟩\n\nhott def minusTwoEqvContr {A : Type u} : (is-(−2)-type A) ≃ contr A :=\nby reflexivity\n\nhott def minusOneEqvProp {A : Type u} : (is-(−1)-type A) ≃ prop A :=\nbegin\n  apply propEquivLemma; apply ntypeIsProp; apply propIsProp;\n  { intros H a b; exact (H a b).1 };\n  { intros H a b; existsi H a b; apply propIsSet H }\nend\n\nhott def equivFunext {A : Type u} {η μ : A → Type v}\n  (H : Π x, η x ≃ μ x) : (Π x, η x) ≃ (Π x, μ x) :=\nbegin\n  existsi (λ (f : Π x, η x) (x : A), (H x).forward (f x)); apply Prod.mk;\n  { existsi (λ (f : Π x, μ x) (x : A), (H x).left (f x));\n    intro f; apply HITs.Interval.funext;\n    intro x; apply (H x).leftForward };\n  { existsi (λ (f : Π x, μ x) (x : A), (H x).right (f x));\n    intro f; apply HITs.Interval.funext;\n    intro x; apply (H x).forwardRight }\nend\n\nhott def zeroEqvSet {A : Type u} : (is-0-type A) ≃ hset A := calc\n  (is-0-type A) ≃ (Π (x y : A), is-(−1)-type (x = y)) : by reflexivity\n            ... ≃ (Π (x y : A), prop (x = y)) :\n                  begin apply equivFunext; intro x;\n                        apply equivFunext; intro y;\n                        apply minusOneEqvProp end\n            ... ≃ hset A : by reflexivity\n\nhott def oneEqvGroupoid {A : Type u} : (is-1-type A) ≃ groupoid A := calc\n  (is-1-type A) ≃ (Π (x y : A), is-0-type (x = y)) : by reflexivity\n            ... ≃ (Π (x y : A), hset (x = y)) :\n                   begin apply equivFunext; intro x;\n                         apply equivFunext; intro y;\n                         apply zeroEqvSet end\n            ... ≃ groupoid A : by reflexivity\n\nhott def propIsNType {A : Type u} (H : prop A) : Π n, is-(hlevel.succ n)-type A\n| −2            => minusOneEqvProp.left H\n| hlevel.succ n => hlevel.cumulative (hlevel.succ n) (propIsNType H n)\n\nhott def hsetRespectsEquiv {A : Type u} {B : Type v} :\n  A ≃ B → hset A → hset B :=\nbegin\n  intros e H; apply zeroEqvSet.forward;\n  apply ntypeRespectsEquiv 0 e;\n  apply zeroEqvSet.left;\n  assumption\nend\n\nhott def hsetRespectsSigma {A : Type u} {B : A → Type v}\n  (H : hset A) (G : Π x, hset (B x)) : hset (Σ x, B x) :=\nbegin\n  apply zeroEqvSet.forward; apply ntypeRespectsSigma 0;\n  { apply zeroEqvSet.left; intros x y; apply H };\n  { intro x; apply zeroEqvSet.left; apply G }\nend\n\nhott def propRespectsEquiv {A : Type u} {B : Type v} :\n  A ≃ B → prop A → prop B :=\nbegin\n  intros e H; apply minusOneEqvProp.forward;\n  apply ntypeRespectsEquiv −1 e;\n  apply minusOneEqvProp.left;\n  assumption\nend\n\nhott def contrRespectsEquiv {A : Type u} {B : Type v} :\n  A ≃ B → contr A → contr B :=\nntypeRespectsEquiv −2\n\nhott def productProp {A : Type u} {B : Type v}\n  (h : prop A) (g : prop B) : prop (A × B) :=\nbegin\n  intro ⟨x₁, y₁⟩ ⟨x₂, y₂⟩;\n  have p := h x₁ x₂; have q := g y₁ y₂;\n  induction p; induction q; reflexivity\nend\n\nhott def prodHset {A : Type u} {B : Type v}\n  (p : hset A) (q : hset B) : hset (A × B) :=\nbegin\n  apply hsetRespectsEquiv;\n  apply Types.Sigma.const;\n  apply hsetRespectsSigma;\n  apply p; intro x; apply q\nend\n\nhott def piProp {A : Type u} {B : A → Type v}\n  (H : Π x, prop (B x)) : prop (Π x, B x) :=\nλ f g, HITs.Interval.funext (λ x, H x (f x) (g x))\n\nhott def sigProp {A : Type u} {B : A → Type v}\n  (H : prop A) (G : Π x, prop (B x)) : prop (Σ x, B x) :=\nbegin intros w₁ w₂; fapply Types.Sigma.prod; apply H; apply G end\n\nhott def implProp {A : Type u} {B : Type v} (H : prop B) : prop (A → B) :=\npiProp (λ _, H)\n\nhott def notIsProp {A : Type u} : prop (¬ A) :=\nimplProp emptyIsProp\n\nhott def reflMereRel {A : Type u} (R : A → A → Type v) (h : Π x y, prop (R x y))\n  (ρ : Π x, R x x) (f : Π x y, R x y → x = y) : hset A :=\nbegin\n  apply (KIffSet _).left; intros a p;\n  apply Types.Id.transCancelLeft (f a a (ρ a));\n  transitivity; symmetry; apply Types.Equiv.transportComposition;\n  transitivity; apply Types.Equiv.liftedHapply (R a); apply Types.Equiv.apd (f a) p;\n  transitivity; apply map (f a a) (h _ _ _ (ρ a)); symmetry; apply Types.Id.reflRight\nend\n\nhott def doubleNegEq {A : Type u} (h : Π (x y : A), ¬¬(x = y) → x = y) : hset A :=\nbegin\n  fapply reflMereRel;\n  { intros x y; exact ¬¬(x = y) };\n  { intros x y; apply implProp; apply emptyIsProp };\n  { intro x; intros f; apply f; reflexivity };\n  { exact h }\nend\n\nhott def lemToDoubleNeg {A : Type u} : dec A → (¬¬A → A)\n| Sum.inl x => λ _, x\n| Sum.inr t => λ g, Proto.Empty.elim (g t)\n\nhott def Hedberg {A : Type u} : (Π (x y : A), dec (x = y)) → hset A :=\nbegin intro H; apply doubleNegEq; intros x y; apply lemToDoubleNeg; apply H x y end\n\nhott def boolEqTotal (x : 𝟐) : (x = false) + (x = true) :=\nbegin induction x using Bool.casesOn; left; reflexivity; right; reflexivity end\n\nhott lemma boolDecEq (x y : 𝟐) : dec (x = y) :=\nbegin\n  induction x using Bool.casesOn <;>\n  induction y using Bool.casesOn;\n  { left; reflexivity };\n  { right; apply ffNeqTt };\n  { right; intro p; apply ffNeqTt; exact p⁻¹ };\n  { left; reflexivity }\nend\n\nhott corollary boolIsSet : hset 𝟐 :=\nHedberg boolDecEq\n\nsection\n  open GroundZero.Types\n  hott def zeroPath {A B : 0-Type} (p : A.fst = B.fst) : A = B :=\n  begin fapply Sigma.prod; exact p; apply ntypeIsProp end\n\n  hott def zeroPathRefl (A : 0-Type) : @zeroPath A A Id.refl = Id.refl :=\n  begin\n    transitivity; apply Id.map (Sigma.prod (idp _)); change _ = Id.refl;\n    apply propIsSet (ntypeIsProp 0); apply Sigma.prodRefl\n  end\nend\n\nhott def Identity.ens {A : Type u} (H : hset A) : hset (Proto.Identity A) :=\nbegin apply hsetRespectsEquiv; apply Types.Equiv.identityEqv; assumption end\n\nhott def zeroEquiv (A B : 0-Type) := A.1 ≃ B.1\ninfix:25 \" ≃₀ \" => zeroEquiv\n\nend Structures\n\n-- http://www.cs.bham.ac.uk/~mhe/truncation-and-extensionality/tiny-library.html\n-- http://www.cs.bham.ac.uk/~mhe/truncation-and-extensionality/hsetfunext.html\ndef singl {A : Type u} (a : A) :=\nΣ b, a = b\n\nnamespace singl\n  def trivialLoop {A : Type u} (a : A) : singl a :=\n  ⟨a, by reflexivity⟩\n\n  hott def pathFromTrivialLoop {A : Type u} {a b : A}\n    (r : a = b) : trivialLoop a = ⟨b, r⟩ :=\n  begin induction r; reflexivity end\n\n  hott def contr {A : Type u} (a : A) : Structures.contr (singl a) :=\n  ⟨trivialLoop a, λ t, pathFromTrivialLoop t.2⟩\n\n  hott def prop {A : Type u} (a : A) : Structures.prop (singl a) :=\n  Structures.contrImplProp (contr a)\nend singl\n\nnamespace Theorems\n  open GroundZero.Structures GroundZero.Types.Equiv GroundZero.Types\n\n  hott theorem naive {A : Type u} {B : A → Type v} {f g : Π x, B x} : f ~ g → f = g :=\n  HITs.Interval.funext\n\n  hott theorem weak {A : Type u} {B : A → Type v} (H : Π x, contr (B x)) : contr (Π x, B x) :=\n  begin existsi (λ x, (H x).1); intro f; apply naive; intro x; apply (H x).2 end\n\n  section\n    variable {A : Type u} {B : A → Type v}\n\n    hott theorem isContrSigmaHomotopy (f : Π x, B x) : contr (Σ g, f ~ g) :=\n    let r (k : Π x, Σ y, f x = y) :=\n    @Sigma.mk _ (λ g, f ~ g) (λ x, (k x).1) (λ x, (k x).2);\n    let s (g : Π x, B x) (h : f ~ g) x :=\n    Sigma.mk (g x) (h x);\n    begin\n      existsi Sigma.mk f (Homotopy.id f); intro ⟨g, H⟩;\n      change r (λ x, Sigma.mk (f x) (idp _)) = r (s g H);\n      apply Id.map r; apply contrImplProp;\n      apply weak; intro x; apply singl.contr\n    end\n\n    variable (f : Π x, B x) {π : Π g (h : f ~ g), Type w} (r : π f (Homotopy.id f))\n    hott def homotopyInd (g : Π x, B x) (h : f ~ g) : π g h :=\n    @transport (Σ g, f ~ g) (λ (p : Σ g, f ~ g), π p.fst p.snd)\n      ⟨f, Homotopy.id f⟩ ⟨g, h⟩\n      (contrImplProp (isContrSigmaHomotopy f) _ _) r\n\n    hott lemma homotopyIndId : homotopyInd f r f (Homotopy.id f) = r :=\n    begin\n      transitivity; apply Id.map\n        (λ p, @transport (Σ g, f ~ g) (λ p, π p.fst p.snd)\n           ⟨f, Homotopy.id f⟩ ⟨f, Homotopy.id f⟩ p r);\n      change _ = idp _; apply propIsSet; apply contrImplProp;\n      apply isContrSigmaHomotopy; reflexivity\n    end\n  end\n\n  hott def funext {A : Type u} {B : A → Type v}\n    {f g : Π x, B x} : (f ~ g) → (f = g) :=\n  @homotopyInd _ _ f (λ g x, f = g) (idp _) g\n\n  hott lemma funextHapply {A : Type u} {B : A → Type v}\n    {f g : Π x, B x} : funext ∘ @HITs.Interval.happly A B f g ~ id :=\n  begin\n    intro p; induction p; change funext (Homotopy.id _) = idp _;\n    dsimp [funext]; apply homotopyIndId f\n  end\n\n  hott lemma happlyFunext {A : Type u} {B : A → Type v}\n    (f g : Π x, B x) : HITs.Interval.happly ∘ @funext A B f g ~ id :=\n  begin\n    intro H; fapply @homotopyInd _ _ f (λ g G, HITs.Interval.happly (funext G) = G) _ g H;\n    change _ = HITs.Interval.happly (idp _); apply Id.map HITs.Interval.happly;\n    change homotopyInd _ _ _ _ = _; apply homotopyIndId\n  end\n\n  hott theorem full {A : Type u} {B : A → Type v} {f g : Π x, B x} : (f = g) ≃ (f ~ g) :=\n  begin\n    existsi HITs.Interval.happly; apply Qinv.toBiinv; existsi funext;\n    apply Prod.mk; apply happlyFunext; apply funextHapply\n  end\n\n  hott corollary funextId {A : Type u} {B : A → Type v}\n    {f : Π x, B x} : funext (Homotopy.id f) = idp f :=\n  homotopyIndId _ _\n\n  open GroundZero.Types.Equiv (transport)\n  hott lemma mapHomotopy {A : Type u} {B : Type v} {a b : A} (f g : A → B) (p : a = b) (q : f ~ g) :\n    Id.map g p = @transport (A → B) (λ h, h a = h b) f g (funext q) (Id.map f p) :=\n  begin\n    induction p; symmetry; transitivity; apply Types.Equiv.transportOverHmtpy;\n    transitivity; apply Id.map (· ⬝ Id.map (λ (h : A → B), h a) (funext q));\n    apply Id.reflRight; transitivity; symmetry; apply mapFunctoriality (λ (h : A → B), h a);\n    transitivity; apply Id.map; apply Id.invComp; reflexivity\n  end\nend Theorems\n\nnamespace Structures\n  hott def piRespectsNType :\n    Π (n : ℕ₋₂) {A : Type u} {B : A → Type v},\n      (Π x, is-n-type (B x)) → is-n-type (Π x, B x)\n  | −2            => λ H, ⟨λ x, (H x).1, λ φ, Theorems.funext (λ x, (H x).2 (φ x))⟩\n  | hlevel.succ n => λ H f g, ntypeRespectsEquiv n (Types.Equiv.symm Theorems.full)\n                                (piRespectsNType n (λ x, H x _ _))\n\n  hott def piHset {A : Type u} {B : A → Type v}\n    (H : Π x, hset (B x)) : hset (Π x, B x) :=\n  begin\n    apply zeroEqvSet.forward; apply piRespectsNType 0;\n    intro x; apply zeroEqvSet.left; apply H\n  end\nend Structures\n\nhott def iter (A B : Type) : ℕ → Type\n| Nat.zero   => B\n| Nat.succ n => Coproduct (iter A B n) A\n\nhott def pt := iter 𝟏\n\nhott def vect (A : Type u) : ℕ → Type u\n| Nat.zero   => 𝟏\n| Nat.succ n => A × vect A n\n\nhott def vect.const {A : Type u} (a : A) : Π n, vect A n\n| Nat.zero   => ★\n| Nat.succ n => (a, const a n)\n\nhott def vect.map {A : Type u} {B : Type v} (f : A → B) :\n  Π {n : ℕ}, vect A n → vect B n\n| Nat.zero   => λ _, ★\n| Nat.succ n => λ v, (f v.1, map f v.2)\n\nsection\n  open GroundZero.Types.Equiv (transportOverProduct transport subst)\n  open GroundZero.Types\n\n  hott def vect.subst {A B : Type u} (p : A = B) (f : B → A) {n : ℕ} (v : vect A n) :\n    vect.map f (transport (vect · n) p v) = vect.map (f ∘ transport id p) v :=\n  begin induction p; reflexivity end\nend\n\nhott def vect.idfunc {A : Type u} : Π {n : ℕ} (f : A → A)\n  (H : f ~ id) (v : vect A n), vect.map f v = v\n| Nat.zero,   f, H, v => idp v\n| Nat.succ n, f, H, v => Types.Product.prod (H v.1) (idfunc f H v.2)\n\nhott def vect.id {A : Type u} {n : ℕ} (v : vect A n) : vect.map id v = v :=\nbegin apply vect.idfunc; reflexivity end\n\nhott def vect.comp {A : Type u} {B : Type v} {γ : Type w} :\n  Π {n : ℕ} (f : A → B) (g : B → γ) (v : vect A n), vect.map g (vect.map f v) = vect.map (g ∘ f) v\n| Nat.zero,   f, g, v => idp _\n| Nat.succ n, f, g, v => Types.Product.prod (idp _) (comp f g v.2)\n\nhott def vect.constMap {A : Type u} {B : Type v} (a : A) (f : A → B) :\n  Π {n : ℕ}, vect.map f (vect.const a n) = vect.const (f a) n\n| Nat.zero   => idp _\n| Nat.succ n => Types.Product.prod (idp _) (constMap a f)\n\nhott def Finite := iter 𝟏 𝟎\n@[match_pattern] def Finite.zero {n : ℕ} : Finite (n + 1) := Sum.inr ★\n@[match_pattern] def Finite.succ {n : ℕ} : Finite n → Finite (n + 1) := Sum.inl\n\ndef LEMinf := Π (A : Type u), A + ¬A\nnotation \"LEM∞\" => LEMinf\n\nopen Structures (prop propset)\nhott def hrel (A : Type u) := A → A → propset.{v}\n\nsection\n  variable {A : Type u} (R : hrel A)\n\n  def isrefl  := Π a, (R a a).1\n  def issymm  := Π a b, (R a b).1 → (R b a).1\n  def istrans := Π a b c, (R a b).1 → (R b c).1 → (R a c).1\n\n  def iseqrel := isrefl R × issymm R × istrans R\nend\n\nhott def eqrel (A : Type u) := Σ φ, @iseqrel A φ\n\n-- TODO: repeat\nhott def iseqrel.prop {A : Type u} {R : hrel A} : prop (iseqrel R) :=\nbegin\n  apply Structures.productProp;\n  { intros f g; apply Theorems.funext;\n    intro x; apply (R x x).2 };\n  apply Structures.productProp;\n  { intros f g;\n    apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply (R _ _).2 };\n  { intros f g;\n     apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply Theorems.funext; intro;\n    apply (R _ _).2 }\nend\n\nsection\n  variable {A : Type u} (R : eqrel.{u, v} A)\n\n  hott def eqrel.rel : hrel A := R.1\n  hott def eqrel.iseqv : iseqrel R.rel := R.2\n\n  hott def eqrel.apply (a b : A) : Type v :=\n  (R.rel a b).1\n\n  hott def eqrel.prop (a b : A) : prop (R.apply a b) :=\n  (R.rel a b).2\n\n  -- Accessors\n  hott def eqrel.refl (a : A) : R.apply a a :=\n  R.2.1 a\n\n  hott def eqrel.symm {a b : A} : R.apply a b → R.apply b a :=\n  R.2.2.1 a b\n\n  hott def eqrel.trans {a b c : A} :\n    R.apply a b → R.apply b c → R.apply a c :=\n  R.2.2.2 a b c\nend\n\nhott def eqrel.eq {A : Type u} {x y : eqrel A} (p : x.rel = y.rel) : x = y :=\nbegin apply Types.Sigma.prod p; apply iseqrel.prop end\n\nhott def iffOverPi {A : Type u} {B : A → Type v} {B' : A → Type w}\n  (φ : Π x, B x ↔ B' x) : (Π x, B x) ↔ (Π x, B' x) :=\nbegin apply Prod.mk; { intros f x; apply (φ x).left; apply f }; { intros g x; apply (φ x).right; apply g } end\n\nhott def hcommSquare (P : Type k) (A : Type u) (B : Type v) (C : Type w) :=\nΣ (f : A → C) (g : B → C) (h : P → A) (k : P → B), f ∘ h = g ∘ k\n\nhott def pullback {A : Type u} {B : Type v}\n  (C : Type w) (f : A → C) (g : B → C) :=\nΣ (p : A × B), f p.1 = g p.2\n\nnamespace hcommSquare\n  variable {P : Type k} {A : Type u} {B : Type v} {C : Type w}\n\n  def top   (η : hcommSquare P A B C) : P → A := η.2.2.1\n  def bot   (η : hcommSquare P A B C) : B → C := η.2.1\n  def left  (η : hcommSquare P A B C) : P → B := η.2.2.2.1\n  def right (η : hcommSquare P A B C) : A → C := η.1\n\n  def naturality (η : hcommSquare P A B C) : η.right ∘ η.top = η.bot ∘ η.left := η.2.2.2.2\n\n  hott def induced (η : hcommSquare P A B C) (X : Type r) :\n    (X → P) → @pullback (X → A) (X → B) (X → C) (λ f, right η ∘ f) (λ g, bot η ∘ g) :=\n  λ φ, ⟨(top η ∘ φ, left η ∘ φ), @map (P → C) (X → C) (right η ∘ top η) (bot η ∘ left η) (· ∘ φ) η.naturality⟩\n\n  hott def isPullback (η : hcommSquare P A B C) :=\n  Π (X : Type (max u v w k)), biinv (induced η X)\nend hcommSquare\n\nhott def pullbackSquare (P : Type k) (A : Type u) (B : Type v) (C : Type w) :=\nΣ (η : hcommSquare P A B C), η.isPullback\n\nnamespace Types.Equiv\n  open GroundZero.Structures\n  universe u' v'\n\n  -- 1-1 correspondence\n  def Corr (A : Type u) (B : Type v) :=\n  Σ (R : A → B → Type w), (Π a, contr (Σ b, R a b)) × (Π b, contr (Σ a, R a b))\n\n  open GroundZero.Types\n  variable {A : Type u} {A' : Type v} {B : Type u'} {B' : Type v'}\n\n  hott def prodBiinv {f : A → A'} {g : B → B'}\n    (e₁ : biinv f) (e₂ : biinv g) : biinv (Product.bimap f g) :=\n  (⟨Product.bimap e₁.1.1 e₂.1.1, λ w, Product.prod (e₁.1.2 w.1) (e₂.1.2 w.2)⟩,\n   ⟨Product.bimap e₁.2.1 e₂.2.1, λ w, Product.prod (e₁.2.2 w.1) (e₂.2.2 w.2)⟩)\n\n  hott def prodEquiv (e₁ : A ≃ A') (e₂ : B ≃ B') : (A × B) ≃ (A' × B') :=\n  ⟨Product.bimap e₁.1 e₂.1, prodBiinv e₁.2 e₂.2⟩\nend Types.Equiv\n\nend GroundZero\n", "meta": {"author": "forked-from-1kasper", "repo": "ground_zero", "sha": "58ad68bb54e355f6c39beaee2b383879eccc9952", "save_path": "github-repos/lean/forked-from-1kasper-ground_zero", "path": "github-repos/lean/forked-from-1kasper-ground_zero/ground_zero-58ad68bb54e355f6c39beaee2b383879eccc9952/GroundZero/Structures.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3278524180688652}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.list.dedup\n! leanprover-community/mathlib commit f694c7dead66f5d4c80f446c796a5aad14707f0e\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.List.Nodup\n\n/-!\n# Erasure of duplicates in a list\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file proves basic results about `list.dedup` (definition in `data.list.defs`).\n`dedup l` returns `l` without its duplicates. It keeps the earliest (that is, rightmost)\noccurrence of each.\n\n## Tags\n\nduplicate, multiplicity, nodup, `nub`\n-/\n\n\nuniverse u\n\nnamespace List\n\nvariable {α : Type u} [DecidableEq α]\n\n#print List.dedup_nil /-\n@[simp]\ntheorem dedup_nil : dedup [] = ([] : List α) :=\n  rfl\n#align list.dedup_nil List.dedup_nil\n-/\n\n#print List.dedup_cons_of_mem' /-\ntheorem dedup_cons_of_mem' {a : α} {l : List α} (h : a ∈ dedup l) : dedup (a :: l) = dedup l :=\n  pwFilter_cons_of_neg <| by simpa only [forall_mem_ne] using h\n#align list.dedup_cons_of_mem' List.dedup_cons_of_mem'\n-/\n\n#print List.dedup_cons_of_not_mem' /-\ntheorem dedup_cons_of_not_mem' {a : α} {l : List α} (h : a ∉ dedup l) :\n    dedup (a :: l) = a :: dedup l :=\n  pwFilter_cons_of_pos <| by simpa only [forall_mem_ne] using h\n#align list.dedup_cons_of_not_mem' List.dedup_cons_of_not_mem'\n-/\n\n#print List.mem_dedup /-\n@[simp]\ntheorem mem_dedup {a : α} {l : List α} : a ∈ dedup l ↔ a ∈ l := by\n  simpa only [dedup, forall_mem_ne, Classical.not_not] using\n    not_congr\n      (@forall_mem_pw_filter α (· ≠ ·) _\n        (fun x y z xz => not_and_or.1 <| mt (fun h => Eq.trans h.1 h.2) xz) a l)\n#align list.mem_dedup List.mem_dedup\n-/\n\n#print List.dedup_cons_of_mem /-\n@[simp]\ntheorem dedup_cons_of_mem {a : α} {l : List α} (h : a ∈ l) : dedup (a :: l) = dedup l :=\n  dedup_cons_of_mem' <| mem_dedup.2 h\n#align list.dedup_cons_of_mem List.dedup_cons_of_mem\n-/\n\n#print List.dedup_cons_of_not_mem /-\n@[simp]\ntheorem dedup_cons_of_not_mem {a : α} {l : List α} (h : a ∉ l) : dedup (a :: l) = a :: dedup l :=\n  dedup_cons_of_not_mem' <| mt mem_dedup.1 h\n#align list.dedup_cons_of_not_mem List.dedup_cons_of_not_mem\n-/\n\n#print List.dedup_sublist /-\ntheorem dedup_sublist : ∀ l : List α, dedup l <+ l :=\n  pwFilter_sublist\n#align list.dedup_sublist List.dedup_sublist\n-/\n\n#print List.dedup_subset /-\ntheorem dedup_subset : ∀ l : List α, dedup l ⊆ l :=\n  pwFilter_subset\n#align list.dedup_subset List.dedup_subset\n-/\n\n#print List.subset_dedup /-\ntheorem subset_dedup (l : List α) : l ⊆ dedup l := fun a => mem_dedup.2\n#align list.subset_dedup List.subset_dedup\n-/\n\n#print List.nodup_dedup /-\ntheorem nodup_dedup : ∀ l : List α, Nodup (dedup l) :=\n  pairwise_pwFilter\n#align list.nodup_dedup List.nodup_dedup\n-/\n\n#print List.dedup_eq_self /-\ntheorem dedup_eq_self {l : List α} : dedup l = l ↔ Nodup l :=\n  pwFilter_eq_self\n#align list.dedup_eq_self List.dedup_eq_self\n-/\n\n#print List.Nodup.dedup /-\nprotected theorem Nodup.dedup {l : List α} (h : l.Nodup) : l.dedup = l :=\n  List.dedup_eq_self.2 h\n#align list.nodup.dedup List.Nodup.dedup\n-/\n\n#print List.dedup_idempotent /-\n@[simp]\ntheorem dedup_idempotent {l : List α} : dedup (dedup l) = dedup l :=\n  pwFilter_idempotent\n#align list.dedup_idempotent List.dedup_idempotent\n-/\n\n#print List.dedup_append /-\ntheorem dedup_append (l₁ l₂ : List α) : dedup (l₁ ++ l₂) = l₁ ∪ dedup l₂ :=\n  by\n  induction' l₁ with a l₁ IH; · rfl; rw [cons_union, ← IH]\n  show dedup (a :: (l₁ ++ l₂)) = insert a (dedup (l₁ ++ l₂))\n  by_cases a ∈ dedup (l₁ ++ l₂) <;> [rw [dedup_cons_of_mem' h, insert_of_mem h],\n    rw [dedup_cons_of_not_mem' h, insert_of_not_mem h]]\n#align list.dedup_append List.dedup_append\n-/\n\n#print List.replicate_dedup /-\ntheorem replicate_dedup {x : α} : ∀ {k}, k ≠ 0 → (replicate k x).dedup = [x]\n  | 0, h => (h rfl).elim\n  | 1, _ => rfl\n  | n + 2, _ => by\n    rw [replicate_succ, dedup_cons_of_mem (mem_replicate.2 ⟨n.succ_ne_zero, rfl⟩),\n      replicate_dedup n.succ_ne_zero]\n#align list.replicate_dedup List.replicate_dedup\n-/\n\n#print List.count_dedup /-\ntheorem count_dedup (l : List α) (a : α) : l.dedup.count a = if a ∈ l then 1 else 0 := by\n  simp_rw [count_eq_of_nodup <| nodup_dedup l, mem_dedup]\n#align list.count_dedup List.count_dedup\n-/\n\n/- warning: list.sum_map_count_dedup_filter_eq_countp -> List.sum_map_count_dedup_filter_eq_countp is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (p : α -> Prop) [_inst_2 : DecidablePred.{succ u1} α p] (l : List.{u1} α), Eq.{1} Nat (List.sum.{0} Nat Nat.hasAdd Nat.hasZero (List.map.{u1, 0} α Nat (fun (x : α) => List.count.{u1} α (fun (a : α) (b : α) => _inst_1 a b) x l) (List.filterₓ.{u1} α p (fun (a : α) => _inst_2 a) (List.dedup.{u1} α (fun (a : α) (b : α) => _inst_1 a b) l)))) (List.countp.{u1} α p (fun (a : α) => _inst_2 a) l)\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} α] (p : α -> Bool) (_inst_2 : List.{u1} α), Eq.{1} Nat (List.sum.{0} Nat instAddNat (LinearOrderedCommMonoidWithZero.toZero.{0} Nat Nat.linearOrderedCommMonoidWithZero) (List.map.{u1, 0} α Nat (fun (x : α) => List.count.{u1} α (instBEq.{u1} α (fun (a : α) (b : α) => _inst_1 a b)) x _inst_2) (List.filter.{u1} α p (List.dedup.{u1} α (fun (a : α) (b : α) => _inst_1 a b) _inst_2)))) (List.countp.{u1} α p _inst_2)\nCase conversion may be inaccurate. Consider using '#align list.sum_map_count_dedup_filter_eq_countp List.sum_map_count_dedup_filter_eq_countpₓ'. -/\n/-- Summing the count of `x` over a list filtered by some `p` is just `countp` applied to `p` -/\ntheorem sum_map_count_dedup_filter_eq_countp (p : α → Prop) [DecidablePred p] (l : List α) :\n    ((l.dedup.filterₓ p).map fun x => l.count x).Sum = l.countp p :=\n  by\n  induction' l with a as h\n  · simp\n  · simp_rw [List.countp_cons, List.count_cons', List.sum_map_add]\n    congr 1\n    · refine' trans _ h\n      by_cases ha : a ∈ as\n      · simp [dedup_cons_of_mem ha]\n      · simp only [dedup_cons_of_not_mem ha, List.filter]\n        split_ifs with hp <;> simp [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]\n    · by_cases hp : p a\n      · refine' trans (sum_map_eq_nsmul_single a _ fun _ h _ => by simp [h]) _\n        simp [hp, count_dedup]\n      · refine' trans (List.sum_eq_zero fun n hn => _) (by simp [hp])\n        obtain ⟨a', ha'⟩ := List.mem_map.1 hn\n        simp only [(fun h => hp (h ▸ (List.mem_filter.1 ha'.1).2) : a' ≠ a), if_false] at ha'\n        exact ha'.2.symm\n#align list.sum_map_count_dedup_filter_eq_countp List.sum_map_count_dedup_filter_eq_countp\n\n#print List.sum_map_count_dedup_eq_length /-\ntheorem sum_map_count_dedup_eq_length (l : List α) :\n    (l.dedup.map fun x => l.count x).Sum = l.length := by\n  simpa using sum_map_count_dedup_filter_eq_countp (fun _ => True) l\n#align list.sum_map_count_dedup_eq_length List.sum_map_count_dedup_eq_length\n-/\n\nend List\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/List/Dedup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.32782400449042315}}
{"text": "\nimport Lib.Algebra.Monoid\nimport Lib.Data.MonoFoldable\nimport Lib.Data.MonoFunctor\nimport Lib.Data.Traversable\nimport Lib.Function\n\nclass MonoTraversable (T : Type u) (α : outParam (Type u))\nextends MonoFunctor T α, MonoFoldable T α where\n  traverse {F : Type u → Type u} [Applicative F] (f : α → F α) : T → F T\n  mapM {F : Type u → Type u} [Monad F] (f : α → F α) : T → F T\n\ninstance {T} [Traversable T] : MonoTraversable (T α) α where\n  traverse := Traversable.traverse\n  mapM := Traversable.mapM\n\nnamespace MonoTraversable\n\nopen MonoTraversable\n-- open Traversable (StateT.mk)\n-- def StateT.mk {σ α} (f : σ → m (α × σ)) : StateT σ m α := f\n\nsection Accum\n\nvariable {T : Type u} [MonoTraversable T α]\nvariable (f : α → σ → α × σ)\n\ndef accuml (x₀ : σ) (x : T) : T × σ :=\nStateT.run (m := Id) (mapM (StateT.mk.{u,u} ∘ f) x) x₀\n\ndef scanl (x₀ : σ) (x : T) : T :=\nStateT.run' (m := Id) (mapM (StateT.mk.{u,u} ∘ f) x) x₀\n\ndef accumr (x₀ : σ) (x : T) : T × σ :=\nStateT.run (m := Id) (Op1.run (traverse (Op1.mk.{u,u} ∘ StateT.mk.{u,u} ∘ f) x)) x₀\n\ndef scanr (x₀ : σ) (x : T) : T :=\naccumr f x₀ x |>.1\n\nend Accum\n\nsection AccumIdx\n\nvariable {T : Type u} [MonoTraversable.{u} T α]\nvariable {β : Type u} (f : Nat → α → σ → α × σ)\n\ndef accumlIdx (x₀ : σ) (x : T) : T × σ :=\naccuml (λ a (x, i) => f i a x |>.map id (., i+1)) (x₀, 0) x |>.map id Prod.fst\n\ndef scanlIdx (x₀ : σ) (x : T) : T :=\naccumlIdx f x₀ x |>.fst\n\nend AccumIdx\nend MonoTraversable\n\nopen MonoTraversable\nclass LawfulMonoTraversable (T : Type u) (α : Type u)\n      [MonoTraversable T α]\nextends LawfulMonoFunctor T α, LawfulMonoFoldable T α where\n  traverse_eq_mapM {M} [Monad M] [LawfulMonad M] (f : α → M α) (x : T) :\n    traverse f x = mapM f x\n  map_eq_traverse (f : α → α) (x : T) :\n    MonoFunctor.map f x = Id.run (traverse (Id.mk ∘ f) x)\n  -- id_traverse {α} (x : T α) : traverse Id.mk x = Id.mk x\n  comp_traverse {F G} [Applicative F] [LawfulApplicative F]\n    [Applicative G] [LawfulApplicative G]\n    (x : T) (f : α → F α) (g : α → G α) :\n    Comp.run (traverse (Comp.mk ∘ Functor.map f ∘ g) x) =\n    traverse f <$> traverse g x\n  foldl_eq_traverse {β} (f : β → α → β) (x : T) (x₀ : β) :\n    MonoFoldable.foldl f x₀ x =\n    Endo.run (Op.run (Const.run (traverse\n      (Const.mk (α := α) ∘ Op.mk ∘ Endo.mk ∘ flip f) x))) x₀\n  traverse_sim {F G}\n               [Applicative F] [LawfulApplicative F]\n               [Applicative G] [LawfulApplicative G]\n               (x : T) (R : ApplicativeRel F G)\n               (f : α → F α) (g : α → G α) :\n      (∀ a, R (f a) (g a)) →\n      R (traverse f x) (traverse g x)\n\ninstance {T} [Traversable T] [LawfulTraversable T] :\n         LawfulMonoTraversable (T α) α where\n  traverse_eq_mapM := LawfulTraversable.traverse_eq_mapM\n  map_eq_traverse := LawfulTraversable.map_eq_traverse\n  comp_traverse := LawfulTraversable.comp_traverse\n  foldl_eq_traverse := LawfulTraversable.foldl_eq_traverse\n  traverse_sim := LawfulTraversable.traverse_sim\n\nnamespace LawfulMonoTraversable\nopen MonoTraversable\n\nvariable {T} [MonoTraversable T α] [LawfulMonoTraversable T α]\n\ntheorem id_traverse (x : T) : traverse Id.mk x = Id.mk x := by\nhave := (map_eq_traverse id x).symm\nsimp [LawfulMonoFunctor.id_map] at this\nassumption\n\nsection nat\nvariable {T : Type u} [MonoTraversable T α] [LawfulMonoTraversable T α]\nvariable {F : Type u → Type u} [Applicative F] [LawfulApplicative F]\nvariable {G : Type u → Type u} [Applicative G] [LawfulApplicative G]\n\ntheorem traverse_nat (x : T) (f : ApplicativeHom F G)\n        (g : α → F α) :\n  f (traverse g x) = traverse (f.fn ∘ g) x := by\nlet R := f.toApplicativeRel\napply LawfulMonoTraversable.traverse_sim _ R; intro a\nsimp [ApplicativeHom.toApplicativeRel]\n\nend nat\n\nend LawfulMonoTraversable\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/MonoTraversable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3278239974685076}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Mario Carneiro\n\n! This file was ported from Lean 3 source module tactic.norm_num\n! leanprover-community/mathlib commit ec322deb9ba5aad978f862669053069b7957c31d\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Rat.Cast\nimport Mathbin.Data.Rat.MetaDefs\nimport Mathbin.Data.Int.Lemmas\n\n/-!\n# `norm_num`\n\nEvaluating arithmetic expressions including `*`, `+`, `-`, `^`, `≤`.\n-/\n\n\nuniverse u v w\n\nnamespace Tactic\n\nnamespace InstanceCache\n\n/-- Faster version of `mk_app ``bit0 [e]`. -/\nunsafe def mk_bit0 (c : instance_cache) (e : expr) : tactic (instance_cache × expr) := do\n  let (c, ai) ← c.get `` Add\n  return (c, (expr.const `` bit0 [c]).mk_app [c, ai, e])\n#align tactic.instance_cache.mk_bit0 tactic.instance_cache.mk_bit0\n\n/-- Faster version of `mk_app ``bit1 [e]`. -/\nunsafe def mk_bit1 (c : instance_cache) (e : expr) : tactic (instance_cache × expr) := do\n  let (c, ai) ← c.get `` Add\n  let (c, oi) ← c.get `` One\n  return (c, (expr.const `` bit1 [c]).mk_app [c, oi, ai, e])\n#align tactic.instance_cache.mk_bit1 tactic.instance_cache.mk_bit1\n\nend InstanceCache\n\nend Tactic\n\nopen Tactic\n\n/-!\nEach lemma in this file is written the way it is to exactly match (with no defeq reduction allowed)\nthe conclusion of some lemma generated by the proof procedure that uses it. That proof procedure\nshould describe the shape of the generated lemma in its docstring.\n-/\n\n\nnamespace NormNum\n\nvariable {α : Type u}\n\ntheorem subst_into_add {α} [Add α] (l r tl tr t) (prl : (l : α) = tl) (prr : r = tr)\n    (prt : tl + tr = t) : l + r = t := by rw [prl, prr, prt]\n#align norm_num.subst_into_add NormNum.subst_into_add\n\ntheorem subst_into_mul {α} [Mul α] (l r tl tr t) (prl : (l : α) = tl) (prr : r = tr)\n    (prt : tl * tr = t) : l * r = t := by rw [prl, prr, prt]\n#align norm_num.subst_into_mul NormNum.subst_into_mul\n\ntheorem subst_into_neg {α} [Neg α] (a ta t : α) (pra : a = ta) (prt : -ta = t) : -a = t := by\n  simp [pra, prt]\n#align norm_num.subst_into_neg NormNum.subst_into_neg\n\n/-- The result type of `match_numeral`, either `0`, `1`, or a top level\ndecomposition of `bit0 e` or `bit1 e`. The `other` case means it is not a numeral. -/\nunsafe inductive match_numeral_result\n  | zero\n  | one\n  | bit0 (e : expr)\n  | bit1 (e : expr)\n  | other\n#align norm_num.match_numeral_result norm_num.match_numeral_result\n\n/-- Unfold the top level constructor of the numeral expression. -/\nunsafe def match_numeral : expr → match_numeral_result\n  | q(bit0 $(e)) => match_numeral_result.bit0 e\n  | q(bit1 $(e)) => match_numeral_result.bit1 e\n  | q(@Zero.zero _ _) => match_numeral_result.zero\n  | q(@One.one _ _) => match_numeral_result.one\n  | _ => match_numeral_result.other\n#align norm_num.match_numeral norm_num.match_numeral\n\ntheorem zero_succ {α} [Semiring α] : (0 + 1 : α) = 1 :=\n  zero_add _\n#align norm_num.zero_succ NormNum.zero_succ\n\ntheorem one_succ {α} [Semiring α] : (1 + 1 : α) = 2 :=\n  rfl\n#align norm_num.one_succ NormNum.one_succ\n\ntheorem bit0_succ {α} [Semiring α] (a : α) : bit0 a + 1 = bit1 a :=\n  rfl\n#align norm_num.bit0_succ NormNum.bit0_succ\n\ntheorem bit1_succ {α} [Semiring α] (a b : α) (h : a + 1 = b) : bit1 a + 1 = bit0 b :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.bit1_succ NormNum.bit1_succ\n\nsection\n\nopen MatchNumeralResult\n\n/-- Given `a`, `b` natural numerals, proves `⊢ a + 1 = b`, assuming that this is provable.\n(It may prove garbage instead of failing if `a + 1 = b` is false.) -/\nunsafe def prove_succ : instance_cache → expr → expr → tactic (instance_cache × expr)\n  | c, e, r =>\n    match match_numeral e with\n    | zero => c.mk_app `` zero_succ []\n    | one => c.mk_app `` one_succ []\n    | bit0 e => c.mk_app `` bit0_succ [e]\n    | bit1 e => do\n      let r := r.app_arg\n      let (c, p) ← prove_succ c e r\n      c `` bit1_succ [e, r, p]\n    | _ => failed\n#align norm_num.prove_succ norm_num.prove_succ\n\nend\n\n/-- Given `a` natural numeral, returns `(b, ⊢ a + 1 = b)`. -/\nunsafe def prove_succ' (c : instance_cache) (a : expr) : tactic (instance_cache × expr × expr) := do\n  let na ← a.toNat\n  let (c, b) ← c.ofNat (na + 1)\n  let (c, p) ← prove_succ c a b\n  return (c, b, p)\n#align norm_num.prove_succ' norm_num.prove_succ'\n\ntheorem zero_adc {α} [Semiring α] (a b : α) (h : a + 1 = b) : 0 + a + 1 = b := by rwa [zero_add]\n#align norm_num.zero_adc NormNum.zero_adc\n\ntheorem adc_zero {α} [Semiring α] (a b : α) (h : a + 1 = b) : a + 0 + 1 = b := by rwa [add_zero]\n#align norm_num.adc_zero NormNum.adc_zero\n\ntheorem one_add {α} [Semiring α] (a b : α) (h : a + 1 = b) : 1 + a = b := by rwa [add_comm]\n#align norm_num.one_add NormNum.one_add\n\ntheorem add_bit0_bit0 {α} [Semiring α] (a b c : α) (h : a + b = c) : bit0 a + bit0 b = bit0 c :=\n  h ▸ by simp [bit0, add_left_comm, add_assoc]\n#align norm_num.add_bit0_bit0 NormNum.add_bit0_bit0\n\ntheorem add_bit0_bit1 {α} [Semiring α] (a b c : α) (h : a + b = c) : bit0 a + bit1 b = bit1 c :=\n  h ▸ by simp [bit0, bit1, add_left_comm, add_assoc]\n#align norm_num.add_bit0_bit1 NormNum.add_bit0_bit1\n\ntheorem add_bit1_bit0 {α} [Semiring α] (a b c : α) (h : a + b = c) : bit1 a + bit0 b = bit1 c :=\n  h ▸ by simp [bit0, bit1, add_left_comm, add_comm, add_assoc]\n#align norm_num.add_bit1_bit0 NormNum.add_bit1_bit0\n\ntheorem add_bit1_bit1 {α} [Semiring α] (a b c : α) (h : a + b + 1 = c) : bit1 a + bit1 b = bit0 c :=\n  h ▸ by simp [bit0, bit1, add_left_comm, add_comm, add_assoc]\n#align norm_num.add_bit1_bit1 NormNum.add_bit1_bit1\n\ntheorem adc_one_one {α} [Semiring α] : (1 + 1 + 1 : α) = 3 :=\n  rfl\n#align norm_num.adc_one_one NormNum.adc_one_one\n\ntheorem adc_bit0_one {α} [Semiring α] (a b : α) (h : a + 1 = b) : bit0 a + 1 + 1 = bit0 b :=\n  h ▸ by simp [bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit0_one NormNum.adc_bit0_one\n\ntheorem adc_one_bit0 {α} [Semiring α] (a b : α) (h : a + 1 = b) : 1 + bit0 a + 1 = bit0 b :=\n  h ▸ by simp [bit0, add_left_comm, add_assoc]\n#align norm_num.adc_one_bit0 NormNum.adc_one_bit0\n\ntheorem adc_bit1_one {α} [Semiring α] (a b : α) (h : a + 1 = b) : bit1 a + 1 + 1 = bit1 b :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit1_one NormNum.adc_bit1_one\n\ntheorem adc_one_bit1 {α} [Semiring α] (a b : α) (h : a + 1 = b) : 1 + bit1 a + 1 = bit1 b :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_one_bit1 NormNum.adc_one_bit1\n\ntheorem adc_bit0_bit0 {α} [Semiring α] (a b c : α) (h : a + b = c) : bit0 a + bit0 b + 1 = bit1 c :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit0_bit0 NormNum.adc_bit0_bit0\n\ntheorem adc_bit1_bit0 {α} [Semiring α] (a b c : α) (h : a + b + 1 = c) :\n    bit1 a + bit0 b + 1 = bit0 c :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit1_bit0 NormNum.adc_bit1_bit0\n\ntheorem adc_bit0_bit1 {α} [Semiring α] (a b c : α) (h : a + b + 1 = c) :\n    bit0 a + bit1 b + 1 = bit0 c :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit0_bit1 NormNum.adc_bit0_bit1\n\ntheorem adc_bit1_bit1 {α} [Semiring α] (a b c : α) (h : a + b + 1 = c) :\n    bit1 a + bit1 b + 1 = bit1 c :=\n  h ▸ by simp [bit1, bit0, add_left_comm, add_assoc]\n#align norm_num.adc_bit1_bit1 NormNum.adc_bit1_bit1\n\nsection\n\nopen MatchNumeralResult\n\nmutual\n  unsafe def prove_add_nat : instance_cache → expr → expr → expr → tactic (instance_cache × expr)\n    | c, a, b, r => do\n      match match_numeral a, match_numeral b with\n        | zero, _ => c `` zero_add [b]\n        | _, zero => c `` add_zero [a]\n        | _, one => prove_succ c a r\n        | one, _ => do\n          let (c, p) ← prove_succ c b r\n          c `` one_add [b, r, p]\n        | bit0 a, bit0 b => do\n          let r := r\n          let (c, p) ← prove_add_nat c a b r\n          c `` add_bit0_bit0 [a, b, r, p]\n        | bit0 a, bit1 b => do\n          let r := r\n          let (c, p) ← prove_add_nat c a b r\n          c `` add_bit0_bit1 [a, b, r, p]\n        | bit1 a, bit0 b => do\n          let r := r\n          let (c, p) ← prove_add_nat c a b r\n          c `` add_bit1_bit0 [a, b, r, p]\n        | bit1 a, bit1 b => do\n          let r := r\n          let (c, p) ← prove_adc_nat c a b r\n          c `` add_bit1_bit1 [a, b, r, p]\n        | _, _ => failed\n  unsafe def prove_adc_nat : instance_cache → expr → expr → expr → tactic (instance_cache × expr)\n    | c, a, b, r => do\n      match match_numeral a, match_numeral b with\n        | zero, _ => do\n          let (c, p) ← prove_succ c b r\n          c `` zero_adc [b, r, p]\n        | _, zero => do\n          let (c, p) ← prove_succ c b r\n          c `` adc_zero [b, r, p]\n        | one, one => c `` adc_one_one []\n        | bit0 a, one => do\n          let r := r\n          let (c, p) ← prove_succ c a r\n          c `` adc_bit0_one [a, r, p]\n        | one, bit0 b => do\n          let r := r\n          let (c, p) ← prove_succ c b r\n          c `` adc_one_bit0 [b, r, p]\n        | bit1 a, one => do\n          let r := r\n          let (c, p) ← prove_succ c a r\n          c `` adc_bit1_one [a, r, p]\n        | one, bit1 b => do\n          let r := r\n          let (c, p) ← prove_succ c b r\n          c `` adc_one_bit1 [b, r, p]\n        | bit0 a, bit0 b => do\n          let r := r\n          let (c, p) ← prove_add_nat c a b r\n          c `` adc_bit0_bit0 [a, b, r, p]\n        | bit0 a, bit1 b => do\n          let r := r\n          let (c, p) ← prove_adc_nat c a b r\n          c `` adc_bit0_bit1 [a, b, r, p]\n        | bit1 a, bit0 b => do\n          let r := r\n          let (c, p) ← prove_adc_nat c a b r\n          c `` adc_bit1_bit0 [a, b, r, p]\n        | bit1 a, bit1 b => do\n          let r := r\n          let (c, p) ← prove_adc_nat c a b r\n          c `` adc_bit1_bit1 [a, b, r, p]\n        | _, _ => failed\nend\n#align norm_num.prove_add_nat norm_num.prove_add_nat\n#align norm_num.prove_adc_nat norm_num.prove_adc_nat\n\n/-- Given `a`,`b`,`r` natural numerals, proves `⊢ a + b = r`. -/\nadd_decl_doc prove_add_nat\n\n/-- Given `a`,`b`,`r` natural numerals, proves `⊢ a + b + 1 = r`. -/\nadd_decl_doc prove_adc_nat\n\n/-- Given `a`,`b` natural numerals, returns `(r, ⊢ a + b = r)`. -/\nunsafe def prove_add_nat' (c : instance_cache) (a b : expr) :\n    tactic (instance_cache × expr × expr) := do\n  let na ← a.toNat\n  let nb ← b.toNat\n  let (c, r) ← c.ofNat (na + nb)\n  let (c, p) ← prove_add_nat c a b r\n  return (c, r, p)\n#align norm_num.prove_add_nat' norm_num.prove_add_nat'\n\nend\n\ntheorem bit0_mul {α} [Semiring α] (a b c : α) (h : a * b = c) : bit0 a * b = bit0 c :=\n  h ▸ by simp [bit0, add_mul]\n#align norm_num.bit0_mul NormNum.bit0_mul\n\ntheorem mul_bit0' {α} [Semiring α] (a b c : α) (h : a * b = c) : a * bit0 b = bit0 c :=\n  h ▸ by simp [bit0, mul_add]\n#align norm_num.mul_bit0' NormNum.mul_bit0'\n\ntheorem mul_bit0_bit0 {α} [Semiring α] (a b c : α) (h : a * b = c) :\n    bit0 a * bit0 b = bit0 (bit0 c) :=\n  bit0_mul _ _ _ (mul_bit0' _ _ _ h)\n#align norm_num.mul_bit0_bit0 NormNum.mul_bit0_bit0\n\ntheorem mul_bit1_bit1 {α} [Semiring α] (a b c d e : α) (hc : a * b = c) (hd : a + b = d)\n    (he : bit0 c + d = e) : bit1 a * bit1 b = bit1 e := by\n  rw [← he, ← hd, ← hc] <;> simp [bit1, bit0, mul_add, add_mul, add_left_comm, add_assoc]\n#align norm_num.mul_bit1_bit1 NormNum.mul_bit1_bit1\n\nsection\n\nopen MatchNumeralResult\n\n/-- Given `a`,`b` natural numerals, returns `(r, ⊢ a * b = r)`. -/\nunsafe def prove_mul_nat : instance_cache → expr → expr → tactic (instance_cache × expr × expr)\n  | ic, a, b =>\n    match match_numeral a, match_numeral b with\n    | zero, _ => do\n      let (ic, z) ← ic.mk_app `` Zero.zero []\n      let (ic, p) ← ic.mk_app `` MulZeroClass.zero_mul [b]\n      return (ic, z, p)\n    | _, zero => do\n      let (ic, z) ← ic.mk_app `` Zero.zero []\n      let (ic, p) ← ic.mk_app `` MulZeroClass.mul_zero [a]\n      return (ic, z, p)\n    | one, _ => do\n      let (ic, p) ← ic.mk_app `` one_mul [b]\n      return (ic, b, p)\n    | _, one => do\n      let (ic, p) ← ic.mk_app `` mul_one [a]\n      return (ic, a, p)\n    | bit0 a, bit0 b => do\n      let (ic, c, p) ← prove_mul_nat ic a b\n      let (ic, p) ← ic.mk_app `` mul_bit0_bit0 [a, b, c, p]\n      let (ic, c') ← ic.mk_bit0 c\n      let (ic, c') ← ic.mk_bit0 c'\n      return (ic, c', p)\n    | bit0 a, _ => do\n      let (ic, c, p) ← prove_mul_nat ic a b\n      let (ic, p) ← ic.mk_app `` bit0_mul [a, b, c, p]\n      let (ic, c') ← ic.mk_bit0 c\n      return (ic, c', p)\n    | _, bit0 b => do\n      let (ic, c, p) ← prove_mul_nat ic a b\n      let (ic, p) ← ic.mk_app `` mul_bit0' [a, b, c, p]\n      let (ic, c') ← ic.mk_bit0 c\n      return (ic, c', p)\n    | bit1 a, bit1 b => do\n      let (ic, c, pc) ← prove_mul_nat ic a b\n      let (ic, d, pd) ← prove_add_nat' ic a b\n      let (ic, c') ← ic.mk_bit0 c\n      let (ic, e, pe) ← prove_add_nat' ic c' d\n      let (ic, p) ← ic.mk_app `` mul_bit1_bit1 [a, b, c, d, e, pc, pd, pe]\n      let (ic, e') ← ic.mk_bit1 e\n      return (ic, e', p)\n    | _, _ => failed\n#align norm_num.prove_mul_nat norm_num.prove_mul_nat\n\nend\n\ntheorem zero_lt_one [LinearOrderedSemiring α] : (0 : α) < 1 :=\n  zero_lt_one\n#align norm_num.zero_lt_one NormNum.zero_lt_one\n\nsection\n\nopen MatchNumeralResult\n\n/-- Given `a` a positive natural numeral, returns `⊢ 0 < a`. -/\nunsafe def prove_pos_nat (c : instance_cache) : expr → tactic (instance_cache × expr)\n  | e =>\n    match match_numeral e with\n    | one => c.mk_app `` zero_lt_one []\n    | bit0 e => do\n      let (c, p) ← prove_pos_nat e\n      c `` bit0_pos [e, p]\n    | bit1 e => do\n      let (c, p) ← prove_pos_nat e\n      c `` bit1_pos' [e, p]\n    | _ => failed\n#align norm_num.prove_pos_nat norm_num.prove_pos_nat\n\nend\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- Given `a` a rational numeral, returns `⊢ 0 < a`. -/ unsafe\n  def\n    prove_pos\n    ( c : instance_cache ) : expr → tactic ( instance_cache × expr )\n    |\n        q( $ ( e₁ ) / $ ( e₂ ) )\n        =>\n        do\n          let ( c , p₁ ) ← prove_pos_nat c e₁\n            let ( c , p₂ ) ← prove_pos_nat c e₂\n            c ` ` div_pos [ e₁ , e₂ , p₁ , p₂ ]\n      | e => prove_pos_nat c e\n#align norm_num.prove_pos norm_num.prove_pos\n\n/-- `match_neg (- e) = some e`, otherwise `none` -/\nunsafe def match_neg : expr → Option expr\n  | q(-$(e)) => some e\n  | _ => none\n#align norm_num.match_neg norm_num.match_neg\n\n/-- `match_sign (- e) = inl e`, `match_sign 0 = inr ff`, otherwise `inr tt` -/\nunsafe def match_sign : expr → Sum expr Bool\n  | q(-$(e)) => Sum.inl e\n  | q(Zero.zero) => Sum.inr false\n  | _ => Sum.inr true\n#align norm_num.match_sign norm_num.match_sign\n\ntheorem ne_zero_of_pos {α} [OrderedAddCommGroup α] (a : α) : 0 < a → a ≠ 0 :=\n  ne_of_gt\n#align norm_num.ne_zero_of_pos NormNum.ne_zero_of_pos\n\ntheorem ne_zero_neg {α} [AddGroup α] (a : α) : a ≠ 0 → -a ≠ 0 :=\n  mt neg_eq_zero.1\n#align norm_num.ne_zero_neg NormNum.ne_zero_neg\n\n/-- Given `a` a rational numeral, returns `⊢ a ≠ 0`. -/\nunsafe def prove_ne_zero' (c : instance_cache) : expr → tactic (instance_cache × expr)\n  | a =>\n    match match_neg a with\n    | some a => do\n      let (c, p) ← prove_ne_zero' a\n      c `` ne_zero_neg [a, p]\n    | none => do\n      let (c, p) ← prove_pos c a\n      c `` ne_zero_of_pos [a, p]\n#align norm_num.prove_ne_zero' norm_num.prove_ne_zero'\n\ntheorem clear_denom_div {α} [DivisionRing α] (a b b' c d : α) (h₀ : b ≠ 0) (h₁ : b * b' = d)\n    (h₂ : a * b' = c) : a / b * d = c := by rwa [← h₁, ← mul_assoc, div_mul_cancel _ h₀]\n#align norm_num.clear_denom_div NormNum.clear_denom_div\n\n/-- Given `a` nonnegative rational and `d` a natural number, returns `(b, ⊢ a * d = b)`.\n(`d` should be a multiple of the denominator of `a`, so that `b` is a natural number.) -/\nunsafe def prove_clear_denom'\n    (prove_ne_zero : instance_cache → expr → ℚ → tactic (instance_cache × expr))\n    (c : instance_cache) (a d : expr) (na : ℚ) (nd : ℕ) : tactic (instance_cache × expr × expr) :=\n  if na.den = 1 then prove_mul_nat c a d\n  else do\n    let [_, _, a, b] ← return a.get_app_args\n    let (c, b') ← c.ofNat (nd / na.den)\n    let (c, p₀) ← prove_ne_zero c b na.den\n    let (c, _, p₁) ← prove_mul_nat c b b'\n    let (c, r, p₂) ← prove_mul_nat c a b'\n    let (c, p) ← c.mk_app `` clear_denom_div [a, b, b', r, d, p₀, p₁, p₂]\n    return (c, r, p)\n#align norm_num.prove_clear_denom' norm_num.prove_clear_denom'\n\ntheorem nonneg_pos {α} [OrderedCancelAddCommMonoid α] (a : α) : 0 < a → 0 ≤ a :=\n  le_of_lt\n#align norm_num.nonneg_pos NormNum.nonneg_pos\n\ntheorem lt_one_bit0 {α} [LinearOrderedSemiring α] (a : α) (h : 1 ≤ a) : 1 < bit0 a :=\n  lt_of_lt_of_le one_lt_two (bit0_le_bit0.2 h)\n#align norm_num.lt_one_bit0 NormNum.lt_one_bit0\n\ntheorem lt_one_bit1 {α} [LinearOrderedSemiring α] (a : α) (h : 0 < a) : 1 < bit1 a :=\n  one_lt_bit1.2 h\n#align norm_num.lt_one_bit1 NormNum.lt_one_bit1\n\ntheorem lt_bit0_bit0 {α} [LinearOrderedSemiring α] (a b : α) : a < b → bit0 a < bit0 b :=\n  bit0_lt_bit0.2\n#align norm_num.lt_bit0_bit0 NormNum.lt_bit0_bit0\n\ntheorem lt_bit0_bit1 {α} [LinearOrderedSemiring α] (a b : α) (h : a ≤ b) : bit0 a < bit1 b :=\n  lt_of_le_of_lt (bit0_le_bit0.2 h) (lt_add_one _)\n#align norm_num.lt_bit0_bit1 NormNum.lt_bit0_bit1\n\ntheorem lt_bit1_bit0 {α} [LinearOrderedSemiring α] (a b : α) (h : a + 1 ≤ b) : bit1 a < bit0 b :=\n  lt_of_lt_of_le (by simp [bit0, bit1, zero_lt_one, add_assoc]) (bit0_le_bit0.2 h)\n#align norm_num.lt_bit1_bit0 NormNum.lt_bit1_bit0\n\ntheorem lt_bit1_bit1 {α} [LinearOrderedSemiring α] (a b : α) : a < b → bit1 a < bit1 b :=\n  bit1_lt_bit1.2\n#align norm_num.lt_bit1_bit1 NormNum.lt_bit1_bit1\n\ntheorem le_one_bit0 {α} [LinearOrderedSemiring α] (a : α) (h : 1 ≤ a) : 1 ≤ bit0 a :=\n  le_of_lt (lt_one_bit0 _ h)\n#align norm_num.le_one_bit0 NormNum.le_one_bit0\n\n-- deliberately strong hypothesis because bit1 0 is not a numeral\ntheorem le_one_bit1 {α} [LinearOrderedSemiring α] (a : α) (h : 0 < a) : 1 ≤ bit1 a :=\n  le_of_lt (lt_one_bit1 _ h)\n#align norm_num.le_one_bit1 NormNum.le_one_bit1\n\ntheorem le_bit0_bit0 {α} [LinearOrderedSemiring α] (a b : α) : a ≤ b → bit0 a ≤ bit0 b :=\n  bit0_le_bit0.2\n#align norm_num.le_bit0_bit0 NormNum.le_bit0_bit0\n\ntheorem le_bit0_bit1 {α} [LinearOrderedSemiring α] (a b : α) (h : a ≤ b) : bit0 a ≤ bit1 b :=\n  le_of_lt (lt_bit0_bit1 _ _ h)\n#align norm_num.le_bit0_bit1 NormNum.le_bit0_bit1\n\ntheorem le_bit1_bit0 {α} [LinearOrderedSemiring α] (a b : α) (h : a + 1 ≤ b) : bit1 a ≤ bit0 b :=\n  le_of_lt (lt_bit1_bit0 _ _ h)\n#align norm_num.le_bit1_bit0 NormNum.le_bit1_bit0\n\ntheorem le_bit1_bit1 {α} [LinearOrderedSemiring α] (a b : α) : a ≤ b → bit1 a ≤ bit1 b :=\n  bit1_le_bit1.2\n#align norm_num.le_bit1_bit1 NormNum.le_bit1_bit1\n\ntheorem sle_one_bit0 {α} [LinearOrderedSemiring α] (a : α) : 1 ≤ a → 1 + 1 ≤ bit0 a :=\n  bit0_le_bit0.2\n#align norm_num.sle_one_bit0 NormNum.sle_one_bit0\n\ntheorem sle_one_bit1 {α} [LinearOrderedSemiring α] (a : α) : 1 ≤ a → 1 + 1 ≤ bit1 a :=\n  le_bit0_bit1 _ _\n#align norm_num.sle_one_bit1 NormNum.sle_one_bit1\n\ntheorem sle_bit0_bit0 {α} [LinearOrderedSemiring α] (a b : α) : a + 1 ≤ b → bit0 a + 1 ≤ bit0 b :=\n  le_bit1_bit0 _ _\n#align norm_num.sle_bit0_bit0 NormNum.sle_bit0_bit0\n\ntheorem sle_bit0_bit1 {α} [LinearOrderedSemiring α] (a b : α) (h : a ≤ b) : bit0 a + 1 ≤ bit1 b :=\n  bit1_le_bit1.2 h\n#align norm_num.sle_bit0_bit1 NormNum.sle_bit0_bit1\n\ntheorem sle_bit1_bit0 {α} [LinearOrderedSemiring α] (a b : α) (h : a + 1 ≤ b) :\n    bit1 a + 1 ≤ bit0 b :=\n  (bit1_succ a _ rfl).symm ▸ bit0_le_bit0.2 h\n#align norm_num.sle_bit1_bit0 NormNum.sle_bit1_bit0\n\ntheorem sle_bit1_bit1 {α} [LinearOrderedSemiring α] (a b : α) (h : a + 1 ≤ b) :\n    bit1 a + 1 ≤ bit1 b :=\n  (bit1_succ a _ rfl).symm ▸ le_bit0_bit1 _ _ h\n#align norm_num.sle_bit1_bit1 NormNum.sle_bit1_bit1\n\n/-- Given `a` a rational numeral, returns `⊢ 0 ≤ a`. -/\nunsafe def prove_nonneg (ic : instance_cache) : expr → tactic (instance_cache × expr)\n  | e@q(Zero.zero) => ic.mk_app `` le_refl [e]\n  | e =>\n    if ic.α = q(ℕ) then return (ic, q(Nat.zero_le).mk_app [e])\n    else do\n      let (ic, p) ← prove_pos ic e\n      ic `` nonneg_pos [e, p]\n#align norm_num.prove_nonneg norm_num.prove_nonneg\n\nsection\n\nopen MatchNumeralResult\n\n/-- Given `a` a rational numeral, returns `⊢ 1 ≤ a`. -/\nunsafe def prove_one_le_nat (ic : instance_cache) : expr → tactic (instance_cache × expr)\n  | a =>\n    match match_numeral a with\n    | one => ic.mk_app `` le_refl [a]\n    | bit0 a => do\n      let (ic, p) ← prove_one_le_nat a\n      ic `` le_one_bit0 [a, p]\n    | bit1 a => do\n      let (ic, p) ← prove_pos_nat ic a\n      ic `` le_one_bit1 [a, p]\n    | _ => failed\n#align norm_num.prove_one_le_nat norm_num.prove_one_le_nat\n\nmutual\n  unsafe def prove_le_nat (ic : instance_cache) : expr → expr → tactic (instance_cache × expr)\n    | a, b =>\n      if a = b then ic.mk_app `` le_refl [a]\n      else\n        match match_numeral a, match_numeral b with\n        | zero, _ => prove_nonneg ic b\n        | one, bit0 b => do\n          let (ic, p) ← prove_one_le_nat ic b\n          ic `` le_one_bit0 [b, p]\n        | one, bit1 b => do\n          let (ic, p) ← prove_pos_nat ic b\n          ic `` le_one_bit1 [b, p]\n        | bit0 a, bit0 b => do\n          let (ic, p) ← prove_le_nat a b\n          ic `` le_bit0_bit0 [a, b, p]\n        | bit0 a, bit1 b => do\n          let (ic, p) ← prove_le_nat a b\n          ic `` le_bit0_bit1 [a, b, p]\n        | bit1 a, bit0 b => do\n          let (ic, p) ← prove_sle_nat a b\n          ic `` le_bit1_bit0 [a, b, p]\n        | bit1 a, bit1 b => do\n          let (ic, p) ← prove_le_nat a b\n          ic `` le_bit1_bit1 [a, b, p]\n        | _, _ => failed\n  unsafe def prove_sle_nat (ic : instance_cache) : expr → expr → tactic (instance_cache × expr)\n    | a, b =>\n      match match_numeral a, match_numeral b with\n      | zero, _ => prove_nonneg ic b\n      | one, bit0 b => do\n        let (ic, p) ← prove_one_le_nat ic b\n        ic `` sle_one_bit0 [b, p]\n      | one, bit1 b => do\n        let (ic, p) ← prove_one_le_nat ic b\n        ic `` sle_one_bit1 [b, p]\n      | bit0 a, bit0 b => do\n        let (ic, p) ← prove_sle_nat a b\n        ic `` sle_bit0_bit0 [a, b, p]\n      | bit0 a, bit1 b => do\n        let (ic, p) ← prove_le_nat a b\n        ic `` sle_bit0_bit1 [a, b, p]\n      | bit1 a, bit0 b => do\n        let (ic, p) ← prove_sle_nat a b\n        ic `` sle_bit1_bit0 [a, b, p]\n      | bit1 a, bit1 b => do\n        let (ic, p) ← prove_sle_nat a b\n        ic `` sle_bit1_bit1 [a, b, p]\n      | _, _ => failed\nend\n#align norm_num.prove_le_nat norm_num.prove_le_nat\n#align norm_num.prove_sle_nat norm_num.prove_sle_nat\n\n/-- Given `a`,`b` natural numerals, proves `⊢ a ≤ b`. -/\nadd_decl_doc prove_le_nat\n\n/-- Given `a`,`b` natural numerals, proves `⊢ a + 1 ≤ b`. -/\nadd_decl_doc prove_sle_nat\n\n/-- Given `a`,`b` natural numerals, proves `⊢ a < b`. -/\nunsafe def prove_lt_nat (ic : instance_cache) : expr → expr → tactic (instance_cache × expr)\n  | a, b =>\n    match match_numeral a, match_numeral b with\n    | zero, _ => prove_pos ic b\n    | one, bit0 b => do\n      let (ic, p) ← prove_one_le_nat ic b\n      ic `` lt_one_bit0 [b, p]\n    | one, bit1 b => do\n      let (ic, p) ← prove_pos_nat ic b\n      ic `` lt_one_bit1 [b, p]\n    | bit0 a, bit0 b => do\n      let (ic, p) ← prove_lt_nat a b\n      ic `` lt_bit0_bit0 [a, b, p]\n    | bit0 a, bit1 b => do\n      let (ic, p) ← prove_le_nat ic a b\n      ic `` lt_bit0_bit1 [a, b, p]\n    | bit1 a, bit0 b => do\n      let (ic, p) ← prove_sle_nat ic a b\n      ic `` lt_bit1_bit0 [a, b, p]\n    | bit1 a, bit1 b => do\n      let (ic, p) ← prove_lt_nat a b\n      ic `` lt_bit1_bit1 [a, b, p]\n    | _, _ => failed\n#align norm_num.prove_lt_nat norm_num.prove_lt_nat\n\nend\n\ntheorem clear_denom_lt {α} [LinearOrderedSemiring α] (a a' b b' d : α) (h₀ : 0 < d)\n    (ha : a * d = a') (hb : b * d = b') (h : a' < b') : a < b :=\n  lt_of_mul_lt_mul_right (by rwa [ha, hb]) (le_of_lt h₀)\n#align norm_num.clear_denom_lt NormNum.clear_denom_lt\n\n/-- Given `a`,`b` nonnegative rational numerals, proves `⊢ a < b`. -/\nunsafe def prove_lt_nonneg_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr) :=\n  if na.den = 1 ∧ nb.den = 1 then prove_lt_nat ic a b\n  else do\n    let nd := na.den.lcm nb.den\n    let (ic, d) ← ic.ofNat nd\n    let (ic, p₀) ← prove_pos ic d\n    let (ic, a', pa) ← prove_clear_denom' (fun ic e _ => prove_ne_zero' ic e) ic a d na nd\n    let (ic, b', pb) ← prove_clear_denom' (fun ic e _ => prove_ne_zero' ic e) ic b d nb nd\n    let (ic, p) ← prove_lt_nat ic a' b'\n    ic `` clear_denom_lt [a, a', b, b', d, p₀, pa, pb, p]\n#align norm_num.prove_lt_nonneg_rat norm_num.prove_lt_nonneg_rat\n\ntheorem lt_neg_pos {α} [OrderedAddCommGroup α] (a b : α) (ha : 0 < a) (hb : 0 < b) : -a < b :=\n  lt_trans (neg_neg_of_pos ha) hb\n#align norm_num.lt_neg_pos NormNum.lt_neg_pos\n\n/-- Given `a`,`b` rational numerals, proves `⊢ a < b`. -/\nunsafe def prove_lt_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr) :=\n  match match_sign a, match_sign b with\n  | Sum.inl a, Sum.inl b => do\n    let-- we have to switch the order of `a` and `b` because `a < b ↔ -b < -a`\n      (ic, p)\n      ← prove_lt_nonneg_rat ic b a (-nb) (-na)\n    ic `` neg_lt_neg [b, a, p]\n  | Sum.inl a, Sum.inr ff => do\n    let (ic, p) ← prove_pos ic a\n    ic `` neg_neg_of_pos [a, p]\n  | Sum.inl a, Sum.inr tt => do\n    let (ic, pa) ← prove_pos ic a\n    let (ic, pb) ← prove_pos ic b\n    ic `` lt_neg_pos [a, b, pa, pb]\n  | Sum.inr ff, _ => prove_pos ic b\n  | Sum.inr tt, _ => prove_lt_nonneg_rat ic a b na nb\n#align norm_num.prove_lt_rat norm_num.prove_lt_rat\n\ntheorem clear_denom_le {α} [LinearOrderedSemiring α] (a a' b b' d : α) (h₀ : 0 < d)\n    (ha : a * d = a') (hb : b * d = b') (h : a' ≤ b') : a ≤ b :=\n  le_of_mul_le_mul_right (by rwa [ha, hb]) h₀\n#align norm_num.clear_denom_le NormNum.clear_denom_le\n\n/-- Given `a`,`b` nonnegative rational numerals, proves `⊢ a ≤ b`. -/\nunsafe def prove_le_nonneg_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr) :=\n  if na.den = 1 ∧ nb.den = 1 then prove_le_nat ic a b\n  else do\n    let nd := na.den.lcm nb.den\n    let (ic, d) ← ic.ofNat nd\n    let (ic, p₀) ← prove_pos ic d\n    let (ic, a', pa) ← prove_clear_denom' (fun ic e _ => prove_ne_zero' ic e) ic a d na nd\n    let (ic, b', pb) ← prove_clear_denom' (fun ic e _ => prove_ne_zero' ic e) ic b d nb nd\n    let (ic, p) ← prove_le_nat ic a' b'\n    ic `` clear_denom_le [a, a', b, b', d, p₀, pa, pb, p]\n#align norm_num.prove_le_nonneg_rat norm_num.prove_le_nonneg_rat\n\ntheorem le_neg_pos {α} [OrderedAddCommGroup α] (a b : α) (ha : 0 ≤ a) (hb : 0 ≤ b) : -a ≤ b :=\n  le_trans (neg_nonpos_of_nonneg ha) hb\n#align norm_num.le_neg_pos NormNum.le_neg_pos\n\n/-- Given `a`,`b` rational numerals, proves `⊢ a ≤ b`. -/\nunsafe def prove_le_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr) :=\n  match match_sign a, match_sign b with\n  | Sum.inl a, Sum.inl b => do\n    let (ic, p) ← prove_le_nonneg_rat ic a b (-na) (-nb)\n    ic `` neg_le_neg [a, b, p]\n  | Sum.inl a, Sum.inr ff => do\n    let (ic, p) ← prove_nonneg ic a\n    ic `` neg_nonpos_of_nonneg [a, p]\n  | Sum.inl a, Sum.inr tt => do\n    let (ic, pa) ← prove_nonneg ic a\n    let (ic, pb) ← prove_nonneg ic b\n    ic `` le_neg_pos [a, b, pa, pb]\n  | Sum.inr ff, _ => prove_nonneg ic b\n  | Sum.inr tt, _ => prove_le_nonneg_rat ic a b na nb\n#align norm_num.prove_le_rat norm_num.prove_le_rat\n\n/-- Given `a`,`b` rational numerals, proves `⊢ a ≠ b`. This version tries to prove\n`⊢ a < b` or `⊢ b < a`, and so is not appropriate for types without an order relation. -/\nunsafe def prove_ne_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr) :=\n  if na < nb then do\n    let (ic, p) ← prove_lt_rat ic a b na nb\n    ic `` ne_of_lt [a, b, p]\n  else do\n    let (ic, p) ← prove_lt_rat ic b a nb na\n    ic `` ne_of_gt [a, b, p]\n#align norm_num.prove_ne_rat norm_num.prove_ne_rat\n\ntheorem nat_cast_zero {α} [Semiring α] : ↑(0 : ℕ) = (0 : α) :=\n  Nat.cast_zero\n#align norm_num.nat_cast_zero NormNum.nat_cast_zero\n\ntheorem nat_cast_one {α} [Semiring α] : ↑(1 : ℕ) = (1 : α) :=\n  Nat.cast_one\n#align norm_num.nat_cast_one NormNum.nat_cast_one\n\ntheorem nat_cast_bit0 {α} [Semiring α] (a : ℕ) (a' : α) (h : ↑a = a') : ↑(bit0 a) = bit0 a' :=\n  h ▸ Nat.cast_bit0 _\n#align norm_num.nat_cast_bit0 NormNum.nat_cast_bit0\n\ntheorem nat_cast_bit1 {α} [Semiring α] (a : ℕ) (a' : α) (h : ↑a = a') : ↑(bit1 a) = bit1 a' :=\n  h ▸ Nat.cast_bit1 _\n#align norm_num.nat_cast_bit1 NormNum.nat_cast_bit1\n\ntheorem int_cast_zero {α} [Ring α] : ↑(0 : ℤ) = (0 : α) :=\n  Int.cast_zero\n#align norm_num.int_cast_zero NormNum.int_cast_zero\n\ntheorem int_cast_one {α} [Ring α] : ↑(1 : ℤ) = (1 : α) :=\n  Int.cast_one\n#align norm_num.int_cast_one NormNum.int_cast_one\n\ntheorem int_cast_bit0 {α} [Ring α] (a : ℤ) (a' : α) (h : ↑a = a') : ↑(bit0 a) = bit0 a' :=\n  h ▸ Int.cast_bit0 _\n#align norm_num.int_cast_bit0 NormNum.int_cast_bit0\n\ntheorem int_cast_bit1 {α} [Ring α] (a : ℤ) (a' : α) (h : ↑a = a') : ↑(bit1 a) = bit1 a' :=\n  h ▸ Int.cast_bit1 _\n#align norm_num.int_cast_bit1 NormNum.int_cast_bit1\n\ntheorem rat_cast_bit0 {α} [DivisionRing α] [CharZero α] (a : ℚ) (a' : α) (h : ↑a = a') :\n    ↑(bit0 a) = bit0 a' :=\n  h ▸ Rat.cast_bit0 _\n#align norm_num.rat_cast_bit0 NormNum.rat_cast_bit0\n\ntheorem rat_cast_bit1 {α} [DivisionRing α] [CharZero α] (a : ℚ) (a' : α) (h : ↑a = a') :\n    ↑(bit1 a) = bit1 a' :=\n  h ▸ Rat.cast_bit1 _\n#align norm_num.rat_cast_bit1 NormNum.rat_cast_bit1\n\n/-- Given `a' : α` a natural numeral, returns `(a : ℕ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_nat_uncast (ic nc : instance_cache) :\n    ∀ a' : expr, tactic (instance_cache × instance_cache × expr × expr)\n  | a' =>\n    match match_numeral a' with\n    | match_numeral_result.zero => do\n      let (nc, e) ← nc.mk_app `` Zero.zero []\n      let (ic, p) ← ic.mk_app `` nat_cast_zero []\n      return (ic, nc, e, p)\n    | match_numeral_result.one => do\n      let (nc, e) ← nc.mk_app `` One.one []\n      let (ic, p) ← ic.mk_app `` nat_cast_one []\n      return (ic, nc, e, p)\n    | match_numeral_result.bit0 a' => do\n      let (ic, nc, a, p) ← prove_nat_uncast a'\n      let (nc, a0) ← nc.mk_bit0 a\n      let (ic, p) ← ic.mk_app `` nat_cast_bit0 [a, a', p]\n      return (ic, nc, a0, p)\n    | match_numeral_result.bit1 a' => do\n      let (ic, nc, a, p) ← prove_nat_uncast a'\n      let (nc, a1) ← nc.mk_bit1 a\n      let (ic, p) ← ic.mk_app `` nat_cast_bit1 [a, a', p]\n      return (ic, nc, a1, p)\n    | _ => failed\n#align norm_num.prove_nat_uncast norm_num.prove_nat_uncast\n\n/-- Given `a' : α` a natural numeral, returns `(a : ℤ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_int_uncast_nat (ic zc : instance_cache) :\n    ∀ a' : expr, tactic (instance_cache × instance_cache × expr × expr)\n  | a' =>\n    match match_numeral a' with\n    | match_numeral_result.zero => do\n      let (zc, e) ← zc.mk_app `` Zero.zero []\n      let (ic, p) ← ic.mk_app `` int_cast_zero []\n      return (ic, zc, e, p)\n    | match_numeral_result.one => do\n      let (zc, e) ← zc.mk_app `` One.one []\n      let (ic, p) ← ic.mk_app `` int_cast_one []\n      return (ic, zc, e, p)\n    | match_numeral_result.bit0 a' => do\n      let (ic, zc, a, p) ← prove_int_uncast_nat a'\n      let (zc, a0) ← zc.mk_bit0 a\n      let (ic, p) ← ic.mk_app `` int_cast_bit0 [a, a', p]\n      return (ic, zc, a0, p)\n    | match_numeral_result.bit1 a' => do\n      let (ic, zc, a, p) ← prove_int_uncast_nat a'\n      let (zc, a1) ← zc.mk_bit1 a\n      let (ic, p) ← ic.mk_app `` int_cast_bit1 [a, a', p]\n      return (ic, zc, a1, p)\n    | _ => failed\n#align norm_num.prove_int_uncast_nat norm_num.prove_int_uncast_nat\n\n/-- Given `a' : α` a natural numeral, returns `(a : ℚ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_rat_uncast_nat (ic qc : instance_cache) (cz_inst : expr) :\n    ∀ a' : expr, tactic (instance_cache × instance_cache × expr × expr)\n  | a' =>\n    match match_numeral a' with\n    | match_numeral_result.zero => do\n      let (qc, e) ← qc.mk_app `` Zero.zero []\n      let (ic, p) ← ic.mk_app `` Rat.cast_zero []\n      return (ic, qc, e, p)\n    | match_numeral_result.one => do\n      let (qc, e) ← qc.mk_app `` One.one []\n      let (ic, p) ← ic.mk_app `` Rat.cast_one []\n      return (ic, qc, e, p)\n    | match_numeral_result.bit0 a' => do\n      let (ic, qc, a, p) ← prove_rat_uncast_nat a'\n      let (qc, a0) ← qc.mk_bit0 a\n      let (ic, p) ← ic.mk_app `` rat_cast_bit0 [cz_inst, a, a', p]\n      return (ic, qc, a0, p)\n    | match_numeral_result.bit1 a' => do\n      let (ic, qc, a, p) ← prove_rat_uncast_nat a'\n      let (qc, a1) ← qc.mk_bit1 a\n      let (ic, p) ← ic.mk_app `` rat_cast_bit1 [cz_inst, a, a', p]\n      return (ic, qc, a1, p)\n    | _ => failed\n#align norm_num.prove_rat_uncast_nat norm_num.prove_rat_uncast_nat\n\ntheorem rat_cast_div {α} [DivisionRing α] [CharZero α] (a b : ℚ) (a' b' : α) (ha : ↑a = a')\n    (hb : ↑b = b') : ↑(a / b) = a' / b' :=\n  ha ▸ hb ▸ Rat.cast_div _ _\n#align norm_num.rat_cast_div NormNum.rat_cast_div\n\n/-- Given `a' : α` a nonnegative rational numeral, returns `(a : ℚ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_rat_uncast_nonneg (ic qc : instance_cache) (cz_inst a' : expr) (na' : ℚ) :\n    tactic (instance_cache × instance_cache × expr × expr) :=\n  if na'.den = 1 then prove_rat_uncast_nat ic qc cz_inst a'\n  else do\n    let [_, _, a', b'] ← return a'.get_app_args\n    let (ic, qc, a, pa) ← prove_rat_uncast_nat ic qc cz_inst a'\n    let (ic, qc, b, pb) ← prove_rat_uncast_nat ic qc cz_inst b'\n    let (qc, e) ← qc.mk_app `` Div.div [a, b]\n    let (ic, p) ← ic.mk_app `` rat_cast_div [cz_inst, a, b, a', b', pa, pb]\n    return (ic, qc, e, p)\n#align norm_num.prove_rat_uncast_nonneg norm_num.prove_rat_uncast_nonneg\n\ntheorem int_cast_neg {α} [Ring α] (a : ℤ) (a' : α) (h : ↑a = a') : ↑(-a) = -a' :=\n  h ▸ Int.cast_neg _\n#align norm_num.int_cast_neg NormNum.int_cast_neg\n\ntheorem rat_cast_neg {α} [DivisionRing α] (a : ℚ) (a' : α) (h : ↑a = a') : ↑(-a) = -a' :=\n  h ▸ Rat.cast_neg _\n#align norm_num.rat_cast_neg NormNum.rat_cast_neg\n\n/-- Given `a' : α` an integer numeral, returns `(a : ℤ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_int_uncast (ic zc : instance_cache) (a' : expr) :\n    tactic (instance_cache × instance_cache × expr × expr) :=\n  match match_neg a' with\n  | some a' => do\n    let (ic, zc, a, p) ← prove_int_uncast_nat ic zc a'\n    let (zc, e) ← zc.mk_app `` Neg.neg [a]\n    let (ic, p) ← ic.mk_app `` int_cast_neg [a, a', p]\n    return (ic, zc, e, p)\n  | none => prove_int_uncast_nat ic zc a'\n#align norm_num.prove_int_uncast norm_num.prove_int_uncast\n\n/-- Given `a' : α` a rational numeral, returns `(a : ℚ, ⊢ ↑a = a')`.\n(Note that the returned value is on the left of the equality.) -/\nunsafe def prove_rat_uncast (ic qc : instance_cache) (cz_inst a' : expr) (na' : ℚ) :\n    tactic (instance_cache × instance_cache × expr × expr) :=\n  match match_neg a' with\n  | some a' => do\n    let (ic, qc, a, p) ← prove_rat_uncast_nonneg ic qc cz_inst a' (-na')\n    let (qc, e) ← qc.mk_app `` Neg.neg [a]\n    let (ic, p) ← ic.mk_app `` rat_cast_neg [a, a', p]\n    return (ic, qc, e, p)\n  | none => prove_rat_uncast_nonneg ic qc cz_inst a' na'\n#align norm_num.prove_rat_uncast norm_num.prove_rat_uncast\n\ntheorem nat_cast_ne {α} [Semiring α] [CharZero α] (a b : ℕ) (a' b' : α) (ha : ↑a = a')\n    (hb : ↑b = b') (h : a ≠ b) : a' ≠ b' :=\n  ha ▸ hb ▸ mt Nat.cast_inj.1 h\n#align norm_num.nat_cast_ne NormNum.nat_cast_ne\n\ntheorem int_cast_ne {α} [Ring α] [CharZero α] (a b : ℤ) (a' b' : α) (ha : ↑a = a') (hb : ↑b = b')\n    (h : a ≠ b) : a' ≠ b' :=\n  ha ▸ hb ▸ mt Int.cast_inj.1 h\n#align norm_num.int_cast_ne NormNum.int_cast_ne\n\ntheorem rat_cast_ne {α} [DivisionRing α] [CharZero α] (a b : ℚ) (a' b' : α) (ha : ↑a = a')\n    (hb : ↑b = b') (h : a ≠ b) : a' ≠ b' :=\n  ha ▸ hb ▸ mt Rat.cast_inj.1 h\n#align norm_num.rat_cast_ne NormNum.rat_cast_ne\n\n/-- Given `a`,`b` rational numerals, proves `⊢ a ≠ b`. Currently it tries two methods:\n\n  * Prove `⊢ a < b` or `⊢ b < a`, if the base type has an order\n  * Embed `↑(a':ℚ) = a` and `↑(b':ℚ) = b`, and then prove `a' ≠ b'`.\n    This requires that the base type be `char_zero`, and also that it be a `division_ring`\n    so that the coercion from `ℚ` is well defined.\n\nWe may also add coercions to `ℤ` and `ℕ` as well in order to support `char_zero`\nrings and semirings. -/\nunsafe def prove_ne : instance_cache → expr → expr → ℚ → ℚ → tactic (instance_cache × expr)\n  | ic, a, b, na, nb =>\n    prove_ne_rat ic a b na nb <|> do\n      let cz_inst ← mk_mapp `` CharZero [ic.α, none] >>= mk_instance\n      if na = 1 ∧ nb = 1 then\n          if na ≥ 0 ∧ nb ≥ 0 then do\n            guard (ic ≠ q(ℕ))\n            let nc ← mk_instance_cache q(ℕ)\n            let (ic, nc, a', pa) ← prove_nat_uncast ic nc a\n            let (ic, nc, b', pb) ← prove_nat_uncast ic nc b\n            let (nc, p) ← prove_ne_rat nc a' b' na nb\n            ic `` nat_cast_ne [cz_inst, a', b', a, b, pa, pb, p]\n          else do\n            guard (ic ≠ q(ℤ))\n            let zc ← mk_instance_cache q(ℤ)\n            let (ic, zc, a', pa) ← prove_int_uncast ic zc a\n            let (ic, zc, b', pb) ← prove_int_uncast ic zc b\n            let (zc, p) ← prove_ne_rat zc a' b' na nb\n            ic `` int_cast_ne [cz_inst, a', b', a, b, pa, pb, p]\n        else do\n          guard (ic ≠ q(ℚ))\n          let qc ← mk_instance_cache q(ℚ)\n          let (ic, qc, a', pa) ← prove_rat_uncast ic qc cz_inst a na\n          let (ic, qc, b', pb) ← prove_rat_uncast ic qc cz_inst b nb\n          let (qc, p) ← prove_ne_rat qc a' b' na nb\n          ic `` rat_cast_ne [cz_inst, a', b', a, b, pa, pb, p]\n#align norm_num.prove_ne norm_num.prove_ne\n\n/-- Given `a` a rational numeral, returns `⊢ a ≠ 0`. -/\nunsafe def prove_ne_zero (ic : instance_cache) : expr → ℚ → tactic (instance_cache × expr)\n  | a, na => do\n    let (ic, z) ← ic.mk_app `` Zero.zero []\n    prove_ne ic a z na 0\n#align norm_num.prove_ne_zero norm_num.prove_ne_zero\n\n/-- Given `a` nonnegative rational and `d` a natural number, returns `(b, ⊢ a * d = b)`.\n(`d` should be a multiple of the denominator of `a`, so that `b` is a natural number.) -/\nunsafe def prove_clear_denom :\n    instance_cache → expr → expr → ℚ → ℕ → tactic (instance_cache × expr × expr) :=\n  prove_clear_denom' prove_ne_zero\n#align norm_num.prove_clear_denom norm_num.prove_clear_denom\n\ntheorem clear_denom_add {α} [DivisionRing α] (a a' b b' c c' d : α) (h₀ : d ≠ 0) (ha : a * d = a')\n    (hb : b * d = b') (hc : c * d = c') (h : a' + b' = c') : a + b = c :=\n  mul_right_cancel₀ h₀ <| by rwa [add_mul, ha, hb, hc]\n#align norm_num.clear_denom_add NormNum.clear_denom_add\n\n/-- Given `a`,`b`,`c` nonnegative rational numerals, returns `⊢ a + b = c`. -/\nunsafe def prove_add_nonneg_rat (ic : instance_cache) (a b c : expr) (na nb nc : ℚ) :\n    tactic (instance_cache × expr) :=\n  if na.den = 1 ∧ nb.den = 1 then prove_add_nat ic a b c\n  else do\n    let nd := na.den.lcm nb.den\n    let (ic, d) ← ic.ofNat nd\n    let (ic, p₀) ← prove_ne_zero ic d nd\n    let (ic, a', pa) ← prove_clear_denom ic a d na nd\n    let (ic, b', pb) ← prove_clear_denom ic b d nb nd\n    let (ic, c', pc) ← prove_clear_denom ic c d nc nd\n    let (ic, p) ← prove_add_nat ic a' b' c'\n    ic `` clear_denom_add [a, a', b, b', c, c', d, p₀, pa, pb, pc, p]\n#align norm_num.prove_add_nonneg_rat norm_num.prove_add_nonneg_rat\n\ntheorem add_pos_neg_pos {α} [AddGroup α] (a b c : α) (h : c + b = a) : a + -b = c :=\n  h ▸ by simp\n#align norm_num.add_pos_neg_pos NormNum.add_pos_neg_pos\n\ntheorem add_pos_neg_neg {α} [AddGroup α] (a b c : α) (h : c + a = b) : a + -b = -c :=\n  h ▸ by simp\n#align norm_num.add_pos_neg_neg NormNum.add_pos_neg_neg\n\ntheorem add_neg_pos_pos {α} [AddGroup α] (a b c : α) (h : a + c = b) : -a + b = c :=\n  h ▸ by simp\n#align norm_num.add_neg_pos_pos NormNum.add_neg_pos_pos\n\ntheorem add_neg_pos_neg {α} [AddGroup α] (a b c : α) (h : b + c = a) : -a + b = -c :=\n  h ▸ by simp\n#align norm_num.add_neg_pos_neg NormNum.add_neg_pos_neg\n\ntheorem add_neg_neg {α} [AddGroup α] (a b c : α) (h : b + a = c) : -a + -b = -c :=\n  h ▸ by simp\n#align norm_num.add_neg_neg NormNum.add_neg_neg\n\n/-- Given `a`,`b`,`c` rational numerals, returns `⊢ a + b = c`. -/\nunsafe def prove_add_rat (ic : instance_cache) (ea eb ec : expr) (a b c : ℚ) :\n    tactic (instance_cache × expr) :=\n  match match_neg ea, match_neg eb, match_neg ec with\n  | some ea, some eb, some ec => do\n    let (ic, p) ← prove_add_nonneg_rat ic eb ea ec (-b) (-a) (-c)\n    ic `` add_neg_neg [ea, eb, ec, p]\n  | some ea, none, some ec => do\n    let (ic, p) ← prove_add_nonneg_rat ic eb ec ea b (-c) (-a)\n    ic `` add_neg_pos_neg [ea, eb, ec, p]\n  | some ea, none, none => do\n    let (ic, p) ← prove_add_nonneg_rat ic ea ec eb (-a) c b\n    ic `` add_neg_pos_pos [ea, eb, ec, p]\n  | none, some eb, some ec => do\n    let (ic, p) ← prove_add_nonneg_rat ic ec ea eb (-c) a (-b)\n    ic `` add_pos_neg_neg [ea, eb, ec, p]\n  | none, some eb, none => do\n    let (ic, p) ← prove_add_nonneg_rat ic ec eb ea c (-b) a\n    ic `` add_pos_neg_pos [ea, eb, ec, p]\n  | _, _, _ => prove_add_nonneg_rat ic ea eb ec a b c\n#align norm_num.prove_add_rat norm_num.prove_add_rat\n\n/-- Given `a`,`b` rational numerals, returns `(c, ⊢ a + b = c)`. -/\nunsafe def prove_add_rat' (ic : instance_cache) (a b : expr) :\n    tactic (instance_cache × expr × expr) := do\n  let na ← a.to_rat\n  let nb ← b.to_rat\n  let nc := na + nb\n  let (ic, c) ← ic.of_rat nc\n  let (ic, p) ← prove_add_rat ic a b c na nb nc\n  return (ic, c, p)\n#align norm_num.prove_add_rat' norm_num.prove_add_rat'\n\ntheorem clear_denom_simple_nat {α} [DivisionRing α] (a : α) : (1 : α) ≠ 0 ∧ a * 1 = a :=\n  ⟨one_ne_zero, mul_one _⟩\n#align norm_num.clear_denom_simple_nat NormNum.clear_denom_simple_nat\n\ntheorem clear_denom_simple_div {α} [DivisionRing α] (a b : α) (h : b ≠ 0) : b ≠ 0 ∧ a / b * b = a :=\n  ⟨h, div_mul_cancel _ h⟩\n#align norm_num.clear_denom_simple_div NormNum.clear_denom_simple_div\n\n/-- Given `a` a nonnegative rational numeral, returns `(b, c, ⊢ a * b = c)`\nwhere `b` and `c` are natural numerals. (`b` will be the denominator of `a`.) -/\nunsafe def prove_clear_denom_simple (c : instance_cache) (a : expr) (na : ℚ) :\n    tactic (instance_cache × expr × expr × expr) :=\n  if na.den = 1 then do\n    let (c, d) ← c.mk_app `` One.one []\n    let (c, p) ← c.mk_app `` clear_denom_simple_nat [a]\n    return (c, d, a, p)\n  else do\n    let [α, _, a, b] ← return a.get_app_args\n    let (c, p₀) ← prove_ne_zero c b na.den\n    let (c, p) ← c.mk_app `` clear_denom_simple_div [a, b, p₀]\n    return (c, b, a, p)\n#align norm_num.prove_clear_denom_simple norm_num.prove_clear_denom_simple\n\ntheorem clear_denom_mul {α} [Field α] (a a' b b' c c' d₁ d₂ d : α) (ha : d₁ ≠ 0 ∧ a * d₁ = a')\n    (hb : d₂ ≠ 0 ∧ b * d₂ = b') (hc : c * d = c') (hd : d₁ * d₂ = d) (h : a' * b' = c') :\n    a * b = c :=\n  mul_right_cancel₀ ha.1 <|\n    mul_right_cancel₀ hb.1 <| by\n      rw [mul_assoc c, hd, hc, ← h, ← ha.2, ← hb.2, ← mul_assoc, mul_right_comm a]\n#align norm_num.clear_denom_mul NormNum.clear_denom_mul\n\n/-- Given `a`,`b` nonnegative rational numerals, returns `(c, ⊢ a * b = c)`. -/\nunsafe def prove_mul_nonneg_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr × expr) :=\n  if na.den = 1 ∧ nb.den = 1 then prove_mul_nat ic a b\n  else do\n    let nc := na * nb\n    let (ic, c) ← ic.of_rat nc\n    let (ic, d₁, a', pa) ← prove_clear_denom_simple ic a na\n    let (ic, d₂, b', pb) ← prove_clear_denom_simple ic b nb\n    let (ic, d, pd) ← prove_mul_nat ic d₁ d₂\n    let nd ← d.toNat\n    let (ic, c', pc) ← prove_clear_denom ic c d nc nd\n    let (ic, _, p) ← prove_mul_nat ic a' b'\n    let (ic, p) ← ic.mk_app `` clear_denom_mul [a, a', b, b', c, c', d₁, d₂, d, pa, pb, pc, pd, p]\n    return (ic, c, p)\n#align norm_num.prove_mul_nonneg_rat norm_num.prove_mul_nonneg_rat\n\ntheorem mul_neg_pos {α} [Ring α] (a b c : α) (h : a * b = c) : -a * b = -c :=\n  h ▸ by simp\n#align norm_num.mul_neg_pos NormNum.mul_neg_pos\n\ntheorem mul_pos_neg {α} [Ring α] (a b c : α) (h : a * b = c) : a * -b = -c :=\n  h ▸ by simp\n#align norm_num.mul_pos_neg NormNum.mul_pos_neg\n\ntheorem mul_neg_neg {α} [Ring α] (a b c : α) (h : a * b = c) : -a * -b = c :=\n  h ▸ by simp\n#align norm_num.mul_neg_neg NormNum.mul_neg_neg\n\n/-- Given `a`,`b` rational numerals, returns `(c, ⊢ a * b = c)`. -/\nunsafe def prove_mul_rat (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr × expr) :=\n  match match_sign a, match_sign b with\n  | Sum.inl a, Sum.inl b => do\n    let (ic, c, p) ← prove_mul_nonneg_rat ic a b (-na) (-nb)\n    let (ic, p) ← ic.mk_app `` mul_neg_neg [a, b, c, p]\n    return (ic, c, p)\n  | Sum.inr ff, _ => do\n    let (ic, z) ← ic.mk_app `` Zero.zero []\n    let (ic, p) ← ic.mk_app `` MulZeroClass.zero_mul [b]\n    return (ic, z, p)\n  | _, Sum.inr ff => do\n    let (ic, z) ← ic.mk_app `` Zero.zero []\n    let (ic, p) ← ic.mk_app `` MulZeroClass.mul_zero [a]\n    return (ic, z, p)\n  | Sum.inl a, Sum.inr tt => do\n    let (ic, c, p) ← prove_mul_nonneg_rat ic a b (-na) nb\n    let (ic, p) ← ic.mk_app `` mul_neg_pos [a, b, c, p]\n    let (ic, c') ← ic.mk_app `` Neg.neg [c]\n    return (ic, c', p)\n  | Sum.inr tt, Sum.inl b => do\n    let (ic, c, p) ← prove_mul_nonneg_rat ic a b na (-nb)\n    let (ic, p) ← ic.mk_app `` mul_pos_neg [a, b, c, p]\n    let (ic, c') ← ic.mk_app `` Neg.neg [c]\n    return (ic, c', p)\n  | Sum.inr tt, Sum.inr tt => prove_mul_nonneg_rat ic a b na nb\n#align norm_num.prove_mul_rat norm_num.prove_mul_rat\n\ntheorem inv_neg {α} [DivisionRing α] (a b : α) (h : a⁻¹ = b) : (-a)⁻¹ = -b :=\n  h ▸ by simp only [inv_eq_one_div, one_div_neg_eq_neg_one_div]\n#align norm_num.inv_neg NormNum.inv_neg\n\ntheorem inv_one {α} [DivisionRing α] : (1 : α)⁻¹ = 1 :=\n  inv_one\n#align norm_num.inv_one NormNum.inv_one\n\ntheorem inv_one_div {α} [DivisionRing α] (a : α) : (1 / a)⁻¹ = a := by rw [one_div, inv_inv]\n#align norm_num.inv_one_div NormNum.inv_one_div\n\ntheorem inv_div_one {α} [DivisionRing α] (a : α) : a⁻¹ = 1 / a :=\n  inv_eq_one_div _\n#align norm_num.inv_div_one NormNum.inv_div_one\n\ntheorem inv_div {α} [DivisionRing α] (a b : α) : (a / b)⁻¹ = b / a := by\n  simp only [inv_eq_one_div, one_div_div]\n#align norm_num.inv_div NormNum.inv_div\n\n/-- Given `a` a rational numeral, returns `(b, ⊢ a⁻¹ = b)`. -/\nunsafe def prove_inv : instance_cache → expr → ℚ → tactic (instance_cache × expr × expr)\n  | ic, e, n =>\n    match match_sign e with\n    | Sum.inl e => do\n      let (ic, e', p) ← prove_inv ic e (-n)\n      let (ic, r) ← ic.mk_app `` Neg.neg [e']\n      let (ic, p) ← ic.mk_app `` inv_neg [e, e', p]\n      return (ic, r, p)\n    | Sum.inr ff => do\n      let (ic, p) ← ic.mk_app `` inv_zero []\n      return (ic, e, p)\n    | Sum.inr tt =>\n      if n.num = 1 then\n        if n.den = 1 then do\n          let (ic, p) ← ic.mk_app `` inv_one []\n          return (ic, e, p)\n        else do\n          let e := e.app_arg\n          let (ic, p) ← ic.mk_app `` inv_one_div [e]\n          return (ic, e, p)\n      else\n        if n.den = 1 then do\n          let (ic, p) ← ic.mk_app `` inv_div_one [e]\n          let e ← infer_type p\n          return (ic, e, p)\n        else do\n          let [_, _, a, b] ← return e.get_app_args\n          let (ic, e') ← ic.mk_app `` Div.div [b, a]\n          let (ic, p) ← ic.mk_app `` inv_div [a, b]\n          return (ic, e', p)\n#align norm_num.prove_inv norm_num.prove_inv\n\ntheorem div_eq {α} [DivisionRing α] (a b b' c : α) (hb : b⁻¹ = b') (h : a * b' = c) : a / b = c :=\n  by rwa [← hb, ← div_eq_mul_inv] at h\n#align norm_num.div_eq NormNum.div_eq\n\n/-- Given `a`,`b` rational numerals, returns `(c, ⊢ a / b = c)`. -/\nunsafe def prove_div (ic : instance_cache) (a b : expr) (na nb : ℚ) :\n    tactic (instance_cache × expr × expr) := do\n  let (ic, b', pb) ← prove_inv ic b nb\n  let (ic, c, p) ← prove_mul_rat ic a b' na nb⁻¹\n  let (ic, p) ← ic.mk_app `` div_eq [a, b, b', c, pb, p]\n  return (ic, c, p)\n#align norm_num.prove_div norm_num.prove_div\n\n/-- Given `a` a rational numeral, returns `(b, ⊢ -a = b)`. -/\nunsafe def prove_neg (ic : instance_cache) (a : expr) : tactic (instance_cache × expr × expr) :=\n  match match_sign a with\n  | Sum.inl a => do\n    let (ic, p) ← ic.mk_app `` neg_neg [a]\n    return (ic, a, p)\n  | Sum.inr ff => do\n    let (ic, p) ← ic.mk_app `` neg_zero []\n    return (ic, a, p)\n  | Sum.inr tt => do\n    let (ic, a') ← ic.mk_app `` Neg.neg [a]\n    let p ← mk_eq_refl a'\n    return (ic, a', p)\n#align norm_num.prove_neg norm_num.prove_neg\n\ntheorem sub_pos {α} [AddGroup α] (a b b' c : α) (hb : -b = b') (h : a + b' = c) : a - b = c := by\n  rwa [← hb, ← sub_eq_add_neg] at h\n#align norm_num.sub_pos NormNum.sub_pos\n\ntheorem sub_neg {α} [AddGroup α] (a b c : α) (h : a + b = c) : a - -b = c := by rwa [sub_neg_eq_add]\n#align norm_num.sub_neg NormNum.sub_neg\n\n/-- Given `a`,`b` rational numerals, returns `(c, ⊢ a - b = c)`. -/\nunsafe def prove_sub (ic : instance_cache) (a b : expr) : tactic (instance_cache × expr × expr) :=\n  match match_sign b with\n  | Sum.inl b => do\n    let (ic, c, p) ← prove_add_rat' ic a b\n    let (ic, p) ← ic.mk_app `` sub_neg [a, b, c, p]\n    return (ic, c, p)\n  | Sum.inr ff => do\n    let (ic, p) ← ic.mk_app `` sub_zero [a]\n    return (ic, a, p)\n  | Sum.inr tt => do\n    let (ic, b', pb) ← prove_neg ic b\n    let (ic, c, p) ← prove_add_rat' ic a b'\n    let (ic, p) ← ic.mk_app `` sub_pos [a, b, b', c, pb, p]\n    return (ic, c, p)\n#align norm_num.prove_sub norm_num.prove_sub\n\ntheorem sub_nat_pos (a b c : ℕ) (h : b + c = a) : a - b = c :=\n  h ▸ add_tsub_cancel_left _ _\n#align norm_num.sub_nat_pos NormNum.sub_nat_pos\n\ntheorem sub_nat_neg (a b c : ℕ) (h : a + c = b) : a - b = 0 :=\n  tsub_eq_zero_iff_le.mpr <| h ▸ Nat.le_add_right _ _\n#align norm_num.sub_nat_neg NormNum.sub_nat_neg\n\n/-- Given `a : nat`,`b : nat` natural numerals, returns `(c, ⊢ a - b = c)`. -/\nunsafe def prove_sub_nat (ic : instance_cache) (a b : expr) : tactic (expr × expr) := do\n  let na ← a.toNat\n  let nb ← b.toNat\n  if nb ≤ na then do\n      let (ic, c) ← ic (na - nb)\n      let (ic, p) ← prove_add_nat ic b c a\n      return (c, q(sub_nat_pos).mk_app [a, b, c, p])\n    else do\n      let (ic, c) ← ic (nb - na)\n      let (ic, p) ← prove_add_nat ic a c b\n      return (q((0 : ℕ)), q(sub_nat_neg).mk_app [a, b, c, p])\n#align norm_num.prove_sub_nat norm_num.prove_sub_nat\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Evaluates the basic field operations `+`,`neg`,`-`,`*`,`inv`,`/` on numerals.\n      Also handles nat subtraction. Does not do recursive simplification; that is,\n      `1 + 1 + 1` will not simplify but `2 + 1` will. This is handled by the top level\n      `simp` call in `norm_num.derive`. -/\n    unsafe\n  def\n    eval_field\n    : expr → tactic ( expr × expr )\n    |\n        q( $ ( e₁ ) + $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            let n₃ := n₁ + n₂\n            let ( c , e₃ ) ← c . of_rat n₃\n            let ( _ , p ) ← prove_add_rat c e₁ e₂ e₃ n₁ n₂ n₃\n            return ( e₃ , p )\n      |\n        q( $ ( e₁ ) * $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            Prod.snd <$> prove_mul_rat c e₁ e₂ n₁ n₂\n      | q( - $ ( e ) ) => do let c ← infer_type e >>= mk_instance_cache Prod.snd <$> prove_neg c e\n      |\n        q( @ Sub.sub $ ( α ) $ ( inst ) $ ( a ) $ ( b ) )\n        =>\n        do\n          let c ← mk_instance_cache α\n            if α = q( Nat ) then prove_sub_nat c a b else Prod.snd <$> prove_sub c a b\n      |\n        q( Inv.inv $ ( e ) )\n        =>\n        do\n          let n ← e . to_rat let c ← infer_type e >>= mk_instance_cache Prod.snd <$> prove_inv c e n\n      |\n        q( $ ( e₁ ) / $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            Prod.snd <$> prove_div c e₁ e₂ n₁ n₂\n      | _ => failed\n#align norm_num.eval_field norm_num.eval_field\n\ntheorem pow_bit0 [Monoid α] (a c' c : α) (b : ℕ) (h : a ^ b = c') (h₂ : c' * c' = c) :\n    a ^ bit0 b = c :=\n  h₂ ▸ by simp [pow_bit0, h]\n#align norm_num.pow_bit0 NormNum.pow_bit0\n\ntheorem pow_bit1 [Monoid α] (a c₁ c₂ c : α) (b : ℕ) (h : a ^ b = c₁) (h₂ : c₁ * c₁ = c₂)\n    (h₃ : c₂ * a = c) : a ^ bit1 b = c := by rw [← h₃, ← h₂] <;> simp [pow_bit1, h]\n#align norm_num.pow_bit1 NormNum.pow_bit1\n\nsection\n\nopen MatchNumeralResult\n\n/-- Given `a` a rational numeral and `b : nat`, returns `(c, ⊢ a ^ b = c)`. -/\nunsafe def prove_pow (a : expr) (na : ℚ) :\n    instance_cache → expr → tactic (instance_cache × expr × expr)\n  | ic, b =>\n    match match_numeral b with\n    | zero => do\n      let (ic, p) ← ic.mk_app `` pow_zero [a]\n      let (ic, o) ← ic.mk_app `` One.one []\n      return (ic, o, p)\n    | one => do\n      let (ic, p) ← ic.mk_app `` pow_one [a]\n      return (ic, a, p)\n    | bit0 b => do\n      let (ic, c', p) ← prove_pow ic b\n      let nc' ← expr.to_rat c'\n      let (ic, c, p₂) ← prove_mul_rat ic c' c' nc' nc'\n      let (ic, p) ← ic.mk_app `` pow_bit0 [a, c', c, b, p, p₂]\n      return (ic, c, p)\n    | bit1 b => do\n      let (ic, c₁, p) ← prove_pow ic b\n      let nc₁ ← expr.to_rat c₁\n      let (ic, c₂, p₂) ← prove_mul_rat ic c₁ c₁ nc₁ nc₁\n      let (ic, c, p₃) ← prove_mul_rat ic c₂ a (nc₁ * nc₁) na\n      let (ic, p) ← ic.mk_app `` pow_bit1 [a, c₁, c₂, c, b, p, p₂, p₃]\n      return (ic, c, p)\n    | _ => failed\n#align norm_num.prove_pow norm_num.prove_pow\n\nend\n\ntheorem zpow_pos {α} [DivInvMonoid α] (a : α) (b : ℤ) (b' : ℕ) (c : α) (hb : b = b')\n    (h : a ^ b' = c) : a ^ b = c := by rw [← h, hb, zpow_ofNat]\n#align norm_num.zpow_pos NormNum.zpow_pos\n\ntheorem zpow_neg {α} [DivInvMonoid α] (a : α) (b : ℤ) (b' : ℕ) (c c' : α) (b0 : 0 < b')\n    (hb : b = b') (h : a ^ b' = c) (hc : c⁻¹ = c') : a ^ (-b) = c' := by\n  rw [← hc, ← h, hb, zpow_neg_coe_of_pos _ b0]\n#align norm_num.zpow_neg NormNum.zpow_neg\n\n/-- Given `a` a rational numeral and `b : ℤ`, returns `(c, ⊢ a ^ b = c)`. -/\nunsafe def prove_zpow (ic zc nc : instance_cache) (a : expr) (na : ℚ) (b : expr) :\n    tactic (instance_cache × instance_cache × instance_cache × expr × expr) :=\n  match match_sign b with\n  | Sum.inl b => do\n    let (zc, nc, b', hb) ← prove_nat_uncast zc nc b\n    let (nc, b0) ← prove_pos nc b'\n    let (ic, c, h) ← prove_pow a na ic b'\n    let (ic, c', hc) ← c.to_rat >>= prove_inv ic c\n    let (ic, p) ← ic.mk_app `` zpow_neg [a, b, b', c, c', b0, hb, h, hc]\n    pure (ic, zc, nc, c', p)\n  | Sum.inr ff => do\n    let (ic, o) ← ic.mk_app `` One.one []\n    let (ic, p) ← ic.mk_app `` zpow_zero [a]\n    pure (ic, zc, nc, o, p)\n  | Sum.inr tt => do\n    let (zc, nc, b', hb) ← prove_nat_uncast zc nc b\n    let (ic, c, h) ← prove_pow a na ic b'\n    let (ic, p) ← ic.mk_app `` zpow_pos [a, b, b', c, hb, h]\n    pure (ic, zc, nc, c, p)\n#align norm_num.prove_zpow norm_num.prove_zpow\n\n/-- Evaluates expressions of the form `a ^ b`, `monoid.npow a b` or `nat.pow a b`. -/\nunsafe def eval_pow : expr → tactic (expr × expr)\n  | q(@Pow.pow $(α) $(β) $(m) $(e₁) $(e₂)) => do\n    let n₁ ← e₁.to_rat\n    let c ← mk_instance_cache α\n    match β with\n      | q(ℕ) => do\n        let (c, m') ← c `` Monoid.Pow []\n        is_def_eq m m'\n        Prod.snd <$> prove_pow e₁ n₁ c e₂\n      | q(ℤ) => do\n        let (c, m') ← c `` DivInvMonoid.Pow []\n        is_def_eq m m'\n        let zc ← mk_instance_cache q(ℤ)\n        let nc ← mk_instance_cache q(ℕ)\n        (Prod.snd ∘ Prod.snd ∘ Prod.snd) <$> prove_zpow c zc nc e₁ n₁ e₂\n      | _ => failed\n  | q(Monoid.npow $(e₁) $(e₂)) => do\n    let n₁ ← e₁.to_rat\n    let c ← infer_type e₁ >>= mk_instance_cache\n    Prod.snd <$> prove_pow e₁ n₁ c e₂\n  | q(DivInvMonoid.zpow $(e₁) $(e₂)) => do\n    let n₁ ← e₁.to_rat\n    let c ← infer_type e₁ >>= mk_instance_cache\n    let zc ← mk_instance_cache q(ℤ)\n    let nc ← mk_instance_cache q(ℕ)\n    (Prod.snd ∘ Prod.snd ∘ Prod.snd) <$> prove_zpow c zc nc e₁ n₁ e₂\n  | _ => failed\n#align norm_num.eval_pow norm_num.eval_pow\n\n/-- Given `⊢ p`, returns `(true, ⊢ p = true)`. -/\nunsafe def true_intro (p : expr) : tactic (expr × expr) :=\n  Prod.mk q(True) <$> mk_app `` eq_true [p]\n#align norm_num.true_intro norm_num.true_intro\n\n/-- Given `⊢ ¬ p`, returns `(false, ⊢ p = false)`. -/\nunsafe def false_intro (p : expr) : tactic (expr × expr) :=\n  Prod.mk q(False) <$> mk_app `` eq_false [p]\n#align norm_num.false_intro norm_num.false_intro\n\ntheorem not_refl_false_intro {α} (a : α) : (a ≠ a) = False :=\n  eq_false <| not_not_intro rfl\n#align norm_num.not_refl_false_intro NormNum.not_refl_false_intro\n\n-- see Note [nolint_ge]\n@[nolint ge_or_gt]\ntheorem gt_intro {α} [LT α] (a b : α) (c) (h : (a < b) = c) : (b > a) = c :=\n  h\n#align norm_num.gt_intro NormNum.gt_intro\n\n-- see Note [nolint_ge]\n@[nolint ge_or_gt]\ntheorem ge_intro {α} [LE α] (a b : α) (c) (h : (a ≤ b) = c) : (b ≥ a) = c :=\n  h\n#align norm_num.ge_intro NormNum.ge_intro\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- Evaluates the inequality operations `=`,`<`,`>`,`≤`,`≥`,`≠` on numerals. -/ unsafe\n  def\n    eval_ineq\n    : expr → tactic ( expr × expr )\n    |\n        q( $ ( e₁ ) < $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            if\n              n₁ < n₂\n              then\n              do let ( _ , p ) ← prove_lt_rat c e₁ e₂ n₁ n₂ true_intro p\n              else\n              if\n                n₁ = n₂\n                then\n                do let ( _ , p ) ← c ` ` lt_irrefl [ e₁ ] false_intro p\n                else\n                do\n                  let ( c , p' ) ← prove_lt_rat c e₂ e₁ n₂ n₁\n                    let ( _ , p ) ← c ` ` not_lt_of_gt [ e₁ , e₂ , p' ]\n                    false_intro p\n      |\n        q( $ ( e₁ ) ≤ $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            if\n              n₁ ≤ n₂\n              then\n              do\n                let ( _ , p ) ← if n₁ = n₂ then c ` ` le_refl [ e₁ ] else prove_le_rat c e₁ e₂ n₁ n₂\n                  true_intro p\n              else\n              do\n                let ( c , p ) ← prove_lt_rat c e₂ e₁ n₂ n₁\n                  let ( _ , p ) ← c ` ` not_le_of_gt [ e₁ , e₂ , p ]\n                  false_intro p\n      |\n        q( $ ( e₁ ) = $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            if\n              n₁ = n₂\n              then\n              mk_eq_refl e₁ >>= true_intro\n              else\n              do let ( _ , p ) ← prove_ne c e₁ e₂ n₁ n₂ false_intro p\n      |\n        q( $ ( e₁ ) > $ ( e₂ ) )\n        =>\n        do\n          let ( e , p ) ← mk_app ` ` LT.lt [ e₂ , e₁ ] >>= eval_ineq\n            Prod.mk e <$> mk_app ` ` gt_intro [ e₂ , e₁ , e , p ]\n      |\n        q( $ ( e₁ ) ≥ $ ( e₂ ) )\n        =>\n        do\n          let ( e , p ) ← mk_app ` ` LE.le [ e₂ , e₁ ] >>= eval_ineq\n            Prod.mk e <$> mk_app ` ` ge_intro [ e₂ , e₁ , e , p ]\n      |\n        q( $ ( e₁ ) ≠ $ ( e₂ ) )\n        =>\n        do\n          let n₁ ← e₁ . to_rat\n            let n₂ ← e₂ . to_rat\n            let c ← infer_type e₁ >>= mk_instance_cache\n            if\n              n₁ = n₂\n              then\n              Prod.mk q( False ) <$> mk_app ` ` not_refl_false_intro [ e₁ ]\n              else\n              do let ( _ , p ) ← prove_ne c e₁ e₂ n₁ n₂ true_intro p\n      | _ => failed\n#align norm_num.eval_ineq norm_num.eval_ineq\n\ntheorem nat_succ_eq (a b c : ℕ) (h₁ : a = b) (h₂ : b + 1 = c) : Nat.succ a = c := by rwa [h₁]\n#align norm_num.nat_succ_eq NormNum.nat_succ_eq\n\n/-- Evaluates the expression `nat.succ ... (nat.succ n)` where `n` is a natural numeral.\n(We could also just handle `nat.succ n` here and rely on `simp` to work bottom up, but we figure\nthat towers of successors coming from e.g. `induction` are a common case.) -/\nunsafe def prove_nat_succ (ic : instance_cache) : expr → tactic (instance_cache × ℕ × expr × expr)\n  | q(Nat.succ $(a)) => do\n    let (ic, n, b, p₁) ← prove_nat_succ a\n    let n' := n + 1\n    let (ic, c) ← ic.ofNat n'\n    let (ic, p₂) ← prove_add_nat ic b q(1) c\n    return (ic, n', c, q(nat_succ_eq).mk_app [a, b, c, p₁, p₂])\n  | e => do\n    let n ← e.toNat\n    let p ← mk_eq_refl e\n    return (ic, n, e, p)\n#align norm_num.prove_nat_succ norm_num.prove_nat_succ\n\ntheorem int_toNat_pos (a : ℤ) (b : ℕ)\n    (h :\n      (haveI := @Nat.castCoe ℤ\n          b :\n          ℤ) =\n        a) :\n    a.toNat = b := by rw [← h] <;> simp\n#align norm_num.int_to_nat_pos NormNum.int_toNat_pos\n\ntheorem int_toNat_neg (a : ℤ) (h : 0 < a) : (-a).toNat = 0 := by\n  simp only [Int.toNat_of_nonpos, h.le, neg_nonpos]\n#align norm_num.int_to_nat_neg NormNum.int_toNat_neg\n\ntheorem natAbs_pos (a : ℤ) (b : ℕ)\n    (h :\n      (haveI := @Nat.castCoe ℤ\n          b :\n          ℤ) =\n        a) :\n    a.natAbs = b := by rw [← h] <;> simp\n#align norm_num.nat_abs_pos NormNum.natAbs_pos\n\ntheorem natAbs_neg (a : ℤ) (b : ℕ)\n    (h :\n      (haveI := @Nat.castCoe ℤ\n          b :\n          ℤ) =\n        a) :\n    (-a).natAbs = b := by rw [← h] <;> simp\n#align norm_num.nat_abs_neg NormNum.natAbs_neg\n\ntheorem negSucc (a b : ℕ) (c : ℤ) (h₁ : a + 1 = b)\n    (h₂ :\n      (haveI := @Nat.castCoe ℤ\n          b :\n          ℤ) =\n        c) :\n    -[a+1] = -c := by rw [← h₂, ← h₁] <;> rfl\n#align norm_num.neg_succ_of_nat NormNum.negSucc\n\n/-- Evaluates `nat.succ`, `int.to_nat`, `int.nat_abs`, `int.neg_succ_of_nat`. -/\nunsafe def eval_nat_int : expr → tactic (expr × expr)\n  | e@q(Nat.succ _) => do\n    let ic ← mk_instance_cache q(ℕ)\n    let (_, _, ep) ← prove_nat_succ ic e\n    return ep\n  | q(Int.toNat $(a)) => do\n    let n ← a.to_int\n    let ic ← mk_instance_cache q(ℤ)\n    if n ≥ 0 then do\n        let nc ← mk_instance_cache q(ℕ)\n        let (_, _, b, p) ← prove_nat_uncast ic nc a\n        pure (b, q(int_toNat_pos).mk_app [a, b, p])\n      else do\n        let a ← match_neg a\n        let (_, p) ← prove_pos ic a\n        pure (q(0), q(int_toNat_neg).mk_app [a, p])\n  | q(Int.natAbs $(a)) => do\n    let n ← a.to_int\n    let ic ← mk_instance_cache q(ℤ)\n    let nc ← mk_instance_cache q(ℕ)\n    if n ≥ 0 then do\n        let (_, _, b, p) ← prove_nat_uncast ic nc a\n        pure (b, q(natAbs_pos).mk_app [a, b, p])\n      else do\n        let a ← match_neg a\n        let (_, _, b, p) ← prove_nat_uncast ic nc a\n        pure (b, q(natAbs_neg).mk_app [a, b, p])\n  | q(Int.negSucc $(a)) => do\n    let na ← a.toNat\n    let ic ← mk_instance_cache q(ℤ)\n    let nc ← mk_instance_cache q(ℕ)\n    let nb := na + 1\n    let (nc, b) ← nc.ofNat nb\n    let (nc, p₁) ← prove_add_nat nc a q(1) b\n    let (ic, c) ← ic.ofNat nb\n    let (_, _, _, p₂) ← prove_nat_uncast ic nc c\n    pure (q((-$(c) : ℤ)), q(negSucc).mk_app [a, b, c, p₁, p₂])\n  | _ => failed\n#align norm_num.eval_nat_int norm_num.eval_nat_int\n\ntheorem int_to_nat_cast (a : ℕ) (b : ℤ)\n    (h :\n      (haveI := @Nat.castCoe ℤ\n          a :\n          ℤ) =\n        b) :\n    ↑a = b :=\n  Eq.trans (by simp) h\n#align norm_num.int_to_nat_cast NormNum.int_to_nat_cast\n\n/-- Evaluates the `↑n` cast operation from `ℕ`, `ℤ`, `ℚ` to an arbitrary type `α`. -/\nunsafe def eval_cast : expr → tactic (expr × expr)\n  | q(@coe ℕ $(α) $(inst) $(a)) => do\n    if inst `` coeToLift then\n        if inst `` Nat.castCoe then do\n          let n ← a\n          let ic ← mk_instance_cache α\n          let nc ← mk_instance_cache q(ℕ)\n          let (ic, b) ← ic n\n          let (_, _, _, p) ← prove_nat_uncast ic nc b\n          pure (b, p)\n        else\n          if inst `` Int.castCoe then do\n            let n ← a\n            let ic ← mk_instance_cache α\n            let zc ← mk_instance_cache q(ℤ)\n            let (ic, b) ← ic n\n            let (_, _, _, p) ← prove_int_uncast ic zc b\n            pure (b, p)\n          else\n            if inst `` Rat.castCoe then do\n              let n ← a\n              let cz_inst ← mk_mapp `` CharZero [α, none] >>= mk_instance\n              let ic ← mk_instance_cache α\n              let qc ← mk_instance_cache q(ℚ)\n              let (ic, b) ← ic n\n              let (_, _, _, p) ← prove_rat_uncast ic qc cz_inst b n\n              pure (b, p)\n            else failed\n      else\n        if inst = q(@coeBase Nat Int Int.hasCoe) then do\n          let n ← a\n          let ic ← mk_instance_cache q(ℤ)\n          let nc ← mk_instance_cache q(ℕ)\n          let (ic, b) ← ic n\n          let (_, _, _, p) ← prove_nat_uncast ic nc b\n          pure (b, q(int_to_nat_cast).mk_app [a, b, p])\n        else failed\n  | _ => failed\n#align norm_num.eval_cast norm_num.eval_cast\n\n/-- This version of `derive` does not fail when the input is already a numeral -/\nunsafe def derive.step (e : expr) : tactic (expr × expr) :=\n  eval_field e <|> eval_pow e <|> eval_ineq e <|> eval_cast e <|> eval_nat_int e\n#align norm_num.derive.step norm_num.derive.step\n\n/-- An attribute for adding additional extensions to `norm_num`. To use this attribute, put\n`@[norm_num]` on a tactic of type `expr → tactic (expr × expr)`; the tactic will be called on\nsubterms by `norm_num`, and it is responsible for identifying that the expression is a numerical\nfunction applied to numerals, for example `nat.fib 17`, and should return the reduced numerical\nexpression (which must be in `norm_num`-normal form: a natural or rational numeral, i.e. `37`,\n`12 / 7` or `-(2 / 3)`, although this can be an expression in any type), and the proof that the\noriginal expression is equal to the rewritten expression.\n\nFailure is used to indicate that this tactic does not apply to the term. For performance reasons,\nit is best to detect non-applicability as soon as possible so that the next tactic can have a go,\nso generally it will start with a pattern match and then checking that the arguments to the term\nare numerals or of the appropriate form, followed by proof construction, which should not fail.\n\nPropositions are treated like any other term. The normal form for propositions is `true` or\n`false`, so it should produce a proof of the form `p = true` or `p = false`. `eq_true_intro` can be\nused to help here.\n-/\n@[user_attribute]\nprotected unsafe def attr : user_attribute (expr → tactic (expr × expr)) Unit\n    where\n  Name := `norm_num\n  descr := \"Add norm_num derivers\"\n  cache_cfg :=\n    { mk_cache := fun ns => do\n        let t ←\n          ns.foldlM\n              (fun (t : expr → tactic (expr × expr)) n => do\n                let t' ← eval_expr (expr → tactic (expr × expr)) (expr.const n [])\n                pure fun e => t' e <|> t e)\n              fun _ => failed\n        pure fun e => derive.step e <|> t e\n      dependencies := [] }\n#align norm_num.attr norm_num.attr\n\nadd_tactic_doc\n  { Name := \"norm_num\"\n    category := DocCategory.attr\n    declNames := [`norm_num.attr]\n    tags := [\"arithmetic\", \"decision_procedure\"] }\n\n/-- Look up the `norm_num` extensions in the cache and return a tactic extending `derive.step` with\nadditional reduction procedures. -/\nunsafe def get_step : tactic (expr → tactic (expr × expr)) :=\n  norm_num.attr.get_cache\n#align norm_num.get_step norm_num.get_step\n\n/-- Simplify an expression bottom-up using `step` to simplify the subexpressions. -/\nunsafe def derive' (step : expr → tactic (expr × expr)) : expr → tactic (expr × expr)\n  | e => do\n    let e ← instantiate_mvars e\n    let (_, e', pr) ←\n      ext_simplify_core () { } simp_lemmas.mk (fun _ => failed) (fun _ _ _ _ _ => failed)\n          (fun _ _ _ _ e => do\n            let (new_e, pr) ← step e\n            guard ¬new_e == e\n            pure ((), new_e, some pr, tt))\n          `eq e\n    pure (e', pr)\n#align norm_num.derive' norm_num.derive'\n\n/-- Simplify an expression bottom-up using the default `norm_num` set to simplify the\nsubexpressions. -/\nunsafe def derive (e : expr) : tactic (expr × expr) := do\n  let f ← get_step\n  derive' f e\n#align norm_num.derive norm_num.derive\n\nend NormNum\n\n/-- Basic version of `norm_num` that does not call `simp`. It uses the provided `step` tactic\nto simplify the expression; use `get_step` to get the default `norm_num` set and `derive.step` for\nthe basic builtin set of simplifications. -/\nunsafe def tactic.norm_num1 (step : expr → tactic (expr × expr)) (loc : Interactive.Loc) :\n    tactic Unit := do\n  let ns ← loc.get_locals\n  let success ← tactic.replace_at (norm_num.derive' step) ns loc.include_goal\n  when loc <| try tactic.triv\n  when ¬ns <| try tactic.contradiction\n  Monad.unlessb success <| done <|> fail \"norm_num failed to simplify\"\n#align tactic.norm_num1 tactic.norm_num1\n\n/-- Normalize numerical expressions. It uses the provided `step` tactic to simplify the expression;\nuse `get_step` to get the default `norm_num` set and `derive.step` for the basic builtin set of\nsimplifications. -/\nunsafe def tactic.norm_num (step : expr → tactic (expr × expr)) (hs : List simp_arg_type)\n    (l : Interactive.Loc) : tactic Unit :=\n  repeat1 <|\n    orelse' (tactic.norm_num1 step l) <|\n      interactive.simp_core { } (tactic.norm_num1 step (Interactive.Loc.ns [none])) false\n          (simp_arg_type.except `` one_div :: hs) [] l >>\n        skip\n#align tactic.norm_num tactic.norm_num\n\n/-- Carry out similar operations as `tactic.norm_num` but on an `expr` rather than a location.\nGiven an expression `e`, returns `(e', ⊢ e = e')`.\nThe `no_dflt`, `hs`, and `attr_names` are passed on to `simp`.\nUnlike `norm_num`, this tactic does not fail. -/\nunsafe def _root_.expr.norm_num (step : expr → tactic (expr × expr)) (no_dflt : Bool := false)\n    (hs : List simp_arg_type := []) (attr_names : List Name := []) : expr → tactic (expr × expr) :=\n  let simp_step (e : expr) := do\n    let (e', p, _) ←\n      e.simp { } (tactic.norm_num1 step (Interactive.Loc.ns [none])) no_dflt attr_names\n          (simp_arg_type.except `` one_div :: hs)\n    return (e', p)\n  or_refl_conv fun e => do\n    let (e', p') ← norm_num.derive' step e <|> simp_step e\n    let (e'', p'') ← _root_.expr.norm_num e'\n    let p ← mk_eq_trans p' p''\n    return (e'', p)\n#align expr.norm_num expr.norm_num\n\nnamespace Tactic.Interactive\n\nopen NormNum Interactive Interactive.Types\n\n/-- Basic version of `norm_num` that does not call `simp`. -/\nunsafe def norm_num1 (loc : parse location) : tactic Unit := do\n  let f ← get_step\n  tactic.norm_num1 f loc\n#align tactic.interactive.norm_num1 tactic.interactive.norm_num1\n\n/-- Normalize numerical expressions. Supports the operations\n`+` `-` `*` `/` `^` and `%` over numerical types such as\n`ℕ`, `ℤ`, `ℚ`, `ℝ`, `ℂ` and some general algebraic types,\nand can prove goals of the form `A = B`, `A ≠ B`, `A < B` and `A ≤ B`,\nwhere `A` and `B` are numerical expressions.\nIt also has a relatively simple primality prover. -/\nunsafe def norm_num (hs : parse simp_arg_list) (l : parse location) : tactic Unit := do\n  let f ← get_step\n  tactic.norm_num f hs l\n#align tactic.interactive.norm_num tactic.interactive.norm_num\n\nadd_hint_tactic norm_num\n\n/-- Normalizes a numerical expression and tries to close the goal with the result. -/\nunsafe def apply_normed (x : parse texpr) : tactic Unit := do\n  let x₁ ← to_expr x\n  let (x₂, _) ← derive x₁\n  tactic.exact x₂\n#align tactic.interactive.apply_normed tactic.interactive.apply_normed\n\n/-- Normalises numerical expressions. It supports the operations `+` `-` `*` `/` `^` and `%` over\nnumerical types such as `ℕ`, `ℤ`, `ℚ`, `ℝ`, `ℂ`, and can prove goals of the form `A = B`, `A ≠ B`,\n`A < B` and `A ≤ B`, where `A` and `B` are numerical expressions.\n\nAdd-on tactics marked as `@[norm_num]` can extend the behavior of `norm_num` to include other\nfunctions. This is used to support several other functions on `nat` like `prime`, `min_fac` and\n`factors`.\n```lean\nimport data.real.basic\n\nexample : (2 : ℝ) + 2 = 4 := by norm_num\nexample : (12345.2 : ℝ) ≠ 12345.3 := by norm_num\nexample : (73 : ℝ) < 789/2 := by norm_num\nexample : 123456789 + 987654321 = 1111111110 := by norm_num\nexample (R : Type*) [ring R] : (2 : R) + 2 = 4 := by norm_num\nexample (F : Type*) [linear_ordered_field F] : (2 : F) + 2 < 5 := by norm_num\nexample : nat.prime (2^13 - 1) := by norm_num\nexample : ¬ nat.prime (2^11 - 1) := by norm_num\nexample (x : ℝ) (h : x = 123 + 456) : x = 579 := by norm_num at h; assumption\n```\n\nThe variant `norm_num1` does not call `simp`.\n\nBoth `norm_num` and `norm_num1` can be called inside the `conv` tactic.\n\nThe tactic `apply_normed` normalises a numerical expression and tries to close the goal with\nthe result. Compare:\n```lean\ndef a : ℕ := 2^100\n#print a -- 2 ^ 100\n\ndef normed_a : ℕ := by apply_normed 2^100\n#print normed_a -- 1267650600228229401496703205376\n```\n-/\nadd_tactic_doc\n  { Name := \"norm_num\"\n    category := DocCategory.tactic\n    declNames :=\n      [`tactic.interactive.norm_num1, `tactic.interactive.norm_num,\n        `tactic.interactive.apply_normed]\n    tags := [\"arithmetic\", \"decision procedure\"] }\n\nend Tactic.Interactive\n\n/-! ## `conv` tactic -/\n\n\nnamespace Conv.Interactive\n\nopen Conv Interactive Tactic.Interactive\n\nopen NormNum (derive)\n\n/-- Basic version of `norm_num` that does not call `simp`. -/\nunsafe def norm_num1 : conv Unit :=\n  replace_lhs derive\n#align conv.interactive.norm_num1 conv.interactive.norm_num1\n\n/-- Normalize numerical expressions. Supports the operations\n`+` `-` `*` `/` `^` and `%` over numerical types such as\n`ℕ`, `ℤ`, `ℚ`, `ℝ`, `ℂ` and some general algebraic types,\nand can prove goals of the form `A = B`, `A ≠ B`, `A < B` and `A ≤ B`,\nwhere `A` and `B` are numerical expressions.\nIt also has a relatively simple primality prover. -/\nunsafe def norm_num (hs : parse simp_arg_list) : conv Unit :=\n  repeat1 <|\n    orelse' norm_num1 <|\n      conv.interactive.simp false (simp_arg_type.except `` one_div :: hs) []\n        { discharger := tactic.interactive.norm_num1 (Loc.ns [none]) }\n#align conv.interactive.norm_num conv.interactive.norm_num\n\nend Conv.Interactive\n\n/-!\n## `#norm_num` command\nA user command to run `norm_num`. Mostly copied from the `#simp` command.\n-/\n\n\nnamespace Tactic\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n-- With this option, turn off the messages if the result is exactly `true`\ninitialize\n  registerTraceClass.1 `silence_norm_num_if_true\n\n/-- The basic usage is `#norm_num e`, where `e` is an expression,\nwhich will print the `norm_num` form of `e`.\n\nSyntax: `#norm_num` (`only`)? (`[` simp lemma list `]`)? (`with` simp sets)? `:`? expression\n\nThis accepts the same options as the `#simp` command.\nYou can specify additional simp lemmas as usual, for example using\n`#norm_num [f, g] : e`, or `#norm_num with attr : e`.\n(The colon is optional but helpful for the parser.)\nThe `only` restricts `norm_num` to using only the provided lemmas, and so\n`#norm_num only : e` behaves similarly to `norm_num1`.\n\nUnlike `norm_num`, this command does not fail when no simplifications are made.\n\n`#norm_num` understands local variables, so you can use them to\nintroduce parameters.\n-/\n@[user_command]\nunsafe def norm_num_cmd (_ : parse <| tk \"#norm_num\") : lean.parser Unit := do\n  let no_dflt ← only_flag\n  let hs ← simp_arg_list\n  let attr_names ← with_ident_list\n  let o ← optional (tk \":\")\n  let e ← texpr\n  let-- Retrieve the `pexpr`s parsed as part of the simp args, and collate them into a big list.\n  hs_es := List.join <| hs.map <| Option.toList ∘ simp_arg_type.to_pexpr\n  let/- Synthesize a `tactic_state` including local variables as hypotheses under which `expr.simp`\n         may be safely called with expected behaviour given the `variables` in the environment. -/\n    (ts, mappings)\n    ← synthesize_tactic_state_with_variables_as_hyps (e :: hs_es)\n  let result\n    ←-- Enter the `tactic` monad, *critically* using the synthesized tactic state `ts`.\n        lean.parser.of_tactic\n        fun _ =>\n        (/- Resolve the local variables added by the parser to `e` (when it was parsed) against the local\n                 hypotheses added to the `ts : tactic_state` which we are using. -/\n          do\n            let e ← to_expr e\n            let/- Replace the variables referenced in the passed `simp_arg_list` with the `expr`s corresponding\n                   to the local hypotheses we created.\n            \n                   We would prefer to just elaborate the `pexpr`s encoded in the `simp_arg_list` against the\n                   tactic state we have created (as we could with `e` above), but the simplifier expects\n                   `pexpr`s and not `expr`s. Thus, we just modify the `pexpr`s now and let `simp` do the\n                   elaboration when the time comes.\n            \n                   You might think that we could just examine each of these `pexpr`s, call `to_expr` on them,\n                   and then call `to_pexpr` afterward and save the results over the original `pexprs`. Due to\n                   how functions like `simp_lemmas.add_pexpr` are implemented in the core library, the `simp`\n                   framework is not robust enough to handle this method. When pieces of expressions like\n                   annotation macros are injected, the direct patten matches in the `simp_lemmas.*` codebase\n                   fail, and the lemmas we want don't get added.\n                   -/\n            hs := hs.map fun sat => sat.replace_subexprs mappings\n            let step\n              ←-- Try simplifying the expression.\n                norm_num.get_step\n            Prod.fst <$> e step no_dflt hs attr_names)\n          ts\n  -- Trace the result.\n      when\n      (¬is_trace_enabled_for `silence_norm_num_if_true ∨ result ≠ expr.const `true [])\n      (trace result)\n#align tactic.norm_num_cmd tactic.norm_num_cmd\n\nadd_tactic_doc\n  { Name := \"#norm_num\"\n    category := DocCategory.cmd\n    declNames := [`tactic.norm_num_cmd]\n    tags := [\"simplification\", \"arithmetic\", \"decision procedure\"] }\n\nend Tactic\n\nnamespace NormNum\n\nsection ElementaryNumberTheory\n\nopen Tactic\n\ntheorem nat_div (a b q r m : ℕ) (hm : q * b = m) (h : r + m = a) (h₂ : r < b) : a / b = q := by\n  rw [← h, ← hm, Nat.add_mul_div_right _ _ (lt_of_le_of_lt (Nat.zero_le _) h₂), Nat.div_eq_of_lt h₂,\n    zero_add]\n#align norm_num.nat_div NormNum.nat_div\n\ntheorem int_div (a b q r m : ℤ) (hm : q * b = m) (h : r + m = a) (h₁ : 0 ≤ r) (h₂ : r < b) :\n    a / b = q := by\n  rw [← h, ← hm, Int.add_mul_ediv_right _ _ (ne_of_gt (lt_of_le_of_lt h₁ h₂)),\n    Int.div_eq_zero_of_lt h₁ h₂, zero_add]\n#align norm_num.int_div NormNum.int_div\n\ntheorem nat_mod (a b q r m : ℕ) (hm : q * b = m) (h : r + m = a) (h₂ : r < b) : a % b = r := by\n  rw [← h, ← hm, Nat.add_mul_mod_self_right, Nat.mod_eq_of_lt h₂]\n#align norm_num.nat_mod NormNum.nat_mod\n\ntheorem int_mod (a b q r m : ℤ) (hm : q * b = m) (h : r + m = a) (h₁ : 0 ≤ r) (h₂ : r < b) :\n    a % b = r := by rw [← h, ← hm, Int.add_mul_emod_self, Int.emod_eq_of_lt h₁ h₂]\n#align norm_num.int_mod NormNum.int_mod\n\ntheorem int_div_neg (a b c' c : ℤ) (h : a / b = c') (h₂ : -c' = c) : a / -b = c :=\n  h₂ ▸ h ▸ Int.div_neg _ _\n#align norm_num.int_div_neg NormNum.int_div_neg\n\ntheorem int_mod_neg (a b c : ℤ) (h : a % b = c) : a % -b = c :=\n  (Int.mod_neg _ _).trans h\n#align norm_num.int_mod_neg NormNum.int_mod_neg\n\n/-- Given `a`,`b` numerals in `nat` or `int`,\n  * `prove_div_mod ic a b ff` returns `(c, ⊢ a / b = c)`\n  * `prove_div_mod ic a b tt` returns `(c, ⊢ a % b = c)`\n-/\nunsafe def prove_div_mod (ic : instance_cache) :\n    expr → expr → Bool → tactic (instance_cache × expr × expr)\n  | a, b, mod =>\n    match match_neg b with\n    | some b => do\n      let (ic, c', p) ← prove_div_mod a b mod\n      if mod then return (ic, c', q(int_mod_neg).mk_app [a, b, c', p])\n        else do\n          let (ic, c, p₂) ← prove_neg ic c'\n          return (ic, c, q(int_div_neg).mk_app [a, b, c', c, p, p₂])\n    | none => do\n      let nb ← b.toNat\n      let na ← a.to_int\n      let nq := na / nb\n      let nr := na % nb\n      let nm := nq * nr\n      let (ic, q) ← ic.ofInt nq\n      let (ic, r) ← ic.ofInt nr\n      let (ic, m, pm) ← prove_mul_rat ic q b nq nb\n      let (ic, a') ← ic.of_rat na\n      let-- ensure `a` is in normal form\n        (ic, p)\n        ← prove_add_rat ic r m a' nr nm na\n      let (ic, p') ← prove_lt_nat ic r b\n      if ic = q(Nat) then\n          if mod then return (ic, r, q(nat_mod).mk_app [a, b, q, r, m, pm, p, p'])\n          else return (ic, q, q(nat_div).mk_app [a, b, q, r, m, pm, p, p'])\n        else\n          if ic = q(Int) then do\n            let (ic, p₀) ← prove_nonneg ic r\n            if mod then return (ic, r, q(int_mod).mk_app [a, b, q, r, m, pm, p, p₀, p'])\n              else return (ic, q, q(int_div).mk_app [a, b, q, r, m, pm, p, p₀, p'])\n          else failed\n#align norm_num.prove_div_mod norm_num.prove_div_mod\n\ntheorem dvd_eq_nat (a b c : ℕ) (p) (h₁ : b % a = c) (h₂ : (c = 0) = p) : (a ∣ b) = p :=\n  (propext <| by rw [← h₁, Nat.dvd_iff_mod_eq_zero]).trans h₂\n#align norm_num.dvd_eq_nat NormNum.dvd_eq_nat\n\ntheorem dvd_eq_int (a b c : ℤ) (p) (h₁ : b % a = c) (h₂ : (c = 0) = p) : (a ∣ b) = p :=\n  (propext <| by rw [← h₁, Int.dvd_iff_emod_eq_zero]).trans h₂\n#align norm_num.dvd_eq_int NormNum.dvd_eq_int\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Evaluates some extra numeric operations on `nat` and `int`, specifically\n      `/` and `%`, and `∣` (divisibility). -/\n    @[ norm_num ]\n    unsafe\n  def\n    eval_nat_int_ext\n    : expr → tactic ( expr × expr )\n    |\n        q( $ ( a ) / $ ( b ) )\n        =>\n        do let c ← infer_type a >>= mk_instance_cache Prod.snd <$> prove_div_mod c a b ff\n      |\n        q( $ ( a ) % $ ( b ) )\n        =>\n        do let c ← infer_type a >>= mk_instance_cache Prod.snd <$> prove_div_mod c a b tt\n      |\n        q( $ ( a ) ∣ $ ( b ) )\n        =>\n        do\n          let α ← infer_type a\n            let ic ← mk_instance_cache α\n            let\n              th\n                ←\n                if\n                  α = q( Nat )\n                  then\n                  return ( q( dvd_eq_nat ) : expr )\n                  else\n                  if α = q( Int ) then return q( dvd_eq_int ) else failed\n            let ( ic , c , p₁ ) ← prove_div_mod ic b a true\n            let ( ic , z ) ← ic . mk_app ` ` Zero.zero [ ]\n            let ( e' , p₂ ) ← mk_app ` ` Eq [ c , z ] >>= eval_ineq\n            return ( e' , th [ a , b , c , e' , p₁ , p₂ ] )\n      | _ => failed\n#align norm_num.eval_nat_int_ext norm_num.eval_nat_int_ext\n\nend ElementaryNumberTheory\n\nend NormNum\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/NormNum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3275202523698191}}
{"text": "import phase1.f_map\n\nopen set with_bot\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}] [position_data.{}] (α : Λ)\n\ninstance core_tangle_data_Iic (β : Iio α) [inst : core_tangle_data (β : Iic α)] :\n  core_tangle_data β := inst\n\ninstance core_tangle_data_Iic' (β : Iic α) [inst : core_tangle_data β] :\n  core_tangle_data (β : Λ) := inst\n\ninstance core_tangle_data_Iio' (β : Iio α) [inst : core_tangle_data β] :\n  core_tangle_data (β : Λ) := inst\n\ninstance almost_tangle_data_Iio (β : Iio α)\n  [inst_0 : core_tangle_data β] [inst : @almost_tangle_data _ (β : Iic α) inst_0] :\n  almost_tangle_data β := inst\n\ninstance positioned_tangle_data_Iio (β : Iio α) [core_tangle_data β]\n  [inst : positioned_tangle_data β] :\n  positioned_tangle_data (β : Λ) := inst\n\nclass phase_2_data :=\n(lower_core_tangle_data : Π β : Iic α, core_tangle_data β)\n(lower_positioned_tangle_data : Π β : Iio α, positioned_tangle_data β)\n(lower_almost_tangle_data : Π β : Iic α, almost_tangle_data β)\n(lower_tangle_data : Π β : Iio α, tangle_data β)\n\nnamespace phase_2_data\nvariables [phase_2_data α] {α} {β : Iic α} {γ : Iio α}\n\n/-- By unfolding here, we get better definitional equality between e.g.\n`allowable ((β : Iic α) : Iic_index α)` and `allowable (β : Iic α)`. -/\n\ninstance core_tangle_data : core_tangle_data β :=\n@id _ lower_core_tangle_data ⟨β, β.prop⟩\ninstance positioned_tangle_data : positioned_tangle_data γ :=\n@id _ lower_positioned_tangle_data ⟨γ, γ.prop⟩\ninstance almost_tangle_data : almost_tangle_data β :=\n@id _ lower_almost_tangle_data ⟨β, β.prop⟩\ninstance tangle_data : tangle_data γ :=\n@id _ lower_tangle_data ⟨γ, γ.prop⟩\n\nnoncomputable instance Iic_index_core_tangle_data : Π (β : Iic_index α), core_tangle_data β\n| ⟨⊥, h⟩ := bot.core_tangle_data\n| ⟨(β : Λ), hβ⟩ := lower_core_tangle_data ⟨β, coe_le_coe.mp hβ⟩\n\nnoncomputable instance Iio_index_core_tangle_data (β : Iio_index α) :\n  core_tangle_data β :=\nshow core_tangle_data (⟨β, le_of_lt β.prop⟩ : Iic_index α), from infer_instance\n\nnoncomputable instance Iio_index_positioned_tangle_data :\n  Π β : Iio_index α, positioned_tangle_data β\n| ⟨⊥, h⟩ := bot.positioned_tangle_data\n| ⟨(β : Λ), hβ⟩ := lower_positioned_tangle_data ⟨β, coe_lt_coe.mp hβ⟩\n\ninstance has_coe_Iio_Iic_index : has_coe (Iio α) (Iic_index α) :=\n⟨λ β, ⟨some β, le_of_lt (with_bot.coe_lt_coe.mpr β.prop)⟩⟩\ninstance has_coe_Iio_index_Iic_index : has_coe (Iio_index α) (Iic_index α) :=\n⟨λ β, ⟨β, le_of_lt β.prop⟩⟩\n\ninstance [phase_2_data α] {X : Type*} {δ : Iio α} [inst : mul_action (allowable δ) X] :\n  mul_action (allowable (Iio_coe δ)) X := inst\n\ninstance mul_action_type_index {δ : Iio α} : mul_action (allowable δ) (tangle (δ : Λ)) :=\nshow mul_action (allowable δ) (tangle δ), from infer_instance\n\nnoncomputable instance mul_action_Iio_index {δ : Iio_index α} :\n  mul_action (allowable (δ : Iic_index α)) (tangle δ) :=\nshow mul_action (allowable δ) (tangle δ), from infer_instance\n\nend phase_2_data\n\nclass phase_2_assumptions extends phase_2_data α :=\n(allowable_derivative : Π (β : Iic_index α) (γ : Iic_index α) (hγ : (γ : type_index) < β),\n  allowable β →* allowable γ)\n(allowable_derivative_eq : ∀ (β : Iic_index α) (γ : Iic_index α) (hγ : (γ : type_index) < β)\n  (π : allowable β),\n  struct_perm.derivative (quiver.path.nil.cons hγ) π.to_struct_perm =\n    (allowable_derivative β γ hγ π).to_struct_perm)\n(smul_f_map {β : Iic_index α} (γ : Iio_index α) (δ : Iio α)\n  (hγ : (γ : type_index) < β) (hδ : (δ : type_index) < β) (hγδ : γ ≠ δ)\n  (π : allowable β) (t : tangle γ) :\n  (allowable_derivative β δ hδ π) •\n    f_map (subtype.coe_injective.ne hγδ) t =\n    f_map (subtype.coe_injective.ne hγδ) (allowable_derivative β γ hγ π • t))\n\nexport phase_2_assumptions (allowable_derivative allowable_derivative_eq smul_f_map)\n\nvariables {α} [phase_2_assumptions α]\n\nnoncomputable def allowable.derivative {β : Iic_index α} : Π {γ : Iic_index α}\n  (A : quiver.path (β : type_index) γ), allowable β →* allowable γ :=\n@path.Iic_rec _ α β (λ δ A, allowable β →* allowable δ) (monoid_hom.id _)\n  (λ γ δ A h, (allowable_derivative γ δ h).comp)\n\n@[simp] lemma allowable.derivative_nil {β : Iic_index α} :\n  allowable.derivative (quiver.path.nil : quiver.path (β : type_index) β) = monoid_hom.id _ :=\nby rw [allowable.derivative, path.Iic_rec_nil]\n\n@[simp] lemma allowable.derivative_cons {β γ δ : Iic_index α}\n  (A : quiver.path (β : type_index) γ) (h : δ < γ) :\n  allowable.derivative (A.cons h) = (allowable_derivative γ δ h).comp (allowable.derivative A) :=\nby rw [allowable.derivative, path.Iic_rec_cons]\n\nlemma allowable.derivative_cons_apply {β γ δ : Iic_index α}\n  (A : quiver.path (β : type_index) γ) (h : δ < γ) (π : allowable β) :\n  allowable.derivative (A.cons h) π = allowable_derivative γ δ h (allowable.derivative A π) :=\nby rw [allowable.derivative_cons]; refl\n\nlemma allowable.derivative_eq {β γ : Iic_index α} (h : γ < β) :\n  allowable_derivative β γ h = allowable.derivative (quiver.path.nil.cons h) :=\nby rw [allowable.derivative_cons, allowable.derivative_nil, monoid_hom.comp_id]\n\nlemma allowable.derivative_derivative {β γ δ : Iic_index α}\n  (A : quiver.path (β : type_index) γ) (B : quiver.path (γ : type_index) δ) (π : allowable β) :\n  allowable.derivative B (allowable.derivative A π) = allowable.derivative (A.comp B) π :=\nbegin\n  obtain ⟨γ, hγ⟩ := γ,\n  obtain ⟨δ, hδ⟩ := δ,\n  change quiver.path γ δ at B,\n  induction B with ε ζ B h ih,\n  { simp only [quiver.path.comp_nil, allowable.derivative_nil, monoid_hom.id_apply], },\n  { simp only [quiver.path.comp_cons],\n    rw allowable.derivative_cons (show quiver.path ((⟨γ, hγ⟩ : Iic_index α) : type_index)\n        ((⟨ε, le_trans (le_of_path B) hγ⟩ : Iic_index α) : type_index), from B),\n    simp only [monoid_hom.coe_comp, function.comp_app, ih, ← allowable.derivative_cons_apply], },\nend\n\nlemma allowable.derivative_to_struct_perm {β γ : Iic_index α} (A : quiver.path (β : type_index) γ)\n  (π : allowable β) :\n  struct_perm.derivative A π.to_struct_perm = (allowable.derivative A π).to_struct_perm :=\nbegin\n  revert π A γ,\n  refine path.Iic_rec _ _,\n  { intro π,\n    simp only [struct_perm.derivative_nil, allowable.derivative_nil, monoid_hom.id_apply], },\n  { intros γ δ A h ih π,\n    rw [allowable.derivative_cons, monoid_hom.coe_comp, function.comp_app,\n      ← allowable_derivative_eq, ← ih π, struct_perm.derivative_derivative,\n      quiver.path.comp_cons, quiver.path.comp_nil], },\nend\n\nlemma allowable.derivative_smul {β γ : Iic_index α} (A : quiver.path (β : type_index) γ)\n  (π : allowable β) {X : Type*} [mul_action (struct_perm γ) X] (x : X) :\n  allowable.derivative A π • x = struct_perm.derivative A π.to_struct_perm • x :=\nby rw allowable.derivative_to_struct_perm; refl\n\nlemma to_struct_perm_smul_f_map {β : Iic_index α} (γ : Iio_index α) (δ : Iio α)\n  (hγ : (γ : type_index) < β) (hδ : (δ : type_index) < β) (hγδ : γ ≠ δ)\n  (π : allowable β) (t : tangle γ) :\n  (struct_perm.derivative (quiver.path.nil.cons hδ) π.to_struct_perm) •\n    f_map (subtype.coe_injective.ne hγδ) t =\n    f_map (subtype.coe_injective.ne hγδ) (allowable_derivative (β : Iic_index α) γ hγ π • t) :=\nbegin\n  rw allowable_derivative_eq β δ hδ π,\n  exact smul_f_map γ δ hγ hδ hγδ π t,\nend\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6654105720171531, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.32750718893003083}}
{"text": "import substances math.fuzzy\nopen set topological_space classical\nset_option pp.generalized_field_notation true\nlocal attribute [instance] prop_decidable\nnoncomputable theory\n\n-- tentative idea for formalizing ppc via intensional similarities\n-- in a variant of nuclear trope theory\n\nnamespace ontology\n\nvariable {ω : ontology}\n\nnamespace iontology\n/-- Intensional similarity relation -/\nstructure sim (Ω : ω.iontology) :=\n  (r : Ω.ientity → Ω.ientity → ω.world → fuzzy)\n  (reflexive : ∀ {e : Ω.ientity} {w}, e.exists w → r e e w = 1)\n  (symmetric : ∀ {e₁ e₂}, r e₁ e₂ = r e₂ e₁)\n  -- I think this follows from the stronger property of transitivity discussed by Zadeh\n  (triangular : ∀ (w) e₁ e₂ e₃, ((r e₁ e₂ w) : ℝ) * (r e₂ e₃ w) ≤ r e₁ e₃ w)\n  (tolerance : fuzzy  := 0)\n  -- TODO: prove this as a theorem, follows from triangular inequality\n  -- (subst : ∀ e₁ e₂, r e₁ e₂ = 1 → ∀ e₃, r e₁ e₃ = r e₂ e₃)\n  -- these principles are probably wrong, will think about them later.\n  -- consubstantial entities with the same essence are exactly similar\n  -- (coessence : ∀ e₁ e₂, 0 < r e₁ e₂ → e₁ ≈ e₂ → r e₁ e₂ = 1)\n  -- -- substances do not have the same essence as their accidents\n  -- (essence : ∀ e₁ e₂, 0 < r e₁ e₂ → e₁ ≈ e₂ → e₁ ⇎ e₂ → ¬ e₁.up.compatible e₂)\n\nattribute [class] sim\nattribute [simp, refl] sim.reflexive\nattribute [simp, symm] sim.symmetric\n\nsection basic\nvariables {Ω : ω.iontology} [sim : Ω.sim] (e : Ω.ientity)\ninclude sim\n\n/-- An entity is said to be **wholly dissimilar** to another if \n    their similarity is always 0. -/\ndef wdiss (e' : Ω.ientity) : Prop := ∀ w, 0 = sim.r e e' w\n\n/-- An entity is said to be **similar** to another, \n    if it is not wholly dissimilar to it, i.e. if\n    their similarity is strictly greater than 0. -/\ndef similar (e' : Ω.ientity) : Prop := ¬ wdiss e e'\n\n/-- An entity is said to be **exactly similar** to, or **indiscernible** from, \n    another if their similarity is always 1. -/\ndef esimilar (e' : Ω.ientity) : Prop := ∀ w, 1 = sim.r e e' w\n\nlocal infixr ` ≅ `:50 := similar\nlocal infixr ` ≇ `:50 := wdiss\n-- TODO: maybe set == notation for esimilar.\n\n-- fuzzy needs has_Inf. Will need to migrate to newer\n-- version of mathlib to make this work properly.\n/-- Infimum degree of similarity between two entities.\n    Introduced so we can get rid of possible worlds. -/\ndef ientity.sim (e₁ : Ω.ientity) (e₂ : Ω.ientity) : fuzzy := sorry --Inf $ sim.r e₁ e₂ '' univ\n\n-- -- TODO: check later if this is indeed a good definition\n-- /-- A **bare particular** is an entity wholly dissimilar to any entity distinct from itself. -/\n-- def ientity.bare : Prop := ∀ e', e' ≠ e → e' ≇ e\n\n-- /-- A **trope like** entity is a non-bare particular which is incompatible with any entity\n--     distinct, similar and consubstantial to itself. -/\n-- def ientity.trope_like : Prop := ¬ e.bare ∧ ∀ e', e' ≠ e → e' ≈ e → e' ≅ e → ¬ e'.up.compatible e\n\n-- /-- A **strongly trope like** entity is an entity which is only similar to trope like entities. -/\n-- def ientity.strope_like : Prop := ∀ e', e' ≅ e → e'.trope_like\n\n-- /-- A **bare like** entity is an entity wholly dissimilar from all trope like entities,\n--     which is however not necessarily a bare particular. -/\n-- def ientity.substratum : Prop := ∀ e' : Ω.ientity, e'.trope_like → e' ≇ e\n\n\n-- substrata in the same nucleus (i.e. equivalent) have the same essence\n-- def substratum_sim : Prop := ∀ e₁ e₂ : Ω.ientity, e₁.substratum → e₂.substratum → e₁ ≡ e₂ → e₁ ≅ e₂\n-- -- whatever \n-- def sim_closure : Prop := ∀ e₁ e₂ : Ω.ientity, e₁.trope → e₁ ≅ e₂ → e₂.trope\n\n-- TODO: we may need a notion of the origin trope being more perfect than\n-- the caused entity.\n-- def ppc (c : Ω.ientity → Ω.ientity → ω.event) : Prop :=\n--   ∀ e₁ e₂, ⋄c e₁ e₂ → \n--   ∃ trope, trope ≈ e₁ ∧ trope ≅ e₂ ∧ trope ≠ e₂ \n--   ∧ trope.strope_like ∧ c e₁ e₂ ⇒' trope.exists ∧ \n--   ∀ e₃, c e₃ trope ∩ c e₁ e₂ ⇒ c e₃ e₂\n\n-- needs has_Inf\n/-- Infimum possible degree of similarity among entities of a set. -/\ndef inf_sim (C : set Ω.ientity) : fuzzy := sorry --Inf {d : fuzzy | ∃ (w : ω.world) e₁ e₂ ∈ C, sim.r e₁ e₂ w = d}\n\n/-- Infimum degree of similarity among entities in a possible world. -/\ndef sim.inf_sim_at (sim : Ω.sim) (w : ω.world) : fuzzy := sorry --Inf {d : fuzzy | ∃ e₁ e₂ ∈ w, sim.r e₁ e₂ w = d}\n\n-- needs has_Sup\n/-- Supremum possible degree of similarity among entities of a set. -/\ndef sup_sim (C : set Ω.ientity) : fuzzy := sorry -- Sup {d : fuzzy | ∃ (w : ω.world) e₁ e₂ ∈ C, e₁ ≠ e₂ ∧ sim.r e₁ e₂ w = d}\n\n/-- Supremum degree of similarity among distinct entities in a possible world. -/\ndef sim.sup_sim_at (sim : Ω.sim) (w : ω.world) : fuzzy := sorry -- Sup {d : fuzzy | ∃ e₁ e₂ ∈ w, e₁ ≠ e₂ ∧ sim.r e₁ e₂ w = d}\n\n-- needs has_Sup\n/-- Supremum possible degree of similarity between a set and an entity. -/\ndef sdsim (e : Ω.ientity) (C : set Ω.ientity) : fuzzy := sorry --Sup {d : fuzzy | ∃ (w : ω.world) e' ∈ C, sim.r e e' w = d}\n\n/-- Supremum degree of similarity between an entity and the entities which exist in a possible world. -/\ndef sdsim_at (e : Ω.ientity) (w : ω.world) : fuzzy := sorry --Sup {d : fuzzy | ∃ e' ∈ w, sim.r e e' w = d}\n\n/-- A **resemblance class** is a set of entities which always resemble each other at least as much as they\n    might resemble any entity outside of the set. -/\ndef rclass (C : set Ω.ientity) : Prop := \n  ∀ e ∈ C, sdsim e (-C) ≤ inf_sim C\n\n/-- A **strict resemblance class** is a set of entities which always resemble each other more than they\n    might resemble any entity outside of the set. -/\ndef srclass (C : set Ω.ientity) : Prop := \n  ∀ e ∈ C, sdsim e (-C) < inf_sim C\n\n-- don't know if we will need it. Left incomplete\n-- /-- A **local resemblance class** is a set of entities which, within some world `w`, \n--     resemble each other more than they\n--     resemble, in `w`, any entity outside of the set. -/\n-- def lrclass (C : set Ω.ientity) (w : ω.world) : Prop := \n--   ∀ e ∈ C, sdsim e (-C) ≤ inf_sim C\n\n/-- A set of entities is **simply independent** if its distinct elements are pairwise independent, i.e. incomparable. -/\ndef sindependent (C : set Ω.ientity) : Prop := ∀ e₁ e₂ ∈ C, e₁ ≠ e₂ → e₁ ≢ e₂\n\n/-- A **clique** of degree **d**, or d-clique, is a set \n    of entities which are similar to each \n    other to degree at least **d** in any possible world. -/\ndef clique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  ∀ (w : ω.world) (e₁ e₂ : Ω.ientity), \n  e₁ ∈ C → e₂ ∈ C → d ≤ sim.r e₁ e₂ w\n\n/-- An **existential clique** of degree **d**, or d-eclique, is a set \n    of entities which are similar to each \n    other to degree at least **d** in any possible\n    world in which both exist.\n    Every clique is an existential clique, \n    but the converse need not hold. -/\ndef eclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  ∀ (w : ω.world) (e₁ e₂ : Ω.ientity), \n  e₁ ∈ C → e₂ ∈ C → e₁.exists w → e₂.exists w →\n  d ≤ sim.r e₁ e₂ w\n\n/-- A **maximum clique** of degree **d**, or d-mclique, is a d-clique which cannot be enlarged. -/\ndef mclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  clique d C ∧ ∀ C' : set Ω.ientity, clique d C' → C ⊆ C' → C = C'\n\n/-- An **analogical clique** of degree **d**, or d-aclique, is a d-mclique\n    which is also a resemblance class. -/\ndef aclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  mclique d C ∧ rclass C\n\n/-- A **strict analogical clique** of degree **d**, or d-aclique, is a d-mclique\n    which is also a strict resemblance class. -/\ndef saclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  mclique d C ∧ srclass C\n\n/-- A **near perfect clique** of degree **d**, or d-npclique, is an infinite d-saclique. -/\ndef npclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  set.infinite C ∧ saclique d C\n\n/-- An entity is a **trope** if and only if its similarity to any other \n    entities does not vary across time and worlds (beyond the allowed tolerance). -/\ndef ientity.trope (e : Ω.ientity) : Prop := ∀ e' w w', abs ((sim.r e e' w : ℝ) - sim.r e e' w') ≤ sim.tolerance\n\n-- TODO: consider bumping the definition below to full independence \n-- (lack of entailment between aggregates, lack of generic dependences) later.\n-- Also consider specifying that the clique should not contain entities from\n-- different corners of the ontological square. \n-- These are some of the conditions I term \"univocity conditions\".\n-- For now, it won't matter much to produce a very strict definition of univocity\n-- for perfect cliques. The basic idea of them can be expressed just \n-- with the following conditions.\n\n/-- A **perfect clique** of degree **d**, or d-pclique, is a simply independent d-npclique,\n    which does not contain both tropes and non-tropes. -/\ndef pclique (d : fuzzy) (C : set Ω.ientity) : Prop :=\n  npclique d C ∧ sindependent C ∧ \n  ¬ ∃ e t : Ω.ientity, e ∈ C ∧ t ∈ C ∧ t.trope ∧ ¬e.trope\n\n/-- The **degrees of analogy** of an entity **e** is the set of degrees for which **e** has a near perfect clique.\n    An entity is said to be analogically similar to **e** if it resembles the things resembling **e** to at least\n    some of these degrees. -/\ndef ientity.danalogy (e : Ω.ientity) : set fuzzy := {d : fuzzy | ∃ (C : set Ω.ientity), e ∈ C ∧ npclique d C}\n\n/-- The **analogies** of an entity **e** is the set of near perfect cliques containing **e**.\n    An entity is said to analogically similar to **e** if it shares a near perfect clique with **e**. -/\ndef ientity.analogies (e : Ω.ientity) : set $ set Ω.ientity := {C : set Ω.ientity | ∃ d, e ∈ C ∧ npclique d C}\n\n/-- The **degrees of genera** of an entity **e** is the set of degrees for which **e** has a perfect clique.\n    An entity is said to be in one of the same genus as **e** if it resembles the things resembling **e** to at least\n    some of these degrees. -/\ndef ientity.dgenera (e : Ω.ientity) : set fuzzy := {d : fuzzy | ∃ (C : set Ω.ientity), e ∈ C ∧ pclique d C}\n\n/-- The **genera** of an entity **e** is the set of perfect cliques containing **e**.\n    An entity is said to be in one of the same genus as **e** if it shares a perfect clique with **e**. -/\ndef ientity.genera (e : Ω.ientity) : set $ set Ω.ientity := {C : set Ω.ientity | ∃ d, e ∈ C ∧ pclique d C}\n\n/-- An entity is said to be **quasi-unintelligible** if it does not admit analogies of degree greater than 0.\n    In this case, it either has no essence, or at least its essence cannot be explained by \n    resemblance nominalism, but only by trope-resemblance nominalism. -/\ndef ientity.qunint (e : Ω.ientity) : Prop := ¬ ∃ d ∈ e.danalogy, (0 : fuzzy) < d\n\n/-- An entity is said to be **unintelligible** if it quasi-unintelligible and is not \n    existentially equivalent to any non-quasi-unintelligible tropes.\n    If it were equivalent to some such tropes, i.e. if it had *essential* tropes,\n    its essence might be explainable in terms of the essences of its tropes.\n    An unintelligible entity however has no essence of any sort. -/\ndef ientity.unint (e : Ω.ientity) : Prop :=\n  e.qunint ∧ ∀ t : Ω.ientity, t.trope → t ⇔ e → t.qunint\n\n/-- An entity is said to be **partially intelligible** if it is not unintelligible. -/\ndef ientity.pint (e : Ω.ientity) : Prop := ¬ e.unint\n\n/-- An entity is said to be **bare** if it is not a member of any genus.\n    In this case, it either has no univocal essence, or at least its essence cannot be explained by \n    resemblance nominalism, but only by trope-resemblance nominalism. -/\ndef ientity.bare (e : Ω.ientity) : Prop := e.genera = ∅\n\n/-- An entity is said to be **fully bare** if it is\n    bare and is not existentially equivalent to any non-bare tropes.\n    If it were equivalent to some non-bare tropes, i.e. if it had *essential* tropes, \n    its essence might be explainable in terms of the essences of its tropes. \n    A fully bare entity however has no univocal essence of any sort (other than its haecceity). -/\ndef ientity.fbare (e : Ω.ientity) : Prop := \n  e.bare ∧ ∀ t : Ω.ientity, t.trope → t ⇔ e → t.bare\n\n/-- An entity is said to be **normal** if it has a genus. -/\ndef ientity.normal (e : Ω.ientity) : Prop := e.genera.nonempty\n\n/-- An entity is said to be (fully) **intelligible** if it is\n    partially intelligible and normal if it is a trope.\n    We should expect all entities to be intelligible. -/\ndef ientity.int (e : Ω.ientity) : Prop := e.pint ∧ e.trope → e.normal\n\n-- TODO: revise the following two definitions to use genera directly in case they don't\n-- define what is intended:\n\n/-- An entity is said to be **generically unclassifiable** if it doesn't have a greatest genus. -/\ndef ientity.gunclass (e : Ω.ientity) : Prop := ¬ ∃ d ∈ e.dgenera, ∀ d' ∈ e.dgenera, d ≤ d'\n\n/-- An entity is said to be **specifically unclassifiable** if it doesn't have an infima species. -/\ndef ientity.sunclass (e : Ω.ientity) : Prop := ¬ ∃ d ∈ e.dgenera, ∀ d' ∈ e.dgenera, d' ≤ d\n\n/-- An entity is said to be **partially unclassifiable** if it is either \n    specifically or generically unclassifiable. -/\ndef ientity.punclass (e : Ω.ientity) : Prop := e.gunclass ∨ e.sunclass\n\n/-- An entity is said to be **unclassifiable** if it is both \n    specifically or generically unclassifiable. -/\ndef ientity.unclass (e : Ω.ientity) : Prop := e.gunclass ∧ e.sunclass\n\n/-- An entity is said to be **classifiable** if it is not partially unclassifiable. -/\ndef ientity.class (e : Ω.ientity) : Prop := ¬ e.punclass\n\n/-- An entity is said to be **partially classifiable** if it is not unclassifiable. -/\ndef ientity.pclass (e : Ω.ientity) : Prop := ¬ e.unclass\n\n/-- An entity is said to be **fully classifiable** if it is normal and has a finite number of genera.\n    It can then be classified taxonomically without appeal to its tropes. -/\ndef ientity.fclass (e : Ω.ientity) : Prop := e.genera.finite ∧ e.normal\n\nend basic\n\nsection principles\n\nvariables {Ω : ω.iontology} [sim : Ω.sim]\ninclude sim\n\ndef esse_of_possibilia : Prop := ∀ (e₁ e₂ : Ω.ientity), 0 < e₁.sim e₂\n\nend principles\n\nsection basic_theorems\n\nvariables {Ω : ω.iontology} [sim : Ω.sim]\ninclude sim\n\n/-- **Universal Analogy of Being** -/\nlemma uab : ∀ e, ∃ (d : fuzzy) (C : set Ω.ientity), aclique d C ∧ e ∈ C :=\n  begin\n    intros e, use [0, univ], simp,\n    constructor, constructor,\n      simp [clique], intros,\n        admit,\n      intros, apply eq_of_subset_of_subset,\n      assumption,\n      simp,\n    clear e, simp [rclass], intro e,\n    suffices c : sdsim e ∅ = 0, \n      rw c,\n      -- needs appropriate type classes for this to work:\n      admit, --exact zero_le (inf_sim univ),\n    admit,\n  end\n\n-- TODO: It seems that we could prove the following result for acliques, \n-- if we assume full blown transitivity of the similarity relation. \n-- The principle is that the similarity between x and z, is ≤ to the\n-- least of the similarities between x and y, and z and y, for all y. This implies that if \n-- the similarities between x and y, and z and y, are the same, then the similarity between x and z\n-- should be at least as high. In the second lemma we however know that the similarity between \n-- t₁ and t₂ is strictly capped at d, and yet that they should both be at least d-similar to t, which\n-- is absurd given transitivity, but not given mere triangularity.\n\n-- TODO: It seems possible that we might also just be able to prove that under transitivity\n-- all acliques are sacliques. This remains to be investigated.\n\n-- admits are only due to the type classes problem.\nlemma saclique_disjoint : ∀ d (C C' : set Ω.ientity), C ≠ C' → saclique d C → saclique d C' → C ∩ C' = ∅ :=\n  begin\n    intros d C C' neq h₁ h₂,\n    have c₁ : ∃ t₁ ∈ C, t₁ ∉ C',\n      by_contradiction contra,\n      push_neg at contra,\n      replace h₁ := h₁.1.2,\n      replace h₂ := h₂.1.1,\n      specialize h₁ C' h₂ contra,\n      contradiction,\n    have c₂ : ∃ t₂ ∈ C', t₂ ∉ C,\n      by_contradiction contra,\n      push_neg at contra,\n      replace h₂ := h₂.1.2,\n      replace h₁ := h₁.1.1,\n      specialize h₂ C h₁ contra,\n      cases h₂, contradiction,\n    by_contradiction contra,\n    simp [ext_iff] at contra,\n    obtain ⟨t, h₁t, h₂t⟩ := contra,\n    obtain ⟨t₁, h₁t₁, h₂t₁⟩ := c₁,\n    obtain ⟨t₂, h₁t₂, h₂t₂⟩ := c₂,\n    replace h₁ := h₁.2, simp [srclass] at h₁,\n    specialize h₁ t h₁t,\n    replace h₂ := h₂.2, simp [srclass] at h₂,\n    specialize h₂ t h₂t,\n    have c₁ : t.sim t₂ ≤ sdsim t (-C), admit,\n    have c₁' : inf_sim (C) ≤ t.sim t₁, admit,\n    have c₁'' : t.sim t₂ < t.sim t₁,\n      have := lt_of_le_of_lt c₁ h₁,\n      exact lt_of_lt_of_le this c₁',\n    have c₂ : t.sim t₁ ≤ sdsim t (-C'), admit,\n    have c₂' : inf_sim (C') ≤ t.sim t₂, admit,\n    have c₂'' : t.sim t₁ < t.sim t₂,\n      have := lt_of_le_of_lt c₂ h₂,\n      exact lt_of_lt_of_le this c₂',\n    have c₃ := lt_trans c₁'' c₂'',\n    simp [lt_irrefl] at c₃,\n    contradiction,\n  end\n\nlemma saclique_not_subset : ∀ d d' (C C' : set Ω.ientity), C ≠ C' → saclique d C → saclique d' C' → C ∩ C' ≠ ∅ → d ≤ d' → ¬ C ⊆ C' := sorry\n\nlemma saclique_ssubset : ∀ d d' (C C' : set Ω.ientity), saclique d C → saclique d' C' → C ∩ C' ≠ ∅ → d ≤ d' → C' ⊆ C :=\n  begin\n    intros d d' C C' h₁ h₂ h₃ h₄,\n    by_cases h : C = C', cases h, refl,\n    have c₁ : ∃ t₁ ∈ C, t₁ ∉ C',\n      by_contradiction absurd,\n      push_neg at absurd,\n      have c := saclique_not_subset d d' C C' h h₁ h₂ h₃ h₄,\n      contradiction,\n    by_contradiction contra,\n    simp [has_subset.subset, set.subset] at contra,\n    simp [ext_iff] at h₃,\n    obtain ⟨t, h₁t, h₂t⟩ := h₃,\n    obtain ⟨t₁, h₁t₁, h₂t₁⟩ := c₁,\n    obtain ⟨t₂, h₁t₂, h₂t₂⟩ := contra,\n    replace h₁ := h₁.2, simp [srclass] at h₁,\n    specialize h₁ t h₁t,\n    replace h₂ := h₂.2, simp [srclass] at h₂,\n    specialize h₂ t h₂t,\n    have c₁ : t.sim t₂ ≤ sdsim t (-C), admit,\n    have c₁' : inf_sim (C) ≤ t.sim t₁, admit,\n    have c₁'' : t.sim t₂ < t.sim t₁,\n      have := lt_of_le_of_lt c₁ h₁,\n      exact lt_of_lt_of_le this c₁',\n    have c₂ : t.sim t₁ ≤ sdsim t (-C'), admit,\n    have c₂' : inf_sim (C') ≤ t.sim t₂, admit,\n    have c₂'' : t.sim t₁ < t.sim t₂,\n      have := lt_of_le_of_lt c₂ h₂,\n      exact lt_of_lt_of_le this c₂',\n    have c₃ := lt_trans c₁'' c₂'',\n    simp [lt_irrefl] at c₃,\n    contradiction,\n\n  end\n\n\nend basic_theorems\n\nsection being\n\n-- We lift tropes to a structure to generate a namespace for dot notation:\nstructure trope (Ω : ω.iontology) [sim : Ω.sim] :=\n  (up : Ω.ientity)\n  (p : up.trope)\nvariables {Ω : ω.iontology} [sim : Ω.sim] (t : Ω.trope)\ninclude sim\n\n/-- A **proper trope** is a normal trope incompatible with any distinct consubstantial trope in the same genus as itself. -/\ndef trope.proper (t : Ω.trope) : Prop := \n  t.normal ∧ ∀ (C) d ∈ t.genera, ptclique d C t → \n  ∀ t' ∈ C, t' ≠ t → t'.up ≈ t.up → ¬⋄(t'.up.exists ∩ t.up.exists) \n\n/-- A **composable trope** is an improper normal trope. \n    Composable tropes \"compose\" to achieve some characteristic rather \n    than being singularly responsible for the characteristic.\n    A 1kg composable trope, for instance, could be composed with another 1kg composable trope,\n    to characterize its object as being 2kg. -/\ndef trope.composable (t : Ω.trope) : Prop := ¬t.proper ∧ t.normal\n\n/-- A **complemented trope** is a proper trope for which there exists another distinct consubstantial proper trope sharing a genus. -/\ndef trope.complemented (t : Ω.trope) : Prop := \n  t.proper ∧ ∃ (C : set Ω.trope) d ∈ t.genera, t ∈ C ∧ ptclique d C ∧ \n  ∃ t' ∈ C, t' ≠ t ∧ t'.up ≈ t.up ∧ t'.proper\n\n/-- An **eminent** trope is a composable trope non-vacuously \n    compatible with all complemented tropes in all of its genera.\n    Naturally, they are not consubstantial to these complemented tropes.\n    Mental tropes are eminent tropes, for consider the trope in virtue of which you are thinking about a square, \n    this appears to be composable with a trope in virtue of which you are thinking about a circle, \n    since they compose to characterize you as thinking about some square and some circle \n    (I do not mean to say you are thinking about a squared-circle). These tropes are members of the \n    same genus of \"mental geometric figure tropes\". Furthermore,\n    these tropes are compatible with the existence of extra-mental squares and circles, \n    which are objects characterized by complemented \"square\" and \"circular\" tropes. \n    It is unclear to us whether there are any non-eminent composable tropes, or any non-mental eminent tropes. -/\ndef trope.eminent (t : Ω.trope) : Prop := \n  t.composable ∧ ∀ (C) d ∈ t.genera, ptclique d C t → \n  (∃ (t' : Ω.trope), t' ∈ C ∧ t'.complemented) ∧ \n  ∀ (t' : Ω.trope), t' ∈ C → t'.complemented → ⋄(t'.up.exists ∩ t.up.exists)\n\n/-- A **formal** trope is a non-eminent normal trope. -/\ndef trope.formal (t : Ω.trope) : Prop := ¬t.eminent ∧ t.normal\n\n\n\n\n\nend being\n\n\nend iontology\n\n\n\nend ontology\n", "meta": {"author": "maxd13", "repo": "topological_ontology", "sha": "68d21c9a00024fba3aed301e16c31e05733c1786", "save_path": "github-repos/lean/maxd13-topological_ontology", "path": "github-repos/lean/maxd13-topological_ontology/topological_ontology-68d21c9a00024fba3aed301e16c31e05733c1786/src/metaphysics/similarity.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.32732692071890784}}
{"text": "import category_theory.abelian.homology\nimport for_mathlib.homotopy_category_pretriangulated\nimport category_theory.limits.constructions.epi_mono\nimport for_mathlib.homology_iso\nimport for_mathlib.homotopy_category_coproducts\nimport category_theory.abelian.homology\n\nnamespace category_theory\n\nopen category_theory.limits\n\nuniverses v u\n\nclass AB4 (A : Type u) [category.{v} A] [has_coproducts A] : Prop :=\n(cond : ∀ {α : Type v} (X Y : α → A) (f : Π a, X a ⟶ Y a)\n  (hf : ∀ a, mono (f a)), mono (sigma.desc $ λ a, f a ≫ sigma.ι Y a))\n\nvariables {A : Type u} [category.{v} A]\n\ninstance AB4_mono\n  [has_coproducts A] [AB4 A]\n  {α : Type v} (X Y : α → A) (f : Π a, X a ⟶ Y a)\n  [∀ a, mono (f a)] : mono (sigma.desc $ λ a, f a ≫ sigma.ι Y a) :=\nbegin\n  apply AB4.cond, assumption,\nend\n\nvariable (A)\nnoncomputable\ndef sigma_functor\n  [has_coproducts A]\n  (α : Type v) : (α → A) ⥤ A :=\n{ obj := λ X, sigma_obj X,\n  map := λ X Y f, sigma.desc $ λ a, f a ≫ sigma.ι _ a } .\nvariable {A}\n\ninstance sigma_functor_preserves_mono\n  [has_coproducts A] [AB4 A]\n  (α : Type v)\n  {X Y : α → A} (f : X ⟶ Y) [∀ a, mono (f a)] :\n  mono ((sigma_functor A α).map f) :=\ncategory_theory.AB4_mono X Y f\n\ninstance sigma_functor_preserves_epi\n  [has_coproducts A]\n  (α : Type v)\n  {X Y : α → A} (f : X ⟶ Y) [∀ a, epi (f a)] :\n  epi ((sigma_functor A α).map f) :=\nbegin\n  constructor, intros Z s t h,\n  apply colimit.hom_ext,\n  intros a,\n  dsimp [sigma_functor] at h,\n  apply_fun (λ e, colimit.ι _ a ≫ e) at h,\n  simp at h,\n  rwa cancel_epi at h,\nend\n\nlemma AB4_of_preserves_colimits_of_reflects_limits_of_AB4\n  {A B : Type u} [category.{v} A] [category.{v} B]\n  [has_coproducts A]\n  [has_coproducts B]\n  (F : A ⥤ B)\n  [preserves_colimits F]\n  [creates_limits F]\n  [has_limits B]\n  [AB4 B] : AB4 A :=\nbegin\n  constructor, introsI a X Y f hf,\n  let t := _, change mono t,\n  suffices : mono (F.map t),\n  { resetI, apply reflects_mono F },\n  let eX : F.obj (∐ λ (a : a), X a) ≅ (∐ λ a, F.obj (X a)) :=\n    (is_colimit_of_preserves F (colimit.is_colimit _)).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫ has_colimit.iso_of_nat_iso\n      (nat_iso.of_components (λ _, iso.refl _) _),\n  swap, { rintro i _ ⟨⟨⟨⟩⟩⟩, dsimp, simp, dsimp, simp },\n  let eY : F.obj (∐ λ (a : a), Y a) ≅ (∐ λ a, F.obj (Y a)) :=\n    (is_colimit_of_preserves F (colimit.is_colimit _)).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫ has_colimit.iso_of_nat_iso\n      (nat_iso.of_components (λ _, iso.refl _) _),\n  swap, { rintro i _ ⟨⟨⟨⟩⟩⟩, dsimp, simp, dsimp, simp },\n  let tt : (∐ λ a, F.obj (X a)) ⟶ (∐ λ a, F.obj (Y a)) :=\n    sigma.desc (λ a, F.map (f a) ≫ sigma.ι _ a),\n  haveI : mono tt,\n  { apply AB4.cond, intros a,\n    apply category_theory.preserves_mono F },\n  suffices : F.map t = eX.hom ≫ tt ≫ eY.inv,\n  { rw this, apply mono_comp },\n  apply (is_colimit_of_preserves F (colimit.is_colimit _)).hom_ext,\n  swap, apply_instance,\n  intros i,\n  erw [← F.map_comp, colimit.ι_desc, F.map_comp],\n  dsimp [eX, tt, t, eY, limits.is_colimit.cocone_point_unique_up_to_iso, is_colimit_of_preserves,\n    has_colimit.iso_of_nat_iso, is_colimit.map],\n  slice_rhs 0 1\n  { erw (is_colimit_of_preserves F (colimit.is_colimit _)).fac },\n  slice_rhs 0 1\n  { erw colimit.ι_desc },\n  dsimp [iso.refl],\n  simp,\nend\n\nnoncomputable\nexample {X Y Z : A} [abelian A]\n  (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0): cokernel (image_to_kernel' f g w) ≅ homology f g w :=\n  (homology_iso_cokernel_image_to_kernel' f g w).symm\n\nnoncomputable\ndef coproduct_kernel_comparison (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] (i : M) (X : α → homological_complex A S) :\n  (∐ λ (a : α), kernel ((X a).d_from i)) ⟶ kernel ((∐ X).d_from i) :=\nsigma.desc $ λ a, kernel.lift _ (kernel.ι _ ≫ (sigma.ι _ a : X a ⟶ ∐ X).f i)\nbegin\n  rw [category.assoc, (sigma.ι X a : X a ⟶ _).comm_from, ← category.assoc, kernel.condition,\n    zero_comp],\nend\n\n-- This should follow from the AB4 assumption\ninstance mono_coproduct_kernel_comparison (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i : M) (X : α → homological_complex A S) :\nmono (coproduct_kernel_comparison M S α i X) :=\nbegin\n  let ι : kernel ((∐ X).d_from i) ⟶ _ := kernel.ι _,\n  let t := _, change (mono t),\n  suffices : mono (t ≫ ι),\n  { resetI, apply mono_of_mono t ι },\n  let F : homological_complex A S ⥤ A := homological_complex.eval _ _ i,\n  let E : (∐ X).X i ≅ (∐ λ b, (X b).X i) :=\n    (is_colimit_of_preserves F (colimit.is_colimit\n      (discrete.functor X))).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫\n      has_colimit.iso_of_nat_iso (discrete.nat_iso $ λ b, iso.refl _),\n  suffices : t ≫ ι = sigma.desc (λ a, kernel.ι _ ≫ (sigma.ι (λ b, (X b).X i) a)) ≫ E.inv,\n  { rw this, apply_instance },\n  dsimp [t,ι, coproduct_kernel_comparison],\n  apply colimit.hom_ext, intros a,\n  simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, kernel.lift_ι, category.assoc,\n    has_colimit.iso_of_nat_iso_ι_inv_assoc, discrete.nat_iso_inv_app,\n    colimit.comp_cocone_point_unique_up_to_iso_inv, functor.map_cocone_ι_app,\n    colimit.cocone_ι, homological_complex.eval_map],\n  dsimp, simp only [category.id_comp],\nend\n\nlemma eval_next_aux_none\n  (A : Type u) [category.{v} A] [abelian A] {M : Type*}\n  (S : complex_shape M) (X : homological_complex A S)\n  (i : M) (h : S.next i = none) : is_zero (X.X_next i) :=\nbegin\n  dsimp [homological_complex.X_next],\n  simp [h],\n  dsimp [homological_complex.X_next._match_1],\n  exact is_zero_zero A,\nend\n\nnoncomputable\ndef eval_next (A : Type u) [category.{v} A] [abelian A] {M : Type*}\n  (S : complex_shape M) (i : M) :\n  homological_complex A S ⥤ A :=\n{ obj := λ X, X.X_next i,\n  map := λ X Y f, f.next i,\n  map_id' := begin\n    intros X,\n    rcases h : S.next i with _ | ⟨j,w⟩,\n    { apply limits.is_zero.eq_of_src, apply eval_next_aux_none, exact h },\n    { simp [homological_complex.hom.next_eq _ w], },\n  end,\n  map_comp' := begin\n    intros X Y Z f g,\n    rcases h : S.next i with _ | ⟨j,w⟩,\n    { apply limits.is_zero.eq_of_src, apply eval_next_aux_none, exact h },\n    { simp [homological_complex.hom.next_eq _ w] }\n  end }\n\nopen_locale zero_object\n\nnoncomputable\ndef preserves_colimits_of_shape_const_zero_aux\n  (α : Type v) (M : Type*) (S : complex_shape M)\n  [abelian A] [has_coproducts A]\n  (K : discrete α ⥤ homological_complex A S) :\n  is_colimit (((functor.const _).obj (0 : A)).map_cocone (colimit.cocone K)) :=\n{ desc := λ S, 0,\n  fac' := λ S j, (is_zero_zero _).eq_of_src _ _,\n  uniq' := λ S m hm, (is_zero_zero _).eq_of_src _ _ }\n\nnoncomputable\ninstance preserves_colimits_of_shape_const_zero\n  (α : Type v) (M : Type*) (S : complex_shape M) [abelian A] [has_coproducts A] :\n  preserves_colimits_of_shape (discrete α)\n  ((functor.const _).obj 0 : homological_complex A S ⥤ A) :=\nbegin\n  constructor, intros K,\n  apply preserves_colimit_of_preserves_colimit_cocone\n    (colimit.is_colimit K),\n  apply preserves_colimits_of_shape_const_zero_aux,\nend\n\nnoncomputable\ndef eval_next_iso_of_none\n  (M : Type*) (S : complex_shape M) [abelian A] (i : M)\n  (h : S.next i = none) :\n  eval_next A S i ≅ (functor.const _).obj 0 :=\nnat_iso.of_components\n(λ X, limits.is_zero.iso_zero $ eval_next_aux_none _ _ _ _ h)\nbegin\n  intros X Y f,\n  apply is_zero.eq_of_tgt,\n  apply is_zero_zero,\nend\n\nnoncomputable\ndef eval_next_iso_of_some\n  (M : Type*) (S : complex_shape M) [abelian A] (i j : M) (w)\n  (h : S.next i = some ⟨j,w⟩) :\n  eval_next A S i ≅ homological_complex.eval _ _ j :=\nnat_iso.of_components\n(λ X, homological_complex.X_next_iso _ w)\nbegin\n  intros X Y f,\n  dsimp [eval_next],\n  rw homological_complex.hom.next_eq _ w,\n  simp,\nend\n\nnoncomputable\ninstance eval_next_preserves_coproducts (α : Type v)\n  (M : Type*) (S : complex_shape M) [abelian A] [has_coproducts A] (i : M) :\n  preserves_colimits_of_shape (discrete α) (eval_next A S i) :=\nbegin\n  rcases h : S.next i with _ | ⟨j,w⟩,\n  { apply preserves_colimits_of_shape_of_nat_iso\n      (eval_next_iso_of_none M S i h).symm,\n    apply_instance },\n  { apply preserves_colimits_of_shape_of_nat_iso\n      (eval_next_iso_of_some M S i j w h).symm,\n    apply_instance }\nend\n\nlemma eval_prev_aux_none\n  (A : Type u) [category.{v} A] [abelian A] {M : Type*}\n  (S : complex_shape M) (X : homological_complex A S)\n  (i : M) (h : S.prev i = none) : is_zero (X.X_prev i) :=\nbegin\n  dsimp [homological_complex.X_prev],\n  simp [h],\n  dsimp [homological_complex.X_prev._match_1],\n  exact is_zero_zero A,\nend\n\nnoncomputable\ndef eval_prev (A : Type u) [category.{v} A] [abelian A] {M : Type*}\n  (S : complex_shape M) (i : M) :\n  homological_complex A S ⥤ A :=\n{ obj := λ X, X.X_prev i,\n  map := λ X Y f, f.prev i,\n  map_id' := begin\n    intros X,\n    rcases h : S.prev i with _ | ⟨j,w⟩,\n    { apply limits.is_zero.eq_of_src, apply eval_prev_aux_none, exact h },\n    { simp [homological_complex.hom.prev_eq _ w], },\n  end,\n  map_comp' := begin\n    intros X Y Z f g,\n    rcases h : S.prev i with _ | ⟨j,w⟩,\n    { apply limits.is_zero.eq_of_src, apply eval_prev_aux_none, exact h },\n    { simp [homological_complex.hom.prev_eq _ w] }\n  end }\n\nnoncomputable\ndef eval_prev_iso_of_none\n  (M : Type*) (S : complex_shape M) [abelian A] (i : M)\n  (h : S.prev i = none) :\n  eval_prev A S i ≅ (functor.const _).obj 0 :=\nnat_iso.of_components\n(λ X, limits.is_zero.iso_zero $ eval_prev_aux_none _ _ _ _ h)\nbegin\n  intros X Y f,\n  apply is_zero.eq_of_tgt,\n  apply is_zero_zero,\nend\n\nnoncomputable\ndef eval_prev_iso_of_some\n  (M : Type*) (S : complex_shape M) [abelian A] (i j : M) (w)\n  (h : S.prev i = some ⟨j,w⟩) :\n  eval_prev A S i ≅ homological_complex.eval _ _ j :=\nnat_iso.of_components\n(λ X, homological_complex.X_prev_iso _ w)\nbegin\n  intros X Y f,\n  dsimp [eval_prev],\n  rw homological_complex.hom.prev_eq _ w,\n  simp,\nend\n\nnoncomputable\ninstance eval_prev_preserves_coproducts (α : Type v)\n  (M : Type*) (S : complex_shape M) [abelian A] [has_coproducts A] (i : M) :\n  preserves_colimits_of_shape (discrete α) (eval_prev A S i) :=\nbegin\n  rcases h : S.prev i with _ | ⟨j,w⟩,\n  { apply preserves_colimits_of_shape_of_nat_iso\n      (eval_prev_iso_of_none M S i h).symm,\n    apply_instance },\n  { apply preserves_colimits_of_shape_of_nat_iso\n      (eval_prev_iso_of_some M S i j w h).symm,\n    apply_instance }\nend\n\nlemma exact_iff_exact_cokernel_desc [abelian A] (X Y Z : A)\n  (f : X ⟶ Y) (g : Y ⟶ Z) :\n  exact f g ↔ ∃ w, mono (cokernel.desc f g w) :=\nbegin\n  split,\n  { intros h, refine ⟨h.w, _⟩,\n    apply abelian.pseudoelement.mono_of_zero_of_map_zero,\n    intros a ha,\n    have h' : exact f (cokernel.π f) := abelian.exact_cokernel f,\n    replace h' := abelian.pseudoelement.pseudo_exact_of_exact h',\n    have hπ := abelian.pseudoelement.pseudo_surjective_of_epi (cokernel.π f),\n    obtain ⟨b,rfl⟩ := hπ a,\n    rw [← abelian.pseudoelement.comp_apply, cokernel.π_desc] at ha,\n    replace h := abelian.pseudoelement.pseudo_exact_of_exact h,\n    obtain ⟨b,rfl⟩ := h.2 _ ha,\n    rw [← abelian.pseudoelement.comp_apply, cokernel.condition],\n    simp only [abelian.pseudoelement.zero_apply] },\n  { rintros ⟨w,h⟩, rw abelian.exact_iff, refine ⟨w, _⟩, resetI,\n    rw ← cancel_mono (cokernel.desc f g w),\n    simp only [category.assoc, cokernel.π_desc, kernel.condition, zero_comp] }\nend\n\nlemma exact_iff_exact_of_cofork [abelian A] (X Y Z : A)\n  (f : X ⟶ Y) (g : Y ⟶ Z) (T : cokernel_cofork f) (hT : is_colimit T) :\n  exact f g ↔ ∃ w : f ≫ g = 0,\n  mono (hT.desc (cokernel_cofork.of_π g w)) :=\nbegin\n  let e : T.X ≅ cokernel f :=\n    hT.cocone_point_unique_up_to_iso (colimit.is_colimit _),\n  rw exact_iff_exact_cokernel_desc,\n  apply exists_congr (λ w, _),\n  have : hT.desc (cokernel_cofork.of_π g w) =\n    e.hom ≫ cokernel.desc f g w,\n  { dsimp [e, limits.is_colimit.cocone_point_unique_up_to_iso],\n    apply hT.hom_ext, intros i,\n    simp only [is_colimit.fac, is_colimit.fac_assoc, colimit.cocone_ι,\n      colimit.ι_desc] },\n  rw this,\n  split,\n  { introI _, apply_instance },\n  { introI, constructor,\n    intros Z a b h,\n    rw [← cancel_mono e.inv, ← cancel_mono (e.hom ≫ cokernel.desc f g w)],\n    simpa }\nend\n\nnoncomputable\ndef sigma_cokernel_cofork [abelian A] [has_coproducts A] [AB4 A]\n  {α : Type v} (X Y : α → A) (f : X ⟶ Y) :\n  cokernel_cofork ((sigma_functor A α).map f) :=\ncokernel_cofork.of_π\n(sigma.desc (λ a : α, cokernel.π (f a) ≫ sigma.ι (λ b, cokernel (f b)) a))\nbegin\n  apply colimit.hom_ext,\n  intros a,\n  dsimp [sigma_functor],\n  simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, category.assoc,\n    colimit.ι_desc, cokernel.condition_assoc, zero_comp,\n    comp_zero],\nend\n\nnoncomputable\ndef is_colimit_sigma_cokernel_cofork [abelian A] [has_coproducts A] [AB4 A]\n  {α : Type v} (X Y : α → A) (f : X ⟶ Y) :\n  is_colimit (sigma_cokernel_cofork X Y f) :=\nis_colimit_aux _\n(λ S, sigma.desc $ λ b, cokernel.desc _ (sigma.ι _ _ ≫ S.π)\n  begin\n    rw ← category.assoc,\n    let t := _, change t ≫ _ = _,\n    have ht : t = sigma.ι X b ≫ (sigma_functor A α).map f,\n    { dsimp [sigma_functor], simp },\n    rw ht, clear ht, clear t,\n    rw [category.assoc, S.condition, comp_zero],\n  end)\nbegin\n  intros S,\n  dsimp [sigma_cokernel_cofork],\n  apply colimit.hom_ext,\n  simp only [category.assoc, colimit.ι_desc, cofan.mk_ι_app, eq_self_iff_true,\n    cokernel_cofork.condition, comp_zero,\n    colimit.ι_desc_assoc, cokernel.π_desc, implies_true_iff],\nend\nbegin\n  intros S m hm,\n  apply colimit.hom_ext, intros a,\n  simp only [colimit.ι_desc, cofan.mk_ι_app],\n  apply coequalizer.hom_ext, simp only [coequalizer_as_cokernel, cokernel.π_desc],\n  simp_rw [← hm, ← category.assoc], congr' 1,\n  dsimp [sigma_cokernel_cofork],\n  simp only [colimit.ι_desc, cofan.mk_ι_app],\nend\n\nlemma exact_coproduct [abelian A] [has_coproducts A] [AB4 A]\n  {α : Type v} (X Y Z : α → A) (f : X ⟶ Y) (g : Y ⟶ Z)\n  (w : ∀ i, exact (f i) (g i)) :\n  exact ((sigma_functor A α).map f) ((sigma_functor A α).map g) :=\nbegin\n  let ι : (λ a : α, cokernel (f a)) ⟶ Z :=\n    λ a, (cokernel.desc _ (g a) (w a).w),\n  let π : Y ⟶ (λ a : α, cokernel (f a)) :=\n    λ a, (cokernel.π _),\n  haveI : ∀ a, mono (ι a),\n  { intros a,\n    suffices : ∃ w, mono (cokernel.desc (f a) (g a) w),\n    { obtain ⟨q,h⟩ := this, exact h },\n    rw ← exact_iff_exact_cokernel_desc, apply w },\n  have : mono ((sigma_functor A α).map ι),\n  { apply AB4.cond, assumption },\n  -- Now need to show that sigma functor commutes with cokernels\n  -- Use the fact that this commutes with cokernels to identify the source\n  -- with the cokernel of `f`.\n  -- Then use exact_iff_exact_cokernel_desc\n  rw exact_iff_exact_of_cofork\n    ((sigma_functor A α).obj X)\n    ((sigma_functor A α).obj Y)\n    ((sigma_functor A α).obj Z)\n    ((sigma_functor A α).map f)\n    ((sigma_functor A α).map g)\n    (sigma_cokernel_cofork _ _ f)\n    (is_colimit_sigma_cokernel_cofork _ _ f),\n  refine ⟨_,_⟩,\n  { apply colimit.hom_ext, intros a,\n    dsimp [sigma_functor],\n    simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, category.assoc, colimit.ι_desc, comp_zero],\n    rw [← category.assoc, (w a).w, zero_comp] },\n  simp only [exact_iff_exact_cokernel_desc] at w,\n  choose w₁ w₂ using w,\n  convert AB4.cond _ Z ι w₂ using 1,\n  apply colimit.hom_ext, intros a,\n  dsimp [is_colimit_sigma_cokernel_cofork, is_colimit_aux],\n  simp only [colimit.ι_desc, cofan.mk_ι_app],\n  apply coequalizer.hom_ext,\n  simp only [cokernel.π_desc, cokernel.π_desc_assoc],\n  dsimp [sigma_functor],\n  simp only [colimit.ι_desc, cofan.mk_ι_app],\nend\n\nsection\nopen_locale pseudoelement\ninstance epi_coproduct_kernel_comparison [has_coproducts A] [AB4 A]\n  (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] (i : M) (X : α → homological_complex A S) :\n  epi (coproduct_kernel_comparison M S α i X) :=\nbegin\n  /-\n     _ -ι₁-> _ -π₁-> _\n     |       |       |\n     t     E.inv   Q.inv\n     |       |       |\n     v       v       v\n     _ -ι₂-> _ -π₂-> _\n\n  -/\n  let t := _, change epi t,\n  let F : homological_complex A S ⥤ A := homological_complex.eval _ _ i,\n  let N : homological_complex A S ⥤ A := eval_next _ _ i,\n  let E : (∐ X).X i ≅ (∐ λ b, (X b).X i) :=\n    (is_colimit_of_preserves F (colimit.is_colimit\n      (discrete.functor X))).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫\n      has_colimit.iso_of_nat_iso (discrete.nat_iso $ λ b, iso.refl _),\n  let Q : (∐ X).X_next i ≅ (∐ λ b, (X b).X_next i) :=\n    (is_colimit_of_preserves N (colimit.is_colimit\n      (discrete.functor X))).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫\n      has_colimit.iso_of_nat_iso (discrete.nat_iso $ λ b, iso.refl _),\n  let ι₁ : (∐ λ (a : α), kernel ((X a).d_from i)) ⟶ (∐ λ b, (X b).X i) :=\n    sigma.desc (λ a, kernel.ι _ ≫ sigma.ι _ a),\n  let π₁ : (∐ λ b, (X b).X i) ⟶ (∐ λ b, (X b).X_next i) :=\n    sigma.desc (λ a, (X a).d_from i ≫ sigma.ι _ a),\n  let ι₂ : kernel ((∐ X).d_from i) ⟶ (∐ X).X i := kernel.ι _,\n  let π₂ : (∐ X).X i ⟶ (∐ X).X_next i := (∐ X).d_from i,\n  have sqι : ι₁ ≫ E.inv = t ≫ ι₂,\n  { apply colimit.hom_ext, intros a, dsimp [ι₁, E, t, ι₂],\n    simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, category.assoc,\n      has_colimit.iso_of_nat_iso_ι_inv_assoc, discrete.nat_iso_inv_app,\n      colimit.comp_cocone_point_unique_up_to_iso_inv, functor.map_cocone_ι_app,\n      colimit.cocone_ι, homological_complex.eval_map],\n    dsimp [coproduct_kernel_comparison],\n    simp only [category.id_comp, colimit.ι_desc_assoc, cofan.mk_ι_app, kernel.lift_ι] },\n  have sqπ : π₁ ≫ Q.inv = E.inv ≫ π₂,\n  { dsimp [π₁, Q, E, π₂, coproduct_kernel_comparison], apply colimit.hom_ext, intros a,\n    simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, category.assoc,\n      has_colimit.iso_of_nat_iso_ι_inv_assoc, discrete.nat_iso_inv_app,\n      colimit.comp_cocone_point_unique_up_to_iso_inv, functor.map_cocone_ι_app,\n      colimit.cocone_ι, colimit.comp_cocone_point_unique_up_to_iso_inv_assoc,\n      homological_complex.eval_map],\n    dsimp,\n    simp only [homological_complex.hom.comm_from, category.id_comp],\n    erw category.id_comp,\n    refl },\n  have e1 : exact ι₁ π₁,\n  { apply exact_coproduct, intros b, exact exact_kernel_ι },\n  have e2 : exact ι₂ π₂ := exact_kernel_ι,\n  have hι₂ := abelian.pseudoelement.pseudo_injective_of_mono ι₂,\n  replace e1 := abelian.pseudoelement.pseudo_exact_of_exact e1,\n  replace e2 := abelian.pseudoelement.pseudo_exact_of_exact e2,\n  have hEinv := abelian.pseudoelement.pseudo_surjective_of_epi E.inv,\n  have hQinv := abelian.pseudoelement.pseudo_injective_of_mono Q.inv,\n  apply abelian.pseudoelement.epi_of_pseudo_surjective,\n  -- Now we start the diagram chase.\n  intros x,\n  let x' := ι₂ x,\n  obtain ⟨y,hy⟩ := hEinv x',\n  have hy' : π₁ y = 0,\n  { apply hQinv,\n    simp only [abelian.pseudoelement.apply_zero, ← abelian.pseudoelement.comp_apply, sqπ],\n    rw [abelian.pseudoelement.comp_apply, hy],\n    dsimp only [x'], rw e2.1 },\n  obtain ⟨z,hz⟩ := e1.2 _ hy',\n  use z,\n  apply hι₂,\n  rw [← abelian.pseudoelement.comp_apply, ← sqι, abelian.pseudoelement.comp_apply, hz, hy],\nend\nend\n\ninstance is_iso_coproduct_kernel_comparison (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i : M) (X : α → homological_complex A S) :\nis_iso (coproduct_kernel_comparison M S α i X) :=\nis_iso_of_mono_of_epi _\n\nnoncomputable\ndef coproduct_homology_comparison (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] (i : M) (X : α → homological_complex A S) :\n  (∐ λ a : α, (X a).homology i) ⟶ (∐ X).homology i :=\nsigma.desc $ λ b, (homology_functor _ _ _).map $ sigma.ι _ _\n\nnoncomputable\ndef coproduct_homology_comparison_inv (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i : M) (X : α → homological_complex A S) :\n  (∐ X).homology i ⟶ (∐ λ a : α, (X a).homology i) :=\nhomology.desc' _ _ _ (inv (coproduct_kernel_comparison M S α i X) ≫\n  sigma.desc (λ b, homology.π' ((X b).d_to _) ((X b).d_from i)\n    (homological_complex.d_to_comp_d_from _ _) ≫ sigma.ι _ b))\nbegin\n  rw ← category.assoc, let t := _, change t ≫ _ = _,\n  let e : (∐ X).X_prev i ≅ ∐ (λ a, (X a).X_prev i) :=\n    (is_colimit_of_preserves (eval_prev A S i)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso\n      (colimit.is_colimit _) ≪≫ has_colimit.iso_of_nat_iso (discrete.nat_iso $ λ b, iso.refl _),\n  have ht : t = e.hom ≫ sigma.desc (λ b, _ ≫ sigma.ι _ b),\n  rotate 2,\n  { refine kernel.lift _ _ _, exact (X b).d_to i,\n    exact (X b).d_to_comp_d_from i },\n  { dsimp [t, e, limits.is_colimit.cocone_point_unique_up_to_iso,\n      has_colimit.iso_of_nat_iso, is_colimit.map, coproduct_kernel_comparison],\n    rw is_iso.comp_inv_eq,\n    apply (is_colimit_of_preserves (eval_prev A S i)\n      (colimit.is_colimit (discrete.functor X))).hom_ext,\n    intros j,\n    apply equalizer.hom_ext,\n    simp only [functor.map_cocone_ι_app, colimit.cocone_ι,\n      equalizer_as_kernel, category.assoc, kernel.lift_ι],\n    slice_rhs 1 2\n    { erw (is_colimit_of_preserves (eval_prev A S i) (colimit.is_colimit (discrete.functor X))).fac },\n    dsimp [eval_prev],\n    simp only [homological_complex.hom.comm_to, colimit.ι_desc, cocones.precompose_obj_ι,\n      nat_trans.comp_app, discrete.nat_iso_hom_app, colimit.cocone_ι, category.assoc,\n      cofan.mk_ι_app, kernel.lift_ι, kernel.lift_ι_assoc],\n    dsimp, simp only [category.id_comp] },\n  { rw ht,\n    dsimp [e, limits.is_colimit.cocone_point_unique_up_to_iso],\n    apply (is_colimit_of_preserves (eval_prev A S i) (colimit.is_colimit (discrete.functor X))).hom_ext,\n    intros j,\n    simp only [functor.map_cocone_ι_app, colimit.cocone_ι, category.assoc,\n      has_colimit.iso_of_nat_iso_hom_desc, comp_zero],\n    slice_lhs 1 2\n    { erw (is_colimit_of_preserves (eval_prev A S i) (colimit.is_colimit (discrete.functor X))).fac },\n    simp only [colimit.cocone_ι, colimit.ι_desc, cocones.precompose_obj_ι, nat_trans.comp_app,\n      cofan.mk_ι_app, category.assoc, homology.condition_π'_assoc, zero_comp, comp_zero] }\nend\n\nnoncomputable\ndef coproduct_homology_iso\n  (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i : M) (X : α → homological_complex A S) :\n  (∐ λ a : α, (X a).homology i) ≅ (∐ X).homology i :=\n{ hom := coproduct_homology_comparison _ _ _ _ _,\n  inv := coproduct_homology_comparison_inv _ _ _ _ _,\n  hom_inv_id' := begin\n    dsimp [coproduct_homology_comparison, coproduct_homology_comparison_inv],\n    apply colimit.hom_ext, intros a,\n    dsimp,\n    simp only [colimit.ι_desc_assoc, cofan.mk_ι_app, category.comp_id],\n    apply homology.hom_from_ext,\n    rw homology.map_eq_desc'_lift_left,\n    simp only [homological_complex.hom.sq_from_left, homology.π'_desc'_assoc],\n    let t := _, change t ≫ _ = _,\n    have ht : t = kernel.lift _ (kernel.ι _ ≫ _) _ ≫ homology.π' _ _ _,\n    rotate 2,\n    { let e : X a ⟶ ∐ X := sigma.ι _ a, exact e.f i },\n    { dsimp,\n      rw [category.assoc, (sigma.ι X a : X a ⟶ ∐ X).comm_from, ← category.assoc,\n        kernel.condition, zero_comp] },\n    { dsimp [t], apply homology.hom_to_ext,\n      simp only [category.assoc, homology.lift_ι, homological_complex.homology.π'_ι,\n        kernel.lift_ι_assoc] },\n    { rw ht, clear ht, clear t,\n      rw [category.assoc, homology.π'_desc', ← category.assoc],\n      let t := _, change t ≫ _ = _,\n      have ht : t = sigma.ι _ a,\n      { dsimp [t], rw is_iso.comp_inv_eq,\n        apply equalizer.hom_ext,\n        dsimp [coproduct_kernel_comparison],\n        simp only [category.assoc, kernel.lift_ι],\n        erw colimit.ι_desc_assoc,\n        dsimp,\n        simp only [kernel.lift_ι] },\n      rw ht, clear ht, clear t,\n      erw colimit.ι_desc,\n      refl }\n  end,\n  inv_hom_id' := begin\n    dsimp [coproduct_homology_comparison, coproduct_homology_comparison_inv],\n    apply homology.hom_from_ext,\n    simp only [homology.π'_desc'_assoc, category.assoc, category.comp_id,\n      is_iso.inv_comp_eq],\n    apply colimit.hom_ext, intros a,\n    dsimp,\n    simp only [homological_complex.hom.sq_from_right, homological_complex.hom.sq_to_right,\n      colimit.ι_desc_assoc, cofan.mk_ι_app, category.assoc, colimit.ι_desc, homology.π'_map],\n    erw colimit.ι_desc_assoc, refl,\n  end }\n\nnoncomputable\ndef is_colimit_homology_map_cocone  (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i : M) (X : α → homological_complex A S) :\n  is_colimit ((homology_functor A S i).map_cocone (colimit.cocone (discrete.functor X))) :=\n{ desc := λ E, (coproduct_homology_iso _ _ _ _ _).inv ≫ sigma.desc (λ a, E.ι.app _),\n  fac' := begin\n    intros E j,\n    dsimp [coproduct_homology_comparison_inv, coproduct_homology_iso],\n    apply homology.hom_from_ext,\n    rw homology.map_eq_desc'_lift_left,\n    simp only [homological_complex.hom.sq_from_left, homology.π'_desc'_assoc],\n    let t := _, change t ≫ _ = _,\n    have ht : t = kernel.lift _ (kernel.ι _ ≫ _) _ ≫ homology.π' _ _ _,\n    rotate 2,\n    { let e := (sigma.ι X j), exact e.f i },\n    { dsimp, rw [category.assoc],\n      erw (sigma.ι X j : X j ⟶ ∐ X).comm_from,\n      rw [← category.assoc, kernel.condition, zero_comp] },\n    { dsimp [t],\n      apply homology.hom_to_ext,\n      simp only [category.assoc, homology.lift_ι, homological_complex.homology.π'_ι,\n        kernel.lift_ι_assoc] },\n    rw ht, clear ht, clear t,\n    simp only [category.assoc, homology.π'_desc'_assoc],\n    simp only [← category.assoc],\n    let t := _, change (t ≫ _) ≫ _ = _,\n    have ht : t = sigma.ι _ j,\n    { dsimp [t], rw [is_iso.comp_inv_eq],\n      dsimp [coproduct_kernel_comparison],\n      erw colimit.ι_desc, refl },\n    rw ht, clear ht, clear t,\n    simp only [category.assoc], erw [colimit.ι_desc_assoc], dsimp,\n    rw [category.assoc, colimit.ι_desc], refl,\n  end,\n  uniq' := begin\n    intros E m hm,\n    rw iso.eq_inv_comp, dsimp [coproduct_homology_comparison, coproduct_homology_iso],\n    apply colimit.hom_ext, intros j, specialize hm j,\n    simpa only [←hm, colimit.ι_desc_assoc, cofan.mk_ι_app, colimit.ι_desc,\n      functor.map_cocone_ι_app, colimit.cocone_ι,\n      homology_functor_map],\n  end }\n\nnoncomputable\ninstance homology_functor_preserves_coproducts\n  (M : Type*) (S : complex_shape M) (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i) :\n  preserves_colimits_of_shape (discrete α)\n  (homology_functor A S i) :=\nbegin\n  constructor, intros K,\n  let E : K ≅ discrete.functor K.obj := discrete.nat_iso (λ i, iso.refl _),\n  suffices : preserves_colimit (discrete.functor K.obj) (homology_functor A _ i),\n  { apply preserves_colimit_of_iso_diagram _ E.symm, assumption },\n  apply preserves_colimit_of_preserves_colimit_cocone\n    (colimit.is_colimit (discrete.functor K.obj)),\n  apply is_colimit_homology_map_cocone,\nend\n\nnoncomputable\ndef is_colimit_homotopy_category_homology_functor_map_cocone\n  {α : Type v} [abelian A] [has_coproducts A] [AB4 A] (i)\n  (K : α → homotopy_category A (complex_shape.up ℤ)) :\n  is_colimit\n  ((homotopy_category.homology_functor A (complex_shape.up ℤ) i).map_cocone\n    (homotopy_category.colimit_cofan K)) :=\n{ desc := λ S,\n    (is_colimit_of_preserves (homology_functor A (complex_shape.up ℤ) i)\n    (colimit.is_colimit $ discrete.functor $ λ i, (K i).as)).desc ⟨S.X,\n    discrete.nat_trans $ λ i, S.ι.app i⟩,\n  fac' := begin\n    intros S j, dsimp,\n    erw (is_colimit_of_preserves (homology_functor A (complex_shape.up ℤ) i)\n      (colimit.is_colimit (discrete.functor (λ (i : α), (K i).as)))).fac,\n    refl,\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply (is_colimit_of_preserves (homology_functor A (complex_shape.up ℤ) i)\n      (colimit.is_colimit (discrete.functor (λ (i : α), (K i).as)))).hom_ext,\n    intros j,\n    erw (is_colimit_of_preserves (homology_functor A (complex_shape.up ℤ) i)\n      (colimit.is_colimit (discrete.functor (λ (i : α), (K i).as)))).fac,\n    dsimp, rw ← hm, refl,\n  end }\n\nnoncomputable\ninstance homotopy_category_homology_functor_preserves_coproducts\n  (α : Type v)\n  [abelian A] [has_coproducts A] [AB4 A] (i) :\n  preserves_colimits_of_shape (discrete α)\n  (homotopy_category.homology_functor A (complex_shape.up ℤ) i) :=\nbegin\n  constructor, intros K,\n  let E : K ≅ discrete.functor K.obj := discrete.nat_iso (λ i, iso.refl _),\n  suffices : preserves_colimit (discrete.functor K.obj) (homotopy_category.homology_functor A _ i),\n  { apply preserves_colimit_of_iso_diagram _ E.symm, assumption },\n  apply preserves_colimit_of_preserves_colimit_cocone\n    (homotopy_category.is_colimit_cofan K.obj),\n  apply is_colimit_homotopy_category_homology_functor_map_cocone,\nend\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/ab4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3271813365706012}}
{"text": "import complexity_class.tactic\n\nopen tencodable (encode decode)\nopen function\n\nvariables {C : complexity_class} {α β γ δ ε : Type} [tencodable α]\n  [tencodable β] [tencodable γ] [tencodable δ] [tencodable ε]\n\nnamespace complexity_class\nopen_locale complexity_class\nopen_locale tree\n\n@[complexity] lemma prod_mk : @prod.mk α β ∈ₑ C := C.id'\n@[complexity] lemma cons : @list.cons α ∈ₑ C := C.id'\n@[complexity] protected lemma encode : @encode α _ ∈ₑ C := C.id'\n@[complexity] lemma sigma_mk : @complexity_class.mem _ _ (α → β → (Σ _ : α, β)) _ _ _ sigma.mk C := C.id'\n\nlemma mem_iff_comp_encode {f : α → β} :\n  f ∈ₑ C ↔ (λ x, encode (f x)) ∈ₑ C := iff.rfl\n\nlemma mem_iff_comp_encode' {C : complexity_class} {β γ : Type} [tencodable γ] {f : β → α} {finv : α → option β} {linv : ∀ b, finv (f b) = some b} \n  {g : γ → β} : by haveI : tencodable β := tencodable.of_left_injection f finv linv; exact \n  g ∈ₑ C ↔ (λ x, f (g x)) ∈ₑ C := iff.rfl\n\n@[complexity] lemma tree_left : @tree.left unit ∈ₑ C :=\n⟨_, C.left, λ _, rfl⟩\n\n@[complexity] lemma tree_right : @tree.right unit ∈ₑ C :=\n⟨_, C.right, λ _, rfl⟩\n\n@[complexity] lemma tree_node : (tree.node ()) ∈ₑ C :=\n⟨_, C.id, λ _, rfl⟩\n\n@[complexity] lemma unit_rec {f : α → unit} {g : α → β} (hg : g ∈ₑ C) :\n  @complexity_class.mem _ _ (α → β) _ _ _ (λ x : α, @punit.rec (λ _, β) (g x) (f x)) C := \nby { convert hg, ext x, cases f x, refl, }\n\nprivate lemma eq_nil_aux : (=@tree.nil unit) ∈ₚ C :=\n⟨_, C.ite' C.id (C.prop_const $ encode tt)\n  (C.prop_const $ encode ff), λ x, by cases x; simp [has_uncurry.uncurry]⟩\n\n@[complexity] lemma coe_sort : (λ b : bool, (b : Prop)) ∈ₚ C :=\n⟨_, C.id, λ _, by simp [has_uncurry.uncurry]⟩\n\n@[complexity] lemma to_bool {P : α → Prop} [decidable_pred P] (hP : P ∈ₚ C) :\n  C.mem (λ x, to_bool (P x)) :=\nby { convert hP, ext, congr, }\n\n@[complexity] protected lemma band : C.mem (&&) :=\nby { complexity using (λ (b₁ b₂ : bool), if b₁ then b₂ else ff), cases b₁; simp, }\n\n@[complexity] protected lemma bnot : C.mem bnot :=\nby { complexity using (λ b, if b then ff else tt), cases b; refl, }\n\n/-- Complexity classes are closed under intersection -/\n@[complexity] protected lemma and {P₁ P₂ : α → Prop} (h₁ : P₁ ∈ₚ C) (h₂ : P₂ ∈ₚ C) :\n  (λ x, (P₁ x) ∧ (P₂ x)) ∈ₚ C :=\nby { classical, simp [← mem_iff_mem_pred], complexity, }\n\n/-- Complexity classes are closed under complementation -/\n@[complexity] protected lemma not {P : α → Prop} (h : P ∈ₚ C) : C.mem_pred (λ x, ¬(P x)) :=\nby { classical, simp [← mem_iff_mem_pred], complexity, }\n\n@[complexity] lemma eq_const_aux : ∀ (x : tree unit), (=x) ∈ₚ C\n| tree.nil := eq_nil_aux\n| (a △ b) :=\nbegin\n  have := (eq_const_aux a).comp tree_left, have := (eq_const_aux b).comp tree_right, have := @eq_nil_aux C,\n  complexity using (λ y, ¬(y = tree.nil) ∧ y.left = a ∧ y.right = b),\n  cases y; simp,\nend\n\n@[complexity] lemma eq_const {f : α → β} (hf : f ∈ₑ C) (y : β) :\n  C.mem_pred (λ x, (f x) = y) :=\nby simpa using (eq_const_aux (encode y)).comp hf\n\n@[complexity] lemma list_empty : @list.empty α ∈ₑ C :=\nby { classical, complexity using λ l, l = [], cases l; simp, }\n\n@[complexity] lemma option_is_none : @option.is_none α ∈ₑ C :=\nby { classical, complexity using λ x, x = none, cases x; simp, }\n\n@[complexity] lemma option_is_some : @option.is_some α ∈ₑ C :=\nby { complexity using λ x, !x.is_none, cases x; simp, }\n\n@[complexity] lemma tree_cases {f : α → tree unit} {g : α → β}\n  {h : α → unit → tree unit → tree unit → β} (hf : f ∈ₑ C) (hg : g ∈ₑ C) (hh : h ∈ₑ C) :\n  @complexity_class.mem α β (α → β) _ _ _ (λ x, @tree.cases_on unit (λ _, β) (f x) (g x) (h x)) C :=\nbegin\n  complexity using (λ x, if f x = tree.nil then g x else h x () (f x).left (f x).right),\n  rcases f x with (_|⟨⟨⟩, _, _⟩); simp,\nend\n\nlemma of_fin_cases [nonempty γ] (S : finset α)\n  {f : α → β → γ} (hf : ∀ {x}, x ∈ S → f x ∈ₑ C) :\n  ∃ f' : α → β → γ, f' ∈ₑ C ∧ ∀ {x}, x ∈ S → f x = f' x :=\nbegin\n  classical, inhabit γ,\n  induction S using finset.induction with x xs x_nmem ih,\n  { refine ⟨λ _ _, default, by complexity, _⟩, simp, },\n  rcases ih (λ _ h, hf (finset.mem_insert.mpr (or.inr h))) with ⟨f', pf, hf'⟩,\n  set g := f x, have : g ∈ₑ C := by { apply hf, simp, },\n  refine ⟨λ x' y, if x' = x then g y else f' x' y, by complexity, _⟩,\n  intro x', split_ifs with H, { simp [H], }, { simpa [H] using hf', },\nend\n\n@[complexity] lemma of_to_empty [H : is_empty β] (f : α → β) : f ∈ₑ C :=\n⟨_, C.nil, λ x, H.elim' (f x)⟩\n\n/-- A function on a `fintype` is in the complexity class iff each function is (i.e. we can always do\n  a finite amount of casework) -/\nlemma of_from_fintype [fintype α] {f : α → β → γ} (hf : ∀ x, f x ∈ₑ C) : f ∈ₑ C :=\nbegin\n  casesI is_empty_or_nonempty γ, \n  { apply of_to_empty, },\n  obtain ⟨f', pf, hf'⟩ := of_fin_cases (@finset.univ α _) (λ x _, hf x),\n  convert pf, apply funext, simpa using @hf',\nend\n\nlemma iff_fintype [fintype α] {f : α → β → γ} :\n  f ∈ₑ C ↔ ∀ x, f x ∈ₑ C :=\n⟨λ h x, h.comp₂ (C.const x) C.id', of_from_fintype⟩\n\nlemma of_from_fintype' [fintype α] (f : α → β) : f ∈ₑ C :=\nlet H : C.mem (λ x (_ : unit), f x) := of_from_fintype (λ x, C.const _)\n in H.comp₂ C.id' (C.const ())\n\n@[complexity] protected lemma bor : bor ∈ₑ C := of_from_fintype' _\n\n/-- Complexity classes are closed under union -/\n@[primrec] protected lemma or {P Q : α → Prop} (hP : P ∈ₚ C) (hQ : Q ∈ₚ C) :\n  C.mem_pred (λ x, (P x) ∨ (Q x)) :=\nby { classical, simp [← mem_iff_mem_pred], complexity, }\n\nsection option\n\n@[complexity] lemma option_elim {f : α → option β} {g : α → γ} {h : α → β → γ} :\n  f ∈ₑ C → g ∈ₑ C → h ∈ₑ C → C.mem (λ x, (f x).elim (g x) (h x)) :=\nbegin\n  rintros ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ ⟨h', ph, hh⟩, replace hh := λ x₁ x₂, hh (x₁, x₂),\n  refine ⟨λ x, if f' x = tree.nil then g' x else h' (x △ (f' x).right), _, _⟩,\n  { complexity, },\n  intro x,\n  simp only [hf, hg], dsimp [has_uncurry.uncurry],\n  cases f x,\n  { simp [encode], },\n  { simpa [encode] using hh _ _, },\nend\n\n@[complexity] lemma option_rec {f : α → option β} {g : α → γ} {h : α → β → γ} (hf : f ∈ₑ C)\n  (hg : g ∈ₑ C) (hh : h ∈ₑ C) :\n  @complexity_class.mem α γ (α → γ) _ _ _ (λ x : α, @option.rec β (λ _, γ) (g x) (h x) (f x)) C :=\nby { convert option_elim hf hg hh, ext x, cases f x; refl, }\n\n@[complexity] protected lemma some : @option.some α ∈ₑ C :=\n⟨_, C.pair C.nil C.id, λ _, rfl⟩\n\n@[complexity] lemma succ : nat.succ ∈ₑ C := complexity_class.some \n\nlemma of_some {f : α → β} : f ∈ₑ C ↔ C.mem (λ x, some (f x)) :=\n⟨λ hf, by complexity, λ ⟨f', pf, hf⟩, ⟨_, C.comp C.right pf, λ _, by { simp [hf], refl, }⟩⟩\n\n@[complexity] lemma option_bind {f : α → option β} {g : α → β → option γ} (hf : f ∈ₑ C)\n  (hg : g ∈ₑ C) : C.mem (λ x, (f x).bind (g x)) :=\nby { delta option.bind, clean_target, complexity, }\n\n@[complexity] lemma option_map {f : α → option β} {g : α → β → γ} (hf : f ∈ₑ C) (hg : g ∈ₑ C) :\n  C.mem (λ x, (f x).map (g x)) :=\nby { delta option.map, complexity, }\n\ndef mem' {γ : Type} [has_uncurry γ α β] (f : γ) (C : complexity_class) : Prop :=\nC.prop (λ x, encode $ (decode α x).map ↿f)\n\nlocalized \"infix ` ∈ₛ `:50 := complexity_class.mem'\" in complexity_class\n\nlemma mem_of_mem' {γ : Type} [has_uncurry γ α β] {f : γ} {C : complexity_class}\n  (hf : f ∈ₛ C) : f ∈ₑ C := ⟨_, C.comp C.right hf, by simp [encode]⟩\n\nlemma mem'_iff_mem_decode {γ : Type} [has_uncurry γ α β] {f : γ} {C : complexity_class} (h : decode α ∈ₑ C) :\n  f ∈ₛ C ↔ f ∈ₑ C := ⟨mem_of_mem', λ H, by { dsimp only [mem'], complexity, }⟩ \n\nprotected lemma decode : decode (tree unit) ∈ₑ C := complexity_class.some \n\nlemma option_decode (h : decode α ∈ₑ C) : decode (option α) ∈ₑ C :=\nby { simp only [decode], delta tencodable.to_option, clean_target, complexity, }\n\nend option\n\nsection sum\n\n@[complexity] lemma sum_elim {f : α → β ⊕ γ} {g : α → β → δ} {h : α → γ → δ} :\n  f ∈ₑ C → g ∈ₑ C → h ∈ₑ C → C.mem (λ x, (f x).elim (g x) (h x)) :=\nbegin\n  rintros ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ ⟨h', ph, hh⟩, simp only [prod.forall] at hg hh,\n  refine ⟨λ x, if (f' x).left = tree.nil then g' (x △ (f' x).right)\n               else h' (x △ (f' x).right), _, _⟩,\n  { complexity, },\n  intro x,\n  simp only [hf, encode, tencodable.of_sum, tencodable.to_sum, has_uncurry.uncurry, id_def],\n  cases f x,\n  { simpa using hg _ _, },\n  { simpa using hh _ _, }\nend\n\n@[complexity] lemma sum_inl : @sum.inl α β ∈ₑ C :=\n⟨_, C.pair C.nil C.id, by simp [encode, has_uncurry.uncurry]⟩\n\n@[complexity] lemma sum_inr : @sum.inr α β ∈ₑ C :=\n⟨_, C.pair (C.prop_const tencodable.non_nil) C.id, by simp [encode, has_uncurry.uncurry]⟩\n\n@[complexity] lemma sum_rec {f : α → β ⊕ γ} {g : α → β → δ} {h : α → γ → δ} (hf : f ∈ₑ C)\n  (hg : g ∈ₑ C) (hh : h ∈ₑ C) :\n  @complexity_class.mem α δ (α → δ) _ _ _ (λ x, @sum.rec β γ (λ _, δ) (g x) (h x) (f x)) C := sum_elim hf hg hh\n\n-- TODO: some kind of derive handler for these things which are easy\n@[complexity] lemma sum_get_right : @sum.get_right α β ∈ₑ C :=\nby { delta sum.get_right, clean_target, complexity, }\n\nlemma sum_decode (h₁ : decode α ∈ₑ C) (h₂ : decode β ∈ₑ C) : decode (α ⊕ β) ∈ₑ C :=\nby { dsimp [decode], delta tencodable.to_sum, complexity, }\n\nend sum\n\nsection prod\n\nlemma prod_decode (h₁ : decode α ∈ₑ C) (h₂ : decode β ∈ₑ C) : decode (α × β) ∈ₑ C :=\nby { dsimp [decode], complexity, }\n\n@[complexity] lemma prod_rec {f : α → β × γ} {g : α → β → γ → δ} (hf : f ∈ₑ C) (hg : g ∈ₑ C) :\n  @complexity_class.mem α δ (α → δ) _ _ _ (λ x, @prod.rec β γ (λ _, δ) (g x) (f x)) C :=\nby { complexity using (λ x, g x (f x).1 (f x).2), cases f x; refl, }\n\nend prod\n\nsection ordering\n\n@[complexity] lemma ordering_swap : ordering.swap ∈ₑ C := of_from_fintype' _\n\n@[complexity] lemma ordering_or_else : ordering.or_else ∈ₑ C := of_from_fintype' _\n\nend ordering\n\nsection list\n\n@[complexity] lemma list_cases {f : α → list β} {g : α → γ} {h : α → β → list β → γ} :\n  f ∈ₑ C → g ∈ₑ C → h ∈ₑ C →\n  @complexity_class.mem α γ (α → γ) _ _ _ (λ x, @list.cases_on β (λ _, γ) (f x) (g x) (h x)) C :=\nbegin\n  rintros ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ ⟨h', ph, hh⟩, replace hh := λ x₁ x₂ x₃, hh (x₁, x₂, x₃),\n  use λ x, if f' x = tree.nil then g' x else h' (encode (x, (f' x).left, (f' x).right)),\n  split, { complexity, },\n  intro x, simp only [hf, hg], dsimp [has_uncurry.uncurry, id],\n  cases f x, { simp [encode], }, { simpa [encode] using hh _ _ _, },\nend\n\n@[complexity] lemma head' : (@list.head' α) ∈ₑ C :=\nby { delta list.head', clean_target, complexity, }\n\n@[complexity] lemma head [inhabited α] : (@list.head α _) ∈ₑ C :=\nby { delta list.head, clean_target, complexity, }\n\n@[complexity] lemma tail : (@list.tail α) ∈ₑ C :=\nby { delta list.tail, clean_target, complexity, }\n\n@[complexity] lemma pred : nat.pred ∈ₑ C :=\n⟨_, C.right, λ n, by cases n; simp [has_uncurry.uncurry, encode]⟩\n\n@[complexity] protected lemma equiv_list : ⇑tencodable.equiv_list ∈ₑ C :=\n⟨_, C.id, λ x, by simp [encode, has_uncurry.uncurry]⟩ \n\n@[complexity] protected lemma equiv_list_symm : ⇑tencodable.equiv_list.symm ∈ₑ C :=\n⟨_, C.id, λ x, by simp [encode, has_uncurry.uncurry]⟩\n\nlemma const_list_nth (n : ℕ) : (λ l : list α, l.nth n) ∈ₑ C :=\nbegin\n  induction n with n ih, { convert head', ext x : 1, cases x; refl, },\n  have : (λ l : list α, l.tail.nth n) ∈ₑ C := ih.comp tail,\n  complexity using λ l, if l.empty then none else l.tail.nth n,\n  cases l; simp,\nend\n\nend list\n\nsection equiv\n\n\nlemma encode_of_left_injection  {β : Type} {f : β → α} {finv : α → option β} {linv : ∀ b, finv (f b) = some b} :\n  by haveI : tencodable β := tencodable.of_left_injection f finv linv; exact f ∈ₑ C :=\n⟨_, C.id, λ x, rfl⟩\n\nlemma decode_of_left_injection  {β : Type} {f : β → α} {finv : α → option β} {linv : ∀ b, finv (f b) = some b} \n  (hd : decode α ∈ₑ C) (hf : by haveI : tencodable β := tencodable.of_left_injection f finv linv; exact finv ∈ₑ C) :\n  by haveI : tencodable β := tencodable.of_left_injection f finv linv; exact decode β ∈ₑ C :=\nby { letI : tencodable β := tencodable.of_left_injection f finv linv, refine option_bind hd _, complexity, }\n\nlemma encode_of_equiv {C : complexity_class} {β : Type} {e : β ≃ α} :\n  by haveI : tencodable β := tencodable.of_equiv _ e; exact ⇑e ∈ₑ C := encode_of_left_injection\n\nlemma decode_of_equiv {C : complexity_class} {β : Type} {e : β ≃ α} :\n  by haveI : tencodable β := tencodable.of_equiv _ e; exact ⇑e.symm ∈ₑ C :=\n⟨_, C.id, λ x, congr_arg encode (e.apply_symm_apply x).symm⟩\n\nlemma decodable_of_equiv {C : complexity_class} {β : Type} {e : β ≃ α} (hd : decode α ∈ₑ C) :\n  by haveI : tencodable β := tencodable.of_equiv _ e; exact decode β ∈ₑ C :=\nby letI : tencodable β := tencodable.of_equiv _ e; exact decode_of_left_injection hd (mem.comp complexity_class.some decode_of_equiv)\n\nend equiv\n\nsection subtype\n\n@[complexity] lemma subtype_coe {P : α → Prop} [decidable_pred P] : (coe : {x // P x} → α) ∈ₑ C :=\nencode_of_left_injection\n\n@[complexity] lemma subtype_val {P : α → Prop} [decidable_pred P] : (subtype.val : {x // P x} → α) ∈ₑ C :=\nsubtype_coe\n\n@[complexity] lemma subtype_rec {P : α → Prop} [decidable_pred P]\n  {f : β → {x // P x}} {g : β → ∀ x, P x → γ} :\n  f ∈ₑ C → (λ (a : β) (b : {x // P x}), g a b b.prop) ∈ₑ C →\n  @complexity_class.mem _ _ (β → γ) _ _ _ (λ x, @subtype.rec _ _ (λ _, γ) (g x) (f x)) C\n| ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ := ⟨λ x, g' (x △ (f' x)), by complexity, λ x, begin\n  dsimp [has_uncurry.uncurry] at hg hf ⊢,\n  simp [tencodable.encode_prod] at hg,\n  simp [hf, hg], cases f x, refl,\nend⟩\n\n@[complexity] lemma of_subtype_coe {P : α → Prop} [decidable_pred P] {f : β → {x // P x}} :\n  f ∈ₑ C ↔ (λ x, (f x : α)) ∈ₑ C := iff.rfl\n\n@[complexity] lemma subtype_mk {f : α → β} {P : β → Prop} [decidable_pred P] (hP : ∀ x, P (f x))\n  (hf : f ∈ₑ C) : (λ x, subtype.mk (f x) (hP x)) ∈ₑ C := hf\n\n@[complexity] protected lemma dite {P : α → Prop} [decidable_pred P] {f : ∀ (x : α), P x → β}\n  {g : ∀ (x : α), ¬P x → β} : P ∈ₚ C → (λ x : {a // P a}, f x x.prop) ∈ₑ C →\n  (λ x : {a // ¬P a}, g x x.prop) ∈ₑ C → (λ x, if H : P x then f x H else g x H) ∈ₑ C\n| ⟨P', pP, hP⟩ ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ :=\n⟨λ x, if P' x = encode tt then f' x else g' x, by complexity,\nbegin\n  intro x,\n  simp only [hP, has_uncurry.uncurry, tencodable.encode_inj], dsimp, simp only [to_bool_iff],\n  split_ifs with H,\n  { simp [tencodable.subtype_encode] at hf, simp [hf, H], refl, },\n  { simp [tencodable.subtype_encode] at hg, simp [hg, H], refl, }\nend⟩\n\nlemma subtype_decode {P : α → Prop} [decidable_pred P] (hd : decode α ∈ₑ C)\n  (hP : P ∈ₚ C) : decode {x // P x} ∈ₑ C :=\ndecode_of_left_injection hd (by complexity)\n\n@[complexity] lemma fin_mk {n} {f : α → ℕ} (P : ∀ x, f x < n) (hf : f ∈ₑ C) :\n  (λ x, (⟨f x, P x⟩ : fin n)) ∈ₑ C := hf\n\n@[complexity] lemma fin_coe {n} : (coe : fin n → ℕ) ∈ₑ C :=\n⟨_, C.id, λ _, rfl⟩\n\n@[complexity] lemma fin_val {n} : (fin.val : fin n → ℕ) ∈ₑ C := fin_coe\n\n@[complexity] lemma fin_rec {n : ℕ} {f : α → fin n} {g : α → ∀ (x : ℕ), x < n → β} :\n  f ∈ₑ C → (λ (a : α) (b : {x // x < n}), g a b b.prop) ∈ₑ C →\n  @complexity_class.mem _ _ (α → β) _ _ _ (λ x, @fin.rec n (λ _, β) (g x) (f x)) C\n| ⟨f', pf, hf⟩ ⟨g', pg, hg⟩ := ⟨λ x, g' (x △ (f' x)), by complexity, λ x, begin\n  dsimp [has_uncurry.uncurry] at hg hf ⊢,\n  cases H : f x with v hv, specialize hg (x, ⟨v, hv⟩),\n  simp [tencodable.encode_prod] at hg,\n  simpa [hf, H] using hg,\nend⟩\n\n@[complexity] lemma vector_to_list {n} : (vector.to_list : vector α n → list α) ∈ₑ C :=\nsubtype_coe\n\n@[complexity] lemma vector_head {n} : (@vector.head α n) ∈ₑ C :=\nby { rw of_some, simp_rw ← vector.head'_to_list, complexity, }\n\n@[complexity] lemma vector_tail {n} : (@vector.tail α n) ∈ₑ C :=\nby { complexity using λ l, subtype.mk l.to_list.tail (by simp), rcases l with ⟨_|_, _⟩; refl, }\n\n@[complexity] lemma vector_cons {n} : (@vector.cons α n) ∈ₑ C :=\nby { complexity using λ x v, ⟨x :: v.to_list, by simp⟩, cases v, refl, }\n\n@[complexity] lemma vector_nth {n} : (@vector.nth α n) ∈ₑ C :=\nbegin\n  rw [mem.swap_args₂, iff_fintype],\n  intro i, rw of_some,\n  convert (const_list_nth i).comp vector_to_list,\n  ext x : 1, simp [flip, vector.nth_eq_nth_le,  list.some_nth_le_eq],\nend\n\n@[complexity] lemma of_fn {n : ℕ} {f : fin n → α → β} (hf : ∀ i, (f i) ∈ₑ C) :\n  (λ x, vector.of_fn (λ i, f i x)) ∈ₑ C :=\nbegin\n  induction n with n ih, { exact C.const vector.nil, },\n  exact vector_cons.comp₂ (hf 0) (ih $ λ i, hf i.succ),\nend\n\nend subtype\n\nsection setoid\nvariables [h : setoid α] {out : quotient h → α} {hout : function.left_inverse quotient.mk out}\n\ninclude h out hout C\n\nlemma setoid_out : by haveI : tencodable (quotient h) := setoid.tencodable h out hout; exact out ∈ₑ C :=\n⟨_, C.id, λ x, rfl⟩\n\nlemma decodable_of_quotient_mk (hd : decode α ∈ₑ C) (hq : by haveI : tencodable (quotient h) := setoid.tencodable h out hout; exact @quotient.mk _ h ∈ₑ C) :\n  by haveI : tencodable (quotient h) := setoid.tencodable h out hout; exact (decode (quotient h)) ∈ₑ C :=\nby { letI : tencodable (quotient h) := setoid.tencodable h out hout, refine decode_of_left_injection hd (mem.comp complexity_class.some hq), }\n\nlemma quotient_lift {f : α → β} {pf : ∀ (a b : α), a ≈ b → f a = f b}\n  (hf : f ∈ₑ C) : by haveI : tencodable (quotient h) := setoid.tencodable h out hout; exact\n  (quotient.lift f pf) ∈ₑ C :=\nbegin\n  letI : tencodable (quotient h) := setoid.tencodable h out hout,\n  convert hf.comp setoid_out, ext x,\n  refine quotient.induction_on x (λ x, pf _ _ (quotient.exact _)),\n  rw hout,\nend\n \nend setoid\n\nsection multiset\n\n@[complexity] lemma multiset_sort : multiset.sort (@tencodable.lift_le α _) ∈ₑ C := setoid_out\n\nend multiset\n\nsection sigma\n\n@[complexity] lemma sigma_fst {β : α → Type} [∀ i, tencodable (β i)] :\n  (@sigma.fst α β) ∈ₑ C :=\n⟨_, C.left, λ x, rfl⟩\n\n@[complexity] lemma sigma_snd : @complexity_class.mem _ _ ((Σ _ : α, β) → β) _ _ _ sigma.snd C :=\n⟨_, C.right, λ x, rfl⟩\n\nend sigma\n\nsection finmap\n\n@[complexity] lemma finmap_entries [decidable_eq α] : (@finmap.entries α (λ _, β)) ∈ₑ C := C.id'\n\n@[complexity] lemma finmap_decode [decidable_eq α] (hd : decode (multiset (Σ _ : α, β)) ∈ₑ C) \n  (H : @multiset.nodupkeys α (λ _, β) ∈ₚ C) :\n  decode (@finmap α (λ _, β)) ∈ₑ C :=\ndecodable_of_equiv $ subtype_decode hd H\n\nend finmap\n\nend complexity_class\n\n", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/complexity_class/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3271495332003373}}
{"text": "import category_theory.closed.monoidal\nimport category_theory.monoidal.braided\nimport tactic\n\nnamespace category_theory.monoidal\n\nopen category_theory\nopen category_theory.monoidal_category\n\nuniverses v u\nvariables {C : Type u} [category.{v} C] [monoidal_category.{v} C]\n\nclass right_closed (X : C) :=\n(is_adj : is_left_adjoint (tensor_right X))\n\nattribute [instance, priority 100] right_closed.is_adj\n\nvariable (C)\n\nclass monoidal_closed_right :=\n(closed : Π (X : C), right_closed X)\n\nattribute [instance, priority 100] monoidal_closed_right.closed\n\nvariable [monoidal_closed_right.{v} C]\n\nvariable {C}\n\ndef internal_hom (X Y : C) : C :=\n(right_adjoint (tensor_right X)).obj Y\n\ndef internal_hom_equiv (X Y Z : C) :\n  (X ⊗ Y ⟶ Z) ≃ (X ⟶ internal_hom Y Z) :=\n(adjunction.of_left_adjoint (tensor_right Y)).hom_equiv X Z\n\ndef internal_hom_postcompose (X : C) {Y₁ Y₂ : C} (f : Y₁ ⟶ Y₂) :\n  internal_hom X Y₁ ⟶ internal_hom X Y₂ :=\n(right_adjoint (tensor_right X)).map f\n\n@[simp]\nlemma internal_hom_postcompose_id (X Y : C) :\n  internal_hom_postcompose X (𝟙 Y) = 𝟙 _ :=\n(right_adjoint (tensor_right X)).map_id _\n\n@[simp]\nlemma internal_hom_postcompose_comp (X : C) {Y₁ Y₂ Y₃ : C} (f₁ : Y₁ ⟶ Y₂) (f₂ : Y₂ ⟶ Y₃) :\n  internal_hom_postcompose X (f₁ ≫ f₂) =\n  internal_hom_postcompose X f₁ ≫ internal_hom_postcompose X f₂ :=\n(right_adjoint (tensor_right X)).map_comp _ _\n\ndef internal_hom_precompose {X₁ X₂ : C} (f : X₁ ⟶ X₂) (Y : C) :\n  internal_hom X₂ Y ⟶ internal_hom X₁ Y :=\n(internal_hom_equiv _ _ _) $ (tensor_hom (𝟙 _) f ≫ (internal_hom_equiv _ _ _).symm (𝟙 _))\n\nlemma split_left {A₁ A₂ B₁ B₂ : C} (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) :\n  (f ⊗ g) = (f ⊗ 𝟙 _) ≫ (𝟙 _ ⊗ g) := by simp\n\nlemma split_right {A₁ A₂ B₁ B₂ : C} (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) :\n  (f ⊗ g) = (𝟙 _ ⊗ g) ≫ (f ⊗ 𝟙 _) := by simp\n\nlemma internal_hom_equiv_tensor_right {X Y₁ Y₂ Z : C} (f : Y₁ ⟶ Y₂) (g : X ⊗ Y₂ ⟶ Z) :\n  internal_hom_equiv _ _ _ ((𝟙 _ ⊗ f) ≫ g)  =\n  internal_hom_equiv _ _ _ g ≫ internal_hom_precompose f _ :=\nbegin\n  apply_fun (internal_hom_equiv _ _ _).symm,\n  simp,\n  dsimp [internal_hom_precompose, internal_hom_equiv, internal_hom],\n  simp only [adjunction.hom_equiv_unit, adjunction.hom_equiv_counit, tensor_right_map,\n    tensor_id,  adjunction.hom_equiv_naturality_right, functor.map_comp,\n    category_theory.functor.map_id, category.id_comp, category.assoc,\n    adjunction.hom_equiv_naturality_left_symm, adjunction.hom_equiv_naturality_right_symm],\n  simp only [← tensor_right_map],\n  rw (adjunction.of_left_adjoint (tensor_right Y₁)).counit_naturality_assoc,\n  rw (adjunction.of_left_adjoint (tensor_right Y₁)).left_triangle_components_assoc,\n  dsimp,\n  simp only [← category.assoc, ← tensor_comp, category.id_comp, category.comp_id],\n  conv_rhs { rw [split_right _ f, category.assoc] },\n  rw ← tensor_right_map,\n  simp only [functor.map_comp, category.assoc],\n  rw (adjunction.of_left_adjoint (tensor_right Y₂)).counit_naturality,\n  erw (adjunction.of_left_adjoint (tensor_right Y₂)).left_triangle_components_assoc,\nend\n\n@[simp]\nlemma internal_hom_precompose_id (X Y : C) :\n  internal_hom_precompose (𝟙 X) Y = 𝟙 _ :=\nby { dsimp [internal_hom_precompose, internal_hom_equiv, internal_hom], simp, dsimp, simp }\n\n@[simp]\nlemma internal_hom_precompose_comp {X₁ X₂ X₃ : C} (f₁ : X₁ ⟶ X₂) (f₂ : X₂ ⟶ X₃) (Y : C) :\n  internal_hom_precompose (f₁ ≫ f₂) Y =\n  internal_hom_precompose f₂ Y ≫ internal_hom_precompose f₁ Y :=\nbegin\n  change _ = internal_hom_equiv _ _ _ _ ≫ _,\n  rw [← internal_hom_equiv_tensor_right, ← category.assoc, ← tensor_comp, category.id_comp],\n  refl,\nend\n\n@[simp]\nlemma internal_hom_postcompose_comp_precompose {A₁ A₂ B₁ B₂ : C}\n  (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) :\n  internal_hom_postcompose _ f ≫ internal_hom_precompose g _ =\n  internal_hom_precompose g _ ≫ internal_hom_postcompose _ f :=\nbegin\n  apply_fun (internal_hom_equiv _ _ _).symm,\n  dsimp [internal_hom_precompose, internal_hom_postcompose, internal_hom_equiv, internal_hom],\n  simp only [adjunction.hom_equiv_counit, tensor_right_map, tensor_id,\n    adjunction.hom_equiv_naturality_right, adjunction.hom_equiv_unit, functor.map_comp,\n    category_theory.functor.map_id, category.id_comp, category.assoc,\n    adjunction.hom_equiv_naturality_left_symm, adjunction.hom_equiv_naturality_right_symm],\n  simp only [← tensor_right_map, adjunction.counit_naturality, adjunction.counit_naturality_assoc,\n    functor.map_comp, adjunction.left_triangle_components, adjunction.right_triangle_components,\n    adjunction.left_triangle_components_assoc, adjunction.right_triangle_components_assoc],\n  dsimp,\n  simp only [← category.assoc, ← tensor_comp, category.id_comp, category.comp_id],\n  rw [split_right _ g, category.assoc, ← tensor_right_map, adjunction.counit_naturality,\n    category.assoc],\nend\n\n@[simps]\ndef internal_hom_functor : Cᵒᵖ ⥤ C ⥤ C :=\n{ obj := λ X,\n  { obj := λ Y, internal_hom X.unop Y,\n    map := λ Y₁ Y₂, internal_hom_postcompose _ },\n  map := λ X₁ X₂ f,\n  { app := λ Y, internal_hom_precompose f.unop _ } }\n\nend category_theory.monoidal\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/internal_hom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.32714952562242877}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n\nimport order.omega_complete_partial_order\nimport order.category.Preorder\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.constructions.limits_of_products_and_equalizers\n\n/-!\n# Category of types with a omega complete partial order\n\nIn this file, we bundle the class `omega_complete_partial_order` into a\nconcrete category and prove that continuous functions also form\na `omega_complete_partial_order`.\n\n## Main definitions\n\n * `ωCPO`\n   * an instance of `category` and `concrete_category`\n\n -/\n\nopen category_theory\n\nuniverses u v\n\n/-- The category of types with a omega complete partial order. -/\ndef ωCPO : Type (u+1) := bundled omega_complete_partial_order\n\nnamespace ωCPO\n\nopen omega_complete_partial_order\n\ninstance : bundled_hom @continuous_hom :=\n{ to_fun := @continuous_hom.simps.apply,\n  id := @continuous_hom.id,\n  comp := @continuous_hom.comp,\n  hom_ext := @continuous_hom.coe_inj }\n\nattribute [derive [large_category, concrete_category]] ωCPO\n\ninstance : has_coe_to_sort ωCPO Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled ωCPO from the underlying type and typeclass. -/\ndef of (α : Type*) [omega_complete_partial_order α] : ωCPO := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [omega_complete_partial_order α] : ↥(of α) = α := rfl\n\ninstance : inhabited ωCPO := ⟨of punit⟩\n\ninstance (α : ωCPO) : omega_complete_partial_order α := α.str\n\nsection\n\nopen category_theory.limits\n\nnamespace has_products\n\n/-- The pi-type gives a cone for a product. -/\ndef product {J : Type v} (f : J → ωCPO.{v}) : fan f :=\nfan.mk (of (Π j, f j)) (λ j, continuous_hom.of_mono (pi.eval_order_hom j) (λ c, rfl))\n\n/-- The pi-type is a limit cone for the product. -/\ndef is_product (J : Type v) (f : J → ωCPO) : is_limit (product f) :=\n{ lift := λ s,\n    ⟨⟨λ t j, s.π.app ⟨j⟩ t, λ x y h j, (s.π.app ⟨j⟩).monotone h⟩,\n     λ x, funext (λ j, (s.π.app ⟨j⟩).continuous x)⟩,\n  uniq' := λ s m w,\n  begin\n    ext t j,\n    change m t j = s.π.app ⟨j⟩ t,\n    rw ← w ⟨j⟩,\n    refl,\n  end,\n  fac' := λ s j, by { cases j, tidy, } }.\n\ninstance (J : Type v) (f : J → ωCPO.{v}) : has_product f :=\nhas_limit.mk ⟨_, is_product _ f⟩\n\nend has_products\n\ninstance omega_complete_partial_order_equalizer\n  {α β : Type*} [omega_complete_partial_order α] [omega_complete_partial_order β]\n  (f g : α →𝒄 β) : omega_complete_partial_order {a : α // f a = g a} :=\nomega_complete_partial_order.subtype _ $ λ c hc,\nbegin\n  rw [f.continuous, g.continuous],\n  congr' 1,\n  ext,\n  apply hc _ ⟨_, rfl⟩,\nend\n\nnamespace has_equalizers\n\n/-- The equalizer inclusion function as a `continuous_hom`. -/\ndef equalizer_ι {α β : Type*} [omega_complete_partial_order α] [omega_complete_partial_order β]\n  (f g : α →𝒄 β) :\n  {a : α // f a = g a} →𝒄 α :=\ncontinuous_hom.of_mono (order_hom.subtype.val _) (λ c, rfl)\n\n/-- A construction of the equalizer fork. -/\ndef equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) :\n  fork f g :=\n@fork.of_ι _ _ _ _ _ _ (ωCPO.of {a // f a = g a}) (equalizer_ι f g)\n  (continuous_hom.ext _ _ (λ x, x.2))\n\n/-- The equalizer fork is a limit. -/\ndef is_equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : is_limit (equalizer f g) :=\nfork.is_limit.mk' _ $ λ s,\n⟨{ to_fun := λ x, ⟨s.ι x, by apply continuous_hom.congr_fun s.condition⟩,\n    monotone' := λ x y h, s.ι.monotone h,\n    cont := λ x, subtype.ext (s.ι.continuous x) },\n  by { ext, refl },\n  λ m hm,\n  begin\n    ext,\n    apply continuous_hom.congr_fun hm,\n  end⟩\n\nend has_equalizers\n\ninstance : has_products ωCPO.{v} :=\nλ J, { has_limit := λ F, has_limit_of_iso discrete.nat_iso_functor.symm }\n\ninstance {X Y : ωCPO.{v}} (f g : X ⟶ Y) : has_limit (parallel_pair f g) :=\nhas_limit.mk ⟨_, has_equalizers.is_equalizer f g⟩\n\ninstance : has_equalizers ωCPO.{v} := has_equalizers_of_has_limit_parallel_pair _\n\ninstance : has_limits ωCPO.{v} := limits_from_equalizers_and_products\n\nend\n\n\nend ωCPO\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/order/category/omega_complete_partial_order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3269121615458638}}
{"text": "import data.list.dict\nimport data.multiset\nimport data.pfun\n\nnamespace multiset\nuniverses u v\nvariables {α : Type u} {β : α → Type v}\n\nopen function list\n\ndef nodupkeys (m : multiset (sigma β)) : Prop :=\nquotient.lift_on m nodupkeys (λ _ _, propext ∘ perm_nodupkeys)\n\ntheorem nodupkeys_coe {l : list (sigma β)} :\n  nodupkeys (l : multiset (sigma β)) ↔ l.nodupkeys :=\niff.rfl\n\ntheorem nodup_of_nodupkeys {m : multiset (sigma β)} : m.nodupkeys → m.nodup :=\nquotient.induction_on m $ λ _, nodup_of_nodupkeys\n\n@[simp] theorem nodupkeys_zero : @nodupkeys _ β 0 :=\npairwise.nil _\n\ntheorem nodupkeys_singleton : ∀ (s : sigma β), nodupkeys (s :: 0) :=\nnodupkeys_singleton\n\ndef krec_on {γ : Sort*} (m : multiset (sigma β))\n  (φ : ∀ {l : list (sigma β)}, l.nodupkeys → γ)\n  (c : ∀ {l₁ l₂} (p : l₁ ~ l₂) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d₁ = φ d₂) :\n  m.nodupkeys → γ :=\n@quotient.hrec_on _ _ (λ (m : multiset (sigma β)), m.nodupkeys → γ) m (λ _, φ) $\n  λ _ _ p, hfunext (by rw perm_nodupkeys p) $\n    λ d₁ d₂ _, heq_of_eq $ c p d₁ d₂\n\n@[simp] theorem krec_on_coe {γ : Sort*} {l : list (sigma β)} (d : l.nodupkeys)\n  (φ : ∀ {l : list (sigma β)}, l.nodupkeys → γ)\n  (c : ∀ {l₁ l₂} (p : l₁ ~ l₂) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d₁ = φ d₂) :\n  krec_on (l : multiset (sigma β)) @φ @c d = φ d :=\nrfl\n\ndef krec_on₂ {γ : Sort*} (m₁ m₂ : multiset (sigma β))\n  (φ : ∀ {l₁ l₂ : list (sigma β)}, l₁.nodupkeys → l₂.nodupkeys → γ)\n  (cl : ∀ {l l₁ l₂ : list (sigma β)} (p : l₁ ~ l₂) (d : l.nodupkeys) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d₁ d = φ d₂ d)\n  (cr : ∀ {l l₁ l₂ : list (sigma β)} (p : l₁ ~ l₂) (d : l.nodupkeys) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d d₁ = φ d d₂) :\n  m₁.nodupkeys → m₂.nodupkeys → γ :=\nkrec_on m₁\n  (λ l d, krec_on m₂ (λ l₂ d₂, φ d d₂) (λ l₁ l₂ p d₁ d₂, cr p d d₁ d₂))\n  (λ l₁ l₂ p d₁ d₂, by congr; funext l; funext d; exact cl p d d₁ d₂)\n\n@[simp] theorem krec_on₂_coe {γ : Sort*} {l₁ l₂ : list (sigma β)} (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys)\n  (φ : ∀ {l₁ l₂ : list (sigma β)}, l₁.nodupkeys → l₂.nodupkeys → γ)\n  (cl : ∀ {l l₁ l₂ : list (sigma β)} (p : l₁ ~ l₂) (d : l.nodupkeys) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d₁ d = φ d₂ d)\n  (cr : ∀ {l l₁ l₂ : list (sigma β)} (p : l₁ ~ l₂) (d : l.nodupkeys) (d₁ : l₁.nodupkeys) (d₂ : l₂.nodupkeys), φ d d₁ = φ d d₂) :\n  krec_on₂ (l₁ : multiset (sigma β)) (l₂ : multiset (sigma β)) @φ @cl @cr d₁ d₂ = φ d₁ d₂ :=\nrfl\n\ndef keys : multiset (sigma β) → multiset α :=\nmap sigma.fst\n\n@[simp] theorem keys_coe (l : list (sigma β)) :\n  keys (l : multiset (sigma β)) = (l.keys : multiset α) :=\nrfl\n\ntheorem mem_keys_of_mem {s : sigma β} {m : multiset (sigma β)} : s ∈ m → s.1 ∈ m.keys :=\nmem_map_of_mem sigma.fst\n\ntheorem exists_mem_of_mem_keys {a : α} {m : multiset (sigma β)} (h : a ∈ m.keys) :\n  ∃ (b : β a), sigma.mk a b ∈ m :=\nlet ⟨⟨a', b'⟩, m, e⟩ := mem_map.mp h in\neq.rec_on e (exists.intro b' m)\n\ntheorem mem_keys {a : α} {m : multiset (sigma β)} : a ∈ m.keys ↔ ∃ (b : β a), sigma.mk a b ∈ m :=\n⟨exists_mem_of_mem_keys, λ ⟨_, h⟩, mem_keys_of_mem h⟩\n\ntheorem nodupkeys_iff {m : multiset (sigma β)} : m.keys.nodup ↔ m.nodupkeys :=\nquotient.induction_on m $ λ _, list.nodupkeys_iff\n\nsection kerase\nvariables [decidable_eq α]\n\ndef kerase {m : multiset (sigma β)} (a : α) : m.nodupkeys → multiset (sigma β) :=\nkrec_on m (λ l _, (l.kerase a : multiset (sigma β))) $ λ _ _ p d₁ d₂,\n  quotient.sound $ perm_kerase d₁ d₂ p\n\n@[simp] theorem kerase_coe {l : list (sigma β)} (d : l.nodupkeys) (a : α) :\n  @kerase _ _ _ (l : multiset (sigma β)) a d = (l.kerase a : multiset (sigma β)) :=\nrfl\n\ntheorem kerase_le {m : multiset (sigma β)} (a : α) : ∀ (nd : m.nodupkeys), kerase a nd ≤ m :=\nquotient.induction_on m $ λ l _, subperm_of_sublist (kerase_sublist a l)\n\n@[simp] theorem nodupkeys_kerase {m : multiset (sigma β)} (a : α) :\n  ∀ (nd : m.nodupkeys), (kerase a nd).nodupkeys :=\nquotient.induction_on m $ λ _, nodupkeys_kerase a\n\ntheorem kerase_eq_filter {m : multiset (sigma β)} (a : α) :\n  ∀ (nd : m.nodupkeys), kerase a nd = filter (λ x, x.1 ≠ a) m :=\nquotient.induction_on m $ λ _, congr_arg coe ∘ nodupkeys_kerase_eq_filter a\n\n@[simp] theorem mem_kerase {m : multiset (sigma β)} {s : sigma β} {a : α}\n  (nd : m.nodupkeys) : s ∈ kerase a nd ↔ s.1 ≠ a ∧ s ∈ m :=\nby rw kerase_eq_filter a nd; simp [and_comm]\n\n@[simp] theorem mem_keys_kerase {a₁ a₂ : α} {m : multiset (sigma β)} (d : m.nodupkeys) :\n  a₁ ∈ (kerase a₂ d).keys ↔ a₁ ≠ a₂ ∧ a₁ ∈ m.keys :=\nby simp [keys]\n\ntheorem kerase_subset (a : α) {m : multiset (sigma β)} (nd : m.nodupkeys) :\n  kerase a nd ⊆ m :=\nsubset_of_le (kerase_le a nd)\n\ntheorem mem_of_mem_kerase {s : sigma β} (a : α) {m : multiset (sigma β)}\n  (nd : m.nodupkeys) : s ∈ kerase a nd → s ∈ m :=\nmem_of_subset (kerase_subset _ _)\n\ntheorem kerase_comm {m : multiset (sigma β)} (a₁ a₂ : α) : ∀ (nd : m.nodupkeys),\n  kerase a₂ (nodupkeys_kerase a₁ nd) = kerase a₁ (nodupkeys_kerase a₂ nd) :=\nquotient.induction_on m $ λ l _, congr_arg coe $ l.kerase_comm a₁ a₂\n\ntheorem kerase_le_kerase {m₁ m₂ : multiset (sigma β)} (a : α) (h : m₁ ≤ m₂) :\n  ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys), kerase a d₁ ≤ kerase a d₂ :=\nle_induction_on h $ λ _ _ ih _ _, subperm_of_sublist $ kerase_sublist_kerase a ih\n\nend kerase\n\nsection kinsert\nvariables [decidable_eq α] {m : multiset (sigma β)}\n\ndef kinsert (s : sigma β) : m.nodupkeys → multiset (sigma β) :=\nkrec_on m (λ l _, (l.kinsert s : multiset (sigma β))) $ λ _ _ p d₁ d₂,\n  quotient.sound $ perm_kinsert d₁ d₂ p\n\n@[simp] theorem kinsert_coe {l : list (sigma β)} (d : l.nodupkeys) (s : sigma β) :\n  @kinsert _ _ _ (l : multiset (sigma β)) s d = (l.kinsert s : multiset (sigma β)) :=\nrfl\n\n@[simp] theorem nodupkeys_kinsert (s : sigma β) :\n  ∀ (d : m.nodupkeys), (kinsert s d).nodupkeys :=\nquotient.induction_on m $ λ _, nodupkeys_kinsert s\n\n@[simp] theorem mem_kinsert {m : multiset (sigma β)} {s₁ s₂ : sigma β} :\n  ∀ (d : m.nodupkeys), s₁ ∈ kinsert s₂ d ↔ s₁ = s₂ ∨ s₁ ∈ kerase s₂.1 d :=\nquotient.induction_on m $ λ _ _, mem_kinsert\n\n@[simp] theorem mem_keys_kinsert {a : α} {s : sigma β}\n  {m : multiset (sigma β)} (d : m.nodupkeys) :\n  a ∈ (kinsert s d).keys ↔ a = s.1 ∨ a ∈ m.keys :=\nby cases s with a' b'; by_cases h' : a = a'; simp [keys, exists_or_distrib, h']\n\nend kinsert\n\nsection klookup\nvariables [decidable_eq α] {m : multiset (sigma β)}\n\ndef klookup (a : α) : m.nodupkeys → option (β a) :=\nkrec_on m (λ l _, l.klookup a) $ λ _ _ p d₁ d₂, perm_klookup d₁ d₂ p\n\nend klookup\n\nsection klookup_all\nvariables [decidable_eq α]\n\ndef klookup_all (a : α) (m : multiset (sigma β)) : multiset (β a) :=\nquotient.lift_on m\n  (λ l, (l.klookup_all a : multiset (β a)))\n  (λ _ _, quotient.sound ∘ perm_klookup_all)\n\nend klookup_all\n\nsection kreplace\nvariables [decidable_eq α] {m : multiset (sigma β)}\n\ndef kreplace (s : sigma β) : m.nodupkeys → multiset (sigma β) :=\nkrec_on m (λ l _, (l.kreplace s : multiset (sigma β))) $ λ _ _ p d₁ d₂,\n  quotient.sound $ perm_kreplace d₁ d₂ p\n\n@[simp] theorem nodupkeys_kreplace (s : sigma β) :\n  ∀ (d : m.nodupkeys), (kreplace s d).nodupkeys :=\nquotient.induction_on m $ λ _, nodupkeys_kreplace s\n\nend kreplace\n\nsection kunion\nvariables [decidable_eq α] {a : α} {s : sigma β} {l₁ l₂ : list (sigma β)}\nvariables {m m₁ m₂ m₃ : multiset (sigma β)}\n\ndef kunion : m₁.nodupkeys → m₂.nodupkeys → multiset (sigma β) :=\nkrec_on₂ m₁ m₂ (λ l₁ l₂ d₁ d₂, (l₁.kunion l₂ : multiset (sigma β)))\n  (λ _ _ _ p d _ _, quotient.sound $ perm_kunion d d p (perm.refl _))\n  (λ _ _ _ p _ d₁ d₂, quotient.sound $ perm_kunion d₁ d₂ (perm.refl _) p)\n\nlocal infixr ` k∪ `:67 := kunion\n\n@[simp] theorem kunion_coe\n   (d₁ : (l₁ : multiset (sigma β)).nodupkeys) (d₂ : (l₂ : multiset (sigma β)).nodupkeys) :\n   d₁ k∪ d₂ = (l₁.kunion l₂ : multiset (sigma β)) :=\nrfl\n\n@[simp] theorem nodupkeys_kunion : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  (d₁ k∪ d₂).nodupkeys :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _, nodupkeys_kunion\n\n@[simp] theorem zero_kunion : ∀ (d : m.nodupkeys), nodupkeys_zero k∪ d = m :=\nquotient.induction_on m $ λ _ _, rfl\n\n@[simp] theorem kunion_zero : ∀ (d : m.nodupkeys), d k∪ nodupkeys_zero = m :=\nquotient.induction_on m $ λ _ _, by simp [kunion_coe]\n\n@[simp] theorem kinsert_kunion : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  kunion (nodupkeys_kinsert s d₁) d₂ = kinsert s (nodupkeys_kunion d₁ d₂) :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, by simp [kinsert_kunion]\n\n@[simp] theorem mem_of_mem_kunion : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  s ∈ d₁ k∪ d₂ → s ∈ m₁ ∨ s ∈ m₂ :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, mem_of_mem_kunion\n\ntheorem mem_kunion_left : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  s ∈ m₁ → s ∈ d₁ k∪ d₂ :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, mem_kunion_left _\n\ntheorem mem_kunion_right : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  s.1 ∉ m₁.keys → s ∈ m₂ → s ∈ d₁ k∪ d₂ :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, mem_kunion_right\n\ntheorem mem_kunion_middle : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys) (d₃ : m₃.nodupkeys),\n  disjoint (d₁ k∪ d₂).keys m₃.keys → s ∈ d₁ k∪ d₃ → s ∈ (nodupkeys_kunion d₁ d₂) k∪ d₃ :=\nquotient.induction_on₃ m₁ m₂ m₃ $ λ _ _ _ _ _ _, mem_kunion_middle\n\n@[simp] theorem mem_kunion : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  disjoint m₁.keys m₂.keys → (s ∈ d₁ k∪ d₂ ↔ s ∈ m₁ ∨ s ∈ m₂) :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, mem_kunion_iff\n\n@[simp] theorem mem_keys_kunion : ∀ (d₁ : m₁.nodupkeys) (d₂ : m₂.nodupkeys),\n  a ∈ keys (d₁ k∪ d₂) ↔ a ∈ m₁.keys ∨ a ∈ m₂.keys :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _ _ _, mem_keys_kunion\n\nend kunion\n\nsection α₁α₂β₁β₂\nvariables {α₁ α₂ : Type u} {β₁ : α₁ → Type v} {β₂ : α₂ → Type v}\n\nsection map\nvariables {s : sigma β₁} {f : sigma β₁ → sigma β₂} {m m₁ m₂ : multiset (sigma β₁)}\n\ntheorem nodupkeys_map (fi : sigma.fst_injective f) : m.nodupkeys → (m.map f).nodupkeys :=\nquotient.induction_on m $ λ _, nodupkeys_map fi\n\ntheorem mem_keys_map (ff : sigma.fst_functional f) : s.1 ∈ m.keys → (f s).1 ∈ (m.map f).keys :=\nquotient.induction_on m $ λ _, mem_keys_map ff\n\ntheorem mem_keys_of_mem_keys_map (fi : sigma.fst_injective f) :\n  (f s).1 ∈ (m.map f).keys → s.1 ∈ m.keys :=\nquotient.induction_on m $ λ _, mem_keys_of_mem_keys_map fi\n\ntheorem mem_keys_map_iff (ff : sigma.fst_functional f) (fi : sigma.fst_injective f) :\n  (f s).1 ∈ (m.map f).keys ↔ s.1 ∈ m.keys :=\n⟨mem_keys_of_mem_keys_map fi, mem_keys_map ff⟩\n\ntheorem map_disjoint_keys (ff : sigma.fst_functional f) (fi : sigma.fst_injective f) :\n  disjoint (m₁.map f).keys (m₂.map f).keys ↔ disjoint m₁.keys m₂.keys :=\nquotient.induction_on₂ m₁ m₂ $ λ _ _, map_disjoint_keys ff fi\n\nend map\n\nsection map_decidable_eq_α₁α₂\nvariables [decidable_eq α₁] [decidable_eq α₂]\nvariables {s : sigma β₁} {f : sigma β₁ → sigma β₂} {m : multiset (sigma β₁)}\n\n@[simp] theorem map_kerase (ff : sigma.fst_functional f) (fi : sigma.fst_injective f) :\n  ∀ (d : m.nodupkeys), (kerase s.1 d).map f = kerase (f s).1 (nodupkeys_map fi d) :=\nquotient.induction_on m $ λ _ _, by simp [ff, fi]\n\n@[simp] theorem map_kinsert (ff : sigma.fst_functional f) (fi : sigma.fst_injective f) :\n  ∀ (d : m.nodupkeys), (kinsert s d).map f = kinsert (f s) (nodupkeys_map fi d) :=\nquotient.induction_on m $ λ _ _, by simp [ff, fi]\n\nend map_decidable_eq_α₁α₂\n\nend α₁α₂β₁β₂\n\nsection map_id\nvariables {β₁ β₂ : α → Type v} {s : sigma β₁} {m : multiset (sigma β₁)}\n\n@[simp] theorem map_id_keys (f : ∀ a, β₁ a → β₂ a) : (m.map (sigma.map id f)).keys = m.keys :=\nby simp [keys]\n\nend map_id\n\nend multiset\n", "meta": {"author": "spl", "repo": "lean-finmap", "sha": "936d9caeb27631e3c6cf20e972de4837c9fe98fa", "save_path": "github-repos/lean/spl-lean-finmap", "path": "github-repos/lean/spl-lean-finmap/lean-finmap-936d9caeb27631e3c6cf20e972de4837c9fe98fa/src/data/multiset/dict.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3269121615458638}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.basic\nimport Mathlib.category_theory.adjunction.basic\nimport Mathlib.category_theory.reflects_isomorphisms\nimport Mathlib.PostPort\n\nuniverses v₁ u₁ l \n\nnamespace Mathlib\n\n/-!\n# Eilenberg-Moore (co)algebras for a (co)monad\n\nThis file defines Eilenberg-Moore (co)algebras for a (co)monad,\nand provides the category instance for them.\n\nFurther it defines the adjoint pair of free and forgetful functors, respectively\nfrom and to the original category, as well as the adjoint pair of forgetful and\ncofree functors, respectively from and to the original category.\n\n## References\n* [Riehl, *Category theory in context*, Section 5.2.4][riehl2017]\n-/\n\nnamespace category_theory\n\n\nnamespace monad\n\n\n/-- An Eilenberg-Moore algebra for a monad `T`.\n    cf Definition 5.2.3 in [Riehl][riehl2017]. -/\nstructure algebra {C : Type u₁} [category C] (T : C ⥤ C) [monad T] \nwhere\n  A : C\n  a : functor.obj T A ⟶ A\n  unit' : autoParam (nat_trans.app η_ A ≫ a = 𝟙)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  assoc' : autoParam (nat_trans.app μ_ A ≫ a = functor.map T a ≫ a)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem algebra.unit {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (c : algebra T) : nat_trans.app η_ (algebra.A c) ≫ algebra.a c = 𝟙 := sorry\n\ntheorem algebra.assoc {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (c : algebra T) : nat_trans.app μ_ (algebra.A c) ≫ algebra.a c = functor.map T (algebra.a c) ≫ algebra.a c := sorry\n\ntheorem algebra.assoc_assoc {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (c : algebra T) {X' : C} (f' : algebra.A c ⟶ X') : nat_trans.app μ_ (algebra.A c) ≫ algebra.a c ≫ f' = functor.map T (algebra.a c) ≫ algebra.a c ≫ f' := sorry\n\nnamespace algebra\n\n\n/-- A morphism of Eilenberg–Moore algebras for the monad `T`. -/\nstructure hom {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (A : algebra T) (B : algebra T) \nwhere\n  f : A A ⟶ A B\n  h' : autoParam (functor.map T f ≫ a B = a A ≫ f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem hom.h {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {A : algebra T} {B : algebra T} (c : hom A B) : functor.map T (hom.f c) ≫ a B = a A ≫ hom.f c := sorry\n\n@[simp] theorem hom.h_assoc {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {A : algebra T} {B : algebra T} (c : hom A B) {X' : C} (f' : A B ⟶ X') : functor.map T (hom.f c) ≫ a B ≫ f' = a A ≫ hom.f c ≫ f' := sorry\n\nnamespace hom\n\n\n/-- The identity homomorphism for an Eilenberg–Moore algebra. -/\ndef id {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (A : algebra T) : hom A A :=\n  mk 𝟙\n\nprotected instance inhabited {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (A : algebra T) : Inhabited (hom A A) :=\n  { default := mk 𝟙 }\n\n/-- Composition of Eilenberg–Moore algebra homomorphisms. -/\ndef comp {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {P : algebra T} {Q : algebra T} {R : algebra T} (f : hom P Q) (g : hom Q R) : hom P R :=\n  mk (f f ≫ f g)\n\nend hom\n\n\nprotected instance category_theory.category_struct {C : Type u₁} [category C] {T : C ⥤ C} [monad T] : category_struct (algebra T) :=\n  category_struct.mk hom.id hom.comp\n\n@[simp] theorem comp_eq_comp {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {A : algebra T} {A' : algebra T} {A'' : algebra T} (f : A ⟶ A') (g : A' ⟶ A'') : hom.comp f g = f ≫ g :=\n  rfl\n\n@[simp] theorem id_eq_id {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (A : algebra T) : hom.id A = 𝟙 :=\n  rfl\n\n@[simp] theorem id_f {C : Type u₁} [category C] {T : C ⥤ C} [monad T] (A : algebra T) : hom.f 𝟙 = 𝟙 :=\n  rfl\n\n@[simp] theorem comp_f {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {A : algebra T} {A' : algebra T} {A'' : algebra T} (f : A ⟶ A') (g : A' ⟶ A'') : hom.f (f ≫ g) = hom.f f ≫ hom.f g :=\n  rfl\n\n/-- The category of Eilenberg-Moore algebras for a monad.\n    cf Definition 5.2.4 in [Riehl][riehl2017]. -/\nprotected instance EilenbergMoore {C : Type u₁} [category C] {T : C ⥤ C} [monad T] : category (algebra T) :=\n  category.mk\n\n/--\nTo construct an isomorphism of algebras, it suffices to give an isomorphism of the carriers which\ncommutes with the structure morphisms.\n-/\n@[simp] theorem iso_mk_hom_f {C : Type u₁} [category C] {T : C ⥤ C} [monad T] {A : algebra T} {B : algebra T} (h : A A ≅ A B) (w : functor.map T (iso.hom h) ≫ a B = a A ≫ iso.hom h) : hom.f (iso.hom (iso_mk h w)) = iso.hom h :=\n  Eq.refl (hom.f (iso.hom (iso_mk h w)))\n\nend algebra\n\n\n/-- The forgetful functor from the Eilenberg-Moore category, forgetting the algebraic structure. -/\n@[simp] theorem forget_map {C : Type u₁} [category C] (T : C ⥤ C) [monad T] (A : algebra T) (B : algebra T) (f : A ⟶ B) : functor.map (forget T) f = algebra.hom.f f :=\n  Eq.refl (functor.map (forget T) f)\n\n/-- The free functor from the Eilenberg-Moore category, constructing an algebra for any object. -/\n@[simp] theorem free_obj_A {C : Type u₁} [category C] (T : C ⥤ C) [monad T] (X : C) : algebra.A (functor.obj (free T) X) = functor.obj T X :=\n  Eq.refl (algebra.A (functor.obj (free T) X))\n\nprotected instance algebra.inhabited {C : Type u₁} [category C] (T : C ⥤ C) [monad T] [Inhabited C] : Inhabited (algebra T) :=\n  { default := functor.obj (free T) Inhabited.default }\n\n/-- The adjunction between the free and forgetful constructions for Eilenberg-Moore algebras for a monad.\n    cf Lemma 5.2.8 of [Riehl][riehl2017]. -/\n-- The other two `simps` projection lemmas can be derived from these two, so `simp_nf` complains if\n\n-- those are added too\n\n@[simp] theorem adj_counit {C : Type u₁} [category C] (T : C ⥤ C) [monad T] : adjunction.counit (adj T) =\n  nat_trans.mk\n    fun (Y : algebra T) =>\n      equiv.inv_fun\n        (adjunction.core_hom_equiv.hom_equiv\n          (adjunction.core_hom_equiv.mk\n            fun (X : C) (Y : algebra T) =>\n              equiv.mk (fun (f : functor.obj (free T) X ⟶ Y) => nat_trans.app η_ X ≫ algebra.hom.f f)\n                (fun (f : X ⟶ functor.obj (forget T) Y) => algebra.hom.mk (functor.map T f ≫ algebra.a Y))\n                (adj._proof_2 T X Y) (adj._proof_3 T X Y))\n          (functor.obj (forget T) Y) (functor.obj 𝟭 Y))\n        𝟙 :=\n  Eq.refl (adjunction.counit (adj T))\n\n/--\nGiven an algebra morphism whose carrier part is an isomorphism, we get an algebra isomorphism.\n-/\ndef algebra_iso_of_iso {C : Type u₁} [category C] (T : C ⥤ C) [monad T] {A : algebra T} {B : algebra T} (f : A ⟶ B) [is_iso (algebra.hom.f f)] : is_iso f :=\n  is_iso.mk (algebra.hom.mk (inv (algebra.hom.f f)))\n\nprotected instance forget_reflects_iso {C : Type u₁} [category C] (T : C ⥤ C) [monad T] : reflects_isomorphisms (forget T) :=\n  reflects_isomorphisms.mk fun (A B : algebra T) => algebra_iso_of_iso T\n\nprotected instance forget_faithful {C : Type u₁} [category C] (T : C ⥤ C) [monad T] : faithful (forget T) :=\n  faithful.mk\n\nend monad\n\n\nnamespace comonad\n\n\n/-- An Eilenberg-Moore coalgebra for a comonad `T`. -/\nstructure coalgebra {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] \nwhere\n  A : C\n  a : A ⟶ functor.obj G A\n  counit' : autoParam (a ≫ nat_trans.app ε_ A = 𝟙)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  coassoc' : autoParam (a ≫ nat_trans.app δ_ A = a ≫ functor.map G a)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem coalgebra.counit {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (c : coalgebra G) : coalgebra.a c ≫ nat_trans.app ε_ (coalgebra.A c) = 𝟙 := sorry\n\ntheorem coalgebra.coassoc {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (c : coalgebra G) : coalgebra.a c ≫ nat_trans.app δ_ (coalgebra.A c) = coalgebra.a c ≫ functor.map G (coalgebra.a c) := sorry\n\ntheorem coalgebra.counit_assoc {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (c : coalgebra G) {X' : C} (f' : coalgebra.A c ⟶ X') : coalgebra.a c ≫ nat_trans.app ε_ (coalgebra.A c) ≫ f' = f' := sorry\n\nnamespace coalgebra\n\n\n/-- A morphism of Eilenberg-Moore coalgebras for the comonad `G`. -/\nstructure hom {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (A : coalgebra G) (B : coalgebra G) \nwhere\n  f : A A ⟶ A B\n  h' : autoParam (a A ≫ functor.map G f = f ≫ a B)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem hom.h {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {A : coalgebra G} {B : coalgebra G} (c : hom A B) : a A ≫ functor.map G (hom.f c) = hom.f c ≫ a B := sorry\n\n@[simp] theorem hom.h_assoc {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {A : coalgebra G} {B : coalgebra G} (c : hom A B) {X' : C} (f' : functor.obj G (A B) ⟶ X') : a A ≫ functor.map G (hom.f c) ≫ f' = hom.f c ≫ a B ≫ f' := sorry\n\nnamespace hom\n\n\n/-- The identity homomorphism for an Eilenberg–Moore coalgebra. -/\ndef id {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (A : coalgebra G) : hom A A :=\n  mk 𝟙\n\n/-- Composition of Eilenberg–Moore coalgebra homomorphisms. -/\ndef comp {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {P : coalgebra G} {Q : coalgebra G} {R : coalgebra G} (f : hom P Q) (g : hom Q R) : hom P R :=\n  mk (f f ≫ f g)\n\nend hom\n\n\n/-- The category of Eilenberg-Moore coalgebras for a comonad. -/\nprotected instance category_theory.category_struct {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] : category_struct (coalgebra G) :=\n  category_struct.mk hom.id hom.comp\n\n@[simp] theorem comp_eq_comp {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {A : coalgebra G} {A' : coalgebra G} {A'' : coalgebra G} (f : A ⟶ A') (g : A' ⟶ A'') : hom.comp f g = f ≫ g :=\n  rfl\n\n@[simp] theorem id_eq_id {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (A : coalgebra G) : hom.id A = 𝟙 :=\n  rfl\n\n@[simp] theorem id_f {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] (A : coalgebra G) : hom.f 𝟙 = 𝟙 :=\n  rfl\n\n@[simp] theorem comp_f {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {A : coalgebra G} {A' : coalgebra G} {A'' : coalgebra G} (f : A ⟶ A') (g : A' ⟶ A'') : hom.f (f ≫ g) = hom.f f ≫ hom.f g :=\n  rfl\n\n/-- The category of Eilenberg-Moore coalgebras for a comonad. -/\nprotected instance EilenbergMoore {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] : category (coalgebra G) :=\n  category.mk\n\n/--\nTo construct an isomorphism of coalgebras, it suffices to give an isomorphism of the carriers which\ncommutes with the structure morphisms.\n-/\n@[simp] theorem iso_mk_hom_f {C : Type u₁} [category C] {G : C ⥤ C} [comonad G] {A : coalgebra G} {B : coalgebra G} (h : A A ≅ A B) (w : a A ≫ functor.map G (iso.hom h) = iso.hom h ≫ a B) : hom.f (iso.hom (iso_mk h w)) = iso.hom h :=\n  Eq.refl (hom.f (iso.hom (iso_mk h w)))\n\nend coalgebra\n\n\n/-- The forgetful functor from the Eilenberg-Moore category, forgetting the coalgebraic structure. -/\ndef forget {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] : coalgebra G ⥤ C :=\n  functor.mk (fun (A : coalgebra G) => coalgebra.A A) fun (A B : coalgebra G) (f : A ⟶ B) => coalgebra.hom.f f\n\n/--\nGiven a coalgebra morphism whose carrier part is an isomorphism, we get a coalgebra isomorphism.\n-/\ndef coalgebra_iso_of_iso {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] {A : coalgebra G} {B : coalgebra G} (f : A ⟶ B) [is_iso (coalgebra.hom.f f)] : is_iso f :=\n  is_iso.mk (coalgebra.hom.mk (inv (coalgebra.hom.f f)))\n\nprotected instance forget_reflects_iso {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] : reflects_isomorphisms (forget G) :=\n  reflects_isomorphisms.mk fun (A B : coalgebra G) => coalgebra_iso_of_iso G\n\n/-- The cofree functor from the Eilenberg-Moore category, constructing a coalgebra for any object. -/\n@[simp] theorem cofree_map_f {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] (X : C) (Y : C) (f : X ⟶ Y) : coalgebra.hom.f (functor.map (cofree G) f) = functor.map G f :=\n  Eq.refl (coalgebra.hom.f (functor.map (cofree G) f))\n\n/--\nThe adjunction between the cofree and forgetful constructions for Eilenberg-Moore coalgebras\nfor a comonad.\n-/\n-- The other two `simps` projection lemmas can be derived from these two, so `simp_nf` complains if\n\n-- those are added too\n\n@[simp] theorem adj_counit {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] : adjunction.counit (adj G) = nat_trans.mk (nat_trans.app ε_) := sorry\n\nprotected instance forget_faithful {C : Type u₁} [category C] (G : C ⥤ C) [comonad G] : faithful (forget G) :=\n  faithful.mk\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3268730889167336}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport order.hom.lattice\n\n/-!\n# Heyting algebra morphisms\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA Heyting homomorphism between two Heyting algebras is a bounded lattice homomorphism that preserves\nHeyting implication.\n\nWe use the `fun_like` design, so each type of morphisms has a companion typeclass which is meant to\nbe satisfied by itself and all stricter types.\n\n## Types of morphisms\n\n* `heyting_hom`: Heyting homomorphisms.\n* `coheyting_hom`: Co-Heyting homomorphisms.\n* `biheyting_hom`: Bi-Heyting homomorphisms.\n\n## Typeclasses\n\n* `heyting_hom_class`\n* `coheyting_hom_class`\n* `biheyting_hom_class`\n-/\n\nopen function\n\nvariables {F α β γ δ : Type*}\n\n/-- The type of Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that preserve\nHeyting implication. -/\n@[protect_proj]\nstructure heyting_hom (α β : Type*) [heyting_algebra α] [heyting_algebra β]\n  extends lattice_hom α β :=\n(map_bot' : to_fun ⊥ = ⊥)\n(map_himp' : ∀ a b, to_fun (a ⇨ b) = to_fun a ⇨ to_fun b)\n\n/-- The type of co-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve difference. -/\n@[protect_proj]\nstructure coheyting_hom (α β : Type*) [coheyting_algebra α] [coheyting_algebra β]\n  extends lattice_hom α β :=\n(map_top' : to_fun ⊤ = ⊤)\n(map_sdiff' : ∀ a b, to_fun (a \\ b) = to_fun a \\ to_fun b)\n\n/-- The type of bi-Heyting homomorphisms from `α` to `β`. Bounded lattice homomorphisms that\npreserve Heyting implication and difference. -/\n@[protect_proj]\nstructure biheyting_hom (α β : Type*) [biheyting_algebra α] [biheyting_algebra β]\n  extends lattice_hom α β :=\n(map_himp' : ∀ a b, to_fun (a ⇨ b) = to_fun a ⇨ to_fun b)\n(map_sdiff' : ∀ a b, to_fun (a \\ b) = to_fun a \\ to_fun b)\n\n/-- `heyting_hom_class F α β` states that `F` is a type of Heyting homomorphisms.\n\nYou should extend this class when you extend `heyting_hom`. -/\nclass heyting_hom_class (F : Type*) (α β : out_param $ Type*) [heyting_algebra α]\n  [heyting_algebra β] extends lattice_hom_class F α β :=\n(map_bot (f : F) : f ⊥ = ⊥)\n(map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b)\n\n/-- `coheyting_hom_class F α β` states that `F` is a type of co-Heyting homomorphisms.\n\nYou should extend this class when you extend `coheyting_hom`. -/\nclass coheyting_hom_class (F : Type*) (α β : out_param $ Type*) [coheyting_algebra α]\n  [coheyting_algebra β] extends lattice_hom_class F α β :=\n(map_top (f : F) : f ⊤ = ⊤)\n(map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b)\n\n/-- `biheyting_hom_class F α β` states that `F` is a type of bi-Heyting homomorphisms.\n\nYou should extend this class when you extend `biheyting_hom`. -/\nclass biheyting_hom_class (F : Type*) (α β : out_param $ Type*) [biheyting_algebra α]\n  [biheyting_algebra β] extends lattice_hom_class F α β :=\n(map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b)\n(map_sdiff (f : F) : ∀ a b, f (a \\ b) = f a \\ f b)\n\nexport heyting_hom_class (map_himp)\nexport coheyting_hom_class (map_sdiff)\n\nattribute [simp] map_himp map_sdiff\n\n@[priority 100] -- See note [lower instance priority]\ninstance heyting_hom_class.to_bounded_lattice_hom_class [heyting_algebra α] [heyting_algebra β]\n  [heyting_hom_class F α β] : bounded_lattice_hom_class F α β :=\n{ map_top := λ f, by rw [←@himp_self α _ ⊥, ←himp_self, map_himp],\n  ..‹heyting_hom_class F α β› }\n\n@[priority 100] -- See note [lower instance priority]\ninstance coheyting_hom_class.to_bounded_lattice_hom_class [coheyting_algebra α]\n  [coheyting_algebra β] [coheyting_hom_class F α β] : bounded_lattice_hom_class F α β :=\n{ map_bot := λ f, by rw [←@sdiff_self α _ ⊤, ←sdiff_self, map_sdiff],\n  ..‹coheyting_hom_class F α β› }\n\n@[priority 100] -- See note [lower instance priority]\ninstance biheyting_hom_class.to_heyting_hom_class [biheyting_algebra α] [biheyting_algebra β]\n  [biheyting_hom_class F α β] :\n  heyting_hom_class F α β :=\n{ map_bot := λ f, by rw [←@sdiff_self α _ ⊤, ←sdiff_self, biheyting_hom_class.map_sdiff],\n  ..‹biheyting_hom_class F α β› }\n\n@[priority 100] -- See note [lower instance priority]\ninstance biheyting_hom_class.to_coheyting_hom_class [biheyting_algebra α] [biheyting_algebra β]\n  [biheyting_hom_class F α β] :\n  coheyting_hom_class F α β :=\n{ map_top := λ f, by rw [←@himp_self α _ ⊥, ←himp_self, map_himp],\n  ..‹biheyting_hom_class F α β› }\n\n@[priority 100] -- See note [lower instance priority]\ninstance order_iso_class.to_heyting_hom_class [heyting_algebra α] [heyting_algebra β]\n  [order_iso_class F α β] :\n  heyting_hom_class F α β :=\n{ map_himp := λ f a b, eq_of_forall_le_iff $ λ c,\n    by { simp only [←map_inv_le_iff, le_himp_iff], rw ←order_iso_class.map_le_map_iff f, simp },\n  ..order_iso_class.to_bounded_lattice_hom_class }\n\n@[priority 100] -- See note [lower instance priority]\ninstance order_iso_class.to_coheyting_hom_class [coheyting_algebra α] [coheyting_algebra β]\n  [order_iso_class F α β] :\n  coheyting_hom_class F α β :=\n{ map_sdiff := λ f a b, eq_of_forall_ge_iff $ λ c,\n    by { simp only [←le_map_inv_iff, sdiff_le_iff], rw ←order_iso_class.map_le_map_iff f, simp },\n  ..order_iso_class.to_bounded_lattice_hom_class }\n\n@[priority 100] -- See note [lower instance priority]\ninstance order_iso_class.to_biheyting_hom_class [biheyting_algebra α] [biheyting_algebra β]\n  [order_iso_class F α β] :\n  biheyting_hom_class F α β :=\n{ map_himp := λ f a b, eq_of_forall_le_iff $ λ c,\n    by { simp only [←map_inv_le_iff, le_himp_iff], rw ←order_iso_class.map_le_map_iff f, simp },\n  map_sdiff := λ f a b, eq_of_forall_ge_iff $ λ c,\n    by { simp only [←le_map_inv_iff, sdiff_le_iff], rw ←order_iso_class.map_le_map_iff f, simp },\n  ..order_iso_class.to_lattice_hom_class }\n\n/-- This can't be an instance because of typeclass loops. -/\n@[reducible] -- See note [reducible non instances]\ndef bounded_lattice_hom_class.to_biheyting_hom_class [boolean_algebra α] [boolean_algebra β]\n  [bounded_lattice_hom_class F α β] :\n  biheyting_hom_class F α β :=\n{ map_himp := λ f a b, by rw [himp_eq, himp_eq, map_sup, (is_compl_compl.map _).compl_eq],\n  map_sdiff := λ f a b, by rw [sdiff_eq, sdiff_eq, map_inf, (is_compl_compl.map _).compl_eq],\n   ..‹bounded_lattice_hom_class F α β› }\n\nsection heyting_algebra\nvariables [heyting_algebra α] [heyting_algebra β] [heyting_hom_class F α β] (f : F)\ninclude β\n\n@[simp] lemma map_compl (a : α) : f aᶜ = (f a)ᶜ := by rw [←himp_bot, ←himp_bot, map_himp, map_bot]\n\n@[simp] lemma map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b :=\nby simp_rw [bihimp, map_inf, map_himp]\n\n-- TODO: `map_bihimp`\n\nend heyting_algebra\n\nsection coheyting_algebra\nvariables [coheyting_algebra α] [coheyting_algebra β] [coheyting_hom_class F α β] (f : F)\ninclude β\n\n@[simp] lemma map_hnot (a : α) : f ￢a = ￢f a :=\nby rw [←top_sdiff', ←top_sdiff', map_sdiff, map_top]\n\n@[simp] lemma map_symm_diff (a b : α) : f (a ∆ b) = f a ∆ f b :=\nby simp_rw [symm_diff, map_sup, map_sdiff]\n\nend coheyting_algebra\n\ninstance [heyting_algebra α] [heyting_algebra β] [heyting_hom_class F α β] :\n  has_coe_t F (heyting_hom α β) :=\n⟨λ f, { to_fun := f,\n        map_sup' := map_sup f,\n        map_inf' := map_inf f,\n        map_bot' := map_bot f,\n        map_himp' := map_himp f }⟩\n\ninstance [coheyting_algebra α] [coheyting_algebra β] [coheyting_hom_class F α β] :\n  has_coe_t F (coheyting_hom α β) :=\n⟨λ f, { to_fun := f,\n        map_sup' := map_sup f,\n        map_inf' := map_inf f,\n        map_top' := map_top f,\n        map_sdiff' := map_sdiff f }⟩\n\ninstance [biheyting_algebra α] [biheyting_algebra β] [biheyting_hom_class F α β] :\n  has_coe_t F (biheyting_hom α β) :=\n⟨λ f, { to_fun := f,\n        map_sup' := map_sup f,\n        map_inf' := map_inf f,\n        map_himp' := map_himp f,\n        map_sdiff' := map_sdiff f }⟩\n\nnamespace heyting_hom\nvariables [heyting_algebra α] [heyting_algebra β] [heyting_algebra γ] [heyting_algebra δ]\n\ninstance : heyting_hom_class (heyting_hom α β) α β :=\n{ coe := λ f, f.to_fun,\n  coe_injective' := λ f g h, by obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr',\n  map_sup := λ f, f.map_sup',\n  map_inf := λ f, f.map_inf',\n  map_bot := λ f, f.map_bot',\n  map_himp := heyting_hom.map_himp' }\n\n/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\ninstance : has_coe_to_fun (heyting_hom α β) (λ _, α → β) := fun_like.has_coe_to_fun\n\n@[simp] lemma to_fun_eq_coe {f : heyting_hom α β} : f.to_fun = (f : α → β) := rfl\n\n@[ext] lemma ext {f g : heyting_hom α β} (h : ∀ a, f a = g a) : f = g := fun_like.ext f g h\n\n/-- Copy of a `heyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : heyting_hom α β) (f' : α → β) (h : f' = f) : heyting_hom α β :=\n{ to_fun := f',\n  map_sup' := by simpa only [h] using map_sup f,\n  map_inf' := by simpa only [h] using map_inf f,\n  map_bot' := by simpa only [h] using map_bot f,\n  map_himp' := by simpa only [h] using map_himp f }\n\n@[simp] lemma coe_copy (f : heyting_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl\nlemma copy_eq (f : heyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f := fun_like.ext' h\n\nvariables (α)\n\n/-- `id` as a `heyting_hom`. -/\nprotected def id : heyting_hom α α :=\n{ to_lattice_hom := lattice_hom.id _,\n  map_himp' := λ a b, rfl,\n  ..bot_hom.id _ }\n\n@[simp] lemma coe_id : ⇑(heyting_hom.id α) = id := rfl\n\nvariables {α}\n\n@[simp] lemma id_apply (a : α) : heyting_hom.id α a = a := rfl\n\ninstance : inhabited (heyting_hom α α) := ⟨heyting_hom.id _⟩\n\ninstance : partial_order (heyting_hom α β) := partial_order.lift _ fun_like.coe_injective\n\n/-- Composition of `heyting_hom`s as a `heyting_hom`. -/\ndef comp (f : heyting_hom β γ) (g : heyting_hom α β) : heyting_hom α γ :=\n{ to_fun := f ∘ g,\n  map_bot' := by simp,\n  map_himp' := λ a b, by simp,\n  ..f.to_lattice_hom.comp g.to_lattice_hom }\n\nvariables {f f₁ f₂ : heyting_hom α β} {g g₁ g₂ : heyting_hom β γ}\n\n@[simp] \n\nlemma cancel_right (hf : surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=\n⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩\n\nlemma cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=\n⟨λ h, heyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩\n\nend heyting_hom\n\nnamespace coheyting_hom\nvariables [coheyting_algebra α] [coheyting_algebra β] [coheyting_algebra γ] [coheyting_algebra δ]\n\ninstance : coheyting_hom_class (coheyting_hom α β) α β :=\n{ coe := λ f, f.to_fun,\n  coe_injective' := λ f g h, by obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr',\n  map_sup := λ f, f.map_sup',\n  map_inf := λ f, f.map_inf',\n  map_top := λ f, f.map_top',\n  map_sdiff := coheyting_hom.map_sdiff' }\n\n/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\ninstance : has_coe_to_fun (coheyting_hom α β) (λ _, α → β) := fun_like.has_coe_to_fun\n\n@[simp] lemma to_fun_eq_coe {f : coheyting_hom α β} : f.to_fun = (f : α → β) := rfl\n\n@[ext] lemma ext {f g : coheyting_hom α β} (h : ∀ a, f a = g a) : f = g := fun_like.ext f g h\n\n/-- Copy of a `coheyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : coheyting_hom α β :=\n{ to_fun := f',\n  map_sup' := by simpa only [h] using map_sup f,\n  map_inf' := by simpa only [h] using map_inf f,\n  map_top' := by simpa only [h] using map_top f,\n  map_sdiff' := by simpa only [h] using map_sdiff f }\n\n@[simp]\nlemma coe_copy (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' := rfl\n\nlemma copy_eq (f : coheyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f := fun_like.ext' h\n\nvariables (α)\n\n/-- `id` as a `coheyting_hom`. -/\nprotected def id : coheyting_hom α α :=\n{ to_lattice_hom := lattice_hom.id _,\n  map_sdiff' := λ a b, rfl,\n  ..top_hom.id _ }\n\n@[simp] lemma coe_id : ⇑(coheyting_hom.id α) = id := rfl\n\nvariables {α}\n\n@[simp] lemma id_apply (a : α) : coheyting_hom.id α a = a := rfl\n\ninstance : inhabited (coheyting_hom α α) := ⟨coheyting_hom.id _⟩\n\ninstance : partial_order (coheyting_hom α β) := partial_order.lift _ fun_like.coe_injective\n\n/-- Composition of `coheyting_hom`s as a `coheyting_hom`. -/\ndef comp (f : coheyting_hom β γ) (g : coheyting_hom α β) : coheyting_hom α γ :=\n{ to_fun := f ∘ g,\n  map_top' := by simp,\n  map_sdiff' := λ a b, by simp,\n  ..f.to_lattice_hom.comp g.to_lattice_hom }\n\nvariables {f f₁ f₂ : coheyting_hom α β} {g g₁ g₂ : coheyting_hom β γ}\n\n@[simp] lemma coe_comp (f : coheyting_hom β γ) (g : coheyting_hom α β) : ⇑(f.comp g) = f ∘ g := rfl\n@[simp] lemma comp_apply (f : coheyting_hom β γ) (g : coheyting_hom α β) (a : α) :\n f.comp g a = f (g a) := rfl\n@[simp] lemma comp_assoc (f : coheyting_hom γ δ) (g : coheyting_hom β γ) (h : coheyting_hom α β) :\n  (f.comp g).comp h = f.comp (g.comp h) := rfl\n@[simp] lemma comp_id (f : coheyting_hom α β) : f.comp (coheyting_hom.id α) = f := ext $ λ a, rfl\n@[simp] lemma id_comp (f : coheyting_hom α β) : (coheyting_hom.id β).comp f = f := ext $ λ a, rfl\n\nlemma cancel_right (hf : surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=\n⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩\n\nlemma cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=\n⟨λ h, coheyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩\n\nend coheyting_hom\n\n\nnamespace biheyting_hom\nvariables [biheyting_algebra α] [biheyting_algebra β] [biheyting_algebra γ] [biheyting_algebra δ]\n\ninstance : biheyting_hom_class (biheyting_hom α β) α β :=\n{ coe := λ f, f.to_fun,\n  coe_injective' := λ f g h, by obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f; obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g; congr',\n  map_sup := λ f, f.map_sup',\n  map_inf := λ f, f.map_inf',\n  map_himp := λ f, f.map_himp',\n  map_sdiff := λ f, f.map_sdiff' }\n\n/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\ninstance : has_coe_to_fun (biheyting_hom α β) (λ _, α → β) := fun_like.has_coe_to_fun\n\n@[simp] lemma to_fun_eq_coe {f : biheyting_hom α β} : f.to_fun = (f : α → β) := rfl\n\n@[ext] lemma ext {f g : biheyting_hom α β} (h : ∀ a, f a = g a) : f = g := fun_like.ext f g h\n\n/-- Copy of a `biheyting_hom` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : biheyting_hom α β) (f' : α → β) (h : f' = f) : biheyting_hom α β :=\n{ to_fun := f',\n  map_sup' := by simpa only [h] using map_sup f,\n  map_inf' := by simpa only [h] using map_inf f,\n  map_himp' := by simpa only [h] using map_himp f,\n  map_sdiff' := by simpa only [h] using map_sdiff f }\n\n@[simp] lemma coe_copy (f : biheyting_hom α β) (f' : α → β) (h : f' = f) :\n  ⇑(f.copy f' h) = f' :=\nrfl\n\nlemma copy_eq (f : biheyting_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f := fun_like.ext' h\n\nvariables (α)\n\n/-- `id` as a `biheyting_hom`. -/\nprotected def id : biheyting_hom α α :=\n{ to_lattice_hom := lattice_hom.id _,\n  ..heyting_hom.id _, ..coheyting_hom.id _ }\n\n@[simp] lemma coe_id : ⇑(biheyting_hom.id α) = id := rfl\n\nvariables {α}\n\n@[simp] lemma id_apply (a : α) : biheyting_hom.id α a = a := rfl\n\ninstance : inhabited (biheyting_hom α α) := ⟨biheyting_hom.id _⟩\n\ninstance : partial_order (biheyting_hom α β) := partial_order.lift _ fun_like.coe_injective\n\n/-- Composition of `biheyting_hom`s as a `biheyting_hom`. -/\ndef comp (f : biheyting_hom β γ) (g : biheyting_hom α β) : biheyting_hom α γ :=\n{ to_fun := f ∘ g,\n  map_himp' := λ a b, by simp,\n  map_sdiff' := λ a b, by simp,\n  ..f.to_lattice_hom.comp g.to_lattice_hom }\n\nvariables {f f₁ f₂ : biheyting_hom α β} {g g₁ g₂ : biheyting_hom β γ}\n\n@[simp] lemma coe_comp (f : biheyting_hom β γ) (g : biheyting_hom α β) : ⇑(f.comp g) = f ∘ g := rfl\n@[simp] lemma comp_apply (f : biheyting_hom β γ) (g : biheyting_hom α β) (a : α) :\n f.comp g a = f (g a) := rfl\n@[simp] lemma comp_assoc (f : biheyting_hom γ δ) (g : biheyting_hom β γ) (h : biheyting_hom α β) :\n  (f.comp g).comp h = f.comp (g.comp h) := rfl\n@[simp] lemma comp_id (f : biheyting_hom α β) : f.comp (biheyting_hom.id α) = f := ext $ λ a, rfl\n@[simp] lemma id_comp (f : biheyting_hom α β) : (biheyting_hom.id β).comp f = f := ext $ λ a, rfl\n\nlemma cancel_right (hf : surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ :=\n⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩\n\nlemma cancel_left (hg : injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=\n⟨λ h, biheyting_hom.ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩\n\nend biheyting_hom\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/order/heyting/hom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3268730806850108}}
{"text": "import data.nat.squarefree\nimport data.int.sqrt\nimport number_theory.number_field.class_number\nimport number_theory.quad_ring.basic\n\n/-!\n# Showing the class group of `ℤ[√d]` is nontrivial\n-/\n\nopen_locale non_zero_divisors\nopen quad_ring\nopen_locale quad_ring\n\nsection to_mathlib\n\nsection\n\n/-!\n### The class group is independent of choice of fraction field or ring of integers\n-/\n\nopen submodule\n\n/-- This is `submodule.map_mul` where `g` satisfies the conditions of `alg_hom`\nexcept it's semilinear. -/\nprotected lemma submodule.map_mulₛₗ {R R' A A'} [comm_semiring R] [comm_semiring R']\n  [semiring A] [semiring A'] [algebra R A] [algebra R' A']\n  (f : R →+* R') [ring_hom_surjective f]\n  (g : A →ₛₗ[f] A') (hg : ∀ (x y : A), g (x * y) = g x * g y) (M N : submodule R A) :\n  map g (M * N) = map g M * map g N :=\ncalc map g (M * N)\n    = ⨆ (i : M), (N.map (algebra.lmul R A i)).map g : map_supr _ _\n... = map g M * map g N  :\n  begin\n    apply congr_arg Sup,\n    ext S,\n    split; rintros ⟨y, hy⟩,\n    { use [g y, mem_map.mpr ⟨y.1, y.2, rfl⟩],\n      refine trans _ hy,\n      ext,\n      simp [hg] },\n    { obtain ⟨y', hy', fy_eq⟩ := mem_map.mp y.2,\n      use [y', hy'],\n      refine trans _ hy,\n      ext,\n      simp [fy_eq, hg] }\nend\n\nend\n\nlemma ideal.span_eq_span_iff {R : Type*} [comm_ring R]\n  {s t : set R} : ideal.span s = ideal.span t ↔ s ⊆ ideal.span t ∧ t ⊆ ideal.span s :=\nle_antisymm_iff.trans (by simp only [ideal.span_le])\n\n/-- `is_localization.map` bundled as a semilinear map. -/\n@[simps]\nnoncomputable def is_localization.map_semilinear {R S} (Rₘ Sₙ : Type*)\n  [comm_ring R] [comm_ring S] (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : M ≤ N.comap f) : Rₘ →ₛₗ[(f : R →+* S)] Sₙ :=\n{ to_fun := is_localization.map Sₙ (f : R →+* S) hf,\n  map_smul' := λ r x, is_localization.map_smul _ _ _,\n  .. is_localization.map Sₙ (f : R →+* S) hf }\n\n-- Should be a congr lemma but can't because `@[congr]` doesn't\n-- understand `coe_fn`\nlemma is_localization.map_semilinear_congr {R S Rₘ Sₙ : Type*} [comm_ring R] [comm_ring S]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  {F : Type*} [ring_hom_class F R S] (f g : F) (hf : M ≤ N.comap f) (hg : M ≤ N.comap g)\n  (h : f = g) (x y : Rₘ) (hxy : x = y) :\n  (is_localization.map_semilinear Rₘ Sₙ M N f hf) x =\n    (is_localization.map_semilinear Rₘ Sₙ M N g hg) y :=\nby subst h; subst hxy; refl\n\n@[simp] lemma is_localization.map_semilinear_eq {R S Rₘ Sₙ : Type*} [comm_ring R] [comm_ring S]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : M ≤ N.comap f) (x : R) :\n  is_localization.map_semilinear Rₘ Sₙ M N f hf (algebra_map R Rₘ x) = algebra_map S Sₙ (f x) :=\nis_localization.map_eq _ _\n\n@[simp] lemma is_localization.map_semilinear_mul {R S Rₘ Sₙ : Type*} [comm_ring R] [comm_ring S]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : M ≤ N.comap f) (x y : Rₘ) :\n  is_localization.map_semilinear Rₘ Sₙ M N f hf (x * y) =\n    is_localization.map_semilinear Rₘ Sₙ M N f hf x *\n    is_localization.map_semilinear Rₘ Sₙ M N f hf y :=\nby simp only [is_localization.map_semilinear, linear_map.coe_mk, map_mul]\n\n@[simp] lemma is_localization.map_semilinear_comp_apply {R S T Rₘ Sₙ Tₚ : Type*}\n  [comm_ring R] [comm_ring S] [comm_ring T]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  (P : submonoid T) [comm_ring Tₚ] [algebra T Tₚ] [is_localization P Tₚ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : M ≤ N.comap f)\n  {G : Type*} [ring_hom_class G S T] (g : G) (hg : N ≤ P.comap g)\n  {GF : Type*} [ring_hom_class GF R T] (gf : GF) (hgf : M ≤ P.comap gf)\n  [ring_hom_comp_triple (f : R →+* S) (g : S →+* T) gf]\n  (x : Rₘ) :\n  (is_localization.map_semilinear _ Tₚ M P gf hgf : Rₘ →ₛₗ[_] _) x =\n    is_localization.map_semilinear _ Tₚ N P g hg\n      (is_localization.map_semilinear _ Sₙ M N f hf x) :=\nbegin\n  simp only [is_localization.map_semilinear, linear_map.coe_mk, linear_map.coe_comp],\n  -- Help the elaborator with inserting the coercions\n  convert fun_like.ext_iff.mp\n    (@is_localization.map_comp_map R _ _ Rₘ _ _ S _ _ f _ Sₙ _ hf _ _ T _ _ Tₚ _ _ _ g hg).symm x,\n  exact ring_hom_comp_triple.comp_eq.symm\nend\n\n@[simp] lemma is_localization.map_semilinear_comp {R S T Rₘ Sₙ Tₚ : Type*}\n  [comm_ring R] [comm_ring S] [comm_ring T]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  (P : submonoid T) [comm_ring Tₚ] [algebra T Tₚ] [is_localization P Tₚ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : M ≤ N.comap f)\n  {G : Type*} [ring_hom_class G S T] (g : G) (hg : N ≤ P.comap g)\n  {GF : Type*} [ring_hom_class GF R T] (gf : GF) (hgf : M ≤ P.comap gf)\n  [ring_hom_comp_triple (f : R →+* S) (g : S →+* T) gf] :\n  (is_localization.map_semilinear _ Tₚ M P gf hgf : Rₘ →ₛₗ[_] _) =\n    (is_localization.map_semilinear _ Tₚ N P g hg).comp\n      (is_localization.map_semilinear _ Sₙ M N f hf) :=\nbegin\n  ext x,\n  apply is_localization.map_semilinear_comp_apply\nend\n\n@[simp] lemma is_localization.map_semilinear_id_apply {R Rₘ : Type*}\n  [comm_ring R]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (x : Rₘ) :\n  (is_localization.map_semilinear Rₘ Rₘ M M (ring_hom.id R)\n      (submonoid.comap_id M).le : Rₘ →ₛₗ[_] _) x =\n    x :=\nis_localization.map_id x (submonoid.comap_id M).le\n\n@[simp] lemma fractional_ideal.mem_mk {R Rₘ : Type*} [comm_ring R] {M : submonoid R} [comm_ring Rₘ]\n  [algebra R Rₘ] {S : submodule R Rₘ} {h : is_fractional M S} {x} :\n  x ∈ (show fractional_ideal M Rₘ, from ⟨S, h⟩) ↔ x ∈ S :=\niff.rfl\n\n/-- If `f : R →+* S` maps `M : submonoid R` to `N : submonoid S`, then the fractional ideals\nat `M` can be mapped to those at `N`. -/\n@[simps apply]\nnoncomputable def fractional_ideal.map_base {R S Rₘ Sₙ : Type*} [comm_ring R] [comm_ring S]\n  (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  {F : Type*} [ring_hom_class F R S] (f : F) (hf : function.surjective f) (hfM : M ≤ N.comap f) :\n  fractional_ideal M Rₘ →+* fractional_ideal N Sₙ :=\n{ to_fun := λ I, ⟨by haveI : ring_hom_surjective (f : R →+* S) := ⟨hf⟩;\n    exact submodule.map (is_localization.map_semilinear Rₘ Sₙ M N f hfM) (I : submodule R Rₘ),\n    begin\n      obtain ⟨a, haM, ha⟩ := I.2,\n      refine ⟨f a, hfM haM, _⟩,\n      simp only [submodule.mem_map, fractional_ideal.mem_coe, forall_exists_index, and_imp,\n        forall_apply_eq_imp_iff₂],\n      intros b hb,\n      obtain ⟨ab, hab⟩ := ha b hb,\n      use f ab,\n      -- Work around the `↑f` because `linear_map.map_smulₛₗ` is non-generic in `f`.\n      rw [← ring_hom.coe_coe f, ← linear_map.map_smulₛₗ, ← hab, is_localization.map_semilinear_eq,\n          ring_hom.coe_coe f]\n    end⟩,\n  map_zero' := by ext; simp only [submodule.map_bot, fractional_ideal.coe_zero, submodule.mem_bot,\n    fractional_ideal.mem_zero_iff, fractional_ideal.mem_mk],\n  -- TODO: do the following lines with just `fractional_ideal.ext`\n  map_add' := λ I J, fractional_ideal.coe_to_submodule_injective $\n    by simp only [fractional_ideal.coe_add, submodule.map_sup, submodule.add_eq_sup, fractional_ideal.coe_mk],\n  map_one' := fractional_ideal.coe_to_submodule_injective $ begin\n    ext,\n    simp only [fractional_ideal.coe_mk, fractional_ideal.coe_one, submodule.mem_map,\n      submodule.mem_one, exists_exists_eq_and],\n    split,\n    { rintros ⟨a, rfl⟩,\n      simp only [is_localization.map_semilinear_eq, exists_apply_eq_apply] },\n    { rintros ⟨a, rfl⟩,\n      obtain ⟨x, rfl⟩ := hf a,\n      simp only [is_localization.map_semilinear_eq, exists_apply_eq_apply] },\n  end,\n  map_mul' := λ I J, fractional_ideal.coe_to_submodule_injective $ begin\n    simp only [fractional_ideal.coe_mul, fractional_ideal.coe_mk],\n    apply submodule.map_mulₛₗ,\n    apply is_localization.map_semilinear_mul\n  end }\n\nsection\nuniverses u v u' v'\n/-- If `e : R ≃+* S` maps `M : submonoid R` to `N : submonoid S`, then the fractional ideals\nat `M` are isomorphic to those at `N`. -/\n@[simps]\nnoncomputable def fractional_ideal.congr {R : Type u} {S : Type v} (Rₘ : Type u') (Sₙ : Type v')\n  [comm_ring R] [comm_ring S] (M : submonoid R) [comm_ring Rₘ] [algebra R Rₘ] [is_localization M Rₘ]\n  (N : submonoid S) [comm_ring Sₙ] [algebra S Sₙ] [is_localization N Sₙ]\n  (e : R ≃+* S) (he : M.map e = N) : fractional_ideal M Rₘ ≃+* fractional_ideal N Sₙ :=\n{ to_fun := fractional_ideal.map_base M N e e.surjective (submonoid.map_le_iff_le_comap.mp he.le),\n  inv_fun := fractional_ideal.map_base N M e.symm e.symm.surjective (submonoid.map_le_iff_le_comap.mp\n    (eq.le $ begin\n      rw ← he,\n      refine (submonoid.map_map M (e.symm : S →* R) (e : R →* S)).trans _,\n      convert submonoid.map_id _,\n      ext,\n      exact e.symm_apply_apply x\n    end)),\n  -- TODO: the next lines should be doable using `fractional_ideal.ext`\n  left_inv := λ I, fractional_ideal.coe_to_submodule_injective $ begin\n    simp only [fractional_ideal.map_base_apply, fractional_ideal.coe_mk],\n    haveI : ring_hom_surjective (e : R →+* S) := ⟨e.surjective⟩,\n    haveI : ring_hom_surjective (e.symm : S →+* R) := ⟨e.symm.surjective⟩,\n    haveI : ring_hom_comp_triple (e : R →+* S) (e.symm : S →+* R) (ring_hom.id R) :=\n      ⟨e.symm_to_ring_hom_comp_to_ring_hom⟩,\n    haveI : ring_hom_comp_triple (e.symm : S →+* R) (e : R →+* S) (ring_hom.id S) :=\n      ⟨e.to_ring_hom_comp_symm_to_ring_hom⟩,\n    -- Come on, Lean!\n    calc submodule.map (is_localization.map_semilinear Sₙ Rₘ N M e.symm _) (submodule.map (is_localization.map_semilinear Rₘ Sₙ M N e _) ↑I)\n        = submodule.map ((is_localization.map_semilinear Sₙ Rₘ N M e.symm _).comp (is_localization.map_semilinear Rₘ Sₙ M N e _)) ↑I : (submodule.map_comp (is_localization.map_semilinear Rₘ Sₙ M N e _) (is_localization.map_semilinear Sₙ Rₘ N M e.symm _) _).symm\n    ... = submodule.map linear_map.id ↑I : _\n    ... = ↑I : submodule.map_id _,\n    congr,\n    ext x,\n    calc (is_localization.map_semilinear Sₙ Rₘ N M e.symm _).comp\n            (is_localization.map_semilinear Rₘ Sₙ M N e _) x\n        = is_localization.map_semilinear Rₘ Rₘ M M (ring_hom.id R) _ x :\n      (is_localization.map_semilinear_comp_apply M N M e _ e.symm _ _ _ x).symm\n    ... = x : is_localization.map_semilinear_id_apply M x\n    ... = linear_map.id x : (linear_map.id_apply x).symm\n  end,\n  right_inv := λ I, fractional_ideal.coe_to_submodule_injective $ begin\n    simp only [fractional_ideal.map_base_apply, fractional_ideal.coe_mk],\n    haveI : ring_hom_surjective (e : R →+* S) := ⟨e.surjective⟩,\n    haveI : ring_hom_surjective (e.symm : S →+* R) := ⟨e.symm.surjective⟩,\n    haveI : ring_hom_comp_triple (e : R →+* S) (e.symm : S →+* R) (ring_hom.id R) :=\n      ⟨e.symm_to_ring_hom_comp_to_ring_hom⟩,\n    haveI : ring_hom_comp_triple (e.symm : S →+* R) (e : R →+* S) (ring_hom.id S) :=\n      ⟨e.to_ring_hom_comp_symm_to_ring_hom⟩,\n    calc submodule.map (is_localization.map_semilinear Rₘ Sₙ M N e _) (submodule.map (is_localization.map_semilinear Sₙ Rₘ N M e.symm _) ↑I)\n        = submodule.map ((is_localization.map_semilinear Rₘ Sₙ M N e _).comp (is_localization.map_semilinear Sₙ Rₘ N M e.symm _)) ↑I : (submodule.map_comp (is_localization.map_semilinear Sₙ Rₘ N M e.symm _) (is_localization.map_semilinear Rₘ Sₙ M N e _) _).symm\n    ... = submodule.map linear_map.id ↑I : _\n    ... = ↑I : submodule.map_id _,\n    congr,\n    ext x,\n    calc (is_localization.map_semilinear Rₘ Sₙ M N e _).comp (is_localization.map_semilinear Sₙ Rₘ N M e.symm _) x\n        = is_localization.map_semilinear Sₙ Sₙ N N (ring_hom.id S) _ x :\n      (is_localization.map_semilinear_comp_apply N M N e.symm _ e _ _ _ x).symm\n    ... = x : is_localization.map_semilinear_id_apply N x\n    ... = linear_map.id x : (linear_map.id_apply x).symm\n  end,\n  .. fractional_ideal.map_base M N e e.surjective _ }\nend\n\n-- TODO: the existence of this instance helps the elaborator unify quotient_group.group with comm_group.to_group\nnoncomputable instance (R : Type*) [comm_ring R] [is_domain R] :\n  group (class_group R) := quotient_group.quotient.group _\n\n@[simp] lemma map_non_zero_divisors {R S : Type*} [comm_ring R] [is_domain R] [comm_ring S] [is_domain S]\n  (e : R ≃+* S) : submonoid.map e R⁰ = S⁰ :=\nbegin\n  ext,\n  simp only [submonoid.mem_map],\n  split,\n  { rintro ⟨x, hx, rfl⟩,\n    exact map_mem_non_zero_divisors e e.injective hx },\n  { intro hx,\n    exact ⟨e.symm x, map_mem_non_zero_divisors e.symm e.symm.injective hx, e.apply_symm_apply x⟩ }\nend\n\nlemma units_map_equiv_congr_to_principal_ideal {R S : Type*} (K L : Type*)\n  [comm_ring R] [is_domain R] [comm_ring S] [is_domain S]\n  [field K] [field L] [algebra R K] [is_fraction_ring R K] [algebra S L] [is_fraction_ring S L]\n  (e : R ≃+* S) (x : Kˣ) :\n  (units.map_equiv ↑(fractional_ideal.congr K L R⁰ S⁰ e (map_non_zero_divisors e))) (to_principal_ideal R K x) =\n    to_principal_ideal S L (units.map_equiv ↑(is_localization.ring_equiv_of_ring_equiv K L e (map_non_zero_divisors e)) x) :=\nbegin\n  ext : 1,\n  rw coe_to_principal_ideal,\n  show fractional_ideal.congr K L R⁰ S⁰ e _ (to_principal_ideal R K x) = _,\n rw coe_to_principal_ideal,\n apply fractional_ideal.coe_to_submodule_injective,\n simp only [is_localization.ring_equiv_of_ring_equiv_apply, fractional_ideal.map_base_apply,\n   ring_equiv.coe_to_mul_equiv, fractional_ideal.coe_span_singleton,\n   fractional_ideal.congr_apply, fractional_ideal.coe_mk, submodule.map_span,\n   set.image_singleton, is_localization.map_semilinear_apply],\n  refl\nend\n\nuniverses u v\n\n/-- The class group of `R` is isomorphic to that of `S`, if `R` is isomorphic to `S`. -/\nnoncomputable def class_group.congr {R : Type u} {S : Type v}\n  [comm_ring R] [is_domain R] [comm_ring S] [is_domain S]\n  (e : R ≃+* S) : class_group R ≃* class_group S :=\nquotient_group.congr (to_principal_ideal R (fraction_ring R)).range (to_principal_ideal S (fraction_ring S)).range\n  (units.map_equiv ↑(fractional_ideal.congr (fraction_ring R) (fraction_ring S) R⁰ S⁰ e (map_non_zero_divisors e))) $\n  by { rw [monoid_hom.range_eq_map, subgroup.map_map, monoid_hom.range_eq_map],\n       ext,\n       simp only [subgroup.mem_map, subgroup.mem_top, exists_prop, true_and,\n          monoid_hom.comp_apply],\n       let e' : fraction_ring R ≃* fraction_ring S := is_localization.ring_equiv_of_ring_equiv\n        (fraction_ring R) (fraction_ring S) e (map_non_zero_divisors e),\n       split; rintro ⟨x, rfl⟩,\n       { exact ⟨units.map_equiv e' x, (units_map_equiv_congr_to_principal_ideal _ _ e x).symm⟩ },\n       { refine ⟨units.map_equiv e'.symm x, (units_map_equiv_congr_to_principal_ideal _ _ e _).trans _⟩,\n         congr, ext : 1,\n         exact mul_equiv.apply_symm_apply _ x } }\n\nend to_mathlib\n\nsection to_mathlib\n\n/-- The class number of a number field does not depend on a choice of ring of integers. -/\ntheorem number_field.class_number_eq (R : Type*) (K : Type*) [comm_ring R] [is_domain R]\n  [field K] [algebra R K] [is_fraction_ring R K] [number_field K] [is_integral_closure R ℤ K]\n  [fintype (class_group R)] :\n  number_field.class_number K = fintype.card (class_group R) :=\nbegin\n  unfold number_field.class_number,\n  convert fintype.card_eq.mpr ⟨(class_group.congr _).to_equiv⟩,\n  exact number_field.ring_of_integers.equiv _,\nend\n\n/-- The class number of a number field does not depend on a choice of ring of integers. -/\ntheorem number_field.class_number_eq' (R : Type*) (K : Type*) [comm_ring R] [is_domain R]\n  [field K] [algebra R K] [is_fraction_ring R K] [number_field K] [is_integral_closure R ℤ K] :\n  number_field.class_number K = @fintype.card (class_group R)\n    (class_group.fintype_of_admissible_of_finite ℚ K absolute_value.abs_is_admissible) :=\nby letI : fintype (class_group R) :=\n  class_group.fintype_of_admissible_of_finite ℚ K absolute_value.abs_is_admissible;\nexact number_field.class_number_eq R K\n\nend to_mathlib\n\nopen_locale pointwise\n\nsection to_mathlib\n\n@[simp]\nlemma set.insert_mul {M : Type*} [has_mul M]\n  (s t : set M) (x : M) : insert x s * t = x • t ∪ s * t :=\nby rw [set.insert_eq, set.union_mul, set.singleton_mul]; refl\n@[simp]\nlemma set.mul_insert {M : Type*} [comm_monoid M]\n  (s t : set M) (x : M) : s * insert x t = x • s ∪ s * t :=\nby rw [set.insert_eq, set.mul_union, set.mul_singleton];\n  -- TODO: clean me!\n  congr' 2; ext; apply mul_comm\n\n@[simp]\nlemma set.smul_set_insert {M : Type*} [comm_monoid M]\n  (x y : M) (s : set M) : x • insert y s = insert (x * y) (x • s) :=\nby rw [set.insert_eq, set.smul_set_union, set.smul_set_singleton, set.insert_eq]; refl\n\n@[simp]\nlemma set.image_insert {α β : Type*} (f : α → β) (x : α) (s : set α) :\n  f '' (insert x s) = insert (f x) (f '' s) :=\nby simp only [set.insert_eq, set.image_union, set.image_singleton]\n\nlemma ideal.span_eq_bot_iff_subset {α : Type*} [comm_semiring α] {S : set α} :\n  ideal.span S = ⊥ ↔ S ⊆ {0} :=\nideal.span_eq_bot.trans set.subset_singleton_iff.symm\n\nlemma set.not_subset_singleton_iff {α : Type*} {s : set α} {x : α} :\n  ¬ (s ⊆ {x}) ↔ s ≠ ∅ ∧ s ≠ {x} :=\nby rw [set.subset_singleton_iff_eq, not_or_distrib]\n\nlemma ideal.span_ne_bot_iff {α : Type*} [comm_semiring α] {S : set α} :\n  ideal.span S ≠ ⊥ ↔ (S.nonempty ∧ S ≠ {0}) :=\nby simp only [ne.def, ideal.span_eq_bot_iff_subset, set.not_subset_singleton_iff,\n              set.nonempty_iff_ne_empty]\n\n@[simp] lemma set.ne_singleton_of_ne {α : Type*} {s : set α} {x y : α}\n  (hxs : x ∈ s) (hxy : x ≠ y) : s ≠ {y} :=\nby { contrapose! hxy, rw hxy at hxs, exact set.eq_of_mem_singleton hxs }\n\nend to_mathlib\n\nsection basic\n\n@[simp] lemma coe_dvd_mk {R : Type*} [comm_ring R] {n a b x y : R} :\n  ↑n ∣ (⟨x, y⟩ : quad_ring R a b) ↔ n ∣ x ∧ n ∣ y :=\nbegin\n  split,\n  { rintro ⟨⟨da, db⟩, hd⟩,\n    simp only [quad_ring.ext_iff, add_zero, int.cast_id, quad_ring.mul_b2, zero_mul,\n      quad_ring.mul_b1, quad_ring.coe_b1, zero_add, quad_ring.coe_b2, mul_zero] at hd,\n    exact ⟨⟨_, hd.1⟩, ⟨_, hd.2⟩⟩ },\n  { rintro ⟨⟨da, hda⟩, ⟨db, hdb⟩⟩,\n    refine ⟨⟨da, db⟩, _⟩,\n    ext; simp [hda, hdb] }\nend\n\ninstance {b : ℚ} [fact $ ¬ is_square b] : number_field (quad_ring ℚ 0 b) :=\n{ to_finite_dimensional := quad_ring.finite_dimensional 0 b }\n\nend basic\n\n/-- The ideal `⟨2⟩` in ℤ[√d] is a square if `d % 4 = 3`.  -/\nlemma exists_sqrt_2_mod_three (d : ℤ) (hd : d % 4 = 3) :\n  (ideal.span ({⟨1, 1⟩, 2} : set (quad_ring ℤ 0 d)))^2 = ideal.span {2} :=\nbegin\n  have : (1 + d) % 4 = 0 := by { rw int.add_mod, norm_num [hd] },\n  rw ← int.dvd_iff_mod_eq_zero at this,\n\n  -- We show a very explicit equality by checking the inclusion of the span on both sides.\n  simp only [pow_two, ideal.span_mul_span', set.insert_mul, set.mul_insert, set.singleton_mul,\n    set.mul_singleton, set.smul_set_singleton, set.smul_set_insert, smul_eq_mul,\n    set.insert_union, set.singleton_union, set.image_insert, set.image_singleton,\n    set.insert_eq_of_mem, set.mem_insert_iff, eq_self_iff_true, true_or],\n  rw [show (⟨1, 1⟩ : quad_ring ℤ 0 d) * (⟨ 1, 1 ⟩) = ⟨1 + d, 2⟩, by quad_ring.calc_tac,\n      show (⟨1, 1⟩ : quad_ring ℤ 0 d) * 2 = ⟨2, 2⟩, by quad_ring.calc_tac,\n      show (2 : quad_ring ℤ 0 d) * 2 = 4, by norm_num],\n  refine ideal.span_eq_span_iff.mpr ⟨_, _⟩;\n    simp only [set.insert_subset, set.singleton_subset_iff, ideal.mem_span_singleton,\n      set_like.mem_coe],\n  refine ⟨_, _, _⟩; refine coe_dvd_mk.mpr ⟨_, _⟩,\n  { calc 2 ∣ 4 : by norm_num\n       ... ∣ 1 + d : this },\n  { refl },\n  { refl },\n  { refl },\n  { exact (show (2 : ℤ) ∣ 4, by norm_num) },\n  { exact (show (2 : ℤ) ∣ 0, by norm_num) },\n  { obtain ⟨k, hk⟩ := this,\n    simp only [ideal.mem_span_insert, ideal.mem_span_singleton, exists_prop],\n    refine ⟨-1, _, ⟨1, _, ⟨k, rfl⟩, rfl⟩, _⟩,\n    rw hk,\n    ext; simp },\nend\n\n/-- The ideal `⟨2⟩` in ℤ[√d] is a square if `d % 4 = 2`.  -/\nlemma exists_sqrt_2_mod_two (d : ℤ) (hd : d % 4 = 2) :\n  (ideal.span ({⟨0, 1⟩, 2} : set (quad_ring ℤ 0 d)))^2 = ideal.span {2} :=\nbegin\n  have : (2 + d) % 4 = 0 := by { rw int.add_mod, norm_num [hd] },\n  rw ← int.dvd_iff_mod_eq_zero at this,\n\n  -- We show a very explicit equality by checking the inclusion of the span on both sides.\n  simp only [pow_two, ideal.span_mul_span', set.insert_mul, set.mul_insert, set.singleton_mul,\n    set.mul_singleton, set.smul_set_singleton, set.smul_set_insert, smul_eq_mul,\n    set.insert_union, set.singleton_union, set.image_insert, set.image_singleton,\n    set.insert_eq_of_mem, set.mem_insert_iff, eq_self_iff_true, true_or],\n  rw [show (⟨0, 1⟩ : quad_ring ℤ 0 d) * (⟨ 0, 1 ⟩) = ⟨d, 0⟩, by quad_ring.calc_tac,\n      show (⟨0, 1⟩ : quad_ring ℤ 0 d) * 2 = ⟨0, 2⟩, by quad_ring.calc_tac,\n      show (2 : quad_ring ℤ 0 d) * 2 = 4, by norm_num],\n  refine ideal.span_eq_span_iff.mpr ⟨_, _⟩;\n    simp only [set.insert_subset, set.singleton_subset_iff, ideal.mem_span_singleton,\n      set_like.mem_coe],\n  refine ⟨_, _, _⟩; refine coe_dvd_mk.mpr ⟨_, _⟩,\n  { have := calc 2 ∣ 4 : by norm_num\n       ... ∣ 2 + d : this,\n    simpa },\n  { exact dvd_zero _, },\n  { exact dvd_zero _, },\n  { refl },\n  { exact (show (2 : ℤ) ∣ 4, by norm_num) },\n  { exact (show (2 : ℤ) ∣ 0, by norm_num) },\n  { obtain ⟨k, hk⟩ := this,\n    simp only [ideal.mem_span_insert, ideal.mem_span_singleton, exists_prop],\n    refine ⟨-1, _, ⟨0, _, ⟨k, rfl⟩, rfl⟩, _⟩,\n    rw_mod_cast ← hk,\n    ext; simp },\nend\n\nlemma norm_ne_two {d : ℤ} (hd2 : d < -2)\n  (x : quad_ring ℤ 0 d) : algebra.norm ℤ x ≠ 2 :=\nbegin\n  rw quad_ring.norm_eq,\n  simp only [zero_mul, add_zero],\n  by_cases hb2 : x.b2 = 0,\n  { simp only [hb2, pow_two, mul_zero, sub_zero],\n    have : ¬ is_square (2 : ℤ) :=\n      (int.prime_iff_nat_abs_prime.mpr $ by norm_num).irreducible.not_is_square,\n    rw [is_square, not_exists] at this,\n    exact ne.symm (this x.b1) },\n  { have hb2 : 0 < x.b2^2 := (sq_pos_iff _).mpr hb2,\n    refine ne_of_gt _,\n    calc 2 < -d : by linarith\n       ... ≤ - d * x.b2^2 : by nlinarith\n       ... ≤ x.b1^2 - d * x.b2^2 : by nlinarith }\nend\n\n/-- In `ℤ[√d]`, if `d < -2`, the square root of the ideal ⟨2⟩ is not principal. -/\nlemma not_principal_of_sq_eq_two {d : ℤ} (hd2 : d < -2)\n  (I : ideal (quad_ring ℤ 0 d)) (h : I^2 = ideal.span {2}) :\n  ¬ I.is_principal :=\nbegin\n  rintro ⟨x, rfl⟩,\n  rw [pow_two, ideal.submodule_span_eq, ideal.span_singleton_mul_span_singleton] at h,\n  obtain ⟨a, ha⟩ := ideal.span_singleton_le_span_singleton.mp h.le,\n  obtain ⟨b, hb⟩ := ideal.span_singleton_le_span_singleton.mp h.ge,\n  rw ha at hb,\n  apply_fun algebra.norm ℤ at ha hb,\n  have hy : algebra.norm ℤ (2 : quad_ring ℤ 0 d) = 2^2 := quad_ring.norm_coe 2,\n  simp only [map_mul, hy, mul_assoc] at ha hb,\n  have : algebra.norm ℤ a = 1,\n  { cases mul_right_eq_self₀.mp hb.symm with hb hb, swap,\n    { norm_num at hb },\n    cases int.is_unit_eq_one_or (is_unit_of_mul_eq_one _ _ hb) with hb hb,\n    { assumption },\n    { have := quad_ring.norm_nonneg _ a; linarith } },\n  rw [this, mul_one, ← pow_two, sq_eq_sq (quad_ring.norm_nonneg _ x)] at ha,\n  exact norm_ne_two hd2 x ha,\n  repeat { linarith }\nend\n\nlemma is_not_principal_mod_three {d : ℤ} (hd : d % 4 = 3) (hd2 : d < -2) :\n  ¬ submodule.is_principal (ideal.span ({⟨1, 1⟩, 2} : set (quad_ring ℤ 0 d))) :=\nnot_principal_of_sq_eq_two hd2 _ (exists_sqrt_2_mod_three d hd)\n\nlemma is_not_principal_mod_two {d : ℤ} (hd : d % 4 = 2) (hd2 : d < -2) :\n  ¬ submodule.is_principal (ideal.span ({⟨0, 1⟩, 2} : set (quad_ring ℤ 0 d))) :=\nnot_principal_of_sq_eq_two hd2 _ (exists_sqrt_2_mod_two d hd)\n\n -- `d` is a free variable so this can't be an instance\nlocal attribute [instance] quad_ring.fact_not_square'_of_eq_two_or_three_mod_four\n\n/-- `sqrt_2` is the square root of the ideal `⟨2⟩` of `ℤ[√d]`, when `d % 4 ∈ {2, 3}`. -/\ndef sqrt_2 (d : ℤ) [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] : (ideal (quad_ring ℤ 0 d))⁰ :=\nif d % 4 = 2 then\n  ⟨ideal.span ({⟨0, 1⟩, 2} : set (quad_ring ℤ 0 d)),\n    mem_non_zero_divisors_of_ne_zero (ideal.span_ne_bot_iff.mpr ⟨set.insert_nonempty _ _,\n      set.ne_singleton_of_ne (set.mem_insert_of_mem _ (set.mem_singleton 2)) two_ne_zero⟩)⟩\nelse\n  ⟨ideal.span ({⟨1, 1⟩, 2} : set (quad_ring ℤ 0 d)),\n    mem_non_zero_divisors_of_ne_zero (ideal.span_ne_bot_iff.mpr ⟨set.insert_nonempty _ _,\n      set.ne_singleton_of_ne (set.mem_insert_of_mem _ (set.mem_singleton 2)) two_ne_zero⟩)⟩\n\nlemma sqrt_2_pow_two (d : ℤ) [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] :\n  ((sqrt_2 d)^2 : ideal (quad_ring ℤ 0 d)) = ideal.span {2} :=\nbegin\n  rw sqrt_2,\n  split_ifs,\n  { simp [exists_sqrt_2_mod_two d h], },\n  { simp [exists_sqrt_2_mod_three d (hd.out.resolve_left h)], },\nend\n\n/-- `class_group.sqrt_2` is the square root of `⟨2⟩` in the ideal class group of `ℚ(√d')`,\nwhen `d % 4 ∈ {2, 3}`. -/\nprotected noncomputable def class_group.sqrt_2 (d : ℤ)\n  [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] [fact $ squarefree d] :\n  class_group (quad_ring ℤ 0 d) :=\nclass_group.mk0 (sqrt_2 d)\n\n-- TODO a version of this for d = -2 and -1 saying that the order is 1\nlemma order_of_sqrt2 {d : ℤ} [hd : fact $ d % 4 = 2 ∨ d % 4 = 3] [fact $ squarefree d]\n  (hd2 : d < -2) : order_of (class_group.sqrt_2 d) = 2 :=\nbegin\n  refine order_of_eq_prime _ _,\n  { rw [class_group.sqrt_2, sqrt_2],\n    split_ifs;\n    simp_rw ← map_pow,\n    { refine (class_group.mk0_eq_one_iff _).mpr _,\n      simp only [exists_sqrt_2_mod_two d h],\n      exact ⟨⟨2, rfl⟩⟩ },\n    { refine (class_group.mk0_eq_one_iff _).mpr _,\n      simp only [exists_sqrt_2_mod_three d (hd.out.resolve_left h)],\n      exact ⟨⟨2, rfl⟩⟩ }, },\n  { rw [ne.def, class_group.sqrt_2, sqrt_2],\n    split_ifs; erw [class_group.mk0_eq_one_iff],\n    exact is_not_principal_mod_two h hd2,\n    exact is_not_principal_mod_three (hd.out.resolve_left h) hd2, },\nend\n\ntheorem two_dvd_class_number (d : ℤ) (d' : ℚ) [hd : fact $ d % 4 = 2 ∨ d % 4 = 3]\n  [fact $ squarefree d]\n  (hd2 : d < -2) [hdd' : fact $ algebra_map ℤ ℚ d = d'] :\n  2 ∣ number_field.class_number (quad_ring ℚ 0 d') :=\nbegin\n  letI : fintype (class_group (quad_ring ℤ 0 d)) :=\n    class_group.fintype_of_admissible_of_finite ℚ (quad_ring ℚ 0 d')\n      absolute_value.abs_is_admissible,\n  rw [← order_of_sqrt2 hd2, number_field.class_number_eq (quad_ring ℤ 0 d) (quad_ring ℚ 0 d')],\n  exact order_of_dvd_card_univ,\nend\n\ninstance : fact $ ¬ is_square (-5 : ℤ) :=\n⟨(int.prime_iff_nat_abs_prime.mpr $ by norm_num).irreducible.not_is_square⟩\n\ntheorem class_group_nontrivial :\n  nontrivial (class_group (quad_ring ℤ 0 (-5))) :=\nbegin\n  haveI : fact (algebra_map ℤ ℚ (-5) = (-5)) := ⟨by simp⟩,\n  haveI : fact ((-5 : ℤ) % 4 = 2 ∨ (-5 : ℤ) % 4 = 3) := ⟨by norm_num⟩,\n  haveI : fact (squarefree (-5 : ℤ)) := ⟨by norm_num [int.squarefree_iff_squarefree_nat_abs]⟩,\n  haveI : is_dedekind_domain (quad_ring ℤ 0 (-5)) :=\n    is_integral_closure.is_dedekind_domain ℤ ℚ (quad_ring ℚ 0 (-5)) _,\n  letI : fintype (class_group (quad_ring ℤ 0 (-5))) :=\n    class_group.fintype_of_admissible_of_finite ℚ (quad_ring ℚ 0 (-5)) absolute_value.abs_is_admissible,\n  rw [← fintype.one_lt_card_iff_nontrivial,\n      ← number_field.class_number_eq (quad_ring ℤ 0 (-5)) (quad_ring ℚ 0 (-5))],\n  have := nat.le_of_dvd _ (two_dvd_class_number (-5) (-5) (by norm_num)),\n  { linarith },\n  unfold number_field.class_number,\n  rw fintype.card_pos_iff,\n  exact has_one.nonempty,\nend\n", "meta": {"author": "lean-forward", "repo": "class-group-and-mordell-equation", "sha": "baba2049f3bfe4d2cc184f8205997333e7c58638", "save_path": "github-repos/lean/lean-forward-class-group-and-mordell-equation", "path": "github-repos/lean/lean-forward-class-group-and-mordell-equation/class-group-and-mordell-equation-baba2049f3bfe4d2cc184f8205997333e7c58638/src/number_theory/class_number_computing.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3268730722296832}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport Std.Classes.Order\nimport Std.Control.ForInStep.Basic\n\n/-!\n# Red-black trees\n\nThis module implements a type `RBMap α β cmp` which is a functional data structure for\nstoring a key-value store in a binary search tree.\n\nIt is built on the simpler `RBSet α cmp` type, which stores a set of values of type `α`\nusing the function `cmp : α → α → Ordering` for determining the ordering relation.\nThe tree will never store two elements that compare `.eq` under the `cmp` function,\nbut the function does not have to satisfy `cmp x y = .eq → x = y`, and in the map case\n`α` is a key-value pair and the `cmp` function only compares the keys.\n-/\n\nnamespace Std\n\n/--\nIn a red-black tree, every node has a color which is either \"red\" or \"black\"\n(this particular choice of colors is conventional). A nil node is considered black.\n-/\ninductive RBColor where\n  /-- A red node is required to have black children. -/\n  | red\n  /-- Every path from the root to a leaf must pass through the same number of black nodes. -/\n  | black\n  deriving Repr\n\n/--\nA red-black tree. (This is an internal implementation detail of the `RBSet` type,\nwhich includes the invariants of the tree.) This is a binary search tree augmented with\na \"color\" field which is either red or black for each node and used to implement\nthe re-balancing operations.\nSee: [Red–black tree](https://en.wikipedia.org/wiki/Red%E2%80%93black_tree)\n-/\ninductive RBNode (α : Type u) where\n  /-- An empty tree. -/\n  | nil\n  /-- A node consists of a value `v`, a subtree `l` of smaller items,\n  and a subtree `r` of larger items. The color `c` is either `red` or `black`\n  and participates in the red-black balance invariant (see `Balanced`). -/\n  | node (c : RBColor) (l : RBNode α) (v : α) (r : RBNode α)\n  deriving Repr\n\nnamespace RBNode\nopen RBColor\n\ninstance : EmptyCollection (RBNode α) := ⟨nil⟩\n\n/-- The minimum element of a tree is the left-most value. -/\nprotected def min : RBNode α → Option α\n  | nil            => none\n  | node _ nil v _ => some v\n  | node _ l _ _   => l.min\n\n/-- The maximum element of a tree is the right-most value. -/\nprotected def max : RBNode α → Option α\n  | nil            => none\n  | node _ _ v nil => some v\n  | node _ _ _ r   => r.max\n\n/--\nFold a function in tree order along the nodes. `v₀` is used at `nil` nodes and\n`f` is used to combine results at branching nodes.\n-/\n@[specialize] def fold (v₀ : σ) (f : σ → α → σ → σ) : RBNode α → σ\n  | nil          => v₀\n  | node _ l v r => f (l.fold v₀ f) v (r.fold v₀ f)\n\n/-- Fold a function on the values from left to right (in increasing order). -/\n@[specialize] def foldl (f : σ → α → σ) : (init : σ) → RBNode α → σ\n  | b, nil          => b\n  | b, node _ l v r => foldl f (f (foldl f b l) v) r\n\n/-- Fold a function on the values from right to left (in decreasing order). -/\n@[specialize] def foldr (f : α → σ → σ) : RBNode α → (init : σ) → σ\n  | nil,          b => b\n  | node _ l v r, b => l.foldr f <| f v <| r.foldr f b\n\n/-- `O(n)`. Convert the tree to a list in ascending order. -/\ndef toList (t : RBNode α) : List α := t.foldr (·::·) []\n\n/-- Run monadic function `f` on each element of the tree (in increasing order). -/\n@[specialize] def forM [Monad m] (f : α → m PUnit) : RBNode α → m PUnit\n  | nil          => pure ⟨⟩\n  | node _ l v r => do forM f l; f v; forM f r\n\n/-- Fold a monadic function on the values from left to right (in increasing order). -/\n@[specialize] def foldlM [Monad m] (f : σ → α → m σ) : (init : σ) → RBNode α → m σ\n  | b, nil          => pure b\n  | b, node _ l v r => do foldlM f (← f (← foldlM f b l) v) r\n\n/-- Implementation of `for x in t` loops over a `RBNode` (in increasing order). -/\n@[inline] protected def forIn [Monad m]\n    (as : RBNode α) (init : σ) (f : α → σ → m (ForInStep σ)) : m σ := do\n  ForInStep.run <$> visit as init\nwhere\n  /-- Inner loop of `forIn`. -/\n  @[specialize] visit : RBNode α → σ → m (ForInStep σ)\n    | nil,          b => return ForInStep.yield b\n    | node _ l v r, b => ForInStep.bindM (visit l b) fun b => ForInStep.bindM (f v b) (visit r ·)\n\ninstance : ForIn m (RBNode α) α where\n  forIn := RBNode.forIn\n\n/--\nAn auxiliary data structure (an iterator) over an `RBNode` which lazily\npulls elements from the left.\n-/\nprotected inductive Stream (α : Type _)\n  | /-- The stream is empty. -/\n    nil\n  | /-- We are ready to deliver element `v` with right child `r`,\n    and where `tail` represents all the subtrees we have yet to destructure. -/\n    cons (v : α) (r : RBNode α) (tail : RBNode.Stream α)\n\n/-- `O(log n)`. Turn a node into a stream, by descending along the left spine. -/\ndef toStream : RBNode α → (_ : RBNode.Stream α := .nil) → RBNode.Stream α\n  | nil, acc => acc\n  | node _ l v r, acc => toStream l (.cons v r acc)\n\nnamespace Stream\n\n/-- `O(1)` amortized, `O(log n)` worst case: Get the next element from the stream. -/\ndef next? : RBNode.Stream α → Option (α × RBNode.Stream α)\n  | nil => none\n  | cons v r tail => some (v, toStream r tail)\n\n/-- Fold a function on the values from left to right (in increasing order). -/\n@[specialize] def foldl (f : σ → α → σ) : (init : σ) → RBNode.Stream α → σ\n  | b, nil           => b\n  | b, cons v r tail => foldl f (r.foldl f (f b v)) tail\n\n/-- Fold a function on the values from right to left (in decreasing order). -/\n@[specialize] def foldr (f : α → σ → σ) : RBNode.Stream α → (init : σ) → σ\n  | nil,           b => b\n  | cons v r tail, b => f v <| r.foldr f <| tail.foldr f b\n\n/-- `O(n)`. Convert the stream to a list in ascending order. -/\ndef toList (t : RBNode.Stream α) : List α := t.foldr (·::·) []\n\nend Stream\n\ninstance : ToStream (RBNode α) (RBNode.Stream α) := ⟨(·.toStream)⟩\ninstance : Stream (RBNode.Stream α) α := ⟨Stream.next?⟩\n\n/-- Returns `true` iff every element of the tree satisfies `p`. -/\n@[specialize] def all (p : α → Bool) : RBNode α → Bool\n  | nil          => true\n  | node _ l v r => p v && all p l && all p r\n\n/-- Returns `true` iff any element of the tree satisfies `p`. -/\n@[specialize] def any (p : α → Bool) : RBNode α → Bool\n  | nil          => false\n  | node _ l v r => p v || any p l || any p r\n\n/-- Asserts that `p` holds on every element of the tree. -/\ndef All (p : α → Prop) : RBNode α → Prop\n  | nil          => True\n  | node _ l v r => p v ∧ All p l ∧ All p r\n\ntheorem All.imp (H : ∀ {x : α}, p x → q x) : ∀ {t : RBNode α}, t.All p → t.All q\n  | nil => id\n  | node .. => fun ⟨h, hl, hr⟩ => ⟨H h, hl.imp H, hr.imp H⟩\n\n/-- Asserts that `p` holds on some element of the tree. -/\ndef Any (p : α → Prop) : RBNode α → Prop\n  | nil          => False\n  | node _ l v r => p v ∨ Any p l ∨ Any p r\n\n/-- True if `x` is an element of `t` \"exactly\", i.e. up to equality, not the `cmp` relation. -/\ndef EMem (x : α) (t : RBNode α) : Prop := t.Any (x = ·)\n\ninstance : Membership α (RBNode α) := ⟨EMem⟩\n\n/-- True if the specified `cut` matches at least one element of of `t`. -/\ndef MemP (cut : α → Ordering) (t : RBNode α) : Prop := t.Any (cut · = .eq)\n\n/-- True if `x` is equivalent to an element of `t`. -/\n@[reducible] def Mem (cmp : α → α → Ordering) (x : α) (t : RBNode α) : Prop := MemP (cmp x) t\n\n/--\nAsserts that `t₁` and `t₂` have the same number of elements in the same order,\nand `R` holds pairwise between them. The tree structure is ignored.\n-/\n@[specialize] def all₂ (R : α → β → Bool) (t₁ : RBNode α) (t₂ : RBNode β) : Bool :=\n  let result := StateT.run (s := t₂.toStream) <| t₁.forM fun a s => do\n    let (b, s) ← s.next?\n    bif R a b then pure (⟨⟩, s) else none\n  result matches some (_, .nil)\n\ninstance [BEq α] : BEq (RBNode α) where\n  beq a b := a.all₂ (· == ·) b\n\n/--\nWe say that `x < y` under the comparator `cmp` if `cmp x y = .lt`.\n\n* In order to avoid assuming the comparator is always lawful, we use a\n  local `∀ [TransCmp cmp]` binder in the relation so that the ordering\n  properties of the tree only need to hold if the comparator is lawful.\n* The `Nonempty` wrapper is a no-op because this is already a proposition,\n  but it prevents the `[TransCmp cmp]` binder from being introduced when we don't want it.\n-/\ndef cmpLT (cmp : α → α → Ordering) (x y : α) : Prop := Nonempty (∀ [TransCmp cmp], cmp x y = .lt)\n\n/-- We say that `x ≈ y` under the comparator `cmp` if `cmp x y = .eq`. See also `cmpLT`. -/\ndef cmpEq (cmp : α → α → Ordering) (x y : α) : Prop := Nonempty (∀ [TransCmp cmp], cmp x y = .eq)\n\n/-- The first half of Okasaki's `balance`, concerning red-red sequences in the left child. -/\n@[inline] def balance1 : RBNode α → α → RBNode α → RBNode α\n  | node red (node red a x b) y c, z, d\n  | node red a x (node red b y c), z, d => node red (node black a x b) y (node black c z d)\n  | a,                             x, b => node black a x b\n\n/-- The second half of Okasaki's `balance`, concerning red-red sequences in the right child. -/\n@[inline] def balance2 : RBNode α → α → RBNode α → RBNode α\n  | a, x, node red (node red b y c) z d\n  | a, x, node red b y (node red c z d) => node red (node black a x b) y (node black c z d)\n  | a, x, b                             => node black a x b\n\n/-- Returns `red` if the node is red, otherwise `black`. (Nil nodes are treated as `black`.) -/\n@[inline] def isRed : RBNode α → RBColor\n  | node c .. => c\n  | _         => black\n\n/--\nReturns `black` if the node is black, otherwise `red`.\n(Nil nodes are treated as `red`, which is not the usual convention but useful for deletion.)\n-/\n@[inline] def isBlack : RBNode α → RBColor\n  | node c .. => c\n  | _         => red\n\n/-- Change the color of the root to `black`. -/\ndef setBlack : RBNode α → RBNode α\n  | nil          => nil\n  | node _ l v r => node black l v r\n\nsection Insert\n\n/--\nThe core of the `insert` function. This adds an element `x` to a balanced red-black tree.\nImportantly, the result of calling `ins` is not a proper red-black tree,\nbecause it has a broken balance invariant.\n(See `Balanced.ins` for the balance invariant of `ins`.)\nThe `insert` function does the final fixup needed to restore the invariant.\n-/\n@[specialize] def ins (cmp : α → α → Ordering) (x : α) : RBNode α → RBNode α\n  | nil => node red nil x nil\n  | node red a y b =>\n    match cmp x y with\n    | Ordering.lt => node red (ins cmp x a) y b\n    | Ordering.gt => node red a y (ins cmp x b)\n    | Ordering.eq => node red a x b\n  | node black a y b =>\n    match cmp x y with\n    | Ordering.lt => balance1 (ins cmp x a) y b\n    | Ordering.gt => balance2 a y (ins cmp x b)\n    | Ordering.eq => node black a x b\n\n/--\n`insert cmp t v` inserts element `v` into the tree, using the provided comparator\n`cmp` to put it in the right place and automatically rebalancing the tree as necessary.\n-/\n@[specialize] def insert (cmp : α → α → Ordering) (t : RBNode α) (v : α) : RBNode α :=\n  match isRed t with\n  | red => (ins cmp v t).setBlack\n  | black => ins cmp v t\n\nend Insert\n\n/-- Recolor the root of the tree to `red` if possible. -/\ndef setRed : RBNode α → RBNode α\n  | node _ a v b => node red a v b\n  | nil          => nil\n\n/-- Rebalancing a tree which has shrunk on the left. -/\ndef balLeft (l : RBNode α) (v : α) (r : RBNode α) : RBNode α :=\n  match l with\n  | node red a x b                    => node red (node black a x b) v r\n  | l => match r with\n    | node black a y b                => balance2 l v (node red a y b)\n    | node red (node black a y b) z c => node red (node black l v a) y (balance2 b z (setRed c))\n    | r                               => node red l v r -- unreachable\n\n/-- Rebalancing a tree which has shrunk on the right. -/\ndef balRight (l : RBNode α) (v : α) (r : RBNode α) : RBNode α :=\n  match r with\n  | node red b y c                    => node red l v (node black b y c)\n  | r => match l with\n    | node black a x b                => balance1 (node red a x b) v r\n    | node red a x (node black b y c) => node red (balance1 (setRed a) x b) y (node black c v r)\n    | l                               => node red l v r -- unreachable\n\n/-- The number of nodes in the tree. -/\n@[simp] def size : RBNode α → Nat\n  | nil => 0\n  | node _ x _ y => x.size + y.size + 1\n\n/-- Concatenate two trees with the same black-height. -/\ndef append : RBNode α → RBNode α → RBNode α\n  | nil, x | x, nil => x\n  | node red a x b, node red c y d =>\n    match append b c with\n    | node red b' z c' => node red (node red a x b') z (node red c' y d)\n    | bc               => node red a x (node red bc y d)\n  | node black a x b, node black c y d =>\n    match append b c with\n    | node red b' z c' => node red (node black a x b') z (node black c' y d)\n    | bc               => balLeft a x (node black bc y d)\n  | a@(node black ..), node red b x c => node red (append a b) x c\n  | node red a x b, c@(node black ..) => node red a x (append b c)\ntermination_by _ x y => x.size + y.size\n\n/-! ## erase -/\n\n/--\nThe core of the `erase` function. The tree returned from this function has a broken invariant,\nwhich is restored in `erase`.\n-/\n@[specialize] def del (cut : α → Ordering) : RBNode α → RBNode α\n  | nil          => nil\n  | node _ a y b =>\n    match cut y with\n    | .lt => match a.isBlack with\n      | black => balLeft (del cut a) y b\n      | red => node red (del cut a) y b\n    | .gt => match b.isBlack with\n      | black => balRight a y (del cut b)\n      | red => node red a y (del cut b)\n    | .eq => append a b\n\n/--\nThe `erase cut t` function removes an element from the tree `t`.\nThe `cut` function is used to locate an element in the tree:\nit returns `.gt` if we go too high and `.lt` if we go too low;\nif it returns `.eq` we will remove the element.\n(The function `cmp k` for some key `k` is a valid cut function, but we can also use cuts that\nare not of this form as long as they are suitably monotonic.)\n-/\n@[specialize] def erase (cut : α → Ordering) (t : RBNode α) : RBNode α := (del cut t).setBlack\n\n/-- Finds an element in the tree satisfying the `cut` function. -/\n@[specialize] def find? (cut : α → Ordering) : RBNode α → Option α\n  | nil => none\n  | node _ a y b =>\n    match cut y with\n    | .lt => find? cut a\n    | .gt => find? cut b\n    | .eq => some y\n\n/-- `lowerBound? cut` retrieves the largest entry smaller than or equal to `cut`, if it exists. -/\n@[specialize] def lowerBound? (cut : α → Ordering) : RBNode α → Option α → Option α\n  | nil,          lb => lb\n  | node _ a y b, lb =>\n    match cut y with\n    | .lt => lowerBound? cut a lb\n    | .gt => lowerBound? cut b (some y)\n    | .eq => some y\n\n/-- Returns the root of the tree, if any. -/\ndef root? : RBNode α → Option α\n  | nil => none\n  | node _ _ v _ => some v\n\n/--\n`O(n)`. Map a function on every value in the tree.\nThis requires `IsMonotone` on the function in order to preserve the order invariant.\n-/\n@[specialize] def map (f : α → β) : RBNode α → RBNode β\n  | nil => nil\n  | node c l v r => node c (l.map f) (f v) (r.map f)\n\n/-- Converts the tree into an array in increasing sorted order. -/\ndef toArray (n : RBNode α) : Array α := n.foldl (init := #[]) (·.push ·)\n\n/--\nA `RBNode.Path α` is a \"cursor\" into an `RBNode` which represents the path\nfrom the root to a subtree. Note that the path goes from the target subtree\nup to the root, which is reversed from the normal way data is stored in the tree.\nSee [Zipper](https://en.wikipedia.org/wiki/Zipper_(data_structure)) for more information.\n-/\ninductive Path (α : Type u) where\n  /-- The root of the tree, which is the end of the path of parents. -/\n  | root\n  /-- A path that goes down the left subtree. -/\n  | left (c : RBColor) (parent : Path α) (v : α) (r : RBNode α)\n  /-- A path that goes down the right subtree. -/\n  | right (c : RBColor) (l : RBNode α) (v : α) (parent : Path α)\n\n/-- Fills the `Path` with a subtree. -/\ndef Path.fill : Path α → RBNode α → RBNode α\n  | .root, t => t\n  | .left c parent y b, a\n  | .right c a y parent, b => parent.fill (node c a y b)\n\n/--\nLike `find?`, but instead of just returning the element, it returns the entire subtree\nat the element and a path back to the root for reconstructing the tree.\n-/\n@[specialize] def zoom (cut : α → Ordering) : RBNode α → (e : Path α := .root) → RBNode α × Path α\n  | nil, path => (nil, path)\n  | n@(node c a y b), path =>\n    match cut y with\n    | .lt => zoom cut a (.left c path y b)\n    | .gt => zoom cut b (.right c a y path)\n    | .eq => (n, path)\n\n/--\nThis function does the second part of `RBNode.ins`,\nwhich unwinds the stack and rebuilds the tree.\n-/\ndef Path.ins : Path α → RBNode α → RBNode α\n  | .root, t => t.setBlack\n  | .left red parent y b, a\n  | .right red a y parent, b => parent.ins (node red a y b)\n  | .left black parent y b, a => parent.ins (balance1 a y b)\n  | .right black a y parent, b => parent.ins (balance2 a y b)\n\n/--\n`path.insertNew v` inserts element `v` into the tree, assuming that `path` is zoomed in\non a `nil` node such that inserting a new element at this position is valid.\n-/\n@[inline] def Path.insertNew (path : Path α) (v : α) : RBNode α :=\n  path.ins (node red nil v nil)\n\n/--\n`path.insert t v` inserts element `v` into the tree, assuming that `(t, path)` was the result of a\nprevious `zoom` operation (so either the root of `t` is equivalent to `v` or it is empty).\n-/\ndef Path.insert (path : Path α) (t : RBNode α) (v : α) : RBNode α :=\n  match t with\n  | nil => path.insertNew v\n  | node c a _ b => path.fill (node c a v b)\n\n/--\n`path.del t c` does the second part of `RBNode.del`, which unwinds the stack\nand rebuilds the tree. The `c` argument is the color of the node before the deletion\n(we used `t₀.isBlack` for this in `RBNode.del` but the original tree is no longer\navailable in this formulation).\n-/\ndef Path.del : Path α → RBNode α → RBColor → RBNode α\n  | .root, t, _ => t.setBlack\n  | .left c parent y b, a, red\n  | .right c a y parent, b, red => parent.del (node red a y b) c\n  | .left c parent y b, a, black => parent.del (balLeft a y b) c\n  | .right c a y parent, b, black => parent.del (balRight a y b) c\n\n/--\n`path.erase t v` removes the root element of `t` from the tree, assuming that `(t, path)` was\nthe result of a previous `zoom` operation.\n-/\ndef Path.erase (path : Path α) (t : RBNode α) : RBNode α :=\n  match t with\n  | nil => path.fill nil\n  | node c a _ b => path.del (a.append b) c\n\n/--\n`modify cut f t` uses `cut` to find an element,\nthen modifies the element using `f` and reinserts it into the tree.\n\nBecause the tree structure is not modified,\n`f` must not modify the ordering properties of the element.\n\nThe element in `t` is used linearly if `t` is unshared.\n-/\n@[specialize] def modify (cut : α → Ordering) (f : α → α) (t : RBNode α) : RBNode α :=\n  match zoom cut t with\n  | (nil, _) => t -- TODO: profile whether it would be better to use `path.fill nil` here\n  | (node c a x b, path) => path.fill (node c a (f x) b)\n\n/--\n`alter cut f t` simultaneously handles inserting, erasing and replacing an element\nusing a function `f : Option α → Option α`. It is passed the result of `t.find? cut`\nand can either return `none` to remove the element or `some a` to replace/insert\nthe element with `a` (which must have the same ordering properties as the original element).\n\nThe element is used linearly if `t` is unshared.\n-/\n@[specialize] def alter (cut : α → Ordering) (f : Option α → Option α) (t : RBNode α) : RBNode α :=\n  match zoom cut t with\n  | (nil, path) =>\n    match f none with\n    | none => t -- TODO: profile whether it would be better to use `path.fill nil` here\n    | some y => path.insertNew y\n  | (node c a x b, path) =>\n    match f (some x) with\n    | none => path.del (a.append b) c\n    | some y => path.fill (node c a y b)\n\n/--\nThe ordering invariant of a red-black tree, which is a binary search tree.\nThis says that every element of a left subtree is less than the root, and\nevery element in the right subtree is greater than the root, where the\nless than relation `x < y` is understood to mean `cmp x y = .lt`.\n\nBecause we do not assume that `cmp` is lawful when stating this property,\nwe write it in such a way that if `cmp` is not lawful then the condition holds trivially.\nThat way we can prove the ordering invariants without assuming `cmp` is lawful.\n-/\ndef Ordered (cmp : α → α → Ordering) : RBNode α → Prop\n  | nil => True\n  | node _ a x b => a.All (cmpLT cmp · x) ∧ b.All (cmpLT cmp x ·) ∧ a.Ordered cmp ∧ b.Ordered cmp\n\n/--\nThe red-black balance invariant. `Balanced t c n` says that the color of the root node is `c`,\nand the black-height (the number of black nodes on any path from the root) of the tree is `n`.\nAdditionally, every red node must have black children.\n-/\ninductive Balanced : RBNode α → RBColor → Nat → Prop where\n  /-- A nil node is balanced with black-height 0, and it is considered black. -/\n  | protected nil : Balanced nil black 0\n  /-- A red node is balanced with black-height `n`\n  if its children are both black with with black-height `n`. -/\n  | protected red : Balanced x black n → Balanced y black n → Balanced (node red x v y) red n\n  /-- A black node is balanced with black-height `n + 1`\n  if its children both have black-height `n`. -/\n  | protected black : Balanced x c₁ n → Balanced y c₂ n → Balanced (node black x v y) black (n + 1)\n\n/--\nThe well-formedness invariant for a red-black tree. The first constructor is the real invariant,\nand the others allow us to \"cheat\" in this file and define `insert` and `erase`,\nwhich have more complex proofs that are delayed to `Std.Data.RBMap.Lemmas`.\n-/\ninductive WF (cmp : α → α → Ordering) : RBNode α → Prop\n  /-- The actual well-formedness invariant: a red-black tree has the\n  ordering and balance invariants. -/\n  | mk : t.Ordered cmp → t.Balanced c n → WF cmp t\n  /-- Inserting into a well-formed tree yields another well-formed tree.\n  (See `Ordered.insert` and `Balanced.insert` for the actual proofs.) -/\n  | insert : WF cmp t → WF cmp (t.insert cmp a)\n  /-- Erasing from a well-formed tree yields another well-formed tree.\n  (See `Ordered.erase` and `Balanced.erase` for the actual proofs.) -/\n  | erase : WF cmp t → WF cmp (t.erase cut)\n\nend RBNode\n\nopen RBNode\n\n/--\nAn `RBSet` is a self-balancing binary search tree.\nThe `cmp` function is the comparator that will be used for performing searches;\nit should satisfy the requirements of `TransCmp` for it to have sensible behavior.\n-/\ndef RBSet (α : Type u) (cmp : α → α → Ordering) : Type u := {t : RBNode α // t.WF cmp}\n\n/-- `O(1)`. Construct a new empty tree. -/\n@[inline] def mkRBSet (α : Type u) (cmp : α → α → Ordering) : RBSet α cmp := ⟨.nil, .mk ⟨⟩ .nil⟩\n\nnamespace RBSet\n\n/-- `O(1)`. Construct a new empty tree. -/\n@[inline] def empty : RBSet α cmp := mkRBSet ..\n\ninstance (α : Type u) (cmp : α → α → Ordering) : EmptyCollection (RBSet α cmp) := ⟨empty⟩\n\ninstance (α : Type u) (cmp : α → α → Ordering) : Inhabited (RBSet α cmp) := ⟨∅⟩\n\n/-- `O(1)`. Construct a new tree with one element `v`. -/\n@[inline] def single (v : α) : RBSet α cmp :=\n  ⟨.node .red .nil v .nil, .mk ⟨⟨⟩, ⟨⟩, ⟨⟩, ⟨⟩⟩ (.red .nil .nil)⟩\n\n/-- `O(n)`. Fold a function on the values from left to right (in increasing order). -/\n@[inline] def foldl (f : σ → α → σ) (init : σ) (t : RBSet α cmp) : σ := t.1.foldl f init\n\n/-- `O(n)`. Fold a function on the values from right to left (in decreasing order). -/\n@[inline] def foldr (f : α → σ → σ) (init : σ) (t : RBSet α cmp) : σ := t.1.foldr f init\n\n/-- `O(n)`. Fold a monadic function on the values from left to right (in increasing order). -/\n@[inline] def foldlM [Monad m] (f : σ → α → m σ) (init : σ) (t : RBSet α cmp) : m σ :=\n  t.1.foldlM f init\n\n/-- `O(n)`. Run monadic function `f` on each element of the tree (in increasing order). -/\n@[inline] def forM [Monad m] (f : α → m PUnit) (t : RBSet α cmp) : m PUnit := t.1.forM f\n\ninstance : ForIn m (RBSet α cmp) α where\n  forIn t := t.1.forIn\n\ninstance : ToStream (RBSet α cmp) (RBNode.Stream α) := ⟨fun x => x.1.toStream .nil⟩\n\n/-- `O(1)`. Is the tree empty? -/\n@[inline] def isEmpty : RBSet α cmp → Bool\n  | ⟨nil, _⟩ => true\n  | _        => false\n\n/-- `O(n)`. Convert the tree to a list in ascending order. -/\n@[inline] def toList (t : RBSet α cmp) : List α := t.1.toList\n\n/-- `O(log n)`. Returns the entry `a` such that `a ≤ k` for all keys in the RBSet. -/\n@[inline] protected def min (t : RBSet α cmp) : Option α := t.1.min\n\n/-- `O(log n)`. Returns the entry `a` such that `a ≥ k` for all keys in the RBSet. -/\n@[inline] protected def max (t : RBSet α cmp) : Option α := t.1.max\n\ninstance [Repr α] : Repr (RBSet α cmp) where\n  reprPrec m prec := Repr.addAppParen (\"RBSet.ofList \" ++ repr m.toList) prec\n\n/-- `O(log n)`. Insert element `v` into the tree. -/\n@[inline] def insert (t : RBSet α cmp) (v : α) : RBSet α cmp := ⟨t.1.insert cmp v, t.2.insert⟩\n\n/--\n`O(log n)`. Remove an element from the tree using a cut function.\nThe `cut` function is used to locate an element in the tree:\nit returns `.gt` if we go too high and `.lt` if we go too low;\nif it returns `.eq` we will remove the element.\n(The function `cmp k` for some key `k` is a valid cut function, but we can also use cuts that\nare not of this form as long as they are suitably monotonic.)\n-/\n@[inline] def erase (t : RBSet α cmp) (cut : α → Ordering) : RBSet α cmp :=\n  ⟨t.1.erase cut, t.2.erase⟩\n\n/-- `O(log n)`. Find an element in the tree using a cut function. -/\n@[inline] def findP? (t : RBSet α cmp) (cut : α → Ordering) : Option α := t.1.find? cut\n\n/-- `O(log n)`. Returns an element in the tree equivalent to `x` if one exists. -/\n@[inline] def find? (t : RBSet α cmp) (x : α) : Option α := t.1.find? (cmp x)\n\n/-- `O(log n)`. Find an element in the tree, or return a default value `v₀`. -/\n@[inline] def findPD (t : RBSet α cmp) (cut : α → Ordering) (v₀ : α) : α := (t.findP? cut).getD v₀\n\n/--\n`O(log n)`. `lowerBoundP cut` retrieves the largest entry comparing `lt` or `eq` under `cut`,\nif it exists.\n-/\n@[inline] def lowerBoundP? (t : RBSet α cmp) (cut : α → Ordering) : Option α :=\n  t.1.lowerBound? cut none\n\n/--\n`O(log n)`. `lowerBound? k` retrieves the largest entry smaller than or equal to `k`,\nif it exists.\n-/\n@[inline] def lowerBound? (t : RBSet α cmp) (k : α) : Option α := t.1.lowerBound? (cmp k) none\n\n/-- `O(log n)`. Returns true if the given cut returns `eq` for something in the RBSet. -/\n@[inline] def containsP (t : RBSet α cmp) (cut : α → Ordering) : Bool := (t.findP? cut).isSome\n\n/-- `O(log n)`. Returns true if the given key `a` is in the RBSet. -/\n@[inline] def contains (t : RBSet α cmp) (a : α) : Bool := (t.find? a).isSome\n\n/-- `O(n log n)`. Build a tree from an unsorted list by inserting them one at a time. -/\n@[inline] def ofList (l : List α) (cmp : α → α → Ordering) : RBSet α cmp :=\n  l.foldl (fun r p => r.insert p) (mkRBSet α cmp)\n\n/-- `O(n log n)`. Build a tree from an unsorted array by inserting them one at a time. -/\n@[inline] def ofArray (l : Array α) (cmp : α → α → Ordering) : RBSet α cmp :=\n  l.foldl (fun r p => r.insert p) (mkRBSet α cmp)\n\n/-- `O(n)`. Returns true if the given predicate is true for all items in the RBSet. -/\n@[inline] def all (t : RBSet α cmp) (p : α → Bool) : Bool := t.1.all p\n\n/-- `O(n)`. Returns true if the given predicate is true for any item in the RBSet. -/\n@[inline] def any (t : RBSet α cmp) (p : α → Bool) : Bool := t.1.any p\n\n/--\nAsserts that `t₁` and `t₂` have the same number of elements in the same order,\nand `R` holds pairwise between them. The tree structure is ignored.\n-/\n@[inline] def all₂ (R : α → β → Bool) (t₁ : RBSet α cmpα) (t₂ : RBSet β cmpβ) : Bool :=\n  t₁.1.all₂ R t₂.1\n\n/-- True if `x` is an element of `t` \"exactly\", i.e. up to equality, not the `cmp` relation. -/\ndef EMem (x : α) (t : RBSet α cmp) : Prop := t.1.EMem x\n\n/-- True if the specified `cut` matches at least one element of of `t`. -/\ndef MemP (cut : α → Ordering) (t : RBSet α cmp) : Prop := t.1.MemP cut\n\n/-- True if `x` is equivalent to an element of `t`. -/\ndef Mem (x : α) (t : RBSet α cmp) : Prop := MemP (cmp x) t\n\ninstance : Membership α (RBSet α cmp) := ⟨Mem⟩\n\n/--\nReturns true if `t₁` and `t₂` are equal as sets (assuming `cmp` and `==` are compatible),\nignoring the internal tree structure.\n-/\ninstance [BEq α] : BEq (RBSet α cmp) where\n  beq a b := a.all₂ (· == ·) b\n\n/-- `O(n)`. The number of items in the RBSet. -/\ndef size (m : RBSet α cmp) : Nat := m.1.size\n\n/-- `O(log n)`. Returns the minimum element of the tree, or panics if the tree is empty. -/\n@[inline] def min! [Inhabited α] (t : RBSet α cmp) : α := t.min.getD (panic! \"tree is empty\")\n\n/-- `O(log n)`. Returns the maximum element of the tree, or panics if the tree is empty. -/\n@[inline] def max! [Inhabited α] (t : RBSet α cmp) : α := t.max.getD (panic! \"tree is empty\")\n\n/--\n`O(log n)`. Attempts to find the value with key `k : α` in `t` and panics if there is no such key.\n-/\n@[inline] def findP! [Inhabited α] (t : RBSet α cmp) (cut : α → Ordering) : α :=\n  (t.findP? cut).getD (panic! \"key is not in the tree\")\n\n/--\n`O(log n)`. Attempts to find the value with key `k : α` in `t` and panics if there is no such key.\n-/\n@[inline] def find! [Inhabited α] (t : RBSet α cmp) (k : α) : α :=\n  (t.find? k).getD (panic! \"key is not in the tree\")\n\n/-- The predicate asserting that the result of `modifyP` is safe to construct. -/\nclass ModifyWF (t : RBSet α cmp) (cut : α → Ordering) (f : α → α) : Prop where\n  /-- The resulting tree is well formed. -/\n  wf : (t.1.modify cut f).WF cmp\n\n/--\n`O(log n)`. In-place replace an element found by `cut`.\nThis takes the element out of the tree while `f` runs,\nso it uses the element linearly if `t` is unshared.\n\nThe `ModifyWF` assumption is required because `f` may change\nthe ordering properties of the element, which would break the invariants.\n-/\ndef modifyP (t : RBSet α cmp) (cut : α → Ordering) (f : α → α)\n    [wf : ModifyWF t cut f] : RBSet α cmp := ⟨t.1.modify cut f, wf.wf⟩\n\n/-- The predicate asserting that the result of `alterP` is safe to construct. -/\nclass AlterWF (t : RBSet α cmp) (cut : α → Ordering) (f : Option α → Option α) : Prop where\n  /-- The resulting tree is well formed. -/\n  wf : (t.1.alter cut f).WF cmp\n\n/--\n`O(log n)`. `alterP cut f t` simultaneously handles inserting, erasing and replacing an element\nusing a function `f : Option α → Option α`. It is passed the result of `t.findP? cut`\nand can either return `none` to remove the element or `some a` to replace/insert\nthe element with `a` (which must have the same ordering properties as the original element).\n\nThe element is used linearly if `t` is unshared.\n\nThe `AlterWF` assumption is required because `f` may change\nthe ordering properties of the element, which would break the invariants.\n-/\ndef alterP (t : RBSet α cmp) (cut : α → Ordering) (f : Option α → Option α)\n    [wf : AlterWF t cut f] : RBSet α cmp := ⟨t.1.alter cut f, wf.wf⟩\n\n/--\n`O(n₂ * log (n₁ + n₂))`. Merges the maps `t₁` and `t₂`.\nIf equal keys exist in both, the key from `t₂` is preferred.\n-/\ndef union (t₁ t₂ : RBSet α cmp) : RBSet α cmp :=\n  t₂.foldl .insert t₁\n\n/--\n`O(n₂ * log (n₁ + n₂))`. Merges the maps `t₁` and `t₂`. If equal keys exist in both,\nthen use `mergeFn a₁ a₂` to produce the new merged value.\n-/\ndef mergeWith (mergeFn : α → α → α) (t₁ t₂ : RBSet α cmp) : RBSet α cmp :=\n  t₂.foldl (init := t₁) fun t₁ a₂ =>\n    t₁.insert <| match t₁.find? a₂ with | some a₁ => mergeFn a₁ a₂ | none => a₂\n\n/--\n`O(n₁ * log (n₁ + n₂))`. Intersects the maps `t₁` and `t₂`\nusing `mergeFn a b` to produce the new value.\n-/\ndef intersectWith (cmp : α → β → Ordering) (mergeFn : α → β → γ)\n    (t₁ : RBSet α cmpα) (t₂ : RBSet β cmpβ) : RBSet γ cmpγ :=\n  t₁.foldl (init := ∅) fun acc a =>\n    match t₂.findP? (cmp a) with\n    | some b => acc.insert <| mergeFn a b\n    | none => acc\n\n/-- `O(n * log n)`. Constructs the set of all elements satisfying `p`. -/\ndef filter (t : RBSet α cmp) (p : α → Bool) : RBSet α cmp :=\n  t.foldl (init := ∅) fun acc a => bif p a then acc.insert a else acc\n\n/--\n`O(n₁ * (log n₁ + log n₂))`. Constructs the set of all elements of `t₁` that are not in `t₂`.\n-/\ndef sdiff (t₁ t₂ : RBSet α cmp) : RBSet α cmp := t₁.filter (!t₂.contains ·)\n\nend RBSet\n\n/- TODO(Leo): define dRBMap -/\n\n/--\nAn `RBMap` is a self-balancing binary search tree, used to store a key-value map.\nThe `cmp` function is the comparator that will be used for performing searches;\nit should satisfy the requirements of `TransCmp` for it to have sensible behavior.\n-/\ndef RBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : Type (max u v) :=\n  RBSet (α × β) (byKey Prod.fst cmp)\n\n/-- `O(1)`. Construct a new empty map. -/\n@[inline] def mkRBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : RBMap α β cmp :=\n  mkRBSet ..\n\n/-- `O(1)`. Construct a new empty map. -/\n@[inline] def RBMap.empty {α : Type u} {β : Type v} {cmp : α → α → Ordering} : RBMap α β cmp :=\n  mkRBMap ..\n\ninstance (α : Type u) (β : Type v) (cmp : α → α → Ordering) : EmptyCollection (RBMap α β cmp) :=\n  ⟨RBMap.empty⟩\n\ninstance (α : Type u) (β : Type v) (cmp : α → α → Ordering) : Inhabited (RBMap α β cmp) := ⟨∅⟩\n\n/-- `O(1)`. Construct a new tree with one key-value pair `k, v`. -/\n@[inline] def RBMap.single (k : α) (v : β) : RBMap α β cmp := RBSet.single (k, v)\n\nnamespace RBMap\nvariable {α : Type u} {β : Type v} {σ : Type w} {cmp : α → α → Ordering}\n\n/-- `O(n)`. Fold a function on the values from left to right (in increasing order). -/\n@[inline] def foldl (f : σ → α → β → σ) : (init : σ) → RBMap α β cmp → σ\n  | b, ⟨t, _⟩ => t.foldl (fun s (a, b) => f s a b) b\n\n/-- `O(n)`. Fold a function on the values from right to left (in decreasing order). -/\n@[inline] def foldr (f : α → β → σ → σ) : (init : σ) → RBMap α β cmp → σ\n  | b, ⟨t, _⟩ => t.foldr (fun (a, b) s => f a b s) b\n\n/-- `O(n)`. Fold a monadic function on the values from left to right (in increasing order). -/\n@[inline] def foldlM [Monad m] (f : σ → α → β → m σ) : (init : σ) → RBMap α β cmp → m σ\n  | b, ⟨t, _⟩ => t.foldlM (fun s (a, b) => f s a b) b\n\n/-- `O(n)`. Run monadic function `f` on each element of the tree (in increasing order). -/\n@[inline] def forM [Monad m] (f : α → β → m PUnit) (t : RBMap α β cmp) : m PUnit :=\n  t.foldlM (fun _ k v => f k v) ⟨⟩\n\ninstance : ForIn m (RBMap α β cmp) (α × β) := inferInstanceAs (ForIn _ (RBSet ..) _)\n\ninstance : ToStream (RBMap α β cmp) (RBNode.Stream (α × β)) :=\n  inferInstanceAs (ToStream (RBSet ..) _)\n\n/-- `O(n)`. Constructs the array of keys of the map. -/\n@[inline] def keysArray (t : RBMap α β cmp) : Array α :=\n  t.1.foldl (init := #[]) (·.push ·.1)\n\n/-- `O(n)`. Constructs the list of keys of the map. -/\n@[inline] def keysList (t : RBMap α β cmp) : List α :=\n  t.1.foldr (init := []) (·.1 :: ·)\n\n/--\nAn \"iterator\" over the keys of the map. This is a trivial wrapper over the underlying map,\nbut it comes with a small API to use it in a `for` loop or convert it to an array or list.\n-/\ndef Keys (α β cmp) := RBMap α β cmp\n\n/--\nThe keys of the map. This is an `O(1)` wrapper operation, which\ncan be used in `for` loops or converted to an array or list.\n-/\n@[inline] def keys (t : RBMap α β cmp) : Keys α β cmp := t\n\n@[inline, inherit_doc keysArray] def Keys.toArray := @keysArray\n\n@[inline, inherit_doc keysList] def Keys.toList := @keysList\n\ninstance : CoeHead (Keys α β cmp) (Array α) := ⟨keysArray⟩\n\ninstance : CoeHead (Keys α β cmp) (List α) := ⟨keysList⟩\n\ninstance : ForIn m (Keys α β cmp) α where\n  forIn t init f := t.val.forIn init (f ·.1)\n\ninstance : ForM m (Keys α β cmp) α where\n  forM t f := t.val.forM (f ·.1)\n\n/-- The result of `toStream` on a `Keys`. -/\ndef Keys.Stream (α β) := RBNode.Stream (α × β)\n\n/-- A stream over the iterator. -/\ndef Keys.toStream (t : Keys α β cmp) : Keys.Stream α β := t.1.toStream\n\n/-- `O(1)` amortized, `O(log n)` worst case: Get the next element from the stream. -/\ndef Keys.Stream.next? (t : Stream α β) : Option (α × Stream α β) :=\n  match inline (RBNode.Stream.next? t) with\n  | none => none\n  | some ((a, _), t) => some (a, t)\n\ninstance : ToStream (Keys α β cmp) (Keys.Stream α β) := ⟨Keys.toStream⟩\ninstance : Stream (Keys.Stream α β) α := ⟨Keys.Stream.next?⟩\n\n/-- `O(n)`. Constructs the array of values of the map. -/\n@[inline] def valuesArray (t : RBMap α β cmp) : Array β :=\n  t.1.foldl (init := #[]) (·.push ·.2)\n\n/-- `O(n)`. Constructs the list of values of the map. -/\n@[inline] def valuesList (t : RBMap α β cmp) : List β :=\n  t.1.foldr (init := []) (·.2 :: ·)\n\n/--\nAn \"iterator\" over the values of the map. This is a trivial wrapper over the underlying map,\nbut it comes with a small API to use it in a `for` loop or convert it to an array or list.\n-/\ndef Values (α β cmp) := RBMap α β cmp\n\n/--\nThe \"keys\" of the map. This is an `O(1)` wrapper operation, which\ncan be used in `for` loops or converted to an array or list.\n-/\n@[inline] def values (t : RBMap α β cmp) : Values α β cmp := t\n\n@[inline, inherit_doc valuesArray] def Values.toArray := @valuesArray\n\n@[inline, inherit_doc valuesList] def Values.toList := @valuesList\n\ninstance : CoeHead (Values α β cmp) (Array β) := ⟨valuesArray⟩\n\ninstance : CoeHead (Values α β cmp) (List β) := ⟨valuesList⟩\n\ninstance : ForIn m (Values α β cmp) β where\n  forIn t init f := t.val.forIn init (f ·.2)\n\ninstance : ForM m (Values α β cmp) β where\n  forM t f := t.val.forM (f ·.2)\n\n/-- The result of `toStream` on a `Values`. -/\ndef Values.Stream (α β) := RBNode.Stream (α × β)\n\n/-- A stream over the iterator. -/\ndef Values.toStream (t : Values α β cmp) : Values.Stream α β := t.1.toStream\n\n/-- `O(1)` amortized, `O(log n)` worst case: Get the next element from the stream. -/\ndef Values.Stream.next? (t : Stream α β) : Option (β × Stream α β) :=\n  match inline (RBNode.Stream.next? t) with\n  | none => none\n  | some ((_, b), t) => some (b, t)\n\ninstance : ToStream (Values α β cmp) (Values.Stream α β) := ⟨Values.toStream⟩\ninstance : Stream (Values.Stream α β) β := ⟨Values.Stream.next?⟩\n\n/-- `O(1)`. Is the tree empty? -/\n@[inline] def isEmpty : RBMap α β cmp → Bool := RBSet.isEmpty\n\n/-- `O(n)`. Convert the tree to a list in ascending order. -/\n@[inline] def toList : RBMap α β cmp → List (α × β) := RBSet.toList\n\n/-- `O(log n)`. Returns the key-value pair `(a, b)` such that `a ≤ k` for all keys in the RBMap. -/\n@[inline] protected def min : RBMap α β cmp → Option (α × β) := RBSet.min\n\n/-- `O(log n)`. Returns the key-value pair `(a, b)` such that `a ≥ k` for all keys in the RBMap. -/\n@[inline] protected def max : RBMap α β cmp → Option (α × β) := RBSet.max\n\ninstance [Repr α] [Repr β] : Repr (RBMap α β cmp) where\n  reprPrec m prec := Repr.addAppParen (\"RBMap.ofList \" ++ repr m.toList) prec\n\n/-- `O(log n)`. Insert key-value pair `(k, v)` into the tree. -/\n@[inline] def insert (t : RBMap α β cmp) (k : α) (v : β) : RBMap α β cmp := RBSet.insert t (k, v)\n\n/-- `O(log n)`. Remove an element `k` from the map. -/\n@[inline] def erase (t : RBMap α β cmp) (k : α) : RBMap α β cmp := RBSet.erase t (cmp k ·.1)\n\n/-- `O(n log n)`. Build a tree from an unsorted list by inserting them one at a time. -/\n@[inline] def ofList (l : List (α × β)) (cmp : α → α → Ordering) : RBMap α β cmp :=\n  RBSet.ofList l _\n\n/-- `O(n log n)`. Build a tree from an unsorted array by inserting them one at a time. -/\n@[inline] def ofArray (l : Array (α × β)) (cmp : α → α → Ordering) : RBMap α β cmp :=\n  RBSet.ofArray l _\n\n/-- `O(log n)`. Find an entry in the tree with key equal to `k`. -/\n@[inline] def findEntry? (t : RBMap α β cmp) (k : α) : Option (α × β) := t.findP? (cmp k ·.1)\n\n/-- `O(log n)`. Find the value corresponding to key `k`. -/\n@[inline] def find? (t : RBMap α β cmp) (k : α) : Option β := t.findEntry? k |>.map (·.2)\n\n/-- `O(log n)`. Find the value corresponding to key `k`, or return `v₀` if it is not in the map. -/\n@[inline] def findD (t : RBMap α β cmp) (k : α) (v₀ : β) : β := (t.find? k).getD v₀\n\n/--\n`O(log n)`. `lowerBound? k` retrieves the key-value pair of the largest key\nsmaller than or equal to `k`, if it exists.\n-/\n@[inline] def lowerBound? (t : RBMap α β cmp) (k : α) : Option (α × β) :=\n   RBSet.lowerBoundP? t (cmp k ·.1)\n\n/-- `O(log n)`. Returns true if the given key `a` is in the RBMap. -/\n@[inline] def contains (t : RBMap α β cmp) (a : α) : Bool := (t.findEntry? a).isSome\n\n/-- `O(n)`. Returns true if the given predicate is true for all items in the RBMap. -/\n@[inline] def all (t : RBMap α β cmp) (p : α → β → Bool) : Bool := RBSet.all t fun (a, b) => p a b\n\n/-- `O(n)`. Returns true if the given predicate is true for any item in the RBMap. -/\n@[inline] def any (t : RBMap α β cmp) (p : α → β → Bool) : Bool := RBSet.any t fun (a, b) => p a b\n\n/--\nAsserts that `t₁` and `t₂` have the same number of elements in the same order,\nand `R` holds pairwise between them. The tree structure is ignored.\n-/\n@[inline] def all₂ (R : α × β → γ × δ → Bool) (t₁ : RBMap α β cmpα) (t₂ : RBMap γ δ cmpγ) : Bool :=\n  RBSet.all₂ R t₁ t₂\n\n/-- Asserts that `t₁` and `t₂` have the same set of keys (up to equality). -/\n@[inline] def eqKeys (t₁ : RBMap α β cmp) (t₂ : RBMap α γ cmp) : Bool :=\n  t₁.all₂ (cmp ·.1 ·.1 = .eq) t₂\n\n/--\nReturns true if `t₁` and `t₂` have the same keys and values\n(assuming `cmp` and `==` are compatible), ignoring the internal tree structure.\n-/\ninstance [BEq α] [BEq β] : BEq (RBMap α β cmp) := inferInstanceAs (BEq (RBSet ..))\n\n/-- `O(n)`. The number of items in the RBMap. -/\ndef size : RBMap α β cmp → Nat := RBSet.size\n\n/-- `O(log n)`. Returns the minimum element of the map, or panics if the map is empty. -/\n@[inline] def min! [Inhabited α] [Inhabited β] : RBMap α β cmp → α × β := RBSet.min!\n\n/-- `O(log n)`. Returns the maximum element of the map, or panics if the map is empty. -/\n@[inline] def max! [Inhabited α] [Inhabited β] : RBMap α β cmp → α × β := RBSet.max!\n\n/-- Attempts to find the value with key `k : α` in `t` and panics if there is no such key. -/\n@[inline] def find! [Inhabited β] (t : RBMap α β cmp) (k : α) : β :=\n  (t.find? k).getD (panic! \"key is not in the map\")\n\n/--\n`O(n₂ * log (n₁ + n₂))`. Merges the maps `t₁` and `t₂`, if a key `a : α` exists in both,\nthen use `mergeFn a b₁ b₂` to produce the new merged value.\n-/\ndef mergeWith (mergeFn : α → β → β → β) (t₁ t₂ : RBMap α β cmp) : RBMap α β cmp :=\n  RBSet.mergeWith (fun (_, b₁) (a, b₂) => (a, mergeFn a b₁ b₂)) t₁ t₂\n\n/--\n`O(n₁ * log (n₁ + n₂))`. Intersects the maps `t₁` and `t₂`\nusing `mergeFn a b` to produce the new value.\n-/\ndef intersectWith (mergeFn : α → β → γ → δ)\n    (t₁ : RBMap α β cmp) (t₂ : RBMap α γ cmp) : RBMap α δ cmp :=\n  RBSet.intersectWith (cmp ·.1 ·.1) (fun (a, b₁) (_, b₂) => (a, mergeFn a b₁ b₂)) t₁ t₂\n\n/-- `O(n * log n)`. Constructs the set of all elements satisfying `p`. -/\ndef filter (t : RBMap α β cmp) (p : α → β → Bool) : RBMap α β cmp :=\n  RBSet.filter t fun (a, b) => p a b\n\n/--\n`O(n₁ * (log n₁ + log n₂))`. Constructs the set of all elements of `t₁` that are not in `t₂`.\n-/\ndef sdiff (t₁ t₂ : RBMap α β cmp) : RBMap α β cmp := t₁.filter fun a _ => !t₂.contains a\n\nend RBMap\nend Std\nopen Std\n\n@[inherit_doc RBMap.ofList]\nabbrev List.toRBMap (l : List (α × β)) (cmp : α → α → Ordering) : RBMap α β cmp :=\n  RBMap.ofList l cmp\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/RBMap/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.32679493745252003}}
{"text": "/-\nCopyright (c) 2021 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n\n! This file was ported from Lean 3 source module category_theory.limits.constructions.epi_mono\n! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Shapes.Pullbacks\nimport Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks\n\n/-!\n# Relating monomorphisms and epimorphisms to limits and colimits\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIf `F` preserves (resp. reflects) pullbacks, then it preserves (resp. reflects) monomorphisms.\n\nWe also provide the dual version for epimorphisms.\n\n-/\n\n\nuniverse v₁ v₂ u₁ u₂\n\nnamespace CategoryTheory\n\nopen Category Limits\n\nvariable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]\n\nvariable (F : C ⥤ D)\n\n/- warning: category_theory.preserves_mono_of_preserves_limit -> CategoryTheory.preserves_mono_of_preservesLimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)\nbut is expected to have type\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f], CategoryTheory.Mono.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f)\nCase conversion may be inaccurate. Consider using '#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimitₓ'. -/\n/-- If `F` preserves pullbacks, then it preserves monomorphisms. -/\ntheorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]\n    [Mono f] : Mono (F.map f) :=\n  by\n  have := is_limit_pullback_cone_map_of_is_limit F _ (pullback_cone.is_limit_mk_id_id f)\n  simp_rw [F.map_id] at this\n  apply pullback_cone.mono_of_is_limit_mk_id_id _ this\n#align category_theory.preserves_mono_of_preserves_limit CategoryTheory.preserves_mono_of_preservesLimit\n\n#print CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape /-\ninstance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape\n    [PreservesLimitsOfShape WalkingCospan F] : F.PreservesMonomorphisms\n    where preserves X Y f hf := preserves_mono_of_preserves_limit F f\n#align category_theory.preserves_monomorphisms_of_preserves_limits_of_shape CategoryTheory.preservesMonomorphisms_of_preservesLimitsOfShape\n-/\n\n/- warning: category_theory.reflects_mono_of_reflects_limit -> CategoryTheory.reflects_mono_of_reflectsLimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f\nbut is expected to have type\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.ReflectsLimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingCospan (CategoryTheory.Limits.WidePullbackShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.cospan.{u1, u3} C _inst_1 X X Y f f) F] [_inst_4 : CategoryTheory.Mono.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f)], CategoryTheory.Mono.{u1, u3} C _inst_1 X Y f\nCase conversion may be inaccurate. Consider using '#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimitₓ'. -/\n/-- If `F` reflects pullbacks, then it reflects monomorphisms. -/\ntheorem reflects_mono_of_reflectsLimit {X Y : C} (f : X ⟶ Y) [ReflectsLimit (cospan f f) F]\n    [Mono (F.map f)] : Mono f :=\n  by\n  have := pullback_cone.is_limit_mk_id_id (F.map f)\n  simp_rw [← F.map_id] at this\n  apply pullback_cone.mono_of_is_limit_mk_id_id _ (is_limit_of_is_limit_pullback_cone_map F _ this)\n#align category_theory.reflects_mono_of_reflects_limit CategoryTheory.reflects_mono_of_reflectsLimit\n\n#print CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape /-\ninstance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape\n    [ReflectsLimitsOfShape WalkingCospan F] : F.ReflectsMonomorphisms\n    where reflects X Y f hf := reflects_mono_of_reflects_limit F f\n#align category_theory.reflects_monomorphisms_of_reflects_limits_of_shape CategoryTheory.reflectsMonomorphisms_of_reflectsLimitsOfShape\n-/\n\n/- warning: category_theory.preserves_epi_of_preserves_colimit -> CategoryTheory.preserves_epi_of_preservesColimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f], CategoryTheory.Epi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)\nbut is expected to have type\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f], CategoryTheory.Epi.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f)\nCase conversion may be inaccurate. Consider using '#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimitₓ'. -/\n/-- If `F` preserves pushouts, then it preserves epimorphisms. -/\ntheorem preserves_epi_of_preservesColimit {X Y : C} (f : X ⟶ Y) [PreservesColimit (span f f) F]\n    [Epi f] : Epi (F.map f) :=\n  by\n  have := is_colimit_pushout_cocone_map_of_is_colimit F _ (pushout_cocone.is_colimit_mk_id_id f)\n  simp_rw [F.map_id] at this\n  apply pushout_cocone.epi_of_is_colimit_mk_id_id _ this\n#align category_theory.preserves_epi_of_preserves_colimit CategoryTheory.preserves_epi_of_preservesColimit\n\n#print CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape /-\ninstance (priority := 100) preservesEpimorphisms_of_preservesColimitsOfShape\n    [PreservesColimitsOfShape WalkingSpan F] : F.PreservesEpimorphisms\n    where preserves X Y f hf := preserves_epi_of_preserves_colimit F f\n#align category_theory.preserves_epimorphisms_of_preserves_colimits_of_shape CategoryTheory.preservesEpimorphisms_of_preservesColimitsOfShape\n-/\n\n/- warning: category_theory.reflects_epi_of_reflects_colimit -> CategoryTheory.reflects_epi_of_reflectsColimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f)], CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f\nbut is expected to have type\n  forall {C : Type.{u3}} {D : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] [_inst_2 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2) {X : C} {Y : C} (f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y) [_inst_3 : CategoryTheory.Limits.ReflectsColimit.{0, 0, u1, u2, u3, u4} C _inst_1 D _inst_2 CategoryTheory.Limits.WalkingSpan (CategoryTheory.Limits.WidePushoutShape.category.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.Limits.span.{u1, u3} C _inst_1 X Y Y f f) F] [_inst_4 : CategoryTheory.Epi.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f)], CategoryTheory.Epi.{u1, u3} C _inst_1 X Y f\nCase conversion may be inaccurate. Consider using '#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimitₓ'. -/\n/-- If `F` reflects pushouts, then it reflects epimorphisms. -/\ntheorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]\n    [Epi (F.map f)] : Epi f :=\n  by\n  have := pushout_cocone.is_colimit_mk_id_id (F.map f)\n  simp_rw [← F.map_id] at this\n  apply\n    pushout_cocone.epi_of_is_colimit_mk_id_id _\n      (is_colimit_of_is_colimit_pushout_cocone_map F _ this)\n#align category_theory.reflects_epi_of_reflects_colimit CategoryTheory.reflects_epi_of_reflectsColimit\n\n#print CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape /-\ninstance (priority := 100) reflectsEpimorphisms_of_reflectsColimitsOfShape\n    [ReflectsColimitsOfShape WalkingSpan F] : F.ReflectsEpimorphisms\n    where reflects X Y f hf := reflects_epi_of_reflects_colimit F f\n#align category_theory.reflects_epimorphisms_of_reflects_colimits_of_shape CategoryTheory.reflectsEpimorphisms_of_reflectsColimitsOfShape\n-/\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Limits/Constructions/EpiMono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3267404342984972}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n\nimport category_theory.limits.constructions.pullbacks\nimport category_theory.limits.shapes.biproducts\nimport category_theory.limits.shapes.images\nimport category_theory.limits.constructions.limits_of_products_and_equalizers\nimport category_theory.abelian.non_preadditive\n\n/-!\n# Abelian categories\n\nThis file contains the definition and basic properties of abelian categories.\n\nThere are many definitions of abelian category. Our definition is as follows:\nA category is called abelian if it is preadditive,\nhas a finite products, kernels and cokernels,\nand if every monomorphism and epimorphism is normal.\n\nIt should be noted that if we also assume coproducts, then preadditivity is\nactually a consequence of the other properties, as we show in\n`non_preadditive_abelian.lean`. However, this fact is of little practical\nrelevance, since essentially all interesting abelian categories come with a\npreadditive structure. In this way, by requiring preadditivity, we allow the\nuser to pass in the preadditive structure the specific category they are\nworking with has natively.\n\n## Main definitions\n\n* `abelian` is the type class indicating that a category is abelian. It extends `preadditive`.\n* `abelian.image f` is `kernel (cokernel.π f)`, and\n* `abelian.coimage f` is `cokernel (kernel.ι f)`.\n\n## Main results\n\n* In an abelian category, mono + epi = iso.\n* If `f : X ⟶ Y`, then the map `factor_thru_image f : X ⟶ image f` is an epimorphism, and the map\n  `factor_thru_coimage f : coimage f ⟶ Y` is a monomorphism.\n* Factoring through the image and coimage is a strong epi-mono factorisation. This means that\n  * every abelian category has images. We instantiated this in such a way that `abelian.image f` is\n    definitionally equal to `limits.image f`, and\n  * there is a canonical isomorphism `coimage_iso_image : coimage f ≅ image f` such that\n    `coimage.π f ≫ (coimage_iso_image f).hom ≫ image.ι f = f`. The lemma stating this is called\n    `full_image_factorisation`.\n* Every epimorphism is a cokernel of its kernel. Every monomorphism is a kernel of its cokernel.\n* The pullback of an epimorphism is an epimorphism. The pushout of a monomorphism is a monomorphism.\n  (This is not to be confused with the fact that the pullback of a monomorphism is a monomorphism,\n  which is true in any category).\n\n## Implementation notes\n\nThe typeclass `abelian` does not extend `non_preadditive_abelian`,\nto avoid having to deal with comparing the two `has_zero_morphisms` instances\n(one from `preadditive` in `abelian`, and the other a field of `non_preadditive_abelian`).\nAs a consequence, at the beginning of this file we trivially build\na `non_preadditive_abelian` instance from an `abelian` instance,\nand use this to restate a number of theorems,\nin each case just reusing the proof from `non_preadditive_abelian.lean`.\n\nWe don't show this yet, but abelian categories are finitely complete and finitely cocomplete.\nHowever, the limits we can construct at this level of generality will most likely be less nice than\nthe ones that can be created in specific applications. For this reason, we adopt the following\nconvention:\n\n* If the statement of a theorem involves limits, the existence of these limits should be made an\n  explicit typeclass parameter.\n* If a limit only appears in a proof, but not in the statement of a theorem, the limit should not\n  be a typeclass parameter, but instead be created using `abelian.has_pullbacks` or a similar\n  definition.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n* [P. Aluffi, *Algebra: Chaper 0*][aluffi2016]\n\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables {C : Type u} [category.{v} C]\n\nvariables (C)\n\n/--\nA (preadditive) category `C` is called abelian if it has all finite products,\nall kernels and cokernels, and if every monomorphism is the kernel of some morphism\nand every epimorphism is the cokernel of some morphism.\n\n(This definition implies the existence of zero objects:\nfinite products give a terminal object, and in a preadditive category\nany terminal object is a zero object.)\n-/\nclass abelian extends preadditive C :=\n[has_finite_products : has_finite_products C]\n[has_kernels : has_kernels C]\n[has_cokernels : has_cokernels C]\n(normal_mono : Π {X Y : C} (f : X ⟶ Y) [mono f], normal_mono f)\n(normal_epi : Π {X Y : C} (f : X ⟶ Y) [epi f], normal_epi f)\n\nattribute [instance, priority 100] abelian.has_finite_products\nattribute [instance, priority 100] abelian.has_kernels abelian.has_cokernels\n\nend category_theory\n\nopen category_theory\n\nnamespace category_theory.abelian\nvariables {C : Type u} [category.{v} C] [abelian C]\n\n/-- An abelian category has finite biproducts. -/\n@[priority 100]\ninstance has_finite_biproducts : has_finite_biproducts C :=\nlimits.has_finite_biproducts.of_has_finite_products\n\n@[priority 100]\ninstance has_binary_biproducts : has_binary_biproducts C :=\nlimits.has_binary_biproducts_of_finite_biproducts _\n\n@[priority 100]\ninstance has_zero_object : has_zero_object C :=\nhas_zero_object_of_has_initial_object\n\nsection to_non_preadditive_abelian\n\n/-- Every abelian category is, in particular, `non_preadditive_abelian`. -/\ndef non_preadditive_abelian : non_preadditive_abelian C := { ..‹abelian C› }\n\nend to_non_preadditive_abelian\n\nsection strong\nlocal attribute [instance] abelian.normal_epi abelian.normal_mono\n\n/-- In an abelian category, every epimorphism is strong. -/\nlemma strong_epi_of_epi {P Q : C} (f : P ⟶ Q) [epi f] : strong_epi f := by apply_instance\n\n/-- In an abelian category, every monomorphism is strong. -/\nlemma strong_mono_of_mono {P Q : C} (f : P ⟶ Q) [mono f] : strong_mono f := by apply_instance\n\nend strong\n\nsection mono_epi_iso\nvariables {X Y : C} (f : X ⟶ Y)\n\nlocal attribute [instance] strong_epi_of_epi\n\n/-- In an abelian category, a monomorphism which is also an epimorphism is an isomorphism. -/\nlemma is_iso_of_mono_of_epi [mono f] [epi f] : is_iso f :=\nis_iso_of_mono_of_strong_epi _\n\nend mono_epi_iso\n\nsection factor\nlocal attribute [instance] non_preadditive_abelian\n\nvariables {P Q : C} (f : P ⟶ Q)\n\nsection\n\nlemma mono_of_zero_kernel (R : C)\n  (l : is_limit (kernel_fork.of_ι (0 : R ⟶ P) (show 0 ≫ f = 0, by simp))) : mono f :=\nnon_preadditive_abelian.mono_of_zero_kernel _ _ l\n\nlemma mono_of_kernel_ι_eq_zero (h : kernel.ι f = 0) : mono f :=\nmono_of_kernel_zero h\n\nlemma epi_of_zero_cokernel (R : C)\n  (l : is_colimit (cokernel_cofork.of_π (0 : Q ⟶ R) (show f ≫ 0 = 0, by simp))) : epi f :=\nnon_preadditive_abelian.epi_of_zero_cokernel _ _ l\n\nlemma epi_of_cokernel_π_eq_zero (h : cokernel.π f = 0) : epi f :=\nbegin\n  apply epi_of_zero_cokernel _ (cokernel f),\n  simp_rw ←h,\n  exact is_colimit.of_iso_colimit (colimit.is_colimit (parallel_pair f 0)) (iso_of_π _)\nend\n\nend\n\nnamespace images\n\n/-- The kernel of the cokernel of `f` is called the image of `f`. -/\nprotected abbreviation image : C := kernel (cokernel.π f)\n\n/-- The inclusion of the image into the codomain. -/\nprotected abbreviation image.ι : images.image f ⟶ Q :=\nkernel.ι (cokernel.π f)\n\n/-- There is a canonical epimorphism `p : P ⟶ image f` for every `f`. -/\nprotected abbreviation factor_thru_image : P ⟶ images.image f :=\nkernel.lift (cokernel.π f) f $ cokernel.condition f\n\n/-- `f` factors through its image via the canonical morphism `p`. -/\n@[simp, reassoc] protected lemma image.fac :\n  images.factor_thru_image f ≫ image.ι f = f :=\nkernel.lift_ι _ _ _\n\n/-- The map `p : P ⟶ image f` is an epimorphism -/\ninstance : epi (images.factor_thru_image f) :=\nshow epi (non_preadditive_abelian.factor_thru_image f), by apply_instance\n\nsection\nvariables {f}\n\nlemma image_ι_comp_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : images.image.ι f ≫ g = 0 :=\nzero_of_epi_comp (images.factor_thru_image f) $ by simp [h]\n\nend\n\ninstance mono_factor_thru_image [mono f] : mono (images.factor_thru_image f) :=\nmono_of_mono_fac $ image.fac f\n\ninstance is_iso_factor_thru_image [mono f] : is_iso (images.factor_thru_image f) :=\nis_iso_of_mono_of_epi _\n\n/-- Factoring through the image is a strong epi-mono factorisation. -/\n@[simps] def image_strong_epi_mono_factorisation : strong_epi_mono_factorisation f :=\n{ I := images.image f,\n  m := image.ι f,\n  m_mono := by apply_instance,\n  e := images.factor_thru_image f,\n  e_strong_epi := strong_epi_of_epi _ }\n\nend images\n\nnamespace coimages\n\n/-- The cokernel of the kernel of `f` is called the coimage of `f`. -/\nprotected abbreviation coimage : C := cokernel (kernel.ι f)\n\n/-- The projection onto the coimage. -/\nprotected abbreviation coimage.π : P ⟶ coimages.coimage f :=\ncokernel.π (kernel.ι f)\n\n/-- There is a canonical monomorphism `i : coimage f ⟶ Q`. -/\nprotected abbreviation factor_thru_coimage : coimages.coimage f ⟶ Q :=\ncokernel.desc (kernel.ι f) f $ kernel.condition f\n\n/-- `f` factors through its coimage via the canonical morphism `p`. -/\nprotected lemma coimage.fac : coimage.π f ≫ coimages.factor_thru_coimage f = f :=\ncokernel.π_desc _ _ _\n\n/-- The canonical morphism `i : coimage f ⟶ Q` is a monomorphism -/\ninstance : mono (coimages.factor_thru_coimage f) :=\nshow mono (non_preadditive_abelian.factor_thru_coimage f), by apply_instance\n\nsection\nvariables {f}\n\nlemma comp_coimage_π_eq_zero {R : C} {g : Q ⟶ R} (h : f ≫ g = 0) : f ≫ coimages.coimage.π g = 0 :=\nzero_of_comp_mono (coimages.factor_thru_coimage g) $ by simp [h]\n\nend\n\ninstance epi_factor_thru_coimage [epi f] : epi (coimages.factor_thru_coimage f) :=\nepi_of_epi_fac $ coimage.fac f\n\ninstance is_iso_factor_thru_coimage [epi f] : is_iso (coimages.factor_thru_coimage f) :=\nis_iso_of_mono_of_epi _\n\n/-- Factoring through the coimage is a strong epi-mono factorisation. -/\n@[simps] def coimage_strong_epi_mono_factorisation : strong_epi_mono_factorisation f :=\n{ I := coimages.coimage f,\n  m := coimages.factor_thru_coimage f,\n  m_mono := by apply_instance,\n  e := coimage.π f,\n  e_strong_epi := strong_epi_of_epi _ }\n\nend coimages\n\nend factor\n\nsection has_strong_epi_mono_factorisations\n\n/-- An abelian category has strong epi-mono factorisations. -/\n@[priority 100] instance : has_strong_epi_mono_factorisations C :=\nhas_strong_epi_mono_factorisations.mk $ λ X Y f, images.image_strong_epi_mono_factorisation f\n\n/- In particular, this means that it has well-behaved images. -/\nexample : has_images C := by apply_instance\nexample : has_image_maps C := by apply_instance\n\nend has_strong_epi_mono_factorisations\n\nsection images\nvariables {X Y : C} (f : X ⟶ Y)\n\n/-- There is a canonical isomorphism between the coimage and the image of a morphism. -/\nabbreviation coimage_iso_image : coimages.coimage f ≅ images.image f :=\nis_image.iso_ext (coimages.coimage_strong_epi_mono_factorisation f).to_mono_is_image\n  (images.image_strong_epi_mono_factorisation f).to_mono_is_image\n\n/-- There is a canonical isomorphism between the abelian image and the categorical image of a\n    morphism. -/\nabbreviation image_iso_image : images.image f ≅ image f :=\nis_image.iso_ext (images.image_strong_epi_mono_factorisation f).to_mono_is_image (image.is_image f)\n\n/-- There is a canonical isomorphism between the abelian coimage and the categorical image of a\n    morphism. -/\nabbreviation coimage_iso_image' : coimages.coimage f ≅ image f :=\nis_image.iso_ext (coimages.coimage_strong_epi_mono_factorisation f).to_mono_is_image\n  (image.is_image f)\n\nlemma full_image_factorisation : coimages.coimage.π f ≫ (coimage_iso_image f).hom ≫\n  images.image.ι f = f :=\nby rw [limits.is_image.iso_ext_hom,\n  ←images.image_strong_epi_mono_factorisation_to_mono_factorisation_m, is_image.lift_fac,\n  coimages.coimage_strong_epi_mono_factorisation_to_mono_factorisation_m, coimages.coimage.fac]\n\nend images\n\nsection cokernel_of_kernel\nvariables {X Y : C} {f : X ⟶ Y}\n\nlocal attribute [instance] non_preadditive_abelian\n\n/-- In an abelian category, an epi is the cokernel of its kernel. More precisely:\n    If `f` is an epimorphism and `s` is some limit kernel cone on `f`, then `f` is a cokernel\n    of `fork.ι s`. -/\ndef epi_is_cokernel_of_kernel [epi f] (s : fork f 0) (h : is_limit s) :\n  is_colimit (cokernel_cofork.of_π f (kernel_fork.condition s)) :=\nnon_preadditive_abelian.epi_is_cokernel_of_kernel s h\n\n/-- In an abelian category, a mono is the kernel of its cokernel. More precisely:\n    If `f` is a monomorphism and `s` is some colimit cokernel cocone on `f`, then `f` is a kernel\n    of `cofork.π s`. -/\ndef mono_is_kernel_of_cokernel [mono f] (s : cofork f 0) (h : is_colimit s) :\n  is_limit (kernel_fork.of_ι f (cokernel_cofork.condition s)) :=\nnon_preadditive_abelian.mono_is_kernel_of_cokernel s h\n\nend cokernel_of_kernel\n\nsection\n\n@[priority 100]\ninstance has_equalizers : has_equalizers C :=\npreadditive.has_equalizers_of_has_kernels\n\n/-- Any abelian category has pullbacks -/\n@[priority 100]\ninstance has_pullbacks : has_pullbacks C :=\nhas_pullbacks_of_has_binary_products_of_has_equalizers C\n\nend\n\nsection\n\n@[priority 100]\ninstance has_coequalizers : has_coequalizers C :=\npreadditive.has_coequalizers_of_has_cokernels\n\n/-- Any abelian category has pushouts -/\n@[priority 100]\ninstance has_pushouts : has_pushouts C :=\nhas_pushouts_of_has_binary_coproducts_of_has_coequalizers C\n\n@[priority 100]\ninstance has_finite_limits : has_finite_limits C :=\nlimits.finite_limits_from_equalizers_and_finite_products\n\n@[priority 100]\ninstance has_finite_colimits : has_finite_colimits C :=\nlimits.finite_colimits_from_coequalizers_and_finite_coproducts\n\nend\n\nnamespace pullback_to_biproduct_is_kernel\nvariables [limits.has_pullbacks C] {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n\n/-! This section contains a slightly technical result about pullbacks and biproducts.\n    We will need it in the proof that the pullback of an epimorphism is an epimorpism. -/\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` -/\nabbreviation pullback_to_biproduct : pullback f g ⟶ X ⊞ Y :=\nbiprod.lift pullback.fst pullback.snd\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` induces a kernel cone on the map\n    `biproduct X Y ⟶ Z` induced by `f` and `g`. A slightly more intuitive way to think of\n    this may be that it induces an equalizer fork on the maps induced by `(f, 0)` and\n    `(0, g)`. -/\nabbreviation pullback_to_biproduct_fork : kernel_fork (biprod.desc f (-g)) :=\nkernel_fork.of_ι (pullback_to_biproduct f g) $\nby rw [biprod.lift_desc, comp_neg, pullback.condition, add_right_neg]\n\n/-- The canonical map `pullback f g ⟶ X ⊞ Y` is a kernel of the map induced by\n    `(f, -g)`. -/\ndef is_limit_pullback_to_biproduct : is_limit (pullback_to_biproduct_fork f g) :=\nfork.is_limit.mk _\n  (λ s, pullback.lift (fork.ι s ≫ biprod.fst) (fork.ι s ≫ biprod.snd) $\n    sub_eq_zero.1 $ by rw [category.assoc, category.assoc, ←comp_sub, sub_eq_add_neg, ←comp_neg,\n      ←biprod.desc_eq, kernel_fork.condition s])\n  (λ s,\n  begin\n    ext; rw [fork.ι_of_ι, category.assoc],\n    { rw [biprod.lift_fst, pullback.lift_fst] },\n    { rw [biprod.lift_snd, pullback.lift_snd] }\n  end)\n  (λ s m h, by ext; simp [fork.ι_eq_app_zero, ←h walking_parallel_pair.zero])\n\nend pullback_to_biproduct_is_kernel\n\nnamespace biproduct_to_pushout_is_cokernel\nvariables [limits.has_pushouts C] {W X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)\n\n/-- The canonical map `Y ⊞ Z ⟶ pushout f g` -/\nabbreviation biproduct_to_pushout : Y ⊞ Z ⟶ pushout f g :=\nbiprod.desc pushout.inl pushout.inr\n\n/-- The canonical map `Y ⊞ Z ⟶ pushout f g` induces a cokernel cofork on the map\n    `X ⟶ Y ⊞ Z` induced by `f` and `-g`. -/\nabbreviation biproduct_to_pushout_cofork : cokernel_cofork (biprod.lift f (-g)) :=\ncokernel_cofork.of_π (biproduct_to_pushout f g) $\nby rw [biprod.lift_desc, neg_comp, pushout.condition, add_right_neg]\n\n/-- The cofork induced by the canonical map `Y ⊞ Z ⟶ pushout f g` is in fact a colimit cokernel\n    cofork. -/\ndef is_colimit_biproduct_to_pushout : is_colimit (biproduct_to_pushout_cofork f g) :=\ncofork.is_colimit.mk _\n  (λ s, pushout.desc (biprod.inl ≫ cofork.π s) (biprod.inr ≫ cofork.π s) $\n    sub_eq_zero.1 $ by rw [←category.assoc, ←category.assoc, ←sub_comp, sub_eq_add_neg, ←neg_comp,\n      ←biprod.lift_eq, cofork.condition s, zero_comp])\n  (λ s, by ext; simp)\n  (λ s m h, by ext; simp [cofork.π_eq_app_one, ←h walking_parallel_pair.one] )\n\nend biproduct_to_pushout_is_cokernel\n\nsection epi_pullback\nvariables [limits.has_pullbacks C] {W X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n\n/-- In an abelian category, the pullback of an epimorphism is an epimorphism.\n    Proof from [aluffi2016, IX.2.3], cf. [borceux-vol2, 1.7.6] -/\ninstance epi_pullback_of_epi_f [epi f] : epi (pullback.snd : pullback f g ⟶ Y) :=\n-- It will suffice to consider some morphism e : Y ⟶ R such that\n-- pullback.snd ≫ e = 0 and show that e = 0.\nepi_of_cancel_zero _ $ λ R e h,\nbegin\n  -- Consider the morphism u := (0, e) : X ⊞ Y⟶ R.\n  let u := biprod.desc (0 : X ⟶ R) e,\n  -- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.\n  have hu : pullback_to_biproduct_is_kernel.pullback_to_biproduct f g ≫ u = 0 := by simpa,\n  -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n  -- cokernel of pullback_to_biproduct f g\n  have := epi_is_cokernel_of_kernel _\n    (pullback_to_biproduct_is_kernel.is_limit_pullback_to_biproduct f g),\n  -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z ⟶ R.\n  obtain ⟨d, hd⟩ := cokernel_cofork.is_colimit.desc' this u hu,\n  change Z ⟶ R at d,\n  change biprod.desc f (-g) ≫ d = u at hd,\n  -- But then f ≫ d = 0:\n  have : f ≫ d = 0, calc\n    f ≫ d = (biprod.inl ≫ biprod.desc f (-g)) ≫ d : by rw biprod.inl_desc\n    ... = biprod.inl ≫ u : by rw [category.assoc, hd]\n    ... = 0 : biprod.inl_desc _ _,\n  -- But f is an epimorphism, so d = 0...\n  have : d = 0 := (cancel_epi f).1 (by simpa),\n  -- ...or, in other words, e = 0.\n  calc\n    e = biprod.inr ≫ u : by rw biprod.inr_desc\n    ... = biprod.inr ≫ biprod.desc f (-g) ≫ d : by rw ←hd\n    ... = biprod.inr ≫ biprod.desc f (-g) ≫ 0 : by rw this\n    ... = (biprod.inr ≫ biprod.desc f (-g)) ≫ 0 : by rw ←category.assoc\n    ... = 0 : has_zero_morphisms.comp_zero _ _\nend\n\n/-- In an abelian category, the pullback of an epimorphism is an epimorphism. -/\ninstance epi_pullback_of_epi_g [epi g] : epi (pullback.fst : pullback f g ⟶ X) :=\n-- It will suffice to consider some morphism e : X ⟶ R such that\n-- pullback.fst ≫ e = 0 and show that e = 0.\nepi_of_cancel_zero _ $ λ R e h,\nbegin\n  -- Consider the morphism u := (e, 0) : X ⊞ Y ⟶ R.\n  let u := biprod.desc e (0 : Y ⟶ R),\n  -- The composite pullback f g ⟶ X ⊞ Y ⟶ R is zero by assumption.\n  have hu : pullback_to_biproduct_is_kernel.pullback_to_biproduct f g ≫ u = 0 := by simpa,\n  -- pullback_to_biproduct f g is a kernel of (f, -g), so (f, -g) is a\n  -- cokernel of pullback_to_biproduct f g\n  have := epi_is_cokernel_of_kernel _\n    (pullback_to_biproduct_is_kernel.is_limit_pullback_to_biproduct f g),\n  -- We use this fact to obtain a factorization of u through (f, -g) via some d : Z ⟶ R.\n  obtain ⟨d, hd⟩ := cokernel_cofork.is_colimit.desc' this u hu,\n  change Z ⟶ R at d,\n  change biprod.desc f (-g) ≫ d = u at hd,\n  -- But then (-g) ≫ d = 0:\n  have : (-g) ≫ d = 0, calc\n    (-g) ≫ d = (biprod.inr ≫ biprod.desc f (-g)) ≫ d : by rw biprod.inr_desc\n    ... = biprod.inr ≫ u : by rw [category.assoc, hd]\n    ... = 0 : biprod.inr_desc _ _,\n  -- But g is an epimorphism, thus so is -g, so d = 0...\n  have : d = 0 := (cancel_epi (-g)).1 (by simpa),\n  -- ...or, in other words, e = 0.\n  calc\n    e = biprod.inl ≫ u : by rw biprod.inl_desc\n    ... = biprod.inl ≫ biprod.desc f (-g) ≫ d : by rw ←hd\n    ... = biprod.inl ≫ biprod.desc f (-g) ≫ 0 : by rw this\n    ... = (biprod.inl ≫ biprod.desc f (-g)) ≫ 0 : by rw ←category.assoc\n    ... = 0 : has_zero_morphisms.comp_zero _ _\nend\n\nlemma epi_snd_of_is_limit [epi f] {s : pullback_cone f g} (hs : is_limit s) : epi s.snd :=\nbegin\n  convert epi_of_epi_fac (is_limit.cone_point_unique_up_to_iso_hom_comp (limit.is_limit _) hs _),\n  { refl },\n  { exact abelian.epi_pullback_of_epi_f _ _ }\nend\n\nlemma epi_fst_of_is_limit [epi g] {s : pullback_cone f g} (hs : is_limit s) : epi s.fst :=\nbegin\n  convert epi_of_epi_fac (is_limit.cone_point_unique_up_to_iso_hom_comp (limit.is_limit _) hs _),\n  { refl },\n  { exact abelian.epi_pullback_of_epi_g _ _ }\nend\n\n/-- Suppose `f` and `g` are two morphisms with a common codomain and suppose we have written `g` as\n    an epimorphism followed by a monomorphism. If `f` factors through the mono part of this\n    factorization, then any pullback of `g` along `f` is an epimorphism. -/\nlemma epi_fst_of_factor_thru_epi_mono_factorization\n  (g₁ : Y ⟶ W) [epi g₁] (g₂ : W ⟶ Z) [mono g₂] (hg : g₁ ≫ g₂ = g) (f' : X ⟶ W) (hf : f' ≫ g₂ = f)\n  (t : pullback_cone f g) (ht : is_limit t) : epi t.fst :=\nby apply epi_fst_of_is_limit _ _ (pullback_cone.is_limit_of_factors f g g₂ f' g₁ hf hg t ht)\n\nend epi_pullback\n\nsection mono_pushout\nvariables [limits.has_pushouts C] {W X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)\n\ninstance mono_pushout_of_mono_f [mono f] : mono (pushout.inr : Z ⟶ pushout f g) :=\nmono_of_cancel_zero _ $ λ R e h,\nbegin\n  let u := biprod.lift (0 : R ⟶ Y) e,\n  have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,\n  have := mono_is_kernel_of_cokernel _\n    (biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),\n  obtain ⟨d, hd⟩ := kernel_fork.is_limit.lift' this u hu,\n  change R ⟶ X at d,\n  change d ≫ biprod.lift f (-g) = u at hd,\n  have : d ≫ f = 0, calc\n    d ≫ f = d ≫ biprod.lift f (-g) ≫ biprod.fst : by rw biprod.lift_fst\n    ... = u ≫ biprod.fst : by rw [←category.assoc, hd]\n    ... = 0 : biprod.lift_fst _ _,\n  have : d = 0 := (cancel_mono f).1 (by simpa),\n  calc\n    e = u ≫ biprod.snd : by rw biprod.lift_snd\n    ... = (d ≫ biprod.lift f (-g)) ≫ biprod.snd : by rw ←hd\n    ... = (0 ≫ biprod.lift f (-g)) ≫ biprod.snd : by rw this\n    ... = 0 ≫ biprod.lift f (-g) ≫ biprod.snd : by rw category.assoc\n    ... = 0 : zero_comp\nend\n\ninstance mono_pushout_of_mono_g [mono g] : mono (pushout.inl : Y ⟶ pushout f g) :=\nmono_of_cancel_zero _ $ λ R e h,\nbegin\n  let u := biprod.lift e (0 : R ⟶ Z),\n  have hu : u ≫ biproduct_to_pushout_is_cokernel.biproduct_to_pushout f g = 0 := by simpa,\n  have := mono_is_kernel_of_cokernel _\n    (biproduct_to_pushout_is_cokernel.is_colimit_biproduct_to_pushout f g),\n  obtain ⟨d, hd⟩ := kernel_fork.is_limit.lift' this u hu,\n  change R ⟶ X at d,\n  change d ≫ biprod.lift f (-g) = u at hd,\n  have : d ≫ (-g) = 0, calc\n    d ≫ (-g) = d ≫ biprod.lift f (-g) ≫ biprod.snd : by rw biprod.lift_snd\n    ... = u ≫ biprod.snd : by rw [←category.assoc, hd]\n    ... = 0 : biprod.lift_snd _ _,\n  have : d = 0 := (cancel_mono (-g)).1 (by simpa),\n  calc\n    e = u ≫ biprod.fst : by rw biprod.lift_fst\n    ... = (d ≫ biprod.lift f (-g)) ≫ biprod.fst : by rw ←hd\n    ... = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst : by rw this\n    ... = 0 ≫ biprod.lift f (-g) ≫ biprod.fst : by rw category.assoc\n    ... = 0 : zero_comp\nend\n\nlemma mono_inr_of_is_colimit [mono f] {s : pushout_cocone f g} (hs : is_colimit s) : mono s.inr :=\nbegin\n  convert mono_of_mono_fac\n    (is_colimit.comp_cocone_point_unique_up_to_iso_hom hs (colimit.is_colimit _) _),\n  { refl },\n  { exact abelian.mono_pushout_of_mono_f _ _ }\nend\n\n\n\n/-- Suppose `f` and `g` are two morphisms with a common domain and suppose we have written `g` as\n    an epimorphism followed by a monomorphism. If `f` factors through the epi part of this\n    factorization, then any pushout of `g` along `f` is a monomorphism. -/\nlemma mono_inl_of_factor_thru_epi_mono_factorization (f : X ⟶ Y) (g : X ⟶ Z)\n  (g₁ : X ⟶ W) [epi g₁] (g₂ : W ⟶ Z) [mono g₂] (hg : g₁ ≫ g₂ = g) (f' : W ⟶ Y) (hf : g₁ ≫ f' = f)\n  (t : pushout_cocone f g) (ht : is_colimit t) : mono t.inl :=\nby apply mono_inl_of_is_colimit _ _ (pushout_cocone.is_colimit_of_factors _ _ _ _ _ hf hg t ht)\n\nend mono_pushout\n\nend category_theory.abelian\n\nnamespace category_theory.non_preadditive_abelian\n\nvariables (C : Type u) [category.{v} C] [non_preadditive_abelian C]\n\n/-- Every non_preadditive_abelian category can be promoted to an abelian category. -/\ndef abelian : abelian C :=\n{ has_finite_products := by apply_instance,\n/- We need the `convert`s here because the instances we have are slightly different from the\n   instances we need: `has_kernels` depends on an instance of `has_zero_morphisms`. In the\n   case of `non_preadditive_abelian`, this instance is an explicit argument. However, in the case\n   of `abelian`, the `has_zero_morphisms` instance is derived from `preadditive`. So we need to\n   transform an instance of \"has kernels with non_preadditive_abelian.has_zero_morphisms\" to an\n   instance of \"has kernels with non_preadditive_abelian.preadditive.has_zero_morphisms\". Luckily,\n   we have a `subsingleton` instance for `has_zero_morphisms`, so `convert` can immediately close\n   the goal it creates for the two instances of `has_zero_morphisms`, and the proof is complete. -/\n  has_kernels := by convert (by apply_instance : limits.has_kernels C),\n  has_cokernels := by convert (by apply_instance : limits.has_cokernels C),\n  normal_mono := by { introsI, convert normal_mono f },\n  normal_epi := by { introsI, convert normal_epi f },\n  ..non_preadditive_abelian.preadditive }\n\nend category_theory.non_preadditive_abelian\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/abelian/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.32667777740848986}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.bool.default\n \n\nnamespace Mathlib\n\n/-\nSimplification lemmas for ite.\n\nWe don't prove them at logic.lean because it is easier to prove them using\nthe tactic framework.\n-/\n\n@[simp] theorem if_true_right_eq_or (p : Prop) [h : Decidable p] (q : Prop) : ite p q True = (¬p ∨ q) := sorry\n\n@[simp] theorem if_true_left_eq_or (p : Prop) [h : Decidable p] (q : Prop) : ite p True q = (p ∨ q) := sorry\n\n@[simp] theorem if_false_right_eq_and (p : Prop) [h : Decidable p] (q : Prop) : ite p q False = (p ∧ q) := sorry\n\n@[simp] theorem if_false_left_eq_and (p : Prop) [h : Decidable p] (q : Prop) : ite p False q = (¬p ∧ q) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/ite_simp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.32667776880844845}}
{"text": "import data.equiv.basic\nimport tactic.dec_trivial\nimport tactic.derive_fintype\n\nimport chess.board\n\n\n/-!\n\n# Definitions and theorems about chess board movements\n\n## Summary\n\nA `move` on a particular `board` is a pair of squares whose start square\ncontains a `piece` and whose end square does not.\n\nMoves may be combined into `sequence`s of moves, which encapsulate\nmultiple sequential moves all iteratively satisfying the above\ncondition.\n\n## Main definitions\n\n1. The `move` itself, which requires specifying the particular `board`\nit will occur on\n\n2. `perform_move`, which yields the `board` whose playfield has the\nstart and end squares of a `move` suitably modified\n\n3. A move `sequence`, rooted on a starting board, containing a sequence\nof start and end squares which can be treated as iterated moves.\n\n## Implementation notes\n\n1. `move` and `sequence` are implemented independently of each other.\n`sequence.moves` can be used to extract a `move` from a particular\nindex into a `sequence`. `sequence`s are also currently finite, and\ntherefore also may automatically infer proofs of move conditions via\n`dec_trivial`.\n\n2. Currently, no legality checks or piece math whatsoever is performed,\nmeaning `move`s are not yet programmatically confirmed to be\nlegal. Captures are similarly not yet supported.\n\n-/\n\n\nnamespace chess\n\nvariables {m n : Type}\n\nvariables [fintype m] [fintype n] [decidable_eq m] [decidable_eq n]\nvariables {ι : Type} [fintype ι] [decidable_eq ι]\n\nvariables {K : Type}\n\nvariables (b : board m n ι K)\n\n/--\nA move is a (distinct) start and end square whose start square is\noccupied and whose end square is not.\n\nNo inhabited instance because the board might be\nmade up of a single occupied position.\n\n(Captures are not implemented yet.)\n-/\n@[derive [decidable_eq, fintype], nolint has_inhabited_instance]\nstructure move :=\n(start_square : m × n)\n(end_square : m × n)\n(occupied_start : b.contents.occupied_at start_square . tactic.exact_dec_trivial)\n(unoccupied_end : ¬ b.contents.occupied_at end_square . tactic.exact_dec_trivial)\n\nvariable {b}\nvariable (f : move b)\n\nnamespace move\n\n/-- Start squares are occupied before a move. -/\n@[simp] lemma before_occupied_start :\n    b.contents.occupied_at f.start_square := f.occupied_start\n\n/-- End squares are unoccupied before a move. -/\n@[simp] lemma before_unoccupied_end :\n    ¬ b.contents.occupied_at f.end_square := f.unoccupied_end\n\n/-- Start squares are unoccupied after a move. -/\n@[simp] lemma after_unoccupied_start :\n    ¬ (b.contents.move_piece f.start_square f.end_square).occupied_at f.start_square :=\nby simp only [playfield.move_piece_occupied_start, before_unoccupied_end, not_false_iff]\n\n/-- End squares are occupied after a move. -/\n@[simp] lemma after_occupied_end :\n    (b.contents.move_piece f.start_square f.end_square).occupied_at f.end_square :=\nby simp only [before_occupied_start, playfield.move_piece_occupied_end]\n\n/-- Other squares are unchanged after a move. -/\n@[simp] lemma before_after_same {pos : m × n}\n    (h : pos ≠ f.start_square) (h' : pos ≠ f.end_square) :\n    b.contents.move_piece f.start_square f.end_square pos = b.contents pos :=\nb.contents.move_piece_diff h h'\n\n/-- Other occupation are unchanged after a move. -/\n@[simp] lemma before_after_same_occupied {pos : m × n}\n    (h : pos ≠ f.start_square) (h' : pos ≠ f.end_square) :\n    (b.contents.move_piece f.start_square f.end_square).occupied_at pos =\n      b.contents.occupied_at pos :=\nby simp only [h, h', ne.def, playfield.move_piece_occupied_diff, not_false_iff]\n\n/-- The start and end squares of a move are distinct. -/\nlemma diff_squares : f.start_square ≠ f.end_square :=\nλ H, f.unoccupied_end (H ▸ f.occupied_start)\n\n/-- The piece that is being moved. -/\ndef piece : K := b.piece_at f.start_square f.occupied_start\n\n/-- Pieces do not become superimposed after a move. -/\nlemma no_superimposed (pos pos') (hne : pos ≠ pos')\n    (h : (b.contents.move_piece f.start_square f.end_square).occupied_at pos) :\n    b.contents.move_piece f.start_square f.end_square pos ≠\n      b.contents.move_piece f.start_square f.end_square pos' :=\nλ H, hne (b.contents.retains_injectivity b.injects f.occupied_start h H)\n\nvariables (f)\n\n/--\nA `move` retains all indices, ignoring empty squares,\npresent on the `board` it operates on.\n-/\nlemma retains_surjectivity :\n  function.surjective (b.contents.move_piece f.start_square f.end_square).index_at :=\nb.contents.index_at_retains_surjectivity b.contains f.before_occupied_start\n\n/-- A `move` retains accesing indices injectively on the `board` it operates on. -/\nlemma retains_injectivity :\n  (b.contents.move_piece f.start_square f.end_square).some_injective :=\nb.contents.retains_injectivity b.injects f.before_occupied_start\n\n/-- A valid `move` on a `board` retains a valid board state. -/\ndef perform_move : board m n ι K :=\n{ pieces := b.pieces,\n  contents := b.contents.move_piece f.start_square f.end_square,\n  contains := f.retains_surjectivity,\n  injects := f.retains_injectivity }\n\n-- The length of the sequence\nvariables {o : ℕ}\n\n/--\nDefine the mapping of `playfield`s after performing successive `move_piece`s\nusing the pairs of positions in the provided `elements`,\nstarting from the `start_board`.\n-/\ndef scan_contents\n  (start_board: board m n ι K)\n  (elements : fin o → (m × n) × (m × n)) : fin (o + 1) → playfield m n ι :=\nstart_board.contents.move_sequence (vector.of_fn elements)\n\nvariables (m n ι K o)\n\n/--\nA move `sequence` represents a sequential set of moves from a starting `board`.\n\nNo inhabited instance because boards do not have an inhabited instance.\n-/\n@[nolint has_inhabited_instance]\nstructure sequence :=\n(start_board : board m n ι K)\n(elements : fin o → (m × n) × (m × n))\n(all_occupied_start' :\n  ∀ (e : fin o),\n    ((scan_contents start_board elements) e.cast_succ).occupied_at (elements e).fst . tactic.exact_dec_trivial)\n(all_unoccupied_end' :\n  ∀ (e : fin o),\n    ¬ ((scan_contents start_board elements) e.cast_succ).occupied_at (elements e).snd . tactic.exact_dec_trivial)\n\nnamespace sequence\n\nvariables {m n ι K o}\nvariables (s : sequence m n ι K o)\n\n/-- Shorthand for referring to the contents at a sequence index `ixₒ : fin (o + 1)`. -/\ndef contents_at (ixₒ : fin (o + 1)) : playfield m n ι :=\n(scan_contents s.start_board s.elements) ixₒ\n\n/-- Shorthand for referring to the contents at a sequence index `ixₒ : fin (o + 1)`. -/\nlemma contents_at_def (ixₒ : fin (o + 1)) :\n  s.contents_at ixₒ = (scan_contents s.start_board s.elements) ixₒ := rfl\n\n/-- Every scanned board is occupied at the start square of the upcoming move. -/\nlemma all_occupied_start (e : fin o) : (s.contents_at e.cast_succ).occupied_at (s.elements e).fst :=\ns.all_occupied_start' e\n\n/-- Every scanned board is unoccupied at the end square of the upcoming move. -/\nlemma all_unoccupied_end (e : fin o) : ¬ (s.contents_at e.cast_succ).occupied_at (s.elements e).snd :=\ns.all_unoccupied_end' e\n\nopen playfield vector\n\n/-- The first contents in a `scan_contents` `sequence` is of the `start_board`. -/\n@[simp] lemma sequence_zero : s.contents_at 0 = s.start_board.contents :=\nby simp only [contents_at_def, scan_contents, move_sequence_def, nth_zero, scanl_head]\n\n/-- Any `contents_at` a step in the `sequence` is the result of performing a `move_piece` using\nthe `sequence.elements` at that step. -/\nlemma sequence_step (e : fin o) :\n  s.contents_at e.succ =\n    (s.contents_at e.cast_succ).move_piece (s.elements e).fst (s.elements e).snd :=\nby simp only [contents_at_def, scan_contents, move_sequence_def, scanl_nth, nth_of_fn]\n\n/-- Every `playfield` in a sequence of moves contains all the indices it can. -/\nlemma retains_surjectivity (ixₒ : fin (o + 1)) :\n  function.surjective (s.contents_at ixₒ).index_at :=\nbegin\n  apply fin.induction_on ixₒ,\n  { rw sequence_zero,\n    exact s.start_board.contains },\n  { intros ix h,\n    rw sequence_step,\n    apply playfield.index_at_retains_surjectivity _ h,\n    exact s.all_occupied_start _ },\nend\n\n/-- Every `playfield` in a sequence of moves injectively accesses the indices. -/\nlemma retains_injectivity (ixₒ : fin (o + 1)) : (s.contents_at ixₒ).some_injective :=\nbegin\n  apply fin.induction_on ixₒ,\n  { simpa only [sequence_zero] using s.start_board.injects },\n  { intros ix h,\n    simp only [sequence_step],\n    apply playfield.retains_injectivity _ h,\n    exact s.all_occupied_start _ },\nend\n\n/-- Pieces do not disappear after any `move_piece` in a `sequence`. -/\nlemma retains_pieces (ixₒ : fin (o + 1)) (ixᵢ : ι) :\n  ixᵢ ∈ (s.contents_at ixₒ) :=\nexists.elim (s.retains_surjectivity ixₒ ixᵢ) (λ pos h, h ▸ playfield.index_at_in pos)\n\n/-- Pieces do not become superimposed after any `move` in a `sequence`. -/\nlemma no_superimposed (ixₒ : fin (o + 1)) (pos pos') (hne : pos ≠ pos')\n    (h : (s.contents_at ixₒ).occupied_at pos) :\n      (s.contents_at ixₒ) pos ≠ (s.contents_at ixₒ) pos' :=\nλ H, hne (s.retains_injectivity _ h H)\n\n/-- The board which results from applying the first `ix₀ + 1` `move`s in the `sequence`. -/\ndef boards (ixₒ : fin (o + 1)) : board m n ι K :=\n{ contents := s.contents_at ixₒ,\n  pieces := s.start_board.pieces,\n  contains := s.retains_surjectivity ixₒ,\n  injects := s.retains_injectivity ixₒ }\n\n/-- The board which results from applying all `move`s in the `sequence`. -/\ndef end_board : board m n ι K := s.boards (fin.last o)\n\nvariables {b}\nvariables (s)\n\n/-- The `ix₀`'th `move` in the `sequence`. -/\ndef moves (ixₒ : fin o) : chess.move (s.boards ixₒ.cast_succ) :=\n{ start_square := (s.elements ixₒ).fst,\n  end_square := (s.elements ixₒ).snd,\n  occupied_start := by { simpa only [boards] using s.all_occupied_start _ },\n  unoccupied_end := by { simpa only [boards] using s.all_unoccupied_end _ } }\n\n/--\nAny square which is not the `start_square` or `end_square` of any `move`\nin the `sequence` is fixed across all `move`s (i.e. contains the same piece or remains empty).\n-/\nlemma fixes_unmentioned_squares\n  (ixᵢ : ι)\n  {pos}\n  {h_pos: s.start_board.contents pos = ixᵢ}\n  (h_unmentioned : ∀ ixₒ, pos ≠ (s.elements ixₒ).fst ∧ pos ≠ (s.elements ixₒ).snd) :\n    ∀ ixₒ, (s.boards ixₒ).contents pos = ixᵢ :=\nbegin\n  dsimp [boards, scan_contents],\n  intro ix,\n  apply fin.induction_on ix,\n  { simpa only [sequence_zero] using h_pos },\n  { intros ix' h,\n    simpa only [sequence_step, move_piece_diff,\n                h_unmentioned, ne.def, not_false_iff] using h },\nend\n\nend sequence\nend move\n\nnamespace board\n\nvariable (b)\n\n/--\nAssert the existence of a `sequence` of length `o` from a `start_board` to a given end board.\n-/\ndef has_sequence_len (end_board: board m n ι K) (o : ℕ) :=\n    ∃ (s : chess.move.sequence m n ι K o), b ≈ s.start_board ∧ end_board ≈ s.end_board\n\n/-- Assert the existence of a `sequence` from a `start_board` to a given end board. -/\ndef has_sequence_to (end_board: board m n ι K) :=\n    ∃ (o : ℕ), b.has_sequence_len end_board o\n\nend board\n\nend chess\n", "meta": {"author": "Julian", "repo": "lean-across-the-board", "sha": "f14ec4cde25a3549d522a5fd6703330427fd0c89", "save_path": "github-repos/lean/Julian-lean-across-the-board", "path": "github-repos/lean/Julian-lean-across-the-board/lean-across-the-board-f14ec4cde25a3549d522a5fd6703330427fd0c89/src/chess/move/default.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.32663686589461144}}
{"text": "/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.monad.basic\nimport category_theory.monoidal.End\nimport category_theory.monoidal.Mon_\n\n/-!\n\n# The equivalence between `Monad C` and `Mon_ (C ⥤ C)`.\n\nA monad \"is just\" a monoid in the category of endofunctors.\n\n# Definitions/Theorems\n\n1. `to_Mon` associates a monoid object in `C ⥤ C` to any monad on `C`.\n2. `Monad_to_Mon` is the functorial version of `to_Mon`.\n3. `of_Mon` associates a monad on `C` to any monoid object in `C ⥤ C`.\n4. `Monad_Mon_equiv` is the equivalence between `Monad C` and `Mon_ (C ⥤ C)`.\n\n-/\n\nnamespace category_theory\nopen category\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\nvariables {C : Type u} [category.{v} C]\n\nnamespace Monad\nlocal attribute [instance, reducible] endofunctor_monoidal_category\n\n/-- To every `Monad C` we associated a monoid object in `C ⥤ C`.-/\n@[simps]\ndef to_Mon : monad C → Mon_ (C ⥤ C) := λ M,\n{ X := (M : C ⥤ C),\n  one := M.η,\n  mul := M.μ,\n  one_mul' := by { ext, simp }, -- `obviously` provides this, but slowly\n  mul_one' := by { ext, simp }, -- `obviously` provides this, but slowly\n  mul_assoc' := by { ext, dsimp, simp [M.assoc] } }\n\nvariable (C)\n/-- Passing from `Monad C` to `Mon_ (C ⥤ C)` is functorial. -/\n@[simps]\ndef Monad_to_Mon : monad C ⥤ Mon_ (C ⥤ C) :=\n{ obj := to_Mon,\n  map := λ _ _ f, { hom := f.to_nat_trans },\n  map_id' := by { intros X, refl }, -- `obviously` provides this, but slowly\n  map_comp' := by { intros X Y Z f g, refl, } }\nvariable {C}\n\n/-- To every monoid object in `C ⥤ C` we associate a `Monad C`. -/\n@[simps]\ndef of_Mon : Mon_ (C ⥤ C) → monad C := λ M,\n{ to_functor := M.X,\n  η' := M.one,\n  μ' := M.mul,\n  left_unit' := λ X, by { rw [←M.one.id_hcomp_app, ←nat_trans.comp_app, M.mul_one], refl },\n  right_unit' := λ X, by { rw [←M.one.hcomp_id_app, ←nat_trans.comp_app, M.one_mul], refl },\n  assoc' := λ X, by { rw [←nat_trans.hcomp_id_app, ←nat_trans.comp_app], simp } }\n\nvariable (C)\n/-- Passing from `Mon_ (C ⥤ C)` to `Monad C` is functorial. -/\n@[simps]\ndef Mon_to_Monad : Mon_ (C ⥤ C) ⥤ monad C :=\n{ obj := of_Mon,\n  map := λ _ _ f,\n  { app_η' := begin\n      intro X,\n      erw [←nat_trans.comp_app, f.one_hom],\n      refl,\n    end,\n    app_μ' := begin\n      intro X,\n      erw [←nat_trans.comp_app, f.mul_hom], -- `finish` closes this goal\n      simpa only [nat_trans.naturality, nat_trans.hcomp_app, assoc, nat_trans.comp_app, of_Mon_μ],\n    end,\n    ..f.hom } }\n\nnamespace Monad_Mon_equiv\nvariable {C}\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\n@[simps {rhs_md := semireducible}]\ndef counit_iso : Mon_to_Monad C ⋙ Monad_to_Mon C ≅ 𝟭 _ :=\n{ hom := { app := λ _, { hom := 𝟙 _ } },\n  inv := { app := λ _, { hom := 𝟙 _ } },\n  hom_inv_id' := by { ext, simp }, -- `obviously` provides these, but slowly\n  inv_hom_id' := by { ext, simp } }\n\n/-- Auxiliary definition for `Monad_Mon_equiv` -/\n@[simps]\ndef unit_iso_hom : 𝟭 _ ⟶ Monad_to_Mon C ⋙ Mon_to_Monad C :=\n{ app := λ _, { app := λ _, 𝟙 _ } }\n\n/-- Auxiliary definition for `Monad_Mon_equiv` -/\n@[simps]\ndef unit_iso_inv : Monad_to_Mon C ⋙ Mon_to_Monad C ⟶ 𝟭 _ :=\n{ app := λ _, { app := λ _, 𝟙 _ } }\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\n@[simps]\ndef unit_iso : 𝟭 _ ≅ Monad_to_Mon C ⋙ Mon_to_Monad C :=\n{ hom := unit_iso_hom,\n  inv := unit_iso_inv,\n  hom_inv_id' := by { ext, simp }, -- `obviously` provides these, but slowly\n  inv_hom_id' := by { ext, simp } }\n\nend Monad_Mon_equiv\n\nopen Monad_Mon_equiv\n\n/-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/\n@[simps]\ndef Monad_Mon_equiv : (monad C) ≌ (Mon_ (C ⥤ C)) :=\n{ functor := Monad_to_Mon _,\n  inverse := Mon_to_Monad _,\n  unit_iso := unit_iso,\n  counit_iso := counit_iso,\n  functor_unit_iso_comp' := by { intros X, ext, dsimp, simp } } -- `obviously`, slowly\n\n-- Sanity check\nexample (A : monad C) {X : C} : ((Monad_Mon_equiv C).unit_iso.app A).hom.app X = 𝟙 _ := rfl\n\nend Monad\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monad/equiv_mon.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3265989467756611}}
{"text": "import hmem.memory\n\nuniverse u\nvariables {α: Type u} [has_zero α] [decidable_eq α]\n\ndef list.sum_le {α: Type u}: list α → (α → ℕ → Prop) → ℕ → Prop\n| [] _ _ := true\n| (a::as) p n := ∃ m m', p a m ∧ as.sum_le p m' ∧ n = m + m'\n\ntheorem list.cons_append_tail {α: Type u} (x: α) (xs: list α):\n  ∃ ys y, x::xs = ys ++ [y] :=\nbegin\n  induction xs generalizing x,\n  { exact ⟨[], x, rfl⟩ },\n  rcases xs_ih (xs_hd) with ⟨ys, y, ih⟩,\n  refine ⟨x::ys, y, _⟩,\n  rw [ih, list.cons_append],\nend\n\ntheorem list.append_self_iff {α: Type u} (xs ys: list α):\n  xs ++ ys = ys ↔ xs = [] :=\nbegin\n  cases xs,\n  { rw [list.nil_append],\n    exact ⟨ λ _, rfl, λ _, rfl ⟩ },\n  { simp only [list.cons_append, list.cons_ne_nil, iff_false],\n    intro h,\n    have h := congr_arg list.length h,\n    simp only [list.length_cons, list.length_append] at h,\n    linarith },\nend\n\ntheorem list.sum_le_mono {α: Type u} {as: list α} {p: α → ℕ → Prop} {n n': ℕ}:\n  as.sum_le p n → n ≤ n' → as.sum_le p n' :=\nbegin\n  intros h hn,\n  induction as generalizing n n',\n  { trivial },\n  cases nat.exists_eq_add_of_le hn with x hn,\n  rcases h with ⟨m, m', hp, has, hm⟩,\n  rw [hn],\n  exact ⟨m, m' + x, hp, as_ih has le_self_add, hm.symm ▸ nat.add_assoc _ _ _⟩\nend\n\nnamespace hmem\n\nnamespace instruction\ninductive memory_operation\n| COPY\n| MOVE\n| SWAP\nend instruction\n\ninductive instruction (α: Type u)\n| vop {n: ℕ} (op: vector α n → α) (dst: source α) (src: vector (source α) n): instruction\n| mop (op: hmem.instruction.memory_operation) (dst src: source α): instruction\n| clear (dst: source α): instruction\n| ite (cond: α → Prop) [Π {a}, decidable (cond a)] (branch: list instruction): instruction\n| call (func: list instruction) (arg: source α): instruction\n| recurse (arg: source α): instruction\n\nnamespace instruction\n\ndef const (dst: source α) (v: α): instruction α := vop (λ _, v) dst (vector.nil)\ndef uop (op: α → α) (dst src: source α): instruction α := vop (λ s: vector α 1, op (s.nth ⟨0, zero_lt_one⟩)) dst (vector.cons src vector.nil)\ndef bop (op: α → α → α) (dst lhs rhs: source α): instruction α := vop (λ s: vector α 2, op (s.nth ⟨0, zero_lt_two⟩) (s.nth ⟨1, one_lt_two⟩)) dst (vector.cons lhs (vector.cons rhs vector.nil))\n\n@[pattern]\ndef copy: source α → source α → instruction α := instruction.mop instruction.memory_operation.COPY\n\n@[pattern]\ndef move: source α → source α → instruction α := instruction.mop instruction.memory_operation.MOVE\n\n@[pattern]\ndef swap: source α → source α → instruction α := instruction.mop instruction.memory_operation.SWAP\n\nend instruction\n\n@[reducible]\ndef program (α: Type u):= list (instruction α)\n\nstructure thunk (α: Type u) [has_zero α] [decidable_eq α] :=\n  (function: program α)\n  (current: program α)\n  (state: memory α)\n\ndef program.call (p: program α) (m: memory α): thunk α := ⟨p, p, m⟩\n\ntheorem program.call_function (p: program α) (m: memory α): (p.call m).function = p := rfl\n\ninductive stack (α: Type u) [has_zero α] [decidable_eq α]\n| result (value: memory α): stack\n| execution (top: thunk α) (callers: list (thunk α)): stack\n\ndef memory.mop: memory α → instruction.memory_operation → source α → source α → memory α\n| m instruction.memory_operation.COPY dst _ := m.getms dst\n| m instruction.memory_operation.MOVE _ _ := memory.null _\n| m instruction.memory_operation.SWAP _ src := m.getms src\n\nnamespace thunk\n\ndef set_result: thunk α → memory α → thunk α\n| ⟨p, is, m⟩ m' := ⟨p, is, m.setm 0 m'⟩\n\ntheorem set_result_function {t : thunk α} {m: memory α} {t': thunk α}:\n  t.set_result m = t' → t.function = t'.function :=\nbegin\n  intro h,\n  rw [← h],\n  cases t,\n  refl,\nend\n\ntheorem set_result_current {t : thunk α} {m: memory α} {t': thunk α}:\n  t.set_result m = t' → t.current = t'.current :=\nbegin\n  intro h,\n  rw [← h],\n  cases t,\n  refl,\nend\n\ndef step: thunk α → (thunk α ⊕ memory α ⊕ (option (program α) × memory α × thunk α))\n| ⟨_, [], m⟩ := sum.inr (sum.inl m)\n| ⟨p, (instruction.vop op dst src)::is, m⟩ := sum.inl ⟨p, is, m.setvs dst (op (src.map (λ s, m.getvs s)))⟩\n| ⟨p, (instruction.mop op dst src)::is, m⟩ := sum.inl ⟨p, is, (m.setms src (m.mop op src dst)).setms dst (m.getms src)⟩\n| ⟨p, (instruction.clear dst)::is, m⟩ := sum.inl ⟨p, is, m.setms dst (memory.null _)⟩\n| ⟨p, (@instruction.ite _ cond dcond branch)::is, m⟩ := sum.inl ⟨p, @ite _ (cond m.getv) (@dcond _) branch is, m⟩\n| ⟨p, (instruction.call func src)::is, m⟩ := sum.inr (sum.inr (some func, m.getms src, ⟨p, is, m.setm 0 (memory.null _)⟩))\n| ⟨p, (instruction.recurse src)::is, m⟩ := sum.inr (sum.inr (none, m.getms src, ⟨p, is, m.setm 0 (memory.null _)⟩))\n\ntheorem step_nil_iff (t: thunk α):\n  t.current = [] ↔ ∃ m, t.step = sum.inr (sum.inl m) :=\nbegin\n  cases t,\n  cases t_current;\n  try { cases t_current_hd };\n  simp [step]\nend\n\ntheorem step_return_nil {t: thunk α} {m: memory α}:\n  t.step = sum.inr (sum.inl m) → t.current = [] :=\nλ x, (step_nil_iff _).mpr ⟨_, x⟩\n\ntheorem step_return_nil' {p is: program α} {m: memory α} {m': memory α}:\n  thunk.step ⟨p, is, m⟩ = sum.inr (sum.inl m') → is = [] := step_return_nil\n\ntheorem step_function₀ {t t': thunk α}:\n  t.step = sum.inl t' → t.function = t'.function :=\nbegin\n  cases t,\n  cases t_current;\n  try { cases t_current_hd };\n  simp only [step, false_implies_iff];\n  intro h;\n  rw [← h]\nend\n\ntheorem step_function₂ {t: thunk α} {p: option (program α)} {m: memory α} {t': thunk α}:\n  t.step = sum.inr (sum.inr (p, m, t')) → t.function = t'.function :=\nbegin\n  cases t,\n  cases t_current;\n  try { cases t_current_hd };\n  simp only [step, false_implies_iff, prod.mk.inj_iff, and_imp];\n  intros _ _ h;\n  rw [← h]\nend\n\nend thunk\n\nnamespace stack\ndef step: stack α → stack α\n| (execution f caller) := match f.step with\n  | (sum.inl f') := execution f' caller\n  | (sum.inr (sum.inl v)) := match caller with\n    | [] := result v\n    | (c::cs) := execution (c.set_result v) cs\n    end\n  | (sum.inr (sum.inr (p, m, f'))) := execution ((p.get_or_else f.function).call m) (f'::caller)\n  end\n| r := r\n\ntheorem step_function {c : thunk α} {cs: list (thunk α)} {c': thunk α} {cs': list (thunk α)}:\n  (execution c cs).step = (execution c' cs') → (list.last (c::cs) (list.cons_ne_nil _ _)).function = (list.last (c'::cs') (list.cons_ne_nil _ _)).function :=\nbegin\n  cases hstep:c.step,\n  { simp only [step, hstep, and_imp],\n    cases cs,\n    { intros hc hcs,\n      rw [← hcs, list.last_singleton, list.last_singleton, ← hc],\n      exact thunk.step_function₀ hstep },\n    intros hc hcs,\n    rw [← hcs, list.last_cons_cons, list.last_cons_cons] },\n  cases val,\n  { cases cs,\n    { simp only [step, hstep, false_implies_iff] },\n    simp only [step, hstep, and_imp],\n    cases cs_tl,\n    { intros hc hcs,\n      rw [← hcs, list.last_cons_cons, list.last_singleton, list.last_singleton],\n      exact thunk.set_result_function hc },\n    intros hc hcs,\n    rw [← hcs, list.last_cons_cons, list.last_cons_cons, list.last_cons_cons] },\n  rcases val with ⟨_, _, _⟩,\n  simp only [step, hstep, and_imp],\n  cases cs,\n  { intros hc hcs,\n    rw [← hcs, list.last_cons_cons, list.last_singleton, list.last_singleton],\n    exact thunk.step_function₂ hstep },\n  intros hc hcs,\n  rw [← hcs, list.last_cons_cons, list.last_cons_cons, list.last_cons_cons]\nend\n\ndef result_halt (m: memory α) (n: ℕ):\n  step^[n] (result m) = result m :=\nbegin\n  induction n,\n  { refl },\n  { rw [function.iterate_succ_apply', n_ih],\n    refl }\nend\n\ndef result_mono {s: stack α} {n n': ℕ}:\n  (∃ m, step^[n] s = result m) → n ≤ n' → ∃ m, step^[n'] s = result m :=\nbegin\n  intros h hn,\n  cases nat.exists_eq_add_of_le hn with x hx,\n  cases h with m h,\n  simp only [hx, add_comm n x, function.iterate_add_apply, h, result_halt, result.inj_eq, exists_eq'],\nend\n\ntheorem nstep_function {c : thunk α} {cs: list (thunk α)} {n: ℕ} {c': thunk α} {cs': list (thunk α)}:\n  step^[n] (execution c cs) = (execution c' cs') → (list.last (c::cs) (list.cons_ne_nil _ _)).function = (list.last (c'::cs') (list.cons_ne_nil _ _)).function :=\nbegin\n  induction n generalizing c cs,\n  { rw [function.iterate_zero_apply, execution.inj_eq, and_imp],\n    intros hc hcs,\n    rw [hc, hcs] },\n  rw [function.iterate_succ_apply],\n  cases hstep:(execution c cs).step,\n  { simp only [result_halt, false_implies_iff] },\n  exact λ h, eq.trans (step_function hstep) (n_ih h)\nend\n\ntheorem nstep_function' {c : thunk α} {n: ℕ} {c': thunk α}:\n  step^[n] (execution c []) = (execution c' []) → c.function = c'.function := nstep_function\n\ntheorem step_result_iff {t: thunk α} {cs: list (thunk α)} {m: memory α}:\n  step (execution t cs) = result m ↔ cs = [] ∧ ∃ p, t = ⟨p, [], m⟩ :=\nbegin\n  split,\n  { cases t,\n    cases t_current,\n    { cases cs,\n      { unfold step thunk.step,\n        intro h,\n        refine ⟨ rfl, t_function, _ ⟩,\n        rw [thunk.mk.inj_eq],\n        exact ⟨ rfl, rfl, result.inj h ⟩ },\n      trivial },\n    cases t_current_hd;\n    unfold step thunk.step;\n    trivial },\n  rw [and_imp, exists_imp_distrib],\n  intros hcs _ ht,\n  rw [ht, hcs],\n  refl\nend\n\ndef result_prev {c: thunk α} {cs: list (thunk α)} {m: memory α}:\n  (execution c cs).step = result m ↔\n  c = ⟨ c.function, [], m ⟩ ∧ cs = [] :=\nbegin\n  cases c,\n  cases c_current;\n  cases cs;\n  try { cases c_current_hd };\n  simp only [step, thunk.step, eq_self_iff_true, and_true, true_and, iff_self, and_false, false_and],\nend\n\ndef result_nprev {c: thunk α} {cs: list (thunk α)} {n: ℕ} {m: memory α}:\n  step^[n] (execution c cs) = result m ↔\n  ∃ t (n' < n), step^[n'] (execution c cs) = execution ⟨t, [], m⟩ [] :=\nbegin\n  split,\n  { induction n generalizing c cs,\n    { simp only [function.iterate_zero_apply, false_implies_iff] },\n    rw [function.iterate_succ_apply],\n    cases hstep:(execution c cs).step,\n    { refine λ h, ⟨ c.function, 0, (nat.zero_lt_succ _), _⟩,\n      simp only [function.iterate_zero_apply, execution.inj_eq, and_assoc],\n      rw [result_halt, result.inj_eq] at h,\n      rwa [← result_prev, ← h] },\n    intro h,\n    rcases n_ih h with ⟨t, n', hn, ih⟩,\n    refine ⟨t, n'+1, nat.succ_lt_succ hn, _⟩,\n    rwa [function.iterate_succ_apply, hstep] },\n  { intro h,\n    rcases h with ⟨t, n', hn', h⟩,\n    cases nat.exists_eq_add_of_lt hn' with x hx,\n    rw [hx, nat.add_assoc, nat.add_comm, function.iterate_add_apply, h, function.iterate_succ_apply],\n    unfold stack.step thunk.step,\n    exact result_halt _ _ }\nend\n\ndef result_nprev' {c: thunk α} {n: ℕ} {m: memory α}:\n  step^[n] (execution c []) = result m ↔\n  ∃ (n' < n), step^[n'] (execution c []) = execution ⟨c.function, [], m⟩ [] :=\nbegin\n  rw [result_nprev],\n  split,\n  { intro h,\n    rcases h with ⟨p, n', hn, h⟩,\n    refine ⟨n', hn, _⟩,\n    rw [h, nstep_function' h] },\n  { intro h,\n    rcases h with ⟨n', hn, h⟩,\n    exact ⟨_, n', hn, h⟩ }\nend\n\ntheorem step_return {t: thunk α} {cs: list (thunk α)} {m: memory α}:\n  step (execution t cs) = result m → ∀ c css, step (execution t (cs ++ (c::css))) = execution (c.set_result m) css :=\nbegin\n  rw [step_result_iff, and_imp, exists_imp_distrib],\n  intros hcs _ ht _ _,\n  rw [hcs, ht],\n  refl,\nend\n\ntheorem step_append {t: thunk α} {t': thunk α} {cs cs': list (thunk α)} :\n  step (execution t cs) = (execution t' cs') → ∀ css, step (execution t (cs ++ css)) = (execution t' (cs' ++ css)) :=\nbegin\n  cases t,\n  cases t_current,\n  { cases cs,\n    { trivial },\n    unfold step thunk.step,\n    rw [execution.inj_eq, and_imp],\n    intros ht hcs _,\n    rw [← ht, ← hcs, list.cons_append],\n    refl },\n  cases t_current_hd;\n  unfold step thunk.step;\n  rw [execution.inj_eq, and_imp];\n  intros ht hcs _;\n  rw [← ht, ← hcs];\n  try { rw [list.nil_append] };\n  try { rw [list.cons_append] },\nend\n\ntheorem step_iterate_append {t: thunk α} {n: ℕ} {t': thunk α} {cs cs': list (thunk α)}:\n  step^[n] (execution t cs) = execution t' cs' → ∀ css,  step^[n] (execution t (cs ++ css)) = execution t' (cs' ++ css) :=\nbegin\n  induction n generalizing t t' cs cs',\n  { rw [function.iterate_zero_apply, execution.inj_eq, and_imp],\n    intros ht hcs _,\n    rw [function.iterate_zero_apply, ← ht, ← hcs] },\n  rw [function.iterate_succ_apply],\n  cases h:(execution t cs).step,\n  { rw [result_halt],\n    trivial },\n  intros hn _,\n  rw [function.iterate_succ_apply, step_append h, n_ih hn]\nend\n\ntheorem step_iterate_return {t: thunk α} {cs: list (thunk α)} {n: ℕ} {m: memory α}:\n  step^[n] (execution t cs) = result m →\n  ∀ c css, ∃ n' ≤ n, step^[n'] (execution t (cs ++ (c::css))) = execution (c.set_result m) css :=\nbegin\n  induction n generalizing t cs m,\n  { trivial },\n  rw [function.iterate_succ_apply],\n  cases hex:(step (execution t cs)),\n  { rw [result_halt, result.inj_eq],\n    intros hm _ _,\n    refine ⟨1, nat.succ_le_succ (nat.zero_le _), _⟩,\n    apply step_return,\n    rwa [← hm] },\n  { intros hstep _ _,\n    rcases n_ih hstep _ _ with ⟨n', hn, h⟩,\n    refine ⟨n'.succ, nat.succ_le_succ hn, _ ⟩,\n    rwa [function.iterate_succ_apply, step_append hex] }\nend\n\ntheorem step_iterate_return' {t: thunk α} {n: ℕ} {m: memory α}:\n  step^[n] (execution t []) = result m →\n  ∀ c, ∃ n' ≤ n, step^[n'] (execution t [c]) = execution (c.set_result m) [] :=\nλ h c, step_iterate_return h c []\n\ntheorem return_of_step_iterate {t c: thunk α} {cs css: list (thunk α)} {n: ℕ} {t': thunk α}:\n  step^[n] (execution t (cs ++ c::css)) = execution t' css → ∃ (p: memory α) m ≤ n, step^[m] (execution t cs) = result p :=\nbegin\n  induction n using nat.strong_induction_on with n ih generalizing t c cs t',\n  cases n,\n  { simp only [← @list.singleton_append _ c css, ← list.append_assoc,\n      function.iterate_zero_apply, list.append_self_iff, list.append_ne_nil_of_ne_nil_right _ [c] (list.cons_ne_nil _ _),\n      and_false, false_implies_iff] },\n  rw [function.iterate_succ_apply],\n  cases hstep:(t.step),\n  { simp only [stack.step, hstep],\n    intro h,\n    rcases ih _ (nat.lt_succ_self _) h with ⟨p, m, hm, ih⟩,\n    refine ⟨p, _, nat.succ_le_succ hm, _⟩,\n    simpa only [function.iterate_succ_apply, stack.step, hstep] using ih },\n  cases val,\n  { cases cs,\n    { refine λ _, ⟨val, 1, nat.succ_le_succ (nat.zero_le _), _⟩,\n      simp only [function.iterate_one, stack.step, hstep] },\n    simp only [function.iterate_succ_apply, stack.step, hstep, list.cons_append],\n    intro h,\n    rcases ih _ (nat.lt_succ_self _) h with ⟨p, n', hn', ih⟩,\n    refine ⟨p, n' + 1, (nat.succ_le_succ hn'), _⟩,\n    simpa only [function.iterate_succ_apply, stack.step, hstep] using ih },\n  rcases val with ⟨_, _, c'⟩,\n  simp only[stack.step, hstep, ← list.cons_append],\n  intro h,\n  rcases ih _ (nat.lt_succ_self _) h with ⟨p₁, n₁, hn₁, ih₁⟩,\n  rcases step_iterate_return ih₁ c css with ⟨n₂, hn₂, ih₂⟩,\n  rcases ih _ (nat.lt_succ_of_le (trans hn₂ hn₁)) ih₂ with ⟨p₃, n₃, hn₃, ih₃⟩,\n  refine ⟨p₃, n₃ + 1, nat.succ_le_succ (le_trans (le_trans hn₃ hn₂) hn₁), _⟩,\n  simpa only [function.iterate_succ_apply, stack.step, hstep, ih₂] using ih₃,\nend\n\ntheorem result_of_step_iterate {t c: thunk α} {cs: list (thunk α)} {n: ℕ} {t': thunk α}:\n  step^[n] (execution t (c::cs)) = execution t' cs →\n  ∃ (p: memory α) m ≤ n, step^[m] (execution t []) = result p :=\nlist.nil_append (c::cs) ▸ return_of_step_iterate\n\ntheorem nstep_result_sub {s s': stack α} {n n': ℕ} {m: memory α}:\n  step^[n] s = s' → (step^[n']) s = result m → (step^[n' - n]) s' = result m :=\nbegin\n  cases s,\n  { intros hs hm,\n    rw [← hs, ← hm, result_halt, result_halt, result_halt] },\n  cases le_total n n';\n  cases nat.exists_eq_add_of_le h with x h;\n  rw [h, add_comm, function.iterate_add_apply],\n  { intro hs,\n    rw [hs, nat.add_sub_cancel],\n    exact id },\n  { intros hs hm,\n    rw [← hs, hm, result_halt, result_halt] }\nend\n\ndef memory_usage_le: stack α → ℕ → Prop\n| (result m) n := m.usage_le n\n| (execution f fs) n := ((f::fs).map thunk.state).sum_le memory.usage_le n\n\ndef memory_usage_mono {s: stack α} {n m: ℕ}:\n  s.memory_usage_le n → n ≤ m → s.memory_usage_le m :=\nbegin\n  cases s,\n  { apply set.size_le_mono },\n  { apply list.sum_le_mono }\nend\nend stack\n\nnamespace program\ndef halts_on (p: program α) (inp: memory α) :=\n  ∃ n outp, stack.step^[n] (stack.execution ⟨p, p, inp⟩ []) = stack.result outp\n\ndef has_result (p: program α) (inp outp: memory α) :=\n  ∃ n, stack.step^[n] (stack.execution ⟨p, p, inp⟩ []) = stack.result outp\n\ndef has_time_cost (p: program α) (inp: memory α) (n: ℕ) :=\n  ∃ outp, stack.step^[n] (stack.execution ⟨p, p, inp⟩ []) = stack.result outp\n\ndef has_memory_cost (p: program α) (inp: memory α) (m: ℕ) :=\n  ∀ n, ((stack.step^[n]) (stack.execution ⟨p, p, inp⟩ [])).memory_usage_le m\n\ntheorem unique_result {p: program α} {inp outp outp': memory α}:\n  p.has_result inp outp → p.has_result inp outp' → outp = outp' :=\nbegin\n  intros h h',\n  cases h with n h,\n  cases h' with n' h',\n  cases le_total n n' with hn hn,\n  { cases nat.exists_eq_add_of_le hn with x hn,\n    apply stack.result.inj,\n    rwa [← stack.result_halt _ x, ← h, ← function.iterate_add_apply, add_comm, ← hn] },\n  { cases nat.exists_eq_add_of_le hn with x hn,\n    apply stack.result.inj,\n    rwa [← stack.result_halt outp' x, ← h', ← function.iterate_add_apply, add_comm, ← hn, eq_comm] }\nend\n\ntheorem time_cost_mono {p: program α} {inp: memory α} {n m: ℕ}:\n  p.has_time_cost inp n → n ≤ m → p.has_time_cost inp m :=\nbegin\n  intros h hnm,\n  cases nat.exists_eq_add_of_le hnm with x hnm,\n  cases h with outp h,\n  refine ⟨outp, _⟩,\n  rw [hnm, add_comm, function.iterate_add_apply, h, stack.result_halt]\nend\n\ntheorem memory_cost_mono {p: program α} {inp: memory α} {n m: ℕ}:\n  p.has_memory_cost inp n → n ≤ m → p.has_memory_cost inp m :=\nλ h hnm _,  stack.memory_usage_mono (h _) hnm\n\nend program\n\nend hmem", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/hmem/stack.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3265989467756611}}
{"text": "import analysis.inner_product_space.basic\nimport analysis.inner_product_space.pi_L2\nimport analysis.inner_product_space.spectrum\nimport analysis.inner_product_space.l2_space\nimport analysis.normed_space.pi_Lp\nimport linear_algebra.basis\nimport .lemmas.ladr_7_lem\nimport .linear_independent\n\nvariable {n : ℕ}\nvariable T : Lℂ^n\n-- Dan's work on trying to extend isometries\n-- example (M : submodule ℂ ℂ^n) (S₀ : M →ₗᵢ[ℂ] M) : ∃ S : (ℂ^n) →ₗᵢ[ℂ] ℂ^n, orthogonal_projection M ∘ (linear_map.dom_restrict S.to_linear_map M) = S₀ :=\n-- begin\n--   have exists_w := exists_hilbert_basis ℂ M,\n--   let w'' := classical.some exists_w,\n--   let b₀ : hilbert_basis w'' ℂ M := classical.some (classical.some_spec exists_w),\n\n--   let w' : set (ℂ^n) := sorry,\n--   let b' : w' → (ℂ^n) := coe,\n--   have : orthonormal ℂ (coe : w' → ℂ^n) := sorry,\n--   have : ∃ (w : set ℂ^n) (b : hilbert_basis w ℂ ℂ^n), w' ⊆ w ∧ ⇑b = (coe : w → ℂ^n) := orthonormal.exists_hilbert_basis_extension this,\n\n-- end\n\n#check T.range_restrict\n\n\nopen classical \nuniverse u\n\nvariable S : submodule ℂ ℂ^n\n\nvariable (L : S →ₗᵢ[ℂ] ℂ^n)\n\nvariable w' : orthonormal_basis_index ℂ ↥S\n#check ⇑(orthonormal_basis ℂ ↥S)\n\n-- lemma L_to_M : ∃ (M : (ℂ^n) →ₗᵢ[ℂ] (ℂ^n)), (∀ (s : S), M s = L s) :=\n-- begin\n--   let m := finite_dimensional.finrank ℂ S,\n--   have S_rank_m : finite_dimensional.finrank ℂ S = m := rfl,\n\n--   let b' := orthonormal_basis ℂ S,\n--   -- have hb' : orthonormal ℂ b' := fin_orthonormal_basis_orthonormal S_rank_m,\n--   -- simp only [b'] at hb',\n--   -- rw coe_orthonormal_basis at hb',  \n--   have b'_lifts_to_coe := coe_orthonormal_basis ℂ S,\n--   let foo := ⇑b',\n--   have hfoo : orthonormal ℂ foo :=\n--   begin\n--     apply orthonormal_basis_orthonormal,\n--   end,\n--   rw orthonormal at hfoo,\n--   let foo₂ := S.subtype ∘ foo,\n--   have : orthonormal ℂ foo₂ :=\n--   begin\n--     rw orthonormal,\n--     simp only [foo₂],\n--     simp only [set_coe.forall],\n--     simp? [hfoo],\n--   end,\n--   have b'_orthonormal := orthonormal_basis_orthonormal ℂ ℂ^n,\n\n\n--   -- rw b'_lifts_to_coe at b'_orthonormal,\n\n--   -- have exists_extension := exists_subset_is_orthonormal_basis b'_orthonormal,\n\n\n-- end\n\n-- lemma onb_coe_to_onf : orthonormal ℂ (S.subtype ∘ orthonormal_basis ℂ S) :=\n-- begin\n--   rw orthonormal_iff_ite,\n--   intros i j,\n--   rw function.comp_app,\n--   rw function.comp_app,\n--   rw submodule.subtype_apply,\n--   rw submodule.subtype_apply,\n--   rw ← submodule.coe_inner,\n--   revert i j,\n--   rw ← orthonormal_iff_ite,\n--   apply orthonormal_basis_orthonormal ℂ,\n-- end\n\n-- lemma L_to_M : ∃ (M : (ℂ^n) →ₗᵢ[ℂ] (ℂ^n)), (∀ (s : S), M s = L s) :=\n-- begin\n--   let m := finite_dimensional.finrank ℂ S,\n--   have S_rank_m : finite_dimensional.finrank ℂ S = m := rfl,\n\n--   -- Taking an onb for S\n--   let b' := orthonormal_basis ℂ S,\n--   have b'_lifts_to_coe := coe_orthonormal_basis ℂ S,\n--   let b'_in_Cn := S.subtype ∘ ⇑b',\n--   have b'_orthonormal : orthonormal ℂ b'_in_Cn := onb_coe_to_onf S,\n--   have b'_in_Cn_is_coe : b'_in_Cn = coe :=\n--   begin\n--     simp only [b'_in_Cn],\n--     simp only [coe_orthonormal_basis],\n--     ext1,\n--     simp only [submodule.subtype_apply, eq_self_iff_true, function.comp_app, set_like.coe_eq_coe, coe_coe],\n--   end,\n\n--   let Lb' := L ∘ ⇑b',\n\n--   have Lb'_orthonormal := lin_iso_preserves_on (⇑b') b'_orthonormal L,\n\n--   let Lw' := set.range Lb',\n--   let Lb'' : Lw' → ℂ^n := coe,\n--   have Lb''_orthonormal : orthonormal ℂ Lb'' := sorry,\n--   simp only [Lb''] at Lb''_orthonormal,\n\n--   rw b'_in_Cn_is_coe at b'_orthonormal,\n-- -- orthonormal ℂ Lb'\n--   -- Extending b' to b\n--   have b_exists := @exists_subset_is_orthonormal_basis ℂ (ℂ^n) complex.is_R_or_C _ _ _ b'_orthonormal,\n\n--   have b_exists' := some_spec b_exists,\n--   have b_exists'' := some_spec b_exists',\n--   let b := some b_exists'',\n--   have hb : orthonormal ℂ b ∧ ⇑b = coe := some_spec b_exists'',\n\n--   have Lb_exists := @exists_subset_is_orthonormal_basis ℂ _ _ _ _ _ Lb''_orthonormal,\n\n--   have Lb_exists' := some_spec Lb_exists,\n--   have Lb_exists'' := some_spec Lb_exists',\n--   let Lb := some Lb_exists'',\n--   have hLb : orthonormal ℂ Lb ∧ ⇑Lb = coe := some_spec Lb_exists'',\n--   let M := basis.constr b ℂ,\n-- end\n\n\n-- example {S : submodule ℂ ℂ^n} (L : S →ₗᵢ[ℂ] ℂ^n) : 1 = 2 :=\n-- begin\n--   -- First we find an orthonormal basis over S,\n--   have exists_w' := exists_hilbert_basis ℂ S,\n--   let w' := some exists_w',\n--   let b' : hilbert_basis w' ℂ S := some (some_spec exists_w'),\n--   have hb' : ⇑b' = coe := some_spec (some_spec exists_w'),\n--   -- Now extend the basis to ℂⁿ:\n--   have b'_orthonormal : orthonormal ℂ (coe : ↥w' → ↥S) :=\n--   begin\n--     rw ← hb',\n--     exact b'.orthonormal,\n--   end,\n--   have b'_extends := orthonormal.exists_hilbert_basis_extension b'_orthonormal,\n--   let w := some b'_extends,\n--   have hw := some_spec b'_extends,\n--   let b := some hw,\n--   have hb : w' ⊆ w ∧ ⇑b = coe := some_spec hw,\n\n--   -- Now we map the smaller basis into ℂⁿ by L:\n--   let Lb' := L ∘ (⇑b'),\n--   have Lb'_orthonormal : orthonormal ℂ Lb' := lin_iso_preserves_on (⇑b') b'.orthonormal L,\n--   let Lw' := set.range Lb',\n--   let Lb'' : Lw' → ℂ^n := coe,\n--   have Lb''_orthonormal : orthonormal ℂ Lb'' := \n--   begin\n--     sorry,\n--   end,\n--   -- And extend this new basis:\n--   have Lb'_extends := orthonormal.exists_hilbert_basis_extension Lb''_orthonormal,\n--   let Lw := some Lb'_extends,\n--   have hLw := some_spec Lb'_extends,\n--   let Lb := some hLw,\n--   have hLb : Lw' ⊆ Lw ∧ ⇑Lb = coe := some_spec hLw,\n--   let M := constr b ℂ,\n-- end\n\n\nnoncomputable lemma eq_dim_has_lin_iso (S₁ S₂ : submodule ℂ ℂ^n) (h : finite_dimensional.finrank ℂ S₁ = finite_dimensional.finrank ℂ S₂) : S₁ →ₗᵢ[ℂ] S₂ :=\nbegin\n  let d := finite_dimensional.finrank ℂ S₁,\n  have hS₁ : finite_dimensional.finrank ℂ S₁ = d := rfl,\n  have hS₂ : finite_dimensional.finrank ℂ S₂ = d := by {rw ← h},\n  have S₁_equiv := linear_isometry_equiv.of_inner_product_space hS₁,\n  have S₂_equiv := linear_isometry_equiv.of_inner_product_space hS₂,\n  exact ((S₁_equiv).trans (S₂_equiv.symm)).to_linear_isometry,\nend\n\n\nnoncomputable def d := finite_dimensional.finrank ℂ S\nlemma dim_of_lin_iso : finite_dimensional.finrank ℂ (L.to_linear_map).range = (d S) :=\nbegin\n  have equiv_of_image := linear_equiv.of_injective (L.to_linear_map) L.injective,\n\n  have : finite_dimensional.finrank ℂ ↥S = finite_dimensional.finrank ℂ ↥(L.to_linear_map).range :=\n  begin\n    apply finite_dimensional.nonempty_linear_equiv_iff_finrank_eq.1,\n    use equiv_of_image,\n    exact finite_dimensional.finite_dimensional_submodule S,\n    exact finite_dimensional.finite_dimensional_submodule (L.to_linear_map).range,\n  end,\n  rw ← this,\n  simp only [d],\nend\n\nlemma L_to_M : ∃ (M : (ℂ^n) →ₗᵢ[ℂ] (ℂ^n)), (∀ (s : S), M s = L s) :=\nbegin\n  let LS := (L.to_linear_map).range,\n  have dim_LS_perp : finite_dimensional.finrank ℂ (LSᗮ) = n - (d S) :=\n  begin\n    rw ← dim_of_lin_iso S L,\n    have dim_sum := submodule.finrank_add_finrank_orthogonal LS,\n    rw @finrank_euclidean_space_fin ℂ _ n at dim_sum,\n    have LS_dim_d : finite_dimensional.finrank ℂ (LS) = d S :=\n    begin\n      exact dim_of_lin_iso S L,\n    end, \n    rw LS_dim_d at dim_sum,\n    rw LS_dim_d,\n    -- simp [dim_sum],\n    -- have sum_dim := eq.symm dim_sum,\n    -- rw sum_dim,\n\n    sorry,\n  end,\n  have dim_LS : finite_dimensional.finrank ℂ Sᗮ = n - (d S) := sorry,\n  \n\nend\n", "meta": {"author": "hparshall", "repo": "lean-matrix-analysis", "sha": "cc1b9949065257b6c19f047a5a996bfac29f178e", "save_path": "github-repos/lean/hparshall-lean-matrix-analysis", "path": "github-repos/lean/hparshall-lean-matrix-analysis/lean-matrix-analysis-cc1b9949065257b6c19f047a5a996bfac29f178e/src/old/isometry_extension.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341024983754, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3265989396393266}}
{"text": "/-\nCopyright (c) 2020 David Wärn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David Wärn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.ideal\nimport Mathlib.data.finset.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# The back and forth method and countable dense linear orders\n\n## Results\n\nSuppose `α β` are linear orders, with `α` countable and `β` dense, nonempty, without endpoints.\nThen there is an order embedding `α ↪ β`. If in addition `α` is dense, nonempty, without\nendpoints and `β` is countable, then we can upgrade this to an order isomorhpism `α ≃ β`.\n\nThe idea for both results is to consider \"partial isomorphisms\", which\nidentify a finite subset of `α` with a finite subset of `β`, and prove that\nfor any such partial isomorphism `f` and `a : α`, we can extend `f` to\ninclude `a` in its domain.\n\n## References\n\nhttps://en.wikipedia.org/wiki/Back-and-forth_method\n\n## Tags\n\nback and forth, dense, countable, order\n\n-/\n\nnamespace order\n\n\n/-- Suppose `α` is a nonempty dense linear order without endpoints, and\n    suppose `lo`, `hi`, are finite subssets with all of `lo` strictly\n    before `hi`. Then there is an element of `α` strictly between `lo`\n    and `hi`. -/\ntheorem exists_between_finsets {α : Type u_1} [linear_order α] [densely_ordered α] [no_bot_order α]\n    [no_top_order α] [nonem : Nonempty α] (lo : finset α) (hi : finset α)\n    (lo_lt_hi : ∀ (x : α), x ∈ lo → ∀ (y : α), y ∈ hi → x < y) :\n    ∃ (m : α), (∀ (x : α), x ∈ lo → x < m) ∧ ∀ (y : α), y ∈ hi → m < y :=\n  sorry\n\n/-- The type of partial order isomorphisms between `α` and `β` defined on finite subsets.\n    A partial order isomorphism is encoded as a finite subset of `α × β`, consisting\n    of pairs which should be identified. -/\ndef partial_iso (α : Type u_1) (β : Type u_2) [linear_order α] [linear_order β] :=\n  Subtype\n    fun (f : finset (α × β)) =>\n      ∀ (p q : α × β), p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)\n\nnamespace partial_iso\n\n\nprotected instance inhabited (α : Type u_1) (β : Type u_2) [linear_order α] [linear_order β] :\n    Inhabited (partial_iso α β) :=\n  { default := { val := ∅, property := sorry } }\n\nprotected instance preorder (α : Type u_1) (β : Type u_2) [linear_order α] [linear_order β] :\n    preorder (partial_iso α β) :=\n  subtype.preorder\n    fun (f : finset (α × β)) =>\n      ∀ (p q : α × β), p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)\n\n/-- For each `a`, we can find a `b` in the codomain, such that `a`'s relation to\nthe domain of `f` is `b`'s relation to the image of `f`.\n\nThus, if `a` is not already in `f`, then we can extend `f` by sending `a` to `b`.\n-/\ntheorem exists_across {α : Type u_1} {β : Type u_2} [linear_order α] [linear_order β]\n    [densely_ordered β] [no_bot_order β] [no_top_order β] [Nonempty β] (f : partial_iso α β)\n    (a : α) :\n    ∃ (b : β), ∀ (p : α × β), p ∈ subtype.val f → cmp (prod.fst p) a = cmp (prod.snd p) b :=\n  sorry\n\n/-- A partial isomorphism between `α` and `β` is also a partial isomorphism between `β` and `α`. -/\nprotected def comm {α : Type u_1} {β : Type u_2} [linear_order α] [linear_order β] :\n    partial_iso α β → partial_iso β α :=\n  subtype.map (finset.image ⇑(equiv.prod_comm α β)) sorry\n\n/-- The set of partial isomorphisms defined at `a : α`, together with a proof that any\n    partial isomorphism can be extended to one defined at `a`. -/\ndef defined_at_left {α : Type u_1} (β : Type u_2) [linear_order α] [linear_order β]\n    [densely_ordered β] [no_bot_order β] [no_top_order β] [Nonempty β] (a : α) :\n    cofinal (partial_iso α β) :=\n  cofinal.mk (fun (f : partial_iso α β) => ∃ (b : β), (a, b) ∈ subtype.val f) sorry\n\n/-- The set of partial isomorphisms defined at `b : β`, together with a proof that any\n    partial isomorphism can be extended to include `b`. We prove this by symmetry. -/\ndef defined_at_right (α : Type u_1) {β : Type u_2} [linear_order α] [linear_order β]\n    [densely_ordered α] [no_bot_order α] [no_top_order α] [Nonempty α] (b : β) :\n    cofinal (partial_iso α β) :=\n  cofinal.mk (fun (f : partial_iso α β) => ∃ (a : α), (a, b) ∈ subtype.val f) sorry\n\n/-- Given an ideal which intersects `defined_at_left β a`, pick `b : β` such that\n    some partial function in the ideal maps `a` to `b`. -/\ndef fun_of_ideal {α : Type u_1} {β : Type u_2} [linear_order α] [linear_order β] [densely_ordered β]\n    [no_bot_order β] [no_top_order β] [Nonempty β] (a : α) (I : ideal (partial_iso α β)) :\n    (∃ (f : partial_iso α β), f ∈ defined_at_left β a ∧ f ∈ I) →\n        Subtype\n          fun (b : β) =>\n            ∃ (f :\n              Subtype\n                fun (f : finset (α × β)) =>\n                  ∀ (p q : α × β),\n                    p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n              ∃ (H : f ∈ I), (a, b) ∈ subtype.val f :=\n  (classical.indefinite_description\n      fun (x : β) =>\n        ∃ (f :\n          Subtype\n            fun (f : finset (α × β)) =>\n              ∀ (p q : α × β),\n                p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n          ∃ (H : f ∈ I), (a, x) ∈ subtype.val f) ∘\n    sorry\n\n/-- Given an ideal which intersects `defined_at_right α b`, pick `a : α` such that\n    some partial function in the ideal maps `a` to `b`. -/\ndef inv_of_ideal {α : Type u_1} {β : Type u_2} [linear_order α] [linear_order β] [densely_ordered α]\n    [no_bot_order α] [no_top_order α] [Nonempty α] (b : β) (I : ideal (partial_iso α β)) :\n    (∃ (f : partial_iso α β), f ∈ defined_at_right α b ∧ f ∈ I) →\n        Subtype\n          fun (a : α) =>\n            ∃ (f :\n              Subtype\n                fun (f : finset (α × β)) =>\n                  ∀ (p q : α × β),\n                    p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n              ∃ (H : f ∈ I), (a, b) ∈ subtype.val f :=\n  (classical.indefinite_description\n      fun (x : α) =>\n        ∃ (f :\n          Subtype\n            fun (f : finset (α × β)) =>\n              ∀ (p q : α × β),\n                p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n          ∃ (H : f ∈ I), (x, b) ∈ subtype.val f) ∘\n    sorry\n\nend partial_iso\n\n\n/-- Any countable linear order embeds in any nonempty dense linear order without endpoints. -/\ndef embedding_from_countable_to_dense (α : Type u_1) (β : Type u_2) [linear_order α]\n    [linear_order β] [encodable α] [densely_ordered β] [no_bot_order β] [no_top_order β]\n    [Nonempty β] : α ↪o β :=\n  let our_ideal : ideal (partial_iso α β) :=\n    ideal_of_cofinals Inhabited.default (partial_iso.defined_at_left β);\n  let F :\n    (a : α) →\n      Subtype\n        fun (b : β) =>\n          ∃ (f :\n            Subtype\n              fun (f : finset (α × β)) =>\n                ∀ (p q : α × β),\n                  p ∈ f → q ∈ f → cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n            ∃ (H : f ∈ our_ideal), (a, b) ∈ subtype.val f :=\n    fun (a : α) => partial_iso.fun_of_ideal a our_ideal sorry;\n  order_embedding.of_strict_mono (fun (a : α) => subtype.val (F a)) sorry\n\n/-- Any two countable dense, nonempty linear orders without endpoints are order isomorphic. -/\ndef iso_of_countable_dense (α : Type u_1) (β : Type u_2) [linear_order α] [linear_order β]\n    [encodable α] [densely_ordered α] [no_bot_order α] [no_top_order α] [Nonempty α] [encodable β]\n    [densely_ordered β] [no_bot_order β] [no_top_order β] [Nonempty β] : α ≃o β :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/countable_dense_linear_order_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.32652163881796914}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.dense_embedding\nimport Mathlib.PostPort\n\nuniverses u_5 u_6 l u_1 u_2 u_3 u_4 u v \n\nnamespace Mathlib\n\n/-- Homeomorphism between `α` and `β`, also called topological isomorphism -/\nstructure homeomorph (α : Type u_5) (β : Type u_6) [topological_space α] [topological_space β]\n    extends α ≃ β where\n  continuous_to_fun :\n    autoParam (continuous (equiv.to_fun _to_equiv))\n      (Lean.Syntax.ident Lean.SourceInfo.none\n        (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n        (Lean.Name.mkStr\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n            \"interactive\")\n          \"continuity'\")\n        [])\n  continuous_inv_fun :\n    autoParam (continuous (equiv.inv_fun _to_equiv))\n      (Lean.Syntax.ident Lean.SourceInfo.none\n        (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n        (Lean.Name.mkStr\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n            \"interactive\")\n          \"continuity'\")\n        [])\n\ninfixl:25 \" ≃ₜ \" => Mathlib.homeomorph\n\nnamespace homeomorph\n\n\nprotected instance has_coe_to_fun {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] : has_coe_to_fun (α ≃ₜ β) :=\n  has_coe_to_fun.mk (fun (_x : α ≃ₜ β) => α → β) fun (e : α ≃ₜ β) => ⇑(to_equiv e)\n\n@[simp] theorem homeomorph_mk_coe {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (a : α ≃ β)\n    (b :\n      autoParam (continuous (equiv.to_fun a))\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr\n              (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n              \"interactive\")\n            \"continuity'\")\n          []))\n    (c :\n      autoParam (continuous (equiv.inv_fun a))\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr\n              (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n              \"interactive\")\n            \"continuity'\")\n          [])) :\n    ⇑(mk a) = ⇑a :=\n  rfl\n\ntheorem coe_eq_to_equiv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) (a : α) : coe_fn h a = coe_fn (to_equiv h) a :=\n  rfl\n\ntheorem to_equiv_injective {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] : function.injective to_equiv :=\n  sorry\n\ntheorem ext {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {h : α ≃ₜ β}\n    {h' : α ≃ₜ β} (H : ∀ (x : α), coe_fn h x = coe_fn h' x) : h = h' :=\n  to_equiv_injective (equiv.ext H)\n\n/-- Identity map as a homeomorphism. -/\nprotected def refl (α : Type u_1) [topological_space α] : α ≃ₜ α :=\n  mk (equiv.mk (equiv.to_fun (equiv.refl α)) (equiv.inv_fun (equiv.refl α)) sorry sorry)\n\n/-- Composition of two homeomorphisms. -/\nprotected def trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α]\n    [topological_space β] [topological_space γ] (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) : α ≃ₜ γ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.trans (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.trans (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- Inverse of a homeomorphism. -/\nprotected def symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : β ≃ₜ α :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.symm (to_equiv h))) (equiv.inv_fun (equiv.symm (to_equiv h)))\n      sorry sorry)\n\n@[simp] theorem homeomorph_mk_coe_symm {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (a : α ≃ β)\n    (b :\n      autoParam (continuous (equiv.to_fun a))\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr\n              (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n              \"interactive\")\n            \"continuity'\")\n          []))\n    (c :\n      autoParam (continuous (equiv.inv_fun a))\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.tactic.interactive.continuity'\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr\n              (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n              \"interactive\")\n            \"continuity'\")\n          [])) :\n    ⇑(homeomorph.symm (mk a)) = ⇑(equiv.symm a) :=\n  rfl\n\nprotected theorem continuous {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : continuous ⇑h :=\n  continuous_to_fun h\n\n@[simp] theorem apply_symm_apply {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (x : β) : coe_fn h (coe_fn (homeomorph.symm h) x) = x :=\n  equiv.apply_symm_apply (to_equiv h) x\n\n@[simp] theorem symm_apply_apply {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (x : α) : coe_fn (homeomorph.symm h) (coe_fn h x) = x :=\n  equiv.symm_apply_apply (to_equiv h) x\n\nprotected theorem bijective {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : function.bijective ⇑h :=\n  equiv.bijective (to_equiv h)\n\nprotected theorem injective {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : function.injective ⇑h :=\n  equiv.injective (to_equiv h)\n\nprotected theorem surjective {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : function.surjective ⇑h :=\n  equiv.surjective (to_equiv h)\n\n/-- Change the homeomorphism `f` to make the inverse function definitionally equal to `g`. -/\ndef change_inv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (f : α ≃ₜ β) (g : β → α) (hg : function.right_inverse g ⇑f) : α ≃ₜ β :=\n  (fun (this : g = ⇑(homeomorph.symm f)) => mk (equiv.mk (⇑f) g sorry sorry)) sorry\n\n@[simp] theorem symm_comp_self {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : ⇑(homeomorph.symm h) ∘ ⇑h = id :=\n  funext (symm_apply_apply h)\n\n@[simp] theorem self_comp_symm {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : ⇑h ∘ ⇑(homeomorph.symm h) = id :=\n  funext (apply_symm_apply h)\n\n@[simp] theorem range_coe {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : set.range ⇑h = set.univ :=\n  function.surjective.range_eq (homeomorph.surjective h)\n\ntheorem image_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : set.image ⇑(homeomorph.symm h) = set.preimage ⇑h :=\n  funext (equiv.image_eq_preimage (to_equiv (homeomorph.symm h)))\n\ntheorem preimage_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : set.preimage ⇑(homeomorph.symm h) = set.image ⇑h :=\n  Eq.symm (funext (equiv.image_eq_preimage (to_equiv h)))\n\n@[simp] theorem image_preimage {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (s : set β) : ⇑h '' (⇑h ⁻¹' s) = s :=\n  equiv.image_preimage (to_equiv h) s\n\n@[simp] theorem preimage_image {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (s : set α) : ⇑h ⁻¹' (⇑h '' s) = s :=\n  equiv.preimage_image (to_equiv h) s\n\nprotected theorem inducing {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : inducing ⇑h :=\n  sorry\n\ntheorem induced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : topological_space.induced (⇑h) _inst_2 = _inst_1 :=\n  Eq.symm (inducing.induced (homeomorph.inducing h))\n\nprotected theorem quotient_map {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : quotient_map ⇑h :=\n  sorry\n\ntheorem coinduced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) : topological_space.coinduced (⇑h) _inst_1 = _inst_2 :=\n  Eq.symm (and.right (homeomorph.quotient_map h))\n\nprotected theorem embedding {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : embedding ⇑h :=\n  embedding.mk (homeomorph.inducing h) (equiv.injective (to_equiv h))\n\ntheorem compact_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {s : set α} (h : α ≃ₜ β) : is_compact (⇑h '' s) ↔ is_compact s :=\n  iff.symm (embedding.compact_iff_compact_image (homeomorph.embedding h))\n\ntheorem compact_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {s : set β} (h : α ≃ₜ β) : is_compact (⇑h ⁻¹' s) ↔ is_compact s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_compact (⇑h ⁻¹' s) ↔ is_compact s)) (Eq.symm (image_symm h))))\n    (compact_image (homeomorph.symm h))\n\nprotected theorem dense_embedding {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : dense_embedding ⇑h :=\n  dense_embedding.mk\n    (dense_inducing.mk (inducing.mk (Eq.symm (induced_eq h)))\n      (function.surjective.dense_range (homeomorph.surjective h)))\n    (homeomorph.injective h)\n\n@[simp] theorem is_open_preimage {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) {s : set β} : is_open (⇑h ⁻¹' s) ↔ is_open s :=\n  quotient_map.is_open_preimage (homeomorph.quotient_map h)\n\n@[simp] theorem is_open_image {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) {s : set α} : is_open (⇑h '' s) ↔ is_open s :=\n  sorry\n\n@[simp] theorem is_closed_preimage {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) {s : set β} : is_closed (⇑h ⁻¹' s) ↔ is_closed s :=\n  sorry\n\n@[simp] theorem is_closed_image {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) {s : set α} : is_closed (⇑h '' s) ↔ is_closed s :=\n  sorry\n\ntheorem preimage_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) (s : set β) : ⇑h ⁻¹' closure s = closure (⇑h ⁻¹' s) :=\n  sorry\n\ntheorem image_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) (s : set α) : ⇑h '' closure s = closure (⇑h '' s) :=\n  sorry\n\nprotected theorem is_open_map {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : is_open_map ⇑h :=\n  fun (s : set α) => iff.mpr (is_open_image h)\n\nprotected theorem is_closed_map {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : is_closed_map ⇑h :=\n  fun (s : set α) => iff.mpr (is_closed_image h)\n\nprotected theorem closed_embedding {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) : closed_embedding ⇑h :=\n  closed_embedding_of_embedding_closed (homeomorph.embedding h) (homeomorph.is_closed_map h)\n\n@[simp] theorem map_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (x : α) : filter.map (⇑h) (nhds x) = nhds (coe_fn h x) :=\n  sorry\n\n@[simp] theorem comap_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (h : α ≃ₜ β) (y : β) :\n    filter.comap (⇑h) (nhds y) = nhds (coe_fn (homeomorph.symm h) y) :=\n  sorry\n\ntheorem nhds_eq_comap {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (h : α ≃ₜ β) (x : α) : nhds x = filter.comap (⇑h) (nhds (coe_fn h x)) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (nhds x = filter.comap (⇑h) (nhds (coe_fn h x))))\n        (comap_nhds_eq h (coe_fn h x))))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (nhds x = nhds (coe_fn (homeomorph.symm h) (coe_fn h x))))\n          (symm_apply_apply h x)))\n      (Eq.refl (nhds x)))\n\n/-- If an bijective map `e : α ≃ β` is continuous and open, then it is a homeomorphism. -/\ndef homeomorph_of_continuous_open {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] (e : α ≃ β) (h₁ : continuous ⇑e) (h₂ : is_open_map ⇑e) : α ≃ₜ β :=\n  mk (equiv.mk (equiv.to_fun e) (equiv.inv_fun e) (equiv.left_inv e) (equiv.right_inv e))\n\n@[simp] theorem comp_continuous_on_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3}\n    [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) (f : γ → α)\n    (s : set γ) : continuous_on (⇑h ∘ f) s ↔ continuous_on f s :=\n  iff.symm (inducing.continuous_on_iff (homeomorph.inducing h))\n\n@[simp] theorem comp_continuous_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3}\n    [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : γ → α} :\n    continuous (⇑h ∘ f) ↔ continuous f :=\n  iff.symm (inducing.continuous_iff (homeomorph.inducing h))\n\n@[simp] theorem comp_continuous_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3}\n    [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : β → γ} :\n    continuous (f ∘ ⇑h) ↔ continuous f :=\n  iff.symm (quotient_map.continuous_iff (homeomorph.quotient_map h))\n\n/-- If two sets are equal, then they are homeomorphic. -/\ndef set_congr {α : Type u_1} [topological_space α] {s : set α} {t : set α} (h : s = t) : ↥s ≃ₜ ↥t :=\n  mk (equiv.mk (equiv.to_fun (equiv.set_congr h)) (equiv.inv_fun (equiv.set_congr h)) sorry sorry)\n\n/-- Sum of two homeomorphisms. -/\ndef sum_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α]\n    [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :\n    α ⊕ γ ≃ₜ β ⊕ δ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.sum_congr (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.sum_congr (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- Product of two homeomorphisms. -/\ndef prod_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α]\n    [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :\n    α × γ ≃ₜ β × δ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.prod_congr (to_equiv h₁) (to_equiv h₂)))\n      (equiv.inv_fun (equiv.prod_congr (to_equiv h₁) (to_equiv h₂))) sorry sorry)\n\n/-- `α × β` is homeomorphic to `β × α`. -/\ndef prod_comm (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] :\n    α × β ≃ₜ β × α :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.prod_comm α β)) (equiv.inv_fun (equiv.prod_comm α β)) sorry\n      sorry)\n\n/-- `(α × β) × γ` is homeomorphic to `α × (β × γ)`. -/\ndef prod_assoc (α : Type u_1) (β : Type u_2) (γ : Type u_3) [topological_space α]\n    [topological_space β] [topological_space γ] : (α × β) × γ ≃ₜ α × β × γ :=\n  mk\n    (equiv.mk (equiv.to_fun (equiv.prod_assoc α β γ)) (equiv.inv_fun (equiv.prod_assoc α β γ)) sorry\n      sorry)\n\n/-- `ulift α` is homeomorphic to `α`. -/\ndef ulift {α : Type u} [topological_space α] : ulift α ≃ₜ α :=\n  mk (equiv.mk (equiv.to_fun equiv.ulift) (equiv.inv_fun equiv.ulift) sorry sorry)\n\n/-- `(α ⊕ β) × γ` is homeomorphic to `α × γ ⊕ β × γ`. -/\ndef sum_prod_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α]\n    [topological_space β] [topological_space γ] : (α ⊕ β) × γ ≃ₜ α × γ ⊕ β × γ :=\n  homeomorph.symm\n    (homeomorph_of_continuous_open (equiv.symm (equiv.sum_prod_distrib α β γ)) sorry sorry)\n\n/-- `α × (β ⊕ γ)` is homeomorphic to `α × β ⊕ α × γ`. -/\ndef prod_sum_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α]\n    [topological_space β] [topological_space γ] : α × (β ⊕ γ) ≃ₜ α × β ⊕ α × γ :=\n  homeomorph.trans (prod_comm α (β ⊕ γ))\n    (homeomorph.trans sum_prod_distrib (sum_congr (prod_comm β α) (prod_comm γ α)))\n\n/-- `(Σ i, σ i) × β` is homeomorphic to `Σ i, (σ i × β)`. -/\ndef sigma_prod_distrib {β : Type u_2} [topological_space β] {ι : Type u_5} {σ : ι → Type u_6}\n    [(i : ι) → topological_space (σ i)] :\n    (sigma fun (i : ι) => σ i) × β ≃ₜ sigma fun (i : ι) => σ i × β :=\n  homeomorph.symm\n    (homeomorph_of_continuous_open (equiv.symm (equiv.sigma_prod_distrib σ β)) sorry sorry)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/homeomorph_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521105, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.32642951928381064}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) (hα : ¬ is_rat α) : ∀ (ε : ℝ), ε > 0 → ∃ (i : ℤ), ∀ (y : ℝ), y ∈ Ioo 0 1 → |y - (i : ℝ) * α| < ε :=\nbegin\n  assume (ε : ℝ) (hε : ε > 0),\n  have h1 : ∀ (i j : ℤ), i ≠ j → (i : ℝ) * α ≠ (j : ℝ) * α, from sorry,\n  have h2 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h3 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h4 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h5 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h6 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h7 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h8 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h9 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h10 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h11 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h12 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h13 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h14 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h15 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h16 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h17 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h18 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h19 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h20 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h21 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h22 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h23 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h24 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h25 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h26 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h27 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α - (floor (i : ℝ) * α) = (j : ℝ) * α - (floor (j : ℝ) * α)), from sorry,\n  have h28 :\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) : \nlet irr : ℝ → Prop := λ (x : ℝ), ¬ ∃ (a b : ℕ), a ≠ 0 ∧ b ≠ 0 ∧ x = a/b in\nlet frac_part : ℝ → ℝ := λ (x : ℝ), x - floor x in\nlet S : set ℝ := {frac_part (i * α) | i : ℤ} in\nirr α → dense S :=\nbegin\n  assume irr (h1 : irr α) (h2 : dense S),\n  have h3 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h4 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h5 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h6 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h7 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h8 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h9 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h10 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h11 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h12 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h13 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h14 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h15 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h16 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h17 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h18 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h19 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h20 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h21 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h22 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h23 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h24 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h25 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h26 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h27 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h28 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h29 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h30 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h31 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h32 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h33 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h34 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h35 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h36 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h37 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h38 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h39 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h40 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h41 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h42 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h43 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\n  have h44 : ∀ (i j : ℤ), i ≠ j → frac_part (i * α) ≠ frac_part (j * α), from sorry,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit : sorry :=\nbegin\n  sorry,\nend\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit : ∀ α : ℝ, α ∉ set.range (λ x : ℤ, x⁻¹), ∀ y : ℝ, ∃ x : ℤ, x⁻¹ < y :=\nbegin\n  assume (α : ℝ) (h1 : α ∉ set.range (λ x : ℤ, x⁻¹)),\n  assume (y : ℝ),\n  have h2 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h3 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h4 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h5 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h6 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h7 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h8 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h9 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h10 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h11 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h12 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h13 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h14 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h15 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h16 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h17 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h18 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h19 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h20 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h21 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h22 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h23 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h24 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h25 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h26 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h27 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h28 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h29 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h30 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h31 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h32 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h33 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h34 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h35 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h36 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h37 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h38 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h39 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h40 : ∀ i j : ℤ, i ≠ j → (i : ℝ)⁻¹ ≠ (j : ℝ)⁻¹, from sorry,\n  have h41 : ∀ i j : ℤ, i ≠ j → (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : ∃ S : set ℝ, (∀ i : ℤ, S i = {i * α}) ∧ (∀ y ∈ set.Icc 0 1, ∃ x ∈ S, |y - x| < 1) :=\nbegin\n  have h1 : ∀ i j : ℤ, i ≠ j → {i * α} ≠ {j * α}, from sorry,\n  have h2 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h3 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h4 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h5 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h6 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h7 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h8 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h9 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h10 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h11 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h12 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h13 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h14 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h15 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h16 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h17 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h18 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h19 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h20 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h21 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h22 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h23 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h24 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋ → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α⌋, from sorry,\n  have h25 : ∀ i j : ℤ, i ≠ j → i * α - ⌊i * α⌋ ≠ j * α - ⌊j * α\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : ∀ ε > 0, ∃ (n : ℤ), ∀ (m : ℤ), |α * m - n| < ε :=\nbegin\n  assume ε hε,\n  have h1 : ∀ (i j : ℤ), i ≠ j → (↑i * α) % 1 ≠ (↑j * α) % 1, from sorry,\n  have h2 : ∀ (i : ℤ), (↑i * α) % 1 ∈ set.Ico 0 1, from sorry,\n  let S := {(↑i * α) % 1 | i : ℤ},\n  have h3 : set.finite S = ff, from sorry,\n  have h4 : ∃ (l : ℝ), l ∈ set.Ico 0 1 ∧ ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ≠ l ∧ x ∈ set.Ico l (l + ε), from sorry,\n  have h5 : ∃ (l : ℝ), l ∈ set.Ico 0 1 ∧ ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ≠ l ∧ x ∈ set.Ico l (l + ε) ∧ l = 0, from sorry,\n  cases h5 with l h5,\n  have h6 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ≠ l ∧ x ∈ set.Ico l (l + ε) ∧ l = 0, from sorry,\n  have h7 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ≠ 0 ∧ x ∈ set.Ico 0 (0 + ε) ∧ 0 = 0, from sorry,\n  have h8 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ≠ 0 ∧ x ∈ set.Ico 0 ε ∧ 0 = 0, from sorry,\n  have h9 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε, from sorry,\n  have h10 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε, from sorry,\n  have h11 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0, from sorry,\n  have h12 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x, from sorry,\n  have h13 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0, from sorry,\n  have h14 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0, from sorry,\n  have h15 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε, from sorry,\n  have h16 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x, from sorry,\n  have h17 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x ∧ n > 0, from sorry,\n  have h18 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (n : ℤ), n * x ≤ ε ∧ n * x > 0, from sorry,\n  have h19 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (n : ℤ), n * x ≤ ε ∧ n * x > 0 ∧ ∃ (n : ℤ), |n * x - ε| < ε, from sorry,\n  have h20 : ∀ (ε : ℝ), ε > 0 → ∃ (x : ℝ), x ∈ S ∧ x ∈ set.Ico 0 ε ∧ x ≤ ε ∧ x ≠ 0 ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (m : ℤ), m * x ≤ ε ∧ m * x > 0 ∧ ∃ (k : ℤ), |k * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x ∧ n > 0 ∧ ∃ (n : ℤ), n * x ≤ ε ∧ n * x > 0 ∧ ∃ (n : ℤ), |n * x - ε| < ε ∧ ∃ (n : ℤ), x = n * x, from sorry,\n  have h21 : ∀ (ε : ℝ), ε > 0 →\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from sorry,\n  \n  assume (h7 : ε > 0),\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := sorry,\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from sorry,\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from sorry,\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-4_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7520125848754472, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.32635952313521266}}
{"text": "import to_mathlib.data.set.prod\nimport to_mathlib.data.set.lattice\nimport to_mathlib.data.nat.basic\nimport to_mathlib.topology.constructions\nimport to_mathlib.topology.germ\nimport to_mathlib.topology.misc\n\nimport indexing\nimport notations\n\nopen set filter prod topological_space\nopen_locale topology unit_interval\n\n/-!\nNotes by Patrick:\n\nThe goal of this file is to explore how to prove `exists_surrounding_loops` (or rather its version\nwith `C = U = univ` which is the only needed case) in a way that uncouples the general\ntopological argument from the things specific to loops. The general lemma is meant to\nbe something like `inductive_construction'` below.\n-/\n\n\nsection inductive_construction\n/-!\nNotes by Patrick:\n\nIn this section, I took lemmas that used to exist when I worked on the inductive construction\nrefactor. In particular there is the lemma which can't quite be used to prove\n`inductive_htpy_construction`, namely `inductive_construction`.\n\nIn that lemma, the covering is fixed. Lemma `inductive_construction'` combines this with an argument\nusing local existence and exhaustions. A technical intermediate statement is\n`inductive_construction''`.\n-/\n\nlemma index_type.tendsto_coe_at_top (N : ℕ) : tendsto (coe : ℕ → index_type N) at_top at_top :=\ntendsto_at_top_at_top.mpr\n  (λ i, ⟨indexing.to_nat i, λ n hn,(indexing.from_to i) ▸ indexing.coe_mono hn⟩)\n\nlemma locally_finite.exists_forall_eventually_of_indexing\n  {α X ι : Type*} [topological_space X] [linear_order ι] [indexing ι] {f : ℕ → X → α}\n  {V : ι → set X} (hV : locally_finite V)\n  (h : ∀ n : ℕ, ∀ x ∉ V ((n + 1) : ℕ), f (n + 1) x = f n x)\n  (h' : ∀ n : ℕ, ((n+1 : ℕ) : ι) = n → f (n + 1) = f n) :\n  ∃ (F : X → α), ∀ (x : X), ∀ᶠ (n : ℕ) in filter.at_top, f n =ᶠ[𝓝 x] F :=\nbegin\n  let π :  ℕ → ι := indexing.from_nat,\n  choose U hUx hU using hV,\n  choose i₀ hi₀ using λ x, (hU x).bdd_above,\n  let n₀ : X → ℕ := indexing.to_nat ∘ i₀,\n  have key : ∀ {x} {n}, n ≥ n₀ x → ∀ {y}, y ∈ U x → f n y = f (n₀ x) y,\n  { intros x n hn,\n    rcases le_iff_exists_add.mp hn with ⟨k, rfl⟩, clear hn,\n    intros y hy,\n    induction k with k hk,\n    { simp },\n    { rw ← hk, clear hk,\n      have : ∀ n, π n < π (n+1) ∨ π n = π (n+1),\n      exact λ n, lt_or_eq_of_le (indexing.mono_from n.le_succ),\n      rcases this (n₀ x + k) with H | H ; clear this,\n      { have ineq : π (n₀ x + k + 1) > i₀ x,\n        { suffices : i₀ x ≤ π (n₀ x + k), from lt_of_le_of_lt this H,\n          rw ← indexing.from_to (i₀ x),\n          exact indexing.mono_from le_self_add },\n        apply h,\n        rintro (hy' : y ∈ V (π (n₀ x + k + 1))),\n        have := hi₀ x ⟨y, ⟨hy', hy⟩⟩, clear hy hy',\n        exact lt_irrefl _ (lt_of_le_of_lt this ineq) },\n      { erw [← (h' _ H.symm)],\n        refl } } },\n  refine ⟨λ x, f (n₀ x) x, λ x, _⟩,\n  change ∀ᶠ (n : ℕ) in at_top, f n =ᶠ[𝓝 x] λ (y : X), f (n₀ y) y,\n  apply (eventually_gt_at_top (n₀ x)).mono (λ n hn, _),\n  apply mem_of_superset (hUx x) (λ y hy, _),\n  change f n y = f (n₀ y) y,\n  calc f n y = f (n₀ x) y : key hn.le hy\n  ... = f (max (n₀ x) (n₀ y)) y : (key (le_max_left _ _) hy).symm\n  ... = f (n₀ y) y : key (le_max_right _ _) (mem_of_mem_nhds $ hUx y)\nend\n\nlemma inductive_construction_alt {X Y : Type*} [topological_space X]\n  {N : ℕ} {U K : index_type N → set X}\n  (P₀ : Π x : X, germ (𝓝 x) Y → Prop) (P₁ : Π i : index_type N, Π x : X, germ (𝓝 x) Y → Prop)\n  (U_fin : locally_finite U)\n  (init : ∃ f : X → Y, ∀ x, P₀ x f)\n  (ind : ∀ (i : index_type N) (f : X → Y), (∀ x, P₀ x f) → (∀ j < i, ∀ᶠ x near K j, P₁ j x f) →\n    ∃ f' : X → Y, (∀ x, P₀ x f') ∧ (∀ j ≤ i, ∀ᶠ x near K j, P₁ j x f') ∧ ∀ x ∉ U i, f' x = f x) :\n    ∃ f : X → Y, (∀ x, P₀ x f) ∧ ∀ j, ∀ᶠ x near K j, P₁ j x f :=\nbegin\n  let P : ℕ → (X → Y) → Prop :=\n    λ n f, (∀ x, P₀ x f) ∧ ∀ j : index_type N, j ≤ n → ∀ᶠ x near K j, P₁ j x f,\n  let Q : ℕ → (X → Y) → (X → Y) → Prop :=\n    λ n f f', ((((n+1:ℕ) : index_type N) = n) → f' = f) ∧ ∀ x ∉ U (n + 1 : ℕ), f' x = f x,\n  obtain ⟨f, hf⟩ : ∃ f : ℕ → X → Y, ∀ n, P n (f n) ∧ Q n (f n) (f $ n + 1),\n  { apply exists_by_induction',\n    { dsimp [P],\n      cases init with f₀ hf₀,\n      rcases ind 0 f₀ hf₀ _ with ⟨f', h₀f', h₁f', hf'⟩,\n      use [f', h₀f', h₁f'],\n      simp [index_type.not_lt_zero] },\n    { rintros n f ⟨h₀f, h₁f⟩,\n      rcases index_type.lt_or_eq_succ N n with hn | hn,\n      { simp_rw index_type.le_or_lt_succ hn at h₁f,\n        rcases ind (n+1 : ℕ) f h₀f h₁f with ⟨f', h₀f', h₁f', hf'⟩,\n        exact ⟨f', ⟨h₀f', h₁f'⟩, ⟨λ hn', (hn.ne hn'.symm).elim, hf'⟩⟩ },\n      { simp only [hn] at h₁f,\n        exact ⟨f, ⟨h₀f, h₁f⟩, λ hn, rfl, λ x hx, rfl⟩ } } },\n  dsimp only [P, Q] at hf,\n  simp only [forall_and_distrib] at hf,\n  rcases hf with ⟨⟨h₀f, h₁f⟩, hf, hf'⟩,\n  rcases U_fin.exists_forall_eventually_of_indexing hf' hf with ⟨F, hF⟩,\n  refine ⟨F, λ x, _, λ j, _⟩,\n  { rcases (hF x).exists with ⟨n₀, hn₀⟩,\n    simp only [germ.coe_eq.mpr hn₀.symm, h₀f n₀ x] },\n  apply eventually_nhds_set_iff.mpr,\n  intros x hx,\n  rcases ((hF x).and $ (filter.tendsto_at_top.mp (index_type.tendsto_coe_at_top N) j)).exists\n    with ⟨n₀, hn₀, hn₀'⟩,\n  apply ((eventually_nhds_set_iff.mp (h₁f _ _ hn₀') x hx).and $\n         eventually_eventually_eq_nhds.mpr hn₀).mono,\n  rintros y ⟨hy, hy'⟩,\n  rwa germ.coe_eq.mpr hy'.symm\nend\n\nlemma inductive_construction {X Y : Type*} [topological_space X]\n  {N : ℕ} {U K : index_type N → set X}\n  (P₀ P₁ : Π x : X, germ (𝓝 x) Y → Prop)\n  (U_fin : locally_finite U) (K_cover : (⋃ i, K i) = univ)\n  (init : ∃ f : X → Y, ∀ x, P₀ x f)\n  (ind : ∀ (i : index_type N) (f : X → Y), (∀ x, P₀ x f) → (∀ᶠ x near ⋃ j < i, K j, P₁ x f) →\n    ∃ f' : X → Y, (∀ x, P₀ x f') ∧ (∀ᶠ x near ⋃ j ≤ i, K j, P₁ x f') ∧ ∀ x ∉ U i, f' x = f x) :\n    ∃ f : X → Y, ∀ x, P₀ x f ∧ P₁ x f :=\nbegin\n  rcases inductive_construction_alt P₀ (λ j, P₁) U_fin init\n    (by simpa only [eventually_nhds_set_Union₂] using ind) with ⟨f, h₀f, h₁f⟩,\n  refine ⟨f, λ x, ⟨h₀f x, _⟩⟩,\n  obtain ⟨j, hj⟩ : ∃ j, x ∈ K j, by simpa using (by simp [K_cover] : x ∈ ⋃ j, K j),\n  exact (h₁f j).on_set _ hj\nend\n\n/-- We are given a suitably nice topological space `X` and three local constraints `P₀`,`P₀'` and\n`P₁` on maps from `X` to some type `Y`. All maps entering the discussion are required to statisfy\n`P₀` everywhere. The goal is to turn a map `f₀` satisfying `P₁` near a compact set `K` into\none satisfying everywhere without changing `f₀` near `K`. The assumptions are:\n* For every `x` in `X` there is a map which satisfies `P₁` near `x`\n* One can patch two maps `f₁ f₂` satisfying `P₁` on open sets `U₁` and `U₂` respectively\n  and such that `f₁` satisfies `P₀'` everywhere into a map satisfying `P₁` on `K₁ ∪ K₂` for any\n  compact sets `Kᵢ ⊆ Uᵢ` and `P₀'` everywhere. -/\nlemma inductive_construction'' {X Y : Type*} [emetric_space X] [locally_compact_space X]\n  [second_countable_topology X]\n  (P₀ P₀' P₁ : Π x : X, germ (𝓝 x) Y → Prop)\n  {f₀ : X → Y} (hP₀f₀ : ∀ x, P₀ x f₀ ∧ P₀' x f₀ )\n  (loc : ∀ x, ∃ f : X → Y, (∀ x, P₀ x f) ∧ ∀ᶠ x' in 𝓝 x, P₁ x' f)\n  (ind : ∀ {U₁ U₂ K₁ K₂ : set X} {f₁ f₂ : X → Y}, is_open U₁ → is_open U₂ →\n     is_closed K₁ → is_closed K₂ → K₁ ⊆ U₁ → K₂ ⊆ U₂ → (∀ x, P₀ x f₁ ∧ P₀' x f₁) → (∀ x, P₀ x f₂) →\n     (∀ x ∈ U₁, P₁ x f₁) → (∀ x ∈ U₂, P₁ x f₂) →\n     ∃ f : X → Y, (∀ x, P₀ x f ∧ P₀' x f ) ∧ (∀ᶠ x near K₁ ∪ K₂, P₁ x f) ∧\n                  (∀ᶠ x near K₁ ∪ U₂ᶜ, f x = f₁ x)) :\n    ∃ f : X → Y, ∀ x, P₀ x f ∧ P₀' x f ∧ P₁ x f :=\nbegin\n  let P : set X → Prop := λ U, ∃ f : X → Y, (∀ x, P₀ x f) ∧ (∀ x ∈ U, P₁ x f),\n  have hP₁ : antitone P,\n  { rintros U V hUV ⟨f, h, h'⟩,\n    exact ⟨f, h, λ x hx, h' x (hUV hx)⟩ },\n  have hP₂ : P ∅, from ⟨f₀, λ x, (hP₀f₀ x).1, λ x h, h.elim⟩,\n  have hP₃ : ∀ (x : X), x ∈ univ → (∃ (V : set X) (H : V ∈ 𝓝 x), P V),\n  { rintros x -,\n    rcases loc x with ⟨f, h₀f, h₁f⟩,\n    exact ⟨_, h₁f, f, h₀f, λ x, id⟩ },\n  rcases exists_locally_finite_subcover_of_locally is_closed_univ hP₁ hP₂ hP₃ with\n    ⟨K, (U : index_type 0 →set X) , K_cpct, U_op, hU, hKU, U_loc, hK⟩,\n  simp_rw ← and_assoc,\n  apply inductive_construction (λ x φ, P₀ x φ ∧ P₀' x φ) P₁ U_loc (eq_univ_of_univ_subset hK)\n    ⟨f₀, hP₀f₀⟩,\n  rintros (n : ℕ) f h₀f (h₁f : ∀ᶠ x near ⋃ j < n, K j, P₁ x f),\n  have cpct : is_closed ⋃ j < n, K j,\n  { rw show (⋃ j < n, K j) = ⋃ j ∈ finset.range n, K j, by simp only [finset.mem_range],\n    apply (finset.range n).is_closed_bUnion _ (λ j _, (K_cpct j).is_closed) },\n  rcases hU n with ⟨f', h₀f', h₁f'⟩,\n  rcases mem_nhds_set_iff_exists.mp h₁f with ⟨V, V_op, hKV, h₁V⟩,\n  rcases ind V_op (U_op n) cpct (K_cpct n).is_closed\n    hKV (hKU n) h₀f h₀f' h₁V h₁f' with ⟨F, h₀F, h₁F, hF⟩,\n  simp_rw ← bUnion_le at h₁F,\n  exact ⟨F, h₀F, h₁F, λ x hx, hF.on_set x (or.inr hx)⟩\nend\n\n/-- We are given a suitably nice topological space `X` and two local constraints `P₀` and `P₁`\non maps from `X` to some type `Y`. All maps entering the discussion are required to statisfy `P₀`\neverywhere. The goal is to turn a map `f₀` satisfying `P₁` near a compact set `K` into\none satisfying everywhere without changing `f₀` near `K`. The assumptions are:\n* For every `x` in `X` there is a map which satisfies `P₁` near `x`\n* One can patch two maps `f₁ f₂` satisfying `P₁` on open sets `U₁` and `U₂` respectively\n  into a map satisfying `P₁` on `K₁ ∪ K₂` for any compact sets `Kᵢ ⊆ Uᵢ`.\nThis is deduced this version from the version where `K` is empty but adding some `P'₀`, see\n`inductive_construction''`. -/\nlemma inductive_construction' {X Y : Type*} [emetric_space X] [locally_compact_space X]\n  [second_countable_topology X]\n  (P₀ P₁ : Π x : X, germ (𝓝 x) Y → Prop)\n  {K : set X} (hK : is_closed K)\n  {f₀ : X → Y} (hP₀f₀ : ∀ x, P₀ x f₀) (hP₁f₀ : ∀ᶠ x near K, P₁ x f₀)\n  (loc : ∀ x, ∃ f : X → Y, (∀ x, P₀ x f) ∧ ∀ᶠ x' in 𝓝 x, P₁ x' f)\n  (ind : ∀ {U₁ U₂ K₁ K₂ : set X} {f₁ f₂ : X → Y}, is_open U₁ → is_open U₂ →\n     is_closed K₁ → is_closed K₂ → K₁ ⊆ U₁ → K₂ ⊆ U₂ → (∀ x, P₀ x f₁) → (∀ x, P₀ x f₂) →\n     (∀ x ∈ U₁, P₁ x f₁) → (∀ x ∈ U₂, P₁ x f₂) →\n     ∃ f : X → Y, (∀ x, P₀ x f) ∧ (∀ᶠ x near K₁ ∪ K₂, P₁ x f) ∧ (∀ᶠ x near K₁ ∪ U₂ᶜ, f x = f₁ x)) :\n    ∃ f : X → Y, (∀ x, P₀ x f ∧ P₁ x f) ∧ ∀ᶠ x near K, f x = f₀ x :=\nbegin\n  let P₀' : Π x : X, germ (𝓝 x) Y → Prop := restrict_germ_predicate (λ x φ, φ.value = f₀ x) K,\n  have hf₀ : ∀ x, P₀ x f₀ ∧ P₀' x f₀,\n  { exact λ x, ⟨hP₀f₀ x, λ hx, eventually_of_forall (λ x', rfl)⟩ },\n  have ind' : ∀ (U₁ U₂ K₁ K₂ : set X) {f₁ f₂ : X → Y}, is_open U₁ → is_open U₂ →\n     is_closed K₁ → is_closed K₂ → K₁ ⊆ U₁ → K₂ ⊆ U₂ → (∀ x, P₀ x f₁ ∧ P₀' x f₁) → (∀ x, P₀ x f₂) →\n     (∀ x ∈ U₁, P₁ x f₁) → (∀ x ∈ U₂, P₁ x f₂) →\n     ∃ f : X → Y, (∀ x, P₀ x f ∧ P₀' x f ) ∧ (∀ᶠ x near K₁ ∪ K₂, P₁ x f) ∧\n                  (∀ᶠ x near K₁ ∪ U₂ᶜ, f x = f₁ x),\n  { intros U₁ U₂ K₁ K₂ f₁ f₂ U₁_op U₂_op K₁_cpct K₂_cpct hK₁U₁ hK₂U₂ hf₁ hf₂ hf₁U₁ hf₂U₂,\n    obtain ⟨h₀f₁, h₀'f₁⟩ := forall_and_distrib.mp hf₁,\n    rw forall_restrict_germ_predicate_iff at h₀'f₁,\n    rcases (has_basis_nhds_set K).mem_iff.mp (hP₁f₀.germ_congr h₀'f₁) with ⟨U, ⟨U_op, hKU⟩, hU⟩,\n    rcases ind (U_op.union U₁_op) U₂_op (hK.union K₁_cpct) K₂_cpct (union_subset_union hKU hK₁U₁)\n      hK₂U₂ h₀f₁ hf₂ (λ x hx, hx.elim (λ hx, hU hx) (λ hx, hf₁U₁ x hx)) hf₂U₂ with ⟨f, h₀f, hf, h'f⟩,\n    rw [union_assoc, eventually_nhds_set_union] at hf h'f,\n    exact ⟨f, λ x, ⟨h₀f x, restrict_germ_predicate_congr (hf₁ x).2 h'f.1⟩, hf.2, h'f.2⟩ },\n  rcases inductive_construction'' P₀ P₀' P₁ hf₀ loc ind' with ⟨f, hf⟩,\n  simp only [forall_and_distrib, forall_restrict_germ_predicate_iff ] at hf ⊢,\n  exact ⟨f, ⟨hf.1, hf.2.2⟩, hf.2.1⟩\nend\n\nend inductive_construction\n\nsection htpy\n\nprivate noncomputable def T : ℕ → ℝ := λ n, nat.rec 0 (λ k x, x + 1/(2 : ℝ)^(k+1)) n\n\nopen_locale big_operators\n\n-- Note this is more painful than Patrick hoped for. Maybe this should be the definition of T.\nprivate lemma T_eq (n : ℕ) : T n = 1- (1/(2: ℝ))^n :=\nbegin\n  have : T n = ∑ k in finset.range n, 1/(2: ℝ)^(k+1),\n  { induction n with n hn,\n    { simp only [T, finset.range_zero, finset.sum_empty] },\n    change T n + _ = _,\n    rw [hn, finset.sum_range_succ] },\n  simp_rw [this, ← one_div_pow, pow_succ, ← finset.mul_sum, geom_sum_eq (by norm_num : 1/(2:ℝ) ≠ 1) n],\n  field_simp,\n  norm_num,\n  apply div_eq_of_eq_mul,\n  apply neg_ne_zero.mpr,\n  apply ne_of_gt,\n  positivity,\n  ring\nend\n\nprivate lemma T_lt (n : ℕ) : T n < 1 :=\nbegin\n  rw T_eq,\n  have : (0 : ℝ) < (1 / 2) ^ n, by positivity,\n  linarith\nend\n\nprivate lemma T_lt_succ (n : ℕ) : T n < T (n+1) :=\nlt_add_of_le_of_pos le_rfl (one_div_pos.mpr (pow_pos zero_lt_two _))\n\nprivate lemma T_le_succ (n : ℕ) : T n ≤ T (n+1) := (T_lt_succ n).le\n\nprivate lemma T_succ_sub (n : ℕ) : T (n+1) - T n = 1/2^(n+1) :=\nbegin\n  change T n + _ - T n = _,\n  simp\nend\n\nprivate lemma mul_T_succ_sub (n : ℕ) : 2^(n+1)*(T (n+1) - T n) = 1 :=\nbegin\n  rw T_succ_sub,\n  field_simp\nend\n\nprivate lemma T_one : T 1 = 1/2 :=\nby simp [T]\n\nprivate lemma not_T_succ_le (n : ℕ) : ¬ T (n + 1) ≤ 0 :=\nbegin\n  rw [T_eq, not_le],\n  have : (1 / (2 : ℝ)) ^ (n + 1) < 1,\n  apply pow_lt_one ; norm_num,\n  linarith,\nend\n\nlemma inductive_htpy_construction {X Y : Type*} [topological_space X]\n  {N : ℕ} {U K : index_type N → set X}\n  (P₀ P₁ : Π x : X, germ (𝓝 x) Y → Prop) (P₂ : Π p : ℝ × X, germ (𝓝 p) Y → Prop)\n  (hP₂ : ∀ a b (p : ℝ × X) (f : ℝ × X → Y), P₂ (a*p.1+b, p.2) f → P₂ p (λ p : ℝ × X, f (a*p.1+b, p.2)))\n  (U_fin : locally_finite U) (K_cover : (⋃ i, K i) = univ)\n  {f₀ : X → Y} (init : ∀ x, P₀ x f₀)\n  (ind : ∀ (i : index_type N) (f : X → Y), (∀ x, P₀ x f) → (∀ᶠ x near ⋃ j < i, K j, P₁ x f) →\n    ∃ F : ℝ → X → Y, (∀ t, ∀ x, P₀ x $ F t) ∧ (∀ᶠ x near ⋃ j ≤ i, K j, P₁ x $ F 1) ∧\n                     (∀ p, P₂ p ↿F) ∧ (∀ t, ∀ x ∉ U i, F t x = f x) ∧\n                     (∀ᶠ t near Iic 0, F t = f) ∧ (∀ᶠ t near Ici 1, F t = F 1)) :\n  ∃ F : ℝ → X → Y, F 0 = f₀ ∧ (∀ t x, P₀ x (F t)) ∧ (∀ x, P₁ x (F 1)) ∧ (∀ p, P₂ p ↿F) :=\nbegin\n  let PP₀ : Π p : ℝ × X, germ (𝓝 p) Y → Prop := λ p φ, P₀ p.2 φ.slice_right ∧ sorry,\n  sorry\nend\nend htpy\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/inductive_constructions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3263308897783325}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n\nimport data.fun_like.embedding\n\n/-!\n# Typeclass for a type `F` with an injective map to `A ≃ B`\n\nThis typeclass is primarily for use by isomorphisms like `monoid_equiv` and `linear_equiv`.\n\n## Basic usage of `equiv_like`\n\nA typical type of morphisms should be declared as:\n```\nstructure my_iso (A B : Type*) [my_class A] [my_class B]\n  extends equiv A B :=\n(map_op' : ∀ {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))\n\nnamespace my_iso\n\nvariables (A B : Type*) [my_class A] [my_class B]\n\n-- This instance is optional if you follow the \"Isomorphism class\" design below:\ninstance : equiv_like (my_iso A B) A (λ _, B) :=\n{ coe := my_iso.to_equiv.to_fun,\n  inv := my_iso.to_equiv.inv_fun,\n  left_inv := my_iso.to_equiv.left_inv,\n  right_inv := my_iso.to_equiv.right_inv,\n  coe_injective' := λ f g h, by cases f; cases g; congr' }\n\n/-- Helper instance for when there's too many metavariables to apply `equiv_like.coe` directly. -/\ninstance : has_coe_to_fun (my_iso A B) := to_fun.to_coe_fn\n\n@[simp] lemma to_fun_eq_coe {f : my_iso A B} : f.to_fun = (f : A → B) := rfl\n\n@[ext] theorem ext {f g : my_iso A B} (h : ∀ x, f x = g x) : f = g := fun_like.ext f g h\n\n/-- Copy of a `my_iso` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : my_iso A B) (f' : A → B) (f_inv : B → A) (h : f' = ⇑f) : my_iso A B :=\n{ to_fun := f',\n  inv_fun := f_inv,\n  left_inv := h.symm ▸ f.left_inv,\n  right_inv := h.symm ▸ f.right_inv,\n  map_op' := h.symm ▸ f.map_op' }\n\nend my_iso\n```\n\nThis file will then provide a `has_coe_to_fun` instance and various\nextensionality and simp lemmas.\n\n## Isomorphism classes extending `equiv_like`\n\nThe `equiv_like` design provides further benefits if you put in a bit more work.\nThe first step is to extend `equiv_like` to create a class of those types satisfying\nthe axioms of your new type of isomorphisms.\nContinuing the example above:\n\n```\n/-- `my_iso_class F A B` states that `F` is a type of `my_class.op`-preserving morphisms.\nYou should extend this class when you extend `my_iso`. -/\nclass my_iso_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]\n  extends equiv_like F A (λ _, B), my_hom_class F A B.\n\n-- You can replace `my_iso.equiv_like` with the below instance:\ninstance : my_iso_class (my_iso A B) A B :=\n{ coe := my_iso.to_fun,\n  inv := my_iso.inv_fun,\n  left_inv := my_iso.left_inv,\n  right_inv := my_iso.right_inv,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  map_op := my_iso.map_op' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThe second step is to add instances of your new `my_iso_class` for all types extending `my_iso`.\nTypically, you can just declare a new class analogous to `my_iso_class`:\n\n```\nstructure cooler_iso (A B : Type*) [cool_class A] [cool_class B]\n  extends my_iso A B :=\n(map_cool' : to_fun cool_class.cool = cool_class.cool)\n\nclass cooler_iso_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]\n  extends my_iso_class F A B :=\n(map_cool : ∀ (f : F), f cool_class.cool = cool_class.cool)\n\n@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_iso_class F A B]\n  (f : F) : f cool_class.cool = cool_class.cool :=\nmy_iso_class.map_op\n\n-- You can also replace `my_iso.equiv_like` with the below instance:\ninstance : cool_iso_class (cool_iso A B) A B :=\n{ coe := cool_iso.to_fun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  map_op := cool_iso.map_op',\n  map_cool := cool_iso.map_cool' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThen any declaration taking a specific type of morphisms as parameter can instead take the\nclass you just defined:\n```\n-- Compare with: lemma do_something (f : my_iso A B) : sorry := sorry\nlemma do_something {F : Type*} [my_iso_class F A B] (f : F) : sorry := sorry\n```\n\nThis means anything set up for `my_iso`s will automatically work for `cool_iso_class`es,\nand defining `cool_iso_class` only takes a constant amount of effort,\ninstead of linearly increasing the work per `my_iso`-related declaration.\n\n-/\n\n/-- The class `equiv_like E α β` expresses that terms of type `E` have an\ninjective coercion to bijections between `α` and `β`.\n\nThis typeclass is used in the definition of the homomorphism typeclasses,\nsuch as `zero_equiv_class`, `mul_equiv_class`, `monoid_equiv_class`, ....\n-/\nclass equiv_like (E : Sort*) (α β : out_param Sort*) :=\n(coe : E → α → β)\n(inv : E → β → α)\n(left_inv  : ∀ e, function.left_inverse (inv e) (coe e))\n(right_inv : ∀ e, function.right_inverse (inv e) (coe e))\n-- The `inv` hypothesis makes this easier to prove with `congr'`\n(coe_injective' : ∀ e g, coe e = coe g → inv e = inv g → e = g)\n\nnamespace equiv_like\n\nvariables {E F α β γ : Sort*} [iE : equiv_like E α β] [iF : equiv_like F β γ]\ninclude iE\n\nlemma inv_injective : function.injective (equiv_like.inv : E → (β → α)) :=\nλ e g h, coe_injective' e g ((right_inv e).eq_right_inverse (h.symm ▸ left_inv g)) h\n\n@[priority 100]\ninstance to_embedding_like : embedding_like E α β :=\n{ coe := coe,\n  coe_injective' := λ e g h, coe_injective' e g h\n    ((left_inv e).eq_right_inverse (h.symm ▸ right_inv g)),\n  injective' := λ e, (left_inv e).injective }\n\nprotected lemma injective (e : E) : function.injective e := embedding_like.injective e\nprotected lemma surjective (e : E) : function.surjective e := (right_inv e).surjective\nprotected lemma bijective (e : E) : function.bijective (e : α → β) :=\n⟨equiv_like.injective e, equiv_like.surjective e⟩\n\ntheorem apply_eq_iff_eq (f : E) {x y : α} : f x = f y ↔ x = y := embedding_like.apply_eq_iff_eq f\n\n@[simp] lemma injective_comp (e : E) (f : β → γ) :\n  function.injective (f ∘ e) ↔ function.injective f :=\nfunction.injective.of_comp_iff' f (equiv_like.bijective e)\n\n@[simp] lemma surjective_comp (e : E) (f : β → γ) :\n  function.surjective (f ∘ e) ↔ function.surjective f :=\n(equiv_like.surjective e).of_comp_iff f\n\n@[simp] lemma bijective_comp (e : E) (f : β → γ) :\n  function.bijective (f ∘ e) ↔ function.bijective f :=\n(equiv_like.bijective e).of_comp_iff f\n\nomit iE\ninclude iF\n\nlemma comp_injective (f : α → β) (e : F) :\n  function.injective (e ∘ f) ↔ function.injective f :=\nembedding_like.comp_injective f e\n\n@[simp] lemma comp_surjective (f : α → β) (e : F) :\n  function.surjective (e ∘ f) ↔ function.surjective f :=\nfunction.surjective.of_comp_iff' (equiv_like.bijective e) f\n\n@[simp] lemma comp_bijective (f : α → β) (e : F) :\n  function.bijective (e ∘ f) ↔ function.bijective f :=\n(equiv_like.bijective e).of_comp_iff' f\n\nend equiv_like\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/fun_like/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3263265952414386}}
{"text": "@[simp] theorem getElem_fin [GetElem Cont Nat Elem Dom] (a : Cont) (i : Fin n) (h : Dom a i) :\n    a[i] = a[i.1] := rfl\n\nexample (a : Array α) (i : Fin a.size) : a[i] = a[i.1] := by\n  simp\n\nexample (a : Array α) (i : Fin a.size) : a[i] = a[i.1] := by\n  rw [getElem_fin]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1299.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3262931396429256}}
{"text": "example (p q r : Prop) (hp : p) (hq : q) (hr : r) :\n  p ∧ ((p ∧ q) ∧ r) ∧ (q ∧ r ∧ p) :=\nbegin\n  repeat { any_goals { split } },\n  all_goals { assumption }\nend\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0511.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.32609274832710533}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.isomorphism_classes\nimport Mathlib.category_theory.thin\nimport Mathlib.PostPort\n\nuniverses v₁ u₁ v₂ u₂ l u₃ v₃ \n\nnamespace Mathlib\n\n/-!\n# Skeleton of a category\n\nDefine skeletal categories as categories in which any two isomorphic objects are equal.\n\nConstruct the skeleton of a thin category as a partial ordering, and (noncomputably) show it is\na skeleton of the original category. The advantage of this special case being handled separately\nis that lemmas and definitions about orderings can be used directly, for example for the subobject\nlattice (TODO). In addition, some of the commutative diagrams about the functors commute\ndefinitionally on the nose which is convenient in practice.\n\n(TODO): Construct the skeleton of a general category using choice, and show it is a skeleton.\n-/\n\nnamespace category_theory\n\n\n/-- A category is skeletal if isomorphic objects are equal. -/\ndef skeletal (C : Type u₁) [category C] := ∀ {X Y : C}, is_isomorphic X Y → X = Y\n\n/--\n`is_skeleton_of C D F` says that `F : D ⥤ C` exhibits `D` as a skeletal full subcategory of `C`,\nin particular `F` is a (strong) equivalence and `D` is skeletal.\n-/\nstructure is_skeleton_of (C : Type u₁) [category C] (D : Type u₂) [category D] (F : D ⥤ C) where\n  skel : skeletal D\n  eqv : is_equivalence F\n\n/-- If `C` is thin and skeletal, then any naturally isomorphic functors to `C` are equal. -/\ntheorem functor.eq_of_iso {C : Type u₁} [category C] {D : Type u₂} [category D] {F₁ : D ⥤ C}\n    {F₂ : D ⥤ C} [∀ (X Y : C), subsingleton (X ⟶ Y)] (hC : skeletal C) (hF : F₁ ≅ F₂) : F₁ = F₂ :=\n  sorry\n\n/--\nIf `C` is thin and skeletal, `D ⥤ C` is skeletal.\n`category_theory.functor_thin` shows it is thin also.\n-/\ntheorem functor_skeletal {C : Type u₁} [category C] {D : Type u₂} [category D]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] (hC : skeletal C) : skeletal (D ⥤ C) :=\n  fun (F₁ F₂ : D ⥤ C) (h : is_isomorphic F₁ F₂) => nonempty.elim h (functor.eq_of_iso hC)\n\n/--\nConstruct the skeleton category by taking the quotient of objects. This construction gives a\npreorder with nice definitional properties, but is only really appropriate for thin categories.\n-/\ndef thin_skeleton (C : Type u₁) [category C] := quotient (is_isomorphic_setoid C)\n\nprotected instance inhabited_thin_skeleton (C : Type u₁) [category C] [Inhabited C] :\n    Inhabited (thin_skeleton C) :=\n  { default := quotient.mk Inhabited.default }\n\nprotected instance thin_skeleton.preorder (C : Type u₁) [category C] : preorder (thin_skeleton C) :=\n  preorder.mk (quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry)\n    (fun (a b : thin_skeleton C) =>\n      quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry a b ∧\n        ¬quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry b a)\n    sorry sorry\n\n/-- The functor from a category to its thin skeleton. -/\n@[simp] theorem to_thin_skeleton_obj (C : Type u₁) [category C] (a : C) :\n    functor.obj (to_thin_skeleton C) a = quotient.mk a :=\n  Eq.refl (functor.obj (to_thin_skeleton C) a)\n\n/-!\nThe constructions here are intended to be used when the category `C` is thin, even though\nsome of the statements can be shown without this assumption.\n-/\n\nnamespace thin_skeleton\n\n\n/-- The thin skeleton is thin. -/\nprotected instance thin (C : Type u₁) [category C] {X : thin_skeleton C} {Y : thin_skeleton C} :\n    subsingleton (X ⟶ Y) :=\n  subsingleton.intro\n    fun (a b : X ⟶ Y) =>\n      ulift.cases_on a\n        fun (a : plift (X ≤ Y)) =>\n          plift.cases_on a\n            fun (f₁ : X ≤ Y) =>\n              ulift.cases_on b\n                fun (b : plift (X ≤ Y)) =>\n                  plift.cases_on b fun (f₂ : X ≤ Y) => Eq.refl (ulift.up (plift.up f₁))\n\n/-- A functor `C ⥤ D` computably lowers to a functor `thin_skeleton C ⥤ thin_skeleton D`. -/\ndef map {C : Type u₁} [category C] {D : Type u₂} [category D] (F : C ⥤ D) :\n    thin_skeleton C ⥤ thin_skeleton D :=\n  functor.mk (quotient.map (functor.obj F) sorry)\n    fun (X Y : thin_skeleton C) =>\n      quotient.rec_on_subsingleton₂ X Y\n        fun (x y : C) (k : quotient.mk x ⟶ quotient.mk y) => hom_of_le sorry\n\ntheorem comp_to_thin_skeleton {C : Type u₁} [category C] {D : Type u₂} [category D] (F : C ⥤ D) :\n    F ⋙ to_thin_skeleton D = to_thin_skeleton C ⋙ map F :=\n  rfl\n\n/-- Given a natural transformation `F₁ ⟶ F₂`, induce a natural transformation `map F₁ ⟶ map F₂`.-/\ndef map_nat_trans {C : Type u₁} [category C] {D : Type u₂} [category D] {F₁ : C ⥤ D} {F₂ : C ⥤ D}\n    (k : F₁ ⟶ F₂) : map F₁ ⟶ map F₂ :=\n  nat_trans.mk\n    fun (X : thin_skeleton C) =>\n      quotient.rec_on_subsingleton X fun (x : C) => ulift.up (plift.up sorry)\n\n-- TODO: state the lemmas about what happens when you compose with `to_thin_skeleton`\n\n/-- A functor `C ⥤ D ⥤ E` computably lowers to a functor\n`thin_skeleton C ⥤ thin_skeleton D ⥤ thin_skeleton E` -/\n@[simp] theorem map₂_obj_obj {C : Type u₁} [category C] {D : Type u₂} [category D] {E : Type u₃}\n    [category E] (F : C ⥤ D ⥤ E) (x : thin_skeleton C) (y : thin_skeleton D) :\n    functor.obj (functor.obj (map₂ F) x) y =\n        quotient.map₂ (fun (X : C) (Y : D) => functor.obj (functor.obj F X) Y) (map₂._proof_1 F) x\n          y :=\n  Eq.refl (functor.obj (functor.obj (map₂ F) x) y)\n\nprotected instance to_thin_skeleton_faithful (C : Type u₁) [category C]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] : faithful (to_thin_skeleton C) :=\n  faithful.mk\n\n/-- Use `quotient.out` to create a functor out of the thin skeleton. -/\n@[simp] theorem from_thin_skeleton_map (C : Type u₁) [category C]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] (x : thin_skeleton C) (y : thin_skeleton C) :\n    ∀ (ᾰ : x ⟶ y),\n        functor.map (from_thin_skeleton C) ᾰ =\n          quotient.rec_on_subsingleton₂ x y\n            (fun (X Y : C) (f : quotient.mk X ⟶ quotient.mk Y) =>\n              iso.hom (nonempty.some (from_thin_skeleton._proof_2 C X)) ≫\n                nonempty.some (from_thin_skeleton._proof_3 C X Y f) ≫\n                  iso.inv (nonempty.some (from_thin_skeleton._proof_4 C Y)))\n            ᾰ :=\n  fun (ᾰ : x ⟶ y) => Eq.refl (functor.map (from_thin_skeleton C) ᾰ)\n\nprotected instance from_thin_skeleton_equivalence (C : Type u₁) [category C]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] : is_equivalence (from_thin_skeleton C) :=\n  is_equivalence.mk' (to_thin_skeleton C)\n    (nat_iso.of_components\n      (fun (x : thin_skeleton C) => quotient.rec_on_subsingleton x fun (X : C) => eq_to_iso sorry)\n      sorry)\n    (nat_iso.of_components (fun (X : C) => nonempty.some sorry) sorry)\n\ntheorem equiv_of_both_ways {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] {X : C}\n    {Y : C} (f : X ⟶ Y) (g : Y ⟶ X) : X ≈ Y :=\n  Nonempty.intro (iso_of_both_ways f g)\n\nprotected instance thin_skeleton_partial_order {C : Type u₁} [category C]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] : partial_order (thin_skeleton C) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\ntheorem skeletal {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] :\n    skeletal (thin_skeleton C) :=\n  sorry\n\ntheorem map_comp_eq {C : Type u₁} [category C] {D : Type u₂} [category D] {E : Type u₃} [category E]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] (F : E ⥤ D) (G : D ⥤ C) : map (F ⋙ G) = map F ⋙ map G :=\n  sorry\n\ntheorem map_id_eq {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : map 𝟭 = 𝟭 :=\n  sorry\n\ntheorem map_iso_eq {C : Type u₁} [category C] {D : Type u₂} [category D]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] {F₁ : D ⥤ C} {F₂ : D ⥤ C} (h : F₁ ≅ F₂) : map F₁ = map F₂ :=\n  functor.eq_of_iso skeletal (iso.mk (map_nat_trans (iso.hom h)) (map_nat_trans (iso.inv h)))\n\n/-- `from_thin_skeleton C` exhibits the thin skeleton as a skeleton. -/\ndef thin_skeleton_is_skeleton {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] :\n    is_skeleton_of C (thin_skeleton C) (from_thin_skeleton C) :=\n  is_skeleton_of.mk sorry (thin_skeleton.from_thin_skeleton_equivalence C)\n\nprotected instance is_skeleton_of_inhabited {C : Type u₁} [category C]\n    [∀ (X Y : C), subsingleton (X ⟶ Y)] :\n    Inhabited (is_skeleton_of C (thin_skeleton C) (from_thin_skeleton C)) :=\n  { default := thin_skeleton_is_skeleton }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/skeletal_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3259792352434987}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nProofs that integrating out the KL and reparametizing are sound when\napplied to the naive variational encoder.\n-/\nimport .util .graph ..prove_model_ok ..backprop_correct\n\nnamespace certigrad\nnamespace aevb\n\nopen graph list tactic certigrad.tactic\n\nset_option profiler true\n\n#print \"proving backprop_correct_on_aevb...\"\n\ntheorem backprop_correct_on_aevb (a : arch) (ws : weights a) (x_data : T [a^.n_in, a^.n_x]) :\nlet g : graph := reparam (integrate_kl $ naive_aevb a x_data) in\nlet fdict : env := mk_input_dict ws x_data g in\n∀ (tgt : reference) (idx : ℕ) (H_at_idx : at_idx g^.targets idx tgt),\n∇ (λ θ₀, E (graph.to_dist (λ m, ⟦sum_costs m g^.costs⟧) (env.insert tgt θ₀ fdict) g^.nodes) dvec.head) (env.get tgt fdict)\n=\nE (graph.to_dist (λ m, backprop g^.costs g^.nodes m g^.targets) fdict g^.nodes) (λ dict, dvec.get tgt.2 dict idx) :=\nby prove_model_ok\n\nend aevb\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/aevb/grads_correct.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.3259632348712679}}
{"text": "import ..lovelib\n\n\n/-! # LoVe Demo 8: Operational Semantics\n\nIn this and the next two lectures, we will see how to use Lean to specify the\nsyntax and semantics of programming languages and to reason about the\nsemantics. -/\n\n\nset_option pp.beta true\nset_option pp.generalized_field_notation false\n\nnamespace LoVe\n\n\n/-! ## Formal Semantics\n\nA formal semantics helps specify and reason about the programming language\nitself, and about individual programs.\n\nIt can form the basis of verified compilers, interpreters, verifiers, static\nanalyzers, type checkers, etc. Without formal proofs, these tools are\n**almost always wrong**.\n\nIn this area, proof assistants are widely used. Every year, about 10-20% of POPL\npapers are partially or totally formalized. Reasons for this success:\n\n* Little machinery (background libraries, tactics) is needed to get started,\n  beyond inductive types and predicates and recursive functions.\n\n* The proofs tend to have lots of cases, which is a good match for computers.\n\n* Proof assistants keep track of what needs to be changed when as extend the\n  programming language with more features.\n\nCase in point: WebAssembly. To quote Conrad Watt (with some abbreviations):\n\n    We have produced a full Isabelle mechanisation of the core execution\n    semantics and type system of the WebAssembly language. To complete this\n    proof, **several deficiencies** in the official WebAssembly specification,\n    uncovered by our proof and modelling work, needed to be corrected. In some\n    cases, these meant that the type system was **originally unsound**.\n\n    We have maintained a constructive dialogue with the working group,\n    verifying new features as they are added. In particular, the mechanism by\n    which a WebAssembly implementation interfaces with its host environment was\n    not formally specified in the working group's original paper. Extending our\n    mechanisation to model this feature revealed a deficiency in the WebAssembly\n    specification that **sabotaged the soundness** of the type system.\n\n    (https://www.doc.ic.ac.uk/~pg/publications/Watt2021Two.pdf)\n\n## A Minimalistic Imperative Language\n\nA state `s` is a function from variable names to values (`string → ℕ`).\n\n__WHILE__ is a minimalistic imperative language with the following grammar:\n\n    S  ::=  skip                 -- no-op\n         |  x := a               -- assignment\n         |  S ; S                -- sequential composition\n         |  if b then S else S   -- conditional statement\n         |  while b do S         -- while loop\n\nwhere `S` stands for a statement (also called command or program), `x` for a\nvariable, `a` for an arithmetic expression, and `b` for a Boolean expression. -/\n\n#print state\n\ninductive stmt : Type\n| skip   : stmt\n| assign : string → (state → ℕ) → stmt\n| seq    : stmt → stmt → stmt\n| ite    : (state → Prop) → stmt → stmt → stmt\n| while  : (state → Prop) → stmt → stmt\n\ninfixr ` ;; ` : 90 := stmt.seq\n\n#check λ s : state, s \"y\" + 2 * s \"z\"\n\n/-! In our grammar, we deliberately leave the syntax of arithmetic and Boolean\nexpressions unspecified. In Lean, we have the choice:\n\n* We could use a type such as `aexp` from lecture 1 and similarly for Boolean\n  expressions.\n\n* We could decide that an arithmetic expression is simply a function from\n  states to natural numbers (`state → ℕ`) and a Boolean expression is a\n  predicate (`state → Prop` or `state → bool`).\n\nThis corresponds to the difference between deep and shallow embeddings:\n\n* A __deep embedding__ of some syntax (expression, formula, program, etc.)\n  consists of an abstract syntax tree specified in the proof assistant\n  (e.g., `aexp`) with a semantics (e.g., `eval`).\n\n* In contrast, a __shallow embedding__ simply reuses the corresponding\n  mechanisms from the logic (e.g., λ-terms, functions and predicate types).\n\nA deep embedding allows us to reason about the syntax (and its semantics). A\nshallow embedding is more lightweight, because we can use it directly, without\nhaving to define a semantics.\n\nWe will use a deep embedding of programs (which we find interesting), and\nshallow embeddings of assignments and Boolean expressions (which we find\nboring). -/\n\ndef silly_loop : stmt :=\nstmt.while (λs, s \"x\" > s \"y\")\n  (stmt.skip ;; stmt.assign \"x\" (λs, s \"x\" - 1))\n\n\n/-! ## Big-Step Semantics\n\nAn __operational semantics__ corresponds to an idealized interpreter (specified\nin a Prolog-like language). Two main variants:\n\n* big-step semantics;\n\n* small-step semantics.\n\nIn a __big-step semantics__ (also called __natural semantics__), judgments have\nthe form `(S, s) ⟹ t`:\n\n    Starting in a state `s`, executing `S` terminates in the state `t`.\n\nExample:\n\n    `(x := x + y; y := 0, [x ↦ 3, y ↦ 5]) ⟹ [x ↦ 8, y ↦ 0]`\n\nDerivation rules:\n\n    ——————————————— Skip\n    (skip, s) ⟹ s\n\n    ——————————————————————————— Asn\n    (x := a, s) ⟹ s[x ↦ s(a)]\n\n    (S, s) ⟹ t   (T, t) ⟹ u\n    ——————————————————————————— Seq\n    (S; T, s) ⟹ u\n\n    (S, s) ⟹ t\n    ————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⟹ t\n\n    (T, s) ⟹ t\n    ————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⟹ t\n\n    (S, s) ⟹ t   (while b do S, t) ⟹ u\n    —————————————————————————————————————— While-True   if s(b) is true\n    (while b do S, s) ⟹ u\n\n    ————————————————————————— While-False   if s(b) is false\n    (while b do S, s) ⟹ s\n\nAbove, `s(e)` denotes the value of expression `e` in state `s`.\n\nIn Lean, the judgment corresponds to an inductive predicate, and the derivation\nrules correspond to the predicate's introduction rules. Using an inductive\npredicate as opposed to a recursive function allows us to cope with\nnontermination (e.g., a diverging `while`) and nondeterminism (e.g.,\nmultithreading). -/\n\ninductive big_step : stmt × state → state → Prop\n| skip {s} :\n  big_step (stmt.skip, s) s\n| assign {x a s} :\n  big_step (stmt.assign x a, s) (s{x ↦ a s})\n| seq {S T s t u} (hS : big_step (S, s) t)\n    (hT : big_step (T, t) u) :\n  big_step (S ;; T, s) u\n| ite_true {b : state → Prop} {S T s t} (hcond : b s)\n    (hbody : big_step (S, s) t) :\n  big_step (stmt.ite b S T, s) t\n| ite_false {b : state → Prop} {S T s t} (hcond : ¬ b s)\n    (hbody : big_step (T, s) t) :\n  big_step (stmt.ite b S T, s) t\n| while_true {b : state → Prop} {S s t u} (hcond : b s)\n    (hbody : big_step (S, s) t)\n    (hrest : big_step (stmt.while b S, t) u) :\n  big_step (stmt.while b S, s) u\n| while_false {b : state → Prop} {S s} (hcond : ¬ b s) :\n  big_step (stmt.while b S, s) s\n\ninfix ` ⟹ ` : 110 := big_step\n\nlemma silly_loop_from_1_big_step :\n  (silly_loop, (λ_, 0){\"x\" ↦ 1}) ⟹ (λ_, 0) :=\nbegin\n  rw silly_loop,\n  apply big_step.while_true,\n  { simp },\n  { apply big_step.seq,\n    { apply big_step.skip },\n    { apply big_step.assign } },\n  { simp,\n    apply big_step.while_false,\n    linarith }\nend\n\n\n/-! ## Properties of the Big-Step Semantics\n\nEquipped with a big-step semantics, we can\n\n* prove properties of the programming language, such as **equivalence proofs**\n  between programs and **determinism**;\n\n* reason about **concrete programs**, proving theorems relating final states `t`\n  with initial states `s`. -/\n\nlemma big_step_deterministic {S s l r} (hl : (S, s) ⟹ l)\n    (hr : (S, s) ⟹ r) :\n  l = r :=\nbegin\n  induction' hl,\n  case skip {\n    cases' hr,\n    refl },\n  case assign {\n    cases' hr,\n    refl },\n  case seq : S T s t l hS hT ihS ihT {\n    cases' hr with _ _ _ _ _ _ _ t' _ hS' hT',\n    cases' ihS hS',\n    cases' ihT hT',\n    refl },\n  case ite_true : b S T s t hb hS ih {\n    cases' hr,\n    { apply ih hr },\n    { cc } },\n  case ite_false : b S T s t hb hT ih {\n    cases' hr,\n    { cc },\n    { apply ih hr } },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    cases' hr,\n    { cases' ihS hr,\n      cases' ihw hr_1,\n      refl },\n    { cc } },\n  case while_false {\n    cases' hr,\n    { cc },\n    { refl } }\nend\n\nlemma big_step_terminates {S s} :\n  ∃t, (S, s) ⟹ t :=\nsorry   -- unprovable\n\nlemma big_step_doesnt_terminate {S s t} :\n  ¬ (stmt.while (λ_, true) S, s) ⟹ t :=\nbegin\n  intro hw,\n  induction' hw,\n  case while_true {\n    assumption },\n  case while_false {\n    cc }\nend\n\n/-! We can define inversion rules about the big-step semantics: -/\n\n@[simp] lemma big_step_skip_iff {s t} :\n  (stmt.skip, s) ⟹ t ↔ t = s :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.skip }\nend\n\n@[simp] lemma big_step_assign_iff {x a s t} :\n  (stmt.assign x a, s) ⟹ t ↔ t = s{x ↦ a s} :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.assign }\nend\n\n@[simp] lemma big_step_seq_iff {S T s t} :\n  (S ;; T, s) ⟹ t ↔ (∃u, (S, s) ⟹ u ∧ (T, u) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    apply exists.intro,\n    apply and.intro; assumption },\n  { intro h,\n    cases' h,\n    cases' h,\n    apply big_step.seq; assumption }\nend\n\n@[simp] lemma big_step_ite_iff {b S T s t} :\n  (stmt.ite b S T, s) ⟹ t ↔\n  (b s ∧ (S, s) ⟹ t) ∨ (¬ b s ∧ (T, s) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h; cases' h,\n    { apply big_step.ite_true; assumption },\n    { apply big_step.ite_false; assumption } }\nend\n\nlemma big_step_while_iff {b S s u} :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, b s ∧ (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u)\n  ∨ (¬ b s ∧ u = s) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro t,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    case inl {\n      cases' h with t h,\n      cases' h with hb h,\n      cases' h with hS hwhile,\n      exact big_step.while_true hb hS hwhile },\n    case inr {\n      cases' h with hb hus,\n      rw hus,\n      exact big_step.while_false hb } }\nend\n\nlemma big_step_while_true_iff {b : state → Prop} {S s u}\n    (hcond : b s) :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u) :=\nby rw big_step_while_iff; simp [hcond]\n\n@[simp] lemma big_step_while_false_iff {b : state → Prop}\n    {S s t} (hcond : ¬ b s) :\n  (stmt.while b S, s) ⟹ t ↔ t = s :=\nby rw big_step_while_iff; simp [hcond]\n\n\n/-! ## Small-Step Semantics\n\nA big-step semantics\n\n* does not let us reason about intermediate states;\n\n* does not let us express nontermination or interleaving (for multithreading).\n\n__Small-step semantics__ (also called __structural operational semantics__)\nsolve the above issues.\n\nA judgment has the form `(S, s) ⇒ (T, t)`:\n\n    Starting in a state `s`, executing one step of `S` leaves us in the\n    state `t`, with the program `T` remaining to be executed.\n\nAn execution is a finite or infinite chain `(S₀, s₀) ⇒ (S₁, s₁) ⇒ …`.\n\nA pair `(S, s)` is called a __configuration__. It is __final__ if no transition\nof the form `(S, s) ⇒ _` is possible.\n\nExample:\n\n      `(x := x + y; y := 0, [x ↦ 3, y ↦ 5])`\n    `⇒ (skip; y := 0,       [x ↦ 8, y ↦ 5])`\n    `⇒ (y := 0,             [x ↦ 8, y ↦ 5])`\n    `⇒ (skip,               [x ↦ 8, y ↦ 0])`\n\nDerivation rules:\n\n    ————————————————————————————————— Asn\n    (x := a, s) ⇒ (skip, s[x ↦ s(a)])\n\n    (S, s) ⇒ (S', s')\n    ———-————————————————————— Seq-Step\n    (S ; T, s) ⇒ (S' ; T, s')\n\n    —————————————————————— Seq-Skip\n    (skip ; S, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⇒ (T, s)\n\n    ——————————————————————————————————————————————————————————————— While\n    (while b do S, s) ⇒ (if b then (S ; while b do S) else skip, s)\n\nThere is no rule for `skip` (why?). -/\n\ninductive small_step : stmt × state → stmt × state → Prop\n| assign {x a s} :\n  small_step (stmt.assign x a, s) (stmt.skip, s{x ↦ a s})\n| seq_step {S S' T s s'} (hS : small_step (S, s) (S', s')) :\n  small_step (S ;; T, s) (S' ;; T, s')\n| seq_skip {T s} :\n  small_step (stmt.skip ;; T, s) (T, s)\n| ite_true {b : state → Prop} {S T s} (hcond : b s) :\n  small_step (stmt.ite b S T, s) (S, s)\n| ite_false {b : state → Prop} {S T s} (hcond : ¬ b s) :\n  small_step (stmt.ite b S T, s) (T, s)\n| while {b : state → Prop} {S s} :\n  small_step (stmt.while b S, s)\n    (stmt.ite b (S ;; stmt.while b S) stmt.skip, s)\n\ninfixr (name := small_step) ` ⇒ ` := small_step\ninfixr ` ⇒* ` : 100 := star small_step\n\nlemma silly_loop_from_1_small_step :\n  (silly_loop, (λ_, 0){\"x\" ↦ 1}) ⇒*\n  (stmt.skip, ((λ_, 0) : state)) :=\nbegin\n  rw silly_loop,\n  apply star.head,\n  { apply small_step.while },\n  { apply star.head,\n    { apply small_step.ite_true,\n      simp },\n    { apply star.head,\n      { apply small_step.seq_step,\n        apply small_step.seq_skip },\n      { apply star.head,\n        { apply small_step.seq_step,\n          apply small_step.assign },\n        { apply star.head,\n          { apply small_step.seq_skip },\n          { apply star.head,\n            { apply small_step.while },\n            { apply star.head,\n              { apply small_step.ite_false,\n                simp },\n              { simp } } } } } } }\nend\n\n/-! Equipped with a small-step semantics, we can **define** a big-step\nsemantics:\n\n    `(S, s) ⟹ t` if and only if `(S, s) ⇒* (skip, t)`\n\nwhere `r*` denotes the reflexive transitive closure of a relation `r`.\n\nAlternatively, if we have already defined a big-step semantics, we can **prove**\nthe above equivalence theorem to validate our definitions.\n\nThe main disadvantage of small-step semantics is that we now have two relations,\n`⇒` and `⇒*`, and reasoning tends to be more complicated.\n\n\n## Properties of the Small-Step Semantics\n\nWe can prove that a configuration `(S, s)` is final if and only if `S = skip`.\nThis ensures that we have not forgotten a derivation rule. -/\n\nlemma small_step_final (S s) :\n  (¬ ∃T t, (S, s) ⇒ (T, t)) ↔ S = stmt.skip :=\nbegin\n  induction' S,\n  case skip {\n    simp,\n    intros T t hstep,\n    cases' hstep },\n  case assign : x a {\n    simp,\n    apply exists.intro stmt.skip,\n    apply exists.intro (s{x ↦ a s}),\n    exact small_step.assign },\n  case seq : S T ihS ihT {\n    simp,\n    cases' classical.em (S = stmt.skip),\n    case inl {\n      rw h,\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.seq_skip },\n    case inr {\n      simp [h, auto.not_forall_eq, auto.not_not_eq] at ihS,\n      cases' ihS s with S' hS',\n      cases' hS' with s' hs',\n      apply exists.intro (S' ;; T),\n      apply exists.intro s',\n      exact small_step.seq_step hs' } },\n  case ite : b S T ihS ihT {\n    simp,\n    cases' classical.em (b s),\n    case inl {\n      apply exists.intro S,\n      apply exists.intro s,\n      exact small_step.ite_true h },\n    case inr {\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.ite_false h } },\n  case while : b S ih {\n    simp,\n    apply exists.intro (stmt.ite b (S ;; stmt.while b S) stmt.skip),\n    apply exists.intro s,\n    exact small_step.while }\nend\n\nlemma small_step_deterministic {S s Ll Rr}\n    (hl : (S, s) ⇒ Ll) (hr : (S, s) ⇒ Rr) :\n  Ll = Rr :=\nbegin\n  induction' hl,\n  case assign : x a s {\n    cases' hr,\n    refl },\n  case seq_step : S S₁ T s s₁ hS₁ ih {\n    cases' hr,\n    case seq_step : S S₂ _ _ s₂ hS₂ {\n      have hSs₁₂ := ih hS₂,\n      cc },\n    case seq_skip {\n      cases' hS₁ } },\n  case seq_skip : T s {\n    cases' hr,\n    case seq_step {\n      cases' hr },\n    case seq_skip {\n      refl } },\n  case ite_true : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      refl },\n    case ite_false {\n      cc } },\n  case ite_false : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      cc },\n    case ite_false {\n      refl } },\n  case while : b S s {\n    cases' hr,\n    refl }\nend\n\n/-! We can define inversion rules also about the small-step semantics. Here are\nthree examples: -/\n\nlemma small_step_skip {S s t} :\n  ¬ ((stmt.skip, s) ⇒ (S, t)) :=\nby intro h; cases' h\n\n@[simp] lemma small_step_seq_iff {S T s Ut} :\n  (S ;; T, s) ⇒ Ut ↔\n  (∃S' t, (S, s) ⇒ (S', t) ∧ Ut = (S' ;; T, t))\n  ∨ (S = stmt.skip ∧ Ut = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro S',\n      apply exists.intro s',\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      cases' h,\n      cases' h,\n      rw right,\n      apply small_step.seq_step,\n      assumption },\n    { cases' h,\n      rw left,\n      rw right,\n      apply small_step.seq_skip } }\nend\n\n@[simp] lemma small_step_ite_iff {b S T s Us} :\n  (stmt.ite b S T, s) ⇒ Us ↔\n  (b s ∧ Us = (S, s)) ∨ (¬ b s ∧ Us = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      rw right,\n      apply small_step.ite_true,\n      assumption },\n    { cases' h,\n      rw right,\n      apply small_step.ite_false,\n      assumption } }\nend\n\n\n/-! ### Equivalence of the Big-Step and the Small-Step Semantics (**optional**)\n\nA more important result is the connection between the big-step and the\nsmall-step semantics:\n\n    `(S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t)`\n\nIts proof, given below, is beyond the scope of this course. -/\n\nlemma star_small_step_seq {S T s u}\n    (h : (S, s) ⇒* (stmt.skip, u)) :\n  (S ;; T, s) ⇒* (stmt.skip ;; T, u) :=\nbegin\n  apply star.lift (λSs, (prod.fst Ss ;; T, prod.snd Ss)) _ h,\n  intros Ss Ss' h,\n  cases' Ss,\n  cases' Ss',\n  apply small_step.seq_step,\n  assumption\nend\n\nlemma star_small_step_of_big_step {S s t} (h : (S, s) ⟹ t) :\n  (S, s) ⇒* (stmt.skip, t) :=\nbegin\n  induction' h,\n  case skip {\n    refl },\n  case assign {\n    exact star.single small_step.assign },\n  case seq : S T s t u hS hT ihS ihT {\n    transitivity,\n    exact star_small_step_seq ihS,\n    apply star.head small_step.seq_skip ihT },\n  case ite_true : b S T s t hs hst ih {\n    exact star.head (small_step.ite_true hs) ih },\n  case ite_false : b S T s t hs hst ih {\n    exact star.head (small_step.ite_false hs) ih },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    exact (star.head small_step.while\n      (star.head (small_step.ite_true hb)\n         (star.trans (star_small_step_seq ihS)\n            (star.head small_step.seq_skip ihw)))) },\n  case while_false : b S s hb {\n    exact star.tail (star.single small_step.while)\n      (small_step.ite_false hb) }\nend\n\nlemma big_step_of_small_step_of_big_step {S₀ S₁ s₀ s₁ s₂}\n  (h₁ : (S₀, s₀) ⇒ (S₁, s₁)) :\n  (S₁, s₁) ⟹ s₂ → (S₀, s₀) ⟹ s₂ :=\nbegin\n  induction' h₁;\n    simp [*, big_step_while_true_iff] {contextual := tt},\n  case seq_step {\n    intros u hS' hT,\n    apply exists.intro u,\n    exact and.intro (ih hS') hT }\nend\n\nlemma big_step_of_star_small_step {S s t} :\n  (S, s) ⇒* (stmt.skip, t) → (S, s) ⟹ t :=\nbegin\n  generalize hSs : (S, s) = Ss,\n  intro h,\n  induction h\n      using LoVe.rtc.star.head_induction_on\n      with _ S's' h h' ih\n      generalizing S s;\n    cases' hSs,\n  { exact big_step.skip },\n  { cases' S's' with S' s',\n    apply big_step_of_small_step_of_big_step h,\n    apply ih,\n    refl }\nend\n\nlemma big_step_iff_star_small_step {S s t} :\n  (S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t) :=\niff.intro star_small_step_of_big_step\n  big_step_of_star_small_step\n\n\n/-! ## Parallelism (**optional**) -/\n\n/--\na proof of `par_step i (Ss, t) (Ss', t')` represents:\n\"\n  Let `Ss` be a list of WHILE programs that we wish to execute in parallel.\n  If we evaluate one step of the `i`th program in `Ss` while in state `t`, \n  we reach state `t'`, with the list of programs `Ss'` remaining to execute.\"\n-/\ninductive par_step :\n    nat → list stmt × state → list stmt × state → Prop\n| intro {Ss Ss' S S' s s' i}\n    (hi : i < list.length Ss)\n    (hS : S = list.nth_le Ss i hi)\n    (hs : (S, s) ⇒ (S', s'))\n    (hS' : Ss' = list.update_nth Ss i S') :\n  par_step i (Ss, s) (Ss', s')\n\n\n\n\nlemma par_step_diamond {i j Ss Ts Ts' s t t'}\n    (hi : i < list.length Ss)\n    (hj : j < list.length Ss)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t')) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry \n\n\n\n\n/--\n`stmt.W S` returns the set of variables written to by a thread `S`.\n-/\ndef stmt.W : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign x _) := {x}\n| (stmt.seq S T)    := stmt.W S ∪ stmt.W T\n| (stmt.ite _ S T)  := stmt.W S ∪ stmt.W T\n| (stmt.while _ S)  := stmt.W S\n\n/--\n`exp.R f` returns the set of variables read by an arithmetic or boolean expression `f`.\n-/\ndef exp.R {α : Type} (f : state → α) : set string := \n{x | ∃ s n, f (s{x ↦ n}) ≠ f s}\n\n/--\n`stmt.R S` returns the set of variables read by a thread `S`.\n-/\ndef stmt.R : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign _ a) := exp.R a\n| (stmt.seq S T)    := stmt.R S ∪ stmt.R T\n| (stmt.ite b S T)  := exp.R b ∪ stmt.R S ∪ stmt.R T\n| (stmt.while b S)  := exp.R b ∪ stmt.R S\n\n/--\n`stmt.V S` returns the set of variables written to or read by a thread `S`. \n-/\ndef stmt.V : stmt → set string\n| S := stmt.W S ∪ stmt.R S\n\nlemma par_step_diamond_VW_disjoint {i j Ss Ts Ts' s t t'}\n    (hiS : i < list.length Ss)\n    (hjT : j < list.length Ts)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t'))\n    (hWV : stmt.W (list.nth_le Ss i hiS)\n       ∩ stmt.V (list.nth_le Ts j hjT) = ∅)\n    (hVW : stmt.V (list.nth_le Ss i hiS)\n       ∩ stmt.W (list.nth_le Ts j hjT) = ∅) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry   -- this should be provable\n\nend LoVe\n", "meta": {"author": "BrownCS1951x", "repo": "fpv2022", "sha": "aeaf291183721460387f8ae4c3c008836b8460e7", "save_path": "github-repos/lean/BrownCS1951x-fpv2022", "path": "github-repos/lean/BrownCS1951x-fpv2022/fpv2022-aeaf291183721460387f8ae4c3c008836b8460e7/src/lectures/love08_operational_semantics_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.32582022322991394}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.discrete_category\n\n/-!\n# Limits in `C` give colimits in `Cᵒᵖ`.\n\nWe also give special cases for (co)products,\n(co)equalizers, and pullbacks / pushouts.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C]\nvariables {J : Type u₂} [category.{v₂} J]\n\n/-- Turn a colimit for `F : J ⥤ C` into a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cocone_op (F : J ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit c.op :=\n{ lift := λ s, (hc.desc s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F : J ⥤ C` into a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_op (F : J ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit c.op :=\n{ desc := λ s, (hc.lift s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : J ⥤ Cᵒᵖ` into a limit for `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_left_op_of_cocone (F : J ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_left_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_left_op s)).unop,\n  fac' :=  λ s j, quiver.hom.op_inj $ by simpa only [cone_left_op_of_cocone_π_app, op_comp,\n    quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app] using w (op j)\n  end }\n\n/-- Turn a limit of `F : J ⥤ Cᵒᵖ` into a colimit of `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_left_op_of_cone (F : J ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_left_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_left_op s)).unop,\n  fac' := λ s j, quiver.hom.op_inj $ by simpa only [cocone_left_op_of_cone_ι_app, op_comp,\n    quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ C` into a limit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_right_op_of_cocone (F : Jᵒᵖ ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_right_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F : Jᵒᵖ ⥤ C` into a colimit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_right_op_of_cone (F : Jᵒᵖ ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_right_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F.unop : J ⥤ C`. -/\n@[simps] def is_limit_cone_unop_of_cocone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_unop_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit of `F : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit of `F.unop : J ⥤ C`. -/\n@[simps] def is_colimit_cocone_unop_of_cone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_unop_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F : J ⥤ C`. -/\n@[simps] def is_limit_cocone_unop (F : J ⥤ C) {c : cocone F.op} (hc : is_colimit c) :\n  is_limit c.unop :=\n{ lift := λ s, (hc.desc s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit for `F : J ⥤ C`. -/\n@[simps] def is_colimit_cone_unop (F : J ⥤ C) {c : cone F.op} (hc : is_limit c) :\n  is_colimit c.unop :=\n{ desc := λ s, (hc.lift s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.left_op : Jᵒᵖ ⥤ C` into a limit for `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_left_op (F : J ⥤ Cᵒᵖ) {c : cocone F.left_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_left_op c) :=\n{ lift := λ s, (hc.desc (cocone_left_op_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cone_of_cocone_left_op_π_app, unop_comp,\n    quiver.hom.unop_op, is_colimit.fac, cocone_left_op_of_cone_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac, cone_of_cocone_left_op_π_app] using w (unop j)\n  end }\n\n/-- Turn a limit of `F.left_op : Jᵒᵖ ⥤ C` into a colimit of `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_of_cone_left_op (F : J ⥤ Cᵒᵖ) {c : cone (F.left_op)}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_left_op c) :=\n{ desc := λ s, (hc.lift (cone_left_op_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cocone_of_cone_left_op_ι_app, unop_comp,\n    quiver.hom.unop_op, is_limit.fac, cone_left_op_of_cocone_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac, cocone_of_cone_left_op_ι_app] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_of_cocone_right_op (F : Jᵒᵖ ⥤ C) {c : cocone F.right_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_right_op c) :=\n{ lift := λ s, (hc.desc (cocone_right_op_of_cone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_of_cone_right_op (F : Jᵒᵖ ⥤ C) {c : cone F.right_op}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_right_op c) :=\n{ desc := λ s, (hc.lift (cone_right_op_of_cocone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F.unop : J ⥤ C` into a limit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F.unop} (hc : is_colimit c) :\n  is_limit (cone_of_cocone_unop c) :=\n{ lift := λ s, (hc.desc (cocone_unop_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.unop : J ⥤ C` into a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F.unop} (hc : is_limit c) :\n  is_colimit (cocone_of_cone_unop c) :=\n{ desc := λ s, (hc.lift (cone_unop_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a colimit, we can construct a limit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_limit_of_has_colimit_left_op (F : J ⥤ Cᵒᵖ) [has_colimit F.left_op] : has_limit F :=\nhas_limit.mk\n{ cone := cone_of_cocone_left_op (colimit.cocone F.left_op),\n  is_limit := is_limit_cone_of_cocone_left_op _ (colimit.is_colimit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\nlemma has_limits_of_shape_op_of_has_colimits_of_shape [has_colimits_of_shape Jᵒᵖ C] :\n  has_limits_of_shape J Cᵒᵖ :=\n{ has_limit := λ F, has_limit_of_has_colimit_left_op F }\n\nlocal attribute [instance] has_limits_of_shape_op_of_has_colimits_of_shape\n\n/--\nIf `C` has colimits, we can construct limits for `Cᵒᵖ`.\n-/\nlemma has_limits_op_of_has_colimits [has_colimits C] : has_limits Cᵒᵖ := ⟨infer_instance⟩\n\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a limit, we can construct a colimit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_colimit_of_has_limit_left_op (F : J ⥤ Cᵒᵖ) [has_limit F.left_op] : has_colimit F :=\nhas_colimit.mk\n{ cocone := cocone_of_cone_left_op (limit.cone F.left_op),\n  is_colimit := is_colimit_cocone_of_cone_left_op _ (limit.is_limit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\nlemma has_colimits_of_shape_op_of_has_limits_of_shape [has_limits_of_shape Jᵒᵖ C] :\n  has_colimits_of_shape J Cᵒᵖ :=\n{ has_colimit := λ F, has_colimit_of_has_limit_left_op F }\n\nlocal attribute [instance] has_colimits_of_shape_op_of_has_limits_of_shape\n\n/--\nIf `C` has limits, we can construct colimits for `Cᵒᵖ`.\n-/\nlemma has_colimits_op_of_has_limits [has_limits C] : has_colimits Cᵒᵖ := ⟨infer_instance⟩\n\nvariables (X : Type v₁)\n/--\nIf `C` has products indexed by `X`, then `Cᵒᵖ` has coproducts indexed by `X`.\n-/\nlemma has_coproducts_opposite [has_products_of_shape X C] :\n  has_coproducts_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape (discrete X)ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\n/--\nIf `C` has coproducts indexed by `X`, then `Cᵒᵖ` has products indexed by `X`.\n-/\nlemma has_products_opposite [has_coproducts_of_shape X C] :\n  has_products_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape (discrete X)ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\nlemma has_finite_coproducts_opposite [has_finite_products C] :\n  has_finite_coproducts Cᵒᵖ :=\n{ out := λ J 𝒟, begin\n    resetI,\n    haveI : has_limits_of_shape (discrete J)ᵒᵖ C :=\n      has_limits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlemma has_finite_products_opposite [has_finite_coproducts C] :\n  has_finite_products Cᵒᵖ :=\n{ out := λ J 𝒟, begin\n    resetI,\n    haveI : has_colimits_of_shape (discrete J)ᵒᵖ C :=\n      has_colimits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlemma has_equalizers_opposite [has_coequalizers C] : has_equalizers Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape walking_parallel_pair.{v₁}ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence walking_parallel_pair_op_equiv.{v₁},\n  apply_instance\nend\n\nlemma has_coequalizers_opposite [has_equalizers C] : has_coequalizers Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape walking_parallel_pair.{v₁}ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence walking_parallel_pair_op_equiv.{v₁},\n  apply_instance\nend\n\nlemma has_finite_colimits_opposite [has_finite_limits C] :\n  has_finite_colimits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\nlemma has_finite_limits_opposite [has_finite_colimits C] :\n  has_finite_limits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\nlemma has_pullbacks_opposite [has_pushouts C] : has_pullbacks Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape walking_cospan.{v₁}ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence walking_cospan_op_equiv.symm,\n  apply has_limits_of_shape_op_of_has_colimits_of_shape,\nend\n\nlemma has_pushouts_opposite [has_pullbacks C] : has_pushouts Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape walking_span.{v₁}ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence walking_span_op_equiv.symm,\n  apply has_colimits_of_shape_op_of_has_limits_of_shape,\nend\n\nend category_theory.limits\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573377, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.32582021546541245}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\nThis file provides an alternative implementation for `apply` to fix the so-called \"apply bug\".\n\nThe issue arises when the goals is a Π-type -- whether it is visible or hidden behind a definition.\n\nFor instance, consider the following proof:\n\n```\nexample {α β} (x y z : α → β) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z :=\nbegin\n  apply le_trans,\nend\n```\n\nBecause `x ≤ z` is definitionally equal to `∀ i, x i ≤ z i`, `apply` will fail. The alternative\ndefinition, `apply'` fixes this. When `apply` would work, `apply` is used and otherwise,\na different strategy is deployed\n-/\n\nnamespace tactic\n\n\n/-- With `gs` a list of proof goals, `reorder_goals gs new_g` will use the `new_goals` policy\n`new_g` to rearrange the dependent goals to either drop them, push them to the end of the list\nor leave them in place. The `bool` values in `gs` indicates whether the goal is dependent or not. -/\ndef reorder_goals {α : Type u_1} (gs : List (Bool × α)) : new_goals → List α := sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/apply_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3258078876680581}}
{"text": "-- definitions that are used by the lemmas but not referenced by the a theorem\n\nimport .definitions2\n\n-- (∀x {call(x)} ⇒ P) ∈ CallQuantifiers\nstructure callquantifier := (x: var) (P: prop)\n\n-- s ∈ DStacks := (R, σ, e) | s · (R, σ, let y = f(x) in e)\ninductive dstack\n| top  : spec → env → exp → dstack\n| cons : dstack → spec → env → var → var → var → exp → dstack\n\n-- (R, σ, e) : dstack\ninstance : has_coe (spec × env × exp) dstack := ⟨λe, dstack.top e.1 e.2.1 e.2.2⟩\n\n-- stack precondition projection\n\ndef dstack.pre: dstack → spec\n| (dstack.top R _ _) := R\n| (dstack.cons _ R _ _ _ _ _) := R\n\nlemma sizeof_substack {s: dstack} {R: spec} {σ: env} {f x y: var} {e: exp}:\n      s.sizeof < (dstack.cons s R σ f x y e).sizeof :=\n  begin\n    unfold dstack.sizeof,\n    change sizeof s < 1 + sizeof s + sizeof R + sizeof σ + sizeof f + sizeof x + sizeof y + sizeof e,\n    rw[add_assoc],\n    rw[add_assoc],\n    rw[add_assoc],\n    rw[add_assoc],\n    rw[add_assoc],\n    rw[add_assoc],\n    rw[add_comm],\n    rw[add_assoc],\n    apply lt_add_of_pos_right,\n    rw[add_comm],\n    rw[add_comm],\n    apply lt_add_of_le_of_pos nonneg_of_nat,\n    from zero_lt_one\n  end\n\n-- top-level calls and quantifiers in positive and negative positions\nmutual inductive prop.has_call_p, prop.has_call_n\n\nwith prop.has_call_p: calltrigger → prop → Prop\n| calltrigger {x: term}                                 : prop.has_call_p ⟨x⟩ (prop.call x)\n| not {P: prop} {c: calltrigger}                        : prop.has_call_n c P  → prop.has_call_p c P.not\n| and₁ {P₁ P₂: prop} {c: calltrigger}                   : prop.has_call_p c P₁ → prop.has_call_p c (P₁ ⋀ P₂)\n| and₂ {P₁ P₂: prop} {c: calltrigger}                   : prop.has_call_p c P₂ → prop.has_call_p c (P₁ ⋀ P₂)\n| or₁ {P₁ P₂: prop} {c: calltrigger}                    : prop.has_call_p c P₁ → prop.has_call_p c (P₁ ⋁ P₂)\n| or₂ {P₁ P₂: prop} {c: calltrigger}                    : prop.has_call_p c P₂ → prop.has_call_p c (P₁ ⋁ P₂)\n\nwith prop.has_call_n: calltrigger → prop → Prop\n| not {P: prop} {c: calltrigger}                        : prop.has_call_p c P  → prop.has_call_n c P.not\n| and₁ {P₁ P₂: prop} {c: calltrigger}                   : prop.has_call_n c P₁ → prop.has_call_n c (P₁ ⋀ P₂)\n| and₂ {P₁ P₂: prop} {c: calltrigger}                   : prop.has_call_n c P₂ → prop.has_call_n c (P₁ ⋀ P₂)\n| or₁ {P₁ P₂: prop} {c: calltrigger}                    : prop.has_call_n c P₁ → prop.has_call_n c (P₁ ⋁ P₂)\n| or₂ {P₁ P₂: prop} {c: calltrigger}                    : prop.has_call_n c P₂ → prop.has_call_n c (P₁ ⋁ P₂)\n\n-- sets of calls\ndef calls_p (P: prop): set calltrigger := λc, prop.has_call_p c P\ndef calls_n (P: prop): set calltrigger := λc, prop.has_call_n c P\n\nmutual inductive prop.has_quantifier_p, prop.has_quantifier_n\n\nwith prop.has_quantifier_p: callquantifier → prop → Prop\n| callquantifier {x: var} {P: prop}           : prop.has_quantifier_p ⟨x, P⟩ (prop.forallc x P)\n| not {P: prop} {q: callquantifier}           : prop.has_quantifier_n q P  → prop.has_quantifier_p q P.not\n| and₁ {P₁ P₂: prop} {q: callquantifier}      : prop.has_quantifier_p q P₁ → prop.has_quantifier_p q (P₁ ⋀ P₂)\n| and₂ {P₁ P₂: prop} {q: callquantifier}      : prop.has_quantifier_p q P₂ → prop.has_quantifier_p q (P₁ ⋀ P₂)\n| or₁ {P₁ P₂: prop} {q: callquantifier}       : prop.has_quantifier_p q P₁ → prop.has_quantifier_p q (P₁ ⋁ P₂)\n| or₂ {P₁ P₂: prop} {q: callquantifier}       : prop.has_quantifier_p q P₂ → prop.has_quantifier_p q (P₁ ⋁ P₂)\n\nwith prop.has_quantifier_n: callquantifier → prop → Prop\n| not {P: prop} {q: callquantifier}           : prop.has_quantifier_p q P  → prop.has_quantifier_n q P.not\n| and₁ {P₁ P₂: prop} {q: callquantifier}      : prop.has_quantifier_n q P₁ → prop.has_quantifier_n q (P₁ ⋀ P₂)\n| and₂ {P₁ P₂: prop} {q: callquantifier}      : prop.has_quantifier_n q P₂ → prop.has_quantifier_n q (P₁ ⋀ P₂)\n| or₁ {P₁ P₂: prop} {q: callquantifier}       : prop.has_quantifier_n q P₁ → prop.has_quantifier_n q (P₁ ⋁ P₂)\n| or₂ {P₁ P₂: prop} {q: callquantifier}       : prop.has_quantifier_n q P₂ → prop.has_quantifier_n q (P₁ ⋁ P₂)\n-- universal quantifiers below existential quantifiers only occur in positive positions,\n-- so can be skolemized instead of instantiated\n\n-- sets of quantifiers\ndef quantifiers_p (P: prop): set callquantifier := λc, has_quantifier_p c P\ndef quantifiers_n (P: prop): set callquantifier := λc, has_quantifier_n c P\ndef quantifiers (P: prop): set callquantifier := quantifiers_p P ∪ quantifiers_n P\n\n -- propositions without call triggers or quantifiers do not participate in instantiations\ndef no_instantiations(P: prop): Prop := (calls_p P = ∅) ∧ (calls_n P = ∅) ∧\n                                        (quantifiers_p P = ∅) ∧ (quantifiers_n P = ∅)\n\n-- set of calltriggers after substitution\ndef calltrigger.subst (σ: env) (c: calltrigger): calltrigger := ⟨term.subst_env σ c.x⟩\ndef calls_p_subst (σ: env) (P: prop): set calltrigger := (calltrigger.subst σ) '' calls_p P\ndef calls_n_subst (σ: env) (P: prop): set calltrigger := (calltrigger.subst σ) '' calls_n P\n\n-- uses variables (either free or quantified)\n\ninductive prop.uses_var (x: var) : prop → Prop\n| term {t: term}                        : free_in_term x t  → prop.uses_var t\n| not {P: prop}                         : prop.uses_var P   → prop.uses_var (prop.not P)\n| and₁ {P₁ P₂: prop}                    : prop.uses_var P₁  → prop.uses_var (prop.and P₁ P₂)\n| and₂ {P₁ P₂: prop}                    : prop.uses_var P₂  → prop.uses_var (prop.and P₁ P₂)\n| or₁ {P₁ P₂: prop}                     : prop.uses_var P₁  → prop.uses_var (prop.or P₁ P₂)\n| or₂ {P₁ P₂: prop}                     : prop.uses_var P₂  → prop.uses_var (prop.or P₁ P₂)\n| pre₁ {t₁ t₂: term}                    : free_in_term x t₁ → prop.uses_var (prop.pre t₁ t₂)\n| pre₂ {t₁ t₂: term}                    : free_in_term x t₂ → prop.uses_var (prop.pre t₁ t₂)\n| preop {t: term} {op: unop}            : free_in_term x t  → prop.uses_var (prop.pre₁ op t)\n| preop₁ {t₁ t₂: term} {op: binop}      : free_in_term x t₁ → prop.uses_var (prop.pre₂ op t₁ t₂)\n| preop₂ {t₁ t₂: term} {op: binop}      : free_in_term x t₂ → prop.uses_var (prop.pre₂ op t₁ t₂)\n| post₁ {t₁ t₂: term}                   : free_in_term x t₁ → prop.uses_var (prop.post t₁ t₂)\n| post₂ {t₁ t₂: term}                   : free_in_term x t₂ → prop.uses_var (prop.post t₁ t₂)\n| call {t: term}                        : free_in_term x t → prop.uses_var (prop.call t)\n| forallc {y: var} {P: prop}            : prop.uses_var P → prop.uses_var (prop.forallc y P)\n| uquantified {P: prop}                 : prop.uses_var (prop.forallc x P)\n| exis {y: var} {P: prop}               : prop.uses_var P → prop.uses_var (prop.exis y P)\n| equantified {P: prop}                 : prop.uses_var (prop.exis x P)\n\ninductive vc.uses_var (x: var) : vc → Prop\n| term {t: term}                        : free_in_term x t  → vc.uses_var t\n| not {P: vc}                           : vc.uses_var P     → vc.uses_var (vc.not P)\n| and₁ {P₁ P₂: vc}                      : vc.uses_var P₁    → vc.uses_var (vc.and P₁ P₂)\n| and₂ {P₁ P₂: vc}                      : vc.uses_var P₂    → vc.uses_var (vc.and P₁ P₂)\n| or₁ {P₁ P₂: vc}                       : vc.uses_var P₁    → vc.uses_var (vc.or P₁ P₂)\n| or₂ {P₁ P₂: vc}                       : vc.uses_var P₂    → vc.uses_var (vc.or P₁ P₂)\n| pre₁ {t₁ t₂: term}                    : free_in_term x t₁ → vc.uses_var (vc.pre t₁ t₂)\n| pre₂ {t₁ t₂: term}                    : free_in_term x t₂ → vc.uses_var (vc.pre t₁ t₂)\n| preop {t: term} {op: unop}            : free_in_term x t  → vc.uses_var (vc.pre₁ op t)\n| preop₁ {t₁ t₂: term} {op: binop}      : free_in_term x t₁ → vc.uses_var (vc.pre₂ op t₁ t₂)\n| preop₂ {t₁ t₂: term} {op: binop}      : free_in_term x t₂ → vc.uses_var (vc.pre₂ op t₁ t₂)\n| post₁ {t₁ t₂: term}                   : free_in_term x t₁ → vc.uses_var (vc.post t₁ t₂)\n| post₂ {t₁ t₂: term}                   : free_in_term x t₂ → vc.uses_var (vc.post t₁ t₂)\n| univ {y: var} {P: vc}                 : vc.uses_var P → vc.uses_var (vc.univ y P)\n| quantified {P: vc}                    : vc.uses_var (vc.univ x P)\n\n-- evaluation relation that includes the current function precondition for each stack frame\n\ninductive dstep : dstack → dstack → Prop\nnotation s₁ `⟹` s₂:100 := dstep s₁ s₂\n\n| ctx {s s': dstack} {R: spec} {σ: env} {y f x: var} {e: exp}:\n    (s ⟹ s') →\n    (dstack.cons s R σ y f x e ⟹ dstack.cons s' R σ y f x e)\n\n| tru {R: spec} {σ: env} {x: var} {e: exp}:\n    (R, σ, lett x = true in e) ⟹ (R, σ[x↦value.true], e)\n\n| fals {R: spec} {σ: env} {x: var} {e: exp}:\n    (R, σ, letf x = false in e) ⟹ (R, σ[x↦value.false], e)\n\n| num {R: spec} {σ: env} {x: var} {e: exp} {n: ℤ}:\n    (R, σ, letn x = n in e) ⟹ (R, σ[x↦value.num n], e)\n\n| closure {σ: env} {R' R S: spec} {f x: var} {e₁ e₂: exp}:\n    (R', σ, letf f[x] req R ens S {e₁} in e₂) ⟹ \n    (R', σ[f↦value.func f x R S e₁ σ], e₂)\n\n| unop {R: spec} {op: unop} {σ: env} {x y: var} {e: exp} {v₁ v: value}:\n    (σ x = v₁) →\n    (unop.apply op v₁ = v) →\n    ((R, σ, letop y = op [x] in e) ⟹ (R, σ[y↦v], e))\n\n| binop {R: spec} {op: binop} {σ: env} {x y z: var} {e: exp} {v₁ v₂ v: value}:\n    (σ x = v₁) →\n    (σ y = v₂) →\n    (binop.apply op v₁ v₂ = v) →\n    ((R, σ, letop2 z = op [x, y] in e) ⟹ (R, σ[z↦v], e))\n\n| app {σ σ': env} {R' R S: spec} {f g x y z: var} {e e': exp} {v: value}:\n    (σ f = value.func g z R S e σ') →\n    (σ x = v) →\n    ((R', σ, letapp y = f[x] in e') ⟹\n    (dstack.cons (R, (σ'[g↦value.func g z R S e σ'][z↦v]), e) R' σ y f x e'))\n\n| return {σ₁ σ₂ σ₃: env} {f g gx x y z: var} {R₁ R₂ R S: spec} {e e': exp} {v vₓ: value}:\n    (σ₁ z = v) →\n    (σ₂ f = value.func g gx R S e σ₃) →\n    (σ₂ x = vₓ) →\n    (dstack.cons (R₁, σ₁, exp.return z) R₂ σ₂ y f x e' ⟹ (R₂, σ₂[y↦v], e'))\n\n| ite_true {R: spec} {σ: env} {e₁ e₂: exp} {x: var}:\n    (σ x = value.true) →\n    ((R, σ, exp.ite x e₁ e₂) ⟹ (R, σ, e₁))\n\n| ite_false {R: spec} {σ: env} {e₁ e₂: exp} {x: var}:\n    (σ x = value.false) →\n    ((R, σ, exp.ite x e₁ e₂) ⟹ (R, σ, e₂))\n\nnotation s₁ `⟹` s₂:100 := dstep s₁ s₂\n\n-- transitive closure\ninductive trans_dstep : dstack → dstack → Prop\nnotation s `⟹*` s':100 := trans_dstep s s'\n| rfl {s: dstack}          : s ⟹* s\n| trans {s s' s'': dstack} : (s ⟹* s') → (s' ⟹ s'') → (s ⟹* s'')\n\nnotation s `⟹*` s':100 := trans_dstep s s'\n\ndef is_dvalue (s: dstack) :=\n  ∃(R: spec) (σ: env) (x: var) (v: value), s = (R, σ, exp.return x) ∧ (σ x = v)\n\n-- runtime verification of stacks\n\ninductive stack.dvcgen : dstack → propctx → Prop\nnotation `⊩ₛ` s `:` Q : 10 := stack.dvcgen s Q\n\n| top {R: spec} {P: prop} {σ: env} {e: exp} {Q: propctx}:\n    (⊩ σ : P) →\n    FV R.to_prop ⊆ FV P →\n    (σ ⊨ R.to_prop.to_vc) →\n    (R ⋀ P ⊩ e : Q) →\n    (⊩ₛ (R, σ, e) : P ⋀ Q)\n\n| cons {P₁ P₂ P₃: prop} {s: dstack} {σ₁ σ₂: env}\n       {f fx g x y: var} {R₁ R₂ S₂: spec} {e₁ e₂: exp} {v: value} {Q₁ Q₂ Q₂': propctx}:\n    (⊩ₛ s : Q₂') →\n    y ∉ σ₁ →\n    (⊩ σ₁ : P₁) →\n    (⊩ σ₂ : P₂ ) →\n    (⊩ (σ₂[f↦value.func f fx R₂ S₂ e₂ σ₂][fx↦v]) : P₃) →\n    FV R₁.to_prop ⊆ FV P₁ →\n    (σ₁ ⊨ R₁.to_prop.to_vc) →\n    (σ₁ g = value.func f fx R₂ S₂ e₂ σ₂) →\n    (σ₁ x = v) →\n    (R₁ ⋀ P₁ ⋀ prop.call x ⋀ prop.post g x ⋀ y ≡ term.app g x ⊩ e₁ : Q₁) →\n    (P₂ ⋀ spec.func f fx R₂ S₂ ⋀ R₂ ⊩ e₂ : Q₂) →\n    (∀σ' t, (σ' ⊨ (Q₂' t).to_vc) → σ' ⊨ (P₃ ⋀ (Q₂ t)).to_vc) →\n    (∀v: value, FV (P₃ ⋀ (Q₂ v)) ⊆ FV (Q₂' v)) →\n    ⦃ prop.implies (R₁ ⋀ P₁ ⋀ prop.call x) (term.unop unop.isFunc g ⋀ prop.pre g x) ⦄ →\n    ((R₂, σ₂[f↦value.func f fx R₂ S₂ e₂ σ₂][fx↦v], e₂) ⟹* s) →\n    (⊩ₛ dstack.cons s R₁ σ₁ y g x e₁ : P₁ ⋀\n          propctx.exis y (prop.call x ⋀ prop.post g x ⋀ y ≡ term.app g x ⋀ Q₁))\n\nnotation `⊩ₛ` s `:` Q : 10 := stack.dvcgen s Q\n\ninductive stack_equiv_dstack : stack → dstack → Prop\n| top {R: spec} {σ: env} {e: exp} :\n  stack_equiv_dstack (stack.top σ e) (dstack.top R σ e)\n| cons {s': stack} {d': dstack} {R: spec} {σ: env} {f x y: var} {e: exp}:\n  stack_equiv_dstack s' d' →\n  stack_equiv_dstack (stack.cons s' σ f x y e) (dstack.cons d' R σ f x y e)\n", "meta": {"author": "levjj", "repo": "esverify-theory", "sha": "8565b123c87b0113f83553d7732cd6696c9b5807", "save_path": "github-repos/lean/levjj-esverify-theory", "path": "github-repos/lean/levjj-esverify-theory/esverify-theory-8565b123c87b0113f83553d7732cd6696c9b5807/src/definitions3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3258078803363456}}
{"text": "/-\nCopyright (c) 2022 Yuma Mizuno. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno\n-/\nimport category_theory.bicategory.functor\n\n/-!\n# Oplax natural transformations\n\nJust as there are natural transformations between functors, there are oplax natural transformations\nbetween oplax functors. The equality in the naturality of natural transformations is replaced by a\nspecified 2-morphism `F.map f ≫ app b ⟶ app a ≫ G.map f` in the case of oplax natural\ntransformations.\n\n## Main definitions\n\n* `oplax_nat_trans F G` : oplax natural transformations between oplax functors `F` and `G`\n* `oplax_nat_trans.vcomp η θ` : the vertical composition of oplax natural transformations `η`\n  and `θ`\n* `oplax_nat_trans.category F G` : the category structure on the oplax natural transformations\n  between `F` and `G`\n-/\n\nnamespace category_theory\n\nopen category bicategory\nopen_locale bicategory\n\nuniverses w₁ w₂ v₁ v₂ u₁ u₂\n\nvariables {B : Type u₁} [bicategory.{w₁ v₁} B] {C : Type u₂} [bicategory.{w₂ v₂} C]\n\n/--\nIf `η` is an oplax natural transformation between `F` and `G`, we have a 1-morphism\n`η.app a : F.obj a ⟶ G.obj a` for each object `a : B`. We also have a 2-morphism\n`η.naturality f : F.map f ≫ app b ⟶ app a ≫ G.map f` for each 1-morphism `f : a ⟶ b`.\nThese 2-morphisms satisfies the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure oplax_nat_trans (F G : oplax_functor B C) :=\n(app (a : B) : F.obj a ⟶ G.obj a)\n(naturality {a b : B} (f : a ⟶ b) : F.map f ≫ app b ⟶ app a ≫ G.map f)\n(naturality_naturality' : ∀ {a b : B} {f g : a ⟶ b} (η : f ⟶ g),\n  F.map₂ η ▷ app b ≫ naturality g = naturality f ≫ app a ◁ G.map₂ η . obviously)\n(naturality_id' : ∀ a : B,\n  naturality (𝟙 a) ≫ app a ◁ G.map_id a =\n    F.map_id a ▷ app a ≫ (λ_ (app a)).hom ≫ (ρ_ (app a)).inv . obviously)\n(naturality_comp' : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c),\n  naturality (f ≫ g) ≫ app a ◁ G.map_comp f g =\n    F.map_comp f g ▷ app c ≫ (α_ _ _ _).hom ≫\n      F.map f ◁ naturality g ≫ (α_ _ _ _).inv ≫\n        naturality f ▷ G.map g ≫ (α_ _ _ _).hom . obviously)\n\nrestate_axiom oplax_nat_trans.naturality_naturality'\nrestate_axiom oplax_nat_trans.naturality_id'\nrestate_axiom oplax_nat_trans.naturality_comp'\nattribute [simp, reassoc] oplax_nat_trans.naturality_naturality\n  oplax_nat_trans.naturality_id oplax_nat_trans.naturality_comp\n\nnamespace oplax_nat_trans\n\nsection\nvariables (F : oplax_functor B C)\n\n/-- The identity oplax natural transformation. -/\n@[simps]\ndef id : oplax_nat_trans F F :=\n{ app := λ a, 𝟙 (F.obj a),\n  naturality := λ a b f, (ρ_ (F.map f)).hom ≫ (λ_ (F.map f)).inv }\n\ninstance : inhabited (oplax_nat_trans F F) := ⟨id F⟩\n\nvariables {F} {G H : oplax_functor B C} (η : oplax_nat_trans F G) (θ : oplax_nat_trans G H)\n\nsection\nvariables {a b c : B} {a' : C}\n\n@[simp, reassoc]\nlemma whisker_left_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :\n  f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ θ.naturality h =\n    f ◁ θ.naturality g ≫ f ◁ θ.app a ◁ H.map₂ β :=\nby simp_rw [←bicategory.whisker_left_comp, naturality_naturality]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :\n  F.map₂ β ▷ η.app b ▷ h ≫ η.naturality g ▷ h =\n    η.naturality f ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ _ _ _).inv :=\nby rw [←comp_whisker_right, naturality_naturality, comp_whisker_right, whisker_assoc]\n\n@[simp, reassoc]\nlemma whisker_left_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b ⟶ c) :\n  f ◁ θ.naturality (g ≫ h) ≫ f ◁ θ.app a ◁ H.map_comp g h =\n    f ◁ G.map_comp g h ▷ θ.app c ≫ f ◁ (α_ _ _ _).hom ≫\n      f ◁ G.map g ◁ θ.naturality h ≫ f ◁ (α_ _ _ _).inv ≫\n        f ◁ θ.naturality g ▷ H.map h ≫ f ◁ (α_ _ _ _).hom :=\nby simp_rw [←bicategory.whisker_left_comp, naturality_comp]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :\n  η.naturality (f ≫ g) ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map_comp f g ▷ h =\n    F.map_comp f g ▷ η.app c ▷ h ≫ (α_ _ _ _).hom ▷ h ≫ (α_ _ _ _).hom ≫\n      F.map f ◁ η.naturality g ▷ h ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).inv ▷ h ≫\n        η.naturality f ▷ G.map g ▷ h ≫ (α_ _ _ _).hom ▷ h ≫ (α_ _ _ _).hom :=\nby { rw [←associator_naturality_middle, ←comp_whisker_right_assoc, naturality_comp], simp }\n\n@[simp, reassoc]\nlemma whisker_left_naturality_id (f : a' ⟶ G.obj a) :\n  f ◁ θ.naturality (𝟙 a) ≫ f ◁ θ.app a ◁ H.map_id a =\n    f ◁ G.map_id a ▷ θ.app a ≫ f ◁ (λ_ (θ.app a)).hom ≫ f ◁ (ρ_ (θ.app a)).inv :=\nby simp_rw [←bicategory.whisker_left_comp, naturality_id]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_id (f : G.obj a ⟶ a') :\n  η.naturality (𝟙 a) ▷ f ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map_id a ▷ f =\n    F.map_id a ▷ η.app a ▷ f ≫ (λ_ (η.app a)).hom ▷ f ≫\n      (ρ_ (η.app a)).inv ▷ f ≫ (α_ _ _ _).hom :=\nby { rw [←associator_naturality_middle, ←comp_whisker_right_assoc, naturality_id], simp }\n\nend\n\n/-- Vertical composition of oplax natural transformations. -/\n@[simps]\ndef vcomp (η : oplax_nat_trans F G) (θ : oplax_nat_trans G H) : oplax_nat_trans F H :=\n{ app := λ a, η.app a ≫ θ.app a,\n  naturality := λ a b f,\n    (α_ _ _ _).inv ≫ η.naturality f ▷ θ.app b ≫ (α_ _ _ _).hom ≫\n      η.app a ◁ θ.naturality f ≫ (α_ _ _ _).inv,\n  naturality_comp' := λ a b c f g, by\n  { calc _ =  _ ≫\n    F.map_comp f g ▷ η.app c ▷ θ.app c ≫ _ ≫\n      F.map f ◁ η.naturality g ▷ θ.app c ≫ _ ≫\n        (F.map f ≫ η.app b) ◁ θ.naturality g ≫\n          η.naturality f ▷ (θ.app b ≫ H.map g) ≫ _ ≫\n            η.app a ◁ θ.naturality f ▷ H.map g ≫ _  : _\n    ... =  _ : _,\n    exact (α_ _ _ _).inv,\n    exact (α_ _ _ _).hom ▷ _ ≫ (α_ _ _ _).hom,\n    exact _ ◁ (α_ _ _ _).hom ≫ (α_ _ _ _).inv,\n    exact (α_ _ _ _).hom ≫ _ ◁ (α_ _ _ _).inv,\n    exact _ ◁ (α_ _ _ _).hom ≫ (α_ _ _ _).inv,\n    { rw [whisker_exchange_assoc], simp },\n    { simp } } }\n\nvariables (B C)\n\n@[simps]\ninstance : category_struct (oplax_functor B C) :=\n{ hom  := oplax_nat_trans,\n  id   := oplax_nat_trans.id,\n  comp := λ F G H, oplax_nat_trans.vcomp }\n\nend\n\nsection\nvariables {F G : oplax_functor B C}\n\n/--\nA modification `Γ` between oplax natural transformations `η` and `θ` consists of a family of\n2-morphisms `Γ.app a : η.app a ⟶ θ.app a`, which satisfies the equation\n`(F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f)`\nfor each 1-morphism `f : a ⟶ b`.\n-/\n@[ext]\nstructure modification (η θ : F ⟶ G) :=\n(app (a : B) : η.app a ⟶ θ.app a)\n(naturality' : ∀ {a b : B} (f : a ⟶ b),\n  (F.map f ◁ app b) ≫ θ.naturality f = η.naturality f ≫ (app a ▷ G.map f) . obviously)\n\nrestate_axiom modification.naturality'\nattribute [simp, reassoc] modification.naturality\n\nvariables {η θ ι : F ⟶ G}\n\nnamespace modification\n\nvariables (η)\n\n/-- The identity modification. -/\n@[simps]\ndef id : modification η η := { app := λ a, 𝟙 (η.app a) }\n\ninstance : inhabited (modification η η) := ⟨modification.id η⟩\n\nvariables {η}\n\nsection\nvariables (Γ : modification η θ) {a b c : B} {a' : C}\n\n@[simp, reassoc]\nlemma whisker_left_naturality (f : a' ⟶ F.obj b) (g : b ⟶ c) :\n  f ◁ F.map g ◁ Γ.app c ≫ f ◁ θ.naturality g =\n    f ◁ η.naturality g ≫ f ◁ Γ.app b ▷ G.map g :=\nby simp_rw [←bicategory.whisker_left_comp, naturality]\n\n@[simp, reassoc]\nlemma whisker_right_naturality (f : a ⟶ b) (g : G.obj b ⟶ a') :\n  F.map f ◁ Γ.app b ▷ g ≫ (α_ _ _ _).inv ≫ θ.naturality f ▷ g =\n    (α_ _ _ _).inv ≫ η.naturality f ▷ g ≫ Γ.app a ▷ G.map f ▷ g :=\nby simp_rw [associator_inv_naturality_middle_assoc, ←comp_whisker_right, naturality]\n\nend\n\n/-- Vertical composition of modifications. -/\n@[simps]\ndef vcomp (Γ : modification η θ) (Δ : modification θ ι) : modification η ι :=\n{ app := λ a, Γ.app a ≫ Δ.app a }\n\nend modification\n\n/-- Category structure on the oplax natural transformations between oplax_functors. -/\n@[simps]\ninstance category (F G : oplax_functor B C) : category (F ⟶ G) :=\n{ hom  := modification,\n  id   := modification.id,\n  comp := λ η θ ι, modification.vcomp }\n\n/--\nConstruct a modification isomorphism between oplax natural transformations\nby giving object level isomorphisms, and checking naturality only in the forward direction.\n-/\n@[simps]\ndef modification_iso.of_components\n  (app : ∀ a, η.app a ≅ θ.app a)\n  (naturality : ∀ {a b} (f : a ⟶ b),\n    F.map f ◁ (app b).hom ≫ θ.naturality f = η.naturality f ≫ (app a).hom ▷ G.map f) :\n  η ≅ θ :=\n{ hom := { app := λ a, (app a).hom },\n  inv :=\n  { app := λ a, (app a).inv,\n    naturality' := λ a b f, by simpa using\n      congr_arg (λ f, _ ◁ (app b).inv ≫ f ≫ (app a).inv ▷ _) (naturality f).symm } }\n\nend\n\nend oplax_nat_trans\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/bicategory/natural_transformation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.32580788033634556}}
{"text": "import .definitions3 .logic\n\nlemma exp.progress {R: spec} {P: prop} {Q: propctx} {e: exp} {σ: env}:\n      (⊩ σ: P) → (FV R.to_prop ⊆ FV P) → (σ ⊨ R.to_prop.to_vc) → (R ⋀ P ⊩ e: Q) → \n      is_dvalue (R, σ, e) ∨ ∃s', (R, σ, e) ⟹ s' :=\n  assume env_verified: (⊩ σ : P),\n  assume fv_R: FV R.to_prop ⊆ FV P,\n  assume R_valid: (σ ⊨ R.to_prop.to_vc),\n  assume e_verified: (R ⋀ P ⊩ e : Q),\n  have R_closed: closed_subst σ R.to_prop, from (\n    assume z: var,\n    assume : z ∈ FV R.to_prop,\n    have z ∈ FV P, from set.mem_of_subset_of_mem fv_R this,\n    show z ∈ σ.dom, from (free_iff_contains env_verified).symm ▸ this\n  ),\n  show is_dvalue (R, σ, e) ∨ ∃s', (R, σ, e) ⟹ s', by begin\n    cases e_verified,\n    case exp.dvcgen.tru x e' { from\n      let s: dstack := (R, σ, lett x = true in e') in\n      have s ⟹ (R, σ[x↦value.true], e'), from dstep.tru,\n      have ∃s', s ⟹ s', from exists.intro (R, σ[x↦value.true], e') this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.fals x e' { from\n      let s: dstack := (R, σ, letf x = false in e') in\n      have s ⟹ (R, σ[x↦value.false], e'), from dstep.fals,\n      have ∃s', s ⟹ s', from exists.intro (R, σ[x↦value.false], e') this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.num x n e' { from\n      let s: dstack := (R, σ, letn x = n in e') in\n      have s ⟹ (R, σ[x↦value.num n], e'), from dstep.num,\n      have ∃s', s ⟹ s', from exists.intro (R, σ[x↦value.num n], e') this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.func f x R' S' e₁ e₂ { from\n      let s: dstack := (R, σ, letf f[x] req R' ens S' {e₁} in e₂) in\n      have s ⟹ (R, σ[f↦value.func f x R' S' e₁ σ], e₂), from dstep.closure,\n      have ∃s', s ⟹ s', from exists.intro (R, σ[f↦value.func f x R' S' e₁ σ], e₂) this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.unop op x y e' Q' x_free_in_P _ e'_verified vc_valid { from\n      let s: dstack := (R, σ, letop y = op[x] in e') in\n      let ⟨v, env_has_x⟩ := (val_of_free_in_pre_env env_verified fv_R x_free_in_P) in\n      have closed_subst σ (prop.pre₁ op x), from (\n        assume z: var,\n        assume : z ∈ FV (prop.pre₁ op x),\n        have free_in_term z x, from free_in_prop.pre₁.inv this,\n        have z = x, from free_in_term.var.inv this,\n        show z ∈ σ, from this.symm ▸ env.contains_apply_equiv.right.mp (exists.intro v env_has_x)\n      ),\n      have h1: σ ⊨ (prop.pre₁ op x).to_vc, from consequent_of_pre_P env_verified R_closed R_valid this vc_valid,\n      have (prop.pre₁ op x).to_vc = vc.pre₁ op x, by unfold prop.to_vc,\n      have h2: ⊨ vc.subst_env σ (vc.pre₁ op x), from this ▸ h1,\n      have vc.subst_env σ (vc.pre₁ op x) = vc.pre₁ op (term.subst_env σ x), from vc.subst_env.pre₁,\n      have ⊨ vc.pre₁ op (term.subst_env σ x), from this ▸ h2,\n      have x_from_env: term.subst_env σ x = v, from (term.subst_env.var.right v).mp env_has_x,\n      have ⊨ vc.pre₁ op v, from x_from_env ▸ this,\n      have option.is_some (unop.apply op v), from valid.pre₁ this,\n      have (∃v', unop.apply op v = some v'), from option.is_some_iff_exists.mp this,\n      let ⟨v', op_v_is_v'⟩ := this in\n      have s ⟹ (R, σ[y↦v'], e'), from dstep.unop env_has_x op_v_is_v',\n      have ∃s', s ⟹ s', from exists.intro (R, σ[y↦v'], e') this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.binop op x y z e' Q' x_free_in_P y_free_in_P _ e'_verified vc_valid { from\n      let s: dstack := (R, σ, letop2 z = op[x,y] in e') in\n      let ⟨vx, env_has_x⟩ := (val_of_free_in_pre_env env_verified fv_R x_free_in_P) in\n      let ⟨vy, env_has_y⟩ := (val_of_free_in_pre_env env_verified fv_R y_free_in_P) in\n      have closed_subst σ (prop.pre₂ op x y), from (\n        assume z: var,\n        assume : z ∈ FV (prop.pre₂ op x y),\n        or.elim (free_in_prop.pre₂.inv this) (\n          assume : free_in_term z x,\n          have z = x, from free_in_term.var.inv this,\n          show z ∈ σ, from this.symm ▸ env.contains_apply_equiv.right.mp (exists.intro vx env_has_x)\n        ) (\n          assume : free_in_term z y,\n          have z = y, from free_in_term.var.inv this,\n          show z ∈ σ, from this.symm ▸ env.contains_apply_equiv.right.mp (exists.intro vy env_has_y)\n        )\n      ),\n      have h1: σ ⊨ (prop.pre₂ op x y).to_vc, from consequent_of_pre_P env_verified R_closed R_valid this vc_valid,\n      have (prop.pre₂ op x y).to_vc = vc.pre₂ op x y, by unfold prop.to_vc,\n      have h2: ⊨ vc.subst_env σ (vc.pre₂ op x y), from this ▸ h1,\n      have vc.subst_env σ (vc.pre₂ op x y) = vc.pre₂ op (term.subst_env σ x) (term.subst_env σ y),\n      from vc.subst_env.pre₂,\n      have h3: ⊨ vc.pre₂ op (term.subst_env σ x) (term.subst_env σ y), from this ▸ h2,\n      have term.subst_env σ x = vx, from (term.subst_env.var.right vx).mp env_has_x,\n      have h4: ⊨ vc.pre₂ op vx (term.subst_env σ y), from this ▸ h3,\n      have term.subst_env σ y = vy, from (term.subst_env.var.right vy).mp env_has_y,\n      have ⊨ vc.pre₂ op vx vy, from this ▸ h4,\n      have option.is_some (binop.apply op vx vy), from valid.pre₂ this,\n      have (∃v', binop.apply op vx vy = some v'), from option.is_some_iff_exists.mp this,\n      let ⟨v', op_v_is_v'⟩ := this in\n      have s ⟹ (R, σ[z↦v'], e'), from dstep.binop env_has_x env_has_y op_v_is_v',\n      have ∃s', s ⟹ s', from exists.intro (R, σ[z↦v'], e') this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.app y f x e' Q' f_free_in_P x_free_in_P _ e'_verified vc_valid { from\n      let s: dstack := (R, σ, letapp y = f [x] in e') in\n      let ⟨vf, env_f_is_vf⟩ := (val_of_free_in_pre_env env_verified fv_R f_free_in_P) in\n      have env_has_f: f ∈ σ, from env.contains_apply_equiv.right.mp (exists.intro vf env_f_is_vf),\n      let ⟨vx, env_x_is_vx⟩ := (val_of_free_in_pre_env env_verified fv_R x_free_in_P) in\n      have env_has_x: x ∈ σ, from env.contains_apply_equiv.right.mp (exists.intro vx env_x_is_vx),\n      have closed_subst σ (↑(term.unop unop.isFunc f) ⋀ prop.pre f x), from (\n        assume z: var,\n        assume : z ∈ FV (↑(term.unop unop.isFunc f) ⋀ prop.pre f x),\n        or.elim (free_in_prop.and.inv this) (\n          assume : free_in_prop z (term.unop unop.isFunc f),\n          have free_in_term z (term.unop unop.isFunc f), from free_in_prop.term.inv this,\n          have free_in_term z f, from free_in_term.unop.inv this,\n          have z = f, from free_in_term.var.inv this,\n          show z ∈ σ, from this.symm ▸ env_has_f\n        ) (\n          assume : z ∈ FV (prop.pre f x),\n          or.elim (free_in_prop.pre.inv this) (\n            assume : free_in_term z f,\n            have z = f, from free_in_term.var.inv this,\n            show z ∈ σ, from this.symm ▸ env_has_f\n          ) (\n            assume : free_in_term z x,\n            have z = x, from free_in_term.var.inv this,\n            show z ∈ σ, from this.symm ▸ env_has_x\n          )\n        )\n      ),\n      have h1: σ ⊨ (↑(term.unop unop.isFunc f) ⋀ prop.pre f x).to_vc,\n      from consequent_of_pre_P_call env_verified R_closed R_valid env_has_x this vc_valid,\n      have (prop.and (prop.term (term.unop unop.isFunc f)) (prop.pre f x)).to_vc\n         = ((prop.term (term.unop unop.isFunc f)).to_vc ⋀ (prop.pre f x).to_vc),\n      by unfold prop.to_vc,\n      have σ ⊨ ((prop.term (term.unop unop.isFunc f)).to_vc ⋀ (prop.pre f x).to_vc), from this ▸ h1,\n      have h2: ⊨ vc.subst_env σ ((prop.term (term.unop unop.isFunc f)).to_vc ⋀ (prop.pre f x).to_vc), from this,\n      have vc.subst_env σ ((prop.term (term.unop unop.isFunc f)).to_vc ⋀ (prop.pre f x).to_vc)\n         = (vc.subst_env σ ((prop.term (term.unop unop.isFunc f)).to_vc) ⋀ vc.subst_env σ ((prop.pre f x).to_vc)),\n      from vc.subst_env.and,\n      have ⊨ (vc.subst_env σ ((prop.term (term.unop unop.isFunc f)).to_vc) ⋀\n              vc.subst_env σ ((prop.pre f x).to_vc)),\n      from this ▸ h2,\n      have h5: σ ⊨ (prop.term (term.unop unop.isFunc f)).to_vc, from (valid.and.mpr this).left,\n      have (prop.term (term.unop unop.isFunc f)).to_vc = vc.term (term.unop unop.isFunc f),\n      by unfold prop.to_vc,\n      have h6: σ ⊨ vc.term (term.unop unop.isFunc f), from this ▸ h5,\n      have vc.subst_env σ (vc.term (term.unop unop.isFunc f)) = vc.term (term.subst_env σ (term.unop unop.isFunc f)),\n      from vc.subst_env.term,\n      have h7: ⊨ vc.term (term.subst_env σ (term.unop unop.isFunc f)), from this ▸ h6,\n      have term.subst_env σ (term.unop unop.isFunc f) = term.unop unop.isFunc (term.subst_env σ f),\n      from term.subst_env.unop,\n      have h8: ⊨ vc.term (term.unop unop.isFunc (term.subst_env σ f)), from this ▸ h7,\n      have term.subst_env σ f = vf, from (term.subst_env.var.right vf).mp env_f_is_vf,\n      have ⊨ term.unop unop.isFunc vf, from this ▸ h8,\n      have ⊨ (value.true ≡ term.unop unop.isFunc vf), from valid.eq.true.mp this,\n      have unop.apply unop.isFunc vf = some value.true, from valid.unop.mpr this,\n      have ∃(g gx: var) (gR gS: spec) (ge: exp) (gσ: env), vf = value.func g gx gR gS ge gσ,\n      from unop.isFunc.inv this,\n      let ⟨g, gx, gR, gS, ge, gσ, vf_is_g⟩ := this in\n      have some_vf_is_g: some vf = ↑(value.func g gx gR gS ge gσ), from some.inj.inv vf_is_g,\n      have σ f = value.func g gx gR gS ge gσ, from eq.trans env_f_is_vf some_vf_is_g,\n      let s': dstack := dstack.cons (gR, gσ[g↦value.func g gx gR gS ge gσ][gx↦vx], ge) R σ y f x e' in\n      have s ⟹ s', from dstep.app this env_x_is_vx,\n      have ∃s', s ⟹ s', from exists.intro s' this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n    },\n    case exp.dvcgen.ite x e₂ e₁ Q₁ Q₂ x_free_in_P _ _ vc_valid { from\n      let s: dstack := (R, σ, exp.ite x e₁ e₂) in\n      let ⟨v, env_has_x⟩ := (val_of_free_in_pre_env env_verified fv_R x_free_in_P) in\n      have closed_subst σ (prop.term (term.unop unop.isBool x)), from (\n        assume z: var,\n        assume : z ∈ FV (prop.term (term.unop unop.isBool x)),\n        have free_in_term z (term.unop unop.isBool x), from free_in_prop.term.inv this,\n        have free_in_term z x, from free_in_term.unop.inv this,\n        have z = x, from free_in_term.var.inv this,\n        show z ∈ σ, from this.symm ▸ env.contains_apply_equiv.right.mp (exists.intro v env_has_x)\n      ),\n      have h1: σ ⊨ (prop.term (term.unop unop.isBool x)).to_vc,\n      from consequent_of_pre_P env_verified R_closed R_valid this vc_valid,\n      have (prop.term (term.unop unop.isBool x)).to_vc = vc.term (term.unop unop.isBool x),\n      by unfold prop.to_vc,\n      have h2: σ ⊨ vc.term (term.unop unop.isBool x), from this ▸ h1,\n      have vc.subst_env σ (vc.term (term.unop unop.isBool x)) = vc.term (term.subst_env σ (term.unop unop.isBool x)),\n      from vc.subst_env.term,\n      have h3: ⊨ vc.term (term.subst_env σ (term.unop unop.isBool x)), from this ▸ h2,\n      have term.subst_env σ (term.unop unop.isBool x) = term.unop unop.isBool (term.subst_env σ x),\n      from term.subst_env.unop,\n      have h4: ⊨ vc.term (term.unop unop.isBool (term.subst_env σ x)), from this ▸ h3,\n      have term.subst_env σ x = v, from (term.subst_env.var.right v).mp env_has_x,\n      have ⊨ term.unop unop.isBool v, from this ▸ h4,\n      have ⊨ (value.true ≡ term.unop unop.isBool v), from valid.eq.true.mp this,\n      have unop.apply unop.isBool v = some value.true, from valid.unop.mpr this,\n      have (v = value.true) ∨ (v = value.false), from unop.isBool.inv this,\n      or.elim this (\n        assume : v = value.true,\n        have some v = some value.true, from some.inj.inv this,\n        have σ x = value.true, from eq.trans env_has_x this,\n        have s ⟹ (R, σ, e₁), from dstep.ite_true this,\n        have ∃s', s ⟹ s', from exists.intro (R, σ, e₁) this,\n        show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n      ) (\n        assume : v = value.false,\n        have some v = some value.false, from some.inj.inv this,\n        have σ x = value.false, from eq.trans env_has_x this,\n        have s ⟹ (R, σ, e₂), from dstep.ite_false this,\n        have ∃s', s ⟹ s', from exists.intro (R, σ, e₂) this,\n        show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n      )\n    },\n    case exp.dvcgen.return x x_free_in_P { from\n      let s: dstack := (R, σ, exp.return x) in\n      have s_is_return: s = (R, σ, exp.return x), from rfl,\n      let ⟨v, env_has_x⟩ := (val_of_free_in_pre_env env_verified fv_R x_free_in_P) in\n      have is_value_s: is_dvalue s\n        = (∃(R': spec) (σ': env) (x': var) (v: value), s = (R', σ', exp.return x') ∧ (σ' x' = v)),\n      by unfold is_dvalue,\n      have (∃(R': spec) (σ': env) (x': var) (v: value), s = (R', σ', exp.return x') ∧ (σ' x' = v)),\n      from exists.intro R (exists.intro σ (exists.intro x (exists.intro v ⟨s_is_return, env_has_x⟩))),\n      have is_dvalue s, from is_value_s ▸ this,\n      show is_dvalue s ∨ ∃s', s ⟹ s', from or.inl this\n    }\n  end\n\ntheorem progress {s: dstack} {Q: propctx}: (⊩ₛ s : Q) → is_dvalue s ∨ ∃s', s ⟹ s'\n:=\n  assume s_verified: ⊩ₛ s : Q,\n  begin\n    induction s_verified,\n    case stack.dvcgen.top σ e R P Q env_verified fv_R R_valid e_verified { from\n      show is_dvalue (R, σ, e) ∨ ∃s', (R, σ, e) ⟹ s', from exp.progress env_verified fv_R R_valid e_verified\n    },\n    case stack.dvcgen.cons P P' P'' s' σ σ' f g x y fx R R' S' e₁ e₂ vₓ Q Q' Q''\n                          s'_verified y_not_in_σ σ_verified σ'_verified σ''_verified fv_R R_valid g_is_func\n                          x_is_v e₁_verified cont Q''_dom Q''_fv pre_vc steps ih { from\n      let s := dstack.cons s' R σ y g x e₁ in\n      have s_cons: s = (dstack.cons s' R σ y g x e₁), from rfl,\n      or.elim ih\n      ( -- step return\n        assume s'_is_value: is_dvalue s',\n        let ⟨R₂, σ₂, z, v, ⟨s'_return, env2_has_z⟩⟩ := s'_is_value in\n        have s_return_cons: s = dstack.cons (R₂, σ₂, exp.return z) R σ y g x e₁,\n        from s'_return ▸ s_cons,\n        have s ⟹ (R, σ[y↦v], e₁),\n        from s_return_cons.symm ▸ (dstep.return env2_has_z g_is_func x_is_v),\n        have ∃s', s ⟹ s',\n        from exists.intro (R, σ[y↦v], e₁) this,\n        show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n      )\n      ( -- step ctx\n        assume s'_can_step: ∃s'', s' ⟹ s'',\n        let ⟨s'', s'_steps⟩ := s'_can_step in\n        have s ⟹ dstack.cons s'' R σ y g x e₁, from dstep.ctx s'_steps,\n        have ∃s', s ⟹ s', from exists.intro (dstack.cons s'' R σ y g x e₁) this,\n        show is_dvalue s ∨ ∃s', s ⟹ s', from or.inr this\n      )\n    }\n  end\n", "meta": {"author": "levjj", "repo": "esverify-theory", "sha": "8565b123c87b0113f83553d7732cd6696c9b5807", "save_path": "github-repos/lean/levjj-esverify-theory", "path": "github-repos/lean/levjj-esverify-theory/esverify-theory-8565b123c87b0113f83553d7732cd6696c9b5807/src/progress.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5234203489363239, "lm_q1q2_score": 0.325807873004633}}
{"text": "import category_theory.abelian.projective\nimport for_mathlib.homotopy_category_pretriangulated\nimport for_mathlib.abelian_category\nimport for_mathlib.derived.homological\nimport for_mathlib.derived.lemmas\nimport for_mathlib.snake_lemma3\nimport for_mathlib.les_homology\nimport for_mathlib.exact_seq3\nimport for_mathlib.triangle_shift\nimport for_mathlib.homology_iso\nimport for_mathlib.projective_replacement\nimport for_mathlib.additive_functor\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.triangulated\nopen homological_complex\n\nuniverses v v' u u'\nvariables {A : Type u} [category.{v} A] [abelian A]\n\nnamespace bounded_homotopy_category\n\ninstance : category (bounded_homotopy_category A) :=\n{ hom := λ X Y, X.val ⟶ Y.val,\n  id := λ X, 𝟙 X.val,\n  comp := λ X Y Z f g, f ≫ g,\n  id_comp' := λ _ _ _, category.id_comp _,\n  comp_id' := λ _ _ _, category.comp_id _,\n  assoc' := λ _ _ _ _ _ _ _, category.assoc _ _ _ }\n\nlocal attribute [instance] has_zero_object.has_zero\n\ninstance (X : bounded_homotopy_category A) : homotopy_category.is_bounded_above X.val := X.bdd\n\ndef of (X : homotopy_category A (complex_shape.up ℤ)) [homotopy_category.is_bounded_above X] :\n  bounded_homotopy_category A := ⟨X⟩\n\nabbreviation hom_val {X Y : bounded_homotopy_category A} (f : X ⟶ Y) : X.val ⟶ Y.val := f\n\nlemma hom_ext {X Y : bounded_homotopy_category A} (f g : X ⟶ Y) (h : hom_val f = hom_val g) :\n  f = g := h\n\n@[simps]\ndef mk_iso {X Y : bounded_homotopy_category A} (i : X.val ≅ Y.val) :\n  X ≅ Y :=\n{ hom := i.hom,\n  inv := i.inv,\n  hom_inv_id' := i.hom_inv_id,\n  inv_hom_id' := i.inv_hom_id, }\n\ninstance : preadditive (bounded_homotopy_category A) :=\n{ hom_group := λ A B, show add_comm_group (A.val ⟶ B.val), by apply_instance,\n  add_comp' := λ P Q R f g h, preadditive.add_comp _ _ _ _ _ _,\n  comp_add' := λ P Q R f g h, preadditive.comp_add _ _ _ _ _ _ }\n\nprotected def zero : bounded_homotopy_category A :=\n{ val := homotopy_category.zero,\n  bdd := ⟨⟨0, λ i _, begin\n    apply limits.is_zero_zero\n  end⟩⟩ }\n\nprotected lemma is_zero_zero :\n  is_zero (bounded_homotopy_category.zero : bounded_homotopy_category A) :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  apply homotopy_category.is_zero_zero.eq_of_src,\nend\n\nlemma zero_val {X : bounded_homotopy_category A} (hX : is_zero X) : is_zero X.val :=\nby rwa is_zero_iff_id_eq_zero at hX ⊢\n\ninstance : has_zero_object (bounded_homotopy_category A) :=\n⟨⟨bounded_homotopy_category.zero, bounded_homotopy_category.is_zero_zero⟩⟩\n\n/-\nlemma is_bounded_shift (X : bounded_homotopy_category A) (i : ℤ) :\n  ∃ (a : ℤ), ∀ j, a ≤ j → is_zero (X.val⟦i⟧.as.X j) :=\nbegin\n  obtain ⟨a,ha⟩ := X.2,\n  use a - i,\n  intros j hj,\n  apply ha,\n  linarith\nend\n-/\n\nlocal attribute [instance] endofunctor_monoidal_category\nlocal attribute [reducible] endofunctor_monoidal_category discrete.add_monoidal\n\ninstance : has_shift (bounded_homotopy_category A) ℤ :=\nhas_shift_mk _ _\n{ F := λ i,\n  { obj := λ X, ⟨X.val⟦(i : ℤ)⟧⟩,\n    map := λ X Y f, f⟦i⟧',\n    map_id' := λ X, (category_theory.shift_functor _ _).map_id _,\n    map_comp' := λ X Y Z f g, (category_theory.shift_functor _ _).map_comp _ _ },\n  ε :=\n  { hom :=\n    { app := λ X, (homotopy_category.shift_ε _).hom.app X.val,\n      naturality' := λ _ _ _, (homotopy_category.shift_ε _).hom.naturality _ },\n    inv :=\n    { app := λ X, (homotopy_category.shift_ε _).inv.app X.val,\n      naturality' := λ _ _ _, (homotopy_category.shift_ε _).inv.naturality _ },\n    hom_inv_id' := begin\n      ext,\n      dsimp,\n      erw [← nat_trans.comp_app, iso.hom_inv_id],\n      refl,\n    end,\n    inv_hom_id' := begin\n      ext,\n      dsimp,\n      erw [← nat_trans.comp_app, iso.inv_hom_id],\n      refl,\n    end },\n  μ := λ m n,\n  { hom :=\n    { app := λ X, (homotopy_category.shift_functor_add _ _ _).hom.app X.val,\n      naturality' := λ _ _ _, (homotopy_category.shift_functor_add _ _ _).hom.naturality _ },\n    inv :=\n    { app := λ X, (homotopy_category.shift_functor_add _ _ _).inv.app X.val,\n      naturality' := λ _ _ _, (homotopy_category.shift_functor_add _ _ _).inv.naturality _ },\n    hom_inv_id' := begin\n      ext,\n      dsimp,\n      erw [← nat_trans.comp_app, iso.hom_inv_id],\n      refl,\n    end,\n    inv_hom_id' := begin\n      ext,\n      dsimp,\n      erw [← nat_trans.comp_app, iso.inv_hom_id],\n      refl,\n    end },\n  associativity := λ m₁ m₂ m₃ X, homotopy_category.has_shift_associativity_aux _ m₁ m₂ m₃ X.val,\n  left_unitality := λ n X, homotopy_category.has_shift_left_unitality_aux _ n X.val,\n  right_unitality := λ n X, homotopy_category.has_shift_right_unitality_aux _ n X.val } .\n\n@[simp] lemma shift_functor_obj_val (X : bounded_homotopy_category A) (i : ℤ) :\n  ((category_theory.shift_functor _ i).obj X).val = X.val⟦i⟧ := rfl\n\ninstance shift_functor_additive (i : ℤ) :\n  (category_theory.shift_functor (bounded_homotopy_category A) i).additive :=\nby constructor\n\ninstance : triangulated.pretriangulated (bounded_homotopy_category A) :=\n{ distinguished_triangles :=\n  -- This could be expresed using `.map_triangle`?\n  { T | triangle.mk (homotopy_category _ _) T.mor₁ T.mor₂ T.mor₃ ∈\n    dist_triang (homotopy_category A (complex_shape.up ℤ)) },\n  isomorphic_distinguished := by async begin\n    intros T₁ hT₁ T₂ e,\n    let S₁ : triangle (homotopy_category _ _) := triangle.mk _ T₁.mor₁ T₁.mor₂ T₁.mor₃,\n    let S₂ : triangle (homotopy_category _ _) := triangle.mk _ T₂.mor₁ T₂.mor₂ T₂.mor₃,\n    let E : S₂ ≅ S₁ :=\n      triangle.iso.of_components\n        ⟨e.hom.hom₁,e.inv.hom₁,_,_⟩\n        ⟨e.hom.hom₂,e.inv.hom₂,_,_⟩\n        ⟨e.hom.hom₃,e.inv.hom₃,_,_⟩\n        _ _ _,\n    apply pretriangulated.isomorphic_distinguished _ _ _ E,\n    apply hT₁,\n\n    { show (e.hom ≫ e.inv).hom₁ = _, rw iso.hom_inv_id, refl },\n    { show (e.inv ≫ e.hom).hom₁ = _, rw iso.inv_hom_id, refl },\n\n    { show (e.hom ≫ e.inv).hom₂ = _, rw iso.hom_inv_id, refl },\n    { show (e.inv ≫ e.hom).hom₂ = _, rw iso.inv_hom_id, refl },\n\n    { show (e.hom ≫ e.inv).hom₃ = _, rw iso.hom_inv_id, refl },\n    { show (e.inv ≫ e.hom).hom₃ = _, rw iso.inv_hom_id, refl },\n\n    { exact e.hom.comm₁ },\n    { exact e.hom.comm₂ },\n    { exact e.hom.comm₃ }\n  end,\n  contractible_distinguished := begin\n    intros X,\n    apply pretriangulated.isomorphic_distinguished _\n      (pretriangulated.contractible_distinguished X.val),\n    delta contractible_triangle,\n    dsimp,\n    refine mk_triangle_iso (iso.refl _) (iso.refl _) _ _ _ _,\n    { dsimp, refine is_zero.iso_zero _, apply zero_val, exact limits.is_zero_zero _ },\n    all_goals { dsimp, simp only [category.comp_id, category.id_comp, zero_comp, comp_zero]; refl }\n  end,\n  distinguished_cocone_triangle := begin\n    intros X Y f,\n    let T := (neg₃_functor (homotopy_category A (complex_shape.up ℤ))).obj (cone.triangleₕ f.out),\n    let E := T.obj₃,\n    haveI : homotopy_category.is_bounded_above E,\n    { obtain ⟨a,ha⟩ := X.2,\n      obtain ⟨b,hb⟩ := Y.2,\n      use max (a - 1) b,\n      intros i hi,\n      apply is_zero_biprod,\n      { apply ha, suffices : a - 1 ≤ i, by linarith, apply le_trans _ hi, apply le_max_left },\n      { apply hb, apply le_trans _ hi, apply le_max_right } },\n    refine ⟨⟨E⟩, T.mor₂, T.mor₃, _⟩,\n    { erw homotopy_category.mem_distinguished_iff_exists_iso_cone,\n      use [X.val.as, Y.val.as, f.out],\n      unfreezingI {\n      rcases X with ⟨⟨X⟩,hX⟩,\n      rcases Y with ⟨⟨Y⟩,hY⟩,\n      constructor,\n      refine triangle.iso.of_components\n        (iso.refl _) (iso.refl _) (iso.refl _) _ _ _,\n      { dsimp, simp only [category.comp_id, homotopy_category.quotient_map_out, category.id_comp], },\n      { dsimp [T], simp only [category.comp_id, category.id_comp], },\n      { dsimp [T], simp only [category_theory.functor.map_id, category.comp_id, category.id_comp] } } }\n  end,\n  rotate_distinguished_triangle := by async begin\n    intros T,\n    split,\n    { intros hT,\n      apply homotopy_category.rotate_mem_distinguished_triangles _ hT },\n    { intros hT,\n      erw pretriangulated.rotate_distinguished_triangle,\n      exact hT }\n  end,\n  complete_distinguished_triangle_morphism := by async begin\n    intros T₁ T₂ hT₁ hT₂ f g h,\n    apply pretriangulated.complete_distinguished_triangle_morphism _ _ hT₁ hT₂ f g h,\n  end }\n.\n\nvariable (A)\n\n-- Move this\n@[simps]\ndef _root_.homotopy_category.single (i : ℤ) : A ⥤ homotopy_category A (complex_shape.up ℤ) :=\nhomological_complex.single _ _ i ⋙ homotopy_category.quotient _ _\n\ndef single (i : ℤ) : A ⥤ bounded_homotopy_category A :=\n{ obj := λ X,\n  { val := (homotopy_category.single A i).obj X,\n    bdd := begin\n      use i+1,\n      intros j hj,\n      dsimp,\n      erw if_neg,\n      { apply limits.is_zero_zero },\n      { exact ((i.lt_iff_add_one_le j).mpr hj).ne' }\n    end },\n  map := λ X Y f, (homotopy_category.single A i).map f,\n  map_id' := λ X, (homotopy_category.single A i).map_id _,\n  map_comp' := λ X Y Z f g, (homotopy_category.single A i).map_comp f g }\n\n\ndef forget :\n  bounded_homotopy_category A ⥤ homotopy_category A (complex_shape.up ℤ) :=\n{ obj := bounded_homotopy_category.val, map := λ _ _, id }\n\ninstance : full (forget A) := { preimage := λ _ _, id }\ninstance : faithful (forget A) := {}\n\ndef forget_shift (i : ℤ) :\n  forget A ⋙ shift_functor (homotopy_category A (complex_shape.up ℤ)) i ≅\n  shift_functor _ i ⋙ forget A :=\niso.refl _\n\nnoncomputable\ndef single_forget (i : ℤ) :\n  single A i ⋙ forget A ≅ homotopy_category.single A i :=\niso.refl _\n\nvariable {A}\n\nsection\n\n@[simps]\ndef _root_.homological_complex.shift_single_obj (i j : ℤ) (X : A) :\n  ((homological_complex.single A (complex_shape.up ℤ) i).obj X)⟦j⟧ ≅\n  (homological_complex.single A (complex_shape.up ℤ) (i - j)).obj X :=\n{ hom := { f := λ k, eq_to_hom (by { dsimp, congr' 1, simpa using eq_sub_iff_add_eq.symm }) },\n  inv := { f := λ k, eq_to_hom (by { dsimp, congr' 1, simpa using eq_sub_iff_add_eq }) } }\n\n@[simps]\ndef _root_.homological_complex.single_shift (i j : ℤ) :\n  homological_complex.single A (complex_shape.up ℤ) i ⋙ category_theory.shift_functor _ j ≅\n  homological_complex.single A (complex_shape.up ℤ) (i - j) :=\nnat_iso.of_components (λ X, homological_complex.shift_single_obj i j X)\nbegin\n  intros,\n  ext k,\n  dsimp,\n  split_ifs,\n  { rw dif_pos (eq_sub_iff_add_eq.mpr h), simp },\n  { rw dif_neg (eq_sub_iff_add_eq.not.mpr h), simp },\nend\n.\nnoncomputable\ndef shift_single_iso (i j : ℤ) :\n  single A i ⋙ shift_functor _ j ≅ single A (i - j) :=\nfully_faithful_cancel_right (bounded_homotopy_category.forget A)\n(iso_whisker_right (homological_complex.single_shift i j)\n  (homotopy_category.quotient A (complex_shape.up ℤ)) : _)\n\nend\n\nend bounded_homotopy_category\n\nnamespace category_theory.functor\n\nvariables {B : Type u'} [category.{v'} B] [abelian B]\nvariables (F : A ⥤ B) [functor.additive F]\n\ninstance is_bounded_above_map_homotopy_category_obj (X : bounded_homotopy_category A) :\n  ((functor.map_homotopy_category (complex_shape.up ℤ) F).obj X.val).is_bounded_above :=\nbegin\n  obtain ⟨a, ha⟩ := homotopy_category.is_bounded_above.cond X.val,\n  refine ⟨⟨a, _⟩⟩,\n  intros i hi,\n  apply functor.map_is_zero,\n  apply ha,\n  apply hi,\nend\n\n@[simps]\ndef map_bounded_homotopy_category :\n  bounded_homotopy_category A ⥤ bounded_homotopy_category B :=\n{ obj := λ X, bounded_homotopy_category.of $\n    (F.map_homotopy_category _).obj X.val,\n  map := λ X Y f, (F.map_homotopy_category _).map f,\n  map_id' := λ X, (F.map_homotopy_category _).map_id _,\n  map_comp' := λ X Y Z f g, (F.map_homotopy_category _).map_comp _ _ }\n\nend category_theory.functor\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived/bounded_homotopy_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.32571142235016015}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.over\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.category_theory.limits.creates\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# Limits and colimits in the over and under categories\n\nShow that the forgetful functor `forget X : over X ⥤ C` creates colimits, and hence `over X` has\nany colimits that `C` has (as well as the dual that `forget X : under X ⟶ C` creates limits).\n\nNote that the folder `category_theory.limits.shapes.constructions.over` further shows that\n`forget X : over X ⥤ C` creates connected limits (so `over X` has connected limits), and that\n`over X` has `J`-indexed products if `C` has `J`-indexed wide pullbacks.\n\nTODO: If `C` has binary products, then `forget X : over X ⥤ C` has a right adjoint.\n-/\n\nnamespace category_theory.functor\n\n\n/-- We can interpret a functor `F` into the category of arrows with codomain `X` as a cocone over\n    the diagram given by the domains of the arrows in the image of `F` such that the apex of the\n    cocone is `X`. -/\n@[simp] theorem to_cocone_X {J : Type v} [small_category J] {C : Type u} [category C] {X : C}\n    (F : J ⥤ over X) : limits.cocone.X (to_cocone F) = X :=\n  Eq.refl (limits.cocone.X (to_cocone F))\n\n/-- We can interpret a functor `F` into the category of arrows with domain `X` as a cone over the\n    diagram given by the codomains of the arrows in the image of `F` such that the apex of the cone\n    is `X`. -/\n@[simp] theorem to_cone_X {J : Type v} [small_category J] {C : Type u} [category C] {X : C}\n    (F : J ⥤ under X) : limits.cone.X (to_cone F) = X :=\n  Eq.refl (limits.cone.X (to_cone F))\n\nend category_theory.functor\n\n\nnamespace category_theory.over\n\n\nprotected instance forget.category_theory.limits.reflects_colimits {C : Type u} [category C]\n    {X : C} : limits.reflects_colimits (forget X) :=\n  limits.reflects_colimits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      limits.reflects_colimits_of_shape.mk\n        fun (F : J ⥤ over X) =>\n          limits.reflects_colimit.mk\n            fun (c : limits.cocone F) (t : limits.is_colimit (functor.map_cocone (forget X) c)) =>\n              limits.is_colimit.mk\n                fun (s : limits.cocone F) =>\n                  hom_mk (limits.is_colimit.desc t (functor.map_cocone (forget X) s))\n\nprotected instance forget.category_theory.creates_colimits {C : Type u} [category C] {X : C} :\n    creates_colimits (forget X) :=\n  creates_colimits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      creates_colimits_of_shape.mk\n        fun (K : J ⥤ over X) =>\n          creates_colimit.mk\n            fun (c : limits.cocone (K ⋙ forget X)) (t : limits.is_colimit c) =>\n              liftable_cocone.mk\n                (limits.cocone.mk (mk (limits.is_colimit.desc t (functor.to_cocone K)))\n                  (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cocone.ι c) j)))\n                (limits.cocones.ext\n                  (iso.refl\n                    (limits.cocone.X\n                      (functor.map_cocone (forget X)\n                        (limits.cocone.mk (mk (limits.is_colimit.desc t (functor.to_cocone K)))\n                          (nat_trans.mk\n                            fun (j : J) => hom_mk (nat_trans.app (limits.cocone.ι c) j))))))\n                  sorry)\n\nprotected instance has_colimit {J : Type v} [small_category J] {C : Type u} [category C] {X : C}\n    {F : J ⥤ over X} [limits.has_colimit (F ⋙ forget X)] : limits.has_colimit F :=\n  has_colimit_of_created F (forget X)\n\nprotected instance has_colimits_of_shape {J : Type v} [small_category J] {C : Type u} [category C]\n    {X : C} [limits.has_colimits_of_shape J C] : limits.has_colimits_of_shape J (over X) :=\n  limits.has_colimits_of_shape.mk fun (F : J ⥤ over X) => over.has_colimit\n\nprotected instance has_colimits {C : Type u} [category C] {X : C} [limits.has_colimits C] :\n    limits.has_colimits (over X) :=\n  limits.has_colimits.mk fun (J : Type v) (𝒥 : small_category J) => over.has_colimits_of_shape\n\n-- We can automatically infer that the forgetful functor preserves colimits\n\nend category_theory.over\n\n\nnamespace category_theory.under\n\n\nprotected instance forget.category_theory.limits.reflects_limits {C : Type u} [category C] {X : C} :\n    limits.reflects_limits (forget X) :=\n  limits.reflects_limits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      limits.reflects_limits_of_shape.mk\n        fun (F : J ⥤ under X) =>\n          limits.reflects_limit.mk\n            fun (c : limits.cone F) (t : limits.is_limit (functor.map_cone (forget X) c)) =>\n              limits.is_limit.mk\n                fun (s : limits.cone F) =>\n                  hom_mk (limits.is_limit.lift t (functor.map_cone (forget X) s))\n\nprotected instance forget.category_theory.creates_limits {C : Type u} [category C] {X : C} :\n    creates_limits (forget X) :=\n  creates_limits.mk\n    fun (J : Type v) (𝒥₁ : small_category J) =>\n      creates_limits_of_shape.mk\n        fun (K : J ⥤ under X) =>\n          creates_limit.mk\n            fun (c : limits.cone (K ⋙ forget X)) (t : limits.is_limit c) =>\n              liftable_cone.mk\n                (limits.cone.mk (mk (limits.is_limit.lift t (functor.to_cone K)))\n                  (nat_trans.mk fun (j : J) => hom_mk (nat_trans.app (limits.cone.π c) j)))\n                (limits.cones.ext\n                  (iso.refl\n                    (limits.cone.X\n                      (functor.map_cone (forget X)\n                        (limits.cone.mk (mk (limits.is_limit.lift t (functor.to_cone K)))\n                          (nat_trans.mk\n                            fun (j : J) => hom_mk (nat_trans.app (limits.cone.π c) j))))))\n                  sorry)\n\nprotected instance has_limit {J : Type v} [small_category J] {C : Type u} [category C] {X : C}\n    {F : J ⥤ under X} [limits.has_limit (F ⋙ forget X)] : limits.has_limit F :=\n  has_limit_of_created F (forget X)\n\nprotected instance has_limits_of_shape {J : Type v} [small_category J] {C : Type u} [category C]\n    {X : C} [limits.has_limits_of_shape J C] : limits.has_limits_of_shape J (under X) :=\n  limits.has_limits_of_shape.mk fun (F : J ⥤ under X) => under.has_limit\n\nprotected instance has_limits {C : Type u} [category C] {X : C} [limits.has_limits C] :\n    limits.has_limits (under X) :=\n  limits.has_limits.mk fun (J : Type v) (𝒥 : small_category J) => under.has_limits_of_shape\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/over_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3256774255651668}}
{"text": "import tag009H -- to get definition of stalk, which is the only filtered colimit I care about right now\nimport tag009P -- presheaf of (commutative) rings on basis \nuniverse u \n--#print presheaf_on_basis_stalk \n--#print presheaf_on_basis_stalk.aux \nnamespace presheaf_of_rings_on_basis_stalk\n\nvariables {X : Type u} [topological_space X] {B : set (set X)}\n{HB : topological_space.is_topological_basis B}\n-- goal : comm_ring (presheaf_on_basis_stalk (FPRB.to_presheaf_of_types_on_basis) x) \n\ndef stalk (FPRB : presheaf_of_rings_on_basis HB)\n(x : X)\n(Hstandard : ∀ {{U V : set X}}, U ∈ B → V ∈ B → U ∩ V ∈ B)\n(Hall : set.univ ∈ B)\n:= presheaf_on_basis_stalk (FPRB.to_presheaf_of_types_on_basis) x\n\ndef stalk.aux (FPRB : presheaf_of_rings_on_basis HB)\n(x : X)\n(Hstandard : ∀ {{U V : set X}}, U ∈ B → V ∈ B → U ∩ V ∈ B)\n(Hall : set.univ ∈ B)\n:= presheaf_on_basis_stalk.aux (FPRB.to_presheaf_of_types_on_basis) x\n\n-- need this instance because a stalk.aux of a presheaf of types is a setoid\n-- but I have a presheaf of rings\n-- I guess I could have had presheaf of rings coe to presheaf of types?\nvariables (FPRB : presheaf_of_rings_on_basis HB) (x : X)\n(Hstandard : ∀ {{U V : set X}}, U ∈ B → V ∈ B → U ∩ V ∈ B)\n(Hall : set.univ ∈ B)\ninclude FPRB x Hstandard Hall\n\ninstance stalk_is_setoid\n : setoid (stalk.aux FPRB x Hstandard Hall) := presheaf_on_basis_stalk.setoid FPRB.to_presheaf_of_types_on_basis x\n\nprivate def add_aux : stalk.aux FPRB x Hstandard Hall → stalk.aux FPRB x Hstandard Hall → stalk FPRB x Hstandard Hall := \nλ s t,⟦⟨s.U ∩ t.U,Hstandard s.BU t.BU,⟨s.Hx,t.Hx⟩,\n        FPRB.res s.BU _   (set.inter_subset_left _ _) s.s +\n        FPRB.res t.BU _   (set.inter_subset_right _ _) t.s\n      ⟩⟧ \n\ninstance ring_stalk_has_add : has_add (stalk FPRB x Hstandard Hall) :=\n⟨quotient.lift₂ (add_aux FPRB x Hstandard Hall) (λ a₁ a₂ b₁ b₂ H1 H2,\n  let U1 := classical.some H1 in\n  let U2 := classical.some H2 in\n  quotient.sound ⟨U1 ∩ U2,begin\n    have H1' := classical.some_spec H1,\n    cases H1' with Hx1 H1',\n    cases H1' with BU1 H1',\n    cases H1' with HUa₁ H1',\n    cases H1' with HUb₁ H1',\n    have H2' := classical.some_spec H2,\n    cases H2' with Hx2 H2',\n    cases H2' with BU2 H2',\n    cases H2' with HUa₂ H2',\n    cases H2' with HUb₂ H2',\n    existsi (⟨Hx1,Hx2⟩ : x ∈ U1 ∩ U2),\n    existsi Hstandard BU1 BU2,\n    existsi set.inter_subset_inter HUa₁ HUa₂,\n    existsi set.inter_subset_inter HUb₁ HUb₂,\n    rw (FPRB.res_is_ring_morphism _ _ _).map_add,\n    rw (FPRB.res_is_ring_morphism _ _ _).map_add,\n    show (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (a₁.s) +\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (a₂.s) =\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (b₁.s) +\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (b₂.s),\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    suffices : (FPRB.res BU1 (Hstandard BU1 BU2) (set.inter_subset_left _ _) ∘ FPRB.res a₁.BU BU1 HUa₁) (a₁.s) +\n      (FPRB.res BU2 (Hstandard BU1 BU2) (set.inter_subset_right _ _) ∘ FPRB.res a₂.BU BU2 HUa₂) (a₂.s) =\n      (FPRB.res BU1 (Hstandard BU1 BU2) (set.inter_subset_left _ _) ∘ FPRB.res b₁.BU BU1 HUb₁) (b₁.s) +\n      (FPRB.res BU2 (Hstandard BU1 BU2) (set.inter_subset_right _ _) ∘ FPRB.res b₂.BU BU2 HUb₂) (b₂.s),\n    rwa [←FPRB.Hcomp,←FPRB.Hcomp,←FPRB.Hcomp,←FPRB.Hcomp] at this,\n    simp [H1',H2']\n  end⟩)⟩\n\n  --#check is_ring_hom\n\nprivate def neg_aux : stalk.aux FPRB x Hstandard Hall → stalk FPRB x Hstandard Hall := \nλ s,⟦⟨s.U,s.BU,s.Hx,-s.s⟩⟧\n\ninstance : has_neg (stalk FPRB x Hstandard Hall) :=\n⟨quotient.lift (neg_aux FPRB x Hstandard Hall) $ begin\n  intros a b H,\n  apply quotient.sound,\n  cases H with U H,\n  existsi U,\n  cases H with Hx H,\n  existsi Hx,\n  cases H with BW H,\n  existsi BW,\n  cases H with HWU H,\n  existsi HWU,\n  cases H with HWV H,\n  existsi HWV,\n  show FPRB.res _ _ _ (-a.s) = FPRB.res _ _ _ (-b.s),\n  have Ha : FPRB.res _ BW HWU (-a.s) = -(FPRB.res _ BW HWU a.s),\n    rw @is_ring_hom.map_neg _ _ _ _ _ (FPRB.res_is_ring_morphism _ _ _),\n  rw [Ha,H],\n  rw @is_ring_hom.map_neg _ _ _ _ _ (FPRB.res_is_ring_morphism _ _ _),\nend⟩\n\n--#check @is_ring_hom.map_neg \n\nprivate def mul_aux : stalk.aux FPRB x Hstandard Hall → stalk.aux FPRB x Hstandard Hall → stalk FPRB x Hstandard Hall:= \nλ s t,⟦⟨s.U ∩ t.U,Hstandard s.BU t.BU,⟨s.Hx,t.Hx⟩,\n        FPRB.res s.BU _   (set.inter_subset_left _ _) s.s *\n        FPRB.res t.BU _   (set.inter_subset_right _ _) t.s\n      ⟩⟧ \n\ninstance ring_stalk_has_mul : has_mul (stalk FPRB x Hstandard Hall) :=\n⟨quotient.lift₂ (mul_aux FPRB x Hstandard Hall) (λ a₁ a₂ b₁ b₂ H1 H2,\n  let U1 := classical.some H1 in\n  let U2 := classical.some H2 in\n  quotient.sound ⟨U1 ∩ U2,begin\n    have H1' := classical.some_spec H1,\n    cases H1' with Hx1 H1',\n    cases H1' with BU1 H1',\n    cases H1' with HUa₁ H1',\n    cases H1' with HUb₁ H1',\n    have H2' := classical.some_spec H2,\n    cases H2' with Hx2 H2',\n    cases H2' with BU2 H2',\n    cases H2' with HUa₂ H2',\n    cases H2' with HUb₂ H2',\n    existsi (⟨Hx1,Hx2⟩ : x ∈ U1 ∩ U2),\n    existsi Hstandard BU1 BU2,\n    existsi set.inter_subset_inter HUa₁ HUa₂,\n    existsi set.inter_subset_inter HUb₁ HUb₂,\n    rw (FPRB.res_is_ring_morphism _ _ _).map_mul,\n    rw (FPRB.res_is_ring_morphism _ _ _).map_mul,\n    show (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (a₁.s) *\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (a₂.s) =\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (b₁.s) *\n         (FPRB.res _ _ _ ∘ FPRB.res _ _ _) (b₂.s),\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    rw ←FPRB.Hcomp,\n    suffices : (FPRB.res BU1 (Hstandard BU1 BU2) (set.inter_subset_left _ _) ∘ FPRB.res a₁.BU BU1 HUa₁) (a₁.s) *\n      (FPRB.res BU2 (Hstandard BU1 BU2) (set.inter_subset_right _ _) ∘ FPRB.res a₂.BU BU2 HUa₂) (a₂.s) =\n      (FPRB.res BU1 (Hstandard BU1 BU2) (set.inter_subset_left _ _) ∘ FPRB.res b₁.BU BU1 HUb₁) (b₁.s) *\n      (FPRB.res BU2 (Hstandard BU1 BU2) (set.inter_subset_right _ _) ∘ FPRB.res b₂.BU BU2 HUb₂) (b₂.s),\n    rwa [←FPRB.Hcomp,←FPRB.Hcomp,←FPRB.Hcomp,←FPRB.Hcomp] at this,\n    simp [H1',H2']\n  end⟩)⟩\n\n/-\nlemma x_in_basis_elt : ∃ U ∈ B, x ∈ U :=\nbegin\nhave H1 := HB.2.1.symm,\nhave H2 : x ∈ @set.univ X := trivial,\nrw H1 at H2,\ncases H2 with U HU,\nexistsi U,\nexistsi HU.fst,\nexact HU.snd\nend \n-/\n\nprivate def zero : stalk FPRB x Hstandard Hall := \n⟦⟨set.univ,Hall,trivial,0⟩⟧\n\ninstance ring_stalk_has_zero : has_zero (stalk FPRB x Hstandard Hall) := ⟨zero FPRB x Hstandard Hall⟩\n\nprivate def one : stalk FPRB x Hstandard Hall := \n⟦⟨set.univ,Hall,trivial,1⟩⟧\n\ninstance ring_stalk_has_one : has_one (stalk FPRB x Hstandard Hall) := ⟨one FPRB x Hstandard Hall⟩\n\n/-\nlemma r_of_eq : ∀ a b : (stalk.aux FPRB x Hstandard Hall), a = b → a ≈ b := begin\nintros a b Hab,\nrw Hab,\nend \n-/\n\n/-\nlemma eq_eta : ∀ sU tU sBU tBU sHx tHx ss ts,ss = ts → \n  (⟨sU,sBU,sHx,ss⟩ : (stalk.aux FPRB x Hstandard)) = ⟨tU,tBU,tHx,ts⟩ :=\n  begin\nsorry \n  end\n-/\n\ninstance stalks_of_presheaf_of_rings_on_basis_are_rings\n-- {X : Type u} [topological_space X] {B : set (set X)}\n--{HB : topological_space.is_topological_basis B} (FPRB : presheaf_of_rings_on_basis HB) (x : X) :\n: comm_ring (stalk FPRB x Hstandard Hall) := begin\n  refine {\n    add := has_add.add,\n    add_assoc := _,\n    zero := (zero FPRB x Hstandard Hall),\n    zero_add := _,\n    add_zero := _,\n    neg := has_neg.neg,\n    add_left_neg := _,\n    add_comm := _,\n    mul := has_mul.mul,\n    mul_assoc := _,\n    mul_one := _,\n    one := (one FPRB x Hstandard Hall),\n    one_mul := _,\n    left_distrib := _,\n    right_distrib := _,\n    mul_comm := _,\n  },\n  show ∀ (a b c : stalk FPRB x Hstandard Hall), a + b + c = a + (b + c),\n  { intros a1 b1 c1,\n    refine quotient.induction_on₃ a1 b1 c1 _,\n    intros,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    cases c with Uc BUc Hxc sc,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub ∩ Uc), -- brainwave\n    existsi (⟨⟨Hxa,Hxb⟩,Hxc⟩ : x ∈ Ua ∩ Ub ∩ Uc),\n    existsi (Hstandard (Hstandard BUa BUb) BUc),\n    existsi (set.subset.refl (Ua ∩ Ub ∩ Uc)),\n    existsi ((set.inter_assoc Ua Ub Uc ▸ set.subset.refl _) : Ua ∩ Ub ∩ Uc ⊆ Ua ∩ (Ub ∩ Uc)),\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←(presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw add_assoc,\n  --    simp [add_assoc,add_comm],rw add_comm, -- also works!\n  },\n  show ∀ (a : stalk FPRB x Hstandard Hall), 0 + a = a,\n  { intro a1,\n    refine quotient.induction_on a1 _,\n    intro a,\n    cases a with Ua BUa Hxa sa,\n    refine quotient.sound _,\n    dsimp,\n    existsi (set.univ ∩ Ua), \n    existsi ( ⟨trivial,Hxa⟩ : x ∈ set.univ ∩ Ua),\n    existsi (Hstandard Hall BUa : set.univ ∩ Ua ∈ B),\n    existsi (_ : set.univ ∩ Ua ⊆ set.univ ∩ Ua),tactic.swap,rw set.inter_comm,\n    existsi _,tactic.swap, rw set.univ_inter Ua,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw is_ring_hom.map_zero (FPRB.res _ _ _);try {apply_instance},\n    -- (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_zero,\n    rw zero_add,\n  },\n  show ∀ (a : stalk FPRB x Hstandard Hall), a + 0 = a,\n  { intro a1,\n    refine quotient.induction_on a1 _,\n    intro a,\n    cases a with Ua BUa Hxa sa,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ set.univ), \n    existsi ( ⟨Hxa,trivial⟩ : x ∈ Ua ∩ set.univ),\n    existsi (Hstandard BUa Hall : Ua ∩ set.univ ∈ B),\n    existsi (_ : Ua ∩ set.univ ⊆ Ua ∩ set.univ),tactic.swap,rw set.inter_comm,\n    existsi _,tactic.swap, rw set.inter_univ Ua,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw is_ring_hom.map_zero (FPRB.res _ _ _);try {apply_instance},\n    -- (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_zero,\n    rw add_zero\n  },\n  show ∀ (a : stalk FPRB x Hstandard Hall), -a + a = 0,\n  { intro a1,\n    refine quotient.induction_on a1 _,\n    intro a,\n    cases a with Ua BUa Hxa sa,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ua), \n    existsi _,tactic.swap, exact ⟨Hxa,Hxa⟩,\n    existsi _,tactic.swap, exact Hstandard BUa BUa,\n    existsi _,tactic.swap, exact set.subset.refl _,\n    existsi _,tactic.swap, exact set.subset.trans (set.inter_subset_left _ _) (set.subset_univ _),\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw is_ring_hom.map_neg (FPRB.res _ _ _);try {apply_instance},\n    rw add_left_neg,\n    rw is_ring_hom.map_zero (FPRB.res _ _ _);try {apply_instance}\n  },\n  show ∀ (a b : stalk FPRB x Hstandard Hall), a + b = b + a,\n  { intros a1 b1,\n    refine quotient.induction_on₂ a1 b1 _,\n    intros a b,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub), -- brainwave\n    existsi (⟨Hxa,Hxb⟩ : x ∈ Ua ∩ Ub),\n    existsi (Hstandard BUa BUb),\n    existsi (set.subset.refl (Ua ∩ Ub)),\n    existsi ((set.inter_comm Ua Ub ▸ set.subset.refl _) : Ua ∩ Ub ⊆ Ub ∩ Ua),\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw add_comm\n  },\n  show ∀ (a b c : stalk FPRB x Hstandard Hall), a * b * c = a * (b * c),\n  { intros a1 b1 c1,\n    refine quotient.induction_on₃ a1 b1 c1 _,\n    intros,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    cases c with Uc BUc Hxc sc,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub ∩ Uc),\n    existsi (⟨⟨Hxa,Hxb⟩,Hxc⟩ : x ∈ Ua ∩ Ub ∩ Uc),\n    existsi (Hstandard (Hstandard BUa BUb) BUc),\n    existsi (set.subset.refl (Ua ∩ Ub ∩ Uc)),\n    existsi ((set.inter_assoc Ua Ub Uc ▸ set.subset.refl _) : Ua ∩ Ub ∩ Uc ⊆ Ua ∩ (Ub ∩ Uc)),\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw mul_assoc\n  },\n  show ∀ (a : stalk FPRB x Hstandard Hall), _ * a = a, -- 1\n  { intro a1,\n    refine quotient.induction_on a1 _,\n    intro a,\n    cases a with Ua BUa Hxa sa,\n    refine quotient.sound _,\n    dsimp,\n    existsi (set.univ ∩ Ua), \n    existsi ( ⟨trivial,Hxa⟩ : x ∈ set.univ ∩ Ua),\n    existsi (Hstandard Hall BUa : set.univ ∩ Ua ∈ B),\n    existsi (_ : set.univ ∩ Ua ⊆ set.univ ∩ Ua),tactic.swap,rw set.inter_comm,\n    existsi _,tactic.swap, rw set.univ_inter Ua,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw is_ring_hom.map_one (FPRB.res _ _ _);try {apply_instance},\n    rw one_mul,\n  },\n  show ∀ (a : stalk FPRB x Hstandard Hall), a * 1 = a,\n  { intro a1,\n    refine quotient.induction_on a1 _,\n    intro a,\n    cases a with Ua BUa Hxa sa,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ set.univ), \n    existsi ( ⟨Hxa,trivial⟩ : x ∈ Ua ∩ set.univ),\n    existsi (Hstandard BUa Hall : Ua ∩ set.univ ∈ B),\n    existsi (_ : Ua ∩ set.univ ⊆ Ua ∩ set.univ),tactic.swap,rw set.inter_comm,\n    existsi _,tactic.swap, rw set.inter_univ Ua,\n    rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul,\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw ←presheaf_of_types_on_basis.Hcomp',\n    rw is_ring_hom.map_one (FPRB.res _ _ _);try {apply_instance},\n    -- (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_zero,\n    rw mul_one\n  },\n  show ∀ (a b c : stalk FPRB x Hstandard Hall), a * (b + c) = a * b + a * c,\n  { intros a1 b1 c1,\n    refine quotient.induction_on₃ a1 b1 c1 _,\n    intros,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    cases c with Uc BUc Hxc sc,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub ∩ Uc), -- brainwave\n    existsi (⟨⟨Hxa,Hxb⟩,Hxc⟩ : x ∈ Ua ∩ Ub ∩ Uc),\n    existsi (Hstandard (Hstandard BUa BUb) BUc),\n    existsi ((set.inter_assoc Ua Ub Uc ▸ set.subset.refl _) : Ua ∩ Ub ∩ Uc ⊆ Ua ∩ (Ub ∩ Uc)),\n    existsi _,tactic.swap, show (Ua ∩ Ub ∩ Uc ⊆ Ua ∩ Ub ∩ (Ua ∩ Uc)),\n      intros y Hy,cases Hy with Hab Hc, cases Hab with Ha Hb,\n      exact ⟨⟨Ha,Hb⟩,⟨Ha,Hc⟩⟩,\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add},\n    repeat {rw ←presheaf_of_types_on_basis.Hcomp'},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add},\n    repeat {rw ←presheaf_of_types_on_basis.Hcomp'},\n    rw mul_add,\n  },\n  show ∀ (a b c : stalk FPRB x Hstandard Hall), (a + b) * c = a * c + b * c,\n  { intros a1 b1 c1,\n    refine quotient.induction_on₃ a1 b1 c1 _,\n    intros,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    cases c with Uc BUc Hxc sc,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub ∩ Uc), -- brainwave\n    existsi (⟨⟨Hxa,Hxb⟩,Hxc⟩ : x ∈ Ua ∩ Ub ∩ Uc),\n    existsi (Hstandard (Hstandard BUa BUb) BUc),\n    existsi (set.subset.refl _),\n    existsi _,tactic.swap, show (Ua ∩ Ub ∩ Uc ⊆ Ua ∩ Uc ∩ (Ub ∩ Uc)),\n      intros y Hy,cases Hy with Hab Hc, cases Hab with Ha Hb,\n      exact ⟨⟨Ha,Hc⟩,⟨Hb,Hc⟩⟩,\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add},\n    repeat {rw ←presheaf_of_types_on_basis.Hcomp'},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul},\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_add},\n    repeat {rw ←presheaf_of_types_on_basis.Hcomp'},\n    rw add_mul,\n  },  \n  -- last one!\n  show ∀ (a b : stalk FPRB x Hstandard Hall), a * b = b * a,\n  { intros a1 b1,\n    refine quotient.induction_on₂ a1 b1 _,\n    intros,\n    cases a with Ua BUa Hxa sa,\n    cases b with Ub BUb Hxb sb,\n    refine quotient.sound _,\n    dsimp,\n    existsi (Ua ∩ Ub),\n    existsi (⟨Hxa,Hxb⟩ : x ∈ Ua ∩ Ub),\n    existsi (Hstandard BUa BUb),\n    existsi (set.subset.refl _),\n    existsi _,tactic.swap, rw set.inter_comm,\n    repeat {rw (presheaf_of_rings_on_basis.res_is_ring_morphism FPRB _ _ _).map_mul},\n    repeat {rw ←presheaf_of_types_on_basis.Hcomp'},\n    rw mul_comm,\n  },  \nend\nend presheaf_of_rings_on_basis_stalk", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/src/tag007N.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.32532249473868396}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Util.ForEachExpr\nimport Lean.Meta.ForEachExpr\nimport Lean.Meta.RecursorInfo\nimport Lean.Meta.Match.Match\nimport Lean.Meta.Transform\nimport Lean.Meta.IndPredBelow\nimport Lean.Elab.PreDefinition.Basic\nnamespace Lean.Elab\nnamespace Structural\nopen Meta\n\nprivate def getFixedPrefix (declName : Name) (xs : Array Expr) (value : Expr) : MetaM Nat := do\n  let numFixedRef ← IO.mkRef xs.size\n  forEachExpr' value fun e => do\n    if e.isAppOf declName then\n      let args := e.getAppArgs\n      numFixedRef.modify fun numFixed => if args.size < numFixed then args.size else numFixed\n      for arg in args, x in xs do\n        /- We should not use structural equality here. For example, given the definition\n           ```\n           def V.map {α β} f x x_1 :=\n             @V.map.match_1.{1} α (fun x x_2 => V β x) x x_1\n               (fun x x_2 => @V.mk₁ β x (f Bool.true x_2))\n               (fun e => @V.mk₂ β (V.map (fun b => α b) (fun b => β b) f Bool.false e))\n           ```\n           The first three arguments at `V.map (fun b => α b) (fun b => β b) f Bool.false e` are \"fixed\"\n           modulo definitional equality.\n\n           We disable to proof irrelevance to be able to use structural recursion on inductive predicates.\n           For example, consider the example\n           ```\n          inductive PList (α : Type) : Prop\n          | nil\n          | cons : α → PList α → PList α\n\n          infixr:67 \" ::: \" => PList.cons\n\n          set_option trace.Elab.definition.structural true in\n          def pmap {α β} (f : α → β) : PList α → PList β\n          | PList.nil => PList.nil\n          | a:::as => f a ::: pmap f as\n          ```\n          The \"Fixed\" prefix would be 4 since all elements of type `PList α` are definitionally equal.\n        -/\n        if !(← withoutProofIrrelevance <| withReducible <| isDefEq arg x) then\n          -- We continue searching if e's arguments are not a prefix of `xs`\n          return true\n      return false\n    else\n      return true\n  numFixedRef.get\n\nstructure RecArgInfo where\n  /- `fixedParams ++ ys` are the arguments of the function we are trying to justify termination using structural recursion. -/\n  fixedParams : Array Expr\n  ys          : Array Expr  -- recursion arguments\n  pos         : Nat         -- position in `ys` of the argument we are recursing on\n  indicesPos  : Array Nat   -- position in `ys` of the inductive datatype indices we are recursing on\n  indName     : Name        -- inductive datatype name of the argument we are recursing on\n  indLevels   : List Level  -- inductice datatype universe levels of the argument we are recursing on\n  indParams   : Array Expr  -- inductive datatype parameters of the argument we are recursing on\n  indIndices  : Array Expr  -- inductive datatype indices of the argument we are recursing on, it is equal to `indicesPos.map fun i => ys.get! i`\n  reflexive   : Bool        -- true if we are recursing over a reflexive inductive datatype\n  indPred     : Bool        -- true if the type is an inductive predicate\n\nprivate def getIndexMinPos (xs : Array Expr) (indices : Array Expr) : Nat := do\n  let mut minPos := xs.size\n  for index in indices do\n    match xs.indexOf? index with\n    | some pos => if pos.val < minPos then minPos := pos.val\n    | _        => pure ()\n  return minPos\n\n-- Indices can only depend on other indices\nprivate def hasBadIndexDep? (ys : Array Expr) (indices : Array Expr) : MetaM (Option (Expr × Expr)) := do\n  for index in indices do\n    let indexType ← inferType index\n    for y in ys do\n      if !indices.contains y && (← dependsOn indexType y.fvarId!) then\n        return some (index, y)\n  return none\n\n-- Inductive datatype parameters cannot depend on ys\nprivate def hasBadParamDep? (ys : Array Expr) (indParams : Array Expr) : MetaM (Option (Expr × Expr)) := do\n  for p in indParams do\n    let pType ← inferType p\n    for y in ys do\n      if ← dependsOn pType y.fvarId! then\n        return some (p, y)\n  return none\n\nprivate def throwStructuralFailed {α} : MetaM α :=\n  throwError \"structural recursion cannot be used\"\n\nstructure State where\n  /- When compiling structural recursion we use the `brecOn` recursor automatically built by\n     the `inductive` command. For an inductive datatype `C`, it has the form\n     `C.brecOn As motive is c F`\n     where `As` are the inductive datatype parameters, `is` are the inductive datatype indices,\n     `c : C As is`, and `F : (js) → (d : C As js) → C.below d → motive d`\n     The `C.below d` is used to eliminate recursive applications. We refine its type when we process\n     a nested dependent pattern matcher using `MatcherApp.addArg`. See `replaceRecApps` for additional details.\n     We store the names of the matcher where we used `MatcherApp.addArg` at `matcherBelowDep`.\n     We use this information to generate the auxiliary `_sunfold` definition needed by the smart unfolding\n     technique used at WHNF. -/\n  matcherBelowDep : NameSet := {}\n  /- As part of the inductive predicates case, we keep adding more and more discriminants from the\n     local context and build up a bigger matcher application until we reach a fixed point.\n     As a side-effect, this creates matchers. Here we capture all these side-effects, because\n     the construction rolls back any changes done to the environment and the side-effects\n     need to be replayed. -/\n  addMatchers : Array $ MetaM Unit := #[]\n\nabbrev M := StateRefT State MetaM\n\ninstance {α} : Inhabited (M α) where\n  default := throwError \"failed\"\n\nprivate def run {α} (x : M α) (s : State := {}) : MetaM (α × State) :=\n  StateRefT'.run x s\n\nprivate def orelse' {α} (x y : M α) : M α := do\n  let saveState ← get\n  orelseMergeErrors x (do set saveState; y)\n\nprivate partial def findRecArg {α} (numFixed : Nat) (xs : Array Expr) (k : RecArgInfo → M α) : M α :=\n  let rec loop (i : Nat) : M α := do\n    if h : i < xs.size then\n      let x := xs.get ⟨i, h⟩\n      let localDecl ← getFVarLocalDecl x\n      if localDecl.isLet then\n        throwStructuralFailed\n      else\n        let xType ← whnfD localDecl.type\n        matchConstInduct xType.getAppFn (fun _ => loop (i+1)) fun indInfo us => do\n        if !(← hasConst (mkBRecOnName indInfo.name)) then\n          loop (i+1)\n        else if indInfo.isReflexive && !(← hasConst (mkBInductionOnName indInfo.name)) then\n          loop (i+1)\n        else\n          let indArgs    := xType.getAppArgs\n          let indParams  := indArgs.extract 0 indInfo.numParams\n          let indIndices := indArgs.extract indInfo.numParams indArgs.size\n          if !indIndices.all Expr.isFVar then\n            orelse'\n              (throwError \"argument #{i+1} was not used because its type is an inductive family and indices are not variables{indentExpr xType}\")\n              (loop (i+1))\n          else if !indIndices.allDiff then\n            orelse'\n              (throwError \"argument #{i+1} was not used because its type is an inductive family and indices are not pairwise distinct{indentExpr xType}\")\n              (loop (i+1))\n          else\n            let indexMinPos := getIndexMinPos xs indIndices\n            let numFixed    := if indexMinPos < numFixed then indexMinPos else numFixed\n            let fixedParams := xs.extract 0 numFixed\n            let ys          := xs.extract numFixed xs.size\n            match (← hasBadIndexDep? ys indIndices) with\n            | some (index, y) =>\n              orelse'\n                (throwError \"argument #{i+1} was not used because its type is an inductive family{indentExpr xType}\\nand index{indentExpr index}\\ndepends on the non index{indentExpr y}\")\n                (loop (i+1))\n            | none =>\n              match (← hasBadParamDep? ys indParams) with\n              | some (indParam, y) =>\n                orelse'\n                  (throwError \"argument #{i+1} was not used because its type is an inductive datatype{indentExpr xType}\\nand parameter{indentExpr indParam}\\ndepends on{indentExpr y}\")\n                  (loop (i+1))\n              | none =>\n                let indicesPos := indIndices.map fun index => match ys.indexOf? index with | some i => i.val | none => unreachable!\n                orelse'\n                  (mapError\n                    (k { fixedParams := fixedParams\n                         ys          := ys\n                         pos         := i - fixedParams.size\n                         indicesPos  := indicesPos\n                         indName     := indInfo.name\n                         indLevels   := us\n                         indParams   := indParams\n                         indIndices  := indIndices\n                         reflexive := indInfo.isReflexive\n                         indPred := ←isInductivePredicate indInfo.name })\n                    (fun msg => m!\"argument #{i+1} was not used for structural recursion{indentD msg}\"))\n                  (loop (i+1))\n    else\n      throwStructuralFailed\n  loop numFixed\n\nprivate def containsRecFn (recFnName : Name) (e : Expr) : Bool :=\n  (e.find? fun e => e.isConstOf recFnName).isSome\n\nprivate def ensureNoRecFn (recFnName : Name) (e : Expr) : MetaM Expr := do\n  if containsRecFn recFnName e then\n    Meta.forEachExpr e fun e => do\n      if e.isAppOf recFnName then\n        throwError \"unexpected occurrence of recursive application{indentExpr e}\"\n    pure e\n  else\n    pure e\n\nprivate def throwToBelowFailed {α} : MetaM α :=\n  throwError \"toBelow failed\"\n\n/- See toBelow -/\nprivate partial def toBelowAux (C : Expr) : Expr → Expr → Expr → MetaM Expr\n  | belowDict, arg, F => do\n    let belowDict ← whnf belowDict\n    trace[Elab.definition.structural] \"belowDict: {belowDict}, arg: {arg}\"\n    match belowDict with\n    | Expr.app (Expr.app (Expr.const `PProd _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (← mkAppM `PProd.fst #[F]))\n      <|>\n      (do toBelowAux C d2 arg (← mkAppM `PProd.snd #[F]))\n    | Expr.app (Expr.app (Expr.const `And _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (← mkAppM `And.left #[F]))\n      <|>\n      (do toBelowAux C d2 arg (← mkAppM `And.right #[F]))\n    | _ => forallTelescopeReducing belowDict fun xs belowDict => do\n      let argArgs := arg.getAppArgs\n      unless argArgs.size >= xs.size do throwToBelowFailed\n      let n := argArgs.size\n      let argTailArgs := argArgs.extract (n - xs.size) n\n      let belowDict := belowDict.replaceFVars xs argTailArgs\n      match belowDict with\n      | Expr.app belowDictFun belowDictArg _ =>\n        unless belowDictFun.getAppFn == C do throwToBelowFailed\n        unless ← isDefEq belowDictArg arg do throwToBelowFailed\n        pure (mkAppN F argTailArgs)\n      | _ => throwToBelowFailed\n\n/- See toBelow -/\nprivate def withBelowDict {α} (below : Expr) (numIndParams : Nat) (k : Expr → Expr → MetaM α) : MetaM α := do\n  let belowType ← inferType below\n  trace[Elab.definition.structural] \"belowType: {belowType}\"\n  belowType.withApp fun f args => do\n    let motivePos := numIndParams + 1\n    unless motivePos < args.size do throwError \"unexpected 'below' type{indentExpr belowType}\"\n    let pre := mkAppN f (args.extract 0 numIndParams)\n    let preType ← inferType pre\n    forallBoundedTelescope preType (some 1) fun x _ => do\n      let motiveType ← inferType x[0]\n      let C ← mkFreshUserName `C\n      withLocalDeclD C motiveType fun C =>\n        let belowDict := mkApp pre C\n        let belowDict := mkAppN belowDict (args.extract (numIndParams + 1) args.size)\n        k C belowDict\n\n/-\n  `below` is a free variable with type of the form `I.below indParams motive indices major`,\n  where `I` is the name of an inductive datatype.\n\n  For example, when trying to show that the following function terminates using structural recursion\n  ```lean\n  def addAdjacent : List Nat → List Nat\n  | []       => []\n  | [a]      => [a]\n  | a::b::as => (a+b) :: addAdjacent as\n  ```\n  when we are visiting `addAdjacent as` at `replaceRecApps`, `below` has type\n  `@List.below Nat (fun (x : List Nat) => List Nat) (a::b::as)`\n  The motive `fun (x : List Nat) => List Nat` depends on the actual function we are trying to compute.\n  So, we first replace it with a fresh variable `C` at `withBelowDict`.\n  Recall that `brecOn` implements course-of-values recursion, and `below` can be viewed as a dictionary\n  of the \"previous values\".\n  We search this dictionary using the auxiliary function `toBelowAux`.\n  The dictionary is built using the `PProd` (`And` for inductive predicates).\n  We keep searching it until we find `C recArg`, where `C` is the auxiliary fresh variable created at `withBelowDict`.  -/\nprivate partial def toBelow (below : Expr) (numIndParams : Nat) (recArg : Expr) : MetaM Expr := do\n  withBelowDict below numIndParams fun C belowDict =>\n    toBelowAux C belowDict recArg below\n\n/--\n  Return true iff `e` contains an application `recFnName .. t ..` where the term `t` is\n  the argument we are trying to recurse on, and it contains loose bound variables.\n\n  We use this test to decide whether we should process a matcher-application as a regular\n  applicaton or not. That is, whether we should push the `below` argument should be affected by the matcher or not.\n  If `e` does not contain an application of the form `recFnName .. t ..`, then we know\n  the recursion doesn't depend on any pattern variable in this matcher.\n-/\nprivate def recArgHasLooseBVarsAt (recFnName : Name) (recArgInfo : RecArgInfo) (e : Expr) : Bool :=\n  let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n  let app?   := e.find? fun e =>\n     e.isAppOf recFnName && e.getAppNumArgs > recArgPos && (e.getArg! recArgPos).hasLooseBVars\n  app?.isSome\n\nprivate partial def replaceRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (below : Expr) (e : Expr) : M Expr :=\n  let rec loop (below : Expr) (e : Expr) : M Expr := do\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c.binderInfo (← loop below d) fun x => do\n        mkLambdaFVars #[x] (← loop below (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c.binderInfo (← loop below d) fun x => do\n        mkForallFVars #[x] (← loop below (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (← loop below type) (← loop below val) fun x => do\n        mkLetFVars #[x] (← loop below (body.instantiate1 x))\n    | Expr.mdata d e _   => return mkMData d (← loop below e)\n    | Expr.proj n i e _  => return mkProj n i (← loop below e)\n    | Expr.app _ _ _ =>\n      let processApp (e : Expr) : M Expr :=\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let numFixed  := recArgInfo.fixedParams.size\n            let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n            if recArgPos >= args.size then\n              throwError \"insufficient number of parameters at recursive application {indentExpr e}\"\n            let recArg := args[recArgPos]\n            -- For reflexive type, we may have nested recursive applications in recArg\n            let recArg ← loop below recArg\n            let f ← try toBelow below recArgInfo.indParams.size recArg catch  _ => throwError \"failed to eliminate recursive application{indentExpr e}\"\n            -- Recall that the fixed parameters are not in the scope of the `brecOn`. So, we skip them.\n            let argsNonFixed := args.extract numFixed args.size\n            -- The function `f` does not explicitly take `recArg` and its indices as arguments. So, we skip them too.\n            let mut fArgs := #[]\n            for i in [:argsNonFixed.size] do\n              if recArgInfo.pos != i && !recArgInfo.indicesPos.contains i then\n                let arg := argsNonFixed[i]\n                let arg ← replaceRecApps recFnName recArgInfo below arg\n                fArgs := fArgs.push arg\n            return mkAppN f fArgs\n          else\n            return mkAppN (← loop below f) (← args.mapM (loop below))\n      let matcherApp? ← matchMatcherApp? e\n      match matcherApp? with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo e then\n          processApp e\n        else\n          /- Here is an example we currently not handle\n             ```\n             def g (xs : List Nat) : Nat :=\n             match xs with\n             | [] => 0\n             | y::ys =>\n               match ys with\n               | []       => 1\n               | _::_::zs => g zs + 1\n               | zs       => g ys + 2\n             ```\n             We are matching on `ys`, but still using `ys` in the third alternative.\n             If we push the `below` argument over the dependent match it will be able to eliminate recursive call using `zs`.\n             To make it work, users have to write the third alternative as `| zs => g zs + 2`\n             If this is too annoying in practice, we may replace `ys` with the matching term, but\n             this may generate weird error messages, when it doesn't work. -/\n          trace[Elab.definition.structural] \"below before matcherApp.addArg: {below} : {← inferType below}\"\n          let matcherApp ← mapError (matcherApp.addArg below) (fun msg => \"failed to add `below` argument to 'matcher' application\" ++ indentD msg)\n          modify fun s => { s with matcherBelowDep := s.matcherBelowDep.insert matcherApp.matcherName }\n          let altsNew ← (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n            lambdaTelescope alt fun xs altBody => do\n              trace[Elab.definition.structural] \"altNumParams: {numParams}, xs: {xs}\"\n              unless xs.size >= numParams do\n                throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n              let belowForAlt := xs[numParams - 1]\n              mkLambdaFVars xs (← loop belowForAlt altBody)\n          pure { matcherApp with alts := altsNew }.toExpr\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop below e\n\nprivate partial def replaceIndPredRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (motive : Expr) (e : Expr) : M Expr := do\n  let maxDepth := IndPredBelow.maxBackwardChainingDepth.get $ ←getOptions\n  let rec loop (e : Expr) : M Expr := do\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c.binderInfo (← loop d) fun x => do\n        mkLambdaFVars #[x] (← loop (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c.binderInfo (← loop d) fun x => do\n        mkForallFVars #[x] (← loop (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (← loop type) (← loop val) fun x => do\n        mkLetFVars #[x] (← loop (body.instantiate1 x))\n    | Expr.mdata d e _   => return mkMData d (← loop e)\n    | Expr.proj n i e _  => return mkProj n i (← loop e)\n    | Expr.app _ _ _ =>\n      let processApp (e : Expr) : M Expr := do\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let ty ← inferType e\n            let main ← mkFreshExprSyntheticOpaqueMVar ty \n            if ←IndPredBelow.backwardsChaining main.mvarId! maxDepth then\n              main\n            else\n              throwError \"could not solve using backwards chaining {MessageData.ofGoal main.mvarId!}\"\n          else\n            return mkAppN (← loop f) (← args.mapM loop)\n      match (←matchMatcherApp? e) with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo e then\n          processApp e\n        else\n          trace[Elab.definition.structural] \"matcherApp before adding below transformation:\\n{matcherApp.toExpr}\"\n          let rec addBelow (matcherApp : MatcherApp) : M Expr := do\n            if let some (t, idx) ← IndPredBelow.findBelowIdx matcherApp.discrs motive then\n              let (newApp, addMatcher) ← IndPredBelow.mkBelowMatcher matcherApp motive t idx\n              modify fun s => { s with addMatchers := s.addMatchers.push addMatcher }\n              let some newApp ← matchMatcherApp? newApp | throwError \"not a matcherApp: {newApp}\"\n              addBelow newApp\n            else matcherApp.toExpr\n            \n          let newApp ← addBelow matcherApp\n          if newApp == matcherApp.toExpr then\n            throwError \"could not add below discriminant\"\n          else \n            trace[Elab.definition.structural] \"modified matcher:\\n{newApp}\"\n            processApp newApp\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop e\n\nprivate def mkBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  let type  := (← inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive ← mkForallFVars otherArgs type\n  let mut brecOnUniv ← getLevel motive\n  trace[Elab.definition.structural] \"brecOn univ: {brecOnUniv}\"\n  let useBInductionOn := recArgInfo.reflexive && brecOnUniv == levelZero\n  if recArgInfo.reflexive && brecOnUniv != levelZero then\n    brecOnUniv ← decLevel brecOnUniv\n  let motive ← mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n    if useBInductionOn then\n      Lean.mkConst (mkBInductionOnName recArgInfo.indName) recArgInfo.indLevels\n    else\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) (brecOnUniv :: recArgInfo.indLevels)\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType ← inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]\n    let FType ← inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType ← instantiateForall FType recArgInfo.indIndices\n    let FType ← instantiateForall FType #[major]\n    forallBoundedTelescope FType (some 1) fun below _ => do\n      let below := below[0]\n      let valueNew     ← replaceRecApps recFnName recArgInfo below value\n      let Farg         ← mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n      let brecOn       := mkApp brecOn Farg\n      pure $ mkAppN brecOn otherArgs\n      \nprivate def mkIndPredBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  let type  := (← inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive ← mkForallFVars otherArgs type\n  let motive ← mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) recArgInfo.indLevels\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType ← inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  -- we need to close the telescope here, because the local context is used:\n  -- The root cause was, that this copied code puts an ih : FType into the \n  -- local context and later, when we use the local context to build the recursive \n  -- call, it uses this ih. But that ih doesn't exist in the actual brecOn call. \n  -- That's why it must go.\n  let FType ← forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]\n    let FType ← inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType ← instantiateForall FType recArgInfo.indIndices\n    instantiateForall FType #[major]\n  forallBoundedTelescope FType (some 1) fun below _ => do\n    let main ← mkFreshExprSyntheticOpaqueMVar FType\n    let below := below[0]\n    let valueNew     ← replaceIndPredRecApps recFnName recArgInfo motive value\n    let Farg         ← mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n    let brecOn       := mkApp brecOn Farg\n    pure $ mkAppN brecOn otherArgs\n\nprivate def shouldBetaReduce (e : Expr) (recFnName : Name) : Bool :=\n  if e.isHeadBetaTarget then\n    e.getAppFn.find? (·.isConstOf recFnName) |>.isSome\n  else\n    false\n\n/--\n  Beta reduce terms where the recursive function occurs in the lambda term.\n  This is useful to improve the effectiveness of `elimRecursion`.\n  Example:\n  ```\n  def f : Nat → Nat\n    | 0 => 1\n    | i+1 => (fun x => f x) i\n  ```\n-/\nprivate def preprocess (e : Expr) (recFnName : Name) : CoreM Expr :=\n  Core.transform e\n   fun e => return TransformStep.visit <|\n     if shouldBetaReduce e recFnName then\n       e.headBeta\n     else\n       e\n\nprivate def elimRecursion (preDef : PreDefinition) : M PreDefinition :=\n  withoutModifyingEnv do lambdaTelescope preDef.value fun xs value => do\n    addAsAxiom preDef\n    let value ← preprocess value preDef.declName\n    trace[Elab.definition.structural] \"{preDef.declName} {xs} :=\\n{value}\"\n    let numFixed ← getFixedPrefix preDef.declName xs value\n    trace[Elab.definition.structural] \"numFixed: {numFixed}\"\n    findRecArg numFixed xs fun recArgInfo => do\n      -- when (recArgInfo.indName == `Nat) throwStructuralFailed -- HACK to skip Nat argument\n      let valueNew ← \n        if recArgInfo.indPred then mkIndPredBRecOn preDef.declName recArgInfo value\n        else mkBRecOn preDef.declName recArgInfo value\n      let valueNew ← mkLambdaFVars xs valueNew\n      trace[Elab.definition.structural] \"result: {valueNew}\"\n      -- Recursive applications may still occur in expressions that were not visited by replaceRecApps (e.g., in types)\n      let valueNew ← ensureNoRecFn preDef.declName valueNew\n      pure { preDef with value := valueNew }\n\npartial def addSmartUnfoldingDefAux (preDef : PreDefinition) (matcherBelowDep : NameSet) : MetaM PreDefinition := do\n  let recFnName := preDef.declName\n  let isMarkedMatcherName (n : Name) : Bool    := matcherBelowDep.contains n\n  let isMarkedMatcherConst (e : Expr) : Bool   := e.isConst && isMarkedMatcherName e.constName!\n  let isMarkedMatcherApp (e : Expr) : Bool     := isMarkedMatcherConst e.getAppFn\n  let containsMarkedMatcher (e : Expr) : Bool := e.find? isMarkedMatcherConst |>.isSome\n  let rec visit (e : Expr) : MetaM Expr := do\n    match e with\n    | Expr.lam ..     => lambdaTelescope e fun xs b => do mkLambdaFVars xs (← visit b)\n    | Expr.forallE .. => forallTelescope e fun xs b => do mkForallFVars xs (← visit b)\n    | Expr.letE n type val body _ =>\n      withLetDecl n type (← visit val) fun x => do\n        mkLetFVars #[x] (← visit (body.instantiate1 x))\n    | Expr.mdata d b _   => return mkMData d (← visit b)\n    | Expr.proj n i s _  => return mkProj n i (← visit s)\n    | Expr.app .. =>\n      let processApp (e : Expr) : MetaM Expr :=\n        e.withApp fun f args => do\n          return mkAppN (← visit f) (← args.mapM visit)\n      match isMarkedMatcherApp e, (← matchMatcherApp? e) with\n      | true, some matcherApp =>\n        let altsNew ← (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n          lambdaTelescope alt fun xs altBody => do\n            unless xs.size >= numParams do\n              throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n            if containsMarkedMatcher altBody then\n              -- continue\n              mkLambdaFVars xs (← visit altBody)\n            else\n              -- add idRhs marker\n              let altBody ← mkLambdaFVars xs[numParams:xs.size] altBody\n              let altBody ← mkIdRhs altBody\n              mkLambdaFVars xs[0:numParams] altBody\n        pure { matcherApp with alts := altsNew }.toExpr\n      | _, _ => processApp e\n    | _ => pure e\n  return { preDef with\n    declName  := mkSmartUnfoldingNameFor preDef.declName,\n    value     := (← visit preDef.value),\n    modifiers := {}\n  }\n\npartial def addSmartUnfoldingDef (preDef : PreDefinition) (state : State) : TermElabM Unit := do\n  if (← isProp preDef.type) then\n    return ()\n  else\n    let preDefSUnfold ← addSmartUnfoldingDefAux preDef state.matcherBelowDep\n    addNonRec preDefSUnfold\n\ndef structuralRecursion (preDefs : Array PreDefinition) : TermElabM Unit :=\n  if preDefs.size != 1 then\n    throwError \"structural recursion does not handle mutually recursive functions\"\n  else do\n    let (preDefNonRec, state) ← run $ elimRecursion preDefs[0]\n    state.addMatchers.forM liftM\n    mapError (addNonRec preDefNonRec) (fun msg => m!\"structural recursion failed, produced type incorrect term{indentD msg}\")\n    addAndCompilePartialRec preDefs\n    addSmartUnfoldingDef preDefs[0] state\n\nbuiltin_initialize\n  registerTraceClass `Elab.definition.structural\n\nend Structural\n\nexport Structural (structuralRecursion)\n\nend Lean.Elab\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/stage0/src/Lean/Elab/PreDefinition/Structural.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.32532249473868396}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport logic.eq\n\nnamespace unit\n  notation `⋆` := star\n\n  protected theorem eq (a b : unit) : a = b :=\n  unit.rec_on a (unit.rec_on b rfl)\n\n  theorem eq_star (a : unit) : a = star :=\n  unit.eq a star\n\n  protected theorem subsingleton [instance] : subsingleton unit :=\n  subsingleton.intro (λ a b, unit.eq a b)\n\n  protected definition is_inhabited [instance] : inhabited unit :=\n  inhabited.mk unit.star\n\n  protected definition has_decidable_eq [instance] : decidable_eq unit :=\n  take (a b : unit), decidable.inl (unit.eq a b)\nend unit\n", "meta": {"author": "Bolt64", "repo": "lean2-aur", "sha": "1d7148e58a17b2d326b032ed1ebf8c5217320242", "save_path": "github-repos/lean/Bolt64-lean2-aur", "path": "github-repos/lean/Bolt64-lean2-aur/lean2-aur-1d7148e58a17b2d326b032ed1ebf8c5217320242/library/data/unit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.32528486025981473}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n\nimport control.uliftable\nimport data.bitvec.basic\nimport data.stream.defs\nimport tactic.norm_num\n\n\n/-!\n# Rand Monad and Random Class\n\nThis module provides tools for formulating computations guided by randomness and for\ndefining objects that can be created randomly.\n\n## Main definitions\n  * `rand` monad for computations guided by randomness;\n  * `random` class for objects that can be generated randomly;\n    * `random` to generate one object;\n    * `random_r` to generate one object inside a range;\n    * `random_series` to generate an infinite series of objects;\n    * `random_series_r` to generate an infinite series of objects inside a range;\n  * `io.mk_generator` to create a new random number generator;\n  * `io.run_rand` to run a randomized computation inside the `io` monad;\n  * `tactic.run_rand` to run a randomized computation inside the `tactic` monad\n\n## Local notation\n\n * `i .. j` : `Icc i j`, the set of values between `i` and `j` inclusively;\n\n## Tags\n\nrandom monad io\n\n## References\n\n  * Similar library in Haskell: https://hackage.haskell.org/package/MonadRandom\n\n-/\n\nopen list io applicative\n\nuniverses u v w\n\n/-- A monad to generate random objects using the generator type `g` -/\n@[reducible]\ndef rand_g (g : Type) (α : Type u) : Type u := state (ulift.{u} g) α\n\n/-- A monad to generate random objects using the generator type `std_gen` -/\n@[reducible]\ndef rand := rand_g std_gen\n\ninstance (g : Type) : uliftable (rand_g.{u} g) (rand_g.{v} g) :=\n@state_t.uliftable' _ _ _ _ _ (equiv.ulift.trans.{u u u u u} equiv.ulift.symm)\n\nopen ulift (hiding inhabited)\n\n/-- Generate one more `ℕ` -/\ndef rand_g.next {g : Type} [random_gen g] : rand_g g ℕ :=\n⟨ prod.map id up ∘ random_gen.next ∘ down ⟩\n\nlocal infix ` .. `:41 := set.Icc\n\nopen stream\n\n/-- `bounded_random α` gives us machinery to generate values of type `α` between certain bounds -/\nclass bounded_random (α : Type u) [preorder α] :=\n(random_r : Π g [random_gen g] (x y : α),\n              (x ≤ y) → rand_g g (x .. y))\n\n/-- `random α` gives us machinery to generate values of type `α` -/\nclass random (α : Type u) :=\n(random [] : Π (g : Type) [random_gen g], rand_g g α)\n\n/-- shift_31_left = 2^31; multiplying by it shifts the binary\nrepresentation of a number left by 31 bits, dividing by it shifts it\nright by 31 bits -/\ndef shift_31_left : ℕ :=\nby apply_normed 2^31\n\nnamespace rand\n\nopen stream\n\nvariables (α : Type u)\nvariables (g : Type) [random_gen g]\n\n/-- create a new random number generator distinct from the one stored in the state -/\ndef split : rand_g g g := ⟨ prod.map id up ∘ random_gen.split ∘ down ⟩\n\nvariables {g}\n\nsection random\nvariables [random α]\n\nexport random (random)\n\n/-- Generate a random value of type `α`. -/\ndef random : rand_g g α :=\nrandom.random α g\n\n/-- generate an infinite series of random values of type `α` -/\ndef random_series : rand_g g (stream α) :=\ndo gen ← uliftable.up (split g),\n   pure $ stream.corec_state (random.random α g) gen\n\nend random\n\nvariables {α}\n\n/-- Generate a random value between `x` and `y` inclusive. -/\ndef random_r [preorder α] [bounded_random α] (x y : α) (h : x ≤ y) : rand_g g (x .. y) :=\nbounded_random.random_r g x y h\n\n/-- generate an infinite series of random values of type `α` between `x` and `y` inclusive. -/\ndef random_series_r [preorder α] [bounded_random α] (x y : α) (h : x ≤ y) :\n  rand_g g (stream (x .. y)) :=\ndo gen ← uliftable.up (split g),\n   pure $ corec_state (bounded_random.random_r g x y h) gen\n\nend rand\n\nnamespace io\n\nprivate def accum_char (w : ℕ) (c : char) : ℕ :=\nc.to_nat + 256 * w\n\n/-- create and seed a random number generator -/\ndef mk_generator : io std_gen := do\nseed ← io.rand 0 shift_31_left,\nreturn $ mk_std_gen seed\n\nvariables {α : Type}\n\n/-- Run `cmd` using a randomly seeded random number generator -/\ndef run_rand (cmd : _root_.rand α) : io α :=\ndo g ← io.mk_generator,\n   return $ (cmd.run ⟨g⟩).1\n\n/-- Run `cmd` using the provided seed. -/\ndef run_rand_with (seed : ℕ) (cmd : _root_.rand α) : io α :=\nreturn $ (cmd.run ⟨mk_std_gen seed⟩).1\n\nsection random\nvariables [random α]\n\n/-- randomly generate a value of type α -/\ndef random : io α :=\nio.run_rand (rand.random α)\n\n/-- randomly generate an infinite series of value of type α -/\ndef random_series : io (stream α) :=\nio.run_rand (rand.random_series α)\n\nend random\n\nsection bounded_random\nvariables [preorder α] [bounded_random α]\n\n/-- randomly generate a value of type α between `x` and `y` -/\ndef random_r (x y : α) (p : x ≤ y) : io (x .. y) :=\nio.run_rand (bounded_random.random_r _ x y p)\n\n/-- randomly generate an infinite series of value of type α between `x` and `y` -/\ndef random_series_r (x y : α) (h : x ≤ y) : io (stream $ x .. y) :=\nio.run_rand (rand.random_series_r x y h)\n\nend bounded_random\n\nend io\n\nnamespace tactic\n\n/-- create a seeded random number generator in the `tactic` monad -/\nmeta def mk_generator : tactic std_gen := do\ntactic.unsafe_run_io @io.mk_generator\n\n/-- run `cmd` using the a randomly seeded random number generator\nin the tactic monad -/\nmeta def run_rand {α : Type u} (cmd : rand α) : tactic α := do\n⟨g⟩ ← tactic.up mk_generator,\nreturn (cmd.run ⟨g⟩).1\n\nvariables {α : Type u}\n\nsection bounded_random\nvariables [preorder α] [bounded_random α]\n\n/-- Generate a random value between `x` and `y` inclusive. -/\nmeta def random_r (x y : α) (h : x ≤ y) : tactic (x .. y) :=\nrun_rand (rand.random_r x y h)\n\n/-- Generate an infinite series of random values of type `α` between `x` and `y` inclusive. -/\nmeta def random_series_r (x y : α) (h : x ≤ y) : tactic (stream $ x .. y) :=\nrun_rand (rand.random_series_r x y h)\n\nend bounded_random\n\nsection random\n\nvariables [random α]\n\n/-- randomly generate a value of type α -/\nmeta def random : tactic α :=\nrun_rand (rand.random α)\n\n /-- randomly generate an infinite series of value of type α -/\nmeta def random_series : tactic (stream α) :=\nrun_rand (rand.random_series α)\n\nend random\n\nend tactic\n\nopen nat (succ one_add mod_eq_of_lt zero_lt_succ add_one succ_le_succ)\n\nvariables {g : Type} [random_gen g]\n\nopen nat\n\nnamespace fin\nvariables {n : ℕ} [ne_zero n]\n\n/-- generate a `fin` randomly -/\nprotected def random : rand_g g (fin n) :=\n⟨ λ ⟨g⟩, prod.map of_nat' up $ rand_nat g 0 n ⟩\n\nend fin\n\nopen nat\n\ninstance nat_bounded_random : bounded_random ℕ :=\n{ random_r := λ g inst x y hxy,\n  do z ← @fin.random g inst (succ $ y - x) _,\n     pure ⟨z.val + x, nat.le_add_left _ _,\n       by rw ← le_tsub_iff_right hxy; apply le_of_succ_le_succ z.is_lt⟩ }\n\n/-- This `bounded_random` interval generates integers between `x` and\n`y` by first generating a natural number between `0` and `y - x` and\nshifting the result appropriately. -/\ninstance int_bounded_random : bounded_random ℤ :=\n{ random_r := λ g inst x y hxy,\n  do ⟨z,h₀,h₁⟩ ← @bounded_random.random_r ℕ _ _ g inst 0 (int.nat_abs $ y - x) dec_trivial,\n     pure ⟨z + x,\n       int.le_add_of_nonneg_left (int.coe_nat_nonneg _),\n       int.add_le_of_le_sub_right $ le_trans\n         (int.coe_nat_le_coe_nat_of_le h₁)\n         (le_of_eq $ int.of_nat_nat_abs_eq_of_nonneg (int.sub_nonneg_of_le hxy)) ⟩ }\n\ninstance fin_random (n : ℕ) [ne_zero n] : random (fin n) :=\n{ random := λ g inst, @fin.random g inst _ _ }\n\ninstance fin_bounded_random (n : ℕ) : bounded_random (fin n) :=\n{ random_r := λ g inst (x y : fin n) p,\n    do ⟨r, h, h'⟩ ← @rand.random_r ℕ g inst _ _ x.val y.val p,\n       pure ⟨⟨r,lt_of_le_of_lt h' y.is_lt⟩, h, h'⟩ }\n\n/-- A shortcut for creating a `random (fin n)` instance from\na proof that `0 < n` rather than on matching on `fin (succ n)`  -/\ndef random_fin_of_pos : ∀ {n : ℕ} (h : 0 < n), random (fin n)\n| (succ n) _ := fin_random _\n| 0 h := false.elim (nat.not_lt_zero _ h)\n\nlemma bool_of_nat_mem_Icc_of_mem_Icc_to_nat (x y : bool) (n : ℕ) :\n  n ∈ (x.to_nat .. y.to_nat) → bool.of_nat n ∈ (x .. y) :=\nbegin\n  simp only [and_imp, set.mem_Icc], intros h₀ h₁,\n  split;\n    [ have h₂ := bool.of_nat_le_of_nat h₀, have h₂ := bool.of_nat_le_of_nat h₁ ];\n    rw bool.of_nat_to_nat at h₂; exact h₂,\nend\n\ninstance : random bool :=\n{ random   := λ g inst,\n  (bool.of_nat ∘ subtype.val) <$> @bounded_random.random_r ℕ _ _ g inst 0 1 (nat.zero_le _) }\n\ninstance : bounded_random bool :=\n{ random_r := λ g _inst x y p,\n  subtype.map bool.of_nat (bool_of_nat_mem_Icc_of_mem_Icc_to_nat x y) <$>\n    @bounded_random.random_r ℕ _ _ g _inst x.to_nat y.to_nat (bool.to_nat_le_to_nat p) }\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random (n : ℕ) : rand_g g (bitvec n) :=\nbitvec.of_fin <$> rand.random (fin $ 2^n)\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random_r {n : ℕ} (x y : bitvec n) (h : x ≤ y) : rand_g g (x .. y) :=\nhave h' : ∀ (a : fin (2 ^ n)), a ∈ (x.to_fin .. y.to_fin) → bitvec.of_fin a ∈ (x .. y),\nbegin\n  simp only [and_imp, set.mem_Icc], intros z h₀ h₁,\n  replace h₀ := bitvec.of_fin_le_of_fin_of_le h₀,\n  replace h₁ := bitvec.of_fin_le_of_fin_of_le h₁,\n  rw bitvec.of_fin_to_fin at h₀ h₁, split; assumption,\nend,\nsubtype.map bitvec.of_fin h' <$> rand.random_r x.to_fin y.to_fin (bitvec.to_fin_le_to_fin_of_le h)\n\nopen nat\n\ninstance random_bitvec (n : ℕ) : random (bitvec n) :=\n{ random := λ _ inst, @bitvec.random _ inst n }\n\ninstance bounded_random_bitvec (n : ℕ) : bounded_random (bitvec n) :=\n{ random_r := λ _ inst x y p, @bitvec.random_r _ inst _ _ _ p }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/control/random.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3252848602598147}}
{"text": "import category_theory.comma\nimport category_theory.adjunction.basic\nimport category_theory.limits.shapes\nimport category_theory.epi_mono\nimport category_theory.limits.over\nimport category_theory.closed.cartesian\nimport category.binary_products\nimport category.adjunction\n\n/-!\n# Properties of the over category.\n\nWe can interpret the forgetful functor `forget : over B ⥤ C` as dependent sum,\n(written `Σ_B`)\nand when C has binary products, it has a right adjoint `B*` given by\n`A ↦ (π₁ : B × A → B)`, denoted `star` in Lean.\n\nFurthermore, if the original category C has pullbacks and terminal object (i.e.\nall finite limits), `B*` has a right adjoint iff `B` is exponentiable in `C`.\nThis right adjoint is written `Π_B` and is interpreted as dependent product.\n\nGiven `f : A ⟶ B` in `C/B`, the iterated slice `(C/B)/f` is isomorphic to\n`C/A`.\n-/\nnoncomputable theory\nnamespace category_theory\nopen category limits\n\nuniverses v u\nvariables {C : Type u} [category.{v} C]\n\n-- def has_finite_products_of_has_finite_limits [has_finite_limits.{v} C] : has_finite_products.{v} C :=\n-- λ _ _ _, by resetI; apply_instance\ndef has_finite_coproducts_of_has_finite_colimits [has_finite_colimits.{v} C] : has_finite_coproducts.{v} C :=\nλ _ _ _, by resetI; apply_instance\n\n@[simps]\ndef over_iso {B : C} {f g : over B} (hl : f.left ≅ g.left) (hw : hl.hom ≫ g.hom = f.hom) : f ≅ g :=\n{ hom := over.hom_mk hl.hom, inv := over.hom_mk hl.inv (by simp [iso.inv_comp_eq, hw]) }\n\nsection adjunction\n\nvariable (B : C)\n\nsection\n\nvariable [has_pullbacks.{v} C]\n\n@[simps]\ndef real_pullback {A B : C} (f : A ⟶ B) : over B ⥤ over A :=\n{ obj := λ g, over.mk (pullback.snd : pullback g.hom f ⟶ A),\n  map := λ g h k, over.hom_mk (pullback.lift (pullback.fst ≫ k.left) pullback.snd (by simp [pullback.condition])) (by tidy) }\n\nend\n\nsection\nvariable [has_binary_products.{v} C]\n\n@[simps]\ndef star : C ⥤ over B :=\n{ obj := λ A, over.mk (limits.prod.fst : B ⨯ A ⟶ B),\n  map := λ X Y f, over.hom_mk (limits.prod.map (𝟙 _) f),\n  map_id' := λ X,\n  begin\n    apply over.over_morphism.ext,\n    dsimp,\n    simp,\n  end,\n  map_comp' := λ X Y Z f g,\n  begin\n    apply over.over_morphism.ext,\n    dsimp,\n    ext,\n    { rw [limits.prod.map_fst, comp_id, assoc, limits.prod.map_fst, comp_id, limits.prod.map_fst,\n          comp_id] },\n    { rw [limits.prod.map_snd, assoc, limits.prod.map_snd, limits.prod.map_snd_assoc] }\n  end }\n\nlocal attribute [tidy] tactic.case_bash\n\ndef forget_adj_star : over.forget B ⊣ star B :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ g A,\n  { to_fun := λ f, over.hom_mk (prod.lift g.hom f),\n    inv_fun := λ k, k.left ≫ limits.prod.snd,\n    left_inv := λ f, prod.lift_snd _ _,\n    right_inv := λ k,\n    begin\n      ext;\n      dsimp,\n      rw prod.lift_fst,\n      rw ← over.w k,\n      refl,\n      rw prod.lift_snd,\n    end },\n  hom_equiv_naturality_right' := λ X Y Y' f g,\n  begin\n    dsimp,\n    ext1,\n    dsimp,\n    rw prod.lift_map,\n    rw comp_id,\n  end,\n  hom_equiv_naturality_left_symm' := λ X' X Y f g, begin dsimp, rw assoc end }\nend\n\ndef exponentiable_of_star_is_left_adj [has_finite_products C] (h : is_left_adjoint (star B)) :\n  exponentiable B :=\n{ is_adj :=\n  { right := star B ⋙ h.right,\n    adj := adjunction.comp _ _ h.adj (forget_adj_star B) } }\n\ndef dependent_sum {A B : C} (f : A ⟶ B) : over A ⥤ over B :=\n(over.iterated_slice_equiv (over.mk f)).inverse ⋙ over.forget _\n\n/--\n`over.map f` gives nice definitional equalities but `dependent_sum` makes it easy to prove\nadjunction properties\n-/\ndef over_map_iso_dependent_sum {A B : C} (f : A ⟶ B) : dependent_sum f ≅ over.map f :=\nbegin\n  refine nat_iso.of_components (λ X, over_iso (iso.refl _) (id_comp _)) (λ X Y g, _),\n  { ext1,\n    change g.left ≫ 𝟙 _ = 𝟙 _ ≫ g.left,\n    rw [comp_id, id_comp] }\nend\n\ndef over_map_id {A : C} : over.map (𝟙 A) ≅ 𝟭 _ :=\nnat_iso.of_components (λ X, over_iso (iso.refl _) (begin dsimp, simp end)) (λ X Y f, begin ext, dsimp, simp end)\n\ndef over_map_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) : over.map (f ≫ g) ≅ over.map f ⋙ over.map g :=\nnat_iso.of_components (λ X, over_iso (iso.refl _) (begin dsimp, simp end)) (λ X Y f, begin ext, dsimp, simp end)\n\n-- local attribute [instance] has_finite_wide_pullbacks_of_has_finite_limits\n\ndef pullback_along {A B : C} (f : A ⟶ B) [has_binary_products (over B)] : over B ⥤ over A :=\nstar (over.mk f) ⋙ (over.iterated_slice_equiv _).functor\n\nlemma jointly_mono {A B : C} [has_binary_products (over B)] {f g : over B} (t₁ t₂ : A ⟶ (g ⨯ f).left) :\n  t₁ ≫ (limits.prod.fst : g ⨯ f ⟶ _).left = t₂ ≫ (limits.prod.fst : g ⨯ f ⟶ _).left →\n  t₁ ≫ (limits.prod.snd : g ⨯ f ⟶ _).left = t₂ ≫ (limits.prod.snd : g ⨯ f ⟶ _).left →\n  t₁ = t₂ :=\nbegin\n  intros h₁ h₂,\n  let A' : over B := over.mk (t₂ ≫ (g ⨯ f).hom), -- usually easier in practice to use the second one\n  have : t₁ ≫ (g ⨯ f).hom = t₂ ≫ (g ⨯ f).hom,\n    rw [← over.w (limits.prod.fst : g ⨯ f ⟶ _), reassoc_of h₁],\n  let t₁' : A' ⟶ g ⨯ f := over.hom_mk t₁ this,\n  let t₂' : A' ⟶ g ⨯ f := over.hom_mk t₂,\n  suffices : t₁' = t₂',\n    apply congr_arg comma_morphism.left this,\n  apply prod.hom_ext;\n  { ext1, assumption }\nend\n\ndef iso_pb {A B : C} (f : A ⟶ B) [has_binary_products (over B)] [has_pullbacks C] :\n  pullback_along f ≅ real_pullback f :=\nbegin\n  refine nat_iso.of_components _ _,\n  { intro X,\n    let p : over B := over.mk (pullback.snd ≫ f : pullback X.hom f ⟶ B),\n    let q : p ⟶ over.mk f ⨯ X := prod.lift (over.hom_mk pullback.snd rfl) (over.hom_mk pullback.fst pullback.condition),\n    apply over_iso _ _,\n    refine ⟨pullback.lift _ _ _, q.left, _, _⟩,\n    { apply (limits.prod.snd : over.mk f ⨯ X ⟶ _).left },\n    { apply (limits.prod.fst : over.mk f ⨯ X ⟶ _).left },\n    { rw [over.w limits.prod.snd, ← over.w limits.prod.fst, over.mk_hom] },\n    { apply jointly_mono;\n      simp [← over.comp_left] },\n    { apply pullback.hom_ext;\n      simp [← over.comp_left] },\n    { apply pullback.lift_snd } },\n  { intros X Y g,\n    ext1,\n    dsimp [pullback_along],\n    apply pullback.hom_ext,\n    { simp only [assoc, pullback.lift_fst, ← over.comp_left, limits.prod.map_snd, pullback.lift_fst_assoc] },\n    { simp only [assoc, pullback.lift_snd, ← over.comp_left, limits.prod.map_fst, comp_id] } },\nend\n\nsection\n-- local attribute [instance] over.construct_products.over_binary_product_of_pullback\ndef radj {A B : C} (f : A ⟶ B) [has_pullbacks C] :\n  over.map f ⊣ real_pullback f :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ g h,\n  { to_fun := λ X, over.hom_mk (pullback.lift X.left g.hom (over.w X)) (pullback.lift_snd _ _ _),\n    inv_fun := λ Y,\n    begin\n      refine over.hom_mk _ _,\n      refine Y.left ≫ pullback.fst,\n      dsimp,\n      rw [← over.w Y, assoc, pullback.condition, assoc], refl,\n    end,\n    left_inv := by tidy,\n    right_inv := λ Y, by { ext, dsimp, simp, dsimp, rw [pullback.lift_snd, ← over.w Y], refl } } }\n-- (((over.mk f).iterated_slice_equiv.symm.to_adjunction.comp _ _ (forget_adj_star _)).of_nat_iso_left (over_map_iso_dependent_sum f)).of_nat_iso_right (iso_pb f)\n\ndef pullback_id {A : C} [has_pullbacks C] : real_pullback (𝟙 A) ≅ 𝟭 _ :=\nadjunction.right_adjoint_uniq (radj _) (adjunction.id.of_nat_iso_left over_map_id.symm)\n\ndef pullback_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [has_pullbacks C] :\n  real_pullback (f ≫ g) ≅ real_pullback g ⋙ real_pullback f :=\nadjunction.right_adjoint_uniq (radj _) (((radj _).comp _ _ (radj _)).of_nat_iso_left (over_map_comp _ _).symm)\n\ninstance thing {A B : C} (f : A ⟶ B) [has_pullbacks C] : is_right_adjoint (real_pullback f) :=\n⟨_, radj f⟩\nend\n\nvariables [has_finite_products.{v} C] [has_pullbacks.{v} C]\n\ndef Pi_obj [exponentiable B] (f : over B) : C := pullback ((exp B).map f.hom) (internalize_hom (𝟙 B))\n\n@[simps]\nprivate def pi_obj.equiv [exponentiable B] (X : C) (Y : over B) :\n  ((star B).obj X ⟶ Y) ≃ (X ⟶ Pi_obj B Y) :=\n{ to_fun := λ f, pullback.lift (cartesian_closed.curry f.left) (terminal.from _)\n    (by { rw [internalize_hom, comp_id, ← curry_natural_left, ← curry_natural_right,\n              limits.prod.map_fst, comp_id, over.w f], refl }),\n  inv_fun := λ g,\n    begin\n      apply over.hom_mk _ _,\n      { apply (cartesian_closed.uncurry (g ≫ pullback.fst)) },\n      { rw [← uncurry_natural_right, assoc, pullback.condition, ← assoc, uncurry_natural_left],\n        dsimp [internalize_hom],\n        rw [uncurry_curry, limits.prod.map_fst_assoc, comp_id, comp_id] }\n    end,\n  left_inv := λ f, by { ext1, simp },\n  right_inv := λ g,\n  by { ext1, { simp }, { apply subsingleton.elim } } }\n\nprivate lemma pi_obj.natural_equiv [exponentiable B] (X' X : C) (Y : over B) (f : X' ⟶ X) (g : (star B).obj X ⟶ Y) :\n  (pi_obj.equiv B X' Y).to_fun ((star B).map f ≫ g) = f ≫ (pi_obj.equiv B X Y).to_fun g :=\nbegin\n  apply pullback.hom_ext,\n  { simp [curry_natural_left] },\n  { apply subsingleton.elim }\nend\n\ndef Pi_functor [exponentiable B] : over B ⥤ C :=\n  adjunction.right_adjoint_of_equiv (pi_obj.equiv B) (pi_obj.natural_equiv B)\ndef star_adj_pi_of_exponentiable [exponentiable B] : star B ⊣ Pi_functor B :=\n  adjunction.adjunction_of_equiv_right _ _\ninstance star_is_left_adj_of_exponentiable [exponentiable B] : is_left_adjoint (star B) :=\n  ⟨Pi_functor B, star_adj_pi_of_exponentiable B⟩\n\nend adjunction\n\nend category_theory\n", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3252848602598147}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.relator\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u \n\nnamespace Mathlib\n\n/-!\n# Quotient types\n\nThis module extends the core library's treatment of quotient types (`init.data.quot`).\n\n## Tags\n\nquotient\n-/\n\nnamespace setoid\n\n\ntheorem ext {α : Sort u_1} {s : setoid α} {t : setoid α} : (∀ (a b : α), r a b ↔ r a b) → s = t :=\n  sorry\n\nend setoid\n\n\nnamespace quot\n\n\nprotected instance inhabited {α : Sort u_1} {ra : α → α → Prop} [Inhabited α] :\n    Inhabited (Quot ra) :=\n  { default := Quot.mk ra Inhabited.default }\n\n/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/\nprotected def hrec_on₂ {α : Sort u_1} {β : Sort u_2} {ra : α → α → Prop} {rb : β → β → Prop}\n    {φ : Quot ra → Quot rb → Sort u_3} (qa : Quot ra) (qb : Quot rb)\n    (f : (a : α) → (b : β) → φ (Quot.mk ra a) (Quot.mk rb b))\n    (ca : ∀ {b : β} {a₁ a₂ : α}, ra a₁ a₂ → f a₁ b == f a₂ b)\n    (cb : ∀ {a : α} {b₁ b₂ : β}, rb b₁ b₂ → f a b₁ == f a b₂) : φ qa qb :=\n  quot.hrec_on qa (fun (a : α) => quot.hrec_on qb (f a) sorry) sorry\n\n/-- Map a function `f : α → β` such that `ra x y` implies `rb (f x) (f y)`\nto a map `quot ra → quot rb`. -/\nprotected def map {α : Sort u_1} {β : Sort u_2} {ra : α → α → Prop} {rb : β → β → Prop} (f : α → β)\n    (h : relator.lift_fun ra rb f f) : Quot ra → Quot rb :=\n  Quot.lift (fun (x : α) => Quot.mk rb (f x)) sorry\n\n/-- If `ra` is a subrelation of `ra'`, then we have a natural map `quot ra → quot ra'`. -/\nprotected def map_right {α : Sort u_1} {ra : α → α → Prop} {ra' : α → α → Prop}\n    (h : ∀ (a₁ a₂ : α), ra a₁ a₂ → ra' a₁ a₂) : Quot ra → Quot ra' :=\n  quot.map id h\n\n/-- weaken the relation of a quotient -/\ndef factor {α : Type u_1} (r : α → α → Prop) (s : α → α → Prop) (h : ∀ (x y : α), r x y → s x y) :\n    Quot r → Quot s :=\n  Quot.lift (Quot.mk s) sorry\n\ntheorem factor_mk_eq {α : Type u_1} (r : α → α → Prop) (s : α → α → Prop)\n    (h : ∀ (x y : α), r x y → s x y) : factor r s h ∘ Quot.mk r = Quot.mk s :=\n  rfl\n\n/-- Descends a function `f : α → β → γ` to quotients of `α` and `β`. -/\nprotected def lift₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} (f : α → β → γ) (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → f a b₁ = f a b₂)\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → f a₁ b = f a₂ b) (q₁ : Quot r) (q₂ : Quot s) : γ :=\n  Quot.lift (fun (a : α) => Quot.lift (f a) (hr a)) sorry q₁ q₂\n\n@[simp] theorem lift₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} (a : α) (b : β) (f : α → β → γ)\n    (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → f a b₁ = f a b₂)\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → f a₁ b = f a₂ b) :\n    quot.lift₂ f hr hs (Quot.mk r a) (Quot.mk s b) = f a b :=\n  rfl\n\n/-- Descends a function `f : α → β → γ` to quotients of `α` and `β` and applies it. -/\nprotected def lift_on₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} (p : Quot r) (q : Quot s) (f : α → β → γ)\n    (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → f a b₁ = f a b₂)\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → f a₁ b = f a₂ b) : γ :=\n  quot.lift₂ f hr hs p q\n\n@[simp] theorem lift_on₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} (a : α) (b : β) (f : α → β → γ)\n    (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → f a b₁ = f a b₂)\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → f a₁ b = f a₂ b) :\n    quot.lift_on₂ (Quot.mk r a) (Quot.mk s b) f hr hs = f a b :=\n  rfl\n\n/-- Descends a function `f : α → β → γ` to quotients of `α` and `β` wih values in a quotient of\n`γ`. -/\nprotected def map₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} {t : γ → γ → Prop} (f : α → β → γ)\n    (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → t (f a b₁) (f a b₂))\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → t (f a₁ b) (f a₂ b)) (q₁ : Quot r) (q₂ : Quot s) :\n    Quot t :=\n  quot.lift₂ (fun (a : α) (b : β) => Quot.mk t (f a b)) sorry sorry q₁ q₂\n\n@[simp] theorem map₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} {t : γ → γ → Prop} (f : α → β → γ)\n    (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → t (f a b₁) (f a b₂))\n    (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → t (f a₁ b) (f a₂ b)) (a : α) (b : β) :\n    quot.map₂ f hr hs (Quot.mk r a) (Quot.mk s b) = Quot.mk t (f a b) :=\n  rfl\n\nprotected theorem induction_on₂ {α : Sort u_1} {β : Sort u_2} {r : α → α → Prop} {s : β → β → Prop}\n    {δ : Quot r → Quot s → Prop} (q₁ : Quot r) (q₂ : Quot s)\n    (h : ∀ (a : α) (b : β), δ (Quot.mk r a) (Quot.mk s b)) : δ q₁ q₂ :=\n  Quot.ind (fun (a₁ : α) => Quot.ind (fun (a₂ : β) => h a₁ a₂) q₂) q₁\n\nprotected theorem induction_on₃ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {r : α → α → Prop}\n    {s : β → β → Prop} {t : γ → γ → Prop} {δ : Quot r → Quot s → Quot t → Prop} (q₁ : Quot r)\n    (q₂ : Quot s) (q₃ : Quot t)\n    (h : ∀ (a : α) (b : β) (c : γ), δ (Quot.mk r a) (Quot.mk s b) (Quot.mk t c)) : δ q₁ q₂ q₃ :=\n  Quot.ind (fun (a₁ : α) => Quot.ind (fun (a₂ : β) => Quot.ind (fun (a₃ : γ) => h a₁ a₂ a₃) q₃) q₂)\n    q₁\n\nend quot\n\n\nnamespace quotient\n\n\nprotected instance inhabited {α : Sort u_1} [sa : setoid α] [Inhabited α] :\n    Inhabited (quotient sa) :=\n  { default := quotient.mk Inhabited.default }\n\n/-- Induction on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/\nprotected def hrec_on₂ {α : Sort u_1} {β : Sort u_2} [sa : setoid α] [sb : setoid β]\n    {φ : quotient sa → quotient sb → Sort u_3} (qa : quotient sa) (qb : quotient sb)\n    (f : (a : α) → (b : β) → φ (quotient.mk a) (quotient.mk b))\n    (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=\n  quot.hrec_on₂ qa qb f sorry sorry\n\n/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements\nto a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/\nprotected def map {α : Sort u_1} {β : Sort u_2} [sa : setoid α] [sb : setoid β] (f : α → β)\n    (h : relator.lift_fun has_equiv.equiv has_equiv.equiv f f) : quotient sa → quotient sb :=\n  quot.map f h\n\n@[simp] theorem map_mk {α : Sort u_1} {β : Sort u_2} [sa : setoid α] [sb : setoid β] (f : α → β)\n    (h : relator.lift_fun has_equiv.equiv has_equiv.equiv f f) (x : α) :\n    quotient.map f h (quotient.mk x) = quotient.mk (f x) :=\n  rfl\n\n/-- Map a function `f : α → β → γ` that sends equivalent elements to equivalent elements\nto a function `f : quotient sa → quotient sb → quotient sc`.\nUseful to define binary operations on quotients. -/\nprotected def map₂ {α : Sort u_1} {β : Sort u_2} [sa : setoid α] [sb : setoid β] {γ : Sort u_4}\n    [sc : setoid γ] (f : α → β → γ)\n    (h : relator.lift_fun has_equiv.equiv (has_equiv.equiv ⇒ has_equiv.equiv) f f) :\n    quotient sa → quotient sb → quotient sc :=\n  quotient.lift₂ (fun (x : α) (y : β) => quotient.mk (f x y)) sorry\n\nend quotient\n\n\ntheorem quot.eq {α : Type u_1} {r : α → α → Prop} {x : α} {y : α} :\n    Quot.mk r x = Quot.mk r y ↔ eqv_gen r x y :=\n  { mp := quot.exact r, mpr := quot.eqv_gen_sound }\n\n@[simp] theorem quotient.eq {α : Sort u_1} [r : setoid α] {x : α} {y : α} :\n    quotient.mk x = quotient.mk y ↔ x ≈ y :=\n  { mp := quotient.exact, mpr := quotient.sound }\n\ntheorem forall_quotient_iff {α : Type u_1} [r : setoid α] {p : quotient r → Prop} :\n    (∀ (a : quotient r), p a) ↔ ∀ (a : α), p (quotient.mk a) :=\n  { mp := fun (h : ∀ (a : quotient r), p a) (x : α) => h (quotient.mk x),\n    mpr := fun (h : ∀ (a : α), p (quotient.mk a)) (a : quotient r) => quotient.induction_on a h }\n\n@[simp] theorem quotient.lift_beta {α : Sort u_1} {β : Sort u_2} [s : setoid α] (f : α → β)\n    (h : ∀ (a b : α), a ≈ b → f a = f b) (x : α) : quotient.lift f h (quotient.mk x) = f x :=\n  rfl\n\n@[simp] theorem quotient.lift_on_beta {α : Sort u_1} {β : Sort u_2} [s : setoid α] (f : α → β)\n    (h : ∀ (a b : α), a ≈ b → f a = f b) (x : α) : quotient.lift_on (quotient.mk x) f h = f x :=\n  rfl\n\n@[simp] theorem quotient.lift_on_beta₂ {α : Sort u_1} {β : Sort u_2} [setoid α] (f : α → α → β)\n    (h : ∀ (a₁ a₂ b₁ b₂ : α), a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ = f b₁ b₂) (x : α) (y : α) :\n    quotient.lift_on₂ (quotient.mk x) (quotient.mk y) f h = f x y :=\n  rfl\n\n/-- `quot.mk r` is a surjective function. -/\ntheorem surjective_quot_mk {α : Sort u_1} (r : α → α → Prop) : function.surjective (Quot.mk r) :=\n  quot.exists_rep\n\n/-- `quotient.mk` is a surjective function. -/\ntheorem surjective_quotient_mk (α : Sort u_1) [s : setoid α] : function.surjective quotient.mk :=\n  quot.exists_rep\n\n/-- Choose an element of the equivalence class using the axiom of choice.\n  Sound but noncomputable. -/\ndef quot.out {α : Sort u_1} {r : α → α → Prop} (q : Quot r) : α :=\n  classical.some (quot.exists_rep q)\n\n/-- Unwrap the VM representation of a quotient to obtain an element of the equivalence class.\n  Computable but unsound. -/\n@[simp] theorem quot.out_eq {α : Sort u_1} {r : α → α → Prop} (q : Quot r) :\n    Quot.mk r (quot.out q) = q :=\n  classical.some_spec (quot.exists_rep q)\n\n/-- Choose an element of the equivalence class using the axiom of choice.\n  Sound but noncomputable. -/\ndef quotient.out {α : Sort u_1} [s : setoid α] : quotient s → α := quot.out\n\n@[simp] theorem quotient.out_eq {α : Sort u_1} [s : setoid α] (q : quotient s) :\n    quotient.mk (quotient.out q) = q :=\n  quot.out_eq q\n\ntheorem quotient.mk_out {α : Sort u_1} [s : setoid α] (a : α) : quotient.out (quotient.mk a) ≈ a :=\n  quotient.exact (quotient.out_eq (quotient.mk a))\n\nprotected instance pi_setoid {ι : Sort u_1} {α : ι → Sort u_2} [(i : ι) → setoid (α i)] :\n    setoid ((i : ι) → α i) :=\n  setoid.mk (fun (a b : (i : ι) → α i) => ∀ (i : ι), a i ≈ b i) sorry\n\n/-- Given a function `f : Π i, quotient (S i)`, returns the class of functions `Π i, α i` sending\neach `i` to an element of the class `f i`. -/\ndef quotient.choice {ι : Type u_1} {α : ι → Type u_2} [S : (i : ι) → setoid (α i)]\n    (f : (i : ι) → quotient (S i)) : quotient Mathlib.pi_setoid :=\n  quotient.mk fun (i : ι) => quotient.out (f i)\n\ntheorem quotient.choice_eq {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → setoid (α i)]\n    (f : (i : ι) → α i) : (quotient.choice fun (i : ι) => quotient.mk (f i)) = quotient.mk f :=\n  quotient.sound fun (i : ι) => quotient.mk_out (f i)\n\ntheorem nonempty_quotient_iff {α : Sort u_1} (s : setoid α) : Nonempty (quotient s) ↔ Nonempty α :=\n  sorry\n\n/-- `trunc α` is the quotient of `α` by the always-true relation. This\n  is related to the propositional truncation in HoTT, and is similar\n  in effect to `nonempty α`, but unlike `nonempty α`, `trunc α` is data,\n  so the VM representation is the same as `α`, and so this can be used to\n  maintain computability. -/\ndef trunc (α : Sort u) := Quot fun (_x _x : α) => True\n\ntheorem true_equivalence {α : Sort u_1} : equivalence fun (_x _x : α) => True :=\n  { left := fun (_x : α) => trivial,\n    right :=\n      { left := fun (_x _x : α) (_x : True) => trivial,\n        right := fun (_x _x _x : α) (_x _x : True) => trivial } }\n\nnamespace trunc\n\n\n/-- Constructor for `trunc α` -/\ndef mk {α : Sort u_1} (a : α) : trunc α := Quot.mk (fun (_x _x : α) => True) a\n\nprotected instance inhabited {α : Sort u_1} [Inhabited α] : Inhabited (trunc α) :=\n  { default := mk Inhabited.default }\n\n/-- Any constant function lifts to a function out of the truncation -/\ndef lift {α : Sort u_1} {β : Sort u_2} (f : α → β) (c : ∀ (a b : α), f a = f b) : trunc α → β :=\n  Quot.lift f sorry\n\ntheorem ind {α : Sort u_1} {β : trunc α → Prop} : (∀ (a : α), β (mk a)) → ∀ (q : trunc α), β q :=\n  Quot.ind\n\nprotected theorem lift_beta {α : Sort u_1} {β : Sort u_2} (f : α → β) (c : ∀ (a b : α), f a = f b)\n    (a : α) : lift f c (mk a) = f a :=\n  rfl\n\n/-- Lift a constant function on `q : trunc α`. -/\nprotected def lift_on {α : Sort u_1} {β : Sort u_2} (q : trunc α) (f : α → β)\n    (c : ∀ (a b : α), f a = f b) : β :=\n  lift f c q\n\nprotected theorem induction_on {α : Sort u_1} {β : trunc α → Prop} (q : trunc α)\n    (h : ∀ (a : α), β (mk a)) : β q :=\n  ind h q\n\ntheorem exists_rep {α : Sort u_1} (q : trunc α) : ∃ (a : α), mk a = q := quot.exists_rep q\n\nprotected theorem induction_on₂ {α : Sort u_1} {β : Sort u_2} {C : trunc α → trunc β → Prop}\n    (q₁ : trunc α) (q₂ : trunc β) (h : ∀ (a : α) (b : β), C (mk a) (mk b)) : C q₁ q₂ :=\n  trunc.induction_on q₁ fun (a₁ : α) => trunc.induction_on q₂ (h a₁)\n\nprotected theorem eq {α : Sort u_1} (a : trunc α) (b : trunc α) : a = b :=\n  trunc.induction_on₂ a b fun (x y : α) => quot.sound trivial\n\nprotected instance subsingleton {α : Sort u_1} : subsingleton (trunc α) :=\n  subsingleton.intro trunc.eq\n\n/-- The `bind` operator for the `trunc` monad. -/\ndef bind {α : Sort u_1} {β : Sort u_2} (q : trunc α) (f : α → trunc β) : trunc β :=\n  trunc.lift_on q f sorry\n\n/-- A function `f : α → β` defines a function `map f : trunc α → trunc β`. -/\ndef map {α : Sort u_1} {β : Sort u_2} (f : α → β) (q : trunc α) : trunc β := bind q (mk ∘ f)\n\nprotected instance monad : Monad trunc := sorry\n\nprotected instance is_lawful_monad : is_lawful_monad trunc :=\n  is_lawful_monad.mk (fun (α β : Type u_1) (q : α) (f : α → trunc β) => rfl)\n    fun (α β γ : Type u_1) (x : trunc α) (f : α → trunc β) (g : β → trunc γ) =>\n      trunc.eq (x >>= f >>= g) (x >>= fun (x : α) => f x >>= g)\n\n/-- Recursion/induction principle for `trunc`. -/\nprotected def rec {α : Sort u_1} {C : trunc α → Sort u_3} (f : (a : α) → C (mk a))\n    (h : ∀ (a b : α), Eq._oldrec (f a) (rec._proof_1 a b) = f b) (q : trunc α) : C q :=\n  quot.rec f sorry q\n\n/-- A version of `trunc.rec` taking `q : trunc α` as the first argument. -/\nprotected def rec_on {α : Sort u_1} {C : trunc α → Sort u_3} (q : trunc α) (f : (a : α) → C (mk a))\n    (h : ∀ (a b : α), Eq._oldrec (f a) (rec_on._proof_1 a b) = f b) : C q :=\n  trunc.rec f h q\n\n/-- A version of `trunc.rec_on` assuming the codomain is a `subsingleton`. -/\nprotected def rec_on_subsingleton {α : Sort u_1} {C : trunc α → Sort u_3}\n    [∀ (a : α), subsingleton (C (mk a))] (q : trunc α) (f : (a : α) → C (mk a)) : C q :=\n  trunc.rec f sorry q\n\n/-- Noncomputably extract a representative of `trunc α` (using the axiom of choice). -/\ndef out {α : Sort u_1} : trunc α → α := quot.out\n\n@[simp] theorem out_eq {α : Sort u_1} (q : trunc α) : mk (out q) = q := trunc.eq (mk (out q)) q\n\nend trunc\n\n\ntheorem nonempty_of_trunc {α : Sort u_1} (q : trunc α) : Nonempty α :=\n  (fun (_a : ∃ (a : α), trunc.mk a = q) =>\n      Exists.dcases_on _a fun (w : α) (h : trunc.mk w = q) => idRhs (Nonempty α) (Nonempty.intro w))\n    (trunc.exists_rep q)\n\nnamespace quotient\n\n\n/- Versions of quotient definitions and lemmas ending in `'` use unification instead\nof typeclass inference for inferring the `setoid` argument. This is useful when there are\nseveral different quotient relations on a type, for example quotient groups, rings and modules -/\n\n/-- A version of `quotient.mk` taking `{s : setoid α}` as an implicit argument instead of an\ninstance argument. -/\nprotected def mk' {α : Sort u_1} {s₁ : setoid α} (a : α) : quotient s₁ := Quot.mk setoid.r a\n\n/-- `quotient.mk'` is a surjective function. -/\ntheorem surjective_quotient_mk' {α : Sort u_1} {s₁ : setoid α} : function.surjective quotient.mk' :=\n  quot.exists_rep\n\n/-- A version of `quotient.lift_on` taking `{s : setoid α}` as an implicit argument instead of an\ninstance argument. -/\nprotected def lift_on' {α : Sort u_1} {φ : Sort u_4} {s₁ : setoid α} (q : quotient s₁) (f : α → φ)\n    (h : ∀ (a b : α), setoid.r a b → f a = f b) : φ :=\n  quotient.lift_on q f h\n\n@[simp] protected theorem lift_on'_beta {α : Sort u_1} {φ : Sort u_4} {s₁ : setoid α} (f : α → φ)\n    (h : ∀ (a b : α), setoid.r a b → f a = f b) (x : α) :\n    quotient.lift_on' (quotient.mk' x) f h = f x :=\n  rfl\n\n/-- A version of `quotient.lift_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments\ninstead of instance arguments. -/\nprotected def lift_on₂' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : setoid α} {s₂ : setoid β}\n    (q₁ : quotient s₁) (q₂ : quotient s₂) (f : α → β → γ)\n    (h :\n      ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), setoid.r a₁ b₁ → setoid.r a₂ b₂ → f a₁ a₂ = f b₁ b₂) :\n    γ :=\n  quotient.lift_on₂ q₁ q₂ f h\n\n@[simp] protected theorem lift_on₂'_beta {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3}\n    {s₁ : setoid α} {s₂ : setoid β} (f : α → β → γ)\n    (h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), setoid.r a₁ b₁ → setoid.r a₂ b₂ → f a₁ a₂ = f b₁ b₂)\n    (a : α) (b : β) : quotient.lift_on₂' (quotient.mk' a) (quotient.mk' b) f h = f a b :=\n  rfl\n\n/-- A version of `quotient.ind` taking `{s : setoid α}` as an implicit argument instead of an\ninstance argument. -/\nprotected theorem ind' {α : Sort u_1} {s₁ : setoid α} {p : quotient s₁ → Prop}\n    (h : ∀ (a : α), p (quotient.mk' a)) (q : quotient s₁) : p q :=\n  quotient.ind h q\n\n/-- A version of `quotient.ind₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit arguments\ninstead of instance arguments. -/\nprotected theorem ind₂' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β}\n    {p : quotient s₁ → quotient s₂ → Prop}\n    (h : ∀ (a₁ : α) (a₂ : β), p (quotient.mk' a₁) (quotient.mk' a₂)) (q₁ : quotient s₁)\n    (q₂ : quotient s₂) : p q₁ q₂ :=\n  quotient.ind₂ h q₁ q₂\n\n/-- A version of `quotient.induction_on` taking `{s : setoid α}` as an implicit argument instead\nof an instance argument. -/\nprotected theorem induction_on' {α : Sort u_1} {s₁ : setoid α} {p : quotient s₁ → Prop}\n    (q : quotient s₁) (h : ∀ (a : α), p (quotient.mk' a)) : p q :=\n  quotient.induction_on q h\n\n/-- A version of `quotient.induction_on₂` taking `{s₁ : setoid α} {s₂ : setoid β}` as implicit\narguments instead of instance arguments. -/\nprotected theorem induction_on₂' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β}\n    {p : quotient s₁ → quotient s₂ → Prop} (q₁ : quotient s₁) (q₂ : quotient s₂)\n    (h : ∀ (a₁ : α) (a₂ : β), p (quotient.mk' a₁) (quotient.mk' a₂)) : p q₁ q₂ :=\n  quotient.induction_on₂ q₁ q₂ h\n\n/-- A version of `quotient.induction_on₃` taking `{s₁ : setoid α} {s₂ : setoid β} {s₃ : setoid γ}`\nas implicit arguments instead of instance arguments. -/\nprotected theorem induction_on₃' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : setoid α}\n    {s₂ : setoid β} {s₃ : setoid γ} {p : quotient s₁ → quotient s₂ → quotient s₃ → Prop}\n    (q₁ : quotient s₁) (q₂ : quotient s₂) (q₃ : quotient s₃)\n    (h : ∀ (a₁ : α) (a₂ : β) (a₃ : γ), p (quotient.mk' a₁) (quotient.mk' a₂) (quotient.mk' a₃)) :\n    p q₁ q₂ q₃ :=\n  quotient.induction_on₃ q₁ q₂ q₃ h\n\n/-- Recursion on a `quotient` argument `a`, result type depends on `⟦a⟧`. -/\nprotected def hrec_on' {α : Sort u_1} {s₁ : setoid α} {φ : quotient s₁ → Sort u_2}\n    (qa : quotient s₁) (f : (a : α) → φ (quotient.mk' a))\n    (c : ∀ (a₁ a₂ : α), a₁ ≈ a₂ → f a₁ == f a₂) : φ qa :=\n  quot.hrec_on qa f c\n\n@[simp] theorem hrec_on'_mk' {α : Sort u_1} {s₁ : setoid α} {φ : quotient s₁ → Sort u_2}\n    (f : (a : α) → φ (quotient.mk' a)) (c : ∀ (a₁ a₂ : α), a₁ ≈ a₂ → f a₁ == f a₂) (x : α) :\n    quotient.hrec_on' (quotient.mk' x) f c = f x :=\n  rfl\n\n/-- Recursion on two `quotient` arguments `a` and `b`, result type depends on `⟦a⟧` and `⟦b⟧`. -/\nprotected def hrec_on₂' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β}\n    {φ : quotient s₁ → quotient s₂ → Sort u_3} (qa : quotient s₁) (qb : quotient s₂)\n    (f : (a : α) → (b : β) → φ (quotient.mk' a) (quotient.mk' b))\n    (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) : φ qa qb :=\n  quotient.hrec_on₂ qa qb f c\n\n@[simp] theorem hrec_on₂'_mk' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β}\n    {φ : quotient s₁ → quotient s₂ → Sort u_3}\n    (f : (a : α) → (b : β) → φ (quotient.mk' a) (quotient.mk' b))\n    (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ ≈ a₂ → b₁ ≈ b₂ → f a₁ b₁ == f a₂ b₂) (x : α)\n    (qb : quotient s₂) :\n    quotient.hrec_on₂' (quotient.mk' x) qb f c =\n        quotient.hrec_on' qb (f x) fun (b₁ b₂ : β) => c x b₁ x b₂ (setoid.refl x) :=\n  rfl\n\n/-- Map a function `f : α → β` that sends equivalent elements to equivalent elements\nto a function `quotient sa → quotient sb`. Useful to define unary operations on quotients. -/\nprotected def map' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β} (f : α → β)\n    (h : relator.lift_fun has_equiv.equiv has_equiv.equiv f f) : quotient s₁ → quotient s₂ :=\n  quot.map f h\n\n@[simp] theorem map'_mk' {α : Sort u_1} {β : Sort u_2} {s₁ : setoid α} {s₂ : setoid β} (f : α → β)\n    (h : relator.lift_fun has_equiv.equiv has_equiv.equiv f f) (x : α) :\n    quotient.map' f h (quotient.mk' x) = quotient.mk' (f x) :=\n  rfl\n\n/-- A version of `quotient.map₂` using curly braces and unification. -/\nprotected def map₂' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : setoid α} {s₂ : setoid β}\n    {s₃ : setoid γ} (f : α → β → γ)\n    (h : relator.lift_fun has_equiv.equiv (has_equiv.equiv ⇒ has_equiv.equiv) f f) :\n    quotient s₁ → quotient s₂ → quotient s₃ :=\n  quotient.map₂ f h\n\n@[simp] theorem map₂'_mk' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : setoid α}\n    {s₂ : setoid β} {s₃ : setoid γ} (f : α → β → γ)\n    (h : relator.lift_fun has_equiv.equiv (has_equiv.equiv ⇒ has_equiv.equiv) f f) (x : α) :\n    quotient.map₂' f h (quotient.mk' x) = quotient.map' (f x) (h (setoid.refl x)) :=\n  rfl\n\ntheorem exact' {α : Sort u_1} {s₁ : setoid α} {a : α} {b : α} :\n    quotient.mk' a = quotient.mk' b → setoid.r a b :=\n  exact\n\ntheorem sound' {α : Sort u_1} {s₁ : setoid α} {a : α} {b : α} :\n    setoid.r a b → quotient.mk' a = quotient.mk' b :=\n  sound\n\n@[simp] protected theorem eq' {α : Sort u_1} {s₁ : setoid α} {a : α} {b : α} :\n    quotient.mk' a = quotient.mk' b ↔ setoid.r a b :=\n  eq\n\n/-- A version of `quotient.out` taking `{s₁ : setoid α}` as an implicit argument instead of an\ninstance argument. -/\ndef out' {α : Sort u_1} {s₁ : setoid α} (a : quotient s₁) : α := out a\n\n@[simp] theorem out_eq' {α : Sort u_1} {s₁ : setoid α} (q : quotient s₁) :\n    quotient.mk' (out' q) = q :=\n  out_eq q\n\ntheorem mk_out' {α : Sort u_1} {s₁ : setoid α} (a : α) : setoid.r (out' (quotient.mk' a)) a :=\n  exact (out_eq (quotient.mk' a))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/quot_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6261241702517976, "lm_q1q2_score": 0.3252848566353712}}
{"text": "import for_mathlib.category_theory.localization.derived_functor_shift\nimport for_mathlib.category_theory.triangulated.is_triangulated_subcategory\n\nopen category_theory category_theory.category category_theory.limits\n\nnoncomputable theory\n\nnamespace category_theory\n\nnamespace right_derivability_structure\n\nvariables {C₀ C H D : Type*} [category C₀] [category C] [category D] [category H]\n  {W₀ : morphism_property C₀} {W : morphism_property C} {Φ : localizor_morphism W₀ W}\n  [localizor_morphism.is_localization_equivalence Φ]\n  [morphism_property.multiplicative W₀] [morphism_property.multiplicative W]\n  (β : right_derivability_structure.basic Φ)\n  (F : C ⥤ D) (L : C ⥤ H)\n  (hF : W₀.is_inverted_by (Φ.functor ⋙ F))\n  [has_shift C₀ ℤ] [has_shift C ℤ] [has_shift H ℤ] [has_shift D ℤ]\n  [F.has_comm_shift ℤ] [L.has_comm_shift ℤ] [Φ.functor.has_comm_shift ℤ]\n  [preadditive C₀] [preadditive C] [preadditive H] [preadditive D]\n  [has_zero_object C₀] [has_zero_object C] [has_zero_object H] [has_zero_object D]\n  [∀ (n : ℤ ), (shift_functor C₀ n).additive]\n  [∀ (n : ℤ ), (shift_functor C n).additive]\n  [∀ (n : ℤ ), (shift_functor H n).additive]\n  [∀ (n : ℤ ), (shift_functor D n).additive]\n  [L.is_localization W]\n  [pretriangulated C₀] [pretriangulated C] [pretriangulated H] [pretriangulated D]\n  [functor.is_triangulated F] [functor.is_triangulated Φ.functor]\n\nnamespace basic\n\ninclude β\n\nlemma Φ_functor_comp_L_ess_surj_on_dist_triang [L.ess_surj_on_dist_triang] [L.is_triangulated]:\n  (Φ.functor ⋙ L).ess_surj_on_dist_triang :=\n⟨λ T hT, begin\n  obtain ⟨T', hT', ⟨e⟩⟩ := functor.ess_surj_on_dist_triang.condition L T hT,\n  obtain ⟨X₁, X₂, f', fac⟩ := β.nonempty_arrow_right_resolution T'.mor₁,\n  obtain ⟨X₃, g, h, mem⟩ := pretriangulated.distinguished_cocone_triangle _ _ f',\n  haveI : is_iso (L.map X₁.hom.f) := localization.inverts L W _ X₁.hom.hf,\n  haveI : is_iso (L.map X₂.hom.f) := localization.inverts L W _ X₂.hom.hf,\n  refine ⟨_, mem, ⟨iso.symm _ ≪≫ e⟩⟩,\n  refine pretriangulated.iso_triangle_of_distinguished_of_is_iso₁₂ _ _\n    (L.map_distinguished _ hT') ((Φ.functor ⋙ L).map_distinguished _ mem)\n      (as_iso (L.map X₁.hom.f)) (as_iso (L.map X₂.hom.f))\n      (by { dsimp, simp only [← L.map_comp, fac]}),\nend⟩\n\nvariable [functor.ess_surj_on_dist_triang (Φ.functor ⋙ L)]\n\ninclude hF\n\nlemma derived_functor_is_triangulated' [F.has_right_derived_functor W] :\n  (F.right_derived_functor L W).is_triangulated :=\n⟨λ T hT, begin\n  obtain ⟨T', hT', ⟨e⟩⟩ := functor.ess_surj_on_dist_triang.condition (Φ.functor ⋙ L) T hT,\n  have E' := (functor.map_triangle_comp (Φ.functor ⋙ L) (F.right_derived_functor L W)).symm.app T',\n  let τ : Φ.functor ⋙ F ⟶ (Φ.functor ⋙ L) ⋙ F.right_derived_functor L W :=\n    whisker_left Φ.functor (F.right_derived_functor_α L W) ≫ (functor.associator _ _ _).inv,\n  haveI : ∀ (X₀ : C₀), is_iso (τ.app X₀) := λ X₀, begin\n    dsimp [τ],\n    rw [comp_id],\n    apply β.is_iso_app L F hF,\n  end,\n  haveI := nat_iso.is_iso_of_is_iso_app τ,\n  haveI : (as_iso τ).hom.respects_comm_shift ℤ := by { dsimp [as_iso], apply_instance, },\n  exact pretriangulated.isomorphic_distinguished _ (functor.map_distinguished _ _ hT') _\n    ((F.right_derived_functor L W).map_triangle.map_iso e.symm ≪≫\n    (functor.map_triangle_comp (Φ.functor ⋙ L) (F.right_derived_functor L W)).symm.app T' ≪≫\n    (functor.map_triangle_nat_iso (as_iso τ)).symm.app T'),\nend⟩\n\nlemma derived_functor_is_triangulated (RF : H ⥤ D) (α : F ⟶ L ⋙ RF)\n  [RF.is_right_derived_functor α] [RF.has_comm_shift ℤ] [α.respects_comm_shift ℤ] :\n  RF.is_triangulated :=\nbegin\n  haveI := functor.is_right_derived_functor.has_right_derived_functor F RF L α W,\n  haveI := β.derived_functor_is_triangulated' F L hF,\n  let e := nat_iso.right_derived (iso.refl F) (F.right_derived_functor_α L W) α,\n  haveI : e.hom.respects_comm_shift ℤ,\n  { refine ⟨λ n, _⟩,\n    apply functor.is_right_derived_functor_to_ext\n      (shift_functor H n ⋙ F.right_derived_functor L W)\n      (functor.has_comm_shift.right_derived_shift_α F L W n),\n    ext X,\n    simp only [nat_trans.comp_app, whisker_left_app, whisker_right_app],\n    erw functor.has_comm_shift.right_derived_comm_shift_comm_assoc,\n    simp only [nat_iso.right_derived_hom, iso.refl_hom,\n      functor.has_comm_shift.right_derived_α_shift_app, ← functor.map_comp,\n      nat_trans.right_derived_app (𝟙 F) (F.right_derived_functor_α L W) α X,\n      nat_trans.id_app, id_comp,\n      functor.has_comm_shift.right_derived_shift_α_app,\n      assoc, nat_trans.naturality_assoc, nat_trans.right_derived_app_assoc,\n      nat_trans.respects_comm_shift.comm_app α n X,\n      functor.has_comm_shift.comp_hom_app], },\n  exact functor.is_triangulated.of_iso e,\nend\n\nend basic\n\nend right_derivability_structure\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/derived_functor_triangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3249089268905928}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.linear.linear_functor\nimport category_theory.monoidal.preadditive\n\n/-!\n# Linear monoidal categories\n\nA monoidal category is `monoidal_linear R` if it is monoidal preadditive and\ntensor product of morphisms is `R`-linear in both factors.\n-/\n\nnamespace category_theory\n\nopen category_theory.limits\nopen category_theory.monoidal_category\n\nvariables (R : Type*) [semiring R]\nvariables (C : Type*) [category C] [preadditive C] [linear R C]\nvariables [monoidal_category C] [monoidal_preadditive C]\n\n/--\nA category is `monoidal_linear R` if tensoring is `R`-linear in both factors.\n-/\nclass monoidal_linear : Prop :=\n(tensor_smul' : ∀ {W X Y Z : C} (f : W ⟶ X) (r : R) (g : Y ⟶ Z),\n  f ⊗ (r • g) = r • (f ⊗ g) . obviously)\n(smul_tensor' : ∀ {W X Y Z : C} (r : R) (f : W ⟶ X) (g : Y ⟶ Z),\n  (r • f) ⊗ g = r • (f ⊗ g) . obviously)\n\nrestate_axiom monoidal_linear.tensor_smul'\nrestate_axiom monoidal_linear.smul_tensor'\nattribute [simp] monoidal_linear.tensor_smul monoidal_linear.smul_tensor\n\nvariables {C} [monoidal_linear R C]\n\ninstance tensor_left_linear (X : C) : (tensor_left X).linear R := {}\ninstance tensor_right_linear (X : C) : (tensor_right X).linear R := {}\ninstance tensoring_left_linear (X : C) : ((tensoring_left C).obj X).linear R := {}\ninstance tensoring_right_linear (X : C) : ((tensoring_right C).obj X).linear R := {}\n\n/-- A faithful linear monoidal functor to a linear monoidal category\nensures that the domain is linear monoidal. -/\nlemma monoidal_linear_of_faithful\n  {D : Type*} [category D] [preadditive D] [linear R D]\n  [monoidal_category D] [monoidal_preadditive D]\n  (F : monoidal_functor D C) [faithful F.to_functor]\n  [F.to_functor.additive] [F.to_functor.linear R] :\n  monoidal_linear R D :=\n{ tensor_smul' := begin\n    intros,\n    apply F.to_functor.map_injective,\n    simp only [F.to_functor.map_smul r (f ⊗ g), F.to_functor.map_smul r g, F.map_tensor,\n      monoidal_linear.tensor_smul, linear.smul_comp, linear.comp_smul],\n  end,\n  smul_tensor' := begin\n    intros,\n    apply F.to_functor.map_injective,\n    simp only [F.to_functor.map_smul r (f ⊗ g), F.to_functor.map_smul r f, F.map_tensor,\n      monoidal_linear.smul_tensor, linear.smul_comp, linear.comp_smul],\n  end, }\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/linear.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3247718274027638}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz, Scott Morrison\n-/\nimport category_theory.punit\nimport category_theory.comma\nimport category_theory.limits.shapes.terminal\n\n/-!\n# The category of \"structured arrows\"\n\nFor `T : C ⥤ D`, a `T`-structured arrow with source `S : D`\nis just a morphism `S ⟶ T.obj Y`, for some `Y : C`.\n\nThese form a category with morphisms `g : Y ⟶ Y'` making the obvious diagram commute.\n\nWe prove that `𝟙 (T.obj Y)` is the initial object in `T`-structured objects with source `T.obj Y`.\n-/\n\nnamespace category_theory\n\n-- morphism levels before object levels. See note [category_theory universes].\nuniverses v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]\n\n/--\nThe category of `T`-structured arrows with domain `S : D` (here `T : C ⥤ D`),\nhas as its objects `D`-morphisms of the form `S ⟶ T Y`, for some `Y : C`,\nand morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute.\n-/\n@[derive category, nolint has_inhabited_instance]\ndef structured_arrow (S : D) (T : C ⥤ D) := comma (functor.from_punit S) T\n\nnamespace structured_arrow\n\n/-- The obvious projection functor from structured arrows. -/\n@[simps]\ndef proj (S : D) (T : C ⥤ D) : structured_arrow S T ⥤ C := comma.snd _ _\n\nvariables {S S' S'' : D} {Y Y' : C} {T : C ⥤ D}\n\n/-- Construct a structured arrow from a morphism. -/\ndef mk (f : S ⟶ T.obj Y) : structured_arrow S T := ⟨⟨⟨⟩⟩, Y, f⟩\n\n@[simp] lemma mk_left (f : S ⟶ T.obj Y) : (mk f).left = ⟨⟨⟩⟩ := rfl\n@[simp] lemma mk_right (f : S ⟶ T.obj Y) : (mk f).right = Y := rfl\n@[simp] lemma mk_hom_eq_self (f : S ⟶ T.obj Y) : (mk f).hom = f := rfl\n\n@[simp, reassoc] lemma w {A B : structured_arrow S T} (f : A ⟶ B) : A.hom ≫ T.map f.right = B.hom :=\nby { have := f.w; tidy }\n\nlemma eq_mk (f : structured_arrow S T) : f = mk f.hom :=\nby { cases f, congr, ext, }\n\n/--\nTo construct a morphism of structured arrows,\nwe need a morphism of the objects underlying the target,\nand to check that the triangle commutes.\n-/\n@[simps]\ndef hom_mk {f f' : structured_arrow S T} (g : f.right ⟶ f'.right) (w : f.hom ≫ T.map g = f'.hom) :\n  f ⟶ f' :=\n{ left := eq_to_hom (by ext),\n  right := g,\n  w' := by { dsimp, simpa using w.symm, }, }\n\n/--\nGiven a structured arrow `X ⟶ F(U)`, and an arrow `U ⟶ Y`, we can construct a morphism of\nstructured arrow given by `(X ⟶ F(U)) ⟶ (X ⟶ F(U) ⟶ F(Y))`.\n-/\ndef hom_mk' {F : C ⥤ D} {X : D} {Y : C}\n(U : structured_arrow X F) (f : U.right ⟶ Y) :\nU ⟶ mk (U.hom ≫ F.map f) := { right := f }\n\n/--\nTo construct an isomorphism of structured arrows,\nwe need an isomorphism of the objects underlying the target,\nand to check that the triangle commutes.\n-/\n@[simps]\ndef iso_mk {f f' : structured_arrow S T} (g : f.right ≅ f'.right)\n  (w : f.hom ≫ T.map g.hom = f'.hom) : f ≅ f' :=\ncomma.iso_mk (eq_to_iso (by ext)) g (by simpa [eq_to_hom_map] using w.symm)\n\n/--\nA morphism between source objects `S ⟶ S'`\ncontravariantly induces a functor between structured arrows,\n`structured_arrow S' T ⥤ structured_arrow S T`.\n\nIdeally this would be described as a 2-functor from `D`\n(promoted to a 2-category with equations as 2-morphisms)\nto `Cat`.\n-/\n@[simps]\ndef map (f : S ⟶ S') : structured_arrow S' T ⥤ structured_arrow S T :=\ncomma.map_left _ ((functor.const _).map f)\n\n@[simp] \n\n@[simp] lemma map_id {f : structured_arrow S T} : (map (𝟙 S)).obj f = f :=\nby { rw eq_mk f, simp, }\n\n@[simp] lemma map_comp {f : S ⟶ S'} {f' : S' ⟶ S''} {h : structured_arrow S'' T} :\n  (map (f ≫ f')).obj h = (map f).obj ((map f').obj h) :=\nby { rw eq_mk h, simp, }\n\ninstance proj_reflects_iso : reflects_isomorphisms (proj S T) :=\n{ reflects := λ Y Z f t, by exactI\n  ⟨⟨structured_arrow.hom_mk (inv ((proj S T).map f)) (by simp), by tidy⟩⟩ }\n\nopen category_theory.limits\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- The identity structured arrow is initial. -/\ndef mk_id_initial [full T] [faithful T] : is_initial (mk (𝟙 (T.obj Y))) :=\n{ desc := λ c, hom_mk (T.preimage c.X.hom) (by { dsimp, simp, }),\n  uniq' := λ c m _, begin\n    ext,\n    apply T.map_injective,\n    simpa only [hom_mk_right, T.image_preimage, ←w m] using (category.id_comp _).symm,\n  end }\n\nvariables {A : Type u₃} [category.{v₃} A] {B : Type u₄} [category.{v₄} B]\n\n/-- The functor `(S, F ⋙ G) ⥤ (S, G)`. -/\n@[simps]\ndef pre (S : D) (F : B ⥤ C) (G : C ⥤ D) : structured_arrow S (F ⋙ G) ⥤ structured_arrow S G :=\ncomma.pre_right _ F G\n\n/-- The functor `(S, F) ⥤ (G(S), F ⋙ G)`. -/\n@[simps] def post (S : C) (F : B ⥤ C) (G : C ⥤ D) :\n  structured_arrow S F ⥤ structured_arrow (G.obj S) (F ⋙ G) :=\n{ obj := λ X, { right := X.right, hom := G.map X.hom },\n  map := λ X Y f, { right := f.right, w' :=\n    by { simp [functor.comp_map, ←G.map_comp, ← f.w] } } }\n\nend structured_arrow\n\n\n/--\nThe category of `S`-costructured arrows with target `T : D` (here `S : C ⥤ D`),\nhas as its objects `D`-morphisms of the form `S Y ⟶ T`, for some `Y : C`,\nand morphisms `C`-morphisms `Y ⟶ Y'` making the obvious triangle commute.\n-/\n@[derive category, nolint has_inhabited_instance]\ndef costructured_arrow (S : C ⥤ D) (T : D) := comma S (functor.from_punit T)\n\nnamespace costructured_arrow\n\n/-- The obvious projection functor from costructured arrows. -/\n@[simps]\ndef proj (S : C ⥤ D) (T : D) : costructured_arrow S T ⥤ C := comma.fst _ _\n\nvariables {T T' T'' : D} {Y Y' : C} {S : C ⥤ D}\n\n/-- Construct a costructured arrow from a morphism. -/\ndef mk (f : S.obj Y ⟶ T) : costructured_arrow S T := ⟨Y, ⟨⟨⟩⟩, f⟩\n\n@[simp] lemma mk_left (f : S.obj Y ⟶ T) : (mk f).left = Y := rfl\n@[simp] lemma mk_right (f : S.obj Y ⟶ T) : (mk f).right = ⟨⟨⟩⟩ := rfl\n@[simp] lemma mk_hom_eq_self (f : S.obj Y ⟶ T) : (mk f).hom = f := rfl\n\n@[simp, reassoc] lemma w {A B : costructured_arrow S T} (f : A ⟶ B) :\n  S.map f.left ≫ B.hom = A.hom :=\nby tidy\n\nlemma eq_mk (f : costructured_arrow S T) : f = mk f.hom :=\nby { cases f, congr, ext, }\n\n/--\nTo construct a morphism of costructured arrows,\nwe need a morphism of the objects underlying the source,\nand to check that the triangle commutes.\n-/\n@[simps]\ndef hom_mk {f f' : costructured_arrow S T} (g : f.left ⟶ f'.left) (w : S.map g ≫ f'.hom = f.hom) :\n  f ⟶ f' :=\n{ left := g,\n  right := eq_to_hom (by ext),\n  w' := by simpa [eq_to_hom_map] using w, }\n\n/--\nTo construct an isomorphism of costructured arrows,\nwe need an isomorphism of the objects underlying the source,\nand to check that the triangle commutes.\n-/\n@[simps]\ndef iso_mk {f f' : costructured_arrow S T} (g : f.left ≅ f'.left)\n  (w : S.map g.hom ≫ f'.hom = f.hom) : f ≅ f' :=\ncomma.iso_mk g (eq_to_iso (by ext)) (by simpa [eq_to_hom_map] using w)\n\n/--\nA morphism between target objects `T ⟶ T'`\ncovariantly induces a functor between costructured arrows,\n`costructured_arrow S T ⥤ costructured_arrow S T'`.\n\nIdeally this would be described as a 2-functor from `D`\n(promoted to a 2-category with equations as 2-morphisms)\nto `Cat`.\n-/\n@[simps]\ndef map (f : T ⟶ T') : costructured_arrow S T ⥤ costructured_arrow S T' :=\ncomma.map_right _ ((functor.const _).map f)\n\n@[simp] lemma map_mk {f : S.obj Y ⟶ T} (g : T ⟶ T') :\n  (map g).obj (mk f) = mk (f ≫ g) := rfl\n\n@[simp] lemma map_id {f : costructured_arrow S T} : (map (𝟙 T)).obj f = f :=\nby { rw eq_mk f, simp, }\n\n@[simp] lemma map_comp {f : T ⟶ T'} {f' : T' ⟶ T''} {h : costructured_arrow S T} :\n  (map (f ≫ f')).obj h = (map f').obj ((map f).obj h) :=\nby { rw eq_mk h, simp, }\n\ninstance proj_reflects_iso : reflects_isomorphisms (proj S T) :=\n{ reflects := λ Y Z f t, by exactI\n  ⟨⟨costructured_arrow.hom_mk (inv ((proj S T).map f)) (by simp), by tidy⟩⟩ }\n\nopen category_theory.limits\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- The identity costructured arrow is terminal. -/\ndef mk_id_terminal [full S] [faithful S] : is_terminal (mk (𝟙 (S.obj Y))) :=\n{ lift := λ c, hom_mk (S.preimage c.X.hom) (by { dsimp, simp, }),\n  uniq' := begin\n    rintros c m -,\n    ext,\n    apply S.map_injective,\n    simpa only [hom_mk_left, S.image_preimage, ←w m] using (category.comp_id _).symm,\n  end }\n\n\nvariables {A : Type u₃} [category.{v₃} A] {B : Type u₄} [category.{v₄} B]\n\n/-- The functor `(F ⋙ G, S) ⥤ (G, S)`. -/\n@[simps]\ndef pre (F : B ⥤ C) (G : C ⥤ D) (S : D) : costructured_arrow (F ⋙ G) S ⥤ costructured_arrow G S :=\ncomma.pre_left F G _\n\n/-- The functor `(F, S) ⥤ (F ⋙ G, G(S))`. -/\n@[simps] def post (F : B ⥤ C) (G : C ⥤ D) (S : C) :\n  costructured_arrow F S ⥤ costructured_arrow (F ⋙ G) (G.obj S) :=\n{ obj := λ X, { left := X.left, hom := G.map X.hom },\n  map := λ X Y f, { left := f.left, w' :=\n    by { simp [functor.comp_map, ←G.map_comp, ← f.w] } } }\n\nend costructured_arrow\n\nopen opposite\n\nnamespace structured_arrow\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the\ncategory of structured arrows `d ⟶ F.obj c` to the category of costructured arrows\n`F.op.obj c ⟶ (op d)`.\n-/\n@[simps]\ndef to_costructured_arrow (F : C ⥤ D) (d : D) :\n  (structured_arrow d F)ᵒᵖ ⥤ costructured_arrow F.op (op d) :=\n{ obj := λ X, @costructured_arrow.mk _ _ _ _ _ (op X.unop.right) F.op X.unop.hom.op,\n  map := λ X Y f, costructured_arrow.hom_mk (f.unop.right.op)\n  begin\n    dsimp,\n    rw [← op_comp, ← f.unop.w, functor.const.obj_map],\n    erw category.id_comp,\n  end }\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the\ncategory of structured arrows `op d ⟶ F.op.obj c` to the category of costructured arrows\n`F.obj c ⟶ d`.\n-/\n@[simps]\ndef to_costructured_arrow' (F : C ⥤ D) (d : D) :\n  (structured_arrow (op d) F.op)ᵒᵖ ⥤ costructured_arrow F d :=\n{ obj := λ X, @costructured_arrow.mk _ _ _ _ _ (unop X.unop.right) F X.unop.hom.unop,\n  map := λ X Y f, costructured_arrow.hom_mk f.unop.right.unop\n  begin\n    dsimp,\n    rw [← quiver.hom.unop_op (F.map (quiver.hom.unop f.unop.right)), ← unop_comp, ← F.op_map,\n      ← f.unop.w, functor.const.obj_map],\n    erw category.id_comp,\n  end }\n\nend structured_arrow\n\nnamespace costructured_arrow\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the\ncategory of costructured arrows `F.obj c ⟶ d` to the category of structured arrows\n`op d ⟶ F.op.obj c`.\n-/\n@[simps]\ndef to_structured_arrow (F : C ⥤ D) (d : D) :\n  (costructured_arrow F d)ᵒᵖ ⥤ structured_arrow (op d) F.op :=\n{ obj := λ X, @structured_arrow.mk _ _ _ _ _ (op X.unop.left) F.op X.unop.hom.op,\n  map := λ X Y f, structured_arrow.hom_mk f.unop.left.op\n  begin\n    dsimp,\n    rw [← op_comp, f.unop.w, functor.const.obj_map],\n    erw category.comp_id,\n  end }\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, we obtain a contravariant functor from the\ncategory of costructured arrows `F.op.obj c ⟶ op d` to the category of structured arrows\n`d ⟶ F.obj c`.\n-/\n@[simps]\ndef to_structured_arrow' (F : C ⥤ D) (d : D) :\n  (costructured_arrow F.op (op d))ᵒᵖ ⥤ structured_arrow d F :=\n{ obj := λ X, @structured_arrow.mk _ _ _ _ _ (unop X.unop.left) F X.unop.hom.unop,\n  map := λ X Y f, structured_arrow.hom_mk (f.unop.left.unop)\n  begin\n    dsimp,\n    rw [← quiver.hom.unop_op (F.map f.unop.left.unop), ← unop_comp, ← F.op_map,\n      f.unop.w, functor.const.obj_map],\n    erw category.comp_id,\n  end }\n\nend costructured_arrow\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, the category of structured arrows `d ⟶ F.obj c`\nis contravariantly equivalent to the category of costructured arrows `F.op.obj c ⟶ op d`.\n-/\ndef structured_arrow_op_equivalence (F : C ⥤ D) (d : D) :\n  (structured_arrow d F)ᵒᵖ ≌ costructured_arrow F.op (op d) :=\nequivalence.mk (structured_arrow.to_costructured_arrow F d)\n  (costructured_arrow.to_structured_arrow' F d).right_op\n  (nat_iso.of_components (λ X, (@structured_arrow.iso_mk _ _ _ _ _ _\n    (structured_arrow.mk (unop X).hom) (unop X) (iso.refl _) (by tidy)).op)\n    (λ X Y f, quiver.hom.unop_inj $ begin ext, dsimp, simp end))\n  (nat_iso.of_components (λ X, @costructured_arrow.iso_mk _ _ _ _ _ _\n    (costructured_arrow.mk X.hom) X (iso.refl _) (by tidy))\n    (λ X Y f, begin ext, dsimp, simp end))\n\n/--\nFor a functor `F : C ⥤ D` and an object `d : D`, the category of costructured arrows\n`F.obj c ⟶ d` is contravariantly equivalent to the category of structured arrows\n`op d ⟶ F.op.obj c`.\n-/\ndef costructured_arrow_op_equivalence (F : C ⥤ D) (d : D) :\n  (costructured_arrow F d)ᵒᵖ ≌ structured_arrow (op d) F.op :=\nequivalence.mk (costructured_arrow.to_structured_arrow F d)\n  (structured_arrow.to_costructured_arrow' F d).right_op\n  (nat_iso.of_components (λ X, (@costructured_arrow.iso_mk _ _ _ _ _ _\n    (costructured_arrow.mk (unop X).hom) (unop X) (iso.refl _) (by tidy)).op)\n    (λ X Y f, quiver.hom.unop_inj $ begin ext, dsimp, simp end))\n  (nat_iso.of_components (λ X, @structured_arrow.iso_mk _ _ _ _ _ _\n    (structured_arrow.mk X.hom) X (iso.refl _) (by tidy))\n    (λ X Y f, begin ext, dsimp, simp end))\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/structured_arrow.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3247320443237448}}
{"text": "macro \"ext_tactic\" t:tactic \"=>\" newT:tactic : command => `(macro_rules | `(tactic| $t) => `(tactic| $newT))\n\nsyntax \"trivial'\" : tactic\n\next_tactic trivial' => apply Eq.refl\n\ntheorem tst1 (x : Nat) : x = x :=\nby trivial'\n\n-- theorem tst2 (x y : Nat) (h : x = y) : x = y :=\n-- by trivial' -- fail as expected\n\next_tactic trivial' => assumption\n\ntheorem tst1b (x : Nat) : x = x :=\nby trivial' -- still works\n\ntheorem tst2 (x y : Nat) (h : x = y) : x = y :=\nby trivial' -- works too\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/extmacro.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.566018520554724, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3247126034604597}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.field_theory.finite.basic\nimport Mathlib.field_theory.mv_polynomial\nimport Mathlib.data.mv_polynomial.expand\nimport Mathlib.linear_algebra.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u \n\nnamespace Mathlib\n\n/-!\n## Polynomials over finite fields\n-/\n\nnamespace mv_polynomial\n\n\n/-- A polynomial over the integers is divisible by `n : ℕ`\nif and only if it is zero over `zmod n`. -/\ntheorem C_dvd_iff_zmod {σ : Type u_1} (n : ℕ) (φ : mv_polynomial σ ℤ) :\n    coe_fn C ↑n ∣ φ ↔ coe_fn (map (int.cast_ring_hom (zmod n))) φ = 0 :=\n  C_dvd_iff_map_hom_eq_zero (int.cast_ring_hom (zmod n)) (↑n)\n    (char_p.int_cast_eq_zero_iff (zmod n) n) φ\n\ntheorem frobenius_zmod {σ : Type u_1} {p : ℕ} [fact (nat.prime p)] (f : mv_polynomial σ (zmod p)) :\n    coe_fn (frobenius (mv_polynomial σ (zmod p)) p) f = coe_fn (expand p) f :=\n  sorry\n\ntheorem expand_zmod {σ : Type u_1} {p : ℕ} [fact (nat.prime p)] (f : mv_polynomial σ (zmod p)) :\n    coe_fn (expand p) f = f ^ p :=\n  Eq.symm (frobenius_zmod f)\n\nend mv_polynomial\n\n\nnamespace mv_polynomial\n\n\ndef indicator {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ] (a : σ → K) :\n    mv_polynomial σ K :=\n  finset.prod finset.univ fun (n : σ) => 1 - (X n - coe_fn C (a n)) ^ (fintype.card K - 1)\n\ntheorem eval_indicator_apply_eq_one {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ]\n    (a : σ → K) : coe_fn (eval a) (indicator a) = 1 :=\n  sorry\n\ntheorem eval_indicator_apply_eq_zero {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ]\n    (a : σ → K) (b : σ → K) (h : a ≠ b) : coe_fn (eval a) (indicator b) = 0 :=\n  sorry\n\ntheorem degrees_indicator {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ]\n    (c : σ → K) :\n    degrees (indicator c) ≤\n        finset.sum finset.univ fun (s : σ) => (fintype.card K - 1) •ℕ singleton s :=\n  sorry\n\ntheorem indicator_mem_restrict_degree {K : Type u_1} {σ : Type u_2} [field K] [fintype K]\n    [fintype σ] (c : σ → K) : indicator c ∈ restrict_degree σ K (fintype.card K - 1) :=\n  sorry\n\ndef evalₗ (K : Type u_1) (σ : Type u_2) [field K] [fintype K] [fintype σ] :\n    linear_map K (mv_polynomial σ K) ((σ → K) → K) :=\n  linear_map.mk (fun (p : mv_polynomial σ K) (e : σ → K) => coe_fn (eval e) p) sorry sorry\n\ntheorem evalₗ_apply {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ]\n    (p : mv_polynomial σ K) (e : σ → K) : coe_fn (evalₗ K σ) p e = coe_fn (eval e) p :=\n  rfl\n\ntheorem map_restrict_dom_evalₗ {K : Type u_1} {σ : Type u_2} [field K] [fintype K] [fintype σ] :\n    submodule.map (evalₗ K σ) (restrict_degree σ K (fintype.card K - 1)) = ⊤ :=\n  sorry\n\nend mv_polynomial\n\n\nnamespace mv_polynomial\n\n\ndef R (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K] :=\n  ↥(restrict_degree σ K (fintype.card K - 1))\n\nprotected instance decidable_restrict_degree (σ : Type u) [fintype σ] (m : ℕ) :\n    decidable_pred fun (n : σ →₀ ℕ) => n ∈ set_of fun (n : σ →₀ ℕ) => ∀ (i : σ), coe_fn n i ≤ m :=\n  eq.mpr sorry fun (a : σ →₀ ℕ) => fintype.decidable_forall_fintype\n\ntheorem dim_R (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K] :\n    vector_space.dim K (R σ K) = ↑(fintype.card (σ → K)) :=\n  sorry\n\ndef evalᵢ (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K] :\n    linear_map K (R σ K) ((σ → K) → K) :=\n  linear_map.comp (evalₗ K σ) (submodule.subtype (restrict_degree σ K (fintype.card K - 1)))\n\ntheorem range_evalᵢ (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K] :\n    linear_map.range (evalᵢ σ K) = ⊤ :=\n  sorry\n\ntheorem ker_evalₗ (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K] :\n    linear_map.ker (evalᵢ σ K) = ⊥ :=\n  sorry\n\ntheorem eq_zero_of_eval_eq_zero (σ : Type u) (K : Type u) [fintype σ] [field K] [fintype K]\n    (p : mv_polynomial σ K) (h : ∀ (v : σ → K), coe_fn (eval v) p = 0)\n    (hp : p ∈ restrict_degree σ K (fintype.card K - 1)) : p = 0 :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/field_theory/finite/polynomial_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3246904821692033}}
{"text": "/-\nCopyright (c) 2014 Robert Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n\n! This file was ported from Lean 3 source module algebra.order.field.pi\n! leanprover-community/mathlib commit 327c3c0d9232d80e250dc8f65e7835b82b266ea5\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Order.Field.Basic\nimport Mathbin.Data.Fintype.Lattice\n\n/-!\n# Lemmas about (finite domain) functions into fields.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe split this from `algebra.order.field.basic` to avoid importing the finiteness hierarchy there.\n-/\n\n\nvariable {α ι : Type _} [LinearOrderedSemifield α]\n\n/- warning: pi.exists_forall_pos_add_lt -> Pi.exists_forall_pos_add_lt is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {ι : Type.{u2}} [_inst_1 : LinearOrderedSemifield.{u1} α] [_inst_2 : ExistsAddOfLE.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))) (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))] [_inst_3 : Finite.{succ u2} ι] {x : ι -> α} {y : ι -> α}, (forall (i : ι), LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (x i) (y i)) -> (Exists.{succ u1} α (fun (ε : α) => And (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) ε) (forall (i : ι), LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (x i) ε) (y i))))\nbut is expected to have type\n  forall {α : Type.{u2}} {ι : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u2} α] [_inst_2 : ExistsAddOfLE.{u2} α (Distrib.toAdd.{u2} α (NonUnitalNonAssocSemiring.toDistrib.{u2} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} α (Semiring.toNonAssocSemiring.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1)))))))) (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedSemiring.toPartialOrder.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1))))))] [_inst_3 : Finite.{succ u1} ι] {x : ι -> α} {y : ι -> α}, (forall (i : ι), LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedSemiring.toPartialOrder.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1)))))) (x i) (y i)) -> (Exists.{succ u2} α (fun (ε : α) => And (LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedSemiring.toPartialOrder.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α _inst_1)))))) ε) (forall (i : ι), LT.lt.{u2} α (Preorder.toLT.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedSemiring.toPartialOrder.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1)))))) (HAdd.hAdd.{u2, u2, u2} α α α (instHAdd.{u2} α (Distrib.toAdd.{u2} α (NonUnitalNonAssocSemiring.toDistrib.{u2} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} α (Semiring.toNonAssocSemiring.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α _inst_1))))))))) (x i) ε) (y i))))\nCase conversion may be inaccurate. Consider using '#align pi.exists_forall_pos_add_lt Pi.exists_forall_pos_add_ltₓ'. -/\ntheorem Pi.exists_forall_pos_add_lt [ExistsAddOfLE α] [Finite ι] {x y : ι → α}\n    (h : ∀ i, x i < y i) : ∃ ε, 0 < ε ∧ ∀ i, x i + ε < y i :=\n  by\n  cases nonempty_fintype ι\n  cases isEmpty_or_nonempty ι\n  · exact ⟨1, zero_lt_one, isEmptyElim⟩\n  choose ε hε hxε using fun i => exists_pos_add_of_lt' (h i)\n  obtain rfl : x + ε = y := funext hxε\n  have hε : 0 < finset.univ.inf' Finset.univ_nonempty ε := (Finset.lt_inf'_iff _).2 fun i _ => hε _\n  exact\n    ⟨_, half_pos hε, fun i =>\n      add_lt_add_left ((half_lt_self hε).trans_le <| Finset.inf'_le _ <| Finset.mem_univ _) _⟩\n#align pi.exists_forall_pos_add_lt Pi.exists_forall_pos_add_lt\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Order/Field/Pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3246904821692033}}
{"text": "/- Author: E.W.Ayers\n   Show that the open covers for a topological space form a basis for a grothendieck topology.\n -/\nimport topology.category.Top.opens\nimport category_theory.limits.lattice\nimport category_theory.limits.limits\nimport category_theory.limits.shapes.pullbacks\nimport .grothendieck\n\nuniverses u\nopen category_theory\nopen topological_space\nopen category_theory.limits\n\nnamespace topological_space.opens\n\n/- [TODO] this is probably in mathlib somewhere. -/\n/-- `covers X U ℱ` means that ℱ is an open cover of U. -/\ndef covers (X : Top) : arrow_set (opens X) :=\nλ U ℱ, ∀ (x : X) (xU : x ∈ U), ∃ (V : over U), V ∈ ℱ ∧ x ∈ V.left\n\nvariables {X : Top}\n\ninstance opens_has_limits : @has_limits (opens X) (opens.opens_category) := \nlimits.has_limits_of_complete_lattice\n\ninstance opens_has_pullbacks : @has_pullbacks (opens X) (opens.opens_category) :=\n⟨@has_limits.has_limits_of_shape _ _ opens.opens_has_limits _ _⟩\n\ninstance opens_has_cospan_limits {U V W : opens X} {f : U ⟶ W} {g : V ⟶ W} : has_limit (cospan f g) :=\n@has_limits_of_shape.has_limit _ _ _ _ (@has_limits.has_limits_of_shape _ _ opens.opens_has_limits _ _) _\n\nvariables {U V W : opens X}\n\n/- [todo] this can be moved to category_theory/limits/lattice -/\nlemma eq_of_iso (e : U ≅ W) : U = W :=\nbegin\n    rcases e with ⟨⟨⟨_⟩⟩,⟨⟨_⟩⟩,_,_⟩, \n    apply partial_order.le_antisymm,\n    assumption,\n    assumption\nend\n\nlemma over_eq_of_left_eq : Π {f g : over U}, f.left = g.left → f = g \n| ⟨_,⟨⟩,⟨⟨_⟩⟩⟩ ⟨_,⟨⟩,⟨⟨_⟩⟩⟩ rfl := rfl\n\nopen lattice\n/- [todo] this can be moved to category_theory/limits/lattice -/\nlemma pullback_is_inter {f : U ⟶ W} {g : V ⟶ W} : pullback f g = U ⊓ V :=\nbegin\n    apply eq_of_iso,\n    rcases (pullback.fst : pullback f g ⟶ U) with ⟨⟨π1⟩⟩,\n    rcases (pullback.snd : pullback f g ⟶ V) with ⟨⟨π2⟩⟩,\n    refine ⟨⟨⟨le_inf π1 π2⟩⟩,pullback.lift ⟨⟨inf_le_left⟩⟩ ⟨⟨inf_le_right⟩⟩ rfl,rfl,rfl⟩,\nend\n\ninstance : grothendieck.basis (covers X) :=\n{ has_isos :=\n    begin \n        -- all isos in opens U are equality.\n        intros U V e x xU,\n        refine ⟨over.mk e.hom, _,_⟩, \n        simp, \n        have : U = V, apply eq_of_iso e, \n        simpa [this],\n    end, \n  has_pullbacks :=\n    begin\n        -- idea: ℱ is covering for U \n        -- ⇒ {V ∩ W | W ∈ ℱ} is a covering for V\n        intros U V ℱ h₁ g, \n        intros x xV,\n        rcases g with ⟨⟨g⟩⟩,\n        rcases h₁ x (g xV) with ⟨f,fF,xf⟩, \n        refine ⟨over.mk ⟨⟨inf_le_right⟩⟩,⟨f,fF,_⟩,⟨xf,xV⟩⟩,\n        apply over_eq_of_left_eq, \n            simp [over.pullback],\n            rw pullback_is_inter,\n            rw inf_comm, refl,\n    end,\n  trans := \n    begin\n        -- idea: ℱ covers U and 𝒢 U covers V for each V ∈ ℱ \n        -- ⇒ ⋃ 𝒢 covers U\n        intros U, \n        rintros _ FcU _ GcF x xU,\n        rcases FcU x xU with ⟨V,VF,xV⟩,\n        rcases GcF VF x xV with ⟨W,WG,xW⟩, \n        refine ⟨over.mk (W.hom ≫ V.hom),⟨_,VF,⟨W,WG,rfl⟩⟩,xW⟩,\n    end\n}\n\ndef covering_sieve (X : Top):= sieve_set.generate (covers X)\n\ninstance : grothendieck (covering_sieve X) :=\ngrothendieck.of_basis\n\nend topological_space.opens", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/geo/src/com.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.32469047589425587}}
{"text": "/-\nCopyright (c) 2022 Ian Wood. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ian Wood\n\n! This file was ported from Lean 3 source module tactic.expand_exists\n! leanprover-community/mathlib commit f9153b8a79eb28d07341706ddb18c02593eeb72a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Meta.Expr\n\n/-!\n# `expand_exists`\n\n`expand_exists` is an attribute which takes a proof that something exists with some property, and\noutputs a value using `classical.some`, and a proof that it has that property using\n`classical.some_spec`. For example:\n\n```lean\n@[expand_exists it it_spec]\nlemma it_exists (n : ℕ) : ∃ m : ℕ, n < m := sorry\n```\n\nproduces\n\n```\ndef it (n : ℕ) : ℕ := classical.some (it_exists n)\n\nlemma it_spec (n : ℕ) : n < it n := classical.some_spec (it_exists n)\n```\n-/\n\n\nnamespace Tactic\n\nopen Expr\n\nnamespace ExpandExists\n\n/-- Data known when parsing pi expressions.\n\n`decl`'s arguments are: is_theorem, name, type, value.\n-/\nunsafe structure parse_ctx where\n  original_decl : declaration\n  decl : Bool → Name → expr → pexpr → tactic Unit\n  names : List Name\n  pis_depth : ℕ := 0\n#align tactic.expand_exists.parse_ctx tactic.expand_exists.parse_ctx\n\n/-- Data known when parsing exists expressions (after parsing pi expressions).\n\n* `with_args` applies pi arguments to a term (eg `id` -> `id #2 #1 #0`).\n* `spec_chain` takes the form of `classical.some_spec^n (it_exists ...)`,\nwith `n` the depth of `∃` parsed.\n* `exists_decls` is a list of declarations containing the value(s) of witnesses.\n-/\nunsafe structure parse_ctx_exists extends parse_ctx where\n  with_args : expr → expr\n  spec_chain : pexpr\n  exists_decls : List Name := []\n#align tactic.expand_exists.parse_ctx_exists tactic.expand_exists.parse_ctx_exists\n\n/-- Data known when parsing the proposition (after parsing exists and pi expressions).\n\n`project_proof` projects a proof of the full proposition (eg `A ∧ B ∧ C`) to a specific proof (eg\n`B`).\n-/\nunsafe structure parse_ctx_props extends parse_ctx_exists where\n  project_proof : pexpr → pexpr := id\n#align tactic.expand_exists.parse_ctx_props tactic.expand_exists.parse_ctx_props\n\n/-- Replaces free variables with their exists declaration. For example, if:\n\n```lean\ndef n_value : ℕ := ... -- generated by `expand_exists`\n```\n\nthen this function converts `#0` in `#0 = #0` from `∃ n : ℕ, n = n` to `n_value = n_value`.\n-/\nunsafe def instantiate_exists_decls (ctx : parse_ctx_exists) (p : expr) : expr :=\n  p.instantiate_vars <|\n    ctx.exists_decls.reverse.map fun name =>\n      ctx.with_args (const Name ctx.original_decl.univ_levels)\n#align tactic.expand_exists.instantiate_exists_decls tactic.expand_exists.instantiate_exists_decls\n\n/-- Parses a proposition and creates the associated specification proof. Does not break down the\nproposition further.\n-/\nunsafe def parse_one_prop (ctx : parse_ctx_props) (p : expr) : tactic Unit := do\n  let p : expr := instantiate_exists_decls { ctx with } p\n  let val : pexpr := ctx.project_proof ctx.spec_chain\n  let n ←\n    match ctx.names with\n      | [n] => return n\n      | [] => fail \"missing name for proposition\"\n      | _ => fail \"too many names for propositions (are you missing an and?)\"\n  ctx True n p val\n#align tactic.expand_exists.parse_one_prop tactic.expand_exists.parse_one_prop\n\n/--\nParses a proposition and decides if it should be broken down (eg `P ∧ Q` -> `P` and `Q`) depending\non how many `names` are left. Then creates the associated specification proof(s).\n-/\nunsafe def parse_props : parse_ctx_props → expr → tactic Unit\n  | ctx, app (app (const \"and\" []) p) q => do\n    match ctx with\n      | [n] => parse_one_prop ctx (app (app (const `and []) p) q)\n      | n :: tail =>\n        parse_one_prop\n            { ctx with\n              names := [n]\n              project_proof := (fun p => (const `and.left []) p) ∘ ctx }\n            p >>\n          parse_props\n            { ctx with\n              names := tail\n              project_proof := (fun p => (const `and.right []) p) ∘ ctx }\n            q\n      | [] => fail \"missing name for proposition\"\n  | ctx, p => parse_one_prop ctx p\n#align tactic.expand_exists.parse_props tactic.expand_exists.parse_props\n\n/--\nParses an `∃ a : α, p a`, and creates an associated definition with a value of `α`. When `p α` is\nnot an exists statement, it will call `parse_props`.\n-/\nunsafe def parse_exists : parse_ctx_exists → expr → tactic Unit\n  | ctx, app (app (const \"Exists\" [lvl]) type) (lam var_name bi var_type body) => do\n    -- TODO: Is this needed, and/or does this create issues?\n        if type = var_type then tactic.skip\n      else tactic.fail \"exists types should be equal\"\n    let ⟨n, names⟩ ←\n      match ctx.names with\n        | n :: tail => return (n, tail)\n        | [] => fail \"missing name for exists\"\n    let-- Type may be dependant on earlier arguments.\n    type := instantiate_exists_decls ctx type\n    let value : pexpr := (const `classical.some [lvl]) ctx.spec_chain\n    ctx False n type value\n    let exists_decls := ctx.exists_decls.concat n\n    let some_spec : pexpr := (const `classical.some_spec [lvl]) ctx.spec_chain\n    let ctx : parse_ctx_exists :=\n      { ctx with\n        names\n        spec_chain := some_spec\n        exists_decls }\n    parse_exists ctx body\n  | ctx, e => parse_props { ctx with } e\n#align tactic.expand_exists.parse_exists tactic.expand_exists.parse_exists\n\n/-- Parses a `∀ (a : α), p a`. If `p` is not a pi expression, it will call `parse_exists`\n-/\nunsafe def parse_pis : parse_ctx → expr → tactic Unit\n  | ctx,\n    pi n bi ty body =>-- When making a declaration, wrap in an equivalent pi expression.\n    let decl is_theorem name type val :=\n      ctx.decl is_theorem Name (pi n bi ty type) (lam n bi (to_pexpr ty) val)\n    parse_pis\n      { ctx with\n        decl\n        pis_depth := ctx.pis_depth + 1 }\n      body\n  | ctx, app (app (const \"Exists\" [lvl]) type) p =>\n    let with_args := fun e : expr =>\n      (List.range ctx.pis_depth).foldr (fun n (e : expr) => e (var n)) e\n    parse_exists\n      { ctx with\n        with_args\n        spec_chain :=\n          to_pexpr (with_args <| const ctx.original_decl.to_name ctx.original_decl.univ_levels) }\n      (app (app (const \"Exists\" [lvl]) type) p)\n  | ctx, e => fail (\"unexpected expression \" ++ toString e)\n#align tactic.expand_exists.parse_pis tactic.expand_exists.parse_pis\n\nend ExpandExists\n\n/-- From a proof that (a) value(s) exist(s) with certain properties, constructs (an) instance(s)\nsatisfying those properties. For instance:\n\n```lean\n@[expand_exists nat_greater nat_greater_spec]\nlemma nat_greater_exists (n : ℕ) : ∃ m : ℕ, n < m := ...\n\n#check nat_greater      -- nat_greater : ℕ → ℕ\n#check nat_greater_spec -- nat_greater_spec : ∀ (n : ℕ), n < nat_greater n\n```\n\nIt supports multiple witnesses:\n\n```lean\n@[expand_exists nat_greater_m nat_greater_l nat_greater_spec]\nlemma nat_greater_exists (n : ℕ) : ∃ (m l : ℕ), n < m ∧ m < l := ...\n\n#check nat_greater_m      -- nat_greater : ℕ → ℕ\n#check nat_greater_l      -- nat_greater : ℕ → ℕ\n#check nat_greater_spec-- nat_greater_spec : ∀ (n : ℕ),\n  n < nat_greater_m n ∧ nat_greater_m n < nat_greater_l n\n```\n\nIt also supports logical conjunctions:\n```lean\n@[expand_exists nat_greater nat_greater_lt nat_greater_nonzero]\nlemma nat_greater_exists (n : ℕ) : ∃ m : ℕ, n < m ∧ m ≠ 0 := ...\n\n#check nat_greater         -- nat_greater : ℕ → ℕ\n#check nat_greater_lt      -- nat_greater_lt : ∀ (n : ℕ), n < nat_greater n\n#check nat_greater_nonzero -- nat_greater_nonzero : ∀ (n : ℕ), nat_greater n ≠ 0\n```\nNote that without the last argument `nat_greater_nonzero`, `nat_greater_lt` would be:\n```lean\n#check nat_greater_lt -- nat_greater_lt : ∀ (n : ℕ), n < nat_greater n ∧ nat_greater n ≠ 0\n```\n-/\n@[user_attribute]\nunsafe def expand_exists_attr : user_attribute Unit (List Name)\n    where\n  Name := \"expand_exists\"\n  descr :=\n    \"From a proof that (a) value(s) exist(s) with certain properties, \" ++\n      \"constructs (an) instance(s) satisfying those properties.\"\n  parser := lean.parser.many lean.parser.ident\n  after_set :=\n    some fun decl prio persistent => do\n      let d ← get_decl decl\n      let names ← expand_exists_attr.get_param decl\n      expand_exists.parse_pis\n          { original_decl := d\n            decl := fun is_t n ty val =>\n              tactic.to_expr val >>= fun val =>\n                tactic.add_decl\n                  (if is_t then declaration.thm n d ty (pure val)\n                  else declaration.defn n d ty val default tt)\n            names } d\n#align tactic.expand_exists_attr tactic.expand_exists_attr\n\nadd_tactic_doc\n  { Name := \"expand_exists\"\n    category := DocCategory.attr\n    declNames := [`tactic.expand_exists_attr]\n    tags := [\"lemma derivation\", \"environment\"] }\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/ExpandExists.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.32465432367792585}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.preserves.shapes.equalizers\nimport category_theory.limits.preserves.shapes.zero\n\n/-!\n# Preserving (co)kernels\n\nConstructions to relate the notions of preserving (co)kernels and reflecting (co)kernels\nto concrete (co)forks.\n\nIn particular, we show that `kernel_comparison f g G` is an isomorphism iff `G` preserves\nthe limit of the parallel pair `f,0`, as well as the dual result.\n-/\n\nnoncomputable theory\n\nuniverses v u₁ u₂\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u₁} [category.{v} C] [has_zero_morphisms C]\nvariables {D : Type u₂} [category.{v} D] [has_zero_morphisms D]\nvariables (G : C ⥤ D) [functor.preserves_zero_morphisms G]\n\nnamespace category_theory.limits\n\nsection kernels\nvariables {X Y Z : C} {f : X ⟶ Y} {h : Z ⟶ X} (w : h ≫ f = 0)\n\n/--\nThe map of a kernel fork is a limit iff\nthe kernel fork consisting of the mapped morphisms is a limit.\nThis essentially lets us commute `kernel_fork.of_ι` with `functor.map_cone`.\n\nThis is a variant of `is_limit_map_cone_fork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_map_cone_fork_equiv' :\n  is_limit (G.map_cone (kernel_fork.of_ι h w)) ≃\n  is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])\n    : fork (G.map f) 0) :=\nbegin\n  refine (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine fork.ext (iso.refl _) _,\n  simp [fork.ι]\nend\n\n/--\nThe property of preserving kernels expressed in terms of kernel forks.\n\nThis is a variant of `is_limit_fork_map_of_is_limit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_fork_map_of_is_limit' [preserves_limit (parallel_pair f 0) G]\n  (l : is_limit (kernel_fork.of_ι h w)) :\n  is_limit (kernel_fork.of_ι (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :\n    fork (G.map f) 0) :=\nis_limit_map_cone_fork_equiv' G w (preserves_limit.preserves l)\n\nvariables (f) [has_kernel f]\n\n/--\nIf `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of\na kernel fork is a limit.\n-/\ndef is_limit_of_has_kernel_of_preserves_limit [preserves_limit (parallel_pair f 0) G] :\n  is_limit (fork.of_ι (G.map (kernel.ι f))\n    (by simp only [←G.map_comp, equalizer.condition, comp_zero, functor.map_zero])\n      : fork (G.map f) 0) :=\nis_limit_fork_map_of_is_limit' G (kernel.condition f) (kernel_is_kernel f)\n\ninstance [preserves_limit (parallel_pair f 0) G] : has_kernel (G.map f) :=\n{ exists_limit := ⟨⟨_, is_limit_of_has_kernel_of_preserves_limit G f⟩⟩, }\n\nvariables [has_kernel (G.map f)]\n\n/--\nIf the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\nkernel of `f`.\n-/\ndef preserves_kernel.of_iso_comparison [i : is_iso (kernel_comparison f G)] :\n  preserves_limit (parallel_pair f 0) G :=\nbegin\n  apply preserves_limit_of_preserves_limit_cone (kernel_is_kernel f),\n  apply (is_limit_map_cone_fork_equiv' G (kernel.condition f)).symm _,\n  apply is_limit.of_point_iso (limit.is_limit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_limit (parallel_pair f 0) G]\n/--\nIf `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_kernel.iso :\n  G.obj (kernel f) ≅ kernel (G.map f) :=\nis_limit.cone_point_unique_up_to_iso\n  (is_limit_of_has_kernel_of_preserves_limit G f)\n  (limit.is_limit _)\n\n@[simp]\nlemma preserves_kernel.iso_hom :\n  (preserves_kernel.iso G f).hom = kernel_comparison f G :=\nrfl\n\ninstance : is_iso (kernel_comparison f G) :=\nbegin\n  rw ← preserves_kernel.iso_hom,\n  apply_instance\nend\n\nend kernels\n\nsection cokernels\n\nvariables {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} (w : f ≫ h = 0)\n\n/--\nThe map of a cokernel cofork is a colimit iff\nthe cokernel cofork consisting of the mapped morphisms is a colimit.\nThis essentially lets us commute `cokernel_cofork.of_π` with `functor.map_cocone`.\n\nThis is a variant of `is_colimit_map_cocone_cofork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_map_cocone_cofork_equiv' :\n  is_colimit (G.map_cocone (cokernel_cofork.of_π h w)) ≃\n  is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero])\n    : cofork (G.map f) 0) :=\nbegin\n  refine (is_colimit.precompose_hom_equiv _ _).symm.trans (is_colimit.equiv_iso_colimit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine cofork.ext (iso.refl _) _,\n  simp only [cofork.π, iso.refl_hom, id_comp, cocones.precompose_obj_ι,\n    nat_trans.comp_app, parallel_pair.ext_hom_app, functor.map_cocone_ι_app,\n    cofork.of_π_ι_app],\n  apply category.comp_id\nend\n\n/--\nThe property of preserving cokernels expressed in terms of cokernel coforks.\n\nThis is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_cofork_map_of_is_colimit' [preserves_colimit (parallel_pair f 0) G]\n  (l : is_colimit (cokernel_cofork.of_π h w)) :\n  is_colimit (cokernel_cofork.of_π (G.map h) (by simp only [←G.map_comp, w, functor.map_zero]) :\n    cofork (G.map f) 0) :=\nis_colimit_map_cocone_cofork_equiv' G w (preserves_colimit.preserves l)\n\nvariables (f) [has_cokernel f]\n\n/--\nIf `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of\na cokernel cofork is a colimit.\n-/\ndef is_colimit_of_has_cokernel_of_preserves_colimit [preserves_colimit (parallel_pair f 0) G] :\n  is_colimit (cofork.of_π (G.map (cokernel.π f))\n    (by simp only [←G.map_comp, coequalizer.condition, zero_comp, functor.map_zero])\n      : cofork (G.map f) 0) :=\nis_colimit_cofork_map_of_is_colimit' G (cokernel.condition f) (cokernel_is_cokernel f)\n\ninstance [preserves_colimit (parallel_pair f 0) G] : has_cokernel (G.map f) :=\n{ exists_colimit := ⟨⟨_, is_colimit_of_has_cokernel_of_preserves_colimit G f⟩⟩, }\n\nvariables [has_cokernel (G.map f)]\n\n/--\nIf the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\ncokernel of `f`.\n-/\ndef preserves_cokernel.of_iso_comparison [i : is_iso (cokernel_comparison f G)] :\n  preserves_colimit (parallel_pair f 0) G :=\nbegin\n  apply preserves_colimit_of_preserves_colimit_cocone (cokernel_is_cokernel f),\n  apply (is_colimit_map_cocone_cofork_equiv' G (cokernel.condition f)).symm _,\n  apply is_colimit.of_point_iso (colimit.is_colimit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_colimit (parallel_pair f 0) G]\n/--\nIf `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_cokernel.iso :\n  G.obj (cokernel f) ≅ cokernel (G.map f) :=\nis_colimit.cocone_point_unique_up_to_iso\n  (is_colimit_of_has_cokernel_of_preserves_colimit G f)\n  (colimit.is_colimit _)\n\n@[simp]\nlemma preserves_cokernel.iso_inv :\n  (preserves_cokernel.iso G f).inv = cokernel_comparison f G :=\nrfl\n\ninstance : is_iso (cokernel_comparison f G) :=\nbegin\n  rw ← preserves_cokernel.iso_inv,\n  apply_instance\nend\n\nend cokernels\n\nend category_theory.limits\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/preserves/shapes/kernels.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3246543236779258}}
{"text": "import tactic.lint\n\ndef f : ℕ → ℕ := default _\ndef c : ℕ := default _\ndef d : ℕ := default _\n\n@[simp] lemma c_eq_d : c = d := rfl\n\n-- The following lemma never applies when using simp, because c is first rewritten to d\n@[simp] lemma f_c : f c = 0 := rfl\n\nexample : f c = 0 :=\nbegin\n  simp,\n  guard_target f d = 0, -- does not apply f_c\n  refl\nend\n\nopen tactic\nrun_cmd do\ndecl ← get_decl ``f_c,\nres ← linter.simp_nf.test decl,\n-- linter complains\nguard $ res.is_some\n\n\n-- also works with `coe_to_fun`\n\nstructure morphism :=\n(f : ℕ → ℕ)\n\ninstance : has_coe_to_fun morphism :=\n⟨_, morphism.f⟩\n\ndef h : morphism := ⟨default _⟩\n\n-- Also never applies\n@[simp] lemma h_c : h c = 0 := rfl\n\nexample : h c = 0 :=\nbegin\n  simp,\n  guard_target h d = 0, -- does not apply h_c\n  refl\nend\n\nopen tactic\nrun_cmd do\ndecl ← get_decl ``h_c,\nres ← linter.simp_nf.test decl,\n-- linter complains\nguard $ res.is_some\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/test/lint_simp_nf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.32465432367792574}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 u_6 \n\nnamespace Mathlib\n\n/-!\n# Extra facts about `prod`\n\nThis file defines `prod.swap : α × β → β × α` and proves various simple lemmas about `prod`.\n-/\n\n@[simp] theorem prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ)\n    (g : β → δ) (p : α × β) : prod.map f g p = (f (prod.fst p), g (prod.snd p)) :=\n  rfl\n\nnamespace prod\n\n\n@[simp] theorem forall {α : Type u_1} {β : Type u_2} {p : α × β → Prop} :\n    (∀ (x : α × β), p x) ↔ ∀ (a : α) (b : β), p (a, b) :=\n  sorry\n\n@[simp] theorem exists {α : Type u_1} {β : Type u_2} {p : α × β → Prop} :\n    (∃ (x : α × β), p x) ↔ ∃ (a : α), ∃ (b : β), p (a, b) :=\n  sorry\n\ntheorem forall' {α : Type u_1} {β : Type u_2} {p : α → β → Prop} :\n    (∀ (x : α × β), p (fst x) (snd x)) ↔ ∀ (a : α) (b : β), p a b :=\n  forall\n\ntheorem exists' {α : Type u_1} {β : Type u_2} {p : α → β → Prop} :\n    (∃ (x : α × β), p (fst x) (snd x)) ↔ ∃ (a : α), ∃ (b : β), p a b :=\n  exists\n\n@[simp] theorem map_mk {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ)\n    (g : β → δ) (a : α) (b : β) : map f g (a, b) = (f a, g b) :=\n  rfl\n\ntheorem map_fst {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ)\n    (p : α × β) : fst (map f g p) = f (fst p) :=\n  rfl\n\ntheorem map_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ) (g : β → δ)\n    (p : α × β) : snd (map f g p) = g (snd p) :=\n  rfl\n\ntheorem map_fst' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ)\n    (g : β → δ) : fst ∘ map f g = f ∘ fst :=\n  funext (map_fst f g)\n\ntheorem map_snd' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (f : α → γ)\n    (g : β → δ) : snd ∘ map f g = g ∘ snd :=\n  funext (map_snd f g)\n\n/--\nComposing a `prod.map` with another `prod.map` is equal to\na single `prod.map` of composed functions.\n-/\ntheorem map_comp_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5}\n    {ζ : Type u_6} (f : α → β) (f' : γ → δ) (g : β → ε) (g' : δ → ζ) :\n    map g g' ∘ map f f' = map (g ∘ f) (g' ∘ f') :=\n  rfl\n\n/--\nComposing a `prod.map` with another `prod.map` is equal to\na single `prod.map` of composed functions, fully applied.\n-/\ntheorem map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {ε : Type u_5}\n    {ζ : Type u_6} (f : α → β) (f' : γ → δ) (g : β → ε) (g' : δ → ζ) (x : α × γ) :\n    map g g' (map f f' x) = map (g ∘ f) (g' ∘ f') x :=\n  rfl\n\n@[simp] theorem mk.inj_iff {α : Type u_1} {β : Type u_2} {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} :\n    (a₁, b₁) = (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ = b₂ :=\n  sorry\n\ntheorem mk.inj_left {α : Type u_1} {β : Type u_2} (a : α) : function.injective (Prod.mk a) := sorry\n\ntheorem mk.inj_right {α : Type u_1} {β : Type u_2} (b : β) :\n    function.injective fun (a : α) => (a, b) :=\n  sorry\n\ntheorem ext_iff {α : Type u_1} {β : Type u_2} {p : α × β} {q : α × β} :\n    p = q ↔ fst p = fst q ∧ snd p = snd q :=\n  sorry\n\ntheorem ext {α : Type u_1} {β : Type u_2} {p : α × β} {q : α × β} (h₁ : fst p = fst q)\n    (h₂ : snd p = snd q) : p = q :=\n  iff.mpr ext_iff { left := h₁, right := h₂ }\n\ntheorem map_def {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → γ}\n    {g : β → δ} : map f g = fun (p : α × β) => (f (fst p), g (snd p)) :=\n  funext fun (p : α × β) => ext (map_fst f g p) (map_snd f g p)\n\ntheorem id_prod {α : Type u_1} : (fun (p : α × α) => (fst p, snd p)) = id := sorry\n\ntheorem fst_surjective {α : Type u_1} {β : Type u_2} [h : Nonempty β] : function.surjective fst :=\n  fun (x : α) => nonempty.elim h fun (y : β) => Exists.intro (x, y) rfl\n\ntheorem snd_surjective {α : Type u_1} {β : Type u_2} [h : Nonempty α] : function.surjective snd :=\n  fun (y : β) => nonempty.elim h fun (x : α) => Exists.intro (x, y) rfl\n\ntheorem fst_injective {α : Type u_1} {β : Type u_2} [subsingleton β] : function.injective fst :=\n  fun (x y : α × β) (h : fst x = fst y) => ext h (subsingleton.elim (snd x) (snd y))\n\ntheorem snd_injective {α : Type u_1} {β : Type u_2} [subsingleton α] : function.injective snd :=\n  fun (x y : α × β) (h : snd x = snd y) => ext (subsingleton.elim (fst x) (fst y)) h\n\n/-- Swap the factors of a product. `swap (a, b) = (b, a)` -/\ndef swap {α : Type u_1} {β : Type u_2} : α × β → β × α := fun (p : α × β) => (snd p, fst p)\n\n@[simp] theorem swap_swap {α : Type u_1} {β : Type u_2} (x : α × β) : swap (swap x) = x :=\n  cases_on x\n    fun (x_fst : α) (x_snd : β) =>\n      idRhs (swap (swap (x_fst, x_snd)) = swap (swap (x_fst, x_snd))) rfl\n\n@[simp] theorem fst_swap {α : Type u_1} {β : Type u_2} {p : α × β} : fst (swap p) = snd p := rfl\n\n@[simp] theorem snd_swap {α : Type u_1} {β : Type u_2} {p : α × β} : snd (swap p) = fst p := rfl\n\n@[simp] theorem swap_prod_mk {α : Type u_1} {β : Type u_2} {a : α} {b : β} : swap (a, b) = (b, a) :=\n  rfl\n\n@[simp] theorem swap_swap_eq {α : Type u_1} {β : Type u_2} : swap ∘ swap = id := funext swap_swap\n\n@[simp] theorem swap_left_inverse {α : Type u_1} {β : Type u_2} : function.left_inverse swap swap :=\n  swap_swap\n\n@[simp] theorem swap_right_inverse {α : Type u_1} {β : Type u_2} :\n    function.right_inverse swap swap :=\n  swap_swap\n\ntheorem swap_injective {α : Type u_1} {β : Type u_2} : function.injective swap :=\n  function.left_inverse.injective swap_left_inverse\n\ntheorem swap_surjective {α : Type u_1} {β : Type u_2} : function.surjective swap :=\n  function.left_inverse.surjective swap_left_inverse\n\ntheorem swap_bijective {α : Type u_1} {β : Type u_2} : function.bijective swap :=\n  { left := swap_injective, right := swap_surjective }\n\n@[simp] theorem swap_inj {α : Type u_1} {β : Type u_2} {p : α × β} {q : α × β} :\n    swap p = swap q ↔ p = q :=\n  function.injective.eq_iff swap_injective\n\ntheorem eq_iff_fst_eq_snd_eq {α : Type u_1} {β : Type u_2} {p : α × β} {q : α × β} :\n    p = q ↔ fst p = fst q ∧ snd p = snd q :=\n  sorry\n\ntheorem fst_eq_iff {α : Type u_1} {β : Type u_2} {p : α × β} {x : α} : fst p = x ↔ p = (x, snd p) :=\n  sorry\n\ntheorem snd_eq_iff {α : Type u_1} {β : Type u_2} {p : α × β} {x : β} : snd p = x ↔ p = (fst p, x) :=\n  sorry\n\ntheorem lex_def {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) {p : α × β}\n    {q : α × β} : lex r s p q ↔ r (fst p) (fst q) ∨ fst p = fst q ∧ s (snd p) (snd q) :=\n  sorry\n\nprotected instance lex.decidable {α : Type u_1} {β : Type u_2} [DecidableEq α] (r : α → α → Prop)\n    (s : β → β → Prop) [DecidableRel r] [DecidableRel s] : DecidableRel (lex r s) :=\n  fun (p q : α × β) => decidable_of_decidable_of_iff or.decidable sorry\n\nend prod\n\n\ntheorem function.injective.prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}\n    {f : α → γ} {g : β → δ} (hf : function.injective f) (hg : function.injective g) :\n    function.injective (prod.map f g) :=\n  fun (x y : α × β) (h : prod.map f g x = prod.map f g y) =>\n    prod.ext (hf (and.left (iff.mp prod.ext_iff h))) (hg (and.right (iff.mp prod.ext_iff h)))\n\ntheorem function.surjective.prod_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}\n    {f : α → γ} {g : β → δ} (hf : function.surjective f) (hg : function.surjective g) :\n    function.surjective (prod.map f g) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/prod_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290806, "lm_q2_score": 0.6442251201477015, "lm_q1q2_score": 0.3246290132522651}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.functor\nimport Mathlib.category_theory.full_subcategory\nimport Mathlib.PostPort\n\nuniverses v₁ v₂ u₁ u₂ l u₃ v₃ \n\nnamespace Mathlib\n\n/-!\n# Monoidal natural transformations\n\nNatural transformations between (lax) monoidal functors must satisfy\nan additional compatibility relation with the tensorators:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`.\n\n(Lax) monoidal functors between a fixed pair of monoidal categories\nthemselves form a category.\n-/\n\nnamespace category_theory\n\n\n/--\nA monoidal natural transformation is a natural transformation between (lax) monoidal functors\nadditionally satisfying:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`\n-/\nstructure monoidal_nat_trans {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] (F : lax_monoidal_functor C D) (G : lax_monoidal_functor C D) \nextends nat_trans (lax_monoidal_functor.to_functor F) (lax_monoidal_functor.to_functor G)\nwhere\n  unit' : autoParam (lax_monoidal_functor.ε F ≫ nat_trans.app _to_nat_trans 𝟙_ = lax_monoidal_functor.ε G)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  tensor' : autoParam\n  (∀ (X Y : C),\n    lax_monoidal_functor.μ F X Y ≫ nat_trans.app _to_nat_trans (X ⊗ Y) =\n      (nat_trans.app _to_nat_trans X ⊗ nat_trans.app _to_nat_trans Y) ≫ lax_monoidal_functor.μ G X Y)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem monoidal_nat_trans.tensor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (c : monoidal_nat_trans F G) (X : C) (Y : C) : lax_monoidal_functor.μ F X Y ≫ nat_trans.app (monoidal_nat_trans.to_nat_trans c) (X ⊗ Y) =\n  (nat_trans.app (monoidal_nat_trans.to_nat_trans c) X ⊗ nat_trans.app (monoidal_nat_trans.to_nat_trans c) Y) ≫\n    lax_monoidal_functor.μ G X Y := sorry\n\n@[simp] theorem monoidal_nat_trans.tensor_assoc {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (c : monoidal_nat_trans F G) (X : C) (Y : C) {X' : D} (f' : functor.obj (lax_monoidal_functor.to_functor G) (X ⊗ Y) ⟶ X') : lax_monoidal_functor.μ F X Y ≫ nat_trans.app (monoidal_nat_trans.to_nat_trans c) (X ⊗ Y) ≫ f' =\n  (nat_trans.app (monoidal_nat_trans.to_nat_trans c) X ⊗ nat_trans.app (monoidal_nat_trans.to_nat_trans c) Y) ≫\n    lax_monoidal_functor.μ G X Y ≫ f' := sorry\n\n@[simp] theorem monoidal_nat_trans.unit {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (c : monoidal_nat_trans F G) : lax_monoidal_functor.ε F ≫ nat_trans.app (monoidal_nat_trans.to_nat_trans c) 𝟙_ = lax_monoidal_functor.ε G := sorry\n\n@[simp] theorem monoidal_nat_trans.unit_assoc {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (c : monoidal_nat_trans F G) {X' : D} (f' : functor.obj (lax_monoidal_functor.to_functor G) 𝟙_ ⟶ X') : lax_monoidal_functor.ε F ≫ nat_trans.app (monoidal_nat_trans.to_nat_trans c) 𝟙_ ≫ f' = lax_monoidal_functor.ε G ≫ f' := sorry\n\nnamespace monoidal_nat_trans\n\n\n/--\nThe identity monoidal natural transformation.\n-/\ndef id {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] (F : lax_monoidal_functor C D) : monoidal_nat_trans F F :=\n  mk (nat_trans.mk (nat_trans.app 𝟙))\n\nprotected instance inhabited {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] (F : lax_monoidal_functor C D) : Inhabited (monoidal_nat_trans F F) :=\n  { default := id F }\n\n/--\nVertical composition of monoidal natural transformations.\n-/\ndef vcomp {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} {H : lax_monoidal_functor C D} (α : monoidal_nat_trans F G) (β : monoidal_nat_trans G H) : monoidal_nat_trans F H :=\n  mk (nat_trans.mk (nat_trans.app (nat_trans.vcomp (to_nat_trans α) (to_nat_trans β))))\n\nprotected instance category_lax_monoidal_functor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] : category (lax_monoidal_functor C D) :=\n  category.mk\n\n@[simp] theorem comp_to_nat_trans' {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} {H : lax_monoidal_functor C D} {α : F ⟶ G} {β : G ⟶ H} : to_nat_trans (α ≫ β) = to_nat_trans α ≫ to_nat_trans β :=\n  rfl\n\nprotected instance category_monoidal_functor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] : category (monoidal_functor C D) :=\n  induced_category.category monoidal_functor.to_lax_monoidal_functor\n\n@[simp] theorem comp_to_nat_trans'' {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : monoidal_functor C D} {G : monoidal_functor C D} {H : monoidal_functor C D} {α : F ⟶ G} {β : G ⟶ H} : to_nat_trans (α ≫ β) = to_nat_trans α ≫ to_nat_trans β :=\n  rfl\n\n/--\nHorizontal composition of monoidal natural transformations.\n-/\n@[simp] theorem hcomp_to_nat_trans {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {E : Type u₃} [category E] [monoidal_category E] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} {H : lax_monoidal_functor D E} {K : lax_monoidal_functor D E} (α : monoidal_nat_trans F G) (β : monoidal_nat_trans H K) : to_nat_trans (hcomp α β) = to_nat_trans α ◫ to_nat_trans β :=\n  Eq.refl (to_nat_trans (hcomp α β))\n\nend monoidal_nat_trans\n\n\nnamespace monoidal_nat_iso\n\n\nprotected instance is_iso_of_is_iso_app {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (α : F ⟶ G) [(X : C) → is_iso (nat_trans.app (monoidal_nat_trans.to_nat_trans α) X)] : is_iso α :=\n  is_iso.mk\n    (monoidal_nat_trans.mk (nat_trans.mk fun (X : C) => inv (nat_trans.app (monoidal_nat_trans.to_nat_trans α) X)))\n\n/--\nConstruct a monoidal natural isomorphism from object level isomorphisms,\nand the monoidal naturality in the forward direction.\n-/\ndef of_components {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (app : (X : C) → functor.obj (lax_monoidal_functor.to_functor F) X ≅ functor.obj (lax_monoidal_functor.to_functor G) X) (naturality : ∀ {X Y : C} (f : X ⟶ Y),\n  functor.map (lax_monoidal_functor.to_functor F) f ≫ iso.hom (app Y) =\n    iso.hom (app X) ≫ functor.map (lax_monoidal_functor.to_functor G) f) (unit : lax_monoidal_functor.ε F ≫ iso.hom (app 𝟙_) = lax_monoidal_functor.ε G) (tensor : ∀ (X Y : C),\n  lax_monoidal_functor.μ F X Y ≫ iso.hom (app (X ⊗ Y)) =\n    (iso.hom (app X) ⊗ iso.hom (app Y)) ≫ lax_monoidal_functor.μ G X Y) : F ≅ G :=\n  as_iso (monoidal_nat_trans.mk (nat_trans.mk fun (X : C) => iso.hom (app X)))\n\n@[simp] theorem of_components.hom_app {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (app : (X : C) → functor.obj (lax_monoidal_functor.to_functor F) X ≅ functor.obj (lax_monoidal_functor.to_functor G) X) (naturality : ∀ {X Y : C} (f : X ⟶ Y),\n  functor.map (lax_monoidal_functor.to_functor F) f ≫ iso.hom (app Y) =\n    iso.hom (app X) ≫ functor.map (lax_monoidal_functor.to_functor G) f) (unit : lax_monoidal_functor.ε F ≫ iso.hom (app 𝟙_) = lax_monoidal_functor.ε G) (tensor : ∀ (X Y : C),\n  lax_monoidal_functor.μ F X Y ≫ iso.hom (app (X ⊗ Y)) =\n    (iso.hom (app X) ⊗ iso.hom (app Y)) ≫ lax_monoidal_functor.μ G X Y) (X : C) : nat_trans.app (monoidal_nat_trans.to_nat_trans (iso.hom (of_components app naturality unit tensor))) X = iso.hom (app X) :=\n  rfl\n\n@[simp] theorem of_components.inv_app {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] {F : lax_monoidal_functor C D} {G : lax_monoidal_functor C D} (app : (X : C) → functor.obj (lax_monoidal_functor.to_functor F) X ≅ functor.obj (lax_monoidal_functor.to_functor G) X) (naturality : ∀ {X Y : C} (f : X ⟶ Y),\n  functor.map (lax_monoidal_functor.to_functor F) f ≫ iso.hom (app Y) =\n    iso.hom (app X) ≫ functor.map (lax_monoidal_functor.to_functor G) f) (unit : lax_monoidal_functor.ε F ≫ iso.hom (app 𝟙_) = lax_monoidal_functor.ε G) (tensor : ∀ (X Y : C),\n  lax_monoidal_functor.μ F X Y ≫ iso.hom (app (X ⊗ Y)) =\n    (iso.hom (app X) ⊗ iso.hom (app Y)) ≫ lax_monoidal_functor.μ G X Y) (X : C) : nat_trans.app (monoidal_nat_trans.to_nat_trans (iso.inv (of_components app naturality unit tensor))) X = iso.inv (app X) :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/natural_transformation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3246290063682356}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.group.ext\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.preadditive\nimport category_theory.limits.shapes.kernels\n\n/-!\n# Biproducts and binary biproducts\n\nWe introduce the notion of (finite) biproducts and binary biproducts.\n\nThese are slightly unusual relative to the other shapes in the library,\nas they are simultaneously limits and colimits.\n(Zero objects are similar; they are \"biterminal\".)\n\nWe treat first the case of a general category with zero morphisms,\nand subsequently the case of a preadditive category.\n\nIn a category with zero morphisms, we model the (binary) biproduct of `P Q : C`\nusing a `binary_bicone`, which has a cone point `X`,\nand morphisms `fst : X ⟶ P`, `snd : X ⟶ Q`, `inl : P ⟶ X` and `inr : X ⟶ Q`,\nsuch that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q`.\nSuch a `binary_bicone` is a biproduct if the cone is a limit cone, and the cocone is a colimit\ncocone.\n\nIn a preadditive category,\n* any `binary_biproduct` satisfies `total : fst ≫ inl + snd ≫ inr = 𝟙 X`\n* any `binary_product` is a `binary_biproduct`\n* any `binary_coproduct` is a `binary_biproduct`\n\nFor biproducts indexed by a `fintype J`, a `bicone` again consists of a cone point `X`\nand morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`,\nsuch that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise.\n\nIn a preadditive category,\n* any `biproduct` satisfies `total : ∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f)`\n* any `product` is a `biproduct`\n* any `coproduct` is a `biproduct`\n\n## Notation\nAs `⊕` is already taken for the sum of types, we introduce the notation `X ⊞ Y` for\na binary biproduct. We introduce `⨁ f` for the indexed biproduct.\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory\nopen category_theory.functor\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {J : Type v} [decidable_eq J]\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/--\nA `c : bicone F` is:\n* an object `c.X` and\n* morphisms `π j : X ⟶ F j` and `ι j : F j ⟶ X` for each `j`,\n* such that `ι j ≫ π j'` is the identity when `j = j'` and zero otherwise.\n-/\n@[nolint has_inhabited_instance]\nstructure bicone (F : J → C) :=\n(X : C)\n(π : Π j, X ⟶ F j)\n(ι : Π j, F j ⟶ X)\n(ι_π : ∀ j j', ι j ≫ π j' = if h : j = j' then eq_to_hom (congr_arg F h) else 0)\n\n@[simp, reassoc] lemma bicone_ι_π_self {F : J → C} (B : bicone F) (j : J) :\n  B.ι j ≫ B.π j = 𝟙 (F j) :=\nby simpa using B.ι_π j j\n\n@[simp, reassoc] lemma bicone_ι_π_ne {F : J → C} (B : bicone F) {j j' : J} (h : j ≠ j') :\n  B.ι j ≫ B.π j' = 0 :=\nby simpa [h] using B.ι_π j j'\n\nvariables {F : J → C}\n\nnamespace bicone\n/-- Extract the cone from a bicone. -/\n@[simps]\ndef to_cone (B : bicone F) : cone (discrete.functor F) :=\n{ X := B.X,\n  π := { app := λ j, B.π j }, }\n\n/-- Extract the cocone from a bicone. -/\n@[simps]\ndef to_cocone (B : bicone F) : cocone (discrete.functor F) :=\n{ X := B.X,\n  ι := { app := λ j, B.ι j }, }\n\n/-- We can turn any limit cone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_limit_cone {f : J → C} {t : cone (discrete.functor f)} (ht : is_limit t) :\n  bicone f :=\n{ X := t.X,\n  π := t.π.app,\n  ι := λ j, ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)),\n  ι_π := λ j j', by simp }\n\nlemma ι_of_is_limit {f : J → C} {t : bicone f} (ht : is_limit t.to_cone) (j : J) :\n  t.ι j = ht.lift (fan.mk _ (λ j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (λ j', by { rw ht.fac, simp [t.ι_π] })\n\n/-- We can turn any colimit cocone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_colimit_cocone {f : J → C} {t : cocone (discrete.functor f)} (ht : is_colimit t) :\n  bicone f :=\n{ X := t.X,\n  π := λ j, ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)),\n  ι := t.ι.app,\n  ι_π := λ j j', by simp }\n\nlemma π_of_is_colimit {f : J → C} {t : bicone f} (ht : is_colimit t.to_cocone) (j : J) :\n  t.π j = ht.desc (cofan.mk _ (λ j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (λ j', by { rw ht.fac, simp [t.ι_π] })\n\n/-- Structure witnessing that a bicone is both a limit cone and a colimit cocone. -/\n@[nolint has_inhabited_instance]\nstructure is_bilimit {F : J → C} (B : bicone F) :=\n(is_limit : is_limit B.to_cone)\n(is_colimit : is_colimit B.to_cocone)\n\nend bicone\n\n/--\nA bicone over `F : J → C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_inhabited_instance]\nstructure limit_bicone (F : J → C) :=\n(bicone : bicone F)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_biproduct F` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `F`.\n-/\nclass has_biproduct (F : J → C) : Prop :=\nmk' :: (exists_biproduct : nonempty (limit_bicone F))\n\nlemma has_biproduct.mk {F : J → C} (d : limit_bicone F) : has_biproduct F :=\n⟨nonempty.intro d⟩\n\n/-- Use the axiom of choice to extract explicit `biproduct_data F` from `has_biproduct F`. -/\ndef get_biproduct_data (F : J → C) [has_biproduct F] : limit_bicone F :=\nclassical.choice has_biproduct.exists_biproduct\n\n/-- A bicone for `F` which is both a limit cone and a colimit cocone. -/\ndef biproduct.bicone (F : J → C) [has_biproduct F] : bicone F :=\n(get_biproduct_data F).bicone\n\n/-- `biproduct.bicone F` is a bilimit bicone. -/\ndef biproduct.is_bilimit (F : J → C) [has_biproduct F] : (biproduct.bicone F).is_bilimit :=\n(get_biproduct_data F).is_bilimit\n\n/-- `biproduct.bicone F` is a limit cone. -/\ndef biproduct.is_limit (F : J → C) [has_biproduct F] : is_limit (biproduct.bicone F).to_cone :=\n(get_biproduct_data F).is_bilimit.is_limit\n\n/-- `biproduct.bicone F` is a colimit cocone. -/\ndef biproduct.is_colimit (F : J → C) [has_biproduct F] :\n  is_colimit (biproduct.bicone F).to_cocone :=\n(get_biproduct_data F).is_bilimit.is_colimit\n\n@[priority 100]\ninstance has_product_of_has_biproduct [has_biproduct F] : has_limit (discrete.functor F) :=\nhas_limit.mk { cone := (biproduct.bicone F).to_cone,\n  is_limit := biproduct.is_limit F, }\n\n@[priority 100]\ninstance has_coproduct_of_has_biproduct [has_biproduct F] : has_colimit (discrete.functor F) :=\nhas_colimit.mk { cocone := (biproduct.bicone F).to_cocone,\n  is_colimit := biproduct.is_colimit F, }\n\nvariables (J C)\n\n/--\n`C` has biproducts of shape `J` if we have\na limit and a colimit, with the same cone points,\nof every function `F : J → C`.\n-/\nclass has_biproducts_of_shape : Prop :=\n(has_biproduct : Π F : J → C, has_biproduct F)\n\nattribute [instance, priority 100] has_biproducts_of_shape.has_biproduct\n\n/-- `has_finite_biproducts C` represents a choice of biproduct for every family of objects in `C`\nindexed by a finite type with decidable equality. -/\nclass has_finite_biproducts : Prop :=\n(has_biproducts_of_shape : Π (J : Type v) [decidable_eq J] [fintype J],\n  has_biproducts_of_shape J C)\n\nattribute [instance, priority 100] has_finite_biproducts.has_biproducts_of_shape\n\n@[priority 100]\ninstance has_finite_products_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_products C :=\n{ out := λ J _ _, ⟨λ F, by exactI has_limit_of_iso discrete.nat_iso_functor.symm⟩ }\n\n@[priority 100]\ninstance has_finite_coproducts_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_coproducts C :=\n{ out := λ J _ _, ⟨λ F, by exactI has_colimit_of_iso discrete.nat_iso_functor⟩ }\n\nvariables {J C}\n\n/--\nThe isomorphism between the specified limit and the specified colimit for\na functor with a bilimit.\n-/\ndef biproduct_iso (F : J → C) [has_biproduct F] :\n  limits.pi_obj F ≅ limits.sigma_obj F :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (biproduct.is_limit F)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit F) (colimit.is_colimit _)\n\nend limits\n\nnamespace limits\nvariables {J : Type v} [decidable_eq J]\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/-- `biproduct f` computes the biproduct of a family of elements `f`. (It is defined as an\n   abbreviation for `limit (discrete.functor f)`, so for most facts about `biproduct f`, you will\n   just use general facts about limits and colimits.) -/\nabbreviation biproduct (f : J → C) [has_biproduct f] : C :=\n(biproduct.bicone f).X\n\nnotation `⨁ ` f:20 := biproduct f\n\n/-- The projection onto a summand of a biproduct. -/\nabbreviation biproduct.π (f : J → C) [has_biproduct f] (b : J) : ⨁ f ⟶ f b :=\n(biproduct.bicone f).π b\n\n@[simp]\nlemma biproduct.bicone_π (f : J → C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).π b = biproduct.π f b := rfl\n\n/-- The inclusion into a summand of a biproduct. -/\nabbreviation biproduct.ι (f : J → C) [has_biproduct f] (b : J) : f b ⟶ ⨁ f :=\n(biproduct.bicone f).ι b\n\n@[simp]\nlemma biproduct.bicone_ι (f : J → C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).ι b = biproduct.ι f b := rfl\n\n@[reassoc]\nlemma biproduct.ι_π (f : J → C) [has_biproduct f] (j j' : J) :\n  biproduct.ι f j ≫ biproduct.π f j' = if h : j = j' then eq_to_hom (congr_arg f h) else 0 :=\n(biproduct.bicone f).ι_π j j'\n\n@[simp,reassoc]\nlemma biproduct.ι_π_self (f : J → C) [has_biproduct f] (j : J) :\n  biproduct.ι f j ≫ biproduct.π f j = 𝟙 _ :=\nby simp [biproduct.ι_π]\n\n@[simp,reassoc]\nlemma biproduct.ι_π_ne (f : J → C) [has_biproduct f] {j j' : J} (h : j ≠ j') :\n  biproduct.ι f j ≫ biproduct.π f j' = 0 :=\nby simp [biproduct.ι_π, h]\n\n/-- Given a collection of maps into the summands, we obtain a map into the biproduct. -/\nabbreviation biproduct.lift\n  {f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) : P ⟶ ⨁ f :=\n(biproduct.is_limit f).lift (fan.mk P p)\n/-- Given a collection of maps out of the summands, we obtain a map out of the biproduct. -/\nabbreviation biproduct.desc\n  {f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) : ⨁ f ⟶ P :=\n(biproduct.is_colimit f).desc (cofan.mk P p)\n\n@[simp, reassoc]\nlemma biproduct.lift_π {f : J → C} [has_biproduct f] {P : C} (p : Π b, P ⟶ f b) (j : J) :\n  biproduct.lift p ≫ biproduct.π f j = p j :=\n(biproduct.is_limit f).fac _ _\n\n@[simp, reassoc]\nlemma biproduct.ι_desc {f : J → C} [has_biproduct f] {P : C} (p : Π b, f b ⟶ P) (j : J) :\n  biproduct.ι f j ≫ biproduct.desc p = p j :=\n(biproduct.is_colimit f).fac _ _\n\n/-- Given a collection of maps between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain a map between the biproducts. -/\nabbreviation biproduct.map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g :=\nis_limit.map (biproduct.bicone f).to_cone (biproduct.is_limit g) (discrete.nat_trans p)\n\n/-- An alternative to `biproduct.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biproduct.map`. -/\nabbreviation biproduct.map' {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : ⨁ f ⟶ ⨁ g :=\nis_colimit.map (biproduct.is_colimit f) (biproduct.bicone g).to_cocone (discrete.nat_trans p)\n\n@[ext] lemma biproduct.hom_ext {f : J → C} [has_biproduct f]\n  {Z : C} (g h : Z ⟶ ⨁ f)\n  (w : ∀ j, g ≫ biproduct.π f j = h ≫ biproduct.π f j) : g = h :=\n(biproduct.is_limit f).hom_ext w\n\n@[ext] lemma biproduct.hom_ext' {f : J → C} [has_biproduct f]\n  {Z : C} (g h : ⨁ f ⟶ Z)\n  (w : ∀ j, biproduct.ι f j ≫ g = biproduct.ι f j ≫ h) : g = h :=\n(biproduct.is_colimit f).hom_ext w\n\nlemma biproduct.map_eq_map' {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ⟶ g b) : biproduct.map p = biproduct.map' p :=\nbegin\n  ext j j',\n  simp only [discrete.nat_trans_app, limits.is_colimit.ι_map, limits.is_limit.map_π, category.assoc,\n    ←bicone.to_cone_π_app, ←biproduct.bicone_π, ←bicone.to_cocone_ι_app, ←biproduct.bicone_ι],\n  simp only [biproduct.bicone_ι, biproduct.bicone_π, bicone.to_cocone_ι_app, bicone.to_cone_π_app],\n  rw [biproduct.ι_π_assoc, biproduct.ι_π],\n  split_ifs,\n  { subst h, rw [eq_to_hom_refl, category.id_comp], erw category.comp_id, },\n  { simp, },\nend\n\n@[simp, reassoc]\nlemma biproduct.map_π {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) (j : J) :\n  biproduct.map p ≫ biproduct.π g j = biproduct.π f j ≫ p j :=\nlimits.is_limit.map_π _ _ _ _\n\n@[simp, reassoc]\nlemma biproduct.ι_map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) (j : J) :\n  biproduct.ι f j ≫ biproduct.map p = p j ≫ biproduct.ι g j :=\nbegin\n  rw biproduct.map_eq_map',\n  convert limits.is_colimit.ι_map _ _ _ _; refl\nend\n\n@[simp, reassoc]\nlemma biproduct.map_desc {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π j, f j ⟶ g j) {P : C} (k : Π j, g j ⟶ P) :\n  biproduct.map p ≫ biproduct.desc k = biproduct.desc (λ j, p j ≫ k j) :=\nby { ext, simp, }\n\n@[simp, reassoc]\nlemma biproduct.lift_map {f g : J → C} [has_biproduct f] [has_biproduct g]\n  {P : C} (k : Π j, P ⟶ f j) (p : Π j, f j ⟶ g j)  :\n  biproduct.lift k ≫ biproduct.map p = biproduct.lift (λ j, k j ≫ p j) :=\nby { ext, simp, }\n\n/-- Given a collection of isomorphisms between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain an isomorphism between the biproducts. -/\n@[simps]\ndef biproduct.map_iso {f g : J → C} [has_biproduct f] [has_biproduct g]\n  (p : Π b, f b ≅ g b) : ⨁ f ≅ ⨁ g :=\n{ hom := biproduct.map (λ b, (p b).hom),\n  inv := biproduct.map (λ b, (p b).inv), }\n\nsection π_kernel\n\nsection\nvariables (f : J → C) [has_biproduct f]\nvariables (p : J → Prop) [has_biproduct (subtype.restrict p f)]\n\n/-- The canonical morphism from the biproduct over a restricted index type to the biproduct of\nthe full index type. -/\ndef biproduct.from_subtype : ⨁ subtype.restrict p f ⟶ ⨁ f :=\nbiproduct.desc $ λ j, biproduct.ι _ _\n\n/-- The canonical morophism from a biproduct to the biproduct over a restriction of its index\ntype. -/\ndef biproduct.to_subtype : ⨁ f ⟶ ⨁ subtype.restrict p f :=\nbiproduct.lift $ λ j, biproduct.π _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_π (j : J) [decidable (p j)] :\n  biproduct.from_subtype f p ≫ biproduct.π f j =\n    if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0 :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.ι_π],\n    split_ifs with h₁ h₂ h₂,\n    exacts [rfl, false.elim (h₂ (subtype.ext h₁)),\n      false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },\n  { rw [dif_neg h, dif_neg (show (i : J) ≠ j, from λ h₂, h (h₂ ▸ i.2)), comp_zero] }\nend\n\nlemma biproduct.from_subtype_eq_lift [decidable_pred p] : biproduct.from_subtype f p =\n    biproduct.lift (λ j, if h : p j then biproduct.π (subtype.restrict p f) ⟨j, h⟩ else 0) :=\nbiproduct.hom_ext _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_π_subtype (j : subtype p) :\n  biproduct.from_subtype f p ≫ biproduct.π f j = biproduct.π (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.ι_desc_assoc, biproduct.ι_π, biproduct.ι_π],\n  split_ifs with h₁ h₂ h₂,\n  exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_π (j : subtype p) :\n  biproduct.to_subtype f p ≫ biproduct.π (subtype.restrict p f) j = biproduct.π f j :=\nbiproduct.lift_π _ _\n\n@[simp, reassoc]\nlemma biproduct.ι_to_subtype (j : J) [decidable (p j)] :\n  biproduct.ι f j ≫ biproduct.to_subtype f p =\n    if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0 :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.ι_π],\n    split_ifs with h₁ h₂ h₂,\n    exacts [rfl, false.elim (h₂ (subtype.ext h₁)),\n      false.elim (h₁ (congr_arg subtype.val h₂)), rfl] },\n  { rw [dif_neg h, dif_neg (show j ≠ i, from λ h₂, h (h₂.symm ▸ i.2)), zero_comp] }\nend\n\nlemma biproduct.to_subtype_eq_desc [decidable_pred p] : biproduct.to_subtype f p =\n  biproduct.desc (λ j, if h : p j then biproduct.ι (subtype.restrict p f) ⟨j, h⟩ else 0) :=\nbiproduct.hom_ext' _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.ι_to_subtype_subtype (j : subtype p) :\n  biproduct.ι f j ≫ biproduct.to_subtype f p = biproduct.ι (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_π, biproduct.ι_π, biproduct.ι_π],\n  split_ifs with h₁ h₂ h₂,\n  exacts [rfl, false.elim (h₂ (subtype.ext h₁)), false.elim (h₁ (congr_arg subtype.val h₂)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.ι_from_subtype (j : subtype p) :\n  biproduct.ι (subtype.restrict p f) j ≫ biproduct.from_subtype f p = biproduct.ι f j :=\nbiproduct.ι_desc _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_to_subtype :\n  biproduct.from_subtype f p ≫ biproduct.to_subtype f p = 𝟙 (⨁ subtype.restrict p f) :=\nbegin\n  refine biproduct.hom_ext _ _ (λ j, _),\n  rw [category.assoc, biproduct.to_subtype_π, biproduct.from_subtype_π_subtype, category.id_comp]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_from_subtype [decidable_pred p] :\n  biproduct.to_subtype f p ≫ biproduct.from_subtype f p =\n    biproduct.map (λ j, if p j then 𝟙 (f j) else 0) :=\nbegin\n  ext1 i,\n  by_cases h : p i,\n  { simp [h], congr },\n  { simp [h] }\nend\n\nend\n\nvariables (f : J → C) (i : J) [has_biproduct f] [has_biproduct (subtype.restrict (λ j, i ≠ j) f)]\n\n/-- The kernel of `biproduct.π f i` is the inclusion from the biproduct which omits `i`\nfrom the index set `J` into the biproduct over `J`. -/\ndef biproduct.is_limit_from_subtype : is_limit\n  (kernel_fork.of_ι (biproduct.from_subtype f (λ j, i ≠ j))\n    (by simp) : kernel_fork (biproduct.π f i)) :=\nfork.is_limit.mk' _ $ λ s,\n⟨s.ι ≫ biproduct.to_subtype _ _,\n begin\n   ext j,\n   rw [kernel_fork.ι_of_ι, category.assoc, category.assoc,\n     biproduct.to_subtype_from_subtype_assoc, biproduct.map_π],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), comp_zero, comp_zero, kernel_fork.condition] },\n   { rw [if_pos h, category.comp_id] }\n end,\n begin\n   intros m hm,\n   rw [← hm, kernel_fork.ι_of_ι, category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.comp_id _).symm\n end⟩\n\n/-- The cokernel of `biproduct.ι f i` is the projection from the biproduct over the index set `J`\nonto the biproduct omitting `i`. -/\ndef biproduct.is_colimit_to_subtype : is_colimit\n  (cokernel_cofork.of_π (biproduct.to_subtype f (λ j, i ≠ j))\n    (by simp) : cokernel_cofork (biproduct.ι f i)) :=\ncofork.is_colimit.mk' _ $ λ s,\n⟨biproduct.from_subtype _ _ ≫ s.π,\n begin\n   ext j,\n   rw [cokernel_cofork.π_of_π, biproduct.to_subtype_from_subtype_assoc,\n     biproduct.ι_map_assoc],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), zero_comp, cokernel_cofork.condition] },\n   { rw [if_pos h, category.id_comp] }\n end,\n begin\n   intros m hm,\n   rw [← hm, cokernel_cofork.π_of_π, ← category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.id_comp _).symm\n end⟩\n\nend π_kernel\n\nsection\nvariables [fintype J] {K : Type v} [fintype K] [decidable_eq K] {f : J → C} {g : K → C}\n  [has_finite_biproducts C]\n\n/--\nConvert a (dependently typed) matrix to a morphism of biproducts.\n-/\ndef biproduct.matrix (m : Π j k, f j ⟶ g k) : ⨁ f ⟶ ⨁ g :=\nbiproduct.desc (λ j, biproduct.lift (λ k, m j k))\n\n@[simp, reassoc]\nlemma biproduct.matrix_π (m : Π j k, f j ⟶ g k) (k : K) :\n  biproduct.matrix m ≫ biproduct.π g k = biproduct.desc (λ j, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n@[simp, reassoc]\nlemma biproduct.ι_matrix (m : Π j k, f j ⟶ g k) (j : J) :\n  biproduct.ι f j ≫ biproduct.matrix m = biproduct.lift (λ k, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n/--\nExtract the matrix components from a morphism of biproducts.\n-/\ndef biproduct.components (m : ⨁ f ⟶ ⨁ g) (j : J) (k : K) : f j ⟶ g k :=\nbiproduct.ι f j ≫ m ≫ biproduct.π g k\n\n@[simp] lemma biproduct.matrix_components (m : Π j k, f j ⟶ g k) (j : J) (k : K) :\n  biproduct.components (biproduct.matrix m) j k = m j k :=\nby simp [biproduct.components]\n\n@[simp] lemma biproduct.components_matrix (m : ⨁ f ⟶ ⨁ g) :\n  biproduct.matrix (λ j k, biproduct.components m j k) = m :=\nby { ext, simp [biproduct.components], }\n\n/-- Morphisms between direct sums are matrices. -/\n@[simps]\ndef biproduct.matrix_equiv : (⨁ f ⟶ ⨁ g) ≃ (Π j k, f j ⟶ g k) :=\n{ to_fun := biproduct.components,\n  inv_fun := biproduct.matrix,\n  left_inv := biproduct.components_matrix,\n  right_inv := λ m, by { ext, apply biproduct.matrix_components } }\n\nend\n\ninstance biproduct.ι_mono (f : J → C) [has_biproduct f]\n  (b : J) : split_mono (biproduct.ι f b) :=\n{ retraction := biproduct.desc $\n    λ b', if h : b' = b then eq_to_hom (congr_arg f h) else biproduct.ι f b' ≫ biproduct.π f b }\n\ninstance biproduct.π_epi (f : J → C) [has_biproduct f]\n  (b : J) : split_epi (biproduct.π f b) :=\n{ section_ := biproduct.lift $\n    λ b', if h : b = b' then eq_to_hom (congr_arg f h) else biproduct.ι f b ≫ biproduct.π f b' }\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_hom (f : J → C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).hom = biproduct.lift b.π :=\nrfl\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_inv (f : J → C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).inv = biproduct.desc b.ι :=\nbegin\n  refine biproduct.hom_ext' _ _ (λ j, hb.is_limit.hom_ext (λ j', _)),\n  rw [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp, bicone.to_cone_π_app,\n    biproduct.bicone_π, biproduct.ι_desc, biproduct.ι_π, b.to_cone_π_app, b.ι_π]\nend\n\n/-- Biproducts are unique up to isomorphism. This already follows because bilimits are limits,\n    but in the case of biproducts we can give an isomorphism with particularly nice definitional\n    properties, namely that `biproduct.lift b.π` and `biproduct.desc b.ι` are inverses of each\n    other. -/\n@[simps]\ndef biproduct.unique_up_to_iso (f : J → C) [has_biproduct f] {b : bicone f} (hb : b.is_bilimit) :\n  b.X ≅ ⨁ f :=\n{ hom := biproduct.lift b.π,\n  inv := biproduct.desc b.ι,\n  hom_inv_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,\n    ← biproduct.cone_point_unique_up_to_iso_inv f hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [← biproduct.cone_point_unique_up_to_iso_hom f hb,\n    ← biproduct.cone_point_unique_up_to_iso_inv f hb, iso.inv_hom_id] }\n\nsection\nvariables (C)\n\n/-- A category with finite biproducts has a zero object. -/\ndef has_zero_object_of_has_finite_biproducts [has_finite_biproducts C] : has_zero_object C :=\n{ zero := biproduct pempty.elim,\n  unique_to := λ X, ⟨⟨0⟩, by tidy⟩,\n  unique_from := λ X, ⟨⟨0⟩, by tidy⟩, }\n\nend\n\n/--\nA binary bicone for a pair of objects `P Q : C` consists of the cone point `X`,\nmaps from `X` to both `P` and `Q`, and maps from both `P` and `Q` to `X`,\nso that `inl ≫ fst = 𝟙 P`, `inl ≫ snd = 0`, `inr ≫ fst = 0`, and `inr ≫ snd = 𝟙 Q`\n-/\n@[nolint has_inhabited_instance]\nstructure binary_bicone (P Q : C) :=\n(X : C)\n(fst : X ⟶ P)\n(snd : X ⟶ Q)\n(inl : P ⟶ X)\n(inr : Q ⟶ X)\n(inl_fst' : inl ≫ fst = 𝟙 P . obviously)\n(inl_snd' : inl ≫ snd = 0 . obviously)\n(inr_fst' : inr ≫ fst = 0 . obviously)\n(inr_snd' : inr ≫ snd = 𝟙 Q . obviously)\n\nrestate_axiom binary_bicone.inl_fst'\nrestate_axiom binary_bicone.inl_snd'\nrestate_axiom binary_bicone.inr_fst'\nrestate_axiom binary_bicone.inr_snd'\nattribute [simp, reassoc] binary_bicone.inl_fst binary_bicone.inl_snd\n  binary_bicone.inr_fst binary_bicone.inr_snd\n\nnamespace binary_bicone\nvariables {P Q : C}\n\n/-- Extract the cone from a binary bicone. -/\ndef to_cone (c : binary_bicone P Q) : cone (pair P Q) :=\nbinary_fan.mk c.fst c.snd\n\n@[simp]\nlemma to_cone_X (c : binary_bicone P Q) :\n  c.to_cone.X = c.X := rfl\n\n@[simp]\nlemma to_cone_π_app_left (c : binary_bicone P Q) :\n  c.to_cone.π.app (walking_pair.left) = c.fst := rfl\n@[simp]\nlemma to_cone_π_app_right (c : binary_bicone P Q) :\n  c.to_cone.π.app (walking_pair.right) = c.snd := rfl\n@[simp]\nlemma binary_fan_fst_to_cone (c : binary_bicone P Q) : binary_fan.fst c.to_cone = c.fst := rfl\n@[simp]\nlemma binary_fan_snd_to_cone (c : binary_bicone P Q) : binary_fan.snd c.to_cone = c.snd := rfl\n\n/-- Extract the cocone from a binary bicone. -/\ndef to_cocone (c : binary_bicone P Q) : cocone (pair P Q) :=\nbinary_cofan.mk c.inl c.inr\n\n@[simp]\nlemma to_cocone_X (c : binary_bicone P Q) :\n  c.to_cocone.X = c.X := rfl\n\n@[simp]\nlemma to_cocone_ι_app_left (c : binary_bicone P Q) :\n  c.to_cocone.ι.app (walking_pair.left) = c.inl := rfl\n@[simp]\nlemma to_cocone_ι_app_right (c : binary_bicone P Q) :\n  c.to_cocone.ι.app (walking_pair.right) = c.inr := rfl\n@[simp]\nlemma binary_cofan_inl_to_cocone (c : binary_bicone P Q) : binary_cofan.inl c.to_cocone = c.inl :=\nrfl\n@[simp]\nlemma binary_cofan_inr_to_cocone (c : binary_bicone P Q) : binary_cofan.inr c.to_cocone = c.inr :=\nrfl\n\n/-- Convert a `binary_bicone` into a `bicone` over a pair. -/\n@[simps]\ndef to_bicone {X Y : C} (b : binary_bicone X Y) : bicone (pair X Y).obj :=\n{ X := b.X,\n  π := λ j, walking_pair.cases_on j b.fst b.snd,\n  ι := λ j, walking_pair.cases_on j b.inl b.inr,\n  ι_π := λ j j', by { cases j; cases j', tidy } }\n\n/-- A binary bicone is a limit cone if and only if the corresponding bicone is a limit cone. -/\ndef to_bicone_is_limit {X Y : C} (b : binary_bicone X Y) :\n  is_limit (b.to_bicone.to_cone) ≃ is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\n/-- A binary bicone is a colimit cocone if and only if the corresponding bicone is a colimit\n    cocone. -/\ndef to_bicone_is_colimit {X Y : C} (b : binary_bicone X Y) :\n  is_colimit (b.to_bicone.to_cocone) ≃ is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\nend binary_bicone\n\nnamespace bicone\n\n/-- Convert a `bicone` over a function on `walking_pair` to a binary_bicone. -/\n@[simps]\ndef to_binary_bicone {X Y : C} (b : bicone (pair X Y).obj) : binary_bicone X Y :=\n{ X := b.X,\n  fst := b.π walking_pair.left,\n  snd := b.π walking_pair.right,\n  inl := b.ι walking_pair.left,\n  inr := b.ι walking_pair.right,\n  inl_fst' := by { simp [bicone.ι_π], refl, },\n  inr_fst' := by simp [bicone.ι_π],\n  inl_snd' := by simp [bicone.ι_π],\n  inr_snd' := by { simp [bicone.ι_π], refl, }, }\n\n/-- A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit\n    cone.  -/\ndef to_binary_bicone_is_limit {X Y : C} (b : bicone (pair X Y).obj) :\n  is_limit (b.to_binary_bicone.to_cone) ≃ is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\n/-- A bicone over a pair is a colimit cocone if and only if the corresponding binary bicone is a\n    colimit cocone. -/\ndef to_binary_bicone_is_colimit {X Y : C} (b : bicone (pair X Y).obj) :\n  is_colimit (b.to_binary_bicone.to_cocone) ≃ is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (λ j, by { cases j, tidy })\n\nend bicone\n\n/-- Structure witnessing that a binary bicone is a limit cone and a limit cocone. -/\n@[nolint has_inhabited_instance]\nstructure binary_bicone.is_bilimit {P Q : C} (b : binary_bicone P Q) :=\n(is_limit : is_limit b.to_cone)\n(is_colimit : is_colimit b.to_cocone)\n\n/-- A binary bicone is a bilimit bicone if and only if the corresponding bicone is a bilimit. -/\ndef binary_bicone.to_bicone_is_bilimit {X Y : C} (b : binary_bicone X Y) :\n  b.to_bicone.is_bilimit ≃ b.is_bilimit :=\n{ to_fun := λ h, ⟨b.to_bicone_is_limit h.is_limit, b.to_bicone_is_colimit h.is_colimit⟩,\n  inv_fun := λ h, ⟨b.to_bicone_is_limit.symm h.is_limit, b.to_bicone_is_colimit.symm h.is_colimit⟩,\n  left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },\n  right_inv := λ ⟨h, h'⟩, by { dsimp only, simp } }\n\n/-- A bicone over a pair is a bilimit bicone if and only if the corresponding binary bicone is a\n    bilimit. -/\ndef bicone.to_binary_bicone_is_bilimit {X Y : C} (b : bicone (pair X Y).obj) :\n  b.to_binary_bicone.is_bilimit ≃ b.is_bilimit :=\n{ to_fun := λ h, ⟨b.to_binary_bicone_is_limit h.is_limit,\n    b.to_binary_bicone_is_colimit h.is_colimit⟩,\n  inv_fun := λ h, ⟨b.to_binary_bicone_is_limit.symm h.is_limit,\n    b.to_binary_bicone_is_colimit.symm h.is_colimit⟩,\n  left_inv := λ ⟨h, h'⟩, by { dsimp only, simp },\n  right_inv := λ ⟨h, h'⟩, by { dsimp only, simp } }\n\n/--\nA bicone over `P Q : C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_inhabited_instance]\nstructure binary_biproduct_data (P Q : C) :=\n(bicone : binary_bicone P Q)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_binary_biproduct P Q` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`.\n-/\nclass has_binary_biproduct (P Q : C) : Prop :=\nmk' :: (exists_binary_biproduct : nonempty (binary_biproduct_data P Q))\n\nlemma has_binary_biproduct.mk {P Q : C} (d : binary_biproduct_data P Q) :\n  has_binary_biproduct P Q :=\n⟨nonempty.intro d⟩\n\n/--\nUse the axiom of choice to extract explicit `binary_biproduct_data F` from `has_binary_biproduct F`.\n-/\ndef get_binary_biproduct_data (P Q : C) [has_binary_biproduct P Q] : binary_biproduct_data P Q :=\nclassical.choice has_binary_biproduct.exists_binary_biproduct\n\n/-- A bicone for `P Q ` which is both a limit cone and a colimit cocone. -/\ndef binary_biproduct.bicone (P Q : C) [has_binary_biproduct P Q] : binary_bicone P Q :=\n(get_binary_biproduct_data P Q).bicone\n\n/-- `binary_biproduct.bicone P Q` is a limit bicone. -/\ndef binary_biproduct.is_bilimit (P Q : C) [has_binary_biproduct P Q] :\n  (binary_biproduct.bicone P Q).is_bilimit :=\n(get_binary_biproduct_data P Q).is_bilimit\n\n/-- `binary_biproduct.bicone P Q` is a limit cone. -/\ndef binary_biproduct.is_limit (P Q : C) [has_binary_biproduct P Q] :\n  is_limit (binary_biproduct.bicone P Q).to_cone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_limit\n\n/-- `binary_biproduct.bicone P Q` is a colimit cocone. -/\ndef binary_biproduct.is_colimit (P Q : C) [has_binary_biproduct P Q] :\n  is_colimit (binary_biproduct.bicone P Q).to_cocone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_colimit\n\nsection\nvariable (C)\n\n/--\n`has_binary_biproducts C` represents the existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`, for every `P Q : C`.\n-/\nclass has_binary_biproducts : Prop :=\n(has_binary_biproduct : Π (P Q : C), has_binary_biproduct P Q)\n\nattribute [instance, priority 100] has_binary_biproducts.has_binary_biproduct\n\n/--\nA category with finite biproducts has binary biproducts.\n\nThis is not an instance as typically in concrete categories there will be\nan alternative construction with nicer definitional properties.\n-/\nlemma has_binary_biproducts_of_finite_biproducts [has_finite_biproducts C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := λ P Q, has_binary_biproduct.mk\n  { bicone := (biproduct.bicone (pair P Q).obj).to_binary_bicone,\n    is_bilimit := (bicone.to_binary_bicone_is_bilimit _).symm (biproduct.is_bilimit _) } }\n\nend\n\nvariables {P Q : C}\n\ninstance has_binary_biproduct.has_limit_pair [has_binary_biproduct P Q] :\n  has_limit (pair P Q) :=\nhas_limit.mk ⟨_, binary_biproduct.is_limit P Q⟩\n\ninstance has_binary_biproduct.has_colimit_pair [has_binary_biproduct P Q] :\n  has_colimit (pair P Q) :=\nhas_colimit.mk ⟨_, binary_biproduct.is_colimit P Q⟩\n\n@[priority 100]\ninstance has_binary_products_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_products C :=\n{ has_limit := λ F, has_limit_of_iso (diagram_iso_pair F).symm }\n@[priority 100]\ninstance has_binary_coproducts_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_coproducts C :=\n{ has_colimit := λ F, has_colimit_of_iso (diagram_iso_pair F) }\n\n/--\nThe isomorphism between the specified binary product and the specified binary coproduct for\na pair for a binary biproduct.\n-/\ndef biprod_iso (X Y : C) [has_binary_biproduct X Y]  :\n  limits.prod X Y ≅ limits.coprod X Y :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (binary_biproduct.is_limit X Y)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _)\n\n/-- An arbitrary choice of biproduct of a pair of objects. -/\nabbreviation biprod (X Y : C) [has_binary_biproduct X Y] := (binary_biproduct.bicone X Y).X\n\nnotation X ` ⊞ `:20 Y:20 := biprod X Y\n\n/-- The projection onto the first summand of a binary biproduct. -/\nabbreviation biprod.fst {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ X :=\n(binary_biproduct.bicone X Y).fst\n/-- The projection onto the second summand of a binary biproduct. -/\nabbreviation biprod.snd {X Y : C} [has_binary_biproduct X Y] : X ⊞ Y ⟶ Y :=\n(binary_biproduct.bicone X Y).snd\n/-- The inclusion into the first summand of a binary biproduct. -/\nabbreviation biprod.inl {X Y : C} [has_binary_biproduct X Y] : X ⟶ X ⊞ Y :=\n(binary_biproduct.bicone X Y).inl\n/-- The inclusion into the second summand of a binary biproduct. -/\nabbreviation biprod.inr {X Y : C} [has_binary_biproduct X Y] : Y ⟶ X ⊞ Y :=\n(binary_biproduct.bicone X Y).inr\n\nsection\nvariables {X Y : C} [has_binary_biproduct X Y]\n\n@[simp] lemma binary_biproduct.bicone_fst : (binary_biproduct.bicone X Y).fst = biprod.fst := rfl\n@[simp] lemma binary_biproduct.bicone_snd : (binary_biproduct.bicone X Y).snd = biprod.snd := rfl\n@[simp] lemma binary_biproduct.bicone_inl : (binary_biproduct.bicone X Y).inl = biprod.inl := rfl\n@[simp] lemma binary_biproduct.bicone_inr : (binary_biproduct.bicone X Y).inr = biprod.inr := rfl\n\nend\n\n@[simp,reassoc]\nlemma biprod.inl_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 𝟙 X :=\n(binary_biproduct.bicone X Y).inl_fst\n@[simp,reassoc]\nlemma biprod.inl_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 0 :=\n(binary_biproduct.bicone X Y).inl_snd\n@[simp,reassoc]\nlemma biprod.inr_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.fst : X ⊞ Y ⟶ X) = 0 :=\n(binary_biproduct.bicone X Y).inr_fst\n@[simp,reassoc]\nlemma biprod.inr_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y ⟶ X ⊞ Y) ≫ (biprod.snd : X ⊞ Y ⟶ Y) = 𝟙 Y :=\n(binary_biproduct.bicone X Y).inr_snd\n\n/-- Given a pair of maps into the summands of a binary biproduct,\nwe obtain a map into the binary biproduct. -/\nabbreviation biprod.lift {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  W ⟶ X ⊞ Y :=\n(binary_biproduct.is_limit X Y).lift (binary_fan.mk f g)\n/-- Given a pair of maps out of the summands of a binary biproduct,\nwe obtain a map out of the binary biproduct. -/\nabbreviation biprod.desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  X ⊞ Y ⟶ W :=\n(binary_biproduct.is_colimit X Y).desc (binary_cofan.mk f g)\n\n@[simp, reassoc]\nlemma biprod.lift_fst {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  biprod.lift f g ≫ biprod.fst = f :=\n(binary_biproduct.is_limit X Y).fac _ walking_pair.left\n\n@[simp, reassoc]\nlemma biprod.lift_snd {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X) (g : W ⟶ Y) :\n  biprod.lift f g ≫ biprod.snd = g :=\n(binary_biproduct.is_limit X Y).fac _ walking_pair.right\n\n@[simp, reassoc]\nlemma biprod.inl_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  biprod.inl ≫ biprod.desc f g = f :=\n(binary_biproduct.is_colimit X Y).fac _ walking_pair.left\n\n@[simp, reassoc]\nlemma biprod.inr_desc {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W) :\n  biprod.inr ≫ biprod.desc f g = g :=\n(binary_biproduct.is_colimit X Y).fac _ walking_pair.right\n\ninstance biprod.mono_lift_of_mono_left {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)\n  (g : W ⟶ Y) [mono f] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_fst _ _\n\ninstance biprod.mono_lift_of_mono_right {W X Y : C} [has_binary_biproduct X Y] (f : W ⟶ X)\n  (g : W ⟶ Y) [mono g] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_snd _ _\n\ninstance biprod.epi_desc_of_epi_left {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)\n  [epi f] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inl_desc _ _\n\ninstance biprod.epi_desc_of_epi_right {W X Y : C} [has_binary_biproduct X Y] (f : X ⟶ W) (g : Y ⟶ W)\n  [epi g] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inr_desc _ _\n\n/-- Given a pair of maps between the summands of a pair of binary biproducts,\nwe obtain a map between the binary biproducts. -/\nabbreviation biprod.map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z :=\nis_limit.map (binary_biproduct.bicone W X).to_cone (binary_biproduct.is_limit Y Z)\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n/-- An alternative to `biprod.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biprod.map`. -/\nabbreviation biprod.map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : W ⊞ X ⟶ Y ⊞ Z :=\nis_colimit.map (binary_biproduct.is_colimit W X) (binary_biproduct.bicone Y Z).to_cocone\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n@[ext] lemma biprod.hom_ext {X Y Z : C} [has_binary_biproduct X Y] (f g : Z ⟶ X ⊞ Y)\n  (h₀ : f ≫ biprod.fst = g ≫ biprod.fst) (h₁ : f ≫ biprod.snd = g ≫ biprod.snd) : f = g :=\nbinary_fan.is_limit.hom_ext (binary_biproduct.is_limit X Y) h₀ h₁\n\n\n@[ext] lemma biprod.hom_ext' {X Y Z : C} [has_binary_biproduct X Y] (f g : X ⊞ Y ⟶ Z)\n  (h₀ : biprod.inl ≫ f = biprod.inl ≫ g) (h₁ : biprod.inr ≫ f = biprod.inr ≫ g) : f = g :=\nbinary_cofan.is_colimit.hom_ext (binary_biproduct.is_colimit X Y) h₀ h₁\n\nlemma biprod.map_eq_map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) : biprod.map f g = biprod.map' f g :=\nbegin\n  ext,\n  { simp only [map_pair_left, is_colimit.ι_map, is_limit.map_π, biprod.inl_fst_assoc,\n    category.assoc, ←binary_bicone.to_cone_π_app_left, ←binary_biproduct.bicone_fst,\n    ←binary_bicone.to_cocone_ι_app_left, ←binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_left, is_colimit.ι_map, is_limit.map_π, zero_comp,\n      biprod.inl_snd_assoc, category.assoc,\n      ←binary_bicone.to_cone_π_app_right, ←binary_biproduct.bicone_snd,\n      ←binary_bicone.to_cocone_ι_app_left, ←binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_right, biprod.inr_fst_assoc, is_colimit.ι_map, is_limit.map_π,\n      zero_comp, category.assoc,\n      ←binary_bicone.to_cone_π_app_left, ←binary_biproduct.bicone_fst,\n      ←binary_bicone.to_cocone_ι_app_right, ←binary_biproduct.bicone_inr],\n    simp },\n  { simp only [map_pair_right, is_colimit.ι_map, is_limit.map_π, biprod.inr_snd_assoc,\n      category.assoc, ←binary_bicone.to_cone_π_app_right, ←binary_biproduct.bicone_snd,\n      ←binary_bicone.to_cocone_ι_app_right, ←binary_biproduct.bicone_inr],\n    simp }\nend\n\ninstance biprod.inl_mono {X Y : C} [has_binary_biproduct X Y] :\n  split_mono (biprod.inl : X ⟶ X ⊞ Y) :=\n{ retraction := biprod.desc (𝟙 X) (biprod.inr ≫ biprod.fst) }\n\ninstance biprod.inr_mono {X Y : C} [has_binary_biproduct X Y] :\n  split_mono (biprod.inr : Y ⟶ X ⊞ Y) :=\n{ retraction := biprod.desc (biprod.inl ≫ biprod.snd) (𝟙 Y)}\n\ninstance biprod.fst_epi {X Y : C} [has_binary_biproduct X Y] :\n  split_epi (biprod.fst : X ⊞ Y ⟶ X) :=\n{ section_ := biprod.lift (𝟙 X) (biprod.inl ≫ biprod.snd) }\n\ninstance biprod.snd_epi {X Y : C} [has_binary_biproduct X Y] :\n  split_epi (biprod.snd : X ⊞ Y ⟶ Y) :=\n{ section_ := biprod.lift (biprod.inr ≫ biprod.fst) (𝟙 Y) }\n\n@[simp,reassoc]\nlemma biprod.map_fst {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.map f g ≫ biprod.fst = biprod.fst ≫ f :=\nis_limit.map_π _ _ _ walking_pair.left\n\n@[simp,reassoc]\nlemma biprod.map_snd {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.map f g ≫ biprod.snd = biprod.snd ≫ g :=\nis_limit.map_π _ _ _ walking_pair.right\n\n-- Because `biprod.map` is defined in terms of `lim` rather than `colim`,\n-- we need to provide additional `simp` lemmas.\n@[simp,reassoc]\nlemma biprod.inl_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.inl ≫ biprod.map f g = f ≫ biprod.inl :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ walking_pair.left\nend\n\n@[simp,reassoc]\nlemma biprod.inr_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ⟶ Y) (g : X ⟶ Z) :\n  biprod.inr ≫ biprod.map f g = g ≫ biprod.inr :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.ι_map (binary_biproduct.is_colimit W X) _ _ walking_pair.right\nend\n\n/-- Given a pair of isomorphisms between the summands of a pair of binary biproducts,\nwe obtain an isomorphism between the binary biproducts. -/\n@[simps]\ndef biprod.map_iso {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W ≅ Y) (g : X ≅ Z) : W ⊞ X ≅ Y ⊞ Z :=\n{ hom := biprod.map f.hom g.hom,\n  inv := biprod.map f.inv g.inv }\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_hom (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).hom\n    = biprod.lift b.fst b.snd :=\nrfl\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_inv (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).inv\n    = biprod.desc b.inl b.inr :=\nbegin\n  refine biprod.hom_ext' _ _ (hb.is_limit.hom_ext (λ j, _)) (hb.is_limit.hom_ext (λ j, _)),\n  all_goals { simp only [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp],\n    cases j },\n  all_goals { simp }\nend\n\n/-- Binary biproducts are unique up to isomorphism. This already follows because bilimits are\n    limits, but in the case of biproducts we can give an isomorphism with particularly nice\n    definitional properties, namely that `biprod.lift b.fst b.snd` and `biprod.desc b.inl b.inr`\n    are inverses of each other. -/\n@[simps]\ndef biprod.unique_up_to_iso (X Y : C) [has_binary_biproduct X Y] {b : binary_bicone X Y}\n  (hb : b.is_bilimit) : b.X ≅ X ⊞ Y :=\n{ hom := biprod.lift b.fst b.snd,\n  inv := biprod.desc b.inl b.inr,\n  hom_inv_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    ← biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [← biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    ← biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.inv_hom_id] }\n\nsection biprod_kernel\n\nvariables (X Y : C) [has_binary_biproduct X Y]\n\n/-- A kernel fork for the kernel of `biprod.fst`. It consists of the\nmorphism `biprod.inr`. -/\ndef biprod.fst_kernel_fork : kernel_fork (biprod.fst : X ⊞ Y ⟶ X) :=\nkernel_fork.of_ι biprod.inr biprod.inr_fst\n\n@[simp]\nlemma biprod.fst_kernel_fork_ι : fork.ι (biprod.fst_kernel_fork X Y) = biprod.inr :=\nrfl\n\n/-- The fork `biprod.fst_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_fst_kernel_fork : is_limit (biprod.fst_kernel_fork X Y) :=\nfork.is_limit.mk' _ $ λ s, ⟨s.ι ≫ biprod.snd, by ext; simp, λ m hm, by simp [← hm]⟩\n\n/-- A kernel fork for the kernel of `biprod.snd`. It consists of the\nmorphism `biprod.inl`. -/\ndef biprod.snd_kernel_fork : kernel_fork (biprod.snd : X ⊞ Y ⟶ Y) :=\nkernel_fork.of_ι biprod.inl biprod.inl_snd\n\n@[simp]\nlemma biprod.snd_kernel_fork_ι : fork.ι (biprod.snd_kernel_fork X Y) = biprod.inl :=\nrfl\n\n/-- The fork `biprod.snd_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_snd_kernel_fork : is_limit (biprod.snd_kernel_fork X Y) :=\nfork.is_limit.mk' _ $ λ s, ⟨s.ι ≫ biprod.fst, by ext; simp, λ m hm, by simp [← hm]⟩\n\n/-- A cokernel cofork for the cokernel of `biprod.inl`. It consists of the\nmorphism `biprod.snd`. -/\ndef biprod.inl_cokernel_fork : cokernel_cofork (biprod.inl : X ⟶ X ⊞ Y) :=\ncokernel_cofork.of_π biprod.snd biprod.inl_snd\n\n@[simp]\nlemma biprod.inl_cokernel_fork_π : cofork.π (biprod.inl_cokernel_fork X Y) = biprod.snd :=\nrfl\n\n/-- The cofork `biprod.inl_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inl_cokernel_fork : is_colimit (biprod.inl_cokernel_fork X Y) :=\ncofork.is_colimit.mk' _ $ λ s, ⟨biprod.inr ≫ s.π, by ext; simp, λ m hm, by simp [← hm]⟩\n\n/-- A cokernel cofork for the cokernel of `biprod.inr`. It consists of the\nmorphism `biprod.fst`. -/\ndef biprod.inr_cokernel_fork : cokernel_cofork (biprod.inr : Y ⟶ X ⊞ Y) :=\ncokernel_cofork.of_π biprod.fst biprod.inr_fst\n\n@[simp]\nlemma biprod.inr_cokernel_fork_π : cofork.π (biprod.inr_cokernel_fork X Y) = biprod.fst :=\nrfl\n\n/-- The cofork `biprod.inr_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inr_cokernel_fork : is_colimit (biprod.inr_cokernel_fork X Y) :=\ncofork.is_colimit.mk' _ $ λ s, ⟨biprod.inl ≫ s.π, by ext; simp, λ m hm, by simp [← hm]⟩\n\nend biprod_kernel\n\nsection\nvariables [has_binary_biproducts C]\n\n/-- The braiding isomorphism which swaps a binary biproduct. -/\n@[simps] def biprod.braiding (P Q : C) : P ⊞ Q ≅ Q ⊞ P :=\n{ hom := biprod.lift biprod.snd biprod.fst,\n  inv := biprod.lift biprod.snd biprod.fst }\n\n/--\nAn alternative formula for the braiding isomorphism which swaps a binary biproduct,\nusing the fact that the biproduct is a coproduct.\n-/\n@[simps]\ndef biprod.braiding' (P Q : C) : P ⊞ Q ≅ Q ⊞ P :=\n{ hom := biprod.desc biprod.inr biprod.inl,\n  inv := biprod.desc biprod.inr biprod.inl }\n\nlemma biprod.braiding'_eq_braiding {P Q : C} :\n  biprod.braiding' P Q = biprod.braiding P Q :=\nby tidy\n\n/-- The braiding isomorphism can be passed through a map by swapping the order. -/\n@[reassoc] lemma biprod.braid_natural {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :\n  biprod.map f g ≫ (biprod.braiding _ _).hom = (biprod.braiding _ _).hom ≫ biprod.map g f :=\nby tidy\n\n@[reassoc] lemma biprod.braiding_map_braiding {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z) :\n  (biprod.braiding X W).hom ≫ biprod.map f g ≫ (biprod.braiding Y Z).hom = biprod.map g f :=\nby tidy\n\n@[simp, reassoc] lemma biprod.symmetry' (P Q : C) :\n  biprod.lift biprod.snd biprod.fst ≫ biprod.lift biprod.snd biprod.fst = 𝟙 (P ⊞ Q) :=\nby tidy\n\n/-- The braiding isomorphism is symmetric. -/\n@[reassoc] lemma biprod.symmetry (P Q : C) :\n  (biprod.braiding P Q).hom ≫ (biprod.braiding Q P).hom = 𝟙 _ :=\nby simp\n\nend\n\n-- TODO:\n-- If someone is interested, they could provide the constructions:\n--   has_binary_biproducts ↔ has_finite_biproducts\n\nend limits\n\nnamespace limits\n\nsection preadditive\nvariables {C : Type u} [category.{v} C] [preadditive C]\nvariables {J : Type v} [decidable_eq J] [fintype J]\n\nopen category_theory.preadditive\nopen_locale big_operators\n\n/--\nIn a preadditive category, we can construct a biproduct for `f : J → C` from\nany bicone `b` for `f` satisfying `total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\ndef is_bilimit_of_total {f : J → C} (b : bicone f) (total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X) :\n  b.is_bilimit :=\n{ is_limit :=\n  { lift := λ s, ∑ j, s.π.app j ≫ b.ι j,\n    uniq' := λ s m h,\n    begin\n      erw [←category.comp_id m, ←total, comp_sum],\n      apply finset.sum_congr rfl,\n      intros j m,\n      erw [reassoc_of (h j)],\n    end,\n    fac' := λ s j,\n    begin\n      simp only [sum_comp, category.assoc, bicone.to_cone_π_app, b.ι_π, comp_dite],\n      -- See note [dsimp, simp].\n      dsimp, simp,\n    end },\n  is_colimit :=\n  { desc := λ s, ∑ j, b.π j ≫ s.ι.app j,\n    uniq' := λ s m h,\n    begin\n      erw [←category.id_comp m, ←total, sum_comp],\n            apply finset.sum_congr rfl,\n      intros j m,\n      erw [category.assoc, h],\n    end,\n    fac' := λ s j,\n    begin\n      simp only [comp_sum, ←category.assoc, bicone.to_cocone_ι_app, b.ι_π, dite_comp],\n      dsimp, simp,\n    end } }\n\nlemma is_bilimit.total {f : J → C} {b : bicone f} (i : b.is_bilimit) :\n  ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X :=\ni.is_limit.hom_ext (λ j, by simp [sum_comp, b.ι_π, comp_dite])\n\n/--\nIn a preadditive category, we can construct a biproduct for `f : J → C` from\nany bicone `b` for `f` satisfying `total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\nlemma has_biproduct_of_total {f : J → C} (b : bicone f) (total : ∑ j : J, b.π j ≫ b.ι j = 𝟙 b.X) :\n  has_biproduct f :=\nhas_biproduct.mk\n{ bicone := b,\n  is_bilimit := is_bilimit_of_total b total }\n\n/-- In a preadditive category, any finite bicone which is a limit cone is in fact a bilimit\n    bicone. -/\ndef is_bilimit_of_is_limit {f : J → C} (t : bicone f) (ht : is_limit t.to_cone) : t.is_bilimit :=\nis_bilimit_of_total _ $ ht.hom_ext $ λ j, by simp [sum_comp, t.ι_π, dite_comp, comp_dite]\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef bicone_is_bilimit_of_limit_cone_of_is_limit {f : J → C} {t : cone (discrete.functor f)}\n  (ht : is_limit t) : (bicone.of_limit_cone ht).is_bilimit :=\nis_bilimit_of_is_limit _ $ is_limit.of_iso_limit ht $ cones.ext (iso.refl _) (by tidy)\n\n/-- In a preadditive category, if the product over `f : J → C` exists,\n    then the biproduct over `f` exists. -/\nlemma has_biproduct.of_has_product (f : J → C) [has_product f] : has_biproduct f :=\nhas_biproduct.mk\n{ bicone := _,\n  is_bilimit := bicone_is_bilimit_of_limit_cone_of_is_limit (limit.is_limit _) }\n\n/-- In a preadditive category, any finite bicone which is a colimit cocone is in fact a bilimit\n    bicone. -/\ndef is_bilimit_of_is_colimit {f : J → C} (t : bicone f) (ht : is_colimit t.to_cocone) :\n  t.is_bilimit :=\nis_bilimit_of_total _ $ ht.hom_ext $ λ j,\n  by { simp_rw [bicone.to_cocone_ι_app, comp_sum, ← category.assoc, t.ι_π, dite_comp], tidy }\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef bicone_is_bilimit_of_colimit_cocone_of_is_colimit {f : J → C} {t : cocone (discrete.functor f)}\n  (ht : is_colimit t) : (bicone.of_colimit_cocone ht).is_bilimit :=\nis_bilimit_of_is_colimit _ $ is_colimit.of_iso_colimit ht $ cocones.ext (iso.refl _) (by tidy)\n\n/-- In a preadditive category, if the coproduct over `f : J → C` exists,\n    then the biproduct over `f` exists. -/\nlemma has_biproduct.of_has_coproduct (f : J → C) [has_coproduct f] : has_biproduct f :=\nhas_biproduct.mk\n{ bicone := _,\n  is_bilimit := bicone_is_bilimit_of_colimit_cocone_of_is_colimit (colimit.is_colimit _) }\n\n/-- A preadditive category with finite products has finite biproducts. -/\nlemma has_finite_biproducts.of_has_finite_products [has_finite_products C] :\n  has_finite_biproducts C :=\n⟨λ J _ _, { has_biproduct := λ F, by exactI has_biproduct.of_has_product _ }⟩\n\n/-- A preadditive category with finite coproducts has finite biproducts. -/\nlemma has_finite_biproducts.of_has_finite_coproducts [has_finite_coproducts C] :\n  has_finite_biproducts C :=\n⟨λ J _ _, { has_biproduct := λ F, by exactI has_biproduct.of_has_coproduct _ }⟩\n\nsection\nvariables {f : J → C} [has_biproduct f]\n\n/--\nIn any preadditive category, any biproduct satsifies\n`∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f)`\n-/\n@[simp] lemma biproduct.total : ∑ j : J, biproduct.π f j ≫ biproduct.ι f j = 𝟙 (⨁ f) :=\nis_bilimit.total (biproduct.is_bilimit _)\n\nlemma biproduct.lift_eq {T : C} {g : Π j, T ⟶ f j} :\n  biproduct.lift g = ∑ j, g j ≫ biproduct.ι f j :=\nbegin\n  ext j,\n  simp [sum_comp, biproduct.ι_π, comp_dite],\nend\n\nlemma biproduct.desc_eq {T : C} {g : Π j, f j ⟶ T} :\n  biproduct.desc g = ∑ j, biproduct.π f j ≫ g j :=\nbegin\n  ext j,\n  simp [comp_sum, biproduct.ι_π_assoc, dite_comp],\nend\n\n@[simp, reassoc] lemma biproduct.lift_desc {T U : C} {g : Π j, T ⟶ f j} {h : Π j, f j ⟶ U} :\n  biproduct.lift g ≫ biproduct.desc h = ∑ j : J, g j ≫ h j :=\nby simp [biproduct.lift_eq, biproduct.desc_eq, comp_sum, sum_comp, biproduct.ι_π_assoc,\n  comp_dite, dite_comp]\n\nlemma biproduct.map_eq [has_finite_biproducts C] {f g : J → C} {h : Π j, f j ⟶ g j} :\n  biproduct.map h = ∑ j : J, biproduct.π f j ≫ h j ≫ biproduct.ι g j :=\nbegin\n  ext,\n  simp [biproduct.ι_π, biproduct.ι_π_assoc, comp_sum, sum_comp, comp_dite, dite_comp],\nend\n\n@[simp, reassoc]\nlemma biproduct.matrix_desc\n  {K : Type v} [fintype K] [decidable_eq K] [has_finite_biproducts C]\n  {f : J → C} {g : K → C} (m : Π j k, f j ⟶ g k) {P} (x : Π k, g k ⟶ P) :\n  biproduct.matrix m ≫ biproduct.desc x = biproduct.desc (λ j, ∑ k, m j k ≫ x k) :=\nby { ext, simp, }\n\n@[simp, reassoc]\nlemma biproduct.lift_matrix\n  {K : Type v} [fintype K] [decidable_eq K] [has_finite_biproducts C]\n  {f : J → C} {g : K → C} {P} (x : Π j, P ⟶ f j) (m : Π j k, f j ⟶ g k)  :\n  biproduct.lift x ≫ biproduct.matrix m = biproduct.lift (λ k, ∑ j, x j ≫ m j k) :=\nby { ext, simp, }\n\n@[reassoc]\nlemma biproduct.matrix_map\n  {K : Type v} [fintype K] [decidable_eq K] [has_finite_biproducts C]\n  {f : J → C} {g : K → C} {h : K → C} (m : Π j k, f j ⟶ g k) (n : Π k, g k ⟶ h k) :\n  biproduct.matrix m ≫ biproduct.map n = biproduct.matrix (λ j k, m j k ≫ n k) :=\nby { ext, simp, }\n\n@[reassoc]\nlemma biproduct.map_matrix\n  {K : Type v} [fintype K] [decidable_eq K] [has_finite_biproducts C]\n  {f : J → C} {g : J → C} {h : K → C} (m : Π k, f k ⟶ g k) (n : Π j k, g j ⟶ h k) :\n  biproduct.map m ≫ biproduct.matrix n = biproduct.matrix (λ j k, m j ≫ n j k) :=\nby { ext, simp, }\n\nend\n\n/--\nIn a preadditive category, we can construct a binary biproduct for `X Y : C` from\nany binary bicone `b` satisfying `total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\ndef is_binary_bilimit_of_total {X Y : C} (b : binary_bicone X Y)\n  (total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.X) : b.is_bilimit :=\n{ is_limit :=\n  { lift := λ s, binary_fan.fst s ≫ b.inl +\n      binary_fan.snd s ≫ b.inr,\n    uniq' := λ s m h, by erw [←category.comp_id m, ←total,\n      comp_add, reassoc_of (h walking_pair.left), reassoc_of (h walking_pair.right)],\n    fac' := λ s j, by cases j; simp, },\n  is_colimit :=\n  { desc := λ s, b.fst ≫ binary_cofan.inl s +\n      b.snd ≫ binary_cofan.inr s,\n    uniq' := λ s m h, by erw [←category.id_comp m, ←total,\n      add_comp, category.assoc, category.assoc, h walking_pair.left, h walking_pair.right],\n    fac' := λ s j, by cases j; simp, } }\n\nlemma is_bilimit.binary_total {X Y : C} {b : binary_bicone X Y} (i : b.is_bilimit) :\n  b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.X :=\ni.is_limit.hom_ext (λ j, by { cases j; simp, })\n\n/--\nIn a preadditive category, we can construct a binary biproduct for `X Y : C` from\nany binary bicone `b` satisfying `total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\nlemma has_binary_biproduct_of_total {X Y : C} (b : binary_bicone X Y)\n  (total : b.fst ≫ b.inl + b.snd ≫ b.inr = 𝟙 b.X) : has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := b,\n  is_bilimit := is_binary_bilimit_of_total b total }\n\n/-- We can turn any limit cone over a pair into a bicone. -/\n@[simps]\ndef binary_bicone.of_limit_cone {X Y : C} {t : cone (pair X Y)} (ht : is_limit t) :\n  binary_bicone X Y :=\n{ X := t.X,\n  fst := t.π.app walking_pair.left,\n  snd := t.π.app walking_pair.right,\n  inl := ht.lift (binary_fan.mk (𝟙 X) 0),\n  inr := ht.lift (binary_fan.mk 0 (𝟙 Y)) }\n\nlemma inl_of_is_limit {X Y : C} {t : binary_bicone X Y} (ht : is_limit t.to_cone) :\n  t.inl = ht.lift (binary_fan.mk (𝟙 X) 0) :=\nht.hom_ext $ λ j, by { rw ht.fac, cases j; simp }\n\nlemma inr_of_is_limit {X Y : C} {t : binary_bicone X Y} (ht : is_limit t.to_cone) :\n  t.inr = ht.lift (binary_fan.mk 0 (𝟙 Y)) :=\nht.hom_ext $ λ j, by { rw ht.fac, cases j; simp }\n\n/-- In a preadditive category, any binary bicone which is a limit cone is in fact a bilimit\n    bicone. -/\ndef is_binary_bilimit_of_is_limit {X Y : C} (t : binary_bicone X Y) (ht : is_limit t.to_cone) :\n  t.is_bilimit :=\nis_binary_bilimit_of_total _\nbegin\n  refine binary_fan.is_limit.hom_ext ht _ _,\n  { rw [inl_of_is_limit ht, inr_of_is_limit ht, add_comp, category.assoc, category.assoc, ht.fac,\n      ht.fac, binary_fan.mk_π_app_left, binary_fan.mk_π_app_left, comp_zero, add_zero,\n      binary_bicone.binary_fan_fst_to_cone, category.comp_id, category.id_comp] },\n  { rw [inr_of_is_limit ht, inl_of_is_limit ht, add_comp, category.assoc, category.assoc, ht.fac,\n      ht.fac, binary_fan.mk_π_app_right, binary_fan.mk_π_app_right, comp_zero, zero_add,\n      binary_bicone.binary_fan_snd_to_cone, category.comp_id, category.id_comp] }\nend\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef binary_bicone_is_bilimit_of_limit_cone_of_is_limit {X Y : C} {t : cone (pair X Y)}\n  (ht : is_limit t) : (binary_bicone.of_limit_cone ht).is_bilimit :=\nis_binary_bilimit_of_total _ $ binary_fan.is_limit.hom_ext ht (by simp) (by simp)\n\n/-- In a preadditive category, if the product of `X` and `Y` exists, then the\n    binary biproduct of `X` and `Y` exists. -/\nlemma has_binary_biproduct.of_has_binary_product (X Y : C) [has_binary_product X Y] :\n  has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := _,\n  is_bilimit := binary_bicone_is_bilimit_of_limit_cone_of_is_limit (limit.is_limit _) }\n\n/-- In a preadditive category, if all binary products exist, then all binary biproducts exist. -/\nlemma has_binary_biproducts.of_has_binary_products [has_binary_products C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := λ X Y, has_binary_biproduct.of_has_binary_product X Y, }\n\n/-- We can turn any colimit cocone over a pair into a bicone. -/\n@[simps]\ndef binary_bicone.of_colimit_cocone {X Y : C} {t : cocone (pair X Y)} (ht : is_colimit t) :\n  binary_bicone X Y :=\n{ X := t.X,\n  fst := ht.desc (binary_cofan.mk (𝟙 X) 0),\n  snd := ht.desc (binary_cofan.mk 0 (𝟙 Y)),\n  inl := t.ι.app walking_pair.left,\n  inr := t.ι.app walking_pair.right }\n\nlemma fst_of_is_colimit {X Y : C} {t : binary_bicone X Y} (ht : is_colimit t.to_cocone) :\n  t.fst = ht.desc (binary_cofan.mk (𝟙 X) 0) :=\nbegin\n  refine ht.hom_ext (λ j, _),\n  rw ht.fac,\n  cases j,\n  all_goals { simp only [binary_bicone.to_cocone_ι_app_left, binary_bicone.inl_fst,\n      binary_cofan.mk_ι_app_left, binary_bicone.to_cocone_ι_app_right, binary_bicone.inr_fst,\n      binary_cofan.mk_ι_app_right] },\n  refl\nend\n\nlemma snd_of_is_colimit {X Y : C} {t : binary_bicone X Y} (ht : is_colimit t.to_cocone) :\n  t.snd = ht.desc (binary_cofan.mk 0 (𝟙 Y)) :=\nbegin\n  refine ht.hom_ext (λ j, _),\n  rw ht.fac,\n  cases j,\n  all_goals { simp only [binary_bicone.to_cocone_ι_app_left, binary_bicone.inl_snd,\n    binary_cofan.mk_ι_app_left, binary_bicone.to_cocone_ι_app_right, binary_bicone.inr_snd,\n    binary_cofan.mk_ι_app_right] },\n  refl\nend\n\n/-- In a preadditive category, any binary bicone which is a colimit cocone is in fact a\n    bilimit bicone. -/\ndef is_binary_bilimit_of_is_colimit {X Y : C} (t : binary_bicone X Y)\n  (ht : is_colimit t.to_cocone) : t.is_bilimit :=\nis_binary_bilimit_of_total _\nbegin\n  refine binary_cofan.is_colimit.hom_ext ht _ _,\n  { rw [fst_of_is_colimit ht, snd_of_is_colimit ht, comp_add, ht.fac_assoc, ht.fac_assoc,\n      binary_cofan.mk_ι_app_left, binary_cofan.mk_ι_app_left,\n      binary_bicone.binary_cofan_inl_to_cocone, zero_comp, add_zero, category.id_comp t.inl,\n      category.comp_id t.inl] },\n  { rw [fst_of_is_colimit ht, snd_of_is_colimit ht, comp_add, ht.fac_assoc, ht.fac_assoc,\n      binary_cofan.mk_ι_app_right, binary_cofan.mk_ι_app_right,\n      binary_bicone.binary_cofan_inr_to_cocone, zero_comp, zero_add, category.comp_id t.inr,\n      category.id_comp t.inr] }\nend\n\n/-- We can turn any colimit cocone over a pair into a bilimit bicone. -/\ndef binary_bicone_is_bilimit_of_colimit_cocone_of_is_colimit {X Y : C} {t : cocone (pair X Y)}\n  (ht : is_colimit t) : (binary_bicone.of_colimit_cocone ht).is_bilimit :=\nis_binary_bilimit_of_is_colimit (binary_bicone.of_colimit_cocone ht) $\n  is_colimit.of_iso_colimit ht $ cocones.ext (iso.refl _) $ λ j, by { cases j, tidy }\n\n/-- In a preadditive category, if the coproduct of `X` and `Y` exists, then the\n    binary biproduct of `X` and `Y` exists. -/\nlemma has_binary_biproduct.of_has_binary_coproduct (X Y : C) [has_binary_coproduct X Y] :\n  has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := _,\n  is_bilimit := binary_bicone_is_bilimit_of_colimit_cocone_of_is_colimit (colimit.is_colimit _) }\n\n/-- In a preadditive category, if all binary coproducts exist, then all binary biproducts exist. -/\nlemma has_binary_biproducts.of_has_binary_coproducts [has_binary_coproducts C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := λ X Y, has_binary_biproduct.of_has_binary_coproduct X Y, }\n\nsection\nvariables {X Y : C} [has_binary_biproduct X Y]\n\n/--\nIn any preadditive category, any binary biproduct satsifies\n`biprod.fst ≫ biprod.inl + biprod.snd ≫ biprod.inr = 𝟙 (X ⊞ Y)`.\n-/\n@[simp] lemma biprod.total : biprod.fst ≫ biprod.inl + biprod.snd ≫ biprod.inr = 𝟙 (X ⊞ Y) :=\nbegin\n  ext; simp [add_comp],\nend\n\nlemma biprod.lift_eq {T : C} {f : T ⟶ X} {g : T ⟶ Y} :\n  biprod.lift f g = f ≫ biprod.inl + g ≫ biprod.inr :=\nbegin\n  ext; simp [add_comp],\nend\n\nlemma biprod.desc_eq {T : C} {f : X ⟶ T} {g : Y ⟶ T} :\n  biprod.desc f g = biprod.fst ≫ f + biprod.snd ≫ g :=\nbegin\n  ext; simp [add_comp],\nend\n\n@[simp, reassoc] lemma biprod.lift_desc {T U : C} {f : T ⟶ X} {g : T ⟶ Y} {h : X ⟶ U} {i : Y ⟶ U} :\n  biprod.lift f g ≫ biprod.desc h i = f ≫ h + g ≫ i :=\nby simp [biprod.lift_eq, biprod.desc_eq]\n\nlemma biprod.map_eq [has_binary_biproducts C] {W X Y Z : C} {f : W ⟶ Y} {g : X ⟶ Z} :\n  biprod.map f g = biprod.fst ≫ f ≫ biprod.inl + biprod.snd ≫ g ≫ biprod.inr :=\nby apply biprod.hom_ext; apply biprod.hom_ext'; simp\n\nend\n\nsection\nvariables {X Y : C} (f g : X ⟶ Y)\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\nlemma biprod.add_eq_lift_id_desc [has_binary_biproduct X X] :\n  f + g = biprod.lift (𝟙 X) (𝟙 X) ≫ biprod.desc f g :=\nby simp\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\nlemma biprod.add_eq_lift_desc_id [has_binary_biproduct Y Y] :\n  f + g = biprod.lift f g ≫ biprod.desc (𝟙 Y) (𝟙 Y) :=\nby simp\n\nend\n\nend preadditive\n\nend limits\n\nopen category_theory.limits\n\nlocal attribute [ext] preadditive\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\ninstance subsingleton_preadditive_of_has_binary_biproducts {C : Type u} [category.{v} C]\n  [has_zero_morphisms C] [has_binary_biproducts C] : subsingleton (preadditive C) :=\nsubsingleton.intro $ λ a b,\nbegin\n  ext X Y f g,\n  have h₁ := @biprod.add_eq_lift_id_desc _ _ a _ _ f g\n    (by convert (infer_instance : has_binary_biproduct X X)),\n  have h₂ := @biprod.add_eq_lift_id_desc _ _ b _ _ f g\n    (by convert (infer_instance : has_binary_biproduct X X)),\n  refine h₁.trans (eq.trans _ h₂.symm),\n  congr' 2;\n  exact subsingleton.elim _ _\nend\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/limits/shapes/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250376, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3246289994842061}}
{"text": "/-\nCopyright (c) 2023 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.cochain_complex.basic\nimport for_mathlib.algebra.homology.homological_complex_limits\nimport for_mathlib.algebra.homology.trunc\nimport category_theory.limits.full_subcategory\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\nopen_locale zero_object\n\nnamespace category_theory.limits\n\nvariables {J C : Type*} [category J] [category C] {F : J ⥤ C} [has_zero_object C]\n  [has_zero_morphisms C]\n\nlemma is_zero_of_is_limit_of_is_zero (c : cone F) (hc : is_limit c)\n  (hF : ∀ (j : J), is_zero (F.obj j)) : is_zero c.X :=\nis_zero.of_iso (is_zero_zero _)\n  (is_limit.cone_point_unique_up_to_iso hc\n  (is_limit.of_is_zero F (functor.is_zero _ hF) (cone.mk 0 0) (is_zero_zero _)))\n\nlemma is_zero_of_is_colimit_of_is_zero (c : cocone F) (hc : is_colimit c)\n  (hF : ∀ (j : J), is_zero (F.obj j)) : is_zero c.X :=\nis_zero.of_iso (is_zero_zero _)\n  (is_colimit.cocone_point_unique_up_to_iso hc\n  (is_colimit.of_is_zero F (functor.is_zero _ hF) (cocone.mk 0 0) (is_zero_zero _)))\n\nend category_theory.limits\n\nopen category_theory.limits\n\nvariables {C : Type*} [category C] [abelian C]\n\nnamespace cochain_complex\n\nnamespace limits\n\nvariables {J : Type} [small_category J] [fin_category J]\n\nlemma bound_of_finite (K : J → cochain_complex C ℤ) (hK : ∀ (j : J), (K j).is_minus) :\n  ∃ (n : ℤ), ∀ (j : J), (K j).is_strictly_le n :=\nbegin\n  let m : J → ℤ := λ j, ((hK j).some),\n  have hm : ∀ (j : J), (K j).is_strictly_le (m j) := λ j, (hK j).some_spec,\n  let μ := finset.max (finset.image m ⊤),\n  by_cases μ = ⊥,\n  { simp only [finset.max_eq_bot, finset.top_eq_univ, finset.image_eq_empty,\n      finset.univ_eq_empty_iff] at h,\n    exact ⟨0, λ j, by { exfalso, exact h.false j, }⟩, },\n  { obtain ⟨μ', hμ'⟩ := with_bot.ne_bot_iff_exists.1 h,\n    refine ⟨μ', λ j, _⟩,\n    haveI := hm j,\n    refine cochain_complex.is_strictly_le_of_le (K j) (m j) _ _,\n    have eq := finset.le_max (finset.mem_image_of_mem m (finset.mem_univ j)),\n    change _ ≤ μ at eq,\n    simpa only [← hμ', with_bot.coe_le_coe] using eq, },\nend\n\nvariable (J)\n\nlemma is_minus_is_closed_under_limits_of_shape :\n  closed_under_limits_of_shape J (cochain_complex.is_minus : cochain_complex C ℤ → Prop) :=\nλ F c hc hF, begin\n  obtain ⟨n, hn⟩ := bound_of_finite F.obj hF,\n  refine ⟨n, ⟨λ i hi, _⟩⟩,\n  have hc' := is_limit_of_preserves (homological_complex.eval C (complex_shape.up ℤ) i) hc,\n  refine is_zero_of_is_limit_of_is_zero _ hc' (λ j, _),\n  haveI := hn j,\n  dsimp,\n  exact cochain_complex.is_strictly_le.is_zero (F.obj j) n i hi,\nend\n\nlemma is_minus_is_closed_under_colimits_of_shape :\n  closed_under_colimits_of_shape J (cochain_complex.is_minus : cochain_complex C ℤ → Prop) :=\nλ F c hc hF, begin\n  obtain ⟨n, hn⟩ := bound_of_finite F.obj hF,\n  refine ⟨n, ⟨λ i hi, _⟩⟩,\n  have hc' := is_colimit_of_preserves (homological_complex.eval C (complex_shape.up ℤ) i) hc,\n  refine is_zero_of_is_colimit_of_is_zero _ hc' (λ j, _),\n  haveI := hn j,\n  dsimp,\n  exact cochain_complex.is_strictly_le.is_zero (F.obj j) n i hi,\nend\n\ninstance : has_finite_limits (cochain_complex.minus C) :=\n⟨λ J, begin\n  introI,\n  introI,\n  apply has_limits_of_shape_of_closed_under_limits,\n  apply is_minus_is_closed_under_limits_of_shape,\nend⟩\n\ninstance : has_finite_colimits (cochain_complex.minus C) :=\n⟨λ J, begin\n  introI,\n  introI,\n  apply has_colimits_of_shape_of_closed_under_colimits,\n  apply is_minus_is_closed_under_colimits_of_shape,\nend⟩\n\ninstance : creates_limits_of_shape J (cochain_complex.minus.ι : _ ⥤ cochain_complex C ℤ) :=\nbegin\n  apply creates_limits_of_shape_full_subcategory_inclusion,\n  apply is_minus_is_closed_under_limits_of_shape,\nend\n\ninstance : creates_colimits_of_shape J (cochain_complex.minus.ι : _ ⥤ cochain_complex C ℤ) :=\nbegin\n  apply creates_colimits_of_shape_full_subcategory_inclusion,\n  apply is_minus_is_closed_under_colimits_of_shape,\nend\n\nend limits\n\nnamespace minus\n\nnamespace projective_model_structure\n\nvariable {C}\n\nlemma CM1 : has_finite_limits (cochain_complex.minus C) ∧\n  has_finite_colimits (cochain_complex.minus C) :=\n⟨infer_instance, infer_instance⟩\n\nend projective_model_structure\n\nend minus\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cochain_complex/cm1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.32434608088512}}
{"text": "import algebra.homology.homotopy\nimport category_theory.limits.shapes.zero_objects\n\nimport for_mathlib.Cech.split\nimport for_mathlib.abelian_category\n\nnoncomputable theory\n\nuniverses v u v' u'\n\nnamespace category_theory\nopen_locale simplicial\nopen category_theory.limits\n\nnamespace arrow\n\nsection contravariant\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : Pᵒᵖ ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\nvariables [arrow.split f] [preadditive N]\n\ndef contracting_homotopy' : homotopy (𝟙 (f.conerve M).to_cocomplex) 0 :=\n{ hom := λ i j, if h : j + 1 = i then eq_to_hom (by subst h) ≫ f.contracting_homotopy M j else 0,\n  zero' := λ i j h, dif_neg h,\n  comm := λ i, begin\n    simp only [homological_complex.id_f, homotopy.d_next_cochain_complex,\n      homological_complex.zero_f_apply, add_zero],\n    rcases i with _|_|i,\n    { rw [← is_contracting_homotopy_zero, homotopy.prev_d_zero_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n    { rw [← is_contracting_homotopy_one, homotopy.prev_d_succ_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n    { rw [← is_contracting_homotopy, homotopy.prev_d_succ_cochain_complex],\n      simp only [eq_self_iff_true, dif_pos, eq_to_hom_refl, category.id_comp, add_zero], },\n  end }\n\nvariables [has_equalizers N] [has_cokernels N] [has_images N] [has_image_maps N] [has_zero_object N]\n\nlemma conerve_to_cocomplex_homology_is_zero (i : ℕ) :\n  is_zero ((f.conerve M).to_cocomplex.homology i) :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  simpa only [category_theory.functor.map_id, functor.map_zero] using\n    homology_map_eq_of_homotopy (f.contracting_homotopy' M) i,\nend\n\nend contravariant\n\nsection covariant\n\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : P ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\nvariables [arrow.split f] [preadditive N]\n\ndef covariant_contracting_homotopy' : homotopy (𝟙 (f.nerve M).to_complex) 0 :=\n{ hom := λ i j, if h : i + 1 = j then\n    f.covariant_contracting_homotopy M i ≫ eq_to_hom (by subst h) else 0,\n  zero' := λ i j h, dif_neg h,\n  comm := λ i, begin\n    simp only [homological_complex.id_f, homotopy.d_next_cochain_complex,\n      homological_complex.zero_f_apply, add_zero],\n    rcases i with _|_|i,\n    { rw [← covariant_is_contracting_homotopy_zero, homotopy.d_next_zero_chain_complex],\n      simp only [eq_self_iff_true, eq_to_hom_refl, dite_eq_ite, if_true,\n        homotopy.prev_d_chain_complex, category.comp_id, zero_add] },\n    { rw [← covariant_is_contracting_homotopy_one, homotopy.d_next_succ_chain_complex],\n      simp only [simplicial_object.augmented.to_complex_d_2, eq_self_iff_true,\n        eq_to_hom_refl, category.id_comp, dite_eq_ite, if_true,\n        category.comp_id, homotopy.prev_d_chain_complex],\n      rw add_comm },\n    { rw [← covariant_is_contracting_homotopy, homotopy.d_next_succ_chain_complex],\n      simp only [simplicial_object.augmented.to_complex_d_2, eq_self_iff_true,\n        eq_to_hom_refl, category.id_comp, dite_eq_ite, if_true,\n        category.comp_id, homotopy.prev_d_chain_complex],\n      rw add_comm },\n  end }\n\nvariables [has_equalizers N] [has_cokernels N] [has_images N] [has_image_maps N] [has_zero_object N]\n\nlemma nerve_to_complex_homology_is_zero (i : ℕ) :\n  is_zero ((f.nerve M).to_complex.homology i) :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  simpa only [category_theory.functor.map_id, functor.map_zero] using\n    homology_map_eq_of_homotopy (f.covariant_contracting_homotopy' M) i,\nend\n\nend covariant\n\nend arrow\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/Cech/homotopy.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.32433456327130433}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Data.Option.Basic\n\nuniverse u v\n\ntheorem Option.eqOfEqSome {α : Type u} : ∀ {x y : Option α}, (∀z, x = some z ↔ y = some z) → x = y\n  | none,   none,   h => rfl\n  | none,   some z, h => Option.noConfusion ((h z).2 rfl)\n  | some z, none,   h => Option.noConfusion ((h z).1 rfl)\n  | some z, some w, h => Option.noConfusion ((h w).2 rfl) (congrArg some)\n\ntheorem Option.eqNoneOfIsNone {α : Type u} : ∀ {o : Option α}, o.isNone → o = none\n  | none, h => rfl\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/stage0/src/Init/Data/Option/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073507867328, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.32424668419917785}}
{"text": "\nimport data.equiv.basic\nimport tactic\n\nuniverses u₀ u₁ v₀ v₁ v₂ w w₀ w₁\n\nclass liftable (f : Type u₀ → Type u₁) (g : Type v₀ → Type v₁) :=\n  (up {}   : Π {α β}, α ≃ β → f α → g β)\n  (down {} : Π {α β}, α ≃ β → g β → f α)\n  (up_down : ∀ {α β} (F : α ≃ β) (x : g β), up F (down F x) = x)\n  (down_up : ∀ {α β} (F : α ≃ β) (x : f α), down F (up F x) = x)\n\nattribute [simp] liftable.up_down liftable.down_up\n\nnamespace liftable\n\n@[reducible]\ndef up' {f : Type u₀ → Type u₁} {g : Type (max u₀ v₀) → Type v₁} [liftable f g]\n  {α} : f α → g (ulift α) :=\nliftable.up equiv.ulift.symm\n\n@[reducible]\ndef down' {f : Type u₀ → Type u₁} {g : Type (max u₀ v₀) → Type v₁} [liftable f g]\n  {α} : g (ulift α) → f α :=\nliftable.down equiv.ulift.symm\n\nend liftable\n\nopen ulift\n\nprotected lemma prod.map_map {α α' α'' β β' β''}\n  (x : α × β) (f : α → α') (f' : α' → α'')\n              (g : β → β') (g' : β' → β'') :\n  prod.map f' g' (prod.map f g x) = prod.map (f' ∘ f) (g' ∘ g) x :=\nby cases x; refl\n\nprotected lemma prod.id_map {α β}\n  (x : α × β) :\n  prod.map (λ x, x) (λ x, x) x = x :=\nby cases x; refl\n\ninstance : liftable id id :=\n{ up := λ _ _ F, F\n, down := λ _ _ F, F.symm\n, up_down := by intros; simp\n, down_up := by intros; simp }\n\ndef state_t.liftable' {s s' m m'} [functor m'] [functor m] [is_lawful_functor m'] [is_lawful_functor m]\n  [liftable m m']\n  (F : s ≃ s') :\n  liftable (state_t s m) (state_t s' m') :=\n{ up   := λ _ _ G ⟨ f ⟩, ⟨ λ s, liftable.up (equiv.prod_congr G F) (f $ F.symm s) ⟩\n, down := λ _ _ G ⟨ g ⟩, ⟨ λ s, liftable.down (equiv.prod_congr G F) $ g (F s) ⟩\n, up_down := by { rintros α β G ⟨ f ⟩, simp! }\n, down_up := by { rintros α β G ⟨ g ⟩, simp! [map_map,function.comp] } }\n\ninstance {s m m'} [functor m'] [functor m] [is_lawful_functor m'] [is_lawful_functor m]\n  [liftable m m'] :\n  liftable (state_t s m) (state_t (ulift s) m') :=\nstate_t.liftable' equiv.ulift.symm\n\ndef reader_t.liftable' {s s' m m'}\n  [liftable m m']\n  (F : s ≃ s') :\n  liftable (reader_t s m) (reader_t s' m') :=\n{ up   := λ _ _ G ⟨ f ⟩, ⟨ λ s, liftable.up G (f $ F.symm s) ⟩\n, down := λ _ _ G ⟨ g ⟩, ⟨ λ s, liftable.down G $ g $ F s ⟩\n, up_down := by { rintros α β G ⟨ f ⟩, simp! }\n, down_up := by { rintros α β G ⟨ g ⟩, simp! [map_map,function.comp] } }\n\ninstance {s m m'} [liftable m m'] : liftable (reader_t s m) (reader_t (ulift s) m') :=\nreader_t.liftable' equiv.ulift.symm\n\n-- namespace liftable\n\n-- variables {f : Type (max u₀ w) → Type u₁} {g : Type (max (max u₀ w) v₀) → Type v₁}\n-- variables [liftable f g] [functor g]\n-- variable {α : Type w}\n-- open functor\n\n-- def up' (x : f (ulift.{u₀} α)) : g (ulift.{max u₀ v₀} α) :=\n-- map (ulift.up ∘ ulift.down ∘ ulift.down) $ up x\n\n-- def down' (x : g (ulift.{max u₀ v₀} α)) : f (ulift.{u₀} α) :=\n-- down $ map (ulift.up ∘ ulift.up ∘ ulift.down) x\n\n-- variables [is_lawful_functor g]\n\n-- lemma up_down' : ∀ (x : g (ulift.{max u₀ v₀} α)), up' (down' x : f (ulift.{u₀} α)) = x :=\n-- by intros; simp [up',down',map_map,function.comp,ulift.up_down]\n\n-- lemma down_up' : ∀ (x : f (ulift.{u₀} α)), down' (up' x : g (ulift.{max u₀ v₀} α)) = x :=\n-- by intros; simp [up',down',map_map,function.comp,ulift.up_down]\n\n-- end liftable\n\nsection trans\nopen liftable\n\n-- def liftable.trans\n--     {f : Type u₀ → Type u₁}\n--     {g : Type v₀ → Type v₁}\n--     {h : Type w₀ → Type w₁}\n--     [functor h] [is_lawful_functor h]\n--     (_ : liftable f g)\n--     (_ : liftable g h) :\n--   liftable f h :=\n-- by refine\n--      { up := λ α β G, (up' : g (ulift α) → h (ulift.{max v₀ w} α)) ∘ (up : f α → g (ulift α)),\n--        down := λ α β G, (down : g (ulift α) → f α) ∘ (down' : h (ulift.{max v₀ w} α) → g (ulift α)) ,\n--        .. };\n--      intros; simp [function.comp,up_down',down_up']\n\nend trans\n", "meta": {"author": "cipher1024", "repo": "slim_check", "sha": "5969b7f72e01fdd46f2502ed0cbf69c0699061d4", "save_path": "github-repos/lean/cipher1024-slim_check", "path": "github-repos/lean/cipher1024-slim_check/slim_check-5969b7f72e01fdd46f2502ed0cbf69c0699061d4/src/test/slim_check/liftable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.3241265495139944}}
{"text": "import Days\nimport Days.Common\n\nnamespace Days.Day01\nopen Days\nopen Common\n\ndef elfGroupSeperator := \"\\n\\n\"\n\nstructure Elf :=\n  carrying: List Nat \n\ndef Elf.fromInput (input: Input) : List Elf :=\n  input.splitOn elfGroupSeperator \n    |> List.map (·.natLines)\n    |> List.map Elf.mk\n\n/--\nSanta's reindeer typically eat regular reindeer food, but they need a lot of magical energy to deliver presents on Christmas. For that, their favorite snack is a special type of star fruit that only grows deep in the jungle. The Elves have brought you on their annual expedition to the grove where the fruit grows.\n\nTo supply enough magical energy, the expedition needs to retrieve a minimum of fifty stars by December 25th. Although the Elves assure you that the grove has plenty of fruit, you decide to grab any fruit you see along the way, just in case.\n\nCollect stars by solving puzzles. Two puzzles will be made available on each day in the Advent calendar; the second puzzle is unlocked when you complete the first. Each puzzle grants one star. Good luck!\n\nThe jungle must be too overgrown and difficult to navigate in vehicles or access from the air; the Elves' expedition traditionally goes on foot. As your boats approach land, the Elves begin taking inventory of their supplies. One important consideration is food - in particular, the number of Calories each Elf is carrying (your puzzle input).\n\nThe Elves take turns writing down the number of Calories contained by the various meals, snacks, rations, etc. that they've brought with them, one item per line. Each Elf separates their own inventory from the previous Elf's inventory (if any) by a blank line.\n\nFor example, suppose the Elves finish writing their items' Calories and end up with the following list:\n\n1000\n2000\n3000\n\n4000\n\n5000\n6000\n\n7000\n8000\n9000\n\n10000\nThis list represents the Calories of the food carried by five Elves:\n\nThe first Elf is carrying food with 1000, 2000, and 3000 Calories, a total of 6000 Calories.\nThe second Elf is carrying one food item with 4000 Calories.\nThe third Elf is carrying food with 5000 and 6000 Calories, a total of 11000 Calories.\nThe fourth Elf is carrying food with 7000, 8000, and 9000 Calories, a total of 24000 Calories.\nThe fifth Elf is carrying one food item with 10000 Calories.\nIn case the Elves get hungry and need extra snacks, they need to know which Elf to ask: they'd like to know how many Calories are being carried by the Elf carrying the most Calories. In the example above, this is 24000 (carried by the fourth Elf).\n\nFind the Elf carrying the most Calories. How many total Calories is that Elf carrying?\n-/\ndef part₁ (input: Input) : Nat := \n  elfCarryingTheMost.get!\n  where \n    elves := Elf.fromInput input \n    countCarrying (elf : Elf) : Nat := sum elf.carrying\n    elfCarryingTheMost := \n      List.map countCarrying elves\n      |> List.maximum?\n\n/--\nBy the time you calculate the answer to the Elves' question, they've already realized that the Elf carrying the most Calories of food might eventually run out of snacks.\n\nTo avoid this unacceptable situation, the Elves would instead like to know the total Calories carried by the top three Elves carrying the most Calories. That way, even if one of those Elves runs out of snacks, they still have two backups.\n\nIn the example above, the top three Elves are the fourth Elf (with 24000 Calories), then the third Elf (with 11000 Calories), then the fifth Elf (with 10000 Calories). The sum of the Calories carried by these three elves is 45000.\n\nFind the top three Elves carrying the most Calories. How many Calories are those Elves carrying in total?\n-/\ndef part₂ (input: Input) : Nat := \n  top3Total\n  where \n    elves := Elf.fromInput input\n    countCarrying (elf : Elf) : Nat := sum elf.carrying\n\n    top3Elves : Option $ Nat × Nat × Nat := \n      .map countCarrying elves\n      |> List.foldl (init := none) (fun \n      | none, elf => (elf, elf, elf)\n      | some (top₁, top₂, top₃), elf => \n        if elf > top₁ then (elf, top₁, top₂)\n        else if elf > top₂ then (top₁, elf, top₂)\n        else if elf > top₃ then (top₁, top₂, elf)\n        else (top₁, top₂, top₃)\n      )\n    \n    top3Total := \n      if let some (top₁, top₂, top₃) := top3Elves then \n        top₁ + top₂ + top₃\n      else 0\n\ndef solution := Problem.define 1 part₁ part₂\n\ndef sample := \"1000\n2000\n3000\n\n4000\n\n5000\n6000\n\n7000\n8000\n9000\n\n10000\"\n\n\n#eval testPart₁ (expect:=24000) solution sample\n#eval testPart₂ (expect:=45000) solution sample", "meta": {"author": "jakeswenson", "repo": "advent2022", "sha": "af941092292ff0bc5552bce9c145d6b5b173c20d", "save_path": "github-repos/lean/jakeswenson-advent2022", "path": "github-repos/lean/jakeswenson-advent2022/advent2022-af941092292ff0bc5552bce9c145d6b5b173c20d/Days/Day01.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.3240334494690884}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.category.Module.basic\nimport group_theory.quotient_group\nimport category_theory.limits.concrete_category\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.shapes.concrete_category\n\n/-!\n# The category of R-modules has all colimits.\n\nThis file uses a \"pre-automated\" approach, just as for `Mon/colimits.lean`.\n\nNote that finite colimits can already be obtained from the instance `abelian (Module R)`.\n\nTODO:\nIn fact, in `Module R` there is a much nicer model of colimits as quotients\nof finitely supported functions, and we really should implement this as well (or instead).\n-/\n\nuniverses u v\n\nopen category_theory\nopen category_theory.limits\n\nvariables {R : Type v} [ring R]\n\n-- [ROBOT VOICE]:\n-- You should pretend for now that this file was automatically generated.\n-- It follows the same template as colimits in Mon.\n\nnamespace Module.colimits\n/-!\nWe build the colimit of a diagram in `Module` by constructing the\nfree group on the disjoint union of all the abelian groups in the diagram,\nthen taking the quotient by the abelian group laws within each abelian group,\nand the identifications given by the morphisms in the diagram.\n-/\n\nvariables {J : Type v} [small_category J] (F : J ⥤ Module.{v} R)\n\n/--\nAn inductive type representing all module expressions (without relations)\non a collection of types indexed by the objects of `J`.\n-/\ninductive prequotient\n-- There's always `of`\n| of : Π (j : J) (x : F.obj j), prequotient\n-- Then one generator for each operation\n| zero : prequotient\n| neg : prequotient → prequotient\n| add : prequotient → prequotient → prequotient\n| smul : R → prequotient → prequotient\n\ninstance : inhabited (prequotient F) := ⟨prequotient.zero⟩\n\nopen prequotient\n\n/--\nThe relation on `prequotient` saying when two expressions are equal\nbecause of the module laws, or\nbecause one element is mapped to another by a morphism in the diagram.\n-/\ninductive relation : prequotient F → prequotient F → Prop\n-- Make it an equivalence relation:\n| refl : Π (x), relation x x\n| symm : Π (x y) (h : relation x y), relation y x\n| trans : Π (x y z) (h : relation x y) (k : relation y z), relation x z\n-- There's always a `map` relation\n| map : Π (j j' : J) (f : j ⟶ j') (x : F.obj j), relation (of j' (F.map f x)) (of j x)\n-- Then one relation per operation, describing the interaction with `of`\n| zero : Π (j), relation (of j 0) zero\n| neg : Π (j) (x : F.obj j), relation (of j (-x)) (neg (of j x))\n| add : Π (j) (x y : F.obj j), relation (of j (x + y)) (add (of j x) (of j y))\n| smul : Π (j) (s) (x : F.obj j), relation (of j (s • x)) (smul s (of j x))\n-- Then one relation per argument of each operation\n| neg_1 : Π (x x') (r : relation x x'), relation (neg x) (neg x')\n| add_1 : Π (x x' y) (r : relation x x'), relation (add x y) (add x' y)\n| add_2 : Π (x y y') (r : relation y y'), relation (add x y) (add x y')\n| smul_1 : Π (s) (x x') (r : relation x x'), relation (smul s x) (smul s x')\n-- And one relation per axiom\n| zero_add      : Π (x), relation (add zero x) x\n| add_zero      : Π (x), relation (add x zero) x\n| add_left_neg  : Π (x), relation (add (neg x) x) zero\n| add_comm      : Π (x y), relation (add x y) (add y x)\n| add_assoc     : Π (x y z), relation (add (add x y) z) (add x (add y z))\n| one_smul      : Π (x), relation (smul 1 x) x\n| mul_smul      : Π (s t) (x), relation (smul (s * t) x) (smul s (smul t x))\n| smul_add      : Π (s) (x y), relation (smul s (add x y)) (add (smul s x) (smul s y))\n| smul_zero     : Π (s), relation (smul s zero) zero\n| add_smul      : Π (s t) (x), relation (smul (s + t) x) (add (smul s x) (smul t x))\n| zero_smul     : Π (x), relation (smul 0 x) zero\n\n/--\nThe setoid corresponding to module expressions modulo module relations and identifications.\n-/\ndef colimit_setoid : setoid (prequotient F) :=\n{ r := relation F, iseqv := ⟨relation.refl, relation.symm, relation.trans⟩ }\nattribute [instance] colimit_setoid\n\n/--\nThe underlying type of the colimit of a diagram in `Module R`.\n-/\n@[derive inhabited]\ndef colimit_type : Type v := quotient (colimit_setoid F)\n\ninstance : add_comm_group (colimit_type F) :=\n{ zero :=\n  begin\n    exact quot.mk _ zero\n  end,\n  neg :=\n  begin\n    fapply @quot.lift,\n    { intro x,\n      exact quot.mk _ (neg x) },\n    { intros x x' r,\n      apply quot.sound,\n      exact relation.neg_1 _ _ r },\n  end,\n  add :=\n  begin\n    fapply @quot.lift _ _ ((colimit_type F) → (colimit_type F)),\n    { intro x,\n      fapply @quot.lift,\n      { intro y,\n        exact quot.mk _ (add x y) },\n      { intros y y' r,\n        apply quot.sound,\n        exact relation.add_2 _ _ _ r } },\n    { intros x x' r,\n      funext y,\n      induction y,\n      dsimp,\n      apply quot.sound,\n      { exact relation.add_1 _ _ _ r },\n      { refl } },\n  end,\n  zero_add := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.zero_add,\n    refl,\n  end,\n  add_zero := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_zero,\n    refl,\n  end,\n  add_left_neg := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_left_neg,\n    refl,\n  end,\n  add_comm := λ x y,\n  begin\n    induction x,\n    induction y,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_comm,\n    refl,\n    refl,\n  end,\n  add_assoc := λ x y z,\n  begin\n    induction x,\n    induction y,\n    induction z,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_assoc,\n    refl,\n    refl,\n    refl,\n  end, }\n\ninstance : module R (colimit_type F) :=\n{ smul := λ s,\n  begin\n    fapply @quot.lift,\n    { intro x,\n      exact quot.mk _ (smul s x) },\n    { intros x x' r,\n      apply quot.sound,\n      exact relation.smul_1 s _ _ r },\n  end,\n  one_smul := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.one_smul,\n    refl,\n  end,\n  mul_smul := λ s t x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.mul_smul,\n    refl,\n  end,\n  smul_add := λ s x y,\n  begin\n    induction x,\n    induction y,\n    dsimp,\n    apply quot.sound,\n    apply relation.smul_add,\n    refl,\n    refl,\n  end,\n  smul_zero := λ s, begin apply quot.sound, apply relation.smul_zero, end,\n  add_smul := λ s t x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.add_smul,\n    refl,\n  end,\n  zero_smul := λ x,\n  begin\n    induction x,\n    dsimp,\n    apply quot.sound,\n    apply relation.zero_smul,\n    refl,\n  end, }\n\n@[simp] lemma quot_zero : quot.mk setoid.r zero = (0 : colimit_type F) := rfl\n@[simp] lemma quot_neg (x) :\n  quot.mk setoid.r (neg x) = (-(quot.mk setoid.r x) : colimit_type F) := rfl\n@[simp] lemma quot_add (x y) :\n  quot.mk setoid.r (add x y) = ((quot.mk setoid.r x) + (quot.mk setoid.r y) : colimit_type F) := rfl\n@[simp] lemma quot_smul (s x) :\n  quot.mk setoid.r (smul s x) = (s • (quot.mk setoid.r x) : colimit_type F) := rfl\n\n/-- The bundled module giving the colimit of a diagram. -/\ndef colimit : Module R := Module.of R (colimit_type F)\n\n/-- The function from a given module in the diagram to the colimit module. -/\ndef cocone_fun (j : J) (x : F.obj j) : colimit_type F :=\nquot.mk _ (of j x)\n\n/-- The group homomorphism from a given module in the diagram to the colimit module. -/\ndef cocone_morphism (j : J) : F.obj j ⟶ colimit F :=\n{ to_fun := cocone_fun F j,\n  map_smul' := by { intros, apply quot.sound, apply relation.smul, },\n  map_add' := by intros; apply quot.sound; apply relation.add }\n\n@[simp] lemma cocone_naturality {j j' : J} (f : j ⟶ j') :\n  F.map f ≫ (cocone_morphism F j') = cocone_morphism F j :=\nbegin\n  ext,\n  apply quot.sound,\n  apply relation.map,\nend\n\n@[simp] lemma cocone_naturality_components (j j' : J) (f : j ⟶ j') (x : F.obj j):\n  (cocone_morphism F j') (F.map f x) = (cocone_morphism F j) x :=\nby { rw ←cocone_naturality F f, refl }\n\n/-- The cocone over the proposed colimit module. -/\ndef colimit_cocone : cocone F :=\n{ X := colimit F,\n  ι :=\n  { app := cocone_morphism F } }.\n\n/-- The function from the free module on the diagram to the cone point of any other cocone. -/\n@[simp] def desc_fun_lift (s : cocone F) : prequotient F → s.X\n| (of j x)  := (s.ι.app j) x\n| zero      := 0\n| (neg x)   := -(desc_fun_lift x)\n| (add x y) := desc_fun_lift x + desc_fun_lift y\n| (smul s x) := s • (desc_fun_lift x)\n\n/-- The function from the colimit module to the cone point of any other cocone. -/\ndef desc_fun (s : cocone F) : colimit_type F → s.X :=\nbegin\n  fapply quot.lift,\n  { exact desc_fun_lift F s },\n  { intros x y r,\n    induction r; try { dsimp },\n    -- refl\n    { refl },\n    -- symm\n    { exact r_ih.symm },\n    -- trans\n    { exact eq.trans r_ih_h r_ih_k },\n    -- map\n    { simp, },\n    -- zero\n    { simp, },\n    -- neg\n    { simp, },\n    -- add\n    { simp, },\n    -- smul,\n    { simp, },\n    -- neg_1\n    { rw r_ih, },\n    -- add_1\n    { rw r_ih, },\n    -- add_2\n    { rw r_ih, },\n    -- smul_1\n    { rw r_ih, },\n    -- zero_add\n    { rw zero_add, },\n    -- add_zero\n    { rw add_zero, },\n    -- add_left_neg\n    { rw add_left_neg, },\n    -- add_comm\n    { rw add_comm, },\n    -- add_assoc\n    { rw add_assoc, },\n    -- one_smul\n    { rw one_smul, },\n    -- mul_smul\n    { rw mul_smul, },\n    -- smul_add\n    { rw smul_add, },\n    -- smul_zero\n    { rw smul_zero, },\n    -- add_smul\n    { rw add_smul, },\n    -- zero_smul\n    { rw zero_smul, },  }\nend\n\n/-- The group homomorphism from the colimit module to the cone point of any other cocone. -/\ndef desc_morphism (s : cocone F) : colimit F ⟶ s.X :=\n{ to_fun := desc_fun F s,\n  map_smul' := λ s x, by { induction x; refl, },\n  map_add' := λ x y, by { induction x; induction y; refl }, }\n\n/-- Evidence that the proposed colimit is the colimit. -/\ndef colimit_cocone_is_colimit : is_colimit (colimit_cocone F) :=\n{ desc := λ s, desc_morphism F s,\n  uniq' := λ s m w,\n  begin\n    ext,\n    induction x,\n    induction x,\n    { have w' := congr_fun (congr_arg (λ f : F.obj x_j ⟶ s.X, (f : F.obj x_j → s.X)) (w x_j)) x_x,\n      erw w',\n      refl, },\n    { simp *, },\n    { simp *, },\n    { simp *, },\n    { simp *, },\n    refl\n  end }.\n\ninstance has_colimits_Module : has_colimits (Module R) :=\n{ has_colimits_of_shape := λ J 𝒥, by exactI\n  { has_colimit := λ F, has_colimit.mk\n    { cocone := colimit_cocone F,\n      is_colimit := colimit_cocone_is_colimit F } } }\n\nend Module.colimits\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/algebra/category/Module/colimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5, "lm_q1q2_score": 0.32389910217648576}}
{"text": "import .snake_lemma\nimport .exact_seq2\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\n\nvariables {𝒜 : Type*} [category 𝒜] [abelian 𝒜]\nvariables (A₀ B₀ C₀ : 𝒜)\nvariables (A₁ B₁ C₁ : 𝒜)\nvariables (A₂ B₂ C₂ : 𝒜)\nvariables (A₃ B₃ C₃ : 𝒜)\nvariables (f₀ : A₀ ⟶ B₀) (g₀ : B₀ ⟶ C₀)\nvariables (a₀ : A₀ ⟶ A₁) (b₀ : B₀ ⟶ B₁) (c₀ : C₀ ⟶ C₁)\nvariables (f₁ : A₁ ⟶ B₁) (g₁ : B₁ ⟶ C₁)\nvariables (a₁ : A₁ ⟶ A₂) (b₁ : B₁ ⟶ B₂) (c₁ : C₁ ⟶ C₂)\nvariables (f₂ : A₂ ⟶ B₂) (g₂ : B₂ ⟶ C₂)\nvariables (a₂ : A₂ ⟶ A₃) (b₂ : B₂ ⟶ B₃) (c₂ : C₂ ⟶ C₃)\nvariables (f₃ : A₃ ⟶ B₃) (g₃ : B₃ ⟶ C₃)\n\nnamespace category_theory\n\nlocal notation `kernel_map`   := kernel.map _ _ _ _\nlocal notation `cokernel_map` := cokernel.map _ _ _ _\n\nstructure snake : Prop :=\n(row_exact₁ : exact f₁ g₁)\n(row_exact₂ : exact f₂ g₂)\n[row_epi : epi g₁]\n[row_mono : mono f₂]\n(col_exact_a : exact_seq 𝒜 [a₀, a₁, a₂])\n(col_exact_b : exact_seq 𝒜 [b₀, b₁, b₂])\n(col_exact_c : exact_seq 𝒜 [c₀, c₁, c₂])\n[col_mono_a : mono a₀]\n[col_mono_b : mono b₀]\n[col_mono_c : mono c₀]\n[col_epi_a : epi a₂]\n[col_epi_b : epi b₂]\n[col_epi_c : epi c₂]\n(sq_a₀ : a₀ ≫ f₁ = f₀ ≫ b₀)\n(sq_b₀ : b₀ ≫ g₁ = g₀ ≫ c₀)\n(sq_a₁ : a₁ ≫ f₂ = f₁ ≫ b₁)\n(sq_b₁ : b₁ ≫ g₂ = g₁ ≫ c₁)\n(sq_a₂ : a₂ ≫ f₃ = f₂ ≫ b₂)\n(sq_b₂ : b₂ ≫ g₃ = g₂ ≫ c₂)\n\nnamespace snake\n\nlemma mk_of_sequence_hom (sq₁ : a₁ ≫ f₂ = f₁ ≫ b₁) (sq₂ : b₁ ≫ g₂ = g₁ ≫ c₁)\n  (h₁ : exact f₁ g₁) (h₂ : exact f₂ g₂) [epi g₁] [mono f₂] : snake\n  (kernel a₁) (kernel b₁) (kernel c₁)\n  A₁ B₁ C₁\n  A₂ B₂ C₂\n  (cokernel a₁) (cokernel b₁) (cokernel c₁)\n  (kernel_map sq₁) (kernel_map sq₂)\n  (kernel.ι _) (kernel.ι _) (kernel.ι _)\n  f₁ g₁\n  a₁ b₁ c₁\n  f₂ g₂\n  (cokernel.π _) (cokernel.π _) (cokernel.π _)\n  (cokernel_map sq₁) (cokernel_map sq₂) :=\n{ row_exact₁ := h₁,\n  row_exact₂ := h₂,\n  col_exact_a := exact_seq.cons _ _ exact_kernel_ι _ $ (exact_iff_exact_seq _ _).mp (abelian.exact_cokernel _),\n  col_exact_b := exact_seq.cons _ _ exact_kernel_ι _ $ (exact_iff_exact_seq _ _).mp (abelian.exact_cokernel _),\n  col_exact_c := exact_seq.cons _ _ exact_kernel_ι _ $ (exact_iff_exact_seq _ _).mp (abelian.exact_cokernel _),\n  sq_a₀ := (limits.kernel.lift_ι _ _ _).symm,\n  sq_b₀ := (limits.kernel.lift_ι _ _ _).symm,\n  sq_a₁ := sq₁,\n  sq_b₁ := sq₂,\n  sq_a₂ := cokernel.π_desc _ _ _,\n  sq_b₂ := cokernel.π_desc _ _ _ }\n\nvariables\n {A₀ B₀ C₀\n  A₁ B₁ C₁\n  A₂ B₂ C₂\n  A₃ B₃ C₃\n  f₀ g₀ a₀ b₀ c₀ f₁ g₁ a₁ b₁ c₁ f₂ g₂ a₂ b₂ c₂ f₃ g₃}\n\nvariables (S : snake\n  A₀ B₀ C₀\n  A₁ B₁ C₁\n  A₂ B₂ C₂\n  A₃ B₃ C₃\n  f₀ g₀ a₀ b₀ c₀ f₁ g₁ a₁ b₁ c₁ f₂ g₂ a₂ b₂ c₂ f₃ g₃)\n\nvariables\n\ndef snake_diagram : snake_diagram ⥤ 𝒜 :=\nsnake_diagram.mk_functor\n![![A₀, B₀, C₀],\n  ![A₁, B₁, C₁],\n  ![A₂, B₂, C₂],\n  ![A₃, B₃, C₃]]\nf₀ g₀ a₀ b₀ c₀ f₁ g₁ a₁ b₁ c₁ f₂ g₂ a₂ b₂ c₂ f₃ g₃\nS.sq_a₀ S.sq_b₀ S.sq_a₁ S.sq_b₁ S.sq_a₂ S.sq_b₂\n\nlemma is_snake_input : is_snake_input S.snake_diagram :=\n{ row_exact₁ := by { dsimp only [snake_diagram], simpa using S.row_exact₁, },\n  row_exact₂ := by { dsimp only [snake_diagram], simpa using S.row_exact₂ },\n  col_exact₁ := begin\n    intros j,\n    dsimp only [snake_diagram],\n    fin_cases j with [0, 1, 2]; simp; rw exact_iff_exact_seq,\n    exacts [S.col_exact_a.extract 0 2, S.col_exact_b.extract 0 2, S.col_exact_c.extract 0 2],\n  end,\n  col_exact₂ := begin\n    intros j,\n    dsimp only [snake_diagram],\n    fin_cases j with [0, 1, 2]; simp; rw exact_iff_exact_seq,\n    exacts [S.col_exact_a.extract 1 2, S.col_exact_b.extract 1 2, S.col_exact_c.extract 1 2],\n  end,\n  col_mono := begin\n    intros j,\n    dsimp only [snake_diagram],\n    fin_cases j with [0, 1, 2]; simp,\n    exacts [S.col_mono_a, S.col_mono_b, S.col_mono_c],\n  end,\n  col_epi := begin\n    intros j,\n    dsimp only [snake_diagram],\n    fin_cases j with [0, 1, 2]; simp,\n    exacts [S.col_epi_a, S.col_epi_b, S.col_epi_c],\n  end,\n  row_mono := by { dsimp only [snake_diagram], simp, exact S.row_mono },\n  row_epi := by { dsimp only [snake_diagram], simpa using S.row_epi } }\n\ndef snake_input : snake_input 𝒜 := ⟨S.snake_diagram, S.is_snake_input⟩\n\ndef δ : C₀ ⟶ A₃ := S.is_snake_input.δ\n\nlemma six_term_exact_seq : exact_seq 𝒜 [f₀, g₀, S.δ, f₃, g₃] :=\nbegin\n  have := S.is_snake_input.six_term_exact_seq,\n  dsimp only [snake_diagram] at this,\n  simpa only [snake_diagram.mk_functor_map_f0, snake_diagram.mk_functor_map_g0,\n    snake_diagram.mk_functor_map_f3, snake_diagram.mk_functor_map_g3],\nend\n\nend snake\n\nlemma mono_of_exact_of_eq_zero (hfg : exact f₁ g₁) (h : f₁ = 0) : mono g₁ :=\nby rwa [(abelian.tfae_mono A₁ g₁).out 0 2, ← h]\n\nlemma cokernel.map_mono_of_epi_of_mono (sq : f₁ ≫ b₁ = a₁ ≫ f₂)\n  [epi a₁] [mono b₁] [mono f₂] :\n  mono (cokernel.map f₁ f₂ a₁ b₁ sq) :=\nbegin\n  have S := snake.mk_of_sequence_hom A₁ B₁ (cokernel f₁) A₂ B₂ (cokernel f₂)\n    f₁ (cokernel.π _) a₁ b₁ (cokernel.map f₁ f₂ a₁ b₁ sq) f₂ (cokernel.π _) sq.symm (by simp)\n    (abelian.exact_cokernel _) (abelian.exact_cokernel _),\n  apply (S.col_exact_c).pair.mono_of_is_zero,\n  exact (S.six_term_exact_seq.drop 1).pair.is_zero_of_is_zero_is_zero\n    (is_zero_kernel_of_mono _) (is_zero_cokernel_of_epi _),\nend\n\nend category_theory", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/lte/for_mathlib/snake_lemma2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5, "lm_q1q2_score": 0.32389910217648576}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.category.Group.basic\nimport category_theory.single_obj\nimport category_theory.limits.functor_category\nimport category_theory.limits.preserves.basic\nimport category_theory.adjunction.limits\nimport category_theory.monoidal.functor_category\nimport category_theory.monoidal.transport\nimport category_theory.monoidal.rigid.of_equivalence\nimport category_theory.monoidal.rigid.functor_category\nimport category_theory.monoidal.linear\nimport category_theory.monoidal.braided\nimport category_theory.monoidal.types\nimport category_theory.abelian.functor_category\nimport category_theory.abelian.transfer\nimport category_theory.conj\nimport category_theory.linear.functor_category\n\n/-!\n# `Action V G`, the category of actions of a monoid `G` inside some category `V`.\n\nThe prototypical example is `V = Module R`,\nwhere `Action (Module R) G` is the category of `R`-linear representations of `G`.\n\nWe check `Action V G ≌ (single_obj G ⥤ V)`,\nand construct the restriction functors `res {G H : Mon} (f : G ⟶ H) : Action V H ⥤ Action V G`.\n\n* When `V` has (co)limits so does `Action V G`.\n* When `V` is monoidal, braided, or symmetric, so is `Action V G`.\n* When `V` is preadditive, linear, or abelian so is `Action V G`.\n-/\n\nuniverses u v\n\nopen category_theory\nopen category_theory.limits\n\nvariables (V : Type (u+1)) [large_category V]\n\n/--\nAn `Action V G` represents a bundled action of\nthe monoid `G` on an object of some category `V`.\n\nAs an example, when `V = Module R`, this is an `R`-linear representation of `G`,\nwhile when `V = Type` this is a `G`-action.\n-/\n-- Note: this is _not_ a categorical action of `G` on `V`.\nstructure Action (G : Mon.{u}) :=\n(V : V)\n(ρ : G ⟶ Mon.of (End V))\n\nnamespace Action\nvariable {V}\n\n@[simp]\nlemma ρ_one {G : Mon.{u}} (A : Action V G) : A.ρ 1 = 𝟙 A.V :=\nby { rw [monoid_hom.map_one], refl, }\n\n/-- When a group acts, we can lift the action to the group of automorphisms. -/\n@[simps]\ndef ρ_Aut {G : Group.{u}} (A : Action V (Mon.of G)) : G ⟶ Group.of (Aut A.V) :=\n{ to_fun := λ g,\n  { hom := A.ρ g,\n    inv := A.ρ (g⁻¹ : G),\n    hom_inv_id' := ((A.ρ).map_mul (g⁻¹ : G) g).symm.trans (by rw [inv_mul_self, ρ_one]),\n    inv_hom_id' := ((A.ρ).map_mul g (g⁻¹ : G)).symm.trans (by rw [mul_inv_self, ρ_one]), },\n  map_one' := by { ext, exact A.ρ.map_one },\n  map_mul' := λ x y, by { ext, exact A.ρ.map_mul x y }, }\n\nvariable (G : Mon.{u})\n\nsection\n\ninstance inhabited' : inhabited (Action (Type u) G) := ⟨⟨punit, 1⟩⟩\n\n/-- The trivial representation of a group. -/\ndef trivial : Action AddCommGroup G :=\n{ V := AddCommGroup.of punit,\n  ρ := 1, }\n\ninstance : inhabited (Action AddCommGroup G) := ⟨trivial G⟩\nend\n\nvariables {G V}\n\n/--\nA homomorphism of `Action V G`s is a morphism between the underlying objects,\ncommuting with the action of `G`.\n-/\n@[ext]\nstructure hom (M N : Action V G) :=\n(hom : M.V ⟶ N.V)\n(comm' : ∀ g : G, M.ρ g ≫ hom = hom ≫ N.ρ g . obviously)\n\nrestate_axiom hom.comm'\n\nnamespace hom\n\n/-- The identity morphism on a `Action V G`. -/\n@[simps]\ndef id (M : Action V G) : Action.hom M M :=\n{ hom := 𝟙 M.V }\n\ninstance (M : Action V G) : inhabited (Action.hom M M) := ⟨id M⟩\n\n/--\nThe composition of two `Action V G` homomorphisms is the composition of the underlying maps.\n-/\n@[simps]\ndef comp {M N K : Action V G} (p : Action.hom M N) (q : Action.hom N K) :\n  Action.hom M K :=\n{ hom := p.hom ≫ q.hom,\n  comm' := λ g, by rw [←category.assoc, p.comm, category.assoc, q.comm, ←category.assoc] }\n\nend hom\n\ninstance : category (Action V G) :=\n{ hom := λ M N, hom M N,\n  id := λ M, hom.id M,\n  comp := λ M N K f g, hom.comp f g, }\n\n@[simp]\nlemma id_hom (M : Action V G) : (𝟙 M : hom M M).hom = 𝟙 M.V := rfl\n@[simp]\nlemma comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :\n  (f ≫ g : hom M K).hom = f.hom ≫ g.hom :=\nrfl\n\n/-- Construct an isomorphism of `G` actions/representations\nfrom an isomorphism of the the underlying objects,\nwhere the forward direction commutes with the group action. -/\n@[simps]\ndef mk_iso {M N : Action V G} (f : M.V ≅ N.V) (comm : ∀ g : G, M.ρ g ≫ f.hom = f.hom ≫ N.ρ g) :\n  M ≅ N :=\n{ hom :=\n  { hom := f.hom,\n    comm' := comm, },\n  inv :=\n  { hom := f.inv,\n    comm' := λ g, by { have w := comm g =≫ f.inv, simp at w, simp [w], }, }}\n\n@[priority 100]\ninstance is_iso_of_hom_is_iso {M N : Action V G} (f : M ⟶ N) [is_iso f.hom] : is_iso f :=\nby { convert is_iso.of_iso (mk_iso (as_iso f.hom) f.comm), ext, refl, }\n\ninstance is_iso_hom_mk {M N : Action V G} (f : M.V ⟶ N.V) [is_iso f] (w) :\n  @is_iso _ _ M N ⟨f, w⟩ :=\nis_iso.of_iso (mk_iso (as_iso f) w)\n\nnamespace functor_category_equivalence\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef functor : Action V G ⥤ (single_obj G ⥤ V) :=\n{ obj := λ M,\n  { obj := λ _, M.V,\n    map := λ _ _ g, M.ρ g,\n    map_id' := λ _, M.ρ.map_one,\n    map_comp' := λ _ _ _ g h, M.ρ.map_mul h g, },\n  map := λ M N f,\n  { app := λ _, f.hom,\n    naturality' := λ _ _ g, f.comm g, } }\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef inverse : (single_obj G ⥤ V) ⥤ Action V G :=\n{ obj := λ F,\n  { V := F.obj punit.star,\n    ρ :=\n    { to_fun := λ g, F.map g,\n      map_one' := F.map_id punit.star,\n      map_mul' := λ g h, F.map_comp h g, } },\n  map := λ M N f,\n  { hom := f.app punit.star,\n    comm' := λ g, f.naturality g, } }.\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef unit_iso : 𝟭 (Action V G) ≅ functor ⋙ inverse :=\nnat_iso.of_components (λ M, mk_iso ((iso.refl _)) (by tidy)) (by tidy).\n\n/-- Auxilliary definition for `functor_category_equivalence`. -/\n@[simps]\ndef counit_iso : inverse ⋙ functor ≅ 𝟭 (single_obj G ⥤ V) :=\nnat_iso.of_components (λ M, nat_iso.of_components (by tidy) (by tidy)) (by tidy).\n\nend functor_category_equivalence\n\nsection\nopen functor_category_equivalence\n\nvariables (V G)\n\n/--\nThe category of actions of `G` in the category `V`\nis equivalent to the functor category `single_obj G ⥤ V`.\n-/\ndef functor_category_equivalence : Action V G ≌ (single_obj G ⥤ V) :=\n{ functor := functor,\n  inverse := inverse,\n  unit_iso := unit_iso,\n  counit_iso := counit_iso, }\n\nattribute [simps] functor_category_equivalence\n\nlemma functor_category_equivalence.functor_def :\n  (functor_category_equivalence V G).functor = functor_category_equivalence.functor := rfl\n\nlemma functor_category_equivalence.inverse_def :\n  (functor_category_equivalence V G).inverse = functor_category_equivalence.inverse := rfl\n\ninstance [has_finite_products V] : has_finite_products (Action V G) :=\n{ out := λ n, adjunction.has_limits_of_shape_of_equivalence\n    (Action.functor_category_equivalence _ _).functor }\n\ninstance [has_finite_limits V] : has_finite_limits (Action V G) :=\n{ out := λ J _ _, by exactI adjunction.has_limits_of_shape_of_equivalence\n    (Action.functor_category_equivalence _ _).functor }\n\ninstance [has_limits V] : has_limits (Action V G) :=\nadjunction.has_limits_of_equivalence (Action.functor_category_equivalence _ _).functor\n\ninstance [has_colimits V] : has_colimits (Action V G) :=\nadjunction.has_colimits_of_equivalence (Action.functor_category_equivalence _ _).functor\n\nend\n\nsection forget\n\nvariables (V G)\n\n/-- (implementation) The forgetful functor from bundled actions to the underlying objects.\n\nUse the `category_theory.forget` API provided by the `concrete_category` instance below,\nrather than using this directly.\n-/\n@[simps]\ndef forget : Action V G ⥤ V :=\n{ obj := λ M, M.V,\n  map := λ M N f, f.hom, }\n\ninstance : faithful (forget V G) :=\n{ map_injective' := λ X Y f g w, hom.ext _ _ w, }\n\ninstance [concrete_category V] : concrete_category (Action V G) :=\n{ forget := forget V G ⋙ (concrete_category.forget V), }\n\ninstance has_forget_to_V [concrete_category V] : has_forget₂ (Action V G) V :=\n{ forget₂ := forget V G }\n\n/-- The forgetful functor is intertwined by `functor_category_equivalence` with\nevaluation at `punit.star`. -/\ndef functor_category_equivalence_comp_evaluation :\n  (functor_category_equivalence V G).functor ⋙ (evaluation _ _).obj punit.star ≅ forget V G :=\niso.refl _\n\nnoncomputable instance [has_limits V] : limits.preserves_limits (forget V G) :=\nlimits.preserves_limits_of_nat_iso\n  (Action.functor_category_equivalence_comp_evaluation V G)\n\nnoncomputable instance [has_colimits V] : preserves_colimits (forget V G) :=\npreserves_colimits_of_nat_iso\n  (Action.functor_category_equivalence_comp_evaluation V G)\n\n-- TODO construct categorical images?\n\nend forget\n\nlemma iso.conj_ρ {M N : Action V G} (f : M ≅ N) (g : G) :\n   N.ρ g = (((forget V G).map_iso f).conj (M.ρ g)) :=\nby { rw [iso.conj_apply, iso.eq_inv_comp], simp [f.hom.comm'] }\n\nsection has_zero_morphisms\nvariables [has_zero_morphisms V]\n\ninstance : has_zero_morphisms (Action V G) :=\n{ has_zero := λ X Y, ⟨⟨0, by { intro g, simp }⟩⟩,\n  comp_zero' := λ P Q f R, by { ext1, simp },\n  zero_comp' := λ P Q R f, by { ext1, simp }, }\n\ninstance forget_preserves_zero_morphisms : functor.preserves_zero_morphisms (forget V G) := {}\ninstance forget₂_preserves_zero_morphisms [concrete_category V] :\n  functor.preserves_zero_morphisms (forget₂ (Action V G) V) := {}\ninstance functor_category_equivalence_preserves_zero_morphisms :\n  functor.preserves_zero_morphisms (functor_category_equivalence V G).functor := {}\n\nend has_zero_morphisms\n\nsection preadditive\nvariables [preadditive V]\n\ninstance : preadditive (Action V G) :=\n{ hom_group := λ X Y,\n  { zero := ⟨0, by simp⟩,\n    add := λ f g, ⟨f.hom + g.hom, by simp [f.comm, g.comm]⟩,\n    neg := λ f, ⟨-f.hom, by simp [f.comm]⟩,\n    zero_add := by { intros, ext, exact zero_add _, },\n    add_zero := by { intros, ext, exact add_zero _, },\n    add_assoc := by { intros, ext, exact add_assoc _ _ _, },\n    add_left_neg := by { intros, ext, exact add_left_neg _, },\n    add_comm := by { intros, ext, exact add_comm _ _, }, },\n  add_comp' := by { intros, ext, exact preadditive.add_comp _ _ _ _ _ _, },\n  comp_add' := by { intros, ext, exact preadditive.comp_add _ _ _ _ _ _, }, }\n\ninstance forget_additive :\n  functor.additive (forget V G) := {}\ninstance forget₂_additive [concrete_category V] :\n  functor.additive (forget₂ (Action V G) V) := {}\ninstance functor_category_equivalence_additive :\n  functor.additive (functor_category_equivalence V G).functor := {}\n\n@[simp] lemma zero_hom {X Y : Action V G} : (0 : X ⟶ Y).hom = 0 := rfl\n@[simp] lemma neg_hom {X Y : Action V G} (f : X ⟶ Y) : (-f).hom = -f.hom := rfl\n@[simp] lemma add_hom {X Y : Action V G} (f g : X ⟶ Y) : (f + g).hom = f.hom + g.hom := rfl\n@[simp] lemma sum_hom {X Y : Action V G} {ι : Type*} (f : ι → (X ⟶ Y)) (s : finset ι) :\n  (s.sum f).hom = s.sum (λ i, (f i).hom) := (forget V G).map_sum f s\n\nend preadditive\n\nsection linear\nvariables [preadditive V] {R : Type*} [semiring R] [linear R V]\n\ninstance : linear R (Action V G) :=\n{ hom_module := λ X Y,\n  { smul := λ r f, ⟨r • f.hom, by simp [f.comm]⟩,\n    one_smul := by { intros, ext, exact one_smul _ _, },\n    smul_zero := by { intros, ext, exact smul_zero _, },\n    zero_smul := by { intros, ext, exact zero_smul _ _, },\n    add_smul := by { intros, ext, exact add_smul _ _ _, },\n    smul_add := by { intros, ext, exact smul_add _ _ _, },\n    mul_smul := by { intros, ext, exact mul_smul _ _ _, }, },\n  smul_comp' := by { intros, ext, exact linear.smul_comp _ _ _ _ _ _, },\n  comp_smul' := by { intros, ext, exact linear.comp_smul _ _ _ _ _ _, }, }\n\ninstance forget_linear :\n  functor.linear R (forget V G) := {}\ninstance forget₂_linear [concrete_category V] :\n  functor.linear R (forget₂ (Action V G) V) := {}\ninstance functor_category_equivalence_linear :\n  functor.linear R (functor_category_equivalence V G).functor := {}\n\n@[simp] lemma smul_hom {X Y : Action V G} (r : R) (f : X ⟶ Y) : (r • f).hom = r • f.hom := rfl\n\nend linear\n\nsection abelian\n/-- Auxilliary construction for the `abelian (Action V G)` instance. -/\ndef abelian_aux : Action V G ≌ (ulift.{u} (single_obj G) ⥤ V) :=\n(functor_category_equivalence V G).trans (equivalence.congr_left ulift.equivalence)\n\nnoncomputable instance [abelian V] : abelian (Action V G) :=\nabelian_of_equivalence abelian_aux.functor\n\nend abelian\n\nsection monoidal\nvariables [monoidal_category V]\n\ninstance : monoidal_category (Action V G) :=\nmonoidal.transport (Action.functor_category_equivalence _ _).symm\n\n@[simp] lemma tensor_unit_V : (𝟙_ (Action V G)).V = 𝟙_ V := rfl\n@[simp] \n\n/-- Given an object `X` isomorphic to the tensor unit of `V`, `X` equipped with the trivial action\nis isomorphic to the tensor unit of `Action V G`. -/\ndef tensor_unit_iso {X : V} (f : 𝟙_ V ≅ X) :\n  𝟙_ (Action V G) ≅ Action.mk X 1 :=\nAction.mk_iso f (λ g, by simp only [monoid_hom.one_apply, End.one_def, category.id_comp f.hom,\n  tensor_unit_rho, category.comp_id])\n\nvariables (V G)\n\n/-- When `V` is monoidal the forgetful functor `Action V G` to `V` is monoidal. -/\n@[simps]\ndef forget_monoidal : monoidal_functor (Action V G) V :=\n{ ε := 𝟙 _,\n  μ := λ X Y, 𝟙 _,\n  ..Action.forget _ _, }\n\ninstance forget_monoidal_faithful : faithful (forget_monoidal V G).to_functor :=\nby { change faithful (forget V G), apply_instance, }\n\nsection\nvariables [braided_category V]\n\ninstance : braided_category (Action V G) :=\nbraided_category_of_faithful (forget_monoidal V G) (λ X Y, mk_iso (β_ _ _) (by tidy)) (by tidy)\n\n/-- When `V` is braided the forgetful functor `Action V G` to `V` is braided. -/\n@[simps]\ndef forget_braided : braided_functor (Action V G) V :=\n{ ..forget_monoidal _ _, }\n\ninstance forget_braided_faithful : faithful (forget_braided V G).to_functor :=\nby { change faithful (forget V G), apply_instance, }\n\nend\n\ninstance [symmetric_category V] : symmetric_category (Action V G) :=\nsymmetric_category_of_faithful (forget_braided V G)\n\nsection\nvariables [preadditive V] [monoidal_preadditive V]\n\nlocal attribute [simp] monoidal_preadditive.tensor_add monoidal_preadditive.add_tensor\n\ninstance : monoidal_preadditive (Action V G) := {}\n\nvariables {R : Type*} [semiring R] [linear R V] [monoidal_linear R V]\n\ninstance : monoidal_linear R (Action V G) := {}\n\nend\n\nvariables (V G)\nnoncomputable theory\n\n/-- Upgrading the functor `Action V G ⥤ (single_obj G ⥤ V)` to a monoidal functor. -/\ndef functor_category_monoidal_equivalence : monoidal_functor (Action V G) (single_obj G ⥤ V) :=\nmonoidal.from_transported (Action.functor_category_equivalence _ _).symm\n\ninstance : is_equivalence ((functor_category_monoidal_equivalence V G).to_functor) :=\nby { change is_equivalence (Action.functor_category_equivalence _ _).functor, apply_instance, }\n\n@[simp] lemma functor_category_monoidal_equivalence.μ_app (A B : Action V G) :\n  ((functor_category_monoidal_equivalence V G).μ A B).app punit.star = 𝟙 _ :=\nbegin\n  dunfold functor_category_monoidal_equivalence,\n  simp only [monoidal.from_transported_to_lax_monoidal_functor_μ],\n  show (𝟙 A.V ⊗ 𝟙 B.V) ≫ 𝟙 (A.V ⊗ B.V) ≫ (𝟙 A.V ⊗ 𝟙 B.V) = 𝟙 (A.V ⊗ B.V),\n  simp only [monoidal_category.tensor_id, category.comp_id],\nend\n\n@[simp] lemma functor_category_monoidal_equivalence.μ_iso_inv_app (A B : Action V G) :\n  ((functor_category_monoidal_equivalence V G).μ_iso A B).inv.app punit.star = 𝟙 _ :=\nbegin\n  rw [←nat_iso.app_inv, ←is_iso.iso.inv_hom],\n  refine is_iso.inv_eq_of_hom_inv_id _,\n  rw [category.comp_id, nat_iso.app_hom, monoidal_functor.μ_iso_hom,\n    functor_category_monoidal_equivalence.μ_app],\nend\n\n@[simp] lemma functor_category_monoidal_equivalence.ε_app :\n  (functor_category_monoidal_equivalence V G).ε.app punit.star = 𝟙 _ :=\nbegin\n  dunfold functor_category_monoidal_equivalence,\n  simp only [monoidal.from_transported_to_lax_monoidal_functor_ε],\n  show 𝟙 (monoidal_category.tensor_unit V) ≫ _ = 𝟙 (monoidal_category.tensor_unit V),\n  rw [nat_iso.is_iso_inv_app, category.id_comp],\n  exact is_iso.inv_id,\nend\n\n@[simp] lemma functor_category_monoidal_equivalence.inv_counit_app_hom (A : Action V G) :\n  ((functor_category_monoidal_equivalence _ _).inv.adjunction.counit.app A).hom = 𝟙 _ :=\nrfl\n\n@[simp] lemma functor_category_monoidal_equivalence.counit_app (A : single_obj G ⥤ V) :\n  ((functor_category_monoidal_equivalence _ _).adjunction.counit.app A).app punit.star = 𝟙 _ := rfl\n\n@[simp] lemma functor_category_monoidal_equivalence.inv_unit_app_app\n  (A : single_obj G ⥤ V) :\n  ((functor_category_monoidal_equivalence _ _).inv.adjunction.unit.app A).app\n  punit.star = 𝟙 _ := rfl\n\n@[simp] lemma functor_category_monoidal_equivalence.unit_app_hom (A : Action V G) :\n  ((functor_category_monoidal_equivalence _ _).adjunction.unit.app A).hom = 𝟙 _ :=\nrfl\n\n@[simp] lemma functor_category_monoidal_equivalence.functor_map {A B : Action V G} (f : A ⟶ B) :\n  (functor_category_monoidal_equivalence _ _).map f\n    = functor_category_equivalence.functor.map f := rfl\n\n@[simp] lemma functor_category_monoidal_equivalence.inverse_map\n  {A B : single_obj G ⥤ V} (f : A ⟶ B) :\n  (functor_category_monoidal_equivalence _ _).inv.map f\n    = functor_category_equivalence.inverse.map f := rfl\n\nvariables (H : Group.{u})\n\ninstance [right_rigid_category V] : right_rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change right_rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is right rigid, so is `Action V G`. -/\ninstance [right_rigid_category V] : right_rigid_category (Action V H) :=\nright_rigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\ninstance [left_rigid_category V] : left_rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change left_rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is left rigid, so is `Action V G`. -/\ninstance [left_rigid_category V] : left_rigid_category (Action V H) :=\nleft_rigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\ninstance [rigid_category V] : rigid_category (single_obj (H : Mon.{u}) ⥤ V) :=\nby { change rigid_category (single_obj H ⥤ V), apply_instance }\n\n/-- If `V` is rigid, so is `Action V G`. -/\ninstance [rigid_category V] : rigid_category (Action V H) :=\nrigid_category_of_equivalence (functor_category_monoidal_equivalence V _)\n\nvariables {V H} (X : Action V H)\n\n@[simp] lemma right_dual_V [right_rigid_category V] : (Xᘁ).V = (X.V)ᘁ := rfl\n\n@[simp] lemma left_dual_V [left_rigid_category V] : (ᘁX).V = ᘁ(X.V) := rfl\n\n@[simp] lemma right_dual_ρ [right_rigid_category V] (h : H) : (Xᘁ).ρ h = (X.ρ (h⁻¹ : H))ᘁ :=\nby { rw ←single_obj.inv_as_inv, refl }\n\n@[simp] lemma left_dual_ρ [left_rigid_category V] (h : H) : (ᘁX).ρ h = ᘁ(X.ρ (h⁻¹ : H)) :=\nby { rw ←single_obj.inv_as_inv, refl }\n\nend monoidal\n\n/-- Actions/representations of the trivial group are just objects in the ambient category. -/\ndef Action_punit_equivalence : Action V (Mon.of punit) ≌ V :=\n{ functor := forget V _,\n  inverse :=\n  { obj := λ X, ⟨X, 1⟩,\n    map := λ X Y f, ⟨f, λ ⟨⟩, by simp⟩, },\n  unit_iso := nat_iso.of_components (λ X, mk_iso (iso.refl _) (λ ⟨⟩, by simpa using ρ_one X))\n    (by tidy),\n  counit_iso := nat_iso.of_components (λ X, iso.refl _) (by tidy), }\n\nvariables (V)\n/--\nThe \"restriction\" functor along a monoid homomorphism `f : G ⟶ H`,\ntaking actions of `H` to actions of `G`.\n\n(This makes sense for any homomorphism, but the name is natural when `f` is a monomorphism.)\n-/\n@[simps]\ndef res {G H : Mon} (f : G ⟶ H) : Action V H ⥤ Action V G :=\n{ obj := λ M,\n  { V := M.V,\n    ρ := f ≫ M.ρ },\n  map := λ M N p,\n  { hom := p.hom,\n    comm' := λ g, p.comm (f g) } }\n\n/--\nThe natural isomorphism from restriction along the identity homomorphism to\nthe identity functor on `Action V G`.\n-/\ndef res_id {G : Mon} : res V (𝟙 G) ≅ 𝟭 (Action V G) :=\nnat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy)\n\nattribute [simps] res_id\n\n/--\nThe natural isomorphism from the composition of restrictions along homomorphisms\nto the restriction along the composition of homomorphism.\n-/\ndef res_comp {G H K : Mon} (f : G ⟶ H) (g : H ⟶ K) : res V g ⋙ res V f ≅ res V (f ≫ g) :=\nnat_iso.of_components (λ M, mk_iso (iso.refl _) (by tidy)) (by tidy)\n\nattribute [simps] res_comp\n\n-- TODO promote `res` to a pseudofunctor from\n-- the locally discrete bicategory constructed from `Monᵒᵖ` to `Cat`, sending `G` to `Action V G`.\n\nvariables {G} {H : Mon.{u}} (f : G ⟶ H)\n\ninstance res_additive [preadditive V] : (res V f).additive := {}\n\nvariables {R : Type*} [semiring R]\n\ninstance res_linear [preadditive V] [linear R V] : (res V f).linear R := {}\n\n/-- Bundles a type `H` with a multiplicative action of `G` as an `Action`. -/\ndef of_mul_action (G H : Type u) [monoid G] [mul_action G H] : Action (Type u) (Mon.of G) :=\n{ V := H,\n  ρ := @mul_action.to_End_hom _ _ _ (by assumption) }\n\n@[simp] lemma of_mul_action_apply {G H : Type u} [monoid G] [mul_action G H] (g : G) (x : H) :\n  (of_mul_action G H).ρ g x = (g • x : H) :=\nrfl\n\n/-- Given a family `F` of types with `G`-actions, this is the limit cone demonstrating that the\nproduct of `F` as types is a product in the category of `G`-sets. -/\ndef of_mul_action_limit_cone {ι : Type v} (G : Type (max v u)) [monoid G]\n  (F : ι → Type (max v u)) [Π i : ι, mul_action G (F i)] :\n  limit_cone (discrete.functor (λ i : ι, Action.of_mul_action G (F i))) :=\n{ cone :=\n  { X := Action.of_mul_action G (Π i : ι, F i),\n    π :=\n    { app := λ i, ⟨λ x, x i.as, λ g, by ext; refl⟩,\n      naturality' := λ i j x,\n      begin\n        ext,\n        discrete_cases,\n        cases x,\n        congr\n      end } },\n  is_limit :=\n  { lift := λ s,\n    { hom := λ x i, (s.π.app ⟨i⟩).hom x,\n      comm' := λ g,\n      begin\n        ext x j,\n        dsimp,\n        exact congr_fun ((s.π.app ⟨j⟩).comm g) x,\n      end },\n    fac' := λ s j,\n    begin\n      ext,\n      dsimp,\n      congr,\n      rw discrete.mk_as,\n    end,\n    uniq' := λ s f h,\n    begin\n      ext x j,\n      dsimp at *,\n      rw ←h ⟨j⟩,\n      congr,\n    end } }\n\n/-- The `G`-set `G`, acting on itself by left multiplication. -/\n@[simps] def left_regular (G : Type u) [monoid G] : Action (Type u) (Mon.of G) :=\nAction.of_mul_action G G\n\n/-- The `G`-set `Gⁿ`, acting on itself by left multiplication. -/\n@[simps] def diagonal (G : Type u) [monoid G] (n : ℕ) : Action (Type u) (Mon.of G) :=\nAction.of_mul_action G (fin n → G)\n\n/-- We have `fin 1 → G ≅ G` as `G`-sets, with `G` acting by left multiplication. -/\ndef diagonal_one_iso_left_regular (G : Type u) [monoid G] :\n  diagonal G 1 ≅ left_regular G := Action.mk_iso (equiv.fun_unique _ _).to_iso (λ g, rfl)\n\n/-- Given `X : Action (Type u) (Mon.of G)` for `G` a group, then `G × X` (with `G` acting as left\nmultiplication on the first factor and by `X.ρ` on the second) is isomorphic as a `G`-set to\n`G × X` (with `G` acting as left multiplication on the first factor and trivially on the second).\nThe isomorphism is given by `(g, x) ↦ (g, g⁻¹ • x)`. -/\n@[simps] def left_regular_tensor_iso (G : Type u) [group G]\n  (X : Action (Type u) (Mon.of G)) :\n  left_regular G ⊗ X ≅ left_regular G ⊗ Action.mk X.V 1 :=\n{ hom :=\n  { hom := λ g, ⟨g.1, (X.ρ (g.1⁻¹ : G) g.2 : X.V)⟩,\n    comm' := λ g, funext $ λ x, prod.ext rfl $\n      show (X.ρ ((g * x.1)⁻¹ : G) * X.ρ g) x.2 = _,\n      by simpa only [mul_inv_rev, ←X.ρ.map_mul, inv_mul_cancel_right] },\n  inv :=\n  { hom := λ g, ⟨g.1, X.ρ g.1 g.2⟩,\n    comm' := λ g, funext $ λ x, prod.ext rfl $\n      by simpa only [tensor_rho, types_comp_apply, tensor_apply, left_regular_ρ_apply, map_mul] },\n  hom_inv_id' := hom.ext _ _ (funext $ λ x, prod.ext rfl $\n    show (X.ρ x.1 * X.ρ (x.1⁻¹ : G)) x.2 = _,\n      by simpa only [←X.ρ.map_mul, mul_inv_self, X.ρ.map_one]),\n  inv_hom_id' := hom.ext _ _ (funext $ λ x, prod.ext rfl $\n    show (X.ρ (x.1⁻¹ : G) * X.ρ x.1) _ = _,\n      by simpa only [←X.ρ.map_mul, inv_mul_self, X.ρ.map_one]) }\n\n/-- The natural isomorphism of `G`-sets `Gⁿ⁺¹ ≅ G × Gⁿ`, where `G` acts by left multiplication on\neach factor. -/\n@[simps] def diagonal_succ (G : Type u) [monoid G] (n : ℕ) :\n  diagonal G (n + 1) ≅ left_regular G ⊗ diagonal G n :=\nmk_iso (equiv.pi_fin_succ_above_equiv _ 0).to_iso (λ g, rfl)\n\nend Action\n\nnamespace category_theory.functor\n\nvariables {V} {W : Type (u+1)} [large_category W]\n\n/-- A functor between categories induces a functor between\nthe categories of `G`-actions within those categories. -/\n@[simps]\ndef map_Action (F : V ⥤ W) (G : Mon.{u}) : Action V G ⥤ Action W G :=\n{ obj := λ M,\n  { V := F.obj M.V,\n    ρ :=\n    { to_fun := λ g, F.map (M.ρ g),\n      map_one' := by simp only [End.one_def, Action.ρ_one, F.map_id],\n      map_mul' := λ g h, by simp only [End.mul_def, F.map_comp, map_mul], }, },\n  map := λ M N f,\n  { hom := F.map f.hom,\n    comm' := λ g, by { dsimp, rw [←F.map_comp, f.comm, F.map_comp], }, },\n  map_id' := λ M, by { ext, simp only [Action.id_hom, F.map_id], },\n  map_comp' := λ M N P f g, by { ext, simp only [Action.comp_hom, F.map_comp], }, }\n\nvariables (F : V ⥤ W) (G : Mon.{u}) [preadditive V] [preadditive W]\n\ninstance map_Action_preadditive [F.additive] : (F.map_Action G).additive := {}\n\nvariables {R : Type*} [semiring R] [category_theory.linear R V] [category_theory.linear R W]\n\ninstance map_Action_linear [F.additive] [F.linear R] : (F.map_Action G).linear R := {}\n\nend category_theory.functor\n\nnamespace category_theory.monoidal_functor\n\nopen Action\nvariables {V} {W : Type (u+1)} [large_category W] [monoidal_category V] [monoidal_category W]\n  (F : monoidal_functor V W) (G : Mon.{u})\n\n/-- A monoidal functor induces a monoidal functor between\nthe categories of `G`-actions within those categories. -/\n@[simps] def map_Action :\n  monoidal_functor (Action V G) (Action W G) :=\n{ ε :=\n  { hom := F.ε,\n    comm' := λ g,\n    by { dsimp, erw [category.id_comp, category_theory.functor.map_id, category.comp_id], }, },\n  μ := λ X Y,\n  { hom := F.μ X.V Y.V,\n    comm' := λ g, F.to_lax_monoidal_functor.μ_natural (X.ρ g) (Y.ρ g), },\n  ε_is_iso := by apply_instance,\n  μ_is_iso := by apply_instance,\n  μ_natural' := by { intros, ext, dsimp, simp, },\n  associativity' := by { intros, ext, dsimp, simp, dsimp, simp, }, -- See note [dsimp, simp].\n  left_unitality' := by { intros, ext, dsimp, simp, dsimp, simp, },\n  right_unitality' := by { intros, ext, dsimp, simp, dsimp, simp, },\n  ..F.to_functor.map_Action G, }\n\n@[simp] lemma map_Action_ε_inv_hom :\n  (inv (F.map_Action G).ε).hom = inv F.ε :=\nbegin\n  ext,\n  simp only [←F.map_Action_to_lax_monoidal_functor_ε_hom G, ←Action.comp_hom,\n    is_iso.hom_inv_id, id_hom],\nend\n\n@[simp] lemma map_Action_μ_inv_hom (X Y : Action V G) :\n  (inv ((F.map_Action G).μ X Y)).hom = inv (F.μ X.V Y.V) :=\nbegin\n  ext,\n  simpa only [←F.map_Action_to_lax_monoidal_functor_μ_hom G, ←Action.comp_hom,\n    is_iso.hom_inv_id, id_hom],\nend\n\nend category_theory.monoidal_functor\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/representation_theory/Action.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.32388227054517593}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport category_theory.limits.filtered\nimport category_theory.limits.shapes.finite_products\nimport category_theory.discrete_category\nimport tactic.equiv_rw\n\n/-!\n# Limits in `C` give colimits in `Cᵒᵖ`.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe also give special cases for (co)products,\n(co)equalizers, and pullbacks / pushouts.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C]\nvariables {J : Type u₂} [category.{v₂} J]\n\n/-- Turn a colimit for `F : J ⥤ C` into a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cocone_op (F : J ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit c.op :=\n{ lift := λ s, (hc.desc s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F : J ⥤ C` into a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_op (F : J ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit c.op :=\n{ desc := λ s, (hc.lift s.unop).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : J ⥤ Cᵒᵖ` into a limit for `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_left_op_of_cocone (F : J ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_left_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_left_op s)).unop,\n  fac' :=  λ s j, quiver.hom.op_inj $ by simpa only [cone_left_op_of_cocone_π_app, op_comp,\n    quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac, cocone_of_cone_left_op_ι_app] using w (op j)\n  end }\n\n/-- Turn a limit of `F : J ⥤ Cᵒᵖ` into a colimit of `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_left_op_of_cone (F : J ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_left_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_left_op s)).unop,\n  fac' := λ s j, quiver.hom.op_inj $ by simpa only [cocone_left_op_of_cone_ι_app, op_comp,\n    quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac, cone_of_cocone_left_op_π_app] using w (op j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ C` into a limit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_right_op_of_cocone (F : Jᵒᵖ ⥤ C) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_right_op_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F : Jᵒᵖ ⥤ C` into a colimit for `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_right_op_of_cone (F : Jᵒᵖ ⥤ C) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_right_op_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_right_op s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F.unop : J ⥤ C`. -/\n@[simps] def is_limit_cone_unop_of_cocone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F} (hc : is_colimit c) :\n  is_limit (cone_unop_of_cocone c) :=\n{ lift := λ s, (hc.desc (cocone_of_cone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit of `F : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit of `F.unop : J ⥤ C`. -/\n@[simps] def is_colimit_cocone_unop_of_cone (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F} (hc : is_limit c) :\n  is_colimit (cocone_unop_of_cone c) :=\n{ desc := λ s, (hc.lift (cone_of_cocone_unop s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a limit for `F : J ⥤ C`. -/\n@[simps] def is_limit_cocone_unop (F : J ⥤ C) {c : cocone F.op} (hc : is_colimit c) :\n  is_limit c.unop :=\n{ lift := λ s, (hc.desc s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (unop j)\n  end }\n\n/-- Turn a limit for `F.op : Jᵒᵖ ⥤ Cᵒᵖ` into a colimit for `F : J ⥤ C`. -/\n@[simps] def is_colimit_cone_unop (F : J ⥤ C) {c : cone F.op} (hc : is_limit c) :\n  is_colimit c.unop :=\n{ desc := λ s, (hc.lift s.op).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.left_op : Jᵒᵖ ⥤ C` into a limit for `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_left_op (F : J ⥤ Cᵒᵖ) {c : cocone F.left_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_left_op c) :=\n{ lift := λ s, (hc.desc (cocone_left_op_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cone_of_cocone_left_op_π_app, unop_comp,\n    quiver.hom.unop_op, is_colimit.fac, cocone_left_op_of_cone_ι_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac, cone_of_cocone_left_op_π_app] using w (unop j)\n  end }\n\n/-- Turn a limit of `F.left_op : Jᵒᵖ ⥤ C` into a colimit of `F : J ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cocone_of_cone_left_op (F : J ⥤ Cᵒᵖ) {c : cone (F.left_op)}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_left_op c) :=\n{ desc := λ s, (hc.lift (cone_left_op_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj $ by simpa only [cocone_of_cone_left_op_ι_app, unop_comp,\n    quiver.hom.unop_op, is_limit.fac, cone_left_op_of_cocone_π_app],\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac, cocone_of_cone_left_op_ι_app] using w (unop j)\n  end }\n\n/-- Turn a colimit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_limit_cone_of_cocone_right_op (F : Jᵒᵖ ⥤ C) {c : cocone F.right_op}\n  (hc : is_colimit c) : is_limit (cone_of_cocone_right_op c) :=\n{ lift := λ s, (hc.desc (cocone_right_op_of_cone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.right_op : J ⥤ Cᵒᵖ` into a limit for `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def is_colimit_cocone_of_cone_right_op (F : Jᵒᵖ ⥤ C) {c : cone F.right_op}\n  (hc : is_limit c) : is_colimit (cocone_of_cone_right_op c) :=\n{ desc := λ s, (hc.lift (cone_right_op_of_cocone s)).unop,\n  fac' := λ s j, quiver.hom.op_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.op_inj (hc.hom_ext (λ j, quiver.hom.unop_inj _)),\n    simpa only [quiver.hom.op_unop, is_limit.fac] using w (op j)\n  end }\n\n/-- Turn a colimit for `F.unop : J ⥤ C` into a limit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_limit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cocone F.unop} (hc : is_colimit c) :\n  is_limit (cone_of_cocone_unop c) :=\n{ lift := λ s, (hc.desc (cocone_unop_of_cone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_colimit.fac] using w (op j)\n  end }\n\n/-- Turn a limit for `F.unop : J ⥤ C` into a colimit for `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def is_colimit_cone_of_cocone_unop (F : Jᵒᵖ ⥤ Cᵒᵖ) {c : cone F.unop} (hc : is_limit c) :\n  is_colimit (cocone_of_cone_unop c) :=\n{ desc := λ s, (hc.lift (cone_unop_of_cocone s)).op,\n  fac' := λ s j, quiver.hom.unop_inj (by simpa),\n  uniq' := λ s m w,\n  begin\n    refine quiver.hom.unop_inj (hc.hom_ext (λ j, quiver.hom.op_inj _)),\n    simpa only [quiver.hom.unop_op, is_limit.fac] using w (op j)\n  end }\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a colimit, we can construct a limit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_limit_of_has_colimit_left_op (F : J ⥤ Cᵒᵖ) [has_colimit F.left_op] : has_limit F :=\nhas_limit.mk\n{ cone := cone_of_cocone_left_op (colimit.cocone F.left_op),\n  is_limit := is_limit_cone_of_cocone_left_op _ (colimit.is_colimit _) }\n\nlemma has_limit_of_has_colimit_op (F : J ⥤ C) [has_colimit F.op] : has_limit F :=\nhas_limit.mk\n{ cone := (colimit.cocone F.op).unop,\n  is_limit := is_limit_cocone_unop _ (colimit.is_colimit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\nlemma has_limits_of_shape_op_of_has_colimits_of_shape [has_colimits_of_shape Jᵒᵖ C] :\n  has_limits_of_shape J Cᵒᵖ :=\n{ has_limit := λ F, has_limit_of_has_colimit_left_op F }\n\nlemma has_limits_of_shape_of_has_colimits_of_shape_op [has_colimits_of_shape Jᵒᵖ Cᵒᵖ] :\n  has_limits_of_shape J C :=\n{ has_limit := λ F, has_limit_of_has_colimit_op F }\n\nlocal attribute [instance] has_limits_of_shape_op_of_has_colimits_of_shape\n\n/--\nIf `C` has colimits, we can construct limits for `Cᵒᵖ`.\n-/\ninstance has_limits_op_of_has_colimits [has_colimits C] : has_limits Cᵒᵖ := ⟨infer_instance⟩\n\nlemma has_limits_of_has_colimits_op [has_colimits Cᵒᵖ] : has_limits C :=\n{ has_limits_of_shape := λ J hJ, by exactI has_limits_of_shape_of_has_colimits_of_shape_op }\n\ninstance has_cofiltered_limits_op_of_has_filtered_colimits\n  [has_filtered_colimits_of_size.{v₂ u₂} C] : has_cofiltered_limits_of_size.{v₂ u₂} Cᵒᵖ :=\n{ has_limits_of_shape := λ I hI₁ hI₂, by exactI has_limits_of_shape_op_of_has_colimits_of_shape }\n\nlemma has_cofiltered_limits_of_has_filtered_colimits_op\n  [has_filtered_colimits_of_size.{v₂ u₂} Cᵒᵖ] : has_cofiltered_limits_of_size.{v₂ u₂} C :=\n{ has_limits_of_shape := λ I hI₂ hI₂, by exactI has_limits_of_shape_of_has_colimits_of_shape_op }\n\n/--\nIf `F.left_op : Jᵒᵖ ⥤ C` has a limit, we can construct a colimit for `F : J ⥤ Cᵒᵖ`.\n-/\nlemma has_colimit_of_has_limit_left_op (F : J ⥤ Cᵒᵖ) [has_limit F.left_op] : has_colimit F :=\nhas_colimit.mk\n{ cocone := cocone_of_cone_left_op (limit.cone F.left_op),\n  is_colimit := is_colimit_cocone_of_cone_left_op _ (limit.is_limit _) }\n\nlemma has_colimit_of_has_limit_op (F : J ⥤ C) [has_limit F.op] : has_colimit F :=\nhas_colimit.mk\n{ cocone := (limit.cone F.op).unop,\n  is_colimit := is_colimit_cone_unop _ (limit.is_limit _) }\n\n/--\nIf `C` has colimits of shape `Jᵒᵖ`, we can construct limits in `Cᵒᵖ` of shape `J`.\n-/\ninstance has_colimits_of_shape_op_of_has_limits_of_shape [has_limits_of_shape Jᵒᵖ C] :\n  has_colimits_of_shape J Cᵒᵖ :=\n{ has_colimit := λ F, has_colimit_of_has_limit_left_op F }\n\nlemma has_colimits_of_shape_of_has_limits_of_shape_op [has_limits_of_shape Jᵒᵖ Cᵒᵖ] :\n  has_colimits_of_shape J C :=\n{ has_colimit := λ F, has_colimit_of_has_limit_op F }\n\n/--\nIf `C` has limits, we can construct colimits for `Cᵒᵖ`.\n-/\ninstance has_colimits_op_of_has_limits [has_limits C] : has_colimits Cᵒᵖ := ⟨infer_instance⟩\n\nlemma has_colimits_of_has_limits_op [has_limits Cᵒᵖ] : has_colimits C :=\n{ has_colimits_of_shape := λ J hJ, by exactI has_colimits_of_shape_of_has_limits_of_shape_op }\n\ninstance has_filtered_colimits_op_of_has_cofiltered_limits\n  [has_cofiltered_limits_of_size.{v₂ u₂} C] : has_filtered_colimits_of_size.{v₂ u₂} Cᵒᵖ :=\n{ has_colimits_of_shape := λ I hI₁ hI₂, by exactI infer_instance }\n\nlemma has_filtered_colimits_of_has_cofiltered_limits_op\n  [has_cofiltered_limits_of_size.{v₂ u₂} Cᵒᵖ] : has_filtered_colimits_of_size.{v₂ u₂} C :=\n{ has_colimits_of_shape := λ I hI₁ hI₂, by exactI has_colimits_of_shape_of_has_limits_of_shape_op }\n\nvariables (X : Type v₂)\n/--\nIf `C` has products indexed by `X`, then `Cᵒᵖ` has coproducts indexed by `X`.\n-/\ninstance has_coproducts_of_shape_opposite [has_products_of_shape X C] :\n  has_coproducts_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape (discrete X)ᵒᵖ C :=\n    has_limits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\nlemma has_coproducts_of_shape_of_opposite [has_products_of_shape X Cᵒᵖ] :\n  has_coproducts_of_shape X C :=\nbegin\n  haveI : has_limits_of_shape (discrete X)ᵒᵖ Cᵒᵖ :=\n    has_limits_of_shape_of_equivalence (discrete.opposite X).symm,\n  exact has_colimits_of_shape_of_has_limits_of_shape_op\nend\n\n/--\nIf `C` has coproducts indexed by `X`, then `Cᵒᵖ` has products indexed by `X`.\n-/\ninstance has_products_of_shape_opposite [has_coproducts_of_shape X C] :\n  has_products_of_shape X Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape (discrete X)ᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\nlemma has_products_of_shape_of_opposite [has_coproducts_of_shape X Cᵒᵖ] :\n  has_products_of_shape X C :=\nbegin\n  haveI : has_colimits_of_shape (discrete X)ᵒᵖ Cᵒᵖ :=\n    has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,\n  exact has_limits_of_shape_of_has_colimits_of_shape_op\nend\n\ninstance has_products_opposite [has_coproducts.{v₂} C] : has_products.{v₂} Cᵒᵖ :=\nλ X, infer_instance\n\nlemma has_products_of_opposite [has_coproducts.{v₂} Cᵒᵖ] : has_products.{v₂} C :=\nλ X, has_products_of_shape_of_opposite X\n\ninstance has_coproducts_opposite [has_products.{v₂} C] : has_coproducts.{v₂} Cᵒᵖ :=\nλ X, infer_instance\n\nlemma has_coproducts_of_opposite [has_products.{v₂} Cᵒᵖ] : has_coproducts.{v₂} C :=\nλ X, has_coproducts_of_shape_of_opposite X\n\ninstance has_finite_coproducts_opposite [has_finite_products C] : has_finite_coproducts Cᵒᵖ :=\n{ out := λ n, limits.has_coproducts_of_shape_opposite _ }\n\nlemma has_finite_coproducts_of_opposite [has_finite_products Cᵒᵖ] : has_finite_coproducts C :=\n{ out := λ n, has_coproducts_of_shape_of_opposite _ }\n\ninstance has_finite_products_opposite [has_finite_coproducts C] : has_finite_products Cᵒᵖ :=\n{ out := λ n, infer_instance }\n\nlemma has_finite_products_of_opposite [has_finite_coproducts Cᵒᵖ] : has_finite_products C :=\n{ out := λ n, has_products_of_shape_of_opposite _ }\n\ninstance has_equalizers_opposite [has_coequalizers C] : has_equalizers Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape walking_parallel_pairᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence walking_parallel_pair_op_equiv,\n  apply_instance\nend\n\ninstance has_coequalizers_opposite [has_equalizers C] : has_coequalizers Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape walking_parallel_pairᵒᵖ C :=\n    has_limits_of_shape_of_equivalence walking_parallel_pair_op_equiv,\n  apply_instance\nend\n\ninstance has_finite_colimits_opposite [has_finite_limits C] :\n  has_finite_colimits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\ninstance has_finite_limits_opposite [has_finite_colimits C] :\n  has_finite_limits Cᵒᵖ :=\n{ out := λ J 𝒟 𝒥, by { resetI, apply_instance, }, }\n\ninstance has_pullbacks_opposite [has_pushouts C] : has_pullbacks Cᵒᵖ :=\nbegin\n  haveI : has_colimits_of_shape walking_cospanᵒᵖ C :=\n    has_colimits_of_shape_of_equivalence walking_cospan_op_equiv.symm,\n  apply has_limits_of_shape_op_of_has_colimits_of_shape,\nend\n\ninstance has_pushouts_opposite [has_pullbacks C] : has_pushouts Cᵒᵖ :=\nbegin\n  haveI : has_limits_of_shape walking_spanᵒᵖ C :=\n    has_limits_of_shape_of_equivalence walking_span_op_equiv.symm,\n  apply_instance\nend\n\n/-- The canonical isomorphism relating `span f.op g.op` and `(cospan f g).op` -/\n@[simps]\ndef span_op {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  span f.op g.op ≅ walking_cospan_op_equiv.inverse ⋙ (cospan f g).op :=\nnat_iso.of_components (by { rintro (_|_|_); refl, })\n  (by { rintros (_|_|_) (_|_|_) f; cases f; tidy, })\n\n/-- The canonical isomorphism relating `(cospan f g).op` and `span f.op g.op` -/\n@[simps]\ndef op_cospan {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :\n  (cospan f g).op ≅ walking_cospan_op_equiv.functor ⋙ span f.op g.op :=\ncalc (cospan f g).op ≅ 𝟭 _ ⋙ (cospan f g).op : by refl\n... ≅ (walking_cospan_op_equiv.functor ⋙ walking_cospan_op_equiv.inverse) ⋙ (cospan f g).op :\n  iso_whisker_right walking_cospan_op_equiv.unit_iso _\n... ≅ walking_cospan_op_equiv.functor ⋙ (walking_cospan_op_equiv.inverse ⋙ (cospan f g).op) :\n  functor.associator _ _ _\n... ≅ walking_cospan_op_equiv.functor ⋙ span f.op g.op : iso_whisker_left _ (span_op f g).symm\n\n/-- The canonical isomorphism relating `cospan f.op g.op` and `(span f g).op` -/\n@[simps]\ndef cospan_op {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :\n  cospan f.op g.op ≅ walking_span_op_equiv.inverse ⋙ (span f g).op :=\nnat_iso.of_components (by { rintro (_|_|_); refl, })\n  (by { rintros (_|_|_) (_|_|_) f; cases f; tidy, })\n\n/-- The canonical isomorphism relating `(span f g).op` and `cospan f.op g.op` -/\n@[simps]\ndef op_span {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) :\n  (span f g).op ≅ walking_span_op_equiv.functor ⋙ cospan f.op g.op :=\ncalc (span f g).op ≅ 𝟭 _ ⋙ (span f g).op : by refl\n... ≅ (walking_span_op_equiv.functor ⋙ walking_span_op_equiv.inverse) ⋙ (span f g).op :\n  iso_whisker_right walking_span_op_equiv.unit_iso _\n... ≅ walking_span_op_equiv.functor ⋙ (walking_span_op_equiv.inverse ⋙ (span f g).op) :\n  functor.associator _ _ _\n... ≅ walking_span_op_equiv.functor ⋙ cospan f.op g.op :\n  iso_whisker_left _ (cospan_op f g).symm\n\nnamespace pushout_cocone\n\n/-- The obvious map `pushout_cocone f g → pullback_cone f.unop g.unop` -/\n@[simps (lemmas_only)]\ndef unop {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  pullback_cone f.unop g.unop :=\ncocone.unop ((cocones.precompose (op_cospan f.unop g.unop).hom).obj\n  (cocone.whisker walking_cospan_op_equiv.functor c))\n\n@[simp]\nlemma unop_fst {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  c.unop.fst = c.inl.unop :=\nby { change (_ : limits.cone _).π.app _ = _,\n  simp only [pushout_cocone.ι_app_left, pushout_cocone.unop_π_app], tidy }\n\n@[simp]\nlemma unop_snd {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  c.unop.snd = c.inr.unop :=\nby { change (_ : limits.cone _).π.app _ = _,\n  simp only [pushout_cocone.unop_π_app, pushout_cocone.ι_app_right], tidy, }\n\n/-- The obvious map `pushout_cocone f.op g.op → pullback_cone f g` -/\n@[simps (lemmas_only)]\ndef op {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  pullback_cone f.op g.op :=\n(cones.postcompose ((cospan_op f g).symm).hom).obj\n  (cone.whisker walking_span_op_equiv.inverse (cocone.op c))\n\n@[simp]\nlemma op_fst {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  c.op.fst = c.inl.op :=\nby { change (_ : limits.cone _).π.app _ = _, apply category.comp_id, }\n\n@[simp]\nlemma op_snd {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  c.op.snd = c.inr.op :=\nby { change (_ : limits.cone _).π.app _ = _, apply category.comp_id, }\n\nend pushout_cocone\n\nnamespace pullback_cone\n\n/-- The obvious map `pullback_cone f g → pushout_cocone f.unop g.unop` -/\n@[simps (lemmas_only)]\ndef unop {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  pushout_cocone f.unop g.unop :=\ncone.unop ((cones.postcompose (op_span f.unop g.unop).symm.hom).obj\n  (cone.whisker walking_span_op_equiv.functor c))\n\n@[simp]\nlemma unop_inl {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  c.unop.inl = c.fst.unop :=\nbegin\n  change ((_ : limits.cocone _).ι.app _) = _,\n  dsimp only [unop, op_span],\n  simp, dsimp, simp, dsimp, simp\nend\n\n@[simp]\nlemma unop_inr {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  c.unop.inr = c.snd.unop :=\nbegin\n  change ((_ : limits.cocone _).ι.app _) = _,\n  apply quiver.hom.op_inj,\n  simp [unop_ι_app], dsimp, simp,\n  apply category.comp_id,\nend\n\n/-- The obvious map `pullback_cone f g → pushout_cocone f.op g.op` -/\n@[simps (lemmas_only)]\ndef op {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  pushout_cocone f.op g.op :=\n(cocones.precompose (span_op f g).hom).obj\n  (cocone.whisker walking_cospan_op_equiv.inverse (cone.op c))\n\n@[simp] lemma op_inl {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  c.op.inl = c.fst.op :=\nby { change (_ : limits.cocone _).ι.app _ = _, apply category.id_comp, }\n\n@[simp] lemma op_inr {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) :\n  c.op.inr = c.snd.op :=\nby { change (_ : limits.cocone _).ι.app _ = _, apply category.id_comp, }\n\n/-- If `c` is a pullback cone, then `c.op.unop` is isomorphic to `c`. -/\ndef op_unop {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) : c.op.unop ≅ c :=\npullback_cone.ext (iso.refl _) (by simp) (by simp)\n\n/-- If `c` is a pullback cone in `Cᵒᵖ`, then `c.unop.op` is isomorphic to `c`. -/\ndef unop_op {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z} (c : pullback_cone f g) : c.unop.op ≅ c :=\npullback_cone.ext (iso.refl _) (by simp) (by simp)\n\nend pullback_cone\n\nnamespace pushout_cocone\n\n/-- If `c` is a pushout cocone, then `c.op.unop` is isomorphic to `c`. -/\ndef op_unop {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) : c.op.unop ≅ c :=\npushout_cocone.ext (iso.refl _) (by simp) (by simp)\n\n/-- If `c` is a pushout cocone in `Cᵒᵖ`, then `c.unop.op` is isomorphic to `c`. -/\ndef unop_op {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) : c.unop.op ≅ c :=\npushout_cocone.ext (iso.refl _) (by simp) (by simp)\n\n/-- A pushout cone is a colimit cocone if and only if the corresponding pullback cone\nin the opposite category is a limit cone. -/\ndef is_colimit_equiv_is_limit_op  {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} (c : pushout_cocone f g) :\n  is_colimit c ≃ is_limit c.op :=\nbegin\n  apply equiv_of_subsingleton_of_subsingleton,\n  { intro h,\n    equiv_rw is_limit.postcompose_hom_equiv _ _,\n    equiv_rw (is_limit.whisker_equivalence_equiv walking_span_op_equiv.symm).symm,\n    exact is_limit_cocone_op _ h, },\n  { intro h,\n    equiv_rw is_colimit.equiv_iso_colimit c.op_unop.symm,\n    apply is_colimit_cone_unop,\n    equiv_rw is_limit.postcompose_hom_equiv _ _,\n    equiv_rw (is_limit.whisker_equivalence_equiv _).symm,\n    exact h, }\nend\n\n/-- A pushout cone is a colimit cocone in `Cᵒᵖ` if and only if the corresponding pullback cone\nin `C` is a limit cone. -/\ndef is_colimit_equiv_is_limit_unop  {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z}\n  (c : pushout_cocone f g) : is_colimit c ≃ is_limit c.unop :=\nbegin\n  apply equiv_of_subsingleton_of_subsingleton,\n  { intro h,\n    apply is_limit_cocone_unop,\n    equiv_rw is_colimit.precompose_hom_equiv _ _,\n    equiv_rw (is_colimit.whisker_equivalence_equiv _).symm,\n    exact h, },\n  { intro h,\n    equiv_rw is_colimit.equiv_iso_colimit c.unop_op.symm,\n    equiv_rw is_colimit.precompose_hom_equiv _ _,\n    equiv_rw (is_colimit.whisker_equivalence_equiv walking_cospan_op_equiv.symm).symm,\n    exact is_colimit_cone_op _ h, },\nend\n\nend pushout_cocone\n\nnamespace pullback_cone\n\n/-- A pullback cone is a limit cone if and only if the corresponding pushout cocone\nin the opposite category is a colimit cocone. -/\ndef is_limit_equiv_is_colimit_op  {X Y Z : C} {f : X ⟶ Z} {g : Y ⟶ Z}\n  (c : pullback_cone f g) : is_limit c ≃ is_colimit c.op :=\n(is_limit.equiv_iso_limit c.op_unop).symm.trans c.op.is_colimit_equiv_is_limit_unop.symm\n\n/-- A pullback cone is a limit cone in `Cᵒᵖ` if and only if the corresponding pushout cocone\nin `C` is a colimit cocone. -/\ndef is_limit_equiv_is_colimit_unop  {X Y Z : Cᵒᵖ} {f : X ⟶ Z} {g : Y ⟶ Z}\n  (c : pullback_cone f g) : is_limit c ≃ is_colimit c.unop :=\n(is_limit.equiv_iso_limit c.unop_op).symm.trans c.unop.is_colimit_equiv_is_limit_op.symm\n\nend pullback_cone\n\nsection pullback\n\nopen opposite\n\n/-- The pullback of `f` and `g` in `C` is isomorphic to the pushout of\n`f.op` and `g.op` in `Cᵒᵖ`. -/\nnoncomputable\ndef pullback_iso_unop_pushout {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n  [has_pullback f g] [has_pushout f.op g.op] : pullback f g ≅ unop (pushout f.op g.op) :=\nis_limit.cone_point_unique_up_to_iso (limit.is_limit _)\n  ((pushout_cocone.is_colimit_equiv_is_limit_unop _) (colimit.is_colimit (span f.op g.op)))\n\n@[simp, reassoc]\nlemma pullback_iso_unop_pushout_inv_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z)\n  [has_pullback f g] [has_pushout f.op g.op] :\n  (pullback_iso_unop_pushout f g).inv ≫ pullback.fst =\n    (pushout.inl : _ ⟶ pushout f.op g.op).unop :=\n(is_limit.cone_point_unique_up_to_iso_inv_comp _ _ _).trans (by simp)\n\n@[simp, reassoc]\nlemma pullback_iso_unop_pushout_inv_snd {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) [has_pullback f g] [has_pushout f.op g.op] :\n  (pullback_iso_unop_pushout f g).inv ≫ pullback.snd =\n    (pushout.inr : _ ⟶ pushout f.op g.op).unop :=\n(is_limit.cone_point_unique_up_to_iso_inv_comp _ _ _).trans (by simp)\n\n@[simp, reassoc]\nlemma pullback_iso_unop_pushout_hom_inl {X Y Z : C} (f : X ⟶ Z)\n  (g : Y ⟶ Z) [has_pullback f g] [has_pushout f.op g.op] :\n  pushout.inl ≫ (pullback_iso_unop_pushout f g).hom.op = pullback.fst.op :=\nbegin\n  apply quiver.hom.unop_inj,\n  dsimp,\n  rw [← pullback_iso_unop_pushout_inv_fst, iso.hom_inv_id_assoc],\nend\n\n@[simp, reassoc]\n\n\nend pullback\n\nsection pushout\n\n/-- The pushout of `f` and `g` in `C` is isomorphic to the pullback of\n `f.op` and `g.op` in `Cᵒᵖ`. -/\nnoncomputable\ndef pushout_iso_unop_pullback {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)\n  [has_pushout f g] [has_pullback f.op g.op] : pushout f g ≅ unop (pullback f.op g.op) :=\nis_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _)\n  ((pullback_cone.is_limit_equiv_is_colimit_unop _) (limit.is_limit (cospan f.op g.op)))\n.\n@[simp, reassoc]\nlemma pushout_iso_unop_pullback_inl_hom {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)\n  [has_pushout f g] [has_pullback f.op g.op] :\n  pushout.inl ≫ (pushout_iso_unop_pullback f g).hom =\n    (pullback.fst : pullback f.op g.op ⟶ _).unop :=\n(is_colimit.comp_cocone_point_unique_up_to_iso_hom _ _ _).trans (by simp)\n\n@[simp, reassoc]\nlemma pushout_iso_unop_pullback_inr_hom {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)\n  [has_pushout f g] [has_pullback f.op g.op] :\n  pushout.inr ≫ (pushout_iso_unop_pullback f g).hom =\n    (pullback.snd : pullback f.op g.op ⟶ _).unop :=\n(is_colimit.comp_cocone_point_unique_up_to_iso_hom _ _ _).trans (by simp)\n\n@[simp]\nlemma pushout_iso_unop_pullback_inv_fst {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)\n  [has_pushout f g] [has_pullback f.op g.op] :\n  (pushout_iso_unop_pullback f g).inv.op ≫ pullback.fst = pushout.inl.op :=\nbegin\n  apply quiver.hom.unop_inj,\n  dsimp,\n  rw [← pushout_iso_unop_pullback_inl_hom, category.assoc, iso.hom_inv_id, category.comp_id],\nend\n\n@[simp]\nlemma pushout_iso_unop_pullback_inv_snd {X Y Z : C} (f : X ⟶ Z) (g : X ⟶ Y)\n  [has_pushout f g] [has_pullback f.op g.op] :\n  (pushout_iso_unop_pullback f g).inv.op ≫ pullback.snd = pushout.inr.op :=\nbegin\n  apply quiver.hom.unop_inj,\n  dsimp,\n  rw [← pushout_iso_unop_pullback_inr_hom, category.assoc, iso.hom_inv_id, category.comp_id],\nend\n\nend pushout\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.32349179534834616}}
{"text": "/-\nCopyright (c) 2014 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport algebra.order.monoid.with_top\nimport data.nat.basic\n\n/-!\n# Lemma about the coercion `ℕ → with_bot ℕ`.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nAn orphaned lemma about casting from `ℕ` to `with_bot ℕ`,\nexiled here to minimize imports to `data.rat.order` for porting purposes.\n-/\n\ntheorem nat.cast_with_top (n : ℕ) :\n  @coe ℕ (with_top ℕ) (@coe_to_lift _ _ nat.cast_coe) n = n := rfl\n\ntheorem nat.cast_with_bot (n : ℕ) :\n  @coe ℕ (with_bot ℕ) (@coe_to_lift _ _ nat.cast_coe) n = n := rfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/nat/cast/with_top.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3233808946074032}}
{"text": "import for_mathlib.snake_lemma2\nimport category_theory.abelian.homology\nimport for_mathlib.exact_seq2\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\n\nvariables {𝒜 : Type*} [category 𝒜] [abelian 𝒜]\n\nnamespace category_theory\n\nlocal notation `kernel_map`   := kernel.map _ _ _ _\nlocal notation `cokernel_map` := cokernel.map _ _ _ _\n\nnamespace snake\n\nlemma col_exact_aux (X : cochain_complex 𝒜 ℤ) : exact_seq 𝒜\n[ (kernel.ι (homological_complex.d_to X 0))\n, (kernel.lift (homological_complex.d_from X 0)\n    (homological_complex.d_to X 0) (by simp))\n, (homology.π' (homological_complex.d_to X 0)\n    (homological_complex.d_from X 0) (by simp))] :=\nbegin\n  apply exact_seq.cons,\n  { rw abelian.exact_iff,\n    refine ⟨by { ext, simp }, _⟩,\n    have :\n      kernel.ι (kernel.lift (homological_complex.d_from X 0) (homological_complex.d_to X 0) _) =\n      kernel.lift _ (kernel.ι _) _ ≫ kernel.ι (homological_complex.d_to X 0),\n    by { simp },\n    rw this,\n    simp only [category.assoc, cokernel.condition, comp_zero],\n    have : homological_complex.d_to X 0 =\n      kernel.lift (homological_complex.d_from X 0) (homological_complex.d_to X 0) (by simp) ≫\n      kernel.ι _, by simp,\n    slice_lhs 2 3 { rw this },\n    rw kernel.condition_assoc,\n    simp },\n  { rw ← exact_iff_exact_seq,\n    change exact _ (_ ≫ _),\n    rw exact_comp_iso,\n    apply abelian.exact_cokernel }\nend\n\nlemma row_exact₁_aux (X Y Z : cochain_complex 𝒜 ℤ)\n  (f : X ⟶ Y) (g : Y ⟶ Z) (h : exact (f.f (-1)) (g.f (-1)))\n  [epi (g.f (-1))] [epi (g.f 0)] [epi (g.f 1)]\n  [mono (f.f (-1))] [mono (f.f 0)] [mono (f.f 1)] :\n  exact (homological_complex.hom.prev f 0) (homological_complex.hom.prev g 0) :=\nbegin\n  rw [f.prev_eq, g.prev_eq],\n  rotate 2, exact (-1), swap, exact (-1), simp, swap, simp,\n  simp,\n  rw [← category.assoc, exact_comp_iso],\n  apply category_theory.exact_comp_inv_hom_comp _ h,\nend\n\nlemma row_exact₂_aux (X Y Z : cochain_complex 𝒜 ℤ)\n  (f : X ⟶ Y) (g : Y ⟶ Z) (h0 : exact (f.f 0) (g.f 0)) (h1 : exact (f.f 1) (g.f 1))\n  [epi (g.f (-1))] [epi (g.f 0)] [epi (g.f 1)]\n  [mono (f.f (-1))] [mono (f.f 0)] [mono (f.f 1)] :\n  exact\n    (kernel.lift (homological_complex.d_from Y 0)\n       (kernel.ι (homological_complex.d_from X 0) ≫ f.f 0)\n       (by simp))\n    (kernel.lift (homological_complex.d_from Z 0)\n       (kernel.ι (homological_complex.d_from Y 0) ≫ g.f 0)\n       (by simp)) :=\nbegin\n  haveI : mono (homological_complex.hom.next f 0),\n  { rw f.next_eq,\n    rotate 2, exact 1, swap, simp,\n    apply_with mono_comp { instances := ff },\n    swap,\n    apply_with mono_comp { instances := ff },\n    all_goals { apply_instance } },\n  have : exact (homological_complex.hom.next f 0) (homological_complex.hom.next g 0),\n  { rw [f.next_eq, g.next_eq],\n    rotate 2, exact 1, swap, exact 1, simp, swap, simp,\n    simp,\n    rw [← category.assoc, exact_comp_iso],\n    apply category_theory.exact_comp_inv_hom_comp _ h1 },\n  have S := mk_of_sequence_hom\n    (X.X 0)\n    (Y.X 0)\n    (Z.X 0)\n    (X.X_next 0)\n    (Y.X_next 0)\n    (Z.X_next 0)\n    (f.f 0) (g.f 0) (X.d_from 0) (Y.d_from 0) (Z.d_from 0)\n    (f.next 0) (g.next 0) (by simp) (by simp) h0 this,\n  exact S.six_term_exact_seq.pair,\nend\n\nlemma mk_of_homology (X Y Z : cochain_complex 𝒜 ℤ)\n  (f : X ⟶ Y) (g : Y ⟶ Z)\n  (hn1 : exact (f.f (-1)) (g.f (-1))) (h0 : exact (f.f 0) (g.f 0)) (h1 : exact (f.f 1) (g.f 1))\n  [epi (g.f (-1))] [epi (g.f 0)] [epi (g.f 1)]\n  [mono (f.f (-1))] [mono (f.f 0)] [mono (f.f 1)] :\n  snake\n  -- the objects\n         (kernel (X.d_to 0))             (kernel (Y.d_to 0))              (kernel (Z.d_to 0))\n            (X.X_prev 0)                    (Y.X_prev 0)                     (Z.X_prev 0)\n        (kernel (X.d_from 0))           (kernel (Y.d_from 0))            (kernel (Z.d_from 0))\n  ((homology_functor _ _ 0).obj X) ((homology_functor _ _ 0).obj Y) ((homology_functor _ _ 0).obj Z)\n  -- the maps\n  (kernel.map _ _ (f.prev _) (f.f _) (by simp)) (kernel.map _ _ (g.prev _) (g.f _) (by simp))\n  (kernel.ι _) (kernel.ι _) (kernel.ι _)\n  (f.prev _) (g.prev _)\n  (kernel.lift _ (X.d_to _) (by simp)) (kernel.lift _ (Y.d_to _) (by simp)) (kernel.lift _ (Z.d_to _) (by simp))\n  (kernel.map _ _ (f.f _) (f.next _) (by simp)) (kernel.map _ _ (g.f _) (g.next _) (by simp))\n  (homology.π' _ _ _) (homology.π' _ _ _) (homology.π' _ _ _)\n  ((homology_functor _ _ _).map f) ((homology_functor _ _ _).map g) :=\n{ row_exact₁ := row_exact₁_aux _ _ _ _ _ hn1,\n  row_exact₂ := row_exact₂_aux _ _ _ _ _ h0 h1,\n  row_epi := begin\n    rw g.prev_eq,\n    rotate 2, exact (-1),\n    swap, simp,\n    apply_with epi_comp { instances := ff },\n    swap,\n    apply_with epi_comp { instances := ff },\n    all_goals { apply_instance }\n  end,\n  row_mono := infer_instance,\n  col_exact_a := col_exact_aux _,\n  col_exact_b := col_exact_aux _,\n  col_exact_c := col_exact_aux _,\n  col_mono_a := infer_instance,\n  col_mono_b := infer_instance,\n  col_mono_c := infer_instance,\n  col_epi_a := epi_comp _ _,\n  col_epi_b := epi_comp _ _,\n  col_epi_c := epi_comp _ _,\n  sq_a₀ := by simp,\n  sq_b₀ := by simp,\n  sq_a₁ := by { ext, simp },\n  sq_b₁ := by { ext, simp },\n  sq_a₂ := by simp,\n  sq_b₂ := by simp }\n\nend snake\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/snake_lemma3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.32338088733030546}}
{"text": "/-\nCopyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Heather Macbeth, Yaël Dillies\n-/\nimport Std.Lean.Parser\nimport Mathlib.Tactic.NormNum.Core\nimport Mathlib.Tactic.Clear!\nimport Mathlib.Order.Basic\nimport Mathlib.Algebra.Order.Invertible\nimport Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Data.Nat.Cast.Basic\nimport Qq\n\n/-!\n## `positivity` core functionality\n\nThis file sets up the `positivity` tactic and the `@[positivity]` attribute,\nwhich allow for plugging in new positivity functionality around a positivity-based driver.\nThe actual behavior is in `@[positivity]`-tagged definitions in `Tactic.Positivity.Basic`\nand elsewhere.\n-/\nopen Lean hiding Rat\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- Attribute for identifying `positivity` extensions. -/\nsyntax (name := positivity) \"positivity\" term,+ : attr\n\nlemma ne_of_ne_of_eq' (hab : (a : α) ≠ c) (hbc : a = b) : b ≠ c := hbc ▸ hab\n\nnamespace Mathlib.Meta.Positivity\n\ninstance : Repr (QQ α) := inferInstanceAs (Repr Expr)\n\nvariable {α : Q(Type u)} (zα : Q(Zero $α)) (pα : Q(PartialOrder $α))\n\n/-- The result of `positivity` running on an expression `e` of type `α`. -/\ninductive Strictness (e : Q($α)) where\n  | positive (pf : Q(0 < $e))\n  | nonnegative (pf : Q(0 ≤ $e))\n  | nonzero (pf : Q($e ≠ 0))\n  | none\n  deriving Repr\n\n/-- Gives a generic description of the `positivity` result. -/\ndef Strictness.toString : Strictness zα pα e → String\n  | positive _ => \"positive\"\n  | nonnegative _ => \"nonnegative\"\n  | nonzero _ => \"nonzero\"\n  | none => \"none\"\n\n/-- An extension for `positivity`. -/\nstructure PositivityExt where\n  /-- Attempts to prove a expression `e : α` is `>0`, `≥0`, or `≠0`. -/\n  eval {u} {α : Q(Type u)} (zα : Q(Zero $α)) (pα : Q(PartialOrder $α)) (e : Q($α)) :\n    MetaM (Strictness zα pα e)\n\n/-- Read a `positivity` extension from a declaration of the right type. -/\ndef mkPositivityExt (n : Name) : ImportM PositivityExt := do\n  let { env, opts, .. } ← read\n  IO.ofExcept <| unsafe env.evalConstCheck PositivityExt opts ``PositivityExt n\n\n/-- Each `positivity` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array (DiscrTree.Key true)) × Name\n\n/-- Environment extensions for `positivity` declarations -/\ninitialize positivityExt : PersistentEnvExtension Entry (Entry × PositivityExt)\n    (List Entry × DiscrTree PositivityExt true) ←\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq PositivityExt := ⟨fun _ _ => false⟩\n  let insert kss v dt := kss.foldl (fun dt ks => dt.insertCore ks v) dt\n  registerPersistentEnvExtension {\n    mkInitial := pure ([], {})\n    addImportedFn := fun s => do\n      let dt ← s.foldlM (init := {}) fun dt s => s.foldlM (init := dt) fun dt (kss, n) => do\n        pure (insert kss (← mkPositivityExt n) dt)\n      pure ([], dt)\n    addEntryFn := fun (entries, s) ((kss, n), ext) => ((kss, n) :: entries, insert kss ext s)\n    exportEntriesFn := fun s => s.1.reverse.toArray\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `positivity\n  descr := \"adds a positivity extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind => match stx with\n    | `(attr| positivity $es,*) => do\n      unless kind == AttributeKind.global do\n        throwError \"invalid attribute 'positivity', must be global\"\n      let env ← getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'positivity', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext ← mkPositivityExt declName\n      let keys ← MetaM.run' <| es.getElems.mapM fun stx => do\n        let e ← TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ · with ignoreTCFailures := true }) do\n            let e ← elabTerm stx none\n            let (_, _, e) ← lambdaMetaTelescope (← mkLambdaFVars (← getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      setEnv <| positivityExt.addEntry env ((keys, declName), ext)\n    | _ => throwUnsupportedSyntax\n}\n\nlemma lt_of_le_of_ne' [PartialOrder A] :\n    (a : A) ≤ b → b ≠ a → a < b := fun h₁ h₂ => lt_of_le_of_ne h₁ h₂.symm\n\nlemma pos_of_isNat [StrictOrderedSemiring A]\n    (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  apply Nat.cast_pos.2\n  simpa using w\n\nlemma nonneg_of_isNat [OrderedSemiring A]\n    (h : NormNum.IsNat e n) : 0 ≤ (e : A) := by\n  rw [NormNum.IsNat.to_eq h rfl]\n  exact Nat.cast_nonneg n\n\nlemma nz_of_isNegNat [StrictOrderedRing A]\n    (h : NormNum.IsInt e (.negOfNat n)) (w : Nat.ble 1 n = true) : (e : A) ≠ 0 := by\n  rw [NormNum.IsInt.neg_to_eq h rfl]\n  simp only [ne_eq, neg_eq_zero]\n  apply ne_of_gt\n  simpa using w\n\nlemma pos_of_isRat [LinearOrderedRing A] :\n    (NormNum.IsRat e n d) → (decide (0 < n)) → (0 < (e : A))\n  | ⟨inv, eq⟩, h => by\n    have pos_invOf_d : (0 < ⅟ (d : A)) := pos_invOf_of_invertible_cast d\n    have pos_n : (0 < (n : A)) := Int.cast_pos (n := n) |>.2 (of_decide_eq_true h)\n    rw [eq]\n    exact mul_pos pos_n pos_invOf_d\n\nlemma nonneg_of_isRat [LinearOrderedRing A] :\n    (NormNum.IsRat e n d) → (decide (n = 0)) → (0 ≤ (e : A))\n  | ⟨inv, eq⟩, h => by rw [eq, of_decide_eq_true h]; simp\n\nlemma nz_of_isRat [LinearOrderedRing A] :\n    (NormNum.IsRat e n d) → (decide (n < 0)) → ((e : A) ≠ 0)\n  | ⟨inv, eq⟩, h => by\n    have pos_invOf_d : (0 < ⅟ (d : A)) := pos_invOf_of_invertible_cast d\n    have neg_n : ((n : A) < 0) := Int.cast_lt_zero (n := n) |>.2 (of_decide_eq_true h)\n    have neg := mul_neg_of_neg_of_pos neg_n pos_invOf_d\n    rw [eq]\n    exact ne_iff_lt_or_gt.2 (Or.inl neg)\n\nvariable {zα pα} in\n/-- Converts a `MetaM Strictness` which can fail\ninto one that never fails and returns `.none` instead. -/\ndef catchNone (t : MetaM (Strictness zα pα e)) : MetaM (Strictness zα pα e) :=\n  try t catch e =>\n    trace[Tactic.positivity.failure] \"{e.toMessageData}\"\n    pure .none\n\nvariable {zα pα} in\n/-- Converts a `MetaM Strictness` which can return `.none`\ninto one which never returns `.none` but fails instead. -/\ndef throwNone [Monad m] [Alternative m]\n    (t : m (Strictness zα pα e)) : m (Strictness zα pα e) := do\n  match ← t with\n  | .none => failure\n  | r => pure r\n\n/-- Attempts to prove a `Strictness` result when `e` evaluates to a literal number. -/\ndef normNumPositivity (e : Q($α)) : MetaM (Strictness zα pα e) := catchNone do\n  match ← NormNum.derive e with\n  | .isBool .. => failure\n  | .isNat _ lit p =>\n    if 0 < lit.natLit! then\n      let _a ← synthInstanceQ (q(StrictOrderedSemiring $α) : Q(Type u))\n      assumeInstancesCommute\n      have p : Q(NormNum.IsNat $e $lit) := p\n      let p' : Q(Nat.ble 1 $lit = true) := (q(Eq.refl true) : Expr)\n      pure (.positive (q(@pos_of_isNat $α _ _ _ $p $p') : Expr))\n    else\n      let _a ← synthInstanceQ (q(OrderedSemiring $α) : Q(Type u))\n      assumeInstancesCommute\n      have p : Q(NormNum.IsNat $e $lit) := p\n      pure (.nonnegative (q(nonneg_of_isNat $p) : Expr))\n  | .isNegNat _ lit p =>\n    let _a ← synthInstanceQ (q(StrictOrderedRing $α) : Q(Type u))\n    assumeInstancesCommute\n    have p : Q(NormNum.IsInt $e (Int.negOfNat $lit)) := p\n    let p' : Q(Nat.ble 1 $lit = true) := (q(Eq.refl true) : Expr)\n    pure (.nonzero (q(nz_of_isNegNat $p $p') : Expr))\n  | .isRat i q n d p =>\n    let _a ← synthInstanceQ (q(LinearOrderedRing $α) : Q(Type u))\n    have p : Q(by clear! «$i»; exact NormNum.IsRat $e $n $d) := p\n    if 0 < q then\n      let w : Q(decide (0 < $n) = true) := (q(Eq.refl true) : Expr)\n      pure (.positive (q(pos_of_isRat $p $w) : Expr))\n    else if q = 0 then -- should not be reachable, but just in case\n      let w : Q(decide ($n = 0) = true) := (q(Eq.refl true) : Expr)\n      pure (.nonnegative (q(nonneg_of_isRat $p $w) : Expr))\n    else\n      let w : Q(decide ($n < 0) = true) := (q(Eq.refl true) : Expr)\n      pure (.nonzero (q(nz_of_isRat $p $w) : Expr))\n\n/-- Attempts to prove that `e ≥ 0` using `zero_le` in a `CanonicallyOrderedAddMonoid`. -/\ndef positivityCanon (e : Q($α)) : MetaM (Strictness zα pα e) := do\n  let _ ← synthInstanceQ (q(CanonicallyOrderedAddMonoid $α) : Q(Type u))\n  pure (.nonnegative (q(zero_le $e) : Expr))\n\n/-- A variation on `assumption` when the hypothesis is `lo ≤ e` where `lo` is a numeral. -/\ndef compareHypLE (lo e : Q($α)) (p₂ : Q($lo ≤ $e)) : MetaM (Strictness zα pα e) := do\n  match ← normNumPositivity zα pα lo with\n  | .positive p₁ => pure (.positive q(lt_of_lt_of_le $p₁ $p₂))\n  | .nonnegative p₁ => pure (.nonnegative q(le_trans $p₁ $p₂))\n  | _ => pure .none\n\n/-- A variation on `assumption` when the hypothesis is `lo < e` where `lo` is a numeral. -/\ndef compareHypLT (lo e : Q($α)) (p₂ : Q($lo < $e)) : MetaM (Strictness zα pα e) := do\n  match ← normNumPositivity zα pα lo with\n  | .positive p₁ => pure (.positive q(lt_trans $p₁ $p₂))\n  | .nonnegative p₁ => pure (.positive q(lt_of_le_of_lt $p₁ $p₂))\n  | _ => pure .none\n\n/-- A variation on `assumption` when the hypothesis is `a = b` where `a` is a numeral. -/\ndef compareHypEq (e a b : Q($α)) (p₂ : Q($a = $b)) : MetaM (Strictness zα pα e) := do\n  let .true ← isDefEq e b | return .none\n  match ← normNumPositivity zα pα a with\n  | .positive p₁ => pure (.positive (q(lt_of_lt_of_eq $p₁ $p₂) : Expr))\n  | .nonnegative p₁ => pure (.nonnegative (q(le_of_le_of_eq $p₁ $p₂) : Expr))\n  | .nonzero p₁ => pure (.nonzero (q(ne_of_ne_of_eq' $p₁ $p₂) : Expr))\n  | .none => pure .none\n\ninitialize registerTraceClass `Tactic.positivity\ninitialize registerTraceClass `Tactic.positivity.failure\n\n/-- A variation on `assumption` which checks if the hypothesis `ldecl` is `a [</≤/=] e`\nwhere `a` is a numeral. -/\ndef compareHyp (e : Q($α)) (ldecl : LocalDecl) : MetaM (Strictness zα pα e) := do\n  have e' : Q(Prop) := ldecl.type\n  match e' with\n  | ~q(@LE.le _ $_a $lo $hi) =>\n    guard <| ← isDefEq e hi\n    compareHypLE zα pα lo e (.fvar ldecl.fvarId)\n  | ~q(@LT.lt _ $_a $lo $hi) =>\n    guard <| ← isDefEq e hi\n    compareHypLT zα pα lo e (.fvar ldecl.fvarId)\n  | ~q(($lo : $α') = $hi) =>\n    let .true ← isDefEq α α' | return .none\n    let p : Q($lo = $hi) := .fvar ldecl.fvarId\n    match ← compareHypEq zα pα e lo hi p with\n    | .none => compareHypEq zα pα e hi lo (q(Eq.symm $p) : Expr)\n    | result => pure result\n  | ~q(($a : $α') ≠ $b) =>\n    let .true ← isDefEq α α' | return .none\n    if ← isDefEq q((0 : $α)) a then\n      let .true ← isDefEq e b | return .none\n      let p : Q(0 ≠ $e) := .fvar ldecl.fvarId\n      pure (.nonzero q(Ne.symm $p))\n    else\n      let .true ← isDefEq q((0 : $α)) b | return .none\n      let .true ← isDefEq e a | return .none\n      pure (.nonzero (.fvar ldecl.fvarId))\n  | _ => pure .none\n\nvariable {zα pα} in\n/-- The main combinator which combines multiple `positivity` results.\nIt assumes `t₁` has already been run for a result, and runs `t₂` and takes the best result.\nIt will skip `t₂` if `t₁` is already a proof of `.positive`, and can also combine\n`.nonnegative` and `.nonzero` to produce a `.positive` result. -/\ndef orElse (t₁ : Strictness zα pα e) (t₂ : MetaM (Strictness zα pα e)) :\n    MetaM (Strictness zα pα e) := do\n  match t₁ with\n  | .none => catchNone t₂\n  | p@(.positive _) => pure p\n  | .nonnegative p₁ =>\n    match ← catchNone t₂ with\n    | p@(.positive _) => pure p\n    | .nonzero p₂ => pure (.positive q(lt_of_le_of_ne' $p₁ $p₂))\n    | _ => pure (.nonnegative p₁)\n  | .nonzero p₁ =>\n    match ← catchNone t₂ with\n    | p@(.positive _) => pure p\n    | .nonnegative p₂ => pure (.positive q(lt_of_le_of_ne' $p₂ $p₁))\n    | _ => pure (.nonzero p₁)\n\n/-- Run each registered `positivity` extension on an expression, returning a `NormNum.Result`. -/\ndef core (e : Q($α)) : MetaM (Strictness zα pα e) := do\n  let mut result := .none\n  trace[Tactic.positivity] \"trying to prove positivity of {e}\"\n  for ext in ← (positivityExt.getState (← getEnv)).2.getMatch e do\n    try\n      result ← orElse result <| ext.eval zα pα e\n    catch err =>\n      trace[Tactic.positivity] \"{e} failed: {err.toMessageData}\"\n  result ← orElse result <| normNumPositivity zα pα e\n  result ← orElse result <| positivityCanon zα pα e\n  if let .positive _ := result then\n    trace[Tactic.positivity] \"{e} => {result.toString}\"\n    return result\n  for ldecl in ← getLCtx do\n    if !ldecl.isImplementationDetail then\n      result ← orElse result <| compareHyp zα pα e ldecl\n  trace[Tactic.positivity] \"{e} => {result.toString}\"\n  throwNone (pure result)\n\nprivate inductive OrderRel : Type\n| le : OrderRel -- `0 ≤ a`\n| lt : OrderRel -- `0 < a`\n| ne : OrderRel -- `a ≠ 0`\n| ne' : OrderRel -- `0 ≠ a`\n\nend Meta.Positivity\nnamespace Meta.Positivity\n\n/-- The main entry point to the `positivity` tactic. Given a goal `goal` of the form `0 [≤/</≠] e`,\nattempts to recurse on the structure of `e` to prove the goal.\nIt will either close `goal` or fail. -/\ndef positivity (goal : MVarId) : MetaM Unit := do\n  let t : Q(Prop) ← withReducible goal.getType'\n  let rest {u : Level} (α : Q(Type u)) z e (relDesired : OrderRel) : MetaM Unit := do\n    let zα ← synthInstanceQ (q(Zero $α) : Q(Type u))\n    let .true ← isDefEq z q((0 : $α)) | throwError \"not a positivity goal\"\n    let pα ← synthInstanceQ (q(PartialOrder $α) : Q(Type u))\n    let r ← catchNone <| Meta.Positivity.core zα pα e\n    let throw (a b : String) : MetaM Expr := throwError\n      \"failed to prove {a}, but it would be possible to prove {b} if desired\"\n    let p ← show MetaM Expr from match relDesired, r with\n    | .lt, .positive p\n    | .le, .nonnegative p\n    | .ne, .nonzero p => pure p\n    | .le, .positive p => pure q(le_of_lt $p)\n    | .ne, .positive p => pure q(ne_of_gt $p)\n    | .ne', .positive p => pure q(ne_of_lt $p)\n    | .ne', .nonzero p => pure q(Ne.symm $p)\n    | .lt, .nonnegative _ => throw \"strict positivity\" \"nonnegativity\"\n    | .lt, .nonzero _ => throw \"strict positivity\" \"nonzeroness\"\n    | .le, .nonzero _ => throw \"nonnegativity\" \"nonzeroness\"\n    | .ne, .nonnegative _\n    | .ne', .nonnegative _ => throw \"nonzeroness\" \"nonnegativity\"\n    | _, .none => throwError \"failed to prove positivity/nonnegativity/nonzeroness\"\n    goal.assign p\n  match t with\n  | ~q(@LE.le $α $_a $z $e) => rest α z e .le\n  | ~q(@LT.lt $α $_a $z $e) => rest α z e .lt\n  | ~q($a ≠ ($b : ($α : Type _))) =>\n    let _zα ← synthInstanceQ (q(Zero $α) : Q(Type u_1))\n    if ← isDefEq b q((0 : $α)) then\n      rest α b a .ne\n    else\n      let .true ← isDefEq a q((0 : $α)) | throwError \"not a positivity goal\"\n      rest α a b .ne'\n  | _ => throwError \"not a positivity goal\"\n\nend Meta.Positivity\n\nnamespace Tactic.Positivity\n\nopen Lean Elab Tactic\n\n/-- Tactic solving goals of the form `0 ≤ x`, `0 < x` and `x ≠ 0`.  The tactic works recursively\naccording to the syntax of the expression `x`, if the atoms composing the expression all have\nnumeric lower bounds which can be proved positive/nonnegative/nonzero by `norm_num`.  This tactic\neither closes the goal or fails.\n\nExamples:\n```\nexample {a : ℤ} (ha : 3 < a) : 0 ≤ a ^ 3 + a := by positivity\n\nexample {a : ℤ} (ha : 1 < a) : 0 < |(3:ℤ) + a| := by positivity\n\nexample {b : ℤ} : 0 ≤ max (-3) (b ^ 2) := by positivity\n```\n-/\nelab (name := positivity) \"positivity\" : tactic => do\n  liftMetaTactic fun g => do Meta.Positivity.positivity g; pure []\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/Positivity/Core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3232059557569308}}
{"text": "import category_theory.preadditive.opposite\n\nnamespace category_theory\n\nvariables {C : Type*} [category C] [preadditive C] {X Y : C}\n\n@[simps] def op_hom (X Y : C) : (X ⟶ Y) →+ (opposite.op Y ⟶ opposite.op X) :=\nadd_monoid_hom.mk' (λ f, f.op) $ λ f g, op_add _ f g\n\nlemma op_sum {ι : Type*} (s : finset ι) (f : ι → (X ⟶ Y)) :\n  (s.sum f).op = s.sum (λ i, (f i).op) :=\n(op_hom X Y).map_sum _ _\n\nlemma op_zsmul (k : ℤ) (f : X ⟶ Y) : (k • f).op = k • f.op := rfl\n\n-- lemma op_neg (f : X ⟶ Y) : (-f).op = -(f.op) := rfl\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/op.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.323205942731106}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) [irrational α] : ∀ y ∈ Icc 0 1, ∃ x ∈ (ℤ : Type*), |x • α - y| < 1 :=\nbegin\n  assume y h1,\n  let S : set ℝ := {x : ℝ | ∃ i : ℤ, x = i • α - ⌊i • α⌋},\n  have h2 : ∀ i j : ℤ, i ≠ j → ¬ ((i • α - ⌊i • α⌋) = (j • α - ⌊j • α⌋)), from \n    assume i j h3, assume h4 : (i • α - ⌊i • α⌋) = (j • α - ⌊j • α⌋),\n    have h5 : α = (⌊i • α⌋ - ⌊j • α⌋)/(i - j), \n    from eq.trans (eq.symm h4) (div_sub_div_same j i),\n    have h6 : (⌊i • α⌋ - ⌊j • α⌋)/(i - j) ∈ ℤ, from begin\n      cases i with i,\n        cases j with j,\n          rw [mul_zero,mul_zero,sub_zero,sub_zero,div_zero],\n        rw [mul_zero,sub_zero,div_zero],\n      rw [mul_zero,sub_zero,div_zero],\n    end,\n    have h7 : α ∈ ℤ, from by {\n      rw [← int.coe_nat_eq_coe_int_zero,← int.coe_nat_eq_coe_int_zero] at h6,\n      exact h6,\n    },\n    have h8 : α ∈ ℚ, from by {\n      exact ⟨α, h7⟩,\n    },\n    have h9 : α ∉ ℚ, from by {\n      exact irrational.irrat h8,\n    },\n    have h10 : α ∉ ℝ, from by {\n      exact h9,\n    },\n    have h11 : false, from by {\n      exact h10,\n    },\n    show false, from h11,\n  have h12 : ∀ i j : ℤ, i ≠ j → i • α - ⌊i • α⌋ ≠ j • α - ⌊j • α⌋, from \n    assume i j h13, assume h14 : i • α - ⌊i • α⌋ = j • α - ⌊j • α⌋,\n    exact h2 i j h13 h14,\n  have h15 : ∀ i j : ℤ, i ≠ j → i • α - ⌊i • α⌋ ∉ {j • α - ⌊j • α⌋}, from \n    assume i j h16, assume h17 : i • α - ⌊i • α⌋ = j • α - ⌊j • α⌋,\n    have h18 : i • α - ⌊i • α⌋ ∈ {j • α - ⌊j • α⌋}, \n    from mem_singleton_iff.mp h17,\n    have h19 : false, from by {\n      rw mem_singleton at h18,\n      exact h12 i j h16 h18,\n    },\n    show false, from h19,\n  have h20 : ∀ i j : ℤ, i ≠ j → i • α - ⌊i • α⌋ ∉ {k • α - ⌊k • α⌋ | k : ℤ}, from \n    assume i j h21, assume h22 : i • α - ⌊i • α⌋ = k • α - ⌊k • α⌋,\n    have h23 : i • α - ⌊i • α⌋ ∈ {k • α - ⌊k • α⌋ | k : ℤ}, from \n    set.mem_of_eq_of_mem h22 (set.mem_univ k),\n    have h24 : false, from by {\n      rw mem_set_of_eq at h23,\n      exact h12 i k h21 h23,\n    },\n    show false, from h24,\n  have h25 : ∀ i j : ℤ, i ≠ j → i • α - ⌊i • α⌋ ∉ {k • α - ⌊k • α⌋}, from \n    assume i j h26, assume h27 : i • α - ⌊i • α⌋ = k • α - ⌊k • α⌋,\n    have h28 : i • α - ⌊i • α⌋ ∈ {k • α - ⌊k • α⌋}, from \n    set.mem_of_eq_of_mem h27 (set.mem_univ k),\n    have h29 : false, from by {\n      rw mem_singleton at h28,\n      exact h12 i k h26 h28,\n    },\n    show false, from h29,\n  have h30 : ∀ i j : ℤ, i ≠ j → i • α - ⌊i • α⌋ ∉ set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from \n    assume i j h31, assume h32 : i • α - ⌊i • α⌋ = k • α - ⌊k • α⌋,\n    have h33 : i • α - ⌊i • α⌋ ∈ set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from \n    eq.symm h32 ▸ set.mem_range k,\n    have h34 : false, from by {\n      rw set.mem_range at h33,\n      exact h12 i k h31 h33,\n    },\n    show false, from h34,\n  have h31 : S = set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from set.ext (λ (x : ℝ),\n    have h32 : x ∈ S ↔ x ∈ set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from iff.intro (\n      assume h33 : x ∈ S,\n      have h34 : ∃ (i : ℤ), x = i • α - ⌊i • α⌋, from h33,\n      have h35 : ∃ (i : ℤ), x = k • α - ⌊k • α⌋, from h34,\n      have h36 : ∃ (i : ℤ), x = k • α - ⌊k • α⌋, from h35,\n      show x ∈ set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from h36,\n    ) (\n      assume h33 : x ∈ set.range (λ (k : ℤ), k • α - ⌊k • α⌋),\n      have h34 : ∃ (i : ℤ), x = k • α - ⌊k • α⌋, from h33,\n      have h35 : ∃ (i : ℤ), x = i • α - ⌊i • α⌋, from h34,\n      have h36 : ∃ (i : ℤ), x = i • α - ⌊i • α⌋, from h35,\n      show x ∈ S, from h36\n    ),\n    show x ∈ S ↔ x ∈ set.range (λ (k : ℤ), k • α - ⌊k • α⌋), from h32),\n  have h32 : ∀ i : ℤ, i • α - ⌊i • α⌋ ∉ S, from assume i : ℤ,\n    have h33 : ∀ j : ℤ, i ≠ j → i • α\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit {α : Type*} [linear_ordered_field α]\n  (a : α) (h : ¬ is_rational a) : ∀ y : α, ∃ x ∈ set.range (λ n : ℤ, n • a), x ≠ y :=\nbegin\n  assume y : α,\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  -- $$i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,$$\n  have h1 : ∀ i j : ℤ, i ≠ j → floor (i • a) ≠ floor (j • a), from \n    by {\n      assume i j : ℤ,\n      assume h_neq : i ≠ j,\n      assume h_eq : floor (i • a) = floor (j • a),\n      show false, from h (rat.of_fractions i j h_neq h_eq),\n    },\n\n  -- Hence, $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  have h2 : ∀ i : ℤ, set.mem (fractional_part (i • a)) (set.range (λ i : ℤ, fractional_part (i • a))), from \n    by {\n      assume i : ℤ,\n      show set.mem (fractional_part (i • a)) (set.range (λ i : ℤ, fractional_part (i • a))), from by {\n        use i,\n        show fractional_part (i • a) = fractional_part (i • a), from rfl,\n      },\n    },\n\n  -- By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.\n  have h3 : ∃ x : α, x ≠ y ∧ ∀ ε > 0, ∃ N : ℕ, ∀ n : ℕ, n > N → |x - (fractional_part ((n : ℤ) • a))| < ε, from \n    by {\n      have h4 := (set.bounded_of_bdd_above (set.range (λ i : ℤ, fractional_part (i • a)))),\n      have h5 := (set.has_sup_finite_or_not_finite (set.range (λ i : ℤ, fractional_part (i • a)))),\n      have h6 := (set.finite.not_infinite h4 h5),\n      use y,\n      split,\n      {\n        assume h7,\n        exact h6 (by {\n          rw h7,\n          show set.finite (set.range (λ (i : ℤ), fractional_part (i • a))), from h4,\n        }),\n      },\n      {\n        assume ε,\n        assume h7,\n        have h8 := (set.has_sup_finite_or_not_finite (set.range (λ i : ℤ, fractional_part (i • a)))),\n        have h9 := (set.finite.not_infinite h4 h8),\n        have h10 := (set.finite_or_infinite h8),\n        have h11 := (set.finite_or_infinite_of_mem (set.range (λ i : ℤ, fractional_part (i • a))) y (h2 0)),\n        rcases h10 with h12 | h13,\n        {\n          have h14 := (set.finite_or_infinite_of_mem (set.range (λ i : ℤ, fractional_part (i • a))) y (h2 0)),\n          have h15 := (set.exists_sup_of_finite h14),\n          have h16 : (set.range (λ i : ℤ, fractional_part (i • a))) ⊆ set.Ico (0 : α) 1, from set.range_subset_iff.mpr (λ i, set.mem_Ico.mpr ⟨by linarith, by linarith⟩),\n          rcases h15 with ⟨x, h17, h18⟩,\n          use x,\n          split,\n          {\n            intro h19,\n            show false, from h17 (by {\n              rw h19,\n              show set.finite (set.range (λ (i : ℤ), fractional_part (i • a))), from h12,\n            }),\n          },\n          {\n            assume h19,\n            have h20 := (set.exists_sup_of_finite h11),\n            rcases h20 with ⟨x', h21, h22⟩,\n            have h23 := (set.subset_iff.mpr h16).mp h21,\n            have h24 := (set.subset_iff.mpr h16).mp h22,\n            have h25 := (set.Ico_subset_of_subset_of_subset h23 h24 h19),\n            have h26 := (set.mem_Ico.mp h25),\n            have h27 := (set.subset_iff.mpr h16).mp h22,\n            have h28 := (set.subset_iff.mpr h16).mp h21,\n            have h29 := (set.subset_iff.mpr h16).mp h22,\n            have h30 := (set.subset_iff.mpr h16).mp h21,\n            have h31 := (set.subset_iff.mpr h16).mp h22,\n            have h32 := (set.subset_iff.mpr h16).mp h21,\n            have h33 := (set.subset_iff.mpr h16).mp h22,\n            have h34 := (set.subset_iff.mpr h16).mp h21,\n            have h35 := (set.subset_iff.mpr h16).mp h22,\n            have h36 := (set.subset_iff.mpr h16).mp h21,\n            have h37 := (set.subset_iff.mpr h16).mp h22,\n            have h38 := (set.subset_iff.mpr h16).mp h21,\n            have h39 := (set.subset_iff.mpr h16).mp h22,\n            have h40 := (set.subset_iff.mpr h16).mp h21,\n            have h41 := (set.subset_iff.mpr h16).mp h22,\n            have h42 := (set.subset_iff.mpr h16).mp h21,\n            have h43 := (set.subset_iff.mpr h16).mp h22,\n            have h44 := (set.subset_iff.mpr h16).mp h21,\n            have h45 := (set.subset_iff.mpr h16).mp h22,\n            have h46 := (set.subset_iff.mpr h16).mp h21,\n            have h47 := (set.subset_iff.mpr h16).mp h22,\n            have h48 := (set.subset_iff.mpr h16).mp h21,\n            have h49 := (set.subset_iff.mpr h16).mp h22,\n            have h50 := (set.subset_iff.mpr h16).mp h21,\n            have h51 := (set.subset_iff.mpr h16).mp h22,\n            have h52 := (set.subset_iff.mpr h16).mp h21,\n            have h53 := (set.subset_iff.mpr h16).mp h22,\n            have h54 := (set.subset_iff.mpr h16).mp h21,\n            have h55 := (set.subset_iff.mpr h16).mp h22,\n            have h56 := (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit {α : Type*} [linear_ordered_field α] (a : α) (h : ¬ is_rat a) :\n  ∀ (y : α), ∃ (x : α), 0 ≤ x ∧ x < 1 ∧ |x - y| < 1 :=\nbegin\n  --Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$.\n  assume y,\n  have h1 : ∀ i j, i ≠ j → ¬ (a*i - ⌊a*i⌋ = a*j - ⌊a*j⌋), from by {\n    assume i j h2, intro h3,\n    have h4 := eq_rat_div_iff (a*i - ⌊a*i⌋) (a*j - ⌊a*j⌋) (i-j),\n    rw [h3, h4] at h, exact h,\n  },\n  \n  --If this were not true, then\n  --$$\n  --i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n  --$$\n  --which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$.\n  --Hence,\n  --$$\n  --S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n  --$$\n  --is an infinite subset of $\\left[0,1\\right]$.\n  have h2 : ∀ i j, i ≠ j → a * i - ⌊a * i⌋ ≠ a * j - ⌊a * j⌋, from by {\n    assume i j h3,\n    intro h4,\n    have h5 : a = (⌊a*i⌋ - ⌊a*j⌋)/(i-j), from eq_rat_div_iff (a*i - ⌊a*i⌋) (a*j - ⌊a*j⌋) (i-j) h4,\n    rw [h5, ← rat_of_int_eq_rat_of_int] at h,\n    exact h,\n  },\n\n  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.\n  have h3 : ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ ∃ (x : ℝ), 0 ≤ x ∧ x < 1 ∧ |x - y| < 1, from by\n  {\n    have h4 : ∃ (x : ℝ), 0 ≤ x ∧ x < 1 ∧ ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ |x - y| < 1, from by {\n      have h5 := Bolzano_Weierstrass (λ (n : ℕ), a * n - ⌊a * n⌋),\n      have h6 : ∀ (n : ℕ), 0 ≤ a * n - ⌊a * n⌋ ∧ a * n - ⌊a * n⌋ < 1, from by {\n        assume n,\n        have h7 := (floor_le_iff a).2 (le_refl _),\n        have h8 := (floor_lt_iff a).2 (lt_add_one _),\n        split; linarith,\n      },\n      have h9 : ∀ (n : ℕ), ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ |a * n - ⌊a * n⌋ - y| < 1, from by {\n        assume n,\n        have h10 := h5 (a * n - ⌊a * n⌋) (h6 n).1 (h6 n).2,\n        cases h10 with y h11,\n        use y,\n        have h12 := (h11 y).1,\n        split,\n        exact h12.1,\n        exact h12.2,\n        exact (h11 y).2,\n      },\n      have h13 : ∃ (x : ℝ), 0 ≤ x ∧ x < 1 ∧ ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ |x - y| < 1, from\n        exists.intro 0 (and.intro (by linarith) (and.intro zero_lt_one (exists.intro 0 (and.intro (by linarith) (and.intro zero_lt_one (by linarith))))))\n        ,\n      show ∃ (x : ℝ), 0 ≤ x ∧ x < 1 ∧ ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ |x - y| < 1, from\n        exists.elim (nat.find_min h13 h9) (λ (N : ℕ) (h14 : 0 ≤ a * N - ⌊a * N⌋ ∧ a * N - ⌊a * N⌋ < 1 ∧ ∃ (y : ℝ), 0 ≤ y ∧ y < 1 ∧ |a * N - ⌊a * N⌋ - y| < 1),\n        use a * N - ⌊a * N⌋,\n        split,\n        exact h14.1,\n        exact h14.2,\n        exact h14.3,\n    },\n    cases h4 with x h5,\n    use x,\n    have h6 : ∀ (y : ℝ), 0 ≤ y ∧ y < 1 → ∃ (x : ℝ), 0 ≤ x ∧ x < 1 ∧ |x - y| < 1, from by {\n      assume y h7,\n      use x,\n      split,\n      exact h5.1,\n      exact h5.2,\n      exact h5.3,\n    },\n    have h8 : ∀ (y : ℝ), 0 ≤ y ∧ y < 1, from by {\n      assume y,\n      have h9 : 0 ≤ y ∧ y < 1 ∨ 1 ≤ y ∧ y < 2, from le_total y 1,\n      cases h9,\n      exact h9,\n      have h10 : 2 ≤ y, from and.left h9,\n      have h11 : y < y + 1, from add_lt_add_right (by linarith) 1,\n      have h12 : y < y + 2, from add_lt_add_right h11 2,\n      have h13 : 2 ≤ y + 2, from by linarith,\n      have h14 : y ≤ y + 2, from by linarith,\n      have h15 : 2 ≤ y + 1, from by linarith,\n      have h16 : y ≤ y + 1, from by linarith,\n      have h17 := h6 y,\n      have h18 := h17 (and.intro h14 h11),\n      have h19 := h6 (y + 1),\n      have h20 := h19 (and.intro h15 h12),\n      have h21 := h6 (y + 2),\n      have h22 := h21 (and.intro h16 h13),\n      cases h18 with x h23,\n      cases h20 with x1 h24,\n      cases h22 with x2 h25,\n      have h26 : y < x1, from by linarith,\n      have h27 : x1 < x2, from by linarith,\n      have h28 : x < x1, from by linarith,\n      have h29 : x1 < x2, from by linarith,\n      have h30 : y < x, from by linarith,\n      have h31 : x2 < y + 2, from by linarith,\n      have h32 := h6 (y + 1),\n      have h33 := h32 (and.intro h15 h31),\n      cases h33 with x3 h34,\n      have h35 : x1 < x3, from by linarith,\n      have h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : \nlet frac_part : ℝ → ℝ := λ (x : ℝ), x - x.nat_abs in\ndense_subset (Icc 0 1) (range (frac_part ∘ (λ (n : ℤ), n * α))) :=\nbegin\n  assume frac_part,\n  have h1 : ∀ i j : ℤ, i ≠ j → frac_part (i * α) ≠ frac_part (j * α), \n  from by {\n    assume i j : ℤ,\n    assume h : i ≠ j,\n    assume h2 : frac_part (i * α) = frac_part (j * α),\n    have h3 : (i * α) - (i * α).nat_abs = (j * α) - (j * α).nat_abs, from by rw h2,\n    have h4 : (i * α) - (i * α).nat_abs = (j * α) - (j * α).nat_abs, from by rw h2,\n    have h5 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h4,\n    have h6 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h4,\n    have h7 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h6,\n    have h8 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h6,\n    have h9 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h8,\n    have h10 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h8,\n    have h11 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h10,\n    have h12 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h10,\n    have h13 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h12,\n    have h14 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h12,\n    have h15 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h14,\n    have h16 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h14,\n    have h17 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h16,\n    have h18 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h16,\n    have h19 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h18,\n    have h20 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h18,\n    have h21 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h20,\n    have h22 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h20,\n    have h23 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h22,\n    have h24 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h22,\n    have h25 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h24,\n    have h26 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h24,\n    have h27 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h26,\n    have h28 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h26,\n    have h29 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h28,\n    have h30 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h28,\n    have h31 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h30,\n    have h32 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h30,\n    have h33 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h32,\n    have h34 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h32,\n    have h35 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h34,\n    have h36 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h34,\n    have h37 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h36,\n    have h38 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h36,\n    have h39 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h38,\n    have h40 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h38,\n    have h41 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h40,\n    have h42 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h40,\n    have h43 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h42,\n    have h44 : (i * α) - (i * α).nat_abs = ((j * α) - (j * α).nat_abs), from by rw h42,\n\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : ¬(α ∈ ℚ)) : \n  ∀ y ∈ Icc 0 1, ∃ x ∈ (set.range (λ n : ℤ, n • α % 1)), |x - y| < 1 :=\nbegin\n  assume y h1,\n  --$\\alpha$ is an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. \n  have h2 : ∀ i j : ℤ, (i ≠ j) → (frac (i • α) ≠ frac (j • α)), from\n    assume (i j : ℤ) (h3 : i ≠ j),\n    have h4 : (i • α) - (i • α % 1) = frac (i • α), from frac_eq_sub_floor,\n    have h5 : (j • α) - (j • α % 1) = frac (j • α), from frac_eq_sub_floor,\n    have h6 : (i • α % 1) = (j • α % 1), from by {\n      rw [← h4, ← h5],\n      linarith,\n    },\n    have h7 : i • α = j • α, from by {\n      rw [← sub_eq_zero (i • α) (i • α % 1), ← sub_eq_zero (j • α) (j • α % 1)],\n      rw h6,\n    },\n    have h8 : i = j, from by {\n      rw ← h7,\n      exact mul_right_cancel hα,\n    },\n    show frac (i • α) ≠ frac (j • α), from by {\n      rw h8,\n      linarith,\n    },\n  \n  --If this were not true, then\n  --$$\n  --i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n  --$$\n  --which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n  --$$\n  --S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n  --$$\n  --is an infinite subset of $\\left[0,1\\right]$.\n  have h9 : ∀ i : ℤ, (frac (i • α) ∈ Icc 0 1), from\n    assume i : ℤ,\n    have h10 : (i • α) % 1 ∈ Icc 0 1, from\n      have h11 : (i • α) % 1 ∈ Ioo 0 1, from\n        have h12 : 0 ≤ (i • α) % 1, from\n          rw [← add_zero ((i • α) % 1)],\n          exact add_le_add_right (floor_le ((i • α) % 1)) _,\n        have h13 : (i • α) % 1 < 1, from\n          calc (i • α) % 1 = (i • α) - (i • α % 1) : by rw frac_eq_sub_floor\n          ... = i • α - floor (i • α) : by rw ← floor_eq_of_ge h12\n          ... < i • α : by linarith\n          ... ≤ 1 : by {\n            rw ← mul_one i,\n            apply mul_le_one_of_nonneg_of_le_one_right,\n            exact le_of_lt (lt_of_lt_of_le zero_lt_one hα),\n          },\n        show (i • α) % 1 ∈ Ioo 0 1, from by {split, exact h12, exact h13},\n      have h14 : 0 ≤ (i • α) % 1, from Ioo.left h11,\n      have h15 : (i • α) % 1 < 1, from Ioo.right h11,\n      show (i • α) % 1 ∈ Icc 0 1, from ⟨h14, h15⟩,\n    have h16 : 0 ≤ frac (i • α), from\n      have h17 : (i • α) % 1 = frac (i • α), from frac_eq_sub_floor,\n      rw h17 at h10,\n      exact Icc.left h10,\n    have h18 : frac (i • α) < 1, from\n      have h17 : (i • α) % 1 = frac (i • α), from frac_eq_sub_floor,\n      rw h17 at h10,\n      exact Icc.right h10,\n    show frac (i • α) ∈ Icc 0 1, from ⟨h16, h18⟩,\n  have h19 : (∀ i j : ℤ, i ≠ j → frac (i • α) ≠ frac (j • α)), from h2,\n  have h20 : ∀ i : ℤ, frac (i • α) ≠ 0, from\n    assume i : ℤ,\n    have h21 : frac (i • α) = frac (i • α % 1), from\n      have h22 : (i • α) = (i • α % 1) + (i • α % 1), from\n        calc (i • α) = (i • α % 1) + (i • α % 1) + (i • α - (i • α % 1) - (i • α % 1)) : by rw [sub_add_cancel, add_sub_of_le (floor_le ((i • α) % 1))]\n        ... = (i • α % 1) + (i • α % 1) + (floor ((i • α) % 1)) : by rw floor_eq_of_ge (le_of_lt (lt_of_lt_of_le zero_lt_one hα))\n        ... = (i • α % 1) + (i • α % 1) + (floor (i • α)) : by rw floor_eq_of_ge (le_of_lt (lt_of_lt_of_le zero_lt_one hα))\n        ... = (i • α % 1) + (i • α % 1) + 0 : by rw floor_eq_of_ge (le_of_lt (lt_of_lt_of_le zero_lt_one hα))\n        ... = (i • α % 1) + (i • α % 1) : by rw zero_add,\n      rw h22,\n      rw mod_add_div (i • α % 1),\n      rw div_eq_of_lt (lt_of_lt_of_le zero_lt_one hα),\n      rw add_zero,\n    rw h21,\n    exact (ne_iff_lt_and_gt.mpr (ne_zero_of_ne_zero_of_ne hα h19)).left,\n  have h21 : ∀ i : ℤ, frac (i • α) < 1, from\n    assume i : ℤ,\n    have h22 : (i • α) % 1 < 1, from\n      have h23 : (i • α) % 1 = frac (i • α), from frac_eq_sub_floor,\n      rw h23 at h10,\n      exact Icc.right h10,\n    have h24 : frac (i • α) = (i • α % 1), from frac_eq_sub_floor,\n    rw h24 at h22,\n    exact h22,\n  have h22 : ∀ i : ℤ, frac (i • α) ∈ Ioo 0 1, from\n    assume i : ℤ,\n    have h23 : frac (i • α) = (i • α % 1), from frac_eq_sub_floor,\n    rw h23 at h10,\n    exact h10,\n  have h23 : (∀ i : ℤ, ∃ j : ℤ, frac (i • α) = (frac (j • α))), from\n\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) :\nlet I := {n : ℤ | n ∈ (range (set.range (λ (n : ℤ), n * α)))} in\n∀ y : ℝ, 0 ≤ y ∧ y ≤ 1 → ∃ x : ℝ, 0 ≤ x ∧ x ≤ 1 ∧ ∀ ε > 0, ∃ N : ℤ, N ∈ I ∧ |x - (N * α)%R| < ε :=\nbegin\n  assume I,\n  assume y,\n  assume h1,\n  have h2 : ∀ i j : ℤ, i ≠ j → (i * α)%R - (i * α)%R.floor ≠ (j * α)%R - (j * α)%R.floor, \n  from by {\n    assume i j,\n    assume h3,\n    have h4 : (i * α)%R - (i * α)%R.floor = (j * α)%R - (j * α)%R.floor, from h3,\n    have h5 : α = ((i * α)%R.floor - (j * α)%R.floor) / (i - j), from h4.symm,\n    have h6 : is_rat α, from by {apply is_rat_div_of_rat, assumption,simp,},\n    exact hα h6,\n  },\n  have h3 : ∀ (i : ℤ), ∃! (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))),\n  from by {\n    assume i,\n    have h4 : (i * α)%R - (i * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))), \n    from by {\n      use i,\n      have h5 : (i * α)%R - (i * α)%R.floor = (i * α)%R - (i * α)%R.floor, from rfl,\n      exact h5,\n    },\n    use (i * α)%R - (i * α)%R.floor,\n    split,\n    {\n      exact h4,\n    },\n    {\n      assume x,\n      assume h6,\n      have h7 : ∀ (j : ℤ), j ≠ i → (i * α)%R - (i * α)%R.floor ≠ (j * α)%R - (j * α)%R.floor, from h2 i,\n      exact h7 i (λ h8, h6 (h8.symm ▸ h4)),\n    }\n  },\n  have h4 : ∀ (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))) → ∃ (i : ℤ), i * α - i * α.floor = x,\n  from by {\n    assume x,\n    assume h5,\n    have h6 : (x : ℝ) ∈ (range (set.range (λ (n : ℤ), n * α))), from h5,\n    cases h3 x.floor with i h7,\n    have h8 : i * α - i * α.floor = x, from h7.property.1 h6,\n    use i,\n    exact h8,\n  },\n  have h5 : ∀ (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))) → ∃ (i : ℤ), i ∈ I,\n  from by {\n    assume x,\n    assume h6,\n    cases h4 x h6 with i h7,\n    use i,\n    have h8 : (i * α)%R - (i * α)%R.floor = x, from h7,\n    have h9 : (i * α)%R - (i * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))), from h8,\n    exact h9,\n  },\n  have h6 : ∃ (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))) ∧ ∀ (ε : ℝ), ε > 0 → ∃ (N : ℤ), N ∈ I ∧ |x - (N * α)%R| < ε,\n  from by {\n    have h7 : ∀ (y : ℝ), 0 ≤ y ∧ y ≤ 1 → ∃ (x : ℝ), 0 ≤ x ∧ x ≤ 1 ∧ ∀ (ε : ℝ), ε > 0 → ∃ (N : ℤ), N ∈ I ∧ |x - (N * α)%R| < ε,\n    from by {\n      assume y,\n      assume h8,\n      cases h8 with h9 h10,\n      have h11 : ∃ (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))) ∧ ∀ (ε : ℝ), ε > 0 → ∃ (N : ℤ), N ∈ I ∧ |x - (N * α)%R| < ε,\n      from by {\n        have h12 : ∀ (y : ℝ), 0 < y → ∃ (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))) ∧ ∀ (ε : ℝ), ε > 0 → ∃ (N : ℤ), N ∈ I ∧ |x - (N * α)%R| < ε,\n        from by {\n          assume y,\n          assume h13,\n          have h14 : ∃ (i : ℤ), ∃ (j : ℤ), i ≠ j ∧ (i * α)%R - (i * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))) ∧ (j * α)%R - (j * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))) ∧ |((i * α)%R - (i * α)%R.floor) - ((j * α)%R - (j * α)%R.floor)| < y,\n          from by {\n            have h15 : ∃ (i : ℤ), (i * α)%R - (i * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))),\n            from by {\n              use 0,\n              have h16 : (0 * α)%R - (0 * α)%R.floor = (0 * α)%R - (0 * α)%R.floor, from rfl,\n              exact h16,\n            },\n            cases h15 with i h16,\n            have h17 : ∃! (x : ℝ), x ∈ (range (set.range (λ (n : ℤ), n * α))), from h3 i,\n            have h18 : ∃ (j : ℤ), j ≠ i ∧ (j * α)%R - (j * α)%R.floor ∈ (range (set.range (λ (n : ℤ), n * α))),\n            from by {\n              cases h17 with i' h17,\n              cases h17.property with h17 h18,\n              cases h17 with j h19,\n              cases h19 with h19 h20,\n              use j,\n              split,\n              {\n                exact h19.symm,\n              },\n              {\n                have h21 : (j * α)%R - (j * α)%R.floor = (j * α)%R - (j * α)%R.floor, from rfl,\n                exact h21,\n              }\n            },\n            cases h18 with j h19,\n            cases h19 with h19 h20,\n            use i,\n            use j,\n            split,\n            {\n              exact h19,\n            },\n            {\n              split,\n              {\n                exact h16,\n             \nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) [irrational α] : ∃ (S : set ℝ), (∀ n : ℕ, n ∈ S) ∧ (∀ n : ℕ, n ∉ S) :=\nbegin\n  --Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. \n  have h1 : ∀ i j : ℤ, i ≠ j → ¬ (fract α i = fract α j),\n  from by {\n    assume i j h2,\n    assume h3 : (fract α i) = (fract α j),\n    have h4 : α = (int_part α i - int_part α j)/(i-j), from by {rw [h3,fract_int_part_difference],ring},\n    have h5 : α ∈ ℚ, from by {apply int.eq_one_of_dvd, rw [← int.cast_dvd,← int.cast_dvd, h4], exact dvd_sub (dvd_mul_of_dvd_right (dvd_refl _) (int.cast_ne_zero.1 (ne_of_gt (int.cast_pos.2 (pos_of_ne_zero h2)))).symm) (dvd_mul_of_dvd_right (dvd_refl _) (int.cast_ne_zero.1 (ne_of_gt (int.cast_pos.2 (pos_of_ne_zero h2)))).symm)},\n    exact absurd h5 irrational.irrational_is_not_rational,\n  },\n\n  --If this were not true, then $i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor$, which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. \n  have h6 : ∀ i j : ℤ, i ≠ j → (int_part α i - int_part α j)/(i-j) ∉ ℚ,\n  from by {\n    assume i j h7,\n    assume h8 : (int_part α i - int_part α j)/(i-j) ∈ ℚ,\n    have h9 : (int_part α i - int_part α j)%Z = 0, from by {rw ← int.cast_inj, rw ← int.cast_inj, rw [← int.cast_dvd,← int.cast_dvd, h8], exact dvd_sub (dvd_mul_of_dvd_right (dvd_refl _) (int.cast_ne_zero.1 (ne_of_gt (int.cast_pos.2 (pos_of_ne_zero h7)))).symm) (dvd_mul_of_dvd_right (dvd_refl _) (int.cast_ne_zero.1 (ne_of_gt (int.cast_pos.2 (pos_of_ne_zero h7)))).symm)},\n    have h10 : (int_part α i - int_part α j) = 0, from by {rw h9, exact int.coe_nat_zero},\n    have h11 : (fract α i) = (fract α j), from by {rw [h10,fract_int_part_difference],ring},\n    exact absurd h11 (h1 i j h7),\n  },\n\n  --Hence, $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  have h12 : ∃ S : set ℤ, (∀ n : ℕ, n ∈ S) ∧ (∀ n : ℕ, n ∉ S), \n  from by {exact exists_infinite_set_of_distinct_fractional_parts α h1},\n\n  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. \n  have h13 : ∃ (S : set ℝ), (∀ n : ℕ, n ∈ S) ∧ (∀ n : ℕ, n ∉ S) := by {\n    cases h12 with S h14,\n    use (fract α '' S),\n    split,\n    {\n      intros n h15,\n      have h16 := set.mem_image_of_mem _ h15,\n      exact h16.right,\n    },\n    {\n      intros n h17,\n      have h18 := set.mem_image_of_mem _ h17,\n      exact h18.left,\n    },\n  },\n\n  --One can thus find pairs of elements of $S$ that are arbitrarily close. \n  cases h13 with S h19,\n  have h20 : ∃ (δ : ℝ), (0 < δ) ∧ (∀ x y : ℝ, (x ∈ S) ∧ (y ∈ S) ∧ (x ≠ y) → (|x - y| < δ)), from by {apply set.has_limit_point_iff.mp h19,},\n\n  --Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n  have h21 : set.has_limit_point (λ (x : ℝ), (|x| : ℝ)) S 0, from by {\n    cases h20 with δ h20,\n    rw [set.has_limit_point_iff],\n    use δ,\n    split,\n    {exact h20.left},\n    {\n      assume ε h22,\n      cases h20.right (ε/2) h22 with x h23,\n      cases h23 with y h23,\n      use x,\n      use y,\n      split,\n      {\n        intros h24,\n        have h25 := h23.left h24,\n        have h26 := h23.right h24,\n        have h27 := h23.right h24,\n        have h28 := h23.left h24,\n        have h29 := h23.left h24,\n        split,\n        {\n          have h30 : |x| < ε/2, from by {rw ← h25, rw ← abs_of_nonneg, exact le_of_lt h20.left},\n          have h31 : (ε/2) ≤ ε, from by {apply div_le_self, exact h20.left},\n          exact le_trans h30 h31,\n        },\n        {\n          have h32 : |y| < ε/2, from by {rw ← h26, rw ← abs_of_nonneg, exact le_of_lt h20.left},\n          have h33 : (ε/2) ≤ ε, from by {apply div_le_self, exact h20.left},\n          exact le_trans h32 h33,\n        },\n        {\n          have h34 : |x - y| < ε/2, from by {rw ← h27, exact h28},\n          have h35 : (ε/2) ≤ ε, from by {apply div_le_self, exact h20.left},\n          exact le_trans h34 h35,\n        },\n        {\n          have h36 : |y - x| < ε/2, from by {rw ← h28, exact h29},\n          have h37 : (ε/2) ≤ ε, from by {apply div_le_self, exact h20.left},\n          exact le_trans h36 h37,\n        },\n      },\n      {\n        rintros h38,\n        have h39 : |x| < ε/2, from by {rw ← h25, rw ← abs_of_nonneg, exact le_of_lt h20.left},\n        have h40 : (ε/2)\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : \n  (∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α) = ((j : ℝ) * α)) ∧ dense ((λ i : ℤ, i * α) '' univ) :=\nbegin\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  -- $$\n  -- i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n  -- $$\n  -- which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. \n  have h1 : ∀ (i j : ℤ), i ≠ j → ¬ ((i : ℝ) * α) = ((j : ℝ) * α), from by {\n    assume (i j : ℤ) (hij : i ≠ j),\n    assume h2 : (i : ℝ) * α = (j : ℝ) * α,\n    have h3 : (i : ℝ) * α = ((j : ℝ) * α - (i : ℝ) * α), from by {\n      rw h2, ring, },\n    have h4 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h5 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h6 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h7 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h8 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h9 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h10 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h11 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h12 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h13 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h14 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h15 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h16 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h17 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h18 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h19 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h20 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h21 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h22 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h23 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h24 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h25 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h26 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h27 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h28 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h29 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h30 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h31 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h32 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h33 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h34 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h35 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from by {\n      rw h2, ring, },\n    have h36 : (i : ℝ) * α = (j : ℝ) * α - (i : ℝ) * α, from\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < ε ↔ ((x0 - l) < ε) ∧ ((l - x0) < ε), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro ⟨ S_1, S_2 ⟩, \n    split; linarith, \n    rintro ⟨ S_3, S_4 ⟩, \n    split; linarith,\n    },\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : ε > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 ε h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 ε h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/lean_proof_with_comments-4_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.3229511171090674}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Patrick Massot, Scott Morrison, Mario Carneiro\n\nimport category_theory.concrete_category\nimport category_theory.full_subcategory\nimport category_theory.functor_category\nimport category_theory.adjunction\nimport category_theory.limits.types\nimport category_theory.natural_isomorphism\nimport category_theory.eq_to_hom\nimport topology.basic\nimport topology.opens\nimport order.galois_connection\n\nopen category_theory\nopen category_theory.nat_iso\nopen topological_space\n\nuniverse u\n\nnamespace category_theory.instances\n\n/-- The category of topological spaces and continuous maps. -/\n@[reducible] def Top : Type (u+1) := bundled topological_space\n\ninstance (x : Top) : topological_space x := x.str\n\nnamespace Top\ninstance : concrete_category @continuous := ⟨@continuous_id, @continuous.comp⟩\n\n-- local attribute [class] continuous\n-- instance {R S : Top} (f : R ⟶ S) : continuous (f : R → S) := f.2\n\nsection\nopen category_theory.limits\n\nvariables {J : Type u} [small_category J]\n\ndef limit (F : J ⥤ Top.{u}) : cone F :=\n{ X := ⟨limit (F ⋙ forget), ⨆ j, (F.obj j).str.induced (limit.π (F ⋙ forget) j)⟩,\n  π :=\n  { app := λ j, ⟨limit.π (F ⋙ forget) j, continuous_iff_induced_le.mpr (lattice.le_supr _ j)⟩,\n    naturality' := λ j j' f, subtype.eq ((limit.cone (F ⋙ forget)).π.naturality f) } }\n\ndef limit_is_limit (F : J ⥤ Top.{u}) : is_limit (limit F) :=\nby refine is_limit.of_faithful forget (limit.is_limit _) (λ s, ⟨_, _⟩) (λ s, rfl);\n   exact continuous_iff_le_coinduced.mpr (lattice.supr_le $ λ j,\n     induced_le_iff_le_coinduced.mpr $ continuous_iff_le_coinduced.mp (s.π.app j).property)\n\ninstance : has_limits.{u} Top.{u} :=\nλ J 𝒥 F, by exactI { cone := limit F, is_limit := limit_is_limit F }\n\ninstance : preserves_limits (forget : Top.{u} ⥤ Type u) :=\nλ J 𝒥 F, by exactI preserves_limit_of_preserves_limit_cone\n  (limit.is_limit F) (limit.is_limit (F ⋙ forget))\n\ndef colimit (F : J ⥤ Top.{u}) : cocone F :=\n{ X := ⟨colimit (F ⋙ forget), ⨅ j, (F.obj j).str.coinduced (colimit.ι (F ⋙ forget) j)⟩,\n  ι :=\n  { app := λ j, ⟨colimit.ι (F ⋙ forget) j, continuous_iff_le_coinduced.mpr (lattice.infi_le _ j)⟩,\n    naturality' := λ j j' f, subtype.eq ((colimit.cocone (F ⋙ forget)).ι.naturality f) } }\n\ndef colimit_is_colimit (F : J ⥤ Top.{u}) : is_colimit (colimit F) :=\nby refine is_colimit.of_faithful forget (colimit.is_colimit _) (λ s, ⟨_, _⟩) (λ s, rfl);\n   exact continuous_iff_induced_le.mpr (lattice.le_infi $ λ j,\n     induced_le_iff_le_coinduced.mpr $ continuous_iff_le_coinduced.mp (s.ι.app j).property)\n\ninstance : has_colimits.{u} Top.{u} :=\nλ J 𝒥 F, by exactI { cocone := colimit F, is_colimit := colimit_is_colimit F }\n\ninstance : preserves_colimits (forget : Top.{u} ⥤ Type u) :=\nλ J 𝒥 F, by exactI preserves_colimit_of_preserves_colimit_cocone\n  (colimit.is_colimit F) (colimit.is_colimit (F ⋙ forget))\n\nend\n\ndef discrete : Type u ⥤ Top.{u} :=\n{ obj := λ X, ⟨X, ⊤⟩,\n  map := λ X Y f, ⟨f, continuous_top⟩ }\n\ndef trivial : Type u ⥤ Top.{u} :=\n{ obj := λ X, ⟨X, ⊥⟩,\n  map := λ X Y f, ⟨f, continuous_bot⟩ }\n\ndef adj₁ : adjunction discrete forget :=\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, f,\n    inv_fun := λ f, ⟨f, continuous_top⟩,\n    left_inv := by tidy,\n    right_inv := by tidy },\n  unit := { app := λ X, id },\n  counit := { app := λ X, ⟨id, continuous_top⟩ } }\n\ndef adj₂ : adjunction forget trivial :=\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ⟨f, continuous_bot⟩,\n    inv_fun := λ f, f,\n    left_inv := by tidy,\n    right_inv := by tidy },\n  unit := { app := λ X, ⟨id, continuous_bot⟩ },\n  counit := { app := λ X, id } }\n\nend Top\n\nvariables {X : Top.{u}}\n\ninstance : small_category (opens X) := by apply_instance\n\ndef nbhd (x : X.α) := { U : opens X // x ∈ U }\ndef nbhds (x : X.α) : small_category (nbhd x) := begin unfold nbhd, apply_instance end\n\nend category_theory.instances\n\nopen category_theory.instances\n\nnamespace topological_space.opens\n\n/-- `opens.map f` gives the functor from open sets in Y to open set in X,\n    given by taking preimages under f. -/\ndef map\n  {X Y : Top.{u}} (f : X ⟶ Y) : opens Y ⥤ opens X :=\n{ obj := λ U, ⟨ f.val ⁻¹' U, f.property _ U.property ⟩,\n  map := λ U V i, ⟨ ⟨ λ a b, i.down.down b ⟩ ⟩ }.\n\n@[simp] lemma map_id_obj (X : Top.{u}) (U : opens X) : (map (𝟙 X)).obj U = U := by tidy\n\n@[simp] def map_id (X : Top.{u}) : map (𝟙 X) ≅ functor.id (opens X) :=\n{ hom := { app := λ U, 𝟙 U },\n  inv := { app := λ U, 𝟙 U } }\n\n-- We could make f g implicit here, but it's nice to be able to see when\n-- they are the identity (often!)\ndef map_iso {X Y : Top.{u}} (f g : X ⟶ Y) (h : f = g) : map f ≅ map g :=\nnat_iso.of_components (λ U, eq_to_iso (congr_fun (congr_arg _ (congr_arg _ h)) _) ) (by obviously)\n\n@[simp] def map_iso_id {X : Top.{u}} (h) : map_iso (𝟙 X) (𝟙 X) h = iso.refl (map _) := rfl\n\nend topological_space.opens\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/instances/topological_spaces.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.322842102689051}}
{"text": "import category_theory.limits.shapes.finite_products\nimport category_theory.preadditive.additive_functor\nimport category_theory.limits.preserves.shapes.binary_products\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category\n\nnamespace limits\n\nvariables {C : Type*} [category C] [preadditive C]\n\nvariables (X Y : C) [has_binary_coproduct X Y] [has_binary_product X Y]\n\ndef coprod_iso_prod :\n  X ⨿ Y ≅ X ⨯ Y :=\n{ hom := prod.lift (coprod.desc (𝟙 X) 0) (coprod.desc 0 (𝟙 Y)),\n  inv := limits.prod.fst ≫ coprod.inl + limits.prod.snd ≫ coprod.inr, }\n\n@[simp, reassoc]\nlemma coprod_inl_coprod_iso_prod_hom :\n  coprod.inl ≫ (coprod_iso_prod X Y).hom = prod.lift (𝟙 X) 0 :=\nbegin\n  dsimp [coprod_iso_prod],\n  simp only [prod.comp_lift, coprod.inl_desc],\nend\n\n@[simp, reassoc]\nlemma coprod_inr_coprod_iso_prod_hom :\n  coprod.inr ≫ (coprod_iso_prod X Y).hom = prod.lift 0 (𝟙 Y) :=\nbegin\n  dsimp [coprod_iso_prod],\n  simp only [prod.comp_lift, coprod.inr_desc],\nend\n\nend limits\n\nopen limits\n\nvariables (C : Type*) [category C] [has_finite_products C]\n\nstructure add_comm_group_object :=\n(X : C)\n(zero : ⊤_ C ⟶ X)\n(add : prod X X ⟶ X)\n(neg : X ⟶ X)\n(add_assoc' : prod.lift (limits.prod.map (𝟙 X) limits.prod.fst ≫ add) (limits.prod.snd ≫ limits.prod.snd) ≫ add =\n  (limits.prod.map (𝟙 X) add) ≫ add)\n(add_zero' : (prod.lift (terminal.from X ≫ zero) (𝟙 X)) ≫ add = 𝟙 X)\n(comm' : prod.lift limits.prod.snd limits.prod.fst ≫ add = add)\n(add_left_neg' : prod.lift neg (𝟙 X) ≫ add = terminal.from X ≫ zero)\n\nvariable {C}\n\ndef add_comm_group_object.add_comm_group_hom\n  (A : C) (G : add_comm_group_object C) : add_comm_group (A ⟶ G.X) :=\nbegin\n  let zero : A ⟶ G.X := terminal.from A ≫ G.zero,\n  let add := λ (g₁ g₂ : A ⟶ G.X), prod.lift g₁ g₂ ≫ G.add,\n  have zero_add : ∀ (g : A ⟶ G.X), add zero g = g,\n  { intro g,\n    dsimp [add, zero],\n    have h := g ≫= G.add_zero',\n    simp only [← assoc, comp_id] at h,\n    convert h,\n    ext,\n    { simp only [prod.comp_lift, prod.lift_fst, ← assoc],\n      congr' 1, },\n    { tidy, }, },\n  have comm : ∀ (g₁ g₂ : A ⟶ G.X), add g₁ g₂ = add g₂ g₁,\n  { intros g₁ g₂,\n    dsimp [add],\n    have h := prod.lift g₁ g₂ ≫= G.comm',\n    simp only [← assoc] at h,\n    convert h.symm,\n    tidy, },\n  exact\n  { add := add,\n    add_assoc := λ g₁ g₂ g₃, begin\n      dsimp,\n      have h := prod.lift g₁ (prod.lift g₂ g₃) ≫= G.add_assoc',\n      simp only [← assoc] at h,\n      convert h,\n      tidy,\n    end,\n    add_comm := comm,\n    zero := zero,\n    neg := λ g, g ≫ G.neg,\n    zero_add := zero_add,\n    add_zero := λ g, begin\n      change add g zero = g,\n      rw [comm, zero_add],\n    end,\n    add_left_neg := λ g, begin\n      change prod.lift (g ≫ G.neg) g ≫ G.add = terminal.from A ≫ G.zero,\n      have h := g ≫= G.add_left_neg',\n      simp only [← assoc] at h,\n      convert h,\n      tidy,\n    end, },\nend\n\n\nnamespace add_comm_group_object\n\nlemma add_comm_group_object_add [preadditive C] (G : add_comm_group_object C) :\n  G.add = limits.prod.fst + limits.prod.snd :=\nbegin\n  haveI : has_finite_biproducts C := limits.has_finite_biproducts.of_has_finite_products,\n  let s₀ := terminal.from G.X ≫ G.zero,\n  let s₁ := coprod.inl ≫ (coprod_iso_prod G.X G.X).hom ≫ G.add,\n  let s₂ := coprod.inr ≫ (coprod_iso_prod G.X G.X).hom ≫ G.add,\n  have hs₀ : ∃ t₀, t₀ = s₀ := ⟨s₀, rfl⟩,\n  have hs₁ : ∃ t₁, t₁ = s₁ := ⟨s₁, rfl⟩,\n  have hs₂ : ∃ t₂, t₂ = s₂ := ⟨s₂, rfl⟩,\n  rcases hs₀ with ⟨t₀, ht₀⟩,\n  rcases hs₁ with ⟨t₁, ht₁⟩,\n  rcases hs₂ with ⟨t₂, ht₂⟩,\n  have Gadd : G.add = limits.prod.fst ≫ t₁ + limits.prod.snd ≫ t₂,\n  { rw [ht₁, ht₂],\n    rw ← cancel_epi (coprod_iso_prod G.X G.X).hom,\n    ext,\n    { simp only [s₁, coprod_inl_coprod_iso_prod_hom_assoc, preadditive.comp_add,\n        prod.lift_fst_assoc, id_comp, prod.lift_snd_assoc, zero_comp, add_zero], },\n    { simp only [s₂, coprod_inr_coprod_iso_prod_hom_assoc, preadditive.comp_add,\n        prod.lift_fst_assoc, zero_comp, prod.lift_snd_assoc, id_comp, zero_add], }, },\n  have h₀ : t₁ ≫ t₁ = t₁ := by simpa only [Gadd, preadditive.comp_add, prod.map_fst_assoc,\n    id_comp, prod.map_snd_assoc, prod.lift_fst_assoc, preadditive.add_comp, assoc,\n    prod.lift_snd_assoc, zero_comp, add_zero]\n      using prod.lift (𝟙 G.X) (prod.lift 0 0) ≫= G.add_assoc',\n  have h₁ : t₁ = t₂ := by simpa only [Gadd, preadditive.comp_add,\n    prod.lift_fst_assoc, prod.lift_snd_assoc, id_comp, zero_comp,\n    add_zero, zero_add] using prod.lift 0 (𝟙 G.X) ≫= G.comm',\n  subst h₁,\n  have h₂ : t₀ ≫ t₁ + t₁ = 𝟙 G.X := by simpa only [ht₀, ← assoc, Gadd,\n    preadditive.comp_add, prod.lift_fst_assoc, prod.lift_snd_assoc, id_comp] using G.add_zero',\n  have h₃ : G.neg ≫ t₁ + t₁ = t₀ := by simpa only [ht₀, Gadd, preadditive.comp_add,\n    prod.lift_fst_assoc, prod.lift_snd_assoc, id_comp] using G.add_left_neg',\n  have h₄ : t₀ + t₁ = 𝟙 G.X,\n  { rw [← h₃, preadditive.add_comp, h₀, assoc, h₀, h₃] at h₂,\n    exact h₂, },\n  have h₅ : t₀ = 𝟙 G.X -t₁ := by simp only [← h₄, add_sub_cancel],\n  have h₆ : t₁ = 𝟙 G.X := by simpa only [h₀, h₅, preadditive.sub_comp,\n    id_comp, sub_self, zero_add] using h₂,\n  subst h₆,\n  simpa only [comp_id] using Gadd,\nend\n\nlocal attribute [instance] add_comm_group_object.add_comm_group_hom\n\nlemma add_eq {A : C} {G : add_comm_group_object C} (g₁ g₂ : A ⟶ G.X) :\n  g₁ + g₂ = prod.lift g₁ g₂ ≫ G.add := rfl\n\nlemma comp_add {A A': C} (f : A ⟶ A') {G : add_comm_group_object C}\n  (g₁ g₂ : A' ⟶ G.X) : f ≫ (g₁ + g₂) = f ≫ g₁ + f ≫ g₂ :=\nby simp only [add_eq, prod.comp_lift_assoc]\n\nvariable {C}\n\ndef hom (G₁ G₂ : add_comm_group_object C) :=\n{ f : G₁.X ⟶ G₂.X // G₁.add ≫ f = limits.prod.map f f ≫ G₂.add }\n\n@[simps]\ndef hom.id (G : add_comm_group_object C) : hom G G :=\n⟨𝟙 G.X, by tidy⟩\n\n@[simps]\ndef hom.comp {G₁ G₂ G₃ : add_comm_group_object C} (f : hom G₁ G₂) (g : hom G₂ G₃) :\n  hom G₁ G₃ :=\n⟨ f.1 ≫ g.1, begin\n  slice_lhs 1 2 { rw f.2,},\n  rw [assoc, g.2, prod.map_map_assoc],\nend⟩\n\ninstance : category (add_comm_group_object C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := λ X Y Z, hom.comp, }\n\n@[simp]\nlemma id_val (G : add_comm_group_object C) : subtype.val (𝟙 G) = 𝟙 G.X := rfl\n\n@[simp]\nlemma comp_val {G₁ G₂ G₃ : add_comm_group_object C} (f : G₁ ⟶ G₂) (g : G₂ ⟶ G₃) :\n  (f ≫ g).1 = f.1 ≫ g.1 := rfl\n\n@[ext]\nlemma hom_ext {G₁ G₂ : add_comm_group_object C} (f g : G₁ ⟶ G₂) (h : f.1 = g.1) : f = g := by tidy\n\nvariable (C)\n\n@[simps]\ndef forget : add_comm_group_object C ⥤ C :=\n{ obj := λ G, G.X,\n  map := λ G₁ G₂ f, f.1, }\n\nvariables {C} (F : C ⥤ add_comm_group_object C) (e : F ⋙ forget C ≅ 𝟭 C)\n\nlemma add_comp {A : C} {G G' : add_comm_group_object C} (f₁ f₂ : A ⟶ G.X) (g : G ⟶ G') :\n  (f₁ + f₂) ≫ g.1 = f₁ ≫ g.1 + f₂ ≫ g.1 :=\nby simp only [add_eq, assoc, g.2, prod.lift_map_assoc]\n\nnamespace preadditive_of\n\ninclude e\n\ndef hom_group (X Y : C) : add_comm_group (X ⟶ Y) :=\nbegin\n  let add : (X ⟶ Y) → (X ⟶ Y) → (X ⟶ Y) :=\n    λ f₁ f₂, ((f₁ ≫ e.inv.app Y : X ⟶ (F.obj Y).X) + (f₂ ≫ e.inv.app Y : X ⟶ (F.obj Y).X)) ≫ e.hom.app Y,\n  have add_comm : ∀ (f₁ f₂), add f₁ f₂ = add f₂ f₁ := λ f₁ f₂, begin\n    dsimp [add],\n    rw add_comm,\n  end,\n  let neg : (X ⟶ Y) → (X ⟶ Y) :=\n    λ f, (-(f ≫ e.inv.app Y : X ⟶ (F.obj Y).X)) ≫ e.hom.app Y,\n  exact\n  { add := add,\n    add_comm := add_comm,\n    add_assoc := λ f₁ f₂ f₃, begin\n      change add (add _ _ ) _ = add _ (add _ _ ),\n      dsimp [add],\n      simp only [assoc, iso.hom_inv_id_app],\n      dsimp,\n      rw [comp_id, comp_id, add_assoc],\n    end,\n    zero := (0 : X ⟶ (F.obj Y).X) ≫ e.hom.app Y,\n    zero_add := λ f, begin\n      change add _ _ = _,\n      dsimp [add],\n      simp only [assoc, iso.hom_inv_id_app],\n      dsimp,\n      rw [comp_id, zero_add, assoc, iso.inv_hom_id_app],\n      dsimp,\n      rw comp_id,\n    end,\n    add_zero := λ f, begin\n      change add _ _ = _,\n      dsimp [add],\n      simp only [assoc, iso.hom_inv_id_app],\n      dsimp,\n      rw [comp_id, add_zero, assoc, iso.inv_hom_id_app],\n      dsimp,\n      rw comp_id,\n    end,\n    neg := neg,\n    add_left_neg := λ f, begin\n      change add (neg f) f = _,\n      dsimp [add, neg],\n      simp only [assoc, iso.hom_inv_id_app],\n      dsimp,\n      simpa only [comp_id, add_left_neg],\n    end, },\nend\n\nlemma add_comp_inv_app {X Y : C} (f₁ f₂ : X ⟶ Y) :\n  (hom_group F e X Y).add f₁ f₂ ≫ e.inv.app Y =\n    (f₁ ≫ e.inv.app Y + f₂ ≫ e.inv.app Y : X ⟶ (F.obj Y).X) :=\nbegin\n  rw [← cancel_mono (e.hom.app Y), assoc, iso.inv_hom_id_app],\n  dsimp,\n  simpa only [comp_id],\nend\n\nend preadditive_of\n\ndef preadditive_of : preadditive C :=\n{ hom_group := λ X Y, preadditive_of.hom_group F e X Y,\n  comp_add' := λ X₁ X₂ X₃ f g₁ g₂, begin\n    change f ≫ (preadditive_of.hom_group F e _ _).add _ _ =\n      (preadditive_of.hom_group F e _ _).add _ _,\n    rw ← cancel_mono (e.inv.app X₃),\n    dsimp,\n    simp only [preadditive_of.add_comp_inv_app, comp_add, assoc],\n  end,\n  add_comp' := λ X₁ X₂ X₃ f₁ f₂ g, begin\n    change (preadditive_of.hom_group F e _ _).add _ _ ≫ g =\n      (preadditive_of.hom_group F e _ _).add _ _,\n    rw ← cancel_mono (e.inv.app X₃),\n    dsimp,\n    have hg := e.inv.naturality g,\n    simp only [functor.id_map, functor.comp_map, forget_map] at hg,\n    simp only [preadditive_of.add_comp_inv_app, assoc, hg,\n      reassoc_of (preadditive_of.add_comp_inv_app F e f₁ f₂)],\n    simp only [← assoc],\n    apply add_comp,\n  end, }\n\nend add_comm_group_object\n\nnamespace preadditive\n\nvariable (C)\n\n@[simps]\ndef to_add_comm_group_object [preadditive C] : C ⥤ add_comm_group_object C :=\n{ obj := λ X,\n  { X := X,\n    zero := 0,\n    add := limits.prod.fst + limits.prod.snd,\n    neg := -𝟙 X,\n    add_assoc' := begin\n      simp only [comp_add, limits.prod.map_fst, comp_id, limits.prod.map_snd, prod.lift_fst, prod.lift_snd],\n      apply add_assoc,\n    end,\n    add_zero' := by tidy,\n    comm' := by simpa only [comp_add, prod.lift_fst, prod.lift_snd] using add_comm _ _,\n    add_left_neg' := by simp, },\n  map := λ X Y f, ⟨f, by simp⟩, }\n\n@[simps]\ndef to_add_comm_group_object_comp_forget_iso [preadditive C] :\n  (to_add_comm_group_object C) ⋙ add_comm_group_object.forget C ≅ 𝟭 C := iso.refl _\n\ninstance : reflects_isomorphisms (add_comm_group_object.forget C) :=\n⟨λ G₁ G₂ f hf, begin\n  haveI : is_iso f.1 := hf,\n  refine ⟨⟨⟨inv f.1, _⟩, _, _⟩⟩,\n  { simp only [← cancel_mono f.1, ← cancel_epi (limits.prod.map f.1 f.1), f.2, assoc,\n    is_iso.inv_hom_id, comp_id, prod.map_map_assoc, is_iso.hom_inv_id, prod.map_id_id, id_comp], },\n  { apply add_comm_group_object.hom_ext,\n    exact is_iso.hom_inv_id f.1, },\n  { apply add_comm_group_object.hom_ext,\n    exact is_iso.inv_hom_id f.1, },\nend⟩\n\nvariable {C}\n\nlemma add_eq_of_add_comm_group [hC : preadditive C] {X Y : C} {G : add_comm_group_object C} (e : G.X ≅ Y)\n  (f₁ f₂ : X ⟶ Y) : f₁ + f₂ = prod.lift (f₁ ≫ e.inv) (f₂ ≫ e.inv) ≫ G.add ≫ e.hom :=\nby simp only [G.add_comm_group_object_add, add_comp, comp_add, prod.lift_fst_assoc, assoc,\n    iso.inv_hom_id, comp_id, prod.lift_snd_assoc]\n\nend preadditive\n\nnamespace functor\n\nvariables {C} {D : Type*} [category D]\n  [has_finite_products D] (F : C ⥤ D)\n  [hF₀ : preserves_limit (functor.empty.{0} C) F]\n  [hF₂ : preserves_limits_of_shape (discrete (walking_pair)) F]\n\ninclude F hF₀ hF₂\n\nlemma preserves_limit_pair_compatibility₁ {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :\n  limits.prod.map (F.map f₁) (F.map f₂) = (preserves_limit_pair.iso F X₁ X₂).inv ≫\n    F.map (limits.prod.map f₁ f₂) ≫ (preserves_limit_pair.iso F Y₁ Y₂).hom :=\nbegin\n  rw [← cancel_epi ((preserves_limit_pair.iso F X₁ X₂).hom), iso.hom_inv_id_assoc],\n  ext,\n  { simp only [preserves_limit_pair.iso_hom, assoc, limits.prod.map_fst, prod_comparison_fst_assoc,\n      prod_comparison_fst, ← F.map_comp], },\n  { simp only [preserves_limit_pair.iso_hom, assoc, limits.prod.map_snd, prod_comparison_snd_assoc,\n      prod_comparison_snd, ← F.map_comp], },\nend\n\nlemma preserves_limit_pair_compatibility₂ {X₁ X₂ : C} :\n  limits.prod.lift (limits.prod.snd : F.obj X₁ ⨯ F.obj X₂ ⟶ F.obj X₂) (limits.prod.fst : F.obj X₁ ⨯ F.obj X₂ ⟶ F.obj X₁)\n  = (preserves_limit_pair.iso F X₁ X₂).inv ≫\n  F.map (limits.prod.lift (limits.prod.snd : X₁ ⨯ X₂ ⟶ X₂) (limits.prod.fst : X₁ ⨯ X₂ ⟶ X₁)) ≫\n    (preserves_limit_pair.iso F X₂ X₁).hom :=\nbegin\n  rw [← cancel_epi ((preserves_limit_pair.iso F X₁ X₂).hom), iso.hom_inv_id_assoc],\n  ext,\n  { simp only [preserves_limit_pair.iso_hom, prod_comparison_snd, prod_comparison_fst,\n      prod.lift_fst, assoc, ← F.map_comp], },\n  { simp only [preserves_limit_pair.iso_hom, prod_comparison_snd, prod_comparison_fst,\n      prod.lift_snd, assoc, ← F.map_comp], },\nend\n\nlemma preserves_limit_pair_compatibility₃ {X Y₁ Y₂ : C} (f₁ : X ⟶ Y₁) (f₂ : X ⟶ Y₂) :\n  limits.prod.lift (F.map f₁) (F.map f₂) =\n    F.map (limits.prod.lift f₁ f₂) ≫ (preserves_limit_pair.iso F Y₁ Y₂).hom :=\nbegin\n  ext,\n  { simp only [prod.lift_fst, preserves_limit_pair.iso_hom, assoc, prod_comparison_fst,\n      ← F.map_comp], },\n  { simp only [prod.lift_snd, preserves_limit_pair.iso_hom, assoc, prod_comparison_snd,\n      ← F.map_comp], },\nend\n\nlemma preserves_limit_pair_compatibility_fst (X₁ X₂ : C) :\n  limits.prod.fst = (preserves_limit_pair.iso F X₁ X₂).inv ≫ F.map limits.prod.fst  :=\nby rw [← cancel_epi (preserves_limit_pair.iso F X₁ X₂).hom,\n    iso.hom_inv_id_assoc, preserves_limit_pair.iso_hom, prod_comparison_fst]\n\nlemma preserves_limit_pair_compatibility_snd (X₁ X₂ : C) :\n  limits.prod.snd = (preserves_limit_pair.iso F X₁ X₂).inv ≫ F.map limits.prod.snd  :=\nby rw [← cancel_epi (preserves_limit_pair.iso F X₁ X₂).hom,\n    iso.hom_inv_id_assoc, preserves_limit_pair.iso_hom, prod_comparison_snd]\n\nlemma preserves_terminal_compatibility (X : C) :\n  terminal.from (F.obj X) = F.map (terminal.from X) ≫ (preserves_terminal.iso F).hom :=\nsubsingleton.elim _ _\n\n@[simps]\ndef map_add_comm_group_object.obj (G : add_comm_group_object C) : add_comm_group_object D :=\n{ X := F.obj G.X,\n  zero := (preserves_terminal.iso F).inv ≫ F.map G.zero,\n  add := (preserves_limit_pair.iso F G.X G.X).inv ≫ F.map G.add,\n  neg := F.map G.neg,\n  add_assoc' := begin\n    rw ← cancel_epi (limits.prod.map (𝟙 (F.obj G.X)) (preserves_limit_pair.iso F G.X G.X).hom),\n    rw ← cancel_epi (preserves_limit_pair.iso F G.X (G.X ⨯ G.X)).hom,\n    convert F.congr_map G.add_assoc',\n    { simp only [F.map_comp, ← assoc],\n      congr' 1,\n      simp only [assoc, ← cancel_mono (preserves_limit_pair.iso F G.X G.X).hom,\n        iso.inv_hom_id, comp_id],\n      ext,\n      { rw [assoc, assoc, assoc, prod.lift_fst, prod.map_map_assoc, comp_id,\n          preserves_limit_pair_compatibility_fst, iso.hom_inv_id_assoc,\n          ← F.map_id, preserves_limit_pair_compatibility₁, assoc, assoc,\n          iso.hom_inv_id_assoc, iso.hom_inv_id_assoc, ← F.map_comp,\n          ← F.map_comp, prod.lift_fst], },\n      { rw [assoc, assoc, assoc, prod.lift_snd, prod.map_snd_assoc,\n          preserves_limit_pair_compatibility_snd, assoc, iso.hom_inv_id_assoc,\n          preserves_limit_pair_compatibility_snd, iso.hom_inv_id_assoc,\n          ← F.map_comp, ← F.map_comp, prod.lift_snd], }, },\n    { rw [prod.map_map_assoc, id_comp, iso.hom_inv_id_assoc, F.map_comp, ← assoc, ← assoc],\n      conv_lhs { rw ← F.map_id, },\n      simp only [preserves_limit_pair_compatibility₁, assoc, iso.hom_inv_id_assoc], },\n  end,\n  add_zero' := begin\n    simp only [preserves_terminal_compatibility, assoc, iso.hom_inv_id_assoc, ← F.map_comp],\n    nth_rewrite 0 ← F.map_id,\n    rw [preserves_limit_pair_compatibility₃, assoc, iso.hom_inv_id_assoc, ← F.map_comp,\n      G.add_zero', F.map_id],\n  end,\n  comm' := by simp only [preserves_limit_pair_compatibility₂, assoc,\n    iso.hom_inv_id_assoc, ← F.map_comp, G.comm'],\n  add_left_neg' := begin\n    rw [← F.map_id, preserves_limit_pair_compatibility₃, assoc, iso.hom_inv_id_assoc,\n      ← F.map_comp, G.add_left_neg', preserves_terminal_compatibility, assoc,\n      iso.hom_inv_id_assoc, F.map_comp],\n  end, }\n\n@[simps]\ndef map_add_comm_group_object :\n  add_comm_group_object C ⥤ add_comm_group_object D :=\n{ obj := λ G, map_add_comm_group_object.obj F G,\n  map := λ G₁ G₂ f, ⟨F.map f.1, by simp only [map_add_comm_group_object.obj_add, assoc,\n    ← F.map_comp, f.2, preserves_limit_pair_compatibility₁, assoc, iso.hom_inv_id_assoc]⟩, }\n\nlemma map_add_comm_group_object_add_compatibility {X : C} (G : add_comm_group_object C)\n  (f₁ f₂ : X ⟶ G.X) : F.map (prod.lift f₁ f₂ ≫ G.add) =\n  prod.lift (F.map f₁) (F.map f₂) ≫ (F.map_add_comm_group_object.obj G).add :=\nbegin\n  dsimp [map_add_comm_group_object.obj],\n  simp only [assoc, preserves_limit_pair_compatibility₃, iso.hom_inv_id_assoc, ← F.map_comp],\nend\n\nlemma additive_of_preserves_binary_products [hC : preadditive C] [hD : preadditive D] : F.additive :=\n⟨λ X Y f₁ f₂, begin\n  let G := (preadditive.to_add_comm_group_object C).obj Y,\n  let e : G.X ≅ Y := iso.refl _,\n  let G' := F.map_add_comm_group_object.obj G,\n  let e' : G'.X ≅ F.obj Y := iso.refl _,\n  rw preadditive.add_eq_of_add_comm_group e,\n  rw preadditive.add_eq_of_add_comm_group e',\n  dsimp [e, e'],\n  repeat { erw comp_id, },\n  exact map_add_comm_group_object_add_compatibility F G f₁ f₂,\nend⟩\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/group_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358548398982, "lm_q2_score": 0.5039061705290806, "lm_q1q2_score": 0.32282036031599703}}
{"text": "/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.finsupp.basic\nimport Mathlib.linear_algebra.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 u_3 u_5 u_6 \n\nnamespace Mathlib\n\n/-!\n# Properties of the semimodule `α →₀ M`\n\nGiven an `R`-semimodule `M`, the `R`-semimodule structure on `α →₀ M` is defined in\n`data.finsupp.basic`.\n\nIn this file we define `finsupp.supported s` to be the set `{f : α →₀ M | f.support ⊆ s}`\ninterpreted as a submodule of `α →₀ M`. We also define `linear_map` versions of various maps:\n\n* `finsupp.lsingle a : M →ₗ[R] ι →₀ M`: `finsupp.single a` as a linear map;\n\n* `finsupp.lapply a : (ι →₀ M) →ₗ[R] M`: the map `λ f, f a` as a linear map;\n\n* `finsupp.lsubtype_domain (s : set α) : (α →₀ M) →ₗ[R] (s →₀ M)`: restriction to a subtype as a\n  linear map;\n\n* `finsupp.restrict_dom`: `finsupp.filter` as a linear map to `finsupp.supported s`;\n\n* `finsupp.lsum`: `finsupp.sum` or `finsupp.lift_add_hom` as a `linear_map`;\n\n* `finsupp.total α M R (v : ι → M)`: sends `l : ι → R` to the linear combination of `v i` with\n  coefficients `l i`;\n\n* `finsupp.total_on`: a restricted version of `finsupp.total` with domain `finsupp.supported R R s`\n  and codomain `submodule.span R (v '' s)`;\n\n* `finsupp.supported_equiv_finsupp`: a linear equivalence between the functions `α →₀ M` supported\n  on `s` and the functions `s →₀ M`;\n\n* `finsupp.lmap_domain`: a linear map version of `finsupp.map_domain`;\n\n* `finsupp.dom_lcongr`: a `linear_equiv` version of `finsupp.dom_congr`;\n\n* `finsupp.congr`: if the sets `s` and `t` are equivalent, then `supported M R s` is equivalent to\n  `supported M R t`;\n\n* `finsupp.lcongr`: a `linear_equiv`alence between `α →₀ M` and `β →₀ N` constructed using `e : α ≃\n  β` and `e' : M ≃ₗ[R] N`.\n\n## Tags\n\nfunction with finite support, semimodule, linear algebra\n-/\n\nnamespace finsupp\n\n\n/-- Interpret `finsupp.single a` as a linear map. -/\ndef lsingle {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (a : α) : linear_map R M (α →₀ M) :=\n  linear_map.mk (add_monoid_hom.to_fun (single_add_hom a)) sorry sorry\n\n/-- Two `R`-linear maps from `finsupp X M` which agree on each `single x y` agree everywhere. -/\ntheorem lhom_ext {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    {φ : linear_map R (α →₀ M) N} {ψ : linear_map R (α →₀ M) N}\n    (h : ∀ (a : α) (b : M), coe_fn φ (single a b) = coe_fn ψ (single a b)) : φ = ψ :=\n  linear_map.to_add_monoid_hom_injective (add_hom_ext h)\n\n/-- Two `R`-linear maps from `finsupp X M` which agree on each `single x y` agree everywhere.\n\nWe formulate this fact using equality of linear maps `φ.comp (lsingle a)` and `ψ.comp (lsingle a)`\nso that the `ext` tactic can apply a type-specific extensionality lemma to prove equality of these\nmaps. E.g., if `M = R`, then it suffices to verify `φ (single a 1) = ψ (single a 1)`. -/\ntheorem lhom_ext' {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    {φ : linear_map R (α →₀ M) N} {ψ : linear_map R (α →₀ M) N}\n    (h : ∀ (a : α), linear_map.comp φ (lsingle a) = linear_map.comp ψ (lsingle a)) : φ = ψ :=\n  lhom_ext fun (a : α) => linear_map.congr_fun (h a)\n\n/-- Interpret `λ (f : α →₀ M), f a` as a linear map. -/\ndef lapply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (a : α) : linear_map R (α →₀ M) M :=\n  linear_map.mk (add_monoid_hom.to_fun (apply_add_hom a)) sorry sorry\n\n/-- Interpret `finsupp.subtype_domain s` as a linear map. -/\ndef lsubtype_domain {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (s : set α) : linear_map R (α →₀ M) (↥s →₀ M) :=\n  linear_map.mk (subtype_domain fun (x : α) => x ∈ s) sorry sorry\n\ntheorem lsubtype_domain_apply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (f : α →₀ M) :\n    coe_fn (lsubtype_domain s) f = subtype_domain (fun (x : α) => x ∈ s) f :=\n  rfl\n\n@[simp] theorem lsingle_apply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (a : α) (b : M) : coe_fn (lsingle a) b = single a b :=\n  rfl\n\n@[simp] theorem lapply_apply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (a : α) (f : α →₀ M) : coe_fn (lapply a) f = coe_fn f a :=\n  rfl\n\n@[simp] theorem ker_lsingle {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (a : α) : linear_map.ker (lsingle a) = ⊥ :=\n  linear_map.ker_eq_bot_of_injective (single_injective a)\n\ntheorem lsingle_range_le_ker_lapply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (t : set α) (h : disjoint s t) :\n    (supr fun (a : α) => supr fun (H : a ∈ s) => linear_map.range (lsingle a)) ≤\n        infi fun (a : α) => infi fun (H : a ∈ t) => linear_map.ker (lapply a) :=\n  sorry\n\ntheorem infi_ker_lapply_le_bot {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] : (infi fun (a : α) => linear_map.ker (lapply a)) ≤ ⊥ :=\n  sorry\n\ntheorem supr_lsingle_range {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] : (supr fun (a : α) => linear_map.range (lsingle a)) = ⊤ :=\n  sorry\n\ntheorem disjoint_lsingle_lsingle {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (t : set α) (hs : disjoint s t) :\n    disjoint (supr fun (a : α) => supr fun (H : a ∈ s) => linear_map.range (lsingle a))\n        (supr fun (a : α) => supr fun (H : a ∈ t) => linear_map.range (lsingle a)) :=\n  sorry\n\ntheorem span_single_image {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set M) (a : α) :\n    submodule.span R (single a '' s) = submodule.map (lsingle a) (submodule.span R s) :=\n  sorry\n\n/-- `finsupp.supported M R s` is the `R`-submodule of all `p : α →₀ M` such that `p.support ⊆ s`. -/\ndef supported {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] (s : set α) : submodule R (α →₀ M) :=\n  submodule.mk (set_of fun (p : α →₀ M) => ↑(support p) ⊆ s) sorry sorry sorry\n\ntheorem mem_supported {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {s : set α} (p : α →₀ M) : p ∈ supported M R s ↔ ↑(support p) ⊆ s :=\n  iff.rfl\n\ntheorem mem_supported' {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {s : set α} (p : α →₀ M) :\n    p ∈ supported M R s ↔ ∀ (x : α), ¬x ∈ s → coe_fn p x = 0 :=\n  sorry\n\ntheorem single_mem_supported {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {s : set α} {a : α} (b : M) (h : a ∈ s) :\n    single a b ∈ supported M R s :=\n  set.subset.trans support_single_subset (iff.mpr finset.singleton_subset_set_iff h)\n\ntheorem supported_eq_span_single {α : Type u_1} (R : Type u_4) [semiring R] (s : set α) :\n    supported R R s = submodule.span R ((fun (i : α) => single i 1) '' s) :=\n  sorry\n\n/-- Interpret `finsupp.filter s` as a linear map from `α →₀ M` to `supported M R s`. -/\ndef restrict_dom {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] (s : set α) : linear_map R (α →₀ M) ↥(supported M R s) :=\n  linear_map.cod_restrict (supported M R s)\n    (linear_map.mk (filter fun (_x : α) => _x ∈ s) sorry sorry) sorry\n\n@[simp] theorem restrict_dom_apply {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (l : α →₀ M) :\n    ↑(coe_fn (restrict_dom M R s) l) = filter (fun (_x : α) => _x ∈ s) l :=\n  rfl\n\ntheorem restrict_dom_comp_subtype {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) :\n    linear_map.comp (restrict_dom M R s) (submodule.subtype (supported M R s)) = linear_map.id :=\n  sorry\n\ntheorem range_restrict_dom {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) : linear_map.range (restrict_dom M R s) = ⊤ :=\n  sorry\n\ntheorem supported_mono {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {s : set α} {t : set α} (st : s ⊆ t) : supported M R s ≤ supported M R t :=\n  fun (l : α →₀ M) (h : l ∈ supported M R s) => set.subset.trans h st\n\n@[simp] theorem supported_empty {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] : supported M R ∅ = ⊥ :=\n  sorry\n\n@[simp] theorem supported_univ {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] : supported M R set.univ = ⊤ :=\n  iff.mpr eq_top_iff fun (l : α →₀ M) (_x : l ∈ ⊤) => set.subset_univ ↑(support l)\n\ntheorem supported_Union {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] {δ : Type u_3} (s : δ → set α) :\n    supported M R (set.Union fun (i : δ) => s i) = supr fun (i : δ) => supported M R (s i) :=\n  sorry\n\ntheorem supported_union {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (t : set α) :\n    supported M R (s ∪ t) = supported M R s ⊔ supported M R t :=\n  sorry\n\ntheorem supported_Inter {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] {ι : Type u_3} (s : ι → set α) :\n    supported M R (set.Inter fun (i : ι) => s i) = infi fun (i : ι) => supported M R (s i) :=\n  sorry\n\ntheorem supported_inter {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) (t : set α) :\n    supported M R (s ∩ t) = supported M R s ⊓ supported M R t :=\n  sorry\n\ntheorem disjoint_supported_supported {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] {s : set α} {t : set α} (h : disjoint s t) :\n    disjoint (supported M R s) (supported M R t) :=\n  sorry\n\ntheorem disjoint_supported_supported_iff {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [nontrivial M] {s : set α} {t : set α} :\n    disjoint (supported M R s) (supported M R t) ↔ disjoint s t :=\n  sorry\n\n/-- Interpret `finsupp.restrict_support_equiv` as a linear equivalence between\n`supported M R s` and `s →₀ M`. -/\ndef supported_equiv_finsupp {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] (s : set α) :\n    linear_equiv R (↥(supported M R s)) (↥s →₀ M) :=\n  let F : ↥(supported M R s) ≃ (↥s →₀ M) := restrict_support_equiv s M;\n  equiv.to_linear_equiv F sorry\n\n/-- Lift a family of linear maps `M →ₗ[R] N` indexed by `x : α` to a linear map from `α →₀ M` to\n`N` using `finsupp.sum`. This is an upgraded version of `finsupp.lift_add_hom`.\nWe define this as an additive equivalence. For a commutative `R`, this equivalence can be\nupgraded further to a linear equivalence. -/\ndef lsum {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N] :\n    (α → linear_map R M N) ≃+ linear_map R (α →₀ M) N :=\n  add_equiv.mk\n    (fun (F : α → linear_map R M N) =>\n      linear_map.mk (fun (d : α →₀ M) => sum d fun (i : α) => ⇑(F i)) sorry sorry)\n    (fun (F : linear_map R (α →₀ M) N) (x : α) => linear_map.comp F (lsingle x)) sorry sorry sorry\n\n@[simp] theorem coe_lsum {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    (f : α → linear_map R M N) :\n    ⇑(coe_fn lsum f) = fun (d : α →₀ M) => sum d fun (i : α) => ⇑(f i) :=\n  rfl\n\ntheorem lsum_apply {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    (f : α → linear_map R M N) (l : α →₀ M) :\n    coe_fn (coe_fn lsum f) l = sum l fun (b : α) => ⇑(f b) :=\n  rfl\n\ntheorem lsum_single {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    (f : α → linear_map R M N) (i : α) (m : M) :\n    coe_fn (coe_fn lsum f) (single i m) = coe_fn (f i) m :=\n  sum_single_index (linear_map.map_zero (f i))\n\ntheorem lsum_symm_apply {α : Type u_1} {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N]\n    (f : linear_map R (α →₀ M) N) (x : α) :\n    coe_fn (add_equiv.symm lsum) f x = linear_map.comp f (lsingle x) :=\n  rfl\n\n/-- Interpret `finsupp.lmap_domain` as a linear map. -/\ndef lmap_domain {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {α' : Type u_5} (f : α → α') : linear_map R (α →₀ M) (α' →₀ M) :=\n  linear_map.mk (map_domain f) sorry map_domain_smul\n\n@[simp] theorem lmap_domain_apply {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} (f : α → α') (l : α →₀ M) :\n    coe_fn (lmap_domain M R f) l = map_domain f l :=\n  rfl\n\n@[simp] theorem lmap_domain_id {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] : lmap_domain M R id = linear_map.id :=\n  linear_map.ext fun (l : α →₀ M) => map_domain_id\n\ntheorem lmap_domain_comp {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} {α'' : Type u_6} (f : α → α')\n    (g : α' → α'') :\n    lmap_domain M R (g ∘ f) = linear_map.comp (lmap_domain M R g) (lmap_domain M R f) :=\n  linear_map.ext fun (l : α →₀ M) => map_domain_comp\n\ntheorem supported_comap_lmap_domain {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} (f : α → α') (s : set α') :\n    supported M R (f ⁻¹' s) ≤ submodule.comap (lmap_domain M R f) (supported M R s) :=\n  sorry\n\ntheorem lmap_domain_supported {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} [Nonempty α] (f : α → α') (s : set α) :\n    submodule.map (lmap_domain M R f) (supported M R s) = supported M R (f '' s) :=\n  sorry\n\ntheorem lmap_domain_disjoint_ker {α : Type u_1} (M : Type u_2) (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} (f : α → α') {s : set α}\n    (H : ∀ (a b : α), a ∈ s → b ∈ s → f a = f b → a = b) :\n    disjoint (supported M R s) (linear_map.ker (lmap_domain M R f)) :=\n  sorry\n\n/-- Interprets (l : α →₀ R) as linear combination of the elements in the family (v : α → M) and\n    evaluates this linear combination. -/\nprotected def total (α : Type u_1) (M : Type u_2) (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] (v : α → M) : linear_map R (α →₀ R) M :=\n  coe_fn lsum fun (i : α) => linear_map.smul_right linear_map.id (v i)\n\ntheorem total_apply {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {v : α → M} (l : α →₀ R) :\n    coe_fn (finsupp.total α M R v) l = sum l fun (i : α) (a : R) => a • v i :=\n  rfl\n\ntheorem total_apply_of_mem_supported {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {v : α → M} {l : α →₀ R} {s : finset α}\n    (hs : l ∈ supported R R ↑s) :\n    coe_fn (finsupp.total α M R v) l = finset.sum s fun (i : α) => coe_fn l i • v i :=\n  sorry\n\n@[simp] theorem total_single {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {v : α → M} (c : R) (a : α) :\n    coe_fn (finsupp.total α M R v) (single a c) = c • v a :=\n  sorry\n\ntheorem total_unique {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] [unique α] (l : α →₀ R) (v : α → M) :\n    coe_fn (finsupp.total α M R v) l = coe_fn l Inhabited.default • v Inhabited.default :=\n  sorry\n\ntheorem total_range {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {v : α → M} (h : function.surjective v) :\n    linear_map.range (finsupp.total α M R v) = ⊤ :=\n  sorry\n\ntheorem range_total {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {v : α → M} :\n    linear_map.range (finsupp.total α M R v) = submodule.span R (set.range v) :=\n  sorry\n\ntheorem lmap_domain_total {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} {M' : Type u_6} [add_comm_monoid M']\n    [semimodule R M'] {v : α → M} {v' : α' → M'} (f : α → α') (g : linear_map R M M')\n    (h : ∀ (i : α), coe_fn g (v i) = v' (f i)) :\n    linear_map.comp (finsupp.total α' M' R v') (lmap_domain R R f) =\n        linear_map.comp g (finsupp.total α M R v) :=\n  sorry\n\ntheorem total_emb_domain {α : Type u_1} (R : Type u_4) [semiring R] {α' : Type u_5} {M' : Type u_6}\n    [add_comm_monoid M'] [semimodule R M'] {v' : α' → M'} (f : α ↪ α') (l : α →₀ R) :\n    coe_fn (finsupp.total α' M' R v') (emb_domain f l) =\n        coe_fn (finsupp.total α M' R (v' ∘ ⇑f)) l :=\n  sorry\n\ntheorem total_map_domain {α : Type u_1} (R : Type u_4) [semiring R] {α' : Type u_5} {M' : Type u_6}\n    [add_comm_monoid M'] [semimodule R M'] {v' : α' → M'} (f : α → α') (hf : function.injective f)\n    (l : α →₀ R) :\n    coe_fn (finsupp.total α' M' R v') (map_domain f l) = coe_fn (finsupp.total α M' R (v' ∘ f)) l :=\n  sorry\n\ntheorem span_eq_map_total {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {v : α → M} (s : set α) :\n    submodule.span R (v '' s) = submodule.map (finsupp.total α M R v) (supported R R s) :=\n  sorry\n\ntheorem mem_span_iff_total {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {v : α → M} {s : set α} {x : M} :\n    x ∈ submodule.span R (v '' s) ↔\n        ∃ (l : α →₀ R), ∃ (H : l ∈ supported R R s), coe_fn (finsupp.total α M R v) l = x :=\n  sorry\n\n/-- `finsupp.total_on M v s` interprets `p : α →₀ R` as a linear combination of a\nsubset of the vectors in `v`, mapping it to the span of those vectors.\n\nThe subset is indicated by a set `s : set α` of indices.\n-/\nprotected def total_on (α : Type u_1) (M : Type u_2) (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] (v : α → M) (s : set α) :\n    linear_map R ↥(supported R R s) ↥(submodule.span R (v '' s)) :=\n  linear_map.cod_restrict (submodule.span R (v '' s))\n    (linear_map.comp (finsupp.total α M R v) (submodule.subtype (supported R R s))) sorry\n\ntheorem total_on_range {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {v : α → M} (s : set α) : linear_map.range (finsupp.total_on α M R v s) = ⊤ :=\n  sorry\n\ntheorem total_comp {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R] [add_comm_monoid M]\n    [semimodule R M] {α' : Type u_5} {v : α → M} (f : α' → α) :\n    finsupp.total α' M R (v ∘ f) = linear_map.comp (finsupp.total α M R v) (lmap_domain R R f) :=\n  sorry\n\ntheorem total_comap_domain {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {α' : Type u_5} {v : α → M} (f : α → α') (l : α' →₀ R)\n    (hf : set.inj_on f (f ⁻¹' ↑(support l))) :\n    coe_fn (finsupp.total α M R v) (comap_domain f l hf) =\n        finset.sum (finset.preimage (support l) f hf) fun (i : α) => coe_fn l (f i) • v i :=\n  sorry\n\ntheorem total_on_finset {α : Type u_1} {M : Type u_2} (R : Type u_4) [semiring R]\n    [add_comm_monoid M] [semimodule R M] {s : finset α} {f : α → R} (g : α → M)\n    (hf : ∀ (a : α), f a ≠ 0 → a ∈ s) :\n    coe_fn (finsupp.total α M R g) (on_finset s f hf) = finset.sum s fun (x : α) => f x • g x :=\n  sorry\n\n/-- An equivalence of domains induces a linear equivalence of finitely supported functions. -/\nprotected def dom_lcongr {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {α₁ : Type u_1} {α₂ : Type u_3} (e : α₁ ≃ α₂) :\n    linear_equiv R (α₁ →₀ M) (α₂ →₀ M) :=\n  add_equiv.to_linear_equiv (finsupp.dom_congr e) sorry\n\n@[simp] theorem dom_lcongr_single {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {α₁ : Type u_1} {α₂ : Type u_3} (e : α₁ ≃ α₂) (i : α₁) (m : M) :\n    coe_fn (finsupp.dom_lcongr e) (single i m) = single (coe_fn e i) m :=\n  sorry\n\n/-- An equivalence of sets induces a linear equivalence of `finsupp`s supported on those sets. -/\ndef congr {α : Type u_1} {M : Type u_2} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {α' : Type u_3} (s : set α) (t : set α') (e : ↥s ≃ ↥t) :\n    linear_equiv R ↥(supported M R s) ↥(supported M R t) :=\n  linear_equiv.trans (supported_equiv_finsupp s)\n    (linear_equiv.trans (finsupp.dom_lcongr e) (linear_equiv.symm (supported_equiv_finsupp t)))\n\n/-- An equivalence of domain and a linear equivalence of codomain induce a linear equivalence of the\ncorresponding finitely supported functions. -/\ndef lcongr {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R] [add_comm_monoid M]\n    [semimodule R M] [add_comm_monoid N] [semimodule R N] {ι : Type u_1} {κ : Type u_5} (e₁ : ι ≃ κ)\n    (e₂ : linear_equiv R M N) : linear_equiv R (ι →₀ M) (κ →₀ N) :=\n  linear_equiv.trans (finsupp.dom_lcongr e₁)\n    (linear_equiv.mk (map_range (⇑e₂) (linear_equiv.map_zero e₂)) sorry sorry\n      (map_range ⇑(linear_equiv.symm e₂) sorry) sorry sorry)\n\n@[simp] theorem lcongr_single {M : Type u_2} {N : Type u_3} {R : Type u_4} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N] {ι : Type u_1}\n    {κ : Type u_5} (e₁ : ι ≃ κ) (e₂ : linear_equiv R M N) (i : ι) (m : M) :\n    coe_fn (lcongr e₁ e₂) (single i m) = single (coe_fn e₁ i) (coe_fn e₂ m) :=\n  sorry\n\nend finsupp\n\n\ntheorem linear_map.map_finsupp_total {R : Type u_1} {M : Type u_2} {N : Type u_3} [semiring R]\n    [add_comm_monoid M] [semimodule R M] [add_comm_monoid N] [semimodule R N] (f : linear_map R M N)\n    {ι : Type u_4} {g : ι → M} (l : ι →₀ R) :\n    coe_fn f (coe_fn (finsupp.total ι M R g) l) = coe_fn (finsupp.total ι N R (⇑f ∘ g)) l :=\n  sorry\n\ntheorem submodule.exists_finset_of_mem_supr {R : Type u_1} {M : Type u_2} [semiring R]\n    [add_comm_monoid M] [semimodule R M] {ι : Type u_3} (p : ι → submodule R M) {m : M}\n    (hm : m ∈ supr fun (i : ι) => p i) :\n    ∃ (s : finset ι), m ∈ supr fun (i : ι) => supr fun (H : i ∈ s) => p i :=\n  sorry\n\ntheorem mem_span_finset {R : Type u_1} {M : Type u_2} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {s : finset M} {x : M} :\n    x ∈ submodule.span R ↑s ↔ ∃ (f : M → R), (finset.sum s fun (i : M) => f i • i) = x :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/finsupp_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3227963716692408}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel\n-/\nimport analysis.normed_space.dual\nimport data.set.intervals.disjoint\nimport measure_theory.measure.haar_lebesgue\nimport analysis.calculus.extend_deriv\nimport measure_theory.function.locally_integrable\nimport measure_theory.integral.set_integral\nimport measure_theory.integral.vitali_caratheodory\n\n/-!\n# Integral over an interval\n\nIn this file we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ` if `a ≤ b` and\n`-∫ x in Ioc b a, f x ∂μ` if `b ≤ a`. We prove a few simple properties and several versions of the\n[fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus).\n\nRecall that its first version states that the function `(u, v) ↦ ∫ x in u..v, f x` has derivative\n`(δu, δv) ↦ δv • f b - δu • f a` at `(a, b)` provided that `f` is continuous at `a` and `b`,\nand its second version states that, if `f` has an integrable derivative on `[a, b]`, then\n`∫ x in a..b, f' x = f b - f a`.\n\n## Main statements\n\n### FTC-1 for Lebesgue measure\n\nWe prove several versions of FTC-1, all in the `interval_integral` namespace. Many of them follow\nthe naming scheme `integral_has(_strict?)_(f?)deriv(_within?)_at(_of_tendsto_ae?)(_right|_left?)`.\nThey formulate FTC in terms of `has(_strict?)_(f?)deriv(_within?)_at`.\nLet us explain the meaning of each part of the name:\n\n* `_strict` means that the theorem is about strict differentiability;\n* `f` means that the theorem is about differentiability in both endpoints; incompatible with\n  `_right|_left`;\n* `_within` means that the theorem is about one-sided derivatives, see below for details;\n* `_of_tendsto_ae` means that instead of continuity the theorem assumes that `f` has a finite limit\n  almost surely as `x` tends to `a` and/or `b`;\n* `_right` or `_left` mean that the theorem is about differentiability in the right (resp., left)\n  endpoint.\n\nWe also reformulate these theorems in terms of `(f?)deriv(_within?)`. These theorems are named\n`(f?)deriv(_within?)_integral(_of_tendsto_ae?)(_right|_left?)` with the same meaning of parts of the\nname.\n\n### One-sided derivatives\n\nTheorem `integral_has_fderiv_within_at_of_tendsto_ae` states that `(u, v) ↦ ∫ x in u..v, f x` has a\nderivative `(δu, δv) ↦ δv • cb - δu • ca` within the set `s × t` at `(a, b)` provided that `f` tends\nto `ca` (resp., `cb`) almost surely at `la` (resp., `lb`), where possible values of `s`, `t`, and\ncorresponding filters `la`, `lb` are given in the following table.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n\nWe use a typeclass `FTC_filter` to make Lean automatically find `la`/`lb` based on `s`/`t`. This way\nwe can formulate one theorem instead of `16` (or `8` if we leave only non-trivial ones not covered\nby `integral_has_deriv_within_at_of_tendsto_ae_(left|right)` and\n`integral_has_fderiv_at_of_tendsto_ae`). Similarly,\n`integral_has_deriv_within_at_of_tendsto_ae_right` works for both one-sided derivatives using the\nsame typeclass to find an appropriate filter.\n\n### FTC for a locally finite measure\n\nBefore proving FTC for the Lebesgue measure, we prove a few statements that can be seen as FTC for\nany measure. The most general of them,\n`measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae`, states the following. Let `(la, la')`\nbe an `FTC_filter` pair of filters around `a` (i.e., `FTC_filter a la la'`) and let `(lb, lb')` be\nan `FTC_filter` pair of filters around `b`. If `f` has finite limits `ca` and `cb` almost surely at\n`la'` and `lb'`, respectively, then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)` as `ua` and `va` tend to `la` while\n`ub` and `vb` tend to `lb`.\n\n### FTC-2 and corollaries\n\nWe use FTC-1 to prove several versions of FTC-2 for the Lebesgue measure, using a similar naming\nscheme as for the versions of FTC-1. They include:\n* `interval_integral.integral_eq_sub_of_has_deriv_right_of_le` - most general version, for functions\n  with a right derivative\n* `interval_integral.integral_eq_sub_of_has_deriv_at'` - version for functions with a derivative on\n  an open set\n* `interval_integral.integral_deriv_eq_sub'` - version that is easiest to use when computing the\n  integral of a specific function\n\nWe then derive additional integration techniques from FTC-2:\n* `interval_integral.integral_mul_deriv_eq_deriv_mul` - integration by parts\n* `interval_integral.integral_comp_mul_deriv''` - integration by substitution\n\nMany applications of these theorems can be found in the file `analysis.special_functions.integrals`.\n\nNote that the assumptions of FTC-2 are formulated in the form that `f'` is integrable. To use it in\na context with the stronger assumption that `f'` is continuous, one can use\n`continuous_on.interval_integrable` or `continuous_on.integrable_on_Icc` or\n`continuous_on.integrable_on_interval`.\n\n## Implementation notes\n\n### Avoiding `if`, `min`, and `max`\n\nIn order to avoid `if`s in the definition, we define `interval_integrable f μ a b` as\n`integrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ`. For any `a`, `b` one of these\nintervals is empty and the other coincides with `set.interval_oc a b = set.Ioc (min a b) (max a b)`.\n\nSimilarly, we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`.\nAgain, for any `a`, `b` one of these integrals is zero, and the other gives the expected result.\n\nThis way some properties can be translated from integrals over sets without dealing with\nthe cases `a ≤ b` and `b ≤ a` separately.\n\n### Choice of the interval\n\nWe use integral over `set.interval_oc a b = set.Ioc (min a b) (max a b)` instead of one of the other\nthree possible intervals with the same endpoints for two reasons:\n\n* this way `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ` holds whenever\n  `f` is integrable on each interval; in particular, it works even if the measure `μ` has an atom\n  at `b`; this rules out `set.Ioo` and `set.Icc` intervals;\n* with this definition for a probability measure `μ`, the integral `∫ x in a..b, 1 ∂μ` equals\n  the difference $F_μ(b)-F_μ(a)$, where $F_μ(a)=μ(-∞, a]$ is the\n  [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function)\n  of `μ`.\n\n### `FTC_filter` class\n\nAs explained above, many theorems in this file rely on the typeclass\n`FTC_filter (a : ℝ) (l l' : filter ℝ)` to avoid code duplication. This typeclass combines four\nassumptions:\n\n- `pure a ≤ l`;\n- `l' ≤ 𝓝 a`;\n- `l'` has a basis of measurable sets;\n- if `u n` and `v n` tend to `l`, then for any `s ∈ l'`, `Ioc (u n) (v n)` is eventually included\n  in `s`.\n\nThis typeclass has the following “real” instances: `(a, pure a, ⊥)`, `(a, 𝓝[≥] a, 𝓝[>] a)`,\n`(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`.\nFurthermore, we have the following instances that are equal to the previously mentioned instances:\n`(a, 𝓝[{a}] a, ⊥)` and `(a, 𝓝[univ] a, 𝓝[univ] a)`.\nWhile the difference between `Ici a` and `Ioi a` doesn't matter for theorems about Lebesgue measure,\nit becomes important in the versions of FTC about any locally finite measure if this measure has an\natom at one of the endpoints.\n\n### Combining one-sided and two-sided derivatives\n\nThere are some `FTC_filter` instances where the fact that it is one-sided or\ntwo-sided depends on the point, namely `(x, 𝓝[Icc a b] x, 𝓝[Icc a b] x)`\n(resp. `(x, 𝓝[[a, b]] x, 𝓝[[a, b]] x)`, where `[a, b] = set.interval a b`),\nwith `x ∈ Icc a b` (resp. `x ∈ [a, b]`).\nThis results in a two-sided derivatives for `x ∈ Ioo a b` and one-sided derivatives for\n`x ∈ {a, b}`. Other instances could be added when needed (in that case, one also needs to add\ninstances for `filter.is_measurably_generated` and `filter.tendsto_Ixx_class`).\n\n## Tags\n\nintegral, fundamental theorem of calculus, FTC-1, FTC-2, change of variables in integrals\n-/\n\nnoncomputable theory\nopen topological_space (second_countable_topology)\nopen measure_theory set classical filter function\n\nopen_locale classical topological_space filter ennreal big_operators interval\n\nvariables {ι 𝕜 E F : Type*} [normed_group E]\n\n/-!\n### Integrability at an interval\n-/\n\n/-- A function `f` is called *interval integrable* with respect to a measure `μ` on an unordered\ninterval `a..b` if it is integrable on both intervals `(a, b]` and `(b, a]`. One of these\nintervals is always empty, so this property is equivalent to `f` being integrable on\n`(min a b, max a b]`. -/\ndef interval_integrable (f : ℝ → E) (μ : measure ℝ) (a b : ℝ) :=\nintegrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ\n\nsection\n\nvariables {f : ℝ → E} {a b : ℝ} {μ : measure ℝ}\n\n/-- A function is interval integrable with respect to a given measure `μ` on `a..b` if and\n  only if it is integrable on `interval_oc a b` with respect to `μ`. This is an equivalent\n  defintion of `interval_integrable`. -/\nlemma interval_integrable_iff : interval_integrable f μ a b ↔ integrable_on f (Ι a b) μ :=\nby rw [interval_oc_eq_union, integrable_on_union, interval_integrable]\n\n/-- If a function is interval integrable with respect to a given measure `μ` on `a..b` then\n  it is integrable on `interval_oc a b` with respect to `μ`. -/\nlemma interval_integrable.def (h : interval_integrable f μ a b) : integrable_on f (Ι a b) μ :=\ninterval_integrable_iff.mp h\n\nlemma interval_integrable_iff_integrable_Ioc_of_le (hab : a ≤ b) :\n  interval_integrable f μ a b ↔ integrable_on f (Ioc a b) μ :=\nby rw [interval_integrable_iff, interval_oc_of_le hab]\n\n/-- If a function is integrable with respect to a given measure `μ` then it is interval integrable\n  with respect to `μ` on `interval a b`. -/\nlemma measure_theory.integrable.interval_integrable (hf : integrable f μ) :\n  interval_integrable f μ a b :=\n⟨hf.integrable_on, hf.integrable_on⟩\n\nlemma measure_theory.integrable_on.interval_integrable (hf : integrable_on f [a, b] μ) :\n  interval_integrable f μ a b :=\n⟨measure_theory.integrable_on.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_interval),\n measure_theory.integrable_on.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_interval')⟩\n\nlemma interval_integrable_const_iff {c : E} :\n  interval_integrable (λ _, c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ :=\nby simp only [interval_integrable_iff, integrable_on_const]\n\n@[simp] lemma interval_integrable_const [is_locally_finite_measure μ] {c : E} :\n  interval_integrable (λ _, c) μ a b :=\ninterval_integrable_const_iff.2 $ or.inr measure_Ioc_lt_top\n\nend\n\nnamespace interval_integrable\n\nsection\n\nvariables {f : ℝ → E} {a b c d : ℝ} {μ ν : measure ℝ}\n\n@[symm] lemma symm (h : interval_integrable f μ a b) : interval_integrable f μ b a :=\nh.symm\n\n@[refl] lemma refl : interval_integrable f μ a a :=\nby split; simp\n\n@[trans] lemma trans (hab : interval_integrable f μ a b) (hbc : interval_integrable f μ b c) :\n  interval_integrable f μ a c :=\n⟨(hab.1.union hbc.1).mono_set Ioc_subset_Ioc_union_Ioc,\n  (hbc.2.union hab.2).mono_set Ioc_subset_Ioc_union_Ioc⟩\n\nlemma trans_iterate {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, interval_integrable f μ (a k) (a $ k+1)) :\n  interval_integrable f μ (a 0) (a n) :=\nbegin\n  induction n with n hn,\n  { simp },\n  { exact (hn (λ k hk, hint k (hk.trans n.lt_succ_self))).trans (hint n n.lt_succ_self) }\nend\n\nlemma neg (h : interval_integrable f μ a b) : interval_integrable (-f) μ a b :=\n⟨h.1.neg, h.2.neg⟩\n\nlemma norm (h : interval_integrable f μ a b) :\n  interval_integrable (λ x, ∥f x∥) μ a b  :=\n⟨h.1.norm, h.2.norm⟩\n\nlemma abs {f : ℝ → ℝ} (h : interval_integrable f μ a b) :\n  interval_integrable (λ x, |f x|) μ a b  :=\nh.norm\n\nlemma mono (hf : interval_integrable f ν a b) (h1 : [c, d] ⊆ [a, b]) (h2 : μ ≤ ν) :\n  interval_integrable f μ c d :=\ninterval_integrable_iff.mpr $ hf.def.mono\n  (interval_oc_subset_interval_oc_of_interval_subset_interval h1) h2\n\nlemma mono_set (hf : interval_integrable f μ a b) (h : [c, d] ⊆ [a, b]) :\n  interval_integrable f μ c d :=\nhf.mono h rfl.le\n\nlemma mono_measure (hf : interval_integrable f ν a b) (h : μ ≤ ν) :\n  interval_integrable f μ a b :=\nhf.mono rfl.subset h\n\nlemma mono_set_ae (hf : interval_integrable f μ a b) (h : Ι c d ≤ᵐ[μ] Ι a b) :\n  interval_integrable f μ c d :=\ninterval_integrable_iff.mpr $ hf.def.mono_set_ae h\n\nlemma mono_fun [normed_group F] {g : ℝ → F}\n  (hf : interval_integrable f μ a b) (hgm : ae_strongly_measurable g (μ.restrict (Ι a b)))\n  (hle : (λ x, ∥g x∥) ≤ᵐ[μ.restrict (Ι a b)] (λ x, ∥f x∥)) : interval_integrable g μ a b :=\ninterval_integrable_iff.2 $ hf.def.integrable.mono hgm hle\n\nlemma mono_fun' {g : ℝ → ℝ} (hg : interval_integrable g μ a b)\n  (hfm : ae_strongly_measurable f (μ.restrict (Ι a b)))\n  (hle : (λ x, ∥f x∥) ≤ᵐ[μ.restrict (Ι a b)] g) : interval_integrable f μ a b :=\ninterval_integrable_iff.2 $ hg.def.integrable.mono' hfm hle\n\nprotected lemma ae_strongly_measurable (h : interval_integrable f μ a b) :\n  ae_strongly_measurable f (μ.restrict (Ioc a b)):=\nh.1.ae_strongly_measurable\n\nprotected lemma ae_strongly_measurable' (h : interval_integrable f μ a b) :\n  ae_strongly_measurable f (μ.restrict (Ioc b a)):=\nh.2.ae_strongly_measurable\n\nend\n\nvariables {f g : ℝ → E} {a b : ℝ} {μ : measure ℝ}\n\nlemma smul [normed_field 𝕜] [normed_space 𝕜 E]\n  {f : ℝ → E} {a b : ℝ} {μ : measure ℝ} (h : interval_integrable f μ a b) (r : 𝕜) :\n  interval_integrable (r • f) μ a b :=\n⟨h.1.smul r, h.2.smul r⟩\n\n@[simp] \n\n@[simp] lemma sub (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  interval_integrable (λ x, f x - g x) μ a b :=\n⟨hf.1.sub hg.1, hf.2.sub hg.2⟩\n\nlemma mul_continuous_on {f g : ℝ → ℝ}\n  (hf : interval_integrable f μ a b) (hg : continuous_on g [a, b]) :\n  interval_integrable (λ x, f x * g x) μ a b :=\nbegin\n  rw interval_integrable_iff at hf ⊢,\n  exact hf.mul_continuous_on_of_subset hg measurable_set_Ioc is_compact_interval Ioc_subset_Icc_self\nend\n\nlemma continuous_on_mul {f g : ℝ → ℝ} (hf : interval_integrable f μ a b)\n  (hg : continuous_on g [a, b]) :\n  interval_integrable (λ x, g x * f x) μ a b :=\nby simpa [mul_comm] using hf.mul_continuous_on hg\n\nend interval_integrable\n\nsection\n\nvariables {μ : measure ℝ} [is_locally_finite_measure μ]\n\nlemma continuous_on.interval_integrable {u : ℝ → E} {a b : ℝ}\n  (hu : continuous_on u (interval a b)) : interval_integrable u μ a b :=\n(continuous_on.integrable_on_Icc hu).interval_integrable\n\nlemma continuous_on.interval_integrable_of_Icc {u : ℝ → E} {a b : ℝ} (h : a ≤ b)\n  (hu : continuous_on u (Icc a b)) : interval_integrable u μ a b :=\ncontinuous_on.interval_integrable ((interval_of_le h).symm ▸ hu)\n\n/-- A continuous function on `ℝ` is `interval_integrable` with respect to any locally finite measure\n`ν` on ℝ. -/\nlemma continuous.interval_integrable {u : ℝ → E} (hu : continuous u) (a b : ℝ) :\n  interval_integrable u μ a b :=\nhu.continuous_on.interval_integrable\n\nend\n\nsection\n\nvariables {μ : measure ℝ} [is_locally_finite_measure μ] [conditionally_complete_linear_order E]\n  [order_topology E] [second_countable_topology E]\n\nlemma monotone_on.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : monotone_on u (interval a b)) :\n  interval_integrable u μ a b :=\nbegin\n  rw interval_integrable_iff,\n  exact (hu.integrable_on_compact is_compact_interval).mono_set Ioc_subset_Icc_self,\nend\n\nlemma antitone_on.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : antitone_on u (interval a b)) :\n  interval_integrable u μ a b :=\n@monotone_on.interval_integrable (order_dual E) _ _ _ _ _ _ _ _ _ hu\n\nlemma monotone.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : monotone u) :\n  interval_integrable u μ a b :=\n(hu.monotone_on _).interval_integrable\n\nlemma antitone.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : antitone u) :\n  interval_integrable u μ a b :=\n(hu.antitone_on _).interval_integrable\n\nend\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : ℝ → E` has a finite limit at `l' ⊓ μ.ae`. Then `f` is interval integrable on\n`u..v` provided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter ℝ` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\nlemma filter.tendsto.eventually_interval_integrable_ae {f : ℝ → E} {μ : measure ℝ}\n  {l l' : filter ℝ}  (hfm : strongly_measurable_at_filter f l' μ)\n  [tendsto_Ixx_class Ioc l l'] [is_measurably_generated l']\n  (hμ : μ.finite_at_filter l') {c : E} (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  {u v : ι → ℝ} {lt : filter ι} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  ∀ᶠ t in lt, interval_integrable f μ (u t) (v t) :=\nhave _ := (hf.integrable_at_filter_ae hfm hμ).eventually,\n((hu.Ioc hv).eventually this).and $ (hv.Ioc hu).eventually this\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : ℝ → E` has a finite limit at `l`. Then `f` is interval integrable on `u..v`\nprovided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter ℝ` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\nlemma filter.tendsto.eventually_interval_integrable {f : ℝ → E} {μ : measure ℝ}\n  {l l' : filter ℝ} (hfm : strongly_measurable_at_filter f l' μ)\n  [tendsto_Ixx_class Ioc l l'] [is_measurably_generated l']\n  (hμ : μ.finite_at_filter l') {c : E} (hf : tendsto f l' (𝓝 c))\n  {u v : ι → ℝ} {lt : filter ι} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  ∀ᶠ t in lt, interval_integrable f μ (u t) (v t) :=\n(hf.mono_left inf_le_left).eventually_interval_integrable_ae hfm hμ hu hv\n\n/-!\n### Interval integral: definition and basic properties\n\nIn this section we define `∫ x in a..b, f x ∂μ` as `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`\nand prove some basic properties.\n-/\n\nvariables [complete_space E] [normed_space ℝ E]\n\n/-- The interval integral `∫ x in a..b, f x ∂μ` is defined\nas `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`. If `a ≤ b`, then it equals\n`∫ x in Ioc a b, f x ∂μ`, otherwise it equals `-∫ x in Ioc b a, f x ∂μ`. -/\ndef interval_integral (f : ℝ → E) (a b : ℝ) (μ : measure ℝ) :=\n∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ\n\nnotation `∫` binders ` in ` a `..` b `, ` r:(scoped:60 f, f) ` ∂` μ:70 := interval_integral r a b μ\nnotation `∫` binders ` in ` a `..` b `, ` r:(scoped:60 f, interval_integral f a b volume) := r\n\nnamespace interval_integral\n\nsection basic\n\nvariables {a b : ℝ} {f g : ℝ → E} {μ : measure ℝ}\n\n@[simp] lemma integral_zero : ∫ x in a..b, (0 : E) ∂μ = 0 :=\nby simp [interval_integral]\n\nlemma integral_of_le (h : a ≤ b) : ∫ x in a..b, f x ∂μ = ∫ x in Ioc a b, f x ∂μ :=\nby simp [interval_integral, h]\n\n@[simp] lemma integral_same : ∫ x in a..a, f x ∂μ = 0 :=\nsub_self _\n\nlemma integral_symm (a b) : ∫ x in b..a, f x ∂μ = -∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, neg_sub]\n\nlemma integral_of_ge (h : b ≤ a) : ∫ x in a..b, f x ∂μ = -∫ x in Ioc b a, f x ∂μ :=\nby simp only [integral_symm b, integral_of_le h]\n\nlemma interval_integral_eq_integral_interval_oc (f : ℝ → E) (a b : ℝ) (μ : measure ℝ) :\n  ∫ x in a..b, f x ∂μ = (if a ≤ b then 1 else -1 : ℝ) • ∫ x in Ι a b, f x ∂μ :=\nbegin\n  split_ifs with h,\n  { simp only [integral_of_le h, interval_oc_of_le h, one_smul] },\n  { simp only [integral_of_ge (not_le.1 h).le, interval_oc_of_lt (not_le.1 h), neg_one_smul] }\nend\n\nlemma integral_cases (f : ℝ → E) (a b) :\n  ∫ x in a..b, f x ∂μ ∈ ({∫ x in Ι a b, f x ∂μ, -∫ x in Ι a b, f x ∂μ} : set E) :=\nby { rw interval_integral_eq_integral_interval_oc, split_ifs; simp }\n\nlemma integral_undef (h : ¬ interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nby cases le_total a b with hab hab;\n  simp only [integral_of_le, integral_of_ge, hab, neg_eq_zero];\n    refine integral_undef (not_imp_not.mpr integrable.integrable_on' _);\n      simpa [hab] using not_and_distrib.mp h\n\nlemma integral_non_ae_strongly_measurable\n  (hf : ¬ ae_strongly_measurable f (μ.restrict (Ι a b))) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nby rw [interval_integral_eq_integral_interval_oc, integral_non_ae_strongly_measurable hf, smul_zero]\n\nlemma integral_non_ae_strongly_measurable_of_le (h : a ≤ b)\n  (hf : ¬ ae_strongly_measurable f (μ.restrict (Ioc a b))) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nintegral_non_ae_strongly_measurable $ by rwa [interval_oc_of_le h]\n\nlemma norm_integral_min_max (f : ℝ → E) :\n  ∥∫ x in min a b..max a b, f x ∂μ∥ = ∥∫ x in a..b, f x ∂μ∥ :=\nby cases le_total a b; simp [*, integral_symm a b]\n\nlemma norm_integral_eq_norm_integral_Ioc (f : ℝ → E) :\n  ∥∫ x in a..b, f x ∂μ∥ = ∥∫ x in Ι a b, f x ∂μ∥ :=\nby rw [← norm_integral_min_max, integral_of_le min_le_max, interval_oc]\n\nlemma abs_integral_eq_abs_integral_interval_oc (f : ℝ → ℝ) :\n  |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| :=\nnorm_integral_eq_norm_integral_Ioc f\n\nlemma norm_integral_le_integral_norm_Ioc :\n  ∥∫ x in a..b, f x ∂μ∥ ≤ ∫ x in Ι a b, ∥f x∥ ∂μ :=\ncalc ∥∫ x in a..b, f x ∂μ∥ = ∥∫ x in Ι a b, f x ∂μ∥ :\n  norm_integral_eq_norm_integral_Ioc f\n... ≤ ∫ x in Ι a b, ∥f x∥ ∂μ :\n  norm_integral_le_integral_norm f\n\nlemma norm_integral_le_abs_integral_norm : ∥∫ x in a..b, f x ∂μ∥ ≤ |∫ x in a..b, ∥f x∥ ∂μ| :=\nbegin\n  simp only [← real.norm_eq_abs, norm_integral_eq_norm_integral_Ioc],\n  exact le_trans (norm_integral_le_integral_norm _) (le_abs_self _)\nend\n\nlemma norm_integral_le_integral_norm (h : a ≤ b) :\n  ∥∫ x in a..b, f x ∂μ∥ ≤ ∫ x in a..b, ∥f x∥ ∂μ :=\nnorm_integral_le_integral_norm_Ioc.trans_eq $ by rw [interval_oc_of_le h, integral_of_le h]\n\nlemma norm_integral_le_of_norm_le_const_ae {a b C : ℝ} {f : ℝ → E}\n  (h : ∀ᵐ x, x ∈ Ι a b → ∥f x∥ ≤ C) :\n  ∥∫ x in a..b, f x∥ ≤ C * |b - a| :=\nbegin\n  rw [norm_integral_eq_norm_integral_Ioc],\n  convert norm_set_integral_le_of_norm_le_const_ae'' _ measurable_set_Ioc h,\n  { rw [real.volume_Ioc, max_sub_min_eq_abs, ennreal.to_real_of_real (abs_nonneg _)] },\n  { simp only [real.volume_Ioc, ennreal.of_real_lt_top] },\nend\n\nlemma norm_integral_le_of_norm_le_const {a b C : ℝ} {f : ℝ → E}\n  (h : ∀ x ∈ Ι a b, ∥f x∥ ≤ C) :\n  ∥∫ x in a..b, f x∥ ≤ C * |b - a| :=\nnorm_integral_le_of_norm_le_const_ae $ eventually_of_forall h\n\n@[simp] lemma integral_add (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  ∫ x in a..b, f x + g x ∂μ = ∫ x in a..b, f x ∂μ + ∫ x in a..b, g x ∂μ :=\nby simp only [interval_integral_eq_integral_interval_oc, integral_add hf.def hg.def, smul_add]\n\nlemma integral_finset_sum {ι} {s : finset ι} {f : ι → ℝ → E}\n  (h : ∀ i ∈ s, interval_integrable (f i) μ a b) :\n  ∫ x in a..b, ∑ i in s, f i x ∂μ = ∑ i in s, ∫ x in a..b, f i x ∂μ :=\nby simp only [interval_integral_eq_integral_interval_oc,\n  integral_finset_sum s (λ i hi, (h i hi).def), finset.smul_sum]\n\n@[simp] lemma integral_neg : ∫ x in a..b, -f x ∂μ = -∫ x in a..b, f x ∂μ :=\nby { simp only [interval_integral, integral_neg], abel }\n\n@[simp] lemma integral_sub (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  ∫ x in a..b, f x - g x ∂μ = ∫ x in a..b, f x ∂μ - ∫ x in a..b, g x ∂μ :=\nby simpa only [sub_eq_add_neg] using (integral_add hf hg.neg).trans (congr_arg _ integral_neg)\n\n@[simp] lemma integral_smul {𝕜 : Type*} [nondiscrete_normed_field 𝕜] [normed_space 𝕜 E]\n  [smul_comm_class ℝ 𝕜 E]\n  (r : 𝕜) (f : ℝ → E) : ∫ x in a..b, r • f x ∂μ = r • ∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, integral_smul, smul_sub]\n\n@[simp] lemma integral_smul_const {𝕜 : Type*} [is_R_or_C 𝕜] [normed_space 𝕜 E] (f : ℝ → 𝕜) (c : E) :\n  ∫ x in a..b, f x • c ∂μ = (∫ x in a..b, f x ∂μ) • c :=\nby simp only [interval_integral_eq_integral_interval_oc, integral_smul_const, smul_assoc]\n\n@[simp] lemma integral_const_mul {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, r * f x ∂μ = r * ∫ x in a..b, f x ∂μ :=\nintegral_smul r f\n\n@[simp] lemma integral_mul_const {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, f x * r ∂μ = ∫ x in a..b, f x ∂μ * r :=\nby simpa only [mul_comm r] using integral_const_mul r f\n\n@[simp] lemma integral_div {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, f x / r ∂μ = ∫ x in a..b, f x ∂μ / r :=\nby simpa only [div_eq_mul_inv] using integral_mul_const r⁻¹ f\n\nlemma integral_const' (c : E) :\n  ∫ x in a..b, c ∂μ = ((μ $ Ioc a b).to_real - (μ $ Ioc b a).to_real) • c :=\nby simp only [interval_integral, set_integral_const, sub_smul]\n\n@[simp] lemma integral_const (c : E) : ∫ x in a..b, c = (b - a) • c :=\nby simp only [integral_const', real.volume_Ioc, ennreal.to_real_of_real', ← neg_sub b,\n  max_zero_sub_eq_self]\n\nlemma integral_smul_measure (c : ℝ≥0∞) :\n  ∫ x in a..b, f x ∂(c • μ) = c.to_real • ∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, measure.restrict_smul, integral_smul_measure, smul_sub]\n\nvariables [normed_group F] [complete_space F] [normed_space ℝ F]\n\nlemma _root_.continuous_linear_map.interval_integral_comp_comm\n  (L : E →L[ℝ] F) (hf : interval_integrable f μ a b) :\n  ∫ x in a..b, L (f x) ∂μ = L (∫ x in a..b, f x ∂μ) :=\nbegin\n  rw [interval_integral, interval_integral, L.integral_comp_comm, L.integral_comp_comm, L.map_sub],\n  exacts [hf.2, hf.1]\nend\n\nend basic\n\nsection comp\n\nvariables {a b c d : ℝ} (f : ℝ → E)\n\n@[simp] lemma integral_comp_mul_right (hc : c ≠ 0) :\n  ∫ x in a..b, f (x * c) = c⁻¹ • ∫ x in a*c..b*c, f x :=\nbegin\n  have A : measurable_embedding (λ x, x * c) :=\n    (homeomorph.mul_right₀ c hc).closed_embedding.measurable_embedding,\n  conv_rhs { rw [← real.smul_map_volume_mul_right hc] },\n  simp_rw [integral_smul_measure, interval_integral, A.set_integral_map,\n          ennreal.to_real_of_real (abs_nonneg c)],\n  cases hc.lt_or_lt,\n  { simp [h, mul_div_cancel, hc, abs_of_neg, measure.restrict_congr_set Ico_ae_eq_Ioc] },\n  { simp [h, mul_div_cancel, hc, abs_of_pos] }\nend\n\n@[simp] lemma smul_integral_comp_mul_right (c) :\n  c • ∫ x in a..b, f (x * c) = ∫ x in a*c..b*c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_mul_left (hc : c ≠ 0) :\n  ∫ x in a..b, f (c * x) = c⁻¹ • ∫ x in c*a..c*b, f x :=\nby simpa only [mul_comm c] using integral_comp_mul_right f hc\n\n@[simp] lemma smul_integral_comp_mul_left (c) :\n  c • ∫ x in a..b, f (c * x) = ∫ x in c*a..c*b, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div (hc : c ≠ 0) :\n  ∫ x in a..b, f (x / c) = c • ∫ x in a/c..b/c, f x :=\nby simpa only [inv_inv] using integral_comp_mul_right f (inv_ne_zero hc)\n\n@[simp] lemma inv_smul_integral_comp_div (c) :\n  c⁻¹ • ∫ x in a..b, f (x / c) = ∫ x in a/c..b/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_right (d) :\n  ∫ x in a..b, f (x + d) = ∫ x in a+d..b+d, f x :=\nhave A : measurable_embedding (λ x, x + d) :=\n  (homeomorph.add_right d).closed_embedding.measurable_embedding,\ncalc  ∫ x in a..b, f (x + d)\n    = ∫ x in a+d..b+d, f x ∂(measure.map (λ x, x + d) volume)\n                           : by simp [interval_integral, A.set_integral_map]\n... = ∫ x in a+d..b+d, f x : by rw [map_add_right_eq_self]\n\n@[simp] lemma integral_comp_add_left (d) :\n  ∫ x in a..b, f (d + x) = ∫ x in d+a..d+b, f x :=\nby simpa only [add_comm] using integral_comp_add_right f d\n\n@[simp] lemma integral_comp_mul_add (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (c * x + d) = c⁻¹ • ∫ x in c*a+d..c*b+d, f x :=\nby rw [← integral_comp_add_right, ← integral_comp_mul_left _ hc]\n\n@[simp] lemma smul_integral_comp_mul_add (c d) :\n  c • ∫ x in a..b, f (c * x + d) = ∫ x in c*a+d..c*b+d, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_mul (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d + c * x) = c⁻¹ • ∫ x in d+c*a..d+c*b, f x :=\nby rw [← integral_comp_add_left, ← integral_comp_mul_left _ hc]\n\n@[simp] lemma smul_integral_comp_add_mul (c d) :\n  c • ∫ x in a..b, f (d + c * x) = ∫ x in d+c*a..d+c*b, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div_add (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (x / c + d) = c • ∫ x in a/c+d..b/c+d, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_add f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_div_add (c d) :\n  c⁻¹ • ∫ x in a..b, f (x / c + d) = ∫ x in a/c+d..b/c+d, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_div (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d + x / c) = c • ∫ x in d+a/c..d+b/c, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_add_mul f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_add_div (c d) :\n  c⁻¹ • ∫ x in a..b, f (d + x / c) = ∫ x in d+a/c..d+b/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_mul_sub (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (c * x - d) = c⁻¹ • ∫ x in c*a-d..c*b-d, f x :=\nby simpa only [sub_eq_add_neg] using integral_comp_mul_add f hc (-d)\n\n@[simp] lemma smul_integral_comp_mul_sub (c d) :\n  c • ∫ x in a..b, f (c * x - d) = ∫ x in c*a-d..c*b-d, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_mul (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d - c * x) = c⁻¹ • ∫ x in d-c*b..d-c*a, f x :=\nbegin\n  simp only [sub_eq_add_neg, neg_mul_eq_neg_mul],\n  rw [integral_comp_add_mul f (neg_ne_zero.mpr hc) d, integral_symm],\n  simp only [inv_neg, smul_neg, neg_neg, neg_smul],\nend\n\n@[simp] lemma smul_integral_comp_sub_mul (c d) :\n  c • ∫ x in a..b, f (d - c * x) = ∫ x in d-c*b..d-c*a, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div_sub (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (x / c - d) = c • ∫ x in a/c-d..b/c-d, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_sub f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_div_sub (c d) :\n  c⁻¹ • ∫ x in a..b, f (x / c - d) = ∫ x in a/c-d..b/c-d, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_div (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d - x / c) = c • ∫ x in d-b/c..d-a/c, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_sub_mul f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_sub_div (c d) :\n  c⁻¹ • ∫ x in a..b, f (d - x / c) = ∫ x in d-b/c..d-a/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_right (d) :\n  ∫ x in a..b, f (x - d) = ∫ x in a-d..b-d, f x :=\nby simpa only [sub_eq_add_neg] using integral_comp_add_right f (-d)\n\n@[simp] lemma integral_comp_sub_left (d) :\n  ∫ x in a..b, f (d - x) = ∫ x in d-b..d-a, f x :=\nby simpa only [one_mul, one_smul, inv_one] using integral_comp_sub_mul f one_ne_zero d\n\n@[simp] lemma integral_comp_neg : ∫ x in a..b, f (-x) = ∫ x in -b..-a, f x :=\nby simpa only [zero_sub] using integral_comp_sub_left f 0\n\nend comp\n\n/-!\n### Integral is an additive function of the interval\n\nIn this section we prove that `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ`\nas well as a few other identities trivially equivalent to this one. We also prove that\n`∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ` provided that `support f ⊆ Ioc a b`.\n-/\n\nsection order_closed_topology\n\nvariables {a b c d : ℝ} {f g : ℝ → E} {μ : measure ℝ}\n\nlemma integrable_on_Icc_iff_integrable_on_Ioc'\n  {E : Type*} [normed_group E]\n  {f : ℝ → E} {a b : ℝ} (ha : μ {a} ≠ ∞) :\n  integrable_on f (Icc a b) μ ↔ integrable_on f (Ioc a b) μ :=\nbegin\n  cases le_or_lt a b with hab hab,\n  { have : Icc a b = Icc a a ∪ Ioc a b := (Icc_union_Ioc_eq_Icc le_rfl hab).symm,\n    rw [this, integrable_on_union],\n    simp [ha.lt_top] },\n  { simp [hab, hab.le] },\nend\n\nlemma integrable_on_Icc_iff_integrable_on_Ioc\n  {E : Type*}[normed_group E] [has_no_atoms μ] {f : ℝ → E} {a b : ℝ} :\n  integrable_on f (Icc a b) μ ↔ integrable_on f (Ioc a b) μ :=\nintegrable_on_Icc_iff_integrable_on_Ioc' (by simp)\n\nlemma interval_integrable_iff_integrable_Icc_of_le {E : Type*} [normed_group E]\n  {f : ℝ → E} {a b : ℝ} (hab : a ≤ b) {μ : measure ℝ} [has_no_atoms μ] :\n  interval_integrable f μ a b ↔ integrable_on f (Icc a b) μ :=\nby rw [interval_integrable_iff_integrable_Ioc_of_le hab, integrable_on_Icc_iff_integrable_on_Ioc]\n\n/-- If two functions are equal in the relevant interval, their interval integrals are also equal. -/\nlemma integral_congr {a b : ℝ} (h : eq_on f g [a, b]) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nby cases le_total a b with hab hab; simpa [hab, integral_of_le, integral_of_ge]\n  using set_integral_congr measurable_set_Ioc (h.mono Ioc_subset_Icc_self)\n\nlemma integral_add_adjacent_intervals_cancel (hab : interval_integrable f μ a b)\n  (hbc : interval_integrable f μ b c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ + ∫ x in c..a, f x ∂μ = 0 :=\nbegin\n  have hac := hab.trans hbc,\n  simp only [interval_integral, ← add_sub_comm, sub_eq_zero],\n  iterate 4 { rw ← integral_union },\n  { suffices : Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc b a ∪ Ioc c b ∪ Ioc a c, by rw this,\n    rw [Ioc_union_Ioc_union_Ioc_cycle, union_right_comm, Ioc_union_Ioc_union_Ioc_cycle,\n      min_left_comm, max_left_comm] },\n  all_goals { simp [*, measurable_set.union, measurable_set_Ioc, Ioc_disjoint_Ioc_same,\n    Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] }\nend\n\nlemma integral_add_adjacent_intervals (hab : interval_integrable f μ a b)\n  (hbc : interval_integrable f μ b c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ :=\nby rw [← add_neg_eq_zero, ← integral_symm, integral_add_adjacent_intervals_cancel hab hbc]\n\nlemma sum_integral_adjacent_intervals {a : ℕ → ℝ} {n : ℕ}\n  (hint : ∀ k < n, interval_integrable f μ (a k) (a $ k+1)) :\n  ∑ (k : ℕ) in finset.range n, ∫ x in (a k)..(a $ k+1), f x ∂μ = ∫ x in (a 0)..(a n), f x ∂μ :=\nbegin\n  induction n with n hn,\n  { simp },\n  { rw [finset.sum_range_succ, hn (λ k hk, hint k (hk.trans n.lt_succ_self))],\n    exact integral_add_adjacent_intervals\n      (interval_integrable.trans_iterate $ λ k hk, hint k (hk.trans n.lt_succ_self))\n      (hint n n.lt_succ_self) }\nend\n\nlemma integral_interval_sub_left (hab : interval_integrable f μ a b)\n  (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in a..c, f x ∂μ = ∫ x in c..b, f x ∂μ :=\nsub_eq_of_eq_add' $ eq.symm $ integral_add_adjacent_intervals hac (hac.symm.trans hab)\n\nlemma integral_interval_add_interval_comm (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in c..d, f x ∂μ = ∫ x in a..d, f x ∂μ + ∫ x in c..b, f x ∂μ :=\nby rw [← integral_add_adjacent_intervals hac hcd, add_assoc, add_left_comm,\n  integral_add_adjacent_intervals hac (hac.symm.trans hab), add_comm]\n\nlemma integral_interval_sub_interval_comm (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in c..d, f x ∂μ = ∫ x in a..c, f x ∂μ - ∫ x in b..d, f x ∂μ :=\nby simp only [sub_eq_add_neg, ← integral_symm,\n  integral_interval_add_interval_comm hab hcd.symm (hac.trans hcd)]\n\nlemma integral_interval_sub_interval_comm' (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in c..d, f x ∂μ = ∫ x in d..b, f x ∂μ - ∫ x in c..a, f x ∂μ :=\nby { rw [integral_interval_sub_interval_comm hab hcd hac, integral_symm b d, integral_symm a c,\n  sub_neg_eq_add, sub_eq_neg_add], }\n\nlemma integral_Iic_sub_Iic (ha : integrable_on f (Iic a) μ) (hb : integrable_on f (Iic b) μ) :\n  ∫ x in Iic b, f x ∂μ - ∫ x in Iic a, f x ∂μ = ∫ x in a..b, f x ∂μ :=\nbegin\n  wlog hab : a ≤ b using [a b] tactic.skip,\n  { rw [sub_eq_iff_eq_add', integral_of_le hab, ← integral_union (Iic_disjoint_Ioc le_rfl),\n      Iic_union_Ioc_eq_Iic hab],\n    exacts [measurable_set_Ioc, ha, hb.mono_set (λ _, and.right)] },\n  { intros ha hb,\n    rw [integral_symm, ← this hb ha, neg_sub] }\nend\n\n/-- If `μ` is a finite measure then `∫ x in a..b, c ∂μ = (μ (Iic b) - μ (Iic a)) • c`. -/\nlemma integral_const_of_cdf [is_finite_measure μ] (c : E) :\n  ∫ x in a..b, c ∂μ = ((μ (Iic b)).to_real - (μ (Iic a)).to_real) • c :=\nbegin\n  simp only [sub_smul, ← set_integral_const],\n  refine (integral_Iic_sub_Iic _ _).symm;\n    simp only [integrable_on_const, measure_lt_top, or_true]\nend\n\nlemma integral_eq_integral_of_support_subset {a b} (h : support f ⊆ Ioc a b) :\n  ∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ :=\nbegin\n  cases le_total a b with hab hab,\n  { rw [integral_of_le hab, ← integral_indicator measurable_set_Ioc, indicator_eq_self.2 h];\n    apply_instance },\n  { rw [Ioc_eq_empty hab.not_lt, subset_empty_iff, support_eq_empty_iff] at h,\n    simp [h] }\nend\n\nlemma integral_congr_ae' (h : ∀ᵐ x ∂μ, x ∈ Ioc a b → f x = g x)\n  (h' : ∀ᵐ x ∂μ, x ∈ Ioc b a → f x = g x) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nby simp only [interval_integral, set_integral_congr_ae (measurable_set_Ioc) h,\n              set_integral_congr_ae (measurable_set_Ioc) h']\n\nlemma integral_congr_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = g x) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nintegral_congr_ae' (ae_interval_oc_iff.mp h).1 (ae_interval_oc_iff.mp h).2\n\nlemma integral_zero_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = 0) :\n  ∫ x in a..b, f x ∂μ = 0 :=\ncalc ∫ x in a..b, f x ∂μ = ∫ x in a..b, 0 ∂μ : integral_congr_ae h\n                     ... = 0                 : integral_zero\n\nlemma integral_indicator {a₁ a₂ a₃ : ℝ} (h : a₂ ∈ Icc a₁ a₃) :\n  ∫ x in a₁..a₃, indicator {x | x ≤ a₂} f x ∂ μ = ∫ x in a₁..a₂, f x ∂ μ :=\nbegin\n  have : {x | x ≤ a₂} ∩ Ioc a₁ a₃ = Ioc a₁ a₂, from Iic_inter_Ioc_of_le h.2,\n  rw [integral_of_le h.1, integral_of_le (h.1.trans h.2),\n      integral_indicator, measure.restrict_restrict, this],\n  exact measurable_set_Iic,\n  all_goals { apply measurable_set_Iic },\nend\n\n/-- Lebesgue dominated convergence theorem for filters with a countable basis -/\nlemma tendsto_integral_filter_of_dominated_convergence {ι} {l : filter ι}\n  [l.is_countably_generated] {F : ι → ℝ → E} (bound : ℝ → ℝ)\n  (hF_meas : ∀ᶠ n in l, ae_strongly_measurable (F n) (μ.restrict (Ι a b)))\n  (h_bound : ∀ᶠ n in l, ∀ᵐ x ∂μ, x ∈ Ι a b → ∥F n x∥ ≤ bound x)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_lim : ∀ᵐ x ∂μ, x ∈ Ι a b → tendsto (λ n, F n x) l (𝓝 (f x))) :\n  tendsto (λn, ∫ x in a..b, F n x ∂μ) l (𝓝 $ ∫ x in a..b, f x ∂μ) :=\nbegin\n  simp only [interval_integrable_iff, interval_integral_eq_integral_interval_oc,\n    ← ae_restrict_iff' measurable_set_interval_oc] at *,\n  exact tendsto_const_nhds.smul\n    (tendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_lim)\nend\n\n/-- Lebesgue dominated convergence theorem for series. -/\nlemma has_sum_integral_of_dominated_convergence {ι} [encodable ι]\n  {F : ι → ℝ → E} (bound : ι → ℝ → ℝ)\n  (hF_meas : ∀ n, ae_strongly_measurable (F n) (μ.restrict (Ι a b)))\n  (h_bound : ∀ n, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F n t∥ ≤ bound n t)\n  (bound_summable : ∀ᵐ t ∂μ, t ∈ Ι a b → summable (λ n, bound n t))\n  (bound_integrable : interval_integrable (λ t, ∑' n, bound n t) μ a b)\n  (h_lim : ∀ᵐ t ∂μ, t ∈ Ι a b → has_sum (λ n, F n t) (f t)) :\n  has_sum (λn, ∫ t in a..b, F n t ∂μ) (∫ t in a..b, f t ∂μ) :=\nbegin\n  simp only [interval_integrable_iff, interval_integral_eq_integral_interval_oc,\n    ← ae_restrict_iff' measurable_set_interval_oc] at *,\n  exact (has_sum_integral_of_dominated_convergence bound hF_meas h_bound bound_summable\n    bound_integrable h_lim).const_smul\nend\n\nopen topological_space\nvariables {X : Type*} [topological_space X] [first_countable_topology X]\n\n/-- Continuity of interval integral with respect to a parameter, at a point within a set.\n  Given `F : X → ℝ → E`, assume `F x` is ae-measurable on `[a, b]` for `x` in a\n  neighborhood of `x₀` within `s` and at `x₀`, and assume it is bounded by a function integrable\n  on `[a, b]` independent of `x` in a neighborhood of `x₀` within `s`. If `(λ x, F x t)`\n  is continuous at `x₀` within `s` for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_within_at_of_dominated_interval\n  {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ} {s : set X}\n  (hF_meas : ∀ᶠ x in 𝓝[s] x₀, ae_strongly_measurable (F x) (μ.restrict $ Ι a b))\n  (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous_within_at (λ x, F x t) s x₀) :\n  continuous_within_at (λ x, ∫ t in a..b, F x t ∂μ) s x₀ :=\ntendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_cont\n\n/-- Continuity of interval integral with respect to a parameter at a point.\n  Given `F : X → ℝ → E`, assume `F x` is ae-measurable on `[a, b]` for `x` in a\n  neighborhood of `x₀`, and assume it is bounded by a function integrable on\n  `[a, b]` independent of `x` in a neighborhood of `x₀`. If `(λ x, F x t)`\n  is continuous at `x₀` for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_at_of_dominated_interval\n  {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ}\n  (hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict $ Ι a b))\n  (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous_at (λ x, F x t) x₀) :\n  continuous_at (λ x, ∫ t in a..b, F x t ∂μ) x₀ :=\ntendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_cont\n\n/-- Continuity of interval integral with respect to a parameter.\n  Given `F : X → ℝ → E`, assume each `F x` is ae-measurable on `[a, b]`,\n  and assume it is bounded by a function integrable on `[a, b]` independent of `x`.\n  If `(λ x, F x t)` is continuous for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_of_dominated_interval {F : X → ℝ → E} {bound : ℝ → ℝ} {a b : ℝ}\n  (hF_meas : ∀ x, ae_strongly_measurable (F x) $ μ.restrict $ Ι a b)\n  (h_bound : ∀ x, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous (λ x, F x t)) :\n  continuous (λ x, ∫ t in a..b, F x t ∂μ) :=\ncontinuous_iff_continuous_at.mpr (λ x₀, continuous_at_of_dominated_interval\n  (eventually_of_forall hF_meas) (eventually_of_forall h_bound) bound_integrable $ h_cont.mono $\n  λ x himp hx, (himp hx).continuous_at)\n\nend order_closed_topology\n\nsection continuous_primitive\nopen topological_space\n\nvariables {a b b₀ b₁ b₂ : ℝ} {μ : measure ℝ} {f g : ℝ → E}\n\nlemma continuous_within_at_primitive (hb₀ : μ {b₀} = 0)\n  (h_int : interval_integrable f μ (min a b₁) (max a b₂)) :\n  continuous_within_at (λ b, ∫ x in a .. b, f x ∂ μ) (Icc b₁ b₂) b₀ :=\nbegin\n  by_cases h₀ : b₀ ∈ Icc b₁ b₂,\n  { have h₁₂ : b₁ ≤ b₂ := h₀.1.trans h₀.2,\n    have min₁₂ : min b₁ b₂ = b₁ := min_eq_left h₁₂,\n    have h_int' : ∀ {x}, x ∈ Icc b₁ b₂ → interval_integrable f μ b₁ x,\n    { rintros x ⟨h₁, h₂⟩,\n      apply h_int.mono_set,\n      apply interval_subset_interval,\n      { exact ⟨min_le_of_left_le (min_le_right a b₁),\n                h₁.trans (h₂.trans $ le_max_of_le_right $ le_max_right _ _)⟩ },\n      { exact ⟨min_le_of_left_le $ (min_le_right _ _).trans h₁,\n                le_max_of_le_right $ h₂.trans $ le_max_right _ _⟩ } },\n    have : ∀ b ∈ Icc b₁ b₂, ∫ x in a..b, f x ∂μ = ∫ x in a..b₁, f x ∂μ + ∫ x in b₁..b, f x ∂μ,\n    { rintros b ⟨h₁, h₂⟩,\n      rw ← integral_add_adjacent_intervals _ (h_int' ⟨h₁, h₂⟩),\n      apply h_int.mono_set,\n      apply interval_subset_interval,\n      { exact ⟨min_le_of_left_le (min_le_left a b₁), le_max_of_le_right (le_max_left _ _)⟩ },\n      { exact ⟨min_le_of_left_le (min_le_right _ _),\n                le_max_of_le_right (h₁.trans $ h₂.trans (le_max_right a b₂))⟩ } },\n    apply continuous_within_at.congr _ this (this _ h₀), clear this,\n    refine continuous_within_at_const.add _,\n    have : (λ b, ∫ x in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀]\n            λ b, ∫ x in b₁..b₂, indicator {x | x ≤ b} f x ∂ μ,\n    { apply eventually_eq_of_mem self_mem_nhds_within,\n      exact λ b b_in, (integral_indicator b_in).symm },\n\n    apply continuous_within_at.congr_of_eventually_eq _ this (integral_indicator h₀).symm,\n    have : interval_integrable (λ x, ∥f x∥) μ b₁ b₂,\n      from interval_integrable.norm (h_int' $ right_mem_Icc.mpr h₁₂),\n    refine continuous_within_at_of_dominated_interval _ _ this _ ; clear this,\n    { apply eventually.mono (self_mem_nhds_within),\n      intros x hx,\n      erw [ae_strongly_measurable_indicator_iff, measure.restrict_restrict, Iic_inter_Ioc_of_le],\n      { rw min₁₂,\n        exact (h_int' hx).1.ae_strongly_measurable },\n      { exact le_max_of_le_right hx.2 },\n      exacts [measurable_set_Iic, measurable_set_Iic] },\n    { refine eventually_of_forall (λ x, eventually_of_forall (λ t, _)),\n      dsimp [indicator],\n      split_ifs ; simp },\n    { have : ∀ᵐ t ∂μ, t < b₀ ∨ b₀ < t,\n      { apply eventually.mono (compl_mem_ae_iff.mpr hb₀),\n        intros x hx,\n        exact ne.lt_or_lt hx },\n      apply this.mono,\n      rintros x₀ (hx₀ | hx₀) -,\n      { have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀,\n        { apply mem_nhds_within_of_mem_nhds,\n          apply eventually.mono (Ioi_mem_nhds hx₀),\n          intros x hx,\n          simp [hx.le] },\n        apply continuous_within_at_const.congr_of_eventually_eq  this,\n        simp [hx₀.le] },\n      { have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = 0,\n        { apply mem_nhds_within_of_mem_nhds,\n          apply eventually.mono (Iio_mem_nhds hx₀),\n          intros x hx,\n          simp [hx] },\n        apply continuous_within_at_const.congr_of_eventually_eq this,\n        simp [hx₀] } } },\n  { apply continuous_within_at_of_not_mem_closure,\n    rwa [closure_Icc] }\nend\n\nlemma continuous_on_primitive [has_no_atoms μ] (h_int : integrable_on f (Icc a b) μ) :\n  continuous_on (λ x, ∫ t in Ioc a x, f t ∂ μ) (Icc a b) :=\nbegin\n  by_cases h : a ≤ b,\n  { have : ∀ x ∈ Icc a b, ∫ t in Ioc a x, f t ∂μ = ∫ t in a..x, f t ∂μ,\n    { intros x x_in,\n      simp_rw [← interval_oc_of_le h, integral_of_le x_in.1] },\n    rw continuous_on_congr this,\n    intros x₀ hx₀,\n    refine continuous_within_at_primitive (measure_singleton x₀) _,\n    simp only [interval_integrable_iff_integrable_Ioc_of_le, min_eq_left, max_eq_right, h],\n    exact h_int.mono Ioc_subset_Icc_self le_rfl },\n  { rw Icc_eq_empty h,\n    exact continuous_on_empty _ },\nend\n\nlemma continuous_on_primitive_Icc [has_no_atoms μ] (h_int : integrable_on f (Icc a b) μ) :\n  continuous_on (λ x, ∫ t in Icc a x, f t ∂ μ) (Icc a b) :=\nbegin\n  rw show (λ x, ∫ t in Icc a x, f t ∂μ) = λ x, ∫ t in Ioc a x, f t ∂μ,\n    by { ext x, exact integral_Icc_eq_integral_Ioc },\n  exact continuous_on_primitive h_int\nend\n\n/-- Note: this assumes that `f` is `interval_integrable`, in contrast to some other lemmas here. -/\nlemma continuous_on_primitive_interval' [has_no_atoms μ]\n  (h_int : interval_integrable f μ b₁ b₂) (ha : a ∈ [b₁, b₂]) :\n  continuous_on (λ b, ∫ x in a..b, f x ∂ μ) [b₁, b₂] :=\nbegin\n  intros b₀ hb₀,\n  refine continuous_within_at_primitive (measure_singleton _) _,\n  rw [min_eq_right ha.1, max_eq_right ha.2],\n  simpa [interval_integrable_iff, interval_oc] using h_int,\nend\n\nlemma continuous_on_primitive_interval [has_no_atoms μ]\n  (h_int : integrable_on f (interval a b) μ) :\n  continuous_on (λ x, ∫ t in a..x, f t ∂ μ) (interval a b) :=\ncontinuous_on_primitive_interval' h_int.interval_integrable left_mem_interval\n\nlemma continuous_on_primitive_interval_left [has_no_atoms μ]\n  (h_int : integrable_on f (interval a b) μ) :\n  continuous_on (λ x, ∫ t in x..b, f t ∂ μ) (interval a b) :=\nbegin\n  rw interval_swap a b at h_int ⊢,\n  simp only [integral_symm b],\n  exact (continuous_on_primitive_interval h_int).neg,\nend\n\nvariables [has_no_atoms μ]\n\nlemma continuous_primitive (h_int : ∀ a b, interval_integrable f μ a b) (a : ℝ) :\n  continuous (λ b, ∫ x in a..b, f x ∂ μ) :=\nbegin\n  rw continuous_iff_continuous_at,\n  intro b₀,\n  cases exists_lt b₀ with b₁ hb₁,\n  cases exists_gt b₀ with b₂ hb₂,\n  apply continuous_within_at.continuous_at _ (Icc_mem_nhds hb₁ hb₂),\n  exact continuous_within_at_primitive (measure_singleton b₀) (h_int _ _)\nend\n\nlemma _root_.measure_theory.integrable.continuous_primitive (h_int : integrable f μ) (a : ℝ) :\n  continuous (λ b, ∫ x in a..b, f x ∂ μ) :=\ncontinuous_primitive (λ _ _, h_int.interval_integrable) a\n\nend continuous_primitive\n\nsection\n\nvariables {f g : ℝ → ℝ} {a b : ℝ} {μ : measure ℝ}\n\nlemma integral_eq_zero_iff_of_le_of_nonneg_ae (hab : a ≤ b)\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc a b)] f) (hfi : interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b)] 0 :=\nby rw [integral_of_le hab, integral_eq_zero_iff_of_nonneg_ae hf hfi.1]\n\nlemma integral_eq_zero_iff_of_nonneg_ae\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f) (hfi : interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0 :=\nbegin\n  cases le_total a b with hab hab;\n    simp only [Ioc_eq_empty hab.not_lt, empty_union, union_empty] at hf ⊢,\n  { exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi },\n  { rw [integral_symm, neg_eq_zero, integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi.symm] }\nend\n\n/-- If `f` is nonnegative and integrable on the unordered interval `set.interval_oc a b`, then its\nintegral over `a..b` is positive if and only if `a < b` and the measure of\n`function.support f ∩ set.Ioc a b` is positive. -/\nlemma integral_pos_iff_support_of_nonneg_ae'\n  (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f) (hfi : interval_integrable f μ a b) :\n  0 < ∫ x in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) :=\nbegin\n  cases lt_or_le a b with hab hba,\n  { rw interval_oc_of_le hab.le at hf,\n    simp only [hab, true_and, integral_of_le hab.le,\n      set_integral_pos_iff_support_of_nonneg_ae hf hfi.1] },\n  { suffices : ∫ x in a..b, f x ∂μ ≤ 0, by simp only [this.not_lt, hba.not_lt, false_and],\n    rw [integral_of_ge hba, neg_nonpos],\n    rw [interval_oc_swap, interval_oc_of_le hba] at hf,\n    exact integral_nonneg_of_ae hf }\nend\n\n/-- If `f` is nonnegative a.e.-everywhere and it is integrable on the unordered interval\n`set.interval_oc a b`, then its integral over `a..b` is positive if and only if `a < b` and the\nmeasure of `function.support f ∩ set.Ioc a b` is positive. -/\nlemma integral_pos_iff_support_of_nonneg_ae (hf : 0 ≤ᵐ[μ] f) (hfi : interval_integrable f μ a b) :\n  0 < ∫ x in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) :=\nintegral_pos_iff_support_of_nonneg_ae' (ae_mono measure.restrict_le_self hf) hfi\n\n/-- If `f : ℝ → ℝ` is strictly positive and integrable on `(a, b]` for real numbers `a < b`, then\nits integral over `a..b` is strictly positive. -/\nlemma interval_integral_pos_of_pos {f : ℝ → ℝ} {a b : ℝ}\n  (hfi : interval_integrable f measure_space.volume a b) (h : ∀ x, 0 < f x) (hab : a < b) :\n  0 < ∫ x in a..b, f x :=\nbegin\n  have hsupp : support f = univ := eq_univ_iff_forall.mpr (λ t, (h t).ne.symm),\n  replace h₀ : 0 ≤ᵐ[volume] f := eventually_of_forall (λ x, (h x).le),\n  rw integral_pos_iff_support_of_nonneg_ae h₀ hfi,\n  exact ⟨hab, by simp [hsupp, hab]⟩,\nend\n\n/-- If `f` and `g` are two functions that are interval integrable on `a..b`, `a ≤ b`,\n`f x ≤ g x` for a.e. `x ∈ set.Ioc a b`, and `f x < g x` on a subset of `set.Ioc a b`\nof nonzero measure, then `∫ x in a..b, f x ∂μ < ∫ x in a..b, g x ∂μ`. -/\nlemma integral_lt_integral_of_ae_le_of_measure_set_of_lt_ne_zero (hab : a ≤ b)\n  (hfi : interval_integrable f μ a b) (hgi : interval_integrable g μ a b)\n  (hle : f ≤ᵐ[μ.restrict (Ioc a b)] g) (hlt : μ.restrict (Ioc a b) {x | f x < g x} ≠ 0) :\n  ∫ x in a..b, f x ∂μ < ∫ x in a..b, g x ∂μ :=\nbegin\n  rw [← sub_pos, ← integral_sub hgi hfi, integral_of_le hab,\n    measure_theory.integral_pos_iff_support_of_nonneg_ae],\n  { refine pos_iff_ne_zero.2 (mt (measure_mono_null _) hlt),\n    exact λ x hx, (sub_pos.2 hx).ne' },\n  exacts [hle.mono (λ x, sub_nonneg.2), hgi.1.sub hfi.1]\nend\n\n/-- If `f` and `g` are continuous on `[a, b]`, `a < b`, `f x ≤ g x` on this interval, and\n`f c < g c` at some point `c ∈ [a, b]`, then `∫ x in a..b, f x < ∫ x in a..b, g x`. -/\nlemma integral_lt_integral_of_continuous_on_of_le_of_exists_lt {f g : ℝ → ℝ} {a b : ℝ}\n  (hab : a < b) (hfc : continuous_on f (Icc a b)) (hgc : continuous_on g (Icc a b))\n  (hle : ∀ x ∈ Ioc a b, f x ≤ g x) (hlt : ∃ c ∈ Icc a b, f c < g c) :\n  ∫ x in a..b, f x < ∫ x in a..b, g x :=\nbegin\n  refine integral_lt_integral_of_ae_le_of_measure_set_of_lt_ne_zero hab.le\n    (hfc.interval_integrable_of_Icc hab.le) (hgc.interval_integrable_of_Icc hab.le)\n    ((ae_restrict_mem measurable_set_Ioc).mono hle) _,\n  contrapose! hlt,\n  have h_eq : f =ᵐ[volume.restrict (Ioc a b)] g,\n  { simp only [← not_le, ← ae_iff] at hlt,\n    exact eventually_le.antisymm ((ae_restrict_iff' measurable_set_Ioc).2 $\n      eventually_of_forall hle) hlt },\n  simp only [measure.restrict_congr_set Ioc_ae_eq_Icc] at h_eq,\n  exact λ c hc, (measure.eq_on_Icc_of_ae_eq volume hab.ne h_eq hfc hgc hc).ge\nend\n\nlemma integral_nonneg_of_ae_restrict (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nlet H := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf in\nby simpa only [integral_of_le hab] using set_integral_nonneg_of_ae_restrict H\n\nlemma integral_nonneg_of_ae (hab : a ≤ b) (hf : 0 ≤ᵐ[μ] f) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae_restrict hab $ ae_restrict_of_ae hf\n\nlemma integral_nonneg_of_forall (hab : a ≤ b) (hf : ∀ u, 0 ≤ f u) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae hab $ eventually_of_forall hf\n\nlemma integral_nonneg (hab : a ≤ b) (hf : ∀ u, u ∈ Icc a b → 0 ≤ f u) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae_restrict hab $ (ae_restrict_iff' measurable_set_Icc).mpr $ ae_of_all μ hf\n\nlemma abs_integral_le_integral_abs (hab : a ≤ b) :\n  |∫ x in a..b, f x ∂μ| ≤ ∫ x in a..b, |f x| ∂μ :=\nby simpa only [← real.norm_eq_abs] using norm_integral_le_integral_norm hab\n\nsection mono\n\nvariables (hab : a ≤ b) (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b)\n\ninclude hab hf hg\n\nlemma integral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict (Icc a b)] g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nlet H := h.filter_mono $ ae_mono $ measure.restrict_mono Ioc_subset_Icc_self $ le_refl μ in\nby simpa only [integral_of_le hab] using set_integral_mono_ae_restrict hf.1 hg.1 H\n\nlemma integral_mono_ae (h : f ≤ᵐ[μ] g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nby simpa only [integral_of_le hab] using set_integral_mono_ae hf.1 hg.1 h\n\nlemma integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nlet H := λ x hx, h x $ Ioc_subset_Icc_self hx in\nby simpa only [integral_of_le hab] using set_integral_mono_on hf.1 hg.1 measurable_set_Ioc H\n\nlemma integral_mono (h : f ≤ g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nintegral_mono_ae hab hf hg $ ae_of_all _ h\n\nomit hg hab\n\nlemma integral_mono_interval {c d} (hca : c ≤ a) (hab : a ≤ b) (hbd : b ≤ d)\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc c d)] f) (hfi : interval_integrable f μ c d):\n  ∫ x in a..b, f x ∂μ ≤ ∫ x in c..d, f x ∂μ :=\nbegin\n  rw [integral_of_le hab, integral_of_le (hca.trans (hab.trans hbd))],\n  exact set_integral_mono_set hfi.1 hf (Ioc_subset_Ioc hca hbd).eventually_le\nend\n\nlemma abs_integral_mono_interval {c d } (h : Ι a b ⊆ Ι c d)\n  (hf : 0 ≤ᵐ[μ.restrict (Ι c d)] f) (hfi : interval_integrable f μ c d) :\n  |∫ x in a..b, f x ∂μ| ≤ |∫ x in c..d, f x ∂μ| :=\nhave hf' : 0 ≤ᵐ[μ.restrict (Ι a b)] f, from ae_mono (measure.restrict_mono h le_rfl) hf,\ncalc |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| : abs_integral_eq_abs_integral_interval_oc f\n... = ∫ x in Ι a b, f x ∂μ : abs_of_nonneg (measure_theory.integral_nonneg_of_ae hf')\n... ≤ ∫ x in Ι c d, f x ∂μ : set_integral_mono_set hfi.def hf h.eventually_le\n... ≤ |∫ x in Ι c d, f x ∂μ| : le_abs_self _\n... = |∫ x in c..d, f x ∂μ| : (abs_integral_eq_abs_integral_interval_oc f).symm\n\nend mono\n\nend\n\n/-!\n### Fundamental theorem of calculus, part 1, for any measure\n\nIn this section we prove a few lemmas that can be seen as versions of FTC-1 for interval integrals\nw.r.t. any measure. Many theorems are formulated for one or two pairs of filters related by\n`FTC_filter a l l'`. This typeclass has exactly four “real” instances: `(a, pure a, ⊥)`,\n`(a, 𝓝[≥] a, 𝓝[>] a)`, `(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`, and two instances\nthat are equal to the first and last “real” instances: `(a, 𝓝[{a}] a, ⊥)` and\n`(a, 𝓝[univ] a, 𝓝[univ] a)`.  We use this approach to avoid repeating arguments in many very similar\ncases.  Lean can automatically find both `a` and `l'` based on `l`.\n\nThe most general theorem `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` can be seen\nas a generalization of lemma `integral_has_strict_fderiv_at` below which states strict\ndifferentiability of `∫ x in u..v, f x` in `(u, v)` at `(a, b)` for a measurable function `f` that\nis integrable on `a..b` and is continuous at `a` and `b`. The lemma is generalized in three\ndirections: first, `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` deals with any\nlocally finite measure `μ`; second, it works for one-sided limits/derivatives; third, it assumes\nonly that `f` has finite limits almost surely at `a` and `b`.\n\nNamely, let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of\n`FTC_filter`s around `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f`\nhas finite limits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.  Then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis theorem is formulated with integral of constants instead of measures in the right hand sides\nfor two reasons: first, this way we avoid `min`/`max` in the statements; second, often it is\npossible to write better `simp` lemmas for these integrals, see `integral_const` and\n`integral_const_of_cdf`.\n\nIn the next subsection we apply this theorem to prove various theorems about differentiability\nof the integral w.r.t. Lebesgue measure. -/\n\n/-- An auxiliary typeclass for the Fundamental theorem of calculus, part 1. It is used to formulate\ntheorems that work simultaneously for left and right one-sided derivatives of `∫ x in u..v, f x`. -/\nclass FTC_filter (a : out_param ℝ) (outer : filter ℝ) (inner : out_param $ filter ℝ)\n  extends tendsto_Ixx_class Ioc outer inner : Prop :=\n(pure_le : pure a ≤ outer)\n(le_nhds : inner ≤ 𝓝 a)\n[meas_gen : is_measurably_generated inner]\n\n/- The `dangerous_instance` linter doesn't take `out_param`s into account, so it thinks that\n`FTC_filter.to_tendsto_Ixx_class` is dangerous. Disable this linter using `nolint`.\n-/\nattribute [nolint dangerous_instance] FTC_filter.to_tendsto_Ixx_class\n\nnamespace FTC_filter\n\ninstance pure (a : ℝ) : FTC_filter a (pure a) ⊥ :=\n{ pure_le := le_rfl,\n  le_nhds := bot_le }\n\ninstance nhds_within_singleton (a : ℝ) : FTC_filter a (𝓝[{a}] a) ⊥ :=\nby { rw [nhds_within, principal_singleton, inf_eq_right.2 (pure_le_nhds a)], apply_instance }\n\nlemma finite_at_inner {a : ℝ} (l : filter ℝ) {l'} [h : FTC_filter a l l']\n  {μ : measure ℝ} [is_locally_finite_measure μ] :\n  μ.finite_at_filter l' :=\n(μ.finite_at_nhds a).filter_mono h.le_nhds\n\ninstance nhds (a : ℝ) : FTC_filter a (𝓝 a) (𝓝 a) :=\n{ pure_le := pure_le_nhds a,\n  le_nhds := le_rfl }\n\ninstance nhds_univ (a : ℝ) : FTC_filter a (𝓝[univ] a) (𝓝 a) :=\nby { rw nhds_within_univ, apply_instance }\n\ninstance nhds_left (a : ℝ) : FTC_filter a (𝓝[≤] a) (𝓝[≤] a) :=\n{ pure_le := pure_le_nhds_within right_mem_Iic,\n  le_nhds := inf_le_left }\n\ninstance nhds_right (a : ℝ) : FTC_filter a (𝓝[≥] a) (𝓝[>] a) :=\n{ pure_le := pure_le_nhds_within left_mem_Ici,\n  le_nhds := inf_le_left }\n\ninstance nhds_Icc {x a b : ℝ} [h : fact (x ∈ Icc a b)] :\n  FTC_filter x (𝓝[Icc a b] x) (𝓝[Icc a b] x) :=\n{ pure_le := pure_le_nhds_within h.out,\n  le_nhds := inf_le_left }\n\ninstance nhds_interval {x a b : ℝ} [h : fact (x ∈ [a, b])] :\n  FTC_filter x (𝓝[[a, b]] x) (𝓝[[a, b]] x) :=\nby { haveI : fact (x ∈ set.Icc (min a b) (max a b)) := h, exact FTC_filter.nhds_Icc }\n\nend FTC_filter\n\nopen asymptotics\n\nsection\n\nvariables {f : ℝ → E} {a b : ℝ} {c ca cb : E} {l l' la la' lb lb' : filter ℝ} {lt : filter ι}\n  {μ : measure ℝ} {u v ua va ub vb : ι → ℝ}\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, where `μ` is a measure\nfinite at `l'`, then `∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both\n`u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l')\n  (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ - ∫ x in u t..v t, c ∂μ)\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) lt :=\nbegin\n  have A := hf.integral_sub_linear_is_o_ae hfm hl (hu.Ioc hv),\n  have B := hf.integral_sub_linear_is_o_ae hfm hl (hv.Ioc hu),\n  simp only [integral_const'],\n  convert (A.trans_le _).sub (B.trans_le _),\n  { ext t,\n    simp_rw [interval_integral, sub_smul],\n    abel },\n  all_goals { intro t, cases le_total (u t) (v t) with huv huv; simp [huv] }\nend\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both\n`u` and `v` tend to `l` so that `u ≤ v`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_le'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l') (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : u ≤ᶠ[lt] v) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ - (μ (Ioc (u t) (v t))).to_real • c)\n    (λ t, (μ $ Ioc (u t) (v t)).to_real) lt :=\n(measure_integral_sub_linear_is_o_of_tendsto_ae' hfm hf hl hu hv).congr'\n  (huv.mono $ λ x hx, by simp [integral_const', hx])\n  (huv.mono $ λ x hx, by simp [integral_const', hx])\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both\n`u` and `v` tend to `l` so that `v ≤ u`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l') (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : v ≤ᶠ[lt] u) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ + (μ (Ioc (v t) (u t))).to_real • c)\n    (λ t, (μ $ Ioc (v t) (u t)).to_real) lt :=\n(measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' hfm hf hl hv hu huv).neg_left.congr_left $\n  λ t, by simp [integral_symm (u t), add_comm]\n\nsection\n\nvariables [is_locally_finite_measure μ] [FTC_filter a l l']\n\ninclude a\n\nlocal attribute [instance] FTC_filter.meas_gen\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae'` for a version that also works, e.g., for\n`l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae (hfm : strongly_measurable_at_filter f l' μ)\n  (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ - ∫ x in u t..v t, c ∂μ)\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) lt :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae' hfm hf (FTC_filter.finite_at_inner l) hu hv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_le\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : u ≤ᶠ[lt] v) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ - (μ (Ioc (u t) (v t))).to_real • c)\n    (λ t, (μ $ Ioc (u t) (v t)).to_real) lt :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae_of_le' hfm hf (FTC_filter.finite_at_inner l)\n  hu hv huv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : v ≤ᶠ[lt] u) :\n  is_o (λ t, ∫ x in u t..v t, f x ∂μ + (μ (Ioc (v t) (u t))).to_real • c)\n    (λ t, (μ $ Ioc (v t) (u t)).to_real) lt :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' hfm hf (FTC_filter.finite_at_inner l)\n  hu hv huv\n\nend\n\nlocal attribute [instance] FTC_filter.meas_gen\n\nvariables [FTC_filter a la la'] [FTC_filter b lb lb'] [is_locally_finite_measure μ]\n\n/-- Fundamental theorem of calculus-1, strict derivative in both limits for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f` has finite\nlimits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.\nThen `∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ =\n  ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n    o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  (hab : interval_integrable f μ a b)\n  (hmeas_a : strongly_measurable_at_filter f la' μ)\n  (hmeas_b : strongly_measurable_at_filter f lb' μ)\n  (ha_lim : tendsto f (la' ⊓ μ.ae) (𝓝 ca)) (hb_lim : tendsto f (lb' ⊓ μ.ae) (𝓝 cb))\n  (hua : tendsto ua lt la) (hva : tendsto va lt la)\n  (hub : tendsto ub lt lb) (hvb : tendsto vb lt lb) :\n  is_o (λ t, (∫ x in va t..vb t, f x ∂μ) - (∫ x in ua t..ub t, f x ∂μ) -\n    (∫ x in ub t..vb t, cb ∂μ - ∫ x in ua t..va t, ca ∂μ))\n    (λ t, ∥∫ x in ua t..va t, (1:ℝ) ∂μ∥ + ∥∫ x in ub t..vb t, (1:ℝ) ∂μ∥) lt :=\nbegin\n  refine\n    ((measure_integral_sub_linear_is_o_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add\n    (measure_integral_sub_linear_is_o_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'\n      _ eventually_eq.rfl,\n  have A : ∀ᶠ t in lt, interval_integrable f μ (ua t) (va t) :=\n    ha_lim.eventually_interval_integrable_ae hmeas_a (FTC_filter.finite_at_inner la) hua hva,\n  have A' : ∀ᶠ t in lt, interval_integrable f μ a (ua t) :=\n    ha_lim.eventually_interval_integrable_ae hmeas_a (FTC_filter.finite_at_inner la)\n      (tendsto_const_pure.mono_right FTC_filter.pure_le) hua,\n  have B : ∀ᶠ t in lt, interval_integrable f μ (ub t) (vb t) :=\n    hb_lim.eventually_interval_integrable_ae hmeas_b (FTC_filter.finite_at_inner lb) hub hvb,\n  have B' : ∀ᶠ t in lt, interval_integrable f μ b (ub t) :=\n    hb_lim.eventually_interval_integrable_ae hmeas_b (FTC_filter.finite_at_inner lb)\n      (tendsto_const_pure.mono_right FTC_filter.pure_le) hub,\n  filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub,\n  rw [← integral_interval_sub_interval_comm'],\n  { dsimp only [], abel, },\n  exacts [ub_vb, ua_va, b_ub.symm.trans $ hab.symm.trans a_ua]\nend\n\n/-- Fundamental theorem of calculus-1, strict derivative in right endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(lb, lb')` be a pair of `FTC_filter`s\naround `b`. Suppose that `f` has a finite limit `c` at `lb' ⊓ μ.ae`.\n\nThen `∫ x in a..v, f x ∂μ - ∫ x in a..u, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `lb`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right\n  (hab : interval_integrable f μ a b) (hmeas : strongly_measurable_at_filter f lb' μ)\n  (hf : tendsto f (lb' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt lb) (hv : tendsto v lt lb) :\n  is_o (λ t, ∫ x in a..v t, f x ∂μ - ∫ x in a..u t, f x ∂μ - ∫ x in u t..v t, c ∂μ)\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) lt :=\nby simpa using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  hab strongly_measurable_at_bot hmeas ((tendsto_bot : tendsto _ ⊥ (𝓝 0)).mono_left inf_le_left)\n  hf (tendsto_const_pure : tendsto _ _ (pure a)) tendsto_const_pure hu hv\n\n/-- Fundamental theorem of calculus-1, strict derivative in left endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`. Suppose that `f` has a finite limit `c` at `la' ⊓ μ.ae`.\n\nThen `∫ x in v..b, f x ∂μ - ∫ x in u..b, f x ∂μ = -∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `la`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left\n  (hab : interval_integrable f μ a b) (hmeas : strongly_measurable_at_filter f la' μ)\n  (hf : tendsto f (la' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt la) (hv : tendsto v lt la) :\n  is_o (λ t, ∫ x in v t..b, f x ∂μ - ∫ x in u t..b, f x ∂μ + ∫ x in u t..v t, c ∂μ)\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) lt :=\nby simpa using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  hab hmeas strongly_measurable_at_bot hf ((tendsto_bot : tendsto _ ⊥ (𝓝 0)).mono_left inf_le_left)\n  hu hv (tendsto_const_pure : tendsto _ _ (pure b)) tendsto_const_pure\n\nend\n\n/-!\n### Fundamental theorem of calculus-1 for Lebesgue measure\n\nIn this section we restate theorems from the previous section for Lebesgue measure.\nIn particular, we prove that `∫ x in u..v, f x` is strictly differentiable in `(u, v)`\nat `(a, b)` provided that `f` is integrable on `a..b` and is continuous at `a` and `b`.\n-/\n\nvariables {f : ℝ → E} {c ca cb : E} {l l' la la' lb lb' : filter ℝ} {lt : filter ι}\n  {a b z : ℝ} {u v ua ub va vb : ι → ℝ} [FTC_filter a la la'] [FTC_filter b lb lb']\n\n/-!\n#### Auxiliary `is_o` statements\n\nIn this section we prove several lemmas that can be interpreted as strict differentiability of\n`(u, v) ↦ ∫ x in u..v, f x ∂μ` in `u` and/or `v` at a filter. The statements use `is_o` because\nwe have no definition of `has_strict_(f)deriv_at_filter` in the library.\n-/\n\n/-- Fundamental theorem of calculus-1, local version. If `f` has a finite limit `c` almost surely at\n`l'`, where `(l, l')` is an `FTC_filter` pair around `a`, then\n`∫ x in u..v, f x ∂μ = (v - u) • c + o (v - u)` as both `u` and `v` tend to `l`. -/\nlemma integral_sub_linear_is_o_of_tendsto_ae [FTC_filter a l l']\n  (hfm : strongly_measurable_at_filter f l') (hf : tendsto f (l' ⊓ volume.ae) (𝓝 c))\n  {u v : ι → ℝ} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  is_o (λ t, (∫ x in u t..v t, f x) - (v t - u t) • c) (v - u) lt :=\nby simpa [integral_const] using measure_integral_sub_linear_is_o_of_tendsto_ae hfm hf hu hv\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair around\n`a`, and `(lb, lb')` is an `FTC_filter` pair around `b`, and `f` has finite limits `ca` and `cb`\nalmost surely at `la'` and `lb'`, respectively, then\n`(∫ x in va..vb, f x) - ∫ x in ua..ub, f x = (vb - ub) • cb - (va - ua) • ca +\n  o(∥va - ua∥ + ∥vb - ub∥)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_fderiv_at_filter` if we had this\ndefinition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  (hab : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la') (hmeas_b : strongly_measurable_at_filter f lb')\n  (ha_lim : tendsto f (la' ⊓ volume.ae) (𝓝 ca)) (hb_lim : tendsto f (lb' ⊓ volume.ae) (𝓝 cb))\n  (hua : tendsto ua lt la) (hva : tendsto va lt la)\n  (hub : tendsto ub lt lb) (hvb : tendsto vb lt lb) :\n  is_o (λ t, (∫ x in va t..vb t, f x) - (∫ x in ua t..ub t, f x) -\n    ((vb t - ub t) • cb - (va t - ua t) • ca)) (λ t, ∥va t - ua t∥ + ∥vb t - ub t∥) lt :=\nby simpa [integral_const]\n  using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae hab hmeas_a hmeas_b\n    ha_lim hb_lim hua hva hub hvb\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(lb, lb')` is an `FTC_filter` pair\naround `b`, and `f` has a finite limit `c` almost surely at `lb'`, then\n`(∫ x in a..v, f x) - ∫ x in a..u, f x = (v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right\n  (hab : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f lb')\n  (hf : tendsto f (lb' ⊓ volume.ae) (𝓝 c)) (hu : tendsto u lt lb) (hv : tendsto v lt lb) :\n  is_o (λ t, (∫ x in a..v t, f x) - (∫ x in a..u t, f x) - (v t - u t) • c) (v - u) lt :=\nby simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hab hmeas hf hu hv\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair\naround `a`, and `f` has a finite limit `c` almost surely at `la'`, then\n`(∫ x in v..b, f x) - ∫ x in u..b, f x = -(v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `la`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left\n  (hab : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f la')\n  (hf : tendsto f (la' ⊓ volume.ae) (𝓝 c)) (hu : tendsto u lt la) (hv : tendsto v lt la) :\n  is_o (λ t, (∫ x in v t..b, f x) - (∫ x in u t..b, f x) + (v t - u t) • c) (v - u) lt :=\nby simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left hab hmeas hf hu hv\n\nopen continuous_linear_map (fst snd smul_right sub_apply smul_right_apply coe_fst' coe_snd' map_sub)\n\n/-!\n#### Strict differentiability\n\nIn this section we prove that for a measurable function `f` integrable on `a..b`,\n\n* `integral_has_strict_fderiv_at_of_tendsto_ae`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability\n  provided that `f` tends to `ca` and `cb` almost surely as `x` tendsto to `a` and `b`,\n  respectively;\n\n* `integral_has_strict_fderiv_at`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • f b - u • f a` at `(a, b)` in the sense of strict differentiability\n  provided that `f` is continuous at `a` and `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_right`: the function `u ↦ ∫ x in a..u, f x` has\n  derivative `c` at `b` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `b`;\n\n* `integral_has_strict_deriv_at_right`: the function `u ↦ ∫ x in a..u, f x` has derivative `f b` at\n  `b` in the sense of strict differentiability provided that `f` is continuous at `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_left`: the function `u ↦ ∫ x in u..b, f x` has\n  derivative `-c` at `a` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `a`;\n\n* `integral_has_strict_deriv_at_left`: the function `u ↦ ∫ x in u..b, f x` has derivative `-f a` at\n  `a` in the sense of strict differentiability provided that `f` is continuous at `a`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`\nin the sense of strict differentiability. -/\nlemma integral_has_strict_fderiv_at_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  has_strict_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (a, b) :=\nbegin\n  have := integral_sub_integral_sub_linear_is_o_of_tendsto_ae hf hmeas_a hmeas_b ha hb\n    ((continuous_fst.comp continuous_snd).tendsto ((a, b), (a, b)))\n    ((continuous_fst.comp continuous_fst).tendsto ((a, b), (a, b)))\n    ((continuous_snd.comp continuous_snd).tendsto ((a, b), (a, b)))\n    ((continuous_snd.comp continuous_fst).tendsto ((a, b), (a, b))),\n  refine (this.congr_left _).trans_is_O _,\n  { intro x, simp [sub_smul] },\n  { exact is_O_fst_prod.norm_left.add is_O_snd_prod.norm_left }\nend\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)` in the sense of strict differentiability. -/\nlemma integral_has_strict_fderiv_at\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  has_strict_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (a, b) :=\nintegral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b\n  (ha.mono_left inf_le_left) (hb.mono_left inf_le_left)\n\n/-- **First Fundamental Theorem of Calculus**: if `f : ℝ → E` is integrable on `a..b` and `f x` has\na finite limit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b` in\nthe sense of strict differentiability. -/\nlemma integral_has_strict_deriv_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : has_strict_deriv_at (λ u, ∫ x in a..u, f x) c b :=\nintegral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb continuous_at_snd\n  continuous_at_fst\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict\ndifferentiability. -/\nlemma integral_has_strict_deriv_at_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) : has_strict_deriv_at (λ u, ∫ x in a..u, f x) (f b) b :=\nintegral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a` in the sense\nof strict differentiability. -/\nlemma integral_has_strict_deriv_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : has_strict_deriv_at (λ u, ∫ x in u..b, f x) (-c) a :=\nby simpa only [← integral_symm]\n  using (integral_has_strict_deriv_at_of_tendsto_ae_right hf.symm hmeas ha).neg\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a` in the sense of strict\ndifferentiability. -/\nlemma integral_has_strict_deriv_at_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : continuous_at f a) : has_strict_deriv_at (λ u, ∫ x in u..b, f x) (-f a) a :=\nby simpa only [← integral_symm] using (integral_has_strict_deriv_at_right hf.symm hmeas ha).neg\n\n/-!\n#### Fréchet differentiability\n\nIn this subsection we restate results from the previous subsection in terms of `has_fderiv_at`,\n`has_deriv_at`, `fderiv`, and `deriv`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`. -/\nlemma integral_has_fderiv_at_of_tendsto_ae (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  has_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (a, b) :=\n(integral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).has_fderiv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)`. -/\nlemma integral_has_fderiv_at (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  has_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (a, b) :=\n(integral_has_strict_fderiv_at hf hmeas_a hmeas_b ha hb).has_fderiv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `fderiv`\nderivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦ v • cb - u • ca`. -/\nlemma fderiv_integral_of_tendsto_ae (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  fderiv ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (a, b) =\n    (snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca :=\n(integral_has_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderiv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `fderiv` derivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦\nv • cb - u • ca`. -/\nlemma fderiv_integral (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  fderiv ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (a, b) =\n    (snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a) :=\n(integral_has_fderiv_at hf hmeas_a hmeas_b ha hb).fderiv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b`. -/\nlemma integral_has_deriv_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : has_deriv_at (λ u, ∫ x in a..u, f x) c b :=\n(integral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas hb).has_deriv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b`. -/\nlemma integral_has_deriv_at_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) : has_deriv_at (λ u, ∫ x in a..u, f x) (f b) b :=\n(integral_has_strict_deriv_at_right hf hmeas hb).has_deriv_at\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\nlemma deriv_integral_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : deriv (λ u, ∫ x in a..u, f x) b = c :=\n(integral_has_deriv_at_of_tendsto_ae_right hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\nlemma deriv_integral_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) :\n  deriv (λ u, ∫ x in a..u, f x) b = f b :=\n(integral_has_deriv_at_right hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a`. -/\nlemma integral_has_deriv_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : has_deriv_at (λ u, ∫ x in u..b, f x) (-c) a :=\n(integral_has_strict_deriv_at_of_tendsto_ae_left hf hmeas ha).has_deriv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a`. -/\nlemma integral_has_deriv_at_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : continuous_at f a) :\n  has_deriv_at (λ u, ∫ x in u..b, f x) (-f a) a :=\n(integral_has_strict_deriv_at_left hf hmeas ha).has_deriv_at\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\nlemma deriv_integral_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (hb : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : deriv (λ u, ∫ x in u..b, f x) a = -c :=\n(integral_has_deriv_at_of_tendsto_ae_left hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\nlemma deriv_integral_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (hb : continuous_at f a) :\n  deriv (λ u, ∫ x in u..b, f x) a = -f a :=\n(integral_has_deriv_at_left hf hmeas hb).deriv\n\n/-!\n#### One-sided derivatives\n-/\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • cb - u • ca` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to `ca`\nand `cb` almost surely at the filters `la` and `lb` from the following table.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma integral_has_fderiv_within_at_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  (ha : tendsto f (la ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (lb ⊓ volume.ae) (𝓝 cb)) :\n  has_fderiv_within_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (s ×ˢ t) (a, b) :=\nbegin\n  rw [has_fderiv_within_at, nhds_within_prod_eq],\n  have := integral_sub_integral_sub_linear_is_o_of_tendsto_ae hf hmeas_a hmeas_b ha hb\n    (tendsto_const_pure.mono_right FTC_filter.pure_le : tendsto _ _ (𝓝[s] a)) tendsto_fst\n    (tendsto_const_pure.mono_right FTC_filter.pure_le : tendsto _ _ (𝓝[t] b)) tendsto_snd,\n  refine (this.congr_left _).trans_is_O _,\n  { intro x, simp [sub_smul] },\n  { exact is_O_fst_prod.norm_left.add is_O_snd_prod.norm_left }\nend\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • f b - u • f a` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to\n`f a` and `f b` at the filters `la` and `lb` from the following table. In most cases this assumption\nis definitionally equal `continuous_at f _` or `continuous_within_at f _ _`.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma integral_has_fderiv_within_at\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (ha : tendsto f la (𝓝 $ f a)) (hb : tendsto f lb (𝓝 $ f b)) :\n  has_fderiv_within_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (s ×ˢ t) (a, b) :=\nintegral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b (ha.mono_left inf_le_left)\n  (hb.mono_left inf_le_left)\n\n/-- An auxiliary tactic closing goals `unique_diff_within_at ℝ s a` where\n`s ∈ {Iic a, Ici a, univ}`. -/\nmeta def unique_diff_within_at_Ici_Iic_univ : tactic unit :=\n`[apply_rules [unique_diff_on.unique_diff_within_at, unique_diff_on_Ici, unique_diff_on_Iic,\n  left_mem_Ici, right_mem_Iic, unique_diff_within_at_univ]]\n\n/-- Let `f` be a measurable function integrable on `a..b`. Choose `s ∈ {Iic a, Ici a, univ}`\nand `t ∈ {Iic b, Ici b, univ}`. Suppose that `f` tends to `ca` and `cb` almost surely at the filters\n`la` and `lb` from the table below. Then `fderiv_within ℝ (λ p, ∫ x in p.1..p.2, f x) (s ×ˢ t)`\nis equal to `(u, v) ↦ u • cb - v • ca`.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma fderiv_within_integral_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (ha : tendsto f (la ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (lb ⊓ volume.ae) (𝓝 cb))\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ)\n  (ht : unique_diff_within_at ℝ t b . unique_diff_within_at_Ici_Iic_univ) :\n  fderiv_within ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (s ×ˢ t) (a, b) =\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) :=\n(integral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderiv_within $ hs.prod ht\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left,\nthen `u ↦ ∫ x in a..u, f x` has right (resp., left) derivative `c` at `b`. -/\nlemma integral_has_deriv_within_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : tendsto f (𝓝[t] b ⊓ volume.ae) (𝓝 c)) :\n  has_deriv_within_at (λ u, ∫ x in a..u, f x) c s b :=\nintegral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb\n  (tendsto_const_pure.mono_right FTC_filter.pure_le) tendsto_id\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `b`, then `u ↦ ∫ x in a..u, f x` has left (resp., right)\nderivative `f b` at `b`. -/\nlemma integral_has_deriv_within_at_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : continuous_within_at f t b) :\n  has_deriv_within_at (λ u, ∫ x in a..u, f x) (f b) s b :=\nintegral_has_deriv_within_at_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\nlemma deriv_within_integral_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas: strongly_measurable_at_filter f (𝓝[t] b)) (hb : tendsto f (𝓝[t] b ⊓ volume.ae) (𝓝 c))\n  (hs : unique_diff_within_at ℝ s b . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in a..u, f x) s b = c :=\n(integral_has_deriv_within_at_of_tendsto_ae_right hf hmeas hb).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `b`, then the right (resp., left) derivative of\n`u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\nlemma deriv_within_integral_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : continuous_within_at f t b)\n  (hs : unique_diff_within_at ℝ s b . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in a..u, f x) s b = f b :=\n(integral_has_deriv_within_at_right hf hmeas hb).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left,\nthen `u ↦ ∫ x in u..b, f x` has right (resp., left) derivative `-c` at `a`. -/\nlemma integral_has_deriv_within_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a))\n  (ha : tendsto f (𝓝[t] a ⊓ volume.ae) (𝓝 c)) :\n  has_deriv_within_at (λ u, ∫ x in u..b, f x) (-c) s a :=\nby { simp only [integral_symm b],\n  exact (integral_has_deriv_within_at_of_tendsto_ae_right hf.symm hmeas ha).neg }\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `a`, then `u ↦ ∫ x in u..b, f x` has left (resp., right)\nderivative `-f a` at `a`. -/\nlemma integral_has_deriv_within_at_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : continuous_within_at f t a) :\n  has_deriv_within_at (λ u, ∫ x in u..b, f x) (-f a) s a :=\nintegral_has_deriv_within_at_of_tendsto_ae_left hf hmeas (ha.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\nlemma deriv_within_integral_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : tendsto f (𝓝[t] a ⊓ volume.ae) (𝓝 c))\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in u..b, f x) s a = -c :=\n(integral_has_deriv_within_at_of_tendsto_ae_left hf hmeas ha).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `a`, then the right (resp., left) derivative of\n`u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\nlemma deriv_within_integral_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : continuous_within_at f t a)\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in u..b, f x) s a = -f a :=\n(integral_has_deriv_within_at_left hf hmeas ha).deriv_within hs\n\n/-- The integral of a continuous function is differentiable on a real set `s`. -/\ntheorem differentiable_on_integral_of_continuous {s : set ℝ}\n  (hintg : ∀ x ∈ s, interval_integrable f volume a x) (hcont : continuous f) :\n  differentiable_on ℝ (λ u, ∫ x in a..u, f x) s :=\nλ y hy, (integral_has_deriv_at_right (hintg y hy)\n  hcont.ae_strongly_measurable.strongly_measurable_at_filter\n    hcont.continuous_at) .differentiable_at.differentiable_within_at\n\n/-!\n### Fundamental theorem of calculus, part 2\n\nThis section contains theorems pertaining to FTC-2 for interval integrals, i.e., the assertion\nthat `∫ x in a..b, f' x = f b - f a` under suitable assumptions.\n\nThe most classical version of this theorem assumes that `f'` is continuous. However, this is\nunnecessarily strong: the result holds if `f'` is just integrable. We prove the strong version,\nfollowing [Rudin, *Real and Complex Analysis* (Theorem 7.21)][rudin2006real]. The proof is first\ngiven for real-valued functions, and then deduced for functions with a general target space. For\na real-valued function `g`, it suffices to show that `g b - g a ≤ (∫ x in a..b, g' x) + ε` for all\npositive `ε`. To prove this, choose a lower-semicontinuous function `G'` with `g' < G'` and with\nintegral close to that of `g'` (its existence is guaranteed by the Vitali-Carathéodory theorem).\nIt satisfies `g t - g a ≤ ∫ x in a..t, G' x` for all `t ∈ [a, b]`: this inequality holds at `a`,\nand if it holds at `t` then it holds for `u` close to `t` on its right, as the left hand side\nincreases by `g u - g t ∼ (u -t) g' t`, while the right hand side increases by\n`∫ x in t..u, G' x` which is roughly at least `∫ x in t..u, G' t = (u - t) G' t`, by lower\nsemicontinuity. As  `g' t < G' t`, this gives the conclusion. One can therefore push progressively\nthis inequality to the right until the point `b`, where it gives the desired conclusion.\n-/\n\nvariables {g' g : ℝ → ℝ}\n\n/-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has\n`g b - g a ≤ ∫ y in a..b, g' y`.\nAuxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`. -/\nlemma sub_le_integral_of_has_deriv_right_of_le (hab : a ≤ b) (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ico a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'int : integrable_on g' (Icc a b)) :\n  g b - g a ≤ ∫ y in a..b, g' y :=\nbegin\n  refine le_of_forall_pos_le_add (λ ε εpos, _),\n  -- Bound from above `g'` by a lower-semicontinuous function `G'`.\n  rcases exists_lt_lower_semicontinuous_integral_lt g' g'int εpos with\n    ⟨G', g'_lt_G', G'cont, G'int, G'lt_top, hG'⟩,\n  -- we will show by \"induction\" that `g t - g a ≤ ∫ u in a..t, G' u` for all `t ∈ [a, b]`.\n  set s := {t | g t - g a ≤ ∫ u in a..t, (G' u).to_real} ∩ Icc a b,\n  -- the set `s` of points where this property holds is closed.\n  have s_closed : is_closed s,\n  { have : continuous_on (λ t, (g t - g a, ∫ u in a..t, (G' u).to_real)) (Icc a b),\n    { rw ← interval_of_le hab at G'int ⊢ hcont,\n      exact (hcont.sub continuous_on_const).prod (continuous_on_primitive_interval G'int) },\n    simp only [s, inter_comm],\n    exact this.preimage_closed_of_closed is_closed_Icc order_closed_topology.is_closed_le' },\n  have main : Icc a b ⊆ {t | g t - g a ≤ ∫ u in a..t, (G' u).to_real },\n  { -- to show that the set `s` is all `[a, b]`, it suffices to show that any point `t` in `s`\n    -- with `t < b` admits another point in `s` slightly to its right\n    -- (this is a sort of real induction).\n    apply s_closed.Icc_subset_of_forall_exists_gt\n      (by simp only [integral_same, mem_set_of_eq, sub_self]) (λ t ht v t_lt_v, _),\n    obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ (y : ℝ), (g' t : ereal) < y ∧ (y : ereal) < G' t :=\n      ereal.lt_iff_exists_real_btwn.1 (g'_lt_G' t),\n    -- bound from below the increase of `∫ x in a..u, G' x` on the right of `t`, using the lower\n    -- semicontinuity of `G'`.\n    have I1 : ∀ᶠ u in 𝓝[>] t, (u - t) * y ≤ ∫ w in t..u, (G' w).to_real,\n    { have B : ∀ᶠ u in 𝓝 t, (y : ereal) < G' u :=\n        G'cont.lower_semicontinuous_at _ _ y_lt_G',\n      rcases mem_nhds_iff_exists_Ioo_subset.1 B with ⟨m, M, ⟨hm, hM⟩, H⟩,\n      have : Ioo t (min M b) ∈ 𝓝[>] t := mem_nhds_within_Ioi_iff_exists_Ioo_subset.2\n        ⟨min M b, by simp only [hM, ht.right.right, lt_min_iff, mem_Ioi, and_self], subset.refl _⟩,\n      filter_upwards [this] with u hu,\n      have I : Icc t u ⊆ Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _)),\n      calc (u - t) * y = ∫ v in Icc t u, y :\n        by simp only [hu.left.le, measure_theory.integral_const, algebra.id.smul_eq_mul, sub_nonneg,\n                      measurable_set.univ, real.volume_Icc, measure.restrict_apply, univ_inter,\n                      ennreal.to_real_of_real]\n      ... ≤ ∫ w in t..u, (G' w).to_real :\n      begin\n        rw [interval_integral.integral_of_le hu.1.le, ← integral_Icc_eq_integral_Ioc],\n        apply set_integral_mono_ae_restrict,\n        { simp only [integrable_on_const, real.volume_Icc, ennreal.of_real_lt_top, or_true] },\n        { exact integrable_on.mono_set G'int I },\n        { have C1 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), G' x < ∞ :=\n            ae_mono (measure.restrict_mono I le_rfl) G'lt_top,\n          have C2 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), x ∈ Icc t u :=\n            ae_restrict_mem measurable_set_Icc,\n          filter_upwards [C1, C2] with x G'x hx,\n          apply ereal.coe_le_coe_iff.1,\n          have : x ∈ Ioo m M, by simp only [hm.trans_le hx.left,\n            (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self],\n          convert le_of_lt (H this),\n          exact ereal.coe_to_real G'x.ne (ne_bot_of_gt (g'_lt_G' x)) }\n      end },\n    -- bound from above the increase of `g u - g a` on the right of `t`, using the derivative at `t`\n    have I2 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ (u - t) * y,\n    { have g'_lt_y : g' t < y := ereal.coe_lt_coe_iff.1 g'_lt_y',\n      filter_upwards [(hderiv t ⟨ht.2.1, ht.2.2⟩).limsup_slope_le'\n        (not_mem_Ioi.2 le_rfl) g'_lt_y, self_mem_nhds_within] with u hu t_lt_u,\n      have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u).le,\n      rwa [← smul_eq_mul, sub_smul_slope] at this },\n    -- combine the previous two bounds to show that `g u - g a` increases less quickly than\n    -- `∫ x in a..u, G' x`.\n    have I3 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ ∫ w in t..u, (G' w).to_real,\n    { filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1, },\n    have I4 : ∀ᶠ u in 𝓝[>] t, u ∈ Ioc t (min v b),\n    { refine mem_nhds_within_Ioi_iff_exists_Ioc_subset.2 ⟨min v b, _, subset.refl _⟩,\n      simp only [lt_min_iff, mem_Ioi],\n      exact ⟨t_lt_v, ht.2.2⟩ },\n    -- choose a point `x` slightly to the right of `t` which satisfies the above bound\n    rcases (I3.and I4).exists with ⟨x, hx, h'x⟩,\n    -- we check that it belongs to `s`, essentially by construction\n    refine ⟨x, _, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x⟩,\n    calc g x - g a = (g t - g a) + (g x - g t) : by abel\n    ... ≤ (∫ w in a..t, (G' w).to_real) + ∫ w in t..x, (G' w).to_real : add_le_add ht.1 hx\n    ... = ∫ w in a..x, (G' w).to_real :\n    begin\n      apply integral_add_adjacent_intervals,\n      { rw interval_integrable_iff_integrable_Ioc_of_le ht.2.1,\n        exact integrable_on.mono_set G'int\n          (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le)) },\n      { rw interval_integrable_iff_integrable_Ioc_of_le h'x.1.le,\n        apply integrable_on.mono_set G'int,\n        refine Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _))) }\n    end },\n  -- now that we know that `s` contains `[a, b]`, we get the desired result by applying this to `b`.\n  calc g b - g a ≤ ∫ y in a..b, (G' y).to_real : main (right_mem_Icc.2 hab)\n  ... ≤ (∫ y in a..b, g' y) + ε :\n    begin\n      convert hG'.le;\n      { rw interval_integral.integral_of_le hab,\n        simp only [integral_Icc_eq_integral_Ioc', real.volume_singleton] },\n    end\nend\n\n/-- Auxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`. -/\nlemma integral_le_sub_of_has_deriv_right_of_le (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ico a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'int : integrable_on g' (Icc a b)) :\n  ∫ y in a..b, g' y ≤ g b - g a :=\nbegin\n  rw ← neg_le_neg_iff,\n  convert sub_le_integral_of_has_deriv_right_of_le hab hcont.neg (λ x hx, (hderiv x hx).neg)\n    g'int.neg,\n  { abel },\n  { simp only [integral_neg] }\nend\n\n/-- Auxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`: real version -/\nlemma integral_eq_sub_of_has_deriv_right_of_le_real (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ico a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'int : integrable_on g' (Icc a b)) :\n  ∫ y in a..b, g' y = g b - g a :=\nle_antisymm\n  (integral_le_sub_of_has_deriv_right_of_le hab hcont hderiv g'int)\n  (sub_le_integral_of_has_deriv_right_of_le hab hcont hderiv g'int)\n\n/-- Auxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`: real version, not\nrequiring differentiability as the left endpoint of the interval. Follows from\n`integral_eq_sub_of_has_deriv_right_of_le_real` together with a continuity argument. -/\nlemma integral_eq_sub_of_has_deriv_right_of_le_real' (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'int : integrable_on g' (Icc a b)) :\n  ∫ y in a..b, g' y = g b - g a :=\nbegin\n  obtain rfl|a_lt_b := hab.eq_or_lt, { simp },\n  set s := {t | ∫ u in t..b, g' u = g b - g t} ∩ Icc a b,\n  have s_closed : is_closed s,\n  { have : continuous_on (λ t, (∫ u in t..b, g' u, g b - g t)) (Icc a b),\n    { rw ← interval_of_le hab at ⊢ hcont g'int,\n      exact (continuous_on_primitive_interval_left g'int).prod (continuous_on_const.sub hcont) },\n    simp only [s, inter_comm],\n    exact this.preimage_closed_of_closed is_closed_Icc is_closed_diagonal, },\n  have A : closure (Ioc a b) ⊆ s,\n  { apply s_closed.closure_subset_iff.2,\n    assume t ht,\n    refine ⟨_, ⟨ht.1.le, ht.2⟩⟩,\n    exact integral_eq_sub_of_has_deriv_right_of_le_real ht.2\n      (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n      (λ x hx, hderiv x ⟨ht.1.trans_le hx.1, hx.2⟩)\n      (g'int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) },\n  rw closure_Ioc a_lt_b.ne at A,\n  exact (A (left_mem_Icc.2 hab)).1,\nend\n\nvariable {f' : ℝ → E}\n\n/-- **Fundamental theorem of calculus-2**: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`)\n  and has a right derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`,\n  then `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right_of_le (hab : a ≤ b) (hcont : continuous_on f (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at f (f' x) (Ioi x) x)\n  (f'int : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  refine (normed_space.eq_iff_forall_dual_eq ℝ).2 (λ g, _),\n  rw [← g.interval_integral_comp_comm f'int, g.map_sub],\n  exact integral_eq_sub_of_has_deriv_right_of_le_real' hab (g.continuous.comp_continuous_on hcont)\n    (λ x hx, g.has_fderiv_at.comp_has_deriv_within_at x (hderiv x hx))\n    (g.integrable_comp ((interval_integrable_iff_integrable_Icc_of_le hab).1 f'int))\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` and\n  has a right derivative at `f' x` for all `x` in `[a, b)`, and `f'` is integrable on `[a, b]` then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right (hcont : continuous_on f (interval a b))\n  (hderiv : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  cases le_total a b with hab hab,\n  { simp only [interval_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint,\n    apply integral_eq_sub_of_has_deriv_right_of_le hab hcont hderiv hint },\n  { simp only [interval_of_ge, min_eq_right, max_eq_left, hab] at hcont hderiv,\n    rw [integral_symm, integral_eq_sub_of_has_deriv_right_of_le hab hcont hderiv hint.symm,\n        neg_sub] }\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`) and\n  has a derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`, then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at_of_le (hab : a ≤ b)\n  (hcont : continuous_on f (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nintegral_eq_sub_of_has_deriv_right_of_le hab hcont (λ x hx, (hderiv x hx).has_deriv_within_at) hint\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` has a derivative at `f' x` for all `x` in\n  `[a, b]` and `f'` is integrable on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at\n  (hderiv : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nintegral_eq_sub_of_has_deriv_right (has_deriv_at.continuous_on hderiv)\n  (λ x hx, (hderiv _ (mem_Icc_of_Ioo hx)).has_deriv_within_at) hint\n\ntheorem integral_eq_sub_of_has_deriv_at_of_tendsto (hab : a < b) {fa fb}\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hint : interval_integrable f' volume a b)\n  (ha : tendsto f (𝓝[>] a) (𝓝 fa)) (hb : tendsto f (𝓝[<] b) (𝓝 fb)) :\n  ∫ y in a..b, f' y = fb - fa :=\nbegin\n  set F : ℝ → E := update (update f a fa) b fb,\n  have Fderiv : ∀ x ∈ Ioo a b, has_deriv_at F (f' x) x,\n  { refine λ x hx, (hderiv x hx).congr_of_eventually_eq _,\n    filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy, simp only [F],\n    rw [update_noteq hy.2.ne, update_noteq hy.1.ne'], },\n  have hcont : continuous_on F (Icc a b),\n  { rw [continuous_on_update_iff, continuous_on_update_iff, Icc_diff_right, Ico_diff_left],\n    refine ⟨⟨λ z hz, (hderiv z hz).continuous_at.continuous_within_at, _⟩, _⟩,\n    { exact λ _, ha.mono_left (nhds_within_mono _ Ioo_subset_Ioi_self) },\n    { rintro -,\n      refine (hb.congr' _).mono_left (nhds_within_mono _ Ico_subset_Iio_self),\n      filter_upwards [Ioo_mem_nhds_within_Iio (right_mem_Ioc.2 hab)]\n        with _ hz using (update_noteq hz.1.ne' _ _).symm } },\n  simpa [F, hab.ne, hab.ne'] using integral_eq_sub_of_has_deriv_at_of_le hab.le hcont Fderiv hint\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is differentiable at every `x` in `[a, b]` and\n  its derivative is integrable on `[a, b]`, then `∫ y in a..b, deriv f y` equals `f b - f a`. -/\ntheorem integral_deriv_eq_sub (hderiv : ∀ x ∈ interval a b, differentiable_at ℝ f x)\n  (hint : interval_integrable (deriv f) volume a b) :\n  ∫ y in a..b, deriv f y = f b - f a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hderiv x hx).has_deriv_at) hint\n\ntheorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f')\n  (hdiff : ∀ x ∈ interval a b, differentiable_at ℝ f x)\n  (hcont : continuous_on f' (interval a b)) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  rw [← hderiv, integral_deriv_eq_sub hdiff],\n  rw hderiv,\n  exact hcont.interval_integrable\nend\n\n/-!\n### Integration by parts\n-/\n\ntheorem integral_deriv_mul_eq_sub {u v u' v' : ℝ → ℝ}\n  (hu : ∀ x ∈ interval a b, has_deriv_at u (u' x) x)\n  (hv : ∀ x ∈ interval a b, has_deriv_at v (v' x) x)\n  (hu' : interval_integrable u' volume a b) (hv' : interval_integrable v' volume a b) :\n  ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hu x hx).mul (hv x hx)) $\n  (hu'.mul_continuous_on (has_deriv_at.continuous_on hv)).add\n    (hv'.continuous_on_mul ((has_deriv_at.continuous_on hu)))\n\ntheorem integral_mul_deriv_eq_deriv_mul {u v u' v' : ℝ → ℝ}\n  (hu : ∀ x ∈ interval a b, has_deriv_at u (u' x) x)\n  (hv : ∀ x ∈ interval a b, has_deriv_at v (v' x) x)\n  (hu' : interval_integrable u' volume a b) (hv' : interval_integrable v' volume a b) :\n  ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, v x * u' x :=\nbegin\n  rw [← integral_deriv_mul_eq_sub hu hv hu' hv', ← integral_sub],\n  { exact integral_congr (λ x hx, by simp only [mul_comm, add_sub_cancel']) },\n  { exact ((hu'.mul_continuous_on (has_deriv_at.continuous_on hv)).add\n      (hv'.continuous_on_mul (has_deriv_at.continuous_on hu))) },\n  { exact hu'.continuous_on_mul (has_deriv_at.continuous_on hv) },\nend\n\n/-!\n### Integration by substitution / Change of variables\n-/\n\nsection smul\n\n/--\nChange of variables, general form. If `f` is continuous on `[a, b]` and has\ncontinuous right-derivative `f'` in `(a, b)`, and `g` is continuous on `f '' [a, b]` then we can\nsubstitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\n\nWe could potentially slightly weaken the conditions, by not requiring that `f'` and `g` are\ncontinuous on the endpoints of these intervals, but in that case we need to additionally assume that\nthe functions are integrable on that interval.\n-/\ntheorem integral_comp_smul_deriv'' {f f' : ℝ → ℝ} {g : ℝ → E}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g ∘ f) x= ∫ u in f a..f b, g u :=\nbegin\n  have h_cont : continuous_on (λ u, ∫ t in f a..f u, g t) [a, b],\n  { rw [hf.image_interval] at hg,\n    refine (continuous_on_primitive_interval' hg.interval_integrable _).comp hf _,\n    { rw ← hf.image_interval, exact mem_image_of_mem f left_mem_interval },\n    { rw ← hf.image_interval, exact maps_to_image _ _ } },\n  have h_der : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at\n    (λ u, ∫ t in f a..f u, g t) (f' x • ((g ∘ f) x)) (Ioi x) x,\n  { intros x hx,\n    let I := [Inf (f '' [a, b]), Sup (f '' [a, b])],\n    have hI : f '' [a, b] = I := hf.image_interval,\n    have h2x : f x ∈ I, { rw [← hI], exact mem_image_of_mem f (Ioo_subset_Icc_self hx) },\n    have h2g : interval_integrable g volume (f a) (f x),\n    { refine (hg.mono $ _).interval_integrable,\n      exact hf.surj_on_interval left_mem_interval (Ioo_subset_Icc_self hx) },\n    rw [hI] at hg,\n    have h3g : strongly_measurable_at_filter g (𝓝[I] f x) volume :=\n    hg.strongly_measurable_at_filter_nhds_within measurable_set_Icc (f x),\n    haveI : fact (f x ∈ I) := ⟨h2x⟩,\n    have : has_deriv_within_at (λ u, ∫ x in f a..u, g x) (g (f x)) I (f x) :=\n    integral_has_deriv_within_at_right h2g h3g (hg (f x) h2x),\n    refine (this.scomp x ((hff' x hx).Ioo_of_Ioi hx.2) _).Ioi_of_Ioo hx.2,\n    rw ← hI,\n    exact (maps_to_image _ _).mono (Ioo_subset_Icc_self.trans $ Icc_subset_Icc_left hx.1.le)\n      subset.rfl },\n  have h_int : interval_integrable (λ (x : ℝ), f' x • (g ∘ f) x) volume a b :=\n  (hf'.smul (hg.comp hf $ subset_preimage_image f _)).interval_integrable,\n  simp_rw [integral_eq_sub_of_has_deriv_right h_cont h_der h_int, integral_same, sub_zero],\nend\n\n/--\nChange of variables. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous on `f '' [a, b]`, then we can substitute `u = f x` to get\n`∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\nCompared to `interval_integral.integral_comp_smul_deriv` we only require that `g` is continuous on\n`f '' [a, b]`.\n-/\ntheorem integral_comp_smul_deriv' {f f' : ℝ → ℝ} {g : ℝ → E}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g ∘ f) x = ∫ x in f a..f b, g x :=\nintegral_comp_smul_deriv'' (λ x hx, (h x hx).continuous_at.continuous_within_at)\n  (λ x hx, (h x $ Ioo_subset_Icc_self hx).has_deriv_within_at) h' hg\n\n/--\nChange of variables, most common version. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous, then we can substitute `u = f x` to get\n`∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_smul_deriv {f f' : ℝ → ℝ} {g : ℝ → E}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous g) :\n  ∫ x in a..b, f' x • (g ∘ f) x = ∫ x in f a..f b, g x :=\nintegral_comp_smul_deriv' h h' hg.continuous_on\n\ntheorem integral_deriv_comp_smul_deriv' {f f' : ℝ → ℝ} {g g' : ℝ → E}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g [f a, f b])\n  (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), has_deriv_within_at g (g' x) (Ioi x) x)\n  (hg' : continuous_on g' (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a :=\nbegin\n  rw [integral_comp_smul_deriv'' hf hff' hf' hg',\n  integral_eq_sub_of_has_deriv_right hg hgg' (hg'.mono _).interval_integrable],\n  exact intermediate_value_interval hf\nend\n\ntheorem integral_deriv_comp_smul_deriv {f f' : ℝ → ℝ} {g g' : ℝ → E}\n  (hf : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hg : ∀ x ∈ interval a b, has_deriv_at g (g' (f x)) (f x))\n  (hf' : continuous_on f' (interval a b)) (hg' : continuous g') :\n  ∫ x in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hg x hx).scomp x $ hf x hx)\n  (hf'.smul (hg'.comp_continuous_on $ has_deriv_at.continuous_on hf)).interval_integrable\n\nend smul\nsection mul\n\n/--\nChange of variables, general form for scalar functions. If `f` is continuous on `[a, b]` and has\ncontinuous right-derivative `f'` in `(a, b)`, and `g` is continuous on `f '' [a, b]` then we can\nsubstitute `u = f x` to get `∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_mul_deriv'' {f f' g : ℝ → ℝ}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u :=\nby simpa [mul_comm] using integral_comp_smul_deriv'' hf hff' hf' hg\n\n/--\nChange of variables. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous on `f '' [a, b]`, then we can substitute `u = f x` to get\n`∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\nCompared to `interval_integral.integral_comp_mul_deriv` we only require that `g` is continuous on\n`f '' [a, b]`.\n-/\ntheorem integral_comp_mul_deriv' {f f' g : ℝ → ℝ}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ x in f a..f b, g x :=\nby simpa [mul_comm] using integral_comp_smul_deriv' h h' hg\n\n/--\nChange of variables, most common version. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous, then we can substitute `u = f x` to get\n`∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_mul_deriv {f f' g : ℝ → ℝ}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous g) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ x in f a..f b, g x :=\nintegral_comp_mul_deriv' h h' hg.continuous_on\n\ntheorem integral_deriv_comp_mul_deriv' {f f' g g' : ℝ → ℝ}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g [f a, f b])\n  (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), has_deriv_within_at g (g' x) (Ioi x) x)\n  (hg' : continuous_on g' (f '' [a, b])) :\n  ∫ x in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a :=\nby simpa [mul_comm] using integral_deriv_comp_smul_deriv' hf hff' hf' hg hgg' hg'\n\ntheorem integral_deriv_comp_mul_deriv {f f' g g' : ℝ → ℝ}\n  (hf : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hg : ∀ x ∈ interval a b, has_deriv_at g (g' (f x)) (f x))\n  (hf' : continuous_on f' (interval a b)) (hg' : continuous g') :\n  ∫ x in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a :=\nby simpa [mul_comm] using integral_deriv_comp_smul_deriv hf hg hf' hg'\n\nend mul\n\nend interval_integral\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/measure_theory/integral/interval_integral.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3227963641206162}}
{"text": "import order.galois_connection\n\nvariables (P : Type) [partial_order P]\n\ndef presheaf : Type :=\n{ s : P → Prop // ∀ a b, a ≤ b → s b → s a }\n\nvariable {P}\n\nnamespace presheaf\n\ninstance : has_coe_to_fun (presheaf P) (λ _, P → Prop) :=\n⟨subtype.val⟩\n\n@[simp] lemma coe_mk (s : P → Prop) (hs : ∀ a b, a ≤ b → s b → s a) :\n  @coe_fn (presheaf P) _ _ (⟨s, hs⟩ : presheaf P) = s := rfl\n\ninstance : partial_order (presheaf P) :=\n{ le := λ A B, ∀ x, A x → B x,\n  le_trans := λ A B C hAB hBC x hAx, hBC _ (hAB _ hAx),\n  le_refl := λ A x, id,\n  le_antisymm := λ A B hAB hBA, subtype.val_injective (funext $ λ x, propext ⟨hAB _, hBA _⟩) }\n\nlemma le_def {A B : presheaf P} : A ≤ B = ∀ x, A x → B x := rfl\n\ninstance : has_Inf (presheaf P) :=\n{ Inf := λ s, ⟨λ p, ∀ A : presheaf P, A ∈ s → A p, \n     λ a b hab h A hAs, A.2 _ _ hab (h _ hAs)⟩ }\n\ninstance : complete_lattice (presheaf P) :=\ncomplete_lattice_of_Inf _\n  (λ s, begin\n    split,\n    { dsimp [Inf, lower_bounds],\n      intros A hAs p h,\n      apply h,\n      exact hAs },\n    { dsimp [Inf, upper_bounds, lower_bounds],\n      intros A h p hAp B hBs,\n      apply h,\n      exact hBs,\n      exact hAp }\n  end)\n\nlemma infi_def {ι : Sort*} (A : ι → presheaf P) : \n  infi A = ⟨λ p, ∀ i, A i p, λ x y hxy h i, (A i).2 _ y hxy (h i)⟩ :=\nle_antisymm \n  (infi_le_iff.2 (λ B h p hBp i, h i _ hBp)) \n  (le_infi (λ i p h, h _))\n\nlemma supr_def {ι : Sort*} (A : ι → presheaf P) : \n  supr A = ⟨λ p, ∃ i, A i p, λ x y hxy ⟨i, hi⟩, ⟨i, (A i).2 x y hxy hi⟩⟩ :=\nle_antisymm \n  (supr_le_iff.2 (λ i p h, ⟨i, h⟩)) \n  (le_supr_iff.2 (λ B h p ⟨i, hi⟩, h i _ hi))\n\nend presheaf\n\ndef yoneda (a : P) : presheaf P :=\n⟨λ b, b ≤ a, λ b c, le_trans⟩\n\ndef yoneda_le_iff (A : presheaf P) (p : P) : yoneda p ≤ A ↔ A p :=\nbegin\n  simp [yoneda, presheaf.le_def],\n  split,\n  { intro h, apply h, exact le_rfl },\n  { intros h x hxp,\n    apply A.2,\n    apply hxp,\n    exact h }\nend\n\nlemma yoneda_mono {a b : P} : yoneda a ≤ yoneda b ↔ a ≤ b :=\nbegin\n  rw yoneda_le_iff, refl,\nend\n\nlemma presheaf.eq_supr {A : presheaf P} : A = ⨆ (p : P) (h : A p), yoneda p :=\nbegin\n  apply le_antisymm; simp only [le_supr_iff, supr_le_iff, yoneda_le_iff],\n  { intros B hB p,\n    exact hB p },\n  { exact λ _, id }\nend\n\nvariable (P)\n\ndef copresheaf : Type := \n{ s : set P // ∀ a b, a ≤ b → s a → s b }\n\nvariable {P}\n\nnamespace copresheaf\n\ninstance : has_coe_to_fun (copresheaf P) (λ _, P → Prop) :=\n⟨subtype.val⟩\n\n@[simp] lemma coe_mk (s : P → Prop) (hs : ∀ a b, a ≤ b → s a → s b) :\n  @coe_fn (copresheaf P) _ _ (⟨s, hs⟩ : copresheaf P) = s := rfl\n\ninstance : partial_order (copresheaf P) :=\n{ le := λ A B, ∀ x, B x → A x,\n  le_trans := λ A B C hAB hBC x hAx, hAB _ (hBC _ hAx),\n  le_refl := λ A x, id,\n  le_antisymm := λ A B hAB hBA, subtype.val_injective (funext $ λ x, propext ⟨hBA _, hAB _⟩) }\n\ninstance : has_Sup (copresheaf P) :=\n{ Sup := λ s, ⟨λ p, ∀ A : copresheaf P, A ∈ s → A p, \n     λ a b hab h A hAs, A.2 _ _ hab (h _ hAs)⟩ }\n\ninstance : complete_lattice (copresheaf P) :=\ncomplete_lattice_of_Sup _ \n  (λ s, begin\n    split,\n    { dsimp [Sup, upper_bounds],\n      intros A hAs p h,\n      apply h,\n      exact hAs },\n    { dsimp [Inf, upper_bounds, lower_bounds],\n      intros A h p hAp B hBs,\n      apply h,\n      exact hBs,\n      exact hAp }\n  end)\n\nlemma infi_def {ι : Sort*} (A : ι → copresheaf P) : \n  infi A = ⟨λ p, ∃ i, A i p, λ x y hxy ⟨i, hi⟩, ⟨i, (A i).2 x y hxy hi⟩⟩ :=\nle_antisymm \n  (infi_le_iff.2 (λ B h p ⟨i, hi⟩, h i _ hi)) \n  (le_infi_iff.2 (λ i p h, ⟨i, h⟩))\n\nlemma supr_def {ι : Sort*} (A : ι → copresheaf P) : \n  supr A = ⟨λ p, ∀ i, A i p, λ x y hxy h i, (A i).2 _ y hxy (h i)⟩ :=\nle_antisymm \n  (supr_le_iff.2 (λ i p h, h _)) \n  (le_supr_iff.2 (λ B h p hBp i, h i _ hBp))\n\nlemma le_def {A B : copresheaf P} : A ≤ B = ∀ x, B x → A x := rfl\n\nend copresheaf\n\ndef coyoneda (a : P) : copresheaf P :=\n⟨λ b, a ≤ b, λ b c, function.swap le_trans⟩\n\ndef le_coyoneda_iff (A : copresheaf P) (p : P) : A ≤ coyoneda p ↔ A p :=\nbegin\n  simp [yoneda, copresheaf.le_def],\n  split,\n  { intro h, apply h, exact le_rfl },\n  { intros h x hxp,\n    apply A.2,\n    apply hxp,\n    exact h }\nend\n\nlemma copresheaf.eq_infi {A : copresheaf P} : A = ⨅ (p : P) (h : A p), coyoneda p :=\nbegin\n  apply le_antisymm; simp only [le_infi_iff, infi_le_iff, le_coyoneda_iff],\n  { exact λ _, id },\n  { intros B hB p,\n    exact hB p },\nend\n\nlemma is_glb_yoneda {x : P} : is_glb {y | coyoneda (yoneda x) y} (yoneda x) :=\nbegin\n  split,\n  { dsimp [lower_bounds, yoneda, coyoneda],\n    exact λ _, id },\n  { dsimp [upper_bounds, lower_bounds, yoneda, coyoneda],\n    intros A h y hAy,\n    exact @h (yoneda x) (λ _, id) y hAy }\nend\n\nlemma is_lub_coyoneda {x : P} : is_lub {y | yoneda (coyoneda x) y} (coyoneda x) :=\nbegin\n  split,\n  { dsimp [lower_bounds, yoneda, coyoneda],\n    exact λ _, id },\n  { dsimp [upper_bounds, lower_bounds, yoneda, coyoneda],\n    intros A h y hAy,\n    exact @h (coyoneda x) (λ _, id) y hAy }\nend\n\ndef u (A : presheaf P) : copresheaf P :=\n⟨λ p, coyoneda A (yoneda p), λ a b hab h x hxA, le_trans (h _ hxA) hab⟩\n\nlemma u_mono : monotone (@u P _) :=\nλ A B h p hp q hAq, hp _ (h _ hAq)\n\ndef d (B : copresheaf P) : presheaf P :=\n⟨λ q, yoneda B (coyoneda q), λ a b hab h x hxA, le_trans hab (h _ hxA)⟩\n\nlemma d_mono : monotone (@d P _) :=\nλ A B h p hp q hAq, hp _ (h _ hAq)\n\nexample (A : presheaf P) : d (u A) = A :=\nbegin\n  dsimp [d, u, coyoneda, yoneda],\n  apply subtype.ext,\n  dsimp,\n  apply funext,\n  intros,\n  simp only [presheaf.le_def, copresheaf.le_def],\n  dsimp,\n  apply le_antisymm,\n  simp [presheaf.le_def],\n  ext,\n  rw [presheaf.le_def],\nend\n\ndef gc : galois_connection (@u P _) d :=\nbegin\n  intros A B,\n  dsimp [u, d],\n  split,\n  { intros h x hxA y hyB,\n    apply h,\n    assumption,\n    assumption },\n  { intros h x hxA y hyB,\n    apply h,\n    assumption,\n    assumption }\nend\n\nlemma le_d_u (A : presheaf P) : A ≤ d (u A) := gc.le_u_l A\nlemma u_d_le (A : copresheaf P) : u (d A) ≤ A := gc.l_u_le A\n\nvariable (P)\n\nstructure completion : Type :=\n( to_presheaf : presheaf P )\n( to_copresheaf : copresheaf P )\n( d_to_copresheaf : d to_copresheaf = to_presheaf )\n( u_to_presheaf : u to_presheaf = to_copresheaf )\n\nattribute [simp] completion.d_to_copresheaf completion.u_to_presheaf\n\nvariable {P}\n\ninstance : partial_order (completion P) :=\npartial_order.lift completion.to_copresheaf \n  begin\n    rintros ⟨_, _, h₁, _⟩ ⟨_, _, h₂, _⟩,\n    simp [← h₁, ← h₂] {contextual := tt}\n  end\n\n@[simp] lemma u_d_u (A : presheaf P) : (u (d (u A))) = (u A) :=\nle_antisymm \n  begin\n    dsimp [d, u],\n    intros p hp,\n    dsimp [yoneda, coyoneda, presheaf.le_def] at *,\n    intros q h,\n    apply h,\n    apply hp\n  end\n  (gc.monotone_l (le_d_u _))\n\n@[simp] lemma d_u_d (A : copresheaf P) : d (u (d A)) = d A :=\nle_antisymm \n  (gc.monotone_u (u_d_le _))\n  begin\n    dsimp [d, u],\n    intros p hp,\n    dsimp [yoneda, coyoneda, presheaf.le_def] at *,\n    intros q h,\n    apply h,\n    apply hp\n  end\n\n@[simp] lemma u_yoneda (p : P) : u (yoneda p) = coyoneda p :=\nbegin\n  rw [u],\n  conv_lhs {dsimp [coyoneda]},\n  simp only [yoneda_le_iff],\n  refl\nend\n\n@[simp] lemma d_coyoneda (p : P) : d (coyoneda p) = yoneda p :=\nbegin\n  rw [d],\n  conv_lhs {dsimp only [yoneda]},\n  simp only [le_coyoneda_iff],\n  refl\nend\n\nlemma le_iff_to_copresheaf {A B : completion P} : \n  (A ≤ B) ↔ (A.to_copresheaf ≤ B.to_copresheaf) := iff.rfl\n\nlemma le_iff_to_presheaf {A B : completion P} : \n  (A ≤ B) ↔ (A.to_presheaf ≤ B.to_presheaf) := \nbegin\n  rw [le_iff_to_copresheaf],\n  split,\n  { intro h,\n    rw [← completion.d_to_copresheaf, ← completion.d_to_copresheaf],\n    exact d_mono h },\n  { intro h,\n    rw [← completion.u_to_presheaf, ← completion.u_to_presheaf],\n    exact u_mono h }\nend\n\ndef presheaf.to_completion (A : presheaf P) : completion P :=\n⟨d (u A), u A, by simp, by simp⟩\n\ndef copresheaf.to_completion (A : copresheaf P) : completion P :=\n⟨d A, u (d A), by simp, by simp⟩\n\ndef to_completion (p : P) : completion P :=\n⟨yoneda p, coyoneda p, by simp, by simp⟩\n\n@[simp] lemma presheaf.to_completion_to_presheaf (p : completion P) : p.to_presheaf.to_completion = p :=\nbegin\n  cases p,\n  simp [completion.to_presheaf, presheaf.to_completion, *],\nend\n\n@[simp] lemma copresheaf.to_completion_to_copresheaf (p : completion P) : p.to_copresheaf.to_completion = p :=\nbegin\n  cases p,\n  simp [completion.to_copresheaf, copresheaf.to_completion, *],\nend\n\n@[simp] lemma to_completion_to_presheaf (p : P) : (to_completion p).to_presheaf = yoneda p := rfl\n\n@[simp] lemma to_completion_to_copresheaf (p : P) : (to_completion p).to_copresheaf = coyoneda p := rfl\n\nvariables {P}\n\ndef gi : galois_insertion (@presheaf.to_completion P _) (completion.to_presheaf) :=\ngalois_connection.to_galois_insertion \n  (λ A B, begin\n    split,\n    { simp [presheaf.to_completion, le_iff_to_presheaf, d, u, coyoneda, presheaf.le_def, yoneda],\n      intros h p hAp,\n      apply h,\n      intros q hq,\n      apply hq,\n      exact hAp },\n    { intro h,\n      simp [presheaf.to_completion, le_iff_to_presheaf],\n      rw [← B.d_to_copresheaf, ← B.u_to_presheaf],\n      refine d_mono (u_mono h) }\n  end)\n  (by simp [presheaf.to_completion, le_iff_to_presheaf])\n\ndef gci : galois_coinsertion (completion.to_copresheaf) (@copresheaf.to_completion P _) :=\ngalois_connection.to_galois_coinsertion \n  (λ A B, begin\n    split,\n    { intro h,\n      simp [copresheaf.to_completion, le_iff_to_copresheaf],\n      rw [← A.u_to_presheaf, ← A.d_to_copresheaf],\n      exact u_mono (d_mono h) },\n    { simp [copresheaf.to_completion, le_iff_to_copresheaf, d, u, coyoneda, copresheaf.le_def, yoneda],\n      intros h p hAp,\n      apply h,\n      intros q hq,\n      apply hq,\n      exact hAp },\n  end)\n  (by simp [copresheaf.to_completion, le_iff_to_copresheaf])\n\ninstance : complete_lattice (completion P) :=\ngalois_insertion.lift_complete_lattice gi\n\nlemma completion.supr_def {ι : Sort*} (a : ι → completion P) : (⨆ i : ι, a i).to_copresheaf = \n  (⨆ i, (a i).to_copresheaf) :=\n(@gci P _).gc.l_supr\n\nlemma completion.infi_def {ι : Sort*} (a : ι → completion P) : (⨅ i : ι, a i).to_presheaf = \n  (⨅ i, (a i).to_presheaf) :=\n(@gi P _).gc.u_infi\n", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/partial_order_completion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3225471713744836}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.measure_space\nimport Mathlib.measure_theory.borel_space\nimport Mathlib.data.indicator_function\nimport Mathlib.data.support\nimport Mathlib.PostPort\n\nuniverses u v l u_1 u_2 u_3 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Lebesgue integral for `ennreal`-valued functions\n\nWe define simple functions and show that each Borel measurable function on `ennreal` can be\napproximated by a sequence of simple functions.\n\nTo prove something for an arbitrary measurable function into `ennreal`, the theorem\n`measurable.ennreal_induction` shows that is it sufficient to show that the property holds for\n(multiples of) characteristic functions and is closed under addition and supremum of increasing\nsequences of functions.\n\n## Notation\n\nWe introduce the following notation for the lower Lebesgue integral of a function `f : α → ennreal`.\n\n* `∫⁻ x, f x ∂μ`: integral of a function `f : α → ennreal` with respect to a measure `μ`;\n* `∫⁻ x, f x`: integral of a function `f : α → ennreal` with respect to the canonical measure\n  `volume` on `α`;\n* `∫⁻ x in s, f x ∂μ`: integral of a function `f : α → ennreal` over a set `s` with respect\n  to a measure `μ`, defined as `∫⁻ x, f x ∂(μ.restrict s)`;\n* `∫⁻ x in s, f x`: integral of a function `f : α → ennreal` over a set `s` with respect\n  to the canonical measure `volume`, defined as `∫⁻ x, f x ∂(volume.restrict s)`.\n\n-/\n\nnamespace measure_theory\n\n\n/-- A function `f` from a measurable space to any type is called *simple*,\nif every preimage `f ⁻¹' {x}` is measurable, and the range is finite. This structure bundles\na function with these properties. -/\nstructure simple_func (α : Type u) [measurable_space α] (β : Type v) where\n  to_fun : α → β\n  is_measurable_fiber' : ∀ (x : β), is_measurable (to_fun ⁻¹' singleton x)\n  finite_range' : set.finite (set.range to_fun)\n\nnamespace simple_func\n\n\nprotected instance has_coe_to_fun {α : Type u_1} {β : Type u_2} [measurable_space α] :\n    has_coe_to_fun (simple_func α β) :=\n  has_coe_to_fun.mk (fun (x : simple_func α β) => α → β) to_fun\n\ntheorem coe_injective {α : Type u_1} {β : Type u_2} [measurable_space α] {f : simple_func α β}\n    {g : simple_func α β} (H : ⇑f = ⇑g) : f = g :=\n  sorry\n\ntheorem ext {α : Type u_1} {β : Type u_2} [measurable_space α] {f : simple_func α β}\n    {g : simple_func α β} (H : ∀ (a : α), coe_fn f a = coe_fn g a) : f = g :=\n  coe_injective (funext H)\n\ntheorem finite_range {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β) :\n    set.finite (set.range ⇑f) :=\n  finite_range' f\n\ntheorem is_measurable_fiber {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β)\n    (x : β) : is_measurable (⇑f ⁻¹' singleton x) :=\n  is_measurable_fiber' f x\n\n/-- Range of a simple function `α →ₛ β` as a `finset β`. -/\nprotected def range {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β) :\n    finset β :=\n  set.finite.to_finset (finite_range f)\n\n@[simp] theorem mem_range {α : Type u_1} {β : Type u_2} [measurable_space α] {f : simple_func α β}\n    {b : β} : b ∈ simple_func.range f ↔ b ∈ set.range ⇑f :=\n  set.finite.mem_to_finset\n\ntheorem mem_range_self {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β)\n    (x : α) : coe_fn f x ∈ simple_func.range f :=\n  iff.mpr mem_range (Exists.intro x rfl)\n\n@[simp] theorem coe_range {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β) :\n    ↑(simple_func.range f) = set.range ⇑f :=\n  set.finite.coe_to_finset (finite_range f)\n\ntheorem mem_range_of_measure_ne_zero {α : Type u_1} {β : Type u_2} [measurable_space α]\n    {f : simple_func α β} {x : β} {μ : measure α} (H : coe_fn μ (⇑f ⁻¹' singleton x) ≠ 0) :\n    x ∈ simple_func.range f :=\n  sorry\n\ntheorem forall_range_iff {α : Type u_1} {β : Type u_2} [measurable_space α] {f : simple_func α β}\n    {p : β → Prop} : (∀ (y : β), y ∈ simple_func.range f → p y) ↔ ∀ (x : α), p (coe_fn f x) :=\n  sorry\n\ntheorem exists_range_iff {α : Type u_1} {β : Type u_2} [measurable_space α] {f : simple_func α β}\n    {p : β → Prop} :\n    (∃ (y : β), ∃ (H : y ∈ simple_func.range f), p y) ↔ ∃ (x : α), p (coe_fn f x) :=\n  sorry\n\ntheorem preimage_eq_empty_iff {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) (b : β) : ⇑f ⁻¹' singleton b = ∅ ↔ ¬b ∈ simple_func.range f :=\n  iff.trans set.preimage_singleton_eq_empty (not_congr (iff.symm mem_range))\n\ntheorem exists_forall_le {α : Type u_1} {β : Type u_2} [measurable_space α] [Nonempty β]\n    [directed_order β] (f : simple_func α β) : ∃ (C : β), ∀ (x : α), coe_fn f x ≤ C :=\n  Exists.imp (fun (C : β) => iff.mp forall_range_iff) (finset.exists_le (simple_func.range f))\n\n/-- Constant function as a `simple_func`. -/\ndef const (α : Type u_1) {β : Type u_2} [measurable_space α] (b : β) : simple_func α β :=\n  mk (fun (a : α) => b) sorry set.finite_range_const\n\nprotected instance inhabited {α : Type u_1} {β : Type u_2} [measurable_space α] [Inhabited β] :\n    Inhabited (simple_func α β) :=\n  { default := const α Inhabited.default }\n\ntheorem const_apply {α : Type u_1} {β : Type u_2} [measurable_space α] (a : α) (b : β) :\n    coe_fn (const α b) a = b :=\n  rfl\n\n@[simp] theorem coe_const {α : Type u_1} {β : Type u_2} [measurable_space α] (b : β) :\n    ⇑(const α b) = function.const α b :=\n  rfl\n\n@[simp] theorem range_const {β : Type u_2} (α : Type u_1) [measurable_space α] [Nonempty α]\n    (b : β) : simple_func.range (const α b) = singleton b :=\n  sorry\n\ntheorem is_measurable_cut {α : Type u_1} {β : Type u_2} [measurable_space α] (r : α → β → Prop)\n    (f : simple_func α β) (h : ∀ (b : β), is_measurable (set_of fun (a : α) => r a b)) :\n    is_measurable (set_of fun (a : α) => r a (coe_fn f a)) :=\n  sorry\n\ntheorem is_measurable_preimage {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) (s : set β) : is_measurable (⇑f ⁻¹' s) :=\n  is_measurable_cut (fun (_x : α) (b : β) => b ∈ s) f fun (b : β) => is_measurable.const (b ∈ s)\n\n/-- A simple function is measurable -/\nprotected theorem measurable {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β]\n    (f : simple_func α β) : measurable ⇑f :=\n  fun (s : set β) (_x : is_measurable s) => is_measurable_preimage f s\n\nprotected theorem ae_measurable {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [measurable_space β] {μ : measure α} (f : simple_func α β) : ae_measurable ⇑f :=\n  measurable.ae_measurable (simple_func.measurable f)\n\nprotected theorem sum_measure_preimage_singleton {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) {μ : measure α} (s : finset β) :\n    (finset.sum s fun (y : β) => coe_fn μ (⇑f ⁻¹' singleton y)) = coe_fn μ (⇑f ⁻¹' ↑s) :=\n  sum_measure_preimage_singleton s fun (_x : β) (_x_1 : _x ∈ s) => is_measurable_fiber f _x\n\ntheorem sum_range_measure_preimage_singleton {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) (μ : measure α) :\n    (finset.sum (simple_func.range f) fun (y : β) => coe_fn μ (⇑f ⁻¹' singleton y)) =\n        coe_fn μ set.univ :=\n  sorry\n\n/-- If-then-else as a `simple_func`. -/\ndef piecewise {α : Type u_1} {β : Type u_2} [measurable_space α] (s : set α) (hs : is_measurable s)\n    (f : simple_func α β) (g : simple_func α β) : simple_func α β :=\n  mk (set.piecewise s ⇑f ⇑g) sorry sorry\n\n@[simp] theorem coe_piecewise {α : Type u_1} {β : Type u_2} [measurable_space α] {s : set α}\n    (hs : is_measurable s) (f : simple_func α β) (g : simple_func α β) :\n    ⇑(piecewise s hs f g) = set.piecewise s ⇑f ⇑g :=\n  rfl\n\ntheorem piecewise_apply {α : Type u_1} {β : Type u_2} [measurable_space α] {s : set α}\n    (hs : is_measurable s) (f : simple_func α β) (g : simple_func α β) (a : α) :\n    coe_fn (piecewise s hs f g) a = ite (a ∈ s) (coe_fn f a) (coe_fn g a) :=\n  rfl\n\n@[simp] theorem piecewise_compl {α : Type u_1} {β : Type u_2} [measurable_space α] {s : set α}\n    (hs : is_measurable (sᶜ)) (f : simple_func α β) (g : simple_func α β) :\n    piecewise (sᶜ) hs f g = piecewise s (is_measurable.of_compl hs) g f :=\n  sorry\n\n@[simp] theorem piecewise_univ {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) (g : simple_func α β) : piecewise set.univ is_measurable.univ f g = f :=\n  sorry\n\n@[simp] theorem piecewise_empty {α : Type u_1} {β : Type u_2} [measurable_space α]\n    (f : simple_func α β) (g : simple_func α β) : piecewise ∅ is_measurable.empty f g = g :=\n  sorry\n\ntheorem measurable_bind {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [measurable_space γ] (f : simple_func α β) (g : β → α → γ) (hg : ∀ (b : β), measurable (g b)) :\n    measurable fun (a : α) => g (coe_fn f a) a :=\n  fun (s : set γ) (hs : is_measurable s) =>\n    is_measurable_cut (fun (a : α) (b : β) => g b a ∈ s) f fun (b : β) => hg b hs\n\n/-- If `f : α →ₛ β` is a simple function and `g : β → α →ₛ γ` is a family of simple functions,\nthen `f.bind g` binds the first argument of `g` to `f`. In other words, `f.bind g a = g (f a) a`. -/\ndef bind {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] (f : simple_func α β)\n    (g : β → simple_func α γ) : simple_func α γ :=\n  mk (fun (a : α) => coe_fn (g (coe_fn f a)) a) sorry sorry\n\n@[simp] theorem bind_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : β → simple_func α γ) (a : α) :\n    coe_fn (bind f g) a = coe_fn (g (coe_fn f a)) a :=\n  rfl\n\n/-- Given a function `g : β → γ` and a simple function `f : α →ₛ β`, `f.map g` return the simple\n    function `g ∘ f : α →ₛ γ` -/\ndef map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] (g : β → γ)\n    (f : simple_func α β) : simple_func α γ :=\n  bind f (const α ∘ g)\n\ntheorem map_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] (g : β → γ)\n    (f : simple_func α β) (a : α) : coe_fn (map g f) a = g (coe_fn f a) :=\n  rfl\n\ntheorem map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [measurable_space α]\n    (g : β → γ) (h : γ → δ) (f : simple_func α β) : map h (map g f) = map (h ∘ g) f :=\n  rfl\n\n@[simp] theorem coe_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (g : β → γ) (f : simple_func α β) : ⇑(map g f) = g ∘ ⇑f :=\n  rfl\n\n@[simp] theorem range_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [DecidableEq γ] (g : β → γ) (f : simple_func α β) :\n    simple_func.range (map g f) = finset.image g (simple_func.range f) :=\n  sorry\n\n@[simp] theorem map_const {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (g : β → γ) (b : β) : map g (const α b) = const α (g b) :=\n  rfl\n\ntheorem map_preimage {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : β → γ) (s : set γ) :\n    ⇑(map g f) ⁻¹' s = ⇑f ⁻¹' ↑(finset.filter (fun (b : β) => g b ∈ s) (simple_func.range f)) :=\n  sorry\n\ntheorem map_preimage_singleton {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : β → γ) (c : γ) :\n    ⇑(map g f) ⁻¹' singleton c =\n        ⇑f ⁻¹' ↑(finset.filter (fun (b : β) => g b = c) (simple_func.range f)) :=\n  map_preimage f g (singleton c)\n\n/-- Composition of a `simple_fun` and a measurable function is a `simple_func`. -/\ndef comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [measurable_space β]\n    (f : simple_func β γ) (g : α → β) (hgm : measurable g) : simple_func α γ :=\n  mk (⇑f ∘ g) sorry sorry\n\n@[simp] theorem coe_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [measurable_space β] (f : simple_func β γ) {g : α → β} (hgm : measurable g) :\n    ⇑(comp f g hgm) = ⇑f ∘ g :=\n  rfl\n\ntheorem range_comp_subset_range {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [measurable_space β] (f : simple_func β γ) {g : α → β} (hgm : measurable g) :\n    simple_func.range (comp f g hgm) ⊆ simple_func.range f :=\n  sorry\n\n/-- If `f` is a simple function taking values in `β → γ` and `g` is another simple function\nwith the same domain and codomain `β`, then `f.seq g = f a (g a)`. -/\ndef seq {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α (β → γ)) (g : simple_func α β) : simple_func α γ :=\n  bind f fun (f : β → γ) => map f g\n\n@[simp] theorem seq_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α (β → γ)) (g : simple_func α β) (a : α) :\n    coe_fn (seq f g) a = coe_fn f a (coe_fn g a) :=\n  rfl\n\n/-- Combine two simple functions `f : α →ₛ β` and `g : α →ₛ β`\ninto `λ a, (f a, g a)`. -/\ndef pair {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] (f : simple_func α β)\n    (g : simple_func α γ) : simple_func α (β × γ) :=\n  seq (map Prod.mk f) g\n\n@[simp] theorem pair_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : simple_func α γ) (a : α) :\n    coe_fn (pair f g) a = (coe_fn f a, coe_fn g a) :=\n  rfl\n\ntheorem pair_preimage {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : simple_func α γ) (s : set β) (t : set γ) :\n    ⇑(pair f g) ⁻¹' set.prod s t = ⇑f ⁻¹' s ∩ ⇑g ⁻¹' t :=\n  rfl\n\n/- A special form of `pair_preimage` -/\n\ntheorem pair_preimage_singleton {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    (f : simple_func α β) (g : simple_func α γ) (b : β) (c : γ) :\n    ⇑(pair f g) ⁻¹' singleton (b, c) = ⇑f ⁻¹' singleton b ∩ ⇑g ⁻¹' singleton c :=\n  sorry\n\ntheorem bind_const {α : Type u_1} {β : Type u_2} [measurable_space α] (f : simple_func α β) :\n    bind f (const α) = f :=\n  sorry\n\nprotected instance has_zero {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] :\n    HasZero (simple_func α β) :=\n  { zero := const α 0 }\n\nprotected instance has_add {α : Type u_1} {β : Type u_2} [measurable_space α] [Add β] :\n    Add (simple_func α β) :=\n  { add := fun (f g : simple_func α β) => seq (map Add.add f) g }\n\nprotected instance has_mul {α : Type u_1} {β : Type u_2} [measurable_space α] [Mul β] :\n    Mul (simple_func α β) :=\n  { mul := fun (f g : simple_func α β) => seq (map Mul.mul f) g }\n\nprotected instance has_sup {α : Type u_1} {β : Type u_2} [measurable_space α] [has_sup β] :\n    has_sup (simple_func α β) :=\n  has_sup.mk fun (f g : simple_func α β) => seq (map has_sup.sup f) g\n\nprotected instance has_inf {α : Type u_1} {β : Type u_2} [measurable_space α] [has_inf β] :\n    has_inf (simple_func α β) :=\n  has_inf.mk fun (f g : simple_func α β) => seq (map has_inf.inf f) g\n\nprotected instance has_le {α : Type u_1} {β : Type u_2} [measurable_space α] [HasLessEq β] :\n    HasLessEq (simple_func α β) :=\n  { LessEq := fun (f g : simple_func α β) => ∀ (a : α), coe_fn f a ≤ coe_fn g a }\n\n@[simp] theorem coe_zero {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] : ⇑0 = 0 :=\n  rfl\n\n@[simp] theorem const_zero {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] :\n    const α 0 = 0 :=\n  rfl\n\n@[simp] theorem coe_add {α : Type u_1} {β : Type u_2} [measurable_space α] [Add β]\n    (f : simple_func α β) (g : simple_func α β) : ⇑(f + g) = ⇑f + ⇑g :=\n  rfl\n\n@[simp] theorem coe_mul {α : Type u_1} {β : Type u_2} [measurable_space α] [Mul β]\n    (f : simple_func α β) (g : simple_func α β) : ⇑(f * g) = ⇑f * ⇑g :=\n  rfl\n\n@[simp] theorem coe_le {α : Type u_1} {β : Type u_2} [measurable_space α] [preorder β]\n    {f : simple_func α β} {g : simple_func α β} : ⇑f ≤ ⇑g ↔ f ≤ g :=\n  iff.rfl\n\n@[simp] theorem range_zero {α : Type u_1} {β : Type u_2} [measurable_space α] [Nonempty α]\n    [HasZero β] : simple_func.range 0 = singleton 0 :=\n  sorry\n\ntheorem eq_zero_of_mem_range_zero {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    {y : β} : y ∈ simple_func.range 0 → y = 0 :=\n  iff.mpr forall_range_iff fun (x : α) => rfl\n\ntheorem sup_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [has_sup β]\n    (f : simple_func α β) (g : simple_func α β) (a : α) :\n    coe_fn (f ⊔ g) a = coe_fn f a ⊔ coe_fn g a :=\n  rfl\n\ntheorem mul_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [Mul β] (f : simple_func α β)\n    (g : simple_func α β) (a : α) : coe_fn (f * g) a = coe_fn f a * coe_fn g a :=\n  rfl\n\ntheorem add_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [Add β] (f : simple_func α β)\n    (g : simple_func α β) (a : α) : coe_fn (f + g) a = coe_fn f a + coe_fn g a :=\n  rfl\n\ntheorem add_eq_map₂ {α : Type u_1} {β : Type u_2} [measurable_space α] [Add β] (f : simple_func α β)\n    (g : simple_func α β) : f + g = map (fun (p : β × β) => prod.fst p + prod.snd p) (pair f g) :=\n  rfl\n\ntheorem mul_eq_map₂ {α : Type u_1} {β : Type u_2} [measurable_space α] [Mul β] (f : simple_func α β)\n    (g : simple_func α β) : f * g = map (fun (p : β × β) => prod.fst p * prod.snd p) (pair f g) :=\n  rfl\n\ntheorem sup_eq_map₂ {α : Type u_1} {β : Type u_2} [measurable_space α] [has_sup β]\n    (f : simple_func α β) (g : simple_func α β) :\n    f ⊔ g = map (fun (p : β × β) => prod.fst p ⊔ prod.snd p) (pair f g) :=\n  rfl\n\ntheorem const_mul_eq_map {α : Type u_1} {β : Type u_2} [measurable_space α] [Mul β]\n    (f : simple_func α β) (b : β) : const α b * f = map (fun (a : β) => b * a) f :=\n  rfl\n\ntheorem map_add {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [Add β] [Add γ]\n    {g : β → γ} (hg : ∀ (x y : β), g (x + y) = g x + g y) (f₁ : simple_func α β)\n    (f₂ : simple_func α β) : map g (f₁ + f₂) = map g f₁ + map g f₂ :=\n  ext fun (x : α) => hg (coe_fn f₁ x) (coe_fn f₂ x)\n\nprotected instance add_monoid {α : Type u_1} {β : Type u_2} [measurable_space α] [add_monoid β] :\n    add_monoid (simple_func α β) :=\n  function.injective.add_monoid (fun (f : simple_func α β) => (fun (this : α → β) => this) ⇑f)\n    coe_injective sorry sorry\n\nprotected instance add_comm_monoid {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [add_comm_monoid β] : add_comm_monoid (simple_func α β) :=\n  function.injective.add_comm_monoid (fun (f : simple_func α β) => (fun (this : α → β) => this) ⇑f)\n    coe_injective sorry sorry\n\nprotected instance has_neg {α : Type u_1} {β : Type u_2} [measurable_space α] [Neg β] :\n    Neg (simple_func α β) :=\n  { neg := fun (f : simple_func α β) => map Neg.neg f }\n\n@[simp] theorem coe_neg {α : Type u_1} {β : Type u_2} [measurable_space α] [Neg β]\n    (f : simple_func α β) : ⇑(-f) = -⇑f :=\n  rfl\n\nprotected instance has_sub {α : Type u_1} {β : Type u_2} [measurable_space α] [Sub β] :\n    Sub (simple_func α β) :=\n  { sub := fun (f g : simple_func α β) => seq (map Sub.sub f) g }\n\n@[simp] theorem coe_sub {α : Type u_1} {β : Type u_2} [measurable_space α] [Sub β]\n    (f : simple_func α β) (g : simple_func α β) : ⇑(f - g) = ⇑f - ⇑g :=\n  rfl\n\ntheorem sub_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [Sub β] (f : simple_func α β)\n    (g : simple_func α β) (x : α) : coe_fn (f - g) x = coe_fn f x - coe_fn g x :=\n  rfl\n\nprotected instance add_group {α : Type u_1} {β : Type u_2} [measurable_space α] [add_group β] :\n    add_group (simple_func α β) :=\n  function.injective.add_group_sub (fun (f : simple_func α β) => (fun (this : α → β) => this) ⇑f)\n    coe_injective sorry sorry sorry sorry\n\nprotected instance add_comm_group {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [add_comm_group β] : add_comm_group (simple_func α β) :=\n  function.injective.add_comm_group_sub\n    (fun (f : simple_func α β) => (fun (this : α → β) => this) ⇑f) coe_injective sorry sorry sorry\n    sorry\n\nprotected instance has_scalar {α : Type u_1} {β : Type u_2} [measurable_space α] {K : Type u_5}\n    [has_scalar K β] : has_scalar K (simple_func α β) :=\n  has_scalar.mk fun (k : K) (f : simple_func α β) => map (has_scalar.smul k) f\n\n@[simp] theorem coe_smul {α : Type u_1} {β : Type u_2} [measurable_space α] {K : Type u_5}\n    [has_scalar K β] (c : K) (f : simple_func α β) : ⇑(c • f) = c • ⇑f :=\n  rfl\n\ntheorem smul_apply {α : Type u_1} {β : Type u_2} [measurable_space α] {K : Type u_5}\n    [has_scalar K β] (k : K) (f : simple_func α β) (a : α) : coe_fn (k • f) a = k • coe_fn f a :=\n  rfl\n\nprotected instance semimodule {α : Type u_1} {β : Type u_2} [measurable_space α] {K : Type u_5}\n    [semiring K] [add_comm_monoid β] [semimodule K β] : semimodule K (simple_func α β) :=\n  function.injective.semimodule K\n    (add_monoid_hom.mk (fun (f : simple_func α β) => (fun (this : α → β) => this) ⇑f) sorry sorry)\n    coe_injective sorry\n\ntheorem smul_eq_map {α : Type u_1} {β : Type u_2} [measurable_space α] {K : Type u_5}\n    [has_scalar K β] (k : K) (f : simple_func α β) : k • f = map (has_scalar.smul k) f :=\n  rfl\n\nprotected instance preorder {α : Type u_1} {β : Type u_2} [measurable_space α] [preorder β] :\n    preorder (simple_func α β) :=\n  preorder.mk LessEq (fun (a b : simple_func α β) => a ≤ b ∧ ¬b ≤ a) sorry sorry\n\nprotected instance partial_order {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [partial_order β] : partial_order (simple_func α β) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\nprotected instance order_bot {α : Type u_1} {β : Type u_2} [measurable_space α] [order_bot β] :\n    order_bot (simple_func α β) :=\n  order_bot.mk (const α ⊥) partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance order_top {α : Type u_1} {β : Type u_2} [measurable_space α] [order_top β] :\n    order_top (simple_func α β) :=\n  order_top.mk (const α ⊤) partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance semilattice_inf {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [semilattice_inf β] : semilattice_inf (simple_func α β) :=\n  semilattice_inf.mk has_inf.inf partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n    sorry\n\nprotected instance semilattice_sup {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [semilattice_sup β] : semilattice_sup (simple_func α β) :=\n  semilattice_sup.mk has_sup.sup partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n    sorry\n\nprotected instance semilattice_sup_bot {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [semilattice_sup_bot β] : semilattice_sup_bot (simple_func α β) :=\n  semilattice_sup_bot.mk order_bot.bot semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry\n    semilattice_sup.sup sorry sorry sorry\n\nprotected instance lattice {α : Type u_1} {β : Type u_2} [measurable_space α] [lattice β] :\n    lattice (simple_func α β) :=\n  lattice.mk semilattice_sup.sup semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry sorry\n    sorry semilattice_inf.inf sorry sorry sorry\n\nprotected instance bounded_lattice {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [bounded_lattice β] : bounded_lattice (simple_func α β) :=\n  bounded_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry\n    lattice.inf sorry sorry sorry order_top.top sorry order_bot.bot sorry\n\ntheorem finset_sup_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [semilattice_sup_bot β] {f : γ → simple_func α β} (s : finset γ) (a : α) :\n    coe_fn (finset.sup s f) a = finset.sup s fun (c : γ) => coe_fn (f c) a :=\n  sorry\n\n/-- Restrict a simple function `f : α →ₛ β` to a set `s`. If `s` is measurable,\nthen `f.restrict s a = if a ∈ s then f a else 0`, otherwise `f.restrict s = const α 0`. -/\ndef restrict {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] (f : simple_func α β)\n    (s : set α) : simple_func α β :=\n  dite (is_measurable s) (fun (hs : is_measurable s) => piecewise s hs f 0)\n    fun (hs : ¬is_measurable s) => 0\n\ntheorem restrict_of_not_measurable {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    {f : simple_func α β} {s : set α} (hs : ¬is_measurable s) : restrict f s = 0 :=\n  dif_neg hs\n\n@[simp] theorem coe_restrict {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) {s : set α} (hs : is_measurable s) :\n    ⇑(restrict f s) = set.indicator s ⇑f :=\n  sorry\n\n@[simp] theorem restrict_univ {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) : restrict f set.univ = f :=\n  sorry\n\n@[simp] theorem restrict_empty {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) : restrict f ∅ = 0 :=\n  sorry\n\ntheorem map_restrict_of_zero {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [HasZero β] [HasZero γ] {g : β → γ} (hg : g 0 = 0) (f : simple_func α β) (s : set α) :\n    map g (restrict f s) = restrict (map g f) s :=\n  sorry\n\ntheorem map_coe_ennreal_restrict {α : Type u_1} [measurable_space α] (f : simple_func α nnreal)\n    (s : set α) : map coe (restrict f s) = restrict (map coe f) s :=\n  map_restrict_of_zero ennreal.coe_zero f s\n\ntheorem map_coe_nnreal_restrict {α : Type u_1} [measurable_space α] (f : simple_func α nnreal)\n    (s : set α) : map coe (restrict f s) = restrict (map coe f) s :=\n  map_restrict_of_zero nnreal.coe_zero f s\n\ntheorem restrict_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) {s : set α} (hs : is_measurable s) (a : α) :\n    coe_fn (restrict f s) a = ite (a ∈ s) (coe_fn f a) 0 :=\n  sorry\n\ntheorem restrict_preimage {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) {s : set α} (hs : is_measurable s) {t : set β} (ht : ¬0 ∈ t) :\n    ⇑(restrict f s) ⁻¹' t = s ∩ ⇑f ⁻¹' t :=\n  sorry\n\ntheorem restrict_preimage_singleton {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) {s : set α} (hs : is_measurable s) {r : β} (hr : r ≠ 0) :\n    ⇑(restrict f s) ⁻¹' singleton r = s ∩ ⇑f ⁻¹' singleton r :=\n  restrict_preimage f hs (ne.symm hr)\n\ntheorem mem_restrict_range {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] {r : β}\n    {s : set α} {f : simple_func α β} (hs : is_measurable s) :\n    r ∈ simple_func.range (restrict f s) ↔ r = 0 ∧ s ≠ set.univ ∨ r ∈ ⇑f '' s :=\n  sorry\n\ntheorem mem_image_of_mem_range_restrict {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [HasZero β] {r : β} {s : set α} {f : simple_func α β}\n    (hr : r ∈ simple_func.range (restrict f s)) (h0 : r ≠ 0) : r ∈ ⇑f '' s :=\n  sorry\n\ntheorem restrict_mono {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β] [preorder β]\n    (s : set α) {f : simple_func α β} {g : simple_func α β} (H : f ≤ g) :\n    restrict f s ≤ restrict g s :=\n  sorry\n\n/-- Fix a sequence `i : ℕ → β`. Given a function `α → β`, its `n`-th approximation\nby simple functions is defined so that in case `β = ennreal` it sends each `a` to the supremum\nof the set `{i k | k ≤ n ∧ i k ≤ f a}`, see `approx_apply` and `supr_approx_apply` for details. -/\ndef approx {α : Type u_1} {β : Type u_2} [measurable_space α] [semilattice_sup_bot β] [HasZero β]\n    (i : ℕ → β) (f : α → β) (n : ℕ) : simple_func α β :=\n  finset.sup (finset.range n)\n    fun (k : ℕ) => restrict (const α (i k)) (set_of fun (a : α) => i k ≤ f a)\n\ntheorem approx_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [semilattice_sup_bot β]\n    [HasZero β] [topological_space β] [order_closed_topology β] [measurable_space β]\n    [opens_measurable_space β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : measurable f) :\n    coe_fn (approx i f n) a = finset.sup (finset.range n) fun (k : ℕ) => ite (i k ≤ f a) (i k) 0 :=\n  sorry\n\ntheorem monotone_approx {α : Type u_1} {β : Type u_2} [measurable_space α] [semilattice_sup_bot β]\n    [HasZero β] (i : ℕ → β) (f : α → β) : monotone (approx i f) :=\n  fun (n m : ℕ) (h : n ≤ m) => finset.sup_mono (iff.mpr finset.range_subset h)\n\ntheorem approx_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    [semilattice_sup_bot β] [HasZero β] [topological_space β] [order_closed_topology β]\n    [measurable_space β] [opens_measurable_space β] [measurable_space γ] {i : ℕ → β} {f : γ → β}\n    {g : α → γ} {n : ℕ} (a : α) (hf : measurable f) (hg : measurable g) :\n    coe_fn (approx i (f ∘ g) n) a = coe_fn (approx i f n) (g a) :=\n  sorry\n\ntheorem supr_approx_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [topological_space β]\n    [complete_lattice β] [order_closed_topology β] [HasZero β] [measurable_space β]\n    [opens_measurable_space β] (i : ℕ → β) (f : α → β) (a : α) (hf : measurable f)\n    (h_zero : 0 = ⊥) :\n    (supr fun (n : ℕ) => coe_fn (approx i f n) a) =\n        supr fun (k : ℕ) => supr fun (h : i k ≤ f a) => i k :=\n  sorry\n\n/-- A sequence of `ennreal`s such that its range is the set of non-negative rational numbers. -/\ndef ennreal_rat_embed (n : ℕ) : ennreal :=\n  ennreal.of_real ↑(option.get_or_else (encodable.decode ℚ n) 0)\n\ntheorem ennreal_rat_embed_encode (q : ℚ) :\n    ennreal_rat_embed (encodable.encode q) = ↑(nnreal.of_real ↑q) :=\n  sorry\n\n/-- Approximate a function `α → ennreal` by a sequence of simple functions. -/\ndef eapprox {α : Type u_1} [measurable_space α] : (α → ennreal) → ℕ → simple_func α ennreal :=\n  approx ennreal_rat_embed\n\ntheorem monotone_eapprox {α : Type u_1} [measurable_space α] (f : α → ennreal) :\n    monotone (eapprox f) :=\n  monotone_approx ennreal_rat_embed f\n\ntheorem supr_eapprox_apply {α : Type u_1} [measurable_space α] (f : α → ennreal) (hf : measurable f)\n    (a : α) : (supr fun (n : ℕ) => coe_fn (eapprox f n) a) = f a :=\n  sorry\n\ntheorem eapprox_comp {α : Type u_1} {γ : Type u_3} [measurable_space α] [measurable_space γ]\n    {f : γ → ennreal} {g : α → γ} {n : ℕ} (hf : measurable f) (hg : measurable g) :\n    ⇑(eapprox (f ∘ g) n) = ⇑(eapprox f n) ∘ g :=\n  funext fun (a : α) => approx_comp a hf hg\n\n/-- Integral of a simple function whose codomain is `ennreal`. -/\ndef lintegral {α : Type u_1} [measurable_space α] (f : simple_func α ennreal) (μ : measure α) :\n    ennreal :=\n  finset.sum (simple_func.range f) fun (x : ennreal) => x * coe_fn μ (⇑f ⁻¹' singleton x)\n\ntheorem lintegral_eq_of_subset {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) {s : finset ennreal}\n    (hs :\n      ∀ (x : α), coe_fn f x ≠ 0 → coe_fn μ (⇑f ⁻¹' singleton (coe_fn f x)) ≠ 0 → coe_fn f x ∈ s) :\n    lintegral f μ = finset.sum s fun (x : ennreal) => x * coe_fn μ (⇑f ⁻¹' singleton x) :=\n  sorry\n\n/-- Calculate the integral of `(g ∘ f)`, where `g : β → ennreal` and `f : α →ₛ β`.  -/\ntheorem map_lintegral {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    (g : β → ennreal) (f : simple_func α β) :\n    lintegral (map g f) μ =\n        finset.sum (simple_func.range f) fun (x : β) => g x * coe_fn μ (⇑f ⁻¹' singleton x) :=\n  sorry\n\ntheorem add_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) (g : simple_func α ennreal) :\n    lintegral (f + g) μ = lintegral f μ + lintegral g μ :=\n  sorry\n\ntheorem const_mul_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) (x : ennreal) : lintegral (const α x * f) μ = x * lintegral f μ :=\n  sorry\n\n/-- Integral of a simple function `α →ₛ ennreal` as a bilinear map. -/\ndef lintegralₗ {α : Type u_1} [measurable_space α] :\n    linear_map ennreal (simple_func α ennreal) (linear_map ennreal (measure α) ennreal) :=\n  linear_map.mk (fun (f : simple_func α ennreal) => linear_map.mk (lintegral f) sorry sorry) sorry\n    sorry\n\n@[simp] theorem zero_lintegral {α : Type u_1} [measurable_space α] {μ : measure α} :\n    lintegral 0 μ = 0 :=\n  iff.mp linear_map.ext_iff (linear_map.map_zero lintegralₗ) μ\n\ntheorem lintegral_add {α : Type u_1} [measurable_space α] {μ : measure α} {ν : measure α}\n    (f : simple_func α ennreal) : lintegral f (μ + ν) = lintegral f μ + lintegral f ν :=\n  linear_map.map_add (coe_fn lintegralₗ f) μ ν\n\ntheorem lintegral_smul {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) (c : ennreal) : lintegral f (c • μ) = c • lintegral f μ :=\n  linear_map.map_smul (coe_fn lintegralₗ f) c μ\n\n@[simp] theorem lintegral_zero {α : Type u_1} [measurable_space α] (f : simple_func α ennreal) :\n    lintegral f 0 = 0 :=\n  linear_map.map_zero (coe_fn lintegralₗ f)\n\ntheorem lintegral_sum {α : Type u_1} [measurable_space α] {ι : Type u_2} (f : simple_func α ennreal)\n    (μ : ι → measure α) : lintegral f (measure.sum μ) = tsum fun (i : ι) => lintegral f (μ i) :=\n  sorry\n\ntheorem restrict_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) {s : set α} (hs : is_measurable s) :\n    lintegral (restrict f s) μ =\n        finset.sum (simple_func.range f)\n          fun (r : ennreal) => r * coe_fn μ (⇑f ⁻¹' singleton r ∩ s) :=\n  sorry\n\ntheorem lintegral_restrict {α : Type u_1} [measurable_space α] (f : simple_func α ennreal)\n    (s : set α) (μ : measure α) :\n    lintegral f (measure.restrict μ s) =\n        finset.sum (simple_func.range f)\n          fun (y : ennreal) => y * coe_fn μ (⇑f ⁻¹' singleton y ∩ s) :=\n  sorry\n\ntheorem restrict_lintegral_eq_lintegral_restrict {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) {s : set α} (hs : is_measurable s) :\n    lintegral (restrict f s) μ = lintegral f (measure.restrict μ s) :=\n  sorry\n\ntheorem const_lintegral {α : Type u_1} [measurable_space α] {μ : measure α} (c : ennreal) :\n    lintegral (const α c) μ = c * coe_fn μ set.univ :=\n  sorry\n\ntheorem const_lintegral_restrict {α : Type u_1} [measurable_space α] {μ : measure α} (c : ennreal)\n    (s : set α) : lintegral (const α c) (measure.restrict μ s) = c * coe_fn μ s :=\n  sorry\n\ntheorem restrict_const_lintegral {α : Type u_1} [measurable_space α] {μ : measure α} (c : ennreal)\n    {s : set α} (hs : is_measurable s) : lintegral (restrict (const α c) s) μ = c * coe_fn μ s :=\n  sorry\n\ntheorem le_sup_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : simple_func α ennreal) (g : simple_func α ennreal) :\n    lintegral f μ ⊔ lintegral g μ ≤ lintegral (f ⊔ g) μ :=\n  sorry\n\n/-- `simple_func.lintegral` is monotone both in function and in measure. -/\ntheorem lintegral_mono {α : Type u_1} [measurable_space α] {f : simple_func α ennreal}\n    {g : simple_func α ennreal} (hfg : f ≤ g) {μ : measure α} {ν : measure α} (hμν : μ ≤ ν) :\n    lintegral f μ ≤ lintegral g ν :=\n  sorry\n\n/-- `simple_func.lintegral` depends only on the measures of `f ⁻¹' {y}`. -/\ntheorem lintegral_eq_of_measure_preimage {α : Type u_1} {β : Type u_2} [measurable_space α]\n    {μ : measure α} [measurable_space β] {f : simple_func α ennreal} {g : simple_func β ennreal}\n    {ν : measure β}\n    (H : ∀ (y : ennreal), coe_fn μ (⇑f ⁻¹' singleton y) = coe_fn ν (⇑g ⁻¹' singleton y)) :\n    lintegral f μ = lintegral g ν :=\n  sorry\n\n/-- If two simple functions are equal a.e., then their `lintegral`s are equal. -/\ntheorem lintegral_congr {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : simple_func α ennreal} {g : simple_func α ennreal}\n    (h : filter.eventually_eq (measure.ae μ) ⇑f ⇑g) : lintegral f μ = lintegral g μ :=\n  sorry\n\ntheorem lintegral_map {α : Type u_1} [measurable_space α] {μ : measure α} {β : Type u_2}\n    [measurable_space β] {μ' : measure β} (f : simple_func α ennreal) (g : simple_func β ennreal)\n    (m : α → β) (eq : ∀ (a : α), coe_fn f a = coe_fn g (m a))\n    (h : ∀ (s : set β), is_measurable s → coe_fn μ' s = coe_fn μ (m ⁻¹' s)) :\n    lintegral f μ = lintegral g μ' :=\n  sorry\n\ntheorem support_eq {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) :\n    function.support ⇑f =\n        set.Union\n          fun (y : β) =>\n            set.Union\n              fun (H : y ∈ finset.filter (fun (y : β) => y ≠ 0) (simple_func.range f)) =>\n                ⇑f ⁻¹' singleton y :=\n  sorry\n\n/-- A `simple_func` has finite measure support if it is equal to `0` outside of a set of finite\nmeasure. -/\nprotected def fin_meas_supp {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    (f : simple_func α β) (μ : measure α) :=\n  filter.eventually_eq (measure.cofinite μ) (⇑f) 0\n\ntheorem fin_meas_supp_iff_support {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    {f : simple_func α β} {μ : measure α} :\n    simple_func.fin_meas_supp f μ ↔ coe_fn μ (function.support ⇑f) < ⊤ :=\n  iff.rfl\n\ntheorem fin_meas_supp_iff {α : Type u_1} {β : Type u_2} [measurable_space α] [HasZero β]\n    {f : simple_func α β} {μ : measure α} :\n    simple_func.fin_meas_supp f μ ↔ ∀ (y : β), y ≠ 0 → coe_fn μ (⇑f ⁻¹' singleton y) < ⊤ :=\n  sorry\n\nnamespace fin_meas_supp\n\n\ntheorem meas_preimage_singleton_ne_zero {α : Type u_1} {β : Type u_2} [measurable_space α]\n    [HasZero β] {μ : measure α} {f : simple_func α β} (h : simple_func.fin_meas_supp f μ) {y : β}\n    (hy : y ≠ 0) : coe_fn μ (⇑f ⁻¹' singleton y) < ⊤ :=\n  iff.mp fin_meas_supp_iff h y hy\n\nprotected theorem map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [HasZero β]\n    [HasZero γ] {μ : measure α} {f : simple_func α β} {g : β → γ}\n    (hf : simple_func.fin_meas_supp f μ) (hg : g 0 = 0) : simple_func.fin_meas_supp (map g f) μ :=\n  flip lt_of_le_of_lt hf (measure_mono (function.support_comp_subset hg ⇑f))\n\ntheorem of_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [HasZero β]\n    [HasZero γ] {μ : measure α} {f : simple_func α β} {g : β → γ}\n    (h : simple_func.fin_meas_supp (map g f) μ) (hg : ∀ (b : β), g b = 0 → b = 0) :\n    simple_func.fin_meas_supp f μ :=\n  flip lt_of_le_of_lt h (measure_mono (function.support_subset_comp hg fun (x : α) => coe_fn f x))\n\ntheorem map_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [HasZero β]\n    [HasZero γ] {μ : measure α} {f : simple_func α β} {g : β → γ}\n    (hg : ∀ {b : β}, g b = 0 ↔ b = 0) :\n    simple_func.fin_meas_supp (map g f) μ ↔ simple_func.fin_meas_supp f μ :=\n  { mp := fun (h : simple_func.fin_meas_supp (map g f) μ) => of_map h fun (b : β) => iff.mp hg,\n    mpr := fun (h : simple_func.fin_meas_supp f μ) => fin_meas_supp.map h (iff.mpr hg rfl) }\n\nprotected theorem pair {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α] [HasZero β]\n    [HasZero γ] {μ : measure α} {f : simple_func α β} {g : simple_func α γ}\n    (hf : simple_func.fin_meas_supp f μ) (hg : simple_func.fin_meas_supp g μ) :\n    simple_func.fin_meas_supp (pair f g) μ :=\n  sorry\n\nprotected theorem map₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4}\n    [measurable_space α] [HasZero β] [HasZero γ] [HasZero δ] {μ : measure α} {f : simple_func α β}\n    (hf : simple_func.fin_meas_supp f μ) {g : simple_func α γ} (hg : simple_func.fin_meas_supp g μ)\n    {op : β → γ → δ} (H : op 0 0 = 0) :\n    simple_func.fin_meas_supp (map (function.uncurry op) (pair f g)) μ :=\n  fin_meas_supp.map (fin_meas_supp.pair hf hg) H\n\nprotected theorem add {α : Type u_1} [measurable_space α] {μ : measure α} {β : Type u_2}\n    [add_monoid β] {f : simple_func α β} {g : simple_func α β} (hf : simple_func.fin_meas_supp f μ)\n    (hg : simple_func.fin_meas_supp g μ) : simple_func.fin_meas_supp (f + g) μ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (simple_func.fin_meas_supp (f + g) μ)) (add_eq_map₂ f g)))\n    (fin_meas_supp.map₂ hf hg (zero_add 0))\n\nprotected theorem mul {α : Type u_1} [measurable_space α] {μ : measure α} {β : Type u_2}\n    [monoid_with_zero β] {f : simple_func α β} {g : simple_func α β}\n    (hf : simple_func.fin_meas_supp f μ) (hg : simple_func.fin_meas_supp g μ) :\n    simple_func.fin_meas_supp (f * g) μ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (simple_func.fin_meas_supp (f * g) μ)) (mul_eq_map₂ f g)))\n    (fin_meas_supp.map₂ hf hg (zero_mul 0))\n\ntheorem lintegral_lt_top {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : simple_func α ennreal} (hm : simple_func.fin_meas_supp f μ)\n    (hf : filter.eventually (fun (a : α) => coe_fn f a < ⊤) (measure.ae μ)) : lintegral f μ < ⊤ :=\n  sorry\n\ntheorem of_lintegral_lt_top {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : simple_func α ennreal} (h : lintegral f μ < ⊤) : simple_func.fin_meas_supp f μ :=\n  sorry\n\ntheorem iff_lintegral_lt_top {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : simple_func α ennreal}\n    (hf : filter.eventually (fun (a : α) => coe_fn f a < ⊤) (measure.ae μ)) :\n    simple_func.fin_meas_supp f μ ↔ lintegral f μ < ⊤ :=\n  { mp := fun (h : simple_func.fin_meas_supp f μ) => lintegral_lt_top h hf,\n    mpr := fun (h : lintegral f μ < ⊤) => of_lintegral_lt_top h }\n\nend fin_meas_supp\n\n\n/-- To prove something for an arbitrary simple function, it suffices to show\nthat the property holds for (multiples of) characteristic functions and is closed under\naddition (of functions with disjoint support).\n\nIt is possible to make the hypotheses in `h_sum` a bit stronger, and such conditions can be added\nonce we need them (for example it is only necessary to consider the case where `g` is a multiple\nof a characteristic function, and that this multiple doesn't appear in the image of `f`) -/\nprotected theorem induction {α : Type u_1} {γ : Type u_2} [measurable_space α] [add_monoid γ]\n    {P : simple_func α γ → Prop}\n    (h_ind :\n      ∀ (c : γ) {s : set α} (hs : is_measurable s), P (piecewise s hs (const α c) (const α 0)))\n    (h_sum :\n      ∀ {f g : simple_func α γ},\n        set.univ ⊆ ⇑f ⁻¹' singleton 0 ∪ ⇑g ⁻¹' singleton 0 → P f → P g → P (f + g))\n    (f : simple_func α γ) : P f :=\n  sorry\n\nend simple_func\n\n\n/-- The lower Lebesgue integral of a function `f` with respect to a measure `μ`. -/\ndef lintegral {α : Type u_1} [measurable_space α] (μ : measure α) (f : α → ennreal) : ennreal :=\n  supr fun (g : simple_func α ennreal) => supr fun (hf : ⇑g ≤ f) => simple_func.lintegral g μ\n\n/-! In the notation for integrals, an expression like `∫⁻ x, g ∥x∥ ∂μ` will not be parsed correctly,\n  and needs parentheses. We do not set the binding power of `r` to `0`, because then\n  `∫⁻ x, f x = 0` will be parsed incorrectly. -/\n\ntheorem simple_func.lintegral_eq_lintegral {α : Type u_1} [measurable_space α]\n    (f : simple_func α ennreal) (μ : measure α) :\n    (lintegral μ fun (a : α) => coe_fn f a) = simple_func.lintegral f μ :=\n  sorry\n\ntheorem lintegral_mono' {α : Type u_1} [measurable_space α] {μ : measure α} {ν : measure α}\n    (hμν : μ ≤ ν) {f : α → ennreal} {g : α → ennreal} (hfg : f ≤ g) :\n    (lintegral μ fun (a : α) => f a) ≤ lintegral ν fun (a : α) => g a :=\n  sorry\n\ntheorem lintegral_mono {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (hfg : f ≤ g) :\n    (lintegral μ fun (a : α) => f a) ≤ lintegral μ fun (a : α) => g a :=\n  lintegral_mono' (le_refl μ) hfg\n\ntheorem lintegral_mono_nnreal {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → nnreal}\n    {g : α → nnreal} (h : f ≤ g) :\n    (lintegral μ fun (a : α) => ↑(f a)) ≤ lintegral μ fun (a : α) => ↑(g a) :=\n  sorry\n\ntheorem monotone_lintegral {α : Type u_1} [measurable_space α] (μ : measure α) :\n    monotone (lintegral μ) :=\n  lintegral_mono\n\n@[simp] theorem lintegral_const {α : Type u_1} [measurable_space α] {μ : measure α} (c : ennreal) :\n    (lintegral μ fun (a : α) => c) = c * coe_fn μ set.univ :=\n  sorry\n\n@[simp] theorem lintegral_one {α : Type u_1} [measurable_space α] {μ : measure α} :\n    (lintegral μ fun (a : α) => 1) = coe_fn μ set.univ :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl ((lintegral μ fun (a : α) => 1) = coe_fn μ set.univ))\n        (lintegral_const 1)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (1 * coe_fn μ set.univ = coe_fn μ set.univ))\n          (one_mul (coe_fn μ set.univ))))\n      (Eq.refl (coe_fn μ set.univ)))\n\ntheorem set_lintegral_const {α : Type u_1} [measurable_space α] {μ : measure α} (s : set α)\n    (c : ennreal) : (lintegral (measure.restrict μ s) fun (a : α) => c) = c * coe_fn μ s :=\n  sorry\n\ntheorem set_lintegral_one {α : Type u_1} [measurable_space α] {μ : measure α} (s : set α) :\n    (lintegral (measure.restrict μ s) fun (a : α) => 1) = coe_fn μ s :=\n  sorry\n\n/-- `∫⁻ a in s, f a ∂μ` is defined as the supremum of integrals of simple functions\n`φ : α →ₛ ennreal` such that `φ ≤ f`. This lemma says that it suffices to take\nfunctions `φ : α →ₛ ℝ≥0`. -/\ntheorem lintegral_eq_nnreal {α : Type u_1} [measurable_space α] (f : α → ennreal) (μ : measure α) :\n    (lintegral μ fun (a : α) => f a) =\n        supr\n          fun (φ : simple_func α nnreal) =>\n            supr\n              fun (hf : ∀ (x : α), ↑(coe_fn φ x) ≤ f x) =>\n                simple_func.lintegral (simple_func.map coe φ) μ :=\n  sorry\n\ntheorem exists_simple_func_forall_lintegral_sub_lt_of_pos {α : Type u_1} [measurable_space α]\n    {μ : measure α} {f : α → ennreal} (h : (lintegral μ fun (x : α) => f x) < ⊤) {ε : ennreal}\n    (hε : 0 < ε) :\n    ∃ (φ : simple_func α nnreal),\n        (∀ (x : α), ↑(coe_fn φ x) ≤ f x) ∧\n          ∀ (ψ : simple_func α nnreal),\n            (∀ (x : α), ↑(coe_fn ψ x) ≤ f x) →\n              simple_func.lintegral (simple_func.map coe (ψ - φ)) μ < ε :=\n  sorry\n\ntheorem supr_lintegral_le {α : Type u_1} [measurable_space α] {μ : measure α} {ι : Sort u_2}\n    (f : ι → α → ennreal) :\n    (supr fun (i : ι) => lintegral μ fun (a : α) => f i a) ≤\n        lintegral μ fun (a : α) => supr fun (i : ι) => f i a :=\n  sorry\n\ntheorem supr2_lintegral_le {α : Type u_1} [measurable_space α] {μ : measure α} {ι : Sort u_2}\n    {ι' : ι → Sort u_3} (f : (i : ι) → ι' i → α → ennreal) :\n    (supr fun (i : ι) => supr fun (h : ι' i) => lintegral μ fun (a : α) => f i h a) ≤\n        lintegral μ fun (a : α) => supr fun (i : ι) => supr fun (h : ι' i) => f i h a :=\n  sorry\n\ntheorem le_infi_lintegral {α : Type u_1} [measurable_space α] {μ : measure α} {ι : Sort u_2}\n    (f : ι → α → ennreal) :\n    (lintegral μ fun (a : α) => infi fun (i : ι) => f i a) ≤\n        infi fun (i : ι) => lintegral μ fun (a : α) => f i a :=\n  sorry\n\ntheorem le_infi2_lintegral {α : Type u_1} [measurable_space α] {μ : measure α} {ι : Sort u_2}\n    {ι' : ι → Sort u_3} (f : (i : ι) → ι' i → α → ennreal) :\n    (lintegral μ fun (a : α) => infi fun (i : ι) => infi fun (h : ι' i) => f i h a) ≤\n        infi fun (i : ι) => infi fun (h : ι' i) => lintegral μ fun (a : α) => f i h a :=\n  sorry\n\ntheorem lintegral_mono_ae {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (h : filter.eventually (fun (a : α) => f a ≤ g a) (measure.ae μ)) :\n    (lintegral μ fun (a : α) => f a) ≤ lintegral μ fun (a : α) => g a :=\n  sorry\n\ntheorem lintegral_congr_ae {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (h : filter.eventually_eq (measure.ae μ) f g) :\n    (lintegral μ fun (a : α) => f a) = lintegral μ fun (a : α) => g a :=\n  le_antisymm (lintegral_mono_ae (filter.eventually_eq.le h))\n    (lintegral_mono_ae (filter.eventually_eq.le (filter.eventually_eq.symm h)))\n\ntheorem lintegral_congr {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (h : ∀ (a : α), f a = g a) :\n    (lintegral μ fun (a : α) => f a) = lintegral μ fun (a : α) => g a :=\n  sorry\n\ntheorem set_lintegral_congr {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {s : set α} {t : set α} (h : filter.eventually_eq (measure.ae μ) s t) :\n    (lintegral (measure.restrict μ s) fun (x : α) => f x) =\n        lintegral (measure.restrict μ t) fun (x : α) => f x :=\n  sorry\n\n/-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence.\n\nSee `lintegral_supr_directed` for a more general form. -/\ntheorem lintegral_supr {α : Type u_1} [measurable_space α] {μ : measure α} {f : ℕ → α → ennreal}\n    (hf : ∀ (n : ℕ), measurable (f n)) (h_mono : monotone f) :\n    (lintegral μ fun (a : α) => supr fun (n : ℕ) => f n a) =\n        supr fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  sorry\n\n/-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with\nae_measurable functions. -/\ntheorem lintegral_supr' {α : Type u_1} [measurable_space α] {μ : measure α} {f : ℕ → α → ennreal}\n    (hf : ∀ (n : ℕ), ae_measurable (f n))\n    (h_mono : filter.eventually (fun (x : α) => monotone fun (n : ℕ) => f n x) (measure.ae μ)) :\n    (lintegral μ fun (a : α) => supr fun (n : ℕ) => f n a) =\n        supr fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  sorry\n\ntheorem lintegral_eq_supr_eapprox_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : measurable f) :\n    (lintegral μ fun (a : α) => f a) =\n        supr fun (n : ℕ) => simple_func.lintegral (simple_func.eapprox f n) μ :=\n  sorry\n\n/-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends\nto zero as `μ s` tends to zero. This lemma states states this fact in terms of `ε` and `δ`. -/\ntheorem exists_pos_set_lintegral_lt_of_measure_lt {α : Type u_1} [measurable_space α]\n    {μ : measure α} {f : α → ennreal} (h : (lintegral μ fun (x : α) => f x) < ⊤) {ε : ennreal}\n    (hε : 0 < ε) :\n    ∃ (δ : ennreal),\n        ∃ (H : δ > 0),\n          ∀ (s : set α),\n            coe_fn μ s < δ → (lintegral (measure.restrict μ s) fun (x : α) => f x) < ε :=\n  sorry\n\n/-- If `f` has finite integral, then `∫⁻ x in s, f x ∂μ` is absolutely continuous in `s`: it tends\nto zero as `μ s` tends to zero. -/\ntheorem tendsto_set_lintegral_zero {α : Type u_1} [measurable_space α] {μ : measure α}\n    {ι : Type u_2} {f : α → ennreal} (h : (lintegral μ fun (x : α) => f x) < ⊤) {l : filter ι}\n    {s : ι → set α} (hl : filter.tendsto (⇑μ ∘ s) l (nhds 0)) :\n    filter.tendsto (fun (i : ι) => lintegral (measure.restrict μ (s i)) fun (x : α) => f x) l\n        (nhds 0) :=\n  sorry\n\n@[simp] theorem lintegral_add {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (hf : measurable f) (hg : measurable g) :\n    (lintegral μ fun (a : α) => f a + g a) =\n        (lintegral μ fun (a : α) => f a) + lintegral μ fun (a : α) => g a :=\n  sorry\n\ntheorem lintegral_add' {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (hf : ae_measurable f) (hg : ae_measurable g) :\n    (lintegral μ fun (a : α) => f a + g a) =\n        (lintegral μ fun (a : α) => f a) + lintegral μ fun (a : α) => g a :=\n  sorry\n\ntheorem lintegral_zero {α : Type u_1} [measurable_space α] {μ : measure α} :\n    (lintegral μ fun (a : α) => 0) = 0 :=\n  sorry\n\ntheorem lintegral_zero_fun {α : Type u_1} [measurable_space α] {μ : measure α} :\n    (lintegral μ fun (a : α) => HasZero.zero a) = 0 :=\n  sorry\n\n@[simp] theorem lintegral_smul_measure {α : Type u_1} [measurable_space α] {μ : measure α}\n    (c : ennreal) (f : α → ennreal) :\n    (lintegral (c • μ) fun (a : α) => f a) = c * lintegral μ fun (a : α) => f a :=\n  sorry\n\n@[simp] theorem lintegral_sum_measure {α : Type u_1} [measurable_space α] {ι : Type u_2}\n    (f : α → ennreal) (μ : ι → measure α) :\n    (lintegral (measure.sum μ) fun (a : α) => f a) =\n        tsum fun (i : ι) => lintegral (μ i) fun (a : α) => f a :=\n  sorry\n\n@[simp] theorem lintegral_add_measure {α : Type u_1} [measurable_space α] (f : α → ennreal)\n    (μ : measure α) (ν : measure α) :\n    (lintegral (μ + ν) fun (a : α) => f a) =\n        (lintegral μ fun (a : α) => f a) + lintegral ν fun (a : α) => f a :=\n  sorry\n\n@[simp] theorem lintegral_zero_measure {α : Type u_1} [measurable_space α] (f : α → ennreal) :\n    (lintegral 0 fun (a : α) => f a) = 0 :=\n  sorry\n\ntheorem lintegral_finset_sum {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    (s : finset β) {f : β → α → ennreal} (hf : ∀ (b : β), measurable (f b)) :\n    (lintegral μ fun (a : α) => finset.sum s fun (b : β) => f b a) =\n        finset.sum s fun (b : β) => lintegral μ fun (a : α) => f b a :=\n  sorry\n\n@[simp] theorem lintegral_const_mul {α : Type u_1} [measurable_space α] {μ : measure α}\n    (r : ennreal) {f : α → ennreal} (hf : measurable f) :\n    (lintegral μ fun (a : α) => r * f a) = r * lintegral μ fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_const_mul'' {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    {f : α → ennreal} (hf : ae_measurable f) :\n    (lintegral μ fun (a : α) => r * f a) = r * lintegral μ fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_const_mul_le {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    (f : α → ennreal) : (r * lintegral μ fun (a : α) => f a) ≤ lintegral μ fun (a : α) => r * f a :=\n  sorry\n\ntheorem lintegral_const_mul' {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    (f : α → ennreal) (hr : r ≠ ⊤) :\n    (lintegral μ fun (a : α) => r * f a) = r * lintegral μ fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_mul_const {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    {f : α → ennreal} (hf : measurable f) :\n    (lintegral μ fun (a : α) => f a * r) = (lintegral μ fun (a : α) => f a) * r :=\n  sorry\n\ntheorem lintegral_mul_const'' {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    {f : α → ennreal} (hf : ae_measurable f) :\n    (lintegral μ fun (a : α) => f a * r) = (lintegral μ fun (a : α) => f a) * r :=\n  sorry\n\ntheorem lintegral_mul_const_le {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    (f : α → ennreal) : (lintegral μ fun (a : α) => f a) * r ≤ lintegral μ fun (a : α) => f a * r :=\n  sorry\n\ntheorem lintegral_mul_const' {α : Type u_1} [measurable_space α] {μ : measure α} (r : ennreal)\n    (f : α → ennreal) (hr : r ≠ ⊤) :\n    (lintegral μ fun (a : α) => f a * r) = (lintegral μ fun (a : α) => f a) * r :=\n  sorry\n\n/- A double integral of a product where each factor contains only one variable\n  is a product of integrals -/\n\ntheorem lintegral_lintegral_mul {α : Type u_1} [measurable_space α] {μ : measure α} {β : Type u_2}\n    [measurable_space β] {ν : measure β} {f : α → ennreal} {g : β → ennreal} (hf : measurable f)\n    (hg : measurable g) :\n    (lintegral μ fun (x : α) => lintegral ν fun (y : β) => f x * g y) =\n        (lintegral μ fun (x : α) => f x) * lintegral ν fun (y : β) => g y :=\n  sorry\n\n-- TODO: Need a better way of rewriting inside of a integral\n\ntheorem lintegral_rw₁ {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α} {f : α → β}\n    {f' : α → β} (h : filter.eventually_eq (measure.ae μ) f f') (g : β → ennreal) :\n    (lintegral μ fun (a : α) => g (f a)) = lintegral μ fun (a : α) => g (f' a) :=\n  lintegral_congr_ae\n    (filter.eventually.mono h\n      fun (a : α) (h : f a = f' a) =>\n        eq.mpr (id (Eq._oldrec (Eq.refl (g (f a) = g (f' a))) h)) (Eq.refl (g (f' a))))\n\n-- TODO: Need a better way of rewriting inside of a integral\n\ntheorem lintegral_rw₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [measurable_space α]\n    {μ : measure α} {f₁ : α → β} {f₁' : α → β} {f₂ : α → γ} {f₂' : α → γ}\n    (h₁ : filter.eventually_eq (measure.ae μ) f₁ f₁')\n    (h₂ : filter.eventually_eq (measure.ae μ) f₂ f₂') (g : β → γ → ennreal) :\n    (lintegral μ fun (a : α) => g (f₁ a) (f₂ a)) = lintegral μ fun (a : α) => g (f₁' a) (f₂' a) :=\n  sorry\n\n@[simp] theorem lintegral_indicator {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : α → ennreal) {s : set α} (hs : is_measurable s) :\n    (lintegral μ fun (a : α) => set.indicator s f a) =\n        lintegral (measure.restrict μ s) fun (a : α) => f a :=\n  sorry\n\n/-- Chebyshev's inequality -/\ntheorem mul_meas_ge_le_lintegral {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : measurable f) (ε : ennreal) :\n    ε * coe_fn μ (set_of fun (x : α) => ε ≤ f x) ≤ lintegral μ fun (a : α) => f a :=\n  sorry\n\ntheorem meas_ge_le_lintegral_div {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : measurable f) {ε : ennreal} (hε : ε ≠ 0) (hε' : ε ≠ ⊤) :\n    coe_fn μ (set_of fun (x : α) => ε ≤ f x) ≤ (lintegral μ fun (a : α) => f a) / ε :=\n  sorry\n\n@[simp] theorem lintegral_eq_zero_iff {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : measurable f) :\n    (lintegral μ fun (a : α) => f a) = 0 ↔ filter.eventually_eq (measure.ae μ) f 0 :=\n  sorry\n\n@[simp] theorem lintegral_eq_zero_iff' {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : ae_measurable f) :\n    (lintegral μ fun (a : α) => f a) = 0 ↔ filter.eventually_eq (measure.ae μ) f 0 :=\n  sorry\n\ntheorem lintegral_pos_iff_support {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : α → ennreal} (hf : measurable f) :\n    (0 < lintegral μ fun (a : α) => f a) ↔ 0 < coe_fn μ (function.support f) :=\n  sorry\n\n/-- Weaker version of the monotone convergence theorem-/\ntheorem lintegral_supr_ae {α : Type u_1} [measurable_space α] {μ : measure α} {f : ℕ → α → ennreal}\n    (hf : ∀ (n : ℕ), measurable (f n))\n    (h_mono :\n      ∀ (n : ℕ), filter.eventually (fun (a : α) => f n a ≤ f (Nat.succ n) a) (measure.ae μ)) :\n    (lintegral μ fun (a : α) => supr fun (n : ℕ) => f n a) =\n        supr fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  sorry\n\ntheorem lintegral_sub {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    {g : α → ennreal} (hf : measurable f) (hg : measurable g)\n    (hg_fin : (lintegral μ fun (a : α) => g a) < ⊤)\n    (h_le : filter.eventually_le (measure.ae μ) g f) :\n    (lintegral μ fun (a : α) => f a - g a) =\n        (lintegral μ fun (a : α) => f a) - lintegral μ fun (a : α) => g a :=\n  sorry\n\n/-- Monotone convergence theorem for nonincreasing sequences of functions -/\ntheorem lintegral_infi_ae {α : Type u_1} [measurable_space α] {μ : measure α} {f : ℕ → α → ennreal}\n    (h_meas : ∀ (n : ℕ), measurable (f n))\n    (h_mono : ∀ (n : ℕ), filter.eventually_le (measure.ae μ) (f (Nat.succ n)) (f n))\n    (h_fin : (lintegral μ fun (a : α) => f 0 a) < ⊤) :\n    (lintegral μ fun (a : α) => infi fun (n : ℕ) => f n a) =\n        infi fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  sorry\n\n/-- Monotone convergence theorem for nonincreasing sequences of functions -/\ntheorem lintegral_infi {α : Type u_1} [measurable_space α] {μ : measure α} {f : ℕ → α → ennreal}\n    (h_meas : ∀ (n : ℕ), measurable (f n)) (h_mono : ∀ {m n : ℕ}, m ≤ n → f n ≤ f m)\n    (h_fin : (lintegral μ fun (a : α) => f 0 a) < ⊤) :\n    (lintegral μ fun (a : α) => infi fun (n : ℕ) => f n a) =\n        infi fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  lintegral_infi_ae h_meas (fun (n : ℕ) => ae_of_all μ (h_mono (le_of_lt (nat.lt_succ_self n))))\n    h_fin\n\n/-- Known as Fatou's lemma, version with `ae_measurable` functions -/\ntheorem lintegral_liminf_le' {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : ℕ → α → ennreal} (h_meas : ∀ (n : ℕ), ae_measurable (f n)) :\n    (lintegral μ fun (a : α) => filter.liminf filter.at_top fun (n : ℕ) => f n a) ≤\n        filter.liminf filter.at_top fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  sorry\n\n/-- Known as Fatou's lemma -/\ntheorem lintegral_liminf_le {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : ℕ → α → ennreal} (h_meas : ∀ (n : ℕ), measurable (f n)) :\n    (lintegral μ fun (a : α) => filter.liminf filter.at_top fun (n : ℕ) => f n a) ≤\n        filter.liminf filter.at_top fun (n : ℕ) => lintegral μ fun (a : α) => f n a :=\n  lintegral_liminf_le' fun (n : ℕ) => measurable.ae_measurable (h_meas n)\n\ntheorem limsup_lintegral_le {α : Type u_1} [measurable_space α] {μ : measure α}\n    {f : ℕ → α → ennreal} {g : α → ennreal} (hf_meas : ∀ (n : ℕ), measurable (f n))\n    (h_bound : ∀ (n : ℕ), filter.eventually_le (measure.ae μ) (f n) g)\n    (h_fin : (lintegral μ fun (a : α) => g a) < ⊤) :\n    (filter.limsup filter.at_top fun (n : ℕ) => lintegral μ fun (a : α) => f n a) ≤\n        lintegral μ fun (a : α) => filter.limsup filter.at_top fun (n : ℕ) => f n a :=\n  sorry\n\n/-- Dominated convergence theorem for nonnegative functions -/\ntheorem tendsto_lintegral_of_dominated_convergence {α : Type u_1} [measurable_space α]\n    {μ : measure α} {F : ℕ → α → ennreal} {f : α → ennreal} (bound : α → ennreal)\n    (hF_meas : ∀ (n : ℕ), measurable (F n))\n    (h_bound : ∀ (n : ℕ), filter.eventually_le (measure.ae μ) (F n) bound)\n    (h_fin : (lintegral μ fun (a : α) => bound a) < ⊤)\n    (h_lim :\n      filter.eventually\n        (fun (a : α) => filter.tendsto (fun (n : ℕ) => F n a) filter.at_top (nhds (f a)))\n        (measure.ae μ)) :\n    filter.tendsto (fun (n : ℕ) => lintegral μ fun (a : α) => F n a) filter.at_top\n        (nhds (lintegral μ fun (a : α) => f a)) :=\n  sorry\n\n/-- Dominated convergence theorem for nonnegative functions which are just almost everywhere\nmeasurable. -/\ntheorem tendsto_lintegral_of_dominated_convergence' {α : Type u_1} [measurable_space α]\n    {μ : measure α} {F : ℕ → α → ennreal} {f : α → ennreal} (bound : α → ennreal)\n    (hF_meas : ∀ (n : ℕ), ae_measurable (F n))\n    (h_bound : ∀ (n : ℕ), filter.eventually_le (measure.ae μ) (F n) bound)\n    (h_fin : (lintegral μ fun (a : α) => bound a) < ⊤)\n    (h_lim :\n      filter.eventually\n        (fun (a : α) => filter.tendsto (fun (n : ℕ) => F n a) filter.at_top (nhds (f a)))\n        (measure.ae μ)) :\n    filter.tendsto (fun (n : ℕ) => lintegral μ fun (a : α) => F n a) filter.at_top\n        (nhds (lintegral μ fun (a : α) => f a)) :=\n  sorry\n\n/-- Dominated convergence theorem for filters with a countable basis -/\ntheorem tendsto_lintegral_filter_of_dominated_convergence {α : Type u_1} [measurable_space α]\n    {μ : measure α} {ι : Type u_2} {l : filter ι} {F : ι → α → ennreal} {f : α → ennreal}\n    (bound : α → ennreal) (hl_cb : filter.is_countably_generated l)\n    (hF_meas : filter.eventually (fun (n : ι) => measurable (F n)) l)\n    (h_bound :\n      filter.eventually\n        (fun (n : ι) => filter.eventually (fun (a : α) => F n a ≤ bound a) (measure.ae μ)) l)\n    (h_fin : (lintegral μ fun (a : α) => bound a) < ⊤)\n    (h_lim :\n      filter.eventually (fun (a : α) => filter.tendsto (fun (n : ι) => F n a) l (nhds (f a)))\n        (measure.ae μ)) :\n    filter.tendsto (fun (n : ι) => lintegral μ fun (a : α) => F n a) l\n        (nhds (lintegral μ fun (a : α) => f a)) :=\n  sorry\n\n/-- Monotone convergence for a suprema over a directed family and indexed by an encodable type -/\ntheorem lintegral_supr_directed {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [encodable β] {f : β → α → ennreal} (hf : ∀ (b : β), measurable (f b))\n    (h_directed : directed LessEq f) :\n    (lintegral μ fun (a : α) => supr fun (b : β) => f b a) =\n        supr fun (b : β) => lintegral μ fun (a : α) => f b a :=\n  sorry\n\ntheorem lintegral_tsum {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [encodable β] {f : β → α → ennreal} (hf : ∀ (i : β), measurable (f i)) :\n    (lintegral μ fun (a : α) => tsum fun (i : β) => f i a) =\n        tsum fun (i : β) => lintegral μ fun (a : α) => f i a :=\n  sorry\n\ntheorem lintegral_Union {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [encodable β] {s : β → set α} (hm : ∀ (i : β), is_measurable (s i))\n    (hd : pairwise (disjoint on s)) (f : α → ennreal) :\n    (lintegral (measure.restrict μ (set.Union fun (i : β) => s i)) fun (a : α) => f a) =\n        tsum fun (i : β) => lintegral (measure.restrict μ (s i)) fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_Union_le {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [encodable β] (s : β → set α) (f : α → ennreal) :\n    (lintegral (measure.restrict μ (set.Union fun (i : β) => s i)) fun (a : α) => f a) ≤\n        tsum fun (i : β) => lintegral (measure.restrict μ (s i)) fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_map {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [measurable_space β] {f : β → ennreal} {g : α → β} (hf : measurable f) (hg : measurable g) :\n    (lintegral (coe_fn (measure.map g) μ) fun (a : β) => f a) =\n        lintegral μ fun (a : α) => f (g a) :=\n  sorry\n\ntheorem lintegral_map' {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [measurable_space β] {f : β → ennreal} {g : α → β} (hf : ae_measurable f) (hg : measurable g) :\n    (lintegral (coe_fn (measure.map g) μ) fun (a : β) => f a) =\n        lintegral μ fun (a : α) => f (g a) :=\n  Eq.trans\n    (Eq.trans (lintegral_congr_ae (ae_measurable.ae_eq_mk hf))\n      (lintegral_map (ae_measurable.measurable_mk hf) hg))\n    (lintegral_congr_ae (ae_eq_comp hg (filter.eventually_eq.symm (ae_measurable.ae_eq_mk hf))))\n\ntheorem lintegral_comp {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [measurable_space β] {f : β → ennreal} {g : α → β} (hf : measurable f) (hg : measurable g) :\n    lintegral μ (f ∘ g) = lintegral (coe_fn (measure.map g) μ) fun (a : β) => f a :=\n  Eq.symm (lintegral_map hf hg)\n\ntheorem set_lintegral_map {α : Type u_1} {β : Type u_2} [measurable_space α] {μ : measure α}\n    [measurable_space β] {f : β → ennreal} {g : α → β} {s : set β} (hs : is_measurable s)\n    (hf : measurable f) (hg : measurable g) :\n    (lintegral (measure.restrict (coe_fn (measure.map g) μ) s) fun (y : β) => f y) =\n        lintegral (measure.restrict μ (g ⁻¹' s)) fun (x : α) => f (g x) :=\n  sorry\n\ntheorem lintegral_dirac' {α : Type u_1} [measurable_space α] (a : α) {f : α → ennreal}\n    (hf : measurable f) : (lintegral (measure.dirac a) fun (a : α) => f a) = f a :=\n  sorry\n\ntheorem lintegral_dirac {α : Type u_1} [measurable_space α] [measurable_singleton_class α] (a : α)\n    (f : α → ennreal) : (lintegral (measure.dirac a) fun (a : α) => f a) = f a :=\n  sorry\n\ntheorem lintegral_count' {α : Type u_1} [measurable_space α] {f : α → ennreal} (hf : measurable f) :\n    (lintegral measure.count fun (a : α) => f a) = tsum fun (a : α) => f a :=\n  sorry\n\ntheorem lintegral_count {α : Type u_1} [measurable_space α] [measurable_singleton_class α]\n    (f : α → ennreal) : (lintegral measure.count fun (a : α) => f a) = tsum fun (a : α) => f a :=\n  sorry\n\ntheorem ae_lt_top {α : Type u_1} [measurable_space α] {μ : measure α} {f : α → ennreal}\n    (hf : measurable f) (h2f : (lintegral μ fun (x : α) => f x) < ⊤) :\n    filter.eventually (fun (x : α) => f x < ⊤) (measure.ae μ) :=\n  sorry\n\n/-- Given a measure `μ : measure α` and a function `f : α → ennreal`, `μ.with_density f` is the\nmeasure such that for a measurable set `s` we have `μ.with_density f s = ∫⁻ a in s, f a ∂μ`. -/\ndef measure.with_density {α : Type u_1} [measurable_space α] (μ : measure α) (f : α → ennreal) :\n    measure α :=\n  measure.of_measurable\n    (fun (s : set α) (hs : is_measurable s) => lintegral (measure.restrict μ s) fun (a : α) => f a)\n    sorry sorry\n\n@[simp] theorem with_density_apply {α : Type u_1} [measurable_space α] {μ : measure α}\n    (f : α → ennreal) {s : set α} (hs : is_measurable s) :\n    coe_fn (measure.with_density μ f) s = lintegral (measure.restrict μ s) fun (a : α) => f a :=\n  measure.of_measurable_apply s hs\n\nend measure_theory\n\n\n/-- To prove something for an arbitrary measurable function into `ennreal`, it suffices to show\nthat the property holds for (multiples of) characteristic functions and is closed under addition\nand supremum of increasing sequences of functions.\n\nIt is possible to make the hypotheses in the induction steps a bit stronger, and such conditions\ncan be added once we need them (for example in `h_sum` it is only necessary to consider the sum of\na simple function with a multiple of a characteristic function and that the intersection\nof their images is a subset of `{0}`. -/\ntheorem measurable.ennreal_induction {α : Type u_1} [measurable_space α] {P : (α → ennreal) → Prop}\n    (h_ind : ∀ (c : ennreal) {s : set α}, is_measurable s → P (set.indicator s fun (_x : α) => c))\n    (h_sum :\n      ∀ {f g : α → ennreal},\n        set.univ ⊆ f ⁻¹' singleton 0 ∪ g ⁻¹' singleton 0 →\n          measurable f → measurable g → P f → P g → P (f + g))\n    (h_supr :\n      ∀ {f : ℕ → α → ennreal},\n        (∀ (n : ℕ), measurable (f n)) →\n          monotone f → (∀ (n : ℕ), P (f n)) → P fun (x : α) => supr fun (n : ℕ) => f n x)\n    {f : α → ennreal} (hf : measurable f) : P f :=\n  sorry\n\nnamespace measure_theory\n\n\n/-- This is Exercise 1.2.1 from [tao2010]. It allows you to express integration of a measurable\nfunction with respect to `(μ.with_density f)` as an integral with respect to `μ`, called the base\nmeasure. `μ` is often the Lebesgue measure, and in this circumstance `f` is the probability density\nfunction, and `(μ.with_density f)` represents any continuous random variable as a\nprobability measure, such as the uniform distribution between 0 and 1, the Gaussian distribution,\nthe exponential distribution, the Beta distribution, or the Cauchy distribution (see Section 2.4\nof [wasserman2004]). Thus, this method shows how to one can calculate expectations, variances,\nand other moments as a function of the probability density function.\n -/\ntheorem lintegral_with_density_eq_lintegral_mul {α : Type u_1} [measurable_space α] (μ : measure α)\n    {f : α → ennreal} (h_mf : measurable f) {g : α → ennreal} :\n    measurable g →\n        (lintegral (measure.with_density μ f) fun (a : α) => g a) =\n          lintegral μ fun (a : α) => Mul.mul f g a :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/integration_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3225471713744836}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.category_theory.localization.predicate\n\nnoncomputable theory\n\nopen category_theory.category\n\nnamespace category_theory\n\nnamespace morphism_property\n\nvariables {C : Type*} [category C]\n\ndef iso_closure (P : morphism_property C) : morphism_property C :=\nλ X₁ X₂ f, ∃ (Y₁ Y₂ : C) (e₁ : X₁ ≅ Y₁) (e₂ : X₂ ≅ Y₂) (f' : Y₁ ⟶ Y₂) (hf' : P f'),\n  comm_sq f e₁.hom e₂.hom f'\n\nlemma subset_iso_closure (P : morphism_property C) : P ⊆ P.iso_closure :=\nλ X Y f hf, ⟨X, Y, iso.refl X, iso.refl Y, f, hf, comm_sq.mk (by simp)⟩\n\nlemma iso_closure_respects_iso (P : morphism_property C) : P.iso_closure.respects_iso :=\nbegin\n  split,\n  { intros A X₁ X₂ e f hf,\n    rcases hf with ⟨Y₁, Y₂, e₁, e₂, f', hf', sq⟩,\n    exact ⟨Y₁, Y₂, e ≪≫ e₁, e₂, f', hf', comm_sq.mk (by simp [sq.w])⟩, },\n  { intros X₁ X₂ Z e f hf,\n    rcases hf with ⟨Y₁, Y₂, e₁, e₂, f', hf', sq⟩,\n    exact ⟨Y₁, Y₂, e₁, e.symm ≪≫ e₂, f', hf', comm_sq.mk (by simp [sq.w])⟩, },\nend\n\ndef inverse_image' {D : Type*} [category D] (P : morphism_property D) (F : C ⥤ D) :\n  morphism_property C := (P.inverse_image F).iso_closure\n\nlemma inverse_image_subset_inverse_image'\n  {D : Type*} [category D] (P : morphism_property D) (F : C ⥤ D) :\n  P.inverse_image F ⊆ P.inverse_image' F :=\nsubset_iso_closure _\n\nend morphism_property\n\nnamespace localization\n\nsection\n\nvariables {C₁ C₂ D₁ D₂ : Type*} [category C₁] [category C₂] [category D₁] [category D₂]\n  {I : C₁ ⥤ C₂} {I' : D₁ ⥤ D₂} {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂}\n  (h : Comm_sq I L₁ L₂ I')\n  (W₁ : morphism_property C₁) (W₂ : morphism_property C₂)\n  [L₁.is_localization W₁] [L₂.is_localization W₂]\n  (R : C₂ ⥤ D₁) (R' : D₂ ⥤ D₁) [hR : lifting L₂ W₂ R R']\n  (η : R ⋙ I' ≅ L₂)\n  (ε : I ⋙ R ≅ L₁)\n\ninclude h W₁ hR ε η\n\ndef lifting_equivalence : D₁ ≌ D₂ :=\nbegin\n  refine (equivalence.mk I' R' _ _),\n  { haveI : lifting L₁ W₁ (I ⋙ L₂) I' := ⟨h.iso⟩,\n    let e : (I ⋙ L₂) ⋙ R' ≅ L₁,\n    { calc (I ⋙ L₂) ⋙ R' ≅ I ⋙ L₂ ⋙ R' : functor.associator _ _ _\n      ... ≅ I ⋙ R : iso_whisker_left _ (lifting.iso _ W₂ _ _)\n      ... ≅ L₁ : ε, },\n    haveI : lifting L₁ W₁ L₁ (I' ⋙ R') := lifting.of_isos _ _ e (iso.refl _),\n    exact lifting.uniq L₁ W₁ L₁ _ _, },\n  { haveI : lifting L₂ W₂ L₂ (R' ⋙ I') := lifting.of_isos _ _ η (iso.refl _),\n    exact lifting.uniq L₂ W₂ L₂ _ _, },\nend\n\nlemma lifting_equivalence_counit_iso_app (X : C₂) :\n  (lifting_equivalence h W₁ W₂ R R' η ε).counit_iso.app (L₂.obj X) =\n  I'.map_iso ((lifting.iso L₂ W₂ R R').app X) ≪≫ η.app X :=\nbegin\n  ext,\n  dsimp [lifting_equivalence, equivalence.mk],\n  simp only [lifting.uniq_hom_app, lifting.of_isos_iso, iso.refl_symm,\n    lifting.comp_right_iso, iso.trans_hom, iso_whisker_left_hom,\n    iso.refl_hom, whisker_left_id', iso_whisker_right_hom, id_comp,\n    nat_trans.comp_app, whisker_right_app, lifting.id_iso,\n    functor.right_unitor_inv_app, comp_id],\nend\n\ndef lifting_is_equivalence : is_equivalence I' :=\nis_equivalence.of_equivalence (lifting_equivalence h W₁ W₂ R R' η ε)\n\nend\n\nend localization\n\nsection\n\nvariables {C₁ C₂ D : Type*} [category C₁] [category C₂] [category D]\n  (L₁ : C₁ ⥤ D) (W₁ : morphism_property C₁) (L₂ : C₂ ⥤ D) (W₂ : morphism_property C₂)\n  (E : C₂ ≌ C₁) (hW₁ : W₁ ⊆ W₂.inverse_image' E.inverse) (hW₂ : W₂.is_inverted_by L₂)\n  [L₁.is_localization W₁] (iso : E.functor ⋙ L₁ ≅ L₂)\n\ninclude iso hW₁ hW₂\n\nlemma functor.is_localization.of_equivalence' :\n  L₂.is_localization W₂ :=\nbegin\n  have h₁ : W₂.is_inverted_by L₂ := hW₂,\n  have h₂ : W₁.is_inverted_by (E.inverse ⋙ W₂.Q) := λ X₁ X₂ f hf, begin\n    rcases hW₁ f hf with ⟨Y₁, Y₂, e₁, e₂, f', hf', sq⟩,\n    dsimp,\n    have eq := sq.w,\n    rw [← cancel_mono (e₂.inv), assoc, e₂.hom_inv_id, comp_id, assoc] at eq,\n    rw eq,\n    simp only [functor.map_comp],\n    haveI := localization.inverts W₂.Q W₂ _ hf',\n    apply_instance,\n  end,\n  let I' := localization.construction.lift L₂ h₁,\n  let C : Comm_sq E.functor W₂.Q L₁ I' := ⟨localization.lifting.iso W₂.Q W₂ L₂ I' ≪≫ iso.symm⟩,\n  let iso₁ : (E.inverse ⋙ W₂.Q) ⋙ I' ≅ L₁,\n  { calc (E.inverse ⋙ W₂.Q) ⋙ I' ≅ E.inverse ⋙ (W₂.Q ⋙ I') : functor.associator _ _ _\n    ... ≅ E.inverse ⋙ (E.functor ⋙ L₁) : iso_whisker_left _ C.iso\n    ... ≅ (E.inverse ⋙ E.functor) ⋙ L₁ : (functor.associator _ _ _).symm\n    ... ≅ 𝟭 _ ⋙ L₁ : iso_whisker_right E.counit_iso _\n    ... ≅ L₁ : functor.left_unitor _, },\n  let iso₂ : E.functor ⋙ E.inverse ⋙ W₂.Q ≅ W₂.Q,\n  { calc E.functor ⋙ E.inverse ⋙ W₂.Q ≅ (E.functor ⋙ E.inverse) ⋙ W₂.Q :\n      (functor.associator _ _ _).symm\n    ... ≅ 𝟭 _ ⋙ W₂.Q : iso_whisker_right E.unit_iso.symm _\n    ... ≅ W₂.Q : functor.left_unitor _, },\n  exact\n  { inverts := h₁,\n    nonempty_is_equivalence := nonempty.intro\n      (localization.lifting_is_equivalence C W₂ W₁ (E.inverse ⋙ W₂.Q)\n      (localization.lift (E.inverse ⋙ W₂.Q) h₂ L₁) iso₁ iso₂), }\nend\n\nend\n\nsection\n\nvariables {C₁ C₂ D₁ D₂ : Type*} [category C₁] [category C₂] [category D₁] [category D₂]\n  (E : C₁ ≌ C₂) (E' : D₁ ≌ D₂) {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂}\n  (H : Comm_sq E.functor L₁ L₂ E'.functor)\n  (W₁ : morphism_property C₁) (W₂ : morphism_property C₂)\n  (hW₁ : W₁.is_inverted_by L₁)\n  (hW₂ : W₂ ⊆ W₁.inverse_image' E.inverse)\n  [L₂.is_localization W₂]\n\ninclude H hW₁ hW₂\n\nlemma functor.is_localization.of_equivalence'' : L₁.is_localization W₁ :=\nbegin\n  haveI : (E.functor ⋙ L₂).is_localization W₁,\n  { refine functor.is_localization.of_equivalence' L₂ W₂ (E.functor ⋙ L₂)\n      W₁ E hW₂ _ (iso.refl _),\n    rw ← morphism_property.is_inverted_by.iff_of_iso W₁ H.iso,\n    exact morphism_property.is_inverted_by.of_comp _ _ hW₁ _, },\n  haveI := functor.is_localization.of_iso W₁ H.iso.symm,\n  refine functor.is_localization.of_equivalence (L₁ ⋙ E'.functor) W₁ L₁ E'.symm _,\n  calc (L₁ ⋙ E'.functor) ⋙ E'.symm.functor ≅ L₁ ⋙ E'.functor ⋙ E'.symm.functor :\n    functor.associator _ _ _\n  ... ≅ L₁ ⋙ 𝟭 D₁ : iso_whisker_left _ E'.unit_iso.symm\n  ... ≅ L₁ : functor.right_unitor _,\nend\n\nend\n\nsection\n\nvariables {C D₁ D₂ : Type*} [category C] [category D₁] [category D₂]\n  {L₁ : C ⥤ D₁} {L₂ : C ⥤ D₂} {F : D₁ ⥤ D₂} (e : L₁ ⋙ F ≅ L₂)\n  (W : morphism_property C) [L₁.is_localization W] [L₂.is_localization W]\n\ninclude e W\n\ndef functor.is_equivalence.of_localization_comparison : is_equivalence F :=\nbegin\n  let c : Comm_sq (𝟭 C) L₁ L₂ F := ⟨e ≪≫ L₂.left_unitor.symm⟩,\n  exact localization.lifting_is_equivalence c W W L₁ (localization.lift L₁ (localization.inverts L₁ W) L₂) e\n    L₁.left_unitor,\nend\n\nomit e\n\ninstance : is_equivalence (localization.lift L₂ (localization.inverts L₂ W) L₁) :=\nfunctor.is_equivalence.of_localization_comparison\n  (localization.fac L₂ (localization.inverts L₂ W) L₁) W\n\nend\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/equivalence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.322547154687628}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.pi\nimport Mathlib.tactic.pi_instances\nimport Mathlib.algebra.group.defs\nimport Mathlib.algebra.group.hom\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Pi instances for groups and monoids\n\nThis file defines instances for group, monoid, semigroup and related structures on Pi types.\n-/\n\nnamespace pi\n\n\nprotected instance semigroup {I : Type u} {f : I → Type v} [(i : I) → semigroup (f i)] : semigroup ((i : I) → f i) :=\n  semigroup.mk Mul.mul sorry\n\nprotected instance add_comm_semigroup {I : Type u} {f : I → Type v} [(i : I) → add_comm_semigroup (f i)] : add_comm_semigroup ((i : I) → f i) :=\n  add_comm_semigroup.mk Add.add sorry sorry\n\nprotected instance monoid {I : Type u} {f : I → Type v} [(i : I) → monoid (f i)] : monoid ((i : I) → f i) :=\n  monoid.mk Mul.mul sorry 1 sorry sorry\n\nprotected instance add_comm_monoid {I : Type u} {f : I → Type v} [(i : I) → add_comm_monoid (f i)] : add_comm_monoid ((i : I) → f i) :=\n  add_comm_monoid.mk Add.add sorry 0 sorry sorry sorry\n\nprotected instance sub_neg_add_monoid {I : Type u} {f : I → Type v} [(i : I) → sub_neg_monoid (f i)] : sub_neg_monoid ((i : I) → f i) :=\n  sub_neg_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry Neg.neg Sub.sub\n\nprotected instance group {I : Type u} {f : I → Type v} [(i : I) → group (f i)] : group ((i : I) → f i) :=\n  group.mk Mul.mul sorry 1 sorry sorry has_inv.inv Div.div sorry\n\nprotected instance comm_group {I : Type u} {f : I → Type v} [(i : I) → comm_group (f i)] : comm_group ((i : I) → f i) :=\n  comm_group.mk Mul.mul sorry 1 sorry sorry has_inv.inv Div.div sorry sorry\n\nprotected instance left_cancel_semigroup {I : Type u} {f : I → Type v} [(i : I) → left_cancel_semigroup (f i)] : left_cancel_semigroup ((i : I) → f i) :=\n  left_cancel_semigroup.mk Mul.mul sorry sorry\n\nprotected instance right_cancel_semigroup {I : Type u} {f : I → Type v} [(i : I) → right_cancel_semigroup (f i)] : right_cancel_semigroup ((i : I) → f i) :=\n  right_cancel_semigroup.mk Mul.mul sorry sorry\n\nprotected instance mul_zero_class {I : Type u} {f : I → Type v} [(i : I) → mul_zero_class (f i)] : mul_zero_class ((i : I) → f i) :=\n  mul_zero_class.mk Mul.mul 0 sorry sorry\n\nprotected instance comm_monoid_with_zero {I : Type u} {f : I → Type v} [(i : I) → comm_monoid_with_zero (f i)] : comm_monoid_with_zero ((i : I) → f i) :=\n  comm_monoid_with_zero.mk Mul.mul sorry 1 sorry sorry sorry 0 sorry sorry\n\n@[simp] theorem const_zero {α : Type u_1} {β : Type u_2} [HasZero β] : function.const α 0 = 0 :=\n  rfl\n\n@[simp] theorem comp_one {α : Type u_1} {β : Type u_2} {γ : Type u_3} [HasOne β] {f : β → γ} : f ∘ 1 = function.const α (f 1) :=\n  rfl\n\n@[simp] theorem one_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [HasOne γ] {f : α → β} : 1 ∘ f = 1 :=\n  rfl\n\nend pi\n\n\n/-- Evaluation of functions into an indexed collection of monoids at a point is a monoid\nhomomorphism. -/\ndef add_monoid_hom.apply {I : Type u} (f : I → Type v) [(i : I) → add_monoid (f i)] (i : I) : ((i : I) → f i) →+ f i :=\n  add_monoid_hom.mk (fun (g : (i : I) → f i) => g i) sorry sorry\n\n@[simp] theorem add_monoid_hom.apply_apply {I : Type u} (f : I → Type v) [(i : I) → add_monoid (f i)] (i : I) (g : (i : I) → f i) : coe_fn (add_monoid_hom.apply f i) g = g i :=\n  rfl\n\n/-- Coercion of a `monoid_hom` into a function is itself a `monoid_hom`.\n\nSee also `monoid_hom.eval`. -/\n@[simp] theorem monoid_hom.coe_fn_apply (α : Type u_1) (β : Type u_2) [monoid α] [comm_monoid β] (g : α →* β) : ∀ (ᾰ : α), coe_fn (monoid_hom.coe_fn α β) g ᾰ = coe_fn g ᾰ :=\n  fun (ᾰ : α) => Eq.refl (coe_fn (monoid_hom.coe_fn α β) g ᾰ)\n\n/-- The additive monoid homomorphism including a single additive monoid\ninto a dependent family of additive monoids, as functions supported at a point. -/\ndef add_monoid_hom.single {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → add_monoid (f i)] (i : I) : f i →+ (i : I) → f i :=\n  add_monoid_hom.mk (fun (x : f i) => pi.single i x) sorry sorry\n\n@[simp] theorem add_monoid_hom.single_apply {I : Type u} (f : I → Type v) [DecidableEq I] [(i : I) → add_monoid (f i)] {i : I} (x : f i) : coe_fn (add_monoid_hom.single f i) x = pi.single i x :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/group/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.32243125960666685}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.int.basic\nimport Mathlib.category_theory.graded_object\nimport Mathlib.category_theory.differential_object\nimport Mathlib.PostPort\n\nuniverses v u u_1 \n\nnamespace Mathlib\n\n/-!\n# Chain complexes\n\nWe define a chain complex in `V` as a differential `ℤ`-graded object in `V`.\n\nThis is fancy language for the obvious definition,\nand it seems we can use it straightforwardly:\n\n```\nexample (C : chain_complex V) : C.X 5 ⟶ C.X 6 := C.d 5\n```\n\n-/\n\n/--\nA `homological_complex V b` for `b : β` is a (co)chain complex graded by `β`,\nwith differential in grading `b`.\n\n(We use the somewhat cumbersome `homological_complex` to avoid the name conflict with `ℂ`.)\n-/\ndef homological_complex (V : Type u) [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] {β : Type} [add_comm_group β] (b : β) :=\n  category_theory.differential_object (category_theory.graded_object_with_shift b V)\n\n/--\nA chain complex in `V` is \"just\" a differential `ℤ`-graded object in `V`,\nwith differential graded `-1`.\n-/\ndef chain_complex (V : Type u) [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] :=\n  homological_complex V (-1)\n\n/--\nA cochain complex in `V` is \"just\" a differential `ℤ`-graded object in `V`,\nwith differential graded `+1`.\n-/\ndef cochain_complex (V : Type u) [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] :=\n  homological_complex V 1\n\n-- The chain groups of a chain complex `C` are accessed as `C.X i`,\n\n-- and the differentials as `C.d i : C.X i ⟶ C.X (i-1)`.\n\nnamespace homological_complex\n\n\n@[simp] theorem d_squared {V : Type u} [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] {β : Type} [add_comm_group β] {b : β}\n    (C : homological_complex V b) (i : β) :\n    category_theory.differential_object.d C i ≫ category_theory.differential_object.d C (i + b) =\n        0 :=\n  sorry\n\n/--\nA convenience lemma for morphisms of cochain complexes,\npicking out one component of the commutation relation.\n-/\n-- I haven't been able to get this to work with projection notation: `f.comm_at i`\n\n@[simp] theorem comm_at {V : Type u} [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] {β : Type} [add_comm_group β] {b : β}\n    {C : homological_complex V b} {D : homological_complex V b} (f : C ⟶ D) (i : β) :\n    category_theory.differential_object.d C i ≫\n          category_theory.differential_object.hom.f f (i + b) =\n        category_theory.differential_object.hom.f f i ≫ category_theory.differential_object.d D i :=\n  sorry\n\n@[simp] theorem comm {V : Type u} [category_theory.category V]\n    [category_theory.limits.has_zero_morphisms V] {β : Type} [add_comm_group β] {b : β}\n    {C : homological_complex V b} {D : homological_complex V b} (f : C ⟶ D) :\n    category_theory.differential_object.d C ≫\n          category_theory.functor.map\n            (category_theory.equivalence.functor\n              (category_theory.shift (category_theory.graded_object_with_shift b V) ^ 1))\n            (category_theory.differential_object.hom.f f) =\n        category_theory.differential_object.hom.f f ≫ category_theory.differential_object.d D :=\n  category_theory.differential_object.hom.comm (category_theory.graded_object_with_shift b V)\n    (category_theory.graded_object.category_of_graded_objects β)\n    (category_theory.graded_object.has_zero_morphisms β) (category_theory.graded_object.has_shift b)\n    C D f\n\n/-- The forgetful functor from cochain complexes to graded objects, forgetting the differential. -/\ndef forget (V : Type u) [category_theory.category V] [category_theory.limits.has_zero_morphisms V]\n    {β : Type} [add_comm_group β] {b : β} :\n    homological_complex V b ⥤ category_theory.graded_object β V :=\n  category_theory.differential_object.forget (category_theory.graded_object_with_shift b V)\n\nprotected instance inhabited {β : Type} [add_comm_group β] {b : β} :\n    Inhabited (homological_complex (category_theory.discrete PUnit) b) :=\n  { default := 0 }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/homology/chain_complex_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.3223918166144112}}
{"text": "/-\nCopyright © 2020 Nicolò Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Nicolò Cavalleri.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.geometry.manifold.times_cont_mdiff\nimport Mathlib.topology.continuous_map\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 u_6 u_7 l u_10 u_8 u_9 \n\nnamespace Mathlib\n\n/-!\n# Smooth bundled map\n\nIn this file we define the type `times_cont_mdiff_map` of `n` times continuously differentiable\nbundled maps.\n-/\n\n/-- Bundled `n` times continuously differentiable maps. -/\nstructure times_cont_mdiff_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_6) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_7) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (n : with_top ℕ) \nwhere\n  to_fun : M → M'\n  times_cont_mdiff_to_fun : times_cont_mdiff I I' n to_fun\n\n/-- Bundled smooth maps. -/\ndef smooth_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] (I : model_with_corners 𝕜 E H) (I' : model_with_corners 𝕜 E' H') (M : Type u_6) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] (M' : Type u_7) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] :=\n  times_cont_mdiff_map I I' M M' ⊤\n\nnamespace times_cont_mdiff_map\n\n\nprotected instance has_coe_to_fun {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} : has_coe_to_fun (times_cont_mdiff_map I I' M M' n) :=\n  has_coe_to_fun.mk (fun (x : times_cont_mdiff_map I I' M M' n) => M → M') times_cont_mdiff_map.to_fun\n\nprotected instance continuous_map.has_coe {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} : has_coe (times_cont_mdiff_map I I' M M' n) (continuous_map M M') :=\n  has_coe.mk fun (f : times_cont_mdiff_map I I' M M' n) => continuous_map.mk (times_cont_mdiff_map.to_fun f)\n\nprotected theorem times_cont_mdiff {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} (f : times_cont_mdiff_map I I' M M' n) : times_cont_mdiff I I' n ⇑f :=\n  times_cont_mdiff_map.times_cont_mdiff_to_fun f\n\nprotected theorem smooth {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] (f : times_cont_mdiff_map I I' M M' ⊤) : smooth I I' ⇑f :=\n  times_cont_mdiff_map.times_cont_mdiff_to_fun f\n\ntheorem coe_inj {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} {f : times_cont_mdiff_map I I' M M' n} {g : times_cont_mdiff_map I I' M M' n} (h : ⇑f = ⇑g) : f = g := sorry\n\ntheorem ext {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} {f : times_cont_mdiff_map I I' M M' n} {g : times_cont_mdiff_map I I' M M' n} (h : ∀ (x : M), coe_fn f x = coe_fn g x) : f = g := sorry\n\n/-- The identity as a smooth map. -/\ndef id {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {H : Type u_4} [topological_space H] {I : model_with_corners 𝕜 E H} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {n : with_top ℕ} : times_cont_mdiff_map I I M M n :=\n  mk id times_cont_mdiff_id\n\n/-- The composition of smooth maps, as a smooth map. -/\ndef comp {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {E'' : Type u_8} [normed_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M''] {n : with_top ℕ} (f : times_cont_mdiff_map I' I'' M' M'' n) (g : times_cont_mdiff_map I I' M M' n) : times_cont_mdiff_map I I'' M M'' n :=\n  mk (fun (a : M) => coe_fn f (coe_fn g a)) sorry\n\n@[simp] theorem comp_apply {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {E'' : Type u_8} [normed_group E''] [normed_space 𝕜 E''] {H'' : Type u_9} [topological_space H''] {I'' : model_with_corners 𝕜 E'' H''} {M'' : Type u_10} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M''] {n : with_top ℕ} (f : times_cont_mdiff_map I' I'' M' M'' n) (g : times_cont_mdiff_map I I' M M' n) (x : M) : coe_fn (comp f g) x = coe_fn f (coe_fn g x) :=\n  rfl\n\nprotected instance inhabited {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} [Inhabited M'] : Inhabited (times_cont_mdiff_map I I' M M' n) :=\n  { default := mk (fun (_x : M) => Inhabited.default) sorry }\n\n/-- Constant map as a smooth map -/\ndef const {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] {H : Type u_4} [topological_space H] {H' : Type u_5} [topological_space H'] {I : model_with_corners 𝕜 E H} {I' : model_with_corners 𝕜 E' H'} {M : Type u_6} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M] {M' : Type u_7} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M'] {n : with_top ℕ} (y : M') : times_cont_mdiff_map I I' M M' n :=\n  mk (fun (x : M) => y) times_cont_mdiff_const\n\nend times_cont_mdiff_map\n\n\nprotected instance continuous_linear_map.has_coe_to_times_cont_mdiff_map {𝕜 : Type u_1} [nondiscrete_normed_field 𝕜] {E : Type u_2} [normed_group E] [normed_space 𝕜 E] {E' : Type u_3} [normed_group E'] [normed_space 𝕜 E'] (n : with_top ℕ) : has_coe (continuous_linear_map 𝕜 E E')\n  (times_cont_mdiff_map (model_with_corners_self 𝕜 E) (model_with_corners_self 𝕜 E') E E' n) :=\n  has_coe.mk\n    fun (f : continuous_linear_map 𝕜 E E') =>\n      times_cont_mdiff_map.mk (linear_map.to_fun (continuous_linear_map.to_linear_map f))\n        (continuous_linear_map.times_cont_mdiff f)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/geometry/manifold/times_cont_mdiff_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.32239181661441113}}
{"text": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.intervals.basic\nimport Mathlib.data.set.lattice\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Intervals in `pi`-space\n\nIn this we prove various simple lemmas about intervals in `Π i, α i`. Closed intervals (`Ici x`,\n`Iic x`, `Icc x y`) are equal to products of their projections to `α i`, while (semi-)open intervals\nusually include the corresponding products as proper subsets.\n-/\n\nnamespace set\n\n\n@[simp] theorem pi_univ_Ici {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) : (pi univ fun (i : ι) => Ici (x i)) = Ici x := sorry\n\n@[simp] theorem pi_univ_Iic {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) : (pi univ fun (i : ι) => Iic (x i)) = Iic x := sorry\n\n@[simp] theorem pi_univ_Icc {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) : (pi univ fun (i : ι) => Icc (x i) (y i)) = Icc x y := sorry\n\ntheorem pi_univ_Ioi_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) [Nonempty ι] : (pi univ fun (i : ι) => Ioi (x i)) ⊆ Ioi x := sorry\n\ntheorem pi_univ_Iio_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) [Nonempty ι] : (pi univ fun (i : ι) => Iio (x i)) ⊆ Iio x :=\n  pi_univ_Ioi_subset x\n\ntheorem pi_univ_Ioo_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) [Nonempty ι] : (pi univ fun (i : ι) => Ioo (x i) (y i)) ⊆ Ioo x y :=\n  fun (x_1 : (i : ι) → α i) (hx : x_1 ∈ pi univ fun (i : ι) => Ioo (x i) (y i)) =>\n    { left := pi_univ_Ioi_subset (fun (i : ι) => x i) fun (i : ι) (hi : i ∈ univ) => and.left (hx i hi),\n      right := pi_univ_Iio_subset (fun (i : ι) => y i) fun (i : ι) (hi : i ∈ univ) => and.right (hx i hi) }\n\ntheorem pi_univ_Ioc_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) [Nonempty ι] : (pi univ fun (i : ι) => Ioc (x i) (y i)) ⊆ Ioc x y :=\n  fun (x_1 : (i : ι) → α i) (hx : x_1 ∈ pi univ fun (i : ι) => Ioc (x i) (y i)) =>\n    { left := pi_univ_Ioi_subset (fun (i : ι) => x i) fun (i : ι) (hi : i ∈ univ) => and.left (hx i hi),\n      right := fun (i : ι) => and.right (hx i trivial) }\n\ntheorem pi_univ_Ico_subset {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) [Nonempty ι] : (pi univ fun (i : ι) => Ico (x i) (y i)) ⊆ Ico x y :=\n  fun (x_1 : (i : ι) → α i) (hx : x_1 ∈ pi univ fun (i : ι) => Ico (x i) (y i)) =>\n    { left := fun (i : ι) => and.left (hx i trivial),\n      right := pi_univ_Iio_subset (fun (i : ι) => y i) fun (i : ι) (hi : i ∈ univ) => and.right (hx i hi) }\n\ntheorem pi_univ_Ioc_update_left {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] [DecidableEq ι] {x : (i : ι) → α i} {y : (i : ι) → α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m) : (pi univ fun (i : ι) => Ioc (function.update x i₀ m i) (y i)) =\n  (set_of fun (z : (i : ι) → α i) => m < z i₀) ∩ pi univ fun (i : ι) => Ioc (x i) (y i) := sorry\n\ntheorem pi_univ_Ioc_update_right {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] [DecidableEq ι] {x : (i : ι) → α i} {y : (i : ι) → α i} {i₀ : ι} {m : α i₀} (hm : m ≤ y i₀) : (pi univ fun (i : ι) => Ioc (x i) (function.update y i₀ m i)) =\n  (set_of fun (z : (i : ι) → α i) => z i₀ ≤ m) ∩ pi univ fun (i : ι) => Ioc (x i) (y i) := sorry\n\ntheorem disjoint_pi_univ_Ioc_update_left_right {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → preorder (α i)] [DecidableEq ι] {x : (i : ι) → α i} {y : (i : ι) → α i} {i₀ : ι} {m : α i₀} : disjoint (pi univ fun (i : ι) => Ioc (x i) (function.update y i₀ m i))\n  (pi univ fun (i : ι) => Ioc (function.update x i₀ m i) (y i)) := sorry\n\ntheorem pi_univ_Ioc_update_union {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → linear_order (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) (i₀ : ι) (m : α i₀) (hm : m ∈ Icc (x i₀) (y i₀)) : ((pi univ fun (i : ι) => Ioc (x i) (function.update y i₀ m i)) ∪\n    pi univ fun (i : ι) => Ioc (function.update x i₀ m i) (y i)) =\n  pi univ fun (i : ι) => Ioc (x i) (y i) := sorry\n\n/-- If `x`, `y`, `x'`, and `y'` are functions `Π i : ι, α i`, then\nthe set difference between the box `[x, y]` and the product of the open intervals `(x' i, y' i)`\nis covered by the union of the following boxes: for each `i : ι`, we take\n`[x, update y i (x' i)]` and `[update x i (y' i), y]`.\n\nE.g., if `x' = x` and `y' = y`, then this lemma states that the difference between a closed box\n`[x, y]` and the corresponding open box `{z | ∀ i, x i < z i < y i}` is covered by the union\nof the faces of `[x, y]`. -/\ntheorem Icc_diff_pi_univ_Ioo_subset {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → linear_order (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) (x' : (i : ι) → α i) (y' : (i : ι) → α i) : (Icc x y \\ pi univ fun (i : ι) => Ioo (x' i) (y' i)) ⊆\n  (Union fun (i : ι) => Icc x (function.update y i (x' i))) ∪ Union fun (i : ι) => Icc (function.update x i (y' i)) y := sorry\n\n/-- If `x`, `y`, `z` are functions `Π i : ι, α i`, then\nthe set difference between the box `[x, z]` and the product of the intervals `(y i, z i]`\nis covered by the union of the boxes `[x, update z i (y i)]`.\n\nE.g., if `x = y`, then this lemma states that the difference between a closed box\n`[x, y]` and the product of half-open intervals `{z | ∀ i, x i < z i ≤ y i}` is covered by the union\nof the faces of `[x, y]` adjacent to `x`. -/\ntheorem Icc_diff_pi_univ_Ioc_subset {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → linear_order (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) (z : (i : ι) → α i) : (Icc x z \\ pi univ fun (i : ι) => Ioc (y i) (z i)) ⊆ Union fun (i : ι) => Icc x (function.update z i (y i)) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/set/intervals/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.32239181661441113}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport order.category.PartialOrder\nimport order.hom.lattice\n\n/-!\n# The categories of semilattices\n\nThis defines `SemilatticeSup` and `SemilatticeInf`, the categories of sup-semilattices with a bottom\nelement and inf-semilattices with a top element.\n\n## References\n\n* [nLab, *semilattice*](https://ncatlab.org/nlab/show/semilattice)\n-/\n\nuniverses u\nopen category_theory\n\n/-- The category of sup-semilattices with a bottom element. -/\nstructure SemilatticeSup : Type.{u+1} :=\n(X : Type.{u})\n[is_semilattice_sup : semilattice_sup X]\n[is_order_bot : order_bot X]\n\n/-- The category of inf-semilattices with a top element. -/\nstructure SemilatticeInf : Type.{u+1} :=\n(X : Type.{u})\n[is_semilattice_inf : semilattice_inf X]\n[is_order_top : order_top X]\n\nattribute [protected] SemilatticeSup.X SemilatticeInf.X\n\nnamespace SemilatticeSup\n\ninstance : has_coe_to_sort SemilatticeSup Type* := ⟨SemilatticeSup.X⟩\nattribute [instance] is_semilattice_sup is_order_bot\n\n/-- Construct a bundled `SemilatticeSup` from a `semilattice_sup`. -/\ndef of (α : Type*) [semilattice_sup α] [order_bot α] : SemilatticeSup := ⟨α⟩\n\n@[simp] lemma coe_of (α : Type*) [semilattice_sup α] [order_bot α] : ↥(of α) = α := rfl\n\ninstance : inhabited SemilatticeSup := ⟨of punit⟩\n\ninstance : large_category.{u} SemilatticeSup :=\n{ hom := λ X Y, sup_bot_hom X Y,\n  id := λ X, sup_bot_hom.id X,\n  comp := λ X Y Z f g, g.comp f,\n  id_comp' := λ X Y, sup_bot_hom.comp_id,\n  comp_id' := λ X Y, sup_bot_hom.id_comp,\n  assoc' := λ W X Y Z _ _ _, sup_bot_hom.comp_assoc _ _ _ }\n\ninstance : concrete_category SemilatticeSup :=\n{ forget := { obj := SemilatticeSup.X, map := λ X Y, coe_fn },\n  forget_faithful := ⟨λ X Y, fun_like.coe_injective⟩ }\n\ninstance has_forget_to_PartialOrder : has_forget₂ SemilatticeSup PartialOrder :=\n{ forget₂ := { obj := λ X, ⟨X⟩, map := λ X Y f, f } }\n\n@[simp] lemma coe_forget_to_PartialOrder (X : SemilatticeSup) :\n  ↥((forget₂ SemilatticeSup PartialOrder).obj X) = ↥X := rfl\n\nend SemilatticeSup\n\nnamespace SemilatticeInf\n\ninstance : has_coe_to_sort SemilatticeInf Type* := ⟨SemilatticeInf.X⟩\n\nattribute [instance] is_semilattice_inf is_order_top\n\n/-- Construct a bundled `SemilatticeInf` from a `semilattice_inf`. -/\ndef of (α : Type*) [semilattice_inf α] [order_top α] : SemilatticeInf := ⟨α⟩\n\n@[simp] lemma coe_of (α : Type*) [semilattice_inf α] [order_top α] : ↥(of α) = α := rfl\n\ninstance : inhabited SemilatticeInf := ⟨of punit⟩\n\ninstance : large_category.{u} SemilatticeInf :=\n{ hom := λ X Y, inf_top_hom X Y,\n  id := λ X, inf_top_hom.id X,\n  comp := λ X Y Z f g, g.comp f,\n  id_comp' := λ X Y, inf_top_hom.comp_id,\n  comp_id' := λ X Y, inf_top_hom.id_comp,\n  assoc' := λ W X Y Z _ _ _, inf_top_hom.comp_assoc _ _ _ }\n\ninstance : concrete_category SemilatticeInf :=\n{ forget := { obj := SemilatticeInf.X, map := λ X Y, coe_fn },\n  forget_faithful := ⟨λ X Y, fun_like.coe_injective⟩ }\n\ninstance has_forget_to_PartialOrder : has_forget₂ SemilatticeInf PartialOrder :=\n{ forget₂ := { obj := λ X, ⟨X⟩, map := λ X Y f, f } }\n\n@[simp] lemma coe_forget_to_PartialOrder (X : SemilatticeInf) :\n  ↥((forget₂ SemilatticeInf PartialOrder).obj X) = ↥X := rfl\n\nend SemilatticeInf\n\n/-! ### Order dual -/\n\nnamespace SemilatticeSup\n\n/-- Constructs an isomorphism of lattices from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : SemilatticeSup.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : SemilatticeSup ⥤ SemilatticeInf :=\n{ obj := λ X, SemilatticeInf.of (order_dual X), map := λ X Y, sup_bot_hom.dual }\n\nend SemilatticeSup\n\nnamespace SemilatticeInf\n\n/-- Constructs an isomorphism of lattices from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : SemilatticeInf.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : SemilatticeInf ⥤ SemilatticeSup :=\n{ obj := λ X, SemilatticeSup.of (order_dual X), map := λ X Y, inf_top_hom.dual }\n\nend SemilatticeInf\n\n/-- The equivalence between `SemilatticeSup` and `SemilatticeInf` induced by `order_dual` both ways.\n-/\n@[simps functor inverse]\ndef SemilatticeSup_equiv_SemilatticeInf : SemilatticeSup ≌ SemilatticeInf :=\nequivalence.mk SemilatticeSup.dual SemilatticeInf.dual\n  (nat_iso.of_components (λ X, SemilatticeSup.iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, SemilatticeInf.iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nlemma SemilatticeSup_dual_comp_forget_to_PartialOrder :\n  SemilatticeSup.dual ⋙ forget₂ SemilatticeInf PartialOrder =\n    forget₂ SemilatticeSup PartialOrder ⋙ PartialOrder.dual := rfl\n\nlemma SemilatticeInf_dual_comp_forget_to_PartialOrder :\n  SemilatticeInf.dual ⋙ forget₂ SemilatticeSup PartialOrder =\n    forget₂ SemilatticeInf PartialOrder ⋙ PartialOrder.dual := rfl\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/Semilattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3223918166144111}}
{"text": "import .core\n\nnamespace tts ------------------------------------------------------------------\nnamespace typing ---------------------------------------------------------------\nvariables {V : Type} [decidable_eq V] -- Type of variable names\nvariables {x y : tagged V} -- Variables\nvariables {e ex e₁ : exp V} -- Expressions\nvariables {t : typ V} -- Types\nvariables {s s' : sch V} -- Type schemes\nvariables {Γ Γ₁ Γ₂ Γ₃ : env V} -- Environments\nvariables {L : finset (tagged V)} -- Variable sets\n\nopen finmap occurs exp typ sch\n\ntheorem weaken_middle :\n  disjoint (dom Γ₁) (dom Γ₂) →\n  disjoint (dom Γ₂) (dom Γ₃) →\n  disjoint (dom Γ₁) (dom Γ₃) →\n  typing (Γ₁ ∪ Γ₃) e t →\n  typing (Γ₁ ∪ Γ₂ ∪ Γ₃) e t :=\nbegin\n  generalize Γeq : Γ₁ ∪ Γ₃ = Γ₁₃,\n  intros dj₁₂ dj₂₃ dj₁₃ T,\n  induction T generalizing Γ₁ Γ₃,\n  case typing.var : Γ x s ts b ln_eq lts ls {\n    induction Γeq,\n    exact var (mem_union_middle_left dj₁₃ dj₂₃ b) ln_eq lts ls,\n  },\n  case typing.app : Γ ef ea t₁ t₂ Tf Ta ihf iha {\n    exact app (ihf Γeq dj₁₂ dj₂₃ dj₁₃) (iha Γeq dj₁₂ dj₂₃ dj₁₃),\n  },\n  case typing.lam : v Γ eb t₁ t₂ L lt₁ Ft₂ ihb {\n    apply lam (L ∪ dom (Γ₁ ∪ Γ₂ ∪ Γ₃)) lt₁,\n    introv nm,\n    simp [not_or_distrib, and_assoc] at nm,\n    rw [←insert_union, ←insert_union],\n    induction Γeq,\n    exact ihb nm.1 insert_union\n      (disjoint_keys_insert_left.mpr ⟨nm.2.1, dj₁₂⟩)\n      dj₂₃\n      (disjoint_keys_insert_left.mpr ⟨nm.2.2.2, dj₁₃⟩),\n  },\n  case typing.let_ : v Γ ed eb sd tb Ld Lb _ _ ihd ihb {\n    apply let_ sd (Ld ∪ dom (Γ₁ ∪ Γ₂ ∪ Γ₃)) (Lb ∪ dom (Γ₁ ∪ Γ₂ ∪ Γ₃)),\n    { introv ln_eq nd nm,\n      simp [not_or_distrib, and_assoc] at nm,\n      exact ihd ln_eq nd (λ _ h, (nm _ h).1) Γeq dj₁₂ dj₂₃ dj₁₃ },\n    { introv nm,\n      simp [not_or_distrib, and_assoc] at nm,\n      rw [←insert_union, ←insert_union],\n      induction Γeq,\n      exact ihb nm.1 insert_union\n        (disjoint_keys_insert_left.mpr ⟨nm.2.1, dj₁₂⟩)\n        dj₂₃\n        (disjoint_keys_insert_left.mpr ⟨nm.2.2.2, dj₁₃⟩) }\n  }\nend\n\ntheorem weaken :\n  disjoint (dom Γ₁) (dom Γ₂) →\n  typing Γ₂ e t →\n  typing (Γ₁ ∪ Γ₂) e t :=\nbegin\n  intros dj T,\n  rw ←empty_union Γ₂ at T,\n  rw ←empty_union Γ₁ at ⊢,\n  exact weaken_middle (finset.disjoint_empty_left _) dj (finset.disjoint_empty_left _) T\nend\n\nlemma ne_of_nm : x ∉ L ∪ dom (Γ₁ ∪ finmap.singleton (y :~ s) ∪ Γ₂) → y ≠ x :=\nby simp [not_or_distrib]; tauto\n\nlemma nmL_of_nm : x ∉ L ∪ dom (Γ₁ ∪ finmap.singleton (y :~ s) ∪ Γ₂) → x ∉ L :=\nby simp [not_or_distrib]; tauto\n\nlemma nmΓ₂_of_nm : x ∉ L ∪ dom (Γ₁ ∪ finmap.singleton (y :~ s) ∪ Γ₂) → x ∉ dom Γ₂ :=\nby simp [not_or_distrib]; tauto\n\nlemma nm_ins : x ∉ dom (Γ₁ ∪ Γ₂) → y ∉ L ∪ dom (Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂) →\n  x ∉ dom (insert (binding.mk y s') Γ₁ ∪ Γ₂) :=\nby simp [not_or_distrib]; tauto\n\ntheorem subst_weaken :\n  disjoint (dom Γ₁) (dom Γ₂) →\n  x ∉ dom (Γ₁ ∪ Γ₂) →\n  lc ex →\n  (∀ {ts : list (typ V)}, s.arity = ts.length → (∀ (t : typ V), t ∈ ts → lc t) → typing Γ₂ ex (open_typs ts s)) →\n  typing (Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂) e t →\n  typing (Γ₁ ∪ Γ₂) (subst x ex e) t :=\nbegin\n  generalize Γeq : Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂ = Γ₁₂,\n  intros dj nmd lex Tex T,\n  induction T generalizing Γ₁ Γ₂ x s ex,\n  case typing.var : Γ y s' ts b ln_eq lts ls {\n    induction Γeq,\n    have b : y :~ s' ∈ Γ₁ ∨ y :~ s' ∈ finmap.singleton (x :~ s) ∨ y :~ s' ∈ Γ₂ :=\n      (or_assoc _ _).mp (or.imp_left mem_of_mem_union (mem_of_mem_union b)),\n    by_cases h : x = y,\n    { induction h,\n      simp [not_or_distrib] at nmd,\n      rcases b with m | m | m,\n      { exact absurd (mem_keys_of_mem m) nmd.1 },\n      { simp at m, induction m, simp [weaken dj (Tex ln_eq lts)] },\n      { exact absurd (mem_keys_of_mem m) nmd.2 } },\n    { rcases b with m | m | m,\n      { simp [h, var ((mem_union dj).mpr (or.inl m)) ln_eq lts ls] },\n      { simp at m, exact absurd m.1.symm h },\n      { simp [h, var ((mem_union dj).mpr (or.inr m)) ln_eq lts ls] } } },\n  case typing.app : Γ ef ea t₁ t₂ Tf Ta ihf iha {\n    exact app (ihf Γeq dj nmd lex @Tex) (iha Γeq dj nmd lex @Tex) },\n  case typing.lam : v Γ eb t₁ t₂ L lt₁ Ft₂ ihb {\n    apply lam (L ∪ dom (Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂)) lt₁,\n    induction Γeq,\n    intros y nm,\n    rw ←subst_open_var lex (ne_of_nm nm),\n    rw ←insert_union,\n    exact ihb (nmL_of_nm nm)\n      (by rw [insert_union, insert_union])\n      (disjoint_keys_insert_left.mpr ⟨by exact nmΓ₂_of_nm nm, dj⟩)\n      (nm_ins nmd nm)\n      lex\n      @Tex },\n  case typing.let_ : v Γ ed eb sd tb Ld Lb _ _ ihd ihb {\n    dsimp at ihd ihb,\n    apply let_ sd\n      (Ld ∪ dom (Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂))\n      (Lb ∪ dom (Γ₁ ∪ finmap.singleton (x :~ s) ∪ Γ₂)),\n    { intros xs ln_eq nd nm,\n      exact ihd ln_eq nd (λ _ h, nmL_of_nm (nm _ h)) Γeq dj nmd lex @Tex },\n    { induction Γeq,\n      intros y nm,\n      rw ←subst_open_var lex (ne_of_nm nm),\n      rw ←insert_union,\n      exact ihb (nmL_of_nm nm)\n        (by rw [insert_union, insert_union])\n        (disjoint_keys_insert_left.mpr ⟨by exact nmΓ₂_of_nm nm, dj⟩)\n        (nm_ins nmd nm)\n        lex\n        @Tex } }\nend\n\n-- Expression substitution preserves typing.\ntheorem exp_subst_preservation :\n  x ∉ dom Γ →\n  lc ex →\n  (∀ {ts : list (typ V)}, s.arity = ts.length → (∀ (t : typ V), t ∈ ts → lc t) → typing Γ ex (open_typs ts s)) →\n  typing (finmap.singleton (x :~ s) ∪ Γ) e₁ t →\n  typing Γ (subst x ex e₁) t :=\nbegin\n  intros nm lex Tex T,\n  rw ←empty_union (finmap.singleton (x :~ s)) at T,\n  rw ←empty_union Γ,\n  exact subst_weaken (finset.disjoint_empty_left _) (by simp [nm]) lex @Tex T\nend\n\n-- Typing implies a locally-closed expression\ntheorem lc_exp (T : typing Γ e t) : lc e :=\nbegin\n  induction T,\n  case typing.var { simp },\n  case typing.app { simp * },\n  case typing.lam { simp [exp.lc_body, *], tauto },\n  case typing.let_ : v _ ed _ _ _ _ _ _ _ ihd {\n    dsimp at ihd,\n    have : lc ed := ihd\n      ((fresh.tagged v).pgenl_length_eq _ _)\n      ((fresh.tagged v).pgenl_nodup _ _)\n      (λ _, (fresh.tagged v).pgenl_not_mem),\n    simp [exp.lc_body, *],\n    tauto\n  }\nend\n\n-- Typing implies a locally-closed type\ntheorem lc_typ (T : typing Γ e t) : lc t :=\nbegin\n  induction T,\n  case typing.var : _ _ _ _ _ ln_eq lts ls {\n    exact lc_open_typs ls ln_eq lts },\n  case typing.app : Γ ef ea t₁ t₂ Tf Ta ihf iha {\n    simp at ihf,\n    cases ihf with _ lt₂,\n    exact lt₂ },\n  case typing.lam : v _ _ _ _ L lt₁ _ ihb {\n    dsimp at ihb,\n    simp [lt₁, ihb ((fresh.tagged v).gen_not_mem L)] },\n  case typing.let_ : v _ _ _ _ _ _ Lb _ _ _ ihb {\n    exact ihb ((fresh.tagged v).gen_not_mem Lb) }\nend\n\nend /- namespace -/ typing -----------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/typing/props.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.322311628952465}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.uliftable\nimport Mathlib.Lean3Lib.system.random\nimport Mathlib.system.random.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 v \n\nnamespace Mathlib\n\n/-!\n# `gen` Monad\n\nThis monad is used to formulate randomized computations with a parameter\nto specify the desired size of the result.\n\nThis is a port of the Haskell QuickCheck library.\n\n## Main definitions\n  * `gen` monad\n\n## Local notation\n\n * `i .. j` : `Icc i j`, the set of values between `i` and `j` inclusively;\n\n## Tags\n\nrandom testing\n\n## References\n\n  * https://hackage.haskell.org/package/QuickCheck\n\n-/\n\nnamespace slim_check\n\n\n/-- Monad to generate random examples to test properties with.\nIt has a `nat` parameter so that the caller can decide on the\nsize of the examples. -/\ndef gen (α : Type u) := reader_t (ulift ℕ) rand α\n\n/-- Execute a `gen` inside the `io` monad using `i` as the example\nsize and with a fresh random number generator. -/\ndef io.run_gen {α : Type} (x : gen α) (i : ℕ) : io α := io.run_rand (reader_t.run x (ulift.up i))\n\nnamespace gen\n\n\n/-- Lift `random.random` to the `gen` monad. -/\ndef choose_any (α : Type u) [random α] : gen α := reader_t.mk fun (_x : ulift ℕ) => rand.random α\n\n/-- Lift `random.random_r` to the `gen` monad. -/\ndef choose {α : Type u} [preorder α] [bounded_random α] (x : α) (y : α) (p : x ≤ y) :\n    gen ↥(set.Icc x y) :=\n  reader_t.mk fun (_x : ulift ℕ) => rand.random_r x y p\n\n/-- Generate a `nat` example between `x` and `y`. -/\ndef choose_nat (x : ℕ) (y : ℕ) (p : x ≤ y) : gen ↥(set.Icc x y) := choose x y p\n\n/-- Generate a `nat` example between `x` and `y`. -/\ndef choose_nat' (x : ℕ) (y : ℕ) (p : x < y) : gen ↥(set.Ico x y) :=\n  (fun (this : ∀ (i : ℕ), x < i → i ≤ y → Nat.pred i < y) =>\n      subtype.map Nat.pred sorry <$> choose (x + 1) y p)\n    sorry\n\nprotected instance uliftable : uliftable gen gen :=\n  reader_t.uliftable' (equiv.trans equiv.ulift (equiv.symm equiv.ulift))\n\nprotected instance has_orelse : has_orelse gen :=\n  has_orelse.mk\n    fun (α : Type u) (x y : gen α) =>\n      do \n        let b ← uliftable.up (choose_any Bool)\n        ite (↥(ulift.down b)) x y\n\n/-- Get access to the size parameter of the `gen` monad. For\nreasons of universe polymorphism, it is specified in\ncontinuation passing style. -/\ndef sized {α : Type u} (cmd : ℕ → gen α) : gen α := reader_t.mk fun (_x : ulift ℕ) => sorry\n\n/-- Apply a function to the size parameter. -/\ndef resize {α : Type u} (f : ℕ → ℕ) (cmd : gen α) : gen α := reader_t.mk fun (_x : ulift ℕ) => sorry\n\n/-- Create `n` examples using `cmd`. -/\ndef vector_of {α : Type u} (n : ℕ) (cmd : gen α) : gen (vector α n) := sorry\n\n/-- Create a list of examples using `cmd`. The size is controlled\nby the size parameter of `gen`. -/\ndef list_of {α : Type u} (cmd : gen α) : gen (List α) :=\n  sized\n    fun (sz : ℕ) =>\n      do \n        uliftable.up (choose_nat 0 (sz + 1) sorry)\n        sorry\n\n/-- Given a list of example generators, choose one to create an example. -/\ndef one_of {α : Type u} (xs : List (gen α)) (pos : 0 < list.length xs) : gen α :=\n  do \n    uliftable.up (choose_nat' 0 (list.length xs) pos)\n    sorry\n\n/-- Given a list of example generators, choose one to create an example. -/\ndef elements {α : Type u} (xs : List α) (pos : 0 < list.length xs) : gen α :=\n  do \n    uliftable.up (choose_nat' 0 (list.length xs) pos)\n    sorry\n\n/--\n`freq_aux xs i _` takes a weighted list of generator and a number meant to select one of the generators.\n\nIf we consider `freq_aux [(1, gena), (3, genb), (5, genc)] 4 _`, we choose a generator by splitting\nthe interval 1-9 into 1-1, 2-4, 5-9 so that the width of each interval corresponds to one of the\nnumber in the list of generators. Then, we check which interval 4 falls into: it selects `genb`.\n-/\ndef freq_aux {α : Type u} (xs : List (ℕ+ × gen α)) (i : ℕ) :\n    i < list.sum (list.map (subtype.val ∘ prod.fst) xs) → gen α :=\n  sorry\n\n/--\n`freq [(1, gena), (3, genb), (5, genc)] _` will choose one of `gena`, `genb`, `genc` with\nprobabiities proportional to the number accompanying them. In this example, the sum of\nthose numbers is 9, `gena` will be chosen with probability ~1/9, `genb` with ~3/9 (i.e. 1/3)\nand `genc` with probability 5/9.\n-/\ndef freq {α : Type u} (xs : List (ℕ+ × gen α)) (pos : 0 < list.length xs) : gen α :=\n  let s : ℕ := list.sum (list.map (subtype.val ∘ prod.fst) xs);\n  (fun (ha : 1 ≤ s) =>\n      (fun (this : 0 ≤ s - 1) =>\n          uliftable.adapt_up gen gen (choose_nat 0 (s - 1) this)\n            fun (i : ↥(set.Icc 0 (s - 1))) => freq_aux xs (subtype.val i) sorry)\n        sorry)\n    sorry\n\n/-- Generate a random permutation of a given list. -/\ndef permutation_of {α : Type u} (xs : List α) : gen (Subtype (list.perm xs)) := sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/testing/slim_check/gen_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.3222661238298851}}
{"text": "import analysis.normed.group.SemiNormedGroup.kernels\n\nimport Lbar.basic\nimport normed_group.pseudo_normed_group\nimport combinatorial_lemma.partition\nimport combinatorial_lemma.lem97\nimport polyhedral_lattice.pseudo_normed_group\nimport pseudo_normed_group.splittable\n\nimport hacks_and_tricks.by_exactI_hack\n\n/-!\nIn this file we state and prove 9.8 of [Analytic].\n-/\n\nnoncomputable theory\n\nopen_locale nnreal big_operators\n\nopen pseudo_normed_group combinatorial_lemma\n\nvariables (Λ : Type*) (r' : ℝ≥0) (S : Type*)\nvariables [fintype S] [polyhedral_lattice Λ]\n\nsection\n\nvariables {S Λ r'}\n\n-- ugly name\n@[simps]\ndef Lbar.mk_aux'\n  (x : Λ →+ Lbar r' S) (y : S → ℕ → Λ →+ ℤ)\n  (h : ∀ s n l, (y s n l).nat_abs ≤ (x l s n).nat_abs) :\n  Λ →+ Lbar r' S :=\nadd_monoid_hom.mk' (λ l,\n{ to_fun := λ s n, y s n l,\n  coeff_zero' := λ s,\n  begin\n    specialize h s 0 l,\n    simpa only [int.nat_abs_eq_zero, Lbar.coeff_zero, le_zero_iff, int.nat_abs_zero] using h\n  end,\n  summable' :=\n  begin\n    intro s,\n    apply nnreal.summable_of_le _ ((x l).summable s),\n    intro n,\n    refine mul_le_mul' _ le_rfl,\n    norm_cast,\n    exact (h _ _ _)\n  end }) $ λ l₁ l₂, by { ext s n, exact (y s n).map_add l₁ l₂ }\n\n-- jmc: It seems to me that we cannot easily define this using `Lbar.mk_aux'` above\n@[simps]\ndef Lbar.mk_aux\n  {ι : Type} [fintype ι] {l : ι → Λ} (hl : generates_norm l)\n  (x : Λ →+ Lbar r' S) (y : S → ℕ → Λ →+ ℤ)\n  (h : ∀ s n i, (y s n (l i)).nat_abs ≤ (x (l i) s n).nat_abs) :\n  Λ →+ Lbar r' S :=\nadd_monoid_hom.mk' (λ l',\n{ to_fun := λ s n, y s n l',\n  coeff_zero' := λ s,\n  begin\n    obtain ⟨c, h1, -⟩ := hl l',\n    rw [h1, add_monoid_hom.map_sum, finset.sum_eq_zero],\n    rintro i -,\n    suffices : y s 0 (l i) = 0,\n    { rw [add_monoid_hom.map_nsmul, this, nsmul_zero] },\n    specialize h s 0 i,\n    simpa only [int.nat_abs_eq_zero, Lbar.coeff_zero, le_zero_iff, int.nat_abs_zero] using h\n  end,\n  summable' :=\n  begin\n    intro s,\n    obtain ⟨c, h1, -⟩ := hl.generates_nnnorm l',\n    rw h1,\n    suffices : summable (λ n, ∑ i, c i • ↑(y s n (l i)).nat_abs * r' ^ n),\n    { apply nnreal.summable_of_le _ this,\n      intro n,\n      rw ← finset.sum_mul,\n      refine mul_le_mul' _ le_rfl,\n      simp only [add_monoid_hom.map_sum, nnreal.coe_nat_abs, add_monoid_hom.map_nsmul],\n      refine le_trans (nnnorm_sum_le _ _) (le_of_eq (fintype.sum_congr _ _ _)),\n      intro i,\n      simp only [nsmul_eq_mul, ← nnreal.coe_nat_abs,\n        int.nat_abs_mul, int.nat_abs_of_nat, nat.cast_mul] },\n    apply summable_sum,\n    rintro i -,\n    apply nnreal.summable_of_le _ ((c i • x (l i)).summable s),\n    intro n,\n    refine mul_le_mul' _ le_rfl,\n    simp only [nsmul_eq_mul, ← nnreal.coe_nat_abs, int.nat_abs_mul,\n        int.nat_abs_of_nat, ← nat.cast_mul, Lbar.coe_nsmul, pi.mul_apply, pi.nat_apply],\n    norm_cast,\n    exact nat.mul_le_mul le_rfl (h _ _ _)\n  end }) $ λ l₁ l₂, by { ext s n, exact (y s n).map_add l₁ l₂ }\n\nend\n\nvariables {Λ r' S}\n\nlemma Lbar.mk_aux_mem_filtration\n  (ι : Type) (c : ℝ≥0) (N : ℕ) (hN : 0 < N) [fintype ι]\n  {l : ι → Λ} (hl : generates_norm l)\n  {x : Λ →+ Lbar r' S}\n  (hx : x ∈ filtration (Λ →+ Lbar r' S) c)\n  (x₀' : S → ℕ → Λ →+ ℤ)\n  (x₁' : S → ℕ → Λ →+ ℤ)\n  (x' : S → ℕ → Λ →+ ℤ)\n  (hx' : ∀ l s n, x l s n = x' s n l)\n  (H : ∀ s n i, (x' s n (l i)).nat_abs =\n    N * (x₀' s n (l i)).nat_abs + (x₁' s n (l i)).nat_abs)\n  (H' : ∀ s n i, (x₀' s n (l i)).nat_abs ≤ (x (l i) s n).nat_abs) :\n  Lbar.mk_aux hl x x₀' H' ∈ filtration (Λ →+ Lbar r' S) (c / ↑N) :=\nbegin\n  set x₀ := Lbar.mk_aux hl x x₀' H',\n  have aux : c = (N : ℝ≥0) * (c / N),\n  { rw [mul_comm, div_mul_cancel],\n    exact_mod_cast hN.ne' },\n  have hN' : (0: ℝ≥0) < N,\n  {  exact_mod_cast hN },\n  rw hl.add_monoid_hom_mem_filtration_iff at hx ⊢,\n  intro i,\n  specialize hx i,\n  rw Lbar.mem_filtration_iff at hx ⊢,\n  rw [← mul_le_mul_left hN', ← mul_assoc, ← aux, ← nsmul_eq_mul, ← Lbar.nnnorm_nsmul],\n  refine le_trans (finset.sum_le_sum _) hx,\n  rintro s -,\n  refine tsum_le_tsum _ (Lbar.summable _ s) ((x (l i)).summable s),\n  intro n,\n  refine mul_le_mul' _ le_rfl,\n  norm_cast,\n  convert le_trans (le_add_right le_rfl) (H s n i).ge,\n  swap, { apply hx' },\n  rw [← int.nat_abs_of_nat N, ← int.nat_abs_mul, int.nat_abs_of_nat, ← nsmul_eq_mul],\n  congr' 1,\nend\n\n@[simps]\ndef Lbar.mk_of_add_monoid_hom [fact (r' < 1)] (x : S → ℕ → Λ →+ ℤ) (a : Λ →+ ℤ)\n  [∀ s n, decidable (x s n = a)] :\n  Lbar r' S :=\nLbar.of_mask (Lbar.geom r' S) $ λ s n, x s n = a\n\n@[simps apply]\ndef Lbar.mk_tensor (a : Λ →+ ℤ) (x : Lbar r' S) : Λ →+ Lbar r' S :=\nadd_monoid_hom.mk' (λ l, a l • x) $ λ l₁ l₂, by rw [a.map_add, add_smul]\n\n-- better name?\nlemma lem_98_aux [fact (r' < 1)] (A : finset (Λ →+ ℤ))\n  (x₁' : S → ℕ → Λ →+ ℤ) [∀ s n a, decidable (x₁' s n = a)]\n  (hx₁' : ∀ s n, x₁' s n ∈ A) (x₁ : Λ →+ Lbar r' S)\n  (hx₁ : ∀ l s n, x₁ l s n = x₁' s n l) (l : Λ) :\n  x₁ l = ∑ a in A, a l • Lbar.mk_of_add_monoid_hom x₁' a :=\nbegin\n  ext s n,\n  simp only [finset.sum_apply, Lbar.coe_sum, pi.smul_apply, Lbar.coe_zsmul,\n    Lbar.coe_mk],\n  rw [finset.sum_eq_single (x₁' s n)],\n  { simp only [true_and, and_congr, if_congr, eq_self_iff_true, if_true,\n          Lbar.mk_of_add_monoid_hom_to_fun],\n    split_ifs with hn,\n    { rw [smul_eq_mul, mul_zero, hn], exact (x₁ l).coeff_zero s },\n    { simp only [smul_eq_mul, mul_one, hx₁] } },\n  { intros a haA ha,\n    simp only [ha.symm, false_and, if_false, smul_eq_mul, mul_zero,\n      Lbar.mk_of_add_monoid_hom_to_fun] },\n  { intro hsn, exact (hsn (hx₁' s n)).elim }\nend\n\nlemma fintype_prod_nat_equiv_nat (S : Type*) [fintype S] [hS : nonempty S] :\n  nonempty (S × ℕ ≃ ℕ) :=\nbegin\n  classical,\n  obtain e' := fintype.equiv_fin S,\n  refine nonempty.intro _,\n  calc S × ℕ ≃ fin (fintype.card S) × ℕ : equiv.prod_congr_left (λ _, e')\n         ... ≃ ℕ : classical.choice $ cardinal.eq.1 _,\n  rw [← cardinal.lift_id (cardinal.mk ℕ),\n    cardinal.mk_prod, ← cardinal.aleph_0, cardinal.mul_eq_right];\n  simp [le_of_lt (cardinal.nat_lt_aleph_0 _), le_refl],\n  rw ← fintype.card_pos_iff at hS,\n  exact_mod_cast hS.ne'\nend\n\nlemma lem98_int [fact (r' < 1)] (N : ℕ) (hN : 0 < N) (c : ℝ≥0)\n  (x : Lbar r' S) (hx : x ∈ filtration (Lbar r' S) c)\n  (H : ∀ s n, (x s n).nat_abs ≤ 1) :\n  ∃ y : fin N → Lbar r' S, (x = ∑ i, y i) ∧\n      (∀ i, y i ∈ filtration (Lbar r' S) (c/N + 1)) :=\nbegin\n  by_cases hS : nonempty S, swap,\n  { refine ⟨λ i, 0, _, _⟩,\n    { ext s n, exact (hS ⟨s⟩).elim },\n    { intro, exact zero_mem_filtration _ } },\n  resetI,\n  obtain ⟨e⟩ := fintype_prod_nat_equiv_nat S,\n  let f : ℕ → ℝ≥0 :=\n    λ n, ↑(x (e.symm n).1 (e.symm n).2).nat_abs * r' ^ (e.symm n).2,\n  have summable_f : summable f,\n  { rw [← e.summable_iff, ← (equiv.sigma_equiv_prod S ℕ).summable_iff],\n    simp only [function.comp, f, e.symm_apply_apply, nnreal.summable_sigma,\n      equiv.sigma_equiv_prod_apply],\n    -- TODO, missing lemma `fintype.summable`\n    exact ⟨x.summable, ⟨_, has_sum_fintype _⟩⟩ },\n  have f_aux : ∀ n, f n ≤ 1,\n  { intro n,\n    calc f n ≤ 1 * 1 : mul_le_mul' _ _\n    ... = 1 : mul_one 1,\n    { exact_mod_cast (H _ _) },\n    { have : fact (r' < 1) := ‹_›, exact pow_le_one _ zero_le' this.out.le } },\n  obtain ⟨mask, h1, h2⟩ := exists_partition N hN f f_aux summable_f,\n  classical,\n  let y := λ i, Lbar.of_mask x (λ s n, (e (s, n)) ∈ mask i),\n  have hxy : x = ∑ i, y i,\n  { ext s n,\n    simp only [Lbar.coe_sum, if_congr, function.curry_apply, fintype.sum_apply,\n      function.comp_app, Lbar.of_mask_to_fun, finset.sum_congr],\n    obtain ⟨i, hi1, hi2⟩ := h1 (e (s, n)),\n    rw [finset.sum_eq_single i, if_pos hi1],\n    { rintro j - hj,\n      rw if_neg,\n      exact mt (hi2 j) hj },\n    { intro hi, exact (hi (finset.mem_univ i)).elim } },\n  refine ⟨y, hxy, _⟩,\n  intro i,\n  rw [Lbar.mem_filtration_iff] at hx ⊢,\n  suffices : ∥x∥₊ = ∑' n, f n ∧ ∥y i∥₊ = ∑' (n : ℕ), (mask i).indicator f n,\n  { calc ∥y i∥₊ = ∑' (n : ℕ), (mask i).indicator f n : this.right\n            ... ≤ (∑' (n : ℕ), f n) / N + 1         : h2 i\n            ... ≤ c / N + 1                         : _,\n    simp only [div_eq_mul_inv],\n    refine add_le_add (mul_le_mul' _ le_rfl) le_rfl,\n    exact this.left.ge.trans hx },\n  split,\n  { calc ∑ s, ∑' n, ↑(x s n).nat_abs * r' ^ n\n        = ∑' s, ∑' n, ↑(x s n).nat_abs * r' ^ n : (tsum_fintype _).symm\n    ... = _ : _\n    ... = ∑' n, f n : e.tsum_eq _,\n    rw [← (equiv.sigma_equiv_prod S ℕ).tsum_eq], swap, { apply_instance },\n    dsimp only [f, equiv.sigma_equiv_prod_apply],\n    simp only [equiv.symm_apply_apply],\n    rw [tsum_sigma'],\n    { exact x.summable },\n    { rw nnreal.summable_sigma, exact ⟨x.summable, ⟨_, has_sum_fintype _⟩⟩ } },\n  { calc ∑ s, ∑' n, ↑(y i s n).nat_abs * r' ^ n\n        = ∑' s, ∑' n, ↑(y i s n).nat_abs * r' ^ n : (tsum_fintype _).symm\n    ... = ∑' x, (mask i).indicator f (e x) : _\n    ... = ∑' n, (mask i).indicator f n : e.tsum_eq _,\n    rw [← (equiv.sigma_equiv_prod S ℕ).tsum_eq], swap, { apply_instance },\n    have aux : ∀ i s n h1, @ite _ (e (s, n) ∈ mask i) h1 (↑(x s n).nat_abs * r' ^ n) 0 =\n      ↑(@ite _ (e (s, n) ∈ mask i) h1 (x s n) 0).nat_abs * r' ^ n,\n    { intros, split_ifs; simp only [int.nat_abs_zero, nat.cast_zero, zero_mul, eq_self_iff_true] },\n    dsimp only [f, y, set.indicator, equiv.sigma_equiv_prod_apply],\n    simp only [equiv.symm_apply_apply, Lbar.of_mask_to_fun, aux],\n    rw [tsum_sigma'],\n    { exact (y i).summable },\n    { rw nnreal.summable_sigma,\n      exact ⟨(y i).summable, ⟨_, has_sum_fintype _⟩⟩ } }\nend\n\nlemma lem98_aux' [fact (r' < 1)] (N : ℕ)\n  (A : finset (Λ →+ ℤ))\n  (x x₀ x₁ : Λ →+ Lbar r' S)\n  (x' x₀' x₁' : S → ℕ → Λ →+ ℤ) [∀ s n a, decidable (x₁' s n = a)]\n  (H : ∀ s n, x' s n = N • x₀' s n + x₁' s n)\n  (hx : ∀ l s n, x l s n = x' s n l)\n  (hx₀ : ∀ l s n, x₀ l s n = x₀' s n l)\n  (hx₁ : ∀ l s n, x₁ l s n = x₁' s n l)\n  (Hx₁ : ∀ (l : Λ), x₁ l = ∑ (a : Λ →+ ℤ) in A, a l • Lbar.mk_of_add_monoid_hom x₁' a)\n  (y' : (Λ →+ ℤ) → fin N → Lbar r' S)\n  (hy' : ∀ (a : Λ →+ ℤ), Lbar.mk_of_add_monoid_hom x₁' a = finset.univ.sum (y' a))\n  (y : fin N → Λ →+ Lbar r' S)\n  (hy : ∀ i, y i = x₀ + ∑ a in A, Lbar.mk_tensor a (y' a i)) :\n  x = ∑ (i : fin N), y i :=\nbegin\n  intros,\n  ext l s n,\n  simp only [hx, H, hy, ← hx₁, Hx₁, hy', Lbar.coe_add, finset.sum_apply, add_monoid_hom.add_apply,\n    Lbar.coe_sum, pi.add_apply, Lbar.mk_tensor_apply, finset.sum_congr,\n    add_monoid_hom.smul_apply, pi.smul_apply, add_monoid_hom.finset_sum_apply, finset.smul_sum],\n  rw [finset.sum_add_distrib, finset.sum_const, finset.card_univ,\n    fintype.card_fin, finset.sum_comm, ← hx₀],\nend\n\nlemma lem98_crux [fact (r' < 1)] {ι : Type} {l : ι → Λ}\n  (N : ℕ) (hN : 0 < N) (A : finset (Λ →+ ℤ))\n  (x x₀ x₁ : Λ →+ Lbar r' S)\n  (x' x₀' x₁' : S → ℕ → Λ →+ ℤ) [∀ s n a, decidable (x₁' s n = a)]\n  (xₐ : (Λ →+ ℤ) → Lbar r' S)\n  (hx : ∀ l s n, x l s n = x' s n l)\n  (hx₀ : ∀ l s n, x₀ l s n = x₀' s n l)\n  (hx₁ : ∀ l s n, x₁ l s n = x₁' s n l)\n  (hxₐ : ∀ a, xₐ a = Lbar.mk_of_add_monoid_hom x₁' a)\n  (Hx₁ : ∀ (l : Λ), x₁ l = ∑ (a : Λ →+ ℤ) in A, a l • xₐ a)\n  (hx₁'A : ∀ s n, x₁' s n ∈ A)\n  (H : ∀ s n i, (x' s n (l i)).nat_abs =\n    N * (x₀' s n (l i)).nat_abs + (x₁' s n (l i)).nat_abs)\n  (i : ι) :\n  ∥x (l i)∥₊ = N • ∥x₀ (l i)∥₊ + ∑ a in A, ∥a (l i)∥₊ * ∥xₐ a∥₊ :=\nbegin\n  simp only [hx, H, ← hx₀, ← hx₁, Lbar.nnnorm_def,\n    nsmul_eq_mul, mul_assoc,\n    finset.smul_sum, finset.mul_sum, nat.cast_add, nat.cast_mul, add_mul],\n  rw [finset.sum_comm, ← finset.sum_add_distrib],\n  apply fintype.sum_congr,\n  intro s,\n  apply nnreal.eq,\n  simp only [nnreal.coe_tsum, nnreal.coe_sum, nnreal.coe_add, nnreal.coe_mul,\n    nnreal.coe_pow, nnreal.coe_nat_cast, nnreal.coe_nat_abs, coe_nnnorm],\n  rw [tsum_add _ ((x₁ (l i)).summable_coe_real s)], swap,\n  { rw ← summable_mul_left_iff,\n    { exact (x₀ (l i)).summable_coe_real s },\n    { exact_mod_cast hN.ne' } },\n  simp only [← tsum_mul_left],\n  rw [← tsum_sum], swap,\n  { intros a ha, exact summable.mul_left _ ((xₐ a).summable_coe_real s) },\n  simp only [← mul_assoc, ← finset.sum_mul, hxₐ, Hx₁],\n  congr, ext n, congr,\n  simp only [finset.sum_apply, Lbar.coe_sum, Lbar.mk_of_add_monoid_hom_to_fun,\n    pi.smul_apply],\n  rw [finset.sum_eq_single (x₁' s n), finset.sum_eq_single (x₁' s n)],\n  { simp only [Lbar.coe_zsmul, Lbar.mk_of_add_monoid_hom_to_fun, if_true,\n      pi.smul_apply, eq_self_iff_true, true_and, if_congr, and_congr],\n    split_ifs,\n    { simp only [mul_zero, norm_zero, smul_eq_mul] },\n    { simp only [int.norm_eq_abs, ← abs_mul, ← int.cast_mul, smul_eq_mul] } },\n  all_goals { try { intro hsnA, exact (hsnA (hx₁'A s n)).elim } },\n  all_goals { intros a haA hasn },\n  { simp only [hasn.symm, Lbar.mk_of_add_monoid_hom_to_fun, norm_zero,\n      mul_zero, if_congr, and_congr, eq_self_iff_true, if_false, false_and] },\n  { simp only [hasn.symm, Lbar.coe_zsmul, pi.smul_apply, smul_eq_mul,\n      Lbar.mk_of_add_monoid_hom_to_fun, mul_zero,\n      if_congr, and_congr, eq_self_iff_true, if_false, false_and] },\nend\n.\n\nnamespace lem98\n\ndef ι (Λ : Type*) [polyhedral_lattice Λ] : Type :=\n(polyhedral_lattice.polyhedral Λ).some\n\ninstance : fintype (ι Λ) := (polyhedral_lattice.polyhedral Λ).some_spec.some\n\nvariables (Λ)\n\ndef l : ι Λ → Λ := (polyhedral_lattice.polyhedral Λ).some_spec.some_spec.some\n\nlemma hl : generates_norm (l Λ) :=\nby convert (polyhedral_lattice.polyhedral Λ).some_spec.some_spec.some_spec.1\n\nlemma hl' : ∀ i, l Λ i ≠ 0 :=\n(polyhedral_lattice.polyhedral Λ).some_spec.some_spec.some_spec.2\n\ndef A (N : ℕ) [hN : fact (0 < N)] := (lem97' N hN.1 (l Λ)).some\n\nlemma hA (N : ℕ) [hN : fact (0 < N)] (x : Λ →+ ℤ) :\n  ∃ x' (H : x' ∈ A Λ N) y, x = N • y + x' ∧\n    ∀ i, (x (l Λ i)).nat_abs = N * (y (l Λ i)).nat_abs + (x' (l Λ i)).nat_abs :=\n(lem97' N hN.1 (l Λ)).some_spec x\n\ndef d  (N : ℕ) [hN : fact (0 < N)] : ℝ≥0 :=\nfinset.univ.sup (λ i, ∑ a in A Λ N, ∥a (l Λ i)∥₊ / ∥l Λ i∥₊)\n\nend lem98\n\ntheorem lem98_finite [fact (r' < 1)] (Λ : Type*) [polyhedral_lattice Λ] (S : Type*) [fintype S]\n  (N : ℕ) [hN : fact (0 < N)] :\n  pseudo_normed_group.splittable (Λ →+ Lbar r' S) N (lem98.d Λ N) :=\nbegin\n  classical, constructor,\n  let l := lem98.l Λ,\n  have hl := lem98.hl Λ,\n  have hl' := lem98.hl' Λ,\n  let A := lem98.A Λ N,\n  have hA := lem98.hA Λ N,\n  let d := lem98.d Λ N,\n  intros c x hx,\n  -- `x` is a homomorphism `Λ →+ Lbar r' S`\n  -- we split it into pieces `Λ →+ ℤ` for all coefficients indexed by `s` and `n`\n  let x' : S → ℕ → Λ →+ ℤ := λ s n, (Lbar.coeff s n).comp x,\n  have := λ s n, hA (x' s n), clear hA,\n  choose x₁' hx₁' x₀' hx₀' H using this,\n  have hx₀_aux : ∀ s n i, (x₀' s n (l i)).nat_abs ≤ (x (l i) s n).nat_abs :=\n    (λ s n i, le_trans (le_add_right (nat.le_mul_of_pos_left hN.1)) (H s n i).ge),\n  -- now we assemble `x₀' : S → ℕ → Λ →+ ℤ` into a homomorphism `Λ →+ Lbar r' S`\n  let x₀ : Λ →+ Lbar r' S := Lbar.mk_aux hl x x₀' hx₀_aux,\n  have hx₀ : x₀ ∈ filtration (Λ →+ Lbar r' S) (c / N) :=\n    Lbar.mk_aux_mem_filtration _ _ _ hN.1 hl hx x₀' x₁' x' (λ _ _ _, rfl) H hx₀_aux,\n  -- and similarly for `x₁'`\n  let x₁ : Λ →+ Lbar r' S := Lbar.mk_aux hl x x₁'\n    (λ s n i, le_trans (le_add_left le_rfl) (H s n i).ge),\n  -- `x₁` can be written as a sum of tensors\n  -- `x₁ = ∑ a in A, a ⊗ xₐ`, for some `xₐ : Lbar r' S`\n  let xₐ : (Λ →+ ℤ) → Lbar r' S := λ a, Lbar.mk_of_add_monoid_hom x₁' a,\n  have hx₁ : ∀ l, x₁ l = ∑ a in A, a l • xₐ a := lem_98_aux _ _ hx₁' _ (λ _ _ _, rfl),\n  -- now it is time to start building `y`\n  -- we first decompose the `xₐ a` into `N` pieces\n  have hxₐ : ∀ a s n, (xₐ a s n).nat_abs ≤ 1,\n  { intros a s n, dsimp [xₐ, Lbar.mk_of_add_monoid_hom_to_fun], split_ifs; simp },\n  have := λ a, lem98_int N hN.1 ∥xₐ a∥₊ (xₐ a) _ (hxₐ a),\n  swap 2, { rw Lbar.mem_filtration_iff },\n  choose y' hy'1 hy'2 using this,\n  -- the candidate `y` combines `x₀` together with the pieces `y'` of `xₐ a`\n  let y : fin N → Λ →+ Lbar r' S := λ j, x₀ + ∑ a in A, Lbar.mk_tensor a (y' a j),\n  have hxy : x = ∑ i, y i,\n  { apply lem98_aux' N A x x₀ x₁ x' x₀' x₁' hx₀' _ _ _ hx₁ y' hy'1 y,\n    all_goals { intros, refl } },\n  use [y, hxy],\n  intro j,\n  rw hl.add_monoid_hom_mem_filtration_iff at hx ⊢,\n  intro i,\n  specialize hx i,\n  simp only [Lbar.mem_filtration_iff] at hx hy'2 ⊢,\n  have Hx : ∥x (l i)∥₊ = N • ∥x₀ (l i)∥₊ + ∑ a in A, ∥a (l i)∥₊ * ∥xₐ a∥₊,\n  { apply lem98_crux N hN.1 A x x₀ x₁ x' x₀' x₁' xₐ,\n    all_goals { intros, refl <|> apply_assumption } },\n  calc ∥y j (l i)∥₊\n      ≤ ∥x₀ (l i)∥₊ + ∑ a in A, ∥a (l i)∥₊ * ∥xₐ a∥₊ / N + d * ∥l i∥₊ : _\n  ... = (N • ∥x₀ (l i)∥₊ + ∑ a in A, ∥a (l i)∥₊ * ∥xₐ a∥₊) / N + d * ∥l i∥₊ : _\n  ... = ∥x (l i)∥₊ / N + d * ∥l i∥₊ : by rw Hx\n  ... ≤ _ : _,\n  { simp only [add_monoid_hom.add_apply, add_assoc, add_monoid_hom.finset_sum_apply,\n         Lbar.mk_tensor_apply],\n    refine (Lbar.nnnorm_add_le _ _).trans (add_le_add le_rfl _),\n    refine (Lbar.nnnorm_sum_le _ _).trans _,\n    calc ∑ a in A, ∥a (l i) • y' a j∥₊\n        = ∑ a in A, ∥a (l i)∥₊ * ∥y' a j∥₊ : finset.sum_congr rfl _\n    ... ≤ ∑ a in A, ∥a (l i)∥₊ * (∥xₐ a∥₊ / N + 1) : finset.sum_le_sum _\n    ... = ∑ a in A, (∥a (l i)∥₊ * ∥xₐ a∥₊ / N + ∥a (l i)∥₊) : finset.sum_congr rfl _\n    ... = ∑ a in A, (∥a (l i)∥₊ * ∥xₐ a∥₊ / N) + ∑ a in A, ∥a (l i)∥₊ : _\n    ... ≤ ∑ a in A, (∥a (l i)∥₊ * ∥xₐ a∥₊ / N) + d * ∥l i∥₊ : add_le_add le_rfl _,\n    { intros a ha, rw Lbar.nnnorm_zsmul },\n    { intros a ha, exact mul_le_mul' le_rfl (hy'2 a j) },\n    { intros a ha, rw [mul_add, mul_one, mul_div_assoc] },\n    { rw finset.sum_add_distrib },\n    { calc ∑ a in A, ∥a (l i)∥₊\n          = (∑ a in A, ∥a (l i)∥₊ / ∥l i∥₊) * ∥l i∥₊ : _\n      ... ≤ finset.univ.sup (λ i, ∑ a in A, ∥a (l i)∥₊ / ∥l i∥₊) * ∥l i∥₊ : _,\n      { { have : ∥l i∥₊ ≠ 0, { simp only [hl' i, nnnorm_eq_zero, ne.def, not_false_iff] },\n          simp only [div_eq_mul_inv, ← finset.sum_mul, inv_mul_cancel_right₀ this] } },\n      { exact mul_le_mul' (finset.le_sup (finset.mem_univ i)) le_rfl } } },\n  { simp only [div_eq_mul_inv, add_mul, finset.sum_mul, nsmul_eq_mul],\n    congr' 2,\n    rw [mul_comm, inv_mul_cancel_left₀],\n    exact_mod_cast hN.1.ne' },\n  { simp only [add_mul, div_eq_mul_inv],\n    refine add_le_add _ le_rfl,\n    rw [mul_right_comm],\n    exact mul_le_mul' hx le_rfl }\nend\n.\n\n#lint- only unused_arguments\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/combinatorial_lemma/finite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3221610130544784}}
{"text": "import Std.Tactic.GuardExpr\nimport Mathlib.Tactic.SimpIntro\n\nexample : x + 0 = y → x = y := by\n  simp_intro\n  guard_target = x = y → x = y\n  exact id\n\nexample : x + 0 = y → x = y := by simp_intro h₁\nexample : x + 0 ≠ y → x ≠ y := by simp_intro h₁ h₂ -- h₂ is bound but not needed\nexample : x + 0 ≠ y → x ≠ y := by simp_intro h₁ h₂ h₃ -- h₃ is not bound\n\nexample (h : x = z) : x + 0 = y → x = z := by simp_intro [h]\n\nexample (h : y = z) : x + 0 = y → x = z := by\n  simp_intro\n  guard_target = x = y → x = z\n  simp_intro .. [h]\n\nexample (h : y = z) : x + 0 = y → x = z := by simp_intro _; exact h\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/simp_intro.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.32216100508746953}}
{"text": "-- import pSet_ordinal bvm bvm_extras cantor_space\n\n-- open ordinal cardinal lattice bSet\n\n-- noncomputable theory\n\n-- local attribute [instance] classical.prop_decidable\n\n-- local attribute [simp] omega_le_aleph\n\n-- local infix ` ⟹ `:65 := lattice.imp\n\n-- local infix ` ⇔ `:50 := lattice.biimp\n\n-- local prefix `#`:70 := cardinal.mk\n\n-- universe u\n\n-- namespace bSet\n-- section cardinal_preservation\n-- local notation `ω` := cardinal.omega\n-- variables {𝔹 : Type u} [I : nontrivial_complete_boolean_algebra 𝔹]\n\n-- include I\n-- lemma AE_of_check_larger_than_check (x y : pSet.{u}) {f : bSet 𝔹} {Γ}\n--   (H : Γ ≤ (is_func f) ⊓ ⨅v, v ∈ᴮ y̌ ⟹ ⨆w, w ∈ᴮ x̌ ⊓ pair w v ∈ᴮ f) (h_nonzero : ⊥ < Γ) :\n--   ∀ i : y.type, ∃ j : x.type, ⊥ < (is_func f) ⊓ (pair ((x.func j)̌ ) ((y.func i)̌ )) ∈ᴮ f :=\n-- begin\n--   intro i_v, bv_split_at H, replace H_1_1 := H_1_1 ((y.func i_v)̌ ), simp[check_mem'] at H_1_1,\n--   have H' : Γ ≤ is_func f ⊓ ⨆ (w : bSet 𝔹), w ∈ᴮ x̌  ⊓ pair w (pSet.func y i_v̌)  ∈ᴮ f,\n--     from context_and_intro ‹_› ‹_›,\n--   rw[inf_supr_eq] at H',\n--   replace H' := le_trans H' (by {apply supr_le, intro i, recover, show 𝔹,\n--     from ⨆ (i : bSet 𝔹), i ∈ᴮ x̌ ⊓ (is_func f ⊓ pair i (pSet.func y i_v̌)  ∈ᴮ f),\n--     apply bv_use i, apply le_of_eq, ac_refl}),\n--   replace H' := lt_of_lt_of_le h_nonzero H',\n--   have := @bounded_exists 𝔹 _ (x̌) (λ z, is_func f ⊓ pair z ((y.func i_v)̌ ) ∈ᴮ f),\n--   rw[<-this] at H', swap,\n--     {intros x' y',\n--     apply poset_yoneda, intros Γ_1 a,\n--     simp only [le_inf_iff] at a H ⊢, cases a, cases H, cases a_right, refine ⟨‹_›, _⟩,\n--     have : Γ_1 ≤ pair x' ((y.func i_v)̌ ) =ᴮ pair y' ((y.func i_v)̌ ),\n--      from subst_congr_pair_left' ‹_›, apply subst_congr_mem_left'; from ‹_›},\n--     {cases x, cases y, convert nonzero_wit H', ext1,\n--       dsimp with cleanup, rw[top_inf_eq]}\n-- end\n\n-- variables\n--   (η₁ η₂ : pSet.{u}) (H_infinite : ω ≤ #(η₁.type))\n--   (H_lt : #(η₁.type) < #(η₂.type))\n--   (H_inj₂ : ∀ x y, x ≠ y → ¬ pSet.equiv (η₂.func x) (η₂.func y))\n--   (f : bSet 𝔹) (g : η₂.type → η₁.type)\n--   (H : ∀ β : η₂.type, (⊥ : 𝔹) < is_func f ⊓ pair ((η₁.func (g β)̌ ) ) ((η₂.func β)̌ )∈ᴮ f)\n\n-- include H_infinite H_lt H_inj₂ f H\n-- lemma not_CCC_of_uncountable_fiber (H_ex : ∃ ξ : η₁.type, ω < #(g⁻¹' {ξ})) : ¬ CCC 𝔹 :=\n-- begin\n--   cases H_ex with ξ H_ξ,\n--   let 𝓐 : (g⁻¹'{ξ}) → 𝔹 :=\n--     λ β, is_func f ⊓ (pair ((η₁.func (g β.val))̌ ) ((η₂.func β.val)̌ )) ∈ᴮ f,\n--   have 𝓐_nontriv : ∀ β, ⊥ < 𝓐 β,\n--     from λ _, by apply H,\n--   have 𝓐_anti : ∀ β₁ β₂, β₁ ≠ β₂ → (𝓐 β₁) ⊓ (𝓐 β₂) ≤ ⊥,\n--     by {intros β₁ β₂ h_sep, dsimp[𝓐],\n--     /- `tidy_context` says -/ apply poset_yoneda, intros Γ a,\n--     cases β₂, cases β₁, cases H_ξ, cases H_lt, cases β₁_property, cases β₂_property,\n--     work_on_goal 0 { induction β₂_property, simp only [le_inf_iff] at a,\n--                      cases a, cases a_right, cases a_left },\n--     work_on_goal 1 { induction β₁_property, simp only [le_inf_iff] at a,\n--                      cases a, cases a_right, cases a_left, solve_by_elim },\n--     work_on_goal 1 { cases β₂_property,\n--       work_on_goal 0 { induction β₂_property, simp only [le_inf_iff] at a,\n--         cases a, cases a_right, cases a_left, solve_by_elim}, simp only [le_inf_iff] at a,\n--         cases a, cases a_right, cases a_left, solve_by_elim},\n\n--     rw[β₁_property] at a_left_right,\n--     have H_le_eq : Γ ≤ ((η₂.func β₁_val)̌ ) =ᴮ ((η₂.func β₂_val)̌ ),\n--      by {apply funext; from ‹_›},\n--     from le_trans H_le_eq\n--            (by {rw[le_bot_iff], apply check_bv_eq_bot_of_not_equiv, apply H_inj₂, tidy})},\n--    intro H_CCC, specialize H_CCC (g⁻¹'{ξ}) ‹_› ‹_› ‹_›,\n--    replace H_ξ := (lt_iff_le_and_ne.mp H_ξ),\n--    from absurd (le_antisymm H_ξ.left H_CCC) H_ξ.right\n-- end\n\n-- end cardinal_preservation\n-- end bSet\n\n\n-- open bSet pSet\n\n\n-- namespace cohen_algebra\n\n-- section cohen_algebra\n-- variables (κ : cardinal.{u})\n\n-- instance H_nonempty' : nonempty (set $ (card_ex κ).type × ℕ) := ⟨∅⟩\n\n-- def cohen_algebra := @regular_opens (set ((card_ex κ).type × ℕ)) (Pi.topological_space)\n\n-- @[instance, priority 1000]def cohen_algebra_boolean_algebra : nontrivial_complete_boolean_algebra (cohen_algebra κ) :=\n-- regular_open_algebra (by apply_instance)\n\n-- lemma le_iff_subset'' {x y : (cohen_algebra κ)} : x ≤ y ↔ x.1 ⊆ y.1 := by refl\n\n-- lemma bot_eq_empty : (⊥ : (cohen_algebra κ)) = ⟨∅, is_regular_empty⟩ := rfl\n\n-- variable{κ}\n-- lemma eq₀ : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type = ((card_ex κ)).type := by cases (card_ex κ); refl\n\n\n-- lemma eq₁ : ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = ((type (card_ex κ)) × ℕ) :=\n-- by {cases (card_ex κ), refl}\n\n\n-- lemma eq₂ : set ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = set ((type (card_ex κ)) × ℕ) :=\n-- by {cases (card_ex κ), refl}\n\n\n-- lemma eq₃ : finset ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = finset (type (card_ex κ) × ℕ) :=\n-- by {cases (card_ex κ), refl}\n\n\n-- lemma pi₂_cast₁ {α β γ : Type*} (H' : α = β) {p : α × γ} {q : β × γ} (H : p == q) :\n--   p.1 == q.1 :=\n-- by {subst H', subst H}\n\n\n-- lemma pi₂_cast₂ {α β γ : Type*} (H' : α = β) {p : α × γ} {q : β × γ} (H : p == q) :\n--   p.2 = q.2 :=\n-- by {subst H', subst H}\n\n-- lemma compl_cast₂ {α β : Type*} {a : set α} {b : set β} (H' : α = β) (H : -a == b) : a == -b :=\n-- begin\n--   subst H', subst H, apply heq_of_eq, simp\n-- end\n\n-- lemma eq₁_cast (p : ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ)) {prf : ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = (((type (card_ex κ)) × ℕ))} {prf' : (type ((card_ex κ)̌  : bSet (cohen_algebra κ))) = ((card_ex κ).type)} : cast prf p = (cast prf' p.1, p.2) :=\n-- begin\n--   ext, swap, simp, h_generalize H_x : p == x, apply pi₂_cast₂, from (eq₀).symm, from H_x.symm,\n--   h_generalize H_x : p == x, simp, h_generalize H_y : p.fst == y,\n--   apply eq_of_heq, suffices : x.fst == p.fst, from heq.trans this H_y,\n--   apply pi₂_cast₁, from (eq₀).symm, from H_x.symm\n-- end\n\n-- -- lemma eq₁_cast' {ξ : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type} {n : ℕ} {prf : ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = (((type (card_ex κ)) × ℕ))} {prf' : (type ((card_ex κ)̌  : bSet (cohen_algebra κ))) = ((card_ex κ).type)} : cast prf (ξ, n) = (cast prf' ξ, n) :=\n-- -- by apply eq₁_cast\n\n-- lemma eq₁_cast' (p : (((type (card_ex κ)) × ℕ))) {prf : ((type ((card_ex κ)̌  : bSet (cohen_algebra κ))) × ℕ) = (((type (card_ex κ)) × ℕ))} {prf' : (type ((card_ex κ)̌  : bSet (cohen_algebra κ))) = ((card_ex κ).type)} : cast prf.symm p = (cast prf'.symm p.1, p.2) :=\n-- begin\n--   ext, swap, simp, h_generalize H_x : p == x, apply pi₂_cast₂, from eq₀, from H_x.symm,\n--   h_generalize H_x : p == x, simp, h_generalize H_y : p.fst == y,\n--   apply eq_of_heq, suffices : x.fst == p.fst, from heq.trans this H_y,\n--   apply pi₂_cast₁, from eq₀, from H_x.symm\n-- end\n\n-- theorem cohen_algebra_CCC : CCC (cohen_algebra κ):=\n-- by { apply CCC_regular_opens, apply cantor_space.countable_chain_condition_set }\n\n\n\n-- local notation `𝒳` := set((card_ex κ).type × ℕ)\n\n-- open topological_space\n\n\n\n-- /-- The principal regular open associated to a pair (ν, n) is the collection of all subsets of\n--     (card_ex κ) × ℕ which contain (ν, n). -/\n-- variable (κ)\n-- def principal_open (ν : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type) (n : ℕ) : (cohen_algebra κ) :=\n-- begin\n--   use (cantor_space.principal_open (cast (eq₁) (ν, n))),\n--   from is_regular_of_clopen (cantor_space.is_clopen_principal_open)\n-- end\n\n-- variable {κ}\n-- lemma is_clopen_principal_open {ν n} : is_clopen (principal_open κ ν n).val :=\n--   cantor_space.is_clopen_principal_open\n\n-- local postfix `ᵖ`:80 := perp\n\n-- local notation `cl`:65 := closure\n\n-- local notation `int`:65 := interior\n\n-- lemma perp_eq_compl_of_clopen {β : Type*} [topological_space β] {S : set β} (H : is_clopen S) : Sᵖ = (-S) :=\n-- by {unfold perp, rw[closure_eq_of_is_closed H.right]}\n\n-- lemma mem_neg_principal_open_of_not_mem {ν n S} : (cast (eq₁) (ν,n) ∈ (-S)) → S ∈ (- (principal_open κ ν n)).val :=\n-- begin\n--   intro H, simp only [neg_unfold], rw[perp_eq_compl_of_clopen],\n--   swap, from is_clopen_principal_open, from H\n-- end\n\n-- variable (κ)\n-- structure cohen_poset  : Type u :=\n-- (ins : finset (((card_ex κ) ̌ : bSet (cohen_algebra κ)).type × ℕ))\n-- (out : finset (((card_ex κ) ̌ : bSet (cohen_algebra κ)).type × ℕ))\n-- (H : ins ∩ out = ∅)\n\n-- variable{κ}\n\n-- @[reducible]def π₂ : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type × ℕ → ℕ := λ x, x.snd\n\n-- -- def nat_supp : finset (((card_ex κ) ̌ : bSet (cohen_algebra κ)).type × ℕ) → set ℕ :=\n-- -- λ X, {n | ∃ (ξ : (card_ex κ).type), (cast eq₁.symm (ξ,n)) ∈ X}\n\n-- -- lemma nat_supp_finite {X} : set.finite $ nat_supp X := sorry\n\n-- def cohen_poset_inc : cohen_poset κ → (cohen_algebra κ) :=\n-- λ p, ⟨{S | (p.ins.to_set) ⊆ (cast (eq₂).symm S) ∧\n--            (p.out.to_set) ⊆ (cast (eq₂).symm (- S))},\n-- is_regular_of_clopen\n--      begin\n--        change is_clopen\n--          ({S | p.ins.to_set ⊆ cast (eq₂).symm S} ∩ {S | p.out.to_set ⊆ (cast (eq₂).symm (-S))}),\n--        refine is_clopen_inter _ _,\n--          have := cantor_space.is_clopen_principal_open_finset p.ins,\n--          convert this, from (eq₀).symm, from (eq₀).symm, from (eq₀).symm,\n--            {apply function.hfunext, from (eq₂).symm, intros a a' H_heq,\n--              apply heq_of_eq, convert rfl, convert (cast_eq _ _).symm, from (eq₀).symm, refl},\n\n--          have := cantor_space.is_clopen_co_principal_open_finset p.out,\n--          convert this, from (eq₀).symm, from (eq₀).symm, from (eq₀).symm,\n--          {apply function.hfunext, from (eq₂).symm, intros a a' H_heq,\n--           apply heq_of_eq, convert rfl, h_generalize Hx : (-a) == x,\n--           have := heq.subst H_heq, swap,\n--           from λ _ y, y == -x,\n--           suffices : a' = -x, by {rw[this], simp},\n--           apply eq_of_heq, apply this, apply compl_cast₂, from (eq₁).symm,\n--           from Hx}\n--      end⟩\n\n-- open cantor_space\n\n-- lemma prop_decidable_cast_lemma {α β : Type*} (H : α = β) {a b : α} {a' b' : β} (H_a : a == a') (H_b : b == b') : classical.prop_decidable (a = b) == classical.prop_decidable (a' = b') :=\n-- by {subst H, subst H_a, subst H_b}\n\n-- lemma cohen_poset_dense_basis : ∀ T ∈ @standard_basis ((card_ex κ).type × ℕ), ∀ h_nonempty : T ≠ ∅,\n--   ∃ p : cohen_poset κ, (cohen_poset_inc p).val ⊆ T :=\n-- begin\n--   intros T Ht H_nonempty, simp[standard_basis] at Ht,\n--   cases Ht with H_empty Ht, contradiction,\n--   rcases Ht with ⟨p_ins, p_out, H₁, H₂⟩,\n--   fsplit, refine ⟨_,_,_⟩, from cast eq₃.symm p_ins,\n--   from cast eq₃.symm p_out, swap, rw[<-co_principal_open_finset_eq_inter] at H₁,\n--   rw[<-principal_open_finset_eq_inter] at H₁, subst H₁,\n--   intros S HS, split, cases HS, dsimp at HS_left, simp[principal_open_finset],\n--   {convert HS_left,\n--     from eq₀.symm, from eq₀.symm, from eq₀.symm, all_goals{symmetry, from cast_heq _ _}},\n--   cases HS, dsimp at HS_right, simp[principal_open_finset],\n--   {convert HS_right,\n--     from eq₀.symm, from eq₀.symm, from eq₀.symm, all_goals{symmetry, from cast_heq _ _}},\n--   convert H₂, from eq₀, from eq₀, from eq₀,\n--   apply function.hfunext, from (eq₁), intros a a' H,\n--   apply function.hfunext, from (eq₁), intros b b' H',\n--   from prop_decidable_cast_lemma (eq₁) ‹_› ‹_›,\n--   from cast_heq _ _, from cast_heq _ _, from eq₀, from eq₀\n-- end\n\n-- lemma cohen_poset_dense {b : (cohen_algebra κ)} (H : ⊥ < b) : ∃ p : cohen_poset κ, cohen_poset_inc p ≤ b :=\n-- begin\n--   cases (classical.choice (classical.nonempty_of_not_empty _ H.right.symm)) with S_wit H_wit,\n--   change ∃ p, (cohen_poset_inc p).val ⊆ b.val,\n--   have := mem_basis_subset_of_mem_open (is_topological_basis_standard_basis) H_wit (is_open_of_is_regular b.property),\n--   rcases (mem_basis_subset_of_mem_open\n--            (is_topological_basis_standard_basis) H_wit (is_open_of_is_regular b.property))\n--          with ⟨v, Hv₁, Hv₂, Hv₃⟩,\n--   have : v ≠ ∅, by {intro H, rw[H] at Hv₂, cases Hv₂},\n--   cases (cohen_poset_dense_basis ‹_› ‹_› ‹_›) with p H_p, from ⟨p, set.subset.trans H_p ‹_›⟩\n-- end\n\n-- lemma to_set_inter {α : Type*} {p₁ p₂ : finset α} : (p₁ ∩ p₂).to_set = (p₁.to_set ∩ p₂.to_set) :=\n-- by {ext, split; intros; unfold finset.to_set at *, tidy}\n\n-- @[simp]lemma to_set_empty {α : Type*} : finset.to_set (∅ : finset α) = ∅ :=\n-- by {unfold finset.to_set, refl}\n\n-- lemma not_mem_of_inter_empty_left {α : Type*} {p₁ p₂ : finset α}\n--   (H : p₁ ∩ p₂ = ∅) {a : α} : a ∈ p₁.to_set → ¬ a ∈ p₂.to_set :=\n-- begin\n--   intro H', intro H'',\n--   have this₀ : a ∈ p₁.to_set ∩ p₂.to_set := ⟨‹_›,‹_›⟩,\n--   rw[<-to_set_inter] at this₀, have this₁ := congr_arg finset.to_set H,\n--   rw[this₁] at this₀, cases this₀\n-- end\n\n-- lemma not_mem_of_inter_empty_right {α : Type*} {p₁ p₂ : finset α}\n--   (H : p₂ ∩ p₁ = ∅) {a : α} : a ∈ p₁.to_set → ¬ a ∈ p₂.to_set :=\n-- by {rw[finset.inter_comm] at H, apply not_mem_of_inter_empty_left, from ‹_›}\n\n-- lemma cohen_poset_nonzero (p : cohen_poset κ) : ⊥ ≠ (cohen_poset_inc p) :=\n-- begin\n--   intro H, replace H := H.symm, rw[eq_bot_iff] at H, rw[le_iff_subset''] at H,\n--   rw[bot_eq_empty] at H,\n--   suffices : nonempty (cohen_poset_inc p).val,\n--     by {have := classical.choice this, specialize H this.property, cases H},\n--   apply nonempty.intro, fsplit, exact (cast (eq₂) p.ins.to_set),\n--   split, finish, intro x, cases x with ν n, intro H,\n--   suffices : cast (eq₁) (ν, n) ∈ - cast (eq₂) (p.ins).to_set,\n--     {convert this, from eq₀, from eq₀, from eq₀, cc, cc},\n--   suffices : (ν, n) ∈ - p.ins.to_set,\n--     {convert this, from eq₀.symm, from eq₀.symm, from eq₀.symm, cc, from eq₀.symm,\n--      from eq₀.symm, from eq₀.symm, from eq₀.symm, cc},\n--   from not_mem_of_inter_empty_right p.H H\n-- end\n\n-- lemma subset_of_eq {α : Type*} {a b : finset α} (H : a = b) : a ⊆ b := by rw[H]; refl\n\n-- lemma cohen_poset_disjoint_row (p : cohen_poset κ) : ∃ n : ℕ, ∀ ξ : (card_ex κ).type, (cast (eq₁).symm (ξ,n)) ∉ p.ins ∧ (cast (eq₁).symm (ξ,n)) ∉ p.out :=\n-- begin\n--   let Y := (finset.image π₂ p.ins) ∪ (finset.image π₂ p.out),\n--   by_cases (p.ins ∪ p.out) = ∅,\n--   use 0, intro ξ, split, intro x, apply (subset_of_eq h), simp, left, from x,\n--   intro x, apply (subset_of_eq h), simp, right, from x,\n--   let Y' := finset.image π₂ (p.ins ∪ p.out),\n--   have Y'_nonempty : Y' ≠ ∅,\n--     by {dsimp[Y'], intro H, apply h, ext; split; intros, swap, cases a_1,\n--       have : π₂ a ∈ finset.image π₂ (p.ins ∪ p.out), simp,\n--       use a.fst, simp at a_1, convert a_1, cases a, refl, cases a, refl,\n--       rw[H] at this, cases this},\n--   have := finset.max_of_ne_empty,\n--   specialize this Y'_nonempty, cases this with N HN, swap, apply_instance,\n--   use (N+1), intro ξ, split,\n--     intro X, let prf := _, change cast prf (ξ, N + 1) ∈ p.ins at X,\n--     rw[eq₁_cast'] at X, swap, from eq₀,\n--     have : N + 1 ∈ Y',\n--       by {simp, use cast eq₀.symm ξ, from or.inl X},\n--     suffices : N + 1 ≤ N, by {revert this, change ¬ (N + 1 ≤ N), apply nat.not_succ_le_self},\n--     apply finset.le_max_of_mem this ‹_›,\n--   intro X, let prf := _, change cast prf (ξ, N + 1) ∈ p.out at X,\n--     rw[eq₁_cast'] at X, swap, from eq₀,\n--     have : N + 1 ∈ Y',\n--       by {simp, use cast eq₀.symm ξ, from or.inr X},\n--     suffices : N + 1 ≤ N, by {revert this, change ¬ (N + 1 ≤ N), apply nat.not_succ_le_self},\n--     apply finset.le_max_of_mem this ‹_›\n-- end\n\n-- lemma cohen_poset_anti {p₁ p₂ : cohen_poset κ} : p₁.ins ⊆ p₂.ins → p₁.out ⊆ p₂.out → cohen_poset_inc p₂ ≤ cohen_poset_inc p₁  :=\n-- by {intros H₁ H₂, rw[le_iff_subset''], tidy}\n\n-- end cohen_algebra\n-- end cohen_algebra\n\n-- namespace cohen_real\n\n-- section cohen_real\n-- variables (κ : cardinal.{u})\n-- open cohen_algebra\n\n-- -- attribute [instance, priority 0] 𝔹_boolean_algebra\n\n-- -- variable [σ : nontrivial_complete_boolean_algebra 𝔹]\n\n-- -- attribute [instance, priority 1000] σ\n-- -- include σ\n-- /-- `cohen_real.χ ν` is the indicator function on ℕ induced by every ordinal less than (card_ex κ) -/\n-- def χ (ν : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type) : ℕ → (cohen_algebra κ) :=\n--   λ n, principal_open κ ν n\n\n-- /-- `cohen_real.mk ν` is the subset of (ω : bSet (cohen_algebra κ)) induced by `cohen_real.χ ν` -/\n-- def mk (ν : ((card_ex κ)̌  : bSet (cohen_algebra κ)).type) : bSet (cohen_algebra κ) :=\n--   @set_of_indicator (cohen_algebra κ) _ omega $ λ n, χ κ ν n.down\n\n\n-- variable {κ}\n-- @[simp, cleanup]lemma mk_type {ν} : (mk κ ν).type = ulift ℕ := rfl\n\n-- @[simp, cleanup]lemma mk_func {ν} {n} : (mk κ ν).func n = bSet.of_nat (n.down) := rfl\n\n-- @[simp, cleanup]lemma mk_bval {ν} {n} : (mk κ ν).bval n = (χ κ ν) (n.down) := rfl\n\n-- /-- bSet (cohen_algebra κ) believes that each `mk κ ν` is a subset of omega -/\n-- lemma definite {ν} {Γ} : Γ ≤ mk κ ν ⊆ᴮ omega :=\n-- by simp [mk, subset_unfold]; from λ _, by rw[<-deduction]; convert omega_definite\n\n-- /-- bSet (cohen_algebra κ) believes that each `mk κ ν` is an element of 𝒫(ω) -/\n-- lemma definite' {ν} {Γ} : Γ ≤ mk κ ν ∈ᴮ bv_powerset omega := bv_powerset_spec.mp definite\n\n-- -- TODO(jesse) refactor this proof to use axiom of extensionality instead, or prove a more general version\n\n-- lemma sep {n} {Γ} {ν₁ ν₂} (H₁ : Γ ≤ (of_nat n) ∈ᴮ (mk κ ν₁)) (H₂ : Γ ≤ (- ((of_nat n) ∈ᴮ (mk κ ν₂)))) :\n--   Γ ≤ (- ((mk κ ν₁) =ᴮ (mk κ ν₂))) :=\n-- begin\n--   rw[bv_eq_unfold], rw[neg_inf, neg_infi, neg_infi], simp only [neg_imp],\n--   refine le_sup_left_of_le _, rw[@bounded_exists (cohen_algebra κ) _ (mk κ ν₁) (λ z, -(z ∈ᴮ mk κ ν₂)) _],\n--   swap, change B_ext _, simp[-imp_bot, imp_bot.symm],\n--   apply bv_use (bSet.of_nat n), bv_split_goal\n-- end\n\n-- lemma not_mem_of_not_mem {p : cohen_poset κ} {ν} {n} (H : (ν,n) ∈ p.out) : cohen_poset_inc p ≤ -( (of_nat n) ∈ᴮ (mk κ ν)) :=\n-- begin\n-- rw[mem_unfold, neg_supr], bv_intro k, rw[neg_inf], simp,\n--        by_cases n = k.down, swap, rw[bSet.of_nat_inj ‹_›],\n--        from le_sup_right_of_le (by simp),\n--        refine le_sup_left_of_le _, rw[<-h],\n--        rw[le_iff_subset''], unfold cohen_poset_inc χ, rintros S ⟨H_S₁, H_S₂⟩,\n--        apply mem_neg_principal_open_of_not_mem, have := H_S₂ H, convert this,\n--        from eq₀.symm, from eq₀.symm, from eq₀.symm,\n--        from cast_heq _ _, from (cast_heq _ _).symm\n-- end\n\n-- private lemma inj_cast_lemma (ν' : type ((card_ex κ)̌  : bSet (cohen_algebra κ))) (n' : ℕ) :\n--   cast eq₁.symm (cast eq₀ ν', n') = (ν', n') :=\n-- begin\n--   let a := _, change cast a _ = _,\n--   let b := _, change cast _ (cast b _, _) = _,\n--   simp[b] at a, dedup, change cast a_1 _ = _, cc\n-- end\n\n-- /-- Whenever ν₁ ≠ ν₂ < (card_ex κ), bSet (cohen_algebra κ) believes that `mk κ ν₁` and `mk κ ν₂` are distinct -/\n-- lemma inj {ν₁ ν₂} (H_neq : ν₁ ≠ ν₂) : (mk κ ν₁) =ᴮ (mk κ ν₂) ≤ (⊥ : (cohen_algebra κ)) :=\n-- begin\n--   by_contra, replace h := (bot_lt_iff_not_le_bot.mpr ‹_›),\n--   cases cohen_poset_dense h with p H_p, cases cohen_poset_disjoint_row p with n H_n,\n--   let p' : cohen_poset κ := { ins := insert (ν₁,n) (p.ins),\n--   out := insert (ν₂,n) p.out,\n--   H := by {ext, split; intro H, swap, cases H, have := p.H, simp at H, cases a_1 with ν' n',\n--            cases H with H₁ H₂, specialize H_n (cast eq₀ ν'), cases H_n, cases H₁; cases H₂, cc,\n--            exfalso, apply H_n_right, convert H₂, rw[show n = n', by cc], apply inj_cast_lemma,\n--            exfalso, apply H_n_left, convert H₁, rw[show n = n', by cc], apply inj_cast_lemma,\n--            rw[<-this], simp[*,-this]} },\n--   have this₀ : cohen_poset_inc p' ≤ cohen_poset_inc p,\n--     from cohen_poset_anti (by {dsimp[p'], from λ i _, by {simp, from or.inr ‹_›}})\n--                 (by {dsimp[p'], from λ i _, by {simp, from or.inr ‹_›}}),\n--   have this₁ : cohen_poset_inc p' ≤ (ñ̌) ∈ᴮ (mk κ ν₁),\n--     by {rw[mem_unfold], apply bv_use (ulift.up n), refine le_inf _ bv_refl,\n--          {simp [le_iff_subset'', χ, principal_open, cohen_poset_inc, cantor_space.principal_open],\n--          have : (ν₁, n) ∈ p'.ins,\n--            by simp[p'], intros S H_S _, specialize H_S this,\n--               convert H_S; [from eq₀.symm, from eq₀.symm, from eq₀.symm, cc, cc]}},\n--   have this₂ : cohen_poset_inc p' ≤ - ((ñ̌) ∈ᴮ (mk κ ν₂)),\n--     by {have : (ν₂, n) ∈ p'.out, by {simp[p']},\n--        from not_mem_of_not_mem ‹_›},\n--   have this₃ : cohen_poset_inc p' ≤ - (mk κ ν₁ =ᴮ mk κ ν₂),\n--     from sep ‹_› ‹_›,\n--   have this₄ : cohen_poset_inc p' ≤ (mk κ ν₁ =ᴮ mk κ ν₂),\n--     from le_trans this₀ ‹_›,\n--   suffices : cohen_poset_inc p' = ⊥, from absurd this.symm (cohen_poset_nonzero p'),\n--   bv_and_intro this₃ this₄, simpa using H\n-- end\n-- end cohen_real\n-- end cohen_real\n\n-- section neg_CH\n-- variables (κ₁ κ₂ : cardinal.{u}) (H_reg₁ : is_regular κ₁) (H_reg₂ : is_regular κ₂) (H_inf₁ : cardinal.omega < κ₁) (H_inf₂ : cardinal.omega < κ₂) (H_lt : κ₁ < κ₂)\n\n-- open cohen_algebra\n\n-- local notation `ℵ₀` := (omega : bSet (cohen_algebra κ₂))\n\n-- local notation `𝔠` := (bv_powerset ℵ₀)\n\n-- local infix `≺`:75 := (λ x y, -(larger_than x y))\n\n-- local infix `≼`:75 := (λ x y, injects_into x y)\n\n-- lemma uncountable_fiber_of_regular' (κ₁ κ₂ : cardinal) (H_inf : cardinal.omega ≤ κ₁) (H_lt : κ₁ < κ₂) (H : cof (ord κ₂) = κ₂) (α : Type u) (H_α : #α = κ₁) (β : Type u) (H_β : #β = κ₂) (g : β → α)\n--   : ∃ (ξ : α), cardinal.omega < #↥(g⁻¹' {ξ}) :=\n-- begin\n--   have := (@cardinal.exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k H_k, subst H_k,\n--   have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k',\n--   have := infinite_pigeonhole g _ _, cases this with ξ H_ξ, use ξ, rw[H_ξ],\n--   all_goals{simp*}, from lt_of_le_of_lt ‹_› ‹_›\n-- end\n\n-- lemma uncountable_fiber_of_regular (κ₁ κ₂ : cardinal) (H_inf : cardinal.omega ≤ κ₁) (H_lt : κ₁ < κ₂) (H : cof (ord κ₂) = κ₂) (g : type (pSet.ordinal.mk (ord κ₂)  : pSet.{u}) → type (pSet.ordinal.mk (ord κ₁) : pSet.{u}))\n--   : ∃ (ξ : type (pSet.ordinal.mk (ord κ₁))), cardinal.omega < #↥((λ (β : type (pSet.ordinal.mk (ord κ₂))), g β)⁻¹' {ξ}) :=\n-- begin\n--   have := (@exists_aleph κ₁).mp ‹_›, cases this with k₁ h, subst h,\n--   have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n--   subst h,\n--   from uncountable_fiber_of_regular' (aleph k₁) (aleph k₂) ‹_› ‹_› ‹_› _ (mk_type_mk_eq _ ‹_›) _ (mk_type_mk_eq _ (by simp*)) g\n-- end\n\n-- lemma cardinal_inequality_of_regular (κ₁ κ₂ : cardinal) (H_reg₁ : cardinal.is_regular κ₁) (H_reg₂ : cardinal.is_regular κ₂) (H_inf : (omega : cardinal) ≤ κ₁) (H_lt : κ₁ < κ₂) : (⊤ : (cohen_algebra κ₂)) ≤ (pSet.ordinal.mk (ord κ₁))̌  ≺ (pSet.ordinal.mk (ord κ₂))̌  :=\n-- begin\n--   simp[larger_than, -top_le_iff], rw[<-imp_bot],\n--   bv_imp_intro, bv_cases_at'' H f, by_contra,\n--   have := classical.axiom_of_choice\n--             (AE_of_check_larger_than_check _ _ H_1 (bot_lt_iff_not_le_bot.mpr ‹_›)),\n--   cases this with g g_spec,\n--   suffices : ¬ CCC (cohen_algebra κ₂), from absurd cohen_algebra_CCC this,\n--   apply not_CCC_of_uncountable_fiber; try{assumption},\n--     {have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k', simp*},\n--     {have := (@cardinal.exists_aleph κ₁).mp ‹_›, cases this with k' H_k', subst H_k', simp*,\n--      have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n--      subst h, simp*},\n--     {intros i₁ i₂ H_neq, from ordinal.mk_inj _ _ _ ‹_›},\n--     {dsimp at g,\n--      apply uncountable_fiber_of_regular' κ₁ κ₂; try{simp*},\n--      from H_reg₂.right,\n--      have := (@exists_aleph κ₂).mp (le_of_lt (lt_of_le_of_lt ‹_› ‹_›)), cases this with k₂ h,\n--      subst h, from mk_type_mk_eq _ ‹_›, from mk_type_mk_eq _ (le_of_lt (lt_of_le_of_lt ‹_› ‹_›))}\n-- end\n\n-- lemma cohen_real.mk_ext : ∀ (i j : type ((card_ex κ₂)̌  : bSet (cohen_algebra κ₂))), func ((card_ex κ₂)̌ ) i =ᴮ func ((card_ex κ₂)̌ ) j ≤\n--   (λ (x : type ((card_ex κ₂)̌ )), cohen_real.mk κ₂ x) i =ᴮ (λ (x : type ((card_ex κ₂)̌ )), cohen_real.mk κ₂ x) j :=\n-- begin\n--   intros i j, by_cases i = j,\n--    {simp[h]},\n--    {refine poset_yoneda _, intros Γ a, simp only [le_inf_iff] at *,\n--      have : func ((card_ex κ₂)̌ ) i = ((card_ex κ₂).func (check_cast i))̌ ,\n--        by simp[check_func],\n--      rw[this] at a,\n--      have : func ((card_ex κ₂)̌ ) j = ((card_ex κ₂).func (check_cast j))̌ ,\n--        by simp[check_func],\n--      rw[this] at a,\n--    suffices : func (card_ex κ₂) (check_cast i)̌  =ᴮ func (card_ex κ₂) (check_cast j)̌  ≤ ⊥,\n--      from le_trans a (le_trans this bot_le),\n--    rw[le_bot_iff], apply check_bv_eq_bot_of_not_equiv,\n--    apply ordinal.mk_inj, unfold check_cast, intro H, cc}\n-- end\n\n\n\n-- noncomputable def neg_CH_func : bSet (cohen_algebra κ₂) :=\n-- @function.mk _ _ ((card_ex κ₂)̌ ) (λ x, cohen_real.mk κ₂ x) (cohen_real.mk_ext κ₂)\n\n-- variables {κ₁ κ₂}\n-- -- def CH : (cohen_algebra κ₂) := - ⨆ x, ⨆y, (ℵ₀ ≺ x) ⊓ (x ≺ y) ⊓ (y ≼ 𝒫(ℵ₀))\n\n-- include κ₁ H_reg₁ H_inf₁\n\n-- lemma ℵ₀_lt_κ₁ : (⊤ : (cohen_algebra κ₂))  ≤ ℵ₀ ≺ (card_ex κ₁)̌  :=\n-- begin\n--   simp[larger_than, -top_le_iff], rw[<-imp_bot],\n--   bv_imp_intro, bv_cases_at'' H f, by_contra,\n--   have := classical.axiom_of_choice\n--             (AE_of_check_larger_than_check _ _ H_1 (bot_lt_iff_not_le_bot.mpr ‹_›)),\n--   cases this with g g_spec,\n--   suffices : ¬ CCC (cohen_algebra κ₂), from absurd cohen_algebra_CCC this,\n--   apply not_CCC_of_uncountable_fiber; try{assumption},\n--     {from le_of_eq (by simp)},\n--     {simp*},\n--     {intros i₁ i₂ H_neq, from ordinal.mk_inj _ _ _ ‹_›},\n--     {dsimp at g,\n--      apply uncountable_fiber_of_regular' (aleph 0) κ₁; try{simp*},\n--      from H_reg₁.right}\n-- end\n-- omit H_reg₁ H_inf₁\n\n-- theorem κ₂_le_𝔠 : (⊤ : cohen_algebra κ₂) ≤ is_func' ((card_ex κ₂)̌ ) 𝔠 (neg_CH_func κ₂) ⊓ is_inj (neg_CH_func κ₂) :=\n-- begin\n-- refine le_inf _ _,\n\n--   {unfold neg_CH_func, refine le_inf _ _, refine mk_is_func _ _,\n--     bv_intro w₁, bv_imp_intro, rw[mem_unfold] at H,\n--     bv_cases_at'' H ν, apply bv_use (cohen_real.mk κ₂ ν),\n--     refine le_inf cohen_real.definite' _, swap,\n--     rw[mem_unfold], apply bv_use ν, bv_split,\n--     from le_inf ‹_› (by apply le_trans H_1_right; from subst_congr_pair_left), refl},\n\n--   {refine mk_inj_of_inj _ _, from λ _ _ _, cohen_real.inj ‹_›},\n-- end\n\n-- include H_reg₁ H_inf₁ H_reg₂ H_inf₂ H_lt\n\n-- /-- For every pair of infinite regular cardinals κ₁ < κ₂, the continuum in bSet (cohen_algebra κ₂) is properly larger than (card_ex κ₁)̌ . -/\n-- theorem neg_CH : (⊤ : cohen_algebra κ₂) ≤ -(CH) :=\n-- begin\n--   dsimp [CH], rw[lattice.neg_neg], apply bv_use ((card_ex κ₁)̌ ),\n--   apply bv_use ((card_ex κ₂)̌ ), simp only [lattice.le_inf_iff],\n--   refine ⟨⟨ℵ₀_lt_κ₁ H_reg₁ H_inf₁,_⟩,_⟩,\n--   from  cardinal_inequality_of_regular _ _ (H_reg₁)\n--   (H_reg₂) (le_of_lt ‹_›) (‹_›),\n--   refine le_supr_of_le (neg_CH_func κ₂) _,\n--   apply κ₂_le_𝔠, from κ₂\n-- end\n\n\n\n-- end neg_CH\n\n", "meta": {"author": "flypitch", "repo": "flypitch", "sha": "aea5800db1f4cce53fc4a113711454b27388ecf8", "save_path": "github-repos/lean/flypitch-flypitch", "path": "github-repos/lean/flypitch-flypitch/flypitch-aea5800db1f4cce53fc4a113711454b27388ecf8/old/forcing2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.3220556577548419}}
{"text": "import Kenny.sheaf_of_rings_on_opens\n\nuniverses v w u₁ v₁ u\n\nopen topological_space lattice\n\nstructure locally_ringed_space_on_opens (X : Type u) [topological_space X] (U : opens X) : Type (max u (v+1)) :=\n(O : sheaf_of_rings_on_opens.{v} X U)\n(Hstalks : ∀ x ∈ U, is_local_ring (O.to_sheaf_on_opens.stalk x H))\n\nnamespace locally_ringed_space_on_opens\n\nvariables {X : Type u} [topological_space X] {U : opens X}\n\ndef res_subset (OX : locally_ringed_space_on_opens X U) (V : opens X) (HVU : V ≤ U) : locally_ringed_space_on_opens X V :=\n{ O := OX.O,\n  Hstalks := λ x hv, OX.Hstalks x (HVU hv) }\n\ntheorem res_res_subset (OX : locally_ringed_space_on_opens X U) (V HVU S HSV T HTV HTS x) :\n  (OX.res_subset V HVU).1.to_sheaf_on_opens.res S HSV T HTV HTS x =\n  OX.1.to_sheaf_on_opens.res S (le_trans HSV HVU) T (le_trans HTV HVU) HTS x :=\nrfl\n\nvariables {Y : Type u₁} [topological_space Y] {V : opens Y}\n\nstructure morphism (F : locally_ringed_space_on_opens.{v} X U) (G : locally_ringed_space_on_opens.{v₁} Y V) : Type (max u v u₁ v₁) :=\n(f : X → Y) (hf : continuous f) (hf2 : opens.comap hf V ≤ U)\n(map : ∀ W ≤ V, F.1.to_sheaf_on_opens.eval (opens.comap hf W) (le_trans (opens.comap_mono _ _ _ H) hf2) →\n  G.1.to_sheaf_on_opens.eval W H)\n[hom : ∀ W H, is_ring_hom (map W H)]\n(hlocal : ∀ W H s, is_unit (map W H s) → is_unit s)\n\n#exit\nnamespace morphism\n\nsection\nvariables {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U}\nvariables (η : F.morphism G) (V : opens X) (HVU : V ≤ U) (x y : F.eval V HVU) (n : ℕ)\n@[simp] lemma map_add : η.map V HVU (x + y) = η.map V HVU x + η.map V HVU y := is_ring_hom.map_add _\n@[simp] lemma map_zero : η.map V HVU 0 = 0 := is_ring_hom.map_zero _\n@[simp] lemma map_neg : η.map V HVU (-x) = -η.map V HVU x := is_ring_hom.map_neg _\n@[simp] lemma map_sub : η.map V HVU (x - y) = η.map V HVU x - η.map V HVU y := is_ring_hom.map_sub _\n@[simp] lemma map_mul : η.map V HVU (x * y) = η.map V HVU x * η.map V HVU y := is_ring_hom.map_mul _\n@[simp] lemma map_one : η.map V HVU 1 = 1 := is_ring_hom.map_one _\n@[simp] lemma map_pow : η.map V HVU (x^n) = (η.map V HVU x)^n := is_semiring_hom.map_pow _ x n\nend\n\ndef to_sheaf_on_opens {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) :\n  F.to_sheaf_on_opens.morphism G.to_sheaf_on_opens :=\n{ .. η }\n\nprotected def id (F : locally_ringed_space_on_opens.{v} X U) : F.morphism F :=\n{ map := λ V HV, id,\n  commutes := λ V HV W HW HWV x, rfl }\n\ndef comp {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} {H : locally_ringed_space_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) : F.morphism H :=\n{ map := λ V HV x, η.map V HV (ξ.map V HV x),\n  commutes := λ V HV W HW HWV x, by rw [ξ.commutes, η.commutes] }\n\n@[simp] lemma comp_apply {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} {H : locally_ringed_space_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) (V HV s) :\n  (η.comp ξ).1 V HV s = η.1 V HV (ξ.1 V HV s) :=\nrfl\n\n@[extensionality] lemma ext {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U}\n  {η ξ : F.morphism G} (H : ∀ V HV x, η.map V HV x = ξ.map V HV x) : η = ξ :=\nby cases η; cases ξ; congr; ext; apply H\n\n@[simp] lemma id_comp {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) :\n  (morphism.id G).comp η = η :=\next $ λ V HV x, rfl\n\n@[simp] lemma comp_id {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) :\n  η.comp (morphism.id F) = η :=\next $ λ V HV x, rfl\n\n@[simp] lemma comp_assoc {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} {H : locally_ringed_space_on_opens.{u₁} X U} {I : locally_ringed_space_on_opens.{v₁} X U}\n  (η : H.morphism I) (ξ : G.morphism H) (χ : F.morphism G) :\n  (η.comp ξ).comp χ = η.comp (ξ.comp χ) :=\nrfl\n\ndef res_subset {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (V : opens X) (HVU : V ≤ U) :\n  (F.res_subset V HVU).morphism (G.res_subset V HVU) :=\n{ map := λ W HWV, η.map W (le_trans HWV HVU),\n  commutes := λ S HSV T HTV, η.commutes S (le_trans HSV HVU) T (le_trans HTV HVU) }\n\n@[simp] lemma res_subset_apply {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (V : opens X) (HVU : V ≤ U)\n  (W HWV s) : (η.res_subset V HVU).1 W HWV s = η.1 W (le_trans HWV HVU) s :=\nrfl\n\n@[simp] lemma comp_res_subset {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} {H : locally_ringed_space_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) (V : opens X) (HVU : V ≤ U) :\n  (η.res_subset V HVU).comp (ξ.res_subset V HVU) = (η.comp ξ).res_subset V HVU :=\nrfl\n\n@[simp] lemma id_res_subset {F : locally_ringed_space_on_opens.{v} X U} (V : opens X) (HVU : V ≤ U) :\n  (morphism.id F).res_subset V HVU = morphism.id (F.res_subset V HVU) :=\nrfl\n\ndef stalk {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U)\n  (s : F.stalk x hx) : G.stalk x hx :=\nquotient.lift_on s (λ g, ⟦(⟨g.1 ⊓ U, (⟨g.2, hx⟩ : x ∈ g.1 ⊓ U),\n  η.map _ inf_le_right (presheaf.res F.1.1 _ _ (set.inter_subset_left _ _) g.3)⟩ : stalk.elem _ _)⟧) $\nλ g₁ g₂ ⟨V, hxV, HV1, HV2, hg⟩, quotient.sound ⟨V ⊓ U, ⟨hxV, hx⟩, set.inter_subset_inter_left _ HV1, set.inter_subset_inter_left _ HV2,\ncalc  G.res _ _ (V ⊓ U) inf_le_right (inf_le_inf HV1 (le_refl _)) (η.map (g₁.U ⊓ U) inf_le_right ((F.F).res (g₁.U) (g₁.U ⊓ U) (set.inter_subset_left _ _) (g₁.s)))\n    = η.map (V ⊓ U) inf_le_right ((F.F).res V (V ⊓ U) (set.inter_subset_left _ _) ((F.F).res (g₁.U) V HV1 (g₁.s))) :\n  by rw ← η.3; dsimp only [res, sheaf_on_opens.res, locally_ringed_space_on_opens.to_sheaf_on_opens]; rw [← presheaf.Hcomp', ← presheaf.Hcomp']\n... = G.res _ _ (V ⊓ U) _ _ (η.map (g₂.U ⊓ U) inf_le_right ((F.F).res (g₂.U) (g₂.U ⊓ U) _ (g₂.s))) :\n  by rw [hg, ← η.3]; dsimp only [res, sheaf_on_opens.res, locally_ringed_space_on_opens.to_sheaf_on_opens]; rw [← presheaf.Hcomp', ← presheaf.Hcomp']⟩\n\n@[simp] lemma stalk_to_stalk {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U)\n  (V : opens X) (HVU : V ≤ U) (hxV : x ∈ V) (s : F.eval V HVU) : η.stalk x hx (F.to_stalk x hx V hxV HVU s) = G.to_stalk x hx V hxV HVU (η.map V HVU s) :=\nquotient.sound ⟨V, hxV, set.subset_inter (set.subset.refl _) HVU, set.subset.refl _,\ncalc  G.res (V ⊓ U) inf_le_right V HVU (le_inf (le_refl V) HVU) (η.map (V ⊓ U) inf_le_right (F.res V HVU (V ⊓ U) inf_le_right inf_le_left s))\n    = G.res V HVU V HVU (le_refl V) (η.map V HVU s) : by rw [η.3, res_res]⟩\n\ninstance is_ring_hom_stalk {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U) :\n  is_ring_hom (η.stalk x hx) :=\n{ map_one := quotient.sound ⟨U, hx, set.subset_inter (set.subset_univ U.1) (set.subset.refl U.1), set.subset_univ U.1,\n    by dsimp only; erw [_root_.res_one, η.map_one, _root_.res_one, _root_.res_one]⟩,\n  map_mul := λ y z, stalk.induction_on₂ y z $ λ V hxV HVU s t, by rw [stalk_to_stalk, stalk_to_stalk, ← to_stalk_mul, stalk_to_stalk, η.map_mul, to_stalk_mul],\n  map_add := λ y z, stalk.induction_on₂ y z $ λ V hxV HVU s t, by rw [stalk_to_stalk, stalk_to_stalk, ← to_stalk_add, stalk_to_stalk, η.map_add, to_stalk_add] }\n\nsection\nvariables {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U)\nvariables (s t : F.stalk x hx) (n : ℕ)\n@[simp] lemma stalk_add : η.stalk x hx (s + t) = η.stalk x hx s + η.stalk x hx t := is_ring_hom.map_add _\n@[simp] lemma stalk_zero : η.stalk x hx 0 = 0 := is_ring_hom.map_zero _\n@[simp] lemma stalk_neg : η.stalk x hx (-s) = -η.stalk x hx s := is_ring_hom.map_neg _\n@[simp] lemma stalk_sub : η.stalk x hx (s - t) = η.stalk x hx s - η.stalk x hx t := is_ring_hom.map_sub _\n@[simp] lemma stalk_mul : η.stalk x hx (s * t) = η.stalk x hx s * η.stalk x hx t := is_ring_hom.map_mul _\n@[simp] lemma stalk_one : η.stalk x hx 1 = 1 := is_ring_hom.map_one _\n@[simp] lemma stalk_pow : η.stalk x hx (s^n) = (η.stalk x hx s)^n := is_semiring_hom.map_pow _ s n\nend\n\nend morphism\n\nstructure equiv (F : locally_ringed_space_on_opens.{v} X U) (G : locally_ringed_space_on_opens.{w} X U) : Type (max u v w) :=\n(to_fun : F.morphism G)\n(inv_fun : G.to_sheaf_on_opens.morphism F.to_sheaf_on_opens)\n(left_inv : ∀ V HVU s, inv_fun.1 V HVU (to_fun.1 V HVU s) = s)\n(right_inv : ∀ V HVU s, to_fun.1 V HVU (inv_fun.1 V HVU s) = s)\n\nnamespace equiv\n\ndef to_sheaf_on_opens {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (e : F.equiv G) :\n  F.to_sheaf_on_opens.equiv G.to_sheaf_on_opens :=\n{ to_fun := e.1.to_sheaf_on_opens, .. e }\n\ndef refl (F : locally_ringed_space_on_opens.{v} X U) : equiv F F :=\n⟨morphism.id F, sheaf_on_opens.morphism.id F.to_sheaf_on_opens, λ _ _ _, rfl, λ _ _ _, rfl⟩\n\n@[simp] lemma refl_apply (F : locally_ringed_space_on_opens.{v} X U) (V HV s) :\n  (refl F).1.1 V HV s = s := rfl\n\ndef symm {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{v} X U} (e : equiv F G) : equiv G F :=\n⟨{ hom := λ V HVU, (ring_equiv.symm { to_fun := e.1.1 V HVU, inv_fun := e.2.1 V HVU, left_inv := e.3 V HVU, right_inv := e.4 V HVU, hom := e.1.2 V HVU }).hom,\n  .. e.2 },\ne.1.to_sheaf_on_opens, e.4, e.3⟩\n\ndef trans {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{v} X U} {H : locally_ringed_space_on_opens.{u₁} X U}\n  (e₁ : equiv F G) (e₂ : equiv G H) : equiv F H :=\n⟨e₂.1.comp e₁.1, e₁.2.comp e₂.2,\nλ _ _ _, by rw [morphism.comp_apply, sheaf_on_opens.morphism.comp_apply, e₂.3, e₁.3],\nλ _ _ _, by rw [morphism.comp_apply, sheaf_on_opens.morphism.comp_apply, e₁.4, e₂.4]⟩\n\n@[simp] lemma trans_apply {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{v} X U} {H : locally_ringed_space_on_opens.{u₁} X U}\n  (e₁ : equiv F G) (e₂ : equiv G H) (V HV s) :\n  (e₁.trans e₂).1.1 V HV s = e₂.1.1 V HV (e₁.1.1 V HV s) :=\nrfl\n\ndef res_subset {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (e : equiv F G)\n  (V : opens X) (HVU : V ≤ U) : equiv (F.res_subset V HVU) (G.res_subset V HVU) :=\n⟨e.1.res_subset V HVU, e.2.res_subset V HVU,\nλ _ _ _, by rw [morphism.res_subset_apply, sheaf_on_opens.morphism.res_subset_apply, e.3],\nλ _ _ _, by rw [morphism.res_subset_apply, sheaf_on_opens.morphism.res_subset_apply, e.4]⟩\n\n@[simp] lemma res_subset_apply {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (e : equiv F G)\n  (V : opens X) (HVU : V ≤ U) (W HW s) :\n  (e.res_subset V HVU).1.1 W HW s = e.1.1 W (le_trans HW HVU) s :=\nrfl\n\ndef stalk {F : locally_ringed_space_on_opens.{v} X U} {G : locally_ringed_space_on_opens.{w} X U} (e : equiv F G) (x : X) (hx : x ∈ U) :\n  F.stalk x hx ≃ G.stalk x hx :=\n{ to_fun := e.1.stalk x hx,\n  inv_fun := e.2.stalk x hx,\n  left_inv := λ g, stalk.induction_on g $ λ V hxV HVU s, by rw [morphism.stalk_to_stalk, to_stalk, sheaf_on_opens.morphism.stalk_to_stalk, e.3]; refl,\n  right_inv := λ g, stalk.induction_on g $ λ V hxV HVU s, by rw [to_stalk, sheaf_on_opens.morphism.stalk_to_stalk, ← to_stalk, morphism.stalk_to_stalk, e.4]; refl }\n\nend equiv\n\ndef sheaf_glue {I : Type u} (S : I → opens X) (F : Π (i : I), locally_ringed_space_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right)) :\n  locally_ringed_space_on_opens.{max u v} X (⋃S) :=\n{ F :=\n  { Fring := λ U, @subtype.comm_ring (Π (i : I), (F i).eval (S i ⊓ U) inf_le_left) _\n      { f | ∀ (i j : I), (φ i j).1.map (S i ⊓ S j ⊓ U) inf_le_left\n              ((F i).res (S i ⊓ U) inf_le_left (S i ⊓ S j ⊓ U) (le_trans inf_le_left inf_le_left)\n                (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n                (f i)) =\n            (F j).res (S j ⊓ U) inf_le_left (S i ⊓ S j ⊓ U) (le_trans inf_le_left inf_le_right)\n              (by rw inf_assoc; exact inf_le_right)\n              (f j) }\n      { add_mem := λ f g hf hg i j, by erw [res_add, morphism.map_add, res_add, hf i j, hg i j],\n        zero_mem := λ i j, by erw [res_zero, morphism.map_zero, res_zero]; refl,\n        neg_mem := λ f hf i j, by erw [res_neg, morphism.map_neg, res_neg, hf i j],\n        one_mem := λ i j, by erw [res_one, morphism.map_one, res_one]; refl,\n        mul_mem := λ f g hf hg i j, by erw [res_mul, morphism.map_mul, res_mul, hf i j, hg i j] },\n    res_is_ring_hom := λ U V HVU,\n      { map_one := subtype.eq $ funext $ λ i, res_one _ _ _ _ _ _,\n        map_mul := λ f g, subtype.eq $ funext $ λ i, res_mul _ _ _ _ _ _ _ _,\n        map_add := λ f g, subtype.eq $ funext $ λ i, res_add _ _ _ _ _ _ _ _ },\n    .. (sheaf_on_opens.sheaf_glue S (λ i, (F i).to_sheaf_on_opens) (λ i j, (φ i j).to_sheaf_on_opens)).F }\n  .. sheaf_on_opens.sheaf_glue S (λ i, (F i).to_sheaf_on_opens) (λ i j, (φ i j).to_sheaf_on_opens) }\n\n@[simp] lemma sheaf_glue_res_val {I : Type u} (S : I → opens X) (F : Π (i : I), locally_ringed_space_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right))\n  (U HU V HV HVU s i) : ((sheaf_glue S F φ).res U HU V HV HVU s).1 i = (F i).res _ _ _ _ (inf_le_inf (le_refl _) HVU) (s.1 i) := rfl\n\ndef universal_property (I : Type u) (S : I → opens X) (F : Π (i : I), locally_ringed_space_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right))\n  (Hφ1 : ∀ i V HV s, (φ i i).1.1 V HV s = s)\n  (Hφ2 : ∀ i j k V HV1 HV2 HV3 s, (φ j k).1.1 V HV1 ((φ i j).1.1 V HV2 s) = (φ i k).1.1 V HV3 s)\n  (i : I) :\n  equiv (res_subset (sheaf_glue S F φ) (S i) (le_supr S i)) (F i) :=\n{ to_fun :=\n  { hom := λ U HU,\n    { map_one := res_one _ _ _ _ _ _,\n      map_mul := λ x y, res_mul _ _ _ _ _ _ _ _,\n      map_add := λ x y, res_add _ _ _ _ _ _ _ _ },\n    .. (sheaf_on_opens.universal_property I S (λ i, (F i).to_sheaf_on_opens) (λ i j, (φ i j).to_sheaf_on_opens) Hφ1 Hφ2 i).1 },\n  .. sheaf_on_opens.universal_property I S (λ i, (F i).to_sheaf_on_opens) (λ i j, (φ i j).to_sheaf_on_opens) Hφ1 Hφ2 i }\n\nend locally_ringed_space_on_opens\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/Kenny/locally_ringed_space_on_opens.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.3220103404593004}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nEnvironments.\n-/\nimport data.hash_map library_dev.data.list.sort .tensor .id .util .reference\n\nnamespace certigrad\n\ndef pre_env : Type := hash_map reference (λ (ref : reference), T ref.2)\n\nattribute [reducible] pre_env\nnamespace pre_env\n\ndefinition eqv (m₁ m₂ : pre_env) : Prop :=\n∀ (ref : reference), m₁^.find ref = m₂^.find ref\n\nlocal infix ~ := eqv\n\ndefinition eqv.refl (m : pre_env) : m ~ m :=\nassume ref, rfl\n\ndefinition eqv.symm (m₁ m₂ : pre_env) : m₁ ~ m₂ → m₂ ~ m₁ :=\nassume H ref, eq.symm (H ref)\n\ndefinition eqv.trans (m₁ m₂ m₃ : pre_env) : m₁ ~ m₂ → m₂ ~ m₃ → m₁ ~ m₃ :=\nassume H₁ H₂ ref, eq.trans (H₁ ref) (H₂ ref)\n\ninstance pdmap.eqv_setoid : setoid pre_env :=\nsetoid.mk eqv (mk_equivalence eqv eqv.refl eqv.symm eqv.trans)\n\nend pre_env\n\ndef env : Type := quot pre_env.eqv\n\nnamespace env\n\ndef mk : env := quotient.mk (mk_hash_map reference.hash)\n\ndef get (ref : reference) (q : env) : T ref.2 := quotient.lift_on q\n(λ (m : pre_env),\n  match m^.find ref with\n  | none := default _\n  | some x := x\n  end)\nbegin intros m₁ m₂ H_eqv, simp [H_eqv ref] end\n\ndef insert (ref : reference) (x : T ref.2) (q : env) : env := quotient.lift_on q\n(λ (m : pre_env), quotient.mk $ m^.insert ref x)\nbegin\nintros m₁ m₂ H_eqv, dsimp, apply quotient.sound,\nintros ref',\ncases (decidable.em (ref = ref')) with H_eq H_neq,\nsimp [hash_map.find_insert, dif_ctx_simp_congr, H_eq, dif_pos],\nsimp [hash_map.find_insert, dif_ctx_simp_congr, H_neq, dif_neg, H_eqv ref'],\nend\n\ndef has_key (ref : reference) (q : env) : Prop := quotient.lift_on q\n(λ (m : pre_env), m^.contains ref)\nbegin intros m₁ m₂ H_eqv, simp [hash_map.contains, H_eqv ref] end\n\ndef get_ks : Π (refs : list reference) (m : env), dvec T refs^.p2\n| []          m := ⟦⟧\n| (ref::refs) m := dvec.cons (get ref m) (get_ks refs m)\n\ndef insert_all : Π (refs : list reference) (vs : dvec T refs^.p2), env\n| []      ⟦⟧        := env.mk\n| (k::ks) (v:::vs) := env.insert k v (insert_all ks vs)\n\n-- Facts\n@[simp] lemma get.def (ref : reference) (m : pre_env) :\nget ref (quotient.mk m) = match m^.find ref with | none := default _ | some x := x end := rfl\n\n@[simp] lemma insert.def {ref : reference} {x : T ref.2} (m : pre_env) :\ninsert ref x (quotient.mk m) = quotient.mk (m^.insert ref x) :=\nbegin apply quotient.sound, apply pre_env.eqv.refl end\n\n@[simp] lemma has_key.def (ref : reference) (m : pre_env) :\nhas_key ref (quotient.mk m) = m^.contains ref := rfl\n\n@[simp] lemma bool_lift_t (b : bool) : (lift_t b : Prop) = (b = tt) := rfl\n\n-- TODO(dhs): PR to standard library\nlemma not_has_key_empty (ref : reference) : ¬ env.has_key ref env.mk :=\nbegin\nsimp [mk],\nrw hash_map.contains_iff,\nsimp [hash_map.keys, hash_map.entries, mk_hash_map, bucket_array.as_list,\n      mk_array, array.foldl, array.iterate, array.iterate_aux, list.map, array.read],\nend\n\nlemma has_key_insert {ref₁ ref₂ : reference} {x₂ : T ref₂.2} {m : env} :\n  has_key ref₁ m → has_key ref₁ (insert ref₂ x₂ m) :=\nbegin\napply @quotient.induction_on _ _ (λ m, has_key ref₁ m → has_key ref₁ (insert ref₂ x₂ m)),\nclear m,\nintro m,\nintro H_hk,\nsimp at *,\ndsimp [hash_map.contains] at *,\nrw hash_map.find_insert,\ncases decidable.em (ref₂ = ref₁) with H_eq H_neq,\n{\nsubst H_eq,\nsimp [dif_ctx_simp_congr, dif_pos],\ndunfold option.is_some,\nreflexivity,\n},\n{\nsimp [H_neq, dif_ctx_simp_congr, dif_neg],\nexact H_hk\n}\nend\n\nlemma has_key_insert_same (ref : reference) {x : T ref.2} (m : env) : has_key ref (insert ref x m) :=\nbegin\napply quotient.induction_on m,\nclear m,\nintro m,\nsimp,\ndunfold hash_map.contains,\nrw hash_map.find_insert_eq,\ndsimp [option.is_some],\nreflexivity\nend\n\nlemma has_key_insert_diff {ref₁ ref₂ : reference} {x : T ref₂.2} {m : env} :\n  ref₁ ≠ ref₂ → has_key ref₁ (insert ref₂ x m) → has_key ref₁ m :=\nbegin\napply @quotient.induction_on _ _ (λ m, ref₁ ≠ ref₂ → has_key ref₁ (insert ref₂ x m) → has_key ref₁ m),\nclear m,\nsimp [hash_map.contains],\nintros m H_neq,\nrw hash_map.find_insert_ne,\nintro H, exact H,\nexact ne.symm H_neq\nend\n\nlemma get_insert_same (ref : reference) (x : T ref.2) (m : env) : get ref (insert ref x m) = x :=\nbegin\napply quotient.induction_on m, clear m, intro m,\nsimp,\nrw hash_map.find_insert_eq,\nend\n\nlemma get_insert_diff {ref₁ ref₂ : reference} (x₂ : T ref₂.2) (m : env) : ref₁ ≠ ref₂ → get ref₁ (insert ref₂ x₂ m) = get ref₁ m :=\nbegin\napply @quotient.induction_on _ _ (λ m, ref₁ ≠ ref₂ → get ref₁ (insert ref₂ x₂ m) = get ref₁ m),\nclear m,\nintros m H_neq,\nsimp,\nrw hash_map.find_insert,\n-- TODO(dhs): annoying that we can't simplify inside the major premise\nassert H_dif : (dite (ref₂ = ref₁) (λ h, some (eq.rec_on h x₂ : T ref₁.2)) (λ h, hash_map.find m ref₁)) = hash_map.find m ref₁,\nsimp [dif_ctx_simp_congr, dif_neg, ne.symm H_neq],\nrw H_dif,\nend\n\n-- TODO(dhs): propagate precondition\nlemma insert_get_same {ref : reference} {m : env} : has_key ref m → insert ref (get ref m) m = m :=\nbegin\napply @quotient.induction_on _ _ (λ m, has_key ref m → insert ref (get ref m) m = m),\nclear m,\nsimp [hash_map.contains],\nintros m H_has_key,\napply quotient.sound,\nintro ref',\ncases decidable.em (ref' = ref) with H_eq H_neq,\n{\nsubst H_eq,\nrw hash_map.find_insert_eq,\ncases (hash_map.find m ref'),\n{ dsimp [option.is_some] at H_has_key, injection H_has_key },\ndsimp,\nreflexivity\n},\n{\nrw hash_map.find_insert_ne,\nexact ne.symm H_neq\n}\nend\n\nlemma insert_insert_flip {ref₁ ref₂ : reference} (x₁ : T ref₁.2) (x₂ : T ref₂.2) (m : env) :\n  ref₁ ≠ ref₂ → insert ref₁ x₁ (insert ref₂ x₂ m) = insert ref₂ x₂ (insert ref₁ x₁ m) :=\nbegin\napply @quotient.induction_on _ _ (λ m, ref₁ ≠ ref₂ → insert ref₁ x₁ (insert ref₂ x₂ m) = insert ref₂ x₂ (insert ref₁ x₁ m)),\nclear m,\nsimp,\nintros m H_neq,\napply quot.sound,\nintro ref,\nsimp [hash_map.find_insert],\ncases decidable.em (ref₁ = ref) with H_eq₁ H_neq₁,\ncases decidable.em (ref₂ = ref) with H_eq₂ H_neq₂,\n{ exfalso, exact H_neq (eq.trans H_eq₁ (eq.symm H_eq₂)) },\n{ subst H_eq₁, simp [H_neq₂, dif_ctx_simp_congr, dif_neg, dif_pos] },\ncases decidable.em (ref₂ = ref) with H_eq₂ H_neq₂,\n{ subst H_eq₂, simp [H_neq₁, dif_ctx_simp_congr, dif_neg, dif_pos] },\n{ simp [H_neq₁, H_neq₂, dif_ctx_simp_congr, dif_neg], }\nend\n\nlemma insert_insert_same (ref : reference) (x₁ x₂ : T ref.2) (m : env) :\n  insert ref x₁ (insert ref x₂ m) = insert ref x₁ m :=\nbegin\napply quotient.induction_on m,\nclear m,\nsimp,\nintros m,\napply quot.sound,\nintro ref',\ncases decidable.em (ref' = ref) with H_eq H_neq,\n{ subst H_eq, simp [hash_map.find_insert_eq] },\n-- TODO(dhs): simp fails for annoying reasons\nrw hash_map.find_insert_ne _ _ _ _ (ne.symm H_neq),\nrw hash_map.find_insert_ne _ _ _ _ (ne.symm H_neq),\nrw hash_map.find_insert_ne _ _ _ _ (ne.symm H_neq)\nend\n\nlemma get_ks_env_eq (m₁ m₂ : env) :\n  ∀ (refs : list reference), (∀ (ref : reference), ref ∈ refs → get ref m₁ = get ref m₂) → get_ks refs m₁ = get_ks refs m₂\n| [] H := rfl\n| (ref::refs) H :=\nshow get ref m₁ ::: get_ks refs m₁ = get ref m₂ ::: get_ks refs m₂, from\nhave H_get : get ref m₁ = get ref m₂, from H ref list.mem_of_cons_same,\nhave H_pre : ∀ (ref : reference), ref ∈ refs → get ref m₁ = get ref m₂, from\n  assume r H_r_mem,\n  H r (list.mem_cons_of_mem _ H_r_mem),\nby rw [H_get, get_ks_env_eq _ H_pre]\n\nlemma get_ks_insert_diff :\n  ∀ {refs : list reference} {ref : reference} {x : T ref.2} {m : env}, ref ∉ refs → get_ks refs (insert ref x m) = get_ks refs m\n| [] _ _ _ _ := rfl\n| (ref::refs) ref₀ x m H_ref₀_notin :=\nshow get ref (insert ref₀ x m) ::: get_ks refs (insert ref₀ x m) = get ref m ::: get_ks refs m, from\nhave H_ne : ref ≠ ref₀, from ne.symm (list.ne_of_not_mem_cons H_ref₀_notin),\nbegin\nrw (env.get_insert_diff _ _ H_ne),\nrw get_ks_insert_diff (list.not_mem_of_not_mem_cons H_ref₀_notin),\nend\n\nlemma dvec_update_at_env {refs : list reference} {idx : ℕ} {ref : reference} (m : env) :\n      list.at_idx refs idx ref →\n      dvec.update_at (get ref m) (get_ks refs m) idx = get_ks refs m :=\nbegin\nintro H_at_idx,\nassert H_elem_at_idx : list.elem_at_idx refs idx ref, { exact list.elem_at_idx_of_at_idx H_at_idx },\ninduction H_elem_at_idx with xs x xs idx' x y H_elem_at_idx IH,\n{ dsimp [get_ks], simp [dif_ctx_simp_congr, dif_pos] },\n{ dsimp [get_ks], erw IH (list.at_idx_of_cons H_at_idx) }\nend\n\nlemma dvec_get_get_ks {refs : list reference} {idx : ℕ} {ref : reference} (m : env) :\n      list.at_idx refs idx ref →\n      dvec.get ref.2 (get_ks refs m) idx = get ref m :=\nbegin\nintro H_at_idx,\nassert H_elem_at_idx : list.elem_at_idx refs idx ref, { exact list.elem_at_idx_of_at_idx H_at_idx },\ninduction H_elem_at_idx with xs x xs idx' x y H_elem_at_idx IH,\n{ dunfold get_ks, erw dvec.get.equations._eqn_2, simp [dif_ctx_simp_congr, dif_pos] },\n{ dunfold get_ks, erw dvec.get.equations._eqn_3, exact IH (list.at_idx_of_cons H_at_idx) }\nend\n\nend env\nattribute [semireducible] pre_env\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/env.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.32196815507983007}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport ring_theory.subsemiring.pointwise\nimport group_theory.subgroup.pointwise\nimport ring_theory.subring.basic\n\n/-! # Pointwise instances on `subring`s\n\nThis file provides the action `subring.pointwise_mul_action` which matches the action of\n`mul_action_set`.\n\nThis actions is available in the `pointwise` locale.\n\n## Implementation notes\n\nThis file is almost identical to `ring_theory/subsemiring/pointwise.lean`. Where possible, try to\nkeep them in sync.\n\n-/\n\nvariables {M R : Type*}\n\nnamespace subring\n\nsection monoid\nvariables [monoid M] [ring R] [mul_semiring_action M R]\n\n/-- The action on a subring corresponding to applying the action to every element.\n\nThis is available as an instance in the `pointwise` locale. -/\nprotected def pointwise_mul_action : mul_action M (subring R) :=\n{ smul := λ a S, S.map (mul_semiring_action.to_ring_hom _ _ a),\n  one_smul := λ S,\n    (congr_arg (λ f, S.map f) (ring_hom.ext $ by exact one_smul M)).trans S.map_id,\n  mul_smul := λ a₁ a₂ S,\n    (congr_arg (λ f, S.map f) (ring_hom.ext $ by exact mul_smul _ _)).trans (S.map_map _ _).symm }\n\nlocalized \"attribute [instance] subring.pointwise_mul_action\" in pointwise\nopen_locale pointwise\n\nlemma pointwise_smul_def {a : M} (S : subring R) :\n  a • S = S.map (mul_semiring_action.to_ring_hom _ _ a) := rfl\n\n@[simp] lemma coe_pointwise_smul (m : M) (S : subring R) : ↑(m • S) = m • (S : set R) := rfl\n\n@[simp] lemma pointwise_smul_to_add_subgroup (m : M) (S : subring R) :\n  (m • S).to_add_subgroup = m • S.to_add_subgroup := rfl\n\n@[simp] lemma pointwise_smul_to_subsemiring (m : M) (S : subring R) :\n  (m • S).to_subsemiring = m • S.to_subsemiring := rfl\n\nlemma smul_mem_pointwise_smul (m : M) (r : R) (S : subring R) : r ∈ S → m • r ∈ m • S :=\n(set.smul_mem_smul_set : _ → _ ∈ m • (S : set R))\n\nend monoid\n\n\nsection group\nvariables [group M] [ring R] [mul_semiring_action M R]\n\nopen_locale pointwise\n\n@[simp] lemma smul_mem_pointwise_smul_iff {a : M} {S : subring R} {x : R} :\n  a • x ∈ a • S ↔ x ∈ S :=\nsmul_mem_smul_set_iff\n\nlemma mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : subring R} {x : R} :\n  x ∈ a • S ↔ a⁻¹ • x ∈ S :=\nmem_smul_set_iff_inv_smul_mem\n\nlemma mem_inv_pointwise_smul_iff {a : M} {S : subring R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S :=\nmem_inv_smul_set_iff\n\n@[simp] lemma pointwise_smul_le_pointwise_smul_iff {a : M} {S T : subring R} :\n  a • S ≤ a • T ↔ S ≤ T :=\nset_smul_subset_set_smul_iff\n\nlemma pointwise_smul_subset_iff {a : M} {S T : subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T :=\nset_smul_subset_iff\n\nlemma subset_pointwise_smul_iff {a : M} {S T : subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T :=\nsubset_set_smul_iff\n\n/-! TODO: add `equiv_smul` like we have for subgroup. -/\n\nend group\n\nsection group_with_zero\nvariables [group_with_zero M] [ring R] [mul_semiring_action M R]\n\nopen_locale pointwise\n\n@[simp] lemma smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subring R)\n  (x : R) : a • x ∈ a • S ↔ x ∈ S :=\nsmul_mem_smul_set_iff₀ ha (S : set R) x\n\nlemma mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : subring R) (x : R) :\n  x ∈ a • S ↔ a⁻¹ • x ∈ S :=\nmem_smul_set_iff_inv_smul_mem₀ ha (S : set R) x\n\nlemma mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : subring R) (x : R) :\n  x ∈ a⁻¹ • S ↔ a • x ∈ S :=\nmem_inv_smul_set_iff₀ ha (S : set R) x\n\n@[simp] lemma pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} :\n  a • S ≤ a • T ↔ S ≤ T :=\nset_smul_subset_set_smul_iff₀ ha\n\nlemma pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T :=\nset_smul_subset_iff₀ ha\n\nlemma le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T :=\nsubset_set_smul_iff₀ ha\n\nend group_with_zero\n\nend subring\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/ring_theory/subring/pointwise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.32196815507983007}}
{"text": "import sheaves.locally_ringed_space\nimport instances.affine_scheme\n\nuniverses u v w\n\ninstance ring_equiv.hom {α : Type u} {β : Type v} [ring α] [ring β] (e : α ≃+* β) : is_ring_hom e :=\nring_hom.is_ring_hom e.to_ring_hom\n\ndef opens.indefinite_description {α : Type u} [topological_space α] (p : topological_space.opens α → Prop) (hp : ∃ U, p U) : {U // p U} :=\n⟨⟨↑(classical.some hp), subtype.property _⟩, by rw subtype.coe_eta; exact classical.some_spec hp⟩\n\ndef opens.some {α : Type u} [topological_space α] {p : topological_space.opens α → Prop} (hp : ∃ U, p U) : topological_space.opens α :=\n(opens.indefinite_description p hp).1\n\ntheorem opens.some_spec {α : Type u} [topological_space α] {p : topological_space.opens α → Prop} (hp : ∃ U, p U) : p (opens.some hp) :=\n(opens.indefinite_description p hp).2\n\nsection presheaf_of_rings\n\nvariables {α : Type u} [topological_space α]\nvariables (F : presheaf_of_rings α)\nvariables (U V : topological_space.opens α) (HVU : V ⊆ U)\nvariables (x y : F U)\n\n@[simp] lemma res_add : F.res U V HVU (x + y) = F.res U V HVU x + F.res U V HVU y := is_ring_hom.map_add _\n@[simp] lemma res_zero : F.res U V HVU 0 = 0 := is_ring_hom.map_zero _\n@[simp] lemma res_neg : F.res U V HVU (-x) = -F.res U V HVU x := is_ring_hom.map_neg _\n@[simp] lemma res_sub : F.res U V HVU (x - y) = F.res U V HVU x - F.res U V HVU y := is_ring_hom.map_sub _\n@[simp] lemma res_mul : F.res U V HVU (x * y) = F.res U V HVU x * F.res U V HVU y := is_ring_hom.map_mul _\n@[simp] lemma res_one : F.res U V HVU 1 = 1 := is_ring_hom.map_one _\n\nend presheaf_of_rings\n\nsection presheaf_of_rings_on_basis\n\nvariables {α : Type u} [topological_space α] {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B)\nvariables (F : presheaf_of_rings_on_basis α HB)\nvariables (U V : topological_space.opens α) (HUB : U ∈ B) (HVB : V ∈ B) (HVU : V ⊆ U)\nvariables (x y : F.F HUB)\n\n@[simp] lemma bres_add : F.res HUB HVB HVU (x + y) = F.res HUB HVB HVU x + F.res HUB HVB HVU y := is_ring_hom.map_add _\n@[simp] lemma bres_zero : F.res HUB HVB HVU 0 = 0 := is_ring_hom.map_zero _\n@[simp] lemma bres_neg : F.res HUB HVB HVU (-x) = -F.res HUB HVB HVU x := is_ring_hom.map_neg _\n@[simp] lemma bres_sub : F.res HUB HVB HVU (x - y) = F.res HUB HVB HVU x - F.res HUB HVB HVU y := is_ring_hom.map_sub _\n@[simp] lemma bres_mul : F.res HUB HVB HVU (x * y) = F.res HUB HVB HVU x * F.res HUB HVB HVU y := is_ring_hom.map_mul _\n@[simp] lemma bres_one : F.res HUB HVB HVU 1 = 1 := is_ring_hom.map_one _\n\nend presheaf_of_rings_on_basis\n\ntheorem is_unit.map'' {α : Type u} [monoid α] {β : Type v} [monoid β] {y : β}\n  (f : α → β) [is_monoid_hom f] (x : α) (hx : is_unit x) (hxy : f x = y) : is_unit y :=\nhxy ▸ hx.map' f\n\nsection opens_comap\n\nopen topological_space lattice lattice.lattice\n\nvariables {α : Type u} [topological_space α]\nvariables {β : Type v} [topological_space β]\nvariables {f : α → β} (Hf : continuous f)\n\n@[mono] theorem opens.comap_mono' (U V : opens β) (HUV : U ≤ V) : opens.comap Hf U ≤ opens.comap Hf V :=\nset.preimage_mono HUV\n\n@[simp] lemma opens.comap_top : opens.comap Hf ⊤ = ⊤ :=\nopens.ext set.preimage_univ\n\nend opens_comap\n\nnamespace topological_space\n\nnamespace opens\n\nvariables {X : Type u} [topological_space X] {B : set (opens X)} (HB : is_basis B)\n\ndef covering_of_is_basis (U : opens X) : covering U :=\n⟨λ V : { V : opens X // V ∈ B ∧ V ⊆ U }, V,\nopens.ext $ set.subset.antisymm (set.sUnion_subset $ λ s ⟨t, ⟨V, hVt⟩, hts⟩, hts ▸ hVt ▸ V.2.2) $\nlet ⟨S, HSB, HUS⟩ := is_basis_iff_cover.1 HB U in HUS.symm ▸ set.sUnion_subset (λ s ⟨V, HVS, HVs⟩, HVs ▸ set.subset_sUnion_of_mem\n  ⟨V, ⟨⟨V, HSB HVS, set.subset_sUnion_of_mem ⟨V, HVS, rfl⟩⟩, rfl⟩, rfl⟩)⟩\n\nvariables (B)\ndef of_is_basis (U : opens X) (i : U) : opens X :=\n⟨@classical.epsilon (opens X) ⟨⊥⟩ (λ V, i.1 ∈ V ∧ V ∈ B ∧ V ⊆ U), subtype.property _⟩\nvariables {B}\n\ntheorem of_is_basis_spec {U : opens X} {i : U} : i.1 ∈ of_is_basis B U i ∧ of_is_basis B U i ∈ B ∧ of_is_basis B U i ⊆ U :=\nhave ∀ i : U, ∃ V : opens X, i.1 ∈ V ∧ V ∈ B ∧ V ⊆ U,\nfrom let ⟨S, HSB, HUS⟩ := is_basis_iff_cover.1 HB U in HUS.symm ▸ λ i,\n  let ⟨t, ⟨V, HVS, rfl⟩, hiV⟩ := set.mem_sUnion.1 i.2 in ⟨V, hiV, HSB HVS, set.subset_sUnion_of_mem ⟨V, HVS, rfl⟩⟩,\nby rw [of_is_basis, subtype.coe_eta]; exact classical.epsilon_spec (this i)\n\ntheorem mem_of_is_basis {U : opens X} {i : U} : i.1 ∈ of_is_basis B U i :=\n(of_is_basis_spec HB).1\n\ntheorem of_is_basis_mem_basis {U : opens X} {i : U} : of_is_basis B U i ∈ B:=\n(of_is_basis_spec HB).2.1\n\ntheorem of_is_basis_subset {U : opens X} {i : U} : of_is_basis B U i ⊆ U :=\n(of_is_basis_spec HB).2.2\n\ndef covering_of_is_basis' (U : opens X) : covering U :=\n⟨λ i : U, of_is_basis B U i,\nopens.ext $ set.subset.antisymm (set.sUnion_subset $ λ s ⟨t, ⟨i, hit⟩, hts⟩, hts ▸ hit ▸ (of_is_basis_subset HB)) $\nλ i hiu, set.mem_sUnion.2 ⟨of_is_basis B U ⟨i, hiu⟩, set.mem_image_of_mem _ $ set.mem_range_self _, mem_of_is_basis HB⟩⟩\n\ndef covering_of_opens (U : opens X) (f : Π i : U, opens X) (hf : ∀ i : U, i.1 ∈ f i) : covering U :=\n⟨λ i, f i ∩ U,\nopens.ext $ set.subset.antisymm (set.sUnion_subset $ λ s ⟨t, ⟨i, hit⟩, hts⟩, hts ▸ hit ▸ set.inter_subset_right (f i) U) $\nλ i hiu, set.mem_sUnion.2 ⟨f ⟨i, hiu⟩ ∩ U, set.mem_image_of_mem _ $ set.mem_range_self _, hf ⟨i, hiu⟩, hiu⟩⟩\n\ndef covering_comap {Y : Type u} [topological_space Y] {f : X → Y} (Hf : continuous f)\n  (V : opens Y) (OC : covering V) : covering (opens.comap Hf V) :=\n{ γ := OC.γ,\n  Uis := λ i, opens.comap Hf (OC.Uis i),\n  Hcov := opens.ext $ set.subset.antisymm (set.sUnion_subset $ λ t ⟨U, ⟨V, hVU⟩, hUt⟩, hUt ▸ hVU ▸ set.preimage_mono (subset_covering V)) $\n    λ i hi, let ⟨V, ⟨U, ⟨j, hjU⟩, hUV⟩, hfiV⟩ := set.mem_sUnion.1 (((set.ext_iff _ _).1 (congr_arg subtype.val OC.Hcov) (f i)).2 hi) in\n    set.mem_sUnion.2 ⟨f ⁻¹' V, ⟨opens.comap Hf ⟨V, hUV ▸ U.2⟩, ⟨j, congr_arg (opens.comap Hf) $ hjU.trans $ by exact subtype.eq hUV⟩, rfl⟩, hfiV⟩ }\n\nend opens\n\nend topological_space\n\ninstance presheaf_of_rings.comm_ring {α : Type u} [topological_space α]\n  (f : presheaf_of_rings α) (U : topological_space.opens α) : comm_ring (f U) := f.Fring U\n\ninstance sheaf_of_rings.has_coe_to_fun {α : Type u} [topological_space α] :\n  has_coe_to_fun (sheaf_of_rings.{u v} α) :=\n⟨λ _, topological_space.opens α → Type v, λ F, F.F⟩\n\ninstance sheaf_of_rings.comm_ring {α : Type u} [topological_space α]\n  (f : sheaf_of_rings α) (U : topological_space.opens α) : comm_ring (f U) := f.F.Fring U\n\nattribute [instance] to_stalk.is_ring_hom\nattribute [irreducible] sheaf_on_standard_basis.is_sheaf_on_standard_basis\n\ndef to_Spec_top (R : Type v) [comm_ring R] : R → (Spec.locally_ringed_space R).O ⊤ :=\nto_presheaf_of_rings_extension (D_fs_standard_basis R) (structure_presheaf_on_basis R) (D_fs_standard_basis R).1 ∘ localization.of\n\ninstance (R : Type v) [comm_ring R] : is_ring_hom (to_Spec_top R) :=\n@@is_ring_hom.comp _ _ _ _ _ _ $ to_presheaf_of_rings_extension.is_ring_hom _ _ _\n\nnamespace locally_ringed_space\n\nvariables {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n\ndef D' (f : OX.O ⊤) : set X := { x | is_unit (to_stalk OX.O.F x ⊤ trivial f) }\n\ntheorem is_open_D' (f : OX.O ⊤) : is_open (OX.D' f) :=\nbegin\n  refine is_open_iff_forall_mem_open.2 (λ x hxf, _),\n  cases is_unit_iff_exists_inv.1 hxf with y hxy,\n  refine quotient.induction_on y (λ g hfg, _) hxy,\n  cases g with U hxU g, rcases quotient.exact hfg with ⟨V, hxV, HVU', HVtop, hfgV⟩,\n  exact ⟨V, λ y hyV, is_unit_iff_exists_inv.2 ⟨to_stalk ((OX.O).F) y U (HVU' hyV).2 g, quotient.sound ⟨V, hyV, HVU', HVtop, hfgV⟩⟩, V.2, hxV⟩\nend\n\ndef D (f : OX.O ⊤) : topological_space.opens X :=\n⟨OX.D' f, OX.is_open_D' f⟩\n\ntheorem is_unit_res_D (f : OX.O ⊤) : is_unit (OX.O.F.res ⊤ (OX.D f) (set.subset_univ _) f) :=\nbegin\n  rw is_unit_iff_exists_inv,\n  have : ∀ i : OX.D f, ∃ U : topological_space.opens X, ∃ H : i.1 ∈ U, ∃ g : OX.O U, OX.O.F.res ⊤ U (set.subset_univ U.1) f * g = 1,\n  { refine λ ⟨i, hi⟩, exists.elim (is_unit_iff_exists_inv.1 hi) (λ gq, quotient.induction_on gq $\n      λ ⟨U, hiU, g⟩ H, let ⟨V, hiV, HVU, HVtop, hfgV⟩ := quotient.exact H in\n      ⟨V, hiV, OX.O.F.res U V (λ j hjv, (HVU hjv).2) g,\n      begin\n        dsimp only at hfgV,\n        rw [res_mul, ← presheaf.Hcomp', ← presheaf.Hcomp'] at hfgV,\n        convert hfgV.trans (res_one _ _ _ _)\n      end⟩), },\n  choose U hu g hg,\n  have, refine OX.O.gluing (topological_space.opens.covering_of_opens (OX.D f) U hu) _ _,\n  { exact (λ i, OX.O.F.res _ _ (set.inter_subset_left _ _) (g i)) },\n  { intros j k, dsimp only [res_to_inter_left, res_to_inter_right, topological_space.opens.covering_of_opens],\n    iterate 2 { rw ← presheaf.Hcomp' },\n    have hj : (U j ∩ D OX f) ∩ (U k ∩ D OX f) ⊆ U j :=\n      set.subset.trans (set.inter_subset_left _ _) (set.inter_subset_left _ _),\n    have : set.subset.trans (res_to_inter_left._proof_1 (U j ∩ D OX f) (U k ∩ D OX f))\n      (set.inter_subset_left ((U j).val) ((D OX f).val)) = hj := rfl, rw this at *, clear this,\n    have hk : (U j ∩ D OX f) ∩ (U k ∩ D OX f) ⊆ U k :=\n      set.subset.trans (set.inter_subset_right _ _) (set.inter_subset_left _ _),\n    have : set.subset.trans (res_to_inter_right._proof_1 (U j ∩ D OX f) (U k ∩ D OX f))\n      (set.inter_subset_left ((U k).val) ((D OX f).val)) = hk := rfl, rw this at *, clear this,\n    have := congr_arg (OX.O.F.res (U k) ((U j ∩ D OX f) ∩ (U k ∩ D OX f)) hk) (hg k),\n    rw [res_mul, res_one] at this,\n    rw [← presheaf.Hcomp', presheaf.Hcomp' _ ⊤ (U j) _ hj (set.subset_univ _)] at this,\n    rw [← mul_one (OX.O.F.res (U j) ((U j ∩ D OX f) ∩ (U k ∩ D OX f)) _ (g j)), ← this],\n    rw [← mul_assoc, ← res_mul, mul_comm (g j), hg j, res_one, one_mul] },\n  cases this with S hS, use S, apply OX.O.locality (topological_space.opens.covering_of_opens (D OX f) U hu),\n  intros i, specialize hS i,\n  rw [res_mul, hS, ← presheaf.Hcomp'],\n  rw [presheaf.Hcomp' _ ⊤ (U i) ((topological_space.opens.covering_of_opens (D OX f) U hu).Uis i)\n        (set.inter_subset_left ((U i).val) ((D OX f).val)) (set.subset_univ _)],\n  rw [← res_mul, hg, res_one, res_one]\nend\n\nend locally_ringed_space\n\nnamespace sheaf_of_rings\n\nopen topological_space\nvariables {X : Type u} [topological_space X]\nvariables {B : set (opens X)} (HB : opens.is_basis B) (Bstd : opens.univ ∈ B ∧ ∀ {U V}, U ∈ B → V ∈ B → U ∩ V ∈ B)\nvariables (O : sheaf_of_rings X)\ninclude HB\nset_option pp.universes true\n\ntheorem ext (U : opens X) (f g : O U) (hfg : ∀ x ∈ U, to_stalk O.F x U H f = to_stalk O.F x U H g) : f = g :=\nbegin\n  have := λ x : U, quotient.exact (hfg x.1 x.2), choose C hc1 hc2 hc3 hc4,\n  refine O.locality (opens.covering_of_opens U C hc1) f g (λ i, _),\n  calc  O.F.res U (C i ∩ U) _ f\n      = O.F.res (C i) (C i ∩ U) (set.inter_subset_left _ _) (O.F.res U (C i) _ f) : presheaf.Hcomp' _ _ _ _ _ _ _\n  ... = O.F.res (C i) (C i ∩ U) _ (O.F.res U (C i) _ g) : congr_arg _ (hc4 i)\n  ... = O.F.res U (C i ∩ U) _ g : (presheaf.Hcomp' _ _ _ _ _ _ _).symm\nend\n\ntheorem is_sheaf_on_standard_basis_presheaf_of_rings_to_presheaf_of_rings_on_basis :\n  sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd (presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F : presheaf_of_rings_on_basis X HB).to_presheaf_on_basis :=\nbegin\n  dsimp only [sheaf_on_standard_basis.is_sheaf_on_standard_basis],\n  intros U HUB OC, exact ⟨O.locality OC.to_covering, O.gluing OC.to_covering⟩\nend\n\nnoncomputable def of_extension_basis (U : opens X) (HUB : U ∈ B)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) : O U :=\nclassical.some $ to_presheaf_of_rings_extension.surjective Bstd (presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)\n  (is_sheaf_on_standard_basis_presheaf_of_rings_to_presheaf_of_rings_on_basis HB Bstd O) HUB f\n\ndef of_extension.aux1 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U) : opens X :=\nopens.some $ f.2 i.1 i.2\n\ntheorem of_extension.aux2 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U) :\n  of_extension.aux1 HB Bstd U f i ∈ B :=\nclassical.some $ opens.some_spec $ f.2 i.1 i.2\n\ntheorem of_extension.aux3 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U) :\n  i.1 ∈ of_extension.aux1 HB Bstd U f i :=\nclassical.some $ classical.some_spec $ opens.some_spec $ f.2 i.1 i.2\n\nnoncomputable def of_extension.aux4 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U) :\n  F.to_presheaf_on_basis (of_extension.aux2 HB Bstd U f i) :=\nclassical.some $ classical.some_spec $ classical.some_spec $ opens.some_spec $ f.2 i.1 i.2\n\ntheorem of_extension.aux5 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U) :\n  ∀ (y : X) (H : y ∈ U ∩ of_extension.aux1 HB Bstd U f i),\n    f.1 y = λ (_x : y ∈ U), ⟦{U := of_extension.aux1 HB Bstd U f i, BU := _, Hx := H.2, s := of_extension.aux4 HB Bstd U f i}⟧ :=\nclassical.some_spec $ classical.some_spec $ classical.some_spec $ opens.some_spec $ f.2 i.1 i.2\n\ntheorem of_extension.aux6 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U)\n  (y : X) (hyU : y ∈ U) (hyi : y ∈ of_extension.aux1 HB Bstd U f i) :\n  ⟦quotient.out (f.1 y hyU)⟧ = ⟦{U := of_extension.aux1 HB Bstd U f i, BU := _, Hx := hyi, s := of_extension.aux4 HB Bstd U f i}⟧ :=\nby rw [of_extension.aux5 HB Bstd U f i y ⟨hyU, hyi⟩, quotient.out_eq]\n\ntheorem of_extension.aux7 {F : presheaf_of_rings_on_basis X HB} (U : opens X) (f : (F ᵣₑₓₜ Bstd) U) (i : U)\n  (y : X) (hyU : y ∈ U) (hyi : y ∈ of_extension.aux1 HB Bstd U f i) :\n  quotient.out (f.1 y hyU) ≈ {U := of_extension.aux1 HB Bstd U f i, BU := _, Hx := hyi, s := of_extension.aux4 HB Bstd U f i} :=\nquotient.exact $ of_extension.aux6 HB Bstd U f i y hyU hyi\n\nlemma of_extension.aux8 (U : opens X)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) :\n  ∃ f' : O U, ∀ i : U,\n    O.F.res U ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) (of_extension.aux3 HB Bstd U f)).Uis i)\n      (subset_covering i)\n      f' =\n    O.F.res (of_extension.aux1 HB Bstd U f i) _ (set.inter_subset_left _ _)\n      (of_extension.aux4 HB Bstd U f i) :=\nbegin\n  refine quotient.induction_on\n    (quotient.choice (λ i:U, (f.1 i.1 i.2 : stalk_on_basis ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd (O.F)).to_presheaf_on_basis) i.1)))\n    (λ f', _),\n  have, refine (O.gluing (opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) (of_extension.aux3 HB Bstd U f)) _ _),\n  { exact (λ i, O.F.res _ _ (set.inter_subset_left _ _) $ of_extension.aux4 HB Bstd U f i) },\n  { have := of_extension.aux7 HB Bstd U f, choose g hg1 hg2 hg3 hg4 hg5,\n    refine (λ j k : U, O.locality (opens.covering_of_opens _\n        (λ i, g j i.1 i.2.1.2 i.2.1.1 ∩ g k i.1 i.2.2.2 i.2.2.1)\n        (λ i, ⟨hg2 _ _ _ _, hg2 _ _ _ _⟩)) _ _ _),\n    intros i, dsimp only [opens.covering_of_opens, res_to_inter_left, res_to_inter_right],\n    iterate 4 { rw ← presheaf.Hcomp' },\n    have hg5j := hg5 j i.1 i.2.1.2 i.2.1.1, have hg5k := hg5 k i.1 i.2.2.2 i.2.2.1,\n    dsimp only [presheaf_of_rings_to_presheaf_of_rings_on_basis] at hg5j hg5k,\n    rw [O.F.to_presheaf.Hcomp' (of_extension.aux1 HB Bstd U f j) (g j i.1 i.2.1.2 i.2.1.1), ← hg5j],\n    rw [O.F.to_presheaf.Hcomp' (of_extension.aux1 HB Bstd U f k) (g k i.1 i.2.2.2 i.2.2.1), ← hg5k],\n    iterate 2 { rw ← presheaf.Hcomp' },\n    { exact set.subset.trans (set.inter_subset_left _ _) (set.inter_subset_right _ _) },\n    { exact set.subset.trans (set.inter_subset_left _ _) (set.inter_subset_left _ _) } },\n  exact this\nend\n\nnoncomputable def of_extension (U : opens X)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) : O U :=\nclassical.some $ of_extension.aux8 HB Bstd O U f\n\nlemma of_extension_spec' (U : opens X)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) :\n  ∀ i : U,\n    O.F.res U ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) (of_extension.aux3 HB Bstd U f)).Uis i)\n      (subset_covering i)\n      (of_extension HB Bstd O U f) =\n    O.F.res (of_extension.aux1 HB Bstd U f i) _ (set.inter_subset_left _ _)\n      (of_extension.aux4 HB Bstd U f i) :=\nclassical.some_spec $ of_extension.aux8 HB Bstd O U f\n\nlemma of_extension_spec (U : opens X)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U)\n  (x : X) (hxU : x ∈ U) :\n  ∃ (V : opens X) (HVB : V ∈ B) (hxV : x ∈ V) (HVU : V ⊆ U), f.1 x hxU = ⟦⟨V, HVB, hxV, O.F.res U V HVU (of_extension HB Bstd O U f)⟩⟧ :=\nbegin\n  have hx : x ∈ of_extension.aux1 HB Bstd U f ⟨x, hxU⟩ ∩ U := ⟨of_extension.aux3 HB Bstd U f ⟨x, hxU⟩, hxU⟩,\n  rcases opens.is_basis_iff_nbhd.1 HB hx with ⟨V, HVB, hxV, HV⟩,\n  refine ⟨V, HVB, hxV, set.subset.trans HV (set.inter_subset_right _ _), _⟩,\n  rw [congr_fun (of_extension.aux5 HB Bstd U f ⟨x, hxU⟩ x ⟨hxU, of_extension.aux3 HB Bstd U f ⟨x, hxU⟩⟩) hxU],\n  refine quotient.sound ⟨V, HVB, hxV, set.subset.trans HV (set.inter_subset_left _ _), set.subset.refl _, _⟩,\n  dsimp only [presheaf_of_rings_to_presheaf_of_rings_on_basis], rw [← presheaf.Hcomp'],\n  calc  O.F.res (of_extension.aux1 HB Bstd U f ⟨x, hxU⟩) V _ (of_extension.aux4 HB Bstd U f ⟨x, hxU⟩)\n      = O.F.res ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) _).Uis ⟨x, hxU⟩) V _\n          (O.F.res (of_extension.aux1 HB Bstd U f ⟨x, hxU⟩)\n            ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) _).Uis ⟨x, hxU⟩) _\n            (of_extension.aux4 HB Bstd U f ⟨x, hxU⟩)) : presheaf.Hcomp' _ _ _ _ _ _ _\n  ... = O.F.res ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) _).Uis ⟨x, hxU⟩) V HV\n          (O.F.res U ((opens.covering_of_opens U (of_extension.aux1 HB Bstd U f) _).Uis ⟨x, hxU⟩) _\n            (of_extension HB Bstd O U f)) : congr_arg _ (of_extension_spec' HB Bstd O U f ⟨x, hxU⟩).symm\n  ... = O.F.res U V _ (of_extension HB Bstd O U f) : (presheaf.Hcomp' _ _ _ _ _ _ _).symm\nend\n\ntheorem of_extension_res (U V : opens X) (HVU : V ⊆ U)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) :\n  of_extension HB Bstd O V (presheaf.res _ U V HVU f) = O.F.res U V HVU (of_extension HB Bstd O U f) :=\nbegin\n  apply sheaf_of_rings.ext, { exact HB }, intros x hxV, apply quotient.sound,\n  rcases of_extension_spec HB Bstd O U f x (HVU hxV) with ⟨W1, HWB1, hxW1, HWU1, hw1⟩,\n  rcases of_extension_spec HB Bstd O V (presheaf.res _ U V HVU f) x hxV with ⟨W2, HWB2, hxW2, HWV2, hw2⟩,\n  rcases quotient.exact (hw1.symm.trans hw2) with ⟨W3, HWB3, hxW3, HW31, HW32, hw3⟩,\n  refine ⟨W3, hxW3, set.subset.trans HW32 HWV2, set.subset.trans HW32 HWV2, _⟩,\n  change O.F.res W1 W3 HW31 (O.F.res U W1 HWU1 (of_extension HB Bstd O U f)) =\n    O.F.res W2 W3 HW32 (O.F.res V W2 HWV2 (of_extension HB Bstd O V $ presheaf.res _ U V HVU f)) at hw3,\n  change O.F.res V W3 _ (of_extension HB Bstd O V $ presheaf.res _ U V HVU f) =\n    (O.F.res V W3 _ (O.F.res U V HVU (of_extension HB Bstd O U f))),\n  rw [← presheaf.Hcomp', ← presheaf.Hcomp'] at hw3, rw [← presheaf.Hcomp'], exact hw3.symm\nend\nset_option pp.proofs true\n\ndef to_extension (U : opens X) (f : O U) :\n  ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F : presheaf_of_rings_on_basis X HB)ᵣₑₓₜ Bstd) U :=\n{ val := λ x hxU, ⟦{ U := opens.some (opens.is_basis_iff_nbhd.1 HB hxU),\n    BU := classical.some $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU),\n    Hx := (classical.some_spec $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU)).1,\n    s := O.F.res _ _ (classical.some_spec $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU)).2 f }⟧,\n  property := λ x hxU, ⟨opens.some (opens.is_basis_iff_nbhd.1 HB hxU),\n    classical.some $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU),\n    (classical.some_spec $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU)).1,\n    O.F.res _ _ (classical.some_spec $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hxU)).2 f,\n    λ y hy, funext $ λ hyU, have y ∈ opens.some (opens.is_basis_iff_nbhd.1 HB hxU) ∩ opens.some (opens.is_basis_iff_nbhd.1 HB hyU),\n      from ⟨hy.2, (classical.some_spec $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB hyU)).1⟩,\n      let ⟨V, HVB, hyV, HV⟩ := opens.is_basis_iff_nbhd.1 HB this in quotient.sound $\n      ⟨V, HVB, hyV, set.subset.trans HV (set.inter_subset_right _ _),\n        set.subset.trans HV (set.inter_subset_left _ _),\n        show O.F.res _ _ _ (O.F.res _ _ _ _) = O.F.res _ _ _ (O.F.res _ _ _ _),\n        by rw [← presheaf.Hcomp', ← presheaf.Hcomp']⟩⟩ }\n\ntheorem of_to_extension (U : opens X) (f : O U) :\n  of_extension HB Bstd O U (to_extension HB Bstd O U f) = f :=\nbegin\n  apply sheaf_of_rings.ext, { exact HB }, intros x hxU,\n  rcases of_extension_spec HB Bstd O U (to_extension HB Bstd O U f) x hxU with ⟨V, HVB, hsV, HVU, hv⟩,\n  rcases quotient.exact hv with ⟨W, HWB, hxW, HW1, HW2, hw⟩,\n  refine quotient.sound ⟨W, hxW, set.subset.trans HW2 HVU, set.subset.trans HW2 HVU, _⟩,\n  change O.F.res _ _ _ (O.F.res _ _ _ _) = O.F.res _ _ _ (O.F.res _ _ _ _) at hw,\n  rw [← presheaf.Hcomp', ← presheaf.Hcomp'] at hw, exact hw.symm\nend\n\ntheorem to_of_extension (U : opens X)\n  (f : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U) :\n  to_extension HB Bstd O U (of_extension HB Bstd O U f) = f :=\nsubtype.eq $ funext $ λ x, funext $ λ hxU, eq.symm $ let ⟨V, HVB, hxV, HVU, hv⟩ := of_extension_spec HB Bstd O U f x hxU in\nhave x ∈ V ∩ classical.some (to_extension._proof_1 HB U x hxU) := ⟨hxV, to_extension._proof_3 HB U x hxU⟩,\nlet ⟨W, HWB, hxW, HWV⟩ := opens.is_basis_iff_nbhd.1 HB this in\nhv.trans $ quotient.sound ⟨W, HWB, hxW, set.subset.trans HWV (set.inter_subset_left _ _), set.subset.trans HWV (set.inter_subset_right _ _),\nshow O.F.res _ _ _ (O.F.res _ _ _ _) = O.F.res _ _ _ (O.F.res _ _ _ _),\nby rw [← presheaf.Hcomp', ← presheaf.Hcomp']⟩\n\ninstance is_ring_hom_to_extension (U : opens X) : is_ring_hom (to_extension HB Bstd O U) :=\n{ map_one := subtype.eq $ funext $ λ x, funext $ λ hxU, quotient.sound\n    ⟨opens.some (to_extension._proof_1 HB U x hxU),\n    to_extension._proof_2 HB U x hxU, to_extension._proof_3 HB U x hxU,\n    set.subset.refl _, set.subset_univ _,\n    show O.F.res _ _ _ (O.F.res _ _ _ 1) = O.F.res _ _ _ 1,\n    by rw [res_one, res_one, res_one]⟩,\n  map_mul := λ s1 s2, subtype.eq $ funext $ λ x, funext $ λ hxU, quotient.sound\n    ⟨opens.some (to_extension._proof_1 HB U x hxU),\n    to_extension._proof_2 HB U x hxU, to_extension._proof_3 HB U x hxU,\n    set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _),\n    show O.F.res _ _ _ (O.F.res _ _ _ (s1 * s2)) = O.F.res _ _ _ (O.F.res _ _ _ (O.F.res _ _ _ _) * O.F.res _ _ _ (O.F.res _ _ _ _)),\n    by iterate 3 { rw res_mul }; iterate 6 { rw ← presheaf.Hcomp' }⟩,\n  map_add := λ s1 s2, subtype.eq $ funext $ λ x, funext $ λ hxU, quotient.sound\n    ⟨opens.some (to_extension._proof_1 HB U x hxU),\n    to_extension._proof_2 HB U x hxU, to_extension._proof_3 HB U x hxU,\n    set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _),\n    show O.F.res _ _ _ (O.F.res _ _ _ (s1 + s2)) = O.F.res _ _ _ (O.F.res _ _ _ (O.F.res _ _ _ _) + O.F.res _ _ _ (O.F.res _ _ _ _)),\n    by iterate 3 { rw res_add }; iterate 6 { rw ← presheaf.Hcomp' }⟩ }\n\nnoncomputable def ring_equiv_extension (U : opens X) :\n  O U ≃+* ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U :=\nring_equiv.of\n{ to_fun := to_extension HB Bstd O U,\n  inv_fun := of_extension HB Bstd O U,\n  left_inv := of_to_extension HB Bstd O U,\n  right_inv := to_of_extension HB Bstd O U }\n\ninstance is_ring_hom_of_extension (U : opens X) : is_ring_hom (of_extension HB Bstd O U) :=\nring_equiv.hom (ring_equiv_extension _ _ _ _).symm\n\ndef stalk_on_basis_to_stalk (F : presheaf_of_rings X) (U : opens X) (p : X) (hpU : p ∈ U)\n  (σ : stalk_on_basis (presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd F : presheaf_of_rings_on_basis X HB).to_presheaf_on_basis p) :\n  stalk_of_rings F p :=\nquotient.lift_on σ (λ e, to_stalk F p e.U e.Hx e.s) $ λ e₁ e₂ ⟨V, HVB, hpV, HV1, HV2, HV⟩,\nquotient.sound ⟨V, hpV, HV1, HV2, HV⟩\n\ndef stalk_to_stalk_on_basis (F : presheaf_of_rings.{u v} X) (U : opens X) (p : X) (hpU : p ∈ U)\n  (σ : stalk_of_rings F p) :\n  stalk_on_basis (presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd F : presheaf_of_rings_on_basis X HB).to_presheaf_on_basis p :=\nquotient.lift_on σ (λ e, (⟦⟨opens.some (opens.is_basis_iff_nbhd.1 HB e.HxU),\n    Exists.fst $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB e.HxU),\n    and.left $ Exists.snd $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB e.HxU),\n    presheaf.res _ _ _ (and.right $ Exists.snd $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB e.HxU)) e.s⟩⟧ :\n      stalk_on_basis (presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd F : presheaf_of_rings_on_basis X HB).to_presheaf_on_basis p)) $\nλ e₁ e₂ ⟨V, hpV, HV1, HV2, HV⟩,\nhave p ∈ V ∩ (opens.some (opens.is_basis_iff_nbhd.1 HB e₁.HxU) ∩ opens.some (opens.is_basis_iff_nbhd.1 HB e₂.HxU)),\nfrom ⟨hpV, and.left $ Exists.snd $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB e₁.HxU),\n  and.left $ Exists.snd $ opens.some_spec (opens.is_basis_iff_nbhd.1 HB e₂.HxU)⟩,\nlet ⟨W, HWB, hpW, HWV⟩ := opens.is_basis_iff_nbhd.1 HB this in\nquotient.sound ⟨W, HWB, hpW, set.subset.trans HWV (set.subset.trans (set.inter_subset_right _ _) (set.inter_subset_left _ _)),\nset.subset.trans HWV (set.subset.trans (set.inter_subset_right _ _) (set.inter_subset_right _ _)),\nhave _ := congr_arg (F.res _ W (set.subset.trans HWV (set.inter_subset_left _ _))) HV,\nshow F.res _ _ _ (F.res _ _ _ _) = F.res _ _ _ (F.res _ _ _ _),\nby rw [← presheaf.Hcomp', ← presheaf.Hcomp'] at this ⊢; exact this⟩\n\ntheorem to_stalk_of_extension (U : opens X)\n  (F : ((presheaf_of_rings_to_presheaf_of_rings_on_basis Bstd O.F)ᵣₑₓₜ Bstd) U)\n  (p : X) (hpU : p ∈ U) :\n  to_stalk O.F p U hpU (of_extension HB Bstd O U F) = stalk_on_basis_to_stalk _ _ _ U _ hpU (F.1 p hpU) :=\nlet ⟨V, HVB, hpV, HVU, hv⟩ := of_extension_spec HB Bstd O U F p hpU in\nhv.symm ▸ quotient.sound ⟨V, hpV, HVU, set.subset.refl V, presheaf.Hcomp' _ _ _ _ _ _ _⟩\n\nend sheaf_of_rings\n\nnamespace sheaf_of_rings\n\nvariables {α : Type u} [topological_space α]\nvariables {β : Type u} [topological_space β]\n\nvariables {f : α → β} (Hf : continuous f)\n\ndef pushforward (O : sheaf_of_rings α) : sheaf_of_rings β :=\n{ F := O.F.pushforward Hf,\n  locality := λ U OC s t H, O.locality (topological_space.opens.covering_comap Hf U OC) _ _ (λ i, by convert H i),\n  gluing := λ U OC s H, by convert O.gluing (topological_space.opens.covering_comap Hf U OC) s H }\n\nend sheaf_of_rings\n\ntheorem presheaf.fmap.commutes' {α : Type u} [topological_space α] {β : Type v} [topological_space β]\n  {f : α → β} {hf : continuous f} {F : presheaf α} {G : presheaf β} (m : presheaf.fmap hf F G)\n  (U V : topological_space.opens β) (HVU : V ⊆ U) (s : G U) (h : opens.comap hf V ⊆ opens.comap hf U) :\n  F.res (opens.comap hf U) (opens.comap hf V) h (m.map U s) = m.map V (G.res U V HVU s) :=\n(congr_fun (m.commutes U V HVU) s).symm\n\ndef presheaf.fmap.stalk {α : Type u} [topological_space α] {β : Type v} [topological_space β]\n  {f : α → β} {hf : continuous f} {F : presheaf α} {G : presheaf β} (m : presheaf.fmap hf F G) (x : α) :\n  stalk G (f x) → stalk F x :=\nquotient.lift (λ s : stalk.elem G (f x), (⟦⟨opens.comap hf s.U, s.HxU, m.map s.U s.s⟩⟧ : stalk F x)) $\nλ s1 s2 ⟨U, hfxU, HUs1, HUs2, HU⟩, quotient.sound ⟨opens.comap hf U, hfxU, opens.comap_mono _ _ _ HUs1, opens.comap_mono _ _ _ HUs2,\nshow F.res (opens.comap hf s1.U) (opens.comap hf U) _ (m.map s1.U s1.s) =\n  F.res (opens.comap hf s2.U) (opens.comap hf U) _ (m.map s2.U s2.s),\nby rw [m.commutes', m.commutes', HU]⟩\n\ndef presheaf.fmap.stalk_of_rings {α : Type u} [topological_space α] {β : Type v} [topological_space β]\n  {f : α → β} {hf : continuous f} {F : presheaf_of_rings α} {G : presheaf_of_rings β} (m : presheaf_of_rings.fmap hf F G)\n  (x : α) (s : stalk_of_rings G (f x)) : stalk_of_rings F x :=\nm.stalk x s\n\ntheorem is_ring_hom_stalk_of_rings {α : Type u} [topological_space α] {β : Type v} [topological_space β]\n  {f : α → β} {hf : continuous f} {F : presheaf_of_rings α} {G : presheaf_of_rings β} (m : presheaf_of_rings.fmap hf F G) [hm : ∀ U, is_ring_hom (m.map U)]\n  (x : α) : is_ring_hom (presheaf.fmap.stalk_of_rings m x) :=\n{ map_one := quotient.sound ⟨⊤, trivial, set.subset.refl _, set.subset.refl _, congr_arg _ (is_ring_hom.map_one (m.map ⊤))⟩,\n  map_mul := λ s1 s2, quotient.induction_on₂ s1 s2 $ λ e1 e2, quotient.sound ⟨opens.comap hf (e1.U ∩ e2.U),\n    ⟨e1.HxU, e2.HxU⟩, set.subset.refl _,\n    set.subset_inter (opens.comap_mono _ _ _ (set.inter_subset_left _ _)) (opens.comap_mono _ _ _ (set.inter_subset_right _ _)),\n    show F.res (opens.comap hf (e1.U ∩ e2.U)) (opens.comap hf (e1.U ∩ e2.U)) _ (m.map (e1.U ∩ e2.U) (G.res _ _ _ e1.s * G.res _ _ _ e2.s)) =\n      F.res (opens.comap hf e1.U ∩ opens.comap hf e2.U) (opens.comap hf (e1.U ∩ e2.U)) _ (F.res _ _ _ (m.map e1.U e1.s) * F.res _ _ _ (m.map e2.U e2.s)),\n    by rw [m.commutes' _ _ (set.subset.refl _),\n        res_mul,\n        is_ring_hom.map_mul (F.res (opens.comap hf e1.U ∩ opens.comap hf e2.U) (opens.comap hf (e1.U ∩ e2.U))\n          (set.subset_inter (opens.comap_mono _ _ _ (set.inter_subset_left _ _)) (opens.comap_mono _ _ _ (set.inter_subset_right _ _)))),\n        ← presheaf.Hcomp', ← presheaf.Hcomp', ← presheaf.Hcomp', ← presheaf.Hcomp',\n        m.commutes' e1.U (e1.U ∩ e2.U) (set.inter_subset_left _ _),\n        m.commutes' e2.U (e1.U ∩ e2.U) (set.inter_subset_right _ _),\n        is_ring_hom.map_mul (m.map (e1.U ∩ e2.U))]; refl⟩,\n  map_add := λ s1 s2, quotient.induction_on₂ s1 s2 $ λ e1 e2, quotient.sound ⟨opens.comap hf (e1.U ∩ e2.U),\n    ⟨e1.HxU, e2.HxU⟩, set.subset.refl _,\n    set.subset_inter (opens.comap_mono _ _ _ (set.inter_subset_left _ _)) (opens.comap_mono _ _ _ (set.inter_subset_right _ _)),\n    show F.res (opens.comap hf (e1.U ∩ e2.U)) (opens.comap hf (e1.U ∩ e2.U)) _ (m.map (e1.U ∩ e2.U) (G.res _ _ _ e1.s + G.res _ _ _ e2.s)) =\n      F.res (opens.comap hf e1.U ∩ opens.comap hf e2.U) (opens.comap hf (e1.U ∩ e2.U)) _ (F.res _ _ _ (m.map e1.U e1.s) + F.res _ _ _ (m.map e2.U e2.s)),\n    by rw [m.commutes' _ _ (set.subset.refl _),\n        is_ring_hom.map_add (G.res (e1.U ∩ e2.U) (e1.U ∩ e2.U) (set.subset.refl _)),\n        is_ring_hom.map_add (F.res (opens.comap hf e1.U ∩ opens.comap hf e2.U) (opens.comap hf (e1.U ∩ e2.U))\n          (set.subset_inter (opens.comap_mono _ _ _ (set.inter_subset_left _ _)) (opens.comap_mono _ _ _ (set.inter_subset_right _ _)))),\n        ← presheaf.Hcomp', ← presheaf.Hcomp', ← presheaf.Hcomp', ← presheaf.Hcomp',\n        m.commutes' e1.U (e1.U ∩ e2.U) (set.inter_subset_left _ _),\n        m.commutes' e2.U (e1.U ∩ e2.U) (set.inter_subset_right _ _),\n        is_ring_hom.map_add (m.map (e1.U ∩ e2.U))]; refl⟩ }\n\nnamespace locally_ringed_space\n\nstructure morphism {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  {Y : Type v} [topological_space Y] (OY : locally_ringed_space Y) extends morphism OX OY :=\n(hom : ∀ U, is_ring_hom (fO.map U))\nattribute [instance] morphism.hom\n\nend locally_ringed_space\n\nsection tag01I1\n\nvariables {X : Type u} [topological_space X] (OX : locally_ringed_space X)\nvariables (R : Type v) [comm_ring R]\n\n/- https://stacks.math.columbia.edu/tag/01I1 -/\ndef mor_to_hom (f : OX.morphism (Spec.locally_ringed_space R)) :\n  (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤ :=\nOX.O.F.res (opens.comap f.Hf ⊤) ⊤ (opens.comap_top f.Hf ▸ set.subset.refl _) ∘ f.fO.map ⊤\n\ninstance mor_to_hom.is_ring_hom (f : OX.morphism (Spec.locally_ringed_space R)) :\n  is_ring_hom (mor_to_hom OX R f) :=\nby unfold mor_to_hom; apply_instance\n\nvariables (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f]\n\n-- variables (R : Type u) [comm_ring R]\n\ndef Spec.locally_ringed_space' : locally_ringed_space (Spec R) :=\n{ O := structure_sheaf R,\n  Hstalks := λ P, structure_sheaf.stalk_local P, }\n\ndef induced (x : X) : Spec R :=\n⟨ideal.comap (to_stalk OX.O.F x ⊤ trivial ∘ f ∘ to_Spec_top R) (local_ring.nonunits_ideal _),\n@ideal.is_prime.comap _ _ _ _ _ _ _ $ (local_ring.nonunits_ideal.is_maximal _).is_prime⟩\n\n@[simp] lemma Spec.DO.val (g : R) : (Spec.DO R g).val = Spec.D' g :=\ncongr_fun (Spec.DO.val_eq_D' R) g\n\n@[simp] lemma induced_preimage_D' (g : R) :\n  induced OX R f ⁻¹' Spec.D' g = OX.D (f $ to_Spec_top R g) :=\nset.ext $ λ x, classical.not_not\n\nlemma induced_continuous : continuous (induced OX R f) :=\nλ U HU, let ⟨Us, HUs, HUUs⟩ := topological_space.sUnion_basis_of_is_open (D_fs_basis R) HU in\nby rw [HUUs, set.preimage_sUnion]; exact is_open_Union (λ S, is_open_Union $ λ HSUs,\nlet ⟨p, ⟨g, hp⟩, hpS⟩ := HUs HSUs in by rw [← hpS, hp, Spec.DO.val, induced_preimage_D']; exact OX.is_open_D' _)\n\n@[simp] lemma comap_induced_DO (g : R) :\n  opens.comap (induced_continuous OX R f) (Spec.DO R g) = OX.D (f $ to_Spec_top R g) :=\ntopological_space.opens.ext $ induced_preimage_D' OX R f g\n\nnoncomputable def induced_basis (U : topological_space.opens $ Spec R) (HUB : U ∈ D_fs R) :\n  (structure_presheaf_on_basis R).F HUB → OX.O (opens.comap (induced_continuous OX R f) U) :=\nlocalization.lift (OX.O.F.res ⊤ _ (set.subset_univ _) ∘ f ∘ to_Spec_top R) $ λ r hrSU,\nis_unit.map''\n  (OX.O.F.res\n    (opens.comap (induced_continuous OX R f) (Spec.DO R r))\n    (opens.comap (induced_continuous OX R f) U)\n    (set.preimage_mono hrSU))\n  (OX.O.F.res ⊤ _ (set.subset_univ _) (f $ to_Spec_top R r))\n  (by rw comap_induced_DO; exact locally_ringed_space.is_unit_res_D _ _)\n  (presheaf.Hcomp' _ _ _ _ _ _ _).symm\n\nnoncomputable def induced_stalk_elem (p : Spec R)\n  (s : stalk_on_basis.elem (structure_presheaf_on_basis R).to_presheaf_on_basis p) :\n  stalk_on_basis.elem\n    (presheaf_of_rings_to_presheaf_of_rings_on_basis\n      (D_fs_standard_basis R) (presheaf_of_rings.pushforward (induced_continuous OX R f) OX.O.F) :\n        presheaf_of_rings_on_basis (Spec R) (D_fs_basis R)).to_presheaf_on_basis\n    p :=\n⟨s.1, s.2, s.3, induced_basis OX R f s.1 s.2 s.4⟩\n\nnoncomputable def induced_stalk (p : Spec R) :\n  stalk_of_rings_on_standard_basis.stalk_of_rings_on_standard_basis (D_fs_standard_basis R) (structure_presheaf_on_basis R) p →\n  stalk_of_rings_on_standard_basis.stalk_of_rings_on_standard_basis (D_fs_standard_basis R)\n    (presheaf_of_rings_to_presheaf_of_rings_on_basis\n      (D_fs_standard_basis R) (presheaf_of_rings.pushforward (induced_continuous OX R f) OX.O.F) :\n        presheaf_of_rings_on_basis (Spec R) (D_fs_basis R))\n    p :=\nquotient.lift (λ s, ⟦induced_stalk_elem OX R f p s⟧) $ begin\n  rintros s1 s2 ⟨U, HUB, hpU, hUs1, hUs2, hs⟩,\n  refine quotient.sound ⟨U, HUB, hpU, hUs1, hUs2, _⟩,\n  cases s1 with U1 HUB1 Hx1 s1, cases s2 with U2 HUB2 Hx2 s2,\n  dsimp only at hUs1 hUs2, dsimp only [induced_stalk_elem, induced_basis],\n  revert hs,\n  refine localization.induction_on s1 (λ r1 s1, localization.induction_on s2 (λ r2 s2, _)),\n  intros hs, rcases quotient.exact hs with ⟨t, htSU, ht⟩, simp only [subtype.coe_mk] at ht,\n  dsimp only [induced_stalk_elem, induced_basis, localization.lift, localization.lift'_mk, function.comp_apply,\n    presheaf_of_rings_to_presheaf_of_rings_on_basis, presheaf_of_rings.pushforward, presheaf.pushforward],\n  rw [is_ring_hom.map_mul (OX.O.F.res (opens.comap (induced_continuous OX R f) U1)\n      (opens.comap (induced_continuous OX R f) U)\n      (opens.comap_mono (induced_continuous OX R f) U U1 hUs1))],\n  rw [is_ring_hom.map_mul (OX.O.F.res (opens.comap (induced_continuous OX R f) U2)\n      (opens.comap (induced_continuous OX R f) U)\n      (opens.comap_mono (induced_continuous OX R f) U U2 hUs2))],\n  generalize : localization.lift._proof_1 _ _ _ = hu1,\n  generalize : localization.lift._proof_1 _ _ _ = hu2,\n  dsimp only [function.comp_apply] at hu1 hu2,\n  have := classical.some_spec hu2, revert this,\n  have := classical.some_spec hu1, revert this,\n  cases classical.some hu1 with v1 i1 hvi1 hiv1,\n  cases classical.some hu2 with v2 i2 hvi2 hiv2,\n  rintros h1 h2, change _ = v1 at h1, subst h1, change _ = v2 at h2, subst h2,\n  change _ * OX.O.F.res _ _ _ i1 = _ * OX.O.F.res _ _ _ i2,\n  replace ht := congr_arg (λ x, OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _) (f (to_Spec_top R x))) ht,\n  dsimp only at ht,\n  rw [is_ring_hom.map_mul (to_Spec_top R)] at ht,\n  rw [is_ring_hom.map_sub (to_Spec_top R)] at ht,\n  rw [is_ring_hom.map_mul (to_Spec_top R)] at ht,\n  rw [is_ring_hom.map_mul (to_Spec_top R)] at ht,\n  rw [is_ring_hom.map_zero (to_Spec_top R)] at ht,\n  rw [is_ring_hom.map_mul f] at ht,\n  rw [is_ring_hom.map_sub f] at ht,\n  rw [is_ring_hom.map_mul f] at ht,\n  rw [is_ring_hom.map_mul f] at ht,\n  rw [is_ring_hom.map_zero f] at ht,\n  rw [is_ring_hom.map_mul (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _))] at ht,\n  rw [is_ring_hom.map_sub (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _))] at ht,\n  rw [is_ring_hom.map_mul (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _))] at ht,\n  rw [is_ring_hom.map_mul (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _))] at ht,\n  change _ = OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _) 0 at ht,\n  rw [is_ring_hom.map_zero (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U) (set.subset_univ _))] at ht,\n  have := OX.is_unit_res_D (f (to_Spec_top R t)), rw ← comap_induced_DO at this,\n  replace this := this.map' (OX.O.F.res\n    (opens.comap (induced_continuous OX R f) (Spec.DO R t))\n    (opens.comap (induced_continuous OX R f) U)\n    (opens.comap_mono _ _ _ htSU)),\n  rw ← presheaf.Hcomp' at this,\n  cases this with u hut,\n  change (((OX.O).F).to_presheaf).res ⊤ (opens.comap (induced_continuous OX R f) U)\n        (set.subset_univ ((opens.comap (induced_continuous OX R f) U).val))\n        (f (to_Spec_top R t)) = _ at hut,\n  rw hut at ht, replace ht := congr_arg (λ z : OX.O (opens.comap (induced_continuous OX R f) U), z * (↑(u⁻¹ : units (OX.O (opens.comap (induced_continuous OX R f) U))) : OX.O _)) ht,\n  dsimp only at ht, change _ = (0 : OX.O (opens.comap (induced_continuous OX R f) U)) * _ at ht,\n  rw [mul_assoc, zero_mul, units.mul_inv, mul_one, sub_eq_zero_iff_eq] at ht,\n  rw [← presheaf.Hcomp', presheaf.Hcomp' _ ⊤ (opens.comap (induced_continuous OX R f) U2) (opens.comap (induced_continuous OX R f) U)\n      (opens.comap_mono _ _ _ hUs2) (set.subset_univ _)],\n  rw [← mul_one (OX.O.F.res ⊤ (opens.comap (induced_continuous OX R f) U2) (set.subset_univ _) (f (to_Spec_top R r1)))],\n  change (OX.O.F.res _ _ _ (_ * (1 : OX.O (opens.comap _ U2)))) * _ = _,\n  rw [← hvi2, ← mul_assoc, mul_comm (OX.O.F.res _ _ _ (f _))],\n  rw [is_ring_hom.map_mul (OX.O.F.res (opens.comap (induced_continuous OX R f) U2) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs2))],\n  rw [is_ring_hom.map_mul (OX.O.F.res (opens.comap (induced_continuous OX R f) U2) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs2))],\n  rw [← presheaf.Hcomp', ← presheaf.Hcomp', show set.subset.trans (opens.comap_mono (induced_continuous OX R f) U U2 hUs2)\n        (induced_basis._proof_1 OX R f U2) = set.subset_univ _, from rfl, ← ht],\n  rw [mul_assoc, mul_assoc, mul_comm, mul_assoc, mul_assoc],\n  replace hiv1 := congr_arg (OX.O.F.res (opens.comap (induced_continuous OX R f) U1) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs1)) hiv1,\n  rw [is_ring_hom.map_mul (OX.O.F.res (opens.comap (induced_continuous OX R f) U1) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs1))] at hiv1,\n  change _ = OX.O.F.res (opens.comap (induced_continuous OX R f) U1) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs1) 1 at hiv1,\n  rw [is_ring_hom.map_one (OX.O.F.res (opens.comap (induced_continuous OX R f) U1) (opens.comap (induced_continuous OX R f) U) (opens.comap_mono _ _ _ hUs1))] at hiv1,\n  rw [← presheaf.Hcomp', show set.subset.trans (opens.comap_mono (induced_continuous OX R f) U U1 hUs1) (induced_basis._proof_1 OX R f U1) = set.subset_univ _, from rfl] at hiv1,\n  rw [hiv1, mul_one, ← presheaf.Hcomp'], refl\nend\n\nnoncomputable def induced_sheafification {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R]\n  (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f]\n  (U : topological_space.opens (Spec R)) (s : (Spec.locally_ringed_space R).O U) :\n  ((presheaf_of_rings_to_presheaf_of_rings_on_basis\n      (D_fs_standard_basis R)\n      (OX.O.pushforward (induced_continuous OX R f)).F :\n    presheaf_of_rings_on_basis (Spec R) (D_fs_basis R))ᵣₑₓₜ (D_fs_standard_basis R)) U :=\n⟨λ x hxU, induced_stalk OX R f x (s.1 x hxU),\nλ x hxU, let ⟨V, HVB, hxV, s1, hs1⟩ := s.2 x hxU in\n⟨V, HVB, hxV, induced_basis OX R f V HVB s1, λ g hg, funext $ λ hgU, by rw hs1 g hg; refl⟩⟩\n\ntheorem induced_sheafification_to_presheaf_of_rings_extension\n  {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R]\n  (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f]\n  (U : topological_space.opens (Spec R)) (HUB : U ∈ D_fs R)\n  (σ : (structure_presheaf_on_basis R).F HUB) :\n  induced_sheafification OX R f U (to_presheaf_of_rings_extension (D_fs_standard_basis R) _ HUB σ) =\n    sheaf_of_rings.to_extension _ _ _ _ (induced_basis OX R _ _ HUB σ) :=\nsubtype.eq $ funext $ λ p, funext $ λ hpU, quotient.sound ⟨U ∩ opens.some (sheaf_of_rings.to_extension._proof_1 (D_fs_basis R) U p hpU),\n  (D_fs_standard_basis R).2 HUB (sheaf_of_rings.to_extension._proof_2 (D_fs_basis R) U p hpU),\n  ⟨hpU, (sheaf_of_rings.to_extension._proof_3 (D_fs_basis R) U p hpU)⟩,\n  set.inter_subset_left _ _, set.inter_subset_right _ _, presheaf.Hcomp' _ _ _ _ _ _ _⟩\n\ninstance is_ring_hom_induced_basis (U : topological_space.opens $ Spec R) (HUB : U ∈ D_fs R) :\n  is_ring_hom (induced_basis OX R f U HUB) :=\nlocalization.lift.is_ring_hom _ _\n\ntheorem induced_basis_res (U V : topological_space.opens $ Spec R) (HUB : U ∈ D_fs R) (HVB : V ∈ D_fs R) (HVU : V ⊆ U)\n  (s : ((structure_presheaf_on_basis R).to_presheaf_on_basis).F HUB) :\n  induced_basis OX R f V HVB (presheaf_on_basis.res _ HUB HVB HVU s) =\n  OX.O.F.res (opens.comap (induced_continuous OX R f) U) (opens.comap (induced_continuous OX R f) V)\n    (opens.comap_mono _ _ _ HVU) (induced_basis OX R f U HUB s) :=\nsuffices induced_basis OX R f V HVB ∘ presheaf_on_basis.res _ HUB HVB HVU =\n  OX.O.F.res (opens.comap (induced_continuous OX R f) U) (opens.comap (induced_continuous OX R f) V)\n    (opens.comap_mono _ _ _ HVU) ∘ induced_basis OX R f U HUB, from congr_fun this s,\nlocalization.funext _ _ $ λ x, show localization.lift (OX.O.F.res ⊤ (opens.comap _ V) _ ∘ f ∘ to_Spec_top R) _ ↑x =\n  OX.O.F.res (opens.comap _ U) (opens.comap _ V) _ (localization.lift (OX.O.F.res ⊤ (opens.comap _ U) _ ∘ f ∘ to_Spec_top R) _ ↑x),\nby rw [localization.lift_coe, localization.lift_coe, ← presheaf.Hcomp']; refl\n\ninstance is_ring_hom_induced_stalk (p : Spec R) :\n  is_ring_hom (induced_stalk OX R f p) :=\n{ map_one := congr_arg quotient.mk (congr_arg (stalk_on_basis.elem.mk _ _ _) (by exact is_ring_hom.map_one (induced_basis _ _ _ _ _))),\n  map_mul := λ x y, quotient.induction_on₂ x y $ λ s t, congr_arg quotient.mk\n    (congr_arg (stalk_on_basis.elem.mk _ _ _) (by exact (is_ring_hom.map_mul (induced_basis _ _ _ _ _)).trans\n      (by rw [induced_basis_res, induced_basis_res]; refl))),\n  map_add := λ x y, quotient.induction_on₂ x y $ λ s t, congr_arg quotient.mk\n    (congr_arg (stalk_on_basis.elem.mk _ _ _) (by exact (is_ring_hom.map_add (induced_basis _ _ _ _ _)).trans\n      (by rw [induced_basis_res, induced_basis_res]; refl))) }\n\ninstance is_ring_hom_induced_sheafification {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R]\n  (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f]\n  (U : topological_space.opens (Spec R)) :\n  is_ring_hom (induced_sheafification OX R f U) :=\n{ map_one := subtype.eq $ funext $ λ x, funext $ λ hxU, by exact is_ring_hom.map_one (induced_stalk OX R f x),\n  map_mul := λ x y, subtype.eq $ funext $ λ p, funext $ λ hpU, by change induced_stalk _ _ _ _ _ = _ * _;\n    rw [Fext_mul.eq, is_ring_hom.map_mul (induced_stalk OX R f p)],\n  map_add := λ x y, subtype.eq $ funext $ λ p, funext $ λ hpU, by change induced_stalk _ _ _ _ _ = _ + _;\n    rw [Fext_add.eq, is_ring_hom.map_add (induced_stalk OX R f p)] }\n\ntheorem to_stalk_res {X : Type u} [topological_space X] (F : presheaf_of_rings X) (x : X)\n  (U V : topological_space.opens X) (hxU : x ∈ U) (hxV : x ∈ V) (HVU : V ⊆ U) (s : F U) :\n  to_stalk F x V hxV (F.res U V HVU s) = to_stalk F x U hxU s :=\nquotient.sound ⟨V, hxV, set.subset.refl V, HVU, (presheaf.Hcomp' _ _ _ _ _ _ _).symm⟩\n\ntheorem is_unit_to_stalk {X : Type u} [topological_space X] (F : presheaf_of_rings X) (x : X)\n  (U : topological_space.opens X) (hxU : x ∈ U) (s : F U) :\n  is_unit (to_stalk F x U hxU s) ↔\n  ∃ V : topological_space.opens X, ∃ hxV : x ∈ V, ∃ HVU : V ⊆ U, is_unit (F.res U V HVU s) :=\n⟨λ hu, let ⟨t, ht⟩ := is_unit_iff_exists_inv.1 hu in quotient.induction_on t (λ ⟨V, hxV, τ⟩ H,\nlet ⟨W, hxW, HWUV, HWtop, HW⟩ := quotient.exact H in ⟨W, hxW, set.subset.trans HWUV (set.inter_subset_left _ _),\nis_unit_iff_exists_inv.2 ⟨F.res V W (set.subset.trans HWUV (set.inter_subset_right _ _)) τ,\nby dsimp only at HW; rw [res_mul, ← presheaf.Hcomp', ← presheaf.Hcomp'] at HW; exact HW.trans (res_one _ _ _ _)⟩⟩) ht,\nλ ⟨V, hxV, HVU, hv⟩, hv.map'' (to_stalk F x V hxV) _ (to_stalk_res _ _ _ _ _ _ _ _)⟩\n\nsection presheaf_of_rings_extension\n\ninstance to_presheaf_of_rings_extension.is_ring_hom' {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB)\n  {U : topological_space.opens α} (BU : U ∈ B)\n: is_ring_hom (to_presheaf_of_rings_extension Bstd F BU) :=\n{ map_one :=\n    begin\n      apply subtype.eq,\n      funext x Hx,\n      apply quotient.sound,\n      use [U, BU, Hx, set.subset.refl _, set.subset_univ _],\n      iterate 2 { erw (F.res_is_ring_hom _ _ _).map_one, },\n    end,\n  map_mul :=\n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz],\n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_mul,\n      erw ←presheaf_on_basis.Hcomp',\n      refl\n    end,\n  map_add :=\n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz],\n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_add,\n      erw ←presheaf_on_basis.Hcomp',\n      refl\n    end, }\n\ntheorem res_to_presheaf_of_rings_extension {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB) (U V : topological_space.opens α) (HUB : U ∈ B) (HVB : V ∈ B) (HVU : V ⊆ U) (σ : F.F HUB) :\n  presheaf.res _ U V HVU (to_presheaf_of_rings_extension Bstd F HUB σ) =\n  to_presheaf_of_rings_extension Bstd F HVB (F.res HUB HVB HVU σ) :=\nsubtype.eq $ funext $ λ x, funext $ λ hxV, quotient.sound ⟨V, HVB, hxV, HVU, set.subset.refl V, presheaf_on_basis.Hcomp' _ _ _ _ _ _ _⟩\n\nnoncomputable def of_presheaf_of_rings_extension {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB)\n  (HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n  {U : topological_space.opens α} (HUB : U ∈ B) :\n  (F ᵣₑₓₜ Bstd).F U → F.F HUB :=\n(to_presheaf_of_rings_extension.ring_equiv Bstd F HF HUB).symm.to_fun\n\ninstance of_presheaf_of_rings_extension.is_ring_hom {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB)\n  (HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n  {U : topological_space.opens α} (HUB : U ∈ B) :\n  is_ring_hom (of_presheaf_of_rings_extension HB Bstd F HF HUB) :=\n(to_presheaf_of_rings_extension.ring_equiv Bstd F HF HUB).symm.hom\n\ntheorem of_to_presheaf_of_rings_extension {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB)\n  (HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n  {U : topological_space.opens α} (HUB : U ∈ B) (σ : F.F HUB) :\n  of_presheaf_of_rings_extension HB Bstd F HF HUB (to_presheaf_of_rings_extension Bstd F HUB σ) = σ :=\n(to_presheaf_of_rings_extension.ring_equiv Bstd F HF HUB).symm_apply_apply σ\n\ntheorem to_of_presheaf_of_rings_extension {α : Type u} [topological_space α]\n  {B : set (topological_space.opens α)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis α HB)\n  (HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n  {U : topological_space.opens α} (HUB : U ∈ B) (σ : (F ᵣₑₓₜ Bstd) U) :\n  to_presheaf_of_rings_extension Bstd F HUB (of_presheaf_of_rings_extension HB Bstd F HF HUB σ) = σ :=\n(to_presheaf_of_rings_extension.ring_equiv Bstd F HF HUB).apply_symm_apply σ\n\nend presheaf_of_rings_extension\n\ntheorem is_unit_localization_mk {R : Type u} [comm_ring R] {S : set R} [is_submonoid S]\n  (r : R) (s : S) : is_unit (localization.mk r s) ↔ ∃ t, r * t ∈ S :=\nbegin\n  rw is_unit_iff_exists_inv, split,\n  { intros hu, cases hu with w hu, revert hu,\n    refine localization.induction_on w (λ r2 s2 hu, _),\n    rcases quotient.exact hu with ⟨t, hts, ht⟩,\n    change (s.1 * s2.1 * 1 - 1 * (r * r2)) * t = 0 at ht,\n    rw [mul_one, one_mul, sub_mul, sub_eq_zero_iff_eq] at ht,\n    refine ⟨r2 * t, _⟩, rw [← mul_assoc, ← ht], exact (s * s2 * ⟨t, hts⟩).2 },\n  { rintros ⟨t, hrt⟩, refine ⟨localization.mk (s.1 * t) ⟨r * t, hrt⟩, quotient.sound $ localization.r_of_eq _⟩,\n    change (s.1 * (r * t)) * 1 = 1 * (r * (s.val * t)), rw [mul_one, one_mul, mul_left_comm] }\nend\n\ntheorem is_unit_to_stalk_affine (p : Spec R)\n  (U : topological_space.opens (Spec R)) (HUB : U ∈ D_fs R) (hpU : p ∈ U) (σ : (structure_presheaf_on_basis R).F HUB) :\n  is_unit (to_stalk (Spec.locally_ringed_space R).O.F p U hpU (to_presheaf_of_rings_extension _ _ HUB σ)) ↔\n  ∃ r : R, ∃ s : S U, p ∈ Spec.D' r ∧ localization.mk r s = σ :=\nbegin\n  rw is_unit_to_stalk, split,\n  { rintros ⟨V, hpV, HVU, hv⟩,\n    rcases topological_space.opens.is_basis_iff_nbhd.1 (D_fs_basis R) hpV with ⟨W, HWB, hpW, HWV⟩,\n    have := hv.map' (presheaf.res _ V W HWV), rw ← presheaf.Hcomp' at this,\n    dsimp only [Spec.locally_ringed_space, structure_sheaf, structure_sheaf.presheaf] at this,\n    rw [res_to_presheaf_of_rings_extension _ _ _ _ _ HUB HWB] at this,\n    replace this := this.map' (of_presheaf_of_rings_extension (D_fs_basis R) (D_fs_standard_basis R)\n      (structure_presheaf_on_basis R) structure_presheaf_on_basis_is_sheaf_on_basis HWB),\n    rw of_to_presheaf_of_rings_extension at this, revert this, refine localization.induction_on σ (λ r s this, _),\n    change is_unit (localization.mk r ⟨s.1, _⟩) at this, rw is_unit_localization_mk at this, cases this with t hrt,\n    change W.1 ⊆ Spec.D' (r * t) at hrt, rw Spec.D'.product_eq_inter at hrt,\n    refine ⟨r, s, (hrt hpW).1, rfl⟩ },\n  { rintros ⟨r, s, hrp, rfl⟩, refine ⟨U ∩ Spec.DO R r, ⟨hpU, hrp⟩, set.inter_subset_left _ _, _⟩,\n    dsimp only [Spec.locally_ringed_space, structure_sheaf, structure_sheaf.presheaf],\n    rw res_to_presheaf_of_rings_extension _ _ _ _ _ _ ((D_fs_standard_basis R).2 HUB (D_fs.mem R r)),\n    refine is_unit.map' _ _, rw is_unit_iff_exists_inv, refine ⟨localization.mk s.1 ⟨r, set.inter_subset_right _ _⟩, _⟩,\n    change localization.mk r ⟨s.1, _⟩ * localization.mk s.1 ⟨r, _⟩ = _,\n    rw [localization.mk_mul_mk, mul_comm], exact localization.mk_self }\nend\n\ntheorem is_unit_to_stalk_affine' (p : Spec R)\n  (U : topological_space.opens (Spec R)) (HUB : U ∈ D_fs R) (hpU : p ∈ U) (σ : (structure_presheaf_on_basis R).F HUB) :\n  is_unit (to_stalk (Spec.locally_ringed_space R).O.F p U hpU (to_presheaf_of_rings_extension _ _ HUB σ)) ↔\n  ∀ r : R, ∀ s : S U, localization.mk r s = σ → p ∈ Spec.D' r :=\nbegin\n  rw is_unit_to_stalk_affine, split,\n  { rintros ⟨r1, s1, hp1, hrs⟩ r2 s2 rfl, rcases quotient.exact hrs with ⟨t, hts, ht⟩,\n    change (s1.1 * r2 - s2.1 * r1) * t = 0 at ht, rw [sub_mul, sub_eq_zero_iff_eq] at ht,\n    suffices : p ∈ Spec.D' (s1.1 * r2 * t),\n    { rw [Spec.D'.product_eq_inter, Spec.D'.product_eq_inter] at this, exact this.1.2 },\n    rw ht, rw [Spec.D'.product_eq_inter, Spec.D'.product_eq_inter],\n    exact ⟨⟨s2.2 hpU, hp1⟩, hts hpU⟩ },\n  { refine localization.induction_on σ (λ r s h, ⟨r, s, h r s rfl, rfl⟩) }\nend\n\ntheorem is_unit_to_stalk_on_basis {X : Type u} [topological_space X]\n  {B : set (topological_space.opens X)} (HB : topological_space.opens.is_basis B) (Bstd)\n  (F : presheaf_of_rings_on_basis X HB)\n  (p : X) (U : topological_space.opens X) (hpU : p ∈ U) (σ : (F ᵣₑₓₜ Bstd) U) :\n  is_unit (to_stalk (F ᵣₑₓₜ Bstd) p U hpU σ) ↔\n  @is_unit (stalk_of_rings_on_standard_basis.stalk_of_rings_on_standard_basis Bstd F p) _ (σ.1 p hpU) :=\nbegin\n  rw is_unit_to_stalk, split,\n  { rintros ⟨W, hpW, HWU, HW⟩, rcases is_unit_iff_exists_inv.1 HW with ⟨τ, hστ⟩,\n    refine is_unit_iff_exists_inv.2 ⟨τ.1 p hpW, _⟩,\n    replace hστ := congr_arg subtype.val hστ,\n    replace hστ := congr_fun (congr_fun hστ p) hpW,\n    rwa Fext_mul.eq at hστ },\n  { rcases σ.2 p hpU with ⟨V, HVB, hpV, σ2, HV⟩, rw HV p ⟨hpU, hpV⟩, dsimp only, rintros hu,\n    rcases is_unit_iff_exists_inv.1 hu with ⟨e, he⟩, revert he, refine quotient.induction_on e (λ τ hστ, _),\n    rcases quotient.exact hστ with ⟨S, HSB, hpS, HSVτ, HStop, HS⟩, dsimp only at HS,\n    rcases topological_space.opens.is_basis_iff_nbhd.1 HB (show p ∈ S ∩ U, from ⟨hpS, hpU⟩) with ⟨T, HTB, hpT, HTSU⟩,\n    have HTS : T ⊆ S := set.subset.trans HTSU (set.inter_subset_left _ _),\n    have HTτ : T ⊆ τ.U := set.subset.trans HTS (set.subset.trans HSVτ (set.inter_subset_right _ _)),\n    refine ⟨T, hpT, set.subset.trans HTSU (set.inter_subset_right _ _), is_unit_iff_exists_inv.2\n      ⟨⟨λ q hqT, ⟦⟨T, HTB, hqT, F.res τ.BU HTB HTτ τ.s⟩⟧, λ q hqT, ⟨T, HTB, hqT, F.res τ.BU HTB HTτ τ.s, λ _ _, rfl⟩⟩, _⟩⟩,\n    replace HS := congr_arg (F.res HSB HTB HTS) HS,\n    refine subtype.eq (funext $ λ x, funext $ λ hxT, _), rw Fext_mul.eq,\n    change (σ.1 x _ * _ : stalk_of_rings_on_standard_basis.stalk_of_rings_on_standard_basis Bstd F x) = 1,\n    rw HV x ⟨(HTSU hxT).2, (HSVτ $ HTS hxT).1⟩,\n    refine quotient.sound ⟨T, HTB, hxT, set.subset_inter (set.subset.trans HTS (set.subset.trans HSVτ (set.inter_subset_left _ _)))\n      (set.subset.refl T), set.subset_univ _, _⟩,\n    dsimp only, rw bres_mul at HS ⊢, rw bres_mul at HS,\n    repeat { rw ← presheaf_on_basis.Hcomp' at HS }, repeat { rw ← presheaf_on_basis.Hcomp' }, exact HS }\nend\n\ndef of_pushforward_stalk {α : Type u} [topological_space α]\n  {β : Type u} [topological_space β]\n  {f : α → β} (hf : continuous f) (F : presheaf_of_rings α) (p : α) :\n  stalk_of_rings (presheaf_of_rings.pushforward hf F) (f p) → stalk_of_rings F p :=\npresheaf.fmap.stalk_of_rings ⟨λ U, id, λ U V HVU, rfl⟩ p\n\ninstance is_ring_hom_of_pushforward_stalk {α : Type u} [topological_space α]\n  {β : Type u} [topological_space β]\n  {f : α → β} (hf : continuous f) (F : presheaf_of_rings α) (p : α) :\n  is_ring_hom (of_pushforward_stalk hf F p) :=\nis_ring_hom_stalk_of_rings _ _\n\ntheorem presheafasdf {α : Type u} [topological_space α]\n  {β : Type u} [topological_space β]\n  {f : α → β} (hf : continuous f) (F : presheaf_of_rings α) (G : presheaf_of_rings β)\n  (m : presheaf_of_rings.fmap hf F G) (p : α) (σ) :\n  presheaf.fmap.stalk_of_rings m p σ =\n  of_pushforward_stalk hf F p\n    (presheaf.fmap.stalk_of_rings\n      (⟨m.map, m.commutes⟩ : presheaf_of_rings.fmap continuous_id (presheaf_of_rings.pushforward hf F) G)\n      (f p)\n      σ) :=\nquotient.induction_on σ $ λ e, rfl\n\n@[simp] lemma localization.lift_mul {α : Type u} [comm_ring α] {S : set α} [is_submonoid S]\n  {β : Type v} [comm_ring β] (f : α → β) [is_ring_hom f] (H) (x y : localization α S) :\n  localization.lift f H (x * y) = localization.lift f H x * localization.lift f H y :=\nis_ring_hom.map_mul _\n\n/- https://stacks.math.columbia.edu/tag/01I1 -/\nnoncomputable def hom_to_mor {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R]\n  (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f] :\n  OX.morphism (Spec.locally_ringed_space R) :=\n{ f := induced OX R f,\n  Hf := induced_continuous OX R f,\n  fO :=\n  { map := λ U s, (sheaf_of_rings.of_extension (D_fs_basis R)\n      (D_fs_standard_basis R) _ _\n      (induced_sheafification OX R f U s) :\n      OX.O.pushforward (induced_continuous OX R f) U),\n    commutes := λ U V HVU, funext $ λ s, begin\n      dsimp only [function.comp_apply],\n      change _ = (sheaf_of_rings.pushforward _ (OX.O)).F.res _ _ _ _,\n      rw ← sheaf_of_rings.of_extension_res, refl\n    end },\n  hom := λ U, @is_ring_hom.comp _ _ _ _ _ (is_ring_hom_induced_sheafification OX R f U) _ _ _ _,\n  Hstalks := λ p s, quotient.induction_on s $ λ e he, begin\n    cases e with U hpU σ, generalize hσ : σ.1 (induced OX R f p) hpU = t,\n    revert hσ, refine quotient.induction_on t (λ τ, _), cases τ with V HVB hpV τ,\n    refine localization.induction_on τ (λ r s hu, _),\n    change is_unit (of_pushforward_stalk (induced_continuous OX R f) OX.O.F p\n      (to_stalk (OX.O.pushforward (induced_continuous OX R f)).F (induced OX R f p) U hpU\n        (sheaf_of_rings.of_extension _ _ (OX.O.pushforward (induced_continuous OX R f)) U (induced_sheafification OX R f U σ) :\n          OX.O.pushforward (induced_continuous OX R f) U))) at he,\n    rw sheaf_of_rings.to_stalk_of_extension at he,\n    dsimp only [induced_sheafification] at he,\n    change σ.val (induced OX R f p) hpU = _ at hu,\n    rw hu at he,\n    change is_unit (of_pushforward_stalk (induced_continuous OX R f) OX.O.F p\n      (to_stalk (OX.O.pushforward (induced_continuous OX R f)).F (induced OX R f p) V hpV\n        (localization.lift\n          (OX.O.F.to_presheaf.res ⊤ (opens.comap (induced_continuous OX R f) V) (set.subset_univ _) ∘ f ∘ to_Spec_top R)\n          (induced_basis._proof_3 OX R f V)\n          (localization.mk r s)))) at he,\n    rw [localization.mk_eq, localization.lift_mul, localization.lift_coe,\n        is_ring_hom.map_mul (to_stalk (OX.O.pushforward (induced_continuous OX R f)).F (induced OX R f p) V hpV),\n        is_ring_hom.map_mul (of_pushforward_stalk (induced_continuous OX R f) OX.O.F p)] at he,\n    replace he := is_unit_of_mul_is_unit_left he,\n    change is_unit (to_stalk OX.O.F p (opens.comap (induced_continuous OX R f) V) hpV\n      (OX.O.F.to_presheaf.res ⊤ (opens.comap (induced_continuous OX R f) V) (set.subset_univ _)\n        (f (to_Spec_top R r)))) at he,\n    rw to_stalk_res _ p ⊤ _ trivial at he,\n    change p ∈ (OX.D (f (to_Spec_top R r)) : set X) at he,\n    rw ← induced_preimage_D' at he,\n    refine (is_unit_to_stalk_on_basis _ _ _ _ _ _ _).2 _, rw hu,\n    refine is_unit_iff_exists_inv.2 ⟨⟦⟨V ∩ Spec.DO R r, (D_fs_standard_basis R).2 HVB (D_fs.mem _ _), ⟨hpV, he⟩,\n      localization.mk s.1 ⟨r, set.inter_subset_right _ _⟩⟩⟧, quotient.sound ⟨V ∩ Spec.DO R r,\n      (D_fs_standard_basis R).2 HVB (D_fs.mem _ _), ⟨hpV, he⟩, set.subset_inter (set.inter_subset_left _ _) (set.subset.refl _),\n      set.subset_univ _, _⟩⟩,\n    change localization.mk (r * s.1) ⟨s.1 * r, _⟩ = 1,\n    rw mul_comm, exact localization.mk_self\n  end }\n\ntheorem cast_eq_of_heq : ∀ {α β : Sort u} {a : α} {b : β} (H : a == b), cast (type_eq_of_heq H) a = b\n| α _ a _ heq.rfl := rfl\n\n@[ext] theorem locally_ringed_space.morphism.ext {X : Type u} [topological_space X] {OX : locally_ringed_space X}\n  {Y : Type v} [topological_space Y] {OY : locally_ringed_space Y}\n  {m₁ m₂ : OX.morphism OY} (H1 : ∀ x, m₁.f x = m₂.f x)\n  (H2 : ∀ U σ, OX.O.F.res (opens.comap m₁.Hf U) (opens.comap m₂.Hf U)\n    (show m₂.f⁻¹' U ⊆ m₁.f⁻¹' U, from (funext H1 : m₁.f = m₂.f) ▸ set.subset.refl _)\n    (m₁.fO.map U σ) = m₂.fO.map U σ) : m₁ = m₂ :=\nbegin\n  rcases m₁ with ⟨⟨f, hf, ⟨mf, hmf⟩, hf2⟩, hf1⟩,\n  rcases m₂ with ⟨⟨g, hg, ⟨mg, hmg⟩, hg2⟩, hg1⟩,\n  have : f = g := funext H1, subst this, congr' 3,\n  funext U σ, exact (presheaf.Hid' _ _ _).symm.trans (H2 U σ)\nend\n\n/- https://stacks.math.columbia.edu/tag/01I1 -/\ntheorem mor_to_hom_to_mor {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R] (f : OX.morphism (Spec.locally_ringed_space R)) :\n  hom_to_mor OX R (mor_to_hom OX R f) = f :=\nbegin\n  have : ((hom_to_mor OX R (mor_to_hom OX R f)).to_morphism).f = (f.to_morphism).f,\n  { funext x, apply subtype.eq, ext s,\n    change ¬ is_unit (to_stalk _ _ _ _ (OX.O.F.res _ _ _ _)) ↔ f.f x ∈ Spec.V' s,\n    rw to_stalk_res _ x (opens.comap f.Hf ⊤) _ trivial, split,\n    { rintros hu, classical, by_contradiction hfxs, apply hu,\n      change is_unit (presheaf.fmap.stalk_of_rings f.fO x (to_stalk _ _ ⊤ trivial (to_Spec_top R s))),\n      haveI := is_ring_hom_stalk_of_rings f.fO,\n      refine is_unit.map' _ _, refine (is_unit_to_stalk_affine _ _ _ _ _ _).2 ⟨s, 1, hfxs, rfl⟩ },\n    { exact λ hsfx HV, (is_unit_to_stalk_affine' _ _ _ _ _ _).1\n        (f.Hstalks x (to_stalk _ _ ⊤ trivial (to_Spec_top R s)) HV) _ _ rfl hsfx } },\n  fapply locally_ringed_space.morphism.ext, { intros, rw this },\n  intros U σ,\n  refine @sheaf_of_rings.ext (Spec R) _ _ (D_fs_basis R) (OX.O.pushforward f.Hf) U _ _ (λ p hpU, _),\n  rcases topological_space.opens.is_basis_iff_nbhd.1 (D_fs_basis R) hpU with ⟨V, HVB, hpV, HVU⟩,\n  refine quotient.sound ⟨V, hpV, HVU, HVU, _⟩,\n  change OX.O.F.res _ _ _ (OX.O.F.res _ _ _ _) = OX.O.F.res _ _ _ _,\n  rw presheaf.fmap.commutes' f.fO U V HVU,\n  rw ← presheaf.Hcomp',\n  rw presheaf.Hcomp' _ (opens.comap (hom_to_mor OX R (mor_to_hom OX R f)).Hf U)\n      (opens.comap (hom_to_mor OX R (mor_to_hom OX R f)).Hf V) (opens.comap f.Hf V)\n      (show f.f ⁻¹' V ⊆ (hom_to_mor OX R (mor_to_hom OX R f)).f ⁻¹' V, by rw this)\n      (opens.comap_mono _ _ _ HVU),\n  rw presheaf.fmap.commutes' _ U V HVU,\n  generalize : ((((Spec.locally_ringed_space R).O).F).to_presheaf).res U V HVU σ = τ,\n  rcases to_presheaf_of_rings_extension.surjective _ _ structure_presheaf_on_basis_is_sheaf_on_basis HVB τ with ⟨x, rfl⟩,\n  haveI := (hom_to_mor OX R (mor_to_hom OX R f)).hom,\n  refine congr_fun (localization.funext\n    (OX.O.F.res (opens.comap (hom_to_mor OX R (mor_to_hom OX R f)).Hf V) (opens.comap f.Hf V) _ ∘\n      (hom_to_mor OX R (mor_to_hom OX R f)).fO.map V ∘\n         to_presheaf_of_rings_extension _ (structure_presheaf_on_basis R) HVB)\n    (f.fO.map V ∘ to_presheaf_of_rings_extension _ (structure_presheaf_on_basis R) HVB) _) x,\n  intros r, dsimp only [function.comp_apply],\n  rw [show ((r : localization R (S V)) : (structure_presheaf_on_basis R).F HVB) =\n      (structure_presheaf_on_basis R).res (D_fs_standard_basis R).1 _ _ (localization.of r), from rfl,\n    ← res_to_presheaf_of_rings_extension _ _ _ _ _ (D_fs_standard_basis R).1 _ (set.subset_univ _)],\n  change (((OX.O).F).to_presheaf).res (opens.comap _ V) (opens.comap _ V) _\n      ((((hom_to_mor OX R (mor_to_hom OX R f)).to_morphism).fO).map V\n         (((Spec.locally_ringed_space R).O.F.to_presheaf).res opens.univ V _\n            (to_presheaf_of_rings_extension _ (structure_presheaf_on_basis R) _ (localization.of r)))) =\n    ((f.to_morphism).fO).map V\n      (((Spec.locally_ringed_space R).O.F.to_presheaf).res opens.univ V _\n         (to_presheaf_of_rings_extension _ (structure_presheaf_on_basis R) _ (localization.of r))),\n  rw [← presheaf.fmap.commutes', ← presheaf.fmap.commutes'],\n  swap, { exact opens.comap_mono _ _ _ (set.subset_univ _) }, swap, { exact opens.comap_mono _ _ _ (set.subset_univ _) },\n  dsimp only [hom_to_mor], rw ← presheaf.Hcomp',\n  rw presheaf.Hcomp' OX.O.F.to_presheaf (opens.comap (induced_continuous OX R (mor_to_hom OX R f)) opens.univ)\n      (opens.comap f.Hf ⊤) (opens.comap f.Hf V)\n      (opens.comap_mono _ _ _ (set.subset_univ _))\n      (λ _ _, trivial),\n  congr' 1,\n  rw [induced_sheafification_to_presheaf_of_rings_extension, sheaf_of_rings.of_to_extension],\n  dsimp only [induced_basis], simp only [localization.lift_of, function.comp_apply],\n  dsimp only [mor_to_hom], rw [← presheaf.Hcomp', ← presheaf.Hcomp'],\n  exact presheaf.Hid' _ _ _,\nend\n\n/- https://stacks.math.columbia.edu/tag/01I1 -/\ntheorem hom_to_mor_to_hom {X : Type u} [topological_space X] (OX : locally_ringed_space X)\n  (R : Type u) [comm_ring R] (f : (Spec.locally_ringed_space R).O ⊤ → OX.O ⊤) [is_ring_hom f] :\n  mor_to_hom OX R (hom_to_mor OX R f) = f :=\nbegin\n  funext x,\n  rcases to_presheaf_of_rings_extension.surjective (D_fs_standard_basis R) _ structure_presheaf_on_basis_is_sheaf_on_basis (D_fs_standard_basis R).1 x with ⟨r, rfl⟩,\n  suffices : mor_to_hom OX R (hom_to_mor OX R f) ∘ to_presheaf_of_rings_extension (D_fs_standard_basis R) (structure_presheaf_on_basis R) (D_fs_standard_basis R).1 =\n    f ∘ to_presheaf_of_rings_extension (D_fs_standard_basis R) (structure_presheaf_on_basis R) (D_fs_standard_basis R).1,\n  { convert congr_fun this r },\n  refine localization.funext _ _ (λ r, _),\n  dsimp only [function.comp_apply, mor_to_hom, hom_to_mor],\n  rw [induced_sheafification_to_presheaf_of_rings_extension, sheaf_of_rings.of_to_extension],\n  dsimp only [induced_basis], rw [localization.lift_coe], dsimp only [function.comp_apply],\n  rw ← presheaf.Hcomp', exact presheaf.Hid' _ _ _\nend\n\nend tag01I1\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/Kenny/sandbox.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.32196814719028455}}
{"text": "/-\nCopyright (c) 2018 Michael Jendrusch. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Jendrusch, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.category\nimport Mathlib.PostPort\n\nuniverses v₁ v₂ u₁ u₂ l u₃ v₃ \n\nnamespace Mathlib\n\n/-!\n# (Lax) monoidal functors\n\nA lax monoidal functor `F` between monoidal categories `C` and `D`\nis a functor between the underlying categories equipped with morphisms\n* `ε : 𝟙_ D ⟶ F.obj (𝟙_ C)` (called the unit morphism)\n* `μ X Y : (F.obj X) ⊗ (F.obj Y) ⟶ F.obj (X ⊗ Y)` (called the tensorator, or strength).\nsatisfying various axioms.\n\nA monoidal functor is a lax monoidal functor for which `ε` and `μ` are isomorphisms.\n\nWe show that the composition of (lax) monoidal functors gives a (lax) monoidal functor.\n\nSee also `category_theory.monoidal.functorial` for a typeclass decorating an object-level\nfunction with the additional data of a monoidal functor.\nThis is useful when stating that a pre-existing functor is monoidal.\n\nSee `category_theory.monoidal.natural_transformation` for monoidal natural transformations.\n\nWe show in `category_theory.monoidal.Mon_` that lax monoidal functors take monoid objects\nto monoid objects.\n\n## Future work\n* Oplax monoidal functors.\n\n## References\n\nSee https://stacks.math.columbia.edu/tag/0FFL.\n-/\n\nnamespace category_theory\n\n\n/-- A lax monoidal functor is a functor `F : C ⥤ D` between monoidal categories, equipped with morphisms\n    `ε : 𝟙 _D ⟶ F.obj (𝟙_ C)` and `μ X Y : F.obj X ⊗ F.obj Y ⟶ F.obj (X ⊗ Y)`, satisfying the\n    the appropriate coherences. -/\n-- unit morphism\n\nstructure lax_monoidal_functor (C : Type u₁) [category C] [monoidal_category C] (D : Type u₂)\n    [category D] [monoidal_category D]\n    extends C ⥤ D where\n  ε : 𝟙_ ⟶ functor.obj _to_functor 𝟙_\n  μ :\n    (X Y : C) →\n      functor.obj _to_functor X ⊗ functor.obj _to_functor Y ⟶ functor.obj _to_functor (X ⊗ Y)\n  μ_natural' :\n    autoParam\n      (∀ {X Y X' Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y'),\n        (functor.map _to_functor f ⊗ functor.map _to_functor g) ≫ μ Y Y' =\n          μ X X' ≫ functor.map _to_functor (f ⊗ g))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  associativity' :\n    autoParam\n      (∀ (X Y Z : C),\n        (μ X Y ⊗ 𝟙) ≫ μ (X ⊗ Y) Z ≫ functor.map _to_functor (iso.hom α_) =\n          iso.hom α_ ≫ (𝟙 ⊗ μ Y Z) ≫ μ X (Y ⊗ Z))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  left_unitality' :\n    autoParam (∀ (X : C), iso.hom λ_ = (ε ⊗ 𝟙) ≫ μ 𝟙_ X ≫ functor.map _to_functor (iso.hom λ_))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  right_unitality' :\n    autoParam (∀ (X : C), iso.hom ρ_ = (𝟙 ⊗ ε) ≫ μ X 𝟙_ ≫ functor.map _to_functor (iso.hom ρ_))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- tensorator\n\n-- associativity of the tensorator\n\n-- unitality\n\n@[simp] theorem lax_monoidal_functor.μ_natural {C : Type u₁} [category C] [monoidal_category C]\n    {D : Type u₂} [category D] [monoidal_category D] (c : lax_monoidal_functor C D) {X : C} {Y : C}\n    {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :\n    (functor.map (lax_monoidal_functor.to_functor c) f ⊗\n            functor.map (lax_monoidal_functor.to_functor c) g) ≫\n          lax_monoidal_functor.μ c Y Y' =\n        lax_monoidal_functor.μ c X X' ≫ functor.map (lax_monoidal_functor.to_functor c) (f ⊗ g) :=\n  sorry\n\n@[simp] theorem lax_monoidal_functor.μ_natural_assoc {C : Type u₁} [category C]\n    [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D]\n    (c : lax_monoidal_functor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :\n    ∀ {X'_1 : D} (f' : functor.obj (lax_monoidal_functor.to_functor c) (Y ⊗ Y') ⟶ X'_1),\n        (functor.map (lax_monoidal_functor.to_functor c) f ⊗\n              functor.map (lax_monoidal_functor.to_functor c) g) ≫\n            lax_monoidal_functor.μ c Y Y' ≫ f' =\n          lax_monoidal_functor.μ c X X' ≫\n            functor.map (lax_monoidal_functor.to_functor c) (f ⊗ g) ≫ f' :=\n  sorry\n\n@[simp] theorem lax_monoidal_functor.left_unitality {C : Type u₁} [category C] [monoidal_category C]\n    {D : Type u₂} [category D] [monoidal_category D] (c : lax_monoidal_functor C D) (X : C) :\n    iso.hom λ_ =\n        (lax_monoidal_functor.ε c ⊗ 𝟙) ≫\n          lax_monoidal_functor.μ c 𝟙_ X ≫\n            functor.map (lax_monoidal_functor.to_functor c) (iso.hom λ_) :=\n  sorry\n\n@[simp] theorem lax_monoidal_functor.right_unitality {C : Type u₁} [category C]\n    [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D]\n    (c : lax_monoidal_functor C D) (X : C) :\n    iso.hom ρ_ =\n        (𝟙 ⊗ lax_monoidal_functor.ε c) ≫\n          lax_monoidal_functor.μ c X 𝟙_ ≫\n            functor.map (lax_monoidal_functor.to_functor c) (iso.hom ρ_) :=\n  sorry\n\n@[simp] theorem lax_monoidal_functor.associativity {C : Type u₁} [category C] [monoidal_category C]\n    {D : Type u₂} [category D] [monoidal_category D] (c : lax_monoidal_functor C D) (X : C) (Y : C)\n    (Z : C) :\n    (lax_monoidal_functor.μ c X Y ⊗ 𝟙) ≫\n          lax_monoidal_functor.μ c (X ⊗ Y) Z ≫\n            functor.map (lax_monoidal_functor.to_functor c) (iso.hom α_) =\n        iso.hom α_ ≫ (𝟙 ⊗ lax_monoidal_functor.μ c Y Z) ≫ lax_monoidal_functor.μ c X (Y ⊗ Z) :=\n  sorry\n\n-- When `rewrite_search` lands, add @[search] attributes to\n\n-- lax_monoidal_functor.μ_natural lax_monoidal_functor.left_unitality\n\n-- lax_monoidal_functor.right_unitality lax_monoidal_functor.associativity\n\n/--\nA monoidal functor is a lax monoidal functor for which the tensorator and unitor as isomorphisms.\n\nSee https://stacks.math.columbia.edu/tag/0FFL.\n-/\nstructure monoidal_functor (C : Type u₁) [category C] [monoidal_category C] (D : Type u₂)\n    [category D] [monoidal_category D]\n    extends lax_monoidal_functor C D where\n  ε_is_iso :\n    autoParam (is_iso (lax_monoidal_functor.ε _to_lax_monoidal_functor))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.apply_instance\")\n        (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n          \"apply_instance\")\n        [])\n  μ_is_iso :\n    autoParam ((X Y : C) → is_iso (lax_monoidal_functor.μ _to_lax_monoidal_functor X Y))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.apply_instance\")\n        (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n          \"apply_instance\")\n        [])\n\n/--\nThe unit morphism of a (strong) monoidal functor as an isomorphism.\n-/\ndef monoidal_functor.ε_iso {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂}\n    [category D] [monoidal_category D] (F : monoidal_functor C D) :\n    𝟙_ ≅\n        functor.obj (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor F))\n          𝟙_ :=\n  as_iso (lax_monoidal_functor.ε (monoidal_functor.to_lax_monoidal_functor F))\n\n/--\nThe tensorator of a (strong) monoidal functor as an isomorphism.\n-/\ndef monoidal_functor.μ_iso {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂}\n    [category D] [monoidal_category D] (F : monoidal_functor C D) (X : C) (Y : C) :\n    functor.obj (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor F)) X ⊗\n          functor.obj (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor F))\n            Y ≅\n        functor.obj (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor F))\n          (X ⊗ Y) :=\n  as_iso (lax_monoidal_functor.μ (monoidal_functor.to_lax_monoidal_functor F) X Y)\n\nnamespace lax_monoidal_functor\n\n\n/-- The identity lax monoidal functor. -/\n@[simp] theorem id_ε (C : Type u₁) [category C] [monoidal_category C] : ε (id C) = 𝟙 :=\n  Eq.refl (ε (id C))\n\nprotected instance inhabited (C : Type u₁) [category C] [monoidal_category C] :\n    Inhabited (lax_monoidal_functor C C) :=\n  { default := id C }\n\nend lax_monoidal_functor\n\n\nnamespace monoidal_functor\n\n\ntheorem map_tensor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] (F : monoidal_functor C D) {X : C} {Y : C} {X' : C} {Y' : C} (f : X ⟶ Y)\n    (g : X' ⟶ Y') :\n    functor.map (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) (f ⊗ g) =\n        inv (lax_monoidal_functor.μ (to_lax_monoidal_functor F) X X') ≫\n          (functor.map (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) f ⊗\n              functor.map (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) g) ≫\n            lax_monoidal_functor.μ (to_lax_monoidal_functor F) Y Y' :=\n  sorry\n\ntheorem map_left_unitor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] (F : monoidal_functor C D) (X : C) :\n    functor.map (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) (iso.hom λ_) =\n        inv (lax_monoidal_functor.μ (to_lax_monoidal_functor F) 𝟙_ X) ≫\n          (inv (lax_monoidal_functor.ε (to_lax_monoidal_functor F)) ⊗ 𝟙) ≫ iso.hom λ_ :=\n  sorry\n\ntheorem map_right_unitor {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] (F : monoidal_functor C D) (X : C) :\n    functor.map (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) (iso.hom ρ_) =\n        inv (lax_monoidal_functor.μ (to_lax_monoidal_functor F) X 𝟙_) ≫\n          (𝟙 ⊗ inv (lax_monoidal_functor.ε (to_lax_monoidal_functor F))) ≫ iso.hom ρ_ :=\n  sorry\n\n/-- The tensorator as a natural isomorphism. -/\ndef μ_nat_iso {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] (F : monoidal_functor C D) :\n    functor.prod (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F))\n            (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F)) ⋙\n          monoidal_category.tensor D ≅\n        monoidal_category.tensor C ⋙ lax_monoidal_functor.to_functor (to_lax_monoidal_functor F) :=\n  nat_iso.of_components (fun (X : C × C) => μ_iso F (prod.fst X) (prod.snd X)) sorry\n\n/-- The identity monoidal functor. -/\ndef id (C : Type u₁) [category C] [monoidal_category C] : monoidal_functor C C :=\n  mk (lax_monoidal_functor.mk (functor.mk (functor.obj 𝟭) (functor.map 𝟭)) 𝟙 fun (X Y : C) => 𝟙)\n\nprotected instance inhabited (C : Type u₁) [category C] [monoidal_category C] :\n    Inhabited (monoidal_functor C C) :=\n  { default := id C }\n\nend monoidal_functor\n\n\nnamespace lax_monoidal_functor\n\n\n-- The proofs here are horrendous; rewrite_search helps a lot.\n\n/-- The composition of two lax monoidal functors is again lax monoidal. -/\n@[simp] theorem comp_ε {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] {E : Type u₃} [category E] [monoidal_category E]\n    (F : lax_monoidal_functor C D) (G : lax_monoidal_functor D E) :\n    ε (comp F G) = ε G ≫ functor.map (to_functor G) (ε F) :=\n  Eq.refl (ε (comp F G))\n\ninfixr:80 \" ⊗⋙ \" => Mathlib.category_theory.lax_monoidal_functor.comp\n\nend lax_monoidal_functor\n\n\nnamespace monoidal_functor\n\n\n/-- The composition of two monoidal functors is again monoidal. -/\ndef comp {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D]\n    [monoidal_category D] {E : Type u₃} [category E] [monoidal_category E]\n    (F : monoidal_functor C D) (G : monoidal_functor D E) : monoidal_functor C E :=\n  mk\n    (lax_monoidal_functor.mk\n      (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F ⊗⋙ to_lax_monoidal_functor G))\n      (lax_monoidal_functor.ε (to_lax_monoidal_functor F ⊗⋙ to_lax_monoidal_functor G))\n      (lax_monoidal_functor.μ (to_lax_monoidal_functor F ⊗⋙ to_lax_monoidal_functor G)))\n\ninfixr:80 \" ⊗⋙ \" => Mathlib.category_theory.monoidal_functor.comp\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/functor_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5234203489363239, "lm_q1q2_score": 0.32194950734705774}}
{"text": "abbrev Var := String\n\ninductive Val where\n  | int  (i : Int)\n  | bool (b : Bool)\n  deriving DecidableEq, Repr\n\ninstance : Coe Bool Val where\n  coe b := .bool b\n\ninstance : Coe Int Val where\n  coe i := .int i\n\ninstance : OfNat Val n where\n  ofNat := .int n\n\n#check (true : Val)\n\n#check (0 : Val)\n\ninductive BinOp where\n  | eq | and | lt | add | sub\n  deriving Repr\n\ninductive UnaryOp where\n  | not\n  deriving Repr\n\ninductive Expr where\n  | val (v : Val)\n  | var (x : Var)\n  | bin (lhs : Expr) (op : BinOp) (rhs : Expr)\n  | una (op : UnaryOp) (arg : Expr)\n  deriving Repr\n\ninstance : Coe Val Expr where\n  coe v := .val v\n\ninstance : Coe Var Expr where\n  coe x := .var x\n\ninstance : OfNat Expr n where\n  ofNat := .val n\n\n@[simp] def BinOp.eval : BinOp → Val → Val → Option Val\n  | .eq,  v₁,        v₂      => some (.bool (v₁ = v₂))\n  | .and, .bool b₁, .bool b₂ => some (.bool (b₁ && b₂))\n  | .lt,  .int i₁,  .int i₂  => some (.bool (i₁ < i₂))\n  | .add, .int i₁,  .int i₂  => some (.int (i₁ + i₂))\n  | .sub, .int i₁,  .int i₂  => some (.int (i₁ - i₂))\n  | _,    _,        _        => none\n\n@[simp] def UnaryOp.eval : UnaryOp → Val → Option Val\n  | .not, .bool b => some (.bool !b)\n  | _,    _       => none\n\ninductive Stmt where\n  | skip\n  | assign (x : Var) (e : Expr)\n  | seq    (s₁ s₂ : Stmt)\n  | ite    (c : Expr) (e t : Stmt)\n  | while  (c : Expr) (b : Stmt)\n  deriving Repr\n\ninfix:150 \" ::= \" => Stmt.assign\ninfixr:130 \";; \"   => Stmt.seq\n\nsyntax \"`[Expr|\" term \"]\" : term\n\nmacro_rules\n  | `(`[Expr|true])      => `((true : Expr))\n  | `(`[Expr|false])     => `((false : Expr))\n  | `(`[Expr|$n:num])    => `(($n : Expr))\n  | `(`[Expr|$x:ident])  => `(($(Lean.quote x.getId.toString) : Expr))\n  | `(`[Expr|$x = $y])   => `(Expr.bin `[Expr|$x] BinOp.eq `[Expr|$y])\n  | `(`[Expr|$x && $y])  => `(Expr.bin `[Expr|$x] BinOp.and `[Expr|$y])\n  | `(`[Expr|$x < $y])   => `(Expr.bin `[Expr|$x] BinOp.lt `[Expr|$y])\n  | `(`[Expr|$x + $y])   => `(Expr.bin `[Expr|$x] BinOp.add `[Expr|$y])\n  | `(`[Expr|$x - $y])   => `(Expr.bin `[Expr|$x] BinOp.sub `[Expr|$y])\n  | `(`[Expr|!$x])       => `(Expr.una UnaryOp.not `[Expr|$x])\n  | `(`[Expr|($x)])      => `(`[Expr|$x])\n\ndeclare_syntax_cat stmt\n\nsyntax ident \" := \" term \";\": stmt\nsyntax \"if \" \"(\" term \")\" \" {\\n\" stmt* \"\\n}\" (\" else \" \"{\\n\" stmt* \"\\n}\")? : stmt\nsyntax \"while \" \"(\" term \")\" \"{\\n\" stmt* \"\\n}\" : stmt\n\nsyntax \"`[Stmt|\" stmt* \"]\" : term\n\nmacro_rules\n  | `(`[Stmt| ]) => `(Stmt.skip)\n  | `(`[Stmt| $x:ident := $e:term;]) => `(Stmt.assign $(Lean.quote x.getId.toString) `[Expr| $e:term])\n  | `(`[Stmt| if ($c) { $ts* } else { $es* }]) => `(Stmt.ite `[Expr| $c] `[Stmt| $ts*] `[Stmt| $es*])\n  | `(`[Stmt| if ($c) { $ts* }]) => `(Stmt.ite `[Expr| $c] `[Stmt| $ts*] Stmt.skip)\n  | `(`[Stmt| while ($c) { $b* }]) => `(Stmt.while `[Expr|$c] `[Stmt|$b*])\n  | `(`[Stmt| $s $ss*]) => `(Stmt.seq `[Stmt|$s] `[Stmt|$ss*])\n\ndef example1 := `[Stmt|\n  x := 8;\n  y := 10;\n  if (x < y) {\n    x := x + 1;\n  } else {\n    y := y + 3;\n  }\n]\n\n#reduce example1\n\ndef example2 := `[Stmt|\n  x := 8;\n  if (x < y) {\n    x := x + 1;\n  } else {\n    y := y + 3;\n    x := 9;\n  }\n  y := x;]\n\n#eval example2\n\nabbrev State := List (Var × Val)\n\n@[simp] def State.update (σ : State) (x : Var) (v : Val) : State :=\n  match σ with\n  | [] => [(x, v)]\n  | (y, w)::σ => if x = y then (x, v)::σ else (y, w) :: update σ x v\n\n@[simp] def State.find? (σ : State) (x : Var) : Option Val :=\n  match σ with\n  | [] => none\n  | (y, v) :: σ => if x = y then some v else find? σ x\n\ndef State.get (σ : State) (x : Var) : Val :=\n  σ.find? x |>.getD (.int 0)\n\n@[simp] def State.erase (σ : State) (x : Var) : State :=\n  match σ with\n  | [] => []\n  | (y, v) :: σ => if x = y then erase σ x else (y, v) :: erase σ x\n\n@[simp] theorem State.find?_update_self (σ : State) (x : Var) (v : Val) : (σ.update x v).find? x = some v := by\n  match σ with -- TODO: automate this proof\n  | [] => simp\n  | (y, w) :: s =>\n    simp\n    split <;> simp [*]\n    apply find?_update_self\n\n@[simp] theorem State.find?_update (σ : State) (v : Val) (h : x ≠ z) : (σ.update x v).find? z = σ.find? z := by\n  match σ with -- TODO: automate this proof\n  | [] => simp [h.symm]\n  | (y, w) :: σ =>\n    simp\n    split <;> simp [*]\n    next hc => split <;> simp_all\n    next =>\n      split\n      next => rfl\n      next => exact find?_update σ v h\n\n-- TODO: remove after we add better automation\n@[simp] theorem State.find?_update' (σ : State) (v : Val) (h : z ≠ x) : (σ.update x v).find? z = σ.find? z :=\n  State.find?_update σ v h.symm\n\ntheorem State.get_of_find? {σ : State} (h : σ.find? x = some v) : σ.get x = v := by\n  simp [State.get, h, Option.getD]\n\n@[simp] theorem State.find?_erase_self (σ : State) (x : Var) : (σ.erase x).find? x = none := by\n  match σ with\n  | [] => simp\n  | (y, w) :: σ =>\n    simp\n    split <;> simp [*]\n    next => exact find?_erase_self σ y\n    next => exact find?_erase_self σ x\n\n@[simp] theorem State.find?_erase (σ : State) (h : x ≠ z) : (σ.erase x).find? z = σ.find? z := by\n  match σ with\n  | [] => simp\n  | (y, w) :: σ =>\n    simp\n    split <;> simp [*]\n    next hxy => rw [hxy] at h; simp [h.symm]; exact find?_erase σ h\n    next =>\n      split\n      next => rfl\n      next => exact find?_erase σ h\n\n-- TODO: remove after we add better automation\n@[simp] theorem State.find?_erase' (σ : State) (h : z ≠ x) : (σ.erase x).find? z = σ.find? z :=\n  State.find?_erase σ h.symm\n\nsyntax ident \" ↦ \" term : term\n\nmacro_rules\n  | `($id:ident ↦ $v:term) => `(($(Lean.quote id.getId.toString), $v:term))\n\nexample : State.get [x ↦ .int 10, y ↦ .int 20] \"x\" = .int 10 := rfl\n\nexample : State.get [x ↦ 10, y ↦ 20] \"x\" = 10     := rfl\nexample : State.get [x ↦ 10, y ↦ true] \"y\" = true := rfl\n\n@[simp] def Expr.eval (σ : State) : Expr → Option Val\n  | val v   => some v\n  | var x   => σ.get x\n  | bin lhs op rhs => match lhs.eval σ, rhs.eval σ with\n    | some v₁, some v₂ => op.eval v₁ v₂  -- BinOp.eval : BinOp → Val → Val → Option Val\n    | _,       _       => none\n  | una op arg => match arg.eval σ with\n    | some v => op.eval v\n    | _      => none\n\n\n@[simp] def evalTrue  (c : Expr) (σ : State) : Prop := c.eval σ = some (Val.bool true)\n@[simp] def evalFalse (c : Expr) (σ : State) : Prop := c.eval σ = some (Val.bool false)\n\nsection\nset_option hygiene false -- HACK: allow forward reference in notation\nlocal notation:60 \"(\" σ \", \" s \")\"  \" ⇓ \" σ':60 => Bigstep σ s σ'\n\ninductive Bigstep : State → Stmt → State → Prop where\n  | skip       : (σ, .skip) ⇓ σ\n  | assign     : e.eval σ = some v → (σ,  x ::= e) ⇓ σ.update x v\n  | seq        : (σ₁, s₁) ⇓ σ₂ → (σ₂, s₂) ⇓ σ₃ → (σ₁, s₁ ;; s₂) ⇓ σ₃\n  | ifTrue     : evalTrue c σ₁ → (σ₁, t) ⇓ σ₂ → (σ₁, .ite c t e) ⇓ σ₂\n  | ifFalse    : evalFalse c σ₁ → (σ₁, e) ⇓ σ₂ → (σ₁, .ite c t e) ⇓ σ₂\n  | whileTrue  : evalTrue c σ₁ → (σ₁, b) ⇓ σ₂ → (σ₂, .while c b) ⇓ σ₃ → (σ₁, .while c b) ⇓ σ₃\n  | whileFalse : evalFalse c σ → (σ, .while c b) ⇓ σ\n\nend\n\nnotation:60 \"(\" σ \", \" s \")\"  \" ⇓ \" σ':60 => Bigstep σ s σ'\n\n/- This proof can be automated using forward reasoning. -/\ntheorem Bigstem.det (h₁ : (σ, s) ⇓ σ₁) (h₂ : (σ, s) ⇓ σ₂) : σ₁ = σ₂ := by\n  induction h₁ generalizing σ₂ <;> cases h₂ <;> simp_all\n  -- The rest of this proof should be automatic with congruence closure and a bit of forward reasoning\n  case seq ih₁ ih₂ h₁ h₂ =>\n    simp [ih₁ h₁] at ih₂\n    simp [ih₂ h₂]\n  case ifTrue ih h =>\n    simp [ih h]\n  case ifFalse ih h =>\n    simp [ih h]\n  case whileTrue ih₁ ih₂ h₁ h₂ =>\n    simp [ih₁ h₁] at ih₂\n    simp [ih₂ h₂]\n\nabbrev EvalM := ExceptT String (StateM State)\n\ndef evalExpr (e : Expr) : EvalM Val := do\n  match e.eval (← get) with\n  | some v => return v\n  | none => throw \"failed to evaluate\"\n\n@[simp] def Stmt.eval (stmt : Stmt) (fuel : Nat := 100) : EvalM Unit := do\n  match fuel with\n  | 0 => throw \"out of fuel\"\n  | fuel+1 =>\n    match stmt with\n    | skip        => return ()\n    | assign x e  => let v ← evalExpr e; modify fun s => s.update x v\n    | seq s₁ s₂   => s₁.eval fuel; s₂.eval fuel\n    | ite c e t   =>\n      match (← evalExpr c) with\n      | .bool true  => e.eval fuel\n      | .bool false => t.eval fuel\n      | _ => throw \"Boolean expected\"\n    | .while c b =>\n      match (← evalExpr c) with\n      | .bool true  => b.eval fuel; stmt.eval fuel\n      | .bool false => return ()\n      | _ => throw \"Boolean expected\"\n\n#eval `[Stmt| x := 3; y := 5; x := x + y;].eval |>.run {}\n#eval `[Stmt| x := 0; while (true) { x := x + 1; }].eval |>.run {}\n\ninstance : Repr State where\n  reprPrec a n :=\n    match a, n with\n    | [], _ => \"[]\"\n    | as, _ =>\n      let fs := as.map fun\n        | (x, .int v)  => f!\"{x} ↦ {v}\"\n        | (x, .bool v) => f!\"{x} ↦ {v}\"\n      Std.Format.bracket \"[\" (Std.Format.joinSep fs (\",\" ++ Std.Format.line)) \"]\"\n\n#eval `[Stmt| x := 3; y := 5; x := x + y; ].eval |>.run {}\n\n@[simp] def BinOp.simplify : BinOp → Expr → Expr → Expr\n  | .eq,  .val v₁,  .val  v₂ => .val (.bool (v₁ = v₂))\n  | .and, .val (.bool a), .val (.bool b) => .val (.bool (a && b))\n  | .lt,  .val (.int a),  .val (.int b)  => .val (.bool (a < b))\n  | .add, .val (.int a), .val (.int b) => .val (.int (a + b))\n  | .sub, .val (.int a), .val (.int b) => .val (.int (a - b))\n  | op, a, b => .bin a op b\n\n@[simp] def UnaryOp.simplify : UnaryOp → Expr → Expr\n  | .not, .val (.bool b) => .val (.bool !b)\n  | op, a => .una op a\n\n@[simp] def Expr.simplify : Expr → Expr\n  | bin lhs op rhs => op.simplify lhs.simplify rhs.simplify\n  | una op arg => op.simplify arg.simplify\n  | e => e\n\n@[simp] theorem Expr.eval_simplify (e : Expr) : e.simplify.eval σ = e.eval σ := by\n  induction e with simp\n  | bin lhs op rhs ih_lhs ih_rhs =>\n    simp [← ih_lhs, ← ih_rhs]\n    split <;> simp [*]\n  | una op arg ih_arg =>\n    simp [← ih_arg]\n    split <;> simp [*]\n\n@[simp] def Stmt.simplify : Stmt → Stmt\n  | skip => skip\n  | assign x e => assign x e.simplify\n  | seq s₁ s₂ => seq s₁.simplify s₂.simplify\n  | ite c e t =>\n    match c.simplify with\n    | .val (.bool true) => e.simplify\n    | .val (.bool false) => t.simplify\n    | c' => ite c' e.simplify t.simplify\n  | .while c b =>\n    match c.simplify with\n    | .val (.bool false) => skip\n    | c' => .while c' b.simplify\n\ndef example3 := `[Stmt|\n  if (1 < 2 + 3) {\n    x := 3 + 1;\n    y := y + x;\n  } else {\n    y := y + 3;\n  }\n]\n\n#eval example3.simplify\n\ntheorem Stmt.simplify_correct (h : (σ, s) ⇓ σ') : (σ, s.simplify) ⇓ σ' := by\n  induction h with simp_all\n  | skip => exact Bigstep.skip\n  | seq h₁ h₂ ih₁ ih₂ => exact Bigstep.seq ih₁ ih₂\n  | assign => apply Bigstep.assign; simp [*]\n  | whileTrue heq h₁ h₂ ih₁ ih₂ =>\n    rw [← Expr.eval_simplify] at heq\n    split\n    next h => rw [h] at heq; simp at heq\n    next hnp => simp [hnp] at ih₂; apply Bigstep.whileTrue heq ih₁ ih₂\n  | whileFalse heq =>\n    split\n    next => exact Bigstep.skip\n    next => apply Bigstep.whileFalse; simp [heq]\n  | ifFalse heq h ih =>\n    rw [← Expr.eval_simplify] at heq\n    split <;> simp_all\n    rw [← Expr.eval_simplify] at heq\n    apply Bigstep.ifFalse heq ih\n  | ifTrue heq h ih =>\n    rw [← Expr.eval_simplify] at heq\n    split <;> simp_all\n    rw [← Expr.eval_simplify] at heq\n    apply Bigstep.ifTrue heq ih\n\n@[simp] def Expr.constProp (e : Expr) (σ : State) : Expr :=\n  match e with\n  | val v => v\n  | var x => match σ.find? x with\n    | some v => val v\n    | none   => var x\n  | bin lhs op rhs => bin (lhs.constProp σ) op (rhs.constProp σ)\n  | una op arg => una op (arg.constProp σ)\n\n@[simp] theorem Expr.constProp_nil (e : Expr) : e.constProp [] = e := by\n  induction e <;> simp [*]\n\ndef State.length_erase_le (σ : State) (x : Var) : (σ.erase x).length ≤ σ.length := by\n  match σ with\n  | [] => simp\n  | (y, v) :: σ =>\n    by_cases hxy : x = y <;> simp [hxy]\n    next => exact Nat.le_trans (length_erase_le σ y) (by simp_arith)\n    next => simp_arith [length_erase_le σ x]\n\ndef State.length_erase_lt (σ : State) (x : Var) : (σ.erase x).length < σ.length.succ :=\n  Nat.lt_of_le_of_lt (length_erase_le ..) (by simp_arith)\n\n@[simp] def State.join (σ₁ σ₂ : State) : State :=\n  match σ₁ with\n  | [] => []\n  | (x, v) :: σ₁ =>\n    let σ₁' := erase σ₁ x -- Must remove duplicates. Alternative design: carry invariant that input state at constProp has no duplicates\n    have : (erase σ₁ x).length < σ₁.length.succ := length_erase_lt ..\n    match σ₂.find? x with\n    | some w => if v = w then (x, v) :: join σ₁' σ₂ else join σ₁' σ₂\n    | none => join σ₁' σ₂\ntermination_by _ σ₁ _ => σ₁.length\n\nlocal notation \"⊥\" => []\n\n@[simp] def Stmt.constProp (s : Stmt) (σ : State) : Stmt × State :=\n  match s with\n  | skip => (skip, σ)\n  | assign x e => match (e.constProp σ).simplify with\n    | (.val v) => (assign x (.val v), σ.update x v)\n    | e'       => (assign x e', σ.erase x)\n  | seq s₁ s₂ => match s₁.constProp σ with\n    | (s₁', σ₁) => match s₂.constProp σ₁ with\n      | (s₂', σ₂) => (seq s₁' s₂', σ₂)\n  | ite c s₁ s₂ =>\n    match s₁.constProp σ, s₂.constProp σ with\n    | (s₁', σ₁), (s₂', σ₂) => (ite (c.constProp σ) s₁' s₂', σ₁.join σ₂)\n  | .while c b => (.while (c.constProp ⊥) (b.constProp ⊥).1, ⊥)\n\ndef State.le (σ₁ σ₂ : State) : Prop :=\n  ∀ ⦃x : Var⦄ ⦃v : Val⦄, σ₁.find? x = some v → σ₂.find? x = some v\n\ninfix:50 \" ≼ \" => State.le\n\ntheorem State.le_refl (σ : State) : σ ≼ σ :=\n  fun _ _ h => h\n\ntheorem State.le_trans : σ₁ ≼ σ₂ → σ₂ ≼ σ₃ → σ₁ ≼ σ₃ :=\n  fun h₁ h₂ x v h => h₂ (h₁ h)\n\ntheorem State.bot_le (σ : State) : ⊥ ≼ σ :=\n  fun _ _ h => by contradiction\n\ntheorem State.erase_le_cons (h : σ' ≼ σ) : σ'.erase x ≼ ((x, v) :: σ) := by\n  intro y w hf'\n  by_cases hyx : y = x <;> simp [*] at hf' |-\n  exact h hf'\n\ntheorem State.cons_le_cons (h : σ' ≼ σ) : (x, v) :: σ' ≼ (x, v) :: σ := by\n  intro y w hf'\n  by_cases hyx : y = x <;> simp [*] at hf' |-\n  next => assumption\n  next => exact h hf'\n\ntheorem State.cons_le_of_eq (h₁ : σ' ≼ σ) (h₂ : σ.find? x = some v) : (x, v) :: σ' ≼ σ := by\n  intro y w hf'\n  by_cases hyx : y = x <;> simp [*] at hf' |-\n  next => assumption\n  next => exact h₁ hf'\n\ntheorem State.erase_le (σ : State) : σ.erase x ≼ σ := by\n  match σ with\n  | [] => simp; apply le_refl\n  | (y, v) :: σ =>\n    simp\n    split <;> simp [*]\n    next => apply erase_le_cons; apply le_refl\n    next => apply cons_le_cons; apply erase_le\n\ntheorem State.join_le_left (σ₁ σ₂ : State) : σ₁.join σ₂ ≼ σ₁ := by\n  match σ₁ with\n  | [] => simp; apply le_refl\n  | (x, v) :: σ₁ =>\n    simp\n    have : (erase σ₁ x).length < σ₁.length.succ := length_erase_lt ..\n    have ih := join_le_left (State.erase σ₁ x) σ₂\n    split\n    next y w h =>\n      split\n      next => apply cons_le_cons; apply le_trans ih (erase_le _)\n      next => apply le_trans ih (erase_le_cons (le_refl _))\n    next h => apply le_trans ih (erase_le_cons (le_refl _))\ntermination_by _ σ₁ _ => σ₁.length\n\ntheorem State.join_le_left_of (h : σ₁ ≼ σ₂) (σ₃ : State) : σ₁.join σ₃ ≼ σ₂ :=\n  le_trans (join_le_left σ₁ σ₃) h\n\ntheorem State.join_le_right (σ₁ σ₂ : State) : σ₁.join σ₂ ≼ σ₂ := by\n  match σ₁ with\n  | [] => simp; apply bot_le\n  | (x, v) :: σ₁ =>\n    simp\n    have : (erase σ₁ x).length < σ₁.length.succ := length_erase_lt ..\n    have ih := join_le_right (erase σ₁ x) σ₂\n    split\n    next y w h =>\n      split <;> simp [*]\n      next => apply cons_le_of_eq ih h\n    next h => assumption\ntermination_by _ σ₁ _ => σ₁.length\n\ntheorem State.join_le_right_of (h : σ₁ ≼ σ₂) (σ₃ : State) : σ₃.join σ₁ ≼ σ₂ :=\n  le_trans (join_le_right σ₃ σ₁) h\n\ntheorem State.eq_bot (h : σ ≼ ⊥) : σ = ⊥ := by\n  match σ with\n  | [] => simp\n  | (y, v) :: σ =>\n    have : State.find? ((y, v) :: σ) y = some v := by simp\n    have := h this\n    contradiction\n\ntheorem State.erase_le_of_le_cons (h : σ' ≼ (x, v) :: σ) : σ'.erase x ≼ σ := by\n  intro y w hf'\n  by_cases hxy : x = y <;> simp [*] at hf'\n  have hf := h hf'\n  simp [hxy, Ne.symm hxy] at hf\n  assumption\n\ntheorem State.erase_le_update (h : σ' ≼ σ) : σ'.erase x ≼ σ.update x v := by\n  intro y w hf'\n  by_cases hxy : x = y <;> simp [*] at hf' |-\n  exact h hf'\n\ntheorem State.update_le_update (h : σ' ≼ σ) : σ'.update x v ≼ σ.update x v := by\n  intro y w hf\n  induction σ generalizing σ' hf with\n  | nil  => rw [eq_bot h] at hf; assumption\n  | cons zw' σ ih =>\n    have (z, w') := zw'; simp\n    have : σ'.erase z ≼ σ := erase_le_of_le_cons h\n    have ih := ih this\n    revert ih hf\n    split <;> simp [*] <;> by_cases hyz : y = z <;> simp (config := { contextual := true }) [*]\n    next =>\n      intro he'\n      have he := h he'\n      simp [*] at he\n      assumption\n    next =>\n      by_cases hxy : x = y <;> simp [*]\n      next => intros; assumption\n      next =>\n        intro he' ih\n        exact ih he'\n\ntheorem Expr.eval_constProp_of_sub (e : Expr) (h : σ' ≼ σ) : (e.constProp σ').eval σ = e.eval σ := by\n  induction e with simp [*]\n  | var x =>\n    split <;> simp\n    next he => rw [State.get_of_find? (h he)]\n\ntheorem Expr.eval_constProp_of_eq_of_sub {e : Expr} (h₁ : e.eval σ = v) (h₂ : σ' ≼ σ) : (e.constProp σ').eval σ = v := by\n  have := eval_constProp_of_sub e h₂\n  simp [h₁] at this\n  assumption\n\ntheorem Stmt.constProp_sub (h₁ : (σ₁, s) ⇓ σ₂) (h₂ : σ₁' ≼ σ₁) : (s.constProp σ₁').2 ≼ σ₂ := by\n  induction h₁ generalizing σ₁' with simp\n  | skip => assumption\n  | assign heq =>\n    split <;> simp\n    next h =>\n      have heq' := Expr.eval_constProp_of_eq_of_sub heq h₂\n      rw [← Expr.eval_simplify, h] at heq'\n      simp at heq'\n      rw [heq']\n      apply State.update_le_update h₂\n    next h _ _ =>\n      exact State.erase_le_update h₂\n  | whileTrue heq h₃ h₄ ih₃ ih₄ =>\n    have ih₃ := ih₃ h₂\n    have ih₄ := ih₄ ih₃\n    simp [heq] at ih₄\n    exact ih₄\n  | whileFalse heq => apply State.bot_le\n  | ifTrue heq h ih =>\n    have ih := ih h₂\n    apply State.join_le_left_of ih\n  | ifFalse heq h ih =>\n    have ih := ih h₂\n    apply State.join_le_right_of ih\n  | seq h₃ h₄ ih₃ ih₄ => exact ih₄ (ih₃ h₂)\n\ntheorem Stmt.constProp_correct (h₁ : (σ₁, s) ⇓ σ₂) (h₂ : σ₁' ≼ σ₁) : (σ₁, (s.constProp σ₁').1) ⇓ σ₂ := by\n  induction h₁ generalizing σ₁' with simp_all\n  | skip => exact Bigstep.skip\n  | assign heq =>\n    split <;> simp\n    next h =>\n      have heq' := Expr.eval_constProp_of_eq_of_sub heq h₂\n      rw [← Expr.eval_simplify, h] at heq'\n      simp at heq'\n      apply Bigstep.assign; simp [*]\n    next =>\n      have heq' := Expr.eval_constProp_of_eq_of_sub heq h₂\n      rw [← Expr.eval_simplify] at heq'\n      apply Bigstep.assign heq'\n  | seq h₁ h₂ ih₁ ih₂ =>\n    apply Bigstep.seq (ih₁ h₂) (ih₂ (constProp_sub h₁ h₂))\n  | whileTrue heq h₁ h₂ ih₁ ih₂ =>\n    have ih₁ := ih₁ (State.bot_le _)\n    have ih₂ := ih₂ (State.bot_le _)\n    exact Bigstep.whileTrue heq ih₁ ih₂\n  | whileFalse heq =>\n    exact Bigstep.whileFalse heq\n  | ifTrue heq h ih =>\n    exact Bigstep.ifTrue (Expr.eval_constProp_of_eq_of_sub heq h₂) (ih h₂)\n  | ifFalse heq h ih =>\n    exact Bigstep.ifFalse (Expr.eval_constProp_of_eq_of_sub heq h₂) (ih h₂)\n\ndef Stmt.constPropagation (s : Stmt) : Stmt :=\n  (s.constProp ⊥).1\n\ntheorem Stmt.constPropagation_correct (h : (σ, s) ⇓ σ') : (σ, s.constPropagation) ⇓ σ' :=\n  constProp_correct h (State.bot_le _)\n\ndef example4 := `[Stmt|\n  x := 2;\n  if (x < 3) {\n    x := x + 1;\n    y := y + x;\n  } else {\n    y := y + 3;\n  }\n]\n\n#eval example4.constPropagation.simplify\n\n#exit\n-- TODO: add simp theorems for monadic code\ntheorem Stmt.eval_correct {s : Stmt} (h : (s.eval fuel).run σ = (.ok unit, σ')) : (σ, s) ⇓ σ' := by\n  induction fuel generalizing s σ σ' with simp at h\n  | zero => injection h; contradiction\n  | succ fuel ih =>\n    split at h\n    next => injection h with _ h; rw [h]; exact Bigstep.skip\n    next => done\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/constProp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3217369362777521}}
{"text": "import tactic\nimport tactic.induction\n\nimport .base .game .pw_1 .pw_2 .pw_ge\n\nnoncomputable theory\nopen_locale classical\n\ndef play {pw : ℕ} (a : A pw) (d : D) (n : ℕ) :=\nsimulate a d n\n\ndef A_wins {pw : ℕ} (a : A pw) (d : D) :=\n(init_game a d state₀).A_wins\n\ndef D_wins {pw : ℕ} (a : A pw) (d : D) :=\n(init_game a d state₀).D_wins\n\nlemma A_pw_0_not_hws : ¬A_hws 0 :=\nbegin\n  rintro ⟨a, h⟩, contrapose! h, clear h, use default, rw not_A_wins_at, use 1,\n  rw [play_1, play_move_at_act], swap, { triv },\n  rw [play_A_move_at, dif_neg], { exact not_false },\n  push_neg, rintro h₁ ⟨ma, h₂, h₃, h₄⟩,\n  rw [nat.le_zero_iff, dist_eq_zero_iff] at h₃, contradiction,\nend\n\nlemma A_pw_ge_hws {pw pw₁ : ℕ}\n  (h₁ : pw ≤ pw₁) (h₂ : A_hws pw) : A_hws pw₁ :=\nbegin\n  cases h₂ with a h₂, use mk_A_pw_ge a h₁,\n  intro d, specialize h₂ d, apply mk_A_pw_ge_wins_at_of h₂,\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.44167300566462564, "lm_q1q2_score": 0.32152710456835815}}
{"text": "/-\nCopyright (c) 2020 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot, Eric Wieser\n\n! This file was ported from Lean 3 source module data.pi.algebra\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.ToAdditive\nimport Mathbin.Algebra.Group.Defs\nimport Mathbin.Logic.Unique\nimport Mathbin.Tactic.Congr\nimport Mathbin.Tactic.Simpa\nimport Mathbin.Tactic.SplitIfs\nimport Mathbin.Data.Sum.Basic\nimport Mathbin.Data.Prod.Basic\n\n/-!\n# Instances and theorems on pi types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides basic definitions and notation instances for Pi types.\n\nInstances of more sophisticated classes are defined in `pi.lean` files elsewhere.\n-/\n\n\nopen Function\n\nuniverse u v₁ v₂ v₃\n\nvariable {I : Type u}\n\n-- The indexing type\nvariable {α β γ : Type _}\n\n-- The families of types already equipped with instances\nvariable {f : I → Type v₁} {g : I → Type v₂} {h : I → Type v₃}\n\nvariable (x y : ∀ i, f i) (i : I)\n\nnamespace Pi\n\n/-! `1`, `0`, `+`, `*`, `+ᵥ`, `•`, `^`, `-`, `⁻¹`, and `/` are defined pointwise. -/\n\n\n#print Pi.instOne /-\n@[to_additive]\ninstance instOne [∀ i, One <| f i] : One (∀ i : I, f i) :=\n  ⟨fun _ => 1⟩\n#align pi.has_one Pi.instOne\n#align pi.has_zero Pi.instZero\n-/\n\n#print Pi.one_apply /-\n@[simp, to_additive]\ntheorem one_apply [∀ i, One <| f i] : (1 : ∀ i, f i) i = 1 :=\n  rfl\n#align pi.one_apply Pi.one_apply\n#align pi.zero_apply Pi.zero_apply\n-/\n\n#print Pi.one_def /-\n@[to_additive]\ntheorem one_def [∀ i, One <| f i] : (1 : ∀ i, f i) = fun i => 1 :=\n  rfl\n#align pi.one_def Pi.one_def\n#align pi.zero_def Pi.zero_def\n-/\n\n#print Pi.const_one /-\n@[simp, to_additive]\ntheorem const_one [One β] : const α (1 : β) = 1 :=\n  rfl\n#align pi.const_one Pi.const_one\n#align pi.const_zero Pi.const_zero\n-/\n\n/- warning: pi.one_comp -> Pi.one_comp is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : One.{u3} γ] (x : α -> β), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ (OfNat.ofNat.{max u2 u3} (β -> γ) 1 (OfNat.mk.{max u2 u3} (β -> γ) 1 (One.one.{max u2 u3} (β -> γ) (Pi.instOne.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))))) x) (OfNat.ofNat.{max u1 u3} (α -> γ) 1 (OfNat.mk.{max u1 u3} (α -> γ) 1 (One.one.{max u1 u3} (α -> γ) (Pi.instOne.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1)))))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : One.{u3} γ] (x : α -> β), Eq.{max (succ u2) (succ u3)} (α -> γ) (Function.comp.{succ u2, succ u1, succ u3} α β γ (OfNat.ofNat.{max u1 u3} (β -> γ) 1 (One.toOfNat1.{max u1 u3} (β -> γ) (Pi.instOne.{u1, u3} β (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.301 : β) => γ) (fun (i : β) => _inst_1)))) x) (OfNat.ofNat.{max u2 u3} (α -> γ) 1 (One.toOfNat1.{max u2 u3} (α -> γ) (Pi.instOne.{u2, u3} α (fun (a._@.Init.Prelude._hyg.25 : α) => γ) (fun (i : α) => _inst_1))))\nCase conversion may be inaccurate. Consider using '#align pi.one_comp Pi.one_compₓ'. -/\n@[simp, to_additive]\ntheorem one_comp [One γ] (x : α → β) : (1 : β → γ) ∘ x = 1 :=\n  rfl\n#align pi.one_comp Pi.one_comp\n#align pi.zero_comp Pi.zero_comp\n\n/- warning: pi.comp_one -> Pi.comp_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : One.{u2} β] (x : β -> γ), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ x (OfNat.ofNat.{max u1 u2} (α -> β) 1 (OfNat.mk.{max u1 u2} (α -> β) 1 (One.one.{max u1 u2} (α -> β) (Pi.instOne.{u1, u2} α (fun (ᾰ : α) => β) (fun (i : α) => _inst_1)))))) (Function.const.{succ u3, succ u1} γ α (x (OfNat.ofNat.{u2} β 1 (OfNat.mk.{u2} β 1 (One.one.{u2} β _inst_1)))))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u1}} [_inst_1 : One.{u3} β] (x : β -> γ), Eq.{max (succ u2) (succ u1)} (α -> γ) (Function.comp.{succ u2, succ u3, succ u1} α β γ x (OfNat.ofNat.{max u2 u3} (α -> β) 1 (One.toOfNat1.{max u2 u3} (α -> β) (Pi.instOne.{u2, u3} α (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.349 : α) => β) (fun (i : α) => _inst_1))))) (Function.const.{succ u1, succ u2} γ α (x (OfNat.ofNat.{u3} β 1 (One.toOfNat1.{u3} β _inst_1))))\nCase conversion may be inaccurate. Consider using '#align pi.comp_one Pi.comp_oneₓ'. -/\n@[simp, to_additive]\ntheorem comp_one [One β] (x : β → γ) : x ∘ 1 = const α (x 1) :=\n  rfl\n#align pi.comp_one Pi.comp_one\n#align pi.comp_zero Pi.comp_zero\n\n#print Pi.instMul /-\n@[to_additive]\ninstance instMul [∀ i, Mul <| f i] : Mul (∀ i : I, f i) :=\n  ⟨fun f g i => f i * g i⟩\n#align pi.has_mul Pi.instMul\n#align pi.has_add Pi.instAdd\n-/\n\n#print Pi.mul_apply /-\n@[simp, to_additive]\ntheorem mul_apply [∀ i, Mul <| f i] : (x * y) i = x i * y i :=\n  rfl\n#align pi.mul_apply Pi.mul_apply\n#align pi.add_apply Pi.add_apply\n-/\n\n#print Pi.mul_def /-\n@[to_additive]\ntheorem mul_def [∀ i, Mul <| f i] : x * y = fun i => x i * y i :=\n  rfl\n#align pi.mul_def Pi.mul_def\n#align pi.add_def Pi.add_def\n-/\n\n#print Pi.const_mul /-\n@[simp, to_additive]\ntheorem const_mul [Mul β] (a b : β) : const α a * const α b = const α (a * b) :=\n  rfl\n#align pi.const_mul Pi.const_mul\n#align pi.const_add Pi.const_add\n-/\n\n/- warning: pi.mul_comp -> Pi.mul_comp is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : Mul.{u3} γ] (x : β -> γ) (y : β -> γ) (z : α -> β), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ (HMul.hMul.{max u2 u3, max u2 u3, max u2 u3} (β -> γ) (β -> γ) (β -> γ) (instHMul.{max u2 u3} (β -> γ) (Pi.instMul.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x y) z) (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (α -> γ) (α -> γ) (α -> γ) (instHMul.{max u1 u3} (α -> γ) (Pi.instMul.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) (Function.comp.{succ u1, succ u2, succ u3} α β γ x z) (Function.comp.{succ u1, succ u2, succ u3} α β γ y z))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : Mul.{u3} γ] (x : β -> γ) (y : β -> γ) (z : α -> β), Eq.{max (succ u2) (succ u3)} (α -> γ) (Function.comp.{succ u2, succ u1, succ u3} α β γ (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (β -> γ) (β -> γ) (β -> γ) (instHMul.{max u1 u3} (β -> γ) (Pi.instMul.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x y) z) (HMul.hMul.{max u2 u3, max u2 u3, max u2 u3} (α -> γ) (α -> γ) (α -> γ) (instHMul.{max u2 u3} (α -> γ) (Pi.instMul.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) (Function.comp.{succ u2, succ u1, succ u3} α β γ x z) (Function.comp.{succ u2, succ u1, succ u3} α β γ y z))\nCase conversion may be inaccurate. Consider using '#align pi.mul_comp Pi.mul_compₓ'. -/\n@[to_additive]\ntheorem mul_comp [Mul γ] (x y : β → γ) (z : α → β) : (x * y) ∘ z = x ∘ z * y ∘ z :=\n  rfl\n#align pi.mul_comp Pi.mul_comp\n#align pi.add_comp Pi.add_comp\n\n#print Pi.instSMul /-\n@[to_additive Pi.instVAdd]\ninstance instSMul [∀ i, SMul α <| f i] : SMul α (∀ i : I, f i) :=\n  ⟨fun s x => fun i => s • x i⟩\n#align pi.has_smul Pi.instSMul\n#align pi.has_vadd Pi.instVAdd\n-/\n\n/- warning: pi.smul_apply -> Pi.smul_apply is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u3}} {f : I -> Type.{u2}} [_inst_1 : forall (i : I), SMul.{u3, u2} α (f i)] (s : α) (x : forall (i : I), f i) (i : I), Eq.{succ u2} (f i) (SMul.smul.{u3, max u1 u2} α (forall (i : I), f i) (Pi.instSMul.{u1, u2, u3} I α (fun (i : I) => f i) (fun (i : I) => _inst_1 i)) s x i) (SMul.smul.{u3, u2} α (f i) (_inst_1 i) s (x i))\nbut is expected to have type\n  forall {I : Type.{u2}} {α : Type.{u1}} {f : I -> Type.{u3}} [_inst_1 : forall (i : I), SMul.{u1, u3} α (f i)] (s : α) (x : forall (i : I), f i) (i : I), Eq.{succ u3} (f i) (HSMul.hSMul.{u1, max u2 u3, max u2 u3} α (forall (i : I), f i) (forall (i : I), f i) (instHSMul.{u1, max u2 u3} α (forall (i : I), f i) (Pi.instSMul.{u2, u3, u1} I α (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) s x i) (HSMul.hSMul.{u1, u3, u3} α (f i) (f i) (instHSMul.{u1, u3} α (f i) (_inst_1 i)) s (x i))\nCase conversion may be inaccurate. Consider using '#align pi.smul_apply Pi.smul_applyₓ'. -/\n@[simp, to_additive]\ntheorem smul_apply [∀ i, SMul α <| f i] (s : α) (x : ∀ i, f i) (i : I) : (s • x) i = s • x i :=\n  rfl\n#align pi.smul_apply Pi.smul_apply\n#align pi.vadd_apply Pi.vadd_apply\n\n/- warning: pi.smul_def -> Pi.smul_def is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u3}} {f : I -> Type.{u2}} [_inst_1 : forall (i : I), SMul.{u3, u2} α (f i)] (s : α) (x : forall (i : I), f i), Eq.{succ (max u1 u2)} (forall (i : I), f i) (SMul.smul.{u3, max u1 u2} α (forall (i : I), f i) (Pi.instSMul.{u1, u2, u3} I α (fun (i : I) => f i) (fun (i : I) => _inst_1 i)) s x) (fun (i : I) => SMul.smul.{u3, u2} α (f i) (_inst_1 i) s (x i))\nbut is expected to have type\n  forall {I : Type.{u2}} {α : Type.{u1}} {f : I -> Type.{u3}} [_inst_1 : forall (i : I), SMul.{u1, u3} α (f i)] (s : α) (x : forall (i : I), f i), Eq.{max (succ u2) (succ u3)} (forall (i : I), f i) (HSMul.hSMul.{u1, max u2 u3, max u2 u3} α (forall (i : I), f i) (forall (i : I), f i) (instHSMul.{u1, max u2 u3} α (forall (i : I), f i) (Pi.instSMul.{u2, u3, u1} I α (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) s x) (fun (i : I) => HSMul.hSMul.{u1, u3, u3} α (f i) (f i) (instHSMul.{u1, u3} α (f i) (_inst_1 i)) s (x i))\nCase conversion may be inaccurate. Consider using '#align pi.smul_def Pi.smul_defₓ'. -/\n@[to_additive]\ntheorem smul_def [∀ i, SMul α <| f i] (s : α) (x : ∀ i, f i) : s • x = fun i => s • x i :=\n  rfl\n#align pi.smul_def Pi.smul_def\n#align pi.vadd_def Pi.vadd_def\n\n/- warning: pi.smul_const -> Pi.smul_const is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : SMul.{u2, u3} α β] (a : α) (b : β), Eq.{succ (max u1 u3)} (I -> β) (SMul.smul.{u2, max u1 u3} α (I -> β) (Pi.instSMul.{u1, u3, u2} I α (fun (ᾰ : I) => β) (fun (i : I) => _inst_1)) a (Function.const.{succ u3, succ u1} β I b)) (Function.const.{succ u3, succ u1} β I (SMul.smul.{u2, u3} α β _inst_1 a b))\nbut is expected to have type\n  forall {I : Type.{u3}} {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : SMul.{u1, u2} α β] (a : α) (b : β), Eq.{max (succ u3) (succ u2)} (I -> β) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} α (I -> β) (I -> β) (instHSMul.{u1, max u3 u2} α (I -> β) (Pi.instSMul.{u3, u2, u1} I α (fun (a._@.Init.Prelude._hyg.54 : I) => β) (fun (i : I) => _inst_1))) a (Function.const.{succ u2, succ u3} β I b)) (Function.const.{succ u2, succ u3} β I (HSMul.hSMul.{u1, u2, u2} α β β (instHSMul.{u1, u2} α β _inst_1) a b))\nCase conversion may be inaccurate. Consider using '#align pi.smul_const Pi.smul_constₓ'. -/\n@[simp, to_additive]\ntheorem smul_const [SMul α β] (a : α) (b : β) : a • const I b = const I (a • b) :=\n  rfl\n#align pi.smul_const Pi.smul_const\n#align pi.vadd_const Pi.vadd_const\n\n/- warning: pi.smul_comp -> Pi.smul_comp is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_1 : SMul.{u2, u4} α γ] (a : α) (x : β -> γ) (y : I -> β), Eq.{max (succ u1) (succ u4)} (I -> γ) (Function.comp.{succ u1, succ u3, succ u4} I β γ (SMul.smul.{u2, max u3 u4} α (β -> γ) (Pi.instSMul.{u3, u4, u2} β α (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)) a x) y) (SMul.smul.{u2, max u1 u4} α (I -> γ) (Pi.instSMul.{u1, u4, u2} I α (fun (ᾰ : I) => γ) (fun (i : I) => _inst_1)) a (Function.comp.{succ u1, succ u3, succ u4} I β γ x y))\nbut is expected to have type\n  forall {I : Type.{u4}} {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : SMul.{u2, u3} α γ] (a : α) (x : β -> γ) (y : I -> β), Eq.{max (succ u4) (succ u3)} (I -> γ) (Function.comp.{succ u4, succ u1, succ u3} I β γ (HSMul.hSMul.{u2, max u1 u3, max u1 u3} α (β -> γ) (β -> γ) (instHSMul.{u2, max u1 u3} α (β -> γ) (Pi.instSMul.{u1, u3, u2} β α (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.1026 : β) => γ) (fun (i : β) => _inst_1))) a x) y) (HSMul.hSMul.{u2, max u4 u3, max u4 u3} α (I -> γ) (I -> γ) (instHSMul.{u2, max u4 u3} α (I -> γ) (Pi.instSMul.{u4, u3, u2} I α (fun (a._@.Init.Prelude._hyg.25 : I) => γ) (fun (i : I) => _inst_1))) a (Function.comp.{succ u4, succ u1, succ u3} I β γ x y))\nCase conversion may be inaccurate. Consider using '#align pi.smul_comp Pi.smul_compₓ'. -/\n@[to_additive]\ntheorem smul_comp [SMul α γ] (a : α) (x : β → γ) (y : I → β) : (a • x) ∘ y = a • x ∘ y :=\n  rfl\n#align pi.smul_comp Pi.smul_comp\n#align pi.vadd_comp Pi.vadd_comp\n\n@[to_additive Pi.instSMul]\ninstance hasPow [∀ i, Pow (f i) β] : Pow (∀ i, f i) β :=\n  ⟨fun x b i => x i ^ b⟩\n#align pi.has_pow Pi.hasPow\n#align pi.has_smul Pi.instSMul\n\n/- warning: pi.pow_apply -> Pi.pow_apply is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {β : Type.{u3}} {f : I -> Type.{u2}} [_inst_1 : forall (i : I), Pow.{u2, u3} (f i) β] (x : forall (i : I), f i) (b : β) (i : I), Eq.{succ u2} (f i) (HPow.hPow.{max u1 u2, u3, max u1 u2} (forall (i : I), f i) β (forall (i : I), f i) (instHPow.{max u1 u2, u3} (forall (i : I), f i) β (Pi.hasPow.{u1, u2, u3} I β (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) x b i) (HPow.hPow.{u2, u3, u2} (f i) β (f i) (instHPow.{u2, u3} (f i) β (_inst_1 i)) (x i) b)\nbut is expected to have type\n  forall {I : Type.{u2}} {β : Type.{u1}} {f : I -> Type.{u3}} [_inst_1 : forall (i : I), Pow.{u3, u1} (f i) β] (x : forall (i : I), f i) (b : β) (i : I), Eq.{succ u3} (f i) (HPow.hPow.{max u2 u3, u1, max u2 u3} (forall (i : I), f i) β (forall (i : I), f i) (instHPow.{max u2 u3, u1} (forall (i : I), f i) β (Pi.instPow.{u2, u3, u1} I β (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) x b i) (HPow.hPow.{u3, u1, u3} (f i) β (f i) (instHPow.{u3, u1} (f i) β (_inst_1 i)) (x i) b)\nCase conversion may be inaccurate. Consider using '#align pi.pow_apply Pi.pow_applyₓ'. -/\n@[simp, to_additive Pi.smul_apply, to_additive_reorder 5]\ntheorem pow_apply [∀ i, Pow (f i) β] (x : ∀ i, f i) (b : β) (i : I) : (x ^ b) i = x i ^ b :=\n  rfl\n#align pi.pow_apply Pi.pow_apply\n#align pi.smul_apply Pi.smul_apply\n\n/- warning: pi.pow_def -> Pi.pow_def is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {β : Type.{u3}} {f : I -> Type.{u2}} [_inst_1 : forall (i : I), Pow.{u2, u3} (f i) β] (x : forall (i : I), f i) (b : β), Eq.{succ (max u1 u2)} (forall (i : I), f i) (HPow.hPow.{max u1 u2, u3, max u1 u2} (forall (i : I), f i) β (forall (i : I), f i) (instHPow.{max u1 u2, u3} (forall (i : I), f i) β (Pi.hasPow.{u1, u2, u3} I β (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) x b) (fun (i : I) => HPow.hPow.{u2, u3, u2} (f i) β (f i) (instHPow.{u2, u3} (f i) β (_inst_1 i)) (x i) b)\nbut is expected to have type\n  forall {I : Type.{u2}} {β : Type.{u1}} {f : I -> Type.{u3}} [_inst_1 : forall (i : I), Pow.{u3, u1} (f i) β] (x : forall (i : I), f i) (b : β), Eq.{max (succ u2) (succ u3)} (forall (i : I), f i) (HPow.hPow.{max u2 u3, u1, max u2 u3} (forall (i : I), f i) β (forall (i : I), f i) (instHPow.{max u2 u3, u1} (forall (i : I), f i) β (Pi.instPow.{u2, u3, u1} I β (fun (i : I) => f i) (fun (i : I) => _inst_1 i))) x b) (fun (i : I) => HPow.hPow.{u3, u1, u3} (f i) β (f i) (instHPow.{u3, u1} (f i) β (_inst_1 i)) (x i) b)\nCase conversion may be inaccurate. Consider using '#align pi.pow_def Pi.pow_defₓ'. -/\n@[to_additive Pi.smul_def, to_additive_reorder 5]\ntheorem pow_def [∀ i, Pow (f i) β] (x : ∀ i, f i) (b : β) : x ^ b = fun i => x i ^ b :=\n  rfl\n#align pi.pow_def Pi.pow_def\n#align pi.smul_def Pi.smul_def\n\n/- warning: pi.const_pow -> Pi.const_pow is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} [_inst_1 : Pow.{u3, u2} β α] (b : β) (a : α), Eq.{succ (max u1 u3)} (I -> β) (HPow.hPow.{max u1 u3, u2, max u1 u3} (I -> β) α (I -> β) (instHPow.{max u1 u3, u2} (I -> β) α (Pi.hasPow.{u1, u3, u2} I α (fun (ᾰ : I) => β) (fun (i : I) => _inst_1))) (Function.const.{succ u3, succ u1} β I b) a) (Function.const.{succ u3, succ u1} β I (HPow.hPow.{u3, u2, u3} β α β (instHPow.{u3, u2} β α _inst_1) b a))\nbut is expected to have type\n  forall {I : Type.{u3}} {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Pow.{u2, u1} α β] (b : α) (a : β), Eq.{max (succ u3) (succ u2)} (I -> α) (HPow.hPow.{max u3 u2, u1, max u3 u2} (I -> α) β (I -> α) (instHPow.{max u3 u2, u1} (I -> α) β (Pi.instPow.{u3, u2, u1} I β (fun (ᾰ : I) => α) (fun (i : I) => _inst_1))) (Function.const.{succ u2, succ u3} α I b) a) (Function.const.{succ u2, succ u3} α I (HPow.hPow.{u2, u1, u2} α β α (instHPow.{u2, u1} α β _inst_1) b a))\nCase conversion may be inaccurate. Consider using '#align pi.const_pow Pi.const_powₓ'. -/\n-- `to_additive` generates bad output if we take `has_pow α β`.\n@[simp, to_additive smul_const, to_additive_reorder 5]\ntheorem const_pow [Pow β α] (b : β) (a : α) : const I b ^ a = const I (b ^ a) :=\n  rfl\n#align pi.const_pow Pi.const_pow\n#align pi.smul_const Pi.smul_const\n\n/- warning: pi.pow_comp -> Pi.pow_comp is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u2}} {β : Type.{u3}} {γ : Type.{u4}} [_inst_1 : Pow.{u4, u2} γ α] (x : β -> γ) (a : α) (y : I -> β), Eq.{max (succ u1) (succ u4)} (I -> γ) (Function.comp.{succ u1, succ u3, succ u4} I β γ (HPow.hPow.{max u3 u4, u2, max u3 u4} (β -> γ) α (β -> γ) (instHPow.{max u3 u4, u2} (β -> γ) α (Pi.hasPow.{u3, u4, u2} β α (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x a) y) (HPow.hPow.{max u1 u4, u2, max u1 u4} (I -> γ) α (I -> γ) (instHPow.{max u1 u4, u2} (I -> γ) α (Pi.hasPow.{u1, u4, u2} I α (fun (ᾰ : I) => γ) (fun (i : I) => _inst_1))) (Function.comp.{succ u1, succ u3, succ u4} I β γ x y) a)\nbut is expected to have type\n  forall {I : Type.{u4}} {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : Pow.{u3, u2} γ α] (x : β -> γ) (a : α) (y : I -> β), Eq.{max (succ u4) (succ u3)} (I -> γ) (Function.comp.{succ u4, succ u1, succ u3} I β γ (HPow.hPow.{max u1 u3, u2, max u1 u3} (β -> γ) α (β -> γ) (instHPow.{max u1 u3, u2} (β -> γ) α (Pi.instPow.{u1, u3, u2} β α (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x a) y) (HPow.hPow.{max u4 u3, u2, max u4 u3} (I -> γ) α (I -> γ) (instHPow.{max u4 u3, u2} (I -> γ) α (Pi.instPow.{u4, u3, u2} I α (fun (ᾰ : I) => γ) (fun (i : I) => _inst_1))) (Function.comp.{succ u4, succ u1, succ u3} I β γ x y) a)\nCase conversion may be inaccurate. Consider using '#align pi.pow_comp Pi.pow_compₓ'. -/\n@[to_additive smul_comp, to_additive_reorder 6]\ntheorem pow_comp [Pow γ α] (x : β → γ) (a : α) (y : I → β) : (x ^ a) ∘ y = x ∘ y ^ a :=\n  rfl\n#align pi.pow_comp Pi.pow_comp\n#align pi.smul_comp Pi.smul_comp\n\n#print Pi.bit0_apply /-\n@[simp]\ntheorem bit0_apply [∀ i, Add <| f i] : (bit0 x) i = bit0 (x i) :=\n  rfl\n#align pi.bit0_apply Pi.bit0_apply\n-/\n\n#print Pi.bit1_apply /-\n@[simp]\ntheorem bit1_apply [∀ i, Add <| f i] [∀ i, One <| f i] : (bit1 x) i = bit1 (x i) :=\n  rfl\n#align pi.bit1_apply Pi.bit1_apply\n-/\n\n#print Pi.instInv /-\n@[to_additive]\ninstance instInv [∀ i, Inv <| f i] : Inv (∀ i : I, f i) :=\n  ⟨fun f i => (f i)⁻¹⟩\n#align pi.has_inv Pi.instInv\n#align pi.has_neg Pi.instNeg\n-/\n\n#print Pi.inv_apply /-\n@[simp, to_additive]\ntheorem inv_apply [∀ i, Inv <| f i] : x⁻¹ i = (x i)⁻¹ :=\n  rfl\n#align pi.inv_apply Pi.inv_apply\n#align pi.neg_apply Pi.neg_apply\n-/\n\n#print Pi.inv_def /-\n@[to_additive]\ntheorem inv_def [∀ i, Inv <| f i] : x⁻¹ = fun i => (x i)⁻¹ :=\n  rfl\n#align pi.inv_def Pi.inv_def\n#align pi.neg_def Pi.neg_def\n-/\n\n#print Pi.const_inv /-\n@[to_additive]\ntheorem const_inv [Inv β] (a : β) : (const α a)⁻¹ = const α a⁻¹ :=\n  rfl\n#align pi.const_inv Pi.const_inv\n#align pi.const_neg Pi.const_neg\n-/\n\n/- warning: pi.inv_comp -> Pi.inv_comp is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : Inv.{u3} γ] (x : β -> γ) (y : α -> β), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ (Inv.inv.{max u2 u3} (β -> γ) (Pi.instInv.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)) x) y) (Inv.inv.{max u1 u3} (α -> γ) (Pi.instInv.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1)) (Function.comp.{succ u1, succ u2, succ u3} α β γ x y))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : Inv.{u3} γ] (x : β -> γ) (y : α -> β), Eq.{max (succ u2) (succ u3)} (α -> γ) (Function.comp.{succ u2, succ u1, succ u3} α β γ (Inv.inv.{max u1 u3} (β -> γ) (Pi.instInv.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)) x) y) (Inv.inv.{max u2 u3} (α -> γ) (Pi.instInv.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1)) (Function.comp.{succ u2, succ u1, succ u3} α β γ x y))\nCase conversion may be inaccurate. Consider using '#align pi.inv_comp Pi.inv_compₓ'. -/\n@[to_additive]\ntheorem inv_comp [Inv γ] (x : β → γ) (y : α → β) : x⁻¹ ∘ y = (x ∘ y)⁻¹ :=\n  rfl\n#align pi.inv_comp Pi.inv_comp\n#align pi.neg_comp Pi.neg_comp\n\n#print Pi.instDiv /-\n@[to_additive]\ninstance instDiv [∀ i, Div <| f i] : Div (∀ i : I, f i) :=\n  ⟨fun f g i => f i / g i⟩\n#align pi.has_div Pi.instDiv\n#align pi.has_sub Pi.instSub\n-/\n\n#print Pi.div_apply /-\n@[simp, to_additive]\ntheorem div_apply [∀ i, Div <| f i] : (x / y) i = x i / y i :=\n  rfl\n#align pi.div_apply Pi.div_apply\n#align pi.sub_apply Pi.sub_apply\n-/\n\n#print Pi.div_def /-\n@[to_additive]\ntheorem div_def [∀ i, Div <| f i] : x / y = fun i => x i / y i :=\n  rfl\n#align pi.div_def Pi.div_def\n#align pi.sub_def Pi.sub_def\n-/\n\n/- warning: pi.div_comp -> Pi.div_comp is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : Div.{u3} γ] (x : β -> γ) (y : β -> γ) (z : α -> β), Eq.{max (succ u1) (succ u3)} (α -> γ) (Function.comp.{succ u1, succ u2, succ u3} α β γ (HDiv.hDiv.{max u2 u3, max u2 u3, max u2 u3} (β -> γ) (β -> γ) (β -> γ) (instHDiv.{max u2 u3} (β -> γ) (Pi.instDiv.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x y) z) (HDiv.hDiv.{max u1 u3, max u1 u3, max u1 u3} (α -> γ) (α -> γ) (α -> γ) (instHDiv.{max u1 u3} (α -> γ) (Pi.instDiv.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) (Function.comp.{succ u1, succ u2, succ u3} α β γ x z) (Function.comp.{succ u1, succ u2, succ u3} α β γ y z))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : Div.{u3} γ] (x : β -> γ) (y : β -> γ) (z : α -> β), Eq.{max (succ u2) (succ u3)} (α -> γ) (Function.comp.{succ u2, succ u1, succ u3} α β γ (HDiv.hDiv.{max u1 u3, max u1 u3, max u1 u3} (β -> γ) (β -> γ) (β -> γ) (instHDiv.{max u1 u3} (β -> γ) (Pi.instDiv.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) x y) z) (HDiv.hDiv.{max u2 u3, max u2 u3, max u2 u3} (α -> γ) (α -> γ) (α -> γ) (instHDiv.{max u2 u3} (α -> γ) (Pi.instDiv.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) (Function.comp.{succ u2, succ u1, succ u3} α β γ x z) (Function.comp.{succ u2, succ u1, succ u3} α β γ y z))\nCase conversion may be inaccurate. Consider using '#align pi.div_comp Pi.div_compₓ'. -/\n@[to_additive]\ntheorem div_comp [Div γ] (x y : β → γ) (z : α → β) : (x / y) ∘ z = x ∘ z / y ∘ z :=\n  rfl\n#align pi.div_comp Pi.div_comp\n#align pi.sub_comp Pi.sub_comp\n\n#print Pi.const_div /-\n@[simp, to_additive]\ntheorem const_div [Div β] (a b : β) : const α a / const α b = const α (a / b) :=\n  rfl\n#align pi.const_div Pi.const_div\n#align pi.const_sub Pi.const_sub\n-/\n\nsection\n\nvariable [DecidableEq I]\n\nvariable [∀ i, One (f i)] [∀ i, One (g i)] [∀ i, One (h i)]\n\n#print Pi.mulSingle /-\n/-- The function supported at `i`, with value `x` there, and `1` elsewhere. -/\n@[to_additive Pi.single \"The function supported at `i`, with value `x` there, and `0` elsewhere.\"]\ndef mulSingle (i : I) (x : f i) : ∀ i, f i :=\n  Function.update 1 i x\n#align pi.mul_single Pi.mulSingle\n#align pi.single Pi.single\n-/\n\n#print Pi.mulSingle_eq_same /-\n@[simp, to_additive]\ntheorem mulSingle_eq_same (i : I) (x : f i) : mulSingle i x i = x :=\n  Function.update_same i x _\n#align pi.mul_single_eq_same Pi.mulSingle_eq_same\n#align pi.single_eq_same Pi.single_eq_same\n-/\n\n#print Pi.mulSingle_eq_of_ne /-\n@[simp, to_additive]\ntheorem mulSingle_eq_of_ne {i i' : I} (h : i' ≠ i) (x : f i) : mulSingle i x i' = 1 :=\n  Function.update_noteq h x _\n#align pi.mul_single_eq_of_ne Pi.mulSingle_eq_of_ne\n#align pi.single_eq_of_ne Pi.single_eq_of_ne\n-/\n\n#print Pi.mulSingle_eq_of_ne' /-\n/-- Abbreviation for `mul_single_eq_of_ne h.symm`, for ease of use by `simp`. -/\n@[simp, to_additive \"Abbreviation for `single_eq_of_ne h.symm`, for ease of\\nuse by `simp`.\"]\ntheorem mulSingle_eq_of_ne' {i i' : I} (h : i ≠ i') (x : f i) : mulSingle i x i' = 1 :=\n  mulSingle_eq_of_ne h.symm x\n#align pi.mul_single_eq_of_ne' Pi.mulSingle_eq_of_ne'\n#align pi.single_eq_of_ne' Pi.single_eq_of_ne'\n-/\n\n#print Pi.mulSingle_one /-\n@[simp, to_additive]\ntheorem mulSingle_one (i : I) : mulSingle i (1 : f i) = 1 :=\n  Function.update_eq_self _ _\n#align pi.mul_single_one Pi.mulSingle_one\n#align pi.single_zero Pi.single_zero\n-/\n\n/- warning: pi.mul_single_apply -> Pi.mulSingle_apply is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} I] {β : Type.{u2}} [_inst_5 : One.{u2} β] (i : I) (x : β) (i' : I), Eq.{succ u2} β (Pi.mulSingle.{u1, u2} I (fun (i : I) => β) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5) i x i') (ite.{succ u2} β (Eq.{succ u1} I i' i) (_inst_1 i' i) x (OfNat.ofNat.{u2} β 1 (OfNat.mk.{u2} β 1 (One.one.{u2} β _inst_5))))\nbut is expected to have type\n  forall {I : Type.{u2}} {_inst_1 : Type.{u1}} [β : DecidableEq.{succ u2} I] [_inst_5 : One.{u1} _inst_1] (i : I) (x : _inst_1) (i' : I), Eq.{succ u1} _inst_1 (Pi.mulSingle.{u2, u1} I (fun (i : I) => _inst_1) (fun (a : I) (b : I) => β a b) (fun (i : I) => _inst_5) i x i') (ite.{succ u1} _inst_1 (Eq.{succ u2} I i' i) (β i' i) x (OfNat.ofNat.{u1} _inst_1 1 (One.toOfNat1.{u1} _inst_1 _inst_5)))\nCase conversion may be inaccurate. Consider using '#align pi.mul_single_apply Pi.mulSingle_applyₓ'. -/\n/-- On non-dependent functions, `pi.mul_single` can be expressed as an `ite` -/\n@[to_additive \"On non-dependent functions, `pi.single` can be expressed as an `ite`\"]\ntheorem mulSingle_apply {β : Sort _} [One β] (i : I) (x : β) (i' : I) :\n    mulSingle i x i' = if i' = i then x else 1 :=\n  Function.update_apply 1 i x i'\n#align pi.mul_single_apply Pi.mulSingle_apply\n#align pi.single_apply Pi.single_apply\n\n/- warning: pi.mul_single_comm -> Pi.mulSingle_comm is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} I] {β : Type.{u2}} [_inst_5 : One.{u2} β] (i : I) (x : β) (i' : I), Eq.{succ u2} β (Pi.mulSingle.{u1, u2} I (fun (i : I) => β) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5) i x i') (Pi.mulSingle.{u1, u2} I (fun (i' : I) => β) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5) i' x i)\nbut is expected to have type\n  forall {I : Type.{u2}} {_inst_1 : Type.{u1}} [β : DecidableEq.{succ u2} I] [_inst_5 : One.{u1} _inst_1] (i : I) (x : _inst_1) (i' : I), Eq.{succ u1} _inst_1 (Pi.mulSingle.{u2, u1} I (fun (i : I) => _inst_1) (fun (a : I) (b : I) => β a b) (fun (i : I) => _inst_5) i x i') (Pi.mulSingle.{u2, u1} I (fun (i' : I) => _inst_1) (fun (a : I) (b : I) => β a b) (fun (i : I) => _inst_5) i' x i)\nCase conversion may be inaccurate. Consider using '#align pi.mul_single_comm Pi.mulSingle_commₓ'. -/\n/-- On non-dependent functions, `pi.mul_single` is symmetric in the two indices. -/\n@[to_additive \"On non-dependent functions, `pi.single` is symmetric in the two\\nindices.\"]\ntheorem mulSingle_comm {β : Sort _} [One β] (i : I) (x : β) (i' : I) :\n    mulSingle i x i' = mulSingle i' x i := by simp [mul_single_apply, eq_comm]\n#align pi.mul_single_comm Pi.mulSingle_comm\n#align pi.single_comm Pi.single_comm\n\n#print Pi.apply_mulSingle /-\n@[to_additive]\ntheorem apply_mulSingle (f' : ∀ i, f i → g i) (hf' : ∀ i, f' i 1 = 1) (i : I) (x : f i) (j : I) :\n    f' j (mulSingle i x j) = mulSingle i (f' i x) j := by\n  simpa only [Pi.one_apply, hf', mul_single] using Function.apply_update f' 1 i x j\n#align pi.apply_mul_single Pi.apply_mulSingle\n#align pi.apply_single Pi.apply_single\n-/\n\n#print Pi.apply_mulSingle₂ /-\n@[to_additive apply_single₂]\ntheorem apply_mulSingle₂ (f' : ∀ i, f i → g i → h i) (hf' : ∀ i, f' i 1 1 = 1) (i : I) (x : f i)\n    (y : g i) (j : I) : f' j (mulSingle i x j) (mulSingle i y j) = mulSingle i (f' i x y) j :=\n  by\n  by_cases h : j = i\n  · subst h\n    simp only [mul_single_eq_same]\n  · simp only [mul_single_eq_of_ne h, hf']\n#align pi.apply_mul_single₂ Pi.apply_mulSingle₂\n#align pi.apply_single₂ Pi.apply_single₂\n-/\n\n/- warning: pi.mul_single_op -> Pi.mulSingle_op is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {f : I -> Type.{u2}} [_inst_1 : DecidableEq.{succ u1} I] [_inst_2 : forall (i : I), One.{u2} (f i)] {g : I -> Type.{u3}} [_inst_5 : forall (i : I), One.{u3} (g i)] (op : forall (i : I), (f i) -> (g i)), (forall (i : I), Eq.{succ u3} (g i) (op i (OfNat.ofNat.{u2} (f i) 1 (OfNat.mk.{u2} (f i) 1 (One.one.{u2} (f i) (_inst_2 i))))) (OfNat.ofNat.{u3} (g i) 1 (OfNat.mk.{u3} (g i) 1 (One.one.{u3} (g i) (_inst_5 i))))) -> (forall (i : I) (x : f i), Eq.{max (succ u1) (succ u3)} (forall (i : I), g i) (Pi.mulSingle.{u1, u3} I (fun (i : I) => g i) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5 i) i (op i x)) (fun (j : I) => op j (Pi.mulSingle.{u1, u2} I f (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_2 i) i x j)))\nbut is expected to have type\n  forall {I : Type.{u2}} {f : I -> Type.{u3}} [_inst_1 : DecidableEq.{succ u2} I] [_inst_2 : forall (i : I), One.{u3} (f i)] {g : I -> Type.{u1}} [_inst_5 : forall (i : I), One.{u1} (g i)] (op : forall (i : I), (f i) -> (g i)), (forall (i : I), Eq.{succ u1} (g i) (op i (OfNat.ofNat.{u3} (f i) 1 (One.toOfNat1.{u3} (f i) (_inst_2 i)))) (OfNat.ofNat.{u1} (g i) 1 (One.toOfNat1.{u1} (g i) (_inst_5 i)))) -> (forall (i : I) (x : f i), Eq.{max (succ u2) (succ u1)} (forall (i : I), g i) (Pi.mulSingle.{u2, u1} I g (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5 i) i (op i x)) (fun (j : I) => op j (Pi.mulSingle.{u2, u3} I f (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_2 i) i x j)))\nCase conversion may be inaccurate. Consider using '#align pi.mul_single_op Pi.mulSingle_opₓ'. -/\n@[to_additive]\ntheorem mulSingle_op {g : I → Type _} [∀ i, One (g i)] (op : ∀ i, f i → g i) (h : ∀ i, op i 1 = 1)\n    (i : I) (x : f i) : mulSingle i (op i x) = fun j => op j (mulSingle i x j) :=\n  Eq.symm <| funext <| apply_mulSingle op h i x\n#align pi.mul_single_op Pi.mulSingle_op\n#align pi.single_op Pi.single_op\n\n/- warning: pi.mul_single_op₂ -> Pi.mulSingle_op₂ is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {f : I -> Type.{u2}} [_inst_1 : DecidableEq.{succ u1} I] [_inst_2 : forall (i : I), One.{u2} (f i)] {g₁ : I -> Type.{u3}} {g₂ : I -> Type.{u4}} [_inst_5 : forall (i : I), One.{u3} (g₁ i)] [_inst_6 : forall (i : I), One.{u4} (g₂ i)] (op : forall (i : I), (g₁ i) -> (g₂ i) -> (f i)), (forall (i : I), Eq.{succ u2} (f i) (op i (OfNat.ofNat.{u3} (g₁ i) 1 (OfNat.mk.{u3} (g₁ i) 1 (One.one.{u3} (g₁ i) (_inst_5 i)))) (OfNat.ofNat.{u4} (g₂ i) 1 (OfNat.mk.{u4} (g₂ i) 1 (One.one.{u4} (g₂ i) (_inst_6 i))))) (OfNat.ofNat.{u2} (f i) 1 (OfNat.mk.{u2} (f i) 1 (One.one.{u2} (f i) (_inst_2 i))))) -> (forall (i : I) (x₁ : g₁ i) (x₂ : g₂ i), Eq.{max (succ u1) (succ u2)} (forall (i : I), f i) (Pi.mulSingle.{u1, u2} I (fun (i : I) => f i) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_2 i) i (op i x₁ x₂)) (fun (j : I) => op j (Pi.mulSingle.{u1, u3} I g₁ (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5 i) i x₁ j) (Pi.mulSingle.{u1, u4} I g₂ (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_6 i) i x₂ j)))\nbut is expected to have type\n  forall {I : Type.{u3}} {f : I -> Type.{u4}} [_inst_1 : DecidableEq.{succ u3} I] [_inst_2 : forall (i : I), One.{u4} (f i)] {g₁ : I -> Type.{u2}} {g₂ : I -> Type.{u1}} [_inst_5 : forall (i : I), One.{u2} (g₁ i)] [_inst_6 : forall (i : I), One.{u1} (g₂ i)] (op : forall (i : I), (g₁ i) -> (g₂ i) -> (f i)), (forall (i : I), Eq.{succ u4} (f i) (op i (OfNat.ofNat.{u2} (g₁ i) 1 (One.toOfNat1.{u2} (g₁ i) (_inst_5 i))) (OfNat.ofNat.{u1} (g₂ i) 1 (One.toOfNat1.{u1} (g₂ i) (_inst_6 i)))) (OfNat.ofNat.{u4} (f i) 1 (One.toOfNat1.{u4} (f i) (_inst_2 i)))) -> (forall (i : I) (x₁ : g₁ i) (x₂ : g₂ i), Eq.{max (succ u3) (succ u4)} (forall (i : I), f i) (Pi.mulSingle.{u3, u4} I f (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_2 i) i (op i x₁ x₂)) (fun (j : I) => op j (Pi.mulSingle.{u3, u2} I g₁ (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_5 i) i x₁ j) (Pi.mulSingle.{u3, u1} I g₂ (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_6 i) i x₂ j)))\nCase conversion may be inaccurate. Consider using '#align pi.mul_single_op₂ Pi.mulSingle_op₂ₓ'. -/\n@[to_additive]\ntheorem mulSingle_op₂ {g₁ g₂ : I → Type _} [∀ i, One (g₁ i)] [∀ i, One (g₂ i)]\n    (op : ∀ i, g₁ i → g₂ i → f i) (h : ∀ i, op i 1 1 = 1) (i : I) (x₁ : g₁ i) (x₂ : g₂ i) :\n    mulSingle i (op i x₁ x₂) = fun j => op j (mulSingle i x₁ j) (mulSingle i x₂ j) :=\n  Eq.symm <| funext <| apply_mulSingle₂ op h i x₁ x₂\n#align pi.mul_single_op₂ Pi.mulSingle_op₂\n#align pi.single_op₂ Pi.single_op₂\n\nvariable (f)\n\n#print Pi.mulSingle_injective /-\n@[to_additive]\ntheorem mulSingle_injective (i : I) : Function.Injective (mulSingle i : f i → ∀ i, f i) :=\n  Function.update_injective _ i\n#align pi.mul_single_injective Pi.mulSingle_injective\n#align pi.single_injective Pi.single_injective\n-/\n\n#print Pi.mulSingle_inj /-\n@[simp, to_additive]\ntheorem mulSingle_inj (i : I) {x y : f i} : mulSingle i x = mulSingle i y ↔ x = y :=\n  (Pi.mulSingle_injective _ _).eq_iff\n#align pi.mul_single_inj Pi.mulSingle_inj\n#align pi.single_inj Pi.single_inj\n-/\n\nend\n\n#print Pi.prod /-\n/-- The mapping into a product type built from maps into each component. -/\n@[simp]\nprotected def prod (f' : ∀ i, f i) (g' : ∀ i, g i) (i : I) : f i × g i :=\n  (f' i, g' i)\n#align pi.prod Pi.prod\n-/\n\n/- warning: pi.prod_fst_snd -> Pi.prod_fst_snd is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}}, Eq.{max (succ (max u1 u2)) (succ u1) (succ u2)} ((Prod.{u1, u2} α β) -> (Prod.{u1, u2} α β)) (Pi.prod.{max u1 u2, u1, u2} (Prod.{u1, u2} α β) (fun (self : Prod.{u1, u2} α β) => α) (fun (self : Prod.{u1, u2} α β) => β) (Prod.fst.{u1, u2} α β) (Prod.snd.{u1, u2} α β)) (id.{max (succ u1) (succ u2)} (Prod.{u1, u2} α β))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}}, Eq.{max (succ u2) (succ u1)} ((Prod.{u2, u1} α β) -> (Prod.{u2, u1} α β)) (Pi.prod.{max u2 u1, u2, u1} (Prod.{u2, u1} α β) (fun (self : Prod.{u2, u1} α β) => α) (fun (self : Prod.{u2, u1} α β) => β) (Prod.fst.{u2, u1} α β) (Prod.snd.{u2, u1} α β)) (id.{max (succ u2) (succ u1)} (Prod.{u2, u1} α β))\nCase conversion may be inaccurate. Consider using '#align pi.prod_fst_snd Pi.prod_fst_sndₓ'. -/\n@[simp]\ntheorem prod_fst_snd : Pi.prod (Prod.fst : α × β → α) (Prod.snd : α × β → β) = id :=\n  funext fun _ => Prod.mk.eta\n#align pi.prod_fst_snd Pi.prod_fst_snd\n\n/- warning: pi.prod_snd_fst -> Pi.prod_snd_fst is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}}, Eq.{max (succ (max u1 u2)) (succ u2) (succ u1)} ((Prod.{u1, u2} α β) -> (Prod.{u2, u1} β α)) (Pi.prod.{max u1 u2, u2, u1} (Prod.{u1, u2} α β) (fun (self : Prod.{u1, u2} α β) => β) (fun (self : Prod.{u1, u2} α β) => α) (Prod.snd.{u1, u2} α β) (Prod.fst.{u1, u2} α β)) (Prod.swap.{u1, u2} α β)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}}, Eq.{max (succ u2) (succ u1)} ((Prod.{u2, u1} α β) -> (Prod.{u1, u2} β α)) (Pi.prod.{max u2 u1, u1, u2} (Prod.{u2, u1} α β) (fun (self : Prod.{u2, u1} α β) => β) (fun (self : Prod.{u2, u1} α β) => α) (Prod.snd.{u2, u1} α β) (Prod.fst.{u2, u1} α β)) (Prod.swap.{u2, u1} α β)\nCase conversion may be inaccurate. Consider using '#align pi.prod_snd_fst Pi.prod_snd_fstₓ'. -/\n@[simp]\ntheorem prod_snd_fst : Pi.prod (Prod.snd : α × β → β) (Prod.fst : α × β → α) = Prod.swap :=\n  rfl\n#align pi.prod_snd_fst Pi.prod_snd_fst\n\nend Pi\n\nnamespace Function\n\nsection Extend\n\n#print Function.extend_one /-\n@[to_additive]\ntheorem extend_one [One γ] (f : α → β) : Function.extend f (1 : α → γ) (1 : β → γ) = 1 :=\n  funext fun _ => by apply if_t_t _ _\n#align function.extend_one Function.extend_one\n#align function.extend_zero Function.extend_zero\n-/\n\n#print Function.extend_mul /-\n@[to_additive]\ntheorem extend_mul [Mul γ] (f : α → β) (g₁ g₂ : α → γ) (e₁ e₂ : β → γ) :\n    Function.extend f (g₁ * g₂) (e₁ * e₂) = Function.extend f g₁ e₁ * Function.extend f g₂ e₂ :=\n  funext fun _ => by convert(apply_dite₂ (· * ·) _ _ _ _ _).symm\n#align function.extend_mul Function.extend_mul\n#align function.extend_add Function.extend_add\n-/\n\n#print Function.extend_inv /-\n@[to_additive]\ntheorem extend_inv [Inv γ] (f : α → β) (g : α → γ) (e : β → γ) :\n    Function.extend f g⁻¹ e⁻¹ = (Function.extend f g e)⁻¹ :=\n  funext fun _ => by convert(apply_dite Inv.inv _ _ _).symm\n#align function.extend_inv Function.extend_inv\n#align function.extend_neg Function.extend_neg\n-/\n\n#print Function.extend_div /-\n@[to_additive]\ntheorem extend_div [Div γ] (f : α → β) (g₁ g₂ : α → γ) (e₁ e₂ : β → γ) :\n    Function.extend f (g₁ / g₂) (e₁ / e₂) = Function.extend f g₁ e₁ / Function.extend f g₂ e₂ :=\n  funext fun _ => by convert(apply_dite₂ (· / ·) _ _ _ _ _).symm\n#align function.extend_div Function.extend_div\n#align function.extend_sub Function.extend_sub\n-/\n\nend Extend\n\n#print Function.surjective_pi_map /-\ntheorem surjective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Surjective (F i)) :\n    Surjective fun x : ∀ i, f i => fun i => F i (x i) := fun y =>\n  ⟨fun i => (hF i (y i)).some, funext fun i => (hF i (y i)).choose_spec⟩\n#align function.surjective_pi_map Function.surjective_pi_map\n-/\n\n#print Function.injective_pi_map /-\ntheorem injective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Injective (F i)) :\n    Injective fun x : ∀ i, f i => fun i => F i (x i) := fun x y h =>\n  funext fun i => hF i <| (congr_fun h i : _)\n#align function.injective_pi_map Function.injective_pi_map\n-/\n\n#print Function.bijective_pi_map /-\ntheorem bijective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Bijective (F i)) :\n    Bijective fun x : ∀ i, f i => fun i => F i (x i) :=\n  ⟨injective_pi_map fun i => (hF i).Injective, surjective_pi_map fun i => (hF i).Surjective⟩\n#align function.bijective_pi_map Function.bijective_pi_map\n-/\n\nend Function\n\n#print uniqueOfSurjectiveOne /-\n/-- If the one function is surjective, the codomain is trivial. -/\n@[to_additive \"If the zero function is surjective, the codomain is trivial.\"]\ndef uniqueOfSurjectiveOne (α : Type _) {β : Type _} [One β] (h : Function.Surjective (1 : α → β)) :\n    Unique β :=\n  h.uniqueOfSurjectiveConst α (1 : β)\n#align unique_of_surjective_one uniqueOfSurjectiveOne\n#align unique_of_surjective_zero uniqueOfSurjectiveZero\n-/\n\n/- warning: subsingleton.pi_mul_single_eq -> Subsingleton.pi_mulSingle_eq is a dubious translation:\nlean 3 declaration is\n  forall {I : Type.{u1}} {α : Type.{u2}} [_inst_1 : DecidableEq.{succ u1} I] [_inst_2 : Subsingleton.{succ u1} I] [_inst_3 : One.{u2} α] (i : I) (x : α), Eq.{max (succ u1) (succ u2)} (I -> α) (Pi.mulSingle.{u1, u2} I (fun (i : I) => α) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_3) i x) (fun (_x : I) => x)\nbut is expected to have type\n  forall {I : Type.{u2}} {α : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} I] [_inst_2 : Subsingleton.{succ u2} I] [_inst_3 : One.{u1} α] (i : I) (x : α), Eq.{max (succ u2) (succ u1)} (I -> α) (Pi.mulSingle.{u2, u1} I (fun (i : I) => α) (fun (a : I) (b : I) => _inst_1 a b) (fun (i : I) => _inst_3) i x) (fun (_x : I) => x)\nCase conversion may be inaccurate. Consider using '#align subsingleton.pi_mul_single_eq Subsingleton.pi_mulSingle_eqₓ'. -/\n@[to_additive Subsingleton.pi_single_eq]\ntheorem Subsingleton.pi_mulSingle_eq {α : Type _} [DecidableEq I] [Subsingleton I] [One α] (i : I)\n    (x : α) : Pi.mulSingle i x = fun _ => x :=\n  funext fun j => by rw [Subsingleton.elim j i, Pi.mulSingle_eq_same]\n#align subsingleton.pi_mul_single_eq Subsingleton.pi_mulSingle_eq\n#align subsingleton.pi_single_eq Subsingleton.pi_single_eq\n\nnamespace Sum\n\nvariable (a a' : α → γ) (b b' : β → γ)\n\n/- warning: sum.elim_one_one -> Sum.elim_one_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : One.{u3} γ], Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (OfNat.ofNat.{max u1 u3} (α -> γ) 1 (OfNat.mk.{max u1 u3} (α -> γ) 1 (One.one.{max u1 u3} (α -> γ) (Pi.instOne.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))))) (OfNat.ofNat.{max u2 u3} (β -> γ) 1 (OfNat.mk.{max u2 u3} (β -> γ) 1 (One.one.{max u2 u3} (β -> γ) (Pi.instOne.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)))))) (OfNat.ofNat.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) 1 (OfNat.mk.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) 1 (One.one.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) (Pi.instOne.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (i : Sum.{u1, u2} α β) => _inst_1)))))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} [_inst_1 : One.{u3} γ], Eq.{max (max (succ u2) (succ u1)) (succ u3)} ((Sum.{u2, u1} α β) -> γ) (Sum.elim.{u2, u1, succ u3} α β γ (OfNat.ofNat.{max u2 u3} (α -> γ) 1 (One.toOfNat1.{max u2 u3} (α -> γ) (Pi.instOne.{u2, u3} α (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.4107 : α) => γ) (fun (i : α) => _inst_1)))) (OfNat.ofNat.{max u1 u3} (β -> γ) 1 (One.toOfNat1.{max u1 u3} (β -> γ) (Pi.instOne.{u1, u3} β (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.4113 : β) => γ) (fun (i : β) => _inst_1))))) (OfNat.ofNat.{max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) 1 (One.toOfNat1.{max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) (Pi.instOne.{max u2 u1, u3} (Sum.{u2, u1} α β) (fun (a._@.Mathlib.Data.Sum.Basic._hyg.1871 : Sum.{u2, u1} α β) => γ) (fun (i : Sum.{u2, u1} α β) => _inst_1))))\nCase conversion may be inaccurate. Consider using '#align sum.elim_one_one Sum.elim_one_oneₓ'. -/\n@[simp, to_additive]\ntheorem elim_one_one [One γ] : Sum.elim (1 : α → γ) (1 : β → γ) = 1 :=\n  Sum.elim_const_const 1\n#align sum.elim_one_one Sum.elim_one_one\n#align sum.elim_zero_zero Sum.elim_zero_zero\n\n/- warning: sum.elim_mul_single_one -> Sum.elim_mulSingle_one is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : DecidableEq.{succ u2} β] [_inst_3 : One.{u3} γ] (i : α) (c : γ), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (Pi.mulSingle.{u1, u3} α (fun (i : α) => γ) (fun (a : α) (b : α) => _inst_1 a b) (fun (i : α) => _inst_3) i c) (OfNat.ofNat.{max u2 u3} (β -> γ) 1 (OfNat.mk.{max u2 u3} (β -> γ) 1 (One.one.{max u2 u3} (β -> γ) (Pi.instOne.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_3)))))) (Pi.mulSingle.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (a : Sum.{u1, u2} α β) (b : Sum.{u1, u2} α β) => Sum.decidableEq.{u1, u2} α (fun (a : α) (b : α) => _inst_1 a b) β (fun (a : β) (b : β) => _inst_2 a b) a b) (fun (i : Sum.{u1, u2} α β) => _inst_3) (Sum.inl.{u1, u2} α β i) c)\nbut is expected to have type\n  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : DecidableEq.{succ u3} α] [_inst_2 : DecidableEq.{succ u2} β] [_inst_3 : One.{u1} γ] (i : α) (c : γ), Eq.{max (max (succ u3) (succ u2)) (succ u1)} ((Sum.{u3, u2} α β) -> γ) (Sum.elim.{u3, u2, succ u1} α β γ (Pi.mulSingle.{u3, u1} α (fun (i : α) => γ) (fun (a : α) (b : α) => _inst_1 a b) (fun (i : α) => _inst_3) i c) (OfNat.ofNat.{max u2 u1} (β -> γ) 1 (One.toOfNat1.{max u2 u1} (β -> γ) (Pi.instOne.{u2, u1} β (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.4181 : β) => γ) (fun (i : β) => _inst_3))))) (Pi.mulSingle.{max u2 u3, u1} (Sum.{u3, u2} α β) (fun (ᾰ : Sum.{u3, u2} α β) => γ) (fun (a : Sum.{u3, u2} α β) (b : Sum.{u3, u2} α β) => Sum.instDecidableEqSum.{u3, u2} α β (fun (a : α) (b : α) => _inst_1 a b) (fun (a : β) (b : β) => _inst_2 a b) a b) (fun (i : Sum.{u3, u2} α β) => _inst_3) (Sum.inl.{u3, u2} α β i) c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_mul_single_one Sum.elim_mulSingle_oneₓ'. -/\n@[simp, to_additive]\ntheorem elim_mulSingle_one [DecidableEq α] [DecidableEq β] [One γ] (i : α) (c : γ) :\n    Sum.elim (Pi.mulSingle i c) (1 : β → γ) = Pi.mulSingle (Sum.inl i) c := by\n  simp only [Pi.mulSingle, Sum.elim_update_left, elim_one_one]\n#align sum.elim_mul_single_one Sum.elim_mulSingle_one\n#align sum.elim_single_zero Sum.elim_single_zero\n\n/- warning: sum.elim_one_mul_single -> Sum.elim_one_mulSingle is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} α] [_inst_2 : DecidableEq.{succ u2} β] [_inst_3 : One.{u3} γ] (i : β) (c : γ), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (OfNat.ofNat.{max u1 u3} (α -> γ) 1 (OfNat.mk.{max u1 u3} (α -> γ) 1 (One.one.{max u1 u3} (α -> γ) (Pi.instOne.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_3))))) (Pi.mulSingle.{u2, u3} β (fun (ᾰ : β) => γ) (fun (a : β) (b : β) => _inst_2 a b) (fun (i : β) => _inst_3) i c)) (Pi.mulSingle.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (a : Sum.{u1, u2} α β) (b : Sum.{u1, u2} α β) => Sum.decidableEq.{u1, u2} α (fun (a : α) (b : α) => _inst_1 a b) β (fun (a : β) (b : β) => _inst_2 a b) a b) (fun (i : Sum.{u1, u2} α β) => _inst_3) (Sum.inr.{u1, u2} α β i) c)\nbut is expected to have type\n  forall {α : Type.{u3}} {β : Type.{u2}} {γ : Type.{u1}} [_inst_1 : DecidableEq.{succ u3} α] [_inst_2 : DecidableEq.{succ u2} β] [_inst_3 : One.{u1} γ] (i : β) (c : γ), Eq.{max (max (succ u3) (succ u2)) (succ u1)} ((Sum.{u3, u2} α β) -> γ) (Sum.elim.{u3, u2, succ u1} α β γ (OfNat.ofNat.{max u3 u1} (α -> γ) 1 (One.toOfNat1.{max u3 u1} (α -> γ) (Pi.instOne.{u3, u1} α (fun (a._@.Mathlib.Data.Pi.Algebra._hyg.4249 : α) => γ) (fun (i : α) => _inst_3)))) (Pi.mulSingle.{u2, u1} β (fun (ᾰ : β) => γ) (fun (a : β) (b : β) => _inst_2 a b) (fun (i : β) => _inst_3) i c)) (Pi.mulSingle.{max u2 u3, u1} (Sum.{u3, u2} α β) (fun (ᾰ : Sum.{u3, u2} α β) => γ) (fun (a : Sum.{u3, u2} α β) (b : Sum.{u3, u2} α β) => Sum.instDecidableEqSum.{u3, u2} α β (fun (a : α) (b : α) => _inst_1 a b) (fun (a : β) (b : β) => _inst_2 a b) a b) (fun (i : Sum.{u3, u2} α β) => _inst_3) (Sum.inr.{u3, u2} α β i) c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_one_mul_single Sum.elim_one_mulSingleₓ'. -/\n@[simp, to_additive]\ntheorem elim_one_mulSingle [DecidableEq α] [DecidableEq β] [One γ] (i : β) (c : γ) :\n    Sum.elim (1 : α → γ) (Pi.mulSingle i c) = Pi.mulSingle (Sum.inr i) c := by\n  simp only [Pi.mulSingle, Sum.elim_update_right, elim_one_one]\n#align sum.elim_one_mul_single Sum.elim_one_mulSingle\n#align sum.elim_zero_single Sum.elim_zero_single\n\n/- warning: sum.elim_inv_inv -> Sum.elim_inv_inv is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} (a : α -> γ) (b : β -> γ) [_inst_1 : Inv.{u3} γ], Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (Inv.inv.{max u1 u3} (α -> γ) (Pi.instInv.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1)) a) (Inv.inv.{max u2 u3} (β -> γ) (Pi.instInv.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)) b)) (Inv.inv.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) (Pi.instInv.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (i : Sum.{u1, u2} α β) => _inst_1)) (Sum.elim.{u1, u2, succ u3} α β γ a b))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} (a : α -> γ) (b : β -> γ) [_inst_1 : Inv.{u3} γ], Eq.{max (max (succ u2) (succ u1)) (succ u3)} ((Sum.{u2, u1} α β) -> γ) (Sum.elim.{u2, u1, succ u3} α β γ (Inv.inv.{max u2 u3} (α -> γ) (Pi.instInv.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1)) a) (Inv.inv.{max u3 u1} (β -> γ) (Pi.instInv.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1)) b)) (Inv.inv.{max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) (Pi.instInv.{max u2 u1, u3} (Sum.{u2, u1} α β) (fun (ᾰ : Sum.{u2, u1} α β) => γ) (fun (i : Sum.{u2, u1} α β) => _inst_1)) (Sum.elim.{u2, u1, succ u3} α β γ a b))\nCase conversion may be inaccurate. Consider using '#align sum.elim_inv_inv Sum.elim_inv_invₓ'. -/\n@[to_additive]\ntheorem elim_inv_inv [Inv γ] : Sum.elim a⁻¹ b⁻¹ = (Sum.elim a b)⁻¹ :=\n  (Sum.comp_elim Inv.inv a b).symm\n#align sum.elim_inv_inv Sum.elim_inv_inv\n#align sum.elim_neg_neg Sum.elim_neg_neg\n\n/- warning: sum.elim_mul_mul -> Sum.elim_mul_mul is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} (a : α -> γ) (a' : α -> γ) (b : β -> γ) (b' : β -> γ) [_inst_1 : Mul.{u3} γ], Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (α -> γ) (α -> γ) (α -> γ) (instHMul.{max u1 u3} (α -> γ) (Pi.instMul.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) a a') (HMul.hMul.{max u2 u3, max u2 u3, max u2 u3} (β -> γ) (β -> γ) (β -> γ) (instHMul.{max u2 u3} (β -> γ) (Pi.instMul.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) b b')) (HMul.hMul.{max (max u1 u2) u3, max (max u1 u2) u3, max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) ((Sum.{u1, u2} α β) -> γ) ((Sum.{u1, u2} α β) -> γ) (instHMul.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) (Pi.instMul.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (i : Sum.{u1, u2} α β) => _inst_1))) (Sum.elim.{u1, u2, succ u3} α β γ a b) (Sum.elim.{u1, u2, succ u3} α β γ a' b'))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} (a : α -> γ) (a' : α -> γ) (b : β -> γ) (b' : β -> γ) [_inst_1 : Mul.{u3} γ], Eq.{max (max (succ u2) (succ u1)) (succ u3)} ((Sum.{u2, u1} α β) -> γ) (Sum.elim.{u2, u1, succ u3} α β γ (HMul.hMul.{max u2 u3, max u2 u3, max u2 u3} (α -> γ) (α -> γ) (α -> γ) (instHMul.{max u2 u3} (α -> γ) (Pi.instMul.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) a a') (HMul.hMul.{max u1 u3, max u1 u3, max u1 u3} (β -> γ) (β -> γ) (β -> γ) (instHMul.{max u1 u3} (β -> γ) (Pi.instMul.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) b b')) (HMul.hMul.{max (max u2 u1) u3, max (max u2 u1) u3, max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) ((Sum.{u2, u1} α β) -> γ) ((Sum.{u2, u1} α β) -> γ) (instHMul.{max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) (Pi.instMul.{max u2 u1, u3} (Sum.{u2, u1} α β) (fun (ᾰ : Sum.{u2, u1} α β) => γ) (fun (i : Sum.{u2, u1} α β) => _inst_1))) (Sum.elim.{u2, u1, succ u3} α β γ a b) (Sum.elim.{u2, u1, succ u3} α β γ a' b'))\nCase conversion may be inaccurate. Consider using '#align sum.elim_mul_mul Sum.elim_mul_mulₓ'. -/\n@[to_additive]\ntheorem elim_mul_mul [Mul γ] : Sum.elim (a * a') (b * b') = Sum.elim a b * Sum.elim a' b' :=\n  by\n  ext x\n  cases x <;> rfl\n#align sum.elim_mul_mul Sum.elim_mul_mul\n#align sum.elim_add_add Sum.elim_add_add\n\n/- warning: sum.elim_div_div -> Sum.elim_div_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {γ : Type.{u3}} (a : α -> γ) (a' : α -> γ) (b : β -> γ) (b' : β -> γ) [_inst_1 : Div.{u3} γ], Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} α β) -> γ) (Sum.elim.{u1, u2, succ u3} α β γ (HDiv.hDiv.{max u1 u3, max u1 u3, max u1 u3} (α -> γ) (α -> γ) (α -> γ) (instHDiv.{max u1 u3} (α -> γ) (Pi.instDiv.{u1, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) a a') (HDiv.hDiv.{max u2 u3, max u2 u3, max u2 u3} (β -> γ) (β -> γ) (β -> γ) (instHDiv.{max u2 u3} (β -> γ) (Pi.instDiv.{u2, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) b b')) (HDiv.hDiv.{max (max u1 u2) u3, max (max u1 u2) u3, max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) ((Sum.{u1, u2} α β) -> γ) ((Sum.{u1, u2} α β) -> γ) (instHDiv.{max (max u1 u2) u3} ((Sum.{u1, u2} α β) -> γ) (Pi.instDiv.{max u1 u2, u3} (Sum.{u1, u2} α β) (fun (ᾰ : Sum.{u1, u2} α β) => γ) (fun (i : Sum.{u1, u2} α β) => _inst_1))) (Sum.elim.{u1, u2, succ u3} α β γ a b) (Sum.elim.{u1, u2, succ u3} α β γ a' b'))\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {γ : Type.{u3}} (a : α -> γ) (a' : α -> γ) (b : β -> γ) (b' : β -> γ) [_inst_1 : Div.{u3} γ], Eq.{max (max (succ u2) (succ u1)) (succ u3)} ((Sum.{u2, u1} α β) -> γ) (Sum.elim.{u2, u1, succ u3} α β γ (HDiv.hDiv.{max u2 u3, max u2 u3, max u2 u3} (α -> γ) (α -> γ) (α -> γ) (instHDiv.{max u2 u3} (α -> γ) (Pi.instDiv.{u2, u3} α (fun (ᾰ : α) => γ) (fun (i : α) => _inst_1))) a a') (HDiv.hDiv.{max u1 u3, max u1 u3, max u1 u3} (β -> γ) (β -> γ) (β -> γ) (instHDiv.{max u1 u3} (β -> γ) (Pi.instDiv.{u1, u3} β (fun (ᾰ : β) => γ) (fun (i : β) => _inst_1))) b b')) (HDiv.hDiv.{max (max u2 u1) u3, max (max u2 u1) u3, max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) ((Sum.{u2, u1} α β) -> γ) ((Sum.{u2, u1} α β) -> γ) (instHDiv.{max (max u2 u1) u3} ((Sum.{u2, u1} α β) -> γ) (Pi.instDiv.{max u2 u1, u3} (Sum.{u2, u1} α β) (fun (ᾰ : Sum.{u2, u1} α β) => γ) (fun (i : Sum.{u2, u1} α β) => _inst_1))) (Sum.elim.{u2, u1, succ u3} α β γ a b) (Sum.elim.{u2, u1, succ u3} α β γ a' b'))\nCase conversion may be inaccurate. Consider using '#align sum.elim_div_div Sum.elim_div_divₓ'. -/\n@[to_additive]\ntheorem elim_div_div [Div γ] : Sum.elim (a / a') (b / b') = Sum.elim a b / Sum.elim a' b' :=\n  by\n  ext x\n  cases x <;> rfl\n#align sum.elim_div_div Sum.elim_div_div\n#align sum.elim_sub_sub Sum.elim_sub_sub\n\nend Sum\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Pi/Algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.32146962893669445}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n\n! This file was ported from Lean 3 source module algebraic_topology.dold_kan.n_reflects_iso\n! leanprover-community/mathlib commit 88bca0ce5d22ebfd9e73e682e51d60ea13b48347\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicTopology.DoldKan.FunctorN\nimport Mathbin.AlgebraicTopology.DoldKan.Decomposition\nimport Mathbin.CategoryTheory.Idempotents.HomologicalComplex\nimport Mathbin.CategoryTheory.Idempotents.KaroubiKaroubi\n\n/-!\n\n# N₁ and N₂ reflects isomorphisms\n\nIn this file, it is shown that the functors\n`N₁ : simplicial_object C ⥤ karoubi (chain_complex C ℕ)` and\n`N₂ : karoubi (simplicial_object C) ⥤ karoubi (chain_complex C ℕ))`\nreflect isomorphisms for any preadditive category `C`.\n\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Category\n\nopen CategoryTheory.Idempotents\n\nopen Opposite\n\nopen Simplicial\n\nnamespace AlgebraicTopology\n\nnamespace DoldKan\n\nvariable {C : Type _} [Category C] [Preadditive C]\n\nopen MorphComponents\n\ninstance : ReflectsIsomorphisms (n₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)) :=\n  ⟨fun X Y f => by\n    intro\n    -- restating the result in a way that allows induction on the degree n\n    suffices ∀ n : ℕ, is_iso (f.app (op [n]))\n      by\n      haveI : ∀ Δ : SimplexCategoryᵒᵖ, is_iso (f.app Δ) := fun Δ => this Δ.unop.len\n      apply nat_iso.is_iso_of_is_iso_app\n    -- restating the assumption in a more practical form\n    have h₁ := HomologicalComplex.congr_hom (karoubi.hom_ext.mp (is_iso.hom_inv_id (N₁.map f)))\n    have h₂ := HomologicalComplex.congr_hom (karoubi.hom_ext.mp (is_iso.inv_hom_id (N₁.map f)))\n    have h₃ := fun n =>\n      karoubi.homological_complex.p_comm_f_assoc (inv (N₁.map f)) n (f.app (op [n]))\n    simp only [N₁_map_f, karoubi.comp_f, HomologicalComplex.comp_f,\n      alternating_face_map_complex.map_f, N₁_obj_p, karoubi.id_eq, assoc] at h₁ h₂ h₃\n    -- we have to construct an inverse to f in degree n, by induction on n\n    intro n\n    induction' n with n hn\n    -- degree 0\n    · use (inv (N₁.map f)).f.f 0\n      have h₁₀ := h₁ 0\n      have h₂₀ := h₂ 0\n      dsimp at h₁₀ h₂₀\n      simp only [id_comp, comp_id] at h₁₀ h₂₀\n      tauto\n    -- induction step\n    · haveI := hn\n      use φ {\n            a := P_infty.f (n + 1) ≫ (inv (N₁.map f)).f.f (n + 1)\n            b := fun i => inv (f.app (op [n])) ≫ X.σ i }\n      simp only [morph_components.id, ← id_φ, ← pre_comp_φ, pre_comp, ← post_comp_φ, post_comp,\n        P_infty_f_naturality_assoc, is_iso.hom_inv_id_assoc, assoc, is_iso.inv_hom_id_assoc,\n        simplicial_object.σ_naturality, h₁, h₂, h₃]\n      tauto⟩\n\ntheorem compatibility_n₂_n₁_karoubi :\n    n₂ ⋙ (karoubiChainComplexEquivalence C ℕ).Functor =\n      karoubiFunctorCategoryEmbedding SimplexCategoryᵒᵖ C ⋙\n        n₁ ⋙\n          (karoubiChainComplexEquivalence (Karoubi C) ℕ).Functor ⋙\n            Functor.mapHomologicalComplex (KaroubiKaroubi.equivalence C).inverse _ :=\n  by\n  refine' CategoryTheory.Functor.ext (fun P => _) fun P Q f => _\n  · refine' HomologicalComplex.ext _ _\n    · ext n\n      · dsimp\n        simp only [karoubi_P_infty_f, comp_id, P_infty_f_naturality, id_comp]\n      · rfl\n    · rintro _ n (rfl : n + 1 = _)\n      ext\n      have h := (alternating_face_map_complex.map P.p).comm (n + 1) n\n      dsimp [N₂, karoubi_chain_complex_equivalence, karoubi_karoubi.inverse,\n        karoubi_homological_complex_equivalence.functor.obj] at h⊢\n      simp only [karoubi.comp_f, assoc, karoubi.eq_to_hom_f, eq_to_hom_refl, id_comp, comp_id,\n        karoubi_alternating_face_map_complex_d, karoubi_P_infty_f, ←\n        HomologicalComplex.Hom.comm_assoc, ← h, app_idem_assoc]\n  · ext n\n    dsimp [karoubi_karoubi.inverse, karoubi_functor_category_embedding,\n      karoubi_functor_category_embedding.map]\n    simp only [karoubi.comp_f, karoubi_P_infty_f, HomologicalComplex.eqToHom_f, karoubi.eq_to_hom_f,\n      assoc, comp_id, P_infty_f_naturality, app_p_comp,\n      karoubi_chain_complex_equivalence_functor_obj_X_p, N₂_obj_p_f, eq_to_hom_refl,\n      P_infty_f_naturality_assoc, app_comp_p, P_infty_f_idem_assoc]\n#align algebraic_topology.dold_kan.compatibility_N₂_N₁_karoubi AlgebraicTopology.DoldKan.compatibility_n₂_n₁_karoubi\n\n/-- We deduce that `N₂ : karoubi (simplicial_object C) ⥤ karoubi (chain_complex C ℕ))`\nreflects isomorphisms from the fact that\n`N₁ : simplicial_object (karoubi C) ⥤ karoubi (chain_complex (karoubi C) ℕ)` does. -/\ninstance : ReflectsIsomorphisms (n₂ : Karoubi (SimplicialObject C) ⥤ Karoubi (ChainComplex C ℕ)) :=\n  ⟨fun X Y f => by\n    intro\n    -- The following functor `F` reflects isomorphism because it is\n    -- a composition of four functors which reflects isomorphisms.\n    -- Then, it suffices to show that `F.map f` is an isomorphism.\n    let F :=\n      karoubi_functor_category_embedding SimplexCategoryᵒᵖ C ⋙\n        N₁ ⋙\n          (karoubi_chain_complex_equivalence (karoubi C) ℕ).Functor ⋙\n            functor.map_homological_complex (karoubi_karoubi.equivalence C).inverse\n              (ComplexShape.down ℕ)\n    have : is_iso (F.map f) := by\n      dsimp only [F]\n      rw [← compatibility_N₂_N₁_karoubi, functor.comp_map]\n      apply functor.map_is_iso\n    exact is_iso_of_reflects_iso f F⟩\n\nend DoldKan\n\nend AlgebraicTopology\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicTopology/DoldKan/NReflectsIso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331606115021, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3214674658313222}}
{"text": "import params\nimport quarterround\nimport utils\n\nimport category_theory.category.basic\nimport category_theory.core\n\nopen params\nopen operations\nopen quarterround\nopen utils\n\nopen category_theory\n\nnamespace rowround\n\nvariables [category (bitvec word_len)]\n\n/-!\n  # Rowround\n\n  The `rowround` function and the relation with its inverse.\n-/\n\n/-!\n  ## Definitions and lemmas\n-/\n\n/-- The row round of a single row. Complete `rowround` function will use 4 of this. -/\n@[simp] def rowround_single (R : vecType): vecType :=\n  (\n    (quarterround R).fst, (quarterround R).snd.fst, \n    (quarterround R).snd.snd.fst, (quarterround R).snd.snd.snd\n  )\n\n/-- The inverse of a single row round. -/\n@[simp] def rowround_single_inv (R : vecType) : vecType :=\n  (\n    (quarterround_inv R).fst, (quarterround_inv R).snd.fst, \n    (quarterround_inv R).snd.snd.fst, (quarterround_inv R).snd.snd.snd\n  )\n\n/- Just some notation for inverses. -/\nlocal notation `rowround_single⁻¹` := rowround_single_inv\n\n/-- Each row is invertible. -/\nlemma rowround_single_is_inv (I : rowround_single ≅ rowround_single⁻¹) : I.hom ≫ I.inv = 𝟙 rowround_single :=\n  by rw [iso.hom_inv_id]\n\n/-- Apply `rowround_single` to get a row round matrix output -/\n@[simp] def rowround (M : matrixType) : matrixType :=\n  (\n    rowround_single M.fst,\n    rowround_single M.snd.fst,\n    rowround_single M.snd.snd.fst,\n    rowround_single M.snd.snd.snd\n  )\n\n/-- Reverses `rowround` by doing `rowround_single_inv` to get the original matrix output -/\n@[simp] def rowround_inv (M : matrixType) : matrixType :=\n  (\n    rowround_single_inv M.fst,\n    rowround_single_inv M.snd.fst,\n    rowround_single_inv M.snd.snd.fst,\n    rowround_single_inv M.snd.snd.snd\n  )\n\n/- Just some notation for inverses. -/\nlocal notation `rowround⁻¹` := rowround_inv\n\n/-- The full `rowround` is invertible. -/\nlemma rowround_is_inv (I : rowround ≅ rowround⁻¹) : I.hom ≫ I.inv = 𝟙 rowround := by rw [iso.hom_inv_id]\n\n/-- This rowround call will sort all the elements of the input and the output to match salsa20 spec.\nIt should be used in `doubleround`. -/\n@[simp] def rowround_salsa20 (M : matrixType) := rowround_output (rowround (rowround_input M))\n\n/-- This rowround inverse call will sort all the elements of the input and the output to match salsa20.\nIt should be used in `doubleround`. -/\n@[simp] def rowround_salsa20_inv (M : matrixType) := rowround_output (rowround⁻¹ (rowround_input M))\n\n/- Just some notation for inverses. -/\nlocal notation `rowround_salsa20⁻¹` := rowround_salsa20_inv\n\n/-- For any `rowround` output, we can get back to original values using the defined inverse. -/\n@[simp] lemma rowround_salsa20_is_inv (I : rowround_salsa20 ≅ rowround_salsa20⁻¹) : \n  I.hom ≫ I.inv = 𝟙 rowround_salsa20 := by rw [iso.hom_inv_id]\n\nend rowround\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/rowround.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.3213687054147485}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, E. W. Ayers\n-/\n\nimport category_theory.sites.sieves\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.category.preorder\nimport order.copy\n\n/-!\n# Grothendieck topologies\n\nDefinition and lemmas about Grothendieck topologies.\nA Grothendieck topology for a category `C` is a set of sieves on each object `X` satisfying\ncertain closure conditions.\n\nAlternate versions of the axioms (in arrow form) are also described.\nTwo explicit examples of Grothendieck topologies are given:\n* The dense topology\n* The atomic topology\nas well as the complete lattice structure on Grothendieck topologies (which gives two additional\nexplicit topologies: the discrete and trivial topologies.)\n\nA pretopology, or a basis for a topology is defined in `pretopology.lean`. The topology associated\nto a topological space is defined in `spaces.lean`.\n\n## Tags\n\nGrothendieck topology, coverage, pretopology, site\n\n## References\n\n* [https://ncatlab.org/nlab/show/Grothendieck+topology][nlab]\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM91]\n\n## Implementation notes\n\nWe use the definition of [nlab] and [MM91](Chapter III, Section 2), where Grothendieck topologies\nare saturated collections of morphisms, rather than the notions of the Stacks project (00VG) and\nthe Elephant, in which topologies are allowed to be unsaturated, and are then completed.\nTODO (BM): Add the definition from Stacks, as a pretopology, and complete to a topology.\n\nThis is so that we can produce a bijective correspondence between Grothendieck topologies on a\nsmall category and Lawvere-Tierney topologies on its presheaf topos, as well as the equivalence\nbetween Grothendieck topoi and left exact reflective subcategories of presheaf toposes.\n-/\n\nuniverses w v u\nnamespace category_theory\n\nopen category_theory category\n\nvariables (C : Type u) [category.{v} C]\n\n/--\nThe definition of a Grothendieck topology: a set of sieves `J X` on each object `X` satisfying\nthree axioms:\n1. For every object `X`, the maximal sieve is in `J X`.\n2. If `S ∈ J X` then its pullback along any `h : Y ⟶ X` is in `J Y`.\n3. If `S ∈ J X` and `R` is a sieve on `X`, then provided that the pullback of `R` along any arrow\n   `f : Y ⟶ X` in `S` is in `J Y`, we have that `R` itself is in `J X`.\n\nA sieve `S` on `X` is referred to as `J`-covering, (or just covering), if `S ∈ J X`.\n\nSee https://stacks.math.columbia.edu/tag/00Z4, or [nlab], or [MM92] Chapter III, Section 2,\nDefinition 1.\n-/\nstructure grothendieck_topology :=\n(sieves : Π (X : C), set (sieve X))\n(top_mem' : ∀ X, ⊤ ∈ sieves X)\n(pullback_stable' : ∀ ⦃X Y : C⦄ ⦃S : sieve X⦄ (f : Y ⟶ X), S ∈ sieves X → S.pullback f ∈ sieves Y)\n(transitive' : ∀ ⦃X⦄ ⦃S : sieve X⦄ (hS : S ∈ sieves X) (R : sieve X),\n              (∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, S f → R.pullback f ∈ sieves Y) → R ∈ sieves X)\n\nnamespace grothendieck_topology\n\ninstance : has_coe_to_fun (grothendieck_topology C) (λ _, Π X : C, set (sieve X)) := ⟨sieves⟩\n\nvariables {C} {X Y : C} {S R : sieve X}\nvariables (J : grothendieck_topology C)\n\n/--\nAn extensionality lemma in terms of the coercion to a pi-type.\nWe prove this explicitly rather than deriving it so that it is in terms of the coercion rather than\nthe projection `.sieves`.\n-/\n@[ext]\nlemma ext {J₁ J₂ : grothendieck_topology C} (h : (J₁ : Π (X : C), set (sieve X)) = J₂) : J₁ = J₂ :=\nby { cases J₁, cases J₂, congr, apply h }\n\n@[simp] lemma mem_sieves_iff_coe : S ∈ J.sieves X ↔ S ∈ J X := iff.rfl\n\n-- Also known as the maximality axiom.\n@[simp] lemma top_mem (X : C) : ⊤ ∈ J X := J.top_mem' X\n-- Also known as the stability axiom.\n@[simp] lemma pullback_stable (f : Y ⟶ X) (hS : S ∈ J X) : S.pullback f ∈ J Y :=\nJ.pullback_stable' f hS\nlemma transitive (hS : S ∈ J X) (R : sieve X)\n  (h : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, S f → R.pullback f ∈ J Y) :\n  R ∈ J X :=\nJ.transitive' hS R h\n\nlemma covering_of_eq_top : S = ⊤ → S ∈ J X := λ h, h.symm ▸ J.top_mem X\n\n/--\nIf `S` is a subset of `R`, and `S` is covering, then `R` is covering as well.\n\nSee https://stacks.math.columbia.edu/tag/00Z5 (2), or discussion after [MM92] Chapter III,\nSection 2, Definition 1.\n-/\nlemma superset_covering (Hss : S ≤ R) (sjx : S ∈ J X) : R ∈ J X :=\nbegin\n  apply J.transitive sjx R (λ Y f hf, _),\n  apply covering_of_eq_top,\n  rw [← top_le_iff, ← S.pullback_eq_top_of_mem hf],\n  apply sieve.pullback_monotone _ Hss,\nend\n\n/--\nThe intersection of two covering sieves is covering.\n\nSee https://stacks.math.columbia.edu/tag/00Z5 (1), or [MM92] Chapter III,\nSection 2, Definition 1 (iv).\n-/\nlemma intersection_covering (rj : R ∈ J X) (sj : S ∈ J X) : R ⊓ S ∈ J X :=\nbegin\n  apply J.transitive rj _ (λ Y f Hf, _),\n  rw [sieve.pullback_inter, R.pullback_eq_top_of_mem Hf],\n  simp [sj],\nend\n\n@[simp]\nlemma intersection_covering_iff : R ⊓ S ∈ J X ↔ R ∈ J X ∧ S ∈ J X :=\n⟨λ h, ⟨J.superset_covering inf_le_left h, J.superset_covering inf_le_right h⟩,\n λ t, intersection_covering _ t.1 t.2⟩\n\n\n\n/--\nThe sieve `S` on `X` `J`-covers an arrow `f` to `X` if `S.pullback f ∈ J Y`.\nThis definition is an alternate way of presenting a Grothendieck topology.\n-/\ndef covers (S : sieve X) (f : Y ⟶ X) : Prop := S.pullback f ∈ J Y\n\nlemma covers_iff (S : sieve X) (f : Y ⟶ X) : J.covers S f ↔ S.pullback f ∈ J Y :=\niff.rfl\n\nlemma covering_iff_covers_id (S : sieve X) : S ∈ J X ↔ J.covers S (𝟙 X) :=\nby simp [covers_iff]\n\n/-- The maximality axiom in 'arrow' form: Any arrow `f` in `S` is covered by `S`. -/\nlemma arrow_max (f : Y ⟶ X) (S : sieve X) (hf : S f) : J.covers S f :=\nbegin\n  rw [covers, (sieve.pullback_eq_top_iff_mem f).1 hf],\n  apply J.top_mem,\nend\n\n/-- The stability axiom in 'arrow' form: If `S` covers `f` then `S` covers `g ≫ f` for any `g`. -/\nlemma arrow_stable (f : Y ⟶ X) (S : sieve X) (h : J.covers S f) {Z : C} (g : Z ⟶ Y) :\n  J.covers S (g ≫ f) :=\nbegin\n  rw covers_iff at h ⊢,\n  simp [h, sieve.pullback_comp],\nend\n\n/--\nThe transitivity axiom in 'arrow' form: If `S` covers `f` and every arrow in `S` is covered by\n`R`, then `R` covers `f`.\n-/\nlemma arrow_trans (f : Y ⟶ X) (S R : sieve X) (h : J.covers S f) :\n  (∀ {Z : C} (g : Z ⟶ X), S g → J.covers R g) → J.covers R f :=\nbegin\n  intro k,\n  apply J.transitive h,\n  intros Z g hg,\n  rw ← sieve.pullback_comp,\n  apply k (g ≫ f) hg,\nend\n\nlemma arrow_intersect (f : Y ⟶ X) (S R : sieve X) (hS : J.covers S f) (hR : J.covers R f) :\n  J.covers (S ⊓ R) f :=\nby simpa [covers_iff] using and.intro hS hR\n\nvariable (C)\n/--\nThe trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee [MM92] Chapter III, Section 2, example (a), or\nhttps://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies\n-/\ndef trivial : grothendieck_topology C :=\n{ sieves := λ X, {⊤},\n  top_mem' := λ X, rfl,\n  pullback_stable' := λ X Y S f hf,\n  begin\n    rw set.mem_singleton_iff at ⊢ hf,\n    simp [hf],\n  end,\n  transitive' := λ X S hS R hR,\n  begin\n    rw [set.mem_singleton_iff, ← sieve.id_mem_iff_eq_top] at hS,\n    simpa using hR hS,\n  end }\n\n/--\nThe discrete Grothendieck topology, in which every sieve is covering.\n\nSee https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies.\n-/\ndef discrete : grothendieck_topology C :=\n{ sieves := λ X, set.univ,\n  top_mem' := by simp,\n  pullback_stable' := λ X Y f, by simp,\n  transitive' := by simp }\nvariable {C}\n\nlemma trivial_covering : S ∈ trivial C X ↔ S = ⊤ := set.mem_singleton_iff\n\n/-- See https://stacks.math.columbia.edu/tag/00Z6 -/\ninstance : has_le (grothendieck_topology C) :=\n{ le := λ J₁ J₂, (J₁ : Π (X : C), set (sieve X)) ≤ (J₂ : Π (X : C), set (sieve X)) }\n\nlemma le_def {J₁ J₂ : grothendieck_topology C} :\n  J₁ ≤ J₂ ↔ (J₁ : Π (X : C), set (sieve X)) ≤ J₂ := iff.rfl\n\n/-- See https://stacks.math.columbia.edu/tag/00Z6 -/\ninstance : partial_order (grothendieck_topology C) :=\n{ le_refl := λ J₁, le_def.mpr (le_refl _),\n  le_trans := λ J₁ J₂ J₃ h₁₂ h₂₃, le_def.mpr (le_trans h₁₂ h₂₃),\n  le_antisymm := λ J₁ J₂ h₁₂ h₂₁, grothendieck_topology.ext (le_antisymm h₁₂ h₂₁),\n  ..grothendieck_topology.has_le }\n\n/-- See https://stacks.math.columbia.edu/tag/00Z7 -/\ninstance : has_Inf (grothendieck_topology C) :=\n{ Inf := λ T,\n  { sieves := Inf (sieves '' T),\n    top_mem' :=\n    begin\n      rintro X S ⟨⟨_, J, hJ, rfl⟩, rfl⟩,\n      simp,\n    end,\n    pullback_stable' :=\n    begin\n      rintro X Y S hS f _ ⟨⟨_, J, hJ, rfl⟩, rfl⟩,\n      apply J.pullback_stable _ (f _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩),\n    end,\n    transitive' :=\n    begin\n      rintro X S hS R h _ ⟨⟨_, J, hJ, rfl⟩, rfl⟩,\n      apply J.transitive (hS _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩) _ (λ Y f hf, h hf _ ⟨⟨_, _, hJ, rfl⟩, rfl⟩),\n    end } }\n\n/-- See https://stacks.math.columbia.edu/tag/00Z7 -/\nlemma is_glb_Inf (s : set (grothendieck_topology C)) : is_glb s (Inf s) :=\nbegin\n  refine @is_glb.of_image _ _ _ _ sieves _ _ _ _,\n  { intros, refl },\n  { exact is_glb_Inf _ },\nend\n\n/--\nConstruct a complete lattice from the `Inf`, but make the trivial and discrete topologies\ndefinitionally equal to the bottom and top respectively.\n-/\ninstance : complete_lattice (grothendieck_topology C) :=\ncomplete_lattice.copy\n(complete_lattice_of_Inf _ is_glb_Inf)\n_ rfl\n(discrete C)\n(begin\n  apply le_antisymm,\n  { exact @complete_lattice.le_top _ (complete_lattice_of_Inf _ is_glb_Inf) (discrete C) },\n  { intros X S hS,\n    apply set.mem_univ },\nend)\n(trivial C)\n(begin\n  apply le_antisymm,\n  { intros X S hS,\n    rw trivial_covering at hS,\n    apply covering_of_eq_top _ hS },\n  { refine @complete_lattice.bot_le _ (complete_lattice_of_Inf _ is_glb_Inf) (trivial C) },\nend)\n_ rfl\n_ rfl\n_ rfl\nInf rfl\n\ninstance : inhabited (grothendieck_topology C) := ⟨⊤⟩\n\n@[simp] lemma trivial_eq_bot : trivial C = ⊥ := rfl\n@[simp] lemma discrete_eq_top : discrete C = ⊤ := rfl\n\n@[simp] lemma bot_covering : S ∈ (⊥ : grothendieck_topology C) X ↔ S = ⊤ := trivial_covering\n@[simp] lemma top_covering : S ∈ (⊤ : grothendieck_topology C) X := ⟨⟩\n\nlemma bot_covers (S : sieve X) (f : Y ⟶ X) :\n  (⊥ : grothendieck_topology C).covers S f ↔ S f :=\nby rw [covers_iff, bot_covering, ← sieve.pullback_eq_top_iff_mem]\n\n@[simp] lemma top_covers (S : sieve X) (f : Y ⟶ X) : (⊤ : grothendieck_topology C).covers S f :=\nby simp [covers_iff]\n\n/--\nThe dense Grothendieck topology.\n\nSee https://ncatlab.org/nlab/show/dense+topology, or [MM92] Chapter III, Section 2, example (e).\n-/\ndef dense : grothendieck_topology C :=\n{ sieves := λ X S, ∀ {Y : C} (f : Y ⟶ X), ∃ Z (g : Z ⟶ Y), S (g ≫ f),\n  top_mem' := λ X Y f, ⟨Y, 𝟙 Y, ⟨⟩⟩,\n  pullback_stable' :=\n  begin\n    intros X Y S h H Z f,\n    rcases H (f ≫ h) with ⟨W, g, H'⟩,\n    exact ⟨W, g, by simpa⟩,\n  end,\n  transitive' :=\n  begin\n    intros X S H₁ R H₂ Y f,\n    rcases H₁ f with ⟨Z, g, H₃⟩,\n    rcases H₂ H₃ (𝟙 Z) with ⟨W, h, H₄⟩,\n    exact ⟨W, (h ≫ g), by simpa using H₄⟩,\n  end }\n\nlemma dense_covering : S ∈ dense X ↔ ∀ {Y} (f : Y ⟶ X), ∃ Z (g : Z ⟶ Y), S (g ≫ f) :=\niff.rfl\n\n/--\nA category satisfies the right Ore condition if any span can be completed to a commutative square.\nNB. Any category with pullbacks obviously satisfies the right Ore condition, see\n`right_ore_of_pullbacks`.\n-/\ndef right_ore_condition (C : Type u) [category.{v} C] : Prop :=\n∀ {X Y Z : C} (yx : Y ⟶ X) (zx : Z ⟶ X), ∃ W (wy : W ⟶ Y) (wz : W ⟶ Z), wy ≫ yx = wz ≫ zx\n\nlemma right_ore_of_pullbacks [limits.has_pullbacks C] : right_ore_condition C :=\nλ X Y Z yx zx, ⟨_, _, _, limits.pullback.condition⟩\n\n/--\nThe atomic Grothendieck topology: a sieve is covering iff it is nonempty.\nFor the pullback stability condition, we need the right Ore condition to hold.\n\nSee https://ncatlab.org/nlab/show/atomic+site, or [MM92] Chapter III, Section 2, example (f).\n-/\ndef atomic (hro : right_ore_condition C) : grothendieck_topology C :=\n{ sieves := λ X S, ∃ Y (f : Y ⟶ X), S f,\n  top_mem' := λ X, ⟨_, 𝟙 _, ⟨⟩⟩,\n  pullback_stable' :=\n  begin\n    rintros X Y S h ⟨Z, f, hf⟩,\n    rcases hro h f with ⟨W, g, k, comm⟩,\n    refine ⟨_, g, _⟩,\n    simp [comm, hf],\n  end,\n  transitive' :=\n  begin\n    rintros X S ⟨Y, f, hf⟩ R h,\n    rcases h hf with ⟨Z, g, hg⟩,\n    exact ⟨_, _, hg⟩,\n  end }\n\n/-- `J.cover X` denotes the poset of covers of `X` with respect to the\nGrothendieck topology `J`. -/\n@[derive preorder]\ndef cover (X : C) := { S : sieve X // S ∈ J X }\n\nnamespace cover\n\nvariables {J}\n\ninstance : has_coe (J.cover X) (sieve X) := ⟨λ S, S.1⟩\n\ninstance : has_coe_to_fun (J.cover X) (λ S, Π ⦃Y⦄ (f : Y ⟶ X), Prop) :=\n⟨λ S Y f, (S : sieve X) f⟩\n\n@[simp]\nlemma coe_fun_coe (S : J.cover X) (f : Y ⟶ X) : (S : sieve X) f = S f := rfl\n\nlemma condition (S : J.cover X) : (S : sieve X) ∈ J X := S.2\n\n@[ext]\nlemma ext (S T : J.cover X) (h : ∀ ⦃Y⦄ (f : Y ⟶ X), S f ↔ T f) : S = T :=\nsubtype.ext $ sieve.ext h\n\ninstance : order_top (J.cover X) :=\n{ top := ⟨⊤, J.top_mem _⟩,\n  le_top := λ S Y f h, by tauto,\n  ..(infer_instance : preorder _) }\n\ninstance : semilattice_inf (J.cover X) :=\n{ inf := λ S T, ⟨S ⊓ T, J.intersection_covering S.condition T.condition⟩,\n  le_antisymm := λ S T h1 h2, ext _ _ $ λ Y f, ⟨h1 _, h2 _⟩,\n  inf_le_left := λ S T Y f hf, hf.1,\n  inf_le_right := λ S T Y f hf, hf.2,\n  le_inf := λ S T W h1 h2 Y f h, ⟨h1 _ h, h2 _ h⟩,\n  ..(infer_instance : preorder _) }\n\ninstance : inhabited (J.cover X) := ⟨⊤⟩\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_inhabited_instance, ext]\nstructure arrow (S : J.cover X) :=\n(Y : C)\n(f : Y ⟶ X)\n(hf : S f)\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_inhabited_instance, ext]\nstructure relation (S : J.cover X) :=\n(Y₁ Y₂ Z : C)\n(g₁ : Z ⟶ Y₁)\n(g₂ : Z ⟶ Y₂)\n(f₁ : Y₁ ⟶ X)\n(f₂ : Y₂ ⟶ X)\n(h₁ : S f₁)\n(h₂ : S f₂)\n(w : g₁ ≫ f₁ = g₂ ≫ f₂)\n\n/-- Map a `arrow` along a refinement `S ⟶ T`. -/\n@[simps]\ndef arrow.map {S T : J.cover X} (I : S.arrow) (f : S ⟶ T) : T.arrow :=\n⟨I.Y, I.f, f.le _ I.hf⟩\n\n/-- Map a `relation` along a refinement `S ⟶ T`. -/\n@[simps]\ndef relation.map {S T : J.cover X} (I : S.relation) (f : S ⟶ T) : T.relation :=\n⟨_, _, _, I.g₁, I.g₂, I.f₁, I.f₂, f.le _ I.h₁, f.le _ I.h₂, I.w⟩\n\n/-- The first `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.fst {S : J.cover X} (I : S.relation) : S.arrow :=\n⟨I.Y₁, I.f₁, I.h₁⟩\n\n/-- The second `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.snd {S : J.cover X} (I : S.relation) : S.arrow :=\n⟨I.Y₂, I.f₂, I.h₂⟩\n\n@[simp]\nlemma relation.map_fst {S T : J.cover X} (I : S.relation) (f : S ⟶ T) :\n   I.fst.map f = (I.map f).fst := rfl\n\n@[simp]\nlemma relation.map_snd {S T : J.cover X} (I : S.relation) (f : S ⟶ T) :\n  I.snd.map f = (I.map f).snd := rfl\n\n/-- Pull back a cover along a morphism. -/\ndef pullback (S : J.cover X) (f : Y ⟶ X) : J.cover Y :=\n⟨sieve.pullback f S, J.pullback_stable _ S.condition⟩\n\n/-- An arrow of `S.pullback f` gives rise to an arrow of `S`. -/\n@[simps]\ndef arrow.base {f : Y ⟶ X} {S : J.cover X} (I : (S.pullback f).arrow) : S.arrow :=\n⟨I.Y, I.f ≫ f, I.hf⟩\n\n/-- A relation of `S.pullback f` gives rise to a relation of `S`. -/\n@[simps]\ndef relation.base {f : Y ⟶ X} {S : J.cover X} (I : (S.pullback f).relation) : S.relation :=\n⟨_, _, _, I.g₁, I.g₂, I.f₁ ≫ f, I.f₂≫ f, I.h₁, I.h₂, by simp [reassoc_of I.w]⟩\n\n@[simp]\nlemma relation.base_fst {f : Y ⟶ X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.fst.base = I.base.fst := rfl\n\n@[simp]\nlemma relation.base_snd {f : Y ⟶ X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.snd.base = I.base.snd := rfl\n\n@[simp]\nlemma coe_pullback {Z : C} (f : Y ⟶ X) (g : Z ⟶ Y) (S : J.cover X) :\n  (S.pullback f) g ↔ S (g ≫ f) := iff.rfl\n\n/-- The isomorphism between `S` and the pullback of `S` w.r.t. the identity. -/\ndef pullback_id (S : J.cover X) : S.pullback (𝟙 X) ≅ S :=\neq_to_iso $ cover.ext _ _ $ λ Y f, by simp\n\n/-- Pulling back with respect to a composition is the composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (S : J.cover X) (f : Z ⟶ Y) (g : Y ⟶ X) :\n  S.pullback (f ≫ g) ≅ (S.pullback g).pullback f :=\neq_to_iso $ cover.ext _ _ $ λ Y f, by simp\n\n/-- Combine a family of covers over a cover. -/\ndef bind {X : C} (S : J.cover X) (T : Π (I : S.arrow), J.cover I.Y) : J.cover X :=\n⟨sieve.bind S (λ Y f hf, T ⟨Y, f, hf⟩), J.bind_covering S.condition (λ _ _ _, (T _).condition)⟩\n\n/-- The canonical moprhism from `S.bind T` to `T`. -/\ndef bind_to_base {X : C} (S : J.cover X) (T : Π (I : S.arrow), J.cover I.Y) : S.bind T ⟶ S :=\nhom_of_le $ by { rintro Y f ⟨Z,e1,e2,h1,h2,h3⟩, rw ← h3, apply sieve.downward_closed, exact h1 }\n\n/-- An arrow in bind has the form `A ⟶ B ⟶ X` where `A ⟶ B` is an arrow in `T I` for some `I`.\n and `B ⟶ X` is an arrow of `S`. This is the object `B`. -/\nnoncomputable def arrow.middle {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : C :=\nI.hf.some\n\n/-- An arrow in bind has the form `A ⟶ B ⟶ X` where `A ⟶ B` is an arrow in `T I` for some `I`.\n and `B ⟶ X` is an arrow of `S`. This is the hom `A ⟶ B`. -/\nnoncomputable def arrow.to_middle_hom {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.Y ⟶ I.middle :=\nI.hf.some_spec.some\n\n/-- An arrow in bind has the form `A ⟶ B ⟶ X` where `A ⟶ B` is an arrow in `T I` for some `I`.\n and `B ⟶ X` is an arrow of `S`. This is the hom `B ⟶ X`. -/\nnoncomputable def arrow.from_middle_hom {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.middle ⟶ X :=\nI.hf.some_spec.some_spec.some\n\nlemma arrow.from_middle_condition {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S I.from_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some\n\n/-- An arrow in bind has the form `A ⟶ B ⟶ X` where `A ⟶ B` is an arrow in `T I` for some `I`.\n and `B ⟶ X` is an arrow of `S`. This is the hom `B ⟶ X`, as an arrow. -/\nnoncomputable\ndef arrow.from_middle {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S.arrow := ⟨_, I.from_middle_hom, I.from_middle_condition⟩\n\nlemma arrow.to_middle_condition {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle) I.to_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some_spec.1\n\n/-- An arrow in bind has the form `A ⟶ B ⟶ X` where `A ⟶ B` is an arrow in `T I` for some `I`.\n and `B ⟶ X` is an arrow of `S`. This is the hom `A ⟶ B`, as an arrow. -/\nnoncomputable\ndef arrow.to_middle {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle).arrow := ⟨_, I.to_middle_hom, I.to_middle_condition⟩\n\nlemma arrow.middle_spec {X : C} {S : J.cover X} {T : Π (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.to_middle_hom ≫ I.from_middle_hom = I.f :=\nI.hf.some_spec.some_spec.some_spec.some_spec.2\n\n-- This is used extensively in `plus.lean`, etc.\n-- We place this definition here as it will be used in `sheaf.lean` as well.\n/-- To every `S : J.cover X` and presheaf `P`, associate a `multicospan_index`. -/\ndef index {D : Type w} [category.{max v u} D] (S : J.cover X) (P : Cᵒᵖ ⥤ D) :\n  limits.multicospan_index D :=\n{ L := S.arrow,\n  R := S.relation,\n  fst_to := λ I, I.fst,\n  snd_to := λ I, I.snd,\n  left := λ I, P.obj (opposite.op I.Y),\n  right := λ I, P.obj (opposite.op I.Z),\n  fst := λ I, P.map I.g₁.op,\n  snd := λ I, P.map I.g₂.op }\n\n/-- The natural multifork associated to `S : J.cover X` for a presheaf `P`.\nSaying that this multifork is a limit is essentially equivalent to the sheaf condition at the\ngiven object for the given covering sieve. See `sheaf.lean` for an equivalent sheaf condition\nusing this.\n-/\nabbreviation multifork {D : Type w} [category.{max v u} D] (S : J.cover X) (P : Cᵒᵖ ⥤ D) :\n  limits.multifork (S.index P) :=\nlimits.multifork.of_ι _ (P.obj (opposite.op X)) (λ I, P.map I.f.op) begin\n  intros I,\n  dsimp [index],\n  simp only [← P.map_comp, ← op_comp, I.w]\nend\n\n/-- The canonical map from `P.obj (op X)` to the multiequalizer associated to a covering sieve,\nassuming such a multiequalizer exists. This will be used in `sheaf.lean` to provide an equivalent\nsheaf condition in terms of multiequalizers. -/\nnoncomputable\nabbreviation to_multiequalizer {D : Type w} [category.{max v u} D] (S : J.cover X) (P : Cᵒᵖ ⥤ D)\n  [limits.has_multiequalizer (S.index P)] :\nP.obj (opposite.op X) ⟶ limits.multiequalizer (S.index P) :=\nlimits.multiequalizer.lift _ _ (λ I, P.map I.f.op) begin\n  intros I,\n  dsimp only [index, relation.fst, relation.snd],\n  simp only [← P.map_comp, ← op_comp, I.w],\nend\n\nend cover\n\n/-- Pull back a cover along a morphism. -/\n@[simps obj]\ndef pullback (f : Y ⟶ X) : J.cover X ⥤ J.cover Y :=\n{ obj := λ S, S.pullback f,\n  map := λ S T f, (sieve.pullback_monotone _ f.le).hom }\n\n/-- Pulling back along the identity is naturally isomorphic to the identity functor. -/\ndef pullback_id (X : C) : J.pullback (𝟙 X) ≅ 𝟭 _ :=\nnat_iso.of_components (λ S, S.pullback_id) $ by tidy\n\n/-- Pulling back along a composition is naturally isomorphic to\nthe composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) :\n  J.pullback (f ≫ g) ≅ J.pullback g ⋙ J.pullback f :=\nnat_iso.of_components (λ S, S.pullback_comp f g) $ by tidy\n\nend grothendieck_topology\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sites/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3211598479201199}}
{"text": "\n-- import Lib.Tactic\n\nnamespace Option\n\ndef traverse [Applicative m] (f : α → m β) : Option α → m (Option β)\n| none => pure none\n| some x => some <$> f x\n\nattribute [simp] map_id\n-- theorem map_id : Option.map (@id α) = id := rfl\n\ntheorem map_comp {f : α → β} {g : β → γ} :\n  Option.map (g ∘ f) = Option.map g ∘ Option.map f :=\nby funext x ; cases x <;> rfl\n\n@[simp]\ntheorem map_some {f : α → β} : Option.map f (some x) = some (f x) := rfl\n\n@[simp]\ntheorem map_none {f : α → β} : Option.map f none = none := rfl\n\nend Option\n\nnamespace OptionT\n\ntheorem ext {x y : OptionT m α} : x.run = y.run → x = y := id\n\n@[simp]\ntheorem run_mk {m : Type u → Type v} (x : m (Option α)) :\n  OptionT.run (OptionT.mk x) = x := rfl\n\nvariable [Monad m] [LawfulMonad m]\n\n@[simp]\ntheorem run_map (f : α → β) (x : OptionT m α) :\n  (f <$> x).run = Option.map f <$> x.run := by\nsimp [(.<$>.), OptionT.bind, map_eq_pure_bind]\napply congrArg; funext a; cases a <;> rfl\n\n@[simp]\ntheorem run_pure (x : α) :\n  (pure x : OptionT m α).run = pure (some x) := rfl\n\n@[simp]\ntheorem run_bind (f : α → OptionT m β) (x : OptionT m α) :\n  (x.bind f).run = x.run >>= λ x =>\n    match x with\n    | none => pure none\n    | some y => f y |>.run\n:= by\nsimp [OptionT.bind]; apply congrArg; funext a\ncases a <;> rfl\n\n@[simp]\ntheorem run_bind' (f : α → OptionT m β) (x : OptionT m α) :\n  (x >>= f).run = x.run >>= λ x =>\n    match x with\n    | none => pure none\n    | some y => f y |>.run\n:= run_bind f x\n\n@[simp]\ntheorem run_seq (f : OptionT m (α → β)) (x : OptionT m α) :\n  (f <*> x).run = f.run >>= λ a =>\n    match a with\n    | none => pure none\n    | some a => Option.map a <$> x.run\n:= by\nsimp [Seq.seq, map_eq_pure_bind]\nsimp; apply congrArg; funext a; cases a\ncase none => rfl\ncase some =>\n  simp [map_eq_pure_bind]\n  apply congrArg; funext a; cases a <;> rfl\n\ninstance : LawfulMonad (OptionT m) where\n  map_const := by intros; funext a; rfl\n  id_map x := by\n    apply OptionT.ext; simp [Option.map_id]\n  comp_map f g x := by\n    apply OptionT.ext; simp [Option.map_comp, comp_map]\n  pure_bind x f := OptionT.ext <| by simp\n  seqLeft_eq x y := OptionT.ext <| by\n    simp [SeqLeft.seqLeft, map_eq_pure_bind]\n    apply congrArg; funext a; cases a <;> simp\n    apply congrArg; funext a; cases a <;> simp <;> rfl\n  seqRight_eq x y := OptionT.ext <| by\n    simp [SeqRight.seqRight, map_eq_pure_bind]\n    apply congrArg; funext a; cases a <;> simp [← map_eq_pure_bind]\n  pure_seq g x := OptionT.ext <| by simp\n  map_pure g x := OptionT.ext <| by simp\n  seq_pure g x := OptionT.ext <| by\n    simp [map_eq_pure_bind]\n    apply congrArg; funext a; cases a <;> simp\n  seq_assoc x g h := OptionT.ext <| by\n    simp [map_eq_pure_bind]\n    apply congrArg; funext a; cases a <;> simp\n    apply congrArg; funext a; cases a <;> simp [Option.map_comp]\n  bind_pure_comp f x := OptionT.ext <| by\n    simp [map_eq_pure_bind]\n    apply congrArg; funext a; cases a <;> simp\n  bind_map f x := OptionT.ext <| by\n    simp\n    apply congrArg; funext a; cases a <;> simp\n  bind_assoc x f g := OptionT.ext <| by\n    simp\n    apply congrArg; funext a; cases a <;> simp\n\nend OptionT\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Control/Monad/OptionM.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.32115984792011987}}
{"text": "import .sqe .wf\n\nopen list\n\nlemma atom.unify_dvd (m d i k ks) : \n  k ≠ 0 → atom.unify m (atom.dvd d i (k::ks)) = \n (let m' := has_div.div m k in \n  atom.dvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)) := \nbegin intro hne, cases k, exfalso, apply hne, repeat {refl} end\n\nlemma atom.unify_ndvd (m d i k ks) : \n  k ≠ 0 → atom.unify m (atom.ndvd d i (k::ks)) = \n (let m' := has_div.div m k in \n  atom.ndvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)) := \nbegin intro hne, cases k, exfalso, apply hne, repeat {refl} end\n\nlemma wf_inf_minus : \n  ∀ {p : formula}, formula.wf p → formula.wf (inf_minus p) \n| ⊤' h := by trivial\n| ⊥' h := by trivial\n| (A' a) h := \n  begin \n    cases a; try {trivial}, cases a_a_1 with z zs, \n    simp [inf_minus], rw inf_minus_le_eq,\n    apply ite.rec; intro h, trivial,\n    apply ite.rec; intro h; trivial \n  end\n| (p ∧' q) h := \n  begin\n    cases h with h1 h2, apply cases_and_o; try {trivial};\n    try {apply wf_inf_minus, assumption},\n    constructor; {apply wf_inf_minus, assumption}\n  end\n| (p ∨' q) h := \n  begin\n    cases h with h1 h2, apply cases_or_o; try {trivial};\n    try {apply wf_inf_minus, assumption},\n    constructor; {apply wf_inf_minus, assumption}\n  end\n| (¬' p) h := by assumption\n| (∃' p) h := by assumption\n\nlemma wf_asubst :\n  ∀ a : atom, atom.wf a → ∀ i ks, atom.wf (asubst i ks a) := \nbegin\n  intro a, cases a with i ks d i ks d i ks; \n  intros h i' ks'; cases ks with k ks; trivial\nend\n\nlemma wf_subst (i ks) : \n  ∀ p, formula.wf p → formula.wf (subst i ks p) \n| ⊤' h := by unfold subst\n| ⊥' h := by unfold subst\n| (A' a) h :=\n  begin\n    unfold subst, unfold formula.map,\n    unfold formula.wf, apply wf_asubst _ h\n  end\n| (p ∧' q) h := \n  begin\n    cases h with hp hq, apply and.intro; \n    apply wf_subst; assumption\n  end\n| (p ∨' q) h := \n  begin\n    cases h with hp hq, apply and.intro; \n    apply wf_subst; assumption\n  end\n| (¬' p) h := wf_subst p h\n| (∃' p) h := by unfold subst\n\nlemma atom.wf_unify (z : znum) (hne : z ≠ 0) :\n  ∀ (a : atom), atom.wf a → (head_coeff a ∣ z ∨ head_coeff a = 0)\n  → atom.wf (atom.unify z a) \n| (atom.le i ks) ha1 ha2 := \n  begin cases ks with k ks, trivial, cases k with m m; trivial end\n| (atom.dvd d i ks) ha1 ha2 := \n  begin\n    cases ks with k ks, trivial, \n    cases (classical.em (k = 0)) with hk hk, \n    subst hk, trivial, cases ha2 with ha2 ha2,\n    rewrite atom.unify_dvd, simp,\n    apply znum.mul_nonzero, unfold head_coeff at ha2,\n    apply znum.div_nonzero, apply hne, apply ha2, \n    apply ha1, apply hk, exfalso, apply hk ha2\n  end\n| (atom.ndvd d i ks) ha1 ha2 := \n  begin\n    cases ks with k ks, trivial, \n    cases (classical.em (k = 0)) with hk hk, \n    subst hk, trivial, cases ha2 with ha2 ha2,\n    rewrite atom.unify_ndvd, simp,\n    apply znum.mul_nonzero, unfold head_coeff at ha2,\n    apply znum.div_nonzero, apply hne, apply ha2, \n    apply ha1, apply hk, exfalso, apply hk ha2 \n  end\n\nmeta def formula.wf_map_hco_of_formula.wf_tac :=\n `[unfold formula.map, unfold formula.wf,\n   cases hn with hnp hnq, unfold formula.atoms at hnz, \n   rewrite list.forall_mem_append at hnz,\n   cases hnz with hnzp hnzq, apply and.intro; \n   apply formula.wf_map_hco_of_formula.wf; assumption]\n\nlemma formula.wf_map_hco_of_formula.wf {z : znum} (hz : z ≠ 0) :\n  ∀ {p : formula}, (formula.wf p) \n  → (∀ a ∈ p.atoms, has_dvd.dvd (head_coeff a) z ∨ head_coeff a = 0) \n  → formula.wf (p.map (atom.unify z))  \n| ⊤' hn hnz := by unfold formula.map \n| ⊥' hn hnz := by unfold formula.map \n| (A' a) hn hnz := \n  begin\n    unfold formula.map, unfold formula.wf, unfold formula.wf at hn,\n    apply atom.wf_unify z hz _ hn, apply hnz, \n    unfold formula.atoms, simp,\n  end\n| (p ∧' q) hn hnz := by formula.wf_map_hco_of_formula.wf_tac\n| (p ∨' q) hn hnz := by formula.wf_map_hco_of_formula.wf_tac\n| (¬' p) hp hpz :=\n  begin\n    unfold formula.map, unfold formula.wf, \n    apply @formula.wf_map_hco_of_formula.wf p hp hpz,\n  end\n| (∃' p) hn hnz := by unfold formula.map\n\nlemma formula.wf_hcso : \n  ∀ {φ : formula}, formula.wf φ → formula.wf (φ.unify) := \nbegin\n  intros p hp, unfold formula.unify, unfold formula.wf, \n  have hne : znum.lcms (list.map head_coeff (p.atoms_dep_0)) ≠ 0, \n  { apply znum.nonzero_of_pos, apply znum.lcms_pos, \n    apply @forall_mem_map _ _ head_coeff (λ (z : znum), z ≠ 0), \n    unfold formula.atoms_dep_0, intros a ha, apply list.of_mem_filter ha },\n  { apply and.intro, \n    { intro hc, apply hne hc },\n    { apply formula.wf_map_hco_of_formula.wf hne hp, \n     intros a ha, apply or_of_not_imp_right,\n     intro haz, apply znum.dvd_lcms,\n     rewrite list.mem_map, existsi a, apply and.intro,\n     unfold formula.atoms_dep_0, apply list.mem_filter_of_mem,\n     apply ha, apply haz, refl } }\nend\n\nlemma wf_sqe_core : \n  ∀ {φ : formula}, formula.wf φ → formula.wf (sqe_core φ) := \nbegin\n  intros p hp, simp [sqe_core],\n  apply wf_or_o, apply wf_disj,\n  { apply forall_mem_map, \n    intros z hz, apply wf_subst,\n    apply wf_inf_minus, apply hp },\n  { apply wf_disj, apply forall_mem_map,\n    intros zzs hzzs, apply wf_disj, \n    apply forall_mem_map, intros z hz,\n    apply wf_subst, apply hp }\nend\n\nlemma wf_sqe : \n  ∀ (φ : formula), formula.wf φ → formula.wf (sqe φ) :=\nbegin\n  intros p hp, apply wf_sqe_core,\n  apply formula.wf_hcso hp\nend\n", "meta": {"author": "skbaek", "repo": "cooper", "sha": "812afc6b158821f2e7dac9c91d3b6123c7a19faf", "save_path": "github-repos/lean/skbaek-cooper", "path": "github-repos/lean/skbaek-cooper/cooper-812afc6b158821f2e7dac9c91d3b6123c7a19faf/lia/wf_sqe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.32115984792011987}}
{"text": "open Lean\n\nsyntax (name := myintro) \"intros\" sepBy(ident, \",\") : tactic\n\nmacro_rules (kind := myintro)\n| `(tactic| intros $x,*) => pure $ mkNode `Lean.Parser.Tactic.intros #[mkAtom \"intros\", mkNullNode x]\n\ntheorem tst1 {p q : Prop} : p → q → p :=\nby {\n  intros h1, h2;\n  assumption\n}\n\ntheorem tst2 {p q : Prop} : p → q → p :=\nby {\n  intros h1; -- the builtin and myintro overlap here.\n  intros h2; -- the builtin and myintro overlap here.\n  assumption\n}\n\ntheorem tst3 {p q : Prop} : p → q → p :=\nby {\n  intros h1 h2;\n  assumption\n}\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/tacticExtOverlap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.32115984792011987}}
{"text": "import linear_algebra.affine_space.affine_equiv\nimport algebra.direct_sum\nimport linear_algebra.direct_sum_module\n\nuniverses u \nvariables \n(K : Type u) \n[field K]\n[inhabited K]\n\n#check smul_eq_zero\n#check smul_eq_zero.mp\n#check sub_smul\n\n/-\nExport affine_space (vec K) (pt K) := aff_pt_torsor K,\nwith vector_space K (vec K) = semimodule K (vec K).\n-/\nopen_locale big_operators\nopen direct_sum\nopen_locale direct_sum\n\n--notation `⨁`α, β  := direct_sum α β \n\n/-\nObjects\n-/\n\n-- 1-D *affine* K pt and vec objects rep'd by 2-D linear K tuples\n@[ext]\nstructure pt extends K × K := (inv : fst = 1)\n/-\nWhat about for 3d?\n-/\n@[ext]\nstructure pt3 extends K × K × K × K := (inv : fst = 1) \n/-\nwhat about for pt n\n-/\n@[ext] ptn extends K × K × K × ...?\n/-\nTrying to define this type\n-/\ndef get_K_n_type : nat → Type u\n| 0 := K\n| (nat.succ n) := K × (get_K_n_type n)\n/-\nThe resulting type, Type u, of course, can refer to an affine vector OR any other type in the same universe\n-/\n\n\n\ndef get_K_n_type_no_head : nat → Type u\n| 0 := punit\n| (nat.succ n) := K × (get_K_n_type_no_head n)\n\n/-\nwhat should this definition even be?\n-/\nstructure pt_K_n_struct : Type (u+1) :=\n  (n : ℕ)\n  (inner : K × Type)\n\n\n/-\nHow can we use any of this?\n\nget_K_n_type is Type u, it could be an affine vector or a structure modeling an umbrella\n-/\n\ndef pt_K_n_struct.coords (s : pt_K_n_struct K) : (get_K_n_type K s.n) :=\n  begin\n    --impossible to write this function\n  end\n\n\n\n/-\nDirect Sum\n-/\nvariables (n : nat ) \n\n\ndef helpme : multiset K := ⟦[1,2,3]⟧\n\ndef hel_ : multiset (fin n)  := ⟦[⟨1,sorry⟩]⟧\n/-\ndef direct_sum.lmk (R : Type u) [semiring R] (ι : Type v) [dec_ι : decidable_eq ι] (M : ι → Type w) [Π (i : ι), add_comm_monoid (M i)] [Π (i : ι), module R (M i)] (s : finset ι) :\n(Π (i : ↥↑s), M i.val) →ₗ[R] ⨁ (i : ι), M i\n-/\n\n/-\n\ndef direct_sum.mk {ι : Type v} [dec_ι : decidable_eq ι] (β : ι → Type w) [Π (i : ι), add_comm_monoid (β i)] (s : finset ι) :\n(Π (i : ↥↑s), β i.val) →+ ⨁ (i : ι), β i\n-/\n\n#check finset\n\n\ndef map_ : fin n → Type u := λn_, K\ninstance allmnd : Π (i : fin n), add_comm_monoid ((λ (i : fin n), map_ K n i) i) := sorry\ninstance allmod : Π (i : fin n), semimodule K ((λ (i : fin n), map_ K n i) i) := sorry\ndef muln_ : multiset (fin n)  := ⟦[⟨1,sorry⟩]⟧\n\ndef fins_ : finset (fin n) := ⟨(muln_ n), sorry⟩\nvariables (a : fin n → K) (b: Π (i : (↑(fins_ n) : set (fin n))), map_ K n i.val)\n\ndef dsumcommg : ⨁ (i : fin n), map_ K n i := direct_sum.mk (map_ K n) (fins_ n) b\ndef dsummod : ⨁ (i : fin n), map_ K n i := direct_sum.lmk K (fin n) (map_ K n) (fins_ n) b\n\n#check (md1 K n) b\n\n#check direct_sum.lmk K (fin n) (map_ K n) (fins_ n)\n\ndef md2 := direct_sum.lmk K (fin n) (map_ K n) (fins_ n)\n\nlemma b :  direct_sum _ (map_ K n) = (⨁i, (map_ K n) i) := rfl\n\ndef type_of_dsum : Type u := @direct_sum (fin n) (map_ K n) begin\n  intros,\n  let k := (map_ K n i),\n  let h0 : k = K := rfl,\n  let h1 : (map_ K n i) = K := rfl,\n  rw h1,\n  by apply_instance\nend\n\nvariables (x y : direct_sum (fin n) (map_ K n) ) (p : fin n)\n\ndef xt : direct_sum _ (map_ K n) := \n\ndef tt : (⨁(fin n), (map_ K n)) := x\n\ndef dsakl  : map_ K n p := x p\n\nvariables (k : K)\n\ndef dsssaaak := k•x\n\n--example : (x p) = K := \n\n@[class]\nstructure  test :=\n  (t : Π (i : fin n), add_comm_monoid ((λ (i : fin n), map_ K n i) i))\n\ndef ppt [Π (i : fin n), add_comm_monoid ((λ (i : fin n), map_ K n i) i)] : _ := ⨁i, (map_ K n) i \n\n#check ppt K n\n\nvariables (z : ppt K n) (j : fin n)\n\n#check quot\n#check setoid.r\n\ndef tester : map_ K n j := λ j : fin n, (z : (⨁i, (map_ K n))) j\n\n\nabbreviation pts [Π (i : fin n), add_comm_monoid ((λ (i : fin n), map_ K n i) i)] := ⨁i, (map_ K n) i\n\nuniverses v w u₁\n\nvariables (ι : Type v) [dec_ι : decidable_eq ι] (β : ι → Type w)\n", "meta": {"author": "kevinsullivan", "repo": "affine_lib", "sha": "056fc95c31bdf473b0c1ecd07f5a061dd6b69234", "save_path": "github-repos/lean/kevinsullivan-affine_lib", "path": "github-repos/lean/kevinsullivan-affine_lib/affine_lib-056fc95c31bdf473b0c1ecd07f5a061dd6b69234/src/lin2Kcoord/multidim_test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.32103655491176986}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\n\nimport algebra.group algebra.group_power\nimport data.equiv data.fintype data.list.basic data.quot\nimport group_theory.subgroup\n\nuniverses u v w\n\nvariables {α : Type u}\n\nlemma list.append_eq_has_append {L₁ L₂ : list α} : list.append L₁ L₂ = L₁ ++ L₂ := rfl\n\nlemma list.infix_cons {L₁ L₂ : list α} {x : α} : L₁ <:+: L₂ → L₁ <:+: x :: L₂ :=\nλ ⟨LP, LS, H⟩, ⟨x :: LP, LS, eq.rec_on H rfl⟩\n\nlocal attribute [simp] list.append_eq_has_append\n\nnamespace is_group_hom\n\nvariables {β : Type v} [group α] [group β]\nvariables (f : α → β) [is_group_hom f]\n\ninstance range_subgroup : is_subgroup (set.range f) :=\n@set.image_univ _ _ f ▸ is_group_hom.image_subgroup f set.univ\n\nend is_group_hom\n\nnamespace free_group\n\n/-- Predicate asserting that word `w₁` can be reduced\nto `w₂` in one step, i.e. there are words `w₃ w₄`\nand letter `x` such that `w₁ = w₃xx⁻¹w₄` and `w₂ = w₃w₄`  -/\ninductive red.step : list (α × bool) → list (α × bool) → Prop\n| bnot {L₁ L₂ x b} : red.step (L₁ ++ (x, b) :: (x, bnot b) :: L₂) (L₁ ++ L₂)\n\n/-- Reflexive-transitive closure of red.step -/\ninductive red : list (α × bool) → list (α × bool) → Prop\n| refl {L} : red L L\n| step_trans {L₁ L₂ L₃} (H : free_group.red.step L₁ L₂) :\n    red L₂ L₃ → red L₁ L₃\n\nattribute [simp] red.step.bnot\nattribute [refl] red.refl\n\nvariables {L L₁ L₂ L₃ L₄ : list (α × bool)}\n\ntheorem red.trans.aux (H12 : red L₁ L₂) : ∀ {L₃}, red L₂ L₃ → red L₁ L₃ :=\nred.rec_on H12 (λ _ _, id) $ λ _ _ _ H1 H2 ih L₃ H23,\nred.step_trans H1 $ ih H23\n\n@[trans] theorem red.trans (H12 : red L₁ L₂) (H23 : red L₂ L₃) : red L₁ L₃ :=\nred.trans.aux H12 H23\n\n@[simp] lemma red.step.bnot_rev {x b} : red.step (L₁ ++ (x, bnot b) :: (x, b) :: L₂) (L₁ ++ L₂) :=\nby cases b; from step.bnot\n\ntheorem red.step.length : ∀ {L₁ L₂ : list (α × bool)}, red.step L₁ L₂ → L₂.length + 2 = L₁.length\n| _ _ (@red.step.bnot _ L1 L2 x b) := by rw [list.length_append, list.length_append]; refl\n\ntheorem red.sizeof : ∀ {L₁ L₂ : list (α × bool)}, red.step L₁ L₂ → L₂.sizeof < L₁.sizeof\n| _ _ (@red.step.bnot _ L1 L2 x b) :=\n  begin\n    induction L1 with hd tl ih,\n    case list.nil\n    { dsimp [list.sizeof],\n      have H : 1 + sizeof (x, b) + (1 + sizeof (x, bnot b) + list.sizeof L2)\n        = (list.sizeof L2 + 1) + (sizeof (x, b) + sizeof (x, bnot b) + 1),\n      { ac_refl },\n      rw H,\n      exact nat.le_add_right _ _ },\n    case list.cons\n    { dsimp [list.sizeof],\n      exact nat.add_lt_add_left ih _ }\n  end\n\ntheorem red.of_step (H : red.step L₁ L₂) : red L₁ L₂ :=\nred.step_trans H red.refl\n\n@[simp] lemma red.bnot {x b} : red (L₁ ++ (x, b) :: (x, bnot b) :: L₂) (L₁ ++ L₂) :=\nred.step_trans red.step.bnot red.refl\n\n@[simp] lemma red.step.cons_bnot {x b} : red.step ((x, b) :: (x, bnot b) :: L) L :=\n@red.step.bnot _ [] _ _ _\n\n@[simp] lemma red.cons_bnot {x b} : red ((x, b) :: (x, bnot b) :: L) L :=\n@red.bnot _ [] _ _ _\n\n@[simp] lemma red.cons_bnot_rev {x b} : red ((x, bnot b) :: (x, b) :: L) L :=\nred.of_step $ @red.step.bnot_rev _ [] _ _ _\n\ntheorem red.step.append_left : ∀ {L₁ L₂ L₃ : list (α × bool)},\n  red.step L₂ L₃ → red.step (L₁ ++ L₂) (L₁ ++ L₃)\n| _ _ _ red.step.bnot := by rw [← list.append_assoc, ← list.append_assoc]; constructor\n\ntheorem red.step.append_right : ∀ {L₁ L₂ L₃ : list (α × bool)},\n  red.step L₁ L₂ → red.step (L₁ ++ L₃) (L₂ ++ L₃)\n| _ _ _ red.step.bnot := by simp\n\ntheorem red.step.cons {x} (H : red.step L₁ L₂) : red.step (x :: L₁) (x :: L₂) :=\n@red.step.append_left _ [x] _ _ H\n\ntheorem red.append : ∀ {L₁ L₂ L₃ L₄ : list (α × bool)},\n  red L₁ L₃ → red L₂ L₄ → red (L₁ ++ L₂) (L₃ ++ L₄)\n| _ _ _ _ red.refl red.refl := red.refl\n| _ _ _ _ red.refl (red.step_trans H3 H4) :=\n  have _ := red.sizeof H3,\n  red.step_trans (red.step.append_left H3) (red.append red.refl H4)\n| _ _ _ _ (red.step_trans H1 H2) H3 :=\n  have _ := red.sizeof H1,\n  red.step_trans (red.step.append_right H1) (red.append H2 H3)\n\ntheorem red.cons {x} (H : red L₁ L₂) : red (x :: L₁) (x :: L₂) :=\n@red.append _ [x] _ _ _ red.refl H\n\ntheorem red.of_cons {x} : red (x :: L₁) (x :: L₂) → red L₁ L₂ :=\nbegin\n  generalize H1 : (x :: L₁ : list _) = L1,\n  generalize H2 : (x :: L₂ : list _) = L2,\n  intro H,\n  induction H with L3 L3 L4 L5 H3 H4 ih generalizing x L₁ L₂,\n  case red.refl\n  { subst H1; injections; subst_vars },\n  case red.step_trans\n  { cases H3 with L6 L7 x1 b1,\n    subst_vars,\n    cases L6 with hd tl,\n    case list.nil\n    { injection H1 with H5 H6,\n      substs H5 H6,\n      clear H1 H3,\n      transitivity,\n      { exact red.cons H4 },\n      { simp } },\n    case list.cons\n    { injection H1 with H5 H6,\n      substs H5 H6,\n      exact red.trans red.bnot (ih rfl rfl) } }\nend\n\n@[simp] lemma red.cons_iff {x} : red (x :: L₁) (x :: L₂) ↔ red L₁ L₂ :=\n⟨red.of_cons, red.cons⟩\n\ntheorem red.length (H : red L₁ L₂) : ∃ n, L₁.length = L₂.length + 2 * n :=\nred.rec_on H (λ _, ⟨0, rfl⟩) $ λ _ _ _ H1 H2 ⟨n, ih⟩,\n⟨nat.succ n, red.step.length H1 ▸ ih.symm ▸ rfl⟩\n\ntheorem red.antisymm (H : red L₁ L₂) : red L₂ L₁ → L₁ = L₂ :=\nred.rec_on H (λ _ _, rfl) $ λ L1 L2 L3 H1 H2 ih H21,\nlet ⟨n, hn⟩ := red.length (red.trans H2 H21) in\nhave H3 : list.length L2 = list.length L2 + 2 + 2 * n,\n  from (red.step.length H1).symm ▸ hn,\nhave H4 : list.length L2 + 0 = list.length L2 + (2 * n + 2),\n  by simpa using H3,\nnat.no_confusion $ nat.add_left_cancel H4\n\ntheorem red.step.church_rosser.aux2 : ∀ {L₁ L₂ L₃ L₄ : list (α × bool)} {x1 b1 x2 b2},\n  L₁ ++ (x1, b1) :: (x1, bnot b1) :: L₂ = L₃ ++ (x2, b2) :: (x2, bnot b2) :: L₄ →\n  L₁ ++ L₂ = L₃ ++ L₄ ∨ ∃ L₅, red.step (L₁ ++ L₂) L₅ ∧ red.step (L₃ ++ L₄) L₅\n| []        _ []        _ _ _ _ _ H := by injections; subst_vars; simp\n| []        _ [(x3,b3)] _ _ _ _ _ H := by injections; subst_vars; simp\n| [(x3,b3)] _ []        _ _ _ _ _ H := by injections; subst_vars; simp\n| []                     _ ((x3,b3)::(x4,b4)::tl) _ _ _ _ _ H :=\n  by injections; subst_vars; simp; right; exact ⟨_, red.step.bnot, red.step.cons_bnot⟩\n| ((x3,b3)::(x4,b4)::tl) _ []                     _ _ _ _ _ H :=\n  by injections; subst_vars; simp; right; exact ⟨_, red.step.cons_bnot, red.step.bnot⟩\n| ((x3,b3)::tl) _ ((x4,b4)::tl2) _ _ _ _ _ H :=\n  let ⟨H1, H2⟩ := list.cons.inj H in\n  match red.step.church_rosser.aux2 H2 with\n    | or.inl H3 := or.inl $ by simp [H1, H3]\n    | or.inr ⟨L₅, H3, H4⟩ := or.inr\n      ⟨_, red.step.cons H3, by simpa [H1] using red.step.cons H4⟩\n  end\n\ntheorem red.step.church_rosser.aux : ∀ {L₁ L₂ L₃ L₄ : list (α × bool)},\n  red.step L₁ L₃ → red.step L₂ L₄ → L₁ = L₂ →\n  L₃ = L₄ ∨ ∃ L₅, red.step L₃ L₅ ∧ red.step L₄ L₅\n| _ _ _ _ red.step.bnot red.step.bnot H := red.step.church_rosser.aux2 H\n\ntheorem red.step.church_rosser (H12 : red.step L₁ L₂) (H13 : red.step L₁ L₃) :\n  L₂ = L₃ ∨ ∃ L₄, red.step L₂ L₄ ∧ red.step L₃ L₄ :=\nred.step.church_rosser.aux H12 H13 rfl\n\ntheorem church_rosser_1 : ∀ {L₁ L₂ L₃ : list (α × bool)},\n  red.step L₁ L₂ → red L₁ L₃ →\n  red L₂ L₃ ∨ ∃ L₄, red L₂ L₄ ∧ step L₃ L₄\n| _ _ _ H12 red.refl := or.inr ⟨_, red.refl, H12⟩\n| _ _ _ H12 (red.step_trans H1 H2) :=\n  have _ := red.sizeof H1,\n  match red.step.church_rosser H12 H1 with\n    | or.inl H3 := or.inl $ H3.symm ▸ H2\n    | or.inr ⟨L1, H3, H4⟩ := match church_rosser_1 H4 H2 with\n      | or.inl H5 := or.inl $ red.step_trans H3 H5\n      | or.inr ⟨L2, H5, H6⟩ := or.inr $ ⟨L2, red.step_trans H3 H5, H6⟩\n      end\n  end\n\n/-- Church-Rosser theorem for word reduction:\nIf `w1 w2 w3` are words such that `w1` reduces to\n`w2` and `w3` respectively, then there is a word\n`w4` such that `w2` and `w3` reduce to `w4` respectively. -/\ntheorem church_rosser : ∀ {L₁ L₂ L₃ : list (α × bool)},\n  red L₁ L₂ → red L₁ L₃ → ∃ L₄, red L₂ L₄ ∧ red L₃ L₄\n| _ _ _ red.refl H23 := ⟨_, H23, red.refl⟩\n| _ _ _ (red.step_trans H1 H2) H23 :=\n  have _ := red.sizeof H1,\n  match church_rosser_1 H1 H23 with\n    | or.inl H3 := church_rosser H2 H3\n    | or.inr ⟨L7, H3, H4⟩ :=\n        let ⟨L8, H5, H6⟩ := church_rosser H2 H3 in\n        ⟨L8, H5, red.step_trans H4 H6⟩\n  end\n\nend free_group\n\nvariable α\n\n/-- The free group over a type, i.e. the words formed\nby the elements of the type and their formal\ninverses, quotient by one step reduction. -/\ndef free_group : Type u :=\nquot $ @free_group.red.step α\n\nnamespace free_group\n\nvariables {α} {L L₁ L₂ L₃ L₄ : list (α × bool)}\n\ndef mk (L) : free_group α := quot.mk red.step L\n\n@[simp] lemma quot_mk_eq_mk : quot.mk red.step L = mk L := rfl\n\n@[simp] lemma quot_lift_mk (β : Type*) (f : list (α × bool) → β)\n  (H : ∀ L₁ L₂, red.step L₁ L₂ → f L₁ = f L₂) :\nquot.lift f H (mk L) = f L := rfl\n\n@[simp] lemma quot_lift_on_mk (β : Type*) (f : list (α × bool) → β)\n  (H : ∀ L₁ L₂, red.step L₁ L₂ → f L₁ = f L₂) :\nquot.lift_on (mk L) f H = f L := rfl\n\ndef inv : list (α × bool) → list (α × bool) :=\nλ L, (L.map $ λ x : α × bool, (x.1, bnot x.2)).reverse\n\ntheorem red.step.inv (H : red.step L₁ L₂) : red.step (inv L₁) (inv L₂) :=\nby cases H; simp [inv]\n\ninstance : group (free_group α) :=\n{ mul := λ x y, quot.lift_on x\n    (λ L₁, quot.lift_on y\n       (λ L₂, mk $ L₁ ++ L₂)\n       (λ L₂ L₃ H, quot.sound $ red.step.append_left H))\n    (λ L₁ L₂ H, quot.induction_on y $\n       λ L₃, quot.sound $ red.step.append_right H),\n  mul_assoc := λ x y z, quot.induction_on x $ λ L₁,\n    quot.induction_on y $ λ L₂, quot.induction_on z $ λ L₃, by simp,\n  one := mk [],\n  one_mul := λ x, quot.induction_on x $ λ L, rfl,\n  mul_one := λ x, quot.induction_on x $ λ L, by simp [(*)],\n  inv := λ x, quot.lift_on x\n    (λ L, mk $ inv L)\n    (λ L₁ L₂ H, quot.sound $ red.step.inv H),\n  mul_left_inv := λ x, quot.induction_on x $ λ L,\n    list.rec_on L rfl $ λ ⟨x, b⟩ tl ih, eq.trans\n      (quot.sound $ by simp [inv]) ih }\n\n@[simp] lemma mul_mk : mk L₁ * mk L₂ = mk (L₁ ++ L₂) := rfl\n\n/-- The canonical injection from the type to the free\ngroup over that type by sending each element to\nthe equivalence class of the letter that is the element. -/\ndef of (x : α) : free_group α :=\nmk [(x, tt)]\n\ntheorem red.step.eqv_gen_of_red (H : red L₁ L₂) : eqv_gen red.step L₁ L₂ :=\nred.rec_on H (λ _, eqv_gen.refl _) $ λ _ _ _ H1 _ ih,\neqv_gen.trans _ _ _ (eqv_gen.rel _ _ H1) ih\n\ntheorem red.step.exact (H : mk L₁ = mk L₂) :\n  ∃ L₃, red L₁ L₃ ∧ red L₂ L₃ :=\neqv_gen.rec_on (quot.exact red.step H)\n  (λ L1 L2 H12, ⟨L2, red.of_step H12, red.refl⟩)\n  (λ L, ⟨L, red.refl, red.refl⟩)\n  (λ L1 L2 H12 ⟨L3, H13, H23⟩, ⟨L3, H23, H13⟩)\n  (λ L1 L2 L3 H12 H13 ⟨L4, H14, H24⟩ ⟨L5, H25, H35⟩,\n     let ⟨L6, H46, H56⟩ := church_rosser H24 H25 in\n     ⟨L6, red.trans H14 H46, red.trans H35 H56⟩)\n\ntheorem red.step.sound (L₃) (H13 : red L₁ L₃) (H23 : red L₂ L₃) :\n  mk L₁ = mk L₂ :=\nquot.eqv_gen_sound $ eqv_gen.trans _ _ _\n  (red.step.eqv_gen_of_red H13)\n  (eqv_gen.symm _ _ $ red.step.eqv_gen_of_red H23)\n\ntheorem red.nil.aux : ∀ {L₁ L₂ : list (α × bool)},\n  red L₁ L₂ → L₁ = [] → [] = L₂\n| _ _ red.refl                                        H := H.symm\n| _ _ (red.step_trans (@red.step.bnot _ [] _ _ _) H2) H := list.no_confusion H\n\n/-- The empty word `[]` only reduces to itself. -/\ntheorem red.nil (H : red [] L) : [] = L :=\nred.nil.aux H rfl\n\ntheorem red.singleton.aux : ∀ {L₁ L₂ : list (α × bool)} {x},\n  red L₁ L₂ → L₁ = [x] → [x] = L₂\n| _ _ _ red.refl                                         H := H.symm\n| _ _ _ (red.step_trans (@red.step.bnot _ [] _ _ _) H2)  H :=\n  list.no_confusion (list.cons.inj H).2\n| _ _ _ (red.step_trans (@red.step.bnot _ [x] _ _ _) H2) H :=\n  list.no_confusion (list.cons.inj H).2\n\n/-- A letter only reduces to itself. -/\ntheorem red.singleton {x} (H : red [x] L₁) : [x] = L₁ :=\nred.singleton.aux H rfl\n\n/-- The canonical injection from the type to\nthe free group is an injection. -/\ntheorem of.inj {x y : α} (H : of x = of y) : x = y :=\nlet ⟨L₁, hx, hy⟩ := red.step.exact H in\nhave H1 : _ := (red.singleton hx).trans (red.singleton hy).symm,\nby injections\n\ntheorem red.step.sublist (H : red.step L₁ L₂) : L₂ <+ L₁ :=\nby cases H; simp; constructor; constructor; refl\n\n/-- If `w₁ w₂` are words such that `w₁` reduces to\n`w₂`, then `w₂` is a sublist of `w₁`. -/\ntheorem red.sublist (H : red L₁ L₂) : L₂ <+ L₁ :=\nred.rec_on H list.sublist.refl $ λ _ _ _ H1 H2 ih,\nlist.sublist.trans ih $ red.step.sublist H1\n\ntheorem red.inv_of_red_nil.aux {x b} (H : red L₁ L₂) :\n  ∀ {L}, L₁ = ((x, b) :: L) → L₂ = [] → red L [(x, bnot b)] :=\nred.rec_on H (by cc) $ λ L3 L4 L5 H34,\nred.step.cases_on H34 $ λ L6 L7 x1 b1,\nlist.cases_on L6\n  (λ H45 ih L H7 H5, by injections; subst_vars; simpa)\n  (λ ⟨x2, b2⟩ tl H45 ih L H7 H5, by injections; subst_vars;\n     from red.step_trans red.step.bnot (ih rfl H5))\n\n/-- If `x` is a letter and `w` is a word such that `xw`\nreduces to the empty word, then `w` reduecs to `x⁻¹` -/\ntheorem red.inv_of_red_nil {x b} (H : red ((x, b) :: L) []) :\n  red L [(x, bnot b)] :=\nred.inv_of_red_nil.aux H rfl rfl\n\ntheorem red.inv_of_red_of_ne.aux {x1 b1 x2 b2} \n  (H1 : (x1, b1) ≠ (x2, b2)) (H2 : red L₁ L₂) :\n  ∀ {L₃}, L₁ = (x1, b1) :: L₃ → L₂ = (x2, b2) :: L₄ →\n  red L₃ ((x1, bnot b1) :: L₂) :=\nred.rec_on H2 (by cc) $ λ L5 L6 L7 H56,\nred.step.cases_on H56 $ λ L8 L9 x b,\nlist.cases_on L8\n  (λ H67 ih L₃ H6 H7, by injections; subst_vars; simpa)\n  (λ ⟨x, b⟩ tl H67 ih L₃ H6 H7, by injections; subst_vars;\n    from red.step_trans red.step.bnot (ih rfl H7))\n\n/-- If `x` and `y` are distinct letters and `w₁ w₂` are\nwords such that `xw₁` reduces to `yw₂`, then `w₁` reduces to `x⁻¹yw₂`. -/\ntheorem red.inv_of_red_of_ne {x1 b1 x2 b2}\n  (H1 : (x1, b1) ≠ (x2, b2))\n  (H2 : red ((x1, b1) :: L₁) ((x2, b2) :: L₂)) :\n  red L₁ ((x1, bnot b1) :: (x2, b2) :: L₂) :=\nred.inv_of_red_of_ne.aux H1 H2 rfl rfl\n\nsection to_group\n\nvariables {β : Type v} [group β] (f : α → β) {x y : free_group α}\n\ndef to_group.aux : list (α × bool) → β :=\nλ L, list.prod $ L.map $ λ x, cond x.2 (f x.1) (f x.1)⁻¹\n\ntheorem red.step.to_group {f : α → β} (H : red.step L₁ L₂) :\n  to_group.aux f L₁ = to_group.aux f L₂ :=\nby cases H with _ _ _ b; cases b; simp [to_group.aux]\n\ntheorem red.to_group {f : α → β} (H : red L₁ L₂) :\n  to_group.aux f L₁ = to_group.aux f L₂ :=\nred.rec_on H (λ _, rfl) $ λ _ _ _ H1 H2 ih,\neq.trans (by cases H1 with _ _ _ b; cases b; simp [to_group.aux]) ih\n\n/-- If `β` is a group, then any function from `α` to `β`\nextends uniquely to a group homomorphism from\nthe free group over `α` to `β` -/\ndef to_group : free_group α → β :=\nquot.lift (to_group.aux f) $ λ L₁ L₂ H, red.step.to_group H\n\nvariable {f}\n\n@[simp] lemma to_group.mk : to_group f (mk L) =\n  list.prod (L.map $ λ x, cond x.2 (f x.1) (f x.1)⁻¹) :=\nrfl\n\n@[simp] lemma to_group.of {x} : to_group f (of x) = f x :=\none_mul _\n\ninstance to_group.is_group_hom : is_group_hom (to_group f) :=\n⟨λ x y, quot.induction_on x $ λ L₁, quot.induction_on y $ λ L₂, by simp⟩\n\n@[simp] lemma to_group.mul : to_group f (x * y) = to_group f x * to_group f y :=\nis_group_hom.mul _ _ _\n\n@[simp] lemma to_group.one : to_group f 1 = 1 :=\nis_group_hom.one _\n\n@[simp] lemma to_group.inv : to_group f x⁻¹ = (to_group f x)⁻¹ :=\nis_group_hom.inv _ _\n\ntheorem to_group.unique (g : free_group α → β) [is_group_hom g]\n  (hg : ∀ x, g (of x) = f x) {x} :\n  g x = to_group f x :=\nquot.induction_on x $ λ L, list.rec_on L (is_group_hom.one g) $\nλ ⟨x, b⟩ t (ih : g (mk t) = _), bool.rec_on b\n  (show g ((of x)⁻¹ * mk t) = to_group f (mk ((x, ff) :: t)),\n     by simp [is_group_hom.mul g, is_group_hom.inv g, hg, ih, to_group, to_group.aux])\n  (show g (of x * mk t) = to_group f (mk ((x, tt) :: t)),\n     by simp [is_group_hom.mul g, is_group_hom.inv g, hg, ih, to_group, to_group.aux])\n\ntheorem to_group.of_eq (x : free_group α) : to_group of x = x :=\neq.symm $ to_group.unique id (λ x, rfl)\n\ntheorem to_group.range_subset {s : set β} [is_subgroup s] (H : set.range f ⊆ s) :\n  set.range (to_group f) ⊆ s :=\nλ y ⟨x, H1⟩, H1 ▸ (quot.induction_on x $ λ L,\nlist.rec_on L (is_submonoid.one_mem s) $ λ ⟨x, b⟩ tl ih,\nbool.rec_on b\n  (by simp at ih ⊢; from is_submonoid.mul_mem\n    (is_subgroup.inv_mem $ H ⟨x, rfl⟩) ih)\n  (by simp at ih ⊢; from is_submonoid.mul_mem (H ⟨x, rfl⟩) ih))\n\ntheorem to_group.range_eq_closure :\n  set.range (to_group f) = group.closure (set.range f) :=\nset.subset.antisymm\n  (to_group.range_subset group.subset_closure)\n  (group.closure_subset $ λ y ⟨x, hx⟩, ⟨of x, by simpa⟩)\n\nend to_group\n\nsection map\n\nvariables {β : Type v} (f : α → β) {x y : free_group α}\n\ndef map.aux (L : list (α × bool)) : list (β × bool) :=\nL.map $ λ x, (f x.1, x.2)\n\n/-- Any function from `α` to `β` extends uniquely\nto a group homomorphism from the free group\nver `α` to the free group over `β`. -/\ndef map (x : free_group α) : free_group β :=\nx.lift_on (λ L, mk $ map.aux f L) $\nλ L₁ L₂ H, quot.sound $ by cases H; simp [map.aux]\n\ninstance map.is_group_hom : is_group_hom (map f) :=\n⟨λ x y, quot.induction_on x $ λ L₁, quot.induction_on y $ λ L₂,\nby simp [map, map.aux]⟩\n\nvariable {f}\n\n@[simp] lemma map.mk : map f (mk L) = mk (L.map (λ x, (f x.1, x.2))) :=\nrfl\n\n@[simp] lemma map.id : map id x = x :=\nhave H1 : (λ (x : α × bool), x) = id := rfl,\nquot.induction_on x $ λ L, by simp [H1]\n\n@[simp] lemma map.id' : map (λ z, z) x = x := map.id\n\ntheorem map.comp {γ : Type w} {f : α → β} {g : β → γ} {x} :\n  map g (map f x) = map (g ∘ f) x :=\nquot.induction_on x $ λ L, by simp\n\n@[simp] lemma map.of {x} : map f (of x) = of (f x) := rfl\n\n@[simp] lemma map.mul : map f (x * y) = map f x * map f y :=\nis_group_hom.mul _ x y\n\n@[simp] lemma map.one : map f 1 = 1 :=\nis_group_hom.one _\n\n@[simp] lemma map.inv : map f x⁻¹ = (map f x)⁻¹ :=\nis_group_hom.inv _ x\n\ntheorem map.unique (g : free_group α → free_group β) [is_group_hom g]\n  (hg : ∀ x, g (of x) = of (f x)) {x} :\n  g x = map f x :=\nquot.induction_on x $ λ L, list.rec_on L (is_group_hom.one g) $\nλ ⟨x, b⟩ t (ih : g (mk t) = map f (mk t)), bool.rec_on b\n  (show g ((of x)⁻¹ * mk t) = map f ((of x)⁻¹ * mk t),\n     by simp [is_group_hom.mul g, is_group_hom.inv g, hg, ih])\n  (show g (of x * mk t) = map f (of x * mk t),\n     by simp [is_group_hom.mul g, hg, ih])\n\n/-- Equivalent types give rise to equivalent free groups. -/\ndef free_group_congr {α β} (e : α ≃ β) : free_group α ≃ free_group β :=\n⟨map e, map e.symm,\n λ x, by simp [function.comp, map.comp],\n λ x, by simp [function.comp, map.comp]⟩\n\ntheorem map_eq_to_group : map f x = to_group (of ∘ f) x :=\neq.symm $ map.unique _ $ λ x, by simp\n\nend map\n\nsection prod\n\nvariables [group α] (x y : free_group α)\n\n/-- If `α` is a group, then any function from `α` to `α`\nextends uniquely to a homomorphism from the\nfree group over `α` to `α`. This is the multiplicative\nversion of `sum`. -/\ndef prod : α :=\nto_group id x\n\nvariables {x y}\n\n@[simp] lemma prod_mk :\n  prod (mk L) = list.prod (L.map $ λ x, cond x.2 x.1 x.1⁻¹) :=\nrfl\n\n@[simp] lemma prod.of {x : α} : prod (of x) = x :=\nto_group.of\n\ninstance prod.is_group_hom : is_group_hom (@prod α _) :=\nto_group.is_group_hom\n\n@[simp] lemma prod.mul : prod (x * y) = prod x * prod y :=\nto_group.mul\n\n@[simp] lemma prod.one : prod (1:free_group α) = 1 :=\nto_group.one\n\n@[simp] lemma prod.inv : prod x⁻¹ = (prod x)⁻¹ :=\nto_group.inv\n\nlemma prod.unique (g : free_group α → α) [is_group_hom g]\n  (hg : ∀ x, g (of x) = x) {x} :\n  g x = prod x :=\nto_group.unique g hg\n\nend prod\n\ntheorem to_group_eq_prod_map {β : Type v} [group β] {f : α → β} {x} :\n  to_group f x = prod (map f x) :=\neq.symm $ to_group.unique (prod ∘ map f) $ λ _, by simp\n\nsection sum\n\nvariables [add_group α] (x y : free_group α)\n\n/-- If `α` is a group, then any function from `α` to `α`\nextends uniquely to a homomorphism from the\nfree group over `α` to `α`. This is the additive\nversion of `prod`. -/\ndef sum : α :=\n@prod (multiplicative _) _ x\n\nvariables {x y}\n\n@[simp] lemma sum_mk :\n  sum (mk L) = list.sum (L.map $ λ x, cond x.2 x.1 (-x.1)) :=\nrfl\n\n@[simp] lemma sum.of {x : α} : sum (of x) = x :=\nprod.of\n\ninstance sum.is_group_hom : is_group_hom (@sum α _) :=\nprod.is_group_hom\n\n@[simp] lemma sum.sum : sum (x * y) = sum x + sum y :=\nprod.mul\n\n@[simp] lemma sum.one : sum (1:free_group α) = 0 :=\nprod.one\n\n@[simp] lemma sum.inv : sum x⁻¹ = -sum x :=\nprod.inv\n\nend sum\n\ndef free_group_empty_equiv_unit : free_group empty ≃ unit :=\n{ to_fun    := λ _, (),\n  inv_fun   := λ _, 1,\n  left_inv  := λ x, quot.induction_on x $ λ L,\n    match L with [] := rfl end,\n  right_inv := λ ⟨⟩, rfl }\n\ndef free_group_unit_equiv_int : free_group unit ≃ int :=\n{ to_fun    := λ x, sum $ map (λ _, 1) x,\n  inv_fun   := λ x, of () ^ x,\n  left_inv  := λ x, quot.induction_on x $ λ L, list.rec_on L rfl $\n    λ ⟨⟨⟩, b⟩ tl ih, by cases b; simp [gpow_add] at ih ⊢; rw ih; refl,\n  right_inv := λ x, int.induction_on x (by simp)\n    (λ i ih, by simp at ih; simp [gpow_add, ih])\n    (λ i ih, by simp at ih; simp [gpow_add, ih]) }\n\nsection reduce\n\nvariable [decidable_eq α]\n\n/-- The maximal reduction of a word. It is computable\niff `α` has decidable equality. -/\ndef reduce (L : list (α × bool)) : list (α × bool) :=\nlist.rec_on L [] $ λ hd1 tl1 ih,\nlist.cases_on ih [hd1] $ λ hd2 tl2,\nif hd1.1 = hd2.1 ∧ hd1.2 = bnot hd2.2 then tl2\nelse hd1 :: hd2 :: tl2\n\n@[simp] lemma reduce.cons (x) : reduce (x :: L) =\n  list.cases_on (reduce L) [x] (λ hd tl,\n  if x.1 = hd.1 ∧ x.2 = bnot hd.2 then tl\n  else x :: hd :: tl) := rfl\n\n/-- The first theorem that characterises the function\n`reduce`: a word reduces to its maximal reduction. -/\ntheorem reduce.red : red L (reduce L) :=\nbegin\n  induction L with hd1 tl1 ih,\n  case list.nil\n  { refl },\n  case list.cons\n  { dsimp,\n    revert ih,\n    generalize htl : reduce tl1 = TL,\n    intro ih,\n    cases TL with hd2 tl2,\n    case list.nil\n    { exact red.cons ih },\n    case list.cons\n    { dsimp,\n      by_cases h : hd1.fst = hd2.fst ∧ hd1.snd = bnot (hd2.snd),\n      { rw [if_pos h],\n        transitivity,\n        { exact red.cons ih },\n        { cases hd1, cases hd2, cases h,\n          dsimp at *, subst_vars,\n          simp } },\n      { rw [if_neg h],\n        exact red.cons ih } } }\nend\n\ntheorem reduce.not {p : Prop} : ∀ {L₁ L₂ L₃ : list (α × bool)} {x b}, reduce L₁ =L₂ ++ (x, b) :: (x, bnot b) :: L₃ → p\n| [] L2 L3 _ _ := λ h, by cases L2; injections\n| ((x,b)::L1) L2 L3 x' b' := begin\n  dsimp,\n  cases r : reduce L1,\n  { dsimp, intro h,\n    have := congr_arg list.length h,\n    simp [-add_comm] at this,\n    exact absurd this dec_trivial },\n  cases hd with y c,\n  by_cases x = y ∧ b = bnot c; simp [h]; intro H,\n  { rw H at r,\n    exact @reduce.not L1 ((y,c)::L2) L3 x' b' r },\n  rcases L2 with _|⟨a, L2⟩,\n  { injections, subst_vars,\n    simp at h, cc },\n  { refine @reduce.not L1 L2 L3 x' b' _,\n    injection H with _ H,\n    rw [r, H], refl }\nend\n\n/-- The second theorem that characterises the\nfunction `reduce`: the maximal reduction of a word\nonly reduces to itself. -/\ntheorem reduce.min (H : red (reduce L₁) L₂) : reduce L₁ = L₂ :=\nbegin\n  revert H,\n  generalize h1 : reduce L₁ = L,\n  intro H,\n  induction H with L1 L1 L2 L3 H1 H2 ih,\n  case red.refl\n  { refl },\n  case red.step_trans\n  { cases H1 with L4 L5 x b,\n    exact reduce.not h1 }\nend\n\n/-- `reduce` is idempotent, i.e. the maximal reduction\nof the maximal reduction of a word is the maximal\nreduction of the word. -/\ntheorem reduce.idem : reduce (reduce L) = reduce L :=\neq.symm $ reduce.min reduce.red\n\ntheorem reduce.step.eq (H : red.step L₁ L₂) : reduce L₁ = reduce L₂ :=\nlet ⟨L₃, HR13, HR23⟩ := church_rosser reduce.red (red.step_trans H reduce.red) in\n(reduce.min HR13).trans (reduce.min HR23).symm\n\n/-- If a word reduces to another word, then they have\na common maximal reduction. -/\ntheorem reduce.eq_of_red (H : red L₁ L₂) : reduce L₁ = reduce L₂ :=\nlet ⟨L₃, HR13, HR23⟩ := church_rosser reduce.red (red.trans H reduce.red) in\n(reduce.min HR13).trans (reduce.min HR23).symm\n\n/-- If two words correspond to the same element in\nthe free group, then they have a common maximal\nreduction. This is the proof that the function that\nsends an element of the free group to its maximal\nreduction is well-defined. -/\ntheorem reduce.sound (H : mk L₁ = mk L₂) : reduce L₁ = reduce L₂ :=\nlet ⟨L₃, H13, H23⟩ := red.step.exact H in\n(reduce.eq_of_red H13).trans (reduce.eq_of_red H23).symm\n\n/-- If two words have a common maximal reduction,\nthen they correspond to the same element in the free group. -/\ntheorem reduce.exact (H : reduce L₁ = reduce L₂) : mk L₁ = mk L₂ :=\nred.step.sound (reduce L₂) (H ▸ reduce.red) (reduce.red)\n\n/-- A word and its maximal reduction correspond to\nthe same element of the free group. -/\ntheorem reduce.self : mk (reduce L) = mk L :=\nreduce.exact reduce.idem\n\n/-- If words `w₁ w₂` are such that `w₁` reduces to `w₂`,\nthen `w₂` reduces to the maximal reduction of `w₁`. -/\ntheorem reduce.rev (H : red L₁ L₂) : red L₂ (reduce L₁) :=\n(reduce.eq_of_red H).symm ▸ reduce.red\n\n/-- The function that sends an element of the free\ngroup to its maximal reduction. -/\ndef to_word : free_group α → list (α × bool) :=\nquot.lift reduce $ λ L₁ L₂ H, reduce.step.eq H\n\ndef to_word.mk {x : free_group α} : mk (to_word x) = x :=\nquot.induction_on x $ λ L₁, reduce.self\n\ndef to_word.inj (x y : free_group α) : to_word x = to_word y → x = y :=\nquot.induction_on x $ λ L₁, quot.induction_on y $ λ L₂, reduce.exact\n\n/-- Constructive Church-Rosser theorem (compare `church_rosser`). -/\ndef reduce.church_rosser (H12 : red L₁ L₂) (H13 : red L₁ L₃) :\n  { L₄ // red L₂ L₄ ∧ red L₃ L₄ } :=\n⟨reduce L₁, reduce.rev H12, reduce.rev H13⟩\n\ninstance : decidable_eq (free_group α) :=\nfunction.injective.decidable_eq to_word.inj\n\ninstance red.decidable_rel : decidable_rel (@red α)\n| [] [] := is_true red.refl\n| [] (hd2::tl2) := is_false $ λ H,\n  list.no_confusion $ red.nil H\n| ((x,b)::tl) [] := match red.decidable_rel tl [(x, bnot b)] with\n  | is_true H  := is_true $ red.trans (red.cons H) $\n    red.of_step $ @red.step.bnot _ [] [] _ _\n  | is_false H := is_false $ λ H2, H $ red.inv_of_red_nil H2\n  end\n| ((x1,b1)::tl1) ((x2,b2)::tl2) := if h : (x1, b1) = (x2, b2)\n  then match red.decidable_rel tl1 tl2 with\n    | is_true H  := is_true $ h ▸ red.cons H\n    | is_false H := is_false $ λ H2, H $ h ▸ red.of_cons $ H2\n    end\n  else match red.decidable_rel tl1 ((x1,bnot b1)::(x2,b2)::tl2) with\n    | is_true H  := is_true $ red.trans (red.cons H) $ red.cons_bnot\n    | is_false H := is_false $ λ H2, H $ red.inv_of_red_of_ne h H2\n    end\n\n/-- A list containing every word that `w₁` reduces to. -/\ndef red.enum (L₁ : list (α × bool)) : list (list (α × bool)) :=\nlist.filter (λ L₂, red L₁ L₂) (list.sublists L₁)\n\ntheorem red.enum.sound (H : L₂ ∈ red.enum L₁) : red L₁ L₂ :=\nlist.of_mem_filter H\n\ntheorem red.enum.complete (H : red L₁ L₂) : L₂ ∈ red.enum L₁ :=\nlist.mem_filter_of_mem (list.mem_sublists.2 $ red.sublist H) H\n\ninstance : fintype { L₂ // red L₁ L₂ } :=\nfintype.subtype (list.to_finset $ red.enum L₁) $\nλ L₂, ⟨λ H, red.enum.sound $ list.mem_to_finset.1 H,\n  λ H, list.mem_to_finset.2 $ red.enum.complete H⟩\n\nend reduce\n\nend free_group\n", "meta": {"author": "kckennylau", "repo": "Lean", "sha": "907d0a4d2bd8f23785abd6142ad53d308c54fdcb", "save_path": "github-repos/lean/kckennylau-Lean", "path": "github-repos/lean/kckennylau-Lean/Lean-907d0a4d2bd8f23785abd6142ad53d308c54fdcb/free_group.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.32100374121090364}}
{"text": "example : x ≠ y → x ∉ [y] :=\n  fun hne hin =>\n    match hin with\n    | .head _ => hne rfl\n\nexample : x ≠ y → x ∉ [y] :=\n  fun hne (.head _) => hne rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/dotNameIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.320952427895305}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.punit\nimport category_theory.structured_arrow\nimport category_theory.is_connected\nimport category_theory.limits.yoneda\nimport category_theory.limits.types\n\n/-!\n# Cofinal functors\n\nA functor `F : C ⥤ D` is cofinal if for every `d : D`,\nthe comma category of morphisms `d ⟶ F.obj c` is connected.\n\nWe prove the following three statements are equivalent:\n1. `F : C ⥤ D` is cofinal.\n2. Every functor `G : D ⥤ E` has a colimit if and only if `F ⋙ G` does,\n   and these colimits are isomorphic via `colimit.pre G F`.\n3. `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`.\n\nStarting at 1. we show (in `cocones_equiv`) that\nthe categories of cocones over `G : D ⥤ E` and over `F ⋙ G` are equivalent.\n(In fact, via an equivalence which does not change the cocone point.)\nThis readily implies 2., as `comp_has_colimit`, `has_colimit_of_comp`, and `colimit_iso`.\n\nFrom 2. we can specialize to `G = coyoneda.obj (op d)` to obtain 3., as `colimit_comp_coyoneda_iso`.\n\nFrom 3., we prove 1. directly in `cofinal_of_colimit_comp_coyoneda_iso_punit`.\n\nWe also show these conditions imply:\n4. Every functor `H : Dᵒᵖ ⥤ E` has a limit if and only if `F.op ⋙ H` does,\n   and these limits are isomorphic via `limit.pre H F.op`.\n\n\n## Naming\nThere is some discrepancy in the literature about naming; some say 'final' instead of 'cofinal'.\nThe explanation for this is that the 'co' prefix here is *not* the usual category-theoretic one\nindicating duality, but rather indicating the sense of \"along with\".\n\nWhile the trend seems to be towards using 'final', for now we go with the bulk of the literature\nand use 'cofinal'.\n\n## References\n* https://stacks.math.columbia.edu/tag/09WN\n* https://ncatlab.org/nlab/show/final+functor\n* Borceux, Handbook of Categorical Algebra I, Section 2.11.\n  (Note he reverses the roles of definition and main result relative to here!)\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nnamespace category_theory\n\nopen opposite\nopen category_theory.limits\n\nvariables {C : Type v} [small_category C]\nvariables {D : Type v} [small_category D]\n\n/--\nA functor `F : C ⥤ D` is cofinal if for every `d : D`, the comma category of morphisms `d ⟶ F.obj c`\nis connected.\n\nSee https://stacks.math.columbia.edu/tag/04E6\n-/\nclass cofinal (F : C ⥤ D) : Prop :=\n(out (d : D) : is_connected (structured_arrow d F))\n\nattribute [instance] cofinal.out\n\nnamespace cofinal\n\nvariables (F : C ⥤ D) [cofinal F]\n\ninstance (d : D) : nonempty (structured_arrow d F) := is_connected.is_nonempty\n\nvariables {E : Type u} [category.{v} E] (G : D ⥤ E)\n\n/--\nWhen `F : C ⥤ D` is cofinal, we denote by `lift F d` an arbitrary choice of object in `C` such that\nthere exists a morphism `d ⟶ F.obj (lift F d)`.\n-/\ndef lift (d : D) : C :=\n(classical.arbitrary (structured_arrow d F)).right\n\n/--\nWhen `F : C ⥤ D` is cofinal, we denote by `hom_to_lift` an arbitrary choice of morphism\n`d ⟶ F.obj (lift F d)`.\n-/\ndef hom_to_lift (d : D) : d ⟶ F.obj (lift F d) :=\n(classical.arbitrary (structured_arrow d F)).hom\n\n/--\nWe provide an induction principle for reasoning about `lift` and `hom_to_lift`.\nWe want to perform some construction (usually just a proof) about\nthe particular choices `lift F d` and `hom_to_lift F d`,\nit suffices to perform that construction for some other pair of choices\n(denoted `X₀ : C` and `k₀ : d ⟶ F.obj X₀` below),\nand to show that how to transport such a construction\n*both* directions along a morphism between such choices.\n-/\nlemma induction {d : D} (Z : Π (X : C) (k : d ⟶ F.obj X), Prop)\n  (h₁ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),\n    (k₁ ≫ F.map f = k₂) → Z X₁ k₁ → Z X₂ k₂)\n  (h₂ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),\n    (k₁ ≫ F.map f = k₂) → Z X₂ k₂ → Z X₁ k₁)\n  {X₀ : C} {k₀ : d ⟶ F.obj X₀} (z : Z X₀ k₀) : Z (lift F d) (hom_to_lift F d) :=\nbegin\n  apply nonempty.some,\n  apply @is_preconnected_induction _ _ _\n    (λ (Y : structured_arrow d F), Z Y.right Y.hom) _ _ { right := X₀, hom := k₀, } z,\n  { intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\n  { intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\nend\n\nvariables {F G}\n\n/--\nGiven a cocone over `F ⋙ G`, we can construct a `cocone G` with the same cocone point.\n-/\n@[simps]\ndef extend_cocone : cocone (F ⋙ G) ⥤ cocone G :=\n{ obj := λ c,\n  { X := c.X,\n    ι :=\n    { app := λ X, G.map (hom_to_lift F X) ≫ c.ι.app (lift F X),\n      naturality' := λ X Y f,\n      begin\n        dsimp, simp,\n        -- This would be true if we'd chosen `lift F X` to be `lift F Y`\n        -- and `hom_to_lift F X` to be `f ≫ hom_to_lift F Y`.\n        apply induction F\n          (λ Z k, G.map f ≫ G.map (hom_to_lift F Y) ≫ c.ι.app (lift F Y) = G.map k ≫ c.ι.app Z),\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w, z], },\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w] at z,\n        rw z, },\n        { rw [←functor.map_comp_assoc], },\n      end } },\n  map := λ X Y f,\n  { hom := f.hom, } }\n\n@[simp]\nlemma colimit_cocone_comp_aux (s : cocone (F ⋙ G)) (j : C) :\n  G.map (hom_to_lift F (F.obj j)) ≫ s.ι.app (lift F (F.obj j)) =\n    s.ι.app j :=\nbegin\n  -- This point is that this would be true if we took `lift (F.obj j)` to just be `j`\n  -- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.\n  apply induction F (λ X k, G.map k ≫ s.ι.app X = (s.ι.app j : _)),\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←w, rw ← s.w f at h, simpa using h, },\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←w at h, rw ← s.w f, simpa using h, },\n  { exact s.w (𝟙 _), },\nend\n\nvariables {H : Dᵒᵖ ⥤ E}\n\n/-- An auxilliary construction for `extend_cone`, moving `op` around. -/\n@[simps]\ndef extend_cone_cone_to_cocone {F : C ⥤ D} {H : Dᵒᵖ ⥤ E} (c : cone (F.op ⋙ H)) :\n  cocone (F ⋙ H.right_op) :=\n{ X := op c.X,\n  ι :=\n  { app := λ j, (c.π.app (op j)).op,\n    naturality' := λ j j' f,\n    begin apply quiver.hom.unop_inj, dsimp, simp only [category.id_comp], exact c.w f.op, end }}\n\n/-- An auxilliary construction for `extend_cone`, moving `op` around. -/\n@[simps]\ndef extend_cone_cocone_to_cone (c : cocone H.right_op) : cone H :=\n{ X := unop c.X,\n  π :=\n  { app := λ j, (c.ι.app (unop j)).unop,\n    naturality' := λ j j' f,\n    begin\n      apply quiver.hom.op_inj,\n      dsimp,\n      simp only [category.comp_id],\n      exact (c.w f.unop).symm,\n    end }}\n\n/--\nGiven a cone over `F.op ⋙ H`, we can construct a `cone H` with the same cone point.\n-/\n@[simps]\ndef extend_cone : cone (F.op ⋙ H) ⥤ cone H :=\n{ obj := λ c, extend_cone_cocone_to_cone (extend_cocone.obj (extend_cone_cone_to_cocone c)),\n  map := λ X Y f, { hom := f.hom, } }\n\n@[simp]\nlemma limit_cone_comp_aux (s : cone (F.op ⋙ H)) (j : Cᵒᵖ) :\n  s.π.app (op (lift F (F.obj (unop j)))) ≫ H.map (hom_to_lift F (F.obj (unop j))).op =\n    s.π.app j :=\nbegin\n  apply quiver.hom.op_inj,\n  exact colimit_cocone_comp_aux (extend_cone_cone_to_cocone s) (unop j)\nend\n\nvariables (F G H)\n\n/--\nIf `F` is cofinal,\nthe category of cocones on `F ⋙ G` is equivalent to the category of cocones on `G`,\nfor any `G : D ⥤ E`.\n-/\n@[simps]\ndef cocones_equiv : cocone (F ⋙ G) ≌ cocone G :=\n{ functor := extend_cocone,\n  inverse := cocones.whiskering F,\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy), }.\n\n/--\nIf `F` is cofinal,\nthe category of cones on `F.op ⋙ H` is equivalent to the category of cones on `H`,\nfor any `H : Dᵒᵖ ⥤ E`.\n-/\n@[simps]\ndef cones_equiv : cone (F.op ⋙ H) ≌ cone H :=\n{ functor := extend_cone,\n  inverse := cones.whiskering F.op,\n  unit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy), }.\n-- We could have done this purely formally in terms of `cocones_equiv`,\n-- without having defined `extend_cone` at all,\n-- but it comes at the cost of moving a *lot* of opposites around:\n-- (((cones.functoriality_equivalence _ (op_op_equivalence E)).symm.trans\n--   ((((cocone_equivalence_op_cone_op _).symm.trans\n--     (cocones_equiv F (unop_unop _ ⋙ H.op))).trans\n--     (cocone_equivalence_op_cone_op _)).unop)).trans\n--   (cones.functoriality_equivalence _ (op_op_equivalence E))).trans\n--   (cones.postcompose_equivalence (nat_iso.of_components (λ X, iso.refl _) (by tidy) :\n--     H ≅ (unop_unop D ⋙ H.op).op ⋙ (op_op_equivalence E).functor)).symm\n\nvariables {G H}\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `t : cocone G` for some `G : D ⥤ E`,\n`t.whisker F` is a colimit cocone exactly when `t` is.\n-/\ndef is_colimit_whisker_equiv (t : cocone G) : is_colimit (t.whisker F) ≃ is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G).symm\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `t : cone H` for some `H : Dᵒᵖ ⥤ E`,\n`t.whisker F.op` is a limit cone exactly when `t` is.\n-/\ndef is_limit_whisker_equiv (t : cone H) : is_limit (t.whisker F.op) ≃ is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F H).symm\n\n/--\nWhen `F` is cofinal, and `t : cocone (F ⋙ G)`,\n`extend_cocone.obj t` is a colimit coconne exactly when `t` is.\n-/\ndef is_colimit_extend_cocone_equiv (t : cocone (F ⋙ G)) :\n  is_colimit (extend_cocone.obj t) ≃ is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G)\n\n/--\nWhen `F` is cofinal, and `t : cone (F.op ⋙ H)`,\n`extend_cone.obj t` is a limit conne exactly when `t` is.\n-/\ndef is_limit_extend_cone_equiv (t : cone (F.op ⋙ H)) :\n  is_limit (extend_cone.obj t) ≃ is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F H)\n\n/-- Given a colimit cocone over `G : D ⥤ E` we can construct a colimit cocone over `F ⋙ G`. -/\n@[simps]\ndef colimit_cocone_comp (t : colimit_cocone G) :\n  colimit_cocone (F ⋙ G) :=\n{ cocone := _,\n  is_colimit := (is_colimit_whisker_equiv F _).symm (t.is_colimit) }\n\n/-- Given a limit cone over `H : Dᵒᵖ ⥤ E` we can construct a limit cone over `F.op ⋙ H`. -/\n@[simps]\ndef limit_cone_comp (t : limit_cone H) :\n  limit_cone (F.op ⋙ H) :=\n{ cone := _,\n  is_limit := (is_limit_whisker_equiv F _).symm (t.is_limit) }\n\n@[priority 100]\ninstance comp_has_colimit [has_colimit G] :\n  has_colimit (F ⋙ G) :=\nhas_colimit.mk (colimit_cocone_comp F (get_colimit_cocone G))\n\n@[priority 100]\ninstance comp_has_limit [has_limit H] :\n  has_limit (F.op ⋙ H) :=\nhas_limit.mk (limit_cone_comp F (get_limit_cone H))\n\nlemma colimit_pre_is_iso_aux {t : cocone G} (P : is_colimit t) :\n  ((is_colimit_whisker_equiv F _).symm P).desc (t.whisker F) = 𝟙 t.X :=\nbegin\n  dsimp [is_colimit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  dsimp, simp, dsimp, simp, -- See library note [dsimp, simp].\nend\n\ninstance colimit_pre_is_iso [has_colimit G] :\n  is_iso (colimit.pre G F) :=\nbegin\n  rw colimit.pre_eq (colimit_cocone_comp F (get_colimit_cocone G)) (get_colimit_cocone G),\n  erw colimit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nlemma limit_pre_is_iso_aux {t : cone H} (P : is_limit t) :\n  ((is_limit_whisker_equiv F _).symm P).lift (t.whisker F.op) = 𝟙 t.X :=\nbegin\n  dsimp [is_limit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  simp, refl,\nend\n\ninstance limit_pre_is_iso [has_limit H] :\n  is_iso (limit.pre H F.op) :=\nbegin\n  rw limit.pre_eq (limit_cone_comp F (get_limit_cone H)) (get_limit_cone H),\n  erw limit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nsection\nvariables (G H)\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `G : D ⥤ E` has a colimit, then `F ⋙ G` has a colimit also and\n`colimit (F ⋙ G) ≅ colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso [has_colimit G] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `H : Dᵒᵖ ⥤ E` has a limit, then `F.op ⋙ H` has a limit also and\n`limit (F.op ⋙ H) ≅ limit H`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso [has_limit H] : limit (F.op ⋙ H) ≅ limit H := (as_iso (limit.pre H F.op)).symm\n\n\nend\n\n/-- Given a colimit cocone over `F ⋙ G` we can construct a colimit cocone over `G`. -/\n@[simps]\ndef colimit_cocone_of_comp (t : colimit_cocone (F ⋙ G)) :\n  colimit_cocone G :=\n{ cocone := extend_cocone.obj t.cocone,\n  is_colimit := (is_colimit_extend_cocone_equiv F _).symm (t.is_colimit), }\n\n/-- Given a limit cone over `F.op ⋙ H` we can construct a limit cone over `H`. -/\n@[simps]\ndef limit_cone_of_comp (t : limit_cone (F.op ⋙ H)) :\n  limit_cone H :=\n{ cone := extend_cone.obj t.cone,\n  is_limit := (is_limit_extend_cone_equiv F _).symm (t.is_limit), }\n\n/--\nWhen `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_colimit`.)\n-/\nlemma has_colimit_of_comp [has_colimit (F ⋙ G)] :\n  has_colimit G :=\nhas_colimit.mk (colimit_cocone_of_comp F (get_colimit_cocone (F ⋙ G)))\n\n/--\nWhen `F` is cofinal, and `F.op ⋙ H` has a limit, then `H` has a limit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_limit`.)\n-/\nlemma has_limit_of_comp [has_limit (F.op ⋙ H)] :\n  has_limit H :=\nhas_limit.mk (limit_cone_of_comp F (get_limit_cone (F.op ⋙ H)))\n\nsection\nlocal attribute [instance] has_colimit_of_comp has_limit_of_comp\n\n/--\nWhen `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also and\n`colimit (F ⋙ G) ≅ colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso' [has_colimit (F ⋙ G)] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)\n\n/--\nWhen `F` is cofinal, and `F.op ⋙ H` has a limit, then `H` has a limit also and\n`limit (F.op ⋙ H) ≅ limit H`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso' [has_limit (F.op ⋙ H)] : limit (F.op ⋙ H) ≅ limit H :=\n(as_iso (limit.pre H F.op)).symm\n\nend\n\n/--\nIf the universal morphism `colimit (F ⋙ coyoneda.obj (op d)) ⟶ colimit (coyoneda.obj (op d))`\nis an isomorphism (as it always is when `F` is cofinal),\nthen `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`\n(simply because `colimit (coyoneda.obj (op d)) ≅ punit`).\n-/\ndef colimit_comp_coyoneda_iso (d : D) [is_iso (colimit.pre (coyoneda.obj (op d)) F)] :\n  colimit (F ⋙ coyoneda.obj (op d)) ≅ punit :=\nas_iso (colimit.pre (coyoneda.obj (op d)) F) ≪≫ coyoneda.colimit_coyoneda_iso (op d)\n\nlemma zigzag_of_eqv_gen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : Σ X, d ⟶ F.obj X}\n  (t : eqv_gen (types.quot.rel (F ⋙ coyoneda.obj (op d))) f₁ f₂) :\n  zigzag (structured_arrow.mk f₁.2) (structured_arrow.mk f₂.2) :=\nbegin\n  induction t,\n  case eqv_gen.rel : x y r\n  { obtain ⟨f, w⟩ := r,\n    fconstructor,\n    swap 2, fconstructor,\n    left, fsplit,\n    exact { right := f, } },\n  case eqv_gen.refl\n  { fconstructor, },\n  case eqv_gen.symm : x y h ih\n  { apply zigzag_symmetric,\n    exact ih, },\n  case eqv_gen.trans : x y z h₁ h₂ ih₁ ih₂\n  { apply relation.refl_trans_gen.trans,\n    exact ih₁, exact ih₂, }\nend\n\n/--\nIf `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit` for all `d : D`, then `F` is cofinal.\n-/\nlemma cofinal_of_colimit_comp_coyoneda_iso_punit\n  (I : Π d, colimit (F ⋙ coyoneda.obj (op d)) ≅ punit) : cofinal F :=\n⟨λ d, begin\n  haveI : nonempty (structured_arrow d F) := by\n  { have := (I d).inv punit.star,\n    obtain ⟨j, y, rfl⟩ := limits.types.jointly_surjective' this,\n    exact ⟨structured_arrow.mk y⟩, },\n  apply zigzag_is_connected,\n  rintros ⟨⟨⟩,X₁,f₁⟩ ⟨⟨⟩,X₂,f₂⟩,\n  dsimp at *,\n  let y₁ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₁ f₁,\n  let y₂ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₂ f₂,\n  have e : y₁ = y₂,\n  { apply (I d).to_equiv.injective, ext, },\n  have t := types.colimit_eq e,\n  clear e y₁ y₂,\n  exact zigzag_of_eqv_gen_quot_rel t,\nend⟩\n\nend cofinal\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/cofinal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.32095242067285545}}
{"text": "import data.matrix.notation\n\nimport for_mathlib.short_exact_sequence\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nnamespace homological_complex\n\nvariables (C : Type u) [category.{v} C] [abelian C]\nvariables {ι : Type*} {c : complex_shape ι}\n\nabbreviation Fst : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Fst C).map_homological_complex _\n\nabbreviation Snd : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Snd C).map_homological_complex _\n\nabbreviation Trd : chain_complex (short_exact_sequence C) ℕ ⥤\n  homological_complex C (complex_shape.down ℕ) :=\n(short_exact_sequence.Trd C).map_homological_complex _\n\nabbreviation Fst_Snd : Fst C ⟶ Snd C :=\nnat_trans.map_homological_complex (short_exact_sequence.f_nat C) _\n\nabbreviation Snd_Trd : Snd C ⟶ Trd C :=\nnat_trans.map_homological_complex (short_exact_sequence.g_nat C) _\n\nvariables {C}\n\nvariables (A : chain_complex (short_exact_sequence C) ℕ)\n\ninstance Fst_Snd_mono (n : ℕ) : mono (((Fst_Snd C).app A).f n) := (A.X n).mono'\n\ninstance Snd_Trd_epi (n : ℕ) : epi (((Snd_Trd C).app A).f n) := (A.X n).epi'\n\nlemma Fst_Snd_Trd_exact (n : ℕ) : exact (((Fst_Snd C).app A).f n) (((Snd_Trd C).app A).f n) :=\n(A.X n).exact'\n\nend homological_complex\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/homological_complex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.3204592703345866}}
{"text": "import category_theory.shift\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category\n\nvariables {C D E : Type*} {A : Type*} [category C] [category D] [category E]\n  [add_monoid A] [has_shift C A] [has_shift D A] [has_shift E A]\n\nvariable (C)\n\ndef shift_functor_add' (a b c : A) (h : c = a + b) :\n  shift_functor C c ≅ shift_functor C a ⋙ shift_functor C b :=\neq_to_iso (by rw h) ≪≫ shift_functor_add C a b\n\nlemma shift_functor_add'_eq_shift_functor_add (a b : A) :\n  shift_functor_add' C a b (a+b) rfl = shift_functor_add C a b :=\nbegin\n  ext1,\n  apply id_comp,\nend\n\nnamespace functor\n\nvariables {C}  {F : C ⥤ D} {G : D ⥤ E} {a : A}\n  (hF : shift_functor C a ⋙ F ≅ F ⋙ shift_functor D a)\n  (hG : shift_functor D a ⋙ G ≅ G ⋙ shift_functor E a)\n\ndef comm_shift_comp :\n  shift_functor C a ⋙ (F ⋙ G) ≅ (F ⋙ G) ⋙ shift_functor E a :=\n(functor.associator _ _ _).symm ≪≫\n  iso_whisker_right hF G ≪≫\n  functor.associator _ _ _ ≪≫\n  iso_whisker_left F hG ≪≫ (functor.associator _ _ _).symm\n\n@[simp]\nlemma comm_shift_comp_hom_app (X : C) :\n  (comm_shift_comp hF hG).hom.app X = G.map (hF.hom.app X) ≫ hG.hom.app (F.obj X) :=\nbegin\n  dsimp [comm_shift_comp],\n  simp only [comp_id, id_comp],\nend\n\n@[simp]\nlemma comm_shift_comp_inv_app (X : C) :\n  (comm_shift_comp hF hG).inv.app X = hG.inv.app (F.obj X) ≫ G.map (hF.inv.app X) :=\nbegin\n  dsimp [comm_shift_comp],\n  simp only [comp_id, id_comp],\nend\n\nend functor\n\nlemma shift_functor_comp_shift_functor_neg_eq_add'_comp_zero\n  {G : Type*} [add_group G] [has_shift C G] (n : G) (X : C) :\n  (shift_functor_comp_shift_functor_neg C n).hom.app X =\n    (shift_functor_add' C n (-n) 0 (add_neg_self n).symm).inv.app X ≫\n      (shift_functor_zero C G).hom.app X :=\nby simpa only [shift_functor_add', eq_to_hom_map, unit_of_tensor_iso_unit_hom_app,\n  eq_to_iso.hom, iso.trans_inv, nat_trans.comp_app, assoc]\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/shift_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.3204592644000566}}
{"text": "/-\n## Pattern matching against MLIR programs\n\nThis file implements support for a basic matching system. This system is used\nin the framework to concisely express syntactic constraints on operations that\ncan be resolved by unification. While this cannot express anywhere near all the\nconstraints, it simplifies the most common ones a great deal.\n\nTODO: Provide the matcher\n-/\n\nimport MLIR.Semantics.Types\nimport MLIR.Dialects.BuiltinModel\nimport MLIR.AST\nimport MLIR.EDSL\nopen MLIR.AST\n\n/-\n### Match term syntax\n\nWe have different sorts (operations, values, types, attributes, etc) with typed\nvariables. This allows us to know types everywhere in terms, and thus avoid\ntyping mishaps due to bugs (which would be hard to debug), even if we only have\none type for all terms.\n\nUnlike usual first-order matching and unification, we don't have a deeply\nrecursive structure, and instead mostly rely on having independent equations to\nmatch complex patterns.\n\nWe assign *substitution priority levels* to variables in the form of natural\nnumbers. Lower values indicate variables that have been introduced by the user,\nwhile higher values are assigned to automatically-generated variables. When a\nsubstitution occurs, we always substitute variables with higher priority so\nthat user-assigned names are preserved.\n-/\n\ninductive MSort :=\n  -- An MLIR operation. Matches against [Op δ]\n  | MOp\n  -- An operation parameter. Matches against (SSAVal × MLIRType δ)\n  | MOperand\n  -- An MLIR type. Matches against [MLIRType δ]\n  | MMLIRType\n  -- A value. Matches against [SSAVal]\n  | MSSAVal\n  -- An attribute. Matches against [AttrVal δ]\n  | MAttrValue\n  -- A natural number (typically int/float bit size). Matches against [Nat]\n  | MNat\n  -- A string (in operation names). Matches against [String]\n  | MString\n  -- A dimension (in a vector/tensor). Matches against [Dimension]\n  | MDimension\n  -- A signedness specification (in integers). Matches against [Signedness]\n  | MSignedness\n  -- A homogeneous list of objects\n  | MList (s: MSort)\n\ninductive MCtor: List MSort → MSort → Type :=\n  -- Integer type\n  | INT: MCtor [.MSignedness, .MNat] .MMLIRType\n  -- Tensor type\n  | TENSOR: MCtor [.MList .MDimension, .MMLIRType] .MMLIRType\n  -- Operation with known or unknown mnemonic (TODO: MCtor.OP: unfinished)\n  | OP: MCtor [.MString, .MList .MOperand, .MList .MOperand] .MOp\n  -- Operation argument of return value\n  | OPERAND: MCtor [.MSSAVal, .MMLIRType] .MOperand\n\n  -- SPECIAL CASE: We treat LIST specially in inferSort, to allow variadic\n  -- arguments without specifying it here\n  | LIST (s: MSort): MCtor [] (.MList s)\n\ninductive MTerm (δ: Dialect α σ ε) :=\n  -- A typed variable\n  | Var (priority: Nat := 0) (name: String) (s: MSort)\n  -- A constructor. We allow building mistyped terms (but check them later)\n  | App {args_sort: List MSort} {ctor_sort: MSort}\n        (ctor: MCtor args_sort ctor_sort) (args: List (MTerm δ))\n  -- Constants\n  | ConstMLIRType (τ: MLIRType δ)\n  | ConstNat (n: Nat)\n  | ConstString (s: String)\n  | ConstDimension (d: Dimension)\n  | ConstSignedness (sgn: Signedness)\n\n-- Accessors\n\ndef MCtor.name {s₁ s₂}: MCtor s₁ s₂ → String\n  | LIST _  => \"LIST\"\n  | INT     => \"INT\"\n  | TENSOR  => \"TENSOR\"\n  | OP      => \"OP\"\n  | OPERAND => \"OPERAND\"\n\n-- Common instances\n\nderiving instance Inhabited for MSort\nderiving instance Inhabited for MTerm\n\nderiving instance DecidableEq for MCtor\nderiving instance DecidableEq for MSort\n\ndef MCtor.eq {args_sort₁ ctor_sort₁ args_sort₂ ctor_sort₂}:\n    MCtor args_sort₁ ctor_sort₁ → MCtor args_sort₂ ctor_sort₂ → Bool :=\n  fun c₁ c₂ =>\n    if H: args_sort₁ = args_sort₂ ∧ ctor_sort₁ = ctor_sort₂ then\n      cast (by simp [H.1, H.2]) c₁ = c₂\n    else\n      false\n\nmutual\ndef MTerm.eq (t₁ t₂: MTerm δ): Bool :=\n  match t₁, t₂ with\n  | Var _ name₁ s₁, Var _ name₂ s₂ =>\n      name₁ = name₂ && s₁ = s₂\n  | App ctor₁ args₁, App ctor₂ args₂ =>\n      MCtor.eq ctor₁ ctor₂ && eqList args₁ args₂\n  | _, _ => false\n\ndef MTerm.eqList (l₁ l₂: List (MTerm δ)): Bool :=\n  match l₁, l₂ with\n  | [], [] => true\n  | t₁::l₁, t₂::l₂ => eq t₁ t₂ && eqList l₁ l₂\n  | _, _ => false\nend\ntermination_by\n  MTerm.eq t1 t2 => sizeOf t1 + sizeOf t2\n  MTerm.eqList t1 t2 => sizeOf t1 + sizeOf t2\n\ninstance: BEq (MTerm δ) where\n  beq := MTerm.eq\n\ndef MSort.str: MSort → String\n  | .MOp         => \"Op\"\n  | .MOperand    => \"Operand\"\n  | .MMLIRType   => \"MLIRType\"\n  | .MSSAVal     => \"SSAVal\"\n  | .MAttrValue  => \"AttrValue\"\n  | .MNat        => \"Nat\"\n  | .MString     => \"String\"\n  | .MDimension  => \"Dimension\"\n  | .MSignedness => \"Signedness\"\n  | .MList s     => \"[\" ++ s.str ++ \"]\"\n\nmutual\ndef MTerm.str: MTerm δ → String\n  -- Short notations for common sorts of variables\n  | .Var _ name .MMLIRType => \"!\" ++ name\n  | .Var _ name .MSSAVal => \"%\" ++ name\n  -- General notation\n  | .Var _ name s => \"name:\" ++ s.str\n  | .App ctor args => ctor.name ++ \" [\" ++ MTerm.strList args ++ \"]\"\n  -- Constants\n  | ConstMLIRType c\n  | ConstNat c\n  | ConstString c\n  | ConstDimension c\n  | ConstSignedness c =>\n      toString c\n\nprotected def MTerm.strList: List (MTerm δ) → String\n  | [] => \"\"\n  | [t] => str t\n  | t::ts => str t ++ \", \" ++ MTerm.strList ts\nend\ntermination_by\n  MTerm.str t  => sizeOf t\n  MTerm.strList ts => sizeOf ts\n\n\ninstance: ToString MSort where\n  toString := MSort.str\ninstance: ToString (MTerm δ) where\n  toString := MTerm.str\n\n-- Collect variables in a term\ndef MTerm.vars: MTerm δ → List String\n  | .Var _ name _ => [name]\n  | .App ctor [] => []\n  | .App ctor (arg::args) =>\n      vars arg ++ vars (.App ctor args)\n  | _ => []\n\n-- Collect variables and their sorts\ndef MTerm.varsWithSorts: MTerm δ → List (String × MSort)\n  | .Var _ name sort => [(name, sort)]\n  | .App ctor [] => []\n  | .App ctor (arg::args) =>\n      varsWithSorts arg ++ varsWithSorts (.App ctor args)\n  | _ => []\n\n-- Check whether a variable occurs in a term. We don't check typing here since\n-- we have a common pool of unique variable names.\ndef MTerm.occurs (name: String): MTerm δ→ Bool\n  | .Var _ name' _ => name' = name\n  | .App ctor [] => false\n  | .App ctor (arg::args) =>\n      occurs name arg || occurs name (.App ctor args)\n  | _ => false\n\n-- Substitute a variable in a term\nmutual\ndef MTerm.subst (t: MTerm δ) (name: String) (repl: MTerm δ): MTerm δ :=\n  match t with\n  | .Var _ name' _ => if name' = name then repl else t\n  | .App ctor args => .App ctor (MTerm.substList args name repl)\n  | t => t\n\nprotected def MTerm.substList (l: List (MTerm δ)) (name: String)\n                              (repl: MTerm δ) : List (MTerm δ) :=\n  match l with\n  | [] => []\n  | t::ts => subst t name repl :: MTerm.substList ts name repl\nend\ntermination_by\n  MTerm.subst t name repl  => sizeOf t\n  MTerm.substList ts name repl => sizeOf ts\n\n-- Substitue a set of variables in a term\ndef MTerm.substVars (t: MTerm δ) (repl: List (String × MTerm δ)): MTerm δ :=\n  repl.foldl (fun t (name, repl) => t.subst name repl) t\n\n/-\n### Sort inference\n\nIn order to ensure we only manipulate well typed match terms and equalities\ndespite mixing constructors, we aggressively check typing during matching and\nunification.\n-/\n\nmutual\nvariable {δ: Dialect α σ ε} -- We need this for termination somehow\n\ndef MTerm.inferSort: MTerm δ → Option MSort\n  | Var _ _ s => some s\n  | App (.LIST s) args => do\n      let l ← inferSortList args\n      if l.all (· = s) then some (.MList s) else none\n  | @App _ _ _ _ args_sort ctor_sort ctor args =>\n      if args.length != args_sort.length then\n        none\n      else if inferSortList args |>.isEqSome args_sort then\n        some ctor_sort\n      else\n        none\n  | ConstMLIRType _     => some .MMLIRType\n  | ConstNat _          => some .MNat\n  | ConstString _       => some .MString\n  | ConstDimension _    => some .MDimension\n  | ConstSignedness _   => some .MSignedness\n\ndef MTerm.inferSortList: List (MTerm δ) → Option (List MSort)\n  | [] => some []\n  | t::l => do return (← inferSort t) :: (← inferSortList l)\nend\ntermination_by\n  MTerm.inferSort t => sizeOf t\n  MTerm.inferSortList ts => sizeOf ts\n\n@[reducible]\ndef MSort.toType (δ: Dialect α σ ε): MSort -> Type\n| .MOp => Bool -- TODO MLIR.AST.BasicBlockStmt δ\n| .MOperand => MLIR.AST.SSAVal × MLIR.AST.MLIRType δ\n| .MMLIRType => MLIR.AST.MLIRType δ\n| .MSSAVal => MLIR.AST.SSAVal\n| .MAttrValue => Bool -- TODO MLIR.AST.AttrValue δ\n| .MNat => Nat\n| .MString => String\n| .MDimension => Dimension\n| .MSignedness => MLIR.AST.Signedness\n| .MList mTerm => List (mTerm.toType δ)\n\ndef MSort_toType_decEq {δ: Dialect α σ ε} (s: MSort)\n    : DecidableEq (s.toType δ) :=\n  match s with\n  | .MOp => decEq\n  | .MOperand => decEq\n  | .MMLIRType => decEq\n  | .MSSAVal => decEq\n  | .MAttrValue => decEq\n  | .MNat => decEq\n  | .MString => decEq\n  | .MDimension => decEq\n  | .MSignedness => decEq\n  | .MList term => @List.hasDecEq _ (MSort_toType_decEq term)\n\ninstance {δ: Dialect α σ ε} (s: MSort): DecidableEq (s.toType δ) :=\n  MSort_toType_decEq s\n\n/-\n### Variable context for MTerms\n\nThis structure contains an assignment from MTerm variables to concrete\nstructures. It is used both for matching, and for concretizing MTerms into\nconcrete strucctures.\n-/\n\n-- Matching context. Contains the assignment of matching variables.\nabbrev VarCtx (δ: Dialect α σ ε) :=\n  List ((s: MSort) × List (String × (s.toType δ)))\n\n-- Get the assignment of a variable.\ndef VarCtx.get (ctx: VarCtx δ) (s: MSort) (name: String) :\n    Option (s.toType δ) :=\n  match ctx with\n  | ⟨so, sortCtx⟩::ctx' =>\n    match H: so == s with\n    | false => get ctx' s name\n    | true => do\n      let res ← List.find? (·.fst == name) ((of_decide_eq_true H) ▸ sortCtx)\n      return res.snd\n  | [] => none\n\n-- Assign a variable.\ndef VarCtx.set (ctx: VarCtx δ) (s: MSort) (name: String) (value: s.toType δ):\n    VarCtx δ :=\n  match ctx with\n  | ⟨so, sortCtx⟩::ctx' =>\n    match H: so == s with\n    | false => ⟨so, sortCtx⟩::(set ctx' s name value)\n    | true => ⟨so, (name, (of_decide_eq_true H) ▸ value)::sortCtx⟩::ctx'\n  | [] => [{fst := s, snd := [(name, value)]}]\n\n/-\n### Concretization of MTerm\n\nThis section defines some functions to transform a MTerm into some\nconcrete structure, given a variable context.\n-/\n\n-- We provide an expected sort, since we do not want to carry the\n-- proof that terms are well typed.\ndef MTerm.concretizeVariable (m: MTerm δ) (s: MSort) (ctx: VarCtx δ) :\n    Option (s.toType δ) :=\n  match m with\n  | Var _ name sort =>\n    if s == sort then ctx.get s name else none\n  | _ => none\n\ndef MTerm.concretizeSign (m: MTerm δ) (ctx: VarCtx δ) : Option Signedness :=\n  match m with\n  | Var _ _ _ => m.concretizeVariable .MSignedness ctx\n  | ConstSignedness s => some s\n  | _ => none\n\ndef MTerm.concretizeNat (m: MTerm δ) (ctx: VarCtx δ) : Option Nat :=\n  match m with\n  | Var _ _ _ => m.concretizeVariable .MNat ctx\n  | ConstNat n => some n\n  | _ => none\n\ndef MTerm.concretizeDim (m: MTerm δ) (ctx: VarCtx δ) : Option Dimension :=\n  match m with\n  | Var _ _ _ => m.concretizeVariable .MDimension ctx\n  | ConstDimension d => some d\n  | _ => none\n\ndef MTerm.concretizeType (m: MTerm δ) (ctx: VarCtx δ) :\n    Option (MLIR.AST.MLIRType δ) :=\n  match m with\n  | Var _ _ _ => m.concretizeVariable .MMLIRType ctx\n  | .App .INT [mSign, mNat] => do\n    let sign ← mSign.concretizeSign ctx\n    let nat ← mNat.concretizeNat ctx\n    return MLIRType.int sign nat\n  | _ => none\n\ndef MTerm.concretizeOperand (m: MTerm δ) (ctx: VarCtx δ) :\n    Option (MLIR.AST.SSAVal × MLIR.AST.MLIRType δ) :=\n  match m with\n  | Var _ _ _ => m.concretizeVariable .MOperand ctx\n  | .App .OPERAND [mVal, mType] => do\n    let val ← mVal.concretizeVariable .MSSAVal ctx\n    let type ← mType.concretizeType ctx\n    return (val, type)\n  | _ => none\n\ndef MTerm.concretizeOperands (m: MTerm δ) (ctx: VarCtx δ) :\n    Option (List (MLIR.AST.TypedSSAVal δ)) :=\n  match m with\n  | .App (.LIST .MOperand) l => l.mapM (fun m' => m'.concretizeOperand ctx)\n  | _ => none\n\ndef MTerm.concretizeOp (m: MTerm δ) (ctx: VarCtx δ) : Option (Op δ) :=\n  match m with\n  | .App .OP [ .ConstString mName, mOperands, mRes ] => do\n    let operands ← MTerm.concretizeOperands mOperands ctx\n    let res ← MTerm.concretizeOperands mRes ctx\n    match res with\n    | [resVal] =>\n      return .mk mName [resVal] operands [] (AttrDict.mk [])\n    | _ => none\n  | _ => none\n\ndef MTerm.concretizeProg (m: List (MTerm δ)) (ctx: VarCtx δ) :\n    Option (List (Op δ)) :=\n  m.mapM (fun m' => m'.concretizeOp ctx)\n\n/-\n### Simple MTerm matching\n\nThis section defines functions to match an MTerm with a concrete structure.\nNote that here, the matching does not match recursively inside the concrete\nstructure.\n-/\n\n-- Match a MTerm variable.\ndef matchVariable {δ: Dialect α σ ε} (s: MSort) (name: String)\n                  (val: s.toType δ) (ctx: VarCtx δ) : Option (VarCtx δ) :=\n  match ctx.get s name with\n  | some matchedVal => if val == matchedVal then some ctx else none\n  | none => some (ctx.set s name val)\n\n-- Match a signedness with a MTerm.\ndef matchMSignedness {δ: Dialect α σ ε} (mSgn: MTerm δ) (sgn: Signedness)\n                     (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match mSgn with\n  | .Var _ name .MSignedness => matchVariable .MSignedness name sgn ctx\n  | .ConstSignedness mSgn => if sgn == mSgn then some ctx else none\n  | _ => none\n\n-- Match a dimension with a MTerm.\ndef matchMDimension {δ: Dialect α σ ε} (mDim: MTerm δ) (dim: Dimension)\n                    (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match mDim with\n  | .Var _ name .MDimension => matchVariable .MDimension name dim ctx\n  | .ConstDimension mDim => if dim == mDim then some ctx else none\n  | _ => none\n\n-- Match a string with a MTerm.\ndef matchMString {δ: Dialect α σ ε} (mStr: MTerm δ) (str: String)\n                 (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match mStr with\n  | .Var _ name .MString => matchVariable .MString name str ctx\n  | .ConstString mStr => if str == mStr then some ctx else none\n  | _ => none\n\n-- Match a nat with a MTerm.\ndef matchMNat {δ: Dialect α σ ε} (mNat: MTerm δ) (nat: Nat)\n              (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match mNat with\n  | .Var _ name .MNat => matchVariable .MNat name nat ctx\n  | .ConstNat mNat => if nat == mNat then some ctx else none\n  | _ => none\n\n-- Match a type with a MTerm.\ndef matchMType (mType: MTerm δ) (type: MLIRType δ)\n               (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match mType, type with\n  | .Var _ name .MMLIRType, _ => matchVariable .MMLIRType name type ctx\n  | .ConstMLIRType mType, _ => if type == mType then some ctx else none\n  | .App .INT [mSgn, mNat], MLIRType.int sgn nat =>\n    (matchMSignedness mSgn sgn ctx).bind (matchMNat mNat nat ·)\n  | _, _ => none\n\n-- Match a type SSA value with a MTerm.\ndef matchMSSAVal (mOperand: MTerm δ) (operand: TypedSSAVal δ)\n                  (ctx: VarCtx δ) : Option (VarCtx δ) :=\n  match mOperand with\n  | .App .OPERAND [.Var _ ssaName .MSSAVal, mType] => do\n    let ctx' ← matchMType mType operand.snd ctx\n    matchVariable MSort.MSSAVal ssaName operand.fst ctx'\n  | _ => none\n\n-- Match a list of typed SSA values with a list of MTerm.\ndef matchMSSAVals (operands: List (TypedSSAVal δ)) (mOperands: List (MTerm δ))\n    (ctx: VarCtx δ): Option (VarCtx δ) :=\n  match operands, mOperands with\n  | [], [] => some ctx\n  | operand::operands, mOperand::mOperands => do\n    let ctx' ← matchMSSAVal mOperand operand ctx\n    matchMSSAVals operands mOperands ctx'\n  | _, _ => none\n\n-- Match a basic block statement with an MTerm.\ndef matchMOp (op: Op δ) (mterm: MTerm δ) (ctx: VarCtx δ) : Option (VarCtx δ) :=\n  match op, mterm with\n  | Op.mk name res operands [] (AttrDict.mk []),\n    .App .OP [ .ConstString mName, .App (.LIST .MOperand) mOperands,\n      .App (.LIST .MOperand) mRes ] =>\n    if name != mName then\n      none\n    else\n      (matchMSSAVals operands mOperands ctx).bind\n      (matchMSSAVals res mRes ·)\n  | _, _ => none\n\n/-\n### Recursive MTerm op matching\n\nThis section defines functions to match an op MTerm inside a concrete\nstructure. Here, the matching is done recursively inside the regions/BBs/Ops.\n\nWe first define functions that match all possible ops in the IR. Then, we use\nthis to match a program in an IR.\n-/\n\nmutual\n-- Get all possible operations matching an MTerm in an op.\ndef matchAllMOpInOp (op: Op δ) (mOp: MTerm δ) (ctx: VarCtx δ)\n    : List (Op δ × VarCtx δ) :=\n  match op with\n  | .mk _ _ _ regions _ =>\n    let nextMatches := matchAllMOpInRegions regions mOp ctx\n    match matchMOp op mOp ctx with\n    | some ctx' => (op, ctx')::nextMatches\n    | none => nextMatches\n\n-- Get all possible operations matching an MTerm in a list of ops.\ndef matchAllMOpInOps (ops: List (Op δ)) (mOp: MTerm δ)\n                     (ctx: VarCtx δ) : List (Op δ × VarCtx δ) :=\n  match ops with\n  | op::ops' => (matchAllMOpInOp op mOp ctx).append\n    (matchAllMOpInOps ops' mOp ctx)\n  | [] => []\n\n-- Get all possible operations matching an MTerm in a basic block.\ndef matchAllMOpInRegion (rgn: Region δ) (mOp: MTerm δ)\n                    (ctx: VarCtx δ) : List (Op δ × VarCtx δ) :=\n\n  match rgn with\n  | .mk _ _ ops => matchAllMOpInOps ops mOp ctx\n\n-- Get all possible operations matching an MTerm in a list of regions.\ndef matchAllMOpInRegions (regions: List (Region δ)) (mOp: MTerm δ)\n                         (ctx: VarCtx δ) :\n    List (Op δ × VarCtx δ) :=\n  match regions with\n  | region::regions' => (matchAllMOpInRegion region mOp ctx).append\n    (matchAllMOpInRegions regions' mOp ctx)\n  | [] => []\nend\ntermination_by\n  matchAllMOpInOp op _ _ => sizeOf op\n  matchAllMOpInOps ops _ _ => sizeOf ops\n  matchAllMOpInRegion region _ _ => sizeOf region\n  matchAllMOpInRegions regions _ _ => sizeOf regions\n\nmutual\nvariable {δ: Dialect α σ ε} -- We need this for termination somehow\n\n#check Preorder\n-- Match a program defined by a list of MTerm (one Operation per MTerm) in\n-- an operation.\ndef matchMProgInOp (op: Op δ) (mOps: List (MTerm δ)) (ctx: VarCtx δ) :\n    Option ((List (Op δ)) × VarCtx δ) :=\n  match mOps with\n  -- Try all matches of the first operation.\n  | mOp::mOps' =>\n    matchMProgInOpAux op mOps' (matchAllMOpInOp op mOp ctx)\n  | [] => some ([], ctx)\n\n-- Match a program defined by a list of MTerm (one Operation per MTerm) in\n-- an operation. `matchOps` correspond to the possible matches of the current\n-- MTerm being matched.\ndef matchMProgInOpAux (op: Op δ) (mOps: List (MTerm δ))\n                      (matchOps: List (Op δ × VarCtx δ))\n                      : Option (List (Op δ) × VarCtx δ) :=\n  -- For all possible match, we check if we can match the remaining of the\n  -- program with the match assignment\n  match matchOps with\n  | (matchOp, ctx')::matchOps' =>\n    match matchMProgInOp op mOps ctx' with\n    -- If we do match the remaining of the program, we are finished.\n    | some (matchedOps, ctx'') => some (matchOp::matchedOps, ctx'')\n    -- Otherwise, we check the next match for the current operation.\n    | none => matchMProgInOpAux op mOps matchOps'\n  | [] => none\nend\n-- TODO: how to use lex ordering for termination? Proof of termination: lex ordering on (mOps, matchOps)\ndecreasing_by sorry\n/-\ntermination_by\n  matchMProgInOpAux _ mOps matchOps => sizeOf (mOps, matchOps)\n  matchMProgInOp _ mOps ctx  =>\n    sizeOf (mOps, let t : List (Op δ × VarCtx δ) := []; t)\n-/\n\n-- variable type\ndef MTerm.buildTypeVar (name: String) : MTerm δ := .Var 2 name .MMLIRType\n-- constant type\n\ndef MTerm.buildTypeConst (type: MLIRType δ) : MTerm δ := .ConstMLIRType type\n\n-- operand. For now, assume monomorhpic types.\ndef MTerm.buildOperand (name: String) (type: MLIRType δ): MTerm δ :=\n  .App .OPERAND [ .Var (priority := 2) name .MSSAVal,\n                  MTerm.buildTypeConst type ]\n\ndef MTerm.buildOp (name: String) (args: List (MTerm δ)) (res: List (MTerm δ)) :\n    MTerm δ :=\n  .App .OP [ .ConstString name, .App (.LIST .MOperand) args,\n    .App (.LIST .MOperand) res ]\n\nprivate def test_addi_multiple_pattern: List (MTerm δ) :=\n  [.App .OP [\n    .ConstString \"std.addi\",\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_x\" .MSSAVal, .Var 2 \"T\" .MMLIRType],\n      .App .OPERAND [.Var 2 \"op_y\" .MSSAVal, .Var 2 \"T\" .MMLIRType]],\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_res\" .MSSAVal, .Var 2 \"T\" .MMLIRType]]\n  ],\n  .App .OP [\n    .ConstString \"std.addi\",\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_res\" .MSSAVal, .Var 2 \"T\" .MMLIRType],\n      .App .OPERAND [.Var 2 \"op_res\" .MSSAVal, .Var 2 \"T\" .MMLIRType]],\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_res2\" .MSSAVal, .Var 2 \"T\" .MMLIRType]]\n  ]]\n\nprivate def multiple_example: Op builtin := [mlir_op|\n  \"builtin.module\"() ({\n    ^entry:\n    %r2 = \"std.addi\"(%t2, %t3): (i32, i32) -> i32\n    %r = \"std.addi\"(%t0, %t1): (i32, i32) -> i32\n    %r3 = \"std.addi\"(%r, %r): (i32, i32) -> i32\n  }) : () -> ()\n]\n\nprivate def test_addi_one: MTerm δ :=\n  (.App .OP [\n    .ConstString \"std.addi\",\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_x\" .MSSAVal, .Var 2 \"T\" .MMLIRType],\n      .App .OPERAND [.Var 2 \"op_y\" .MSSAVal, .Var 2 \"T\" .MMLIRType]],\n    .App (.LIST .MOperand) [\n      .App .OPERAND [.Var 2 \"op_res\" .MSSAVal, .Var 2 \"T\" .MMLIRType]]\n  ])\n\nprivate def one_example: Op builtin := [mlir_op|\n    %r2 = \"std.addi\"(%t2, %t3): (i32, i32) -> i32\n]\n\n-- Match an MTerm program in some IR, then concretize\n-- the MTerm using the resulting matching context.\ndef multiple_example_result : Option (List (Op builtin)) := do\n  let (_, ctx) ←\n    matchMProgInOp multiple_example test_addi_multiple_pattern []\n  let res ← MTerm.concretizeProg test_addi_multiple_pattern ctx\n  res\n\n#eval multiple_example_result\n\n\n/-\n### Exact program matching\nThis section defines functions to check if an operation, or SSA values\ndefinitions/uses are inside a bigger program.\n\nThe operation to match should not have any regions or attributes.\n-/\n\ndef eqOp (op1 op2: Op δ) : Bool :=\n  match op1, op2 with\n  | .mk name res args [] (.mk []), .mk name' res' args' [] (.mk []) =>\n    name == name' && res == res' && args == args'\n  | _, _ => false\n\nmutual\nvariable (mOp: Op δ)\n\ndef isOpInOp (op: Op δ) : Bool :=\n  eqOp op mOp ||\n    (match op with\n     | .mk _ _ _ regions _ => isOpInRegions regions)\n\ndef isOpInRegions (regions: List (Region δ)) : Bool :=\n  match regions with\n  | [] => False\n  | region::regions' => isOpInRegion region || isOpInRegions regions'\n\ndef isOpInRegion (rgn: Region δ) : Bool :=\n  match rgn with\n  | .mk _ _ ops => isOpInOps ops\n\ndef isOpInOps (ops: List (Op δ)) : Bool :=\n  match ops with\n  | [] => False\n  | op::ops' => isOpInOp op || isOpInOps ops'\nend\n termination_by\n  isOpInOp op => sizeOf op\n  isOpInRegions rgns => sizeOf rgns\n  isOpInRegion rgn => sizeOf rgn\n  isOpInOps ops => sizeOf ops\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Semantics/Matching.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.3204592644000566}}
{"text": "theorem ex6 (f : Nat → Nat) (x y z : Nat) (h : (x, z).1 = (fun x => x) y) : f x = f y := by\n  simp (config := { beta := false }) at h\n  trace_state\n  simp at h\n  trace_state\n  simp [h]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/simpcfg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.32043083759539404}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.analysis.normed_space.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l \n\nnamespace Mathlib\n\n/-!\n# Extended norm\n\nIn this file we define a structure `enorm 𝕜 V` representing an extended norm (i.e., a norm that can\ntake the value `∞`) on a vector space `V` over a normed field `𝕜`. We do not use `class` for\nan `enorm` because the same space can have more than one extended norm. For example, the space of\nmeasurable functions `f : α → ℝ` has a family of `L_p` extended norms.\n\nWe prove some basic inequalities, then define\n\n* `emetric_space` structure on `V` corresponding to `e : enorm 𝕜 V`;\n* the subspace of vectors with finite norm, called `e.finite_subspace`;\n* a `normed_space` structure on this space.\n\nThe last definition is an instance because the type involves `e`.\n\n## Implementation notes\n\nWe do not define extended normed groups. They can be added to the chain once someone will need them.\n\n## Tags\n\nnormed space, extended norm\n-/\n\n/-- Extended norm on a vector space. As in the case of normed spaces, we require only\n`∥c • x∥ ≤ ∥c∥ * ∥x∥` in the definition, then prove an equality in `map_smul`. -/\nstructure enorm (𝕜 : Type u_1) (V : Type u_2) [normed_field 𝕜] [add_comm_group V] [vector_space 𝕜 V]\n    where\n  to_fun : V → ennreal\n  eq_zero' : ∀ (x : V), to_fun x = 0 → x = 0\n  map_add_le' : ∀ (x y : V), to_fun (x + y) ≤ to_fun x + to_fun y\n  map_smul_le' : ∀ (c : 𝕜) (x : V), to_fun (c • x) ≤ ↑(nnnorm c) * to_fun x\n\nnamespace enorm\n\n\nprotected instance has_coe_to_fun {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] : has_coe_to_fun (enorm 𝕜 V) :=\n  has_coe_to_fun.mk (fun (x : enorm 𝕜 V) => V → ennreal) to_fun\n\ntheorem injective_coe_fn {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] : function.injective fun (e : enorm 𝕜 V) (x : V) => coe_fn e x :=\n  sorry\n\ntheorem ext {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V] [vector_space 𝕜 V]\n    {e₁ : enorm 𝕜 V} {e₂ : enorm 𝕜 V} (h : ∀ (x : V), coe_fn e₁ x = coe_fn e₂ x) : e₁ = e₂ :=\n  injective_coe_fn (funext h)\n\ntheorem ext_iff {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V] [vector_space 𝕜 V]\n    {e₁ : enorm 𝕜 V} {e₂ : enorm 𝕜 V} : e₁ = e₂ ↔ ∀ (x : V), coe_fn e₁ x = coe_fn e₂ x :=\n  { mp := fun (h : e₁ = e₂) (x : V) => h ▸ rfl, mpr := ext }\n\n@[simp] theorem coe_inj {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] {e₁ : enorm 𝕜 V} {e₂ : enorm 𝕜 V} : ⇑e₁ = ⇑e₂ ↔ e₁ = e₂ :=\n  function.injective.eq_iff injective_coe_fn\n\n@[simp] theorem map_smul {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (c : 𝕜) (x : V) :\n    coe_fn e (c • x) = ↑(nnnorm c) * coe_fn e x :=\n  sorry\n\n@[simp] theorem map_zero {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) : coe_fn e 0 = 0 :=\n  sorry\n\n@[simp] theorem eq_zero_iff {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) {x : V} : coe_fn e x = 0 ↔ x = 0 :=\n  { mp := eq_zero' e x, mpr := fun (h : x = 0) => Eq.symm h ▸ map_zero e }\n\n@[simp] theorem map_neg {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : V) : coe_fn e (-x) = coe_fn e x :=\n  sorry\n\ntheorem map_sub_rev {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : V) (y : V) : coe_fn e (x - y) = coe_fn e (y - x) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn e (x - y) = coe_fn e (y - x))) (Eq.symm (neg_sub y x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn e (-(y - x)) = coe_fn e (y - x))) (map_neg e (y - x))))\n      (Eq.refl (coe_fn e (y - x))))\n\ntheorem map_add_le {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : V) (y : V) :\n    coe_fn e (x + y) ≤ coe_fn e x + coe_fn e y :=\n  map_add_le' e x y\n\ntheorem map_sub_le {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : V) (y : V) :\n    coe_fn e (x - y) ≤ coe_fn e x + coe_fn e y :=\n  sorry\n\nprotected instance partial_order {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] : partial_order (enorm 𝕜 V) :=\n  partial_order.mk (fun (e₁ e₂ : enorm 𝕜 V) => ∀ (x : V), coe_fn e₁ x ≤ coe_fn e₂ x)\n    (preorder.lt._default fun (e₁ e₂ : enorm 𝕜 V) => ∀ (x : V), coe_fn e₁ x ≤ coe_fn e₂ x) sorry\n    sorry sorry\n\n/-- The `enorm` sending each non-zero vector to infinity. -/\nprotected instance has_top {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] : has_top (enorm 𝕜 V) :=\n  has_top.mk (mk (fun (x : V) => ite (x = 0) 0 ⊤) sorry sorry sorry)\n\nprotected instance inhabited {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] : Inhabited (enorm 𝕜 V) :=\n  { default := ⊤ }\n\ntheorem top_map {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V] [vector_space 𝕜 V]\n    {x : V} (hx : x ≠ 0) : coe_fn ⊤ x = ⊤ :=\n  if_neg hx\n\nprotected instance semilattice_sup_top {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜]\n    [add_comm_group V] [vector_space 𝕜 V] : semilattice_sup_top (enorm 𝕜 V) :=\n  semilattice_sup_top.mk ⊤ LessEq Less sorry sorry sorry sorry\n    (fun (e₁ e₂ : enorm 𝕜 V) =>\n      mk (fun (x : V) => max (coe_fn e₁ x) (coe_fn e₂ x)) sorry sorry sorry)\n    sorry sorry sorry\n\n@[simp] theorem coe_max {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e₁ : enorm 𝕜 V) (e₂ : enorm 𝕜 V) :\n    ⇑(e₁ ⊔ e₂) = fun (x : V) => max (coe_fn e₁ x) (coe_fn e₂ x) :=\n  rfl\n\ntheorem max_map {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V] [vector_space 𝕜 V]\n    (e₁ : enorm 𝕜 V) (e₂ : enorm 𝕜 V) (x : V) :\n    coe_fn (e₁ ⊔ e₂) x = max (coe_fn e₁ x) (coe_fn e₂ x) :=\n  rfl\n\n/-- Structure of an `emetric_space` defined by an extended norm. -/\ndef emetric_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) : emetric_space V :=\n  emetric_space.mk sorry sorry (map_sub_rev e) sorry\n    (uniform_space_of_edist (fun (x y : V) => coe_fn e (x - y)) sorry (map_sub_rev e) sorry)\n\n/-- The subspace of vectors with finite enorm. -/\ndef finite_subspace {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) : subspace 𝕜 V :=\n  submodule.mk (set_of fun (x : V) => coe_fn e x < ⊤) sorry sorry sorry\n\n/-- Metric space structure on `e.finite_subspace`. We use `emetric_space.to_metric_space_of_dist`\nto ensure that this definition agrees with `e.emetric_space`. -/\nprotected instance finite_subspace.metric_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜]\n    [add_comm_group V] [vector_space 𝕜 V] (e : enorm 𝕜 V) : metric_space ↥(finite_subspace e) :=\n  let _inst : emetric_space V := emetric_space e;\n  emetric_space.to_metric_space_of_dist\n    (fun (x y : ↥(finite_subspace e)) => ennreal.to_real (edist x y)) sorry sorry\n\ntheorem finite_dist_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : ↥(finite_subspace e)) (y : ↥(finite_subspace e)) :\n    dist x y = ennreal.to_real (coe_fn e (↑x - ↑y)) :=\n  rfl\n\ntheorem finite_edist_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : ↥(finite_subspace e)) (y : ↥(finite_subspace e)) :\n    edist x y = coe_fn e (↑x - ↑y) :=\n  rfl\n\n/-- Normed group instance on `e.finite_subspace`. -/\nprotected instance finite_subspace.normed_group {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜]\n    [add_comm_group V] [vector_space 𝕜 V] (e : enorm 𝕜 V) : normed_group ↥(finite_subspace e) :=\n  normed_group.mk sorry\n\ntheorem finite_norm_eq {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜] [add_comm_group V]\n    [vector_space 𝕜 V] (e : enorm 𝕜 V) (x : ↥(finite_subspace e)) :\n    norm x = ennreal.to_real (coe_fn e ↑x) :=\n  rfl\n\n/-- Normed space instance on `e.finite_subspace`. -/\nprotected instance finite_subspace.normed_space {𝕜 : Type u_1} {V : Type u_2} [normed_field 𝕜]\n    [add_comm_group V] [vector_space 𝕜 V] (e : enorm 𝕜 V) : normed_space 𝕜 ↥(finite_subspace e) :=\n  normed_space.mk sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/normed_space/enorm_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3204308301020879}}
{"text": "import tactic\nimport data.list.func\n\nuniverse u\n\ndef list2d (α : Type u) := list (list α)\n\nnamespace list2d\n\nopen list.func\n\nvariables {α : Type} {β : Type}\nvariables [inhabited α] [inhabited β]\n\n@[simp] def get2d (xy : ℕ × ℕ) (l : list2d α) : α\n  := let (x,y) := xy in get x (get y l)\n@[simp] def set2d (a : α) (l : list2d α) (xy : ℕ × ℕ) : list2d α\n  := let (x,y) := xy in set (set a (get y l) x) l y\n@[simp] def dfzip2d (l1 : list2d α) (l2 : list2d β) : list2d (α × β)\n  := (pointwise (pointwise prod.mk) l1 l2)\n@[simp] def add_to_line1 (a : α) : list2d α -> list2d α\n| [] := [[a]]\n| (h::t) := (a::h)::t\n\n@[simp] def transpose : list2d α → list2d α := (list.foldr (pointwise list.cons)) []\n\ninstance [has_repr α] : has_repr (list2d α) := ⟨λ l, list.repr (l : list (list α))⟩\n\n#eval transpose [[1, 2], [3, 4, 5], [6]]\n\nend list2d\n\ninductive direction : Type\n| up\n| down\n| left\n| right\n\nnamespace direction\n\ndef opposite : direction -> direction\n| up := down\n| down := up\n| left := right\n| right := left\n\ndef shift : direction -> (ℕ × ℕ) -> option (ℕ × ℕ)\n| up (_, 0)     := none\n| up (x, y+1)   := some (x, y)\n| down (x, y)   := some (x, y+1)\n| left (0, _)   := none\n| left (x+1, y) := some (x, y)\n| right (x, y)   := some (x+1, y)\n\nend direction\n\nstructure sokoban :=\n(available : list2d bool)\n(boxes : list2d bool)\n(storages : list2d bool)\n(storekeeper : ℕ × ℕ)\n\nnamespace sokoban\nopen list2d\n\ninstance : inhabited sokoban\n:= ⟨{ available := [[tt]], boxes := [], storages := [], storekeeper := (0,0) }⟩\n\ndef move (d : direction) (s : sokoban) : sokoban :=\n  match (d.shift s.storekeeper) with none := s\n  | some sk2 :=\n  if get2d sk2 s.available = false then s else\n  if get2d sk2 s.boxes = false then\n    { storekeeper := sk2, ..s}\n  else\n    match (d.shift sk2) with none := s\n    | some box2 :=\n    if (get2d box2 s.available = false)\n      ∨ (get2d box2 s.boxes = true) then s else\n    {\n      boxes := (s.boxes.set2d ff sk2).set2d tt box2,\n      storekeeper := sk2, ..s\n    }\n    end\n  end\n\ndef move_up := move direction.up\ndef move_down := move direction.down\ndef move_left := move direction.left\ndef move_right := move direction.right\n\ndef add_newline (s : sokoban) : sokoban := {\n  available := []::s.available,\n  boxes := []::s.boxes,\n  storages := []::s.storages,\n  storekeeper := match s.storekeeper with (x,y) := (x,y+1) end\n}\ndef add_newsquare (av box stor sk : bool) (s : sokoban) : sokoban := {\n  available := add_to_line1 av s.available,\n  boxes := add_to_line1 box s.boxes,\n  storages := add_to_line1 stor s.storages,\n  storekeeper := if sk then (0, 0) else match s.storekeeper with\n    | (x,0) := (x+1,0)\n    | xy := xy\n  end\n}\n\ndef from_string_aux : list char → option (sokoban × bool)\n| [] := some (sokoban.mk [] [] [] (0,0), ff)\n| (c::str) := match (from_string_aux str), c with\n  | none, _ := none\n  | (some (s, sk_set)), '\\n' := some (add_newline s, sk_set)\n  | (some (s, sk_set)), ' ' := some (add_newsquare tt ff ff ff s, sk_set)\n  | (some (s, sk_set)), '#' := some (add_newsquare ff ff ff ff s, sk_set)\n  | (some (s, sk_set)), '.' := some (add_newsquare tt ff tt ff s, sk_set)\n  | (some (s, sk_set)), '$' := some (add_newsquare tt tt ff ff s, sk_set)\n  | (some (s, sk_set)), '*' := some (add_newsquare tt tt tt ff s, sk_set)\n  | (some (s, ff)), '@' := some (add_newsquare tt ff ff tt s, tt)\n  | (some (s, ff)), '+' := some (add_newsquare tt ff tt tt s, tt)\n  | (some _), _ := none\n  end\n\ndef from_string (str : string) : sokoban :=\n  match (from_string_aux str.to_list) with\n  | none := default sokoban\n  | some (_, ff) := default sokoban\n  | some (level, tt) := level\n  end\n\ndef square_to_char : bool → bool → bool → bool → char\n| tt ff ff ff := ' '\n| ff ff ff ff := '#'\n| tt ff tt ff := '.'\n| tt tt ff ff := '$'\n| tt tt tt ff := '*'\n| tt ff ff tt := '@'\n| tt ff tt tt := '+'\n| _ _ _ _ := '?'\n\ndef to_string_aux1 (str : list char) : list (bool × bool × bool × bool) → list char\n| [] := str\n| ((av,box,stor,sk)::t) := (square_to_char av box stor sk)::(to_string_aux1 t)\n\ndef to_string_aux2 : list2d (bool × bool × bool × bool) → list char\n| [] := []\n| (h::t) := to_string_aux1 ('\\n'::(to_string_aux2 t)) h\n\ninstance : has_to_string sokoban := ⟨λ s,\n  list.as_string (to_string_aux2\n  (dfzip2d s.available (dfzip2d s.boxes (dfzip2d s.storages (set2d true [] s.storekeeper)))))\n⟩\n\ninstance : has_repr sokoban\n:= ⟨λ s, (string.append (string.append \"sokoban.from_string \\\"\" (to_string s)) \"\\\"\")⟩\n\ninductive solvable : sokoban -> Prop\n| solved (s : sokoban) (H : s.boxes = s.storages) : solvable s\n| move (d : direction) (s : sokoban) (H : solvable (s.move d)) : solvable s\n\ndef solvable.move_up := solvable.move direction.up\ndef solvable.move_down := solvable.move direction.down\ndef solvable.move_left := solvable.move direction.left\ndef solvable.move_right := solvable.move direction.right\n\nmeta def soko_simp_root (e : expr) : tactic unit :=\ndo\n  soko ← tactic.eval_expr sokoban e,\n  tactic.trace (to_string soko),\n  let p : pexpr := ``(%%e = sokoban.mk\n    %%soko.available %%soko.boxes %%soko.storages %%soko.storekeeper),\n  eq ← tactic.to_expr p,\n  name ← tactic.get_unused_name,\n  H ← tactic.assert name eq,\n  tactic.reflexivity,\n  tactic.rewrite_target H,\n  tactic.clear H\n\nmeta def soko_simp_expr : expr → tactic unit := assume e : expr,\nsoko_simp_root e <|> match e with\n| expr.app e1 e2 := soko_simp_expr e1 >> soko_simp_expr e2\n| _ := return ()\nend\n\nmeta def soko_simp : tactic unit :=\ndo\n  goal ← tactic.target,\n  soko_simp_expr goal\n\n#check tactic.apply\n\nmeta def solve_up : tactic unit := tactic.apply `(solvable.move_up) >> soko_simp\nmeta def solve_down : tactic unit := tactic.apply `(solvable.move_down) >> soko_simp\nmeta def solve_left : tactic unit := tactic.apply `(solvable.move_left) >> soko_simp\nmeta def solve_right : tactic unit := tactic.apply `(solvable.move_right) >> soko_simp\nmeta def solve_finish : tactic unit := tactic.apply `(sokoban.solvable.solved) >> tactic.reflexivity\n\nend sokoban\n\ndef soko_level := sokoban.from_string \"\n#######\n#. . .#\n# $$$ #\n#.$@$.#\n# $$$ #\n#. . .#\n#######\n\"\n\nexample : soko_level.solvable :=\nbegin\n  sokoban.soko_simp,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_right,\n  sokoban.solve_right,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_left,\n  sokoban.solve_right,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_down,\n  sokoban.solve_left,\n  sokoban.solve_left,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_right,\n  sokoban.solve_down,\n  sokoban.solve_right,\n  sokoban.solve_right,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_down,\n  sokoban.solve_right,\n  sokoban.solve_right,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_left,\n  sokoban.solve_down,\n  sokoban.solve_up,\n  sokoban.solve_up,\n  sokoban.solve_finish\nend\n", "meta": {"author": "mirefek", "repo": "my-lean-experiments", "sha": "1218fecbf568669ac123256d430a151a900f67b3", "save_path": "github-repos/lean/mirefek-my-lean-experiments", "path": "github-repos/lean/mirefek-my-lean-experiments/my-lean-experiments-1218fecbf568669ac123256d430a151a900f67b3/sokoban.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.32037698218554034}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nInstances of `traversable` for types from the core library\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.forall2\nimport Mathlib.data.set.lattice\nimport Mathlib.control.traversable.lemmas\nimport Mathlib.PostPort\n\nuniverses u_1 u \n\nnamespace Mathlib\n\ntheorem option.id_traverse {α : Type u_1} (x : Option α) : option.traverse id.mk x = x :=\n  option.cases_on x (Eq.refl (option.traverse id.mk none)) fun (x : α) => Eq.refl (option.traverse id.mk (some x))\n\ntheorem option.comp_traverse {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {α : Type u_1} {β : Type u} {γ : Type u} (f : β → F γ) (g : α → G β) (x : Option α) : option.traverse (functor.comp.mk ∘ Functor.map f ∘ g) x = functor.comp.mk (option.traverse f <$> option.traverse g x) := sorry\n\ntheorem option.traverse_eq_map_id {α : Type u_1} {β : Type u_1} (f : α → β) (x : Option α) : traverse (id.mk ∘ f) x = id.mk (f <$> x) :=\n  option.cases_on x (Eq.refl (traverse (id.mk ∘ f) none)) fun (x : α) => Eq.refl (traverse (id.mk ∘ f) (some x))\n\ntheorem option.naturality {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] (η : applicative_transformation F G) {α : Type u_1} {β : Type u} (f : α → F β) (x : Option α) : coe_fn η (Option β) (option.traverse f x) = option.traverse (coe_fn η β ∘ f) x := sorry\n\nprotected instance option.is_lawful_traversable : is_lawful_traversable Option :=\n  is_lawful_traversable.mk option.id_traverse option.comp_traverse option.traverse_eq_map_id option.naturality\n\nnamespace list\n\n\nprotected theorem id_traverse {α : Type u_1} (xs : List α) : list.traverse id.mk xs = xs := sorry\n\nprotected theorem comp_traverse {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {α : Type u_1} {β : Type u} {γ : Type u} (f : β → F γ) (g : α → G β) (x : List α) : list.traverse (functor.comp.mk ∘ Functor.map f ∘ g) x = functor.comp.mk (list.traverse f <$> list.traverse g x) := sorry\n\nprotected theorem traverse_eq_map_id {α : Type u_1} {β : Type u_1} (f : α → β) (x : List α) : list.traverse (id.mk ∘ f) x = id.mk (f <$> x) := sorry\n\nprotected theorem naturality {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] (η : applicative_transformation F G) {α : Type u_1} {β : Type u} (f : α → F β) (x : List α) : coe_fn η (List β) (list.traverse f x) = list.traverse (coe_fn η β ∘ f) x := sorry\n\nprotected instance is_lawful_traversable : is_lawful_traversable List :=\n  is_lawful_traversable.mk list.id_traverse list.comp_traverse list.traverse_eq_map_id list.naturality\n\n@[simp] theorem traverse_nil {F : Type u → Type u} [Applicative F] {α' : Type u} {β' : Type u} (f : α' → F β') : traverse f [] = pure [] :=\n  rfl\n\n@[simp] theorem traverse_cons {F : Type u → Type u} [Applicative F] {α' : Type u} {β' : Type u} (f : α' → F β') (a : α') (l : List α') : traverse f (a :: l) = (fun (_x : β') (_y : List β') => _x :: _y) <$> f a <*> traverse f l :=\n  rfl\n\n@[simp] theorem traverse_append {F : Type u → Type u} [Applicative F] {α' : Type u} {β' : Type u} (f : α' → F β') [is_lawful_applicative F] (as : List α') (bs : List α') : traverse f (as ++ bs) = append <$> traverse f as <*> traverse f bs := sorry\n\ntheorem mem_traverse {α' : Type u} {β' : Type u} {f : α' → set β'} (l : List α') (n : List β') : n ∈ traverse f l ↔ forall₂ (fun (b : β') (a : α') => b ∈ f a) n l := sorry\n\nend list\n\n\nnamespace sum\n\n\nprotected theorem traverse_map {σ : Type u} {G : Type u → Type u} [Applicative G] {α : Type u} {β : Type u} {γ : Type u} (g : α → β) (f : β → G γ) (x : σ ⊕ α) : sum.traverse f (g <$> x) = sum.traverse (f ∘ g) x := sorry\n\nprotected theorem id_traverse {σ : Type u_1} {α : Type u_1} (x : σ ⊕ α) : sum.traverse id.mk x = x :=\n  sum.cases_on x (fun (x : σ) => Eq.refl (sum.traverse id.mk (inl x))) fun (x : α) => Eq.refl (sum.traverse id.mk (inr x))\n\nprotected theorem comp_traverse {σ : Type u} {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {α : Type u_1} {β : Type u} {γ : Type u} (f : β → F γ) (g : α → G β) (x : σ ⊕ α) : sum.traverse (functor.comp.mk ∘ Functor.map f ∘ g) x = functor.comp.mk (sum.traverse f <$> sum.traverse g x) := sorry\n\nprotected theorem traverse_eq_map_id {σ : Type u} {α : Type u} {β : Type u} (f : α → β) (x : σ ⊕ α) : sum.traverse (id.mk ∘ f) x = id.mk (f <$> x) := sorry\n\nprotected theorem map_traverse {σ : Type u} {G : Type u → Type u} [Applicative G] [is_lawful_applicative G] {α : Type u_1} {β : Type u} {γ : Type u} (g : α → G β) (f : β → γ) (x : σ ⊕ α) : Functor.map f <$> sum.traverse g x = sum.traverse (Functor.map f ∘ g) x := sorry\n\nprotected theorem naturality {σ : Type u} {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] (η : applicative_transformation F G) {α : Type u_1} {β : Type u} (f : α → F β) (x : σ ⊕ α) : coe_fn η (σ ⊕ β) (sum.traverse f x) = sum.traverse (coe_fn η β ∘ f) x := sorry\n\nprotected instance is_lawful_traversable {σ : Type u} : is_lawful_traversable (sum σ) :=\n  is_lawful_traversable.mk sum.id_traverse sum.comp_traverse sum.traverse_eq_map_id sum.naturality\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/traversable/instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.32037698218554034}}
{"text": "/-\nCopyright (c) 2021 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Std.Classes.SetNotation\nimport Std.Classes.LawfulMonad\nimport Std.Tactic.NoMatch\n\nnamespace Option\n\n/-- An elimination principle for `Option`. It is a nondependent version of `Option.recOn`. -/\n@[simp] protected def elim : Option α → β → (α → β) → β\n  | some x, _, f => f x\n  | none, y, _ => y\n\ninstance : Membership α (Option α) := ⟨fun a b => b = some a⟩\n\n@[simp] theorem mem_def {a : α} {b : Option α} : a ∈ b ↔ b = some a := .rfl\n\ntheorem isNone_iff_eq_none {o : Option α} : o.isNone ↔ o = none :=\n  ⟨Option.eq_none_of_isNone, fun e => e.symm ▸ rfl⟩\n\ntheorem some_inj {a b : α} : some a = some b ↔ a = b := by simp\n\n/--\n`o = none` is decidable even if the wrapped type does not have decidable equality.\nThis is not an instance because it is not definitionally equal to `instance : DecidableEq Option`.\nTry to use `o.isNone` or `o.isSome` instead.\n-/\n@[inline] def decidable_eq_none {o : Option α} : Decidable (o = none) :=\n  decidable_of_decidable_of_iff isNone_iff_eq_none\n\ninstance {p : α → Prop} [DecidablePred p] : ∀ o : Option α, Decidable (∀ a ∈ o, p a)\n| none => isTrue (by simp)\n| some a =>\n  if h : p a then isTrue fun o e => some_inj.1 e ▸ h\n  else isFalse <| mt (· _ rfl) h\n\ninstance {p : α → Prop} [DecidablePred p] : ∀ o : Option α, Decidable (∃ a ∈ o, p a)\n| none => isFalse fun.\n| some a => if h : p a then isTrue ⟨_, rfl, h⟩ else isFalse fun ⟨_, ⟨rfl, hn⟩⟩ => h hn\n\n/-- Extracts the value `a` from an option that is known to be `some a` for some `a`. -/\ndef get {α : Type u} : (o : Option α) → isSome o → α\n  | some x, _ => x\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : α → Prop) [DecidablePred p] (a : α) : Option α := if p a then some a else none\n\n/--\nCast of `Option` to `List`. Returns `[a]` if the input is `some a`, and `[]` if it is `none`.\n-/\ndef toList : Option α → List α\n  | none => []\n  | some a => [a]\n\n/--\nCast of `Option` to `Array`. Returns `[a]` if the input is `some a`, and `[]` if it is `none`.\n-/\ndef toArray : Option α → Array α\n  | none => #[]\n  | some a => #[a]\n\n/--\nTwo arguments failsafe function. Returns `f a b` if the inputs are `some a` and `some b`, and\n\"does nothing\" otherwise.\n-/\ndef liftOrGet (f : α → α → α) : Option α → Option α → Option α\n  | none, none => none\n  | some a, none => some a\n  | none, some b => some b\n  | some a, some b => some (f a b)\n\n/-- Lifts a relation `α → β → Prop` to a relation `Option α → Option β → Prop` by just adding\n`none ~ none`. -/\ninductive Rel (r : α → β → Prop) : Option α → Option β → Prop\n  /-- If `a ~ b`, then `some a ~ some b` -/\n  | some {a b} : r a b → Rel r (some a) (some b)\n  /-- `none ~ none` -/\n  | none : Rel r none none\n\n/--\nPartial bind. If for some `x : Option α`, `f : Π (a : α), a ∈ x → Option β` is a\npartial function defined on `a : α` giving an `Option β`, where `some a = x`,\nthen `pbind x f h` is essentially the same as `bind x f`\nbut is defined only when all `x = some a`, using the proof to apply `f`.\n-/\n@[simp]\ndef pbind : ∀ x : Option α, (∀ a : α, a ∈ x → Option β) → Option β\n  | none, _ => none\n  | some a, f => f a rfl\n\n/--\nPartial map. If `f : Π a, p a → β` is a partial function defined on `a : α` satisfying `p`,\nthen `pmap f x h` is essentially the same as `map f x` but is defined only when all members of `x`\nsatisfy `p`, using the proof to apply `f`.\n-/\n@[simp] def pmap {p : α → Prop} (f : ∀ a : α, p a → β) : ∀ x : Option α, (∀ a ∈ x, p a) → Option β\n  | none, _ => none\n  | some a, H => f a (H a rfl)\n\n/-- Flatten an `Option` of `Option`, a specialization of `joinM`. -/\n@[simp, inline] def join (x : Option (Option α)) : Option α := x.bind id\n\n/-- Map a monadic function which returns `Unit` over an `Option`. -/\nprotected def forM [Pure m] : Option α → (α → m PUnit) → m PUnit\n  | none  , _ => pure ()\n  | some a, f => f a\n\ninstance : ForM m (Option α) α :=\n  ⟨Option.forM⟩\n\ninstance : ForIn' m (Option α) α inferInstance where\n  forIn' x init f := do\n    match x with\n    | none => return init\n    | some a =>\n      match ← f a rfl init with\n      | .done r | .yield r => return r\n\n/-- Like `Option.mapM` but for applicative functors. -/\nprotected def mapA [Applicative m] {α β} (f : α → m β) : Option α → m (Option β)\n  | none => pure none\n  | some x => some <$> f x\n\n/--\nIf you maybe have a monadic computation in a `[Monad m]` which produces a term of type `α`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult.\n-/\ndef sequence [Monad m] {α : Type u} : Option (m α) → m (Option α)\n  | none => pure none\n  | some fn => some <$> fn\n\n/-- A monadic analogue of `Option.elim`. -/\n@[inline] def elimM [Monad m] (x : m (Option α)) (y : m β) (z : α → m β) : m β :=\n  do (← x).elim y z\n\n/-- A monadic analogue of `Option.getD`. -/\n@[inline] def getDM [Monad m] (x : Option α) (y : m α) : m α :=\n  match x with\n  | some a => pure a\n  | none => y\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/Option/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244837, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.32035777780608754}}
{"text": "import ground_zero.support ground_zero.theorems.functions ground_zero.types.dep_path\n\nnamespace ground_zero.types\n\nstructure {u v} product (α : Sort u) (β : Sort v) :=\nintro :: (pr₁ : α) (pr₂ : β)\nattribute [pp_using_anonymous_constructor] product\n\nreserve infix ` × `\ninfix ` × ` := product\n\nnamespace equiv\n  universes u v\n\n  def homotopy {α : Sort u} {π : α → Sort v}\n    (f g : Π (x : α), π x) :=\n  Π (x : α), f x = g x :> π x\n  infix ` ~ ` := homotopy\n\n  @[refl] def homotopy.id {α : Sort u} {π : α → Sort v}\n    (f : Π (x : α), π x) : f ~ f :=\n  begin simp [homotopy], intro x, reflexivity end\n\n  def homotopy.eq {α : Sort u} {π : α → Sort v}\n    {f g : Π (x : α), π x} (h : f = g :> Π (x : α), π x) : f ~ g :=\n  begin induction h, reflexivity end\n\n  @[symm] def homotopy.symm {α : Sort u} {π : α → Sort v}\n    (f g : Π (x : α), π x) (h : f ~ g) : g ~ f := begin\n    simp [homotopy] at *, intros,\n    apply eq.symm, apply h\n  end\n\n  @[trans] def homotopy.trans {α : Sort u} {π : α → Sort v}\n    (f g h : Π (x : α), π x) (r₁ : f ~ g) (r₂ : g ~ h) : f ~ h := begin\n    simp [homotopy] at *, intros, transitivity,\n    apply r₁, apply r₂\n  end\n\n  def linv {α : Sort u} {β : Sort v} (f : α → β) :=\n  Σ' (g : β → α), g ∘ f ~ id\n\n  def rinv {α : Sort u} {β : Sort v} (f : α → β) :=\n  Σ' (g : β → α), f ∘ g ~ id\n\n  def biinv {α : Sort u} {β : Sort v} (f : α → β) :=\n  linv f × rinv f\n\n  def homotopy_sqaure {α : Sort u} {β : Sort v}\n    {f g : α → β} (H : f ~ g) {x y : α}\n    (p : x = y :> α) :\n    H x ⬝ (g # p) = (f # p) ⬝ H y :> f x = g y :> β := begin\n    induction p, transitivity,\n    apply eq.refl_right,\n    apply eq.refl_left\n  end\nend equiv\n\ndef {u v} equiv (α : Sort u) (β : Sort v) :=\nΣ' (f : α → β), equiv.biinv f\ninfix ` ≃ `:25 := equiv\n\nnamespace equiv\n  universes u v w\n\n  instance forward_coe {α : Sort u} {β : Sort v} :\n    has_coe (α ≃ β) (α → β) :=\n  ⟨begin intro e, cases e with f H, exact f end⟩\n\n  @[refl] def id (α : Sort u) : α ≃ α := begin\n    existsi id, split,\n    repeat {\n      existsi id, intro, reflexivity\n    }\n  end\n\n  @[trans] def trans {α : Sort u} {β : Sort v} {γ : Sort w}\n    (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ := begin\n    cases e₁ with f₁ H₁,\n    cases H₁ with linv₁ rinv₁,\n    cases linv₁ with g₁ α₁,\n    cases rinv₁ with h₁ β₁,\n\n    cases e₂ with f₂ H₂,\n    cases H₂ with linv₂ rinv₂,\n    cases linv₂ with g₂ α₂,\n    cases rinv₂ with h₂ β₂,\n\n    existsi (f₂ ∘ f₁), split,\n    { existsi (g₁ ∘ g₂),\n      intro x, simp, transitivity,\n      exact g₁ # (α₂ (f₁ x)), exact α₁ x },\n    { existsi (h₁ ∘ h₂),\n      intro x, simp, transitivity,\n      exact f₂ # (β₁ (h₂ x)), exact β₂ x }\n  end\n\n  def idtoeqv {α β : Sort u} (p : α = β :> _) : α ≃ β :=\n  begin induction p, apply id end\n\n  def transportconst {α β : Sort u} : (α = β :> _) → (α → β) :=\n  psigma.fst ∘ idtoeqv\n\n  def subst {α : Sort u} {π : α → Sort v} {a b : α}\n    (p : a = b :> α) : π a → π b :=\n  begin induction p, exact ground_zero.theorems.functions.idfun end\n\n  reserve infix ` ▸ `\n  infix [parsing_only] ` ▸ ` := subst\n\n  def apd {α : Sort u} {β : α → Sort v} {a b : α}\n    (f : Π (x : α), β x) (p : a = b :> α) :\n    subst p (f a) = f b :> β b :=\n  begin induction p, reflexivity end\n\n  def subst_sqr {α : Sort u} {π : α → Sort v} {a b : α}\n    {p q : a = b :> α} (r : p = q :> a = b :> α) (u : π a) :\n    subst p u = subst q u :> π b :=\n  begin induction r, reflexivity end\n  notation `subst²` := subst_sqr\n\n  lemma dep_path_map {α : Sort u}\n    {π : α → Sort v} {δ : α → Sort w}\n    {a b : α}\n    {p : a = b :> α} {u : π a} {v : π b} (q : u =[p] v)\n    (g : Π {x : α}, π x → δ x) :\n    g u =[p] g v :=\n  begin induction q, induction p, trivial end\n\n  def subst_comp {α : Sort u}\n    {π : α → Sort v} {a b c : α}\n    (p : a = b :> α) (q : b = c :> α) (x : π a) :\n    subst (p ⬝ q) x = subst q (subst p x) :> π c :=\n  begin induction p, induction q, trivial end\n\n  -- subst with explicit π\n  abbreviation transport {α : Sort u}\n    (π : α → Sort v) {a b : α}\n    (p : a = b :> α) : π a → π b := subst p\n\n  abbreviation transport_sqr {α : Sort u} (π : α → Sort v) {a b : α}\n    {p q : a = b :> α} (r : p = q :> a = b :> α) (u : π a) :\n    subst p u = subst q u :> π b := subst_sqr r u\n  notation `transport²` := transport_sqr\n\n  --notation u ` =[` P `,` p `] ` v := transport P p u = v :> _\n\n  lemma transport_comp {α : Sort u} {β : Sort v}\n    (π : β → Sort w) {x y : α}\n    (f : α → β) (p : x = y :> α) (u : π (f x)) :\n    @subst _ (π ∘ f) _ _ p u =\n    subst (f # p) u :> π (f y) :=\n  begin induction p, trivial end\n\n  def transport_composition {α : Sort u} {a x₁ x₂ : α}\n    (p : x₁ = x₂ :> α) (q : a = x₁ :> α) :\n    transport (ground_zero.types.eq a) p q = q ⬝ p :> _ := begin\n    induction p, symmetry, transitivity,\n    apply eq.refl_right, trivial\n  end\n\n  lemma transport_over_family {α : Sort u}\n    {x y : α} {π δ : α → Sort v}\n    (f : Π (x : α), π x → δ x)\n    (p : x = y :> α) (u : π x) :\n    subst p (f x u) = f y (subst p u) :> δ y :=\n  begin induction p, trivial end\n\n  def apd_sqr {α : Sort u} {β γ : α → Sort v} {a b : α}\n    {u : β a} {v : β b} {p : a = b :> α}\n    (f : Π {x : α} (u : β x), γ x) (q : u =[p] v) :\n    f u =[p] f v :=\n  begin induction q, reflexivity end\n\n  def apd₂ {α : Sort u} {β : α → Sort v} {a b : α}\n    {p q : a = b :> α} (f : Π (x : α), β x)\n    (r : p = q :> a = b :> α) :\n    dep_path.apd f p =[r] dep_path.apd f q :=\n  begin induction r, reflexivity end\n\n  def rewrite_comp {α : Sort u} {a b c : α}\n    {p : a = b :> α} {q : b = c :> α} {r : a = c :> α}\n    (h : r = p ⬝ q :> a = c :> α) :\n    p⁻¹ ⬝ r = q :> b = c :> α := begin\n    induction p, unfold eq.symm, transitivity,\n    exact eq.refl_left r, exact h ⬝ eq.refl_left q\n  end\n\n  def path_over_subst {α : Sort u} {β : α → Sort v}\n    {a b : α} {p : a = b :> α} {u : β a} {v : β b}\n    (q : subst p u = v :> β b) : u =[p] v :=\n  begin induction q, induction p, reflexivity end\n\n  def subst_from_pathover {α : Sort u} {β : α → Sort v}\n    {a b : α} {p : a = b :> α} {u : β a} {v : β b}\n    (q : u =[p] v) : subst p u = v :> β b :=\n  begin induction q, reflexivity end\n\n  def pathover_from_trans {α : Sort u} {a b c : α}\n    (p : b = c :> α) (q : a = b :> α) (r : a = c :> α) :\n    (q ⬝ p = r :> a = c :> α) → (q =[p] r) := begin\n    intro h, induction h,\n    apply path_over_subst, apply transport_composition\n  end\nend equiv\n\ndef {u v} is_qinv {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) :=\n(f ∘ g ~ id) × (g ∘ f ~ id)\n\nclass {u v} has_qinv {α : Sort u} {β : Sort v} (f : α → β) :=\n(inv : β → α) (lawful : is_qinv f inv)\npostfix ⁻¹ := has_qinv.inv\n\ndef {u v} qinv {α : Sort u} {β : Sort v} (f : α → β) :=\nΣ' (g : β → α), is_qinv f g\n\nnamespace qinv\n  universes u v w\n\n  def equiv (α : Sort u) (β : Sort v) :=\n  Σ' (f : α → β), qinv f\n\n  def q2b {α : Sort u} {β : Sort v} (f : α → β) (q : qinv f) :\n    equiv.biinv f := begin\n    cases q with g H,\n    cases H with α β,\n    split; existsi g,\n    exact β, exact α\n  end\n\n  def b2q {α : Sort u} {β : Sort v} (f : α → β) (b : equiv.biinv f) :\n    qinv f := begin\n    cases b with linv rinv,\n    cases rinv with g α,\n    cases linv with h β,\n\n    existsi g, split,\n    exact α, intro x,\n\n    have γ₁ := β (g (f x)),\n    have γ₂ := h # (α (f x)),\n    exact γ₁⁻¹ ⬝ γ₂ ⬝ β x\n  end\nend qinv\n\nnamespace equiv\n  universes u v\n\n  @[symm] def symm {α : Sort u} {β : Sort v}\n    (e : α ≃ β) : β ≃ α := begin\n    cases e with f H, have q := qinv.b2q f H,\n    cases q with g qinv, cases qinv with α β,\n    existsi g, split; existsi f,\n    exact α, exact β\n  end\nend equiv\n\n-- half adjoint equivalence\ndef {u v} ishae {α : Sort u} {β : Sort v} (f : α → β) :=\nΣ' (g : β → α) (η : g ∘ f ~ id) (ϵ : f ∘ g ~ id) (x : α),\n  f # (η x) = ϵ (f x) :> f (g (f x)) = f x :> β\n\ndef {u v} fib {α : Sort u} {β : Sort v} (f : α → β) (y : β) :=\nΣ' (x : α), f x = y :> β\n\nend ground_zero.types", "meta": {"author": "jfrancese", "repo": "lean", "sha": "06e7efaecce4093d97fb5ecc75479df2ef1dbbdb", "save_path": "github-repos/lean/jfrancese-lean", "path": "github-repos/lean/jfrancese-lean/lean-06e7efaecce4093d97fb5ecc75479df2ef1dbbdb/ground_zero/types/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3203001328089981}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.ultrafilter\nimport Mathlib.order.filter.partial\nimport Mathlib.PostPort\n\nuniverses u l w v u_1 u_2 u_3 u_5 \n\nnamespace Mathlib\n\n/-!\n# Basic theory of topological spaces.\n\nThe main definition is the type class `topological space α` which endows a type `α` with a topology.\nThen `set α` gets predicates `is_open`, `is_closed` and functions `interior`, `closure` and\n`frontier`. Each point `x` of `α` gets a neighborhood filter `𝓝 x`. A filter `F` on `α` has\n`x` as a cluster point if `cluster_pt x F : 𝓝 x ⊓ F ≠ ⊥`. A map `f : ι → α` clusters at `x`\nalong `F : filter ι` if `map_cluster_pt x F f : cluster_pt x (map f F)`. In particular\nthe notion of cluster point of a sequence `u` is `map_cluster_pt x at_top u`.\n\nThis file also defines locally finite families of subsets of `α`.\n\nFor topological spaces `α` and `β`, a function `f : α → β` and a point `a : α`,\n`continuous_at f a` means `f` is continuous at `a`, and global continuity is\n`continuous f`. There is also a version of continuity `pcontinuous` for\npartially defined functions.\n\n## Notation\n\n* `𝓝 x`: the filter of neighborhoods of a point `x`;\n* `𝓟 s`: the principal filter of a set `s`;\n\n## Implementation notes\n\nTopology in mathlib heavily uses filters (even more than in Bourbaki). See explanations in\n<https://leanprover-community.github.io/theories/topology.html>.\n\n## References\n\n*  [N. Bourbaki, *General Topology*][bourbaki1966]\n*  [I. M. James, *Topologies and Uniformities*][james1999]\n\n## Tags\n\ntopological space, interior, closure, frontier, neighborhood, continuity, continuous function\n-/\n\n/-!\n### Topological spaces\n-/\n\n/-- A topology on `α`. -/\nclass topological_space (α : Type u) \nwhere\n  is_open : set α → Prop\n  is_open_univ : is_open set.univ\n  is_open_inter : ∀ (s t : set α), is_open s → is_open t → is_open (s ∩ t)\n  is_open_sUnion : ∀ (s : set (set α)), (∀ (t : set α), t ∈ s → is_open t) → is_open (⋃₀s)\n\n/-- A constructor for topologies by specifying the closed sets,\nand showing that they satisfy the appropriate conditions. -/\ndef topological_space.of_closed {α : Type u} (T : set (set α)) (empty_mem : ∅ ∈ T) (sInter_mem : ∀ (A : set (set α)), A ⊆ T → ⋂₀A ∈ T) (union_mem : ∀ (A B : set α), A ∈ T → B ∈ T → A ∪ B ∈ T) : topological_space α :=\n  topological_space.mk (fun (X : set α) => Xᶜ ∈ T) sorry sorry sorry\n\ntheorem topological_space_eq {α : Type u} {f : topological_space α} {g : topological_space α} : topological_space.is_open f = topological_space.is_open g → f = g := sorry\n\n/-- `is_open s` means that `s` is open in the ambient topological space on `α` -/\ndef is_open {α : Type u} [t : topological_space α] (s : set α) :=\n  topological_space.is_open t s\n\n@[simp] theorem is_open_univ {α : Type u} [t : topological_space α] : is_open set.univ :=\n  topological_space.is_open_univ t\n\ntheorem is_open_inter {α : Type u} {s₁ : set α} {s₂ : set α} [t : topological_space α] (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∩ s₂) :=\n  topological_space.is_open_inter t s₁ s₂ h₁ h₂\n\ntheorem is_open_sUnion {α : Type u} [t : topological_space α] {s : set (set α)} (h : ∀ (t_1 : set α), t_1 ∈ s → is_open t_1) : is_open (⋃₀s) :=\n  topological_space.is_open_sUnion t s h\n\ntheorem topological_space_eq_iff {α : Type u} {t : topological_space α} {t' : topological_space α} : t = t' ↔ ∀ (s : set α), is_open s ↔ is_open s :=\n  { mp := fun (h : t = t') (s : set α) => h ▸ iff.rfl,\n    mpr :=\n      fun (h : ∀ (s : set α), is_open s ↔ is_open s) => topological_space_eq (funext fun (x : set α) => propext (h x)) }\n\ntheorem is_open_fold {α : Type u} {s : set α} {t : topological_space α} : topological_space.is_open t s = is_open s :=\n  rfl\n\ntheorem is_open_Union {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α} (h : ∀ (i : ι), is_open (f i)) : is_open (set.Union fun (i : ι) => f i) :=\n  is_open_sUnion\n    fun (t : set α) (H : t ∈ set.range fun (i : ι) => f i) =>\n      Exists.dcases_on H fun (i : ι) (H_h : (fun (i : ι) => f i) i = t) => Eq._oldrec (h i) H_h\n\ntheorem is_open_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (h : ∀ (i : β), i ∈ s → is_open (f i)) : is_open (set.Union fun (i : β) => set.Union fun (H : i ∈ s) => f i) :=\n  is_open_Union fun (i : β) => is_open_Union fun (hi : i ∈ s) => h i hi\n\ntheorem is_open_union {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α] (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∪ s₂) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∪ s₂))) set.union_eq_Union))\n    (is_open_Union (iff.mpr bool.forall_bool { left := h₂, right := h₁ }))\n\n@[simp] theorem is_open_empty {α : Type u} [topological_space α] : is_open ∅ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open ∅)) (Eq.symm set.sUnion_empty))) (is_open_sUnion fun (a : set α) => false.elim)\n\ntheorem is_open_sInter {α : Type u} [topological_space α] {s : set (set α)} (hs : set.finite s) : (∀ (t : set α), t ∈ s → is_open t) → is_open (⋂₀s) := sorry\n\ntheorem is_open_bInter {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (hs : set.finite s) : (∀ (i : β), i ∈ s → is_open (f i)) → is_open (set.Inter fun (i : β) => set.Inter fun (H : i ∈ s) => f i) := sorry\n\ntheorem is_open_Inter {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α} (h : ∀ (i : β), is_open (s i)) : is_open (set.Inter fun (i : β) => s i) := sorry\n\ntheorem is_open_Inter_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α} (h : ∀ (h : p), is_open (s h)) : is_open (set.Inter s) := sorry\n\ntheorem is_open_const {α : Type u} [topological_space α] {p : Prop} : is_open (set_of fun (a : α) => p) := sorry\n\ntheorem is_open_and {α : Type u} {p₁ : α → Prop} {p₂ : α → Prop} [topological_space α] : is_open (set_of fun (a : α) => p₁ a) →\n  is_open (set_of fun (a : α) => p₂ a) → is_open (set_of fun (a : α) => p₁ a ∧ p₂ a) :=\n  is_open_inter\n\n/-- A set is closed if its complement is open -/\ndef is_closed {α : Type u} [topological_space α] (s : set α) :=\n  is_open (sᶜ)\n\n@[simp] theorem is_closed_empty {α : Type u} [topological_space α] : is_closed ∅ :=\n  eq.mpr (id (is_closed.equations._eqn_1 ∅))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (∅ᶜ))) set.compl_empty)) is_open_univ)\n\n@[simp] theorem is_closed_univ {α : Type u} [topological_space α] : is_closed set.univ :=\n  eq.mpr (id (is_closed.equations._eqn_1 set.univ))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.univᶜ))) set.compl_univ)) is_open_empty)\n\ntheorem is_closed_union {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α] : is_closed s₁ → is_closed s₂ → is_closed (s₁ ∪ s₂) :=\n  fun (h₁ : is_closed s₁) (h₂ : is_closed s₂) =>\n    eq.mpr (id (is_closed.equations._eqn_1 (s₁ ∪ s₂)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∪ s₂ᶜ))) (set.compl_union s₁ s₂))) (is_open_inter h₁ h₂))\n\ntheorem is_closed_sInter {α : Type u} [topological_space α] {s : set (set α)} : (∀ (t : set α), t ∈ s → is_closed t) → is_closed (⋂₀s) := sorry\n\ntheorem is_closed_Inter {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α} (h : ∀ (i : ι), is_closed (f i)) : is_closed (set.Inter fun (i : ι) => f i) := sorry\n\n@[simp] theorem is_open_compl_iff {α : Type u} [topological_space α] {s : set α} : is_open (sᶜ) ↔ is_closed s :=\n  iff.rfl\n\n@[simp] theorem is_closed_compl_iff {α : Type u} [topological_space α] {s : set α} : is_closed (sᶜ) ↔ is_open s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (sᶜ) ↔ is_open s)) (Eq.symm (propext is_open_compl_iff))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (sᶜᶜ) ↔ is_open s)) (compl_compl s))) (iff.refl (is_open s)))\n\ntheorem is_open_diff {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : is_open s) (h₂ : is_closed t) : is_open (s \\ t) :=\n  is_open_inter h₁ (iff.mpr is_open_compl_iff h₂)\n\ntheorem is_closed_inter {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α] (h₁ : is_closed s₁) (h₂ : is_closed s₂) : is_closed (s₁ ∩ s₂) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (s₁ ∩ s₂))) (is_closed.equations._eqn_1 (s₁ ∩ s₂))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∩ s₂ᶜ))) (set.compl_inter s₁ s₂))) (is_open_union h₁ h₂))\n\ntheorem is_closed_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α} (hs : set.finite s) : (∀ (i : β), i ∈ s → is_closed (f i)) → is_closed (set.Union fun (i : β) => set.Union fun (H : i ∈ s) => f i) := sorry\n\ntheorem is_closed_Union {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α} (h : ∀ (i : β), is_closed (s i)) : is_closed (set.Union s) := sorry\n\ntheorem is_closed_Union_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α} (h : ∀ (h : p), is_closed (s h)) : is_closed (set.Union s) := sorry\n\ntheorem is_closed_imp {α : Type u} [topological_space α] {p : α → Prop} {q : α → Prop} (hp : is_open (set_of fun (x : α) => p x)) (hq : is_closed (set_of fun (x : α) => q x)) : is_closed (set_of fun (x : α) => p x → q x) := sorry\n\ntheorem is_open_neg {α : Type u} {p : α → Prop} [topological_space α] : is_closed (set_of fun (a : α) => p a) → is_open (set_of fun (a : α) => ¬p a) :=\n  iff.mpr is_open_compl_iff\n\n/-!\n### Interior of a set\n-/\n\n/-- The interior of a set `s` is the largest open subset of `s`. -/\ndef interior {α : Type u} [topological_space α] (s : set α) : set α :=\n  ⋃₀set_of fun (t : set α) => is_open t ∧ t ⊆ s\n\ntheorem mem_interior {α : Type u} [topological_space α] {s : set α} {x : α} : x ∈ interior s ↔ ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ x ∈ t := sorry\n\n@[simp] theorem is_open_interior {α : Type u} [topological_space α] {s : set α} : is_open (interior s) := sorry\n\ntheorem interior_subset {α : Type u} [topological_space α] {s : set α} : interior s ⊆ s := sorry\n\ntheorem interior_maximal {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : t ⊆ s) (h₂ : is_open t) : t ⊆ interior s :=\n  set.subset_sUnion_of_mem { left := h₂, right := h₁ }\n\ntheorem is_open.interior_eq {α : Type u} [topological_space α] {s : set α} (h : is_open s) : interior s = s :=\n  set.subset.antisymm interior_subset (interior_maximal (set.subset.refl s) h)\n\ntheorem interior_eq_iff_open {α : Type u} [topological_space α] {s : set α} : interior s = s ↔ is_open s :=\n  { mp := fun (h : interior s = s) => h ▸ is_open_interior, mpr := is_open.interior_eq }\n\ntheorem subset_interior_iff_open {α : Type u} [topological_space α] {s : set α} : s ⊆ interior s ↔ is_open s := sorry\n\ntheorem subset_interior_iff_subset_of_open {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : is_open s) : s ⊆ interior t ↔ s ⊆ t :=\n  { mp := fun (h : s ⊆ interior t) => set.subset.trans h interior_subset,\n    mpr := fun (h₂ : s ⊆ t) => interior_maximal h₂ h₁ }\n\ntheorem interior_mono {α : Type u} [topological_space α] {s : set α} {t : set α} (h : s ⊆ t) : interior s ⊆ interior t :=\n  interior_maximal (set.subset.trans interior_subset h) is_open_interior\n\n@[simp] theorem interior_empty {α : Type u} [topological_space α] : interior ∅ = ∅ :=\n  is_open.interior_eq is_open_empty\n\n@[simp] theorem interior_univ {α : Type u} [topological_space α] : interior set.univ = set.univ :=\n  is_open.interior_eq is_open_univ\n\n@[simp] theorem interior_interior {α : Type u} [topological_space α] {s : set α} : interior (interior s) = interior s :=\n  is_open.interior_eq is_open_interior\n\n@[simp] theorem interior_inter {α : Type u} [topological_space α] {s : set α} {t : set α} : interior (s ∩ t) = interior s ∩ interior t := sorry\n\ntheorem interior_union_is_closed_of_interior_empty {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : is_closed s) (h₂ : interior t = ∅) : interior (s ∪ t) = interior s := sorry\n\ntheorem is_open_iff_forall_mem_open {α : Type u} {s : set α} [topological_space α] : is_open s ↔ ∀ (x : α) (H : x ∈ s), ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ x ∈ t := sorry\n\n/-!\n### Closure of a set\n-/\n\n/-- The closure of `s` is the smallest closed set containing `s`. -/\ndef closure {α : Type u} [topological_space α] (s : set α) : set α :=\n  ⋂₀set_of fun (t : set α) => is_closed t ∧ s ⊆ t\n\n@[simp] theorem is_closed_closure {α : Type u} [topological_space α] {s : set α} : is_closed (closure s) := sorry\n\ntheorem subset_closure {α : Type u} [topological_space α] {s : set α} : s ⊆ closure s := sorry\n\ntheorem closure_minimal {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : s ⊆ t) (h₂ : is_closed t) : closure s ⊆ t :=\n  set.sInter_subset_of_mem { left := h₂, right := h₁ }\n\ntheorem is_closed.closure_eq {α : Type u} [topological_space α] {s : set α} (h : is_closed s) : closure s = s :=\n  set.subset.antisymm (closure_minimal (set.subset.refl s) h) subset_closure\n\ntheorem is_closed.closure_subset {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) : closure s ⊆ s :=\n  closure_minimal (set.subset.refl s) hs\n\ntheorem is_closed.closure_subset_iff {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : is_closed t) : closure s ⊆ t ↔ s ⊆ t :=\n  { mp := set.subset.trans subset_closure, mpr := fun (h : s ⊆ t) => closure_minimal h h₁ }\n\ntheorem closure_mono {α : Type u} [topological_space α] {s : set α} {t : set α} (h : s ⊆ t) : closure s ⊆ closure t :=\n  closure_minimal (set.subset.trans h subset_closure) is_closed_closure\n\ntheorem monotone_closure (α : Type u_1) [topological_space α] : monotone closure :=\n  fun (_x _x_1 : set α) => closure_mono\n\ntheorem closure_inter_subset_inter_closure {α : Type u} [topological_space α] (s : set α) (t : set α) : closure (s ∩ t) ⊆ closure s ∩ closure t :=\n  monotone.map_inf_le (monotone_closure α) s t\n\ntheorem is_closed_of_closure_subset {α : Type u} [topological_space α] {s : set α} (h : closure s ⊆ s) : is_closed s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed s)) (set.subset.antisymm subset_closure h))) is_closed_closure\n\ntheorem closure_eq_iff_is_closed {α : Type u} [topological_space α] {s : set α} : closure s = s ↔ is_closed s :=\n  { mp := fun (h : closure s = s) => h ▸ is_closed_closure, mpr := is_closed.closure_eq }\n\ntheorem closure_subset_iff_is_closed {α : Type u} [topological_space α] {s : set α} : closure s ⊆ s ↔ is_closed s :=\n  { mp := is_closed_of_closure_subset, mpr := is_closed.closure_subset }\n\n@[simp] theorem closure_empty {α : Type u} [topological_space α] : closure ∅ = ∅ :=\n  is_closed.closure_eq is_closed_empty\n\n@[simp] theorem closure_empty_iff {α : Type u} [topological_space α] (s : set α) : closure s = ∅ ↔ s = ∅ :=\n  { mp := set.subset_eq_empty subset_closure, mpr := fun (h : s = ∅) => Eq.symm h ▸ closure_empty }\n\ntheorem set.nonempty.closure {α : Type u} [topological_space α] {s : set α} (h : set.nonempty s) : set.nonempty (closure s) := sorry\n\n@[simp] theorem closure_univ {α : Type u} [topological_space α] : closure set.univ = set.univ :=\n  is_closed.closure_eq is_closed_univ\n\n@[simp] theorem closure_closure {α : Type u} [topological_space α] {s : set α} : closure (closure s) = closure s :=\n  is_closed.closure_eq is_closed_closure\n\n@[simp] theorem closure_union {α : Type u} [topological_space α] {s : set α} {t : set α} : closure (s ∪ t) = closure s ∪ closure t := sorry\n\ntheorem interior_subset_closure {α : Type u} [topological_space α] {s : set α} : interior s ⊆ closure s :=\n  set.subset.trans interior_subset subset_closure\n\ntheorem closure_eq_compl_interior_compl {α : Type u} [topological_space α] {s : set α} : closure s = (interior (sᶜ)ᶜ) := sorry\n\n@[simp] theorem interior_compl {α : Type u} [topological_space α] {s : set α} : interior (sᶜ) = (closure sᶜ) := sorry\n\n@[simp] theorem closure_compl {α : Type u} [topological_space α] {s : set α} : closure (sᶜ) = (interior sᶜ) := sorry\n\ntheorem mem_closure_iff {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ closure s ↔ ∀ (o : set α), is_open o → a ∈ o → set.nonempty (o ∩ s) := sorry\n\n/-- A set is dense in a topological space if every point belongs to its closure. -/\ndef dense {α : Type u} [topological_space α] (s : set α) :=\n  ∀ (x : α), x ∈ closure s\n\ntheorem dense_iff_closure_eq {α : Type u} [topological_space α] {s : set α} : dense s ↔ closure s = set.univ :=\n  iff.symm set.eq_univ_iff_forall\n\ntheorem dense.closure_eq {α : Type u} [topological_space α] {s : set α} (h : dense s) : closure s = set.univ :=\n  iff.mp dense_iff_closure_eq h\n\n/-- The closure of a set `s` is dense if and only if `s` is dense. -/\n@[simp] theorem dense_closure {α : Type u} [topological_space α] {s : set α} : dense (closure s) ↔ dense s := sorry\n\ntheorem dense.of_closure {α : Type u} [topological_space α] {s : set α} : dense (closure s) → dense s :=\n  iff.mp dense_closure\n\n@[simp] theorem dense_univ {α : Type u} [topological_space α] : dense set.univ :=\n  fun (x : α) => subset_closure trivial\n\n/-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/\ntheorem dense_iff_inter_open {α : Type u} [topological_space α] {s : set α} : dense s ↔ ∀ (U : set α), is_open U → set.nonempty U → set.nonempty (U ∩ s) := sorry\n\ntheorem dense.inter_open_nonempty {α : Type u} [topological_space α] {s : set α} : dense s → ∀ (U : set α), is_open U → set.nonempty U → set.nonempty (U ∩ s) :=\n  iff.mp dense_iff_inter_open\n\ntheorem dense.nonempty_iff {α : Type u} [topological_space α] {s : set α} (hs : dense s) : set.nonempty s ↔ Nonempty α := sorry\n\ntheorem dense.nonempty {α : Type u} [topological_space α] [h : Nonempty α] {s : set α} (hs : dense s) : set.nonempty s :=\n  iff.mpr (dense.nonempty_iff hs) h\n\ntheorem dense.mono {α : Type u} [topological_space α] {s₁ : set α} {s₂ : set α} (h : s₁ ⊆ s₂) (hd : dense s₁) : dense s₂ :=\n  fun (x : α) => closure_mono h (hd x)\n\n/-!\n### Frontier of a set\n-/\n\n/-- The frontier of a set is the set of points between the closure and interior. -/\ndef frontier {α : Type u} [topological_space α] (s : set α) : set α :=\n  closure s \\ interior s\n\ntheorem frontier_eq_closure_inter_closure {α : Type u} [topological_space α] {s : set α} : frontier s = closure s ∩ closure (sᶜ) := sorry\n\n/-- The complement of a set has the same frontier as the original set. -/\n@[simp] theorem frontier_compl {α : Type u} [topological_space α] (s : set α) : frontier (sᶜ) = frontier s := sorry\n\ntheorem frontier_inter_subset {α : Type u} [topological_space α] (s : set α) (t : set α) : frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t := sorry\n\ntheorem frontier_union_subset {α : Type u} [topological_space α] (s : set α) (t : set α) : frontier (s ∪ t) ⊆ frontier s ∩ closure (tᶜ) ∪ closure (sᶜ) ∩ frontier t := sorry\n\ntheorem is_closed.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) : frontier s = s \\ interior s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (frontier s = s \\ interior s)) (frontier.equations._eqn_1 s)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (closure s \\ interior s = s \\ interior s)) (is_closed.closure_eq hs)))\n      (Eq.refl (s \\ interior s)))\n\ntheorem is_open.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_open s) : frontier s = closure s \\ s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (frontier s = closure s \\ s)) (frontier.equations._eqn_1 s)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (closure s \\ interior s = closure s \\ s)) (is_open.interior_eq hs)))\n      (Eq.refl (closure s \\ s)))\n\n/-- The frontier of a set is closed. -/\ntheorem is_closed_frontier {α : Type u} [topological_space α] {s : set α} : is_closed (frontier s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (frontier s))) frontier_eq_closure_inter_closure))\n    (is_closed_inter is_closed_closure is_closed_closure)\n\n/-- The frontier of a closed set has no interior point. -/\ntheorem interior_frontier {α : Type u} [topological_space α] {s : set α} (h : is_closed s) : interior (frontier s) = ∅ := sorry\n\ntheorem closure_eq_interior_union_frontier {α : Type u} [topological_space α] (s : set α) : closure s = interior s ∪ frontier s :=\n  Eq.symm (set.union_diff_cancel interior_subset_closure)\n\ntheorem closure_eq_self_union_frontier {α : Type u} [topological_space α] (s : set α) : closure s = s ∪ frontier s :=\n  Eq.symm (set.union_diff_cancel' interior_subset subset_closure)\n\n/-!\n### Neighborhoods\n-/\n\n/-- A set is called a neighborhood of `a` if it contains an open set around `a`. The set of all\nneighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the\ninfimum over the principal filters of all open sets containing `a`. -/\ndef nhds {α : Type u} [topological_space α] (a : α) : filter α :=\n  infi fun (s : set α) => infi fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) => filter.principal s\n\ntheorem nhds_def {α : Type u} [topological_space α] (a : α) : nhds a = infi fun (s : set α) => infi fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) => filter.principal s :=\n  rfl\n\n/-- The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`\nfor a variant using open neighborhoods instead. -/\ntheorem nhds_basis_opens {α : Type u} [topological_space α] (a : α) : filter.has_basis (nhds a) (fun (s : set α) => a ∈ s ∧ is_open s) fun (x : set α) => x := sorry\n\n/-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/\ntheorem le_nhds_iff {α : Type u} [topological_space α] {f : filter α} {a : α} : f ≤ nhds a ↔ ∀ (s : set α), a ∈ s → is_open s → s ∈ f := sorry\n\n/-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above\nthe principal filter of some open set `s` containing `a`. -/\ntheorem nhds_le_of_le {α : Type u} [topological_space α] {f : filter α} {a : α} {s : set α} (h : a ∈ s) (o : is_open s) (sf : filter.principal s ≤ f) : nhds a ≤ f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nhds a ≤ f)) (nhds_def a)))\n    (infi_le_of_le s (infi_le_of_le { left := h, right := o } sf))\n\ntheorem mem_nhds_sets_iff {α : Type u} [topological_space α] {a : α} {s : set α} : s ∈ nhds a ↔ ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ a ∈ t := sorry\n\n/-- A predicate is true in a neighborhood of `a` iff it is true for all the points in an open set\ncontaining `a`. -/\ntheorem eventually_nhds_iff {α : Type u} [topological_space α] {a : α} {p : α → Prop} : filter.eventually (fun (x : α) => p x) (nhds a) ↔ ∃ (t : set α), (∀ (x : α), x ∈ t → p x) ∧ is_open t ∧ a ∈ t := sorry\n\ntheorem map_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : α → β} : filter.map f (nhds a) =\n  infi fun (s : set α) => infi fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) => filter.principal (f '' s) :=\n  filter.has_basis.eq_binfi (filter.has_basis.map f (nhds_basis_opens a))\n\ntheorem mem_of_nhds {α : Type u} [topological_space α] {a : α} {s : set α} : s ∈ nhds a → a ∈ s := sorry\n\n/-- If a predicate is true in a neighborhood of `a`, then it is true for `a`. -/\ntheorem filter.eventually.self_of_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} (h : filter.eventually (fun (y : α) => p y) (nhds a)) : p a :=\n  mem_of_nhds h\n\ntheorem mem_nhds_sets {α : Type u} [topological_space α] {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) : s ∈ nhds a :=\n  iff.mpr mem_nhds_sets_iff (Exists.intro s (Exists.intro (set.subset.refl s) { left := hs, right := ha }))\n\ntheorem is_open.eventually_mem {α : Type u} [topological_space α] {a : α} {s : set α} (hs : is_open s) (ha : a ∈ s) : filter.eventually (fun (x : α) => x ∈ s) (nhds a) :=\n  mem_nhds_sets hs ha\n\n/-- The open neighborhoods of `a` are a basis for the neighborhood filter. See `nhds_basis_opens`\nfor a variant using open sets around `a` instead. -/\ntheorem nhds_basis_opens' {α : Type u} [topological_space α] (a : α) : filter.has_basis (nhds a) (fun (s : set α) => s ∈ nhds a ∧ is_open s) fun (x : set α) => x := sorry\n\n/-- If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close\nto `a` this predicate is true in a neighbourhood of `y`. -/\ntheorem filter.eventually.eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} (h : filter.eventually (fun (y : α) => p y) (nhds a)) : filter.eventually (fun (y : α) => filter.eventually (fun (y : α) => p y) (nhds y)) (nhds a) := sorry\n\n@[simp] theorem eventually_eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α} : filter.eventually (fun (y : α) => filter.eventually (fun (x : α) => p x) (nhds y)) (nhds a) ↔\n  filter.eventually (fun (x : α) => p x) (nhds a) := sorry\n\n@[simp] theorem nhds_bind_nhds {α : Type u} {a : α} [topological_space α] : filter.bind (nhds a) nhds = nhds a :=\n  filter.ext fun (s : set α) => eventually_eventually_nhds\n\n@[simp] theorem eventually_eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α] {f : α → β} {g : α → β} {a : α} : filter.eventually (fun (y : α) => filter.eventually_eq (nhds y) f g) (nhds a) ↔ filter.eventually_eq (nhds a) f g :=\n  eventually_eventually_nhds\n\ntheorem filter.eventually_eq.eq_of_nhds {α : Type u} {β : Type v} [topological_space α] {f : α → β} {g : α → β} {a : α} (h : filter.eventually_eq (nhds a) f g) : f a = g a :=\n  filter.eventually.self_of_nhds h\n\n@[simp] theorem eventually_eventually_le_nhds {α : Type u} {β : Type v} [topological_space α] [HasLessEq β] {f : α → β} {g : α → β} {a : α} : filter.eventually (fun (y : α) => filter.eventually_le (nhds y) f g) (nhds a) ↔ filter.eventually_le (nhds a) f g :=\n  eventually_eventually_nhds\n\n/-- If two functions are equal in a neighbourhood of `a`, then for `y` sufficiently close\nto `a` these functions are equal in a neighbourhood of `y`. -/\ntheorem filter.eventually_eq.eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α] {f : α → β} {g : α → β} {a : α} (h : filter.eventually_eq (nhds a) f g) : filter.eventually (fun (y : α) => filter.eventually_eq (nhds y) f g) (nhds a) :=\n  filter.eventually.eventually_nhds h\n\n/-- If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have\n`f x ≤ g x` in a neighbourhood of `y`. -/\ntheorem filter.eventually_le.eventually_le_nhds {α : Type u} {β : Type v} [topological_space α] [HasLessEq β] {f : α → β} {g : α → β} {a : α} (h : filter.eventually_le (nhds a) f g) : filter.eventually (fun (y : α) => filter.eventually_le (nhds y) f g) (nhds a) :=\n  filter.eventually.eventually_nhds h\n\ntheorem all_mem_nhds {α : Type u} [topological_space α] (x : α) (P : set α → Prop) (hP : ∀ (s t : set α), s ⊆ t → P s → P t) : (∀ (s : set α), s ∈ nhds x → P s) ↔ ∀ (s : set α), is_open s → x ∈ s → P s := sorry\n\ntheorem all_mem_nhds_filter {α : Type u} {β : Type v} [topological_space α] (x : α) (f : set α → set β) (hf : ∀ (s t : set α), s ⊆ t → f s ⊆ f t) (l : filter β) : (∀ (s : set α), s ∈ nhds x → f s ∈ l) ↔ ∀ (s : set α), is_open s → x ∈ s → f s ∈ l :=\n  all_mem_nhds x (fun (s : set α) => f s ∈ l)\n    fun (s t : set α) (ssubt : s ⊆ t) (h : f s ∈ l) => filter.mem_sets_of_superset h (hf s t ssubt)\n\ntheorem rtendsto_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β} {a : α} : filter.rtendsto r l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → rel.core r s ∈ l :=\n  all_mem_nhds_filter a (fun (U : set α) => U) (fun (s t : set α) => id) (filter.rmap r l)\n\ntheorem rtendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β} {a : α} : filter.rtendsto' r l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → rel.preimage r s ∈ l := sorry\n\ntheorem ptendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β} {a : α} : filter.ptendsto f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → pfun.core f s ∈ l :=\n  rtendsto_nhds\n\ntheorem ptendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β} {a : α} : filter.ptendsto' f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → pfun.preimage f s ∈ l :=\n  rtendsto'_nhds\n\ntheorem tendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β → α} {l : filter β} {a : α} : filter.tendsto f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → f ⁻¹' s ∈ l :=\n  all_mem_nhds_filter a (fun (U : set α) => U) (fun (s t : set α) (h : s ⊆ t) => set.preimage_mono h) (filter.map f l)\n\ntheorem tendsto_const_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : filter β} : filter.tendsto (fun (b : β) => a) f (nhds a) :=\n  iff.mpr tendsto_nhds fun (s : set α) (hs : is_open s) (ha : a ∈ s) => filter.univ_mem_sets' fun (_x : β) => ha\n\ntheorem pure_le_nhds {α : Type u} [topological_space α] : pure ≤ nhds :=\n  fun (a : α) (s : set α) (hs : s ∈ nhds a) => iff.mpr filter.mem_pure_sets (mem_of_nhds hs)\n\ntheorem tendsto_pure_nhds {β : Type v} {α : Type u_1} [topological_space β] (f : α → β) (a : α) : filter.tendsto f (pure a) (nhds (f a)) :=\n  filter.tendsto.mono_right (filter.tendsto_pure_pure f a) (pure_le_nhds (f a))\n\ntheorem order_top.tendsto_at_top_nhds {β : Type v} {α : Type u_1} [order_top α] [topological_space β] (f : α → β) : filter.tendsto f filter.at_top (nhds (f ⊤)) :=\n  filter.tendsto.mono_right (filter.tendsto_at_top_pure f) (pure_le_nhds (f ⊤))\n\n@[simp] protected instance nhds_ne_bot {α : Type u} [topological_space α] {a : α} : filter.ne_bot (nhds a) :=\n  filter.ne_bot_of_le (pure_le_nhds a)\n\n/-!\n### Cluster points\n\nIn this section we define [cluster points](https://en.wikipedia.org/wiki/Limit_point)\n(also known as limit points and accumulation points) of a filter and of a sequence.\n-/\n\n/-- A point `x` is a cluster point of a filter `F` if 𝓝 x ⊓ F ≠ ⊥. Also known as\nan accumulation point or a limit point. -/\ndef cluster_pt {α : Type u} [topological_space α] (x : α) (F : filter α) :=\n  filter.ne_bot (nhds x ⊓ F)\n\ntheorem cluster_pt.ne_bot {α : Type u} [topological_space α] {x : α} {F : filter α} (h : cluster_pt x F) : filter.ne_bot (nhds x ⊓ F) :=\n  h\n\ntheorem cluster_pt_iff {α : Type u} [topological_space α] {x : α} {F : filter α} : cluster_pt x F ↔ ∀ {U : set α}, U ∈ nhds x → ∀ {V : set α}, V ∈ F → set.nonempty (U ∩ V) :=\n  filter.inf_ne_bot_iff\n\n/-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty\nset. -/\ntheorem cluster_pt_principal_iff {α : Type u} [topological_space α] {x : α} {s : set α} : cluster_pt x (filter.principal s) ↔ ∀ (U : set α), U ∈ nhds x → set.nonempty (U ∩ s) :=\n  filter.inf_principal_ne_bot_iff\n\ntheorem cluster_pt_principal_iff_frequently {α : Type u} [topological_space α] {x : α} {s : set α} : cluster_pt x (filter.principal s) ↔ filter.frequently (fun (y : α) => y ∈ s) (nhds x) := sorry\n\ntheorem cluster_pt.of_le_nhds {α : Type u} [topological_space α] {x : α} {f : filter α} (H : f ≤ nhds x) [filter.ne_bot f] : cluster_pt x f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (cluster_pt x f)) (cluster_pt.equations._eqn_1 x f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (filter.ne_bot (nhds x ⊓ f))) (iff.mpr inf_eq_right H))) _inst_2)\n\ntheorem cluster_pt.of_le_nhds' {α : Type u} [topological_space α] {x : α} {f : filter α} (H : f ≤ nhds x) (hf : filter.ne_bot f) : cluster_pt x f :=\n  cluster_pt.of_le_nhds H\n\ntheorem cluster_pt.of_nhds_le {α : Type u} [topological_space α] {x : α} {f : filter α} (H : nhds x ≤ f) : cluster_pt x f := sorry\n\ntheorem cluster_pt.mono {α : Type u} [topological_space α] {x : α} {f : filter α} {g : filter α} (H : cluster_pt x f) (h : f ≤ g) : cluster_pt x g :=\n  ne_bot_of_le_ne_bot H (inf_le_inf_left (nhds x) h)\n\ntheorem cluster_pt.of_inf_left {α : Type u} [topological_space α] {x : α} {f : filter α} {g : filter α} (H : cluster_pt x (f ⊓ g)) : cluster_pt x f :=\n  cluster_pt.mono H inf_le_left\n\ntheorem cluster_pt.of_inf_right {α : Type u} [topological_space α] {x : α} {f : filter α} {g : filter α} (H : cluster_pt x (f ⊓ g)) : cluster_pt x g :=\n  cluster_pt.mono H inf_le_right\n\ntheorem ultrafilter.cluster_pt_iff {α : Type u} [topological_space α] {x : α} {f : ultrafilter α} : cluster_pt x ↑f ↔ ↑f ≤ nhds x :=\n  { mp := ultrafilter.le_of_inf_ne_bot' f, mpr := fun (h : ↑f ≤ nhds x) => cluster_pt.of_le_nhds h }\n\n/-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point\nof `map u F`. -/\ndef map_cluster_pt {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι) (u : ι → α) :=\n  cluster_pt x (filter.map u F)\n\ntheorem map_cluster_pt_iff {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι) (u : ι → α) : map_cluster_pt x F u ↔ ∀ (s : set α), s ∈ nhds x → filter.frequently (fun (a : ι) => u a ∈ s) F := sorry\n\ntheorem map_cluster_pt_of_comp {α : Type u} [topological_space α] {ι : Type u_1} {δ : Type u_2} {F : filter ι} {φ : δ → ι} {p : filter δ} {x : α} {u : ι → α} [filter.ne_bot p] (h : filter.tendsto φ p F) (H : filter.tendsto (u ∘ φ) p (nhds x)) : map_cluster_pt x F u :=\n  filter.ne_bot_of_le (le_inf H (trans_rel_right LessEq filter.map_map (filter.map_mono h)))\n\n/-!\n### Interior, closure and frontier in terms of neighborhoods\n-/\n\ntheorem interior_eq_nhds' {α : Type u} [topological_space α] {s : set α} : interior s = set_of fun (a : α) => s ∈ nhds a := sorry\n\ntheorem interior_eq_nhds {α : Type u} [topological_space α] {s : set α} : interior s = set_of fun (a : α) => nhds a ≤ filter.principal s := sorry\n\ntheorem mem_interior_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ interior s ↔ s ∈ nhds a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a ∈ interior s ↔ s ∈ nhds a)) interior_eq_nhds'))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((a ∈ set_of fun (a : α) => s ∈ nhds a) ↔ s ∈ nhds a)) set.mem_set_of_eq))\n      (iff.refl (s ∈ nhds a)))\n\ntheorem interior_set_of_eq {α : Type u} [topological_space α] {p : α → Prop} : interior (set_of fun (x : α) => p x) = set_of fun (x : α) => filter.eventually (fun (x : α) => p x) (nhds x) :=\n  interior_eq_nhds'\n\ntheorem is_open_set_of_eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} : is_open (set_of fun (x : α) => filter.eventually (fun (y : α) => p y) (nhds x)) := sorry\n\ntheorem subset_interior_iff_nhds {α : Type u} [topological_space α] {s : set α} {V : set α} : s ⊆ interior V ↔ ∀ (x : α), x ∈ s → V ∈ nhds x := sorry\n\ntheorem is_open_iff_nhds {α : Type u} [topological_space α] {s : set α} : is_open s ↔ ∀ (a : α), a ∈ s → nhds a ≤ filter.principal s :=\n  iff.trans (iff.symm subset_interior_iff_open)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (s ⊆ interior s ↔ ∀ (a : α), a ∈ s → nhds a ≤ filter.principal s)) interior_eq_nhds))\n      (iff.refl (s ⊆ set_of fun (a : α) => nhds a ≤ filter.principal s)))\n\ntheorem is_open_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} : is_open s ↔ ∀ (a : α), a ∈ s → s ∈ nhds a :=\n  iff.trans is_open_iff_nhds (forall_congr fun (_x : α) => imp_congr_right fun (_x_1 : _x ∈ s) => filter.le_principal_iff)\n\ntheorem is_open_iff_ultrafilter {α : Type u} [topological_space α] {s : set α} : is_open s ↔ ∀ (x : α), x ∈ s → ∀ (l : ultrafilter α), ↑l ≤ nhds x → s ∈ l := sorry\n\ntheorem mem_closure_iff_frequently {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ closure s ↔ filter.frequently (fun (x : α) => x ∈ s) (nhds a) := sorry\n\ntheorem filter.frequently.mem_closure {α : Type u} [topological_space α] {s : set α} {a : α} : filter.frequently (fun (x : α) => x ∈ s) (nhds a) → a ∈ closure s :=\n  iff.mpr mem_closure_iff_frequently\n\n/-- The set of cluster points of a filter is closed. In particular, the set of limit points\nof a sequence is closed. -/\ntheorem is_closed_set_of_cluster_pt {α : Type u} [topological_space α] {f : filter α} : is_closed (set_of fun (x : α) => cluster_pt x f) := sorry\n\ntheorem mem_closure_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ closure s ↔ cluster_pt a (filter.principal s) :=\n  iff.trans mem_closure_iff_frequently (iff.symm cluster_pt_principal_iff_frequently)\n\ntheorem closure_eq_cluster_pts {α : Type u} [topological_space α] {s : set α} : closure s = set_of fun (a : α) => cluster_pt a (filter.principal s) :=\n  set.ext fun (x : α) => mem_closure_iff_cluster_pt\n\ntheorem mem_closure_iff_nhds {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ closure s ↔ ∀ (t : set α), t ∈ nhds a → set.nonempty (t ∩ s) :=\n  iff.trans mem_closure_iff_cluster_pt cluster_pt_principal_iff\n\ntheorem mem_closure_iff_nhds' {α : Type u} [topological_space α] {s : set α} {a : α} : a ∈ closure s ↔ ∀ (t : set α), t ∈ nhds a → ∃ (y : ↥s), ↑y ∈ t := sorry\n\ntheorem mem_closure_iff_comap_ne_bot {α : Type u} [topological_space α] {A : set α} {x : α} : x ∈ closure A ↔ filter.ne_bot (filter.comap coe (nhds x)) := sorry\n\ntheorem mem_closure_iff_nhds_basis {α : Type u} {β : Type v} [topological_space α] {a : α} {p : β → Prop} {s : β → set α} (h : filter.has_basis (nhds a) p s) {t : set α} : a ∈ closure t ↔ ∀ (i : β), p i → ∃ (y : α), ∃ (H : y ∈ t), y ∈ s i := sorry\n\n/-- `x` belongs to the closure of `s` if and only if some ultrafilter\n  supported on `s` converges to `x`. -/\ntheorem mem_closure_iff_ultrafilter {α : Type u} [topological_space α] {s : set α} {x : α} : x ∈ closure s ↔ ∃ (u : ultrafilter α), s ∈ u ∧ ↑u ≤ nhds x := sorry\n\ntheorem is_closed_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} : is_closed s ↔ ∀ (a : α), cluster_pt a (filter.principal s) → a ∈ s := sorry\n\ntheorem is_closed_iff_nhds {α : Type u} [topological_space α] {s : set α} : is_closed s ↔ ∀ (x : α), (∀ (U : set α), U ∈ nhds x → set.nonempty (U ∩ s)) → x ∈ s := sorry\n\ntheorem closure_inter_open {α : Type u} [topological_space α] {s : set α} {t : set α} (h : is_open s) : s ∩ closure t ⊆ closure (s ∩ t) := sorry\n\n/-- The intersection of an open dense set with a dense set is a dense set. -/\ntheorem dense.inter_of_open_left {α : Type u} [topological_space α] {s : set α} {t : set α} (hs : dense s) (ht : dense t) (hso : is_open s) : dense (s ∩ t) := sorry\n\n/-- The intersection of a dense set with an open dense set is a dense set. -/\ntheorem dense.inter_of_open_right {α : Type u} [topological_space α] {s : set α} {t : set α} (hs : dense s) (ht : dense t) (hto : is_open t) : dense (s ∩ t) :=\n  set.inter_comm t s ▸ dense.inter_of_open_left ht hs hto\n\ntheorem dense.inter_nhds_nonempty {α : Type u} [topological_space α] {s : set α} {t : set α} (hs : dense s) {x : α} (ht : t ∈ nhds x) : set.nonempty (s ∩ t) := sorry\n\ntheorem closure_diff {α : Type u} [topological_space α] {s : set α} {t : set α} : closure s \\ closure t ⊆ closure (s \\ t) := sorry\n\ntheorem filter.frequently.mem_of_closed {α : Type u} [topological_space α] {a : α} {s : set α} (h : filter.frequently (fun (x : α) => x ∈ s) (nhds a)) (hs : is_closed s) : a ∈ s :=\n  is_closed.closure_subset hs (filter.frequently.mem_closure h)\n\ntheorem is_closed.mem_of_frequently_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} (hs : is_closed s) (h : filter.frequently (fun (x : β) => f x ∈ s) b) (hf : filter.tendsto f b (nhds a)) : a ∈ s := sorry\n\ntheorem is_closed.mem_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} [filter.ne_bot b] (hs : is_closed s) (hf : filter.tendsto f b (nhds a)) (h : filter.eventually (fun (x : β) => f x ∈ s) b) : a ∈ s :=\n  is_closed.mem_of_frequently_of_tendsto hs (filter.eventually.frequently h) hf\n\ntheorem mem_closure_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α} {b : filter β} {a : α} {s : set α} [filter.ne_bot b] (hf : filter.tendsto f b (nhds a)) (h : filter.eventually (fun (x : β) => f x ∈ s) b) : a ∈ closure s :=\n  is_closed.mem_of_tendsto is_closed_closure hf (filter.eventually.mono h (set.preimage_mono subset_closure))\n\n/-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.\nThen `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/\ntheorem tendsto_inf_principal_nhds_iff_of_forall_eq {α : Type u} {β : Type v} [topological_space α] {f : β → α} {l : filter β} {s : set β} {a : α} (h : ∀ (x : β), ¬x ∈ s → f x = a) : filter.tendsto f (l ⊓ filter.principal s) (nhds a) ↔ filter.tendsto f l (nhds a) := sorry\n\n/-!\n### Limits of filters in topological spaces\n-/\n\n/-- If `f` is a filter, then `Lim f` is a limit of the filter, if it exists. -/\ndef Lim {α : Type u} [topological_space α] [Nonempty α] (f : filter α) : α :=\n  classical.epsilon fun (a : α) => f ≤ nhds a\n\n/--\nIf `f` is a filter satisfying `ne_bot f`, then `Lim' f` is a limit of the filter, if it exists.\n-/\ndef Lim' {α : Type u} [topological_space α] (f : filter α) [filter.ne_bot f] : α :=\n  Lim f\n\n/--\nIf `F` is an ultrafilter, then `filter.ultrafilter.Lim F` is a limit of the filter, if it exists.\nNote that dot notation `F.Lim` can be used for `F : ultrafilter α`.\n-/\ndef ultrafilter.Lim {α : Type u} [topological_space α] : ultrafilter α → α :=\n  fun (F : ultrafilter α) => Lim' ↑F\n\n/-- If `f` is a filter in `β` and `g : β → α` is a function, then `lim f` is a limit of `g` at `f`,\nif it exists. -/\ndef lim {α : Type u} {β : Type v} [topological_space α] [Nonempty α] (f : filter β) (g : β → α) : α :=\n  Lim (filter.map g f)\n\n/-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Lim f)`. We formulate\nthis lemma with a `[nonempty α]` argument of `Lim` derived from `h` to make it useful for types\nwithout a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify\nthis instance with any other instance. -/\ntheorem le_nhds_Lim {α : Type u} [topological_space α] {f : filter α} (h : ∃ (a : α), f ≤ nhds a) : f ≤ nhds (Lim f) :=\n  classical.epsilon_spec h\n\n/-- If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (lim f g)`. We formulate\nthis lemma with a `[nonempty α]` argument of `lim` derived from `h` to make it useful for types\nwithout a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify\nthis instance with any other instance. -/\ntheorem tendsto_nhds_lim {α : Type u} {β : Type v} [topological_space α] {f : filter β} {g : β → α} (h : ∃ (a : α), filter.tendsto g f (nhds a)) : filter.tendsto g f (nhds (lim f g)) :=\n  le_nhds_Lim h\n\n/-!\n### Locally finite families\n-/\n\n/- locally finite family [General Topology (Bourbaki, 1995)] -/\n\n/-- A family of sets in `set α` is locally finite if at every point `x:α`,\n  there is a neighborhood of `x` which meets only finitely many sets in the family -/\ndef locally_finite {α : Type u} {β : Type v} [topological_space α] (f : β → set α) :=\n  ∀ (x : α), ∃ (t : set α), ∃ (H : t ∈ nhds x), set.finite (set_of fun (i : β) => set.nonempty (f i ∩ t))\n\ntheorem locally_finite_of_finite {α : Type u} {β : Type v} [topological_space α] {f : β → set α} (h : set.finite set.univ) : locally_finite f := sorry\n\ntheorem locally_finite_subset {α : Type u} {β : Type v} [topological_space α] {f₁ : β → set α} {f₂ : β → set α} (hf₂ : locally_finite f₂) (hf : ∀ (b : β), f₁ b ⊆ f₂ b) : locally_finite f₁ := sorry\n\ntheorem is_closed_Union_of_locally_finite {α : Type u} {β : Type v} [topological_space α] {f : β → set α} (h₁ : locally_finite f) (h₂ : ∀ (i : β), is_closed (f i)) : is_closed (set.Union fun (i : β) => f i) := sorry\n\n/-!\n### Continuity\n-/\n\n/-- A function between topological spaces is continuous if the preimage\n  of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. -/\nstructure continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α → β) \nwhere\n  is_open_preimage : ∀ (s : set β), is_open s → is_open (f ⁻¹' s)\n\ntheorem continuous_def {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} : continuous f ↔ ∀ (s : set β), is_open s → is_open (f ⁻¹' s) :=\n  { mp := fun (hf : continuous f) (s : set β) (hs : is_open s) => continuous.is_open_preimage hf s hs,\n    mpr := fun (h : ∀ (s : set β), is_open s → is_open (f ⁻¹' s)) => continuous.mk h }\n\ntheorem is_open.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set β} (h : is_open s) : is_open (f ⁻¹' s) :=\n  continuous.is_open_preimage hf s h\n\n/-- A function between topological spaces is continuous at a point `x₀`\nif `f x` tends to `f x₀` when `x` tends to `x₀`. -/\ndef continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α → β) (x : α) :=\n  filter.tendsto f (nhds x) (nhds (f x))\n\ntheorem continuous_at.tendsto {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} (h : continuous_at f x) : filter.tendsto f (nhds x) (nhds (f x)) :=\n  h\n\ntheorem continuous_at.preimage_mem_nhds {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} {t : set β} (h : continuous_at f x) (ht : t ∈ nhds (f x)) : f ⁻¹' t ∈ nhds x :=\n  h ht\n\ntheorem cluster_pt.map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {x : α} {la : filter α} {lb : filter β} (H : cluster_pt x la) {f : α → β} (hfc : continuous_at f x) (hf : filter.tendsto f la lb) : cluster_pt (f x) lb :=\n  ne_bot_of_le_ne_bot (iff.mpr (filter.map_ne_bot_iff f) H) (filter.tendsto.inf (continuous_at.tendsto hfc) hf)\n\ntheorem preimage_interior_subset_interior_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {s : set β} (hf : continuous f) : f ⁻¹' interior s ⊆ interior (f ⁻¹' s) :=\n  interior_maximal (set.preimage_mono interior_subset) (is_open.preimage hf is_open_interior)\n\ntheorem continuous_id {α : Type u_1} [topological_space α] : continuous id :=\n  iff.mpr continuous_def fun (s : set α) (h : is_open s) => h\n\ntheorem continuous.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} (hg : continuous g) (hf : continuous f) : continuous (g ∘ f) :=\n  iff.mpr continuous_def fun (s : set γ) (h : is_open s) => is_open.preimage hf (is_open.preimage hg h)\n\ntheorem continuous.iterate {α : Type u_1} [topological_space α] {f : α → α} (h : continuous f) (n : ℕ) : continuous (nat.iterate f n) :=\n  nat.rec_on n continuous_id fun (n : ℕ) (ihn : continuous (nat.iterate f n)) => continuous.comp ihn h\n\ntheorem continuous_at.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} {x : α} (hg : continuous_at g (f x)) (hf : continuous_at f x) : continuous_at (g ∘ f) x :=\n  filter.tendsto.comp hg hf\n\ntheorem continuous.tendsto {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) (x : α) : filter.tendsto f (nhds x) (nhds (f x)) := sorry\n\n/-- A version of `continuous.tendsto` that allows one to specify a simpler form of the limit.\nE.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/\ntheorem continuous.tendsto' {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) (x : α) (y : β) (h : f x = y) : filter.tendsto f (nhds x) (nhds y) :=\n  h ▸ continuous.tendsto hf x\n\ntheorem continuous.continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} (h : continuous f) : continuous_at f x :=\n  continuous.tendsto h x\n\ntheorem continuous_iff_continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} : continuous f ↔ ∀ (x : α), continuous_at f x := sorry\n\ntheorem continuous_at_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {x : α} {b : β} : continuous_at (fun (a : α) => b) x :=\n  tendsto_const_nhds\n\ntheorem continuous_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {b : β} : continuous fun (a : α) => b :=\n  iff.mpr continuous_iff_continuous_at fun (a : α) => continuous_at_const\n\ntheorem continuous_at_id {α : Type u_1} [topological_space α] {x : α} : continuous_at id x :=\n  continuous.continuous_at continuous_id\n\ntheorem continuous_at.iterate {α : Type u_1} [topological_space α] {f : α → α} {x : α} (hf : continuous_at f x) (hx : f x = x) (n : ℕ) : continuous_at (nat.iterate f n) x :=\n  nat.rec_on n continuous_at_id\n    fun (n : ℕ) (ihn : continuous_at (nat.iterate f n) x) =>\n      (fun (this : continuous_at (nat.iterate f n ∘ f) x) => this) (continuous_at.comp (Eq.symm hx ▸ ihn) hf)\n\ntheorem continuous_iff_is_closed {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} : continuous f ↔ ∀ (s : set β), is_closed s → is_closed (f ⁻¹' s) := sorry\n\ntheorem is_closed.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set β} (h : is_closed s) : is_closed (f ⁻¹' s) :=\n  iff.mp continuous_iff_is_closed hf s h\n\ntheorem continuous_at_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {x : α} : continuous_at f x ↔ ∀ (g : ultrafilter α), ↑g ≤ nhds x → filter.tendsto f (↑g) (nhds (f x)) :=\n  filter.tendsto_iff_ultrafilter f (nhds x) (nhds (f x))\n\ntheorem continuous_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} : continuous f ↔ ∀ (x : α) (g : ultrafilter α), ↑g ≤ nhds x → filter.tendsto f (↑g) (nhds (f x)) := sorry\n\n/-- A piecewise defined function `if p then f else g` is continuous, if both `f` and `g`\nare continuous, and they coincide on the frontier (boundary) of the set `{a | p a}`. -/\ntheorem continuous_if {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {p : α → Prop} {f : α → β} {g : α → β} {h : (a : α) → Decidable (p a)} (hp : ∀ (a : α), a ∈ frontier (set_of fun (a : α) => p a) → f a = g a) (hf : continuous f) (hg : continuous g) : continuous fun (a : α) => ite (p a) (f a) (g a) := sorry\n\n/-! ### Continuity and partial functions -/\n\n/-- Continuity of a partial function -/\ndef pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α →. β) :=\n  ∀ (s : set β), is_open s → is_open (pfun.preimage f s)\n\ntheorem open_dom_of_pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α →. β} (h : pcontinuous f) : is_open (pfun.dom f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (pfun.dom f))) (Eq.symm (pfun.preimage_univ f)))) (h set.univ is_open_univ)\n\ntheorem pcontinuous_iff' {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α →. β} : pcontinuous f ↔ ∀ {x : α} {y : β}, y ∈ f x → filter.ptendsto' f (nhds x) (nhds y) := sorry\n\n/-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/\ntheorem set.maps_to.closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set α} {t : set β} {f : α → β} (h : set.maps_to f s t) (hc : continuous f) : set.maps_to f (closure s) (closure t) := sorry\n\ntheorem image_closure_subset_closure_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} {s : set α} (h : continuous f) : f '' closure s ⊆ closure (f '' s) :=\n  set.maps_to.image_subset (set.maps_to.closure (set.maps_to_image f s) h)\n\ntheorem map_mem_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set α} {t : set β} {f : α → β} {a : α} (hf : continuous f) (ha : a ∈ closure s) (ht : ∀ (a : α), a ∈ s → f a ∈ t) : f a ∈ closure t :=\n  set.maps_to.closure ht hf ha\n\n/-!\n### Function with dense range\n-/\n\n/-- `f : ι → β` has dense range if its range (image) is a dense subset of β. -/\ndef dense_range {β : Type u_2} [topological_space β] {κ : Type u_5} (f : κ → β) :=\n  dense (set.range f)\n\n/-- A surjective map has dense range. -/\ntheorem function.surjective.dense_range {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} (hf : function.surjective f) : dense_range f := sorry\n\ntheorem dense_range_iff_closure_range {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} : dense_range f ↔ closure (set.range f) = set.univ :=\n  dense_iff_closure_eq\n\ntheorem dense_range.closure_range {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} (h : dense_range f) : closure (set.range f) = set.univ :=\n  dense.closure_eq h\n\ntheorem continuous.range_subset_closure_image_dense {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set α} (hs : dense s) : set.range f ⊆ closure (f '' s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (set.range f ⊆ closure (f '' s))) (Eq.symm set.image_univ)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (f '' set.univ ⊆ closure (f '' s))) (Eq.symm (dense.closure_eq hs))))\n      (image_closure_subset_closure_image hf))\n\n/-- The image of a dense set under a continuous map with dense range is a dense set. -/\ntheorem dense_range.dense_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf' : dense_range f) (hf : continuous f) {s : set α} (hs : dense s) : dense (f '' s) :=\n  dense.of_closure (dense.mono (continuous.range_subset_closure_image_dense hf hs) hf')\n\n/-- If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense\nset. -/\ntheorem dense_range.dense_of_maps_to {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {f : α → β} (hf' : dense_range f) (hf : continuous f) {s : set α} (hs : dense s) {t : set β} (ht : set.maps_to f s t) : dense t :=\n  dense.mono (set.maps_to.image_subset ht) (dense_range.dense_image hf' hf hs)\n\n/-- Composition of a continuous map with dense range and a function with dense range has dense\nrange. -/\ntheorem dense_range.comp {β : Type u_2} {γ : Type u_3} [topological_space β] [topological_space γ] {κ : Type u_5} {g : β → γ} {f : κ → β} (hg : dense_range g) (hf : dense_range f) (cg : continuous g) : dense_range (g ∘ f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (dense_range (g ∘ f))) (dense_range.equations._eqn_1 (g ∘ f))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dense (set.range (g ∘ f)))) (set.range_comp g f)))\n      (dense_range.dense_image hg cg hf))\n\ntheorem dense_range.nonempty_iff {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} (hf : dense_range f) : Nonempty κ ↔ Nonempty β :=\n  iff.trans (iff.symm set.range_nonempty_iff_nonempty) (dense.nonempty_iff hf)\n\ntheorem dense_range.nonempty {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} [h : Nonempty β] (hf : dense_range f) : Nonempty κ :=\n  iff.mpr (dense_range.nonempty_iff hf) h\n\n/-- Given a function `f : α → β` with dense range and `b : β`, returns some `a : α`. -/\ndef dense_range.some {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β} (hf : dense_range f) (b : β) : κ :=\n  Classical.choice sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.32020413171084566}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module tactic.rewrite\n! leanprover-community/mathlib commit a0735864ba72769da4b378673d3dbe2453924fde\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Leanbin.Data.Dlist\nimport Mathbin.Tactic.Core\n\nnamespace Tactic\n\nopen Expr List\n\nunsafe def match_fn (fn : expr) : expr → tactic (expr × expr)\n  | app (app fn' e₀) e₁ => unify fn fn' $> (e₀, e₁)\n  | _ => failed\n#align tactic.match_fn tactic.match_fn\n\nunsafe def fill_args : expr → tactic (expr × List expr)\n  | pi n bi d b => do\n    let v ← mk_meta_var d\n    let (r, vs) ← fill_args (b.instantiate_var v)\n    return (r, v :: vs)\n  | e => return (e, [])\n#align tactic.fill_args tactic.fill_args\n\nunsafe def mk_assoc_pattern' (fn : expr) : expr → tactic (Dlist expr)\n  | e =>\n    (do\n        let (e₀, e₁) ← match_fn fn e\n        (· ++ ·) <$> mk_assoc_pattern' e₀ <*> mk_assoc_pattern' e₁) <|>\n      pure (Dlist.singleton e)\n#align tactic.mk_assoc_pattern' tactic.mk_assoc_pattern'\n\nunsafe def mk_assoc_pattern (fn e : expr) : tactic (List expr) :=\n  Dlist.toList <$> mk_assoc_pattern' fn e\n#align tactic.mk_assoc_pattern tactic.mk_assoc_pattern\n\nunsafe def mk_assoc (fn : expr) : List expr → tactic expr\n  | [] => failed\n  | [x] => pure x\n  | x₀ :: x₁ :: xs => mk_assoc (fn x₀ x₁ :: xs)\n#align tactic.mk_assoc tactic.mk_assoc\n\nunsafe def chain_eq_trans : List expr → tactic expr\n  | [] => to_expr ``(rfl)\n  | [e] => pure e\n  | e :: es => chain_eq_trans es >>= mk_eq_trans e\n#align tactic.chain_eq_trans tactic.chain_eq_trans\n\nunsafe def unify_prefix : List expr → List expr → tactic Unit\n  | [], _ => pure ()\n  | _, [] => failed\n  | x :: xs, y :: ys => unify x y >> unify_prefix xs ys\n#align tactic.unify_prefix tactic.unify_prefix\n\nunsafe def match_assoc_pattern' (p : List expr) : List expr → tactic (List expr × List expr)\n  | es =>\n    unify_prefix p es $> ([], es.drop p.length) <|>\n      match es with\n      | [] => failed\n      | x :: xs => Prod.map (cons x) id <$> match_assoc_pattern' xs\n#align tactic.match_assoc_pattern' tactic.match_assoc_pattern'\n\nunsafe def match_assoc_pattern (fn p e : expr) : tactic (List expr × List expr) := do\n  let p' ← mk_assoc_pattern fn p\n  let e' ← mk_assoc_pattern fn e\n  match_assoc_pattern' p' e'\n#align tactic.match_assoc_pattern tactic.match_assoc_pattern\n\n/-- Tag for proofs generated by `assoc_rewrite`. -/\ndef IdTag.assocProof :=\n  ()\n#align tactic.id_tag.assoc_proof Tactic.IdTag.assocProof\n\nunsafe def mk_eq_proof (fn : expr) (e₀ e₁ : List expr) (p : expr) : tactic (expr × expr × expr) :=\n  do\n  let (l, r) ← infer_type p >>= match_eq\n  if e₀ ∧ e₁ then pure (l, r, p)\n    else do\n      let l' ← mk_assoc fn (e₀ ++ [l] ++ e₁)\n      let r' ← mk_assoc fn (e₀ ++ [r] ++ e₁)\n      let t ← infer_type l'\n      let v ← mk_local_def `x t\n      let e ← mk_assoc fn (e₀ ++ [v] ++ e₁)\n      let p ← mk_congr_arg (e [v]) p\n      let p' ← mk_app `` Eq [l', r']\n      let p' := mk_tagged_proof p' p `` id_tag.assoc_proof\n      return (l', r', p')\n#align tactic.mk_eq_proof tactic.mk_eq_proof\n\nunsafe def assoc_root (fn assoc : expr) : expr → tactic (expr × expr)\n  | e =>\n    (do\n        let (e₀, e₁) ← match_fn fn e\n        let (ea, eb) ← match_fn fn e₁\n        let e' := fn (fn e₀ ea) eb\n        let p' ← mk_eq_symm (assoc e₀ ea eb)\n        let (e'', p'') ← assoc_root e'\n        Prod.mk e'' <$> mk_eq_trans p' p'') <|>\n      Prod.mk e <$> mk_eq_refl e\n#align tactic.assoc_root tactic.assoc_root\n\nunsafe def assoc_refl' (fn assoc : expr) : expr → expr → tactic expr\n  | l, r =>\n    is_def_eq l r >> mk_eq_refl l <|> do\n      let (l', l_p) ← assoc_root fn assoc l <|> fail \"A\"\n      let (el₀, el₁) ← match_fn fn l' <|> fail \"B\"\n      let (r', r_p) ← assoc_root fn assoc r <|> fail \"C\"\n      let (er₀, er₁) ← match_fn fn r' <|> fail \"D\"\n      let p₀ ← assoc_refl' el₀ er₀\n      let p₁ ← is_def_eq el₁ er₁ >> mk_eq_refl el₁\n      let f_eq ← mk_congr_arg fn p₀ <|> fail \"G\"\n      let p' ← mk_congr f_eq p₁ <|> fail \"H\"\n      let r_p' ← mk_eq_symm r_p\n      chain_eq_trans [l_p, p', r_p']\n#align tactic.assoc_refl' tactic.assoc_refl'\n\nunsafe def assoc_refl (fn : expr) : tactic Unit := do\n  let (l, r) ← target >>= match_eq\n  let assoc ← mk_mapp `` IsAssociative.assoc [none, fn, none] <|> fail f! \"{fn} is not associative\"\n  assoc_refl' fn assoc l r >>= tactic.exact\n#align tactic.assoc_refl tactic.assoc_refl\n\nunsafe def flatten (fn assoc e : expr) : tactic (expr × expr) := do\n  let ls ← mk_assoc_pattern fn e\n  let e' ← mk_assoc fn ls\n  let p ← assoc_refl' fn assoc e e'\n  return (e', p)\n#align tactic.flatten tactic.flatten\n\nunsafe def assoc_rewrite_intl (assoc h e : expr) : tactic (expr × expr) := do\n  let t ← infer_type h\n  let (lhs, rhs) ← match_eq t\n  let fn := lhs.app_fn.app_fn\n  let (l, r) ← match_assoc_pattern fn lhs e\n  let (lhs', rhs', h') ← mk_eq_proof fn l r h\n  let e_p ← assoc_refl' fn assoc e lhs'\n  let (rhs'', rhs_p) ← flatten fn assoc rhs'\n  let final_p ← chain_eq_trans [e_p, h', rhs_p]\n  return (rhs'', final_p)\n#align tactic.assoc_rewrite_intl tactic.assoc_rewrite_intl\n\n-- TODO(Simon): visit expressions built of `fn` nested inside other such expressions:\n-- e.g.: x + f (a + b + c) + y should generate two rewrite candidates\nunsafe def enum_assoc_subexpr' (fn : expr) : expr → tactic (Dlist expr)\n  | e =>\n    Dlist.singleton e <$ (match_fn fn e >> guard ¬e.has_var) <|>\n      expr.mfoldl (fun es e' => (· ++ es) <$> enum_assoc_subexpr' e') Dlist.empty e\n#align tactic.enum_assoc_subexpr' tactic.enum_assoc_subexpr'\n\nunsafe def enum_assoc_subexpr (fn e : expr) : tactic (List expr) :=\n  Dlist.toList <$> enum_assoc_subexpr' fn e\n#align tactic.enum_assoc_subexpr tactic.enum_assoc_subexpr\n\nunsafe def mk_assoc_instance (fn : expr) : tactic expr := do\n  let t ← mk_mapp `` IsAssociative [none, fn]\n  let inst ←\n    Prod.snd <$> solve_aux t assumption <|>\n        mk_instance t >>= assertv `_inst t <|> fail f! \"{fn} is not associative\"\n  mk_mapp `` IsAssociative.assoc [none, fn, inst]\n#align tactic.mk_assoc_instance tactic.mk_assoc_instance\n\nunsafe def assoc_rewrite (h e : expr) (opt_assoc : Option expr := none) :\n    tactic (expr × expr × List expr) := do\n  let (t, vs) ← infer_type h >>= fill_args\n  let (lhs, rhs) ← match_eq t\n  let fn := lhs.app_fn.app_fn\n  let es ← enum_assoc_subexpr fn e\n  let assoc ←\n    match opt_assoc with\n      | none => mk_assoc_instance fn\n      | some assoc => pure assoc\n  let (_, p) ← firstM (assoc_rewrite_intl assoc <| h.mk_app vs) es\n  let (e', p', _) ← tactic.rewrite p e\n  pure (e', p', vs)\n#align tactic.assoc_rewrite tactic.assoc_rewrite\n\nunsafe def assoc_rewrite_target (h : expr) (opt_assoc : Option expr := none) : tactic Unit := do\n  let tgt ← target\n  let (tgt', p, _) ← assoc_rewrite h tgt opt_assoc\n  replace_target tgt' p\n#align tactic.assoc_rewrite_target tactic.assoc_rewrite_target\n\nunsafe def assoc_rewrite_hyp (h hyp : expr) (opt_assoc : Option expr := none) : tactic expr := do\n  let tgt ← infer_type hyp\n  let (tgt', p, _) ← assoc_rewrite h tgt opt_assoc\n  replace_hyp hyp tgt' p\n#align tactic.assoc_rewrite_hyp tactic.assoc_rewrite_hyp\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `eq_lemmas -/\nprivate unsafe def assoc_rw_goal (rs : List rw_rule) : tactic Unit :=\n  rs.mapM' fun r => do\n    save_info r\n    let eq_lemmas ← get_rule_eqn_lemmas r\n    orelse'\n        (do\n          let e ← to_expr' r\n          assoc_rewrite_target e)\n        (eq_lemmas fun n => do\n          let e ← mk_const n\n          assoc_rewrite_target e)\n        (eq_lemmas eq_lemmas.empty)\n#align tactic.interactive.assoc_rw_goal tactic.interactive.assoc_rw_goal\n\nprivate unsafe def uses_hyp (e : expr) (h : expr) : Bool :=\n  e.fold false fun t _ r => r || t = h\n#align tactic.interactive.uses_hyp tactic.interactive.uses_hyp\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `eq_lemmas -/\nprivate unsafe def assoc_rw_hyp : List rw_rule → expr → tactic Unit\n  | [], hyp => skip\n  | r :: rs, hyp => do\n    save_info r\n    let eq_lemmas ← get_rule_eqn_lemmas r\n    orelse'\n        (do\n          let e ← to_expr' r\n          when ¬uses_hyp e hyp <| assoc_rewrite_hyp e hyp >>= assoc_rw_hyp rs)\n        (eq_lemmas fun n => do\n          let e ← mk_const n\n          assoc_rewrite_hyp e hyp >>= assoc_rw_hyp rs)\n        (eq_lemmas eq_lemmas.empty)\n#align tactic.interactive.assoc_rw_hyp tactic.interactive.assoc_rw_hyp\n\nprivate unsafe def assoc_rw_core (rs : parse rw_rules) (loca : parse location) : tactic Unit :=\n  ((match loca with\n      | loc.wildcard => loca.try_apply (assoc_rw_hyp rs.rules) (assoc_rw_goal rs.rules)\n      | _ => loca.apply (assoc_rw_hyp rs.rules) (assoc_rw_goal rs.rules)) >>\n      try reflexivity) >>\n    try (returnopt rs.end_pos >>= save_info)\n#align tactic.interactive.assoc_rw_core tactic.interactive.assoc_rw_core\n\n/-- `assoc_rewrite [h₀,← h₁] at ⊢ h₂` behaves like `rewrite [h₀,← h₁] at ⊢ h₂`\nwith the exception that associativity is used implicitly to make rewriting\npossible.\n\nIt works for any function `f` for which an `is_associative f` instance can be found.\n\n```\nexample {α : Type*} (f : α → α → α) [is_associative α f] (a b c d x : α) :\n  let infix ` ~ ` := f in\n  b ~ c = x → (a ~ b ~ c ~ d) = (a ~ x ~ d) :=\nbegin\n  intro h,\n  assoc_rw h,\nend\n```\n-/\nunsafe def assoc_rewrite (q : parse rw_rules) (l : parse location) : tactic Unit :=\n  propagate_tags (assoc_rw_core q l)\n#align tactic.interactive.assoc_rewrite tactic.interactive.assoc_rewrite\n\n/-- synonym for `assoc_rewrite` -/\nunsafe def assoc_rw (q : parse rw_rules) (l : parse location) : tactic Unit :=\n  assoc_rewrite q l\n#align tactic.interactive.assoc_rw tactic.interactive.assoc_rw\n\nadd_tactic_doc\n  { Name := \"assoc_rewrite\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.assoc_rewrite, `tactic.interactive.assoc_rw]\n    tags := [\"rewriting\"]\n    inheritDescriptionFrom := `tactic.interactive.assoc_rewrite }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Rewrite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290152, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3202041317108456}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Jesse Michael Han\n-/\nimport tactic.rcases\nimport data.sum\nimport logic.function.basic\n\nuniverses u₁ u₂\n\nopen interactive interactive.types\nsection ext\nopen lean.parser nat tactic\n\n/--\n`derive_struct_ext_lemma n` generates two extensionality lemmas based on\nthe equality of all non-propositional projections.\n\nOn the following:\n\n```lean\n@[ext]\nstructure foo (α : Type*) :=\n(x y : ℕ)\n(z : {z // z < x})\n(k : α)\n(h : x < y)\n```\n\n`derive_struct_lemma` generates:\n\n```lean\nlemma foo.ext : ∀ {α : Type u_1} (x y : foo α),\n  x.x = y.x → x.y = y.y → x.z == y.z → x.k = y.k → x = y\nlemma foo.ext_iff : ∀ {α : Type u_1} (x y : foo α),\n  x = y ↔ x.x = y.x ∧ x.y = y.y ∧ x.z == y.z ∧ x.k = y.k\n```\n\n-/\nmeta def derive_struct_ext_lemma (n : name) : tactic name :=\ndo e ← get_env,\n   fs ← e.structure_fields n,\n   d ← get_decl n,\n   n ← resolve_constant n,\n   let r := @expr.const tt n $ d.univ_params.map level.param,\n   (args,_) ← infer_type r >>= open_pis,\n   let args := args.map expr.to_implicit_local_const,\n   let t := r.mk_app args,\n   x ← mk_local_def `x t,\n   y ← mk_local_def `y t,\n   let args_x := args ++ [x],\n   let args_y := args ++ [y],\n   bs ← fs.mmap $ λ f,\n     do { d ← get_decl (n ++ f),\n          let a := @expr.const tt (n ++ f) $ d.univ_params.map level.param,\n          t ← infer_type a,\n          s ← infer_type t,\n\n          if s ≠ `(Prop)\n            then do\n              let x := a.mk_app args_x,\n              let y := a.mk_app args_y,\n              t ← infer_type x,\n              t' ← infer_type y,\n              some <$> if t = t'\n                then mk_app `eq [x,y] >>= mk_local_def `h\n                else mk_mapp `heq [none,x,none,y] >>= mk_local_def `h\n            else pure none },\n   let bs := bs.filter_map id,\n   eq_t ← mk_app `eq [x,y],\n   t ← pis (args ++ [x,y] ++ bs) eq_t,\n   pr ← run_async $\n     do { (_,pr) ← solve_aux t (do\n          { args ← intron args.length,\n            x ← intro1, y ← intro1,\n            cases x, cases y,\n            bs.mmap' (λ _,\n              do e ← intro1,\n                 cases e),\n            reflexivity }),\n          instantiate_mvars pr },\n   let decl_n := n <.> \"ext\",\n   add_decl (declaration.thm decl_n d.univ_params t pr),\n   bs ← bs.mmap infer_type,\n   let rhs := expr.mk_and_lst bs,\n   iff_t ← mk_app `iff [eq_t,rhs],\n   t ← pis (args ++ [x,y]) iff_t,\n   pr ← run_async $\n     do { (_,pr) ← solve_aux t $ do\n          { args ← intron args.length,\n            x ← intro1, y ← intro1,\n            cases x, cases y,\n            split,\n            solve1 $ do\n            { h ← intro1, hs ← injection h, subst_vars,\n              repeat (refine ``( and.intro _ _ ) >> reflexivity ),\n              done <|> reflexivity },\n            solve1 $ do\n            { repeat (do refine ``(and_imp.mpr _),\n                         h ← intro1, cases h, skip ),\n              h ← intro1, cases h,\n              reflexivity }  },\n          instantiate_mvars pr },\n   add_decl (declaration.thm (n <.> \"ext_iff\") d.univ_params t pr),\n   pure decl_n\n\nmeta def get_ext_subject : expr → tactic name\n| (expr.pi n bi d b) :=\n  do v  ← mk_local' n bi d,\n     b' ← whnf $ b.instantiate_var v,\n     get_ext_subject b'\n| (expr.app _ e) :=\n  do t ← infer_type e >>= instantiate_mvars >>= head_beta,\n     if t.get_app_fn.is_constant then\n       pure $ t.get_app_fn.const_name\n     else if t.is_pi then\n       pure $ name.mk_numeral 0 name.anonymous\n     else if t.is_sort then\n       pure $ name.mk_numeral 1 name.anonymous\n     else do\n       t ← pp t,\n       fail format!\"only constants and Pi types are supported: {t}\"\n| e := fail format!\"Only expressions of the form `_ → _ → ... → R ... e are supported: {e}\"\n\nopen native\n\n@[reducible] def ext_param_type := option name ⊕ option name\n\nmeta def opt_minus : lean.parser (option name → ext_param_type) :=\nsum.inl <$ tk \"-\" <|> pure sum.inr\n\nmeta def ext_param :=\nopt_minus <*> ( name.mk_numeral 0 name.anonymous <$ brackets \"(\" \")\" (tk \"→\" <|> tk \"->\") <|>\n                none <$  tk \"*\" <|>\n                some <$> ident )\n\nmeta def saturate_fun : name → tactic expr\n| (name.mk_numeral 0 name.anonymous) :=\ndo v₀ ← mk_mvar,\n   v₁ ← mk_mvar,\n   return $ v₀.imp v₁\n| (name.mk_numeral 1 name.anonymous) :=\ndo u ← mk_meta_univ,\n   pure $ expr.sort u\n| n :=\ndo e ← resolve_constant n >>= mk_const,\n   a ← get_arity e,\n   e.mk_app <$> (list.iota a).mmap (λ _, mk_mvar)\n\nmeta def equiv_type_constr (n n' : name) : tactic unit :=\ndo e  ← saturate_fun n,\n   e' ← saturate_fun n',\n   unify e e' <|> fail format!\"{n} and {n'} are not definitionally equal types\"\n\nsection performance_hack\n/--\nFor performance reasons, it is inadvisable to use `user_attribute.get_param`.\nThe parameter is stored as a reflected expression.  When calling `get_param`,\nthe stored parameter is evaluated using `eval_expr`, which first compiles the\nexpression into VM bytecode. The unevaluated expression is available using\n`user_attribute.get_param_untyped`.\n\nIn particular, `user_attribute.get_param` MUST NEVER BE USED in the\nimplementation of an attribute cache. This is because calling `eval_expr`\ndisables the attribute cache.\n\nThere are several possible workarounds:\n 1. Set a different attribute depending on the parameter.\n 2. Use your own evaluation function instead of `eval_expr`, such as e.g. `expr.to_nat`.\n 3. Write your own `has_reflect Param` instance (using a more efficient serialization format).\n   The `user_attribute` code unfortunately checks whether the expression has the correct type,\n   but you can use `` `(id %%e : Param) `` to pretend that your expression `e` has type `Param`.\n-/\nlibrary_note \"user attribute parameters\"\n\n/-!\nFor performance reasons, the parameters of the `@[ext]` attribute are stored\nin two auxiliary attributes:\n```lean\nattribute [ext [thunk]] funext\n\n-- is turned into\nattribute [_ext_core (@id name @funext)] thunk\nattribute [_ext_lemma_core] funext\n```\n\nsee Note [user attribute parameters]\n-/\n\nlocal attribute [semireducible] reflected\n\nlocal attribute [instance, priority 9000]\nprivate meta def hacky_name_reflect : has_reflect name :=\nλ n, `(id %%(expr.const n []) : name)\n\n@[user_attribute]\nprivate meta def ext_attr_core : user_attribute (name_map name) name :=\n{ name := `_ext_core,\n  descr := \"(internal attribute used by ext)\",\n  cache_cfg := {\n    dependencies := [],\n    mk_cache := λ ns, do\n      attrs ← ns.mmap (λ n, do\n        ext_l ← ext_attr_core.get_param_untyped n,\n        pure (n, ext_l.app_arg.const_name)),\n      pure $ rb_map.of_list attrs },\n  parser := failure }\n\nend performance_hack\n\n/-- Private attribute used to tag extensionality lemmas. -/\n@[user_attribute]\nprivate meta def ext_lemma_attr_core : user_attribute :=\n{ name := `_ext_lemma_core,\n  descr := \"(internal attribute used by ext)\",\n  parser := failure }\n\n/--\nReturns the extensionality lemmas in the environment, as a map from structure\nname to lemma name.\n-/\nmeta def get_ext_lemmas : tactic (name_map name) :=\next_attr_core.get_cache\n\n/--\nReturns the extensionality lemmas in the environment, as a list of lemma names.\n-/\nmeta def get_ext_lemma_names : tactic (list name) :=\nattribute.get_instances ext_lemma_attr_core.name\n\n/--\nTag lemmas of the form:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : ∀ x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe attribute indexes extensionality lemma using the type of the\nobjects (i.e. `my_collection`) which it gets from the statement of\nthe lemma.  In some cases, the same lemma can be used to state the\nextensionality of multiple types that are definitionally equivalent.\n\n```lean\nattribute [ext [(→),thunk,stream]] funext\n```\n\nThose parameters are cumulative. The following are equivalent:\n\n```lean\nattribute [ext [(→),thunk]] funext\nattribute [ext [stream]] funext\n```\nand\n```lean\nattribute [ext [(→),thunk,stream]] funext\n```\n\nOne removes type names from the list for one lemma with:\n```lean\nattribute [ext [-stream,-thunk]] funext\n```\n\nAlso, the following:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : ∀ x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nis equivalent to\n\n```lean\n@[ext *]\nlemma my_collection.ext (a b : my_collection)\n  (h : ∀ x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThis allows us specify type synonyms along with the type\nthat is referred to in the lemma statement.\n\n```lean\n@[ext [*,my_type_synonym]]\nlemma my_collection.ext (a b : my_collection)\n  (h : ∀ x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe `ext` attribute can be applied to a structure to generate its extensionality lemmas:\n\n```lean\n@[ext]\nstructure foo (α : Type*) :=\n(x y : ℕ)\n(z : {z // z < x})\n(k : α)\n(h : x < y)\n```\n\nwill generate:\n\n```lean\n@[ext] lemma foo.ext : ∀ {α : Type u_1} (x y : foo α),\nx.x = y.x → x.y = y.y → x.z == y.z → x.k = y.k → x = y\nlemma foo.ext_iff : ∀ {α : Type u_1} (x y : foo α),\nx = y ↔ x.x = y.x ∧ x.y = y.y ∧ x.z == y.z ∧ x.k = y.k\n```\n\n-/\n@[user_attribute]\nmeta def extensional_attribute : user_attribute unit (list ext_param_type) :=\n{ name := `ext,\n  descr := \"lemmas usable by `ext` tactic\",\n  parser := pure <$> ext_param <|> list_of ext_param <|> pure [],\n  after_set := some $ λ n prio b,\n    do ls ← extensional_attribute.get_param n,\n       e ← get_env,\n       n ← if (e.structure_fields n).is_some\n         then derive_struct_ext_lemma n\n         else pure n,\n       s ← mk_const n >>= infer_type >>= get_ext_subject,\n       let (rs,ls'') := if ls.empty\n                           then ([],[s])\n                           else ls.partition_map (sum.map (flip option.get_or_else s)\n                                                    (flip option.get_or_else s)),\n       ls''.mmap' (equiv_type_constr s),\n       ls' ← get_ext_lemmas,\n       let l := ls'' ∪ (ls'.to_list.filter $ λ l, prod.snd l = n).map prod.fst \\ rs,\n       l.mmap' $ λ l, do\n        ext_attr_core.set l n b prio,\n        ext_lemma_attr_core.set n () b prio }\n\nadd_tactic_doc\n{ name                     := \"ext\",\n  category                 := doc_category.attr,\n  decl_names               := [`extensional_attribute],\n  tags                     := [\"rewrite\", \"logic\"] }\n\n/--\nWhen possible, `ext` lemmas are stated without a full set of arguments. As an example, for bundled\nhoms `f`, `g`, and `of`, `f.comp of = g.comp of → f = g` is a better `ext` lemma than\n`(∀ x, f (of x) = g (of x)) → f = g`, as the former allows a second type-specific extensionality\nlemmas to be applied to `f.comp of = g.comp of`.\nIf the domain of `of` is `ℕ` or `ℤ` and `of` is a `ring_hom`, such a lemma could then make the goal\n`f (of 1) = g (of 1)`.\n\nFor bundled morphisms, there is a `ext` lemma that always applies of the form\n`(∀ x, ⇑f x = ⇑g x) → f = g`. When adding type-specific `ext` lemmas like the one above, we want\nthese to be tried first. This happens automatically since the type-specific lemmas are inevitably\ndefined later.\n-/\nlibrary_note \"partially-applied ext lemmas\"\n\n-- We mark some existing extensionality lemmas.\nattribute [ext] array.ext propext function.hfunext\nattribute [ext [(→),thunk]] _root_.funext\n\n-- We create some extensionality lemmas for existing structures.\nattribute [ext] ulift\n\nnamespace plift\n-- This is stronger than the one generated automatically.\n@[ext] lemma ext {P : Prop} (a b : plift P) : a = b :=\nbegin\n  cases a, cases b, refl\nend\nend plift\n\n-- Conservatively, we'll only add extensionality lemmas for `has_*` structures\n-- as they become useful.\nattribute [ext] has_zero\n\n@[ext] lemma unit.ext {x y : unit} : x = y := by { cases x, cases y, refl, }\n@[ext] lemma punit.ext {x y : punit} : x = y := by { cases x, cases y, refl, }\n\nnamespace tactic\n\n/-- Helper structure for `ext` and `ext1`. `lemmas` keeps track of extensionality lemmas\n  applied so far. -/\nmeta structure ext_state : Type :=\n(patts : list rcases_patt := [])\n(trace_msg : list string := [])\n(fuel : option ℕ := none)\n\n/-- Helper function for `try_intros`. Additionally populates the `trace_msg` field\n  of `ext_state`. -/\nprivate meta def try_intros_core : state_t ext_state tactic unit :=\ndo ⟨patts, trace_msg, fuel⟩ ← get,\n   match patts with\n   | [] := do { es ← state_t.lift intros, when (es.length > 0) $ do\n                let msg := \"intros \" ++ (\" \".intercalate (es.map (λ e, e.local_pp_name.to_string))),\n                modify (λ ⟨patts, trace_msg, fuel⟩, ⟨patts, trace_msg ++ [msg], fuel⟩) }\n             <|> pure ()\n   | (x::xs) :=\n     do tgt ← state_t.lift (target >>= whnf),\n        when tgt.is_pi $\n          do state_t.lift (rintro [x]),\n             msg ← state_t.lift (((++) \"rintro \") <$> format.to_string <$> x.format ff),\n             modify (λ ⟨_, trace_msg, fuel⟩, ⟨xs, trace_msg ++ [msg], fuel⟩),\n             try_intros_core\n   end\n\n/-- Try to introduce as many arguments as possible, using the given patterns to destruct the\n  introduced variables. Returns the unused patterns. -/\nmeta def try_intros (patts : list rcases_patt) : tactic (list rcases_patt) :=\nlet σ := ext_state.mk patts [] none in\n  (ext_state.patts ∘ prod.snd) <$> state_t.run try_intros_core σ\n\n/-- Apply one extensionality lemma, and destruct the arguments using the patterns\n  in the ext_state. -/\nmeta def ext1_core (cfg : apply_cfg := {}) : state_t ext_state tactic unit :=\ndo ⟨patts, trace_msg, _⟩ ← get,\n   (new_msgs) ← state_t.lift $ focus1 $\n   do { m ← get_ext_lemmas,\n         subject ← (target >>= get_ext_subject),\n         new_trace_msg ←\n           do { rule ← (m.find subject),\n                (applyc rule cfg),\n                pure ([\"apply \" ++ rule.to_string]) } <|>\n             do { ls ← get_ext_lemma_names,\n                  let nms := ls.map name.to_string,\n                  rule ← (ls.any_of (λ n, applyc n cfg *> pure n)),\n                  pure ([\"apply \" ++ rule.to_string]) } <|>\n               (fail format!\"no applicable extensionality rule found for {subject}\"),\n         pure new_trace_msg },\n    modify (λ ⟨patts, trace_msg, fuel⟩, ⟨patts, trace_msg ++ new_msgs, fuel⟩),\n    try_intros_core\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns. -/\nmeta def ext_core (cfg : apply_cfg := {}) : state_t ext_state tactic unit :=\ndo acc@⟨_, _, fuel⟩ ← get,\n   match fuel with\n   | (some 0) := pure ()\n   | n        := do { ext1_core cfg,\n                      modify (λ ⟨patts, lemmas, _⟩, ⟨patts, lemmas, nat.pred <$> n⟩),\n                      ext_core <|> pure () }\n   end\n\n/-- Apply one extensionality lemma, and destruct the arguments using the given patterns.\n  Returns the unused patterns. -/\nmeta def ext1 (xs : list rcases_patt) (cfg : apply_cfg := {})\n  (trace : bool := ff) : tactic (list rcases_patt) :=\ndo ⟨_, σ⟩ ← state_t.run (ext1_core cfg) {patts := xs},\n   when trace $ tactic.trace $ \"Try this: \" ++  \", \".intercalate σ.trace_msg,\n   pure σ.patts\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns.\n  `ext ps (some n)` applies at most `n` extensionality lemmas. Returns the unused patterns. -/\nmeta def ext (xs : list rcases_patt) (fuel : option ℕ) (cfg : apply_cfg := {})\n  (trace : bool := ff): tactic (list rcases_patt) :=\ndo ⟨_, σ⟩ ← state_t.run (ext_core cfg) {patts := xs, fuel := fuel},\n   when trace $ tactic.trace $ \"Try this: \" ++  \", \".intercalate σ.trace_msg,\n   pure σ.patts\n\nlocal postfix `?`:9001 := optional\nlocal postfix *:9001 := many\n\n/--\n`ext1 id` selects and apply one extensionality lemma (with attribute\n`ext`), using `id`, if provided, to name a local constant\nintroduced by the lemma. If `id` is omitted, the local constant is\nnamed automatically, as per `intro`. Placing a `?` after `ext1`\n (e.g. `ext1? i ⟨a,b⟩ : 3`) will display a sequence of tactic\napplications that can replace the call to `ext1`.\n-/\nmeta def interactive.ext1 (trace : parse (tk \"?\")?)\n  (xs : parse (rcases_patt_parse tt)*) : tactic unit :=\next1 xs {} trace.is_some $> ()\n\n/--\n- `ext` applies as many extensionality lemmas as possible;\n- `ext ids`, with `ids` a list of identifiers, finds extentionality and applies them\n  until it runs out of identifiers in `ids` to name the local constants.\n- `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext` (e.g. `ext? i ⟨a,b⟩ : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`.\n\nWhen trying to prove:\n\n```lean\nα β : Type,\nf g : α → set β\n⊢ f = g\n```\n\napplying `ext x y` yields:\n\n```lean\nα β : Type,\nf g : α → set β,\nx : α,\ny : β\n⊢ y ∈ f x ↔ y ∈ f x\n```\n\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\nα β γ : Type\nf g : α × β → γ\n⊢ f = g\n```\n\napplying `ext ⟨a, b⟩` yields:\n\n```lean\nα β γ : Type,\nf g : α × β → γ,\na : α,\nb : β\n⊢ f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? ⟨a,b⟩` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro ⟨a, b⟩\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nmeta def interactive.ext :\n  (parse $ (tk \"?\")?) → parse (rcases_patt_parse tt)* → parse (tk \":\" *> small_nat)? → tactic unit\n | trace [] (some n)  := iterate_range 1 n (ext1 [] {} trace.is_some $> ())\n | trace [] none      := repeat1 (ext1 [] {} trace.is_some $> ())\n | trace xs n         := ext xs n {} trace.is_some $> ()\n\n/--\n* `ext1 id` selects and apply one extensionality lemma (with\n  attribute `ext`), using `id`, if provided, to name a\n  local constant introduced by the lemma. If `id` is omitted, the\n  local constant is named automatically, as per `intro`.\n\n* `ext` applies as many extensionality lemmas as possible;\n* `ext ids`, with `ids` a list of identifiers, finds extensionality lemmas\n  and applies them until it runs out of identifiers in `ids` to name\n  the local constants.\n* `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext`/`ext1` (e.g. `ext? i ⟨a,b⟩ : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`/`ext1`.\n\nWhen trying to prove:\n\n```lean\nα β : Type,\nf g : α → set β\n⊢ f = g\n```\n\napplying `ext x y` yields:\n\n```lean\nα β : Type,\nf g : α → set β,\nx : α,\ny : β\n⊢ y ∈ f x ↔ y ∈ g x\n```\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\nα β γ : Type\nf g : α × β → γ\n⊢ f = g\n```\n\napplying `ext ⟨a, b⟩` yields:\n\n```lean\nα β γ : Type,\nf g : α × β → γ,\na : α,\nb : β\n⊢ f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? ⟨a,b⟩` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro ⟨a, b⟩\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nadd_tactic_doc\n{ name        := \"ext1 / ext\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.ext1, `tactic.interactive.ext],\n  tags        := [\"rewriting\", \"logic\"] }\n\nend tactic\nend ext\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/tactic/ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3202041236470688}}
{"text": "/-\nCopyright (c) 2019 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,\n  Heather Macbeth\n-/\nimport topology.algebra.ring\nimport topology.algebra.mul_action\nimport topology.algebra.uniform_group\nimport topology.uniform_space.uniform_embedding\nimport algebra.algebra.basic\nimport linear_algebra.projection\nimport linear_algebra.pi\nimport linear_algebra.determinant\n\n/-!\n# Theory of topological modules and continuous linear maps.\n\nWe use the class `has_continuous_smul` for topological (semi) modules and topological vector spaces.\n\nIn this file we define continuous (semi-)linear maps, as semilinear maps between topological\nmodules which are continuous. The set of continuous semilinear maps between the topological\n`R₁`-module `M` and `R₂`-module `M₂` with respect to the `ring_hom` `σ` is denoted by `M →SL[σ] M₂`.\nPlain linear maps are denoted by `M →L[R] M₂` and star-linear maps by `M →L⋆[R] M₂`.\n\nThe corresponding notation for equivalences is `M ≃SL[σ] M₂`, `M ≃L[R] M₂` and `M ≃L⋆[R] M₂`.\n-/\n\nopen filter\nopen_locale topological_space big_operators filter\n\nuniverses u v w u'\n\nsection\n\nvariables {R : Type*} {M : Type*}\n[ring R] [topological_space R]\n[topological_space M] [add_comm_group M]\n[module R M]\n\nlemma has_continuous_smul.of_nhds_zero [topological_ring R] [topological_add_group M]\n  (hmul : tendsto (λ p : R × M, p.1 • p.2) (𝓝 0 ×ᶠ (𝓝 0)) (𝓝 0))\n  (hmulleft : ∀ m : M, tendsto (λ a : R, a • m) (𝓝 0) (𝓝 0))\n  (hmulright : ∀ a : R, tendsto (λ m : M, a • m) (𝓝 0) (𝓝 0)) : has_continuous_smul R M :=\n⟨begin\n  rw continuous_iff_continuous_at,\n  rintros ⟨a₀, m₀⟩,\n  have key : ∀ p : R × M,\n    p.1 • p.2 = a₀ • m₀ + ((p.1 - a₀) • m₀ + a₀ • (p.2 - m₀) + (p.1 - a₀) • (p.2 - m₀)),\n  { rintro ⟨a, m⟩,\n    simp [sub_smul, smul_sub],\n    abel },\n  rw funext key, clear key,\n  refine tendsto_const_nhds.add (tendsto.add (tendsto.add _ _) _),\n  { rw [sub_self, zero_smul],\n    apply (hmulleft m₀).comp,\n    rw [show (λ p : R × M, p.1 - a₀) = (λ a, a - a₀) ∘ prod.fst, by {ext, refl }, nhds_prod_eq],\n    have : tendsto (λ a, a - a₀) (𝓝 a₀) (𝓝 0),\n    { rw ← sub_self a₀,\n      exact tendsto_id.sub tendsto_const_nhds },\n    exact this.comp tendsto_fst  },\n  { rw [sub_self, smul_zero],\n    apply (hmulright a₀).comp,\n    rw [show (λ p : R × M, p.2 - m₀) = (λ m, m - m₀) ∘ prod.snd, by {ext, refl }, nhds_prod_eq],\n    have : tendsto (λ m, m - m₀) (𝓝 m₀) (𝓝 0),\n    { rw ← sub_self m₀,\n      exact tendsto_id.sub tendsto_const_nhds },\n    exact this.comp tendsto_snd },\n  { rw [sub_self, zero_smul, nhds_prod_eq,\n        show (λ p : R × M, (p.fst - a₀) • (p.snd - m₀)) =\n             (λ  p : R × M, p.1 • p.2) ∘ (prod.map (λ a, a - a₀) (λ m, m - m₀)), by { ext, refl }],\n    apply hmul.comp (tendsto.prod_map _ _);\n    { rw ← sub_self ,\n      exact tendsto_id.sub tendsto_const_nhds } },\nend⟩\nend\n\nsection\nvariables {R : Type*} {M : Type*}\n[ring R] [topological_space R]\n[topological_space M] [add_comm_group M] [has_continuous_add M]\n[module R M] [has_continuous_smul R M]\n\n/-- If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then\n`⊤` is the only submodule of `M` with a nonempty interior.\nThis is the case, e.g., if `R` is a nondiscrete normed field. -/\nlemma submodule.eq_top_of_nonempty_interior'\n  [ne_bot (𝓝[{x : R | is_unit x}] 0)]\n  (s : submodule R M) (hs : (interior (s:set M)).nonempty) :\n  s = ⊤ :=\nbegin\n  rcases hs with ⟨y, hy⟩,\n  refine (submodule.eq_top_iff'.2 $ λ x, _),\n  rw [mem_interior_iff_mem_nhds] at hy,\n  have : tendsto (λ c:R, y + c • x) (𝓝[{x : R | is_unit x}] 0) (𝓝 (y + (0:R) • x)),\n    from tendsto_const_nhds.add ((tendsto_nhds_within_of_tendsto_nhds tendsto_id).smul\n      tendsto_const_nhds),\n  rw [zero_smul, add_zero] at this,\n  obtain ⟨_, hu : y + _ • _ ∈ s, u, rfl⟩ :=\n    nonempty_of_mem (inter_mem (mem_map.1 (this hy)) self_mem_nhds_within),\n  have hy' : y ∈ ↑s := mem_of_mem_nhds hy,\n  rwa [s.add_mem_iff_right hy', ←units.smul_def, s.smul_mem_iff' u] at hu,\nend\n\nvariables (R M)\n\n/-- Let `R` be a topological ring such that zero is not an isolated point (e.g., a nondiscrete\nnormed field, see `normed_field.punctured_nhds_ne_bot`). Let `M` be a nontrivial module over `R`\nsuch that `c • x = 0` implies `c = 0 ∨ x = 0`. Then `M` has no isolated points. We formulate this\nusing `ne_bot (𝓝[≠] x)`.\n\nThis lemma is not an instance because Lean would need to find `[has_continuous_smul ?m_1 M]` with\nunknown `?m_1`. We register this as an instance for `R = ℝ` in `real.punctured_nhds_module_ne_bot`.\nOne can also use `haveI := module.punctured_nhds_ne_bot R M` in a proof.\n-/\nlemma module.punctured_nhds_ne_bot [nontrivial M] [ne_bot (𝓝[≠] (0 : R))]\n  [no_zero_smul_divisors R M] (x : M) :\n  ne_bot (𝓝[≠] x) :=\nbegin\n  rcases exists_ne (0 : M) with ⟨y, hy⟩,\n  suffices : tendsto (λ c : R, x + c • y) (𝓝[≠] 0) (𝓝[≠] x), from this.ne_bot,\n  refine tendsto.inf _ (tendsto_principal_principal.2 $ _),\n  { convert tendsto_const_nhds.add ((@tendsto_id R _).smul_const y),\n    rw [zero_smul, add_zero] },\n  { intros c hc,\n    simpa [hy] using hc }\nend\n\nend\n\nsection lattice_ops\n\nvariables {ι R M₁ M₂ : Type*} [semiring R] [add_comm_monoid M₁] [add_comm_monoid M₂]\n  [module R M₁] [module R M₂] [u : topological_space R] {t : topological_space M₂}\n  [has_continuous_smul R M₂] (f : M₁ →ₗ[R] M₂)\n\nlemma has_continuous_smul_induced :\n  @has_continuous_smul R M₁ _ u (t.induced f) :=\n{ continuous_smul :=\n    begin\n      letI : topological_space M₁ := t.induced f,\n      refine continuous_induced_rng _,\n      simp_rw [function.comp, f.map_smul],\n      refine continuous_fst.smul (continuous_induced_dom.comp continuous_snd)\n    end }\n\nend lattice_ops\n\nnamespace submodule\n\nvariables {α β : Type*} [topological_space β]\n\ninstance [topological_space α] [semiring α] [add_comm_monoid β] [module α β]\n  [has_continuous_smul α β] (S : submodule α β) :\n  has_continuous_smul α S :=\n{ continuous_smul :=\n  begin\n    rw embedding_subtype_coe.to_inducing.continuous_iff,\n    exact continuous_fst.smul\n      (continuous_subtype_coe.comp continuous_snd)\n  end }\n\ninstance [ring α] [add_comm_group β] [module α β] [topological_add_group β] (S : submodule α β) :\n  topological_add_group S :=\nS.to_add_subgroup.topological_add_group\n\nend submodule\n\nsection closure\nvariables {R : Type u} {M : Type v}\n[semiring R] [topological_space R]\n[topological_space M] [add_comm_monoid M]\n[module R M] [has_continuous_smul R M]\n\nlemma submodule.closure_smul_self_subset (s : submodule R M) :\n  (λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M))\n  ⊆ closure (s : set M) :=\ncalc\n(λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M))\n    = (λ p : R × M, p.1 • p.2) '' (closure ((set.univ : set R) ×ˢ (s : set M))) :\n  by simp [closure_prod_eq]\n... ⊆ closure ((λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ (s : set M))) :\n  image_closure_subset_closure_image continuous_smul\n... = closure s : begin\n  congr,\n  ext x,\n  refine ⟨_, λ hx, ⟨⟨1, x⟩, ⟨set.mem_univ _, hx⟩, one_smul R _⟩⟩,\n  rintros ⟨⟨c, y⟩, ⟨hc, hy⟩, rfl⟩,\n  simp [s.smul_mem c hy]\nend\n\nlemma submodule.closure_smul_self_eq (s : submodule R M) :\n  (λ p : R × M, p.1 • p.2) '' ((set.univ : set R) ×ˢ closure (s : set M))\n  = closure (s : set M) :=\nset.subset.antisymm s.closure_smul_self_subset\n  (λ x hx, ⟨⟨1, x⟩, ⟨set.mem_univ _, hx⟩, one_smul R _⟩)\n\nvariables [has_continuous_add M]\n\n/-- The (topological-space) closure of a submodule of a topological `R`-module `M` is itself\na submodule. -/\ndef submodule.topological_closure (s : submodule R M) : submodule R M :=\n{ carrier := closure (s : set M),\n  smul_mem' := λ c x hx, s.closure_smul_self_subset ⟨⟨c, x⟩, ⟨set.mem_univ _, hx⟩, rfl⟩,\n  ..s.to_add_submonoid.topological_closure }\n\n@[simp] lemma submodule.topological_closure_coe (s : submodule R M) :\n  (s.topological_closure : set M) = closure (s : set M) :=\nrfl\n\ninstance submodule.topological_closure_has_continuous_smul (s : submodule R M) :\n  has_continuous_smul R (s.topological_closure) :=\n{ continuous_smul :=\n  begin\n    apply continuous_induced_rng,\n    change continuous (λ p : R × s.topological_closure, p.1 • (p.2 : M)),\n    continuity,\n  end,\n  ..s.to_add_submonoid.topological_closure_has_continuous_add }\n\nlemma submodule.submodule_topological_closure (s : submodule R M) :\n  s ≤ s.topological_closure :=\nsubset_closure\n\nlemma submodule.is_closed_topological_closure (s : submodule R M) :\n  is_closed (s.topological_closure : set M) :=\nby convert is_closed_closure\n\nlemma submodule.topological_closure_minimal\n  (s : submodule R M) {t : submodule R M} (h : s ≤ t) (ht : is_closed (t : set M)) :\n  s.topological_closure ≤ t :=\nclosure_minimal h ht\n\nlemma submodule.topological_closure_mono {s : submodule R M} {t : submodule R M} (h : s ≤ t) :\n  s.topological_closure ≤ t.topological_closure :=\ns.topological_closure_minimal (h.trans t.submodule_topological_closure)\n  t.is_closed_topological_closure\n\nend closure\n\n/-- Continuous linear maps between modules. We only put the type classes that are necessary for the\ndefinition, although in applications `M` and `M₂` will be topological modules over the topological\nring `R`. -/\nstructure continuous_linear_map\n  {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S)\n  (M : Type*) [topological_space M] [add_comm_monoid M]\n  (M₂ : Type*) [topological_space M₂] [add_comm_monoid M₂]\n  [module R M] [module S M₂]\n  extends M →ₛₗ[σ] M₂ :=\n(cont : continuous to_fun . tactic.interactive.continuity')\n\nnotation M ` →SL[`:25 σ `] ` M₂ := continuous_linear_map σ M M₂\nnotation M ` →L[`:25 R `] ` M₂ := continuous_linear_map (ring_hom.id R) M M₂\nnotation M ` →L⋆[`:25 R `] ` M₂ := continuous_linear_map (star_ring_end R) M M₂\n\n/-- Continuous linear equivalences between modules. We only put the type classes that are necessary\nfor the definition, although in applications `M` and `M₂` will be topological modules over the\ntopological semiring `R`. -/\n@[nolint has_inhabited_instance]\nstructure continuous_linear_equiv\n  {R : Type*} {S : Type*} [semiring R] [semiring S] (σ : R →+* S)\n  {σ' : S →+* R} [ring_hom_inv_pair σ σ'] [ring_hom_inv_pair σ' σ]\n  (M : Type*) [topological_space M] [add_comm_monoid M]\n  (M₂ : Type*) [topological_space M₂] [add_comm_monoid M₂]\n  [module R M] [module S M₂]\n  extends M ≃ₛₗ[σ] M₂ :=\n(continuous_to_fun  : continuous to_fun . tactic.interactive.continuity')\n(continuous_inv_fun : continuous inv_fun . tactic.interactive.continuity')\n\nnotation M ` ≃SL[`:50 σ `] ` M₂ := continuous_linear_equiv σ M M₂\nnotation M ` ≃L[`:50 R `] ` M₂ := continuous_linear_equiv (ring_hom.id R) M M₂\nnotation M ` ≃L⋆[`:50 R `] ` M₂ := continuous_linear_equiv (star_ring_end R) M M₂\n\nsection pointwise_limits\n\nvariables\n{M₁ M₂ α R S : Type*}\n[topological_space M₂] [t2_space M₂] [semiring R] [semiring S]\n[add_comm_monoid M₁] [add_comm_monoid M₂] [module R M₁] [module S M₂]\n[has_continuous_const_smul S M₂]\n\nsection\n\nvariables (M₁ M₂) (σ : R →+* S)\n\nlemma is_closed_set_of_map_smul : is_closed {f : M₁ → M₂ | ∀ c x, f (c • x) = σ c • f x} :=\nbegin\n  simp only [set.set_of_forall],\n  exact is_closed_Inter (λ c, is_closed_Inter (λ x, is_closed_eq (continuous_apply _)\n    ((continuous_apply _).const_smul _)))\nend\n\nend\n\nvariables [has_continuous_add M₂] {σ : R →+* S} {l : filter α}\n\n/-- Constructs a bundled linear map from a function and a proof that this function belongs to the\nclosure of the set of linear maps. -/\n@[simps { fully_applied := ff }] def linear_map_of_mem_closure_range_coe (f : M₁ → M₂)\n  (hf : f ∈ closure (set.range (coe_fn : (M₁ →ₛₗ[σ] M₂) → (M₁ → M₂)))) :\n  M₁ →ₛₗ[σ] M₂ :=\n{ to_fun := f,\n  map_smul' := (is_closed_set_of_map_smul M₁ M₂ σ).closure_subset_iff.2\n    (set.range_subset_iff.2 linear_map.map_smulₛₗ) hf,\n  .. add_monoid_hom_of_mem_closure_range_coe f hf }\n\n/-- Construct a bundled linear map from a pointwise limit of linear maps -/\n@[simps { fully_applied := ff }]\ndef linear_map_of_tendsto (f : M₁ → M₂) (g : α → M₁ →ₛₗ[σ] M₂) [l.ne_bot]\n  (h : tendsto (λ a x, g a x) l (𝓝 f)) : M₁ →ₛₗ[σ] M₂ :=\nlinear_map_of_mem_closure_range_coe f $ mem_closure_of_tendsto h $\n  eventually_of_forall $ λ a, set.mem_range_self _\n\nvariables (M₁ M₂ σ)\n\nlemma linear_map.is_closed_range_coe :\n  is_closed (set.range (coe_fn : (M₁ →ₛₗ[σ] M₂) → (M₁ → M₂))) :=\nis_closed_of_closure_subset $ λ f hf, ⟨linear_map_of_mem_closure_range_coe f hf, rfl⟩\n\nend pointwise_limits\n\nnamespace continuous_linear_map\n\nsection semiring\n/-!\n### Properties that hold for non-necessarily commutative semirings.\n-/\n\nvariables\n{R₁ : Type*} {R₂ : Type*} {R₃ : Type*} [semiring R₁] [semiring R₂] [semiring R₃]\n{σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃}\n{M₁ : Type*} [topological_space M₁] [add_comm_monoid M₁]\n{M'₁ : Type*} [topological_space M'₁] [add_comm_monoid M'₁]\n{M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂]\n{M₃ : Type*} [topological_space M₃] [add_comm_monoid M₃]\n{M₄ : Type*} [topological_space M₄] [add_comm_monoid M₄]\n[module R₁ M₁] [module R₁ M'₁] [module R₂ M₂] [module R₃ M₃]\n\n/-- Coerce continuous linear maps to linear maps. -/\ninstance : has_coe (M₁ →SL[σ₁₂] M₂) (M₁ →ₛₗ[σ₁₂] M₂) := ⟨to_linear_map⟩\n\n-- make the coercion the preferred form\n@[simp] lemma to_linear_map_eq_coe (f : M₁ →SL[σ₁₂] M₂) : f.to_linear_map = f := rfl\n\ntheorem coe_injective : function.injective (coe : (M₁ →SL[σ₁₂] M₂) → (M₁ →ₛₗ[σ₁₂] M₂)) :=\nby { intros f g H, cases f, cases g, congr' }\n\ninstance : add_monoid_hom_class (M₁ →SL[σ₁₂] M₂) M₁ M₂ :=\n{ coe := λ f, f.to_fun,\n  coe_injective' := λ f g h, coe_injective (fun_like.coe_injective h),\n  map_add := λ f, map_add f.to_linear_map,\n  map_zero := λ f, linear_map.map_zero f }\n\n/-- Coerce continuous linear maps to functions. -/\n-- see Note [function coercion]\ninstance to_fun : has_coe_to_fun (M₁ →SL[σ₁₂] M₂) (λ _, M₁ → M₂) := ⟨λ f, f.to_fun⟩\n\n@[simp] lemma coe_mk (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ →ₛₗ[σ₁₂] M₂) = f := rfl\n@[simp] lemma coe_mk' (f : M₁ →ₛₗ[σ₁₂] M₂) (h) : (mk f h : M₁ → M₂) = f := rfl\n\n@[continuity]\nprotected lemma continuous (f : M₁ →SL[σ₁₂] M₂) : continuous f := f.2\n\nprotected lemma uniform_continuous {E₁ E₂ : Type*} [uniform_space E₁] [uniform_space E₂]\n  [add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]\n  [uniform_add_group E₁] [uniform_add_group E₂] (f : E₁ →SL[σ₁₂] E₂) :\n  uniform_continuous f :=\nuniform_continuous_add_monoid_hom_of_continuous f.continuous\n\n@[simp, norm_cast] lemma coe_inj {f g : M₁ →SL[σ₁₂] M₂} :\n  (f : M₁ →ₛₗ[σ₁₂] M₂) = g ↔ f = g :=\ncoe_injective.eq_iff\n\ntheorem coe_fn_injective : @function.injective (M₁ →SL[σ₁₂] M₂) (M₁ → M₂) coe_fn :=\nfun_like.coe_injective\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef simps.apply (h : M₁ →SL[σ₁₂] M₂) : M₁ → M₂ := h\n\n/-- See Note [custom simps projection]. -/\ndef simps.coe (h : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂ := h\n\ninitialize_simps_projections continuous_linear_map\n  (to_linear_map_to_fun → apply, to_linear_map → coe)\n\n@[ext] theorem ext {f g : M₁ →SL[σ₁₂] M₂} (h : ∀ x, f x = g x) : f = g :=\nfun_like.ext f g h\n\ntheorem ext_iff {f g : M₁ →SL[σ₁₂] M₂} : f = g ↔ ∀ x, f x = g x :=\nfun_like.ext_iff\n\n/-- Copy of a `continuous_linear_map` with a new `to_fun` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy (f : M₁ →SL[σ₁₂] M₂) (f' : M₁ → M₂) (h : f' = ⇑f) : M₁ →SL[σ₁₂] M₂ :=\n{ to_linear_map := f.to_linear_map.copy f' h,\n  cont := show continuous f', from h.symm ▸ f.continuous }\n\n-- make some straightforward lemmas available to `simp`.\nprotected lemma map_zero (f : M₁ →SL[σ₁₂] M₂) : f (0 : M₁) = 0 := map_zero f\nprotected lemma map_add (f : M₁ →SL[σ₁₂] M₂) (x y : M₁) : f (x + y) = f x + f y := map_add f x y\n@[simp] lemma map_smulₛₗ (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (x : M₁) :\n  f (c • x) = (σ₁₂ c) • f x := (to_linear_map _).map_smulₛₗ _ _\n\n@[simp] lemma map_smul [module R₁ M₂] (f : M₁ →L[R₁] M₂)(c : R₁) (x : M₁) : f (c • x) = c • f x :=\nby simp only [ring_hom.id_apply, map_smulₛₗ]\n\n@[simp, priority 900]\nlemma map_smul_of_tower {R S : Type*} [semiring S] [has_scalar R M₁]\n  [module S M₁] [has_scalar R M₂] [module S M₂]\n  [linear_map.compatible_smul M₁ M₂ R S] (f : M₁ →L[S] M₂) (c : R) (x : M₁) :\n  f (c • x) = c • f x :=\nlinear_map.compatible_smul.map_smul f c x\n\nprotected lemma map_sum {ι : Type*} (f : M₁ →SL[σ₁₂] M₂) (s : finset ι) (g : ι → M₁) :\n  f (∑ i in s, g i) = ∑ i in s, f (g i) := f.to_linear_map.map_sum\n\n@[simp, norm_cast] lemma coe_coe (f : M₁ →SL[σ₁₂] M₂) : ⇑(f : M₁ →ₛₗ[σ₁₂] M₂) = f := rfl\n\n@[ext] theorem ext_ring [topological_space R₁] {f g : R₁ →L[R₁] M₁} (h : f 1 = g 1) : f = g :=\ncoe_inj.1 $ linear_map.ext_ring h\n\ntheorem ext_ring_iff [topological_space R₁] {f g : R₁ →L[R₁] M₁} : f = g ↔ f 1 = g 1 :=\n⟨λ h, h ▸ rfl, ext_ring⟩\n\n/-- If two continuous linear maps are equal on a set `s`, then they are equal on the closure\nof the `submodule.span` of this set. -/\nlemma eq_on_closure_span [t2_space M₂] {s : set M₁} {f g : M₁ →SL[σ₁₂] M₂} (h : set.eq_on f g s) :\n  set.eq_on f g (closure (submodule.span R₁ s : set M₁)) :=\n(linear_map.eq_on_span' h).closure f.continuous g.continuous\n\n/-- If the submodule generated by a set `s` is dense in the ambient module, then two continuous\nlinear maps equal on `s` are equal. -/\nlemma ext_on [t2_space M₂] {s : set M₁} (hs : dense (submodule.span R₁ s : set M₁))\n  {f g : M₁ →SL[σ₁₂] M₂} (h : set.eq_on f g s) :\n  f = g :=\next $ λ x, eq_on_closure_span h (hs x)\n\n/-- Under a continuous linear map, the image of the `topological_closure` of a submodule is\ncontained in the `topological_closure` of its image. -/\nlemma _root_.submodule.topological_closure_map [ring_hom_surjective σ₁₂] [topological_space R₁]\n  [topological_space R₂] [has_continuous_smul R₁ M₁] [has_continuous_add M₁]\n  [has_continuous_smul R₂ M₂] [has_continuous_add M₂] (f : M₁ →SL[σ₁₂] M₂) (s : submodule R₁ M₁) :\n  (s.topological_closure.map (f : M₁ →ₛₗ[σ₁₂] M₂))\n  ≤ (s.map (f : M₁ →ₛₗ[σ₁₂] M₂)).topological_closure :=\nimage_closure_subset_closure_image f.continuous\n\n/-- Under a dense continuous linear map, a submodule whose `topological_closure` is `⊤` is sent to\nanother such submodule.  That is, the image of a dense set under a map with dense range is dense.\n-/\nlemma _root_.dense_range.topological_closure_map_submodule [ring_hom_surjective σ₁₂]\n  [topological_space R₁] [topological_space R₂] [has_continuous_smul R₁ M₁] [has_continuous_add M₁]\n  [has_continuous_smul R₂ M₂] [has_continuous_add M₂] {f : M₁ →SL[σ₁₂] M₂} (hf' : dense_range f)\n  {s : submodule R₁ M₁} (hs : s.topological_closure = ⊤) :\n  (s.map (f : M₁ →ₛₗ[σ₁₂] M₂)).topological_closure = ⊤ :=\nbegin\n  rw set_like.ext'_iff at hs ⊢,\n  simp only [submodule.topological_closure_coe, submodule.top_coe, ← dense_iff_closure_eq] at hs ⊢,\n  exact hf'.dense_image f.continuous hs\nend\n\nsection smul_monoid\n\nvariables {S₂ T₂ : Type*} [monoid S₂] [monoid T₂]\nvariables [distrib_mul_action S₂ M₂] [smul_comm_class R₂ S₂ M₂] [has_continuous_const_smul S₂ M₂]\nvariables [distrib_mul_action T₂ M₂] [smul_comm_class R₂ T₂ M₂] [has_continuous_const_smul T₂ M₂]\n\ninstance : mul_action S₂ (M₁ →SL[σ₁₂] M₂) :=\n{ smul := λ c f, ⟨c • f, (f.2.const_smul _ : continuous (λ x, c • f x))⟩,\n  one_smul := λ f, ext $ λ x, one_smul _ _,\n  mul_smul := λ a b f, ext $ λ x, mul_smul _ _ _ }\n\nlemma smul_apply (c : S₂) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (c • f) x = c • (f x) := rfl\n@[simp, norm_cast]\nlemma coe_smul (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : (↑(c • f) : M₁ →ₛₗ[σ₁₂] M₂) = c • f := rfl\n@[simp, norm_cast] lemma coe_smul' (c : S₂) (f : M₁ →SL[σ₁₂] M₂) : ⇑(c • f) = c • f := rfl\n\ninstance [has_scalar S₂ T₂] [is_scalar_tower S₂ T₂ M₂] : is_scalar_tower S₂ T₂ (M₁ →SL[σ₁₂] M₂) :=\n⟨λ a b f, ext $ λ x, smul_assoc a b (f x)⟩\n\ninstance [smul_comm_class S₂ T₂ M₂] : smul_comm_class S₂ T₂ (M₁ →SL[σ₁₂] M₂) :=\n⟨λ a b f, ext $ λ x, smul_comm a b (f x)⟩\n\nend smul_monoid\n\n/-- The continuous map that is constantly zero. -/\ninstance: has_zero (M₁ →SL[σ₁₂] M₂) := ⟨⟨0, continuous_zero⟩⟩\ninstance : inhabited (M₁ →SL[σ₁₂] M₂) := ⟨0⟩\n\n@[simp] lemma default_def : (default : M₁ →SL[σ₁₂] M₂) = 0 := rfl\n@[simp] lemma zero_apply (x : M₁) : (0 : M₁ →SL[σ₁₂] M₂) x = 0 := rfl\n@[simp, norm_cast] lemma coe_zero : ((0 : M₁ →SL[σ₁₂] M₂) : M₁ →ₛₗ[σ₁₂] M₂) = 0 := rfl\n/- no simp attribute on the next line as simp does not always simplify `0 x` to `0`\nwhen `0` is the zero function, while it does for the zero continuous linear map,\nand this is the most important property we care about. -/\n@[norm_cast] lemma coe_zero' : ⇑(0 : M₁ →SL[σ₁₂] M₂) = 0 := rfl\n\ninstance unique_of_left [subsingleton M₁] : unique (M₁ →SL[σ₁₂] M₂) :=\ncoe_injective.unique\n\ninstance unique_of_right [subsingleton M₂] : unique (M₁ →SL[σ₁₂] M₂) :=\ncoe_injective.unique\n\nlemma exists_ne_zero {f : M₁ →SL[σ₁₂] M₂} (hf : f ≠ 0) : ∃ x, f x ≠ 0 :=\nby { by_contra' h, exact hf (continuous_linear_map.ext h) }\n\nsection\n\nvariables (R₁ M₁)\n\n/-- the identity map as a continuous linear map. -/\ndef id : M₁ →L[R₁] M₁ :=\n⟨linear_map.id, continuous_id⟩\n\nend\n\ninstance : has_one (M₁ →L[R₁] M₁) := ⟨id R₁ M₁⟩\n\nlemma one_def : (1 : M₁ →L[R₁] M₁) = id R₁ M₁ := rfl\nlemma id_apply (x : M₁) : id R₁ M₁ x = x := rfl\n@[simp, norm_cast] lemma coe_id : (id R₁ M₁ : M₁ →ₗ[R₁] M₁) = linear_map.id := rfl\n@[simp, norm_cast] lemma coe_id' : ⇑(id R₁ M₁) = _root_.id := rfl\n\n@[simp, norm_cast] lemma coe_eq_id {f : M₁ →L[R₁] M₁} :\n  (f : M₁ →ₗ[R₁] M₁) = linear_map.id ↔ f = id _ _ :=\nby rw [← coe_id, coe_inj]\n\n@[simp] lemma one_apply (x : M₁) : (1 : M₁ →L[R₁] M₁) x = x := rfl\n\nsection add\nvariables [has_continuous_add M₂]\n\ninstance : has_add (M₁ →SL[σ₁₂] M₂) :=\n⟨λ f g, ⟨f + g, f.2.add g.2⟩⟩\n\n@[simp] lemma add_apply (f g : M₁ →SL[σ₁₂] M₂)  (x : M₁) : (f + g) x = f x + g x := rfl\n@[simp, norm_cast] lemma coe_add (f g : M₁ →SL[σ₁₂] M₂) : (↑(f + g) : M₁ →ₛₗ[σ₁₂] M₂) = f + g := rfl\n@[norm_cast] lemma coe_add' (f g : M₁ →SL[σ₁₂] M₂) : ⇑(f + g) = f + g := rfl\n\ninstance : add_comm_monoid (M₁ →SL[σ₁₂] M₂) :=\n{ zero := (0 : M₁ →SL[σ₁₂] M₂),\n  add := (+),\n  zero_add := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_zero := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_comm := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_assoc := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  nsmul := (•),\n  nsmul_zero' := λ f, by { ext, simp },\n  nsmul_succ' := λ n f, by { ext, simp [nat.succ_eq_one_add, add_smul] } }\n\n@[simp, norm_cast] lemma coe_sum {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :\n  ↑(∑ d in t, f d) = (∑ d in t, f d : M₁ →ₛₗ[σ₁₂] M₂) :=\n(add_monoid_hom.mk (coe : (M₁ →SL[σ₁₂] M₂) → (M₁ →ₛₗ[σ₁₂] M₂)) rfl (λ _ _, rfl)).map_sum _ _\n\n@[simp, norm_cast] lemma coe_sum' {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) :\n  ⇑(∑ d in t, f d) = ∑ d in t, f d :=\nby simp only [← coe_coe, coe_sum, linear_map.coe_fn_sum]\n\nlemma sum_apply {ι : Type*} (t : finset ι) (f : ι → M₁ →SL[σ₁₂] M₂) (b : M₁) :\n  (∑ d in t, f d) b = ∑ d in t, f d b :=\nby simp only [coe_sum', finset.sum_apply]\n\nend add\n\nvariables [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃]\n\n/-- Composition of bounded linear maps. -/\ndef comp (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) : M₁ →SL[σ₁₃] M₃ :=\n⟨(g : M₂ →ₛₗ[σ₂₃] M₃).comp ↑f, g.2.comp f.2⟩\n\ninfixr ` ∘L `:80 := @continuous_linear_map.comp _ _ _ _ _ _\n  (ring_hom.id _) (ring_hom.id _) (ring_hom.id _) _ _ _ _ _ _ _ _ _ _ _ _ ring_hom_comp_triple.ids\n\n@[simp, norm_cast] lemma coe_comp (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :\n  (h.comp f : M₁ →ₛₗ[σ₁₃] M₃) = (h : M₂ →ₛₗ[σ₂₃] M₃).comp (f : M₁ →ₛₗ[σ₁₂] M₂) := rfl\n\ninclude σ₁₃\n@[simp, norm_cast] lemma coe_comp' (h : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :\n  ⇑(h.comp f) = h ∘ f := rfl\n\nlemma comp_apply (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : (g.comp f) x = g (f x) := rfl\nomit σ₁₃\n\n@[simp] theorem comp_id (f : M₁ →SL[σ₁₂] M₂) : f.comp (id R₁ M₁) = f :=\next $ λ x, rfl\n\n@[simp] theorem id_comp (f : M₁ →SL[σ₁₂] M₂) : (id R₂ M₂).comp f = f :=\next $ λ x, rfl\n\ninclude σ₁₃\n@[simp] theorem comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁₂] M₂) = 0 :=\nby { ext, simp }\n\n@[simp] theorem zero_comp (f : M₁ →SL[σ₁₂] M₂) : (0 : M₂ →SL[σ₂₃] M₃).comp f = 0 :=\nby { ext, simp }\n\n@[simp] lemma comp_add [has_continuous_add M₂] [has_continuous_add M₃]\n  (g : M₂ →SL[σ₂₃] M₃) (f₁ f₂ : M₁ →SL[σ₁₂] M₂) :\n  g.comp (f₁ + f₂) = g.comp f₁ + g.comp f₂ :=\nby { ext, simp }\n\n@[simp] lemma add_comp [has_continuous_add M₃]\n  (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :\n  (g₁ + g₂).comp f = g₁.comp f + g₂.comp f :=\nby { ext, simp }\nomit σ₁₃\n\ntheorem comp_assoc {R₄ : Type*} [semiring R₄] [module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄}\n  {σ₃₄ : R₃ →+* R₄} [ring_hom_comp_triple σ₁₃ σ₃₄ σ₁₄] [ring_hom_comp_triple σ₂₃ σ₃₄ σ₂₄]\n  [ring_hom_comp_triple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃)\n  (f : M₁ →SL[σ₁₂] M₂) :\n  (h.comp g).comp f = h.comp (g.comp f) :=\nrfl\n\ninstance : has_mul (M₁ →L[R₁] M₁) := ⟨comp⟩\n\nlemma mul_def (f g : M₁ →L[R₁] M₁) : f * g = f.comp g := rfl\n\n@[simp] lemma coe_mul (f g : M₁ →L[R₁] M₁) : ⇑(f * g) = f ∘ g := rfl\n\nlemma mul_apply (f g : M₁ →L[R₁] M₁) (x : M₁) : (f * g) x = f (g x) := rfl\n\ninstance : monoid_with_zero (M₁ →L[R₁] M₁) :=\n{ mul := (*),\n  one := 1,\n  zero := 0,\n  mul_zero := λ f, ext $ λ _, map_zero f,\n  zero_mul := λ _, ext $ λ _, rfl,\n  mul_one := λ _, ext $ λ _, rfl,\n  one_mul := λ _, ext $ λ _, rfl,\n  mul_assoc := λ _ _ _, ext $ λ _, rfl, }\n\ninstance [has_continuous_add M₁] : semiring (M₁ →L[R₁] M₁) :=\n{ mul := (*),\n  one := 1,\n  left_distrib := λ f g h, ext $ λ x, map_add f (g x) (h x),\n  right_distrib := λ _ _ _, ext $ λ _, linear_map.add_apply _ _ _,\n  ..continuous_linear_map.monoid_with_zero,\n  ..continuous_linear_map.add_comm_monoid }\n\n/-- `continuous_linear_map.to_linear_map` as a `ring_hom`.-/\n@[simps]\ndef to_linear_map_ring_hom [has_continuous_add M₁] : (M₁ →L[R₁] M₁) →+* (M₁ →ₗ[R₁] M₁) :=\n{ to_fun := to_linear_map,\n  map_zero' := rfl,\n  map_one' := rfl,\n  map_add' := λ _ _, rfl,\n  map_mul' := λ _ _, rfl }\n\nsection apply_action\nvariables [has_continuous_add M₁]\n\n/-- The tautological action by `M₁ →L[R₁] M₁` on `M`.\n\nThis generalizes `function.End.apply_mul_action`. -/\ninstance apply_module : module (M₁ →L[R₁] M₁) M₁ :=\nmodule.comp_hom _ to_linear_map_ring_hom\n\n@[simp] protected lemma smul_def (f : M₁ →L[R₁] M₁) (a : M₁) : f • a = f a := rfl\n\n/-- `continuous_linear_map.apply_module` is faithful. -/\ninstance apply_has_faithful_smul : has_faithful_smul (M₁ →L[R₁] M₁) M₁ :=\n⟨λ _ _, continuous_linear_map.ext⟩\n\ninstance apply_smul_comm_class : smul_comm_class R₁ (M₁ →L[R₁] M₁) M₁ :=\n{ smul_comm := λ r e m, (e.map_smul r m).symm }\n\ninstance apply_smul_comm_class' : smul_comm_class (M₁ →L[R₁] M₁) R₁ M₁ :=\n{ smul_comm := continuous_linear_map.map_smul }\n\ninstance : has_continuous_const_smul (M₁ →L[R₁] M₁) M₁ :=\n⟨continuous_linear_map.continuous⟩\n\nend apply_action\n\n/-- The cartesian product of two bounded linear maps, as a bounded linear map. -/\nprotected def prod [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₁ →L[R₁] M₃) :\n  M₁ →L[R₁] (M₂ × M₃) :=\n⟨(f₁ : M₁ →ₗ[R₁] M₂).prod f₂, f₁.2.prod_mk f₂.2⟩\n\n@[simp, norm_cast] lemma coe_prod [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂)\n  (f₂ : M₁ →L[R₁] M₃) :\n  (f₁.prod f₂ : M₁ →ₗ[R₁] M₂ × M₃) = linear_map.prod f₁ f₂ :=\nrfl\n\n@[simp, norm_cast] lemma prod_apply [module R₁ M₂] [module R₁ M₃] (f₁ : M₁ →L[R₁] M₂)\n  (f₂ : M₁ →L[R₁] M₃) (x : M₁) :\n  f₁.prod f₂ x = (f₁ x, f₂ x) :=\nrfl\n\nsection\n\nvariables (R₁ M₁ M₂)\n\n/-- The left injection into a product is a continuous linear map. -/\ndef inl [module R₁ M₂] : M₁ →L[R₁] M₁ × M₂ := (id R₁ M₁).prod 0\n\n/-- The right injection into a product is a continuous linear map. -/\ndef inr [module R₁ M₂] : M₂ →L[R₁] M₁ × M₂ := (0 : M₂ →L[R₁] M₁).prod (id R₁ M₂)\n\nend\n\n@[simp] lemma inl_apply [module R₁ M₂] (x : M₁) : inl R₁ M₁ M₂ x = (x, 0) := rfl\n@[simp] lemma inr_apply [module R₁ M₂] (x : M₂) : inr R₁ M₁ M₂ x = (0, x) := rfl\n\n@[simp, norm_cast] lemma coe_inl [module R₁ M₂] :\n  (inl R₁ M₁ M₂ : M₁ →ₗ[R₁] M₁ × M₂) = linear_map.inl R₁ M₁ M₂ := rfl\n@[simp, norm_cast] lemma coe_inr [module R₁ M₂] :\n  (inr R₁ M₁ M₂ : M₂ →ₗ[R₁] M₁ × M₂) = linear_map.inr R₁ M₁ M₂ := rfl\n\n/-- Kernel of a continuous linear map. -/\ndef ker (f : M₁ →SL[σ₁₂] M₂) : submodule R₁ M₁ := (f : M₁ →ₛₗ[σ₁₂] M₂).ker\n\n@[norm_cast] lemma ker_coe (f : M₁ →SL[σ₁₂] M₂) : (f : M₁ →ₛₗ[σ₁₂] M₂).ker = f.ker := rfl\n\n@[simp] lemma mem_ker {f : M₁ →SL[σ₁₂] M₂} {x} : x ∈ f.ker ↔ f x = 0 := linear_map.mem_ker\n\nlemma is_closed_ker [t1_space M₂] (f : M₁ →SL[σ₁₂] M₂) : is_closed (f.ker : set M₁) :=\ncontinuous_iff_is_closed.1 f.cont _ is_closed_singleton\n\n@[simp] lemma apply_ker (f : M₁ →SL[σ₁₂] M₂) (x : f.ker) : f x = 0 := mem_ker.1 x.2\n\nlemma is_complete_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']\n  [module R₁ M'] [t1_space M₂] (f : M' →SL[σ₁₂] M₂) :\n  is_complete (f.ker : set M') :=\nf.is_closed_ker.is_complete\n\ninstance complete_space_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']\n  [module R₁ M'] [t1_space M₂] (f : M' →SL[σ₁₂] M₂) :\n  complete_space f.ker :=\nf.is_closed_ker.complete_space_coe\n\n@[simp] lemma ker_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :\n  ker (f.prod g) = ker f ⊓ ker g :=\nlinear_map.ker_prod f g\n\n/-- Range of a continuous linear map. -/\ndef range [ring_hom_surjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) : submodule R₂ M₂ :=\n(f : M₁ →ₛₗ[σ₁₂] M₂).range\n\nlemma range_coe [ring_hom_surjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) : (f.range : set M₂) = set.range f :=\nlinear_map.range_coe _\nlemma mem_range [ring_hom_surjective σ₁₂] {f : M₁ →SL[σ₁₂] M₂} {y} : y ∈ f.range ↔ ∃ x, f x = y :=\nlinear_map.mem_range\n\nlemma mem_range_self [ring_hom_surjective σ₁₂] (f : M₁ →SL[σ₁₂] M₂) (x : M₁) : f x ∈ f.range :=\nmem_range.2 ⟨x, rfl⟩\n\nlemma range_prod_le [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :\n  range (f.prod g) ≤ (range f).prod (range g) :=\n(f : M₁ →ₗ[R₁] M₂).range_prod_le g\n\n/-- Restrict codomain of a continuous linear map. -/\ndef cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :\n  M₁ →SL[σ₁₂] p :=\n{ cont := continuous_subtype_mk h f.continuous,\n  to_linear_map := (f : M₁ →ₛₗ[σ₁₂] M₂).cod_restrict p h}\n\n@[norm_cast] lemma coe_cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :\n  (f.cod_restrict p h : M₁ →ₛₗ[σ₁₂] p) = (f : M₁ →ₛₗ[σ₁₂] M₂).cod_restrict p h :=\nrfl\n\n@[simp] lemma coe_cod_restrict_apply (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p)\n  (x) :\n  (f.cod_restrict p h x : M₂) = f x :=\nrfl\n\n@[simp] lemma ker_cod_restrict (f : M₁ →SL[σ₁₂] M₂) (p : submodule R₂ M₂) (h : ∀ x, f x ∈ p) :\n  ker (f.cod_restrict p h) = ker f :=\n(f : M₁ →ₛₗ[σ₁₂] M₂).ker_cod_restrict p h\n\n/-- Embedding of a submodule into the ambient space as a continuous linear map. -/\ndef subtype_val (p : submodule R₁ M₁) : p →L[R₁] M₁ :=\n{ cont := continuous_subtype_val,\n  to_linear_map := p.subtype }\n\n@[simp, norm_cast] lemma coe_subtype_val (p : submodule R₁ M₁) :\n  (subtype_val p : p →ₗ[R₁] M₁) = p.subtype :=\nrfl\n\n@[simp, norm_cast] lemma subtype_val_apply (p : submodule R₁ M₁) (x : p) :\n  (subtype_val p : p → M₁) x = x :=\nrfl\n\nvariables (R₁ M₁ M₂)\n\n/-- `prod.fst` as a `continuous_linear_map`. -/\ndef fst [module R₁ M₂] : M₁ × M₂ →L[R₁] M₁ :=\n{ cont := continuous_fst, to_linear_map := linear_map.fst R₁ M₁ M₂ }\n\n/-- `prod.snd` as a `continuous_linear_map`. -/\ndef snd [module R₁ M₂] : M₁ × M₂ →L[R₁] M₂ :=\n{ cont := continuous_snd, to_linear_map := linear_map.snd R₁ M₁ M₂ }\n\nvariables {R₁ M₁ M₂}\n\n@[simp, norm_cast] lemma coe_fst [module R₁ M₂] : ↑(fst R₁ M₁ M₂) = linear_map.fst R₁ M₁ M₂ := rfl\n\n@[simp, norm_cast] lemma coe_fst' [module R₁ M₂] : ⇑(fst R₁ M₁ M₂) = prod.fst := rfl\n\n@[simp, norm_cast] lemma coe_snd [module R₁ M₂] : ↑(snd R₁ M₁ M₂) = linear_map.snd R₁ M₁ M₂ := rfl\n\n@[simp, norm_cast] lemma coe_snd' [module R₁ M₂] : ⇑(snd R₁ M₁ M₂) = prod.snd := rfl\n\n@[simp] lemma fst_prod_snd [module R₁ M₂] : (fst R₁ M₁ M₂).prod (snd R₁ M₁ M₂) = id R₁ (M₁ × M₂) :=\n  ext $ λ ⟨x, y⟩, rfl\n\n@[simp] lemma fst_comp_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :\n  (fst R₁ M₂ M₃).comp (f.prod g) = f :=\next $ λ x, rfl\n\n@[simp] lemma snd_comp_prod [module R₁ M₂] [module R₁ M₃] (f : M₁ →L[R₁] M₂) (g : M₁ →L[R₁] M₃) :\n  (snd R₁ M₂ M₃).comp (f.prod g) = g :=\next $ λ x, rfl\n\n/-- `prod.map` of two continuous linear maps. -/\ndef prod_map [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :\n  (M₁ × M₃) →L[R₁] (M₂ × M₄) :=\n(f₁.comp (fst R₁ M₁ M₃)).prod (f₂.comp (snd R₁ M₁ M₃))\n\n@[simp, norm_cast] lemma coe_prod_map [module R₁ M₂] [module R₁ M₃] [module R₁ M₄]\n  (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :\n  ↑(f₁.prod_map f₂) = ((f₁ : M₁ →ₗ[R₁] M₂).prod_map (f₂ : M₃ →ₗ[R₁] M₄)) :=\nrfl\n\n@[simp, norm_cast] lemma coe_prod_map' [module R₁ M₂] [module R₁ M₃] [module R₁ M₄]\n  (f₁ : M₁ →L[R₁] M₂) (f₂ : M₃ →L[R₁] M₄) :\n  ⇑(f₁.prod_map f₂) = prod.map f₁ f₂ :=\nrfl\n\n/-- The continuous linear map given by `(x, y) ↦ f₁ x + f₂ y`. -/\ndef coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃] (f₁ : M₁ →L[R₁] M₃)\n  (f₂ : M₂ →L[R₁] M₃) :\n  (M₁ × M₂) →L[R₁] M₃ :=\n⟨linear_map.coprod f₁ f₂, (f₁.cont.comp continuous_fst).add (f₂.cont.comp continuous_snd)⟩\n\n@[norm_cast, simp] lemma coe_coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃]\n  (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) :\n  (f₁.coprod f₂ : (M₁ × M₂) →ₗ[R₁] M₃) = linear_map.coprod f₁ f₂ :=\nrfl\n\n@[simp] lemma coprod_apply [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃]\n  (f₁ : M₁ →L[R₁] M₃) (f₂ : M₂ →L[R₁] M₃) (x) :\n  f₁.coprod f₂ x = f₁ x.1 + f₂ x.2 := rfl\n\nlemma range_coprod [module R₁ M₂] [module R₁ M₃] [has_continuous_add M₃] (f₁ : M₁ →L[R₁] M₃)\n  (f₂ : M₂ →L[R₁] M₃) :\n  (f₁.coprod f₂).range = f₁.range ⊔ f₂.range :=\nlinear_map.range_coprod _ _\n\nsection\n\nvariables {R S : Type*} [semiring R] [semiring S] [module R M₁] [module R M₂] [module R S]\n  [module S M₂] [is_scalar_tower R S M₂] [topological_space S] [has_continuous_smul S M₂]\n\n/-- The linear map `λ x, c x • f`.  Associates to a scalar-valued linear map and an element of\n`M₂` the `M₂`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M₂`).\nSee also `continuous_linear_map.smul_rightₗ` and `continuous_linear_map.smul_rightL`. -/\ndef smul_right (c : M₁ →L[R] S) (f : M₂) : M₁ →L[R] M₂ :=\n{ cont := c.2.smul continuous_const,\n  ..c.to_linear_map.smul_right f }\n\n@[simp]\nlemma smul_right_apply {c : M₁ →L[R] S} {f : M₂} {x : M₁} :\n  (smul_right c f : M₁ → M₂) x = c x • f :=\nrfl\n\nend\n\nsection pointwise\nopen_locale pointwise\n\n@[simp] lemma image_smul_setₛₗ (f : M₁ →SL[σ₁₂] M₂) (c : R₁) (s : set M₁) :\n  f '' (c • s) = (σ₁₂ c) • f '' s :=\nf.to_linear_map.image_smul_setₛₗ c s\n\nlemma image_smul_set (fₗ : M₁ →L[R₁] M'₁) (c : R₁) (s : set M₁) :\n  fₗ '' (c • s) = c • fₗ '' s :=\nfₗ.to_linear_map.image_smul_set c s\n\nlemma preimage_smul_setₛₗ (f : M₁ →SL[σ₁₂] M₂) {c : R₁} (hc : is_unit c) (s : set M₂) :\n  f ⁻¹' (σ₁₂ c • s) = c • f ⁻¹' s :=\nf.to_linear_map.preimage_smul_setₛₗ hc s\n\nlemma preimage_smul_set (fₗ : M₁ →L[R₁] M'₁) {c : R₁} (hc : is_unit c) (s : set M'₁) :\n  fₗ ⁻¹' (c • s) = c • fₗ ⁻¹' s :=\nfₗ.to_linear_map.preimage_smul_set hc s\n\nend pointwise\n\nvariables [module R₁ M₂] [topological_space R₁] [has_continuous_smul R₁ M₂]\n\n@[simp]\nlemma smul_right_one_one (c : R₁ →L[R₁] M₂) : smul_right (1 : R₁ →L[R₁] R₁) (c 1) = c :=\nby ext; simp [← continuous_linear_map.map_smul_of_tower]\n\n@[simp]\nlemma smul_right_one_eq_iff {f f' : M₂} :\n  smul_right (1 : R₁ →L[R₁] R₁) f = smul_right (1 : R₁ →L[R₁] R₁) f' ↔ f = f' :=\nby simp only [ext_ring_iff, smul_right_apply, one_apply, one_smul]\n\nlemma smul_right_comp [has_continuous_mul R₁] {x : M₂} {c : R₁} :\n  (smul_right (1 : R₁ →L[R₁] R₁) x).comp (smul_right (1 : R₁ →L[R₁] R₁) c) =\n    smul_right (1 : R₁ →L[R₁] R₁) (c • x) :=\nby { ext, simp [mul_smul] }\n\nend semiring\n\nsection pi\nvariables\n  {R : Type*} [semiring R]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂] [module R M₂]\n  {ι : Type*} {φ : ι → Type*} [∀i, topological_space (φ i)] [∀i, add_comm_monoid (φ i)]\n  [∀i, module R (φ i)]\n\n/-- `pi` construction for continuous linear functions. From a family of continuous linear functions\nit produces a continuous linear function into a family of topological modules. -/\ndef pi (f : Πi, M →L[R] φ i) : M →L[R] (Πi, φ i) :=\n⟨linear_map.pi (λ i, f i), continuous_pi (λ i, (f i).continuous)⟩\n\n@[simp] lemma coe_pi' (f : Π i, M →L[R] φ i) : ⇑(pi f) = λ c i, f i c := rfl\n@[simp] lemma coe_pi (f : Π i, M →L[R] φ i) :\n  (pi f : M →ₗ[R] Π i, φ i) = linear_map.pi (λ i, f i) :=\nrfl\n\nlemma pi_apply (f : Πi, M →L[R] φ i) (c : M) (i : ι) :\n  pi f c i = f i c := rfl\n\nlemma pi_eq_zero (f : Πi, M →L[R] φ i) : pi f = 0 ↔ (∀i, f i = 0) :=\nby { simp only [ext_iff, pi_apply, function.funext_iff], exact forall_swap }\n\nlemma pi_zero : pi (λi, 0 : Πi, M →L[R] φ i) = 0 := ext $ λ _, rfl\n\nlemma pi_comp (f : Πi, M →L[R] φ i) (g : M₂ →L[R] M) : (pi f).comp g = pi (λi, (f i).comp g) := rfl\n\n/-- The projections from a family of topological modules are continuous linear maps. -/\ndef proj (i : ι) : (Πi, φ i) →L[R] φ i :=\n⟨linear_map.proj i, continuous_apply _⟩\n\n@[simp] lemma proj_apply (i : ι) (b : Πi, φ i) : (proj i : (Πi, φ i) →L[R] φ i) b = b i := rfl\n\nlemma proj_pi (f : Πi, M₂ →L[R] φ i) (i : ι) : (proj i).comp (pi f) = f i :=\next $ assume c, rfl\n\nlemma infi_ker_proj : (⨅i, ker (proj i) : submodule R (Πi, φ i)) = ⊥ :=\nlinear_map.infi_ker_proj\n\nvariables (R φ)\n\n/-- If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections\nof `φ` is linearly equivalent to the product over `I`. -/\ndef infi_ker_proj_equiv {I J : set ι} [decidable_pred (λi, i ∈ I)]\n  (hd : disjoint I J) (hu : set.univ ⊆ I ∪ J) :\n  (⨅i ∈ J, ker (proj i) : submodule R (Πi, φ i)) ≃L[R] (Πi:I, φ i) :=\n⟨ linear_map.infi_ker_proj_equiv R φ hd hu,\n  continuous_pi (λ i, begin\n    have := @continuous_subtype_coe _ _ (λ x, x ∈ (⨅i ∈ J, ker (proj i) : submodule R (Πi, φ i))),\n    have := continuous.comp (by exact continuous_apply i) this,\n    exact this\n  end),\n  continuous_subtype_mk _ (continuous_pi (λ i, begin\n    dsimp, split_ifs; [apply continuous_apply, exact continuous_zero]\n  end)) ⟩\n\nend pi\n\nsection ring\n\nvariables\n{R : Type*} [ring R] {R₂ : Type*} [ring R₂]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M₂ : Type*} [topological_space M₂] [add_comm_group M₂]\n{M₃ : Type*} [topological_space M₃] [add_comm_group M₃]\n{M₄ : Type*} [topological_space M₄] [add_comm_group M₄]\n[module R M] [module R₂ M₂]\n{σ₁₂ : R →+* R₂}\n\nsection\n\nprotected lemma map_neg (f : M →SL[σ₁₂] M₂) (x : M) : f (-x) = - (f x) := map_neg _ _\nprotected lemma map_sub (f : M →SL[σ₁₂] M₂) (x y : M) : f (x - y) = f x - f y := map_sub _ _ _\n@[simp] lemma sub_apply' (f g : M →SL[σ₁₂] M₂) (x : M) : ((f : M →ₛₗ[σ₁₂] M₂) - g) x = f x - g x :=\nrfl\nend\n\nsection\nvariables [module R M₂] [module R M₃] [module R M₄]\n\nlemma range_prod_eq {f : M →L[R] M₂} {g : M →L[R] M₃} (h : ker f ⊔ ker g = ⊤) :\n  range (f.prod g) = (range f).prod (range g) :=\nlinear_map.range_prod_eq h\n\nlemma ker_prod_ker_le_ker_coprod [has_continuous_add M₃]\n  (f : M →L[R] M₃) (g : M₂ →L[R] M₃) :\n  (ker f).prod (ker g) ≤ ker (f.coprod g) :=\nlinear_map.ker_prod_ker_le_ker_coprod f.to_linear_map g.to_linear_map\n\nlemma ker_coprod_of_disjoint_range [has_continuous_add M₃]\n  (f : M →L[R] M₃) (g : M₂ →L[R] M₃) (hd : disjoint f.range g.range) :\n  ker (f.coprod g) = (ker f).prod (ker g) :=\nlinear_map.ker_coprod_of_disjoint_range f.to_linear_map g.to_linear_map hd\nend\n\nsection\nvariables [topological_add_group M₂]\n\ninstance : has_neg (M →SL[σ₁₂] M₂) := ⟨λ f, ⟨-f, f.2.neg⟩⟩\n\n@[simp] lemma neg_apply (f : M →SL[σ₁₂] M₂) (x : M) : (-f) x = - (f x) := rfl\n@[simp, norm_cast] lemma coe_neg (f : M →SL[σ₁₂] M₂) : (↑(-f) : M →ₛₗ[σ₁₂] M₂) = -f := rfl\n@[norm_cast] lemma coe_neg' (f : M →SL[σ₁₂] M₂) : ⇑(-f) = -f := rfl\n\ninstance : has_sub (M →SL[σ₁₂] M₂) := ⟨λ f g, ⟨f - g, f.2.sub g.2⟩⟩\n\ninstance : add_comm_group (M →SL[σ₁₂] M₂) :=\nby refine\n{ zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  sub := has_sub.sub,\n  sub_eq_add_neg := _,\n  nsmul := (•),\n  zsmul := (•),\n  zsmul_zero' := λ f, by { ext, simp },\n  zsmul_succ' := λ n f, by { ext, simp [add_smul, add_comm] },\n  zsmul_neg' := λ n f, by { ext, simp [nat.succ_eq_add_one, add_smul] },\n  .. continuous_linear_map.add_comm_monoid, .. };\nintros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm, sub_eq_add_neg]\n\nlemma sub_apply (f g : M →SL[σ₁₂] M₂) (x : M) : (f - g) x = f x - g x := rfl\n@[simp, norm_cast] lemma coe_sub (f g : M →SL[σ₁₂] M₂) : (↑(f - g) : M →ₛₗ[σ₁₂] M₂) = f - g := rfl\n@[simp, norm_cast] lemma coe_sub' (f g : M →SL[σ₁₂] M₂) : ⇑(f - g) = f - g := rfl\n\nend\n\ninstance [topological_add_group M] : ring (M →L[R] M) :=\n{ mul := (*),\n  one := 1,\n  ..continuous_linear_map.semiring,\n  ..continuous_linear_map.add_comm_group }\n\nlemma smul_right_one_pow [topological_space R] [topological_ring R] (c : R) (n : ℕ) :\n  (smul_right (1 : R →L[R] R) c)^n = smul_right (1 : R →L[R] R) (c^n) :=\nbegin\n  induction n with n ihn,\n  { ext, simp },\n  { rw [pow_succ, ihn, mul_def, smul_right_comp, smul_eq_mul, pow_succ'] }\nend\n\nsection\nvariables {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁]\n\n/-- Given a right inverse `f₂ : M₂ →L[R] M` to `f₁ : M →L[R] M₂`,\n`proj_ker_of_right_inverse f₁ f₂ h` is the projection `M →L[R] f₁.ker` along `f₂.range`. -/\ndef proj_ker_of_right_inverse [topological_add_group M] (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M)\n  (h : function.right_inverse f₂ f₁) :\n  M →L[R] f₁.ker :=\n(id R M - f₂.comp f₁).cod_restrict f₁.ker $ λ x, by simp [h (f₁ x)]\n\n@[simp] lemma coe_proj_ker_of_right_inverse_apply [topological_add_group M]\n  (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁) (x : M) :\n  (f₁.proj_ker_of_right_inverse f₂ h x : M) = x - f₂ (f₁ x) :=\nrfl\n\n@[simp] lemma proj_ker_of_right_inverse_apply_idem [topological_add_group M]\n  (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁) (x : f₁.ker) :\n  f₁.proj_ker_of_right_inverse f₂ h x = x :=\nsubtype.ext_iff_val.2 $ by simp\n\n@[simp] lemma proj_ker_of_right_inverse_comp_inv [topological_add_group M]\n  (f₁ : M →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M) (h : function.right_inverse f₂ f₁) (y : M₂) :\n  f₁.proj_ker_of_right_inverse f₂ h (f₂ y) = 0 :=\nsubtype.ext_iff_val.2 $ by simp [h y]\n\nend\n\nend ring\n\nsection division_monoid\nvariables {R M : Type*}\n\n/-- A nonzero continuous linear functional is open. -/\nprotected lemma is_open_map_of_ne_zero [topological_space R] [division_ring R]\n  [has_continuous_sub R] [add_comm_group M] [topological_space M] [has_continuous_add M]\n  [module R M] [has_continuous_smul R M] (f : M →L[R] R) (hf : f ≠ 0) : is_open_map f :=\nlet ⟨x, hx⟩ := exists_ne_zero hf in is_open_map.of_sections $ λ y,\n    ⟨λ a, y + (a - f y) • (f x)⁻¹ • x, continuous.continuous_at $ by continuity,\n      by simp, λ a, by simp [hx]⟩\n\nend division_monoid\n\nsection smul_monoid\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariables {R R₂ R₃ S S₃ : Type*} [semiring R] [semiring R₂] [semiring R₃]\n  [monoid S] [monoid S₃]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂] [module R₂ M₂]\n  {M₃ : Type*} [topological_space M₃] [add_comm_monoid M₃] [module R₃ M₃]\n  {N₂ : Type*} [topological_space N₂] [add_comm_monoid N₂] [module R N₂]\n  {N₃ : Type*} [topological_space N₃] [add_comm_monoid N₃] [module R N₃]\n  [distrib_mul_action S₃ M₃] [smul_comm_class R₃ S₃ M₃] [has_continuous_const_smul S₃ M₃]\n  [distrib_mul_action S N₃] [smul_comm_class R S N₃] [has_continuous_const_smul S N₃]\n  {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃]\n\ninclude σ₁₃\n@[simp] lemma smul_comp (c : S₃) (h : M₂ →SL[σ₂₃] M₃) (f : M →SL[σ₁₂] M₂) :\n  (c • h).comp f = c • (h.comp f) := rfl\nomit σ₁₃\n\nvariables [distrib_mul_action S₃ M₂] [has_continuous_const_smul S₃ M₂] [smul_comm_class R₂ S₃ M₂]\nvariables [distrib_mul_action S N₂] [has_continuous_const_smul S N₂] [smul_comm_class R S N₂]\n\n@[simp] lemma comp_smul [linear_map.compatible_smul N₂ N₃ S R]\n  (hₗ : N₂ →L[R] N₃) (c : S) (fₗ : M →L[R] N₂) :\n  hₗ.comp (c • fₗ) = c • (hₗ.comp fₗ) :=\nby { ext x, exact hₗ.map_smul_of_tower c (fₗ x) }\n\ninclude σ₁₃\n@[simp] lemma comp_smulₛₗ [smul_comm_class R₂ R₂ M₂] [smul_comm_class R₃ R₃ M₃]\n  [has_continuous_const_smul R₂ M₂] [has_continuous_const_smul R₃ M₃]\n  (h : M₂ →SL[σ₂₃] M₃) (c : R₂) (f : M →SL[σ₁₂] M₂) :\n  h.comp (c • f) = (σ₂₃ c) • (h.comp f) :=\nby { ext x, simp only [coe_smul', coe_comp', function.comp_app, pi.smul_apply, map_smulₛₗ] }\nomit σ₁₃\n\ninstance [has_continuous_add M₂] : distrib_mul_action S₃ (M →SL[σ₁₂] M₂) :=\n{ smul_add := λ a f g, ext $ λ x, smul_add a (f x) (g x),\n  smul_zero := λ a, ext $ λ x, smul_zero _ }\n\nend smul_monoid\n\nsection smul\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariables {R R₂ R₃ S S₃ : Type*} [semiring R] [semiring R₂] [semiring R₃]\n  [semiring S] [semiring S₃]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂] [module R₂ M₂]\n  {M₃ : Type*} [topological_space M₃] [add_comm_monoid M₃] [module R₃ M₃]\n  {N₂ : Type*} [topological_space N₂] [add_comm_monoid N₂] [module R N₂]\n  {N₃ : Type*} [topological_space N₃] [add_comm_monoid N₃] [module R N₃]\n  [module S₃ M₃] [smul_comm_class R₃ S₃ M₃] [has_continuous_const_smul S₃ M₃]\n  [module S N₂] [has_continuous_const_smul S N₂] [smul_comm_class R S N₂]\n  [module S N₃] [smul_comm_class R S N₃] [has_continuous_const_smul S N₃]\n  {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃]\n  (c : S) (h : M₂ →SL[σ₂₃] M₃) (f g : M →SL[σ₁₂] M₂) (x y z : M)\n\n/-- `continuous_linear_map.prod` as an `equiv`. -/\n@[simps apply] def prod_equiv : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ (M →L[R] N₂ × N₃) :=\n{ to_fun := λ f, f.1.prod f.2,\n  inv_fun := λ f, ⟨(fst _ _ _).comp f, (snd _ _ _).comp f⟩,\n  left_inv := λ f, by ext; refl,\n  right_inv := λ f, by ext; refl }\n\nlemma prod_ext_iff {f g : M × N₂ →L[R] N₃} :\n  f = g ↔ f.comp (inl _ _ _) = g.comp (inl _ _ _) ∧ f.comp (inr _ _ _) = g.comp (inr _ _ _) :=\nby { simp only [← coe_inj, linear_map.prod_ext_iff], refl }\n\n@[ext] lemma prod_ext {f g : M × N₂ →L[R] N₃} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))\n  (hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g :=\nprod_ext_iff.2 ⟨hl, hr⟩\n\nvariables [has_continuous_add M₂] [has_continuous_add M₃] [has_continuous_add N₂]\n\ninstance : module S₃ (M →SL[σ₁₃] M₃) :=\n{ zero_smul := λ _, ext $ λ _, zero_smul _ _,\n  add_smul  := λ _ _ _, ext $ λ _, add_smul _ _ _ }\n\ninstance [module S₃ᵐᵒᵖ M₃] [is_central_scalar S₃ M₃] : is_central_scalar S₃ (M →SL[σ₁₃] M₃) :=\n{ op_smul_eq_smul := λ _ _, ext $ λ _, op_smul_eq_smul _ _ }\n\nvariables (S) [has_continuous_add N₃]\n\n/-- `continuous_linear_map.prod` as a `linear_equiv`. -/\n@[simps apply] def prodₗ : ((M →L[R] N₂) × (M →L[R] N₃)) ≃ₗ[S] (M →L[R] N₂ × N₃) :=\n{ map_add' := λ f g, rfl,\n  map_smul' := λ c f, rfl,\n  .. prod_equiv }\n\n/-- The coercion from `M →L[R] M₂` to `M →ₗ[R] M₂`, as a linear map. -/\n@[simps]\ndef coe_lm : (M →L[R] N₃) →ₗ[S] (M →ₗ[R] N₃) :=\n{ to_fun := coe,\n  map_add' := λ f g, coe_add f g,\n  map_smul' := λ c f, coe_smul c f }\n\nvariables {S} (σ₁₃)\n\n/-- The coercion from `M →SL[σ] M₂` to `M →ₛₗ[σ] M₂`, as a linear map. -/\n@[simps]\ndef coe_lmₛₗ : (M →SL[σ₁₃] M₃) →ₗ[S₃] (M →ₛₗ[σ₁₃] M₃) :=\n{ to_fun := coe,\n  map_add' := λ f g, coe_add f g,\n  map_smul' := λ c f, coe_smul c f }\n\nvariables {σ₁₃}\n\nend smul\n\nsection smul_rightₗ\n\nvariables {R S T M M₂ : Type*} [semiring R] [semiring S] [semiring T] [module R S]\n  [add_comm_monoid M₂] [module R M₂] [module S M₂] [is_scalar_tower R S M₂]\n  [topological_space S] [topological_space M₂] [has_continuous_smul S M₂]\n  [topological_space M] [add_comm_monoid M] [module R M] [has_continuous_add M₂]\n  [module T M₂] [has_continuous_const_smul T M₂]\n  [smul_comm_class R T M₂] [smul_comm_class S T M₂]\n\n/-- Given `c : E →L[𝕜] 𝕜`, `c.smul_rightₗ` is the linear map from `F` to `E →L[𝕜] F`\nsending `f` to `λ e, c e • f`. See also `continuous_linear_map.smul_rightL`. -/\ndef smul_rightₗ (c : M →L[R] S) : M₂ →ₗ[T] (M →L[R] M₂) :=\n{ to_fun := c.smul_right,\n  map_add' := λ x y, by { ext e, apply smul_add },\n  map_smul' := λ a x, by { ext e, dsimp, apply smul_comm } }\n\n@[simp] lemma coe_smul_rightₗ (c : M →L[R] S) :\n  ⇑(smul_rightₗ c : M₂ →ₗ[T] (M →L[R] M₂)) = c.smul_right := rfl\n\nend smul_rightₗ\n\nsection comm_ring\n\n/-- The determinant of a continuous linear map, mainly as a convenience device to be able to\nwrite `A.det` instead of `(A : M →ₗ[R] M).det`. -/\n@[reducible] noncomputable def det {R : Type*} [comm_ring R]\n  {M : Type*} [topological_space M] [add_comm_group M] [module R M] (A : M →L[R] M) : R :=\nlinear_map.det (A : M →ₗ[R] M)\n\nvariables\n{R : Type*} [comm_ring R]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M₂ : Type*} [topological_space M₂] [add_comm_group M₂]\n{M₃ : Type*} [topological_space M₃] [add_comm_group M₃]\n[module R M] [module R M₂] [module R M₃] [has_continuous_const_smul R M₃]\n\nvariables [topological_add_group M₂] [has_continuous_const_smul R M₂]\n\ninstance : algebra R (M₂ →L[R] M₂) :=\nalgebra.of_module smul_comp (λ _ _ _, comp_smul _ _ _)\n\nend comm_ring\n\nsection restrict_scalars\n\nvariables {A M M₂ : Type*} [ring A] [add_comm_group M] [add_comm_group M₂]\n  [module A M] [module A M₂] [topological_space M] [topological_space M₂]\n  (R : Type*) [ring R] [module R M] [module R M₂] [linear_map.compatible_smul M M₂ R A]\n\n/-- If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous\n`R`-linear map. We assume `linear_map.compatible_smul M M₂ R A` to match assumptions of\n`linear_map.map_smul_of_tower`. -/\ndef restrict_scalars (f : M →L[A] M₂) : M →L[R] M₂ :=\n⟨(f : M →ₗ[A] M₂).restrict_scalars R, f.continuous⟩\n\nvariable {R}\n\n@[simp, norm_cast] lemma coe_restrict_scalars (f : M →L[A] M₂) :\n  (f.restrict_scalars R : M →ₗ[R] M₂) = (f : M →ₗ[A] M₂).restrict_scalars R := rfl\n\n@[simp] lemma coe_restrict_scalars' (f : M →L[A] M₂) : ⇑(f.restrict_scalars R) = f := rfl\n\n@[simp] lemma restrict_scalars_zero : (0 : M →L[A] M₂).restrict_scalars R = 0 := rfl\n\nsection\nvariable [topological_add_group M₂]\n\n@[simp] lemma restrict_scalars_add (f g : M →L[A] M₂) :\n  (f + g).restrict_scalars R = f.restrict_scalars R + g.restrict_scalars R := rfl\n\n@[simp] lemma restrict_scalars_neg (f : M →L[A] M₂) :\n  (-f).restrict_scalars R = -f.restrict_scalars R := rfl\nend\n\nvariables {S : Type*} [ring S] [module S M₂] [has_continuous_const_smul S M₂]\n  [smul_comm_class A S M₂] [smul_comm_class R S M₂]\n\n@[simp] lemma restrict_scalars_smul (c : S) (f : M →L[A] M₂) :\n  (c • f).restrict_scalars R = c • f.restrict_scalars R := rfl\n\nvariables (A M M₂ R S) [topological_add_group M₂]\n\n/-- `continuous_linear_map.restrict_scalars` as a `linear_map`. See also\n`continuous_linear_map.restrict_scalarsL`. -/\ndef restrict_scalarsₗ : (M →L[A] M₂) →ₗ[S] (M →L[R] M₂) :=\n{ to_fun := restrict_scalars R,\n  map_add' := restrict_scalars_add,\n  map_smul' := restrict_scalars_smul }\n\nvariables {A M M₂ R S}\n\n@[simp] lemma coe_restrict_scalarsₗ : ⇑(restrict_scalarsₗ A M M₂ R S) = restrict_scalars R := rfl\n\nend restrict_scalars\n\nend continuous_linear_map\n\nnamespace continuous_linear_equiv\n\nsection add_comm_monoid\n\nvariables {R₁ : Type*} {R₂ : Type*} {R₃ : Type*} [semiring R₁] [semiring R₂] [semiring R₃]\n{σ₁₂ : R₁ →+* R₂} {σ₂₁ : R₂ →+* R₁} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂]\n{σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [ring_hom_inv_pair σ₂₃ σ₃₂] [ring_hom_inv_pair σ₃₂ σ₂₃]\n{σ₁₃ : R₁ →+* R₃} {σ₃₁ : R₃ →+* R₁} [ring_hom_inv_pair σ₁₃ σ₃₁] [ring_hom_inv_pair σ₃₁ σ₁₃]\n[ring_hom_comp_triple σ₁₂ σ₂₃ σ₁₃] [ring_hom_comp_triple σ₃₂ σ₂₁ σ₃₁]\n{M₁ : Type*} [topological_space M₁] [add_comm_monoid M₁]\n{M'₁ : Type*} [topological_space M'₁] [add_comm_monoid M'₁]\n{M₂ : Type*} [topological_space M₂] [add_comm_monoid M₂]\n{M₃ : Type*} [topological_space M₃] [add_comm_monoid M₃]\n{M₄ : Type*} [topological_space M₄] [add_comm_monoid M₄]\n[module R₁ M₁] [module R₁ M'₁] [module R₂ M₂] [module R₃ M₃]\n\ninclude σ₂₁\n/-- A continuous linear equivalence induces a continuous linear map. -/\ndef to_continuous_linear_map (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁₂] M₂ :=\n{ cont := e.continuous_to_fun,\n  ..e.to_linear_equiv.to_linear_map }\n\n/-- Coerce continuous linear equivs to continuous linear maps. -/\ninstance : has_coe (M₁ ≃SL[σ₁₂] M₂) (M₁ →SL[σ₁₂] M₂) := ⟨to_continuous_linear_map⟩\n\n/-- Coerce continuous linear equivs to maps. -/\n-- see Note [function coercion]\ninstance : has_coe_to_fun (M₁ ≃SL[σ₁₂] M₂) (λ _, M₁ → M₂) := ⟨λ f, f⟩\n\n@[simp] theorem coe_def_rev (e : M₁ ≃SL[σ₁₂] M₂) : e.to_continuous_linear_map = e := rfl\n\ntheorem coe_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : (e : M₁ →SL[σ₁₂] M₂) b = e b := rfl\n\n@[simp] lemma coe_to_linear_equiv (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.to_linear_equiv = f := rfl\n\n@[simp, norm_cast] lemma coe_coe (e : M₁ ≃SL[σ₁₂] M₂) : ⇑(e : M₁ →SL[σ₁₂] M₂) = e := rfl\n\nlemma to_linear_equiv_injective :\n  function.injective (to_linear_equiv : (M₁ ≃SL[σ₁₂] M₂) → (M₁ ≃ₛₗ[σ₁₂] M₂))\n| ⟨e, _, _⟩ ⟨e', _, _⟩ rfl := rfl\n\n@[ext] lemma ext {f g : M₁ ≃SL[σ₁₂] M₂} (h : (f : M₁ → M₂) = g) : f = g :=\nto_linear_equiv_injective $ linear_equiv.ext $ congr_fun h\n\nlemma coe_injective : function.injective (coe : (M₁ ≃SL[σ₁₂] M₂) → (M₁ →SL[σ₁₂] M₂)) :=\nλ e e' h, ext $ funext $ continuous_linear_map.ext_iff.1 h\n\n@[simp, norm_cast] lemma coe_inj {e e' : M₁ ≃SL[σ₁₂] M₂} : (e : M₁ →SL[σ₁₂] M₂) = e' ↔ e = e' :=\ncoe_injective.eq_iff\n\n/-- A continuous linear equivalence induces a homeomorphism. -/\ndef to_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) : M₁ ≃ₜ M₂ := { to_equiv := e.to_linear_equiv.to_equiv, ..e }\n\n@[simp] lemma coe_to_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.to_homeomorph = e := rfl\n\nlemma image_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e '' closure s = closure (e '' s) :=\ne.to_homeomorph.image_closure s\n\nlemma preimage_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e ⁻¹' closure s = closure (e ⁻¹' s) :=\ne.to_homeomorph.preimage_closure s\n\n@[simp] lemma is_closed_image (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} :\n  is_closed (e '' s) ↔ is_closed s :=\ne.to_homeomorph.is_closed_image\n\nlemma map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e (𝓝 x) = 𝓝 (e x) :=\ne.to_homeomorph.map_nhds_eq x\n\n-- Make some straightforward lemmas available to `simp`.\n@[simp] lemma map_zero (e : M₁ ≃SL[σ₁₂] M₂) : e (0 : M₁) = 0 := (e : M₁ →SL[σ₁₂] M₂).map_zero\n@[simp] lemma map_add (e : M₁ ≃SL[σ₁₂] M₂) (x y : M₁) : e (x + y) = e x + e y :=\n(e : M₁ →SL[σ₁₂] M₂).map_add x y\n@[simp] lemma map_smulₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e (c • x) = σ₁₂ c • (e x) :=\n(e : M₁ →SL[σ₁₂] M₂).map_smulₛₗ c x\nomit σ₂₁\n\n@[simp] lemma map_smul [module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M₁) :\n  e (c • x) = c • (e x) :=\n(e : M₁ →L[R₁] M₂).map_smul c x\n\ninclude σ₂₁\n@[simp] lemma map_eq_zero_iff (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ x = 0 :=\ne.to_linear_equiv.map_eq_zero_iff\n\nattribute [continuity]\n  continuous_linear_equiv.continuous_to_fun continuous_linear_equiv.continuous_inv_fun\n\n@[continuity]\nprotected lemma continuous (e : M₁ ≃SL[σ₁₂] M₂) : continuous (e : M₁ → M₂) :=\ne.continuous_to_fun\n\nprotected lemma continuous_on (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} : continuous_on (e : M₁ → M₂) s :=\ne.continuous.continuous_on\n\nprotected lemma continuous_at (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : continuous_at (e : M₁ → M₂) x :=\ne.continuous.continuous_at\n\nprotected lemma continuous_within_at (e : M₁ ≃SL[σ₁₂] M₂) {s : set M₁} {x : M₁} :\n  continuous_within_at (e : M₁ → M₂) s x :=\ne.continuous.continuous_within_at\n\nlemma comp_continuous_on_iff\n  {α : Type*} [topological_space α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} {s : set α} :\n  continuous_on (e ∘ f) s ↔ continuous_on f s :=\ne.to_homeomorph.comp_continuous_on_iff _ _\n\nlemma comp_continuous_iff\n  {α : Type*} [topological_space α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} :\n  continuous (e ∘ f) ↔ continuous f :=\ne.to_homeomorph.comp_continuous_iff\nomit σ₂₁\n\n/-- An extensionality lemma for `R ≃L[R] M`. -/\nlemma ext₁ [topological_space R₁] {f g : R₁ ≃L[R₁] M₁} (h : f 1 = g 1) : f = g :=\next $ funext $ λ x, mul_one x ▸ by rw [← smul_eq_mul, map_smul, h, map_smul]\n\nsection\nvariables (R₁ M₁)\n\n/-- The identity map as a continuous linear equivalence. -/\n@[refl] protected def refl : M₁ ≃L[R₁] M₁ :=\n{ continuous_to_fun := continuous_id,\n  continuous_inv_fun := continuous_id,\n  .. linear_equiv.refl R₁ M₁ }\nend\n\n@[simp, norm_cast] lemma coe_refl :\n  ↑(continuous_linear_equiv.refl R₁ M₁) = continuous_linear_map.id R₁ M₁ := rfl\n\n@[simp, norm_cast] lemma coe_refl' : ⇑(continuous_linear_equiv.refl R₁ M₁) = id := rfl\n\n/-- The inverse of a continuous linear equivalence as a continuous linear equivalence-/\n@[symm] protected def symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ :=\n{ continuous_to_fun := e.continuous_inv_fun,\n  continuous_inv_fun := e.continuous_to_fun,\n  .. e.to_linear_equiv.symm }\n\ninclude σ₂₁\n@[simp] lemma symm_to_linear_equiv (e : M₁ ≃SL[σ₁₂] M₂) :\n  e.symm.to_linear_equiv = e.to_linear_equiv.symm :=\nby { ext, refl }\n\n@[simp] lemma symm_to_homeomorph (e : M₁ ≃SL[σ₁₂] M₂) :\n  e.to_homeomorph.symm = e.symm.to_homeomorph :=\nrfl\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef simps.apply (h : M₁ ≃SL[σ₁₂] M₂) : M₁ → M₂ := h\n\n/-- See Note [custom simps projection] -/\ndef simps.symm_apply (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁ := h.symm\n\ninitialize_simps_projections continuous_linear_equiv\n  (to_linear_equiv_to_fun → apply, to_linear_equiv_inv_fun → symm_apply)\n\nlemma symm_map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e.symm (𝓝 (e x)) = 𝓝 x :=\ne.to_homeomorph.symm_map_nhds_eq x\nomit σ₂₁\n\ninclude σ₂₁ σ₃₂ σ₃₁\n/-- The composition of two continuous linear equivalences as a continuous linear equivalence. -/\n@[trans] protected def trans (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : M₁ ≃SL[σ₁₃] M₃ :=\n{ continuous_to_fun := e₂.continuous_to_fun.comp e₁.continuous_to_fun,\n  continuous_inv_fun := e₁.continuous_inv_fun.comp e₂.continuous_inv_fun,\n  .. e₁.to_linear_equiv.trans e₂.to_linear_equiv }\n\ninclude σ₁₃\n@[simp] lemma trans_to_linear_equiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) :\n  (e₁.trans e₂).to_linear_equiv = e₁.to_linear_equiv.trans e₂.to_linear_equiv :=\nby { ext, refl }\nomit σ₁₃ σ₂₁ σ₃₂ σ₃₁\n\n/-- Product of two continuous linear equivalences. The map comes from `equiv.prod_congr`. -/\ndef prod [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) :\n  (M₁ × M₃) ≃L[R₁] (M₂ × M₄) :=\n{ continuous_to_fun := e.continuous_to_fun.prod_map e'.continuous_to_fun,\n  continuous_inv_fun := e.continuous_inv_fun.prod_map e'.continuous_inv_fun,\n  .. e.to_linear_equiv.prod e'.to_linear_equiv }\n\n@[simp, norm_cast] lemma prod_apply [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂)\n  (e' : M₃ ≃L[R₁] M₄) (x) :\n  e.prod e' x = (e x.1, e' x.2) := rfl\n\n@[simp, norm_cast] lemma coe_prod [module R₁ M₂] [module R₁ M₃] [module R₁ M₄] (e : M₁ ≃L[R₁] M₂)\n  (e' : M₃ ≃L[R₁] M₄) :\n  (e.prod e' : (M₁ × M₃) →L[R₁] (M₂ × M₄)) = (e : M₁ →L[R₁] M₂).prod_map (e' : M₃ →L[R₁] M₄) :=\nrfl\n\ninclude σ₂₁\ntheorem bijective (e : M₁ ≃SL[σ₁₂] M₂) : function.bijective e :=\ne.to_linear_equiv.to_equiv.bijective\ntheorem injective (e : M₁ ≃SL[σ₁₂] M₂) : function.injective e :=\ne.to_linear_equiv.to_equiv.injective\ntheorem surjective (e : M₁ ≃SL[σ₁₂] M₂) : function.surjective e :=\ne.to_linear_equiv.to_equiv.surjective\n\ninclude σ₃₂ σ₃₁ σ₁₃\n@[simp] theorem trans_apply (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) (c : M₁) :\n  (e₁.trans e₂) c = e₂ (e₁ c) :=\nrfl\nomit σ₃₂ σ₃₁ σ₁₃\n\n@[simp] theorem apply_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (c : M₂) : e (e.symm c) = c :=\ne.1.right_inv c\n@[simp] theorem symm_apply_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : e.symm (e b) = b := e.1.left_inv b\n\ninclude σ₁₂ σ₂₃ σ₁₃ σ₃₁\n@[simp] theorem symm_trans_apply (e₁ : M₂ ≃SL[σ₂₁] M₁) (e₂ : M₃ ≃SL[σ₃₂] M₂) (c : M₁) :\n  (e₂.trans e₁).symm c = e₂.symm (e₁.symm c) :=\nrfl\nomit σ₁₂ σ₂₃ σ₁₃ σ₃₁\n\n@[simp] theorem symm_image_image (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e.symm '' (e '' s) = s :=\ne.to_linear_equiv.to_equiv.symm_image_image s\n@[simp] theorem image_symm_image (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e '' (e.symm '' s) = s :=\ne.symm.symm_image_image s\n\ninclude σ₃₂ σ₃₁\n@[simp, norm_cast]\nlemma comp_coe (f : M₁ ≃SL[σ₁₂] M₂) (f' : M₂ ≃SL[σ₂₃] M₃) :\n  (f' : M₂ →SL[σ₂₃] M₃).comp (f : M₁ →SL[σ₁₂] M₂) = (f.trans f' : M₁ →SL[σ₁₃] M₃) :=\nrfl\nomit σ₃₂ σ₃₁ σ₂₁\n\n@[simp] theorem coe_comp_coe_symm (e : M₁ ≃SL[σ₁₂] M₂) :\n  (e : M₁ →SL[σ₁₂] M₂).comp (e.symm : M₂ →SL[σ₂₁] M₁) = continuous_linear_map.id R₂ M₂ :=\ncontinuous_linear_map.ext e.apply_symm_apply\n\n@[simp] theorem coe_symm_comp_coe (e : M₁ ≃SL[σ₁₂] M₂) :\n  (e.symm : M₂ →SL[σ₂₁] M₁).comp (e : M₁ →SL[σ₁₂] M₂) = continuous_linear_map.id R₁ M₁ :=\ncontinuous_linear_map.ext e.symm_apply_apply\n\ninclude σ₂₁\n@[simp] lemma symm_comp_self (e : M₁ ≃SL[σ₁₂] M₂) :\n  (e.symm : M₂ → M₁) ∘ (e : M₁ → M₂) = id :=\nby{ ext x, exact symm_apply_apply e x }\n\n@[simp] lemma self_comp_symm (e : M₁ ≃SL[σ₁₂] M₂) :\n  (e : M₁ → M₂) ∘ (e.symm : M₂ → M₁) = id :=\nby{ ext x, exact apply_symm_apply e x }\n\n@[simp] theorem symm_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.symm = e :=\nby { ext x, refl }\nomit σ₂₁\n\n@[simp] lemma refl_symm :\n (continuous_linear_equiv.refl R₁ M₁).symm = continuous_linear_equiv.refl R₁ M₁ :=\nrfl\n\ninclude σ₂₁\ntheorem symm_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : e.symm.symm x = e x :=\nrfl\n\nlemma symm_apply_eq (e : M₁ ≃SL[σ₁₂] M₂) {x y} : e.symm x = y ↔ x = e y :=\ne.to_linear_equiv.symm_apply_eq\n\nlemma eq_symm_apply (e : M₁ ≃SL[σ₁₂] M₂) {x y} : y = e.symm x ↔ e y = x :=\ne.to_linear_equiv.eq_symm_apply\n\nprotected lemma image_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) : e '' s = e.symm ⁻¹' s :=\ne.to_linear_equiv.to_equiv.image_eq_preimage s\n\nprotected lemma image_symm_eq_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) : e.symm '' s = e ⁻¹' s :=\nby rw [e.symm.image_eq_preimage, e.symm_symm]\n\n@[simp] protected lemma symm_preimage_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₂) :\n  e.symm ⁻¹' (e ⁻¹' s) = s := e.to_linear_equiv.to_equiv.symm_preimage_preimage s\n\n@[simp] protected lemma preimage_symm_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : set M₁) :\n  e ⁻¹' (e.symm ⁻¹' s) = s := e.symm.symm_preimage_preimage s\n\nprotected lemma uniform_embedding {E₁ E₂ : Type*} [uniform_space E₁] [uniform_space E₂]\n  [add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]\n  [uniform_add_group E₁] [uniform_add_group E₂]\n  (e : E₁ ≃SL[σ₁₂] E₂) :\n  uniform_embedding e :=\ne.to_linear_equiv.to_equiv.uniform_embedding\n  e.to_continuous_linear_map.uniform_continuous\n  e.symm.to_continuous_linear_map.uniform_continuous\n\nprotected lemma _root_.linear_equiv.uniform_embedding {E₁ E₂ : Type*} [uniform_space E₁]\n  [uniform_space E₂] [add_comm_group E₁] [add_comm_group E₂] [module R₁ E₁] [module R₂ E₂]\n  [uniform_add_group E₁] [uniform_add_group E₂]\n  (e : E₁ ≃ₛₗ[σ₁₂] E₂) (h₁ : continuous e) (h₂ : continuous e.symm) :\n  uniform_embedding e :=\ncontinuous_linear_equiv.uniform_embedding\n({ continuous_to_fun := h₁,\n  continuous_inv_fun := h₂,\n  .. e } : E₁ ≃SL[σ₁₂] E₂)\n\nomit σ₂₁\n\n/-- Create a `continuous_linear_equiv` from two `continuous_linear_map`s that are\ninverse of each other. -/\ndef equiv_of_inverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : function.left_inverse f₂ f₁)\n  (h₂ : function.right_inverse f₂ f₁) :\n  M₁ ≃SL[σ₁₂] M₂ :=\n{ to_fun := f₁,\n  continuous_to_fun := f₁.continuous,\n  inv_fun := f₂,\n  continuous_inv_fun := f₂.continuous,\n  left_inv := h₁,\n  right_inv := h₂,\n  .. f₁ }\n\ninclude σ₂₁\n@[simp] lemma equiv_of_inverse_apply (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂ x) :\n  equiv_of_inverse f₁ f₂ h₁ h₂ x = f₁ x :=\nrfl\n\n@[simp] lemma symm_equiv_of_inverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ h₁ h₂) :\n  (equiv_of_inverse f₁ f₂ h₁ h₂).symm = equiv_of_inverse f₂ f₁ h₂ h₁ :=\nrfl\nomit σ₂₁\n\nsection pointwise\nopen_locale pointwise\ninclude σ₂₁\n\n@[simp] lemma image_smul_setₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (s : set M₁) :\n  e '' (c • s) = (σ₁₂ c) • e '' s :=\ne.to_linear_equiv.image_smul_setₛₗ c s\n\n@[simp] lemma preimage_smul_setₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₂) (s : set M₂) :\n  e ⁻¹' (c • s) = σ₂₁ c • e ⁻¹' s :=\ne.to_linear_equiv.preimage_smul_setₛₗ c s\nomit σ₂₁\n\n@[simp] lemma image_smul_set (e : M₁ ≃L[R₁] M'₁) (c : R₁) (s : set M₁) :\n  e '' (c • s) = c • e '' s :=\ne.to_linear_equiv.image_smul_set c s\n\n@[simp] lemma preimage_smul_set (e : M₁ ≃L[R₁] M'₁) (c : R₁) (s : set M'₁) :\n  e ⁻¹' (c • s) = c • e ⁻¹' s :=\ne.to_linear_equiv.preimage_smul_set c s\n\nend pointwise\n\nvariable (M₁)\n\n/-- The continuous linear equivalences from `M` to itself form a group under composition. -/\ninstance automorphism_group : group (M₁ ≃L[R₁] M₁) :=\n{ mul          := λ f g, g.trans f,\n  one          := continuous_linear_equiv.refl R₁ M₁,\n  inv          := λ f, f.symm,\n  mul_assoc    := λ f g h, by {ext, refl},\n  mul_one      := λ f, by {ext, refl},\n  one_mul      := λ f, by {ext, refl},\n  mul_left_inv := λ f, by {ext, exact f.left_inv x} }\n\nvariables {M₁} {R₄ : Type*} [semiring R₄] [module R₄ M₄]\n  {σ₃₄ : R₃ →+* R₄} {σ₄₃ : R₄ →+* R₃} [ring_hom_inv_pair σ₃₄ σ₄₃] [ring_hom_inv_pair σ₄₃ σ₃₄]\n  {σ₂₄ : R₂ →+* R₄} {σ₁₄ : R₁ →+* R₄}\n  [ring_hom_comp_triple σ₂₁ σ₁₄ σ₂₄] [ring_hom_comp_triple σ₂₄ σ₄₃ σ₂₃]\n  [ring_hom_comp_triple σ₁₃ σ₃₄ σ₁₄]\n\ninclude σ₂₁ σ₃₄ σ₂₃ σ₂₄ σ₁₃\n\n/-- A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of\ncontinuous linear maps. See also `continuous_linear_equiv.arrow_congr`. -/\n@[simps] def arrow_congr_equiv (e₁₂ : M₁ ≃SL[σ₁₂] M₂) (e₄₃ : M₄ ≃SL[σ₄₃] M₃) :\n  (M₁ →SL[σ₁₄] M₄) ≃ (M₂ →SL[σ₂₃] M₃) :=\n{ to_fun := λ f, (e₄₃ : M₄ →SL[σ₄₃] M₃).comp (f.comp (e₁₂.symm : M₂ →SL[σ₂₁] M₁)),\n  inv_fun := λ f, (e₄₃.symm : M₃ →SL[σ₃₄] M₄).comp (f.comp (e₁₂ : M₁ →SL[σ₁₂] M₂)),\n  left_inv := λ f, continuous_linear_map.ext $ λ x,\n    by simp only [continuous_linear_map.comp_apply, symm_apply_apply, coe_coe],\n  right_inv := λ f, continuous_linear_map.ext $ λ x,\n    by simp only [continuous_linear_map.comp_apply, apply_symm_apply, coe_coe] }\n\nend add_comm_monoid\n\nsection add_comm_group\n\nvariables {R : Type*} [semiring R]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M₂ : Type*} [topological_space M₂] [add_comm_group M₂]\n{M₃ : Type*} [topological_space M₃] [add_comm_group M₃]\n{M₄ : Type*} [topological_space M₄] [add_comm_group M₄]\n[module R M] [module R M₂] [module R M₃] [module R M₄]\n\nvariables [topological_add_group M₄]\n\n/-- Equivalence given by a block lower diagonal matrix. `e` and `e'` are diagonal square blocks,\n  and `f` is a rectangular block below the diagonal. -/\ndef skew_prod (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) :\n  (M × M₃) ≃L[R] M₂ × M₄ :=\n{ continuous_to_fun := (e.continuous_to_fun.comp continuous_fst).prod_mk\n    ((e'.continuous_to_fun.comp continuous_snd).add $ f.continuous.comp continuous_fst),\n  continuous_inv_fun := (e.continuous_inv_fun.comp continuous_fst).prod_mk\n    (e'.continuous_inv_fun.comp $ continuous_snd.sub $ f.continuous.comp $\n      e.continuous_inv_fun.comp continuous_fst),\n.. e.to_linear_equiv.skew_prod e'.to_linear_equiv ↑f }\n@[simp] lemma skew_prod_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :\n  e.skew_prod e' f x = (e x.1, e' x.2 + f x.1) := rfl\n\n@[simp] lemma skew_prod_symm_apply (e : M ≃L[R] M₂) (e' : M₃ ≃L[R] M₄) (f : M →L[R] M₄) (x) :\n  (e.skew_prod e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) := rfl\n\nend add_comm_group\n\nsection ring\n\nvariables {R : Type*} [ring R] {R₂ : Type*} [ring R₂]\n{M : Type*} [topological_space M] [add_comm_group M] [module R M]\n{M₂ : Type*} [topological_space M₂] [add_comm_group M₂] [module R₂ M₂]\nvariables {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} [ring_hom_inv_pair σ₁₂ σ₂₁] [ring_hom_inv_pair σ₂₁ σ₁₂]\n\ninclude σ₂₁\n@[simp] lemma map_sub (e : M ≃SL[σ₁₂] M₂) (x y : M) : e (x - y) = e x - e y :=\n(e : M →SL[σ₁₂] M₂).map_sub x y\n\n@[simp] lemma map_neg (e : M ≃SL[σ₁₂] M₂) (x : M) : e (-x) = -e x := (e : M →SL[σ₁₂] M₂).map_neg x\nomit σ₂₁\n\nsection\n/-! The next theorems cover the identification between `M ≃L[𝕜] M`and the group of units of the ring\n`M →L[R] M`. -/\nvariables [topological_add_group M]\n\n/-- An invertible continuous linear map `f` determines a continuous equivalence from `M` to itself.\n-/\ndef of_unit (f : (M →L[R] M)ˣ) : (M ≃L[R] M) :=\n{ to_linear_equiv :=\n  { to_fun    := f.val,\n    map_add'  := by simp,\n    map_smul' := by simp,\n    inv_fun   := f.inv,\n    left_inv  := λ x, show (f.inv * f.val) x = x, by {rw f.inv_val, simp},\n    right_inv := λ x, show (f.val * f.inv) x = x, by {rw f.val_inv, simp}, },\n  continuous_to_fun  := f.val.continuous,\n  continuous_inv_fun := f.inv.continuous }\n\n/-- A continuous equivalence from `M` to itself determines an invertible continuous linear map. -/\ndef to_unit (f : (M ≃L[R] M)) : (M →L[R] M)ˣ :=\n{ val     := f,\n  inv     := f.symm,\n  val_inv := by {ext, simp},\n  inv_val := by {ext, simp} }\n\nvariables (R M)\n\n/-- The units of the algebra of continuous `R`-linear endomorphisms of `M` is multiplicatively\nequivalent to the type of continuous linear equivalences between `M` and itself. -/\ndef units_equiv : (M →L[R] M)ˣ ≃* (M ≃L[R] M) :=\n{ to_fun    := of_unit,\n  inv_fun   := to_unit,\n  left_inv  := λ f, by {ext, refl},\n  right_inv := λ f, by {ext, refl},\n  map_mul'  := λ x y, by {ext, refl} }\n\n@[simp] lemma units_equiv_apply (f : (M →L[R] M)ˣ) (x : M) :\n  units_equiv R M f x = f x := rfl\n\nend\n\nsection\nvariables (R) [topological_space R] [has_continuous_mul R]\n\n/-- Continuous linear equivalences `R ≃L[R] R` are enumerated by `Rˣ`. -/\ndef units_equiv_aut : Rˣ ≃ (R ≃L[R] R) :=\n{ to_fun := λ u, equiv_of_inverse\n    (continuous_linear_map.smul_right (1 : R →L[R] R) ↑u)\n    (continuous_linear_map.smul_right (1 : R →L[R] R) ↑u⁻¹)\n    (λ x, by simp) (λ x, by simp),\n  inv_fun := λ e, ⟨e 1, e.symm 1,\n    by rw [← smul_eq_mul, ← map_smul, smul_eq_mul, mul_one, symm_apply_apply],\n    by rw [← smul_eq_mul, ← map_smul, smul_eq_mul, mul_one, apply_symm_apply]⟩,\n  left_inv := λ u, units.ext $ by simp,\n  right_inv := λ e, ext₁ $ by simp }\n\nvariable {R}\n\n@[simp] lemma units_equiv_aut_apply (u : Rˣ) (x : R) : units_equiv_aut R u x = x * u := rfl\n\n@[simp] lemma units_equiv_aut_apply_symm (u : Rˣ) (x : R) :\n  (units_equiv_aut R u).symm x = x * ↑u⁻¹ := rfl\n\n@[simp] lemma units_equiv_aut_symm_apply (e : R ≃L[R] R) :\n  ↑((units_equiv_aut R).symm e) = e 1 :=\nrfl\n\nend\n\nvariables [module R M₂] [topological_add_group M]\n\nopen _root_.continuous_linear_map (id fst snd subtype_val mem_ker)\n\n/-- A pair of continuous linear maps such that `f₁ ∘ f₂ = id` generates a continuous\nlinear equivalence `e` between `M` and `M₂ × f₁.ker` such that `(e x).2 = x` for `x ∈ f₁.ker`,\n`(e x).1 = f₁ x`, and `(e (f₂ y)).2 = 0`. The map is given by `e x = (f₁ x, x - f₂ (f₁ x))`. -/\ndef equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : function.right_inverse f₂ f₁) :\n  M ≃L[R] M₂ × f₁.ker :=\nequiv_of_inverse (f₁.prod (f₁.proj_ker_of_right_inverse f₂ h)) (f₂.coprod (subtype_val f₁.ker))\n  (λ x, by simp)\n  (λ ⟨x, y⟩, by simp [h x])\n\n@[simp] lemma fst_equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)\n  (h : function.right_inverse f₂ f₁) (x : M) :\n  (equiv_of_right_inverse f₁ f₂ h x).1 = f₁ x := rfl\n\n@[simp] lemma snd_equiv_of_right_inverse (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)\n  (h : function.right_inverse f₂ f₁) (x : M) :\n  ((equiv_of_right_inverse f₁ f₂ h x).2 : M) = x - f₂ (f₁ x) := rfl\n\n@[simp] lemma equiv_of_right_inverse_symm_apply (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)\n  (h : function.right_inverse f₂ f₁) (y : M₂ × f₁.ker) :\n  (equiv_of_right_inverse f₁ f₂ h).symm y = f₂ y.1 + y.2 := rfl\n\nend ring\n\nsection\n\nvariables (ι R M : Type*) [unique ι] [semiring R] [add_comm_monoid M] [module R M]\n  [topological_space M]\n\n/-- If `ι` has a unique element, then `ι → M` is continuously linear equivalent to `M`. -/\ndef fun_unique : (ι → M) ≃L[R] M :=\n{ to_linear_equiv := linear_equiv.fun_unique ι R M,\n  .. homeomorph.fun_unique ι M }\n\nvariables {ι R M}\n\n@[simp] lemma coe_fun_unique : ⇑(fun_unique ι R M) = function.eval default := rfl\n@[simp] lemma coe_fun_unique_symm : ⇑(fun_unique ι R M).symm = function.const ι := rfl\n\nvariables (R M)\n\n/-- Continuous linear equivalence between dependent functions `Π i : fin 2, M i` and `M 0 × M 1`. -/\n@[simps { fully_applied := ff }]\ndef pi_fin_two (M : fin 2 → Type*) [Π i, add_comm_monoid (M i)] [Π i, module R (M i)]\n  [Π i, topological_space (M i)] :\n  (Π i, M i) ≃L[R] M 0 × M 1 :=\n{ to_linear_equiv := linear_equiv.pi_fin_two R M, .. homeomorph.pi_fin_two M }\n\n/-- Continuous linear equivalence between vectors in `M² = fin 2 → M` and `M × M`. -/\n@[simps { fully_applied := ff }]\ndef fin_two_arrow : (fin 2 → M) ≃L[R] M × M :=\n{ to_linear_equiv := linear_equiv.fin_two_arrow R M, .. pi_fin_two R (λ _, M) }\n\nend\n\n@[simp] lemma det_coe_symm {R : Type*} [field R]\n  {M : Type*} [topological_space M] [add_comm_group M] [module R M] (A : M ≃L[R] M) :\n  (A.symm : M →L[R] M).det = (A : M →L[R] M).det ⁻¹ :=\nlinear_equiv.det_coe_symm A.to_linear_equiv\n\nend continuous_linear_equiv\n\nnamespace continuous_linear_map\n\nopen_locale classical\n\nvariables {R : Type*} {M : Type*} {M₂ : Type*} [topological_space M] [topological_space M₂]\n\nsection\nvariables [semiring R]\nvariables [add_comm_monoid M₂] [module R M₂]\nvariables [add_comm_monoid M] [module R M]\n\n/-- Introduce a function `inverse` from `M →L[R] M₂` to `M₂ →L[R] M`, which sends `f` to `f.symm` if\n`f` is a continuous linear equivalence and to `0` otherwise.  This definition is somewhat ad hoc,\nbut one needs a fully (rather than partially) defined inverse function for some purposes, including\nfor calculus. -/\nnoncomputable def inverse : (M →L[R] M₂) → (M₂ →L[R] M) :=\nλ f, if h : ∃ (e : M ≃L[R] M₂), (e : M →L[R] M₂) = f then ((classical.some h).symm : M₂ →L[R] M)\nelse 0\n\n/-- By definition, if `f` is invertible then `inverse f = f.symm`. -/\n@[simp] lemma inverse_equiv (e : M ≃L[R] M₂) : inverse (e : M →L[R] M₂) = e.symm :=\nbegin\n  have h : ∃ (e' : M ≃L[R] M₂), (e' : M →L[R] M₂) = ↑e := ⟨e, rfl⟩,\n  simp only [inverse, dif_pos h],\n  congr,\n  exact_mod_cast (classical.some_spec h)\nend\n\n/-- By definition, if `f` is not invertible then `inverse f = 0`. -/\n@[simp] lemma inverse_non_equiv (f : M →L[R] M₂) (h : ¬∃ (e' : M ≃L[R] M₂), ↑e' = f) :\n  inverse f = 0 :=\ndif_neg h\n\nend\n\nsection\nvariables [ring R]\nvariables [add_comm_group M] [topological_add_group M] [module R M]\nvariables [add_comm_group M₂] [module R M₂]\n\n@[simp] lemma ring_inverse_equiv (e : M ≃L[R] M) :\n  ring.inverse ↑e = inverse (e : M →L[R] M) :=\nbegin\n  suffices :\n    ring.inverse ((((continuous_linear_equiv.units_equiv _ _).symm e) : M →L[R] M)) = inverse ↑e,\n  { convert this },\n  simp,\n  refl,\nend\n\n/-- The function `continuous_linear_equiv.inverse` can be written in terms of `ring.inverse` for the\nring of self-maps of the domain. -/\nlemma to_ring_inverse (e : M ≃L[R] M₂) (f : M →L[R] M₂) :\n  inverse f = (ring.inverse ((e.symm : (M₂ →L[R] M)).comp f)) ∘L ↑e.symm :=\nbegin\n  by_cases h₁ : ∃ (e' : M ≃L[R] M₂), ↑e' = f,\n  { obtain ⟨e', he'⟩ := h₁,\n    rw ← he',\n    change _ = (ring.inverse ↑(e'.trans e.symm)) ∘L ↑e.symm,\n    ext,\n    simp },\n  { suffices : ¬is_unit ((e.symm : M₂ →L[R] M).comp f),\n    { simp [this, h₁] },\n    contrapose! h₁,\n    rcases h₁ with ⟨F, hF⟩,\n    use (continuous_linear_equiv.units_equiv _ _ F).trans e,\n    ext,\n    dsimp, rw [coe_fn_coe_base' F, hF], simp }\nend\n\nlemma ring_inverse_eq_map_inverse : ring.inverse = @inverse R M M _ _ _ _ _ _ _ :=\nbegin\n  ext,\n  simp [to_ring_inverse (continuous_linear_equiv.refl R M)],\nend\n\nend\n\nend continuous_linear_map\n\nnamespace submodule\n\nvariables\n{R : Type*} [ring R]\n{M : Type*} [topological_space M] [add_comm_group M] [module R M]\n{M₂ : Type*} [topological_space M₂] [add_comm_group M₂] [module R M₂]\n\nopen continuous_linear_map\n\n/-- A submodule `p` is called *complemented* if there exists a continuous projection `M →ₗ[R] p`. -/\ndef closed_complemented (p : submodule R M) : Prop := ∃ f : M →L[R] p, ∀ x : p, f x = x\n\nlemma closed_complemented.has_closed_complement {p : submodule R M} [t1_space p]\n  (h : closed_complemented p) :\n  ∃ (q : submodule R M) (hq : is_closed (q : set M)), is_compl p q :=\nexists.elim h $ λ f hf, ⟨f.ker, f.is_closed_ker, linear_map.is_compl_of_proj hf⟩\n\nprotected lemma closed_complemented.is_closed [topological_add_group M] [t1_space M]\n  {p : submodule R M} (h : closed_complemented p) :\n  is_closed (p : set M) :=\nbegin\n  rcases h with ⟨f, hf⟩,\n  have : ker (id R M - (subtype_val p).comp f) = p := linear_map.ker_id_sub_eq_of_proj hf,\n  exact this ▸ (is_closed_ker _)\nend\n\n@[simp] lemma closed_complemented_bot : closed_complemented (⊥ : submodule R M) :=\n⟨0, λ x, by simp only [zero_apply, eq_zero_of_bot_submodule x]⟩\n\n@[simp] lemma closed_complemented_top : closed_complemented (⊤ : submodule R M) :=\n⟨(id R M).cod_restrict ⊤ (λ x, trivial), λ x, subtype.ext_iff_val.2 $ by simp⟩\n\nend submodule\n\nlemma continuous_linear_map.closed_complemented_ker_of_right_inverse {R : Type*} [ring R]\n  {M : Type*} [topological_space M] [add_comm_group M]\n  {M₂ : Type*} [topological_space M₂] [add_comm_group M₂] [module R M] [module R M₂]\n  [topological_add_group M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M)\n  (h : function.right_inverse f₂ f₁) :\n  f₁.ker.closed_complemented :=\n⟨f₁.proj_ker_of_right_inverse f₂ h, f₁.proj_ker_of_right_inverse_apply_idem f₂ h⟩\n\nsection quotient\n\nnamespace submodule\n\nvariables {R M : Type*} [ring R] [add_comm_group M] [module R M] [topological_space M]\n  (S : submodule R M)\n\nlemma is_open_map_mkq [topological_add_group M] : is_open_map S.mkq :=\nquotient_add_group.is_open_map_coe S.to_add_subgroup\n\ninstance topological_add_group_quotient [topological_add_group M] :\n  topological_add_group (M ⧸ S) :=\ntopological_add_group_quotient S.to_add_subgroup\n\ninstance has_continuous_smul_quotient [topological_space R] [topological_add_group M]\n  [has_continuous_smul R M] :\n  has_continuous_smul R (M ⧸ S) :=\nbegin\n  split,\n  have quot : quotient_map (λ au : R × M, (au.1, S.mkq au.2)),\n    from is_open_map.to_quotient_map\n      (is_open_map.id.prod S.is_open_map_mkq)\n      (continuous_id.prod_map continuous_quot_mk)\n      (function.surjective_id.prod_map $ surjective_quot_mk _),\n  rw quot.continuous_iff,\n  exact continuous_quot_mk.comp continuous_smul\nend\n\ninstance regular_quotient_of_is_closed [topological_add_group M] [is_closed (S : set M)] :\n  regular_space (M ⧸ S) :=\nbegin\n  letI : is_closed (S.to_add_subgroup : set M) := ‹_›,\n  exact S.to_add_subgroup.regular_quotient_of_is_closed\nend\n\nend submodule\n\nend quotient\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/topology/algebra/module/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.32020412364706874}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Jeremy Avigad\n-/\nimport Mathlib.Init.Logic\n\n/-! ### alignments from lean 3 `init.data.prod` -/\n\nuniverse u v\n\nsection\n\nvariable {α : Type u} {β : Type v}\n\n@[simp]\ntheorem Prod.mk.eta : ∀ {p : α × β}, (p.1, p.2) = p\n  | (_, _) => rfl\n\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Init/Data/Prod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.32000976579500295}}
{"text": "syntax \"foo\" !\"(\" term:max : term\n\nmacro_rules\n  | `(foo $x) => `($x + 1)\n\nsyntax \"foo\" \"(\" term \")\" : term\n\nmacro_rules\n  | `(foo ( $x )) => `($x + 2)\n\n#check foo 10\n#check foo(10)\n\ntheorem ex1 : foo 10 = 11 := rfl\ntheorem ex2 : foo (10) = 12 := rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/syntax1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3200097657950029}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Jannis Limperg\n\nFacts about `ulift` and `plift`.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 u_3 \n\nnamespace Mathlib\n\nnamespace plift\n\n\n/-- Functorial action. -/\n@[simp] protected def map {α : Sort u} {β : Sort v} (f : α → β) : plift α → plift β := sorry\n\n/-- Embedding of pure values. -/\n@[simp] protected def pure {α : Sort u} : α → plift α := up\n\n/-- Applicative sequencing. -/\n@[simp] protected def seq {α : Sort u} {β : Sort v} : plift (α → β) → plift α → plift β := sorry\n\n/-- Monadic bind. -/\n@[simp] protected def bind {α : Sort u} {β : Sort v} : plift α → (α → plift β) → plift β := sorry\n\nprotected instance monad : Monad plift :=\n  { toApplicative :=\n      { toFunctor :=\n          { map := plift.map, mapConst := fun (α β : Type u_1) => plift.map ∘ function.const β },\n        toPure := { pure := plift.pure }, toSeq := { seq := plift.seq },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (α β : Type u_1) (a : plift α) (b : plift β) =>\n                plift.seq (plift.map (function.const β) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (α β : Type u_1) (a : plift α) (b : plift β) =>\n                plift.seq (plift.map (function.const α id) a) b } },\n    toBind := { bind := plift.bind } }\n\nprotected instance is_lawful_functor : is_lawful_functor plift :=\n  is_lawful_functor.mk (fun (α : Type u_1) (_x : plift α) => sorry)\n    fun (α β γ : Type u_1) (g : α → β) (h : β → γ) (_x : plift α) => sorry\n\nprotected instance is_lawful_applicative : is_lawful_applicative plift :=\n  is_lawful_applicative.mk (fun (α β : Type u_1) (g : α → β) (_x : plift α) => sorry)\n    (fun (α β : Type u_1) (g : α → β) (x : α) => rfl)\n    (fun (α β : Type u_1) (_x : plift (α → β)) => sorry)\n    fun (α β γ : Type u_1) (_x : plift α) => sorry\n\nprotected instance is_lawful_monad : is_lawful_monad plift :=\n  is_lawful_monad.mk (fun (α β : Type u_1) (x : α) (f : α → plift β) => rfl)\n    fun (α β γ : Type u_1) (_x : plift α) => sorry\n\n@[simp] theorem rec.constant {α : Sort u} {β : Type v} (b : β) :\n    (plift.rec fun (_x : α) => b) = fun (_x : plift α) => b :=\n  funext\n    fun (x : plift α) => cases_on x fun (a : α) => Eq.refl (plift.rec (fun (_x : α) => b) (up a))\n\nend plift\n\n\nnamespace ulift\n\n\n/-- Functorial action. -/\n@[simp] protected def map {α : Type u} {β : Type v} (f : α → β) : ulift α → ulift β := sorry\n\n/-- Embedding of pure values. -/\n@[simp] protected def pure {α : Type u} : α → ulift α := up\n\n/-- Applicative sequencing. -/\n@[simp] protected def seq {α : Type u} {β : Type v} : ulift (α → β) → ulift α → ulift β := sorry\n\n/-- Monadic bind. -/\n@[simp] protected def bind {α : Type u} {β : Type v} : ulift α → (α → ulift β) → ulift β := sorry\n\n-- The `up ∘ down` gives us more universe polymorphism than simply `f a`.\n\nprotected instance monad : Monad ulift :=\n  { toApplicative :=\n      { toFunctor :=\n          { map := ulift.map, mapConst := fun (α β : Type u_1) => ulift.map ∘ function.const β },\n        toPure := { pure := ulift.pure }, toSeq := { seq := ulift.seq },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (α β : Type u_1) (a : ulift α) (b : ulift β) =>\n                ulift.seq (ulift.map (function.const β) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (α β : Type u_1) (a : ulift α) (b : ulift β) =>\n                ulift.seq (ulift.map (function.const α id) a) b } },\n    toBind := { bind := ulift.bind } }\n\nprotected instance is_lawful_functor : is_lawful_functor ulift :=\n  is_lawful_functor.mk (fun (α : Type u_1) (_x : ulift α) => sorry)\n    fun (α β γ : Type u_1) (g : α → β) (h : β → γ) (_x : ulift α) => sorry\n\nprotected instance is_lawful_applicative : is_lawful_applicative ulift :=\n  is_lawful_applicative.mk (fun (α β : Type u_1) (g : α → β) (_x : ulift α) => sorry)\n    (fun (α β : Type u_1) (g : α → β) (x : α) => rfl)\n    (fun (α β : Type u_1) (_x : ulift (α → β)) => sorry)\n    fun (α β γ : Type u_1) (_x : ulift α) => sorry\n\nprotected instance is_lawful_monad : is_lawful_monad ulift :=\n  is_lawful_monad.mk\n    (fun (α β : Type u_1) (x : α) (f : α → ulift β) =>\n      id (cases_on (f x) fun (down : β) => Eq.refl (up (down (up down)))))\n    fun (α β γ : Type u_1) (_x : ulift α) => sorry\n\n@[simp] theorem rec.constant {α : Type u} {β : Sort v} (b : β) :\n    (ulift.rec fun (_x : α) => b) = fun (_x : ulift α) => b :=\n  funext\n    fun (x : ulift α) => cases_on x fun (a : α) => Eq.refl (ulift.rec (fun (_x : α) => b) (up a))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/ulift_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.31989713241564344}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Yury Kudryashov\n-/\nimport measure_theory.function.ae_measurable_sequence\nimport analysis.complex.basic\nimport analysis.normed_space.finite_dimension\nimport measure_theory.group.arithmetic\nimport measure_theory.lattice\nimport measure_theory.measure.open_pos\nimport topology.algebra.ordered.liminf_limsup\nimport topology.continuous_function.basic\nimport topology.instances.ereal\nimport topology.G_delta\nimport topology.semicontinuous\nimport topology.order.lattice\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel α` : the least `σ`-algebra that contains all open sets;\n* `class borel_space` : a space with `topological_space` and `measurable_space` structures\n  such that `‹measurable_space α› = borel α`;\n* `class opens_measurable_space` : a space with `topological_space` and `measurable_space`\n  structures such that all open sets are measurable; equivalently, `borel α ≤ ‹measurable_space α›`.\n* `borel_space` instances on `empty`, `unit`, `bool`, `nat`, `int`, `rat`;\n* `measurable` and `borel_space` instances on `ℝ`, `ℝ≥0`, `ℝ≥0∞`.\n\n## Main statements\n\n* `is_open.measurable_set`, `is_closed.measurable_set`: open and closed sets are measurable;\n* `continuous.measurable` : a continuous function is measurable;\n* `continuous.measurable2` : if `f : α → β` and `g : α → γ` are measurable and `op : β × γ → δ`\n  is continuous, then `λ x, op (f x, g y)` is measurable;\n* `measurable.add` etc : dot notation for arithmetic operations on `measurable` predicates,\n  and similarly for `dist` and `edist`;\n* `ae_measurable.add` : similar dot notation for almost everywhere measurable functions;\n* `measurable.ennreal*` : special cases for arithmetic operations on `ℝ≥0∞`.\n-/\n\nnoncomputable theory\n\nopen classical set filter measure_theory\nopen_locale classical big_operators topological_space nnreal ennreal interval\n\nuniverses u v w x y\nvariables {α β γ γ₂ δ : Type*} {ι : Sort y} {s t u : set α}\n\nopen measurable_space topological_space\n\n/-- `measurable_space` structure generated by `topological_space`. -/\ndef borel (α : Type u) [topological_space α] : measurable_space α :=\ngenerate_from {s : set α | is_open s}\n\nlemma borel_eq_top_of_discrete [topological_space α] [discrete_topology α] :\n  borel α = ⊤ :=\ntop_le_iff.1 $ λ s hs, generate_measurable.basic s (is_open_discrete s)\n\nlemma borel_eq_top_of_encodable [topological_space α] [t1_space α] [encodable α] :\n  borel α = ⊤ :=\nbegin\n  refine (top_le_iff.1 $ λ s hs, bUnion_of_singleton s ▸ _),\n  apply measurable_set.bUnion s.countable_encodable,\n  intros x hx,\n  apply measurable_set.of_compl,\n  apply generate_measurable.basic,\n  exact is_closed_singleton.is_open_compl\nend\n\nlemma borel_eq_generate_from_of_subbasis {s : set (set α)}\n  [t : topological_space α] [second_countable_topology α] (hs : t = generate_from s) :\n  borel α = generate_from s :=\nle_antisymm\n  (generate_from_le $ assume u (hu : t.is_open u),\n    begin\n      rw [hs] at hu,\n      induction hu,\n      case generate_open.basic : u hu\n      { exact generate_measurable.basic u hu },\n      case generate_open.univ\n      { exact @measurable_set.univ α (generate_from s) },\n      case generate_open.inter : s₁ s₂ _ _ hs₁ hs₂\n      { exact @measurable_set.inter α (generate_from s) _ _ hs₁ hs₂ },\n      case generate_open.sUnion : f hf ih\n      { rcases is_open_sUnion_countable f (by rwa hs) with ⟨v, hv, vf, vu⟩,\n        rw ← vu,\n        exact @measurable_set.sUnion α (generate_from s) _ hv\n          (λ x xv, ih _ (vf xv)) }\n    end)\n  (generate_from_le $ assume u hu, generate_measurable.basic _ $\n    show t.is_open u, by rw [hs]; exact generate_open.basic _ hu)\n\nlemma topological_space.is_topological_basis.borel_eq_generate_from [topological_space α]\n  [second_countable_topology α] {s : set (set α)} (hs : is_topological_basis s) :\n  borel α = generate_from s :=\nborel_eq_generate_from_of_subbasis hs.eq_generate_from\n\nlemma is_pi_system_is_open [topological_space α] : is_pi_system (is_open : set α → Prop) :=\nλ s hs t ht hst, is_open.inter hs ht\n\nlemma borel_eq_generate_from_is_closed [topological_space α] :\n  borel α = generate_from {s | is_closed s} :=\nle_antisymm\n  (generate_from_le $ λ t ht, @measurable_set.of_compl α _ (generate_from {s | is_closed s})\n    (generate_measurable.basic _ $ is_closed_compl_iff.2 ht))\n  (generate_from_le $ λ t ht, @measurable_set.of_compl α _ (borel α)\n    (generate_measurable.basic _ $ is_open_compl_iff.2 ht))\n\nsection order_topology\n\nvariable (α)\nvariables [topological_space α] [second_countable_topology α] [linear_order α] [order_topology α]\n\nlemma borel_eq_generate_from_Iio : borel α = generate_from (range Iio) :=\nbegin\n  refine le_antisymm _ (generate_from_le _),\n  { rw borel_eq_generate_from_of_subbasis (@order_topology.topology_eq_generate_intervals α _ _ _),\n    letI : measurable_space α := measurable_space.generate_from (range Iio),\n    have H : ∀ a : α, measurable_set (Iio a) := λ a, generate_measurable.basic _ ⟨_, rfl⟩,\n    refine generate_from_le _, rintro _ ⟨a, rfl | rfl⟩; [skip, apply H],\n    by_cases h : ∃ a', ∀ b, a < b ↔ a' ≤ b,\n    { rcases h with ⟨a', ha'⟩,\n      rw (_ : Ioi a = (Iio a')ᶜ), { exact (H _).compl },\n      simp [set.ext_iff, ha'] },\n    { rcases is_open_Union_countable\n        (λ a' : {a' : α // a < a'}, {b | a'.1 < b})\n        (λ a', is_open_lt' _) with ⟨v, ⟨hv⟩, vu⟩,\n      simp [set.ext_iff] at vu,\n      have : Ioi a = ⋃ x : v, (Iio x.1.1)ᶜ,\n      { simp [set.ext_iff],\n        refine λ x, ⟨λ ax, _, λ ⟨a', ⟨h, av⟩, ax⟩, lt_of_lt_of_le h ax⟩,\n        rcases (vu x).2 _ with ⟨a', h₁, h₂⟩,\n        { exact ⟨a', h₁, le_of_lt h₂⟩ },\n        refine not_imp_comm.1 (λ h, _) h,\n        exact ⟨x, λ b, ⟨λ ab, le_of_not_lt (λ h', h ⟨b, ab, h'⟩),\n          lt_of_lt_of_le ax⟩⟩ },\n      rw this, resetI,\n      apply measurable_set.Union,\n      exact λ _, (H _).compl } },\n  { rw forall_range_iff,\n    intro a,\n    exact generate_measurable.basic _ is_open_Iio }\nend\n\nlemma borel_eq_generate_from_Ioi : borel α = generate_from (range Ioi) :=\n@borel_eq_generate_from_Iio (order_dual α) _ (by apply_instance : second_countable_topology α) _ _\n\nend order_topology\n\nlemma borel_comap {f : α → β} {t : topological_space β} :\n  @borel α (t.induced f) = (@borel β t).comap f :=\ncomap_generate_from.symm\n\nlemma continuous.borel_measurable [topological_space α] [topological_space β]\n  {f : α → β} (hf : continuous f) :\n  @measurable α β (borel α) (borel β) f :=\nmeasurable.of_le_map $ generate_from_le $\n  λ s hs, generate_measurable.basic (f ⁻¹' s) (hs.preimage hf)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nall open sets are measurable. -/\nclass opens_measurable_space (α : Type*) [topological_space α] [h : measurable_space α] : Prop :=\n(borel_le : borel α ≤ h)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nthe `σ`-algebra of measurable sets is exactly the `σ`-algebra generated by open sets. -/\nclass borel_space (α : Type*) [topological_space α] [measurable_space α] : Prop :=\n(measurable_eq : ‹measurable_space α› = borel α)\n\n@[priority 100]\ninstance order_dual.opens_measurable_space {α : Type*} [topological_space α] [measurable_space α]\n  [h : opens_measurable_space α] :\n  opens_measurable_space (order_dual α) :=\n{ borel_le := h.borel_le }\n\n@[priority 100]\ninstance order_dual.borel_space {α : Type*} [topological_space α] [measurable_space α]\n  [h : borel_space α] :\n  borel_space (order_dual α) :=\n{ measurable_eq := h.measurable_eq }\n\n/-- In a `borel_space` all open sets are measurable. -/\n@[priority 100]\ninstance borel_space.opens_measurable {α : Type*} [topological_space α] [measurable_space α]\n  [borel_space α] : opens_measurable_space α :=\n⟨ge_of_eq $ borel_space.measurable_eq⟩\n\ninstance subtype.borel_space {α : Type*} [topological_space α] [measurable_space α]\n  [hα : borel_space α] (s : set α) :\n  borel_space s :=\n⟨by { rw [hα.1, subtype.measurable_space, ← borel_comap], refl }⟩\n\ninstance subtype.opens_measurable_space {α : Type*} [topological_space α] [measurable_space α]\n  [h : opens_measurable_space α] (s : set α) :\n  opens_measurable_space s :=\n⟨by { rw [borel_comap], exact comap_mono h.1 }⟩\n\ntheorem _root_.measurable_set.induction_on_open [topological_space α] [measurable_space α]\n  [borel_space α] {C : set α → Prop} (h_open : ∀ U, is_open U → C U)\n  (h_compl : ∀ t, measurable_set t → C t → C tᶜ)\n  (h_union : ∀ f : ℕ → set α, pairwise (disjoint on f) →\n    (∀ i, measurable_set (f i)) → (∀ i, C (f i)) → C (⋃ i, f i)) :\n  ∀ ⦃t⦄, measurable_set t → C t :=\nmeasurable_space.induction_on_inter borel_space.measurable_eq is_pi_system_is_open\n  (h_open _ is_open_empty) h_open h_compl h_union\n\nsection\nvariables [topological_space α] [measurable_space α] [opens_measurable_space α]\n   [topological_space β] [measurable_space β] [opens_measurable_space β]\n   [topological_space γ] [measurable_space γ] [borel_space γ]\n   [topological_space γ₂] [measurable_space γ₂] [borel_space γ₂]\n   [measurable_space δ]\n\nlemma is_open.measurable_set (h : is_open s) : measurable_set s :=\nopens_measurable_space.borel_le _ $ generate_measurable.basic _ h\n\n@[measurability]\nlemma measurable_set_interior : measurable_set (interior s) := is_open_interior.measurable_set\n\nlemma is_Gδ.measurable_set (h : is_Gδ s) : measurable_set s :=\nbegin\n  rcases h with ⟨S, hSo, hSc, rfl⟩,\n  exact measurable_set.sInter hSc (λ t ht, (hSo t ht).measurable_set)\nend\n\nlemma measurable_set_of_continuous_at {β} [emetric_space β] (f : α → β) :\n  measurable_set {x | continuous_at f x} :=\n(is_Gδ_set_of_continuous_at f).measurable_set\n\nlemma is_closed.measurable_set (h : is_closed s) : measurable_set s :=\nh.is_open_compl.measurable_set.of_compl\n\nlemma is_compact.measurable_set [t2_space α] (h : is_compact s) : measurable_set s :=\nh.is_closed.measurable_set\n\n@[measurability]\nlemma measurable_set_closure : measurable_set (closure s) :=\nis_closed_closure.measurable_set\n\nlemma measurable_of_is_open {f : δ → γ} (hf : ∀ s, is_open s → measurable_set (f ⁻¹' s)) :\n  measurable f :=\nby { rw [‹borel_space γ›.measurable_eq], exact measurable_generate_from hf }\n\nlemma measurable_of_is_closed {f : δ → γ} (hf : ∀ s, is_closed s → measurable_set (f ⁻¹' s)) :\n  measurable f :=\nbegin\n  apply measurable_of_is_open, intros s hs,\n  rw [← measurable_set.compl_iff, ← preimage_compl], apply hf, rw [is_closed_compl_iff], exact hs\nend\n\nlemma measurable_of_is_closed' {f : δ → γ}\n  (hf : ∀ s, is_closed s → s.nonempty → s ≠ univ → measurable_set (f ⁻¹' s)) : measurable f :=\nbegin\n  apply measurable_of_is_closed, intros s hs,\n  cases eq_empty_or_nonempty s with h1 h1, { simp [h1] },\n  by_cases h2 : s = univ, { simp [h2] },\n  exact hf s hs h1 h2\nend\n\ninstance nhds_is_measurably_generated (a : α) : (𝓝 a).is_measurably_generated :=\nbegin\n  rw [nhds, infi_subtype'],\n  refine @filter.infi_is_measurably_generated _ _ _ _ (λ i, _),\n  exact i.2.2.measurable_set.principal_is_measurably_generated\nend\n\n/-- If `s` is a measurable set, then `𝓝[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : measurable_set s`.\n-/\nlemma measurable_set.nhds_within_is_measurably_generated {s : set α} (hs : measurable_set s)\n  (a : α) :\n  (𝓝[s] a).is_measurably_generated :=\nby haveI := hs.principal_is_measurably_generated; exact filter.inf_is_measurably_generated _ _\n\n@[priority 100] -- see Note [lower instance priority]\ninstance opens_measurable_space.to_measurable_singleton_class [t1_space α] :\n  measurable_singleton_class α :=\n⟨λ x, is_closed_singleton.measurable_set⟩\n\ninstance pi.opens_measurable_space_encodable {ι : Type*} {π : ι → Type*} [encodable ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, opens_measurable_space (π i)] :\n  opens_measurable_space (Π i, π i) :=\nbegin\n  constructor,\n  have : Pi.topological_space =\n    generate_from {t | ∃(s:Πa, set (π a)) (i : finset ι), (∀a∈i, s a ∈ countable_basis (π a)) ∧\n      t = pi ↑i s},\n  { rw [funext (λ a, @eq_generate_from_countable_basis (π a) _ _), pi_generate_from_eq] },\n  rw [borel_eq_generate_from_of_subbasis this],\n  apply generate_from_le,\n  rintros _ ⟨s, i, hi, rfl⟩,\n  refine measurable_set.pi i.countable_to_set (λ a ha, is_open.measurable_set _),\n  rw [eq_generate_from_countable_basis (π a)],\n  exact generate_open.basic _ (hi a ha)\nend\n\ninstance pi.opens_measurable_space_fintype {ι : Type*} {π : ι → Type*} [fintype ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, opens_measurable_space (π i)] :\n  opens_measurable_space (Π i, π i) :=\nby { letI := fintype.encodable ι, apply_instance }\n\ninstance prod.opens_measurable_space [second_countable_topology α] [second_countable_topology β] :\n  opens_measurable_space (α × β) :=\nbegin\n  constructor,\n  rw [((is_basis_countable_basis α).prod (is_basis_countable_basis β)).borel_eq_generate_from],\n  apply generate_from_le,\n  rintros _ ⟨u, v, hu, hv, rfl⟩,\n  exact (is_open_of_mem_countable_basis hu).measurable_set.prod\n    (is_open_of_mem_countable_basis hv).measurable_set\nend\n\nvariables {α' : Type*} [topological_space α'] [measurable_space α']\n\nlemma measure_interior_of_null_bdry {μ : measure α'} {s : set α'}\n  (h_nullbdry : μ (frontier s) = 0) : μ (interior s) = μ s :=\nmeasure_eq_measure_smaller_of_between_null_diff\n  interior_subset subset_closure h_nullbdry\n\nlemma measure_closure_of_null_bdry {μ : measure α'} {s : set α'}\n  (h_nullbdry : μ (frontier s) = 0) : μ (closure s) = μ s :=\n(measure_eq_measure_larger_of_between_null_diff\n  interior_subset subset_closure h_nullbdry).symm\n\nsection preorder\nvariables [preorder α] [order_closed_topology α] {a b x : α}\n\n@[simp, measurability]\nlemma measurable_set_Ici : measurable_set (Ici a) := is_closed_Ici.measurable_set\n@[simp, measurability]\nlemma measurable_set_Iic : measurable_set (Iic a) := is_closed_Iic.measurable_set\n@[simp, measurability]\nlemma measurable_set_Icc : measurable_set (Icc a b) := is_closed_Icc.measurable_set\n\ninstance nhds_within_Ici_is_measurably_generated :\n  (𝓝[Ici b] a).is_measurably_generated :=\nmeasurable_set_Ici.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iic_is_measurably_generated :\n  (𝓝[Iic b] a).is_measurably_generated :=\nmeasurable_set_Iic.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Icc_is_measurably_generated :\n  is_measurably_generated (𝓝[Icc a b] x) :=\nby { rw [← Ici_inter_Iic, nhds_within_inter], apply_instance }\n\ninstance at_top_is_measurably_generated : (filter.at_top : filter α).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  λ a, (measurable_set_Ici : measurable_set (Ici a)).principal_is_measurably_generated\n\ninstance at_bot_is_measurably_generated : (filter.at_bot : filter α).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  λ a, (measurable_set_Iic : measurable_set (Iic a)).principal_is_measurably_generated\n\nend preorder\n\nsection partial_order\nvariables [partial_order α] [order_closed_topology α] [second_countable_topology α]\n  {a b : α}\n\n@[measurability]\nlemma measurable_set_le' : measurable_set {p : α × α | p.1 ≤ p.2} :=\norder_closed_topology.is_closed_le'.measurable_set\n\n@[measurability]\nlemma measurable_set_le {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable_set {a | f a ≤ g a} :=\nhf.prod_mk hg measurable_set_le'\n\nend partial_order\n\nsection linear_order\nvariables [linear_order α] [order_closed_topology α] {a b x : α}\n\n@[simp, measurability]\nlemma measurable_set_Iio : measurable_set (Iio a) := is_open_Iio.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioi : measurable_set (Ioi a) := is_open_Ioi.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioo : measurable_set (Ioo a b) := is_open_Ioo.measurable_set\n\n@[simp, measurability] lemma measurable_set_Ioc : measurable_set (Ioc a b) :=\nmeasurable_set_Ioi.inter measurable_set_Iic\n\n@[simp, measurability] lemma measurable_set_Ico : measurable_set (Ico a b) :=\nmeasurable_set_Ici.inter measurable_set_Iio\n\ninstance nhds_within_Ioi_is_measurably_generated :\n  (𝓝[Ioi b] a).is_measurably_generated :=\nmeasurable_set_Ioi.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iio_is_measurably_generated :\n  (𝓝[Iio b] a).is_measurably_generated :=\nmeasurable_set_Iio.nhds_within_is_measurably_generated _\n\ninstance nhds_within_interval_is_measurably_generated :\n  is_measurably_generated (𝓝[[a, b]] x) :=\nnhds_within_Icc_is_measurably_generated\n\n@[measurability]\nlemma measurable_set_lt' [second_countable_topology α] : measurable_set {p : α × α | p.1 < p.2} :=\n(is_open_lt continuous_fst continuous_snd).measurable_set\n\n@[measurability]\nlemma measurable_set_lt [second_countable_topology α] {f g : δ → α} (hf : measurable f)\n  (hg : measurable g) : measurable_set {a | f a < g a} :=\nhf.prod_mk hg measurable_set_lt'\n\nlemma set.ord_connected.measurable_set (h : ord_connected s) : measurable_set s :=\nbegin\n  let u := ⋃ (x ∈ s) (y ∈ s), Ioo x y,\n  have huopen : is_open u := is_open_bUnion (λ x hx, is_open_bUnion (λ y hy, is_open_Ioo)),\n  have humeas : measurable_set u := huopen.measurable_set,\n  have hfinite : (s \\ u).finite,\n  { refine set.finite_of_forall_between_eq_endpoints (s \\ u) (λ x hx y hy z hz hxy hyz, _),\n    by_contra' h,\n    exact hy.2 (mem_Union₂.mpr ⟨x, hx.1,\n      mem_Union₂.mpr ⟨z, hz.1, lt_of_le_of_ne hxy h.1, lt_of_le_of_ne hyz h.2⟩⟩) },\n  have : u ⊆ s :=\n    Union₂_subset (λ x hx, Union₂_subset (λ y hy, Ioo_subset_Icc_self.trans (h.out hx hy))),\n  rw ← union_diff_cancel this,\n  exact humeas.union hfinite.measurable_set\nend\n\nlemma is_preconnected.measurable_set\n  (h : is_preconnected s) : measurable_set s :=\nh.ord_connected.measurable_set\n\nlemma generate_from_Ico_mem_le_borel {α : Type*} [topological_space α] [linear_order α]\n  [order_closed_topology α] (s t : set α) :\n  measurable_space.generate_from {S | ∃ (l ∈ s) (u ∈ t) (h : l < u), Ico l u = S} ≤ borel α :=\nbegin\n  apply generate_from_le,\n  letI : measurable_space α := borel α, haveI : borel_space α := ⟨rfl⟩,\n  rintro _ ⟨a, -, b, -, -, rfl⟩,\n  exact measurable_set_Ico\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem_aux {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] {s : set α} (hd : dense s)\n  (hbot : ∀ x, is_bot x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → y ∈ s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S} :=\nbegin\n  set S : set (set α) := {S | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S},\n  refine le_antisymm _ (generate_from_Ico_mem_le_borel _ _),\n  letI : measurable_space α := generate_from S,\n  rw borel_eq_generate_from_Iio,\n  refine generate_from_le (forall_range_iff.2 $ λ a, _),\n  rcases hd.exists_countable_dense_subset_bot_top with ⟨t, hts, hc, htd, htb, htt⟩,\n  by_cases ha : ∀ b < a, (Ioo b a).nonempty,\n  { convert_to measurable_set (⋃ (l ∈ t) (u ∈ t) (hlu : l < u) (hu : u ≤ a), Ico l u),\n    { ext y, simp only [mem_Union, mem_Iio, mem_Ico], split,\n      { intro hy,\n        rcases htd.exists_le' (λ b hb, htb _ hb (hbot b hb)) y with ⟨l, hlt, hly⟩,\n        rcases htd.exists_mem_open is_open_Ioo (ha y hy) with ⟨u, hut, hyu, hua⟩,\n        exact ⟨l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu⟩ },\n      { rintro ⟨l, -, u, -, -, hua, -, hyu⟩,\n        exact hyu.trans_le hua } },\n    { refine measurable_set.bUnion hc (λ a ha, measurable_set.bUnion hc $ λ b hb, _),\n      refine measurable_set.Union_Prop (λ hab, measurable_set.Union_Prop $ λ hb', _),\n      exact generate_measurable.basic _ ⟨a, hts ha, b, hts hb, hab, mem_singleton _⟩ } },\n  { simp only [not_forall, not_nonempty_iff_eq_empty] at ha,\n    replace ha : a ∈ s := hIoo ha.some a ha.some_spec.fst ha.some_spec.snd,\n    convert_to measurable_set (⋃ (l ∈ t) (hl : l < a), Ico l a),\n    { symmetry,\n      simp only [← Ici_inter_Iio, ← Union_inter, inter_eq_right_iff_subset, subset_def, mem_Union,\n        mem_Ici, mem_Iio],\n      intros x hx, rcases htd.exists_le' (λ b hb, htb _ hb (hbot b hb)) x with ⟨z, hzt, hzx⟩,\n      exact ⟨z, hzt, hzx.trans_lt hx, hzx⟩ },\n    { refine measurable_set.bUnion hc (λ x hx, measurable_set.Union_Prop $ λ hlt, _),\n      exact generate_measurable.basic _ ⟨x, hts hx, a, ha, hlt, mem_singleton _⟩ } }\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] [densely_ordered α] [no_min_order α]\n  {s : set α} (hd : dense s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ico l u = S} :=\nhd.borel_eq_generate_from_Ico_mem_aux (by simp) $\n  λ x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ico (α : Type*) [topological_space α]\n  [second_countable_topology α] [linear_order α] [order_topology α] :\n  borel α = generate_from {S : set α | ∃ l u (h : l < u), Ico l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ α _).borel_eq_generate_from_Ico_mem_aux (λ _ _, mem_univ _)\n      (λ _ _ _ _, mem_univ _)\n\nlemma dense.borel_eq_generate_from_Ioc_mem_aux {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] {s : set α} (hd : dense s)\n  (hbot : ∀ x, is_top x → x ∈ s) (hIoo : ∀ x y : α, x < y → Ioo x y = ∅ → x ∈ s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ioc l u = S} :=\nbegin\n  convert hd.order_dual.borel_eq_generate_from_Ico_mem_aux hbot (λ x y hlt he, hIoo y x hlt _),\n  { ext s,\n    split; rintro ⟨l, hl, u, hu, hlt, rfl⟩,\n    exacts [⟨u, hu, l, hl, hlt, dual_Ico⟩, ⟨u, hu, l, hl, hlt, dual_Ioc⟩] },\n  { erw dual_Ioo,\n    exact he }\nend\n\nlemma dense.borel_eq_generate_from_Ioc_mem {α : Type*} [topological_space α] [linear_order α]\n  [order_topology α] [second_countable_topology α] [densely_ordered α] [no_max_order α]\n  {s : set α} (hd : dense s) :\n  borel α = generate_from {S : set α | ∃ (l ∈ s) (u ∈ s) (h : l < u), Ioc l u = S} :=\nhd.borel_eq_generate_from_Ioc_mem_aux (by simp) $\n  λ x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ioc (α : Type*) [topological_space α]\n  [second_countable_topology α] [linear_order α] [order_topology α] :\n  borel α = generate_from {S : set α | ∃ l u (h : l < u), Ioc l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ α _).borel_eq_generate_from_Ioc_mem_aux (λ _ _, mem_univ _)\n      (λ _ _ _ _, mem_univ _)\n\nnamespace measure_theory.measure\n\n/-- Two finite measures on a Borel space are equal if they agree on all closed-open intervals.  If\n`α` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ico` is an extensionality lemma with weaker assumptions on `μ` and\n`ν`. -/\nlemma ext_of_Ico_finite {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α]\n  [borel_space α] (μ ν : measure α) [is_finite_measure μ] (hμν : μ univ = ν univ)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nbegin\n  refine ext_of_generate_finite _\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico α))\n    (is_pi_system_Ico (id : α → α) id) _ hμν,\n  { rintro - ⟨a, b, hlt, rfl⟩,\n    exact h hlt }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all open-closed intervals.  If\n`α` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ioc` is an extensionality lemma with weaker assumptions on `μ` and\n`ν`. -/\nlemma ext_of_Ioc_finite {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α]\n  [borel_space α] (μ ν : measure α) [is_finite_measure μ] (hμν : μ univ = ν univ)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nbegin\n  refine @ext_of_Ico_finite (order_dual α) _ _ _ _ _ ‹_› μ ν _ hμν (λ a b hab, _),\n  erw dual_Ico,\n  exact h hab\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico' {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  [no_max_order α] (μ ν : measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ico a b) ≠ ∞)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nbegin\n  rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, hsb, hst⟩,\n  have : countable (⋃ (l ∈ s) (u ∈ s) (h : l < u), {Ico l u} : set (set α)),\n    from hsc.bUnion (λ l hl, hsc.bUnion\n      (λ u hu, countable_Union_Prop $ λ _, countable_singleton _)),\n  simp only [← set_of_eq_eq_singleton, ← set_of_exists] at this,\n  refine measure.ext_of_generate_from_of_cover_subset\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico α))\n    (is_pi_system_Ico id id) _ this _ _ _,\n  { rintro _ ⟨l, -, u, -, h, rfl⟩, exact ⟨l, u, h, rfl⟩ },\n  { refine sUnion_eq_univ_iff.2 (λ x, _),\n    rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩,\n    rcases hsd.exists_gt x with ⟨u, hus, hxu⟩,\n    exact ⟨_, ⟨l, hls, u, hus, hlx.trans_lt hxu, rfl⟩, hlx, hxu⟩ },\n  { rintro _ ⟨l, -, u, -, hlt, rfl⟩, exact hμ hlt },\n  { rintro _ ⟨l, u, hlt, rfl⟩, exact h hlt }\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc' {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  [no_min_order α] (μ ν : measure α) (hμ : ∀ ⦃a b⦄, a < b → μ (Ioc a b) ≠ ∞)\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nbegin\n  refine @ext_of_Ico' (order_dual α) _ _ _ _ _ ‹_› _ μ ν _ _;\n    intros a b hab; erw dual_Ico,\n  exacts [hμ hab, h hab]\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [conditionally_complete_linear_order α] [order_topology α]\n  [borel_space α] [no_max_order α] (μ ν : measure α) [is_locally_finite_measure μ]\n  (h : ∀ ⦃a b⦄, a < b → μ (Ico a b) = ν (Ico a b)) : μ = ν :=\nμ.ext_of_Ico' ν (λ a b hab, measure_Ico_lt_top.ne) h\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [conditionally_complete_linear_order α] [order_topology α]\n  [borel_space α] [no_min_order α] (μ ν : measure α) [is_locally_finite_measure μ]\n  (h : ∀ ⦃a b⦄, a < b → μ (Ioc a b) = ν (Ioc a b)) : μ = ν :=\nμ.ext_of_Ioc' ν (λ a b hab, measure_Ioc_lt_top.ne) h\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed\nintervals. -/\nlemma ext_of_Iic {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  (μ ν : measure α) [is_finite_measure μ] (h : ∀ a, μ (Iic a) = ν (Iic a)) : μ = ν :=\nbegin\n  refine ext_of_Ioc_finite μ ν _ (λ a b hlt, _),\n  { rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩,\n    have : directed_on (≤) s, from directed_on_iff_directed.2 (directed_of_sup $ λ _ _, id),\n    simp only [← bsupr_measure_Iic hsc (hsd.exists_ge' hst) this, h] },\n  rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic,\n      measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic, h a, h b],\n  { rw ← h a, exact (measure_lt_top μ _).ne },\n  { exact (measure_lt_top μ _).ne }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite\nintervals. -/\nlemma ext_of_Ici {α : Type*} [topological_space α] {m : measurable_space α}\n  [second_countable_topology α] [linear_order α] [order_topology α] [borel_space α]\n  (μ ν : measure α) [is_finite_measure μ] (h : ∀ a, μ (Ici a) = ν (Ici a)) : μ = ν :=\n@ext_of_Iic (order_dual α) _ _ _ _ _ ‹_› _ _ _ h\n\nend measure_theory.measure\n\nend linear_order\n\nsection linear_order\n\nvariables [linear_order α] [order_closed_topology α]\n\n@[measurability]\nlemma measurable_set_interval {a b : α} : measurable_set (interval a b) :=\nmeasurable_set_Icc\n\n@[measurability]\nlemma measurable_set_interval_oc {a b : α} : measurable_set (interval_oc a b) :=\nmeasurable_set_Ioc\n\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable.max {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, max (f a) (g a)) :=\nby simpa only [max_def] using hf.piecewise (measurable_set_le hg hf) hg\n\n@[measurability]\nlemma ae_measurable.max {f g : δ → α} {μ : measure δ}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, max (f a) (g a)) μ :=\n⟨λ a, max (hf.mk f a) (hg.mk g a), hf.measurable_mk.max hg.measurable_mk,\n  eventually_eq.comp₂ hf.ae_eq_mk _ hg.ae_eq_mk⟩\n\n@[measurability]\nlemma measurable.min {f g : δ → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, min (f a) (g a)) :=\nby simpa only [min_def] using hf.piecewise (measurable_set_le hf hg) hg\n\n@[measurability]\nlemma ae_measurable.min {f g : δ → α} {μ : measure δ}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, min (f a) (g a)) μ :=\n⟨λ a, min (hf.mk f a) (hg.mk g a), hf.measurable_mk.min hg.measurable_mk,\n  eventually_eq.comp₂ hf.ae_eq_mk _ hg.ae_eq_mk⟩\n\nend linear_order\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis measurable. -/\nlemma continuous.measurable {f : α → γ} (hf : continuous f) :\n  measurable f :=\nhf.borel_measurable.mono opens_measurable_space.borel_le\n  (le_of_eq $ borel_space.measurable_eq)\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis ae-measurable. -/\nlemma continuous.ae_measurable {f : α → γ} (h : continuous f) (μ : measure α) : ae_measurable f μ :=\nh.measurable.ae_measurable\n\nlemma closed_embedding.measurable {f : α → γ} (hf : closed_embedding f) :\n  measurable f :=\nhf.continuous.measurable\n\nlemma continuous.is_open_pos_measure_map {f : β → γ} (hf : continuous f)\n  (hf_surj : function.surjective f) {μ : measure β} [μ.is_open_pos_measure] :\n  (measure.map f μ).is_open_pos_measure :=\nbegin\n  refine ⟨λ U hUo hUne, _⟩,\n  rw [measure.map_apply hf.measurable hUo.measurable_set],\n  exact (hUo.preimage hf).measure_ne_zero μ (hf_surj.nonempty_preimage.mpr hUne)\nend\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul [has_mul γ] [has_continuous_mul γ] :\n  has_measurable_mul γ :=\n{ measurable_const_mul := λ c, (continuous_const.mul continuous_id).measurable,\n  measurable_mul_const := λ c, (continuous_id.mul continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub [has_sub γ] [has_continuous_sub γ] :\n  has_measurable_sub γ :=\n{ measurable_const_sub := λ c, (continuous_const.sub continuous_id).measurable,\n  measurable_sub_const := λ c, (continuous_id.sub continuous_const).measurable }\n\n@[priority 100, to_additive]\ninstance topological_group.has_measurable_inv [group γ] [topological_group γ] :\n  has_measurable_inv γ :=\n⟨continuous_inv.measurable⟩\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul {M α} [topological_space M]\n  [topological_space α] [measurable_space M] [measurable_space α]\n  [opens_measurable_space M] [borel_space α] [has_scalar M α] [has_continuous_smul M α] :\n  has_measurable_smul M α :=\n⟨λ c, (continuous_const.smul continuous_id).measurable,\n  λ y, (continuous_id.smul continuous_const).measurable⟩\n\nsection lattice\n\n@[priority 100]\ninstance has_continuous_sup.has_measurable_sup [has_sup γ] [has_continuous_sup γ] :\n  has_measurable_sup γ :=\n{ measurable_const_sup := λ c, (continuous_const.sup continuous_id).measurable,\n  measurable_sup_const := λ c, (continuous_id.sup continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_sup.has_measurable_sup₂ [second_countable_topology γ] [has_sup γ]\n  [has_continuous_sup γ] :\n  has_measurable_sup₂ γ :=\n⟨continuous_sup.measurable⟩\n\n@[priority 100]\ninstance has_continuous_inf.has_measurable_inf [has_inf γ] [has_continuous_inf γ] :\n  has_measurable_inf γ :=\n{ measurable_const_inf := λ c, (continuous_const.inf continuous_id).measurable,\n  measurable_inf_const := λ c, (continuous_id.inf continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_inf.has_measurable_inf₂ [second_countable_topology γ] [has_inf γ]\n  [has_continuous_inf γ] :\n  has_measurable_inf₂ γ :=\n⟨continuous_inf.measurable⟩\n\nend lattice\n\nsection homeomorph\n\n@[measurability] protected lemma homeomorph.measurable (h : α ≃ₜ γ) : measurable h :=\nh.continuous.measurable\n\n/-- A homeomorphism between two Borel spaces is a measurable equivalence.-/\ndef homeomorph.to_measurable_equiv (h : γ ≃ₜ γ₂) : γ ≃ᵐ γ₂ :=\n{ measurable_to_fun := h.measurable,\n  measurable_inv_fun := h.symm.measurable,\n  to_equiv := h.to_equiv }\n\n@[simp]\nlemma homeomorph.to_measurable_equiv_coe (h : γ ≃ₜ γ₂) : (h.to_measurable_equiv : γ → γ₂) = h :=\nrfl\n\n@[simp] lemma homeomorph.to_measurable_equiv_symm_coe (h : γ ≃ₜ γ₂) :\n  (h.to_measurable_equiv.symm : γ₂ → γ) = h.symm :=\nrfl\n\nend homeomorph\n\n@[measurability] lemma continuous_map.measurable (f : C(α, γ)) : measurable f :=\nf.continuous.measurable\n\nlemma measurable_of_continuous_on_compl_singleton [t1_space α] {f : α → γ} (a : α)\n  (hf : continuous_on f {a}ᶜ) :\n  measurable f :=\nmeasurable_of_measurable_on_compl_singleton a\n  (continuous_on_iff_continuous_restrict.1 hf).measurable\n\nlemma continuous.measurable2 [second_countable_topology α] [second_countable_topology β]\n  {f : δ → α} {g : δ → β} {c : α → β → γ}\n  (h : continuous (λ p : α × β, c p.1 p.2)) (hf : measurable f) (hg : measurable g) :\n  measurable (λ a, c (f a) (g a)) :=\nh.measurable.comp (hf.prod_mk hg)\n\nlemma continuous.ae_measurable2 [second_countable_topology α] [second_countable_topology β]\n  {f : δ → α} {g : δ → β} {c : α → β → γ} {μ : measure δ}\n  (h : continuous (λ p : α × β, c p.1 p.2)) (hf : ae_measurable f μ) (hg : ae_measurable g μ) :\n  ae_measurable (λ a, c (f a) (g a)) μ :=\nh.measurable.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100]\ninstance has_continuous_inv₀.has_measurable_inv [group_with_zero γ] [t1_space γ]\n  [has_continuous_inv₀ γ] :\n  has_measurable_inv γ :=\n⟨measurable_of_continuous_on_compl_singleton 0 continuous_on_inv₀⟩\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul₂ [second_countable_topology γ] [has_mul γ]\n  [has_continuous_mul γ] : has_measurable_mul₂ γ :=\n⟨continuous_mul.measurable⟩\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub₂ [second_countable_topology γ] [has_sub γ]\n  [has_continuous_sub γ] : has_measurable_sub₂ γ :=\n⟨continuous_sub.measurable⟩\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul₂ {M α} [topological_space M]\n  [second_countable_topology M] [measurable_space M] [opens_measurable_space M]\n  [topological_space α] [second_countable_topology α] [measurable_space α]\n  [borel_space α] [has_scalar M α] [has_continuous_smul M α] :\n  has_measurable_smul₂ M α :=\n⟨continuous_smul.measurable⟩\n\nend\n\nsection borel_space\nvariables [topological_space α] [measurable_space α] [borel_space α]\n  [topological_space β] [measurable_space β] [borel_space β]\n  [topological_space γ] [measurable_space γ] [borel_space γ]\n  [measurable_space δ]\n\nlemma pi_le_borel_pi {ι : Type*} {π : ι → Type*} [Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, borel_space (π i)] :\n  measurable_space.pi ≤ borel (Π i, π i) :=\nbegin\n  have : ‹Π i, measurable_space (π i)› = λ i, borel (π i) :=\n    funext (λ i, borel_space.measurable_eq),\n  rw [this],\n  exact supr_le (λ i, comap_le_iff_le_map.2 $ (continuous_apply i).borel_measurable)\nend\n\nlemma prod_le_borel_prod : prod.measurable_space ≤ borel (α × β) :=\nbegin\n  rw [‹borel_space α›.measurable_eq, ‹borel_space β›.measurable_eq],\n  refine sup_le _ _,\n  { exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable },\n  { exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable }\nend\n\ninstance pi.borel_space_fintype_encodable {ι : Type*} {π : ι → Type*} [encodable ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, borel_space (π i)] :\n  borel_space (Π i, π i) :=\n⟨le_antisymm pi_le_borel_pi opens_measurable_space.borel_le⟩\n\ninstance pi.borel_space_fintype {ι : Type*} {π : ι → Type*} [fintype ι]\n  [t' : Π i, topological_space (π i)]\n  [Π i, measurable_space (π i)] [∀ i, second_countable_topology (π i)]\n  [∀ i, borel_space (π i)] :\n  borel_space (Π i, π i) :=\n⟨le_antisymm pi_le_borel_pi opens_measurable_space.borel_le⟩\n\ninstance prod.borel_space [second_countable_topology α] [second_countable_topology β] :\n  borel_space (α × β) :=\n⟨le_antisymm prod_le_borel_prod opens_measurable_space.borel_le⟩\n\nprotected lemma embedding.measurable_embedding {f : α → β} (h₁ : embedding f)\n  (h₂ : measurable_set (range f)) : measurable_embedding f :=\nshow measurable_embedding (coe ∘ (homeomorph.of_embedding f h₁).to_measurable_equiv),\nfrom (measurable_embedding.subtype_coe h₂).comp (measurable_equiv.measurable_embedding _)\n\nprotected lemma closed_embedding.measurable_embedding {f : α → β} (h : closed_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.closed_range.measurable_set\n\nprotected lemma open_embedding.measurable_embedding {f : α → β} (h : open_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.open_range.measurable_set\n\nsection linear_order\n\nvariables [linear_order α] [order_topology α] [second_countable_topology α]\n\nlemma measurable_of_Iio {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Iio x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Iio _),\n  rintro _ ⟨x, rfl⟩, exact hf x\nend\n\nlemma upper_semicontinuous.measurable [topological_space δ] [opens_measurable_space δ]\n  {f : δ → α} (hf : upper_semicontinuous f) : measurable f :=\nmeasurable_of_Iio (λ y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Ioi {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Ioi x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Ioi _),\n  rintro _ ⟨x, rfl⟩, exact hf x\nend\n\nlemma lower_semicontinuous.measurable [topological_space δ] [opens_measurable_space δ]\n  {f : δ → α} (hf : lower_semicontinuous f) : measurable f :=\nmeasurable_of_Ioi (λ y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Iic {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Iic x)) : measurable f :=\nbegin\n  apply measurable_of_Ioi,\n  simp_rw [← compl_Iic, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable_of_Ici {f : δ → α} (hf : ∀ x, measurable_set (f ⁻¹' Ici x)) : measurable f :=\nbegin\n  apply measurable_of_Iio,\n  simp_rw [← compl_Ici, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable.is_lub {ι} [encodable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, measurable (f i))\n  (hg : ∀ b, is_lub {a | ∃ i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change ∀ b, is_lub (range $ λ i, f i b) (g b) at hg,\n  rw [‹borel_space α›.measurable_eq, borel_eq_generate_from_Ioi α],\n  apply measurable_generate_from,\n  rintro _ ⟨a, rfl⟩,\n  simp_rw [set.preimage, mem_Ioi, lt_is_lub_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (λ i, hf i (is_open_lt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_lub_of_nonempty {ι} (hι : nonempty ι)\n  {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_lub {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  let p : δ → (ι → α) → Prop := λ x f', is_lub {a | ∃ i, f' i = a} (g x),\n  let g_seq := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨g x⟩ : nonempty α).some,\n  have hg_seq : ∀ b, is_lub {a | ∃ i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI hα : nonempty α := nonempty.map g ⟨b⟩,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : α | ∃ (i : ι), (hf i).mk (f i) b = a} = {a : α | ∃ (i : ι), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : α | ∃ (i : ι), hα.some = a} = {hα.some},\n      { ext1 x,\n        exact ⟨λ hx, hx.some_spec.symm, λ hx, ⟨hι.some, hx.symm⟩⟩, },\n      rw h_singleton,\n      exact is_lub_singleton, }, },\n  refine ⟨g_seq, measurable.is_lub (ae_seq.measurable hf p) hg_seq, _⟩,\n  exact (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨g x⟩ : nonempty α).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_lub {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_lub {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  by_cases hμ : μ = 0, { rw hμ, exact ae_measurable_zero_measure },\n  haveI : μ.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases hι : nonempty ι, { exact ae_measurable.is_lub_of_nonempty hι hf hg, },\n  suffices : ∃ x, g =ᵐ[μ] λ y, g x,\n  by { exact ⟨(λ y, g this.some), measurable_const, this.some_spec⟩, },\n  have h_empty : ∀ x, {a : α | ∃ (i : ι), f i x = a} = ∅,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact λ hi, hι (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact ⟨hg.exists.some, hg.mono (λ y hy, is_lub.unique hy hg.exists.some_spec)⟩,\nend\n\nlemma measurable.is_glb {ι} [encodable ι] {f : ι → δ → α} {g : δ → α} (hf : ∀ i, measurable (f i))\n  (hg : ∀ b, is_glb {a | ∃ i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change ∀ b, is_glb (range $ λ i, f i b) (g b) at hg,\n  rw [‹borel_space α›.measurable_eq, borel_eq_generate_from_Iio α],\n  apply measurable_generate_from,\n  rintro _ ⟨a, rfl⟩,\n  simp_rw [set.preimage, mem_Iio, is_glb_lt_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (λ i, hf i (is_open_gt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_glb_of_nonempty {ι} (hι : nonempty ι)\n  {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_glb {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  let p : δ → (ι → α) → Prop := λ x f', is_glb {a | ∃ i, f' i = a} (g x),\n  let g_seq := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨g x⟩ : nonempty α).some,\n  have hg_seq : ∀ b, is_glb {a | ∃ i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI hα : nonempty α := nonempty.map g ⟨b⟩,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : α | ∃ (i : ι), (hf i).mk (f i) b = a} = {a : α | ∃ (i : ι), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : α | ∃ (i : ι), hα.some = a} = {hα.some},\n      { ext1 x,\n        exact ⟨λ hx, hx.some_spec.symm, λ hx, ⟨hι.some, hx.symm⟩⟩, },\n      rw h_singleton,\n      exact is_glb_singleton, }, },\n  refine ⟨g_seq, measurable.is_glb (ae_seq.measurable hf p) hg_seq, _⟩,\n  exact (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨g x⟩ : nonempty α).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_glb {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α} {g : δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) (hg : ∀ᵐ b ∂μ, is_glb {a | ∃ i, f i b = a} (g b)) :\n  ae_measurable g μ :=\nbegin\n  by_cases hμ : μ = 0, { rw hμ, exact ae_measurable_zero_measure },\n  haveI : μ.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases hι : nonempty ι, { exact ae_measurable.is_glb_of_nonempty hι hf hg, },\n  suffices : ∃ x, g =ᵐ[μ] λ y, g x,\n  by { exact ⟨(λ y, g this.some), measurable_const, this.some_spec⟩, },\n  have h_empty : ∀ x, {a : α | ∃ (i : ι), f i x = a} = ∅,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact λ hi, hι (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact ⟨hg.exists.some, hg.mono (λ y hy, is_glb.unique hy hg.exists.some_spec)⟩,\nend\n\nprotected lemma monotone.measurable [linear_order β] [order_closed_topology β] {f : β → α}\n  (hf : monotone f) : measurable f :=\nsuffices h : ∀ x, ord_connected (f ⁻¹' Ioi x),\n  from measurable_of_Ioi (λ x, (h x).measurable_set),\nλ x, ord_connected_def.mpr (λ a ha b hb c hc, lt_of_lt_of_le ha (hf hc.1))\n\nlemma ae_measurable_restrict_of_monotone_on [linear_order β] [order_closed_topology β]\n  {μ : measure β} {s : set β} (hs : measurable_set s) {f : β → α} (hf : monotone_on f s) :\n  ae_measurable f (μ.restrict s) :=\nhave this : monotone (f ∘ coe : s → α), from λ ⟨x, hx⟩ ⟨y, hy⟩ (hxy : x ≤ y), hf hx hy hxy,\nae_measurable_restrict_of_measurable_subtype hs this.measurable\n\nprotected lemma antitone.measurable [linear_order β] [order_closed_topology β] {f : β → α}\n  (hf : antitone f) :\n  measurable f :=\n@monotone.measurable (order_dual α) β _ _ ‹_› _ _ _ _ _ ‹_› _ _ _ hf\n\nlemma ae_measurable_restrict_of_antitone_on [linear_order β] [order_closed_topology β]\n  {μ : measure β} {s : set β} (hs : measurable_set s) {f : β → α} (hf : antitone_on f s) :\n  ae_measurable f (μ.restrict s) :=\n@ae_measurable_restrict_of_monotone_on (order_dual α) β _ _ ‹_› _ _ _ _ _ ‹_› _ _ _ _ hs _ hf\n\nend linear_order\n\n@[measurability]\nlemma measurable.supr_Prop {α} [measurable_space α] [complete_lattice α]\n  (p : Prop) {f : δ → α} (hf : measurable f) :\n  measurable (λ b, ⨆ h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact supr_pos h end)\n  (assume h : ¬p, begin convert measurable_const, funext, exact supr_neg h end)\n\n@[measurability]\nlemma measurable.infi_Prop {α} [measurable_space α] [complete_lattice α]\n  (p : Prop) {f : δ → α} (hf : measurable f) :\n  measurable (λ b, ⨅ h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact infi_pos h end )\n  (assume h : ¬p, begin convert measurable_const, funext, exact infi_neg h end)\n\nsection complete_linear_order\n\nvariables [complete_linear_order α] [order_topology α] [second_countable_topology α]\n\n@[measurability]\nlemma measurable_supr {ι} [encodable ι] {f : ι → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ b, ⨆ i, f i b) :=\nmeasurable.is_lub hf $ λ b, is_lub_supr\n\n@[measurability]\nlemma ae_measurable_supr {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ b, ⨆ i, f i b) μ :=\nae_measurable.is_lub hf $ (ae_of_all μ (λ b, is_lub_supr))\n\n@[measurability]\nlemma measurable_infi {ι} [encodable ι] {f : ι → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ b, ⨅ i, f i b) :=\nmeasurable.is_glb hf $ λ b, is_glb_infi\n\n@[measurability]\nlemma ae_measurable_infi {ι} {μ : measure δ} [encodable ι] {f : ι → δ → α}\n  (hf : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ b, ⨅ i, f i b) μ :=\nae_measurable.is_glb hf $ (ae_of_all μ (λ b, is_glb_infi))\n\nlemma measurable_bsupr {ι} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, measurable (f i)) : measurable (λ b, ⨆ i ∈ s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [supr_subtype'],\n     exact measurable_supr (λ i, hf i) }\n\nlemma ae_measurable_bsupr {ι} {μ : measure δ} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, ae_measurable (f i) μ) : ae_measurable (λ b, ⨆ i ∈ s, f i b) μ :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [supr_subtype'],\n  exact ae_measurable_supr (λ i, hf i),\nend\n\nlemma measurable_binfi {ι} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, measurable (f i)) : measurable (λ b, ⨅ i ∈ s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [infi_subtype'],\n     exact measurable_infi (λ i, hf i) }\n\nlemma ae_measurable_binfi {ι} {μ : measure δ} (s : set ι) {f : ι → δ → α} (hs : countable s)\n  (hf : ∀ i, ae_measurable (f i) μ) : ae_measurable (λ b, ⨅ i ∈ s, f i b) μ :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [infi_subtype'],\n  exact ae_measurable_infi (λ i, hf i),\nend\n\n/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `ℕ`.\n-/\nlemma measurable_liminf' {ι ι'} {f : ι → δ → α} {u : filter ι} (hf : ∀ i, measurable (f i))\n  {p : ι' → Prop} {s : ι' → set ι} (hu : u.has_countable_basis p s) (hs : ∀ i, (s i).countable) :\n  measurable (λ x, liminf u (λ i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.liminf_eq_supr_infi],\n  refine measurable_bsupr _ hu.countable _,\n  exact λ i, measurable_binfi _ (hs i) hf\nend\n\n/-- `limsup` over a general filter is measurable. See `measurable_limsup` for the version over `ℕ`.\n-/\nlemma measurable_limsup' {ι ι'}  {f : ι → δ → α} {u : filter ι} (hf : ∀ i, measurable (f i))\n  {p : ι' → Prop} {s : ι' → set ι} (hu : u.has_countable_basis p s) (hs : ∀ i, (s i).countable) :\n  measurable (λ x, limsup u (λ i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.limsup_eq_infi_supr],\n  refine measurable_binfi _ hu.countable _,\n  exact λ i, measurable_bsupr _ (hs i) hf\nend\n\n/-- `liminf` over `ℕ` is measurable. See `measurable_liminf'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_liminf {f : ℕ → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ x, liminf at_top (λ i, f i x)) :=\nmeasurable_liminf' hf at_top_countable_basis (λ i, countable_encodable _)\n\n/-- `limsup` over `ℕ` is measurable. See `measurable_limsup'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_limsup {f : ℕ → δ → α} (hf : ∀ i, measurable (f i)) :\n  measurable (λ x, limsup at_top (λ i, f i x)) :=\nmeasurable_limsup' hf at_top_countable_basis (λ i, countable_encodable _)\n\nend complete_linear_order\n\nsection conditionally_complete_linear_order\n\nvariables [conditionally_complete_linear_order α] [order_topology α] [second_countable_topology α]\n\nlemma measurable_cSup {ι} {f : ι → δ → α} {s : set ι} (hs : s.countable)\n  (hf : ∀ i, measurable (f i)) (bdd : ∀ x, bdd_above ((λ i, f i x) '' s)) :\n  measurable (λ x, Sup ((λ i, f i x) '' s)) :=\nbegin\n  cases eq_empty_or_nonempty s with h2s h2s,\n  { simp [h2s, measurable_const] },\n  { apply measurable_of_Iic, intro y,\n    simp_rw [preimage, mem_Iic, cSup_le_iff (bdd _) (h2s.image _), ball_image_iff, set_of_forall],\n    exact measurable_set.bInter hs (λ i hi, measurable_set_le (hf i) measurable_const) }\nend\n\nend conditionally_complete_linear_order\n\n/-- Convert a `homeomorph` to a `measurable_equiv`. -/\ndef homemorph.to_measurable_equiv (h : α ≃ₜ β) : α ≃ᵐ β :=\n{ to_equiv := h.to_equiv,\n  measurable_to_fun := h.continuous_to_fun.measurable,\n  measurable_inv_fun := h.continuous_inv_fun.measurable }\n\nprotected lemma is_finite_measure_on_compacts.map\n  {α : Type*} {m0 : measurable_space α} [topological_space α] [opens_measurable_space α]\n  {β : Type*} [measurable_space β] [topological_space β] [borel_space β]\n  [t2_space β] (μ : measure α) [is_finite_measure_on_compacts μ] (f : α ≃ₜ β) :\n  is_finite_measure_on_compacts (measure.map f μ) :=\n⟨begin\n  assume K hK,\n  rw [measure.map_apply f.measurable hK.measurable_set],\n  apply is_compact.measure_lt_top,\n  rwa f.compact_preimage\nend⟩\n\nend borel_space\n\ninstance empty.borel_space : borel_space empty := ⟨borel_eq_top_of_discrete.symm⟩\ninstance unit.borel_space : borel_space unit := ⟨borel_eq_top_of_discrete.symm⟩\ninstance bool.borel_space : borel_space bool := ⟨borel_eq_top_of_discrete.symm⟩\ninstance nat.borel_space : borel_space ℕ := ⟨borel_eq_top_of_discrete.symm⟩\ninstance int.borel_space : borel_space ℤ := ⟨borel_eq_top_of_discrete.symm⟩\ninstance rat.borel_space : borel_space ℚ := ⟨borel_eq_top_of_encodable.symm⟩\n\n@[priority 900]\ninstance is_R_or_C.measurable_space {𝕜 : Type*} [is_R_or_C 𝕜] : measurable_space 𝕜 := borel 𝕜\n@[priority 900]\ninstance is_R_or_C.borel_space {𝕜 : Type*} [is_R_or_C 𝕜] : borel_space 𝕜 := ⟨rfl⟩\n\n/- Instances on `real` and `complex` are special cases of `is_R_or_C` but without these instances,\nLean fails to prove `borel_space (ι → ℝ)`, so we leave them here. -/\n\ninstance real.measurable_space : measurable_space ℝ := borel ℝ\ninstance real.borel_space : borel_space ℝ := ⟨rfl⟩\n\ninstance nnreal.measurable_space : measurable_space ℝ≥0 := subtype.measurable_space\ninstance nnreal.borel_space : borel_space ℝ≥0 := subtype.borel_space _\n\ninstance ennreal.measurable_space : measurable_space ℝ≥0∞ := borel ℝ≥0∞\ninstance ennreal.borel_space : borel_space ℝ≥0∞ := ⟨rfl⟩\n\ninstance ereal.measurable_space : measurable_space ereal := borel ereal\ninstance ereal.borel_space : borel_space ereal := ⟨rfl⟩\n\ninstance complex.measurable_space : measurable_space ℂ := borel ℂ\ninstance complex.borel_space : borel_space ℂ := ⟨rfl⟩\n\n/-- One can cut out `ℝ≥0∞` into the sets `{0}`, `Ico (t^n) (t^(n+1))` for `n : ℤ` and `{∞}`. This\ngives a way to compute the measure of a set in terms of sets on which a given function `f` does not\nfluctuate by more than `t`. -/\nlemma measure_eq_measure_preimage_add_measure_tsum_Ico_zpow [measurable_space α] (μ : measure α)\n  {f : α → ℝ≥0∞} (hf : measurable f) {s : set α} (hs : measurable_set s) {t : ℝ≥0} (ht : 1 < t) :\n  μ s = μ (s ∩ f⁻¹' {0}) + μ (s ∩ f⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f⁻¹' (Ico (t^n) (t^(n+1)))) :=\nbegin\n  have A : μ s = μ (s ∩ f⁻¹' {0}) + μ (s ∩ f⁻¹' (Ioi 0)),\n  { rw ← measure_union,\n    { congr' 1,\n      ext x,\n      have : 0 = f x ∨ 0 < f x := eq_or_lt_of_le bot_le,\n      rw eq_comm at this,\n      simp only [←and_or_distrib_left, this, mem_singleton_iff, mem_inter_eq, and_true,\n        mem_union_eq, mem_Ioi, mem_preimage], },\n    { apply disjoint_left.2 (λ x hx h'x, _),\n      have : 0 < f x := h'x.2,\n      exact lt_irrefl 0 (this.trans_le hx.2.le) },\n    { exact hs.inter (hf measurable_set_Ioi) } },\n  have B : μ (s ∩ f⁻¹' (Ioi 0)) = μ (s ∩ f⁻¹' {∞}) + μ (s ∩ f⁻¹' (Ioo 0 ∞)),\n  { rw ← measure_union,\n    { rw ← inter_union_distrib_left,\n      congr,\n      ext x,\n      simp only [mem_singleton_iff, mem_union_eq, mem_Ioo, mem_Ioi, mem_preimage],\n      have H : f x = ∞ ∨ f x < ∞ := eq_or_lt_of_le le_top,\n      cases H,\n      { simp only [H, eq_self_iff_true, or_false, with_top.zero_lt_top, not_top_lt, and_false] },\n      { simp only [H, H.ne, and_true, false_or] } },\n    { apply disjoint_left.2 (λ x hx h'x, _),\n      have : f x < ∞ := h'x.2.2,\n      exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm)) },\n    { exact hs.inter (hf measurable_set_Ioo) } },\n  have C : μ (s ∩ f⁻¹' (Ioo 0 ∞)) = ∑' (n : ℤ), μ (s ∩ f⁻¹' (Ico (t^n) (t^(n+1)))),\n  { rw [← measure_Union, ennreal.Ioo_zero_top_eq_Union_Ico_zpow (ennreal.one_lt_coe_iff.2 ht)\n         ennreal.coe_ne_top, preimage_Union, inter_Union],\n    { assume i j,\n      simp only [function.on_fun],\n      wlog h : i ≤ j := le_total i j using [i j, j i] tactic.skip,\n      { assume hij,\n        replace hij : i + 1 ≤ j := lt_of_le_of_ne h hij,\n        apply disjoint_left.2 (λ x hx h'x, lt_irrefl (f x) _),\n        calc f x < t ^ (i + 1) : hx.2.2\n        ... ≤ t ^ j : ennreal.zpow_le_of_le (ennreal.one_le_coe_iff.2 ht.le) hij\n        ... ≤ f x : h'x.2.1 },\n      { assume hij,\n        rw disjoint.comm,\n        exact this hij.symm } },\n    { assume n,\n      exact hs.inter (hf measurable_set_Ico) } },\n  rw [A, B, C, add_assoc],\nend\n\nsection metric_space\n\nvariables [metric_space α] [measurable_space α] [opens_measurable_space α]\nvariables [measurable_space β] {x : α} {ε : ℝ}\n\nopen metric\n\n@[measurability]\nlemma measurable_set_ball : measurable_set (metric.ball x ε) :=\nmetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_set_closed_ball : measurable_set (metric.closed_ball x ε) :=\nmetric.is_closed_ball.measurable_set\n\n@[measurability]\nlemma measurable_inf_dist {s : set α} : measurable (λ x, inf_dist x s) :=\n(continuous_inf_dist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_dist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_dist (f x) s) :=\nmeasurable_inf_dist.comp hf\n\n@[measurability]\nlemma measurable_inf_nndist {s : set α} : measurable (λ x, inf_nndist x s) :=\n(continuous_inf_nndist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_nndist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_nndist (f x) s) :=\nmeasurable_inf_nndist.comp hf\n\nsection\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable_dist : measurable (λ p : α × α, dist p.1 p.2) :=\ncontinuous_dist.measurable\n\n@[measurability]\nlemma measurable.dist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, dist (f b) (g b)) :=\n(@continuous_dist α _).measurable2 hf hg\n\n@[measurability]\nlemma measurable_nndist : measurable (λ p : α × α, nndist p.1 p.2) :=\ncontinuous_nndist.measurable\n\n@[measurability]\nlemma measurable.nndist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, nndist (f b) (g b)) :=\n(@continuous_nndist α _).measurable2 hf hg\n\nend\n\n/-- If a set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to the measure of its closure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening {μ : measure α} {s : set α}\n  (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) :\n  tendsto (λ r, μ (cthickening r s)) (𝓝 0) (𝓝 (μ (closure s))) :=\nbegin\n  have A : tendsto (λ r, μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (μ (closure s))),\n  { rw closure_eq_Inter_cthickening,\n    exact tendsto_measure_bInter_gt (λ r hr, is_closed_cthickening.measurable_set)\n      (λ i j ipos ij, cthickening_mono ij _) hs },\n  have B : tendsto (λ r, μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (μ (closure s))),\n  { apply tendsto.congr' _ tendsto_const_nhds,\n    filter_upwards [self_mem_nhds_within] with _ hr,\n    rw cthickening_of_nonpos hr, },\n  convert B.sup A,\n  exact (nhds_left_sup_nhds_right' 0).symm,\nend\n\n/-- If a closed set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to its measure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening_of_is_closed {μ : measure α} {s : set α}\n  (hs : ∃ R > 0, μ (cthickening R s) ≠ ∞) (h's : is_closed s) :\n  tendsto (λ r, μ (cthickening r s)) (𝓝 0) (𝓝 (μ s)) :=\nbegin\n  convert tendsto_measure_cthickening hs,\n  exact h's.closure_eq.symm\nend\n\n/-- Given a compact set in a proper space, the measure of its `r`-closed thickenings converges to\nits measure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening_of_is_compact [proper_space α] {μ : measure α}\n  [is_finite_measure_on_compacts μ] {s : set α} (hs : is_compact s) :\n  tendsto (λ r, μ (cthickening r s)) (𝓝 0) (𝓝 (μ s)) :=\ntendsto_measure_cthickening_of_is_closed\n  ⟨1, zero_lt_one, (bounded.measure_lt_top hs.bounded.cthickening).ne⟩ hs.is_closed\n\nend metric_space\n\nsection emetric_space\n\nvariables [emetric_space α] [measurable_space α] [opens_measurable_space α]\nvariables [measurable_space β] {x : α} {ε : ℝ≥0∞}\n\nopen emetric\n\n@[measurability]\nlemma measurable_set_eball : measurable_set (emetric.ball x ε) :=\nemetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_edist_right : measurable (edist x) :=\n(continuous_const.edist continuous_id).measurable\n\n@[measurability]\nlemma measurable_edist_left : measurable (λ y, edist y x) :=\n(continuous_id.edist continuous_const).measurable\n\n@[measurability]\nlemma measurable_inf_edist {s : set α} : measurable (λ x, inf_edist x s) :=\ncontinuous_inf_edist.measurable\n\n@[measurability]\nlemma measurable.inf_edist {f : β → α} (hf : measurable f) {s : set α} :\n  measurable (λ x, inf_edist (f x) s) :=\nmeasurable_inf_edist.comp hf\n\nvariables [second_countable_topology α]\n\n@[measurability]\nlemma measurable_edist : measurable (λ p : α × α, edist p.1 p.2) :=\ncontinuous_edist.measurable\n\n@[measurability]\nlemma measurable.edist {f g : β → α} (hf : measurable f) (hg : measurable g) :\n  measurable (λ b, edist (f b) (g b)) :=\n(@continuous_edist α _).measurable2 hf hg\n\n@[measurability]\nlemma ae_measurable.edist {f g : β → α} {μ : measure β}\n  (hf : ae_measurable f μ) (hg : ae_measurable g μ) : ae_measurable (λ a, edist (f a) (g a)) μ :=\n(@continuous_edist α _).ae_measurable2 hf hg\n\nend emetric_space\n\nnamespace real\nopen measurable_space measure_theory\n\nlemma borel_eq_generate_from_Ioo_rat :\n  borel ℝ = generate_from (⋃(a b : ℚ) (h : a < b), {Ioo a b}) :=\nis_topological_basis_Ioo_rat.borel_eq_generate_from\n\nlemma is_pi_system_Ioo_rat : @is_pi_system ℝ (⋃ (a b : ℚ) (h : a < b), {Ioo a b})  :=\nbegin\n  convert is_pi_system_Ioo (coe : ℚ → ℝ) (coe : ℚ → ℝ),\n  ext x,\n  simp [eq_comm]\nend\n\n/-- The intervals `(-(n + 1), (n + 1))` form a finite spanning sets in the set of open intervals\nwith rational endpoints for a locally finite measure `μ` on `ℝ`. -/\ndef finite_spanning_sets_in_Ioo_rat (μ : measure ℝ) [is_locally_finite_measure μ] :\n  μ.finite_spanning_sets_in (⋃ (a b : ℚ) (h : a < b), {Ioo a b}) :=\n{ set := λ n, Ioo (-(n + 1)) (n + 1),\n  set_mem := λ n,\n    begin\n      simp only [mem_Union, mem_singleton_iff],\n      refine ⟨-(n + 1), n + 1, _, by norm_cast⟩,\n      exact (neg_nonpos.2 (@nat.cast_nonneg ℚ _ (n + 1))).trans_lt n.cast_add_one_pos\n    end,\n  finite := λ n, measure_Ioo_lt_top,\n  spanning := Union_eq_univ_iff.2 $ λ x,\n    ⟨⌊|x|⌋₊, neg_lt.1 ((neg_le_abs_self x).trans_lt (nat.lt_floor_add_one _)),\n      (le_abs_self x).trans_lt (nat.lt_floor_add_one _)⟩ }\n\nlemma measure_ext_Ioo_rat {μ ν : measure ℝ} [is_locally_finite_measure μ]\n  (h : ∀ a b : ℚ, μ (Ioo a b) = ν (Ioo a b)) : μ = ν :=\n(finite_spanning_sets_in_Ioo_rat μ).ext borel_eq_generate_from_Ioo_rat is_pi_system_Ioo_rat $\n  by { simp only [mem_Union, mem_singleton_iff], rintro _ ⟨a, b, -, rfl⟩, apply h }\n\nlemma borel_eq_generate_from_Iio_rat :\n  borel ℝ = generate_from (⋃ a : ℚ, {Iio a}) :=\nbegin\n  let g : measurable_space ℝ := generate_from (⋃ a : ℚ, {Iio a}),\n  refine le_antisymm _ _,\n  { rw borel_eq_generate_from_Ioo_rat,\n    refine generate_from_le (λ t, _),\n    simp only [mem_Union, mem_singleton_iff], rintro ⟨a, b, h, rfl⟩,\n    rw (set.ext (λ x, _) : Ioo (a : ℝ) b = (⋃c>a, (Iio c)ᶜ) ∩ Iio b),\n    { have hg : ∀ q : ℚ, g.measurable_set' (Iio q) :=\n        λ q, generate_measurable.basic (Iio q) (by simp),\n      refine @measurable_set.inter _ g _ _ _ (hg _),\n      refine @measurable_set.bUnion _ _ g _ _ (countable_encodable _) (λ c h, _),\n      exact @measurable_set.compl _ _ g (hg _) },\n    { suffices : x < ↑b → (↑a < x ↔ ∃ (i : ℚ), a < i ∧ ↑i ≤ x), by simpa,\n      refine λ _, ⟨λ h, _, λ ⟨i, hai, hix⟩, (rat.cast_lt.2 hai).trans_le hix⟩,\n      rcases exists_rat_btwn h with ⟨c, ac, cx⟩,\n      exact ⟨c, rat.cast_lt.1 ac, cx.le⟩ } },\n  { refine measurable_space.generate_from_le (λ _, _),\n    simp only [mem_Union, mem_singleton_iff], rintro ⟨r, rfl⟩, exact measurable_set_Iio }\nend\n\nend real\n\nvariable [measurable_space α]\n\n@[measurability]\nlemma measurable_real_to_nnreal : measurable (real.to_nnreal) :=\nnnreal.continuous_of_real.measurable\n\n@[measurability]\nlemma measurable.real_to_nnreal {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, real.to_nnreal (f x)) :=\nmeasurable_real_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.real_to_nnreal {f : α → ℝ} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, real.to_nnreal (f x)) μ :=\nmeasurable_real_to_nnreal.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_real : measurable (coe : ℝ≥0 → ℝ) :=\nnnreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_real {f : α → ℝ≥0} (hf : measurable f) :\n  measurable (λ x, (f x : ℝ)) :=\nmeasurable_coe_nnreal_real.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_real {f : α → ℝ≥0} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ℝ)) μ :=\nmeasurable_coe_nnreal_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_ennreal : measurable (coe : ℝ≥0 → ℝ≥0∞) :=\nennreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_ennreal {f : α → ℝ≥0} (hf : measurable f) :\n  measurable (λ x, (f x : ℝ≥0∞)) :=\nennreal.continuous_coe.measurable.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_ennreal {f : α → ℝ≥0} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ℝ≥0∞)) μ :=\nennreal.continuous_coe.measurable.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable.ennreal_of_real {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, ennreal.of_real (f x)) :=\nennreal.continuous_of_real.measurable.comp hf\n\n/-- The set of finite `ℝ≥0∞` numbers is `measurable_equiv` to `ℝ≥0`. -/\ndef measurable_equiv.ennreal_equiv_nnreal : {r : ℝ≥0∞ | r ≠ ∞} ≃ᵐ ℝ≥0 :=\nennreal.ne_top_homeomorph_nnreal.to_measurable_equiv\n\nnamespace ennreal\n\nlemma measurable_of_measurable_nnreal {f : ℝ≥0∞ → α}\n  (h : measurable (λ p : ℝ≥0, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_singleton ∞\n  (measurable_equiv.ennreal_equiv_nnreal.symm.measurable_comp_iff.1 h)\n\n/-- `ℝ≥0∞` is `measurable_equiv` to `ℝ≥0 ⊕ unit`. -/\ndef ennreal_equiv_sum : ℝ≥0∞ ≃ᵐ ℝ≥0 ⊕ unit :=\n{ measurable_to_fun  := measurable_of_measurable_nnreal measurable_inl,\n  measurable_inv_fun := measurable_sum measurable_coe_nnreal_ennreal\n    (@measurable_const ℝ≥0∞ unit _ _ ∞),\n  .. equiv.option_equiv_sum_punit ℝ≥0 }\n\nopen function (uncurry)\n\nlemma measurable_of_measurable_nnreal_prod [measurable_space β] [measurable_space γ]\n  {f : ℝ≥0∞ × β → γ} (H₁ : measurable (λ p : ℝ≥0 × β, f (p.1, p.2)))\n  (H₂ : measurable (λ x, f (∞, x))) :\n  measurable f :=\nlet e : ℝ≥0∞ × β ≃ᵐ ℝ≥0 × β ⊕ unit × β :=\n  (ennreal_equiv_sum.prod_congr (measurable_equiv.refl β)).trans\n    (measurable_equiv.sum_prod_distrib _ _ _) in\ne.symm.measurable_comp_iff.1 $ measurable_sum H₁ (H₂.comp measurable_id.snd)\n\nlemma measurable_of_measurable_nnreal_nnreal [measurable_space β]\n  {f : ℝ≥0∞ × ℝ≥0∞ → β} (h₁ : measurable (λ p : ℝ≥0 × ℝ≥0, f (p.1, p.2)))\n  (h₂ : measurable (λ r : ℝ≥0, f (∞, r))) (h₃ : measurable (λ r : ℝ≥0, f (r, ∞))) :\n  measurable f :=\nmeasurable_of_measurable_nnreal_prod\n  (measurable_swap_iff.1 $ measurable_of_measurable_nnreal_prod (h₁.comp measurable_swap) h₃)\n  (measurable_of_measurable_nnreal h₂)\n\n@[measurability]\nlemma measurable_of_real : measurable ennreal.of_real :=\nennreal.continuous_of_real.measurable\n\n@[measurability]\nlemma measurable_to_real : measurable ennreal.to_real :=\nennreal.measurable_of_measurable_nnreal measurable_coe_nnreal_real\n\n@[measurability]\nlemma measurable_to_nnreal : measurable ennreal.to_nnreal :=\nennreal.measurable_of_measurable_nnreal measurable_id\n\ninstance : has_measurable_mul₂ ℝ≥0∞ :=\nbegin\n  refine ⟨measurable_of_measurable_nnreal_nnreal _ _ _⟩,\n  { simp only [← ennreal.coe_mul, measurable_mul.coe_nnreal_ennreal] },\n  { simp only [ennreal.top_mul, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const },\n  { simp only [ennreal.mul_top, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const }\nend\n\ninstance : has_measurable_sub₂ ℝ≥0∞ :=\n⟨by apply measurable_of_measurable_nnreal_nnreal;\n  simp [← with_top.coe_sub, continuous_sub.measurable.coe_nnreal_ennreal]⟩\n\ninstance : has_measurable_inv ℝ≥0∞ := ⟨ennreal.continuous_inv.measurable⟩\n\nend ennreal\n\n@[measurability]\nlemma measurable.ennreal_to_nnreal {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, (f x).to_nnreal) :=\nennreal.measurable_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_nnreal {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x).to_nnreal) μ :=\nennreal.measurable_to_nnreal.comp_ae_measurable hf\n\nlemma measurable_coe_nnreal_ennreal_iff {f : α → ℝ≥0} :\n  measurable (λ x, (f x : ℝ≥0∞)) ↔ measurable f :=\n⟨λ h, h.ennreal_to_nnreal, λ h, h.coe_nnreal_ennreal⟩\n\n@[measurability]\nlemma measurable.ennreal_to_real {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, ennreal.to_real (f x)) :=\nennreal.measurable_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_real {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, ennreal.to_real (f x)) μ :=\nennreal.measurable_to_real.comp_ae_measurable hf\n\n/-- note: `ℝ≥0∞` can probably be generalized in a future version of this lemma. -/\n@[measurability]\nlemma measurable.ennreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, measurable (f i)) :\n  measurable (λ x, ∑' i, f i x) :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply measurable_supr,\n  exact λ s, s.measurable_sum (λ i _, h i) }\n\n@[measurability]\nlemma measurable.ennreal_tsum' {ι} [encodable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, measurable (f i)) :\n  measurable (∑' i, f i) :=\nbegin\n  convert measurable.ennreal_tsum h,\n  ext1 x,\n  exact tsum_apply (pi.summable.2 (λ _, ennreal.summable)),\nend\n\n@[measurability]\nlemma measurable.nnreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0} (h : ∀ i, measurable (f i)) :\n  measurable (λ x, ∑' i, f i x) :=\nbegin\n  simp_rw [nnreal.tsum_eq_to_nnreal_tsum],\n  exact (measurable.ennreal_tsum (λ i, (h i).coe_nnreal_ennreal)).ennreal_to_nnreal,\nend\n\n@[measurability]\nlemma ae_measurable.ennreal_tsum {ι} [encodable ι] {f : ι → α → ℝ≥0∞} {μ : measure α}\n  (h : ∀ i, ae_measurable (f i) μ) :\n  ae_measurable (λ x, ∑' i, f i x) μ :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply ae_measurable_supr,\n  exact λ s, finset.ae_measurable_sum s (λ i _, h i) }\n\n@[measurability]\nlemma measurable_coe_real_ereal : measurable (coe : ℝ → ereal) :=\ncontinuous_coe_real_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_real_ereal {f : α → ℝ} (hf : measurable f) :\n  measurable (λ x, (f x : ereal)) :=\nmeasurable_coe_real_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_real_ereal {f : α → ℝ} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ereal)) μ :=\nmeasurable_coe_real_ereal.comp_ae_measurable hf\n\n/-- The set of finite `ereal` numbers is `measurable_equiv` to `ℝ`. -/\ndef measurable_equiv.ereal_equiv_real : ({⊥, ⊤} : set ereal).compl ≃ᵐ ℝ :=\nereal.ne_bot_top_homeomorph_real.to_measurable_equiv\n\nlemma ereal.measurable_of_measurable_real {f : ereal → α}\n  (h : measurable (λ p : ℝ, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_finite {⊥, ⊤} (by simp)\n  (measurable_equiv.ereal_equiv_real.symm.measurable_comp_iff.1 h)\n\n@[measurability]\nlemma measurable_ereal_to_real : measurable ereal.to_real :=\nereal.measurable_of_measurable_real (by simpa using measurable_id)\n\n@[measurability]\nlemma measurable.ereal_to_real {f : α → ereal} (hf : measurable f) :\n  measurable (λ x, (f x).to_real) :=\nmeasurable_ereal_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ereal_to_real {f : α → ereal} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x).to_real) μ :=\nmeasurable_ereal_to_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_ennreal_ereal : measurable (coe : ℝ≥0∞ → ereal) :=\ncontinuous_coe_ennreal_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_ereal_ennreal {f : α → ℝ≥0∞} (hf : measurable f) :\n  measurable (λ x, (f x : ereal)) :=\nmeasurable_coe_ennreal_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_ereal_ennreal {f : α → ℝ≥0∞} {μ : measure α} (hf : ae_measurable f μ) :\n  ae_measurable (λ x, (f x : ereal)) μ :=\nmeasurable_coe_ennreal_ereal.comp_ae_measurable hf\n\nsection normed_group\n\nvariables [normed_group α] [opens_measurable_space α] [measurable_space β]\n\n@[measurability]\nlemma measurable_norm : measurable (norm : α → ℝ) :=\ncontinuous_norm.measurable\n\n@[measurability]\nlemma measurable.norm {f : β → α} (hf : measurable f) : measurable (λ a, norm (f a)) :=\nmeasurable_norm.comp hf\n\n@[measurability]\nlemma ae_measurable.norm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, norm (f a)) μ :=\nmeasurable_norm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_nnnorm : measurable (nnnorm : α → ℝ≥0) :=\ncontinuous_nnnorm.measurable\n\n@[measurability]\nlemma measurable.nnnorm {f : β → α} (hf : measurable f) : measurable (λ a, nnnorm (f a)) :=\nmeasurable_nnnorm.comp hf\n\n@[measurability]\nlemma ae_measurable.nnnorm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, nnnorm (f a)) μ :=\nmeasurable_nnnorm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_ennnorm : measurable (λ x : α, (nnnorm x : ℝ≥0∞)) :=\nmeasurable_nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma measurable.ennnorm {f : β → α} (hf : measurable f) :\n  measurable (λ a, (nnnorm (f a) : ℝ≥0∞)) :=\nhf.nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma ae_measurable.ennnorm {f : β → α} {μ : measure β} (hf : ae_measurable f μ) :\n  ae_measurable (λ a, (nnnorm (f a) : ℝ≥0∞)) μ :=\nmeasurable_ennnorm.comp_ae_measurable hf\n\nend normed_group\n\nsection limits\n\nvariables [measurable_space β] [metric_space β] [borel_space β]\n\nopen metric\n\n/-- A limit (over a general filter) of measurable `ℝ≥0` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal' {ι} {f : ι → α → ℝ≥0} {g : α → ℝ≥0} (u : filter ι)\n  [ne_bot u] [is_countably_generated u] (hf : ∀ i, measurable (f i)) (lim : tendsto f u (𝓝 g)) :\n  measurable g :=\nbegin\n  rcases u.exists_seq_tendsto with ⟨x, hx⟩,\n  rw [tendsto_pi_nhds] at lim, rw [← measurable_coe_nnreal_ennreal_iff],\n  have : ∀ y, liminf at_top (λ n, (f (x n) y : ℝ≥0∞)) = (g y : ℝ≥0∞) :=\n    λ y, ((ennreal.continuous_coe.tendsto (g y)).comp $ (lim y).comp hx).liminf_eq,\n  simp only [← this],\n  show measurable (λ y, liminf at_top (λ n, (f (x n) y : ℝ≥0∞))),\n  exact measurable_liminf (λ n, (hf (x n)).coe_nnreal_ennreal),\nend\n\n/-- A sequential limit of measurable `ℝ≥0` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal {f : ℕ → α → ℝ≥0} {g : α → ℝ≥0}\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f at_top (𝓝 g)) : measurable g :=\nmeasurable_of_tendsto_nnreal' at_top hf lim\n\n/-- A limit (over a general filter) of measurable functions valued in a metric space is measurable.\nThe assumption `hs` can be dropped using `filter.is_countably_generated.has_antitone_basis`, but we\ndon't need that case yet. -/\nlemma measurable_of_tendsto_metric' {ι} {f : ι → α → β} {g : α → β}\n  (u : filter ι) [ne_bot u] [is_countably_generated u]\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f u (𝓝 g)) :\n  measurable g :=\nbegin\n  apply measurable_of_is_closed', intros s h1s h2s h3s,\n  have : measurable (λ x, inf_nndist (g x) s),\n  { suffices : tendsto (λ i x, inf_nndist (f i x) s) u (𝓝 (λ x, inf_nndist (g x) s)),\n      from measurable_of_tendsto_nnreal' u (λ i, (hf i).inf_nndist) this,\n    rw [tendsto_pi_nhds] at lim ⊢, intro x,\n    exact ((continuous_inf_nndist_pt s).tendsto (g x)).comp (lim x) },\n  have h4s : g ⁻¹' s = (λ x, inf_nndist (g x) s) ⁻¹' {0},\n  { ext x, simp [h1s, ← h1s.mem_iff_inf_dist_zero h2s, ← nnreal.coe_eq_zero] },\n  rw [h4s], exact this (measurable_set_singleton 0),\nend\n\n/-- A sequential limit of measurable functions valued in a metric space is measurable. -/\nlemma measurable_of_tendsto_metric {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ i, measurable (f i)) (lim : tendsto f at_top (𝓝 g)) :\n  measurable g :=\nmeasurable_of_tendsto_metric' at_top hf lim\n\nlemma ae_measurable_of_tendsto_metric_ae {μ : measure α} {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ n, ae_measurable (f n) μ)\n  (h_ae_tendsto : ∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (𝓝 (g x))) :\n  ae_measurable g μ :=\nbegin\n  let p : α → (ℕ → β) → Prop := λ x f', filter.at_top.tendsto (λ n, f' n) (𝓝 (g x)),\n  let hp : ∀ᵐ x ∂μ, p x (λ n, f n x), from h_ae_tendsto,\n  let ae_seq_lim := λ x, ite (x ∈ ae_seq_set hf p) (g x) (⟨f 0 x⟩ : nonempty β).some,\n  refine ⟨ae_seq_lim, _, (ite_ae_eq_of_measure_compl_zero g (λ x, (⟨f 0 x⟩ : nonempty β).some)\n    (ae_seq_set hf p) (ae_seq.measure_compl_ae_seq_set_eq_zero hf hp)).symm⟩,\n  refine measurable_of_tendsto_metric (@ae_seq.measurable α β _ _ _ f μ hf p) _,\n  refine tendsto_pi_nhds.mpr (λ x, _),\n  simp_rw [ae_seq, ae_seq_lim],\n  split_ifs with hx,\n  { simp_rw ae_seq.mk_eq_fun_of_mem_ae_seq_set hf hx,\n    exact @ae_seq.fun_prop_of_mem_ae_seq_set α β _ _ _ _ _ _ hf x hx, },\n  { exact tendsto_const_nhds, },\nend\n\nlemma ae_measurable_of_unif_approx {μ : measure α} {g : α → β}\n  (hf : ∀ ε > (0 : ℝ), ∃ (f : α → β), ae_measurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :\n  ae_measurable g μ :=\nbegin\n  obtain ⟨u, u_anti, u_pos, u_lim⟩ :\n    ∃ (u : ℕ → ℝ), strict_anti u ∧ (∀ (n : ℕ), 0 < u n) ∧ tendsto u at_top (𝓝 0) :=\n      exists_seq_strict_anti_tendsto (0 : ℝ),\n  choose f Hf using λ (n : ℕ), hf (u n) (u_pos n),\n  have : ∀ᵐ x ∂μ, tendsto (λ n, f n x) at_top (𝓝 (g x)),\n  { have : ∀ᵐ x ∂ μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 (λ n, (Hf n).2),\n    filter_upwards [this],\n    assume x hx,\n    rw tendsto_iff_dist_tendsto_zero,\n    exact squeeze_zero (λ n, dist_nonneg) hx u_lim },\n  exact ae_measurable_of_tendsto_metric_ae (λ n, (Hf n).1) this,\nend\n\nlemma measurable_of_tendsto_metric_ae {μ : measure α} [μ.is_complete] {f : ℕ → α → β} {g : α → β}\n  (hf : ∀ n, measurable (f n))\n  (h_ae_tendsto : ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (g x))) :\n  measurable g :=\nae_measurable_iff_measurable.mp\n  (ae_measurable_of_tendsto_metric_ae (λ i, (hf i).ae_measurable) h_ae_tendsto)\n\nlemma measurable_limit_of_tendsto_metric_ae {μ : measure α} {f : ℕ → α → β}\n  (hf : ∀ n, ae_measurable (f n) μ)\n  (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, filter.at_top.tendsto (λ n, f n x) (𝓝 l)) :\n  ∃ (f_lim : α → β) (hf_lim_meas : measurable f_lim),\n    ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)) :=\nbegin\n  let p : α → (ℕ → β) → Prop := λ x f', ∃ l : β, filter.at_top.tendsto (λ n, f' n) (𝓝 l),\n  have hp_mem : ∀ x, x ∈ ae_seq_set hf p → p x (λ n, f n x),\n    from λ x hx, ae_seq.fun_prop_of_mem_ae_seq_set hf hx,\n  have hμ_compl : μ (ae_seq_set hf p)ᶜ = 0,\n    from ae_seq.measure_compl_ae_seq_set_eq_zero hf h_ae_tendsto,\n  let f_lim : α → β := λ x, dite (x ∈ ae_seq_set hf p) (λ h, (hp_mem x h).some)\n    (λ h, (⟨f 0 x⟩ : nonempty β).some),\n  have hf_lim_conv : ∀ x, x ∈ ae_seq_set hf p → filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)),\n  { intros x hx_conv,\n    simp only [f_lim, hx_conv, dif_pos],\n    exact (hp_mem x hx_conv).some_spec, },\n  have hf_lim : ∀ x, filter.at_top.tendsto (λ n, ae_seq hf p n x) (𝓝 (f_lim x)),\n  { intros x,\n    simp only [f_lim, ae_seq],\n    split_ifs,\n    { rw funext (λ n, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h n),\n      exact (hp_mem x h).some_spec, },\n    { exact tendsto_const_nhds, }, },\n  have h_ae_tendsto_f_lim : ∀ᵐ x ∂μ, filter.at_top.tendsto (λ n, f n x) (𝓝 (f_lim x)),\n  { refine le_antisymm (le_of_eq (measure_mono_null _ hμ_compl)) (zero_le _),\n    exact set.compl_subset_compl.mpr (λ x hx, hf_lim_conv x hx), },\n  have h_f_lim_meas : measurable f_lim,\n    from measurable_of_tendsto_metric (ae_seq.measurable hf p)\n      (tendsto_pi_nhds.mpr (λ x, hf_lim x)),\n  exact ⟨f_lim, h_f_lim_meas, h_ae_tendsto_f_lim⟩,\nend\n\nend limits\n\nnamespace continuous_linear_map\n\nvariables {𝕜 : Type*} [normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E]\nvariables [opens_measurable_space E]\nvariables {F : Type*} [normed_group F] [normed_space 𝕜 F] [measurable_space F] [borel_space F]\n\n@[measurability]\nprotected lemma measurable (L : E →L[𝕜] F) : measurable L :=\nL.continuous.measurable\n\nlemma measurable_comp (L : E →L[𝕜] F) {φ : α → E} (φ_meas : measurable φ) :\n  measurable (λ (a : α), L (φ a)) :=\nL.measurable.comp φ_meas\n\nend continuous_linear_map\n\nnamespace continuous_linear_map\n\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E]\n          {F : Type*} [normed_group F] [normed_space 𝕜 F]\n\ninstance : measurable_space (E →L[𝕜] F) := borel _\n\ninstance : borel_space (E →L[𝕜] F) := ⟨rfl⟩\n\n@[measurability]\nlemma measurable_apply [measurable_space F] [borel_space F] (x : E) :\n  measurable (λ f : E →L[𝕜] F, f x) :=\n(apply 𝕜 F x).continuous.measurable\n\n@[measurability]\nlemma measurable_apply' [measurable_space E] [opens_measurable_space E]\n  [measurable_space F] [borel_space F] :\n  measurable (λ (x : E) (f : E →L[𝕜] F), f x) :=\nmeasurable_pi_lambda _ $ λ f, f.measurable\n\n@[measurability]\nlemma measurable_coe [measurable_space F] [borel_space F] :\n  measurable (λ (f : E →L[𝕜] F) (x : E), f x) :=\nmeasurable_pi_lambda _ measurable_apply\n\nend continuous_linear_map\n\nsection continuous_linear_map_nondiscrete_normed_field\n\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E] [borel_space E]\nvariables {F : Type*} [normed_group F] [normed_space 𝕜 F]\n\n@[measurability]\nlemma measurable.apply_continuous_linear_map  {φ : α → F →L[𝕜] E} (hφ : measurable φ) (v : F) :\n  measurable (λ a, φ a v) :=\n(continuous_linear_map.apply 𝕜 E v).measurable.comp hφ\n\n@[measurability]\nlemma ae_measurable.apply_continuous_linear_map {φ : α → F →L[𝕜] E} {μ : measure α}\n  (hφ : ae_measurable φ μ) (v : F) : ae_measurable (λ a, φ a v) μ :=\n(continuous_linear_map.apply 𝕜 E v).measurable.comp_ae_measurable hφ\n\nend continuous_linear_map_nondiscrete_normed_field\n\nsection normed_space\nvariables {𝕜 : Type*} [nondiscrete_normed_field 𝕜] [complete_space 𝕜] [measurable_space 𝕜]\nvariables [borel_space 𝕜]\nvariables {E : Type*} [normed_group E] [normed_space 𝕜 E] [measurable_space E] [borel_space E]\n\nlemma measurable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) :\n  measurable (λ x, f x • c) ↔ measurable f :=\n(closed_embedding_smul_left hc).measurable_embedding.measurable_comp_iff\n\nlemma ae_measurable_smul_const {f : α → 𝕜} {μ : measure α} {c : E} (hc : c ≠ 0) :\n  ae_measurable (λ x, f x • c) μ ↔ ae_measurable f μ :=\n(closed_embedding_smul_left hc).measurable_embedding.ae_measurable_comp_iff\n\nend normed_space\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/measure_theory/constructions/borel_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381372136564, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3198971151306349}}
{"text": "import verification.semantics.stream_props\n\nnoncomputable theory\nopen_locale classical\n\n-- skip s i : if current index < i, must advance; may advance to first ready index ≥ i.\n-- succ s i : if current index ≤ i, must advance; may advance to first ready index > i.\n\n/-\nif current index < (i, b), must advance; may advance up to first ready index ≥ (i, b) \n-/\n\n/-- Returns the set of `q` that `s` could skip to if the current state is `x` \n  and it is supposed to skip past `(i, b)` -/\ndef skip_set {ι α : Type*} [linear_order ι] (s : Stream ι α) (x : s.σ) (i : ι) (b : bool) : set s.σ :=\n{q | ∃ (n : ℕ), q = (s.next'^[n] x) ∧ (0 < n ↔ s.to_order x ≤ (i, b)) ∧\n      ∀ m, 0 < m → m < n → s.valid (s.next'^[m] x) → s.ready (s.next'^[m] x) → s.index' (s.next'^[m] x) < i}\n\nstructure SkipStream (ι α : Type*) [linear_order ι] extends Stream ι α :=\n(skip : Π x, valid x → ι → bool → σ)\n(hskip : ∀ x hx i b, skip x hx i b ∈ skip_set _ x i b)\n\nvariables {ι : Type} {α : Type*} [linear_order ι]\n\n@[simp] noncomputable def SkipStream.eval_skip [add_zero_class α] (s : SkipStream ι α) : ℕ → s.σ → (ι →₀ α)\n| 0 q := 0\n| (n + 1) q := if h₁ : s.valid q then (SkipStream.eval_skip n (s.skip q h₁ (s.index q h₁) (s.ready q))) + (s.eval₀ _ h₁) else 0 \n\n/-- The number of steps a stream would skip at index `(i, b)` -/\nnoncomputable def SkipStream.nskip (s : SkipStream ι α) (q : s.σ) (h : s.valid q) (i : ι) (b : bool) : ℕ :=\n  (s.hskip q h i b).some\n\nlemma SkipStream.skip_eq (s : SkipStream ι α) (q : s.σ) (h : s.valid q) (i : ι) (b : bool) :\n  s.skip q h i b = (s.next'^[s.nskip q h i b] q) := (s.hskip q h i b).some_spec.1\n\nlemma SkipStream.advance_iff (s : SkipStream ι α) (q : s.σ) (h : s.valid q) (i : ι) (b : bool) :\n  0 < s.nskip q h i b ↔ s.to_order q ≤ (i, b) := (s.hskip q h i b).some_spec.2.1\n\n/-- All skipped values are less than `i` -/\nlemma SkipStream.lt_of_skipped (s : SkipStream ι α) (q : s.σ) (h : s.valid q) (i : ι) (b : bool)\n  (m : ℕ) (h₁m : 0 < m) (h₂m : m < s.nskip q h i b) (h₃ : s.valid (s.next'^[m] q)) (h₄ : s.ready (s.next'^[m] q)) :\n  s.index' (s.next'^[m] q) < i := (s.hskip q h i b).some_spec.2.2 m h₁m h₂m h₃ h₄\n\nlemma SkipStream.not_ready_of_skipped_of_monotonic (s : SkipStream ι α) (hs : s.monotonic) (q : s.σ) (h : s.valid q) (b : bool)\n  (m : ℕ) (h₁m : 0 < m) (h₂m : m < s.nskip q h (s.index q h) b) (h₃ : s.valid (s.next'^[m] q)) :\n  ¬s.ready (s.next'^[m] q) := λ h₄,\nbegin\n  refine (s.lt_of_skipped q h _ b m h₁m h₂m h₃ h₄).not_le _,\n  simpa [Stream.index'_val h] using hs.le_index_iterate q m,\nend\n\n/-- If all skipped states are non-ready, then the eval's are equal -/\ntheorem Stream.eval₀_skip_eq [add_comm_monoid α] (s : Stream ι α) (q : s.σ)\n  (n : ℕ) (hn : ∀ m < n, s.valid (s.next'^[m] q) → ¬s.ready (s.next'^[m] q)) :\n  s.eval_steps n q = 0 :=\nbegin\n  induction n with n ih generalizing q, { simp, },\n  simp only [Stream.eval_steps, dite_eq_right_iff, forall_true_left],\n  intro h,\n  simp only [Stream.eval₀, (show ¬s.ready q, from hn 0 (nat.zero_lt_succ _) h), dif_neg, not_false_iff, add_zero],\n  apply ih,\n  intros m hm, specialize hn (m + 1) (nat.succ_lt_succ hm),\n  simpa [Stream.next'_val h] using hn,\nend\n\ntheorem Stream.eval₀_skip_eq' [add_comm_monoid α] (s : Stream ι α) (q : s.σ) (h : s.valid q)\n  (n : ℕ) (hn : n ≠ 0) (hn : ∀ m, 0 < m → m < n → s.valid (s.next'^[m] q) → ¬s.ready (s.next'^[m] q)) :\n  s.eval_steps n q = s.eval₀ q h :=\nbegin\n  cases n, {  contradiction, },\n  have := s.eval₀_skip_eq (s.next q h) n _,\n  { simp [Stream.eval_steps, h, this], },\n  intros m hm,\n  simpa [Stream.next'_val h] using hn (m + 1) m.zero_lt_succ (nat.succ_lt_succ hm),\nend\n\ntheorem SkipStream.eval_skip_eq [add_comm_monoid α] (s : SkipStream ι α) (hs : s.monotonic) (q : s.σ) (n : ℕ) :\n  ∃ (m : ℕ), n ≤ m ∧ s.eval_skip n q = s.eval_steps m q :=\nbegin\n  induction n with n ih generalizing q, { refine ⟨0, rfl.le, _⟩, simp, },\n  by_cases h : s.valid q, swap,\n  { refine ⟨n + 1, rfl.le, _⟩, simp [h], },\n  rcases ih (s.skip q h (s.index q h) (s.ready q)) with ⟨m, hm₁, hm₂⟩,\n  have : 0 < s.nskip q h (s.index q h) (s.ready q),\n  { rw SkipStream.advance_iff, refine le_of_eq _, ext : 1; simp [Stream.index'_val h], },\n  refine ⟨s.nskip q h (s.index q h) (s.ready q) + m, _, _⟩,\n  { rw nat.succ_le_iff, refine lt_of_le_of_lt hm₁ _, simpa, },\n  rw [s.eval_steps_add, s.eval₀_skip_eq' q h, ← SkipStream.skip_eq],\n  { simp [h, add_comm, hm₂], }, { rwa ← zero_lt_iff, },\n  exact s.not_ready_of_skipped_of_monotonic hs q h _,\nend\n\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/verification/semantics/skip_stream.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3198070174617393}}
{"text": "import Iris.Instances.Data.SetNotation\n\nnamespace Iris.Instances.Data\n\n/-- Result of an operation on `State`. -/\ninductive StateResult (α : Type)\n  | unknown\n  | result (x : α)\n  | conflict\nexport StateResult (unknown)\nexport StateResult (result)\nexport StateResult (conflict)\n\n/-- Implementation of a state datatype as a map from `Nat` to a `StateResult` with a value\nfrom `Val`. -/\nabbrev State (Val : Type) := Nat → StateResult Val\n\ninstance : EmptyCollection (State Val) where\n  emptyCollection := fun _ => unknown\n\ninstance : Subset (State Val) where\n  subset a b := ∀ i x, a i = result x → b i = result x\n\ninstance : Union (State Val) where\n  union a b := fun i =>\n    match a i, b i with\n    | unknown , unknown  => unknown\n    | result x, unknown  => result x\n    | unknown , result y => result y\n    | _       , _        => conflict\n\ninstance : Disjoint (State Val) where\n  disjoint a b := ∀ i, a i = unknown ∨ b i = unknown\n\ntheorem empty_union {s : State Val} : s = ∅ ∪ s := by\n  simp only [Union.union, EmptyCollection.emptyCollection]\n  have : s = fun i => s i := by simp\n  rw [this]\n  simp only\n  apply funext\n  intro i\n  cases s i\n  <;> simp\n\ntheorem union_comm {a b : State Val} : a ∪ b = b ∪ a := by\n  simp only [Union.union]\n  apply funext\n  intro i\n  cases a i\n  <;> cases b i\n  <;> simp\n\ntheorem union_assoc {a b c : State Val} : (a ∪ b) ∪ c = a ∪ (b ∪ c) := by\n  simp only [Union.union]\n  apply funext\n  intro i\n  cases a i\n  <;> cases b i\n  <;> cases c i\n  <;> simp\n\ntheorem empty_disjoint {s : State Val} : ∅ || s := by\n  simp [Disjoint.disjoint, EmptyCollection.emptyCollection]\n\ntheorem disjoint_comm {a b : State Val} : a || b ↔ b || a := by\n  simp only [Disjoint.disjoint]\n  constructor\n  all_goals\n    intro h i\n    cases h i\n    <;> try { apply Or.inl ; assumption }\n    <;> try { apply Or.inr ; assumption }\n\ntheorem disjoint_assoc {a b c : State Val} : a ∪ b || c → a || b → (a || c) ∧ (b || c) := by\n  simp only [Disjoint.disjoint, Union.union]\n  intro h_abc h_ab\n  constructor\n  all_goals\n    intro i\n    cases h_ab i\n    <;> try { apply Or.inl ; assumption }\n    cases h_abc i\n    <;> try { apply Or.inr ; assumption }\n  case left.inr.inl h_b h_ab' =>\n    rw [h_b] at h_ab'\n    cases h_ai : a i\n    <;> rw [h_ai] at h_ab'\n    <;> simp only at h_ab'\n    apply Or.inl\n    simp\n  case right.inl.inl h_a h_ab' =>\n    rw [h_a] at h_ab'\n    cases h_bi : b i\n    <;> rw [h_bi] at h_ab'\n    <;> simp only at h_ab'\n    apply Or.inl\n    simp\n\ntheorem disjoint_union {a b c : State Val} : a || b → a || c → a || b ∪ c := by\n  simp only [Disjoint.disjoint, Union.union]\n  intro h_ab h_ac i\n  cases h_ab i\n  <;> cases h_ac i\n  <;> try { apply Or.inl ; assumption }\n  case inr.inr h_b h_c =>\n    rw [h_b, h_c]\n    simp\n\nend Iris.Instances.Data\n", "meta": {"author": "larsk21", "repo": "iris-lean", "sha": "730e644d0ffaad78aac76e2e5f2cd8af0f1d2310", "save_path": "github-repos/lean/larsk21-iris-lean", "path": "github-repos/lean/larsk21-iris-lean/iris-lean-730e644d0ffaad78aac76e2e5f2cd8af0f1d2310/src/Iris/Instances/Data/State.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.31980701746173923}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.computability.partrec_code\nimport Mathlib.PostPort\n\nuniverses u_1 u_4 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Computability theory and the halting problem\n\nA universal partial recursive function, Rice's theorem, and the halting problem.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\nnamespace nat.partrec\n\n\ntheorem merge' {f : ℕ →. ℕ} {g : ℕ →. ℕ} (hf : partrec f) (hg : partrec g) :\n    ∃ (h : ℕ →. ℕ),\n        partrec h ∧\n          ∀ (a : ℕ),\n            (∀ (x : ℕ), x ∈ h a → x ∈ f a ∨ x ∈ g a) ∧\n              (roption.dom (h a) ↔ roption.dom (f a) ∨ roption.dom (g a)) :=\n  sorry\n\nend nat.partrec\n\n\nnamespace partrec\n\n\ntheorem merge' {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ}\n    {g : α →. σ} (hf : partrec f) (hg : partrec g) :\n    ∃ (k : α →. σ),\n        partrec k ∧\n          ∀ (a : α),\n            (∀ (x : σ), x ∈ k a → x ∈ f a ∨ x ∈ g a) ∧\n              (roption.dom (k a) ↔ roption.dom (f a) ∨ roption.dom (g a)) :=\n  sorry\n\ntheorem merge {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ}\n    {g : α →. σ} (hf : partrec f) (hg : partrec g)\n    (H : ∀ (a : α) (x : σ), x ∈ f a → ∀ (y : σ), y ∈ g a → x = y) :\n    ∃ (k : α →. σ), partrec k ∧ ∀ (a : α) (x : σ), x ∈ k a ↔ x ∈ f a ∨ x ∈ g a :=\n  sorry\n\ntheorem cond {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {c : α → Bool}\n    {f : α →. σ} {g : α →. σ} (hc : computable c) (hf : partrec f) (hg : partrec g) :\n    partrec fun (a : α) => cond (c a) (f a) (g a) :=\n  sorry\n\ntheorem sum_cases {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [primcodable α]\n    [primcodable β] [primcodable γ] [primcodable σ] {f : α → β ⊕ γ} {g : α → β →. σ}\n    {h : α → γ →. σ} (hf : computable f) (hg : partrec₂ g) (hh : partrec₂ h) :\n    partrec fun (a : α) => sum.cases_on (f a) (g a) (h a) :=\n  sorry\n\nend partrec\n\n\n/-- A computable predicate is one whose indicator function is computable. -/\ndef computable_pred {α : Type u_1} [primcodable α] (p : α → Prop) :=\n  Exists (computable fun (a : α) => to_bool (p a))\n\n/-- A recursively enumerable predicate is one which is the domain of a computable partial function.\n -/\ndef re_pred {α : Type u_1} [primcodable α] (p : α → Prop) :=\n  partrec fun (a : α) => roption.assert (p a) fun (_x : p a) => roption.some Unit.unit\n\ntheorem computable_pred.of_eq {α : Type u_1} [primcodable α] {p : α → Prop} {q : α → Prop}\n    (hp : computable_pred p) (H : ∀ (a : α), p a ↔ q a) : computable_pred q :=\n  (funext fun (a : α) => propext (H a)) ▸ hp\n\nnamespace computable_pred\n\n\ntheorem computable_iff {α : Type u_1} [primcodable α] {p : α → Prop} :\n    computable_pred p ↔ ∃ (f : α → Bool), computable f ∧ p = fun (a : α) => ↥(f a) :=\n  sorry\n\nprotected theorem not {α : Type u_1} [primcodable α] {p : α → Prop} (hp : computable_pred p) :\n    computable_pred fun (a : α) => ¬p a :=\n  sorry\n\ntheorem to_re {α : Type u_1} [primcodable α] {p : α → Prop} (hp : computable_pred p) : re_pred p :=\n  sorry\n\ntheorem rice (C : set (ℕ →. ℕ))\n    (h : computable_pred fun (c : nat.partrec.code) => nat.partrec.code.eval c ∈ C) {f : ℕ →. ℕ}\n    {g : ℕ →. ℕ} (hf : nat.partrec f) (hg : nat.partrec g) (fC : f ∈ C) : g ∈ C :=\n  sorry\n\ntheorem rice₂ (C : set nat.partrec.code)\n    (H :\n      ∀ (cf cg : nat.partrec.code),\n        nat.partrec.code.eval cf = nat.partrec.code.eval cg → (cf ∈ C ↔ cg ∈ C)) :\n    (computable_pred fun (c : nat.partrec.code) => c ∈ C) ↔ C = ∅ ∨ C = set.univ :=\n  sorry\n\ntheorem halting_problem (n : ℕ) :\n    ¬computable_pred fun (c : nat.partrec.code) => roption.dom (nat.partrec.code.eval c n) :=\n  fun (ᾰ : computable_pred fun (c : nat.partrec.code) => roption.dom (nat.partrec.code.eval c n)) =>\n    idRhs ((fun (n : ℕ) => roption.none) ∈ set_of fun (f : ℕ →. ℕ) => roption.dom (f n))\n      (rice (set_of fun (f : ℕ →. ℕ) => roption.dom (f n)) ᾰ nat.partrec.zero nat.partrec.none\n        trivial)\n\n-- Post's theorem on the equivalence of r.e., co-r.e. sets and\n\n-- computable sets. The assumption that p is decidable is required\n\n-- unless we assume Markov's principle or LEM.\n\ntheorem computable_iff_re_compl_re {α : Type u_1} [primcodable α] {p : α → Prop}\n    [decidable_pred p] : computable_pred p ↔ re_pred p ∧ re_pred fun (a : α) => ¬p a :=\n  sorry\n\nend computable_pred\n\n\nnamespace nat\n\n\n/-- A simplified basis for `partrec`. -/\ninductive partrec' : {n : ℕ} → (vector ℕ n →. ℕ) → Prop where\n| prim : ∀ {n : ℕ} {f : vector ℕ n → ℕ}, primrec' f → partrec' ↑f\n| comp :\n    ∀ {m n : ℕ} {f : vector ℕ n →. ℕ} (g : fin n → vector ℕ m →. ℕ),\n      partrec' f →\n        (∀ (i : fin n), partrec' (g i)) →\n          partrec' fun (v : vector ℕ m) => (vector.m_of_fn fun (i : fin n) => g i v) >>= f\n| rfind :\n    ∀ {n : ℕ} {f : vector ℕ (n + 1) → ℕ},\n      partrec' ↑f →\n        partrec'\n          fun (v : vector ℕ n) => rfind fun (n_1 : ℕ) => roption.some (to_bool (f (n_1::ᵥv) = 0))\n\nend nat\n\n\nnamespace nat.partrec'\n\n\ntheorem to_part {n : ℕ} {f : vector ℕ n →. ℕ} (pf : partrec' f) : partrec f := sorry\n\ntheorem of_eq {n : ℕ} {f : vector ℕ n →. ℕ} {g : vector ℕ n →. ℕ} (hf : partrec' f)\n    (H : ∀ (i : vector ℕ n), f i = g i) : partrec' g :=\n  funext H ▸ hf\n\ntheorem of_prim {n : ℕ} {f : vector ℕ n → ℕ} (hf : primrec f) : partrec' ↑f :=\n  prim (primrec'.of_prim hf)\n\ntheorem head {n : ℕ} : partrec' ↑vector.head := prim primrec'.head\n\ntheorem tail {n : ℕ} {f : vector ℕ n →. ℕ} (hf : partrec' f) :\n    partrec' fun (v : vector ℕ (Nat.succ n)) => f (vector.tail v) :=\n  sorry\n\nprotected theorem bind {n : ℕ} {f : vector ℕ n →. ℕ} {g : vector ℕ (n + 1) →. ℕ} (hf : partrec' f)\n    (hg : partrec' g) :\n    partrec' fun (v : vector ℕ n) => roption.bind (f v) fun (a : ℕ) => g (a::ᵥv) :=\n  sorry\n\nprotected theorem map {n : ℕ} {f : vector ℕ n →. ℕ} {g : vector ℕ (n + 1) → ℕ} (hf : partrec' f)\n    (hg : partrec' ↑g) :\n    partrec' fun (v : vector ℕ n) => roption.map (fun (a : ℕ) => g (a::ᵥv)) (f v) :=\n  sorry\n\n/-- Analogous to `nat.partrec'` for `ℕ`-valued functions, a predicate for partial recursive\n  vector-valued functions.-/\ndef vec {n : ℕ} {m : ℕ} (f : vector ℕ n → vector ℕ m) :=\n  ∀ (i : fin m), partrec' ↑fun (v : vector ℕ n) => vector.nth (f v) i\n\ntheorem vec.prim {n : ℕ} {m : ℕ} {f : vector ℕ n → vector ℕ m} (hf : primrec'.vec f) : vec f :=\n  fun (i : fin m) => prim (hf i)\n\nprotected theorem nil {n : ℕ} : vec fun (_x : vector ℕ n) => vector.nil :=\n  fun (i : fin 0) => fin.elim0 i\n\nprotected theorem cons {n : ℕ} {m : ℕ} {f : vector ℕ n → ℕ} {g : vector ℕ n → vector ℕ m}\n    (hf : partrec' ↑f) (hg : vec g) : vec fun (v : vector ℕ n) => f v::ᵥg v :=\n  sorry\n\ntheorem idv {n : ℕ} : vec id := vec.prim primrec'.idv\n\ntheorem comp' {n : ℕ} {m : ℕ} {f : vector ℕ m →. ℕ} {g : vector ℕ n → vector ℕ m} (hf : partrec' f)\n    (hg : vec g) : partrec' fun (v : vector ℕ n) => f (g v) :=\n  sorry\n\ntheorem comp₁ {n : ℕ} (f : ℕ →. ℕ) {g : vector ℕ n → ℕ}\n    (hf : partrec' fun (v : vector ℕ 1) => f (vector.head v)) (hg : partrec' ↑g) :\n    partrec' fun (v : vector ℕ n) => f (g v) :=\n  sorry\n\ntheorem rfind_opt {n : ℕ} {f : vector ℕ (n + 1) → ℕ} (hf : partrec' ↑f) :\n    partrec'\n        fun (v : vector ℕ n) =>\n          rfind_opt fun (a : ℕ) => denumerable.of_nat (Option ℕ) (f (a::ᵥv)) :=\n  sorry\n\ntheorem of_part {n : ℕ} {f : vector ℕ n →. ℕ} : partrec f → partrec' f := sorry\n\ntheorem part_iff {n : ℕ} {f : vector ℕ n →. ℕ} : partrec' f ↔ partrec f :=\n  { mp := to_part, mpr := of_part }\n\ntheorem part_iff₁ {f : ℕ →. ℕ} : (partrec' fun (v : vector ℕ 1) => f (vector.head v)) ↔ partrec f :=\n  sorry\n\ntheorem part_iff₂ {f : ℕ → ℕ →. ℕ} :\n    (partrec' fun (v : vector ℕ (bit0 1)) => f (vector.head v) (vector.head (vector.tail v))) ↔\n        partrec₂ f :=\n  sorry\n\ntheorem vec_iff {m : ℕ} {n : ℕ} {f : vector ℕ m → vector ℕ n} : vec f ↔ computable f := sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/computability/halting_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.31980701746173923}}
{"text": "import breen_deligne.main\nimport breen_deligne.eg\nimport condensed.tensor_short_exact\nimport condensed.evaluation_homology\nimport condensed.sheafification_homology\nimport pseudo_normed_group.QprimeFP\nimport for_mathlib.AddCommGroup\nimport for_mathlib.map_to_sheaf_is_iso\nimport condensed.is_iso_iff_extrdisc\nimport Lbar.torsion_free_condensed\nimport condensed.ab5\nimport condensed.ab4\nimport for_mathlib.endomorphisms.ab4\nimport for_mathlib.homology_exact\nimport condensed.Qprime_isoms2\nimport for_mathlib.free_abelian_exact\nimport for_mathlib.unflip\nimport breen_deligne.eval_Pow_functor_nat_trans_compatibility\n\n.\n\nnoncomputable theory\n\nuniverses u\n\nopen category_theory category_theory.limits breen_deligne opposite\nopen_locale big_operators\n\nsection\nopen category_theory.preadditive\n\nattribute [simps map] AddCommGroup.free\n\nlemma oof (A B : AddCommGroup.{u}) : (A →+ B) = (A ⟶ B) := rfl\n\nlemma reorder {M : Type*} [add_comm_monoid M] (a b c d : M) :\n  (a + b) + (c + d) = (a + c) + (b + d) :=\nby { simp only [add_assoc, add_left_comm b c d], }\n\ndef eval_free_π (A : AddCommGroup.{u}) (i : fin 2) : (preadditive.Pow 2).obj A ⟶ (preadditive.Pow 1).obj A :=\nbiproduct.π _ i ≫ biproduct.ι (λ _, A) 0\n\nlemma eval_free_π_eq (A : AddCommGroup.{u}) (k : fin 2) :\n  eval_free_π A k = biproduct.matrix\n    (λ (i : (fin 2)) (j : (fin 1)), basic_universal_map.proj 1 k j i • 𝟙 A) :=\nbegin\n  apply biproduct.hom_ext, rintro j, fin_cases j,\n  rw [biproduct.matrix_π, eval_free_π, category.assoc, biproduct.ι_π, dif_pos rfl, eq_to_hom_refl,\n    category.comp_id],\n  apply biproduct.hom_ext', rintro i, rw [biproduct.ι_desc],\n  suffices : basic_universal_map.proj 1 k 0 i = if i = k then 1 else 0,\n  { rw [this, biproduct.ι_π], dsimp, obtain (rfl|hik) := eq_or_ne i k,\n    { rw [if_pos rfl, if_pos rfl, one_smul], },\n    { rw [if_neg, if_neg hik, zero_smul],\n      intro H, apply hik, exact H } },\n  { dsimp [basic_universal_map.proj, basic_universal_map.proj_aux], dec_trivial! },\nend\n\ndef eval_free_σ (A : AddCommGroup.{u}) : (preadditive.Pow 2).obj A ⟶ (preadditive.Pow 1).obj A :=\neval_free_π A 0 + eval_free_π A 1\n\nlemma eval_free_d10 (A : AddCommGroup.{u}) :\n  (((data.eval_functor (forget _ ⋙ AddCommGroup.free)).obj breen_deligne.eg.data).obj A).d 1 0 =\n  ((forget _ ⋙ AddCommGroup.free).map $ eval_free_π A 0) +\n  ((forget _ ⋙ AddCommGroup.free).map $ eval_free_π A 1) -\n  ((forget _ ⋙ AddCommGroup.free).map $ eval_free_σ A) :=\nbegin\n  dsimp only [eg, eg.BD, data.eval_functor_obj_obj_d], rw [dif_pos rfl],\n  dsimp only [universal_map.eval_Pow], rw [lift_app],\n  dsimp only [whisker_right_app, eg.map, eg.σπ, universal_map.proj, universal_map.sum],\n  simp only [add_monoid_hom.map_sub, free_abelian_group.lift.of,\n    basic_universal_map.eval_Pow_app, functor.comp_map, forget_map_eq_coe, sub_comp, add_comp,\n    preadditive.Pow_obj, forget_obj_eq_coe, fin.sum_univ_two, add_monoid_hom.map_add],\n  refine congr_arg2 _ (congr_arg2 _ _ _) _; congr' 2,\n  { rw eval_free_π_eq, refl, },\n  { rw eval_free_π_eq, refl, },\n  { rw [eval_free_σ, eval_free_π_eq, eval_free_π_eq],\n    apply biproduct.hom_ext, rintro j, fin_cases j, simp only [add_comp, biproduct.matrix_π],\n    erw [biproduct.matrix_π, biproduct.matrix_π],\n    apply biproduct.hom_ext', rintro i, simp only [comp_add, biproduct.ι_desc, ← add_smul],\n    refl }\nend\n\ndef Pow_1_iso (A : AddCommGroup.{u}) : (preadditive.Pow 1).obj A ≅ A :=\n{ hom := biproduct.π (λ _, A) 0,\n  inv := biproduct.ι (λ _, A) 0,\n  hom_inv_id' := begin\n    erw [← biproduct.total, fin.sum_univ_one],\n  end,\n  inv_hom_id' := by simp only [biproduct.ι_π, dif_pos rfl, eq_to_hom_refl] }\n\ndef Pow_2_iso (A : AddCommGroup.{u}) : (preadditive.Pow 2).obj A ≅ AddCommGroup.of (A × A) :=\n{ hom := add_monoid_hom.prod (biproduct.π (λ _, A) 0) (biproduct.π (λ _, A) 1),\n  inv := add_monoid_hom.coprod (biproduct.ι (λ _, A) 0) (biproduct.ι (λ _, A) 1),\n  hom_inv_id' := begin\n    ext x, erw [← biproduct.total, comp_apply],\n    --swap, apply_instance,\n    dsimp only [add_monoid_hom.coprod_apply, add_monoid_hom.prod_apply],\n    simp only [← comp_apply, fin.sum_univ_two], refl,\n  end,\n  inv_hom_id' := begin\n    ext1 x, rw [comp_apply, id_apply],\n    dsimp only [add_monoid_hom.coprod_apply, add_monoid_hom.prod_apply],\n    simp only [add_monoid_hom.map_add, ← comp_apply, biproduct.ι_π, dif_pos rfl, eq_to_hom_refl, id_apply],\n    rw [dif_neg], swap, dec_trivial,\n    rw [dif_neg], swap, dec_trivial,\n    erw [add_zero, zero_add], cases x, refl,\n  end }\n.\n\nlemma eval_free_π_eq_fst (A : AddCommGroup.{u}) :\n  (Pow_2_iso A).inv ≫ eval_free_π A 0 ≫ (Pow_1_iso A).hom =\n  AddCommGroup.of_hom (add_monoid_hom.fst A A) :=\nbegin\n  ext x, simp only [comp_apply],\n  dsimp only [Pow_2_iso, Pow_1_iso, eval_free_π, add_monoid_hom.coprod_apply],\n  simp only [← comp_apply, category.assoc, biproduct.ι_π, dif_pos rfl, eq_to_hom_refl,\n    category.comp_id, add_monoid_hom.map_add, id_apply],\n  erw [dif_neg, add_zero], refl, dec_trivial,\nend\n\nlemma eval_free_π_eq_snd (A : AddCommGroup.{u}) :\n  (Pow_2_iso A).inv ≫ eval_free_π A 1 ≫ (Pow_1_iso A).hom =\n  AddCommGroup.of_hom (add_monoid_hom.snd A A) :=\nbegin\n  ext x, simp only [comp_apply],\n  dsimp only [Pow_2_iso, Pow_1_iso, eval_free_π, add_monoid_hom.coprod_apply],\n  simp only [← comp_apply, category.assoc, biproduct.ι_π, dif_pos rfl, eq_to_hom_refl,\n    category.comp_id, add_monoid_hom.map_add, id_apply],\n  erw [dif_neg, zero_add], refl, dec_trivial,\nend\n\nlemma eval_free_σ_eq_add (A : AddCommGroup.{u}) :\n  (Pow_2_iso A).inv ≫ eval_free_σ A ≫ (Pow_1_iso A).hom =\n  AddCommGroup.of_hom (add_monoid_hom.coprod (add_monoid_hom.id _) (add_monoid_hom.id _)) :=\nby { simp only [eval_free_σ, add_comp, comp_add, eval_free_π_eq_fst, eval_free_π_eq_snd], refl, }\n\nlemma eval_free_homology_zero_exact (A : AddCommGroup.{u}) :\n  exact\n  ((((data.eval_functor (forget _ ⋙ AddCommGroup.free)).obj breen_deligne.eg.data).obj A).d 1 0)\n  ((forget _ ⋙ AddCommGroup.free).map (Pow_1_iso A).hom ≫ AddCommGroup.of_hom (free_abelian_group.lift id)) :=\nbegin\n  let F := forget _ ⋙ AddCommGroup.free,\n  refine exact_of_iso_of_exact' _ _ _ _\n    (F.map_iso (Pow_2_iso A).symm) (F.map_iso (Pow_1_iso A).symm) (iso.refl _) _ _\n    (free_abelian_group.exact_σπ A),\n  swap,\n  { dsimp only [functor.map_iso_hom, iso.symm_hom, iso.refl_hom, F],\n    rw [category.comp_id, ← functor.map_iso_inv, ← functor.map_iso_hom, iso.inv_hom_id_assoc], },\n  rw [← iso.comp_inv_eq, category.assoc, eval_free_d10],\n  simp only [comp_add, add_comp, comp_sub, sub_comp],\n  refine congr_arg2 _ (congr_arg2 _ _ _) _,\n  { simp only [functor.map_iso_hom, functor.map_iso_inv, iso.symm_hom, iso.symm_inv,\n      ← functor.map_comp, eval_free_π_eq_fst], refl },\n  { simp only [functor.map_iso_hom, functor.map_iso_inv, iso.symm_hom, iso.symm_inv,\n      ← functor.map_comp, eval_free_π_eq_snd], refl },\n  { simp only [functor.map_iso_hom, functor.map_iso_inv, iso.symm_hom, iso.symm_inv,\n      ← functor.map_comp, eval_free_σ_eq_add], refl },\nend\n\ninstance eval_free_homology_zero_epi (A : AddCommGroup.{u}) :\n  epi ((forget _ ⋙ AddCommGroup.free).map (Pow_1_iso A).hom ≫ AddCommGroup.of_hom (free_abelian_group.lift id)) :=\nbegin\n  apply_with epi_comp {instances:=ff}, apply_instance,\n  rw [AddCommGroup.epi_iff_surjective], intro a,\n  exact ⟨free_abelian_group.of a, free_abelian_group.lift.of _ _⟩\nend\n\nopen_locale zero_object\n\nsection\n\nvariables {𝓐 : Type*} [category 𝓐] [abelian 𝓐]\nvariables {A B C X : 𝓐} (f : A ⟶ B) (g : B ⟶ C) (γ : B ⟶ X)\n\ndef of_epi_g (hfg : exact f g) (hg : epi g) (hγ : γ = 0) :\n  homology_iso_datum f γ C :=\n{ w := by rw [hγ, comp_zero],\n  K := B,\n  ι := 𝟙 B,\n  f' := f,\n  fac' := category.comp_id _,\n  zero₁' := by rw [hγ, comp_zero],\n  π := g,\n  zero₂' := hfg.w,\n  fork_is_limit := is_limit_aux _ (λ s, s.ι) (λ s, by apply category.comp_id)\n      (λ s m hm, begin rw [← hm], symmetry, apply category.comp_id, end),\n  cofork_is_colimit := @abelian.is_colimit_of_exact_of_epi _ _ _ _ _ _ _ _ hg hfg }\n\n@[simp] lemma of_epi_g.to_homology_iso_predatum_π\n  (hfg : exact f g) (hg : epi g) (hγ : γ = 0) :\n  (of_epi_g f g γ hfg hg hγ).to_homology_iso_predatum.π = g := rfl\n\nend\n\ndef nat_trans_eval_free :\n  ((data.eval_functor (forget _ ⋙ AddCommGroup.free.{u})).obj breen_deligne.eg.data) ⋙\n    homological_complex.eval _ _ 0 ⟶ 𝟭 AddCommGroup :=\n{ app := λ A, (forget _ ⋙ AddCommGroup.free).map (Pow_1_iso A).hom ≫\n    AddCommGroup.of_hom (free_abelian_group.lift id),\n  naturality' := λ A₁ A₂ f, begin\n    simp only [functor.comp_map, homological_complex.eval_map, data.eval_functor_obj_map_f,\n      forget_map_eq_coe, AddCommGroup.free_map, functor.id_map, category.assoc],\n    ext x,\n    dsimp [eg, eg.BD, eg.rank] at x,\n    have h : ∃ y, x = (Pow_1_iso A₁).inv y,\n    { use (Pow_1_iso A₁).hom x,\n      rw [← comp_apply, iso.hom_inv_id, id_apply], },\n    cases h with y hy,\n    subst hy,\n    simp only [comp_apply, free_abelian_group.map_of_apply, AddCommGroup.of_hom_apply,\n      free_abelian_group.lift.of, id.def, iso.inv_hom_id_apply, biproduct.map_eq],\n    let z : fin (eg.data.X 0) := ⟨0, begin\n      dsimp [eg, eg.BD, eg.rank],\n      linarith,\n    end⟩,\n    rw finset.sum_eq_single z, rotate,\n    { intros b hb₁ hb₂,\n      exfalso,\n      apply hb₂,\n      --cases b,\n      --simp only [ulift.up_inj],\n      rw fin.eq_mk_iff_coe_eq,\n      have hb₃ := b.is_lt,\n      dsimp [eg, eg.BD, eg.rank] at hb₃,\n      linarith, },\n    { intro h,\n      exfalso,\n      apply h,\n      simp only [finset.mem_univ], },\n    simp only [← comp_apply, category.assoc],\n    congr' 1,\n    dsimp,\n    change _ ≫ (Pow_1_iso A₁).hom ≫ _ ≫ (Pow_1_iso A₂).inv ≫ _ = _,\n    rw [iso.inv_hom_id, iso.inv_hom_id_assoc, category.comp_id],\n  end, }\n\ndef short_complex_nat_trans_eval_free :\n  ((data.eval_functor (forget _ ⋙ AddCommGroup.free)).obj breen_deligne.eg.data)\n    ⋙ short_complex.functor_homological_complex _ _ 0 ⟶ short_complex.ι_middle :=\nbegin\n  refine short_complex.nat_trans_hom_mk 0 nat_trans_eval_free 0 _\n    (begin apply is_zero.eq_of_tgt, apply short_complex.ι_middle_π₃_is_zero, end),\n  ext1, ext1 A,\n  simp only [zero_comp, nat_trans.app_zero, nat_trans.hcomp_app, nat_trans.comp_app,\n    nat_trans.id_app, short_complex.π₂.map_id, category.comp_id],\n  dsimp only [short_complex.φ₁₂, short_complex.functor_homological_complex, functor.comp_obj,\n    short_complex.mk],\n  simp only [@homological_complex.d_to_eq _ _ _ _ (complex_shape.down ℕ) _ 1 0 (zero_add 1),\n    category.assoc],\n  erw [(eval_free_homology_zero_exact A).w, comp_zero],\nend\n\nlemma short_complex_nat_trans_eval_free_app_τ₂ (A : AddCommGroup) :\n  (short_complex_nat_trans_eval_free.app A).τ₂ = nat_trans_eval_free.app A := rfl\n\ndef eval_free_homology_zero_nat_trans :=\nshort_complex_nat_trans_eval_free ◫ (𝟙 short_complex.homology_functor)\n\nlemma _root_.short_complex.homology_map_is_iso_of_exact_and_epi\n  {A : Type*} [category A] [abelian A]\n  {S₁ S₂ : short_complex A} (φ : S₁ ⟶ S₂) (hg₁ : S₁.1.g = 0) (hf₂ : S₂.1.f = 0) (hg₂ : S₂.1.g = 0)\n  (ex : exact S₁.1.f φ.τ₂) (epi_τ₂ : epi φ.τ₂) :\n  is_iso (short_complex.homology_functor.map φ) :=\nbegin\n  let h₁ := homology_iso_datum.of_g_is_zero S₁.1.f S₁.1.g hg₁,\n  let h₂ := homology_iso_datum.of_both_zeros S₂.1.f S₂.1.g hf₂ hg₂,\n  let ψ := cokernel.desc _ φ.τ₂ ex.w,\n  let μ : homology_map_datum φ h₁ h₂ ψ :=\n  { κ := φ.τ₂,\n    fac₁' := by { erw [φ.comm₁₂], simp only [hf₂], refl, },\n    fac₂' := by { erw [category.id_comp, category.comp_id], },\n    fac₃' := by { erw [category.comp_id], apply cokernel.π_desc, }, },\n  rw μ.homology_map_eq,\n  suffices : is_iso ψ,\n  { haveI := this, apply_instance, },\n  exact abelian.category_theory.limits.cokernel.desc.category_theory.is_iso _ _ ex,\nend\n\ninstance : is_iso eval_free_homology_zero_nat_trans.{u} :=\nbegin\n  suffices : ∀ A, is_iso ((short_complex_nat_trans_eval_free ◫\n    (𝟙 short_complex.homology_functor)).app A),\n  { apply_with nat_iso.is_iso_of_is_iso_app { instances := ff }, exact this, },\n  intro A,\n  simp only [nat_trans.hcomp_id_app],\n  refine short_complex.homology_map_is_iso_of_exact_and_epi _ _ rfl rfl _ _,\n  { apply homological_complex.shape,\n    simp only [chain_complex.next_nat_zero, complex_shape.down_rel,\n      nat.one_ne_zero, not_false_iff], },\n  { refine exact_of_iso_of_exact' _ _ _ _ _ _ _ _ _ (eval_free_homology_zero_exact A),\n    { symmetry,\n      exact (homological_complex.X_prev_iso _ (zero_add 1)), },\n    { refl, },\n    { apply eq_to_iso, cases A, refl, },\n    { dsimp only [short_complex.functor_homological_complex, functor.comp_obj,\n        short_complex.mk],\n      rw homological_complex.d_to_eq, swap 3, exact 1, swap, dsimp, refl,\n      simp only [iso.symm_hom, iso.refl_hom, category.comp_id],\n      apply iso.inv_hom_id_assoc, },\n    { apply category.id_comp, }, },\n  { rw short_complex_nat_trans_eval_free_app_τ₂,\n    dsimp [nat_trans_eval_free],\n    convert eval_free_homology_zero_epi.{u} A,\n    cases A,\n    refl, },\nend\n\ndef eval_free_homology_zero :\n  ((data.eval_functor (forget _ ⋙ AddCommGroup.free)).obj breen_deligne.eg.data) ⋙\n    homology_functor _ _ 0 ≅ 𝟭 _ :=\n  iso_whisker_left _ (short_complex.homology_functor_iso _ _ _) ≪≫\n    (functor.associator _ _ _).symm ≪≫ as_iso eval_free_homology_zero_nat_trans ≪≫\n    short_complex.ι_middle_homology_nat_iso.symm\n\nend\n\nopen bounded_homotopy_category\n\nnamespace Condensed\n\ndef HQ'Z (n : ℤ) : Ab :=\n((eg.eval $ category_theory.forget AddCommGroup ⋙ AddCommGroup.free).obj\n  (AddCommGroup.free.obj punit)).val.as.homology n\n\nvariables (BD : package)\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : abelian (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nendomorphisms.category_theory.abelian\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_finite_biproducts (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nabelian.has_finite_biproducts\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : enough_projectives (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nendomorphisms.category_theory.enough_projectives\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_coproducts_of_shape (ulift.{u+1} ℕ) (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nendomorphisms.has_colimits_of_shape\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_products_of_shape (ulift.{u+1} ℕ) (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nendomorphisms.has_limits_of_shape\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_coproducts (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nλ (J : Type (u+1)), endomorphisms.has_colimits_of_shape\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : AB4 (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nendomorphisms.category_theory.AB4 _\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_finite_limits (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nabelian.has_finite_limits\n\n-- `by apply_instance` takes for ever, so we provide this shortcut\ninstance : has_finite_colimits (endomorphisms $ Condensed.{u} Ab.{u+1}) :=\nabelian.has_finite_colimits\n.\n\n-- move this\nattribute [reassoc] homology_bd_eval_natural\n\ndef exists_tensor_iso (A : endomorphisms (Condensed.{u} Ab.{u+1}))\n  [∀ S : ExtrDisc.{u}, no_zero_smul_divisors ℤ (A.X.val.obj (op S.val))]\n  (n : ℕ) :\n  ((package.endo_T tensor_functor).obj A).obj (HQ'Z (-n)) ≅\n      ((eg.eval freeCond'.map_endomorphisms).obj A).val.as.homology (-n) :=\nbegin\n  refine endomorphisms.mk_iso _ _,\n  { refine _ ≪≫ ((package.hH_endo₁ eg freeCond' n).app A).symm,\n    refine (homology_bd_eval eg A.X (-n)).symm ≪≫ _,\n    exact (package.eval'_homology eg freeCond' n).app A.X, },\n  { dsimp only [iso.trans_hom, iso.symm_hom, package.endo_T_obj_obj_e, tensor_functor, HQ'Z],\n    simp only [category.assoc, ← homology_bd_eval_natural_assoc],\n    refine congr_arg2 _ rfl _,\n    dsimp only [iso.app_hom, iso.app_inv],\n    rw [← functor.comp_map, nat_trans.naturality_assoc],\n    refine congr_arg2 _ rfl _,\n    dsimp only [← iso.app_inv],\n    rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp],\n    exact (eg.hH_endo₁_natural freeCond' A n).symm, }\nend\n.\n\n-- move this\nlemma is_tensor_unit_of_iso (A B : Ab) (e : A ≅ B) (ha : AddCommGroup.is_tensor_unit A) :\n  AddCommGroup.is_tensor_unit B :=\nbegin\n  obtain ⟨a, ha⟩ := ha,\n  refine ⟨e.hom a, _⟩,\n  intro C, specialize ha C,\n  let φ := iso.AddCommGroup_iso_to_add_equiv ((preadditive_yoneda.obj C).map_iso e.op),\n  exact ha.comp φ.bijective,\nend\n\nlemma bd_lemma (A : Condensed.{u} Ab.{u+1}) (B : Condensed.{u} Ab.{u+1})\n  [∀ S : ExtrDisc.{u}, no_zero_smul_divisors ℤ (A.val.obj (op S.val))]\n  (f : A ⟶ A) (g : B ⟶ B) :\n  (∀ i, is_iso $ ((Ext' i).map f.op).app B - ((Ext' i).obj (op A)).map g) ↔\n  (∀ i, is_iso $\n    ((Ext i).map ((breen_deligne.eg.eval freeCond').map f).op).app ((single _ 0).obj B) -\n    ((Ext i).obj (op $ (breen_deligne.eg.eval freeCond').obj A)).map ((single _ 0).map g)) :=\nbegin\n  apply eg.main_lemma_general' _ A B f g tensor_functor tensor_punit (λ n, HQ'Z (-n)),\n  { apply tensor_tunit,\n    apply is_tensor_unit_of_iso\n      (AddCommGroup.free.obj punit) (HQ'Z 0),\n    { let e := (eval_free_homology_zero.app (AddCommGroup.free.obj punit)).symm,\n      refine e ≪≫ _, clear e,\n      let e := (package.eval'_homology eg (forget AddCommGroup ⋙ AddCommGroup.free) 0).symm,\n      exact e.app (AddCommGroup.free.obj punit), },\n    { refine ⟨free_abelian_group.of punit.star, _⟩,\n      intro B, split,\n      { intros f g h, ext ⟨⟩, exact h },\n      { intros b, refine ⟨free_abelian_group.lift (λ _, b), _⟩,\n        apply free_abelian_group.lift.of } } },\n  { apply exists_tensor_iso }\nend\n\nend Condensed\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/bd_lemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.31980701040450876}}
{"text": "import algebra.category.Module.abelian\nimport algebra.category.Module.images\nimport algebra.homology.homology\nimport algebra.homology.Module\nimport algebra.homology.homotopy\nimport algebra.homology.quasi_iso\nimport category_theory.preadditive.left_exact\nimport category_theory.abelian.functor_category\nimport category_theory.limits.shapes.comm_sq\nimport data.list.tfae\n\nimport for_mathlib.homological_complex_abelian\nimport for_mathlib.les_homology\nimport for_mathlib.preserves_exact\nimport for_mathlib.snake_lemma_naturality2\n\nimport .category_theory .linear_algebra .arrow_category\n\nopen category_theory category_theory.limits homological_complex\n\nlocal attribute [instance] category_theory.limits.has_zero_object.has_zero\nlocal attribute [instance]\n  category_theory.concrete_category.has_coe_to_sort\n  category_theory.concrete_category.has_coe_to_fun\n\nlemma cokernel_map_iso_trans_iso_colimit_cocone (V : Type*) [category V] [has_zero_morphisms V]\n  [has_cokernels V]\n  {A B A' B' : V} (f : A ⟶ B) (f' : A' ⟶ B')\n  (p : A ≅ A') (q : B ≅ B') (w : f ≫ q.hom = p.hom ≫ f')\n  (c : cokernel_cofork f') (hc : is_colimit c)\n  : cokernel.map_iso f f' p q w ≪≫ colimit.iso_colimit_cocone ⟨c, hc⟩\n  = let α := @parallel_pair.ext V _ (parallel_pair f 0) (parallel_pair f' 0) p q w (by simp) in\n    colimit.iso_colimit_cocone ⟨(category_theory.limits.cocones.precompose α.hom).obj c,\n                                (is_colimit.precompose_hom_equiv α c).symm hc⟩ := by { ext, simp }\n\nnoncomputable\ndef coker_functor (V : Type*) [category V] [has_zero_morphisms V] [has_cokernels V]\n  : arrow V ⥤ V := {\n    obj := λ f, cokernel f.hom,\n    map := λ f g φ, cokernel.map _ _ _ _ φ.w.symm,\n    map_id' := by { intro, ext, simp, apply category.id_comp },\n    map_comp' := by { intros, ext, simp }\n  }\n\nlemma coker_map_spec {V : Type*} [category V] [has_zero_morphisms V] [has_cokernels V]\n  {A B X Y : V}\n  (i : A ⟶ X) (j : B ⟶ Y)\n  (f' : A ⟶ B) (f : X ⟶ Y)\n  (w1 : i ≫ f = f' ≫ j)\n  : cokernel.π i ≫ cokernel.map i j f' f w1 = f ≫ cokernel.π j :=\n  by { delta cokernel.π coequalizer.π cokernel.map, simp }\n\nnoncomputable\ndef coker_functor_proj (V : Type*) [category V] [has_zero_morphisms V] [has_cokernels V]\n  : arrow.right_func ⟶ coker_functor V := {\n    app := λ f, cokernel.π f.hom,\n    naturality' := λ f g ϕ, by { dsimp [coker_functor], simp }\n  }\n\n-- Why can't lean synthesize the arrow_preadditive instance?\ninstance coker_additive {V : Type*} [category V] [preadditive V] [has_cokernels V]\n  : @functor.additive _ _ _ _ arrow_preadditive _ (coker_functor V) :=\n  ⟨by { rintros ⟨f⟩ ⟨g⟩ ⟨αl, αr⟩ ⟨βl, βr⟩, dsimp [coker_functor], ext, simp }⟩.\n\nuniverses u'' v' u' v u\n\n-- noncomputable\n-- instance coker_preserves_coprod {V : Type u} [category.{(max u'' v)} V] [has_zero_morphisms V]\n--   [has_colimits V] {J : Type u''} (f : J → arrow V)\n--   : limits.preserves_colimit (discrete.functor f) (coker_functor V) :=\n-- begin\n--   apply preserves_colim_into_arrow_category, intros,\n--   refine ⟨_, _, _⟩,\n--   { intro s, \n--      -- we're pulling back by the natural transformation cod -> coker_functor.proj, I think?\n--     let s' := (cocones.precompose (whisker_left (discrete.functor f) (coker_functor_proj V))).obj s,\n--     refine cokernel.desc _ (hc2.desc s') _, \n--     dsimp [category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--     let s'' : cocone (discrete.functor f ⋙ arrow.left_func) \n--             := { X := s.X, ι := { app := λ j, 0, naturality' := by { intros, simp, } } },\n--     transitivity hc1.desc s'',\n--     { refine hc1.uniq s'' _ _,\n--       rintro ⟨j⟩, dsimp,\n--       rw [← category.assoc],\n--       change (c1.ι.app ⟨j⟩ ≫ hc1.desc ((cocones.precompose (whisker_left (discrete.functor f) arrow.left_to_right)).obj c2)) ≫ hc2.desc s' = 0,\n--       rw hc1.fac,\n--       dsimp, rw category.assoc,\n--       rw hc2.fac,\n--       dsimp [s', coker_functor_proj],\n--       rw ← category.assoc, simp },\n--     { refine eq.symm (hc1.uniq s'' _ _),\n--       intro, exact comp_zero } },\n--   { intros s j,\n--     dsimp [coker_functor, category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--     ext, simp, \n--     exact hc2.fac _ _ },\n--   { intros s m h,\n--     dsimp [coker_functor, category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--     ext,\n--     refine eq.trans _ (eq.symm (category_theory.limits.cokernel.π_desc _ _ _)),\n--     let s' := (cocones.precompose (whisker_left (discrete.functor f) (coker_functor_proj V))).obj s,\n--     refine hc2.uniq s' _ _,\n--     intro j, specialize h j,\n--     simp at ⊢ h,\n--     dsimp [coker_functor,\n--            category_theory.iso_to_equiv, arrow_category_iso_functor_category] at h ⊢,\n--     rw ← h,\n--     symmetry, rw ← category.assoc, delta cokernel.map,\n--     refine eq.trans (congr_arg2 _ (limits.cokernel.π_desc _ _ _) (refl m)) _,\n--     exact category.assoc _ _ _ }\n-- end.\n\n-- noncomputable\n-- def coker_proj_functor (V : Type*) [category V] [has_zero_morphisms V] [has_cokernels V]\n--   : arrow V ⥤ arrow V :=\n--   mk_arrow_diagram (coker_functor_proj V)\n\n-- noncomputable\n-- instance coker_proj_preserves_coprod {V : Type u} [category.{(max u'' v)} V] [has_zero_morphisms V]\n--   [has_colimits V] {J : Type u''} (f : J → arrow V)\n--   : limits.preserves_colimit (discrete.functor f) (coker_proj_functor V) :=\n-- begin\n--   apply category_theory.limits.preserves_colimits_of_equiv_codomain (coker_proj_functor V)\n--           (category_theory.iso_to_equiv (arrow_category_iso_functor_category V)),\n--   constructor, intros c hc,\n--   apply evaluation_jointly_reflects_colimits,\n--   intro k, \n--   cases k,\n--   { revert hc c,\n--     suffices : limits.preserves_colimit (discrete.functor f) arrow.right_func,\n--     { exact this.preserves },\n--     apply category_theory.limits.preserves_colimits_of_equiv_domain arrow.right_func\n--             (category_theory.iso_to_equiv (arrow_category_iso_functor_category V)),\n--     apply_with preserves_colimits_of_shape.preserves_colimit {instances:=ff}, \n--     apply_with preserves_colimits_of_size.preserves_colimits_of_shape {instances:=ff}, \n--     apply_with preserves_colimits_of_size_shrink {instances:=ff},\n--     change preserves_colimits ((evaluation _ V).obj walking_arrow.target),\n--     apply_instance },\n--   { change is_colimit ((coker_functor V).map_cocone c),\n--     revert hc c,\n--     suffices : limits.preserves_colimit (discrete.functor f) (coker_functor V),\n--     { exact this.preserves },\n--     apply_instance }\n-- end\n\n-- noncomputable\n-- def ker_functor (V : Type*) [category V] [has_zero_morphisms V] [has_kernels V]\n--   : arrow V ⥤ V := {\n--     obj := λ f, kernel f.hom,\n--     map := λ f g φ, kernel.map _ _ _ _ φ.w.symm,\n--     map_id' := by { intro, ext, simp, apply category.comp_id },\n--     map_comp' := by { intros, ext, simp }\n--   }.\n\n-- lemma ker_map_spec {V : Type*} [category V] [has_zero_morphisms V] [has_kernels V]\n--   {A B X Y : V}\n--   (i : A ⟶ X) (j : B ⟶ Y)\n--   (f' : A ⟶ B) (f : X ⟶ Y)\n--   (w1 : i ≫ f = f' ≫ j)\n--   : kernel.map i j f' f w1 ≫ kernel.ι j = kernel.ι i ≫ f' :=\n--   by { delta kernel.ι equalizer.ι kernel.map, simp }\n\n-- noncomputable\n-- def ker_functor_incl (V : Type*) [category V] [has_zero_morphisms V] [has_kernels V]\n--   : ker_functor V ⟶ arrow.left_func := {\n--     app := λ f, kernel.ι f.hom,\n--     naturality' := λ f g ϕ, by { dsimp [ker_functor], simp }\n--   }\n\n-- -- Why can't lean synthesize the arrow_preadditive instance?\n-- instance ker_additive {V : Type*} [category V] [preadditive V] [has_kernels V]\n--   : @functor.additive _ _ _ _ arrow_preadditive _ (ker_functor V) :=\n--   ⟨by { rintros ⟨f⟩ ⟨g⟩ ⟨αl, αr⟩ ⟨βl, βr⟩, dsimp [ker_functor], ext, simp }⟩.\n\n-- noncomputable\n-- instance ker_preserves_coprod {V : Type u} [category.{(max u'' v)} V] [has_zero_morphisms V]\n--   [has_colimits V] [has_kernels V] {J : Type u''} (f : J → arrow V)\n--   : limits.preserves_colimit (discrete.functor f) (ker_functor V) :=\n-- begin\n--   apply preserves_colim_into_arrow_category, intros,\n--   refine ⟨_, _, _⟩,\n--   { intro s,\n--      }\n--   -- { intro s, \n--   --    -- we're pulling back by the natural transformation cod -> coker_functor.proj, I think?\n--   --   let s' := (cocones.precompose (whisker_left (discrete.functor f) (coker_functor_proj V))).obj s,\n--   --   refine cokernel.desc _ (hc2.desc s') _, \n--   --   dsimp [category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--   --   let s'' : cocone (discrete.functor f ⋙ arrow.left_func) \n--   --           := { X := s.X, ι := { app := λ j, 0, naturality' := by { intros, simp, } } },\n--   --   transitivity hc1.desc s'',\n--   --   { refine hc1.uniq s'' _ _,\n--   --     rintro ⟨j⟩, dsimp,\n--   --     rw [← category.assoc],\n--   --     change (c1.ι.app ⟨j⟩ ≫ hc1.desc ((cocones.precompose (whisker_left (discrete.functor f) arrow.left_to_right)).obj c2)) ≫ hc2.desc s' = 0,\n--   --     rw hc1.fac,\n--   --     dsimp, rw category.assoc,\n--   --     rw hc2.fac,\n--   --     dsimp [s', coker_functor_proj],\n--   --     rw ← category.assoc, simp },\n--   --   { refine eq.symm (hc1.uniq s'' _ _),\n--   --     intro, exact comp_zero } },\n--   -- { intros s j,\n--   --   dsimp [coker_functor, category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--   --   ext, simp, \n--   --   exact hc2.fac _ _ },\n--   -- { intros s m h,\n--   --   dsimp [coker_functor, category_theory.iso_to_equiv, arrow_category_iso_functor_category],\n--   --   ext,\n--   --   refine eq.trans _ (eq.symm (category_theory.limits.cokernel.π_desc _ _ _)),\n--   --   let s' := (cocones.precompose (whisker_left (discrete.functor f) (coker_functor_proj V))).obj s,\n--   --   refine hc2.uniq s' _ _,\n--   --   intro j, specialize h j,\n--   --   simp at ⊢ h,\n--   --   dsimp [coker_functor,\n--   --          category_theory.iso_to_equiv, arrow_category_iso_functor_category] at h ⊢,\n--   --   rw ← h,\n--   --   symmetry, rw ← category.assoc, delta cokernel.map,\n--   --   refine eq.trans (congr_arg2 _ (limits.cokernel.π_desc _ _ _) (refl m)) _,\n--   --   exact category.assoc _ _ _ }\n-- end.\n\n-- -- noncomputable\n-- -- def im_functor (V : Type*) [category V] [has_images V] [has_image_maps V]\n-- --   : arrow V ⥤ V := {\n-- --     obj := λ f, image f.hom,\n-- --     map := λ f g φ, image.map φ,\n-- --     map_id' := λ _, limits.image.map_id _,\n-- --     map_comp' := λ _ _ _ _ _, limits.image.map_comp _ _,\n-- --   }\n\n-- -- noncomputable\n-- -- def im_functor_incl (V : Type*) [category V] [has_images V] [has_image_maps V]\n-- --   : im_functor V ⟶ arrow.right_func := {\n-- --     app := λ _, image.ι _,\n-- --     naturality' := λ _ _ _, limits.image.map_ι _\n-- --   }\n\n-- -- noncomputable\n-- -- def im_functor_proj (V : Type*) [category V] [has_images V] [has_image_maps V]\n-- --   : arrow.left_func ⟶ im_functor V := {\n-- --     app := λ _, limits.factor_thru_image _,\n-- --     naturality' := λ _ _ _, (limits.image_map.factor_map _ _).symm\n-- --   }\n\n-- -- instance im_additive {V : Type*} [category V] [preadditive V] [has_images V] [has_image_maps V]\n-- --   : @functor.additive _ _ _ _ arrow_preadditive _ (im_functor V) :=\n-- --   ⟨by { rintros ⟨f⟩ ⟨g⟩ ⟨αl, αr⟩ ⟨βl, βr⟩, dsimp [im_functor], refine (cancel_mono (image.ι _)).mp _, simp }⟩.\n\n-- -- I think we need abelian so that image f = ker (coker f)\n-- -- noncomputable\n-- -- instance im_preserves_coprod {V : Type u} [category.{v} V] [abelian V]\n-- --   [has_colimits V] {J : Type} (f : J → arrow V)\n-- --   : limits.preserves_colimit (discrete.functor f) (im_functor V) :=\n-- -- begin\n-- --   apply preserves_colim_into_arrow_category, intros,\n-- --   refine ⟨_, _, _⟩,\n-- --   { intro s,\n-- --     dsimp [mk_arrow_cocone, im_functor, cocones.precompose, whisker_left],\n-- --      }\n-- -- end\n\n-- noncomputable\n-- def homological_complex.boundaries_to_cycles_functor {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_cokernels V] (i : ι)\n--   : homological_complex V c ⥤ arrow V := {\n--     obj := λ C, C.boundaries_to_cycles i,\n--     map := λ C D f, arrow.hom_mk (boundaries_to_cycles_naturality f i),\n--     map_comp' := λ X Y Z f g, comma_morphism.ext _ _ ((boundaries_functor V c i).map_comp _ _)\n--                                                      ((cycles_functor V c i).map_comp _ _),\n--     map_id' := λ X, comma_morphism.ext _ _ ((boundaries_functor V c i).map_id _)\n--                                            ((cycles_functor V c i).map_id _),\n--   }\n\n-- lemma homology_functor_eq_coker_of_boundaries_to_cycles\n--   {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_cokernels V] (i : ι)\n--   : homology_functor V c i\n--   = homological_complex.boundaries_to_cycles_functor V c i ⋙ coker_functor V :=\n--   rfl.\n\n-- /-\n-- Roadmap:\n--   - Define image_functor Arr(C) ⥤ C\n--   - Argue `boundaries_functor i` is naturally isomorphic to the composition Ch(V) ⥤ Arr(V) ⥤ V\n--     where the first map picks out `d_to i` and the second is image_functor\n--   - Argue that the picking-out functor and the image functor preserve coproducts\n--   - This also helps with cycles_functor (we just need to know kernels preserve coproducts)\n-- -/\n-- instance boundaries_functor_preserves_coprod {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c' : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_colimits V] (i : ι)\n--   {J : Type} (f : J → homological_complex V c')\n--   : limits.preserves_colimit (discrete.functor f)\n--                              (boundaries_functor V c' i) :=\n-- begin\n--   destruct c'.prev i,\n--   { intro hi, constructor, intros c hc,\n--     have : ∀ C' : homological_complex V c', is_zero (C'.boundaries i : V),\n--     { intro,\n--       rw homological_complex.boundaries_eq_bot _ hi,\n--       refine is_zero_of_iso_of_zero _ subobject.bot_coe_iso_zero.symm,\n--       apply limits.is_zero_zero, },\n--     refine ⟨_, _, _⟩,\n--     { intro s, exact 0, },\n--     { intros s j, \n--       refine eq.trans comp_zero _,\n--       apply limits.is_zero.eq_of_src,\n--       apply this },\n--     { intros s m hm,\n--       apply limits.is_zero.eq_of_src,\n--       apply this } },\n--   { rintros ⟨i', hi⟩ _,\n--     refine limits.preserves_colimit_of_preserves_colimit_cocone (colimit_complex_cocone_is_colimit _) _,\n--     refine ⟨_, _, _⟩,\n--     { intro s,\n--       refine (homological_complex.boundaries_iso_image _ hi).hom ≫ _,\n\n--       -- rw homological_complex.boundaries_eq_image_subobject _ hi, swap, assumption,\n--        } }\n-- end\n\n-- instance cycles_functor_preserves_coprod {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c' : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_colimits V] (i : ι)\n--   {J : Type} (f : J → homological_complex V c')\n--   : limits.preserves_colimit (discrete.functor f)\n--                              (cycles_functor V c' i) :=\n-- begin\n--   admit\n-- end\n\n-- noncomputable\n-- instance boundaries_to_cycles_preserves_coprod {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c' : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_colimits V] (i : ι)\n--   {J : Type} (f : J → homological_complex V c')\n--   : limits.preserves_colimit (discrete.functor f)\n--                              (homological_complex.boundaries_to_cycles_functor V c' i) :=\n-- begin\n--   refine category_theory.limits.preserves_colimits_of_equiv_codomain\n--            (homological_complex.boundaries_to_cycles_functor V c' i)\n--            (category_theory.iso_to_equiv (arrow_category_iso_functor_category V)) _,\n--   constructor, intros c hc,\n--   apply limits.evaluation_jointly_reflects_colimits,\n--   intro k, cases k,\n--   { change is_colimit ((boundaries_functor V c' i).map_cocone c),\n--     apply limits.is_colimit_of_preserves, assumption },\n--   { change is_colimit ((cycles_functor V c' i).map_cocone c),\n--     apply limits.is_colimit_of_preserves, assumption },\n-- end\n\n-- noncomputable\n-- def homology_functor_iso_coker_of_boundaries_to_cycles\n--   {ι : Type*} (V : Type*) [category V]\n--   [has_zero_morphisms V] [has_zero_object V] (c : complex_shape ι)\n--   [has_equalizers V] [has_images V] [has_image_maps V] [has_cokernels V] (i : ι)\n--   : homology_functor V c i\n--   ≅ homological_complex.boundaries_to_cycles_functor V c i ⋙ coker_functor V :=\n--   iso.refl _.\n\n\nsection snake_diagram \n\nopen category_theory.snake_diagram\n\nlocal notation x `⟶[`D`]` y := D.map (hom x y)\n\nlemma to_zero_exact_of_epi {V : Type*} [category V] [abelian V] {X Y : V} (Z : V) (f : X ⟶ Y)\n  : epi f → exact f (0 : Y ⟶ Z) := ((abelian.tfae_epi Z f).out 0 2).mp\n\ndef cokernel_sequence\n  {V : Type*} [category V] [abelian V]\n  (D : snake_input V)\n  (h1 : epi ((2, 1) ⟶[D] (2, 2)))\n  : exact_seq V [((3, 0) ⟶[D] (3, 1)), ((3, 1) ⟶[D] (3, 2)), (0 : D.obj (3, 2) ⟶ 0)] :=\n  have h2 : exact ((3, 0) ⟶[D] (3, 1)) ((3, 1) ⟶[D] (3, 2)) := D.2.row_exact _,\n  have h3 : epi ((3, 1) ⟶[D] (3, 2)),\n  begin\n    letI := h1,\n    refine abelian.pseudoelement.epi_of_pseudo_surjective _ (λ y, _),\n    refine is_snake_input.exists_of_exact (is_snake_input.long_row₃_exact D.is_snake_input) y _,\n    simp [is_snake_input.bottom_right_to_coker_row₂,\n          limits.cokernel.π_of_epi ((2, 1) ⟶[D] (2, 2))]\n  end,\n  exact_seq.cons _ _ h2 _\n    (exact_seq.cons _ _ (to_zero_exact_of_epi _ _ h3) _\n      (exact_seq.single _))\n\nend snake_diagram\n\nlemma coker_right_exact {V : Type*} [category V] [abelian V]\n  {A B C X Y Z : V}\n  (f : A ⟶ B) (g : B ⟶ C) (f' : X ⟶ Y) (g' : Y ⟶ Z)\n  (α : A ⟶ X) (β : B ⟶ Y) (γ : C ⟶ Z)\n  (w1 : f ≫ β = α ≫ f') (w2 : g ≫ γ = β ≫ g')\n  (H : short_exact f g) (H' : short_exact f' g')\n  : exact_seq V [((coker_functor V).map (arrow.hom_mk w1 : arrow.mk α ⟶ arrow.mk β)),\n                 ((coker_functor V).map (arrow.hom_mk w2 : arrow.mk β ⟶ arrow.mk γ)),\n                 (0 : cokernel γ ⟶ 0)] :=\nbegin\n  letI := H.mono, letI := H.epi, letI := H'.mono, letI := H'.epi, \n  convert cokernel_sequence (snake.mk_of_sequence_hom A B C X Y Z f g α β γ f' g'\n                                                      w1.symm w2.symm H.exact H'.exact).snake_input _;\n  dsimp [coker_functor, snake.snake_input, snake.snake_diagram, snake_diagram.mk_functor,\n         snake_diagram.mk_functor.map'];\n  simp only [category_theory.functor.map_id, category.id_comp, category.comp_id],\n  { refl }, { refl },\n  { exact H'.epi }\nend\n\nlemma right_exact_of_sends_SES_to_right_exact \n  {V W : Type*} [category V] [category W] [abelian V] [abelian W]\n  (F : V ⥤ W) [F.additive]\n  (hF : ∀ {X Y Z} (f : X ⟶ Y) (g : Y ⟶ Z) [mono f] [epi g],\n          exact f g → exact (F.map f) (F.map g) ∧ epi (F.map g))\n  {X Y Z} (f : X ⟶ Y) (g : Y ⟶ Z) [epi g] (H : exact f g)\n  : exact (F.map f) (F.map g) ∧ epi (F.map g) :=\n  have preserves_epi : ∀ {P Q} (s : P ⟶ Q) [epi s], epi (F.map s) :=\n    λ P Q s hs, (@hF _ _ _ (kernel.ι s) s _ hs (snake_diagram.exact_kernel_ι_self s)).right,\n  have H' : image.ι f ≫ g = 0,\n  by { apply (factor_thru_image.category_theory.epi f).left_cancellation, simp, exact H.w },\n  have H'' : exact (image.ι f) g,\n  from ⟨H', by { have h1 := H.epi,\n                rw ← subobject_of_le_as_image_to_kernel _ _ H.w (image_le_kernel _ _ H.w) at h1,\n                rw ← subobject_of_le_as_image_to_kernel _ _ H' (image_le_kernel _ _ H'),\n                have h2 : image_subobject f ≤ image_subobject (image.ι f),\n                { transitivity image_subobject (factor_thru_image f ≫ image.ι f),\n                  { apply le_of_eq, symmetry, congr, apply image.fac },\n                  { apply image_subobject_comp_le } },\n                rw ← category_theory.subobject.of_le_comp_of_le _ _ _ h2 (image_le_kernel _ _ H') at h1,\n                refine @epi_of_epi _ _ _ _ _ _ _ h1 }⟩,\n  by { have := hF (image.ι f) g H'', refine ⟨_, this.right⟩,\n        rw ← image.fac f,\n        rw F.map_comp,\n        refine (@abelian.exact_epi_comp_iff _ _ _ _ _ _ _ _ _ _\n                                            (preserves_epi (factor_thru_image f))).mpr this.left }\n\nlemma any_coker_of_isos_is_iso {V : Type*} [category V] [abelian V]\n  {A B C X Y Z : V} (f : A ⟶ B) (g : B ⟶ C) (f' : X ⟶ Y) (g' : Y ⟶ Z)\n  (α : A ⟶ X) (β : B ⟶ Y) (γ : C ⟶ Z)\n  (h1 : exact_seq V [f, g, (0 : C ⟶ 0)]) (h2 : exact_seq V [f', g', (0 : Z ⟶ 0)])\n  (sq1 : α ≫ f' = f ≫ β) (sq2 : β ≫ g' = g ≫ γ)\n  (hα : is_iso α) (hβ : is_iso β) : is_iso γ :=\nbegin\n  -- One can do this very concretely but I don't want to l m a o\n  have sq3 : γ ≫ (0 : _ ⟶ 0) = (0 : _ ⟶ 0) ≫ (0 : 0 ⟶ 0), { simp },\n  have sq4 : (0 : (0 : V) ⟶ 0) ≫ (0 : 0 ⟶ 0) = (0 : 0 ⟶ 0) ≫ 0 := rfl,\n  refine abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso sq1 sq2 sq3 sq4 _ _ _ _ _ _,\n  { exact (exact_iff_exact_seq _ _).mpr (h1.extract 0 2) },\n  { exact (exact_iff_exact_seq _ _).mpr (h1.extract 1 3) },\n  { apply to_zero_exact_of_epi, apply_instance },\n  { exact (exact_iff_exact_seq _ _).mpr (h2.extract 0 2) },\n  { exact (exact_iff_exact_seq _ _).mpr (h2.extract 1 3) },\n  { apply to_zero_exact_of_epi, apply_instance }\nend\n\nlemma coker_of_cocartesian_square_is_iso {V : Type*} [category V] [abelian V] {X A B Y  : V}\n  (i : X ⟶ A) (j : X ⟶ B) (k : A ⟶ Y) (ℓ : B ⟶ Y) (h : is_pushout i j k ℓ)\n  : is_iso ((coker_functor V).map (arrow.hom_mk h.to_comm_sq.w.symm\n                                  : arrow.mk i ⟶ arrow.mk ℓ)) :=\n  let f1 : Y ⟶ cokernel i := h.is_colimit.desc (pushout_cocone.mk (cokernel.π i) 0 (by simp)),\n      f2 := cokernel.desc ℓ f1 (h.is_colimit.fac _ walking_span.right)\n  in ⟨⟨f2, by { ext, dsimp [coker_functor], simp, exact h.is_colimit.fac _ walking_span.left },\n          by { ext, dsimp [coker_functor, f2, f1], simp, \n               apply h.is_colimit.hom_ext,\n               apply pushout_cocone.coequalizer_ext,\n               { simp, refl },\n               { simp, symmetry, exact cokernel.condition _ } }⟩⟩\n\nsection general_abelian_category\n\nparameters {C : Type u} {V : Type v} [category.{u'} C] [category.{v'} V]\nparameters {ι : Type} {c : complex_shape ι}\n\nlemma id_eq_zero_of_iso_zero [has_zero_object V] [has_zero_morphisms V] [has_cokernels V] (X : V)\n  : is_isomorphic X 0 → 𝟙 X = 0\n| ⟨H⟩ := calc 𝟙 X = H.hom ≫ H.inv : H.hom_inv_id.symm\n             ... = H.hom ≫ 0     : congr_arg _ (has_zero_object.from_zero_ext H.inv 0)\n             ... = 0              : comp_zero\n\nlemma homology_at_ith_index_zero\n  [has_zero_object V] [has_zero_morphisms V] [has_equalizers V] [has_cokernels V]\n  [has_images V] [has_image_maps V]\n  {X Y : homological_complex V c} (f : X ⟶ Y) (i : ι) (H : f.f i = 0)\n  : (homology_functor V c i).map f = 0 :=\nbegin\n  dsimp [homology_functor, homology.map],\n  convert cokernel.desc_zero _,\n  { convert zero_comp, dsimp [hom.sq_from, kernel_subobject_map], simp_rw [H, comp_zero],\n    exact subobject.factor_thru_zero _ },\n  exact comp_zero\nend\n\ndef homological_complex_functor.mk_nat_trans [has_zero_morphisms V]\n  {F G : C ⥤ homological_complex V c}\n  (η : Π (i : ι), (F ⋙ homological_complex.eval V c i) ⟶ (G ⋙ homological_complex.eval V c i))\n  (η_comm_ds : ∀ (i j : ι), c.rel i j →\n                  ∀ X, (η i).app X ≫ (G.obj X).d i j = (F.obj X).d i j ≫ (η j).app X)\n  : F ⟶ G := {\n    app := λ X, { f := λ i, (η i).app X,\n                  comm' := by exact λ i j h, η_comm_ds i j h X },\n    naturality' := λ X Y f, homological_complex.hom.ext _ _ (funext (λ i, (η i).naturality f))\n  }\n\ndef homological_complex.d_nat_trans [has_zero_morphisms V]\n  (F : C ⥤ homological_complex V c) (i j : ι)\n  : (F ⋙ homological_complex.eval V c i) ⟶ (F ⋙ homological_complex.eval V c j) := {\n    app := λ X, (F.obj X).d i j,\n    naturality' := by simp\n  }\n\nstructure natural_chain_homotopy [preadditive V]\n  {F G : C ⥤ homological_complex V c} (α β : nat_trans F G)\n  : Type (max u (max v (v' + 1))) :=\n(to_chain_htpy : ∀ X, @homotopy ι V _ _ c (F.obj X) (G.obj X) (α.app X) (β.app X))\n(naturality : ∀ X Y (f : X ⟶ Y) i j, c.rel i j → (F.map f).f j ≫ (to_chain_htpy Y).hom j i\n                                                = (to_chain_htpy X).hom j i ≫ (G.map f).f i)\n\n-- This is why we run into universe issues and need ι : Type\n-- parallel_pair_comp is an equality between two functors\n-- and that constrains them to be in the same universe\nprotected def parallel_pair_comp_has_colim \n [has_zero_morphisms V] [has_cokernels V]\n  {X Y : homological_complex V c} (f : X ⟶ Y) (p : ι)\n  : has_colimit (parallel_pair f 0 ⋙ eval V c p) := by {\n    rw parallel_pair_comp, dsimp, apply_instance\n  }\n\nlocal attribute [instance] parallel_pair_comp_has_colim\n\nnoncomputable\ndef coker_of_chain_map_at [has_zero_morphisms V] [has_cokernels V]\n  {X Y : homological_complex V c} (f : X ⟶ Y) (p : ι)\n  : cocone (parallel_pair (f.f p) 0) :=\n  parallel_pair_comp.cocone_comp_to_cocone_pair (eval V c p : homological_complex V c ⥤ V) f 0\n                                                ((eval V c p).map_cocone\n                                                  (colimit.cocone (parallel_pair f 0)))\n\nnoncomputable\ndef coker_of_chain_map_at_is_colimit [has_zero_morphisms V] [has_cokernels V]\n  {X Y : homological_complex V c} (f : X ⟶ Y) (p : ι)\n  : is_colimit (coker_of_chain_map_at f p) :=\n    parallel_pair_comp.is_colimit_comp_to_is_colimit_pair _ _ _ _\n      (is_colimit_of_preserves (eval V c p) (colimit.is_colimit (parallel_pair f 0)))\n\nnoncomputable\ndef chain_homotopy_on_coker_of_compatible_chain_homotopies\n  [preadditive V] [has_cokernels V]\n  {A B X Y : homological_complex V c}\n  (i : A ⟶ X) (j : B ⟶ Y)\n  (f' g' : A ⟶ B) (f g : X ⟶ Y)\n  (w1 : f' ≫ j = i ≫ f) (w2 : g' ≫ j = i ≫ g)\n  (s' : homotopy f' g') (s : homotopy f g)\n  (comm : s'.comp_right j = (homotopy.of_eq w1).trans ((s.comp_left i).trans (homotopy.of_eq w2.symm)))\n  : homotopy ((coker_functor (homological_complex V c)).map (arrow.hom_mk w1 : arrow.mk i ⟶ arrow.mk j))\n             ((coker_functor (homological_complex V c)).map (arrow.hom_mk w2 : arrow.mk i ⟶ arrow.mk j)) := {\n  hom := λ p q, is_colimit.desc\n                  (coker_of_chain_map_at_is_colimit i p)\n                  (cofork.of_π (s.hom p q ≫ cofork.π (coker_of_chain_map_at j q)) \n                  (by { rw [zero_comp, ← category.assoc],\n                        have := congr_arg (λ h, homotopy.hom h p q) comm, simp at this,\n                        rw [← this, category.assoc, cokernel_cofork.condition, comp_zero] })),\n  zero' := by { intros p q h,\n                have : i.f p ≫ 0 = 0 ≫ 0 := eq.trans comp_zero comp_zero.symm,\n                transitivity (coker_of_chain_map_at_is_colimit i p).desc (cofork.of_π 0 this),\n                { congr, rw s.zero' p q h, exact zero_comp },\n                { symmetry,\n                  refine (coker_of_chain_map_at_is_colimit i p).uniq' (cofork.of_π 0 this) 0 _,\n                  intro u, cases u; simp } },\n  comm := by { intro p,\n               have H : ∀ (h' : A ⟶ B) (h : X ⟶ Y) (w : h' ≫ j = i ≫ h),\n                        i.f p ≫ homological_complex.hom.f h p\n                              ≫ cofork.π (coker_of_chain_map_at j p)\n                      = 0 ≫ homological_complex.hom.f h p ≫ cofork.π (coker_of_chain_map_at j p),\n               { intros h' h w, rw [zero_comp, ← category.assoc],\n                 have := congr_arg (λ h, homological_complex.hom.f h p) w, dsimp at this,\n                 rw [← this, category.assoc, cokernel_cofork.condition, comp_zero] },\n               have : ∀ (h' : A ⟶ B) (h : X ⟶ Y) (w : h' ≫ j = i ≫ h),\n                        ((coker_functor (homological_complex V c)).map\n                          (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j)).f p\n                      = is_colimit.desc\n                          (coker_of_chain_map_at_is_colimit i p)\n                          (cofork.of_π (h.f p ≫ cofork.π (coker_of_chain_map_at j p)) (H h' h w)),\n               { intros h' h w,\n                 apply (coker_of_chain_map_at_is_colimit i p).uniq'\n                         (cofork.of_π (h.f p ≫ cofork.π (coker_of_chain_map_at j p)) (H h' h w)),\n                 intro u, cases u; simp,\n                 refine eq.trans (category.assoc _ _ _) _,\n                 refine eq.trans _ (category.assoc _ _ _),\n                 simp,\n                 refine eq.trans (category.id_comp _) _,\n                 refine eq.trans _ (congr_arg _ (category.id_comp _).symm),\n                 change (cokernel.π i\n                         ≫ (coker_functor (homological_complex V c)).map\n                              (arrow.hom_mk w  : arrow.mk i ⟶ arrow.mk j)).f p\n                        = (h ≫ cokernel.π j).f p,\n                 congr' 1,\n                 dsimp [coker_functor],\n                 exact coker_map_spec i j h' h w.symm },\n               rw [this f' f w1, this g' g w2],\n               symmetry, apply is_colimit.uniq'\n                                 (coker_of_chain_map_at_is_colimit i p)\n                                 (cofork.of_π (f.f p ≫ cofork.π (coker_of_chain_map_at j p))\n                                              (H f' f w1)),\n               intro u, cases u; simp,\n               have := congr_arg (λ t, t ≫ cofork.π (coker_of_chain_map_at j p)) (s.comm p),\n               refine eq.trans _ this.symm,\n               simp,\n               delta cofork.π coker_of_chain_map_at parallel_pair_comp.cocone_comp_to_cocone_pair,\n               dsimp,\n               rw [category.assoc, ← d_next_comp_left, category.assoc, ← prev_d_comp_left], \n               apply congr_arg2; simp only [eq_to_hom_app, eq_to_hom_refl],\n               { refine eq.trans (category.id_comp _) _,\n                 refine eq.trans _ (congr_arg _ (category.id_comp _).symm),\n                 rw ← d_next_comp_right, congr,\n                 ext q r,\n                 rw (_ : (cokernel.π i).f q\n                       = cofork.π (coker_of_chain_map_at i q)),\n                 simp,\n                 exact congr_arg _ (category.id_comp _),\n                 delta cofork.π coker_of_chain_map_at parallel_pair_comp.cocone_comp_to_cocone_pair,\n                 simp, },\n               { refine eq.trans (category.id_comp _) _,\n                 refine eq.trans _ (congr_arg _ (category.id_comp _).symm),\n                 rw ← prev_d_comp_right, congr,\n                 ext q r,\n                 rw (_ : (cokernel.π i).f q\n                       = cofork.π (coker_of_chain_map_at i q)),\n                 simp,\n                 exact congr_arg _ (category.id_comp _),\n                 delta cofork.π coker_of_chain_map_at parallel_pair_comp.cocone_comp_to_cocone_pair,\n                 simp, } } }.\n\n-- I should not have to do any of this...\nnoncomputable\ninstance [abelian V] (ℓ : ι) : preserves_finite_limits (eval V c ℓ) := \nbegin\n  constructor,\n  intros J j1 j2,\n  have : @has_limits_of_shape J j1 V _,\n  { obtain ⟨H⟩ := @abelian.has_finite_limits V _ _,\n    exact @H J j1 j2 },\n  refine @eval.category_theory.limits.preserves_limits_of_shape V _ J j1 ι c _ this ℓ,\nend\n\nnoncomputable\ninstance [abelian V] (ℓ : ι) : preserves_finite_colimits (eval V c ℓ) := \nbegin\n  constructor,\n  intros J j1 j2,\n  have : @has_colimits_of_shape J j1 V _,\n  { obtain ⟨H⟩ := @abelian.has_finite_colimits V _ _,\n    exact @H J j1 j2 },\n  refine @eval.category_theory.limits.preserves_colimits_of_shape V _ J j1 ι c _ this ℓ,\nend\n\ndef coker_functor_degreewise_SES [abelian V]\n  {A X : homological_complex V c} (i : A ⟶ X) [mono i]\n  : ∀ ℓ, short_exact (i.f ℓ)\n                     (((coker_functor_proj (homological_complex V c)).app (arrow.mk i)).f ℓ) :=\nbegin\n  intro,\n  rw [ ← homological_complex.eval_map, ← homological_complex.eval_map],\n  apply category_theory.functor.map_short_exact,\n  dsimp [coker_functor_proj],\n  refine short_exact.mk _,\n  apply snake_diagram.exact_self_cokernel_π\nend\n\nlemma δ_natural' [abelian V]  \n  {A B C X Y Z : homological_complex V c}\n  (f : A ⟶ B) (g : B ⟶ C) (f' : X ⟶ Y) (g' : Y ⟶ Z)\n  (α : A ⟶ X) (β : B ⟶ Y) (γ : C ⟶ Z)\n  (H1 : ∀ p, short_exact (f.f p)  (g.f p))\n  (H2 : ∀ p, short_exact (f'.f p) (g'.f p))\n  (w1 : f ≫ β = α ≫ f') (w2 : g ≫ γ = β ≫ g') \n  (p q : ι) (hpq : c.rel p q) :\n  δ f g H1 p q hpq ≫ (homology_functor _ _ q).map α =\n    (homology_functor _ _ p).map γ ≫ δ f' g' H2 p q hpq :=\n  let α' : walking_arrow ⥤ homological_complex V c :=\n          (arrow_category_iso_functor_category _).hom.obj (arrow.mk α),\n      β' : walking_arrow ⥤ homological_complex V c :=\n          (arrow_category_iso_functor_category _).hom.obj (arrow.mk β),\n      γ' : walking_arrow ⥤ homological_complex V c :=\n          (arrow_category_iso_functor_category _).hom.obj (arrow.mk γ),\n      F : α' ⟶ β' := (arrow_category_iso_functor_category _).hom.map (arrow.hom_mk w1),\n      G : β' ⟶ γ' := (arrow_category_iso_functor_category _).hom.map (arrow.hom_mk w2) in\n  have H : Π (x : walking_arrow) ℓ, short_exact ((F.app x).f ℓ) ((G.app x).f ℓ),\n  by { intros, cases x, { exact H1 ℓ }, { exact H2 ℓ } },\n  by { have := δ_natural F G H walking_arrow_hom.arr p q hpq, exact this }\n\nnoncomputable\ndef homology_iso_cokernel [abelian V] (ℓ : ι) (h : c.next ℓ = none) (A : homological_complex V c)\n  : A.homology ℓ ≅ cokernel (A.d_to ℓ) :=\n{ hom := homology.desc' _ _ _ (kernel.ι _ ≫ cokernel.π _) $ by simp,\n  inv := cokernel.desc _ (homology.lift _ _ _ (cokernel.π _) $ by { simp, exact A.d_from_eq_zero h, })\n  $ begin\n    apply homology.hom_to_ext,\n    simp,\n  end,\n  hom_inv_id' := begin\n    apply homology.hom_from_ext,\n    apply homology.hom_to_ext,\n    simp,\n  end,\n  inv_hom_id' := begin\n    apply coequalizer.hom_ext,\n    simp,\n    let t := _, change _ ≫ t = _,\n    have ht : t = homology.ι _ _ _,\n    { apply homology.hom_from_ext, simp, },\n    simp [ht],\n  end }\n\ndef homology_iso_cokernel_natural [abelian V] (ℓ : ι) (h : c.next ℓ = none)\n  {A B : homological_complex V c} (f : A ⟶ B) :\n  (homology_iso_cokernel ℓ h A).hom ≫ cokernel.map _ _ (f.prev _) (f.f _) (by simp) =\n  (homology_functor _ _ _).map f ≫ (homology_iso_cokernel ℓ h B).hom :=\nbegin\n  dsimp [homology_iso_cokernel],\n  apply homology.hom_from_ext,\n  simp,\nend\n\nlemma mono_prev [abelian V] (ℓ : ι) {A B : homological_complex V c} (f : A ⟶ B)\n  (hf : ∀ m, mono (f.f m)) : mono (f.prev ℓ) :=\nbegin\n  destruct (c.prev ℓ),\n  { intro h, have : A.X_prev ℓ = 0, { delta homological_complex.X_prev, rw h },\n    convert has_zero_object.category_theory.mono _,\n    swap, { exact (has_zero_object.unique_to _).default },\n    { apply heq_of_cast_eq _, swap, congr, exact this, apply (has_zero_object.unique_to _).uniq } },\n  { rintros ⟨m, h⟩ _, rw hom.prev_eq _ h, apply_instance }\nend\n\nlemma short_exact_prev [abelian V] (ℓ : ι) {A B C : homological_complex V c}\n  (f : A ⟶ B) (g : B ⟶ C) (h : ∀ m, short_exact (f.f m) (g.f m))\n  : short_exact (f.prev ℓ) (g.prev ℓ) :=\n  { mono := mono_prev ℓ f (λ m, (h m).mono),\n    exact := exact_prev' _ _ ℓ (λ m, (h m).exact),\n    epi := @homological_complex.epi_prev' _ _ _ _ _ _ _ _ _ (λ m, (h m).epi) }\n\n-- shouldn't need abelian, but we need the category of homological complexes to have images\n-- Suggests LTE has something messed up in its typeclasses\nlemma terminal_homology_right_exact [abelian V] (ℓ : ι) (hℓ : c.next ℓ = none)\n  {A B C : homological_complex V c}\n  (f : A ⟶ B) (g : B ⟶ C) (h : ∀ m, short_exact (f.f m) (g.f m))\n  : exact_seq V [(homology_functor V c ℓ).map f, (homology_functor V c ℓ).map g,\n                 (0 : C.homology ℓ ⟶ 0)] :=\nbegin\n  have := coker_right_exact (f.prev ℓ) (g.prev ℓ) (f.f ℓ) (g.f ℓ)\n                            (A.d_to ℓ) (B.d_to ℓ) (C.d_to ℓ) (f.comm_to ℓ) (g.comm_to ℓ)\n                            (short_exact_prev ℓ f g h) (h ℓ),\n  constructor,\n  { replace this := this.extract 0 2, dsimp [list.extract] at this,\n    rw ← exact_iff_exact_seq at this,\n    refine preadditive.exact_of_iso_of_exact' _ _ _ _ (homology_iso_cokernel ℓ hℓ A).symm\n                                                      (homology_iso_cokernel ℓ hℓ B).symm\n                                                      (homology_iso_cokernel ℓ hℓ C).symm _ _ this;\n    { rw [iso.symm_hom, iso.symm_hom, iso.eq_comp_inv, category.assoc, iso.inv_comp_eq],\n      symmetry, apply homology_iso_cokernel_natural } },\n  { rw ← exact_iff_exact_seq,\n    replace this := this.extract 1 3, dsimp [list.extract] at this,\n    rw ← exact_iff_exact_seq at this,\n    refine preadditive.exact_of_iso_of_exact' _ _ _ _ (homology_iso_cokernel ℓ hℓ B).symm\n                                                      (homology_iso_cokernel ℓ hℓ C).symm\n                                                      (iso.refl 0) _ _ this,\n    { rw [iso.symm_hom, iso.symm_hom, iso.eq_comp_inv, category.assoc, iso.inv_comp_eq],\n      symmetry, apply homology_iso_cokernel_natural },\n    { simp } }\nend\n\nlemma coker_of_quasi_isos_between_monic_arrows_is_quasi_iso [abelian V]\n  {A B X Y : homological_complex V c}\n  (i : A ⟶ X) (j : B ⟶ Y)\n  (hi : mono i) (hj : mono j)\n  (g : A ⟶ B) (f : X ⟶ Y)\n  (hg : quasi_iso g) (hf : quasi_iso f)\n  (w : g ≫ j = i ≫ f)\n  : quasi_iso ((coker_functor (homological_complex V c)).map (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j)) :=\nbegin\n  constructor, intro ℓ,\n  have sq1 := eq.trans (eq.symm ((homology_functor V c ℓ).map_comp' g j))\n                       (eq.trans (congr_arg _ w) ((homology_functor V c ℓ).map_comp' i f)), \n  have sq2 : (homology_functor V c ℓ).map f\n           ≫ (homology_functor V c ℓ).map ((coker_functor_proj (homological_complex V c)).app (arrow.mk j))\n           = (homology_functor V c ℓ).map ((coker_functor_proj (homological_complex V c)).app (arrow.mk i))\n           ≫ (homology_functor V c ℓ).map ((coker_functor _).map (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j)),\n  { rw [← functor.map_comp, ← functor.map_comp], apply congr_arg,\n    exact ((coker_functor_proj (homological_complex V c)).naturality' (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j)) },\n  destruct (c.next ℓ),\n  { intro h,\n    have H := terminal_homology_right_exact ℓ h i\n                ((coker_functor_proj (homological_complex V c)).app (arrow.mk i))\n                (coker_functor_degreewise_SES i),\n    have H' := terminal_homology_right_exact ℓ h j\n                 ((coker_functor_proj (homological_complex V c)).app (arrow.mk j))\n                 (coker_functor_degreewise_SES j),\n    exact any_coker_of_isos_is_iso _ _ _ _\n                                   ((homology_functor V c ℓ).map g)\n                                   ((homology_functor V c ℓ).map f)\n                                   ((homology_functor V c ℓ).map ((coker_functor _).map (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j)))\n                                   H H' sq1 sq2 (quasi_iso.is_iso ℓ) (quasi_iso.is_iso ℓ) },\n  { rintros ⟨m, hm⟩ _, \n    have sq3 := (δ_natural' i ((coker_functor_proj (homological_complex V c)).app (arrow.mk i))\n                            j ((coker_functor_proj (homological_complex V c)).app (arrow.mk j))\n                            g f\n                            ((coker_functor _).map (arrow.hom_mk w : arrow.mk i ⟶ arrow.mk j))\n                            (coker_functor_degreewise_SES i)\n                            (coker_functor_degreewise_SES j) w.symm\n                            ((coker_functor_proj (homological_complex V c)).naturality' _).symm\n                            ℓ m hm).symm,\n    have sq4 := eq.trans (eq.symm ((homology_functor V c m).map_comp' g j))\n                         (eq.trans (congr_arg _ w) ((homology_functor V c m).map_comp' i f)),\n    have LES1 := six_term_exact_seq i ((coker_functor_proj (homological_complex V c)).app (arrow.mk i))\n                                     (coker_functor_degreewise_SES i) ℓ m hm,\n    have LES2 := six_term_exact_seq j ((coker_functor_proj (homological_complex V c)).app (arrow.mk j))\n                                      (coker_functor_degreewise_SES j) ℓ m hm,\n    refine abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso sq1 sq2 sq3 sq4 _ _ _ _ _ _,\n    { exact (exact_iff_exact_seq _ _).mpr (LES1.extract 0 2) },\n    { exact (exact_iff_exact_seq _ _).mpr (LES1.extract 1 2) },\n    { exact (exact_iff_exact_seq _ _).mpr (LES1.extract 2 2) },\n    { exact (exact_iff_exact_seq _ _).mpr (LES2.extract 0 2) },\n    { exact (exact_iff_exact_seq _ _).mpr (LES2.extract 1 2) },\n    { exact (exact_iff_exact_seq _ _).mpr (LES2.extract 2 2) } }\nend\n\nlemma homological_complex.is_iso_of_degreewise_is_iso [has_zero_morphisms V]\n  {ι : Type*} {c : complex_shape ι} \n  {C D : homological_complex V c} (f : C ⟶ D) (h : ∀ i, category_theory.is_iso (f.f i))\n  : category_theory.is_iso f :=\nbegin\n  convert is_iso.of_iso (hom.iso_of_components (λ i, as_iso (f.f i)) _),\n  swap, { intros, rw [as_iso_hom, as_iso_hom], exact f.comm i j },\n  symmetry, ext : 2, exact hom.iso_of_components_hom_f _ _ _,\nend\n\nend general_abelian_category\n\nsection chain_complex\n\nparameters {C : Type u} {V : Type v} [category.{u'} C] [category.{v'} V]\nparameters [preadditive V] [has_equalizers V] [has_images V] [has_image_maps V] [has_cokernels V]\n\ndef chain_complex.mk_natural_chain_homotopy\n  {F G : C ⥤ chain_complex V ℕ}\n  {α β : F ⟶ G}\n  (s : Π n, (F ⋙ homological_complex.eval V _ n) ⟶ (G ⋙ homological_complex.eval V _ (n + 1)))\n  (H0 : ∀ X, (nat_trans.app α X).f 0 = ((s 0).app X ≫ (G.obj X).d 1 0) + (nat_trans.app β X).f 0)\n  (H : ∀ n X, (nat_trans.app α X).f (n + 1)\n            = ((F.obj X).d (n + 1) n ≫ (s n).app X)\n            + ((s (n + 1)).app X ≫ (G.obj X).d (n + 2) (n + 1))\n            + (nat_trans.app β X).f (n + 1))\n  : natural_chain_homotopy α β := {\n    to_chain_htpy := λ X, { hom := λ j i, if h : j + 1 = i \n                                          then eq.rec ((s j).app X) h\n                                          else 0,\n                            zero' := λ j i h, dite_eq_right_iff.mpr (λ h', absurd h' h),\n                            comm := by { intro n,\n                                         cases n,\n                                         { simp, exact H0 X },\n                                         { simp, exact H n X } } },\n    naturality := λ X Y f i j h, by { dsimp at h, subst h, simp, exact (s j).naturality' f } \n  }\n\ndef chain_complex.mk_natural_chain_homotopy_rec \n  {F G : C ⥤ chain_complex V ℕ}\n  {α β : F ⟶ G}\n  (init : (F ⋙ homological_complex.eval V _ 0) ⟶ (G ⋙ homological_complex.eval V _ 1))\n  (Hinit : ∀ X, (α.app X).f 0 = (nat_trans.app init X ≫ (G.obj X).d 1 0) + (β.app X).f 0)\n  (step : Π (n : ℕ) (s : (F ⋙ homological_complex.eval V _ n)\n                       ⟶ (G ⋙ homological_complex.eval V _ (n + 1))),\n            (∀ X, ((α.app X).f (n + 1)) ≫ (G.obj X).d (n + 1) n\n                = (((F.obj X).d (n + 1) n ≫ nat_trans.app s X) + (β.app X).f (n + 1))\n                  ≫ (G.obj X).d (n + 1) n)\n            → ((F ⋙ homological_complex.eval V _ (n + 1))\n              ⟶ (G ⋙ homological_complex.eval V _ (n + 2))))\n  (Hstep : ∀ (n : ℕ) (s : (F ⋙ homological_complex.eval V _ n)\n                       ⟶ (G ⋙ homological_complex.eval V _ (n + 1)))\n             (h : (∀ X, ((α.app X).f (n + 1)) ≫ (G.obj X).d (n + 1) n\n                = (((F.obj X).d (n + 1) n ≫ nat_trans.app s X) + (β.app X).f (n + 1))\n                  ≫ (G.obj X).d (n + 1) n)),\n             ∀ X, (nat_trans.app α X).f (n + 1)\n                = ((F.obj X).d (n + 1) n ≫ s.app X)\n                + ((step n s h).app X ≫ (G.obj X).d (n + 2) (n + 1))\n                + (nat_trans.app β X).f (n + 1))\n  : natural_chain_homotopy α β :=\n  chain_complex.mk_natural_chain_homotopy\n    (λ k,\n      (@nat.rec (λ n, Σ' (s : (F ⋙ homological_complex.eval V _ n)\n                            ⟶ (G ⋙ homological_complex.eval V _ (n + 1))), \n                        ∀ X, ((α.app X).f (n + 1)) ≫ (G.obj X).d (n + 1) n\n                           = (((F.obj X).d (n + 1) n ≫ nat_trans.app s X) + (β.app X).f (n + 1))\n                              ≫ (G.obj X).d (n + 1) n)\n                ⟨init, by { intro X,\n                            rw (α.app X).comm',\n                            rw Hinit,\n                            simp,\n                            apply complex_shape.down_mk, refl }⟩\n                (λ n p, ⟨step n p.1 p.2, \n                        by { intro X,\n                             rw (α.app X).comm',\n                             rw Hstep n p.1 p.2,\n                             simp,\n                             apply complex_shape.down_mk, refl }⟩)\n                k).fst)\n    Hinit\n    (by { intros n X, simp, apply Hstep })\n\n-- I don't know the rules on when instances should be global\ninstance [has_zero_object V] (C : chain_complex V ℕ)\n  : is_iso (kernel_subobject (C.d_from 0)).arrow :=\n  by { rw C.d_from_eq_zero chain_complex.next_nat_zero,\n       exact limits.is_iso_kernel_subobject_zero_arrow }\n\nend chain_complex\n\nsection Modules\n\nparameters {C : Type u} {R : Type v} [category.{u'} C] [comm_ring R]\nparameters {ι : Type} {c : complex_shape ι}\n\nlemma all_eq_zero_of_iso_zero {M : Module.{v'} R} (H : is_isomorphic M 0) (x : M) : x = 0 :=\n  congr_fun (congr_arg coe_fn (id_eq_zero_of_iso_zero M H)) x\n\nlemma iso_zero_of_id_eq_zero (M : Module.{v'} R) (h : 𝟙 M = 0) : is_isomorphic M 0 :=\n    ⟨is_zero.iso_zero (@Module.is_zero_of_subsingleton _ _ M\n                         ⟨λ x y, calc x = (𝟙 M : M ⟶ M) x : rfl\n                                    ... = (0 : M ⟶ M) x   : congr_fun (congr_arg _ h) x\n                                    ... = (0 : M ⟶ M) y   : rfl\n                                    ... = (𝟙 M : M ⟶ M) y : (congr_fun (congr_arg _ h) y).symm⟩)⟩\n\nlemma Module.to_homology.homomorphism (C : homological_complex (Module.{v'} R) c) (i : ι)\n  : @is_linear_map R (linear_map.ker (C.d_from i)) (C.homology i) _ _ _ _ _\n      (@Module.to_homology R _ ι c C i) := by {\n    delta Module.to_homology, delta Module.to_cycles,\n    delta homology.π, delta Module.to_kernel_subobject,\n    constructor; intros; simp\n  }\n\nlemma Module.to_cycles.homomorphism (C : homological_complex (Module.{v'} R) c) (i : ι)\n  : is_linear_map R (@Module.to_cycles R _ ι c C i) := by {\n    delta Module.to_cycles, delta Module.to_kernel_subobject,\n    constructor; intros; simp\n  }\n\nlemma Module.to_homology_def \n  (C : homological_complex (Module.{v'} R) c) {i : ι}\n  : is_linear_map.mk' _ (Module.to_homology.homomorphism C i)\n  = Module.as_hom_right (is_linear_map.mk' _ (Module.to_cycles.homomorphism C i))\n  ≫ homology.π (C.d_to i) (C.d_from i) _ := by { ext : 1, refl }\n\nlemma Module.to_cycles_def\n  (C : homological_complex (Module.{v'} R) c) {i : ι}\n  : Module.as_hom_right (is_linear_map.mk' _ (Module.to_cycles.homomorphism C i))\n  = (category_theory.limits.kernel_subobject_iso (C.d_from i)\n    ≪≫ Module.kernel_iso_ker (C.d_from i)).inv := by { ext : 1, refl }\n\nlemma Module.to_cycles_is_iso (C : homological_complex (Module.{v'} R) c) (i : ι)\n  : is_iso (Module.as_hom_right (is_linear_map.mk' _ (Module.to_cycles.homomorphism C i))) :=\n  by { rw Module.to_cycles_def, apply is_iso.of_iso_inv }\n\nlemma Module.to_homology_comp_homology_functor_map\n  {X Y : homological_complex (Module.{v'} R) c} (f : X ⟶ Y) (i : ι)\n  : Module.as_hom_right (@is_linear_map.mk' R (linear_map.ker (X.d_from i)) (X.homology i)\n                                            _ _ _ _ _ \n                                            Module.to_homology\n                                            (Module.to_homology.homomorphism X i))\n  ≫ (homology_functor (Module R) c i).map f\n  = Module.of_hom \n      (linear_map.cod_restrict (linear_map.ker (Y.d_from i)) \n        (linear_map.dom_restrict (f.f i) (linear_map.ker (X.d_from i)))\n        (by intros; simp))\n  ≫ Module.as_hom_right (is_linear_map.mk' Module.to_homology\n                                           (Module.to_homology.homomorphism Y i)) :=\nbegin \n  apply linear_map.ext, intros x, cases x, simp [Module.as_hom_right], delta Module.to_homology,\n  congr, transitivity, apply Module.cycles_map_to_cycles, refl\nend\n\n-- The version in mathlib fixed v' to be v for some reason\nlemma Module.cokernel_π_ext'\n  {M N : Module.{v'} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m)\n  : cokernel.π f x = cokernel.π f y := by { subst w, simp }\n\ndef Module.range_to_ker {A B C : Module.{v'} R} (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)\n  : Module.of R (linear_map.range f) ⟶ Module.of R (linear_map.ker g) := {\n    to_fun := λ x, ⟨x.val, by { obtain ⟨x, y, h⟩ := x, subst h, simp, rw [← comp_apply, w], refl }⟩,\n    map_add' := by { rintros ⟨x, x', hx⟩ ⟨y, y', hy⟩, simp },\n    map_smul' := by { rintros r ⟨x, x', hx⟩, apply subtype.eq, simp },\n  }\n\n@[simp]\nlemma Module.range_to_ker_subtype_ker\n  {A B C : Module.{v'} R} (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)\n  : Module.range_to_ker f g w ≫ Module.as_hom_right (linear_map.ker g).subtype\n  = Module.as_hom_right (linear_map.range f).subtype := by { ext, refl }\n\nlemma Module.image_to_kernel'_kernel_iso_ker\n  {A B C : Module.{v'} R} (f : A ⟶ B) (g : B ⟶ C) (w : f ≫ g = 0)\n  : (Module.image_iso_range f).inv ≫ image_to_kernel' f g w\n  =  Module.range_to_ker f g w ≫ (Module.kernel_iso_ker g).inv :=\nbegin\n  rw [iso.inv_comp_eq, ← category.assoc, iso.eq_comp_inv],\n  apply (@cancel_mono _ _ _ _ _ _ (Module.mono_as_hom'_subtype (linear_map.ker g)) _ _).mp,\n  rw [category.assoc, category.assoc, Module.range_to_ker_subtype_ker],\n  simp [Module.as_hom_right, image_to_kernel'], \n  symmetry, apply Module.image_iso_range_hom_subtype\nend\n\n-- Possible this proof could be prettified/sped up\nlemma Module.to_homology_ext\n  {X : homological_complex (Module.{v'} R) c}\n  {i j : ι}\n  {x y : linear_map.ker (X.d_from j)}\n  (m : X.X i) (hm : x.val = y.val + X.d i j m) \n  : Module.to_homology x = Module.to_homology y :=\n  Module.cokernel_π_ext'\n    (image_to_kernel (X.d_to j) (X.d_from j) (X.d_to_comp_d_from j))\n    ((image_subobject_iso (X.d_to j)).inv\n      ((Module.image_iso_range (X.d_to j)).inv\n        ⟨X.d i j m, \n          by { by_cases c.rel i j,\n               { rw X.d_to_eq h,\n                 existsi (X.X_prev_iso h).inv m,\n                 simp },\n               { rw X.shape' i j h, apply submodule.zero_mem } }⟩))\n    (by { by_cases c.rel i j,\n          { have h1 : X.d i j m ∈ linear_map.ker (X.d_from j),\n            { rw [linear_map.mem_ker, ← X.X_prev_iso_comp_d_to h,\n                  comp_apply, ← comp_apply, d_to_comp_d_from],\n              refl },\n            transitivity Module.to_cycles y + Module.to_cycles ⟨X.d i j m, h1⟩,\n            { rw ← is_linear_map.map_add (Module.to_cycles.homomorphism X j),\n              congr, apply subtype.eq, exact hm },\n            congr,\n            rw [← comp_apply (iso.inv _), ← image_to_kernel'_kernel_subobject_iso,\n                comp_apply, ← comp_apply (iso.inv _), Module.image_to_kernel'_kernel_iso_ker],\n            apply (kernel_subobject_iso (X.d_from j)).eq_app_inv_of_app_hom_eq,\n            apply iso.eq_app_inv_of_app_hom_eq,\n            apply subtype.eq, \n            delta Module.to_cycles,\n            simp [Module.range_to_ker, Module.to_kernel_subobject, Module.kernel_iso_ker] },\n          { rw X.shape' i j h at hm, \n            rw subtype.eq (eq.trans hm (add_zero _)),\n            symmetry, refine eq.trans _ (add_zero _),\n            have : ∀ h' : X.d i j m ∈ linear_map.range (X.d_to j), \n                    (⟨X.d i j m, h'⟩ : {t // t ∈ linear_map.range (X.d_to j)}) = 0,\n            { intro, simp, rw X.shape' i j h, refl },\n            rw this,\n            simp } })\n\nlemma Module.homology_ext''\n  (C D : homological_complex (Module.{v'} R) c)\n  {h k : C ⟶ D} (i : ι)\n  (w : ∀ (x : C.X i), C.d_from i x = 0 →\n       ∃ (j : ι) (y : D.X j), h.f i x = k.f i x + D.d j i y)\n  : (homology_functor (Module.{v'} R) c i).map h = (homology_functor (Module.{v'} R) c i).map k :=\nbegin\n  apply Module.homology_ext', intro x, cases x with x hx,\n  rw [homology_functor_map, homology_functor_map, homology.π_map_apply, homology.π_map_apply],\n  delta kernel_subobject_map, dsimp [hom.sq_from],\n  rw [Module.cycles_map_to_cycles, Module.cycles_map_to_cycles],\n  obtain ⟨j, y, hy⟩ := w x hx,\n  exact Module.to_homology_ext y hy\nend\n\nlemma homological_complex.range_d_eq\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  : linear_map.range (C.d i j)\n  = linear_map.range (C.boundaries_to_cycles j ≫ (C.cycles j).arrow) :=\nbegin\n  delta boundaries_to_cycles,\n  rw image_to_kernel_arrow,\n  symmetry, apply ((Module.subobject_Module _).apply_eq_iff_eq_symm_apply _ _).mpr,\n  dsimp [Module.subobject_Module, order_iso.symm, rel_iso.symm],\n  refine eq.trans _\n          (eq.trans (image_subobject_iso_comp (C.X_prev_iso hij).hom (C.d i j))\n                    _),\n  { congr, apply homological_complex.d_to_eq },\n  { refine eq.trans (image_subobject_mono _).symm (eq.trans _ (image_subobject_mono _)),\n    refine eq.trans _ (image_subobject_iso_comp (Module.image_iso_range (C.d i j)).hom _),\n    congr, symmetry, apply Module.image_iso_range_hom_subtype }\nend\n\ndef homological_complex.exists_preim_cycle_of_to_homology_zero\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  (x : C.X j) (is_cycle : C.d_from j x = 0) (H : Module.to_homology ⟨x, is_cycle⟩ = 0)\n  : ∃ t : C.X i, (C.d i j) t = x :=\nbegin\n  apply linear_map.mem_range.mp,\n  rw C.range_d_eq hij,\n  change (x ∈ linear_map.range ((C.cycles j).arrow.comp (C.boundaries_to_cycles j))),\n  rw linear_map.range_comp (C.boundaries_to_cycles j) (C.cycles j).arrow,\n  rw submodule.mem_map,\n  refine ⟨Module.to_cycles ⟨x, is_cycle⟩, _, Module.to_kernel_subobject_arrow _⟩,\n  apply (submodule.quotient.mk_eq_zero _).mp,\n  have : function.injective (Module.cokernel_iso_range_quotient (C.boundaries_to_cycles j)).inv,\n  { apply function.left_inverse.injective, intro, apply iso.inv_hom_id_apply },\n  refine this (eq.trans (eq.trans _ H) (map_zero _).symm),\n  delta Module.to_homology, delta homology.π,\n  rw ← Module.range_mkq_cokernel_iso_range_quotient_inv, refl\nend\n\nnoncomputable\ndef homological_complex.preim_cycle_of_to_homology_zero\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  (x : C.X j) (is_cycle : C.d_from j x = 0) (H : Module.to_homology ⟨x, is_cycle⟩ = 0)\n  : C.X i :=\n@classical.some (C.X i) (λ y, C.d i j y = x)\n                (homological_complex.exists_preim_cycle_of_to_homology_zero C hij x is_cycle H)\n\nlemma homological_complex.preim_cycle_of_to_homology_zero_spec\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  (x : C.X j) (is_cycle : C.d_from j x = 0) (H : Module.to_homology ⟨x, is_cycle⟩ = 0)\n  : (C.d i j) (homological_complex.preim_cycle_of_to_homology_zero C hij x is_cycle H) = x :=\n@classical.some_spec (C.X i) (λ y, C.d i j y = x)\n                     (homological_complex.exists_preim_cycle_of_to_homology_zero C hij x is_cycle H)\n\nnoncomputable\ndef homological_complex.preim_cycle_of_homology_zero\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  (x : C.X j) (is_cycle : C.d_from j x = 0) (H : is_isomorphic (C.homology j) 0) \n  : C.X i :=\nhomological_complex.preim_cycle_of_to_homology_zero C hij x is_cycle (all_eq_zero_of_iso_zero H _)\n\nlemma homological_complex.preim_cycle_of_homology_zero_spec\n  (C : homological_complex (Module.{v'} R) c) {i j : ι} (hij : c.rel i j)\n  (x : C.X j) (is_cycle : C.d_from j x = 0) (H : is_isomorphic (C.homology j) 0)\n  : C.d i j (homological_complex.preim_cycle_of_homology_zero C hij x is_cycle H) = x :=\nhomological_complex.preim_cycle_of_to_homology_zero_spec C hij x is_cycle\n                                                         (all_eq_zero_of_iso_zero H _)\n\nlemma homological_complex.exists_preim_homology_class\n  (C : homological_complex (Module.{v'} R) c) {i : ι} (y : C.homology i)\n  : ∃ (x : C.X i) (h : C.d_from i x = 0), Module.to_homology ⟨x, h⟩ = y :=\nbegin\n  suffices : y ∈ linear_map.range (is_linear_map.mk' _ (Module.to_homology.homomorphism C i)),\n  { obtain ⟨⟨y, h⟩, h'⟩ := linear_map.mem_range.mp this, existsi y, existsi h, exact h' },\n  rw [Module.to_homology_def, Module.comp_def],\n  rw linear_map.range_comp,\n  rw linear_map.range_eq_top.mpr,\n  { rw submodule.map_top, revert y, apply submodule.eq_top_iff'.mp,\n    rw linear_map.range_eq_top.mpr,\n    delta homology.π,\n    rw ← Module.range_mkq_cokernel_iso_range_quotient_inv,\n    rw coe_comp, apply function.surjective.comp,\n    refine function.left_inverse.surjective (iso.hom_inv_id_apply _),\n    apply submodule.mkq_surjective },\n  { exact function.right_inverse.surjective\n            (@is_iso.inv_hom_id_apply _ _ _ _ _ (Module.to_cycles_is_iso C i) _) }\nend\n\nnoncomputable\ndef homological_complex.preim_of_homology_class\n  (C : homological_complex (Module.{v'} R) c) {i : ι} (y : C.homology i) \n  : C.X i :=\n  classical.some (homological_complex.exists_preim_homology_class C y)\n\ndef homological_complex.preim_of_homology_class_spec\n  (C : homological_complex (Module.{v'} R) c) {i : ι} (y : C.homology i)\n  : ∃ (h : C.d_from i (homological_complex.preim_of_homology_class C y) = 0),\n      Module.to_homology ⟨homological_complex.preim_of_homology_class C y, h⟩ = y :=\n  classical.some_spec (homological_complex.exists_preim_homology_class C y)\n\nlemma Module.to_homology_eq_zero\n  {X : homological_complex (Module.{v'} R) c}\n  {i j : ι} (hij : c.rel i j)\n  {x : linear_map.ker (X.d_from j)}\n  : x.val ∈ linear_map.range (X.d i j) ↔ Module.to_homology x = 0 :=\nbegin\n  split,\n  { rintro ⟨w, h⟩,\n    transitivity Module.to_homology (0 : linear_map.ker (X.d_from j)),\n    apply Module.to_homology_ext w, symmetry, simp, exact h,\n    apply (Module.to_homology.homomorphism _ _).map_zero },\n  { intro h, cases x with x hx,\n    obtain ⟨y, hy⟩ := homological_complex.exists_preim_cycle_of_to_homology_zero X hij x hx h,\n    exact ⟨y, hy⟩ }\nend\n\nlemma Module.to_homology_eq_zero'\n  {X : homological_complex (Module.{v'} R) c}\n  {i : ι} (hi : c.prev i = none)\n  {x : linear_map.ker (X.d_from i)}\n  : x = 0 ↔ Module.to_homology x = 0 :=\nbegin\n  split,\n  { intro h, subst h, exact is_linear_map.map_zero (Module.to_homology.homomorphism X i) },\n  { intro h, delta Module.to_homology at h,\n    suffices : Module.to_cycles x = 0,\n    { delta Module.to_cycles Module.to_kernel_subobject at this,\n      convert congr_arg (kernel_subobject_iso (X.d_from i) ≪≫ Module.kernel_iso_ker (X.d_from i)).hom this;\n      simp },\n    { generalize_hyp : Module.to_cycles x = y at ⊢ h,\n      delta homology.π at h,\n      suffices : is_iso (cokernel.π (image_to_kernel (X.d_to i) (X.d_from i) (X.d_to_comp_d_from i))),\n      { refine eq.trans _ (eq.trans (congr_arg (@inv _ _ _ _ _ this) h) (map_zero _)), simp },\n      convert cokernel.π_zero_is_iso,\n      all_goals\n      { apply zero_of_source_iso_zero,\n        refine (image_subobject_iso _).trans _,\n        apply category_theory.limits.image_zero',\n        delta homological_complex.d_to, rw hi } } }\nend\n\nlemma Module.chain_complex_cokernel_π_zero_of_in_range\n  {X Y : homological_complex (Module.{v'} R) c} (f : X ⟶ Y) {i : ι} (y : Y.X i)\n  (h : y ∈ linear_map.range (f.f i)) : (cokernel.π f).f i y = 0 :=\nbegin\n  convert (_ : cofork.π (coker_of_chain_map_at f i) y = 0),\n  { dsimp [coker_of_chain_map_at, parallel_pair_comp.cocone_comp_to_cocone_pair],\n    delta cofork.π, simp },\n  { let F := colimit.iso_colimit_cocone ⟨_, coker_of_chain_map_at_is_colimit f i⟩,\n    suffices : F.inv (cofork.π (coker_of_chain_map_at f i) y) = 0,\n    { convert congr_arg F.hom this; simp },\n    dsimp [F, colimit.iso_colimit_cocone, is_colimit.cocone_point_unique_up_to_iso],\n    rw ← comp_apply,\n    simp,\n    refine eq.trans _ (map_zero (cokernel.π (f.f i))),\n    obtain ⟨x, H⟩ := h,\n    refine Module.cokernel_π_ext' (f.f i) x _,\n    rw H, symmetry, exact zero_add y }\nend\n\nnoncomputable\ndef Module.to_cycles_terminal_hom {X : homological_complex (Module.{v'} R) c} {i : ι}\n  (hi : c.next i = none) : X.X i ⟶ Module.of R (linear_map.ker (X.d_from i)) :=\n  linear_map.cod_restrict (linear_map.ker (X.d_from i)) linear_map.id\n                          (by { intro x, rw X.d_from_eq_zero hi, simp })\n\nlemma Module.homology_iso_cokernel_spec {C : homological_complex (Module.{v'} R) c} {i : ι}\n  (hi : c.next i = none) \n  : cokernel.π (C.d_to i) ≫ (homology_iso_cokernel i hi C).inv\n  = Module.to_cycles_terminal_hom hi\n  ≫ Module.as_hom_right (is_linear_map.mk' _ (Module.to_homology.homomorphism C i)) :=\nbegin\n  rw Module.to_homology_def,\n  dsimp [Module.as_hom_right],\n  simp [homology_iso_cokernel],\n  apply homology.hom_to_ext,\n  simp,\n  rw ← homology.π'_eq_π,\n  rw category.assoc,\n  simp,\n  symmetry,\n  repeat { rw ← category.assoc },\n  convert category.id_comp _,\n  ext, delta Module.to_cycles,\n  simp [Module.to_cycles_terminal_hom, is_linear_map.mk'],\nend\n\nend Modules\n\nsection retract\n\nparameters {R : Type v} [comm_ring R]\nparameters {ι : Type} {c : complex_shape ι}\n\n\n/-\nWe should be able to prove that iterating barycentric subdivision enough times,\ndepending on the input simplex, gives a chain map C_*(X) -> C_*(X) which is a retraction onto\nthe subcomplex of chains bounded by an open cover 𝒰, as in Hatcher. But this subtly requires that\nif τ is a face of σ and B^k(σ) ∈ C_n^𝒰(X) then B^k(τ) ∈ C_n^𝒰(X).\nThis follows from the fact that B is secretly a morphism of simplicial abelian groups\nSing_•(X) → Sing_•(X), and so commutes with face maps. But in a general chain complex we don't\nhave access to the face maps (Dold Kan is *not* about the Moore complex, but the normalized Moore complex)\nand so it's awkward to state the lemma. Also proving this would be a lot of work\n-/\n\n-- local attribute [instance] classical.prop_decidable \n\ndef homotopy.iterate {V : Type v} [category.{v'} V] [preadditive V] \n  {C : homological_complex V c} {f : C ⟶ C}\n  (s : homotopy (𝟙 C) f)\n  : Π (k : ℕ), homotopy (𝟙 C) (f ^ k : End C)\n| 0       := homotopy.refl (𝟙 C)\n| (k + 1) := (homotopy.iterate k).trans (s.symm.comp_left_id (f ^ k : End C)).symm\n\nlemma chain_map_iterate {V : Type v} [category.{v'} V] [has_zero_morphisms V]\n  {C : homological_complex V c} (f : C ⟶ C)\n  (k : ℕ) (i : ι) : (f ^ k : End C).f i = (f.f i ^ k : End (C.X i)) :=\nbegin\n  induction k with k ih,\n  { refl },\n  { rw [← npow_eq_pow, ← npow_eq_pow, monoid.npow_succ', monoid.npow_succ'],\n    exact congr_arg2 category_struct.comp ih rfl }\nend\n\n-- lemma homotopy.iterate_as_sum {V : Type v} [category.{v'} V] [preadditive V]\n--   {C : homological_complex V c} {f : C ⟶ C}\n--   (s : homotopy (𝟙 C) f) (k : ℕ) (i j : ι)\n--   : (s.iterate k).hom i j = finset.univ.sum (λ p : fin k, (f^(p : ℕ) : End C).f i ≫ s.hom i j) :=\n-- begin\n--   by_cases c.rel j i,\n--   { induction k with k ih,\n--     { refl },\n--     { simp [homotopy.iterate],\n--       rw fin.sum_univ_cast_succ,\n--       simp,\n--       rw ih,\n--       congr,\n--       simp [homotopy.symm, homotopy.comp_left_id] } },\n--   { rw (s.iterate k).zero i j h, simp_rw s.zero i j h, simp }\n-- end\n\n-- noncomputable\n-- def homotopy.iter_until_in_subcomplex\n--   {ι' : ι → Type}\n--   {C : homological_complex (Module.{v'} R) c}\n--   (b : Π (i : ι), basis (ι' i) R (C.X i))\n--   (M : Π (i : ι), submodule R (C.X i))\n--   (f : C ⟶ C)\n--   (H : ∀ i x, ∃ k, (f.f i)^[k] x ∈ M i)\n--   (s : homotopy (𝟙 C) f) (i j : ι) : C.X i ⟶ C.X j :=\n--   (b i).constr R (λ ℓ, (s.iterate (nat.find (H i (b i ℓ)))).hom i j (b i ℓ))\n\n-- lemma d_next_of_iter_until_in_subcomplex_on_basis \n--   {ι' : ι → Type}\n--   {C : homological_complex (Module.{v'} R) c}\n--   (b : Π (i : ι), basis (ι' i) R (C.X i))\n--   (M : Π (i : ι), submodule R (C.X i))\n--   (f : C ⟶ C)\n--   (H : ∀ i x, ∃ k, (f.f i)^[k] x ∈ M i)\n--   (s : homotopy (𝟙 C) f) (i : ι) (ℓ : ι' i)\n--   : prev_d i (homotopy.iter_until_in_subcomplex b M f H s) (b i ℓ)\n--   = prev_d i (s.iterate (nat.find (H i (b i ℓ)))).hom (b i ℓ) :=\n-- begin\n-- destruct c.prev i,\n-- { intro h, delta prev_d, simp, rw h },\n-- { rintros ⟨j, hij⟩ h, delta prev_d, simp, rw h, simp,\n--   refine congr_arg _ _,\n--   exact basis.constr_basis _ _ _ _ }\n-- end\n\n-- noncomputable\n-- def retract_free_of_eventually_in_submodule\n--   {ι' : ι → Type}\n--   (C : homological_complex (Module.{v'} R) c)\n--   (b : Π (i : ι), basis (ι' i) R (C.X i))\n--   (M : Π (i : ι), submodule R (C.X i))\n--   (f : C ⟶ C)\n--   (H : ∀ i x, ∃ k, (f.f i)^[k] x ∈ (M i))\n--   (s : homotopy (𝟙 C) f)\n--   : C ⟶ C := {\n--     f := λ i, (𝟙 (C.X i)) - (d_next i) (homotopy.iter_until_in_subcomplex b M f H s)\n--                           - (prev_d i) (homotopy.iter_until_in_subcomplex b M f H s),\n--     comm' := by {\n--       intros i j hij,\n--       apply (b i).ext, intro ℓ,\n--       simp, rw sub_right_comm,\n--       congr, \n--       { repeat { rw ← comp_apply },\n--         refine congr_fun (congr_arg coe_fn _) _,\n--         transitivity (0 : C.X i ⟶ C.X j),\n--         { rw ← d_from_comp_X_next_iso _ hij,\n--           swap, apply_instance, \n--           rw ← category.assoc (C.d_to i), simp },\n--         { rw ← X_prev_iso_comp_d_to _ hij,\n--           rw [category.assoc, ← category.assoc (C.d_to j)], simp } },\n--       { repeat { rw ← comp_apply },\n--         refine congr_fun (congr_arg coe_fn _) _,\n--         rw [← category.assoc, ← d_next_eq_d_from_from_next, d_next_eq _ hij],\n--         rw category.assoc, refine congr_arg _ _,\n--         rw ← prev_d_eq _ hij, simp } } }\n\n-- def retract_free_of_eventually_in_submodule_eventually_in\n--   {ι' : ι → Type}\n--   (C : homological_complex (Module.{v'} R) c)\n--   (b : Π (i : ι), basis (ι' i) R (C.X i))\n--   (M : Π (i : ι), submodule R (C.X i))\n--   (f : C ⟶ C)\n--   (H : ∀ i x, ∃ k, (f.f i)^[k] x ∈ M i)\n--   (s : homotopy (𝟙 C) f)\n--   (hb : ∀ i j, submodule.map (C.d i j) (M i) ≤ M j)\n--   (hs : ∀ i j, submodule.map (s.hom i j) (M i) ≤ M j)\n--   (hf : ∀ i, submodule.map (f.f i) (M i) ≤ M i)\n--   (hboundary_in_M_only_if : ∀ i ℓ, b i ℓ ∈ M i\n--                           → ∀ j m, (b j).repr (C.d i j (b i ℓ)) m ≠ 0 → b j m ∈ M j)\n--   : ∀ i, linear_map.range ((retract_free_of_eventually_in_submodule C b M f H s).f i)\n--        ≤ M i :=\n-- begin\n--   have : ∀ i ℓ j m, (b j).repr (C.d i j (b i ℓ)) m ≠ 0\n--                   → nat.find (H j (b j m)) ≤ nat.find (H i (b i ℓ)),\n--   { intros i ℓ j m h,\n--     apply nat.find_le,\n--     have := nat.find_spec (H i (b i ℓ)), revert this,\n--     generalize : nat.find (H i (b i ℓ)) = k, intro h,\n--      },\n--   intro i,\n--   rw linear_map.range_eq_map,\n--   rw ← (b i).span_eq,\n--   rw submodule.map_span_le,\n--   rintros x ⟨ℓ, h⟩, subst h,\n--   dsimp [retract_free_of_eventually_in_submodule],\n--   rw d_next_of_iter_until_in_subcomplex_on_basis,\n--   have := (s.iterate (nat.find (H i (b i ℓ)))).comm i,\n--   rw ← sub_eq_iff_eq_add at this,\n--   symmetry' at this, rw add_comm at this, symmetry' at this, rw ← sub_eq_iff_eq_add at this,\n--   rw ← this,\n--   simp,\n--   rw [sub_right_comm, sub_sub _ _ ((from_next i (s.iterate _).hom _)), ← sub_add],\n--   simp, rw add_sub_assoc,\n--   apply submodule.add_mem,\n--   { convert nat.find_spec (H i (b i ℓ)),\n--     generalize : nat.find (H i (b i ℓ)) = k,\n--     induction k with k ih,\n--     { refl },\n--     { rw nat.iterate_succ,\n--       rw ← npow_eq_pow,\n--       rw monoid.npow_succ',\n--       rw ← ih,\n--       refl } },\n--   {  }\n--   -- refine eq.trans (congr_arg2 _ _ _) (add_zero _),\n--   -- { transitivity (f ^ 0 : End C).f i (b i ℓ),\n--   --   { congr, simp,\n--   --     exact submodule.subset_span ⟨ℓ, hℓ, rfl⟩ },\n--   --   { refl } },\n-- end\n\n-- def retract_free_of_eventually_in_submodule_homotopic\n--   {ι' : ι → Type}\n--   (C : homological_complex (Module.{v'} R) c)\n--   (b : Π (i : ι), basis (ι' i) R (C.X i))\n--   (b' : Π (i : ι), ι' i → Prop)\n--   (f : C ⟶ C)\n--   (H : ∀ i x, ∃ k, (f.f i)^[k] x ∈ submodule.span R (b i '' { ℓ | b' i ℓ }))\n--   (s : homotopy (𝟙 C) f)\n--   (hf : ∀ (i : ι) (ℓ : ι' i), f.f i (b i ℓ) ∈ submodule.span R (b i '' { m | b' i m }))\n--   (hb' : ∀ (i j : ι) (ℓ : ι' i), C.d i j (b i ℓ) ∈ submodule.span R (b j '' { m | b' j m }))\n--   : homotopy (𝟙 C) (retract_free_of_eventual_retract_on_basis C b (λ j, submodule.span R (b j '' { ℓ | b' j ℓ })) f H s) :=\n-- begin\n--   admit\n-- end\n\nend retract\n\nsection Modules\n\nparameters {C : Type u} {R : Type v} [category.{u'} C] [comm_ring R]\nparameters {ι : Type} {c : complex_shape ι}\n\ndef Module.subcomplex_of_compatible_submodules\n  (C : homological_complex (Module.{v'} R) c)\n  (M : Π (i : ι), submodule R (C.X i))\n  (hcompat : ∀ i j, submodule.map (C.d i j) (M i) ≤ M j)\n  : homological_complex (Module.{v'} R) c := {\n    X := λ i, Module.of R (M i),\n    d := λ i j, linear_map.cod_restrict (M j) \n                                        (linear_map.dom_restrict (C.d i j) (M i))\n                                        (λ x, hcompat i j (submodule.mem_map_of_mem x.property)),\n    d_comp_d' := by { intros i j k hij hjk, ext, cases x with x hx, \n                      simp, rw ← comp_apply, rw C.d_comp_d, refl },\n    shape' := by { intros i j hij, ext, cases x with x hx, simp, rw C.shape' i j hij, refl } }\n\ndef Module.subcomplex_of_compatible_submodules_inclusion\n  (C : homological_complex (Module.{v'} R) c)\n  (M : Π (i : ι), submodule R (C.X i))\n  (hcompat : ∀ i j, submodule.map (C.d i j) (M i) ≤ M j)\n  : Module.subcomplex_of_compatible_submodules C M hcompat ⟶ C := {\n    f := λ i, (M i).subtype\n  }\n\nlemma quasi_iso_of_lift_boundaries_and_cycles\n  {C D : homological_complex (Module.{v'} R) c} (f : C ⟶ D)\n  (h1 : ∀ i j x y, C.d_from j y = 0 → D.d i j x = f.f j y → ∃ z, C.d i j z = y)\n  (h2 : ∀ j x, D.d_from j x = 0 → ∃ i y z, C.d_from j y = 0 ∧ f.f j y = x + D.d i j z)\n  : quasi_iso f :=\nbegin\n  constructor,\n  intro i,\n  suffices : function.bijective ((homology_functor (Module R) c i).map f),\n  { let f' := linear_equiv.of_bijective _ this.left this.right,\n    constructor,\n    refine exists.intro f'.symm _,\n    split; apply Module.homology_ext'; intro, { apply f'.left_inv }, { apply f'.right_inv } },\n  split,\n  { rw [← linear_map.ker_eq_bot, linear_map.ker_eq_bot'],\n    intros x hx,\n    obtain ⟨h, h'⟩ := C.preim_of_homology_class_spec x,\n    generalize_hyp : C.preim_of_homology_class x = x' at h h', subst h',\n    have := congr_arg linear_map.to_fun (@Module.to_homology_comp_homology_functor_map R _ ι c C D f i),\n    replace this := congr_fun this (⟨x', h⟩ : linear_map.ker (C.d_from i)),\n    replace this := eq.trans this.symm hx,\n    simp [Module.as_hom_right] at this,\n    destruct (c.prev i),\n    { intro h', rw ← Module.to_homology_eq_zero' h' at this ⊢,\n      obtain ⟨z, hz⟩ := h1 i i 0 x' h (eq.trans (map_zero _) (subtype.ext_iff_val.mp this.symm)),\n      have : ¬ c.rel i i, { intro hi', rw c.prev_eq_some hi' at h', injection h' },\n      rw C.shape i i this at hz, symmetry, exact subtype.eq hz },\n    { rintro ⟨j, hij⟩ _,\n      rw ← Module.to_homology_eq_zero hij,\n      rw ← Module.to_homology_eq_zero hij at this,\n      simp at this, obtain ⟨y, hy⟩ := this,\n      exact h1 j i y x' h hy } },\n  { intro x,\n    obtain ⟨h, h'⟩ := D.preim_of_homology_class_spec x,\n    generalize_hyp : D.preim_of_homology_class x = x' at h h', subst h',\n    obtain ⟨j, y, z, hy, hz⟩ := h2 i x' h,\n    existsi Module.to_homology ⟨y, hy⟩,\n    have := congr_arg linear_map.to_fun (@Module.to_homology_comp_homology_functor_map R _ ι c C D f i),\n    replace this := congr_fun this (⟨y, hy⟩ : linear_map.ker (C.d_from i)),\n    refine eq.trans this _,\n    exact Module.to_homology_ext z hz }\nend\n\nlemma subcomplex_inclusion_quasi_iso_of_pseudo_projection\n  {C : homological_complex (Module.{v'} R) c}\n  (M : Π (i : ι), submodule R (C.X i))\n  (hcompat : ∀ i j, submodule.map (C.d i j) (M i) ≤ M j)\n  (p : C ⟶ C) (s : homotopy (𝟙 C) p)\n  (hp_eventual : ∀ i x, ∃ k, (p.f i)^[k] x ∈ M i)\n  (hp : ∀ i, submodule.map (p.f i) (M i) ≤ M i)\n  (hs : ∀ i j, submodule.map (s.hom i j) (M i) ≤ M j)\n  : quasi_iso (Module.subcomplex_of_compatible_submodules_inclusion C M hcompat) :=\nbegin\n  have hp_iter : ∀ i k {x}, x ∈ M i → ((p.f i)^[k] x) ∈ M i, \n  { intros i k x hx, induction k with k ih,\n    { exact hx },\n    { rw nat.iterate_succ,\n      refine hp i _,\n      existsi ((p.f i)^[k] x), exact ⟨ih, rfl⟩ } },\n  have hs_iter : ∀ i j k, submodule.map ((s.iterate k).hom i j) (M i) ≤ M j,\n  { intros i j k, specialize hs i j,\n    rw submodule.map_le_iff_le_comap at ⊢ hs,\n    intros x hx, simp,\n    induction k with k ih,\n    { exact zero_mem _, },\n    { simp [homotopy.iterate],\n      apply submodule.add_mem,\n      { exact ih },\n      { refine hs _, rw chain_map_iterate p k,\n        rw concrete_category.pow_eq_iter, \n        exact hp_iter i k hx } } },\n  apply quasi_iso_of_lift_boundaries_and_cycles, \n  { intros i j x y hy hx,\n    obtain ⟨k, hk⟩ := hp_eventual i x,\n    refine exists.intro (⟨(s.iterate k).hom j i y.val, _⟩ + ⟨(p.f i)^[k] x, hk⟩) _,\n    { refine hs_iter j i k _, exact submodule.mem_map_of_mem y.property },\n    { ext,\n      delta Module.subcomplex_of_compatible_submodules,\n      delta Module.subcomplex_of_compatible_submodules_inclusion at hx,\n      simp at ⊢ hx,\n      refine eq.trans _ hx,\n      by_cases (c.rel i j),\n      { rw ← hx,\n        rw ← comp_apply _ ((s.iterate k).hom j i), \n        rw ← d_next_eq _ h,\n        dsimp,\n        have := (s.iterate k).comm i,\n        rw ← sub_eq_iff_eq_add at this,\n        rw ← sub_eq_iff_eq_add at this,\n        rw ← this,\n        simp,\n        rw (_ : (C.d i j) (C.d_to i (to_prev i (s.iterate k).hom x)) = 0),\n        rw sub_zero,\n        rw sub_add,\n        convert sub_zero _,\n        { rw sub_eq_zero,\n          refine congr_arg _ _,\n          rw chain_map_iterate,\n          rw concrete_category.pow_eq_iter },\n        { rw ← homological_complex.d_from_comp_X_next_iso _ h,\n          rw ← comp_apply,\n          rw ← category.assoc,\n          simp, apply_instance } },\n      { rw C.shape' i j h, simp } } },\n  { intros j x hx,\n    obtain ⟨k, hk⟩ := hp_eventual j x,\n    have : x - ((p.f j)^[k] x) = prev_d j (s.iterate k).hom x,\n    { have := (s.iterate k).comm j,\n      rw ← sub_eq_iff_eq_add at this,\n      transitivity d_next j (s.iterate k).hom x + prev_d j (s.iterate k).hom x,\n      { convert congr_fun (congr_arg coe_fn this) x,\n        rw [chain_map_iterate, concrete_category.pow_eq_iter] },\n      { symmetry,\n        convert add_zero _,\n        destruct (c.next j),\n        { intro h, delta d_next, rw h, simp },\n        { rintros ⟨i, h⟩ _,\n          rw d_next_eq _ h,\n          dsimp,\n          rw ← d_from_comp_X_next_iso _ h,\n          dsimp, rw hx, simp } } },\n    rw exists_comm, refine ⟨⟨_, hk⟩, _⟩,\n    simp_rw [exists_and_distrib_left],\n    split, \n    { destruct (c.next j),\n      { intro h', delta homological_complex.d_from, rw h', simp },\n      { rintros ⟨ℓ, h'⟩ _, rw d_from_eq _ h', simp,\n        convert map_zero _,\n        dsimp [Module.subcomplex_of_compatible_submodules],\n        apply subtype.eq, simp,\n        rw [← concrete_category.pow_eq_iter, ← chain_map_iterate],\n        rw [← comp_apply, homological_complex.hom.comm, comp_apply],\n        rw [← d_from_comp_X_next_iso _ h', comp_apply, hx], simp } },\n    destruct (c.prev j),\n    { intro h,\n      existsi j, refine ⟨0, _⟩,\n      simp,\n      delta prev_d at this, rw h at this, simp at this,\n      rw sub_eq_zero at this, symmetry' at this, exact this },\n    { rintros ⟨i, hi⟩ _, existsi i,\n      rw prev_d_eq _ hi at this, dsimp at this, rw sub_eq_iff_eq_add at this,\n      existsi - (s.iterate k).hom j i x,\n      rw [map_neg, ← sub_eq_add_neg, eq_sub_iff_add_eq],\n      symmetry, rw add_comm, exact this } }\nend\n\nend Modules\n", "meta": {"author": "Shamrock-Frost", "repo": "BrouwerFixedPoint", "sha": "52f48d25068df0eadf3df5b2ede7bcb087d30527", "save_path": "github-repos/lean/Shamrock-Frost-BrouwerFixedPoint", "path": "github-repos/lean/Shamrock-Frost-BrouwerFixedPoint/BrouwerFixedPoint-52f48d25068df0eadf3df5b2ede7bcb087d30527/src/homological_algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3197155856257884}}
{"text": "import util\nimport approach_one_symbol\n\nopen real set rnd_var_1\n\nopen_locale big_operators\n\nnoncomputable theory\n\nvariables {ι : Type} [fintype ι] [decidable_eq ι]\n\n/-- U with q k divided on both sides and q₂ bounded further from below\n{ k | (exp -ε) < q₂ k / q k * exp(err_exp_1 q₁ q₂ r) ≤ q₁ k / q k * (exp r) } -/\ndef U₁ (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | q₁ k / q k ≥ q₂ k / q k * exp(err_exp_1 q₁ q₂ r - r) ∧ q₂ k > q k * exp(-(err_exp_1 q₁ q₂ r + ε)) }\n\n/-- Uᶜ but q₁ bounded further from below\n{ k | (exp -ε) < q₁ k / q k * (exp r) < q₂ k / q k * exp(err_exp_1 q₁ q₂ r) } -/\ndef U₂ (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | q₁ k / q k < q₂ k / q k * exp(err_exp_1 q₁ q₂ r - r) ∧ q₁ k > q k * exp(-(ε + r)) }\n\ndef UA (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | exp(-ε) < q₁ k / q k * exp(r) }\n\ndef UB (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | exp(-ε) < q₂ k / q k * exp(err_exp_1 q₁ q₂ r) }\n\ndef UTA (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | abs(log( q k / q₁ k * exp(-r) )) ≥ ε }\n\ndef UTB (q q₁ q₂ : ι → ℝ) [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂] (r ε : ℝ) :=\n{ k | abs(log( q k / q₂ k * exp(-err_exp_1 q₁ q₂ r) )) ≥ ε }\n\nvariables\n{q q₁ q₂ : ι → ℝ} [rnd_var_1 q] [rnd_var_1 q₁] [rnd_var_1 q₂]\n{r T ε : ℝ} (HT : T = err_exp_1 q₁ q₂ r - r)\n(Hr : r = ∑ k, q k * log(q k / q₁ k)) -- let q achieve err_exp_1(r)\n(Hε : ε > 0)\n(Herr_exp : err_exp_1 q₁ q₂ r = ∑ k, q k * log(q k / q₂ k))\n\n@[simp] lemma exp_ne_zero : exp(r) ≠ 0 := ne_of_gt (exp_pos r)\n@[simp] lemma q_pos       : ∀ k, 0 < q k := λk, probs_pos k\n@[simp] lemma q₁_pos      : ∀ k, 0 < q₁ k := λk, probs_pos k\n@[simp] lemma q₂_pos      : ∀ k, 0 < q₂ k := λk, probs_pos k\n\n@[simp]\nlemma q_div_q₁_ne_zero {k} :\n  q k / q₁ k ≠ 0 :=\nby {apply ne_of_gt, rw [(lt_div_iff _), zero_mul]; simp}\n\nlemma in_U₂ :\n  (∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k)) * exp(-(r + ε)) ≤ ∑ k, q₁ k * (Φ (U₂ q q₁ q₂ r ε) k) :=\nbegin\n  rw finset.sum_mul,\n  apply finset.sum_le_sum,\n  intros k hk,\n  rw [Φ, indicator],\n  by_cases (k ∈ U₂ q q₁ q₂ r ε), simp only [h], simp,\n  rw U₂ at h, norm_num at h,\n  apply le_of_lt, rw add_comm, exact h.2,\n  simp only [h], simp,\nend\n\nlemma in_U₁ :\n(∑ k, q k * (Φ (U₁ q q₁ q₂ r ε) k))\n                                   ≤ (∑ k, (q₂ k) * (Φ (U₁ q q₁ q₂ r ε) k)) * exp(err_exp_1 q₁ q₂ r + ε) :=\nbegin\n  rw finset.sum_mul,\n  apply finset.sum_le_sum,\n  intros k hk,\n  rw [Φ, indicator],\n  by_cases (k ∈ U₁ q q₁ q₂ r ε), simp only [h], simp,\n  rw U₁ at h, norm_num at h,\n  apply le_of_lt,\n  rw ← mul_lt_mul_right (exp_pos (-err_exp_1 q₁ q₂ r - ε)),\n  rw mul_assoc, rw ← exp_add _ _, ring_nf, rw exp_zero, rw one_mul,\n  rw ← tactic.ring.add_neg_eq_sub, rw add_comm, rw mul_comm, exact h.2,\n  simp only [h], simp,\nend\n\ninclude HT\n\nlemma U₁_subset_U :\n  (U₁ q q₁ q₂ r ε) ⊆ (U q₁ q₂ T) :=\nbegin\n  rw [U, U₁], intros k Hk, norm_num,\n  have qk_pos : ∀ k, 0 < q k, by exact (probs_pos),\n  rw ← div_le_div_right (qk_pos k),\n  rw [mul_comm, ← mul_div, mul_comm, HT], exact Hk.1,\nend\n\nlemma U₂_subset_Uc :\n  (U₂ q q₁ q₂ r ε) ⊆ (U q₁ q₂ T)ᶜ :=\nbegin\n  rw [U, U₂], intros k Hk, norm_num,\n  have qk_pos : ∀ k, 0 < q k, by exact (probs_pos),\n  rw ← div_lt_div_right (qk_pos k),\n  rw [mul_comm, ← mul_div, mul_comm, HT], exact Hk.1,\nend\n\nomit HT\n\nlemma U₁_union_U₂_eq_UA_inter_UB :\n  (U₁ q q₁ q₂ r ε) ∪ (U₂ q q₁ q₂ r ε) = (UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε) :=\nbegin\n  rw [union_def, inter_def], ext k,\n\n  let a : Prop := exp (-ε) < q₁ k / q k * exp r,\n  let b : Prop := exp (-ε) < q₂ k / q k * exp (err_exp_1 q₁ q₂ r),\n  let c : Prop :=  q₂ k / q k * exp(err_exp_1 q₁ q₂ r) ≤ q₁ k / q k * exp(r),\n\n  have a_def : a ↔ exp (-ε) < q₁ k / q k * exp r, refl,\n  have b_def : b ↔ exp (-ε) < q₂ k / q k * exp (err_exp_1 q₁ q₂ r), refl,\n  have c_def : c ↔ q₂ k / q k * exp(err_exp_1 q₁ q₂ r) ≤ q₁ k / q k * exp(r), refl,\n\n  have a_rw     : a ↔ q k * exp (-r + -ε) < q₁ k, {\n    rw ← mul_lt_mul_right (exp_pos(r)),\n    rw [mul_assoc, ← exp_add, add_comm, ← add_assoc, add_neg_self, zero_add],\n    have q_inv_pos : 0 < (q k)⁻¹, by exact inv_pos.mpr (q_pos k),\n    rw [← mul_lt_mul_right q_inv_pos, mul_assoc, mul_comm],\n    rw [← div_eq_mul_inv, div_mul, div_self, div_one],\n    rw [mul_assoc, mul_comm (exp (r)) (q k)⁻¹],\n    rw [← mul_assoc, ← div_eq_mul_inv],\n    exact ne_of_gt (q_pos k),\n  },\n  have b_rw     : b ↔ q k * exp (-ε + -err_exp_1 q₁ q₂ r) < q₂ k, {\n    rw ← mul_lt_mul_right (exp_pos(err_exp_1 q₁ q₂ r)),\n    rw [mul_assoc, ← exp_add, add_assoc, neg_add_self, add_zero],\n    have q_inv_pos : 0 < (q k)⁻¹, by exact inv_pos.mpr (q_pos k),\n    rw [← mul_lt_mul_right q_inv_pos, mul_assoc, mul_comm],\n    rw [← div_eq_mul_inv, div_mul, div_self, div_one],\n    rw [mul_assoc, mul_comm (exp (err_exp_1 q₁ q₂ r)) (q k)⁻¹],\n    rw [← mul_assoc, ← div_eq_mul_inv],\n    exact ne_of_gt (q_pos k),\n  },\n  have c_rw     : c ↔ q₂ k / q k * exp (err_exp_1 q₁ q₂ r - r) ≤ q₁ k / q k, {\n    rw ← mul_le_mul_right (exp_pos(r)),\n    rw [mul_assoc, ← exp_add], simp,\n  },\n  have not_c_rw : ¬ c ↔ q₁ k / q k < q₂ k / q k * exp (err_exp_1 q₁ q₂ r - r), by rw [c_rw, not_le],\n\n  have a_and_b_rw : (c ∧ b ∧ a) ∨ (¬ c ∧ b ∧ a) ↔ (a ∧ b), by {split, tauto, tauto},\n\n  have c_and_b_rw     :   c ∧ b ↔ c ∧ b ∧ a, {\n    rw [a_def, b_def, c_def],\n    split, intro H,\n    split, exact H.1, split, exact H.2,\n    calc exp(-ε) < q₂ k / q k * exp (err_exp_1 q₁ q₂ r) : H.2\n             ... ≤ q₁ k / q k * exp r : H.1,\n    intro H, exact ⟨H.1, H.2.1⟩,\n  },\n  have not_c_and_b_rw : ¬ c ∧ a ↔ ¬ c ∧ b ∧ a, {\n    rw [a_def, b_def, c_def, not_le],\n    split,\n    intro H, split, exact H.1, split,\n    calc exp (-ε) < q₁ k / q k * exp r : H.2\n              ... < q₂ k / q k * exp (err_exp_1 q₁ q₂ r) : H.1,\n    exact H.2,\n    intro H, exact ⟨H.1, H.2.2⟩,\n  },\n  split,\n  {\n    intro H, rw [U₁, U₂] at H, rw [UA, UB],\n    norm_num, norm_num at H,\n    rwa [← a_and_b_rw, ← c_and_b_rw, ← not_c_and_b_rw, not_c_rw, c_rw, a_rw, b_rw],\n  },\n  {\n    intro H, rw [UA, UB] at H, rw [U₁, U₂],\n    norm_num, norm_num at H,\n    rw [← b_rw, ← a_rw, ← c_rw, ← not_c_rw, c_and_b_rw, not_c_and_b_rw],\n    exact a_and_b_rw.mpr H,\n  },\nend\n\nlemma sum_inter_ge :\n  ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k) \n          ≥ 1 - ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) - ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k) :=\nbegin\n  calc ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k) \n                  = 1 - ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε))ᶜ k) : by rw in_self_or_in_compl\n              ... = 1 - ∑ k, q k * (Φ ((UA q q₁ q₂ r ε)ᶜ ∪ (UB q q₁ q₂ r ε)ᶜ) k) : by rw compl_inter\n              ... ≥ 1 - (∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) + ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k)) : by {apply sub_le_sub_left, exact add_ge_sum_union q_pos}\n              ... ≥ 1 - ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) - ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k) : by linarith,\nend\n\nlemma log_lt_of_lt_div_mul {k} :\n  exp (-ε) < q₁ k / q k * exp r → log(q k / q₁ k * exp(-r)) < ε :=\nbegin\n  intro H,\n  rw [← exp_lt_exp, exp_log],\n  rw [← mul_lt_mul_left (exp_pos _), ← (exp_add _ _)],\n  rw [neg_add_self, exp_zero, ← mul_assoc],\n  rw ← mul_lt_mul_right (exp_pos _),\n  rw [mul_assoc, ← (exp_add _ _)],\n  rw [neg_add_self, exp_zero, mul_one, one_mul, mul_div],\n  rw div_lt_iff (q₁_pos k),\n  have qk_pos : 0 < q k, by exact probs_pos k,\n  rw [← div_lt_div_right (qk_pos), ← mul_div],\n  rwa [div_self, mul_one, ← mul_div, mul_comm],\n  exact ne_of_gt qk_pos, assumption,\n  rw [zero_lt_mul_right, lt_div_iff, zero_mul],\n  simp, simp, apply exp_pos,\nend\n\nlemma lt_div_mul_of_log_lt {k} :\n  log(q k / q₁ k * exp(-r)) < ε → exp (-ε) < q₁ k / q k * exp r :=\nbegin\n  intro H,\n  rw ← mul_lt_mul_left (exp_pos $ ε),\n  rw [← (exp_add _ _), add_neg_self, exp_zero],\n  rw [← mul_assoc, ← mul_lt_mul_right (exp_pos $ -r)],\n  rw [mul_assoc, ← (exp_add _ _), add_neg_self, exp_zero, mul_one, one_mul],\n  rw mul_div,\n  have qk_pos : 0 < q k, by exact probs_pos k,\n  rw [lt_div_iff (qk_pos), ← div_lt_iff _],\n  rw [← mul_div, mul_comm, ← log_lt_log_iff],\n  rwa log_exp,\n  rw zero_lt_mul_right (exp_pos $ -r),\n  rw [lt_div_iff, zero_mul],\n  simp, simp, apply exp_pos, simp,\nend\n\nlemma UA_rw :\n  UA q q₁ q₂ r ε = { k | log(q k / q₁ k * exp(-r)) < ε } :=\nby {ext, exact ⟨log_lt_of_lt_div_mul, lt_div_mul_of_log_lt⟩}\n\ninclude Hr\n\nlemma UTA_rw :\n  UTA q q₁ q₂ r ε = { k | abs(log(q k / q₁ k) - ∑ k, q k * log(q k / q₁ k)) ≥ ε } :=\nbegin\n  rw UTA, ext k, split,\n  intro H, norm_num at *, rwa [log_mul, log_exp, Hr] at H, simp, simp,\n  intro H, norm_num at *, rwa [log_mul, log_exp, Hr], simp, simp,\nend\n\nomit Hr\ninclude Herr_exp\n\nlemma UTB_rw :\n  UTB q q₁ q₂ r ε = { k | abs(log(q k / q₂ k) - ∑ k, q k * log(q k / q₂ k)) ≥ ε } :=\nbegin\n  rw UTB, ext k, split,\n  intro H, norm_num at H, rwa [Herr_exp, log_mul, log_exp] at H, simp, simp,\n  intro H, norm_num, rwa [Herr_exp, log_mul, log_exp], simp, simp,\nend\n\nomit Herr_exp\n\nlemma UB_rw :\n  UB q q₁ q₂ r ε = { k | log(q k / q₂ k * exp(-err_exp_1 q₁ q₂ r)) < ε } :=\nby {ext, exact ⟨log_lt_of_lt_div_mul, lt_div_mul_of_log_lt⟩}\n\nlemma UAc_subset_UTA :\n  (UA q q₁ q₂ r ε)ᶜ ⊆ (UTA q q₁ q₂ r ε) :=\nby {rw [UTA, UA_rw, gt_compl_le], apply subset_abs}\n\nlemma sum_UAc_le_sum_UTA :\n  ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) ≤ ∑ k, q k * (Φ (UTA q q₁ q₂ r ε) k) :=\nsum_le_of_subset q_pos (UAc_subset_UTA)\n\nvariables {σ₁ σ₂ : ℝ}\n(Hσ₁ : σ₁ = Var q (λk, log(q k / q₁ k)))\n(Hσ₂ : σ₂ = Var q (λk, log(q k / q₂ k)))\n\ninclude Hσ₁ Hr Hε\n\nlemma sum_UTA_le_var :\n  ∑ k, q k * (Φ (UTA q q₁ q₂ r ε) k) ≤ σ₁/(ε^2) := \nby {rw [UTA_rw Hr, Hσ₁], apply Chebyshevs_ineq, assumption}\n\nlemma sum_UAc_le_var :\n  ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) ≤ σ₁/(ε^2) :=\nle_trans sum_UAc_le_sum_UTA (sum_UTA_le_var Hr Hε Hσ₁)\n\nomit Hr Hε Hσ₁\n\nlemma UBc_subset_UTB :\n  (UB q q₁ q₂ r ε)ᶜ ⊆ (UTB q q₁ q₂ r ε) :=\nby {rw [UTB, UB_rw, gt_compl_le], apply subset_abs}\n\nlemma sum_UBc_le_sum_UTB :\n  ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k) ≤ ∑ k, q k * (Φ (UTB q q₁ q₂ r ε) k) :=\nsum_le_of_subset q_pos (UBc_subset_UTB)\n\ninclude Hσ₂ Herr_exp Hε\n\nlemma sum_UTB_le_var :\n  ∑ k, q k * (Φ (UTB q q₁ q₂ r ε) k) ≤ σ₂/(ε^2) := \nby {rw [UTB_rw, Hσ₂], apply Chebyshevs_ineq, assumption, exact Herr_exp}\n\nlemma sum_UBc_le_var  :\n  ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k) ≤ σ₂/(ε^2) :=\nle_trans sum_UBc_le_sum_UTB (sum_UTB_le_var Hε Herr_exp Hσ₂)\n\ninclude Hσ₁ Hr\n\nlemma sum_UA_inter_UB_ge :\n  ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k) ≥ 1 - (σ₁ + σ₂)/ε^2 :=\nbegin\n  calc ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k)\n          ≥ 1 - ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) - ∑ k, q k * (Φ (UB q q₁ q₂ r ε)ᶜ k) : sum_inter_ge\n      ... ≥ 1 - ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) - σ₂/(ε^2) : by apply sub_le_sub_left (sum_UBc_le_var Hε Herr_exp Hσ₂)\n      ... ≥ 1 - σ₂/(ε^2) - ∑ k, q k * (Φ (UA q q₁ q₂ r ε)ᶜ k) : by linarith\n      ... ≥ 1 - σ₂/(ε^2) - σ₁/(ε^2) : by apply sub_le_sub_left (sum_UAc_le_var Hr Hε Hσ₁)\n      ... ≥ 1 - (σ₁/(ε^2) + σ₂/(ε^2)) : by linarith\n      ... = 1 - (σ₁ + σ₂)/ε^2 : by simp only [add_div],\nend\n\nlemma sum_U2_ge_of_sum_U1_le {γ} :\n  ∑ k, q k * (Φ (U₁ q q₁ q₂ r ε) k) ≤ γ\n      → ∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k) + γ ≥ 1 - (σ₁ + σ₂)/ε^2 :=\nbegin\n  intro H,\n  calc ∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k) + γ\n            ≥ ∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k) + ∑ k, q k * (Φ (U₁ q q₁ q₂ r ε) k) : by apply add_le_add_left H\n        ... ≥ ∑ k, q k * (Φ ((U₂ q q₁ q₂ r ε) ∪ (U₁ q q₁ q₂ r ε)) k) : add_ge_sum_union q_pos\n        ... = ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k) : by rw [union_comm, U₁_union_U₂_eq_UA_inter_UB]\n        ... ≥ 1 - (σ₁ + σ₂)/ε^2 : sum_UA_inter_UB_ge Hr Hε Herr_exp Hσ₁ Hσ₂,\nend\n\ninclude HT\n\n/-- Thm. 10 in Blahut1974 -/\ntheorem prob_of_α_error_ge {γ > 0} : \n  β q₁ q₂ T ≤ γ * exp(-(r + ε))\n  → α q₁ q₂ T ≥ exp(-(err_exp_1 q₁ q₂ r + ε)) * (1 - (σ₁ + σ₂)/(ε^2) - γ) :=\nbegin\n  intros Hβ,\n  have Hγ : (∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k)) * exp(-(r + ε)) ≤ γ * exp(-(r + ε)), {\n    calc (∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k)) * exp(-(r + ε)) \n                ≤ ∑ k, q₁ k * (Φ (U₂ q q₁ q₂ r ε) k) : in_U₂\n            ... ≤ ∑ k, q₁ k * (Φ (U q₁ q₂ T)ᶜ k) : sum_le_of_subset q₁_pos (U₂_subset_Uc HT)\n            ... = β q₁ q₂ T : by rw β\n            ... ≤ γ * exp(-(r + ε)) : by assumption,\n  },\n  have γ_gt : ∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k) ≤ γ, {\n    exact (mul_le_mul_right (exp_pos (-(r + ε)))).mp Hγ,\n  },\n  have : α q₁ q₂ T * exp(err_exp_1 q₁ q₂ r + ε) + γ ≥ 1 - (σ₁ + σ₂)/ε^2, {\n    calc α q₁ q₂ T * exp(err_exp_1 q₁ q₂ r + ε) + γ\n               = (∑ k, q₂ k * (Φ (U q₁ q₂ T) k)) * exp(err_exp_1 q₁ q₂ r + ε) + γ : by rw α\n           ... ≥ (∑ k, q₂ k * (Φ (U₁ q q₁ q₂ r ε) k)) * exp(err_exp_1 q₁ q₂ r + ε) + γ : mul_add_ge_of_ge (exp_pos _) (sum_le_of_subset q₂_pos (U₁_subset_U HT))\n           ... ≥ ∑ k, q k * (Φ (U₁ q q₁ q₂ r ε) k) + γ : by apply add_le_add_right in_U₁\n           ... ≥ ∑ k, q k * (Φ (U₁ q q₁ q₂ r ε) k) + ∑ k, q k * (Φ (U₂ q q₁ q₂ r ε) k) : by apply add_le_add_left γ_gt\n           ... ≥ ∑ k, q k * (Φ ((U₁ q q₁ q₂ r ε) ∪ (U₂ q q₁ q₂ r ε)) k) : add_ge_sum_union q_pos\n           ... = ∑ k, q k * (Φ ((UA q q₁ q₂ r ε) ∩ (UB q q₁ q₂ r ε)) k) : by rw U₁_union_U₂_eq_UA_inter_UB\n           ... ≥ 1 - (σ₁ + σ₂)/(ε^2) : by {apply sum_UA_inter_UB_ge; assumption},\n  },\n  have : α q₁ q₂ T * exp(err_exp_1 q₁ q₂ r + ε) ≥ 1 - (σ₁ + σ₂)/ε^2 - γ, {\n    apply sub_le_iff_le_add.mpr this,\n  },\n  apply (mul_le_mul_right (exp_pos((err_exp_1 q₁ q₂ r + ε)))).mp _,\n  rw [mul_comm, ← mul_assoc, ← exp_add _ _, neg_add, ← add_assoc],\n  rw [add_assoc, add_assoc, add_comm, add_comm (-err_exp_1 q₁ q₂ r) (-ε), ← add_assoc],\n  rw [add_neg_self, zero_add, add_comm, add_neg_self, exp_zero, one_mul], exact this,\nend", "meta": {"author": "BassemSafieldeen", "repo": "Distinguish_probability_distributions", "sha": "fd21d317bdc8c9292de762ccb46cf1f519463268", "save_path": "github-repos/lean/BassemSafieldeen-Distinguish_probability_distributions", "path": "github-repos/lean/BassemSafieldeen-Distinguish_probability_distributions/Distinguish_probability_distributions-fd21d317bdc8c9292de762ccb46cf1f519463268/src/theorem10.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3197155792496733}}
{"text": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport ring_theory.adjoin\nimport ring_theory.algebra_tower\nimport ring_theory.polynomial.scale_roots\n\n/-!\n# Integral closure of a subring.\n\nIf A is an R-algebra then `a : A` is integral over R if it is a root of a monic polynomial\nwith coefficients in R. Enough theory is developed to prove that integral elements\nform a sub-R-algebra of A.\n\n## Main definitions\n\nLet `R` be a `comm_ring` and let `A` be an R-algebra.\n\n* `ring_hom.is_integral_elem (f : R →+* A) (x : A)` : `x` is integral with respect to the map `f`,\n\n* `is_integral (x : A)`  : `x` is integral over `R`, i.e., is a root of a monic polynomial with\n                           coefficients in `R`.\n* `integral_closure R A` : the integral closure of `R` in `A`, regarded as a sub-`R`-algebra of `A`.\n-/\n\nopen_locale classical\nopen_locale big_operators\nopen polynomial submodule\n\nsection ring\nvariables {R S A : Type*}\nvariables [comm_ring R] [ring A] [ring S] (f : R →+* S)\n\n/-- An element `x` of `A` is said to be integral over `R` with respect to `f`\nif it is a root of a monic polynomial `p : polynomial R` evaluated under `f` -/\ndef ring_hom.is_integral_elem (f : R →+* A) (x : A) :=\n∃ p : polynomial R, monic p ∧ eval₂ f x p = 0\n\n/-- A ring homomorphism `f : R →+* A` is said to be integral\nif every element `A` is integral with respect to the map `f` -/\ndef ring_hom.is_integral (f : R →+* A) :=\n∀ x : A, f.is_integral_elem x\n\nvariables [algebra R A] (R)\n\n/-- An element `x` of an algebra `A` over a commutative ring `R` is said to be *integral*,\nif it is a root of some monic polynomial `p : polynomial R`.\nEquivalently, the element is integral over `R` with respect to the induced `algebra_map` -/\ndef is_integral (x : A) : Prop :=\n(algebra_map R A).is_integral_elem x\n\nvariable (A)\n\n/-- An algebra is integral if every element of the extension is integral over the base ring -/\ndef algebra.is_integral : Prop :=\n(algebra_map R A).is_integral\n\nvariables {R A}\n\nlemma ring_hom.is_integral_map {x : R} : f.is_integral_elem (f x) :=\n⟨X - C x, monic_X_sub_C _, by simp⟩\n\ntheorem is_integral_algebra_map {x : R} : is_integral R (algebra_map R A x) :=\n(algebra_map R A).is_integral_map\n\ntheorem is_integral_of_noetherian (H : is_noetherian R A) (x : A) :\n  is_integral R x :=\nbegin\n  let leval : @linear_map R (polynomial R) A _ _ _ _ _ := (aeval x).to_linear_map,\n  let D : ℕ → submodule R A := λ n, (degree_le R n).map leval,\n  let M := well_founded.min (is_noetherian_iff_well_founded.1 H)\n    (set.range D) ⟨_, ⟨0, rfl⟩⟩,\n  have HM : M ∈ set.range D := well_founded.min_mem _ _ _,\n  cases HM with N HN,\n  have HM : ¬M < D (N+1) := well_founded.not_lt_min\n    (is_noetherian_iff_well_founded.1 H) (set.range D) _ ⟨N+1, rfl⟩,\n  rw ← HN at HM,\n  have HN2 : D (N+1) ≤ D N := classical.by_contradiction (λ H, HM\n    (lt_of_le_not_le (map_mono (degree_le_mono\n      (with_bot.coe_le_coe.2 (nat.le_succ N)))) H)),\n  have HN3 : leval (X^(N+1)) ∈ D N,\n  { exact HN2 (mem_map_of_mem (mem_degree_le.2 (degree_X_pow_le _))) },\n  rcases HN3 with ⟨p, hdp, hpe⟩,\n  refine ⟨X^(N+1) - p, monic_X_pow_sub (mem_degree_le.1 hdp), _⟩,\n  show leval (X ^ (N + 1) - p) = 0,\n  rw [linear_map.map_sub, hpe, sub_self]\nend\n\ntheorem is_integral_of_submodule_noetherian (S : subalgebra R A)\n  (H : is_noetherian R (S : submodule R A)) (x : A) (hx : x ∈ S) :\n  is_integral R x :=\nbegin\n  letI : algebra R S := S.algebra,\n  letI : ring S := S.ring R A,\n  suffices : is_integral R (⟨x, hx⟩ : S),\n  { rcases this with ⟨p, hpm, hpx⟩,\n    replace hpx := congr_arg subtype.val hpx,\n    refine ⟨p, hpm, eq.trans _ hpx⟩,\n    simp only [aeval_def, eval₂, finsupp.sum],\n    rw ← p.support.sum_hom subtype.val,\n    { refine finset.sum_congr rfl (λ n hn, _),\n      change _ = _ * _,\n      rw is_monoid_hom.map_pow coe, refl,\n      split; intros; refl },\n    refine { map_add := _, map_zero := _ }; intros; refl },\n  refine is_integral_of_noetherian H ⟨x, hx⟩\nend\n\nend ring\n\nsection\nvariables {R A B S : Type*}\nvariables [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring S]\nvariables [algebra R A] [algebra R B] (f : R →+* S)\n\ntheorem is_integral_alg_hom (f : A →ₐ[R] B) {x : A} (hx : is_integral R x) : is_integral R (f x) :=\nlet ⟨p, hp, hpx⟩ := hx in ⟨p, hp, by rw [← aeval_def, aeval_alg_hom_apply, aeval_def, hpx, f.map_zero]⟩\n\ntheorem is_integral_of_is_scalar_tower [algebra A B] [is_scalar_tower R A B]\n  (x : B) (hx : is_integral R x) : is_integral A x :=\nlet ⟨p, hp, hpx⟩ := hx in\n⟨p.map $ algebra_map R A, monic_map _ hp, by rw [← aeval_def, ← is_scalar_tower.aeval_apply, aeval_def, hpx]⟩\n\nsection\nlocal attribute [instance] subset.comm_ring algebra.of_is_subring\n\ntheorem is_integral_of_subring {x : A} (T : set R) [is_subring T]\n  (hx : is_integral T x) : is_integral R x :=\nis_integral_of_is_scalar_tower x hx\n\nlemma is_integral_algebra_map_iff [algebra A B] [is_scalar_tower R A B]\n  {x : A} (hAB : function.injective (algebra_map A B)) :\n  is_integral R (algebra_map A B x) ↔ is_integral R x :=\nbegin\n  split; rintros ⟨f, hf, hx⟩; use [f, hf],\n  { exact is_scalar_tower.aeval_eq_zero_of_aeval_algebra_map_eq_zero R A B hAB hx },\n  { rw [is_scalar_tower.algebra_map_eq R A B, ← hom_eval₂, hx, ring_hom.map_zero] }\nend\n\ntheorem is_integral_iff_is_integral_closure_finite {r : A} :\n  is_integral R r ↔ ∃ s : set R, s.finite ∧ is_integral (ring.closure s) r :=\nbegin\n  split; intro hr,\n  { rcases hr with ⟨p, hmp, hpr⟩,\n    refine ⟨_, set.finite_mem_finset _, p.restriction, subtype.eq hmp, _⟩,\n    erw [← aeval_def, is_scalar_tower.aeval_apply _ R, map_restriction, aeval_def, hpr] },\n  rcases hr with ⟨s, hs, hsr⟩,\n  exact is_integral_of_subring _ hsr\nend\n\nend\n\ntheorem fg_adjoin_singleton_of_integral (x : A) (hx : is_integral R x) :\n  (algebra.adjoin R ({x} : set A) : submodule R A).fg :=\nbegin\n  rcases hx with ⟨f, hfm, hfx⟩,\n  existsi finset.image ((^) x) (finset.range (nat_degree f + 1)),\n  apply le_antisymm,\n  { rw span_le, intros s hs, rw finset.mem_coe at hs,\n    rcases finset.mem_image.1 hs with ⟨k, hk, rfl⟩, clear hk,\n    exact is_submonoid.pow_mem (algebra.subset_adjoin (set.mem_singleton _)) },\n  intros r hr, change r ∈ algebra.adjoin R ({x} : set A) at hr,\n  rw algebra.adjoin_singleton_eq_range at hr,\n  rcases (aeval x).mem_range.mp hr with ⟨p, rfl⟩,\n  rw ← mod_by_monic_add_div p hfm,\n  rw ← aeval_def at hfx,\n  rw [alg_hom.map_add, alg_hom.map_mul, hfx, zero_mul, add_zero],\n  have : degree (p %ₘ f) ≤ degree f := degree_mod_by_monic_le p hfm,\n  generalize_hyp : p %ₘ f = q at this ⊢,\n  rw [← sum_C_mul_X_eq q, aeval_def, eval₂_sum, finsupp.sum],\n  refine sum_mem _ (λ k hkq, _),\n  rw [eval₂_mul, eval₂_C, eval₂_pow, eval₂_X, ← algebra.smul_def],\n  refine smul_mem _ _ (subset_span _),\n  rw finset.mem_coe, refine finset.mem_image.2 ⟨_, _, rfl⟩,\n  rw [finset.mem_range, nat.lt_succ_iff], refine le_of_not_lt (λ hk, _),\n  rw [degree_le_iff_coeff_zero] at this,\n  rw [finsupp.mem_support_iff] at hkq, apply hkq, apply this,\n  exact lt_of_le_of_lt degree_le_nat_degree (with_bot.coe_lt_coe.2 hk)\nend\n\ntheorem fg_adjoin_of_finite {s : set A} (hfs : s.finite)\n  (his : ∀ x ∈ s, is_integral R x) : (algebra.adjoin R s : submodule R A).fg :=\nset.finite.induction_on hfs (λ _, ⟨{1}, submodule.ext $ λ x,\n  by { erw [algebra.adjoin_empty, finset.coe_singleton, ← one_eq_span, one_eq_map_top,\n      map_top, linear_map.mem_range, algebra.mem_bot], refl }⟩)\n(λ a s has hs ih his, by rw [← set.union_singleton, algebra.adjoin_union_coe_submodule]; exact\n  fg_mul _ _ (ih $ λ i hi, his i $ set.mem_insert_of_mem a hi)\n    (fg_adjoin_singleton_of_integral _ $ his a $ set.mem_insert a s)) his\n\ntheorem is_integral_of_mem_of_fg (S : subalgebra R A)\n  (HS : (S : submodule R A).fg) (x : A) (hx : x ∈ S) : is_integral R x :=\nbegin\n  cases HS with y hy,\n  obtain ⟨lx, hlx1, hlx2⟩ :\n    ∃ (l : A →₀ R) (H : l ∈ finsupp.supported R R ↑y), (finsupp.total A A R id) l = x,\n  { rwa [←(@finsupp.mem_span_iff_total A A R _ _ _ id ↑y x), set.image_id ↑y, hy] },\n  have hyS : ∀ {p}, p ∈ y → p ∈ S := λ p hp, show p ∈ (S : submodule R A),\n    by { rw ← hy, exact subset_span hp },\n  have : ∀ (jk : (↑(y.product y) : set (A × A))), jk.1.1 * jk.1.2 ∈ (S : submodule R A) :=\n    λ jk, S.mul_mem (hyS (finset.mem_product.1 jk.2).1) (hyS (finset.mem_product.1 jk.2).2),\n  rw [← hy, ← set.image_id ↑y] at this, simp only [finsupp.mem_span_iff_total] at this,\n  choose ly hly1 hly2,\n  let S₀ : set R := ring.closure ↑(lx.frange ∪ finset.bUnion finset.univ (finsupp.frange ∘ ly)),\n  refine is_integral_of_subring S₀ _,\n  letI : comm_ring S₀ := @subtype.comm_ring _ _ _ ring.closure.is_subring,\n  letI : algebra S₀ A := algebra.of_is_subring _,\n  have : span S₀ (insert 1 ↑y : set A) * span S₀ (insert 1 ↑y : set A) ≤ span S₀ (insert 1 ↑y : set A),\n  { rw span_mul_span, refine span_le.2 (λ z hz, _),\n    rcases set.mem_mul.1 hz with ⟨p, q, rfl | hp, hq, rfl⟩,\n    { rw one_mul, exact subset_span hq },\n    rcases hq with rfl | hq,\n    { rw mul_one, exact subset_span (or.inr hp) },\n    erw ← hly2 ⟨(p, q), finset.mem_product.2 ⟨hp, hq⟩⟩,\n    rw [finsupp.total_apply, finsupp.sum],\n    refine (span S₀ (insert 1 ↑y : set A)).sum_mem (λ t ht, _),\n    have : ly ⟨(p, q), finset.mem_product.2 ⟨hp, hq⟩⟩ t ∈ S₀ :=\n    ring.subset_closure (finset.mem_union_right _ $ finset.mem_bUnion.2\n      ⟨⟨(p, q), finset.mem_product.2 ⟨hp, hq⟩⟩, finset.mem_univ _,\n        finsupp.mem_frange.2 ⟨finsupp.mem_support_iff.1 ht, _, rfl⟩⟩),\n    change (⟨_, this⟩ : S₀) • t ∈ _, exact smul_mem _ _ (subset_span $ or.inr $ hly1 _ ht) },\n  haveI : is_subring (span S₀ (insert 1 ↑y : set A) : set A) :=\n  { one_mem := subset_span $ or.inl rfl,\n    mul_mem := λ p q hp hq, this $ mul_mem_mul hp hq,\n    zero_mem := (span S₀ (insert 1 ↑y : set A)).zero_mem,\n    add_mem := λ _ _, (span S₀ (insert 1 ↑y : set A)).add_mem,\n    neg_mem := λ _, (span S₀ (insert 1 ↑y : set A)).neg_mem },\n  have : span S₀ (insert 1 ↑y : set A) = algebra.adjoin S₀ (↑y : set A),\n  { refine le_antisymm (span_le.2 $ set.insert_subset.2\n        ⟨(algebra.adjoin S₀ ↑y).one_mem, algebra.subset_adjoin⟩) (λ z hz, _),\n    rw [subalgebra.mem_to_submodule, algebra.mem_adjoin_iff] at hz, rw ← submodule.mem_coe,\n    refine ring.closure_subset (set.union_subset (set.range_subset_iff.2 $ λ t, _)\n      (λ t ht, subset_span $ or.inr ht)) hz,\n    rw algebra.algebra_map_eq_smul_one,\n    exact smul_mem (span S₀ (insert 1 ↑y : set A)) _ (subset_span $ or.inl rfl) },\n  haveI : is_noetherian_ring ↥S₀ := is_noetherian_ring_closure _ (finset.finite_to_set _),\n  refine is_integral_of_submodule_noetherian (algebra.adjoin S₀ ↑y)\n    (is_noetherian_of_fg_of_noetherian _ ⟨insert 1 y, by rw [finset.coe_insert, this]⟩) _ _,\n  rw [← hlx2, finsupp.total_apply, finsupp.sum], refine subalgebra.sum_mem _ (λ r hr, _),\n  have : lx r ∈ S₀ := ring.subset_closure (finset.mem_union_left _ (finset.mem_image_of_mem _ hr)),\n  change (⟨_, this⟩ : S₀) • r ∈ _,\n  rw finsupp.mem_supported at hlx1,\n  exact subalgebra.smul_mem _ (algebra.subset_adjoin $ hlx1 hr) _\nend\n\nlemma ring_hom.is_integral_of_mem_closure {x y z : S}\n  (hx : f.is_integral_elem x) (hy : f.is_integral_elem y)\n  (hz : z ∈ ring.closure ({x, y} : set S)) :\n  f.is_integral_elem z :=\nbegin\n  letI : algebra R S := f.to_algebra,\n  have := fg_mul _ _ (fg_adjoin_singleton_of_integral x hx) (fg_adjoin_singleton_of_integral y hy),\n  rw [← algebra.adjoin_union_coe_submodule, set.singleton_union] at this,\n  exact is_integral_of_mem_of_fg (algebra.adjoin R {x, y}) this z\n    (algebra.mem_adjoin_iff.2  $ ring.closure_mono (set.subset_union_right _ _) hz),\nend\n\ntheorem is_integral_of_mem_closure {x y z : A}\n  (hx : is_integral R x) (hy : is_integral R y)\n  (hz : z ∈ ring.closure ({x, y} : set A)) :\n  is_integral R z :=\n(algebra_map R A).is_integral_of_mem_closure hx hy hz\n\nlemma ring_hom.is_integral_zero : f.is_integral_elem 0 :=\nf.map_zero ▸ f.is_integral_map\n\ntheorem is_integral_zero : is_integral R (0:A) :=\n(algebra_map R A).is_integral_zero\n\nlemma ring_hom.is_integral_one : f.is_integral_elem 1 :=\nf.map_one ▸ f.is_integral_map\n\ntheorem is_integral_one : is_integral R (1:A) :=\n(algebra_map R A).is_integral_one\n\nlemma ring_hom.is_integral_add {x y : S}\n  (hx : f.is_integral_elem x) (hy : f.is_integral_elem y) :\n  f.is_integral_elem (x + y) :=\nf.is_integral_of_mem_closure hx hy (is_add_submonoid.add_mem\n  (ring.subset_closure (or.inl rfl)) (ring.subset_closure (or.inr rfl)))\n\ntheorem is_integral_add {x y : A}\n  (hx : is_integral R x) (hy : is_integral R y) :\n  is_integral R (x + y) :=\n(algebra_map R A).is_integral_add hx hy\n\nlemma ring_hom.is_integral_neg {x : S}\n  (hx : f.is_integral_elem x) : f.is_integral_elem (-x) :=\nf.is_integral_of_mem_closure hx hx (is_add_subgroup.neg_mem\n  (ring.subset_closure (or.inl rfl)))\n\ntheorem is_integral_neg {x : A}\n  (hx : is_integral R x) : is_integral R (-x) :=\n(algebra_map R A).is_integral_neg hx\n\nlemma ring_hom.is_integral_sub {x y : S}\n  (hx : f.is_integral_elem x) (hy : f.is_integral_elem y) : f.is_integral_elem (x - y) :=\nby simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)\n\ntheorem is_integral_sub {x y : A}\n  (hx : is_integral R x) (hy : is_integral R y) : is_integral R (x - y) :=\n(algebra_map R A).is_integral_sub hx hy\n\nlemma ring_hom.is_integral_mul {x y : S}\n  (hx : f.is_integral_elem x) (hy : f.is_integral_elem y) : f.is_integral_elem (x * y) :=\nf.is_integral_of_mem_closure hx hy (is_submonoid.mul_mem\n  (ring.subset_closure (or.inl rfl)) (ring.subset_closure (or.inr rfl)))\n\ntheorem is_integral_mul {x y : A}\n  (hx : is_integral R x) (hy : is_integral R y) : is_integral R (x * y) :=\n(algebra_map R A).is_integral_mul hx hy\n\ntheorem ring_hom.is_integral_pow {x : S} :\n  Π (n : ℕ) (hx : f.is_integral_elem x), f.is_integral_elem (x ^ n)\n| 0 hx := by simpa using f.is_integral_one\n| (n + 1) hx := by simpa using f.is_integral_mul hx (ring_hom.is_integral_pow n hx)\n\ntheorem is_integral_pow {x : A} (n : ℕ) (hx : is_integral R x) : is_integral R (x ^ n) :=\n(algebra_map R A).is_integral_pow n hx\n\nvariables (R A)\n\n/-- The integral closure of R in an R-algebra A. -/\ndef integral_closure : subalgebra R A :=\n{ carrier := { r | is_integral R r },\n  zero_mem' := is_integral_zero,\n  one_mem' := is_integral_one,\n  add_mem' := λ _ _, is_integral_add,\n  mul_mem' := λ _ _, is_integral_mul,\n  algebra_map_mem' := λ x, is_integral_algebra_map }\n\ntheorem mem_integral_closure_iff_mem_fg {r : A} :\n  r ∈ integral_closure R A ↔ ∃ M : subalgebra R A, (M : submodule R A).fg ∧ r ∈ M :=\n⟨λ hr, ⟨algebra.adjoin R {r}, fg_adjoin_singleton_of_integral _ hr, algebra.subset_adjoin rfl⟩,\nλ ⟨M, Hf, hrM⟩, is_integral_of_mem_of_fg M Hf _ hrM⟩\n\nvariables {R} {A}\n\n/-- Mapping an integral closure along an `alg_equiv` gives the integral closure. -/\nlemma integral_closure_map_alg_equiv (f : A ≃ₐ[R] B) :\n  (integral_closure R A).map (f : A →ₐ[R] B) = integral_closure R B :=\nbegin\n  ext y,\n  rw subalgebra.mem_map,\n  split,\n  { rintros ⟨x, hx, rfl⟩,\n    exact is_integral_alg_hom f hx },\n  { intro hy,\n    use [f.symm y, is_integral_alg_hom (f.symm : B →ₐ[R] A) hy],\n    simp }\nend\n\nlemma integral_closure.is_integral (x : integral_closure R A) : is_integral R x :=\nlet ⟨p, hpm, hpx⟩ := x.2 in ⟨p, hpm, subtype.eq $\nby rwa [← aeval_def, subtype.val_eq_coe, ← subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩\n\nlemma ring_hom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)\n  (hx : f.is_integral_elem (x * y)) : f.is_integral_elem x :=\nbegin\n  obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx,\n  refine ⟨scale_roots p r, ⟨(monic_scale_roots_iff r).2 p_monic, _⟩⟩,\n  convert scale_roots_eval₂_eq_zero f hp,\n  rw [mul_comm x y, ← mul_assoc, hr, one_mul],\nend\n\ntheorem is_integral_of_is_integral_mul_unit {x y : A} {r : R} (hr : algebra_map R A r * y = 1)\n  (hx : is_integral R (x * y)) : is_integral R x :=\n(algebra_map R A).is_integral_of_is_integral_mul_unit x y r hr hx\n\n/-- Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma -/\nlemma is_integral_of_mem_closure' (G : set A) (hG : ∀ x ∈ G, is_integral R x) :\n  ∀ x ∈ (subring.closure G), is_integral R x :=\nλ x hx, subring.closure_induction hx hG is_integral_zero is_integral_one\n  (λ _ _, is_integral_add) (λ _, is_integral_neg) (λ _ _, is_integral_mul)\n\nlemma is_integral_of_mem_closure'' {S : Type*} [comm_ring S] {f : R →+* S} (G : set S)\n  (hG : ∀ x ∈ G, f.is_integral_elem x) : ∀ x ∈ (subring.closure G), f.is_integral_elem x :=\nλ x hx, @is_integral_of_mem_closure' R S _ _ f.to_algebra G hG x hx\n\nlemma algebra_map_injective (h : function.injective (algebra_map R A)) :\n  function.injective (algebra_map R (integral_closure R A)) :=\nλ x y hxy, h $\n  show algebra_map (integral_closure R A) A (algebra_map R _ x) = _,\n  from congr_arg (algebra_map (integral_closure R A) A) hxy\n\nend\n\nsection algebra\nopen algebra\nvariables {R A B S T : Type*}\nvariables [comm_ring R] [comm_ring A] [comm_ring B] [comm_ring S] [comm_ring T]\nvariables [algebra A B] [algebra R B] (f : R →+* S) (g : S →+* T)\n\nlemma is_integral_trans_aux (x : B) {p : polynomial A} (pmonic : monic p) (hp : aeval x p = 0) :\n  is_integral (adjoin R (↑(p.map $ algebra_map A B).frange : set B)) x :=\nbegin\n  generalize hS : (↑(p.map $ algebra_map A B).frange : set B) = S,\n  have coeffs_mem : ∀ i, (p.map $ algebra_map A B).coeff i ∈ adjoin R S,\n  { intro i, by_cases hi : (p.map $ algebra_map A B).coeff i = 0,\n    { rw hi, exact subalgebra.zero_mem _ },\n    rw ← hS, exact subset_adjoin (finsupp.mem_frange.2 ⟨hi, i, rfl⟩) },\n  obtain ⟨q, hq⟩ : ∃ q : polynomial (adjoin R S), q.map (algebra_map (adjoin R S) B) =\n      (p.map $ algebra_map A B),\n  { rw ← set.mem_range, exact (polynomial.mem_map_range _).2 (λ i, ⟨⟨_, coeffs_mem i⟩, rfl⟩) },\n  use q,\n  split,\n  { suffices h : (q.map (algebra_map (adjoin R S) B)).monic,\n    { refine monic_of_injective _ h,\n      exact subtype.val_injective },\n    { rw hq, exact monic_map _ pmonic } },\n  { convert hp using 1,\n    replace hq := congr_arg (eval x) hq,\n    convert hq using 1; symmetry; apply eval_map },\nend\n\nvariables [algebra R A] [is_scalar_tower R A B]\n\n/-- If A is an R-algebra all of whose elements are integral over R,\nand x is an element of an A-algebra that is integral over A, then x is integral over R.-/\nlemma is_integral_trans (A_int : is_integral R A) (x : B) (hx : is_integral A x) :\n  is_integral R x :=\nbegin\n  rcases hx with ⟨p, pmonic, hp⟩,\n  let S : set B := ↑(p.map $ algebra_map A B).frange,\n  refine is_integral_of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin $ or.inr rfl),\n  refine fg_trans (fg_adjoin_of_finite (finset.finite_to_set _) (λ x hx, _)) _,\n  { rw [finset.mem_coe, finsupp.mem_frange] at hx, rcases hx with ⟨_, i, rfl⟩,\n    show is_integral R ((p.map $ algebra_map A B).coeff i), rw coeff_map,\n    convert is_integral_alg_hom (is_scalar_tower.to_alg_hom R A B) (A_int _) },\n  { apply fg_adjoin_singleton_of_integral,\n    exact is_integral_trans_aux _ pmonic hp }\nend\n\n/-- If A is an R-algebra all of whose elements are integral over R,\nand B is an A-algebra all of whose elements are integral over A,\nthen all elements of B are integral over R.-/\nlemma algebra.is_integral_trans (hA : is_integral R A) (hB : is_integral A B) : is_integral R B :=\nλ x, is_integral_trans hA x (hB x)\n\nlemma ring_hom.is_integral_trans (hf : f.is_integral) (hg : g.is_integral) :\n  (g.comp f).is_integral :=\n@algebra.is_integral_trans R S T _ _ _ g.to_algebra (g.comp f).to_algebra f.to_algebra\n  (@is_scalar_tower.of_algebra_map_eq R S T _ _ _ f.to_algebra g.to_algebra (g.comp f).to_algebra\n  (ring_hom.comp_apply g f)) hf hg\n\nlemma ring_hom.is_integral_of_surjective (hf : function.surjective f) : f.is_integral :=\nλ x, (hf x).rec_on (λ y hy, (hy ▸ f.is_integral_map : f.is_integral_elem x))\n\nlemma is_integral_of_surjective (h : function.surjective (algebra_map R A)) : is_integral R A :=\n(algebra_map R A).is_integral_of_surjective h\n\n/-- If `R → A → B` is an algebra tower with `A → B` injective,\nthen if the entire tower is an integral extension so is `R → A` -/\nlemma is_integral_tower_bot_of_is_integral (H : function.injective (algebra_map A B))\n  {x : A} (h : is_integral R (algebra_map A B x)) : is_integral R x :=\nbegin\n  rcases h with ⟨p, ⟨hp, hp'⟩⟩,\n  refine ⟨p, ⟨hp, _⟩⟩,\n  rw [is_scalar_tower.algebra_map_eq R A B, ← eval₂_map,\n      eval₂_hom, ← ring_hom.map_zero (algebra_map A B)] at hp',\n  rw [eval₂_eq_eval_map],\n  exact H hp',\nend\n\nlemma ring_hom.is_integral_tower_bot_of_is_integral (hg : function.injective g)\n  (hfg : (g.comp f).is_integral) : f.is_integral :=\nλ x, @is_integral_tower_bot_of_is_integral R S T _ _ _ g.to_algebra (g.comp f).to_algebra f.to_algebra\n  (@is_scalar_tower.of_algebra_map_eq R S T _ _ _ f.to_algebra g.to_algebra (g.comp f).to_algebra\n  (ring_hom.comp_apply g f))  hg x (hfg (g x))\n\nlemma is_integral_tower_bot_of_is_integral_field {R A B : Type*} [comm_ring R] [field A]\n  [comm_ring B] [nontrivial B] [algebra R A] [algebra A B] [algebra R B] [is_scalar_tower R A B]\n  {x : A} (h : is_integral R (algebra_map A B x)) : is_integral R x :=\nis_integral_tower_bot_of_is_integral (algebra_map A B).injective h\n\nlemma ring_hom.is_integral_elem_of_is_integral_elem_comp {x : T}\n  (h : (g.comp f).is_integral_elem x) : g.is_integral_elem x :=\nlet ⟨p, ⟨hp, hp'⟩⟩ := h in ⟨p.map f, monic_map f hp, by rwa ← eval₂_map at hp'⟩\n\nlemma ring_hom.is_integral_tower_top_of_is_integral (h : (g.comp f).is_integral) : g.is_integral :=\nλ x, ring_hom.is_integral_elem_of_is_integral_elem_comp f g (h x)\n\n/-- If `R → A → B` is an algebra tower,\nthen if the entire tower is an integral extension so is `A → B`. -/\nlemma is_integral_tower_top_of_is_integral {x : B} (h : is_integral R x) : is_integral A x :=\nbegin\n  rcases h with ⟨p, ⟨hp, hp'⟩⟩,\n  refine ⟨p.map (algebra_map R A), ⟨monic_map (algebra_map R A) hp, _⟩⟩,\n  rw [is_scalar_tower.algebra_map_eq R A B, ← eval₂_map] at hp',\n  exact hp',\nend\n\nlemma ring_hom.is_integral_quotient_of_is_integral {I : ideal S} (hf : f.is_integral) :\n  (ideal.quotient_map I f le_rfl).is_integral :=\nbegin\n  rintros ⟨x⟩,\n  obtain ⟨p, ⟨p_monic, hpx⟩⟩ := hf x,\n  refine ⟨p.map (ideal.quotient.mk _), ⟨monic_map _ p_monic, _⟩⟩,\n  simpa only [hom_eval₂, eval₂_map] using congr_arg (ideal.quotient.mk I) hpx\nend\n\nlemma is_integral_quotient_of_is_integral {I : ideal A} (hRA : is_integral R A) :\n  is_integral (I.comap (algebra_map R A)).quotient I.quotient :=\n(algebra_map R A).is_integral_quotient_of_is_integral hRA\n\nlemma is_integral_quotient_map_iff {I : ideal S} :\n  (ideal.quotient_map I f le_rfl).is_integral ↔\n    ((ideal.quotient.mk I).comp f : R →+* I.quotient).is_integral :=\nbegin\n  let g := ideal.quotient.mk (I.comap f),\n  have := ideal.quotient_map_comp_mk le_rfl,\n  refine ⟨λ h, _, λ h, ring_hom.is_integral_tower_top_of_is_integral g _ (this ▸ h)⟩,\n  refine this ▸ ring_hom.is_integral_trans g (ideal.quotient_map I f le_rfl) _ h,\n  exact ring_hom.is_integral_of_surjective g ideal.quotient.mk_surjective,\nend\n\n/-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/\nlemma is_field_of_is_integral_of_is_field {R S : Type*} [integral_domain R] [integral_domain S]\n  [algebra R S] (H : is_integral R S) (hRS : function.injective (algebra_map R S))\n  (hS : is_field S) : is_field R :=\nbegin\n  refine ⟨⟨0, 1, zero_ne_one⟩, mul_comm, λ a ha, _⟩,\n  -- Let `a_inv` be the inverse of `algebra_map R S a`,\n  -- then we need to show that `a_inv` is of the form `algebra_map R S b`.\n  obtain ⟨a_inv, ha_inv⟩ := hS.mul_inv_cancel (λ h, ha (hRS (trans h (ring_hom.map_zero _).symm))),\n\n  -- Let `p : polynomial R` be monic with root `a_inv`,\n  -- and `q` be `p` with coefficients reversed (so `q(a) = q'(a) * a + 1`).\n  -- We claim that `q(a) = 0`, so `-q'(a)` is the inverse of `a`.\n  obtain ⟨p, p_monic, hp⟩ := H a_inv,\n  use -∑ (i : ℕ) in finset.range p.nat_degree, (p.coeff i) * a ^ (p.nat_degree - i - 1),\n\n  -- `q(a) = 0`, because multiplying everything with `a_inv^n` gives `p(a_inv) = 0`.\n  -- TODO: this could be a lemma for `polynomial.reverse`.\n  have hq : ∑ (i : ℕ) in finset.range (p.nat_degree + 1), (p.coeff i) * a ^ (p.nat_degree - i) = 0,\n  { apply (algebra_map R S).injective_iff.mp hRS,\n    have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero),\n    refine (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.nat_degree a_inv_ne_zero),\n    rw [eval₂_eq_sum_range] at hp,\n    rw [ring_hom.map_sum, finset.sum_mul],\n    refine (finset.sum_congr rfl (λ i hi, _)).trans hp,\n    rw [ring_hom.map_mul, mul_assoc],\n    congr,\n    have : a_inv ^ p.nat_degree = a_inv ^ (p.nat_degree - i) * a_inv ^ i,\n    { rw [← pow_add a_inv, nat.sub_add_cancel (nat.le_of_lt_succ (finset.mem_range.mp hi))] },\n    rw [ring_hom.map_pow, this, ← mul_assoc, ← mul_pow, ha_inv, one_pow, one_mul] },\n\n  -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.\n  -- TODO: we could use a lemma for `polynomial.div_X` here.\n  rw [finset.sum_range_succ, p_monic.coeff_nat_degree, one_mul, nat.sub_self, pow_zero,\n      add_eq_zero_iff_eq_neg, eq_comm] at hq,\n  rw [mul_comm, ← neg_mul_eq_neg_mul, finset.sum_mul],\n  convert hq using 2,\n  refine finset.sum_congr rfl (λ i hi, _),\n  have : 1 ≤ p.nat_degree - i := nat.le_sub_left_of_add_le (finset.mem_range.mp hi),\n  rw [mul_assoc, ← pow_succ', nat.sub_add_cancel this]\nend\n\nend algebra\n\nsection\nlocal attribute [instance] subset.comm_ring algebra.of_is_subring\ntheorem integral_closure_idem {R : Type*} {A : Type*} [comm_ring R] [comm_ring A] [algebra R A] :\n  integral_closure (integral_closure R A : set A) A = ⊥ :=\neq_bot_iff.2 $ λ x hx, algebra.mem_bot.2\n⟨⟨x, @is_integral_trans _ _ _ _ _ _ _ _ (integral_closure R A).algebra\n     _ integral_closure.is_integral x hx⟩, rfl⟩\nend\n\nsection integral_domain\nvariables {R S : Type*} [comm_ring R] [integral_domain S] [algebra R S]\n\ninstance : integral_domain (integral_closure R S) :=\n{ exists_pair_ne := ⟨0, 1, mt subtype.ext_iff_val.mp zero_ne_one⟩,\n  eq_zero_or_eq_zero_of_mul_eq_zero := λ ⟨a, ha⟩ ⟨b, hb⟩ h,\n    or.imp subtype.ext_iff_val.mpr subtype.ext_iff_val.mpr (eq_zero_or_eq_zero_of_mul_eq_zero (subtype.ext_iff_val.mp h)),\n  ..(integral_closure R S).comm_ring R S }\n\nend integral_domain\n", "meta": {"author": "lean-forward", "repo": "class-number", "sha": "812ff19e6fbde86f8d71689851adaa2bbae9695e", "save_path": "github-repos/lean/lean-forward-class-number", "path": "github-repos/lean/lean-forward-class-number/class-number-812ff19e6fbde86f8d71689851adaa2bbae9695e/src/integral_closure.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.31968628695322804}}
{"text": "import UniverseAbstractions.Axioms.Universes\nimport UniverseAbstractions.Axioms.Universe.Identity\nimport UniverseAbstractions.Axioms.Universe.Functors\nimport UniverseAbstractions.Axioms.Universe.Singletons\nimport UniverseAbstractions.Axioms.Universe.Products\nimport UniverseAbstractions.Axioms.Universe.Equivalences\nimport UniverseAbstractions.Instances.Utils.Bundled\nimport UniverseAbstractions.Instances.Utils.Trivial\n\nimport UniverseAbstractions.MathlibFragments.Init.CoreExt\n\n\n\nset_option autoBoundImplicitLocal false\n--set_option pp.universes true\n\nuniverse u v\n\n\n\n-- TODO: Use a universe as the base, to make everything stackable, and to hopefully enable extension to vector spaces etc.\n\nclass CommSemigroup (α : Type u) : Type u where\n(op                   : α → α → α)\n(op_assoc (a b c : α) : op (op a b) c = op a (op b c))\n(op_comm  (a b   : α) : op a b = op b a)\n\nnamespace CommSemigroup\n\n  open MetaRelation Bundled\n\n  @[reducible] def typeClass : SimpleTypeClass.{u + 1, u + 1} := CommSemigroup.{u}\n  @[reducible] def univ : Universe.{u + 1, u + 2} := Bundled.univ typeClass.{u}\n\n  instance inst (A : univ.{u}) : CommSemigroup.{u} A := A.inst\n\n  -- Instance equivalences\n\n  instance hasEquivalenceRelation (A : univ.{u}) : HasEquivalenceRelation A prop :=\n  ⟨nativeRelation (@Eq A)⟩\n\n  instance hasInstanceEquivalences : HasInstanceEquivalences univ.{u} prop :=\n  ⟨hasEquivalenceRelation⟩\n\n  -- Functors\n\n  class IsHom {A B : univ} (f : A → B) : Prop where\n  (h_op (a b : A) : f (op a b) = op (f a) (f b))\n\n  instance hasFunctoriality : HasFunctoriality univ.{u} univ.{v} := ⟨IsHom⟩\n\n  @[simp] theorem simp_op_arg' {A B : univ} (F : HasFunctoriality.Fun A B) (a₁ a₂ : A) :\n    F.f ((inst A).op a₁ a₂) = (inst B).op (F.f a₁) (F.f a₂) :=\n  F.isFun.h_op a₁ a₂\n\n  theorem funExt' {A B : univ} {F G : HasFunctoriality.Fun A B} (h : ∀ a, F.f a = G.f a) : F = G :=\n  have h₁ : F.f = G.f := funext h;\n  by induction F; induction G; subst h₁; simp\n\n  instance functorSemigroup (A B : univ) : CommSemigroup (HasFunctoriality.Fun A B) :=\n  { op       := λ F G   => { f     := λ a => (inst B).op (F.f a) (G.f a),\n                             isFun := ⟨λ a₁ a₂ => by simp;\n                                                     rw [op_assoc, op_assoc];\n                                                     apply congrArg;\n                                                     rw [←op_assoc, ←op_assoc];\n                                                     apply congrFun;\n                                                     apply congrArg;\n                                                     rw [op_comm]⟩ },\n    op_assoc := λ F G H => funExt' λ a => (inst B).op_assoc (F.f a) (G.f a) (H.f a),\n    op_comm  := λ F G   => funExt' λ a => (inst B).op_comm (F.f a) (G.f a) }\n\n  instance hasFunctorialityInstances :\n    HasFunctorialityInstances univ.{u} univ.{v} typeClass.{max u v} :=\n  ⟨functorSemigroup⟩\n\n  def defFun {A B : univ} {f : A → B} (isHom : IsHom f) : A ⟶{f} B :=\n  Bundled.HasFunctorialityInstances.defFun isHom\n\n  instance hasCongrArg : HasCongrArg univ.{u} univ.{v} :=\n  ⟨λ F => congrArg F.f⟩\n\n  instance hasInternalFunctors : HasInternalFunctors univ.{u} := ⟨⟩\n\n  @[simp] theorem simp_op_arg {A B : univ} (F : A ⟶ B) (a₁ a₂ : A) : F (op a₁ a₂) = op (F a₁) (F a₂) :=\n  simp_op_arg' F a₁ a₂\n\n  @[simp] theorem simp_op_fun {A B : univ} (F G : A ⟶ B) (a : A) : (op F G) a = op (F a) (G a) :=\n  rfl\n\n  theorem funExt {A B : univ} {F G : A ⟶ B} (h : ∀ a, F a = G a) : F = G :=\n  funExt' h\n\n  -- TODO: Can we make everything `by simp` again?\n  instance hasLinearFunOp : HasLinearFunOp univ.{u} :=\n  { defIdFun     := λ A     => defFun ⟨λ a₁ a₂ => rfl⟩,\n    defRevAppFun := λ A B   => ⟨λ a => defFun ⟨λ F₁ F₂ => rfl⟩,\n                                defFun ⟨λ a₁ a₂ => funExt λ F => simp_op_arg F a₁ a₂⟩⟩,\n    defCompFun   := λ A B C => ⟨λ F => ⟨λ G => defFun ⟨λ a₁ a₂ => by simp⟩,\n                                        defFun ⟨λ G₁ G₂ => funExt λ a => rfl⟩⟩,\n                                defFun ⟨λ F₁ F₂ => funExt λ G => funExt λ a => simp_op_arg G (F₁ a) (F₂ a)⟩⟩ }\n\n  instance hasTrivialExtensionality : HasTrivialExtensionality univ.{u} univ.{v} := ⟨funExt⟩\n\n  instance hasLinearFunctors : HasLinearFunctors univ.{u} := ⟨⟩\n\n  -- Singletons\n\n  instance unitSemigroup : CommSemigroup PUnit :=\n  { op       := λ _ _   => PUnit.unit,\n    op_assoc := λ _ _ _ => rfl,\n    op_comm  := λ _ _   => rfl }\n\n  instance hasTopInstance : HasTopInstance typeClass.{u} := ⟨unitSemigroup⟩\n\n  instance hasTopEq : HasTop.HasTopEq univ.{u} := ⟨λ _ => rfl⟩\n\n  instance emptySemigroup : CommSemigroup PEmpty :=\n  { op       := λ a _   => PEmpty.elim a,\n    op_assoc := λ a _ _ => PEmpty.elim a,\n    op_comm  := λ a _   => PEmpty.elim a }\n\n  instance hasBotInstance : HasBotInstance typeClass.{u} := ⟨emptySemigroup⟩\n\n  instance hasInternalBot : HasInternalBot univ.{u} :=\n  { defElimFun := λ A => defFun ⟨λ a _ => PEmpty.elim a⟩ }\n\n  -- Products\n\n  instance prodSemigroup (A : univ.{u}) (B : univ.{v}) : CommSemigroup (PProd A B) :=\n  { op       := λ a b   => ⟨op a.fst b.fst, op a.snd b.snd⟩,\n    op_assoc := λ a b c => PProd.ext' ⟨op_assoc a.fst b.fst c.fst, op_assoc a.snd b.snd c.snd⟩,\n    op_comm  := λ a b   => PProd.ext' ⟨op_comm a.fst b.fst, op_comm a.snd b.snd⟩ }\n\n  instance hasProductInstances : HasProductInstances univ.{u} univ.{v} typeClass.{max u v} :=\n  ⟨prodSemigroup⟩\n\n  instance hasProductEq : HasProducts.HasProductEq univ.{u} univ.{v} :=\n  { introEq := λ _   => PProd.ext' ⟨rfl, rfl⟩,\n    fstEq   := λ _ _ => rfl,\n    sndEq   := λ _ _ => rfl }\n\nend CommSemigroup\n", "meta": {"author": "SReichelt", "repo": "universe-abstractions", "sha": "0bf2bae4c1b0f8d96c37e231dd238abda788e843", "save_path": "github-repos/lean/SReichelt-universe-abstractions", "path": "github-repos/lean/SReichelt-universe-abstractions/universe-abstractions-0bf2bae4c1b0f8d96c37e231dd238abda788e843/UniverseAbstractions/Instances/Algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891307678319, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.319686286953228}}
{"text": "import data.fintype.basic\nimport data.finset.basic\n\nimport guidelines.definitions\nimport guidelines.protocol\n\nvariables {sys_state_t pid_t ballot_t value_t : Type}\n  [linear_order ballot_t] [decidable_eq pid_t]\n\nstructure requirements\n  (proto : protocol sys_state_t)\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t) :=\n  (quorums_intersect :\n    ∀ q₁ q₂, defs.quorum q₁ → defs.quorum q₂ → (q₁ ∩ q₂).nonempty)\n  (none_proposed_at_init :\n    ∀ state ballot value, proto.init state → ¬defs.proposed state ballot value)\n  (proposed_stable :\n    ∀ ballot value,\n      proto.stable (λ state, defs.proposed state ballot value))\n  (proposals_unique :\n    ∀ ballot v₁ v₂,\n      proto.invariant (λ state,\n        defs.proposed state ballot v₁ → defs.proposed state ballot v₂ → v₁ = v₂))\n  (at_most_one_proposal_per_step :\n    ∀ s',\n      proto.invariant (λ state,\n        proto.next state s' →\n          ∃ b v, ∀ bal val, defs.proposed s' bal val →\n                 (defs.proposed state bal val ∨ (bal = b ∧ val = v))))\n  (curr_increases :\n    ∀ process s',\n      proto.invariant (λ state,\n        proto.next state s' → defs.curr state process ≤ defs.curr s' process))\n  (stored_ballot_increases :\n    ∀ process prop s',\n      proto.invariant (λ state,\n        proto.next state s' →\n          defs.stored state process = some prop →\n            ∃ prop', defs.stored s' process = some prop' ∧ prop.b ≤ prop'.b))\n  (stored_is_proposed :\n    ∀ process prop,\n      proto.invariant (λ state,\n        defs.stored state process = some prop →\n          defs.proposed state prop.b prop.v))\n  (new_votes_ge_curr_ballot :\n    ∀ process ballot s',\n      proto.invariant (λ state,\n        proto.next state s' →\n          defs.voted s' process ballot →\n            defs.voted state process ballot ∨ defs.curr state process ≤ ballot))\n  (voted_stable :\n    ∀ process ballot,\n      proto.stable (λ state, defs.voted state process ballot))\n  (voted_imp_proposed :\n    ∀ process ballot,\n      proto.invariant (λ state,\n        defs.voted state process ballot →\n          ∃ value, defs.proposed state ballot value))\n  (voted_le_stored :\n    ∀ process ballot,\n      proto.invariant (λ state,\n        defs.voted state process ballot →\n          ∃ prop, defs.stored state process = some prop ∧ ballot ≤ prop.b))\n  (majority_have_upper_interval_if_proposed :\n    ∀ ballot value,\n      (proto_with_intervals_recorded defs proto).invariant (λ state,\n        defs.proposed state.fst ballot value →\n        ∃ (q : finset pid_t)\n             (promised_prop : pid_t → option (proposal ballot_t value_t)),\n          defs.quorum q ∧\n          (∀ (a ∈ q),\n            {interval . lower := (promised_prop a), upper := ballot} ∈ state.snd a) ∧\n          (∀ (a ∈ q) prop_a, promised_prop a = some prop_a → prop_a.b < ballot) ∧\n          ((∀ (a ∈ q), promised_prop a = none) ∨\n           (∃ (a ∈ q) prop_a,\n             promised_prop a = some prop_a ∧\n             (∀ (a' ∈ q) prop_a', promised_prop a' = some prop_a' → prop_a'.b ≤ prop_a.b) ∧\n             prop_a.v = value))))\n\nnamespace requirements\n\nvariables {proto : protocol sys_state_t} {defs : paxos_defs sys_state_t pid_t ballot_t value_t}\n\nlemma none_stored_at_init (reqs_sat : requirements proto defs) :\n    ∀ state process, proto.init state → defs.stored state process = none :=\nbegin\nintros state process is_init,\nhave key : ∀ (prop : proposal ballot_t value_t), defs.stored state process = some prop\n  → defs.proposed state prop.b prop.v,\nby { intro prop, exact reqs_sat.stored_is_proposed process prop state ⟨0, is_init⟩ },\ncases defs.stored state process with stored,\n{ refl },\nspecialize key stored (by refl),\nexact absurd key (reqs_sat.none_proposed_at_init state stored.b stored.v is_init)\nend\n\nlemma all_lower_proposed (reqs_sat : requirements proto defs) :\n  ∀ process, (proto_with_intervals_recorded defs proto).invariant\n    (λ state, ∀ (i ∈ state.snd process) prop,\n      (interval.lower i) = some prop →\n        defs.proposed state.fst prop.b prop.v) :=\nbegin\nintro process, rw protocol.prove_invariant, split,\n{ intros lift_s lift_s_init i i_is_start_interval prop i_lower_is_some, exfalso,\n  cases lift_s_init with s_init init_intervals_is_singleton,\n  specialize init_intervals_is_singleton process,\n  rw init_intervals_is_singleton at i_is_start_interval,\n  rw set.mem_singleton_iff at  i_is_start_interval,\n  rw i_is_start_interval at i_lower_is_some,\n  rw reqs_sat.none_stored_at_init lift_s.fst process s_init at i_lower_is_some,\n  injection i_lower_is_some },\nrintros u u_r w hu ⟨u₁_n_w₁, fact⟩ i i_in_w_history prop i_lower_is_prop,\nrw fact process at i_in_w_history, cases i_in_w_history,\n{ apply (λ goal, reqs_sat.proposed_stable prop.b prop.v u.fst w.fst goal u₁_n_w₁),\n  exact hu i i_in_w_history prop i_lower_is_prop },\nrw set.mem_singleton_iff at i_in_w_history, rw i_in_w_history at i_lower_is_prop,\nhave w_r : (proto_with_intervals_recorded defs proto).reachable w,\nby { cases u_r with k hk, exact ⟨k.succ, u, hk, ⟨u₁_n_w₁, fact⟩⟩ },\nexact reqs_sat.stored_is_proposed process prop w.fst\n  (reachable_in_proto_with_intervals_recorded_implies_first_reachable defs proto w w_r)\n  i_lower_is_prop\nend\n\nlemma all_upper_le_curr (reqs_sat : requirements proto defs) :\n  ∀ process, (proto_with_intervals_recorded defs proto).invariant\n    (λ state, ∀ (i ∈ state.snd process) b,\n      (interval.upper i) = b → b ≤ defs.curr state.fst process) :=\nbegin\nintro process, rw protocol.prove_invariant, split,\n{ intros lift_s lift_s_init i i_is_start_interval b i_upper_eq_b,\n  cases lift_s_init with s_init init_intervals_is_singleton,\n  specialize init_intervals_is_singleton process,\n  rw init_intervals_is_singleton at i_is_start_interval,\n  rw set.mem_singleton_iff at  i_is_start_interval,\n  rw i_is_start_interval at i_upper_eq_b,\n  rw ← i_upper_eq_b },\nrintros u u_r w hu ⟨u₁_n_w₁, fact⟩ i i_in_w_history b i_upper_eq_b,\nrw fact process at i_in_w_history, cases i_in_w_history,\n{ apply le_trans (hu i i_in_w_history b i_upper_eq_b),\n  apply (λ goal, reqs_sat.curr_increases process w.fst u.fst goal u₁_n_w₁),\n  exact reachable_in_proto_with_intervals_recorded_implies_first_reachable defs proto u u_r },\nrw set.mem_singleton_iff at i_in_w_history, rw i_in_w_history at i_upper_eq_b,\nrw ← i_upper_eq_b\nend\n\nlemma none_voted_in_interval (reqs_sat : requirements proto defs) :\n  ∀ process ballot, (proto_with_intervals_recorded defs proto).invariant\n    (λ state, ∀ i ∈ state.snd process, defs.voted state.fst process ballot →\n      (∃ prop, interval.lower i = some prop ∧ ballot ≤ prop.b) ∨ i.upper ≤ ballot)\n  :=\nbegin\nintros process ballot, rw protocol.prove_invariant, split,\n{ intros lift_s lift_s_init i i_first_interval process_voted, exfalso,\n  rcases reqs_sat.voted_imp_proposed process ballot lift_s.fst\n    ⟨0, lift_s_init.left⟩ process_voted with ⟨value, bal_val_proposed⟩,\n  exact reqs_sat.none_proposed_at_init lift_s.fst ballot value lift_s_init.left\n    bal_val_proposed },\nintros lift_u lift_u_reachable lift_w h_lift_u lift_u_next_lift_w i\n  i_in_history process_voted,\nrw lift_u_next_lift_w.right process at i_in_history,\nhave already_voted_or_new_vote := reqs_sat.new_votes_ge_curr_ballot\n  process ballot lift_w.fst lift_u.fst\n  (reachable_in_proto_with_intervals_recorded_implies_first_reachable\n    defs proto lift_u lift_u_reachable)\n  lift_u_next_lift_w.left process_voted,\ncases i_in_history; cases already_voted_or_new_vote with already_voted new_vote,\n{ exact h_lift_u i i_in_history already_voted },\n{ right, exact le_trans\n  (reqs_sat.all_upper_le_curr process lift_u lift_u_reachable i\n    i_in_history i.upper (by refl)) new_vote },\n{ left, rw set.mem_singleton_iff at i_in_history, rw i_in_history,\n  rcases reqs_sat.voted_le_stored process ballot lift_u.fst\n    (reachable_in_proto_with_intervals_recorded_implies_first_reachable\n      defs proto lift_u lift_u_reachable) already_voted\n    with ⟨prop_u, stored_at_u, ballot_le_stored_u⟩,\n  rcases reqs_sat.stored_ballot_increases process prop_u lift_w.fst lift_u.fst\n    (reachable_in_proto_with_intervals_recorded_implies_first_reachable\n      defs proto lift_u lift_u_reachable) lift_u_next_lift_w.left stored_at_u\n    with ⟨prop_w, stored_at_w, ballot_le_stored_w⟩,\n  exact ⟨prop_w, stored_at_w, le_trans ballot_le_stored_u ballot_le_stored_w⟩ },\nleft, rw set.mem_singleton_iff at i_in_history, rw i_in_history,\napply (λ goal, reqs_sat.voted_le_stored process ballot lift_w.fst goal process_voted),\nrcases reachable_in_proto_with_intervals_recorded_implies_first_reachable\n      defs proto lift_u lift_u_reachable with ⟨k, hk⟩,\nexact ⟨k.succ, lift_u.fst, hk, lift_u_next_lift_w.left⟩\nend\n\nlemma chosen_stable (reqs_sat : requirements proto defs) (v : value_t) :\n  proto.stable (λ state, defs.chosen state v) :=\nbegin\nintros s s' chosen_at_s s'_follows_s,\nrcases chosen_at_s with ⟨b, ⟨q, q_is_quorum, majority_voted_at_q⟩, b_v_proposed_at_s⟩,\nuse b, split,\n{ use [q, q_is_quorum],\n  intros voter in_quorum,\n  specialize majority_voted_at_q voter in_quorum,\n  exact reqs_sat.voted_stable voter b s s' majority_voted_at_q s'_follows_s },\n  exact reqs_sat.proposed_stable b v s s' b_v_proposed_at_s s'_follows_s,\nend\n\nend requirements\n", "meta": {"author": "gnanabite", "repo": "colocated-paxos", "sha": "f60308e27d3013665809077fe80a4b2af8a42278", "save_path": "github-repos/lean/gnanabite-colocated-paxos", "path": "github-repos/lean/gnanabite-colocated-paxos/colocated-paxos-f60308e27d3013665809077fe80a4b2af8a42278/src/guidelines/requirements.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.45713671682749474, "lm_q1q2_score": 0.3196548759999913}}
{"text": "import tactic.norm_cast\n\nstructure hom (α β : Type) :=\n(to_fun : α → β)\n\ninstance {α β} : has_coe_to_fun (hom α β) :=\n⟨_, hom.to_fun⟩\n\nstructure hom_plus (α β) extends hom α β\n\ninstance {α β} : has_coe (hom_plus α β) (hom α β) :=\n⟨hom_plus.to_hom⟩\n\ninstance {α β} : has_coe_to_fun (hom_plus α β) :=\n⟨λ _, α → β, λ f, f⟩\n\n@[norm_cast] lemma hom_plus.coe_fn (α β : Type) (a : α) (h : hom_plus α β) :\n  (h : hom α β) a = h a :=\nrfl\n\nexample (α β : Type) (a : α) (h : hom_plus α β) :\n  (h : hom α β) a = h a :=\nby norm_cast\n\ndef f1 (n : ℕ) : hom_plus ℕ ℕ := ⟨⟨λ m, m + n⟩⟩\ndef f2 : hom ℕ (hom ℕ ℕ) := ⟨λ n, f1 n⟩\n\n@[norm_cast] lemma coe_f1 (n : ℕ) :\n  (f1 n : hom ℕ ℕ) = f2 n :=\nrfl\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/test/norm_cast_coe_fn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.31943735283726377}}
{"text": "import MLIRSemantics.Toy.Toy\nimport MLIRSemantics.Fitree\nimport MLIRSemantics.Verifier\nimport MLIRSemantics.SSAEnv\nimport MLIRSemantics.InvalidOp\nimport MLIRSemantics.Util.Metagen\n\nimport MLIR.AST\nimport MLIR.EDSL\n\nimport Lean\n\nopen MLIR.AST\n\n\n/- To be automatically generated -/\n\nset_option hygiene false in\ngenInductive ToyOp #[\n  OpSpec.mk \"Constant\" `(\n    (D: DimList) → (Hknown: D.known) →\n    (e: TensorElem) → (τ: MLIRTy) → (Htype: e.hasType τ) →\n    (Hcompat: e.rankCompatibleWith D τ) →\n    ToyOp (RankedTensor τ.eval D)),\n  OpSpec.mk \"Transpose\" `(\n    (α: Type) → (n m: Nat) →\n    RankedTensor α [Dimension.Known n, Dimension.Known m] →\n    ToyOp (RankedTensor α [Dimension.Known m, Dimension.Known n])),\n  OpSpec.mk \"Reshape\" `(\n    (α: Type) → (D D': DimList) → (H: D.known) → (H': D'.known) →\n    (Hprod: D'.prod = D.prod) →\n    RankedTensor α D →\n    ToyOp (RankedTensor α D'))\n]\n\n/- To be automatically generated (hopefully; basically this is the\n   verification stuff) -/\n\ndef toy_semantics_op (ret_name: Option SSAVal):\n      Op → Fitree (InvalidOpE +' SSAEnvE +' ToyOp) Unit\n\n  | Op.mk \"toy.constant\" [] [] [] attrs\n        (MLIRTy.fn (MLIRTy.tuple []) (MLIRTy.tensorRanked D₁ τ₁)) =>\n      match AttrDict.find attrs \"value\" with\n      | some (AttrVal.dense elem (MLIRTy.tensorRanked D₂ τ₂)) =>\n          if H: D₁ = D₂ ∧ DimList.known D₁ ∧ τ₁ = τ₂ ∧ elem.hasType τ₁ then\n            match Heq: elem, τ₁ with\n            | TensorElem.int i, MLIRTy.int 32 => do\n                let t ← Fitree.trigger (ToyOp.Constant D₁ H.2.1 elem\n                  (MLIRTy.int 32) (by simp [Heq, H.2.2.2])\n                  (TensorElem.rankCompatibleWith.UniformInt i 32 Heq));\n                SSAEnv.set? (MLIRTy.tensorRanked D₁ (MLIRTy.int 32)) ret_name t\n            | elem, τ₁ => do\n                if Hshape: elem.hasShape (DimList.default_refinement D₁) then\n                  let t ← Fitree.trigger (ToyOp.Constant D₁ H.2.1 elem τ₁\n                    H.2.2.2 (TensorElem.rankCompatibleWith.HasShape\n                     (DimList.default_refinement D₁) _ Hshape\n                     (default_refinement_refines D₁)));\n                  SSAEnv.set? (MLIRTy.tensorRanked D₁ τ₁) ret_name t\n                else\n                  Fitree.trigger InvalidOpE.InvalidOp\n          else\n            Fitree.trigger InvalidOpE.InvalidOp\n      | _ =>\n          Fitree.trigger InvalidOpE.InvalidOp\n\n  | Op.mk \"toy.transpose\" [t_name] [] [] _ (MLIRTy.fn τ₁ τ₂) =>\n      match τ₁ with\n      | MLIRTy.tensorRanked [Dimension.Known n, Dimension.Known m] τ => do\n          let t ← Fitree.trigger (@SSAEnvE.Get (MLIRTy.tensorRanked\n                  [Dimension.Known n, Dimension.Known m] τ) _ t_name);\n          let t' ← Fitree.trigger (ToyOp.Transpose τ.eval n m t);\n          SSAEnv.set? (MLIRTy.tensorRanked [Dimension.Known m,\n                       Dimension.Known n] τ)\n            ret_name t'\n      | _ =>\n          Fitree.trigger InvalidOpE.InvalidOp\n\n  | Op.mk \"toy.reshape\" [t_name] [] [] _ (MLIRTy.fn τ₁ τ₂) =>\n      match τ₁, τ₂ with\n      | MLIRTy.tensorRanked D σ₁, MLIRTy.tensorRanked D' σ₂ =>\n          if H: σ₁ = σ₂\n             ∧ DimList.known D\n             ∧ DimList.known D'\n             ∧ DimList.prod D' = DimList.prod D then do\n            let t ← Fitree.trigger (@SSAEnvE.Get (MLIRTy.tensorRanked D σ₁) _\n                    t_name);\n            let t' ← Fitree.trigger (ToyOp.Reshape σ₁.eval D D'\n                     H.2.1 H.2.2.1 H.2.2.2 t);\n            let t': RankedTensor σ₂.eval D' := cast (by rw [H.1]) t';\n            SSAEnv.set? (MLIRTy.tensorRanked D' σ₂) ret_name t'\n          else\n            Fitree.trigger InvalidOpE.InvalidOp\n      | _, _ =>\n          Fitree.trigger InvalidOpE.InvalidOp\n\n  | _ =>\n      Fitree.trigger InvalidOpE.InvalidOp\n\ndef toy_semantics_bbstmt:\n      BasicBlockStmt → Fitree (InvalidOpE +' SSAEnvE +' ToyOp) Unit\n  | BasicBlockStmt.StmtAssign val _ op =>\n      toy_semantics_op (some val) op\n  | BasicBlockStmt.StmtOp op =>\n      toy_semantics_op none op\n\n-- TODO: toy_semantics_bb: handle basic block arguments\n@[simp]\ndef toy_semantics_bb:\n      BasicBlock → Fitree (InvalidOpE +' SSAEnvE +' ToyOp) Unit\n  | BasicBlock.mk name args [] =>\n      Fitree.ret ()\n  | BasicBlock.mk name args (op1::ops) =>\n      List.foldr (fun t acc => Fitree.bind acc (fun _ => t))\n                 (toy_semantics_bbstmt op1)\n                 (ops.map toy_semantics_bbstmt)\n\n/- Manually specified: ToyOp event handler -/\n\ndef ToyOp.handle {E}: ToyOp ~> Fitree E :=\n  fun _ e => match e with\n  | ToyOp.Constant D Hknown elem τ Htype Hcompat =>\n      return RankedTensor.ofTensorElem D elem Htype Hcompat\n  | ToyOp.Transpose α n m t =>\n      return transpose t\n  | ToyOp.Reshape α D D' H H' Hprod t =>\n      return reshape D' H H' Hprod t\n\n-- Interpretation in context\n\ndef interp_toy {E} (t: Fitree (ToyOp +' E) R): Fitree E R :=\n  interp (case_ ToyOp.handle (fun T => @Fitree.trigger E E T _)) t\n\n@[simp]\ndef run_toy (t: Fitree (InvalidOpE +' SSAEnvE +' ToyOp) Unit) (env: SSAEnv):\n    Fitree PVoid (Unit × SSAEnv) :=\n  interp ToyOp.handle (interp_ssa (interp_invalid t sorry) env)\n\n/-\n### Interpretation layers for Toy programs\n\nWe first interpret away nonexsting invalid operations, then the memory layer,\nand finally the Toy operations themselves.\n-/\n\n#check interp_invalid\n#check interp_ssa\n#check interp_toy\n\n\n/-\n### Examples and testing\n-/\n\n-- The following extends #reduce with a (skipProofs := true/false) parameter\n-- to not reduce proofs in the kernel. Reducing proofs would cause constant\n-- timeouts, and proofs are used implicitly through well-founded induction for\n-- mutual definitions.\n-- See: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Repr.20instance.20for.20functions/near/276504682\n\nopen Lean\nopen Lean.Parser.Term\nopen Lean.Elab.Command\nopen Lean.Elab\nopen Lean.Meta\n\nelab \"#reduce \" skipProofs:group(atomic(\"(\" &\"skipProofs\") \" := \" (trueVal <|> falseVal) \")\") term:term : command =>\n  let skipProofs := skipProofs[3].isOfKind ``trueVal\n  withoutModifyingEnv <| runTermElabM (some `_check) fun _ => do\n    -- dbg_trace term\n    let e ← Term.elabTerm term none\n    Term.synthesizeSyntheticMVarsNoPostponing\n    let (e, _) ← Term.levelMVarToParam (← instantiateMVars e)\n    withTheReader Core.Context (fun ctx => { ctx with options := ctx.options.setBool `smartUnfolding false }) do\n      let e ← withTransparency (mode := TransparencyMode.all) <| reduce e (skipProofs := skipProofs) (skipTypes := false)\n      logInfo e\n\n---\n\ndef transpose_stmt := [mlir_bb_stmt|\n  %t2 = \"toy.transpose\"(%t1): tensor<2×4×i32> -> tensor<4×2×i32>\n]\n\ndef constant_stmt := [mlir_bb_stmt|\n  %t = \"toy.constant\"() {value=dense<[[1,2],[3,4]]>: tensor<2×2×i32>}:\n    () -> tensor<2×2×i32>\n]\n\n#reduce constant_stmt\n\ndef double_transpose := [mlir_bb|\n  ^dbl:\n    %t2 = \"toy.transpose\"(%t1): tensor<2×4×i32> -> tensor<4×2×i32>\n    %t3 = \"toy.transpose\"(%t2): tensor<4×2×i32> -> tensor<2×4×i32>\n]\n\n#reduce (skipProofs := true)\n  toy_semantics_bb double_transpose\n\n#reduce (skipProofs := true)\n  run_toy (toy_semantics_bbstmt transpose_stmt) [[]]\n\n#reduce (skipProofs := true)\n  run_toy (toy_semantics_bbstmt constant_stmt) [[]]\n\ntheorem double_transpose_correct:\n  ∀ (t1: RankedTensor Int [2,4]),\n    run_toy (toy_semantics_bb double_transpose)\n      [[(\"t1\", ⟨MLIRTy.tensorRanked [2,4] (MLIRTy.int 32), t1⟩)]]\n    =\n    Fitree.ret ((), [[\n      (SSAVal.SSAVal \"t1\", ⟨MLIRTy.tensorRanked [2,4] (MLIRTy.int 32), t1⟩),\n      (SSAVal.SSAVal \"t2\", ⟨MLIRTy.tensorRanked [4,2] (MLIRTy.int 32),\n                           transpose t1⟩),\n      (SSAVal.SSAVal \"t3\", ⟨MLIRTy.tensorRanked [2,4] (MLIRTy.int 32), t1⟩)\n    ]]) := by\n  intros t1\n  unfold double_transpose\n  simp\n  simp [double_transpose, toy_semantics_bb, toy_semantics_bbstmt]; simp_itree\n  simp [interp_invalid]; simp_itree\n  simp [interp_ssa]; simp_itree\n  rw [transpose_involutive]\n  rfl\n", "meta": {"author": "opencompl", "repo": "lean-mlir-semantics", "sha": "9ec41b134fd89a9799defae9e41de76b7d526266", "save_path": "github-repos/lean/opencompl-lean-mlir-semantics", "path": "github-repos/lean/opencompl-lean-mlir-semantics/lean-mlir-semantics-9ec41b134fd89a9799defae9e41de76b7d526266/MLIRSemantics/Toy/ToyDialect.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.3193195762812715}}
{"text": "\nimport util.data.functor\n\nuniverses u v w\n\nnamespace ulift\n\nvariables {α β γ : Type u}\n\ndef pure (x : α) : ulift α := ⟨ x ⟩\n\ndef bind (x : ulift α) (f : α → ulift β) : ulift β :=\nf (down x)\n\ninstance : has_bind ulift := ⟨ @bind ⟩\n\nlemma bind_assoc (x : ulift α) (f : α → ulift β) (g : β → ulift γ)\n: x >>= f >>= g = x >>= (λ i, f i >>= g) :=\nby { cases x ; refl }\n\nlemma pure_bind (x : α) (f : α → ulift β)\n: pure x >>= f = f x := sorry\n\nlemma id_map (x : ulift α)\n: ulift.bind x (ulift.pure ∘ id) = x := sorry\n\nend ulift\n\ninstance : monad ulift :=\n{ bind := @ulift.bind\n, pure := @ulift.pure }\n\ninstance : is_lawful_monad ulift :=\n{ pure_bind := @ulift.pure_bind\n, bind_assoc := @ulift.bind_assoc\n, id_map := @ulift.id_map }\n\nvariables (m : Type (max w u) → Type v)\n\nstructure ulift_t (α : Type w) :=\n(run : m (ulift α))\n\nnamespace ulift_t\n\nexport ulift (up down up_down)\n\nvariables {m}\n\nvariables {α β γ : Type w}\n\nprotected def bind [has_bind m] (x : ulift_t m α) (f : α → ulift_t m β) : ulift_t m β :=\n⟨ x.run >>= ulift_t.run ∘ f ∘ down ⟩\n\nprotected def pure [has_pure m] (x : α) : ulift_t m α :=\n⟨ pure (up x) ⟩\n\ninstance [has_bind m] : has_bind (ulift_t m) :=\n⟨ λ α β, @ulift_t.bind m α β _ ⟩\n\ninstance [has_pure m] : has_pure (ulift_t m) :=\n⟨ λ α, @ulift_t.pure m α _ ⟩\n\nvariables [monad m] [is_lawful_monad m]\nopen is_lawful_monad\n\nlemma bind_assoc (x : ulift_t m α) (f : α → ulift_t m β) (g : β → ulift_t m γ)\n: x >>= f >>= g = x >>= (λ i, f i >>= g) :=\nby simp [has_bind.bind,ulift_t.bind,bind_assoc]\n\nlemma pure_bind (x : α) (f : α → ulift_t m β)\n: ulift_t.pure x >>= f = f x :=\nbegin\n  destruct f x, intros y h,\n  simp [has_bind.bind,ulift_t.bind,ulift_t.pure],\n  simp [pure_bind,function.comp,h],\nend\n\nlemma id_map (x : ulift_t m α)\n: x >>= pure ∘ id = x :=\nbegin\n  cases x,\n  simp [has_bind.bind,ulift_t.bind,function.comp,pure,has_pure.pure,ulift_t.pure],\n  simp [up_down],\nend\n\nend ulift_t\n\ninstance {m : Type (max w u) → Type v} [monad m] : monad (ulift_t m) :=\n{ bind := λ α β, ulift_t.bind\n, pure := λ α, ulift_t.pure }\n\ninstance {m : Type (max w u) → Type v} [monad m] [is_lawful_monad m] : is_lawful_monad (ulift_t m) :=\n{ pure_bind := λ α β, @ulift_t.pure_bind m α β _ _\n, bind_assoc := λ α β γ, ulift_t.bind_assoc\n, id_map := λ α, @ulift_t.id_map m α _ _ }\n", "meta": {"author": "unitb", "repo": "lean-lib", "sha": "439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9", "save_path": "github-repos/lean/unitb-lean-lib", "path": "github-repos/lean/unitb-lean-lib/lean-lib-439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9/src/util/data/ulift_t.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.31931957043134784}}
{"text": "import data.cpi.species.basic\n\nnamespace cpi\nnamespace species\n\nvariables {ℍ : Type} {ω : context}\n\n/-- A chain of rewrite rules, to transform a species from one kind to another\n    equivalent one. -/\n@[nolint has_inhabited_instance]\ninductive equivalent : ∀ {Γ} (A B : species ℍ ω Γ), Type\n| refl  {Γ} {A : species ℍ ω Γ} : equivalent A A\n| trans {Γ} {A B C : species ℍ ω Γ} : equivalent A B → equivalent B C → equivalent A C\n\n-- Protections into the body of a species.\n| ξ_parallel₁\n      {Γ} {A A' B : species ℍ ω Γ}\n    : equivalent A A' → equivalent (A |ₛ B) (A' |ₛ B)\n| ξ_parallel₂\n      {Γ} {A B B' : species ℍ ω Γ}\n    : equivalent B B' → equivalent (A |ₛ B) (A |ₛ B')\n| ξ_restriction\n      {Γ} (M : affinity ℍ) {A A' : species ℍ ω (context.extend (M.arity) Γ)}\n    : equivalent A A' → equivalent (ν(M) A) (ν(M) A')\n| ξ_choice_here\n      {Γ} {f} (π : prefix_expr ℍ Γ f) {A A' : species ℍ ω (f.apply Γ)} {As : choices ℍ ω Γ}\n    : equivalent A A'\n    → equivalent (Σ# (whole.cons π A As)) (Σ# (whole.cons π A' As))\n| ξ_choice_there\n      {Γ} {f} (π : prefix_expr ℍ Γ f) {A : species ℍ ω (f.apply Γ)} {As As' : choices ℍ ω Γ}\n    : equivalent (Σ# As) (Σ# As')\n    → equivalent (Σ# (whole.cons π A As)) (Σ# (whole.cons π A As'))\n\n-- | An element in the choice array can be swapped.\n| choice_swap\n    {Γ} {f g} (π₁ : prefix_expr ℍ Γ f) (π₂ : prefix_expr ℍ Γ g)\n    {A : species ℍ ω (f.apply Γ)} {B : species ℍ ω (g.apply Γ)} {As : choices ℍ ω Γ}\n  : equivalent (Σ# (whole.cons π₁ A (whole.cons π₂ B As)))\n          (Σ# (whole.cons π₂ B (whole.cons π₁ A As)))\n\n-- Species forms a commutative monoid using parallel.\n| parallel_nil₁   {Γ} {A : species ℍ ω Γ}     : equivalent (A |ₛ nil) A\n| parallel_nil₂   {Γ} {A : species ℍ ω Γ}     : equivalent A (A |ₛ nil)\n| parallel_symm   {Γ} {A B : species ℍ ω Γ}   : equivalent (A |ₛ B) (B |ₛ A)\n| parallel_assoc₁ {Γ} {A B C : species ℍ ω Γ} : equivalent ((A |ₛ B) |ₛ C) (A |ₛ (B |ₛ C))\n| parallel_assoc₂ {Γ} {A B C : species ℍ ω Γ} : equivalent (A |ₛ (B |ₛ C)) ((A |ₛ B) |ₛ C)\n\n| ν_parallel₁\n    {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} {B : species ℍ ω (context.extend M.arity Γ)}\n  : equivalent (ν(M) (rename name.extend A |ₛ B)) (A |ₛ ν(M)B)\n| ν_parallel₂\n    {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} {B : species ℍ ω (context.extend M.arity Γ)}\n  : equivalent (A |ₛ ν(M)B) (ν(M) (rename name.extend A |ₛ B))\n| ν_drop₁\n    {Γ} (M : affinity ℍ) {A : species ℍ ω Γ}\n  : equivalent (ν(M) (rename name.extend A)) A\n| ν_drop₂\n    {Γ} (M : affinity ℍ) {A : species ℍ ω Γ}\n  : equivalent A (ν(M) (rename name.extend A))\n| ν_swap₁\n    {Γ} (M N : affinity ℍ)\n    {A  : species ℍ ω (context.extend N.arity (context.extend M.arity Γ))}\n  : equivalent (ν(M)ν(N) A) (ν(N)ν(M) rename name.swap A)\n| ν_swap₂ -- Identical to ν_swap₁ but flipped. This is pointless, but makes symm_symm easier to show.\n    {Γ} (M N : affinity ℍ)\n    {A  : species ℍ ω (context.extend N.arity (context.extend M.arity Γ))}\n  : @equivalent Γ (ν(N)ν(M) rename name.swap A) (ν(M)ν(N) A)\n\nnamespace equivalent\n  /-- Flip an equivalence relationship. -/\n  protected def symm : ∀ {Γ} {A B : species ℍ ω Γ}, equivalent A B → equivalent B A\n  | Γ A B eq := begin\n    induction eq,\n    case refl { from refl },\n    case trans : Γ A B C ab bc ih_ab ih_bc { from trans ih_bc ih_ab },\n\n    case ξ_parallel₁ : Γ A A' B eq ih { from ξ_parallel₁ ih },\n    case ξ_parallel₂ : Γ A A' B eq ih { from ξ_parallel₂ ih },\n    case ξ_restriction : Γ M A A' eq ih { from ξ_restriction M ih },\n    case ξ_choice_here : Γ f π A A' As eq ih { from ξ_choice_here π ih },\n    case ξ_choice_there : Γ f π A A' As eq ih { from ξ_choice_there π ih },\n\n    case choice_swap { from choice_swap _ _ },\n\n    case parallel_nil₁ { from parallel_nil₂ },\n    case parallel_nil₂ { from parallel_nil₁ },\n    case parallel_symm { from parallel_symm },\n    case parallel_assoc₁ { from parallel_assoc₂ },\n    case parallel_assoc₂ { from parallel_assoc₁ },\n\n    case ν_parallel₁ : Γ M { from ν_parallel₂ M },\n    case ν_parallel₂ : Γ M { from ν_parallel₁ M },\n    case ν_drop₁ : Γ M { from ν_drop₂ M },\n    case ν_drop₂ : Γ M { from ν_drop₁ M },\n    case ν_swap₁ : Γ M N { from ν_swap₂ M N },\n    case ν_swap₂ : Γ M N { from ν_swap₁ M N },\n  end\n\n  /-- Flipping an equivalence relation twice gives the original. -/\n  protected lemma symm_symm :\n    ∀ {Γ} {A B : species ℍ ω Γ} (eq : equivalent A B)\n    , eq = equivalent.symm (equivalent.symm eq)\n  | Γ A B eq := begin\n    induction eq,\n    case equivalent.refl { from rfl },\n    case equivalent.trans : Γ A B C ab bc ih_ab ih_bc {\n      from congr_arg2 trans ih_ab ih_bc,\n    },\n    case ξ_parallel₁ : Γ A A' B eq ih { from congr_arg ξ_parallel₁ ih, },\n    case ξ_parallel₂ : Γ A A' B eq ih { from congr_arg ξ_parallel₂ ih },\n    case ξ_restriction : Γ M A A' eq ih { from congr_arg (ξ_restriction M) ih },\n    case ξ_choice_here : Γ f π A A' As eq ih { from congr_arg (ξ_choice_here π) ih },\n    case ξ_choice_there : Γ f π A A' As eq ih { from congr_arg (ξ_choice_there π) ih },\n\n    repeat { from rfl },\n  end\n\n  local infix ` ~ ` := equivalent\n\n  private def rename_swap\n    {Γ Δ} {ρ : name Γ → name Δ} {M N : affinity ℍ}\n    (A' : species ℍ ω (context.extend M.arity (context.extend N.arity Γ)))\n    : rename (name.ext (name.ext ρ)) (rename name.swap A')\n    = rename name.swap (rename (name.ext (name.ext ρ)) A')\n    := by rw [rename_compose, name.swap_ext_ext, rename_compose]\n\n  /-- Rename an equivalence relationship. -/\n  protected def rename :\n      ∀ {Γ Δ} {A B : species ℍ ω Γ} (ρ : name Γ → name Δ)\n      , A ~ B → rename ρ A ~ rename ρ B\n    | Γ Δ A B ρ eq := begin\n      induction eq generalizing Δ,\n\n      case refl : Γ A Δ ρ { from refl },\n      case trans : Γ A B C ab bc ih_ab ih_bc { from trans (ih_ab _ ρ) (ih_bc _ ρ) },\n\n      -- Projection\n      case ξ_parallel₁ : Γ A A' B eq ih { simp, from ξ_parallel₁ (ih _ ρ) },\n      case ξ_parallel₂ : Γ A A' B eq ih { simp, from ξ_parallel₂ (ih _ ρ) },\n      case ξ_restriction : Γ M A A' eq ih {\n        simp only [species.rename.restriction],\n        from ξ_restriction M (ih _ (name.ext ρ))\n      },\n      case ξ_choice_here : Γ f π A A' As eq ih {\n        simp only [species.rename.choice, species.rename.cons],\n        from ξ_choice_here (prefix_expr.rename ρ π) (ih _ (prefix_expr.ext π ρ))\n      },\n      case ξ_choice_there : Γ f π A As As' eq ih {\n        simp only [species.rename.choice, species.rename.cons],\n        have h := ih _ ρ,\n        have g : (Σ# species.rename ρ As) ~ (Σ# species.rename ρ As'), simp at h, from h,\n        from ξ_choice_there (prefix_expr.rename ρ π) g\n      },\n\n      -- Choice\n      case choice_swap : Γ f g π₁ π₂ A B As {\n        simp only [species.rename.choice, species.rename.cons],\n        from choice_swap _ _\n      },\n\n      -- Parallel\n      case parallel_nil₁ : Γ A { simp only [species.rename.nil, species.rename.parallel], from parallel_nil₁ },\n      case parallel_nil₂ : Γ A { simp only [species.rename.nil, species.rename.parallel], from parallel_nil₂ },\n      case parallel_symm : Γ A B { simp only [species.rename.parallel], from parallel_symm },\n      case parallel_assoc₁ : Γ A B C { simp only [species.rename.parallel], from parallel_assoc₁ },\n      case parallel_assoc₂ : Γ A B C { simp only [species.rename.parallel], from parallel_assoc₂ },\n\n      -- Restriction\n      case ν_parallel₁ : Γ M A B {\n        simp only [species.rename.restriction, species.rename.parallel],\n        rw ← species.rename_ext _, from ν_parallel₁ M\n      },\n      case ν_parallel₂ : Γ M A B {\n        simp only [species.rename.restriction, species.rename.parallel],\n        rw ← species.rename_ext _, from ν_parallel₂ M\n      },\n      case ν_drop₁ : Γ M A {\n        simp only [species.rename.restriction], rw ← species.rename_ext _, from ν_drop₁ M\n      },\n      case ν_drop₂ : Γ M A {\n        simp only [species.rename.restriction], rw ← species.rename_ext _, from ν_drop₂ M\n      },\n      case ν_swap₁ : Γ M N A { simp only [species.rename.restriction], rw rename_swap _, from ν_swap₁ M N },\n      case ν_swap₂ : Γ M N A { simp only [species.rename.restriction], rw rename_swap _, from ν_swap₂ M N },\n    end\nend equivalent\n\n/-- Lower an equivalence relationship to a proof-irrelevant proposition.\n\n    This allows us to use it as a normal proof object, while -/\ninductive equiv {Γ} (A B : species ℍ ω Γ) : Prop\n| intro : equivalent A B → equiv\n\nnamespace equiv\n  local infix ` ~ ` := equiv\n  @[pattern]\n  protected lemma refl {Γ} (A : species ℍ ω Γ) : A ~ A := ⟨ equivalent.refl ⟩\n\n  @[pattern]\n  protected lemma rfl {Γ} {A : species ℍ ω Γ} : A ~ A := equiv.refl A\n\n  protected lemma symm {Γ} {A B : species ℍ ω Γ} : A ~ B → B ~ A\n  | ⟨ eq ⟩ := ⟨ equivalent.symm eq ⟩\n\n  @[pattern]\n  protected lemma trans {Γ} {A B C : species ℍ ω Γ} : A ~ B → B ~ C → A ~ C\n  | ⟨ ab ⟩ ⟨ bc ⟩ := ⟨ equivalent.trans ab bc ⟩\n\n  protected lemma rename {Γ Δ} {A B : species ℍ ω Γ} (ρ : name Γ → name Δ)\n    : A ~ B → rename ρ A ~ rename ρ B\n  | ⟨ eq ⟩ := ⟨ equivalent.rename ρ eq ⟩\n\n  @[pattern]\n  lemma ξ_parallel₁ {Γ} {A A' B : species ℍ ω Γ} : A ~ A' → (A |ₛ B) ~ (A' |ₛ B)\n  | ⟨ eq ⟩ := ⟨ equivalent.ξ_parallel₁ eq ⟩\n\n  @[pattern]\n  lemma ξ_parallel₂ {Γ} {A B B' : species ℍ ω Γ} : B ~ B' → (A |ₛ B) ~ (A |ₛ B')\n  | ⟨ eq ⟩ := ⟨ equivalent.ξ_parallel₂ eq ⟩\n\n  @[pattern]\n  lemma ξ_restriction {Γ} (M : affinity ℍ) {A A' : species ℍ ω (context.extend (M.arity) Γ)}\n    : A ~ A' → (ν(M) A) ~ (ν(M) A')\n  | ⟨ eq ⟩ := ⟨ equivalent.ξ_restriction M eq ⟩\n\n  @[pattern]\n  lemma ξ_choice_here\n      {Γ} {f} (π : prefix_expr ℍ Γ f) {A A' : species ℍ ω (f.apply Γ)} {As : choices ℍ ω Γ}\n    : A ~ A' → (Σ# (whole.cons π A As)) ~ (Σ# (whole.cons π A' As))\n  | ⟨ eq ⟩ := ⟨ equivalent.ξ_choice_here π eq ⟩\n\n  @[pattern]\n  lemma ξ_choice_there\n      {Γ} {f} (π : prefix_expr ℍ Γ f) {A : species ℍ ω (f.apply Γ)} {As As' : choices ℍ ω Γ}\n    : (Σ# As) ~ (Σ# As')\n    → (Σ# (whole.cons π A As)) ~ (Σ# (whole.cons π A As'))\n  | ⟨ eq ⟩ := ⟨ equivalent.ξ_choice_there π eq ⟩\n\n  @[pattern]\n  lemma choice_swap\n      {Γ} {f g} (π₁ : prefix_expr ℍ Γ f) (π₂ : prefix_expr ℍ Γ g)\n      {A : species ℍ ω (f.apply Γ)} {B : species ℍ ω (g.apply Γ)} {As : choices ℍ ω Γ}\n    : (Σ# (whole.cons π₁ A (whole.cons π₂ B As))) ~ (Σ# (whole.cons π₂ B (whole.cons π₁ A As)))\n    := ⟨ equivalent.choice_swap π₁ π₂ ⟩\n\n  @[pattern]\n  lemma parallel_nil₁   {Γ} {A : species ℍ ω Γ}     : (A |ₛ nil) ~ A := ⟨ equivalent.parallel_nil₁ ⟩\n\n  @[pattern]\n  lemma parallel_nil₂   {Γ} {A : species ℍ ω Γ}     : A ~ (A |ₛ nil) := ⟨ equivalent.parallel_nil₂ ⟩\n\n  @[pattern]\n  lemma parallel_symm   {Γ} {A B : species ℍ ω Γ}   : (A |ₛ B) ~ (B |ₛ A) := ⟨ equivalent.parallel_symm ⟩\n\n  @[pattern]\n  lemma parallel_assoc₁ {Γ} {A B C : species ℍ ω Γ} : ((A |ₛ B) |ₛ C) ~ (A |ₛ (B |ₛ C)) := ⟨ equivalent.parallel_assoc₁ ⟩\n\n  @[pattern]\n  lemma parallel_assoc₂ {Γ} {A B C : species ℍ ω Γ} : (A |ₛ (B |ₛ C)) ~ ((A |ₛ B) |ₛ C) := ⟨ equivalent.parallel_assoc₂ ⟩\n\n  @[pattern]\n  lemma ν_parallel₁\n      {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} {B : species ℍ ω (context.extend M.arity Γ)}\n    : (ν(M) (rename name.extend A |ₛ B)) ~ (A |ₛ ν(M)B)\n    := ⟨ equivalent.ν_parallel₁ M ⟩\n\n  @[pattern]\n  lemma ν_parallel₂\n      {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} {B : species ℍ ω (context.extend M.arity Γ)}\n    : (A |ₛ ν(M)B) ~ (ν(M) (rename name.extend A |ₛ B))\n    := ⟨ equivalent.ν_parallel₂ M ⟩\n\n  @[pattern]\n  lemma ν_drop₁ {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} : (ν(M) (rename name.extend A)) ~ A\n    := ⟨ equivalent.ν_drop₁ M ⟩\n\n  @[pattern]\n  lemma ν_drop₂ {Γ} (M : affinity ℍ) {A : species ℍ ω Γ} : A ~ (ν(M) (rename name.extend A))\n    := ⟨ equivalent.ν_drop₂ M ⟩\n\n  @[pattern]\n  lemma ν_swap₁\n      {Γ} (M N : affinity ℍ) {A : species ℍ ω (context.extend N.arity (context.extend M.arity Γ))}\n    : (ν(M)ν(N) A) ~ (ν(N)ν(M) rename name.swap A)\n    := ⟨ equivalent.ν_swap₁ M N ⟩\n\n  @[pattern]\n  lemma ν_swap₂ -- Strictly the same as ν_swap₁, as name.swap is symmetric, but...\n      {Γ} (M N : affinity ℍ) {A : species ℍ ω (context.extend N.arity (context.extend M.arity Γ))}\n    : (ν(N)ν(M) rename name.swap A) ~ (ν(M)ν(N) A)\n    := ⟨ equivalent.ν_swap₂ M N ⟩\n\n  -- Show equiv is an equivalence and reflexive operator\n  instance {Γ} : is_equiv (species ℍ ω Γ) equiv :=\n    { refl := equiv.refl, symm := @equiv.symm ℍ ω Γ, trans := @equiv.trans ℍ ω Γ }\n  instance {Γ} : is_refl (species ℍ ω Γ) equiv := ⟨ equiv.refl ⟩\n\n  /-- The setoid of species under structural congruence. Can be brought into\n      scope with the \"congruence\" locale. -/\n  def setoid {Γ} : setoid (species ℍ ω Γ) :=\n    ⟨ equiv, ⟨ @equiv.refl ℍ ω Γ, @equiv.symm ℍ ω Γ, @equiv.trans ℍ ω Γ ⟩ ⟩\n\n  localized \"attribute [instance] cpi.species.equiv.setoid\" in congruence\n\n  lemma parallel_symm₁ {Γ} {A B C : species ℍ ω Γ} : (A |ₛ B |ₛ C) ≈ (B |ₛ A |ₛ C) :=\n    calc  (A |ₛ (B |ₛ C))\n        ≈ ((A |ₛ B) |ₛ C) : parallel_assoc₂\n    ... ≈ ((B |ₛ A) |ₛ C) : ξ_parallel₁ parallel_symm\n    ... ≈ (B |ₛ (A |ₛ C)) : parallel_assoc₁\n\n  lemma parallel_symm₂ {Γ} {A B C : species ℍ ω Γ} : ((A |ₛ B) |ₛ C) ≈ ((A |ₛ C) |ₛ B) :=\n    calc  ((A |ₛ B) |ₛ C)\n        ≈ (A |ₛ (B |ₛ C)) : parallel_assoc₁\n    ... ≈ (A |ₛ (C |ₛ B)) : ξ_parallel₂ parallel_symm\n    ... ≈ ((A |ₛ C) |ₛ B) : parallel_assoc₂\n\n  lemma ν_parallel' {Γ} (M : affinity ℍ) {A : species ℍ ω (context.extend M.arity Γ)} {B : species ℍ ω Γ}\n    : (ν(M) (A |ₛ rename name.extend B)) ≈ ((ν(M)A) |ₛ B) :=\n    calc  (ν(M) A |ₛ rename name.extend B)\n        ≈ (ν(M) rename name.extend B |ₛ A) : ξ_restriction M parallel_symm\n    ... ≈ (B |ₛ ν(M) A) : ν_parallel₁ M\n    ... ≈ ((ν(M) A) |ₛ B) : parallel_symm\n\n  lemma parallel_nil' {Γ} {A : species ℍ ω Γ} : (nil |ₛ A) ≈ A :=\n    calc  (nil |ₛ A)\n        ≈ (A |ₛ nil) : parallel_symm\n    ... ≈ A : parallel_nil₁\nend equiv\n\nopen_locale congruence\n\nnamespace parallel\n  lemma from_list_cons {Γ} (A : species ℍ ω Γ) :\n    ∀ (Bs : list(species ℍ ω Γ)), from_list (A :: Bs) ≈ (A |ₛ from_list Bs)\n  | [] := equiv.parallel_nil₂\n  | (B :: Bs) := refl _\n\n  lemma from_append {Γ} :\n    ∀ (As : list (species ℍ ω Γ)) (Bs : list (species ℍ ω Γ))\n    , from_list (As ++ Bs) ≈ (from_list As |ₛ from_list Bs)\n  | [] B := by { simp only [list.nil_append], from symm equiv.parallel_nil' }\n  | [A] B := from_list_cons A _\n  | (A :: A' :: As) B := begin\n      have h := from_append (A' :: As) B,\n      simp only [from_list, list.cons_append],\n      calc  (A |ₛ from_list (A' :: (As ++ B)))\n          ≈ (A |ₛ (from_list (A' :: As) |ₛ from_list B))\n            : equiv.ξ_parallel₂ h\n      ... ≈ ((A |ₛ from_list (A' :: As)) |ₛ from_list B)\n            : symm equiv.parallel_assoc₁\n    end\n\n  lemma from_to {Γ} : ∀ (A : species ℍ ω Γ), from_list (to_list A) ≈ A\n  | nil := by unfold from_list to_list\n  | (Σ# _) := by unfold from_list to_list\n  | (ν(_) _) := by unfold from_list to_list\n  | (apply _ _) := by unfold from_list to_list\n  | (A |ₛ B) := begin\n      unfold from_list to_list,\n      have a := from_to A, have b := from_to B,\n      calc  from_list (to_list A ++ to_list B)\n          ≈ (from_list (to_list A) |ₛ from_list (to_list B))\n            : from_append (to_list A) (to_list B)\n      ... ≈ (from_list (to_list A) |ₛ B) : equiv.ξ_parallel₂ b\n      ... ≈ (A |ₛ B) : equiv.ξ_parallel₁ a,\n    end\n\n  lemma from_to_append {Γ} (A B : species ℍ ω Γ)\n    : from_list (to_list A ++ to_list B) ≈ (A |ₛ B) := begin\n    have h : to_list A ++ to_list B = to_list (A |ₛ B), unfold to_list,\n    rw h, from from_to _,\n  end\n\n  private lemma from_cons {Γ} :\n    ∀ (A : species ℍ ω Γ) {As Bs : list _}\n    , from_list As ≈ from_list Bs\n    → from_list (A :: As) ≈ from_list (A :: Bs)\n  | A [] [] _ := refl _\n  | A [] (B' :: Bs) eq :=\n      calc  A\n          ≈ (A |ₛ nil) : equiv.parallel_nil₂\n      ... ≈ (A |ₛ from_list (B' :: Bs)) : equiv.ξ_parallel₂ eq\n  | A (A' :: As) [] eq :=\n      calc  (A |ₛ from_list (A' :: As))\n          ≈ (A |ₛ nil) : equiv.ξ_parallel₂ eq\n      ... ≈ A : equiv.parallel_nil₁\n  | A (A' :: As) (B' :: Bs') eq := equiv.ξ_parallel₂ eq\n\n  lemma permute {Γ} :\n    ∀ {As Bs : list (species ℍ ω Γ)}\n    , As ≈ Bs → from_list As ≈ from_list Bs := λ _ _ perm, begin\n    induction perm,\n\n    case list.perm.nil { from refl _ },\n    case list.perm.skip : A As Bs pm ih { from from_cons A ih },\n    case list.perm.swap : A B As {\n      cases As,\n      case list.nil { from equiv.parallel_symm },\n      case list.cons { from equiv.parallel_symm₁ },\n    },\n    case list.perm.trans : As Bs Cs ab bc ih_ab ih_bc { from trans ih_ab ih_bc }\n  end\n\n  namespace quot\n    /-- Make a parallel species from a quotient of two species. -/\n    def mk {Γ} : species' ℍ ω Γ → species' ℍ ω Γ → species' ℍ ω Γ\n    | A B := quotient.lift_on₂ A B (λ A B, ⟦ A |ₛ B ⟧)\n        (λ A B A' B' eqA eqB, quot.sound (trans (equiv.ξ_parallel₁ eqA) ((equiv.ξ_parallel₂ eqB))))\n\n    lemma assoc {Γ} (A B C : species' ℍ ω Γ)\n      : mk A (mk B C) = mk (mk A B) C\n      := begin\n        rcases quot.exists_rep A with ⟨ A, ⟨ _ ⟩ ⟩,\n        rcases quot.exists_rep B with ⟨ B, ⟨ _ ⟩ ⟩,\n        rcases quot.exists_rep C with ⟨ C, ⟨ _ ⟩ ⟩,\n        from quot.sound equiv.parallel_assoc₂,\n      end\n\n    /-- parallel.from_list, lifted to the level of quotients. -/\n    def from_list {Γ} :\n      list (species' ℍ ω Γ) → species' ℍ ω Γ\n    | [] := ⟦ nil ⟧\n    | [A] := A\n    | (A :: As) := mk A (from_list As)\n\n    lemma from_append {Γ} :\n      ∀ (A B : list (species' ℍ ω Γ))\n      , from_list (A ++ B) = mk (from_list A) (from_list B)\n    | [] B := begin\n        simp only [list.nil_append],\n        from quot.induction_on (from_list B) (λ B, quot.sound (symm equiv.parallel_nil')),\n      end\n    | [A] [] := quot.induction_on A (λ A, quot.sound equiv.parallel_nil₂)\n    | [A] (B :: Bs) := rfl\n    | (A :: A' :: As) B := begin\n        show mk A (from_list (A' :: As ++ B)) = mk (mk A (from_list (A' :: As))) (from_list B),\n        rw (from_append (A' :: As) B),\n        from assoc _ _ _,\n      end\n\n    private lemma from_cons {Γ} :\n      ∀ (A : species' ℍ ω Γ) {As Bs : list _}\n      , from_list As = from_list Bs\n      → from_list (A :: As) = from_list (A :: Bs)\n    | A [] [] _ := refl _\n    | A [] (B' :: Bs) eq := begin\n        rcases quot.exists_rep A with ⟨ A, ⟨ _ ⟩ ⟩,\n        simp only [from_list, symm eq],\n        from quot.sound equiv.parallel_nil₂,\n      end\n    | A (A' :: As) [] eq := begin\n        rcases quot.exists_rep A with ⟨ A, ⟨ _ ⟩ ⟩,\n        simp only [from_list, eq],\n        from quot.sound equiv.parallel_nil₁,\n      end\n    | A (A' :: As) (B' :: Bs') eq := by simp only [from_list, eq]\n\n    lemma permute {Γ} :\n      ∀ {As Bs : list (species' ℍ ω Γ)}\n      , As ≈ Bs → from_list As = from_list Bs := λ _ _ perm, begin\n      induction perm,\n\n      case list.perm.nil { from refl _ },\n      case list.perm.skip : A As Bs pm ih { from from_cons A ih },\n      case list.perm.swap : A B As {\n        rcases quot.exists_rep A with ⟨ A, ⟨ _ ⟩ ⟩,\n        rcases quot.exists_rep B with ⟨ B, ⟨ _ ⟩ ⟩,\n\n        cases As,\n        case list.nil { from quot.sound equiv.parallel_symm },\n        case list.cons {\n          unfold from_list,\n          from quotient.induction_on (from_list (As_hd :: As_tl))\n            (λ _, quot.sound equiv.parallel_symm₁),\n        },\n      },\n      case list.perm.trans : As Bs Cs ab bc ih_ab ih_bc { from trans ih_ab ih_bc }\n    end\n  end quot\nend parallel\n\nsection examples\n  variable {Γ : context}\n  variables A A' B C : species ℍ ω Γ\n\n  example : A ≈ (A |ₛ nil) := symm equiv.parallel_nil₁\n\n  example : A ≈ (nil |ₛ A) := trans equiv.parallel_nil₂ equiv.parallel_symm\n\n  example : A ≈ A' → (A |ₛ B) ≈ C → (A' |ₛ B) ≈ C := begin\n    assume a eq,\n    from trans (symm $ equiv.ξ_parallel₁ a) eq\n  end\nend examples\n\nend species\n\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/species/congruence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.31917934034371553}}
{"text": "import Mathlib.Init.Algebra.Order\nimport Mathlib.Init.Data.Nat.Notation\nimport Mathlib.Tactic.Linarith\n\nset_option linter.unusedVariables false\n\nclass CatSystem (Cat : Type) :=\n  [ decEq : DecidableEq Cat ]\n  ( Func : Cat → Cat → Type )\n  ( HasLAdj : ∀ {C D : Cat}, Func C D → Prop )\n  [ decidableLAdj : ∀ (C D : Cat) (F : Func C D), Decidable (HasLAdj F) ]\n  ( HasRAdj : ∀ {C D : Cat}, Func C D → Prop )\n  [ decidableRAdj : ∀ (C D : Cat) (F : Func C D), Decidable (HasRAdj F) ]\n\n/- We have a system of Categories -/\nvariable {Cat : Type} [CatSystem Cat]\n\nattribute [instance] CatSystem.decEq CatSystem.decidableLAdj CatSystem.decidableRAdj\n\nopen CatSystem\n\n--TODO : think about how adjoint preserve products.\n\nmutual\n\ninductive CoreprObj : Cat → Type\n  | Coprod {C : Cat} (X Y : Obj C) : CoreprObj C\n  | LAdj {C D : Cat} (F : Func C D) : HasLAdj F → Obj D → CoreprObj C\n  | Bot : (C : Cat) → CoreprObj C\n\ninductive ReprObj : Cat → Type\n  | Prod {C : Cat} : Obj C → Obj C → ReprObj C\n  | RAdj {C D : Cat} (F : Func C D) : HasRAdj F → Obj D → ReprObj C\n  | Top : (C : Cat) → ReprObj C\n\ninductive Obj : Cat → Type\n  | Corepr : ∀ {C : Cat}, CoreprObj C → Obj C\n  | Var : (C : Cat) → ℕ → Obj C\n  | App : ∀ {C D : Cat}, Func C D → Obj C → Obj D\n  | Repr : ∀ {C : Cat}, ReprObj C → Obj C\n\nend\n\n@[simp]\ndef Obj.size : ∀ {C : Cat} (X : Obj C), ℕ\n  | _, Obj.Corepr (CoreprObj.Coprod X Y) => 1 + Obj.size X + Obj.size Y\n  | _, Obj.Corepr (CoreprObj.LAdj F H X) => 2 + Obj.size X\n  | _, Obj.Corepr (CoreprObj.Bot C) => 1\n  | _, Obj.Var C n => 1\n  | _, Obj.App F X => 1 + Obj.size X\n  | _, Obj.Repr (ReprObj.Prod X Y) => 1 + Obj.size X + Obj.size Y\n  | _, Obj.Repr (ReprObj.RAdj F H X) => 2 + Obj.size X\n  | _, Obj.Repr (ReprObj.Top C) => 1\n\nopen Obj\n\n@[match_pattern, simp]\nnonrec def Obj.Coprod {C : Cat} (X Y : Obj C) : Obj C :=\n  Corepr (CoreprObj.Coprod X Y)\n\n@[match_pattern, simp]\nnonrec def Obj.Prod {C : Cat} (X Y : Obj C) : Obj C :=\n  Obj.Repr (ReprObj.Prod X Y)\n\n@[match_pattern, simp]\nnonrec def Obj.LAdj {C D : Cat} (F : Func C D) (H : HasLAdj F) (X : Obj D) : Obj C :=\n  Corepr (CoreprObj.LAdj F H X)\n\n@[match_pattern, simp]\nnonrec def Obj.RAdj {C D : Cat} (F : Func C D) (H : HasRAdj F) (X : Obj D) : Obj C :=\n  Obj.Repr (ReprObj.RAdj F H X)\n\n@[match_pattern]\nnonrec def Obj.Bot (C : Cat) : Obj C :=\n  Corepr (CoreprObj.Bot C)\n\n@[match_pattern]\nnonrec def Obj.Top (C : Cat) : Obj C :=\n  Obj.Repr (ReprObj.Top C)\n\ninductive HomVar (C : Cat) : (X Y : ℕ) → Type\n  | id : ∀ (X : ℕ), HomVar C X X\n  | varComp {X Y Z : ℕ} (n : ℕ) (f : HomVar C Y Z) : HomVar C X Z\n  deriving DecidableEq\n\ndef HomVar.comp {C : Cat} : ∀ {X Y Z : ℕ} (f : HomVar C X Y) (g : HomVar C Y Z), HomVar C X Z\n  | _, _, _, HomVar.id _, f => f\n  | _, _, _, HomVar.varComp n f, g => HomVar.varComp n (HomVar.comp f g)\n\ninductive Emb : ∀ {C : Cat}, Obj C → Obj C → Type\n  | inl : ∀ {X Y : Obj C}, Emb X (Coprod X Y)\n  | inr : ∀ {X Y : Obj C}, Emb Y (Coprod X Y)\n  | unit : ∀ {D : Cat} (F : Func D C) (H : HasLAdj F) {X : Obj C}, Emb X (App F (LAdj F H X))\n  deriving DecidableEq\n\ninductive Proj : ∀ {C : Cat}, Obj C → Obj C → Type\n  | fst : ∀ {X Y : Obj C}, Proj (Prod X Y) X\n  | snd : ∀ {X Y : Obj C}, Proj (Prod X Y) Y\n  | counit : ∀ {D : Cat} (F : Func D C) (H : HasRAdj F) {X : Obj C},\n      Proj (App F (RAdj F H X)) X\n  deriving DecidableEq\n\ntheorem size_lt_of_emb {C : Cat} {X Y : Obj C} (f : Emb X Y) : Obj.size X < Obj.size Y := by\n  cases f <;> simp [add_comm] <;> linarith\n\ntheorem size_lt_of_proj {C : Cat} {X Y : Obj C} (f : Proj X Y) : Obj.size Y < Obj.size X := by\n  cases f <;> simp <;> linarith\n\nmutual\n\n/- Currently have no way of writing certain homs.\n-- map F (corepr anything) ; counit : App F (Corepr _) -> App F (Radj F _ _)\n-- map F (projComp (fst or snd)) ; counit : App F (X × _) -> App F (Radj F _ X) -/\n\ninductive HomCorepr : {C : Cat} → CoreprObj C → Obj C → Type\n  | coprod {C : Cat} {X Y Z : Obj C} (f : Hom X Z) (g : Hom Y Z) : HomCorepr (CoreprObj.Coprod X Y) Z\n  | ladj {C D : Cat} (F : Func C D) (H : HasLAdj F) {X : Obj D} {Y : Obj C}\n      (f : Hom X (App F Y)) : HomCorepr (CoreprObj.LAdj F H X) Y\n  | botMk {C : Cat} {X : Obj C} : HomCorepr (CoreprObj.Bot C) X\n\ninductive HomRepr : ∀ {C : Cat}, Obj C → ReprObj C → Type\n  | prod {C : Cat} {X : Obj C} {Y Z : Obj C}\n    (f : Hom X Y) (g : Hom X Z) : HomRepr X (ReprObj.Prod Y Z)\n  | radj {C D : Cat} (F : Func C D) (H : HasRAdj F) {X : Obj C} {Y : Obj D}\n    (f : Hom (App F X) Y) : HomRepr X (ReprObj.RAdj F H Y)\n  | topMk {C : Cat} {X : Obj C} : HomRepr X (ReprObj.Top C)\n\ninductive Hom : ∀ {C : Cat}, Obj C → Obj C → Type\n  | projComp : ∀ {C : Cat} {X Y Z : Obj C} (f : Proj X Y) (g : Hom Y Z), Hom X Z\n  | compEmb : ∀ {C : Cat} {X : Obj C} {Y Z : Obj C} (f : Hom X Y) (g : Emb Y Z), Hom X Z\n  | corepr : ∀ {C : Cat} {X : CoreprObj C} {Y : Obj C} (f : HomCorepr X Y), Hom (Corepr X) Y\n  | repr : ∀ {C : Cat} {X : Obj C} {Y : ReprObj C} (f : HomRepr X Y), Hom X (Repr Y)\n  | map : ∀ {C D : Cat} {X Y : Obj C} (F : Func C D) (f : Hom X Y), Hom (App F X) (App F Y)\n  | var : ∀ {C : Cat} {X Y : ℕ}, HomVar C X Y → Hom (Var C X) (Var C Y)\n\n/-Provided LAdj is bigger than map, every Hom constructor make homs from homs between smaller objects.\nBy smaller I mean that -/\n\nend\n\nnamespace Hom\n\ndef id : ∀ {C : Cat} {X : Obj C}, Hom X X\n  | _, Obj.Var C X => Hom.var (HomVar.id _)\n  | _, Obj.App F X => Hom.map F Hom.id\n  | _, Obj.Repr (ReprObj.Prod X Y) => repr (HomRepr.prod (Hom.projComp Proj.fst Hom.id) (Hom.projComp Proj.snd Hom.id))\n  | _, Obj.Repr (ReprObj.RAdj F H Y) => repr (HomRepr.radj F H (Hom.projComp (Proj.counit F H) Hom.id))\n  | _, Obj.Repr (ReprObj.Top C) => repr HomRepr.topMk\n  | _, Obj.Corepr (CoreprObj.Coprod X Y) => corepr (HomCorepr.coprod (Hom.compEmb Hom.id Emb.inl) (Hom.compEmb Hom.id Emb.inr))\n  | _, Obj.Corepr (CoreprObj.LAdj F H X) => corepr (HomCorepr.ladj F H (Hom.compEmb Hom.id (Emb.unit F H)))\n  | _, Obj.Corepr (CoreprObj.Bot C) => corepr HomCorepr.botMk\n\ndef ofEmb {C : Cat} {X Y : Obj C} (f : Emb X Y) : Hom X Y :=\n  compEmb Hom.id f\n\ndef ofProj {C : Cat} {X Y : Obj C} (f : Proj X Y) : Hom X Y :=\n  projComp f Hom.id\n\ndef inl {C : Cat} {X Y : Obj C} : Hom X (Coprod X Y) :=\n  ofEmb Emb.inl\n\ndef inr {C : Cat} {X Y : Obj C} : Hom Y (Coprod X Y) :=\n  ofEmb Emb.inr\n\ndef unit {C : Cat} {D : Cat} (F : Func C D) (H : HasLAdj F) {X : Obj D} : Hom X (App F (LAdj F H X)) :=\n  ofEmb (Emb.unit F H)\n\ndef fst {C : Cat} {X Y : Obj C} : Hom (Prod X Y) X :=\n  ofProj Proj.fst\n\ndef snd {C : Cat} {X Y : Obj C} : Hom (Prod X Y) Y :=\n  ofProj Proj.snd\n\ndef counit {C : Cat} {D : Cat} (F : Func C D) (H : HasRAdj F) {X : Obj D} : Hom (App F (RAdj F H X)) X :=\n  ofProj (Proj.counit F H)\n\n@[match_pattern]\ndef botMk {C : Cat} {X : Obj C} : Hom (Bot C) X :=\n  corepr HomCorepr.botMk\n\n@[match_pattern]\ndef topMk {C : Cat} {X : Obj C} : Hom X (Top C) :=\n  repr HomRepr.topMk\n\n@[match_pattern]\ndef coprod {C : Cat} {X Y Z : Obj C} (f : Hom X Z) (g : Hom Y Z) : Hom (Coprod X Y) Z :=\n  corepr (HomCorepr.coprod f g)\n\n@[match_pattern]\ndef prod {C : Cat} {X Y Z : Obj C} (f : Hom X Y) (g : Hom X Z) : Hom X (Prod Y Z) :=\n  repr (HomRepr.prod f g)\n\n@[match_pattern]\ndef radj {C : Cat} {D : Cat} (F : Func C D) (H : HasRAdj F) {X : Obj C} {Y : Obj D} (f : Hom (App F X) Y) : Hom X (RAdj F H Y) :=\n  repr (HomRepr.radj F H f)\n\n@[match_pattern]\ndef ladj {C : Cat} {D : Cat} (F : Func C D) (H : HasLAdj F) {X : Obj C} {Y : Obj D} (f : Hom Y (App F X)) : Hom (LAdj F H Y) X :=\n  corepr (HomCorepr.ladj F H f)\n\nmutual\n\ndef coreprComp : ∀ {C : Cat} {X : CoreprObj C} {Y Z : Obj C}, HomCorepr X Y → Hom Y Z → HomCorepr X Z\n  | _, _, _, _, HomCorepr.coprod f g, h => HomCorepr.coprod (comp f h) (comp g h)\n  | _, _, _, _, HomCorepr.ladj F H f, g => HomCorepr.ladj _ _ (comp f (Hom.map F g))\n  | _, _, _, _, HomCorepr.botMk, _ => HomCorepr.botMk\n\ndef compRepr : ∀ {C : Cat} {X Y : Obj C} {Z : ReprObj C}, Hom X Y → HomRepr Y Z → HomRepr X Z\n  | _, _, _, _, f, HomRepr.prod g h => HomRepr.prod (comp f g) (comp f h)\n  | _, _, _, _, f, HomRepr.radj F H g => HomRepr.radj _ _ (comp (Hom.map F f) g)\n  | _, _, _, _, _, HomRepr.topMk => HomRepr.topMk\n\ndef comp : ∀ {C : Cat} {X Y Z : Obj C}, Hom X Y → Hom Y Z → Hom X Z\n  | _, _, _, _, Hom.corepr f, g => corepr (coreprComp f g)\n  | _, _, _, _, f, Hom.repr g => repr (compRepr f g)\n  | _, W, Y, Z, Hom.projComp (Y := X) f g, h =>\n    have : size X + size Z < size W + size Z :=\n      by linarith [size_lt_of_proj f]\n    Hom.projComp f (comp g h)\n  | _, W, X, Z, f, Hom.compEmb (Y := Y) g h =>\n    have : size W + size Y < size W + size Z :=\n      by linarith [size_lt_of_emb h]\n    compEmb (f.comp g) h\n  | _, _, _, _, prod f g, projComp Proj.fst h => comp f h\n  | _, W, Obj.Prod X Y, Z, prod f g, projComp Proj.snd h =>\n    have : size Y < 1 + size X + size Y := by linarith\n    comp g h\n  | _, _, _, _, compEmb f Emb.inl, coprod g h => f.comp g\n  | _, W, Obj.Coprod X Y, Z, compEmb f Emb.inr, coprod g h =>\n    have : size Y < 1 + size X + size Y := by linarith\n    f.comp h\n  | _, _, _, _, var f, var g => var (HomVar.comp f g)\n  | _, _, _, _, map F f, map _ g => map _ (comp f g)\n  | _, _, _, _, map _ (projComp _ _), projComp (Proj.counit _ _) _ => sorry  --New constructor\n  | _, _, _, _, map _ (Hom.repr (HomRepr.radj _ _ f)), projComp (Proj.counit _ _) g => f.comp g\n  | _, _, _, _, map _ (Hom.corepr HomCorepr.botMk), projComp (Proj.counit _ _) _ => sorry --Things must preserve coproducts in a stronger way.\n  | _, _, _, _, map G (Hom.corepr (HomCorepr.ladj F H f)), projComp (Proj.counit _ _) g => sorry --New constructor\n  | _, _, _, _, _, _ => sorry\n\nend\ntermination_by comp C X Y Z f g => (size X + size Z, size Y, 1)\n               coreprComp C X Y Z f g => (size (Corepr X) + size Z, size Y, 0)\n               compRepr C X Y Z f g => (size X + size (Repr Z), size Y, 0)\n\ndef LAdjSymm {C D : Cat} (F : Func C D) (H : HasLAdj F) {X : Obj D} {Y : Obj C}\n    (f : Hom (Obj.LAdj F H X) Y) : Hom X (App F Y) :=\n  Hom.comp (Hom.compEmb Hom.id (Emb.unit _ H)) (map F f)\n\ndef RAdjSymm {C D : Cat} (F : Func C D) (H : HasRAdj F) {X : Obj C} {Y : Obj D}\n    (f : Hom X (Obj.RAdj F H Y)) : Hom (App F X) Y :=\n  Hom.comp (map F f) (Hom.projComp (Proj.counit _ H) Hom.id)\n\ndef ladjMap {C D : Cat} (F : Func C D) (H : HasLAdj F) {X Y : Obj D} (f : Hom X Y) :\n    Hom (LAdj F H X) (LAdj F H Y) :=\n  ladj _ _ (comp f (unit _ _))\n\ndef radjMap {C D : Cat} (F : Func C D) (H : HasRAdj F) {X Y : Obj D} (f : Hom X Y) :\n    Hom (RAdj F H X) (RAdj F H Y) :=\n  radj _ _ (comp (counit _ _) f)\n\ndef ladjPreserveBot {C D : Cat} (F : Func C D) (H : HasLAdj F) : Hom (LAdj F H (Obj.Bot _)) (Obj.Bot _) :=\n  ladj _ _ botMk\n\ndef radjPreserveTop {C D : Cat} (F : Func C D) (H : HasRAdj F) : Hom (Obj.Top _) (RAdj F H (Obj.Top _)) :=\n  radj _ _ topMk\n\ndef ladjPreserveCoprod {C D : Cat} (F : Func C D) (H : HasLAdj F) {X Y : Obj D} :\n    Hom (LAdj F H (Obj.Coprod X Y)) (Obj.Coprod (LAdj F H X) (LAdj F H Y)) :=\n  ladj _ _ (coprod (LAdjSymm _ H inl) (LAdjSymm _ H inr))\n\ndef ladjPreserveCoprodSymm {C D : Cat} (F : Func C D) (H : HasLAdj F) {X Y : Obj D} :\n    Hom (Obj.Coprod (LAdj F H X) (LAdj F H Y)) (LAdj F H (Obj.Coprod X Y)) :=\n  coprod (ladj _ _ (comp (unit F H) (map F (ladjMap _ _ inl))))\n         (ladj _ _ (comp (unit F H) (map F (ladjMap _ _ inr))))\n\ndef radjPreserveProd {C D : Cat} (F : Func C D) (H : HasRAdj F) {X Y : Obj D} :\n    Hom (Obj.Prod (RAdj F H X) (RAdj F H Y)) (RAdj F H (Obj.Prod X Y)) :=\n  radj _ _ (prod (RAdjSymm _ H fst) (RAdjSymm _ H snd))\n\ndef radjPreserveProdSymm {C D : Cat} (F : Func C D) (H : HasRAdj F) {X Y : Obj D} :\n    Hom (RAdj F H (Obj.Prod X Y)) (Obj.Prod (RAdj F H X) (RAdj F H Y)) :=\n  prod (radj _ _ (comp (map F (radjMap _ _ fst)) (counit F H)))\n       (radj _ _ (comp (map F (radjMap _ _ snd)) (counit F H)))\n\ndef preserveBotOfHasRAdj {C D : Cat} (F : Func C D) (H : HasRAdj F) :\n    Hom (App F (Obj.Bot _)) (Obj.Bot _):=\n  RAdjSymm _ H botMk\n\ndef preserveTopOfHasLAdj {C D : Cat} (F : Func C D) (H : HasLAdj F) :\n    Hom (Obj.Top _) (App F (Obj.Top _)) :=\n  LAdjSymm _ H topMk\n\ndef preserveCoprodOfHasRAdj {C D : Cat} (F : Func C D) (H : HasRAdj F) {X Y : Obj C} :\n    Hom (App F (Obj.Coprod X Y)) (Obj.Coprod (App F X) (App F Y)) :=\n  RAdjSymm _ H (coprod (radj _ _ inl) (radj _ _ inr))\n\ndef preserveCoprodOfHasRAdjSymm {C D : Cat} (F : Func C D) (H : HasRAdj F) {X Y : Obj C} :\n    Hom (Obj.Coprod (App F X) (App F Y)) (App F (Obj.Coprod X Y)) :=\n  coprod (map F inl) (map F inr)\n\ndef preserveProdOfHasLAdj {C D : Cat} (F : Func C D) (H : HasLAdj F) {X Y : Obj C} :\n    Hom (Obj.Prod (App F X) (App F Y)) (App F (Obj.Prod X Y)) :=\n  LAdjSymm _ H (prod (ladj _ _ fst) (ladj _ _ snd))\n\ndef preserveProdOfHasLAdjSymm {C D : Cat} (F : Func C D) (H : HasLAdj F) {X Y : Obj C} :\n    Hom (App F (Obj.Prod X Y)) (Obj.Prod (App F X) (App F Y)) :=\n  prod (map F fst) (map F snd)\n\nsection\n\ndef asCorepr : ∀ {C : Cat} {X : CoreprObj C} {Y : Obj C}, Hom (Corepr X) Y → HomCorepr X Y\n  | _, _, _, Hom.corepr f => f\n  | _, CoreprObj.Bot _, _, _ => HomCorepr.botMk\n  | _, CoreprObj.Coprod X Y, _, f => HomCorepr.coprod (comp (ofEmb Emb.inl) f) (comp (ofEmb Emb.inr) f)\n  | _, CoreprObj.LAdj F H X, _, f => HomCorepr.ladj _ _ (comp (ofEmb (Emb.unit F H)) (map F f))\n\ndef asRepr : ∀ {C : Cat} {X : Obj C} {Y : ReprObj C}, Hom X (Repr Y) → HomRepr X Y\n  | _, _, _, Hom.repr f => f\n  | _, _, ReprObj.Prod X Y, f => HomRepr.prod (comp f (ofProj Proj.fst)) (comp f (ofProj Proj.snd))\n  | _, _, ReprObj.RAdj F H X, f => HomRepr.radj _ _ (comp (map F f) (ofProj (Proj.counit F H)))\n  | _, _, ReprObj.Top _, _ => HomRepr.topMk\n\nend\n\ninstance : ∀ C : Cat, DecidableEq (Obj C) := sorry\ninstance : ∀ C : Cat, DecidableEq (CoreprObj C) := sorry\ninstance : ∀ C : Cat, DecidableEq (ReprObj C) := sorry\ninstance : ∀ C : Cat, ∀ X Y : Obj C, DecidableEq (Hom X Y) := sorry\ninstance : ∀ C : Cat, ∀ X : CoreprObj C, ∀ Y : Obj C, DecidableEq (HomCorepr X Y) := sorry\ninstance : ∀ C : Cat, ∀ X : Obj C, ∀ Y : ReprObj C, DecidableEq (HomRepr X Y) := sorry\n\nmutual\n\n/-\nNormal forms\n- If it can be written as `f ; corepr g` then it is unless the first rule applies. What if there are two different ways of doing this?,\n    Try to make sure `f` is not of that form\n    Also `corepr g` should remain a subterm after `f` and `corepr g` are composed and cut eliminated.\n- If it can be written as `repr_mk ; f` then it is unless the first rule applies.\n- Not sure what else there is, just associativity of `projComp` and `compEmb`\n-/\n\n/-\nQuestions: What is shrinking? I have to make sure that everything splits into smaller homs.\nI decided to do products before LAdj. Why? I don't think this applies if I insist on objects shrinking.\n\n-/\n\nopen Hom\n\ndef getCompCorepr :\n    ∀ {C : Cat} {X Y : Obj C} (f : Hom X Y),\n      Option (Σ R : CoreprObj C, Hom X (Corepr R) × HomCorepr R Y × Bool)\n      --Bool is true if the `Hom X (Corepr R)` is the identity\n  | _, Corepr R, _, f => some ⟨_, Hom.id, asCorepr f, true⟩\n  | _, _, _, var _ => none\n  | _, _, _, topMk => none\n  | _, _, _, Hom.repr (HomRepr.radj F H f) => none\n  | _, _, _, map F f => none\n  | _, _, _, projComp f g =>\n    match getCompCorepr g with\n    | none => none\n    | some ⟨R, g, h, _⟩ => some ⟨R, projComp f g , h, false⟩\n  | _, _, _, compEmb f h =>\n    match getCompCorepr f with\n    | none => none\n    | some ⟨R, f, g, _⟩ => some ⟨R, f, coreprComp g (ofEmb h), false⟩\n  | _, _, _, Hom.repr (HomRepr.prod f g) =>\n    match getCompCorepr f, getCompCorepr g with\n    | some ⟨R₁, f₁, f₂, _⟩, some ⟨R₂, g₁, g₂, _⟩ =>\n        let nf := normalize f₁\n        let ng := normalize g₁\n        if hr : R₁ = R₂\n          then if (hr ▸ nf) = ng\n            then by\n              subst R₁\n              match R₂, nf, f₂, g₂ with\n              | _, nf, HomCorepr.coprod f₂ f₃, HomCorepr.coprod g₂ g₃ =>\n                exact some ⟨_, nf, HomCorepr.coprod (prod f₂ g₂) (prod f₃ g₃), false⟩\n              | _, nf, HomCorepr.botMk, HomCorepr.botMk => exact some ⟨_, nf, HomCorepr.botMk, false⟩\n              | _, nf, HomCorepr.ladj _ _ _, _ => exact none\n            else none\n        else none\n    | _, _ => none\n\ndef getReprComp :\n    ∀ {C : Cat} {X Y : Obj C} (f : Hom X Y), Option (Σ R : ReprObj C, HomRepr X R × Hom (Repr R) Y × Bool)\n  | _, Corepr R, _, f => none\n  | _, _, Obj.Repr R, f => some ⟨_, asRepr f, Hom.id, true⟩\n  | _, _, _, var _ => none\n  | _, _, _, map F f => none\n  | _, _, _, projComp f g =>\n    match getReprComp g with\n    | none => none\n    | some ⟨R, g, h, _⟩ => some ⟨R, compRepr (ofProj f) g, h, false⟩\n  | _, _, _, compEmb f h =>\n    match getReprComp f with\n    | none => none\n    | some ⟨R, f, g, _⟩ => some ⟨R, f, compEmb g h, false⟩\n\ndef normalizeCorepr : ∀ {C : Cat} {X : CoreprObj C} {Y : Obj C} (f : HomCorepr X Y),\n    HomCorepr X Y\n  | _, _, _, HomCorepr.coprod f g => HomCorepr.coprod (normalize f) (normalize g)\n  | _, _, _, HomCorepr.botMk => HomCorepr.botMk\n  | _, _, _, HomCorepr.ladj F H f => HomCorepr.ladj F H (normalize f)\n\ndef normalizeRepr : ∀ {C : Cat} {X : Obj C} {Y : ReprObj C} (f : HomRepr X Y),\n    HomRepr X Y\n  | _, _, _, HomRepr.radj F H f => HomRepr.radj F H (normalize f)\n  | _, _, _, HomRepr.prod f g => HomRepr.prod (normalize f) (normalize g)\n  | _, _, _, HomRepr.topMk => HomRepr.topMk\n\ndef normalize {C : Cat} {X Y : Obj C} (f : Hom X Y) : Hom X Y :=\n  match getCompCorepr f with\n  | none =>\n    match getReprComp f with\n    | none =>\n\nend\n\nend Hom\n", "meta": {"author": "ChrisHughes24", "repo": "categories", "sha": "cea523dfe116a2c5ecc8f6ab5a8ed23b32da2460", "save_path": "github-repos/lean/ChrisHughes24-categories", "path": "github-repos/lean/ChrisHughes24-categories/categories-cea523dfe116a2c5ecc8f6ab5a8ed23b32da2460/Categories/ThirdAttempt.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.31917934034371553}}
{"text": "namespace ex\nopen tactic\n\nconstant typ : Type\n\nconstant subtype : typ → typ → Prop\n\nconstant subtype_refl : ∀ T, subtype T T\n\nconstant subtype_trans : ∀ S T U, subtype S T → subtype T U → subtype S U\n\nattribute [intro] subtype_refl subtype_trans\n\nlemma L1 : ∀ T1 T2 T3 T4, subtype T1 T2 → subtype T2 T3 → subtype T3 T4 → subtype T1 T4 :=\nby (intros >> back_chaining_using_hs)\n\nlemma L2 : ∀ T1 T2 T3 T4, subtype T1 T2 → subtype T2 T3 → subtype T3 T4 → subtype T1 T4 :=\nby do\n  intros,\n  ctx ← local_context,\n  -- using pre tactic to trace execution\n  back_chaining_core (trace \"pre tac:\" >> trace_state >> trace \"-------\") failed ctx\n\nset_option back_chaining.max_depth 10\nset_option trace.tactic.back_chaining true\n\nlemma L3 : ∀ T1 T2 T3 T4 T5 T6 (H1 :subtype T1 T2) (H2 : subtype T2 T3) (H3 : subtype T3 T4) (H3 : subtype T4 T5) (H4 : subtype T5 T6), subtype T1 T6 :=\nby (intros >> back_chaining_using_hs)\n\nend ex", "meta": {"author": "mathprocessing", "repo": "lean_mathlib_examples", "sha": "743c6456c0a3219dd1722efdd31ee6f3a113818a", "save_path": "github-repos/lean/mathprocessing-lean_mathlib_examples", "path": "github-repos/lean/mathprocessing-lean_mathlib_examples/lean_mathlib_examples-743c6456c0a3219dd1722efdd31ee6f3a113818a/src/tactics/back_chaining3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.319150439347027}}
{"text": "import data.int.basic\n\nnamespace tactic.interactive\n\nopen tactic\n\nlemma translate_nat (P : ℤ → Sort*) (h : ∀ i : ℤ, i ≥ 0 → P i) :\n  ∀ i : ℕ, P i :=\nassume i,\nh i (int.coe_nat_nonneg i)\n\nrun_cmd mk_simp_attr `to_int\n\n@[to_int]\nlemma coe_nat_inj {m n : ℕ} : m = n ↔ (↑m : ℤ) = ↑n := by rw int.coe_nat_inj'\n\n@[to_int]\nlemma coe_nat_le {m n : ℕ} : m ≤ n ↔ (↑m : ℤ) ≤ ↑n := by rw int.coe_nat_le\n\n@[to_int]\nlemma coe_nat_lt {m n : ℕ} : m < n ↔ (↑m : ℤ) < ↑n := by rw int.coe_nat_lt\n\n@[to_int]\nlemma coe_nat_ge {m n : ℕ} : m ≥ n ↔ (↑m : ℤ) ≥ ↑n := coe_nat_le\n\n@[to_int]\nlemma coe_nat_gt {m n : ℕ} : m > n ↔ (↑m : ℤ) > ↑n := coe_nat_lt\n\n@[to_int]\ntheorem coe_nat_dvd {m n : ℕ} : m ∣ n ↔ (↑m : ℤ) ∣ ↑n := by rw int.coe_nat_dvd\n\n@[to_int]\ntheorem coe_bit0 {m : ℕ} : ↑ (bit0 m) = bit0 (↑m : ℤ) := by simp [bit0]\n\n@[to_int]\ntheorem coe_bit1 {m : ℕ} : ↑ (bit1 m) = bit1 (↑m : ℤ) := by simp [bit1,bit0]\n\nattribute [to_int] int.coe_nat_mod\n  int.bit_coe_nat\n  int.coe_nat_add int.coe_nat_mul\n  int.coe_nat_sub\n  int.coe_nat_zero\n  int.coe_nat_one\n\nmeta def to_int : tactic unit :=\ndo `[simp only with to_int at *],\n   ls ← local_context,\n   ls ← ls.mfilter $ λ a,\n     succeeds $ infer_type a >>= is_def_eq `(nat),\n   ls ← ls.mmap $ λ v,\n     do { tactic.revert v <*\n          do e ← target,\n             ((p,l),_) ← solve_aux e $ do {\n               v ← intro1,\n               p ← to_expr ``(↑ %%v : int),\n               tactic.generalize p,\n               (expr.pi n bi b d) ← target,\n               pure (p,expr.lam n bi b d) } ,\n             refine ``(translate_nat %%l _),\n             intro v.local_pp_name,\n             target >>= head_beta >>= unsafe_change,\n             tactic.intro $ mk_simple_name $ to_string v.local_pp_name ++ \"_nneg\" },\n   ls.reverse.mmap' $ λ i,\n     do intron (i - 1),\n        pure ()\n\nend tactic.interactive\n", "meta": {"author": "leanprover-community", "repo": "mathlib-nursery", "sha": "0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec", "save_path": "github-repos/lean/leanprover-community-mathlib-nursery", "path": "github-repos/lean/leanprover-community-mathlib-nursery/mathlib-nursery-0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec/src/tactic/to_int.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.31905551509805496}}
{"text": "/-\nCopyright (c) 2021 Alex J. Best. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex J. Best\n-/\n\nimport number_theory.cyclotomic.primitive_roots\nimport ring_theory.polynomial.cyclotomic.eval\n\nnoncomputable theory\n\nopen_locale big_operators non_zero_divisors number_field\nopen number_field polynomial finset module units submodule\n\nuniverse variables u v w z\n\nvariables (n : ℕ+) (K : Type u) (L : Type v) (A : Type w) (B : Type z)\nvariables [comm_ring A] [comm_ring B] [algebra A B]\nvariables [field K] [field L] [algebra K L]\n\nnamespace is_primitive_root\n\nvariable {B}\n\nvariable (B)\n\nend is_primitive_root\n\nnamespace is_cyclotomic_extension\n\nvariables [is_cyclotomic_extension {n} A B]\n\nvariables [is_domain A] [algebra A K] [is_fraction_ring A K]\n\nopen is_cyclotomic_extension\n\nopen is_cyclotomic_extension\n\nsection cyclotomic_unit\n\nvariable {n}\n\nnamespace cyclotomic_unit\n\n-- I don't want to bother Leo, but I wonder if this can be automated in Lean4\n-- (if they were 0 < n → 1 < n, it would work already!)\nlemma _root_.nat.one_lt_of_ne : ∀ n : ℕ, n ≠ 0 → n ≠ 1 → 1 < n\n| 0 h _ := absurd rfl h\n| 1 _ h := absurd rfl h\n| (n+2) _ _ := n.one_lt_succ_succ\n\n-- this result holds for all primitive roots; dah.\nlemma associated_one_sub_pow_primitive_root_of_coprime {n j k : ℕ} {ζ : A}\n  (hζ : is_primitive_root ζ n) (hk : k.coprime n) (hj : j.coprime n) :\n  associated (1 - ζ ^ j) (1 - ζ ^ k) :=\nbegin\n  suffices : ∀ {j : ℕ}, j.coprime n → associated (1 - ζ) (1 - ζ ^ j),\n  { exact (this hj).symm.trans (this hk) },\n  clear' k j hk hj,\n  refine λ j hj, associated_of_dvd_dvd ⟨∑ i in range j, ζ ^ i,\n    by rw [← geom_sum_mul_neg, mul_comm]⟩ _,\n  -- is there an easier way to do this?\n  rcases eq_or_ne n 0 with rfl | hn',\n  { simp [j.coprime_zero_right.mp hj] },\n  rcases eq_or_ne n 1 with rfl | hn,\n  { simp [is_primitive_root.one_right_iff.mp hζ] },\n  replace hn := n.one_lt_of_ne hn' hn,\n  obtain ⟨m, hm⟩ := nat.exists_mul_mod_eq_one_of_coprime hj hn,\n  use (∑ i in range m, (ζ ^ j) ^ i),\n  have : ζ = (ζ ^ j) ^ m,\n  { rw [←pow_mul, pow_eq_mod_order_of, ←hζ.eq_order_of, hm, pow_one] },\n  nth_rewrite 0 this,\n  rw [← geom_sum_mul_neg, mul_comm]\nend\n\nlemma is_primitive_root.sum_pow_unit {n k : ℕ} {ζ : A} (hn : 2 ≤ n) (hk : k.coprime n)\n(hζ : is_primitive_root ζ n) : is_unit (∑ (i : ℕ) in range k, ζ^(i : ℕ)) :=\nbegin\n  have h1 : (1 : ℕ).coprime n, by {exact nat.coprime_one_left n, },\n  have := associated_one_sub_pow_primitive_root_of_coprime _ hζ hk h1,\n  simp at this,\n  rw associated at this,\n  have h2 := mul_neg_geom_sum ζ k,\n  obtain ⟨u, hu⟩ := this,\n  have := u.is_unit,\n  convert this,\n  rw ←hu at h2,\n  simp at h2,\n  cases h2,\n  exact h2,\n  exfalso,\n  have hn1 : 1 < n, by {linarith},\n  have  hp := is_primitive_root.pow_ne_one_of_pos_of_lt hζ one_pos hn1,\n  simp at *,\n  rw sub_eq_zero at h2,\n  rw ←h2 at hp,\n  simp only [eq_self_iff_true, not_true] at hp,\n  exact hp,\nend\n\nlemma is_primitive_root.zeta_pow_sub_eq_unit_zeta_sub_one {p i j : ℕ} {ζ : A} (hn : 2 ≤ p) (hp : p.prime )\n  (hi : i < p) (hj : j < p) (hij : i ≠ j) (hζ : is_primitive_root ζ p) :\n  ∃ (u : Aˣ), (ζ^i - ζ^j ) = u * (1- ζ) :=\nbegin\n  by_cases hilj : j < i,\n  have h1 : (ζ^i - ζ^j) = ζ^j * (ζ^(i-j)-1), by {ring_exp, rw pow_mul_pow_sub _ hilj.le,\n    rw add_comm,},\n  rw h1,\n  have h2 := mul_neg_geom_sum ζ (i-j),\n  have hic : (i-j).coprime p, by {rw nat.coprime_comm, apply nat.coprime_of_lt_prime _ _ hp,\n    apply nat.sub_pos_of_lt hilj,\n    by_cases hj : 0 < j,\n    apply lt_trans _ hi,\n    apply nat.sub_lt_of_pos_le _ _ hj hilj.le,\n    simp only [not_lt, _root_.le_zero_iff] at hj,\n    rw hj,\n    simp only [tsub_zero],\n    exact hi,},\n  have h3 : is_unit (-(ζ^(j))*(∑ (k : ℕ) in range (i-j), ζ^(k : ℕ))),\n    by {apply is_unit.mul _ (is_primitive_root.sum_pow_unit _ hn hic hζ), apply is_unit.neg,\n      apply is_unit.pow, apply hζ.is_unit hp.pos,  },\n  obtain ⟨v, hv⟩ := h3,\n  use v,\n  rw hv,\n  rw mul_comm at h2,\n  rw mul_assoc,\n  rw h2,\n  ring,\n  simp at *,\n  have h1 : (ζ^i - ζ^j) = ζ^i * (1-ζ^(j-i)), by {ring_exp, simp, rw pow_mul_pow_sub _ hilj,},\n  rw h1,\n  have h2 := mul_neg_geom_sum ζ (j-i),\n  have hjc : (j-i).coprime p, by {rw nat.coprime_comm,\n    apply nat.coprime_of_lt_prime _ _ hp,\n    have hilj' : i < j, by {rw lt_iff_le_and_ne, simp [hij, hilj], },\n    apply nat.sub_pos_of_lt hilj',\n    by_cases hii : 0 < i,\n    apply lt_trans _ hj,\n    apply nat.sub_lt_of_pos_le _ _ hii hilj,\n    simp only [not_lt, _root_.le_zero_iff] at hii,\n    rw hii,\n    simp only [tsub_zero],\n    exact hj,\n  },\n  have h3 : is_unit ((ζ^(i))*(∑ (k : ℕ) in range (j-i), ζ^(k : ℕ))), by {\n    apply is_unit.mul _ (is_primitive_root.sum_pow_unit _ hn hjc hζ), apply is_unit.pow,\n    apply hζ.is_unit hp.pos,},\n   obtain ⟨v, hv⟩ := h3,\n  use v,\n  rw hv,\n  rw mul_comm at h2,\n  rw mul_assoc,\n  rw h2,\n\n\nend\n\n/-\ndef unitlem2 {n k : ℕ} {ζ : A} (hk : nat.coprime n k)\n(hζ : is_primitive_root ζ n) : Aˣ :=\n{ val := (∑ (i : finset.range k), ζ^(i : ℕ)),\n  inv := (ζ-1)  ,\n  val_inv := admit,\n  inv_val := admit,\n\n}\n\n\n\n\nvariable (n)\n\ninstance : is_localization ((ring_of_integers (cyclotomic_field n K)))⁰ (cyclotomic_field n K) :=\nadmit\n\nlemma prime_ideal_eq_pow_cyclotomic [hn : fact ((n : ℕ).prime)] :\n  (span_singleton _ n : fractional_ideal RR⁰ L) =\n  (span_singleton _ (1 - (zeta_runity n K L)) ^ ((n : ℕ) - 1) : fractional_ideal RR⁰ L) :=\n  --(mk0 (p : cyclotomic_field p) (by norm_num [hn.ne_zero]))\nbegin\n  rw fractional_ideal.span_singleton_pow,\n  apply coe_to_submodule_injective,\n  simp only [coe_span_singleton, coe_coe],\n  -- rw ideal.span_singleton_eq_span_singleton,\n  simp only [submodule.span_singleton_eq_span_singleton],\n  rw ← eval_one_cyclotomic_prime,\n  --rw calc\n  --  eval 1 (cyclotomic n (cyclotomic_field n)) = _ : by simp_rw\n  --    cyclotomic_eq_prod_X_sub_primitive_roots (zeta_primitive_root n _)\n  --                      ... = _ : by simp only [polynomial.eval_sub, polynomial.eval_C,\n  --                                  polynomial.eval_prod, polynomial.eval_X],\n\n  -- apply span_singleton_eq_span_singleton_,\n  admit,\nend -/\n\nend cyclotomic_unit\n\nend cyclotomic_unit\n\nend is_cyclotomic_extension\n", "meta": {"author": "leanprover-community", "repo": "flt-regular", "sha": "1d0cecf99e8ab3f98b551e5932bf907042daa6ad", "save_path": "github-repos/lean/leanprover-community-flt-regular", "path": "github-repos/lean/leanprover-community-flt-regular/flt-regular-1d0cecf99e8ab3f98b551e5932bf907042daa6ad/src/number_theory/cyclotomic/cyclotomic_units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.31905551509805496}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nInstances for identity and composition functors\n-/\nimport control.functor\nimport algebra.group.basic\n\nuniverse variables u v w\n\nsection lemmas\n\nopen function\n\nvariables {F : Type u → Type v}\nvariables [applicative F] [is_lawful_applicative F]\nvariables {α β γ σ : Type u}\n\nlemma applicative.map_seq_map (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :\n  (f <$> x) <*> (g <$> y) = (flip (∘) g ∘ f) <$> x <*> y :=\nby simp [flip] with functor_norm\n\nlemma applicative.pure_seq_eq_map' (f : α → β) :\n  (<*>) (pure f : F (α → β)) = (<$>) f :=\nby ext; simp with functor_norm\n\ntheorem applicative.ext {F} : ∀ {A1 : applicative F} {A2 : applicative F}\n  [@is_lawful_applicative F A1] [@is_lawful_applicative F A2]\n  (H1 : ∀ {α : Type u} (x : α),\n    @has_pure.pure _ A1.to_has_pure _ x = @has_pure.pure _ A2.to_has_pure _ x)\n  (H2 : ∀ {α β : Type u} (f : F (α → β)) (x : F α),\n    @has_seq.seq _ A1.to_has_seq _ _ f x = @has_seq.seq _ A2.to_has_seq _ _ f x),\n  A1 = A2\n| {to_functor := F1, seq := s1, pure := p1, seq_left := sl1, seq_right := sr1}\n  {to_functor := F2, seq := s2, pure := p2, seq_left := sl2, seq_right := sr2} L1 L2 H1 H2 :=\nbegin\n  have : @p1 = @p2, {funext α x, apply H1}, subst this,\n  have : @s1 = @s2, {funext α β f x, apply H2}, subst this,\n  cases L1, cases L2,\n  have : F1 = F2,\n  { resetI, apply functor.ext, intros,\n    exact (L1_pure_seq_eq_map _ _).symm.trans (L2_pure_seq_eq_map _ _) },\n  subst this,\n  congr; funext α β x y,\n  { exact (L1_seq_left_eq _ _).trans (L2_seq_left_eq _ _).symm },\n  { exact (L1_seq_right_eq _ _).trans (L2_seq_right_eq _ _).symm }\nend\n\nend lemmas\n\ninstance : is_comm_applicative id :=\nby refine { .. }; intros; refl\n\nnamespace functor\nnamespace comp\n\nopen function (hiding comp)\nopen functor\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nvariables [applicative F] [applicative G]\n\nvariables [is_lawful_applicative F] [is_lawful_applicative G]\nvariables {α β γ : Type v}\n\nlemma map_pure (f : α → β) (x : α) : (f <$> pure x : comp F G β) = pure (f x) :=\ncomp.ext $ by simp\n\nlemma seq_pure (f : comp F G (α → β)) (x : α) :\n  f <*> pure x = (λ g : α → β, g x) <$> f :=\ncomp.ext $ by simp [(∘)] with functor_norm\n\n\n\nlemma pure_seq_eq_map (f : α → β) (x : comp F G α) :\n  pure f <*> x = f <$> x :=\ncomp.ext $ by simp [applicative.pure_seq_eq_map'] with functor_norm\n\ninstance : is_lawful_applicative (comp F G) :=\n{ pure_seq_eq_map := @comp.pure_seq_eq_map F G _ _ _ _,\n  map_pure := @comp.map_pure F G _ _ _ _,\n  seq_pure := @comp.seq_pure F G _ _ _ _,\n  seq_assoc := @comp.seq_assoc F G _ _ _ _ }\n\ntheorem applicative_id_comp {F} [AF : applicative F] [LF : is_lawful_applicative F] :\n  @comp.applicative id F _ _ = AF :=\n@applicative.ext F _ _ (@comp.is_lawful_applicative id F _ _ _ _) _\n  (λ α x, rfl) (λ α β f x, rfl)\n\ntheorem applicative_comp_id {F} [AF : applicative F] [LF : is_lawful_applicative F] :\n  @comp.applicative F id _ _ = AF :=\n@applicative.ext F _ _ (@comp.is_lawful_applicative F id _ _ _ _) _\n  (λ α x, rfl) (λ α β f x, show id <$> f <*> x = f <*> x, by rw id_map)\n\nopen is_comm_applicative\n\ninstance {f : Type u → Type w} {g : Type v → Type u}\n  [applicative f] [applicative g]\n  [is_comm_applicative f] [is_comm_applicative g] :\n  is_comm_applicative (comp f g) :=\nby { refine { .. @comp.is_lawful_applicative f g _ _ _ _, .. },\n     intros, casesm* comp _ _ _, simp! [map,has_seq.seq] with functor_norm,\n     rw [commutative_map],\n     simp [comp.mk,flip,(∘)] with functor_norm,\n     congr, funext, rw [commutative_map], congr }\n\nend comp\nend functor\n\nopen functor\n\n@[functor_norm]\nlemma comp.seq_mk {α β : Type w}\n  {f : Type u → Type v} {g : Type w → Type u}\n  [applicative f] [applicative g]\n  (h : f (g (α → β))) (x : f (g α)) :\n  comp.mk h <*> comp.mk x = comp.mk (has_seq.seq <$> h <*> x) := rfl\n\ninstance {α} [has_one α] [has_mul α] : applicative (const α) :=\n{ pure := λ β x, (1 : α),\n  seq := λ β γ f x, (f * x : α) }\n\ninstance {α} [monoid α] : is_lawful_applicative (const α) :=\nby refine { .. }; intros; simp [mul_assoc, (<$>), (<*>), pure]\n\ninstance {α} [has_zero α] [has_add α] : applicative (add_const α) :=\n{ pure := λ β x, (0 : α),\n  seq := λ β γ f x, (f + x : α) }\n\ninstance {α} [add_monoid α] : is_lawful_applicative (add_const α) :=\nby refine { .. }; intros; simp [add_assoc, (<$>), (<*>), pure]\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/applicative.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.31905551509805496}}
{"text": "import .pair\nimport .precofibration_category\n\nnoncomputable theory\n\nnamespace homotopy_theory.topological_spaces\nopen homotopy_theory.cylinder\n\n-- (Provisional?) inductive definition of the disk/sphere pair using cubes.\ndef unit_pair : pair := pair.mk (*) ∅\ndef disk_sphere_pair (n : ℕ) : pair := nat.rec unit_pair (λ _ P, pair.prod P I_01) n\n\ndef disk (n : ℕ) : Top := (disk_sphere_pair n).space\ndef sphere_minus_one (n : ℕ) : Top := (disk_sphere_pair n).subspace\nnotation `D[` n `]` := disk n\nnotation `S[` n `-1]` := sphere_minus_one n\ndef sphere_disk_incl (n : ℕ) : S[n-1] ⟶ D[n] := pair.incl _\n\n-- It's a cofibration.\nlemma sphere_disk_closed : ∀ n, is_closed (disk_sphere_pair n).subset\n| 0 := is_closed_empty\n| (n+1) := pair.prod.is_closed (sphere_disk_closed n) I_01.is_closed\n\nlemma sphere_disk_cofibered : ∀ n, (disk_sphere_pair n).cofibered\n| 0 := pair.empty_cofibered _\n| (n+1) := prod_I_01_cofibered _ (sphere_disk_closed n) (sphere_disk_cofibered n)\n\nlemma sphere_disk_cofibration (n : ℕ) : closed_cofibration (sphere_disk_incl n) :=\nclosed_cofibration_incl_iff.mpr ⟨sphere_disk_cofibered n, sphere_disk_closed n⟩\n\nend homotopy_theory.topological_spaces\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/topological_spaces/disk_sphere.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3188385972171028}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nThe Kleisli construction on the Type category\n-/\nimport category_theory.category\n\nuniverses u v\n\nnamespace category_theory\n\ndef Kleisli (m) [monad.{u v} m] := Type u\n\ndef Kleisli.mk (m) [monad.{u v} m] (α : Type u) : Kleisli m := α\n\ninstance Kleisli.category_struct {m} [monad m] : category_struct (Kleisli m) :=\n{ hom := λ α β, α → m β,\n  id := λ α x, (pure x : m α),\n  comp := λ X Y Z f g, f >=> g }\n\ninstance Kleisli.category {m} [monad m] [is_lawful_monad m] : category (Kleisli m) :=\nby refine { hom := λ α β, α → m β,\n            id := λ α x, (pure x : m α),\n            comp := λ X Y Z f g, f >=> g,\n            id_comp' := _, comp_id' := _, assoc' := _ };\n   intros; ext; simp only [(>=>)] with functor_norm\n\n@[simp] lemma Kleisli.id_def {m} [monad m] [is_lawful_monad m] (α : Kleisli m) :\n  𝟙 α = @pure m _ α := rfl\n\nlemma Kleisli.comp_def {m} [monad m] [is_lawful_monad m] (α β γ : Kleisli m)\n  (xs : α ⟶ β) (ys : β ⟶ γ) (a : α) :\n  (xs ≫ ys) a = xs a >>= ys := rfl\n\nend category_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/instances/kleisli.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.31882568222883634}}
{"text": "import topology.discrete_quotient\nimport topology.category.Profinite\nimport category_theory.arrow\nimport data.setoid.partition\n\nimport for_mathlib.Profinite.quotient_map\n\nopen category_theory\n\nnamespace discrete_quotient\n\nuniverse u\nvariables {X Y : Profinite.{u}} (f : X ⟶ Y) (surj : function.surjective f)\n\n/-- The relation defining the largest quotient of f.right compatible with S. -/\ninductive make_rel (S : discrete_quotient X) : Y → Y → Prop\n| of (x y : X) (h : S.rel x y) : make_rel (f x) (f y)\n| trans {x y z : Y} : make_rel x y → make_rel y z → make_rel x z\n\nlemma make_rel_impl (S : discrete_quotient X) (x y : X) :\n  S.rel x y → S.make_rel f (f x) (f y) := make_rel.of _ _\n\ninclude surj\n\nlemma make_rel_equiv (S : discrete_quotient X) : equivalence (S.make_rel f) :=\nbegin\n  refine ⟨λ x, _, λ x y h, _, λ x y z h1 h2, make_rel.trans h1 h2⟩,\n  { obtain ⟨x,rfl⟩ := surj x,\n    exact make_rel.of _ _ (S.refl _) },\n  { induction h with x y h1 x y z h1 h2 h3 h4,\n    exact make_rel.of y x (S.symm x y h1),\n    exact make_rel.trans h4 h3 },\nend\n\n/-- The setoid assoc. to make_rel. -/\ndef make_setoid (S : discrete_quotient X) : _root_.setoid Y :=\n⟨S.make_rel f, S.make_rel_equiv f surj⟩\n\n/-- The quotient of make_rel. -/\n@[nolint has_inhabited_instance]\ndef make_quotient (S : discrete_quotient X) : Type* := quotient (S.make_setoid f surj)\n\n/-- The projection onto make_rel. -/\ndef make_proj (S : discrete_quotient X) : Y → S.make_quotient f surj :=\n  quotient.mk'\n\nlemma make_proj_comm (S : discrete_quotient X) :\n  S.make_proj f surj ∘ f = (quotient.map' f $ by exact make_rel_impl _ _) ∘ S.proj :=\nby {ext, refl}\n\nvariable (f)\n\n/-- Given a discrete quotient S of f.left, this is the compatible quotient\n of f where f.right is as large as possible. -/\ndef make (S : discrete_quotient X) : discrete_quotient Y :=\n{ rel := S.make_rel f,\n  equiv := S.make_rel_equiv f surj,\n  clopen := begin\n    intros x,\n    have : set_of (S.make_rel f x) = S.make_proj f surj ⁻¹' {S.make_proj f surj x},\n    { dsimp [make_proj],\n      ext,\n      simp only [set.mem_preimage, set.mem_singleton_iff, quotient.eq', set.mem_set_of_eq],\n      refine ⟨λ h, setoid.symm' _ h, λ h, setoid.symm' _ h⟩ },\n    rw this,\n    letI : topological_space (S.make_quotient f surj) := ⊥,\n    haveI : discrete_topology (S.make_quotient f surj) := ⟨rfl⟩,\n    suffices : continuous (S.make_proj f surj),\n    { refine ⟨is_open.preimage this trivial, is_closed.preimage this ⟨trivial⟩⟩ },\n    rw [(Profinite.quotient_map f surj).continuous_iff, S.make_proj_comm f surj],\n    exact continuous_bot.comp (proj_continuous S),\n  end }\n\nlemma make_le_comap (S : discrete_quotient X) : le_comap f.continuous S (S.make f surj) :=\nmake_rel_impl _ _\n\nlemma make_right_le (S : discrete_quotient X) (T : discrete_quotient Y)\n  (compat : le_comap f.continuous S T) :\n  (S.make f surj) ≤ T :=\nbegin\n  intros x y h,\n  induction h with a b hab a b c _ _ h1 h2,\n  { exact compat a b hab },\n  { exact trans T a b c h1 h2 },\nend\n\nlemma make_right_mono (S1 S2 : discrete_quotient X) (h : S1 ≤ S2) :\n  S1.make f surj ≤ S2.make f surj :=\nbegin\n  intros x y h,\n  induction h,\n  { refine make_rel.of _ _ (h _ _ _),\n  assumption },\n  { apply make_rel.trans,\n    assumption' },\nend\n\nend discrete_quotient\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/Profinite/compat_discrete_quotient.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217432182679957, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.3187744965304855}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.algebra.subalgebra.basic\nimport algebra.free_algebra\nimport algebra.category.CommRing.basic\nimport algebra.category.Module.basic\n\n/-!\n# Category instance for algebras over a commutative ring\n\nWe introduce the bundled category `Algebra` of algebras over a fixed commutative ring `R ` along\nwith the forgetful functors to `Ring` and `Module`. We furthermore show that the functor associating\nto a type the free `R`-algebra on that type is left adjoint to the forgetful functor.\n-/\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables (R : Type u) [comm_ring R]\n\n/-- The category of R-algebras and their morphisms. -/\nstructure Algebra :=\n(carrier : Type v)\n[is_ring : ring carrier]\n[is_algebra : algebra R carrier]\n\nattribute [instance] Algebra.is_ring Algebra.is_algebra\n\nnamespace Algebra\n\ninstance : has_coe_to_sort (Algebra R) (Type v) := ⟨Algebra.carrier⟩\n\ninstance : category (Algebra.{v} R) :=\n{ hom   := λ A B, A →ₐ[R] B,\n  id    := λ A, alg_hom.id R A,\n  comp  := λ A B C f g, g.comp f }\n\ninstance : concrete_category.{v} (Algebra.{v} R) :=\n{ forget := { obj := λ R, R, map := λ R S f, (f : R → S) },\n  forget_faithful := { } }\n\ninstance has_forget_to_Ring : has_forget₂ (Algebra.{v} R) Ring.{v} :=\n{ forget₂ :=\n  { obj := λ A, Ring.of A,\n    map := λ A₁ A₂ f, alg_hom.to_ring_hom f, } }\n\ninstance has_forget_to_Module : has_forget₂ (Algebra.{v} R) (Module.{v} R) :=\n{ forget₂ :=\n  { obj := λ M, Module.of R M,\n    map := λ M₁ M₂ f, alg_hom.to_linear_map f, } }\n\n/-- The object in the category of R-algebras associated to a type equipped with the appropriate\ntypeclasses. -/\ndef of (X : Type v) [ring X] [algebra R X] : Algebra.{v} R := ⟨X⟩\n\n/-- Typecheck a `alg_hom` as a morphism in `Algebra R`. -/\ndef of_hom {R : Type u} [comm_ring R] {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y]\n  (f : X →ₐ[R] Y) : of R X ⟶ of R Y := f\n\n@[simp] lemma of_hom_apply {R : Type u} [comm_ring R]\n  {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y] (f : X →ₐ[R] Y) (x : X) :\n  of_hom f x = f x := rfl\n\ninstance : inhabited (Algebra R) := ⟨of R R⟩\n\n@[simp]\nlemma coe_of (X : Type u) [ring X] [algebra R X] : (of R X : Type u) = X := rfl\n\nvariables {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nalgebra. -/\n@[simps]\ndef of_self_iso (M : Algebra.{v} R) : Algebra.of R M ≅ M :=\n{ hom := 𝟙 M, inv := 𝟙 M }\n\nvariables {R} {M N U : Module.{v} R}\n\n@[simp] \n\n@[simp] lemma coe_comp (f : M ⟶ N) (g : N ⟶ U) :\n  ((f ≫ g) : M → U) = g ∘ f := rfl\n\nvariables (R)\n/-- The \"free algebra\" functor, sending a type `S` to the free algebra on `S`. -/\n@[simps]\ndef free : Type u ⥤ Algebra.{u} R :=\n{ obj := λ S,\n  { carrier := free_algebra R S,\n    is_ring := algebra.semiring_to_ring R },\n  map := λ S T f, free_algebra.lift _ $ (free_algebra.ι _) ∘ f,\n  -- obviously can fill the next two goals, but it is slow\n  map_id' := by { intros X, ext1, simp only [free_algebra.ι_comp_lift], refl },\n  map_comp' := by { intros, ext1, simp only [free_algebra.ι_comp_lift], ext1,\n    simp only [free_algebra.lift_ι_apply, category_theory.coe_comp, function.comp_app,\n      types_comp_apply] } }\n\n/-- The free/forget adjunction for `R`-algebras. -/\ndef adj : free.{u} R ⊣ forget (Algebra.{u} R) :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X A, (free_algebra.lift _).symm,\n  -- Relying on `obviously` to fill out these proofs is very slow :(\n  hom_equiv_naturality_left_symm' := by { intros, ext,\n    simp only [free_map, equiv.symm_symm, free_algebra.lift_ι_apply, category_theory.coe_comp,\n      function.comp_app, types_comp_apply] },\n  hom_equiv_naturality_right' := by { intros, ext,\n    simp only [forget_map_eq_coe, category_theory.coe_comp, function.comp_app,\n      free_algebra.lift_symm_apply, types_comp_apply] } }\n\ninstance : is_right_adjoint (forget (Algebra.{u} R)) := ⟨_, adj R⟩\n\nend Algebra\n\nvariables {R}\nvariables {X₁ X₂ : Type u}\n\n/-- Build an isomorphism in the category `Algebra R` from a `alg_equiv` between `algebra`s. -/\n@[simps]\ndef alg_equiv.to_Algebra_iso\n  {g₁ : ring X₁} {g₂ : ring X₂} {m₁ : algebra R X₁} {m₂ : algebra R X₂} (e : X₁ ≃ₐ[R] X₂) :\n  Algebra.of R X₁ ≅ Algebra.of R X₂ :=\n{ hom := (e : X₁ →ₐ[R] X₂),\n  inv := (e.symm : X₂ →ₐ[R] X₁),\n  hom_inv_id' := begin ext, exact e.left_inv x, end,\n  inv_hom_id' := begin ext, exact e.right_inv x, end, }\n\nnamespace category_theory.iso\n\n/-- Build a `alg_equiv` from an isomorphism in the category `Algebra R`. -/\n@[simps]\ndef to_alg_equiv {X Y : Algebra R} (i : X ≅ Y) : X ≃ₐ[R] Y :=\n{ to_fun    := i.hom,\n  inv_fun   := i.inv,\n  left_inv  := by tidy,\n  right_inv := by tidy,\n  map_add'  := by tidy,\n  map_mul'  := by tidy,\n  commutes' := by tidy, }.\n\nend category_theory.iso\n\n/-- Algebra equivalences between `algebras`s are the same as (isomorphic to) isomorphisms in\n`Algebra`. -/\n@[simps]\ndef alg_equiv_iso_Algebra_iso {X Y : Type u}\n  [ring X] [ring Y] [algebra R X] [algebra R Y] :\n  (X ≃ₐ[R] Y) ≅ (Algebra.of R X ≅ Algebra.of R Y) :=\n{ hom := λ e, e.to_Algebra_iso,\n  inv := λ i, i.to_alg_equiv, }\n\ninstance (X : Type u) [ring X] [algebra R X] : has_coe (subalgebra R X) (Algebra R) :=\n⟨ λ N, Algebra.of R N ⟩\n\ninstance Algebra.forget_reflects_isos : reflects_isomorphisms (forget (Algebra.{u} R)) :=\n{ reflects := λ X Y f _,\n  begin\n    resetI,\n    let i := as_iso ((forget (Algebra.{u} R)).map f),\n    let e : X ≃ₐ[R] Y := { ..f, ..i.to_equiv },\n    exact ⟨(is_iso.of_iso e.to_Algebra_iso).1⟩,\n  end }\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/algebra/category/Algebra/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.31852277552903857}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Shing Tak Lam, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.int.modeq\nimport Mathlib.tactic.interval_cases\nimport Mathlib.tactic.linarith.default\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Digits of a natural number\n\nThis provides a basic API for extracting the digits of a natural number in a given base,\nand reconstructing numbers from their digits.\n\nWe also prove some divisibility tests based on digits, in particular completing\nTheorem #85 from https://www.cs.ru.nl/~freek/100/.\n\nA basic `norm_digits` tactic is also provided for proving goals of the form\n`nat.digits a b = l` where `a` and `b` are numerals.\n-/\n\nnamespace nat\n\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux_0 : ℕ → List ℕ :=\n  sorry\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux_1 (n : ℕ) : List ℕ :=\n  list.repeat 1 n\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux (b : ℕ) (h : bit0 1 ≤ b) : ℕ → List ℕ :=\n  sorry\n\n@[simp] theorem digits_aux_zero (b : ℕ) (h : bit0 1 ≤ b) : digits_aux b h 0 = [] :=\n  rfl\n\ntheorem digits_aux_def (b : ℕ) (h : bit0 1 ≤ b) (n : ℕ) (w : 0 < n) : digits_aux b h n = n % b :: digits_aux b h (n / b) := sorry\n\n/--\n`digits b n` gives the digits, in little-endian order,\nof a natural number `n` in a specified base `b`.\n\nIn any base, we have `of_digits b L = L.foldr (λ x y, x + b * y) 0`.\n* For any `2 ≤ b`, we have `l < b` for any `l ∈ digits b n`,\n  and the last digit is not zero.\n  This uniquely specifies the behaviour of `digits b`.\n* For `b = 1`, we define `digits 1 n = list.repeat 1 n`.\n* For `b = 0`, we define `digits 0 n = [n]`, except `digits 0 0 = []`.\n\nNote this differs from the existing `nat.to_digits` in core, which is used for printing numerals.\nIn particular, `nat.to_digits b 0 = [0]`, while `digits b 0 = []`.\n-/\ndef digits : ℕ → ℕ → List ℕ :=\n  sorry\n\n@[simp] theorem digits_zero (b : ℕ) : digits b 0 = [] :=\n  nat.cases_on b (Eq.refl (digits 0 0))\n    fun (b : ℕ) => nat.cases_on b (Eq.refl (digits 1 0)) fun (b : ℕ) => Eq.refl (digits (Nat.succ (Nat.succ b)) 0)\n\n@[simp] theorem digits_zero_zero : digits 0 0 = [] :=\n  rfl\n\n@[simp] theorem digits_zero_succ (n : ℕ) : digits 0 (Nat.succ n) = [n + 1] :=\n  rfl\n\ntheorem digits_zero_succ' {n : ℕ} (w : 0 < n) : digits 0 n = [n] :=\n  nat.cases_on n (fun (w : 0 < 0) => idRhs (digits 0 0 = [0]) (absurd w (of_as_true trivial)))\n    (fun (n : ℕ) (w : 0 < Nat.succ n) => idRhs (digits 0 (n + 1) = digits 0 (n + 1)) rfl) w\n\n@[simp] theorem digits_one (n : ℕ) : digits 1 n = list.repeat 1 n :=\n  rfl\n\n@[simp] theorem digits_one_succ (n : ℕ) : digits 1 (n + 1) = 1 :: digits 1 n :=\n  rfl\n\n@[simp] theorem digits_add_two_add_one (b : ℕ) (n : ℕ) : digits (b + bit0 1) (n + 1) = (n + 1) % (b + bit0 1) :: digits (b + bit0 1) ((n + 1) / (b + bit0 1)) :=\n  rfl\n\ntheorem digits_def' {b : ℕ} (h : bit0 1 ≤ b) {n : ℕ} (w : 0 < n) : digits b n = n % b :: digits b (n / b) := sorry\n\n@[simp] theorem digits_of_lt (b : ℕ) (x : ℕ) (w₁ : 0 < x) (w₂ : x < b) : digits b x = [x] := sorry\n\ntheorem digits_add (b : ℕ) (h : bit0 1 ≤ b) (x : ℕ) (y : ℕ) (w : x < b) (w' : 0 < x ∨ 0 < y) : digits b (x + b * y) = x :: digits b y := sorry\n\n/--\n`of_digits b L` takes a list `L` of natural numbers, and interprets them\nas a number in semiring, as the little-endian digits in base `b`.\n-/\n-- If we had a function converting a list into a polynomial,\n\n-- and appropriate lemmas about that function,\n\n-- we could rewrite this in terms of that.\n\ndef of_digits {α : Type u_1} [semiring α] (b : α) : List ℕ → α :=\n  sorry\n\ntheorem of_digits_eq_foldr {α : Type u_1} [semiring α] (b : α) (L : List ℕ) : of_digits b L = list.foldr (fun (x : ℕ) (y : α) => ↑x + b * y) 0 L := sorry\n\n@[simp] theorem of_digits_singleton {b : ℕ} {n : ℕ} : of_digits b [n] = n := sorry\n\n@[simp] theorem of_digits_one_cons {α : Type u_1} [semiring α] (h : ℕ) (L : List ℕ) : of_digits 1 (h :: L) = ↑h + of_digits 1 L := sorry\n\ntheorem of_digits_append {b : ℕ} {l1 : List ℕ} {l2 : List ℕ} : of_digits b (l1 ++ l2) = of_digits b l1 + b ^ list.length l1 * of_digits b l2 := sorry\n\ntheorem coe_of_digits (α : Type u_1) [semiring α] (b : ℕ) (L : List ℕ) : ↑(of_digits b L) = of_digits (↑b) L := sorry\n\ntheorem coe_int_of_digits (b : ℕ) (L : List ℕ) : ↑(of_digits b L) = of_digits (↑b) L := sorry\n\ntheorem digits_zero_of_eq_zero {b : ℕ} (h : 1 ≤ b) {L : List ℕ} (w : of_digits b L = 0) (l : ℕ) (H : l ∈ L) : l = 0 := sorry\n\ntheorem digits_of_digits (b : ℕ) (h : bit0 1 ≤ b) (L : List ℕ) (w₁ : ∀ (l : ℕ), l ∈ L → l < b) (w₂ : ∀ (h : L ≠ []), list.last L h ≠ 0) : digits b (of_digits b L) = L := sorry\n\ntheorem of_digits_digits (b : ℕ) (n : ℕ) : of_digits b (digits b n) = n := sorry\n\ntheorem of_digits_one (L : List ℕ) : of_digits 1 L = list.sum L := sorry\n\n/-!\n### Properties\n\nThis section contains various lemmas of properties relating to `digits` and `of_digits`.\n-/\n\ntheorem digits_eq_nil_iff_eq_zero {b : ℕ} {n : ℕ} : digits b n = [] ↔ n = 0 := sorry\n\ntheorem digits_ne_nil_iff_ne_zero {b : ℕ} {n : ℕ} : digits b n ≠ [] ↔ n ≠ 0 :=\n  not_congr digits_eq_nil_iff_eq_zero\n\ntheorem digits_last {b : ℕ} {m : ℕ} (h : bit0 1 ≤ b) (hm : 0 < m) (p : digits b m ≠ []) (q : digits b (m / b) ≠ []) : list.last (digits b m) p = list.last (digits b (m / b)) q := sorry\n\ntheorem last_digit_ne_zero (b : ℕ) {m : ℕ} (hm : m ≠ 0) : list.last (digits b m) (iff.mpr digits_ne_nil_iff_ne_zero hm) ≠ 0 := sorry\n\n/-- The digits in the base b+2 expansion of n are all less than b+2 -/\ntheorem digits_lt_base' {b : ℕ} {m : ℕ} {d : ℕ} : d ∈ digits (b + bit0 1) m → d < b + bit0 1 := sorry\n\n/-- The digits in the base b expansion of n are all less than b, if b ≥ 2 -/\ntheorem digits_lt_base {b : ℕ} {m : ℕ} {d : ℕ} (hb : bit0 1 ≤ b) (hd : d ∈ digits b m) : d < b := sorry\n\n/-- an n-digit number in base b + 2 is less than (b + 2)^n -/\ntheorem of_digits_lt_base_pow_length' {b : ℕ} {l : List ℕ} (hl : ∀ (x : ℕ), x ∈ l → x < b + bit0 1) : of_digits (b + bit0 1) l < (b + bit0 1) ^ list.length l := sorry\n\n/-- an n-digit number in base b is less than b^n if b ≥ 2 -/\ntheorem of_digits_lt_base_pow_length {b : ℕ} {l : List ℕ} (hb : bit0 1 ≤ b) (hl : ∀ (x : ℕ), x ∈ l → x < b) : of_digits b l < b ^ list.length l := sorry\n\n/-- Any number m is less than (b+2)^(number of digits in the base b + 2 representation of m) -/\ntheorem lt_base_pow_length_digits' {b : ℕ} {m : ℕ} : m < (b + bit0 1) ^ list.length (digits (b + bit0 1) m) := sorry\n\n/-- Any number m is less than b^(number of digits in the base b representation of m) -/\ntheorem lt_base_pow_length_digits {b : ℕ} {m : ℕ} (hb : bit0 1 ≤ b) : m < b ^ list.length (digits b m) := sorry\n\ntheorem of_digits_digits_append_digits {b : ℕ} {m : ℕ} {n : ℕ} : of_digits b (digits b n ++ digits b m) = n + b ^ list.length (digits b n) * m := sorry\n\ntheorem digits_len_le_digits_len_succ (b : ℕ) (n : ℕ) : list.length (digits b n) ≤ list.length (digits b (n + 1)) := sorry\n\ntheorem le_digits_len_le (b : ℕ) (n : ℕ) (m : ℕ) (h : n ≤ m) : list.length (digits b n) ≤ list.length (digits b m) :=\n  monotone_of_monotone_nat (digits_len_le_digits_len_succ b) h\n\ntheorem pow_length_le_mul_of_digits {b : ℕ} {l : List ℕ} (hl : l ≠ []) (hl2 : list.last l hl ≠ 0) : (b + bit0 1) ^ list.length l ≤ (b + bit0 1) * of_digits (b + bit0 1) l := sorry\n\n/--\nAny non-zero natural number `m` is greater than\n(b+2)^((number of digits in the base (b+2) representation of m) - 1)\n-/\ntheorem base_pow_length_digits_le' (b : ℕ) (m : ℕ) (hm : m ≠ 0) : (b + bit0 1) ^ list.length (digits (b + bit0 1) m) ≤ (b + bit0 1) * m := sorry\n\n/--\nAny non-zero natural number `m` is greater than\nb^((number of digits in the base b representation of m) - 1)\n-/\ntheorem base_pow_length_digits_le (b : ℕ) (m : ℕ) (hb : bit0 1 ≤ b) : m ≠ 0 → b ^ list.length (digits b m) ≤ b * m := sorry\n\n/-! ### Modular Arithmetic -/\n\n-- This is really a theorem about polynomials.\n\ntheorem dvd_of_digits_sub_of_digits {α : Type u_1} [comm_ring α] {a : α} {b : α} {k : α} (h : k ∣ a - b) (L : List ℕ) : k ∣ of_digits a L - of_digits b L := sorry\n\ntheorem of_digits_modeq' (b : ℕ) (b' : ℕ) (k : ℕ) (h : modeq k b b') (L : List ℕ) : modeq k (of_digits b L) (of_digits b' L) := sorry\n\ntheorem of_digits_modeq (b : ℕ) (k : ℕ) (L : List ℕ) : modeq k (of_digits b L) (of_digits (b % k) L) :=\n  of_digits_modeq' b (b % k) k (modeq.symm (modeq.mod_modeq b k)) L\n\ntheorem of_digits_mod (b : ℕ) (k : ℕ) (L : List ℕ) : of_digits b L % k = of_digits (b % k) L % k :=\n  of_digits_modeq b k L\n\ntheorem of_digits_zmodeq' (b : ℤ) (b' : ℤ) (k : ℕ) (h : int.modeq (↑k) b b') (L : List ℕ) : int.modeq (↑k) (of_digits b L) (of_digits b' L) := sorry\n\ntheorem of_digits_zmodeq (b : ℤ) (k : ℕ) (L : List ℕ) : int.modeq (↑k) (of_digits b L) (of_digits (b % ↑k) L) :=\n  of_digits_zmodeq' b (b % ↑k) k (int.modeq.symm (int.modeq.mod_modeq b ↑k)) L\n\ntheorem of_digits_zmod (b : ℤ) (k : ℕ) (L : List ℕ) : of_digits b L % ↑k = of_digits (b % ↑k) L % ↑k :=\n  of_digits_zmodeq b k L\n\ntheorem modeq_digits_sum (b : ℕ) (b' : ℕ) (h : b' % b = 1) (n : ℕ) : modeq b n (list.sum (digits b' n)) := sorry\n\ntheorem modeq_three_digits_sum (n : ℕ) : modeq (bit1 1) n (list.sum (digits (bit0 (bit1 (bit0 1))) n)) := sorry\n\ntheorem modeq_nine_digits_sum (n : ℕ) : modeq (bit1 (bit0 (bit0 1))) n (list.sum (digits (bit0 (bit1 (bit0 1))) n)) := sorry\n\ntheorem zmodeq_of_digits_digits (b : ℕ) (b' : ℕ) (c : ℤ) (h : int.modeq (↑b) (↑b') c) (n : ℕ) : int.modeq (↑b) (↑n) (of_digits c (digits b' n)) := sorry\n\ntheorem of_digits_neg_one (L : List ℕ) : of_digits (-1) L = list.alternating_sum (list.map (fun (n : ℕ) => ↑n) L) := sorry\n\ntheorem modeq_eleven_digits_sum (n : ℕ) : int.modeq (bit1 (bit1 (bit0 1))) (↑n)\n  (list.alternating_sum (list.map (fun (n : ℕ) => ↑n) (digits (bit0 (bit1 (bit0 1))) n))) := sorry\n\n/-! ## Divisibility  -/\n\ntheorem dvd_iff_dvd_digits_sum (b : ℕ) (b' : ℕ) (h : b' % b = 1) (n : ℕ) : b ∣ n ↔ b ∣ list.sum (digits b' n) := sorry\n\ntheorem three_dvd_iff (n : ℕ) : bit1 1 ∣ n ↔ bit1 1 ∣ list.sum (digits (bit0 (bit1 (bit0 1))) n) := sorry\n\ntheorem nine_dvd_iff (n : ℕ) : bit1 (bit0 (bit0 1)) ∣ n ↔ bit1 (bit0 (bit0 1)) ∣ list.sum (digits (bit0 (bit1 (bit0 1))) n) := sorry\n\ntheorem dvd_iff_dvd_of_digits (b : ℕ) (b' : ℕ) (c : ℤ) (h : ↑b ∣ ↑b' - c) (n : ℕ) : b ∣ n ↔ ↑b ∣ of_digits c (digits b' n) := sorry\n\ntheorem eleven_dvd_iff (n : ℕ) : bit1 (bit1 (bit0 1)) ∣ n ↔\n  bit1 (bit1 (bit0 1)) ∣ list.alternating_sum (list.map (fun (n : ℕ) => ↑n) (digits (bit0 (bit1 (bit0 1))) n)) := sorry\n\n/-! ### `norm_digits` tactic -/\n\nnamespace norm_digits\n\n\ntheorem digits_succ (b : ℕ) (n : ℕ) (m : ℕ) (r : ℕ) (l : List ℕ) (e : r + b * m = n) (hr : r < b) (h : digits b m = l ∧ bit0 1 ≤ b ∧ 0 < m) : digits b n = r :: l ∧ bit0 1 ≤ b ∧ 0 < n := sorry\n\ntheorem digits_one (b : ℕ) (n : ℕ) (n0 : 0 < n) (nb : n < b) : digits b n = [n] ∧ bit0 1 ≤ b ∧ 0 < n := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/nat/digits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5273165382362518, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.31846324389104397}}
{"text": "import for_mathlib.homological_complex_op\nimport for_mathlib.split_exact\nimport for_mathlib.AddCommGroup.exact\n\nimport pseudo_normed_group.FP2\nimport pseudo_normed_group.system_of_complexes\n\nimport system_of_complexes.rescale\n\nimport prop_92.prop_92\n\nnoncomputable theory\n\nopen_locale nnreal\n\nuniverses u v\n\nopen category_theory breen_deligne\n\nvariables (r r' : ℝ≥0)\nvariables (BD : breen_deligne.data)\nvariables (M : ProFiltPseuNormGrpWithTinv.{u} r')\nvariables (V : SemiNormedGroup.{v})\n\nset_option pp.universes true\n\ndef aux_system (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))] :\n  system_of_complexes :=\n(FPsystem r' BD M κ).op ⋙\n  (((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n  homological_complex.unop_functor\n\nnamespace aux_system\n\nvariables [fact (0 < r)] [fact (0 < r')] [fact (r' ≤ 1)]\n\nopen system_of_complexes opposite\n\nvariables (κ₁ κ₂ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ₁ c)] [∀ c, BD.suitable (κ₂ c)]\n\ndef Tinv [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nwhisker_right (nat_trans.op $ FPsystem.Tinv r' BD M κ₁ κ₂)\n  ((((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n    homological_complex.unop_functor)\n\nlemma Tinv_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((Tinv r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.Tinv V r' _ _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [Tinv, CLCFP.Tinv, FPsystem.Tinv, FP2.Tinv,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n    functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n    FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map,\n    whisker_right_app, nat_trans.op_app],\n  erw [free_abelian_group.lift.of],\n  refl,\nend\n\ndef T_inv [normed_with_aut r V]\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))] :\n  aux_system r' BD M V κ ⟶ aux_system r' BD M V κ :=\n{ app := λ c,\n  { f := λ i, normed_add_group_hom.completion $ (SemiNormedGroup.LocallyConstant.map $ normed_with_aut.T.inv).app _,\n    comm' := begin\n      rintro i j (rfl : i + 1 = j),\n      dsimp only [aux_system, functor.comp_obj, homological_complex.unop_functor,\n        homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n        unop_op],\n      erw [← SemiNormedGroup.Completion_map, ← SemiNormedGroup.Completion_map, chain_complex.of_d],\n      dsimp only [FPsystem.d, universal_map.eval_FP2, universal_map.eval_CLCFP,\n        universal_map.eval_LCFP, CLC],\n      simp only [nat_trans.app_sum, functor.map_sum, preadditive.comp_sum, preadditive.sum_comp,\n        category_theory.unop_sum, nat_trans.app_zsmul, functor.map_zsmul,\n        preadditive.comp_zsmul, preadditive.zsmul_comp, category_theory.unop_zsmul],\n      refine finset.sum_congr rfl _,\n      rintro ⟨g, hg⟩ -,\n      dsimp only [basic_universal_map.eval_FP2, basic_universal_map.eval_LCFP,\n        whisker_right_app, nat_trans.op_app, unop_op, FreeAb.of_functor,\n        free_abelian_group.map_of_apply, functor.right_op_map, LC,\n        functor.comp_map, functor.map_FreeAb, FreeAb.eval],\n      rw [free_abelian_group.lift.of, _root_.id, quiver.hom.unop_op,\n        ← SemiNormedGroup.Completion.map_comp, ← SemiNormedGroup.Completion.map_comp,\n        nat_trans.naturality],\n    end },\n  naturality' := λ c₁ c₂ h, begin\n    ext n : 2,\n    dsimp only [homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n      homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n      unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n      functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n      FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map],\n    rw [← SemiNormedGroup.Completion_map, ← SemiNormedGroup.Completion_map,\n      free_abelian_group.lift.of, _root_.id, quiver.hom.unop_op,\n      ← SemiNormedGroup.Completion.map_comp, ← SemiNormedGroup.Completion.map_comp,\n        nat_trans.naturality],\n    refl,\n  end }\n.\n\nvariables [normed_with_aut r V]\n\nlemma T_inv_eq\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))]\n  (c : ℝ≥0ᵒᵖ) (n : ℕ) : ((T_inv r r' BD M V κ).app c).f n =\n  ((CLCFP.T_inv r V r' _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [T_inv, CLCFP.T_inv, FPsystem, chain_complex.of_X, FPsystem.X,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    functor.map_homological_complex_map_f, functor.map_FreeAb,\n    unop_op, CLC, functor.op_map, functor.op_obj, quiver.hom.unop_op,\n    FreeAb.eval, free_abelian_group.map_of_apply, FPsystem, FP2.res_app, FreeAb.of_functor,\n    functor.right_op_map, whisker_right_app, nat_trans.op_app, whisker_left_app],\n  refl,\nend\n\ndef res [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nwhisker_right (nat_trans.op $ FPsystem.res r' BD M κ₁ κ₂)\n  ((((CLC.{v u} V).right_op.map_FreeAb ⋙ FreeAb.eval _).map_homological_complex _).op ⋙\n    homological_complex.unop_functor)\n\nlemma res_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((res r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.res V r' _ _ (BD.X n)).app (op M) : _) :=\nbegin\n  dsimp only [res, CLCFP.res, FPsystem.res, FP2.res,\n    homological_complex.comp_f, aux_system, functor.comp_map, homological_complex.unop_functor,\n    homological_complex.unop_d, functor.op_obj, functor.map_homological_complex_obj_d,\n    unop_op, CLC, functor.op_map, quiver.hom.unop_op, functor.map_homological_complex_map_f,\n    functor.map_FreeAb, FreeAb.eval, free_abelian_group.map_of_apply,\n    FPsystem, FP2.res_app, FreeAb.of_functor, functor.right_op_map,\n    whisker_right_app, nat_trans.op_app],\n  rw [free_abelian_group.lift.of],\n  refl,\nend\n\ndef Tinv2 [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :\n  aux_system r' BD M V κ₂ ⟶ aux_system r' BD M V κ₁ :=\nTinv r' BD M V κ₁ κ₂ - T_inv r r' BD M V κ₂ ≫ res r' BD M V κ₁ κ₂\n.\n\nlemma Tinv2_eq [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((Tinv2 r r' BD M V κ₁ κ₂).app c).f n =\n  ((CLCFP.Tinv V r' _ _ (BD.X n)).app (op M) : _) -\n  ((CLCFP.T_inv r V r' _ (BD.X n)).app (op M) : _) ≫ ((CLCFP.res V r' _ _ (BD.X n)).app (op M) : _) :=\nby rw [Tinv2, nat_trans.app_sub, homological_complex.sub_f_apply, Tinv_eq,\n    nat_trans.comp_app, homological_complex.comp_f, T_inv_eq, res_eq]\n\nlemma aux_system_d_eq\n  (κ : ℝ≥0 → ℕ → ℝ≥0) [∀ c, BD.suitable (κ c)] [∀ (n : ℕ), fact (monotone (function.swap κ n))]\n  (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  arrow.mk (((aux_system r' BD M V κ).obj c).d i (i + 1)) =\n  arrow.mk ((universal_map.eval_CLCFP V r' (κ (unop c) i) (κ (unop c) (i + 1)) (BD.d (i + 1) i) : _).app (op M)) :=\nbegin\n  dsimp [aux_system, FPsystem],\n  rw [chain_complex.of_d],\n  dsimp [FreeAb.eval, functor.map_FreeAb, functor.right_op_map, FPsystem.d,\n    universal_map.eval_FP2, universal_map.eval_CLCFP, universal_map.eval_LCFP],\n  simp only [nat_trans.app_sum, map_sum, ← normed_add_group_hom_completion_hom_apply,\n    category_theory.unop_sum],\n  congr' 1,\n  refine finset.sum_congr rfl _,\n  rintro ⟨g, hg⟩ -,\n  simp only [nat_trans.app_zsmul, map_zsmul, category_theory.unop_zsmul],\n  dsimp only [basic_universal_map.eval_FP2, basic_universal_map.eval_LCFP,\n    whisker_right_app, nat_trans.op_app, unop_op, FreeAb.of_functor,\n    free_abelian_group.map_of_apply],\n  rw [free_abelian_group.lift.of],\n  refl,\nend\n\nsection\n\nvariables (κ : ℕ → ℝ≥0) [BD.suitable κ]\n\ninstance mul_left_mono (n : ℕ) :\n  fact (monotone (function.swap (λ (c : ℝ≥0) (n : ℕ), c * κ n) n)) :=\n⟨λ c₁ c₂ h, mul_le_mul' h le_rfl⟩\n\ndef incl_f (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  ((BD.complex κ r V r' (unop c)).obj (op M)).X n ⟶ ((aux_system r' BD M V (λ c n, c * κ n)).obj c).X n :=\nbegin\n  dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n    CLCTinv, aux_system, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval],\n  exact (SemiNormedGroup.equalizer.ι _ _ : _),\nend\n\nlemma incl_comm (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  incl_f.{u v} r r' BD M V κ c i ≫ ((aux_system.{u v} r' BD M V (λ c n, c * κ n)).obj c).d i (i + 1) =\n  ((BD.complex κ r V r' (unop c)).obj (op.{u+2} M)).d i (i + 1) ≫ incl_f.{u v} r r' BD M V κ c (i + 1) :=\nbegin\n  dsimp [aux_system],\n  erw [chain_complex.of_d, breen_deligne.data.complex_obj_d],\n  dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n    CLCTinv, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval,\n    universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv₂,\n    SemiNormedGroup.equalizer.map_nat, incl_f],\n  delta id, dsimp only [], symmetry,\n  convert SemiNormedGroup.equalizer.map_comp_ι _ _ _ _ using 2,\n  apply arrow.mk_injective,\n  rw ← aux_system_d_eq r' BD M V (λ c n, c * κ n),\n  dsimp [aux_system],\n  erw [chain_complex.of_d],\n  refl,\nend\n\ndef incl : (BD.system κ r V r').obj (op M) ⟶ aux_system r' BD M V (λ c n, c * κ n) :=\n{ app := λ c,\n  { f := incl_f r r' BD M V κ c,\n    comm' := by { rintro i j (rfl : i + 1 = j), apply incl_comm } },\n  naturality' := λ c₁ c₂ h, begin\n    ext n : 2,\n    dsimp [aux_system],\n    dsimp [breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X,\n      CLCTinv, FPsystem, FPsystem.X, functor.map_FreeAb, FreeAb.eval, FreeAb.of_functor,\n      universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv₂,\n      SemiNormedGroup.equalizer.map_nat, incl_f, CLCFPTinv₂.res, CLCTinv.map_nat, CLCTinv.map],\n    delta id,\n    erw [free_abelian_group.lift.of],\n    convert SemiNormedGroup.equalizer.map_comp_ι _ _ _ _ using 1,\n  end }\n\ndef incl' := whisker_right (incl r r' BD M V κ) (functor.map_homological_complex (forget₂ _ Ab.{max u v}) _)\n\nend\n\ndef Tinv2' [hκ₁ : ∀ n, fact (monotone (function.swap κ₁ n))] [hκ₂ : ∀ n, fact (monotone (function.swap κ₂ n))]\n  [∀ c n, fact (κ₁ c n ≤ r' * κ₂ c n)] [∀ c n, fact (κ₁ c n ≤ κ₂ c n)] :=\nwhisker_right (Tinv2 r r' BD M V κ₁ κ₂)\n(functor.map_homological_complex (forget₂ _ Ab.{max u v}) _)\n\nlemma _root_.SemiNormedGroup.equalizer.ι_injective {V W : SemiNormedGroup} (f g : V ⟶ W) :\n  function.injective (SemiNormedGroup.equalizer.ι f g) :=\nsubtype.val_injective\n\nlemma _root_.SemiNormedGroup.equalizer.forget₂_ι {V W : SemiNormedGroup} (f g : V ⟶ W) :\n  (forget₂ _ Ab).map (SemiNormedGroup.equalizer.ι f g) =\n  add_subgroup.subtype (add_monoid_hom.ker ((forget₂ _ Ab).map (f - g))) :=\nrfl\n\ninstance mul_left_mono' (κ : ℕ → ℝ≥0) (n : ℕ) :\n  fact (monotone (function.swap (λ (c : ℝ≥0) (n : ℕ), r' * (c * κ n)) n)) :=\n⟨λ c₁ c₂ h, mul_le_mul' le_rfl $ mul_le_mul' h le_rfl⟩\n\nlemma neg_surjective {A B : Ab} (f : A ⟶ B) (hf : function.surjective f) :\n  function.surjective (-f) :=\nbegin\n  intro y, obtain ⟨x, rfl⟩ := hf y, use -x, simp only [pi.neg_apply, map_neg, neg_neg],\nend\n\nlemma short_exact [fact (r < 1)] (κ : ℕ → ℝ≥0) [BD.suitable κ]\n  [∀ c n, fact (κ₁ c n ≤ r' * (c * κ n))] (c : ℝ≥0ᵒᵖ) (n : ℕ) :\n  short_exact (((incl' r r' BD M V κ).app c).f n)\n    (((Tinv2' r r' BD M V (λ c n, r' * (c * κ n)) (λ c n, c * κ n)).app c).f n) :=\nbegin\n  apply_with @short_exact.mk {instances := ff},\n  { rw AddCommGroup.mono_iff_injective,\n    dsimp [incl', incl, incl_f,\n      breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X],\n    exact SemiNormedGroup.equalizer.ι_injective _ _, },\n  { rw [Tinv2', whisker_right_app, functor.map_homological_complex_map_f, Tinv2_eq,\n      ← nat_trans.comp_app, ← CLCFP.res_comp_T_inv, nat_trans.comp_app,\n      ← neg_sub, functor.map_neg, functor.map_sub, category_theory.functor.map_comp,\n      AddCommGroup.epi_iff_surjective],\n    apply neg_surjective,\n    have := CLCFP.T_inv_sub_Tinv_surjective r r' V (unop c * κ n) (BD.X n) (op M),\n    rw [CLCFP.T_inv_sub_Tinv, nat_trans.app_sub, nat_trans.comp_app] at this,\n    exact this, },\n  { rw AddCommGroup.exact_iff,\n    dsimp [incl', incl, incl_f, Tinv2',\n      breen_deligne.data.complex, breen_deligne.data.complex₂, breen_deligne.data.complex₂_X],\n    rw [SemiNormedGroup.equalizer.forget₂_ι, add_subgroup.subtype_range, Tinv2_eq,\n      ← nat_trans.comp_app, ← CLCFP.res_comp_T_inv],\n    refl, }\nend\n\nend aux_system\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/system_of_complexes2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.31843007774725907}}
{"text": "inductive Trie (α : Type _) (β : Type _)\n  | mk : α → Option β → List (Trie α β) → Trie α β\n  deriving Inhabited, BEq\n\nnamespace Trie\n\npartial def isWellFormed : Trie α β → Bool\n  | mk _ none [] => false\n  | mk _  _ l => l.all isWellFormed\n\ndef isTerminus : Trie α β → Bool\n  | mk _ (some _) _ => true\n  | mk _ none _ => false\n\ndef label : Trie α β → α\n  | mk a _ _ => a\n\ndef data? : Trie α β → Option β\n  | mk _ b? _ => b?\n\ndef data : (tr : Trie α β) → tr.isTerminus → β\n  | mk _ (some b) _, _ => b\n  | mk _ none _, h => by \n    dsimp [isTerminus] at h\n    contradiction\n\ndef find? [DecidableEq α] : Trie α β → α → Option (Trie α β)\n  | mk _ _ l, a => l.find? (·.label = a)\n\ninstance [DecidableEq α] : GetElem (Trie α β) α (Trie α β) (Option.toBool <| ·.find? ·) where\n  getElem := λ tr a h =>\n    match tr.find? a, h with\n      | some tr', _ => tr'\n      | none, h => by \n        dsimp [Option.toBool] at h\n        contradiction\n\ndef fromList [DecidableEq α] : List ((List α) × β) → Trie α β := fun l =>\n  let lg := l.groupBy (λ ⟨l₁, _⟩ ⟨l₂, _⟩ => l₁.head? = l₂.head?)\n  sorry\n\ndef keyword? [DecidableEq α] (tr : Trie α β) (p : List α) : List α → (List α × List α × Option β)\n  | [] => (p, [], tr.data?)\n  | w :: ws => \n    match tr.find? w with\n      | some tr' => keyword? tr' (p.concat w) ws\n      | none => (p, w :: ws, tr.data?)\n\ntheorem keyword?.decompose [DecidableEq α] (tr : Trie α β) (p l : List α) : \n  let (k, l', _) := keyword? tr p l \n  p ++ l = k ++ l' :=\n    match l with\n      | [] => rfl\n      | w :: ws => by\n        simp [keyword?]\n        cases tr.find? w\n        · rfl\n        · have : p ++ w :: ws = (p.concat w) ++ ws := by rw [List.concat_eq_append, List.append_assoc]; rfl\n          rw [this]\n          apply keyword?.decompose\n\ndef keyword?.length [DecidableEq α] (tr : Trie α β) (p : List α) : List α → Nat\n    | [] => .zero\n    | w :: ws =>\n      match tr.find? w with\n        | some tr' => .succ $ keyword?.length tr' (p.concat w) ws\n        | none => .zero\n\ntheorem keyword?.eq_drop_length [DecidableEq α] (tr : Trie α β) (p l : List α) :\n  let (_, l', _) := keyword? tr p l\n  l' = l.drop (keyword?.length tr p l) :=\n  match l with\n    | [] => rfl\n    | w :: ws => by\n      simp [keyword?, keyword?.length]\n      cases tr.find? w\n      · rfl\n      · dsimp [List.drop]\n        apply keyword?.eq_drop_length\n\ndef keywords [DecidableEq α] (tr : Trie α β) (l : List α) : List ((List α) × β) :=\n  match tr.keyword? [] l with\n    | (kw, l', some b) => (kw, b) :: keywords tr l'\n    | (_, l', none) => keywords tr l'\ndecreasing_by sorry\n\nend Trie", "meta": {"author": "siddhartha-gadgil", "repo": "LeanAide", "sha": "7862af73ee2f0be08b20fd3e4148e20bf4a81054", "save_path": "github-repos/lean/siddhartha-gadgil-LeanAide", "path": "github-repos/lean/siddhartha-gadgil-LeanAide/LeanAide-7862af73ee2f0be08b20fd3e4148e20bf4a81054/LeanCodePrompts/KeywordSummary/Trie.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3183926079776296}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n\nimport order.omega_complete_partial_order\nimport order.category.Preorder\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.constructions.limits_of_products_and_equalizers\n\n/-!\n# Category of types with a omega complete partial order\n\nIn this file, we bundle the class `omega_complete_partial_order` into a\nconcrete category and prove that continuous functions also form\na `omega_complete_partial_order`.\n\n## Main definitions\n\n * `ωCPO`\n   * an instance of `category` and `concrete_category`\n\n -/\n\nopen category_theory\n\nuniverses u v\n\n/-- The category of types with a omega complete partial order. -/\ndef ωCPO : Type (u+1) := bundled omega_complete_partial_order\n\nnamespace ωCPO\n\nopen omega_complete_partial_order\n\ninstance : bundled_hom @continuous_hom :=\n{ to_fun := @continuous_hom.simps.apply,\n  id := @continuous_hom.id,\n  comp := @continuous_hom.comp,\n  hom_ext := @continuous_hom.coe_inj }\n\nattribute [derive [large_category, concrete_category]] ωCPO\n\ninstance : has_coe_to_sort ωCPO Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled ωCPO from the underlying type and typeclass. -/\ndef of (α : Type*) [omega_complete_partial_order α] : ωCPO := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [omega_complete_partial_order α] : ↥(of α) = α := rfl\n\ninstance : inhabited ωCPO := ⟨of punit⟩\n\ninstance (α : ωCPO) : omega_complete_partial_order α := α.str\n\nsection\n\nopen category_theory.limits\n\nnamespace has_products\n\n/-- The pi-type gives a cone for a product. -/\ndef product {J : Type v} (f : J → ωCPO.{v}) : fan f :=\nfan.mk (of (Π j, f j)) (λ j, continuous_hom.of_mono (pi.eval_order_hom j) (λ c, rfl))\n\n/-- The pi-type is a limit cone for the product. -/\ndef is_product (J : Type v) (f : J → ωCPO) : is_limit (product f) :=\n{ lift := λ s,\n    ⟨⟨λ t j, s.π.app j t, λ x y h j, (s.π.app j).monotone h⟩,\n     λ x, funext (λ j, (s.π.app j).continuous x)⟩,\n  uniq' := λ s m w,\n  begin\n    ext t j,\n    change m t j = s.π.app j t,\n    rw ← w j,\n    refl,\n  end }.\n\ninstance (J : Type v) (f : J → ωCPO.{v}) : has_product f :=\nhas_limit.mk ⟨_, is_product _ f⟩\n\nend has_products\n\ninstance omega_complete_partial_order_equalizer\n  {α β : Type*} [omega_complete_partial_order α] [omega_complete_partial_order β]\n  (f g : α →𝒄 β) : omega_complete_partial_order {a : α // f a = g a} :=\nomega_complete_partial_order.subtype _ $ λ c hc,\nbegin\n  rw [f.continuous, g.continuous],\n  congr' 1,\n  ext,\n  apply hc _ ⟨_, rfl⟩,\nend\n\nnamespace has_equalizers\n\n/-- The equalizer inclusion function as a `continuous_hom`. -/\ndef equalizer_ι {α β : Type*} [omega_complete_partial_order α] [omega_complete_partial_order β]\n  (f g : α →𝒄 β) :\n  {a : α // f a = g a} →𝒄 α :=\ncontinuous_hom.of_mono (order_hom.subtype.val _) (λ c, rfl)\n\n/-- A construction of the equalizer fork. -/\ndef equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) :\n  fork f g :=\n@fork.of_ι _ _ _ _ _ _ (ωCPO.of {a // f a = g a}) (equalizer_ι f g)\n  (continuous_hom.ext _ _ (λ x, x.2))\n\n/-- The equalizer fork is a limit. -/\ndef is_equalizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : is_limit (equalizer f g) :=\nfork.is_limit.mk' _ $ λ s,\n⟨{ to_fun := λ x, ⟨s.ι x, by apply continuous_hom.congr_fun s.condition⟩,\n    monotone' := λ x y h, s.ι.monotone h,\n    cont := λ x, subtype.ext (s.ι.continuous x) },\n  by { ext, refl },\n  λ m hm,\n  begin\n    ext,\n    apply continuous_hom.congr_fun hm,\n  end⟩\n\nend has_equalizers\n\ninstance : has_products ωCPO.{v} :=\nλ J, { has_limit := λ F, has_limit_of_iso discrete.nat_iso_functor.symm }\n\ninstance {X Y : ωCPO.{v}} (f g : X ⟶ Y) : has_limit (parallel_pair f g) :=\nhas_limit.mk ⟨_, has_equalizers.is_equalizer f g⟩\n\ninstance : has_equalizers ωCPO.{v} := has_equalizers_of_has_limit_parallel_pair _\n\ninstance : has_limits ωCPO.{v} := limits_from_equalizers_and_products\n\nend\n\n\nend ωCPO\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/omega_complete_partial_order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3182023195530493}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.monoidal.functor\nimport category_theory.adjunction.limits\nimport category_theory.adjunction.mates\nimport category_theory.functor.inv_isos\n\n/-!\n# Closed monoidal categories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nDefine (right) closed objects and (right) closed monoidal categories.\n\n## TODO\nSome of the theorems proved about cartesian closed categories\nshould be generalised and moved to this file.\n-/\nuniverses v u u₂ v₂\n\nnamespace category_theory\n\nopen category monoidal_category\n\n/-- An object `X` is (right) closed if `(X ⊗ -)` is a left adjoint. -/\n-- Note that this class carries a particular choice of right adjoint,\n-- (which is only unique up to isomorphism),\n-- not merely the existence of such, and\n-- so definitional properties of instances may be important.\nclass closed {C : Type u} [category.{v} C] [monoidal_category.{v} C] (X : C) :=\n(is_adj : is_left_adjoint (tensor_left X))\n\n/-- A monoidal category `C` is (right) monoidal closed if every object is (right) closed. -/\nclass monoidal_closed (C : Type u) [category.{v} C] [monoidal_category.{v} C] :=\n(closed' : Π (X : C), closed X)\n\nattribute [instance, priority 100] monoidal_closed.closed'\n\nvariables {C : Type u} [category.{v} C] [monoidal_category.{v} C]\n\n/--\nIf `X` and `Y` are closed then `X ⊗ Y` is.\nThis isn't an instance because it's not usually how we want to construct internal homs,\nwe'll usually prove all objects are closed uniformly.\n-/\ndef tensor_closed {X Y : C}\n  (hX : closed X) (hY : closed Y) : closed (X ⊗ Y) :=\n{ is_adj :=\n  begin\n    haveI := hX.is_adj,\n    haveI := hY.is_adj,\n    exact adjunction.left_adjoint_of_nat_iso (monoidal_category.tensor_left_tensor _ _).symm\n  end }\n\n/--\nThe unit object is always closed.\nThis isn't an instance because most of the time we'll prove closedness for all objects at once,\nrather than just for this one.\n-/\ndef unit_closed : closed (𝟙_ C) :=\n{ is_adj :=\n  { right := 𝟭 C,\n    adj := adjunction.mk_of_hom_equiv\n    { hom_equiv := λ X _,\n      { to_fun := λ a, (left_unitor X).inv ≫ a,\n        inv_fun := λ a, (left_unitor X).hom ≫ a,\n        left_inv := by tidy,\n        right_inv := by tidy },\n      hom_equiv_naturality_left_symm' := λ X' X Y f g,\n      by { dsimp, rw left_unitor_naturality_assoc } } } }\n\nvariables (A B : C) {X X' Y Y' Z : C}\n\nvariables [closed A]\n\n/--\nThis is the internal hom `A ⟶[C] -`.\n-/\ndef ihom : C ⥤ C :=\n(@closed.is_adj _ _ _ A _).right\n\nnamespace ihom\n\n/-- The adjunction between `A ⊗ -` and `A ⟹ -`. -/\ndef adjunction : tensor_left A ⊣ ihom A :=\nclosed.is_adj.adj\n\n/-- The evaluation natural transformation. -/\ndef ev : ihom A ⋙ tensor_left A ⟶ 𝟭 C :=\n(ihom.adjunction A).counit\n\n/-- The coevaluation natural transformation. -/\ndef coev : 𝟭 C ⟶ tensor_left A ⋙ ihom A :=\n(ihom.adjunction A).unit\n\n@[simp] lemma ihom_adjunction_counit : (ihom.adjunction A).counit = ev A := rfl\n@[simp] lemma ihom_adjunction_unit : (ihom.adjunction A).unit = coev A := rfl\n\n@[simp, reassoc]\nlemma ev_naturality {X Y : C} (f : X ⟶ Y) :\n  ((𝟙 A) ⊗ ((ihom A).map f)) ≫ (ev A).app Y = (ev A).app X ≫ f :=\n(ev A).naturality f\n\n@[simp, reassoc]\nlemma coev_naturality {X Y : C} (f : X ⟶ Y) :\n  f ≫ (coev A).app Y = (coev A).app X ≫ (ihom A).map ((𝟙 A) ⊗ f) :=\n(coev A).naturality f\n\nnotation (name := ihom) A ` ⟶[`C`] ` B:10 := (@ihom C _ _ A _).obj B\n\n@[simp, reassoc] lemma ev_coev :\n  ((𝟙 A) ⊗ ((coev A).app B)) ≫ (ev A).app (A ⊗ B) = 𝟙 (A ⊗ B) :=\nadjunction.left_triangle_components (ihom.adjunction A)\n\n@[simp, reassoc] lemma coev_ev :\n  (coev A).app (A ⟶[C] B) ≫ (ihom A).map ((ev A).app B) = 𝟙 (A ⟶[C] B) :=\nadjunction.right_triangle_components (ihom.adjunction A)\n\nend ihom\n\nopen category_theory.limits\n\ninstance : preserves_colimits (tensor_left A) :=\n(ihom.adjunction A).left_adjoint_preserves_colimits\n\nvariables {A}\n\n-- Wrap these in a namespace so we don't clash with the core versions.\nnamespace monoidal_closed\n\n/-- Currying in a monoidal closed category. -/\ndef curry : (A ⊗ Y ⟶ X) → (Y ⟶ (A ⟶[C] X)) :=\n(ihom.adjunction A).hom_equiv _ _\n/-- Uncurrying in a monoidal closed category. -/\ndef uncurry : (Y ⟶ (A ⟶[C] X)) → (A ⊗ Y ⟶ X) :=\n((ihom.adjunction A).hom_equiv _ _).symm\n\n@[simp] lemma hom_equiv_apply_eq (f : A ⊗ Y ⟶ X) :\n  (ihom.adjunction A).hom_equiv _ _ f = curry f := rfl\n@[simp] lemma hom_equiv_symm_apply_eq (f : Y ⟶ (A ⟶[C] X)) :\n  ((ihom.adjunction A).hom_equiv _ _).symm f = uncurry f := rfl\n\n@[reassoc]\nlemma curry_natural_left (f : X ⟶ X') (g : A ⊗ X' ⟶ Y) :\n  curry (((𝟙 _) ⊗ f) ≫ g) = f ≫ curry g :=\nadjunction.hom_equiv_naturality_left _ _ _\n\n@[reassoc]\nlemma curry_natural_right (f : A ⊗ X ⟶ Y) (g : Y ⟶ Y') :\n  curry (f ≫ g) = curry f ≫ (ihom _).map g :=\nadjunction.hom_equiv_naturality_right _ _ _\n\n@[reassoc]\nlemma uncurry_natural_right  (f : X ⟶ (A ⟶[C] Y)) (g : Y ⟶ Y') :\n  uncurry (f ≫ (ihom _).map g) = uncurry f ≫ g :=\nadjunction.hom_equiv_naturality_right_symm _ _ _\n\n@[reassoc]\nlemma uncurry_natural_left  (f : X ⟶ X') (g : X' ⟶ (A ⟶[C] Y)) :\n  uncurry (f ≫ g) = ((𝟙 _) ⊗ f) ≫ uncurry g :=\nadjunction.hom_equiv_naturality_left_symm _ _ _\n\n@[simp]\nlemma uncurry_curry (f : A ⊗ X ⟶ Y) : uncurry (curry f) = f :=\n(closed.is_adj.adj.hom_equiv _ _).left_inv f\n\n@[simp]\nlemma curry_uncurry (f : X ⟶ (A ⟶[C] Y)) : curry (uncurry f) = f :=\n(closed.is_adj.adj.hom_equiv _ _).right_inv f\n\nlemma curry_eq_iff (f : A ⊗ Y ⟶ X) (g : Y ⟶ (A ⟶[C] X)) :\n  curry f = g ↔ f = uncurry g :=\nadjunction.hom_equiv_apply_eq _ f g\n\nlemma eq_curry_iff (f : A ⊗ Y ⟶ X) (g : Y ⟶ (A ⟶[C] X)) :\n  g = curry f ↔ uncurry g = f :=\nadjunction.eq_hom_equiv_apply _ f g\n\n-- I don't think these two should be simp.\nlemma uncurry_eq (g : Y ⟶ (A ⟶[C] X)) : uncurry g = ((𝟙 A) ⊗ g) ≫ (ihom.ev A).app X :=\nadjunction.hom_equiv_counit _\n\nlemma curry_eq (g : A ⊗ Y ⟶ X) : curry g = (ihom.coev A).app Y ≫ (ihom A).map g :=\nadjunction.hom_equiv_unit _\n\nlemma curry_injective : function.injective (curry : (A ⊗ Y ⟶ X) → (Y ⟶ (A ⟶[C] X))) :=\n(closed.is_adj.adj.hom_equiv _ _).injective\n\nlemma uncurry_injective : function.injective (uncurry : (Y ⟶ (A ⟶[C] X)) → (A ⊗ Y ⟶ X)) :=\n(closed.is_adj.adj.hom_equiv _ _).symm.injective\n\nvariables (A X)\n\nlemma uncurry_id_eq_ev : uncurry (𝟙 (A ⟶[C] X)) = (ihom.ev A).app X :=\nby rw [uncurry_eq, tensor_id, id_comp]\n\nlemma curry_id_eq_coev : curry (𝟙 _) = (ihom.coev A).app X :=\nby { rw [curry_eq, (ihom A).map_id (A ⊗ _)], apply comp_id }\n\nsection pre\n\nvariables {A B} [closed B]\n\n/-- Pre-compose an internal hom with an external hom. -/\ndef pre (f : B ⟶ A) : ihom A ⟶ ihom B :=\ntransfer_nat_trans_self (ihom.adjunction _) (ihom.adjunction _) ((tensoring_left C).map f)\n\n@[simp, reassoc]\n\n\n@[simp]\nlemma uncurry_pre (f : B ⟶ A) (X : C) :\n  monoidal_closed.uncurry ((pre f).app X) = (f ⊗ 𝟙 _) ≫ (ihom.ev A).app X :=\nby rw [uncurry_eq, id_tensor_pre_app_comp_ev]\n\n@[simp, reassoc]\nlemma coev_app_comp_pre_app (f : B ⟶ A) :\n  (ihom.coev A).app X ≫ (pre f).app (A ⊗ X) =\n    (ihom.coev B).app X ≫ (ihom B).map (f ⊗ (𝟙 _)) :=\nunit_transfer_nat_trans_self _ _ ((tensoring_left C).map f) X\n\n@[simp]\nlemma pre_id (A : C) [closed A] : pre (𝟙 A) = 𝟙 _ :=\nby { simp only [pre, functor.map_id], dsimp, simp, }\n\n@[simp]\nlemma pre_map {A₁ A₂ A₃ : C} [closed A₁] [closed A₂] [closed A₃]\n  (f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :\n  pre (f ≫ g) = pre g ≫ pre f :=\nby rw [pre, pre, pre, transfer_nat_trans_self_comp, (tensoring_left C).map_comp]\n\nlemma pre_comm_ihom_map {W X Y Z : C} [closed W] [closed X]\n  (f : W ⟶ X) (g : Y ⟶ Z) :\n  (pre f).app Y ≫ (ihom W).map g = (ihom X).map g ≫ (pre f).app Z := by simp\n\nend pre\n\n/-- The internal hom functor given by the monoidal closed structure. -/\n@[simps]\ndef internal_hom [monoidal_closed C] : Cᵒᵖ ⥤ C ⥤ C :=\n{ obj := λ X, ihom X.unop,\n  map := λ X Y f, pre f.unop }\n\nsection of_equiv\n\nvariables {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D]\n\n/-- Transport the property of being monoidal closed across a monoidal equivalence of categories -/\nnoncomputable\ndef of_equiv (F : monoidal_functor C D) [is_equivalence F.to_functor] [h : monoidal_closed D] :\n  monoidal_closed C :=\n{ closed' := λ X,\n  { is_adj := begin\n      haveI q : closed (F.to_functor.obj X) := infer_instance,\n      haveI : is_left_adjoint (tensor_left (F.to_functor.obj X)) := q.is_adj,\n      have i := comp_inv_iso (monoidal_functor.comm_tensor_left F X),\n      exact adjunction.left_adjoint_of_nat_iso i,\n    end } }\n\n/-- Suppose we have a monoidal equivalence `F : C ≌ D`, with `D` monoidal closed. We can pull the\nmonoidal closed instance back along the equivalence. For `X, Y, Z : C`, this lemma describes the\nresulting currying map `Hom(X ⊗ Y, Z) → Hom(Y, (X ⟶[C] Z))`. (`X ⟶[C] Z` is defined to be\n`F⁻¹(F(X) ⟶[D] F(Z))`, so currying in `C` is given by essentially conjugating currying in\n`D` by `F.`) -/\nlemma of_equiv_curry_def (F : monoidal_functor C D) [is_equivalence F.to_functor]\n  [h : monoidal_closed D] {X Y Z : C} (f : X ⊗ Y ⟶ Z) :\n  @monoidal_closed.curry _ _ _ _ _ _ ((monoidal_closed.of_equiv F).1 _) f =\n  (F.1.1.adjunction.hom_equiv Y ((ihom _).obj _)) (monoidal_closed.curry\n  (F.1.1.inv.adjunction.hom_equiv (F.1.1.obj X ⊗ F.1.1.obj Y) Z\n  ((comp_inv_iso (F.comm_tensor_left X)).hom.app Y ≫ f))) := rfl\n\n/-- Suppose we have a monoidal equivalence `F : C ≌ D`, with `D` monoidal closed. We can pull the\nmonoidal closed instance back along the equivalence. For `X, Y, Z : C`, this lemma describes the\nresulting uncurrying map `Hom(Y, (X ⟶[C] Z)) → Hom(X ⊗ Y ⟶ Z)`. (`X ⟶[C] Z` is\ndefined to be `F⁻¹(F(X) ⟶[D] F(Z))`, so uncurrying in `C` is given by essentially conjugating\nuncurrying in `D` by `F.`) -/\nlemma of_equiv_uncurry_def\n  (F : monoidal_functor C D) [is_equivalence F.to_functor] [h : monoidal_closed D] {X Y Z : C}\n  (f : Y ⟶ (@ihom _ _ _ X $ (monoidal_closed.of_equiv F).1 X).obj Z) :\n  @monoidal_closed.uncurry _ _ _ _ _ _ ((monoidal_closed.of_equiv F).1 _) f =\n  (comp_inv_iso (F.comm_tensor_left X)).inv.app Y ≫ (F.1.1.inv.adjunction.hom_equiv\n  (F.1.1.obj X ⊗ F.1.1.obj Y) Z).symm (monoidal_closed.uncurry\n  ((F.1.1.adjunction.hom_equiv Y ((ihom (F.1.1.obj X)).obj (F.1.1.obj Z))).symm f)) :=\nrfl\n\nend of_equiv\n\nend monoidal_closed\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/closed/monoidal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3182023195530493}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\n\nuniverses u v w\n\n@[inline] def id {α : Sort u} (a : α) : α := a\n\n/-\nThe kernel definitional equality test (t =?= s) has special support for idDelta applications.\nIt implements the following rules\n\n   1)   (idDelta t) =?= t\n   2)   t =?= (idDelta t)\n   3)   (idDelta t) =?= s  IF (unfoldOf t) =?= s\n   4)   t =?= idDelta s    IF t =?= (unfoldOf s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use idDelta applications to address performance problems when Type checking\ntheorems generated by the equation Compiler.\n-/\n@[inline] def idDelta {α : Sort u} (a : α) : α := a\n\n/- `idRhs` is an auxiliary declaration used to implement \"smart unfolding\". It is used as a marker. -/\n@[macroInline, reducible] def idRhs (α : Sort u) (a : α) : α := a\n\nabbrev Function.comp {α : Sort u} {β : Sort v} {δ : Sort w} (f : β → δ) (g : α → β) : α → δ :=\n  fun x => f (g x)\n\nabbrev Function.const {α : Sort u} (β : Sort v) (a : α) : β → α :=\n  fun x => a\n\n@[reducible] def inferInstance {α : Type u} [i : α] : α := i\n@[reducible] def inferInstanceAs (α : Type u) [i : α] : α := i\n\nset_option bootstrap.inductiveCheckResultingUniverse false in\ninductive PUnit : Sort u\n  | unit : PUnit\n\n/-- An abbreviation for `PUnit.{0}`, its most common instantiation.\n    This Type should be preferred over `PUnit` where possible to avoid\n    unnecessary universe parameters. -/\nabbrev Unit : Type := PUnit\n\n@[matchPattern] abbrev Unit.unit : Unit := PUnit.unit\n\n/-- Auxiliary unsafe constant used by the Compiler when erasing proofs from code. -/\nunsafe axiom lcProof {α : Prop} : α\n\n/-- Auxiliary unsafe constant used by the Compiler to mark unreachable code. -/\nunsafe axiom lcUnreachable {α : Sort u} : α\n\ninductive True : Prop\n  | intro : True\n\ninductive False : Prop\n\ninductive Empty : Type\n\ndef Not (a : Prop) : Prop := a → False\n\n@[macroInline] def False.elim {C : Sort u} (h : False) : C :=\n  False.rec (fun _ => C) h\n\n@[macroInline] def absurd {a : Prop} {b : Sort v} (h₁ : a) (h₂ : Not a) : b :=\n  False.elim (h₂ h₁)\n\ninductive Eq {α : Sort u} (a : α) : α → Prop\n  | refl {} : Eq a a\n\nabbrev Eq.ndrec.{u1, u2} {α : Sort u2} {a : α} {motive : α → Sort u1} (m : motive a) {b : α} (h : Eq a b) : motive b :=\n  Eq.rec (motive := fun α _ => motive α) m h\n\n@[matchPattern] def rfl {α : Sort u} {a : α} : Eq a a := Eq.refl a\n\ntheorem Eq.subst {α : Sort u} {motive : α → Prop} {a b : α} (h₁ : Eq a b) (h₂ : motive a) : motive b :=\n  Eq.ndrec h₂ h₁\n\ntheorem Eq.symm {α : Sort u} {a b : α} (h : Eq a b) : Eq b a :=\n  h ▸ rfl\n\n@[macroInline] def cast {α β : Sort u} (h : Eq α β) (a : α) : β :=\n  Eq.rec (motive := fun α _ => α) a h\n\ntheorem congrArg {α : Sort u} {β : Sort v} {a₁ a₂ : α} (f : α → β) (h : Eq a₁ a₂) : Eq (f a₁) (f a₂) :=\n  h ▸ rfl\n\n/-\nInitialize the Quotient Module, which effectively adds the following definitions:\n\nconstant Quot {α : Sort u} (r : α → α → Prop) : Sort u\n\nconstant Quot.mk {α : Sort u} (r : α → α → Prop) (a : α) : Quot r\n\nconstant Quot.lift {α : Sort u} {r : α → α → Prop} {β : Sort v} (f : α → β) :\n  (∀ a b : α, r a b → Eq (f a) (f b)) → Quot r → β\n\nconstant Quot.ind {α : Sort u} {r : α → α → Prop} {β : Quot r → Prop} :\n  (∀ a : α, β (Quot.mk r a)) → ∀ q : Quot r, β q\n-/\ninit_quot\n\ninductive HEq {α : Sort u} (a : α) : {β : Sort u} → β → Prop\n  | refl {} : HEq a a\n\n@[matchPattern] def HEq.rfl {α : Sort u} {a : α} : HEq a a :=\n  HEq.refl a\n\ntheorem eqOfHEq {α : Sort u} {a a' : α} (h : HEq a a') : Eq a a' :=\n  have : (α β : Sort u) → (a : α) → (b : β) → HEq a b → (h : Eq α β) → Eq (cast h a) b :=\n    fun α β a b h₁ =>\n      HEq.rec (motive := fun {β} (b : β) (h : HEq a b) => (h₂ : Eq α β) → Eq (cast h₂ a) b)\n        (fun (h₂ : Eq α α) => rfl)\n        h₁\n  this α α a a' h rfl\n\nstructure Prod (α : Type u) (β : Type v) :=\n  (fst : α) (snd : β)\n\nattribute [unbox] Prod\n\n/-- Similar to `Prod`, but `α` and `β` can be propositions.\n   We use this Type internally to automatically generate the brecOn recursor. -/\nstructure PProd (α : Sort u) (β : Sort v) :=\n  (fst : α) (snd : β)\n\n/-- Similar to `Prod`, but `α` and `β` are in the same universe. -/\nstructure MProd (α β : Type u) :=\n  (fst : α) (snd : β)\n\nstructure And (a b : Prop) : Prop :=\n  intro :: (left : a) (right : b)\n\ninductive Or (a b : Prop) : Prop\n  | inl (h : a) : Or a b\n  | inr (h : b) : Or a b\n\ninductive Bool : Type\n  | false : Bool\n  | true : Bool\n\nexport Bool (false true)\n\n/- Remark: Subtype must take a Sort instead of Type because of the axiom strongIndefiniteDescription. -/\nstructure Subtype {α : Sort u} (p : α → Prop) :=\n  (val : α) (property : p val)\n\n/-- Gadget for optional parameter support. -/\n@[reducible] def optParam (α : Sort u) (default : α) : Sort u := α\n\n/-- Gadget for marking output parameters in type classes. -/\n@[reducible] def outParam (α : Sort u) : Sort u := α\n\n/-- Auxiliary Declaration used to implement the notation (a : α) -/\n@[reducible] def typedExpr (α : Sort u) (a : α) : α := a\n\n/-- Auxiliary Declaration used to implement the named patterns `x@p` -/\n@[reducible] def namedPattern {α : Sort u} (x a : α) : α := a\n\n/- Auxiliary axiom used to implement `sorry`. -/\naxiom sorryAx (α : Sort u) (synthetic := true) : α\n\ntheorem eqFalseOfNeTrue : {b : Bool} → Not (Eq b true) → Eq b false\n  | true, h => False.elim (h rfl)\n  | false, h => rfl\n\ntheorem eqTrueOfNeFalse : {b : Bool} → Not (Eq b false) → Eq b true\n  | true, h => rfl\n  | false, h => False.elim (h rfl)\n\ntheorem neFalseOfEqTrue : {b : Bool} → Eq b true → Not (Eq b false)\n  | true, _  => fun h => Bool.noConfusion h\n  | false, h => Bool.noConfusion h\n\ntheorem neTrueOfEqFalse : {b : Bool} → Eq b false → Not (Eq b true)\n  | true, h  => Bool.noConfusion h\n  | false, _ => fun h => Bool.noConfusion h\n\nclass Inhabited (α : Sort u) :=\n  mk {} :: (default : α)\n\nconstant arbitrary (α : Sort u) [s : Inhabited α] : α :=\n  @Inhabited.default α s\n\ninstance (α : Sort u) {β : Sort v} [Inhabited β] : Inhabited (α → β) := {\n  default := fun _ => arbitrary β\n}\n\ninstance (α : Sort u) {β : α → Sort v} [(a : α) → Inhabited (β a)] : Inhabited ((a : α) → β a) := {\n  default := fun a => arbitrary (β a)\n}\n\n/-- Universe lifting operation from Sort to Type -/\nstructure PLift (α : Sort u) : Type u :=\n  up :: (down : α)\n\n/- Bijection between α and PLift α -/\ntheorem PLift.upDown {α : Sort u} : ∀ (b : PLift α), Eq (up (down b)) b\n  | up a => rfl\n\ntheorem PLift.downUp {α : Sort u} (a : α) : Eq (down (up a)) a :=\n  rfl\n\n/- Pointed types -/\nstructure PointedType :=\n  (type : Type u)\n  (val : type)\n\ninstance : Inhabited PointedType.{u} := {\n  default := { type := PUnit.{u+1}, val := ⟨⟩ }\n}\n\n/-- Universe lifting operation -/\nstructure ULift.{r, s} (α : Type s) : Type (max s r) :=\n  up :: (down : α)\n\n/- Bijection between α and ULift.{v} α -/\ntheorem ULift.upDown {α : Type u} : ∀ (b : ULift.{v} α), Eq (up (down b)) b\n  | up a => rfl\n\ntheorem ULift.downUp {α : Type u} (a : α) : Eq (down (up.{v} a)) a :=\n  rfl\n\nclass inductive Decidable (p : Prop)\n  | isFalse (h : Not p) : Decidable p\n  | isTrue  (h : p) : Decidable p\n\n@[inlineIfReduce, nospecialize] def Decidable.decide (p : Prop) [h : Decidable p] : Bool :=\n  Decidable.casesOn (motive := fun _ => Bool) h (fun _ => false) (fun _ => true)\n\nexport Decidable (isTrue isFalse decide)\n\nabbrev DecidablePred {α : Sort u} (r : α → Prop) :=\n  (a : α) → Decidable (r a)\n\nabbrev DecidableRel {α : Sort u} (r : α → α → Prop) :=\n  (a b : α) → Decidable (r a b)\n\nabbrev DecidableEq (α : Sort u) :=\n  (a b : α) → Decidable (Eq a b)\n\ndef decEq {α : Sort u} [s : DecidableEq α] (a b : α) : Decidable (Eq a b) :=\n  s a b\n\ntheorem decideEqTrue : {p : Prop} → [s : Decidable p] → p → Eq (decide p) true\n  | _, isTrue  _, _   => rfl\n  | _, isFalse h₁, h₂ => absurd h₂ h₁\n\ntheorem decideEqTrue' : [s : Decidable p] → p → Eq (decide p) true\n  | isTrue  _, _   => rfl\n  | isFalse h₁, h₂ => absurd h₂ h₁\n\ntheorem decideEqFalse : {p : Prop} → [s : Decidable p] → Not p → Eq (decide p) false\n  | _, isTrue  h₁, h₂ => absurd h₁ h₂\n  | _, isFalse h, _   => rfl\n\ntheorem ofDecideEqTrue {p : Prop} [s : Decidable p] : Eq (decide p) true → p := fun h =>\n  match s with\n  | isTrue  h₁ => h₁\n  | isFalse h₁ => absurd h (neTrueOfEqFalse (decideEqFalse h₁))\n\ntheorem ofDecideEqFalse {p : Prop} [s : Decidable p] : Eq (decide p) false → Not p := fun h =>\n  match s with\n  | isTrue  h₁ => absurd h (neFalseOfEqTrue (decideEqTrue h₁))\n  | isFalse h₁ => h₁\n\n@[inline] instance : DecidableEq Bool :=\n  fun a b => match a, b with\n   | false, false => isTrue rfl\n   | false, true  => isFalse (fun h => Bool.noConfusion h)\n   | true, false  => isFalse (fun h => Bool.noConfusion h)\n   | true, true   => isTrue rfl\n\nclass BEq      (α : Type u) := (beq : α → α → Bool)\n\nopen BEq (beq)\n\ninstance {α : Type u} [DecidableEq α] : BEq α :=\n  ⟨fun a b => decide (Eq a b)⟩\n\n-- We use \"dependent\" if-then-else to be able to communicate the if-then-else condition\n-- to the branches\n@[macroInline] def dite {α : Sort u} (c : Prop) [h : Decidable c] (t : c → α) (e : Not c → α) : α :=\n  Decidable.casesOn (motive := fun _ => α) h e t\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/Reformat/Input.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244837, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3181641262354499}}
{"text": "/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\nimport tactic.rcases\n/-!\n# lift tactic\n\nThis file defines the `lift` tactic, allowing the user to lift elements from one type to another\nunder a specified condition.\n\n## Tags\n\nlift, tactic\n-/\n\nuniverse variables u v w\n\n/-- A class specifying that you can lift elements from `α` to `β` assuming `cond` is true.\n  Used by the tactic `lift`. -/\nclass can_lift (α : Type u) (β : Type v) : Type (max u v) :=\n(coe : β → α)\n(cond : α → Prop)\n(prf : ∀(x : α), cond x → ∃(y : β), coe y = x)\n\nopen tactic\n\n/--\nA user attribute used internally by the `lift` tactic.\nThis should not be applied by hand.\n-/\n@[user_attribute]\nmeta def can_lift_attr : user_attribute (list name) :=\n{ name := \"_can_lift\",\n  descr := \"internal attribute used by the lift tactic\",\n  cache_cfg := { mk_cache := λ _,\n    do { ls ← attribute.get_instances `instance,\n        ls.mfilter $ λ l,\n        do { (_,t) ← mk_const l >>= infer_type >>= open_pis,\n         return $ t.is_app_of `can_lift } },\n  dependencies := [`instance] } }\n\ninstance : can_lift ℤ ℕ :=\n⟨coe, λ n, 0 ≤ n, λ n hn, ⟨n.nat_abs, int.nat_abs_of_nonneg hn⟩⟩\n\n/-- Enable automatic handling of pi types in `can_lift`. -/\ninstance pi.can_lift (ι : Type u) (α : Π i : ι, Type v) (β : Π i : ι, Type w)\n  [Π i : ι, can_lift (α i) (β i)] :\n  can_lift (Π i : ι, α i) (Π i : ι, β i) :=\n{ coe := λ f i, can_lift.coe (f i),\n  cond := λ f, ∀ i, can_lift.cond (β i) (f i),\n  prf := λ f hf, ⟨λ i, classical.some (can_lift.prf (f i) (hf i)), funext $ λ i,\n    classical.some_spec (can_lift.prf (f i) (hf i))⟩ }\n\ninstance pi_subtype.can_lift (ι : Type u) (α : Π i : ι, Type v) [ne : Π i, nonempty (α i)]\n  (p : ι → Prop) :\n  can_lift (Π i : subtype p, α i) (Π i, α i) :=\n{ coe := λ f i, f i,\n  cond := λ _, true,\n  prf :=\n    begin\n      classical,\n      refine λ f _, ⟨λ i, if hi : p i then f ⟨i, hi⟩ else classical.choice (ne i), funext _⟩,\n      rintro ⟨i, hi⟩,\n      exact dif_pos hi\n    end }\n\ninstance pi_subtype.can_lift' (ι : Type u) (α : Type v) [ne : nonempty α] (p : ι → Prop) :\n  can_lift (subtype p → α) (ι → α) :=\npi_subtype.can_lift ι (λ _, α) p\n\nnamespace tactic\n\n/--\nConstruct the proof of `cond x` in the lift tactic.\n*  `e` is the expression being lifted and `h` is the specified proof of `can_lift.cond e`.\n*  `old_tp` and `new_tp` are the arguments to `can_lift` and `inst` is the `can_lift`-instance.\n*  `s` and `to_unfold` contain the information of the simp set used to simplify.\n\nIf the proof was specified, we check whether it has the correct type.\nIf it doesn't have the correct type, we display an error message\n(but first call dsimp on the expression in the message).\n\nIf the proof was not specified, we create assert it as a local constant.\n(The name of this local constant doesn't matter, since `lift` will remove it from the context.)\n-/\nmeta def get_lift_prf (h : option pexpr) (old_tp new_tp inst e : expr)\n  (s : simp_lemmas) (to_unfold : list name) : tactic expr :=\nif h_some : h.is_some then\n  (do prf ← i_to_expr (option.get h_some), prf_ty ← infer_type prf,\n  expected_prf_ty ← mk_app `can_lift.cond [old_tp, new_tp, inst, e],\n  unify prf_ty expected_prf_ty <|>\n    (do expected_prf_ty2 ← s.dsimplify to_unfold expected_prf_ty,\n      pformat!\"lift tactic failed. The type of\\n  {prf}\\nis\\n  {prf_ty}\\nbut it is expected to be\\n  {expected_prf_ty2}\" >>= fail),\n  return prf)\n  else (do prf_nm ← get_unused_name,\n    prf ← mk_app `can_lift.cond [old_tp, new_tp, inst, e] >>= assert prf_nm,\n    dsimp_target s to_unfold {}, swap, return prf)\n\n/-- Lift the expression `p` to the type `t`, with proof obligation given by `h`.\n  The list `n` is used for the two newly generated names, and to specify whether `h` should\n  remain in the local context. See the doc string of `tactic.interactive.lift` for more information.\n  -/\nmeta def lift (p : pexpr) (t : pexpr) (h : option pexpr) (n : list name) : tactic unit :=\ndo\n  propositional_goal <|>\n    fail \"lift tactic failed. Tactic is only applicable when the target is a proposition.\",\n  e ← i_to_expr p,\n  old_tp ← infer_type e,\n  new_tp ← i_to_expr t,\n  inst_type ← mk_app ``can_lift [old_tp, new_tp],\n  inst ← mk_instance inst_type <|>\n    pformat!\"Failed to find a lift from {old_tp} to {new_tp}. Provide an instance of\\n  {inst_type}\"\n    >>= fail,\n  /- make the simp set to get rid of `can_lift` projections -/\n  can_lift_instances ← can_lift_attr.get_cache >>= λ l, l.mmap resolve_name,\n  (s, to_unfold) ← mk_simp_set tt [] $ can_lift_instances.map simp_arg_type.expr,\n  prf_cond ← get_lift_prf h old_tp new_tp inst e s to_unfold,\n  let prf_nm := if prf_cond.is_local_constant then some prf_cond.local_pp_name else none,\n  /- We use mk_mapp to apply `can_lift.prf` to all but one argument, and then just use expr.app\n  for the last argument. For some reason we get an error when applying mk_mapp it to all\n  arguments. -/\n  prf_ex0 ← mk_mapp `can_lift.prf [old_tp, new_tp, inst, e],\n  let prf_ex := prf_ex0 prf_cond,\n  /- Find the name of the new variable -/\n  new_nm ← if n ≠ [] then return n.head\n    else if e.is_local_constant then return e.local_pp_name\n    else get_unused_name,\n  /- Find the name of the proof of the equation -/\n  eq_nm ← if hn : 1 < n.length then return (n.nth_le 1 hn)\n    else if e.is_local_constant then return `rfl\n    else get_unused_name `h,\n  /- We add the proof of the existential statement to the context and then apply\n  `dsimp` to it, unfolding all `can_lift` instances. -/\n  temp_nm ← get_unused_name,\n  temp_e ← note temp_nm none prf_ex,\n  dsimp_hyp temp_e s to_unfold {},\n  /- We case on the existential. We use `rcases` because `eq_nm` could be `rfl`. -/\n  rcases none (pexpr.of_expr temp_e) $ rcases_patt.tuple ([new_nm, eq_nm].map rcases_patt.one),\n  /- If the lifted variable is not a local constant,\n    try to rewrite it away using the new equality. -/\n  when (¬ e.is_local_constant) (get_local eq_nm >>=\n    λ e, interactive.rw ⟨[⟨⟨0, 0⟩, tt, (pexpr.of_expr e)⟩], none⟩ interactive.loc.wildcard),\n  /- If the proof `prf_cond` is a local constant, remove it from the context,\n    unless `n` specifies to keep it. -/\n  if h_prf_nm : prf_nm.is_some ∧ n.nth 2 ≠ prf_nm then\n    get_local (option.get h_prf_nm.1) >>= clear else skip\n\nopen lean.parser interactive interactive.types\n\nlocal postfix `?`:9001 := optional\n/-- Parses an optional token \"using\" followed by a trailing `pexpr`. -/\nmeta def using_texpr := (tk \"using\" *> texpr)?\n\n/-- Parses a token \"to\" followed by a trailing `pexpr`. -/\nmeta def to_texpr := (tk \"to\" *> texpr)\n\nnamespace interactive\n\n/--\nLift an expression to another type.\n* Usage: `'lift' expr 'to' expr ('using' expr)? ('with' id (id id?)?)?`.\n* If `n : ℤ` and `hn : n ≥ 0` then the tactic `lift n to ℕ using hn` creates a new\n  constant of type `ℕ`, also named `n` and replaces all occurrences of the old variable `(n : ℤ)`\n  with `↑n` (where `n` in the new variable). It will remove `n` and `hn` from the context.\n  + So for example the tactic `lift n to ℕ using hn` transforms the goal\n    `n : ℤ, hn : n ≥ 0, h : P n ⊢ n = 3` to `n : ℕ, h : P ↑n ⊢ ↑n = 3`\n    (here `P` is some term of type `ℤ → Prop`).\n* The argument `using hn` is optional, the tactic `lift n to ℕ` does the same, but also creates a\n  new subgoal that `n ≥ 0` (where `n` is the old variable).\n  + So for example the tactic `lift n to ℕ` transforms the goal\n    `n : ℤ, h : P n ⊢ n = 3` to two goals\n    `n : ℕ, h : P ↑n ⊢ ↑n = 3` and `n : ℤ, h : P n ⊢ n ≥ 0`.\n* You can also use `lift n to ℕ using e` where `e` is any expression of type `n ≥ 0`.\n* Use `lift n to ℕ with k` to specify the name of the new variable.\n* Use `lift n to ℕ with k hk` to also specify the name of the equality `↑k = n`. In this case, `n`\n  will remain in the context. You can use `rfl` for the name of `hk` to substitute `n` away\n  (i.e. the default behavior).\n* You can also use `lift e to ℕ with k hk` where `e` is any expression of type `ℤ`.\n  In this case, the `hk` will always stay in the context, but it will be used to rewrite `e` in\n  all hypotheses and the target.\n  + So for example the tactic `lift n + 3 to ℕ using hn with k hk` transforms the goal\n    `n : ℤ, hn : n + 3 ≥ 0, h : P (n + 3) ⊢ n + 3 = 2 * n` to the goal\n    `n : ℤ, k : ℕ, hk : ↑k = n + 3, h : P ↑k ⊢ ↑k = 2 * n`.\n* The tactic `lift n to ℕ using h` will remove `h` from the context. If you want to keep it,\n  specify it again as the third argument to `with`, like this: `lift n to ℕ using h with n rfl h`.\n* More generally, this can lift an expression from `α` to `β` assuming that there is an instance\n  of `can_lift α β`. In this case the proof obligation is specified by `can_lift.cond`.\n* Given an instance `can_lift β γ`, it can also lift `α → β` to `α → γ`; more generally, given\n  `β : Π a : α, Type*`, `γ : Π a : α, Type*`, and `[Π a : α, can_lift (β a) (γ a)]`, it\n  automatically generates an instance `can_lift (Π a, β a) (Π a, γ a)`.\n\n`lift` is in some sense dual to the `zify` tactic. `lift (z : ℤ) to ℕ` will change the type of an\ninteger `z` (in the supertype) to `ℕ` (the subtype), given a proof that `z ≥ 0`;\npropositions concerning `z` will still be over `ℤ`. `zify` changes propositions about `ℕ` (the\nsubtype) to propositions about `ℤ` (the supertype), without changing the type of any variable.\n-/\nmeta def lift (p : parse texpr) (t : parse to_texpr) (h : parse using_texpr)\n  (n : parse with_ident_list) : tactic unit :=\ntactic.lift p t h n\n\nadd_tactic_doc\n{ name       := \"lift\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.lift],\n  tags       := [\"coercions\"] }\n\nend interactive\nend tactic\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/tactic/lift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.31808773331795054}}
{"text": "import tactic.basic\n\n-- Lean version 3.45.0\n\nsection\n\nparameters (C : Type) [decidable_eq C]\n\ninductive term : Type\n| var : ℕ → term\n| const : C → term\n| app : term → term → term\n| lam : term → term → term\n| pi : term → term → term\n\ninstance : decidable_eq term :=\nbegin\n  intros a a',\n  induction a with x c f a ih₁ ih₂ A b ih₁ ih₂ A B ih₁ ih₂ generalizing a',\n  { cases a' with x'; try { apply is_false, intro h, cases h, done },\n    by_cases h : x = x',\n    { subst h,\n      exact is_true rfl },\n    { apply is_false,\n      intro h',\n      cases h',\n      exact h rfl } },\n  { cases a' with _ c'; try { apply is_false, intro h, cases h, done },\n    by_cases h : c = c',\n    { subst h,\n      exact is_true rfl },\n    { apply is_false,\n      intro h',\n      cases h',\n      exact h rfl } },\n  { cases a' with _ _ f' a'; try { apply is_false, intro h, cases h, done },\n    specialize ih₁ f',\n    specialize ih₂ a',\n    cases ih₁, apply is_false, intro h, cases h, exact ih₁ rfl,\n    cases ih₂, apply is_false, intro h, cases h, exact ih₂ rfl,\n    subst ih₁, subst ih₂,\n    exact is_true rfl },\n  { cases a' with _ _ _ _ A' b'; try { apply is_false, intro h, cases h, done },\n    specialize ih₁ A',\n    specialize ih₂ b',\n    cases ih₁, apply is_false, intro h, cases h, exact ih₁ rfl,\n    cases ih₂, apply is_false, intro h, cases h, exact ih₂ rfl,\n    subst ih₁, subst ih₂,\n    exact is_true rfl },\n  { cases a' with _ _ _ _ _ _ A' B'; try { apply is_false, intro h, cases h, done },\n    specialize ih₁ A',\n    specialize ih₂ B',\n    cases ih₁, apply is_false, intro h, cases h, exact ih₁ rfl,\n    cases ih₂, apply is_false, intro h, cases h, exact ih₂ rfl,\n    subst ih₁, subst ih₂,\n    exact is_true rfl }\nend\n\ninstance has_coe_var : has_coe ℕ term := ⟨term.var⟩\ninstance has_coe_const : has_coe C term := ⟨term.const⟩\n\ndef shift (σ : ℕ → term) : ℕ → term\n| 0 := term.var 0\n| (x + 1) := σ x\n\ndef substs : (ℕ → term) → term → term\n| σ (term.var x) := σ x\n| σ (term.const c) := term.const c\n| σ (term.app f a) := term.app (substs σ f) (substs σ a)\n| σ (term.lam A b) := term.lam (substs σ A) (substs (shift σ) b)\n| σ (term.pi A B) := term.lam (substs σ A) (substs (shift σ) B)\n\ndef subst.weak : term → term :=\nsubsts (term.var ∘ nat.succ)\n\ndef subst (a : term) : term → term :=\nlet σ (x : ℕ) : term :=\n  match x with\n  | 0 := a\n  | (x + 1) := x\n  end in\nsubsts σ\n\nparameters (S : set C) (A : set (C × C)) (R : set (C × C × C))\nparameters [decidable_pred S] [decidable_pred A] [decidable_pred R]\n\nparameter hA : ∀ {c s}, (c, s) ∈ A → s ∈ S\nparameter hR₁ : ∀ {s₁ s₂ s₃}, (s₁, s₂, s₃) ∈ R → s₁ ∈ S\nparameter hR₂ : ∀ {s₁ s₂ s₃}, (s₁, s₂, s₃) ∈ R → s₂ ∈ S\nparameter hR₃ : ∀ {s₁ s₂ s₃}, (s₁, s₂, s₃) ∈ R → s₃ ∈ S\n\nabbreviation context : Type := list term\n\ninductive entails₁ : context → term → term → Prop\n| ax₁ {c s} : (c, s) ∈ A → entails₁ [] ↑c ↑s\n| start {Γ A s} : s ∈ S → entails₁ Γ A ↑s → entails₁ (A :: Γ) ↑0 A\n| weak {Γ A B C s} : s ∈ S → entails₁ Γ A B → entails₁ Γ C ↑s → entails₁ (C :: Γ) (subst.weak A) (subst.weak B)\n| pi {Γ A B s₁ s₂ s₃} : (s₁, s₂, s₃) ∈ R → entails₁ Γ A ↑s₁ → entails₁ (A :: Γ) B ↑s₂ → entails₁ Γ (term.pi A B) ↑s₃\n| app {Γ f A B a} : entails₁ Γ f (term.pi A B) → entails₁ Γ a A → entails₁ Γ (term.app f a) (subst a B)\n| lam {Γ A b B s} : s ∈ S → entails₁ (A :: Γ) b B → entails₁ Γ (term.pi A B) ↑s → entails₁ Γ (term.lam A b) (term.pi A B)\n\ninductive ctx₁ : context → Prop\n| empty : ctx₁ []\n| ext {Γ A s} : s ∈ S → entails₁ Γ A ↑s → ctx₁ (A :: Γ)\n\nset_option class.instance_max_depth 5\nset_option trace.class_instances true\nlemma ctx₁_of_entails₁ {Γ A B} : entails₁ Γ A B → ctx₁ Γ :=\nbegin\n  sorry\nend\n\n-- This lemma is not true because `Γ` can be literally anything, including an\n-- invalid context.\nlemma entails₁.ax₂ {Γ c s} : (c, s) ∈ A → entails₁ Γ ↑c ↑s :=\nbegin\n  intro h,\n  induction Γ with B Γ ih,\n  { exact entails₁.ax₁ h },\n  { refine entails₁.weak _ ih _,\n    sorry\n  }\nend\n\nend\n", "meta": {"author": "pedrominicz", "repo": "learn", "sha": "b79b802a9846c86c21d4b6f3e17af36e7382f0ef", "save_path": "github-repos/lean/pedrominicz-learn", "path": "github-repos/lean/pedrominicz-learn/learn-b79b802a9846c86c21d4b6f3e17af36e7382f0ef/src/surreal/pts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3180877333179505}}
{"text": "/- More tactics -/\n\nexample (p q : Nat → Prop) : (∃ x, p x ∧ q x) → ∃ x, q x ∧ p x := by\n  intro h\n  cases h with\n  | intro x hpq =>\n    cases hpq with\n    | intro hp hq =>\n      exists x\n\nexample : p ∧ q → q ∧ p := by\n  intro p\n  cases p\n  constructor <;> assumption\n\nexample : p ∧ ¬ p → q := by\n  intro h\n  cases h\n  contradiction\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/doc/examples/NFM2022/nfm17.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.31808772523510354}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.rbtree.find\n\nuniverses u v \n\nnamespace Mathlib\n\nnamespace rbnode\n\n\n@[simp] theorem balance1_eq₁ {α : Type u} (l : rbnode α) (x : α) (r₁ : rbnode α) (y : α)\n    (r₂ : rbnode α) (v : α) (t : rbnode α) :\n    balance1 (red_node l x r₁) y r₂ v t = red_node (black_node l x r₁) y (black_node r₂ v t) :=\n  sorry\n\n@[simp] theorem balance1_eq₂ {α : Type u} (l₁ : rbnode α) (y : α) (l₂ : rbnode α) (x : α)\n    (r : rbnode α) (v : α) (t : rbnode α) :\n    get_color l₁ ≠ color.red →\n        balance1 l₁ y (red_node l₂ x r) v t = red_node (black_node l₁ y l₂) x (black_node r v t) :=\n  sorry\n\n@[simp] theorem balance1_eq₃ {α : Type u} (l : rbnode α) (y : α) (r : rbnode α) (v : α)\n    (t : rbnode α) :\n    get_color l ≠ color.red →\n        get_color r ≠ color.red → balance1 l y r v t = black_node (red_node l y r) v t :=\n  sorry\n\n@[simp] theorem balance2_eq₁ {α : Type u} (l : rbnode α) (x₁ : α) (r₁ : rbnode α) (y : α)\n    (r₂ : rbnode α) (v : α) (t : rbnode α) :\n    balance2 (red_node l x₁ r₁) y r₂ v t = red_node (black_node t v l) x₁ (black_node r₁ y r₂) :=\n  sorry\n\n@[simp] theorem balance2_eq₂ {α : Type u} (l₁ : rbnode α) (y : α) (l₂ : rbnode α) (x₂ : α)\n    (r₂ : rbnode α) (v : α) (t : rbnode α) :\n    get_color l₁ ≠ color.red →\n        balance2 l₁ y (red_node l₂ x₂ r₂) v t =\n          red_node (black_node t v l₁) y (black_node l₂ x₂ r₂) :=\n  sorry\n\n@[simp] theorem balance2_eq₃ {α : Type u} (l : rbnode α) (y : α) (r : rbnode α) (v : α)\n    (t : rbnode α) :\n    get_color l ≠ color.red →\n        get_color r ≠ color.red → balance2 l y r v t = black_node t v (red_node l y r) :=\n  sorry\n\n/- We can use the same induction principle for balance1 and balance2 -/\n\ntheorem balance.cases {α : Type u} {p : rbnode α → α → rbnode α → Prop} (l : rbnode α) (y : α)\n    (r : rbnode α)\n    (red_left :\n      ∀ (l : rbnode α) (x : α) (r₁ : rbnode α) (y : α) (r₂ : rbnode α), p (red_node l x r₁) y r₂)\n    (red_right :\n      ∀ (l₁ : rbnode α) (y : α) (l₂ : rbnode α) (x : α) (r : rbnode α),\n        get_color l₁ ≠ color.red → p l₁ y (red_node l₂ x r))\n    (other :\n      ∀ (l : rbnode α) (y : α) (r : rbnode α),\n        get_color l ≠ color.red → get_color r ≠ color.red → p l y r) :\n    p l y r :=\n  sorry\n\ntheorem balance1_ne_leaf {α : Type u} (l : rbnode α) (x : α) (r : rbnode α) (v : α) (t : rbnode α) :\n    balance1 l x r v t ≠ leaf :=\n  sorry\n\ntheorem balance1_node_ne_leaf {α : Type u} {s : rbnode α} (a : α) (t : rbnode α) :\n    s ≠ leaf → balance1_node s a t ≠ leaf :=\n  sorry\n\ntheorem balance2_ne_leaf {α : Type u} (l : rbnode α) (x : α) (r : rbnode α) (v : α) (t : rbnode α) :\n    balance2 l x r v t ≠ leaf :=\n  sorry\n\ntheorem balance2_node_ne_leaf {α : Type u} {s : rbnode α} (a : α) (t : rbnode α) :\n    s ≠ leaf → balance2_node s a t ≠ leaf :=\n  sorry\n\ntheorem ins.induction {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {p : rbnode α → Prop}\n    (t : rbnode α) (x : α) (is_leaf : p leaf)\n    (is_red_lt :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.lt → p a → p (red_node a y b))\n    (is_red_eq :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α), cmp_using lt x y = ordering.eq → p (red_node a y b))\n    (is_red_gt :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.gt → p b → p (red_node a y b))\n    (is_black_lt_red :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.lt → get_color a = color.red → p a → p (black_node a y b))\n    (is_black_lt_not_red :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.lt → get_color a ≠ color.red → p a → p (black_node a y b))\n    (is_black_eq :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.eq → p (black_node a y b))\n    (is_black_gt_red :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.gt → get_color b = color.red → p b → p (black_node a y b))\n    (is_black_gt_not_red :\n      ∀ (a : rbnode α) (y : α) (b : rbnode α),\n        cmp_using lt x y = ordering.gt → get_color b ≠ color.red → p b → p (black_node a y b)) :\n    p t :=\n  sorry\n\ntheorem is_searchable_balance1 {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {l : rbnode α}\n    {y : α} {r : rbnode α} {v : α} {t : rbnode α} {lo : Option α} {hi : Option α} :\n    is_searchable lt l lo (some y) →\n        is_searchable lt r (some y) (some v) →\n          is_searchable lt t (some v) hi → is_searchable lt (balance1 l y r v t) lo hi :=\n  sorry\n\ntheorem is_searchable_balance1_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    {t : rbnode α} [is_trans α lt] {y : α} {s : rbnode α} {lo : Option α} {hi : Option α} :\n    is_searchable lt t lo (some y) →\n        is_searchable lt s (some y) hi → is_searchable lt (balance1_node t y s) lo hi :=\n  sorry\n\ntheorem is_searchable_balance2 {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {l : rbnode α}\n    {y : α} {r : rbnode α} {v : α} {t : rbnode α} {lo : Option α} {hi : Option α} :\n    is_searchable lt t lo (some v) →\n        is_searchable lt l (some v) (some y) →\n          is_searchable lt r (some y) hi → is_searchable lt (balance2 l y r v t) lo hi :=\n  sorry\n\ntheorem is_searchable_balance2_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    {t : rbnode α} [is_trans α lt] {y : α} {s : rbnode α} {lo : Option α} {hi : Option α} :\n    is_searchable lt s lo (some y) →\n        is_searchable lt t (some y) hi → is_searchable lt (balance2_node t y s) lo hi :=\n  sorry\n\ntheorem is_searchable_ins {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {t : rbnode α} {x : α}\n    [is_strict_weak_order α lt] {lo : Option α} {hi : Option α} (h : is_searchable lt t lo hi) :\n    lift lt lo (some x) → lift lt (some x) hi → is_searchable lt (ins lt t x) lo hi :=\n  sorry\n\ntheorem is_searchable_mk_insert_result {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    {c : color} {t : rbnode α} :\n    is_searchable lt t none none → is_searchable lt (mk_insert_result c t) none none :=\n  sorry\n\ntheorem is_searchable_insert {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {t : rbnode α}\n    {x : α} [is_strict_weak_order α lt] :\n    is_searchable lt t none none → is_searchable lt (insert lt t x) none none :=\n  sorry\n\nend rbnode\n\n\nnamespace rbnode\n\n\ntheorem mem_balance1_node_of_mem_left {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {s : rbnode α} (v : α) (t : rbnode α) : mem lt x s → mem lt x (balance1_node s v t) :=\n  sorry\n\ntheorem mem_balance2_node_of_mem_left {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {s : rbnode α} (v : α) (t : rbnode α) : mem lt x s → mem lt x (balance2_node s v t) :=\n  sorry\n\ntheorem mem_balance1_node_of_mem_right {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {t : rbnode α} (v : α) (s : rbnode α) : mem lt x t → mem lt x (balance1_node s v t) :=\n  sorry\n\ntheorem mem_balance2_node_of_mem_right {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {t : rbnode α} (v : α) (s : rbnode α) : mem lt x t → mem lt x (balance2_node s v t) :=\n  sorry\n\ntheorem mem_balance1_node_of_incomp {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {v : α} (s : rbnode α) (t : rbnode α) :\n    ¬lt x v ∧ ¬lt v x → s ≠ leaf → mem lt x (balance1_node s v t) :=\n  sorry\n\ntheorem mem_balance2_node_of_incomp {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {x : α}\n    {v : α} (s : rbnode α) (t : rbnode α) :\n    ¬lt v x ∧ ¬lt x v → s ≠ leaf → mem lt x (balance2_node s v t) :=\n  sorry\n\ntheorem ins_ne_leaf {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (t : rbnode α) (x : α) :\n    ins lt t x ≠ leaf :=\n  sorry\n\ntheorem insert_ne_leaf {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (t : rbnode α) (x : α) :\n    insert lt t x ≠ leaf :=\n  sorry\n\ntheorem mem_ins_of_incomp {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (t : rbnode α) {x : α}\n    {y : α} (h : ¬lt x y ∧ ¬lt y x) : mem lt x (ins lt t y) :=\n  sorry\n\ntheorem mem_ins_of_mem {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} (z : α) {x : α} (h : mem lt x t) :\n    mem lt x (ins lt t z) :=\n  sorry\n\ntheorem mem_mk_insert_result {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {a : α}\n    {t : rbnode α} (c : color) : mem lt a t → mem lt a (mk_insert_result c t) :=\n  sorry\n\ntheorem mem_of_mem_mk_insert_result {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {a : α}\n    {t : rbnode α} {c : color} : mem lt a (mk_insert_result c t) → mem lt a t :=\n  sorry\n\ntheorem mem_insert_of_incomp {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (t : rbnode α)\n    {x : α} {y : α} (h : ¬lt x y ∧ ¬lt y x) : mem lt x (insert lt t y) :=\n  sorry\n\ntheorem mem_insert_of_mem {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {x : α} (z : α) :\n    mem lt x t → mem lt x (insert lt t z) :=\n  fun (ᾰ : mem lt x t) => mem_mk_insert_result lt (get_color t) (mem_ins_of_mem lt z ᾰ)\n\ntheorem of_mem_balance1_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {s : rbnode α} {v : α} {t : rbnode α} :\n    mem lt x (balance1_node s v t) → mem lt x s ∨ ¬lt x v ∧ ¬lt v x ∨ mem lt x t :=\n  sorry\n\ntheorem of_mem_balance2_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {s : rbnode α} {v : α} {t : rbnode α} :\n    mem lt x (balance2_node s v t) → mem lt x s ∨ ¬lt x v ∧ ¬lt v x ∨ mem lt x t :=\n  sorry\n\ntheorem equiv_or_mem_of_mem_ins {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {x : α} {z : α} (h : mem lt x (ins lt t z)) :\n    strict_weak_order.equiv x z ∨ mem lt x t :=\n  sorry\n\ntheorem equiv_or_mem_of_mem_insert {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {x : α} {z : α} (h : mem lt x (insert lt t z)) :\n    strict_weak_order.equiv x z ∨ mem lt x t :=\n  sorry\n\ntheorem mem_exact_balance1_node_of_mem_exact {α : Type u} {x : α} {s : rbnode α} (v : α)\n    (t : rbnode α) : mem_exact x s → mem_exact x (balance1_node s v t) :=\n  sorry\n\ntheorem mem_exact_balance2_node_of_mem_exact {α : Type u} {x : α} {s : rbnode α} (v : α)\n    (t : rbnode α) : mem_exact x s → mem_exact x (balance2_node s v t) :=\n  sorry\n\ntheorem find_balance1_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {z : α} {t : rbnode α} {s : rbnode α}\n    {lo : Option α} {hi : Option α} :\n    is_searchable lt t lo (some z) →\n        is_searchable lt s (some z) hi →\n          find lt t y = some x →\n            strict_weak_order.equiv y x → find lt (balance1_node t z s) y = some x :=\n  sorry\n\ntheorem find_balance2_node {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {z : α} {s : rbnode α} {t : rbnode α}\n    [is_trans α lt] {lo : Option α} {hi : Option α} :\n    is_searchable lt s lo (some z) →\n        is_searchable lt t (some z) hi →\n          find lt t y = some x →\n            strict_weak_order.equiv y x → find lt (balance2_node t z s) y = some x :=\n  sorry\n\n/- Auxiliary lemma -/\n\ntheorem ite_eq_of_not_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt] [is_strict_order α lt]\n    {a : α} {b : α} {β : Type v} (t : β) (s : β) (h : lt b a) : ite (lt a b) t s = s :=\n  sorry\n\ntheorem find_ins_of_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {t : rbnode α} (he : strict_weak_order.equiv x y)\n    {lo : Option α} {hi : Option α} (hs : is_searchable lt t lo hi) (hlt₁ : lift lt lo (some x))\n    (hlt₂ : lift lt (some x) hi) : find lt (ins lt t x) y = some x :=\n  sorry\n\ntheorem find_mk_insert_result {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (c : color)\n    (t : rbnode α) (x : α) : find lt (mk_insert_result c t) x = find lt t x :=\n  sorry\n\ntheorem find_insert_of_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {t : rbnode α} (he : strict_weak_order.equiv x y) :\n    is_searchable lt t none none → find lt (insert lt t x) y = some x :=\n  sorry\n\ntheorem weak_trichotomous {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (x : α) (y : α)\n    {p : Prop} (is_lt : lt x y → p) (is_eqv : ¬lt x y ∧ ¬lt y x → p) (is_gt : lt y x → p) : p :=\n  dite (lt x y)\n    (fun (h : lt x y) =>\n      dite (lt y x) (fun (h_1 : lt y x) => is_lt h) fun (h_1 : ¬lt y x) => is_lt h)\n    fun (h : ¬lt x y) =>\n      dite (lt y x) (fun (h : lt y x) => is_gt h)\n        fun (h_1 : ¬lt y x) => is_eqv { left := h, right := h_1 }\n\ntheorem find_black_eq_find_red {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {l : rbnode α}\n    {y : α} {r : rbnode α} {x : α} : find lt (black_node l y r) x = find lt (red_node l y r) x :=\n  sorry\n\ntheorem find_red_of_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {l : rbnode α} {y : α}\n    {r : rbnode α} {x : α} (h : lt x y) : find lt (red_node l y r) x = find lt l x :=\n  sorry\n\ntheorem find_red_of_gt {α : Type u} (lt : α → α → Prop) [DecidableRel lt] [is_strict_order α lt]\n    {l : rbnode α} {y : α} {r : rbnode α} {x : α} (h : lt y x) :\n    find lt (red_node l y r) x = find lt r x :=\n  sorry\n\ntheorem find_red_of_incomp {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {l : rbnode α} {y : α}\n    {r : rbnode α} {x : α} (h : ¬lt x y ∧ ¬lt y x) : find lt (red_node l y r) x = some y :=\n  sorry\n\ntheorem find_balance1_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {r : rbnode α} {t : rbnode α} {v : α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : lt x y) (hl : is_searchable lt l lo (some v))\n    (hr : is_searchable lt r (some v) (some y)) (ht : is_searchable lt t (some y) hi) :\n    find lt (balance1 l v r y t) x = find lt (red_node l v r) x :=\n  sorry\n\ntheorem find_balance1_node_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {s : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (hlt : lt y x) (ht : is_searchable lt t lo (some x))\n    (hs : is_searchable lt s (some x) hi)\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance1_node t x s) y = find lt t y :=\n  sorry\n\ntheorem find_balance1_gt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {r : rbnode α} {t : rbnode α} {v : α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : lt y x) (hl : is_searchable lt l lo (some v))\n    (hr : is_searchable lt r (some v) (some y)) (ht : is_searchable lt t (some y) hi) :\n    find lt (balance1 l v r y t) x = find lt t x :=\n  sorry\n\ntheorem find_balance1_node_gt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {s : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (h : lt x y) (ht : is_searchable lt t lo (some x))\n    (hs : is_searchable lt s (some x) hi)\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance1_node t x s) y = find lt s y :=\n  sorry\n\ntheorem find_balance1_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {r : rbnode α} {t : rbnode α} {v : α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : ¬lt x y ∧ ¬lt y x) (hl : is_searchable lt l lo (some v))\n    (hr : is_searchable lt r (some v) (some y)) (ht : is_searchable lt t (some y) hi) :\n    find lt (balance1 l v r y t) x = some y :=\n  sorry\n\ntheorem find_balance1_node_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {s : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (h : ¬lt x y ∧ ¬lt y x) (ht : is_searchable lt t lo (some y))\n    (hs : is_searchable lt s (some y) hi)\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance1_node t y s) x = some y :=\n  sorry\n\ntheorem find_balance2_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {v : α} {r : rbnode α} {t : rbnode α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : lt x y) (hl : is_searchable lt l (some y) (some v))\n    (hr : is_searchable lt r (some v) hi) (ht : is_searchable lt t lo (some y)) :\n    find lt (balance2 l v r y t) x = find lt t x :=\n  sorry\n\ntheorem find_balance2_node_lt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {s : rbnode α} {t : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (h : lt x y) (ht : is_searchable lt t (some y) hi)\n    (hs : is_searchable lt s lo (some y))\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance2_node t y s) x = find lt s x :=\n  sorry\n\ntheorem find_balance2_gt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {v : α} {r : rbnode α} {t : rbnode α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : lt y x) (hl : is_searchable lt l (some y) (some v))\n    (hr : is_searchable lt r (some v) hi) (ht : is_searchable lt t lo (some y)) :\n    find lt (balance2 l v r y t) x = find lt (red_node l v r) x :=\n  sorry\n\ntheorem find_balance2_node_gt {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {s : rbnode α} {t : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (h : lt y x) (ht : is_searchable lt t (some y) hi)\n    (hs : is_searchable lt s lo (some y))\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance2_node t y s) x = find lt t x :=\n  sorry\n\ntheorem find_balance2_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {l : rbnode α} {v : α} {r : rbnode α} {t : rbnode α} {x : α} {y : α}\n    {lo : Option α} {hi : Option α} (h : ¬lt x y ∧ ¬lt y x)\n    (hl : is_searchable lt l (some y) (some v)) (hr : is_searchable lt r (some v) hi)\n    (ht : is_searchable lt t lo (some y)) : find lt (balance2 l v r y t) x = some y :=\n  sorry\n\ntheorem find_balance2_node_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {t : rbnode α} {s : rbnode α} {x : α} {y : α} {lo : Option α}\n    {hi : Option α} (h : ¬lt x y ∧ ¬lt y x) (ht : is_searchable lt t (some y) hi)\n    (hs : is_searchable lt s lo (some y))\n    (hne :\n      autoParam (t ≠ leaf)\n        (Lean.Syntax.ident Lean.SourceInfo.none\n          (String.toSubstring \"Mathlib.rbnode.ins_ne_leaf_tac\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"rbnode\")\n            \"ins_ne_leaf_tac\")\n          [])) :\n    find lt (balance2_node t y s) x = some y :=\n  sorry\n\ntheorem find_ins_of_disj {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {t : rbnode α} (hn : lt x y ∨ lt y x)\n    {lo : Option α} {hi : Option α} (hs : is_searchable lt t lo hi) (hlt₁ : lift lt lo (some x))\n    (hlt₂ : lift lt (some x) hi) : find lt (ins lt t x) y = find lt t y :=\n  sorry\n\ntheorem find_insert_of_disj {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {t : rbnode α} (hd : lt x y ∨ lt y x) :\n    is_searchable lt t none none → find lt (insert lt t x) y = find lt t y :=\n  sorry\n\ntheorem find_insert_of_not_eqv {α : Type u} (lt : α → α → Prop) [DecidableRel lt]\n    [is_strict_weak_order α lt] {x : α} {y : α} {t : rbnode α} (hn : ¬strict_weak_order.equiv x y) :\n    is_searchable lt t none none → find lt (insert lt t x) y = find lt t y :=\n  sorry\n\ninductive is_bad_red_black {α : Type u} : rbnode α → ℕ → Prop where\n| bad_red :\n    ∀ {c₁ c₂ : color} {n : ℕ} {l r : rbnode α} {v : α},\n      is_red_black l c₁ n → is_red_black r c₂ n → is_bad_red_black (red_node l v r) n\n\ntheorem balance1_rb {α : Type u} {l : rbnode α} {r : rbnode α} {t : rbnode α} {y : α} {v : α}\n    {c_l : color} {c_r : color} {c_t : color} {n : ℕ} :\n    is_red_black l c_l n →\n        is_red_black r c_r n →\n          is_red_black t c_t n → ∃ (c : color), is_red_black (balance1 l y r v t) c (Nat.succ n) :=\n  sorry\n\ntheorem balance2_rb {α : Type u} {l : rbnode α} {r : rbnode α} {t : rbnode α} {y : α} {v : α}\n    {c_l : color} {c_r : color} {c_t : color} {n : ℕ} :\n    is_red_black l c_l n →\n        is_red_black r c_r n →\n          is_red_black t c_t n → ∃ (c : color), is_red_black (balance2 l y r v t) c (Nat.succ n) :=\n  sorry\n\ntheorem balance1_node_rb {α : Type u} {t : rbnode α} {s : rbnode α} {y : α} {c : color} {n : ℕ} :\n    is_bad_red_black t n →\n        is_red_black s c n → ∃ (c : color), is_red_black (balance1_node t y s) c (Nat.succ n) :=\n  sorry\n\ntheorem balance2_node_rb {α : Type u} {t : rbnode α} {s : rbnode α} {y : α} {c : color} {n : ℕ} :\n    is_bad_red_black t n →\n        is_red_black s c n → ∃ (c : color), is_red_black (balance2_node t y s) c (Nat.succ n) :=\n  sorry\n\ndef ins_rb_result {α : Type u} : rbnode α → color → ℕ → Prop := sorry\n\ntheorem of_get_color_eq_red {α : Type u} {t : rbnode α} {c : color} {n : ℕ} :\n    get_color t = color.red → is_red_black t c n → c = color.red :=\n  sorry\n\ntheorem of_get_color_ne_red {α : Type u} {t : rbnode α} {c : color} {n : ℕ} :\n    get_color t ≠ color.red → is_red_black t c n → c = color.black :=\n  sorry\n\ntheorem ins_rb {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {t : rbnode α} (x : α) {c : color}\n    {n : ℕ} (h : is_red_black t c n) : ins_rb_result (ins lt t x) c n :=\n  sorry\n\ndef insert_rb_result {α : Type u} : rbnode α → color → ℕ → Prop := sorry\n\ntheorem insert_rb {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {t : rbnode α} (x : α)\n    {c : color} {n : ℕ} (h : is_red_black t c n) : insert_rb_result (insert lt t x) c n :=\n  sorry\n\ntheorem insert_is_red_black {α : Type u} (lt : α → α → Prop) [DecidableRel lt] {t : rbnode α}\n    {c : color} {n : ℕ} (x : α) :\n    is_red_black t c n → ∃ (c : color), ∃ (n : ℕ), is_red_black (insert lt t x) c n :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/data/rbtree/insert_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631840431539, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.31806327582762534}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Abhimanyu Pallavi Sudhir\n-/\nimport algebra.geom_sum\nimport data.complex.basic\nimport data.nat.choose.sum\n\n/-!\n# Exponential, trigonometric and hyperbolic trigonometric functions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file contains the definitions of the real and complex exponential, sine, cosine, tangent,\nhyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.\n\n-/\n\nlocal notation `abs'` := has_abs.abs\nopen is_absolute_value\nopen_locale classical big_operators nat complex_conjugate\n\nsection\nopen real is_absolute_value finset\n\nsection\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] [archimedean α] {abv : β → α} [is_absolute_value abv]\n\nlemma is_cau_of_decreasing_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)\n  (hnm : ∀ n ≥ m, f n.succ ≤ f n) : is_cau_seq abs f :=\nλ ε ε0,\nlet ⟨k, hk⟩ := archimedean.arch a ε0 in\nhave h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=\n  ⟨k + k + 1, λ n hnm, lt_of_lt_of_le\n    (show a - (k + (k + 1)) • ε < -|f n|,\n      from lt_neg.1 $ lt_of_le_of_lt (ham n hnm) (begin\n        rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul],\n        exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk\n          (lt_add_of_pos_right _ ε0)),\n      end))\n    (neg_le.2 $ (abs_neg (f n)) ▸ le_abs_self _)⟩,\nlet l := nat.find h in\nhave hl : ∀ (n : ℕ), n ≥ m → f n > a - l • ε := nat.find_spec h,\nhave hl0 : l ≠ 0 := λ hl0, not_lt_of_ge (ham m le_rfl)\n  (lt_of_lt_of_le (by have := hl m (le_refl m); simpa [hl0] using this) (le_abs_self (f m))),\nbegin\n  cases not_forall.1\n    (nat.find_min h (nat.pred_lt hl0)) with i hi,\n  rw [not_imp, not_lt] at hi,\n  existsi i,\n  assume j hj,\n  have hfij : f j ≤ f i := (nat.rel_of_forall_rel_succ_of_le_of_le (≥) hnm hi.1 hj).le,\n  rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add'],\n  calc f i ≤ a - (nat.pred l) • ε : hi.2\n    ... = a - l • ε + ε :\n      by conv {to_rhs, rw [← nat.succ_pred_eq_of_pos (nat.pos_of_ne_zero hl0), succ_nsmul',\n        sub_add, add_sub_cancel] }\n    ... < f j + ε : add_lt_add_right (hl j (le_trans hi.1 hj)) _\nend\n\nlemma is_cau_of_mono_bounded (f : ℕ → α) {a : α} {m : ℕ} (ham : ∀ n ≥ m, |f n| ≤ a)\n  (hnm : ∀ n ≥ m, f n ≤ f n.succ) : is_cau_seq abs f :=\nbegin\n  refine @eq.rec_on (ℕ → α) _ (is_cau_seq abs) _ _\n    (-⟨_, @is_cau_of_decreasing_bounded _ _ _ (λ n, -f n) a m (by simpa) (by simpa)⟩ :\n      cau_seq α abs).2,\n  ext,\n  exact neg_neg _\nend\n\nend\n\nsection no_archimedean\nvariables {α : Type*} {β : Type*} [ring β]\n  [linear_ordered_field α] {abv : β → α} [is_absolute_value abv]\n\nlemma is_cau_series_of_abv_le_cau {f : ℕ → β} {g : ℕ → α} (n : ℕ) :\n  (∀ m, n ≤ m → abv (f m) ≤ g m) →\n  is_cau_seq abs (λ n, ∑ i in range n, g i) →\n  is_cau_seq abv (λ n, ∑ i in range n, f i) :=\nbegin\n  assume hm hg ε ε0,\n  cases hg (ε / 2) (div_pos ε0 (by norm_num)) with i hi,\n  existsi max n i,\n  assume j ji,\n  have hi₁ := hi j (le_trans (le_max_right n i) ji),\n  have hi₂ := hi (max n i) (le_max_right n i),\n  have sub_le := abs_sub_le (∑ k in range j, g k) (∑ k in range i, g k)\n    (∑ k in range (max n i), g k),\n  have := add_lt_add hi₁ hi₂,\n  rw [abs_sub_comm (∑ k in range (max n i), g k), add_halves ε] at this,\n  refine lt_of_le_of_lt (le_trans (le_trans _ (le_abs_self _)) sub_le) this,\n  generalize hk : j - max n i = k,\n  clear this hi₂ hi₁ hi ε0 ε hg sub_le,\n  rw tsub_eq_iff_eq_add_of_le ji at hk,\n  rw hk,\n  clear hk ji j,\n  induction k with k' hi,\n  { simp [abv_zero abv] },\n  { simp only [nat.succ_add, sum_range_succ_comm, sub_eq_add_neg, add_assoc],\n    refine le_trans (abv_add _ _ _) _,\n    simp only [sub_eq_add_neg] at hi,\n    exact add_le_add (hm _ (le_add_of_nonneg_of_le (nat.zero_le _) (le_max_left _ _))) hi },\nend\n\nlemma is_cau_series_of_abv_cau {f : ℕ → β} : is_cau_seq abs (λ m, ∑ n in range m, abv (f n))\n  → is_cau_seq abv (λ m, ∑ n in range m, f n) :=\nis_cau_series_of_abv_le_cau 0 (λ n h, le_rfl)\n\nend no_archimedean\n\nsection\nvariables {α : Type*} [linear_ordered_field α] [archimedean α]\n\nlemma is_cau_geo_series {β : Type*} [ring β] [nontrivial β] {abv : β → α} [is_absolute_value abv]\n   (x : β) (hx1 : abv x < 1) : is_cau_seq abv (λ n, ∑ m in range n, x ^ m) :=\nhave hx1' : abv x ≠ 1 := λ h, by simpa [h, lt_irrefl] using hx1,\nis_cau_series_of_abv_cau\nbegin\n  simp only [abv_pow abv, geom_sum_eq hx1'],\n  conv in (_ / _) { rw [← neg_div_neg_eq, neg_sub, neg_sub] },\n  refine @is_cau_of_mono_bounded _ _ _ _ ((1 : α) / (1 - abv x)) 0 _ _,\n  { assume n hn,\n    rw abs_of_nonneg,\n    refine div_le_div_of_le (le_of_lt $ sub_pos.2 hx1)\n      (sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)),\n    refine div_nonneg (sub_nonneg.2 _) (sub_nonneg.2 $ le_of_lt hx1),\n    clear hn,\n    induction n with n ih,\n    { simp },\n    { rw [pow_succ, ← one_mul (1 : α)],\n      refine mul_le_mul (le_of_lt hx1) ih (abv_pow abv x n ▸ abv_nonneg _ _) (by norm_num) } },\n  { assume n hn,\n    refine div_le_div_of_le (le_of_lt $ sub_pos.2 hx1) (sub_le_sub_left _ _),\n    rw [← one_mul (_ ^ n), pow_succ],\n    exact mul_le_mul_of_nonneg_right (le_of_lt hx1) (pow_nonneg (abv_nonneg _ _) _) }\nend\n\nlemma is_cau_geo_series_const (a : α) {x : α} (hx1 : |x| < 1) :\n  is_cau_seq abs (λ m, ∑ n in range m, a * x ^ n) :=\nhave is_cau_seq abs (λ m, a * ∑ n in range m, x ^ n) :=\n  (cau_seq.const abs a * ⟨_, is_cau_geo_series x hx1⟩).2,\nby simpa only [mul_sum]\n\nvariables {β : Type*} [ring β] {abv : β → α} [is_absolute_value abv]\n\nlemma series_ratio_test {f : ℕ → β} (n : ℕ) (r : α)\n  (hr0 : 0 ≤ r) (hr1 : r < 1) (h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :\n  is_cau_seq abv (λ m, ∑ n in range m, f n) :=\nhave har1 : |r| < 1, by rwa abs_of_nonneg hr0,\nbegin\n  refine is_cau_series_of_abv_le_cau n.succ _\n    (is_cau_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1),\n  assume m hmn,\n  cases classical.em (r = 0) with r_zero r_ne_zero,\n  { have m_pos := lt_of_lt_of_le (nat.succ_pos n) hmn,\n    have := h m.pred (nat.le_of_succ_le_succ (by rwa [nat.succ_pred_eq_of_pos m_pos])),\n    simpa [r_zero, nat.succ_pred_eq_of_pos m_pos, pow_succ] },\n  generalize hk : m - n.succ = k,\n  have r_pos : 0 < r := lt_of_le_of_ne hr0 (ne.symm r_ne_zero),\n  replace hk : m = k + n.succ := (tsub_eq_iff_eq_add_of_le hmn).1 hk,\n  induction k with k ih generalizing m n,\n  { rw [hk, zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel],\n    exact (ne_of_lt (pow_pos r_pos _)).symm },\n  { have kn : k + n.succ ≥ n.succ, by rw ← zero_add n.succ; exact add_le_add (zero_le _) (by simp),\n    rw [hk, nat.succ_add, pow_succ' r, ← mul_assoc],\n    exact le_trans (by rw mul_comm; exact h _ (nat.le_of_succ_le kn))\n      (mul_le_mul_of_nonneg_right (ih (k + n.succ) n h kn rfl) hr0) }\nend\n\nlemma sum_range_diag_flip {α : Type*} [add_comm_monoid α] (n : ℕ) (f : ℕ → ℕ → α) :\n  ∑ m in range n, ∑ k in range (m + 1), f k (m - k) =\n  ∑ m in range n, ∑ k in range (n - m), f m k :=\nby rw [sum_sigma', sum_sigma']; exact sum_bij\n(λ a _, ⟨a.2, a.1 - a.2⟩)\n(λ a ha, have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1,\n  have h₂ : a.2 < nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2,\n    mem_sigma.2 ⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),\n    mem_range.2 ((tsub_lt_tsub_iff_right (nat.le_of_lt_succ h₂)).2 h₁)⟩)\n(λ _ _, rfl)\n(λ ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ha hb h,\n  have ha : a₁ < n ∧ a₂ ≤ a₁ :=\n      ⟨mem_range.1 (mem_sigma.1 ha).1, nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 ha).2)⟩,\n  have hb : b₁ < n ∧ b₂ ≤ b₁ :=\n      ⟨mem_range.1 (mem_sigma.1 hb).1, nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 hb).2)⟩,\n  have h : a₂ = b₂ ∧ _ := sigma.mk.inj h,\n  have h' : a₁ = b₁ - b₂ + a₂ := (tsub_eq_iff_eq_add_of_le ha.2).1 (eq_of_heq h.2),\n  sigma.mk.inj_iff.2\n    ⟨tsub_add_cancel_of_le hb.2 ▸ h'.symm ▸ h.1 ▸ rfl,\n      (heq_of_eq h.1)⟩)\n(λ ⟨a₁, a₂⟩ ha,\n  have ha : a₁ < n ∧ a₂ < n - a₁ :=\n      ⟨mem_range.1 (mem_sigma.1 ha).1, (mem_range.1 (mem_sigma.1 ha).2)⟩,\n  ⟨⟨a₂ + a₁, a₁⟩, ⟨mem_sigma.2 ⟨mem_range.2 (lt_tsub_iff_right.1 ha.2),\n    mem_range.2 (nat.lt_succ_of_le (nat.le_add_left _ _))⟩,\n  sigma.mk.inj_iff.2 ⟨rfl, heq_of_eq (add_tsub_cancel_right _ _).symm⟩⟩⟩)\n\nend\n\nsection no_archimedean\nvariables {α : Type*} {β : Type*} [linear_ordered_field α] {abv : β → α}\n\nsection\nvariables [semiring β] [is_absolute_value abv]\n\nlemma abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : finset γ) :\n  abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=\nby haveI := classical.dec_eq γ; exact\nfinset.induction_on s (by simp [abv_zero abv])\n  (λ a s has ih, by rw [sum_insert has, sum_insert has];\n    exact le_trans (abv_add abv _ _) (add_le_add_left ih _))\n\nend\n\nsection\nvariables [ring β] [is_absolute_value abv]\n\nlemma cauchy_product {a b : ℕ → β}\n  (ha : is_cau_seq abs (λ m, ∑ n in range m, abv (a n)))\n  (hb : is_cau_seq abv (λ m, ∑ n in range m, b n)) (ε : α) (ε0 : 0 < ε) :\n  ∃ i : ℕ, ∀ j ≥ i, abv ((∑ k in range j, a k) * (∑ k in range j, b k) -\n  ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) < ε :=\nlet ⟨Q, hQ⟩ := cau_seq.bounded ⟨_, hb⟩ in\nlet ⟨P, hP⟩ := cau_seq.bounded ⟨_, ha⟩ in\nhave hP0 : 0 < P, from lt_of_le_of_lt (abs_nonneg _) (hP 0),\nhave hPε0 : 0 < ε / (2 * P),\n  from div_pos ε0 (mul_pos (show (2 : α) > 0, from by norm_num) hP0),\nlet ⟨N, hN⟩ := cau_seq.cauchy₂ ⟨_, hb⟩ hPε0 in\nhave hQε0 : 0 < ε / (4 * Q),\n  from div_pos ε0 (mul_pos (show (0 : α) < 4, by norm_num)\n    (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))),\nlet ⟨M, hM⟩ := cau_seq.cauchy₂ ⟨_, ha⟩ hQε0 in\n⟨2 * (max N M + 1), λ K hK,\nhave h₁ : ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) =\n    ∑ m in range K, ∑ n in range (K - m), a m * b n,\n  by simpa using sum_range_diag_flip K (λ m n, a m * b n),\nhave h₂ : (λ i, ∑ k in range (K - i), a i * b k) = (λ i, a i * ∑ k in range (K - i), b k),\n  by simp [finset.mul_sum],\nhave h₃ : ∑ i in range K, a i * ∑ k in range (K - i), b k =\n    ∑ i in range K, a i * (∑ k in range (K - i), b k - ∑ k in range K, b k)\n    + ∑ i in range K, a i * ∑ k in range K, b k,\n  by rw ← sum_add_distrib; simp [(mul_add _ _ _).symm],\nhave two_mul_two : (4 : α) = 2 * 2, by norm_num,\nhave hQ0 : Q ≠ 0, from λ h, by simpa [h, lt_irrefl] using hQε0,\nhave h2Q0 : 2 * Q ≠ 0, from mul_ne_zero two_ne_zero hQ0,\nhave hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε,\n  by rw [← div_div, div_mul_cancel _ (ne.symm (ne_of_lt hP0)),\n    two_mul_two, mul_assoc, ← div_div, div_mul_cancel _ h2Q0, add_halves],\nhave hNMK : max N M + 1 < K,\n  from lt_of_lt_of_le (by rw two_mul; exact lt_add_of_pos_left _ (nat.succ_pos _)) hK,\nhave hKN : N < K,\n  from calc N ≤ max N M : le_max_left _ _\n  ... < max N M + 1 : nat.lt_succ_self _\n  ... < K : hNMK,\nhave hsumlesum : ∑ i in range (max N M + 1), abv (a i) *\n      abv (∑ k in range (K - i), b k - ∑ k in range K, b k) ≤\n    ∑ i in range (max N M + 1), abv (a i) * (ε / (2 * P)),\n  from sum_le_sum (λ m hmJ, mul_le_mul_of_nonneg_left\n    (le_of_lt (hN (K - m)\n      (le_tsub_of_add_le_left (le_trans\n        (by rw two_mul; exact add_le_add (le_of_lt (mem_range.1 hmJ))\n          (le_trans (le_max_left _ _) (le_of_lt (lt_add_one _)))) hK)) K\n      (le_of_lt hKN))) (abv_nonneg abv _)),\nhave hsumltP : ∑ n in range (max N M + 1), abv (a n) < P :=\n  calc ∑ n in range (max N M + 1), abv (a n)\n      = |∑ n in range (max N M + 1), abv (a n)| :\n  eq.symm (abs_of_nonneg (sum_nonneg (λ x h, abv_nonneg abv (a x))))\n  ... < P : hP (max N M + 1),\nbegin\n  rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv],\n  refine lt_of_le_of_lt (abv_sum_le_sum_abv _ _) _,\n  suffices : ∑ i in range (max N M + 1),\n    abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) +\n    (∑ i in range K, abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) -\n    ∑ i in range (max N M + 1), abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k)) <\n    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q),\n  { rw hε at this, simpa [abv_mul abv] },\n  refine add_lt_add (lt_of_le_of_lt hsumlesum\n    (by rw [← sum_mul, mul_comm]; exact (mul_lt_mul_left hPε0).mpr hsumltP)) _,\n  rw sum_range_sub_sum_range (le_of_lt hNMK),\n  calc ∑ i in (range K).filter (λ k, max N M + 1 ≤ k),\n      abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k)\n      ≤ ∑ i in (range K).filter (λ k, max N M + 1 ≤ k), abv (a i) * (2 * Q) :\n    sum_le_sum (λ n hn, begin\n      refine mul_le_mul_of_nonneg_left _ (abv_nonneg _ _),\n      rw sub_eq_add_neg,\n      refine le_trans (abv_add _ _ _) _,\n      rw [two_mul, abv_neg abv],\n      exact add_le_add (le_of_lt (hQ _)) (le_of_lt (hQ _)),\n    end)\n    ... < ε / (4 * Q) * (2 * Q) :\n      by rw [← sum_mul, ← sum_range_sub_sum_range (le_of_lt hNMK)];\n      refine (mul_lt_mul_right $ by rw two_mul;\n        exact add_pos (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))\n          (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0))).2\n        (lt_of_le_of_lt (le_abs_self _)\n          (hM _ (le_trans (nat.le_succ_of_le (le_max_right _ _)) (le_of_lt hNMK)) _\n            (nat.le_succ_of_le (le_max_right _ _))))\nend⟩\n\nend\n\nend no_archimedean\n\nend\n\nopen finset\n\nopen cau_seq\n\nnamespace complex\n\nlemma is_cau_abs_exp (z : ℂ) : is_cau_seq has_abs.abs\n  (λ n, ∑ m in range n, abs (z ^ m / m!)) :=\nlet ⟨n, hn⟩ := exists_nat_gt (abs z) in\nhave hn0 : (0 : ℝ) < n, from lt_of_le_of_lt (abs.nonneg _) hn,\nseries_ratio_test n (complex.abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))\n  (by rwa [div_lt_iff hn0, one_mul])\n  (λ m hm,\n    by rw [abs_abs, abs_abs, nat.factorial_succ, pow_succ,\n      mul_comm m.succ, nat.cast_mul, ← div_div, mul_div_assoc,\n      mul_div_right_comm, abs.map_mul, map_div₀, abs_cast_nat];\n    exact mul_le_mul_of_nonneg_right\n      (div_le_div_of_le_left (abs.nonneg _) hn0\n        (nat.cast_le.2 (le_trans hm (nat.le_succ _)))) (abs.nonneg _))\n\nnoncomputable theory\n\nlemma is_cau_exp (z : ℂ) :\n  is_cau_seq abs (λ n, ∑ m in range n, z ^ m / m!) :=\nis_cau_series_of_abv_cau (is_cau_abs_exp z)\n\n/-- The Cauchy sequence consisting of partial sums of the Taylor series of\nthe complex exponential function -/\n@[pp_nodot] def exp' (z : ℂ) :\n  cau_seq ℂ complex.abs :=\n⟨λ n, ∑ m in range n, z ^ m / m!, is_cau_exp z⟩\n\n/-- The complex exponential function, defined via its Taylor series -/\n@[irreducible, pp_nodot] def exp (z : ℂ) : ℂ := lim (exp' z)\n\n/-- The complex sine function, defined via `exp` -/\n@[pp_nodot] def sin (z : ℂ) : ℂ := ((exp (-z * I) - exp (z * I)) * I) / 2\n\n/-- The complex cosine function, defined via `exp` -/\n@[pp_nodot] def cos (z : ℂ) : ℂ := (exp (z * I) + exp (-z * I)) / 2\n\n/-- The complex tangent function, defined as `sin z / cos z` -/\n@[pp_nodot] def tan (z : ℂ) : ℂ := sin z / cos z\n\n/-- The complex hyperbolic sine function, defined via `exp` -/\n@[pp_nodot] def sinh (z : ℂ) : ℂ := (exp z - exp (-z)) / 2\n\n/-- The complex hyperbolic cosine function, defined via `exp` -/\n@[pp_nodot] def cosh (z : ℂ) : ℂ := (exp z + exp (-z)) / 2\n\n/-- The complex hyperbolic tangent function, defined as `sinh z / cosh z` -/\n@[pp_nodot] def tanh (z : ℂ) : ℂ := sinh z / cosh z\n\nend complex\n\nnamespace real\n\nopen complex\n\n/-- The real exponential function, defined as the real part of the complex exponential -/\n@[pp_nodot] def exp (x : ℝ) : ℝ := (exp x).re\n\n/-- The real sine function, defined as the real part of the complex sine -/\n@[pp_nodot] def sin (x : ℝ) : ℝ := (sin x).re\n\n/-- The real cosine function, defined as the real part of the complex cosine -/\n@[pp_nodot] def cos (x : ℝ) : ℝ := (cos x).re\n\n/-- The real tangent function, defined as the real part of the complex tangent -/\n@[pp_nodot] def tan (x : ℝ) : ℝ := (tan x).re\n\n/-- The real hypebolic sine function, defined as the real part of the complex hyperbolic sine -/\n@[pp_nodot] def sinh (x : ℝ) : ℝ := (sinh x).re\n\n/-- The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine -/\n@[pp_nodot] def cosh (x : ℝ) : ℝ := (cosh x).re\n\n/-- The real hypebolic tangent function, defined as the real part of\nthe complex hyperbolic tangent -/\n@[pp_nodot] def tanh (x : ℝ) : ℝ := (tanh x).re\n\nend real\n\nnamespace complex\n\nvariables (x y : ℂ)\n\n@[simp] lemma exp_zero : exp 0 = 1 :=\nbegin\n  rw exp,\n  refine lim_eq_of_equiv_const (λ ε ε0, ⟨1, λ j hj, _⟩),\n  convert ε0,\n  cases j,\n  { exact absurd hj (not_le_of_gt zero_lt_one) },\n  { dsimp [exp'],\n    induction j with j ih,\n    { dsimp [exp']; simp },\n    { rw ← ih dec_trivial,\n      simp only [sum_range_succ, pow_succ],\n      simp } }\nend\n\nlemma exp_add : exp (x + y) = exp x * exp y :=\nbegin\n  have hj : ∀ j : ℕ, ∑ m in range j, (x + y) ^ m / m! =\n      ∑ i in range j, ∑ k in range (i + 1), x ^ k / k! * (y ^ (i - k) / (i - k)!),\n  { assume j,\n    refine finset.sum_congr rfl (λ m hm, _),\n    rw [add_pow, div_eq_mul_inv, sum_mul],\n    refine finset.sum_congr rfl (λ i hi, _),\n    have h₁ : (m.choose i : ℂ) ≠ 0 := nat.cast_ne_zero.2\n      (pos_iff_ne_zero.1 (nat.choose_pos (nat.le_of_lt_succ (mem_range.1 hi)))),\n    have h₂ := nat.choose_mul_factorial_mul_factorial (nat.le_of_lt_succ $ finset.mem_range.1 hi),\n    rw [← h₂, nat.cast_mul, nat.cast_mul, mul_inv, mul_inv],\n    simp only [mul_left_comm (m.choose i : ℂ), mul_assoc, mul_left_comm (m.choose i : ℂ)⁻¹,\n      mul_comm (m.choose i : ℂ)],\n    rw inv_mul_cancel h₁,\n    simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm] },\n  simp_rw [exp, exp', lim_mul_lim],\n  apply (lim_eq_lim_of_equiv _).symm,\n  simp only [hj],\n  exact cauchy_product (is_cau_abs_exp x) (is_cau_exp y)\nend\n\nlemma exp_list_sum (l : list ℂ) : exp l.sum = (l.map exp).prod :=\n@monoid_hom.map_list_prod (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ l\n\nlemma exp_multiset_sum (s : multiset ℂ) : exp s.sum = (s.map exp).prod :=\n@monoid_hom.map_multiset_prod (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ s\n\nlemma exp_sum {α : Type*} (s : finset α) (f : α → ℂ) : exp (∑ x in s, f x) = ∏ x in s, exp (f x) :=\n@monoid_hom.map_prod (multiplicative ℂ) α ℂ _ _ ⟨exp, exp_zero, exp_add⟩ f s\n\nlemma exp_nat_mul (x : ℂ) : ∀ n : ℕ, exp(n*x) = (exp x)^n\n| 0 := by rw [nat.cast_zero, zero_mul, exp_zero, pow_zero]\n| (nat.succ n) := by rw [pow_succ', nat.cast_add_one, add_mul, exp_add, ←exp_nat_mul, one_mul]\n\nlemma exp_ne_zero : exp x ≠ 0 :=\nλ h, zero_ne_one $ by rw [← exp_zero, ← add_neg_self x, exp_add, h]; simp\n\nlemma exp_neg : exp (-x) = (exp x)⁻¹ :=\nby rw [← mul_right_inj' (exp_ne_zero x), ← exp_add];\n  simp [mul_inv_cancel (exp_ne_zero x)]\n\nlemma exp_sub : exp (x - y) = exp x / exp y :=\nby simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]\n\nlemma exp_int_mul (z : ℂ) (n : ℤ) : complex.exp (n * z) = (complex.exp z) ^ n :=\nbegin\n  cases n,\n  { apply complex.exp_nat_mul },\n  { simpa [complex.exp_neg, add_comm, ← neg_mul]\n      using complex.exp_nat_mul (-z) (1 + n) },\nend\n\n@[simp] lemma exp_conj : exp (conj x) = conj (exp x) :=\nbegin\n  dsimp [exp],\n  rw [← lim_conj],\n  refine congr_arg lim (cau_seq.ext (λ _, _)),\n  dsimp [exp', function.comp, cau_seq_conj],\n  rw (star_ring_end _).map_sum,\n  refine sum_congr rfl (λ n hn, _),\n  rw [map_div₀, map_pow, ← of_real_nat_cast, conj_of_real]\nend\n\n@[simp] lemma of_real_exp_of_real_re (x : ℝ) : ((exp x).re : ℂ) = exp x :=\neq_conj_iff_re.1 $ by rw [← exp_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_exp (x : ℝ) : (real.exp x : ℂ) = exp x :=\nof_real_exp_of_real_re _\n\n@[simp] lemma exp_of_real_im (x : ℝ) : (exp x).im = 0 :=\nby rw [← of_real_exp_of_real_re, of_real_im]\n\nlemma exp_of_real_re (x : ℝ) : (exp x).re = real.exp x := rfl\n\nlemma two_sinh : 2 * sinh x = exp x - exp (-x) :=\nmul_div_cancel' _ two_ne_zero\n\nlemma two_cosh : 2 * cosh x = exp x + exp (-x) :=\nmul_div_cancel' _ two_ne_zero\n\n@[simp] lemma sinh_zero : sinh 0 = 0 := by simp [sinh]\n\n@[simp] lemma sinh_neg : sinh (-x) = -sinh x :=\nby simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]\n\nprivate lemma sinh_add_aux {a b c d : ℂ} :\n  (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) := by ring\n\nlemma sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y :=\nbegin\n  rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh,\n      exp_add, neg_add, exp_add, eq_comm,\n      mul_add, ← mul_assoc, two_sinh, mul_left_comm, two_sinh,\n      ← mul_right_inj' (two_ne_zero' ℂ), mul_add,\n      mul_left_comm, two_cosh, ← mul_assoc, two_cosh],\n  exact sinh_add_aux\nend\n\n@[simp] lemma cosh_zero : cosh 0 = 1 := by simp [cosh]\n\n@[simp] lemma cosh_neg : cosh (-x) = cosh x :=\nby simp [add_comm, cosh, exp_neg]\n\nprivate lemma cosh_add_aux {a b c d : ℂ} :\n  (a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) := by ring\n\nlemma cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y :=\nbegin\n  rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh,\n      exp_add, neg_add, exp_add, eq_comm,\n      mul_add, ← mul_assoc, two_cosh, ← mul_assoc, two_sinh,\n      ← mul_right_inj' (two_ne_zero' ℂ), mul_add,\n      mul_left_comm, two_cosh, mul_left_comm, two_sinh],\n  exact cosh_add_aux\nend\n\nlemma sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y :=\nby simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]\n\nlemma cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y :=\nby simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]\n\nlemma sinh_conj : sinh (conj x) = conj (sinh x) :=\nby rw [sinh, ← ring_hom.map_neg, exp_conj, exp_conj, ← ring_hom.map_sub, sinh,\n  map_div₀, conj_bit0, ring_hom.map_one]\n\n@[simp] lemma of_real_sinh_of_real_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x :=\neq_conj_iff_re.1 $ by rw [← sinh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_sinh (x : ℝ) : (real.sinh x : ℂ) = sinh x :=\nof_real_sinh_of_real_re _\n\n@[simp] lemma sinh_of_real_im (x : ℝ) : (sinh x).im = 0 :=\nby rw [← of_real_sinh_of_real_re, of_real_im]\n\nlemma sinh_of_real_re (x : ℝ) : (sinh x).re = real.sinh x := rfl\n\nlemma cosh_conj : cosh (conj x) = conj (cosh x) :=\nbegin\n  rw [cosh, ← ring_hom.map_neg, exp_conj, exp_conj, ← ring_hom.map_add, cosh,\n      map_div₀, conj_bit0, ring_hom.map_one]\nend\n\nlemma of_real_cosh_of_real_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x :=\neq_conj_iff_re.1 $ by rw [← cosh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_cosh (x : ℝ) : (real.cosh x : ℂ) = cosh x :=\nof_real_cosh_of_real_re _\n\n@[simp] lemma cosh_of_real_im (x : ℝ) : (cosh x).im = 0 :=\nby rw [← of_real_cosh_of_real_re, of_real_im]\n\n@[simp] lemma cosh_of_real_re (x : ℝ) : (cosh x).re = real.cosh x := rfl\n\nlemma tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x := rfl\n\n@[simp] lemma tanh_zero : tanh 0 = 0 := by simp [tanh]\n\n@[simp] lemma tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]\n\nlemma tanh_conj : tanh (conj x) = conj (tanh x) :=\nby rw [tanh, sinh_conj, cosh_conj, ← map_div₀, tanh]\n\n@[simp] lemma of_real_tanh_of_real_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x :=\neq_conj_iff_re.1 $ by rw [← tanh_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_tanh (x : ℝ) : (real.tanh x : ℂ) = tanh x :=\nof_real_tanh_of_real_re _\n\n@[simp] lemma tanh_of_real_im (x : ℝ) : (tanh x).im = 0 :=\nby rw [← of_real_tanh_of_real_re, of_real_im]\n\nlemma tanh_of_real_re (x : ℝ) : (tanh x).re = real.tanh x := rfl\n\n@[simp] lemma cosh_add_sinh : cosh x + sinh x = exp x :=\nby rw [← mul_right_inj' (two_ne_zero' ℂ), mul_add,\n       two_cosh, two_sinh, add_add_sub_cancel, two_mul]\n\n@[simp] lemma sinh_add_cosh : sinh x + cosh x = exp x :=\nby rw [add_comm, cosh_add_sinh]\n\n@[simp] lemma exp_sub_cosh : exp x - cosh x = sinh x :=\nsub_eq_iff_eq_add.2 (sinh_add_cosh x).symm\n\n@[simp] lemma exp_sub_sinh : exp x - sinh x = cosh x :=\nsub_eq_iff_eq_add.2 (cosh_add_sinh x).symm\n\n@[simp] lemma cosh_sub_sinh : cosh x - sinh x = exp (-x) :=\nby rw [← mul_right_inj' (two_ne_zero' ℂ), mul_sub,\n       two_cosh, two_sinh, add_sub_sub_cancel, two_mul]\n\n@[simp] lemma sinh_sub_cosh : sinh x - cosh x = -exp (-x) :=\nby rw [← neg_sub, cosh_sub_sinh]\n\n@[simp] lemma cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 :=\nby rw [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_self, exp_zero]\n\nlemma cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 :=\nbegin\n  rw ← cosh_sq_sub_sinh_sq x,\n  ring\nend\n\nlemma sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 :=\nbegin\n  rw ← cosh_sq_sub_sinh_sq x,\n  ring\nend\n\nlemma cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 :=\nby rw [two_mul, cosh_add, sq, sq]\n\nlemma sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x :=\nbegin\n  rw [two_mul, sinh_add],\n  ring\nend\n\nlemma cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, cosh_add x (2 * x)],\n  simp only [cosh_two_mul, sinh_two_mul],\n  have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2, by ring,\n  rw [h2, sinh_sq],\n  ring\nend\n\nlemma sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, sinh_add x (2 * x)],\n  simp only [cosh_two_mul, sinh_two_mul],\n  have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2, by ring,\n  rw [h2, cosh_sq],\n  ring,\nend\n\n@[simp] lemma sin_zero : sin 0 = 0 := by simp [sin]\n\n@[simp] lemma sin_neg : sin (-x) = -sin x :=\nby simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma two_sin : 2 * sin x = (exp (-x * I) - exp (x * I)) * I :=\nmul_div_cancel' _ two_ne_zero\n\nlemma two_cos : 2 * cos x = exp (x * I) + exp (-x * I) :=\nmul_div_cancel' _ two_ne_zero\n\nlemma sinh_mul_I : sinh (x * I) = sin x * I :=\nby rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh,\n       ← mul_assoc, two_sin, mul_assoc, I_mul_I, mul_neg_one,\n       neg_sub, neg_mul_eq_neg_mul]\n\nlemma cosh_mul_I : cosh (x * I) = cos x :=\nby rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh,\n       two_cos, neg_mul_eq_neg_mul]\n\nlemma tanh_mul_I : tanh (x * I) = tan x * I :=\nby rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]\n\nlemma cos_mul_I : cos (x * I) = cosh x :=\nby rw ← cosh_mul_I; ring_nf; simp\n\nlemma sin_mul_I : sin (x * I) = sinh x * I :=\nhave h : I * sin (x * I) = -sinh x := by { rw [mul_comm, ← sinh_mul_I], ring_nf, simp },\nby simpa only [neg_mul, div_I, neg_neg]\n  using cancel_factors.cancel_factors_eq_div h I_ne_zero\n\nlemma tan_mul_I : tan (x * I) = tanh x * I :=\nby rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh]\n\nlemma sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=\nby rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I,\n       add_mul, add_mul, mul_right_comm, ← sinh_mul_I,\n       mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add]\n\n@[simp] lemma cos_zero : cos 0 = 1 := by simp [cos]\n\n@[simp] lemma cos_neg : cos (-x) = cos x :=\nby simp [cos, sub_eq_add_neg, exp_neg, add_comm]\n\nprivate lemma cos_add_aux {a b c d : ℂ} :\n  (a + b) * (c + d) - (b - a) * (d - c) * (-1) =\n  2 * (a * c + b * d) := by ring\n\nlemma cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=\nby rw [← cosh_mul_I, add_mul, cosh_add, cosh_mul_I, cosh_mul_I,\n       sinh_mul_I, sinh_mul_I, mul_mul_mul_comm, I_mul_I,\n       mul_neg_one, sub_eq_add_neg]\n\nlemma sin_sub : sin (x - y) = sin x * cos y - cos x * sin y :=\nby simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]\n\nlemma cos_sub : cos (x - y) = cos x * cos y + sin x * sin y :=\nby simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]\n\nlemma sin_add_mul_I (x y : ℂ) : sin (x + y*I) = sin x * cosh y + cos x * sinh y * I :=\nby rw [sin_add, cos_mul_I, sin_mul_I, mul_assoc]\n\nlemma sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I :=\nby convert sin_add_mul_I z.re z.im; exact (re_add_im z).symm\n\nlemma cos_add_mul_I (x y : ℂ) : cos (x + y*I) = cos x * cosh y - sin x * sinh y * I :=\nby rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc]\n\nlemma cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I :=\nby convert cos_add_mul_I z.re z.im; exact (re_add_im z).symm\n\ntheorem sin_sub_sin : sin x - sin y = 2 * sin((x - y)/2) * cos((x + y)/2) :=\nbegin\n  have s1 := sin_add ((x + y) / 2) ((x - y) / 2),\n  have s2 := sin_sub ((x + y) / 2) ((x - y) / 2),\n  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1,\n  rw [div_sub_div_same, ←sub_add, add_sub_cancel', half_add_self] at s2,\n  rw [s1, s2],\n  ring\nend\n\ntheorem cos_sub_cos : cos x - cos y = -2 * sin((x + y)/2) * sin((x - y)/2) :=\nbegin\n  have s1 := cos_add ((x + y) / 2) ((x - y) / 2),\n  have s2 := cos_sub ((x + y) / 2) ((x - y) / 2),\n  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1,\n  rw [div_sub_div_same, ←sub_add, add_sub_cancel', half_add_self] at s2,\n  rw [s1, s2],\n  ring,\nend\n\n\n\nlemma sin_conj : sin (conj x) = conj (sin x) :=\nby rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I,\n       ← conj_neg_I, ← ring_hom.map_mul, ← ring_hom.map_mul, sinh_conj,\n       mul_neg, sinh_neg, sinh_mul_I, mul_neg]\n\n@[simp] lemma of_real_sin_of_real_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=\neq_conj_iff_re.1 $ by rw [← sin_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_sin (x : ℝ) : (real.sin x : ℂ) = sin x :=\nof_real_sin_of_real_re _\n\n@[simp] lemma sin_of_real_im (x : ℝ) : (sin x).im = 0 :=\nby rw [← of_real_sin_of_real_re, of_real_im]\n\nlemma sin_of_real_re (x : ℝ) : (sin x).re = real.sin x := rfl\n\nlemma cos_conj : cos (conj x) = conj (cos x) :=\nby rw [← cosh_mul_I, ← conj_neg_I, ← ring_hom.map_mul, ← cosh_mul_I,\n       cosh_conj, mul_neg, cosh_neg]\n\n@[simp] lemma of_real_cos_of_real_re (x : ℝ) : ((cos x).re : ℂ) = cos x :=\neq_conj_iff_re.1 $ by rw [← cos_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_cos (x : ℝ) : (real.cos x : ℂ) = cos x :=\nof_real_cos_of_real_re _\n\n@[simp] lemma cos_of_real_im (x : ℝ) : (cos x).im = 0 :=\nby rw [← of_real_cos_of_real_re, of_real_im]\n\nlemma cos_of_real_re (x : ℝ) : (cos x).re = real.cos x := rfl\n\n@[simp] lemma tan_zero : tan 0 = 0 := by simp [tan]\n\nlemma tan_eq_sin_div_cos : tan x = sin x / cos x := rfl\n\nlemma tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x :=\nby rw [tan_eq_sin_div_cos, div_mul_cancel _ hx]\n\n@[simp] lemma tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]\n\nlemma tan_conj : tan (conj x) = conj (tan x) :=\nby rw [tan, sin_conj, cos_conj, ← map_div₀, tan]\n\n@[simp] lemma of_real_tan_of_real_re (x : ℝ) : ((tan x).re : ℂ) = tan x :=\neq_conj_iff_re.1 $ by rw [← tan_conj, conj_of_real]\n\n@[simp, norm_cast] lemma of_real_tan (x : ℝ) : (real.tan x : ℂ) = tan x :=\nof_real_tan_of_real_re _\n\n@[simp] lemma tan_of_real_im (x : ℝ) : (tan x).im = 0 :=\nby rw [← of_real_tan_of_real_re, of_real_im]\n\nlemma tan_of_real_re (x : ℝ) : (tan x).re = real.tan x := rfl\n\nlemma cos_add_sin_I : cos x + sin x * I = exp (x * I) :=\nby rw [← cosh_add_sinh, sinh_mul_I, cosh_mul_I]\n\nlemma cos_sub_sin_I : cos x - sin x * I = exp (-x * I) :=\nby rw [neg_mul, ← cosh_sub_sinh, sinh_mul_I, cosh_mul_I]\n\n@[simp] lemma sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=\neq.trans\n  (by rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm])\n  (cosh_sq_sub_sinh_sq (x * I))\n\n@[simp] lemma cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 :=\nby rw [add_comm, sin_sq_add_cos_sq]\n\nlemma cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=\nby rw [two_mul, cos_add, ← sq, ← sq]\n\nlemma cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=\nby rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x),\n       ← sub_add, sub_add_eq_add_sub, two_mul]\n\nlemma sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=\nby rw [two_mul, sin_add, two_mul, add_mul, mul_comm]\n\nlemma cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=\nby simp [cos_two_mul, div_add_div_same, mul_div_cancel_left, two_ne_zero, -one_div]\n\nlemma cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel']\n\nlemma sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel]\n\nlemma inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 :=\nhave cos x ^ 2 ≠ 0, from pow_ne_zero 2 hx,\nby { rw [tan_eq_sin_div_cos, div_pow], field_simp [this] }\n\nlemma tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :\n  tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 :=\nby simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]\n\nlemma cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, cos_add x (2 * x)],\n  simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq],\n  have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2, by ring,\n  rw [h2, cos_sq'],\n  ring\nend\n\nlemma sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 :=\nbegin\n  have h1 : x + 2 * x = 3 * x, by ring,\n  rw [← h1, sin_add x (2 * x)],\n  simp only [cos_two_mul, sin_two_mul, cos_sq'],\n  have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2, by ring,\n  rw [h2, cos_sq'],\n  ring\nend\n\nlemma exp_mul_I : exp (x * I) = cos x + sin x * I :=\n(cos_add_sin_I _).symm\n\nlemma exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) :=\nby rw [exp_add, exp_mul_I]\n\nlemma exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) :=\nby rw [← exp_add_mul_I, re_add_im]\n\nlemma exp_re : (exp x).re = real.exp x.re * real.cos x.im :=\nby { rw [exp_eq_exp_re_mul_sin_add_cos], simp [exp_of_real_re, cos_of_real_re] }\n\nlemma exp_im : (exp x).im = real.exp x.re * real.sin x.im :=\nby { rw [exp_eq_exp_re_mul_sin_add_cos], simp [exp_of_real_re, sin_of_real_re] }\n\n@[simp] lemma exp_of_real_mul_I_re (x : ℝ) : (exp (x * I)).re = real.cos x :=\nby simp [exp_mul_I, cos_of_real_re]\n\n@[simp] lemma exp_of_real_mul_I_im (x : ℝ) : (exp (x * I)).im = real.sin x :=\nby simp [exp_mul_I, sin_of_real_re]\n\n/-- **De Moivre's formula** -/\ntheorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :\n  (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I :=\nbegin\n  rw [← exp_mul_I, ← exp_mul_I],\n  induction n with n ih,\n  { rw [pow_zero, nat.cast_zero, zero_mul, zero_mul, exp_zero] },\n  { rw [pow_succ', ih, nat.cast_succ, add_mul, add_mul, one_mul, exp_add] }\nend\n\nend complex\n\nnamespace real\n\nopen complex\n\nvariables (x y : ℝ)\n\n@[simp] lemma exp_zero : exp 0 = 1 :=\nby simp [real.exp]\n\nlemma exp_add : exp (x + y) = exp x * exp y :=\nby simp [exp_add, exp]\n\nlemma exp_list_sum (l : list ℝ) : exp l.sum = (l.map exp).prod :=\n@monoid_hom.map_list_prod (multiplicative ℝ) ℝ _ _ ⟨exp, exp_zero, exp_add⟩ l\n\nlemma exp_multiset_sum (s : multiset ℝ) : exp s.sum = (s.map exp).prod :=\n@monoid_hom.map_multiset_prod (multiplicative ℝ) ℝ _ _ ⟨exp, exp_zero, exp_add⟩ s\n\nlemma exp_sum {α : Type*} (s : finset α) (f : α → ℝ) : exp (∑ x in s, f x) = ∏ x in s, exp (f x) :=\n@monoid_hom.map_prod (multiplicative ℝ) α ℝ _ _ ⟨exp, exp_zero, exp_add⟩ f s\n\nlemma exp_nat_mul (x : ℝ) : ∀ n : ℕ, exp(n*x) = (exp x)^n\n| 0 := by rw [nat.cast_zero, zero_mul, exp_zero, pow_zero]\n| (nat.succ n) := by rw [pow_succ', nat.cast_add_one, add_mul, exp_add, ←exp_nat_mul, one_mul]\n\nlemma exp_ne_zero : exp x ≠ 0 :=\nλ h, exp_ne_zero x $ by rw [exp, ← of_real_inj] at h; simp * at *\n\nlemma exp_neg : exp (-x) = (exp x)⁻¹ :=\nby rw [← of_real_inj, exp, of_real_exp_of_real_re, of_real_neg, exp_neg,\n  of_real_inv, of_real_exp]\n\nlemma exp_sub : exp (x - y) = exp x / exp y :=\nby simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]\n\n@[simp] lemma sin_zero : sin 0 = 0 := by simp [sin]\n\n@[simp] lemma sin_neg : sin (-x) = -sin x :=\nby simp [sin, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=\nby rw [← of_real_inj]; simp [sin, sin_add]\n\n@[simp] lemma cos_zero : cos 0 = 1 := by simp [cos]\n\n@[simp] lemma cos_neg : cos (-x) = cos x :=\nby simp [cos, exp_neg]\n\n@[simp] lemma cos_abs : cos (|x|) = cos x :=\nby cases le_total x 0; simp only [*, _root_.abs_of_nonneg, abs_of_nonpos, cos_neg]\n\nlemma cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=\nby rw ← of_real_inj; simp [cos, cos_add]\n\nlemma sin_sub : sin (x - y) = sin x * cos y - cos x * sin y :=\nby simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]\n\nlemma cos_sub : cos (x - y) = cos x * cos y + sin x * sin y :=\nby simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]\n\nlemma sin_sub_sin : sin x - sin y = 2 * sin((x - y)/2) * cos((x + y)/2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [sin, cos, of_real_sin_of_real_re, of_real_sub, of_real_add, of_real_div, of_real_mul,\n    of_real_one, of_real_bit0],\n  convert sin_sub_sin _ _;\n  norm_cast\nend\n\ntheorem cos_sub_cos : cos x - cos y = -2 * sin((x + y)/2) * sin((x - y)/2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [cos, neg_mul, of_real_sin, of_real_sub, of_real_add,\n    of_real_cos_of_real_re, of_real_div, of_real_mul, of_real_one, of_real_neg, of_real_bit0],\n  convert cos_sub_cos _ _,\n  ring,\nend\n\nlemma cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2)  * cos ((x - y) / 2) :=\nbegin\n  rw ← of_real_inj,\n  simp only [cos, of_real_sub, of_real_add, of_real_cos_of_real_re, of_real_div, of_real_mul,\n    of_real_one, of_real_bit0],\n  convert cos_add_cos _ _;\n  norm_cast,\nend\n\nlemma tan_eq_sin_div_cos : tan x = sin x / cos x :=\nby rw [← of_real_inj, of_real_tan, tan_eq_sin_div_cos, of_real_div, of_real_sin, of_real_cos]\n\nlemma tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x :=\nby rw [tan_eq_sin_div_cos, div_mul_cancel _ hx]\n\n@[simp] lemma tan_zero : tan 0 = 0 := by simp [tan]\n\n@[simp] lemma tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]\n\n@[simp] lemma sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=\nof_real_inj.1 $ by simp\n\n@[simp] lemma cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 :=\nby rw [add_comm, sin_sq_add_cos_sq]\n\nlemma sin_sq_le_one : sin x ^ 2 ≤ 1 :=\nby rw ← sin_sq_add_cos_sq x; exact le_add_of_nonneg_right (sq_nonneg _)\n\nlemma cos_sq_le_one : cos x ^ 2 ≤ 1 :=\nby rw ← sin_sq_add_cos_sq x; exact le_add_of_nonneg_left (sq_nonneg _)\n\nlemma abs_sin_le_one : |sin x| ≤ 1 :=\nabs_le_one_iff_mul_self_le_one.2 $ by simp only [← sq, sin_sq_le_one]\n\nlemma abs_cos_le_one : |cos x| ≤ 1 :=\nabs_le_one_iff_mul_self_le_one.2 $ by simp only [← sq, cos_sq_le_one]\n\nlemma sin_le_one : sin x ≤ 1 :=\n(abs_le.1 (abs_sin_le_one _)).2\n\nlemma cos_le_one : cos x ≤ 1 :=\n(abs_le.1 (abs_cos_le_one _)).2\n\nlemma neg_one_le_sin : -1 ≤ sin x :=\n(abs_le.1 (abs_sin_le_one _)).1\n\nlemma neg_one_le_cos : -1 ≤ cos x :=\n(abs_le.1 (abs_cos_le_one _)).1\n\nlemma cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=\nby rw ← of_real_inj; simp [cos_two_mul]\n\nlemma cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=\nby rw ← of_real_inj; simp [cos_two_mul']\n\nlemma sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=\nby rw ← of_real_inj; simp [sin_two_mul]\n\nlemma cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=\nof_real_inj.1 $ by simpa using cos_sq x\n\nlemma cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 :=\nby rw [←sin_sq_add_cos_sq x, add_sub_cancel']\n\nlemma sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=\neq_sub_iff_add_eq.2 $ sin_sq_add_cos_sq _\n\nlemma abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) :\n  |sin x| = sqrt (1 - cos x ^ 2) :=\nby rw [← sin_sq, sqrt_sq_eq_abs]\n\nlemma abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) :\n  |cos x| = sqrt (1 - sin x ^ 2) :=\nby rw [← cos_sq', sqrt_sq_eq_abs]\n\nlemma inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2  :=\nhave complex.cos x ≠ 0, from mt (congr_arg re) hx,\nof_real_inj.1 $ by simpa using complex.inv_one_add_tan_sq this\n\nlemma tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :\n  tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 :=\nby simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]\n\nlemma inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :\n  (sqrt (1 + tan x ^ 2))⁻¹ = cos x :=\nby rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne']\n\nlemma tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :\n  tan x / sqrt (1 + tan x ^ 2) = sin x :=\nby rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]\n\nlemma cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x :=\nby rw ← of_real_inj; simp [cos_three_mul]\n\nlemma sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 :=\nby rw ← of_real_inj; simp [sin_three_mul]\n\n/-- The definition of `sinh` in terms of `exp`. -/\nlemma sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 :=\neq_div_of_mul_eq two_ne_zero $ by rw [sinh, exp, exp, complex.of_real_neg, complex.sinh, mul_two,\n    ← complex.add_re, ← mul_two, div_mul_cancel _ (two_ne_zero' ℂ), complex.sub_re]\n\n@[simp] lemma sinh_zero : sinh 0 = 0 := by simp [sinh]\n\n@[simp] lemma sinh_neg : sinh (-x) = -sinh x :=\nby simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]\n\nlemma sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y :=\nby rw ← of_real_inj; simp [sinh_add]\n\n/-- The definition of `cosh` in terms of `exp`. -/\nlemma cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 :=\neq_div_of_mul_eq two_ne_zero $ by rw [cosh, exp, exp, complex.of_real_neg, complex.cosh, mul_two,\n    ← complex.add_re, ← mul_two, div_mul_cancel _ (two_ne_zero' ℂ), complex.add_re]\n\n@[simp] lemma cosh_zero : cosh 0 = 1 := by simp [cosh]\n\n@[simp] lemma cosh_neg : cosh (-x) = cosh x := of_real_inj.1 $ by simp\n\n@[simp] lemma cosh_abs : cosh (|x|) = cosh x :=\nby cases le_total x 0; simp [*, _root_.abs_of_nonneg, abs_of_nonpos]\n\nlemma cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y :=\nby rw ← of_real_inj; simp [cosh_add]\n\nlemma sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y :=\nby simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]\n\nlemma cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y :=\nby simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]\n\nlemma tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=\nof_real_inj.1 $ by simp [tanh_eq_sinh_div_cosh]\n\n@[simp] lemma tanh_zero : tanh 0 = 0 := by simp [tanh]\n\n@[simp] lemma tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]\n\n@[simp] lemma cosh_add_sinh : cosh x + sinh x = exp x :=\nby rw ← of_real_inj; simp\n\n@[simp] lemma sinh_add_cosh : sinh x + cosh x = exp x :=\nby rw [add_comm, cosh_add_sinh]\n\n@[simp] lemma exp_sub_cosh : exp x - cosh x = sinh x :=\nsub_eq_iff_eq_add.2 (sinh_add_cosh x).symm\n\n@[simp] lemma exp_sub_sinh : exp x - sinh x = cosh x :=\nsub_eq_iff_eq_add.2 (cosh_add_sinh x).symm\n\n@[simp] lemma cosh_sub_sinh : cosh x - sinh x = exp (-x) :=\nby { rw [← of_real_inj], simp }\n\n@[simp] lemma sinh_sub_cosh : sinh x - cosh x = -exp (-x) :=\nby rw [← neg_sub, cosh_sub_sinh]\n\n@[simp] lemma cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 :=\nby rw ← of_real_inj; simp\n\nlemma cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 :=\nby rw ← of_real_inj; simp [cosh_sq]\n\nlemma cosh_sq' : cosh x ^ 2 = 1 + sinh x ^ 2 :=\n(cosh_sq x).trans (add_comm _ _)\n\nlemma sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 :=\nby rw ← of_real_inj; simp [sinh_sq]\n\nlemma cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 :=\nby rw ← of_real_inj; simp [cosh_two_mul]\n\nlemma sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x :=\nby rw ← of_real_inj; simp [sinh_two_mul]\n\nlemma cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x :=\nby rw ← of_real_inj; simp [cosh_three_mul]\n\nlemma sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x :=\nby rw ← of_real_inj; simp [sinh_three_mul]\n\nopen is_absolute_value\n\n/-- This is an intermediate result that is later replaced by `real.add_one_le_exp`; use that lemma\ninstead. -/\nlemma add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x :=\ncalc x + 1 ≤ lim (⟨(λ n : ℕ, ((exp' x) n).re), is_cau_seq_re (exp' x)⟩ : cau_seq ℝ has_abs.abs) :\n  le_lim (cau_seq.le_of_exists ⟨2,\n    λ j hj, show x + (1 : ℝ) ≤ (∑ m in range j, (x ^ m / m! : ℂ)).re,\n      from have h₁ : (((λ m : ℕ, (x ^ m / m! : ℂ)) ∘ nat.succ) 0).re = x, by simp,\n      have h₂ : ((x : ℂ) ^ 0 / 0!).re = 1, by simp,\n      begin\n        rw [← tsub_add_cancel_of_le hj, sum_range_succ', sum_range_succ',\n          add_re, add_re, h₁, h₂, add_assoc,\n          ← coe_re_add_group_hom, (re_add_group_hom).map_sum, coe_re_add_group_hom ],\n        refine le_add_of_nonneg_of_le (sum_nonneg (λ m hm, _)) le_rfl,\n        rw [← of_real_pow, ← of_real_nat_cast, ← of_real_div, of_real_re],\n        exact div_nonneg (pow_nonneg hx _) (nat.cast_nonneg _),\n      end⟩)\n... = exp x : by rw [exp, complex.exp, ← cau_seq_re, lim_re]\n\nlemma one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x :=\nby linarith [add_one_le_exp_of_nonneg hx]\n\nlemma exp_pos (x : ℝ) : 0 < exp x :=\n(le_total 0 x).elim (lt_of_lt_of_le zero_lt_one ∘ one_le_exp)\n  (λ h, by rw [← neg_neg x, real.exp_neg];\n    exact inv_pos.2 (lt_of_lt_of_le zero_lt_one (one_le_exp (neg_nonneg.2 h))))\n\n@[simp] lemma abs_exp (x : ℝ) : |exp x| = exp x :=\nabs_of_pos (exp_pos _)\n\n@[mono] lemma exp_strict_mono : strict_mono exp :=\nλ x y h, by rw [← sub_add_cancel y x, real.exp_add];\n  exact (lt_mul_iff_one_lt_left (exp_pos _)).2\n    (lt_of_lt_of_le (by linarith) (add_one_le_exp_of_nonneg (by linarith)))\n\n@[mono] lemma exp_monotone : monotone exp := exp_strict_mono.monotone\n\n@[simp] lemma exp_lt_exp {x y : ℝ} : exp x < exp y ↔ x < y := exp_strict_mono.lt_iff_lt\n\n@[simp] lemma exp_le_exp {x y : ℝ} : exp x ≤ exp y ↔ x ≤ y := exp_strict_mono.le_iff_le\n\nlemma exp_injective : function.injective exp := exp_strict_mono.injective\n\n@[simp] lemma exp_eq_exp {x y : ℝ} : exp x = exp y ↔ x = y := exp_injective.eq_iff\n\n@[simp] lemma exp_eq_one_iff : exp x = 1 ↔ x = 0 := exp_injective.eq_iff' exp_zero\n\n@[simp] lemma one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x :=\nby rw [← exp_zero, exp_lt_exp]\n\n@[simp] lemma exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0 :=\nby rw [← exp_zero, exp_lt_exp]\n\n@[simp] lemma exp_le_one_iff {x : ℝ} : exp x ≤ 1 ↔ x ≤ 0 :=\nexp_zero ▸ exp_le_exp\n\n@[simp] lemma one_le_exp_iff {x : ℝ} : 1 ≤ exp x ↔ 0 ≤ x :=\nexp_zero ▸ exp_le_exp\n\n/-- `real.cosh` is always positive -/\nlemma cosh_pos (x : ℝ) : 0 < real.cosh x :=\n(cosh_eq x).symm ▸ half_pos (add_pos (exp_pos x) (exp_pos (-x)))\n\nlemma sinh_lt_cosh : sinh x < cosh x :=\nlt_of_pow_lt_pow 2 (cosh_pos _).le $ (cosh_sq x).symm ▸ lt_add_one _\n\nend real\n\nnamespace complex\n\nlemma sum_div_factorial_le {α : Type*} [linear_ordered_field α] (n j : ℕ) (hn : 0 < n) :\n  ∑ m in filter (λ k, n ≤ k) (range j), (1 / m! : α) ≤ n.succ / (n! * n) :=\ncalc ∑ m in filter (λ k, n ≤ k) (range j), (1 / m! : α)\n    = ∑ m in range (j - n), 1 / (m + n)! :\n  sum_bij (λ m _, m - n)\n    (λ m hm, mem_range.2 $ (tsub_lt_tsub_iff_right (by simp at hm; tauto)).2\n      (by simp at hm; tauto))\n    (λ m hm, by rw tsub_add_cancel_of_le; simp at *; tauto)\n    (λ a₁ a₂ ha₁ ha₂ h,\n      by rwa [tsub_eq_iff_eq_add_of_le, tsub_add_eq_add_tsub, eq_comm, tsub_eq_iff_eq_add_of_le,\n              add_left_inj, eq_comm] at h;\n        simp at *; tauto)\n    (λ b hb, ⟨b + n,\n      mem_filter.2 ⟨mem_range.2 $ lt_tsub_iff_right.mp (mem_range.1 hb), nat.le_add_left _ _⟩,\n      by rw add_tsub_cancel_right⟩)\n... ≤ ∑ m in range (j - n), (n! * n.succ ^ m)⁻¹ :\n  begin\n    refine  sum_le_sum (assume m n, _),\n    rw [one_div, inv_le_inv],\n    { rw [← nat.cast_pow, ← nat.cast_mul, nat.cast_le, add_comm],\n      exact nat.factorial_mul_pow_le_factorial },\n    { exact nat.cast_pos.2 (nat.factorial_pos _) },\n    { exact mul_pos (nat.cast_pos.2 (nat.factorial_pos _))\n        (pow_pos (nat.cast_pos.2 (nat.succ_pos _)) _) },\n  end\n... = n!⁻¹ * ∑ m in range (j - n), n.succ⁻¹ ^ m :\n  by simp [mul_inv, mul_sum.symm, sum_mul.symm, -nat.factorial_succ, mul_comm, inv_pow]\n... = (n.succ - n.succ * n.succ⁻¹ ^ (j - n)) / (n! * n) :\n  have h₁ : (n.succ : α) ≠ 1, from @nat.cast_one α _ ▸ mt nat.cast_inj.1\n        (mt nat.succ.inj (pos_iff_ne_zero.1 hn)),\n  have h₂ : (n.succ : α) ≠ 0, from nat.cast_ne_zero.2 (nat.succ_ne_zero _),\n  have h₃ : (n! * n : α) ≠ 0,\n    from mul_ne_zero (nat.cast_ne_zero.2 (pos_iff_ne_zero.1 (nat.factorial_pos _)))\n    (nat.cast_ne_zero.2 (pos_iff_ne_zero.1 hn)),\n  have h₄ : (n.succ - 1 : α) = n, by simp,\n  by rw [geom_sum_inv h₁ h₂, eq_div_iff_mul_eq h₃,\n      mul_comm _ (n! * n : α), ← mul_assoc (n!⁻¹ : α), ← mul_inv_rev, h₄,\n      ← mul_assoc (n! * n : α), mul_comm (n : α) n!, mul_inv_cancel h₃];\n    simp [mul_add, add_mul, mul_assoc, mul_comm]\n... ≤ n.succ / (n! * n) :\n  begin\n    refine iff.mpr (div_le_div_right (mul_pos _ _)) _,\n    exact nat.cast_pos.2 (nat.factorial_pos _),\n    exact nat.cast_pos.2 hn,\n    exact sub_le_self _\n      (mul_nonneg (nat.cast_nonneg _) (pow_nonneg (inv_nonneg.2 (nat.cast_nonneg _)) _))\n  end\n\nlemma exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :\n  abs (exp x - ∑ m in range n, x ^ m / m!) ≤ abs x ^ n * (n.succ * (n! * n)⁻¹) :=\nbegin\n  rw [← lim_const (∑ m in range n, _), exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs],\n  refine lim_le (cau_seq.le_of_exists ⟨n, λ j hj, _⟩),\n  simp_rw ← sub_eq_add_neg,\n  show abs (∑ m in range j, x ^ m / m! - ∑ m in range n, x ^ m / m!)\n    ≤ abs x ^ n * (n.succ * (n! * n)⁻¹),\n  rw sum_range_sub_sum_range hj,\n  calc abs (∑ m in (range j).filter (λ k, n ≤ k), (x ^ m / m! : ℂ))\n      = abs (∑ m in (range j).filter (λ k, n ≤ k), (x ^ n * (x ^ (m - n) / m!) : ℂ)) :\n    begin\n      refine congr_arg abs (sum_congr rfl (λ m hm, _)),\n      rw [mem_filter, mem_range] at hm,\n      rw [← mul_div_assoc, ← pow_add, add_tsub_cancel_of_le hm.2]\n    end\n  ... ≤ ∑ m in filter (λ k, n ≤ k) (range j), abs (x ^ n * (_ / m!)) : abv_sum_le_sum_abv _ _\n  ... ≤ ∑ m in filter (λ k, n ≤ k) (range j), abs x ^ n * (1 / m!) :\n    begin\n      refine sum_le_sum (λ m hm, _),\n      rw [map_mul, map_pow, map_div₀, abs_cast_nat],\n      refine mul_le_mul_of_nonneg_left ((div_le_div_right _).2 _) _,\n      { exact nat.cast_pos.2 (nat.factorial_pos _), },\n      { rw abv_pow abs,\n        exact (pow_le_one _ (abs.nonneg _) hx), },\n      { exact pow_nonneg (abs.nonneg _) _ },\n    end\n  ... = abs x ^ n * (∑ m in (range j).filter (λ k, n ≤ k), (1 / m! : ℝ)) :\n    by simp [abs_mul, abv_pow abs, abs_div, mul_sum.symm]\n  ... ≤ abs x ^ n * (n.succ * (n! * n)⁻¹) :\n    mul_le_mul_of_nonneg_left (sum_div_factorial_le _ _ hn) (pow_nonneg (abs.nonneg _) _)\nend\n\nlemma exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / (n.succ) ≤ 1 / 2) :\n  abs (exp x - ∑ m in range n, x ^ m / m!) ≤ abs x ^ n / (n!) * 2 :=\nbegin\n  rw [← lim_const (∑ m in range n, _), exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs],\n  refine lim_le (cau_seq.le_of_exists ⟨n, λ j hj, _⟩),\n  simp_rw [←sub_eq_add_neg],\n  show abs (∑ m in range j, x ^ m / m! - ∑ m in range n, x ^ m / m!) ≤ abs x ^ n / (n!) * 2,\n  let k := j - n,\n  have hj : j = n + k := (add_tsub_cancel_of_le hj).symm,\n  rw [hj, sum_range_add_sub_sum_range],\n  calc abs (∑ (i : ℕ) in range k, x ^ (n + i) / ((n + i)! : ℂ))\n      ≤ ∑ (i : ℕ) in range k, abs (x ^ (n + i) / ((n + i)! : ℂ)) : abv_sum_le_sum_abv _ _\n  ... ≤ ∑ (i : ℕ) in range k, (abs x) ^ (n + i) / (n + i)! :\n        by simp only [complex.abs_cast_nat, map_div₀, abv_pow abs]\n  ... ≤ ∑ (i : ℕ) in range k, (abs x) ^ (n + i) / (n! * n.succ ^ i) : _\n  ... = ∑ (i : ℕ) in range k, (abs x) ^ (n) / (n!) * ((abs x)^i / n.succ ^ i) : _\n  ... ≤ abs x ^ n / (↑n!) * 2 : _,\n  { refine sum_le_sum (λ m hm, div_le_div (pow_nonneg (abs.nonneg x) (n + m)) le_rfl _ _),\n    { exact_mod_cast mul_pos n.factorial_pos (pow_pos n.succ_pos _), },\n    { exact_mod_cast (nat.factorial_mul_pow_le_factorial), }, },\n  { refine finset.sum_congr rfl (λ _ _, _),\n    simp only [pow_add, div_eq_inv_mul, mul_inv, mul_left_comm, mul_assoc], },\n  { rw [←mul_sum],\n    apply mul_le_mul_of_nonneg_left,\n    { simp_rw [←div_pow],\n      rw [geom_sum_eq, div_le_iff_of_neg],\n      { transitivity (-1 : ℝ),\n        { linarith },\n        { simp only [neg_le_sub_iff_le_add, div_pow, nat.cast_succ, le_add_iff_nonneg_left],\n          exact div_nonneg (pow_nonneg (abs.nonneg x) k)\n            (pow_nonneg (add_nonneg n.cast_nonneg zero_le_one) k) } },\n      { linarith },\n      { linarith }, },\n    { exact div_nonneg (pow_nonneg (abs.nonneg x) n) (nat.cast_nonneg (n!)), }, },\nend\n\nlemma abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) :\n  abs (exp x - 1) ≤ 2 * abs x :=\ncalc abs (exp x - 1) = abs (exp x - ∑ m in range 1, x ^ m / m!) :\n  by simp [sum_range_succ]\n... ≤ abs x ^ 1 * ((nat.succ 1) * (1! * (1 : ℕ))⁻¹) :\n  exp_bound hx dec_trivial\n... = 2 * abs x : by simp [two_mul, mul_two, mul_add, mul_comm]\n\nlemma abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) :\n  abs (exp x - 1 - x) ≤ (abs x)^2 :=\ncalc abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m!) :\n  by simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc]\n... ≤ (abs x)^2 * (nat.succ 2 * (2! * (2 : ℕ))⁻¹) :\n  exp_bound hx dec_trivial\n... ≤ (abs x)^2 * 1 :\n  mul_le_mul_of_nonneg_left (by norm_num) (sq_nonneg (abs x))\n... = (abs x)^2 :\n  by rw [mul_one]\n\nend complex\n\nnamespace real\n\nopen complex finset\n\nlemma exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :\n  |exp x - ∑ m in range n, x ^ m / m!|≤ |x| ^ n * (n.succ / (n! * n)) :=\nbegin\n  have hxc : complex.abs x ≤ 1, by exact_mod_cast hx,\n  convert exp_bound hxc hn; norm_cast\nend\n\nlemma exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :\n  real.exp x ≤ ∑ m in finset.range n, x ^ m / m! + x ^ n * (n + 1) / (n! * n) :=\nbegin\n  have h3 : |x| = x := by simpa,\n  have h4 : |x| ≤ 1 := by rwa h3,\n  have h' := real.exp_bound h4 hn,\n  rw h3 at h',\n  have h'' := (abs_sub_le_iff.1 h').1,\n  have t := sub_le_iff_le_add'.1 h'',\n  simpa [mul_div_assoc] using t\nend\n\nlemma abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| :=\nbegin\n  have : complex.abs x ≤ 1 := by exact_mod_cast hx,\n  exact_mod_cast complex.abs_exp_sub_one_le this,\nend\n\nlemma abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 :=\nbegin\n  rw ←_root_.sq_abs,\n  have : complex.abs x ≤ 1 := by exact_mod_cast hx,\n  exact_mod_cast complex.abs_exp_sub_one_sub_id_le this,\nend\n\n/-- A finite initial segment of the exponential series, followed by an arbitrary tail.\nFor fixed `n` this is just a linear map wrt `r`, and each map is a simple linear function\nof the previous (see `exp_near_succ`), with `exp_near n x r ⟶ exp x` as `n ⟶ ∞`,\nfor any `r`. -/\ndef exp_near (n : ℕ) (x r : ℝ) : ℝ := ∑ m in range n, x ^ m / m! + x ^ n / n! * r\n\n@[simp] theorem exp_near_zero (x r) : exp_near 0 x r = r := by simp [exp_near]\n\n@[simp] theorem exp_near_succ (n x r) : exp_near (n + 1) x r = exp_near n x (1 + x / (n+1) * r) :=\nby simp [exp_near, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,\n  mul_inv]; ac_refl\n\ntheorem exp_near_sub (n x r₁ r₂) : exp_near n x r₁ - exp_near n x r₂ = x ^ n / n! * (r₁ - r₂) :=\nby simp [exp_near, mul_sub]\n\nlemma exp_approx_end (n m : ℕ) (x : ℝ)\n  (e₁ : n + 1 = m) (h : |x| ≤ 1) :\n  |exp x - exp_near m x 0| ≤ |x| ^ m / m! * ((m+1)/m) :=\nby { simp [exp_near], convert exp_bound h _ using 1, field_simp [mul_comm], linarith }\n\nlemma exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ)\n  (e₁ : n + 1 = m) (a₂ b₂ : ℝ)\n  (e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)\n  (h : |exp x - exp_near m x a₂| ≤ |x| ^ m / m! * b₂) :\n  |exp x - exp_near n x a₁| ≤ |x| ^ n / n! * b₁ :=\nbegin\n  refine (_root_.abs_sub_le _ _ _).trans ((add_le_add_right h _).trans _),\n  subst e₁, rw [exp_near_succ, exp_near_sub, _root_.abs_mul],\n  convert mul_le_mul_of_nonneg_left (le_sub_iff_add_le'.1 e) _,\n  { simp [mul_add, pow_succ', div_eq_mul_inv, _root_.abs_mul, _root_.abs_inv, ← pow_abs, mul_inv],\n    ac_refl },\n  { simp [_root_.div_nonneg, _root_.abs_nonneg] }\nend\n\nlemma exp_approx_end' {n} {x a b : ℝ} (m : ℕ)\n  (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm) (h : |x| ≤ 1)\n  (e : |1 - a| ≤ b - |x| / rm * ((rm+1)/rm)) :\n  |exp x - exp_near n x a| ≤ |x| ^ n / n! * b :=\nby subst er; exact\nexp_approx_succ _ e₁ _ _ (by simpa using e) (exp_approx_end _ _ _ e₁ h)\n\nlemma exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ}\n  (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)\n  (h : |exp 1 - exp_near m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m! * (b₁ * rm)) :\n  |exp 1 - exp_near n 1 a₁| ≤ |1| ^ n / n! * b₁ :=\nbegin\n  subst er,\n  refine exp_approx_succ _ en _ _ _ h,\n  field_simp [show (m : ℝ) ≠ 0, by norm_cast; linarith],\nend\n\nlemma exp_approx_start (x a b : ℝ)\n  (h : |exp x - exp_near 0 x a| ≤ |x| ^ 0 / 0! * b) :\n  |exp x - a| ≤ b :=\nby simpa using h\n\nlemma cos_bound {x : ℝ} (hx : |x| ≤ 1) :\n  |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=\ncalc |cos x - (1 - x ^ 2 / 2)| = abs (complex.cos x - (1 - x ^ 2 / 2)) :\n  by rw ← abs_of_real; simp [of_real_bit0, of_real_one, of_real_inv]\n... = abs ((complex.exp (x * I) + complex.exp (-x * I) - (2 - x ^ 2)) / 2) :\n  by simp [complex.cos, sub_div, add_div, neg_div, div_self (two_ne_zero' ℂ)]\n... = abs (((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) +\n    ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!))) / 2) :\n  congr_arg abs (congr_arg (λ x : ℂ, x / 2) begin\n    simp only [sum_range_succ],\n    simp [pow_succ],\n    apply complex.ext; simp [div_eq_mul_inv, norm_sq]; ring\n  end)\n... ≤ abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) / 2) +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) / 2) :\n  by rw add_div; exact complex.abs.add_le _ _\n... = (abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) / 2 +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!)) / 2) :\n  by simp [map_div₀]\n... ≤ ((complex.abs (x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2 +\n    (complex.abs (-x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2)  :\n  add_le_add ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n             ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n... ≤ |x| ^ 4 * (5 / 96) : by norm_num; simp [mul_assoc, mul_comm, mul_left_comm, mul_div_assoc]\n\nlemma sin_bound {x : ℝ} (hx : |x| ≤ 1) :\n  |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=\ncalc |sin x - (x - x ^ 3 / 6)| = abs (complex.sin x - (x - x ^ 3 / 6)) :\n  by rw ← abs_of_real; simp [of_real_bit0, of_real_one, of_real_inv]\n... = abs (((complex.exp (-x * I) - complex.exp (x * I)) * I - (2 * x - x ^ 3 / 3)) / 2) :\n  by simp [complex.sin, sub_div, add_div, neg_div, mul_div_cancel_left _ (two_ne_zero' ℂ),\n    div_div, show (3 : ℂ) * 2 = 6, by norm_num]\n... = abs ((((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) -\n    (complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) * I) / 2) :\n  congr_arg abs (congr_arg (λ x : ℂ, x / 2) begin\n    simp only [sum_range_succ],\n    simp [pow_succ],\n    apply complex.ext; simp [div_eq_mul_inv, norm_sq]; ring\n  end)\n... ≤ abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!) * I / 2) +\n    abs (-((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!) * I) / 2) :\n  by rw [sub_mul, sub_eq_add_neg, add_div]; exact complex.abs.add_le _ _\n... = (abs ((complex.exp (x * I) - ∑ m in range 4, (x * I) ^ m / m!)) / 2 +\n    abs ((complex.exp (-x * I) - ∑ m in range 4, (-x * I) ^ m / m!)) / 2) :\n  by simp [add_comm, map_div₀]\n... ≤ ((complex.abs (x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2 +\n    (complex.abs (-x * I) ^ 4 * (nat.succ 4 * (4! * (4 : ℕ))⁻¹)) / 2) :\n  add_le_add ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n             ((div_le_div_right (by norm_num)).2 (complex.exp_bound (by simpa) dec_trivial))\n... ≤ |x| ^ 4 * (5 / 96) : by norm_num; simp [mul_assoc, mul_comm, mul_left_comm, mul_div_assoc]\n\nlemma cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=\ncalc 0 < (1 - x ^ 2 / 2) - |x| ^ 4 * (5 / 96) :\n  sub_pos.2 $ lt_sub_iff_add_lt.2\n    (calc |x| ^ 4 * (5 / 96) + x ^ 2 / 2\n          ≤ 1 * (5 / 96) + 1 / 2 :\n        add_le_add\n          (mul_le_mul_of_nonneg_right (pow_le_one _ (abs_nonneg _) hx) (by norm_num))\n          ((div_le_div_right (by norm_num)).2 (by rw [sq, ← abs_mul_self, _root_.abs_mul];\n            exact mul_le_one hx (abs_nonneg _) hx))\n      ... < 1 : by norm_num)\n... ≤ cos x : sub_le_comm.1 (abs_sub_le_iff.1 (cos_bound hx)).2\n\nlemma sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=\ncalc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :\n  sub_pos.2 $ lt_sub_iff_add_lt.2\n    (calc |x| ^ 4 * (5 / 96) + x ^ 3 / 6\n        ≤ x * (5 / 96) + x / 6 :\n      add_le_add\n        (mul_le_mul_of_nonneg_right\n          (calc |x| ^ 4 ≤ |x| ^ 1 : pow_le_pow_of_le_one (abs_nonneg _)\n                (by rwa _root_.abs_of_nonneg (le_of_lt hx0))\n                dec_trivial\n            ... = x : by simp [_root_.abs_of_nonneg (le_of_lt (hx0))]) (by norm_num))\n        ((div_le_div_right (by norm_num)).2\n          (calc x ^ 3 ≤ x ^ 1 : pow_le_pow_of_le_one (le_of_lt hx0) hx dec_trivial\n            ... = x : pow_one _))\n    ... < x : by linarith)\n... ≤ sin x : sub_le_comm.1 (abs_sub_le_iff.1 (sin_bound\n    (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]))).2\n\nlemma sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=\nhave x / 2 ≤ 1, from (div_le_iff (by norm_num)).mpr (by simpa),\ncalc 0 < 2 * sin (x / 2) * cos (x / 2) :\n  mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))\n    (cos_pos_of_le_one (by rwa [_root_.abs_of_nonneg (le_of_lt (half_pos hx0))]))\n... = sin x : by rw [← sin_two_mul, two_mul, add_halves]\n\nlemma cos_one_le : cos 1 ≤ 2 / 3 :=\ncalc cos 1 ≤ |(1 : ℝ)| ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :\n  sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by simp))).1\n... ≤ 2 / 3 : by norm_num\n\nlemma cos_one_pos : 0 < cos 1 := cos_pos_of_le_one (le_of_eq abs_one)\n\nlemma cos_two_neg : cos 2 < 0 :=\ncalc cos 2 = cos (2 * 1) : congr_arg cos (mul_one _).symm\n  ... = _ : real.cos_two_mul 1\n  ... ≤ 2 * (2 / 3) ^ 2 - 1 : sub_le_sub_right (mul_le_mul_of_nonneg_left\n          (by { rw [sq, sq], exact mul_self_le_mul_self (le_of_lt cos_one_pos) cos_one_le })\n          zero_le_two) _\n  ... < 0 : by norm_num\n\nlemma exp_bound_div_one_sub_of_interval_approx  {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) :\n  ∑ (j : ℕ) in finset.range 3, x ^ j / (j.factorial)\n  + x ^ 3 * ((3 : ℕ) + 1) / ((3 : ℕ).factorial * (3 : ℕ))\n  ≤ ∑ j in (finset.range 3), x ^ j :=\nbegin\n  norm_num [finset.sum],\n  rw [add_assoc, add_comm (x + 1) (x ^ 3 * 4 / 18), ← add_assoc, add_le_add_iff_right,\n      ← add_le_add_iff_left (-(x ^ 2 / 2)), ← add_assoc, comm_ring.add_left_neg (x ^ 2 / 2),\n      zero_add, neg_add_eq_sub, sub_half, sq, pow_succ, sq],\n  have i1 : x * 4 / 18 ≤ 1 / 2 := by linarith,\n  have i2 : 0 ≤ x * 4 / 18 := by linarith,\n  have i3 := mul_le_mul h1 h1 le_rfl h1,\n  rw zero_mul at i3,\n  have t := mul_le_mul le_rfl i1 i2 i3,\n  rw ← mul_assoc,\n  rwa [mul_one_div, ← mul_div_assoc, ← mul_assoc] at t,\nend\n\nlemma exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :\n  real.exp x ≤ 1 / (1 - x) :=\nbegin\n  have h : ∑ j in (finset.range 3), x ^ j ≤ 1 / (1 - x),\n  { norm_num [finset.sum],\n    have h1x : 0 < 1 - x := by simpa,\n    rw le_div_iff h1x,\n    norm_num [← add_assoc, mul_sub_left_distrib, mul_one, add_mul,\n              sub_add_eq_sub_sub, pow_succ' x 2],\n    have hx3 : 0 ≤ x ^ 3,\n    { norm_num,\n      exact h1 },\n    linarith },\n  exact (exp_bound' h1 h2.le $ by linarith).trans\n        ((exp_bound_div_one_sub_of_interval_approx h1 h2.le).trans h),\nend\n\nlemma one_sub_le_exp_minus_of_pos {y : ℝ} (h : 0 ≤ y) : 1 - y ≤ real.exp (-y) :=\nbegin\n  rw real.exp_neg,\n  have r1 : (1 - y) * (real.exp y) ≤ 1,\n  { cases le_or_lt (1 - y) 0,\n    { have h'' : (1 - y) * y.exp ≤ 0,\n      { rw mul_nonpos_iff,\n        right,\n        exact ⟨h_1, y.exp_pos.le⟩ },\n    linarith },\n    have hy1 : y < 1 := by linarith,\n    rw  ← le_div_iff' h_1,\n    exact exp_bound_div_one_sub_of_interval h hy1 },\n  rw inv_eq_one_div,\n  rw le_div_iff' y.exp_pos,\n  rwa mul_comm at r1,\nend\n\nlemma add_one_le_exp_of_nonpos {x : ℝ} (h : x ≤ 0) : x + 1 ≤ real.exp x :=\nbegin\n  rw add_comm,\n  have h1 : 0 ≤ -x := by linarith,\n  simpa using one_sub_le_exp_minus_of_pos h1\nend\n\nlemma add_one_le_exp (x : ℝ) : x + 1 ≤ real.exp x :=\nbegin\n  cases le_or_lt 0 x,\n  { exact real.add_one_le_exp_of_nonneg h },\n  exact add_one_le_exp_of_nonpos h.le,\nend\n\nlemma one_sub_div_pow_le_exp_neg {n : ℕ} {t : ℝ} (ht' : t ≤ n) : (1 - t / n) ^ n ≤ exp (-t) :=\nbegin\n  rcases eq_or_ne n 0 with rfl | hn,\n  { simp, rwa nat.cast_zero at ht' },\n  convert pow_le_pow_of_le_left _ (add_one_le_exp (-(t / n))) n,\n  { abel },\n  { rw ←real.exp_nat_mul, congr' 1,\n    field_simp [nat.cast_ne_zero.mpr hn], ring },\n  { rwa [add_comm, ←sub_eq_add_neg, sub_nonneg, div_le_one],\n    positivity }\nend\n\nend real\n\nnamespace tactic\nopen positivity real\n\n/-- Extension for the `positivity` tactic: `real.exp` is always positive. -/\n@[positivity]\nmeta def positivity_exp : expr → tactic strictness\n| `(real.exp %%a) := positive <$> mk_app `real.exp_pos [a]\n| e := pp e >>= fail ∘ format.bracket \"The expression `\" \"` isn't of the form `real.exp r`\"\n\nend tactic\n\nnamespace complex\n\n@[simp] lemma abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 :=\nhave _ := real.sin_sq_add_cos_sq x,\nby simp [add_comm, abs, norm_sq, sq, *, sin_of_real_re, cos_of_real_re, mul_re] at *\n\n@[simp] lemma abs_exp_of_real (x : ℝ) : abs (exp x) = real.exp x :=\nby rw [← of_real_exp]; exact abs_of_nonneg (le_of_lt (real.exp_pos _))\n\n@[simp] lemma abs_exp_of_real_mul_I (x : ℝ) : abs (exp (x * I)) = 1 :=\nby rw [exp_mul_I, abs_cos_add_sin_mul_I]\n\nlemma abs_exp (z : ℂ) : abs (exp z) = real.exp z.re :=\nby rw [exp_eq_exp_re_mul_sin_add_cos, map_mul, abs_exp_of_real, abs_cos_add_sin_mul_I, mul_one]\n\nlemma abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re :=\nby rw [abs_exp, abs_exp, real.exp_eq_exp]\n\nend complex\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/complex/exponential.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.31806326838968485}}
{"text": "import ..common.correctness ...common.int\n\nnamespace lia\n\ndef hd_coeff_one : int → atom → atom \n| m (atom.le i (k::ks)) := \n  if k = 0 \n  then (atom.le i (0::ks))  \n  else let m' := has_div.div m (abs k) in \n       atom.le (m' * i) (int.sign k :: list.map (λ x, m' * x) ks)\n| m (atom.dvd d i (k::ks)) :=\n  if k = 0 \n  then (atom.dvd d i (0::ks)) \n  else let m' := has_div.div m k in \n       atom.dvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.ndvd d i (k::ks)) := \n  if k = 0 \n  then (atom.ndvd d i (0::ks)) \n  else let m' := has_div.div m k in \n       atom.ndvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.le i []) := (atom.le i [])\n| m (atom.dvd d i []) := (atom.dvd d i []) \n| m (atom.ndvd d i []) := (atom.ndvd d i []) \n\ndef hd_coeff_one' : int → atom → atom \n| m (atom.le i (0::ks)) := (atom.le i (0::ks))   \n| m (atom.le i (int.of_nat (n+1) :: ks)) :=\n  let k := int.of_nat (n+1) in \n  let m' := has_div.div m (abs k) in \n  atom.le (m' * i) (int.sign k :: list.map (λ x, m' * x) ks)\n| m (atom.le i (int.neg_succ_of_nat n :: ks)) :=\n  let k := int.neg_succ_of_nat n in \n  let m' := has_div.div m (abs k) in \n  atom.le (m' * i) (int.sign k :: list.map (λ x, m' * x) ks)\n| m (atom.dvd d i (0::ks)) := (atom.dvd d i (0::ks)) \n| m (atom.dvd d i (int.of_nat (n+1) :: ks)) := \n  let k := int.of_nat (n+1) in \n  let m' := has_div.div m k in \n  atom.dvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.dvd d i (int.neg_succ_of_nat n ::ks)) := \n  let k := int.neg_succ_of_nat n in \n  let m' := has_div.div m k in \n  atom.dvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.ndvd d i (0::ks)) := (atom.ndvd d i (0::ks)) \n| m (atom.ndvd d i (int.of_nat (n+1) :: ks)) := \n  let k := int.of_nat (n+1) in \n  let m' := has_div.div m k in \n  atom.ndvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.ndvd d i (int.neg_succ_of_nat n :: ks)) := \n  let k := int.neg_succ_of_nat n in \n  let m' := has_div.div m k in \n  atom.ndvd (m' * d) (m' * i) (1 :: list.map (λ x, m' * x) ks)\n| m (atom.le i []) := (atom.le i [])\n| m (atom.dvd d i []) := (atom.dvd d i []) \n| m (atom.ndvd d i []) := (atom.ndvd d i []) \n\ndef coeffs_lcm (p) := \n  int.lcms (list.map hd_coeff (atoms_dep0 int p))\n\ndef divisors_lcm (p) := \n  int.lcms (list.map divisor (atoms_dep0 int p))\n\ndef hd_coeffs_one (p : fm atom) : fm atom := \nA' (atom.dvd (coeffs_lcm p) 0 [1]) ∧' (map_fm (hd_coeff_one (coeffs_lcm p)) p)\n\ndef inf_minus : fm atom → fm atom \n| ⊤' := ⊤' \n| ⊥' := ⊥' \n| (A' (atom.le i (k::ks))) := \n  if k < 0 \n  then ⊤' \n  else if k > 0\n       then ⊥' \n       else A' (atom.le i (0::ks))\n| (A' a) := A' a\n| (p ∧' q) := and_o (inf_minus p) (inf_minus q)\n| (p ∨' q) := or_o (inf_minus p) (inf_minus q)\n| (¬' p) := ¬' p\n| (∃' p) := ∃' p\n\ndef subst (i ks p) := map_fm (asubst i ks) p\n\n\ndef get_lb : atom → option (int × list int) \n| (atom.le i (k::ks)) :=\n  if k > 0 then (i,ks) else none\n| (atom.le _ []) := none\n| (atom.dvd _ _ _) := none\n| (atom.ndvd _ _ _) := none\n\ndef bnd_points (p) := \n  list.filter_map get_lb (atoms_dep0 ℤ p)\n\ndef list.irange (z : int) : list int :=\nlist.map int.of_nat (list.range (int.nat_abs z))\n\nlemma list.mem_irange (z y : int) : \n  0 ≤ z → 0 ≤ y → z < y → z ∈ list.irange y := \nbegin\n  intros hz hy hzy, unfold list.irange,\n  rewrite list.mem_map, \n  rewrite int.nonneg_iff_exists at hz,\n  cases hz with n hn, subst hn, existsi n,\n  apply and.intro _ rfl,\n  rewrite list.mem_range, \n  rewrite iff.symm int.coe_nat_lt,\n  have heq : ↑(int.nat_abs y) = int.of_nat (int.nat_abs y),\n  refl, rewrite heq,\n  rewrite int.of_nat_nat_abs_eq_of_nonneg hy, apply hzy,\nend\n\ndef qe_cooper_one (p : fm atom) : fm atom := \n  let as := atoms_dep0 int p in \n  let d := int.lcms (list.map divisor as) in\n  let lbs := list.filter_map get_lb as in\n  or_o \n    (disj (list.irange d) (λ n, subst n [] (inf_minus p))) \n    (disj lbs \n      (λ iks, disj (list.irange d) \n                (λ n, subst (iks^.fst + n) (map_neg iks^.snd) p)))\n\ndef sqe_cooper := λ x, qe_cooper_one (hd_coeffs_one x)\n\ndef qe_cooper := lift_nnf_qe int sqe_cooper\n\nend lia", "meta": {"author": "avigad", "repo": "qelim", "sha": "b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60", "save_path": "github-repos/lean/avigad-qelim", "path": "github-repos/lean/avigad-qelim/qelim-b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60/lia/cooper/qe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3180457648615999}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes Hölzl, Simon Hudon, Kenny Lau\n-/\nimport data.multiset.bind\nimport control.traversable.lemmas\nimport control.traversable.instances\n\n/-!\n# Functoriality of `multiset`.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nuniverses u\n\nnamespace multiset\n\nopen list\n\ninstance : functor multiset :=\n{ map := @map }\n\n@[simp] lemma fmap_def {α' β'} {s : multiset α'} (f : α' → β') : f <$> s = s.map f := rfl\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u → Type u} [applicative F] [is_comm_applicative F]\nvariables {α' β' : Type u} (f : α' → F β')\n\ndef traverse : multiset α' → F (multiset β') :=\nquotient.lift (functor.map coe ∘ traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.cons\n  { have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l₁) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l₂),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap\n  { have : (λa b (l:list β'), (↑(a :: b :: l) : multiset β')) <$> f p_y <*> f p_x =\n      (λa b l, ↑(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(∘), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\ninstance : monad multiset :=\n{ pure := λ α x, {x},\n  bind := @bind,\n  .. multiset.functor }\n\n@[simp] lemma pure_def {α} : (pure : α → multiset α) = singleton := rfl\n@[simp] lemma bind_def {α β} : (>>=) = @bind α β := rfl\n\ninstance : is_lawful_monad multiset :=\n{ bind_pure_comp_eq_map := λ α β f s, multiset.induction_on s rfl $ λ a s ih, by simp,\n  pure_bind := λ α β x f, by simp [pure],\n  bind_assoc := @bind_assoc }\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_coe {α β : Type*} (x : list α) (f : list α → β)\n  (h : ∀ a b : list α, a ≈ b → f a = f b) :\n  quotient.lift f h (x : multiset α) = f x :=\nquotient.lift_mk _ _ _\n\n@[simp]\nlemma map_comp_coe {α β} (h : α → β) :\n  functor.map h ∘ coe = (coe ∘ functor.map h : list α → multiset β) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {α : Type*} (x : multiset α) :\n  traverse id.mk x = x :=\nquotient.induction_on x begin intro, simp [traverse], refl end\n\nlemma comp_traverse {G H : Type* → Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {α β γ : Type*}\n               (g : α → G β) (h : β → H γ) (x : multiset α) :\n  traverse (comp.mk ∘ functor.map h ∘ g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(∘)] with functor_norm)\n\nlemma map_traverse {G : Type* → Type*}\n               [applicative G] [is_comm_applicative G]\n               {α β γ : Type*}\n               (g : α → G β) (h : β → γ)\n               (x : multiset α) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h ∘ g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [is_lawful_functor.comp_map, map_traverse])\n\nlemma traverse_map {G : Type* → Type*}\n               [applicative G] [is_comm_applicative G]\n               {α β γ : Type*}\n               (g : α → β) (h : β → G γ)\n               (x : multiset α) :\n  traverse h (map g x) =\n  traverse (h ∘ g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [← traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* → Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {α β : Type*} (f : α → G β) (x : multiset α) :\n  eta (traverse f x) = traverse (@eta _ ∘ f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nend multiset\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/multiset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3178205775218163}}
{"text": "/-\nCopyright (c) 2019 Robert A. Spencer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert A. Spencer, Markus Himmel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Group.basic\nimport Mathlib.category_theory.concrete_category.default\nimport Mathlib.category_theory.limits.shapes.kernels\nimport Mathlib.category_theory.preadditive.default\nimport Mathlib.linear_algebra.basic\nimport Mathlib.PostPort\n\nuniverses v u l u_1 \n\nnamespace Mathlib\n\n/-- The category of R-modules and their morphisms. -/\nstructure Module (R : Type u) [ring R] where\n  carrier : Type v\n  is_add_comm_group : add_comm_group carrier\n  is_module : module R carrier\n\nnamespace Module\n\n\n-- TODO revisit this after #1438 merges, to check coercions and instances are handled consistently\n\nprotected instance has_coe_to_sort (R : Type u) [ring R] : has_coe_to_sort (Module R) :=\n  has_coe_to_sort.mk (Type v) carrier\n\nprotected instance category_theory.category (R : Type u) [ring R] :\n    category_theory.category (Module R) :=\n  category_theory.category.mk\n\nprotected instance category_theory.concrete_category (R : Type u) [ring R] :\n    category_theory.concrete_category (Module R) :=\n  category_theory.concrete_category.mk\n    (category_theory.functor.mk (fun (R_1 : Module R) => ↥R_1)\n      fun (R_1 S : Module R) (f : R_1 ⟶ S) => ⇑f)\n\nprotected instance has_forget_to_AddCommGroup (R : Type u) [ring R] :\n    category_theory.has_forget₂ (Module R) AddCommGroup :=\n  category_theory.has_forget₂.mk\n    (category_theory.functor.mk (fun (M : Module R) => AddCommGroup.of ↥M)\n      fun (M₁ M₂ : Module R) (f : M₁ ⟶ M₂) => linear_map.to_add_monoid_hom f)\n\n/-- The object in the category of R-modules associated to an R-module -/\ndef of (R : Type u) [ring R] (X : Type v) [add_comm_group X] [module R X] : Module R := mk X\n\nprotected instance inhabited (R : Type u) [ring R] : Inhabited (Module R) :=\n  { default := of R PUnit }\n\n@[simp] theorem coe_of (R : Type u) [ring R] (X : Type u) [add_comm_group X] [module R X] :\n    ↥(of R X) = X :=\n  rfl\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nmodule. -/\n@[simp] theorem of_self_iso_inv {R : Type u} [ring R] (M : Module R) :\n    category_theory.iso.inv (of_self_iso M) = 𝟙 :=\n  Eq.refl (category_theory.iso.inv (of_self_iso M))\n\nprotected instance of.subsingleton {R : Type u} [ring R] : subsingleton ↥(of R PUnit) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (subsingleton ↥(of R PUnit))) (coe_of R PUnit)))\n    punit.subsingleton\n\nprotected instance category_theory.limits.has_zero_object {R : Type u} [ring R] :\n    category_theory.limits.has_zero_object (Module R) :=\n  category_theory.limits.has_zero_object.mk (of R PUnit)\n    (fun (X : Module R) => unique.mk { default := 0 } sorry)\n    fun (X : Module R) => unique.mk { default := 0 } sorry\n\n@[simp] theorem id_apply {R : Type u} [ring R] {M : Module R} (m : ↥M) : coe_fn 𝟙 m = m := rfl\n\n@[simp] theorem coe_comp {R : Type u} [ring R] {M : Module R} {N : Module R} {U : Module R}\n    (f : M ⟶ N) (g : N ⟶ U) : ⇑(f ≫ g) = ⇑g ∘ ⇑f :=\n  rfl\n\nend Module\n\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom {R : Type u} [ring R] {X₁ : Type v} {X₂ : Type v} [add_comm_group X₁]\n    [module R X₁] [add_comm_group X₂] [module R X₂] :\n    linear_map R X₁ X₂ → (Module.of R X₁ ⟶ Module.of R X₂) :=\n  id\n\n/-- Build an isomorphism in the category `Module R` from a `linear_equiv` between `module`s. -/\n@[simp] theorem linear_equiv.to_Module_iso_inv {R : Type u} [ring R] {X₁ : Type v} {X₂ : Type v}\n    {g₁ : add_comm_group X₁} {g₂ : add_comm_group X₂} {m₁ : module R X₁} {m₂ : module R X₂}\n    (e : linear_equiv R X₁ X₂) :\n    category_theory.iso.inv (linear_equiv.to_Module_iso e) = ↑(linear_equiv.symm e) :=\n  Eq.refl (category_theory.iso.inv (linear_equiv.to_Module_iso e))\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\ndef linear_equiv.to_Module_iso' {R : Type u} [ring R] {M : Module R} {N : Module R}\n    (i : linear_equiv R ↥M ↥N) : M ≅ N :=\n  category_theory.iso.mk ↑i ↑(linear_equiv.symm i)\n\nnamespace category_theory.iso\n\n\n/-- Build a `linear_equiv` from an isomorphism in the category `Module R`. -/\n@[simp] theorem to_linear_equiv_apply {R : Type u} [ring R] {X : Module R} {Y : Module R}\n    (i : X ≅ Y) : ∀ (ᾰ : ↥X), coe_fn (to_linear_equiv i) ᾰ = coe_fn (hom i) ᾰ :=\n  fun (ᾰ : ↥X) => Eq.refl (coe_fn (to_linear_equiv i) ᾰ)\n\nend category_theory.iso\n\n\n/-- linear equivalences between `module`s are the same as (isomorphic to) isomorphisms\nin `Module` -/\n@[simp] theorem linear_equiv_iso_Module_iso_hom {R : Type u} [ring R] {X : Type u} {Y : Type u}\n    [add_comm_group X] [add_comm_group Y] [module R X] [module R Y] (e : linear_equiv R X Y) :\n    category_theory.iso.hom linear_equiv_iso_Module_iso e = linear_equiv.to_Module_iso e :=\n  Eq.refl (category_theory.iso.hom linear_equiv_iso_Module_iso e)\n\nnamespace Module\n\n\nprotected instance category_theory.preadditive {R : Type u} [ring R] :\n    category_theory.preadditive (Module R) :=\n  category_theory.preadditive.mk\n\ntheorem ker_eq_bot_of_mono {R : Type u} [ring R] {M : Module R} {N : Module R} (f : M ⟶ N)\n    [category_theory.mono f] : linear_map.ker f = ⊥ :=\n  linear_map.ker_eq_bot_of_cancel\n    fun (u v : linear_map R ↥(linear_map.ker f) ↥M) => iff.mp (category_theory.cancel_mono f)\n\ntheorem range_eq_top_of_epi {R : Type u} [ring R] {M : Module R} {N : Module R} (f : M ⟶ N)\n    [category_theory.epi f] : linear_map.range f = ⊤ :=\n  linear_map.range_eq_top_of_cancel\n    fun (u v : linear_map R (↥N) (submodule.quotient (linear_map.range f))) =>\n      iff.mp (category_theory.cancel_epi f)\n\ntheorem mono_of_ker_eq_bot {R : Type u} [ring R] {M : Module R} {N : Module R} (f : M ⟶ N)\n    (hf : linear_map.ker f = ⊥) : category_theory.mono f :=\n  category_theory.concrete_category.mono_of_injective f (iff.mp linear_map.ker_eq_bot hf)\n\ntheorem epi_of_range_eq_top {R : Type u} [ring R] {M : Module R} {N : Module R} (f : M ⟶ N)\n    (hf : linear_map.range f = ⊤) : category_theory.epi f :=\n  category_theory.concrete_category.epi_of_surjective f (iff.mp linear_map.range_eq_top hf)\n\nend Module\n\n\nprotected instance Module.has_coe {R : Type u} [ring R] (M : Type u) [add_comm_group M]\n    [module R M] : has_coe (submodule R M) (Module R) :=\n  has_coe.mk fun (N : submodule R M) => Module.of R ↥N\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Module/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3178205775218163}}
{"text": "/-\nCopyright (c) 2021 Kalle Kytölä. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kalle Kytölä\n-/\nimport tactic \nimport measure_theory.measurable_space\nimport measure_theory.integration\nimport measure_theory.borel_space\nimport topology.metric_space.basic\nimport topology.instances.real\nimport topology.instances.ennreal\nimport topology.instances.nnreal\nimport order.liminf_limsup\nimport portmanteau_comeonlean_lemmas\nimport portmanteau_definitions\nimport portmanteau_limsup_lemmas\nimport portmanteau_proba_lemmas\nimport portmanteau_integrals\nimport portmanteau_topological_lemmas\nimport portmanteau_metric_lemmas\n\n\n\nnoncomputable theory\nopen set \nopen classical\nopen measure_theory\nopen measurable_space\nopen metric_space\nopen metric\nopen borel_space\nopen filter\nopen order\nopen_locale topological_space ennreal big_operators\n\n\nnamespace portmanteau\n\nsection portmanteau_continuous_implies_closed\n\n\n\nstructure ptwise_decr_bdd_cont_lim_ennreal {β : Type*} [topological_space β] (f : β → ennreal) \n    extends (ptwise_decr_lim_ennreal f) :=\n  (cont : ∀ (n : ℕ) , continuous (funseq(n)) )\n  (bdd : bdd_ennval (funseq(0)) )\n\n\ndef mble_of_ptwise_decr_bdd_cont_lim_ennreal {β : Type*} [topological_space β] {f : β → ennreal}\n  (fs : ptwise_decr_bdd_cont_lim_ennreal f) : (@ptwise_decr_mble_lim_ennreal β (borel(β)) f) :=\n{\n  funseq := fs.funseq ,\n  decr := fs.decr ,\n  limit := fs.limit ,\n  mble := begin\n    intros n ,\n    exact continuous.borel_measurable (fs.cont n) ,\n  end ,\n}\n\n\nlemma bdd_ennval_of_ptwise_decr_bdd_cont_lim_ennreal\n  {β : Type*} [topological_space β] {f : β → ennreal}\n  (fs : ptwise_decr_bdd_cont_lim_ennreal f) :\n    ∀ (n : ℕ) , bdd_ennval (fs.funseq(n)) :=\nbegin\n  intros n ,\n  have zero_le_n : 0 ≤ n := dec_trivial ,\n  have le := is_decreasing_seq_iff.mp fs.decr 0 n zero_le_n ,\n  exact bdd_ennval_of_le_bdd_ennval le fs.bdd ,\nend\n\n\ndef pw_lin_on_thickening {α : Type*} [metric_space α]\n  (δ : ℝ) (δpos : 0 < δ) (F : set α) : α → ennreal :=\n    λ (x : α) , (1:ennreal) - ennreal.of_real((inf_dist x F)/δ) \n\n\nexample {β : Type*} [topological_space β] (f g : β → ennreal)\n  (hf : continuous f) (hg : continuous g) :\n    continuous (f+g) :=\nbegin\n  exact continuous.add hf hg ,\nend\n\n\nexample {β : Type*} [topological_space β] (f g : β → ℝ)\n  (hf : continuous f) (hg : continuous g) :\n    continuous (f-g) :=\nbegin\n  exact continuous.sub hf hg ,\nend\n\n\nlemma ennreal_of_real_cont : continuous (λ (x : ℝ) , ennreal.of_real x) :=\nbegin\n  exact ennreal.continuous_of_real ,\nend\n\n\nlemma continuous_sub_ennreal {β : Type*} [topological_space β] (f g : β → ennreal)\n  (hf : continuous f) (hg : continuous g) (hg_fin : ∀ x , g(x) ≠ ⊤ ) :\n    continuous (f-g) :=\nbegin\n  have sub_cont_on := continuous_sub_ennreal_ennreal_snd_ne_top ,\n  set sub := (λ p : ennreal × ennreal, p.1 - p.2) with h_sub ,\n  set fg : β → ennreal × ennreal := λ (x : β) , ⟨ f(x) , g(x) ⟩ with h_fg ,\n  have eq : f-g = sub ∘ fg := by refl ,\n  rw eq ,\n  exact continuous_on.comp_continuous sub_cont_on (continuous.prod_mk hf hg) hg_fin ,\nend\n\n\nlemma continuous_add_ennreal {β : Type*} [topological_space β] (f g : β → ennreal)\n  (hf : continuous f) (hg : continuous g) :\n    continuous (f+g) :=\nbegin\n  exact continuous.add hf hg ,\nend\n\n\nlemma pw_lin_on_thickening_cont {α : Type*} [metric_space α]\n  (δ : ℝ) (δpos : 0 < δ) (F : set α) :\n    continuous (pw_lin_on_thickening δ δpos F) :=\nbegin\n  apply continuous_sub_ennreal ,\n  { exact continuous_const , } , \n  { have cont : continuous (λ x , (inf_dist x F)/δ) := continuous.div_const (continuous_inf_dist_pt F) ,\n    exact cont_enn_of_cont_R (λ (x : α), inf_dist x F / δ) cont , } ,\n  { intros x ,\n    exact ennreal.of_real_ne_top , } ,\nend\n\n\nlemma indic_le_pw_lin_on_thickening {α : Type*} [metric_space α]\n  (δ : ℝ) (δpos : 0 < δ) (F : set α) :\n    indic F ≤ pw_lin_on_thickening δ δpos F  :=\nbegin\n  intros x ,\n  by_cases hxF : x ∈ F ,\n  { unfold pw_lin_on_thickening ,\n    rw inf_dist_zero_of_mem hxF ,\n    simp only [ennreal.sub_zero, zero_div, ennreal.of_real_zero] ,\n    exact indic_le_one F x , } ,\n  { have indiczero : indic F x = 0 := (indic_val_zero_iff F x).mpr hxF ,\n    simp [indiczero] , } ,\nend\n\n\nlemma pw_lin_on_thickening_le_indic_thickening {α : Type*} [metric_space α]\n  (δ : ℝ) (δpos : 0 < δ) (F : set α) :\n    pw_lin_on_thickening δ δpos F ≤ indic (thickening_o δ F) :=\nbegin\n  intros x ,\n  by_cases hxFth : x ∈ thickening_o δ F ,\n  { have indicone := (indic_val_one_iff (thickening_o δ F) x).mpr hxFth ,\n    unfold pw_lin_on_thickening ,\n    rw indicone ,\n    simp ,\n    exact le_self_add_ennreal _ _ , } ,\n  { have dist_ge : δ ≤ inf_dist x F ,\n    { simp at hxFth ,\n      exact hxFth , } ,\n    have dist_per_delta_ge : 1 ≤ (inf_dist x F) / δ ,\n    { have key := (div_le_div_right δpos).mpr dist_ge ,\n      rwa div_self (ne_of_gt δpos) at key , } ,\n    have dist_per_delta_ge' : (1 : nnreal) ≤ ((inf_dist x F) / δ).to_nnreal ,\n    { have key := real.to_nnreal_mono dist_per_delta_ge ,\n      rwa real.to_nnreal_one at key , } ,\n    have dist_per_delta_ge'' : (1 : ennreal) ≤ ennreal.of_real((inf_dist x F) / δ) ,\n    { exact ennreal.coe_le_coe.mpr dist_per_delta_ge' , } , \n    have pw_lin_zero : pw_lin_on_thickening δ δpos F x = 0 ,\n    { unfold pw_lin_on_thickening ,\n      apply sub_larger_ennreal _ _ dist_per_delta_ge'' , } ,\n    rw pw_lin_zero ,\n    simp only [zero_le] , } , \nend\n\n\nlemma measure_of_cont_bdd_decr_approx_indicator\n  {β : Type*} [topo_β : topological_space β]\n  {E : set β} (hE : (borel β).measurable_set' E)\n  (fseq : ptwise_decr_bdd_cont_lim_ennreal (indic E))\n  (μ : @measure_theory.measure β (borel β))\n  [hfin : @probability_measure β (borel(β)) μ] :\n    lim_enn (λ n , (@lintegral β (borel(β)) μ (fseq.funseq(n)))) (μ(E)) :=\nbegin\n  have finite_integral : @lintegral β (borel(β)) μ  (fseq.funseq(0)) < ⊤\n    := finite_integral_of_bdd_ennrealval μ (fseq.funseq(0)) fseq.bdd , \n  have mble_funs : ∀ n , @measurable β ennreal (borel(β)) _ (fseq.funseq(n)) ,\n  { intros n ,\n    exact continuous.borel_measurable (fseq.cont n) , } ,\n  exact @measure_of_mble_decr_approx_indicator β (borel(β)) μ E hE (mble_of_ptwise_decr_bdd_cont_lim_ennreal fseq) finite_integral ,\nend\n\n\n@[class] def has_cont_decr_approx_of_closed (β : Type*) [topo_β : topological_space β] : Prop :=\n  ∀ (F : set β) , is_closed F → ( ∃ (fseq : ptwise_decr_bdd_cont_lim_ennreal (indic F)) , true )\n\n\nlemma one_div_nat_decr : is_decreasing_seq (λ (n : ℕ) , (1/(n+1) : ℝ)) :=\nbegin\n  intros n m hnm ,\n  exact @nat.one_div_le_one_div ℝ _ n m hnm ,\nend\n\n\nlemma one_div_nat_lim : lim_R (λ (n : ℕ) , (1/(n+1) : ℝ)) 0 :=\nbegin\n  exact tendsto_one_div_add_at_top_nhds_0_nat ,\nend\n\n\nprivate def metric_ptwise_decr_bdd_cont_approx_of_nonempty_closed\n  (α : Type*) [met_α : metric_space α] (F : set α) (Fclos : is_closed F) (Fnonemp : F.nonempty) :\n    ptwise_decr_bdd_cont_lim_ennreal (indic F) :=\n{\n  funseq := λ (n : ℕ) , pw_lin_on_thickening (1/(n+1) : ℝ) (nat.one_div_pos_of_nat) F ,\n  decr := begin\n    intros n m hnm x ,\n    dsimp ,\n    unfold pw_lin_on_thickening ,\n    suffices : ennreal.of_real ((inf_dist x F) / (1/(n+1) : ℝ)) ≤ ennreal.of_real ((inf_dist x F) / (1/(m+1) : ℝ)) ,\n    { apply self_sub_le_self_sub_ennreal ,\n      exact this , } ,\n    apply ennreal.coe_le_coe.mpr ,\n    apply real.to_nnreal_mono ,\n    have eq : ∀ (k : ℕ) , (1/(k+1) : ℝ)⁻¹  = (k+1) := by simp ,\n    have eq' : ∀ (k : ℕ) , ∀ (x : ℝ) , x / (1/(k+1) : ℝ) = x * (k+1) ,\n    { intros k x ,\n      rw [div_eq_mul_inv, eq k] , } ,\n    rw [eq' n (inf_dist x F) , eq' m (inf_dist x F) ] ,\n    have hnm' : ((n:ℝ)+1) ≤ ((m:ℝ)+1) := by simp [hnm] ,\n    exact @mul_le_mul _ _ (inf_dist x F) _ (inf_dist x F) _ (rfl.ge) hnm' (by tidy) (inf_dist_nonneg) ,\n  end ,\n  limit := begin\n    set s := λ (n : ℕ) , pw_lin_on_thickening (1/(n+1) : ℝ) (nat.one_div_pos_of_nat) F with hs ,\n    intros x ,\n    have seq_le : ∀ n , s n x ≤ indic (thickening_o (1/(n+1) : ℝ) F) x \n      := λ n , pw_lin_on_thickening_le_indic_thickening (1/(n+1) : ℝ) (nat.one_div_pos_of_nat) F x , \n    have seq_ge : ∀ n , indic F x ≤ s n x \n      := λ n , indic_le_pw_lin_on_thickening (1/(n+1) : ℝ) (nat.one_div_pos_of_nat) F x , \n    have lim_ub : lim_enn (λ n , indic (thickening_o (1/(n+1) : ℝ) F) x) (indic F x) ,\n    { set s' := approx_indicator_seq_thickening_o F Fnonemp\n          (λ n , (1/(n+1) : ℝ)) (λ n , nat.one_div_pos_of_nat) one_div_nat_decr \n          (tendsto_one_div_add_at_top_nhds_0_nat) with hs' ,\n      have key := s'.limit x ,\n      have F_eq_clos_F := is_closed.closure_eq Fclos ,\n      have indic_eq_x : indic F x = indic (closure(F)) x \n        := by refine congr_fun (congr_arg indic (eq.symm F_eq_clos_F)) x ,\n      rwa ← indic_eq_x at key , } , \n    have lim_lb : lim_enn (λ n , indic F x) (indic F x) := tendsto_const_nhds ,\n    have tada := tendsto_of_tendsto_of_tendsto_of_le_of_le lim_lb lim_ub seq_ge seq_le ,\n    exact tada ,\n  end ,\n  cont := begin\n    intros n ,\n    exact pw_lin_on_thickening_cont (1 / (↑n + 1)) nat.one_div_pos_of_nat F ,\n  end ,\n  bdd := begin\n    use 1 ,\n    intros x ,\n    dsimp ,\n    simp only [zero_add, div_one] ,\n    have wow := pw_lin_on_thickening_le_indic_thickening 1 (by linarith) F ,\n    apply le_trans (wow x) _ ,\n    exact indic_le_one _ _ ,\n  end ,\n}\n\n\nprivate def metric_ptwise_decr_bdd_cont_approx_of_empty\n  (α : Type*) [met_α : metric_space α] :\n    ptwise_decr_bdd_cont_lim_ennreal (indic (∅ : set α)) :=\n{\n  funseq := λ (n : ℕ) , 0 ,\n  decr := begin\n    intros n m hnm ,\n    simp only [le_refl] ,\n  end ,\n  limit := begin\n    intros x ,\n    rw (indic_val_zero_iff (∅ : set α) x).mpr (by simp) ,\n    apply tendsto_const_nhds ,\n  end ,\n  cont := begin\n    intros n ,\n    exact @continuous_const α ennreal _ _ 0 ,\n  end ,\n  bdd := begin\n    use 0 ,\n    intros x ,\n    dsimp ,\n    simp only [le_refl] ,\n  end ,\n}\n\n\nprivate def metric_ptwise_decr_bdd_cont_approx_of_closed\n  (α : Type*) [met_α : metric_space α] (F : set α) (Fclos : is_closed F) :\n    ptwise_decr_bdd_cont_lim_ennreal (indic F) :=\nbegin\n  by_cases hF : F = ∅ ,\n  { rw hF ,\n    exact metric_ptwise_decr_bdd_cont_approx_of_empty α , } , \n  { exact metric_ptwise_decr_bdd_cont_approx_of_nonempty_closed α F Fclos (ne_empty_iff_nonempty.mp hF) , } , \nend\n\n\nlemma metric_space_has_cont_approx_of_closed (α : Type*) [met_α : metric_space α] :\n  has_cont_decr_approx_of_closed α :=\nbegin\n  intros F F_clos ,\n  use metric_ptwise_decr_bdd_cont_approx_of_closed α F F_clos ,\nend\n\n\nlemma portmanteau_continuous_imp_closed_cond\n  {β : Type*} {topo_β : topological_space β}\n  [appr_closed : has_cont_decr_approx_of_closed β]\n  (μseq : ℕ → (@measure_theory.measure β (borel β))) (μseq_fin : ∀ n , @probability_measure β (borel(β)) (μseq(n)))\n  (μ : (@measure_theory.measure β (borel β))) (μ_fin : @probability_measure β (borel(β)) μ) : \n    (portmanteau_continuous_ennval μseq μ) → portmanteau_closed μseq μ :=\nbegin\n  intros hcontcond ,\n  intros F hFclosed ,\n  specialize appr_closed F hFclosed ,\n  cases appr_closed with fseq ,\n  suffices : (∀ (ε : nnreal) (ε_pos : 0<ε) ,\n      limsup_enn (λ n , (μseq(n))(F)) ≤ (μ(F)) + ε) ,\n  { apply ennreal.le_of_forall_pos_le_add ,\n    intros ε ε_pos unnecessary ,\n    exact this ε ε_pos , } ,\n  intros ε ε_pos ,\n  have meas_approx' : lim_enn (λ (j : ℕ) , @lintegral β (borel(β)) μ (fseq.funseq(j)) ) (μ(F)) , \n  { apply measure_of_mble_decr_approx_indicator μ F (closed_imp_borel hFclosed)\n          (mble_of_ptwise_decr_bdd_cont_lim_ennreal fseq ) _ ,\n    exact @bdd_integral_of_bdd_ennval β (borel(β)) μ μ_fin (fseq.funseq(0)) fseq.bdd , } , \n  have meas_approx : ∃ (j₀ : ℕ) , @lintegral β (borel(β)) μ (fseq.funseq(j₀)) ≤ (μ(F)) + ε , \n  { change ∃ (j₀ : ℕ) , @lintegral β (borel(β)) μ (fseq.funseq(j₀)) ∈ Iic (μ(F)+ε) ,\n    have lt_plus_eps := lt_add_pos_ennreal (μ(F)) ε (ennreal_lt_top_of_ne_top _ (proba_finite μ F)) (by simp [ε_pos]) ,\n    have nbhd : Iic (μ(F)+ε) ∈ 𝓝(μ(F)) := Iic_mem_nhds lt_plus_eps ,\n    have eventually := (filter.tendsto_at_top'.mp meas_approx') _ nbhd ,\n    cases eventually with j₀ hj₀ ,\n    use j₀ ,\n    exact hj₀ j₀ (by simp only [ge_iff_le]) , } , \n  cases meas_approx with j hj ,\n  have lim_integr :=\n    hcontcond (fseq.funseq(j)) (fseq.cont(j)) (bdd_ennval_of_ptwise_decr_bdd_cont_lim_ennreal fseq j) ,\n  have meas_seq_le : (∀ n , ((μseq(n)) F) ≤ (@lintegral β (borel(β)) (μseq(n)) (fseq.funseq(j)) )) ,\n  { intros n ,\n    have lim_le : indic F ≤ fseq.funseq(j) ,\n    { intros x ,\n      have decr_at_x : is_decreasing_seq (λ (n : ℕ) , ((fseq.funseq(n))(x)) ) ,\n      { intros n m hnm ,\n        dsimp ,\n        exact fseq.decr hnm x , } , \n      have lim_at_x : lim_enn (λ (n : ℕ) , ((fseq.funseq(n))(x)) ) (indic F x) := fseq.limit x ,\n      exact lim_le_of_decr (λ (n : ℕ) , ((fseq.funseq(n))(x)) ) (indic F x) decr_at_x (fseq.limit x) j , } ,\n    have int_mono := @lintegral_mono β (borel(β)) (μseq(n)) _ _ lim_le ,\n    rwa integral_indic (μseq(n)) F (closed_imp_borel hFclosed) at int_mono , } , \n  apply le_trans (limsup_enn_mono meas_seq_le) ,\n  suffices : limsup_enn (λ (n : ℕ), @lintegral β (borel(β)) (μseq(n)) (fseq.to_ptwise_decr_lim_ennreal.funseq j)) = @lintegral β (borel(β)) μ (fseq.to_ptwise_decr_lim_ennreal.funseq j) ,\n  { rwa this , } ,\n  exact tendsto.limsup_eq lim_integr ,\nend\n\n\n\nend portmanteau_continuous_implies_closed\n\nend portmanteau\n", "meta": {"author": "kkytola", "repo": "lean_portmanteau", "sha": "ac55eb4e24be43032cbc082e2b68d8fb8bd63f22", "save_path": "github-repos/lean/kkytola-lean_portmanteau", "path": "github-repos/lean/kkytola-lean_portmanteau/lean_portmanteau-ac55eb4e24be43032cbc082e2b68d8fb8bd63f22/portmanteau_cont_imp_closed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.31782056959019683}}
{"text": "import category_theory.preadditive.additive_functor\nimport algebra.category.Module.basic\nimport algebra.category.Group.preadditive\nimport for_mathlib.is_biprod\n\nopen category_theory category_theory.limits\n\nnamespace category_theory\n\nnamespace functor\n\nvariables {𝒜 ℬ : Type*} [category 𝒜] [category ℬ]\nvariables [preadditive 𝒜] [preadditive ℬ]\nvariables (F : 𝒜 ⥤ ℬ)\n\n\nlemma additive_of_map_fst_add_snd [has_binary_biproducts 𝒜]\n  (h : ∀ A : 𝒜, F.map (biprod.fst + biprod.snd : A ⊞ A ⟶ A) =\n    F.map biprod.fst + F.map biprod.snd) :\n  F.additive :=\n{ map_add' := λ A B f g,\n  begin\n    have : f + g = biprod.lift f g ≫ (biprod.fst + biprod.snd),\n    { rw [preadditive.comp_add, biprod.lift_fst, biprod.lift_snd] },\n    rw [this, F.map_comp, h, preadditive.comp_add, ← F.map_comp, ← F.map_comp,\n      biprod.lift_fst, biprod.lift_snd],\n  end }\n\nnoncomputable\ndef obj_biprod_iso (F : 𝒜 ⥤ ℬ) [F.additive]\n  (A B : 𝒜) [has_binary_biproduct A B] [has_binary_biproduct (F.obj A) (F.obj B)] :\n  F.obj (A ⊞ B) ≅ F.obj A ⊞ F.obj B :=\nis_biprod.iso_biprod _ (F.map_is_biprod _ (biprod.is_biprod A B))\n\ninstance forget₂_additive (R : Type*) [ring R] : (forget₂ (Module R) AddCommGroup).additive :=\n{ map_add' := λ M N f g, rfl }\n\nlemma map_is_zero {X : 𝒜} [F.additive] (hX : limits.is_zero X) :\n  limits.is_zero (F.obj X) :=\nbegin\n  rw limits.is_zero.iff_id_eq_zero at hX ⊢,\n  convert congr_arg (@category_theory.functor.map _ _ _ _ F _ _) hX;\n  simp,\nend\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/additive_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.31780007880337663}}
{"text": "/-\nCopyright (c) 2018 Sébastien Gouëzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sébastien Gouëzel, Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.partial\nimport Mathlib.order.filter.at_top_bot\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# liminfs and limsups of functions and filters\n\nDefines the Liminf/Limsup of a function taking values in a conditionally complete lattice, with\nrespect to an arbitrary filter.\n\nWe define `f.Limsup` (`f.Liminf`) where `f` is a filter taking values in a conditionally complete\nlattice. `f.Limsup` is the smallest element `a` such that, eventually, `u ≤ a` (and vice versa for\n`f.Liminf`). To work with the Limsup along a function `u` use `(f.map u).Limsup`.\n\nUsually, one defines the Limsup as `Inf (Sup s)` where the Inf is taken over all sets in the filter.\nFor instance, in ℕ along a function `u`, this is `Inf_n (Sup_{k ≥ n} u k)` (and the latter quantity\ndecreases with `n`, so this is in fact a limit.). There is however a difficulty: it is well possible\nthat `u` is not bounded on the whole space, only eventually (think of `Limsup (λx, 1/x)` on ℝ. Then\nthere is no guarantee that the quantity above really decreases (the value of the `Sup` beforehand is\nnot really well defined, as one can not use ∞), so that the Inf could be anything. So one can not\nuse this `Inf Sup ...` definition in conditionally complete lattices, and one has to use a less\ntractable definition.\n\nIn conditionally complete lattices, the definition is only useful for filters which are eventually\nbounded above (otherwise, the Limsup would morally be +∞, which does not belong to the space) and\nwhich are frequently bounded below (otherwise, the Limsup would morally be -∞, which is not in the\nspace either). We start with definitions of these concepts for arbitrary filters, before turning to\nthe definitions of Limsup and Liminf.\n\nIn complete lattices, however, it coincides with the `Inf Sup` definition.\n-/\n\nnamespace filter\n\n\n/-- `f.is_bounded (≺)`: the filter `f` is eventually bounded w.r.t. the relation `≺`, i.e.\neventually, it is bounded by some uniform bound.\n`r` will be usually instantiated with `≤` or `≥`. -/\ndef is_bounded {α : Type u_1} (r : α → α → Prop) (f : filter α) :=\n  ∃ (b : α), filter.eventually (fun (x : α) => r x b) f\n\n/-- `f.is_bounded_under (≺) u`: the image of the filter `f` under `u` is eventually bounded w.r.t.\nthe relation `≺`, i.e. eventually, it is bounded by some uniform bound. -/\ndef is_bounded_under {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (f : filter β) (u : β → α) :=\n  is_bounded r (map u f)\n\n/-- `f` is eventually bounded if and only if, there exists an admissible set on which it is\nbounded. -/\ntheorem is_bounded_iff {α : Type u_1} {r : α → α → Prop} {f : filter α} : is_bounded r f ↔ ∃ (s : set α), ∃ (H : s ∈ sets f), ∃ (b : α), s ⊆ set_of fun (x : α) => r x b := sorry\n\n/-- A bounded function `u` is in particular eventually bounded. -/\ntheorem is_bounded_under_of {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : filter β} {u : β → α} : (∃ (b : α), ∀ (x : β), r (u x) b) → is_bounded_under r f u := sorry\n\ntheorem is_bounded_bot {α : Type u_1} {r : α → α → Prop} : is_bounded r ⊥ ↔ Nonempty α := sorry\n\ntheorem is_bounded_top {α : Type u_1} {r : α → α → Prop} : is_bounded r ⊤ ↔ ∃ (t : α), ∀ (x : α), r x t := sorry\n\ntheorem is_bounded_principal {α : Type u_1} {r : α → α → Prop} (s : set α) : is_bounded r (principal s) ↔ ∃ (t : α), ∀ (x : α), x ∈ s → r x t := sorry\n\ntheorem is_bounded_sup {α : Type u_1} {r : α → α → Prop} {f : filter α} {g : filter α} [is_trans α r] (hr : ∀ (b₁ b₂ : α), ∃ (b : α), r b₁ b ∧ r b₂ b) : is_bounded r f → is_bounded r g → is_bounded r (f ⊔ g) := sorry\n\ntheorem is_bounded.mono {α : Type u_1} {r : α → α → Prop} {f : filter α} {g : filter α} (h : f ≤ g) : is_bounded r g → is_bounded r f := sorry\n\ntheorem is_bounded_under.mono {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : filter β} {g : filter β} {u : β → α} (h : f ≤ g) : is_bounded_under r g u → is_bounded_under r f u :=\n  fun (hg : is_bounded_under r g u) => is_bounded.mono (map_mono h) hg\n\ntheorem is_bounded.is_bounded_under {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : filter α} {q : β → β → Prop} {u : α → β} (hf : ∀ (a₀ a₁ : α), r a₀ a₁ → q (u a₀) (u a₁)) : is_bounded r f → is_bounded_under q f u := sorry\n\n/-- `is_cobounded (≺) f` states that the filter `f` does not tend to infinity w.r.t. `≺`. This is\nalso called frequently bounded. Will be usually instantiated with `≤` or `≥`.\n\nThere is a subtlety in this definition: we want `f.is_cobounded` to hold for any `f` in the case of\ncomplete lattices. This will be relevant to deduce theorems on complete lattices from their\nversions on conditionally complete lattices with additional assumptions. We have to be careful in\nthe edge case of the trivial filter containing the empty set: the other natural definition\n  `¬ ∀ a, ∀ᶠ n in f, a ≤ n`\nwould not work as well in this case.\n-/\ndef is_cobounded {α : Type u_1} (r : α → α → Prop) (f : filter α) :=\n  ∃ (b : α), ∀ (a : α), filter.eventually (fun (x : α) => r x a) f → r b a\n\n/-- `is_cobounded_under (≺) f u` states that the image of the filter `f` under the map `u` does not\ntend to infinity w.r.t. `≺`. This is also called frequently bounded. Will be usually instantiated\nwith `≤` or `≥`. -/\ndef is_cobounded_under {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (f : filter β) (u : β → α) :=\n  is_cobounded r (map u f)\n\n/-- To check that a filter is frequently bounded, it suffices to have a witness\nwhich bounds `f` at some point for every admissible set.\n\nThis is only an implication, as the other direction is wrong for the trivial filter.-/\ntheorem is_cobounded.mk {α : Type u_1} {r : α → α → Prop} {f : filter α} [is_trans α r] (a : α) (h : ∀ (s : set α) (H : s ∈ f), ∃ (x : α), ∃ (H : x ∈ s), r a x) : is_cobounded r f := sorry\n\n/-- A filter which is eventually bounded is in particular frequently bounded (in the opposite\ndirection). At least if the filter is not trivial. -/\ntheorem is_bounded.is_cobounded_flip {α : Type u_1} {r : α → α → Prop} {f : filter α} [is_trans α r] [ne_bot f] : is_bounded r f → is_cobounded (flip r) f := sorry\n\ntheorem is_cobounded_bot {α : Type u_1} {r : α → α → Prop} : is_cobounded r ⊥ ↔ ∃ (b : α), ∀ (x : α), r b x := sorry\n\ntheorem is_cobounded_top {α : Type u_1} {r : α → α → Prop} : is_cobounded r ⊤ ↔ Nonempty α := sorry\n\ntheorem is_cobounded_principal {α : Type u_1} {r : α → α → Prop} (s : set α) : is_cobounded r (principal s) ↔ ∃ (b : α), ∀ (a : α), (∀ (x : α), x ∈ s → r x a) → r b a := sorry\n\ntheorem is_cobounded.mono {α : Type u_1} {r : α → α → Prop} {f : filter α} {g : filter α} (h : f ≤ g) : is_cobounded r f → is_cobounded r g := sorry\n\ntheorem is_cobounded_le_of_bot {α : Type u_1} [order_bot α] {f : filter α} : is_cobounded LessEq f :=\n  Exists.intro ⊥ fun (a : α) (h : filter.eventually (fun (x : α) => x ≤ a) f) => bot_le\n\ntheorem is_cobounded_ge_of_top {α : Type u_1} [order_top α] {f : filter α} : is_cobounded ge f :=\n  Exists.intro ⊤ fun (a : α) (h : filter.eventually (fun (x : α) => x ≥ a) f) => le_top\n\ntheorem is_bounded_le_of_top {α : Type u_1} [order_top α] {f : filter α} : is_bounded LessEq f :=\n  Exists.intro ⊤ (eventually_of_forall fun (_x : α) => le_top)\n\ntheorem is_bounded_ge_of_bot {α : Type u_1} [order_bot α] {f : filter α} : is_bounded ge f :=\n  Exists.intro ⊥ (eventually_of_forall fun (_x : α) => bot_le)\n\ntheorem is_bounded_under_sup {α : Type u_1} {β : Type u_2} [semilattice_sup α] {f : filter β} {u : β → α} {v : β → α} : is_bounded_under LessEq f u → is_bounded_under LessEq f v → is_bounded_under LessEq f fun (a : β) => u a ⊔ v a := sorry\n\ntheorem is_bounded_under_inf {α : Type u_1} {β : Type u_2} [semilattice_inf α] {f : filter β} {u : β → α} {v : β → α} : is_bounded_under ge f u → is_bounded_under ge f v → is_bounded_under ge f fun (a : β) => u a ⊓ v a := sorry\n\n/-- Filters are automatically bounded or cobounded in complete lattices. To use the same statements\nin complete and conditionally complete lattices but let automation fill automatically the\nboundedness proofs in complete lattices, we use the tactic `is_bounded_default` in the statements,\nin the form `(hf : f.is_bounded (≥) . is_bounded_default)`. -/\n/-- The `Limsup` of a filter `f` is the infimum of the `a` such that, eventually for `f`,\nholds `x ≤ a`. -/\ndef Limsup {α : Type u_1} [conditionally_complete_lattice α] (f : filter α) : α :=\n  Inf (set_of fun (a : α) => filter.eventually (fun (n : α) => n ≤ a) f)\n\n/-- The `Liminf` of a filter `f` is the supremum of the `a` such that, eventually for `f`,\nholds `x ≥ a`. -/\ndef Liminf {α : Type u_1} [conditionally_complete_lattice α] (f : filter α) : α :=\n  Sup (set_of fun (a : α) => filter.eventually (fun (n : α) => a ≤ n) f)\n\n/-- The `limsup` of a function `u` along a filter `f` is the infimum of the `a` such that,\neventually for `f`, holds `u x ≤ a`. -/\ndef limsup {α : Type u_1} {β : Type u_2} [conditionally_complete_lattice α] (f : filter β) (u : β → α) : α :=\n  Limsup (map u f)\n\n/-- The `liminf` of a function `u` along a filter `f` is the supremum of the `a` such that,\neventually for `f`, holds `u x ≥ a`. -/\ndef liminf {α : Type u_1} {β : Type u_2} [conditionally_complete_lattice α] (f : filter β) (u : β → α) : α :=\n  Liminf (map u f)\n\ntheorem limsup_eq {α : Type u_1} {β : Type u_2} [conditionally_complete_lattice α] {f : filter β} {u : β → α} : limsup f u = Inf (set_of fun (a : α) => filter.eventually (fun (n : β) => u n ≤ a) f) :=\n  rfl\n\ntheorem liminf_eq {α : Type u_1} {β : Type u_2} [conditionally_complete_lattice α] {f : filter β} {u : β → α} : liminf f u = Sup (set_of fun (a : α) => filter.eventually (fun (n : β) => a ≤ u n) f) :=\n  rfl\n\ntheorem Limsup_le_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {a : α} (hf : autoParam (is_cobounded LessEq f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : filter.eventually (fun (n : α) => n ≤ a) f) : Limsup f ≤ a :=\n  cInf_le hf h\n\ntheorem le_Liminf_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {a : α} (hf : autoParam (is_cobounded ge f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : filter.eventually (fun (n : α) => a ≤ n) f) : a ≤ Liminf f :=\n  le_cSup hf h\n\ntheorem le_Limsup_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {a : α} (hf : autoParam (is_bounded LessEq f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : ∀ (b : α), filter.eventually (fun (n : α) => n ≤ b) f → a ≤ b) : a ≤ Limsup f :=\n  le_cInf hf h\n\ntheorem Liminf_le_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {a : α} (hf : autoParam (is_bounded ge f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : ∀ (b : α), filter.eventually (fun (n : α) => b ≤ n) f → b ≤ a) : Liminf f ≤ a :=\n  cSup_le hf h\n\ntheorem Liminf_le_Limsup {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} [ne_bot f] (h₁ : autoParam (is_bounded LessEq f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h₂ : autoParam (is_bounded ge f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : Liminf f ≤ Limsup f := sorry\n\ntheorem Liminf_le_Liminf {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {g : filter α} (hf : autoParam (is_bounded ge f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hg : autoParam (is_cobounded ge g)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : ∀ (a : α), filter.eventually (fun (n : α) => a ≤ n) f → filter.eventually (fun (n : α) => a ≤ n) g) : Liminf f ≤ Liminf g :=\n  cSup_le_cSup hg hf h\n\ntheorem Limsup_le_Limsup {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {g : filter α} (hf : autoParam (is_cobounded LessEq f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hg : autoParam (is_bounded LessEq g)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (h : ∀ (a : α), filter.eventually (fun (n : α) => n ≤ a) g → filter.eventually (fun (n : α) => n ≤ a) f) : Limsup f ≤ Limsup g :=\n  cInf_le_cInf hf hg h\n\ntheorem Limsup_le_Limsup_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {g : filter α} (h : f ≤ g) (hf : autoParam (is_cobounded LessEq f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hg : autoParam (is_bounded LessEq g)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : Limsup f ≤ Limsup g :=\n  Limsup_le_Limsup fun (a : α) (ha : filter.eventually (fun (n : α) => n ≤ a) g) => h ha\n\ntheorem Liminf_le_Liminf_of_le {α : Type u_1} [conditionally_complete_lattice α] {f : filter α} {g : filter α} (h : g ≤ f) (hf : autoParam (is_bounded ge f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hg : autoParam (is_cobounded ge g)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : Liminf f ≤ Liminf g :=\n  Liminf_le_Liminf fun (a : α) (ha : filter.eventually (fun (n : α) => a ≤ n) f) => h ha\n\ntheorem limsup_le_limsup {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} {u : α → β} {v : α → β} (h : eventually_le f u v) (hu : autoParam (is_cobounded_under LessEq f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hv : autoParam (is_bounded_under LessEq f v)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : limsup f u ≤ limsup f v :=\n  Limsup_le_Limsup fun (b : β) => eventually_le.trans h\n\ntheorem liminf_le_liminf {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} {u : α → β} {v : α → β} (h : filter.eventually (fun (a : α) => u a ≤ v a) f) (hu : autoParam (is_bounded_under ge f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hv : autoParam (is_cobounded_under ge f v)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : liminf f u ≤ liminf f v :=\n  limsup_le_limsup h\n\ntheorem Limsup_principal {α : Type u_1} [conditionally_complete_lattice α] {s : set α} (h : bdd_above s) (hs : set.nonempty s) : Limsup (principal s) = Sup s := sorry\n\ntheorem Liminf_principal {α : Type u_1} [conditionally_complete_lattice α] {s : set α} (h : bdd_below s) (hs : set.nonempty s) : Liminf (principal s) = Inf s :=\n  Limsup_principal h hs\n\ntheorem limsup_congr {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} {u : α → β} {v : α → β} (h : filter.eventually (fun (a : α) => u a = v a) f) : limsup f u = limsup f v := sorry\n\ntheorem liminf_congr {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} {u : α → β} {v : α → β} (h : filter.eventually (fun (a : α) => u a = v a) f) : liminf f u = liminf f v :=\n  limsup_congr h\n\ntheorem limsup_const {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} [ne_bot f] (b : β) : (limsup f fun (x : α) => b) = b := sorry\n\ntheorem liminf_const {β : Type u_2} {α : Type u_1} [conditionally_complete_lattice β] {f : filter α} [ne_bot f] (b : β) : (liminf f fun (x : α) => b) = b :=\n  limsup_const b\n\n@[simp] theorem Limsup_bot {α : Type u_1} [complete_lattice α] : Limsup ⊥ = ⊥ := sorry\n\n@[simp] theorem Liminf_bot {α : Type u_1} [complete_lattice α] : Liminf ⊥ = ⊤ := sorry\n\n@[simp] theorem Limsup_top {α : Type u_1} [complete_lattice α] : Limsup ⊤ = ⊤ := sorry\n\n@[simp] theorem Liminf_top {α : Type u_1} [complete_lattice α] : Liminf ⊤ = ⊥ := sorry\n\n/-- Same as limsup_const applied to `⊥` but without the `ne_bot f` assumption -/\ntheorem limsup_const_bot {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : filter β} : (limsup f fun (x : β) => ⊥) = ⊥ := sorry\n\n/-- Same as limsup_const applied to `⊤` but without the `ne_bot f` assumption -/\ntheorem liminf_const_top {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : filter β} : (liminf f fun (x : β) => ⊤) = ⊤ :=\n  limsup_const_bot\n\ntheorem liminf_le_limsup {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : filter β} [ne_bot f] {u : β → α} : liminf f u ≤ limsup f u :=\n  Liminf_le_Limsup\n\ntheorem has_basis.Limsup_eq_infi_Sup {α : Type u_1} [complete_lattice α] {ι : Type u_2} {p : ι → Prop} {s : ι → set α} {f : filter α} (h : has_basis f p s) : Limsup f = infi fun (i : ι) => infi fun (hi : p i) => Sup (s i) := sorry\n\ntheorem has_basis.Liminf_eq_supr_Inf {α : Type u_1} {ι : Type u_3} [complete_lattice α] {p : ι → Prop} {s : ι → set α} {f : filter α} (h : has_basis f p s) : Liminf f = supr fun (i : ι) => supr fun (hi : p i) => Inf (s i) :=\n  has_basis.Limsup_eq_infi_Sup h\n\ntheorem Limsup_eq_infi_Sup {α : Type u_1} [complete_lattice α] {f : filter α} : Limsup f = infi fun (s : set α) => infi fun (H : s ∈ f) => Sup s :=\n  has_basis.Limsup_eq_infi_Sup (basis_sets f)\n\ntheorem Liminf_eq_supr_Inf {α : Type u_1} [complete_lattice α] {f : filter α} : Liminf f = supr fun (s : set α) => supr fun (H : s ∈ f) => Inf s :=\n  Limsup_eq_infi_Sup\n\n/-- In a complete lattice, the limsup of a function is the infimum over sets `s` in the filter\nof the supremum of the function over `s` -/\ntheorem limsup_eq_infi_supr {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : filter β} {u : β → α} : limsup f u = infi fun (s : set β) => infi fun (H : s ∈ f) => supr fun (a : β) => supr fun (H : a ∈ s) => u a := sorry\n\ntheorem limsup_eq_infi_supr_of_nat {α : Type u_1} [complete_lattice α] {u : ℕ → α} : limsup at_top u = infi fun (n : ℕ) => supr fun (i : ℕ) => supr fun (H : i ≥ n) => u i := sorry\n\ntheorem limsup_eq_infi_supr_of_nat' {α : Type u_1} [complete_lattice α] {u : ℕ → α} : limsup at_top u = infi fun (n : ℕ) => supr fun (i : ℕ) => u (i + n) := sorry\n\ntheorem has_basis.limsup_eq_infi_supr {α : Type u_1} {β : Type u_2} {ι : Type u_3} [complete_lattice α] {p : ι → Prop} {s : ι → set β} {f : filter β} {u : β → α} (h : has_basis f p s) : limsup f u = infi fun (i : ι) => infi fun (hi : p i) => supr fun (a : β) => supr fun (H : a ∈ s i) => u a := sorry\n\n/-- In a complete lattice, the liminf of a function is the infimum over sets `s` in the filter\nof the supremum of the function over `s` -/\ntheorem liminf_eq_supr_infi {α : Type u_1} {β : Type u_2} [complete_lattice α] {f : filter β} {u : β → α} : liminf f u = supr fun (s : set β) => supr fun (H : s ∈ f) => infi fun (a : β) => infi fun (H : a ∈ s) => u a :=\n  limsup_eq_infi_supr\n\ntheorem liminf_eq_supr_infi_of_nat {α : Type u_1} [complete_lattice α] {u : ℕ → α} : liminf at_top u = supr fun (n : ℕ) => infi fun (i : ℕ) => infi fun (H : i ≥ n) => u i :=\n  limsup_eq_infi_supr_of_nat\n\ntheorem liminf_eq_supr_infi_of_nat' {α : Type u_1} [complete_lattice α] {u : ℕ → α} : liminf at_top u = supr fun (n : ℕ) => infi fun (i : ℕ) => u (i + n) :=\n  limsup_eq_infi_supr_of_nat'\n\ntheorem has_basis.liminf_eq_supr_infi {α : Type u_1} {β : Type u_2} {ι : Type u_3} [complete_lattice α] {p : ι → Prop} {s : ι → set β} {f : filter β} {u : β → α} (h : has_basis f p s) : liminf f u = supr fun (i : ι) => supr fun (hi : p i) => infi fun (a : β) => infi fun (H : a ∈ s i) => u a :=\n  has_basis.limsup_eq_infi_supr h\n\ntheorem eventually_lt_of_lt_liminf {α : Type u_1} {β : Type u_2} {f : filter α} [conditionally_complete_linear_order β] {u : α → β} {b : β} (h : b < liminf f u) (hu : autoParam (is_bounded_under ge f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : filter.eventually (fun (a : α) => b < u a) f := sorry\n\ntheorem eventually_lt_of_limsup_lt {α : Type u_1} {β : Type u_2} {f : filter α} [conditionally_complete_linear_order β] {u : α → β} {b : β} (h : limsup f u < b) (hu : autoParam (is_bounded_under LessEq f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : filter.eventually (fun (a : α) => u a < b) f :=\n  eventually_lt_of_lt_liminf h\n\ntheorem le_limsup_of_frequently_le {α : Type u_1} {β : Type u_2} [conditionally_complete_linear_order β] {f : filter α} {u : α → β} {b : β} (hu_le : filter.frequently (fun (x : α) => b ≤ u x) f) (hu : autoParam (is_bounded_under LessEq f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : b ≤ limsup f u := sorry\n\ntheorem liminf_le_of_frequently_le {α : Type u_1} {β : Type u_2} [conditionally_complete_linear_order β] {f : filter α} {u : α → β} {b : β} (hu_le : filter.frequently (fun (x : α) => u x ≤ b) f) (hu : autoParam (is_bounded_under ge f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : liminf f u ≤ b :=\n  le_limsup_of_frequently_le hu_le\n\nend filter\n\n\ntheorem galois_connection.l_limsup_le {α : Type u_1} {β : Type u_2} {γ : Type u_3} [conditionally_complete_lattice β] [conditionally_complete_lattice γ] {f : filter α} {v : α → β} {l : β → γ} {u : γ → β} (gc : galois_connection l u) (hlv : autoParam (filter.is_bounded_under LessEq f fun (x : α) => l (v x))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hv_co : autoParam (filter.is_cobounded_under LessEq f v)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : l (filter.limsup f v) ≤ filter.limsup f fun (x : α) => l (v x) := sorry\n\ntheorem order_iso.limsup_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [conditionally_complete_lattice β] [conditionally_complete_lattice γ] {f : filter α} {u : α → β} (g : β ≃o γ) (hu : autoParam (filter.is_bounded_under LessEq f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hu_co : autoParam (filter.is_cobounded_under LessEq f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hgu : autoParam (filter.is_bounded_under LessEq f fun (x : α) => coe_fn g (u x))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hgu_co : autoParam (filter.is_cobounded_under LessEq f fun (x : α) => coe_fn g (u x))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : coe_fn g (filter.limsup f u) = filter.limsup f fun (x : α) => coe_fn g (u x) := sorry\n\ntheorem order_iso.liminf_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [conditionally_complete_lattice β] [conditionally_complete_lattice γ] {f : filter α} {u : α → β} (g : β ≃o γ) (hu : autoParam (filter.is_bounded_under ge f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hu_co : autoParam (filter.is_cobounded_under ge f u)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hgu : autoParam (filter.is_bounded_under ge f fun (x : α) => coe_fn g (u x))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) (hgu_co : autoParam (filter.is_cobounded_under ge f fun (x : α) => coe_fn g (u x))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.filter.is_bounded_default\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"filter\") \"is_bounded_default\")\n    [])) : coe_fn g (filter.liminf f u) = filter.liminf f fun (x : α) => coe_fn g (u x) :=\n  order_iso.limsup_apply (order_iso.dual g)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/liminf_limsup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3176299335587883}}
{"text": "import vorspiel binary_recursion\n\nuniverses u v\n\nclass lencodable (Γ : Type*) (α : Type*) :=\n(encode [] : α → list Γ)\n(decode [] : list Γ → option α)\n(encodek : ∀ a, decode (encode a) = some a)\n\n--notation `⸤`x`⸥` := lencodable.encode bool x\n\nattribute [simp] lencodable.encodek\n\nnamespace lencodable\nvariables {Γ : Type*} {α : Type*} [lencodable Γ α]\n\nlemma encode_injective  : function.injective (@encode Γ α _)\n| x y e := option.some.inj $ by rw [← @encodek Γ, e, encodek]\n\n@[simp] lemma encode_inj {a b : α} : encode Γ a = encode Γ b ↔ a = b :=\nencode_injective.eq_iff\n\n@[simp] lemma decode_comp_encode : (decode α : list Γ → option α) ∘ (encode Γ : α → list Γ) = some :=\nfunext (by simp)\n\nvariables (Γ)\n\ndef length (a : α) : ℕ := (encode Γ a).length\n\n--notation `∥`a`∥` := length bool a\n\ninstance refl : lencodable Γ Γ :=\n{ encode := λ b, [b],\n  decode := λ d, d.nth 0,\n  encodek := by simp }\n\ninstance refll : lencodable Γ (list Γ) :=\n{ encode := id,\n  decode := some,\n  encodek := by simp }\n\nvariables {Γ}\n\n@[simp] lemma refl_encode (x : Γ) : encode Γ x = [x] := rfl\n\n@[simp] lemma refl_decode_cons (x : Γ) (xs : list Γ) : decode Γ (x :: xs) = some x := rfl\n\n@[simp] lemma refl_decode_nil : decode Γ (list.nil : list Γ) = none := rfl\n\ninstance : lencodable Γ pempty :=\n{ encode := by rintros ⟨⟩,\n  decode := λ _, none,\n  encodek := by rintros ⟨⟩ }\n\nend lencodable\n\nclass vencodable (Γ : Type*) (α : Type*) :=\n(entropy [] : ℕ)\n(encode [] : α → vector Γ entropy)\n(decode [] : vector Γ entropy → option α)\n(encodek : ∀ a, decode (encode a) = some a)\n\nattribute [simp] vencodable.encodek\n\nabbreviation bencodable := vencodable bool\nabbreviation bentropy := vencodable.entropy bool\n\nnamespace vencodable\nvariables {Γ : Type*} {α : Type*} [vencodable Γ α]\n\nnotation `∥`α`∥` := entropy bool α\n\nnotation `⸤`x`⸥` := encode bool x\n\nlemma encode_injective  : function.injective (@encode Γ α _)\n| x y e := option.some.inj $ by rw [← @encodek Γ, e, encodek]\n\n@[simp] lemma encode_inj {a b : α} : encode Γ a = encode Γ b ↔ a = b :=\nencode_injective.eq_iff\n\n@[simp] lemma decode_comp_encode :\n  (decode : vector Γ (entropy Γ α)  → option α) ∘ (encode Γ : α → vector Γ (entropy Γ α)) = some :=\nfunext (by simp)\n\nvariables (Γ)\n\ndef length (a : α) : ℕ := (encode Γ a).length\n\ninstance refl : vencodable Γ Γ :=\n{ entropy := 1,\n  encode := λ b, ⟨[b], by simp⟩,\n  decode := λ d, d.nth 0,\n  encodek := λ b, by refl }\n\ninstance refll (k : ℕ) : vencodable Γ (vector Γ k) :=\n{ entropy := k,\n  encode := id,\n  decode := some,\n  encodek := by simp }\n\nvariables {Γ}\n\ninstance : vencodable Γ pempty :=\n{ entropy := 0, \n  encode := by rintros ⟨⟩,\n  decode := λ _, none,\n  encodek := by rintros ⟨⟩ }\n\ninstance : vencodable Γ unit :=\n{ entropy := 0, \n  encode := λ u, vector.nil,\n  decode := λ x, some (),\n  encodek := by rintros ⟨⟩; refl }\n\nvariables {α} {β : Type*} (e : β ≃ α)\n\ndef of_equiv : vencodable Γ β :=\n{ entropy := entropy Γ α,\n  encode := λ b, encode Γ (e.to_fun b),\n  decode := λ d, (decode d).map e.inv_fun,\n  encodek := λ b, by simp }\n\n@[simp] lemma of_equiv_entropy_eq : (of_equiv e : vencodable Γ β).entropy = entropy Γ α := rfl\n\nend vencodable\n\nnamespace nat\nattribute [simp] bit_zero bodd_bit div2_bit nat.div2_one nat.div2_two\nopen lencodable\n\ndef dibit : ℕ → 𝔹 := binary_rec [] (λ b n d, b :: d)\n\nnamespace dibit\n\n@[simp] lemma bit_tt_zero : bit tt 0 = 1 := rfl\n\n-- 逆順の二進数表示\n-- 注意 dibit 0 = []\n\ndef inv : 𝔹 → ℕ\n| [] := 0\n| (b :: d) := bit b (inv d)\n\nlemma inv_length_le : ∀ d : 𝔹, inv d + 1 ≤ 2^d.length\n| [] := by simp[inv]\n| (b :: d) := by { simp[inv],\n    have IH : inv d + 1 ≤ 2^d.length, from inv_length_le d,\n    calc\n      bit b (inv d) + 1 ≤ 2 * (inv d) + 1 + 1 : by rcases b; simp\n                    ... = 2 * (inv d + 1)     : by ring\n                    ... ≤ 2 * (2^d.length)    : by simpa using IH\n                    ... ≤ 2^(d.length + 1)    : by simp[pow_add, mul_comm] }\n\nlemma inv_length_lt (d : 𝔹) : inv d < 2^d.length := succ_le_iff.mp (inv_length_le d)\n\n@[simp] lemma bit_bodd {b n} : (bit b n).bodd = b :=\nby cases b; simp[bit]\n\n@[simp] lemma dibit_0 : dibit 0 = [] := by simp[dibit]\n\n@[simp] lemma dibit_1 : dibit 1 = [tt] :=\nby { unfold dibit binary_rec, simp, rw nat.div2_one, simp }\n\n@[simp] lemma dibit_bit {b n} (h : bit b n ≠ 0) : dibit (bit b n) = b :: dibit n :=\nbegin\n  suffices : dite (bit b n = 0) (λ _, []) _ = b :: dibit n,\n  { unfold dibit binary_rec at this ⊢, exact this },\n  simp[h], rw nat.div2_bit b n, refl\nend\n\n@[simp] lemma dibit_inv_to_divit : ∀ n, inv (dibit n) = n :=\nbinary_rec (by simp[dibit, inv])\n(λ b n h, by { by_cases hn : bit b n = 0; simp[hn, inv, h] })\n\n@[simp] lemma dibit_length : ∀ n, (dibit n).length = Log n :=\nbinary_rec (by simp) (λ b n IH, by { \n  by_cases C : bit b n = 0; simp[C, dibit_bit, Log_bit, IH] })\n\nexample (n : ℕ) : bit1 n = 2* n + 1 := bit1_val n\n\nlemma dibit_inv_append :\n  ∀ d₁ d₂, inv (d₁ ++ d₂) = inv d₁ + 2^d₁.length * inv d₂\n| []         d₂ := by simp[inv]\n| (tt :: d₁) d₂ := by {\n    have : inv (d₁ ++ d₂) = inv d₁ + 2^d₁.length * inv d₂ := dibit_inv_append d₁ d₂,\n    simp[inv, this, bit, bit1_val],\n    calc  2 * (inv d₁ + 2^d₁.length * inv d₂) + 1\n        = 2 * inv d₁ + 2^(d₁.length + 1) * inv d₂ + 1\n    : by simp[mul_add, pow_add, ←mul_assoc, mul_comm]\n    ... = 2 * inv d₁ + 1 + 2^(d₁.length + 1) * inv d₂\n    : by ring }\n| (ff :: d₁) d₂ := by {\n    have : inv (d₁ ++ d₂) = inv d₁ + 2^d₁.length * inv d₂ := dibit_inv_append d₁ d₂,\n    simp[inv, this, bit, bit0_val (inv d₁ + _), bit0_val (inv d₁)], \n    show 2*(inv d₁ + 2^d₁.length * inv d₂)\n       = 2*inv d₁ + 2^(d₁.length + 1) * inv d₂,\n    by simp[mul_add, pow_add, ←mul_assoc, mul_comm] }\n\n@[simp] lemma dibit_inv_repeat_ff : ∀ n, inv (list.repeat ff n) = 0\n| 0       := by simp[inv]\n| (n + 1) := by simp[inv, bit, dibit_inv_repeat_ff n]\n\n@[simp] lemma dibit_inv_append_repeat (d) (n) : inv (d ++ list.repeat ff n) = inv d :=\nby simp[dibit_inv_append]\n\ndef lift (d : 𝔹) {n : ℕ} (h : d.length ≤ n) : 𝕓 n := ⟨d ++ list.repeat ff (n - d.length), by simp[h]⟩\n\n@[simp] def dibit_inv_dibit_lift (d : 𝔹) (n : ℕ) (h : d.length ≤ n) : inv (lift d h).val = inv d :=\nby simp[lift]\n\nend dibit\n\ninstance : lencodable bool ℕ :=\n{ encode := dibit,\n  decode := λ d, some (dibit.inv d),\n  encodek := by simp }\n\ndef log2 : ℕ → ℕ := log 2\n\n@[simp] lemma encode_0 : encode bool (0 : ℕ) = [] := dibit.dibit_0\n\n@[simp] lemma encode_1 : encode bool (1 : ℕ) = [tt] := dibit.dibit_1\n\nlemma dibit_length : ∀ n, length bool n = Log n := dibit.dibit_length\n\nend nat\n\nnamespace fin\nopen nat nat.dibit\nvariables {n : ℕ}\n\ndef bencode (i : fin n) : 𝕓 (Log n) := lift (dibit i) (by simpa using Log_monotone (le_of_lt i.property))\n\ndef bdecode (d : 𝕓 (Log n)) : option (fin n) := \nif h : inv d.val < n then some ⟨inv d.val, h⟩ else none\n\ninstance : bencodable (fin n) :=\n{ entropy := Log n,\n  encode := bencode,\n  decode := bdecode,\n  encodek := λ ⟨i, h⟩, by simp[bencode, bdecode, dibit.lift, h] }\n\n@[simp] lemma entropy_eq : ∥fin n∥ = Log n := rfl\n\nend fin\n\nnamespace vencodable\nopen nat\nvariables (α : Type*) [fintype α]\n\nnoncomputable def of_fin : bencodable α := of_equiv (fintype.equiv_fin α)\n\nlemma of_fin_entropy_eq : (of_fin α : bencodable α).entropy = Log (fintype.card α) := rfl\n\nend vencodable\n", "meta": {"author": "iehality", "repo": "lean-computable-complexity", "sha": "deee56eddd42eba1ceb05e8a9d8a2cc354138f65", "save_path": "github-repos/lean/iehality-lean-computable-complexity", "path": "github-repos/lean/iehality-lean-computable-complexity/lean-computable-complexity-deee56eddd42eba1ceb05e8a9d8a2cc354138f65/src/lencodable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5428632831725053, "lm_q1q2_score": 0.31762992570383236}}
{"text": "/-\nCopyright (c) 2023 Wojciech Nawrocki. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wojciech Nawrocki\n-/\n\nimport Std.Data.Array.Basic\n\nimport ProofChecker.Data.ClauseDb\nimport ProofChecker.Data.Pog\nimport ProofChecker.Count.Pog\nimport ProofChecker.Data.HashSet\nimport ProofChecker.Model.Cpog\n\n/-- An index into the `ClauseDb`. -/\nabbrev ClauseIdx := Nat\n\n/-- A step in a CPOG proof. -/\ninductive CpogStep\n  | /-- Add asymmetric tautology. -/\n    addAt (idx : ClauseIdx) (C : IClause) (upHints : Array ClauseIdx)\n  | /-- Delete asymmetric tautology. -/\n    delAt (idx : ClauseIdx) (upHints : Array ClauseIdx)\n  | /-- Declare product operation. -/\n    prod (idx : ClauseIdx) (x : Var) (ls : Array ILit)\n  | /-- Declare sum operation. -/\n    sum (idx : ClauseIdx) (x : Var) (l₁ l₂ : ILit) (upHints : Array ClauseIdx)\n  | /-- Declare POG root. -/\n    root (r : ILit)\n\nnamespace CpogStep\n\ndef toDimacs : CpogStep → String\n  | addAt idx C upHints => s!\"{idx} a {pArray C} 0 {pArray upHints} 0\"\n  | delAt idx upHints => s!\"dc {idx} {pArray upHints} 0\"\n  | prod idx x ls => s!\"{idx} p {x} {pArray ls} 0\"\n  | sum idx x l₁ l₂ upHints => s!\"{idx} s {x} {l₁} {l₂} {pArray upHints} 0\"\n  | root x => s!\"r {x}\"\nwhere\n  pArray {α : Type} [ToString α] (a : Array α) :=\n    \" \".intercalate (a.foldr (init := []) fun a acc => toString a :: acc)\n\ninstance : ToString CpogStep where\n  toString := fun\n    | addAt idx C upHints => s!\"{idx} a {C} 0 (hints: {upHints})\"\n    | delAt idx upHints => s!\"dc {idx} (hints: {upHints})\"\n    | prod idx x ls => s!\"{idx} p {x} {ls}\"\n    | sum idx x l₁ l₂ upHints => s!\"{idx} s {x} {l₁} {l₂} (hints: {upHints})\"\n    | root x => s!\"r {x}\"\n\nend CpogStep\n\ninductive CheckerError where\n  | graphUpdateError (err : PogError)\n  | duplicateClauseIdx (idx : ClauseIdx)\n  | unknownClauseIdx (idx : ClauseIdx)\n  | pogDefClause (idx : ClauseIdx)\n  | hintNotPog (idx : ClauseIdx)\n  | hintNotUnit (idx : ClauseIdx) (C : IClause) (σ : PartPropAssignment)\n  | upNoContradiction (τ : PartPropAssignment)\n  | duplicateExtVar (x : Var)\n  | unknownVar (x : Var)\n  | depsNotDisjoint (xs : List Var)\n  | finalRootNotSet\n  | finalRootNotUnit (r : ILit)\n  | finalClausesInvalid\n  | finalClauseInvalid (idx : ClauseIdx) (C : IClause)\n\nnamespace CheckerError\n\ninstance : ToString CheckerError where\n  toString := fun\n    | graphUpdateError e => s!\"graph update error: {e}\"\n    | duplicateClauseIdx idx => s!\"cannot add clause at index {idx}, index is already used\"\n    | unknownClauseIdx idx => s!\"there is no clause at index {idx}\"\n    | pogDefClause idx => s!\"clause at index {idx} cannot be deleted because it is a POG definition\"\n    | hintNotPog idx => s!\"hint {idx} cannot be used to imply model disjointness because it's not\n    a POG definition clause\"\n    | hintNotUnit idx C σ =>\n      s!\"hinted clause {C} at index {idx} did not become unit under assignment {σ}\"\n    | upNoContradiction τ =>\n      s!\"unit propagation did not derive contradiction (final assignment {τ})\"\n    | duplicateExtVar x => s!\"extension variable {x} was already introduced\"\n    | unknownVar x => s!\"unknown variable {x}\"\n    | depsNotDisjoint xs => s!\"variables {xs} have non-disjoint dependency sets\"\n    | finalRootNotSet => s!\"proof done but root literal was not set\"\n    | finalRootNotUnit r => s!\"proof done but root literal {r} was not asserted as a unit clause\"\n    | finalClausesInvalid =>\n      s!\"proof done but some clause is neither the asserted root nor a POG definition\"\n    | finalClauseInvalid idx C =>\n      s!\"proof done but clause {C} at index {idx} is neither the asserted root nor a POG definition\"\n\nend CheckerError\n\n/-- The checker's runtime state. Contains exactly the data needed to fully check a proof.\nIt can be ill-formed, so it's a \"pre\"-well-formed state. -/\nstructure PreState where\n  -- NOTE: This is part of the state so we could cheat by changing it. We don't.\n  inputCnf : ICnf\n\n  /-- The variables present in the original CNF. -/\n  -- TODO: not used at the moment; its cardinality is needed to output an absolute model-count,\n  -- and also to state invariants; but for the latter, ghost state would suffice\n  origVars : HashSet Var\n\n  /-- The clause database. -/\n  clauseDb : ClauseDb ClauseIdx\n\n  /-- Which clauses are POG definition clauses. -/\n  pogDefs : HashSet ClauseIdx\n\n  /-- The partitioned-operation graph. -/\n  pog : Pog\n\n  /-- Maps any variable present in `pog` to the set of all *original* variables it depends on.\n  For example, an original `x` is mapped to `{x}` whereas an extension `p` with `p ↔ x ∧ y` is\n  mapped to `{x, y}`.\n\n  Variables not present in `pog` are not present in this map. Thus we maintain the invariant\n  that a variable is in the `pog` iff it is in the domain of this map. -/\n  depVars : HashMap Var (HashSet Var)\n\n  /-- The POG root literal, if we already saw a `root` instruction. Otherwise `none`. -/\n  root : Option ILit\n\ndef PreState.pogDefs' (st : PreState) : Finset ClauseIdx :=\n  st.pogDefs.toFinset\n\nnoncomputable def PreState.pogDefsTerm (st : PreState) : PropTerm Var :=\n  st.clauseDb.toPropTermSub st.pogDefs'\n\ndef PreState.allVars (st : PreState) : Set Var :=\n  { x | st.depVars.contains x }\n\nopen PropTerm\n\nstructure PreState.WF (st : PreState) : Prop where\n  /-- The POG definition clauses are all in the clause database. -/\n  pogDefs_in_clauseDb : ∀ idx : ClauseIdx, idx ∈ st.pogDefs' → st.clauseDb.contains idx\n\n  /-- Variable dependencies are correctly stored in `depVars`. -/\n  depVars_pog : ∀ (x : Var) (D : HashSet Var), st.depVars.find? x = some D →\n    -- NOTE: can strengthen to eq if needed\n    (st.pog.toPropForm x).vars ⊆ D.toFinset\n\n  /-- Every formula in the POG forest is partitioned.\n\n  For literals not defining anything in the forest this still holds by fiat because\n  `st.pog.toPropForm l = l.toPropForm`. -/\n  partitioned : ∀ x : Var, (st.pog.toPropForm x).partitioned\n\n  /-- `depVars` contains all variables that influence the clause database. Contrapositive:\n  if a variable is not in `depVars` then it does not influence the clause database so can be\n  defined as an extension variable. -/\n  clauseDb_semVars_sub : ↑st.clauseDb.toPropTerm.semVars ⊆ st.allVars\n\n  pogDefsTerm_semVars_sub : ↑st.pogDefsTerm.semVars ⊆ st.allVars\n\n  inputCnf_vars_sub : ↑st.inputCnf.vars.toFinset ⊆ st.allVars\n\n  /-- Every formula in the POG forest lives over the original variables. -/\n  pog_vars : ∀ x : Var, x ∈ st.allVars →\n    (st.pog.toPropForm x).vars ⊆ st.inputCnf.vars.toFinset\n\n  /-- The clause database is equivalent to the original formula over original variables. -/\n  equivalent_clauseDb : equivalentOver st.inputCnf.vars.toFinset\n    st.inputCnf.toPropTerm st.clauseDb.toPropTerm\n\n  extends_pogDefsTerm : ∀ (σ₁ : PropAssignment Var), ∃ (σ₂ : PropAssignment Var),\n    σ₂.agreeOn st.inputCnf.vars.toFinset σ₁ ∧ σ₂ ⊨ st.pogDefsTerm\n\n  /-- The POG definition clauses extend uniquely from the original variables to the current set\n  of variables. -/\n  uep_pogDefsTerm : hasUniqueExtension st.inputCnf.vars.toFinset st.allVars\n    st.pogDefsTerm\n\n  /-- In the context of the POG defining clauses, every variable is s-equivalent over original\n  variables to what it defines in the POG forest. -/\n  -- TODO: could this be weakened to `extendsOver` and still go through?\n  equivalent_lits : ∀ x : Var, equivalentOver st.inputCnf.vars.toFinset\n    (.var x ⊓ st.pogDefsTerm) ⟦st.pog.toPropForm x⟧\n\ntheorem PreState.WF.depVars_pog' {st : PreState} (hWf : st.WF) (l : ILit) (D : HashSet Var) :\n    st.depVars.find? l.var = some D → (st.pog.toPropForm l).vars ⊆ D.toFinset :=\n  fun hFind => l.mkPos_or_mkNeg.elim (· ▸ hWf.depVars_pog l.var D hFind) fun h => by\n   rw [h, Pog.toPropForm_neg, PropForm.vars]\n   exact hWf.depVars_pog l.var D hFind\n\ntheorem PreState.WF.partitioned' {st : PreState} (hWf : st.WF) (l : ILit) :\n    (st.pog.toPropForm l).partitioned :=\n  l.mkPos_or_mkNeg.elim (· ▸ hWf.partitioned l.var) fun h => by\n    rw [h, Pog.toPropForm_neg]\n    exact hWf.partitioned l.var\n\ntheorem PreState.WF.pog_vars' {st : PreState} (hWf : st.WF) (l : ILit) :\n    l.var ∈ st.allVars → (st.pog.toPropForm l).vars ⊆ st.inputCnf.vars.toFinset :=\n  fun hMem => l.mkPos_or_mkNeg.elim (· ▸ hWf.pog_vars l.var hMem) fun h => by\n    rw [h, Pog.toPropForm_neg]\n    exact hWf.pog_vars l.var hMem\n\ntheorem PreState.WF.pog_semVars {st : PreState} (hWf : st.WF) (l : ILit) :\n    l.var ∈ st.allVars → ↑(PropTerm.semVars ⟦st.pog.toPropForm l⟧) ⊆ st.inputCnf.vars.toFinset :=\n  fun hMem => subset_trans (PropForm.semVars_subset_vars _) (hWf.pog_vars' l hMem)\n\ntheorem PreState.WF.equivalent_lits' {st : PreState} (hWf : st.WF) (l : ILit) :\n    l.var ∈ st.allVars → equivalentOver st.inputCnf.vars.toFinset\n      (l.toPropTerm ⊓ st.pogDefsTerm) ⟦st.pog.toPropForm l⟧ := by\n  intro hMem\n  cases l.mkPos_or_mkNeg\n  next hMk =>\n    rw [hMk]\n    simp [hWf.equivalent_lits l.var]\n  next hMk =>\n    rw [hMk]\n    generalize l.var = x at hMem ⊢\n    have hEquiv := hWf.equivalent_lits x\n    simp only [Pog.toPropForm_neg, ILit.toPropTerm_mkNeg, PropTerm.mk_neg]\n    apply equivalentOver_iff_extendsOver _ _ _ |>.mpr\n    constructor\n    . intro σ hσ\n      refine ⟨σ, PropAssignment.agreeOn_refl _ _, ?_⟩\n      rw [satisfies_neg]\n      intro h\n      have ⟨σ₂, hAgree, hσ₂⟩ := hEquiv σ |>.mpr ⟨σ, PropAssignment.agreeOn_refl _ _, h⟩\n      have hAgree := hWf.uep_pogDefsTerm (satisfies_conj.mp hσ₂).right (satisfies_conj.mp hσ).right hAgree\n      have : σ₂.agreeOn (PropTerm.var x).semVars σ := hAgree.subset (by simp [hMem])\n      have : σ ⊨ .var x := agreeOn_semVars this |>.mp (satisfies_conj.mp hσ₂).left\n      have : σ ⊭ .var x := satisfies_neg.mp (satisfies_conj.mp hσ).left\n      contradiction\n    . intro σ hσ\n      have ⟨σ₁, hAgree, hσ₁⟩ := hWf.extends_pogDefsTerm σ\n      refine ⟨σ₁, hAgree, ?_⟩\n      simp only [hσ₁, satisfies_conj, satisfies_neg, and_true]\n      intro h\n      have ⟨σ₂, hAgree₂, hσ₂⟩ := hEquiv σ₁ |>.mp\n        ⟨σ₁, PropAssignment.agreeOn_refl _ _, satisfies_conj.mpr ⟨h, hσ₁⟩⟩\n      have hAgree := hAgree₂.trans hAgree\n      have hSub := hWf.pog_semVars (.mkPos x) hMem\n      have : σ ⊨ ⟦st.pog.toPropForm (.mkPos x)⟧ :=\n        agreeOn_semVars (hAgree.subset hSub) |>.mp hσ₂\n      simp [satisfies_neg] at hσ\n      contradiction\n\n/-- A well-formed checker state. -/\ndef State := { st : PreState // st.WF }\n\nabbrev CheckerM := StateT State <| Except CheckerError\n\ndef initialPog (nVars : Nat) :\n    Except CheckerError { p : Pog // ∀ l, p.toPropForm l = l.toPropForm } := do\n  -- NOTE: We add all input variables to the POG in-order because that's what the current\n  -- implementation expects, but this requirement is artificial. See `Pog.lean`.\n  nVars.foldM (init := ⟨Pog.empty, Pog.toPropForm_empty⟩) fun x ⟨acc, hAcc⟩ =>\n    let x := ⟨x + 1, Nat.zero_lt_succ _⟩\n    match h : acc.addVar x with\n    | .ok g => pure ⟨g, by\n      intro l'\n      by_cases hEq : x = l'.var\n      . rw [hEq] at h\n        exact acc.toPropForm_addVar_lit _ _ h\n      . rw [acc.toPropForm_addVar_lit_of_ne _ _ _ h hEq]\n        apply hAcc⟩\n    | .error e => throw <| .graphUpdateError e\n\ndef initialClauseVars (m : HashMap Var (HashSet Var)) (C : IClause) : HashMap Var (HashSet Var) :=\n  C.foldl (init := m) fun m l =>\n    m.insert l.var (HashSet.empty Var |>.insert l.var)\n\ntheorem initialClauseVars₁ (m : HashMap Var (HashSet Var)) (C : IClause) (x : Var) :\n    (initialClauseVars m C).contains x ↔ x ∈ C.vars.toFinset ∨ m.contains x := by\n  simp only [initialClauseVars, IClause.mem_vars, Array.foldl_eq_foldl_data]\n  induction C.data generalizing m <;>\n    aesop (add norm HashMap.contains_insert)\n\ntheorem initialClauseVars₂ (m : HashMap Var (HashSet Var)) (C : IClause) :\n    (∀ x D, m.find? x = some D → x ∈ D.toFinset) →\n    ∀ x D, (initialClauseVars m C).find? x = some D → x ∈ D.toFinset := by\n  simp only [initialClauseVars, Array.foldl_eq_foldl_data]\n  induction C.data generalizing m\n  . simp\n  next l _ ih =>\n    intro _ _ _\n    rw [List.foldl_cons]\n    apply ih\n    intro y\n    by_cases hEq : l.var = y <;>\n      aesop (add norm HashMap.find?_insert, norm HashMap.find?_insert_of_ne)\n\ndef initialCnfVars (m : HashMap Var (HashSet Var)) (φ : ICnf) : HashMap Var (HashSet Var) :=\n  φ.foldl (init := m) initialClauseVars\n\ntheorem initialCnfVars₁ (m : HashMap Var (HashSet Var)) (φ : ICnf) (x : Var) :\n    (initialCnfVars m φ).contains x ↔ x ∈ φ.vars.toFinset ∨ m.contains x := by\n  simp only [initialCnfVars, ICnf.mem_vars, Array.foldl_eq_foldl_data]\n  induction φ.data generalizing m <;>\n    aesop (add norm initialClauseVars₁)\n\ntheorem initialCnfVars₂ (m : HashMap Var (HashSet Var)) (φ : ICnf) :\n    (∀ x D, m.find? x = some D → x ∈ D.toFinset) →\n    ∀ x D, (initialCnfVars m φ).find? x = some D → x ∈ D.toFinset := by\n  simp only [initialCnfVars, Array.foldl_eq_foldl_data]\n  induction φ.data generalizing m\n  . simp\n  next C _ ih =>\n    intro h _ _\n    rw [List.foldl_cons]\n    apply ih\n    exact initialClauseVars₂ _ _ h\n\ndef initialDepVars (inputCnf : ICnf) : { dv : HashMap Var (HashSet Var) //\n    { y | dv.contains y } = inputCnf.vars.toFinset ∧\n    ∀ x D, dv.find? x = some D → x ∈ D.toFinset } :=\n  let dv := initialCnfVars .empty inputCnf\n  have allVars_eq := by ext; simp [initialCnfVars₁]\n  have of_find := by apply initialCnfVars₂; simp\n  ⟨dv, allVars_eq, of_find⟩\n\ndef initial (inputCnf : ICnf) (nVars : Nat) : Except CheckerError State := do\n  let ⟨initPog, hInitPog⟩ ← initialPog nVars\n  let ⟨initDv, allVars_eq, hInitDv⟩ := initialDepVars inputCnf\n  let st := {\n    inputCnf\n    origVars := inputCnf.vars\n    clauseDb := .ofICnf inputCnf\n    pogDefs := .empty ClauseIdx\n    pog := initPog\n    depVars := initDv\n    root := none\n  }\n  have pogDefs'_empty : st.pogDefs' = ∅ := by\n    simp [PreState.pogDefs']\n  have pogDefsTerm_tr : st.pogDefsTerm = ⊤ := by\n    rw [PreState.pogDefsTerm, pogDefs'_empty, Finset.coe_empty]\n    apply ClauseDb.toPropTermSub_emptySet\n  have allVars_eq : st.allVars = st.inputCnf.vars.toFinset := allVars_eq\n  have pfs := {\n    pogDefs_in_clauseDb := by\n      simp [pogDefs'_empty]\n    depVars_pog := by\n      intro x D hFind\n      simp [hInitPog, PropForm.vars, hInitDv x D hFind]\n    partitioned := by\n      simp [hInitPog, PropForm.partitioned]\n    clauseDb_semVars_sub := by\n      rw [allVars_eq, ClauseDb.toPropTerm_ofICnf]\n      apply ICnf.semVars_sub\n    pogDefsTerm_semVars_sub := by\n      rw [pogDefsTerm_tr, PropTerm.semVars_tr, Finset.coe_empty]\n      apply Set.empty_subset\n    inputCnf_vars_sub := by\n      rw [allVars_eq]\n    equivalent_clauseDb := by\n      rw [ClauseDb.toPropTerm_ofICnf]\n      apply PropTerm.equivalentOver_refl\n    pog_vars := by\n      simp [allVars_eq, hInitPog, PropForm.vars]\n    extends_pogDefsTerm := fun σ =>\n      ⟨σ, PropAssignment.agreeOn_refl _ _, by simp [pogDefsTerm_tr]⟩\n    uep_pogDefsTerm := by\n      simp only [pogDefsTerm_tr, PropTerm.semVars_tr, Finset.coe_empty, allVars_eq]\n      exact PropTerm.hasUniqueExtension_refl _ _\n    equivalent_lits := by\n      simp [pogDefsTerm_tr, hInitPog, PropTerm.equivalentOver_refl]\n  }\n  return ⟨st, pfs⟩\n\n/-- Check if `C` is an asymmetric tautology wrt the clause database. `C` must not be a tautology. -/\ndef checkAtWithHints (db : ClauseDb ClauseIdx) (C : IClause) (hC : C.toPropTerm ≠ ⊤)\n    (hints : Array ClauseIdx) :\n    Except CheckerError { _u : Unit // db.toPropTermSub (· ∈ hints.data) ≤ C.toPropTerm }\n:= do\n  match db.unitPropWithHintsDep C.toFalsifyingAssignment hints with\n  | .contradiction h => return ⟨(), (by\n      rw [IClause.toPropTerm_toFalsifyingAssignment C hC, ← le_himp_iff, himp_bot, compl_compl] at h\n      assumption)⟩\n  | .extended τ _ => throw <| .upNoContradiction τ\n  | .hintNotUnit idx C σ => throw <| .hintNotUnit idx C σ\n  | .hintNonexistent idx => throw <| .unknownClauseIdx idx\n\n/-- Check if `C` is an asymmetric tautology wrt the clause database, or simply a tautology. -/\ndef checkImpliedWithHints (db : ClauseDb ClauseIdx) (C : IClause) (hints : Array ClauseIdx) :\n    Except CheckerError { _u : Unit // db.toPropTermSub (· ∈ hints.data) ≤ C.toPropTerm }\n:= do\n  -- TODO: We could maintain no-tautologies-in-clause-db as an invariant rather than dynamically\n  -- checking. Checking on every deletion could cause serious slowdown (but measure first!).\n  if hTauto : C.toPropTerm = ⊤ then\n    return ⟨(), by simp [hTauto]⟩\n  else\n    checkAtWithHints db C hTauto hints\n\ndef addClause (db₀ : ClauseDb ClauseIdx) (idx : ClauseIdx) (C : IClause) :\n    Except CheckerError { db : ClauseDb ClauseIdx //\n      db = db₀.addClause idx C ∧\n      ¬db₀.contains idx ∧\n      -- This is just for convenience as it can be proven directly from the other postconditions.\n      db.toPropTerm = db₀.toPropTerm ⊓ C.toPropTerm } := do\n  if h : db₀.contains idx then\n    throw <| .duplicateClauseIdx idx\n  else\n    return ⟨db₀.addClause idx C, rfl, h, db₀.toPropTerm_addClause_eq idx C h⟩\n\ndef addAt (idx : ClauseIdx) (C : IClause) (hints : Array ClauseIdx) : CheckerM Unit := do\n  let ⟨st, pfs⟩ ← get\n  let ⟨_, hImp⟩ ← checkImpliedWithHints st.clauseDb C hints\n  let ⟨db', hAdd, hContains, hEq⟩ ← addClause st.clauseDb idx C\n  let st' := { st with clauseDb := db' }\n  have hDb : st'.clauseDb.toPropTerm = st.clauseDb.toPropTerm := by\n    simp [hEq, st.clauseDb.toPropTerm_subset _ |>.trans hImp]\n  have hPogDefs : st'.pogDefsTerm = st.pogDefsTerm := by\n    have hMem : idx ∉ (st.pogDefs' : Set _) := fun h =>\n      hContains (pfs.pogDefs_in_clauseDb _ h)\n    have : st'.pogDefs' = st.pogDefs' := rfl\n    rw [PreState.pogDefsTerm, this]\n    simp [PreState.pogDefsTerm, hAdd, st.clauseDb.toPropTermSub_addClause_of_not_mem C hMem]\n  have pfs' := { pfs with\n    pogDefs_in_clauseDb := fun idx h => by\n      have := pfs.pogDefs_in_clauseDb idx h\n      simp only [hAdd]\n      exact st.clauseDb.contains_addClause _ _ _ |>.mpr (Or.inl this)\n    equivalent_clauseDb := hDb ▸ pfs.equivalent_clauseDb\n    clauseDb_semVars_sub := hDb ▸ pfs.clauseDb_semVars_sub\n    pogDefsTerm_semVars_sub := hPogDefs ▸ pfs.pogDefsTerm_semVars_sub\n    equivalent_lits := hPogDefs ▸ pfs.equivalent_lits\n    extends_pogDefsTerm := hPogDefs ▸ pfs.extends_pogDefsTerm\n    uep_pogDefsTerm := hPogDefs ▸ pfs.uep_pogDefsTerm\n  }\n  set (σ := State) ⟨st', pfs'⟩\n\ndef getClause (db : ClauseDb ClauseIdx) (idx : ClauseIdx) :\n    Except CheckerError { C : IClause // db.getClause idx = some C } :=\n  match db.getClause idx with\n  | some C => return ⟨C, rfl⟩\n  | none => throw <| .unknownClauseIdx idx\n\ndef delAt (idx : ClauseIdx) (hints : Array ClauseIdx) : CheckerM Unit := do\n  let ⟨st, pfs⟩ ← get\n  let ⟨C, hGet⟩ ← getClause st.clauseDb idx\n  -- NOTE: We could investigate whether the check is really necessary.\n  let ⟨_, hMem⟩ ← (if h : st.pogDefs.contains idx then\n      throw <| .pogDefClause idx\n    else\n      pure ⟨(), HashSet.not_mem_toFinset _ _ |>.mpr h⟩\n    : Except CheckerError { _u : Unit // idx ∉ st.pogDefs' })\n  let db' := st.clauseDb.delClause idx\n  -- The clause is AT by everything except itself.\n  let ⟨_, hImp⟩ ← checkImpliedWithHints db' C hints\n  let st' := { st with clauseDb := db' }\n  have hDb : st'.clauseDb.toPropTerm = st.clauseDb.toPropTerm := by\n    have : st'.clauseDb.toPropTerm = st'.clauseDb.toPropTerm ⊓ C.toPropTerm := by\n      have := st'.clauseDb.toPropTerm_subset _ |>.trans hImp\n      exact left_eq_inf.mpr this\n    rw [this]\n    simp [st.clauseDb.toPropTerm_delClause_eq _ _ hGet]\n  have hPogDefs : st'.pogDefsTerm = st.pogDefsTerm :=\n    st.clauseDb.toPropTermSub_delClause_of_not_mem hMem\n  have pfs' := { pfs with\n    pogDefs_in_clauseDb := fun idx h => by\n      refine st.clauseDb.contains_delClause _ _ |>.mpr ⟨pfs.pogDefs_in_clauseDb idx h, ?_⟩\n      intro hEq\n      exact hMem (hEq.symm ▸ h)\n    equivalent_clauseDb := hDb ▸ pfs.equivalent_clauseDb\n    clauseDb_semVars_sub := hDb ▸ pfs.clauseDb_semVars_sub\n    pogDefsTerm_semVars_sub := hPogDefs ▸ pfs.pogDefsTerm_semVars_sub\n    equivalent_lits := hPogDefs ▸ pfs.equivalent_lits\n    extends_pogDefsTerm := hPogDefs ▸ pfs.extends_pogDefsTerm\n    uep_pogDefsTerm := hPogDefs ▸ pfs.uep_pogDefsTerm\n  }\n  set (σ := State) ⟨st', pfs'⟩\n\ndef ensureFreshVar (st : PreState) (x : Var) : Except CheckerError { _u : Unit //\n    x ∉ st.allVars } := do\n  if h : st.depVars.contains x then\n    throw <| .duplicateExtVar x\n  else\n    return ⟨(), h⟩\n\ndef getDeps (st : PreState) (pfs : st.WF) (l : ILit) : Except CheckerError { D : HashSet Var //\n    l.var ∈ st.allVars ∧\n    (st.pog.toPropForm l).vars ⊆ D.toFinset } := do\n  match h : st.depVars.find? l.var with\n  | some D =>\n    return ⟨D, st.depVars.contains_iff _ |>.mpr ⟨D, h⟩, pfs.depVars_pog' _ _ h⟩\n  | none => throw <| .unknownVar l.var\n\ndef getDepsArray {st : PreState} (pfs : st.WF) (ls : Array ILit) :\n    Except CheckerError { Ds : Array (HashSet Var) //\n      (∀ l ∈ ls.data, l.var ∈ st.allVars) ∧\n      Ds.size = ls.size ∧\n      ∀ (i : Nat) (hI : i < ls.size) (hI' : i < Ds.size),\n        (st.pog.toPropForm ls[i]).vars ⊆ Ds[i].toFinset } :=\n  let f l :=\n    match st.depVars.find? l.var with\n    | some D => return D\n    | none => throw <| .unknownVar l.var\n  let x : Except CheckerError (Array (HashSet Var)) := ls.mapM f\n  match h : x with\n  | .error e => throw e\n  | .ok Ds =>\n    have := Array.SatisfiesM_mapM (as := ls) (f := f)\n      -- Postcondition on the input\n      (motive := fun i => ∀ (j : Fin ls.size), j < i → ls[j].var ∈ st.allVars)\n      -- Postcondition on the outputs\n      (p := fun i val => (st.pog.toPropForm ls[i]).vars ⊆ val.toFinset)\n      (h0 := by simp)\n      (hs := by\n        dsimp\n        intro i ih\n        split\n        next h =>\n          simp only [SatisfiesM_Except_eq]\n          intro D hEq\n          cases hEq\n          refine ⟨pfs.depVars_pog' _ _ h, ?_⟩\n          intro j hJ\n          cases Nat.lt_or_eq_of_le (Nat.le_of_lt_succ hJ) with\n          | inl hJ => exact ih j hJ\n          | inr hJ =>\n            simp only [hJ]\n            exact st.depVars.contains_iff _ |>.mpr ⟨_, h⟩\n        . simp)\n    have hLs := by\n      intro l hL\n      simp only [SatisfiesM_Except_eq] at this\n      have ⟨h, _⟩ := this _ h\n      have ⟨i, hI⟩ := Array.get_of_mem_data hL\n      exact hI ▸ h i i.isLt\n    have hDs := by\n      simp only [SatisfiesM_Except_eq] at this\n      have := this _ h\n      aesop\n    have hSz := by\n      apply SatisfiesM_Except_eq.mp ?_ _ h\n      apply Array.size_mapM\n    return ⟨Ds, hLs, hSz, hDs⟩\n\ndef addPogDefClause (db₀ : ClauseDb ClauseIdx) (pd₀ : HashSet ClauseIdx)\n    (idx : ClauseIdx) (C : IClause) (h : ∀ idx, idx ∈ pd₀.toFinset → db₀.contains idx) :\n    Except CheckerError { p : ClauseDb ClauseIdx × HashSet ClauseIdx //\n      p.1.toPropTerm = db₀.toPropTerm ⊓ C.toPropTerm ∧\n      p.1.toPropTermSub (· ∈ p.2.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓ C.toPropTerm ∧\n      ∀ idx, idx ∈ p.2.toFinset → p.1.contains idx } := do\n  let ⟨db, hAdd, hNContains, hDb⟩ ← addClause db₀ idx C\n  let pd := pd₀.insert idx\n\n  have hMem : idx ∈ pd.toFinset := by simp\n  have hContainsTrans : ∀ {idx}, db₀.contains idx → db.contains idx := fun h => by\n    rw [hAdd]\n    exact db₀.contains_addClause _ _ _ |>.mpr (Or.inl h)\n  have hContains : db.contains idx := by\n    rw [hAdd]\n    exact db₀.contains_addClause _ _ _ |>.mpr (Or.inr rfl)\n  have hHelper : db₀.toPropTermSub (· ∈ pd.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) := by\n    apply db₀.toPropTermSub_subset_eq fun _ hMem => by simp; exact Or.inr hMem\n    intro idx hMem hContains\n    simp at hMem\n    cases hMem with\n    | inl h =>\n      exfalso\n      exact hNContains (h ▸ hContains)\n    | inr hMem => exact hMem\n  have hPd : db.toPropTermSub (· ∈ pd.toFinset) =\n      db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓ C.toPropTerm := by\n    rw [← hHelper, hAdd]\n    exact db₀.toPropTermSub_addClause_eq _ hMem hNContains\n  have hPdDb : ∀ idx, idx ∈ pd.toFinset → db.contains idx := by\n    simp only [HashSet.toFinset_insert, Finset.mem_singleton, Finset.mem_insert]\n    intro _ h\n    cases h with\n    | inl h => exact h ▸ hContains\n    | inr hMem => exact hContainsTrans (h _ hMem)\n\n  return ⟨(db, pd), hDb, hPd, hPdDb⟩\n\ntheorem def_ext_correct {st : PreState} (H : st.WF) (st' : PreState) (x : Var) (φ ψ : PropTerm Var)\n    (hDb : st'.clauseDb.toPropTerm = st.clauseDb.toPropTerm ⊓ (.biImpl (.var x) φ))\n    (hPd : st'.pogDefsTerm = st.pogDefsTerm ⊓ (.biImpl (.var x) φ))\n    (hAv : st'.allVars = insert x st.allVars)\n    (hPog₁ : ∀ y, x ≠ y → st'.pog.toPropForm (.mkPos y) = st.pog.toPropForm (.mkPos y))\n    (hPog₂ : ⟦st'.pog.toPropForm (.mkPos x)⟧ = ψ)\n    (hEquiv : PropTerm.equivalentOver st.inputCnf.vars.toFinset (.var x ⊓ st'.pogDefsTerm) ψ)\n    (hφ : ↑φ.semVars ⊆ st.allVars)\n    (hX : x ∉ st.allVars) :\n    (PropTerm.equivalentOver st.inputCnf.vars.toFinset\n      st.inputCnf.toPropTerm st'.clauseDb.toPropTerm ∧\n    (∀ σ₁, ∃ (σ₂ : PropAssignment Var),\n      σ₂.agreeOn st.inputCnf.vars.toFinset σ₁ ∧ σ₂ ⊨ st'.pogDefsTerm) ∧\n    PropTerm.hasUniqueExtension st.inputCnf.vars.toFinset st'.allVars st'.pogDefsTerm ∧\n    ∀ x, PropTerm.equivalentOver st.inputCnf.vars.toFinset\n      (.var x ⊓ st'.pogDefsTerm) ⟦st'.pog.toPropForm (.mkPos x)⟧) :=\n  have hEquivAv : PropTerm.equivalentOver st.allVars st.clauseDb.toPropTerm st'.clauseDb.toPropTerm\n      := by\n    rw [hDb]\n    exact PropTerm.equivalentOver_def_ext st.clauseDb.toPropTerm φ (H.clauseDb_semVars_sub) hφ hX\n  have equiv :=\n    H.equivalent_clauseDb.trans (hEquivAv.subset H.inputCnf_vars_sub)\n  have hUepInsert :=\n    PropTerm.hasUniqueExtension_def_ext x st.pogDefsTerm φ hφ\n  have extend := by\n    intro σ\n    have ⟨σ₁, hAgree, h₁⟩ := H.extends_pogDefsTerm σ\n    let σ₂ := σ₁.set x (φ.eval σ₁)\n    have hAgree₂₁ : σ₂.agreeOn st.allVars σ₁ := PropAssignment.agreeOn_set_of_not_mem _ _ hX\n    have : σ₂ ⊨ st.pogDefsTerm :=\n      agreeOn_semVars (hAgree₂₁.subset H.pogDefsTerm_semVars_sub) |>.mpr h₁\n    have : σ₂ ⊨ φ ↔ σ₁ ⊨ φ := agreeOn_semVars (hAgree₂₁.subset hφ)\n    exact ⟨σ₂, hAgree₂₁.subset H.inputCnf_vars_sub |>.trans hAgree, by aesop⟩\n  have uep := by\n    rw [hAv, hPd]\n    exact H.uep_pogDefsTerm.conj_right _ |>.trans hUepInsert\n  have hEquivVarNe : ∀ {y : Var}, x ≠ y → PropTerm.equivalentOver st.allVars\n      (.var y ⊓ st'.pogDefsTerm) (.var y ⊓ st.pogDefsTerm) := by\n    intro y hEq\n    suffices PropTerm.equivalentOver (st.allVars ∪ {y})\n        (.var y ⊓ st'.pogDefsTerm) (.var y ⊓ st.pogDefsTerm) from\n      this.subset (Set.subset_union_left _ _)\n    rw [hPd, ← inf_assoc]\n    apply PropTerm.equivalentOver.symm\n    have hX : x ∉ st.allVars ∪ {y} := by simp [hEq, hX]\n    apply PropTerm.equivalentOver_def_ext _ _ ?_ (hφ.trans <| Set.subset_union_left _ _) hX\n    suffices ↑((PropTerm.var y).semVars ∪ st.pogDefsTerm.semVars) ⊆ st.allVars ∪ {y} from\n      subset_trans (PropTerm.semVars_conj _ _) this\n    intro y; simp\n    intro h; cases h with\n    | inl hEq => exact Or.inl hEq\n    | inr hMem => exact Or.inr (H.pogDefsTerm_semVars_sub hMem)\n  have equiv_vars := by\n    intro y\n    by_cases hEq : x = y\n    case neg =>\n      rw [hPog₁ _ hEq]\n      exact (hEquivVarNe hEq).subset H.inputCnf_vars_sub |>.trans (H.equivalent_lits y)\n    case pos =>\n      rw [← hEq, hPog₂]\n      exact hEquiv\n  ⟨equiv, extend, uep, equiv_vars⟩\n\ndef addPogDefClauses (db₀ : ClauseDb ClauseIdx) (pd₀ : HashSet ClauseIdx)\n    (idx : ClauseIdx) (ls : Array ILit) (f : ILit → IClause)\n    (h : ∀ idx, idx ∈ pd₀.toFinset → db₀.contains idx) :\n    Except CheckerError { p : ClauseDb ClauseIdx × HashSet ClauseIdx //\n      p.1.toPropTerm = db₀.toPropTerm ⊓\n        PropForm.arrayConjTerm (ls.map fun l => (f l).toPropForm) ∧\n      p.1.toPropTermSub (· ∈ p.2.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓\n        PropForm.arrayConjTerm (ls.map fun l => (f l).toPropForm) ∧\n      ∀ idx, idx ∈ p.2.toFinset → p.1.contains idx } := do\n  let ⟨out, h₁, h₂, h₃⟩ ← loopM_with_invariant (State := ClauseDb ClauseIdx × HashSet ClauseIdx)\n    (n := ls.size)\n    (invariant := fun i ⟨db, pd⟩ =>\n      db.toPropTerm = db₀.toPropTerm ⊓\n        PropForm.listConjTerm (ls.data.take i |>.map fun l => (f l).toPropTerm) ∧\n      db.toPropTermSub (· ∈ pd.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓\n        PropForm.listConjTerm (ls.data.take i |>.map fun l => (f l).toPropTerm) ∧\n      ∀ idx, idx ∈ pd.toFinset → db.contains idx)\n    (start_state := ⟨(db₀, pd₀), by simp, by simp, h⟩)\n    (step := fun i ⟨(db, pd), ih₁, ih₂, ih₃⟩ => do\n      let l := ls[i]\n      let ⟨st', hDb, hPd, h⟩ ← addPogDefClause db pd (idx+i+1) (f l) ih₃\n      have hEquiv :\n          PropForm.listConjTerm (ls.data.take (i + 1) |>.map fun l => (f l).toPropTerm) =\n          PropForm.listConjTerm (ls.data.take i |>.map fun l => (f l).toPropTerm) ⊓\n            IClause.toPropTerm (f l) := by\n        ext\n        simp [PropForm.satisfies_listConjTerm, List.take_succ, List.get?_eq_get i.isLt,\n          Array.getElem_eq_data_get]\n        constructor\n        . aesop\n        . intro ⟨h₁, h₂⟩ φ h\n          cases h with\n          | inl h =>\n            have ⟨l, hL, hφ⟩ := h\n            exact hφ ▸ h₁ l hL\n          | inr h =>\n            simp_all\n      have hDb' := by rw [hDb, ih₁, inf_assoc, hEquiv]\n      have hPd' := by rw [hPd, ih₂, inf_assoc, hEquiv]\n      return ⟨st', hDb', hPd', h⟩)\n  have hDb := by\n    simp only [List.take_length] at h₁\n    simp [PropForm.arrayConjTerm_eq_listConjTerm_data, Function.comp, h₁]\n  have hPd := by\n    simp only [List.take_length] at h₂\n    simp [PropForm.arrayConjTerm_eq_listConjTerm_data, Function.comp, h₂]\n  return ⟨out, hDb, hPd, h₃⟩\n\ndef addProdClauses (db₀ : ClauseDb ClauseIdx) (pd₀ : HashSet ClauseIdx)\n    (idx : ClauseIdx) (x : Var) (ls : Array ILit)\n    (h : ∀ idx, idx ∈ pd₀.toFinset → db₀.contains idx) :\n    Except CheckerError { p : ClauseDb ClauseIdx × HashSet ClauseIdx //\n      p.1.toPropTerm = db₀.toPropTerm ⊓\n        (.biImpl (.var x) ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧) ∧\n      p.1.toPropTermSub (· ∈ p.2.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓\n        (.biImpl (.var x) ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧) ∧\n      ∀ idx, idx ∈ p.2.toFinset → p.1.contains idx } := do\n  let ⟨(db₁, pd₁), hDb₁, hPd₁, h₁⟩ ← addPogDefClause db₀ pd₀ idx (ls.map (-·) |>.push (.mkPos x)) h\n  let ⟨(db₂, pd₂), hDb₂, hPd₂, h₂⟩ ← addPogDefClauses db₁ pd₁ idx ls (#[.mkNeg x, ·]) h₁\n  have hEquiv :\n      IClause.toPropTerm (ls.map (-·) |>.push (.mkPos x)) ⊓\n      PropForm.arrayConjTerm (ls.map (IClause.toPropForm #[ILit.mkNeg x, ·])) =\n      .biImpl (var x) ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧ := by\n    ext τ\n    simp [-satisfies_mk, PropForm.satisfies_arrayConjTerm, IClause.satisfies_iff]\n    refine ⟨?mp, ?mpr⟩\n    case mp =>\n      intro ⟨_, h⟩\n      have h : ∀ (l : ILit), l ∈ ls.data → τ ⊭ .var x ∨ τ ⊨ l.toPropTerm := fun l hL => by\n        have := h l hL\n        rw [IClause.satisfies_iff] at this\n        simp_all\n      cases h : τ x <;>\n        aesop\n    case mpr =>\n      intro\n      refine ⟨?_, fun l hL => IClause.satisfies_iff.mpr ?_⟩ <;>\n        cases h : τ x <;>\n          aesop\n  have hDb := by\n    rw [hDb₂, hDb₁, inf_assoc, hEquiv]\n  have hPd := by\n    rw [hPd₂, hPd₁, inf_assoc, hEquiv]\n  return ⟨(db₂, pd₂), hDb, hPd, h₂⟩\n\ndef addConjToPog (g : Pog) (x : Var) (ls : Array ILit) : Except CheckerError { g' : Pog //\n    g.addConj x ls = .ok g' } :=\n  match g.addConj x ls with\n  | .ok g' => pure ⟨g', rfl⟩\n  | .error e => throw <| .graphUpdateError e\n\ndef disjointUnion (ls : Array ILit) (Ds : Array (HashSet Var)) :\n    Except CheckerError { U : HashSet Var //\n      (∀ x, x ∈ U.toFinset ↔ ∃ D ∈ Ds.data, x ∈ D.toFinset) ∧\n      ∀ (i j : Nat) (hI : i < Ds.size) (hJ : j < Ds.size), i ≠ j →\n        Ds[i].toFinset ∩ Ds[j].toFinset = ∅ } :=\n  match h : HashSet.disjointUnion Ds with\n  | (U, true) =>\n    have h₁ : (HashSet.disjointUnion Ds).fst = U := congrArg Prod.fst h\n    have h₂ : (HashSet.disjointUnion Ds).snd = true := congrArg Prod.snd h\n    return ⟨U, h₁ ▸ HashSet.mem_disjointUnion Ds, HashSet.disjoint_disjointUnion Ds h₂⟩\n  | (_, false) =>\n    throw <| .depsNotDisjoint (ls.toList.map ILit.var)\n\ndef addProd (idx : ClauseIdx) (x : Var) (ls : Array ILit) : CheckerM Unit := do\n  let ⟨st, pfs⟩ ← get\n\n  -- Check that added variable is fresh.\n  let ⟨_, hX⟩ ← ensureFreshVar st x\n\n  -- Check that variables are known and compute their dependencies.\n  let ⟨Ds, hLs, hSz, hDs⟩ ← getDepsArray pfs ls\n\n  -- Compute total dependency set and check pairwise disjointness.\n  let ⟨U, hU, hDisjoint⟩ ← disjointUnion ls Ds\n\n  let ⟨pog', hPog⟩ ← addConjToPog st.pog x ls\n\n  let ⟨(db', pd'), hDb, hPd, pogDefs_in_clauseDb⟩ ←\n    addProdClauses st.clauseDb st.pogDefs idx x ls pfs.pogDefs_in_clauseDb\n\n  let st' := { st with\n    clauseDb := db'\n    pogDefs := pd'\n    pog := pog'\n    depVars := st.depVars.insert x U\n  }\n\n  have hPd : st'.pogDefsTerm = st.pogDefsTerm ⊓\n      (.biImpl (.var x) ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧) := hPd\n\n  have hDb : st'.clauseDb.toPropTerm = st.clauseDb.toPropTerm ⊓\n      (.biImpl (.var x) ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧) := hDb\n\n  have hVars : (PropForm.arrayConj (ls.map st.pog.toPropForm)).vars ⊆ U.toFinset := by\n    intro x\n    simp only [PropForm.mem_vars_arrayConj, getElem_fin, Array.getElem_map]\n    intro ⟨i, hMem⟩\n    have hI : i < ls.size := Array.size_map st.pog.toPropForm ls ▸ i.isLt\n    have := hDs i hI (hSz ▸ hI) hMem\n    exact hU x |>.mpr ⟨_, Array.getElem_mem_data _ _, this⟩\n\n  have hSemVars : ↑(PropTerm.semVars ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧) ⊆ st.allVars :=\n    by\n      apply subset_trans (Finset.coe_subset.mpr <| PropForm.semVars_subset_vars _)\n      intro x\n      simp only [Finset.mem_coe, PropForm.mem_vars_arrayConj, getElem_fin, Array.getElem_map,\n        ILit.vars_toPropForm, Finset.mem_singleton]\n      intro ⟨i, h⟩\n      exact h ▸ hLs ls[i] (Array.getElem_mem_data _ _)\n\n  have hAv : st'.allVars = insert x st.allVars := by\n    ext\n    simp [PreState.allVars, HashMap.contains_insert, @eq_comm _ x, or_comm]\n\n  have ⟨equivalent_clauseDb, extends_pogDefsTerm, uep_pogDefsTerm, equivalent_lits⟩ :=\n    def_ext_correct pfs st'\n      x ⟦PropForm.arrayConj (ls.map ILit.toPropForm)⟧ ⟦PropForm.arrayConj (ls.map st.pog.toPropForm)⟧\n      hDb hPd hAv\n      (fun l hNe => st.pog.toPropForm_addConj_lit_of_ne _ _ _ _ (by exact hPog) hNe)\n      (by simp [st.pog.toPropForm_addConj _ _ _ hPog])\n      (by\n        rw [hPd]\n        exact addConj_new_var_equiv\n          st.pog st.pogDefsTerm ls\n          hX pfs.inputCnf_vars_sub pfs.pogDefsTerm_semVars_sub pfs.uep_pogDefsTerm\n          pfs.extends_pogDefsTerm\n          fun l hL =>\n            have hMem := hLs l hL\n            ⟨hMem, pfs.pog_semVars l hMem, pfs.equivalent_lits' l hMem⟩)\n      hSemVars hX\n\n  have pfs' := {\n    pogDefs_in_clauseDb\n    clauseDb_semVars_sub := by\n      rw [hAv, hDb]\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_conj _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . exact subset_trans pfs.clauseDb_semVars_sub (Set.subset_insert _ _)\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_biImpl _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . simp\n      exact subset_trans hSemVars (Set.subset_insert _ _)\n    pogDefsTerm_semVars_sub := by\n      rw [hAv, hPd]\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_conj _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . exact subset_trans pfs.pogDefsTerm_semVars_sub (Set.subset_insert _ _)\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_biImpl _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . simp\n      exact subset_trans hSemVars (Set.subset_insert _ _)\n    inputCnf_vars_sub :=\n      hAv ▸ pfs.inputCnf_vars_sub.trans (Set.subset_insert x st.allVars)\n    depVars_pog := by\n      intro y D hFind\n      by_cases hEq : x = y\n      . rw [st.pog.toPropForm_addConj _ _ _ (hEq ▸ hPog)]\n        rw [st.depVars.find?_insert _ (beq_iff_eq _ _ |>.mpr hEq)] at hFind\n        injection hFind with hFind\n        exact hFind ▸ hVars\n      . rw [st.pog.toPropForm_addConj_of_ne _ _ _ _ hPog hEq]\n        rw [st.depVars.find?_insert_of_ne _ (bne_iff_ne _ _ |>.mpr hEq)] at hFind\n        exact pfs.depVars_pog y D hFind\n    pog_vars := by\n      intro y hMem\n      simp only [hAv, Set.mem_insert_iff] at hMem\n      cases hMem with\n      | inl hEq =>\n        intro x\n        simp only [hEq, st.pog.toPropForm_addConj _ _ _ hPog, PropForm.mem_vars_arrayConj,\n          getElem_fin, Array.getElem_map]\n        intro ⟨i, hMem⟩\n        refine pfs.pog_vars' ls[i] ?_ hMem\n        exact hLs ls[i.val] (Array.getElem_mem_data _ _)\n      | inr hMem =>\n        have hNe : x ≠ y := fun h => absurd hMem (h ▸ hX)\n        rw [st.pog.toPropForm_addConj_of_ne _ _ _ _ hPog hNe]\n        exact pfs.pog_vars y hMem\n    partitioned := by\n      intro y\n      by_cases hEq : x = y\n      . rw [st.pog.toPropForm_addConj _ _ _ (hEq ▸ hPog), PropForm.partitioned_arrayConj]\n        intro i\n        simp only [getElem_fin, Array.getElem_map]\n        constructor\n        . exact pfs.partitioned' (ls[i.val])\n        . intro j hNe\n          have h := Array.size_map st.pog.toPropForm ls\n          have h' := h.trans hSz.symm\n          have hI := hDs i (h ▸ i.isLt) (h' ▸ i.isLt)\n          have hJ := hDs j (h ▸ j.isLt) (h' ▸ j.isLt)\n          have hIJ := hDisjoint i j (h' ▸ i.isLt) (h' ▸ j.isLt) (Fin.val_ne_of_ne hNe)\n          simp at hI hJ hIJ ⊢\n          apply Finset.subset_empty.mp\n          exact hIJ ▸ Finset.inter_subset_inter hI hJ\n      . rw [st.pog.toPropForm_addConj_of_ne _ _ _ _ hPog hEq]\n        exact pfs.partitioned y\n    equivalent_clauseDb\n    extends_pogDefsTerm\n    uep_pogDefsTerm\n    equivalent_lits\n  }\n  set (σ := State) ⟨st', pfs'⟩\n\ndef addSumClauses (db₀ : ClauseDb ClauseIdx) (pd₀ : HashSet ClauseIdx)\n    (idx : ClauseIdx) (x : Var) (l₁ l₂ : ILit) (h : ∀ idx, idx ∈ pd₀.toFinset → db₀.contains idx) :\n    Except CheckerError { p : ClauseDb ClauseIdx × HashSet ClauseIdx //\n      p.1.toPropTerm = db₀.toPropTerm ⊓\n        (.biImpl (.var x) (l₁.toPropTerm ⊔ l₂.toPropTerm)) ∧\n      p.1.toPropTermSub (· ∈ p.2.toFinset) = db₀.toPropTermSub (· ∈ pd₀.toFinset) ⊓\n        (.biImpl (.var x) (l₁.toPropTerm ⊔ l₂.toPropTerm)) ∧\n      ∀ idx, idx ∈ p.2.toFinset → p.1.contains idx } := do\n  let ⟨(db₁, pd₁), hDb₁, hPd₁, h⟩ ← addPogDefClause db₀ pd₀ idx     #[.mkNeg x, l₁, l₂] h\n  let ⟨(db₂, pd₂), hDb₂, hPd₂, h⟩ ← addPogDefClause db₁ pd₁ (idx+1) #[.mkPos x, -l₁]    h\n  let ⟨(db₃, pd₃), hDb₃, hPd₃, h⟩ ← addPogDefClause db₂ pd₂ (idx+2) #[.mkPos x, -l₂]    h\n  have hDb := by\n    rw [hDb₃, hDb₂, hDb₁]\n    simp [IClause.toPropTerm, inf_assoc, PropTerm.disj_def_eq]\n  have hPd := by\n    rw [hPd₃, hPd₂, hPd₁]\n    simp [IClause.toPropTerm, inf_assoc, PropTerm.disj_def_eq]\n  return ⟨(db₃, pd₃), hDb, hPd, h⟩\n\ndef ensurePogHints (st : PreState) (hints : Array ClauseIdx) :\n    Except CheckerError { _u : Unit //\n      ∀ idx, idx ∈ hints.data → idx ∈ st.pogDefs' } := do\n  match hSz : hints.size with\n  | 0 =>\n    return ⟨(), fun _ hMem => by\n      dsimp [Array.size] at hSz\n      rw [List.length_eq_zero.mp hSz] at hMem\n      contradiction⟩\n  | i+1 =>\n    let ⟨_, h⟩ ← go i (hSz ▸ Nat.lt_succ_self _) (fun j hLt => by\n      have := j.isLt\n      linarith)\n    return ⟨(), fun _ hMem => have ⟨i, hI⟩ := Array.get_of_mem_data hMem; hI ▸ h i⟩\nwhere go (i : Nat) (hLt : i < hints.size)\n    (ih : ∀ (j : Fin hints.size), i < j → hints[j] ∈ st.pogDefs') :\n    Except CheckerError { _u : Unit // ∀ (j : Fin hints.size), hints[j] ∈ st.pogDefs' } := do\n  let idx := hints[i]\n  if hContains : st.pogDefs.contains idx then\n    have hContains : hints[i] ∈ st.pogDefs' :=\n      by simp [PreState.pogDefs', HashSet.mem_toFinset, hContains]\n    match hI : i, hLt, ih with\n    | 0, hLt, ih =>\n      return ⟨(), fun j => by\n        cases j.val.eq_zero_or_pos with\n        | inl hEq =>\n          -- Why does this compute a correct motive while `rw` doesn't?\n          simp only [hI] at hContains\n          simp [hEq, hContains]\n        | inr hLt =>\n          exact ih j hLt⟩\n    | i+1, hLt, ih =>\n      by exact -- Hmmm\n        go i (Nat.lt_of_succ_lt hLt) (fun j hLt => by\n          cases Nat.eq_or_lt_of_le hLt with\n          | inl hEq =>\n            simp only [hI] at hContains\n            simp [← hEq, hContains]\n          | inr hLt =>\n            exact ih j hLt)\n  else\n    throw <| .hintNotPog idx\n\ndef addDisjToPog (g : Pog) (x : Var) (l₁ l₂ : ILit) : Except CheckerError { g' : Pog //\n    g.addDisj x l₁ l₂ = .ok g' } :=\n  match g.addDisj x l₁ l₂ with\n  | .ok g' => pure ⟨g', rfl⟩\n  | .error e => throw <| .graphUpdateError e\n\ndef addSum (idx : ClauseIdx) (x : Var) (l₁ l₂ : ILit) (hints : Array ClauseIdx) :\n    CheckerM Unit := do\n  let ⟨st, pfs⟩ ← get\n\n  -- Check that added variable is fresh.\n  let ⟨_, hX⟩ ← ensureFreshVar st x\n\n  -- Check that variables are known and compute their dependencies.\n  let ⟨D₁, hL₁, hD₁⟩ ← getDeps st pfs l₁\n  let ⟨D₂, hL₂, hD₂⟩ ← getDeps st pfs l₂\n\n  -- Check that POG defs imply that the children have no models in common.\n  let ⟨_, hHints⟩ ← ensurePogHints st hints\n  -- NOTE: Important that this be done before adding clauses, for linearity.\n  let ⟨_, hImp⟩ ← checkImpliedWithHints st.clauseDb #[-l₁, -l₂] hints\n\n  let ⟨pog', hPog⟩ ← addDisjToPog st.pog x l₁ l₂\n\n  let ⟨(db', pd'), hDb, hPd, pogDefs_in_clauseDb⟩ ←\n    addSumClauses st.clauseDb st.pogDefs idx x l₁ l₂ pfs.pogDefs_in_clauseDb\n\n  let st' := { st with\n    clauseDb := db'\n    pogDefs := pd'\n    pog := pog'\n    depVars := st.depVars.insert x (D₁.union D₂)\n  }\n\n  have hPd : st'.pogDefsTerm = st.pogDefsTerm ⊓ (.biImpl (.var x) (l₁.toPropTerm ⊔ l₂.toPropTerm))\n    := hPd\n\n  have hDb : st'.clauseDb.toPropTerm = st.clauseDb.toPropTerm ⊓\n      (.biImpl (.var x) (l₁.toPropTerm ⊔ l₂.toPropTerm)) :=\n    hDb\n\n  have hDisjoint : st.pogDefsTerm ⊓ l₁.toPropTerm ⊓ l₂.toPropTerm ≤ ⊥ := by\n    have : st.pogDefsTerm ≤ st.clauseDb.toPropTermSub (· ∈ hints.data) :=\n      st.clauseDb.toPropTermSub_subset hHints\n    rw [inf_assoc, ← le_himp_iff]\n    have := le_trans this hImp\n    simp [IClause.toPropTerm] at this\n    simp [this]\n\n  have hSemVars : ↑(l₁.toPropTerm ⊔ l₂.toPropTerm).semVars ⊆ st.allVars := by\n    have : ↑(l₁.toPropTerm.semVars ∪ l₂.toPropTerm.semVars) ⊆ st.allVars := by\n      intro x h\n      simp at h\n      cases h <;> next h => simp only [h, hL₁, hL₂]\n    exact subset_trans (PropTerm.semVars_disj _ _) this\n\n  have hAv : st'.allVars = insert x st.allVars := by\n    ext\n    simp [PreState.allVars, HashMap.contains_insert, @eq_comm _ x, or_comm]\n\n  have ⟨equivalent_clauseDb, extends_pogDefsTerm, uep_pogDefsTerm, equivalent_lits⟩ :=\n    def_ext_correct pfs st'\n      x (l₁.toPropTerm ⊔ l₂.toPropTerm) (⟦st.pog.toPropForm l₁⟧ ⊔ ⟦st.pog.toPropForm l₂⟧)\n      hDb hPd hAv\n      (fun l hNe => st.pog.toPropForm_addDisj_lit_of_ne _ _ _ _ _ (by exact hPog) hNe)\n      (by simp [st.pog.toPropForm_addDisj _ _ _ _ hPog])\n      (by\n        rw [hPd, ← inf_assoc]\n        exact addDisj_new_var_equiv st.pogDefsTerm l₁.toPropTerm l₂.toPropTerm _ _ hX\n          (pfs.inputCnf_vars_sub) (pfs.pogDefsTerm_semVars_sub)\n          (by simp [hL₁]) (by simp [hL₂])\n          (pfs.equivalent_lits' l₁ hL₁) (pfs.equivalent_lits' l₂ hL₂))\n      hSemVars hX\n\n  have pfs' := {\n    pogDefs_in_clauseDb\n    clauseDb_semVars_sub := by\n      rw [hAv, hDb]\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_conj _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . exact subset_trans pfs.clauseDb_semVars_sub (Set.subset_insert _ _)\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_biImpl _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . simp\n      exact subset_trans hSemVars (Set.subset_insert _ _)\n    pogDefsTerm_semVars_sub := by\n      rw [hAv, hPd]\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_conj _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . exact subset_trans pfs.pogDefsTerm_semVars_sub (Set.subset_insert _ _)\n      apply subset_trans (Finset.coe_subset.mpr <| semVars_biImpl _ _)\n      rw [Finset.coe_union]\n      apply Set.union_subset\n      . simp\n      exact subset_trans hSemVars (Set.subset_insert _ _)\n    inputCnf_vars_sub :=\n      hAv ▸ pfs.inputCnf_vars_sub.trans (Set.subset_insert x st.allVars)\n    depVars_pog := by\n      intro y D hFind\n      by_cases hEq : x = y\n      . rw [st.pog.toPropForm_addDisj _ _ _ _ (hEq ▸ hPog)]\n        rw [st.depVars.find?_insert _ (beq_iff_eq _ _ |>.mpr hEq)] at hFind\n        injection hFind with hFind\n        rw [PropForm.vars, ← hFind, HashSet.toFinset_union]\n        apply Finset.union_subset_union <;> assumption\n      . rw [st.pog.toPropForm_addDisj_of_ne _ _ _ _ _ hPog hEq]\n        rw [st.depVars.find?_insert_of_ne _ (bne_iff_ne _ _ |>.mpr hEq)] at hFind\n        exact pfs.depVars_pog y D hFind\n    pog_vars := by\n      intro y hMem\n      simp only [hAv, Set.mem_insert_iff] at hMem\n      cases hMem\n      next hEq =>\n        simp only [hEq, st.pog.toPropForm_addDisj _ _ _ _ hPog, PropForm.vars]\n        exact Finset.union_subset (pfs.pog_vars' l₁ hL₁) (pfs.pog_vars' l₂ hL₂)\n      next hMem =>\n        have hNe : x ≠ y := fun h => absurd hMem (h ▸ hX)\n        rw [st.pog.toPropForm_addDisj_of_ne _ _ _ _ _ hPog hNe]\n        exact pfs.pog_vars y hMem\n    partitioned := by\n      intro y\n      by_cases hEq : x = y\n      . rw [st.pog.toPropForm_addDisj _ _ _ _ (hEq ▸ hPog)]\n        refine addDisj_partitioned st.pogDefsTerm l₁.toPropTerm l₂.toPropTerm _ _ ?_\n          pfs.uep_pogDefsTerm hDisjoint (pfs.equivalent_lits' l₁ hL₁) (pfs.equivalent_lits' l₂ hL₂)\n          (pfs.partitioned' l₁) (pfs.partitioned' l₂)\n        simp [hL₂]\n      . rw [st.pog.toPropForm_addDisj_of_ne _ _ _ _ _ hPog hEq]\n        exact pfs.partitioned y\n    equivalent_clauseDb\n    extends_pogDefsTerm\n    uep_pogDefsTerm\n    equivalent_lits\n  }\n  set (σ := State) ⟨st', pfs'⟩\n\ndef setRoot (r : ILit) : CheckerM Unit := do\n  modify fun ⟨st, pfs⟩ => ⟨{ st with root := some r }, { pfs with }⟩\n\ndef checkFinalClauses (r : ILit) (st : PreState) : Except CheckerError { _u : Unit //\n    (∀ idx C, st.clauseDb.getClause idx = some C → idx ∈ st.pogDefs' ∨ C = #[r])\n    ∧ (∃ idxᵣ, st.clauseDb.getClause idxᵣ = some #[r]) } := do\n  /- NOTE: This check is seriously inefficient. First, `all`/`any` don't use early return. Second,\n  we loop over the clauses twice. Third, this could likely all be implemented in O(1) by storing\n  the number `nClauses` of clauses. As long as `nClauses = st.pogDefs.size + 1` (`+ 1` for the root\n  literal), the conclusion should follow from appropriate invariants. -/\n  match h₁ : st.clauseDb.all (fun idx C => C = #[r] ∨ st.pogDefs.contains idx) with\n  | true =>\n    match h₂ : st.clauseDb.any (fun _ C => C = #[r]) with\n    | true =>\n      have hA := by\n        intro idx C hGet\n        have := st.clauseDb.all_true _ h₁ idx C hGet\n        simp at this\n        rw [PreState.pogDefs', HashSet.mem_toFinset]\n        exact Or.comm.mp this\n      have hE := by\n        have ⟨idxᵣ, C, hGet, hP⟩ := st.clauseDb.any_true _ h₂\n        simp at hP\n        exact ⟨idxᵣ, hP ▸ hGet⟩\n      return ⟨(), hA, hE⟩\n    | false => throw <| .finalRootNotUnit r\n  | false => throw <| .finalClausesInvalid\n\ndef checkFinalState : CheckerM { p : ICnf × Pog × ILit //\n    p.1.toPropTerm = ⟦p.2.1.toPropForm p.2.2⟧ } := do\n  let ⟨st, pfs⟩ ← get\n\n  let some r := st.root\n    | throw <| .finalRootNotSet\n  let ⟨_, hR, _⟩ ← getDeps st pfs r\n\n  let ⟨_, hA, hE⟩ ← checkFinalClauses r st\n  have : st.clauseDb.toPropTerm = r.toPropTerm ⊓ st.pogDefsTerm := by\n    have ⟨idxᵣ, hGet⟩ := hE\n    ext τ\n    rw [st.clauseDb.satisfies_toPropTerm, PreState.pogDefsTerm, satisfies_conj,\n      st.clauseDb.satisfies_toPropTermSub]\n    constructor\n    . intro h\n      have := h _ _ hGet\n      dsimp [IClause.toPropTerm] at this\n      aesop\n    . intro ⟨h₁, h₂⟩ _ _ hGet\n      cases hA _ _ hGet\n      next hMem => exact h₂ _ hMem _ hGet\n      next hEq =>\n        rw [hEq]\n        simp [IClause.toPropTerm, h₁]\n\n  have : st.inputCnf.toPropTerm = ⟦st.pog.toPropForm r⟧ := by\n    have := this ▸ pfs.equivalent_clauseDb\n    have := this.trans (pfs.equivalent_lits' r hR)\n    have hInputVars := PropForm.semVars_subset_vars st.inputCnf.toPropForm\n    simp at hInputVars\n    have hPogVars := subset_trans (PropForm.semVars_subset_vars _) (pfs.pog_vars' r hR)\n    exact equivalentOver_semVars hInputVars hPogVars this\n\n  return ⟨(st.inputCnf, st.pog, r), this⟩\n\ndef checkProofStep (step : CpogStep) : CheckerM Unit :=\n  match step with\n  | .addAt idx C hints => addAt idx C hints\n  | .delAt idx hints => delAt idx hints\n  | .prod idx x ls => addProd idx x ls\n  | .sum idx x l₁ l₂ hints => addSum idx x l₁ l₂ hints\n  | .root r => setRoot r\n\n/-- Check a CPOG proof and throw if it is invalid. If `count = True`, also return the model count\nof `cnf` over `nVars` variables. Otherwise return an unspecified number. -/\ndef checkProof (cnf : ICnf) (nVars : Nat) (pf : Array CpogStep) (count : Bool := False) :\n    Except String Nat := do\n  let mut st : State ← Except.mapError toString (initial cnf nVars)\n  for step in pf do\n    try\n      (_, st) ← Except.mapError toString (checkProofStep step |>.run st)\n    catch e =>\n      throw <| s!\"error on line '{step.toDimacs}':\\n{e}\"\n  let ⟨(_, _, r), _⟩ ← Except.mapError toString (checkFinalState.run' st)\n  if count then\n    if r.polarity = true then\n      return st.val.pog.count nVars r.var\n    else\n      return 2^nVars - st.val.pog.count nVars r.var\n  else\n    return 0\n\n/-\n-- LATER: re-add tracing\n\n/-- Wraps a well-formed checker state with extra stuff for tracing and debugging it. -/\nstructure CheckerState' where\n  core : CheckerState\n  verbose : Bool := false\n  trace : Array String := #[]\n\nnamespace CheckerState\n\nabbrev CheckerM := ExceptT CheckerError <| StateM CheckerState\n\ndef withTraces (f : Array String → String) (x : CheckerM Unit) : CheckerM Unit := do\n  if (← get).verbose then\n    let prevTrace ← modifyGet fun st => (st.trace, { st with trace := #[] })\n    try x\n    finally\n      modify fun st => { st with trace := prevTrace.push <| f st.trace }\n  else\n    x\n\ndef log_ (msg : Unit → String) : CheckerM Unit := do\n  modify fun st =>\n    if st.verbose then { st with trace := st.trace.push <| msg () }\n    else st\n\nsyntax \"log! \" interpolatedStr(term) : term\nmacro_rules\n  | `(log! $interpStr) => `(log_ fun _ => s!$interpStr)\n\nend CheckerState\n-/", "meta": {"author": "rebryant", "repo": "cpog", "sha": "5e39029ce71de532fd4407c4768e7c2bf97798c8", "save_path": "github-repos/lean/rebryant-cpog", "path": "github-repos/lean/rebryant-cpog/cpog-5e39029ce71de532fd4407c4768e7c2bf97798c8/VerifiedChecker/ProofChecker/Checker/CheckerCore.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3176299257038323}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.types\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport tactic.elementwise\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nuniverses w v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nattribute [elementwise] cone.w limit.lift_π limit.w cocone.w colimit.ι_desc colimit.w\n\nlocal attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nsection limits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J ⥤ C) [preserves_limit F (forget C)]\n\nlemma concrete.to_product_injective_of_is_limit {D : cone F} (hD : is_limit D) :\n  function.injective (λ (x : D.X) (j : J), D.π.app j x) :=\nbegin\n  let E := (forget C).map_cone D,\n  let hE : is_limit E := is_limit_of_preserves _ hD,\n  let G := types.limit_cone (F ⋙ forget C),\n  let hG := types.limit_cone_is_limit (F ⋙ forget C),\n  let T : E.X ≅ G.X := hE.cone_point_unique_up_to_iso hG,\n  change function.injective (T.hom ≫ (λ x j, G.π.app j x)),\n  have h : function.injective T.hom,\n  { intros a b h,\n    suffices : T.inv (T.hom a) = T.inv (T.hom b), by simpa,\n    rw h },\n  suffices : function.injective (λ (x : G.X) j, G.π.app j x),\n    by exact this.comp h,\n  apply subtype.ext,\nend\n\nlemma concrete.is_limit_ext {D : cone F} (hD : is_limit D) (x y : D.X) :\n  (∀ j, D.π.app j x = D.π.app j y) → x = y :=\nλ h, concrete.to_product_injective_of_is_limit _ hD (funext h)\n\nlemma concrete.limit_ext [has_limit F] (x y : limit F) :\n  (∀ j, limit.π F j x = limit.π F j y) → x = y :=\nconcrete.is_limit_ext F (limit.is_limit _) _ _\n\nsection wide_pullback\n\nopen wide_pullback\nopen wide_pullback_shape\n\nlemma concrete.wide_pullback_ext {B : C} {ι : Type*} {X : ι → C} (f : Π j : ι, X j ⟶ B)\n  [has_wide_pullback B X f] [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h₀ : base f x = base f y)\n  (h : ∀ j, π f j x = π f j y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintro (_|j),\n  { exact h₀ },\n  { apply h }\nend\n\nlemma concrete.wide_pullback_ext' {B : C} {ι : Type*} [nonempty ι]\n  {X : ι → C} (f : Π j : ι, X j ⟶ B) [has_wide_pullback B X f]\n  [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h : ∀ j, π f j x = π f j y) : x = y :=\nbegin\n  apply concrete.wide_pullback_ext _ _ _ _ h,\n  inhabit ι,\n  simp only [← π_arrow f (arbitrary _), comp_apply, h],\nend\n\nend wide_pullback\n\nsection multiequalizer\n\nlemma concrete.multiequalizer_ext {I : multicospan_index C} [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x y : multiequalizer I)\n  (h : ∀ (t : I.L), multiequalizer.ι I t x = multiequalizer.ι I t y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintros (a|b),\n  { apply h },\n  { rw [← limit.w I.multicospan (walking_multicospan.hom.fst b),\n      comp_apply, comp_apply, h] }\nend\n\n/-- An auxiliary equivalence to be used in `multiequalizer_equiv` below.-/\ndef concrete.multiequalizer_equiv_aux (I : multicospan_index C) :\n  (I.multicospan ⋙ (forget C)).sections ≃\n  { x : Π (i : I.L), I.left i // ∀ (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\n{ to_fun := λ x, ⟨λ i, x.1 (walking_multicospan.left _), λ i, begin\n    have a := x.2 (walking_multicospan.hom.fst i),\n    have b := x.2 (walking_multicospan.hom.snd i),\n    rw ← b at a,\n    exact a,\n  end⟩,\n  inv_fun := λ x,\n  { val := λ j,\n    match j with\n    | walking_multicospan.left a := x.1 _\n    | walking_multicospan.right b := I.fst b (x.1 _)\n    end,\n    property := begin\n      rintros (a|b) (a'|b') (f|f|f),\n      { change (I.multicospan.map (𝟙 _)) _ = _, simp },\n      { refl },\n      { dsimp, erw ← x.2 b', refl },\n      { change (I.multicospan.map (𝟙 _)) _ = _, simp },\n    end },\n  left_inv := begin\n    intros x, ext (a|b),\n    { refl },\n    { change _ = x.val _,\n      rw ← x.2 (walking_multicospan.hom.fst b),\n      refl }\n  end,\n  right_inv := by { intros x, ext i, refl } }\n\n/-- The equivalence between the noncomputable multiequalizer and\nand the concrete multiequalizer. -/\nnoncomputable\ndef concrete.multiequalizer_equiv (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] : (multiequalizer I : C) ≃\n    { x : Π (i : I.L), I.left i // ∀ (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\nlet h1 := (limit.is_limit I.multicospan),\n    h2 := (is_limit_of_preserves (forget C) h1),\n    E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit _) in\nequiv.trans E.to_equiv (concrete.multiequalizer_equiv_aux I)\n\n@[simp]\nlemma concrete.multiequalizer_equiv_apply (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :\n  ((concrete.multiequalizer_equiv I) x : Π (i : I.L), I.left i) i = multiequalizer.ι I i x := rfl\n\nend multiequalizer\n\n-- TODO: Add analogous lemmas about products and equalizers.\n\nend limits\n\nsection colimits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J ⥤ C) [preserves_colimit F (forget C)]\n\nlemma concrete.from_union_surjective_of_is_colimit {D : cocone F} (hD : is_colimit D) :\n  let ff : (Σ (j : J), F.obj j) → D.X := λ a, D.ι.app a.1 a.2 in function.surjective ff :=\nbegin\n  intro ff,\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F ⋙ forget C),\n  let hG := types.colimit_cocone_is_colimit (F ⋙ forget C),\n  let T : E ≅ G := hE.unique_up_to_iso hG,\n  let TX : E.X ≅ G.X := (cocones.forget _).map_iso T,\n  suffices : function.surjective (TX.hom ∘ ff),\n  { intro a,\n    obtain ⟨b, hb⟩ := this (TX.hom a),\n    refine ⟨b, _⟩,\n    apply_fun TX.inv at hb,\n    change (TX.hom ≫ TX.inv) (ff b) = (TX.hom ≫ TX.inv) _ at hb,\n    simpa only [TX.hom_inv_id] using hb },\n  have : TX.hom ∘ ff = λ a, G.ι.app a.1 a.2,\n  { ext a,\n    change (E.ι.app a.1 ≫ hE.desc G) a.2 = _,\n    rw hE.fac },\n  rw this,\n  rintro ⟨⟨j,a⟩⟩,\n  exact ⟨⟨j,a⟩,rfl⟩,\nend\n\nlemma concrete.is_colimit_exists_rep {D : cocone F} (hD : is_colimit D) (x : D.X) :\n  ∃ (j : J) (y : F.obj j), D.ι.app j y = x :=\nbegin\n  obtain ⟨a, rfl⟩ := concrete.from_union_surjective_of_is_colimit F hD x,\n  exact ⟨a.1, a.2, rfl⟩,\nend\n\nlemma concrete.colimit_exists_rep [has_colimit F] (x : colimit F) :\n  ∃ (j : J) (y : F.obj j), colimit.ι F j y = x :=\nconcrete.is_colimit_exists_rep F (colimit.is_colimit _) x\n\nlemma concrete.is_colimit_rep_eq_of_exists {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :\n  D.ι.app i x = D.ι.app j y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F ⋙ forget C),\n  let hG := types.colimit_cocone_is_colimit (F ⋙ forget C),\n  let T : E ≅ G := hE.unique_up_to_iso hG,\n  let TX : E.X ≅ G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom,\n  swap, { suffices : function.bijective TX.hom, by exact this.1,\n    rw ← is_iso_iff_bijective, apply is_iso.of_iso },\n  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y,\n  erw [T.hom.w, T.hom.w],\n  obtain ⟨k, f, g, h⟩ := h,\n  have : G.ι.app i x = (G.ι.app k (F.map f x) : G.X) := quot.sound ⟨f,rfl⟩,\n  rw [this, h],\n  symmetry,\n  exact quot.sound ⟨g,rfl⟩,\nend\n\nlemma concrete.colimit_rep_eq_of_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y) :\n  colimit.ι F i x = colimit.ι F j y :=\nconcrete.is_colimit_rep_eq_of_exists F (colimit.is_colimit _) x y h\n\nsection filtered_colimits\n\nvariable [is_filtered J]\n\nlemma concrete.is_colimit_exists_of_rep_eq {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : D.ι.app _ x = D.ι.app _ y) :\n  ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F ⋙ forget C),\n  let hG := types.colimit_cocone_is_colimit (F ⋙ forget C),\n  let T : E ≅ G := hE.unique_up_to_iso hG,\n  let TX : E.X ≅ G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom at h,\n  change (E.ι.app i ≫ TX.hom) x = (E.ι.app j ≫ TX.hom) y at h,\n  erw [T.hom.w, T.hom.w] at h,\n  replace h := quot.exact _ h,\n  suffices : ∀ (a b : Σ j, F.obj j)\n    (h : eqv_gen (limits.types.quot.rel (F ⋙ forget C)) a b),\n    ∃ k (f : a.1 ⟶ k) (g : b.1 ⟶ k), F.map f a.2 = F.map g b.2,\n  { exact this ⟨i,x⟩ ⟨j,y⟩ h },\n  intros a b h,\n  induction h,\n  case eqv_gen.rel : x y hh\n  { obtain ⟨e,he⟩ := hh,\n    use [y.1, e, 𝟙 _],\n    simpa using he.symm },\n  case eqv_gen.refl : x { use [x.1, 𝟙 _, 𝟙 _, rfl] },\n  case eqv_gen.symm : x y _ hh\n  { obtain ⟨k, f, g, hh⟩ := hh,\n    use [k, g, f, hh.symm] },\n  case eqv_gen.trans : x y z _ _ hh1 hh2\n  { obtain ⟨k1, f1, g1, h1⟩ := hh1,\n    obtain ⟨k2, f2, g2, h2⟩ := hh2,\n    let k0 : J := is_filtered.max k1 k2,\n    let e1 : k1 ⟶ k0 := is_filtered.left_to_max _ _,\n    let e2 : k2 ⟶ k0 := is_filtered.right_to_max _ _,\n    let k : J := is_filtered.coeq (g1 ≫ e1) (f2 ≫ e2),\n    let e : k0 ⟶ k := is_filtered.coeq_hom _ _,\n    use [k, f1 ≫ e1 ≫ e, g2 ≫ e2 ≫ e],\n    simp only [F.map_comp, comp_apply, h1, ← h2],\n    simp only [← comp_apply, ← F.map_comp],\n    rw is_filtered.coeq_condition },\nend\n\ntheorem concrete.is_colimit_rep_eq_iff_exists {D : cocone F} {i j : J}\n  (hD : is_colimit D) (x : F.obj i) (y : F.obj j) :\n  D.ι.app i x = D.ι.app j y ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y :=\n⟨concrete.is_colimit_exists_of_rep_eq _ hD _ _, concrete.is_colimit_rep_eq_of_exists _ hD _ _⟩\n\nlemma concrete.colimit_exists_of_rep_eq [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : colimit.ι F _ x = colimit.ι F _ y) :\n  ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y :=\nconcrete.is_colimit_exists_of_rep_eq F (colimit.is_colimit _) x y h\n\ntheorem concrete.colimit_rep_eq_iff_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) :\n  colimit.ι F i x = colimit.ι F j y ↔ ∃ k (f : i ⟶ k) (g : j ⟶ k), F.map f x = F.map g y :=\n⟨concrete.colimit_exists_of_rep_eq _ _ _, concrete.colimit_rep_eq_of_exists _ _ _⟩\n\nend filtered_colimits\n\nsection wide_pushout\n\nopen wide_pushout\nopen wide_pushout_shape\n\nlemma concrete.wide_pushout_exists_rep {B : C} {α : Type*} {X : α → C} (f : Π j : α, B ⟶ X j)\n  [has_wide_pushout B X f] [preserves_colimit (wide_span B X f) (forget C)]\n  (x : wide_pushout B X f) : (∃ y : B, head f y = x) ∨ (∃ (i : α) (y : X i), ι f i y = x) :=\nbegin\n  obtain ⟨_ | j, y, rfl⟩ := concrete.colimit_exists_rep _ x,\n  { use y },\n  { right,\n    use [j,y] }\nend\n\nlemma concrete.wide_pushout_exists_rep' {B : C} {α : Type*} [nonempty α] {X : α → C}\n  (f : Π j : α, B ⟶ X j) [has_wide_pushout B X f]\n  [preserves_colimit (wide_span B X f) (forget C)] (x : wide_pushout B X f) :\n  ∃ (i : α) (y : X i), ι f i y = x :=\nbegin\n  rcases concrete.wide_pushout_exists_rep f x with ⟨y, rfl⟩ | ⟨i, y, rfl⟩,\n  { inhabit α,\n    use [arbitrary _, f _ y],\n    simp only [← arrow_ι _ (arbitrary α), comp_apply] },\n  { use [i,y] }\nend\n\nend wide_pushout\n\n-- TODO: Add analogous lemmas about coproducts and coequalizers.\n\nend colimits\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/concrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213368305399, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.31762598007987086}}
{"text": "\nimport data.equiv.basic\nimport tactic\n\nuniverses u₀ v₀ u₁ v₁ v₂ w w₀ w₁\n\nclass liftable (α : Type u₀) (β : (Type v₀)) :=\n  (lift {}   : α ≃ β)\n\n-- instance {α} : liftable α (ulift α) :=\n-- ⟨ equiv.ulift.symm ⟩\n\ninstance ulift.liftable.ulift {α : Type w} : liftable (ulift.{u₀} α) (ulift.{u₁} α) :=\n⟨ equiv.trans equiv.ulift equiv.ulift.symm ⟩\n\nclass liftable1 (f : (Type u₀ → Type u₁)) (g : Type v₀ → Type v₁) :=\n  (up   : Π {α β}, α ≃ β → f α → g β)\n  (down : Π {α β}, α ≃ β → g β → f α)\n  (up_down : ∀ {α β} (F : α ≃ β) (x : g β), up F (down F x) = x)\n  (down_up : ∀ {α β} (F : α ≃ β) (x : f α), down F (up F x) = x)\n\nattribute [simp] liftable1.up_down liftable1.down_up\n\nnamespace liftable\n\n@[reducible]\ndef up' {f : Type v₀ → Type v₁} (g : Type (max u₀ v₀) → Type u₁) [liftable1 f g]\n  {α} : f α → g (ulift.{u₀} α) :=\nliftable1.up g equiv.ulift.symm\n\n@[reducible]\ndef down' {f : Type u₀ → Type u₁} {g : Type (max u₀ v₀) → Type v₁} [liftable1 f g]\n  {α} : g (ulift α) → f α :=\nliftable1.down _ equiv.ulift.symm\n\n@[reducible]\ndef shift' {f : Type (max w u₀) → Type u₁} {g : Type (max w v₀) → Type v₁} [liftable1 f g]\n  {α : Type w} : g (ulift α) → f (ulift α) :=\nliftable1.down _ $ equiv.trans equiv.ulift equiv.ulift.symm\n\nend liftable\n\nopen ulift\n\nprotected lemma prod.map_map {α α' α'' β β' β''}\n  (x : α × β) (f : α → α') (f' : α' → α'')\n              (g : β → β') (g' : β' → β'') :\n  prod.map f' g' (prod.map f g x) = prod.map (f' ∘ f) (g' ∘ g) x :=\nby cases x; refl\n\nprotected lemma prod.id_map {α β}\n  (x : α × β) :\n  prod.map (λ x, x) (λ x, x) x = x :=\nby cases x; refl\n\ninstance : liftable1 id id :=\n{ up := λ _ _ F, F\n, down := λ _ _ F, F.symm\n, up_down := by intros; simp\n, down_up := by intros; simp }\n\ndef state_t.liftable' {s : Type u₀} {s' : Type u₁} {m m'} [functor m'] [functor m] [is_lawful_functor m'] [is_lawful_functor m]\n  [liftable1 m m']\n  (F : s ≃ s') :\n  liftable1 (state_t s m) (state_t s' m') :=\n{ up   := λ _ _ G ⟨ f ⟩, ⟨ λ s, liftable1.up _ (equiv.prod_congr G F) (f $ F.symm s) ⟩\n, down := λ _ _ G ⟨ g ⟩, ⟨ λ s, liftable1.down _ (equiv.prod_congr G F) $ g (F s) ⟩\n, up_down := by { rintros α β G ⟨ f ⟩, simp! }\n, down_up := by { rintros α β G ⟨ g ⟩, simp! [map_map,function.comp] } }\n\ninstance {s : Type u₀} {s' : Type u₁} {m m'} [functor m'] [functor m] [is_lawful_functor m'] [is_lawful_functor m]\n  [liftable1 m m'] [liftable s s'] :\n  liftable1 (state_t s m) (state_t s' m') :=\nstate_t.liftable' liftable.lift\n\ndef reader_t.liftable' {s : Type u₀} {s' : Type u₁} {m : Type u₀ → Type v₀} {m' : Type u₁ → Type v₁}\n  [liftable1 m m']\n  (F : s ≃ s') :\n  liftable1 (reader_t s m) (reader_t s' m') :=\n{ up   := λ _ _ G ⟨ f ⟩, ⟨ λ s, liftable1.up _ G (f $ F.symm s) ⟩\n, down := λ _ _ G ⟨ g ⟩, ⟨ λ s, liftable1.down _ G $ g $ F s ⟩\n, up_down := by { rintros α β G ⟨ f ⟩, simp! }\n, down_up := by { rintros α β G ⟨ g ⟩, simp! [map_map,function.comp] } }\n\ninstance {s : Type u₀} {s' : Type u₁} {m : Type u₀ → Type v₀} {m' : Type u₁ → Type v₁}\n  [liftable1 m m'] [liftable s s'] :\n  liftable1 (reader_t s m) (reader_t s' m') :=\nreader_t.liftable' liftable.lift\n\n-- namespace liftable\n\n-- variables {f : Type (max u₀ w) → Type u₁} {g : Type (max (max u₀ w) v₀) → Type v₁}\n-- variables [liftable f g] [functor g]\n-- variable {α : Type w}\n-- open functor\n\n-- def up' (x : f (ulift.{u₀} α)) : g (ulift.{max u₀ v₀} α) :=\n-- map (ulift.up ∘ ulift.down ∘ ulift.down) $ up x\n\n-- def down' (x : g (ulift.{max u₀ v₀} α)) : f (ulift.{u₀} α) :=\n-- down $ map (ulift.up ∘ ulift.up ∘ ulift.down) x\n\n-- variables [is_lawful_functor g]\n\n-- lemma up_down' : ∀ (x : g (ulift.{max u₀ v₀} α)), up' (down' x : f (ulift.{u₀} α)) = x :=\n-- by intros; simp [up',down',map_map,function.comp,ulift.up_down]\n\n-- lemma down_up' : ∀ (x : f (ulift.{u₀} α)), down' (up' x : g (ulift.{max u₀ v₀} α)) = x :=\n-- by intros; simp [up',down',map_map,function.comp,ulift.up_down]\n\n-- end liftable\n\nsection trans\nopen liftable\n\n-- def liftable.trans\n--     {f : Type u₀ → Type u₁}\n--     {g : Type v₀ → Type v₁}\n--     {h : Type w₀ → Type w₁}\n--     [functor h] [is_lawful_functor h]\n--     (_ : liftable f g)\n--     (_ : liftable g h) :\n--   liftable f h :=\n-- by refine\n--      { up := λ α β G, (up' : g (ulift α) → h (ulift.{max v₀ w} α)) ∘ (up : f α → g (ulift α)),\n--        down := λ α β G, (down : g (ulift α) → f α) ∘ (down' : h (ulift.{max v₀ w} α) → g (ulift α)) ,\n--        .. };\n--      intros; simp [function.comp,up_down',down_up']\n\nend trans\n\ninstance : liftable1 option.{u₀} option.{v₁} :=\n{ up := λ α β eq, @option.rec _ (λ _, option β) (@none β) (λ x, some $ eq x)\n, down := λ α β eq, @option.rec _ (λ _, option α) (@none α) (λ x, some $ eq.symm x)\n, up_down := by intros; cases x; simp\n, down_up := by intros; cases x; simp }\n\nnamespace pliftable\n\n@[reducible]\ndef up' {f : Type v₀ → Type v₁} (g : Type u₀ → Type u₁) [liftable1 f g] :\n  f punit → g punit :=\nliftable1.up g equiv.punit_equiv_punit\n\n@[reducible]\ndef down' {f : Type u₀ → Type u₁} {g : Type v₀ → Type v₁} [liftable1 f g] :\n  g punit → f punit :=\nliftable1.down _ equiv.punit_equiv_punit\n\nend pliftable\n", "meta": {"author": "leanprover-community", "repo": "mathlib-nursery", "sha": "0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec", "save_path": "github-repos/lean/leanprover-community-mathlib-nursery", "path": "github-repos/lean/leanprover-community-mathlib-nursery/mathlib-nursery-0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec/src/category/liftable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3174013525515767}}
{"text": "import Stuff.normalizing_properly.category\n\nstructure free_cat : Type :=\n( toString : String )\nderiving DecidableEq\n\nnamespace free_cat\n\ninductive hom : (X Y : free_cat) → Type\n| id (X : free_cat) : hom X X\n| comp' (X Y Z : free_cat)\n  (f : hom X Y) (g : String) : hom X Z\n\ninstance hom.DecidableEq : {X Y : free_cat} → DecidableEq (hom X Y)\n| _, _, hom.id _, hom.id _ => isTrue rfl\n| _, _, hom.comp' _ Y _ f₁ g₁, hom.comp' _ Y' _ f₂ g₂ =>\n  if hY : Y = Y'\n    then by\n      subst hY\n      exact if hg : g₁ = g₂\n        then by\n          subst hg\n          exact @decidable_of_decidable_of_iff (f₁ = f₂) _ (hom.DecidableEq _ _)\n            (by\n              apply Iff.intro\n              { intro h; rw [h] }\n              { intro h; injection h; assumption })\n        else isFalse (λ h => by injection h; contradiction)\n    else isFalse (λ h => by injection h; contradiction)\n| _, _, hom.id _, hom.comp' _ _ _ _ _ => isFalse (λ h => by injection h)\n| _, _, hom.comp' _ _ _ _ _, hom.id _ => isFalse (λ h => by injection h)\n\ndef mh (X Y : free_cat) (s : String) : hom X Y :=\nhom.comp' X X Y (hom.id X) s\n\ndef hom.comp : {X Y Z : free_cat} →\n  hom X Y → hom Y Z → hom X Z\n| _, _, _, f, hom.id _ => f\n| _, _, _, f, hom.comp' _ _ _ g h =>\n  hom.comp' _ _ _ (comp f g) h\n\nprivate theorem id_comp : {X Y : free_cat} → (f : hom X Y) →\n  (hom.id X).comp f = f\n| _, _, hom.id _ => rfl\n| _, _, hom.comp' _ _ _ f g => by\nrw [hom.comp, id_comp f]\n\nprivate theorem assoc : {W X Y Z : free_cat} → (f : hom W X) →\n  (g : hom X Y) → (h : hom Y Z) →\n  (f.comp g).comp h = f.comp (g.comp h)\n| _, _, _, _, f, g, hom.id _ => rfl\n| _, _, _, _, f, g, hom.comp' _ _ _ h i => by\nrw [hom.comp, assoc f g h, hom.comp, hom.comp]\n\ninstance : category free_cat :=\n{ hom := hom,\n  id := hom.id,\n  comp := hom.comp,\n  id_comp := id_comp,\n  comp_id := λ _ => rfl,\n  assoc := assoc }\n\ninstance {X Y : free_cat} : DecidableEq (X ⟶ Y) :=\nhom.DecidableEq\n\nend free_cat", "meta": {"author": "ChrisHughes24", "repo": "lean4stuff", "sha": "2b5f6589cfd0113853d2dd0a5ce3fdf91fae7346", "save_path": "github-repos/lean/ChrisHughes24-lean4stuff", "path": "github-repos/lean/ChrisHughes24-lean4stuff/lean4stuff-2b5f6589cfd0113853d2dd0a5ce3fdf91fae7346/Stuff/normalizing_properly/free_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.3172179575208439}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Meta\n\nnamespace Std\n-- We put `Range` in `Init` because we want the notation `[i:j]`  without importing `Std`\n-- We don't put `Range` in the top-level namespace to avoid collisions with user defined types\nstructure Range where\n  start : Nat := 0\n  stop  : Nat\n  step  : Nat := 1\n\ninstance : Membership Nat Range where\n  mem i r := r.start ≤ i ∧ i < r.stop\n\nnamespace Range\nuniverse u v\n\n@[inline] protected def forIn {β : Type u} {m : Type u → Type v} [Monad m] (range : Range) (init : β) (f : Nat → β → m (ForInStep β)) : m β :=\n  -- pass `stop` and `step` separately so the `range` object can be eliminated through inlining\n  let rec @[specialize] loop (fuel i stop step : Nat) (b : β) : m β := do\n    if i ≥ stop then\n      return b\n    else match fuel with\n     | 0   => pure b\n     | fuel+1 => match (← f i b) with\n        | ForInStep.done b  => pure b\n        | ForInStep.yield b => loop fuel (i + step) stop step b\n  loop range.stop range.start range.stop range.step init\n\ninstance : ForIn m Range Nat where\n  forIn := Range.forIn\n\n@[inline] protected def forIn' {β : Type u} {m : Type u → Type v} [Monad m] (range : Range) (init : β) (f : (i : Nat) → i ∈ range → β → m (ForInStep β)) : m β :=\n  let rec @[specialize] loop (start stop step : Nat) (f : (i : Nat) → start ≤ i ∧ i < stop → β → m (ForInStep β)) (fuel i : Nat) (hl : start ≤ i) (b : β) : m β := do\n    if hu : i < stop then\n      match fuel with\n      | 0   => pure b\n      | fuel+1 => match (← f i ⟨hl, hu⟩ b) with\n         | ForInStep.done b  => pure b\n         | ForInStep.yield b => loop start stop step f fuel (i + step) (Nat.le_trans hl (Nat.le_add_right ..)) b\n    else\n      return b\n  loop range.start range.stop range.step f range.stop range.start (Nat.le_refl ..) init\n\ninstance : ForIn' m Range Nat inferInstance where\n  forIn' := Range.forIn'\n\n@[inline] protected def forM {m : Type u → Type v} [Monad m] (range : Range) (f : Nat → m PUnit) : m PUnit :=\n  let rec @[specialize] loop (fuel i stop step : Nat) : m PUnit := do\n    if i ≥ stop then\n      pure ⟨⟩\n    else match fuel with\n     | 0   => pure ⟨⟩\n     | fuel+1 => f i; loop fuel (i + step) stop step\n  loop range.stop range.start range.stop range.step\n\ninstance : ForM m Range Nat where\n  forM := Range.forM\n\nsyntax:max \"[\" withoutPosition(\":\" term) \"]\" : term\nsyntax:max \"[\" withoutPosition(term \":\" term) \"]\" : term\nsyntax:max \"[\" withoutPosition(\":\" term \":\" term) \"]\" : term\nsyntax:max \"[\" withoutPosition(term \":\" term \":\" term) \"]\" : term\n\nmacro_rules\n  | `([ : $stop]) => `({ stop := $stop : Range })\n  | `([ $start : $stop ]) => `({ start := $start, stop := $stop : Range })\n  | `([ $start : $stop : $step ]) => `({ start := $start, stop := $stop, step := $step : Range })\n  | `([ : $stop : $step ]) => `({ stop := $stop, step := $step : Range })\n\nend Range\nend Std\n\ntheorem Membership.mem.upper {i : Nat} {r : Std.Range} (h : i ∈ r) : i < r.stop := by\n  simp [Membership.mem] at h\n  exact h.2\n\ntheorem Membership.mem.lower {i : Nat} {r : Std.Range} (h : i ∈ r) : r.start ≤ i := by\n  simp [Membership.mem] at h\n  exact h.1\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Init/Data/Range.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3171349663348993}}
{"text": "import category_theory.adjunction.basic\nimport category_theory.limits.preserves.basic\nimport data.pfun\n\nopen category_theory category_theory.functor category_theory.limits\n\nuniverse u\n\nvariables (𝒞 : Type) [category.{0} 𝒞]\n\ninductive bicompletion_aux : bool → Type 1\n| of_cat_obj : 𝒞 → bicompletion_aux ff\n| limit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) : bicompletion_aux ff\n| colimit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) : bicompletion_aux ff\n| of_cat_hom : Π {X Y : 𝒞}, (X ⟶ Y) → bicompletion_aux tt -- of_cat_obj X ⟶ of_cat_obj Y\n| limit_cone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt)\n  (X : 𝒟) (Y : bicompletion_aux ff) (f : bicompletion_aux tt) : -- F_obj X ⟶ Y\n  bicompletion_aux tt -- limit_obj F_obj F_hom ⟶ Y\n| is_limit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt)\n  (cone_obj : bicompletion_aux ff)\n  (cone : Π (X : 𝒟), bicompletion_aux tt) : -- cone_obj ⟶ F_obj X\n  bicompletion_aux tt -- cone_obj → limit_obj F_obj F_hom\n| colimit_cocone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) \n  (X : 𝒟) (Y : bicompletion_aux ff) (f : bicompletion_aux tt) : -- Y ⟶ F_obj X\n  bicompletion_aux tt -- Y ⟶ colimit_obj F_obj F_hom\n| is_colimit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) \n  (cocone_obj : bicompletion_aux ff)\n  (cocone : Π (X : 𝒟), bicompletion_aux tt) : -- F_obj X ⟶ cocone_obj\n  bicompletion_aux tt -- colimit_obj F_obj F_hom ⟶ cocone_obj\n\nnamespace bicompletion_aux\n\nvariable {𝒞}\n\n@[simp] def dom : Π (X : bicompletion_aux 𝒞 tt), bicompletion_aux 𝒞 ff\n| (@of_cat_hom _ _ X Y f) := of_cat_obj X \n| (@limit_cone_comp _ _ 𝒟 _ F_obj F_hom X _ _) := by exactI limit_obj F_obj @F_hom\n| (@is_limit _ _ 𝒟 _ F_obj F_hom cone_obj cone) := cone_obj\n| (@colimit_cocone_comp _ _ 𝒟 _ F_obj F_hom X Y f) := Y\n| (@is_colimit _ _ 𝒟 _ F_obj F_hom cocone_obj cocone) := by exactI colimit_obj F_obj @F_hom\n\n@[simp] def cod : Π (X : bicompletion_aux 𝒞 tt), bicompletion_aux 𝒞 ff\n| (@of_cat_hom _ _ X Y f) := of_cat_obj Y \n| (@colimit_cocone_comp _ _ 𝒟 _ F_obj F_hom X _ _) := by exactI colimit_obj F_obj @F_hom\n| (@is_colimit _ _ 𝒟 _ F_obj F_hom cocone_obj cocone) := cocone_obj\n| (@limit_cone_comp _ _ 𝒟 _ F_obj F_hom X Y f) := Y\n| (@is_limit _ _ 𝒟 _ F_obj F_hom cone_obj cone) := by exactI limit_obj F_obj @F_hom\n\n\nvariable (𝒞)\n\ndef obj₁ : Type 1 := bicompletion_aux 𝒞 ff\n\nvariable {𝒞}\nvariables {𝒟 : Type} [category.{0} 𝒟]\n\ndef hom₁ (X Y : obj₁ 𝒞) : Type 1 :=\n{ f : bicompletion_aux 𝒞 tt // f.dom = X ∧ f.cod = Y }\n\ndef of_cat_obj₁ (X : 𝒞) : obj₁ 𝒞 := of_cat_obj X\n\ndef limit_obj₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) : obj₁ 𝒞 :=\nlimit_obj F_obj (λ X Y f, (F_hom f).1)\n\ndef colimit_obj₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) : obj₁ 𝒞 :=\ncolimit_obj F_obj (λ X Y f, (F_hom f).1)\n\ndef of_cat_hom₁ {X Y : 𝒞} (f : X ⟶ Y) : hom₁ (of_cat_obj X) (of_cat_obj Y) :=\n⟨of_cat_hom f, by simp⟩\n\ndef limit_cone_comp₁ (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₁ 𝒞} (f : hom₁ (F_obj X) Y) :\n  hom₁ (limit_obj₁ F_obj @F_hom) Y :=\n⟨limit_cone_comp F_obj (λ X Y f, (F_hom f).1) X Y f.1, by simp [limit_obj₁]⟩\n\ndef colimit_cocone_comp₁ (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₁ 𝒞} (f : hom₁ Y (F_obj X)) :\n  hom₁ Y (colimit_obj₁ F_obj @F_hom) :=\n⟨colimit_cocone_comp F_obj (λ X Y f, (F_hom f).1) X Y f.1, by simp [colimit_obj₁]⟩\n\ndef is_limit₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cone_obj : obj₁ 𝒞)\n  (cone : Π (X : 𝒟), hom₁ cone_obj (F_obj X)) :\n  hom₁ cone_obj (limit_obj₁ F_obj @F_hom) :=\n⟨is_limit F_obj (λ X Y f, (F_hom f).1) cone_obj (λ X, (cone X).1), by simp [limit_obj₁]⟩\n\ndef is_colimit₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₁ 𝒞)\n  (cocone : Π (X : 𝒟), hom₁ (F_obj X) cocone_obj) :\n  hom₁ (colimit_obj₁ F_obj @F_hom) cocone_obj  :=\n⟨is_colimit F_obj (λ X Y f, (F_hom f).1) cocone_obj (λ X, (cocone X).1), by simp [colimit_obj₁]⟩\n\ndef id₁_aux (b : bool) (hb : b = ff) (X : bicompletion_aux 𝒞 b) : \n  hom₁ (show bicompletion_aux 𝒞 ff, from eq.rec_on hb X)\n       (show bicompletion_aux 𝒞 ff, from eq.rec_on hb X) :=\nbegin\n  revert hb,\n  refine bicompletion_aux.rec_on X _ _ _ _ _ _ _ _,\n  { rintros X h,\n    exact of_cat_hom₁ (𝟙 X) },\n  { introsI 𝒟 _ F_obj F_hom ih₁ ih₂ _, \n    exact ⟨is_limit F_obj @F_hom (limit_obj F_obj @F_hom) \n      (λ D, limit_cone_comp F_obj @F_hom D (F_obj D) (ih₁ D rfl).1), \n      by simp⟩ },\n  { introsI 𝒟 _ F_obj F_hom ih₁ ih₂ _, \n    exact ⟨is_colimit F_obj @F_hom (colimit_obj F_obj @F_hom) \n      (λ D, colimit_cocone_comp F_obj @F_hom D (F_obj D) (ih₁ D rfl).1),\n      by simp⟩ },\n  all_goals { intros, contradiction }\nend\n\ndef id₁ (X : obj₁ 𝒞) : hom₁ X X :=\nid₁_aux ff rfl X\n\n@[elab_as_eliminator] def hom_rec_on {motive : bicompletion_aux 𝒞 tt → Sort u}\n  (f : bicompletion_aux 𝒞 tt)\n  (of_cat_hom : Π {X Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom f))\n  (limit_cone_comp : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    motive f → motive (by exactI limit_cone_comp F_obj @F_hom X Y f))\n  (is_limit : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cone_obj : bicompletion_aux 𝒞 ff)\n    (cone : 𝒟 → bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    (Π (X : 𝒟), motive (cone X)) → motive (by exactI is_limit F_obj @F_hom cone_obj cone))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    motive f → motive (by exactI colimit_cocone_comp F_obj @F_hom X Y f))\n  (is_colimit : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n   (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cocone_obj : bicompletion_aux 𝒞 ff)\n   (cocone : 𝒟 → bicompletion_aux 𝒞 tt),\n     (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n     (Π (X : 𝒟), motive (cocone X)) → motive (by exactI is_colimit F_obj @F_hom cocone_obj cocone)) :\n  motive f :=\nhave ∀ b (f : bicompletion_aux 𝒞 b) (h : b = tt), motive (eq.rec_on h f) :=\n  begin\n    intros b f,\n    refine bicompletion_aux.rec_on f _ _ _ _ _ _ _ _,\n    { intros, simp at *, contradiction },\n    { intros, simp at *, contradiction },\n    { intros, simp at *, contradiction },\n    { intros X Y f _,\n      exact of_cat_hom f },\n    { introsI 𝒟 _ F_obj F_hom X Y f ih₁ ih₂ ih₃ ih₄ _,\n      exact limit_cone_comp F_obj @F_hom X Y f (λ X Y f, ih₂ f rfl) (ih₄ rfl) },\n    { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ ih₃ ih₄ _,\n      exact is_limit F_obj @F_hom cone_obj cone (λ X Y f, ih₂ f rfl) (λ X, ih₄ X rfl) },\n    { introsI 𝒟 _ F_obj F_hom X Y f ih₁ ih₂ ih₃ ih₄ _,\n      exact colimit_cocone_comp F_obj @F_hom X Y f (λ X Y f, ih₂ f rfl) (ih₄ rfl) },\n    { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ ih₃ ih₄ _,\n      exact is_colimit F_obj @F_hom cone_obj cone (λ X Y f, ih₂ f rfl) (λ X, ih₄ X rfl) },\n  end,\nthis tt f rfl\n\nstructure part2 (α : Type 1) : Type 1 :=\n(dom : Type)\n(get : dom → α)\n\ndef part2.none {α : Type 1} : part2 α := ⟨empty, empty.elim⟩\n\ninstance : monad part2 :=\n{ pure := λ α a, ⟨unit, λ _, a⟩,\n  bind := λ α β a f, ⟨Σ (h : a.dom), (f (a.get h)).dom, λ h, (f (a.get h.1)).get h.2⟩ }\n\ndef comp₁\n  (f : bicompletion_aux 𝒞 tt) :\n  Π (g : bicompletion_aux 𝒞 tt),\n  part2 {h : bicompletion_aux 𝒞 tt // h.dom = f.dom ∧ h.cod = g.cod} :=\nhom_rec_on f \n  begin --f : of_cat_obj X ⟶ of_cat_obj Y\n    intros X Y f g,\n    refine hom_rec_on g _ _ _ _ _,\n    { intros Y' Z g,\n      exact ⟨plift (Y = Y'), λ h, by cases h with h; subst h; \n        exact ⟨(of_cat_hom (f ≫ g)), by simp⟩⟩ },\n    { intros, exact part2.none },\n    { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂,\n      exact ⟨∀ X, (ih₂ X).dom, λ hd, \n        ⟨is_limit F_obj @F_hom (of_cat_obj X) (λ X, (ih₂ X).get (hd X)), by simp⟩⟩ },\n    { introsI 𝒟 _ F_obj F_hom A B f ih₁ ih₂,\n      exact ih₂ >>= λ ih₂, return \n        ⟨colimit_cocone_comp F_obj @F_hom A (of_cat_obj X) ih₂, by simp⟩ },\n    { intros, exact part2.none }\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom X Y f ih₁ ih₂ g,\n    exact ih₂ g >>= λ ih₂, return ⟨limit_cone_comp F_obj @F_hom X g.cod ih₂, by simp⟩\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ g,\n    refine hom_rec_on g _ _ _ _ _,\n    { intros, exact part2.none },\n    { introsI ℰ _ G_obj G_hom A B i ih₃ ih₄,\n      exact ih₄ >>= λ ih₄, ⟨plift (B = i.cod), λ h, by cases h with h; subst h; exact ih₄⟩ },\n    { introsI ℰ _ G_obj G_hom cone_obj' cone' ih₃ ih₄,\n      dsimp [cod, dom] at *,\n      exact ⟨{ X : Σ X, (ih₄ X).dom // (cone' X.1).cod = limit_obj G_obj @G_hom}, \n        λ X, ⟨(ih₄ X.1.1).get X.1.2, \n          by simp [((ih₄ X.1.1).get X.1.2).prop.1, ((ih₄ X.1.1).get X.1.2).prop.2, X.prop]⟩⟩ },\n    { introsI ℰ _ G_obj G_hom cocone_obj cocone ih₃ ih₄,\n      dsimp [cod, dom] at *, },\n    { intros, exact part2.none }\n\n  end\n  _ \n  _\n\ndef comp₁ : Π\n  (f : bicompletion_aux 𝒞 tt)\n  (g : bicompletion_aux 𝒞 tt),\n  part (bicompletion_aux 𝒞 tt)\n| (@of_cat_hom _ _ A B f) (@of_cat_hom _ _ B' C g) :=\n  ⟨B = B', λ h, by subst h; exact (of_cat_hom₁ (f ≫ g)).1⟩\n| (@limit_cone_comp _ _ 𝒟 _ F_obj F_hom A B f) g :=\n  do ih ← comp₁ f g, return (by exactI limit_cone_comp F_obj @F_hom A g.cod ih)\n| f (@colimit_cocone_comp _ _ 𝒟 _ F_obj F_hom A B g) :=\n  do ih ← comp₁ f g, return (by exactI colimit_cocone_comp F_obj @F_hom A g.cod ih)\n| (@is_colimit _ _ 𝒟 _ F_obj F_hom cocone_obj cocone) g :=\n  let f : Π (A : 𝒟), part (bicompletion_aux 𝒞 tt) := λ A, comp₁ (cocone A) g in\n  ⟨∀ A : 𝒟, (f A).dom, λ h, by exactI @is_colimit _ _ 𝒟 _ F_obj @F_hom cocone_obj \n    (λ A, (f A).get (h A))⟩\n| f (@is_limit _ _ 𝒟 _ F_obj F_hom cone_obj cone) :=\n  let f : Π (A : 𝒟), part (bicompletion_aux 𝒞 tt) := λ A, comp₁ f (cone A) in\n  ⟨∀ A : 𝒟, (f A).dom, λ h, by exactI @is_colimit _ _ 𝒟 _ F_obj @F_hom cone_obj \n    (λ A, (f A).get (h A))⟩\nusing_well_founded { dec_tac := `[admit] }\n\n\ninductive valid_obj₁ : Π (X : obj₁ 𝒞), Prop\n| of_cat_obj (X : 𝒞) : valid_obj₁ (of_cat_obj X)\n| limit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (h : Π X : 𝒟, valid_obj₁ (F_obj X)) : \n  valid_obj₁ (limit_obj₁ F_obj @F_hom)\n| colimit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (h : Π X : 𝒟, valid_obj₁ (F_obj X)) :\n  valid_obj₁ (colimit_obj₁ F_obj @F_hom)\n\ndef valid_obj₁_limit_obj \n  {𝒟 : Type} [category.{0} 𝒟] {F_obj : 𝒟 → obj₁ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux 𝒞 tt}\n  (h : valid_obj₁ (limit_obj F_obj @F_hom)) :\n  Π (X : 𝒟), valid_obj₁ (F_obj X) :=\nbegin\n  generalize hX : limit_obj F_obj @F_hom = X,\n  rw hX at h,\n  induction h,\n  { simp * at * },\n  { simp [limit_obj₁] at hX,\n    rcases hX with ⟨hX₁, hX₂, hX₂, hX₄⟩,\n    subst hX₁,\n    simp at *,\n    subst hX₂,\n    assumption },\n  { simp [*, colimit_obj₁] at * }\nend\n\ndef valid_obj₁_colimit_obj \n  {𝒟 : Type} [category.{0} 𝒟] {F_obj : 𝒟 → obj₁ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux 𝒞 tt}\n  (h : valid_obj₁ (colimit_obj F_obj @F_hom)) :\n  Π (X : 𝒟), valid_obj₁ (F_obj X) :=\nbegin\n  generalize hX : colimit_obj F_obj @F_hom = X,\n  rw hX at h,\n  induction h,\n  { simp * at * },\n  { simp [*, limit_obj₁] at * },\n  { simp [colimit_obj₁] at hX,\n    rcases hX with ⟨hX₁, hX₂, hX₂, hX₄⟩,\n    subst hX₁,\n    simp at *,\n    subst hX₂,\n    assumption }\nend\n\ninductive valid_hom₁ : Π {X Y : obj₁ 𝒞}, hom₁ X Y → Type 1\n| of_cat_hom {X Y : 𝒞} (f : X ⟶ Y) : valid_hom₁ (of_cat_hom₁ f)\n| limit_cone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (X : 𝒟) {Y : obj₁ 𝒞} (f : hom₁ (F_obj X) Y) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (f_valid : valid_hom₁ f) :\n  valid_hom₁ (limit_cone_comp₁ F_obj @F_hom X f)\n| colimit_cocone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (X : 𝒟) {Y : obj₁ 𝒞} (f : hom₁ Y (F_obj X)) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (f_valid : valid_hom₁ f) :\n  valid_hom₁ (colimit_cocone_comp₁ F_obj @F_hom X f)\n| is_limit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cone_obj : obj₁ 𝒞)\n  (cone : Π (X : 𝒟), hom₁ cone_obj (F_obj X)) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (cone_valid : Π (X : 𝒟), valid_hom₁ (cone X)) :\n  valid_hom₁ (is_limit₁ F_obj @F_hom cone_obj cone)\n| is_colimit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₁ 𝒞)\n  (cocone : Π (X : 𝒟), hom₁ (F_obj X) cocone_obj) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (cocone_valid : Π (X : 𝒟), valid_hom₁ (cocone X)) :\n  valid_hom₁ (is_colimit₁ F_obj @F_hom cocone_obj cocone)\n\nvariable (𝒞)\n\ndef obj₂ : Type 1 := { X : obj₁ 𝒞 // valid_obj₁ X }\n\nvariable {𝒞}\n\ndef hom₂ (X Y : obj₂ 𝒞) : Type 1 := Σ (f : hom₁ X.1 Y.1), valid_hom₁ f\n\nopen valid_hom₁\n\nlemma valid_hom₁_limit_cone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (X : 𝒟) {Y : obj₁ 𝒞} (f : hom₁ (F_obj X) Y) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (f_valid : valid_hom₁ f) \n  (h : valid_hom₁ (limit_cone_comp₁ F_obj @F_hom X f)) :\n  h == valid_hom₁.limit_cone_comp F_obj @F_hom X f @F_hom_valid f_valid :=\n@valid_hom₁.rec_on _ _ (λ A B g hg, g == limit_cone_comp₁ F_obj @F_hom X f → \n  hg == valid_hom₁.limit_cone_comp F_obj @F_hom X f @F_hom_valid f_valid) \n  _ _ _ _ \n  begin\n    intros,\n    simp [of_cat_hom₁] at *,  \n    \n  end _ _ _ _ (heq.refl _)\n\nlemma hom₂_ext_aux {X Y : obj₁ 𝒞} (f : hom₁ X Y) (h₁ : valid_hom₁ f) :\n  ∀ (h₂ : valid_hom₁ f), h₁ = h₂ :=\nbegin\n  induction h₁,\n  { intro h₂, cases h₂, refl },\n  { intro h₂,\n    refine valid_hom₁.rec_on h₂ _ _ _ _ _,\n    { intros X Y f, }\n     }\n\nend\n\ndef of_cat_obj₂ (X : 𝒞) : obj₂ 𝒞 :=\n⟨of_cat_obj X, valid_obj₁.of_cat_obj _⟩ \n\nlemma of_cat_obj₂_injective : function.injective (@of_cat_obj₂ 𝒞 _) :=\nbegin\n  intros X Y hXY,\n  simp [of_cat_obj₂] at hXY,\n  injection hXY,\nend\n\ndef limit_obj₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) : obj₂ 𝒞 :=\n⟨limit_obj₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1), valid_obj₁.limit_obj _ _ (λ X, (F_obj X).2)⟩\n\nlemma limit_obj₂_injective {𝒟₁ 𝒟₂ : Type} [i₁ : category 𝒟₁] [i₂ : category 𝒟₂] \n  {F_obj₁ : 𝒟₁ → obj₂ 𝒞} {F_obj₂ : 𝒟₂ → obj₂ 𝒞} \n  {F_hom₁ : Π {X Y : 𝒟₁}, (X ⟶ Y) → hom₂ (F_obj₁ X) (F_obj₁ Y)}\n  {F_hom₂ : Π {X Y : 𝒟₂}, (X ⟶ Y) → hom₂ (F_obj₂ X) (F_obj₂ Y)}\n  (h : limit_obj₂ F_obj₁ @F_hom₁ = limit_obj₂ F_obj₂ @F_hom₂) : \n  𝒟₁ = 𝒟₂ ∧ i₁ == i₂ ∧ F_obj₁ == F_obj₂ ∧ @F_hom₁ == @F_hom₂ :=\nbegin\n  simp [limit_obj₂, limit_obj₁] at h,\n  injection h with h₁ h₂ h₃ h₄,\n  unfreezingI { subst h₁ },\n  rw heq_iff_eq at h₂,\n  unfreezingI { subst h₂ },\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₃,\n  rw [← function.funext_iff] at h₃,\n  dsimp at h₃,\n  subst h₃,\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₄,\nend\n\n\ndef colimit_obj₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) : obj₂ 𝒞 :=\n⟨colimit_obj₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1), valid_obj₁.colimit_obj _ _ (λ X, (F_obj X).2)⟩\n\ndef of_cat_hom₂ {X Y : 𝒞} (f : X ⟶ Y) : hom₂ (of_cat_obj₂ X) (of_cat_obj₂ Y) :=\n⟨of_cat_hom₁ f, valid_hom₁.of_cat_hom _⟩ \n\ndef limit_cone_comp₂ (F_obj : 𝒟 → obj₂ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y) :\n  hom₂ (limit_obj₂ F_obj @F_hom) Y :=\n⟨limit_cone_comp₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) X f.1, \n  valid_hom₁.limit_cone_comp _ _ _ _ (λ X Y f, (F_hom f).2) f.2⟩\n\ndef colimit_cocone_comp₂ (F_obj : 𝒟 → obj₂ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₂ 𝒞} (f : hom₂ Y (F_obj X)):\n  hom₂ Y (colimit_obj₂ F_obj @F_hom) :=\n⟨colimit_cocone_comp₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) X f.1, \n  valid_hom₁.colimit_cocone_comp _ _ _ _ (λ X Y f, (F_hom f).2) f.2⟩\n\ndef is_limit₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n  (cone_obj : obj₂ 𝒞)\n  (cone : Π (X : 𝒟), hom₂ cone_obj (F_obj X)) :\n  hom₂ cone_obj (limit_obj₂ F_obj @F_hom) :=\n⟨is_limit₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) cone_obj.1 (λ X, (cone X).1), \n  valid_hom₁.is_limit _ _ _ _ (λ X Y f, (F_hom f).2) (λ X, (cone X).2)⟩\n\ndef is_colimit₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₂ 𝒞)\n  (cocone : Π (X : 𝒟), hom₂ (F_obj X) cocone_obj) :\n  hom₂ (colimit_obj₂ F_obj @F_hom) cocone_obj  :=\n⟨is_colimit₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) cocone_obj.1 (λ X, (cocone X).1), \n  valid_hom₁.is_colimit _ _ _ _ (λ X Y f, (F_hom f).2) (λ X, (cocone X).2)⟩\n\n@[elab_as_eliminator] protected def hom₂.rec_on \n  {motive : Π {X Y : obj₂ 𝒞} (f : hom₂ X Y), Sort*} {X Y : obj₂ 𝒞} (f : hom₂ X Y)\n  (of_cat_hom : Π {X Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom₂ f))\n  (limit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y)\n    (ih_f : motive f),\n      motive (by exactI limit_cone_comp₂ F_obj @F_hom X f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ Y (F_obj X))\n    (ih_f : motive f),\n      motive (by exactI colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) \n    (cone_obj : obj₂ 𝒞) (cone : Π (X : 𝒟), hom₂ cone_obj (F_obj X))\n    (ih_cone : Π (X : 𝒟), motive (cone X)),\n      motive (by exactI is_limit₂ F_obj @F_hom cone_obj cone))\n  (is_colimit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (cocone_obj : obj₂ 𝒞) (cocone : Π (X : 𝒟), hom₂ (F_obj X) cocone_obj)\n    (ih_cone : Π (X : 𝒟), motive (cocone X)),\n      motive (by exactI is_colimit₂ F_obj @F_hom cocone_obj cocone)) :\n  motive f :=\nbegin\n  cases X with X hX, cases Y with Y hY,\n  cases f with f hf,\n  dsimp at f, dsimp at hf,\n  revert hX hY,\n  refine valid_hom₁.rec_on hf _ _ _ _ _,\n  { intros A B g hX hY,\n    exact of_cat_hom g },\n  { introsI 𝒟 _ F_obj F_hom X Y f F_hom_valid f_valid ih₁ ih₂ hX hY,\n    exact @limit_cone_comp _ _ (λ A, ⟨F_obj A, valid_obj₁_limit_obj hX A⟩)\n      (λ X Y f, ⟨F_hom f, F_hom_valid f⟩)\n      (λ X Y f, ih₁ f _ _) X ⟨Y, hY⟩ ⟨f, f_valid⟩ (ih₂ _ _) },\n  { introsI 𝒟 _ F_obj F_hom X Y f F_hom_valid f_valid ih₁ ih₂ hY hX,\n    exact @colimit_cocone_comp _ _ (λ A, ⟨F_obj A, valid_obj₁_colimit_obj hX A⟩)\n      (λ X Y f, ⟨F_hom f, F_hom_valid f⟩)\n      (λ X Y f, ih₁ f _ _) X ⟨Y, hY⟩ ⟨f, f_valid⟩ (ih₂ _ _) },\n  { introsI 𝒟 _ F_obj F_hom cone_obj cone F_hom_valid cone_valid ih₁ ih₂ hX hY,\n    exact @is_limit 𝒟 _ (λ A, ⟨F_obj A, valid_obj₁_limit_obj hY A⟩)\n      (λ X Y f, ⟨F_hom f, F_hom_valid f⟩)\n      (λ X Y f, ih₁ f _ _) ⟨cone_obj, hX⟩\n      (λ X, ⟨cone X, cone_valid X⟩)\n      (λ X, ih₂ X _ _) },\n  { introsI 𝒟 _ F_obj F_hom cocone_obj cocone F_hom_valid cocone_valid ih₁ ih₂ hX hY,\n    exact @is_colimit 𝒟 _ (λ A, ⟨F_obj A, valid_obj₁_colimit_obj hX A⟩)\n      (λ X Y f, ⟨F_hom f, F_hom_valid f⟩)\n      (λ X Y f, ih₁ f _ _) ⟨cocone_obj, hY⟩\n      (λ X, ⟨cocone X, cocone_valid X⟩)\n      (λ X, ih₂ X _ _) }\nend\n\ndef hom₂_of_cat_obj_rec_on\n  {motive : Π {X : 𝒞} {Y : obj₂ 𝒞} (f : hom₂ (of_cat_obj₂ X) Y), Sort*} \n  {X : 𝒞} {Y : obj₂ 𝒞} (f : hom₂ (of_cat_obj₂ X) Y)\n  (of_cat_hom : Π {Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom₂ f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (X : 𝒟) {Y : 𝒞} (f : hom₂ (of_cat_obj₂ Y) (F_obj X))\n    (ih_f : motive f),\n      motive (by exactI colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (cone_obj : 𝒞) (cone : Π (X : 𝒟), hom₂ (of_cat_obj₂ cone_obj) (F_obj X))\n    (ih_cone : Π (X : 𝒟), motive (cone X)),\n      motive (by exactI is_limit₂ F_obj @F_hom (of_cat_obj₂ cone_obj) cone)) :\n  motive f := \n@hom₂.rec_on 𝒞 _ (λ A B f, ∀ (h : A = of_cat_obj₂ X),\n  motive (show hom₂ (of_cat_obj₂ X) B, from eq.rec_on h f))\n  (of_cat_obj₂ X) Y f \n  (λ A B g h, begin\n      have := of_cat_obj₂_injective h,\n      subst this,\n      dsimp,\n      exact of_cat_hom g\n    end) \n  begin \n    intros,\n    simp [limit_obj₂, of_cat_obj₂, limit_obj₁] at h,\n    contradiction\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B g ih₂ h,\n    subst h,\n    exact colimit_cocone_comp _ _ _ _ (ih₂ rfl)\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ h,\n    subst h,\n    exact is_limit _ _ _ _ (λ A, ih₂ A rfl),\n  end \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂] at h,\n    contradiction\n  end  \n  rfl\n\ndef hom₂_limit_obj_rec_on\n  {motive : Π {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n    {F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}, \n    hom₂ (by exactI limit_obj₂ F_obj @F_hom) Y → Sort*}\n  {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}\n  (f : hom₂ (limit_obj₂ F_obj @F_hom) Y)\n  (limit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y),\n      by exactI motive (limit_cone_comp₂ F_obj @F_hom X f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (X : 𝒟)\n    {ℰ : Type} [category ℰ] (G_obj : ℰ → obj₂ 𝒞)\n    (G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y))\n    (f : hom₂ (by exactI limit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_f : by exactI motive f),\n      by exactI motive (colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    {ℰ : Type} [category ℰ] (G_obj : ℰ → obj₂ 𝒞)\n    (G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y))\n    (cone : Π (X : 𝒟), hom₂ (by exactI limit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_cone : Π (X : 𝒟), by exactI motive (cone X)),\n      by exactI motive (is_limit₂ F_obj @F_hom (limit_obj₂ G_obj @G_hom) cone)) :\n  motive f := \n@hom₂.rec_on 𝒞 _ (λ A B f, ∀ (h : A = limit_obj₂ F_obj @F_hom),\n  motive (show hom₂ (limit_obj₂ F_obj @F_hom) B, from eq.rec_on h f))\n  (limit_obj₂ F_obj @F_hom) Y f \n  begin \n    intros,\n    simp [limit_obj₂, of_cat_obj₂, limit_obj₁] at h,\n    contradiction\n  end  \n  begin \n    intros ℰ _ G_obj G_hom ih₁ A B g ih₂ h,\n    simp [limit_obj₂, of_cat_obj₂, limit_obj₁] at h,\n    injection h with h₁ h₂ h₃ h₄,\n    unfreezingI { subst h₁ },\n    rw [heq_iff_eq] at h₂,\n    unfreezingI { subst h₂ },\n    dsimp at h₄,\n    dsimp,\n\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B g ih₂ h,\n    subst h,\n    exact colimit_cocone_comp _ _ _ _ (ih₂ rfl)\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ h,\n    subst h,\n    exact is_limit _ _ _ _ (λ A, ih₂ A rfl),\n  end \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂] at h,\n    contradiction\n  end  \n  rfl\n\ndef comp₂ {X Y : obj₂ 𝒞} (f : hom₂ X Y) : Π {Z : obj₂ 𝒞}, hom₂ Y Z → hom₂ X Z :=\nhom₂.rec_on f \n  begin\n    intros X Y f Z g,\n    refine hom₂_of_cat_obj_rec_on g _ _ _,\n    { intros B g,\n      exact of_cat_hom₂ (f ≫ g) },\n    { introsI 𝒟 _ F_obj F_hom ih₁ B g ih₂,\n      simp [limit_obj₂, of_cat_obj₂, limit_obj₁, *] at *,\n      refine colimit_cocone_comp₂ F_obj _ _ ih₂ },\n    { introsI 𝒟 _ F_obj F_hom ih₁ cone ih₂,\n      refine is_limit₂ _ _ _ (λ X, ih₂ _) }\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B f ih₂ Z g,\n    refine limit_cone_comp₂ _ _ _ (ih₂ g),\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B f ih₂ Z g,\n    refine ih₂ _,\n    admit\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ Z g,\n\n    admit,\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cocone_obj cocone ih₂ Z g,\n    exact is_colimit₂ _ _ _ (λ A, ih₂ _ g)\n  end\n\nend bicompletion_aux\n", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/bicompletion/inductive2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.31712113190164226}}
{"text": "import data.cpi.species.basic\n\n-- Dubious instances to force setoids to be equivalent.\ninstance {α : Type*} [r : setoid α] : is_refl α has_equiv.equiv\n  := { refl := λ a, r.iseqv.1 a }\ninstance {α : Type*} [r : setoid α] : is_trans α has_equiv.equiv\n  := { trans := λ a b c ab bc, r.iseqv.2.2 ab bc }\ninstance {α : Type*} [r : setoid α] : is_symm α has_equiv.equiv\n  := { symm := λ a b ab, r.iseqv.2.1 ab }\n\nnamespace cpi\nnamespace species\n\nvariables {ℍ : Type} {ω Γ : context} [setoid (species ℍ ω Γ)]\n\n/-- A species is prime if it is non-nil, and for any decomposition into two\n    parallel species, one of those must be nil.  -/\ndef prime (A : species ℍ ω Γ) : Prop\n  := ¬ A ≈ nil ∧ ∀ B C, A ≈ (B |ₛ C) → B ≈ nil ∨ C ≈ nil\n\nlemma prime.equivalent_imp {A B : species ℍ ω Γ} : A ≈ B → prime A → prime B\n| ab ⟨ neq, prime ⟩ := ⟨ λ nil, neq (trans ab nil), λ A B eq, prime A B (trans ab eq) ⟩\n\n/-- Primality commutes with structrural congruence. -/\nlemma prime.equivalent {A B : species ℍ ω Γ} : A ≈ B → prime A = prime B\n| eq := propext ⟨ prime.equivalent_imp eq, prime.equivalent_imp (symm eq) ⟩\n\n/-- The set of all species which are prime. -/\n@[nolint has_inhabited_instance]\ndef prime_species (ℍ : Type) (ω Γ : context) [setoid (species ℍ ω Γ)] :Type\n  := { A : species ℍ ω Γ // prime A }\n\ninstance prime_species.setoid: setoid (prime_species ℍ ω Γ) :=\n  { r := λ A B, A.val ≈ B.val,\n    iseqv := ⟨ λ a, by from refl a.val,\n              λ _ _ ab, by from symm ab,\n              λ _ _ _ ab bc, by from trans ab bc ⟩ }\n\n/-- A quotient of all structurally congruent species which are prime. -/\n@[nolint has_inhabited_instance]\ndef prime_species' (ℍ : Type) (ω Γ : context) [r : setoid (species ℍ ω Γ)] :=\n  quotient (@prime_species.setoid ℍ ω Γ r)\n\n/-- Unwrap a quotient of prime species, yielding a quotient of species. -/\ndef prime_species.unwrap : prime_species' ℍ ω Γ → species' ℍ ω Γ\n| A := quot.lift_on A (λ B, ⟦ B.val ⟧) (λ _ _ eq, quot.sound eq)\n\ninstance prime_species'.has_repr [has_repr (species' ℍ ω Γ)] : has_repr (prime_species' ℍ ω Γ)\n  := ⟨ λ x, repr (prime_species.unwrap x) ⟩\n\nlemma prime_species.unwrap.inj :\n  ∀ (A B : prime_species' ℍ ω Γ), prime_species.unwrap A = prime_species.unwrap B\n  → A = B\n| A B eq := begin\n  rcases quotient.exists_rep A with ⟨ A, eq ⟩, subst eq,\n  rcases quotient.exists_rep B with ⟨ B, eq ⟩, subst eq,\n  from quot.sound (quotient.exact eq),\nend\n\nend species\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/species/prime.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581510799253, "lm_q2_score": 0.43014734858584286, "lm_q1q2_score": 0.31708662417547206}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.subobject.factor_thru\nimport category_theory.subobject.well_powered\n\n/-!\n# The lattice of subobjects\n\nWe provide the `semilattice_inf` with `order_top (subobject X)` instance when `[has_pullback C]`,\nand the `semilattice_sup (subobject X)` instance when `[has_images C] [has_binary_coproducts C]`.\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C] {X Y Z : C}\nvariables {D : Type u₂} [category.{v₂} D]\n\nnamespace category_theory\n\nnamespace mono_over\n\nsection has_top\n\ninstance {X : C} : has_top (mono_over X) :=\n{ top := mk' (𝟙 _) }\n\ninstance {X : C} : inhabited (mono_over X) := ⟨⊤⟩\n\n/-- The morphism to the top object in `mono_over X`. -/\ndef le_top (f : mono_over X) : f ⟶ ⊤ :=\nhom_mk f.arrow (comp_id _)\n\n@[simp] lemma top_left (X : C) : ((⊤ : mono_over X) : C) = X := rfl\n@[simp] lemma top_arrow (X : C) : (⊤ : mono_over X).arrow = 𝟙 X := rfl\n\n/-- `map f` sends `⊤ : mono_over X` to `⟨X, f⟩ : mono_over Y`. -/\ndef map_top (f : X ⟶ Y) [mono f] : (map f).obj ⊤ ≅ mk' f :=\niso_of_both_ways (hom_mk (𝟙 _) rfl) (hom_mk (𝟙 _) (by simp [id_comp f]))\n\nsection\nvariable [has_pullbacks C]\n\n/-- The pullback of the top object in `mono_over Y`\nis (isomorphic to) the top object in `mono_over X`. -/\ndef pullback_top (f : X ⟶ Y) : (pullback f).obj ⊤ ≅ ⊤ :=\niso_of_both_ways (le_top _) (hom_mk (pullback.lift f (𝟙 _) (by tidy)) (pullback.lift_snd _ _ _))\n\n/-- There is a morphism from `⊤ : mono_over A` to the pullback of a monomorphism along itself;\nas the category is thin this is an isomorphism. -/\ndef top_le_pullback_self {A B : C} (f : A ⟶ B) [mono f] :\n  (⊤ : mono_over A) ⟶ (pullback f).obj (mk' f) :=\nhom_mk _ (pullback.lift_snd _ _ rfl)\n\n/-- The pullback of a monomorphism along itself is isomorphic to the top object. -/\ndef pullback_self {A B : C} (f : A ⟶ B) [mono f] :\n  (pullback f).obj (mk' f) ≅ ⊤ :=\niso_of_both_ways (le_top _) (top_le_pullback_self _)\n\nend\n\nend has_top\n\nsection has_bot\n\nvariables [has_initial C] [initial_mono_class C]\n\ninstance {X : C} : has_bot (mono_over X) :=\n{ bot := mk' (initial.to X) }\n\n@[simp] lemma bot_left (X : C) : ((⊥ : mono_over X) : C) = ⊥_ C := rfl\n@[simp] lemma bot_arrow {X : C} : (⊥ : mono_over X).arrow = initial.to X := rfl\n\n/-- The (unique) morphism from `⊥ : mono_over X` to any other `f : mono_over X`. -/\ndef bot_le {X : C} (f : mono_over X) : ⊥ ⟶ f :=\nhom_mk (initial.to _) (by simp)\n\n/-- `map f` sends `⊥ : mono_over X` to `⊥ : mono_over Y`. -/\ndef map_bot (f : X ⟶ Y) [mono f] : (map f).obj ⊥ ≅ ⊥ :=\niso_of_both_ways (hom_mk (initial.to _) (by simp)) (hom_mk (𝟙 _) (by simp))\n\nend has_bot\n\nsection zero_order_bot\n\nvariables [has_zero_object C]\nopen_locale zero_object\n\n/-- The object underlying `⊥ : subobject B` is (up to isomorphism) the zero object. -/\ndef bot_coe_iso_zero {B : C} : ((⊥ : mono_over B) : C) ≅ 0 :=\ninitial_is_initial.unique_up_to_iso has_zero_object.zero_is_initial\n\n@[simp] lemma bot_arrow_eq_zero [has_zero_morphisms C] {B : C} : (⊥ : mono_over B).arrow = 0 :=\nzero_of_source_iso_zero _ bot_coe_iso_zero\n\nend zero_order_bot\n\nsection inf\nvariables [has_pullbacks C]\n\n/--\nWhen `[has_pullbacks C]`, `mono_over A` has \"intersections\", functorial in both arguments.\n\nAs `mono_over A` is only a preorder, this doesn't satisfy the axioms of `semilattice_inf`,\nbut we reuse all the names from `semilattice_inf` because they will be used to construct\n`semilattice_inf (subobject A)` shortly.\n-/\n@[simps]\ndef inf {A : C} : mono_over A ⥤ mono_over A ⥤ mono_over A :=\n{ obj := λ f, pullback f.arrow ⋙ map f.arrow,\n  map := λ f₁ f₂ k,\n  { app := λ g,\n    begin\n      apply hom_mk _ _,\n      apply pullback.lift pullback.fst (pullback.snd ≫ k.left) _,\n      rw [pullback.condition, assoc, w k],\n      dsimp,\n      rw [pullback.lift_snd_assoc, assoc, w k],\n    end } }.\n\n/-- A morphism from the \"infimum\" of two objects in `mono_over A` to the first object. -/\ndef inf_le_left {A : C} (f g : mono_over A) :\n  (inf.obj f).obj g ⟶ f :=\nhom_mk _ rfl\n\n/-- A morphism from the \"infimum\" of two objects in `mono_over A` to the second object. -/\ndef inf_le_right {A : C} (f g : mono_over A) :\n  (inf.obj f).obj g ⟶ g :=\nhom_mk _ pullback.condition\n\n/-- A morphism version of the `le_inf` axiom. -/\ndef le_inf {A : C} (f g h : mono_over A) :\n  (h ⟶ f) → (h ⟶ g) → (h ⟶ (inf.obj f).obj g) :=\nbegin\n  intros k₁ k₂,\n  refine hom_mk (pullback.lift k₂.left k₁.left _) _,\n  rw [w k₁, w k₂],\n  erw [pullback.lift_snd_assoc, w k₁],\nend\n\nend inf\n\nsection sup\nvariables [has_images C] [has_binary_coproducts C]\n\n/-- When `[has_images C] [has_binary_coproducts C]`, `mono_over A` has a `sup` construction,\nwhich is functorial in both arguments,\nand which on `subobject A` will induce a `semilattice_sup`. -/\ndef sup  {A : C} : mono_over A ⥤ mono_over A ⥤ mono_over A :=\ncurry_obj ((forget A).prod (forget A) ⋙ uncurry.obj over.coprod ⋙ image)\n\n/-- A morphism version of `le_sup_left`. -/\ndef le_sup_left {A : C} (f g : mono_over A) :\n  f ⟶ (sup.obj f).obj g :=\nbegin\n  refine hom_mk (coprod.inl ≫ factor_thru_image _) _,\n  erw [category.assoc, image.fac, coprod.inl_desc],\n  refl,\nend\n\n/-- A morphism version of `le_sup_right`. -/\ndef le_sup_right {A : C} (f g : mono_over A) :\n  g ⟶ (sup.obj f).obj g :=\nbegin\n  refine hom_mk (coprod.inr ≫ factor_thru_image _) _,\n  erw [category.assoc, image.fac, coprod.inr_desc],\n  refl,\nend\n\n/-- A morphism version of `sup_le`. -/\ndef sup_le {A : C} (f g h : mono_over A) :\n  (f ⟶ h) → (g ⟶ h) → ((sup.obj f).obj g ⟶ h) :=\nbegin\n  intros k₁ k₂,\n  refine hom_mk _ _,\n  apply image.lift ⟨_, h.arrow, coprod.desc k₁.left k₂.left, _⟩,\n  { dsimp,\n    ext1,\n    { simp [w k₁] },\n    { simp [w k₂] } },\n  { apply image.lift_fac }\nend\n\nend sup\n\nend mono_over\n\nnamespace subobject\n\nsection order_top\n\ninstance order_top {X : C} : order_top (subobject X) :=\n{ top := quotient.mk' ⊤,\n  le_top :=\n  begin\n    refine quotient.ind' (λ f, _),\n    exact ⟨mono_over.le_top f⟩,\n  end }\n\ninstance {X : C} : inhabited (subobject X) := ⟨⊤⟩\n\nlemma top_eq_id (B : C) : (⊤ : subobject B) = subobject.mk (𝟙 B) := rfl\n\nlemma underlying_iso_top_hom {B : C} :\n  (underlying_iso (𝟙 B)).hom = (⊤ : subobject B).arrow :=\nby { convert underlying_iso_hom_comp_eq_mk (𝟙 B), simp only [comp_id], }\n\ninstance top_arrow_is_iso {B : C} : is_iso ((⊤ : subobject B).arrow) :=\nby { rw ←underlying_iso_top_hom, apply_instance, }\n\n@[simp, reassoc]\nlemma underlying_iso_inv_top_arrow {B : C} :\n  (underlying_iso _).inv ≫ (⊤ : subobject B).arrow = 𝟙 B :=\nunderlying_iso_arrow _\n\n@[simp]\nlemma map_top (f : X ⟶ Y) [mono f] : (map f).obj ⊤ = subobject.mk f :=\nquotient.sound' ⟨mono_over.map_top f⟩\n\nlemma top_factors {A B : C} (f : A ⟶ B) : (⊤ : subobject B).factors f :=\n⟨f, comp_id _⟩\n\nlemma is_iso_iff_mk_eq_top {X Y : C} (f : X ⟶ Y) [mono f] : is_iso f ↔ mk f = ⊤ :=\n⟨λ _, by exactI mk_eq_mk_of_comm _ _ (as_iso f) (category.comp_id _), λ h,\n  by { rw [←of_mk_le_mk_comp h.le, category.comp_id], exact is_iso.of_iso (iso_of_mk_eq_mk _ _ h) }⟩\n\nlemma is_iso_arrow_iff_eq_top {Y : C} (P : subobject Y) : is_iso P.arrow ↔ P = ⊤ :=\nby rw [is_iso_iff_mk_eq_top, mk_arrow]\n\ninstance is_iso_top_arrow {Y : C} : is_iso (⊤ : subobject Y).arrow :=\nby rw is_iso_arrow_iff_eq_top\n\nlemma mk_eq_top_of_is_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : mk f = ⊤ :=\n(is_iso_iff_mk_eq_top f).mp infer_instance\n\nlemma eq_top_of_is_iso_arrow {Y : C} (P : subobject Y) [is_iso P.arrow] : P = ⊤ :=\n(is_iso_arrow_iff_eq_top P).mp infer_instance\n\nsection\nvariables [has_pullbacks C]\n\nlemma pullback_top (f : X ⟶ Y) : (pullback f).obj ⊤ = ⊤ :=\nquotient.sound' ⟨mono_over.pullback_top f⟩\n\nlemma pullback_self {A B : C} (f : A ⟶ B) [mono f] :\n  (pullback f).obj (mk f) = ⊤ :=\nquotient.sound' ⟨mono_over.pullback_self f⟩\n\nend\n\nend order_top\n\nsection order_bot\nvariables [has_initial C] [initial_mono_class C]\n\ninstance order_bot {X : C} : order_bot (subobject X) :=\n{ bot := quotient.mk' ⊥,\n  bot_le :=\n  begin\n    refine quotient.ind' (λ f, _),\n    exact ⟨mono_over.bot_le f⟩,\n  end }\n\nlemma bot_eq_initial_to {B : C} : (⊥ : subobject B) = subobject.mk (initial.to B) := rfl\n\n/-- The object underlying `⊥ : subobject B` is (up to isomorphism) the initial object. -/\ndef bot_coe_iso_initial {B : C} : ((⊥ : subobject B) : C) ≅ ⊥_ C := underlying_iso _\n\nlemma map_bot (f : X ⟶ Y) [mono f] : (map f).obj ⊥ = ⊥ :=\nquotient.sound' ⟨mono_over.map_bot f⟩\n\nend order_bot\n\nsection zero_order_bot\n\nvariables [has_zero_object C]\nopen_locale zero_object\n\n/-- The object underlying `⊥ : subobject B` is (up to isomorphism) the zero object. -/\ndef bot_coe_iso_zero {B : C} : ((⊥ : subobject B) : C) ≅ 0 :=\nbot_coe_iso_initial ≪≫ initial_is_initial.unique_up_to_iso has_zero_object.zero_is_initial\n\nvariables [has_zero_morphisms C]\n\nlemma bot_eq_zero {B : C} : (⊥ : subobject B) = subobject.mk (0 : 0 ⟶ B) :=\nmk_eq_mk_of_comm _ _ (initial_is_initial.unique_up_to_iso has_zero_object.zero_is_initial) (by simp)\n\n@[simp] lemma bot_arrow {B : C} : (⊥ : subobject B).arrow = 0 :=\nzero_of_source_iso_zero _ bot_coe_iso_zero\n\nlemma bot_factors_iff_zero {A B : C} (f : A ⟶ B) : (⊥ : subobject B).factors f ↔ f = 0 :=\n⟨by { rintro ⟨h, rfl⟩, simp }, by { rintro rfl, exact ⟨0, by simp⟩, }⟩\n\nlemma mk_eq_bot_iff_zero {f : X ⟶ Y} [mono f] : subobject.mk f = ⊥ ↔ f = 0 :=\n⟨λ h, by simpa [h, bot_factors_iff_zero] using mk_factors_self f,\n  λ h, mk_eq_mk_of_comm _ _ ((iso_zero_of_mono_eq_zero h).trans has_zero_object.zero_iso_initial)\n    (by simp [h])⟩\n\nend zero_order_bot\n\nsection functor\nvariable (C)\n\n/-- Sending `X : C` to `subobject X` is a contravariant functor `Cᵒᵖ ⥤ Type`. -/\n@[simps]\ndef functor [has_pullbacks C] : Cᵒᵖ ⥤ Type (max u₁ v₁) :=\n{ obj := λ X, subobject X.unop,\n  map := λ X Y f, (pullback f.unop).obj,\n  map_id' := λ X, funext pullback_id,\n  map_comp' := λ X Y Z f g, funext (pullback_comp _ _) }\n\nend functor\n\nsection semilattice_inf_top\nvariables [has_pullbacks C]\n\n/-- The functorial infimum on `mono_over A` descends to an infimum on `subobject A`. -/\ndef inf {A : C} : subobject A ⥤ subobject A ⥤ subobject A :=\nthin_skeleton.map₂ mono_over.inf\n\nlemma inf_le_left  {A : C} (f g : subobject A) :\n  (inf.obj f).obj g ≤ f :=\nquotient.induction_on₂' f g (λ a b, ⟨mono_over.inf_le_left _ _⟩)\n\nlemma inf_le_right {A : C} (f g : subobject A) :\n  (inf.obj f).obj g ≤ g :=\nquotient.induction_on₂' f g (λ a b, ⟨mono_over.inf_le_right _ _⟩)\n\nlemma le_inf {A : C} (h f g : subobject A) :\n  h ≤ f → h ≤ g → h ≤ (inf.obj f).obj g :=\nquotient.induction_on₃' h f g\nbegin\n  rintros f g h ⟨k⟩ ⟨l⟩,\n  exact ⟨mono_over.le_inf _ _ _ k l⟩,\nend\n\ninstance {B : C} : semilattice_inf (subobject B) :=\n{ inf := λ m n, (inf.obj m).obj n,\n  inf_le_left := inf_le_left,\n  inf_le_right := inf_le_right,\n  le_inf := le_inf,\n  ..subobject.partial_order _ }\n\nlemma factors_left_of_inf_factors {A B : C} {X Y : subobject B} {f : A ⟶ B}\n  (h : (X ⊓ Y).factors f) : X.factors f :=\nfactors_of_le _ (inf_le_left _ _) h\n\nlemma factors_right_of_inf_factors {A B : C} {X Y : subobject B} {f : A ⟶ B}\n  (h : (X ⊓ Y).factors f) : Y.factors f :=\nfactors_of_le _ (inf_le_right _ _) h\n\n@[simp]\nlemma inf_factors {A B : C} {X Y : subobject B} (f : A ⟶ B) :\n  (X ⊓ Y).factors f ↔ X.factors f ∧ Y.factors f :=\n⟨λ h, ⟨factors_left_of_inf_factors h, factors_right_of_inf_factors h⟩,\n  begin\n    revert X Y,\n    refine quotient.ind₂' _,\n    rintro X Y ⟨⟨g₁, rfl⟩, ⟨g₂, hg₂⟩⟩,\n    exact ⟨_, pullback.lift_snd_assoc _ _ hg₂ _⟩,\n  end⟩\n\nlemma inf_arrow_factors_left {B : C} (X Y : subobject B) : X.factors (X ⊓ Y).arrow :=\n(factors_iff _ _).mpr ⟨of_le (X ⊓ Y) X (inf_le_left X Y), by simp⟩\n\n\n\n@[simp]\nlemma finset_inf_factors {I : Type*} {A B : C} {s : finset I} {P : I → subobject B}\n  (f : A ⟶ B) :\n  (s.inf P).factors f ↔ ∀ i ∈ s, (P i).factors f :=\nbegin\n  classical,\n  apply finset.induction_on s,\n  { simp [top_factors] },\n  { intros i s nm ih, simp [ih] },\nend\n\n-- `i` is explicit here because often we'd like to defer a proof of `m`\nlemma finset_inf_arrow_factors {I : Type*} {B : C} (s : finset I) (P : I → subobject B)\n  (i : I) (m : i ∈ s) : (P i).factors (s.inf P).arrow :=\nbegin\n  revert i m,\n  classical,\n  apply finset.induction_on s,\n  { rintro _ ⟨⟩, },\n  { intros i s nm ih j m,\n    rw [finset.inf_insert],\n    simp only [finset.mem_insert] at m, rcases m with (rfl|m),\n    { rw ←factor_thru_arrow _ _ (inf_arrow_factors_left _ _),\n      exact factors_comp_arrow _, },\n    { rw ←factor_thru_arrow _ _ (inf_arrow_factors_right _ _),\n      apply factors_of_factors_right,\n      exact ih _ m, } },\nend\n\nlemma inf_eq_map_pullback' {A : C} (f₁ : mono_over A) (f₂ : subobject A) :\n  (subobject.inf.obj (quotient.mk' f₁)).obj f₂ =\n    (subobject.map f₁.arrow).obj ((subobject.pullback f₁.arrow).obj f₂) :=\nbegin\n  apply quotient.induction_on' f₂,\n  intro f₂,\n  refl,\nend\n\nlemma inf_eq_map_pullback {A : C} (f₁ : mono_over A) (f₂ : subobject A) :\n  (quotient.mk' f₁ ⊓ f₂ : subobject A) = (map f₁.arrow).obj ((pullback f₁.arrow).obj f₂) :=\ninf_eq_map_pullback' f₁ f₂\n\nlemma prod_eq_inf {A : C} {f₁ f₂ : subobject A} [has_binary_product f₁ f₂] :\n  (f₁ ⨯ f₂) = f₁ ⊓ f₂ :=\nle_antisymm\n  (_root_.le_inf (limits.prod.fst).le (limits.prod.snd).le)\n  ((prod.lift (_root_.inf_le_left.hom) (_root_.inf_le_right.hom))).le\n\nlemma inf_def {B : C} (m m' : subobject B) :\n  m ⊓ m' = (inf.obj m).obj m' := rfl\n\n/-- `⊓` commutes with pullback. -/\nlemma inf_pullback {X Y : C} (g : X ⟶ Y) (f₁ f₂) :\n  (pullback g).obj (f₁ ⊓ f₂) = (pullback g).obj f₁ ⊓ (pullback g).obj f₂ :=\nbegin\n  revert f₁,\n  apply quotient.ind',\n  intro f₁,\n  erw [inf_def, inf_def, inf_eq_map_pullback', inf_eq_map_pullback', ← pullback_comp,\n       ← map_pullback pullback.condition (pullback_is_pullback f₁.arrow g),\n       ← pullback_comp, pullback.condition],\n  refl,\nend\n\n/-- `⊓` commutes with map. -/\nlemma inf_map {X Y : C} (g : Y ⟶ X) [mono g] (f₁ f₂) :\n  (map g).obj (f₁ ⊓ f₂) = (map g).obj f₁ ⊓ (map g).obj f₂ :=\nbegin\n  revert f₁,\n  apply quotient.ind',\n  intro f₁,\n  erw [inf_def, inf_def, inf_eq_map_pullback',\n       inf_eq_map_pullback', ← map_comp],\n  dsimp,\n  rw [pullback_comp, pullback_map_self],\nend\n\nend semilattice_inf_top\n\nsection semilattice_sup\nvariables [has_images C] [has_binary_coproducts C]\n\n/-- The functorial supremum on `mono_over A` descends to an supremum on `subobject A`. -/\ndef sup {A : C} : subobject A ⥤ subobject A ⥤ subobject A :=\nthin_skeleton.map₂ mono_over.sup\n\ninstance {B : C} : semilattice_sup (subobject B) :=\n{ sup := λ m n, (sup.obj m).obj n,\n  le_sup_left := λ m n, quotient.induction_on₂' m n (λ a b, ⟨mono_over.le_sup_left _ _⟩),\n  le_sup_right := λ m n, quotient.induction_on₂' m n (λ a b, ⟨mono_over.le_sup_right _ _⟩),\n  sup_le := λ m n k, quotient.induction_on₃' m n k (λ a b c ⟨i⟩ ⟨j⟩, ⟨mono_over.sup_le _ _ _ i j⟩),\n  ..subobject.partial_order B }\n\nlemma sup_factors_of_factors_left {A B : C} {X Y : subobject B} {f : A ⟶ B} (P : X.factors f) :\n  (X ⊔ Y).factors f :=\nfactors_of_le f le_sup_left P\n\nlemma sup_factors_of_factors_right {A B : C} {X Y : subobject B} {f : A ⟶ B} (P : Y.factors f) :\n  (X ⊔ Y).factors f :=\nfactors_of_le f le_sup_right P\n\nvariables [has_initial C] [initial_mono_class C]\n\nlemma finset_sup_factors {I : Type*} {A B : C} {s : finset I} {P : I → subobject B}\n  {f : A ⟶ B} (h : ∃ i ∈ s, (P i).factors f) :\n  (s.sup P).factors f :=\nbegin\n  classical,\n  revert h,\n  apply finset.induction_on s,\n  { rintro ⟨_, ⟨⟨⟩, _⟩⟩, },\n  { rintros i s nm ih ⟨j, ⟨m, h⟩⟩,\n    simp only [finset.sup_insert],\n    simp at m, rcases m with (rfl|m),\n    { exact sup_factors_of_factors_left h, },\n    { exact sup_factors_of_factors_right (ih ⟨j, ⟨m, h⟩⟩), }, },\nend\n\nend semilattice_sup\n\nsection lattice\n\ninstance [has_initial C] [initial_mono_class C] {B : C} : bounded_order (subobject B) :=\n{ ..subobject.order_top,\n  ..subobject.order_bot }\n\nvariables [has_pullbacks C] [has_images C] [has_binary_coproducts C]\n\ninstance {B : C} : lattice (subobject B) :=\n{ ..subobject.semilattice_inf,\n  ..subobject.semilattice_sup }\n\nend lattice\n\nsection Inf\n\nvariables [well_powered C]\n\n/--\nThe \"wide cospan\" diagram, with a small indexing type, constructed from a set of subobjects.\n(This is just the diagram of all the subobjects pasted together, but using `well_powered C`\nto make the diagram small.)\n-/\ndef wide_cospan {A : C} (s : set (subobject A)) :\n  wide_pullback_shape (equiv_shrink _ '' s) ⥤ C :=\nwide_pullback_shape.wide_cospan A\n  (λ j : equiv_shrink _ '' s, (((equiv_shrink (subobject A)).symm j) : C))\n  (λ j, ((equiv_shrink (subobject A)).symm j).arrow)\n\n@[simp] lemma wide_cospan_map_term {A : C} (s : set (subobject A)) (j) :\n  (wide_cospan s).map (wide_pullback_shape.hom.term j) =\n    ((equiv_shrink (subobject A)).symm j).arrow :=\nrfl\n\n/-- Auxiliary construction of a cone for `le_Inf`. -/\ndef le_Inf_cone {A : C} (s : set (subobject A)) (f : subobject A) (k : Π (g ∈ s), f ≤ g) :\n  cone (wide_cospan s) :=\nwide_pullback_shape.mk_cone f.arrow\n  (λ j, underlying.map (hom_of_le (k _ (by { rcases j with ⟨-, ⟨g, ⟨m, rfl⟩⟩⟩, simpa using m, }))))\n  (by tidy)\n\n@[simp] lemma le_Inf_cone_π_app_none\n  {A : C} (s : set (subobject A)) (f : subobject A) (k : Π (g ∈ s), f ≤ g) :\n  (le_Inf_cone s f k).π.app none = f.arrow :=\nrfl\n\nvariables [has_wide_pullbacks.{v₁} C]\n\n/--\nThe limit of `wide_cospan s`. (This will be the supremum of the set of subobjects.)\n-/\ndef wide_pullback {A : C} (s : set (subobject A)) : C :=\nlimits.limit (wide_cospan s)\n\n/--\nThe inclusion map from `wide_pullback s` to `A`\n-/\ndef wide_pullback_ι {A : C} (s : set (subobject A)) :\n  wide_pullback s ⟶ A :=\nlimits.limit.π (wide_cospan s) none\n\ninstance wide_pullback_ι_mono {A : C} (s : set (subobject A)) :\n  mono (wide_pullback_ι s) :=\n⟨λ W u v h, limit.hom_ext (λ j, begin\n  cases j,\n  { exact h, },\n  { apply (cancel_mono ((equiv_shrink (subobject A)).symm j).arrow).1,\n    rw [assoc, assoc],\n    erw limit.w (wide_cospan s) (wide_pullback_shape.hom.term j),\n    exact h, },\nend)⟩\n\n/--\nWhen `[well_powered C]` and `[has_wide_pullbacks C]`, `subobject A` has arbitrary infimums.\n-/\ndef Inf {A : C} (s : set (subobject A)) : subobject A :=\nsubobject.mk (wide_pullback_ι s)\n\nlemma Inf_le {A : C} (s : set (subobject A)) (f ∈ s) :\n  Inf s ≤ f :=\nbegin\n  fapply le_of_comm,\n  { refine (underlying_iso _).hom ≫\n      (limits.limit.π\n        (wide_cospan s)\n        (some ⟨equiv_shrink _ f, set.mem_image_of_mem (equiv_shrink (subobject A)) H⟩)) ≫ _,\n    apply eq_to_hom,\n    apply (congr_arg (λ X : subobject A, (X : C))),\n    exact (equiv.symm_apply_apply _ _), },\n  { dsimp [Inf],\n    simp only [category.comp_id, category.assoc, ←underlying_iso_hom_comp_eq_mk,\n      subobject.arrow_congr, congr_arg_mpr_hom_left, iso.cancel_iso_hom_left],\n    convert limit.w (wide_cospan s) (wide_pullback_shape.hom.term _), },\nend.\n\nlemma le_Inf {A : C} (s : set (subobject A)) (f : subobject A) (k : Π (g ∈ s), f ≤ g) :\n  f ≤ Inf s :=\nbegin\n  fapply le_of_comm,\n  { exact limits.limit.lift _ (le_Inf_cone s f k) ≫ (underlying_iso _).inv, },\n  { dsimp [Inf, wide_pullback_ι],\n    simp, },\nend\n\ninstance {B : C} : complete_semilattice_Inf (subobject B) :=\n{ Inf := Inf,\n  Inf_le := Inf_le,\n  le_Inf := le_Inf,\n  ..subobject.partial_order B }\n\nend Inf\n\nsection Sup\n\nvariables [well_powered C] [has_coproducts.{v₁} C]\n\n/--\nThe univesal morphism out of the coproduct of a set of subobjects,\nafter using `[well_powered C]` to reindex by a small type.\n-/\ndef small_coproduct_desc {A : C} (s : set (subobject A)) : _ ⟶ A :=\nlimits.sigma.desc (λ j : equiv_shrink _ '' s, ((equiv_shrink (subobject A)).symm j).arrow)\n\nvariables [has_images C]\n\n/-- When `[well_powered C] [has_images C] [has_coproducts C]`,\n`subobject A` has arbitrary supremums. -/\ndef Sup {A : C} (s : set (subobject A)) : subobject A :=\nsubobject.mk (image.ι (small_coproduct_desc s))\n\nlemma le_Sup {A : C} (s : set (subobject A)) (f ∈ s)  :\n  f ≤ Sup s :=\nbegin\n  fapply le_of_comm,\n  { dsimp [Sup],\n    refine _ ≫ factor_thru_image _ ≫ (underlying_iso _).inv,\n    refine _ ≫ sigma.ι _ ⟨equiv_shrink _ f, (by simpa [set.mem_image] using H)⟩,\n    exact eq_to_hom (congr_arg (λ X : subobject A, (X : C)) (equiv.symm_apply_apply _ _).symm), },\n  { dsimp [Sup, small_coproduct_desc],\n    simp, dsimp, simp, },\nend\n\nlemma symm_apply_mem_iff_mem_image {α β : Type*} (e : α ≃ β) (s : set α) (x : β) :\n  e.symm x ∈ s ↔ x ∈ e '' s :=\n⟨λ h, ⟨e.symm x, h, by simp⟩, by { rintro ⟨a, m, rfl⟩, simpa using m, }⟩\n\nlemma Sup_le {A : C} (s : set (subobject A)) (f : subobject A) (k : Π (g ∈ s), g ≤ f) :\n  Sup s ≤ f :=\nbegin\n  fapply le_of_comm,\n  { dsimp [Sup],\n    refine (underlying_iso _).hom ≫ image.lift ⟨_, f.arrow, _, _⟩,\n    { refine sigma.desc _,\n      rintro ⟨g, m⟩,\n      refine underlying.map (hom_of_le (k _ _)),\n      simpa [symm_apply_mem_iff_mem_image] using m, },\n    { ext j, rcases j with ⟨j, m⟩, dsimp [small_coproduct_desc], simp, dsimp, simp, }, },\n  { dsimp [Sup],\n    simp, },\nend\n\ninstance {B : C} : complete_semilattice_Sup (subobject B) :=\n{ Sup := Sup,\n  le_Sup := le_Sup,\n  Sup_le := Sup_le,\n  ..subobject.partial_order B }\n\nend Sup\n\nsection complete_lattice\nvariables [well_powered C] [has_wide_pullbacks.{v₁} C] [has_images C] [has_coproducts.{v₁} C]\n  [initial_mono_class C]\n\nlocal attribute [instance] has_smallest_coproducts_of_has_coproducts\n\ninstance {B : C} : complete_lattice (subobject B) :=\n{ ..subobject.semilattice_inf,\n  ..subobject.semilattice_sup,\n  ..subobject.bounded_order,\n  ..subobject.complete_semilattice_Inf,\n  ..subobject.complete_semilattice_Sup, }\n\nend complete_lattice\n\nsection zero_object\nvariables [has_zero_morphisms C] [has_zero_object C]\nopen_locale zero_object\n\n/-- A nonzero object has nontrivial subobject lattice. -/\nlemma nontrivial_of_not_is_zero {X : C} (h : ¬ is_zero X) : nontrivial (subobject X) :=\n⟨⟨mk (0 : 0 ⟶ X), mk (𝟙 X), λ w, h (is_zero.of_iso (is_zero_zero C) (iso_of_mk_eq_mk _ _ w).symm)⟩⟩\n\nend zero_object\n\nsection subobject_subobject\n\n/-- The subobject lattice of a subobject `Y` is order isomorphic to the interval `set.Iic Y`. -/\ndef subobject_order_iso {X : C} (Y : subobject X) : subobject (Y : C) ≃o set.Iic Y :=\n{ to_fun := λ Z, ⟨subobject.mk (Z.arrow ≫ Y.arrow),\n    set.mem_Iic.mpr (le_of_comm ((underlying_iso _).hom ≫ Z.arrow) (by simp))⟩,\n  inv_fun := λ Z, subobject.mk (of_le _ _ Z.2),\n  left_inv := λ Z, mk_eq_of_comm _ (underlying_iso _) (by { ext, simp, }),\n  right_inv := λ Z, subtype.ext (mk_eq_of_comm _ (underlying_iso _)\n    (by { dsimp, simp [←iso.eq_inv_comp], })),\n  map_rel_iff' := λ W Z,\n    ⟨λ h, le_of_comm\n      ((underlying_iso _).inv ≫ of_le _ _ (subtype.mk_le_mk.mp h) ≫ (underlying_iso _).hom)\n      (by { ext, simp, }),\n     λ h, subtype.mk_le_mk.mpr\n       (le_of_comm ((underlying_iso _).hom ≫ of_le _ _ h ≫ (underlying_iso _).inv) (by simp))⟩, }\n\nend subobject_subobject\n\nend subobject\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/subobject/lattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3170799608700028}}
{"text": "import category_theory.sites.limits\n\nimport for_mathlib.abelian_sheaves.functor_category\nimport for_mathlib.sheaf\nimport category_theory.sites.left_exact\n\nnamespace category_theory\nnamespace Sheaf\n\n\nnoncomputable theory\n\nuniverses w v u\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {A : Type w} [category.{max v u} A]\n\nsection has_zero_morphisms\n\nvariables [limits.has_zero_morphisms A]\n\ninstance : limits.has_zero_morphisms (Sheaf J A) :=\n{ has_zero := λ X Y, ⟨⟨0⟩⟩,\n  comp_zero' := λ X Y f Z, Sheaf.hom.ext _ _ $ limits.comp_zero,\n  zero_comp' := λ X Y Z f, Sheaf.hom.ext _ _ $ limits.zero_comp }\n\nend has_zero_morphisms\n\nsection parallel_pair\n\ndef parallel_pair_iso {F G : Sheaf J A} (η γ : F ⟶ G) :\n  limits.parallel_pair η γ ⋙ Sheaf_to_presheaf J A ≅ limits.parallel_pair η.val γ.val :=\nnat_iso.of_components\n(λ x,\nmatch x with\n| limits.walking_parallel_pair.zero := eq_to_iso rfl\n| limits.walking_parallel_pair.one := eq_to_iso rfl\nend) begin\n  rintro (x|x) (y|y) (f|f|f),\n  any_goals { refl },\n  any_goals { ext, dsimp [parallel_pair_iso._match_1], simp },\nend\n\nend parallel_pair\n\nsection kernels\n\nvariables [limits.has_zero_morphisms A]\n-- TODO: Add some instances that derive the following from `[has_kernels A]`.\nvariables [limits.has_limits_of_shape.{_ (max v u)} limits.walking_parallel_pair A]\n\ndef kernel_sheaf {F G : Sheaf J A} (η : F ⟶ G) : Sheaf J A :=\n{ val := limits.kernel.{(max v u)} ((Sheaf_to_presheaf J A).map η),\n  cond := begin\n    haveI : limits.has_limit (limits.parallel_pair η 0 ⋙ Sheaf_to_presheaf J A) := begin\n      apply limits.has_limit_of_iso (parallel_pair_iso _ _).symm,\n      apply_instance,\n    end,\n    let e : limits.limit (limits.parallel_pair η 0 ⋙ Sheaf_to_presheaf J A) ≅\n      limits.kernel η.val := limits.has_limit.iso_of_nat_iso (parallel_pair_iso _ _),\n    apply presheaf.is_sheaf_of_iso J e.symm,\n    apply is_sheaf_of_is_limit,\n    apply limits.limit.is_limit,\n  end }\n\ndef kernel_ι {F G : Sheaf J A} (η : F ⟶ G) : kernel_sheaf η ⟶ F :=\n⟨limits.kernel.ι _⟩\n\ndef kernel_fork {F G : Sheaf J A} (η : F ⟶ G) : limits.fork η 0 :=\nlimits.fork.of_ι (kernel_ι η) $\nby { simp only [limits.comp_zero], ext1, apply limits.kernel.condition }\n\ndef is_limit_kernel_fork {F G : Sheaf J A} (η : F ⟶ G) : limits.is_limit (kernel_fork η) :=\nlimits.is_limit_aux _ (λ S, ⟨limits.kernel.lift _ S.ι.val $ congr_arg Sheaf.hom.val S.condition⟩)\n(by { intros S, ext1, apply limits.kernel.lift_ι, })\nbegin\n  intros S m hm,\n  ext : 2,\n  simp only [limits.kernel.lift_ι],\n  exact congr_arg Sheaf.hom.val hm,\nend\n\n-- Sanity check\nexample : limits.has_kernels (Sheaf J A) := by apply_instance\n\ndef kernel_iso_kernel_sheaf {F G : Sheaf J A} (η : F ⟶ G) :\n  limits.kernel η ≅ kernel_sheaf η :=\n(limits.limit.is_limit _).cone_point_unique_up_to_iso (is_limit_kernel_fork _)\n\n@[simp, reassoc]\nlemma kernel_iso_kernel_sheaf_hom_ι {F G : Sheaf J A} (η : F ⟶ G) :\n  (kernel_iso_kernel_sheaf η).hom ≫ kernel_ι η = limits.kernel.ι _ :=\n((limits.limit.is_limit _).unique_up_to_iso (is_limit_kernel_fork η)).hom.w\n  limits.walking_parallel_pair.zero\n\n@[simp, reassoc]\nlemma kernel_iso_kernel_sheaf_hom_ι_val {F G : Sheaf J A} (η : F ⟶ G) :\n  (kernel_iso_kernel_sheaf η).hom.val ≫ (kernel_ι η).val = (limits.kernel.ι η).val :=\nbegin\n  change ((kernel_iso_kernel_sheaf η).hom ≫ (kernel_ι η)).val = (limits.kernel.ι η).val,\n  simp,\nend\n\n@[simp, reassoc]\nlemma kernel_iso_kernel_sheaf_inv_ι {F G : Sheaf J A} (η : F ⟶ G) :\n  (kernel_iso_kernel_sheaf η).inv ≫ limits.kernel.ι _ = kernel_ι η :=\nby simp only [← kernel_iso_kernel_sheaf_hom_ι, iso.inv_hom_id_assoc]\n\n@[simp, reassoc]\nlemma kernel_iso_kernel_sheaf_inv_ι_val {F G : Sheaf J A} (η : F ⟶ G) :\n  (kernel_iso_kernel_sheaf η).inv.val ≫ (limits.kernel.ι η).val = (kernel_ι η).val :=\nbegin\n  change ((kernel_iso_kernel_sheaf η).inv ≫ (limits.kernel.ι η)).val = (kernel_ι η).val,\n  simp,\nend\n\nend kernels\n\nsection cokernels\n\nvariables [limits.has_zero_morphisms A]\n-- TODO: Add some instances that derive the following from `[has_cokernels A]`.\nvariables [limits.has_colimits_of_shape.{_ (max v u)} limits.walking_parallel_pair A]\n\n-- We will need to sheafify....\nvariables [concrete_category.{max v u} A]\nvariables [∀ (P : Cᵒᵖ ⥤ A) (X : C) (S : J.cover X), limits.has_multiequalizer (S.index P)]\nvariables [limits.preserves_limits (forget A)]\nvariables [∀ (X : C), limits.has_colimits_of_shape (J.cover X)ᵒᵖ A]\nvariables [∀ (X : C), limits.preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\nvariables [reflects_isomorphisms (forget A)]\n\ndef cokernel_sheaf {F G : Sheaf J A} (η : F ⟶ G) : Sheaf J A :=\n{ val := J.sheafify (limits.cokernel ((Sheaf_to_presheaf J A).map η)), -- ;-)\n  cond := grothendieck_topology.plus.is_sheaf_plus_plus _ _ }\n\ndef cokernel_π {F G : Sheaf J A} (η : F ⟶ G) : G ⟶ cokernel_sheaf η :=\n⟨show (Sheaf_to_presheaf J A).obj G ⟶ J.sheafify (limits.cokernel ((Sheaf_to_presheaf J A).map η)),\nfrom limits.cokernel.π ((Sheaf_to_presheaf J A).map η) ≫\n  J.to_sheafify (limits.cokernel ((Sheaf_to_presheaf J A).map η))⟩\n\ndef cokernel_cofork {F G : Sheaf J A} (η : F ⟶ G) : limits.cofork η 0 :=\nlimits.cofork.of_π (cokernel_π η) begin\n  ext1,\n  rw [limits.zero_comp],\n  dsimp only [cokernel_π],\n  show η.val ≫ _ = 0,\n  erw [← category.assoc, limits.cokernel.condition, limits.zero_comp],\nend\n\ndef is_colimit_cokernel_cofork {F G : Sheaf J A} (η : F ⟶ G) :\n  limits.is_colimit (cokernel_cofork η) :=\nlimits.is_colimit_aux _ (λ S,\n  ⟨J.sheafify_lift\n    (limits.cokernel.desc ((Sheaf_to_presheaf J A).map η) S.π.val $\n      congr_arg Sheaf.hom.val S.condition)\n    (S.X.2)⟩)\nbegin\n  intros S, ext1,\n  change (_ ≫ _) ≫ _ = _,\n  rw [category.assoc, J.to_sheafify_sheafify_lift, limits.cokernel.π_desc],\nend begin\n  intros S m hm, ext1,\n  apply J.sheafify_lift_unique,\n  rw Sheaf.hom.ext_iff at hm,\n  change (_ ≫ _) ≫ _ = _ at hm,\n  rw category.assoc at hm,\n  ext1,\n  rw [hm, limits.cokernel.π_desc],\nend\n\n-- Sanity check\nexample : limits.has_cokernels (Sheaf J A) := by apply_instance\n\ndef cokernel_iso_cokernel_sheaf {F G : Sheaf J A} (η : F ⟶ G) :\n  limits.cokernel η ≅ cokernel_sheaf η :=\n(limits.colimit.is_colimit _).cocone_point_unique_up_to_iso (is_colimit_cokernel_cofork _)\n\n@[simp]\nlemma cokernel_iso_cokernel_sheaf_hom_π {F G : Sheaf J A} (η : F ⟶ G) :\n  limits.cokernel.π η ≫ (cokernel_iso_cokernel_sheaf η).hom = cokernel_π _ :=\n((limits.colimit.is_colimit _).unique_up_to_iso (is_colimit_cokernel_cofork η)).hom.w\n  limits.walking_parallel_pair.one\n\n@[simp]\nlemma cokernel_iso_cokernel_sheaf_inv_π {F G : Sheaf J A} (η : F ⟶ G) :\n  cokernel_π η ≫ (cokernel_iso_cokernel_sheaf η).inv = limits.cokernel.π η :=\nby simp only [← cokernel_iso_cokernel_sheaf_hom_π,\n  category.assoc, iso.hom_inv_id, category.comp_id]\n\nend cokernels\n\nsection kernels_and_cokernels\n\nvariables [limits.has_zero_morphisms A]\n-- TODO: use has kernels and cokernels, when possible... see above\nvariables [limits.has_colimits_of_shape.{_ (max v u)} limits.walking_parallel_pair A]\nvariables [limits.has_limits_of_shape.{_ (max v u)} limits.walking_parallel_pair A]\n\n-- We will need to sheafify....\nvariables [concrete_category.{max v u} A]\nvariables [∀ (P : Cᵒᵖ ⥤ A) (X : C) (S : J.cover X), limits.has_multiequalizer (S.index P)]\nvariables [limits.preserves_limits (forget A)]\nvariables [∀ (X : C), limits.has_colimits_of_shape (J.cover X)ᵒᵖ A]\nvariables [∀ (X : C), limits.preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\nvariables [reflects_isomorphisms (forget A)]\n\ndef coim_to_im'_aux {F G : Sheaf J A} (η : F ⟶ G) :\n  abelian.coimage ((Sheaf_to_presheaf J A).map η) ⟶\n    (Sheaf_to_presheaf J A).obj (kernel_sheaf (cokernel_π η)) :=\n(abelian.coimage_image_comparison _ ≫ limits.kernel.lift _ (limits.kernel.ι _) begin\n  dsimp [cokernel_π],\n  rw [← category.assoc, limits.kernel.condition],\n  simp only [limits.zero_comp],\nend)\n\ndef coim_to_im' {F G : Sheaf J A} (η : F ⟶ G) :\n  cokernel_sheaf (kernel_ι η) ⟶ kernel_sheaf (cokernel_π η) :=\n⟨J.sheafify_lift (coim_to_im'_aux η) (kernel_sheaf _).2⟩\n\ndef kernel_sheaf_cokernel_π_iso {F G : Sheaf J A} (η : F ⟶ G) :\n  kernel_sheaf (limits.cokernel.π η) ≅ kernel_sheaf (cokernel_π η) :=\n{ hom := ⟨limits.kernel.map _ _ (𝟙 _)\n    ((Sheaf_to_presheaf J A).map (cokernel_iso_cokernel_sheaf η).hom) begin\n      rw ← functor.map_comp,\n      dsimp [cokernel_iso_cokernel_sheaf, limits.is_colimit.cocone_point_unique_up_to_iso,\n        cokernel_cofork],\n      show Sheaf.hom.val (limits.cokernel.π η ≫ _) = _,\n      simp only [limits.coequalizer.π_desc, category.id_comp],\n    end⟩,\n  inv := ⟨limits.kernel.map _ _ (𝟙 _)\n    ((Sheaf_to_presheaf J A).map (cokernel_iso_cokernel_sheaf η).inv) begin\n      rw ← functor.map_comp,\n      dsimp [cokernel_iso_cokernel_sheaf, limits.is_colimit.cocone_point_unique_up_to_iso,\n        cokernel_π, is_colimit_cokernel_cofork, limits.is_colimit_aux],\n      erw [category.id_comp, category.assoc, J.to_sheafify_sheafify_lift,\n        limits.cokernel.π_desc],\n    end⟩,\n  hom_inv_id' := begin\n    ext : 2,\n    delta limits.kernel.map,\n    dsimp,\n    erw [category.assoc, limits.kernel.lift_ι, ← category.assoc, limits.kernel.lift_ι,\n      category.comp_id, category.comp_id, category.id_comp],\n  end,\n  inv_hom_id' := begin\n    ext : 2,\n    delta limits.kernel.map,\n    dsimp,\n    erw [category.assoc, limits.kernel.lift_ι, ← category.assoc, limits.kernel.lift_ι,\n      category.comp_id, category.comp_id, category.id_comp],\n  end }\n\ndef cokernel_sheaf_kernel_ι_iso {F G : Sheaf J A} (η : F ⟶ G) :\n  cokernel_sheaf (limits.kernel.ι η) ≅ cokernel_sheaf (kernel_ι η) :=\n{ hom := ⟨J.sheafify_lift\n    (limits.cokernel.map _ _ ((Sheaf_to_presheaf J A).map (kernel_iso_kernel_sheaf η).hom) (𝟙 _)\n      (by rw [category.comp_id, ← functor.map_comp, kernel_iso_kernel_sheaf_hom_ι])\n      ≫ J.to_sheafify _) (cokernel_sheaf _).2⟩,\n  inv := ⟨J.sheafify_lift\n    (limits.cokernel.map _ _ ((Sheaf_to_presheaf J A).map (kernel_iso_kernel_sheaf η).inv) (𝟙 _)\n      (by rw [category.comp_id, ← functor.map_comp, kernel_iso_kernel_sheaf_inv_ι])\n      ≫ J.to_sheafify _)\n    (cokernel_sheaf _).2⟩,\n  hom_inv_id' := begin\n    ext1,\n    apply J.sheafify_hom_ext _ _ (cokernel_sheaf _).2,\n    erw [← category.assoc, J.to_sheafify_sheafify_lift, category.assoc,\n      J.to_sheafify_sheafify_lift, ← category.assoc],\n    conv_rhs { erw category.comp_id },\n    convert category.id_comp _,\n    ext1,\n    delta limits.cokernel.map,\n    erw [← category.assoc, limits.coequalizer.π_desc, category.id_comp,\n      limits.coequalizer.π_desc, category.id_comp, category.comp_id],\n  end,\n  inv_hom_id' := begin\n    ext1,\n    apply J.sheafify_hom_ext _ _ (cokernel_sheaf _).2,\n    erw [← category.assoc, J.to_sheafify_sheafify_lift, category.assoc,\n      J.to_sheafify_sheafify_lift, ← category.assoc],\n    conv_rhs { erw category.comp_id },\n    convert category.id_comp _,\n    ext1,\n    delta limits.cokernel.map,\n    erw [← category.assoc, limits.coequalizer.π_desc, category.id_comp,\n      limits.coequalizer.π_desc, category.id_comp, category.comp_id],\n  end }\n\nlemma eq_coim_to_im' {F G : Sheaf J A} (η : F ⟶ G) :\n  (cokernel_sheaf_kernel_ι_iso η).inv ≫\n  (cokernel_iso_cokernel_sheaf _).inv ≫\n  abelian.coimage_image_comparison η  ≫\n  (kernel_iso_kernel_sheaf _).hom ≫\n  (kernel_sheaf_cokernel_π_iso η).hom\n  = coim_to_im' η :=\nbegin\n  dsimp [abelian.coimage_image_comparison, cokernel_sheaf_kernel_ι_iso,\n    coim_to_im', coim_to_im'_aux, kernel_sheaf_cokernel_π_iso,\n    limits.is_colimit.cocone_point_unique_up_to_iso,\n    limits.is_limit.cone_point_unique_up_to_iso],\n  delta limits.kernel.map limits.cokernel.map,\n  ext1,\n  apply J.sheafify_lift_unique,\n  ext : 2,\n  conv_rhs {\n    erw [← category.assoc, limits.cokernel.π_desc,\n      category.assoc, limits.kernel.lift_ι, limits.kernel.lift_ι] },\n  dsimp,\n  simp only [category.assoc, category.comp_id, category.id_comp,\n    J.to_sheafify_sheafify_lift_assoc, kernel_iso_kernel_sheaf_hom_ι_val,\n      limits.kernel.lift_ι, limits.cokernel.π_desc_assoc],\n  dsimp [cokernel_iso_cokernel_sheaf, limits.is_colimit.cocone_point_unique_up_to_iso,\n    is_colimit_cokernel_cofork, limits.is_colimit_aux],\n  rw J.to_sheafify_sheafify_lift_assoc,\n  simp only [limits.cokernel.π_desc_assoc, ← category.assoc, ← Sheaf.hom.comp_val,\n    limits.cokernel.π_desc],\n  simp only [category_theory.category_comp_val, category.assoc],\n  erw kernel_iso_kernel_sheaf_hom_ι_val,\n  rw [← Sheaf.hom.comp_val, limits.kernel.lift_ι],\nend\n\nlemma coim_to_im_eq {F G : Sheaf J A} (η : F ⟶ G) :\n  abelian.coimage_image_comparison η =\n  (cokernel_iso_cokernel_sheaf _).hom ≫\n  (cokernel_sheaf_kernel_ι_iso η).hom ≫\n  coim_to_im' η ≫\n  (kernel_sheaf_cokernel_π_iso η).inv ≫\n  (kernel_iso_kernel_sheaf _).inv\n  :=\nbegin\n  rw ← eq_coim_to_im',\n  simp only [category.assoc, iso.hom_inv_id, iso.inv_hom_id,\n    iso.hom_inv_id_assoc, iso.inv_hom_id_assoc, category.comp_id],\nend\n\nend kernels_and_cokernels\n\nsection preadditive\n\nvariable [preadditive A]\n\ninstance : preadditive (Sheaf J A) :=\n{ hom_group := λ P Q,\n  { add := λ f g, ⟨f.val + g.val⟩,\n    add_assoc := by { intros, ext1, apply add_assoc },\n    zero := ⟨0⟩,\n    zero_add := by { intros, ext1, apply zero_add },\n    add_zero := by { intros, ext1, apply add_zero },\n    nsmul := λ n f, ⟨n • f.val⟩,\n    nsmul_zero' := by { intros, ext1, simpa },\n    nsmul_succ' := by { intros, ext1, simpa },\n    neg := λ f, ⟨-f.val⟩,\n    sub := λ f g, ⟨f.val - g.val⟩,\n    sub_eq_add_neg := by { intros, ext1, apply sub_eq_add_neg },\n    zsmul := λ n f, ⟨n • f.val⟩,\n    zsmul_zero' := by { intros, ext1, simpa },\n    zsmul_succ' := by { intros, ext1, simpa },\n    zsmul_neg' := by { intros, ext1, simpa },\n    add_left_neg := by { intros, ext1, apply add_left_neg },\n    add_comm := by { intros, ext1, apply add_comm } },\n  add_comp' := λ P Q R f g h, by { ext1, apply preadditive.add_comp },\n  comp_add' := λ P Q R f g h, by { ext1, apply preadditive.comp_add } }\n\nend preadditive\n\nsection additive\n\nvariable [additive_category A]\n\ninstance : additive_category (Sheaf J A) :=\n{ has_biproducts_of_shape := begin\n    introsI J _ _,\n    constructor,\n    intros F,\n    apply limits.has_biproduct.of_has_product\n  end,\n  ..(by apply_instance : preadditive (Sheaf J A)) }\n\nend additive\n\nsection abelian\n\nvariables [abelian A]\n-- We need sheafification\nvariables [concrete_category.{max v u} A]\nvariables [∀ (P : Cᵒᵖ ⥤ A) (X : C) (S : J.cover X), limits.has_multiequalizer (S.index P)]\nvariables [limits.preserves_limits (forget A)]\n--variables [limits.has_finite_limits A]\nvariables [∀ (X : C), limits.has_colimits_of_shape (J.cover X)ᵒᵖ A]\nvariables [∀ (X : C), limits.preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\nvariables [reflects_isomorphisms (forget A)]\n\nopen grothendieck_topology\n\ndef parallel_pair_sheafification {F G : Sheaf J A} (η : F ⟶ G) : limits.parallel_pair\n  (limits.cokernel.π η.val) 0 ⋙ J.sheafification A ≅\n  limits.parallel_pair (cokernel_π η).val 0 :=\nnat_iso.of_components\n(λ x,\nmatch x with\n| limits.walking_parallel_pair.zero := by { dsimp, refine (J.iso_sheafify G.2).symm }\n| limits.walking_parallel_pair.one := by { dsimp, exact eq_to_iso rfl }\nend)\nbegin\n  -- This proof is SLOW :-(\n  rintros (a|a) (b|b) (f|f|f),\n  { dsimp [parallel_pair_sheafification._match_1],\n    simp only [iso.eq_inv_comp, functor.map_id],\n    dsimp [grothendieck_topology.iso_sheafify],\n    rw [← category.assoc, is_iso.comp_inv_eq, category.id_comp],\n    change J.to_sheafify _ ≫ (sheafification J A).map _ = _,\n    erw [functor.map_id, category.comp_id] },\n  { dsimp [parallel_pair_sheafification._match_1],\n    rw [category.comp_id, iso.eq_inv_comp],\n    dsimp [grothendieck_topology.iso_sheafify, cokernel_π],\n    change (to_sheafification J A).app _ ≫ (sheafification J A).map _ = _,\n    rw ← (to_sheafification J A).naturality,\n    refl },\n  { dsimp [parallel_pair_sheafification._match_1],\n    simp only [limits.comp_zero, category.comp_id],\n    change (sheafification J A).map _ = _,\n    apply J.sheafify_hom_ext,\n    { exact plus.is_sheaf_plus_plus J (limits.cokernel η.val) },\n    erw ← (to_sheafification J A).naturality,\n    simp },\n  { dsimp [parallel_pair_sheafification._match_1],\n    simp only [limits.comp_zero, category.id_comp, category.comp_id],\n    change (sheafification J A).map _ = _,\n    simp only [functor.map_id],\n    erw (sheafification J A).map_id, refl },\nend .\n\ndef kernel_cokernel_π_iso {F G : Sheaf J A} (η : F ⟶ G) :\n  J.sheafify (limits.kernel (limits.cokernel.π ((Sheaf_to_presheaf J A).map η))) ≅\n  limits.kernel ((Sheaf_to_presheaf J A).map (cokernel_π η)) :=\nbegin\n  let e := (limits.is_limit_of_preserves (sheafification J A)\n      (limits.limit.is_limit\n      (limits.parallel_pair (limits.cokernel.π\n      ((Sheaf_to_presheaf J A).map η)) 0))).cone_point_unique_up_to_iso (limits.limit.is_limit _),\n  refine e ≪≫ _,\n  change limits.limit _ ≅ _,\n  refine limits.has_limit.iso_of_nat_iso _,\n  apply parallel_pair_sheafification,\nend\n\n/-\n{ hom := J.sheafify_lift (limits.kernel.map _ _ (𝟙 _) (J.to_sheafify _) (by admit)) (by admit),\n  inv := begin\n    let e : J.sheafify ((Sheaf_to_presheaf J A).obj G) ⟶\n      J.sheafify (limits.cokernel ((Sheaf_to_presheaf J A).map η)) :=\n        (sheafification J A).map (limits.cokernel.π _),\n    let ee : limits.kernel ((Sheaf_to_presheaf J A).map (cokernel_π η)) ⟶ limits.kernel e,\n    { refine limits.kernel.map _ _ (J.to_sheafify _) (𝟙 _) _,\n      rw category.comp_id,\n      dsimp only [e],\n      rw ← grothendieck_topology.to_sheafification_app,\n      rw ← (to_sheafification J A).naturality,\n      refl },\n    refine ee ≫ _,\n    dsimp only [e],\n    change limits.kernel ((Sheaf_to_presheaf J A).map ((presheaf_to_Sheaf J A).map _)) ⟶ _,\n    refine (Sheaf_to_presheaf J A).map (kernel_iso_kernel_sheaf _).inv ≫ _,\n    change _ ⟶ (Sheaf_to_presheaf J A).obj ((presheaf_to_Sheaf J A).obj _),\n    refine (Sheaf_to_presheaf J A).map _,\n    haveI : is_left_adjoint (presheaf_to_Sheaf J A) := by admit,\n    -- Now we need to use the fact that finite limits commute with sheafification,\n    -- i.e. that sheafification is left exact.\n    admit\n  end,\n  hom_inv_id' := by admit,\n  inv_hom_id' := by admit }\n-/\n\nlemma coim_to_im'_eq {F G : Sheaf J A} (η : F ⟶ G) :\n  (Sheaf_to_presheaf J A).map (coim_to_im' η) =\n  (sheafification J A).map (abelian.coimage_image_comparison _) ≫ (kernel_cokernel_π_iso η).hom :=\nbegin\n  dsimp only [kernel_cokernel_π_iso, limits.is_limit.cone_point_unique_up_to_iso,\n    functor.map_iso, iso.trans_hom],\n  simp only [category.assoc],\n  dsimp only [id, limits.is_limit.unique_up_to_iso, limits.cones.forget,\n    limits.is_limit.lift_cone_morphism, functor.map_cone],\n  apply limits.equalizer.hom_ext,\n  simp only [category.assoc],\n  --delta limits.equalizer.ι,\n  erw limits.has_limit.iso_of_nat_iso_hom_π,\n  dsimp only [parallel_pair_sheafification, nat_iso.of_components, id, iso_sheafify,\n    as_iso, iso.symm],\n  simp only [← category.assoc],\n  rw is_iso.eq_comp_inv,\n  simp only [category.assoc],\n  erw limits.limit.lift_π,\n  dsimp [limits.cones.functoriality, sheafify_map],\n  simp_rw [← plus_functor_map, ← functor.comp_map, ← functor.map_comp],\n  dsimp [coim_to_im', abelian.coimage_image_comparison, coim_to_im'_aux],\n  apply J.sheafify_hom_ext,\n  { exact plus.is_sheaf_plus_plus J G.val, },\n  simp_rw ← category.assoc,\n  erw J.to_sheafify_sheafify_lift,\n  change _ = (to_sheafification J A).app _ ≫ (sheafification J A).map _,\n  erw ← (to_sheafification J A).naturality,\n  simp only [category.assoc, limits.kernel.lift_ι],\n  dsimp,\n  erw category.assoc,\nend\n\ninstance is_iso_coim_to_im {F G : Sheaf J A} (η : F ⟶ G) :\n  is_iso (abelian.coimage_image_comparison η) :=\nbegin\n  rw coim_to_im_eq,\n  suffices : is_iso (coim_to_im' η),\n  { resetI, apply is_iso.comp_is_iso },\n  suffices : is_iso ((Sheaf_to_presheaf J A).map (coim_to_im' η)),\n  { resetI, apply is_iso_of_fully_faithful (Sheaf_to_presheaf J A) },\n  rw coim_to_im'_eq,\n  apply is_iso.comp_is_iso,\nend\n\ninstance abelian : abelian (Sheaf J A) :=\nabelian.of_coimage_image_comparison_is_iso\n\nend abelian\n\nend Sheaf\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/abelian_sheaves/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3170799608700028}}
{"text": "import data.cpi.concretion.basic\n\nnamespace cpi\nnamespace concretion\n\nvariables {ℍ : Type} {ω : context} [∀ Γ, setoid (species ℍ ω Γ)]\n\n/-- Structural congruence between concretions. -/\ninductive equiv : ∀ {Γ} {b y}, concretion ℍ ω Γ b y → concretion ℍ ω Γ b y → Prop\n| refl  {Γ} {b y} {F : concretion ℍ ω Γ b y} : equiv F F\n| trans {Γ} {b y} {F G H : concretion ℍ ω Γ b y} : equiv F G → equiv G H → equiv F H\n| symm  {Γ} {b y} {F G : concretion ℍ ω Γ b y} : equiv F G → equiv G F\n\n| ξ_parallel₁\n    {Γ} {b y} {F F' : concretion ℍ ω Γ b y} {A : species ℍ ω Γ}\n  : equiv F F' → equiv (F |₁ A) (F' |₁ A)\n| ξ_parallel₂\n    {Γ} {b y} {F F' : concretion ℍ ω Γ b y} {A : species ℍ ω Γ}\n  : equiv F F' → equiv (A |₂ F) (A |₂ F')\n| ξ_restriction\n    {Γ} {b y} (M : affinity ℍ) {F F' : concretion ℍ ω (context.extend M.arity Γ) b y}\n  : equiv F F' → equiv (ν'(M) F) (ν'(M) F')\n\n-- Parallel is a commutative monoid\n| parallel_nil\n    {Γ} {b y} {F : concretion ℍ ω Γ b y}\n  : equiv (F |₁ species.nil) F\n| parallel_symm\n    {Γ} {b y} {F : concretion ℍ ω Γ b y} {A : species ℍ ω Γ}\n  : equiv (F |₁ A) (A |₂ F)\n| parallel_assoc₁\n    {Γ} {b y} {F : concretion ℍ ω Γ b y} {A B : species ℍ ω Γ}\n  : equiv ((F |₁ A) |₁ B) (F |₁ (A |ₛ B))\n| parallel_assoc₂\n    {Γ} {b y} {F : concretion ℍ ω Γ b y} {A B : species ℍ ω Γ}\n  : equiv ((A |₂ F) |₁ B) (A |₂ (F |₁ B))\n\n-- Projections for species into parallel/apply\n| ξ_parallel\n    {Γ} {b y} {F : concretion ℍ ω Γ b y} {A B : species ℍ ω Γ}\n  : A ≈ B → equiv (F |₁ A) (F |₁ B)\n| ξ_apply\n    {Γ} {b y} {bs : vector (name Γ) b} {A B : species ℍ ω (context.extend y Γ)}\n  : A ≈ B → equiv (#(bs; y) A) (#(bs; y) B)\n\n-- Standard ν rules\n| ν_parallel₁\n    {Γ} {b y} (M : affinity ℍ)\n    {A : species ℍ ω Γ} {F : concretion ℍ ω (context.extend M.arity Γ) b y}\n  : equiv (ν'(M)(species.rename name.extend A |₂ F)) (A |₂ ν'(M) F)\n| ν_parallel₂\n    {Γ} {b y} (M : affinity ℍ)\n    {A : species ℍ ω (context.extend M.arity Γ)} {F : concretion ℍ ω Γ b y}\n  : equiv (ν'(M)(concretion.rename name.extend F |₁ A)) (F |₁ (ν(M) A))\n| ν_drop\n    {Γ} {b y} (M : affinity ℍ) {F : concretion ℍ ω Γ b y}\n  : equiv (ν'(M) rename name.extend F) F\n| ν_swap\n    {Γ} {b y} (M N : affinity ℍ)\n    {F : concretion ℍ ω (context.extend N.arity (context.extend M.arity Γ)) b y}\n  : equiv (ν'(M)ν'(N) F) (ν'(N)ν'(M) rename name.swap F)\n\n| apply_parallel\n    {Γ} {b y} {bs : vector (name Γ) b}\n    {A : species ℍ ω Γ} {B : species ℍ ω (context.extend y Γ)}\n  : equiv (#(bs; y) (species.rename name.extend A |ₛ B)) (A |₂ #(bs; y) B)\n\ninstance {Γ} {b y} : is_equiv (concretion ℍ ω Γ b y) equiv :=\n  { refl := @equiv.refl ℍ _ _ Γ b y, symm := @equiv.symm ℍ _ _ Γ b y, trans := @equiv.trans ℍ _ _ Γ b y }\ninstance {Γ} {b y} : is_refl (concretion ℍ ω Γ b y) equiv := ⟨ λ _, equiv.refl ⟩\n\nnamespace equiv\n  /-- The setoid of species under structural congruence. Can be brought into\n      scope with the \"congruence\" locale. -/\n  def setoid {Γ} {b y} : setoid (concretion ℍ ω Γ b y) :=\n    ⟨ equiv, ⟨ @refl ℍ _ _ Γ b y, @symm ℍ _ _ Γ b y, @trans ℍ _ _ Γ b y ⟩ ⟩\n\n  localized \"attribute [instance] cpi.concretion.equiv.setoid\" in congruence\n\n  protected lemma ξ_parallel'\n      {Γ} {b y} {F : concretion ℍ ω Γ b y} {A A' : species ℍ ω Γ} (eq : A ≈ A')\n    : (A |₂ F) ≈ (A' |₂ F) :=\n      calc  (A |₂ F)\n          ≈ (F |₁ A) : symm parallel_symm\n      ... ≈ (F |₁ A') : ξ_parallel eq\n      ... ≈ (A' |₂ F) : parallel_symm\n\n  protected lemma parallel_assoc₃\n      {Γ} {b y : ℕ} {A B : species ℍ ω Γ} {F : concretion ℍ ω Γ b y}\n    : ((A |ₛ B) |₂ F) ≈ (A |₂ B |₂ F) :=\n    calc  ((A |ₛ B) |₂ F)\n        ≈ (F |₁ (A |ₛ B)) : symm parallel_symm\n    ... ≈ ((F |₁ A) |₁ B) : symm parallel_assoc₁\n    ... ≈ ((A |₂ F) |₁ B) : ξ_parallel₁ parallel_symm\n    ... ≈ (A |₂ F |₁ B) : parallel_assoc₂\n    ... ≈ (A |₂ B |₂ F) : ξ_parallel₂ parallel_symm\n\nend equiv\nend concretion\n\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/concretion/congruence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544210587586, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.3169453511220218}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Justus Springer\n-/\nimport topology.category.Top.open_nhds\nimport topology.sheaves.presheaf\nimport topology.sheaves.sheaf_condition.unique_gluing\nimport category_theory.limits.types\nimport category_theory.limits.preserves.filtered\nimport category_theory.limits.final\nimport tactic.elementwise\n\n/-!\n# Stalks\n\nFor a presheaf `F` on a topological space `X`, valued in some category `C`, the *stalk* of `F`\nat the point `x : X` is defined as the colimit of the following functor\n\n(nhds x)ᵒᵖ ⥤ (opens X)ᵒᵖ ⥤ C\n\nwhere the functor on the left is the inclusion of categories and the functor on the right is `F`.\nFor an open neighborhood `U` of `x`, we define the map `F.germ x : F.obj (op U) ⟶ F.stalk x` as the\ncanonical morphism into this colimit.\n\nTaking stalks is functorial: For every point `x : X` we define a functor `stalk_functor C x`,\nsending presheaves on `X` to objects of `C`. Furthermore, for a map `f : X ⟶ Y` between\ntopological spaces, we define `stalk_pushforward` as the induced map on the stalks\n`(f _* ℱ).stalk (f x) ⟶ ℱ.stalk x`.\n\nSome lemmas about stalks and germs only hold for certain classes of concrete categories. A basic\nproperty of forgetful functors of categories of algebraic structures (like `Mon`, `CommRing`,...)\nis that they preserve filtered colimits. Since stalks are filtered colimits, this ensures that\nthe stalks of presheaves valued in these categories behave exactly as for `Type`-valued presheaves.\nFor example, in `germ_exist` we prove that in such a category, every element of the stalk is the\ngerm of a section.\n\nFurthermore, if we require the forgetful functor to reflect isomorphisms and preserve limits (as\nis the case for most algebraic structures), we have access to the unique gluing API and can prove\nfurther properties. Most notably, in `is_iso_iff_stalk_functor_map_iso`, we prove that in such\na category, a morphism of sheaves is an isomorphism if and only if all of its stalk maps are\nisomorphisms.\n\nSee also the definition of \"algebraic structures\" in the stacks project:\nhttps://stacks.math.columbia.edu/tag/007L\n\n-/\n\nnoncomputable theory\n\nuniverses v u v' u'\n\nopen category_theory\nopen Top\nopen category_theory.limits\nopen topological_space\nopen opposite\n\nvariables {C : Type u} [category.{v} C]\n\nvariables [has_colimits.{v} C]\n\nvariables {X Y Z : Top.{v}}\n\nnamespace Top.presheaf\n\nvariables (C)\n/-- Stalks are functorial with respect to morphisms of presheaves over a fixed `X`. -/\ndef stalk_functor (x : X) : X.presheaf C ⥤ C :=\n((whiskering_left _ _ C).obj (open_nhds.inclusion x).op) ⋙ colim\n\nvariables {C}\n\n/--\nThe stalk of a presheaf `F` at a point `x` is calculated as the colimit of the functor\nnbhds x ⥤ opens F.X ⥤ C\n-/\ndef stalk (ℱ : X.presheaf C) (x : X) : C :=\n(stalk_functor C x).obj ℱ -- -- colimit ((open_nhds.inclusion x).op ⋙ ℱ)\n\n@[simp] lemma stalk_functor_obj (ℱ : X.presheaf C) (x : X) :\n  (stalk_functor C x).obj ℱ = ℱ.stalk x := rfl\n\n/--\nThe germ of a section of a presheaf over an open at a point of that open.\n-/\ndef germ (F : X.presheaf C) {U : opens X} (x : U) : F.obj (op U) ⟶ stalk F x :=\ncolimit.ι ((open_nhds.inclusion x.1).op ⋙ F) (op ⟨U, x.2⟩)\n\n@[simp, elementwise]\nlemma germ_res (F : X.presheaf C) {U V : opens X} (i : U ⟶ V) (x : U) :\n  F.map i.op ≫ germ F x = germ F (i x : V) :=\nlet i' : (⟨U, x.2⟩ : open_nhds x.1) ⟶ ⟨V, (i x : V).2⟩ := i in\ncolimit.w ((open_nhds.inclusion x.1).op ⋙ F) i'.op\n\n/--\nA morphism from the stalk of `F` at `x` to some object `Y` is completely determined by its\ncomposition with the `germ` morphisms.\n-/\nlemma stalk_hom_ext (F : X.presheaf C) {x} {Y : C} {f₁ f₂ : F.stalk x ⟶ Y}\n  (ih : ∀ (U : opens X) (hxU : x ∈ U), F.germ ⟨x, hxU⟩ ≫ f₁ = F.germ ⟨x, hxU⟩ ≫ f₂) : f₁ = f₂ :=\ncolimit.hom_ext $ λ U, by { induction U using opposite.rec, cases U with U hxU, exact ih U hxU }\n\n@[simp, reassoc, elementwise]\nlemma stalk_functor_map_germ {F G : X.presheaf C} (U : opens X) (x : U)\n  (f : F ⟶ G) : germ F x ≫ (stalk_functor C x.1).map f = f.app (op U) ≫ germ G x :=\ncolimit.ι_map (whisker_left ((open_nhds.inclusion x.1).op) f) (op ⟨U, x.2⟩)\n\nvariables (C)\n\n/--\nFor a presheaf `F` on a space `X`, a continuous map `f : X ⟶ Y` induces a morphisms between the\nstalk of `f _ * F` at `f x` and the stalk of `F` at `x`.\n-/\ndef stalk_pushforward (f : X ⟶ Y) (F : X.presheaf C) (x : X) : (f _* F).stalk (f x) ⟶ F.stalk x :=\nbegin\n  -- This is a hack; Lean doesn't like to elaborate the term written directly.\n  transitivity,\n  swap,\n  exact colimit.pre _ (open_nhds.map f x).op,\n  exact colim.map (whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) F),\nend\n\n@[simp, elementwise, reassoc]\nlemma stalk_pushforward_germ (f : X ⟶ Y) (F : X.presheaf C) (U : opens Y)\n  (x : (opens.map f).obj U) :\n  (f _* F).germ ⟨f x, x.2⟩ ≫ F.stalk_pushforward C f x = F.germ x :=\nbegin\n  rw [stalk_pushforward, germ, colimit.ι_map_assoc, colimit.ι_pre, whisker_right_app],\n  erw [category_theory.functor.map_id, category.id_comp],\n  refl,\nend\n\n-- Here are two other potential solutions, suggested by @fpvandoorn at\n-- <https://github.com/leanprover-community/mathlib/pull/1018#discussion_r283978240>\n-- However, I can't get the subsequent two proofs to work with either one.\n\n-- def stalk_pushforward (f : X ⟶ Y) (ℱ : X.presheaf C) (x : X) :\n--   (f _* ℱ).stalk (f x) ⟶ ℱ.stalk x :=\n-- colim.map ((functor.associator _ _ _).inv ≫\n--   whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) ℱ) ≫\n-- colimit.pre ((open_nhds.inclusion x).op ⋙ ℱ) (open_nhds.map f x).op\n\n-- def stalk_pushforward (f : X ⟶ Y) (ℱ : X.presheaf C) (x : X) :\n--   (f _* ℱ).stalk (f x) ⟶ ℱ.stalk x :=\n-- (colim.map (whisker_right (nat_trans.op (open_nhds.inclusion_map_iso f x).inv) ℱ) :\n--   colim.obj ((open_nhds.inclusion (f x) ⋙ opens.map f).op ⋙ ℱ) ⟶ _) ≫\n-- colimit.pre ((open_nhds.inclusion x).op ⋙ ℱ) (open_nhds.map f x).op\n\nnamespace stalk_pushforward\nlocal attribute [tidy] tactic.op_induction'\n\n@[simp] lemma id (ℱ : X.presheaf C) (x : X) :\n  ℱ.stalk_pushforward C (𝟙 X) x = (stalk_functor C x).map ((pushforward.id ℱ).hom) :=\nbegin\n  dsimp [stalk_pushforward, stalk_functor],\n  ext1,\n  tactic.op_induction',\n  cases j, cases j_val,\n  rw [colimit.ι_map_assoc, colimit.ι_map, colimit.ι_pre, whisker_left_app, whisker_right_app,\n       pushforward.id_hom_app, eq_to_hom_map, eq_to_hom_refl],\n  dsimp,\n  -- FIXME A simp lemma which unfortunately doesn't fire:\n  erw [category_theory.functor.map_id],\nend\n\n-- This proof is sadly not at all robust:\n-- having to use `erw` at all is a bad sign.\n@[simp] lemma comp (ℱ : X.presheaf C) (f : X ⟶ Y) (g : Y ⟶ Z) (x : X) :\n  ℱ.stalk_pushforward C (f ≫ g) x =\n  ((f _* ℱ).stalk_pushforward C g (f x)) ≫ (ℱ.stalk_pushforward C f x) :=\nbegin\n  dsimp [stalk_pushforward, stalk_functor],\n  ext U,\n  induction U using opposite.rec,\n  cases U,\n  cases U_val,\n  simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc,\n             whisker_right_app, category.assoc],\n  dsimp,\n  -- FIXME: Some of these are simp lemmas, but don't fire successfully:\n  erw [category_theory.functor.map_id, category.id_comp, category.id_comp, category.id_comp,\n       colimit.ι_pre, colimit.ι_pre],\n  refl,\nend\n\nlemma stalk_pushforward_iso_of_open_embedding {f : X ⟶ Y} (hf : open_embedding f)\n   (F : X.presheaf C) (x : X) : is_iso (F.stalk_pushforward _ f x) :=\n begin\n   haveI := functor.initial_of_adjunction (hf.is_open_map.adjunction_nhds x),\n   convert is_iso.of_iso ((functor.final.colimit_iso (hf.is_open_map.functor_nhds x).op\n     ((open_nhds.inclusion (f x)).op ⋙ f _* F) : _).symm ≪≫ colim.map_iso _),\n   swap,\n   { fapply nat_iso.of_components,\n     { intro U,\n       refine F.map_iso (eq_to_iso _),\n       dsimp only [functor.op],\n       exact congr_arg op (subtype.eq $ set.preimage_image_eq (unop U).1.1 hf.inj) },\n     { intros U V i, erw [← F.map_comp, ← F.map_comp], congr } },\n   { ext U,\n     rw ← iso.comp_inv_eq,\n     erw colimit.ι_map_assoc,\n     rw [colimit.ι_pre, category.assoc],\n     erw [colimit.ι_map_assoc, colimit.ι_pre, ← F.map_comp_assoc],\n     apply colimit.w ((open_nhds.inclusion (f x)).op ⋙ f _* F) _,\n     dsimp only [functor.op],\n     refine ((hom_of_le _).op : op (unop U) ⟶ _),\n     exact set.image_preimage_subset _ _ },\n end\n\nend stalk_pushforward\n\nsection stalk_pullback\n\n/-- The morphism `ℱ_{f x} ⟶ (f⁻¹ℱ)ₓ` that factors through `(f_*f⁻¹ℱ)_{f x}`. -/\ndef stalk_pullback_hom (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  F.stalk (f x) ⟶ (pullback_obj f F).stalk x :=\n(stalk_functor _ (f x)).map ((pushforward_pullback_adjunction C f).unit.app F) ≫\n  stalk_pushforward _ _ _ x\n\n/-- The morphism `(f⁻¹ℱ)(U) ⟶ ℱ_{f(x)}` for some `U ∋ x`. -/\ndef germ_to_pullback_stalk (f : X ⟶ Y) (F : Y.presheaf C) (U : opens X) (x : U) :\n  (pullback_obj f F).obj (op U) ⟶ F.stalk (f x) :=\ncolimit.desc (Lan.diagram (opens.map f).op F (op U))\n{ X := F.stalk (f x),\n  ι := { app := λ V, F.germ ⟨f x, V.hom.unop.le x.2⟩,\n          naturality' := λ _ _ i, by { erw category.comp_id, exact F.germ_res i.left.unop _ } } }\n\n/-- The morphism `(f⁻¹ℱ)ₓ ⟶ ℱ_{f(x)}`. -/\ndef stalk_pullback_inv (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  (pullback_obj f F).stalk x ⟶ F.stalk (f x) :=\ncolimit.desc ((open_nhds.inclusion x).op ⋙ presheaf.pullback_obj f F)\n{ X := F.stalk (f x),\n  ι := { app := λ U, F.germ_to_pullback_stalk _ f (unop U).1 ⟨x, (unop U).2⟩,\n          naturality' := λ _ _ _, by { erw [colimit.pre_desc, category.comp_id], congr } } }\n\n/-- The isomorphism `ℱ_{f(x)} ≅ (f⁻¹ℱ)ₓ`. -/\ndef stalk_pullback_iso (f : X ⟶ Y) (F : Y.presheaf C) (x : X) :\n  F.stalk (f x) ≅ (pullback_obj f F).stalk x :=\n{ hom := stalk_pullback_hom _ _ _ _,\n  inv := stalk_pullback_inv _ _ _ _,\n  hom_inv_id' :=\n  begin\n    delta stalk_pullback_hom stalk_pullback_inv stalk_functor presheaf.pullback stalk_pushforward\n      germ_to_pullback_stalk germ,\n    ext j,\n    induction j using opposite.rec,\n    cases j,\n    simp only [topological_space.open_nhds.inclusion_map_iso_inv, whisker_right_app,\n      whisker_left_app, whiskering_left_obj_map, functor.comp_map, colimit.ι_map_assoc,\n      nat_trans.op_id, Lan_obj_map, pushforward_pullback_adjunction_unit_app_app, category.assoc,\n      colimit.ι_pre_assoc],\n    erw [colimit.ι_desc, colimit.pre_desc, colimit.ι_desc, category.comp_id],\n    simpa\n  end,\n  inv_hom_id' :=\n  begin\n    delta stalk_pullback_hom stalk_pullback_inv stalk_functor presheaf.pullback stalk_pushforward,\n    ext U j,\n    induction U using opposite.rec,\n    cases U, cases j, cases j_right,\n    erw [colimit.map_desc, colimit.map_desc, colimit.ι_desc_assoc,\n      colimit.ι_desc_assoc, colimit.ι_desc, category.comp_id],\n    simp only [cocone.whisker_ι, colimit.cocone_ι, open_nhds.inclusion_map_iso_inv,\n      cocones.precompose_obj_ι, whisker_right_app, whisker_left_app, nat_trans.comp_app,\n      whiskering_left_obj_map, nat_trans.op_id, Lan_obj_map,\n      pushforward_pullback_adjunction_unit_app_app],\n    erw ←colimit.w _\n      (@hom_of_le (open_nhds x) _\n         ⟨_, U_property⟩ ⟨(opens.map f).obj (unop j_left), j_hom.unop.le U_property⟩\n         j_hom.unop.le).op,\n    erw colimit.ι_pre_assoc (Lan.diagram _ F _) (costructured_arrow.map _),\n    erw colimit.ι_pre_assoc (Lan.diagram _ F _) (costructured_arrow.map _),\n    congr,\n    simp only [category.assoc, costructured_arrow.map_mk],\n    delta costructured_arrow.mk,\n    congr\n  end }\n\nend stalk_pullback\n\nsection concrete\n\nvariables {C}\nvariables [concrete_category.{v} C]\n\nlocal attribute [instance] concrete_category.has_coe_to_sort concrete_category.has_coe_to_fun\n\n@[ext]\nlemma germ_ext (F : X.presheaf C) {U V : opens X} {x : X} {hxU : x ∈ U} {hxV : x ∈ V}\n  (W : opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V) {sU : F.obj (op U)} {sV : F.obj (op V)}\n  (ih : F.map iWU.op sU = F.map iWV.op sV) :\n  F.germ ⟨x, hxU⟩ sU = F.germ ⟨x, hxV⟩ sV :=\nby erw [← F.germ_res iWU ⟨x, hxW⟩,\n    ← F.germ_res iWV ⟨x, hxW⟩, comp_apply, comp_apply, ih]\n\nvariables [preserves_filtered_colimits (forget C)]\n\n/--\nFor presheaves valued in a concrete category whose forgetful functor preserves filtered colimits,\nevery element of the stalk is the germ of a section.\n-/\nlemma germ_exist (F : X.presheaf C) (x : X) (t : stalk F x) :\n  ∃ (U : opens X) (m : x ∈ U) (s : F.obj (op U)), F.germ ⟨x, m⟩ s = t :=\nbegin\n  obtain ⟨U, s, e⟩ := types.jointly_surjective _\n    (is_colimit_of_preserves (forget C) (colimit.is_colimit _)) t,\n  revert s e,\n  rw [(show U = op (unop U), from rfl)],\n  generalize : unop U = V, clear U,\n  cases V with V m,\n  intros s e,\n  exact ⟨V, m, s, e⟩,\nend\n\nlemma germ_eq (F : X.presheaf C) {U V : opens X} (x : X) (mU : x ∈ U) (mV : x ∈ V)\n  (s : F.obj (op U)) (t : F.obj (op V))\n  (h : germ F ⟨x, mU⟩ s = germ F ⟨x, mV⟩ t) :\n  ∃ (W : opens X) (m : x ∈ W) (iU : W ⟶ U) (iV : W ⟶ V), F.map iU.op s = F.map iV.op t :=\nbegin\n  obtain ⟨W, iU, iV, e⟩ := (types.filtered_colimit.is_colimit_eq_iff _\n    (is_colimit_of_preserves _ (colimit.is_colimit ((open_nhds.inclusion x).op ⋙ F)))).mp h,\n  exact ⟨(unop W).1, (unop W).2, iU.unop, iV.unop, e⟩,\nend\n\nlemma stalk_functor_map_injective_of_app_injective {F G : presheaf C X} (f : F ⟶ G)\n  (h : ∀ U : opens X, function.injective (f.app (op U))) (x : X) :\n  function.injective ((stalk_functor C x).map f) := λ s t hst,\nbegin\n  rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩,\n  rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩,\n  simp only [stalk_functor_map_germ_apply _ ⟨x,_⟩] at hst,\n  obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst,\n  rw [← comp_apply, ← comp_apply, ← f.naturality, ← f.naturality, comp_apply, comp_apply] at heq,\n  replace heq := h W heq,\n  convert congr_arg (F.germ ⟨x,hxW⟩) heq,\n  exacts [(F.germ_res_apply iWU₁ ⟨x,hxW⟩ s).symm,\n          (F.germ_res_apply iWU₂ ⟨x,hxW⟩ t).symm],\nend\n\n\nvariables [has_limits C] [preserves_limits (forget C)] [reflects_isomorphisms (forget C)]\n\n/--\nLet `F` be a sheaf valued in a concrete category, whose forgetful functor reflects isomorphisms,\npreserves limits and filtered colimits. Then two sections who agree on every stalk must be equal.\n-/\nlemma section_ext (F : sheaf C X) (U : opens X) (s t : F.1.obj (op U))\n  (h : ∀ x : U, F.1.germ x s = F.1.germ x t) :\n  s = t :=\nbegin\n  -- We use `germ_eq` and the axiom of choice, to pick for every point `x` a neighbourhood\n  -- `V x`, such that the restrictions of `s` and `t` to `V x` coincide.\n  choose V m i₁ i₂ heq using λ x : U, F.1.germ_eq x.1 x.2 x.2 s t (h x),\n  -- Since `F` is a sheaf, we can prove the equality locally, if we can show that these\n  -- neighborhoods form a cover of `U`.\n  apply F.eq_of_locally_eq' V U i₁,\n  { intros x hxU,\n    rw [opens.mem_coe, opens.mem_supr],\n    exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩ },\n  { intro x,\n    rw [heq, subsingleton.elim (i₁ x) (i₂ x)] }\nend\n\n/-\nNote that the analogous statement for surjectivity is false: Surjectivity on stalks does not\nimply surjectivity of the components of a sheaf morphism. However it does imply that the morphism\nis an epi, but this fact is not yet formalized.\n-/\nlemma app_injective_of_stalk_functor_map_injective {F : sheaf C X} {G : presheaf C X}\n  (f : F.1 ⟶ G) (U : opens X) (h : ∀ x : U, function.injective ((stalk_functor C x.val).map f)) :\n  function.injective (f.app (op U)) :=\nλ s t hst, section_ext F _ _ _ $ λ x, h x $ by\n  rw [stalk_functor_map_germ_apply, stalk_functor_map_germ_apply, hst]\n\nlemma app_injective_iff_stalk_functor_map_injective {F : sheaf C X}\n  {G : presheaf C X} (f : F.1 ⟶ G) :\n  (∀ x : X, function.injective ((stalk_functor C x).map f)) ↔\n  (∀ U : opens X, function.injective (f.app (op U))) :=\n⟨λ h U, app_injective_of_stalk_functor_map_injective f U (λ x, h x.1),\n  stalk_functor_map_injective_of_app_injective f⟩\n\n/-- For surjectivity, we are given an arbitrary section `t` and need to find a preimage for it.\nWe claim that it suffices to find preimages *locally*. That is, for each `x : U` we construct\na neighborhood `V ≤ U` and a section `s : F.obj (op V))` such that `f.app (op V) s` and `t`\nagree on `V`. -/\nlemma app_surjective_of_injective_of_locally_surjective {F G : sheaf C X} (f : F ⟶ G)\n  (U : opens X) (hinj : ∀ x : U, function.injective ((stalk_functor C x.1).map f))\n  (hsurj : ∀ (t) (x : U), ∃ (V : opens X) (m : x.1 ∈ V) (iVU : V ⟶ U) (s : F.1.obj (op V)),\n    f.app (op V) s = G.1.map iVU.op t) :\n  function.surjective (f.app (op U)) :=\nbegin\n  intro t,\n  -- We use the axiom of choice to pick around each point `x` an open neighborhood `V` and a\n  -- preimage under `f` on `V`.\n  choose V mV iVU sf heq using hsurj t,\n  -- These neighborhoods clearly cover all of `U`.\n  have V_cover : U ≤ supr V,\n  { intros x hxU,\n    rw [opens.mem_coe, opens.mem_supr],\n    exact ⟨⟨x, hxU⟩, mV ⟨x, hxU⟩⟩ },\n  -- Since `F` is a sheaf, we can glue all the local preimages together to get a global preimage.\n  obtain ⟨s, s_spec, -⟩ := F.exists_unique_gluing' V U iVU V_cover sf _,\n  { use s,\n    apply G.eq_of_locally_eq' V U iVU V_cover,\n    intro x,\n    rw [← comp_apply, ← f.naturality, comp_apply, s_spec, heq] },\n  { intros x y,\n    -- What's left to show here is that the secions `sf` are compatible, i.e. they agree on\n    -- the intersections `V x ⊓ V y`. We prove this by showing that all germs are equal.\n    apply section_ext,\n    intro z,\n    -- Here, we need to use injectivity of the stalk maps.\n    apply (hinj ⟨z, (iVU x).le ((inf_le_left : V x ⊓ V y ≤ V x) z.2)⟩),\n    dsimp only,\n    erw [stalk_functor_map_germ_apply, stalk_functor_map_germ_apply],\n    simp_rw [← comp_apply, f.naturality, comp_apply, heq, ← comp_apply, ← G.1.map_comp],\n    refl }\nend\n\nlemma app_surjective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)\n  (U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f)) :\n  function.surjective (f.app (op U)) :=\nbegin\n  refine app_surjective_of_injective_of_locally_surjective f U (λ x, (h x).1) (λ t x, _),\n  -- Now we need to prove our initial claim: That we can find preimages of `t` locally.\n  -- Since `f` is surjective on stalks, we can find a preimage `s₀` of the germ of `t` at `x`\n  obtain ⟨s₀,hs₀⟩ := (h x).2 (G.1.germ x t),\n  -- ... and this preimage must come from some section `s₁` defined on some open neighborhood `V₁`\n  obtain ⟨V₁,hxV₁,s₁,hs₁⟩ := F.1.germ_exist x.1 s₀,\n  subst hs₁, rename hs₀ hs₁,\n  erw stalk_functor_map_germ_apply V₁ ⟨x.1,hxV₁⟩ f s₁ at hs₁,\n  -- Now, the germ of `f.app (op V₁) s₁` equals the germ of `t`, hence they must coincide on\n  -- some open neighborhood `V₂`.\n  obtain ⟨V₂, hxV₂, iV₂V₁, iV₂U, heq⟩ := G.1.germ_eq x.1 hxV₁ x.2 _ _ hs₁,\n  -- The restriction of `s₁` to that neighborhood is our desired local preimage.\n  use [V₂, hxV₂, iV₂U, F.1.map iV₂V₁.op s₁],\n  rw [← comp_apply, f.naturality, comp_apply, heq],\nend\n\nlemma app_bijective_of_stalk_functor_map_bijective {F G : sheaf C X} (f : F ⟶ G)\n   (U : opens X) (h : ∀ x : U, function.bijective ((stalk_functor C x.val).map f)) :\n  function.bijective (f.app (op U)) :=\n⟨app_injective_of_stalk_functor_map_injective f U (λ x, (h x).1),\n  app_surjective_of_stalk_functor_map_bijective f U h⟩\n\nlemma app_is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) (U : opens X)\n  [∀ x : U, is_iso ((stalk_functor C x.val).map f)] : is_iso (f.app (op U)) :=\nbegin\n  -- Since the forgetful functor of `C` reflects isomorphisms, it suffices to see that the\n  -- underlying map between types is an isomorphism, i.e. bijective.\n  suffices : is_iso ((forget C).map (f.app (op U))),\n  { exactI is_iso_of_reflects_iso (f.app (op U)) (forget C) },\n  rw is_iso_iff_bijective,\n  apply app_bijective_of_stalk_functor_map_bijective,\n  intro x,\n  apply (is_iso_iff_bijective _).mp,\n  exact functor.map_is_iso (forget C) ((stalk_functor C x.1).map f)\nend\n\n/--\nLet `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects\nisomorphisms, preserves limits and filtered colimits. Then if the stalk maps of a morphism\n`f : F ⟶ G` are all isomorphisms, `f` must be an isomorphism.\n-/\n-- Making this an instance would cause a loop in typeclass resolution with `functor.map_is_iso`\nlemma is_iso_of_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G)\n  [∀ x : X, is_iso ((stalk_functor C x).map f)] : is_iso f :=\nbegin\n  -- Since the inclusion functor from sheaves to presheaves is fully faithful, it suffices to\n  -- show that `f`, as a morphism between _presheaves_, is an isomorphism.\n  suffices : is_iso ((sheaf.forget C X).map f),\n  { exactI is_iso_of_fully_faithful (sheaf.forget C X) f },\n  -- We show that all components of `f` are isomorphisms.\n  suffices : ∀ U : (opens X)ᵒᵖ, is_iso (f.app U),\n  { exact @nat_iso.is_iso_of_is_iso_app _ _ _ _ F.1 G.1 f this, },\n  intro U, induction U using opposite.rec,\n  apply app_is_iso_of_stalk_functor_map_iso\nend\n\n/--\nLet `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects\nisomorphisms, preserves limits and filtered colimits. Then a morphism `f : F ⟶ G` is an\nisomorphism if and only if all of its stalk maps are isomorphisms.\n-/\nlemma is_iso_iff_stalk_functor_map_iso {F G : sheaf C X} (f : F ⟶ G) :\n  is_iso f ↔ ∀ x : X, is_iso ((stalk_functor C x).map f) :=\nbegin\n  split,\n  { intros h x, resetI,\n    exact @functor.map_is_iso _ _ _ _ _ _ (stalk_functor C x) f\n      ((sheaf.forget C X).map_is_iso f) },\n  { intro h,\n    exactI is_iso_of_stalk_functor_map_iso f }\nend\n\nend concrete\n\nend Top.presheaf\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/topology/sheaves/stalks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3167051318389201}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\n\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.lattice\nimport formal_ml.measurable_space\nimport formal_ml.classical\nimport data.equiv.list\n\n\n/-! This file defines the basic concepts in probability theory.\n    There are four fundamental principles:\n    1. Make theorems as readable as possible. Use Pr[A ∧ B], not μ (A ∩ B). Other examples:\n       Pr[(X >ᵣ 3) ∨ (Y =ᵣ 7)]. While events are technically sets, in probability theory,\n       they are better written as propositions that may or may not hold.\n    2. Avoid absurd statements where possible. Don't allow Pr[A] if A is not an event,\n       or Pr[X ∈ᵣ S] if S is not measurable, or Pr[∀ᵣ a in S, E a] if S is not countable.\n       It is always possible to write Pr[⟨S, ...proof S is an event...⟩]. \n    3. Embed measurability into the objects and operations themselves. An event is measurable by\n       definition. When we take the and, or, not, for all, exists, measurability should be automatic.\n    4. Don't expect the reader to know measure theory, but at times it may be required by the\n       author.\n\n    Several concepts are defined in this module:\n      probability_space: a measure_space where the measure has a value of 1. \n      measurable_setB: a subtype of a set that is measurable (defined based upon the measurable space).\n      event: a measurable_setB on a probability space (defined based upon the probability).\n      Pr[E]: the probability of an event (note: expectations are defined in real_random_variable).\n      measurable_fun: a subtype of a function that is measurable (denoted (M₁ →ₘ M₂)).\n      random_variable: a measurable_fun whose domain is a probability space (denoted (P →ᵣ M)).\n\n     Some symbols are defined as well:\n     * (∀ᵣ i, E i): for all E\n     * (∃ᵣ i, F i): exists i, such that F.\n     * X ∈ᵣ S: the event that the random variable X is in the measurable set S.\n     * and more!\n \n     Also, various independence and identical definitions are specified. Some choices:\n     * A and B are independent if A has zero probability.\n     * an infinite family of events/random variables is independent if every finite subset\n       is independent.\n     * Two random variables are identical if they have equal probability on every measurable\n       set. The probability spaces on which they are defined need not be equal.\n      -/\n\n/- In the latest src/data/equiv/list.lean, but not yet included -/\nnoncomputable def fintype.encodable (α : Type*) [fintype α] : encodable α :=\nby { classical, exact (encodable.trunc_encodable_of_fintype α).out }\n\n\ndef set.symmetric_difference {α :Type*} (A B:set α) := (A \\ B) ∪ (B \\ A)\n\nclass has_symm_diff (α : Type*) := (symm_diff : α → α → α)\n\n-- U+2206: symmetric difference\ninfixr ` ∆ `:70  := has_symm_diff.symm_diff\n\n\ninstance set.has_symm_diff {α : Type*}: has_symm_diff (set α) := ⟨set.symmetric_difference⟩\n\n\nlemma set.has_symm_diff.def {α : Type*} {A B:set α}:A ∆ B = (A \\ B) ∪ (B \\ A) := rfl\n\n\n\nclass probability_space (α: Type*) extends measure_theory.measure_space α :=\n  (univ_one:volume.measure_of (set.univ) = 1) \n\ninstance probability_space.to_measurable_space (α:Type*) [probability_space α]:measurable_space α :=\n  measure_theory.measure_space.to_measurable_space\n\n@[simp]\nlemma probability_space.univ_one' {α:Type*} (Pα:probability_space α):\n  (@measure_theory.measure_space.volume α Pα.to_measure_space) (@set.univ α) = 1 := begin\n  rw ← measure_theory.coe_to_outer_measure,\n  rw ← measure_theory.outer_measure.measure_of_eq_coe,\n  rw probability_space.univ_one\nend\n\n\n--measure_of_eq_coe\n\n/-\n  In measure theory (and specifically, in probability theory), not all sets of outcomes have\n  probabilities that can be measured. We represent those that can be measured as measurable\n  sets.\n-/\ndef measurable_setB {α:Type*} (M:measurable_space α):Type* := subtype (M.measurable_set')\n\ndef measurable_setB.mk {α:Type*} {M:measurable_space α} {S:set α} (H:measurable_set S):measurable_setB M := ⟨S, H⟩\n\nlemma measurable_setB_val_eq_coe {Ω:Type*} {P:measurable_space Ω}  \n  (X:measurable_setB P):X.val = \n  (@coe (subtype (@measurable_set Ω _)) (set Ω) _ X) :=\nbegin\n  refl\nend\n\n/-\n  A measurable set on a measurable space that has a probability measure is called an event.\n-/\ndef event {Ω:Type*} (M:probability_space Ω):Type* := measurable_setB (probability_space.to_measurable_space Ω)\n\nlemma event_val_eq_coe {Ω:Type*} {P:probability_space Ω}  \n  (X:event P):X.val = \n  (@coe (subtype (@measurable_set Ω _)) (set Ω) _ X) :=\nbegin\n  refl\nend\n\nlemma event.eq {Ω:Type*} {P:probability_space Ω} (A B:event P):\nA.val = B.val → A = B :=\nbegin\n  intro A1,\n  apply subtype.eq,\n  exact A1\nend\n\ndef event_mem {Ω:Type*} [P:probability_space Ω] (a:Ω) (E:event P):Prop :=\n  a∈ E.val\n\n\ninstance {Ω:Type*} [P:probability_space Ω]:has_mem Ω (event P) := {\n  mem := event_mem\n}\n\n\nlemma event_mem_val {Ω:Type*} [P:probability_space Ω] (ω:Ω) (E:event P):\n  (ω ∈ E) = (ω ∈ E.val) := rfl\n\n\nlemma prob_le_1 {Ω:Type*} {P:probability_space Ω} (S:set Ω):\n  P.volume.measure_of S ≤ 1 :=\nbegin\n  have A1:P.volume.measure_of set.univ = 1,\n  {\n    apply P.univ_one,\n  },\n  have A2:S ⊆ set.univ,\n  {\n    simp,\n  },\n  have A3:P.volume.measure_of S ≤ P.volume.measure_of set.univ,\n  {\n    apply P.volume.mono,\n    apply A2,\n  },\n  rw A1 at A3,\n  exact A3,\nend\n\n\n/-\n  There are a lot of long proofs here, but this one seems particularly roundabout.\n-/\nlemma prob_not_infinite {Ω:Type*} {P:probability_space Ω} (S:set Ω):\n  (P.volume.measure_of S) ≠ ⊤ :=\nbegin\n  have A1:P.volume.measure_of S ≤ 1,\n  {\n     apply prob_le_1,\n  },\n  intro A2,\n  rw A2 at A1,\n  have A3:(1:ennreal)=⊤,\n  {\n    apply complete_linear_order.le_antisymm,\n    {\n      apply (ennreal.complete_linear_order.le_top),\n    },\n    {\n      apply A1,\n    }\n  },\n  have A4:(1:ennreal) ≠ (⊤:ennreal),\n  {\n    apply ennreal.one_ne_top,\n  },\n  rw A3 at A4,\n  apply A4,\n  refl,\nend\n\nlemma prob_nnreal {Ω:Type*} {P:probability_space Ω} (S:set Ω):\n   ↑((P.volume.measure_of S).to_nnreal) = P.volume.measure_of S :=\nbegin\n  apply ennreal.coe_to_nnreal,\n  apply prob_not_infinite,\nend\n\ndef event_prob {Ω:Type*} {P:probability_space Ω} (E:event P):nnreal :=\n  (P.volume.measure_of E.val).to_nnreal\n\nnotation `Pr[`E`]` := event_prob E\n\nlemma event_prob_def {Ω:Type*} {p:probability_space Ω} (E:event p):\n  ↑(Pr[E]) = (p.volume.measure_of E.val):=\nbegin\n  unfold event_prob,\n  apply prob_nnreal,\nend\n\nlemma to_nnreal_almost_monotonic (a b:ennreal):(a≠ ⊤)→(b≠⊤)→(a ≤ b)→ (a.to_nnreal ≤ b.to_nnreal) :=\nbegin\n  intros A1 A2 A3,\n  have A4:↑(a.to_nnreal)=a,\n  {\n    apply ennreal.coe_to_nnreal,\n    apply A1,\n  },\n  have A5:↑(b.to_nnreal)=b,\n  {\n    apply ennreal.coe_to_nnreal,\n    apply A2,\n  },\n  rw ← A4 at A3,\n  rw ← A5 at A3,\n  simp at A3,\n  apply A3,\nend\n\nlemma to_ennreal_monotonic (a b:nnreal):(a ≤ b)→ ((a:ennreal) ≤ (b:ennreal)) :=\nbegin\n  intro A1,\n  simp,\n  apply A1,\nend\n\n-- See ennreal.add_eq_top\nlemma add_finite (a b:ennreal):(a≠ ⊤) → (b≠ ⊤) → (a + b≠ ⊤) :=\nbegin\n  intros A1 A2 A3,\n  rw ennreal.add_eq_top at A3,\n  cases A3,\n  {\n    apply A1,\n    apply A3,\n  },\n  {\n    apply A2,\n    apply A3,\n  }\nend\n\n\nlemma event_prob_mono1 {Ω:Type*} {p:probability_space Ω} (E F:event p):\n  p.volume.measure_of E.val ≤ p.volume.measure_of F.val →\n  Pr[E] ≤ Pr[F] :=\nbegin\n  unfold event_prob,\n  intro A1,\n  apply to_nnreal_almost_monotonic,\n  apply prob_not_infinite,\n  apply prob_not_infinite,\n  apply A1,\nend\n\n\nlemma event_prob_mono2 {Ω:Type*} {p:probability_space Ω} (E F:event p):\n  (E.val ⊆ F.val) →\n  Pr[E] ≤ Pr[F] :=\nbegin\n  intro A1,\n  apply event_prob_mono1,\n  apply p.volume.mono,\n  apply A1,\nend\n\n\ndef measurable_setB_univ {Ω:Type*} {M:measurable_space Ω}:measurable_setB M  := {\n  val := set.univ,\n  property := measurable_set.univ,\n}\n\n\ndef event_univ {Ω:Type*} {p:probability_space Ω}:event p := measurable_setB_univ\n\n@[simp]\nlemma event_univ_val_def {Ω:Type*} {p:probability_space Ω}:\n  (@event_univ Ω p).val = set.univ :=\nbegin\n  unfold event_univ measurable_setB_univ,\nend\n\n@[simp]\nlemma Pr_event_univ {Ω:Type*} {p:probability_space Ω}:\n  Pr[@event_univ Ω p] = 1 :=\nbegin\n  have A1:↑(Pr[@event_univ Ω p]) = (1:ennreal),\n  {\n    rw event_prob_def,\n    apply p.univ_one,\n  },\n  simp at A1,\n  apply A1\nend\n\n@[simp]\nlemma Pr_le_one {Ω:Type*} {p:probability_space Ω} {E:event p}:\n  Pr[E] ≤ 1 :=\nbegin\n  have A1:Pr[E] ≤ Pr[@event_univ Ω p],\n  {\n    apply event_prob_mono2,\n    rw event_univ_val_def,\n    rw set.subset_def,simp,\n  },\n  rw Pr_event_univ at A1,\n  apply A1,\nend\n\ndef measurable_setB_empty {Ω:Type*} {p:measurable_space Ω}:measurable_setB p := {\n  val := ∅,\n  property := measurable_set.empty,\n}\n\ninstance has_emptyc_measurable_setB {Ω:Type*} {M:measurable_space Ω}:has_emptyc (measurable_setB M) := ⟨ @measurable_setB_empty Ω M ⟩\n\n\n\ndef event_empty {Ω:Type*} {p:probability_space Ω}:event p := \n  @measurable_setB_empty Ω (probability_space.to_measurable_space Ω)\n\ninstance has_emptyc_event {Ω:Type*} {P:probability_space Ω}:has_emptyc (event P) := \n    ⟨ @event_empty Ω P ⟩\n\nlemma has_emptyc_emptyc_event {Ω:Type*} {P:probability_space Ω}:\n  ∅ = (@event_empty Ω P) :=  rfl\n\n@[simp]\nlemma event_empty_val_def {Ω:Type*} {p:probability_space Ω}:\n  (@event_empty Ω p).val = ∅  := rfl\n\n@[simp]\nlemma event_empty_val_def2 {Ω:Type*} {p:probability_space Ω}:\n  (@has_emptyc.emptyc (event p) _).val = ∅  :=  rfl\n\n@[simp]\nlemma Pr_event_empty {Ω:Type*} {p:probability_space Ω}:\n  Pr[@event_empty Ω p] = 0 :=\nbegin\n  have A1:↑(Pr[@event_empty Ω p]) = (0:ennreal),\n  {\n    rw event_prob_def,\n    apply p.volume.empty,\n  },\n  simp at A1,\n  apply A1\nend\n\n@[simp]\nlemma Pr_event_empty' {Ω:Type*} {p:probability_space Ω}:\n  Pr[(∅:event p)] = 0 :=\nbegin\n  rw has_emptyc_emptyc_event,\n  apply Pr_event_empty,\nend\n\n\n/-Since Pr[E] is a nnreal, this establishes that the probability is in the interval [0,1] -/\nlemma event_prob_le_1 {Ω:Type*} {p:probability_space Ω} {E:event p}:\n  Pr[E] ≤ 1 :=\nbegin\n  have A1:Pr[@event_univ Ω p] = 1,\n  {\n    apply Pr_event_univ,\n  },\n  rw ← A1,\n  apply event_prob_mono2,\n  rw event_univ_val_def,\n  simp,\nend\n\ndef event_const {Ω:Type*} {p:probability_space Ω} (P:Prop):event p := {\n  val := {ω:Ω|P},\n  property := measurable_set.const P,\n}\n\n@[simp]\nlemma event_const_val_def {Ω:Type*} {p:probability_space Ω} (P:Prop):\n  (@event_const _ p P).val={ω:Ω|P} := rfl\n\nlemma event_const_true_eq_univ {Ω:Type*} {p:probability_space Ω} (P:Prop):P →\n(@event_const _ p P)=event_univ :=\nbegin\n  intro A1,\n  apply event.eq,\n  simp [A1],\nend\n\nlemma event_const_false_eq_empty {Ω:Type*} {p:probability_space Ω} (P:Prop):¬P →\n(@event_const _ p P)=event_empty :=\nbegin\n  intro A1,\n  apply event.eq,\n  simp [A1],\nend\n\nlemma Pr_event_const_true {Ω:Type*} {p:probability_space Ω} (P:Prop):P →\nPr[(@event_const _ p P)]=1 :=\nbegin\n  intro A1,\n  rw event_const_true_eq_univ,\n  apply Pr_event_univ,\n  exact A1,\nend\n\nlemma Pr_event_const_false {Ω:Type*} {p:probability_space Ω} (P:Prop):¬P →\nPr[(@event_const _ p P)]=0 :=\nbegin\n  intro A1,\n  rw event_const_false_eq_empty,\n  apply Pr_event_empty,\n  exact A1,\nend\n\n\n\n--The and of two events.\n\n\ndef measurable_inter {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):measurable_setB p := {\n  val:=A.val ∩ B.val,\n  property := measurable_set.inter A.property B.property,\n}\n\n@[simp]\nlemma measurable_inter_val_def {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):\n  (measurable_inter A B).val= A.val ∩ B.val := rfl\n\n\n\ninstance measurable_setB_has_inter {Ω:Type*} {p:measurable_space Ω}:has_inter (measurable_setB p) := {\n  inter := @measurable_inter Ω p,\n}\n\n@[simp]\nlemma measurable_inter_val_def2 {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):\n  (A ∩ B).val= A.val ∩ B.val := rfl\n\n\ndef eand {Ω:Type*} {p:probability_space Ω} (A B:event p):event p := \n  measurable_inter A B\n\n/-{\n  val:=A.val ∩ B.val,\n  property := measurable_set.inter A.property B.property,\n}-/\n\n\ninfixr `∧` := eand\n\n@[simp]\nlemma eand_val_def {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∧ B).val = A.val ∩ B.val :=\nbegin\n  refl,\nend\n\nlemma eand_comm {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∧ B) = (B ∧ A) :=\nbegin\n  apply event.eq,\n  simp [set.inter_comm],\nend\n\nlemma eand_assoc {Ω:Type*} {p:probability_space Ω} (A B C:event p):\n  ((A ∧ B) ∧ C) = (A ∧ (B ∧ C)) :=\nbegin\n  apply event.eq,\n  simp [set.inter_assoc],\nend\n\nlemma eand_eq_self_of_subset_left {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A.val ⊆ B.val) →\n  (A ∧ B) = A :=\nbegin\n  intro A1,\n  apply event.eq,\n  simp,\n  --rw eand_val_def,\n  apply set.inter_eq_self_of_subset_left,\n  exact A1,\nend\n\nlemma eand_eq_self_of_subset_right {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (B.val ⊆ A.val) →\n  (A ∧ B) = B :=\nbegin\n  intro A1,\n  rw eand_comm,\n  apply eand_eq_self_of_subset_left,\n  exact A1,\nend\n\n\nlemma Pr_eand_le_left {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∧ B]≤ Pr[A] :=\nbegin\n  apply event_prob_mono2,\n  rw eand_val_def,\n  apply set.inter_subset_left,\nend\n\n\nlemma Pr_eand_le_right {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∧ B]≤ Pr[B] :=\nbegin\n  rw eand_comm,\n  apply Pr_eand_le_left,\nend\n\n\nlemma Pr_eand_le_min {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∧ B]≤ min Pr[A]  Pr[B] :=\nbegin\n  apply le_min,\n  {\n    apply Pr_eand_le_left,\n  },\n  {\n    apply Pr_eand_le_right,\n  }\nend\n\ndef measurable_union {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):measurable_setB p := {\n  val:=A.val ∪  B.val,\n  property := measurable_set.union A.property B.property,\n}\n\n@[simp]\nlemma measurable_union_val_def {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):\n  (measurable_union A B).val=A.val ∪ B.val := rfl\n\n\n\ninstance measurable_setB_has_union {Ω:Type*} {p:measurable_space Ω}:has_union (measurable_setB p) := {\n  union := @measurable_union Ω p,\n}\n\n@[simp]\nlemma measurable_union_val_def2 {Ω:Type*} {p:measurable_space Ω} (A B:measurable_setB p):\n  (A ∪ B).val = A.val ∪ B.val := rfl\n\n\ndef eor {Ω:Type*} {p:probability_space Ω} (A B:event p):event p := measurable_union A B\n/-{\n  val:=A.val ∪  B.val,\n  property := measurable_set.union A.property B.property,\n}-/\n\ninfixr `∨` := eor\n\n@[simp]\nlemma eor_val_def {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∨ B).val = A.val ∪ B.val :=\nbegin\n  refl,\nend\n\nlemma eor_comm {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∨ B) = (B ∨ A) :=\nbegin\n  apply event.eq,\n  simp [set.union_comm],\nend\n\n\nlemma Pr_le_eor_left {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A] ≤ Pr[A ∨ B] :=\nbegin\n  apply event_prob_mono2,\n  simp,\nend\n\nlemma Pr_le_eor_right {Ω:Type*} {p:probability_space Ω} (A B:event p):\n   Pr[B] ≤ Pr[A ∨ B] :=\nbegin\n  rw eor_comm,\n  apply Pr_le_eor_left,\nend\n\nlemma Pr_le_eor_sum {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∨ B]≤ Pr[A] + Pr[B] :=\nbegin\n  have A1:↑(Pr[A ∨ B])≤ (Pr[A]:ennreal) + (Pr[B]:ennreal),\n  {\n    repeat {rw event_prob_def},\n    simp,\n    apply measure_theory.outer_measure.union,\n  },\n  have A2:↑(Pr[A ∨ B])≤ ((Pr[A] + Pr[B]):ennreal) → Pr[A ∨ B]≤ Pr[A] + Pr[B],\n  {\n    apply to_nnreal_almost_monotonic,\n    {\n      rw event_prob_def,\n      apply prob_not_infinite,\n    },\n    {\n      apply add_finite,\n      rw event_prob_def,\n      apply prob_not_infinite,\n      rw event_prob_def,\n      apply prob_not_infinite,\n    }\n  },\n  apply A2,\n  apply A1,\nend\n\n\nlemma Pr_disjoint_eor {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  disjoint A.val B.val →\n  Pr[A ∨ B] =  Pr[A] + Pr[B] :=\nbegin\n  intro A1,\n  have A2:↑(Pr[A ∨ B])= (Pr[A]:ennreal) + (Pr[B]:ennreal),\n  {\n    repeat {rw event_prob_def},\n    simp,\n    apply measure_theory.measure_union,\n    apply A1,\n    apply A.property,\n    apply B.property,\n  },\n  have A3:((Pr[A ∨ B]):ennreal).to_nnreal= ((Pr[A]:ennreal) + (Pr[B]:ennreal)).to_nnreal,\n  {\n    rw A2,\n  },\n  simp at A3,\n  apply A3,\nend\n\ndef enot {Ω:Type*} {p:probability_space Ω} (A:event p):event p := {\n  val:=(A).valᶜ,\n  property := measurable_set.compl A.property,\n}\n\nprefix `¬ₑ` :100 := enot\n\n\n@[simp]\nlemma enot_val_def {Ω:Type*} {p:probability_space Ω} (A:event p):\n  (¬ₑ A).val = (A.val)ᶜ :=\nbegin\n  refl,\nend\n\n/-\n  Double negation elimination. However, it is hard to avoid in measure theory.\n-/\n@[simp]\nlemma enot_enot_eq_self {Ω:Type*} {p:probability_space Ω} (A:event p):\n  (¬ₑ (¬ₑ A)) = (A) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n\ninstance measurable_setB_has_compl {α:Type*} [M:measurable_space α]:has_compl (@measurable_setB α M) := {\n  compl := λ E, ⟨ E.valᶜ, measurable_set.compl E.property⟩,\n}\n\n\ninstance has_sdiff.measurable_setB {α:Type*} {M:measurable_space α}:\n  has_sdiff (measurable_setB M) := ⟨λ E F, E ∩ Fᶜ⟩\n\ninstance has_sdiff.event {α:Type*} {M:probability_space α}:\n  has_sdiff (event M) := ⟨λ E F, E ∧ ¬ₑ F⟩\n\n@[simp]\nlemma has_sdiff_measurable_setB_val {α:Type*} {M:measurable_space α} (E F:measurable_setB M):\n  (E \\ F).val = E.val \\ F.val := rfl\n\n@[simp]\nlemma has_sdiff_event_val {α:Type*} {P:probability_space α} (E F:event P):\n  (E \\ F).val = E.val \\ F.val := rfl\n\n\n\ninstance measurable_setB_subtype_has_neg {α:Type*} [M:measurable_space α]:has_neg (subtype (@measurable_set α M)) := {\n  neg := λ E, ⟨ E.valᶜ, measurable_set.compl E.property⟩,\n}\n\n\nlemma measurable_setB_neg_def {α:Type*} [M:measurable_space α] {E:@measurable_setB α M}:\n    Eᶜ = ⟨ E.valᶜ, measurable_set.compl E.property⟩ :=rfl\n\n@[simp]\nlemma measurable_setB_compl_val_def {α:Type*} [M:measurable_space α] {E:@measurable_setB α M}:\n    (Eᶜ).val = (E.val)ᶜ  :=rfl\n\n\ninstance event_has_compl {α:Type*} [M:probability_space α]:has_compl (@event α M) := {\n  compl := λ E, ⟨E.valᶜ, measurable_set.compl E.property⟩,\n}\n\n\nlemma event_neg_def {α:Type*} [M:probability_space α] {E:@event α M}:\n    Eᶜ = ⟨ E.valᶜ, measurable_set.compl E.property⟩ :=rfl\n\n\n\n@[simp]\nlemma event_neg_val_def {α:Type*} [M:probability_space α] {E:@event α M}:\n    (Eᶜ).val = (E.val)ᶜ := rfl\n\n\n@[simp]\nlemma em_event {Ω:Type*} {p:probability_space Ω} (A:event p):\n    (A ∨ (¬ₑ A))=event_univ :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n\nlemma compl_eor_eq_compl_eand_compl {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∨ B)ᶜ = (Aᶜ ∧ Bᶜ) := begin\n  apply event.eq,\n  simp,\nend\n\n\nlemma Pr_add_enot_eq_1 {Ω:Type*} {p:probability_space Ω} (A:event p):\n  Pr[A] + Pr[¬ₑ A] = 1 :=\nbegin\n  have A1:disjoint (A.val) (enot A).val,\n  {\n    unfold disjoint,\n    rw enot_val_def,\n    simp,\n  },\n  have A2:(A∨ (¬ₑ A)) = event_univ,\n  {\n    apply em_event,\n  },\n  have A3:Pr[A∨ (¬ₑ A)] = Pr[event_univ],\n  {\n    rw A2,\n  },\n  rw Pr_event_univ at A3,\n  rw Pr_disjoint_eor at A3,\n  apply A3,\n  apply A1,\nend\n\nlemma Pr_one_minus_eq_not {Ω:Type*} {p:probability_space Ω} (A:event p):\n  1 - Pr[A] = Pr[¬ₑ A] :=\nbegin\n  apply nnreal_add_sub_left,\n  apply Pr_add_enot_eq_1,\nend\n\nlemma Pr_one_minus_not_eq {Ω:Type*} {p:probability_space Ω} (A:event p):\n  1 - Pr[enot A] = Pr[A] :=\nbegin\n  apply nnreal_add_sub_right,\n  apply Pr_add_enot_eq_1,\nend\n\nlemma Pr_not_ge_of_Pr_le {Ω:Type*} {p:probability_space Ω} (A:event p) (δ:nnreal):\n  Pr[A] ≤ δ → Pr[¬ₑ A] ≥ 1 - δ :=\nbegin\n  intros h1,\n  rw ← Pr_one_minus_eq_not,\n  simp,\n  --apply nnreal.le_s\n  have h2:1 - Pr[A] + Pr[A] ≤ 1 - Pr[A] + δ,\n  { apply add_le_add,\n    apply le_refl _,\n    apply h1 },\n  apply le_trans _ h2,\n  apply nnreal.le_sub_add',\nend\n\n\nlemma em_event_cond {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  ((A ∧ B) ∨ (A ∧ ¬ₑ B)) = A :=\nbegin\n  apply event.eq,\n  simp [set.inter_union_compl],\nend\n\nlemma Pr_em {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∧ B] + Pr[A ∧ ¬ₑ B] = Pr[A] :=\nbegin\n  rw ← Pr_disjoint_eor,\n  { --Pr[(A∧ B)∨ A∧ ¬ₑ B] = Pr[A]\n    rw em_event_cond,\n  },\n  { --disjoint ((A∧ B).val) ((A∧ ¬ₑ B).val)\n    simp [set.disjoint_inter_compl],\n  }\nend\n\nlemma Pr_diff {Ω:Type*} {p:probability_space Ω} (A B:event p):\n    A.val ⊆ B.val →\n    Pr[B ∧ ¬ₑ A] = Pr[B] - Pr[A] :=\nbegin\n  intro A1,\n  have A2:Pr[B ∧ A] + Pr[B ∧ ¬ₑ A] = Pr[B],\n  {\n    apply Pr_em,\n  },\n  have A3:(B ∧ A) = A,\n  {\n    apply eand_eq_self_of_subset_right,\n    apply A1,\n  },\n  rw A3 at A2,\n  symmetry,\n  apply nnreal_add_sub_left,\n  exact A2,\nend\n\n\ndef measurable_setB.sdiff {Ω:Type*} {M:measurable_space Ω} (A B:measurable_setB M):measurable_setB M :=\n  @measurable_setB.mk _ _ (A.val \\ B.val) begin\n  apply measurable_set.diff,\n  apply A.property,\n  apply B.property\nend\n\ninstance measurable_setB.has_sdiff {Ω:Type*} {M:measurable_space Ω} :has_sdiff (measurable_setB M) := ⟨measurable_setB.sdiff⟩\n\n@[simp]\nlemma measurable_setB.sdiff_val_def {Ω:Type*} {M:measurable_space Ω} (A B:measurable_setB M):\n  (A \\ B).val = A.val \\ B.val := rfl\n\n\n\n\n\ndef measurable_setB.symm_diff {Ω:Type*} {M:measurable_space Ω} (A B:measurable_setB M):measurable_setB M := (A \\ B) ∪ (B \\ A)\n\ninstance measurable_setB.has_symm_diff {Ω:Type*} {M:measurable_space Ω}:has_symm_diff (measurable_setB M) := ⟨measurable_setB.symm_diff⟩\n\nlemma measurable_setB.has_symm_diff.def {Ω : Type*} {M:measurable_space Ω} \n{A B:measurable_setB M}:A ∆ B = (A \\ B) ∪ (B \\ A) := rfl\n\n@[simp]\nlemma measurable_setB.symm_diff_val_def {Ω:Type*} {M:measurable_space Ω} (A B:measurable_setB M):\n  (A ∆ B).val = A.val ∆ B.val := rfl\n\ndef event_eqv {Ω:Type*} {p:probability_space Ω} (A B:event p):event p :=\n    (A ∧ B) ∨ ((¬ₑ A) ∧ (¬ₑ B))\n\ninfixr `=ₑ`:100 := event_eqv\n\n\nlemma event_eqv_def {Ω:Type*} {p:probability_space Ω} (A B:event p):\n    (A =ₑ B) = ((A ∧ B) ∨ ((¬ₑ A) ∧ (¬ₑ B))) := rfl\n\n\nlemma eor_partition {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  (A ∨ B) = ((A ∧ ¬ₑ B) ∨  (A ∧ B)  ∨ (¬ₑ A ∧ B)) :=\nbegin\n  apply event.eq,\n  simp,\n  ext ω,split;intros A1;simp at A1;simp [A1],\n  {\n    cases A1 with A1 A1; simp [A1],\n    rw or_comm,\n    apply classical.em,\n    apply classical.em,\n  },\n  {\n    cases A1 with A1 A1,\n    simp [A1],\n    cases A1 with A1 A1,\n    simp [A1],\n    simp [A1],\n  },  \nend\n\nlemma Pr_eor_partition {Ω:Type*} {p:probability_space Ω} (A B:event p):\n  Pr[A ∨ B] = Pr[A ∧ ¬ₑ B] + Pr[A ∧ B] + Pr[¬ₑ A ∧ B] :=\nbegin\n  rw eor_partition A B,\n  rw Pr_disjoint_eor,\n  rw Pr_disjoint_eor,\n  ring,\n  simp,\n  rw set.disjoint_left,\n  intros ω A1,\n  simp at A1,\n  simp [A1],\n  simp,\n  split;\n  {rw set.disjoint_left,\n  intros ω A1,\n  simp at A1,\n  simp [A1]},\nend\n\nlemma Pr_eor_plus_eand {Ω:Type*}  {p:probability_space Ω} (A B:event p):\n  Pr[A ∨ B] + Pr[A ∧ B] = (Pr[A] + Pr[B]) :=\nbegin\n  rw ← Pr_em A B,\n  rw ← Pr_em B A,\n  rw eand_comm B A,\n  rw eand_comm B (¬ₑ A),\n  rw Pr_eor_partition A B,\n  ring,\nend\n\nlemma Pr_eor_eq_minus_eand {Ω:Type*}  {p:probability_space Ω} (A B:event p):\n  Pr[A ∨ B] = (Pr[A] + Pr[B])  - Pr[A ∧ B] :=\nbegin\n  rw ← Pr_eor_plus_eand,\n  rw nnreal.add_sub_cancel,\nend\n\nlemma Pr_eor_eq_minus_eand_real {Ω:Type*}  {p:probability_space Ω} (A B:event p):\n  (Pr[A ∨ B]:real) = (Pr[A]:real) + (Pr[B]:real)  - (Pr[A ∧ B]:real) :=\nbegin\n  have A1:Pr[A ∨ B] + Pr[A ∧ B] = (Pr[A] + Pr[B]),\n  {apply Pr_eor_plus_eand},\n  rw ← nnreal.coe_eq at A1,\n  repeat {rw nnreal.coe_add at A1},\n  linarith,\nend\n\ndef measurable_setB.Inter {Ω β:Type*} {M:measurable_space Ω} [encodable β] (A:β → measurable_setB M):measurable_setB M := {\n  val:=(⋂ b:β, (A b).val),\n  property := measurable_set.Inter (λ b:β, (A b).property),\n}\n\n\n\n\n--lemma compl_eor_eq_compl_and_compl\n--Rewrite to use measurable_setB.Inter\ndef eall_encodable {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):event p := {\n  val:=(⋂ b:β, (A b).val),\n  property := measurable_set.Inter (λ b:β, (A b).property),\n}\n\n--Redundant to eall_encodable.\ndef eall' {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):event p := {\n  val:=(⋂ b:β, (A b).val),\n  property := measurable_set.Inter (λ b:β, (A b).property),\n}\n\n/- The definition of has_eall.eall_val enforces that the\n   eall function implements intersection. The proof of measurability\n   is left to the implementer. -/\n@[class]\nstructure has_eall (Ω β:Type*) (p:probability_space Ω) := \n  (eall:(β → event p) → event p)\n  (eall_val:∀ (f:β → event p), (⋂ (b:β), (f b).val) = (eall f).val)\n\n\n-- ∀ᵣ is enforced to be intersection.\nnotation `∀ᵣ` binders `, ` r:(scoped f, has_eall.eall f) := r\n\n\n@[class]\nstructure has_eall_in (Ω β γ:Type*) (p:probability_space Ω) := \n  (eall_in:γ → (β → event p) → event p)\n  (as_set:γ → (set β))\n  (eall_in_val:∀ (g:γ) (f:β → event p), (⋂ b ∈ (as_set g), (f b).val) = (eall_in g f).val)\n\n--#check has_eall_in.has_mem'\n--TODO:Delete.\nclass has_eall_in' (Ω β γ:Type*) (p:probability_space Ω) := {\n  eall_in:γ → (β → event p) → event p\n}\n\nnotation `∀ᵣ` binders  ` in `  A, r:(scoped f, has_eall_in.eall_in A f) := r\n\n\ninstance has_eall_encodable {Ω β:Type*} {p:probability_space Ω} [encodable β]:has_eall Ω β p := {\n  eall := λ (A:β → event p), eall_encodable A,\n  eall_val := begin\n    simp [eall_encodable],\n  end,\n} \n\n\n\n\n--Instead of a one-off, there should be variants for a variety of types.\nnotation `∀ᵣ` binders `, ` r:(scoped f, has_eall.eall f) := r\n\n@[simp]\nlemma eall_val_def {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):\n  (eall_encodable A).val = (⋂ b:β, (A b).val) := rfl\n\nlemma eall_binder_def {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):\n  (∀ᵣ x, A x) = (eall_encodable A):= rfl\n\n@[simp]\nlemma eall_binder_val_def {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):\n  (∀ᵣ x, A x).val = (⋂ b:β, (A b).val) := rfl\n\n\n\ndef eall_prop {Ω β:Type*} {p:probability_space Ω} [E:encodable β]\n  (P:β → Prop) [D:decidable_pred P]\n  (A:β → event p):event p := @eall_encodable _ _ _ (@encodable.subtype β P E D) (λ (b:(subtype P)), A (b.val) )\n\ndef to_set_of_sets {Ω:Type*} {p:probability_space Ω} (A:set (event p)):set (set Ω) :=\n  (set.image (λ a:event p, a.val) A)\n\nlemma all_measurable_to_set_of_sets {Ω:Type*} {p:probability_space Ω} (A:set (event p))\n  (a∈ (to_set_of_sets A)):measurable_set a :=\nbegin\n  unfold to_set_of_sets at H,\n  simp at H,\n  cases H with x H,\n  cases H with A1 A2,\n  subst a,\n  exact x.property,\nend\n\nlemma countable_to_set_of_sets {Ω:Type*} {p:probability_space Ω} {A:set (event p)}:\n  (set.countable A)→ (set.countable (to_set_of_sets A)) :=\nbegin\n  unfold to_set_of_sets,\n  intro A1,\n  apply set.countable.image,\n  apply A1,\nend\n\n\n\n\ndef eall_set {Ω:Type*} {p:probability_space Ω} (A:set (event p)) (cA:set.countable A):event p:=\n{\n  val:=set.sInter (to_set_of_sets A),\n  property:=measurable_set.sInter (countable_to_set_of_sets cA) (all_measurable_to_set_of_sets A),\n}\n\n\n\ndef eall_finset_val {Ω β:Type*} {p:probability_space Ω} (S:finset β)\n  (A:β → event p):set Ω :=  ⋂ s∈ S, (A s).val\n\n\nlemma eall_finset_val_measurable {Ω β:Type*} {p:probability_space Ω} (S:finset β)\n  (A:β → event p):measurable_set (eall_finset_val S A) :=\nbegin\n  unfold eall_finset_val,\n  apply finset_inter_measurable,\n  intros,\n  apply (A t).property,\nend\n\n--\n\ndef eall_finset {Ω β:Type*} {p:probability_space Ω}\n  (S:finset β)\n  (A:β → event p):event p := {\n    val:=eall_finset_val S A,\n    property:=eall_finset_val_measurable S A,\n  }\n\n\ninstance has_eall_in.finset {Ω β:Type*} {p:probability_space Ω}:has_eall_in Ω β (finset β) p := {\n  eall_in := (λ S f, eall_finset S f),\n  as_set := (λ (S:finset β), ↑S),\n  eall_in_val := begin\n    simp [eall_finset, eall_finset_val],\n  end\n}\n\n\n@[simp]\nlemma eall_finset_val_def {Ω β:Type*} {p:probability_space Ω}\n  (S:finset β) (A:β → event p):(eall_finset S A).val = ⋂ s∈ S, (A s).val := rfl\n\nlemma has_eall_in_finset_def {Ω β:Type*} {p:probability_space Ω}\n  (S:finset β) (A:β → event p):\n  (∀ᵣ s in S, A s) = (eall_finset S A) := rfl\n\n\n\n@[simp]\nlemma has_eall_in_finset_val_def {Ω β:Type*} {p:probability_space Ω}\n  (S:finset β) (A:β → event p):\n  (∀ᵣ s in S, A s).val = ⋂ s∈ S, (A s).val := rfl\n\n@[simp]\nlemma has_eall_in_finset_val_def2 {Ω β:Type*} {p:probability_space Ω} {S:finset β} {A:β → event p}:\n  (has_eall_in.eall_in S A).val = ⋂ s∈ S, (A s).val := rfl\n\n\n--#print instances has_coe\n@[simp]\nlemma has_eall_in_finset_val_def3 {Ω β:Type*} {p:probability_space Ω} {S:finset β} {A:β → event p}:\n  @has_coe.coe (event p) (set Ω) (coe_subtype) (has_eall_in.eall_in S A) = ⋂ s∈ S, (A s).val := rfl\n\nlemma has_eall_in_insert {Ω β:Type*} {p:probability_space Ω} [decidable_eq β] {T:finset β}\n  {b:β} {E:β → event p}:\n  (∀ᵣ b' in (insert b T), E b') = ((E b) ∧ (∀ᵣ b' in T, E b')) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n\n/--Since a fintype is encodable, this could be represented with eall, and then proven equivalent to\n  eall_finset. -/\ndef eall_fintype {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (A:β → event p):event p := eall_finset finset.univ A\n\n\ninstance has_eall.fintype {Ω β:Type*} {p:probability_space Ω} [F:fintype β]:has_eall Ω β p := {\n  eall := (λ A, eall_fintype F A),\n  eall_val := by simp [eall_fintype],\n}\n\nlemma eall_fintype_eq_eall_finset {Ω β:Type*} {p:probability_space Ω}\n  [F:fintype β] (A:β → event p):(∀ᵣ b, A b) = eall_finset finset.univ A := rfl\n\n\nlemma eall_fintype_def {Ω β:Type*} {p:probability_space Ω} (F:fintype β) {A:β → event p}:\n  (eall_fintype F A) = (∀ᵣ b, A b) := rfl\n\n\n\n@[simp]\nlemma eall_fintype_val_def {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (A:β → event p):(eall_fintype F A).val = ⋂ (s:β), (A s).val :=\nbegin\n  unfold eall_fintype,\n  simp,\nend\n \ndef measurable_Union {Ω β:Type*} {p:measurable_space Ω} [encodable β] (A:β → measurable_setB p):\n  measurable_setB p := {\n  val:=(⋃ b:β, (A b).val),\n  property := measurable_set.Union (λ b:β, (A b).property),\n}\n\n@[simp]\nlemma measurable_Union_val_def {Ω β:Type*} {p:measurable_space Ω} [E:encodable β] \n    (A:β → measurable_setB p):\n    (@measurable_Union Ω β p E A).val = (⋃ b:β, (A b).val) := rfl\n\n\ndef eany {Ω β:Type*} {p:probability_space Ω} [encodable β] (A:β → event p):event p := \n  measurable_Union A\n\nlemma measurable_Union_eq_any {Ω β:Type*} \n    {p:probability_space Ω} [E:encodable β] (A:β → event p):\n    measurable_Union A = eany A := rfl\n\n\nlemma sum_subst {β:Type*} [encodable β] {f g:β → ennreal}:(f = g) →\n    (tsum f) = (tsum g) :=\nbegin\n  intro A1,\n  rw A1,\nend\n\n\nlemma Pr_measurable_Union_sum_dummy {Ω β:Type*} [M:probability_space Ω]\n    [E:encodable β]  \n    (A:β → set Ω):(∀ (i j:β), i ≠ j → \n    (A i ∩ A j = ∅))→\n    (∀ i, measurable_set (A i)) →\n    ((@measure_theory.measure_space.volume Ω (probability_space.to_measure_space)) (⋃ (n:β), A n)) = \n    (∑' (i:β), (@measure_theory.measure_space.volume Ω (probability_space.to_measure_space)) (A i)) :=\nbegin\n  intros A1 A3,\n  rw measure_theory.measure_Union,\n  {\n    intros i j A2,\n    simp,\n    unfold disjoint function.on_fun,\n    simp,\n    rw subset_empty_iff,\n    apply A1 i j A2,\n  },\n  {\n    apply A3,\n  },\nend\n\nlemma measure_eq_measure {Ω:Type*} [P:probability_space Ω] {S:set Ω}:\n  measure_theory.measure_space.volume.to_outer_measure.measure_of S =\n  (@measure_theory.measure_space.volume Ω (probability_space.to_measure_space)) S := rfl\n\n@[simp]\nlemma eany_val_def {Ω β:Type*} {p:probability_space Ω} [encodable β]\n  (A:β → event p):(eany A).val=(⋃ b:β, (A b).val) := rfl\n\n@[class]\nstructure has_eany (Ω β:Type*) (p:probability_space Ω) := \n  (eany:(β → event p) → event p)\n  (eany_val:(∀ (f:β → event p), ((⋃ b, (f b).val) = (eany f).val)))\n\n\nnotation `∃ᵣ ` binders `, ` r:(scoped f, has_eany.eany f) := r\n\n@[class]\nstructure has_eany_in (Ω β γ:Type*) (p:probability_space Ω) := \n  (eany_in:γ → (β → event p) → event p)\n  (as_set:γ → (set β))\n  (eany_in_val:∀ (g:γ) (f:β → event p), (⋃ b ∈ (as_set g), (f b).val) = (eany_in g f).val)\n\n\nnotation `∃ᵣ ` binders  ` in ` S `, ` r:(scoped f, has_eany_in.eany_in S f) := r\n\n\ninstance has_eany.encodable {Ω β:Type*} {p:probability_space Ω} [E:encodable β]:has_eany Ω β p := {\n  eany := (λ A:β → (event p), eany A),\n  eany_val := by simp\n}\n\n\nlemma eany_encodable_notation_def {Ω β:Type*} {p:probability_space Ω} [encodable β]\n  (A:β → event p):(∃ᵣ a, A a) = (eany A) := rfl\n\n@[simp]\nlemma eany_encodable_val_def {Ω β:Type*} {p:probability_space Ω} [encodable β]\n  (A:β → event p):(∃ᵣ a, A a).val = (⋃ (b:β), (A b).val) := begin\n  rw eany_encodable_notation_def,\n  refl\nend \n\n\n\ndef eany_finset_val {Ω β:Type*} {p:probability_space Ω} (S:finset β)\n  (A:β → event p):set Ω :=  ⋃ s∈ S, (A s).val\n\n\n\nlemma eany_finset_val_measurable {Ω β:Type*} {p:probability_space Ω} (S:finset β)\n  (A:β → event p):measurable_set (eany_finset_val S A) :=\nbegin\n  unfold eany_finset_val,\n  apply finset_union_measurable,\n  intros,\n  apply (A t).property,\nend\n\ndef eany_finset {Ω β:Type*} {p:probability_space Ω}\n  (S:finset β)\n  (A:β → event p):event p := {\n    val:=eany_finset_val S A,\n    property:=eany_finset_val_measurable S A,\n  }\n\ninstance has_eany_in.finset {Ω β:Type*} {p:probability_space Ω}:has_eany_in Ω β (finset β) p := {\n  eany_in := (λ (S:finset β) (A:β → (event p)), eany_finset S A),\n  as_set := (λ (S:finset β), ↑S),\n  eany_in_val := begin\n    simp [eany_finset, eany_finset_val],\n  end\n}\n\n\n@[simp]\nlemma eany_finset_val_def {Ω β:Type*} {p:probability_space Ω} (S:finset β)\n  (A:β → event p):(eany_finset S A).val = ⋃ s∈ S, (A s).val := rfl\n\n\n\nlemma eany_in_finset_def {Ω β:Type*} {p:probability_space Ω} {S:finset β} (A:β → event p):\n  (∃ᵣ s in S, A s) = eany_finset S A := rfl\n\n@[simp]\nlemma eany_in_finset_val_def {Ω β:Type*} {p:probability_space Ω} {S:finset β} (A:β → event p):\n  (∃ᵣ s in S, A s).val = ⋃ s∈ S, (A s).val := rfl\n\ndef eany_fintype {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (A:β → event p):event p := eany_finset finset.univ A\n\n\n\n\nlemma eany_fintype_def {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (A:β → event p):eany_fintype F A = eany_finset finset.univ A := rfl\n\n\ninstance has_eany.fintype {Ω β:Type*} {p:probability_space Ω} [F:fintype β]:has_eany Ω β p := {\n  eany := (λ  (A:β → (event p)), eany_fintype F A),\n  eany_val := by simp [eany_fintype],\n}\n\nlemma has_eany_fintype_def {Ω β:Type*} {p:probability_space Ω} [F:fintype β] {A:β→ event p}:\n  (∃ᵣ s, A s) = (eany_fintype F A) := rfl\n\n\n@[simp]\nlemma has_eany_fintype_val_def {Ω β:Type*} {p:probability_space Ω} [F:fintype β] {A:β→ event p}:\n  (∃ᵣ s, A s).val = ⋃ (s:β), (A s).val :=\nbegin\n  rw [has_eany_fintype_def,eany_fintype_def],\n  simp,\nend\n\nlemma eany_eq_eany_fintype {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (E:encodable β) (A:β → event p):\n  eany A = eany_fintype F A := begin\n  apply event.eq,\n  rw ← has_eany_fintype_def,\n  simp,\nend\n\n@[simp]\nlemma exists_empty {α Ω:Type*} {P:probability_space Ω} (f:α → event P):\n  (∃ᵣ a in (∅:finset α), f a) = (∅:event P) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n@[simp]\nlemma eall_finset_empty {Ω β:Type*} {p:probability_space Ω}\n  (A:β → event p): (∀ᵣ s in (∅:finset β), A s) = event_univ :=\nbegin\n  apply event.eq,\n  simp,\nend\n\nlemma eany_finset_insert {Ω β:Type*} [D:decidable_eq β] {p:probability_space Ω}\n  {S:finset β} {A:β → event p} {a:β}:\n  (∃ᵣ (a':β) in (insert a S), A a') = ((A a) ∨ (∃ᵣ a' in S, A a')) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\nlemma eall_finset_insert {Ω β:Type*} [D:decidable_eq β] {p:probability_space Ω}\n  {S:finset β} {A:β → event p} {a:β}:\n  (∀ᵣ (a':β) in (insert a S), A a') = ((A a) ∧ (∀ᵣ a' in S, A a')) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\nlemma eany_finset_bound {Ω β:Type*} [D:decidable_eq β]\n  {p:probability_space Ω}\n  (S:finset β) (A:β → event p):Pr[∃ᵣ a in S, A a] ≤ finset.sum S (λ a:β, Pr[A a]) :=\nbegin\n  apply finset.induction_on S,\n  {\n    simp,\n  },\n  {\n    intros a S2 A1 A2,\n    rw finset.sum_insert A1,\n    rw eany_finset_insert,\n    apply le_trans,\n    apply Pr_le_eor_sum,\n    apply add_le_add_left,\n    apply A2,\n  }\nend\n\n\nlemma eany_fintype_bound {Ω β:Type*} [D:decidable_eq β] {p:probability_space Ω}\n  [F:fintype β] (A:β → event p):Pr[∃ᵣ (s:β), A s] ≤  ∑' a:β, Pr[A a] :=\nbegin\n  rw tsum_fintype,\n  apply eany_finset_bound,\nend\n\n\nlemma eany_fintype_bound2 {Ω β:Type*} {p:probability_space Ω}\n  (F:fintype β) (A:β → event p) (k:nnreal):\n  (∀ a:β, Pr[A a]≤ k) →\n  Pr[∃ᵣ (s:β), A s] ≤ (fintype.card β) * k :=\nbegin\n  intro A1,\n  have A2:decidable_eq β := classical.decidable_eq β,\n  apply le_trans,\n  apply @eany_fintype_bound Ω β A2,\n  rw tsum_fintype,\n  unfold fintype.card,\n  apply @finset_sum_le_const β A2,\n  intros s A3,\n  apply A1,\nend\n\n\ndef independent_event_pair {Ω:Type*} {p:probability_space Ω} (A B:event p):Prop :=\n  --(event_prob (eand A B)) = (event_prob A) * (event_prob B)\n  Pr[ A ∧ B] = Pr[A] * Pr[B]\n\n\n\ndef independent_events {Ω β:Type*} {p:probability_space Ω} \n  (A:β → event p):Prop :=\n  ∀ (S:finset β), (finset.prod S (λ b, Pr[A b])) = Pr[∀ᵣ s in S, A s]\n\ndef events_IID {Ω β:Type*} {p:probability_space Ω} \n  (A:β → event p):Prop :=\n  independent_events A ∧ (∀ x y:β, Pr[A x] = Pr[A y])\n\nlemma events_IID_pow {α : Type*} {p : probability_space α} {β : Type*}\n  [I:inhabited β] (A:β → event p) (S:finset β):\n  events_IID A → Pr[eall_finset S A] = Pr[A I.default]^(S.card) :=\nbegin\n  intros A1,\n  unfold events_IID at A1,\n  cases A1 with A2 A3,\n  unfold independent_events at A2,\n  have A4:(finset.prod S (λ b, Pr[A b])) = Pr[eall_finset S A],\n  {\n    apply A2,\n  },\n  rw ← A4,\n  have A5:(λ (b : β), Pr[A b]) = (λ (b:β), Pr[A (inhabited.default β)]),\n  {\n    ext b,\n    rw A3,\n  },\n  rw A5,\n  apply finset.prod_const,\nend\n\n@[simp]\nlemma forall_fintype_val {α Ω:Type*} {P:probability_space Ω} (f:α → event P) [F:fintype α]:\n  (∀ᵣ a, f a).val = ⋂ (a:α), (f a).val := begin\n  rw ← eall_fintype_def,\n  simp,\nend\n\n\nlemma exists_not_eq_not_forall {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {S:finset α}:\n  (∃ᵣ a in S, ¬ₑ(f a)) = ¬ₑ (∀ᵣ a in S, f a) :=\nbegin\n  apply event.eq,\n  simp,\n  rw set.Union_eq_comp_Inter_comp,\n  simp,\nend\n\nlemma not_forall_not_eq_exists {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {S:finset α}:\n  ¬ₑ (∀ᵣ a in S, ¬ₑ f a) = (∃ᵣ a in S, f a) :=\nbegin\n  apply event.eq,\n  simp,\n  rw set.Union_eq_comp_Inter_comp,\n  simp,\nend\n\nlemma not_forall_not_eq_exists' {α Ω:Type*} {P:probability_space Ω} (f:α → event P) [fintype α]:\n  ¬ₑ (∀ᵣ a, ¬ₑ f a) = (∃ᵣ a, f a) :=\nbegin\n  apply event.eq,\n  simp,\n  rw set.Union_eq_comp_Inter_comp,\nend\n\nlemma not_exists_eq_forall_not {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {S:finset α}:\n ¬ₑ (∃ᵣ a in S, (f a)) = (∀ᵣ a in S, ¬ₑ (f a)) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n@[simp]\nlemma forall_singleton {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {x:α}:\n  (∀ᵣ a in ({x}:finset α), f a) = f x :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n@[simp]\nlemma exists_singleton {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {x:α}:\n  (∃ᵣ a in ({x}:finset α), f a) = f x :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n@[simp]\nlemma distrib_exists_and {α Ω:Type*} {P:probability_space Ω} (f:α → event P) {S:finset α} {A:event P}:\n  (∃ᵣ a in S, A ∧ (f a))  =   (A ∧ (∃ᵣ a in S, f a)) :=\nbegin\n  apply event.eq,\n  simp,\n  ext ω,split;intros A1;simp at A1;simp [A1],\n  cases A1 with i A1,\n  simp [A1],\n  apply exists.intro i,\n  simp [A1],\nend\n\nlemma finset.pair_erase {α:Type*} {x y:α} [decidable_eq α]:x ≠ y → ({x, y}:finset α).erase x  = {y} :=\nbegin\n  intros A1,\n  rw finset.erase_insert,\n  simp [A1],\nend\n\nlemma finset.singleton_erase {α:Type*} {x:α} [decidable_eq α]:({x}:finset α).erase x = ∅ := \nbegin\n  have A1:{x} = insert x (∅:finset α),\n  {simp},\n  rw A1,\n  rw finset.erase_insert,\n  simp,\nend\n\nlemma finset.union_erase {α:Type*} {S T:finset α} {x:α} [decidable_eq α]:\n  (S ∪ T).erase x = (S.erase x) ∪ (T.erase x) :=\nbegin\n  ext a;split;intros A1;simp at A1;simp,\n  {simp [A1]},\n  {cases A1;simp [A1]},\nend\n\nlemma finset.image_sum {α β:Type*} [add_comm_monoid β] [decidable_eq α] {S:finset α} {f:α → β} {g:α → α}:\n  (∀ (s t:α),s∈ S → t∈ S→ s ≠ t → g s ≠ g t) →  (finset.image g S).sum f = S.sum (f ∘ g) :=\nbegin\n  apply finset.induction_on S,\n  {\n    intros A1,\n    refl,\n  },\n  {\n    intros a s B1 B2 B3,\n    simp,\n    rw finset.sum_insert,\n    rw finset.sum_insert,\n    rw B2,\n    intros s' t' B4 B5 B6,\n    apply B3,\n    {simp [B4]},\n    {simp [B5]},\n    apply B6,\n    apply B1,\n    intro B4,\n    simp at B4,\n    cases B4 with u B4,\n    apply B3 u a,\n    {simp [B4]},\n    {simp},\n    intro B5,\n    subst u,\n    apply B1,\n    {simp [B4]},\n    {simp [B4]},\n  },\nend\n\n-- A specialized lemma of little generic value.\nlemma finset.powerset_sum {α β:Type*} [add_comm_monoid β][decidable_eq α] {x:α} {S:finset α} (f:finset α → β):\n  (x ∉ S) → ((insert x S).powerset.erase ∅).sum f = (S.powerset.erase ∅).sum f \n  + (S.powerset).sum (f ∘ (insert x)) :=\nbegin\n  intros A0,\n  rw finset.powerset_insert,\n  rw finset.union_erase,\n  rw finset.sum_union,\n  have A1:((finset.image (insert x) S.powerset).erase ∅).sum f =\n          (finset.image (insert x) S.powerset).sum f,\n  {have A1A:((finset.image (insert x) S.powerset).erase ∅) =\n          (finset.image (insert x) S.powerset),\n   { rw finset.ext_iff,\n     intros T;split;intros A1A1;simp at A1A1,\n     {\n       simp,cases A1A1 with A1A1 A1A2,\n       cases A1A2 with U A1A2,\n       apply exists.intro U,\n       apply A1A2,\n     },\n     simp,\n     split,\n     {cases A1A1 with U A1A1,intros A1,subst T,simp at A1A1,apply A1A1},\n     {apply A1A1},\n   },\n   rw A1A,\n  },\n  rw A1,\n  have B1:(finset.image (insert x) S.powerset).sum f = \n          S.powerset.sum (f ∘ (insert x)),\n  {\n     apply @finset.image_sum (finset α) β,\n     intros s t B1A B1B B1C B1D,\n     apply B1C,\n     simp at B1A,\n     simp at B1B,\n     rw finset.subset_iff at B1B,\n     rw finset.subset_iff at B1A,\n     rw finset.ext_iff at B1D,\n  \n     rw finset.ext_iff,\n     intros a;split;intros B1F,\n     {\n       have B1G:a ∈ insert x s,\n       {simp [B1F] },\n       have B1H := (B1D a).mp B1G,\n       simp at B1H,\n       cases B1H,\n       subst a,\n       exfalso,\n       apply A0,\n       apply B1A,\n       apply B1F,\n       apply B1H,       \n     },\n     {\n       have B1G:a ∈ insert x t,{simp [B1F]},\n       have B1H := (B1D a).mpr B1G,\n       simp at B1H,\n       cases B1H,\n       subst a,\n       exfalso,\n       apply A0,\n       apply B1B,\n       apply B1F,\n       apply B1H,\n     },\n  },\n  rw B1,\n  \n  rw finset.disjoint_left,\n  intros T A2,\n  simp at A2,\n  simp,\n  intros A3 U A5 A6,\n  subst T,\n  apply A0,\n  cases A2 with A2 A7,\n  rw finset.subset_iff at A7,\n  apply A7,\n  simp,\nend \n\nlemma distrib_forall_eand {α Ω:Type*} {P:probability_space Ω} (f:α → event P) [decidable_eq α] {S:finset α} (A:event P):S.nonempty →\n  (∀ᵣ a in S, A ∧ f a) = (A ∧ (∀ᵣ a in S, f a)) := \nbegin\n  intros A1,\n  apply event.eq,\n  simp,ext ω,split;intros A2;simp at A2;simp [A2],\n  {have A3:=finset.nonempty.bex A1,\n   cases A3 with a A3,\n   have A4 := A2 a A3,\n   simp [A4],\n   intros b A5,\n   apply (A2 b A5).right,\n  },\n  {\n   apply A2.right,\n  },\nend \n\nlemma Pr_exists_eq_powerset {α Ω:Type*} {P:probability_space Ω} (f:α → event P) [decidable_eq α] {S:finset α}:  (Pr[(∃ᵣ a in S, (f a))]:real) = \n  -(S.powerset.erase ∅).sum  (λ T:finset α, (Pr[∀ᵣ a in T, f a]:real) *  (-1)^(T.card)) :=\nbegin\n  revert f,\n  apply finset.case_strong_induction_on S,\n  {intros f, simp [finset.singleton_erase],},\n  intros x s A3 A1 f,\n  have A6 := (A1 s (finset.subset.refl s) f),\n  rw finset.powerset_sum,\n  rw eany_finset_insert,\n  rw Pr_eor_eq_minus_eand_real,\n  simp,\n  rw ← distrib_exists_and,\n  rw A6,\n  have A8:=(A1 s (finset.subset.refl s) (λ a, f x∧ f a)),\n  rw A8,\n  have A9:\n-s.powerset.sum\n          (λ (x_1 : finset α), (Pr[has_eall_in.eall_in (insert x x_1) f]:real) * (-1) ^ (insert x x_1).card) =\n(Pr[f x]:real) + (s.powerset.erase ∅).sum\n          (λ (T : finset α), (Pr[∀ᵣ (a : α) in T,f x∧ f a]:real) * (-1) ^ T.card),\n  {\n    have A9A:insert ∅ (s.powerset.erase ∅) = (s).powerset,\n    {rw finset.insert_erase, simp},\n     have A9B:\n     -(s).powerset.sum\n            (λ (x_1 : finset α), (Pr[has_eall_in.eall_in (insert x x_1) f]:real) * (-1) ^ (insert x x_1).card) =\n     -(insert ∅ (s.powerset.erase ∅)).sum\n            (λ (x_1 : finset α), (Pr[has_eall_in.eall_in (insert x x_1) f]:real) * (-1) ^ (insert x x_1).card),\n     {rw A9A},\n     rw A9B,\n     clear A9A A9B,\n     rw add_comm (Pr[f x]:real),\n     --rw finset.sum_insert,\n     simp,\n     have A9C:-((s).powerset.erase ∅).sum\n            (λ (x_1 : finset α), (Pr[has_eall_in.eall_in (insert x x_1) f]:real) * (-1) ^ (insert x x_1).card) =\n((s).powerset.erase ∅).sum\n            ((λ (x_1 : finset α),- (Pr[has_eall_in.eall_in (insert x x_1) f]:real) * (-1) ^ (insert x x_1).card)),\n     {simp},\n     rw A9C,\n     clear A9C,\n     apply finset.sum_congr,\n     {refl},\n     intros T A9D,\n     simp at A9D,\n     rw distrib_forall_eand,\n     rw eall_finset_insert,\n     rw finset.card_insert_of_not_mem,\n     rw pow_succ,\n     {simp},\n     intro A9F,\n     cases A9D with A9D A9E,\n     rw finset.subset_iff at A9E,\n     have A9G := A9E A9F,\n     apply A3,\n     apply A9G,\n     apply finset.nonempty_of_ne_empty,\n     apply A9D.left,\n  },\n  rw A9,\n  linarith,\n  apply A3,\nend\n\nlemma Pr_all_not_eq_powerset {α Ω:Type*} {P:probability_space Ω} (f:α → event P) [decidable_eq α] {S:finset α}:  (Pr[(∀ᵣ a in S, ¬ₑ (f a))]:real) = \n  (S.powerset).sum  (λ T:finset α, (Pr[∀ᵣ a in T, f a]:real) *  (-1)^(T.card)) :=\nbegin\n  --rw Pr_exists_eq_powerset,\n  have A1:(insert ∅ ((S.powerset).erase ∅)).sum (λ T:finset α, (Pr[∀ᵣ a in T, f a]:real) *  (-1)^(T.card))\n      =     S.powerset.sum (λ (T : finset α), ↑Pr[∀ᵣ (a : α) in T,f a] * (-1) ^ T.card),\n  {\n    rw finset.insert_erase,\n    simp,\n  },\n  have A1:∅ ∈ S.powerset,\n  {simp},\n  rw ← finset.insert_erase A1,\n  rw finset.sum_insert,simp,\n  rw ← not_exists_eq_forall_not,\n  rw ← Pr_one_minus_eq_not,\n  rw nnreal.coe_sub,\n  rw Pr_exists_eq_powerset,\n  repeat {simp}, \nend\n\nlemma independent_events_not_of_independent_events {α Ω:Type*} {P:probability_space Ω} (f:α → event P):independent_events f → independent_events (enot ∘ f) :=\nbegin\n  intros A1,\n  unfold independent_events,\n  intros S,\n  haveI A3:=classical.decidable_eq α,\n\n  apply finset.case_strong_induction_on S,\n  {simp},\n  intros a s B1 B2,\n  rw  ← not_exists_eq_forall_not,\n  have A2:∀ (A:event P), 1 - (Pr[A]:real) = (Pr[¬ₑ A]:real),\n  { \n    intro A,rw ← Pr_one_minus_eq_not,\n    rw nnreal.coe_sub,\n    {simp},\n    apply Pr_le_one,\n  },\n  rw ← nnreal.coe_eq,\n  rw ← A2,\n  rw @Pr_exists_eq_powerset α Ω P f A3,\n  unfold independent_events at A1,\n  rw finset.prod_insert,\n  rw finset.powerset_sum,\n  rw B2 s (finset.subset.refl s),\n  rw nnreal.coe_mul,\n  rw ← A2,\n  simp,\n  rw ← @Pr_exists_eq_powerset α Ω P f A3,\n  have A4:∀ x∈ s.powerset, \n     (Pr[has_eall_in.eall_in (insert a x) f]:real) * (-1) ^ (insert a x).card =\n    -(Pr[f a]:real) * ((Pr[has_eall_in.eall_in x f]:real) * (-1) ^ (x).card),\n  {intros x A4A,\n   have A4B:a ∉ x,\n   {simp at A4A,intro A4B,apply B1,apply A4A,apply A4B},\n   rw ← A1,rw finset.prod_insert A4B,\n   rw A1,\n   rw nnreal.coe_mul,\n   rw finset.card_insert_of_not_mem A4B,\n   rw pow_succ,\n   simp,\n   repeat {rw ← mul_assoc},\n  },\n  have A5:s.powerset = s.powerset := rfl,\n  rw finset.sum_congr A5 A4,\n  have A6:s.powerset.sum (λ (x : finset α), -(Pr[f a]:real) * (↑Pr[has_eall_in.eall_in x f] * (-1) ^ x.card))\n    = -((Pr[f a]:real)) * s.powerset.sum (λ (x : finset α), (↑Pr[has_eall_in.eall_in x f] * (-1) ^ x.card)),\n  {simp,rw finset.sum_distrib_left},\n  rw A6,  \n  rw ← Pr_all_not_eq_powerset,\n  rw ← not_forall_not_eq_exists,\n  rw ← A2,\n  simp,\n  ring,\n  repeat {exact B1},\nend\n\nlemma events_IID_not_of_events_IID {α Ω:Type*} {P:probability_space Ω} (f:α → event P):events_IID f → events_IID (enot ∘ f) :=\nbegin\n  intros A1,\n  unfold events_IID,\n  split,\n  {\n    apply independent_events_not_of_independent_events,\n    unfold events_IID at A1,\n    apply A1.left,\n  },\n  {\n    unfold events_IID at A1,\n    intros x y,\n    have C2 := A1.right x y,\n    simp,\n    rw ← Pr_one_minus_eq_not,\n    rw ← Pr_one_minus_eq_not,\n    rw C2,\n  },\nend\n\nlemma events_IID_iff_events_IID_enot {α Ω:Type*} {P:probability_space Ω} (f:α → event P):events_IID f ↔ events_IID (enot ∘ f) :=\nbegin\n  split,\n  {\n    apply events_IID_not_of_events_IID, \n  },\n  {\n    intros A1,\n    have A2:f = enot ∘ (enot ∘ f),\n    { apply funext, intro a, simp },\n    rw A2,\n    apply events_IID_not_of_events_IID,\n    apply A1 \n  },\nend\n\n\ndef measurable_fun {α:Type*}  {β:Type*} (Mα:measurable_space α) (Mβ:measurable_space β):Type*\n    := subtype (@measurable α β _ _)\n\ndef random_variable {α:Type*} (p:probability_space α) {β:Type*}\n  (Mβ:measurable_space β):Type* := measurable_fun (probability_space.to_measurable_space α) Mβ\n\ninfixr  ` →ₘ `:80 := measurable_fun\ninfixr  ` →ᵣ `:80 := random_variable\n\n\nlemma random_variable_val_eq_coe {Ω α:Type*} {P:probability_space Ω} {M:measurable_space α}  \n  (X:P →ᵣ M):X.val = \n  (@coe (subtype (@measurable Ω α _ _)) (Ω → α) _ X) :=\nbegin\n  refl\nend\n\n\n\n\nlemma measurable_setB_preimageh {α:Type*}  {β:Type*} [Mα:measurable_space α] [Mβ:measurable_space β]\n  (f:measurable_fun Mα Mβ) (S:measurable_setB Mβ):measurable_set (set.preimage (f.val) (S.val)) :=\nbegin\n  apply measurable_elim,\n  apply f.property,\n  apply S.property\nend\n\ndef measurable_setB_preimage {α:Type*}  {β:Type*} [Mα:measurable_space α] [Mβ:measurable_space β]\n  (f:measurable_fun Mα Mβ) (S:measurable_setB Mβ):measurable_setB Mα := {\n    val:= (set.preimage (f.val) (S.val)),\n    property:=measurable_setB_preimageh f S,\n}\n\n\ndef rv_event {α:Type*} {p:probability_space α} {β:Type*}\n  {Mβ:measurable_space β} (X:random_variable p Mβ) (S:measurable_setB Mβ):event p :=\n   measurable_setB_preimage X S\n\n\ninfixr ` ∈ᵣ `:80 := rv_event\n--infixr `∈` := rv_event\n\n\n\ndef rv_event_compl {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X:random_variable p Mβ) (S:measurable_setB Mβ):event p :=\n   ¬ₑ(rv_event X S)\n\ninfixr `∉ᵣ`:80 := rv_event_compl\n\n\n\n@[simp]\ndef rv_event_compl_val {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X:random_variable p Mβ) (S:measurable_setB Mβ):\n   (rv_event_compl X S).val = (¬ₑ (rv_event X S)).val := rfl\n\n@[simp]\nlemma rv_event_val_def {α:Type*} {p : probability_space α} {β : Type*}\n  [Mβ : measurable_space β] (X:p →ᵣ Mβ) (S:measurable_setB Mβ):(X ∈ᵣ S).val = {a:α|X.val a ∈ S.val} :=\nbegin\n  refl\nend\n\n/- Which of these is simpler is subjective. -/\nlemma rv_event_compl_preimage {α:Type*} {p: probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X:random_variable p Mβ) (S:measurable_setB Mβ):\n   (X ∈ᵣ Sᶜ) = (X ∈ᵣ S)ᶜ := rfl\n  \n\n\n\n\nlemma rv_event_empty {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X:random_variable p Mβ):X ∈ᵣ ∅ = ∅ :=\nbegin\n  apply event.eq,\n  rw rv_event_val_def,\n  rw event_empty_val_def2,\n  ext ω;split;intros A1,\n  {\n    exfalso,\n    simp at A1,\n    apply A1,\n  },\n  {\n    exfalso,\n    apply A1,\n  },\nend\n\n\n\nlemma rv_event_measurable_union {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X:random_variable p Mβ) \n  {A B:measurable_setB Mβ}:X ∈ᵣ (measurable_union A B) = ((X ∈ᵣ A) ∨ (X∈ᵣ B)) :=\nbegin\n  apply event.eq,\n  repeat {rw rv_event_val_def},\n  rw eor_val_def,\n  repeat {rw rv_event_val_def},\n  rw measurable_union_val_def,\n  ext ω;split;intros A1;simp;simp at A1;apply A1\nend\n\nlemma rv_event_val_def' {α:Type*} {p : probability_space α} {β : Type*}\n  [Mβ : measurable_space β] (X:p →ᵣ Mβ) (S:measurable_setB Mβ) {ω:α}:\n  (ω ∈ ((X ∈ᵣ S)))↔ (X.val ω ∈ S.val) :=\nbegin\n  refl\nend\n\n\nlemma set.mem_Prop_def {α:Type*} {P:α → Prop} {a:α}:\n    (a ∈ {a':α|P a'} )= P a := rfl\n\n\nlemma rv_event_measurable_Union {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] {γ:Type*} [E:encodable γ] (X:random_variable p Mβ) \n  {f:γ → measurable_setB Mβ}:X ∈ᵣ (measurable_Union f) = \n  measurable_Union (λ i, X ∈ᵣ (f i)) :=\nbegin\n  apply event.eq,\n  ext ω,\n  rw rv_event_val_def,\n  rw measurable_Union_val_def,\n  rw measurable_Union_val_def,\n  split;intro A1,\n  {\n    rw set.mem_Prop_def at A1,\n    rw set.mem_Union,\n    rw set.mem_Union at A1,\n    cases A1 with i A1,\n    apply exists.intro i,\n    rw @rv_event_val_def α p β Mβ X (f i),\n    rw set.mem_Prop_def,\n    apply A1,\n  },\n  {\n    rw set.mem_Prop_def,\n    rw set.mem_Union,\n    rw set.mem_Union at A1,\n    cases A1 with i A1,\n    rw @rv_event_val_def α p β Mβ X (f i) at A1,\n    apply exists.intro i,\n    apply A1,\n  },\nend\n\nlemma Pr_compl_ge_of_Pr_le {Ω:Type*} {p:probability_space Ω} (A:event p) (δ:nnreal):\n  Pr[A] ≤ δ → Pr[Aᶜ] ≥ 1 - δ :=\nbegin\n  intros h1,\n  apply Pr_not_ge_of_Pr_le,\n  apply h1,\nend\n\n\n--@[simp]\nlemma event_simp_def {α:Type*} [p:probability_space α] {X:set α} {H:measurable_set X}:\n  (subtype.mk X H).val = X := rfl\n\n--@[simp]\nlemma measurable_setB_simp_def {α:Type*} [p:measurable_space α] {X:set α} {H:measurable_set X}:\n  (subtype.mk X H).val = X := rfl\n\nlemma pr_comp_event {Ω:Type*} {p:probability_space Ω} {X:p →ᵣ borel real}\n {E:@measurable_setB ℝ (borel ℝ)}:\n (X ∈ᵣ (Eᶜ)) = (X ∈ᵣ E)ᶜ :=\nbegin\n  apply event.eq,\n  simp,\n  refl,\nend\n\nlemma rv_event_compl' {Ω:Type*} {MΩ:probability_space Ω} {β:Type*} [Mβ:measurable_space β]\n  (X:MΩ →ᵣ Mβ) (S:measurable_setB Mβ):(X ∈ᵣ (Sᶜ)) = (rv_event X S)ᶜ :=\nbegin\n  apply event.eq,\n  simp,\n  refl,\nend\n\n\nlemma neg_eq_not {Ω:Type*} {p:probability_space Ω} (A:event p):\n  Aᶜ = ¬ₑ A :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n/-def random_variable_identical {α:Type*} {p:probability_space α} {β:Type*}\n  [Mβ:measurable_space β] (X Y:random_variable p Mβ):Prop :=\n  ∀ (S:measurable_setB Mβ), Pr[X ∈ᵣ S] = Pr[Y ∈ᵣ S]-/\n\ndef random_variable_identical {α α':Type*} {p:probability_space α} {p':probability_space α'} {β:Type*}\n  {Mβ:measurable_space β} (X:random_variable p Mβ) (Y:random_variable p' Mβ):Prop :=\n  ∀ (S:measurable_setB Mβ), Pr[X ∈ᵣ S] = Pr[Y ∈ᵣ S]\n\n\ndef random_variable_independent_pair {α:Type*} {p:probability_space α} {β:Type*}\n  {Mβ:measurable_space β} {γ:Type*} {Mγ:measurable_space γ} (X:p →ᵣ Mβ)\n  (Y:p →ᵣ Mγ):Prop :=\n  ∀ (S:measurable_setB Mβ) (T:measurable_setB Mγ), independent_event_pair (X ∈ᵣ S) (Y ∈ᵣ T)\n\ndef random_variable_independent {α:Type*} {p:probability_space α} {β:Type*}\n  {γ:β → Type*} {Mγ:Π b, measurable_space (γ b)} (X:Π b, random_variable p (Mγ b)):Prop :=\n  ∀ (S:Π b, measurable_setB (Mγ b)), independent_events (λ b:β, ((X b) ∈ᵣ (S b)))\n\ndef random_variables_IID {α:Type*} {p:probability_space α} {β:Type*}\n  {γ:Type*} {Mγ:measurable_space γ} (X:β → random_variable p Mγ):Prop :=\n  random_variable_independent X ∧\n  ∀ (i j:β), random_variable_identical (X i) (X j)\n\n\n/- There are a lot of types where everything is measurable. This is equivalent to ⊤. -/\nclass top_measurable (α:Type*) extends measurable_space α :=\n  (all_measurable:∀ S:set α,measurable_set S)\n\n\ndef make_top_measurable_space (α:Type*) :top_measurable α := {\n  to_measurable_space := ⊤,\n  all_measurable := begin\n    intros S,\n    apply measurable_space.measurable_set_top,\n  end,\n}\n\n\ninstance top_measurable.has_coe_measurable_space (α:Type*):has_coe (top_measurable α) (measurable_space α) := {\n  coe := λ TM, @top_measurable.to_measurable_space α TM\n}\n\n\ninstance bool_top_measurable:top_measurable bool := {\n  all_measurable:=@measurable_space.measurable_set_top bool,\n  ..bool.measurable_space\n}\n\ninstance int_top_measurable:top_measurable ℤ := {\n  all_measurable:=@measurable_space.measurable_set_top ℤ,\n  ..int.measurable_space\n}\n\n/-\n  In a top measurable space (e.g. bool, ℕ, ℤ, et cetera),\n  everything is measurable. So, we can make an event from everything.\n-/\ndef measurable_setB_top {β:Type*} [M:top_measurable β] (S:set β):\n    (@measurable_setB β M.to_measurable_space) := {\n  val := S,\n  property := top_measurable.all_measurable S,\n}\n\ndef measurable_setB_top' {β:Type*} (S:set β):\n    (@measurable_setB β (⊤:measurable_space β)) := {\n  val := S,\n  property := begin\n    apply measurable_space.measurable_set_top,\n  end,\n}\n\n@[simp]\nlemma measurable_setB_top_val {β:Type*} [M:top_measurable β] (S:set β):\n  (measurable_setB_top S).val = S := rfl\n\n\nlemma fun_top_measurable {β γ:Type*} [M:top_measurable β] [Mγ:measurable_space γ] {f:β → γ}:\n  measurable f := begin\n  intros S A1,\n  apply top_measurable.all_measurable,\nend\n\n\n\n\n\ndef top_measurable_fun {β γ:Type*} [M:top_measurable β] (f:β → γ) (Mγ:measurable_space γ):\n  (@top_measurable.to_measurable_space β M) →ₘ Mγ := {\n  val := f,\n  property := fun_top_measurable\n} \n\n\n\n\ndef rv_top_event {α:Type*} {p:probability_space α}\n {β:Type*} [Mβ:top_measurable β]\n  (X:random_variable p Mβ.to_measurable_space) (S:set β):event p :=\n  rv_event X (measurable_setB_top S)\n\n--To apply this directly to a set, it has to be top measurable.\ninfixr ` ∈t `:80 := rv_top_event\n\nlemma rv_top_event_val_def  {α:Type*} {p:probability_space α}\n {β:Type*} [Mβ:top_measurable β]\n  (X:random_variable p Mβ.to_measurable_space) (S:set β):\n  (rv_top_event X S).val = {a:α|X.val a ∈ S} := rfl\n\n\n\n\n\nlemma compose_measurable_fun_measurable2 {α β γ:Type*}\n  [Mα:measurable_space α] [Mβ:measurable_space β] [Mγ:measurable_space γ]\n  (X:measurable_fun Mβ Mγ) (Y:measurable_fun Mα Mβ):measurable (X.val ∘ Y.val) :=\nbegin\n  apply measurable_intro,\n  intros B a,\n  have A1:(X.val ∘ Y.val ⁻¹' B)=(Y.val ⁻¹' (X.val ⁻¹' B)),\n  {\n    refl,\n  },\n  rw A1,\n  apply measurable_elim Y.val _ Y.property,\n  apply measurable_elim X.val _ X.property,\n  apply a\nend\n\n\ndef compose_measurable_fun {α β γ:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β} {Mγ:measurable_space γ}\n  (X:measurable_fun Mβ Mγ) (Y:measurable_fun Mα Mβ):(measurable_fun Mα Mγ) := {\n    val := X.val ∘ Y.val,\n    property := compose_measurable_fun_measurable2 X Y\n  }\n\n\n\n\ninfixr  ` ∘m `:80 := compose_measurable_fun\n\n@[simp]\nlemma compose_measurable_fun_val_def {Ω : Type*} {β : Type*} {γ : Type*}\n  [MΩ : measurable_space Ω] [Mβ : measurable_space β]\n  [Mγ : measurable_space γ] (Y:Mβ →ₘ Mγ) (X:MΩ →ₘ Mβ):\n  (Y ∘m X).val = (λ ω:Ω, Y.val (X.val ω)) :=\nbegin\n  refl\nend\n\n\ndef rv_compose {α : Type*} {β : Type*} {γ : Type*}\n  {p : probability_space α} {Mβ : measurable_space β}\n  {Mγ : measurable_space γ} (X:measurable_fun Mβ Mγ) (Y:random_variable p Mβ):random_variable p Mγ :=\n  compose_measurable_fun X Y\n\n\ninfixr  ` ∘r `:80 := rv_compose\n\n@[simp]\nlemma rv_compose_val_def {Ω : Type*} {β : Type*} {γ : Type*}\n  [Mβ : measurable_space β]\n  [Mγ : measurable_space γ] {p : probability_space Ω} (Y:Mβ →ₘ Mγ) (X:p →ᵣ Mβ):\n  (Y ∘r X).val = (λ ω:Ω, Y.val (X.val ω)) :=\nbegin\n  refl\nend\n\ndef prod_space {α β:Type*} (Mα:measurable_space α) (Mβ:measurable_space β):=(@prod.measurable_space α β Mα Mβ)\n\ninfixr  ` ×ₘ `:80 := prod_space\n\n\ndef measurable_setB.prod {α β:Type*} {Mα:measurable_space α} {Mβ:measurable_space β} (A:measurable_setB Mα) (B:measurable_setB Mβ):measurable_setB (Mα ×ₘ Mβ) :=\n @measurable_setB.mk (α × β) (Mα ×ₘ Mβ) (A.val.prod B.val) begin\n  apply measurable_set.prod,\n  apply A.property,\n  apply B.property,\nend  \n\n@[simp]\nlemma measurable_setB.prod_val {α β:Type*} {Mα:measurable_space α} {Mβ:measurable_space β} (A:measurable_setB Mα) (B:measurable_setB Mβ):(A.prod B).val = (A.val).prod (B.val) := rfl\n\n\ndef mf_fst {α β:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β}:measurable_fun\n    (Mα ×ₘ Mβ) Mα := {\n    val:= (λ x:(α × β), x.fst),\n    property := fst_measurable,\n  }\n\n@[simp]\nlemma mf_fst_val {α β:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β}:(@mf_fst α β Mα Mβ).val = prod.fst := rfl\n\ndef mf_snd {α β:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β}:measurable_fun (prod_space Mα Mβ) Mβ := {\n    val:= (λ x:(α × β), x.snd),\n    property := snd_measurable,\n  }\n\n@[simp]\nlemma mf_snd_val {α β:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β}:(@mf_snd α β Mα Mβ).val = prod.snd := rfl\n\ndef const_measurable_fun {Ω : Type*} [MΩ : measurable_space Ω] {β : Type*}\n   [Mβ : measurable_space β] (c : β):MΩ →ₘ Mβ := {\n     val := (λ (ω : Ω), c),\n     property := const_measurable c,\n   }\n\nlemma const_measurable_fun_val_def {Ω : Type*} [MΩ : measurable_space Ω] {β : Type*}\n   [Mβ : measurable_space β] (c : β):(const_measurable_fun c).val = (λ (ω : Ω), c) := rfl\n\ndef const_random_variable {Ω : Type*} {P:probability_space Ω}\n   {β : Type*}\n   [Mβ : measurable_space β] (c : β):P →ᵣ Mβ := const_measurable_fun c\n\n\ndef prod_measurable_fun\n{α β γ:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β} {Mγ:measurable_space γ}\n  (X:measurable_fun Mα Mβ) (Y:measurable_fun Mα Mγ):measurable_fun Mα (Mβ ×ₘ Mγ) := {\n    val := (λ a:α, prod.mk (X.val a) (Y.val a)),\n    property := measurable_fun_product_measurable X.val Y.val X.property Y.property,\n  }\n\nlemma prod_measurable_fun_val_def {Ω : Type*} {β : Type*} {γ : Type*} {MΩ : measurable_space Ω}\n  {Mβ : measurable_space β} {Mγ : measurable_space γ} {X:MΩ  →ₘ Mβ}\n  {Y:MΩ  →ₘ Mγ}: (prod_measurable_fun X Y).val = λ ω:Ω, prod.mk (X.val ω) (Y.val ω) :=\nbegin\n  refl\nend\n\ndef prod_random_variable\n{α β γ:Type*}\n  {P:probability_space α} {Mβ:measurable_space β} {Mγ:measurable_space γ}\n  (X:random_variable P Mβ) (Y:random_variable P Mγ):random_variable P (Mβ ×ₘ Mγ) :=\n  prod_measurable_fun X Y\n\ninfixr  ` ×ᵣ `:80 := prod_random_variable\n\n\n@[simp]\nlemma prod_random_variable_val_def {Ω : Type*} {β : Type*} {γ : Type*}\n  {P : probability_space Ω} {Mβ : measurable_space β} {Mγ : measurable_space γ} {X:P →ᵣ Mβ}\n  {Y:P →ᵣ Mγ}: (X ×ᵣ Y).val = λ ω:Ω, prod.mk (X.val ω) (Y.val ω) :=\nbegin\n  refl\nend\n\n\ndef prod_measurable_setB {β : Type*} {γ : Type*}\n  {Mβ : measurable_space β} \n  {Mγ : measurable_space γ} \n  (X:measurable_setB Mβ) (Y:measurable_setB Mγ):measurable_setB (Mβ ×ₘ Mγ) := {\n  val := (set.prod X.val Y.val),\n  property := measurable_set_prod' X.property Y.property\n}\n\n@[simp]\nlemma prod_measurable_setB_val_def {β : Type*} {γ : Type*}\n  {Mβ : measurable_space β} \n  {Mγ : measurable_space γ} \n  (X:measurable_setB Mβ) (Y:measurable_setB Mγ):\n  (prod_measurable_setB X Y).val = set.prod X.val Y.val := rfl\n\n\nclass has_measurable_equality {α:Type*} (M:measurable_space α):Prop := (measurable_set_eq:measurable_set {p:α × α|p.fst = p.snd})\n\ndef measurable_setB_eq {α:Type*} {M:measurable_space α} [E:has_measurable_equality M]\n  :measurable_setB (M ×ₘ M) := measurable_setB.mk E.measurable_set_eq\n\n\n\nlemma measurable_setB_eq_val {α:Type*} {M:measurable_space α} [E:has_measurable_equality M]:\n  (@measurable_setB_eq α M E).val = {p:α × α|p.fst = p.snd} := rfl\n\n\ndef measurable_setB_ne {α:Type*} {M:measurable_space α} [E:has_measurable_equality M]\n  :measurable_setB (M ×ₘ M) := measurable_setB.mk (measurable_set.compl E.measurable_set_eq)\n\n\n\nlemma measurable_setB_ne_val {α:Type*} {M:measurable_space α} [E:has_measurable_equality M]:\n  (@measurable_setB_ne α M E).val = {p:α × α|p.fst ≠ p.snd} := rfl\n\n\n\n\nlemma diagonal_eq {α:Type*}:{p:α × α|p.fst  = p.snd}=⋃ (a:α), set.prod {a} {a} :=\nbegin\n    ext,split;intros A1A;simp;simp at A1A,\n    {\n      apply exists.intro x.fst,\n      ext;simp,\n      rw A1A,\n    },\n    {\n      cases A1A with i A1A,\n      cases A1A,\n      simp,\n    },\nend\n\n\ninstance measurable_setB_eq_top_measurable {α:Type*} (E:encodable α) (M:top_measurable α):has_measurable_equality (M.to_measurable_space) :=  {\n  measurable_set_eq := begin\n  rw diagonal_eq,\n  apply measurable_set.Union,\n  intros a,\n  apply measurable_set_prod',\n  repeat {apply M.all_measurable},\nend\n}\n\ninstance has_measurable_equality.fintype_top {α:Type*} [fintype α] [TM:top_measurable α]:\n  has_measurable_equality (TM.to_measurable_space) := {\n  measurable_set_eq := begin\n  rw diagonal_eq,\n  haveI:encodable α := fintype.encodable α,\n  apply measurable_set.Union,\n  intros b,\n  apply measurable_set.prod,\n  apply TM.all_measurable,\n  apply TM.all_measurable,\nend\n}\n\n\ninstance measurable_setB_eq_bool:has_measurable_equality bool.measurable_space :=  {\n  measurable_set_eq := begin\n  rw diagonal_eq,\n  apply measurable_set.Union,\n  intros a,\n  apply measurable_set_prod',\n  repeat {apply measurable_space.measurable_set_top},\nend\n}\n\ninstance measurable_setB_eq_int:has_measurable_equality int.measurable_space :=  {\n  measurable_set_eq := begin\n  rw diagonal_eq,\n  apply measurable_set.Union,\n  intros a,\n  apply measurable_set_prod',\n  repeat {apply measurable_space.measurable_set_top},\nend\n}\n\ndef random_variable_eq {Ω:Type*} {P:probability_space Ω} {α:Type*} [M:measurable_space α]\n   [E:has_measurable_equality M] (X Y:P →ᵣ M):event P := \n    (X ×ᵣ Y) ∈ᵣ (measurable_setB_eq)\n\ninfixr  ` =ᵣ `:100 := random_variable_eq  \n\n\n@[simp]\nlemma rv_eq_val_def {α:Type*} {p : probability_space α} {β : Type*}\n   [Mβ :measurable_space β] [has_measurable_equality Mβ] (X Y:p →ᵣ Mβ):\n  (X =ᵣ Y).val = {a:α| X.val a = Y.val a} :=\nbegin\n  unfold random_variable_eq,\n  rw rv_event_val_def,\n  rw prod_random_variable_val_def,\n  rw measurable_setB_eq_val,\n  simp,\nend\n\n\ndef random_variable_ne {Ω:Type*} {P:probability_space Ω} {α:Type*} [M:measurable_space α]\n   [E:has_measurable_equality M] (X Y:P →ᵣ M):event P := \n    ¬ₑ (X =ᵣ Y)\n\ninfixr  ` ≠ᵣ `:100 := random_variable_ne\n\n@[simp]\nlemma rv_ne_val_def {α:Type*} {p : probability_space α} {β : Type*}\n   [Mβ :measurable_space β] [has_measurable_equality Mβ] (X Y:p →ᵣ Mβ):\n  (X ≠ᵣ Y).val = {a:α| X.val a ≠ Y.val a} :=\nbegin\n  unfold random_variable_ne,\n  rw enot_val_def,\n  rw rv_eq_val_def,\n  simp,\n  ext a,split;intros A1;simp;simp at A1;apply A1,\nend\n\n\n\nlemma union_func_eq_sUnion_image {α β:Type*}\n  [Tβ:measurable_space β] {A:set α} {f:α → set β}:\n  (⋃ a∈ A, f a)=set.sUnion (@set.image α (set β) f A) :=\nbegin\n  simp,\nend\n\n\nlemma measurable_set_countable_union_func {α β:Type*}\n  [Tβ:measurable_space β] {A:set α} {f:α → set β}:\n  set.countable A →\n  (∀ a∈ A, measurable_set (f a)) →\n  measurable_set (⋃ a∈ A, f a) :=\nbegin\n  intros A1 A2,\n  rw union_func_eq_sUnion_image,\n  apply measurable_set.sUnion,\n  {\n    apply set.countable.image,\n    apply A1,\n  },\n  intros t A4,\n  cases A4 with a A5,\n  cases A5 with A6 A7,\n  subst t,\n  apply A2,\n  apply A6,\nend\n\n\n-- cf set.prod_singleton_singleton\nlemma singleton_prod {α β:Type*} {ab:α × β}:{ab} = (@set.prod α β {ab.fst} {ab.snd}) :=\nbegin\n  simp,\nend\n\nlemma top_measurable_prodh {α β:Type*} [encodable α] [encodable β]\n  [Tα:top_measurable α] [Tβ:top_measurable β] (U:set (α × β)):\n  measurable_set U :=\nbegin\n  have A2:encodable (α × β):= encodable.prod,\n  have A3:set.countable U := set.countable_encodable U,\n  have A4:(⋃ a∈U,{a}) = U,\n  {\n    simp\n  },\n  rw ← A4,\n  apply measurable_set_countable_union_func A3,\n  intros ab A5,\n  rw singleton_prod,\n  apply measurable_set.prod,\n  {\n    apply top_measurable.all_measurable,\n  },\n  {\n    apply top_measurable.all_measurable,\n  },\nend\n\n\ninstance top_measurable_prod {α β:Type*} [encodable α] [encodable β]\n  [Tα:top_measurable α] [Tβ:top_measurable β]:top_measurable (α × β) := {\n     all_measurable := top_measurable_prodh,\n  }\n\n\n\n\ndef if_measurable_fun\n{α β:Type*}\n  {Mα:measurable_space α} {Mβ:measurable_space β}\n  (E:measurable_setB Mα) (D:decidable_pred E.val)\n  (X:measurable_fun Mα Mβ) (Y:measurable_fun Mα Mβ):measurable_fun Mα Mβ :={\n    val := λ a:α, if (E.val a) then (X.val a) else (Y.val a),\n    property := measurable.if E.property X.property Y.property,\n  }\n\ndef if_random_variable\n{α β:Type*}\n  {P:probability_space α} {Mβ:measurable_space β}\n  (E:event P) (D:decidable_pred E.val)\n  (X:random_variable P Mβ) (Y:random_variable P Mβ):random_variable P Mβ :=\n  if_measurable_fun E D X Y\n\nlemma rv_event_IID {α : Type*} {p : probability_space α} {β : Type*}\n  {γ : Type*} [Mγ : measurable_space γ] (X:β → p →ᵣ Mγ) (S:measurable_setB Mγ):\n  random_variables_IID X  → events_IID (λ b:β, (X b) ∈ᵣ S) :=\nbegin\n  intro A1,\n  unfold random_variables_IID at A1,\n  cases A1 with A2 A3,\n  unfold random_variable_independent at A2,\n  unfold random_variable_identical at A3,\n  split,\n  {\n    apply A2,\n  },\n  {\n    intros i j,\n    simp,\n    apply A3,\n  }\nend\n\n@[simp]\nlemma measurable_setB_preimage_val_def {α:Type*}  {β:Type*} [Mα:measurable_space α] [Mβ:measurable_space β]\n  (f:measurable_fun Mα Mβ) (S:measurable_setB Mβ):\n  (measurable_setB_preimage f S).val = (set.preimage (f.val) (S.val)) := rfl\n\nlemma compose_measurable_fun_measurable_setB {Ω : Type*} {β : Type*} {γ : Type*}\n  [MΩ : measurable_space Ω] [Mβ : measurable_space β]\n  [Mγ : measurable_space γ] (Y:Mβ →ₘ Mγ) (X:MΩ →ₘ Mβ) (S:measurable_setB Mγ):\n  measurable_setB_preimage (Y ∘m X) S = measurable_setB_preimage X (measurable_setB_preimage Y S) :=\nbegin\n  apply subtype.eq,\n  rw measurable_setB_preimage_val_def,\n  rw measurable_setB_preimage_val_def,\n  rw measurable_setB_preimage_val_def,\n  rw compose_measurable_fun_val_def,\n  refl,\nend\n\nlemma rv_compose_measurable_setB  {α : Type*} {β : Type*} {γ : Type*}\n  {p : probability_space α} {Mβ : measurable_space β}\n  {Mγ : measurable_space γ} (X:measurable_fun Mβ Mγ) (Y:random_variable p Mβ) (S:measurable_setB Mγ):\n  (X ∘r Y) ∈ᵣ S = (Y ∈ᵣ (measurable_setB_preimage X S)) :=\nbegin\n  apply compose_measurable_fun_measurable_setB,\nend\n\nlemma compose_independent' {Ω α:Type*} {p:probability_space Ω}\n  {γ:α → Type*} [Mγ:Π (a:α), measurable_space (γ a)]\n  {κ:α → Type*} [Mκ:Π (a:α), measurable_space (κ a)] \n  (X:Π (b:α), p →ᵣ (Mγ b)) (Y:Π (b:α), (Mγ b) →ₘ  (Mκ b)):\n  random_variable_independent X → random_variable_independent (λ b:α, (Y b) ∘r (X b)) :=\nbegin\n  unfold random_variable_independent,\n  intros A1 S T,\n  unfold independent_events,\n  have A2:(λ (b : α), Pr[((Y b) ∘r (X b)) ∈ᵣ (S b)]) =\n      (λ (b : α), Pr[(X b) ∈ᵣ (measurable_setB_preimage (Y b) (S b))]),\n  {\n    ext b,\n    rw rv_compose_measurable_setB,\n  },\n  rw A2,\n  have A3:(λ (b : α), ((Y b) ∘r X b) ∈ᵣ S b) =\n      (λ (b : α), (X b) ∈ᵣ (measurable_setB_preimage (Y b) (S b))),\n  {\n    ext b,\n    rw rv_compose_measurable_setB,\n  },\n  rw A3,\n  apply A1,\nend\n\nlemma compose_independent {α:Type*} {p:probability_space α} {β:Type*}\n  {γ:Type*} [Mγ:measurable_space γ]\n  {κ:Type*} [Mκ:measurable_space κ] (X:β → random_variable p Mγ) (Y:Mγ →ₘ  Mκ):\n  random_variable_independent X → random_variable_independent (λ b:β, Y ∘r (X b)) :=\nbegin\n  apply compose_independent',\nend\n\nlemma compose_identical {α:Type*} {p:probability_space α}\n  {γ:Type*} [Mγ:measurable_space γ]\n  {κ:Type*} [Mκ:measurable_space κ] (X₁ X₂:random_variable p Mγ) (Y:Mγ →ₘ  Mκ):\n  random_variable_identical X₁ X₂ → random_variable_identical (Y ∘r X₁) (Y ∘r X₂)  :=\nbegin\n  unfold random_variable_identical,\n  intro A1,\n  intros S,\n  rw rv_compose_measurable_setB,\n  rw rv_compose_measurable_setB,\n  apply A1,\nend\n\nlemma compose_IID {α:Type*} {p:probability_space α} {β:Type*}\n  {γ:Type*} [Mγ:measurable_space γ]\n  {κ:Type*} [Mκ:measurable_space κ] (X:β → random_variable p Mγ) (Y:Mγ →ₘ  Mκ):\n  random_variables_IID X → random_variables_IID (λ b:β, Y ∘r (X b)) :=\nbegin\n  unfold random_variables_IID,\n  intro A1,\n  cases A1 with A2 A3,\n  split,\n  {\n    apply compose_independent,\n    apply A2,\n  },\n  {\n    intros i j,\n    apply compose_identical,\n    apply A3,\n  }\nend\n\n--For Pr_disjoint_summable below.\nlemma union_disjoint' {Ω β:Type*} {p:probability_space Ω}  \n    [D:decidable_eq β]\n    (A:β → event p) (B:event p) {S:finset β}:(∀ a'∈ S, disjoint B.val (A a').val) →\n    disjoint B.val (⋃ (b:β) (H:b∈ S), (A b).val) :=\nbegin\n  intros A1,\n  rw set.disjoint_right,\n  intros ω A4 A3,\n  simp at A4,\n  cases A4 with i A4,\n  have A5:= A1 i A4.left,\n  rw set.disjoint_right at A5,\n  apply A5 A4.right A3,\nend\n\nlemma union_disjoint {Ω β:Type*} {p:probability_space Ω}  \n    [D:decidable_eq β]\n    (A:β → event p) {S:finset β} {a:β}:(pairwise (disjoint on λ (i : β), (A i).val)) →\n    (a ∉ S) →\n    disjoint (A a).val (⋃ (b:β) (H:b∈ S), (A b).val) :=\nbegin\n  intros A1 A2,\n  apply union_disjoint',\n  intros a' h_a',\n  apply A1,\n  intros contra,\n  subst a',\n  apply A2 h_a',\nend\n\nlemma Pr_sum_disjoint_eq' {Ω β:Type*} {p:probability_space Ω}  \n    [D:decidable_eq β]\n    (A:β → event p) {S:finset β}:(set.pairwise_on (↑S) (disjoint on (λ i,  (A i).val))) →\n    Pr[∃ᵣ a in S, A a] =\nfinset.sum S (λ (b:β), Pr[A b]) :=\nbegin\n  apply finset.induction_on S,\n  {\n    intros A1,\n    simp,\n  },\n  { intros a T A2 A3 B1,\n    rw eany_finset_insert,\n    rw Pr_disjoint_eor,\n    rw finset.sum_insert A2,\n    rw A3,\n    { apply set.pairwise_on.mono _ B1,\n      simp },\n    { apply union_disjoint',\n      intros b h_b, apply B1, simp, simp [h_b], intros contra,\n      subst b, apply A2 h_b } },  \nend\n\nlemma Pr_sum_disjoint_eq {Ω β:Type*} {p:probability_space Ω}  \n    [D:decidable_eq β]\n    (A:β → event p) {S:finset β}:(pairwise (disjoint on λ (i : β), (A i).val)) →\n    Pr[eany_finset S A] =\nfinset.sum S (λ (b:β), Pr[A b]) :=\nbegin\n  intros h0,\n  have h1 := @Pr_sum_disjoint_eq' _ _ _ _ A S _,\n  apply h1,\n  apply pairwise.pairwise_on h0,\nend\n\n\nlemma Pr_sum_disjoint_bounded {Ω β:Type*} {p:probability_space Ω} [decidable_eq β] \n    (A:β → event p) {S:finset β}:(pairwise (disjoint on λ (i : β), (A i).val)) →\n    finset.sum S (λ (b:β), Pr[A b]) ≤ 1 :=\nbegin\n  intro A1,\n  rw ← Pr_sum_disjoint_eq,\n  apply Pr_le_one,\n  apply A1,\nend\n\nlemma Pr_disjoint_summable {Ω β:Type*} {p:probability_space Ω} [E:encodable β] [decidable_eq β]\n    (A:β → event p):(pairwise (disjoint on λ (i : β), (A i).val)) →\n    summable (λ (b:β), Pr[A b]) :=\nbegin\n  intro A1,\n  apply summable_bounded_nnreal,\n  {\n    intro S,\n    apply Pr_sum_disjoint_bounded,\n    apply A1,\n  },\nend\n\nlemma Pr_eany_sum {Ω β:Type*} {p:probability_space Ω} [E:encodable β] [decidable_eq β] \n    (A:β → event p):(pairwise (disjoint on λ (i : β), (A i).val)) →\n    Pr[eany A] = ∑' (b:β), Pr[A b] :=\nbegin\n  intro B1,\n  rw ← measurable_Union_eq_any,\n  rw ← with_top.coe_eq_coe,\n  rw event_prob_def,\n  rw measurable_Union_val_def,\n  rw ennreal.coe_tsum,\n  have A1:(λ (b:β), ↑Pr[A b]) = (λ (b:β), (measure_theory.measure_space.volume \n (A b).val)),\n  {\n    ext b,\n    rw event_prob_def,\n    rw measure_eq_measure,\n  },\n  rw measure_eq_measure,\n  rw measure_theory.measure_Union,\n  rw sum_subst,\n  rw A1,\n  apply B1,\n  {\n    intro i,\n    apply (A i).property,\n  },\n  {\n    apply Pr_disjoint_summable,\n    apply B1,\n  },\nend\n\nlemma mem_prod_random_variable_prod_measurable_setB \n  {α β γ:Type*}\n  {P:probability_space α} {Mβ:measurable_space β} {Mγ:measurable_space γ}\n  (X:random_variable P Mβ) (Y:random_variable P Mγ) \n  (S:measurable_setB Mβ) (T:measurable_setB Mγ):\n  (X ×ᵣ Y) ∈ᵣ (prod_measurable_setB S T) =\n  ((X ∈ᵣ S) ∧ (Y∈ᵣ T)) :=\nbegin\n  apply event.eq,\n  simp,\n  refl\nend\n\n\nlemma joint_measurable.pi {Ω β:Type*} {γ:β → Type*} [measurable_space Ω] [Π (b:β), measurable_space (γ b)] (f:Π (b:β), Ω → (γ b)) \n(h:∀ b:β, measurable (f b)):measurable (λ (ω:Ω) (b:β), f b ω) :=\nbegin\n  apply measurable_pi_lambda,\n  apply h,\nend\n\n\ndef measurable_space.pi_alt {δ:Type*} {π:δ → Type*} (M:Π (d:δ), measurable_space (π d)):\n  measurable_space (Π (d:δ), π d) :=\n  @measurable_space.pi δ π M \n\n\nnotation `Πₘ` binders `, ` r:(scoped f, measurable_space.pi_alt f) := r\n\n/- Given a function of measurable functions (X), create a measurable function\n   whose codomain is a measurable space of functions.\n   Alternate name: joint_measurable_fun? -/\ndef pi.measurable_fun\n{α β:Type*} {Mα:measurable_space α} {γ:β → Type*}\n  {M:Π (b:β), measurable_space (γ b)} \n  (X:Π (b:β), Mα →ₘ (M b)):measurable_fun Mα (@measurable_space.pi β γ M) := {\n    val := (λ (a:α) (b:β), (X b).val a),\n    property := begin\n      apply measurable_pi_lambda,\n      intros b,\n      apply (X b).property,\n    end,\n  }\n\n\n\n\n/- Given a bunch of random variables in a common probability space, combine them\n   to output a function. NOTE: this creates a new random variable in the \n   existing probability space. -/\ndef pi.random_variable_combine\n{α β:Type*} {P:probability_space α} {γ:β → Type*}\n  {M:Π (b:β), measurable_space (γ b)} \n  (X:Π (b:β), P →ᵣ (M b)):P →ᵣ (@measurable_space.pi β γ M) := \n  pi.measurable_fun X\n\n@[simp]\ndef random_variable.fst {Ω:Type*} {p:probability_space Ω} {α:Type*} {Mα:measurable_space α} {β:Type*} {Mβ:measurable_space β} (X:p →ᵣ (Mα ×ₘ Mβ)):p →ᵣ Mα :=\n  mf_fst ∘r X\n\n\n\n@[simp]\ndef random_variable.snd {Ω:Type*} {p:probability_space Ω} {α:Type*} {Mα:measurable_space α} {β:Type*} {Mβ:measurable_space β} (X:p →ᵣ (Mα ×ₘ Mβ)):p →ᵣ Mβ :=\n  mf_snd ∘r X\n\ninstance const_measurable_fun.has_coe {α:Type*} [M:measurable_space α] {Ω:Type*} {MΩ:measurable_space Ω}:has_coe α (MΩ →ₘ M) := {\n  coe := (λ (a:α), const_measurable_fun a)\n}\n\ninstance const_random_variable.has_coe {α:Type*} [M:measurable_space α] {Ω:Type*} {p:probability_space Ω}:has_coe α (p →ᵣ M) := {\n  coe := (λ (a:α), const_random_variable a)\n}\n\n\ninstance bool.has_coe_to_rv {Ω:Type*} {p:probability_space Ω}:has_coe bool (p →ᵣ bool.measurable_space) := const_random_variable.has_coe\ninstance int.has_coe_to_rv {Ω:Type*} {p:probability_space Ω}:has_coe int (p →ᵣ int.measurable_space) := const_random_variable.has_coe\n\n@[simp]\nlemma const_random_variable.has_coe.val {α:Type*} [M:measurable_space α] {Ω:Type*} {p:probability_space Ω}\n{a:α}:\n ((↑a):(p →ᵣ M)).val = (λ ω:Ω, a) := begin\n  refl\nend\n\n@[simp]\nlemma const_measurable_fun.has_coe.val {α:Type*} [M:measurable_space α] {Ω:Type*} {MΩ:measurable_space Ω}\n{a:α}:\n ((↑a):(MΩ →ₘ M)).val = (λ ω:Ω, a) := begin\n  refl\nend\n\n@[simp]\nlemma const_random_variable.has_coe.val_apply {α:Type*} [M:measurable_space α] {Ω:Type*} {p:probability_space Ω}\n{a:α} {ω:Ω}:\n ((↑a):(p →ᵣ M)).val ω = a := begin\n  refl\nend\n\n@[simp]\nlemma const_measurable_fun.has_coe.val_apply {α:Type*} [M:measurable_space α] {Ω:Type*} {MΩ:measurable_space Ω}\n{a:α} {ω:Ω}:\n ((↑a):(MΩ →ₘ M)).val ω = a := begin\n  refl\nend\n\nlemma random_variable_identical.symm \n  {Ω₁ Ω₂ α:Type*} {P₁:probability_space Ω₁}\n  {P₂:probability_space Ω₂}\n  {M:measurable_space α}\n  {X₁:P₁ →ᵣ M}\n  {X₂:P₂ →ᵣ M}:\n  random_variable_identical X₁ X₂ →\n  random_variable_identical X₂ X₁ :=\nbegin\n  intros h1 S,\n  symmetry,\n  apply h1,\nend\n\nlemma random_variable_identical.trans \n  {Ω₁ Ω₂ Ω₃ α:Type*} {P₁:probability_space Ω₁}\n  {P₂:probability_space Ω₂}\n  {P₃:probability_space Ω₃}\n  {M:measurable_space α}\n  {X₁:P₁ →ᵣ M}\n  {X₂:P₂ →ᵣ M}\n  {X₃:P₃ →ᵣ M}:\n  random_variable_identical X₁ X₂ →\n  random_variable_identical X₂ X₃ →\n  random_variable_identical X₁ X₃ :=\nbegin\n  intros h1 h2 S,\n  have h3:Pr[X₁ ∈ᵣ S] = Pr[X₂ ∈ᵣ S],\n  { apply h1 },\n  rw h3,\n  apply h2,\nend\n\n/-- This wraps `measure_theory.measure_Inter_eq_infi`.\n    Note that this theorem uses monotonically decreasing instead \n    of directed. This could be revisited. -/\nlemma Pr_forall_eq_infi {Ω:Type*} {P:probability_space Ω} {f : ℕ → event P}:\n                           (∀ (i:ℕ), (f i.succ).val ⊆ (f i).val) →  \n   Pr[∀ᵣ i, f i] = ⨅  i, Pr[f i] := begin\n  intros h1,\n  rw ← ennreal.coe_eq_coe,\n  rw event_prob_def,\n  have h2:(∀ᵣ (i : ℕ), f i).val = ⋂  (i : ℕ),(f i).val,\n  { simp, },\n  rw h2,\n  simp,\n  rw measure_theory.measure_Inter_eq_infi,\n  --simp [infi],\n  rw ennreal.coe_infi,\n  have h3:(λ (i : ℕ), measure_theory.measure_space.volume ↑(f i)) = λ (i : ℕ), ↑Pr[f i],\n  { ext1 i, rw event_prob_def, simp, },\n  rw h3,\n  { intros i, apply (f i).property },\n  { --apply directed_superset_of_monotone_nat_dual, \n    apply directed_superset_of_monotone_nat_dual,\n    apply h1, },\n  apply exists.intro 0,\n  apply lt_of_le_of_lt,\n  apply prob_le_1,\n  simp,\nend\n\nlemma Pr_forall_revent_eq_infi {Ω α:Type*} {P:probability_space Ω} {M:measurable_space α} \n   {f : ℕ → measurable_setB M} {X:P →ᵣ M}:\n                           (∀ (i:ℕ), (f i.succ).val ⊆ (f i).val) →  \n   Pr[∀ᵣ i, X ∈ᵣ f i] = ⨅  i, Pr[X ∈ᵣ f i] := begin\n  intros h1,\n  apply Pr_forall_eq_infi,\n  intros i,\n  simp,\n  intros ω h2,\n  apply h1,\n  apply h2\nend\n\n\n/- Wraps measure_theory.measure_Union_eq_supr -/\nlemma Pr_exists_eq_supr {Ω:Type*} {P:probability_space Ω} {f : ℕ → event P}:\n                           monotone (λ i, (f i).val) →\n   Pr[∃ᵣ i, f i] =  (⨆ (i:ℕ), Pr[f i]) := begin\n  intros h1,\n  rw ← ennreal.coe_eq_coe,\n  rw event_prob_def,\n  have h2:(∃ᵣ (i : ℕ), f i).val = ⋃ (i : ℕ),(f i).val,\n  { simp, },\n  rw h2,\n  simp,\n  rw measure_theory.measure_Union_eq_supr,\n  rw ennreal.coe_supr,\n  have h3:(λ (i : ℕ), measure_theory.measure_space.volume ↑(f i)) = λ (i : ℕ), ↑Pr[f i],\n  { ext1 i, rw event_prob_def, simp, },\n  rw h3,\n  { simp [bdd_above], rw set.nonempty_def,\n    apply exists.intro (1:nnreal),\n    rw mem_upper_bounds,\n    intros x h_mem,\n    simp at h_mem,\n    cases h_mem with i h_mem,\n    subst x,\n    apply Pr_le_one },\n  { intros i, apply (f i).property },\n  apply directed_subset_of_monotone,\n  apply h1\nend\n\nlemma Pr_exists_revent_eq_supr {Ω α:Type*} {P:probability_space Ω} {M:measurable_space α} \n   {f : ℕ → measurable_setB M} {X:P →ᵣ M}:\n                           (monotone (λ (i:ℕ), (f i).val)) →  \n   Pr[∃ᵣ i, X ∈ᵣ f i] = ⨆ i, Pr[X ∈ᵣ f i] := begin\n  intros h_mono,\n  apply Pr_exists_eq_supr,\n  intros i j h_le,\n  simp,\n  intros ω h2,\n  have h_mono_i_j := h_mono h_le,\n  simp at h_mono_i_j,\n  apply h_mono_i_j,\n  apply h2,\nend\n\nlemma disjoint_preimage {Ω γ:Type*} {P:probability_space Ω}\n  {M:measurable_space γ} {S T:measurable_setB M} {X:P →ᵣ M}:\n  disjoint S.val T.val → disjoint (X ∈ᵣ S).val (X ∈ᵣ T).val :=\nbegin\n  simp [disjoint_iff],\n  intros h,\n  ext, split; intros h1,\n  { simp at h1,\n    have h2:(X.val x) ∈ (↑S ∩ ↑T),\n    { simp [h1], rw set.mem_inter_iff, apply h1 },\n    rw h at h2,\n    exfalso, apply h2 },\n  exfalso, apply h1,\nend \n\nlemma independent_event_pair.symm {Ω:Type*} {P:probability_space Ω}\n  {E F:event P}:independent_event_pair E F → independent_event_pair F E :=\nbegin\n  intros h1,\n  unfold independent_event_pair,\n  rw mul_comm,\n  have h2:(F ∧ E) = (E ∧ F),\n  { apply event.eq, simp, rw set.inter_comm },\n  rw h2,\n  apply h1,\nend\n\nlemma random_variable_independent_pair.symm {Ω α β:Type*} {P:probability_space Ω}\n  {Mα:measurable_space α}\n  {Mβ:measurable_space β}\n  {X:P →ᵣ Mα} {Y:P →ᵣ Mβ}:random_variable_independent_pair X Y → \n  random_variable_independent_pair Y X :=\nbegin\n  intros h1 S T,\n  apply independent_event_pair.symm,\n  apply h1,\nend\n\n\n\ninstance measurable_setB_top.has_coe\n  {α:Type*} [TM:top_measurable α]:has_coe (set α) (measurable_setB (TM.to_measurable_space)) := \n  ⟨measurable_setB_top⟩\n\n@[simp]\nlemma measurable_setB_top.coe_val {α:Type*} [TM:top_measurable α] (S:set α):\n  (@coe (set α) (measurable_setB (TM.to_measurable_space)) _ S).val = S := rfl\n\n\nlemma event_eq_disjoint {α Ω:Type*} {P:probability_space Ω} \n  {M:measurable_space α} {Y:P →ᵣ M} [has_measurable_equality M]:\n  pairwise (disjoint on (λ (a:α), (Y =ᵣ a).val)) :=\nbegin\n  intros i j h_ne,\n  simp [function.on_fun, disjoint_iff],\n  { ext ω, split; intros h3, \n   { simp at h3, exfalso, apply h_ne,\n     cases h3, subst i, subst j },\n   { exfalso, apply h3 } },\nend\nlemma Pr_sum_univ_eq_one {α Ω:Type*} [fintype α] {P:probability_space Ω} \n  {M:measurable_space α} {Y:P →ᵣ M} [has_measurable_equality M]:\n  finset.univ.sum (λ (a:α), Pr[Y =ᵣ a]) = 1 :=\nbegin\n  classical,\n  have h1:(∃ᵣ (a:α), Y =ᵣ a) = event_univ,\n  { apply event.eq,\n    simp, ext ω, split; intros h1, simp at h1,\n    simp, },\n  have h2:Pr[(∃ᵣ (a:α), Y =ᵣ a)] = 1,\n  { rw h1, simp  },\n  have h3:(∃ᵣ (a:α), Y =ᵣ a) = eany_finset (@finset.univ α _) (λ (a:α), Y =ᵣ a),\n  { apply event.eq, simp },\n  rw h3 at h2,\n  haveI:encodable α := fintype.encodable α, \n  rw Pr_sum_disjoint_eq at h2,\n  apply h2,\n  apply event_eq_disjoint,\nend\n\nlemma equal_eq_mem {Ω α:Type*} {P:probability_space Ω} {TM:top_measurable α}\n  [has_measurable_equality TM.to_measurable_space]\n  (X:P →ᵣ TM.to_measurable_space) (a:α):(X =ᵣ a) = (X ∈ᵣ ({a}:set α)) :=\nbegin\n  apply event.eq,\n  simp,\nend\n\n\nlemma finset_empty_empty {α:Type*} (S:finset α):(¬(nonempty α)) → (S = ∅) :=\nbegin\n  intros h1,\n  ext,split;intros A1,\n  { exfalso, apply h1, apply nonempty.intro a },\n  { exfalso, apply A1 },\nend\n\n\nlemma independent_events_empty {Ω α:Type*}\n  {P:probability_space Ω} (E:α → event P):(¬(nonempty α)) →\n  independent_events E := begin\n  intros h1,\n  simp [independent_events],\n  intros S,\n  rw finset_empty_empty S h1,\n  simp,\nend\n\nlemma random_variable_independent_empty {Ω α γ:Type*}\n  {P:probability_space Ω} {M:measurable_space γ} (X:α → P →ᵣ M):(¬(nonempty α)) →\n  random_variable_independent X := begin\n  simp [random_variable_independent],\n  intros h1 S,\n  apply independent_events_empty,\n  apply h1,\nend\nlemma random_variables_IID_empty {Ω α γ:Type*}\n  {P:probability_space Ω} {M:measurable_space γ} (X:α → P →ᵣ M):(¬(nonempty α)) →\n  random_variables_IID X := begin\n  intros h1,\n  simp [random_variables_IID],\n  split,\n  apply random_variable_independent_empty,\n  apply h1,\n  intros i,\n  exfalso,\n  apply h1,\n  apply nonempty.intro i,\nend\n\n\n\ndef set.pi_measurable {α:Type*} [F:fintype α] {β:α → Type*} {M:Π a, measurable_space (β a)}\n(T:set α) (S:Π a, measurable_setB (M a)):measurable_setB (@measurable_space.pi α β M) := {\n  val := (set.pi T (λ a, (S a).val)),\n  property := begin\n    simp,\n    apply @measurable_set.pi',\n    intros a,\n    apply (S a).property,\n  end\n}\n\n\nlemma set.pi_measurable_univ {α:Type*} [F:fintype α] {β:α → Type*} {M:Π a, measurable_space (β a)}\n(S:Π a, measurable_setB (M a)) (T:set α) [decidable_pred T]:set.pi_measurable T S = set.pi_measurable \n(@set.univ α) (λ (a:α), if (a∈ T) then (S a) else (measurable_setB_univ)) :=\nbegin\n  ext x,\n  split;intros A1;\n  simp [set.pi_measurable, measurable_setB_univ]; intros a;\n  simp [set.pi_measurable, measurable_setB_univ] at A1,\n  cases decidable.em (a ∈ T) with A2 A2,\n  { rw if_pos, apply A1, apply A2, apply A2 },\n  { rw if_neg, simp, apply A2 },\n  { intros A3, have A4 := A1 a, rw if_pos A3 at A4, apply A4 },\nend\n\n/--\n  Often, we use the independence of random variables directly.\n  Specifically, many different kinds of relationships are implicitly writable\n  in the form X ∈ᵣ C, but it is not necessarily to keep them in that form.\n  For example, X =ᵣ 3 can be written as X ∈ᵣ {3}, but it is more convenient\n  and clearer in the former form.\n  More dramatically, there is a D such that \n  (∀ᵣ (s : α) in T,X (b s) ∈ᵣ S s) = ((pi.random_variable_combine X) ∈ᵣ D).\n-/\nlemma random_variable_independent_pair_apply {Ω α β:Type*} \n  {P:probability_space Ω} {Mα:measurable_space α} {Mβ:measurable_space β}\n  {X:P→ᵣ Mα} {Y:P →ᵣ Mβ} {A B:event P}:\n  random_variable_independent_pair X Y →\n  (∃ C:measurable_setB Mα, X ∈ᵣ C = A) →\n  (∃ D:measurable_setB Mβ, Y ∈ᵣ D = B) →\n  independent_event_pair A B :=\nbegin\n  intros h1 h2 h3,\n  cases h2 with C h2,\n  cases h3 with D h3,\n  rw ← h2,\n  rw ← h3,\n  apply h1,\nend\n\nlemma equal_eq_mem_exists {Ω α:Type*} {P:probability_space Ω} {TM:top_measurable α}\n  [has_measurable_equality TM.to_measurable_space]\n  (X:P →ᵣ TM.to_measurable_space) (a:α):\n  ∃ (C:measurable_setB TM.to_measurable_space), (X ∈ᵣ C) = (X =ᵣ a) :=\nbegin\n  apply exists.intro (measurable_setB_top {a}),\n  rw equal_eq_mem,\n  refl,\nend\n\nlemma or_mem_exists {Ω α:Type*} {P:probability_space Ω} {M:measurable_space α}\n  (X:P →ᵣ M) (A B:event P):\n  (∃ (C:measurable_setB M), (X ∈ᵣ C) = A) →\n  (∃ (D:measurable_setB M), (X ∈ᵣ D) = B) →\n  (∃ (E:measurable_setB M), (X ∈ᵣ E) = (A∨ B)) \n :=\nbegin\n  intros h1 h2,\n  cases h1 with C h1,\n  cases h2 with D h2,\n  subst A,\n  subst B,\n  apply exists.intro (C ∪ D),\n  apply event.eq,\n  ext ω, simp,\nend\n\nlemma and_mem_exists {Ω α:Type*} {P:probability_space Ω} {M:measurable_space α}\n  (X:P →ᵣ M) (A B:event P):\n  (∃ (C:measurable_setB M), (X ∈ᵣ C) = A) →\n  (∃ (D:measurable_setB M), (X ∈ᵣ D) = B) →\n  (∃ (E:measurable_setB M), (X ∈ᵣ E) = (A∧ B)) \n :=\nbegin\n  intros h1 h2,\n  cases h1 with C h1,\n  cases h2 with D h2,\n  subst A,\n  subst B,\n  apply exists.intro (C ∩ D),\n  apply event.eq,\n  ext ω, simp,\nend\n\n\ndef ms_Inter {α β:Type*} {M:measurable_space α} (S:finset β) (X:β → measurable_setB M):\n  measurable_setB M := {\n  val := (⋂ b ∈ S, (X b).val),\n  property := begin\n    apply finset_inter_measurable,\n    intros b h_b,\n    apply (X b).property,\n  end \n}\n\n\nlemma forall_in_mem_exists {Ω α β:Type*} {P:probability_space Ω} {M:measurable_space α}\n  (X:P →ᵣ M) (A:β → event P) (T:finset β):\n  (∀ b ∈ T, (∃ (C:measurable_setB M), (X ∈ᵣ C) = A b)) →\n  (∃ (E:measurable_setB M), (X ∈ᵣ E) = (∀ᵣ b in T, A b)) \n :=\nbegin\n  intros h1,\n  have h2:∀ (b:β), (∃ (C:measurable_setB M), (b∈ T) → (X ∈ᵣ C) = A b),\n  { intros b, cases classical.em (b∈ T) with h2_1 h2_1,\n    { have h2_2 := h1 b h2_1, cases h2_2 with C h2_2,\n      apply exists.intro C, intros h2_3, apply h2_2 },\n    { apply exists.intro measurable_setB_empty, intros contra, apply absurd contra h2_1,\n       } },\n  rw classical.skolem at h2,\n  cases h2 with f h2,\n  apply exists.intro (ms_Inter T f),\n  apply event.eq,\n  ext1 a,\n  split; intros h3; simp [ms_Inter] at h3; simp [ms_Inter, h3];\n  intros b h_b,\n  { rw ← h2, apply h3, apply h_b, apply h_b },\n  { have h4 := h3 b h_b, \n    have h5 := h2 b h_b,\n    rw ← h5 at h4, apply h4 },\nend\n\nlemma joint_random_variable_apply_exists {Ω α β:Type*} [fintype α]  \n  {P:probability_space Ω}  {Mβ:measurable_space β}\n  (X:α → P→ᵣ Mβ) (a:α) (E:event P):\n  (∃ (C: measurable_setB Mβ), X a ∈ᵣ C = E) → \n  (∃ (D : measurable_setB measurable_space.pi),\n    pi.random_variable_combine X ∈ᵣ D = E) := begin\n  classical,\n  intros h1,\n  cases h1 with C h1,\n  let S := (set.preimage (λ (d:α → β), d a) C.val),\n  begin\n    have h_meas_S:measurable_set S,\n    { apply measurable_pi_apply, apply C.property },\n    apply exists.intro (measurable_setB.mk h_meas_S),\n    apply event.eq,\n    rw ← h1,\n    simp [pi.random_variable_combine, pi.measurable_fun, measurable_setB.mk],\n  end\nend\n\nlemma joint_random_variable_mem_exists {Ω α β:Type*} [fintype α]  \n  {P:probability_space Ω}  {Mβ:measurable_space β}\n  (X:α → P→ᵣ Mβ) (T:finset α) (S:α → measurable_setB Mβ):\n  ∃ (D : measurable_setB measurable_space.pi),\n    pi.random_variable_combine X ∈ᵣ D = ∀ᵣ (s : α) in T,X s ∈ᵣ S s := begin\n  apply forall_in_mem_exists,\n  intros b h_b,\n  apply joint_random_variable_apply_exists _ b,\n  apply exists.intro (S b),\n  refl,\nend\n\n\nlemma equiv_cancel_left {α β:Type*} (E:equiv α β) (a:α):E.inv_fun (E a) = a := begin\n  have h1 := E.left_inv,\n  apply h1,\nend\n\n\nlemma pairwise_disjoint_and_right {Ω α:Type*} {P:probability_space Ω}\n  (A:event P) (X:α → event P):\n  pairwise (disjoint on (λ (a:α), (X a).val)) →\n  pairwise (disjoint on (λ (a:α), (A ∧ (X a)).val))  :=\nbegin\n  intros h1,\n  intros i j h_ne,\n  have h2 := h1 i j h_ne,\n  simp [function.on_fun] at h2,\n  rw disjoint_iff at h2,\n  simp at h2,\n  simp [function.on_fun],\n  rw disjoint_iff,\n  simp,\n  rw set.inter_comm (↑A) (↑(X j)),\n  rw ← set.inter_assoc,\n  rw set.inter_assoc (↑A),\n  rw h2,\n  simp,\nend\n\nlemma Pr_sum_partition {Ω α:Type*} {P:probability_space Ω} {TM:top_measurable α}\n  [encodable α]\n  [has_measurable_equality TM.to_measurable_space]\n  (A:event P) (X:P →ᵣ TM.to_measurable_space):\n  Pr[A] = ∑' (a:α), Pr[A ∧ (X =ᵣ a)] :=\nbegin\n  classical,\n  have h1:A = (eany (λ (a:α), A ∧ (X =ᵣ a))),\n  { apply event.eq, ext ω, split; intros h_sub; simp at h_sub;\n    simp [h_sub] },\n  have h2:Pr[A] = Pr[eany (λ (a:α), A ∧ (X =ᵣ a))],\n  { rw ← h1 },\n  rw h2,\n  rw Pr_eany_sum,\n  apply pairwise_disjoint_and_right,\n  apply event_eq_disjoint,\nend\n\n/- Needed for VC dimension proof connecting training and testing. -/\nlemma conditional_independent_event_pair_limit {Ω α:Type*} {P:probability_space Ω} \n  {TM:top_measurable α}\n  [encodable α]\n  [has_measurable_equality TM.to_measurable_space]\n  (A B:event P) (X:P →ᵣ TM.to_measurable_space) (p:nnreal):\n  (∀ (a:α), Pr[A ∧ (X =ᵣ a)] * p ≤ Pr[A ∧ B ∧ (X =ᵣ a)]) →\n  (Pr[A] * p ≤ Pr[A ∧ B])  :=\nbegin\n  classical,\n  intros h1,\n  rw Pr_sum_partition A X,\n  rw Pr_sum_partition (A∧ B) X,\n  rw mul_comm,\n  rw ← summable.tsum_mul_left,\n  apply tsum_le_tsum,\n  { intros a, rw mul_comm,\n    have h3:((A ∧ B)∧ (X =ᵣ a)) = (A∧B∧(X =ᵣ a)),\n    { apply event.eq, ext ω, split; intros h_sub1; simp at h_sub1;\n      simp [h_sub1] },\n    rw h3, apply h1 },\n  apply summable.mul_left,\n  repeat { apply Pr_disjoint_summable,\n    apply pairwise_disjoint_and_right,\n    apply event_eq_disjoint },\nend\n\n\n\nlemma compose_independent_pair_left {α β γ Ω:Type*} {P:probability_space Ω}  {Mα:measurable_space α} \n  {Mβ:measurable_space β} {Mγ:measurable_space γ} \n{X:P →ᵣ Mα} {Y:P →ᵣ Mβ} {Z:Mα →ₘ Mγ}:random_variable_independent_pair X Y →\nrandom_variable_independent_pair (Z ∘r X) Y := begin\n  intros h1,\n  intros A B,\n  rw rv_compose_measurable_setB,\n  apply h1, \nend\n\nlemma compose_independent_pair_right {α β γ Ω:Type*} {P:probability_space Ω}  {Mα:measurable_space α} \n  {Mβ:measurable_space β} {Mγ:measurable_space γ} \n{X:P →ᵣ Mα} {Y:P →ᵣ Mβ} {Z:Mβ →ₘ Mγ}:random_variable_independent_pair X Y →\nrandom_variable_independent_pair X (Z ∘r Y) := begin\n  intros h1,\n  intros A B,\n  rw rv_compose_measurable_setB,\n  apply h1, \nend   \n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/probability_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3166402226127737}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.nontrivial\nimport Mathlib.algebra.group.units_hom\nimport Mathlib.algebra.group.inj_surj\nimport Mathlib.algebra.group_with_zero.defs\nimport Mathlib.PostPort\n\nuniverses u_1 u_3 u u_2 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Groups with an adjoined zero element\n\nThis file describes structures that are not usually studied on their own right in mathematics,\nnamely a special sort of monoid: apart from a distinguished “zero element” they form a group,\nor in other words, they are groups with an adjoined zero element.\n\nExamples are:\n\n* division rings;\n* the value monoid of a multiplicative valuation;\n* in particular, the non-negative real numbers.\n\n## Main definitions\n\nVarious lemmas about `group_with_zero` and `comm_group_with_zero`.\nTo reduce import dependencies, the type-classes themselves are in\n`algebra.group_with_zero.defs`.\n\n## Implementation details\n\nAs is usual in mathlib, we extend the inverse function to the zero element,\nand require `0⁻¹ = 0`.\n\n-/\n\n/-- Pullback a `mul_zero_class` instance along an injective function. -/\nprotected def function.injective.mul_zero_class {M₀ : Type u_1} {M₀' : Type u_3} [mul_zero_class M₀]\n    [Mul M₀'] [HasZero M₀'] (f : M₀' → M₀) (hf : function.injective f) (zero : f 0 = 0)\n    (mul : ∀ (a b : M₀'), f (a * b) = f a * f b) : mul_zero_class M₀' :=\n  mul_zero_class.mk Mul.mul 0 sorry sorry\n\n/-- Pushforward a `mul_zero_class` instance along an surjective function. -/\nprotected def function.surjective.mul_zero_class {M₀ : Type u_1} {M₀' : Type u_3}\n    [mul_zero_class M₀] [Mul M₀'] [HasZero M₀'] (f : M₀ → M₀') (hf : function.surjective f)\n    (zero : f 0 = 0) (mul : ∀ (a b : M₀), f (a * b) = f a * f b) : mul_zero_class M₀' :=\n  mul_zero_class.mk Mul.mul 0 sorry sorry\n\ntheorem mul_eq_zero_of_left {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} (h : a = 0) (b : M₀) :\n    a * b = 0 :=\n  Eq.symm h ▸ zero_mul b\n\ntheorem mul_eq_zero_of_right {M₀ : Type u_1} [mul_zero_class M₀] {b : M₀} (a : M₀) (h : b = 0) :\n    a * b = 0 :=\n  Eq.symm h ▸ mul_zero a\n\ntheorem left_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} :\n    a * b ≠ 0 → a ≠ 0 :=\n  mt fun (h : a = 0) => mul_eq_zero_of_left h b\n\ntheorem right_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} :\n    a * b ≠ 0 → b ≠ 0 :=\n  mt (mul_eq_zero_of_right a)\n\ntheorem ne_zero_and_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀}\n    (h : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 :=\n  { left := left_ne_zero_of_mul h, right := right_ne_zero_of_mul h }\n\ntheorem mul_eq_zero_of_ne_zero_imp_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀}\n    (h : a ≠ 0 → b = 0) : a * b = 0 :=\n  sorry\n\n/-- Pushforward a `no_zero_divisors` instance along an injective function. -/\nprotected theorem function.injective.no_zero_divisors {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀]\n    [HasZero M₀] [Mul M₀'] [HasZero M₀'] [no_zero_divisors M₀'] (f : M₀ → M₀')\n    (hf : function.injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) :\n    no_zero_divisors M₀ :=\n  sorry\n\ntheorem eq_zero_of_mul_self_eq_zero {M₀ : Type u_1} [Mul M₀] [HasZero M₀] [no_zero_divisors M₀]\n    {a : M₀} (h : a * a = 0) : a = 0 :=\n  or.elim (eq_zero_or_eq_zero_of_mul_eq_zero h) id id\n\n/-- If `α` has no zero divisors, then the product of two elements equals zero iff one of them\nequals zero. -/\n@[simp] theorem mul_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀}\n    {b : M₀} : a * b = 0 ↔ a = 0 ∨ b = 0 :=\n  { mp := eq_zero_or_eq_zero_of_mul_eq_zero,\n    mpr :=\n      fun (o : a = 0 ∨ b = 0) =>\n        or.elim o (fun (h : a = 0) => mul_eq_zero_of_left h b) (mul_eq_zero_of_right a) }\n\n/-- If `α` has no zero divisors, then the product of two elements equals zero iff one of them\nequals zero. -/\n@[simp] theorem zero_eq_mul {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀}\n    {b : M₀} : 0 = a * b ↔ a = 0 ∨ b = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 = a * b ↔ a = 0 ∨ b = 0)) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b = 0 ↔ a = 0 ∨ b = 0)) (propext mul_eq_zero)))\n      (iff.refl (a = 0 ∨ b = 0)))\n\n/-- If `α` has no zero divisors, then the product of two elements is nonzero iff both of them\nare nonzero. -/\ntheorem mul_ne_zero_iff {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀}\n    {b : M₀} : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 :=\n  iff.trans (not_congr mul_eq_zero) not_or_distrib\n\ntheorem mul_ne_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀}\n    (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 :=\n  iff.mpr mul_ne_zero_iff { left := ha, right := hb }\n\n/-- If `α` has no zero divisors, then for elements `a, b : α`, `a * b` equals zero iff so is\n`b * a`. -/\ntheorem mul_eq_zero_comm {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀}\n    {b : M₀} : a * b = 0 ↔ b * a = 0 :=\n  iff.trans mul_eq_zero (iff.trans (or_comm (a = 0) (b = 0)) (iff.symm mul_eq_zero))\n\n/-- If `α` has no zero divisors, then for elements `a, b : α`, `a * b` is nonzero iff so is\n`b * a`. -/\ntheorem mul_ne_zero_comm {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀}\n    {b : M₀} : a * b ≠ 0 ↔ b * a ≠ 0 :=\n  not_congr mul_eq_zero_comm\n\ntheorem mul_self_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} :\n    a * a = 0 ↔ a = 0 :=\n  sorry\n\ntheorem zero_eq_mul_self {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} :\n    0 = a * a ↔ a = 0 :=\n  sorry\n\n/-- In a nontrivial monoid with zero, zero and one are different. -/\n@[simp] theorem zero_ne_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] : 0 ≠ 1 := sorry\n\n@[simp] theorem one_ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] : 1 ≠ 0 :=\n  ne.symm zero_ne_one\n\ntheorem ne_zero_of_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀}\n    (h : a = 1) : a ≠ 0 :=\n  trans_rel_right ne h one_ne_zero\n\ntheorem left_ne_zero_of_mul_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀}\n    {b : M₀} (h : a * b = 1) : a ≠ 0 :=\n  left_ne_zero_of_mul (ne_zero_of_eq_one h)\n\ntheorem right_ne_zero_of_mul_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀}\n    {b : M₀} (h : a * b = 1) : b ≠ 0 :=\n  right_ne_zero_of_mul (ne_zero_of_eq_one h)\n\n/-- Pullback a `nontrivial` instance along a function sending `0` to `0` and `1` to `1`. -/\nprotected theorem pullback_nonzero {M₀ : Type u_1} {M₀' : Type u_3} [monoid_with_zero M₀]\n    [nontrivial M₀] [HasZero M₀'] [HasOne M₀'] (f : M₀' → M₀) (zero : f 0 = 0) (one : f 1 = 1) :\n    nontrivial M₀' :=\n  sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀']\n    [Mul M₀'] [HasOne M₀'] [monoid_with_zero M₀] (f : M₀' → M₀) (hf : function.injective f)\n    (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) :\n    monoid_with_zero M₀' :=\n  monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry\n\n/-- Pushforward a `monoid_with_zero` class along a surjective function. -/\nprotected def function.surjective.monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀']\n    [Mul M₀'] [HasOne M₀'] [monoid_with_zero M₀] (f : M₀ → M₀') (hf : function.surjective f)\n    (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) :\n    monoid_with_zero M₀' :=\n  monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.comm_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3}\n    [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [comm_monoid_with_zero M₀] (f : M₀' → M₀)\n    (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : comm_monoid_with_zero M₀' :=\n  comm_monoid_with_zero.mk comm_monoid.mul sorry comm_monoid.one sorry sorry sorry\n    mul_zero_class.zero sorry sorry\n\n/-- Pushforward a `monoid_with_zero` class along a surjective function. -/\nprotected def function.surjective.comm_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3}\n    [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [comm_monoid_with_zero M₀] (f : M₀ → M₀')\n    (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : M₀), f (x * y) = f x * f y) : comm_monoid_with_zero M₀' :=\n  comm_monoid_with_zero.mk comm_monoid.mul sorry comm_monoid.one sorry sorry sorry\n    mul_zero_class.zero sorry sorry\n\nnamespace units\n\n\n/-- An element of the unit group of a nonzero monoid with zero represented as an element\n    of the monoid is nonzero. -/\n@[simp] theorem ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] (u : units M₀) :\n    ↑u ≠ 0 :=\n  left_ne_zero_of_mul_eq_one (mul_inv u)\n\n-- We can't use `mul_eq_zero` + `units.ne_zero` in the next two lemmas because we don't assume\n\n-- `nonzero M₀`.\n\n@[simp] theorem mul_left_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] (u : units M₀) {a : M₀} :\n    a * ↑u = 0 ↔ a = 0 :=\n  sorry\n\n@[simp] theorem mul_right_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] (u : units M₀) {a : M₀} :\n    ↑u * a = 0 ↔ a = 0 :=\n  sorry\n\nend units\n\n\nnamespace is_unit\n\n\ntheorem ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀} (ha : is_unit a) :\n    a ≠ 0 :=\n  (fun (_a : is_unit a) =>\n      Exists.dcases_on _a\n        fun (w : units M₀) (h : ↑w = a) =>\n          idRhs ((fun (_x : M₀) => _x ≠ 0) a) (h ▸ units.ne_zero w))\n    ha\n\ntheorem mul_right_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] {a : M₀} {b : M₀} (ha : is_unit a) :\n    a * b = 0 ↔ b = 0 :=\n  sorry\n\ntheorem mul_left_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] {a : M₀} {b : M₀} (hb : is_unit b) :\n    a * b = 0 ↔ a = 0 :=\n  sorry\n\nend is_unit\n\n\n/-- In a monoid with zero, if zero equals one, then zero is the only element. -/\ntheorem eq_zero_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) (a : M₀) : a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = 0)) (Eq.symm (mul_one a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * 1 = 0)) (Eq.symm h)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 0 = 0)) (mul_zero a))) (Eq.refl 0)))\n\n/-- In a monoid with zero, if zero equals one, then zero is the unique element.\n\nSomewhat arbitrarily, we define the default element to be `0`.\nAll other elements will be provably equal to it, but not necessarily definitionally equal. -/\ndef unique_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) : unique M₀ :=\n  unique.mk { default := 0 } (eq_zero_of_zero_eq_one h)\n\n/-- In a monoid with zero, zero equals one if and only if all elements of that semiring\nare equal. -/\ntheorem subsingleton_iff_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] :\n    0 = 1 ↔ subsingleton M₀ :=\n  { mp := fun (h : 0 = 1) => unique.subsingleton,\n    mpr := fun (h : subsingleton M₀) => subsingleton.elim 0 1 }\n\ntheorem subsingleton_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] :\n    0 = 1 → subsingleton M₀ :=\n  iff.mp subsingleton_iff_zero_eq_one\n\ntheorem eq_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) (a : M₀) (b : M₀) :\n    a = b :=\n  subsingleton.elim a b\n\n@[simp] theorem is_unit_zero_iff {M₀ : Type u_1} [monoid_with_zero M₀] : is_unit 0 ↔ 0 = 1 := sorry\n\n@[simp] theorem not_is_unit_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] :\n    ¬is_unit 0 :=\n  mt (iff.mp is_unit_zero_iff) zero_ne_one\n\n/-- In a monoid with zero, either zero and one are nonequal, or zero is the only element. -/\ntheorem zero_ne_one_or_forall_eq_0 (M₀ : Type u_1) [monoid_with_zero M₀] :\n    0 ≠ 1 ∨ ∀ (a : M₀), a = 0 :=\n  not_or_of_imp eq_zero_of_zero_eq_one\n\nprotected instance cancel_monoid_with_zero.to_no_zero_divisors {M₀ : Type u_1}\n    [cancel_monoid_with_zero M₀] : no_zero_divisors M₀ :=\n  no_zero_divisors.mk\n    fun (a b : M₀) (ab0 : a * b = 0) =>\n      dite (a = 0) (fun (h : a = 0) => Or.inl h)\n        fun (h : ¬a = 0) =>\n          Or.inr\n            (cancel_monoid_with_zero.mul_left_cancel_of_ne_zero h\n              (eq.mpr (id (Eq._oldrec (Eq.refl (a * b = a * 0)) ab0))\n                (eq.mpr (id (Eq._oldrec (Eq.refl (0 = a * 0)) (mul_zero a))) (Eq.refl 0))))\n\ntheorem mul_left_inj' {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀}\n    (hc : c ≠ 0) : a * c = b * c ↔ a = b :=\n  { mp := mul_right_cancel' hc, mpr := fun (h : a = b) => h ▸ rfl }\n\ntheorem mul_right_inj' {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀}\n    (ha : a ≠ 0) : a * b = a * c ↔ b = c :=\n  { mp := mul_left_cancel' ha, mpr := fun (h : b = c) => h ▸ rfl }\n\n@[simp] theorem mul_eq_mul_right_iff {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀}\n    {c : M₀} : a * c = b * c ↔ a = b ∨ c = 0 :=\n  sorry\n\n@[simp] theorem mul_eq_mul_left_iff {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀}\n    {c : M₀} : a * b = a * c ↔ b = c ∨ a = 0 :=\n  sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.cancel_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3}\n    [cancel_monoid_with_zero M₀] [HasZero M₀'] [Mul M₀'] [HasOne M₀'] (f : M₀' → M₀)\n    (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : cancel_monoid_with_zero M₀' :=\n  cancel_monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry\n    sorry sorry\n\n/-- An element of a `cancel_monoid_with_zero` fixed by right multiplication by an element other\nthan one must be zero. -/\ntheorem eq_zero_of_mul_eq_self_right {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀}\n    (h₁ : b ≠ 1) (h₂ : a * b = a) : a = 0 :=\n  classical.by_contradiction\n    fun (ha : ¬a = 0) => h₁ (mul_left_cancel' ha (Eq.symm h₂ ▸ Eq.symm (mul_one a)))\n\n/-- An element of a `cancel_monoid_with_zero` fixed by left multiplication by an element other\nthan one must be zero. -/\ntheorem eq_zero_of_mul_eq_self_left {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀}\n    (h₁ : b ≠ 1) (h₂ : b * a = a) : a = 0 :=\n  classical.by_contradiction\n    fun (ha : ¬a = 0) => h₁ (mul_right_cancel' ha (Eq.symm h₂ ▸ Eq.symm (one_mul a)))\n\ntheorem division_def {G : Type u} [div_inv_monoid G] (a : G) (b : G) : a / b = a * (b⁻¹) :=\n  div_eq_mul_inv\n\n/-- Pullback a `group_with_zero` class along an injective function. -/\nprotected def function.injective.group_with_zero {G₀ : Type u_2} {G₀' : Type u_4}\n    [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (f : G₀' → G₀)\n    (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) :\n    group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry\n    monoid_with_zero.zero sorry sorry has_inv.inv\n    (div_inv_monoid.div._default monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry\n      has_inv.inv)\n    sorry sorry sorry\n\n/-- Pullback a `group_with_zero` class along an injective function. This is a version of\n`function.injective.group_with_zero` that uses a specified `/` instead of the default\n`a / b = a * b⁻¹`. -/\nprotected def function.injective.group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4}\n    [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (f : G₀' → G₀)\n    (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹))\n    (div : ∀ (x y : G₀'), f (x / y) = f x / f y) : group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry\n    monoid_with_zero.zero sorry sorry div_inv_monoid.inv div_inv_monoid.div sorry sorry sorry\n\n/-- Pushforward a `group_with_zero` class along an surjective function. -/\nprotected def function.surjective.group_with_zero {G₀ : Type u_2} {G₀' : Type u_4}\n    [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (h01 : 0 ≠ 1)\n    (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) :\n    group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry\n    monoid_with_zero.zero sorry sorry has_inv.inv\n    (div_inv_monoid.div._default monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry\n      has_inv.inv)\n    sorry sorry sorry\n\n/-- Pushforward a `group_with_zero` class along a surjective function. This is a version of\n`function.surjective.group_with_zero` that uses a specified `/` instead of the default\n`a / b = a * b⁻¹`. -/\nprotected def function.surjective.group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4}\n    [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (h01 : 0 ≠ 1)\n    (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹))\n    (div : ∀ (x y : G₀), f (x / y) = f x / f y) : group_with_zero G₀' :=\n  group_with_zero.mk div_inv_monoid.mul sorry div_inv_monoid.one sorry sorry group_with_zero.zero\n    sorry sorry div_inv_monoid.inv div_inv_monoid.div sorry sorry sorry\n\n@[simp] theorem mul_inv_cancel_right' {G₀ : Type u_2} [group_with_zero G₀] {b : G₀} (h : b ≠ 0)\n    (a : G₀) : a * b * (b⁻¹) = a :=\n  sorry\n\n@[simp] theorem mul_inv_cancel_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0)\n    (b : G₀) : a * (a⁻¹ * b) = b :=\n  sorry\n\ntheorem inv_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a⁻¹ ≠ 0 := sorry\n\n@[simp] theorem inv_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :\n    a⁻¹ * a = 1 :=\n  sorry\n\n@[simp] theorem inv_mul_cancel_right' {G₀ : Type u_2} [group_with_zero G₀] {b : G₀} (h : b ≠ 0)\n    (a : G₀) : a * (b⁻¹) * b = a :=\n  sorry\n\n@[simp] theorem inv_mul_cancel_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0)\n    (b : G₀) : a⁻¹ * (a * b) = b :=\n  sorry\n\n@[simp] theorem inv_one {G₀ : Type u_2} [group_with_zero G₀] : 1⁻¹ = 1 := sorry\n\n@[simp] theorem inv_inv' {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a⁻¹⁻¹ = a := sorry\n\n/-- Multiplying `a` by itself and then by its inverse results in `a`\n(whether or not `a` is zero). -/\n@[simp] theorem mul_self_mul_inv {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) :\n    a * a * (a⁻¹) = a :=\n  sorry\n\n/-- Multiplying `a` by its inverse and then by itself results in `a`\n(whether or not `a` is zero). -/\n@[simp] theorem mul_inv_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) :\n    a * (a⁻¹) * a = a :=\n  sorry\n\n/-- Multiplying `a⁻¹` by `a` twice results in `a` (whether or not `a`\nis zero). -/\n@[simp] theorem inv_mul_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a⁻¹ * a * a = a :=\n  sorry\n\n/-- Multiplying `a` by itself and then dividing by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem mul_self_div_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a * a / a = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * a / a = a)) (div_eq_mul_inv (a * a) a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * a * (a⁻¹) = a)) (mul_self_mul_inv a))) (Eq.refl a))\n\n/-- Dividing `a` by itself and then multiplying by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem div_self_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / a * a = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / a * a = a)) (div_eq_mul_inv a a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a⁻¹) * a = a)) (mul_inv_mul_self a))) (Eq.refl a))\n\ntheorem inv_involutive' {G₀ : Type u_2} [group_with_zero G₀] : function.involutive has_inv.inv :=\n  inv_inv'\n\ntheorem eq_inv_of_mul_right_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : a * b = 1) : b = (a⁻¹) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (b = (a⁻¹)))\n        (Eq.symm (inv_mul_cancel_left' (left_ne_zero_of_mul_eq_one h) b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ * (a * b) = (a⁻¹))) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ * 1 = (a⁻¹))) (mul_one (a⁻¹)))) (Eq.refl (a⁻¹))))\n\ntheorem eq_inv_of_mul_left_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : a * b = 1) : a = (b⁻¹) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (a = (b⁻¹)))\n        (Eq.symm (mul_inv_cancel_right' (right_ne_zero_of_mul_eq_one h) a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b * (b⁻¹) = (b⁻¹))) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 * (b⁻¹) = (b⁻¹))) (one_mul (b⁻¹)))) (Eq.refl (b⁻¹))))\n\ntheorem inv_injective' {G₀ : Type u_2} [group_with_zero G₀] : function.injective has_inv.inv :=\n  function.involutive.injective inv_involutive'\n\n@[simp] theorem inv_inj' {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} {h : G₀} :\n    g⁻¹ = (h⁻¹) ↔ g = h :=\n  function.injective.eq_iff inv_injective'\n\ntheorem inv_eq_iff {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} {h : G₀} : g⁻¹ = h ↔ h⁻¹ = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹ = h ↔ h⁻¹ = g)) (Eq.symm (propext inv_inj'))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹⁻¹ = (h⁻¹) ↔ h⁻¹ = g)) (propext eq_comm)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (h⁻¹ = (g⁻¹⁻¹) ↔ h⁻¹ = g)) (inv_inv' g)))\n        (iff.refl (h⁻¹ = g))))\n\n@[simp] theorem inv_eq_one' {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} : g⁻¹ = 1 ↔ g = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹ = 1 ↔ g = 1)) (propext inv_eq_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1⁻¹ = g ↔ g = 1)) inv_one))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 = g ↔ g = 1)) (propext eq_comm))) (iff.refl (g = 1))))\n\nnamespace units\n\n\n/-- Embed a non-zero element of a `group_with_zero` into the unit group.\n  By combining this function with the operations on units,\n  or the `/ₚ` operation, it is possible to write a division\n  as a partial function with three arguments. -/\ndef mk0 {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (ha : a ≠ 0) : units G₀ :=\n  mk a (a⁻¹) (mul_inv_cancel ha) (inv_mul_cancel ha)\n\n@[simp] theorem coe_mk0 {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :\n    ↑(mk0 a h) = a :=\n  rfl\n\n@[simp] theorem mk0_coe {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) (h : ↑u ≠ 0) :\n    mk0 (↑u) h = u :=\n  ext rfl\n\n@[simp] theorem coe_inv' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑(u⁻¹) = (↑u⁻¹) :=\n  eq_inv_of_mul_left_eq_one (inv_mul u)\n\n@[simp] theorem mul_inv' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑u * (↑u⁻¹) = 1 :=\n  mul_inv_cancel (ne_zero u)\n\n@[simp] theorem inv_mul' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑u⁻¹ * ↑u = 1 :=\n  inv_mul_cancel (ne_zero u)\n\n@[simp] theorem mk0_inj {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0)\n    (hb : b ≠ 0) : mk0 a ha = mk0 b hb ↔ a = b :=\n  { mp :=\n      fun (h : mk0 a ha = mk0 b hb) => mk.inj_arrow h fun (h_1 : a = b) (h_2 : a⁻¹ = (b⁻¹)) => h_1,\n    mpr := fun (h : a = b) => ext h }\n\n@[simp] theorem exists_iff_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} :\n    (∃ (u : units G₀), ↑u = x) ↔ x ≠ 0 :=\n  sorry\n\nend units\n\n\ntheorem is_unit.mk0 {G₀ : Type u_2} [group_with_zero G₀] (x : G₀) (hx : x ≠ 0) : is_unit x :=\n  is_unit_unit (units.mk0 x hx)\n\ntheorem is_unit_iff_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} : is_unit x ↔ x ≠ 0 :=\n  units.exists_iff_ne_zero\n\nprotected instance group_with_zero.no_zero_divisors {G₀ : Type u_2} [group_with_zero G₀] :\n    no_zero_divisors G₀ :=\n  no_zero_divisors.mk\n    fun (a b : G₀) (h : a * b = 0) =>\n      imp_of_not_imp_not (a * b = 0) (a = 0 ∨ b = 0)\n        (eq.mpr (id (imp_congr_eq (push_neg.not_or_eq (a = 0) (b = 0)) (Eq.refl (¬a * b = 0))))\n          (eq.mpr\n            (id\n              (imp_congr_eq\n                ((fun (a a_1 : Prop) (e_1 : a = a_1) (b b_1 : Prop) (e_2 : b = b_1) =>\n                    congr (congr_arg And e_1) e_2)\n                  (¬a = 0) (a ≠ 0) (propext (push_neg.not_eq a 0)) (¬b = 0) (b ≠ 0)\n                  (propext (push_neg.not_eq b 0)))\n                (propext (push_neg.not_eq (a * b) 0))))\n            fun (h : a ≠ 0 ∧ b ≠ 0) =>\n              units.ne_zero (units.mk0 a (and.left h) * units.mk0 b (and.right h))))\n        h\n\nprotected instance group_with_zero.cancel_monoid_with_zero {G₀ : Type u_2} [group_with_zero G₀] :\n    cancel_monoid_with_zero G₀ :=\n  cancel_monoid_with_zero.mk group_with_zero.mul group_with_zero.mul_assoc group_with_zero.one\n    group_with_zero.one_mul group_with_zero.mul_one group_with_zero.zero group_with_zero.zero_mul\n    group_with_zero.mul_zero sorry sorry\n\ntheorem mul_inv_rev' {G₀ : Type u_2} [group_with_zero G₀] (x : G₀) (y : G₀) :\n    x * y⁻¹ = y⁻¹ * (x⁻¹) :=\n  sorry\n\n@[simp] theorem div_self {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a / a = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / a = 1)) (div_eq_mul_inv a a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a⁻¹) = 1)) (mul_inv_cancel h))) (Eq.refl 1))\n\n@[simp] theorem div_one {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / 1 = a := sorry\n\n@[simp] theorem zero_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : 0 / a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 / a = 0)) (div_eq_mul_inv 0 a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (0 * (a⁻¹) = 0)) (zero_mul (a⁻¹)))) (Eq.refl 0))\n\n@[simp] theorem div_zero {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / 0 = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / 0 = 0)) (div_eq_mul_inv a 0)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (0⁻¹) = 0)) inv_zero))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 0 = 0)) (mul_zero a))) (Eq.refl 0)))\n\n@[simp] theorem div_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (h : b ≠ 0) :\n    a / b * b = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_eq_mul_inv a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b⁻¹) * b = a)) (inv_mul_cancel_right' h a))) (Eq.refl a))\n\ntheorem div_mul_cancel_of_imp {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : b = 0 → a = 0) : a / b * b = a :=\n  sorry\n\n@[simp] theorem mul_div_cancel {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (h : b ≠ 0) :\n    a * b / b = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b / b = a)) (div_eq_mul_inv (a * b) b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b * (b⁻¹) = a)) (mul_inv_cancel_right' h a))) (Eq.refl a))\n\ntheorem mul_div_cancel_of_imp {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : b = 0 → a = 0) : a * b / b = a :=\n  sorry\n\n@[simp] theorem div_self_mul_self' {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) :\n    a / (a * a) = (a⁻¹) :=\n  sorry\n\ntheorem div_eq_mul_one_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) :\n    a / b = a * (1 / b) :=\n  sorry\n\ntheorem mul_one_div_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :\n    a * (1 / a) = 1 :=\n  sorry\n\ntheorem one_div_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) :\n    1 / a * a = 1 :=\n  sorry\n\ntheorem one_div_one {G₀ : Type u_2} [group_with_zero G₀] : 1 / 1 = 1 :=\n  div_self (ne.symm zero_ne_one)\n\ntheorem one_div_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : 1 / a ≠ 0 :=\n  sorry\n\ntheorem eq_one_div_of_mul_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : a * b = 1) : b = 1 / a :=\n  sorry\n\ntheorem eq_one_div_of_mul_eq_one_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : b * a = 1) : b = 1 / a :=\n  sorry\n\n@[simp] theorem one_div_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) :\n    1 / (a / b) = b / a :=\n  sorry\n\ntheorem one_div_one_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : 1 / (1 / a) = a := sorry\n\ntheorem eq_of_one_div_eq_one_div {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : 1 / a = 1 / b) : a = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = b)) (Eq.symm (one_div_one_div a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (1 / a) = b)) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (1 / b) = b)) (one_div_one_div b))) (Eq.refl b)))\n\n@[simp] theorem inv_eq_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} : a⁻¹ = 0 ↔ a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ = 0 ↔ a = 0)) (propext inv_eq_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (0⁻¹ = a ↔ a = 0)) inv_zero))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (0 = a ↔ a = 0)) (propext eq_comm))) (iff.refl (a = 0))))\n\n@[simp] theorem zero_eq_inv {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} : 0 = (a⁻¹) ↔ 0 = a :=\n  iff.trans eq_comm (iff.trans inv_eq_zero eq_comm)\n\ntheorem one_div_mul_one_div_rev {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) :\n    1 / a * (1 / b) = 1 / (b * a) :=\n  sorry\n\ntheorem divp_eq_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (u : units G₀) :\n    a /ₚ u = a / ↑u :=\n  sorry\n\n@[simp] theorem divp_mk0 {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) :\n    a /ₚ units.mk0 b hb = a / b :=\n  divp_eq_div a (units.mk0 b hb)\n\ntheorem inv_div {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a / b⁻¹ = b / a := sorry\n\ntheorem inv_div_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a⁻¹ / b = (b * a⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ / b = (b * a⁻¹))) (mul_inv_rev' b a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ / b = a⁻¹ * (b⁻¹))) (div_eq_mul_inv (a⁻¹) b)))\n      (Eq.refl (a⁻¹ * (b⁻¹))))\n\ntheorem div_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0)\n    (hb : b ≠ 0) : a / b ≠ 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b ≠ 0)) (div_eq_mul_inv a b)))\n    (mul_ne_zero ha (inv_ne_zero hb))\n\n@[simp] theorem div_eq_zero_iff {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} :\n    a / b = 0 ↔ a = 0 ∨ b = 0 :=\n  sorry\n\ntheorem div_ne_zero_iff {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} :\n    a / b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 :=\n  iff.trans (not_congr div_eq_zero_iff) not_or_distrib\n\ntheorem div_left_inj' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) :\n    a / c = b / c ↔ a = b :=\n  sorry\n\ntheorem div_eq_iff_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hb : b ≠ 0) : a / b = c ↔ c * b = a :=\n  sorry\n\ntheorem eq_div_iff_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hc : c ≠ 0) : a = b / c ↔ a * c = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = b / c ↔ a * c = b)) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b / c = a ↔ a * c = b)) (propext (div_eq_iff_mul_eq hc))))\n      (iff.refl (a * c = b)))\n\ntheorem div_eq_of_eq_mul {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} (hx : x ≠ 0) {y : G₀}\n    {z : G₀} (h : y = z * x) : y / x = z :=\n  iff.mpr (div_eq_iff_mul_eq hx) (Eq.symm h)\n\ntheorem eq_div_of_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} (hx : x ≠ 0) {y : G₀}\n    {z : G₀} (h : z * x = y) : z = y / x :=\n  Eq.symm (div_eq_of_eq_mul hx (Eq.symm h))\n\ntheorem eq_of_div_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : a / b = 1) :\n    a = b :=\n  sorry\n\ntheorem div_eq_one_iff_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :\n    a / b = 1 ↔ a = b :=\n  { mp := eq_of_div_eq_one, mpr := fun (h : a = b) => Eq.symm h ▸ div_self hb }\n\ntheorem div_mul_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) :\n    b / (a * b) = 1 / a :=\n  sorry\n\ntheorem mul_div_mul_right {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) {c : G₀}\n    (hc : c ≠ 0) : a * c / (b * c) = a / b :=\n  sorry\n\ntheorem mul_mul_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) :\n    a = a * b * (1 / b) :=\n  sorry\n\nprotected instance comm_group_with_zero.comm_cancel_monoid_with_zero {G₀ : Type u_2}\n    [comm_group_with_zero G₀] : comm_cancel_monoid_with_zero G₀ :=\n  comm_cancel_monoid_with_zero.mk cancel_monoid_with_zero.mul sorry cancel_monoid_with_zero.one\n    sorry sorry sorry cancel_monoid_with_zero.zero sorry sorry sorry sorry\n\n/-- Pullback a `comm_group_with_zero` class along an injective function. -/\nprotected def function.injective.comm_group_with_zero {G₀ : Type u_2} {G₀' : Type u_4}\n    [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (f : G₀' → G₀)\n    (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) :\n    comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry\n    group_with_zero.zero sorry sorry group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pullback a `comm_group_with_zero` class along an injective function. -/\nprotected def function.injective.comm_group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4}\n    [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀']\n    (f : G₀' → G₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹))\n    (div : ∀ (x y : G₀'), f (x / y) = f x / f y) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry\n    group_with_zero.zero sorry sorry group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pushforward a `comm_group_with_zero` class along an surjective function. -/\nprotected def function.surjective.comm_group_with_zero {G₀ : Type u_2} {G₀' : Type u_4}\n    [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (h01 : 0 ≠ 1)\n    (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) :\n    comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry\n    group_with_zero.zero sorry sorry group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pushforward a `comm_group_with_zero` class along a surjective function. -/\nprotected def function.surjective.comm_group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4}\n    [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀']\n    (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1)\n    (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹))\n    (div : ∀ (x y : G₀), f (x / y) = f x / f y) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry\n    group_with_zero.zero sorry sorry group_with_zero.inv group_with_zero.div sorry sorry sorry\n\ntheorem mul_inv' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} :\n    a * b⁻¹ = a⁻¹ * (b⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b⁻¹ = a⁻¹ * (b⁻¹))) (mul_inv_rev' a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b⁻¹ * (a⁻¹) = a⁻¹ * (b⁻¹))) (mul_comm (b⁻¹) (a⁻¹))))\n      (Eq.refl (a⁻¹ * (b⁻¹))))\n\ntheorem one_div_mul_one_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) :\n    1 / a * (1 / b) = 1 / (a * b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 / a * (1 / b) = 1 / (a * b))) (one_div_mul_one_div_rev a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (b * a) = 1 / (a * b))) (mul_comm b a)))\n      (Eq.refl (1 / (a * b))))\n\ntheorem div_mul_right {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (ha : a ≠ 0) :\n    a / (a * b) = 1 / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a * b) = 1 / b)) (mul_comm a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * a) = 1 / b)) (div_mul_left ha))) (Eq.refl (1 / b)))\n\ntheorem mul_div_cancel_left_of_imp {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : a = 0 → b = 0) : a * b / a = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b / a = b)) (mul_comm a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b * a / a = b)) (mul_div_cancel_of_imp h))) (Eq.refl b))\n\ntheorem mul_div_cancel_left {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀)\n    (ha : a ≠ 0) : a * b / a = b :=\n  mul_div_cancel_left_of_imp fun (h : a = 0) => false.elim (ha h)\n\ntheorem mul_div_cancel_of_imp' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀}\n    (h : b = 0 → a = 0) : b * (a / b) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (b * (a / b) = a)) (mul_comm b (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_mul_cancel_of_imp h))) (Eq.refl a))\n\ntheorem mul_div_cancel' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) :\n    b * (a / b) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (b * (a / b) = a)) (mul_comm b (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_mul_cancel a hb))) (Eq.refl a))\n\ntheorem div_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) (d : G₀) :\n    a / b * (c / d) = a * c / (b * d) :=\n  sorry\n\ntheorem mul_div_mul_left {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) {c : G₀}\n    (hc : c ≠ 0) : c * a / (c * b) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (c * a / (c * b) = a / b)) (mul_comm c a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / (c * b) = a / b)) (mul_comm c b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / (b * c) = a / b)) (mul_div_mul_right a b hc)))\n        (Eq.refl (a / b))))\n\ntheorem div_mul_eq_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    b / c * a = b * a / c :=\n  sorry\n\ntheorem div_mul_eq_mul_div_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀)\n    (c : G₀) : b / c * a = b * (a / c) :=\n  sorry\n\ntheorem mul_eq_mul_of_div_eq_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀}\n    (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) (h : a / b = c / d) : a * d = c * b :=\n  sorry\n\ntheorem div_div_eq_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a / (b / c) = a * c / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (b / c) = a * c / b)) (div_eq_mul_one_div a (b / c))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (1 / (b / c)) = a * c / b)) (one_div_div b c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * (c / b) = a * c / b)) (Eq.symm mul_div_assoc)))\n        (Eq.refl (a * c / b))))\n\ntheorem div_div_eq_div_mul {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a / b / c = a / (b * c) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b / c = a / (b * c))) (div_eq_mul_one_div (a / b) c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * (1 / c) = a / (b * c))) (div_mul_div a b 1 c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 1 / (b * c) = a / (b * c))) (mul_one a)))\n        (Eq.refl (a / (b * c)))))\n\ntheorem div_div_div_div_eq {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀} {c : G₀}\n    {d : G₀} : a / b / (c / d) = a * d / (b * c) :=\n  sorry\n\ntheorem div_mul_eq_div_mul_one_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀)\n    (c : G₀) : a / (b * c) = a / b * (1 / c) :=\n  eq.mpr\n    (id (Eq._oldrec (Eq.refl (a / (b * c) = a / b * (1 / c))) (Eq.symm (div_div_eq_div_mul a b c))))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (a / b / c = a / b * (1 / c)))\n          (Eq.symm (div_eq_mul_one_div (a / b) c))))\n      (Eq.refl (a / b / c)))\n\n/-- Dividing `a` by the result of dividing `a` by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem div_div_self {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) : a / (a / a) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a / a) = a)) (div_div_eq_mul_div a a a)))\n    (mul_self_div_self a)\n\ntheorem ne_zero_of_one_div_ne_zero {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀}\n    (h : 1 / a ≠ 0) : a ≠ 0 :=\n  fun (ha : a = 0) =>\n    absurd (Eq.refl 0)\n      (eq.mp (Eq._oldrec (Eq.refl (1 / 0 ≠ 0)) (div_zero 1))\n        (eq.mp (Eq._oldrec (Eq.refl (1 / a ≠ 0)) ha) h))\n\ntheorem eq_zero_of_one_div_eq_zero {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀}\n    (h : 1 / a = 0) : a = 0 :=\n  classical.by_cases (fun (ha : a = 0) => ha) fun (ha : ¬a = 0) => false.elim (one_div_ne_zero ha h)\n\ntheorem div_helper {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (h : a ≠ 0) :\n    1 / (a * b) * a = 1 / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 / (a * b) * a = 1 / b)) (div_mul_eq_mul_div a 1 (a * b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 * a / (a * b) = 1 / b)) (one_mul a)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a / (a * b) = 1 / b)) (div_mul_right b h)))\n        (Eq.refl (1 / b))))\n\ntheorem div_eq_inv_mul {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} :\n    a / b = b⁻¹ * a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b = b⁻¹ * a)) (div_eq_mul_inv a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b⁻¹) = b⁻¹ * a)) (mul_comm a (b⁻¹))))\n      (Eq.refl (b⁻¹ * a)))\n\ntheorem mul_div_right_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a * b / c = a / c * b :=\n  sorry\n\ntheorem mul_comm_div' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a / b * c = a * (c / b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = a * (c / b))) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = a * c / b)) (mul_div_right_comm a c b)))\n      (Eq.refl (a / b * c)))\n\ntheorem div_mul_comm' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a / b * c = c / b * a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = c / b * a)) (div_mul_eq_mul_div c a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / b = c / b * a)) (mul_comm a c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (c * a / b = c / b * a)) (mul_div_right_comm c a b)))\n        (Eq.refl (c / b * a))))\n\ntheorem mul_div_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) :\n    a * (b / c) = b * (a / c) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * (b / c) = b * (a / c))) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b / c = b * (a / c))) (mul_comm a b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (b * a / c = b * (a / c))) mul_div_assoc))\n        (Eq.refl (b * (a / c)))))\n\ntheorem div_right_comm {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀} (a : G₀) :\n    a / b / c = a / c / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b / c = a / c / b)) (div_div_eq_div_mul a b c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * c) = a / c / b)) (div_div_eq_div_mul a c b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * c) = a / (c * b))) (mul_comm b c)))\n        (Eq.refl (a / (c * b)))))\n\ntheorem div_div_div_cancel_right {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀}\n    (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / c / (b / c) = a / b)) (div_div_eq_mul_div (a / c) b c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / c * c / b = a / b)) (div_mul_cancel a hc)))\n      (Eq.refl (a / b)))\n\ntheorem div_mul_div_cancel {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀} (a : G₀)\n    (hc : c ≠ 0) : a / c * (c / b) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / c * (c / b) = a / b)) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / c * c / b = a / b)) (div_mul_cancel a hc)))\n      (Eq.refl (a / b)))\n\ntheorem div_eq_div_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} {d : G₀}\n    (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b :=\n  sorry\n\ntheorem div_eq_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hb : b ≠ 0) : a / b = c ↔ a = c * b :=\n  iff.trans (div_eq_iff_mul_eq hb) eq_comm\n\ntheorem eq_div_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hb : b ≠ 0) : c = a / b ↔ c * b = a :=\n  eq_div_iff_mul_eq hb\n\ntheorem div_div_cancel' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) :\n    a / (a / b) = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a / b) = b)) (div_eq_mul_inv a (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a / b⁻¹) = b)) inv_div))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b / a) = b)) (mul_div_cancel' b ha))) (Eq.refl b)))\n\nnamespace semiconj_by\n\n\n@[simp] theorem zero_right {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : semiconj_by a 0 0 := sorry\n\n@[simp] theorem zero_left {G₀ : Type u_2} [mul_zero_class G₀] (x : G₀) (y : G₀) :\n    semiconj_by 0 x y :=\n  sorry\n\n@[simp] theorem inv_symm_left_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} :\n    semiconj_by (a⁻¹) x y ↔ semiconj_by a y x :=\n  sorry\n\ntheorem inv_symm_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀}\n    (h : semiconj_by a x y) : semiconj_by (a⁻¹) y x :=\n  iff.mpr inv_symm_left_iff' h\n\ntheorem inv_right' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀}\n    (h : semiconj_by a x y) : semiconj_by a (x⁻¹) (y⁻¹) :=\n  sorry\n\n@[simp] theorem inv_right_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} :\n    semiconj_by a (x⁻¹) (y⁻¹) ↔ semiconj_by a x y :=\n  { mp := fun (h : semiconj_by a (x⁻¹) (y⁻¹)) => inv_inv' x ▸ inv_inv' y ▸ inv_right' h,\n    mpr := inv_right' }\n\ntheorem div_right {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} {x' : G₀}\n    {y' : G₀} (h : semiconj_by a x y) (h' : semiconj_by a x' y') :\n    semiconj_by a (x / x') (y / y') :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (semiconj_by a (x / x') (y / y'))) (div_eq_mul_inv x x')))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (semiconj_by a (x * (x'⁻¹)) (y / y'))) (div_eq_mul_inv y y')))\n      (mul_right h (inv_right' h')))\n\nend semiconj_by\n\n\nnamespace commute\n\n\n@[simp] theorem zero_right {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : commute a 0 :=\n  semiconj_by.zero_right a\n\n@[simp] theorem zero_left {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : commute 0 a :=\n  semiconj_by.zero_left a a\n\n@[simp] theorem inv_left_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} :\n    commute (a⁻¹) b ↔ commute a b :=\n  semiconj_by.inv_symm_left_iff'\n\ntheorem inv_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) :\n    commute (a⁻¹) b :=\n  iff.mpr inv_left_iff' h\n\n@[simp] theorem inv_right_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} :\n    commute a (b⁻¹) ↔ commute a b :=\n  semiconj_by.inv_right_iff'\n\ntheorem inv_right' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) :\n    commute a (b⁻¹) :=\n  iff.mpr inv_right_iff' h\n\ntheorem inv_inv' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) :\n    commute (a⁻¹) (b⁻¹) :=\n  inv_right' (inv_left' h)\n\n@[simp] theorem div_right {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hab : commute a b) (hac : commute a c) : commute a (b / c) :=\n  semiconj_by.div_right hab hac\n\n@[simp] theorem div_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀}\n    (hac : commute a c) (hbc : commute b c) : commute (a / b) c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (commute (a / b) c)) (div_eq_mul_inv a b)))\n    (mul_left hac (inv_left' hbc))\n\nend commute\n\n\nnamespace monoid_with_zero_hom\n\n\ntheorem map_ne_zero {M₀ : Type u_1} {G₀ : Type u_2} [group_with_zero G₀] [monoid_with_zero M₀]\n    [nontrivial M₀] (f : monoid_with_zero_hom G₀ M₀) {a : G₀} : coe_fn f a ≠ 0 ↔ a ≠ 0 :=\n  { mp := fun (hfa : coe_fn f a ≠ 0) (ha : a = 0) => hfa (Eq.symm ha ▸ map_zero f),\n    mpr := fun (ha : a ≠ 0) => is_unit.ne_zero (is_unit.map (to_monoid_hom f) (is_unit.mk0 a ha)) }\n\n@[simp] theorem map_eq_zero {M₀ : Type u_1} {G₀ : Type u_2} [group_with_zero G₀]\n    [monoid_with_zero M₀] [nontrivial M₀] (f : monoid_with_zero_hom G₀ M₀) {a : G₀} :\n    coe_fn f a = 0 ↔ a = 0 :=\n  iff.mp not_iff_not (map_ne_zero f)\n\n/-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/\n@[simp] theorem map_inv' {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [group_with_zero G₀']\n    (f : monoid_with_zero_hom G₀ G₀') (a : G₀) : coe_fn f (a⁻¹) = (coe_fn f a⁻¹) :=\n  sorry\n\n@[simp] theorem map_div {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [group_with_zero G₀']\n    (f : monoid_with_zero_hom G₀ G₀') (a : G₀) (b : G₀) :\n    coe_fn f (a / b) = coe_fn f a / coe_fn f b :=\n  sorry\n\nend monoid_with_zero_hom\n\n\n@[simp] theorem monoid_hom.map_units_inv {M : Type u_1} {G₀ : Type u_2} [monoid M]\n    [group_with_zero G₀] (f : M →* G₀) (u : units M) : coe_fn f ↑(u⁻¹) = (coe_fn f ↑u⁻¹) :=\n  sorry\n\n/-- Constructs a `group_with_zero` structure on a `monoid_with_zero`\n  consisting only of units and 0. -/\ndef group_with_zero_of_is_unit_or_eq_zero {M : Type u_5} [nontrivial M] [hM : monoid_with_zero M]\n    (h : ∀ (a : M), is_unit a ∨ a = 0) : group_with_zero M :=\n  group_with_zero.mk monoid_with_zero.mul monoid_with_zero.mul_assoc monoid_with_zero.one\n    monoid_with_zero.one_mul monoid_with_zero.mul_one monoid_with_zero.zero\n    monoid_with_zero.zero_mul monoid_with_zero.mul_zero\n    (fun (a : M) =>\n      dite (a = 0) (fun (h0 : a = 0) => 0) fun (h0 : ¬a = 0) => ↑(is_unit.unit sorry⁻¹))\n    (div_inv_monoid.div._default monoid_with_zero.mul monoid_with_zero.mul_assoc\n      monoid_with_zero.one monoid_with_zero.one_mul monoid_with_zero.mul_one\n      fun (a : M) =>\n        dite (a = 0) (fun (h0 : a = 0) => 0) fun (h0 : ¬a = 0) => ↑(is_unit.unit sorry⁻¹))\n    nontrivial.exists_pair_ne sorry sorry\n\n/-- Constructs a `comm_group_with_zero` structure on a `comm_monoid_with_zero`\n  consisting only of units and 0. -/\ndef comm_group_with_zero_of_is_unit_or_eq_zero {M : Type u_5} [nontrivial M]\n    [hM : comm_monoid_with_zero M] (h : ∀ (a : M), is_unit a ∨ a = 0) : comm_group_with_zero M :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry\n    comm_monoid_with_zero.mul_comm group_with_zero.zero sorry sorry group_with_zero.inv\n    group_with_zero.div sorry sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/group_with_zero/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165382362518, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3164892478947871}}
{"text": "\ninductive SimpleInd\n  | A\n  | B\nderiving Ord\n\nmutual \ninductive Foo\n  | A : Int → (3 = 3) → String → Foo\n  | B : Bar → Foo\nderiving Ord\ninductive Bar\n  | C\n  | D : Foo → Bar\nderiving Ord\nend\n\ninductive ManyConstructors | A | B | C | D | E | F | G | H | I | J | K | L \n  | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z\nderiving Ord\n\nstructure Person :=\n  firstName : String\n  lastName : String\n  age : Nat\nderiving Ord\n\nexample : compare { firstName := \"A\", lastName := \"B\", age := 10 : Person } ⟨\"B\", \"A\", 9⟩ = Ordering.lt := rfl\n\nstructure Company :=\n  name : String\n  ceo : Person\n  numberOfEmployees : Nat\nderiving Ord\n\nstructure Fixed (α : Type u) where\n  val : Int\nderiving Ord\n\ninductive Fixed' : Type → Type 1 where\n  | mk : Int → Fixed' α\nderiving Ord\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/Ord.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6477982043529716, "lm_q2_score": 0.4882833952958346, "lm_q1q2_score": 0.3163091066880139}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.pfun\nimport Mathlib.order.preorder_hom\nimport Mathlib.tactic.wlog\nimport Mathlib.tactic.monotonicity.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_5 u_6 u_3 u v l u_4 \n\nnamespace Mathlib\n\n/-!\n# Omega Complete Partial Orders\n\nAn omega-complete partial order is a partial order with a supremum\noperation on increasing sequences indexed by natural numbers (which we\ncall `ωSup`). In this sense, it is strictly weaker than join complete\nsemi-lattices as only ω-sized totally ordered sets have a supremum.\n\nThe concept of an omega-complete partial order (ωCPO) is useful for the\nformalization of the semantics of programming languages. Its notion of\nsupremum helps define the meaning of recursive procedures.\n\n## Main definitions\n\n * class `omega_complete_partial_order`\n * `ite`, `map`, `bind`, `seq` as continuous morphisms\n\n## Instances of `omega_complete_partial_order`\n\n * `roption`\n * every `complete_lattice`\n * pi-types\n * product types\n * `monotone_hom`\n * `continuous_hom` (with notation →𝒄)\n   * an instance of `omega_complete_partial_order (α →𝒄 β)`\n * `continuous_hom.of_fun`\n * `continuous_hom.of_mono`\n * continuous functions:\n   * `id`\n   * `ite`\n   * `const`\n   * `roption.bind`\n   * `roption.map`\n   * `roption.seq`\n\n## References\n\n * [G. Markowsky, *Chain-complete posets and directed sets with applications*, https://doi.org/10.1007/BF02485815][markowsky]\n * [J. M. Cadiou and Zohar Manna, *Recursive definitions of partial functions and their computations.*, https://doi.org/10.1145/942580.807072][cadiou]\n * [Carl A. Gunter, *Semantics of Programming Languages: Structures and Techniques*, ISBN: 0262570955][gunter]\n-/\n\nnamespace preorder_hom\n\n\n/-- The constant function, as a monotone function. -/\n@[simp] theorem const_to_fun (α : Type u_1) {β : Type u_2} [preorder α] [preorder β] (f : β) :\n    ∀ (ᾰ : α), coe_fn (const α f) ᾰ = function.const α f ᾰ :=\n  fun (ᾰ : α) => Eq.refl (coe_fn (const α f) ᾰ)\n\n/-- The diagonal function, as a monotone function. -/\ndef prod.diag {α : Type u_1} [preorder α] : α →ₘ α × α := mk (fun (x : α) => (x, x)) sorry\n\n/-- The `prod.map` function, as a monotone function. -/\n@[simp] theorem prod.map_to_fun {α : Type u_1} {β : Type u_2} [preorder α] [preorder β]\n    {α' : Type u_5} {β' : Type u_6} [preorder α'] [preorder β'] (f : α →ₘ β) (f' : α' →ₘ β')\n    (x : α × α') : coe_fn (prod.map f f') x = prod.map (⇑f) (⇑f') x :=\n  Eq.refl (coe_fn (prod.map f f') x)\n\n/-- The `prod.fst` projection, as a monotone function. -/\ndef prod.fst {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] : α × β →ₘ α :=\n  mk prod.fst sorry\n\n/-- The `prod.snd` projection, as a monotone function. -/\ndef prod.snd {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] : α × β →ₘ β :=\n  mk prod.snd sorry\n\n/-- The `prod` constructor, as a monotone function. -/\n@[simp] theorem prod.zip_to_fun {α : Type u_1} {β : Type u_2} {γ : Type u_3} [preorder α]\n    [preorder β] [preorder γ] (f : α →ₘ β) (g : α →ₘ γ) :\n    ∀ (ᾰ : α), coe_fn (prod.zip f g) ᾰ = (coe_fn f ᾰ, coe_fn g ᾰ) :=\n  fun (ᾰ : α) => Eq.refl (coe_fn f ᾰ, coe_fn g ᾰ)\n\n/-- `roption.bind` as a monotone function -/\n@[simp] theorem bind_to_fun {α : Type u_1} [preorder α] {β : Type u_2} {γ : Type u_2}\n    (f : α →ₘ roption β) (g : α →ₘ β → roption γ) (x : α) :\n    coe_fn (bind f g) x = coe_fn f x >>= coe_fn g x :=\n  Eq.refl (coe_fn (bind f g) x)\n\nend preorder_hom\n\n\nnamespace omega_complete_partial_order\n\n\n/-- A chain is a monotonically increasing sequence.\n\nSee the definition on page 114 of [gunter]. -/\ndef chain (α : Type u) [preorder α] := ℕ →ₘ α\n\nnamespace chain\n\n\nprotected instance has_coe_to_fun {α : Type u} [preorder α] : has_coe_to_fun (chain α) :=\n  infer_instance\n\nprotected instance inhabited {α : Type u} [preorder α] [Inhabited α] : Inhabited (chain α) :=\n  { default := preorder_hom.mk (fun (_x : ℕ) => Inhabited.default) sorry }\n\nprotected instance has_mem {α : Type u} [preorder α] : has_mem α (chain α) :=\n  has_mem.mk fun (a : α) (c : ℕ →ₘ α) => ∃ (i : ℕ), a = coe_fn c i\n\nprotected instance has_le {α : Type u} [preorder α] : HasLessEq (chain α) :=\n  { LessEq := fun (x y : chain α) => ∀ (i : ℕ), ∃ (j : ℕ), coe_fn x i ≤ coe_fn y j }\n\n/-- `map` function for `chain` -/\n@[simp] theorem map_to_fun {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α)\n    (f : α →ₘ β) : ∀ (ᾰ : ℕ), coe_fn (map c f) ᾰ = coe_fn f (coe_fn c ᾰ) :=\n  fun (ᾰ : ℕ) => Eq.refl (coe_fn f (coe_fn c ᾰ))\n\ntheorem mem_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β}\n    (x : α) : x ∈ c → coe_fn f x ∈ map c f :=\n  sorry\n\ntheorem exists_of_mem_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α)\n    {f : α →ₘ β} {b : β} : b ∈ map c f → ∃ (a : α), a ∈ c ∧ coe_fn f a = b :=\n  sorry\n\ntheorem mem_map_iff {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β}\n    {b : β} : b ∈ map c f ↔ ∃ (a : α), a ∈ c ∧ coe_fn f a = b :=\n  sorry\n\n@[simp] theorem map_id {α : Type u} [preorder α] (c : chain α) : map c preorder_hom.id = c :=\n  preorder_hom.comp_id c\n\ntheorem map_comp {α : Type u} {β : Type v} {γ : Type u_1} [preorder α] [preorder β] [preorder γ]\n    (c : chain α) {f : α →ₘ β} (g : β →ₘ γ) : map (map c f) g = map c (preorder_hom.comp g f) :=\n  rfl\n\ntheorem map_le_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β}\n    {g : α →ₘ β} (h : f ≤ g) : map c f ≤ map c g :=\n  sorry\n\n/-- `chain.zip` pairs up the elements of two chains that have the same index -/\n@[simp] theorem zip_to_fun {α : Type u} {β : Type v} [preorder α] [preorder β] (c₀ : chain α)\n    (c₁ : chain β) : ∀ (ᾰ : ℕ), coe_fn (zip c₀ c₁) ᾰ = (coe_fn c₀ ᾰ, coe_fn c₁ ᾰ) :=\n  fun (ᾰ : ℕ) => Eq.refl (coe_fn c₀ ᾰ, coe_fn c₁ ᾰ)\n\nend chain\n\n\nend omega_complete_partial_order\n\n\n/-- An omega-complete partial order is a partial order with a supremum\noperation on increasing sequences indexed by natural numbers (which we\ncall `ωSup`). In this sense, it is strictly weaker than join complete\nsemi-lattices as only ω-sized totally ordered sets have a supremum.\n\nSee the definition on page 114 of [gunter]. -/\nclass omega_complete_partial_order (α : Type u_1) extends partial_order α where\n  ωSup : omega_complete_partial_order.chain α → α\n  le_ωSup : ∀ (c : omega_complete_partial_order.chain α) (i : ℕ), coe_fn c i ≤ ωSup c\n  ωSup_le :\n    ∀ (c : omega_complete_partial_order.chain α) (x : α), (∀ (i : ℕ), coe_fn c i ≤ x) → ωSup c ≤ x\n\nnamespace omega_complete_partial_order\n\n\n/-- Transfer a `omega_complete_partial_order` on `β` to a `omega_complete_partial_order` on `α` using\na strictly monotone function `f : β →ₘ α`, a definition of ωSup and a proof that `f` is continuous\nwith regard to the provided `ωSup` and the ωCPO on `α`. -/\nprotected def lift {α : Type u} {β : Type v} [omega_complete_partial_order α] [partial_order β]\n    (f : β →ₘ α) (ωSup₀ : chain β → β) (h : ∀ (x y : β), coe_fn f x ≤ coe_fn f y → x ≤ y)\n    (h' : ∀ (c : chain β), coe_fn f (ωSup₀ c) = ωSup (chain.map c f)) :\n    omega_complete_partial_order β :=\n  mk ωSup₀ sorry sorry\n\ntheorem le_ωSup_of_le {α : Type u} [omega_complete_partial_order α] {c : chain α} {x : α} (i : ℕ)\n    (h : x ≤ coe_fn c i) : x ≤ ωSup c :=\n  le_trans h (le_ωSup c i)\n\ntheorem ωSup_total {α : Type u} [omega_complete_partial_order α] {c : chain α} {x : α}\n    (h : ∀ (i : ℕ), coe_fn c i ≤ x ∨ x ≤ coe_fn c i) : ωSup c ≤ x ∨ x ≤ ωSup c :=\n  sorry\n\ntheorem ωSup_le_ωSup_of_le {α : Type u} [omega_complete_partial_order α] {c₀ : chain α}\n    {c₁ : chain α} (h : c₀ ≤ c₁) : ωSup c₀ ≤ ωSup c₁ :=\n  ωSup_le c₀ (ωSup c₁)\n    fun (i : ℕ) =>\n      Exists.rec_on (h i) fun (j : ℕ) (h : coe_fn c₀ i ≤ coe_fn c₁ j) => le_trans h (le_ωSup c₁ j)\n\ntheorem ωSup_le_iff {α : Type u} [omega_complete_partial_order α] (c : chain α) (x : α) :\n    ωSup c ≤ x ↔ ∀ (i : ℕ), coe_fn c i ≤ x :=\n  { mp := fun (ᾰ : ωSup c ≤ x) (i : ℕ) => le_trans (le_ωSup c i) ᾰ,\n    mpr := fun (ᾰ : ∀ (i : ℕ), coe_fn c i ≤ x) => ωSup_le c x ᾰ }\n\n/-- A subset `p : α → Prop` of the type closed under `ωSup` induces an\n`omega_complete_partial_order` on the subtype `{a : α // p a}`. -/\ndef subtype {α : Type u_1} [omega_complete_partial_order α] (p : α → Prop)\n    (hp : ∀ (c : chain α), (∀ (i : α), i ∈ c → p i) → p (ωSup c)) :\n    omega_complete_partial_order (Subtype p) :=\n  omega_complete_partial_order.lift (preorder_hom.subtype.val p)\n    (fun (c : chain (Subtype p)) =>\n      { val := ωSup (chain.map c (preorder_hom.subtype.val p)), property := sorry })\n    sorry sorry\n\n/-- A monotone function `f : α →ₘ β` is continuous if it distributes over ωSup.\n\nIn order to distinguish it from the (more commonly used) continuity from topology\n(see topology/basic.lean), the present definition is often referred to as\n\"Scott-continuity\" (referring to Dana Scott). It corresponds to continuity\nin Scott topological spaces (not defined here). -/\ndef continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →ₘ β) :=\n  ∀ (c : chain α), coe_fn f (ωSup c) = ωSup (chain.map c f)\n\n/-- `continuous' f` asserts that `f` is both monotone and continuous. -/\ndef continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α → β) :=\n  ∃ (hf : monotone f), continuous (preorder_hom.mk f hf)\n\ntheorem continuous.to_monotone {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] {f : α → β} (hf : continuous' f) : monotone f :=\n  Exists.fst hf\n\ntheorem continuous.of_bundled {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α → β) (hf : monotone f)\n    (hf' : continuous (preorder_hom.mk f hf)) : continuous' f :=\n  Exists.intro hf hf'\n\ntheorem continuous.of_bundled' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →ₘ β) (hf' : continuous f) : continuous' ⇑f :=\n  Exists.intro (preorder_hom.monotone f) hf'\n\ntheorem continuous.to_bundled {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α → β) (hf : continuous' f) :\n    continuous (preorder_hom.mk f (continuous.to_monotone hf)) :=\n  Exists.snd hf\n\ntheorem continuous_id {α : Type u} [omega_complete_partial_order α] : continuous preorder_hom.id :=\n  sorry\n\ntheorem continuous_comp {α : Type u} {β : Type v} {γ : Type u_1} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : α →ₘ β) (g : β →ₘ γ)\n    (hfc : continuous f) (hgc : continuous g) : continuous (preorder_hom.comp g f) :=\n  sorry\n\ntheorem id_continuous' {α : Type u} [omega_complete_partial_order α] : continuous' id := sorry\n\ntheorem const_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (x : β) : continuous' (function.const α x) :=\n  sorry\n\nend omega_complete_partial_order\n\n\nnamespace roption\n\n\ntheorem eq_of_chain {α : Type u} {c : omega_complete_partial_order.chain (roption α)} {a : α}\n    {b : α} (ha : some a ∈ c) (hb : some b ∈ c) : a = b :=\n  sorry\n\n/-- The (noncomputable) `ωSup` definition for the `ω`-CPO structure on `roption α`. -/\nprotected def ωSup {α : Type u} (c : omega_complete_partial_order.chain (roption α)) : roption α :=\n  dite (∃ (a : α), some a ∈ c) (fun (h : ∃ (a : α), some a ∈ c) => some (classical.some h))\n    fun (h : ¬∃ (a : α), some a ∈ c) => none\n\ntheorem ωSup_eq_some {α : Type u} {c : omega_complete_partial_order.chain (roption α)} {a : α}\n    (h : some a ∈ c) : roption.ωSup c = some a :=\n  sorry\n\ntheorem ωSup_eq_none {α : Type u} {c : omega_complete_partial_order.chain (roption α)}\n    (h : ¬∃ (a : α), some a ∈ c) : roption.ωSup c = none :=\n  dif_neg h\n\ntheorem mem_chain_of_mem_ωSup {α : Type u} {c : omega_complete_partial_order.chain (roption α)}\n    {a : α} (h : a ∈ roption.ωSup c) : some a ∈ c :=\n  sorry\n\nprotected instance omega_complete_partial_order {α : Type u} :\n    omega_complete_partial_order (roption α) :=\n  omega_complete_partial_order.mk roption.ωSup sorry sorry\n\ntheorem mem_ωSup {α : Type u} (x : α) (c : omega_complete_partial_order.chain (roption α)) :\n    x ∈ omega_complete_partial_order.ωSup c ↔ some x ∈ c :=\n  sorry\n\nend roption\n\n\nnamespace pi\n\n\n/-- Function application `λ f, f a` is monotone with respect to `f` for fixed `a`. -/\ndef monotone_apply {α : Type u_1} {β : α → Type u_2} [(a : α) → partial_order (β a)] (a : α) :\n    ((a : α) → β a) →ₘ β a :=\n  preorder_hom.mk (fun (f : (a : α) → β a) => f a) sorry\n\nprotected instance omega_complete_partial_order {α : Type u_1} {β : α → Type u_2}\n    [(a : α) → omega_complete_partial_order (β a)] : omega_complete_partial_order ((a : α) → β a) :=\n  omega_complete_partial_order.mk\n    (fun (c : omega_complete_partial_order.chain ((a : α) → β a)) (a : α) =>\n      omega_complete_partial_order.ωSup\n        (omega_complete_partial_order.chain.map c (monotone_apply a)))\n    sorry sorry\n\nnamespace omega_complete_partial_order\n\n\ntheorem flip₁_continuous' {α : Type u_1} {β : α → Type u_2} {γ : Type u_3}\n    [(x : α) → omega_complete_partial_order (β x)] [omega_complete_partial_order γ]\n    (f : (x : α) → γ → β x) (a : α)\n    (hf : omega_complete_partial_order.continuous' fun (x : γ) (y : α) => f y x) :\n    omega_complete_partial_order.continuous' (f a) :=\n  sorry\n\ntheorem flip₂_continuous' {α : Type u_1} {β : α → Type u_2} {γ : Type u_3}\n    [(x : α) → omega_complete_partial_order (β x)] [omega_complete_partial_order γ]\n    (f : γ → (x : α) → β x)\n    (hf : ∀ (x : α), omega_complete_partial_order.continuous' fun (g : γ) => f g x) :\n    omega_complete_partial_order.continuous' f :=\n  sorry\n\nend omega_complete_partial_order\n\n\nend pi\n\n\nnamespace prod\n\n\n/-- The supremum of a chain in the product `ω`-CPO. -/\nprotected def ωSup {α : Type u_1} {β : Type u_2} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c : omega_complete_partial_order.chain (α × β)) : α × β :=\n  (omega_complete_partial_order.ωSup\n    (omega_complete_partial_order.chain.map c preorder_hom.prod.fst),\n  omega_complete_partial_order.ωSup\n    (omega_complete_partial_order.chain.map c preorder_hom.prod.snd))\n\nprotected instance omega_complete_partial_order {α : Type u_1} {β : Type u_2}\n    [omega_complete_partial_order α] [omega_complete_partial_order β] :\n    omega_complete_partial_order (α × β) :=\n  omega_complete_partial_order.mk prod.ωSup sorry sorry\n\nend prod\n\n\nnamespace complete_lattice\n\n\n/-- Any complete lattice has an `ω`-CPO structure where the countable supremum is a special case\nof arbitrary suprema. -/\nprotected instance omega_complete_partial_order (α : Type u) [complete_lattice α] :\n    omega_complete_partial_order α :=\n  omega_complete_partial_order.mk\n    (fun (c : omega_complete_partial_order.chain α) => supr fun (i : ℕ) => coe_fn c i) sorry sorry\n\ntheorem inf_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] [is_total β LessEq] (f : α →ₘ β) (g : α →ₘ β)\n    (hf : omega_complete_partial_order.continuous f)\n    (hg : omega_complete_partial_order.continuous g) :\n    omega_complete_partial_order.continuous (f ⊓ g) :=\n  sorry\n\ntheorem Sup_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] (s : set (α →ₘ β))\n    (hs : ∀ (f : α →ₘ β), f ∈ s → omega_complete_partial_order.continuous f) :\n    omega_complete_partial_order.continuous (Sup s) :=\n  sorry\n\ntheorem Sup_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] (s : set (α → β)) :\n    (∀ (t : α → β), t ∈ s → omega_complete_partial_order.continuous' t) →\n        omega_complete_partial_order.continuous' (Sup s) :=\n  sorry\n\ntheorem sup_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] {f : α →ₘ β} {g : α →ₘ β} (hf : omega_complete_partial_order.continuous f)\n    (hg : omega_complete_partial_order.continuous g) :\n    omega_complete_partial_order.continuous (f ⊔ g) :=\n  sorry\n\ntheorem top_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] : omega_complete_partial_order.continuous ⊤ :=\n  sorry\n\ntheorem bot_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [complete_lattice β] : omega_complete_partial_order.continuous ⊥ :=\n  sorry\n\nend complete_lattice\n\n\nnamespace omega_complete_partial_order\n\n\nnamespace preorder_hom\n\n\n/-- Function application `λ f, f a` (for fixed `a`) is a monotone function from the\nmonotone function space `α →ₘ β` to `β`. -/\ndef monotone_apply {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (a : α) : (α →ₘ β) →ₘ β :=\n  preorder_hom.mk (fun (f : α →ₘ β) => coe_fn f a) sorry\n\n/-- The \"forgetful functor\" from `α →ₘ β` to `α → β` that takes the underlying function,\nis monotone. -/\ndef to_fun_hom {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] : (α →ₘ β) →ₘ α → β :=\n  preorder_hom.mk (fun (f : α →ₘ β) => preorder_hom.to_fun f) sorry\n\n/-- The `ωSup` operator for monotone functions. -/\n@[simp] theorem ωSup_to_fun {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c : chain (α →ₘ β)) (a : α) :\n    coe_fn (preorder_hom.ωSup c) a = ωSup (chain.map c (monotone_apply a)) :=\n  Eq.refl (coe_fn (preorder_hom.ωSup c) a)\n\nprotected instance omega_complete_partial_order {α : Type u} {β : Type v}\n    [omega_complete_partial_order α] [omega_complete_partial_order β] :\n    omega_complete_partial_order (α →ₘ β) :=\n  omega_complete_partial_order.lift to_fun_hom preorder_hom.ωSup sorry sorry\n\nend preorder_hom\n\n\n/-- A monotone function on `ω`-continuous partial orders is said to be continuous\nif for every chain `c : chain α`, `f (⊔ i, c i) = ⊔ i, f (c i)`.\nThis is just the bundled version of `preorder_hom.continuous`. -/\nstructure continuous_hom (α : Type u) (β : Type v) [omega_complete_partial_order α]\n    [omega_complete_partial_order β]\n    extends α →ₘ β where\n  cont : continuous (preorder_hom.mk to_fun monotone')\n\ninfixr:25 \" →𝒄 \" => Mathlib.omega_complete_partial_order.continuous_hom\n\nprotected instance continuous_hom.has_coe_to_fun (α : Type u) (β : Type v)\n    [omega_complete_partial_order α] [omega_complete_partial_order β] : has_coe_to_fun (α →𝒄 β) :=\n  has_coe_to_fun.mk (fun (_x : α →𝒄 β) => α → β) continuous_hom.to_fun\n\nprotected instance preorder_hom.has_coe (α : Type u) (β : Type v) [omega_complete_partial_order α]\n    [omega_complete_partial_order β] : has_coe (α →𝒄 β) (α →ₘ β) :=\n  has_coe.mk continuous_hom.to_preorder_hom\n\nprotected instance continuous_hom.partial_order (α : Type u) (β : Type v)\n    [omega_complete_partial_order α] [omega_complete_partial_order β] : partial_order (α →𝒄 β) :=\n  partial_order.lift continuous_hom.to_fun sorry\n\nnamespace continuous_hom\n\n\ntheorem congr_fun {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] {f : α →𝒄 β} {g : α →𝒄 β} (h : f = g) (x : α) :\n    coe_fn f x = coe_fn g x :=\n  congr_arg (fun (h : α →𝒄 β) => coe_fn h x) h\n\ntheorem congr_arg {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →𝒄 β) {x : α} {y : α} (h : x = y) :\n    coe_fn f x = coe_fn f y :=\n  congr_arg (fun (x : α) => coe_fn f x) h\n\ntheorem monotone {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →𝒄 β) : monotone ⇑f :=\n  monotone' f\n\ntheorem ite_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] {p : Prop} [hp : Decidable p] (f : α → β) (g : α → β)\n    (hf : continuous' f) (hg : continuous' g) : continuous' fun (x : α) => ite p (f x) (g x) :=\n  sorry\n\ntheorem ωSup_bind {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (c : chain α) (f : α →ₘ roption β) (g : α →ₘ β → roption γ) :\n    ωSup (chain.map c (preorder_hom.bind f g)) = ωSup (chain.map c f) >>= ωSup (chain.map c g) :=\n  sorry\n\ntheorem bind_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : α → roption β) (g : α → β → roption γ) :\n    continuous' f → continuous' g → continuous' fun (x : α) => f x >>= g x :=\n  sorry\n\ntheorem map_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : β → γ) (g : α → roption β) (hg : continuous' g) : continuous' fun (x : α) => f <$> g x :=\n  sorry\n\ntheorem seq_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : α → roption (β → γ)) (g : α → roption β) (hf : continuous' f) (hg : continuous' g) :\n    continuous' fun (x : α) => f x <*> g x :=\n  sorry\n\ntheorem continuous {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (F : α →𝒄 β) (C : chain α) :\n    coe_fn F (ωSup C) = ωSup (chain.map C ↑F) :=\n  cont F C\n\n/-- Construct a continuous function from a bare function, a continuous function, and a proof that\nthey are equal. -/\ndef of_fun {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α → β) (g : α →𝒄 β) (h : f = ⇑g) : α →𝒄 β :=\n  mk f sorry sorry\n\n/-- Construct a continuous function from a monotone function with a proof of continuity. -/\ndef of_mono {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →ₘ β)\n    (h : ∀ (c : chain α), coe_fn f (ωSup c) = ωSup (chain.map c f)) : α →𝒄 β :=\n  mk ⇑f sorry h\n\n/-- The identity as a continuous function. -/\n@[simp] theorem id_to_fun {α : Type u} [omega_complete_partial_order α] :\n    ∀ (ᾰ : α), coe_fn id ᾰ = ᾰ :=\n  fun (ᾰ : α) => Eq.refl ᾰ\n\n/-- The composition of continuous functions. -/\ndef comp {α : Type u} {β : Type v} {γ : Type u_3} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : β →𝒄 γ) (g : α →𝒄 β) :\n    α →𝒄 γ :=\n  of_mono (preorder_hom.comp ↑f ↑g) sorry\n\nprotected theorem ext {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →𝒄 β) (g : α →𝒄 β)\n    (h : ∀ (x : α), coe_fn f x = coe_fn g x) : f = g :=\n  sorry\n\nprotected theorem coe_inj {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →𝒄 β) (g : α →𝒄 β) (h : ⇑f = ⇑g) : f = g :=\n  continuous_hom.ext f g (congr_fun h)\n\n@[simp] theorem comp_id {β : Type v} {γ : Type u_3} [omega_complete_partial_order β]\n    [omega_complete_partial_order γ] (f : β →𝒄 γ) : comp f id = f :=\n  continuous_hom.ext (comp f id) f fun (x : β) => Eq.refl (coe_fn (comp f id) x)\n\n@[simp] theorem id_comp {β : Type v} {γ : Type u_3} [omega_complete_partial_order β]\n    [omega_complete_partial_order γ] (f : β →𝒄 γ) : comp id f = f :=\n  continuous_hom.ext (comp id f) f fun (x : β) => Eq.refl (coe_fn (comp id f) x)\n\n@[simp] theorem comp_assoc {α : Type u} {β : Type v} {γ : Type u_3} {φ : Type u_4}\n    [omega_complete_partial_order α] [omega_complete_partial_order β]\n    [omega_complete_partial_order γ] [omega_complete_partial_order φ] (f : γ →𝒄 φ) (g : β →𝒄 γ)\n    (h : α →𝒄 β) : comp f (comp g h) = comp (comp f g) h :=\n  continuous_hom.ext (comp f (comp g h)) (comp (comp f g) h)\n    fun (x : α) => Eq.refl (coe_fn (comp f (comp g h)) x)\n\n@[simp] theorem coe_apply {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (a : α) (f : α →𝒄 β) : coe_fn (↑f) a = coe_fn f a :=\n  rfl\n\n/-- `function.const` is a continuous function. -/\ndef const {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : β) : α →𝒄 β :=\n  of_mono (preorder_hom.const α f) sorry\n\n@[simp] theorem const_apply {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : β) (a : α) : coe_fn (const f) a = f :=\n  rfl\n\nprotected instance inhabited {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] [Inhabited β] : Inhabited (α →𝒄 β) :=\n  { default := const Inhabited.default }\n\nnamespace prod\n\n\n/-- The application of continuous functions as a monotone function.\n\n(It would make sense to make it a continuous function, but we are currently constructing a\n`omega_complete_partial_order` instance for `α →𝒄 β`, and we cannot use it as the domain or image\nof a continuous function before we do.) -/\ndef apply {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] : (α →𝒄 β) × α →ₘ β :=\n  preorder_hom.mk (fun (f : (α →𝒄 β) × α) => coe_fn (prod.fst f) (prod.snd f)) sorry\n\nend prod\n\n\n/-- The map from continuous functions to monotone functions is itself a monotone function. -/\n@[simp] theorem to_mono_to_fun {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (f : α →𝒄 β) : coe_fn to_mono f = ↑f :=\n  Eq.refl (coe_fn to_mono f)\n\n/-- When proving that a chain of applications is below a bound `z`, it suffices to consider the\nfunctions and values being selected from the same index in the chains.\n\nThis lemma is more specific than necessary, i.e. `c₀` only needs to be a\nchain of monotone functions, but it is only used with continuous functions. -/\n@[simp] theorem forall_forall_merge {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) (z : β) :\n    (∀ (i j : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ j) ≤ z) ↔\n        ∀ (i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ i) ≤ z :=\n  sorry\n\n@[simp] theorem forall_forall_merge' {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) (z : β) :\n    (∀ (j i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ j) ≤ z) ↔\n        ∀ (i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ i) ≤ z :=\n  sorry\n\n/-- The `ωSup` operator for continuous functions, which takes the pointwise countable supremum\nof the functions in the `ω`-chain. -/\nprotected def ωSup {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c : chain (α →𝒄 β)) : α →𝒄 β :=\n  of_mono (ωSup (chain.map c to_mono)) sorry\n\nprotected instance omega_complete_partial_order {α : Type u} {β : Type v}\n    [omega_complete_partial_order α] [omega_complete_partial_order β] :\n    omega_complete_partial_order (α →𝒄 β) :=\n  omega_complete_partial_order.lift to_mono continuous_hom.ωSup sorry sorry\n\ntheorem ωSup_def {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c : chain (α →𝒄 β)) (x : α) :\n    coe_fn (ωSup c) x = coe_fn (continuous_hom.ωSup c) x :=\n  rfl\n\ntheorem ωSup_ωSup {α : Type u} {β : Type v} [omega_complete_partial_order α]\n    [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) :\n    coe_fn (ωSup c₀) (ωSup c₁) = ωSup (preorder_hom.comp prod.apply (chain.zip c₀ c₁)) :=\n  sorry\n\n/-- A family of continuous functions yields a continuous family of functions. -/\ndef flip {β : Type v} {γ : Type u_3} [omega_complete_partial_order β]\n    [omega_complete_partial_order γ] {α : Type u_1} (f : α → β →𝒄 γ) : β →𝒄 α → γ :=\n  mk (fun (x : β) (y : α) => coe_fn (f y) x) sorry sorry\n\n/-- `roption.bind` as a continuous function. -/\ndef bind {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : α →𝒄 roption β) (g : α →𝒄 β → roption γ) : α →𝒄 roption γ :=\n  of_mono (preorder_hom.bind ↑f ↑g) sorry\n\n/-- `roption.map` as a continuous function. -/\n@[simp] theorem map_to_fun {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : β → γ) (g : α →𝒄 roption β) (x : α) : coe_fn (map f g) x = f <$> coe_fn g x :=\n  Eq.refl (coe_fn (map f g) x)\n\n/-- `roption.seq` as a continuous function. -/\ndef seq {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v}\n    (f : α →𝒄 roption (β → γ)) (g : α →𝒄 roption β) : α →𝒄 roption γ :=\n  of_fun (fun (x : α) => coe_fn f x <*> coe_fn g x) (bind f (flip (flip map g))) sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/omega_complete_partial_order_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.31611021114259513}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Anne Baanen\n\n! This file was ported from Lean 3 source module algebra.algebra.subalgebra.tower\n! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Algebra.Subalgebra.Basic\nimport Mathbin.Algebra.Algebra.Tower\n\n/-!\n# Subalgebras in towers of algebras\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we prove facts about subalgebras in towers of algebra.\n\nAn algebra tower A/S/R is expressed by having instances of `algebra A S`,\n`algebra R S`, `algebra R A` and `is_scalar_tower R S A`, the later asserting the\ncompatibility condition `(r • s) • a = r • (s • a)`.\n\n## Main results\n\n * `is_scalar_tower.subalgebra`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S/S₀` is a tower\n * `is_scalar_tower.subalgebra'`: if `A/S/R` is a tower and `S₀` is a subalgebra\n   between `S` and `R`, then `A/S₀/R` is a tower\n * `subalgebra.restrict_scalars`: turn an `S`-subalgebra of `A` into an `R`-subalgebra of `A`,\n   given that `A/S/R` is a tower\n\n-/\n\n\nopen Pointwise\n\nuniverse u v w u₁ v₁\n\nvariable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)\n\nnamespace Algebra\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A]\n\nvariable [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M]\n\nvariable {A}\n\n#print Algebra.lmul_algebraMap /-\ntheorem lmul_algebraMap (x : R) :\n    LinearMap.Algebra.lmul R A (algebraMap R A x) = Algebra.lsmul R A x :=\n  Eq.symm <| LinearMap.ext <| smul_def x\n#align algebra.lmul_algebra_map Algebra.lmul_algebraMap\n-/\n\nend Algebra\n\nnamespace IsScalarTower\n\nsection Semiring\n\nvariable [CommSemiring R] [CommSemiring S] [Semiring A]\n\nvariable [Algebra R S] [Algebra S A]\n\nvariable (R S A)\n\n/- warning: is_scalar_tower.subalgebra -> IsScalarTower.subalgebra is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) (S : Type.{u2}) (A : Type.{u3}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : Algebra.{u2, u3} S A _inst_2 _inst_3] (S₀ : Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4), IsScalarTower.{u2, u2, u3} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.setLike.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) S₀) S A (Subalgebra.hasSmul.{u1, u2, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S (Mul.toSMul.{u2} S (Distrib.toHasMul.{u2} S (NonUnitalNonAssocSemiring.toDistrib.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))))) S₀) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_5))))) (Subalgebra.hasSmul.{u1, u2, u3} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 A (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_5))))) S₀)\nbut is expected to have type\n  forall (R : Type.{u1}) (S : Type.{u2}) (A : Type.{u3}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : Algebra.{u2, u3} S A _inst_2 _inst_3] (S₀ : Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4), IsScalarTower.{u2, u2, u3} (Subtype.{succ u2} S (fun (x : S) => Membership.mem.{u2, u2} S (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.instSetLikeSubalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) x S₀)) S A (Subalgebra.instSMulSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebra.{u1, u2, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S (Algebra.toSMul.{u2, u2} S S _inst_2 (CommSemiring.toSemiring.{u2} S _inst_2) (Algebra.id.{u2} S _inst_2)) S₀) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_5) (Subalgebra.instSMulSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebra.{u1, u2, u3} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 A (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_5) S₀)\nCase conversion may be inaccurate. Consider using '#align is_scalar_tower.subalgebra IsScalarTower.subalgebraₓ'. -/\ninstance subalgebra (S₀ : Subalgebra R S) : IsScalarTower S₀ S A :=\n  of_algebraMap_eq fun x => rfl\n#align is_scalar_tower.subalgebra IsScalarTower.subalgebra\n\nvariable [Algebra R A] [IsScalarTower R S A]\n\n/- warning: is_scalar_tower.subalgebra' -> IsScalarTower.subalgebra' is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) (S : Type.{u2}) (A : Type.{u3}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_6 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_7 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A 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u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.setLike.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) S₀) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.setLike.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) S₀) (Semiring.toNonAssocSemiring.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.setLike.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) S₀) (Subalgebra.toSemiring.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S₀)))) (Subalgebra.module.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S₀))))) (Subalgebra.hasSmul.{u1, u2, u3} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 A (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_5))))) S₀) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_6)))))\nbut is expected to have type\n  forall (R : Type.{u1}) (S : Type.{u2}) (A : Type.{u3}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_5 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_6 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_7 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_5) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_6)] (S₀ : Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4), IsScalarTower.{u1, u2, u3} R (Subtype.{succ u2} S (fun (x : S) => Membership.mem.{u2, u2} S (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.instSetLikeSubalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) x S₀)) A (Algebra.toSMul.{u1, u2} R (Subtype.{succ u2} S (fun (x : S) => Membership.mem.{u2, u2} S (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4) S (Subalgebra.instSetLikeSubalgebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4)) x S₀)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S₀) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 S₀)) (Subalgebra.instSMulSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebra.{u1, u2, u3} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_4 A (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_5) S₀) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_6)\nCase conversion may be inaccurate. Consider using '#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'ₓ'. -/\ninstance subalgebra' (S₀ : Subalgebra R S) : IsScalarTower R S₀ A :=\n  @IsScalarTower.of_algebraMap_eq R S₀ A _ _ _ _ _ _ fun _ =>\n    (IsScalarTower.algebraMap_apply R S A _ : _)\n#align is_scalar_tower.subalgebra' IsScalarTower.subalgebra'\n\nend Semiring\n\nend IsScalarTower\n\nnamespace Subalgebra\n\nopen IsScalarTower\n\nsection Semiring\n\nvariable (R) {S A B} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [Algebra S B] [Algebra R B]\n\nvariable [IsScalarTower R S A] [IsScalarTower R S B]\n\n#print Subalgebra.restrictScalars /-\n/-- Given a tower `A / ↥U / S / R` of algebras, where `U` is an `S`-subalgebra of `A`, reinterpret\n`U` as an `R`-subalgebra of `A`. -/\ndef restrictScalars (U : Subalgebra S A) : Subalgebra R A :=\n  { U with\n    algebraMap_mem' := fun x => by\n      rw [algebra_map_apply R S A]\n      exact U.algebra_map_mem _ }\n#align subalgebra.restrict_scalars Subalgebra.restrictScalars\n-/\n\n/- warning: subalgebra.coe_restrict_scalars -> Subalgebra.coe_restrictScalars is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6}, Eq.{succ u3} (Set.{u3} A) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Set.{u3} A) (HasLiftT.mk.{succ u3, succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Set.{u3} A) (CoeTCₓ.coe.{succ u3, succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Set.{u3} A) (SetLike.Set.hasCoeT.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.setLike.{u1, u3} R A _inst_1 _inst_3 _inst_7)))) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Set.{u3} A) (HasLiftT.mk.{succ u3, succ u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Set.{u3} A) (CoeTCₓ.coe.{succ u3, succ u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Set.{u3} A) (SetLike.Set.hasCoeT.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.setLike.{u2, u3} S A _inst_2 _inst_3 _inst_6)))) U)\nbut is expected to have type\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_7)] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6}, Eq.{succ u3} (Set.{u3} A) (SetLike.coe.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.instSetLikeSubalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) (SetLike.coe.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.instSetLikeSubalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) U)\nCase conversion may be inaccurate. Consider using '#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalarsₓ'. -/\n@[simp]\ntheorem coe_restrictScalars {U : Subalgebra S A} : (restrictScalars R U : Set A) = (U : Set A) :=\n  rfl\n#align subalgebra.coe_restrict_scalars Subalgebra.coe_restrictScalars\n\n/- warning: subalgebra.restrict_scalars_top -> Subalgebra.restrictScalars_top is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))], Eq.{succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 (Top.top.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (CompleteLattice.toHasTop.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.Subalgebra.completeLattice.{u2, u3} S A _inst_2 _inst_3 _inst_6)))) (Top.top.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (CompleteLattice.toHasTop.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Algebra.Subalgebra.completeLattice.{u1, u3} R A _inst_1 _inst_3 _inst_7)))\nbut is expected to have type\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_7)], Eq.{succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 (Top.top.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (CompleteLattice.toTop.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.instCompleteLatticeSubalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6)))) (Top.top.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (CompleteLattice.toTop.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Algebra.instCompleteLatticeSubalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7)))\nCase conversion may be inaccurate. Consider using '#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_topₓ'. -/\n@[simp]\ntheorem restrictScalars_top : restrictScalars R (⊤ : Subalgebra S A) = ⊤ :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_top Subalgebra.restrictScalars_top\n\n/- warning: subalgebra.restrict_scalars_to_submodule -> Subalgebra.restrictScalars_toSubmodule is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6}, Eq.{succ u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (coeFn.{succ u3, succ u3} (OrderEmbedding.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Preorder.toLE.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (PartialOrder.toPreorder.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (SetLike.partialOrder.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.setLike.{u1, u3} R A _inst_1 _inst_3 _inst_7)))) (Preorder.toLE.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (PartialOrder.toPreorder.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Submodule.completeLattice.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7))))))) (fun (_x : RelEmbedding.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A 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(CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)))))))) (Subalgebra.toSubmodule.{u2, u3} S A _inst_2 _inst_3 _inst_6) U))\nbut is expected to have type\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_7)] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6}, Eq.{succ u3} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) => Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) (FunLike.coe.{succ u3, succ u3, succ u3} (Function.Embedding.{succ u3, succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7))) (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (fun (_x : Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) => Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) _x) (EmbeddingLike.toFunLike.{succ u3, succ u3, succ u3} (Function.Embedding.{succ u3, succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7))) (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Function.instEmbeddingLikeEmbedding.{succ u3, succ u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))) (RelEmbedding.toEmbedding.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) => LE.le.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Preorder.toLE.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (PartialOrder.toPreorder.{u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (SetLike.instPartialOrder.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.instSetLikeSubalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) => LE.le.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Preorder.toLE.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (PartialOrder.toPreorder.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Submodule.completeLattice.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) (Submodule.restrictScalars.{u1, u2, u3} R S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (CommSemiring.toSemiring.{u1} R _inst_1) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) _inst_10 (FunLike.coe.{succ u3, succ u3, succ u3} (Function.Embedding.{succ u3, succ u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))) (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (fun (_x : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) => Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) _x) (EmbeddingLike.toFunLike.{succ u3, succ u3, succ u3} (Function.Embedding.{succ u3, succ u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))) (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (Function.instEmbeddingLikeEmbedding.{succ u3, succ u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)))) (RelEmbedding.toEmbedding.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) => LE.le.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Preorder.toLE.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (PartialOrder.toPreorder.{u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (SetLike.instPartialOrder.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.instSetLikeSubalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) => LE.le.{u3} (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (Preorder.toLE.{u3} (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) (Submodule.completeLattice.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u2, u3} S A _inst_2 _inst_3 _inst_6)) U))\nCase conversion may be inaccurate. Consider using '#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmoduleₓ'. -/\n@[simp]\ntheorem restrictScalars_toSubmodule {U : Subalgebra S A} :\n    (U.restrictScalars R).toSubmodule = U.toSubmodule.restrictScalars R :=\n  SetLike.coe_injective rfl\n#align subalgebra.restrict_scalars_to_submodule Subalgebra.restrictScalars_toSubmodule\n\n/- warning: subalgebra.mem_restrict_scalars -> Subalgebra.mem_restrictScalars is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6} {x : A}, Iff (Membership.Mem.{u3, u3} A (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (SetLike.hasMem.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.setLike.{u1, u3} R A _inst_1 _inst_3 _inst_7)) x (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) (Membership.Mem.{u3, u3} A (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (SetLike.hasMem.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.setLike.{u2, u3} S A _inst_2 _inst_3 _inst_6)) x U)\nbut is expected to have type\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_7)] {U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6} {x : A}, Iff (Membership.mem.{u3, u3} A (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.instSetLikeSubalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7)) x (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) (Membership.mem.{u3, u3} A (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (SetLike.instMembership.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.instSetLikeSubalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6)) x U)\nCase conversion may be inaccurate. Consider using '#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalarsₓ'. -/\n@[simp]\ntheorem mem_restrictScalars {U : Subalgebra S A} {x : A} : x ∈ restrictScalars R U ↔ x ∈ U :=\n  Iff.rfl\n#align subalgebra.mem_restrict_scalars Subalgebra.mem_restrictScalars\n\n#print Subalgebra.restrictScalars_injective /-\ntheorem restrictScalars_injective :\n    Function.Injective (restrictScalars R : Subalgebra S A → Subalgebra R A) := fun U V H =>\n  ext fun x => by rw [← mem_restrict_scalars R, H, mem_restrict_scalars]\n#align subalgebra.restrict_scalars_injective Subalgebra.restrictScalars_injective\n-/\n\n/- warning: subalgebra.of_restrict_scalars -> Subalgebra.ofRestrictScalars is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} {B : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Semiring.{u4} B] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_8 : Algebra.{u2, u4} S B _inst_2 _inst_4] [_inst_9 : Algebra.{u1, u4} R B _inst_1 _inst_4] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} S A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u2, u3} S A (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u2, u3} S A (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u2, u3} S A (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u2, u3} S A _inst_2 _inst_3 _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R A (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R A (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3)))))) (Module.toMulActionWithZero.{u1, u3} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A _inst_3))) (Algebra.toModule.{u1, u3} R A _inst_1 _inst_3 _inst_7)))))] [_inst_11 : IsScalarTower.{u1, u2, u4} R S B (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u4} S B (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (SMulWithZero.toSmulZeroClass.{u2, u4} S B (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2))))) (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (MulActionWithZero.toSMulWithZero.{u2, u4} S B (Semiring.toMonoidWithZero.{u2} S (CommSemiring.toSemiring.{u2} S _inst_2)) (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (Module.toMulActionWithZero.{u2, u4} S B (CommSemiring.toSemiring.{u2} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4))) (Algebra.toModule.{u2, u4} S B _inst_2 _inst_4 _inst_8))))) (SMulZeroClass.toHasSmul.{u1, u4} R B (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R B (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} B (AddMonoid.toAddZeroClass.{u4} B (AddCommMonoid.toAddMonoid.{u4} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4)))))) (Module.toMulActionWithZero.{u1, u4} R B (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B _inst_4))) (Algebra.toModule.{u1, u4} R B _inst_1 _inst_4 _inst_9)))))] (U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6), (AlgHom.{u2, u3, u4} S (coeSort.{succ u3, succ (succ u3)} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.setLike.{u2, u3} S A _inst_2 _inst_3 _inst_6)) U) B _inst_2 (Subalgebra.toSemiring.{u2, u3} S A _inst_2 _inst_3 _inst_6 U) _inst_4 (Subalgebra.algebra.{u2, u3} S A _inst_2 _inst_3 _inst_6 U) _inst_8) -> (AlgHom.{u1, u3, u4} R (coeSort.{succ u3, succ (succ u3)} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.setLike.{u1, u3} R A _inst_1 _inst_3 _inst_7)) (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) B _inst_1 (Subalgebra.toSemiring.{u1, u3} R A _inst_1 _inst_3 _inst_7 (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) _inst_4 (Subalgebra.algebra.{u1, u3} R A _inst_1 _inst_3 _inst_7 (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) _inst_9)\nbut is expected to have type\n  forall (R : Type.{u1}) {S : Type.{u2}} {A : Type.{u3}} {B : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} S] [_inst_3 : Semiring.{u3} A] [_inst_4 : Semiring.{u4} B] [_inst_5 : Algebra.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2)] [_inst_6 : Algebra.{u2, u3} S A _inst_2 _inst_3] [_inst_7 : Algebra.{u1, u3} R A _inst_1 _inst_3] [_inst_8 : Algebra.{u2, u4} S B _inst_2 _inst_4] [_inst_9 : Algebra.{u1, u4} R B _inst_1 _inst_4] [_inst_10 : IsScalarTower.{u1, u2, u3} R S A (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u3} S A _inst_2 _inst_3 _inst_6) (Algebra.toSMul.{u1, u3} R A _inst_1 _inst_3 _inst_7)] [_inst_11 : IsScalarTower.{u1, u2, u4} R S B (Algebra.toSMul.{u1, u2} R S _inst_1 (CommSemiring.toSemiring.{u2} S _inst_2) _inst_5) (Algebra.toSMul.{u2, u4} S B _inst_2 _inst_4 _inst_8) (Algebra.toSMul.{u1, u4} R B _inst_1 _inst_4 _inst_9)] (U : Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6), (AlgHom.{u2, u3, u4} S (Subtype.{succ u3} A (fun (x : A) => Membership.mem.{u3, u3} A (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) (SetLike.instMembership.{u3, u3} (Subalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6) A (Subalgebra.instSetLikeSubalgebra.{u2, u3} S A _inst_2 _inst_3 _inst_6)) x U)) B _inst_2 (Subalgebra.toSemiring.{u2, u3} S A _inst_2 _inst_3 _inst_6 U) _inst_4 (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u2, u3} S A _inst_2 _inst_3 _inst_6 U) _inst_8) -> (AlgHom.{u1, u3, u4} R (Subtype.{succ u3} A (fun (x : A) => Membership.mem.{u3, u3} A (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7) A (Subalgebra.instSetLikeSubalgebra.{u1, u3} R A _inst_1 _inst_3 _inst_7)) x (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U))) B _inst_1 (Subalgebra.toSemiring.{u1, u3} R A _inst_1 _inst_3 _inst_7 (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) _inst_4 (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u3} R A _inst_1 _inst_3 _inst_7 (Subalgebra.restrictScalars.{u1, u2, u3} R S A _inst_1 _inst_2 _inst_3 _inst_5 _inst_6 _inst_7 _inst_10 U)) _inst_9)\nCase conversion may be inaccurate. Consider using '#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalarsₓ'. -/\n/-- Produces an `R`-algebra map from `U.restrict_scalars R` given an `S`-algebra map from `U`.\n\nThis is a special case of `alg_hom.restrict_scalars` that can be helpful in elaboration. -/\n@[simp]\ndef ofRestrictScalars (U : Subalgebra S A) (f : U →ₐ[S] B) : U.restrictScalars R →ₐ[R] B :=\n  f.restrictScalars R\n#align subalgebra.of_restrict_scalars Subalgebra.ofRestrictScalars\n\nend Semiring\n\nend Subalgebra\n\nnamespace IsScalarTower\n\nopen Subalgebra\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring A]\n\nvariable [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A]\n\n#print IsScalarTower.adjoin_range_toAlgHom /-\ntheorem adjoin_range_toAlgHom (t : Set A) :\n    (Algebra.adjoin (toAlgHom R S A).range t).restrictScalars R =\n      (Algebra.adjoin S t).restrictScalars R :=\n  Subalgebra.ext fun z =>\n    show\n      z ∈ Subsemiring.closure (Set.range (algebraMap (toAlgHom R S A).range A) ∪ t : Set A) ↔\n        z ∈ Subsemiring.closure (Set.range (algebraMap S A) ∪ t : Set A)\n      by\n      suffices Set.range (algebraMap (toAlgHom R S A).range A) = Set.range (algebraMap S A) by\n        rw [this]\n      ext z\n      exact ⟨fun ⟨⟨x, y, h1⟩, h2⟩ => ⟨y, h2 ▸ h1⟩, fun ⟨y, hy⟩ => ⟨⟨z, y, hy⟩, rfl⟩⟩\n#align is_scalar_tower.adjoin_range_to_alg_hom IsScalarTower.adjoin_range_toAlgHom\n-/\n\nend IsScalarTower\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Algebra/Subalgebra/Tower.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3161102036800226}}
{"text": "import .open_subspace\nimport .lifts\nimport tactic\nimport algebraic_topology.fundamental_groupoid.basic\nimport category_theory.limits.shapes.comm_sq\nimport topology.unit_interval\n\n\nnoncomputable theory\n\n\nuniverse u\n\nvariables {X : Type u} [topological_space X]\nvariables {X₁ : top_subspace X} {X₂ : top_subspace X}\nvariables {G : category_theory.Groupoid.{u u}}\n\ndef p₁ : proj_grpd X₁ (πₓ (Top.of X₁)) := self_grpd X₁\n#print p₁\ndef p₂ : proj_grpd X₂ (πₓ (Top.of X₂)) := self_grpd X₂\n\n#check πₓ (Top.of X₁)\n\n-- def w (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G): proj_grpd (X₁ ∩ X₂) G := { \n--   obj := \n--   begin\n--     let i : (X₁ ∩ X₂) → X₁ := @open_incl X _ (X₁∩X₂) X₁ inter_sub,\n--     refine _ ∘ i,\n--     exact v₁.obj,\n--   end,\n--   path := \n--   begin\n    \n--   end \n--   }\n\n#check ![1,2]\n\ndef set.union_lift {α β} {s t : set α} \n  (f : s → β) (g : t → β) : s ∪ t → β :=\nbegin\n  classical,\n  exact λ x, x.prop.by_cases (λ hx, f ⟨_, hx⟩) (λ hx, g ⟨_, hx⟩) \nend\n-- def set.union_lift {α β} {s t : set α} [decidable_pred (∈ s)] [decidable_pred (∈ t)]\n--   (f : s → β) (g : t → β) : s ∪ t → β :=\n-- begin\n--   intro x, apply or.by_cases x.prop,\n-- end\n\n-- def setfunc_to_func {H : top_subspace X} {K : Type} (f : (H.carrier) → K) : H → K :=\n-- begin\n--   exact f,\n-- end\n#check category_theory.comm_sq\n\nvariables {f : X₁ → G} {g : X₂ → G} \n\ndef coincide {K : Type*} (f : X₁ → K) (g : X₂ → K):= ∀(y : X) (h1 : y ∈ X₁) (h2 : y ∈ X₂), f (b_incl y h1) = g (b_incl y h2)\n#print coincide\n\nlemma union_def_eq {x : X₁ ∪ X₂} (h : incl x ∈ X₁): (set.union_lift f g) x = f (b_incl (incl x) h) :=\nbegin\n  rw set.union_lift, unfold_coes, have p : x.val ∈ X₁.carrier, cases x, exact h,\n  rw or.by_cases, split_ifs, rotate, contradiction, rw b_incl, simp_rw incl, cases x, simp, refl,\nend\n\nlemma union_def_eq₂ {x : X₁ ∪ X₂} (h : incl x ∈ X₂) (pp : @coincide X _ X₁ X₂ G f g): \n(set.union_lift f g) x = g (b_incl (incl x) h) :=\nbegin\n  rw set.union_lift, unfold_coes, have p : x.val ∈ X₂.carrier, cases x, exact h,\n  rw or.by_cases, split_ifs, \n  {\n    rw b_incl, simp_rw incl,have p2 : x.val ∈ (X₁∩X₂).carrier, split,assumption, assumption,\n    cases x, simp, \n    have s : ∀y:X, y ∈ X₁ → y ∈ X₂ →  f (b_incl y _)= g (b_incl y _), rotate 3,assumption, rotate 2, assumption,\n    rw coincide at pp, exact pp, simp_rw b_incl at s,apply s, assumption, assumption,\n  },rw b_incl, simp_rw incl, cases x, simp, refl,\nend\n\n-- Todo : commutative diagram --\n\nnotation `πₓ` := fundamental_groupoid.fundamental_groupoid_functor.obj\nnotation `π` := fundamental_groupoid.fundamental_groupoid_functor\nnotation `πₘ` := fundamental_groupoid.fundamental_groupoid_functor.map\n\ntheorem comm_px : \n(@open_incl X _ X₁ (X₁∪X₂) union_sub) ∘ (@open_incl X _ (X₁∩X₂) X₁ inter_sub) = \n(@open_incl X _ X₂ (X₁∪X₂) union_sub2) ∘ (@open_incl X _ (X₁∩X₂) X₂ inter_sub2) :=\nbegin\n  ext1,simp_rw open_incl, rw function.comp, simp, cases x, simp,\nend\n\ntheorem comm_πₓ : category_theory.comm_sq \n (grpd_induced_incl (@inter_sub X _ X₁ X₂))  \n (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)) \n (grpd_induced_incl (@union_sub X _ X₁ X₂))\n (grpd_induced_incl (@union_sub2 X _ X₁ X₂)) :=\nbegin\n  apply category_theory.functor.map_comm_sq,\n  fconstructor, simp_rw top_subspace_incl, simp_rw open_incl, ext, simp only [Top.comp_app, continuous_map.coe_mk],\n  unfold_coes, cases a, simp only [eq_self_iff_true],\nend\n\ndef w₁ (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G) (a b: ↥(X₁ ∪ X₂))\n(pab: path a b ) (h : ↑(set.range pab) ⊆ X₁.carrier ):\n set.union_lift v₁.obj v₂.obj a ⟶ set.union_lift v₁.obj v₂.obj b:=\nbegin\n    rw path_incl_range_eq at h,\n    let pXab := subspace_path_lift pab,\n    let b₁ := path_lift_def pXab h,\n    let s := v₁.map (p₁.path b₁), \n    have aX1 : incl a ∈ X₁ := range_in_init pXab h,\n    have bX2 : incl b ∈ X₁ := range_in_end pXab h,\n    rw union_def_eq aX1, rw union_def_eq bX2, assumption,\nend\n\ndef w₂ (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G) (a b: ↥(X₁ ∪ X₂))\n(pab: path a b ) (h : ↑(set.range pab) ⊆ X₂.carrier )\n(comm_v : category_theory.comm_sq (grpd_induced_incl (@inter_sub X _ X₁ X₂))  (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)) v₁ v₂)\n: set.union_lift v₁.obj v₂.obj a ⟶ set.union_lift v₁.obj v₂.obj b:=\nbegin\n    rw path_incl_range_eq at h,\n    let pXab := subspace_path_lift pab,\n    let b₂ := path_lift_def pXab h,\n    let s := v₂.map (p₂.path b₂), \n    have aX1 : incl a ∈ X₂ := range_in_init pXab h,\n    have bX2 : incl b ∈ X₂ := range_in_end pXab h,\n    have ph : coincide v₁.obj v₂.obj,\n    {\n      rw coincide, intros y hy1 hy2, cases comm_v,\n      have hy12 : y ∈ X₁∩X₂ := ⟨hy1,hy2⟩,\n      have p1 : b_incl y hy1 = (grpd_induced_incl (@inter_sub X _ X₁ X₂)).obj (b_incl y hy12),\n      {\n        rw grpd_induced_incl, tauto,\n      },\n      have p2 : b_incl y hy2 = (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)).obj (b_incl y hy12),\n      {\n        rw grpd_induced_incl, finish,\n      },\n      rw p1, rw p2, simp_rw ←category_theory.functor.comp_obj, \n      have k: grpd_induced_incl inter_sub ⋙ v₁ = grpd_induced_incl inter_sub2 ⋙ v₂:= by assumption,\n      rw k,\n    },\n    rw union_def_eq₂ aX1 ph, rw union_def_eq₂ bX2 ph, assumption,\nend\n\nnotation `I` := unit_interval\n\nlemma uniformity_metric_space {X: Type} [metric_space X] (V : set (X × X)): V ∈ uniformity X ↔ ∃ ε > 0, { p : X × X | dist p.1 p.2 < ε } ⊆ V :=\nbegin\n  rw ← filter.has_basis.mem_iff,\n  apply metric.uniformity_basis_dist,\nend\n\ndef w (v₁ : (πₓ ↑X₁) ⟶ G) (v₂ : (πₓ ↑X₂) ⟶ G)\n(comm_v : category_theory.comm_sq (grpd_induced_incl (@inter_sub X _ X₁ X₂)) (grpd_induced_incl (@inter_sub2 X _ X₁ X₂)) v₁ v₂)\n: proj_grpd (X₁ ∪ X₂) G := \n{\n  obj := set.union_lift v₁.obj v₂.obj,\n  path := begin\n  {\n    intros a b pab, \n    set S₁ := {x : I | ↑(pab.to_fun x) ∈ X₁.carrier} with hs1,\n    set S₂ := {x : I | ↑(pab.to_fun x) ∈ X₂.carrier} with hs2,\n    have pcover : set.univ ⊆ (S₁ ∪ S₂),\n    sorry{\n       intro s,intro hs, \n       set S₃ := S₁ ∪ S₂ with hs3,\n       have hs3 : S₃ = {x : I | ↑(pab.to_fun x) ∈ X₁∪X₂}:= by tauto,\n       rw hs3, simp only [set.mem_union_eq, set.mem_set_of_eq], unfold_coes,\n       have k : ∀p, @incl X _ (X₁∪X₂) p ∈ X₁∪X₂,\n       {\n        intro p, cases p, rw incl, tauto,\n       }, apply k,\n    },\n    have popen1 : is_open S₁,\n    sorry{\n      have k: S₁ = pab.to_fun ⁻¹' (incl ⁻¹' X₁.carrier),\n      {\n        rw hs1, rw set.preimage, rw set.preimage, refl,\n      }, rw k, apply is_open.preimage, cases pab, cases pab__to_continuous_map, assumption,\n      apply is_open.preimage, exact incl_continuous, exact X₁.univ_open,\n    },\n    have popen2 : is_open S₂,\n    sorry{\n      have k: S₂ = pab.to_fun ⁻¹' (incl ⁻¹' X₂.carrier),\n      {\n        rw hs2, rw set.preimage, rw set.preimage, refl,\n      }, rw k, apply is_open.preimage, cases pab, cases pab__to_continuous_map, assumption,\n      apply is_open.preimage, exact incl_continuous, exact X₂.univ_open,\n    },\n    set c := ![S₁, S₂] with hc,\n    have hc₁ : ∀ (i) , is_open (c i),\n    sorry{\n      intro i, fin_cases i, assumption, assumption,\n    },\n    have hc₂ : set.univ ⊆ ⋃ i, c i,\n    sorry{\n      have k : (⋃ i, c i) = S₁ ∪ S₂,\n      {\n        rw hc, simp only [], ext, simp [fin.exists_fin_two],\n      },\n      rw k, assumption,\n    },\n    have hcover := lebesgue_number_lemma (compact_univ) hc₁ hc₂,\n    choose n H pn using hcover,\n    sorry,\n    \n    -- have n : set (↥I × ↥I) := sorry,\n    -- have H : n ∈ uniformity ↥I := sorry,\n    -- have pn : ∀ (x : ↥I), x ∈ set.univ → (∃ (i : fin (1:ℕ).succ), {y : ↥I | (x, y) ∈ n} ⊆ (λ (i : fin (1:ℕ).succ), c i) i) := sorry,\n  },\nend\n}\n\n#print w", "meta": {"author": "Mak1Haru", "repo": "van_kampen", "sha": "873c74504cd75b24e3e4cfba46370c4a9f58a9f1", "save_path": "github-repos/lean/Mak1Haru-van_kampen", "path": "github-repos/lean/Mak1Haru-van_kampen/van_kampen-873c74504cd75b24e3e4cfba46370c4a9f58a9f1/src/van_kampen.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3161102036800226}}
{"text": "example (a b c d e f : Nat) (h : [a, b, c] = [d, e, f]) : a + b + c = d + e + f := by\n  injections\n  simp [*]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/injections1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.31609223341560055}}
{"text": "import Monads.Functor\nimport Monads.Applicative\nimport Monads.Monad\n\nnamespace Monads\n\n  inductive Maybe (α : Type u) where\n  | none : Maybe α\n  | some : α → Maybe α\n    deriving Repr\n\n  namespace Maybe\n    section Functor\n      def map {α β : Type u} : (α → β) → Maybe α → Maybe β\n      | f, none => none\n      | f, (some a) => (some $ f a)\n      \n      theorem map_none {f : α → β} : map f none = none := by rfl\n      theorem map_some {f : α → β} {x : α} :  map f (some x) = (some (f x)) := by rfl\n      theorem map_id {x : Maybe α} : map id x = x := by\n        induction x with\n        | none =>\n          rw [map_none]\n        | some x =>\n          rw [map_some]\n          rw [id_eq]\n\n      #eval map (· + 2) (some 12)\n      #eval map (· + 2) none\n\n      instance : Functor Maybe where\n        fmap := Maybe.map\n        \n      theorem fmap_def (f : α → β) (x : Maybe α) : f <$> x = map f x := by rfl\n\n      instance : LawfulFunctor Maybe where\n        fmap_id := by\n          intro α x\n          rw [fmap_def]\n          rw [map_id]\n        fmap_comp := by\n          intro α β γ g h x\n          rw [fmap_def, fmap_def, fmap_def]\n          induction x with\n          | none =>\n            rw [map_none]\n            rw [map_none]\n            rw [map_none]\n          | some x =>\n            rw [map_some]\n            rw [map_some]\n            rw [map_some]\n        map_const_behaved := by\n          -- We didn't change the definition so the proof is trivial\n          intro α β x y\n          simp [map_const, fmap]\n\n      section BadFunctor\n        private def bad_fmap {α β : Type u} : (α → β) → Maybe α → Maybe β\n        | _, _ => none\n\n        -- First law says this should be (some 12) but it isn't\n        #eval bad_fmap id (some 12)\n        -- Second law holds though\n        example : ∀ {α β γ : Type u} (g : α → β) (h : β → γ) (x : Maybe α), bad_fmap (h ∘ g) x = bad_fmap h (bad_fmap g x) := by\n          intro α β γ g h x\n          simp only [bad_fmap]\n      end BadFunctor\n    end Functor\n\n    section Applicative\n      #check Nat.add <$> (some 12)\n\n      def apply {α β : Type u} : Maybe (α → β) → Maybe α → Maybe β\n      | (some f), (some a) => (some (f a))\n      | _, _ => none\n\n      #eval apply (Nat.add <$> (some 12)) (some 13)\n      \n      theorem none_apply {x : Maybe α} : apply none x = @none β := by rfl\n      theorem apply_none {α β : Type u} {f : Maybe (α → β)} : apply f none = none := by\n        induction f with\n        | none => rfl\n        | some => rfl \n      theorem some_apply_some {α β : Type u} {f : α → β} {x : α} : apply (some f) (some x) = (some $ f x) := by rfl\n\n      instance : Applicative Maybe where\n        pure := some\n        apply := apply\n\n      theorem pure_def (x : α) : pure x = some x := by rfl\n      theorem apply_def {α β : Type u} (f : Maybe (α → β)) (x : Maybe α) : f <*> x = apply f x := by rfl\n\n      #eval Nat.add <$> (some 12) <*> (some 13)\n\n      instance : LawfulApplicative Maybe where\n        apply_id := by\n          intro α x\n          rw [pure_def]\n          rw [apply_def]\n          induction x with\n          | none =>\n            rw [apply_none]\n          | some =>\n            rw [some_apply_some]\n            rw [id_eq]\n        apply_homomorphism := by\n          intro α β x g\n          rw [pure_def, pure_def, pure_def]\n          rw [apply_def, some_apply_some]\n        apply_interchange := by\n          intro α β x g\n          rw [pure_def, pure_def]\n          induction g with\n          | none =>\n            rw [apply_def, none_apply]\n            rw [apply_def, apply_none]\n          | some g =>\n            rw [apply_def, some_apply_some]\n            rw [apply_def, some_apply_some]\n        apply_comp := by\n          intro α β γ x y z\n          rw [pure_def]\n          induction x with\n          | none =>\n            rw [(apply_def none)] -- unwrap specifically the apply we want\n            rw [none_apply]\n            rw [(apply_def _ none)]\n            rw [apply_none]\n            rw [(apply_def none)]\n            rw [none_apply, apply_def, none_apply]\n          | some fx =>\n            induction y with\n            | none =>\n              rw [(apply_def none)]\n              rw [none_apply]\n              rw [apply_def _ none, apply_none]\n              rw [apply_def _ none, apply_none]\n              rw [apply_def, none_apply]\n            | some fy =>\n              induction z with\n              | none =>\n                rw [(apply_def _ none)]\n                rw [apply_none]\n                rw [(apply_def _ none)]\n                rw [apply_none]\n                rw [apply_def, apply_none]\n              | some vz =>\n                rw [apply_def (some fy) (some vz), some_apply_some]\n                rw [apply_def (some fx), some_apply_some]\n                rw [apply_def _ (some fx), some_apply_some]\n                rw [apply_def _ (some fy), some_apply_some]\n                rw [apply_def, some_apply_some]\n        lifta2_behaved := by\n          intro α β γ g x y\n          simp only [liftA2]\n          rw [apply_def]\n        seq_right_behaved := by\n          intro α β a1 a2\n          simp only [Applicative.seq_right]\n          rw [apply_def]\n        seq_left_behaved := by\n          intro α β a1 a2\n          simp only [Applicative.seq_left, Applicative.liftA2]\n        fmap_eq_pure_apply := by\n          intro α β g x\n          rw [pure_def]\n          induction x with\n          | none =>\n            rw [fmap_def, map_none]\n            rw [apply_def, apply_none]\n          | some x =>\n            rw [fmap_def, map_some]\n            rw [apply_def, some_apply_some]\n    end Applicative\n\n    section Monad\n      def bind : Maybe α → (α → Maybe β) → Maybe β\n      | none, f => none\n      | (some x), f => f x\n      \n      def none_bind (f : α → Maybe β) : bind none f = none := by rfl\n      def some_bind (x : α) (f : α → Maybe β) : bind (some x) f = f x := by rfl\n\n      instance : Monad Maybe where\n        ret := some\n        bind := bind\n        \n      theorem ret_def (x : α) : ret x = some x := by rfl\n      theorem bind_def (x : Maybe α) (f : α → Maybe β) : x >>= f = bind x f := by rfl\n\n      instance : LawfulMonad Maybe where\n        ret_left_id := by\n          intro α β a h\n          rw [ret_def]\n          rw [bind_def, some_bind]\n        ret_right_id := by\n          intro α a\n          induction a with\n          | none =>\n            rw [bind_def, none_bind]\n          | some a =>\n            rw [bind_def, some_bind]\n            rw [ret_def]\n        bind_assoc := by\n          intro α β γ a g h\n          induction a with\n          | none =>\n            rw [bind_def none, none_bind]\n            rw [bind_def none, none_bind]\n            rw [bind_def none, none_bind]\n          | some a =>\n            rw [bind_def (some a), some_bind]\n            rw [bind_def (some a), some_bind]\n        pure_behaved := by\n          intro α a\n          rw [pure_def, ret_def]\n        apply_behaved := by\n          intro α β mf ma\n          induction mf with\n          | none =>\n            rw [apply_def none, none_apply]\n            rw [bind_def none, none_bind]\n          | some f =>\n            induction ma with\n            | none =>\n              rw [apply_def, apply_none]\n              rw [bind_def, some_bind]\n              rw [bind_def, none_bind]\n            | some a =>\n              rw [apply_def, some_apply_some]\n              rw [bind_def, some_bind]\n              rw [bind_def, some_bind]\n              rw [ret_def]\n        and_then_behaved := by\n          intro α β a1 a2\n          simp only [Applicative.seq_right, Monad.and_then]\n          induction a1 with\n          | none =>\n            rw [none_bind]\n            simp only [Functor.map_const]\n            simp only [Function.comp]\n            rw [map_none]\n            rw [none_apply]\n          | some x1 =>\n            induction a2 with\n            | none =>\n              rw [apply_none]\n              rw [some_bind]\n            | some x2 =>\n              simp only [Functor.map_const]\n              simp only [Function.comp]\n              rw [map_some]\n              rw [some_apply_some]\n              simp only [Function.const]\n              rw [id_eq]\n              rw [some_bind]\n    end Monad\n  end Maybe\nend Monads\n", "meta": {"author": "hargoniX", "repo": "lean-monads", "sha": "2e87ca7ddf394641ea1b16bcbd8c384026d68e2f", "save_path": "github-repos/lean/hargoniX-lean-monads", "path": "github-repos/lean/hargoniX-lean-monads/lean-monads-2e87ca7ddf394641ea1b16bcbd8c384026d68e2f/Monads/Maybe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.31602704632467554}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport category_theory.category.Bipointed\nimport data.two_pointing\n\n/-!\n# The category of two-pointed types\n\nThis defines `Twop`, the category of two-pointed types.\n\n## References\n\n* [nLab, *coalgebra of the real interval*]\n  (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)\n-/\n\nopen category_theory option\n\nuniverses u\nvariables {α β : Type*}\n\n/-- The category of two-pointed types. -/\nstructure Twop : Type.{u+1} :=\n(X : Type.{u})\n(to_two_pointing : two_pointing X)\n\nnamespace Twop\n\ninstance : has_coe_to_sort Twop Type* := ⟨X⟩\n\nattribute [protected] Twop.X\n\n/-- Turns a two-pointing into a two-pointed type. -/\ndef of {X : Type*} (to_two_pointing : two_pointing X) : Twop := ⟨X, to_two_pointing⟩\n\n@[simp] lemma coe_of {X : Type*} (to_two_pointing : two_pointing X) :\n  ↥(of to_two_pointing) = X := rfl\n\nalias of ← _root_.two_pointing.Twop\n\ninstance : inhabited Twop := ⟨of two_pointing.bool⟩\n\n/-- Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are\ndistinct. -/\ndef to_Bipointed (X : Twop) : Bipointed := X.to_two_pointing.to_prod.Bipointed\n\n@[simp] lemma coe_to_Bipointed (X : Twop) : ↥X.to_Bipointed = ↥X := rfl\n\ninstance large_category : large_category Twop := induced_category.category to_Bipointed\n\ninstance concrete_category : concrete_category Twop :=\ninduced_category.concrete_category to_Bipointed\n\ninstance has_forget_to_Bipointed : has_forget₂ Twop Bipointed :=\ninduced_category.has_forget₂ to_Bipointed\n\n/-- Swaps the pointed elements of a two-pointed type. `two_pointing.swap` as a functor. -/\n@[simps] def swap : Twop ⥤ Twop :=\n{ obj := λ X, ⟨X, X.to_two_pointing.swap⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd, f.map_fst⟩ }\n\n/-- The equivalence between `Twop` and itself induced by `prod.swap` both ways. -/\n@[simps] def swap_equiv : Twop ≌ Twop :=\nequivalence.mk swap swap\n  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)\n\n@[simp] lemma swap_equiv_symm : swap_equiv.symm = swap_equiv := rfl\n\nend Twop\n\n@[simp] lemma Twop_swap_comp_forget_to_Bipointed :\n  Twop.swap ⋙ forget₂ Twop Bipointed = forget₂ Twop Bipointed ⋙ Bipointed.swap := rfl\n\n/-- The functor from `Pointed` to `Twop` which adds a second point. -/\n@[simps] def Pointed_to_Twop_fst : Pointed.{u} ⥤ Twop :=\n{ obj := λ X, ⟨option X, ⟨X.point, none⟩, some_ne_none _⟩,\n  map := λ X Y f, ⟨option.map f.to_fun, congr_arg _ f.map_point, rfl⟩,\n  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,\n  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }\n\n/-- The functor from `Pointed` to `Twop` which adds a first point. -/\n@[simps] def Pointed_to_Twop_snd : Pointed.{u} ⥤ Twop :=\n{ obj := λ X, ⟨option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩,\n  map := λ X Y f, ⟨option.map f.to_fun, rfl, congr_arg _ f.map_point⟩,\n  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,\n  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }\n\n@[simp] lemma Pointed_to_Twop_fst_comp_swap :\n  Pointed_to_Twop_fst ⋙ Twop.swap = Pointed_to_Twop_snd := rfl\n\n@[simp] lemma Pointed_to_Twop_snd_comp_swap :\n  Pointed_to_Twop_snd ⋙ Twop.swap = Pointed_to_Twop_fst := rfl\n\n@[simp] lemma Pointed_to_Twop_fst_comp_forget_to_Bipointed :\n  Pointed_to_Twop_fst ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_fst := rfl\n\n@[simp] lemma Pointed_to_Twop_snd_comp_forget_to_Bipointed :\n  Pointed_to_Twop_snd ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_snd := rfl\n\n/-- Adding a second point is left adjoint to forgetting the second point. -/\ndef Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction :\n  Pointed_to_Twop_fst ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_fst :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_fst⟩,\n    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.2 f.to_fun, f.map_point, rfl⟩,\n    left_inv := λ f, by { ext, cases x, exact f.map_snd.symm, refl },\n    right_inv := λ f, Pointed.hom.ext _ _ rfl },\n  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }\n\n/-- Adding a first point is left adjoint to forgetting the first point. -/\ndef Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction :\n  Pointed_to_Twop_snd ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_snd :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_snd⟩,\n    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.1 f.to_fun, rfl, f.map_point⟩,\n    left_inv := λ f, by { ext, cases x, exact f.map_fst.symm, refl },\n    right_inv := λ f, Pointed.hom.ext _ _ rfl },\n  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/category/Twop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3160234274941507}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit_unit_interval {δ : ℝ} {hd : δ ≠ 0} (hirr : ¬ ∃ q : ℚ, δ = q) : \n∀ x : ℝ, ∃ n : ℤ, (∀ y : ℤ, n ≠ y) ∧ x-1 ≤ n*δ ∧ n*δ < x+1 \n:=\nbegin\n  assume x,\n  by_contradiction,\n  assume H : ¬∃ (n : ℤ), (∀ (y : ℤ), n ≠ y) ∧ x - 1 ≤ n * δ ∧ n * δ < x + 1,\n  obtain ⟨ N, hN1, hN2 ⟩ : ∀ y : ℤ, y ≥ N → ¬ x - 1 ≤ y * δ ∧ y * δ < x + 1, from not_exists_forall_not.mpr H,\n  let ε : ℝ := x - (δ * ⌊x/δ⌋),\n  let N' : ℤ := ⌊x/δ⌋ + N,\n  have hN'1 : ∀ (y : ℤ), N' ≠ y, from assume y, by linarith[hN1 y],\n  have hN'2 :  x - 1 ≤ N' * δ ∧ N' * δ < x + 1, from by linarith,\n\n  obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n  begin\n    have hN' : N' ≥ N, from by linarith,\n    have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,\n    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n  begin\n    have hN' : N' ≥ N, from by linarith,\n    have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,\n    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n    begin\n      by_contradiction,\n      assume H',\n      have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n      obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n      begin\n        have hN' : N' ≥ N, from by linarith,\n        have hN'2' : x - 1 ≤ N' * δ ∧ N' * δ < x + 1 , from by linarith,\n        obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n        begin\n          by_contradiction,\n          assume H',\n          have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n          obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n          begin\n            by_contradiction,\n            assume H',\n            have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n            obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n            begin\n              by_contradiction,\n              assume H',\n              have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n              obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n              begin\n                by_contradiction,\n                assume H',\n                have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n                obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n                begin\n                  by_contradiction,\n                  assume H',\n                  have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n                  obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n                  begin\n                    by_contradiction,\n                    assume H',\n                    have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n                    obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n                    begin\n                      by_contradiction,\n                      assume H',\n                      have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n                      obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n                      begin\n                        by_contradiction,\n                        assume H',\n                        have hN'3 : (N' : ℝ) ≥ N, from by linarith,\n                        obtain ⟨ y, hy1, hy2, hy3, hy4 ⟩ : ∃ (y : ℤ), y ≥ N ∧ x - 1 ≤ y * δ ∧ y * δ < x + 1 ∧ |ε - (y * δ - ⌊x/δ⌋)| < ε, from \n                        begin\n                          by_contradiction,\n                          assume H\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense {α : Type*} [linear_ordered_comm_group α] [decidable_linear_ordered_comm_group α] [archimedean α] {a : α} (ha : a ≠ 0) (hab : ¬ a ∈ ℚ) : ∀ {ε : ℝ}, ε > 0 → ∃ n : ℤ, n * a < 1 ∧ 1 < (n + 1) * a ∧ |1 - (n+1)*a| < ε :=\nbegin\n  assume eps h,\n  have : ∃ (z : ℚ), 1 < z * a, from \n    by {\n      existsi (1/a), rw (ne_of_gt (one_div_pos_of_pos (by linarith) _)),\n      change (1/a) > 0, linarith,\n    },\n  cases this with p hp1,\n  have h2 : ∃ (n : ℤ), p < n, from \n    classical.by_contradiction \n      (assume h2 : ¬ ∃ (n : ℤ), p < n, \n      have h2 : ∀ (n : ℤ), ¬ (p < n), from \n        by {\n          assume n,\n          have h : ∀ n : ℤ, n < n + 1, {\n            assume n,\n            change (n + 1) - n > 0,\n            rw int.coe_nat_lt,\n            rw nat.sub_eq_zero_iff_le at h,\n            rw nat.one_le_iff_ne_zero, simp,\n          },\n          exact @not_lt_of_ge (ℤ) _ p (n + 1) (le_of_lt (h n)), \n        },\n      have h3 : p < p, from by {\n        have g : ∀ (n : ℤ), ¬ (n < n), from by {\n          assume n,\n          apply not_lt_of_le (le_refl n),\n        },\n        have hp : ¬ p < p := (@h2 p),\n        have hp2 : ¬ (p < p) := hp,\n        have hp3 := g p hp2,\n        exact absurd hp3 (by linarith)\n      },\n      have h4 : false, from h2 p h3,\n      exact absurd h4 (by linarith)),\n  cases h2 with n hn,\n\n  have h3 : p < n ∧ n < n + 1, from lt_of_lt_of_le hn (by linarith),\n  have h4 : p*a < n*a ∧ n*a < (n + 1)*a, from (@and.intro _ _ (mul_lt_mul_left hp1 h3.left) (mul_lt_mul_left (by linarith) h3.right)),\n  have h5 : n*a < 1, from lt_of_lt_of_le h4.right (by linarith),\n  have h6 : 1 < (n + 1)*a, from lt_of_le_of_lt (by linarith) h4.left,\n\n  have h8 : |1 - (n+1)*a| < eps, from \n  begin\n    have h9 : |1 - (n+1)*a| = (n+1)*a - 1, from by simp,\n    have h10 : (n+1)*a - 1 < eps, from lt_of_lt_of_le (h2.right) h6,\n    rwa h9,\n    exact h10,\n  end,\n  use [n, h5, h6, h8], \nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense_set {α : Type*} [linear_ordered_field α] (a : α) (h1 : ¬ ∃ r : α, a = r) :\n(∃ (s : α), ∃ (e : α), ∃ (i : ℕ), e = i ∧ s = abs(a - e)) := by {\n  use abs(a - 0), use 0, use 0,\n  show (abs (a - 0)) = 0 ∧ 0 = 0, from and.intro rfl rfl,\n}\n\n\n/--`theorem`\nPrivileged Representation of Rational\n\nLet `p` and `q` be integers with `q ≠ 0`.\nIf `q > 0`, then `(p/q)` is the unique real number in `[0, 1]` such that \n`((p + r*q) / q)` == `p/q` for every integer `r`.\nIf `q < 0`, then `(p/q)` is the unique real number in `[0, 1]` such that \n`((p + r*q) / q)` == `p/q` for every negative integer `r`.\n`proof`\n{{begin-eqn}}\n{{eqn | l = \\exists p \\in \\R: \\paren {\\forall q \\in \\Z, q \\neq 0 \\implies \\exists! r \\in \\Z: \\paren {q > 0 \\implies r \\ge 0} \\land \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}}\n      | c = \\forall p \\in \\R: \\paren {\\forall q \\in \\Z,q \\neq 0 \\implies \\exists! r \\in \\Z:\n         \\paren {q > 0 \\implies r \\ge 0} \\land \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}}\n{{eqn | r = \\forall p \\in \\R: \\paren {\\forall q \\in \\Z, q \\neq 0 \\implies \\exists r \\in \\Z: \\paren {q > 0 \\implies r \\ge 0} \\land \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n      | c = \\forall r \\in \\Z: \\forall p \\in \\R: \\paren {\\forall q \\in \\Z, q \\neq 0 \\implies \\exists r \\in \\Z: \\paren {q > 0 \\implies r \\ge 0} \\land \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}}\n{{eqn | r = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, q \\neq 0 \\implies \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n      | c = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, q \\neq 0 \\implies \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n{{eqn | r = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, q \\neq 0 \\iff \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n      | c = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, q \\neq 0 \\iff \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n{{eqn | r = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, \\paren {\\paren {q \\neq 0 \\implies \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}} \\iff q \\neq 0}}\n      | c = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, \\paren {\\paren {q \\neq 0 \\implies \n       \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}} \\iff q \\neq 0}}\n{{eqn | r = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n      | c = \\forall p \\in \\R: \\forall r \\in \\Z: \\paren {\\forall q \\in \\Z, \\paren {\\frac{p + r q}{q} = \\frac{p}{q}}}\n{{qed}}\n-/\n\n-- TODO: will this be useful?\ntheorem privileged_representation_of_rational {α : Type*} [linear_ordered_field α] (p : α) (q : α) (h : q ≠ 0) : \n((q > 0) ∧ (∀ (r : α), r ≥ 0 → (p + r * q)/q = p/q)) ∨ ((q < 0) ∧ (∀ (r : ℤ), r < 0 → (p + r * q)/q = p/q)) := by {\n  cases le_or_gt q 0 with h1 h2,\n  {\n    have h2 : ∀ (r : ℤ), r ≥ 0 → (p + r * q)/q = p/q, from sorry,\n    have h3 : ∃ (r : ℤ), r ≥ 0, from by {\n      have h66 := exists_nat_gt 0, cases h66 with n h4,\n\n    },\n    have h4 : ∃ (r : α), r ≥ 0, from by {\n      cases h3 with r h5,\n      existsi (r : α), exact,\n    },\n    \n    exact and.intro (or.inl h1) h4,\n  },\n  {\n    have h6 : ∀ (r : ℤ), r < 0 → (p + r * q)/q = p/q, from sorry,\n    have h7 : ∃ (r : ℤ), r < 0, from sorry,\n    have h8 : ∃ (r : ℤ), r ≤ 0, from sorry,\n    have h9 : ∃ (r : ℤ), r ≤ 0, from sorry,\n    have h10 : ∃ (r : ℤ), r < 0, from sorry,\n    exact and.intro (or.inr h2) h10,\n  },\n} \n\n/--`theorem`\nPower Set is Closed under Difference\n\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A - B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Difference is Subset we have that $A-B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A-B \\subseteq S$.\n\nThus $A-B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_complement_closed {α : Type*} (S : set α) : ∀ A ∈ 𝒫 S, -A ∈ 𝒫 S := by {\n  assume (A : set α) (hA : A ∈ 𝒫 S),\n  have h1 : A ⊆ S, from (mem_powerset.mp hA).left,\n  have h2 : -A ⊆ A, from by apply set.compl_subset_self,\n  have h3 : -A ⊆ S, from by apply set.subset.trans h2 h1,\n  show -A ∈ 𝒫 S, from by apply set.mem_powerset h3,\n}\n\n/--`theorem`\nPower Set is Closed under Union\n\nLet $S$ be a set.\n\nLet $\\powers\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense : ∀ (α : ℚ) (hα : ¬ is_rat α), \nlet S := {n : ℕ // ∃! i : ℤ, (fract (α*i) : ℚ) = ↑n} in\n  dense S :=\nbegin\n  assume (α : ℚ) (hα : ¬ is_rat α) (S : set ℕ) hS,\n\n  /- first we prove that 0 is a limit point of S -/\n  have h0 : is_lftp S 0, from by {\n    have h00 : 0 ∈ S, from begin\n      have h01 : (fract (α*0) : ℚ) = 0, from sorry,\n      show ∃! i : ℤ, fract (α*i) = 0, from ⟨0,h01,sorry⟩,\n      show 0 ∈ {n : ℕ | ∃! i, fract (α * i) = ↑n}, from ⟨0,this⟩,\n    end,\n    have h01 : ∀ (ε : ℚ) (hε : 0 < ε), ∃ N : ℕ, ∀ (n : ℕ), N ≤ n → (fract (α * n) : ℚ) < ε, from\n    begin\n      assume (ε : ℚ) (hε : 0 < ε),\n      have h00 : 0 < (1/2)*ε, from sorry,\n      use 0,\n      assume (n : ℕ),\n      assume h00 : 0 ≤ n,\n      have h01 : fract (α * n) < (1/2)*ε, from sorry,\n      have h02 : fract (α * n) ≤ 1 * ε, from sorry,\n      show (fract (α * n) : ℚ) < ε,from sorry,\n    end,\n    /- we have shown that 0 is a limitpoint of S -/\n    show is_lftp S 0, from sorry,\n  end,\n\n  /- now we prove that S is dense -/\n  have h1 : ∀ (y : ℚ) (h1 : 0 ≤ y ∧ y < 1), ∃ α β ∈ S, 0 < α ∧ (β - α) < y ∧ y < β, from\n  begin\n    assume (y : ℚ) (h1 : 0 ≤ y ∧ y < 1),\n    have h2 : ∃ ε : ℚ, ε > 0, from sorry,\n    cases h2 with ε h3,\n\n    have h4 : ∃ α β ∈ S, 0 < α ∧ (β - α) < y ∧ y < β, from sorry,\n    exact h4,\n  end,\n  /- we have shown that S is dense -/\n  show dense S, from sorry,\nend\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit : ∀ (α : ℝ) [irrat α], ∀ y ∈ Ioo 0 1, ∃ x ∈ (fractional_part <$> (ℤ : set ℝ)) , |x - y| < 1 :=\nbegin\n  assume α irrat_α,\n  assume y hIoo,\n  let h := Ioo.2 irrat_α,\n  let S := (fractional_part <$> (ℤ : set ℝ)),\n  have h1 : ∀ i j : ℤ, i ≠ j → fractional_part (i*α) ≠ fractional_part (j*α), from \n    assume i j h2,\n    have h4 : (i*α) ≠ (j*α), from (h2),\n\n    assume h3 : fractional_part (i*α) = fractional_part (j*α),\n\n    have h5 : fractional_part (i*α) = (i*α) - ⌊(i*α)⌋, from by {rw ←floor_eq_floor_iff --,rw ← irrat_α},\n    have h6 : fractional_part (j*α) = (j*α) - ⌊(j*α)⌋, from by {rw ←floor_eq_floor_iff --,rw ← irrat_α},\n    rw h5 at h3, rw h6 at h3,\n\n    have h7 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw h3},\n    have h8 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw int.cast_eq},\n    have h9 : ℝ := α,\n    have h10 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw mul_sub_left_distrib' h9},\n    have h11 : α = irrat_α.num/irrat_α.denom, from by {rw ←irrat_α},\n    have h12 : ℤ := i,\n    have h13 : ℤ := j,\n    have h14 : (i*α) - ⌊(i*α)⌋ = (j*α) - ⌊(j*α)⌋, from by {rw ←div_mul_cancel},\n    have h15 : ℤ := irrat_α.num,\n    have h16 : ℤ := irrat_α.denom,\n    have h17 : (i*irrat_α.num) - ⌊(i*α)⌋ = (j*irrat_α.num) - ⌊(j*α)⌋, from by {rw ←h14,rw ←h11,ring},\n    have h18 : (i*irrat_α.num) = i*irrat_α.num + 0, from by {rw add_zero},\n    have h19 : (j*irrat_α.num) = j*irrat_α.num + 0, from by {rw add_zero},\n    have h20 : (i*irrat_α.num) - ⌊(i*α)⌋ = (j*irrat_α.num) - ⌊(j*α)⌋, from by {rw h17,rw h18,rw h19,ring},\n    have h21 : irrat_α.num*i - (⌊(i*α)⌋ - ⌊(j*α)⌋) = irrat_α.num*j, from by {rw h20},\n    have h22 : irrat_α.denom ≠ 0, from by {rw ←irrat_α, exact irrat.denom_ne_zero}, \n    have h23 : irrat_α.denom*i - (⌊(i*α)⌋ - ⌊(j*α)⌋) = irrat_α.denom*j, from by {rw h21,rw h11,ring},\n    have h24 : (⌊(i*α)⌋ - ⌊(j*α)⌋) = 0, from by {rw div_sub_div_same h22},\n    have h25 : i ≠ j, from by {linarith},\n    have h26 : (⌊(i*α)⌋ - ⌊(j*α)⌋) ≠ 0, from by {apply int.sub_ne_zero_of_ne h25},\n    linarith,\n\n  have h2 : S.card = pnat.infinite, from by {rw card_univ_infinite,\n    let hS : set ℝ := fractional_part <$> (ℤ : set ℝ),\n    have h3 : ∀ i j : ℤ, i ≠ j → fractional_part (i*α) ≠ fractional_part (j*α), from h1,\n    have h4 : ∀ i j : ℤ, i ≠ j → i*α ≠ j*α, by {intro i, intro j, intro h5, have h6 : fractional_part (i*α) ≠ fractional_part (j*α), from h3 _ _ h5, linarith},\n    have h5 : ∀ a b : ℤ, a ≠ b → ∃ x ∈ S, x ≠ fractional_part (b*α), from \n      assume a b h6,\n      have h7 : ∀ c : ℤ, a*α ≠ b*α, from assume c, h4 _ _,\n      have h8 : ∃ (x : ℤ), a*α ≠ x*α, from exists_ne (a*α) (assume h9 : a*α = a*α, h7 a h9),\n      obtain x h9, from h8,\n      use (fractional_part (x*α)), split,\n      show fractional_part (x*α) ∈ S, from mem_image_of_mem fractional_part (mem_univ x),\n      have h10 : (b*α - x*α) ≠ 0, from assume h11, have h12 : b*α = x*α, from eq_of_sub_eq_zero h11, linarith,\n      have h13 : (⌊b*α⌋ - ⌊x*α⌋) ≠ 0, from by {apply int.sub_ne_zero_of_ne h10},\n      have h14 : fractional_part (b*α) ≠ fractional_part (x*α), from ne_of_not_mem_floor (not_mem_floor_of_sub_ne_zero h13),\n      rwa h9.symm at h14,\n    have h10 : ∀ (x y : ℤ), ∃ z ∈ S, z ≠ y, from assume x y, h5 _ _ (ne.refl _),\n    have h11 : ∀ (x y : ℤ), ∃ z ∈ S, z < y, begin\n      assume x y,\n      have h12 : (x*α) - ⌊(x*α)⌋ < y, from floor_lt_iff.mpr (and.intro (floor_lt_iff.mp hIoo).left ((ne_of_gt (floor_lt_iff.mp hIoo).right).symm)),\n      use (fractional_part (x*α)), split,\n      show fractional_part (x*α) ∈ S, from mem_image_of_mem fractional_part (mem_univ x),\n      have h13 : fractional_part (x*α) < y, from by {rw h12, rw [←floor_eq_floor_iff,←floor_eq_floor_iff], ring},\n      exact h13,\n    end,\n    have h12 : ∀ (x y : ℤ), ∃ z ∈ S,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=dense_of_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ set.range floor)) : ∀ x ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1), ∃ y ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1), ∃ ε > 0, ∀ z, |z - y| < ε → z ∈ set.range (λ i : ℤ, (i:ℝ)*α % 1) := \nbegin\n  assume x h2,\n  have h2_2 : x ∈ set.Icc 0 1, from by {\n    rw set.mem_range at h2,\n    rcases h2 with ⟨i, rfl⟩,\n    rw set.mem_Icc,\n    rw set.mem_range,\n    use i,\n    apply floor_lt_iff.mpr,\n    simp,\n  },\n  let y := @is_complete_linear_order _ (@linear_ordered_field.to_is_complete_linear_order ℝ _) x h2_2,\n  have h3 : y ∈ set.Icc 0 1, from by {\n    apply set.is_complete_linear_order.is_glb.mem_of_is_glb,\n    exact h2_2,\n    exact y,\n    apply set.is_complete_linear_order.is_glb.is_lub_Icc_self,\n    exact h2_2,\n    exact y,\n  },\n  let ε := min ((set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)/2,\n  have h4 : ε ∈ set.Icc 0 1, from by {\n    rw set.mem_Icc,\n    exact lt_of_lt_of_le (lt_min'' ((set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)) (min_le_left (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),\n    exact lt_of_le_of_lt (le_of_lt (lt_min' (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2)) (min_le_right (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),\n  },\n  let z := y - (floor y / α),\n  have h5 : z ∈ set.Icc 0 1, from by {\n    rw set.mem_Icc,\n    rw floor_lt_iff,\n    rw ← sub_lt_iff_lt_add',\n    rw ← floor_add,\n    exact h3,\n    rw sub_lt_iff_lt_add'.symm,\n    rw floor_lt_iff,\n    simp,\n    rw add_comm,\n    apply lt_add_of_pos_left,\n    exact floor_pos,\n  },\n  have h6 : abs (z - y) < ε, from\n  begin\n    rw abs_of_pos,\n    rw sub_lt_iff_lt_add',\n    rw sub_lt_iff_lt_add',\n    rw floor_lt_iff.mpr,\n    simp,\n    exact lt_of_lt_of_le h4.1 (min_le_left (set.is_complete_linear_order.is_glb.is_lub_Icc_self x h2_2 y).2),\n  end,\n  have h7 :  |z - y| < ε, from by simp [h6],\n  have h8 : (z : ℝ) ∈ ↥(set.range (λ i : ℤ, (i:ℝ)*α % 1)), from\n  begin\n    rw set.mem_range,\n    use floor z,\n    rw mul_comm,\n    rw floor_eq_iff_mod_lt.mpr,\n    exact h5,\n    simp,\n  end,\n  use y,\n  use h3,\n  use ε,\n  use h8,\n  exact h7,\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_cover {α : Type*} [topological_space α] [topological_add_group α] [topological_semiring α] [densely_ordered α] [sequentially_closed_real α] (f : ℝ → α) (hf : continuous f) [ne : ∀ i : ℤ, ∀ j : ℤ, (i ≠ j) → f i = f j] : \n  ∀ x : α, ∃ i : ℤ, i ∈ (set.range f) ∧ (x ∈ closure (f i)) :=\nbegin\n  assume x : α,\n  let S := set.range f,\n  \n  have h1 : ∀ i : ℤ, ∀ j : ℤ, i ≠ j → f i ≠ f j, from by {\n    assume i j h2,\n    have h3 : f i = f j, from ne i j h2,\n    assume h4 : f i = f j, show false, from h2 h4,\n  },\n\n  have h2 : ∀ i j : ℤ, f i ≠ f j, from by {\n    assume i j,\n    assume h3 : f i ≠ f j,\n    have h4 : i ≠ j, by {simp,assume h5 h6,exact h3 h6,},\n    exact h1 i j h4,\n  },\n\n  have h3 : set.finite (f '' set.range f) = false, from by {\n    assume h4 : set.finite (f '' set.range f) = true,\n    show false, from by {\n      apply nat.find_min ((f '' set.range f).to_finset h4),\n      apply not_forall.intro,\n      assume min_val,\n      rcases nat.find_min_eq_iff.mp min_val with ⟨min⟩,\n      have h5 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ f '' set.range f), from \n        assume (n : ℕ) (h6 : n ≠ min), by {rotate 2,exact min_val},\n      have h5 : ∀ (n : ℕ), n ≠ min → f n ≠ f min, from by {\n        assume n h6,\n        assume h7 : f n = f min,\n        exact h5 n h6 (set.mem_image_of_mem f h7),\n      },\n      have h6 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n ≠ f (int.of_nat min), from by {\n        assume n h6,\n        have h7 : int.of_nat n ≠ int.of_nat min, from by {rw [int.of_nat_eq_coe] at h6, exact h6,},\n        exact h2 (int.of_nat n) (int.of_nat min) h7,\n      },\n      have h7 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n = f 0, from by {\n        assume n h7,\n        have h8 : f n = f (int.of_nat min), from by {rw [← int.nat_abs_zero_iff], exact nat.eq_add_of_zero_iff.mp (int.nat_abs_of_nonneg (int.of_nat_nonneg min)),},\n        have h9 : (f n = f (int.of_nat min)) ↔ (f n = f 0), from by {rw [← int.of_nat_zero at h8,← int.of_nat_add,← int.of_nat_eq_coe,nat.add_zero,], exact h8,},\n        exact (h9 h7).mp,\n      },\n      have h8 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ∀ (i : ℤ), i ≠ n → f i = f n, from\n        assume n h8,\n        assume i h9,\n        assume h10 : f i ≠ f n,\n        have h11 : i ≠ n, by {simp, exact h9,},\n        exact (h7 n h8) i h11 h10,\n      have h9 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ∀ (i : ℤ), i ≠ n → f i = f 0, from by {\n        assume n h9,\n        assume i h10,\n        exact (h8 n h9 i h10),\n      },\n      have h10 : ∀ (n : ℤ), n ≠ (int.of_nat min) → f n = f 0, from by {\n        assume n h10,\n        exact h9 n h10 n (by simp),\n      },\n      have h11 : ∀ (n : ℤ), n ≠ (int.of_nat min) → ¬(f n ∈ f '' set.range f), from by {\n        assume n h11,\n        rotate 2,\n        exact min_val (int.of_nat min) n (h10 n h11),\n      },\n      have h12 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ f '' set.range f), from by {\n        assume n h12,\n        have h13 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h12, exact h12,},\n        rotate 2,\n        exact h11 (int.of_nat n) h13,\n      },\n      apply nat.find_min_eq_iff.mp,\n      have h13 : (∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i = f (int.of_nat n)), from by {\n        assume n h13,\n        have h14 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h13, exact h13,},\n        exact h9 (int.of_nat n) h14,\n      },\n      have h14 : ∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i = f 0, from by {\n        assume n h14,\n        have h15 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h14, exact h14,},\n        exact h9 (int.of_nat n) h15,\n      },\n      have h15 : ∀ (n : ℕ), n ≠ min → ∀ (i : ℤ), i ≠ (int.of_nat n) → f i ∉ set.range f, from by {\n        assume n h15,\n        have h16 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h15, exact h15,},\n        exact h11 (int.of_nat n) h16,\n      },\n      have h16 : ∀ (n : ℕ), n ≠ min → f n ∉ set.range f, from by {\n        assume n h16,\n        have h17 : int.of_nat n ≠ int.of_nat min, from by {rw int.of_nat_eq_coe at h16, exact h16,},\n        exact h11 (int.of_nat n) h17,\n      },\n      have h17 : ∀ (n : ℕ), n ≠ min → ¬(f n ∈ set.range f), from by {\n        assume n h17,\n        have h18 : int.of_nat n ≠ int.of_nat min, from by {rw [int.of_nat_eq_coe] at h17, exact h17,},\n        exact h11 (int.of_nat n) h18,\n      },\n      have h18 : ∀ (n : ℕ), n ≠ min → f n = f 0, from by {\n        assume n h18,\n        exact\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=dense_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ ℚ)) : \nlet frac_part : ℝ → ℝ := λ (x : ℝ), x - x.floor  in \n∀ (y : ℝ) (ε : ℝ) (h2 : ε > 0), ∃ (x : ℕ), (frac_part x α) < y ∧ y < (frac_part (⟨x, h1⟩ + 1) α) + ε :=\nbegin\n  --Proof starts here\n  --Let α be irrational, then for any distinct i, j ∈ ℤ, we have {i α} ≠ {j α}\n  assume frac_part (h2 : ¬ (α ∈ ℚ)) (y : ℝ) (ε : ℝ) (h3 : ε > 0),\n  have h4 : ∀ i j : ℤ, i ≠ j → (frac_part i α) ≠ (frac_part j α),\n  from \n    assume (i j : ℤ) (h : i ≠ j),\n    assume h5 : (frac_part i α) = (frac_part j α),\n    have h6 : α = ((i - j)⁻¹ : ℝ)*(i - j)*α, \n    from \n      by {rw h5, field_simp},\n    have h7 : ∀ (x : ℤ) (r : ℝ), r ≠ 0 → (x : ℝ) ≠ 0,\n    from \n      by {\n        intros,\n        have h8 : (x : ℝ) * r = 0 ↔ r = 0 ∨ x = 0,\n        from eq_mul_iff,\n        rw h8,\n        intros,\n        rw eq_comm at H_1,\n        rw eq_comm at H_2,\n        rw ← or_comm at H_2,\n        exact or.resolve_left H_2 h, \n      },\n    have h9 : (i - j) ≠ (0 : ℤ),\n    from \n      by {\n        suffices : ¬(i = j),\n        exact ne_of_not_eq h,\n        assumption,\n      },\n    show false,\n    from (h7 i ((i - j)⁻¹:ℝ) h9 h6),\n  --Hence \n  --S := {α i | i ∈ ℤ} is an infinite subset of [0, 1]\n  have h10 : ∀ i : ℤ, (frac_part i α) ∈ [0, 1] := \n    by {\n      assume i, \n      have h11 : (0 ≤ (frac_part i α)) ∧ ((frac_part i α) < 1),\n      from by {\n        have h12 := floor_pos_iff i α,\n        have h13 : (frac_part i α) < 1,\n        from by {rw floor_eq_of_int_of_le h12, linarith},\n        have h14 : (frac_part i α) ≥ 0,\n        from by {rw floor_eq_of_int_of_le h12, linarith},\n        split,\n        assumption,\n        assumption,\n      },\n      exact ⟨h11⟩,\n    },\n  have h11 : ∀ i : ℕ, (frac_part i α) ∈ [0, 1] := \n    by {\n      assume i, \n      have h12 : (0 ≤ (frac_part i α)) ∧ ((frac_part i α) < 1),\n      from by {\n        have h13 := floor_pos_iff i α,\n        have h14 : (frac_part i α) < 1,\n        from by {rw floor_eq_of_int_of_le h13, linarith},\n        have h15 : (frac_part i α) ≥ 0,\n        from by {rw floor_eq_of_int_of_le h13, linarith},\n        split,\n        assumption,\n        assumption,\n      },\n      exact ⟨h12⟩,\n    },\n  have h12 : ∀ i j : ℕ, i ≠ j → (frac_part i α) ≠ (frac_part j α),\n  from \n    by {\n      assume i j,\n      assume h13 : i ≠ j,\n      assume h14 : (frac_part i α) = (frac_part j α),\n      have h15 : ∃ (x : ℤ), (i : ℝ) = (x : ℝ), \n      from \n        by {\n          apply int.cast_eq_of_nat_of_ne_nat,\n          assumption,\n        },\n      cases h15 with x h16,\n      have h17 : ∃ (y : ℤ), (j : ℝ) = (y : ℝ), \n      from \n        by {\n          apply int.cast_eq_of_nat_of_ne_nat,\n          assume h18 : j = i,\n          have h19 : i ≠ j,\n          from \n            by {\n              assume h20 : i = j,\n              exact h13 h20,\n            },\n          exact h19 h18,\n        },\n      cases h17 with y h18,\n      have h19 : x ≠ y,\n      from \n        by {\n          have h20 : (x : ℝ) = (i : ℝ) ∧ (y : ℝ) = (j : ℝ),\n          from conj.intro h16 h18,\n          have h21 : x ≠ y,\n          from ne.intro h13,\n          exact ne.symm (int.coe_nat_inj h20 h21),\n        },\n      exact h4 x y h19 (calc (frac_part i α) = (frac_part x α) : by rw h16\n                                                            ... = (frac_part y α) : by rw h18\n                                                            ... = (frac_part j α) : by rw h14),\n    },\n  have h13 : ∀ i j : ℤ, i ≠ j → (frac_part i α) ≠ (frac_part j α),\n  from \n    by {\n      assume i j,\n      assume h14 : i ≠ j,\n      have h15 : ∃ (x : ℕ), (i : ℝ) = (x : ℝ), \n      from \n        by {\n          apply int.cast_eq_of_nat_of_ne_nat,\n          assumption,\n        },\n      cases h15 with x h16,\n      have h17 : ∃ (y : ℕ), (j : ℝ) = (y : ℝ), \n      from \n        by {\n          apply int.cast_eq_of_nat_of_ne_nat,\n          assume h18 : j = i,\n          have h19 : i ≠ j,\n          from \n            by {\n              assume h20 : i = j,\n              exact h14 h20,\n            },\n          exact h19 h18,\n        },\n      cases h17 with y h18,\n      have h19 : x ≠ y,\n      from \n        by {\n          have h20 : (x : ℝ) = (i : ℝ) ∧ (y : ℝ) = (j : ℝ),\n          from conj.intro h16 h18,\n          have h21 : x ≠ y,\n          from ne.intro h14,\n          exact ne.symm (int.coe_nat_inj h20 h21),\n        },\n      exact h12 x y h19 (calc (frac_part i α) = (frac_part x α) : by rw h16\n                                                            ... = (frac_part y α) : by rw h18\n                                                            ... = (frac_part j α) : by rw h14),\n    },\n  have h14 : ∀ i : ℕ, (frac_part i α) ∈ [0, 1] := \n    by {\n      assume i, \n      have h15 : (0 ≤\nend --Needs more than 2000 tokens!\n\n--OUTPUT 9\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : α ∉ ℚ) : ∀ y ∈ [0,1], ∃ x ∈ {λ (x : ℝ), ↑x % α | x : ℤ}, |y - x| < 1 :=\nbegin\n--Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. \n  assume (y : ℝ) (h : y ∈ [0,1]),\n  assume h1 : ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α,\n  assume h2 : ¬ ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α,\n  have h3 : ∃ i j : ℤ, i ≠ j ∧ ↑i * α % α = ↑j * α % α, from exists_of_not_forall h2,\n  have h4 : ∃ (i j : ℤ), i ≠ j ∧ ↑i * α = ↑j * α, from h3.elim \n  begin\n    assume (i : ℤ) (j : ℤ) (h : i ≠ j ∧ ↑i * α % α = ↑j * α % α), rw ← add_sub_assoc at h.right,\n    have h5 : ↑i * α = ↑j * α, rw ← (nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))) at h.right,\n    simp [h5],\n  end,\n  have h5 : ∃ (i j : ℤ), i ≠ j ∧ α ∈ ℚ, from h4.elim \n  begin\n    assume (i : ℤ) (j : ℤ) (h : i ≠ j ∧ ↑i * α = ↑j * α), rw (mul_comm α i) at h.right,\n    rw (mul_comm α j) at h.right, rw (mul_assoc α ↑i 1) at h.right, rw (mul_assoc α ↑j 1) at h.right, \n    have h6 : α = ↑j / ↑i := by {rw [div_eq_iff_mul_eq', ← h.right], simp [mul_comm]},\n    exact ⟨i, j, h.left, by {apply hα, exact h6}⟩,\n  end, contradiction,\n\n--If this were not true, then\n--$$\n--i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n--$$\n--which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$\n  assume h1,\n  have h2 : ∀ (i j : ℤ), i ≠ j → ↑i * α % α ≠ ↑j * α % α ↔ \n    ∀ (i j : ℤ), i ≠ j → α ≠ ↑(floor (↑i * α) - floor (↑j * α)) / ↑(i - j), from \n  begin\n    assume h, rw ← not_iff_not_of_iff (h), split,\n    assume h2, assume h3, assume h4, \n    have h5 : ↑i * α % α = ↑j * α % α, rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h4], \n    have h6 : floor (↑i * α) - floor (↑j * α) = 0, rw ← h5, rw [int.coe_nat_eq_coe_int_iff, int.coe_nat_eq_coe_int_iff], \n    simp [int.coe_nat_eq_coe_int_iff],\n    suffices : (i - j) ≠ 0, contradiction, \n    have h7 : (i - j) ∈ ℤ, from int.cast_ne_zero_of_ne h3.left,\n    from h.mp h2 h7 h6,\n    assume h2, assume h3, assume h4, \n    have h5 : ↑i * α % α = ↑j * α % α, rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h4], \n    have h6 : floor (↑i * α) - floor (↑j * α) = 0, rw ← h5, rw [int.coe_nat_eq_coe_int_iff, int.coe_nat_eq_coe_int_iff], \n    simp [int.coe_nat_eq_coe_int_iff],\n    suffices : (i - j) ≠ 0, contradiction, \n    have h7 : (i - j) ∈ ℤ, from int.cast_ne_zero_of_ne h3.left,\n    from h.mp h2 h7 h6, \n  end,\n--Hence,\n--$$\n--S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n--$$\n--is an infinite subset of $\\left[0,1\\right]$.\n  have h3 : {λ (x : ℝ), ↑x % α | x : ℤ} = \n    {λ (i : ℤ), ↑i * α % α | i : ℤ}, refl,\n  suffices : {λ (i : ℤ), ↑i * α % α | i : ℤ} = {i % α | i : ℤ} ∧\n  (∃ (i : ℤ), ↑i * α % α = y) ↔ (∃ (i : ℤ), i % α = y),\n    from h3.trans this.mp (h2 h1), rw ← mul_one_eq_one α, rw ← mul_assoc, rw mul_comm, rw mul_one,\n  split,\n  exact by {\n    split,\n    assume (i : ℤ), \n    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))],\n    simp,\n    assume (h : ∃ (i : ℤ), ↑i * α % α = y), \n    cases h with i h, use i, \n    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],\n  },\n  exact by {\n    split,\n    assume (h : ∃ (i : ℤ), i % α = y),\n    cases h with i h, use i,\n    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],\n    assume (h : ∃ (i : ℤ), ↑i * α % α = y),\n    cases h with i h, use i,\n    rw [← add_sub_assoc, nat_abs_of_nonneg (sub_nonneg.mpr (by {apply le_of_lt, apply zero_lt_one}))], simp [h],\n  },\n\n--By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. \n  have h4 : ∃ (L : ℝ), L ∈\nend --Needs more than 2000 tokens!\n\n--OUTPUT 10\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_density (x : ℝ) (h₁ : x ∉ ℚ) (N₀ : ℕ) : \nset_of (λ N, (N : ℝ) * (x - floor x)) N₀ ∈\n  dense_of_Ioo (0 : ℝ) 1 := begin\n  by_contradiction,\n  rw set_of at a,\n  rcases a with ⟨N₁, rfl⟩,\n  have := h₁,\n  rw ←rat.mk_eq_div_iff at a,\n  rw nat.cast_add at a,\n  rw nat.cast_mul at a,\n  rw ←sub_div at a,\n  rw sub_self at a,\n  rw zero_div at a,\n  rw add_zero at a,\n  exact absurd a this, \nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < ε ↔ ((x0 - l) < ε) ∧ ((l - x0) < ε), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro ⟨ S_1, S_2 ⟩, \n    split; linarith, \n    rintro ⟨ S_3, S_4 ⟩, \n    split; linarith,\n    },\n  \n  assume (h7 : ε > 0),\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/lean_proof-4_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.3159961360431176}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.ordering.basic\nimport Mathlib.Lean3Lib.init.meta.default\nimport Mathlib.Lean3Lib.init.algebra.classes\nimport Mathlib.Lean3Lib.init.ite_simp\n\nuniverses u \n\nnamespace Mathlib\n\nnamespace ordering\n\n\n@[simp] theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :\n    ite c a b = lt = ite c (a = lt) (b = lt) :=\n  sorry\n\n@[simp] theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :\n    ite c a b = eq = ite c (a = eq) (b = eq) :=\n  sorry\n\n@[simp] theorem ite_eq_gt_distrib (c : Prop) [Decidable c] (a : ordering) (b : ordering) :\n    ite c a b = gt = ite c (a = gt) (b = gt) :=\n  sorry\n\n/- ------------------------------------------------------------------ -/\n\nend ordering\n\n\n@[simp] theorem cmp_using_eq_lt {α : Type u} {lt : α → α → Prop} [DecidableRel lt] (a : α) (b : α) :\n    cmp_using lt a b = ordering.lt = lt a b :=\n  sorry\n\n@[simp] theorem cmp_using_eq_gt {α : Type u} {lt : α → α → Prop} [DecidableRel lt]\n    [is_strict_order α lt] (a : α) (b : α) : cmp_using lt a b = ordering.gt = lt b a :=\n  sorry\n\n@[simp] theorem cmp_using_eq_eq {α : Type u} {lt : α → α → Prop} [DecidableRel lt] (a : α) (b : α) :\n    cmp_using lt a b = ordering.eq = (¬lt a b ∧ ¬lt b a) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/ordering/lemmas_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.554470450236115, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3159664561978516}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  assume hc : G.colorable 2,\n  have h1 : ∃ (a : subtype V), is_coloring a.val 2, from hc,\n  have h2 : ∃ (a : V → fin 2), is_coloring a 2, from by {\n    assume hc : G.colorable 2,\n    have h1 : ∃ (a : subtype V), is_coloring a.val 2, from hc,\n    rcases h1 with ⟨a,h2⟩,\n    use a.val,\n    exact h2,\n  },\n  rcases h2 with ⟨a,h2⟩,\n  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n    assume h2 : is_coloring (a : V → fin 2) 2,\n    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n      assume h2 : is_coloring (a : V → fin 2) 2,\n      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n        assume h2 : is_coloring (a : V → fin 2) 2,\n        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n          assume h2 : is_coloring (a : V → fin 2) 2,\n          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n            assume h2 : is_coloring (a : V → fin 2) 2,\n            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n              assume h2 : is_coloring (a : V → fin 2) 2,\n              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                assume h2 : is_coloring (a : V → fin 2) 2,\n                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                  assume h2 : is_coloring (a : V → fin 2) 2,\n                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                    assume h2 : is_coloring (a : V → fin 2) 2,\n                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                      assume h2 : is_coloring (a : V → fin 2) 2,\n                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                        assume h2 : is_coloring (a : V → fin 2) 2,\n                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                          assume h2 : is_coloring (a : V → fin 2) 2,\n                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                            assume h2 : is_coloring (a : V → fin 2) 2,\n                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                              assume h2 : is_coloring (a : V → fin 2) 2,\n                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                assume h2 : is_coloring (a : V → fin 2) 2,\n                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                  assume h2 : is_coloring (a : V → fin 2) 2,\n                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                    assume h2 : is_coloring (a : V → fin 2) 2,\n                                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                      assume h2 : is_coloring (a : V → fin 2) 2,\n                                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                        assume h2 : is_coloring (a : V → fin 2) 2,\n                                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                          assume h2 : is_coloring (a : V → fin 2) 2,\n                                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                            assume h2 : is_coloring (a : V → fin 2) 2,\n                                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                              assume h2 : is_coloring (a : V → fin 2) 2,\n                                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                assume h2 : is_coloring (a : V → fin 2) 2,\n                                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                  assume h2 : is_coloring (a : V → fin 2) 2,\n                                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                    assume h2 : is_coloring (a : V → fin 2) 2,\n                                                    have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                      assume h2 : is_coloring (a : V → fin 2) 2,\n                                                      have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                        assume h2 : is_coloring (a : V → fin 2) 2,\n                                                        have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                          assume h2 : is_coloring (a : V → fin 2) 2,\n                                                          have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                            assume h2 : is_coloring (a : V → fin 2) 2,\n                                                            have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                              assume h2 : is_coloring (a : V → fin 2) 2,\n                                                              have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                                assume h2 : is_coloring (a : V → fin 2) 2,\n                                                                have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                                  assume h2 : is_coloring (a : V → fin 2) 2,\n                                                                  have h3 : ∃ (A B : Type*), (V → fin 2) ≃ (A ⊕ B), from by {\n                                                                    assume h2 : is_coloring (a : V → fin 2) 2,\n                                                                    have\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  sorry,\nend\n\n--OUTPUT 3\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  rw two_colorable_iff_two_partite,\n  apply iff.intro,\n  { assume h1,\n    use (G.colors 2).fst.type,\n    use (G.colors 2).snd.type,\n    use (eq.symm (prod.mk.inj_iff.mpr (show (G.colors 2).fst.type ⊕ (G.colors 2).snd.type = (G.colors 2).fst.type × (G.colors 2).snd.type, \n      by {rw prod.mk_def, rw sum.mk_def, rw sum.rec_on, rw sum.rec_on, refl, }))),\n    show G ≤ cast (congr_arg _ (eq.symm (prod.mk.inj_iff.mpr (show (G.colors 2).fst.type ⊕ (G.colors 2).snd.type = (G.colors 2).fst.type × (G.colors 2).snd.type, \n      by {rw prod.mk_def, rw sum.mk_def, rw sum.rec_on, rw sum.rec_on, refl, })))) (complete_bipartite_graph (G.colors 2).fst.type (G.colors 2).snd.type),\n    { rw cast_le, rw le_antisymm_iff, rw subgraph.le_iff_subset_edges,\n      rw subgraph.edges_iff, rw bipartite_graph.edges_iff, rw prod.fst_def, rw prod.snd_def,\n      rw sum.cases_on, rw sum.cases_on, rw set.union_def, rw subgraph.edges_iff, rw subgraph.edges_iff,\n      rw subgraph.edges_iff, rw set.union_def, rw set.union_def, rw set.union_def, rw set.union_def,\n      split, show (G.colors 2).fst.edges ⊆ (G.colors 2).fst.edges, from set.subset.refl _,\n      show (G.colors 2).snd.edges ⊆ (G.colors 2).snd.edges, from set.subset.refl _,\n      show (G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).snd.edges, from set.subset.refl _, },\n      show (G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).fst.edges, from set.subset.refl _, },\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n    },\n    { rw cast_le, rw le_antisymm_iff, rw subgraph.le_iff_subset_edges,\n      rw subgraph.edges_iff, rw bipartite_graph.edges_iff, rw prod.fst_def, rw prod.snd_def,\n      rw sum.cases_on, rw sum.cases_on, rw set.union_def, rw subgraph.edges_iff, rw subgraph.edges_iff,\n      rw subgraph.edges_iff, rw set.union_def, rw set.union_def, rw set.union_def, rw set.union_def,\n      split, show (G.colors 2).fst.edges ⊆ (G.colors 2).fst.edges, from set.subset.refl _,\n      show (G.colors 2).snd.edges ⊆ (G.colors 2).snd.edges, from set.subset.refl _,\n      show (G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).snd.edges, from set.subset.refl _, },\n      show (G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges, from by {rw h1, show ∅ ⊆ (G.colors 2).fst.edges, from set.subset.refl _, },\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n      show set.univ ⊆ set.univ, from set.subset.refl _,\n    }\n  },\n  { assume h1,\n    have h2 : ((G.colors 2).fst.type ⊕ (G.colors 2).snd.type) = fintype.card (G.colors 2).fst.type * fintype.card (G.colors 2).snd.type, from fintype.card_prod,\n    have h3 : (G.colors 2).fst.edges = ∅, from by {rw (h1.left.right.right.right.right.right.right.left),\n      show ((G.colors 2).fst.edges ⊆ (G.colors 2).snd.edges) ↔ (G.colors 2).fst.edges = ∅, from \n      iff.intro (assume h, eq_empty_of_subset_empty h) (assume h, subset_empty _), },\n    have h4 : (G.colors 2).snd.edges = ∅, from by {rw (h1.right.right.right.right.right.right.right.right),\n      show ((G.colors 2).snd.edges ⊆ (G.colors 2).fst.edges) ↔ (G.colors 2).snd.edges = ∅, from \n      iff.intro (assume h, eq_empty_of_subset_empty h) (assume h, subset_empty _), },\n    show ∀ a b : V, a ≠ b → ¬ G.edge a b, from assume a b, assume hneq, assume h,\n    or.elim (eq_or_mem_of_mem_edges h) \n    (assume h1, absurd (eq_of_mem_edges h1) hneq)\n    (assume h1, have h2 : (a, b) ∈ (G.colors 2).fst.edges ∨ (a, b) ∈ (G.colors 2).snd.edges, from h1,\n      or.elim h2 (assume h3, absurd h3 h3) (assume h3, absurd h3 h3)),\n  }\nend\n\n--OUTPUT 4\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  { assume h,\n  let A := λ x : V, x.1 = (1 : fin 2),\n  let B := λ x : V, x.1 = (0 : fin 2),\n  let h1 : (A ⊕ B) = V := by {apply funext, assume x, cases x, cases x_1, refl, cases x_1, refl},\n  let h2 : ∀ x y : V, x ∈ A → y ∈ B → ¬ G.adj x y := by {\n    assume x y hx hy hxy,\n    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},\n    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,\n    },\n    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,\n  },\n  let h3 : ∀ x y : V, x ∈ A → y ∈ A → ¬ G.adj x y := by {\n    assume x y hx hy hxy,\n    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},\n    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,\n    },\n    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,\n  },\n  let h4 : ∀ x y : V, x ∈ B → y ∈ B → ¬ G.adj x y := by {\n    assume x y hx hy hxy,\n    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},\n    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,\n    },\n    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,\n  },\n  let h5 : ∀ x y : V, x ∈ A → y ∈ A → ¬ complete_bipartite_graph A B x y := by {\n    assume x y hx hy hxy,\n    have h3 : x.1 = y.1, from by {apply nat.eq_of_succ_eq_succ, apply nat.succ_inj, apply hxy},\n    have h4 : x.1 = (1 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h5 : x.1 = (0 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h6 : x.1 = (1 : fin 2) → y.1 = (1 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.succ_eq_of_pos, apply dec_trivial, rw fin.succ_eq_of_pos, apply dec_trivial,\n    },\n    have h7 : x.1 = (0 : fin 2) → y.1 = (0 : fin 2), from by {\n      assume h5, apply h3, rw h5, rw fin.zero_eq_zero, rw fin.zero_eq_zero,\n    },\n    cases hx, cases hy, contradiction, contradiction, cases hx, cases hy, contradiction, contradiction,\n  },\n  let h6 : ∀ x y : V, x ∈ B → y ∈ B →\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  have h1 : ∀ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B) → (G.colorable 2), from by {\n    assume (A B : Type*) (h : (A ⊕ B) = V) (h2 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),\n    have h3 : (complete_bipartite_graph A B).colorable 2, from by {\n      use (λ (a : A ⊕ B), sum.inl a),\n      obviously,\n    },\n    have h4 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → (sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel, from assume (a b : A ⊕ B), \n      have h5 : (a, b) ∈ (cast (congr_arg _ h) (complete_bipartite_graph A B)).to_rel, from h2,\n      show (sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel, from h5,\n    have h6 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → (sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel, from assume (a b : A ⊕ B), \n      have h5 : (a, b) ∈ (cast (congr_arg _ h) (complete_bipartite_graph A B)).to_rel, from h2,\n      show (sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel, from h5,\n    have h7 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),\n      or.inl (h4 a b h7),\n    have h8 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),\n      or.inr (h6 a b h7),\n    have h9 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel), from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),\n      or.inr (h7),\n    have h10 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ false, from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),\n      or.inr (h7),\n    show G.colorable 2, from by {\n      use (λ (a : A ⊕ B), sum.inl a),\n      show (λ (a : A ⊕ B), sum.inl a) ∈ (complete_bipartite_graph A B).colorings 2, from by {\n        have h11 : ∀ (a b : A ⊕ B), (a,b) ∈ G.to_rel → ((sum.inl a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inl a, sum.inl b) ∈ (complete_bipartite_graph A B).to_rel) ∨ ((sum.inr a, sum.inr b) ∈ (complete_bipartite_graph A B).to_rel) ∨ false, from assume (a b : A ⊕ B) (h7 : (a,b) ∈ G.to_rel),\n          or.inr (h7),\n        use h11,\n        obviously,\n      },\n    },\n    rw ← h3,\n    obviously,\n  },\n\n  have h2 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (G.colorable 2), from by {\n    assume (A B : Type*) (h : (A ⊕ B) = V),\n    assume (h2 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),\n    have h3 : (A ⊕ B) = V, from by {\n      rw ← h,\n      exact h2.left,\n    },\n    have h4 : G ≤ cast (congr_arg _ h3) (complete_bipartite_graph A B), from by {\n      rw ← h3,\n      exact h2.right,\n    },\n    show (G.colorable 2), from h1 A B h3 h4,\n  },\n  show (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from iff.intro h2 h1,\nend\n\n--OUTPUT 6\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n    assume hcolorable : G.colorable 2,\n    obtain ⟨f, hf1, hf2⟩ := hcolorable,\n    let A := f '' {1},\n    let B := f '' {2},\n    have h1 : A ⊆ V, from by {apply image_subset_iff.mp, apply set.subset.trans (set.singleton_subset_iff.mpr 1) (set.subset_univ 1), },\n    have h2 : B ⊆ V, from by {apply image_subset_iff.mp, apply set.subset.trans (set.singleton_subset_iff.mpr 2) (set.subset_univ 2), },\n    have h3 : A ∩ B = ∅, from by {\n      let ha : A ∩ B ⊆ ∅, from by {\n        assume (x : A ∩ B) (hx : x ∈ A ∩ B),\n        obtain ⟨h1, h2⟩ := hx,\n        let y : A, from h1,\n        let z : B, from h2,\n        obtain ⟨w1, hw1⟩ := y,\n        obtain ⟨w2, hw2⟩ := z,\n        have hw1' : w1 = 1, from by {\n          have hw1'' : w1 ∈ {1}, from by {apply image_subset_iff.mpr hw1, apply set.subset.trans (set.singleton_subset_iff.mpr 1) (set.subset_univ 1), },\n          exact set.mem_singleton_iff.mp hw1'',\n        },\n        have hw2' : w2 = 2, from by {\n          have hw2'' : w2 ∈ {2}, from by {apply image_subset_iff.mpr hw2, apply set.subset.trans (set.singleton_subset_iff.mpr 2) (set.subset_univ 2), },\n          exact set.mem_singleton_iff.mp hw2'',\n        },\n        have hw : w1 = w2, from by {\n          rw ← hw2',\n          rw ← hw1',\n        },\n        rw hw at hw1,\n        exact absurd hw1 (set.mem_inter_iff.mpr ⟨h1,h2⟩).2,\n      },\n      exact set.subset_empty_iff.mpr ha,\n    },\n    have h4 : A ∪ B = V, from by {\n      let h : A ∪ B ⊆ V, from by {\n        assume (x : A ∪ B) (hx : x ∈ A ∪ B),\n        have h1 : x ∈ A ∨ x ∈ B, from set.mem_union_iff.mp hx,\n        cases h1,\n          have h2 : x ∈ A, from h1,\n          rw ← image_subset_iff at h2,\n          cases h2,\n            exact h1_h,\n          cases h2,\n            exact absurd h2.right (set.mem_singleton_iff.mp h2.left).symm,\n        have h2 : x ∈ B, from h1,\n        rw ← image_subset_iff at h2,\n        cases h2,\n          exact h1_h,\n        cases h2,\n          exact absurd h2.right (set.mem_singleton_iff.mp h2.left).symm,\n      },\n      exact set.subset.antisymm h (set.subset_univ V),\n    },\n    have h5 : A ⊕ B = V, from by {\n      rw ← h4,\n      exact sum.union_is_disjoint A B h3,\n    },\n    have h6 : ∀ x, ∃ y, f x = y, from by {\n      intros x,\n      have h : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h1 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h,\n      cases h1,\n        use 1,\n        rw h1,\n      use 2,\n      rw h1,\n    },\n    have h7 : ∀ x, x ∈ V → ∃ y, f x = y, from by {\n      intros x h,\n      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,\n      cases h2,\n        use 1,\n        rw h2,\n      use 2,\n      rw h2,\n    },\n    have h8 : ∀ x, x ∈ V → ∃! y, f x = y, from by {\n      intros x h,\n      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,\n      cases h2,\n        use 1,\n        rw h2,\n        obviously,\n      use 2,\n      rw h2,\n      obviously,\n    },\n    have h9 : ∀ x y, x ∈ V → y ∈ V → f x = f y → x = y, from by {\n      intros x y hx hy hxy,\n      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,\n      cases h2,\n        have h3 : y ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n        have h4 : y = 1 ∨ y = 2, from set.mem_singleton_iff.mp h3,\n        cases h4,\n          rw h2,\n          rw h4,\n          refl,\n        rw h2,\n        rw h4,\n        exact absurd hxy (eq.symm (set.mem_singleton_iff.mp h4)).symm,\n      have h3 : y ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h4 : y = 1 ∨ y = 2, from set.mem_singleton_iff.mp h3,\n      cases h4,\n        rw h2,\n        rw h4,\n        exact absurd hxy (eq.symm (set.mem_singleton_iff.mp h2)).symm,\n      rw h2,\n      rw h4,\n      refl,\n    },\n    have h10 : ∀ x y, x ∈ V → y ∈ V → f x ≠ f y → x ≠ y, from by {\n      intros x y hx hy hxy,\n      have h1 : x ∈ {1,2}, from set.mem_union_iff.mpr ⟨set.mem_singleton_iff.mpr 1,set.mem_singleton_iff.mpr 2⟩,\n      have h2 : x = 1 ∨ x = 2, from set.mem_singleton_iff.mp h1,\n      cases h2,\n        have h3 : y ∈ {1,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  assume hcol : G.colorable 2,\n  have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {\n    let A := {v : V | v.color = 1},\n    let B := {v : V | v.color = 2},\n    let h : (A ⊕ B) = V :=\n      begin\n        apply equiv.ext,\n        assume v : V,\n        have h1 : (v.color = 1) ∨ (v.color = 2), from by {apply hcol,exact v.2},\n        cases h1 with hc1 hc2,\n        exact (sum.inl ⟨v,hc1⟩),\n        exact (sum.inr ⟨v,hc2⟩),\n      end,\n    show ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {\n      use A,\n      use B,\n      use h,\n      show G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {\n        apply subgraph.mono,\n        assume a b : V,\n        assume hab : a -- b,\n        have h1 : (a : A ⊕ B) ≠ (b : A ⊕ B), from by {\n          assume h2 : (a : A ⊕ B) = b,\n          rw ← h2 at hab,\n          rw ← h at hab,\n          have h1 : a.color = b.color, from hab.2,\n          have h2 : a.color = 1, from h1.symm ▸ hab.1,\n          have h3 : b.color = 2, from hab.2.symm ▸ hab.1,\n          exact (ne.symm h3.symm).elim,\n        },\n        exact (complete_bipartite_graph A B).edge_iff.2 ⟨h1,hab.2⟩,\n      },\n    },\n  },\n  exact h1,\n\n  assume hbip : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B),\n  have h1 : ∃ (A B : Type*) (h : (A ⊕ B) = V), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from by {\n    let (A B : Type*) (h : (A ⊕ B) = V) (h1 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) := hbip,\n    let h2 : ∀ (v : V), ∃! (a : A ⊕ B), a = v := \n      begin\n        assume v : V,\n        have h2 : ∃! (a : A ⊕ B), a = v, from by {\n          have h3 : v ∈ V, from by {exact (v.2).1},\n          have h4 : v ∈ A ⊕ B, from by {rw ← h, exact h3},\n          cases h4 with ha hb,\n          use sum.inl ha,\n          obviously,\n          use sum.inr hb,\n          obviously,\n        },\n        exact h2,\n      end,\n    use A,\n    use B,\n    use h,\n    use h2,\n  },\n  have h2 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h1},\n  have h3 : ∃ (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact classical.some h2},\n  have h4 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h2,\n  have h5 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact h3.right},\n  have h6 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h5,\n  have h7 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h6},\n  have h8 : V → fin 2, from classical.some h7,\n  have h9 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h7,\n  have h10 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h9,\n  have h11 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h10,\n  have h12 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h11},\n  have h13 : V → fin 2, from classical.some h12,\n  have h14 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h12,\n  have h15 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h14,\n  have h16 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h15},\n  have h17 : V → fin 2, from classical.some h16,\n  have h18 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h16,\n  have h19 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h18,\n  have h20 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h19},\n  have h21 : V → fin 2, from classical.some h20,\n  have h22 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h20,\n  have h23 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h22,\n  have h24 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of_fintype h23},\n  have h25 : V → fin 2, from classical.some h24,\n  have h26 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from some_spec h24,\n  have h27 : ∀ (v : V), ∃! (a : A ⊕ B), a = v, from h26,\n  have h28 : ∃! (f : V → fin 2), ∀ (v : V), ∃! (a : A ⊕ B), a = v, from \n    by {exact exists_unique_of_exists_unique_of\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  { assume h1 : (G.colorable 2),\n    have h2 : ∃ (f : V → fin 2), G.is_coloring f, from h1, \n    cases h2 with f h2,\n    have h3 : (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from by {\n      let A : Type* := {x : V | f x = 0},\n      let B : Type* := {x : V | f x = 1},\n      let h : (A ⊕ B) = V := by {\n        ext x,\n        split,\n        { assume hx : x ∈ A,\n          have hh1 : f x = 0, from hx,\n          show x ∈ (A ⊕ B), from or.inl ⟨x,hh1⟩, },\n        { assume hx : x ∈ B,\n          have hh1 : f x = 1, from hx,\n          show x ∈ (A ⊕ B), from or.inr ⟨x,hh1⟩, },\n        { assume hx : x ∈ (A ⊕ B),\n          cases hx,\n          { assume hx1 : x ∈ A,\n            have hh1 : f x = 0, from hx1,\n            show x ∈ A, from ⟨x,hh1⟩, },\n          { assume hx1 : x ∈ B,\n            have hh1 : f x = 1, from hx1,\n            show x ∈ B, from ⟨x,hh1⟩, }\n        }\n      },\n      have h4 : G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by {\n        simp only [has_edge, subgraph.has_edge, cast, subtype.coe_mk, subtype.coe_mk, complete_bipartite_graph],\n        assume x y hxy,\n        cases hxy,\n        { assume h1,\n          have h2 : f x = f y, from h2 ⟨x,h1⟩ ⟨y,h1⟩,\n          rw h2 at h1,\n          exact h1.elim, },\n        { assume h1,\n          have h2 : f x = f y, from h2 ⟨x,h1⟩ ⟨y,h1⟩,\n          rw h2 at h1,\n          exact h1.elim, },\n      },\n      use [A,B,h,h4],\n    },\n    exact h3,\n  },\n  { assume h1 : (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)),\n    cases h1 with A h1, cases h1 with B h1, cases h1 with h h1,\n    let f : V → fin 2 := λ (x : V), if x ∈ A then 0 else 1,\n    have h2 : G.is_coloring f, from by {\n      simp only [has_edge, subgraph.has_edge, cast, subtype.coe_mk, subtype.coe_mk, complete_bipartite_graph],\n      assume x y hxy,\n      cases hxy,\n      { assume h1,\n        have h2 : f x = f y, from by {rw [f,f], simp only [hxy.left,hxy.right]},\n        have h3 : x ∈ A, from h1.left,\n        have h4 : y ∈ A, from h1.right,\n        rw h2 at h3,\n        exact h3.elim,\n      },\n      { assume h1,\n        have h2 : f x = f y, from by {rw [f,f], simp only [hxy.left,hxy.right]},\n        have h3 : x ∈ B, from h1.left,\n        have h4 : y ∈ B, from h1.right,\n        rw h2 at h3,\n        exact h3.elim,\n      },\n    },\n    exact ⟨f,h2⟩,\n  },\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.31592784653506056}}
{"text": "import homotopy_theory.formal.cylinder.homotopy\nimport homotopy_theory.formal.cylinder.sdr\nimport .definitions\nimport .homotopy_lemmas\nimport .homotopy_equivalences\n\n/-\n\nDold's theorem: Suppose j : A → X and j' : A → X' are cofibrations and\nf : X → X' is a homotopy equivalence with f ∘ j = j'. Then f is a\nhomotopy equivalence under A.\n\n-/\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cofibrations\nsection C\nopen category_theory.has_initial_object\nopen homotopy_theory.cylinder\nopen precofibration_category\nopen I_category\n\nparameters {C : Type u} [category.{v} C] [has_initial_object.{v} C]\n  [has_coproducts.{v} C] [I_category.{v} C]\n\n-- [Kamps & Porter, Lemma I.6.4]. Apparently, we already did most of\n-- the hard work.\nlemma dold_lemma {a x : C} {j : a ⟶ x} (hj : is_cof j) {g : x ⟶ x} (hg : g ∘ j = j)\n  (h : g ≃ 𝟙 x) : ∃ g', g' ∘ j = j ∧ g' ∘ g ≃ 𝟙 x rel j :=\nlet ⟨φ⟩ := h,\n    φ' := φ.congr_right j,\n    ⟨ψ, hψ₁, hψ₂⟩ := hep_cof j hj 0 x (𝟙 x) φ'.H (by rw [φ'.Hi₀, hg]; simp),\n    g' := ψ ∘ i 1 @> x,\n    ψh : homotopy (𝟙 x) g' := { H := ψ, Hi₀ := hψ₁, Hi₁ := rfl } in\nhave g' ∘ j = j, by dsimp [g']; rw [i_nat_assoc, hψ₂, φ'.Hi₁]; simp,\nsuffices H : g' ∘ g ≃ 𝟙 x rel j, from ⟨g', this, H⟩,\nlet ψhg : homotopy g (g' ∘ g) :=\n  @eq.rec_on (x ⟶ x) (𝟙 x ∘ g) (λ f, homotopy f (g' ∘ g)) g (by simp)\n    (ψh.congr_right g) in\nhave ψhg.H ∘ I &> j = φ'.H, begin\n  convert hψ₂ using 1, rw homotopy.eq_rec_on_left,\n  change ψ ∘ I &> g ∘ I &> j = ψ ∘ I &> j,\n  rw [←assoc, ←I.map_comp, hg]\nend,\nequiv_private.f₁_f₂ j hj homotopy.refl_is_rel 0 this rfl\n\nlemma dold_lemma' {a x x' : C} {j : a ⟶ x} (hj : is_cof j) (f : x ⟶ x') (r : x' ⟶ x)\n  (hr : r ∘ f ∘ j = j) (h : r ∘ f ≃ 𝟙 x) : ∃ r', r' ∘ f ∘ j = j ∧ r' ∘ f ≃ 𝟙 x rel j :=\nlet ⟨g', hg'₁, hg'₂⟩ := dold_lemma hj hr h in\n⟨g' ∘ r,\n calc\n  g' ∘ r ∘ f ∘ j = g' ∘ (r ∘ f ∘ j) : by simp\n  ...            = j                : by rw [hr, hg'₁],\n by convert hg'₂ using 1; simp⟩\n\n-- Ugh! We'd like to use `calc` to compose homotopies rel j', but\n-- homotopic_rel.trans has an extra `is_cof j` argument which we have\n-- no way to provide explicitly. So, we locally arrange for the\n-- argument to be provided by the type class system.\nlocal attribute [class] is_cof\n\n@[trans] private lemma homotopic_rel.trans' {a b x : C} {j : a ⟶ b} [hj : is_cof j]\n  {f₀ f₁ f₂ : b ⟶ x} (h₁ : f₀ ≃ f₁ rel j) (h₂ : f₁ ≃ f₂ rel j) : f₀ ≃ f₂ rel j :=\nhomotopic_rel.trans hj h₁ h₂\n\n-- Why is this necessary? doesn't work without `local`\nlocal notation f₀ ` ≃ `:50 f₁:50 := homotopic f₀ f₁\n\n-- [Kamps & Porter, Theorem I.6.3]\nlemma dold_theorem {a x x' : C} {j : a ⟶ x} (hj : is_cof j) {j' : a ⟶ x'} (hj' : is_cof j')\n  {f : x ⟶ x'} (hf : f ∘ j = j') (hef : homotopy_equivalence f) :\n  ∃ h : x' ⟶ x, h ∘ j' = j ∧ h ∘ f ≃ 𝟙 _ rel j ∧ f ∘ h ≃ 𝟙 _ rel j' :=\nlet ⟨f', hf'₁, hf'₂⟩ := homotopy_equivalence_iff.mp hef in\nhave f' ∘ j' ≃ j, from calc\n  f' ∘ j' = f' ∘ f ∘ j : by rw ←hf; simp\n  ...     ≃ j          : by convert hf'₁.congr_right j; simp,\nlet ⟨H⟩ := this,\n    ⟨H', hH'₁, hH'₂⟩ := hep_cof j' hj' 0 x f' H.H H.Hi₀.symm,\n    f'' := H' ∘ i 1 @> x' in\nhave f' ≃ f'', from ⟨⟨H', hH'₁, rfl⟩⟩,\nhave f'' ∘ j' = j, by dsimp [f'']; rw [i_nat_assoc, hH'₂, H.Hi₁],\nlet ⟨h, hh₁, hh₂⟩ :=\n      dold_lemma' hj f f'' (by rw [←assoc, hf, this]) $ calc\n        f'' ∘ f ≃ f' ∘ f : ‹f' ≃ f''›.symm.congr_right f\n        ...     ≃ 𝟙 x    : hf'₁ in\nhave f ∘ h ≃ 𝟙 x', from calc\n  f ∘ h ≃ f ∘ h ∘ (f ∘ f')  : by convert hf'₂.symm.congr_left (f ∘ h) using 1; simp\n  ...   ≃ f ∘ h ∘ (f ∘ f'') : (‹f' ≃ f''›.congr_left f).congr_left (f ∘ h)\n  ...   = f ∘ (h ∘ f) ∘ f'' : by simp\n  ...   ≃ f ∘ 𝟙 x ∘ f''     : (hh₂.forget_rel.congr_left f).congr_right f''\n  ...   ≃ f ∘ f'            : by convert ‹f' ≃ f''›.symm.congr_left f; simp\n  ...   ≃ 𝟙 x'              : hf'₂,\nhave fhj' : f ∘ h ∘ j' = j', by rw [←hf, ←assoc]; congr; simp [hh₁],\nlet ⟨k, hk₁, hk₂⟩ := dold_lemma' hj' h f fhj' this in\nhave hk₂' : k ∘ h ≃ 𝟙 x' rel f ∘ h ∘ j', by convert hk₂; exact fhj',\nhave hh₂' : h ∘ f ≃ 𝟙 x rel h ∘ j', by convert hh₂; rw [←hf]; simp [hh₁],\n⟨h, by rw [←hf]; simp [hh₁], hh₂, calc\n  f ∘ h ≃ (k ∘ h) ∘ (f ∘ h) rel j'  : by convert (hk₂'.congr_right (f ∘ h)).symm hj' using 1; simp\n  ...   = k ∘ (h ∘ f) ∘ h           : by simp\n  ...   ≃ k ∘ (𝟙 x) ∘ h     rel j'  : by convert (hh₂'.congr_left k).congr_right h using 1; refl\n  ...   = k ∘ h                     : by simp\n  ...   ≃ 𝟙 x'              rel j'  : hk₂⟩\n\n-- [Kamps & Porter, Theorem I.6.9]. Apply Dold's theorem to j itself.\nlemma heq_iff_sdr_inclusion {a x : C} {j : a ⟶ x} (hj : is_cof j) :\n  homotopy_equivalence j ↔ is_sdr_inclusion j :=\niff.intro\n  (assume hf,\n    let ⟨h, hh₁, hh₂, hh₃⟩ := dold_theorem (cof_id a) hj (by simp) hf in\n    ⟨⟨h, hh₁, hh₃⟩⟩)\n  (assume ⟨⟨r, h, H⟩⟩, homotopy_equivalence_iff.mpr\n    ⟨r, by convert homotopic.refl (𝟙 a), H.forget_rel⟩)\n\nlemma pushout_is_acof {a x a' x' : C} {j : a ⟶ x} {f : a ⟶ a'} {f' : x ⟶ x'} {j' : a' ⟶ x'}\n  (po : Is_pushout j f f' j') (hj : is_cof j) (hej : homotopy_equivalence j) :\n  homotopy_equivalence j' :=\nhave is_cof j', from pushout_is_cof po hj,\n(heq_iff_sdr_inclusion this).mpr $\npushout_of_sdr_inclusion po (I_preserves_pushout_by_cof hj po) $\n(heq_iff_sdr_inclusion hj).mp hej\n\nend C\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/i_category/dold.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3156939732931831}}
{"text": "import ..lovelib\n\n\n/-! # LoVe Demo 8: Operational Semantics\n\nIn this and the next two lectures, we will see how to use Lean to specify the\nsyntax and semantics of programming languages and to reason about the\nsemantics. -/\n\n\nset_option pp.beta true\nset_option pp.generalized_field_notation false\n\nnamespace LoVe\n\n\n/-! ## Formal Semantics\n\nA formal semantics helps specify and reason about the programming language\nitself, and about individual programs.\n\nIt can form the basis of verified compilers, interpreters, verifiers, static\nanalyzers, type checkers, etc. Without formal proofs, these tools are\n**almost always wrong**.\n\nIn this area, proof assistants are widely used. Every year, about 10-20% of POPL\npapers are partially or totally formalized. Reasons for this success:\n\n* Little machinery (background libraries, tactics) is needed to get started,\n  beyond inductive types and predicates and recursive functions.\n\n* The proofs tend to have lots of cases, which is a good match for computers.\n\n* Proof assistants keep track of what needs to be changed when as extend the\n  programming language with more features.\n\nCase in point: WebAssembly. To quote Conrad Watt (with some abbreviations):\n\n    We have produced a full Isabelle mechanisation of the core execution\n    semantics and type system of the WebAssembly language. To complete this\n    proof, **several deficiencies** in the official WebAssembly specification,\n    uncovered by our proof and modelling work, needed to be corrected. In some\n    cases, these meant that the type system was **originally unsound**.\n\n    We have maintained a constructive dialogue with the working group,\n    verifying new features as they are added. In particular, the mechanism by\n    which a WebAssembly implementation interfaces with its host environment was\n    not formally specified in the working group's original paper. Extending our\n    mechanisation to model this feature revealed a deficiency in the WebAssembly\n    specification that **sabotaged the soundness** of the type system.\n\n\n## A Minimalistic Imperative Language\n\nA state `s` is a function from variable names to values (`string → ℕ`).\n\n__WHILE__ is a minimalistic imperative language with the following grammar:\n\n    S  ::=  skip                 -- no-op\n         |  x := a               -- assignment\n         |  S ; S                -- sequential composition\n         |  if b then S else S   -- conditional statement\n         |  while b do S         -- while loop\n\nwhere `S` stands for a statement (also called command or program), `x` for a\nvariable, `a` for an arithmetic expression, and `b` for a Boolean expression. -/\n\n#print state\n\ninductive stmt : Type\n| skip   : stmt\n| assign : string → (state → ℕ) → stmt\n| seq    : stmt → stmt → stmt\n| ite    : (state → Prop) → stmt → stmt → stmt\n| while  : (state → Prop) → stmt → stmt\n\ninfixr ` ;; ` : 90 := stmt.seq\n\n#check λ s : state, s \"y\" + 2 * s \"z\"\n\n/-! In our grammar, we deliberately leave the syntax of arithmetic and Boolean\nexpressions unspecified. In Lean, we have the choice:\n\n* We could use a type such as `aexp` from lecture 1 and similarly for Boolean\n  expressions.\n\n* We could decide that an arithmetic expression is simply a function from\n  states to natural numbers (`state → ℕ`) and a Boolean expression is a\n  predicate (`state → Prop` or `state → bool`).\n\nThis corresponds to the difference between deep and shallow embeddings:\n\n* A __deep embedding__ of some syntax (expression, formula, program, etc.)\n  consists of an abstract syntax tree specified in the proof assistant\n  (e.g., `aexp`) with a semantics (e.g., `eval`).\n\n* In contrast, a __shallow embedding__ simply reuses the corresponding\n  mechanisms from the logic (e.g., λ-terms, functions and predicate types).\n\nA deep embedding allows us to reason about the syntax (and its semantics). A\nshallow embedding is more lightweight, because we can use it directly, without\nhaving to define a semantics.\n\nWe will use a deep embedding of programs (which we find interesting), and\nshallow embeddings of assignments and Boolean expressions (which we find\nboring). -/\n\ndef silly_loop : stmt :=\nstmt.while (λs, s \"x\" > s \"y\")\n  (stmt.skip ;; stmt.assign \"x\" (λs, s \"x\" - 1))\n\n\n/-! ## Big-Step Semantics\n\nAn __operational semantics__ corresponds to an idealized interpreter (specified\nin a Prolog-like language). Two main variants:\n\n* big-step semantics;\n\n* small-step semantics.\n\nIn a __big-step semantics__ (also called __natural semantics__), judgments have\nthe form `(S, s) ⟹ t`:\n\n    Starting in a state `s`, executing `S` terminates in the state `t`.\n\nExample:\n\n    `(x := x + y; y := 0, [x ↦ 3, y ↦ 5]) ⟹ [x ↦ 8, y ↦ 0]`\n\nDerivation rules:\n\n    ——————————————— Skip\n    (skip, s) ⟹ s\n\n    ——————————————————————————— Asn\n    (x := a, s) ⟹ s[x ↦ s(a)]\n\n    (S, s) ⟹ t   (T, t) ⟹ u\n    ——————————————————————————— Seq\n    (S; T, s) ⟹ u\n\n    (S, s) ⟹ t\n    ————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⟹ t\n\n    (T, s) ⟹ t\n    ————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⟹ t\n\n    (S, s) ⟹ t   (while b do S, t) ⟹ u\n    —————————————————————————————————————— While-True   if s(b) is true\n    (while b do S, s) ⟹ u\n\n    ————————————————————————— While-False   if s(b) is false\n    (while b do S, s) ⟹ s\n\nAbove, `s(e)` denotes the value of expression `e` in state `s`.\n\nIn Lean, the judgment corresponds to an inductive predicate, and the derivation\nrules correspond to the predicate's introduction rules. Using an inductive\npredicate as opposed to a recursive function allows us to cope with\nnontermination (e.g., a diverging `while`) and nondeterminism (e.g.,\nmultithreading). -/\n\ninductive big_step : stmt × state → state → Prop\n| skip {s} :\n  big_step (stmt.skip, s) s\n| assign {x a s} :\n  big_step (stmt.assign x a, s) (s{x ↦ a s})\n| seq {S T s t u} (hS : big_step (S, s) t)\n    (hT : big_step (T, t) u) :\n  big_step (S ;; T, s) u\n| ite_true {b : state → Prop} {S T s t} (hcond : b s)\n    (hbody : big_step (S, s) t) :\n  big_step (stmt.ite b S T, s) t\n| ite_false {b : state → Prop} {S T s t} (hcond : ¬ b s)\n    (hbody : big_step (T, s) t) :\n  big_step (stmt.ite b S T, s) t\n| while_true {b : state → Prop} {S s t u} (hcond : b s)\n    (hbody : big_step (S, s) t)\n    (hrest : big_step (stmt.while b S, t) u) :\n  big_step (stmt.while b S, s) u\n| while_false {b : state → Prop} {S s} (hcond : ¬ b s) :\n  big_step (stmt.while b S, s) s\n\ninfix ` ⟹ ` : 110 := big_step\n\nlemma silly_loop_from_1_big_step :\n  (silly_loop, (λ_, 0){\"x\" ↦ 1}) ⟹ (λ_, 0) :=\nbegin\n  rw silly_loop,\n  apply big_step.while_true,\n  { simp },\n  { apply big_step.seq,\n    { apply big_step.skip },\n    { apply big_step.assign } },\n  { simp,\n    apply big_step.while_false,\n    linarith }\nend\n\n\n/-! ## Properties of the Big-Step Semantics\n\nEquipped with a big-step semantics, we can\n\n* prove properties of the programming language, such as **equivalence proofs**\n  between programs and **determinism**;\n\n* reason about **concrete programs**, proving theorems relating final states `t`\n  with initial states `s`. -/\n\nlemma big_step_deterministic {S s l r} (hl : (S, s) ⟹ l)\n    (hr : (S, s) ⟹ r) :\n  l = r :=\nbegin\n  induction' hl,\n  case skip {\n    cases' hr,\n    refl },\n  case assign {\n    cases' hr,\n    refl },\n  case seq : S T s t l hS hT ihS ihT {\n    cases' hr with _ _ _ _ _ _ _ t' _ hS' hT',\n    cases' ihS hS',\n    cases' ihT hT',\n    refl },\n  case ite_true : b S T s t hb hS ih {\n    cases' hr,\n    { apply ih hr },\n    { cc } },\n  case ite_false : b S T s t hb hT ih {\n    cases' hr,\n    { cc },\n    { apply ih hr } },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    cases' hr,\n    { cases' ihS hr,\n      cases' ihw hr_1,\n      refl },\n    { cc } },\n  case while_false {\n    cases' hr,\n    { cc },\n    { refl } }\nend\n\nlemma big_step_terminates {S s} :\n  ∃t, (S, s) ⟹ t :=\nsorry   -- unprovable\n\nlemma big_step_doesnt_terminate {S s t} :\n  ¬ (stmt.while (λ_, true) S, s) ⟹ t :=\nbegin\n  intro hw,\n  induction' hw,\n  case while_true {\n    assumption },\n  case while_false {\n    cc }\nend\n\n/-! We can define inversion rules about the big-step semantics: -/\n\n@[simp] lemma big_step_skip_iff {s t} :\n  (stmt.skip, s) ⟹ t ↔ t = s :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.skip }\nend\n\n@[simp] lemma big_step_assign_iff {x a s t} :\n  (stmt.assign x a, s) ⟹ t ↔ t = s{x ↦ a s} :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    refl },\n  { intro h,\n    rw h,\n    exact big_step.assign }\nend\n\n@[simp] lemma big_step_seq_iff {S T s t} :\n  (S ;; T, s) ⟹ t ↔ (∃u, (S, s) ⟹ u ∧ (T, u) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    apply exists.intro,\n    apply and.intro; assumption },\n  { intro h,\n    cases' h,\n    cases' h,\n    apply big_step.seq; assumption }\nend\n\n@[simp] lemma big_step_ite_iff {b S T s t} :\n  (stmt.ite b S T, s) ⟹ t ↔\n  (b s ∧ (S, s) ⟹ t) ∨ (¬ b s ∧ (T, s) ⟹ t) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h; cases' h,\n    { apply big_step.ite_true; assumption },\n    { apply big_step.ite_false; assumption } }\nend\n\nlemma big_step_while_iff {b S s u} :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, b s ∧ (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u)\n  ∨ (¬ b s ∧ u = s) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro t,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    case inl {\n      cases' h with t h,\n      cases' h with hb h,\n      cases' h with hS hwhile,\n      exact big_step.while_true hb hS hwhile },\n    case inr {\n      cases' h with hb hus,\n      rw hus,\n      exact big_step.while_false hb } }\nend\n\nlemma big_step_while_true_iff {b : state → Prop} {S s u}\n    (hcond : b s) :\n  (stmt.while b S, s) ⟹ u ↔\n  (∃t, (S, s) ⟹ t ∧ (stmt.while b S, t) ⟹ u) :=\nby rw big_step_while_iff; simp [hcond]\n\n@[simp] lemma big_step_while_false_iff {b : state → Prop}\n    {S s t} (hcond : ¬ b s) :\n  (stmt.while b S, s) ⟹ t ↔ t = s :=\nby rw big_step_while_iff; simp [hcond]\n\n\n/-! ## Small-Step Semantics\n\nA big-step semantics\n\n* does not let us reason about intermediate states;\n\n* does not let us express nontermination or interleaving (for multithreading).\n\n__Small-step semantics__ (also called __structural operational semantics__)\nsolve the above issues.\n\nA judgment has the form `(S, s) ⇒ (T, t)`:\n\n    Starting in a state `s`, executing one step of `S` leaves us in the\n    state `t`, with the program `T` remaining to be executed.\n\nAn execution is a finite or infinite chain `(S₀, s₀) ⇒ (S₁, s₁) ⇒ …`.\n\nA pair `(S, s)` is called a __configuration__. It is __final__ if no transition\nof the form `(S, s) ⇒ _` is possible.\n\nExample:\n\n      `(x := x + y; y := 0, [x ↦ 3, y ↦ 5])`\n    `⇒ (skip; y := 0,       [x ↦ 8, y ↦ 5])`\n    `⇒ (y := 0,             [x ↦ 8, y ↦ 5])`\n    `⇒ (skip,               [x ↦ 8, y ↦ 0])`\n\nDerivation rules:\n\n    ————————————————————————————————— Asn\n    (x := a, s) ⇒ (skip, s[x ↦ s(a)])\n\n    (S, s) ⇒ (S', s')\n    ———-————————————————————— Seq-Step\n    (S ; T, s) ⇒ (S' ; T, s')\n\n    —————————————————————— Seq-Skip\n    (skip ; S, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-True   if s(b) is true\n    (if b then S else T, s) ⇒ (S, s)\n\n    ———————————————————————————————— If-False   if s(b) is false\n    (if b then S else T, s) ⇒ (T, s)\n\n    ——————————————————————————————————————————————————————————————— While\n    (while b do S, s) ⇒ (if b then (S ; while b do S) else skip, s)\n\nThere is no rule for `skip` (why?). -/\n\ninductive small_step : stmt × state → stmt × state → Prop\n| assign {x a s} :\n  small_step (stmt.assign x a, s) (stmt.skip, s{x ↦ a s})\n| seq_step {S S' T s s'} (hS : small_step (S, s) (S', s')) :\n  small_step (S ;; T, s) (S' ;; T, s')\n| seq_skip {T s} :\n  small_step (stmt.skip ;; T, s) (T, s)\n| ite_true {b : state → Prop} {S T s} (hcond : b s) :\n  small_step (stmt.ite b S T, s) (S, s)\n| ite_false {b : state → Prop} {S T s} (hcond : ¬ b s) :\n  small_step (stmt.ite b S T, s) (T, s)\n| while {b : state → Prop} {S s} :\n  small_step (stmt.while b S, s)\n    (stmt.ite b (S ;; stmt.while b S) stmt.skip, s)\n\ninfixr ` ⇒ ` := small_step\ninfixr ` ⇒* ` : 100 := star small_step\n\nlemma silly_loop_from_1_small_step :\n  (silly_loop, (λ_, 0){\"x\" ↦ 1}) ⇒*\n  (stmt.skip, ((λ_, 0) : state)) :=\nbegin\n  rw silly_loop,\n  apply star.head,\n  { apply small_step.while },\n  { apply star.head,\n    { apply small_step.ite_true,\n      simp },\n    { apply star.head,\n      { apply small_step.seq_step,\n        apply small_step.seq_skip },\n      { apply star.head,\n        { apply small_step.seq_step,\n          apply small_step.assign },\n        { apply star.head,\n          { apply small_step.seq_skip },\n          { apply star.head,\n            { apply small_step.while },\n            { apply star.head,\n              { apply small_step.ite_false,\n                simp },\n              { simp } } } } } } }\nend\n\n/-! Equipped with a small-step semantics, we can **define** a big-step\nsemantics:\n\n    `(S, s) ⟹ t` if and only if `(S, s) ⇒* (skip, t)`\n\nwhere `r*` denotes the reflexive transitive closure of a relation `r`.\n\nAlternatively, if we have already defined a big-step semantics, we can **prove**\nthe above equivalence theorem to validate our definitions.\n\nThe main disadvantage of small-step semantics is that we now have two relations,\n`⇒` and `⇒*`, and reasoning tends to be more complicated.\n\n\n## Properties of the Small-Step Semantics\n\nWe can prove that a configuration `(S, s)` is final if and only if `S = skip`.\nThis ensures that we have not forgotten a derivation rule. -/\n\nlemma small_step_final (S s) :\n  (¬ ∃T t, (S, s) ⇒ (T, t)) ↔ S = stmt.skip :=\nbegin\n  induction' S,\n  case skip {\n    simp,\n    intros T t hstep,\n    cases' hstep },\n  case assign : x a {\n    simp,\n    apply exists.intro stmt.skip,\n    apply exists.intro (s{x ↦ a s}),\n    exact small_step.assign },\n  case seq : S T ihS ihT {\n    simp,\n    cases' classical.em (S = stmt.skip),\n    case inl {\n      rw h,\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.seq_skip },\n    case inr {\n      simp [h, auto.not_forall_eq, auto.not_not_eq] at ihS,\n      cases' ihS s with S' hS',\n      cases' hS' with s' hs',\n      apply exists.intro (S' ;; T),\n      apply exists.intro s',\n      exact small_step.seq_step hs' } },\n  case ite : b S T ihS ihT {\n    simp,\n    cases' classical.em (b s),\n    case inl {\n      apply exists.intro S,\n      apply exists.intro s,\n      exact small_step.ite_true h },\n    case inr {\n      apply exists.intro T,\n      apply exists.intro s,\n      exact small_step.ite_false h } },\n  case while : b S ih {\n    simp,\n    apply exists.intro (stmt.ite b (S ;; stmt.while b S) stmt.skip),\n    apply exists.intro s,\n    exact small_step.while }\nend\n\nlemma small_step_deterministic {S s Ll Rr}\n    (hl : (S, s) ⇒ Ll) (hr : (S, s) ⇒ Rr) :\n  Ll = Rr :=\nbegin\n  induction' hl,\n  case assign : x a s {\n    cases' hr,\n    refl },\n  case seq_step : S S₁ T s s₁ hS₁ ih {\n    cases' hr,\n    case seq_step : S S₂ _ _ s₂ hS₂ {\n      have hSs₁₂ := ih hS₂,\n      cc },\n    case seq_skip {\n      cases' hS₁ } },\n  case seq_skip : T s {\n    cases' hr,\n    case seq_step {\n      cases' hr },\n    case seq_skip {\n      refl } },\n  case ite_true : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      refl },\n    case ite_false {\n      cc } },\n  case ite_false : b S T s hcond {\n    cases' hr,\n    case ite_true {\n      cc },\n    case ite_false {\n      refl } },\n  case while : b S s {\n    cases' hr,\n    refl }\nend\n\n/-! We can define inversion rules also about the small-step semantics. Here are\nthree examples: -/\n\nlemma small_step_skip {S s t} :\n  ¬ ((stmt.skip, s) ⇒ (S, t)) :=\nby intro h; cases' h\n\n@[simp] lemma small_step_seq_iff {S T s Ut} :\n  (S ;; T, s) ⇒ Ut ↔\n  (∃S' t, (S, s) ⇒ (S', t) ∧ Ut = (S' ;; T, t))\n  ∨ (S = stmt.skip ∧ Ut = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      apply exists.intro S',\n      apply exists.intro s',\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      cases' h,\n      cases' h,\n      rw right,\n      apply small_step.seq_step,\n      assumption },\n    { cases' h,\n      rw left,\n      rw right,\n      apply small_step.seq_skip } }\nend\n\n@[simp] lemma small_step_ite_iff {b S T s Us} :\n  (stmt.ite b S T, s) ⇒ Us ↔\n  (b s ∧ Us = (S, s)) ∨ (¬ b s ∧ Us = (T, s)) :=\nbegin\n  apply iff.intro,\n  { intro h,\n    cases' h,\n    { apply or.intro_left,\n      cc },\n    { apply or.intro_right,\n      cc } },\n  { intro h,\n    cases' h,\n    { cases' h,\n      rw right,\n      apply small_step.ite_true,\n      assumption },\n    { cases' h,\n      rw right,\n      apply small_step.ite_false,\n      assumption } }\nend\n\n\n/-! ### Equivalence of the Big-Step and the Small-Step Semantics (**optional**)\n\nA more important result is the connection between the big-step and the\nsmall-step semantics:\n\n    `(S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t)`\n\nIts proof, given below, is beyond the scope of this course. -/\n\nlemma star_small_step_seq {S T s u}\n    (h : (S, s) ⇒* (stmt.skip, u)) :\n  (S ;; T, s) ⇒* (stmt.skip ;; T, u) :=\nbegin\n  apply star.lift (λSs, (prod.fst Ss ;; T, prod.snd Ss)) _ h,\n  intros Ss Ss' h,\n  cases' Ss,\n  cases' Ss',\n  apply small_step.seq_step,\n  assumption\nend\n\nlemma star_small_step_of_big_step {S s t} (h : (S, s) ⟹ t) :\n  (S, s) ⇒* (stmt.skip, t) :=\nbegin\n  induction' h,\n  case skip {\n    refl },\n  case assign {\n    exact star.single small_step.assign },\n  case seq : S T s t u hS hT ihS ihT {\n    transitivity,\n    exact star_small_step_seq ihS,\n    apply star.head small_step.seq_skip ihT },\n  case ite_true : b S T s t hs hst ih {\n    exact star.head (small_step.ite_true hs) ih },\n  case ite_false : b S T s t hs hst ih {\n    exact star.head (small_step.ite_false hs) ih },\n  case while_true : b S s t u hb hS hw ihS ihw {\n    exact (star.head small_step.while\n      (star.head (small_step.ite_true hb)\n         (star.trans (star_small_step_seq ihS)\n            (star.head small_step.seq_skip ihw)))) },\n  case while_false : b S s hb {\n    exact star.tail (star.single small_step.while)\n      (small_step.ite_false hb) }\nend\n\nlemma big_step_of_small_step_of_big_step {S₀ S₁ s₀ s₁ s₂}\n  (h₁ : (S₀, s₀) ⇒ (S₁, s₁)) :\n  (S₁, s₁) ⟹ s₂ → (S₀, s₀) ⟹ s₂ :=\nbegin\n  induction' h₁;\n    simp [*, big_step_while_true_iff] {contextual := tt},\n  case seq_step {\n    intros u hS' hT,\n    apply exists.intro u,\n    exact and.intro (ih hS') hT }\nend\n\nlemma big_step_of_star_small_step {S s t} :\n  (S, s) ⇒* (stmt.skip, t) → (S, s) ⟹ t :=\nbegin\n  generalize hSs : (S, s) = Ss,\n  intro h,\n  induction h\n      using LoVe.rtc.star.head_induction_on\n      with _ S's' h h' ih\n      generalizing S s;\n    cases' hSs,\n  { exact big_step.skip },\n  { cases' S's' with S' s',\n    apply big_step_of_small_step_of_big_step h,\n    apply ih,\n    refl }\nend\n\nlemma big_step_iff_star_small_step {S s t} :\n  (S, s) ⟹ t ↔ (S, s) ⇒* (stmt.skip, t) :=\niff.intro star_small_step_of_big_step\n  big_step_of_star_small_step\n\n\n/-! ## Parallelism (**optional**) -/\n\n/--\na proof of `par_step i (Ss, t) (Ss', t')` represents:\n\"\n  Let `Ss` be a list of WHILE programs that we wish to execute in parallel.\n  If we evaluate one step of the `i`th program in `Ss` while in state `t`, \n  we reach state `t'`, with the list of programs `Ss'` remaining to execute.\"\n-/\ninductive par_step :\n    nat → list stmt × state → list stmt × state → Prop\n| intro {Ss Ss' S S' s s' i}\n    (hi : i < list.length Ss)\n    (hS : S = list.nth_le Ss i hi)\n    (hs : (S, s) ⇒ (S', s'))\n    (hS' : Ss' = list.update_nth Ss i S') :\n  par_step i (Ss, s) (Ss', s')\n\n\n\n\nlemma par_step_diamond {i j Ss Ts Ts' s t t'}\n    (hi : i < list.length Ss)\n    (hj : j < list.length Ss)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t')) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry \n\n\n\n\n/--\n`stmt.W S` returns the set of variables written to by a thread `S`.\n-/\ndef stmt.W : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign x _) := {x}\n| (stmt.seq S T)    := stmt.W S ∪ stmt.W T\n| (stmt.ite _ S T)  := stmt.W S ∪ stmt.W T\n| (stmt.while _ S)  := stmt.W S\n\n/--\n`exp.R f` returns the set of variables read by an arithmetic or boolean expression `f`.\n-/\ndef exp.R {α : Type} (f : state → α) : set string := \n{x | ∃ s n, f (s{x ↦ n}) ≠ f s}\n\n/--\n`stmt.R S` returns the set of variables read by a thread `S`.\n-/\ndef stmt.R : stmt → set string\n| stmt.skip         := ∅\n| (stmt.assign _ a) := exp.R a\n| (stmt.seq S T)    := stmt.R S ∪ stmt.R T\n| (stmt.ite b S T)  := exp.R b ∪ stmt.R S ∪ stmt.R T\n| (stmt.while b S)  := exp.R b ∪ stmt.R S\n\n/--\n`stmt.V S` returns the set of variables written to or read by a thread `S`. \n-/\ndef stmt.V : stmt → set string\n| S := stmt.W S ∪ stmt.R S\n\nlemma par_step_diamond_VW_disjoint {i j Ss Ts Ts' s t t'}\n    (hiS : i < list.length Ss)\n    (hjT : j < list.length Ts)\n    (hij : i ≠ j)\n    (hT : par_step i (Ss, s) (Ts, t))\n    (hT' : par_step j (Ss, s) (Ts', t'))\n    (hWV : stmt.W (list.nth_le Ss i hiS)\n       ∩ stmt.V (list.nth_le Ts j hjT) = ∅)\n    (hVW : stmt.V (list.nth_le Ss i hiS)\n       ∩ stmt.W (list.nth_le Ts j hjT) = ∅) :\n  ∃u Us, par_step j (Ts, t) (Us, u) ∧\n    par_step i (Ts', t') (Us, u) :=\nsorry   -- this should be provable\n\nend LoVe\n", "meta": {"author": "BrownCS1951x", "repo": "fpv2021", "sha": "10bdbd92e64fb34115b68794b8ff480468f4dcaa", "save_path": "github-repos/lean/BrownCS1951x-fpv2021", "path": "github-repos/lean/BrownCS1951x-fpv2021/fpv2021-10bdbd92e64fb34115b68794b8ff480468f4dcaa/src/lectures/love08_operational_semantics_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.31556810430549637}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.vector\nimport Mathlib.data.list.nodup\nimport Mathlib.data.list.of_fn\nimport Mathlib.control.applicative\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u u_3 \n\nnamespace Mathlib\n\n/-!\n# Additional theorems about the `vector` type\n\nThis file introduces the infix notation `::ᵥ` for `vector.cons`.\n-/\n\nnamespace vector\n\n\ninfixr:67 \"::ᵥ\" => Mathlib.vector.cons\n\nprotected instance inhabited {n : ℕ} {α : Type u_1} [Inhabited α] : Inhabited (vector α n) :=\n  { default := of_fn fun (_x : fin n) => Inhabited.default }\n\ntheorem to_list_injective {n : ℕ} {α : Type u_1} : function.injective to_list :=\n  subtype.val_injective\n\n/-- Two `v w : vector α n` are equal iff they are equal at every single index. -/\ntheorem ext {n : ℕ} {α : Type u_1} {v : vector α n} {w : vector α n} (h : ∀ (m : fin n), nth v m = nth w m) : v = w := sorry\n\n/-- The empty `vector` is a `subsingleton`. -/\nprotected instance zero_subsingleton {α : Type u_1} : subsingleton (vector α 0) :=\n  subsingleton.intro fun (_x _x_1 : vector α 0) => ext fun (m : fin 0) => fin.elim0 m\n\n@[simp] theorem cons_val {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : subtype.val (a::ᵥv) = a :: subtype.val v := sorry\n\n@[simp] theorem cons_head {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : head (a::ᵥv) = a := sorry\n\n@[simp] theorem cons_tail {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : tail (a::ᵥv) = v := sorry\n\n@[simp] theorem to_list_of_fn {α : Type u_1} {n : ℕ} (f : fin n → α) : to_list (of_fn f) = list.of_fn f := sorry\n\n@[simp] theorem mk_to_list {n : ℕ} {α : Type u_1} (v : vector α n) (h : list.length (to_list v) = n) : { val := to_list v, property := h } = v := sorry\n\n@[simp] theorem to_list_map {n : ℕ} {α : Type u_1} {β : Type u_2} (v : vector α n) (f : α → β) : to_list (map f v) = list.map f (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = n) =>\n      Eq.refl (to_list (map f { val := v_val, property := v_property }))\n\ntheorem nth_eq_nth_le {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) : nth v i =\n  list.nth_le (to_list v) (subtype.val i)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (subtype.val i < list.length (to_list v))) (to_list_length v)))\n      (subtype.property i)) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = n) =>\n      idRhs (nth { val := v_val, property := v_property } i = nth { val := v_val, property := v_property } i) rfl\n\n@[simp] theorem nth_map {n : ℕ} {α : Type u_1} {β : Type u_2} (v : vector α n) (f : α → β) (i : fin n) : nth (map f v) i = f (nth v i) := sorry\n\n@[simp] theorem nth_of_fn {α : Type u_1} {n : ℕ} (f : fin n → α) (i : fin n) : nth (of_fn f) i = f i := sorry\n\n@[simp] theorem of_fn_nth {n : ℕ} {α : Type u_1} (v : vector α n) : of_fn (nth v) = v := sorry\n\n@[simp] theorem nth_tail {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) (i : fin n) : nth (tail v) i = nth v (fin.succ i) := sorry\n\n@[simp] theorem tail_val {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) : subtype.val (tail v) = list.tail (subtype.val v) := sorry\n\n/-- The `tail` of a `nil` vector is `nil`. -/\n@[simp] theorem tail_nil {α : Type u_1} : tail nil = nil :=\n  rfl\n\n/-- The `tail` of a vector made up of one element is `nil`. -/\n@[simp] theorem singleton_tail {α : Type u_1} (v : vector α 1) : tail v = nil :=\n  eq.mpr (id (propext (eq_iff_true_of_subsingleton (tail v) nil))) trivial\n\n@[simp] theorem tail_of_fn {α : Type u_1} {n : ℕ} (f : fin (Nat.succ n) → α) : tail (of_fn f) = of_fn fun (i : fin (Nat.succ n - 1)) => f (fin.succ i) := sorry\n\n/-- The list that makes up a `vector` made up of a single element,\nretrieved via `to_list`, is equal to the list of that single element. -/\n@[simp] theorem to_list_singleton {α : Type u_1} (v : vector α 1) : to_list v = [head v] := sorry\n\n/-- Mapping under `id` does not change a vector. -/\n@[simp] theorem map_id {α : Type u_1} {n : ℕ} (v : vector α n) : map id v = v := sorry\n\ntheorem mem_iff_nth {n : ℕ} {α : Type u_1} {a : α} {v : vector α n} : a ∈ to_list v ↔ ∃ (i : fin n), nth v i = a := sorry\n\ntheorem nodup_iff_nth_inj {n : ℕ} {α : Type u_1} {v : vector α n} : list.nodup (to_list v) ↔ function.injective (nth v) := sorry\n\n@[simp] theorem nth_mem {n : ℕ} {α : Type u_1} (i : fin n) (v : vector α n) : nth v i ∈ to_list v := sorry\n\ntheorem head'_to_list {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) : list.head' (to_list v) = some (head v) := sorry\n\ndef reverse {n : ℕ} {α : Type u_1} (v : vector α n) : vector α n :=\n  { val := list.reverse (to_list v), property := sorry }\n\n/-- The `list` of a vector after a `reverse`, retrieved by `to_list` is equal\nto the `list.reverse` after retrieving a vector's `to_list`. -/\ntheorem to_list_reverse {n : ℕ} {α : Type u_1} {v : vector α n} : to_list (reverse v) = list.reverse (to_list v) :=\n  rfl\n\n@[simp] theorem nth_zero {n : ℕ} {α : Type u_1} (v : vector α (Nat.succ n)) : nth v 0 = head v := sorry\n\n@[simp] theorem head_of_fn {α : Type u_1} {n : ℕ} (f : fin (Nat.succ n) → α) : head (of_fn f) = f 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (head (of_fn f) = f 0)) (Eq.symm (nth_zero (of_fn f)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth (of_fn f) 0 = f 0)) (nth_of_fn f 0))) (Eq.refl (f 0)))\n\n@[simp] theorem nth_cons_zero {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) : nth (a::ᵥv) 0 = a := sorry\n\n/-- Accessing the `nth` element of a vector made up\nof one element `x : α` is `x` itself. -/\n@[simp] theorem nth_cons_nil {α : Type u_1} {ix : fin 1} (x : α) : nth (x::ᵥnil) ix = x := sorry\n\n@[simp] theorem nth_cons_succ {n : ℕ} {α : Type u_1} (a : α) (v : vector α n) (i : fin n) : nth (a::ᵥv) (fin.succ i) = nth v i :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nth (a::ᵥv) (fin.succ i) = nth v i)) (Eq.symm (nth_tail (a::ᵥv) i))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth (tail (a::ᵥv)) i = nth v i)) (tail_cons a v))) (Eq.refl (nth v i)))\n\n/-- The last element of a `vector`, given that the vector is at least one element. -/\ndef last {n : ℕ} {α : Type u_1} (v : vector α (n + 1)) : α :=\n  nth v (fin.last n)\n\n/-- The last element of a `vector`, given that the vector is at least one element. -/\ntheorem last_def {n : ℕ} {α : Type u_1} {v : vector α (n + 1)} : last v = nth v (fin.last n) :=\n  rfl\n\n/-- The `last` element of a vector is the `head` of the `reverse` vector. -/\ntheorem reverse_nth_zero {n : ℕ} {α : Type u_1} {v : vector α (n + 1)} : head (reverse v) = last v := sorry\n\n/--\nConstruct a `vector β (n + 1)` from a `vector α n` by scanning `f : β → α → β`\nfrom the \"left\", that is, from 0 to `fin.last n`, using `b : β` as the starting value.\n-/\ndef scanl {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) : vector β (n + 1) :=\n  { val := list.scanl f b (to_list v), property := sorry }\n\n/-- Providing an empty vector to `scanl` gives the starting value `b : β`. -/\n@[simp] theorem scanl_nil {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) : scanl f b nil = b::ᵥnil :=\n  rfl\n\n/--\nThe recursive step of `scanl` splits a vector `x ::ᵥ v : vector α (n + 1)`\ninto the provided starting value `b : β` and the recursed `scanl`\n`f b x : β` as the starting value.\n\nThis lemma is the `cons` version of `scanl_nth`.\n-/\n@[simp] theorem scanl_cons {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) (x : α) : scanl f b (x::ᵥv) = b::ᵥscanl f (f b x) v := sorry\n\n/--\nThe underlying `list` of a `vector` after a `scanl` is the `list.scanl`\nof the underlying `list` of the original `vector`.\n-/\n@[simp] theorem scanl_val {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) {v : vector α n} : subtype.val (scanl f b v) = list.scanl f b (subtype.val v) := sorry\n\n/--\nThe `to_list` of a `vector` after a `scanl` is the `list.scanl`\nof the `to_list` of the original `vector`.\n-/\n@[simp] theorem to_list_scanl {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) : to_list (scanl f b v) = list.scanl f b (to_list v) :=\n  rfl\n\n/--\nThe recursive step of `scanl` splits a vector made up of a single element\n`x ::ᵥ nil : vector α 1` into a `vector` of the provided starting value `b : β`\nand the mapped `f b x : β` as the last value.\n-/\n@[simp] theorem scanl_singleton {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α 1) : scanl f b v = b::ᵥf b (head v)::ᵥnil := sorry\n\n/--\nThe first element of `scanl` of a vector `v : vector α n`,\nretrieved via `head`, is the starting value `b : β`.\n-/\n@[simp] theorem scanl_head {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) : head (scanl f b v) = b := sorry\n\n/--\nFor an index `i : fin n`, the `nth` element of `scanl` of a\nvector `v : vector α n` at `i.succ`, is equal to the application\nfunction `f : β → α → β` of the `i.cast_succ` element of\n`scanl f b v` and `nth v i`.\n\nThis lemma is the `nth` version of `scanl_cons`.\n-/\n@[simp] theorem scanl_nth {n : ℕ} {α : Type u_1} {β : Type u_2} (f : β → α → β) (b : β) (v : vector α n) (i : fin n) : nth (scanl f b v) (fin.succ i) = f (nth (scanl f b v) (coe_fn fin.cast_succ i)) (nth v i) := sorry\n\ndef m_of_fn {m : Type u → Type u_1} [Monad m] {α : Type u} {n : ℕ} : (fin n → m α) → m (vector α n) :=\n  sorry\n\ntheorem m_of_fn_pure {m : Type u_1 → Type u_2} [Monad m] [is_lawful_monad m] {α : Type u_1} {n : ℕ} (f : fin n → α) : (m_of_fn fun (i : fin n) => pure (f i)) = pure (of_fn f) := sorry\n\ndef mmap {m : Type u → Type u_1} [Monad m] {α : Type u_2} {β : Type u} (f : α → m β) {n : ℕ} : vector α n → m (vector β n) :=\n  sorry\n\n@[simp] theorem mmap_nil {m : Type u_1 → Type u_2} [Monad m] {α : Type u_3} {β : Type u_1} (f : α → m β) : mmap f nil = pure nil :=\n  rfl\n\n@[simp] theorem mmap_cons {m : Type u_1 → Type u_2} [Monad m] {α : Type u_3} {β : Type u_1} (f : α → m β) (a : α) {n : ℕ} (v : vector α n) : mmap f (a::ᵥv) =\n  do \n    let h' ← f a \n    let t' ← mmap f v \n    pure (h'::ᵥt') := sorry\n\n/-- Define `C v` by induction on `v : vector α (n + 1)`, a vector of\nat least one element.\nThis function has two arguments: `h0` handles the base case on `C nil`,\nand `hs` defines the inductive step using `∀ x : α, C v → C (x ::ᵥ v)`. -/\ndef induction_on {α : Type u_1} {n : ℕ} {C : {n : ℕ} → vector α n → Sort u_2} (v : vector α (n + 1)) (h0 : C nil) (hs : {n : ℕ} → {x : α} → {w : vector α n} → C w → C (x::ᵥw)) : C v :=\n  Nat.rec (fun (v : vector α (0 + 1)) => eq.mpr sorry (eq.mpr sorry (hs h0)))\n    (fun (n : ℕ) (hn : (v : vector α (n + 1)) → C v) (v : vector α (Nat.succ n + 1)) => eq.mpr sorry (hs (hn (tail v)))) n\n    v\n\ndef to_array {n : ℕ} {α : Type u_1} : vector α n → array n α :=\n  sorry\n\ndef insert_nth {n : ℕ} {α : Type u_1} (a : α) (i : fin (n + 1)) (v : vector α n) : vector α (n + 1) :=\n  { val := list.insert_nth (↑i) a (subtype.val v), property := sorry }\n\ntheorem insert_nth_val {n : ℕ} {α : Type u_1} {a : α} {i : fin (n + 1)} {v : vector α n} : subtype.val (insert_nth a i v) = list.insert_nth (subtype.val i) a (subtype.val v) :=\n  rfl\n\n@[simp] theorem remove_nth_val {n : ℕ} {α : Type u_1} {i : fin n} {v : vector α n} : subtype.val (remove_nth i v) = list.remove_nth (subtype.val v) ↑i := sorry\n\ntheorem remove_nth_insert_nth {n : ℕ} {α : Type u_1} {a : α} {v : vector α n} {i : fin (n + 1)} : remove_nth i (insert_nth a i v) = v :=\n  subtype.eq (list.remove_nth_insert_nth (subtype.val i) (subtype.val v))\n\ntheorem remove_nth_insert_nth_ne {n : ℕ} {α : Type u_1} {a : α} {v : vector α (n + 1)} {i : fin (n + bit0 1)} {j : fin (n + bit0 1)} (h : i ≠ j) : remove_nth i (insert_nth a j v) = insert_nth a (fin.pred_above i j (ne.symm h)) (remove_nth (fin.pred_above j i h) v) := sorry\n\ntheorem insert_nth_comm {n : ℕ} {α : Type u_1} (a : α) (b : α) (i : fin (n + 1)) (j : fin (n + 1)) (h : i ≤ j) (v : vector α n) : insert_nth b (fin.succ j) (insert_nth a i v) = insert_nth a (coe_fn fin.cast_succ i) (insert_nth b j v) := sorry\n\n/-- `update_nth v n a` replaces the `n`th element of `v` with `a` -/\ndef update_nth {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) (a : α) : vector α n :=\n  { val := list.update_nth (subtype.val v) (subtype.val i) a, property := sorry }\n\n@[simp] theorem nth_update_nth_same {n : ℕ} {α : Type u_1} (v : vector α n) (i : fin n) (a : α) : nth (update_nth v i a) i = a := sorry\n\ntheorem nth_update_nth_of_ne {n : ℕ} {α : Type u_1} {v : vector α n} {i : fin n} {j : fin n} (h : i ≠ j) (a : α) : nth (update_nth v i a) j = nth v j := sorry\n\ntheorem nth_update_nth_eq_if {n : ℕ} {α : Type u_1} {v : vector α n} {i : fin n} {j : fin n} (a : α) : nth (update_nth v i a) j = ite (i = j) a (nth v j) := sorry\n\nend vector\n\n\nnamespace vector\n\n\nprotected def traverse {n : ℕ} {F : Type u → Type u} [Applicative F] {α : Type u} {β : Type u} (f : α → F β) : vector α n → F (vector β n) :=\n  sorry\n\n@[simp] protected theorem traverse_def {n : ℕ} {F : Type u → Type u} [Applicative F] [is_lawful_applicative F] {α : Type u} {β : Type u} (f : α → F β) (x : α) (xs : vector α n) : vector.traverse f (x::ᵥxs) = cons <$> f x <*> vector.traverse f xs :=\n  subtype.cases_on xs\n    fun (xs : List α) (xs_property : list.length xs = n) =>\n      eq.drec (Eq.refl (vector.traverse f (x::ᵥ{ val := xs, property := Eq.refl (list.length xs) }))) xs_property\n\nprotected theorem id_traverse {n : ℕ} {α : Type u} (x : vector α n) : vector.traverse id.mk x = x := sorry\n\nprotected theorem comp_traverse {n : ℕ} {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {α : Type u} {β : Type u} {γ : Type u} (f : β → F γ) (g : α → G β) (x : vector α n) : vector.traverse (functor.comp.mk ∘ Functor.map f ∘ g) x = functor.comp.mk (vector.traverse f <$> vector.traverse g x) := sorry\n\nprotected theorem traverse_eq_map_id {n : ℕ} {α : Type u_1} {β : Type u_1} (f : α → β) (x : vector α n) : vector.traverse (id.mk ∘ f) x = id.mk (map f x) := sorry\n\nprotected theorem naturality {n : ℕ} {F : Type u → Type u} {G : Type u → Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] (η : applicative_transformation F G) {α : Type u} {β : Type u} (f : α → F β) (x : vector α n) : coe_fn η (vector β n) (vector.traverse f x) = vector.traverse (coe_fn η β ∘ f) x := sorry\n\nprotected instance flip.traversable {n : ℕ} : traversable (flip vector n) :=\n  traversable.mk vector.traverse\n\nprotected instance flip.is_lawful_traversable {n : ℕ} : is_lawful_traversable (flip vector n) :=\n  is_lawful_traversable.mk vector.id_traverse vector.comp_traverse vector.traverse_eq_map_id vector.naturality\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/vector2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3155681043054963}}
{"text": "/-\nCopyright (c) 2018 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,\n  Michael Howes\n-/\nimport group_theory.subgroup.basic\nimport deprecated.submonoid\n\n/-!\n# Unbundled subgroups (deprecated)\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file is deprecated, and is no longer imported by anything in mathlib other than other\ndeprecated files, and test files. You should not need to import it.\n\nThis file defines unbundled multiplicative and additive subgroups. Instead of using this file,\nplease use `subgroup G` and `add_subgroup A`, defined in `group_theory.subgroup.basic`.\n\n## Main definitions\n\n`is_add_subgroup (S : set A)` : the predicate that `S` is the underlying subset of an additive\nsubgroup of `A`. The bundled variant `add_subgroup A` should be used in preference to this.\n\n`is_subgroup (S : set G)` : the predicate that `S` is the underlying subset of a subgroup\nof `G`. The bundled variant `subgroup G` should be used in preference to this.\n\n## Tags\n\nsubgroup, subgroups, is_subgroup\n-/\nopen set function\n\nvariables {G : Type*} {H : Type*} {A : Type*} {a a₁ a₂ b c: G}\n\nsection group\nvariables [group G] [add_group A]\n\n/-- `s` is an additive subgroup: a set containing 0 and closed under addition and negation. -/\nstructure is_add_subgroup (s : set A) extends is_add_submonoid s : Prop :=\n(neg_mem {a} : a ∈ s → -a ∈ s)\n\n/-- `s` is a subgroup: a set containing 1 and closed under multiplication and inverse. -/\n@[to_additive]\nstructure is_subgroup (s : set G) extends is_submonoid s : Prop :=\n(inv_mem {a} : a ∈ s → a⁻¹ ∈ s)\n\n@[to_additive]\nlemma is_subgroup.div_mem {s : set G} (hs : is_subgroup s) {x y : G} (hx : x ∈ s) (hy : y ∈ s) :\n  x / y ∈ s :=\nby simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)\n\nlemma additive.is_add_subgroup\n  {s : set G} (hs : is_subgroup s) : @is_add_subgroup (additive G) _ s :=\n@is_add_subgroup.mk (additive G) _ _ (additive.is_add_submonoid hs.to_is_submonoid) $\n  λ _, hs.inv_mem\n\ntheorem additive.is_add_subgroup_iff\n  {s : set G} : @is_add_subgroup (additive G) _ s ↔ is_subgroup s :=\n⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @is_subgroup.mk G _ _ ⟨h₁, @h₂⟩ @h₃,\n  λ h, by exactI additive.is_add_subgroup h⟩\n\nlemma multiplicative.is_subgroup\n  {s : set A} (hs : is_add_subgroup s) : @is_subgroup (multiplicative A) _ s :=\n@is_subgroup.mk (multiplicative A) _ _ (multiplicative.is_submonoid hs.to_is_add_submonoid) $\n  λ _, hs.neg_mem\n\ntheorem multiplicative.is_subgroup_iff\n  {s : set A} : @is_subgroup (multiplicative A) _ s ↔ is_add_subgroup s :=\n⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @is_add_subgroup.mk A _ _ ⟨h₁, @h₂⟩ @h₃,\n  λ h, by exactI multiplicative.is_subgroup h⟩\n\n@[to_additive of_add_neg]\ntheorem is_subgroup.of_div (s : set G)\n  (one_mem : (1:G) ∈ s) (div_mem : ∀{a b:G}, a ∈ s → b ∈ s → a * b⁻¹ ∈ s) :\n  is_subgroup s :=\nhave inv_mem : ∀a, a ∈ s → a⁻¹ ∈ s, from\n  assume a ha,\n  have 1 * a⁻¹ ∈ s, from div_mem one_mem ha,\n  by simpa,\n{ inv_mem := inv_mem,\n  mul_mem := assume a b ha hb,\n    have a * b⁻¹⁻¹ ∈ s, from div_mem ha (inv_mem b hb),\n    by simpa,\n  one_mem := one_mem }\n\ntheorem is_add_subgroup.of_sub (s : set A)\n  (zero_mem : (0:A) ∈ s) (sub_mem : ∀{a b:A}, a ∈ s → b ∈ s → a - b ∈ s) :\n  is_add_subgroup s :=\nis_add_subgroup.of_add_neg s zero_mem\n  (λ x y hx hy, by simpa only [sub_eq_add_neg] using sub_mem hx hy)\n\n@[to_additive]\nlemma is_subgroup.inter {s₁ s₂ : set G} (hs₁ : is_subgroup s₁) (hs₂ : is_subgroup s₂) :\n  is_subgroup (s₁ ∩ s₂) :=\n{ inv_mem := λ x hx, ⟨hs₁.inv_mem hx.1, hs₂.inv_mem hx.2⟩,\n  ..is_submonoid.inter hs₁.to_is_submonoid hs₂.to_is_submonoid}\n\n@[to_additive]\nlemma is_subgroup.Inter {ι : Sort*} {s : ι → set G} (hs : ∀ y : ι, is_subgroup (s y)) :\n  is_subgroup (set.Inter s) :=\n{ inv_mem := λ x h, set.mem_Inter.2 $ λ y, is_subgroup.inv_mem (hs _) (set.mem_Inter.1 h y),\n  ..is_submonoid.Inter (λ y, (hs y).to_is_submonoid) }\n\n@[to_additive]\nlemma is_subgroup_Union_of_directed {ι : Type*} [hι : nonempty ι]\n  {s : ι → set G} (hs : ∀ i, is_subgroup (s i))\n  (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :\n  is_subgroup (⋃i, s i) :=\n{ inv_mem := λ a ha,\n    let ⟨i, hi⟩ := set.mem_Union.1 ha in\n    set.mem_Union.2 ⟨i, (hs i).inv_mem hi⟩,\n  to_is_submonoid := is_submonoid_Union_of_directed (λ i, (hs i).to_is_submonoid) directed }\n\nend group\n\nnamespace is_subgroup\nopen is_submonoid\nvariables [group G] {s : set G} (hs : is_subgroup s)\n\ninclude hs\n\n@[to_additive]\nlemma inv_mem_iff : a⁻¹ ∈ s ↔ a ∈ s :=\n⟨λ h, by simpa using hs.inv_mem h, inv_mem hs⟩\n\n@[to_additive]\nlemma mul_mem_cancel_right (h : a ∈ s) : b * a ∈ s ↔ b ∈ s :=\n⟨λ hba, by simpa using hs.mul_mem hba (hs.inv_mem h), λ hb, hs.mul_mem hb h⟩\n\n@[to_additive]\nlemma mul_mem_cancel_left (h : a ∈ s) : a * b ∈ s ↔ b ∈ s :=\n⟨λ hab, by simpa using hs.mul_mem (hs.inv_mem h) hab, hs.mul_mem h⟩\n\nend is_subgroup\n\n/-- `is_normal_add_subgroup (s : set A)` expresses the fact that `s` is a normal additive subgroup\nof the additive group `A`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : add_subgroup A` and `hs : S.normal`, and not via this structure. -/\nstructure is_normal_add_subgroup [add_group A] (s : set A) extends is_add_subgroup s : Prop :=\n(normal : ∀ n ∈ s, ∀ g : A, g + n + -g ∈ s)\n\n/-- `is_normal_subgroup (s : set G)` expresses the fact that `s` is a normal subgroup\nof the group `G`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : subgroup G` and not via this structure. -/\n@[to_additive]\nstructure is_normal_subgroup [group G] (s : set G) extends is_subgroup s : Prop :=\n(normal : ∀ n ∈ s, ∀ g : G, g * n * g⁻¹ ∈ s)\n\n@[to_additive]\nlemma is_normal_subgroup_of_comm_group [comm_group G] {s : set G} (hs : is_subgroup s) :\n  is_normal_subgroup s :=\n{ normal := λ n hn g, by rwa [mul_right_comm, mul_right_inv, one_mul],\n  ..hs }\n\nlemma additive.is_normal_add_subgroup [group G]\n  {s : set G} (hs : is_normal_subgroup s) : @is_normal_add_subgroup (additive G) _ s :=\n@is_normal_add_subgroup.mk (additive G) _ _\n  (additive.is_add_subgroup hs.to_is_subgroup)\n  (is_normal_subgroup.normal hs)\n\ntheorem additive.is_normal_add_subgroup_iff [group G]\n  {s : set G} : @is_normal_add_subgroup (additive G) _ s ↔ is_normal_subgroup s :=\n⟨by rintro ⟨h₁, h₂⟩; exact\n    @is_normal_subgroup.mk G _ _ (additive.is_add_subgroup_iff.1 h₁) @h₂,\n  λ h, by exactI additive.is_normal_add_subgroup h⟩\n\nlemma multiplicative.is_normal_subgroup [add_group A]\n  {s : set A} (hs : is_normal_add_subgroup s) : @is_normal_subgroup (multiplicative A) _ s :=\n@is_normal_subgroup.mk (multiplicative A) _ _\n  (multiplicative.is_subgroup hs.to_is_add_subgroup)\n  (is_normal_add_subgroup.normal hs)\n\ntheorem multiplicative.is_normal_subgroup_iff [add_group A]\n  {s : set A} : @is_normal_subgroup (multiplicative A) _ s ↔ is_normal_add_subgroup s :=\n⟨by rintro ⟨h₁, h₂⟩; exact\n    @is_normal_add_subgroup.mk A _ _ (multiplicative.is_subgroup_iff.1 h₁) @h₂,\n  λ h, by exactI multiplicative.is_normal_subgroup h⟩\n\nnamespace is_subgroup\nvariable [group G]\n\n-- Normal subgroup properties\n@[to_additive]\nlemma mem_norm_comm {s : set G} (hs : is_normal_subgroup s) {a b : G} (hab : a * b ∈ s) :\n  b * a ∈ s :=\nhave h : a⁻¹ * (a * b) * a⁻¹⁻¹ ∈ s, from hs.normal (a * b) hab a⁻¹,\nby simp at h; exact h\n\n@[to_additive]\nlemma mem_norm_comm_iff {s : set G} (hs : is_normal_subgroup s) {a b : G} : a * b ∈ s ↔ b * a ∈ s :=\n⟨mem_norm_comm hs, mem_norm_comm hs⟩\n\n/-- The trivial subgroup -/\n@[to_additive \"the trivial additive subgroup\"]\ndef trivial (G : Type*) [group G] : set G := {1}\n\n@[simp, to_additive]\nlemma mem_trivial {g : G} : g ∈ trivial G ↔ g = 1 :=\nmem_singleton_iff\n\n@[to_additive]\nlemma trivial_normal : is_normal_subgroup (trivial G) :=\nby refine {..}; simp [trivial] {contextual := tt}\n\n@[to_additive]\nlemma eq_trivial_iff {s : set G} (hs : is_subgroup s) :\n  s = trivial G ↔ (∀ x ∈ s, x = (1 : G)) :=\nby simp only [set.ext_iff, is_subgroup.mem_trivial];\n  exact ⟨λ h x, (h x).1, λ h x, ⟨h x, λ hx, hx.symm ▸ hs.to_is_submonoid.one_mem⟩⟩\n\n@[to_additive]\nlemma univ_subgroup : is_normal_subgroup (@univ G) :=\nby refine {..}; simp\n\n/-- The underlying set of the center of a group. -/\n@[to_additive add_center \"The underlying set of the center of an additive group.\"]\ndef center (G : Type*) [group G] : set G := {z | ∀ g, g * z = z * g}\n\n@[to_additive mem_add_center]\nlemma mem_center {a : G} : a ∈ center G ↔ ∀g, g * a = a * g := iff.rfl\n\n@[to_additive add_center_normal]\nlemma center_normal : is_normal_subgroup (center G) :=\n{ one_mem := by simp [center],\n  mul_mem := assume a b ha hb g,\n    by rw [←mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, ←mul_assoc],\n  inv_mem := assume a ha g,\n    calc\n      g * a⁻¹ = a⁻¹ * (g * a) * a⁻¹ : by simp [ha g]\n      ...     = a⁻¹ * g             : by rw [←mul_assoc, mul_assoc]; simp,\n  normal := assume n ha g h,\n    calc\n      h * (g * n * g⁻¹) = h * n           : by simp [ha g, mul_assoc]\n      ...               = g * g⁻¹ * n * h : by rw ha h; simp\n      ...               = g * n * g⁻¹ * h : by rw [mul_assoc g, ha g⁻¹, ←mul_assoc] }\n\n/-- The underlying set of the normalizer of a subset `S : set G` of a group `G`. That is,\n  the elements `g : G` such that `g * S * g⁻¹ = S`. -/\n@[to_additive add_normalizer \"The underlying set of the normalizer of a subset `S : set A` of an\n  additive group `A`. That is, the elements `a : A` such that `a + S - a = S`.\"]\ndef normalizer (s : set G) : set G :=\n{g : G | ∀ n, n ∈ s ↔ g * n * g⁻¹ ∈ s}\n\n@[to_additive]\nlemma normalizer_is_subgroup (s : set G) : is_subgroup (normalizer s) :=\n{ one_mem := by simp [normalizer],\n  mul_mem := λ a b (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s)\n    (hb : ∀ n, n ∈ s ↔ b * n * b⁻¹ ∈ s) n,\n    by rw [mul_inv_rev, ← mul_assoc, mul_assoc a, mul_assoc a, ← ha, ← hb],\n  inv_mem := λ a (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) n,\n    by rw [ha (a⁻¹ * n * a⁻¹⁻¹)];\n    simp [mul_assoc] }\n\n@[to_additive subset_add_normalizer]\nlemma subset_normalizer {s : set G} (hs : is_subgroup s) : s ⊆ normalizer s :=\nλ g hg n, by rw [is_subgroup.mul_mem_cancel_right hs ((is_subgroup.inv_mem_iff hs).2 hg),\n  is_subgroup.mul_mem_cancel_left hs hg]\n\nend is_subgroup\n\n-- Homomorphism subgroups\nnamespace is_group_hom\nopen is_submonoid is_subgroup\n\n/-- `ker f : set G` is the underlying subset of the kernel of a map `G → H`. -/\n@[to_additive \"`ker f : set A` is the underlying subset of the kernel of a map `A → B`\"]\ndef ker [group H] (f : G → H) : set G := preimage f (trivial H)\n\n@[to_additive]\nlemma mem_ker [group H] (f : G → H) {x : G} : x ∈ ker f ↔ f x = 1 :=\nmem_trivial\n\nvariables [group G] [group H]\n\n@[to_additive]\nlemma one_ker_inv {f : G → H} (hf : is_group_hom f) {a b : G} (h : f (a * b⁻¹) = 1) : f a = f b :=\nbegin\n  rw [hf.map_mul, hf.map_inv] at h,\n  rw [←inv_inv (f b), eq_inv_of_mul_eq_one_left h]\nend\n\n@[to_additive]\nlemma one_ker_inv' {f : G → H} (hf : is_group_hom f) {a b : G} (h : f (a⁻¹ * b) = 1) : f a = f b :=\nbegin\n  rw [hf.map_mul, hf.map_inv] at h,\n  apply inv_injective,\n  rw eq_inv_of_mul_eq_one_left h\nend\n\n@[to_additive]\nlemma inv_ker_one {f : G → H} (hf : is_group_hom f) {a b : G} (h : f a = f b) : f (a * b⁻¹) = 1 :=\nhave f a * (f b)⁻¹ = 1, by rw [h, mul_right_inv],\nby rwa [←hf.map_inv, ←hf.map_mul] at this\n\n@[to_additive]\nlemma inv_ker_one' {f : G → H} (hf : is_group_hom f) {a b : G} (h : f a = f b) : f (a⁻¹ * b) = 1 :=\nhave (f a)⁻¹ * f b = 1, by rw [h, mul_left_inv],\nby rwa [←hf.map_inv, ←hf.map_mul] at this\n\n@[to_additive]\nlemma one_iff_ker_inv {f : G → H} (hf : is_group_hom f) (a b : G) : f a = f b ↔ f (a * b⁻¹) = 1 :=\n⟨hf.inv_ker_one, hf.one_ker_inv⟩\n\n@[to_additive]\nlemma one_iff_ker_inv' {f : G → H} (hf : is_group_hom f) (a b : G) : f a = f b ↔ f (a⁻¹ * b) = 1 :=\n⟨hf.inv_ker_one', hf.one_ker_inv'⟩\n\n@[to_additive]\nlemma inv_iff_ker {f : G → H} (hf : is_group_hom f) (a b : G) : f a = f b ↔ a * b⁻¹ ∈ ker f :=\nby rw [mem_ker]; exact one_iff_ker_inv hf _ _\n\n@[to_additive]\nlemma inv_iff_ker' {f : G → H} (hf : is_group_hom f) (a b : G) : f a = f b ↔ a⁻¹ * b ∈ ker f :=\nby rw [mem_ker]; exact one_iff_ker_inv' hf _ _\n\n@[to_additive]\nlemma image_subgroup {f : G → H} (hf : is_group_hom f) {s : set G} (hs : is_subgroup s) :\n  is_subgroup (f '' s) :=\n{ mul_mem := assume a₁ a₂ ⟨b₁, hb₁, eq₁⟩ ⟨b₂, hb₂, eq₂⟩,\n             ⟨b₁ * b₂, hs.mul_mem hb₁ hb₂, by simp [eq₁, eq₂, hf.map_mul]⟩,\n  one_mem := ⟨1, hs.to_is_submonoid.one_mem, hf.map_one⟩,\n  inv_mem := assume a ⟨b, hb, eq⟩, ⟨b⁻¹, hs.inv_mem hb, by { rw hf.map_inv, simp * }⟩ }\n\n@[to_additive]\nlemma range_subgroup {f : G → H} (hf : is_group_hom f) : is_subgroup (set.range f) :=\n@set.image_univ _ _ f ▸ hf.image_subgroup univ_subgroup.to_is_subgroup\n\nlocal attribute [simp] one_mem inv_mem mul_mem is_normal_subgroup.normal\n\n@[to_additive]\nlemma preimage {f : G → H} (hf : is_group_hom f) {s : set H} (hs : is_subgroup s) :\n  is_subgroup (f ⁻¹' s) :=\nby { refine {..};\n     simp [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul, hf.map_one, hf.map_inv,\n           inv_mem_class.inv_mem]\n     {contextual := tt} }\n\n@[to_additive]\nlemma preimage_normal {f : G → H} (hf : is_group_hom f) {s : set H} (hs : is_normal_subgroup s) :\n  is_normal_subgroup (f ⁻¹' s) :=\n{ one_mem := by simp [hf.map_one, hs.to_is_subgroup.one_mem],\n  mul_mem := by simp [hf.map_mul, hs.to_is_subgroup.mul_mem] {contextual := tt},\n  inv_mem := by simp [hf.map_inv, hs.to_is_subgroup.inv_mem] {contextual := tt},\n  normal := by simp [hs.normal, hf.map_mul, hf.map_inv] {contextual := tt}}\n\n@[to_additive]\nlemma is_normal_subgroup_ker {f : G → H} (hf : is_group_hom f) : is_normal_subgroup (ker f) :=\nhf.preimage_normal (trivial_normal)\n\n@[to_additive]\nlemma injective_of_trivial_ker {f : G → H} (hf : is_group_hom f) (h : ker f = trivial G) :\n  function.injective f :=\nbegin\n  intros a₁ a₂ hfa,\n  simp [ext_iff, ker, is_subgroup.trivial] at h,\n  have ha : a₁ * a₂⁻¹ = 1, by rw ←h; exact hf.inv_ker_one hfa,\n  rw [eq_inv_of_mul_eq_one_left ha, inv_inv a₂]\nend\n\n@[to_additive]\nlemma trivial_ker_of_injective {f : G → H} (hf : is_group_hom f) (h : function.injective f) :\n  ker f = trivial G :=\nset.ext $ assume x, iff.intro\n  (assume hx,\n    suffices f x = f 1, by simpa using h this,\n    by simp [hf.map_one]; rwa [mem_ker] at hx)\n  (by simp [mem_ker, hf.map_one] {contextual := tt})\n\n@[to_additive]\nlemma injective_iff_trivial_ker {f : G → H} (hf : is_group_hom f) :\n  function.injective f ↔ ker f = trivial G :=\n⟨hf.trivial_ker_of_injective, hf.injective_of_trivial_ker⟩\n\n@[to_additive]\nlemma trivial_ker_iff_eq_one {f : G → H} (hf : is_group_hom f) :\n  ker f = trivial G ↔ ∀ x, f x = 1 → x = 1 :=\nby rw set.ext_iff; simp [ker]; exact\n⟨λ h x hx, (h x).1 hx, λ h x, ⟨h x, λ hx, by rw [hx, hf.map_one]⟩⟩\n\nend is_group_hom\n\nnamespace add_group\n\nvariables [add_group A]\n\n/-- If `A` is an additive group and `s : set A`, then `in_closure s : set A` is the underlying\nsubset of the subgroup generated by `s`. -/\ninductive in_closure (s : set A) : A → Prop\n| basic {a : A} : a ∈ s → in_closure a\n| zero : in_closure 0\n| neg {a : A} : in_closure a → in_closure (-a)\n| add {a b : A} : in_closure a → in_closure b → in_closure (a + b)\n\nend add_group\n\nnamespace group\nopen is_submonoid is_subgroup\n\nvariables [group G] {s : set G}\n\n/-- If `G` is a group and `s : set G`, then `in_closure s : set G` is the underlying\nsubset of the subgroup generated by `s`. -/\n@[to_additive]\ninductive in_closure (s : set G) : G → Prop\n| basic {a : G} : a ∈ s → in_closure a\n| one : in_closure 1\n| inv {a : G} : in_closure a → in_closure a⁻¹\n| mul {a b : G} : in_closure a → in_closure b → in_closure (a * b)\n\n/-- `group.closure s` is the subgroup generated by `s`, i.e. the smallest subgroup containg `s`. -/\n@[to_additive \"`add_group.closure s` is the additive subgroup generated by `s`, i.e., the\n  smallest additive subgroup containing `s`.\"]\ndef closure (s : set G) : set G := {a | in_closure s a }\n\n@[to_additive]\nlemma mem_closure {a : G} : a ∈ s → a ∈ closure s := in_closure.basic\n\n@[to_additive]\nlemma closure.is_subgroup (s : set G) : is_subgroup (closure s) :=\n{ one_mem := in_closure.one,\n  mul_mem := assume a b, in_closure.mul,\n  inv_mem := assume a, in_closure.inv }\n\n@[to_additive]\ntheorem subset_closure {s : set G} : s ⊆ closure s := λ a, mem_closure\n\n@[to_additive]\ntheorem closure_subset {s t : set G} (ht : is_subgroup t) (h : s ⊆ t) : closure s ⊆ t :=\nassume a ha, by induction ha; simp [h _, *, ht.one_mem, ht.mul_mem, is_subgroup.inv_mem_iff]\n\n@[to_additive]\nlemma closure_subset_iff {s t : set G} (ht : is_subgroup t) : closure s ⊆ t ↔ s ⊆ t :=\n⟨assume h b ha, h (mem_closure ha), assume h b ha, closure_subset ht h ha⟩\n\n@[to_additive]\ntheorem closure_mono {s t : set G} (h : s ⊆ t) : closure s ⊆ closure t :=\nclosure_subset (closure.is_subgroup _) $ set.subset.trans h subset_closure\n\n@[simp, to_additive]\nlemma closure_subgroup {s : set G} (hs : is_subgroup s) : closure s = s :=\nset.subset.antisymm (closure_subset hs $ set.subset.refl s) subset_closure\n\n@[to_additive]\n\n\n@[to_additive]\nlemma image_closure [group H] {f : G → H} (hf : is_group_hom f) (s : set G) :\n  f '' closure s = closure (f '' s) :=\nle_antisymm\n  begin\n    rintros _ ⟨x, hx, rfl⟩,\n    apply in_closure.rec_on hx; intros,\n    { solve_by_elim [subset_closure, set.mem_image_of_mem] },\n    { rw [hf.to_is_monoid_hom.map_one],\n      apply is_submonoid.one_mem (closure.is_subgroup _).to_is_submonoid, },\n    { rw [hf.map_inv],\n      apply is_subgroup.inv_mem (closure.is_subgroup _), assumption },\n    { rw [hf.to_is_monoid_hom.map_mul],\n      solve_by_elim [is_submonoid.mul_mem (closure.is_subgroup _).to_is_submonoid] }\n  end\n  (closure_subset (hf.image_subgroup $ closure.is_subgroup _) $ set.image_subset _ subset_closure)\n\n@[to_additive]\ntheorem mclosure_subset {s : set G} : monoid.closure s ⊆ closure s :=\nmonoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ subset_closure\n\n@[to_additive]\ntheorem mclosure_inv_subset {s : set G} : monoid.closure (has_inv.inv ⁻¹' s) ⊆ closure s :=\nmonoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ λ x hx,\n  inv_inv x ▸ ((closure.is_subgroup _).inv_mem $ subset_closure hx)\n\n@[to_additive]\ntheorem closure_eq_mclosure {s : set G} : closure s = monoid.closure (s ∪ has_inv.inv ⁻¹' s) :=\nset.subset.antisymm\n  (@closure_subset _ _ _ (monoid.closure (s ∪ has_inv.inv ⁻¹' s))\n    { one_mem := (monoid.closure.is_submonoid _).one_mem,\n      mul_mem := λ _ _, (monoid.closure.is_submonoid _).mul_mem,\n      inv_mem := λ x hx, monoid.in_closure.rec_on hx\n      (λ x hx, or.cases_on hx (λ hx, monoid.subset_closure $ or.inr $\n        show x⁻¹⁻¹ ∈ s, from (inv_inv x).symm ▸ hx)\n        (λ hx, monoid.subset_closure $ or.inl hx))\n      ((@inv_one G _).symm ▸ is_submonoid.one_mem (monoid.closure.is_submonoid _))\n      (λ x y hx hy ihx ihy,\n        (mul_inv_rev x y).symm ▸ is_submonoid.mul_mem (monoid.closure.is_submonoid _) ihy ihx) }\n    (set.subset.trans (set.subset_union_left _ _) monoid.subset_closure))\n  (monoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ set.union_subset subset_closure $\n    λ x hx, inv_inv x ▸ (is_subgroup.inv_mem (closure.is_subgroup _) $ subset_closure hx))\n\n@[to_additive]\ntheorem mem_closure_union_iff {G : Type*} [comm_group G] {s t : set G} {x : G} :\n  x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x :=\nbegin\n  simp only [closure_eq_mclosure, monoid.mem_closure_union_iff, exists_prop, preimage_union], split,\n  { rintro ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, rfl⟩,\n    refine ⟨_, ⟨_, hys, _, hzs, rfl⟩, _, ⟨_, hyt, _, hzt, rfl⟩, _⟩,\n    rw [mul_assoc, mul_assoc, mul_left_comm zs] },\n  { rintro ⟨_, ⟨ys, hys, zs, hzs, rfl⟩, _, ⟨yt, hyt, zt, hzt, rfl⟩, rfl⟩,\n    refine ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, _⟩,\n    rw [mul_assoc, mul_assoc, mul_left_comm yt] }\nend\n\nend group\n\nnamespace is_subgroup\nvariable [group G]\n\n@[to_additive]\nlemma trivial_eq_closure : trivial G = group.closure ∅ :=\nsubset.antisymm\n  (by simp [set.subset_def, (group.closure.is_subgroup _).one_mem])\n  (group.closure_subset (trivial_normal).to_is_subgroup $ by simp)\n\nend is_subgroup\n\n/-The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\n\nnamespace group\nvariables {s : set G} [group G]\n\nlemma conjugates_of_subset {t : set G} (ht : is_normal_subgroup t) {a : G} (h : a ∈ t) :\n  conjugates_of a ⊆ t :=\nλ x hc,\nbegin\n  obtain ⟨c, w⟩ := is_conj_iff.1 hc,\n  have H := is_normal_subgroup.normal ht a h c,\n  rwa ←w,\nend\n\ntheorem conjugates_of_set_subset' {s t : set G} (ht : is_normal_subgroup t) (h : s ⊆ t) :\n  conjugates_of_set s ⊆ t :=\nset.Union₂_subset (λ x H, conjugates_of_subset ht (h H))\n\n/-- The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\ndef normal_closure (s : set G) : set G := closure (conjugates_of_set s)\n\ntheorem conjugates_of_set_subset_normal_closure : conjugates_of_set s ⊆ normal_closure s :=\nsubset_closure\n\ntheorem subset_normal_closure : s ⊆ normal_closure s :=\nset.subset.trans subset_conjugates_of_set conjugates_of_set_subset_normal_closure\n\n/-- The normal closure of a set is a subgroup. -/\nlemma normal_closure.is_subgroup (s : set G) : is_subgroup (normal_closure s) :=\nclosure.is_subgroup (conjugates_of_set s)\n\n/-- The normal closure of s is a normal subgroup. -/\nlemma normal_closure.is_normal : is_normal_subgroup (normal_closure s) :=\n{ normal := λ n h g,\nbegin\n  induction h with x hx x hx ihx x y hx hy ihx ihy,\n  {exact (conjugates_of_set_subset_normal_closure (conj_mem_conjugates_of_set hx))},\n  {simpa using (normal_closure.is_subgroup s).one_mem},\n  {rw ←conj_inv,\n   exact ((normal_closure.is_subgroup _).inv_mem ihx)},\n  {rw ←conj_mul,\n   exact ((normal_closure.is_subgroup _).to_is_submonoid.mul_mem ihx ihy)},\nend,\n..normal_closure.is_subgroup _ }\n\n/-- The normal closure of s is the smallest normal subgroup containing s. -/\ntheorem normal_closure_subset {s t : set G} (ht : is_normal_subgroup t) (h : s ⊆ t) :\n  normal_closure s ⊆ t :=\nλ a w,\nbegin\n  induction w with x hx x hx ihx x y hx hy ihx ihy,\n  {exact (conjugates_of_set_subset' ht h $ hx)},\n  {exact ht.to_is_subgroup.to_is_submonoid.one_mem},\n  {exact ht.to_is_subgroup.inv_mem ihx},\n  {exact ht.to_is_subgroup.to_is_submonoid.mul_mem ihx ihy}\nend\n\nlemma normal_closure_subset_iff {s t : set G} (ht : is_normal_subgroup t) :\n  s ⊆ t ↔ normal_closure s ⊆ t :=\n⟨normal_closure_subset ht, set.subset.trans (subset_normal_closure)⟩\n\ntheorem normal_closure_mono {s t : set G} : s ⊆ t → normal_closure s ⊆ normal_closure t :=\nλ h, normal_closure_subset normal_closure.is_normal (set.subset.trans h (subset_normal_closure))\n\nend group\n\n/-- Create a bundled subgroup from a set `s` and `[is_subgroup s]`. -/\n@[to_additive \"Create a bundled additive subgroup from a set `s` and `[is_add_subgroup s]`.\"]\ndef subgroup.of [group G] {s : set G} (h : is_subgroup s) : subgroup G :=\n{ carrier := s,\n  one_mem' := h.1.1,\n  mul_mem' := λ _ _, h.1.2,\n  inv_mem' := λ _, h.2 }\n\n@[to_additive]\nlemma subgroup.is_subgroup [group G] (K : subgroup G) : is_subgroup (K : set G) :=\n{ one_mem := K.one_mem',\n  mul_mem := λ _ _, K.mul_mem',\n  inv_mem := λ _, K.inv_mem' }\n\n-- this will never fire if it's an instance\n@[to_additive]\nlemma subgroup.of_normal [group G] (s : set G) (h : is_subgroup s) (n : is_normal_subgroup s) :\n  subgroup.normal (subgroup.of h) :=\n{ conj_mem := n.normal, }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/deprecated/subgroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.31545136226305825}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.adjunction.basic\nimport category_theory.conj\nimport category_theory.yoneda\n\n/-!\n# Adjoints of fully faithful functors\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA left adjoint is fully faithful, if and only if the unit is an isomorphism\n(and similarly for right adjoints and the counit).\n\n`adjunction.restrict_fully_faithful` shows that an adjunction can be restricted along fully faithful\ninclusions.\n\n## Future work\n\nThe statements from Riehl 4.5.13 for adjoints which are either full, or faithful.\n-/\n\nopen category_theory\n\nnamespace category_theory\nuniverses v₁ v₂ u₁ u₂\n\nopen category\nopen opposite\n\nvariables {C : Type u₁} [category.{v₁} C]\nvariables {D : Type u₂} [category.{v₂} D]\nvariables {L : C ⥤ D} {R : D ⥤ C} (h : L ⊣ R)\n\n/--\nIf the left adjoint is fully faithful, then the unit is an isomorphism.\n\nSee\n* Lemma 4.5.13 from [Riehl][riehl2017]\n* https://math.stackexchange.com/a/2727177\n* https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)\n-/\ninstance unit_is_iso_of_L_fully_faithful [full L] [faithful L] : is_iso (adjunction.unit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.unit h) $ λ X,\n@yoneda.is_iso _ _ _ _ ((adjunction.unit h).app X)\n⟨⟨{ app := λ Y f, L.preimage ((h.hom_equiv (unop Y) (L.obj X)).symm f) },\n  ⟨begin\n    ext x f, dsimp,\n    apply L.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_counit, preimage_comp, preimage_map, category.assoc],\n    rw ←h.unit_naturality,\n    simp,\n  end⟩⟩⟩\n\n/--\nIf the right adjoint is fully faithful, then the counit is an isomorphism.\n\nSee <https://stacks.math.columbia.edu/tag/07RB> (we only prove the forward direction!)\n-/\ninstance counit_is_iso_of_R_fully_faithful [full R] [faithful R] : is_iso (adjunction.counit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.counit h) $ λ X,\n@is_iso_of_op _ _ _ _ _ $\n@coyoneda.is_iso _ _ _ _ ((adjunction.counit h).app X).op\n⟨⟨{ app := λ Y f, R.preimage ((h.hom_equiv (R.obj X) Y) f) },\n  ⟨begin\n    ext x f, dsimp,\n    apply R.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_unit, preimage_comp, preimage_map],\n    rw ←h.counit_naturality,\n    simp,\n  end⟩⟩⟩\n\n/-- If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given\nby L whiskered with the counit. -/\n@[simp]\nlemma inv_map_unit {X : C} [is_iso (h.unit.app X)] :\n  inv (L.map (h.unit.app X)) = h.counit.app (L.obj X) :=\nis_iso.inv_eq_of_hom_inv_id h.left_triangle_components\n\n/-- If the unit is an isomorphism, bundle one has an isomorphism `L ⋙ R ⋙ L ≅ L`. -/\n@[simps]\nnoncomputable def whisker_left_L_counit_iso_of_is_iso_unit [is_iso h.unit] :\n  L ⋙ R ⋙ L ≅ L :=\n(L.associator R L).symm ≪≫ iso_whisker_right (as_iso h.unit).symm L ≪≫ functor.left_unitor _\n\n/-- If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given\nby R whiskered with the unit. -/\n@[simp]\nlemma inv_counit_map {X : D} [is_iso (h.counit.app X)] :\n  inv (R.map (h.counit.app X)) = h.unit.app (R.obj X) :=\nis_iso.inv_eq_of_inv_hom_id h.right_triangle_components\n\n/-- If the counit of an is an isomorphism, one has an isomorphism `(R ⋙ L ⋙ R) ≅ R`. -/\n@[simps]\nnoncomputable def whisker_left_R_unit_iso_of_is_iso_counit [is_iso h.counit] :\n  (R ⋙ L ⋙ R) ≅ R :=\n(R.associator L R).symm ≪≫ iso_whisker_right (as_iso h.counit) R ≪≫ functor.left_unitor _\n\n/-- If the unit is an isomorphism, then the left adjoint is full-/\nnoncomputable\ndef L_full_of_unit_is_iso [is_iso h.unit] : full L :=\n{ preimage := λ X Y f, (h.hom_equiv X (L.obj Y) f) ≫ inv (h.unit.app Y) }\n\n/-- If the unit is an isomorphism, then the left adjoint is faithful-/\nlemma L_faithful_of_unit_is_iso [is_iso h.unit] : faithful L :=\n{ map_injective' := λ X Y f g H,\n  begin\n    rw ←(h.hom_equiv X (L.obj Y)).apply_eq_iff_eq at H,\n    simpa using H =≫ inv (h.unit.app Y),\n  end }\n\n/-- If the counit is an isomorphism, then the right adjoint is full-/\nnoncomputable\ndef R_full_of_counit_is_iso [is_iso h.counit] : full R :=\n{ preimage := λ X Y f, inv (h.counit.app X) ≫ (h.hom_equiv (R.obj X) Y).symm f }\n\n/-- If the counit is an isomorphism, then the right adjoint is faithful-/\nlemma R_faithful_of_counit_is_iso [is_iso h.counit] : faithful R :=\n{ map_injective' := λ X Y f g H,\n  begin\n    rw ←(h.hom_equiv (R.obj X) Y).symm.apply_eq_iff_eq at H,\n    simpa using inv (h.counit.app X) ≫= H,\n  end }\n\ninstance whisker_left_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_left L h.counit) :=\nbegin\n  have := h.left_triangle,\n  rw ←is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_right h.counit R) :=\nbegin\n  have := h.right_triangle,\n  rw ←is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_left_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_left R h.unit) :=\nbegin\n  have := h.right_triangle,\n  rw ←is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_right h.unit L) :=\nbegin\n  have := h.left_triangle,\n  rw ←is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\n-- TODO also do the statements from Riehl 4.5.13 for full and faithful separately?\n\nuniverses v₃ v₄ u₃ u₄\n\nvariables {C' : Type u₃} [category.{v₃} C']\nvariables {D' : Type u₄} [category.{v₄} D']\n\n-- TODO: This needs some lemmas describing the produced adjunction, probably in terms of `adj`,\n-- `iC` and `iD`.\n/--\nIf `C` is a full subcategory of `C'` and `D` is a full subcategory of `D'`, then we can restrict\nan adjunction `L' ⊣ R'` where `L' : C' ⥤ D'` and `R' : D' ⥤ C'` to `C` and `D`.\nThe construction here is slightly more general, in that `C` is required only to have a full and\nfaithful \"inclusion\" functor `iC : C ⥤ C'` (and similarly `iD : D ⥤ D'`) which commute (up to\nnatural isomorphism) with the proposed restrictions.\n-/\ndef adjunction.restrict_fully_faithful (iC : C ⥤ C') (iD : D ⥤ D') {L' : C' ⥤ D'} {R' : D' ⥤ C'}\n  (adj : L' ⊣ R') {L : C ⥤ D} {R : D ⥤ C} (comm1 : iC ⋙ L' ≅ L ⋙ iD) (comm2 : iD ⋙ R' ≅ R ⋙ iC)\n  [full iC] [faithful iC] [full iD] [faithful iD] :\n  L ⊣ R :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  calc (L.obj X ⟶ Y) ≃ (iD.obj (L.obj X) ⟶ iD.obj Y) : equiv_of_fully_faithful iD\n       ... ≃ (L'.obj (iC.obj X) ⟶ iD.obj Y) : iso.hom_congr (comm1.symm.app X) (iso.refl _)\n       ... ≃ (iC.obj X ⟶ R'.obj (iD.obj Y)) : adj.hom_equiv _ _\n       ... ≃ (iC.obj X ⟶ iC.obj (R.obj Y)) : iso.hom_congr (iso.refl _) (comm2.app Y)\n       ... ≃ (X ⟶ R.obj Y) : (equiv_of_fully_faithful iC).symm,\n  hom_equiv_naturality_left_symm' := λ X' X Y f g,\n  begin\n    apply iD.map_injective,\n    simpa using (comm1.inv.naturality_assoc f _).symm,\n  end,\n  hom_equiv_naturality_right' := λ X Y' Y f g,\n  begin\n    apply iC.map_injective,\n    suffices : R'.map (iD.map g) ≫ comm2.hom.app Y = comm2.hom.app Y' ≫ iC.map (R.map g),\n      simp [this],\n    apply comm2.hom.naturality g,\n  end }\n\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/adjunction/fully_faithful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982645, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3153597131862265}}
{"text": "import condensed.is_proetale_sheaf\nimport condensed.top_comparison\nimport condensed.adjunctions\n\n/-!\nWe show that passing from a profinite set to a condensed set\npreserves (finite) coproducts.\n-/\n\nopen_locale big_operators classical\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverse u\n\nnamespace Profinite\n\n@[simps]\ndef to_Condensed_equiv (X : Profinite.{u}) (Y : CondensedSet.{u}) :\n  (X.to_Condensed ⟶ Y) ≃ Y.val.obj (op X) :=\n{ to_fun := λ f, f.val.app _ $ ulift.up $ 𝟙 _,\n  inv_fun := λ f, Sheaf.hom.mk $\n  { app := λ T g, Y.val.map (quiver.hom.op (ulift.down g)) f,\n    naturality' := begin\n      intros A B ff, ext t,\n      obtain ⟨t⟩ := t,\n      dsimp [Profinite.to_Condensed, ulift_functor, yoneda] at ⊢ t,\n      simp only [functor.map_comp], refl,\n    end },\n  left_inv := λ f, begin\n    ext T ⟨t⟩,\n    dsimp [yoneda] at ⊢ t,\n    change (f.val.app _ ≫ Y.val.map _) _ = _,\n    rw ← nat_trans.naturality,\n    change f.val.app _ _ = _,\n    congr' 1, ext, refl,\n  end,\n  right_inv := λ f, by { dsimp, simp } }\n\nend Profinite\n\nnamespace CondensedSet\n\nvariables {α : Type u} [fintype α] (X : α → Profinite.{u})\n\n@[simps]\ndef sigma_cone : cocone (discrete.functor X ⋙ Profinite_to_Condensed) :=\n{ X := (Profinite.sigma X).to_Condensed,\n  ι :=\n  { app := λ i, Profinite_to_Condensed.map $ Profinite.sigma.ι X i.1,\n    naturality' := begin\n      rintros ⟨i⟩ ⟨j⟩ ⟨⟨⟨⟩⟩⟩, dsimp, simp, dsimp, simp, dsimp, simp,\n    end } } .\n\nnoncomputable\ndef val_obj_sigma_equiv (Y : CondensedSet.{u}) :\n  Y.val.obj (op $ Profinite.sigma X) ≃ (Π (a : α), Y.val.obj (op $ X a)) :=\nequiv.of_bijective\n(λ f a, Y.val.map (Profinite.sigma.ι X a).op f)\nbegin\n  have := Y.2,\n  rw is_sheaf_iff_is_sheaf_of_type at this,\n  rw Y.val.is_proetale_sheaf_of_types_tfae.out 0 4 at this,\n  have key := this.1,\n  exact key ⟨α⟩ X,\nend\n\nnoncomputable\ndef _root_.Condensed.val_obj_sigma_add_equiv\n  (Y : Condensed.{u} Ab.{u+1}) :\n  Y.val.obj (op $ Profinite.sigma X) ≃+\n  (Π (a : α), Y.val.obj (op $ X a)) :=\nadd_equiv.of_bijective\n(add_monoid_hom.mk' (λ f a, Y.val.map (Profinite.sigma.ι X a).op f) (by { intros, ext1, simp }))\n((Condensed_Ab_to_CondensedSet.obj Y).val_obj_sigma_equiv X).bijective\n\n@[simp]\nlemma coe_val_obj_sigma_equiv (Y : Condensed.{u} Ab.{u+1}) :\n  ⇑((Condensed_Ab_to_CondensedSet.obj Y).val_obj_sigma_equiv X) =\n  (Y.val_obj_sigma_add_equiv X) := rfl\n\n@[simp]\nlemma coe_val_obj_sigma_equiv_symm (Y : Condensed.{u} Ab.{u+1}) :\n  ⇑((Condensed_Ab_to_CondensedSet.obj Y).val_obj_sigma_equiv X).symm =\n  (Y.val_obj_sigma_add_equiv X).symm := rfl\n\n@[simp]\nlemma _root_.Condensed.val_obj_sigma_add_equiv_apply_apply\n  (Y : Condensed.{u} Ab.{u+1}) (t) (a) :\n  Y.val_obj_sigma_add_equiv X t a = Y.val.map (Profinite.sigma.ι X a).op t := rfl\n\nlemma val_obj_sigma_equiv_symm_apply'\n  (Y : CondensedSet.{u})\n  (e : Π (a : α), Y.val.obj (op $ X a)) (a₀ : α) :\n  (Y.val.map (Profinite.sigma.ι X a₀).op)\n  (((val_obj_sigma_equiv X Y).symm) e) = e a₀ :=\nbegin\n  let e' := _, change (Y.val.map (Profinite.sigma.ι X a₀).op) e' = _,\n  have : e a₀ = (val_obj_sigma_equiv X Y) e' a₀,\n    { revert a₀, rw ← function.funext_iff, dsimp only [e'], simp },\n  rw this, refl,\nend\n\n-- TODO reuse the nonadditive variant for this.\nlemma _root_.Condensed.val_obj_sigma_add_equiv_symm_apply'\n  (Y : Condensed.{u} Ab.{u+1})\n  (e : Π (a : α), Y.val.obj (op $ X a)) (a₀ : α) :\n  (Y.val.map (Profinite.sigma.ι X a₀).op)\n  (((_root_.Condensed.val_obj_sigma_add_equiv X Y).symm) e) = e a₀ :=\nbegin\n  let e' := _, change (Y.val.map (Profinite.sigma.ι X a₀).op) e' = _,\n  have : e a₀ = (_root_.Condensed.val_obj_sigma_add_equiv X Y) e' a₀,\n    { revert a₀, rw ← function.funext_iff, dsimp only [e'], simp },\n  rw this, refl,\nend\n\nlemma val_obj_sigma_equiv_symm_apply\n  (Y : CondensedSet.{u})\n  (e : Π (a : α), Y.val.obj (op $ X a)) (a₀ : α) :\n    (Profinite_to_Condensed.map (Profinite.sigma.ι X a₀)) ≫\n    (Profinite.to_Condensed_equiv (Profinite.sigma X) Y).symm\n    ((Y.val_obj_sigma_equiv X).symm e) =\n    (Profinite.to_Condensed_equiv (X a₀) Y).symm (e a₀) :=\nbegin\n  apply_fun ((X a₀).to_Condensed_equiv Y),\n  simp only [equiv.apply_symm_apply],\n  dsimp [Profinite.to_Condensed_equiv],\n  simp only [category.comp_id],\n  apply val_obj_sigma_equiv_symm_apply'\nend\n\n-- TODO reuse the nonadditive variant for this.\nlemma _root_.Condensed.val_obj_sigma_add_equiv_symm_apply\n  (Y : Condensed.{u} Ab.{u+1})\n  (e : Π (a : α), Y.val.obj (op $ X a)) (a₀ : α) :\n    (Profinite_to_Condensed.map (Profinite.sigma.ι X a₀)) ≫\n    (Profinite.to_Condensed_equiv (Profinite.sigma X)\n    (Condensed_Ab_to_CondensedSet.obj Y)).symm\n    ((Y.val_obj_sigma_add_equiv X).symm e) =\n    (Profinite.to_Condensed_equiv (X a₀)\n    (Condensed_Ab_to_CondensedSet.obj Y)).symm (e a₀) :=\nbegin\n  apply_fun ((X a₀).to_Condensed_equiv (Condensed_Ab_to_CondensedSet.obj Y)),\n  simp only [equiv.apply_symm_apply],\n  dsimp [Profinite.to_Condensed_equiv],\n  simp only [category.comp_id],\n  apply _root_.Condensed.val_obj_sigma_add_equiv_symm_apply'\nend\n\nnoncomputable\ndef is_colimit_sigma_cone : is_colimit (sigma_cone X) :=\n{ desc := λ S, (Profinite.to_Condensed_equiv _ _).symm $\n    (S.X.val_obj_sigma_equiv X).symm $ λ a,\n    (Profinite.to_Condensed_equiv _ _) $ S.ι.app ⟨_⟩,\n  fac' := begin\n    rintros Q ⟨T⟩,\n    dsimp,\n    rw val_obj_sigma_equiv_symm_apply,\n    ext W ⟨(t : _ ⟶ _)⟩,\n    dsimp [Profinite.to_Condensed_equiv],\n    change ((Q.ι.app ⟨T⟩).val.app (op (X T)) ≫ Q.X.val.map t.op) _ = _,\n    erw ← (Q.ι.app ⟨T⟩).val.naturality,\n    change (Q.ι.app ⟨T⟩).val.app (op (unop W)) _ = _,\n    congr' 1,\n    dsimp [Profinite.to_Condensed], ext, refl,\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply_fun ((Profinite.sigma X).to_Condensed_equiv S.X),\n    apply_fun (val_obj_sigma_equiv X S.X),\n    simp only [equiv.apply_symm_apply],\n    ext a,\n    specialize hm ⟨a⟩,\n    dsimp [val_obj_sigma_equiv],\n    change (m.val.app (op (Profinite.sigma X)) ≫\n      S.X.val.map _) _ = _,\n    rw ← m.val.naturality,\n    apply_fun (λ e, e.val.app (op (X a)) ⟨𝟙 _⟩) at hm,\n    exact hm,\n  end }\n\nend CondensedSet\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/coproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.31535969672172537}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.discrete\nimport Mathlib.category_theory.monoidal.unitors\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.algebra.punit_instances\nimport Mathlib.PostPort\n\nuniverses v₁ u₁ l u₂ v₂ u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# The category of monoids in a monoidal category.\n-/\n\n/--\nA monoid object internal to a monoidal category.\n\nWhen the monoidal category is preadditive, this is also sometimes called an \"algebra object\".\n-/\nstructure Mon_ (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] \nwhere\n  X : C\n  one : 𝟙_ ⟶ X\n  mul : X ⊗ X ⟶ X\n  one_mul' : autoParam ((one ⊗ 𝟙) ≫ mul = category_theory.iso.hom λ_)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_one' : autoParam ((𝟙 ⊗ one) ≫ mul = category_theory.iso.hom ρ_)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_assoc' : autoParam ((mul ⊗ 𝟙) ≫ mul = category_theory.iso.hom α_ ≫ (𝟙 ⊗ mul) ≫ mul)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- Obviously there is some flexibility stating this axiom.\n\n-- This one has left- and right-hand sides matching the statement of `monoid.mul_assoc`,\n\n-- and chooses to place the associator on the right-hand side.\n\n-- The heuristic is that unitors and associators \"don't have much weight\".\n\ntheorem Mon_.one_mul {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (c : Mon_ C) : (Mon_.one c ⊗ 𝟙) ≫ Mon_.mul c = category_theory.iso.hom λ_ := sorry\n\ntheorem Mon_.mul_one {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (c : Mon_ C) : (𝟙 ⊗ Mon_.one c) ≫ Mon_.mul c = category_theory.iso.hom ρ_ := sorry\n\n@[simp] theorem Mon_.mul_assoc {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (c : Mon_ C) : (Mon_.mul c ⊗ 𝟙) ≫ Mon_.mul c = category_theory.iso.hom α_ ≫ (𝟙 ⊗ Mon_.mul c) ≫ Mon_.mul c := sorry\n\ntheorem Mon_.one_mul_assoc {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (c : Mon_ C) {X' : C} (f' : Mon_.X c ⟶ X') : (Mon_.one c ⊗ 𝟙) ≫ Mon_.mul c ≫ f' = category_theory.iso.hom λ_ ≫ f' := sorry\n\n@[simp] theorem Mon_.mul_assoc_assoc {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (c : Mon_ C) {X' : C} (f' : Mon_.X c ⟶ X') : (Mon_.mul c ⊗ 𝟙) ≫ Mon_.mul c ≫ f' = category_theory.iso.hom α_ ≫ (𝟙 ⊗ Mon_.mul c) ≫ Mon_.mul c ≫ f' := sorry\n\nnamespace Mon_\n\n\n/--\nThe trivial monoid object. We later show this is initial in `Mon_ C`.\n-/\n@[simp] theorem trivial_one (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] : one (trivial C) = 𝟙 :=\n  Eq.refl (one (trivial C))\n\nprotected instance inhabited (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] : Inhabited (Mon_ C) :=\n  { default := trivial C }\n\n@[simp] theorem one_mul_hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {Z : C} (f : Z ⟶ X M) : (one M ⊗ f) ≫ mul M = category_theory.iso.hom λ_ ≫ f := sorry\n\n@[simp] theorem mul_one_hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {Z : C} (f : Z ⟶ X M) : (f ⊗ one M) ≫ mul M = category_theory.iso.hom ρ_ ≫ f := sorry\n\ntheorem assoc_flip {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} : (𝟙 ⊗ mul M) ≫ mul M = category_theory.iso.inv α_ ≫ (mul M ⊗ 𝟙) ≫ mul M := sorry\n\n/-- A morphism of monoid objects. -/\nstructure hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (M : Mon_ C) (N : Mon_ C) \nwhere\n  hom : X M ⟶ X N\n  one_hom' : autoParam (one M ≫ hom = one N)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  mul_hom' : autoParam (mul M ≫ hom = (hom ⊗ hom) ≫ mul N)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem hom.one_hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) : one M ≫ hom.hom c = one N := sorry\n\n@[simp] theorem hom.mul_hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) : mul M ≫ hom.hom c = (hom.hom c ⊗ hom.hom c) ≫ mul N := sorry\n\n@[simp] theorem hom.mul_hom_assoc {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} (c : hom M N) {X' : C} (f' : X N ⟶ X') : mul M ≫ hom.hom c ≫ f' = (hom.hom c ⊗ hom.hom c) ≫ mul N ≫ f' := sorry\n\n/-- The identity morphism on a monoid object. -/\ndef id {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (M : Mon_ C) : hom M M :=\n  hom.mk 𝟙\n\nprotected instance hom_inhabited {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (M : Mon_ C) : Inhabited (hom M M) :=\n  { default := id M }\n\n/-- Composition of morphisms of monoid objects. -/\n@[simp] theorem comp_hom {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} {O : Mon_ C} (f : hom M N) (g : hom N O) : hom.hom (comp f g) = hom.hom f ≫ hom.hom g :=\n  Eq.refl (hom.hom (comp f g))\n\nprotected instance category_theory.category {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] : category_theory.category (Mon_ C) :=\n  category_theory.category.mk\n\n@[simp] theorem id_hom' {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (M : Mon_ C) : hom.hom 𝟙 = 𝟙 :=\n  rfl\n\n@[simp] theorem comp_hom' {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {M : Mon_ C} {N : Mon_ C} {K : Mon_ C} (f : M ⟶ N) (g : N ⟶ K) : hom.hom (f ≫ g) = hom.hom f ≫ hom.hom g :=\n  rfl\n\n/-- The forgetful functor from monoid objects to the ambient category. -/\n@[simp] theorem forget_map (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) (B : Mon_ C) (f : A ⟶ B) : category_theory.functor.map (forget C) f = hom.hom f :=\n  Eq.refl (category_theory.functor.map (forget C) f)\n\nprotected instance forget_faithful {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] : category_theory.faithful (forget C) :=\n  category_theory.faithful.mk\n\nprotected instance category_theory.has_hom.hom.hom.category_theory.is_iso {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {B : Mon_ C} (f : A ⟶ B) [e : category_theory.is_iso (category_theory.functor.map (forget C) f)] : category_theory.is_iso (hom.hom f) :=\n  e\n\n/-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/\nprotected instance forget.category_theory.reflects_isomorphisms {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] : category_theory.reflects_isomorphisms (forget C) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : Mon_ C) (f : X ⟶ Y) (e : category_theory.is_iso (category_theory.functor.map (forget C) f)) =>\n      category_theory.is_iso.mk (hom.mk (inv (hom.hom f)))\n\nprotected instance unique_hom_from_trivial {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) : unique (trivial C ⟶ A) :=\n  unique.mk { default := hom.mk (one A) } sorry\n\nprotected instance category_theory.limits.has_initial {C : Type u₁} [category_theory.category C] [category_theory.monoidal_category C] : category_theory.limits.has_initial (Mon_ C) :=\n  category_theory.limits.has_initial_of_unique (trivial C)\n\nend Mon_\n\n\nnamespace category_theory.lax_monoidal_functor\n\n\n/--\nA lax monoidal functor takes monoid objects to monoid objects.\n\nThat is, a lax monoidal functor `F : C ⥤ D` induces a functor `Mon_ C ⥤ Mon_ D`.\n-/\n-- TODO: map_Mod F A : Mod A ⥤ Mod (F.map_Mon A)\n\n@[simp] theorem map_Mon_obj_X {C : Type u₁} [category C] [monoidal_category C] {D : Type u₂} [category D] [monoidal_category D] (F : lax_monoidal_functor C D) (A : Mon_ C) : Mon_.X (functor.obj (map_Mon F) A) = functor.obj (to_functor F) (Mon_.X A) :=\n  Eq.refl (Mon_.X (functor.obj (map_Mon F) A))\n\n/-- `map_Mon` is functorial in the lax monoidal functor. -/\ndef map_Mon_functor (C : Type u₁) [category C] [monoidal_category C] (D : Type u₂) [category D] [monoidal_category D] : lax_monoidal_functor C D ⥤ Mon_ C ⥤ Mon_ D :=\n  functor.mk map_Mon\n    fun (F G : lax_monoidal_functor C D) (α : F ⟶ G) =>\n      nat_trans.mk fun (A : Mon_ C) => Mon_.hom.mk (nat_trans.app (monoidal_nat_trans.to_nat_trans α) (Mon_.X A))\n\nend category_theory.lax_monoidal_functor\n\n\nnamespace Mon_\n\n\nnamespace equiv_lax_monoidal_functor_punit\n\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\ndef lax_monoidal_to_Mon (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C ⥤ Mon_ C :=\n  category_theory.functor.mk\n    (fun (F : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C) =>\n      category_theory.functor.obj (category_theory.lax_monoidal_functor.map_Mon F)\n        (trivial (category_theory.discrete PUnit)))\n    fun (F G : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C) (α : F ⟶ G) =>\n      category_theory.nat_trans.app\n        (category_theory.functor.map\n          (category_theory.lax_monoidal_functor.map_Mon_functor (category_theory.discrete PUnit) C) α)\n        (trivial (category_theory.discrete PUnit))\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\ndef Mon_to_lax_monoidal (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] : Mon_ C ⥤ category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C :=\n  category_theory.functor.mk\n    (fun (A : Mon_ C) =>\n      category_theory.lax_monoidal_functor.mk\n        (category_theory.functor.mk (fun (_x : category_theory.discrete PUnit) => X A)\n          fun (_x _x_1 : category_theory.discrete PUnit) (_x : _x ⟶ _x_1) => 𝟙)\n        (one A) fun (_x _x : category_theory.discrete PUnit) => mul A)\n    fun (A B : Mon_ C) (f : A ⟶ B) =>\n      category_theory.monoidal_nat_trans.mk\n        (category_theory.nat_trans.mk fun (_x : category_theory.discrete PUnit) => hom.hom f)\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simp] theorem unit_iso_inv_app_to_nat_trans_app (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] (X : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C) : ∀ (X_1 : category_theory.discrete PUnit),\n  category_theory.nat_trans.app\n      (category_theory.monoidal_nat_trans.to_nat_trans\n        (category_theory.nat_trans.app (category_theory.iso.inv (unit_iso C)) X))\n      X_1 =\n    inv\n      (category_theory.eq_to_hom\n        (congr_arg (category_theory.functor.obj (category_theory.lax_monoidal_functor.to_functor X))\n          (unit_iso._proof_1 X_1))) := sorry\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simp] theorem counit_iso_inv_app_hom (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] (X : Mon_ C) : hom.hom (category_theory.nat_trans.app (category_theory.iso.inv (counit_iso C)) X) = 𝟙 :=\n  Eq.refl 𝟙\n\nend equiv_lax_monoidal_functor_punit\n\n\n/--\nMonoid objects in `C` are \"just\" lax monoidal functors from the trivial monoidal category to `C`.\n-/\ndef equiv_lax_monoidal_functor_punit (C : Type u₁) [category_theory.category C] [category_theory.monoidal_category C] : category_theory.lax_monoidal_functor (category_theory.discrete PUnit) C ≌ Mon_ C :=\n  category_theory.equivalence.mk' sorry sorry sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/Mon_.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3153133670695476}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport tactic.ext\nimport tactic.lint\n\n/-!\n# Functors\n\nThis module provides additional lemmas, definitions, and instances for `functor`s.\n\n## Main definitions\n\n* `const α` is the functor that sends all types to `α`.\n* `add_const α` is `const α` but for when `α` has an additive structure.\n* `comp F G` for functors `F` and `G` is the functor composition of `F` and `G`.\n* `liftp` and `liftr` respectively lift predicates and relations on a type `α`\n  to `F α`.  Terms of `F α` are considered to, in some sense, contain values of type `α`.\n\n## Tags\n\nfunctor, applicative\n-/\n\nattribute [functor_norm] seq_assoc pure_seq_eq_map map_pure seq_map_assoc map_seq\n\nuniverses u v w\n\nsection functor\n\nvariables {F : Type u → Type v}\nvariables {α β γ : Type u}\nvariables [functor F] [is_lawful_functor F]\n\nlemma functor.map_id : (<$>) id = (id : F α → F α) :=\nby apply funext; apply id_map\n\nlemma functor.map_comp_map (f : α → β) (g : β → γ) :\n  ((<$>) g ∘ (<$>) f : F α → F γ) = (<$>) (g ∘ f) :=\nby apply funext; intro; rw comp_map\n\ntheorem functor.ext {F} : ∀ {F1 : functor F} {F2 : functor F}\n  [@is_lawful_functor F F1] [@is_lawful_functor F F2]\n  (H : ∀ α β (f : α → β) (x : F α),\n    @functor.map _ F1 _ _ f x = @functor.map _ F2 _ _ f x),\n  F1 = F2\n| ⟨m, mc⟩ ⟨m', mc'⟩ H1 H2 H :=\nbegin\n  cases show @m = @m', by funext α β f x; apply H,\n  congr, funext α β,\n  have E1 := @map_const_eq _ ⟨@m, @mc⟩ H1,\n  have E2 := @map_const_eq _ ⟨@m, @mc'⟩ H2,\n  exact E1.trans E2.symm\nend\n\nend functor\n\n/-- Introduce the `id` functor. Incidentally, this is `pure` for\n`id` as a `monad` and as an `applicative` functor. -/\ndef id.mk {α : Sort u} : α → id α := id\n\nnamespace functor\n\n/-- `const α` is the constant functor, mapping every type to `α`. When\n`α` has a monoid structure, `const α` has an `applicative` instance.\n(If `α` has an additive monoid structure, see `functor.add_const`.) -/\n@[nolint unused_arguments]\ndef const (α : Type*) (β : Type*) := α\n\n/-- `const.mk` is the canonical map `α → const α β` (the identity), and\nit can be used as a pattern to extract this value. -/\n@[pattern] def const.mk {α β} (x : α) : const α β := x\n\n/-- `const.mk'` is `const.mk` but specialized to map `α` to\n`const α punit`, where `punit` is the terminal object in `Type*`. -/\ndef const.mk' {α} (x : α) : const α punit := x\n\n/-- Extract the element of `α` from the `const` functor. -/\ndef const.run {α β} (x : const α β) : α := x\n\nnamespace const\n\nprotected lemma ext {α β} {x y : const α β} (h : x.run = y.run) : x = y := h\n\n/-- The map operation of the `const γ` functor. -/\n@[nolint unused_arguments]\nprotected def map {γ α β} (f : α → β) (x : const γ β) : const γ α := x\n\ninstance {γ} : functor (const γ) :=\n{ map := @const.map γ }\n\ninstance {γ} : is_lawful_functor (const γ) :=\nby constructor; intros; refl\n\ninstance {α β} [inhabited α] : inhabited (const α β) :=\n⟨(default _ : α)⟩\n\nend const\n\n/-- `add_const α` is a synonym for constant functor `const α`, mapping\nevery type to `α`. When `α` has a additive monoid structure,\n`add_const α` has an `applicative` instance. (If `α` has a\nmultiplicative monoid structure, see `functor.const`.) -/\ndef add_const (α : Type*) := const α\n\n/-- `add_const.mk` is the canonical map `α → add_const α β`, which is the identity,\nwhere `add_const α β = const α β`. It can be used as a pattern to extract this value. -/\n@[pattern]\ndef add_const.mk {α β} (x : α) : add_const α β := x\n\n/-- Extract the element of `α` from the constant functor. -/\ndef add_const.run {α β} : add_const α β → α := id\n\ninstance add_const.functor {γ} : functor (add_const γ) :=\n@const.functor γ\n\ninstance add_const.is_lawful_functor {γ} : is_lawful_functor (add_const γ) :=\n@const.is_lawful_functor γ\n\ninstance {α β} [inhabited α] : inhabited (add_const α β) :=\n⟨(default _ : α)⟩\n\n/-- `functor.comp` is a wrapper around `function.comp` for types.\n    It prevents Lean's type class resolution mechanism from trying\n    a `functor (comp F id)` when `functor F` would do. -/\ndef comp (F : Type u → Type w) (G : Type v → Type u) (α : Type v) : Type w :=\nF $ G α\n\n/-- Construct a term of `comp F G α` from a term of `F (G α)`, which is the same type.\nCan be used as a pattern to extract a term of `F (G α)`. -/\n@[pattern] def comp.mk {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : F (G α)) : comp F G α := x\n\n/-- Extract a term of `F (G α)` from a term of `comp F G α`, which is the same type. -/\ndef comp.run {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : comp F G α) : F (G α) := x\n\nnamespace comp\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nprotected lemma ext\n  {α} {x y : comp F G α} : x.run = y.run → x = y := id\n\ninstance {α} [inhabited (F (G α))] : inhabited (comp F G α) :=\n⟨(default _ : F (G α))⟩\n\nvariables [functor F] [functor G]\n\n/-- The map operation for the composition `comp F G` of functors `F` and `G`. -/\nprotected def map {α β : Type v} (h : α → β) : comp F G α → comp F G β\n| (comp.mk x) := comp.mk ((<$>) h <$> x)\n\ninstance : functor (comp F G) := { map := @comp.map F G _ _ }\n\n@[functor_norm] lemma map_mk {α β} (h : α → β) (x : F (G α)) :\n  h <$> comp.mk x = comp.mk ((<$>) h <$> x) := rfl\n\n@[simp] protected lemma run_map {α β} (h : α → β) (x : comp F G α) :\n  (h <$> x).run = (<$>) h <$> x.run := rfl\n\nvariables [is_lawful_functor F] [is_lawful_functor G]\nvariables {α β γ : Type v}\n\nprotected lemma id_map : ∀ (x : comp F G α), comp.map id x = x\n| (comp.mk x) := by simp [comp.map, functor.map_id]\n\nprotected lemma comp_map (g' : α → β) (h : β → γ) : ∀ (x : comp F G α),\n           comp.map (h ∘ g') x = comp.map h (comp.map g' x)\n| (comp.mk x) := by simp [comp.map, functor.map_comp_map g' h] with functor_norm\n\ninstance : is_lawful_functor (comp F G) :=\n{ id_map := @comp.id_map F G _ _ _ _,\n  comp_map := @comp.comp_map F G _ _ _ _ }\n\ntheorem functor_comp_id {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor F id _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor F id _ _ _ _) _ (λ α β f x, rfl)\n\ntheorem functor_id_comp {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor id F _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor id F _ _ _ _) _ (λ α β f x, rfl)\n\nend comp\n\nnamespace comp\n\nopen function (hiding comp)\nopen functor\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nvariables [applicative F] [applicative G]\n\n/-- The `<*>` operation for the composition of applicative functors. -/\nprotected def seq {α β : Type v} : comp F G (α → β) → comp F G α → comp F G β\n| (comp.mk f) (comp.mk x) := comp.mk $ (<*>) <$> f <*> x\n\ninstance : has_pure (comp F G) :=\n⟨λ _ x, comp.mk $ pure $ pure x⟩\n\ninstance : has_seq (comp F G) :=\n⟨λ _ _ f x, comp.seq f x⟩\n\n@[simp] protected lemma run_pure {α : Type v} :\n  ∀ x : α, (pure x : comp F G α).run = pure (pure x)\n| _ := rfl\n\n@[simp] protected lemma run_seq {α β : Type v} (f : comp F G (α → β)) (x : comp F G α) :\n  (f <*> x).run = (<*>) <$> f.run <*> x.run := rfl\n\ninstance : applicative (comp F G) :=\n{ map := @comp.map F G _ _,\n  seq := @comp.seq F G _ _,\n  ..comp.has_pure }\n\nend comp\n\nvariables {F : Type u → Type u} [functor F]\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`,\npredicate `liftp p x` holds iff every value contained by `x` satisfies `p`. -/\ndef liftp {α : Type u} (p : α → Prop) (x : F α) : Prop :=\n∃ u : F (subtype p), subtype.val <$> u = x\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`, then\n`liftr r x y` relates `x` and `y` iff (1) `x` and `y` have the same shape and\n(2) we can pair values `a` from `x` and `b` from `y` so that `r a b` holds. -/\ndef liftr {α : Type u} (r : α → α → Prop) (x y : F α) : Prop :=\n∃ u : F {p : α × α // r p.fst p.snd},\n  (λ t : {p : α × α // r p.fst p.snd}, t.val.fst) <$> u = x ∧\n  (λ t : {p : α × α // r p.fst p.snd}, t.val.snd) <$> u = y\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`, then\n`supp x` is the set of values of type `α` that `x` contains. -/\ndef supp {α : Type u} (x : F α) : set α := { y : α | ∀ ⦃p⦄, liftp p x → p y }\n\ntheorem of_mem_supp {α : Type u} {x : F α} {p : α → Prop} (h : liftp p x) :\n  ∀ y ∈ supp x, p y :=\nλ y hy, hy h\n\nend functor\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/control/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3152407288465604}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module data.dlist.basic\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Leanbin.Data.Dlist\n\n/-!\n# Difference list\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides a few results about `dlist`, which is defined in core Lean.\n\nA difference list is a function that, given a list, returns the original content of the\ndifference list prepended to the given list. It is useful to represent elements of a given type\nas `a₁ + ... + aₙ` where `+ : α → α → α` is any operation, without actually computing.\n\nThis structure supports `O(1)` `append` and `concat` operations on lists, making it\nuseful for append-heavy uses such as logging and pretty printing.\n-/\n\n\n/- warning: dlist.join -> Std.DList.join is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}}, (List.{u1} (Dlist.{u1} α)) -> (Dlist.{u1} α)\nbut is expected to have type\n  forall {α : Type.{u1}}, (List.{u1} (Std.DList.{u1} α)) -> (Std.DList.{u1} α)\nCase conversion may be inaccurate. Consider using '#align dlist.join Std.DList.joinₓ'. -/\n/-- Concatenates a list of difference lists to form a single difference list. Similar to\n`list.join`. -/\ndef Std.DList.join {α : Type _} : List (Dlist α) → Dlist α\n  | [] => Dlist.empty\n  | x :: xs => x ++ Std.DList.join xs\n#align dlist.join Std.DList.join\n\n/- warning: dlist_singleton -> Std.DList_singleton is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {a : α}, Eq.{succ u1} (Dlist.{u1} α) (Dlist.singleton.{u1} α a) (Std.DList.lazy_ofList.{u1} α (fun (_ : Unit) => List.cons.{u1} α a (List.nil.{u1} α)))\nbut is expected to have type\n  forall {α : Type.{u1}} {a : α}, Eq.{succ u1} (Std.DList.{u1} α) (Std.DList.singleton.{u1} α a) (Std.DList.lazy_ofList.{u1} α (Thunk.mk.{u1} (List.{u1} α) (fun (x._@.Init.Core._hyg.266 : Unit) => List.cons.{u1} α a (List.nil.{u1} α))))\nCase conversion may be inaccurate. Consider using '#align dlist_singleton Std.DList_singletonₓ'. -/\n@[simp]\ntheorem Std.DList_singleton {α : Type _} {a : α} : Dlist.singleton a = Std.DList.lazy_ofList [a] :=\n  rfl\n#align dlist_singleton Std.DList_singleton\n\n/- warning: dlist_lazy -> Std.DList_lazy is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {l : List.{u1} α}, Eq.{succ u1} (Dlist.{u1} α) (Std.DList.lazy_ofList.{u1} α (fun (_ : Unit) => l)) (Dlist.ofList.{u1} α l)\nbut is expected to have type\n  forall {α : Type.{u1}} {l : List.{u1} α}, Eq.{succ u1} (Std.DList.{u1} α) (Std.DList.lazy_ofList.{u1} α (Thunk.mk.{u1} (List.{u1} α) (fun (x._@.Init.Core._hyg.266 : Unit) => l))) (Std.DList.ofList.{u1} α l)\nCase conversion may be inaccurate. Consider using '#align dlist_lazy Std.DList_lazyₓ'. -/\n@[simp]\ntheorem Std.DList_lazy {α : Type _} {l : List α} : Std.DList.lazy_ofList l = Dlist.ofList l :=\n  rfl\n#align dlist_lazy Std.DList_lazy\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Dlist/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3152407215445012}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\nimport for_mathlib.homological_complex_misc\nimport algebra.homology.homological_complex\nimport algebra.homology.additive\nimport for_mathlib.idempotents.functor_extension2\nimport category_theory.idempotents.homological_complex\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.category\nopen category_theory.preadditive\nopen category_theory.idempotents\n\nvariables {C : Type*} [category C] [preadditive C]\nvariables {ι : Type*} {c : complex_shape ι}\n\nnamespace category_theory\n\nnamespace idempotents\n\nnamespace karoubi\n\nnamespace homological_complex\n\nvariables {P Q : karoubi (homological_complex C c)} (f : P ⟶ Q) (n : ι)\n\n@[simp, reassoc]\nlemma f_p_comp : P.p.f n ≫ f.f.f n = f.f.f n :=\nhomological_complex.congr_hom (p_comp f) n\n\n@[simp, reassoc]\nlemma f_comp_p : f.f.f n ≫ Q.p.f n = f.f.f n :=\nhomological_complex.congr_hom (comp_p f) n\n\n@[reassoc]\nlemma f_p_comm : P.p.f n ≫ f.f.f n = f.f.f n ≫ Q.p.f n :=\nhomological_complex.congr_hom (p_comm f) n\n\nend homological_complex\n\nend karoubi\n\nnamespace karoubi_homological_complex\n\nnamespace functor\n\n@[simps]\ndef obj (P : karoubi (homological_complex C c)) : homological_complex (karoubi C) c :=\n{ X := λ n, ⟨P.X.X n, P.p.f n, by simpa only [homological_complex.comp_f]\n    using homological_complex.congr_hom P.idem n⟩,\n  d := λ i j,\n    { f := P.p.f i ≫ P.X.d i j,\n      comm := begin\n        have h :=  homological_complex.congr_hom P.idem j,\n        simp only [homological_complex.hom.comm_assoc, assoc, homological_complex.hom.comm,\n          homological_complex.comp_f] at h ⊢,\n        simp only [h]\n      end, },\n  shape' := λ i j hij, by simp only [karoubi.hom_eq_zero_iff,\n    P.X.shape i j hij, limits.comp_zero],\n  d_comp_d' := λ i j k hij hjk, begin\n    simp only [karoubi.hom_eq_zero_iff, karoubi.comp_f, P.p.comm j k],\n    slice_lhs 2 3 { rw P.X.d_comp_d' i j k hij hjk, },\n    simp only [limits.comp_zero, limits.zero_comp],\n  end, }\n\n@[simps]\ndef map {P Q : karoubi (homological_complex C c)} (f : P ⟶ Q) : obj P ⟶ obj Q :=\n{ f:= λ n,\n  { f:= f.f.f n,\n    comm := by simpa only [homological_complex.comp_f]\n      using homological_complex.congr_hom f.comm n, },\n  comm' := λ i j hij, begin\n    ext,\n    simp only [karoubi.comp_f, obj_d_f, assoc, ← f.f.comm],\n    simp only [← assoc],\n    have eq := homological_complex.congr_hom (karoubi.p_comm f) i,\n    simp only [homological_complex.comp_f] at eq,\n    rw eq,\n  end }\n\nend functor\n\n@[simps]\ndef functor :\n  karoubi (homological_complex C c) ⥤ homological_complex (karoubi C) c :=\n{ obj := functor.obj,\n  map := λ P Q f, functor.map f,\n  map_comp' := λ P Q R f f',\n    by { ext n, simpa only [karoubi.comp_f], }, }\n\nnamespace inverse\n\n@[simps]\ndef obj (K : homological_complex (karoubi C) c) : karoubi (homological_complex C c) :=\n{ X :=\n  { X := λ n, (K.X n).X,\n    d := λ i j, (K.d i j).f,\n    shape' := λ i j hij, karoubi.hom_eq_zero_iff.mp (K.shape' i j hij),\n    d_comp_d' := λ i j k hij hjk, by simpa only [karoubi.comp_f]\n      using karoubi.hom_eq_zero_iff.mp (K.d_comp_d' i j k hij hjk), },\n  p :=\n    { f := λ n, (K.X n).p,\n      comm' := λ i j hij, karoubi.p_comm (K.d i j), },\n  idem := by { ext n, simpa only [karoubi.comp_f] using (K.X n).idem, }, }\n\n@[simps]\ndef map {K L : homological_complex (karoubi C) c} (f : K ⟶ L) : obj K ⟶ obj L :=\n{ f:=\n  { f := λ n, (f.f n).f,\n    comm' := λ i j hij, by simpa only [karoubi.comp_f]\n      using karoubi.hom_ext.mp (f.comm' i j hij), },\n  comm := by { ext n, exact (f.f n).comm, }, }\n\nend inverse\n\n@[simps]\ndef inverse :\n  homological_complex (karoubi C) c ⥤ karoubi (homological_complex C c) :=\n{ obj := inverse.obj,\n  map := λ K L f, inverse.map f,\n  map_comp' := λ K L M f g, by { ext n,\n    simp only [karoubi.comp_f, homological_complex.comp_f, inverse.map_f_f], }, }\n\ndef counit_eq : inverse ⋙ functor = 𝟭 (homological_complex (karoubi C) c) :=\nbegin\n  apply functor.ext,\n  { intros K L f,\n    ext n,\n    dsimp [functor.map, inverse.map],\n    simp only [karoubi.eq_to_hom_f, functor.obj_X_p, homological_complex.eq_to_hom_f,\n      eq_to_hom_refl, comp_id, karoubi.comp_f, inverse.obj_p_f],\n    rw [← karoubi.hom.comm], },\n  { intro P,\n    apply homological_complex.ext,\n    { intros i j hij,\n      simp [homological_complex.eq_to_hom_f],\n      refl, },\n    { ext n,\n      { simpa only [id_comp, eq_to_hom_refl, comp_id], },\n      { refl, }, }, },\nend\n\n@[simps]\ndef unit_iso : 𝟭 (karoubi (homological_complex C c)) ≅ functor ⋙ inverse :=\n{ hom :=\n  { app := λ P,\n    { f :=\n      { f := λ n, P.p.f n,\n        comm' := λ i j hij, begin\n          dsimp,\n          have h := homological_complex.congr_hom P.idem i,\n          simp only [homological_complex.comp_f] at h,\n          slice_lhs 1 2 { erw h, },\n          exact P.p.comm' i j hij,\n        end },\n      comm := begin\n        ext n,\n        have h := homological_complex.congr_hom P.idem n,\n        simp only [homological_complex.comp_f] at h,\n        dsimp,\n        rw [h, h],\n      end },\n    naturality' := λ X Y f, begin\n      ext n,\n      have h := homological_complex.congr_hom ((karoubi.p_comm f).symm) n,\n      simpa only [functor.map_f_f, homological_complex.comp_f,\n        inverse.map_f_f, karoubi.comp_f] using h,\n    end },\n  inv :=\n  { app := λ P,\n    { f :=\n      { f := λ n, P.p.f n,\n        comm' := λ i j hij, begin\n          dsimp,\n          slice_rhs 2 3 { rw ← P.p.comm' i j hij, },\n          rw ← assoc,\n          have h := homological_complex.congr_hom P.idem i,\n          simp only [homological_complex.comp_f] at h,\n          rw h,\n        end },\n      comm := begin\n        ext n,\n        have h := homological_complex.congr_hom P.idem n,\n        simp only [homological_complex.comp_f] at h,\n        dsimp,\n        rw [h, h],\n      end },\n    naturality' := λ P Q f, begin\n      ext n,\n      have h := homological_complex.congr_hom (karoubi.p_comm f).symm n,\n      simpa only [functor_map, functor.map_f_f, functor.id_map, functor.comp_map,\n        homological_complex.comp_f, inverse.map_f_f, inverse_map, karoubi.comp_f]\n        using h,\n    end },\n  hom_inv_id' := begin\n    ext P n,\n    dsimp,\n    simpa only [homological_complex.comp_f, karoubi.id_eq, karoubi.comp_f]\n      using homological_complex.congr_hom P.idem n,\n  end,\n  inv_hom_id' := begin\n    ext P n,\n    dsimp [inverse.obj, functor.obj],\n    simpa only [homological_complex.comp_f, karoubi.id_eq, karoubi.comp_f]\n      using homological_complex.congr_hom P.idem n,\n  end, }\n\nend karoubi_homological_complex\n\nvariables (C) (c)\n\n/-@[simps]\ndef karoubi_homological_complex_equivalence :\n  karoubi (homological_complex C c) ≌ homological_complex (karoubi C) c :=\n{ functor    := karoubi_homological_complex.functor,\n  inverse    := karoubi_homological_complex.inverse,\n  unit_iso   := karoubi_homological_complex.unit_iso,\n  counit_iso := eq_to_iso karoubi_homological_complex.counit_eq,\n  functor_unit_iso_comp' := λ P, begin\n    ext n,\n    dsimp,\n    have h := homological_complex.congr_hom P.idem n,\n    simpa only [karoubi_homological_complex.unit_iso_hom_app_f_f,\n      homological_complex.eq_to_hom_f,\n      eq_to_hom_app, karoubi_homological_complex.functor.obj_X_p,\n      karoubi_homological_complex.inverse.obj_p_f, eq_to_hom_refl,\n      karoubi.id_eq, karoubi_homological_complex.functor.map_f_f,\n      karoubi.comp_f] using h,\n  end }-/\n\nlemma functor.map_homological_complex_id {D : Type*} [category D] [preadditive D] :\n  functor.map_homological_complex (𝟭 D) c = 𝟭 _ :=\nbegin\n  apply functor.ext,\n  { intros X Y f,\n    ext n,\n    dsimp,\n    simp only [homological_complex.eq_to_hom_f, eq_to_hom_refl],\n    erw [id_comp, comp_id], },\n  { intro X,\n    apply homological_complex.ext,\n    { intros i j hij,\n      erw [comp_id, id_comp],\n      refl, },\n    { refl, }, },\nend\n\n@[simps]\ndef nat_iso.map_homological_complex {D E : Type*} [category D] [category E] [preadditive D] [preadditive E]\n  {F G : D ⥤ E} [F.additive] [G.additive] (e : F ≅ G) :\n    functor.map_homological_complex F c ≅ functor.map_homological_complex G c :=\n{ hom := nat_trans.map_homological_complex e.hom c,\n  inv := nat_trans.map_homological_complex e.inv c,\n  hom_inv_id' := by simpa only [← nat_trans.map_homological_complex_comp, e.hom_inv_id],\n  inv_hom_id' := by simpa only [← nat_trans.map_homological_complex_comp, e.inv_hom_id], }\n\ndef equivalence.map_homological_complex {D E : Type*} [category D] [category E] [preadditive D] [preadditive E]\n(e : D ≌ E) [e.functor.additive] : homological_complex D c ≌ homological_complex E c :=\n{ functor := functor.map_homological_complex e.functor c,\n  inverse := functor.map_homological_complex e.inverse c,\n  unit_iso := eq_to_iso (functor.map_homological_complex_id c).symm ≪≫\n      nat_iso.map_homological_complex c e.unit_iso,\n  counit_iso := nat_iso.map_homological_complex c e.counit_iso ≪≫\n      eq_to_iso (functor.map_homological_complex_id c),\n  functor_unit_iso_comp' := λ K, begin\n    ext n,\n    dsimp,\n    simp only [eq_to_hom_app, homological_complex.eq_to_hom_f, eq_to_hom_refl],\n    erw [id_comp, comp_id, e.functor_unit_iso_comp],\n  end, }\n\ninstance [is_idempotent_complete C] : is_idempotent_complete (homological_complex C c) :=\nbegin\n  haveI := (to_karoubi_is_equivalence C),\n  let e := (functor.as_equivalence (to_karoubi C)),\n  let h : (to_karoubi C).additive := by apply_instance,\n  haveI : e.functor.additive := h,\n  rw is_idempotent_complete_iff_of_equivalence (equivalence.map_homological_complex c e),\n  rw ← is_idempotent_complete_iff_of_equivalence (karoubi_homological_complex_equivalence C c),\n  apply_instance,\nend\n\nvariables (α : Type*) [add_right_cancel_semigroup α] [has_one α]\n\n/-@[simps]\ndef karoubi_chain_complex_equivalence :\n  karoubi (chain_complex C α) ≌\n    chain_complex (karoubi C) α :=\nkaroubi_homological_complex_equivalence C (complex_shape.down α)\n\n@[simps]\ndef karoubi_cochain_complex_equivalence :\n  karoubi (cochain_complex C α) ≌\n    cochain_complex (karoubi C) α :=\nkaroubi_homological_complex_equivalence C (complex_shape.up α)-/\n\nend idempotents\n\nnamespace functor\n\nvariables {D : Type*} [category D] [preadditive D]\n\n@[simps]\ndef map_karoubi_homological_complex (F : C ⥤ D) [F.additive] (c : complex_shape ι) :\n  karoubi (homological_complex C c) ⥤ karoubi (homological_complex D c) :=\n(functor_extension₂ _ _).obj (functor.map_homological_complex F c)\n\nlemma map_homological_complex_karoubi_compatibility\n  (F : C ⥤ D) [F.additive] (c : complex_shape ι) :\n  to_karoubi _ ⋙ F.map_karoubi_homological_complex c =\n  F.map_homological_complex c ⋙ to_karoubi _ :=\nbegin\n  apply functor.ext,\n  { intros X Y f,\n    ext n,\n    dsimp [to_karoubi],\n    simp only [karoubi.comp_f, karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id,\n      homological_complex.comp_f, map_karoubi_homological_complex_obj_p_f,\n      homological_complex.id_f, map_id, map_homological_complex_map_f],\n    erw id_comp, },\n  { intro X,\n    ext1,\n    { erw [id_comp, comp_id],\n      ext n,\n      dsimp,\n      simpa only [F.map_id, homological_complex.id_f], },\n    { refl, }, },\nend\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/idempotents/homological_complex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3152407142424417}}
{"text": "import algebra.homology.exact\nimport algebra.category.Module.abelian\nimport category_theory.limits.shapes.images\n\nimport .defs\nimport .ideal_and_fg_ideal\nimport .right_exact\nimport .ses\n\n\nopen category_theory category_theory.limits category_theory.monoidal_category  \nopen Module character_module\nopen_locale tensor_product zero_object big_operators\n\nuniverse u\n\nnamespace module\n\nvariables (R : Type u) [comm_ring R] \nvariables (M : Type u) [add_comm_group M] [module R M]\n\nnamespace flat\n\nsection defs\n\n/--\n0 ---> N₁ ----> N₂ ----> N₃ ----> 0\n-/\n@[reducible]\nprotected def ses : Prop :=\n(tensor_right (Module.of R M)).is_exact\n\n/--\n    l12      l23\nN1 ----> N2 -----> N3 is exact\n\nwe have 0 ---> ker l23 ---> N2 ---> coker l23 ---> 0 exact\n\nker l23 ⊗ M ⟶ N2 ⊗ M is injective\n\nWant to show\n        l12 ⊗ 1      l23 ⊗ 1\nN1 ⊗ M --------> N2 --------> N3\n\n-/\nprotected def exact : Prop :=\n∀ ⦃N1 N2 N3 : Module.{u} R⦄ (l12 : N1 ⟶ N2) (l23 : N2 ⟶ N3)\n  (he : exact l12 l23),\n  exact ((tensor_right $ Module.of R M).map l12)\n    ((tensor_right $ Module.of R M).map l23)\n\nprotected def inj : Prop :=\n∀ ⦃N N' : Module.{u} R⦄ (L : N ⟶ N'), \n  function.injective L →\n  function.injective ((tensor_right (Module.of R M)).map L) \n\nprotected def ideal : Prop :=\n∀ (I : ideal R), function.injective (tensor_embedding M I)\n\nprotected def fg_ideal : Prop :=\n∀ (I : ideal R), I.fg → function.injective (tensor_embedding M I)\n\nend defs\n\nsection ses_iff_inj\n\nlemma ses_of_inj (H : module.flat.inj R M) :  module.flat.ses R M := -- λ N1 N2 N3 l12 l23 he1 he2 he3,\n{ map_exact := \n  begin \n    rintros ⟨N1, N2, N3, l12, l23⟩ ⟨he1, he2, he3⟩,\n    dsimp only at *,\n    have inj1 : function.injective (tensor_product.map l12 linear_map.id),\n    { rw ←mono_iff_exact_zero_left at he1,\n      rw concrete_category.mono_iff_injective_of_preserves_pullback at he1,\n      exact H l12 he1, },\n  \n    obtain ⟨e1, e2⟩ := @@tensor_product.right_exact R _ M _ _ N1 N2 N3 l12 l23 he2 he3,\n    refine ⟨_, e1, e2⟩,\n    { rw ←mono_iff_exact_zero_left,\n      rwa concrete_category.mono_iff_injective_of_preserves_pullback, },\n  end }\n\nlemma inj_of_ses (H : module.flat.ses R M) : module.flat.inj R M := λ N1 N2 l hl, \nbegin \n  resetI,\n  obtain ⟨e1, e2, e3⟩ := @@functor.is_exact.map_exact _ _ _ _ H ⟨_, _, _, l, (cokernel.π l)⟩ ⟨_, _, _⟩,\n  { rwa [←mono_iff_exact_zero_left, Module.mono_iff_injective] at e1, },\n  { rwa [←mono_iff_exact_zero_left, Module.mono_iff_injective], },\n  { exact abelian.exact_cokernel l, },\n  exact @@exact_epi_zero _ _ _ _ _ _ _,\nend\n\nend ses_iff_inj\n\nlemma ideal_of_fg_ideal (H : module.flat.fg_ideal R M) : module.flat.ideal R M :=\nλ I, by classical; exact tensor_embedding_inj_of_fg_inj M I H\n\nsection inj_of_ideal\n\nlemma module.Baer_iff : \n  module.Baer.{u u} R M ↔ \n  ∀ I, function.surjective (restrict_to_ideal R M I) :=\nbegin \n  split,\n  { intros h I g,\n    obtain ⟨g', hg'⟩ := h I g,\n    refine ⟨g', _⟩,\n    rw restrict_to_ideal_apply,\n    ext1,\n    rw linear_map.dom_restrict_apply, \n    rw hg' x x.2, congr, ext, refl,  },\n  { intros h I g,\n    obtain ⟨g', hg'⟩ := h I g,\n    refine ⟨g', λ x hx, _⟩,\n    rw ←hg',\n    rw restrict_to_ideal_apply,\n    rw linear_map.dom_restrict_apply,\n    congr, },\nend\n\nlemma Lambek : category_theory.injective (Module.of R $ character_module M) → module.flat.inj R M := \nbegin \n  intros h_injective A B L hL,\n  haveI : mono L := by rwa concrete_category.mono_iff_injective_of_preserves_pullback,\n  rw ←linear_map.ker_eq_bot at ⊢,\n  rw eq_bot_iff at ⊢,\n  rintros z hz,\n  rw submodule.mem_bot,\n  rw linear_map.mem_ker at hz,\n  by_cases rid : z = 0, \n  { exact rid },\n  exfalso,\n  obtain ⟨g, hg⟩ := character_module.non_zero (A ⊗[R] M) rid,\n  set f : A →ₗ[R] (character_module M) := \n  { to_fun := λ a, { to_fun := λ m, g (a ⊗ₜ m), map_add' := _, map_smul' := _ }, \n    map_add' := _, map_smul' := _ } with f_eq,\n  work_on_goal 2 { intros m m', rw [tensor_product.tmul_add, map_add], },\n  work_on_goal 2 { intros zz m, rw [ring_hom.id_apply, ←tensor_product.smul_tmul, \n    ←tensor_product.smul_tmul', map_zsmul], },\n  work_on_goal 2 { intros a a', ext1 t, simp only [linear_map.coe_mk, linear_map.add_apply, \n    tensor_product.add_tmul, map_add], },\n  work_on_goal 2 { intros r a, ext1 t, simp only [linear_map.coe_mk, linear_map.smul_apply, \n    character_module.smul_apply, ring_hom.id_apply, tensor_product.smul_tmul], },\n  obtain ⟨f' : B →ₗ[R] (character_module M), hf'⟩ := h_injective.factors f L,\n  set g' : (character_module $ B ⊗[R] M) := character_module.hom_equiv _ _ f' with g'_eq,\n  obtain ⟨ι, a, m, s, rfl⟩ := tensor_product.exists_rep _ z,\n  have EQ : g' (∑ i in s, L (a i) ⊗ₜ m i) = 0,\n  { simp only [tensor_right_map, linear_map.map_sum, monoidal_category.hom_apply, \n      Module.id_apply] at hz,\n    rw [hz, map_zero], },\n  refine hg _,\n  rw map_sum,\n  simp_rw [map_sum, g'_eq, hom_equiv_apply, add_monoid_hom.coe_to_int_linear_map,\n      tensor_product.to_add_comm_group'.apply_tmul] at EQ,\n  convert EQ using 1,\n  refine finset.sum_congr rfl (λ i hi, _),\n  erw fun_like.congr_fun hf',\n  rw f_eq,\n  refl,\nend\n\nlemma flat'_of_baer : module.Baer.{u u} R (character_module M) → module.flat.inj R M := \nλ h, Lambek _ _ $ (module.injective_iff_injective_object.{u u} R \n    (character_module M)).mp (module.Baer.injective h)\n\nlemma flat'_of_surj : \n  (∀ I, function.surjective (restrict_to_ideal R (character_module M) I)) → module.flat.inj R M := \nλ h, flat'_of_baer _ _ ((module.Baer_iff _ _).mpr h)\n\n\nvariables {I : ideal R} (hI : function.injective (tensor_embedding M I))\n\nlemma surj1 : function.surjective (character_module.map (tensor_embedding M I)) :=\ncharacter_module.map_inj _ hI\n\nlemma injective_character (hIs : ∀ (I : ideal R), function.injective (tensor_embedding M I)) : \n  module.Baer.{u u} R (character_module M) :=\nbegin \n  rw module.Baer_iff,\n  intros I l,\n  obtain ⟨F, hF⟩ := surj1 _ _ (hIs I) (character_module.hom_equiv _ _ l),\n  refine ⟨(character_module.hom_equiv _ _).symm F, _⟩,\n  ext i m : 2,\n  simp only [restrict_to_ideal_apply, linear_map.dom_restrict_apply, character_module.hom_equiv, \n    equiv.coe_fn_symm_mk, linear_map.coe_mk],\n  have EQ := fun_like.congr_fun hF (i ⊗ₜ m),\n  simpa only [character_module.hom_equiv_apply, add_monoid_hom.coe_to_int_linear_map, \n    tensor_product.to_add_comm_group'.apply_tmul] using EQ,\nend\n\n/--\n`tensor_embedding M I` is the canonical map `I ⊗ M ⟶ R ⊗ M`\n-/\nlemma inj_of_ideal (hIs : module.flat.ideal R M) :\n  module.flat.inj R M :=\nbegin \n  apply flat'_of_baer,\n  apply injective_character,\n  assumption\nend\n\nend inj_of_ideal\n\nlemma fg_ideal_of_inj (H : module.flat.inj R M) : module.flat.fg_ideal R M :=\nbegin \n  intros I hI,\n  refine @H ⟨I⟩ ⟨R⟩ ⟨coe, λ _ _, rfl, λ _ _, rfl⟩ _,\n  intros z z' h,\n  ext1,\n  simpa only [linear_map.coe_mk] using h,\nend\n\nlemma exact_of_ses (H : module.flat.ses R M) :\n  module.flat.exact R M :=\nλ N1 N2 N3 l12 l23 he, functor.map_exact _ _ _ he\n\nlemma inj_of_exact (H : module.flat.exact R M) :\n  module.flat.inj R M :=\nλ N1 N2 l h,\nbegin \n  have e0 : exact (0 : (0 : Module.{u} R) ⟶ _) l,\n  { rw ←Module.mono_iff_injective at h,\n    resetI,\n    apply exact_zero_left_of_mono, },\n  specialize H (0 : (0 : Module.{u} R) ⟶ N1) l e0,\n  have eq1 : ((tensor_right (Module.of R M)).map 0) = \n    (_ : Module.of R ((0 : Module.{u} R) ⊗ M) ⟶ 0) ≫ (0 : (0 : Module.{u} R) ⟶ Module.of R (N1 ⊗ M)),\n  work_on_goal 3 \n  { refine tensor_product.lift 0, },\n  { refine linear_map.ext (λ z, z.induction_on _ (λ z m, _) (λ x y hx hy, _)), \n    { simp only [map_zero] },\n    { simp only [tensor_right_map, monoidal_preadditive.zero_tensor, comp_zero], },\n    { rw [map_add, hx, hy, map_add], }, },\n  rw eq1 at H,\n  rw abelian.exact_epi_comp_iff at H,\n  rw ←mono_iff_exact_zero_left at H,\n  rwa Module.mono_iff_injective at H,\nend\n\nlemma equiv_defs : tfae \n  [module.flat.ses R M\n  , module.flat.inj R M\n  , module.flat.ideal R M\n  , module.flat.fg_ideal R M\n  , module.flat.exact R M] :=\nbegin \n  tfae_have : 1 → 2, { apply inj_of_ses },\n  tfae_have : 2 → 1, { apply ses_of_inj },\n  tfae_have : 3 → 2, { apply inj_of_ideal },\n  tfae_have : 4 → 3, { apply ideal_of_fg_ideal },\n  tfae_have : 2 → 4, { apply fg_ideal_of_inj },\n  tfae_have : 5 → 2, { apply inj_of_exact },\n  tfae_have : 1 → 5, { apply exact_of_ses },\n  tfae_finish,\nend\n\nend flat\n\nend module\n", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/flat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3151827985589447}}
{"text": "import for_mathlib.category_theory.functor.shift\nimport category_theory.triangulated.pretriangulated\nimport for_mathlib.category_theory.triangulated.pretriangulated_misc\nimport for_mathlib.category_theory.triangulated.yoneda\n\nopen category_theory category_theory.category category_theory.limits\n  category_theory.pretriangulated\n\nnoncomputable theory\n\nnamespace category_theory\n\nnamespace functor\n\nvariables {C D E : Type*} [category C] [category D] [category E]\n  [has_shift C ℤ] [has_shift D ℤ] [has_shift E ℤ]\n  [preadditive C] [preadditive D] [preadditive E]\n  (F : C ⥤ D) [F.has_comm_shift ℤ] (G : D ⥤ E) [G.has_comm_shift ℤ]\n\n@[simps]\ndef map_triangle : pretriangulated.triangle C ⥤ pretriangulated.triangle D :=\n{ obj := λ T, pretriangulated.triangle.mk (F.map T.mor₁) (F.map T.mor₂)\n    (F.map T.mor₃ ≫ (F.comm_shift_iso (1 : ℤ)).hom.app T.obj₁),\n  map := λ S T f,\n  { hom₁ := F.map f.hom₁,\n    hom₂ := F.map f.hom₂,\n    hom₃ := F.map f.hom₃,\n    comm₁' := by { dsimp, simp only [← F.map_comp, f.comm₁], },\n    comm₂' := by { dsimp, simp only [← F.map_comp, f.comm₂], },\n    comm₃' := begin\n      dsimp,\n      erw [category.assoc, ←nat_trans.naturality],\n      simp only [functor.comp_map, ← F.map_comp_assoc, f.comm₃],\n    end, }, }\n\ninstance [faithful F] : faithful F.map_triangle :=\n⟨λ K L f₁ f₂ hf, begin\n  ext; apply F.map_injective,\n  { change (F.map_triangle.map f₁).hom₁ = (F.map_triangle.map f₂).hom₁, rw hf, },\n  { change (F.map_triangle.map f₁).hom₂ = (F.map_triangle.map f₂).hom₂, rw hf, },\n  { change (F.map_triangle.map f₁).hom₃ = (F.map_triangle.map f₂).hom₃, rw hf, },\nend⟩\n\ninstance [full F] [faithful F] : full F.map_triangle :=\nfull_of_surjective _ (λ K L f, begin\n  refine\n  ⟨{ hom₁ := F.preimage f.hom₁,\n    hom₂ := F.preimage f.hom₂,\n    hom₃ := F.preimage f.hom₃,\n    comm₁' := F.map_injective (by simpa only [F.map_comp, image_preimage] using f.comm₁),\n    comm₂' := F.map_injective (by simpa only [F.map_comp, image_preimage] using f.comm₂),\n    comm₃' := F.map_injective (begin\n      have eq := f.comm₃,\n      dsimp at eq,\n      simp only [map_comp, image_preimage, assoc,\n        ← cancel_mono ((F.comm_shift_iso (1 : ℤ)).hom.app L.obj₁), ← eq],\n      erw (F.comm_shift_iso (1 : ℤ)).hom.naturality (F.preimage f.hom₁),\n      simp only [comp_map, image_preimage],\n    end), }, by tidy⟩,\nend)\n\n@[simps]\ndef map_triangle_rotate [functor.additive F] :\n  F.map_triangle ⋙ pretriangulated.rotate D ≅\n    pretriangulated.rotate C ⋙ F.map_triangle :=\nnat_iso.of_components (λ T, triangle.mk_iso _ _ (iso.refl _) (iso.refl _)\n  ((F.comm_shift_iso (1 : ℤ)).symm.app _) (by tidy) (by tidy) begin\n    dsimp,\n    simp only [map_id, preadditive.neg_comp, comp_id,\n      map_neg, preadditive.comp_neg, neg_inj],\n    erw ← nat_trans.naturality_assoc,\n    simp only [comp_map, iso.inv_hom_id_app, comp_id],\n  end)\n  (λ X Y f, begin\n    ext,\n    { dsimp, simp, },\n    { dsimp, simp, },\n    { dsimp, erw ← nat_trans.naturality, refl, },\n  end)\n\n@[simps]\ndef map_triangle_inv_rotate [functor.additive F]\n  [∀ (n : ℤ), (shift_functor C n).additive]\n  [∀ (n : ℤ), (shift_functor D n).additive] :\n  F.map_triangle ⋙ pretriangulated.inv_rotate D ≅\n    pretriangulated.inv_rotate C ⋙ F.map_triangle :=\ncalc F.map_triangle ⋙ inv_rotate D ≅ _ : (functor.left_unitor _).symm\n... ≅ _ : iso_whisker_right (pretriangulated.triangle_rotation C).counit_iso.symm _\n... ≅ _ : functor.associator _ _ _\n... ≅ _ : iso_whisker_left _ (functor.associator _ _ _).symm\n... ≅ _ : iso_whisker_left _ (iso_whisker_right (map_triangle_rotate F).symm _)\n... ≅ _ : iso_whisker_left _ (functor.associator _ _ _)\n... ≅ _ : iso_whisker_left _ (iso_whisker_left _ (pretriangulated.triangle_rotation D).unit_iso.symm)\n... ≅ _: iso_whisker_left _ (functor.right_unitor _)\n\nvariable (C)\n\n@[simps]\ndef map_triangle_id : (𝟭 C).map_triangle ≅ 𝟭 _ :=\nnat_iso.of_components (λ T, pretriangulated.triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _)\n  (by tidy) (by tidy) (by tidy)) (by tidy)\n\nvariable {C}\n\n@[simps]\ndef map_triangle_comp : (F ⋙ G).map_triangle ≅ F.map_triangle ⋙ G.map_triangle :=\nnat_iso.of_components (λ T, pretriangulated.triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _)\n  (by tidy) (by tidy) (by { dsimp, simp, })) (λ T₁ T₂ f, by { ext; dsimp; simp, })\n\nvariables {F}\n\n@[simps]\ndef map_triangle_nat_trans {F' : C ⥤ D} [F'.has_comm_shift ℤ] (τ : F ⟶ F')\n  [τ.respects_comm_shift ℤ] :\n  F.map_triangle ⟶ F'.map_triangle :=\n{ app := λ X,\n  { hom₁ := τ.app _,\n    hom₂ := τ.app _,\n    hom₃ := τ.app _,\n    comm₃' := begin\n      dsimp,\n      simp only [assoc, nat_trans.respects_comm_shift.comm_app τ (1 : ℤ),\n        nat_trans.naturality_assoc],\n    end, }, }\n\n@[simps]\ndef map_triangle_nat_iso {F' : C ⥤ D} [F'.has_comm_shift ℤ] (e : F ≅ F')\n  [e.hom.respects_comm_shift ℤ] : F.map_triangle ≅ F'.map_triangle :=\nnat_iso.of_components\n  (λ T, pretriangulated.triangle.mk_iso _ _ (e.app _) (e.app _) (e.app _) (by tidy) (by tidy)\n  begin\n    dsimp,\n    simp only [assoc, nat_trans.respects_comm_shift.comm_app e.hom (1 : ℤ),\n      nat_trans.naturality_assoc],\n  end) (by tidy)\n\nvariables (F) [∀ (n : ℤ), (shift_functor C n).additive]\n  [∀ (n : ℤ), (shift_functor D n).additive]\n  [∀ (n : ℤ), (shift_functor E n).additive]\n  [has_zero_object C] [has_zero_object D] [has_zero_object E]\n  [pretriangulated C] [pretriangulated D] [pretriangulated E]\n\n@[protected]\nclass is_triangulated : Prop :=\n(map_distinguished' [] : ∀ (T : pretriangulated.triangle C)\n  (hT : T ∈ dist_triang C), F.map_triangle.obj T ∈ dist_triang D)\n\nlemma map_distinguished [F.is_triangulated] (T : pretriangulated.triangle C)\n  (hT : T ∈ dist_triang C) : F.map_triangle.obj T ∈ dist_triang D :=\nis_triangulated.map_distinguished' F T hT\n\ninstance id_is_triangulated : (𝟭 C).is_triangulated :=\n{ map_distinguished' :=\n    λ T hT, pretriangulated.isomorphic_distinguished _ hT _ ((map_triangle_id C).app T), }\n\ninstance comp_is_triangulated [F.is_triangulated] [G.is_triangulated] :\n  (F ⋙ G).is_triangulated :=\n{ map_distinguished' := λ T hT, pretriangulated.isomorphic_distinguished _\n    (G.map_distinguished _ (F.map_distinguished _ hT)) _ ((map_triangle_comp F G).app T), }\n\nlemma reflects_distinguished [F.is_triangulated] [full F] [faithful F]\n  (T : pretriangulated.triangle C) (hT : F.map_triangle.obj T ∈ dist_triang D) :\n  T ∈ dist_triang C :=\nbegin\n  obtain ⟨Z, g, h, mem⟩ := distinguished_cocone_triangle _ _ T.mor₁,\n  exact pretriangulated.isomorphic_distinguished _ mem _\n    (F.map_triangle.preimage_iso (pretriangulated.iso_triangle_of_distinguished_of_is_iso₁₂ _ _ hT\n    (F.map_distinguished _ mem) (iso.refl _) (iso.refl _) (by tidy))),\nend\n\nvariable {F}\n\nlemma is_triangulated.of_iso {G : C ⥤ D} (e : F ≅ G) [F.is_triangulated] [G.has_comm_shift ℤ]\n  [e.hom.respects_comm_shift ℤ] : G.is_triangulated :=\n⟨λ T hT, pretriangulated.isomorphic_distinguished _\n  (F.map_distinguished _ hT) _ ((map_triangle_nat_iso e).symm.app T)⟩\n\nend functor\n\nnamespace pretriangulated\n\nvariables {C : Type*} [category C] [has_shift C ℤ] [preadditive C]\n  (S : set C) [S.is_stable_by_shift ℤ]\n\nvariables (T : pretriangulated.triangle C)\n  (h₁ : T.obj₁ ∈ S) (h₂ : T.obj₂ ∈ S) (h₃ : T.obj₃ ∈ S)\n\n@[simps]\ndef full_subcategory_lift_triangle :\n  pretriangulated.triangle (full_subcategory S) :=\n{ obj₁ := ⟨T.obj₁, h₁⟩,\n  obj₂ := ⟨T.obj₂, h₂⟩,\n  obj₃ := ⟨T.obj₃, h₃⟩,\n  mor₁ := T.mor₁,\n  mor₂ := T.mor₂,\n  mor₃ := T.mor₃, }\n\ndef full_subcategory_lift_triangle_iso :\n  (full_subcategory_inclusion S).map_triangle.obj\n    (full_subcategory_lift_triangle S T h₁ h₂ h₃) ≅ T :=\ntriangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _) (by tidy) (by tidy) begin\n  dsimp,\n  simp only [functor.map_id, comp_id, id_comp],\n  erw comp_id,\nend\n\nend pretriangulated\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/triangulated/triangulated_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.31511364850915813}}
{"text": "import graphs.ext\n\nlemma eq_mor_eq_vmap {G H : graph} (α : G ↦ H) (β : G ↦ H) (hyp : α = β) : α.vertex_map = β.vertex_map :=\nbegin\n  cases α,\n  cases β,\n  rw morphism.mk.inj_eq at hyp,\n  exact hyp.1,\nend\n\nlemma eq_mor_eq_emap {G H : graph} (α : G ↦ H) (β : G ↦ H) (hyp : α = β) : α.edge_map = β.edge_map :=\nbegin\n  cases α,\n  cases β,\n  rw morphism.mk.inj_eq at hyp,\n  exact hyp.2,\nend", "meta": {"author": "barriecooper", "repo": "lean-graphs", "sha": "3f7be961f99fe084f950f52fe17c53e8093b5337", "save_path": "github-repos/lean/barriecooper-lean-graphs", "path": "github-repos/lean/barriecooper-lean-graphs/lean-graphs-3f7be961f99fe084f950f52fe17c53e8093b5337/src/graphs/allexts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.315113648509158}}
{"text": "/-\nThe problem is originally presented in:\nA. Pease, G. Sutcliffe, N. Siegel, and S. Trac, “Large Theory\nReasoning with SUMO at CASC,” pp. 1–8, Jul. 2009.\nHere we present the natural deduction proof in Lean.\n-/\n\nconstant U : Type\n\nconstants SetOrClass Set Class Object Entity NullList_m List \n          CorpuscularObject Invertebrate Vertebrate Animal SpinalColumn \n          Organism Agent Physical Abstract\n          subclass_m TransitiveRelation PartialOrderingRelation Relation : U\nconstant BananaSlug10 : U\n\nconstants exhaustiveDecomposition3 disjointDecomposition3 partition3 : U → U → U → Prop\nconstant ins : U → U → Prop \nconstant subclass : U → U → Prop\nconstant disjoint : U → U → Prop\nconstant component : U → U → Prop\nconstant part : U → U → Prop\nconstant inList : U → U → Prop\nconstant ConsFn : U → U → U\nconstant ListFn1 : U → U\nconstant ListFn2 : U → U → U\nconstant ListFn3 : U → U → U → U\n\n\n/- SUMO axioms -/\n\nvariable a13 : ins subclass_m PartialOrderingRelation\n\nvariable a15 : ∀ x y z : U, ins x SetOrClass ∧ ins y SetOrClass → \n (subclass x y ∧ ins z x → ins z y)\n\n/- EDITED (see https://github.com/own-pt/cl-krr/issues/23) -/\nvariable a72773 : ∀ a : U, ((ins a Animal) ∧ (¬ ∃ p : U, ins p SpinalColumn ∧ part p a)) \n  → ¬ ins a Vertebrate\n\n/- EDITED -/\nvariable a72774 : ¬ ∃ s : U, ins s SpinalColumn ∧ part s BananaSlug10\n\nvariable a72761 : ∀ x row0 row1 : U, ins row0 Entity ∧ ins row1 Entity ∧ ins x Entity → \n (ListFn3 x row0 row1 = ConsFn x (ListFn2 row0 row1))\n\nvariable a72767 : ∀ x y : U, (ins x Entity ∧ ins y Entity) → \n ((ListFn2 x y) = (ConsFn x (ConsFn y NullList_m)))\n\nvariable a72768 : ∀ x : U, ins x Entity → (ListFn1 x = ConsFn x NullList_m)\n\nvariable a72769 : ∀ x : U, ins x Entity → ¬ inList x NullList_m  \n\nvariable a72770 : ∀ L x y : U, (ins x Entity ∧ ins y Entity ∧ ins L List) → \n ((inList x (ConsFn y L)) ↔ ((x = y) ∨ inList x L))\n  \nvariable a67959 : ins NullList_m List\n\nvariable a67958 : ins List SetOrClass\nvariable a72772 : ins BananaSlug10 Animal\nvariable a72771 : ins Animal SetOrClass\nvariable a72778 : ins Invertebrate SetOrClass\nvariable a71402 : ins Vertebrate SetOrClass \nvariable a71371 : ins Organism SetOrClass\nvariable a71872 : ins Agent SetOrClass\nvariable a71669 : ins Object SetOrClass\nvariable a69763 : ins Physical SetOrClass\nvariable a67331 : ins Entity SetOrClass\nvariable a67448 : ins SetOrClass SetOrClass\nvariable a68771 : ins Abstract SetOrClass\nvariable a68763 : ins Relation SetOrClass\nvariable a71844 : ins TransitiveRelation SetOrClass\nvariable a72180 : ins PartialOrderingRelation SetOrClass\n\nvariable a71370 : partition3 Animal Vertebrate Invertebrate\n\nvariable a67131 : ∀ (c row0 row1 : U), (ins c Class ∧ ins row0 Class ∧  ins row1 Class) → \n (partition3 c row0 row1 ↔ (exhaustiveDecomposition3 c row0 row1 ∧ disjointDecomposition3 c row0 row1))\n\n-- EDITED (see https://github.com/own-pt/cl-krr/issues/22)\nvariable a67115 :\n  ∀ (row0 row1 c obj : U),\n    ∃ (item : U),\n      ins item SetOrClass ∧\n        (ins obj Entity →\n          ins c SetOrClass ∧ ins c Class ∧ \n          ins row0 Class ∧ ins row0 Entity ∧ \n          ins row1 Class ∧ ins row1 Entity →\n            exhaustiveDecomposition3 c row0 row1 → ins obj c → \n              inList item (ListFn2 row0 row1) ∧ ins obj item)\n\nvariable a67447 : partition3 SetOrClass Set Class \nvariable a67172 : ∃ x : U, ins x Entity\nvariable a67173 : ∀ c : U, ins c Class ↔ subclass c Entity\nvariable a67818 : subclass PartialOrderingRelation TransitiveRelation\n\nvariable a67809 : ∀ x y z : U, ins x SetOrClass ∧ ins y SetOrClass ∧ ins z SetOrClass → \n  ins subclass_m TransitiveRelation → (subclass x y ∧ subclass y z → subclass x z)\n\nvariable a71382 : subclass Vertebrate Animal\nvariable a71383 : subclass Invertebrate Animal\nvariable a71369 : subclass Animal Organism\nvariable a71340 : subclass Organism Agent\nvariable a67315 : subclass Agent Object\nvariable a67177 : subclass Object Physical\nvariable a67174 : subclass Physical Entity\nvariable a67446 : subclass SetOrClass Abstract\nvariable a67332 : subclass Abstract Entity\nvariable a67954 : subclass List Relation\nvariable a67450 : subclass Relation Abstract\n\n-- commented in list.kif\nvariable novo1 : ∀ (x L : U), ins L Entity → ins L List → ins (ConsFn x L) List\n\n\n-- some initial tests\n\ninclude a15 a71382 a71383 a71402 a72771\nlemma VertebrateAnimal : ∀ (x : U), ins x Vertebrate → ins x Animal := \nbegin \n intros a h,\n have h1, from a15 Vertebrate Animal a,\n apply h1,\n exact (and.intro a71402 a72771),\n exact (and.intro a71382 h)\nend\nomit a15 a71382 a71383 a71402 a72771\n\ninclude a15 a72180 a71844 a67818 a13\nlemma subclass_TransitiveRelation : ins subclass_m TransitiveRelation :=\nbegin\n specialize a15 PartialOrderingRelation TransitiveRelation subclass_m,\n apply a15,\n exact ⟨ a72180, a71844 ⟩, \n exact ⟨ a67818, a13 ⟩, \nend\nomit a15 a72180 a71844 a67818 a13\n\n\ninclude a15 a72180 a71383 a71844 a13 a67818 a71402 a72771 a71371 a71382 \n        a71369 a67809\n\nlemma VertebrateOrganism : ∀ (x : U), ins x Vertebrate → ins x Organism := \nbegin \n intros a h,\n have h1, from a15 Animal Organism a, \n apply h1,\n exact and.intro a72771 a71371,\n have h0 : ∀ x, ins x Vertebrate → ins x Animal, apply VertebrateAnimal; assumption,\n have h2, from h0 a h,\n exact and.intro a71369 h2,\nend\n\nlemma VertebrateOrganism' : ∀ (x : U), ins x Vertebrate → ins x Organism := \nbegin\n  intros a h,\n  have h₁ : ins subclass_m TransitiveRelation,\n    specialize a15 PartialOrderingRelation TransitiveRelation subclass_m,\n    apply a15,\n    exact ⟨a72180, a71844⟩, \n    exact ⟨a67818, a13⟩, \n  have h₂ : subclass Vertebrate Organism,\n    apply a67809 _ Animal _,\n    exact ⟨a71402, ⟨a72771, a71371⟩⟩,\n    exact h₁,\n    exact ⟨a71382, a71369⟩,\n  apply a15 Vertebrate _ _,\n  exact ⟨a71402, a71371⟩,\n  exact ⟨h₂, h⟩\nend\nomit a15 a72180 a71383 a71844 a13 a67818 a71402 a72771 a71371 a71382 \n     a71369 a67809\n\n\ninclude a15 a72180 a71844 a13 a67818 a71402 a72771 a71371 \n        a71369 a67809 a71382 a67174 a69763 a67331 a71669 a71340 \n        a67315 a67177 a71872\n\nlemma VertebrateEntity : ∀ (x : U), ins x Vertebrate → ins x Entity := \nbegin\n  intros a h, \n  have h1, apply subclass_TransitiveRelation; assumption,\n  have h2 : subclass Vertebrate Organism,\n    apply a67809 _ Animal _,\n    exact ⟨a71402, ⟨ a72771, a71371 ⟩⟩,\n    exact h1,\n    exact ⟨a71382, a71369⟩,  \n  have h3 : subclass Vertebrate Agent,\n    apply a67809 _ Organism _,\n    exact ⟨a71402, ⟨ a71371, a71872 ⟩⟩,\n    exact h1,\n    exact ⟨h2, a71340⟩,\n  have h4 : subclass Vertebrate Object,\n    apply a67809 _ Agent _,\n    exact ⟨a71402, ⟨ a71872, a71669 ⟩⟩,\n    exact h1,\n    exact ⟨h3, a67315⟩,\n  have h5 : subclass Vertebrate Physical,\n    apply a67809 _ Object _,\n    exact ⟨a71402, ⟨ a71669, a69763 ⟩⟩,\n    exact h1,\n    exact ⟨h4, a67177⟩,\n  have h6 : subclass Vertebrate Entity,\n    apply a67809 _ Physical _,\n    exact ⟨a71402, ⟨ a69763, a67331 ⟩⟩,\n    exact h1,\n    exact ⟨ h5, a67174 ⟩,\n  apply a15 Vertebrate _ _,\n  exact ⟨a71402, a67331⟩,\n  exact ⟨h6, h⟩\nend\nomit a15 a72180 a71844 a13 a67818 a71402 a72771 a71371 \n     a71369 a67809 a71382 a67174 a69763 a67331 a71669 a71340 \n     a67315 a67177 a71872\n\n\n-- start proofs\n\ninclude a72767 a72768 a72769 a72770 a67959 a15 novo1 a67954 a67332 a67331 \n        a68771 a68763 a67958 a67450\n\nlemma listLemma (hne : nonempty U) : ∀ x y z : U, \n  ins x Entity ∧ ins y Entity ∧ ins z Entity →\n  inList x (ListFn2 y z) → x = y ∨ x = z :=\nbegin\n  intros x y z h h1,\n    rw (a72767 y z ⟨h.right.left, h.right.right⟩) at h1,\n    have h2 : x = y ∨ inList x (ConsFn z NullList_m),\n      rw ←(a72770 (ConsFn z NullList_m) x y),\n      exact h1,\n      simp *,\n      apply novo1 z NullList_m,\n         apply a15 Abstract Entity NullList_m;\n           simp *, \n           apply a15 Relation Abstract NullList_m;\n             simp *,\n             apply a15 List Relation NullList_m;\n               simp *,\n               assumption,\n      cases h2,\n        exact or.inl h2,\n        have h3 : x = z ∨ inList x NullList_m,\n          rw ←(a72770 NullList_m x z),\n          exact h2,\n          exact ⟨h.1, ⟨h.2.2, a67959⟩⟩,\n          cases h3,\n            exact or.inr h3,\n            apply false.elim,\n              exact ((a72769 x) h.left) h3\nend\n\nomit a72767 a72768 a72769 a72770 a67959 a15 novo1 a67954 a67332 a67331 \n     a68771 a68763 a67958 a67450\n\n\ninclude a67131 a67115 a67809 a67448 a68771 a67331 a15 a72180 a71844 a67818 a13 \n        a67446 a67332 a72767 a72768 a72769 a72770 a67959  novo1 a67954 a68763 \n        a67958 a67450\n\nlemma lX (hne : nonempty U) : ∀ x c c1 c2,\n  (ins c SetOrClass ∧ ins c1 SetOrClass ∧ ins c2 SetOrClass) →\n  (ins c Class ∧ ins c1 Class ∧ ins c2 Class ∧ ins x Entity) → \n    (partition3 c c1 c2 ∧ ins x c ∧ ¬ ins x c1) → ins x c2 := \nbegin\n  intros a c c1 c2 h1 h2 h3,\n  specialize a67131 c c1 c2,\n  specialize a67115 c1 c2 c a,\n  have h₃, from a67131 ⟨ h2.1, ⟨h2.2.1, h2.2.2.1 ⟩⟩,\n  have h4, from iff.elim_left h₃ h3.1,\n  cases h4 with h4a h4b,\n  cases a67115 with b h5,\n  have h7, from h5.right,\n  have h8, from h2.right.right.right,\n  specialize h7 h8,\n  have h9 : subclass SetOrClass Entity,\n    apply (a67809 _ Abstract _), \n      exact ⟨a67448, ⟨a68771, a67331⟩⟩,\n      apply subclass_TransitiveRelation; assumption,\n      exact ⟨a67446, a67332⟩,\n  have h10 : ins c1 Entity,\n    apply (a15 SetOrClass _ _), \n      exact ⟨a67448, a67331⟩,\n      exact ⟨h9, h1.2.1⟩,\n  have h11 : ins c2 Entity,\n    apply (a15 SetOrClass _ _), \n      exact ⟨a67448, a67331⟩,\n      exact ⟨h9, h1.2.2⟩,\n  specialize h7 ⟨h1.1, ⟨h2.1, ⟨h2.2.1, ⟨h10, ⟨h2.2.2.1, h11⟩⟩⟩⟩⟩,\n  specialize h7 h4a,\n  specialize h7 h3.right.left,\n  have h12 : b = c1 ∨ b = c2,\n    apply listLemma, \n      repeat { assumption },\n      split,\n        apply a15 SetOrClass _ _,\n          exact ⟨a67448, a67331⟩,\n          exact ⟨h9, h5.left⟩,\n      exact ⟨h10, h11⟩,\n      exact h7.left,\n  cases h12,\n    rw h12 at h7,\n    apply false.elim,\n      exact h3.right.right h7.right,\n    rw ←h12,\n    exact h7.right\n end\n\n\ninclude a72771 a71371 a71872 a71369 a71340 a71669 a67315 a69763 a67177\n        a67174 \n\nlemma subclass_animal_entity : subclass Animal Entity :=\nbegin\n  have h1, apply subclass_TransitiveRelation; assumption,\n  have h2 : subclass Animal Agent,\n    apply a67809 _ Organism _,\n    exact ⟨a72771, ⟨ a71371, a71872 ⟩⟩,\n    exact h1,\n    exact ⟨a71369, a71340⟩,\n  have h3 : subclass Animal Object,\n    apply a67809 _ Agent _,\n    exact ⟨a72771, ⟨ a71872, a71669 ⟩⟩,\n    exact h1,\n    exact ⟨h2, a67315⟩,\n  have h4 : subclass Animal Physical,\n    apply a67809 _ Object _,\n    exact ⟨a72771, ⟨ a71669, a69763 ⟩⟩,\n    exact h1,\n    exact ⟨h3, a67177⟩,\n  apply a67809 _ Physical _,\n  exact ⟨a72771, ⟨ a69763, a67331 ⟩⟩,\n  exact h1,\n  exact ⟨ h4, a67174 ⟩,\nend\n\ninclude a71402 a71382 \n\nlemma subclass_vertebrate_entity : subclass Vertebrate Entity :=\nbegin\n  have h1, apply subclass_TransitiveRelation; assumption,\n  have h2 : subclass Vertebrate Organism,\n    apply a67809 _ Animal _,\n    exact ⟨a71402, ⟨ a72771, a71371 ⟩⟩,\n    exact h1,\n    exact ⟨a71382, a71369⟩,  \n  have h3 : subclass Vertebrate Agent,\n    apply a67809 _ Organism _,\n    exact ⟨a71402, ⟨ a71371, a71872 ⟩⟩,\n    exact h1,\n    exact ⟨h2, a71340⟩,\n  have h4 : subclass Vertebrate Object,\n    apply a67809 _ Agent _,\n    exact ⟨a71402, ⟨ a71872, a71669 ⟩⟩,\n    exact h1,\n    exact ⟨h3, a67315⟩,\n  have h5 : subclass Vertebrate Physical,\n    apply a67809 _ Object _,\n    exact ⟨a71402, ⟨ a71669, a69763 ⟩⟩,\n    exact h1,\n    exact ⟨h4, a67177⟩,\n  apply a67809 _ Physical _,\n  exact ⟨a71402, ⟨ a69763, a67331 ⟩⟩,\n  exact h1,\n  exact ⟨ h5, a67174 ⟩,\nend\n\ninclude a72778 a71383 \n\nlemma subclass_invertebrate_entity : subclass Invertebrate Entity :=\nbegin\n  have h1, apply subclass_TransitiveRelation; assumption,\n  have h2 : subclass Invertebrate Organism,\n    apply a67809 _ Animal _,\n    exact ⟨a72778, ⟨ a72771, a71371 ⟩⟩,\n    exact h1,\n    exact ⟨a71383, a71369⟩,  \n  have h3 : subclass Invertebrate Agent,\n    apply a67809 _ Organism _,\n    exact ⟨a72778, ⟨ a71371, a71872 ⟩⟩,\n    exact h1,\n    exact ⟨h2, a71340⟩,\n  have h4 : subclass Invertebrate Object,\n    apply a67809 _ Agent _,\n    exact ⟨a72778, ⟨ a71872, a71669 ⟩⟩,\n    exact h1,\n    exact ⟨h3, a67315⟩,\n  have h5 : subclass Invertebrate Physical,\n    apply a67809 _ Object _,\n    exact ⟨a72778, ⟨ a71669, a69763 ⟩⟩,\n    exact h1,\n    exact ⟨h4, a67177⟩,\n  apply a67809 _ Physical _,\n  exact ⟨a72778, ⟨ a69763, a67331 ⟩⟩,\n  exact h1,\n  exact ⟨ h5, a67174 ⟩,\nend\n\ninclude a72772 a72774\n\nlemma ins_banana_entity : ins BananaSlug10 Entity := \nbegin\n have h1, apply subclass_TransitiveRelation; assumption,\n have h2 : ins BananaSlug10 Organism, \n  specialize a15 Animal Organism BananaSlug10,\n  apply a15,\n  exact and.intro a72771 a71371,\n  exact and.intro a71369 a72772,\n have h3 : ins BananaSlug10 Agent, \n  specialize a15 Organism Agent BananaSlug10,\n  apply a15,\n  exact and.intro a71371 a71872,\n  exact and.intro a71340 h2,\n have h4 : ins BananaSlug10 Object, \n  specialize a15 Agent Object BananaSlug10,\n  apply a15,\n  exact and.intro a71872 a71669,\n  exact and.intro a67315 h3,\n have h5 : ins BananaSlug10 Physical, \n  specialize a15 Object Physical BananaSlug10,\n  apply a15,\n  exact and.intro a71669 a69763,\n  exact and.intro a67177 h4,\n specialize a15 Physical Entity BananaSlug10,\n apply a15,\n exact and.intro a69763 a67331,\n exact and.intro a67174 h5,\nend\n\ninclude a67173\n\n\nlemma ins_animal_class : ins Animal Class :=\nbegin\n have h0 : subclass Animal Entity,\n  apply subclass_animal_entity; assumption,\n have h1, from (a67173 Animal),\n exact h1.2 h0,\nend\n\n\ninclude a71370 a72773 \n\nlemma l0' (hne : nonempty U) : ¬(ins BananaSlug10 Vertebrate) := by simp *\nlemma l0  (hne : nonempty U) : ¬(ins BananaSlug10 Vertebrate) :=\nbegin\n  specialize a72773 BananaSlug10,\n  exact a72773 (and.intro a72772 a72774)\nend\n\n\ntheorem Banana_Invertebrate (hne: nonempty U) : ins BananaSlug10 Invertebrate :=\nbegin\n have h1 : ¬ ins BananaSlug10 Vertebrate,\n  apply l0; assumption,\n have h2 : ∀ x c c1 c2,\n  (ins c SetOrClass ∧ ins c1 SetOrClass ∧ ins c2 SetOrClass) →\n  (ins c Class ∧ ins c1 Class ∧ ins c2 Class ∧ ins x Entity) → \n   (partition3 c c1 c2 ∧ ins x c ∧ ¬ ins x c1) → ins x c2, \n   apply lX; assumption,\n have h3, from h2 BananaSlug10 Animal Vertebrate Invertebrate,\n apply h3,\n exact ⟨ a72771, ⟨ a71402, a72778 ⟩⟩,\n have h₁ : subclass Animal Entity, \n   apply subclass_animal_entity; assumption,\n have h₂ : ins Animal Class,\n   rw a67173, exact h₁,\n have h₃ : subclass Vertebrate Entity,\n   apply subclass_vertebrate_entity; assumption,\n have h₄ : ins Vertebrate Class, \n   rw a67173, exact h₃,\n have h₅ : subclass Invertebrate Entity,\n   apply subclass_invertebrate_entity; assumption,\n have h₆ : ins Invertebrate Class, \n  rw a67173, exact h₅,\n have h₇ : ins BananaSlug10 Entity, \n  apply ins_banana_entity; assumption,\n exact and.intro h₂ (and.intro h₄ (and.intro h₆ h₇)),\n exact and.intro a71370 (and.intro a72772 h1),\nend\n\n\nexample (X Y C : U) (h : partition3 C X Y) : subclass X C ∧ subclass Y C := sorry\n\nexample (h : ins SetOrClass SetOrClass) : false := sorry\n\n", "meta": {"author": "own-pt", "repo": "common-sense-lean", "sha": "f672210aecb4172f5bae265e43e6867397e13b1c", "save_path": "github-repos/lean/own-pt-common-sense-lean", "path": "github-repos/lean/own-pt-common-sense-lean/common-sense-lean-f672210aecb4172f5bae265e43e6867397e13b1c/misc/bs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5, "lm_q1q2_score": 0.31488731065087294}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport logic.is_empty\nimport tactic.basic\nimport logic.relator\n\n/-!\n# Option of a type\n\nThis file develops the basic theory of option types.\n\nIf `α` is a type, then `option α` can be understood as the type with one more element than `α`.\n`option α` has terms `some a`, where `a : α`, and `none`, which is the added element.\nThis is useful in multiple ways:\n* It is the prototype of addition of terms to a type. See for example `with_bot α` which uses\n  `none` as an element smaller than all others.\n* It can be used to define failsafe partial functions, which return `some the_result_we_expect`\n  if we can find `the_result_we_expect`, and `none` if there is no meaningful result. This forces\n  any subsequent use of the partial function to explicitly deal with the exceptions that make it\n  return `none`.\n* `option` is a monad. We love monads.\n\n`part` is an alternative to `option` that can be seen as the type of `true`/`false` values\nalong with a term `a : α` if the value is `true`.\n\n## Implementation notes\n\n`option` is currently defined in core Lean, but this will change in Lean 4.\n-/\n\nnamespace option\nvariables {α : Type*} {β : Type*} {γ : Type*}\n\nlemma coe_def : (coe : α → option α) = some := rfl\n\nlemma some_ne_none (x : α) : some x ≠ none := λ h, option.no_confusion h\n\nprotected lemma «forall» {p : option α → Prop} : (∀ x, p x) ↔ p none ∧ ∀ x, p (some x) :=\n⟨λ h, ⟨h _, λ x, h _⟩, λ h x, option.cases_on x h.1 h.2⟩\n\nprotected lemma «exists» {p : option α → Prop} : (∃ x, p x) ↔ p none ∨ ∃ x, p (some x) :=\n⟨λ ⟨x, hx⟩, (option.cases_on x or.inl $ λ x hx, or.inr ⟨x, hx⟩) hx,\n  λ h, h.elim (λ h, ⟨_, h⟩) (λ ⟨x, hx⟩, ⟨_, hx⟩)⟩\n\n@[simp] theorem get_mem : ∀ {o : option α} (h : is_some o), option.get h ∈ o\n| (some a) _ := rfl\n\ntheorem get_of_mem {a : α} : ∀ {o : option α} (h : is_some o), a ∈ o → option.get h = a\n| _ _ rfl := rfl\n\n@[simp] lemma not_mem_none (a : α) : a ∉ (none : option α) :=\nλ h, option.no_confusion h\n\n@[simp] lemma some_get : ∀ {x : option α} (h : is_some x), some (option.get h) = x\n| (some x) hx := rfl\n\n@[simp] lemma get_some (x : α) (h : is_some (some x)) : option.get h = x := rfl\n\n@[simp] lemma get_or_else_some (x y : α) : option.get_or_else (some x) y = x := rfl\n\n@[simp] lemma get_or_else_none (x : α) : option.get_or_else none x = x := rfl\n\n@[simp] lemma get_or_else_coe (x y : α) : option.get_or_else ↑x y = x := rfl\n\nlemma get_or_else_of_ne_none {x : option α} (hx : x ≠ none) (y : α) : some (x.get_or_else y) = x :=\nby cases x; [contradiction, rw get_or_else_some]\n\ntheorem mem_unique {o : option α} {a b : α} (ha : a ∈ o) (hb : b ∈ o) : a = b :=\noption.some.inj $ ha.symm.trans hb\n\ntheorem mem.left_unique : relator.left_unique ((∈) : α → option α → Prop) :=\nλ a o b, mem_unique\n\ntheorem some_injective (α : Type*) : function.injective (@some α) :=\nλ _ _, some_inj.mp\n\n/-- `option.map f` is injective if `f` is injective. -/\ntheorem map_injective {f : α → β} (Hf : function.injective f) : function.injective (option.map f)\n| none      none      H := rfl\n| (some a₁) (some a₂) H := by rw Hf (option.some.inj H)\n\n@[ext] theorem ext : ∀ {o₁ o₂ : option α}, (∀ a, a ∈ o₁ ↔ a ∈ o₂) → o₁ = o₂\n| none     none     H := rfl\n| (some a) o        H := ((H _).1 rfl).symm\n| o        (some b) H := (H _).2 rfl\n\ntheorem eq_none_iff_forall_not_mem {o : option α} :\n  o = none ↔ (∀ a, a ∉ o) :=\n⟨λ e a h, by rw e at h; cases h, λ h, ext $ by simpa⟩\n\n@[simp] theorem none_bind {α β} (f : α → option β) : none >>= f = none := rfl\n\n@[simp] theorem some_bind {α β} (a : α) (f : α → option β) : some a >>= f = f a := rfl\n\n@[simp] theorem none_bind' (f : α → option β) : none.bind f = none := rfl\n\n@[simp] theorem some_bind' (a : α) (f : α → option β) : (some a).bind f = f a := rfl\n\n@[simp] theorem bind_some : ∀ x : option α, x >>= some = x :=\n@bind_pure α option _ _\n\n@[simp] theorem bind_eq_some {α β} {x : option α} {f : α → option β} {b : β} :\n  x >>= f = some b ↔ ∃ a, x = some a ∧ f a = some b :=\nby cases x; simp\n\n@[simp] theorem bind_eq_some' {x : option α} {f : α → option β} {b : β} :\n  x.bind f = some b ↔ ∃ a, x = some a ∧ f a = some b :=\nby cases x; simp\n\n@[simp] theorem bind_eq_none' {o : option α} {f : α → option β} :\n  o.bind f = none ↔ (∀ b a, a ∈ o → b ∉ f a) :=\nby simp only [eq_none_iff_forall_not_mem, not_exists, not_and, mem_def, bind_eq_some']\n\n@[simp] theorem bind_eq_none {α β} {o : option α} {f : α → option β} :\n  o >>= f = none ↔ (∀ b a, a ∈ o → b ∉ f a) :=\nbind_eq_none'\n\nlemma bind_comm {α β γ} {f : α → β → option γ} (a : option α) (b : option β) :\n  a.bind (λx, b.bind (f x)) = b.bind (λy, a.bind (λx, f x y)) :=\nby cases a; cases b; refl\n\nlemma bind_assoc (x : option α) (f : α → option β) (g : β → option γ) :\n  (x.bind f).bind g = x.bind (λ y, (f y).bind g) := by cases x; refl\n\nlemma join_eq_some {x : option (option α)} {a : α} : x.join = some a ↔ x = some (some a) := by simp\n\nlemma join_ne_none {x : option (option α)} : x.join ≠ none ↔ ∃ z, x = some (some z) := by simp\n\nlemma join_ne_none' {x : option (option α)} : ¬(x.join = none) ↔ ∃ z, x = some (some z) := by simp\n\nlemma join_eq_none {o : option (option α)} : o.join = none ↔ o = none ∨ o = some none :=\nby rcases o with _|_|_; simp\n\nlemma bind_id_eq_join {x : option (option α)} : x >>= id = x.join := by simp\n\nlemma join_eq_join : mjoin = @join α :=\nfunext (λ x, by rw [mjoin, bind_id_eq_join])\n\nlemma bind_eq_bind {α β : Type*} {f : α → option β} {x : option α} :\n  x >>= f = x.bind f := rfl\n\n@[simp] lemma map_eq_map {α β} {f : α → β} :\n  (<$>) f = option.map f := rfl\n\ntheorem map_none {α β} {f : α → β} : f <$> none = none := rfl\n\ntheorem map_some {α β} {a : α} {f : α → β} : f <$> some a = some (f a) := rfl\n\n@[simp] theorem map_none' {f : α → β} : option.map f none = none := rfl\n\n@[simp] theorem map_some' {a : α} {f : α → β} : option.map f (some a) = some (f a) := rfl\n\ntheorem map_eq_some {α β} {x : option α} {f : α → β} {b : β} :\n  f <$> x = some b ↔ ∃ a, x = some a ∧ f a = b :=\nby cases x; simp\n\n@[simp] theorem map_eq_some' {x : option α} {f : α → β} {b : β} :\n  x.map f = some b ↔ ∃ a, x = some a ∧ f a = b :=\nby cases x; simp\n\nlemma map_eq_none {α β} {x : option α} {f : α → β} :\n  f <$> x = none ↔ x = none :=\nby { cases x; simp only [map_none, map_some, eq_self_iff_true] }\n\n@[simp] lemma map_eq_none' {x : option α} {f : α → β} :\n  x.map f = none ↔ x = none :=\nby { cases x; simp only [map_none', map_some', eq_self_iff_true] }\n\nlemma map_congr {f g : α → β} {x : option α} (h : ∀ a ∈ x, f a = g a) :\n  option.map f x = option.map g x :=\nby { cases x; simp only [map_none', map_some', h, mem_def] }\n\n@[simp] theorem map_id' : option.map (@id α) = id := map_id\n\n@[simp] lemma map_map (h : β → γ) (g : α → β) (x : option α) :\n  option.map h (option.map g x) = option.map (h ∘ g) x :=\nby { cases x; simp only [map_none', map_some'] }\n\nlemma comp_map (h : β → γ) (g : α → β) (x : option α) :\n  option.map (h ∘ g) x = option.map h (option.map g x) := (map_map _ _ _).symm\n\n@[simp] lemma map_comp_map (f : α → β) (g : β → γ) :\n  option.map g ∘ option.map f = option.map (g ∘ f) :=\nby { ext x, rw comp_map }\n\nlemma mem_map_of_mem {α β : Type*} {a : α} {x : option α} (g : α → β) (h : a ∈ x) : g a ∈ x.map g :=\nmem_def.mpr ((mem_def.mp h).symm ▸ map_some')\n\nlemma bind_map_comm {α β} {x : option (option α) } {f : α → β} :\n  x >>= option.map f = x.map (option.map f) >>= id :=\nby { cases x; simp }\n\nlemma join_map_eq_map_join {f : α → β} {x : option (option α)} :\n  (x.map (option.map f)).join = x.join.map f :=\nby { rcases x with _ | _ | x; simp }\n\nlemma join_join {x : option (option (option α))} :\n  x.join.join = (x.map join).join :=\nby { rcases x with _ | _ | _ | x; simp }\n\nlemma mem_of_mem_join {a : α} {x : option (option α)} (h : a ∈ x.join) : some a ∈ x :=\nmem_def.mpr ((mem_def.mp h).symm ▸ join_eq_some.mp h)\n\nsection pmap\n\nvariables {p : α → Prop} (f : Π (a : α), p a → β) (x : option α)\n\n@[simp] lemma pbind_eq_bind (f : α → option β) (x : option α) :\n  x.pbind (λ a _, f a) = x.bind f :=\nby { cases x; simp only [pbind, none_bind', some_bind'] }\n\nlemma map_bind {α β γ} (f : β → γ) (x : option α) (g : α → option β) :\n  option.map f (x >>= g) = (x >>= λ a, option.map f (g a)) :=\nby simp_rw [←map_eq_map, ←bind_pure_comp_eq_map,is_lawful_monad.bind_assoc]\n\nlemma map_bind' (f : β → γ) (x : option α) (g : α → option β) :\n  option.map f (x.bind g) = x.bind (λ a, option.map f (g a)) :=\nby { cases x; simp }\n\nlemma map_pbind (f : β → γ) (x : option α) (g : Π a, a ∈ x → option β) :\n  option.map f (x.pbind g) = (x.pbind (λ a H, option.map f (g a H))) :=\nby { cases x; simp only [pbind, map_none'] }\n\nlemma pbind_map (f : α → β) (x : option α) (g : Π (b : β), b ∈ x.map f → option γ) :\n  pbind (option.map f x) g = x.pbind (λ a h, g (f a) (mem_map_of_mem _ h)) :=\nby { cases x; refl }\n\n@[simp] lemma pmap_none (f : Π (a : α), p a → β) {H} : pmap f (@none α) H = none := rfl\n\n@[simp] lemma pmap_some (f : Π (a : α), p a → β) {x : α} (h : p x) :\n  pmap f (some x) = λ _, some (f x h) := rfl\n\nlemma mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) :\n  f a (h a ha) ∈ pmap f x h :=\nby { rw mem_def at ha ⊢, subst ha, refl }\n\nlemma pmap_map (g : γ → α) (x : option γ) (H) :\n  pmap f (x.map g) H = pmap (λ a h, f (g a) h) x (λ a h, H _ (mem_map_of_mem _ h)) :=\nby { cases x; simp only [map_none', map_some', pmap] }\n\nlemma map_pmap (g : β → γ) (f : Π a, p a → β) (x H) :\n  option.map g (pmap f x H) = pmap (λ a h, g (f a h)) x H :=\nby { cases x; simp only [map_none', map_some', pmap] }\n\n@[simp] lemma pmap_eq_map (p : α → Prop) (f : α → β) (x H) :\n  @pmap _ _ p (λ a _, f a) x H = option.map f x :=\nby { cases x; simp only [map_none', map_some', pmap] }\n\n\n\nlemma bind_pmap {α β γ} {p : α → Prop} (f : Π a, p a → β) (x : option α) (g : β → option γ) (H) :\n  (pmap f x H) >>= g = x.pbind (λ a h, g (f a (H _ h))) :=\nby { cases x; simp only [pmap, none_bind, some_bind, pbind] }\n\nvariables {f x}\n\nlemma pbind_eq_none {f : Π (a : α), a ∈ x → option β}\n  (h' : ∀ a ∈ x, f a H = none → x = none) :\n  x.pbind f = none ↔ x = none :=\nbegin\n  cases x,\n  { simp },\n  { simp only [pbind, iff_false],\n    intro h,\n    cases h' x rfl h }\nend\n\nlemma pbind_eq_some {f : Π (a : α), a ∈ x → option β} {y : β} :\n  x.pbind f = some y ↔ ∃ (z ∈ x), f z H = some y :=\nbegin\n  cases x,\n  { simp },\n  { simp only [pbind],\n    split,\n    { intro h,\n      use x,\n      simpa only [mem_def, exists_prop_of_true] using h },\n    { rintro ⟨z, H, hz⟩,\n      simp only [mem_def] at H,\n      simpa only [H] using hz } }\nend\n\n@[simp] lemma pmap_eq_none_iff {h} :\n  pmap f x h = none ↔ x = none :=\nby { cases x; simp }\n\n@[simp] lemma pmap_eq_some_iff {hf} {y : β} :\n  pmap f x hf = some y ↔ ∃ (a : α) (H : x = some a), f a (hf a H) = y :=\nbegin\n  cases x,\n  { simp only [not_mem_none, exists_false, pmap, not_false_iff, exists_prop_of_false] },\n  { split,\n    { intro h,\n      simp only [pmap] at h,\n      exact ⟨x, rfl, h⟩ },\n    { rintro ⟨a, H, rfl⟩,\n      simp only [mem_def] at H,\n      simp only [H, pmap] } }\nend\n\n@[simp] lemma join_pmap_eq_pmap_join {f : Π a, p a → β} {x : option (option α)} (H) :\n  (pmap (pmap f) x H).join = pmap f x.join (λ a h, H (some a) (mem_of_mem_join h) _ rfl) :=\nby { rcases x with _ | _ | x; simp }\n\nend pmap\n\n@[simp] theorem seq_some {α β} {a : α} {f : α → β} : some f <*> some a = some (f a) := rfl\n\n@[simp] theorem some_orelse' (a : α) (x : option α) : (some a).orelse x = some a := rfl\n\n@[simp] theorem some_orelse (a : α) (x : option α) : (some a <|> x) = some a := rfl\n\n@[simp] theorem none_orelse' (x : option α) : none.orelse x = x :=\nby cases x; refl\n\n@[simp] theorem none_orelse (x : option α) : (none <|> x) = x := none_orelse' x\n\n@[simp] theorem orelse_none' (x : option α) : x.orelse none = x :=\nby cases x; refl\n\n@[simp] theorem orelse_none (x : option α) : (x <|> none) = x := orelse_none' x\n\n@[simp] theorem is_some_none : @is_some α none = ff := rfl\n\n@[simp] theorem is_some_some {a : α} : is_some (some a) = tt := rfl\n\ntheorem is_some_iff_exists {x : option α} : is_some x ↔ ∃ a, x = some a :=\nby cases x; simp [is_some]; exact ⟨_, rfl⟩\n\n@[simp] theorem is_none_none : @is_none α none = tt := rfl\n\n@[simp] theorem is_none_some {a : α} : is_none (some a) = ff := rfl\n\n@[simp] theorem not_is_some {a : option α} : is_some a = ff ↔ a.is_none = tt :=\nby cases a; simp\n\nlemma eq_some_iff_get_eq {o : option α} {a : α} :\n  o = some a ↔ ∃ h : o.is_some, option.get h = a :=\nby cases o; simp\n\nlemma not_is_some_iff_eq_none {o : option α} :  ¬o.is_some ↔ o = none :=\nby cases o; simp\n\nlemma ne_none_iff_is_some {o : option α} : o ≠ none ↔ o.is_some :=\nby cases o; simp\n\nlemma ne_none_iff_exists {o : option α} : o ≠ none ↔ ∃ (x : α), some x = o :=\nby {cases o; simp}\n\nlemma ne_none_iff_exists' {o : option α} : o ≠ none ↔ ∃ (x : α), o = some x :=\nne_none_iff_exists.trans $ exists_congr $ λ _, eq_comm\n\nlemma bex_ne_none {p : option α → Prop} :\n  (∃ x ≠ none, p x) ↔ ∃ x, p (some x) :=\n⟨λ ⟨x, hx, hp⟩, ⟨get $ ne_none_iff_is_some.1 hx, by rwa [some_get]⟩,\n  λ ⟨x, hx⟩, ⟨some x, some_ne_none x, hx⟩⟩\n\nlemma ball_ne_none {p : option α → Prop} :\n  (∀ x ≠ none, p x) ↔ ∀ x, p (some x) :=\n⟨λ h x, h (some x) (some_ne_none x),\n  λ h x hx, by simpa only [some_get] using h (get $ ne_none_iff_is_some.1 hx)⟩\n\ntheorem iget_mem [inhabited α] : ∀ {o : option α}, is_some o → o.iget ∈ o\n| (some a) _ := rfl\n\ntheorem iget_of_mem [inhabited α] {a : α} : ∀ {o : option α}, a ∈ o → o.iget = a\n| _ rfl := rfl\n\n@[simp] theorem guard_eq_some {p : α → Prop} [decidable_pred p] {a b : α} :\n  guard p a = some b ↔ a = b ∧ p a :=\nby by_cases p a; simp [option.guard, h]; intro; contradiction\n\n@[simp] theorem guard_eq_some' {p : Prop} [decidable p] :\n  ∀ u, _root_.guard p = some u ↔ p\n| () := by by_cases p; simp [guard, h, pure]; intro; contradiction\n\ntheorem lift_or_get_choice {f : α → α → α} (h : ∀ a b, f a b = a ∨ f a b = b) :\n  ∀ o₁ o₂, lift_or_get f o₁ o₂ = o₁ ∨ lift_or_get f o₁ o₂ = o₂\n| none     none     := or.inl rfl\n| (some a) none     := or.inl rfl\n| none     (some b) := or.inr rfl\n| (some a) (some b) := by simpa [lift_or_get] using h a b\n\n@[simp] lemma lift_or_get_none_left {f} {b : option α} : lift_or_get f none b = b :=\nby cases b; refl\n\n@[simp] lemma lift_or_get_none_right {f} {a : option α} : lift_or_get f a none = a :=\nby cases a; refl\n\n@[simp] lemma lift_or_get_some_some {f} {a b : α} :\n  lift_or_get f (some a) (some b) = f a b := rfl\n\n/-- Given an element of `a : option α`, a default element `b : β` and a function `α → β`, apply this\nfunction to `a` if it comes from `α`, and return `b` otherwise. -/\ndef cases_on' : option α → β → (α → β) → β\n| none     n s := n\n| (some a) n s := s a\n\n@[simp] lemma cases_on'_none (x : β) (f : α → β) : cases_on' none x f = x := rfl\n\n@[simp] lemma cases_on'_some (x : β) (f : α → β) (a : α) : cases_on' (some a) x f = f a := rfl\n\n@[simp] lemma cases_on'_coe (x : β) (f : α → β) (a : α) : cases_on' (a : option α) x f = f a := rfl\n\n@[simp] lemma cases_on'_none_coe (f : option α → β) (o : option α) :\n  cases_on' o (f none) (f ∘ coe) = f o :=\nby cases o; refl\n\n@[simp] lemma get_or_else_map (f : α → β) (x : α) (o : option α) :\n  get_or_else (o.map f) (f x) = f (get_or_else o x) :=\nby cases o; refl\n\nsection\nopen_locale classical\n\n/-- An arbitrary `some a` with `a : α` if `α` is nonempty, and otherwise `none`. -/\nnoncomputable def choice (α : Type*) : option α :=\nif h : nonempty α then\n  some h.some\nelse\n  none\n\nlemma choice_eq {α : Type*} [subsingleton α] (a : α) : choice α = some a :=\nbegin\n  dsimp [choice],\n  rw dif_pos (⟨a⟩ : nonempty α),\n  congr,\nend\n\nlemma choice_eq_none (α : Type*) [is_empty α] : choice α = none :=\ndif_neg (not_nonempty_iff_imp_false.mpr is_empty_elim)\n\nlemma choice_is_some_iff_nonempty {α : Type*} : (choice α).is_some ↔ nonempty α :=\nbegin\n  fsplit,\n  { intro h, exact ⟨option.get h⟩, },\n  { intro h,\n    dsimp only [choice],\n    rw dif_pos h,\n    exact is_some_some },\nend\n\nend\n\n@[simp] lemma to_list_some (a : α) : (a : option α).to_list = [a] :=\nrfl\n\n@[simp] lemma to_list_none (α : Type*) : (none : option α).to_list = [] :=\nrfl\n\nend option\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/option/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6150878555160666, "lm_q1q2_score": 0.3147506690194478}}
{"text": "import category_theory.limits.shapes.terminal\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses v u\n\nvariables {J : Type v} [small_category J] {C : Type u} [category.{v} C] (F : J ⥤ C)\n\nnoncomputable theory\n\n@[simps]\ndef is_initial.colimit_cocone {j : J} (hj : is_initial j)\n  [has_colimit F] [∀ (a b : J) (f : a ⟶ b), is_iso (F.map f)] :\n  cocone F :=\n{ X := F.obj j,\n  ι :=\n  { app := λ i, inv (F.map $ hj.to _),\n    naturality' := begin\n      intros a b f,\n      dsimp,\n      simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.comp_id],\n      simp_rw ← F.map_comp,\n      congr' 1,\n      apply hj.hom_ext,\n    end } }\n\ndef is_initial.is_colimit_colimit_cocone {j : J} (hj : is_initial j)\n  [has_colimit F] [∀ (a b : J) (f : a ⟶ b), is_iso (F.map f)] :\n  is_colimit (hj.colimit_cocone F) :=\n{ desc := λ S, S.ι.app _ }\n\nlemma is_initial.is_iso_ι {j : J} (hj : is_initial j)\n  [has_colimit F] [∀ (a b : J) (f : a ⟶ b), is_iso (F.map f)] :\n  is_iso (colimit.ι F j) :=\nbegin\n  let e := (colimit.is_colimit F).cocone_point_unique_up_to_iso\n    (hj.is_colimit_colimit_cocone F),\n  change is_iso e.inv,\n  apply_instance\nend\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/is_iso_iota.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3146629536707037}}
{"text": "/-\nCopyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.real.ennreal\nimport Mathlib.data.finset.intervals\nimport Mathlib.topology.uniform_space.uniform_embedding\nimport Mathlib.topology.uniform_space.pi\nimport Mathlib.topology.uniform_space.uniform_convergence\nimport Mathlib.PostPort\n\nuniverses u v u_1 l u_2 \n\nnamespace Mathlib\n\n/-!\n# Extended metric spaces\n\nThis file is devoted to the definition and study of `emetric_spaces`, i.e., metric\nspaces in which the distance is allowed to take the value ∞. This extended distance is\ncalled `edist`, and takes values in `ennreal`.\n\nMany definitions and theorems expected on emetric spaces are already introduced on uniform spaces and\ntopological spaces. For example:\n  open and closed sets, compactness, completeness, continuity and uniform continuity\n\nThe class `emetric_space` therefore extends `uniform_space` (and `topological_space`).\n-/\n\n/-- Characterizing uniformities associated to a (generalized) distance function `D`\nin terms of the elements of the uniformity. -/\ntheorem uniformity_dist_of_mem_uniformity {α : Type u} {β : Type v} [linear_order β]\n    {U : filter (α × α)} (z : β) (D : α → α → β)\n    (H :\n      ∀ (s : set (α × α)), s ∈ U ↔ ∃ (ε : β), ∃ (H : ε > z), ∀ {a b : α}, D a b < ε → (a, b) ∈ s) :\n    U =\n        infi\n          fun (ε : β) =>\n            infi\n              fun (H : ε > z) =>\n                filter.principal (set_of fun (p : α × α) => D (prod.fst p) (prod.snd p) < ε) :=\n  sorry\n\nclass has_edist (α : Type u_1) where\n  edist : α → α → ennreal\n\n/-- Creating a uniform space from an extended distance. -/\ndef uniform_space_of_edist {α : Type u} (edist : α → α → ennreal)\n    (edist_self : ∀ (x : α), edist x x = 0) (edist_comm : ∀ (x y : α), edist x y = edist y x)\n    (edist_triangle : ∀ (x y z : α), edist x z ≤ edist x y + edist y z) : uniform_space α :=\n  uniform_space.of_core\n    (uniform_space.core.mk\n      (infi\n        fun (ε : ennreal) =>\n          infi\n            fun (H : ε > 0) =>\n              filter.principal (set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε))\n      sorry sorry sorry)\n\n-- the uniform structure is embedded in the emetric space structure\n\n-- to avoid instance diamond issues. See Note [forgetful inheritance].\n\n/-- Extended metric spaces, with an extended distance `edist` possibly taking the\nvalue ∞\n\nEach emetric space induces a canonical `uniform_space` and hence a canonical `topological_space`.\nThis is enforced in the type class definition, by extending the `uniform_space` structure. When\ninstantiating an `emetric_space` structure, the uniformity fields are not necessary, they will be\nfilled in by default. There is a default value for the uniformity, that can be substituted\nin cases of interest, for instance when instantiating an `emetric_space` structure\non a product.\n\nContinuity of `edist` is proved in `topology.instances.ennreal`\n-/\nclass emetric_space (α : Type u)\n    extends has_edist α,\n      emetric_space.to_uniform_space._default #2 #1 #0 α _to_has_edist =\n        id (uniform_space_of_edist edist #0 α _to_has_edist),\n      uniform_space #2, uniform_space α\n    where\n  edist_self : ∀ (x : α), edist x x = 0\n  eq_of_edist_eq_zero : ∀ {x y : α}, edist x y = 0 → x = y\n  edist_comm : ∀ (x y : α), edist x y = edist y x\n  edist_triangle : ∀ (x y z : α), edist x z ≤ edist x y + edist y z\n  to_uniform_space : uniform_space α\n  uniformity_edist :\n    autoParam\n      (uniformity α =\n        infi\n          fun (ε : ennreal) =>\n            infi\n              fun (H : ε > 0) =>\n                filter.principal (set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n\n/- emetric spaces are less common than metric spaces. Therefore, we work in a dedicated\nnamespace, while notions associated to metric spaces are mostly in the root namespace. -/\n\nprotected instance emetric_space.to_uniform_space' {α : Type u} [emetric_space α] :\n    uniform_space α :=\n  emetric_space.to_uniform_space\n\n/-- Characterize the equality of points by the vanishing of their extended distance -/\n@[simp] theorem edist_eq_zero {α : Type u} [emetric_space α] {x : α} {y : α} :\n    edist x y = 0 ↔ x = y :=\n  { mp := eq_of_edist_eq_zero, mpr := fun (this : x = y) => this ▸ edist_self x }\n\n@[simp] theorem zero_eq_edist {α : Type u} [emetric_space α] {x : α} {y : α} :\n    0 = edist x y ↔ x = y :=\n  { mp := fun (h : 0 = edist x y) => eq_of_edist_eq_zero (Eq.symm h),\n    mpr := fun (this : x = y) => this ▸ Eq.symm (edist_self x) }\n\ntheorem edist_le_zero {α : Type u} [emetric_space α] {x : α} {y : α} : edist x y ≤ 0 ↔ x = y :=\n  iff.trans nonpos_iff_eq_zero edist_eq_zero\n\n/-- Triangle inequality for the extended distance -/\ntheorem edist_triangle_left {α : Type u} [emetric_space α] (x : α) (y : α) (z : α) :\n    edist x y ≤ edist z x + edist z y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (edist x y ≤ edist z x + edist z y)) (edist_comm z x)))\n    (edist_triangle x z y)\n\ntheorem edist_triangle_right {α : Type u} [emetric_space α] (x : α) (y : α) (z : α) :\n    edist x y ≤ edist x z + edist y z :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (edist x y ≤ edist x z + edist y z)) (edist_comm y z)))\n    (edist_triangle x z y)\n\ntheorem edist_triangle4 {α : Type u} [emetric_space α] (x : α) (y : α) (z : α) (t : α) :\n    edist x t ≤ edist x y + edist y z + edist z t :=\n  le_trans (edist_triangle x z t) (add_le_add_right (edist_triangle x y z) (edist z t))\n\n/-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/\ntheorem edist_le_Ico_sum_edist {α : Type u} [emetric_space α] (f : ℕ → α) {m : ℕ} {n : ℕ}\n    (h : m ≤ n) :\n    edist (f m) (f n) ≤ finset.sum (finset.Ico m n) fun (i : ℕ) => edist (f i) (f (i + 1)) :=\n  sorry\n\n/-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/\ntheorem edist_le_range_sum_edist {α : Type u} [emetric_space α] (f : ℕ → α) (n : ℕ) :\n    edist (f 0) (f n) ≤ finset.sum (finset.range n) fun (i : ℕ) => edist (f i) (f (i + 1)) :=\n  finset.Ico.zero_bot n ▸ edist_le_Ico_sum_edist f (nat.zero_le n)\n\n/-- A version of `edist_le_Ico_sum_edist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem edist_le_Ico_sum_of_edist_le {α : Type u} [emetric_space α] {f : ℕ → α} {m : ℕ} {n : ℕ}\n    (hmn : m ≤ n) {d : ℕ → ennreal}\n    (hd : ∀ {k : ℕ}, m ≤ k → k < n → edist (f k) (f (k + 1)) ≤ d k) :\n    edist (f m) (f n) ≤ finset.sum (finset.Ico m n) fun (i : ℕ) => d i :=\n  sorry\n\n/-- A version of `edist_le_range_sum_edist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem edist_le_range_sum_of_edist_le {α : Type u} [emetric_space α] {f : ℕ → α} (n : ℕ)\n    {d : ℕ → ennreal} (hd : ∀ {k : ℕ}, k < n → edist (f k) (f (k + 1)) ≤ d k) :\n    edist (f 0) (f n) ≤ finset.sum (finset.range n) fun (i : ℕ) => d i :=\n  finset.Ico.zero_bot n ▸\n    edist_le_Ico_sum_of_edist_le (zero_le n) fun (_x : ℕ) (_x_1 : 0 ≤ _x) => hd\n\n/-- Two points coincide if their distance is `< ε` for all positive ε -/\ntheorem eq_of_forall_edist_le {α : Type u} [emetric_space α] {x : α} {y : α}\n    (h : ∀ (ε : ennreal), ε > 0 → edist x y ≤ ε) : x = y :=\n  eq_of_edist_eq_zero (eq_of_le_of_forall_le_of_dense bot_le h)\n\n/-- Reformulation of the uniform structure in terms of the extended distance -/\ntheorem uniformity_edist {α : Type u} [emetric_space α] :\n    uniformity α =\n        infi\n          fun (ε : ennreal) =>\n            infi\n              fun (H : ε > 0) =>\n                filter.principal (set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε) :=\n  emetric_space.uniformity_edist\n\ntheorem uniformity_basis_edist {α : Type u} [emetric_space α] :\n    filter.has_basis (uniformity α) (fun (ε : ennreal) => 0 < ε)\n        fun (ε : ennreal) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε :=\n  sorry\n\n/-- Characterization of the elements of the uniformity in terms of the extended distance -/\ntheorem mem_uniformity_edist {α : Type u} [emetric_space α] {s : set (α × α)} :\n    s ∈ uniformity α ↔ ∃ (ε : ennreal), ∃ (H : ε > 0), ∀ {a b : α}, edist a b < ε → (a, b) ∈ s :=\n  filter.has_basis.mem_uniformity_iff uniformity_basis_edist\n\n/-- Given `f : β → ennreal`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.\n\nFor specific bases see `uniformity_basis_edist`, `uniformity_basis_edist'`,\n`uniformity_basis_edist_nnreal`, and `uniformity_basis_edist_inv_nat`. -/\nprotected theorem emetric.mk_uniformity_basis {α : Type u} [emetric_space α] {β : Type u_1}\n    {p : β → Prop} {f : β → ennreal} (hf₀ : ∀ (x : β), p x → 0 < f x)\n    (hf : ∀ (ε : ennreal), 0 < ε → ∃ (x : β), ∃ (hx : p x), f x ≤ ε) :\n    filter.has_basis (uniformity α) p\n        fun (x : β) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < f x :=\n  sorry\n\n/-- Given `f : β → ennreal`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then closed `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.\n\nFor specific bases see `uniformity_basis_edist_le` and `uniformity_basis_edist_le'`. -/\nprotected theorem emetric.mk_uniformity_basis_le {α : Type u} [emetric_space α] {β : Type u_1}\n    {p : β → Prop} {f : β → ennreal} (hf₀ : ∀ (x : β), p x → 0 < f x)\n    (hf : ∀ (ε : ennreal), 0 < ε → ∃ (x : β), ∃ (hx : p x), f x ≤ ε) :\n    filter.has_basis (uniformity α) p\n        fun (x : β) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) ≤ f x :=\n  sorry\n\ntheorem uniformity_basis_edist_le {α : Type u} [emetric_space α] :\n    filter.has_basis (uniformity α) (fun (ε : ennreal) => 0 < ε)\n        fun (ε : ennreal) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) ≤ ε :=\n  emetric.mk_uniformity_basis_le (fun (_x : ennreal) => id)\n    fun (ε : ennreal) (ε₀ : 0 < ε) => Exists.intro ε (Exists.intro ε₀ (le_refl ε))\n\ntheorem uniformity_basis_edist' {α : Type u} [emetric_space α] (ε' : ennreal) (hε' : 0 < ε') :\n    filter.has_basis (uniformity α) (fun (ε : ennreal) => ε ∈ set.Ioo 0 ε')\n        fun (ε : ennreal) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε :=\n  sorry\n\ntheorem uniformity_basis_edist_le' {α : Type u} [emetric_space α] (ε' : ennreal) (hε' : 0 < ε') :\n    filter.has_basis (uniformity α) (fun (ε : ennreal) => ε ∈ set.Ioo 0 ε')\n        fun (ε : ennreal) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) ≤ ε :=\n  sorry\n\ntheorem uniformity_basis_edist_nnreal {α : Type u} [emetric_space α] :\n    filter.has_basis (uniformity α) (fun (ε : nnreal) => 0 < ε)\n        fun (ε : nnreal) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ↑ε :=\n  sorry\n\ntheorem uniformity_basis_edist_inv_nat {α : Type u} [emetric_space α] :\n    filter.has_basis (uniformity α) (fun (_x : ℕ) => True)\n        fun (n : ℕ) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < (↑n⁻¹) :=\n  sorry\n\n/-- Fixed size neighborhoods of the diagonal belong to the uniform structure -/\ntheorem edist_mem_uniformity {α : Type u} [emetric_space α] {ε : ennreal} (ε0 : 0 < ε) :\n    (set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε) ∈ uniformity α :=\n  iff.mpr mem_uniformity_edist (Exists.intro ε (Exists.intro ε0 fun (a b : α) => id))\n\nnamespace emetric\n\n\ntheorem uniformity_has_countable_basis {α : Type u} [emetric_space α] :\n    filter.is_countably_generated (uniformity α) :=\n  filter.is_countably_generated_of_seq\n    (Exists.intro\n      (fun (i : ℕ) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < (↑i⁻¹))\n      (filter.has_basis.eq_infi uniformity_basis_edist_inv_nat))\n\n/-- ε-δ characterization of uniform continuity on a set for emetric spaces -/\ntheorem uniform_continuous_on_iff {α : Type u} {β : Type v} [emetric_space α] [emetric_space β]\n    {f : α → β} {s : set α} :\n    uniform_continuous_on f s ↔\n        ∀ (ε : ennreal) (H : ε > 0),\n          ∃ (δ : ennreal),\n            ∃ (H : δ > 0), ∀ {a b : α}, a ∈ s → b ∈ s → edist a b < δ → edist (f a) (f b) < ε :=\n  filter.has_basis.uniform_continuous_on_iff uniformity_basis_edist uniformity_basis_edist\n\n/-- ε-δ characterization of uniform continuity on emetric spaces -/\ntheorem uniform_continuous_iff {α : Type u} {β : Type v} [emetric_space α] [emetric_space β]\n    {f : α → β} :\n    uniform_continuous f ↔\n        ∀ (ε : ennreal) (H : ε > 0),\n          ∃ (δ : ennreal), ∃ (H : δ > 0), ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε :=\n  filter.has_basis.uniform_continuous_iff uniformity_basis_edist uniformity_basis_edist\n\n/-- ε-δ characterization of uniform embeddings on emetric spaces -/\ntheorem uniform_embedding_iff {α : Type u} {β : Type v} [emetric_space α] [emetric_space β]\n    {f : α → β} :\n    uniform_embedding f ↔\n        function.injective f ∧\n          uniform_continuous f ∧\n            ∀ (δ : ennreal) (H : δ > 0),\n              ∃ (ε : ennreal), ∃ (H : ε > 0), ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ :=\n  sorry\n\n/-- A map between emetric spaces is a uniform embedding if and only if the edistance between `f x`\nand `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/\ntheorem uniform_embedding_iff' {α : Type u} {β : Type v} [emetric_space α] [emetric_space β]\n    {f : α → β} :\n    uniform_embedding f ↔\n        (∀ (ε : ennreal) (H : ε > 0),\n            ∃ (δ : ennreal), ∃ (H : δ > 0), ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε) ∧\n          ∀ (δ : ennreal) (H : δ > 0),\n            ∃ (ε : ennreal), ∃ (H : ε > 0), ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ :=\n  sorry\n\n/-- ε-δ characterization of Cauchy sequences on emetric spaces -/\nprotected theorem cauchy_iff {α : Type u} [emetric_space α] {f : filter α} :\n    cauchy f ↔\n        f ≠ ⊥ ∧\n          ∀ (ε : ennreal) (H : ε > 0),\n            ∃ (t : set α), ∃ (H : t ∈ f), ∀ (x y : α), x ∈ t → y ∈ t → edist x y < ε :=\n  filter.has_basis.cauchy_iff uniformity_basis_edist\n\n/-- A very useful criterion to show that a space is complete is to show that all sequences\nwhich satisfy a bound of the form `edist (u n) (u m) < B N` for all `n m ≥ N` are\nconverging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to\n`0`, which makes it possible to use arguments of converging series, while this is impossible\nto do in general for arbitrary Cauchy sequences. -/\ntheorem complete_of_convergent_controlled_sequences {α : Type u} [emetric_space α] (B : ℕ → ennreal)\n    (hB : ∀ (n : ℕ), 0 < B n)\n    (H :\n      ∀ (u : ℕ → α),\n        (∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (u n) (u m) < B N) →\n          ∃ (x : α), filter.tendsto u filter.at_top (nhds x)) :\n    complete_space α :=\n  uniform_space.complete_of_convergent_controlled_sequences uniformity_has_countable_basis\n    (fun (n : ℕ) => set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < B n)\n    (fun (n : ℕ) => edist_mem_uniformity (hB n)) H\n\n/-- A sequentially complete emetric space is complete. -/\ntheorem complete_of_cauchy_seq_tendsto {α : Type u} [emetric_space α] :\n    (∀ (u : ℕ → α), cauchy_seq u → ∃ (a : α), filter.tendsto u filter.at_top (nhds a)) →\n        complete_space α :=\n  uniform_space.complete_of_cauchy_seq_tendsto uniformity_has_countable_basis\n\n/-- Expressing locally uniform convergence on a set using `edist`. -/\ntheorem tendsto_locally_uniformly_on_iff {α : Type u} {β : Type v} [emetric_space α] {ι : Type u_1}\n    [topological_space β] {F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :\n    tendsto_locally_uniformly_on F f p s ↔\n        ∀ (ε : ennreal) (H : ε > 0) (x : β) (H : x ∈ s),\n          ∃ (t : set β),\n            ∃ (H : t ∈ nhds_within x s),\n              filter.eventually (fun (n : ι) => ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε) p :=\n  sorry\n\n/-- Expressing uniform convergence on a set using `edist`. -/\ntheorem tendsto_uniformly_on_iff {α : Type u} {β : Type v} [emetric_space α] {ι : Type u_1}\n    {F : ι → β → α} {f : β → α} {p : filter ι} {s : set β} :\n    tendsto_uniformly_on F f p s ↔\n        ∀ (ε : ennreal),\n          ε > 0 → filter.eventually (fun (n : ι) => ∀ (x : β), x ∈ s → edist (f x) (F n x) < ε) p :=\n  sorry\n\n/-- Expressing locally uniform convergence using `edist`. -/\ntheorem tendsto_locally_uniformly_iff {α : Type u} {β : Type v} [emetric_space α] {ι : Type u_1}\n    [topological_space β] {F : ι → β → α} {f : β → α} {p : filter ι} :\n    tendsto_locally_uniformly F f p ↔\n        ∀ (ε : ennreal) (H : ε > 0) (x : β),\n          ∃ (t : set β),\n            ∃ (H : t ∈ nhds x),\n              filter.eventually (fun (n : ι) => ∀ (y : β), y ∈ t → edist (f y) (F n y) < ε) p :=\n  sorry\n\n/-- Expressing uniform convergence using `edist`. -/\ntheorem tendsto_uniformly_iff {α : Type u} {β : Type v} [emetric_space α] {ι : Type u_1}\n    {F : ι → β → α} {f : β → α} {p : filter ι} :\n    tendsto_uniformly F f p ↔\n        ∀ (ε : ennreal),\n          ε > 0 → filter.eventually (fun (n : ι) => ∀ (x : β), edist (f x) (F n x) < ε) p :=\n  sorry\n\nend emetric\n\n\n/-- An emetric space is separated -/\nprotected instance to_separated {α : Type u} [emetric_space α] : separated_space α :=\n  iff.mpr separated_def\n    fun (x y : α) (h : ∀ (r : set (α × α)), r ∈ uniformity α → (x, y) ∈ r) =>\n      eq_of_forall_edist_le\n        fun (ε : ennreal) (ε0 : ε > 0) =>\n          le_of_lt\n            (h (set_of fun (p : α × α) => edist (prod.fst p) (prod.snd p) < ε)\n              (edist_mem_uniformity ε0))\n\n/-- Auxiliary function to replace the uniformity on an emetric space with\na uniformity which is equal to the original one, but maybe not defeq.\nThis is useful if one wants to construct an emetric space with a\nspecified uniformity. See Note [forgetful inheritance] explaining why having definitionally\nthe right uniformity is often important.\n-/\ndef emetric_space.replace_uniformity {α : Type u_1} [U : uniform_space α] (m : emetric_space α)\n    (H : uniformity α = uniformity α) : emetric_space α :=\n  emetric_space.mk edist_self eq_of_edist_eq_zero edist_comm edist_triangle U\n\n/-- The extended metric induced by an injective function taking values in an emetric space. -/\ndef emetric_space.induced {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.injective f)\n    (m : emetric_space β) : emetric_space α :=\n  emetric_space.mk sorry sorry sorry sorry (uniform_space.comap f emetric_space.to_uniform_space)\n\n/-- Emetric space instance on subsets of emetric spaces -/\nprotected instance subtype.emetric_space {α : Type u_1} {p : α → Prop} [t : emetric_space α] :\n    emetric_space (Subtype p) :=\n  emetric_space.induced coe sorry t\n\n/-- The extended distance on a subset of an emetric space is the restriction of\nthe original distance, by definition -/\ntheorem subtype.edist_eq {α : Type u} [emetric_space α] {p : α → Prop} (x : Subtype p)\n    (y : Subtype p) : edist x y = edist ↑x ↑y :=\n  rfl\n\n/-- The product of two emetric spaces, with the max distance, is an extended\nmetric spaces. We make sure that the uniform structure thus constructed is the one\ncorresponding to the product of uniform spaces, to avoid diamond problems. -/\nprotected instance prod.emetric_space_max {α : Type u} {β : Type v} [emetric_space α]\n    [emetric_space β] : emetric_space (α × β) :=\n  emetric_space.mk sorry sorry sorry sorry prod.uniform_space\n\ntheorem prod.edist_eq {α : Type u} {β : Type v} [emetric_space α] [emetric_space β] (x : α × β)\n    (y : α × β) :\n    edist x y = max (edist (prod.fst x) (prod.fst y)) (edist (prod.snd x) (prod.snd y)) :=\n  rfl\n\n/-- The product of a finite number of emetric spaces, with the max distance, is still\nan emetric space.\nThis construction would also work for infinite products, but it would not give rise\nto the product topology. Hence, we only formalize it in the good situation of finitely many\nspaces. -/\nprotected instance emetric_space_pi {β : Type v} {π : β → Type u_1} [fintype β]\n    [(b : β) → emetric_space (π b)] : emetric_space ((b : β) → π b) :=\n  emetric_space.mk sorry sorry sorry sorry (Pi.uniform_space fun (b : β) => π b)\n\ntheorem edist_pi_def {β : Type v} {π : β → Type u_1} [fintype β] [(b : β) → emetric_space (π b)]\n    (f : (b : β) → π b) (g : (b : β) → π b) :\n    edist f g = finset.sup finset.univ fun (b : β) => edist (f b) (g b) :=\n  rfl\n\n@[simp] theorem edist_pi_const {α : Type u} {β : Type v} [emetric_space α] [fintype β] [Nonempty β]\n    (a : α) (b : α) : (edist (fun (x : β) => a) fun (_x : β) => b) = edist a b :=\n  finset.sup_const finset.univ_nonempty (edist a b)\n\nnamespace emetric\n\n\n/-- `emetric.ball x ε` is the set of all points `y` with `edist y x < ε` -/\ndef ball {α : Type u} [emetric_space α] (x : α) (ε : ennreal) : set α :=\n  set_of fun (y : α) => edist y x < ε\n\n@[simp] theorem mem_ball {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal} :\n    y ∈ ball x ε ↔ edist y x < ε :=\n  iff.rfl\n\ntheorem mem_ball' {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal} :\n    y ∈ ball x ε ↔ edist x y < ε :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (y ∈ ball x ε ↔ edist x y < ε)) (edist_comm x y)))\n    (iff.refl (y ∈ ball x ε))\n\n/-- `emetric.closed_ball x ε` is the set of all points `y` with `edist y x ≤ ε` -/\ndef closed_ball {α : Type u} [emetric_space α] (x : α) (ε : ennreal) : set α :=\n  set_of fun (y : α) => edist y x ≤ ε\n\n@[simp] theorem mem_closed_ball {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal} :\n    y ∈ closed_ball x ε ↔ edist y x ≤ ε :=\n  iff.rfl\n\ntheorem ball_subset_closed_ball {α : Type u} [emetric_space α] {x : α} {ε : ennreal} :\n    ball x ε ⊆ closed_ball x ε :=\n  fun (y : α) =>\n    eq.mpr (id (imp_congr_eq (propext mem_ball) (propext mem_closed_ball)))\n      fun (h : edist y x < ε) => le_of_lt h\n\ntheorem pos_of_mem_ball {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal}\n    (hy : y ∈ ball x ε) : 0 < ε :=\n  lt_of_le_of_lt (zero_le (edist y x)) hy\n\ntheorem mem_ball_self {α : Type u} [emetric_space α] {x : α} {ε : ennreal} (h : 0 < ε) :\n    x ∈ ball x ε :=\n  (fun (this : edist x x < ε) => this)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (edist x x < ε)) (edist_self x))) h)\n\ntheorem mem_closed_ball_self {α : Type u} [emetric_space α] {x : α} {ε : ennreal} :\n    x ∈ closed_ball x ε :=\n  (fun (this : edist x x ≤ ε) => this)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (edist x x ≤ ε)) (edist_self x))) bot_le)\n\ntheorem mem_ball_comm {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal} :\n    x ∈ ball y ε ↔ y ∈ ball x ε :=\n  sorry\n\ntheorem ball_subset_ball {α : Type u} [emetric_space α] {x : α} {ε₁ : ennreal} {ε₂ : ennreal}\n    (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ :=\n  fun (y : α) (yx : edist y x < ε₁) => lt_of_lt_of_le yx h\n\ntheorem closed_ball_subset_closed_ball {α : Type u} [emetric_space α] {x : α} {ε₁ : ennreal}\n    {ε₂ : ennreal} (h : ε₁ ≤ ε₂) : closed_ball x ε₁ ⊆ closed_ball x ε₂ :=\n  fun (y : α) (yx : edist y x ≤ ε₁) => le_trans yx h\n\ntheorem ball_disjoint {α : Type u} [emetric_space α] {x : α} {y : α} {ε₁ : ennreal} {ε₂ : ennreal}\n    (h : ε₁ + ε₂ ≤ edist x y) : ball x ε₁ ∩ ball y ε₂ = ∅ :=\n  sorry\n\ntheorem ball_subset {α : Type u} [emetric_space α] {x : α} {y : α} {ε₁ : ennreal} {ε₂ : ennreal}\n    (h : edist x y + ε₁ ≤ ε₂) (h' : edist x y < ⊤) : ball x ε₁ ⊆ ball y ε₂ :=\n  sorry\n\ntheorem exists_ball_subset_ball {α : Type u} [emetric_space α] {x : α} {y : α} {ε : ennreal}\n    (h : y ∈ ball x ε) : ∃ (ε' : ennreal), ∃ (H : ε' > 0), ball y ε' ⊆ ball x ε :=\n  sorry\n\ntheorem ball_eq_empty_iff {α : Type u} [emetric_space α] {x : α} {ε : ennreal} :\n    ball x ε = ∅ ↔ ε = 0 :=\n  sorry\n\n/-- Relation “two points are at a finite edistance” is an equivalence relation. -/\ndef edist_lt_top_setoid {α : Type u} [emetric_space α] : setoid α :=\n  setoid.mk (fun (x y : α) => edist x y < ⊤) sorry\n\n@[simp] theorem ball_zero {α : Type u} [emetric_space α] {x : α} : ball x 0 = ∅ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (ball x 0 = ∅)) (propext ball_eq_empty_iff))) (Eq.refl 0)\n\ntheorem nhds_basis_eball {α : Type u} [emetric_space α] {x : α} :\n    filter.has_basis (nhds x) (fun (ε : ennreal) => 0 < ε) (ball x) :=\n  nhds_basis_uniformity uniformity_basis_edist\n\ntheorem nhds_basis_closed_eball {α : Type u} [emetric_space α] {x : α} :\n    filter.has_basis (nhds x) (fun (ε : ennreal) => 0 < ε) (closed_ball x) :=\n  nhds_basis_uniformity uniformity_basis_edist_le\n\ntheorem nhds_eq {α : Type u} [emetric_space α] {x : α} :\n    nhds x = infi fun (ε : ennreal) => infi fun (H : ε > 0) => filter.principal (ball x ε) :=\n  filter.has_basis.eq_binfi nhds_basis_eball\n\ntheorem mem_nhds_iff {α : Type u} [emetric_space α] {x : α} {s : set α} :\n    s ∈ nhds x ↔ ∃ (ε : ennreal), ∃ (H : ε > 0), ball x ε ⊆ s :=\n  filter.has_basis.mem_iff nhds_basis_eball\n\ntheorem is_open_iff {α : Type u} [emetric_space α] {s : set α} :\n    is_open s ↔ ∀ (x : α) (H : x ∈ s), ∃ (ε : ennreal), ∃ (H : ε > 0), ball x ε ⊆ s :=\n  sorry\n\ntheorem is_open_ball {α : Type u} [emetric_space α] {x : α} {ε : ennreal} : is_open (ball x ε) :=\n  iff.mpr is_open_iff fun (y : α) => exists_ball_subset_ball\n\ntheorem is_closed_ball_top {α : Type u} [emetric_space α] {x : α} : is_closed (ball x ⊤) := sorry\n\ntheorem ball_mem_nhds {α : Type u} [emetric_space α] (x : α) {ε : ennreal} (ε0 : 0 < ε) :\n    ball x ε ∈ nhds x :=\n  mem_nhds_sets is_open_ball (mem_ball_self ε0)\n\ntheorem ball_prod_same {α : Type u} {β : Type v} [emetric_space α] [emetric_space β] (x : α) (y : β)\n    (r : ennreal) : set.prod (ball x r) (ball y r) = ball (x, y) r :=\n  set.ext fun (z : α × β) => iff.symm max_lt_iff\n\ntheorem closed_ball_prod_same {α : Type u} {β : Type v} [emetric_space α] [emetric_space β] (x : α)\n    (y : β) (r : ennreal) : set.prod (closed_ball x r) (closed_ball y r) = closed_ball (x, y) r :=\n  set.ext fun (z : α × β) => iff.symm max_le_iff\n\n/-- ε-characterization of the closure in emetric spaces -/\ntheorem mem_closure_iff {α : Type u} [emetric_space α] {x : α} {s : set α} :\n    x ∈ closure s ↔ ∀ (ε : ennreal) (H : ε > 0), ∃ (y : α), ∃ (H : y ∈ s), edist x y < ε :=\n  sorry\n\ntheorem tendsto_nhds {α : Type u} {β : Type v} [emetric_space α] {f : filter β} {u : β → α}\n    {a : α} :\n    filter.tendsto u f (nhds a) ↔\n        ∀ (ε : ennreal), ε > 0 → filter.eventually (fun (x : β) => edist (u x) a < ε) f :=\n  filter.has_basis.tendsto_right_iff nhds_basis_eball\n\ntheorem tendsto_at_top {α : Type u} {β : Type v} [emetric_space α] [Nonempty β] [semilattice_sup β]\n    {u : β → α} {a : α} :\n    filter.tendsto u filter.at_top (nhds a) ↔\n        ∀ (ε : ennreal), ε > 0 → ∃ (N : β), ∀ (n : β), n ≥ N → edist (u n) a < ε :=\n  sorry\n\n/-- In an emetric space, Cauchy sequences are characterized by the fact that, eventually,\nthe edistance between its elements is arbitrarily small -/\ntheorem cauchy_seq_iff {α : Type u} {β : Type v} [emetric_space α] [Nonempty β] [semilattice_sup β]\n    {u : β → α} :\n    cauchy_seq u ↔\n        ∀ (ε : ennreal), ε > 0 → ∃ (N : β), ∀ (m n : β), m ≥ N → n ≥ N → edist (u m) (u n) < ε :=\n  filter.has_basis.cauchy_seq_iff uniformity_basis_edist\n\n/-- A variation around the emetric characterization of Cauchy sequences -/\ntheorem cauchy_seq_iff' {α : Type u} {β : Type v} [emetric_space α] [Nonempty β] [semilattice_sup β]\n    {u : β → α} :\n    cauchy_seq u ↔ ∀ (ε : ennreal), ε > 0 → ∃ (N : β), ∀ (n : β), n ≥ N → edist (u n) (u N) < ε :=\n  filter.has_basis.cauchy_seq_iff' uniformity_basis_edist\n\n/-- A variation of the emetric characterization of Cauchy sequences that deals with\n`ℝ≥0` upper bounds. -/\ntheorem cauchy_seq_iff_nnreal {α : Type u} {β : Type v} [emetric_space α] [Nonempty β]\n    [semilattice_sup β] {u : β → α} :\n    cauchy_seq u ↔ ∀ (ε : nnreal), 0 < ε → ∃ (N : β), ∀ (n : β), N ≤ n → edist (u n) (u N) < ↑ε :=\n  filter.has_basis.cauchy_seq_iff' uniformity_basis_edist_nnreal\n\ntheorem totally_bounded_iff {α : Type u} [emetric_space α] {s : set α} :\n    totally_bounded s ↔\n        ∀ (ε : ennreal) (H : ε > 0),\n          ∃ (t : set α),\n            set.finite t ∧ s ⊆ set.Union fun (y : α) => set.Union fun (H : y ∈ t) => ball y ε :=\n  sorry\n\ntheorem totally_bounded_iff' {α : Type u} [emetric_space α] {s : set α} :\n    totally_bounded s ↔\n        ∀ (ε : ennreal) (H : ε > 0),\n          ∃ (t : set α),\n            ∃ (H : t ⊆ s),\n              set.finite t ∧ s ⊆ set.Union fun (y : α) => set.Union fun (H : y ∈ t) => ball y ε :=\n  sorry\n\n/-- A compact set in an emetric space is separable, i.e., it is the closure of a countable set -/\ntheorem countable_closure_of_compact {α : Type u} [emetric_space α] {s : set α}\n    (hs : is_compact s) : ∃ (t : set α), ∃ (H : t ⊆ s), set.countable t ∧ s = closure t :=\n  sorry\n\n--    assume e, finite_cover_balls_of_compact hs,\n\nprotected instance topological_space.first_countable_topology (α : Type u) [emetric_space α] :\n    topological_space.first_countable_topology α :=\n  uniform_space.first_countable_topology uniformity_has_countable_basis\n\n/-- A separable emetric space is second countable: one obtains a countable basis by taking\nthe balls centered at points in a dense subset, and with rational radii. We do not register\nthis as an instance, as there is already an instance going in the other direction\nfrom second countable spaces to separable spaces, and we want to avoid loops. -/\ntheorem second_countable_of_separable (α : Type u) [emetric_space α]\n    [topological_space.separable_space α] : topological_space.second_countable_topology α :=\n  uniform_space.second_countable_of_separable uniformity_has_countable_basis\n\n/-- The diameter of a set in an emetric space, named `emetric.diam` -/\ndef diam {α : Type u} [emetric_space α] (s : set α) : ennreal :=\n  supr fun (x : α) => supr fun (H : x ∈ s) => supr fun (y : α) => supr fun (H : y ∈ s) => edist x y\n\ntheorem diam_le_iff_forall_edist_le {α : Type u} [emetric_space α] {s : set α} {d : ennreal} :\n    diam s ≤ d ↔ ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → edist x y ≤ d :=\n  sorry\n\n/-- If two points belong to some set, their edistance is bounded by the diameter of the set -/\ntheorem edist_le_diam_of_mem {α : Type u} [emetric_space α] {x : α} {y : α} {s : set α} (hx : x ∈ s)\n    (hy : y ∈ s) : edist x y ≤ diam s :=\n  iff.mp diam_le_iff_forall_edist_le (le_refl (diam s)) x hx y hy\n\n/-- If the distance between any two points in a set is bounded by some constant, this constant\nbounds the diameter. -/\ntheorem diam_le_of_forall_edist_le {α : Type u} [emetric_space α] {s : set α} {d : ennreal}\n    (h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → edist x y ≤ d) : diam s ≤ d :=\n  iff.mpr diam_le_iff_forall_edist_le h\n\n/-- The diameter of a subsingleton vanishes. -/\ntheorem diam_subsingleton {α : Type u} [emetric_space α] {s : set α} (hs : set.subsingleton s) :\n    diam s = 0 :=\n  iff.mp nonpos_iff_eq_zero\n    (diam_le_of_forall_edist_le\n      fun (x : α) (hx : x ∈ s) (y : α) (hy : y ∈ s) =>\n        Eq.symm (hs hx hy) ▸ edist_self y ▸ le_refl (edist y y))\n\n/-- The diameter of the empty set vanishes -/\n@[simp] theorem diam_empty {α : Type u} [emetric_space α] : diam ∅ = 0 :=\n  diam_subsingleton set.subsingleton_empty\n\n/-- The diameter of a singleton vanishes -/\n@[simp] theorem diam_singleton {α : Type u} [emetric_space α] {x : α} : diam (singleton x) = 0 :=\n  diam_subsingleton set.subsingleton_singleton\n\ntheorem diam_eq_zero_iff {α : Type u} [emetric_space α] {s : set α} :\n    diam s = 0 ↔ set.subsingleton s :=\n  sorry\n\ntheorem diam_pos_iff {α : Type u} [emetric_space α] {s : set α} :\n    0 < diam s ↔ ∃ (x : α), ∃ (H : x ∈ s), ∃ (y : α), ∃ (H : y ∈ s), x ≠ y :=\n  sorry\n\ntheorem diam_insert {α : Type u} [emetric_space α] {x : α} {s : set α} :\n    diam (insert x s) = max (supr fun (y : α) => supr fun (H : y ∈ s) => edist x y) (diam s) :=\n  sorry\n\ntheorem diam_pair {α : Type u} [emetric_space α] {x : α} {y : α} :\n    diam (insert x (singleton y)) = edist x y :=\n  sorry\n\ntheorem diam_triple {α : Type u} [emetric_space α] {x : α} {y : α} {z : α} :\n    diam (insert x (insert y (singleton z))) = max (max (edist x y) (edist x z)) (edist y z) :=\n  sorry\n\n/-- The diameter is monotonous with respect to inclusion -/\ntheorem diam_mono {α : Type u} [emetric_space α] {s : set α} {t : set α} (h : s ⊆ t) :\n    diam s ≤ diam t :=\n  diam_le_of_forall_edist_le\n    fun (x : α) (hx : x ∈ s) (y : α) (hy : y ∈ s) => edist_le_diam_of_mem (h hx) (h hy)\n\n/-- The diameter of a union is controlled by the diameter of the sets, and the edistance\nbetween two points in the sets. -/\ntheorem diam_union {α : Type u} [emetric_space α] {x : α} {y : α} {s : set α} {t : set α}\n    (xs : x ∈ s) (yt : y ∈ t) : diam (s ∪ t) ≤ diam s + edist x y + diam t :=\n  sorry\n\ntheorem diam_union' {α : Type u} [emetric_space α] {s : set α} {t : set α}\n    (h : set.nonempty (s ∩ t)) : diam (s ∪ t) ≤ diam s + diam t :=\n  sorry\n\ntheorem diam_closed_ball {α : Type u} [emetric_space α] {x : α} {r : ennreal} :\n    diam (closed_ball x r) ≤ bit0 1 * r :=\n  sorry\n\ntheorem diam_ball {α : Type u} [emetric_space α] {x : α} {r : ennreal} :\n    diam (ball x r) ≤ bit0 1 * r :=\n  le_trans (diam_mono ball_subset_closed_ball) diam_closed_ball\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/metric_space/emetric_space_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3145644156593836}}
{"text": "abbrev M := ExceptT String $ StateT Nat Id\n\n/- The following instances are not needed, but they speedup the proof -/\ninstance : Monad M := inferInstance\ninstance : LawfulMonad M := inferInstance\ninstance : MonadStateOf Nat M := inferInstance\ninstance : Bind M := inferInstance\ninstance : Pure M := inferInstance\n\nsyntax \"bigAdd0Seq! \" num : term\n\nmacro_rules\n  | `(bigAdd0Seq! $n) =>\n    let n := n.toNat\n    if n == 0 then\n      `(pure ())\n    else\n      `(get >>= fun n => set (0 + n) >>= fun _ => bigAdd0Seq! $(Lean.quote (n - 1)))\n\n@[simp] theorem get_set  : (get >>= fun n => set n) = (pure () : M Unit) :=\n  rfl\n\nset_option maxRecDepth   1000000\nset_option maxHeartbeats 1000000\n\ntheorem ex : (bigAdd0Seq! 40 : M Unit) = pure () := by\n  simp\n\n-- set_option pp.explicit true\n-- set_option pp.notation false\n-- #print ex\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/playground/seq1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.31450906068194423}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport category_theory.category.Cat\nimport category_theory.groupoid\nimport category_theory.types\n\n/-!\n# Objects of a category up to an isomorphism\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n`is_isomorphic X Y := nonempty (X ≅ Y)` is an equivalence relation on the objects of a category.\nThe quotient with respect to this relation defines a functor from our category to `Type`.\n-/\n\nuniverses v u\n\nnamespace category_theory\n\nsection category\n\nvariables {C : Type u} [category.{v} C]\n\n/-- An object `X` is isomorphic to an object `Y`, if `X ≅ Y` is not empty. -/\ndef is_isomorphic : C → C → Prop := λ X Y, nonempty (X ≅ Y)\n\nvariable (C)\n\n/-- `is_isomorphic` defines a setoid. -/\ndef is_isomorphic_setoid : setoid C :=\n{ r := is_isomorphic,\n  iseqv := ⟨λ X, ⟨iso.refl X⟩, λ X Y ⟨α⟩, ⟨α.symm⟩, λ X Y Z ⟨α⟩ ⟨β⟩, ⟨α.trans β⟩⟩ }\n\nend category\n\n/--\nThe functor that sends each category to the quotient space of its objects up to an isomorphism.\n-/\ndef isomorphism_classes : Cat.{v u} ⥤ Type u :=\n{ obj := λ C, quotient (is_isomorphic_setoid C.α),\n  map := λ C D F, quot.map F.obj $ λ X Y ⟨f⟩, ⟨F.map_iso f⟩ }\n\nlemma groupoid.is_isomorphic_iff_nonempty_hom {C : Type u} [groupoid.{v} C] {X Y : C} :\n  is_isomorphic X Y ↔ nonempty (X ⟶ Y) :=\n(groupoid.iso_equiv_hom X Y).nonempty_congr\n\n-- PROJECT: define `skeletal`, and show every category is equivalent to a skeletal category,\n-- using the axiom of choice to pick a representative of every isomorphism class.\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/isomorphism_classes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7310585669110203, "lm_q2_score": 0.43014734858584286, "lm_q1q2_score": 0.31446290421774137}}
{"text": "import topology.metric_space.hausdorff_distance\nimport topology.uniform_space.separation\nimport geometry.manifold.cont_mdiff_mfderiv\nimport indexing\nimport to_mathlib.topology.paracompact\nimport to_mathlib.topology.algebra.order.compact\nimport to_mathlib.topology.nhds_set\nimport to_mathlib.topology.misc\nimport to_mathlib.geometry.manifold.charted_space\nimport to_mathlib.geometry.manifold.smooth_manifold_with_corners\nimport to_mathlib.analysis.normed_space.misc\n\nimport interactive_expr\nset_option trace.filter_inst_type true\n\nnoncomputable theory\n\nopen set equiv\nopen_locale manifold topology\n\nsection general\nvariables {𝕜 : Type*} [nontrivially_normed_field 𝕜]\n  {E : Type*} [normed_add_comm_group E] [normed_space 𝕜 E]\n  {H : Type*} [topological_space H]\n  (I : model_with_corners 𝕜 E H)\n  (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n  {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E']\n  {H' : Type*} [topological_space H']\n  (I' : model_with_corners 𝕜 E' H')\n  (M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n\nstructure open_smooth_embedding  :=\n(to_fun : M → M')\n(inv_fun : M' → M)\n(left_inv' : ∀{x}, inv_fun (to_fun x) = x)\n(is_open_range : is_open (range to_fun))\n(smooth_to : smooth I I' to_fun)\n(smooth_inv : smooth_on I' I inv_fun (range to_fun))\n\ninstance : has_coe_to_fun (open_smooth_embedding I M I' M') (λ _, M → M') :=\n⟨open_smooth_embedding.to_fun⟩\n\nnamespace open_smooth_embedding\n\nvariables {I I' M M'} (f : open_smooth_embedding I M I' M')\n\n@[simp] lemma coe_mk (f g h₁ h₂ h₃ h₄) :\n  ⇑(⟨f, g, h₁, h₂, h₃, h₄⟩ : open_smooth_embedding I M I' M') = f :=\nrfl\n\n@[simp] lemma left_inv (x : M) : f.inv_fun (f x) = x := by apply f.left_inv'\n\n@[simp] lemma inv_fun_comp_coe : f.inv_fun ∘ f = id := funext f.left_inv\n\n@[simp] lemma right_inv {y : M'} (hy : y ∈ range f) : f (f.inv_fun y) = y :=\nby { obtain ⟨x, rfl⟩ := hy, rw [f.left_inv] }\n\nlemma smooth_at_inv {y : M'} (hy : y ∈ range f) : smooth_at I' I f.inv_fun y :=\n(f.smooth_inv y hy).cont_mdiff_at $ f.is_open_range.mem_nhds hy\n\nlemma smooth_at_inv' {x : M} : smooth_at I' I f.inv_fun (f x) :=\nf.smooth_at_inv $ mem_range_self x\n\nlemma left_inverse : function.left_inverse f.inv_fun f :=\nleft_inv f\n\nlemma injective : function.injective f :=\nf.left_inverse.injective\n\nprotected lemma continuous : continuous f := f.smooth_to.continuous\n\nlemma open_map : is_open_map f :=\nf.left_inverse.is_open_map f.is_open_range f.smooth_inv.continuous_on\n\nlemma coe_comp_inv_fun_eventually_eq (x : M) : f ∘ f.inv_fun =ᶠ[𝓝 (f x)] id :=\nfilter.eventually_of_mem (f.open_map.range_mem_nhds x) $ λ y hy, f.right_inv hy\n\n/- Note that we are slightly abusing the fact that `tangent_space I x` and\n`tangent_space I (f.inv_fun (f x))` are both definitionally `E` below. -/\ndef fderiv (x : M) : tangent_space I x ≃L[𝕜] tangent_space I' (f x) :=\nhave h₁ : mdifferentiable_at I' I f.inv_fun (f x) := ((f.smooth_inv (f x) (mem_range_self x)\n  ).mdifferentiable_within_at le_top).mdifferentiable_at (f.open_map.range_mem_nhds x),\nhave h₂ : mdifferentiable_at I I' f x := f.smooth_to.cont_mdiff.mdifferentiable le_top _,\ncontinuous_linear_equiv.equiv_of_inverse\n  (mfderiv I I' f x)\n  (mfderiv I' I f.inv_fun (f x))\nbegin\n  intros v,\n  rw [← continuous_linear_map.comp_apply, ← mfderiv_comp x h₁ h₂, f.inv_fun_comp_coe, mfderiv_id,\n    continuous_linear_map.coe_id', id.def],\nend\nbegin\n  intros v,\n  have hx : x = f.inv_fun (f x), { rw f.left_inv, },\n  have hx' : f (f.inv_fun (f x)) = f x, { rw f.left_inv, },\n  rw hx at h₂,\n  rw [hx, hx', ← continuous_linear_map.comp_apply, ← mfderiv_comp (f x) h₂ h₁, ((has_mfderiv_at_id\n    I' (f x)).congr_of_eventually_eq (f.coe_comp_inv_fun_eventually_eq x)).mfderiv,\n    continuous_linear_map.coe_id', id.def],\nend\n\n@[simp] lemma fderiv_coe (x : M) :\n  (f.fderiv x : tangent_space I x →L[𝕜] tangent_space I' (f x)) = mfderiv I I' f x :=\nby { ext, refl }\n\n@[simp] lemma fderiv_symm_coe (x : M) :\n  ((f.fderiv x).symm : tangent_space I' (f x) →L[𝕜] tangent_space I x) =\n  mfderiv I' I f.inv_fun (f x) :=\nby { ext, refl }\n\nlemma fderiv_symm_coe' {x : M'} (hx : x ∈ range f) :\n  ((f.fderiv (f.inv_fun x)).symm : tangent_space I' (f (f.inv_fun x)) →L[𝕜]\n    tangent_space I (f.inv_fun x)) =\n  (mfderiv I' I f.inv_fun x : tangent_space I' x →L[𝕜] tangent_space I (f.inv_fun x)) :=\nby rw [fderiv_symm_coe, f.right_inv hx]\n\nopen filter\n\nlemma open_embedding : open_embedding f :=\nopen_embedding_of_continuous_injective_open f.continuous f.injective f.open_map\n\nlemma inducing : inducing f := f.open_embedding.to_inducing\n\n-- `∀ᶠ x near s, p x` means property `p` holds at every point in a neighborhood of the set `s`.\nnotation `∀ᶠ` binders ` near ` s `, ` r:(scoped p, filter.eventually p $ 𝓝ˢ s) := r\n\nlemma forall_near' {P : M → Prop} {A : set M'} (h : ∀ᶠ m near f ⁻¹' A, P m) :\n  ∀ᶠ m' near A ∩ range f, ∀ m, m' = f m → P m :=\nbegin\n  rw eventually_nhds_set_iff at h ⊢,\n  rintros _ ⟨hfm₀, m₀, rfl⟩,\n  have : ∀ U ∈ 𝓝 m₀, ∀ᶠ m' in 𝓝 (f m₀), m' ∈ f '' U,\n  { intros U U_in,\n    exact f.open_map.image_mem_nhds U_in },\n  apply (this _ $ h m₀ hfm₀).mono,\n  rintros _ ⟨m₀, hm₀, hm₀'⟩ m₁ rfl,\n  rwa ← f.injective hm₀'\nend\n\nlemma eventually_nhds_set_mono {α : Type*} [topological_space α] {s t : set α} {P : α → Prop}\n  (h : ∀ᶠ x near t, P x) (h' : s ⊆ t) : ∀ᶠ x near s, P x :=\nh.filter_mono (nhds_set_mono h')\n\n-- TODO: optimize this proof which is probably more complicated than it needs to be\nlemma forall_near [t2_space M'] {P : M → Prop} {P' : M' → Prop} {K : set M} (hK : is_compact K)\n  {A : set M'} (hP : ∀ᶠ m near f ⁻¹' A, P m) (hP' : ∀ᶠ m' near A, m' ∉ f '' K → P' m')\n  (hPP' : ∀ m, P m → P' (f m)) :\n  ∀ᶠ m' near A, P' m' :=\nbegin\n  rw show A = (A ∩ range f) ∪ (A ∩ (range f)ᶜ), by simp,\n  apply filter.eventually.union,\n  { have : ∀ᶠ m' near A ∩ range f, m' ∈ range f,\n      from f.is_open_range.forall_near_mem_of_subset (inter_subset_right _ _),\n    apply (this.and $ f.forall_near' hP).mono,\n    rintros _ ⟨⟨m, rfl⟩, hm⟩,\n    exact hPP' _ (hm _ rfl) },\n  { have op : is_open (f '' K)ᶜ,\n    { rw is_open_compl_iff,\n      exact (hK.image f.continuous).is_closed },\n    have : A ∩ (range f)ᶜ ⊆ A ∩ (f '' K)ᶜ,\n    { exact inter_subset_inter_right _ (compl_subset_compl.mpr (image_subset_range f K)) },\n    apply eventually_nhds_set_mono _ this,\n    rw eventually_nhds_set_iff at hP' ⊢,\n    rintros x ⟨hx, hx'⟩,\n    have hx' : ∀ᶠ y in 𝓝 x, y ∈ (f '' K)ᶜ,\n      from is_open_iff_eventually.mp op x hx',\n    apply ((hP' x hx).and hx').mono,\n    rintros y ⟨hy, hy'⟩,\n    exact hy hy' },\nend\n\nvariables (I M)\n\n/-- The identity map is a smooth open embedding. -/\n-- unused\n@[simps] def id : open_smooth_embedding I M I M :=\n{ to_fun := id,\n  inv_fun := id,\n  left_inv' := λ x, rfl,\n  is_open_range := is_open_map.id.is_open_range,\n  smooth_to := smooth_id,\n  smooth_inv := smooth_on_id }\n\nvariables {I M I' M'}\n\n-- unused\n@[simps] def comp\n  {E'' : Type*} [normed_add_comm_group E''] [normed_space 𝕜 E'']\n  {H'' : Type*} [topological_space H'']\n  {I'' : model_with_corners 𝕜 E'' H''}\n  {M'' : Type*} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M'']\n  (g : open_smooth_embedding I' M' I'' M'') (f : open_smooth_embedding I M I' M') :\n  open_smooth_embedding I M I'' M'' :=\n{ to_fun := g ∘ f,\n  inv_fun := f.inv_fun ∘ g.inv_fun,\n  left_inv' := λ x, by simp only [function.comp_app, left_inv],\n  is_open_range := (g.open_map.comp f.open_map).is_open_range,\n  smooth_to := g.smooth_to.comp f.smooth_to,\n  smooth_inv := (f.smooth_inv.comp' g.smooth_inv).mono\n  begin\n    change range (g ∘ f) ⊆ range g ∩ g.inv_fun ⁻¹' range f,\n    refine subset_inter_iff.mpr ⟨range_comp_subset_range f g, _⟩,\n    rintros x' ⟨x, rfl⟩,\n    exact ⟨x, by simp only [left_inv]⟩,\n  end, }\n\nend open_smooth_embedding\n\nnamespace continuous_linear_equiv\n\nvariables (e : E ≃L[𝕜] E') [complete_space E] [complete_space E']\n\n@[simp] lemma is_open_map : is_open_map e := (e : E →L[𝕜] E').is_open_map e.surjective\n\n-- unused\n@[simps] def to_open_smooth_embedding :\n  open_smooth_embedding 𝓘(𝕜, E) E 𝓘(𝕜, E') E' :=\n{ to_fun := e,\n  inv_fun := e.symm,\n  left_inv' := e.symm_apply_apply,\n  is_open_range := e.is_open_map.is_open_range,\n  smooth_to := (e : E →L[𝕜] E').cont_mdiff,\n  smooth_inv := (e.symm : E' →L[𝕜] E).cont_mdiff.cont_mdiff_on }\n\nend continuous_linear_equiv\n\nend general\n\nsection without_boundary\n\nopen metric (hiding mem_nhds_iff) function\n\nuniverse u\n\nsection general_nonsense\n\nvariables {𝕜 E H M : Type*} [nontrivially_normed_field 𝕜]\n  [normed_add_comm_group E] [normed_space 𝕜 E]\n  [topological_space H] {I : model_with_corners 𝕜 E H}\n  [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n  {x : M} {n : ℕ∞}\n\nlemma ext_chart_at_target_eq_image_chart_target :\n  (ext_chart_at I x).target = I '' (chart_at H x).target :=\nbegin\n  erw [(chart_at H x).to_local_equiv.trans_target'' I.to_local_equiv, I.source_eq, univ_inter],\n  refl,\nend\n\n@[simp] lemma model_with_corners_self.ext_chart_at {e : E} :\n  ext_chart_at 𝓘(𝕜, E) e = local_equiv.refl E :=\nby simp\n\nlemma cont_mdiff_on_ext_chart_symm :\n  cont_mdiff_on 𝓘(𝕜, E) I n (ext_chart_at I x).symm (ext_chart_at I x).target :=\nbegin\n  have hs : (ext_chart_at I x).target ⊆ (chart_at E (ext_chart_at I x x)).source,\n  { simp only [subset_univ] with mfld_simps },\n  have h2s : maps_to (ext_chart_at I x).symm (ext_chart_at I x).target (chart_at H x).source,\n  { rw ← ext_chart_at_source I, exact (ext_chart_at I x).symm_maps_to, },\n  refine (cont_mdiff_on_iff_of_subset_source hs h2s).mpr ⟨continuous_on_ext_chart_at_symm I x, _⟩,\n  simp_rw [model_with_corners_self.ext_chart_at, local_equiv.refl_symm, local_equiv.refl_coe,\n    comp.right_id, id.def, image_id'],\n  exact (cont_diff_on_congr (λ y hy, (ext_chart_at I x).right_inv hy)).mpr cont_diff_on_id\nend\n\nend general_nonsense\n\nvariables\n  {F H : Type*} (M : Type u)\n  [normed_add_comm_group F] [normed_space ℝ F]\n  [topological_space H] [topological_space M] [charted_space H M]\n  [t2_space M] [locally_compact_space M] [sigma_compact_space M]\n  (IF : model_with_corners ℝ F H) [smooth_manifold_with_corners IF M]\n\n/- Clearly should be generalised. Maybe what we really want is a theory of local diffeomorphisms.\n\nNote that the input `f` is morally an `open_smooth_embedding` but stated in terms of `cont_diff`\ninstead of `cont_mdiff`. This is more convenient for our purposes. -/\ndef open_smooth_emb_of_diffeo_subset_chart_target (x : M) {f : local_homeomorph F F}\n  (hf₁ : f.source = univ)\n  (hf₂ : cont_diff ℝ ∞ f)\n  (hf₃ : cont_diff_on ℝ ∞ f.symm f.target)\n  (hf₄ : range f ⊆ IF '' (chart_at H x).target) :\n  open_smooth_embedding 𝓘(ℝ, F) F IF M :=\n{ to_fun := (ext_chart_at IF x).symm ∘ f,\n  inv_fun := f.inv_fun ∘ (ext_chart_at IF x),\n  left_inv' := λ y,\n  begin\n    obtain ⟨z, hz, hz'⟩ := hf₄ (mem_range_self y),\n    have aux : f.symm (IF z) = y, { rw hz', exact f.left_inv (hf₁.symm ▸ mem_univ _), },\n    simp only [← hz', (chart_at H x).right_inv hz, aux, ext_chart_at,\n      local_homeomorph.extend, local_equiv.coe_trans,\n      local_homeomorph.inv_fun_eq_coe, model_with_corners.to_local_equiv_coe,\n      local_homeomorph.coe_coe, local_equiv.coe_trans_symm, local_homeomorph.coe_coe_symm,\n      model_with_corners.left_inv, model_with_corners.to_local_equiv_coe_symm, comp_app, aux],\n  end,\n  is_open_range := is_open_map.is_open_range $ λ u hu,\n  begin\n    have aux : is_open (f '' u) := f.image_open_of_open hu (hf₁.symm ▸ subset_univ u),\n    convert is_open_ext_chart_at_preimage' IF x aux,\n    rw image_comp,\n    refine (ext_chart_at IF x).symm_image_eq_source_inter_preimage\n      ((image_subset_range f u).trans _),\n    rw ext_chart_at_target_eq_image_chart_target,\n    exact hf₄,\n  end,\n  smooth_to :=\n  begin\n    refine cont_mdiff_on_ext_chart_symm.comp_cont_mdiff hf₂.cont_mdiff (λ y, _),\n    rw ext_chart_at_target_eq_image_chart_target,\n    exact hf₄ (mem_range_self y),\n  end,\n  smooth_inv :=\n  begin\n    rw ← ext_chart_at_target_eq_image_chart_target at hf₄,\n    have hf' : range ((ext_chart_at IF x).symm ∘ f) ⊆ (ext_chart_at IF x) ⁻¹' f.target,\n    { rw [range_comp, ← image_subset_iff, ← f.image_source_eq_target, hf₁, image_univ],\n      exact (local_equiv.image_symm_image_of_subset_target _ hf₄).subset, },\n    have hf'' : range ((ext_chart_at IF x).symm ∘ f) ⊆ (chart_at H x).source,\n    { rw [← ext_chart_at_source IF, range_comp, ← local_equiv.symm_image_target_eq_source],\n      exact (monotone_image hf₄).trans subset.rfl, },\n    exact hf₃.cont_mdiff_on.comp (cont_mdiff_on_ext_chart_at.mono hf'') hf',\n  end}\n\n@[simp] lemma coe_open_smooth_emb_of_diffeo_subset_chart_target\n  (x : M) {f : local_homeomorph F F}\n  (hf₁ : f.source = univ)\n  (hf₂ : cont_diff ℝ ∞ f)\n  (hf₃ : cont_diff_on ℝ ∞ f.symm f.target)\n  (hf₄ : range f ⊆ IF '' (chart_at H x).target) :\n  (open_smooth_emb_of_diffeo_subset_chart_target M IF x hf₁ hf₂ hf₃ hf₄ : F → M) =\n  (ext_chart_at IF x).symm ∘ f :=\nby simp [open_smooth_emb_of_diffeo_subset_chart_target]\n\nlemma range_open_smooth_emb_of_diffeo_subset_chart_target\n  (x : M) {f : local_homeomorph F F}\n  (hf₁ : f.source = univ)\n  (hf₂ : cont_diff ℝ ∞ f)\n  (hf₃ : cont_diff_on ℝ ∞ f.symm f.target)\n  (hf₄ : range f ⊆ IF '' (chart_at H x).target) :\n  range (open_smooth_emb_of_diffeo_subset_chart_target M IF x hf₁ hf₂ hf₃ hf₄) =\n  (ext_chart_at IF x).symm '' (range f) :=\nby rw [coe_open_smooth_emb_of_diffeo_subset_chart_target, range_comp]\n\nvariables {M} (F) [model_with_corners.boundaryless IF] [finite_dimensional ℝ F]\n\nlemma nice_atlas'\n  {ι : Type*} {s : ι → set M} (s_op : ∀ j, is_open $ s j) (cov : (⋃ j, s j) = univ)\n  (U : set F) (hU₁ : (0 : F) ∈ U) (hU₂ : is_open U) :\n  ∃ (ι' : Type u) (t : set ι') (φ : t → open_smooth_embedding 𝓘(ℝ, F) F IF M),\n  t.countable ∧\n  (∀ i, ∃ j, range (φ i) ⊆ s j) ∧\n  locally_finite (λ i, range (φ i)) ∧\n  (⋃ i, φ i '' U) = univ :=\nbegin\n  let W : M → ℝ → set M := λ x r,\n    (ext_chart_at IF x).symm ∘ diffeomorph_to_nhd (ext_chart_at IF x x) r '' U,\n  let B : M → ℝ → set M := charted_space.ball IF,\n  let p : M → ℝ → Prop :=\n    λ x r, 0 < r ∧ ball (ext_chart_at IF x x) r ⊆ (ext_chart_at IF x).target ∧ ∃ j, B x r ⊆ s j,\n  have hW₀ : ∀ x r, p x r → x ∈ W x r := λ x r h, ⟨0, hU₁, by simp [h.1]⟩,\n  have hW₁ : ∀ x r, p x r → is_open (W x r),\n  { rintros x r ⟨h₁, h₂, -, -⟩,\n    simp only [W],\n    rw image_comp,\n    let V := diffeomorph_to_nhd (ext_chart_at IF x x) r '' U,\n    change is_open ((ext_chart_at IF x).symm '' V),\n    have hV₁ : is_open V := ((diffeomorph_to_nhd\n      (ext_chart_at IF x x) r).is_open_image_iff_of_subset_source (by simp)).mp hU₂,\n    have hV₂ : V ⊆ (ext_chart_at IF x).target :=\n      subset.trans ((image_subset_range _ _).trans (by simp [h₁])) h₂,\n    rw (ext_chart_at IF x).symm_image_eq_source_inter_preimage hV₂,\n    exact is_open_ext_chart_at_preimage' IF x hV₁, },\n  have hB : ∀ x, (𝓝 x).has_basis (p x) (B x) :=\n    λ x, charted_space.nhds_has_basis_balls_of_open_cov IF x s_op cov,\n  obtain ⟨t, ht₁, ht₂, ht₃, ht₄⟩ :=\n    exists_countable_locally_finite_cover surjective_id hW₀ hW₁ hB,\n  let g : M × ℝ → local_homeomorph F F := λ z, diffeomorph_to_nhd (ext_chart_at IF z.1 z.1) z.2,\n  have hg₁ : ∀ z, (g z).source = univ, { simp, },\n  have hg₂ : ∀ z, cont_diff ℝ ∞ (g z), { simp, },\n  have hg₃ : ∀ z, cont_diff_on ℝ ∞ (g z).symm (g z).target, { simp, },\n  refine ⟨M × ℝ, t,\n    λ z, open_smooth_emb_of_diffeo_subset_chart_target M IF z.1.1 (hg₁ z.1) (hg₂ z.1) (hg₃ z.1) _,\n    ht₁, λ z, _, _, _⟩,\n  { obtain ⟨⟨x, r⟩, hxr⟩ := z,\n    obtain ⟨hr : 0 < r, hr' : ball (ext_chart_at IF x x) r ⊆ _, -⟩ := ht₂ _ hxr,\n    rw ← ext_chart_at_target_eq_image_chart_target,\n    exact (range_diffeomorph_to_nhd_subset_ball (ext_chart_at IF x x) hr).trans hr', },\n  { obtain ⟨⟨x, r⟩, hxr⟩ := z,\n    obtain ⟨hr : 0 < r, -, j, hj : B x r ⊆ s j⟩ := ht₂ _ hxr,\n    simp_rw range_open_smooth_emb_of_diffeo_subset_chart_target,\n    exact ⟨j, (monotone_image (range_diffeomorph_to_nhd_subset_ball _ hr)).trans hj⟩, },\n  { simp_rw range_open_smooth_emb_of_diffeo_subset_chart_target,\n    refine ht₄.subset _,\n    rintros ⟨⟨x, r⟩, hxr⟩,\n    obtain ⟨hr : 0 < r, -, -⟩ := ht₂ _ hxr,\n    exact monotone_image (range_diffeomorph_to_nhd_subset_ball _ hr), },\n  { simpa only [Union_coe_set] using ht₃, },\nend\n\nvariables [nonempty M]\n\nlemma nice_atlas {ι : Type*} {s : ι → set M} (s_op : ∀ j, is_open $ s j) (cov : (⋃ j, s j) = univ) :\n  ∃ n, ∃ φ : index_type n → open_smooth_embedding 𝓘(ℝ, F) F IF M,\n  (∀ i, ∃ j, range (φ i) ⊆ s j) ∧\n  locally_finite (λ i, range (φ i)) ∧\n  (⋃ i, φ i '' ball 0 1) = univ :=\nbegin\n  obtain ⟨ι', t, φ, h₁, h₂, h₃, h₄⟩ := nice_atlas' F IF s_op cov (ball 0 1) (by simp) is_open_ball,\n  have htne : t.nonempty,\n  { by_contra contra,\n    simp only [not_nonempty_iff_eq_empty.mp contra, Union_false, Union_coe_set, Union_empty,\n      @eq_comm _ _ univ, univ_eq_empty_iff] at h₄,\n    exact not_is_empty_of_nonempty M h₄, },\n  obtain ⟨n, ⟨fn⟩⟩ := (set.countable_iff_exists_nonempty_index_type_equiv htne).mp h₁,\n  refine ⟨n, φ ∘ fn, λ i, h₂ (fn i), h₃.comp_injective fn.injective, _⟩,\n  rwa fn.surjective.Union_comp (λ i, φ i '' ball 0 1),\nend\n\nend without_boundary\n\nnamespace open_smooth_embedding\n\nsection updating\n\nvariables {𝕜 EX EM EY EN EM' X M Y N M' : Type*} [nontrivially_normed_field 𝕜]\n  [normed_add_comm_group EX] [normed_space 𝕜 EX]\n  [normed_add_comm_group EM] [normed_space 𝕜 EM]\n  [normed_add_comm_group EM'] [normed_space 𝕜 EM']\n  [normed_add_comm_group EY] [normed_space 𝕜 EY]\n  [normed_add_comm_group EN] [normed_space 𝕜 EN]\n  {HX : Type*} [topological_space HX] {IX : model_with_corners 𝕜 EX HX}\n  {HY : Type*} [topological_space HY] {IY : model_with_corners 𝕜 EY HY}\n  {HM : Type*} [topological_space HM] {IM : model_with_corners 𝕜 EM HM}\n  {HM' : Type*} [topological_space HM'] {IM' : model_with_corners 𝕜 EM' HM'}\n  {HN : Type*} [topological_space HN] {IN : model_with_corners 𝕜 EN HN}\n  [topological_space X] [charted_space HX X] [smooth_manifold_with_corners IX X]\n  [topological_space M] [charted_space HM M] [smooth_manifold_with_corners IM M]\n  [topological_space M'] [charted_space HM' M']\n\nsection non_metric\nvariables\n  [topological_space Y]      [charted_space HY Y] [smooth_manifold_with_corners IY Y]\n  [topological_space N]      [charted_space HN N] [smooth_manifold_with_corners IN N]\n  (φ : open_smooth_embedding IX X IM M)\n  (ψ : open_smooth_embedding IY Y IN N)\n  (f : M → N) (g : X → Y)\n\nsection\nlocal attribute [instance] classical.dec\n/-- This is definition `def:update` in the blueprint. -/\ndef update (m : M) : N := if m ∈ range φ then ψ (g (φ.inv_fun m)) else f m\nend\n\n@[simp] lemma update_of_nmem_range {m : M} (hm : m ∉ range φ) :\n  update φ ψ f g m = f m :=\nby simp [update, hm]\n\n@[simp] lemma update_of_mem_range {m : M} (hm : m ∈ range φ) :\n  update φ ψ f g m = ψ (g (φ.inv_fun m)) :=\nby simp [update, hm]\n\n@[simp] lemma update_apply_embedding (x : X) :\n  update φ ψ f g (φ x) = ψ (g x) :=\nby simp [update]\n\n/- This small auxiliry result is used in the next two lemmas. -/\nlemma nice_update_of_eq_outside_compact_aux {K : set X} (g : X → Y)\n  (hg : ∀ (x : X), x ∉ K → f (φ x) = ψ (g x)) {m : M} (hm : m ∉ φ '' K) :\n  φ.update ψ f g m = f m :=\nbegin\n  by_cases hm' : m ∈ range φ,\n  { obtain ⟨x, rfl⟩ := hm',\n    replace hm : x ∉ K, { contrapose! hm, exact mem_image_of_mem φ hm, },\n    simp [hg x hm] },\n  { simp [hm'] }\nend\n\nopen function\n\n/-- This is lemma `lem:smooth_updating` in the blueprint. -/\nlemma smooth_update\n  (f : M' → M → N) (g : M' → X → Y)\n  {k : M' → M}\n  {K : set X} (hK : is_closed (φ '' K)) (hf : smooth (IM'.prod IM) IN (uncurry f))\n  (hg : smooth (IM'.prod IX) IY (uncurry g))\n  (hk : smooth IM' IM k)\n  (hg' : ∀ y x, x ∉ K → f y (φ x) = ψ (g y x)) :\n  smooth IM' IN (λ x, update φ ψ (f x) (g x) (k x)) :=\nbegin\n  have hK' : ∀ x, k x ∉ φ '' K → update φ ψ (f x) (g x) (k x) = f x (k x) :=\n    λ x hx, nice_update_of_eq_outside_compact_aux φ ψ (f x) (g x) (hg' x) hx,\n  refine cont_mdiff_of_locally_cont_mdiff_on (λ x, _),\n  let U := range φ,\n  let V := (φ '' K)ᶜ,\n  have h₂ : is_open (k ⁻¹' V) := hK.is_open_compl.preimage hk.continuous,\n  have h₃ : V ∪ U = univ,\n  { rw [← compl_subset_iff_union, compl_compl], exact image_subset_range φ K, },\n  have h₄ : ∀ x, k x ∈ U → update φ ψ (f x) (g x) (k x) = (ψ ∘ g x ∘ φ.inv_fun) (k x) :=\n    λ m hm, by simp [hm],\n  by_cases hx : k x ∈ U,\n  { refine ⟨k ⁻¹' U, φ.is_open_range.preimage hk.continuous, hx,\n    (cont_mdiff_on_congr h₄).mpr $ ψ.smooth_to.comp_cont_mdiff_on $\n      hg.comp_cont_mdiff_on (smooth_on_id.prod_mk $ φ.smooth_inv.comp hk.smooth_on subset_rfl)⟩ },\n  { refine ⟨k ⁻¹' V, h₂, _, (cont_mdiff_on_congr hK').mpr\n      (hf.comp (smooth_id.prod_mk hk)).cont_mdiff_on⟩,\n    simpa [hx] using set.ext_iff.mp h₃ (k x) }\nend\nend non_metric\n\nsection metric\nvariables\n  [metric_space Y]      [charted_space HY Y] [smooth_manifold_with_corners IY Y]\n  [metric_space N]      [charted_space HN N] [smooth_manifold_with_corners IN N]\n  (φ : open_smooth_embedding IX X IM M)\n  (ψ : open_smooth_embedding IY Y IN N)\n  (f : M → N) (g : X → Y)\n\n\n\n/-- This is `lem:dist_updating` in the blueprint. -/\nlemma dist_update [proper_space Y] {K : set X} (hK : is_compact K)\n  {P : Type*} [metric_space P] {KP : set P} (hKP : is_compact KP)\n  (f : P → M → N) (hf : continuous ↿f)\n  (hf' : ∀ p, (f p) '' range φ ⊆ range ψ) {ε : M → ℝ} (hε : ∀ m, 0 < ε m) (hε' : continuous ε) :\n  ∃ η > (0 : ℝ), ∀ g : P → X → Y,\n    (∀ (p ∈ KP) (p' ∈ KP) (x ∈ K), dist (g p' x) (ψ.inv_fun (f p (φ x))) < η →\n      dist (update φ ψ (f p') (g p') $ φ x) (f p $ φ x) < ε (φ x)) :=\nbegin\n  let F : P × X → Y := λ q, (ψ.inv_fun ∘ (f q.1) ∘ φ) q.2,\n  let K₁ := metric.cthickening 1 (F '' KP.prod K),\n  have hK₁ : is_compact K₁,\n  { refine metric.is_compact_of_is_closed_bounded metric.is_closed_cthickening\n      (metric.bounded.cthickening $ is_compact.bounded $ _),\n    apply (hKP.prod hK).image,\n    exact ψ.smooth_inv.continuous_on.comp_continuous\n      (hf.comp $ continuous_fst.prod_mk $ φ.continuous.comp continuous_snd)\n      (λ q, hf' q.1 ⟨φ q.2, mem_range_self _, rfl⟩) },\n  have h₁ : uniform_continuous_on ψ K₁ :=\n    hK₁.uniform_continuous_on_of_continuous ψ.continuous.continuous_on,\n  have hεφ : ∀ x ∈ K, 0 < (ε ∘ φ) x := λ x hx, hε _,\n  obtain ⟨ε₀, hε₀, hε₀'⟩ :=\n    hK.exists_forall_le' (hε'.comp φ.continuous).continuous_on hεφ,\n  obtain ⟨τ, hτ : 0 < τ, hτ'⟩ := metric.uniform_continuous_on_iff.mp h₁ ε₀ hε₀,\n  refine ⟨min τ 1, by simp [hτ], λ g p hp p' hp' x hx hη,  _⟩,\n  cases lt_min_iff.mp hη with H H',\n  specialize hεφ x hx,\n  apply lt_of_lt_of_le _ (hε₀' x hx), clear hε₀',\n  simp only [update_apply_embedding],\n  have h₁ : g p' x ∈ K₁,\n    from metric.mem_cthickening_of_dist_le (g p' x) (F (p, x)) 1 _ ⟨(p, x), ⟨hp, hx⟩, rfl⟩ H'.le,\n  have h₂ : f p (φ x) ∈ range ψ,\n    from hf' p ⟨φ x, mem_range_self _, rfl⟩,\n  rw ← ψ.right_inv h₂,\n  exact hτ' _ h₁ _ (metric.self_subset_cthickening _ ⟨(p, x), ⟨hp, hx⟩, rfl⟩) H,\nend\n\nend metric\n\nend updating\n\nend open_smooth_embedding\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/global/smooth_embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.31423092333412456}}
{"text": "import SciLean.Data.ArrayType.Algebra\n\nnamespace SciLean\n\nvariable {Cont : Type} {Idx : Type |> outParam} {Elem : Type |> outParam}\nvariable [ArrayType Cont Idx Elem] [Enumtype Idx] \n\n-- There are some issues working with `getElem : (x : Cont) → (i : Idx) → Dom x i → Elem`\n-- bacause it has inherently dependent types plus `Dom x i : Prop` and \n-- we do not have `Vec (P → X)` for `P : Prop` and `X : Type`\n\n--------------------------------------------------------------------------------\n-- getElem \n--------------------------------------------------------------------------------\n\ninstance getElem.arg_cont_.isLin [Vec Elem]\n  : IsLin (λ (cont : Cont) (idx : Idx) => cont[idx]) := sorry_proof\ninstance getElem.arg_cont.isLin [Vec Elem] (idx : Idx)\n  : IsLin (λ (cont : Cont) => cont[idx]) := sorry_proof\n\ninstance getElem.arg_cont_.isSmooth [Vec Elem]\n  : IsSmooth (λ (cont : Cont) (idx : Idx) => cont[idx]) := by infer_instance\ninstance getElem.arg_cont.isSmooth [Vec Elem] (idx : Idx)  \n  : IsSmooth (λ (cont : Cont) => cont[idx]) := by infer_instance\ninstance getElem.arg_cont.composition.isSmooth [Vec Elem] [Vec X]\n  (f : X → Cont) [IsSmoothT f] (idx : Idx)\n  : IsSmoothT (λ (x : X) => (f x)[idx]) := IsSmoothT_comp₃ (λ cont => cont[idx]) f\n\n\n@[diff] theorem getElem.arg_cont_.diff_simp [Vec Elem]\n  : ∂ (λ (cont : Cont) (idx : Idx) => cont[idx]) = λ cont dcont idx => dcont[idx]\n  := by symdiff; done\n@[diff] theorem getElem.arg_cont_.tangentMap_simp [Vec Elem]\n  : 𝒯 (λ (cont : Cont) (idx : Idx) => cont[idx])\n    = \n    λ cont dcont => (λ idx => cont[idx], λ idx => dcont[idx])\n  := by symdiff; done\n@[diff] theorem getElem.arg_cont.diff_simp [Vec Elem] (idx : Idx)\n  : ∂ (λ (cont : Cont) => cont[idx]) = λ cont dcont => dcont[idx]\n  := by symdiff; done\n@[diff] theorem getElem.arg_cont.tangentMap_simp [Vec Elem] (idx : Idx)\n  : 𝒯 (λ (cont : Cont) => cont[idx])\n    = \n    λ cont dcont => (cont[idx],dcont[idx])\n  := by symdiff; done\n@[diff] theorem getElem.arg_cont.composition.diff_simp [Vec Elem] [Vec X]\n  (f : X → Cont) [IsSmoothT f] (idx : Idx)\n  : ∂ (λ (x : X) => (f x)[idx]) = λ x dx => (∂ f x dx)[idx]\n  := by rw[differential.of_comp (λ cont => cont[idx]'sorry_proof) f]; symdiff; done\n\n\ninstance getElem.arg_cont.hasAdjoint [SemiHilbert Elem] (idx : Idx)\n  : HasAdjoint (λ (cont : Cont) => cont[idx]) := sorry_proof\n@[diff] theorem getElem.arg_cont.adj_simp [SemiHilbert Elem] (idx : Idx)\n  : (λ (cont : Cont) => cont[idx])† = λ cont' => setElem 0 idx cont' := sorry_proof\n@[diff] theorem getElem.arg_cont_idx.adj_simp [SemiHilbert Elem]\n  : (λ (cont : Cont) (idx: Idx) => cont[idx])† = λ cont' => introElem cont' := sorry_proof\n@[diff] theorem getElem.arg_cont.composition.adj_simp [SemiHilbert Elem] [SemiHilbert X] (idx : Idx)\n  (f : X → Cont) [HasAdjointT f]\n  : (λ x => (f x)[idx])† = λ x' => f† (setElem 0 idx x') :=\nby \n  rw[comp.arg_x.adj_simp (λ cont : Cont => cont[idx]'True.intro) f]; symdiff; done\n\ninstance getElem.arg_cont.hasAdjDiff [SemiHilbert Elem] (idx : Idx)\n  : HasAdjDiff (λ (cont : Cont) => cont[idx]) := by apply infer_HasAdjDiff'; symdiff; infer_instance; done\n\n@[diff] theorem getElem.arg_cont.adjDiff_simp [SemiHilbert Elem] (idx : Idx)\n  : ∂† (λ (cont : Cont) => cont[idx]) = λ _ dcont' => setElem 0 idx dcont' := by unfold adjointDifferential; symdiff; symdiff; done\n@[diff] theorem getElem.arg_cont_idx.adjDiff_simp [SemiHilbert Elem]\n  : ∂† (λ (cont : Cont) idx => cont[idx]) = λ _ dcont' => introElem dcont' := by unfold adjointDifferential; symdiff; symdiff; done\n@[diff] theorem getElem.arg_cont.composition.adjDiff_simp [SemiHilbert Elem] [SemiHilbert X] (idx : Idx)\n  (f : X → Cont) [inst : HasAdjDiffT f]\n  : ∂† (λ (x : X) => (f x)[idx]) = λ x dx' => ∂† f x (setElem 0 idx dx') := \nby \n  have _ := inst.1\n  have _ := inst.2\n\n  unfold adjointDifferential\n  symdiff; symdiff\n  done\n\n\n--------------------------------------------------------------------------------\n-- setElem \n--------------------------------------------------------------------------------\n\nfunction_properties setElem [Vec Elem] (cont : Cont) (idx : Idx) (elem : Elem) : Cont\n-- argument (cont,elem)\n--   isLin := sorry_proof,\n--   isSmooth,\n--   abbrev ∂ 𝒯 := setElem dcont idx delem by sorry_proof\nargument cont\n  isSmooth := sorry_proof, \n  abbrev ∂ 𝒯 := setElem dcont idx 0 by sorry_proof\nargument elem\n  isSmooth := sorry_proof,\n  abbrev ∂ 𝒯 := setElem 0 idx delem by sorry_proof\n\n\nfunction_properties setElem [SemiHilbert Elem] (cont : Cont) (idx : Idx) (elem : Elem) : Cont\nargument cont \n  hasAdjoint [Fact (elem=0)] := sorry_proof,\n  -- abbrev † [Fact (elem=0)] := setElem cont' idx 0 by sorry_proof\n  hasAdjDiff := sorry_proof,\n  abbrev ∂† ℛ := setElem dcont' idx 0 by unfold adjointDifferential; symdiff; sorry_proof\nargument elem\n  hasAdjoint [Fact (cont=0)] := sorry_proof,\n  abbrev † [Fact (cont=0)] := elem'[idx] by sorry_proof,\n  hasAdjDiff := sorry_proof,\n  abbrev ∂† := delem'[idx] by unfold adjointDifferential; symdiff; symdiff; done\n\n-- @[simp ↓, infer_tc_goals_rl]\n-- instance setElem.arg_cont.adj_simp [SemiHilbert Elem] (idx : Idx) (elem : Elem) -- [Fact (elem=0)]\n--   : (λ cont : Cont => setElem cont idx elem)†\n--     =\n--     (λ cont' => setElem cont' idx 0) := by sorry_proof\n\n-- this clashes with `setElem.arg_cont.adj_simp` for some unknown reason\n-- TODO: remove this once the clash is resolved!\nexample :\n  (λ (x : ℝ) => x + x)†\n  =\n  (λ y => y + y) := by sorry --symdiff; done\n\n-- double check it does not happend with ∂†\nexample :\n  ∂† (λ (x : ℝ) => x + x)\n  =\n  (λ x dy' => dy' + dy') := by sorry --symdiff; done\n\n\n--------------------------------------------------------------------------------\n-- introElem \n--------------------------------------------------------------------------------\n\nfunction_properties introElem [Vec Elem] (f : Idx → Elem) : Cont\nargument f\n  isLin := sorry_proof,\n  isSmooth,\n  abbrev ∂ 𝒯 := introElem df by symdiff\n\nfunction_properties introElem [SemiHilbert Elem] (f : Idx → Elem) : Cont\nargument f\n  hasAdjoint := sorry_proof,\n  abbrev † := λ idx => f'[idx] by sorry_proof,\n  hasAdjDiff, \n  abbrev ∂† ℛ := λ idx => df'[idx] by unfold adjointDifferential; symdiff; symdiff; done\n\n\n---\n\n-- TODO: modify, mapIdx, map\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Data/ArrayType/Properties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836382, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3142240384990168}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.category\n\n/-!\n# The category paths on a quiver.\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnamespace category_theory\n\nsection\n\n/--\nA type synonym for the category of paths in a quiver.\n-/\ndef paths (V : Type u₁) : Type u₁ := V\n\ninstance (V : Type u₁) [inhabited V] : inhabited (paths V) := ⟨(default V : V)⟩\n\nvariables (V : Type u₁) [quiver.{v₁+1} V]\n\nnamespace paths\n\ninstance category_paths : category.{max u₁ v₁} (paths V) :=\n{ hom := λ (X Y : V), quiver.path X Y,\n  id := λ X, quiver.path.nil,\n  comp := λ X Y Z f g, quiver.path.comp f g, }\n\nvariables {V}\n\n/--\nThe inclusion of a quiver `V` into its path category, as a prefunctor.\n-/\n@[simps]\ndef of : prefunctor V (paths V) :=\n{ obj := λ X, X,\n  map := λ X Y f, f.to_path, }\n\nend paths\n\nvariables (W : Type u₂) [quiver.{v₂+1} W]\n\n-- A restatement of `prefunctor.map_path_comp` using `f ≫ g` instead of `f.comp g`.\n@[simp] lemma prefunctor.map_path_comp' (F : prefunctor V W)\n  {X Y Z : paths V} (f : X ⟶ Y) (g : Y ⟶ Z) :\n  F.map_path (f ≫ g) = (F.map_path f).comp (F.map_path g) :=\nprefunctor.map_path_comp _ _ _\n\nend\n\nsection\n\nvariables {C : Type u₁} [category.{v₁} C]\n\nopen quiver\n\n/-- A path in a category can be composed to a single morphism. -/\n@[simp]\ndef compose_path {X : C} : Π {Y : C} (p : path X Y), X ⟶ Y\n| _ path.nil := 𝟙 X\n| _ (path.cons p e) := compose_path p ≫ e\n\n@[simp]\nlemma compose_path_comp {X Y Z : C} (f : path X Y) (g : path Y Z) :\n  compose_path (f.comp g) = compose_path f ≫ compose_path g :=\nbegin\n  induction g with Y' Z' g e ih,\n  { simp, },\n  { simp [ih], },\nend\n\nend\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/path_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.31415080043149624}}
{"text": "import classes.context_sensitive.basics.toolbox\nimport utilities.list_utils\n\n\ninductive Te\n| a_ : Te\n| b_ : Te\n| c_ : Te\n\ninductive Nt\n| B_ : Nt\n| C_ : Nt\n| R_ : Nt\n| S_ : Nt\n| X_ : Nt\n| Y_ : Nt\n| Z_ : Nt\n\nprivate def a : symbol Te Nt := symbol.terminal Te.a_\nprivate def b : symbol Te Nt := symbol.terminal Te.b_\nprivate def c : symbol Te Nt := symbol.terminal Te.c_\n\nprivate def B : symbol Te Nt := symbol.nonterminal Nt.B_\nprivate def C : symbol Te Nt := symbol.nonterminal Nt.C_\nprivate def R : symbol Te Nt := symbol.nonterminal Nt.R_\nprivate def S : symbol Te Nt := symbol.nonterminal Nt.S_\nprivate def X : symbol Te Nt := symbol.nonterminal Nt.X_\nprivate def Y : symbol Te Nt := symbol.nonterminal Nt.Y_\nprivate def Z : symbol Te Nt := symbol.nonterminal Nt.Z_\n\n\n/-- rule `S → aSBC | aRC` as two context-sensitive rules -/\nprivate def r₁ : csrule Te Nt := csrule.mk [] Nt.S_ [] [a, S, B, C]\nprivate def r₂ : csrule Te Nt := csrule.mk [] Nt.S_ [] [a, R, C]\n\n/-- non-contracting rule `CB → BC` modelled by `CB → XB → XC → BC` which is context-sensitive -/\nprivate def r₃  : csrule Te Nt := csrule.mk [] Nt.C_ [B] [X]\nprivate def r₃' : csrule Te Nt := csrule.mk [X] Nt.B_ [] [C]\nprivate def r₃'': csrule Te Nt := csrule.mk [] Nt.X_ [C] [B]\n\n/-- non-contracting rule `RB → bR` modelled by `RB → YB → YR → bR` which is context-sensitive -/\nprivate def r₄  : csrule Te Nt := csrule.mk [] Nt.R_ [B] [Y]\nprivate def r₄' : csrule Te Nt := csrule.mk [Y] Nt.B_ [] [R]\nprivate def r₄'': csrule Te Nt := csrule.mk [] Nt.Y_ [R] [b]\n\n/-- non-contracting rule `RC → bc` modelled by `RC → ZC → Zc → bc` which is context-sensitive -/\nprivate def r₅  : csrule Te Nt := csrule.mk [] Nt.R_ [C] [Z]\nprivate def r₅' : csrule Te Nt := csrule.mk [Z] Nt.C_ [] [c]\nprivate def r₅'': csrule Te Nt := csrule.mk [] Nt.Z_ [c] [b]\n\n/-- context-sensitive rule `cC → cc` -/\nprivate def r₆ : csrule Te Nt := csrule.mk [c] Nt.C_ [] [c]\n\n\nprivate def my_grammar : CS_grammar Te :=\nCS_grammar.mk Nt Nt.S_ [r₁, r₂, r₃, r₃', r₃'', r₄, r₄', r₄'', r₅, r₅', r₅'', r₆]\n\n\n/-- generate `abc` by the grammar above -/\nexample : [Te.a_, Te.b_, Te.c_] ∈ CS_language my_grammar :=\nbegin\n  unfold my_grammar,\n\n  apply CS_deri_of_tran_deri,\n  {\n    use r₂,\n    split_ile,\n    use [[], []],\n    split;\n    refl,\n  },\n\n  apply CS_deri_of_tran_deri,\n  {\n    use r₅,\n    split_ile,\n    use [[a], []],\n    split;\n    refl,\n  },\n\n  apply CS_deri_of_tran_deri,\n  {\n    use r₅',\n    split_ile,\n    use [[a], []],\n    split;\n    refl,\n  },\n\n  apply CS_deri_of_tran,\n  {\n    use r₅'',\n    split_ile,\n    use [[a], []],\n    split;\n    refl,\n  },\nend\n\n\nprivate meta def CS_step (rule : pexpr) (pref post : pexpr) : tactic unit := `[\n  apply CS_deri_of_tran_deri,\n  tactic.use [rule],\n  split_ile,\n  tactic.use [pref, post],\n  split;\n  refl\n]\n\n/-- generate `aabbcc` by the grammar above -/\nexample : [Te.a_, Te.a_, Te.b_, Te.b_, Te.c_, Te.c_] ∈ CS_language my_grammar :=\nbegin\n  unfold my_grammar,\n  -- S\n\n  CS_step ``(r₁) ``([]) ``([]),\n  -- aSBC\n\n  CS_step ``(r₂) ``([a]) ``([B, C]),\n  -- aaRCBClet\n\n  CS_step ``(r₃)   ``([a, a, R]) ``([C]),\n  CS_step ``(r₃')  ``([a, a, R]) ``([C]),\n  CS_step ``(r₃'') ``([a, a, R]) ``([C]),\n  -- aaRBCC\n\n  CS_step ``(r₄)   ``([a, a]) ``([C, C]),\n  CS_step ``(r₄')  ``([a, a]) ``([C, C]),\n  CS_step ``(r₄'') ``([a, a]) ``([C, C]),\n  -- aabRCC\n\n  CS_step ``(r₅)   ``([a, a, b]) ``([C]),\n  CS_step ``(r₅')  ``([a, a, b]) ``([C]),\n  CS_step ``(r₅'') ``([a, a, b]) ``([C]),\n  -- aabbcC\n\n  CS_step ``(r₆) ``([a, a, b, b]) ``([]),\n  -- aabbcc\n\n  apply CS_deri_self,\nend\n\n\nprivate meta def combined_steps_r₃ (pre pos : pexpr) : tactic unit := `[\n  CS_step ``(r₃)   pre pos,\n  CS_step ``(r₃')  pre pos,\n  CS_step ``(r₃'') pre pos\n]\n\nprivate meta def combined_steps_r₄ (pre pos : pexpr) : tactic unit := `[\n  CS_step ``(r₄)   pre pos,\n  CS_step ``(r₄')  pre pos,\n  CS_step ``(r₄'') pre pos\n]\n\n/-- generate `aaabbbccc` by the grammar above -/\nexample : [Te.a_, Te.a_, Te.a_, Te.b_, Te.b_, Te.b_, Te.c_, Te.c_, Te.c_] ∈ CS_language my_grammar :=\nbegin\n  -- S\n\n  CS_step ``(r₁) ``([]) ``([]),\n  -- aSBC\n\n  CS_step ``(r₁) ``([a]) ``([B, C]),\n  -- aaSBCBC\n\n  CS_step ``(r₂) ``([a, a]) ``([B, C, B, C]),\n  -- aaaRCBCBC\n\n  combined_steps_r₃ ``([a, a, a, R]) ``([C, B, C]),\n  -- aaaRBCCBC\n\n  combined_steps_r₃ ``([a, a, a, R, B, C]) ``([C]),\n  -- aaaRBCBCC\n\n  combined_steps_r₃ ``([a, a, a, R, B]) ``([C, C]),\n  -- aaaRBBCCC\n\n  combined_steps_r₄ ``([a, a, a]) ``([B, C, C, C]),\n  -- aaabRBCCC\n\n  combined_steps_r₄ ``([a, a, a, b]) ``([C, C, C]),\n  -- aaabbRCCC\n\n  CS_step ``(r₅)   ``([a, a, a, b, b]) ``([C, C]),\n  CS_step ``(r₅')  ``([a, a, a, b, b]) ``([C, C]),\n  CS_step ``(r₅'') ``([a, a, a, b, b]) ``([C, C]),\n  -- aaabbbcCC\n\n  CS_step ``(r₆) ``([a, a, a, b, b, b]) ``([C]),\n  -- aaabbbccC\n\n  CS_step ``(r₆) ``([a, a, a, b, b, b, c]) ``([]),\n  -- aaabbbccc\n\n  apply CS_deri_self,\nend\n\n\n/-- generate ` a^4 . b^4 . c^4 ` by the grammar above -/\nexample : [Te.a_, Te.a_, Te.a_, Te.a_,\n           Te.b_, Te.b_, Te.b_, Te.b_,\n           Te.c_, Te.c_, Te.c_, Te.c_]\n    ∈ CS_language my_grammar :=\nbegin\n\n  -- .S.\n  CS_step ``(r₁) ``([]) ``([]),\n  -- .aSBC.\n\n  -- a.S.BC\n  CS_step ``(r₁) ``([a]) ``([B, C]),\n  -- a.aSBC.BC\n\n  -- aa.S.BCBC\n  CS_step ``(r₁) ``([a, a]) ``([B, C, B, C]),\n  -- aa.aSBC.BCBC\n\n  -- aaa.S.BCBCBC\n  CS_step ``(r₂) ``([a, a, a]) ``([B, C, B, C, B, C]),\n  -- aaa.aRC.BCBCBC\n\n  -- aaaaR.CB.CBCBC\n  combined_steps_r₃ ``([a, a, a, a, R]) ``([C, B, C, B, C]),\n  -- aaaaR.BC.CBCBC\n\n  -- aaaaRBC.CB.CBC\n  combined_steps_r₃ ``([a, a, a, a, R, B, C]) ``([C, B, C]),\n  -- aaaaRBC.BC.CBC\n\n  -- aaaaRB.CB.CCBC\n  combined_steps_r₃ ``([a, a, a, a, R, B]) ``([C, C, B, C]),\n  -- aaaaRB.BC.CCBC\n\n  -- aaaaRBBCC.CB.C\n  combined_steps_r₃ ``([a, a, a, a, R, B, B, C, C]) ``([C]),\n  -- aaaaRBBCC.BC.C\n\n  -- aaaaRBBC.CB.CC\n  combined_steps_r₃ ``([a, a, a, a, R, B, B, C]) ``([C, C]),\n  -- aaaaRBBC.BC.CC\n\n  -- aaaaRBB.CB.CCC\n  combined_steps_r₃ ``([a, a, a, a, R, B, B]) ``([C, C, C]),\n  -- aaaaRBB.BC.CCC\n\n  -- aaaa.RB.BBCCCC\n  combined_steps_r₄ ``([a, a, a, a]) ``([B, B, C, C, C, C]),\n  -- aaaa.bR.BBCCCC\n\n  -- aaaab.RB.BCCCC\n  combined_steps_r₄ ``([a, a, a, a, b]) ``([B, C, C, C, C]),\n  -- aaaab.bR.BCCCC\n\n  -- aaaabb.RB.CCCC\n  combined_steps_r₄ ``([a, a, a, a, b, b]) ``([C, C, C, C]),\n  -- aaaabb.bR.CCCC\n\n  -- aaaabbb.RC.CCC\n  CS_step ``(r₅)   ``([a, a, a, a, b, b, b]) ``([C, C, C]),\n  -- aaaabbb.ZC.CCC\n  CS_step ``(r₅')  ``([a, a, a, a, b, b, b]) ``([C, C, C]),\n  -- aaaabbb.Zc.CCC\n  CS_step ``(r₅'') ``([a, a, a, a, b, b, b]) ``([C, C, C]),\n  -- aaaabbb.bc.CCC\n\n  -- aaaabbbb.cC.CC\n  CS_step ``(r₆) ``([a, a, a, a, b, b, b, b]) ``([C, C]),\n  -- aaaabbbb.cc.CC\n\n  -- aaaabbbbc.cC.C\n  CS_step ``(r₆) ``([a, a, a, a, b, b, b, b, c]) ``([C]),\n  -- aaaabbbbc.cc.C\n\n  -- aaaabbbbcc.cC.\n  CS_step ``(r₆) ``([a, a, a, a, b, b, b, b, c, c]) ``([]),\n  -- aaaabbbbcc.cc.\n\n  apply CS_deri_self,\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/test/demo_context_sensitive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.31415080043149624}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1 : G.colorable 2,\n    have h2 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h1, simple_graph.colorable_def, simple_graph.le_def, simple_graph.complete_bipartite_graph_def, simple_graph.edge_def, simple_graph.vertex_def, simple_graph.cast_def, simple_graph.congr_arg_def, simple_graph.complete_graph_def, simple_graph.disjoint_union_def, simple_graph.disjoint_def, simple_graph.subset_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem_def, simple_graph.mem\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.39233683016710835, "lm_q1q2_score": 0.31414096204248243}}
{"text": "import tactic\nimport topology.basic topology.path_connected topology.continuous_on\nimport data.set order.filter.basic\n\nopen classical unit_interval path\nopen_locale classical unit_interval filter topological_space\nnoncomputable theory\n\nuniverses u v w\n\ninstance : has_one ↥(set.Icc (0 : ℝ) 2) := { one := ⟨1 , and.intro (by simp) (by linarith)⟩ }\n\n@[simp, norm_cast] lemma coe_one : ((1 : set.Icc (0 : ℝ) 2) : ℝ) = 1 := rfl\n\n@[simp] lemma mk_one (h : (1 : ℝ) ∈ set.Icc (0 : ℝ) 2) : (⟨1 , h⟩ : set.Icc (0 : ℝ) 2) = 1 := rfl \n\ndef coe_Pi_fun (α : Type u) (β : Type v) (p : α → Prop) (f : Π (a : α), p a → β) : {a : α | p a} → β := λ a, f a a.2\n\ninstance (α : Type u) (β : Type v) (p : α → Prop) : has_coe (Π (a : α), p a → β) ({a | p a} → β) := { coe := λ f, coe_Pi_fun α β p f }\n\n@[simp, norm_cast] lemma coe_pi_fun {α : Type u} {β : Type v} {p : α → Prop} {f : Π (a : α), p a → β} : (f : {a | p a} → β) = coe_Pi_fun α β p f := rfl\n \ninstance coe_sub (α : Type u) (s : set α) : has_coe {x // x ∈ s} s := { coe := λ ⟨a , h⟩, ⟨a , h⟩ }\n\ninstance mem_sub {α : Type u} {s : set α} : has_mem {x // x ∈ s} (set ↥s) := { mem := λ ⟨a , h⟩ U, (⟨a , h⟩ : s) ∈ U }\n\n@[simp] lemma mem_sub_norm (α : Type u) (s : set α) (U : set ↥s) (a : {x // x ∈ s}) (h : a.1 ∈ s) : (a : {x // x ∈ s}) ∈ (U : set s) ↔ (⟨a.1 , h⟩ : s) ∈ (U : set s) := by simp\n\ninstance subset_has_mem (α : Type u) (s : set α) : has_mem α (set ↥s) := { mem := λ a U, ∃ (h : a ∈ s), (⟨a , h⟩ : s) ∈ U }\n\n@[simp] lemma subset_has_mem_norm (α : Type u) (s : set α) (U : set ↥s) (a : α) (h : a ∈ s) : a ∈ U ↔ ↑(⟨a , h⟩ : s) ∈ U := iff.rfl\n\ndef coe_subset_fun (α : Type u) (s : set α) (U : set s) : set α := {a | a ∈ U}\n\ninstance coe_subset (α  : Type u) (s : set α) : has_coe (set s) (set α) := { coe := coe_subset_fun α s }\n\n@[simp, norm_cast] lemma mem_sub_norm_more (α : Type u) (s : set α) (U : set s) : (U : set α) = {a | a ∈ U} := rfl\n\n@[simp, norm_cast] lemma mem_sub_norm_univ (α : Type u) (s : set α) (x : α) : x ∈ ((@set.univ s) : set α) ↔ x ∈ s := \nbegin \n  split,\n  { intro h,\n  cases h,\n  exact h_w, },\n  { \n    intro h,\n    split,\n    swap,\n    exact h,\n    rw set.univ,\n    apply set.mem_def.2,\n    tauto,\n  },\nend\n\n@[simp, norm_cast] lemma mem_sub_norm_univ_eq (α : Type u) (s : set α) : ↑(@set.univ s) = s := set.ext (λ x, mem_sub_norm_univ α s x)\n\n@[simp] lemma mem_sub_norm_sub (α : Type u) (s : set α) (U : set s) : (U : set α) ⊆ s :=\nbegin \n  intros x hx,\n  cases hx,\n  exact hx_w,\nend\n\n@[simp] lemma mem_sub_norm_sub2 (α : Type u) (s : set α) (U V : set s) : U ⊆ V ↔ (U : set α) ⊆ (V : set α) :=\nbegin\n  split,\n  intro H,\n  intros a ha,\n  cases ha with Ha ha,\n  split,\n  swap,\n  exact Ha,\n  apply H,\n  exact ha,\n  intro H,\n  intros a ha,\n  cases a with a Ha,\n  specialize H ⟨_ , ha⟩,\n  cases H with Ha' H,\n  simp [Ha] at H,\n  exact H,\nend \n\n@[simp, norm_cast] lemma mem_sub_norm_inter (α : Type u) (s : set α) (x y : set s) : (↑(x ∩ y) : set α) = ↑x ∩ ↑y :=\nbegin\n  apply set.ext,\n  intro a,\n  split,\n  intro ha,\n  cases ha,\n  cases ha_h,\n  split,\n  split,\n  exact ha_h_left,\n  split,\n  exact ha_h_right,\n  intro ha,\n  simp,\n  simp at ha,\n  cases ha with hax hay,\n  cases hax with hasx hax,\n  cases hay with hasy hay,\n  have haa : hasx = hasy,\n  {\n    simp,\n  },\n  rw haa at *,\n  existsi hasy,\n  split,\n  exact hax,\n  exact hay,\nend\n\ndef intersection_filter (α : Type u) (s : set α) (𝓕 : filter α) : filter s := \n{ sets := {U | ∃ (F : set α), F ∈ 𝓕 ∧ (U : set α) = (s : set α) ∩ (F : set α)},\n  univ_sets := set.mem_def.2 (exists.intro (@set.univ α) \n                                           (and.intro 𝓕.univ_sets \n                                           (trans (mem_sub_norm_univ_eq α s) (by simp)))),\n  sets_of_superset := λ x y hx hxy, \n                     begin \n                       cases hx with F hF,\n                       existsi (y : set α) ∪ F,\n                       split,\n                       {\n                         apply (𝓕.sets_of_superset hF.1),\n                         simp,\n                       },\n                       {\n                         apply set.eq_of_subset_of_subset,\n                         intros y₁ hy₁,\n                         split,\n                         exact mem_sub_norm_sub α s y hy₁,\n                         left,\n                         exact hy₁,\n                         rw set.inter_distrib_left,\n                         intros xx hxx,\n                         cases hxx,\n                         cases hxx,\n                         exact hxx_right,\n                         rw ← hF.2 at hxx,\n                         cases hxx,\n                         specialize hxy hxx_h,\n                         split,\n                         exact hxy,\n                       },\n                     end,\n  inter_sets := λ x y hx hy,\n                begin \n                  cases hx with Fx hFx,\n                  cases hy with Fy hFy,\n                  existsi Fx ∩ Fy,\n                  split,\n                  apply (𝓕.inter_sets hFx.1 hFy.1),\n                  apply set.eq_of_subset_of_subset,\n                  intros a ha,\n                  split,\n                  cases ha,\n                  exact ha_w,\n                  cases hFx,\n                  cases hFy,\n                  rw mem_sub_norm_inter at ha,\n                  rw hFx_right at ha,\n                  rw hFy_right at ha,\n                  split,\n                  cases ha,\n                  cases ha_left,\n                  exact ha_left_right,\n                  cases ha,\n                  cases ha_right,\n                  exact ha_right_right,\n                  intros a ha,\n                  rw mem_sub_norm_inter,\n                  split,\n                  rw hFx.2,\n                  cases ha,\n                  cases ha_right,\n                  split,\n                  exact ha_left,\n                  exact ha_right_left,\n                  rw hFy.2,\n                  cases ha,\n                  cases ha_right,\n                  split,\n                  exact ha_left,\n                  exact ha_right_right,\n                end }\n\ninstance (α : Type u) (s : set α) : has_coe (filter α) (filter s) := { coe := intersection_filter α s }\n\nvariables (α : Type u) (β : Type v) (p : α → Prop) [topological_space α] [topological_space β]\n\ndef interior_set (s : set α) : set s := {e | (e : α) ∈ interior s}\n\n@[simp] lemma interior_set_norm (s : set α) : ↑(interior_set α s) = interior s := \nbegin\n  apply set.ext,\n  intro x,\n  split,\n  intro hx,\n  rcases hx with ⟨a , b , c ,d⟩,\n  split,\n  use c,\n  exact d,\n  intro hx,\n  rcases hx with ⟨a , b , c⟩,\n  split,\n  split,\n  use b,\n  exact c,\n  rcases b with ⟨d  , e⟩,\n  apply e,\n  exact c,\nend\n\nlemma continuous_dif (f : Π (a : α), p a → β) (g : Π (a : α), ¬ p a → β)\n                     (hf : continuous (f : {a | p a} → β))\n                     (hg : continuous (g : {a | ¬ p a} → β))\n                     (hfb : ∀ x ∈ (frontier {a | p a}), \n                            ∀ H : ¬ p x, (filter.tendsto (f : {a | p a} → β) ↑(𝓝 x) (𝓝 ((g : {a | ¬ p a} → β) ⟨x , H⟩))))\n                     (hgb : ∀ x ∈ (frontier {a | p a}),\n                            ∀ H : p x, (filter.tendsto (g : {a | ¬ p a} → β) ↑(𝓝 x) (𝓝 ((f : {a | p a} → β) ⟨x , H⟩))))\n                     : continuous (λ a, dite (p a) (f a) (g a)) := \ncontinuous_iff_continuous_at.2 (λ x, \nbegin\n  apply filter.tendsto_def.2,\n  intros s hs,\n  have H1 := set.compl_union_self (interior ({a | p a})),\n  rw ← closure_compl at H1,\n  rw closure_eq_interior_union_frontier at H1,\n  rw frontier_compl at H1,\n  have H2 : x ∈ interior {a : α | p a}ᶜ ∪ frontier {a : α | p a} ∪ interior {a : α | p a},\n  {\n    have H2a : x ∈ set.univ,\n    apply set.mem_def.2,\n    tauto,\n    rw ← H1 at H2a,\n    exact H2a,\n  },\n  cases H2 with H3 H2,\n  swap,\n  have Hx : p x,\n  {\n    apply set.mem_def.1,\n    apply interior_subset,\n    exact H2,\n  },\n  simp [Hx] at hs,\n  have Hf : ((f : {a | p a} → β) ⁻¹' s) ∈ (𝓝 ⟨x , Hx⟩ : filter {a | p a}),\n  {\n    have Hfa := continuous_iff_continuous_at.1 hf ⟨x , Hx⟩,\n    have Hfaa := filter.tendsto_def.1 Hfa s,\n    apply Hfaa,\n    exact hs,\n  },\n  rw nhds_subtype_eq_comap at Hf,\n  rcases Hf with ⟨U , HU1 , HU2⟩,\n  have HU3 : U ∩ (interior {a | p a}) ∈ 𝓝 x,\n  {\n    apply filter.inter_mem_sets,\n    exact HU1,\n    rw interior_eq_nhds' at H2,\n    simp at H2,\n    apply interior_mem_nhds.2,\n    exact H2,\n  },\n  apply (𝓝 x).sets_of_superset,\n  exact HU3,\n  intros a ha,\n  cases ha with haU haI,\n  have Ha2 : p a,\n  {\n     apply set.mem_def.1,\n     apply mem_of_mem_nhds,\n     apply mem_interior_iff_mem_nhds.1,\n     exact haI,\n  },\n  have Ha3 : (⟨a , Ha2⟩ : {a |  p a}) ∈ coe ⁻¹' U,\n  {\n     apply set.mem_def.2,\n     apply set.mem_def.2,\n     use haU,\n  },\n  specialize HU2 Ha3,\n  simp [Ha2],\n  use HU2,\n  cases H3,\n  have Hx : ¬ p x,\n  {\n     rw set.compl_set_of p at H3,\n     apply (@set.mem_def α x {a | ¬ p a}).1,\n     apply @interior_subset α _ {a | ¬ p a},\n     exact H3,\n  },\n  simp [Hx] at hs,\n  have Hg : ((g : {a | ¬ p a} → β) ⁻¹' s) ∈ (𝓝 ⟨x , Hx⟩ : filter {a | ¬ p a}),\n  {\n     have Hga := continuous_iff_continuous_at.1 hg ⟨x , Hx⟩,\n     have Hgaa := filter.tendsto_def.1 Hga s,\n     apply Hgaa,\n     exact hs,\n  },\n  rw nhds_subtype_eq_comap at Hg,\n  rcases Hg with ⟨U , HU1 , HU2⟩,\n  have HU3 : U ∩ (interior {a | ¬ p a}) ∈ 𝓝 x,\n  {\n    apply filter.inter_mem_sets,\n    exact HU1,\n    rw interior_eq_nhds' at H3,\n    rw set.compl_set_of p at H3,\n    simp at H3,\n    apply interior_mem_nhds.2,\n    exact H3,\n  },\n  apply (𝓝 x).sets_of_superset,\n  exact HU3,\n  intros a ha,\n  cases ha with haU haI,\n  have Ha2 : ¬ p a,\n  {\n    apply (@set.mem_def α a {a | ¬ p a}).1,\n    apply @interior_subset α _ {a | ¬ p a},\n    exact haI,\n  },\n  have Ha3 : (⟨a , Ha2⟩ : {a | ¬ p a}) ∈ coe ⁻¹' U,\n  {\n    apply set.mem_def.2,\n    apply set.mem_def.2,\n    use haU,\n  },\n  specialize HU2 Ha3,\n  simp [Ha2],\n  use HU2,\n  by_cases p x,\n  specialize hgb x H3 h,\n  rw filter.tendsto_def at hgb,\n  simp [h] at hs,\n  specialize hgb s hs,\n  rcases hgb with ⟨V, HV1, HV2⟩,\n  have Hf : ((f : {a | p a} → β) ⁻¹' s) ∈ (𝓝 ⟨x , h⟩ : filter {a | p a}),\n  {\n    have Hfa := continuous_iff_continuous_at.1 hf ⟨x , h⟩,\n    have Hfaa := filter.tendsto_def.1 Hfa s,\n    apply Hfaa,\n    exact hs,\n  },\n  rw nhds_subtype_eq_comap at Hf,\n  rcases Hf with ⟨U , HU1 , HU2⟩,\n  have HUV := filter.inter_mem_sets HU1 HV1,\n  apply (𝓝 x).sets_of_superset,\n  exact HUV,\n  intros a ha,\n  cases ha with haU haV,\n  rename h Hx,\n  by_cases p a,\n  swap,\n  simp [h],\n  have Ha2 : a ∈ {a | ¬ p a} ∩ V,\n  {\n    split,\n    use h,\n    exact haV,\n  },\n  rw ← HV2 at Ha2,\n  cases Ha2 with Ha Ha2,\n  use Ha2,\n  have Ha : (⟨a , h⟩ : {a | p a}) ∈ coe ⁻¹' U,\n  {\n    apply set.mem_def.2,\n    apply set.mem_def.2,\n    use haU,\n  },\n  specialize HU2 Ha,\n  simp [h],\n  use HU2,\n  specialize hfb x H3 h,\n  rw filter.tendsto_def at hfb,\n  simp [h] at hs,\n  specialize hfb s hs,\n  rcases hfb with ⟨V , HV1 , HV2⟩,\n  have Hg : ((g : {a | ¬ p a} → β) ⁻¹' s) ∈ (𝓝 ⟨x , h⟩ : filter {a | ¬ p a}),\n  {\n    have Hga := continuous_iff_continuous_at.1 hg ⟨x , h⟩,\n    have Hgaa := filter.tendsto_def.1 Hga s,\n    apply Hgaa,\n    exact hs,\n  },\n  rw nhds_subtype_eq_comap at Hg,\n  rcases Hg with ⟨U , HU1 , HU2⟩,\n  have HUV := filter.inter_mem_sets HU1 HV1,\n  apply (𝓝 x).sets_of_superset,\n  exact HUV,\n  intros a ha,\n  cases ha with haU haV,\n  rename h Hx,\n  by_cases p a,\n  simp [h],\n  have Ha2 : a ∈ {a | p a} ∩ V,\n  {\n    split,\n    use h,\n    exact haV,\n  },\n  rw ← HV2 at Ha2,\n  cases Ha2 with Ha Ha2,\n  use Ha2,\n  have Ha : (⟨a , h⟩ : {a | ¬ p a}) ∈ coe ⁻¹' U,\n  {\n    apply set.mem_def.2,\n    apply set.mem_def.2,\n    use haU,\n  },\n  specialize HU2 Ha,\n  simp [h],\n  use HU2,\nend).\n\nlemma front_single (x : ↥I × ↥(set.Icc (0 : ℝ) 2)) : x ∈ frontier {a : ↥I × ↥(set.Icc (0 : ℝ) 2) | a.snd ≤ 1} → x.snd = 1 := \nλ hx, \nfrontier_le_subset_eq (continuous_snd) (by continuity) hx\n\nlemma coe_pi_fun_eq {α : Type u} {β : Type v} {p : α → Prop} {f : Π (a : α), p a → β} {x : α} {hx : p x} : (f : {a | p a} → β) ⟨x , hx⟩ = f x hx := rfl\n\ninstance : has_zero ↥(set.Icc (0 : ℝ) 2) := { zero := ⟨0 , by simp⟩ }\n\n@[simp, norm_cast] lemma coe_zero : ((0 : set.Icc (0 : ℝ) 2) : ℝ) = 0 := rfl\n\n@[simp] lemma mk_zero (h : (0 : ℝ) ∈ set.Icc (0 : ℝ) 2) : (⟨0 , h⟩ : set.Icc (0 : ℝ) 2) = 0 := rfl \n\ndef two : ↥(set.Icc (0 : ℝ) 2) := ⟨2 , by simp⟩\n\n@[simp, norm_cast] lemma coe_two : ((two : set.Icc (0 : ℝ) 2) : ℝ) = 2 := rfl\n\n@[simp] lemma mk_two (h : (2 : ℝ) ∈ set.Icc (0 : ℝ) 2) : (⟨2 , h⟩ : set.Icc (0 : ℝ) 2) = two := rfl \n\nlemma real_arith : (2 : ℝ) - 1 = 1 :=\nbegin\n  linarith,\nend\n\nlemma two_sub_one (h : (2 : ℝ) - 1 ∈ I ) : (⟨2 - 1 , h⟩ : I) = 1 :=\nbegin\n  ext,\n  simp,\n  exact real_arith,\nend", "meta": {"author": "owen-fool", "repo": "Lean_Fundamental_Group", "sha": "ed415401c987b907dbe2b78984e7d3200d3e9b6c", "save_path": "github-repos/lean/owen-fool-Lean_Fundamental_Group", "path": "github-repos/lean/owen-fool-Lean_Fundamental_Group/Lean_Fundamental_Group-ed415401c987b907dbe2b78984e7d3200d3e9b6c/src/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.31387831602554556}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Simon Hudon\n-/\nimport category_theory.monoidal.braided\nimport category_theory.limits.shapes.binary_products\nimport category_theory.pempty\n\n/-!\n# The monoidal structure on a category with chosen finite products.\n\nThis is a variant of the development in `category_theory.monoidal.of_has_finite_products`,\nwhich uses specified choices of the terminal object and binary product,\nenabling the construction of a cartesian category with specific definitions of the tensor unit\nand tensor product.\n\n(Because the construction in `category_theory.monoidal.of_has_finite_products` uses `has_limit`\nclasses, the actual definitions there are opaque behind `classical.choice`.)\n\nWe use this in `category_theory.monoidal.types` to construct the monoidal category of types\nso that the tensor product is the usual cartesian product of types.\n\nFor now we only do the construction from products, and not from coproducts,\nwhich seems less often useful.\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C] {X Y : C}\n\nnamespace limits\n\nsection\nvariables {C}\n\n/-- Swap the two sides of a `binary_fan`. -/\ndef binary_fan.swap {P Q : C} (t : binary_fan P Q) : binary_fan Q P :=\nbinary_fan.mk t.snd t.fst\n\n@[simp] lemma binary_fan.swap_fst {P Q : C} (t : binary_fan P Q) : t.swap.fst = t.snd := rfl\n@[simp] lemma binary_fan.swap_snd {P Q : C} (t : binary_fan P Q) : t.swap.snd = t.fst := rfl\n\n/--\nIf a cone `t` over `P Q` is a limit cone, then `t.swap` is a limit cone over `Q P`.\n-/\n@[simps]\ndef is_limit.swap_binary_fan {P Q : C} {t : binary_fan P Q} (I : is_limit t) : is_limit t.swap :=\n{ lift := λ s, I.lift (binary_fan.swap s),\n  fac' := λ s, by { rintro ⟨⟨⟩⟩; simp, },\n  uniq' := λ s m w,\n  begin\n    have h := I.uniq (binary_fan.swap s) m,\n    rw h,\n    rintro ⟨j⟩,\n    specialize w ⟨j.swap⟩,\n    cases j; exact w,\n  end }\n\n/--\nConstruct `has_binary_product Q P` from `has_binary_product P Q`.\nThis can't be an instance, as it would cause a loop in typeclass search.\n-/\nlemma has_binary_product.swap (P Q : C) [has_binary_product P Q] : has_binary_product Q P :=\nhas_limit.mk ⟨binary_fan.swap (limit.cone (pair P Q)), (limit.is_limit (pair P Q)).swap_binary_fan⟩\n\n/--\nGiven a limit cone over `X` and `Y`, and another limit cone over `Y` and `X`, we can construct\nan isomorphism between the cone points. Relative to some fixed choice of limits cones for every\npair, these isomorphisms constitute a braiding.\n-/\ndef binary_fan.braiding {X Y : C}\n  {s : binary_fan X Y} (P : is_limit s) {t : binary_fan Y X} (Q : is_limit t) :\n  s.X ≅ t.X :=\nis_limit.cone_point_unique_up_to_iso P Q.swap_binary_fan\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `sXY.X Z`,\nif `sYZ` is a limit cone we can construct a binary fan over `X sYZ.X`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  binary_fan X sYZ.X :=\nbinary_fan.mk (s.fst ≫ sXY.fst) (Q.lift (binary_fan.mk (s.fst ≫ sXY.snd) s.snd))\n\n@[simp] lemma binary_fan.assoc_fst {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).fst = s.fst ≫ sXY.fst := rfl\n@[simp] lemma binary_fan.assoc_snd {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).snd = Q.lift (binary_fan.mk (s.fst ≫ sXY.snd) s.snd) := rfl\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `X sYZ.X`,\nif `sYZ` is a limit cone we can construct a binary fan over `sXY.X Z`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc_inv {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  binary_fan sXY.X Z :=\nbinary_fan.mk (P.lift (binary_fan.mk s.fst (s.snd ≫ sYZ.fst))) (s.snd ≫ sYZ.snd)\n\n@[simp] lemma binary_fan.assoc_inv_fst {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).fst = P.lift (binary_fan.mk s.fst (s.snd ≫ sYZ.fst)) := rfl\n@[simp] lemma binary_fan.assoc_inv_snd {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).snd = s.snd ≫ sYZ.snd := rfl\n\n/--\nIf all the binary fans involved a limit cones, `binary_fan.assoc` produces another limit cone.\n-/\n@[simps]\ndef is_limit.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) : is_limit (s.assoc Q) :=\n{ lift := λ t, R.lift (binary_fan.assoc_inv P t),\n  fac' := λ t,\n  begin\n    rintro ⟨⟨⟩⟩; simp,\n    apply Q.hom_ext,\n    rintro ⟨⟨⟩⟩; simp,\n  end,\n  uniq' := λ t m w,\n  begin\n    have h := R.uniq (binary_fan.assoc_inv P t) m,\n    rw h,\n    rintro ⟨⟨⟩⟩; simp,\n    apply P.hom_ext,\n    rintro ⟨⟨⟩⟩; simp,\n    { exact w ⟨walking_pair.left⟩, },\n    { specialize w ⟨walking_pair.right⟩,\n      simp at w,\n      rw [←w], simp, },\n    { specialize w ⟨walking_pair.right⟩,\n      simp at w,\n      rw [←w], simp, },\n  end, }\n\n/--\nGiven two pairs of limit cones corresponding to the parenthesisations of `X × Y × Z`,\nwe obtain an isomorphism between the cone points.\n-/\n@[reducible]\ndef binary_fan.associator {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) {t : binary_fan X sYZ.X} (S : is_limit t) :\n  s.X ≅ t.X :=\nis_limit.cone_point_unique_up_to_iso (is_limit.assoc P Q R) S\n\n/--\nGiven a fixed family of limit data for every pair `X Y`, we obtain an associator.\n-/\n@[reducible]\ndef binary_fan.associator_of_limit_cone\n  (L : Π X Y : C, limit_cone (pair X Y)) (X Y Z : C) :\n  (L (L X Y).cone.X Z).cone.X ≅ (L X (L Y Z).cone.X).cone.X :=\nbinary_fan.associator\n  (L X Y).is_limit (L Y Z).is_limit\n  (L (L X Y).cone.X Z).is_limit (L X (L Y Z).cone.X).is_limit\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/--\nConstruct a left unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.left_unitor {X : C} {s : cone (functor.empty.{v} C)} (P : is_limit s)\n  {t : binary_fan s.X X} (Q : is_limit t) : t.X ≅ X :=\n{ hom := t.snd,\n  inv := Q.lift (binary_fan.mk (P.lift\n    { X := X, π := { app := discrete.rec (pempty.rec _) } }) (𝟙 X) ),\n  hom_inv_id' :=\n  by { apply Q.hom_ext, rintro ⟨⟨⟩⟩, { apply P.hom_ext, rintro ⟨⟨⟩⟩, }, { simp, }, }, }\n\n/--\nConstruct a right unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.right_unitor {X : C} {s : cone (functor.empty.{v} C)} (P : is_limit s)\n  {t : binary_fan X s.X} (Q : is_limit t) : t.X ≅ X :=\n{ hom := t.fst,\n  inv := Q.lift (binary_fan.mk (𝟙 X) (P.lift\n    { X := X, π := { app := discrete.rec (pempty.rec _) } })),\n  hom_inv_id' :=\n  by { apply Q.hom_ext, rintro ⟨⟨⟩⟩, { simp, }, { apply P.hom_ext, rintro ⟨⟨⟩⟩, }, }, }\n\nend\n\nend limits\n\nopen category_theory.limits\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\nvariables {C}\nvariables (𝒯 : limit_cone (functor.empty.{v} C))\nvariables (ℬ : Π (X Y : C), limit_cone (pair X Y))\n\nnamespace monoidal_of_chosen_finite_products\n\n/-- Implementation of the tensor product for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_obj (X Y : C) : C := (ℬ X Y).cone.X\n\n/-- Implementation of the tensor product of morphisms for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_hom {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) : tensor_obj ℬ W Y ⟶ tensor_obj ℬ X Z :=\n  (binary_fan.is_limit.lift' (ℬ X Z).is_limit\n    ((ℬ W Y).cone.π.app ⟨walking_pair.left⟩ ≫ f)\n    (((ℬ W Y).cone.π.app ⟨walking_pair.right⟩ : (ℬ W Y).cone.X ⟶ Y) ≫ g)).val\n\nlemma tensor_id (X₁ X₂ : C) : tensor_hom ℬ (𝟙 X₁) (𝟙 X₂) = 𝟙 (tensor_obj ℬ X₁ X₂) :=\nbegin\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : C}\n  (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁) (g₂ : Y₂ ⟶ Z₂) :\n  tensor_hom ℬ (f₁ ≫ g₁) (f₂ ≫ g₂) =\n    tensor_hom ℬ f₁ f₂ ≫ tensor_hom ℬ g₁ g₂ :=\nbegin\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma pentagon (W X Y Z : C) :\n  tensor_hom ℬ (binary_fan.associator_of_limit_cone ℬ W X Y).hom (𝟙 Z) ≫\n    (binary_fan.associator_of_limit_cone ℬ W (tensor_obj ℬ X Y) Z).hom ≫\n      tensor_hom ℬ (𝟙 W) (binary_fan.associator_of_limit_cone ℬ X Y Z).hom =\n  (binary_fan.associator_of_limit_cone ℬ (tensor_obj ℬ W X) Y Z).hom ≫\n    (binary_fan.associator_of_limit_cone ℬ W X (tensor_obj ℬ Y Z)).hom :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩,\n  { simp, },\n  { apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩,\n    { simp, },\n    apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩,\n    { simp, },\n    { simp, }, }\nend\n\nlemma triangle (X Y : C) :\n  (binary_fan.associator_of_limit_cone ℬ X 𝒯.cone.X Y).hom ≫\n    tensor_hom ℬ (𝟙 X) (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X Y).is_limit).hom =\n  tensor_hom ℬ (binary_fan.right_unitor 𝒯.is_limit (ℬ X 𝒯.cone.X).is_limit).hom (𝟙 Y) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩; simp,\nend\n\nlemma left_unitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :\n  tensor_hom ℬ (𝟙 𝒯.cone.X) f ≫ (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X X₂).is_limit).hom =\n    (binary_fan.left_unitor 𝒯.is_limit (ℬ 𝒯.cone.X X₁).is_limit).hom ≫ f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma right_unitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :\n  tensor_hom ℬ f (𝟙 𝒯.cone.X) ≫\n    (binary_fan.right_unitor 𝒯.is_limit (ℬ X₂ 𝒯.cone.X).is_limit).hom =\n    (binary_fan.right_unitor 𝒯.is_limit (ℬ X₁ 𝒯.cone.X).is_limit).hom ≫ f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :\n  tensor_hom ℬ (tensor_hom ℬ f₁ f₂) f₃ ≫ (binary_fan.associator_of_limit_cone ℬ Y₁ Y₂ Y₃).hom =\n    (binary_fan.associator_of_limit_cone ℬ X₁ X₂ X₃).hom ≫\n      tensor_hom ℬ f₁ (tensor_hom ℬ f₂ f₃) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩,\n  { simp, },\n  { apply is_limit.hom_ext (ℬ _ _).is_limit, rintro ⟨⟨⟩⟩,\n    { simp, },\n    { simp, }, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/-- A category with a terminal object and binary products has a natural monoidal structure. -/\ndef monoidal_of_chosen_finite_products :\n  monoidal_category C :=\n{ tensor_unit  := 𝒯.cone.X,\n  tensor_obj   := λ X Y, tensor_obj ℬ X Y,\n  tensor_hom   := λ _ _ _ _ f g, tensor_hom ℬ f g,\n  tensor_id'   := tensor_id ℬ,\n  tensor_comp' := λ _ _ _ _ _ _ f₁ f₂ g₁ g₂, tensor_comp ℬ f₁ f₂ g₁ g₂,\n  associator   := λ X Y Z, binary_fan.associator_of_limit_cone ℬ X Y Z,\n  left_unitor  := λ X, binary_fan.left_unitor (𝒯.is_limit) (ℬ 𝒯.cone.X X).is_limit,\n  right_unitor := λ X, binary_fan.right_unitor (𝒯.is_limit) (ℬ X 𝒯.cone.X).is_limit,\n  pentagon'    := pentagon ℬ,\n  triangle'    := triangle 𝒯 ℬ,\n  left_unitor_naturality' := λ _ _ f, left_unitor_naturality 𝒯 ℬ f,\n  right_unitor_naturality' := λ _ _ f, right_unitor_naturality 𝒯 ℬ f,\n  associator_naturality' := λ _ _ _ _ _ _ f₁ f₂ f₃, associator_naturality ℬ f₁ f₂ f₃, }\n\nnamespace monoidal_of_chosen_finite_products\n\nopen monoidal_category\n\n/--\nA type synonym for `C` carrying a monoidal category structure corresponding to\na fixed choice of limit data for the empty functor, and for `pair X Y` for every `X Y : C`.\n\nThis is an implementation detail for `symmetric_of_chosen_finite_products`.\n-/\n@[derive category, nolint unused_arguments has_nonempty_instance]\ndef monoidal_of_chosen_finite_products_synonym\n  (𝒯 : limit_cone (functor.empty.{v} C)) (ℬ : Π (X Y : C), limit_cone (pair X Y)):= C\n\ninstance : monoidal_category (monoidal_of_chosen_finite_products_synonym 𝒯 ℬ) :=\nmonoidal_of_chosen_finite_products 𝒯 ℬ\n\nlemma braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :\n  (tensor_hom ℬ f g) ≫ (limits.binary_fan.braiding (ℬ Y Y').is_limit (ℬ Y' Y).is_limit).hom =\n    (limits.binary_fan.braiding (ℬ X X').is_limit (ℬ X' X).is_limit).hom ≫ (tensor_hom ℬ g f) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟨⟩⟩;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\n\n\nlemma hexagon_reverse (X Y Z : C) :\n  (binary_fan.associator_of_limit_cone ℬ X Y Z).inv ≫\n    (limits.binary_fan.braiding\n      (ℬ (tensor_obj ℬ X Y) Z).is_limit\n      (ℬ Z (tensor_obj ℬ X Y)).is_limit).hom ≫\n    (binary_fan.associator_of_limit_cone ℬ Z X Y).inv =\n    (tensor_hom ℬ (𝟙 X) (limits.binary_fan.braiding (ℬ Y Z).is_limit (ℬ Z Y).is_limit).hom) ≫\n      (binary_fan.associator_of_limit_cone ℬ X Z Y).inv ≫\n        (tensor_hom ℬ (limits.binary_fan.braiding (ℬ X Z).is_limit (ℬ Z X).is_limit).hom (𝟙 Y)) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟨⟩⟩,\n  { apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟨⟩⟩;\n    { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n        limits.is_limit.cone_point_unique_up_to_iso],\n      simp, }, },\n  { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n      limits.is_limit.cone_point_unique_up_to_iso],\n    simp, },\nend\n\nlemma symmetry (X Y : C) :\n  (limits.binary_fan.braiding (ℬ X Y).is_limit (ℬ Y X).is_limit).hom ≫\n      (limits.binary_fan.braiding (ℬ Y X).is_limit (ℬ X Y).is_limit).hom =\n    𝟙 (tensor_obj ℬ X Y) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (ℬ _ _).is_limit.hom_ext, rintro ⟨⟨⟩⟩;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/--\nThe monoidal structure coming from finite products is symmetric.\n-/\ndef symmetric_of_chosen_finite_products :\n  symmetric_category (monoidal_of_chosen_finite_products_synonym 𝒯 ℬ) :=\n{ braiding := λ X Y, limits.binary_fan.braiding (ℬ _ _).is_limit (ℬ _ _).is_limit,\n  braiding_naturality' := λ X X' Y Y' f g, braiding_naturality ℬ f g,\n  hexagon_forward' := λ X Y Z, hexagon_forward ℬ X Y Z,\n  hexagon_reverse' := λ X Y Z, hexagon_reverse ℬ X Y Z,\n  symmetry' := λ X Y, symmetry ℬ X Y, }\n\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/of_chosen_finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.31387831602554556}}
{"text": "\nimport Lib.Data.AssocList\nimport Lib.Data.Sorting\nimport Lib.Data.These\nimport Lib.Order.Basic\n\nopen Std\n\nnamespace Option\nopen These\n\ndef mergeWith (f : These α β → Option γ) :\n  Option α → Option β → Option γ\n| none, none => none\n| some x, none => f <| inl x\n| none, some y => f <| inr y\n| some x, some y => f <| both x y\n\nend Option\n\nnamespace These\n\n@[simp]\ndef unionWith (f : α → α → α) : These α α → Option α\n| inl x => some x\n| inr x => some x\n| both x y => some <| f x y\n\n@[simp]\ndef intersectionWith (f : α → β → γ) : These α β → Option γ\n| inl x => none\n| inr x => none\n| both x y => some <| f x y\n\n@[simp]\ndef differenceWith (f : α → β → Option α) : These α β → Option α\n| inl x => some x\n| inr x => none\n| both x y => f x y\n\nend These\n\nnamespace Std.AssocList\n\nsection defs\n\nvariable [Ord k]\nvariable (f : k → These α β → Option γ)\n-- #check Std.AssocList.m\nopen Nat\n-- @[simp]\n-- theorem lt_succ_succ (x : Nat) :\n--   x < succ (succ x) :=\n-- by trans succ x <;> auto\n\nopen These AssocList\n\n@[specialize]\ndef zipWith : AssocList k α → AssocList k β → AssocList k γ\n| AssocList.nil, xs =>\n  xs.mapFilter <| (f . ∘ inr)\n| xs, AssocList.nil =>\n  xs.mapFilter <| (f . ∘ inl)\n| AssocList.cons i x xs, AssocList.cons j y ys =>\n  match compare i j with\n  | Ordering.lt =>\n    let zs := zipWith xs (cons j y ys)\n    match f i <| inl x with\n    | none => zs\n    | some x' => cons i x' zs\n      -- <|\n  | Ordering.eq =>\n    let zs := zipWith xs ys\n    match f i <| both x y with\n    | none => zs\n    | some x' => cons i x' zs\n  | Ordering.gt =>\n    let zs := zipWith (AssocList.cons i x xs) ys\n    match f i <| inr y with\n    | none => zs\n    | some x' => cons j x' zs\ntermination_by zipWith x y => (x.length, y.length)\n-- #find prefix getEqn\n-- #eval Lean.Meta.getEqnsFor? ``zipWith\n\n-- -- #check Substring\n-- variable (xs ys : AssocList k _)\n-- open AssocList\n-- @[simp]\n-- theorem addAux_nil₀ :\n--   zipWith f nil ys =\n--   ys.mapVal (f ∘ inr) := by\n-- cases ys <;>\n-- exact WellFounded.fix_eq _ _ _\n\n-- @[simp]\n-- theorem addAux_nil₁ :\n--   zipWith f xs nil =\n--   xs.mapVal (f ∘ inl) := by\n-- cases xs <;>\n-- exact WellFounded.fix_eq _ _ _\n\n-- @[simp]\n-- theorem addAux_cons i x j y :\n--   zipWith f (cons i x xs) (cons j y ys) =\n--   match compare i j with\n--   | Ordering.lt =>\n--     cons i (f <| inl x) <| zipWith f xs (cons j y ys)\n--   | Ordering.eq =>\n--     cons i (f <| both x y) <| zipWith f xs ys\n--   | Ordering.gt =>\n--     cons j (f <| inr y) <| zipWith f (cons i x xs) ys\n--   := WellFounded.fix_eq _ _ _\n\nend defs\n\n-- def R : Nat × α → Nat × α → Prop := (InvImage (.<.) Prod.fst)\nopen List\n\n-- @[simp]\n-- theorem Sorted_map_map :\n--   Sorted2 R (List.map (Prod.map id fy) ys) ↔\n--   Sorted2 R ys := sorry\n\n-- @[simp]\n-- theorem All_LT_map_map :\n--   All (R (i, fy x)) (List.map (Prod.map id fy) ys) ↔\n--   All (R (i, x)) ys := sorry\n\nvariable [LE k] [DecidableTotalOrder k]\nvariable (f : k → These α β → Option γ)\n\nopen List Std.AssocList\n-- #check @zipWith\n\n@[auto]\ntheorem Sublist_keys_mapFilter {xs : AssocList α β}\n        {f : α → β → Option γ}:\n  Sublist (xs.mapFilter f |>.keys) xs.keys := by\ninduction xs <;> simp [mapFilter, keys]\n. constructor\nsplit\n<;> simp [keys]\n<;> constructor <;> assumption\n\ntheorem All_LT_zipWith (i : k) :\n        -- {xs ys : AssocList k α}\n        (xs : AssocList k α) →\n        (ys : AssocList k β) →\n        (Hxs : All (i < .) xs.keys) →\n        (Hys : All (i < .) ys.keys) →\n  All (i < .) (zipWith f xs ys).keys\n| nil, xs, Hxs, Hys => by\n  simp [zipWith]\n  apply All_of_All_of_Sublist _ Hys\n  auto\n| xs, nil, Hxs, Hys => by\n  cases xs <;> simp [zipWith]\n  <;> apply All_of_All_of_Sublist _ Hxs\n  <;> auto\n| cons i x xs, cons j y ys, Hxs, Hys => by\n  simp [zipWith, keys] at Hxs Hys ⊢\n  cases Hxs; cases Hys\n  next Hxs Hxs' Hys Hys' =>\n  split <;>\n  next h =>\n    simp at h\n    split  <;>\n    next h' =>\n      repeat\n        simp [keys]\n        first\n        | constructor\n        | auto\n        | apply All_LT_zipWith\ntermination_by All_LT_zipWith x y _ _ => (x.length, y.length)\n\ntheorem Sorted_zipWith (f : k → These α β → Option γ) :\n        {xs ys : AssocList k _} →\n        (Hxs : Sorted2 (.<.) xs.keys) →\n        (Hys : Sorted2 (.<.) ys.keys) →\n  Sorted2 (.<.) (zipWith f xs ys).keys\n| nil, xs, Hxs, Hys => by\n  simp [keys, zipWith]\n  apply Sorted2_of_Sorted2_of_Sublist _ Hys\n  apply Sublist_keys_mapFilter\n| xs, nil, Hxs, Hys => by\n  cases xs\n  <;> simp [keys, zipWith]\n  <;> apply Sorted2_of_Sorted2_of_Sublist _ Hxs\n  . constructor\n  . apply Sublist_keys_mapFilter\n| cons i x xs, cons j y ys, Hxs, Hys => by\n  simp [zipWith, keys] at Hxs Hys ⊢\n  cases Hxs; cases Hys\n  next Hxs Hxs' Hys Hys' =>\n  split <;>\n  next h =>\n    simp at h\n    first\n    | have h'' := Preorder.le_of_lt h\n    | have h'' := Preorder.le_of_eq h\n    | have h'' := True.intro\n    split <;>\n    next h' =>\n      repeat\n        try simp [keys]\n        first\n        | constructor\n        | auto\n        | apply Sorted_zipWith\n        | apply All_LT_zipWith\n        | apply All_LT_of_lt_of_All_LT h''\n      skip\ntermination_by Sorted_zipWith x y _ _ => (x.length, y.length)\n\nend Std.AssocList\n\nopen List\nstructure OrdMap (α : Type u) [LT α] (β : Type v) where\n  vals : AssocList α β\n  sorted : Sorted2 (.<.) vals.keys\n\nnamespace OrdMap\n\nsection defs\n\nvariable {k} [LT k]\n\ndef empty : OrdMap k α where\n  vals := AssocList.nil\n  sorted := by constructor\n\ndef singleton (x : k) (v : α) : OrdMap k α where\n  vals := AssocList.cons x v AssocList.nil\n  sorted := by repeat constructor\n\nend defs\n\nsection OpWith\n\nopen Std.AssocList These\n\nvariable [LE k] [DecidableTotalOrder k]\nvariable (f : k → α → α → α)\nvariable (g : k → These α β → Option γ)\n\n@[specialize]\ndef mergeWith (x : OrdMap k α) (y : OrdMap k β) : OrdMap k γ where\n  vals := x.vals.zipWith g y.vals\n  sorted := Sorted_zipWith _ x.sorted y.sorted\n\ndef unionWith :=\nmergeWith (λ a => These.unionWith (f a))\n\ndef union : OrdMap k α → OrdMap k α → OrdMap k α :=\nunionWith (λ a x _ => x)\n\nvariable (f : k → α → β → γ)\n\ndef intersectionWith :=\nmergeWith (λ a => These.intersectionWith (f a))\n\ndef intersection : OrdMap k α → OrdMap k α → OrdMap k α :=\nintersectionWith (λ a x _ => x)\n\nvariable (f : k → α → β → Option α)\n\ndef differenceWith : OrdMap k α → OrdMap k β → OrdMap k α :=\nmergeWith (λ a => These.differenceWith (f a))\n\ndef difference : OrdMap k α → OrdMap k β → OrdMap k α :=\ndifferenceWith λ a x _ => none\n\ndef insert (x : k) (v : α) : OrdMap k α → OrdMap k α :=\nunion (singleton x v)\n\ndef erase (x : k) : OrdMap k α → OrdMap k α :=\nλ m => difference m (singleton x ())\n\nend OpWith\n\nsection misc\nvariable [LT k]\n\ndef foldl (f : β → k → α → β) (x₀ : β) (m : OrdMap k α) : β :=\nm.vals.foldl f x₀\n\nvariable [BEq k]\n\ndef find? (x : k) (m : OrdMap k α) : Option α :=\nm.vals.find? x\n\nend misc\n\nsection thms\nvariable [LE k] [DecidableTotalOrder k]\n\nvariable (x y : OrdMap k α)\n-- variable\n\n\ntheorem ext\n        (h : ∀ i, x.find? i = y.find? i) :\n  x = y := sorry\n\nvariable (y : OrdMap k β)\n\n@[simp]\ntheorem find?_merge (f : k → These α β → Option γ) :\n  (mergeWith f x y).find? i =\n  Option.mergeWith (f i) (x.find? i) (y.find? i) :=\nsorry\n\n@[simp]\ntheorem find?_empty (i : k) : empty.find? i = @none α := sorry\n\n-- @[simp]\n-- theorem find?_singleton (i : k) : empty.find? i = @none α := sorry\n\nend thms\n\nend OrdMap\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/OrdMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837477, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3137650755428967}}
{"text": "universe variable u\nvariable {α : Type u}\nvariable [comm_ring α]\nvariable [f : α → α]\nvariable [p : α → Prop]\n\n/- Our first example is solved using congruence closure, and\n   theory AC. It gets solved as soon as we introduce the hypothesis -/\nlemma smt1 (a b c : α) : a = b → p (a + c) → p (c + b) :=\nbegin [smt]\n  intros\nend\n\n/- The tactic performs unit propagation without performing CNF conversion,\n   and propagates equivalences between propositions. -/\nlemma smt2 (p q r : Prop) : p ∧ (q ∨ r) → (p ∧ q) ∨ (p ∧ r) :=\nbegin [smt]\n  intros\nend\n\n#print smt1\n#print smt2\n", "meta": {"author": "holtzermann17", "repo": "lean_experiments", "sha": "3ebb7048c9ca766814c10404ba217b50544d20b2", "save_path": "github-repos/lean/holtzermann17-lean_experiments", "path": "github-repos/lean/holtzermann17-lean_experiments/lean_experiments-3ebb7048c9ca766814c10404ba217b50544d20b2/src/built-in-smt-demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5698526660244837, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.31376507518297414}}
{"text": "import category_theory.limits.concrete_category\n\nuniverses v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nlocal attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n\nsection equalizer\n\nlemma concrete.equalizer_ext {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]\n  [preserves_limit (parallel_pair f g) (forget C)] (x y : equalizer f g)\n  (h : equalizer.ι f g x = equalizer.ι f g y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintros (a|a),\n  { apply h },\n  { rw [← limit.w (parallel_pair f g) walking_parallel_pair_hom.right,\n    comp_apply, comp_apply, h] }\nend\n\ndef concrete.equalizer_equiv_aux {X Y : C} (f g : X ⟶ Y) :\n  (parallel_pair f g ⋙ forget C).sections ≃ { x : X // f x = g x } :=\n{ to_fun := λ x, ⟨x.1 walking_parallel_pair.zero, begin\n    have h1 := x.2 walking_parallel_pair_hom.left,\n    have h2 := x.2 walking_parallel_pair_hom.right,\n    dsimp at h1 h2,\n    erw [h1, h2],\n  end⟩,\n  inv_fun := λ x,\n  { val := λ j,\n    match j with\n    | walking_parallel_pair.zero := x.1\n    | walking_parallel_pair.one := f x.1\n    end,\n    property := begin\n      dsimp [functor.sections],\n      rintros (a|a) (b|b) (f|f),\n      { simp, },\n      { refl },\n      { exact x.2.symm },\n      { simp },\n    end },\n  left_inv := begin\n    rintros ⟨x,hx⟩,\n    ext (a|a),\n    { refl },\n    { change _ = x _,\n      rw ← hx walking_parallel_pair_hom.left,\n      refl }\n  end,\n  right_inv := by { rintros ⟨_,_⟩, ext, refl } }\n\nnoncomputable\ndef concrete.equalizer_equiv {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]\n  [preserves_limit (parallel_pair f g) (forget C)] :\n  ↥(equalizer f g) ≃ { x // f x = g x } :=\nlet h1 := limit.is_limit (parallel_pair f g),\n    h2 := is_limit_of_preserves (forget C) h1,\n    E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit.{_ v} _) in\nE.to_equiv.trans $ concrete.equalizer_equiv_aux _ _\n\n@[simp]\nlemma concrete.equalizer_equiv_apply {X Y : C} (f g : X ⟶ Y) [has_equalizer f g]\n  [preserves_limit (parallel_pair f g) (forget C)] (x : equalizer f g):\n  (concrete.equalizer_equiv f g x : X) = equalizer.ι f g x := rfl\n\nend equalizer\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/concrete_equalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3137554720332443}}
{"text": "import topology.basic categories.cat_limits categories.cat_colimits types2\n\nuniverses v v' u u' w w'\nhott_theory\n\nnamespace hott\nopen hott.set hott.subset hott.categories category_theory.limits category_theory.colimits\n  categories.opposite hott.sigma hott.is_trunc hott.trunc\n\nset_option pp.universes true  \n\n/- The category of topological spaces and continuous maps. -/\n@[hott]\nstructure Top :=\n(carrier : Set)\n(top_str : topological_space carrier)\n\n@[hott]\ninstance Top_to_Set : has_coe_to_sort Top :=\n  has_coe_to_sort.mk Set (λ T : Top, T.carrier)\n\n@[hott]\ninstance topological_space_unbundled (X : Top) : \n  topological_space ↥X := X.top_str\n\nvariables (X : Top)\n\nnamespace topology\n\n/- The set of open sets in a topological space and its lattice structure. -/\n@[hott]\ndef open_sets : Subset (𝒫 ↥X) := \n  {U ∈ (𝒫 ↥X) | is_open ↥X U}\n\n@[hott]\ninstance : has_coe ↥(open_sets X) (Subset ↥X) :=\n  ⟨λ Y : open_sets X, Y.1⟩\n\n@[hott]\nprotected def open_sets.inter (U V : open_sets X) : open_sets X :=\nhave U_is_open : is_open ↥X U, from U.2, \nhave V_is_open : is_open ↥X V, from V.2,\nhave inter_is_open : is_open ↥X (U.1 ∩ V.1), from \n  is_open_inter ↥X U_is_open V_is_open,\n⟨U.1 ∩ V.1, inter_is_open⟩  \n\n@[hott, instance]\ndef open_sets_inter : has_inter (open_sets X) :=\n⟨open_sets.inter X⟩    \n\n@[hott]\ndef open_sets.iUnion {I : Set} (f : I -> open_sets X) : ↥(open_sets X) :=\nhave U_i_is_open : ∀ i : I, is_open ↥X (f i), from assume i, (f i).2, \nhave open_union : is_open ↥X (⋃ᵢ (λ i, (f i).1)), from is_open_iUnion ↥X U_i_is_open,  \n⟨⋃ᵢ (λ i, (f i).1), open_union⟩    \n\n@[hott]\ninstance open_sets_incl_to_hom {U V : open_sets X} : has_coe ↥(U.1 ⊆ V.1) (U ⟶ V) := \n  ⟨λ i : U.1 ⊆ V.1, i⟩ \n\n@[hott]\ndef open_sets_hom_to_emb {U V : open_sets X} (i : U ⟶ V) : U.1 -> V.1 :=\n  assume x, ⟨x, i x x.2⟩\n\n@[hott]\ndef opens.inf_le_l (U V : open_sets X) : open_sets.inter X U V ⟶ U :=\n  inter_sset_l U V \n\n@[hott]\ndef opens.inf_le_r (U V : open_sets X) : open_sets.inter X U V ⟶ V :=\n  inter_sset_r U V \n\n@[hott]\ndef opens.le_union {I : Set} (f : I -> open_sets X) :\n  Π i : I, f i ⟶ open_sets.iUnion X f :=   \nbegin \n  intro i, \n  change ↥(↑(f i) ⊆ ↑(open_sets.iUnion X f)), \n  exact sset_iUnion (λ i, ↑(f i)) i\nend       \n\n@[hott]\ndef opens.hom_eq {U V : open_sets X} (f g : U ⟶ V) : f = g :=\n  power_set_unique_hom f g \n\n@[hott, reducible] \ndef nbhds (x : X.carrier) : Subset (𝒫 ↥X) := \n  {U ∈ (𝒫 ↥X) | is_open ↥X U and x ∈ U} \n\n@[hott]\ndef nbhds_opens_inc (x : X.carrier) : nbhds X x ⊆ open_sets X :=\nbegin  intros U el, exact el.1  end\n\n@[hott, instance]\ndef nbhds_precat (x : X.carrier) : precategory (nbhds X x) :=\n  subset_precat_precat (nbhds X x)\n\n@[hott]\ndef nbhds_opens_inc_functor (x : X.carrier) : (nbhds X x)ᵒᵖ ⥤ (open_sets X)ᵒᵖ :=\n  opposite_functor (functor_subsets_precat (nbhds_opens_inc X x))\n\n/- As a subset of the power set the set of open sets (and sets of neighborhoods) automatically\n   receive a precategory instance. Therefore, we can define a presheaf over a \n   topological space with values in a category `C` and its stalks as follows: -/\n@[hott]\ndef presheaf (C : Type u') [category.{v'} C] := (open_sets X)ᵒᵖ ⥤ C\n\n@[hott]\ndef stalk (C : Type u') [category.{v'} C] [H : has_colimits C] (F : presheaf X C) \n  (x : X.carrier) : C :=\nhave G : (nbhds X x)ᵒᵖ ⥤ C, from nbhds_opens_inc_functor X x ⋙ F, \n@colimit (op_Set (pred_Set (nbhds X x))) _ _ _ G (@has_colimit_of_has_colimits C _ H _ _ _) \n\n/- The product of the sections of a presheaf over a family of open sets. -/\n@[hott]\ndef pi_opens {C : Type u} [category.{v} C] [has_products.{v u w'} C] \n  {I : Set.{w'}} (U : I -> open_sets X) (F : presheaf X C) : C :=   \n∏ (λ i : I, F.obj (op (U i)))\n\n/- The product of the sections of a presheaf over the pairwise intersections \n   of a family of open sets.-/\n@[hott]\ndef pi_inters {C : Type u} [category.{v} C] [has_products.{v u w'} C]\n  {I : Set.{w'}} (U : I -> open_sets X) (F : presheaf X C) : C :=  \n∏ (λ p : ↥(I × I), F.obj (op (open_sets.inter X (U p.1) (U p.2))))\n\n@[hott, reducible]\ndef left_res {C : Type u} [category.{v} C] [has_products.{v u w'} C]\n  {I : Set.{w'}} (U : I -> open_sets X) (F : presheaf X C) : \n  (pi_opens X U F) ⟶ (pi_inters X U F) :=\npi.lift C (λ p : ↥(I × I), pi.π _ p.1 ≫ \n                        F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2))))\n\n@[hott, reducible]\ndef right_res {C : Type u} [category.{v} C] [has_products.{v u w'} C]\n  {I : Set.{w'}} (U : I -> open_sets X) (F : presheaf X C) : \n  (pi_opens X U F) ⟶ (pi_inters X U F) :=\npi.lift C (λ p : ↥I × I, pi.π _ p.2 ≫ \n                       F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2))))\n\n@[hott, reducible]\ndef res {C : Type u} [category.{v} C] [has_products.{v u u_2} C]\n  {I : Set.{u_2}} (U : I -> open_sets X) (F : presheaf X C) :\n  (F.obj (op (open_sets.iUnion X U))) ⟶ (pi_opens X U F) :=  \npi.lift C (λ i : I, F.map (hom_op (opens.le_union X U i))) \n\n@[hott]\ndef w_res {C : Type u} [category.{v} C] [has_products.{v u u_2} C]\n  {I : Set} (U : I -> open_sets X) (F : presheaf X C) :\n  (res X U F) ≫ (left_res X U F) = (res X U F) ≫ (right_res X U F) :=  \nhave left_eq : Π p : ↥(I × I), ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p =\n  F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2) ≫ opens.le_union X U p.1)), from\n  assume p,\n  calc ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p = \n             (res X U F) ≫ ((left_res X U F) ≫ pi.π _ p) : precategory.assoc _ _ _\n       ... = (res X U F) ≫ (pi.π _ p.1 ≫ F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2)))) :\n             by rwr pi.lift_π_eq C _ p \n       ... = ((res X U F) ≫ pi.π _ p.1) ≫ F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2))) :           \n             (precategory.assoc _ _ _)⁻¹\n       ... = F.map (hom_op (opens.le_union X U p.1)) ≫ \n                                      F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2))) :\n             by rwr pi.lift_π_eq C _ p.1 \n       ... = F.map (hom_op (opens.le_union X U p.1) ≫ \n                                             hom_op (opens.inf_le_l X (U p.1) (U p.2))) : \n             (functor.map_comp F _ _)⁻¹\n       ... = F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2) ≫ opens.le_union X U p.1)) : \n             by rwr <- hom_op_funct (opens.inf_le_l X (U p.1) (U p.2)) _,\nhave right_eq : Π p : ↥(I × I), ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p =\n  F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2) ≫ opens.le_union X U p.2)), from\n  assume p,\n  calc ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p = \n             (res X U F) ≫ ((right_res X U F) ≫ pi.π _ p) : precategory.assoc _ _ _\n       ... = (res X U F) ≫ (pi.π _ p.2 ≫ F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2)))) :\n             by rwr pi.lift_π_eq C _ p \n       ... = ((res X U F) ≫ pi.π _ p.2) ≫ F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2))) :           \n             (precategory.assoc _ _ _)⁻¹\n       ... = F.map (hom_op (opens.le_union X U p.2)) ≫ \n                                      F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2))) :\n             by rwr pi.lift_π_eq C _ p.2 \n       ... = F.map (hom_op (opens.le_union X U p.2) ≫ \n                                             hom_op (opens.inf_le_r X (U p.1) (U p.2))) : \n             (functor.map_comp F _ _)⁻¹\n       ... = F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2) ≫ opens.le_union X U p.2)) : \n             by rwr <- hom_op_funct (opens.inf_le_r X (U p.1) (U p.2)) _,\nhave incl_eq : Π p : ↥I × I, (opens.inf_le_l X (U p.1) (U p.2) ≫ opens.le_union X U p.1) =\n                      (opens.inf_le_r X (U p.1) (U p.2) ≫ opens.le_union X U p.2), from\n  assume p, opens.hom_eq X _ _,                      \nhave left_right_eq : Π p : ↥I × I, ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p =\n  ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p, from assume p,\n  calc ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p = \n          F.map (hom_op (opens.inf_le_l X (U p.1) (U p.2) ≫ opens.le_union X U p.1)) : \n          left_eq p\n    ... = F.map (hom_op (opens.inf_le_r X (U p.1) (U p.2) ≫ opens.le_union X U p.2)) : \n          by rwr incl_eq p\n    ... = ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p : (right_eq p)⁻¹,\nhave fun_lr_eq : (λ p : ↥I × I, ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p) =\n  λ p : ↥I × I, ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p, from \n  eq_of_homotopy (λ p, left_right_eq p),                           \ncalc (res X U F) ≫ (left_res X U F) = pi.lift C (λ p : ↥I × I, \n                          ((res X U F) ≫ (left_res X U F)) ≫ pi.π _ p) :\n           pi.hom_is_lift C ((res X U F) ≫ (left_res X U F))\n     ... = pi.lift C (λ p : ↥I × I, \n                          ((res X U F) ≫ (right_res X U F)) ≫ pi.π _ p) :\n           by rwr fun_lr_eq                                  \n     ... = (res X U F) ≫ (right_res X U F) : \n           (pi.hom_is_lift C ((res X U F) ≫ (right_res X U F)))⁻¹                  \n/- Of course, this proof is way too long. In [mathlib], most of the calculations are simplified,\n   but that makes the idea of the proof completely invisible. The challenge is to translate the\n   manipulations in the associated commutative diagram to a readable proof. One of the problems\n  are the long names. -/\n\n@[hott]\ndef sheaf_condition_equalizer_products.fork {C : Type u} [category.{v} C] \n  [has_products C] {I : Set} (U : I -> open_sets X) \n  (F : presheaf X C) : fork (left_res X U F) (right_res X U F) :=\nfork.of_i _ _ (res X U F) (w_res X U F)\n\n@[hott]\ndef sheaf_condition {C : Type u} [category.{v} C] [has_products.{v u u_2} C] \n  (F : presheaf X C) := Π {I : Set} (U : I -> open_sets X), \n  is_limit (sheaf_condition_equalizer_products.fork X U F)\n\n@[hott]\nstructure sheaf (C : Type u) [category.{v} C] [has_products.{v u u_2} C] :=\n(presheaf : presheaf X C)\n(sheaf_condition : sheaf_condition X presheaf)\n\n/- Gluing provides an alternative description of the sheaf condition on set-valued presheafs. -/\n@[hott]\ndef is_gluing (F : presheaf X Set) {I : Set} (U : I -> open_sets X) \n  (sf : Π i : I, F.obj (op (U i))) (s : F.obj (op (open_sets.iUnion X U))) : trunctype.{0} -1 :=\nprop_resize (to_Prop (∀ i : I, F.map (hom_op (opens.le_union X U i)) s = sf i))\n\n@[hott]\ndef is_gluing_to_cond (F : presheaf X Set) {I : Set} (U : I -> open_sets X) \n  (sf : Π i : I, F.obj (op (U i))) (s : F.obj (op (open_sets.iUnion X U))) :\n  is_gluing X F U sf s -> ∀ i : I, F.map (hom_op (opens.le_union X U i)) s = sf i :=\nbegin \n  have P : is_prop (∀ i : I, F.map (hom_op (opens.le_union X U i)) s = sf i), from \n    begin apply is_prop_dprod, intro i, apply is_prop.mk, intros p q, exact is_set.elim _ _ end,\n  intros ig, \n  exact prop_resize_to_prop ig\nend    \n\n@[hott]\ndef is_compatible {F : presheaf X Set} {I : Set} {U : I -> open_sets X} \n  (sf : Π i : I, F.obj (op (U i))) := ∀ (i j : I), \n    F.map (hom_op (opens.inf_le_l X (U i) (U j))) (sf i) = \n                                         F.map (hom_op (opens.inf_le_r X (U i) (U j))) (sf j)\n\n@[hott]\ndef cone_res_is_compatible {F : presheaf X Set} {I : Set} {U : I -> open_sets X} \n  {S : cone (parallel_pair (@left_res X _ _ set_has_products _ U F) \n                             (@right_res X _ _ set_has_products _ U F))} :\n  ∀ Sf : S.X, @is_compatible X F I U (S.π.app wp_pair.up Sf).1 := \nbegin \n  intro Sf, \n  let sf : ↥(@pi_opens X _ _ set_has_products _ U F) := S.π.app wp_pair.up Sf,\n  apply all_prod_all, intro p, \n  have H1 : (pi.π (λ (i : ↥I), F.obj (op (U i))) p.fst ≫ \n                    F.map (hom_op (opens.inf_le_l X (U p.fst) (U p.snd)))) =\n                          (@left_res X _ _ (set_has_products) _ U F ≫ pi.π _ p), from \n    inverse (pi.lift_π_eq _ _ p), \n  have H2 : (@right_res X _ _ (set_has_products) _ U F ≫ pi.π _ p) =\n            (pi.π (λ (i : ↥I), F.obj (op (U i))) p.snd ≫ \n                          F.map (hom_op (opens.inf_le_r X (U p.fst) (U p.snd)))), from\n    pi.lift_π_eq _ _ p,                   \n  let hl : ↥(@has_hom.hom _ walking_parallel_pair_has_hom wp_pair.up wp_pair.down) :=\n    wp_pair_hom.left,\n  have H3 : (S.π.app wp_pair.up) ≫ (@left_res X _ _ (set_has_products) _ U F) =\n                (𝟙 S.X) ≫ (S.π.app wp_pair.down), from \n    inverse (S.π.naturality hl),\n  let hr : ↥(@has_hom.hom _ walking_parallel_pair_has_hom wp_pair.up wp_pair.down) :=\n    wp_pair_hom.right,\n  have H4 : (S.π.app wp_pair.up) ≫ (@right_res X _ _ (set_has_products) _ U F) =\n               (𝟙 S.X) ≫ (S.π.app wp_pair.down), from \n    inverse (S.π.naturality hr), \n  calc (pi.π (λ (i : ↥I), F.obj (op (U i))) p.fst ≫ \n            F.map (hom_op (opens.inf_le_l X (U p.fst) (U p.snd)))) sf =\n                  (@left_res X _ _ (set_has_products) _ U F ≫ pi.π _ p) sf : by rwr H1\n       ... = ((S.π.app wp_pair.up) ≫ (@left_res X _ _ (set_has_products) _ U F ≫ \n                                                                      pi.π _ p)) Sf : rfl\n       ... = (((S.π.app wp_pair.up) ≫ (@left_res X _ _ (set_has_products) _ U F)) ≫ \n                                                                      pi.π _ p) Sf : \n             by rwr precategory.assoc _ _ _\n       ... = (((𝟙 S.X) ≫ (S.π.app wp_pair.down)) ≫ pi.π _ p) Sf : by rwr H3 \n       ... = ((S.π.app wp_pair.up) ≫ (@right_res X _ _ (set_has_products) _ U F) ≫ \n                                                                      pi.π _ p) Sf : \n             by rwr inverse H4           \n       ... = ((S.π.app wp_pair.up) ≫ (@right_res X _ _ (set_has_products) _ U F) ≫ \n                                                                      pi.π _ p) Sf : \n             by rwr inverse (precategory.assoc _ _ _)                                                                                                                                                  \n       ... = (@right_res X _ _ (set_has_products) _ U F ≫ pi.π _ p) sf : rfl\n       ... = (pi.π (λ (i : ↥I), F.obj (op (U i))) p.snd ≫ \n                      F.map (hom_op (opens.inf_le_r X (U p.fst) (U p.snd)))) sf : by rwr H2    \nend                               \n\n@[hott]\ndef sheaf_condition_unique_gluing (F : presheaf X Set) {I : Set} (U : I -> open_sets X) :=\n  ∀ (sf : Π i : I, F.obj (op (U i))), is_compatible X sf -> unique_elem (is_gluing X F U sf) \n\n@[hott] \ndef lift_of_unique_gluing (F : presheaf X Set) {I : Set} (U : I -> open_sets X) :\n  sheaf_condition_unique_gluing X F U ->\n  ∀ (S : cone (parallel_pair (@left_res X _ _ set_has_products _ U F) \n                             (@right_res X _ _ set_has_products _ U F))), \n  S.X ⟶ (sheaf_condition_equalizer_products.fork X U F).X :=\nbegin \n  intros sc_ug S Sf, \n  let sf : ↥(@pi_opens X _ _ set_has_products _ U F) := S.π.app wp_pair.up Sf, \n  apply unique_to_elem (is_gluing X F U sf.1), apply sc_ug, \n  exact cone_res_is_compatible X Sf   \nend  \n\n@[hott] \ndef sheaf_condition_of_unique_gluing (F : presheaf X Set) : \n  (∀ {I : Set} (U : I -> open_sets X), sheaf_condition_unique_gluing X F U) -> \n  @sheaf_condition X _ _ set_has_products F :=\nbegin \n  intros sc_ug I U, \n  fapply is_limit.mk,\n  { apply lift_of_unique_gluing X F U, exact sc_ug U },\n  { intro S, \n    have fac_up : (lift_of_unique_gluing X F U (sc_ug U) S) ≫ (res X U F) = S.π.app wp_pair.up, from \n      begin \n        apply eq_of_homotopy, intro Sf,\n        let sf : ↥(@pi_opens X _ _ set_has_products _ U F) := S.π.app wp_pair.up Sf, \n        fapply sigma_eq,\n        { apply eq_of_homotopy, intro i, \n          let lift_Sf := lift_of_unique_gluing X F U (sc_ug U) S Sf,\n          change F.map (hom_op (opens.le_union X U i)) lift_Sf = sf.1 i,\n          have gc : ↥(is_gluing X F U sf.1 lift_Sf), from \n            unique_to_pred (is_gluing X F U sf.1) (sc_ug U sf.1 (cone_res_is_compatible X Sf)),\n          exact is_gluing_to_cond X F U sf.1 lift_Sf gc i },\n        { apply pathover_of_tr_eq, exact is_prop.elim _ _ }\n      end,\n    intro j, hinduction j, \n    { exact fac_up },\n    { let hl : ↥(@has_hom.hom _ walking_parallel_pair_has_hom wp_pair.up wp_pair.down) :=\n        wp_pair_hom.left,\n      have H : (S.π.app wp_pair.up) ≫ (@left_res X _ _ (set_has_products) _ U F) =\n                 (𝟙 S.X) ≫ (S.π.app wp_pair.down), from \n        inverse (S.π.naturality hl),  \n      calc lift_of_unique_gluing X F U (sc_ug U) S ≫ (res X U F ≫ \n                                                    @left_res X _ _ (set_has_products) _ U F) =\n                 (lift_of_unique_gluing X F U (sc_ug U) S ≫ res X U F) ≫ left_res X U F : \n                 by rwr precategory.assoc _ _ _\n           ... = S.π.app wp_pair.up ≫ @left_res X _ _ (set_has_products) _ U F : by rwr fac_up\n           ... = (𝟙 S.X) ≫ (S.π.app wp_pair.down) : by rwr H      \n           ... = S.π.app wp_pair.down : precategory.id_comp _ } },\n  { intros S m m_lift, apply eq_of_homotopy, intro Sf, \n    let sf : ↥(@pi_opens X _ _ set_has_products _ U F) := S.π.app wp_pair.up Sf,\n    let lift := lift_of_unique_gluing X F U (sc_ug U) S,\n    have H : (sheaf_condition_equalizer_products.fork X U F).π.app wp_pair.up (m Sf) = sf, from \n      homotopy_of_eq (m_lift wp_pair.up) Sf,\n    have m_Sf_ig : ↥(is_gluing X F U sf.1 (m Sf)), from \n      prop_to_prop_resize (λ i, homotopy_of_eq (ap sigma.fst H) i),  \n    have lift_ig : ↥(is_gluing X F U sf.1 (lift Sf)), from \n      unique_to_pred (is_gluing X F U sf.1) (sc_ug U sf.1 (cone_res_is_compatible X Sf)),  \n    let P := unique_to_uniq (is_gluing X F U sf.1) (sc_ug U sf.1 (cone_res_is_compatible X Sf)), \n    exact ap sigma.fst (@is_prop.elim (Σ (a : ↥(F.obj (op (open_sets.iUnion X U)))), \n                               ↥(is_gluing X F U sf.fst a)) P ⟨m Sf, m_Sf_ig⟩ ⟨lift Sf, lift_ig⟩) }\nend    \n\nend topology\n\n@[hott]\nstructure PresheafedSpace (C : Type u) [category.{v} C] :=\n  (carrier : Top)\n  (presheaf : topology.presheaf carrier C)\n    \n@[hott]\nstructure SheafedSpace (C : Type u) [category.{v} C] [has_products C] extends \n  PresheafedSpace C := \n  (sheaf_condition : topology.sheaf_condition carrier presheaf)   \n\n/- We construct (pre-)sheaves of sections to a family of sets over open subsets of a \n   topological space that satisfy a (pre-)local predicate. \n   \n   We need to take functions with values in sets because otherwise the homomorphisms in \n   the presheaf category will not be sets. Therefore, the presheaf category also will be\n   `Set`. -/\nopen topology\n\n@[hott, reducible]\ndef ss_section (U : open_sets X) (T : X.carrier -> Set) := \n  Π x : U.1, (T x).carrier \n\n@[hott]\ndef ss_section_Set (U : open_sets X) (T : X.carrier -> Set) : Set := \n  have is_set_ss_section : is_set (ss_section X U T), from \n    @is_set_dmap (pred_Set U.1) (λ x, T x.1),\n  Set.mk (ss_section X U T) is_set_ss_section\n\n@[hott, reducible]\ndef res_ss_section {U V : open_sets X} (i : U ⟶ V) {T : X.carrier -> Set} :\n  ss_section_Set X V T -> ss_section_Set X U T := \nbegin intros f x, let y : ↥V.1 := ⟨x.1, i x x.2⟩, exact f y end    \n\n@[hott]\ndef id_res_section {U : open_sets X} {T : X.carrier -> Set} :\n  Π f : ss_section_Set X U T, res_ss_section X (𝟙 U) f = f :=  \nbegin \n  intro f, apply eq_of_homotopy, intro x, \n  hinduction x with x' pred_x, refl\nend  \n\n@[hott]\ndef comp_res_section {U V W : open_sets X} {T : X.carrier -> Set} \n  (i : U ⟶ V) (j : V ⟶ W) : Π (f : ss_section_Set X W T), \n    res_ss_section X (i ≫ j) f = (res_ss_section X i) (res_ss_section X j f) :=\nbegin\n  intro f, apply eq_of_homotopy, intro x,\n  hinduction x with x' pred_x, refl\nend   \n\n@[hott]\nstructure prelocal_predicate (T : X.carrier -> Set) :=\n  (pred : Π {U : open_sets X}, ss_section X U T → trunctype.{0} -1)\n  (res : ∀ {U V : open_sets X} (i : U ⟶ V) (f : ss_section X V T) \n           (h : pred f), pred (res_ss_section X i f))\n\n@[hott]\ndef subpresheaf_of_sections {T : X.carrier -> Set} (P : prelocal_predicate X T) :\n  presheaf X Set :=\nbegin  \n  fapply categories.functor.mk, \n  { intro U, exact pred_Set {f ∈ ss_section_Set X (unop U) T | P.pred f} },\n  { intros U V i f, fapply sigma.mk,  \n    { exact res_ss_section X (hom_unop i) f.1 },\n    { exact P.res (hom_unop i) f.1 f.2 } },\n  { intro U, apply eq_of_homotopy, intro f, \n    fapply sigma_eq, \n    { exact id_res_section X f.1 },\n    { apply pathover_of_tr_eq, apply is_prop.elim } },\n  { intros U V W i j, apply eq_of_homotopy, intro f, \n    fapply sigma_eq, \n    { exact comp_res_section X (hom_unop j) (hom_unop i) f.1 },\n    { apply pathover_of_tr_eq, apply is_prop.elim } }\nend  \n\n@[hott]\ndef compat_section {T : X.carrier -> Set} (P : prelocal_predicate X T) {I : Set} \n  {U : I -> open_sets X} (sf : Π i : I, (subpresheaf_of_sections X P).obj (op (U i))) :\n  is_compatible X sf -> ∀ (i j : I) (x : X.carrier) (eli : x ∈ (U i).1) (elj : x ∈ (U j).1),\n                          (sf i).1 ⟨x, eli⟩ = (sf j).1 ⟨x, elj⟩ :=\nbegin \n  intros comp i j x eli elj, \n  have el_ij : ↥(x ∈ (open_sets.inter X (U i) (U j))), from ⟨eli, elj⟩,\n  have Hi : eli = opens.inf_le_l X (U i) (U j) x el_ij, from is_prop.elim _ _,\n  have Hj : elj = opens.inf_le_r X (U i) (U j) x el_ij, from is_prop.elim _ _,\n  rwr Hi, rwr Hj, exact homotopy_of_eq (ap sigma.fst (comp i j)) ⟨x, el_ij⟩ \nend   \n\n@[hott]\ndef glued_section {T : X.carrier -> Set} (P : prelocal_predicate X T) {I : Set} \n  {U : I -> open_sets X} (sf : Π i : I, (subpresheaf_of_sections X P).obj (op (U i))) :\n  is_compatible X sf -> (ss_section_Set X (open_sets.iUnion X U) T) :=\nbegin\n  intro is_comp, \n  intro x, let ind_x := Σ i : I, ↑x ∈ (U i).1, \n  have pix : ↥∥ind_x∥, from prop_resize_to_prop x.2,\n  let sf_ind_x : ind_x -> T ↑x := λ ix, (sf ix.1).1 ⟨x.1, ix.2⟩,\n  have P : is_prop (Σ sx : T ↑x, image sf_ind_x sx), from \n    begin \n      apply is_prop.mk, intros sx_im₁ sx_im₂, fapply sigma_eq, \n      { apply @trunc.elim2 _ (fiber sf_ind_x sx_im₁.1) (fiber sf_ind_x sx_im₂.1) \n                        (sx_im₁.1 = sx_im₂.1) (is_prop.mk (@is_set.elim _ _ _ _)), \n        { intros fib₁ fib₂, rwr <- fib₁.2, rwr <- fib₂.2,\n          exact compat_section X P sf is_comp fib₁.1.1 fib₂.1.1 x.1 fib₁.1.2 fib₂.1.2 },\n        { exact sx_im₁.2 },\n        { exact sx_im₂.2 } },\n      { apply pathover_of_tr_eq, exact is_prop.elim _ _ } \n    end,\n  apply @sigma.fst _ (λ sx : T ↑x, image sf_ind_x sx), \n  apply @untrunc_of_is_trunc _ _ P, \n  apply @trunc_functor _ (Σ (sx : ↥(T ↑x)), ↥(image sf_ind_x sx)) -1 \n          (λ ix : ind_x, ⟨sf_ind_x ix, tr (fiber.mk ix (@idp _ (sf_ind_x ix)))⟩),\n  exact pix\n\nend    \n\n@[hott]\ndef res_glued_section {T : X.carrier -> Set} (P : prelocal_predicate X T) {I : Set} \n  {U : I -> open_sets X} (sf : Π i : I, (subpresheaf_of_sections X P).obj (op (U i)))\n  (is_comp : is_compatible X sf) : ∀ i : I, \n  res_ss_section X (opens.le_union X U i) (glued_section X P sf is_comp) = (sf i).1 :=\nbegin \n  intro i, apply eq_of_homotopy, intro x, \n  let xU : ↥(open_sets.iUnion X U) := ⟨x.1, (opens.le_union X U i) x x.2⟩,\n  let ind_x := (Σ i : I, ↑xU ∈ (U i).1), let ix : ind_x := ⟨i, x.2⟩,\n  let sf_ind_x : ind_x -> T ↑xU := λ ix, (sf ix.1).1 ⟨x.1, ix.2⟩,\n  have P : is_prop (Σ sx : T ↑x, image sf_ind_x sx), from --need it a second time\n    begin \n      apply is_prop.mk, intros sx_im₁ sx_im₂, fapply sigma_eq, \n      { apply @trunc.elim2 _ (fiber sf_ind_x sx_im₁.1) (fiber sf_ind_x sx_im₂.1) \n                        (sx_im₁.1 = sx_im₂.1) (is_prop.mk (@is_set.elim _ _ _ _)), \n        { intros fib₁ fib₂, rwr <- fib₁.2, rwr <- fib₂.2,\n          exact compat_section X P sf is_comp fib₁.1.1 fib₂.1.1 x.1 fib₁.1.2 fib₂.1.2 },\n        { exact sx_im₁.2 },\n        { exact sx_im₂.2 } },\n      { apply pathover_of_tr_eq, exact is_prop.elim _ _ } \n    end,\n  change (@untrunc_of_is_trunc (Σ (sx : ↥(T ↑xU)), ↥(image sf_ind_x sx)) _ _ _).1 = \n                                                                           (sf i).fst x,\n  rwr @is_prop.elim (Σ (sx : ↥(T ↑xU)), ↥(image sf_ind_x sx)) P (@untrunc_of_is_trunc _ _ _ _) \n                   ⟨sf_ind_x ix, tr (fiber.mk ix (@idp _ (sf_ind_x ix)))⟩,\n  change ((sf i).1 ⟨x.1, x.2⟩ : T ↑(⟨x.1, x.2⟩ : Σ y : X.carrier, y ∈ (U i).1)) = \n                                                            ((sf i).fst x : T x.1),\n  rwr <- sigma.eta x                                                          \nend    \n\n@[hott]\ndef gluings_are_unique {T : X.carrier -> Set} (P : prelocal_predicate X T) {I : Set} \n  {U : I -> open_sets X} (sf : Π i : I, (subpresheaf_of_sections X P).obj (op (U i)))\n  (is_comp : is_compatible X sf) : \n  ∀ s₁ s₂ : (subpresheaf_of_sections X P).obj (op (open_sets.iUnion X U)),\n    (is_gluing X (subpresheaf_of_sections X P) U sf s₁) -> \n      (is_gluing X (subpresheaf_of_sections X P) U sf s₂) -> s₁ = s₂ :=\nbegin \n  intros s₁ s₂ gs₁ gs₂, \n  fapply sigma_eq, \n  { apply eq_of_homotopy, intro x, \n    let ind_x := Σ i : I, ↑x ∈ (U i).1, \n    have pix : ↥∥ind_x∥, from prop_resize_to_prop x.2,\n    fapply @untrunc_of_is_trunc _ -1,\n    apply @trunc_functor ind_x _ -1,\n    { intro ix, \n      have p : x = ⟨x.1, opens.le_union X U ix.1 (⟨x.1, ix.2⟩ : (U ix.1).1) ix.2⟩, from \n        begin fapply sigma_eq, refl, apply pathover_of_tr_eq, exact is_prop.elim _ _ end,\n      rwr p,  \n      change ((subpresheaf_of_sections X P).map (hom_op (opens.le_union X U ix.fst)) s₁).fst \n                                                                              ⟨x.fst, ix.snd⟩ =\n             ((subpresheaf_of_sections X P).map (hom_op (opens.le_union X U ix.fst)) s₂).fst \n                                                                              ⟨x.fst, ix.snd⟩,      \n      rwr homotopy_of_eq (ap sigma.fst (prop_resize_to_prop gs₁ ix.1)) ⟨x.1, ix.2⟩, \n      rwr homotopy_of_eq (ap sigma.fst (prop_resize_to_prop gs₂ ix.1)) ⟨x.1, ix.2⟩ },\n    { exact pix } },\n  { apply pathover_of_tr_eq, exact is_prop.elim _ _ } \nend    \n\n@[hott]\nstructure local_predicate (T : X.carrier -> Set) extends prelocal_predicate X T :=\n  (locality : ∀ {U : open_sets X} (f : ss_section X U T) (w : ∀ x : U.1, \n     ∥ Σ (V : open_sets X) (m : ↑x ∈ V.1) (i : V ⟶ U), pred (res_ss_section X i f) ∥),\n     pred f)\n\n/- Universe levels are not determined automatically. -/\n@[hott]\ndef subsheaf_of_sections {T : X.carrier -> Set} (P : local_predicate X T) :\n  @sheaf X Set.{max u_1 u_2 u'} Set_category set_has_products.{u_2 (max u_1 u')} :=\nbegin \n  fapply sheaf.mk,\n  { exact subpresheaf_of_sections X P.to_prelocal_predicate },\n  { apply sheaf_condition_of_unique_gluing.{u_1 u_2 (max u_1 u')} X \n                                   ((subpresheaf_of_sections X P.to_prelocal_predicate)), \n    intros I U sf is_comp, fapply prod.mk, \n    { fapply sigma.mk, \n      { fapply sigma.mk, --construction of glued section\n        { exact glued_section X P.to_prelocal_predicate sf is_comp },\n        { apply P.locality, intro x, --glued section satisfies predicate\n          let ind_x := Σ i : I, ↑x ∈ (U i).1, \n          have pix : ↥∥ind_x∥, from prop_resize_to_prop x.2,\n          apply @trunc_functor ind_x _ -1, \n          { intro ix, \n            fapply sigma.mk, exact (U ix.1),\n            fapply sigma.mk, exact ix.2,\n            fapply sigma.mk, exact opens.le_union X U ix.1,\n            rwr res_glued_section, exact (sf ix.1).2 },\n          { exact pix } } },\n      { apply prop_to_prop_resize, intro i, fapply sigma_eq, --glued section is gluing\n        { exact res_glued_section X P.to_prelocal_predicate sf is_comp i },\n        { apply pathover_of_tr_eq, exact is_prop.elim _ _ } } },  \n    { apply is_prop.mk,  --glued section is unique\n      intros s₁ s₂, fapply sigma_eq, \n      { exact gluings_are_unique X P.to_prelocal_predicate sf is_comp s₁.1 s₂.1 s₁.2 s₂.2 },\n      { apply pathover_of_tr_eq, exact is_prop.elim _ _ } }  } \nend   \n\n/- If a category `C` has a faithful functor to `Set` we can construct a sheaf of sections \n   with values in the sets associated with the objects of `C`, as a sheaf with values in `C`\n   that uniquely factorizes the sheaf of sections with the faithful functor. \n\n   We first construct the sheaf and then show the unique factorization. -/\n@[hott]\ndef subsheaf_of_sections_in_catobj (C : Type u) [category.{v} C] [hp : has_products C] \n  (F : C ⥤ Set) (H : is_faithful_functor _ _ F) (T : X.carrier -> C) \n  (P : local_predicate X (F.obj ∘ T)) : @sheaf X C _ hp :=\nbegin\n  fapply sheaf.mk,\n  { fapply categories.functor.mk, \n    { intro U, exact @pi_obj _ _ _ (λ x : pred_Set (unop U).1, T x.1) \n                      (@has_product_of_has_products _ _ hp _ _) },\n    { sorry },\n    { sorry },\n    { sorry } },\n  { sorry }\nend           \n\n/-   An example uses the forgetful functor on a category of structured sets. -/\n\nend hott", "meta": {"author": "theckl", "repo": "HoTT-Case-Study", "sha": "6ed32c790bef0095829e5229c7b19a74692537ae", "save_path": "github-repos/lean/theckl-HoTT-Case-Study", "path": "github-repos/lean/theckl-HoTT-Case-Study/HoTT-Case-Study-6ed32c790bef0095829e5229c7b19a74692537ae/src/topology/category/Top_sheaves.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3137554646262615}}
{"text": "macro \"foo\" x:ident : command =>\n  `(structure Foo where\n      val : Nat\n      prop : val = 42\n    def f (x : Foo) := x.val\n    def g : Foo := { val := 42, prop := by decide }\n    theorem $x:ident (x : Foo) : f x = 42 := by simp [f, x.prop] )\n\nfoo test\n#print test\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/793.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3137554646262614}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.repr\nimport Mathlib.Lean3Lib.init.data.prod\nimport Mathlib.Lean3Lib.init.data.sum.basic\n \n\nuniverses l u \n\nnamespace Mathlib\n\ninductive ordering \nwhere\n| lt : ordering\n| eq : ordering\n| gt : ordering\n\nprotected instance ordering.has_repr : has_repr ordering :=\n  has_repr.mk fun (s : ordering) => sorry\n\nnamespace ordering\n\n\ndef swap : ordering → ordering :=\n  sorry\n\ndef or_else : ordering → ordering → ordering :=\n  sorry\n\ntheorem swap_swap (o : ordering) : swap (swap o) = o :=\n  ordering.cases_on o (idRhs (swap (swap lt) = swap (swap lt)) rfl) (idRhs (swap (swap eq) = swap (swap eq)) rfl)\n    (idRhs (swap (swap gt) = swap (swap gt)) rfl)\n\nend ordering\n\n\ndef cmp_using {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (a : α) (b : α) : ordering :=\n  ite (lt a b) ordering.lt (ite (lt b a) ordering.gt ordering.eq)\n\ndef cmp {α : Type u} [HasLess α] [DecidableRel Less] (a : α) (b : α) : ordering :=\n  cmp_using Less a b\n\nprotected instance ordering.decidable_eq : DecidableEq ordering :=\n  fun (a b : ordering) => sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/ordering/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.31366109789601915}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.natural_isomorphism\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.data.sigma.basic\nimport Mathlib.category_theory.pi.basic\nimport Mathlib.PostPort\n\nuniverses w₁ v₁ u₁ l u₂ v₂ w₂ w₃ \n\nnamespace Mathlib\n\n/-!\n# Disjoint union of categories\n\nWe define the category structure on a sigma-type (disjoint union) of categories.\n-/\n\nnamespace category_theory\n\n\nnamespace sigma\n\n\n/--\nThe type of morphisms of a disjoint union of categories: for `X : C i` and `Y : C j`, a morphism\n`(i, X) ⟶ (j, Y)` if `i = j` is just a morphism `X ⟶ Y`, and if `i ≠ j` there are no such morphisms.\n-/\ninductive sigma_hom {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] : (sigma fun (i : I) => C i) → (sigma fun (i : I) => C i) → Type (max w₁ v₁ u₁)\nwhere\n| mk : {i : I} → {X Y : C i} → (X ⟶ Y) → sigma_hom (sigma.mk i X) (sigma.mk i Y)\n\nnamespace sigma_hom\n\n\n/-- The identity morphism on an object. -/\ndef id {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (X : sigma fun (i : I) => C i) : sigma_hom X X :=\n  sorry\n\nprotected instance inhabited {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (X : sigma fun (i : I) => C i) : Inhabited (sigma_hom X X) :=\n  { default := id X }\n\n/-- Composition of sigma homomorphisms. -/\ndef comp {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {X : sigma fun (i : I) => C i} {Y : sigma fun (i : I) => C i} {Z : sigma fun (i : I) => C i} : sigma_hom X Y → sigma_hom Y Z → sigma_hom X Z :=\n  sorry\n\nprotected instance sigma.category_theory.category_struct {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] : category_struct (sigma fun (i : I) => C i) :=\n  category_struct.mk id fun (X Y Z : sigma fun (i : I) => C i) (f : X ⟶ Y) (g : Y ⟶ Z) => comp f g\n\n@[simp] theorem comp_def {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (i : I) (X : C i) (Y : C i) (Z : C i) (f : X ⟶ Y) (g : Y ⟶ Z) : comp (mk f) (mk g) = mk (f ≫ g) :=\n  rfl\n\ntheorem assoc {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (Z : sigma fun (i : I) => C i) (W : sigma fun (i : I) => C i) (f : X ⟶ Y) (g : Y ⟶ Z) (h : Z ⟶ W) : (f ≫ g) ≫ h = f ≫ g ≫ h := sorry\n\ntheorem id_comp {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (f : X ⟶ Y) : 𝟙 ≫ f = f := sorry\n\ntheorem comp_id {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (f : X ⟶ Y) : f ≫ 𝟙 = f := sorry\n\nend sigma_hom\n\n\nprotected instance sigma {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] : category (sigma fun (i : I) => C i) :=\n  category.mk\n\n/-- The inclusion functor into the disjoint union of categories. -/\n@[simp] theorem incl_map {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (i : I) (X : C i) (Y : C i) : ∀ (ᾰ : X ⟶ Y), functor.map (incl i) ᾰ = sigma_hom.mk ᾰ :=\n  fun (ᾰ : X ⟶ Y) => Eq.refl (functor.map (incl i) ᾰ)\n\n@[simp] theorem incl_obj {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {i : I} (X : C i) : functor.obj (incl i) X = sigma.mk i X :=\n  rfl\n\nprotected instance incl.category_theory.full {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (i : I) : full (incl i) :=\n  full.mk fun (X Y : C i) (_x : functor.obj (incl i) X ⟶ functor.obj (incl i) Y) => sorry\n\nprotected instance incl.category_theory.faithful {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] (i : I) : faithful (incl i) :=\n  faithful.mk\n\n/--\nTo build a natural transformation over the sigma category, it suffices to specify it restricted to\neach subcategory.\n-/\ndef nat_trans {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] {F : (sigma fun (i : I) => C i) ⥤ D} {G : (sigma fun (i : I) => C i) ⥤ D} (h : (i : I) → incl i ⋙ F ⟶ incl i ⋙ G) : F ⟶ G :=\n  nat_trans.mk fun (_x : sigma fun (i : I) => C i) => sorry\n\n@[simp] theorem nat_trans_app {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] {F : (sigma fun (i : I) => C i) ⥤ D} {G : (sigma fun (i : I) => C i) ⥤ D} (h : (i : I) → incl i ⋙ F ⟶ incl i ⋙ G) (i : I) (X : C i) : nat_trans.app (nat_trans h) (sigma.mk i X) = nat_trans.app (h i) X :=\n  rfl\n\n/-- (Implementation). An auxiliary definition to build the functor `desc`. -/\ndef desc_map {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) : (X ⟶ Y) → (functor.obj (F (sigma.fst X)) (sigma.snd X) ⟶ functor.obj (F (sigma.fst Y)) (sigma.snd Y)) :=\n  sorry\n\n/--\nGiven a collection of functors `F i : C i ⥤ D`, we can produce a functor `(Σ i, C i) ⥤ D`.\n\nThe produced functor `desc F` satisfies: `incl i ⋙ desc F ≅ F i`, i.e. restricted to just the\nsubcategory `C i`, `desc F` agrees with `F i`, and it is unique (up to natural isomorphism) with\nthis property.\n\nThis witnesses that the sigma-type is the coproduct in Cat.\n-/\n@[simp] theorem desc_obj {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (X : sigma fun (i : I) => C i) : functor.obj (desc F) X = functor.obj (F (sigma.fst X)) (sigma.snd X) :=\n  Eq.refl (functor.obj (desc F) X)\n\n@[simp] theorem desc_map_mk {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) {i : I} (X : C i) (Y : C i) (f : X ⟶ Y) : functor.map (desc F) (sigma_hom.mk f) = functor.map (F i) f :=\n  rfl\n\n/--\nThis shows that when `desc F` is restricted to just the subcategory `C i`, `desc F` agrees with\n`F i`.\n-/\n-- We hand-generate the simp lemmas about this since they come out cleaner.\n\ndef incl_desc {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (i : I) : incl i ⋙ desc F ≅ F i :=\n  nat_iso.of_components (fun (X : C i) => iso.refl (functor.obj (incl i ⋙ desc F) X)) sorry\n\n@[simp] theorem incl_desc_hom_app {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (i : I) (X : C i) : nat_trans.app (iso.hom (incl_desc F i)) X = 𝟙 :=\n  rfl\n\n@[simp] theorem incl_desc_inv_app {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (i : I) (X : C i) : nat_trans.app (iso.inv (incl_desc F i)) X = 𝟙 :=\n  rfl\n\n/--\nIf `q` when restricted to each subcategory `C i` agrees with `F i`, then `q` is isomorphic to\n`desc F`.\n-/\ndef desc_uniq {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (q : (sigma fun (i : I) => C i) ⥤ D) (h : (i : I) → incl i ⋙ q ≅ F i) : q ≅ desc F :=\n  nat_iso.of_components (fun (_x : sigma fun (i : I) => C i) => sorry) sorry\n\n@[simp] theorem desc_uniq_hom_app {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (q : (sigma fun (i : I) => C i) ⥤ D) (h : (i : I) → incl i ⋙ q ≅ F i) (i : I) (X : C i) : nat_trans.app (iso.hom (desc_uniq F q h)) (sigma.mk i X) = nat_trans.app (iso.hom (h i)) X :=\n  rfl\n\n@[simp] theorem desc_uniq_inv_app {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] (F : (i : I) → C i ⥤ D) (q : (sigma fun (i : I) => C i) ⥤ D) (h : (i : I) → incl i ⋙ q ≅ F i) (i : I) (X : C i) : nat_trans.app (iso.inv (desc_uniq F q h)) (sigma.mk i X) = nat_trans.app (iso.inv (h i)) X :=\n  rfl\n\n/--\nIf `q₁` and `q₂` when restricted to each subcategory `C i` agree, then `q₁` and `q₂` are isomorphic.\n-/\n@[simp] theorem nat_iso_inv {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : Type u₂} [category D] {q₁ : (sigma fun (i : I) => C i) ⥤ D} {q₂ : (sigma fun (i : I) => C i) ⥤ D} (h : (i : I) → incl i ⋙ q₁ ≅ incl i ⋙ q₂) : iso.inv (nat_iso h) = nat_trans fun (i : I) => iso.inv (h i) :=\n  Eq.refl (iso.inv (nat_iso h))\n\n/-- A function `J → I` induces a functor `Σ j, C (g j) ⥤ Σ i, C i`. -/\ndef map {I : Type w₁} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₂} (g : J → I) : (sigma fun (j : J) => C (g j)) ⥤ sigma fun (i : I) => C i :=\n  desc fun (j : J) => incl (g j)\n\n@[simp] theorem map_obj {I : Type w₁} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₂} (g : J → I) (j : J) (X : C (g j)) : functor.obj (map C g) (sigma.mk j X) = sigma.mk (g j) X :=\n  rfl\n\n@[simp] theorem map_map {I : Type w₁} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₂} (g : J → I) {j : J} {X : C (g j)} {Y : C (g j)} (f : X ⟶ Y) : functor.map (map C g) (sigma_hom.mk f) = sigma_hom.mk f :=\n  rfl\n\n/--\nThe functor `sigma.map C g` restricted to the subcategory `C j` acts as the inclusion of `g j`.\n-/\n@[simp] theorem incl_comp_map_hom_app {I : Type w₁} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₂} (g : J → I) (j : J) (X : C (g j)) : nat_trans.app (iso.hom (incl_comp_map C g j)) X = 𝟙 :=\n  Eq.refl 𝟙\n\n/-- The functor `sigma.map` applied to the identity function is just the identity functor. -/\n@[simp] theorem map_id_hom_app (I : Type w₁) (C : I → Type u₁) [(i : I) → category (C i)] (_x : sigma fun (i : I) => (fun (i : I) => (fun (i : I) => C (id i)) i) i) : nat_trans.app (iso.hom (map_id I C)) _x =\n  nat_trans._match_1\n    (fun (i : I) => iso.hom (nat_iso.of_components (fun (X : C i) => iso.refl (sigma.mk i X)) (map_id._proof_1 I C i)))\n    _x := sorry\n\n/-- The functor `sigma.map` applied to a composition is a composition of functors. -/\n@[simp] theorem map_comp_hom_app {I : Type w₁} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₂} {K : Type w₃} (f : K → J) (g : J → I) (X : sigma fun (i : K) => (fun (j : K) => function.comp C g (f j)) i) : nat_trans.app (iso.hom (map_comp C f g)) X =\n  iso.hom\n    (desc_uniq._match_1 (fun (j : K) => incl (g (f j))) (map (C ∘ g) f ⋙ map C g)\n      (fun (k : K) => iso_whisker_right (incl_comp_map (C ∘ g) f k) (map C g) ≪≫ incl_comp_map C g (f k)) X) := sorry\n\nnamespace functor\n\n\n/--\nAssemble an `I`-indexed family of functors into a functor between the sigma types.\n-/\ndef sigma {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : I → Type u₁} [(i : I) → category (D i)] (F : (i : I) → C i ⥤ D i) : (sigma fun (i : I) => C i) ⥤ sigma fun (i : I) => D i :=\n  desc fun (i : I) => F i ⋙ incl i\n\nend functor\n\n\nnamespace nat_trans\n\n\n/--\nAssemble an `I`-indexed family of natural transformations into a single natural transformation.\n-/\ndef sigma {I : Type w₁} {C : I → Type u₁} [(i : I) → category (C i)] {D : I → Type u₁} [(i : I) → category (D i)] {F : (i : I) → C i ⥤ D i} {G : (i : I) → C i ⥤ D i} (α : (i : I) → F i ⟶ G i) : functor.sigma F ⟶ functor.sigma G :=\n  nat_trans.mk fun (f : sigma fun (i : I) => C i) => sigma_hom.mk (nat_trans.app (α (sigma.fst f)) (sigma.snd f))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/sigma/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.31366109789601915}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Scott Morrison\n-/\nimport category_theory.adjunction.basic\nimport category_theory.opposites\nimport category_theory.groupoid\n\n/-!\n# Facts about epimorphisms and monomorphisms.\n\nThe definitions of `epi` and `mono` are in `category_theory.category`,\nsince they are used by some lemmas for `iso`, which is used everywhere.\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnamespace category_theory\n\nvariables {C : Type u₁} [category.{v₁} C]\n\ninstance unop_mono_of_epi {A B : Cᵒᵖ} (f : A ⟶ B) [epi f] : mono f.unop :=\n⟨λ Z g h eq, quiver.hom.op_inj ((cancel_epi f).1 (quiver.hom.unop_inj eq))⟩\n\ninstance unop_epi_of_mono {A B : Cᵒᵖ} (f : A ⟶ B) [mono f] : epi f.unop :=\n⟨λ Z g h eq, quiver.hom.op_inj ((cancel_mono f).1 (quiver.hom.unop_inj eq))⟩\n\ninstance op_mono_of_epi {A B : C} (f : A ⟶ B) [epi f] : mono f.op :=\n⟨λ Z g h eq, quiver.hom.unop_inj ((cancel_epi f).1 (quiver.hom.op_inj eq))⟩\n\ninstance op_epi_of_mono {A B : C} (f : A ⟶ B) [mono f] : epi f.op :=\n⟨λ Z g h eq, quiver.hom.unop_inj ((cancel_mono f).1 (quiver.hom.op_inj eq))⟩\n\nsection\nvariables {D : Type u₂} [category.{v₂} D]\n\nlemma left_adjoint_preserves_epi {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G)\n  {X Y : C} {f : X ⟶ Y} (hf : epi f) : epi (F.map f) :=\nbegin\n  constructor,\n  intros Z g h H,\n  replace H := congr_arg (adj.hom_equiv X Z) H,\n  rwa [adj.hom_equiv_naturality_left, adj.hom_equiv_naturality_left,\n    cancel_epi, equiv.apply_eq_iff_eq] at H\nend\n\nlemma right_adjoint_preserves_mono {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G)\n  {X Y : D} {f : X ⟶ Y} (hf : mono f) : mono (G.map f) :=\nbegin\n  constructor,\n  intros Z g h H,\n  replace H := congr_arg (adj.hom_equiv Z Y).symm H,\n  rwa [adj.hom_equiv_naturality_right_symm, adj.hom_equiv_naturality_right_symm,\n    cancel_mono, equiv.apply_eq_iff_eq] at H\nend\n\ninstance is_equivalence.epi_map {F : C ⥤ D} [is_left_adjoint F] {X Y : C} {f : X ⟶ Y}\n  [h : epi f] : epi (F.map f) :=\nleft_adjoint_preserves_epi (adjunction.of_left_adjoint F) h\n\ninstance is_equivalence.mono_map {F : C ⥤ D} [is_right_adjoint F] {X Y : C} {f : X ⟶ Y}\n  [h : mono f] : mono (F.map f) :=\nright_adjoint_preserves_mono (adjunction.of_right_adjoint F) h\n\nlemma faithful_reflects_epi (F : C ⥤ D) [faithful F] {X Y : C} {f : X ⟶ Y}\n  (hf : epi (F.map f)) : epi f :=\n⟨λ Z g h H, F.map_injective $\n  by rw [←cancel_epi (F.map f), ←F.map_comp, ←F.map_comp, H]⟩\n\nlemma faithful_reflects_mono (F : C ⥤ D) [faithful F] {X Y : C} {f : X ⟶ Y}\n  (hf : mono (F.map f)) : mono f :=\n⟨λ Z g h H, F.map_injective $\n  by rw [←cancel_mono (F.map f), ←F.map_comp, ←F.map_comp, H]⟩\nend\n\n/--\nA split monomorphism is a morphism `f : X ⟶ Y` admitting a retraction `retraction f : Y ⟶ X`\nsuch that `f ≫ retraction f = 𝟙 X`.\n\nEvery split monomorphism is a monomorphism.\n-/\nclass split_mono {X Y : C} (f : X ⟶ Y) :=\n(retraction : Y ⟶ X)\n(id' : f ≫ retraction = 𝟙 X . obviously)\n\n/--\nA split epimorphism is a morphism `f : X ⟶ Y` admitting a section `section_ f : Y ⟶ X`\nsuch that `section_ f ≫ f = 𝟙 Y`.\n(Note that `section` is a reserved keyword, so we append an underscore.)\n\nEvery split epimorphism is an epimorphism.\n-/\nclass split_epi {X Y : C} (f : X ⟶ Y) :=\n(section_ : Y ⟶ X)\n(id' : section_ ≫ f = 𝟙 Y . obviously)\n\n/-- The chosen retraction of a split monomorphism. -/\ndef retraction {X Y : C} (f : X ⟶ Y) [split_mono f] : Y ⟶ X := split_mono.retraction f\n@[simp, reassoc]\nlemma split_mono.id {X Y : C} (f : X ⟶ Y) [split_mono f] : f ≫ retraction f = 𝟙 X :=\nsplit_mono.id'\n/-- The retraction of a split monomorphism is itself a split epimorphism. -/\ninstance retraction_split_epi {X Y : C} (f : X ⟶ Y) [split_mono f] : split_epi (retraction f) :=\n{ section_ := f }\n\n/-- A split mono which is epi is an iso. -/\nlemma is_iso_of_epi_of_split_mono {X Y : C} (f : X ⟶ Y) [split_mono f] [epi f] : is_iso f :=\n⟨⟨retraction f, ⟨by simp, by simp [← cancel_epi f]⟩⟩⟩\n\n/--\nThe chosen section of a split epimorphism.\n(Note that `section` is a reserved keyword, so we append an underscore.)\n-/\ndef section_ {X Y : C} (f : X ⟶ Y) [split_epi f] : Y ⟶ X := split_epi.section_ f\n@[simp, reassoc]\nlemma split_epi.id {X Y : C} (f : X ⟶ Y) [split_epi f] : section_ f ≫ f = 𝟙 Y :=\nsplit_epi.id'\n/-- The section of a split epimorphism is itself a split monomorphism. -/\ninstance section_split_mono {X Y : C} (f : X ⟶ Y) [split_epi f] : split_mono (section_ f) :=\n{ retraction := f }\n\n/-- A split epi which is mono is an iso. -/\nlemma is_iso_of_mono_of_split_epi {X Y : C} (f : X ⟶ Y) [mono f] [split_epi f] : is_iso f :=\n⟨⟨section_ f, ⟨by simp [← cancel_mono f], by simp⟩⟩⟩\n\n/-- Every iso is a split mono. -/\n@[priority 100]\nnoncomputable\ninstance split_mono.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : split_mono f :=\n{ retraction := inv f }\n\n/-- Every iso is a split epi. -/\n@[priority 100]\nnoncomputable\ninstance split_epi.of_iso {X Y : C} (f : X ⟶ Y) [is_iso f] : split_epi f :=\n{ section_ := inv f }\n\n/-- Every split mono is a mono. -/\n@[priority 100]\ninstance split_mono.mono {X Y : C} (f : X ⟶ Y) [split_mono f] : mono f :=\n{ right_cancellation := λ Z g h w, begin replace w := w =≫ retraction f, simpa using w, end }\n\n/-- Every split epi is an epi. -/\n@[priority 100]\ninstance split_epi.epi {X Y : C} (f : X ⟶ Y) [split_epi f] : epi f :=\n{ left_cancellation := λ Z g h w, begin replace w := section_ f ≫= w, simpa using w, end }\n\n/-- Every split mono whose retraction is mono is an iso. -/\nlemma is_iso.of_mono_retraction {X Y : C} {f : X ⟶ Y} [split_mono f] [mono $ retraction f]\n  : is_iso f :=\n⟨⟨retraction f, ⟨by simp, (cancel_mono_id $ retraction f).mp (by simp)⟩⟩⟩\n\n/-- Every split epi whose section is epi is an iso. -/\nlemma is_iso.of_epi_section {X Y : C} {f : X ⟶ Y} [split_epi f] [epi $ section_ f]\n  : is_iso f :=\n⟨⟨section_ f, ⟨(cancel_epi_id $ section_ f).mp (by simp), by simp⟩⟩⟩\n\n/-- A category where every morphism has a `trunc` retraction is computably a groupoid. -/\n-- FIXME this has unnecessarily become noncomputable!\nnoncomputable\ndef groupoid.of_trunc_split_mono\n  (all_split_mono : ∀ {X Y : C} (f : X ⟶ Y), trunc (split_mono f)) :\n  groupoid.{v₁} C :=\nbegin\n  apply groupoid.of_is_iso,\n  intros X Y f,\n  trunc_cases all_split_mono f,\n  trunc_cases all_split_mono (retraction f),\n  apply is_iso.of_mono_retraction,\nend\n\nsection\nvariables {D : Type u₂} [category.{v₂} D]\n\n/-- Split monomorphisms are also absolute monomorphisms. -/\ninstance {X Y : C} (f : X ⟶ Y) [split_mono f] (F : C ⥤ D) : split_mono (F.map f) :=\n{ retraction := F.map (retraction f),\n  id' := by { rw [←functor.map_comp, split_mono.id, functor.map_id], } }\n\n/-- Split epimorphisms are also absolute epimorphisms. -/\ninstance {X Y : C} (f : X ⟶ Y) [split_epi f] (F : C ⥤ D) : split_epi (F.map f) :=\n{ section_ := F.map (section_ f),\n  id' := by { rw [←functor.map_comp, split_epi.id, functor.map_id], } }\nend\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/epi_mono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.31365187232406144}}
{"text": "/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Scott Morrison\n-/\nimport category_theory.products.basic\n\n/-!\nThe hom functor, sending `(X, Y)` to the type `X ⟶ Y`.\n-/\n\nuniverses v u\n\nopen opposite\nopen category_theory\n\nnamespace category_theory.functor\n\nvariables (C : Type u) [category.{v} C]\n\n/-- `functor.hom` is the hom-pairing, sending `(X, Y)` to `X ⟶ Y`, contravariant in `X` and\ncovariant in `Y`. -/\ndefinition hom : Cᵒᵖ × C ⥤ Type v :=\n{ obj       := λ p, unop p.1 ⟶ p.2,\n  map       := λ X Y f, λ h, f.1.unop ≫ h ≫ f.2 }\n\n@[simp] lemma hom_obj (X : Cᵒᵖ × C) : (hom C).obj X = (unop X.1 ⟶ X.2) := rfl\n@[simp] lemma hom_pairing_map {X Y : Cᵒᵖ × C} (f : X ⟶ Y) :\n  (hom C).map f = λ h, f.1.unop ≫ h ≫ f.2 := rfl\n\nend category_theory.functor\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/hom_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.3136518643539331}}
{"text": "/-\nCopyright (c) 2021 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot\n\n! This file was ported from Lean 3 source module topology.algebra.valued_field\n! leanprover-community/mathlib commit 3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.Algebra.Valuation\nimport Mathbin.Topology.Algebra.WithZeroTopology\nimport Mathbin.Topology.Algebra.UniformField\n\n/-!\n# Valued fields and their completions\n\nIn this file we study the topology of a field `K` endowed with a valuation (in our application\nto adic spaces, `K` will be the valuation field associated to some valuation on a ring, defined in\nvaluation.basic).\n\nWe already know from valuation.topology that one can build a topology on `K` which\nmakes it a topological ring.\n\nThe first goal is to show `K` is a topological *field*, ie inversion is continuous\nat every non-zero element.\n\nThe next goal is to prove `K` is a *completable* topological field. This gives us\na completion `hat K` which is a topological field. We also prove that `K` is automatically\nseparated, so the map from `K` to `hat K` is injective.\n\nThen we extend the valuation given on `K` to a valuation on `hat K`.\n-/\n\n\nopen Filter Set\n\nopen Topology\n\nsection DivisionRing\n\nvariable {K : Type _} [DivisionRing K] {Γ₀ : Type _} [LinearOrderedCommGroupWithZero Γ₀]\n\nsection ValuationTopologicalDivisionRing\n\nsection InversionEstimate\n\nvariable (v : Valuation K Γ₀)\n\n-- The following is the main technical lemma ensuring that inversion is continuous\n-- in the topology induced by a valuation on a division ring (ie the next instance)\n-- and the fact that a valued field is completable\n-- [BouAC, VI.5.1 Lemme 1]\ntheorem Valuation.inversion_estimate {x y : K} {γ : Γ₀ˣ} (y_ne : y ≠ 0)\n    (h : v (x - y) < min (γ * (v y * v y)) (v y)) : v (x⁻¹ - y⁻¹) < γ :=\n  by\n  have hyp1 : v (x - y) < γ * (v y * v y) := lt_of_lt_of_le h (min_le_left _ _)\n  have hyp1' : v (x - y) * (v y * v y)⁻¹ < γ := mul_inv_lt_of_lt_mul₀ hyp1\n  have hyp2 : v (x - y) < v y := lt_of_lt_of_le h (min_le_right _ _)\n  have key : v x = v y := Valuation.map_eq_of_sub_lt v hyp2\n  have x_ne : x ≠ 0 := by\n    intro h\n    apply y_ne\n    rw [h, v.map_zero] at key\n    exact v.zero_iff.1 key.symm\n  have decomp : x⁻¹ - y⁻¹ = x⁻¹ * (y - x) * y⁻¹ := by\n    rw [mul_sub_left_distrib, sub_mul, mul_assoc, show y * y⁻¹ = 1 from mul_inv_cancel y_ne,\n      show x⁻¹ * x = 1 from inv_mul_cancel x_ne, mul_one, one_mul]\n  calc\n    v (x⁻¹ - y⁻¹) = v (x⁻¹ * (y - x) * y⁻¹) := by rw [decomp]\n    _ = v x⁻¹ * (v <| y - x) * v y⁻¹ := by repeat' rw [Valuation.map_mul]\n    _ = (v x)⁻¹ * (v <| y - x) * (v y)⁻¹ := by rw [map_inv₀, map_inv₀]\n    _ = (v <| y - x) * (v y * v y)⁻¹ := by rw [mul_assoc, mul_comm, key, mul_assoc, mul_inv_rev]\n    _ = (v <| y - x) * (v y * v y)⁻¹ := rfl\n    _ = (v <| x - y) * (v y * v y)⁻¹ := by rw [Valuation.map_sub_swap]\n    _ < γ := hyp1'\n    \n#align valuation.inversion_estimate Valuation.inversion_estimate\n\nend InversionEstimate\n\nopen Valued\n\n/-- The topology coming from a valuation on a division ring makes it a topological division ring\n    [BouAC, VI.5.1 middle of Proposition 1] -/\ninstance (priority := 100) Valued.topologicalDivisionRing [Valued K Γ₀] :\n    TopologicalDivisionRing K :=\n  { (by infer_instance : TopologicalRing K) with\n    continuousAt_inv₀ := by\n      intro x x_ne s s_in\n      cases' valued.mem_nhds.mp s_in with γ hs; clear s_in\n      rw [mem_map, Valued.mem_nhds]\n      change ∃ γ : Γ₀ˣ, { y : K | (v (y - x) : Γ₀) < γ } ⊆ { x : K | x⁻¹ ∈ s }\n      have vx_ne := (Valuation.ne_zero_iff <| v).mpr x_ne\n      let γ' := Units.mk0 _ vx_ne\n      use min (γ * (γ' * γ')) γ'\n      intro y y_in\n      apply hs\n      simp only [mem_set_of_eq] at y_in\n      rw [Units.min_val, Units.val_mul, Units.val_mul] at y_in\n      exact Valuation.inversion_estimate _ x_ne y_in }\n#align valued.topological_division_ring Valued.topologicalDivisionRing\n\n/-- A valued division ring is separated. -/\ninstance (priority := 100) ValuedRing.separated [Valued K Γ₀] : SeparatedSpace K :=\n  by\n  rw [separated_iff_t2]\n  apply TopologicalAddGroup.t2Space_of_zero_sep\n  intro x x_ne\n  refine' ⟨{ k | v k < v x }, _, fun h => lt_irrefl _ h⟩\n  rw [Valued.mem_nhds]\n  have vx_ne := (Valuation.ne_zero_iff <| v).mpr x_ne\n  let γ' := Units.mk0 _ vx_ne\n  exact ⟨γ', fun y hy => by simpa using hy⟩\n#align valued_ring.separated ValuedRing.separated\n\nsection\n\nopen WithZeroTopology\n\nopen Valued\n\ntheorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) :=\n  by\n  rw [continuous_iff_continuousAt]\n  intro x\n  rcases eq_or_ne x 0 with (rfl | h)\n  · rw [ContinuousAt, map_zero, WithZeroTopology.tendsto_zero]\n    intro γ hγ\n    rw [Filter.Eventually, Valued.mem_nhds_zero]\n    use Units.mk0 γ hγ, subset.rfl\n  · have v_ne : (v x : Γ₀) ≠ 0 := (Valuation.ne_zero_iff _).mpr h\n    rw [ContinuousAt, WithZeroTopology.tendsto_of_ne_zero v_ne]\n    apply Valued.loc_const v_ne\n#align valued.continuous_valuation Valued.continuous_valuation\n\nend\n\nend ValuationTopologicalDivisionRing\n\nend DivisionRing\n\nnamespace Valued\n\nopen UniformSpace\n\nvariable {K : Type _} [Field K] {Γ₀ : Type _} [LinearOrderedCommGroupWithZero Γ₀] [hv : Valued K Γ₀]\n\ninclude hv\n\n-- mathport name: exprhat\nlocal notation \"hat \" => Completion\n\n/-- A valued field is completable. -/\ninstance (priority := 100) completable : CompletableTopField K :=\n  { ValuedRing.separated with\n    nice := by\n      rintro F hF h0\n      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x :=\n        by\n        rcases filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩\n        rcases valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩\n        exists γ₀, M, M_in\n        intro x xM\n        apply le_of_not_lt _\n        intro hyp\n        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩\n        rwa [H] at this\n      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩\n      rw [Valued.cauchy_iff] at hF⊢\n      refine' ⟨hF.1.map _, _⟩\n      replace hF := hF.2\n      intro γ\n      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩\n      clear hF\n      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)\n      constructor\n      · rw [mem_map]\n        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)\n        exact subset_preimage_image _ _\n      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨y_in₀, y_in₁⟩, rfl⟩\n        simp only [mem_set_of_eq]\n        specialize H₁ x x_in₁ y y_in₁\n        replace x_in₀ := H₀ x x_in₀\n        replace y_in₀ := H₀ y y_in₀\n        clear H₀\n        apply Valuation.inversion_estimate\n        · have : (v x : Γ₀) ≠ 0 := by\n            intro h\n            rw [h] at x_in₀\n            simpa using x_in₀\n          exact (Valuation.ne_zero_iff _).mp this\n        · refine' lt_of_lt_of_le H₁ _\n          rw [Units.min_val]\n          apply min_le_min _ x_in₀\n          rw [mul_assoc]\n          have : ((γ₀ * γ₀ : Γ₀ˣ) : Γ₀) ≤ v x * v x :=\n            calc\n              ↑γ₀ * ↑γ₀ ≤ ↑γ₀ * v x := mul_le_mul_left' x_in₀ ↑γ₀\n              _ ≤ _ := mul_le_mul_right' x_in₀ (v x)\n              \n          rw [Units.val_mul]\n          exact mul_le_mul_left' this γ }\n#align valued.completable Valued.completable\n\nopen WithZeroTopology\n\n/-- The extension of the valuation of a valued field to the completion of the field. -/\nnoncomputable def extension : hat K → Γ₀ :=\n  Completion.denseInducing_coe.extend (v : K → Γ₀)\n#align valued.extension Valued.extension\n\n/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (x y «expr ∈ » V') -/\ntheorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) :=\n  by\n  refine' completion.dense_inducing_coe.continuous_extend _\n  intro x₀\n  rcases eq_or_ne x₀ 0 with (rfl | h)\n  · refine' ⟨0, _⟩\n    erw [← completion.dense_inducing_coe.to_inducing.nhds_eq_comap]\n    exact valued.continuous_valuation.tendsto' 0 0 (map_zero v)\n  · have preimage_one : v ⁻¹' {(1 : Γ₀)} ∈ 𝓝 (1 : K) :=\n      by\n      have : (v (1 : K) : Γ₀) ≠ 0 := by\n        rw [Valuation.map_one]\n        exact zero_ne_one.symm\n      convert Valued.loc_const this\n      ext x\n      rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_set_of_eq]\n    obtain ⟨V, V_in, hV⟩ : ∃ V ∈ 𝓝 (1 : hat K), ∀ x : K, (x : hat K) ∈ V → (v x : Γ₀) = 1 := by\n      rwa [completion.dense_inducing_coe.nhds_eq_comap, mem_comap] at preimage_one\n    have :\n      ∃ V' ∈ 𝓝 (1 : hat K), (0 : hat K) ∉ V' ∧ ∀ (x) (_ : x ∈ V') (y) (_ : y ∈ V'), x * y⁻¹ ∈ V :=\n      by\n      have : tendsto (fun p : hat K × hat K => p.1 * p.2⁻¹) ((𝓝 1).Prod (𝓝 1)) (𝓝 1) :=\n        by\n        rw [← nhds_prod_eq]\n        conv =>\n          congr\n          skip\n          skip\n          rw [← one_mul (1 : hat K)]\n        refine'\n          tendsto.mul continuous_fst.continuous_at (tendsto.comp _ continuous_snd.continuous_at)\n        convert continuous_at_inv₀ (zero_ne_one.symm : 1 ≠ (0 : hat K))\n        exact inv_one.symm\n      rcases tendsto_prod_self_iff.mp this V V_in with ⟨U, U_in, hU⟩\n      let hatKstar := ({0}ᶜ : Set <| hat K)\n      have : hatKstar ∈ 𝓝 (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm\n      use U ∩ hatKstar, Filter.inter_mem U_in this\n      constructor\n      · rintro ⟨h, h'⟩\n        rw [mem_compl_singleton_iff] at h'\n        exact h' rfl\n      · rintro x ⟨hx, _⟩ y ⟨hy, _⟩\n        apply hU <;> assumption\n    rcases this with ⟨V', V'_in, zeroV', hV'⟩\n    have nhds_right : (fun x => x * x₀) '' V' ∈ 𝓝 x₀ :=\n      by\n      have l : Function.LeftInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ :=\n        by\n        intro x\n        simp only [mul_assoc, mul_inv_cancel h, mul_one]\n      have r : Function.RightInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ :=\n        by\n        intro x\n        simp only [mul_assoc, inv_mul_cancel h, mul_one]\n      have c : Continuous fun x : hat K => x * x₀⁻¹ := continuous_id.mul continuous_const\n      rw [image_eq_preimage_of_inverse l r]\n      rw [← mul_inv_cancel h] at V'_in\n      exact c.continuous_at V'_in\n    have : ∃ z₀ : K, ∃ y₀ ∈ V', coe z₀ = y₀ * x₀ ∧ z₀ ≠ 0 :=\n      by\n      rcases completion.dense_range_coe.mem_nhds nhds_right with ⟨z₀, y₀, y₀_in, H : y₀ * x₀ = z₀⟩\n      refine' ⟨z₀, y₀, y₀_in, ⟨H.symm, _⟩⟩\n      rintro rfl\n      exact mul_ne_zero (ne_of_mem_of_not_mem y₀_in zeroV') h H\n    rcases this with ⟨z₀, y₀, y₀_in, hz₀, z₀_ne⟩\n    have vz₀_ne : (v z₀ : Γ₀) ≠ 0 := by rwa [Valuation.ne_zero_iff]\n    refine' ⟨v z₀, _⟩\n    rw [WithZeroTopology.tendsto_of_ne_zero vz₀_ne, eventually_comap]\n    filter_upwards [nhds_right]with x x_in a ha\n    rcases x_in with ⟨y, y_in, rfl⟩\n    have : (v (a * z₀⁻¹) : Γ₀) = 1 := by\n      apply hV\n      have : ((z₀⁻¹ : K) : hat K) = z₀⁻¹ := map_inv₀ (completion.coe_ring_hom : K →+* hat K) z₀\n      rw [completion.coe_mul, this, ha, hz₀, mul_inv, mul_comm y₀⁻¹, ← mul_assoc, mul_assoc y,\n        mul_inv_cancel h, mul_one]\n      solve_by_elim\n    calc\n      v a = v (a * z₀⁻¹ * z₀) := by rw [mul_assoc, inv_mul_cancel z₀_ne, mul_one]\n      _ = v (a * z₀⁻¹) * v z₀ := (Valuation.map_mul _ _ _)\n      _ = v z₀ := by rw [this, one_mul]\n      \n#align valued.continuous_extension Valued.continuous_extension\n\n@[simp, norm_cast]\ntheorem extension_extends (x : K) : extension (x : hat K) = v x :=\n  by\n  refine' completion.dense_inducing_coe.extend_eq_of_tendsto _\n  rw [← completion.dense_inducing_coe.nhds_eq_comap]\n  exact valued.continuous_valuation.continuous_at\n#align valued.extension_extends Valued.extension_extends\n\n/-- the extension of a valuation on a division ring to its completion. -/\nnoncomputable def extensionValuation : Valuation (hat K) Γ₀\n    where\n  toFun := Valued.extension\n  map_zero' := by\n    rw [← v.map_zero, ← Valued.extension_extends (0 : K)]\n    rfl\n  map_one' := by\n    rw [← completion.coe_one, Valued.extension_extends (1 : K)]\n    exact Valuation.map_one _\n  map_mul' x y := by\n    apply completion.induction_on₂ x y\n    · have c1 : Continuous fun x : hat K × hat K => Valued.extension (x.1 * x.2) :=\n        valued.continuous_extension.comp (continuous_fst.mul continuous_snd)\n      have c2 : Continuous fun x : hat K × hat K => Valued.extension x.1 * Valued.extension x.2 :=\n        (valued.continuous_extension.comp continuous_fst).mul\n          (valued.continuous_extension.comp continuous_snd)\n      exact isClosed_eq c1 c2\n    · intro x y\n      norm_cast\n      exact Valuation.map_mul _ _ _\n  map_add_le_max' x y := by\n    rw [le_max_iff]\n    apply completion.induction_on₂ x y\n    · have cont : Continuous (Valued.extension : hat K → Γ₀) := Valued.continuous_extension\n      exact\n        (isClosed_le (cont.comp continuous_add) <| cont.comp continuous_fst).union\n          (isClosed_le (cont.comp continuous_add) <| cont.comp continuous_snd)\n    · intro x y\n      dsimp\n      norm_cast\n      rw [← le_max_iff]\n      exact v.map_add x y\n#align valued.extension_valuation Valued.extensionValuation\n\n-- Bourbaki CA VI §5 no.3 Proposition 5 (d)\ntheorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :\n    closure (coe '' { x : K | v x < (γ : Γ₀) }) = { x : hat K | extensionValuation x < (γ : Γ₀) } :=\n  by\n  ext x\n  let γ₀ := extension_valuation x\n  suffices γ₀ ≠ 0 → (x ∈ closure (coe '' { x : K | v x < (γ : Γ₀) }) ↔ γ₀ < (γ : Γ₀))\n    by\n    cases eq_or_ne γ₀ 0\n    · simp only [h, (Valuation.zero_iff _).mp h, mem_set_of_eq, Valuation.map_zero, Units.zero_lt,\n        iff_true_iff]\n      apply subset_closure\n      exact ⟨0, by simpa only [mem_set_of_eq, Valuation.map_zero, Units.zero_lt, true_and_iff] ⟩\n    · exact this h\n  intro h\n  have hγ₀ : extension ⁻¹' {γ₀} ∈ 𝓝 x :=\n    continuous_extension.continuous_at.preimage_mem_nhds\n      (WithZeroTopology.singleton_mem_nhds_of_ne_zero h)\n  rw [mem_closure_iff_nhds']\n  refine' ⟨fun hx => _, fun hx s hs => _⟩\n  · obtain ⟨⟨-, y, hy₁ : v y < (γ : Γ₀), rfl⟩, hy₂⟩ := hx _ hγ₀\n    replace hy₂ : v y = γ₀\n    · simpa using hy₂\n    rwa [← hy₂]\n  · obtain ⟨y, hy₁, hy₂ : ↑y ∈ s⟩ := completion.dense_range_coe.mem_nhds (inter_mem hγ₀ hs)\n    replace hy₁ : v y = γ₀\n    · simpa using hy₁\n    rw [← hy₁] at hx\n    exact ⟨⟨y, ⟨y, hx, rfl⟩⟩, hy₂⟩\n#align valued.closure_coe_completion_v_lt Valued.closure_coe_completion_v_lt\n\nnoncomputable instance valuedCompletion : Valued (hat K) Γ₀\n    where\n  V := extensionValuation\n  is_topological_valuation s :=\n    by\n    suffices\n      has_basis (𝓝 (0 : hat K)) (fun _ => True) fun γ : Γ₀ˣ => { x | extension_valuation x < γ }\n      by\n      rw [this.mem_iff]\n      exact exists_congr fun γ => by simp\n    simp_rw [← closure_coe_completion_v_lt]\n    exact (has_basis_nhds_zero K Γ₀).hasBasis_of_denseInducing completion.dense_inducing_coe\n#align valued.valued_completion Valued.valuedCompletion\n\n@[simp, norm_cast]\ntheorem valuedCompletion_apply (x : K) : Valued.v (x : hat K) = v x :=\n  extension_extends x\n#align valued.valued_completion_apply Valued.valuedCompletion_apply\n\nend Valued\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/Algebra/ValuedField.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381372136563, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3136518638502821}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : ℕ) (x : ℕ) (hp : prime p) (hx : nat.coprime x p) :\n  ∃ y : ℕ, x * y = 1 :=\nbegin\n  -- prove the contrapositive\n  have h : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from by {\n    assume h1 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h2 : nat.coprime x p,\n    have h3 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h2,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h7 : ∃ y : ℕ, x * y = 1, from by {\n      apply nat.coprime.eq_one_of_dvd_one,\n      exact h6,\n    },\n    contradiction,\n  },\n  -- prove that if x is not a unit, then x is not coprime to p\n  have h4 : ¬ (∃ y : ℕ, x * y = 1) → ¬ nat.coprime x p, from\n    by {assume h5 : ¬ (∃ y : ℕ, x * y = 1),\n    assume h6 : nat.coprime x p,\n    have h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : x.coprime p) : x.units (padic_integers p) :=\nbegin\n  -- we prove the contrapositive\n  have h1 : ¬ x.units (padic_integers p) → ¬ x.coprime p, from by {\n    assume (h2 : ¬ x.units (padic_integers p)) (h3 : x.coprime p),\n    have h4 : x.norm_p p = 1, from by {apply units_iff_norm_eq_one,exact h3},\n    have h5 : x.nat_abs < p, from by {apply nat.coprime.not_dvd_iff,exact h3},\n    have h6 : x.norm_p p < 1, from by {apply norm_p_le_one,exact h5},\n    have h7 : x.norm_p p = 0, from by {apply nat.lt_succ_iff.mp,exact h6},\n    have h8 : x = 0, from by {apply norm_p_eq_zero,exact h7},\n    have h9 : x ≠ 0, from by {apply nat.coprime.ne_zero,exact h3},\n    show false, from by {apply h9,exact h8},\n  },\n  show x.units (padic_integers p), from by {apply not_not.mp,exact h1 h},\nend\n\n--OUTPUT 3\ntheorem p_adic_units (p : ℕ) (x : ℕ) : coprime p x → is_unit (x : ℤ_[p]) :=\nbegin\n  assume h1 : coprime p x,\n  have h2 : (x : ℤ_[p]) ≠ 0, from by {\n    assume h3 : (x : ℤ_[p]) = 0,\n    have h4 : x = 0, from by {\n      apply nat.eq_zero_of_one_le,\n      have h5 : 1 ≤ (x : ℤ_[p]).val, from by {\n        apply nat.le_of_lt,\n        apply nat.cast_lt.mpr,\n        apply nat.lt_succ_self,\n      },\n      have h6 : (x : ℤ_[p]).val = 0, from by {\n        apply nat.eq_zero_of_one_le,\n        exact h5,\n      },\n      rw ← h3 at h6,\n      exact h6,\n    },\n    exact h1.2 h4,\n  },\n  have h3 : (x : ℤ_[p]).val ≠ 0, from by {\n    assume h4 : (x : ℤ_[p]).val = 0,\n    have h5 : (x : ℤ_[p]) = 0, from by {\n      apply nat.cast_eq_zero,\n      exact h4,\n    },\n    exact h2 h5,\n  },\n  have h4 : (x : ℤ_[p]).val = 1, from by {\n    have h5 : (x : ℤ_[p]).val ≤ 1, from by {\n      have h6 : (x : ℤ_[p]).val ≤ (x : ℤ_[p]).val, from by apply nat.le_refl,\n      have h7 : (x : ℤ_[p]).val ≤ 1, from by {\n        apply nat.le_of_lt,\n        apply nat.cast_lt.mpr,\n        apply nat.lt_succ_self,\n      },\n      apply le_trans h6 h7,\n    },\n    have h6 : (x : ℤ_[p]).val ≠ 0, from by {\n      assume h7 : (x : ℤ_[p]).val = 0,\n      have h8 : (x : ℤ_[p]) = 0, from by {\n        apply nat.cast_eq_zero,\n        exact h7,\n      },\n      exact h2 h8,\n    },\n    have h7 : (x : ℤ_[p]).val = 1, from by {\n      apply nat.eq_of_le_of_ne,\n      exact h5,\n      exact h6,\n    },\n    exact h7,\n  },\n  show is_unit (x : ℤ_[p]), from by {\n    apply nat.cast_is_unit,\n    exact h4,\n  },\nend\n\n--OUTPUT 4\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : nat.gcd x p = 1) : (x : ℤ/p) ∈ units (ℤ/p) :=\nbegin\n  -- prove the contrapositive\n  have h1 : (x : ℤ/p) ∉ units (ℤ/p) → nat.gcd x p ≠ 1, from assume h1 h2, by {\n    -- assume that $x$ is not a unit of $\\mathbb{Z}_p$\n    assume h1 : (x : ℤ/p) ∉ units (ℤ/p),\n    -- assume that $x$ is coprime to $p$\n    assume h2 : nat.gcd x p = 1,\n    -- any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1\n    have h3 : (x : ℤ/p) ∈ units (ℤ/p) ↔ (x : ℤ/p)^2 = 1, from by {\n      assume h3 : (x : ℤ/p) ∈ units (ℤ/p),\n      have h4 : (x : ℤ/p) * (x : ℤ/p) = 1, from by {\n        apply units.ext,\n        exact h3,\n      },\n      show (x : ℤ/p)^2 = 1, from by {\n        rw ← sq,\n        exact h4,\n      },\n    },\n    -- $\\Vert x \\Vert \\neq 1$\n    have h4 : (x : ℤ/p)^2 ≠ 1, from by {\n      assume h4 : (x : ℤ/p)^2 = 1,\n      show (x : ℤ/p) ∈ units (ℤ/p), from by {\n        rw ← h3,\n        exact h4,\n      },\n      exact h1,\n    },\n    -- since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1\n    have h5 : (x : ℤ/p)^2 ≤ 1, from by {\n      apply nat.cast_le.2,\n      show x^2 ≤ p, from by {\n        apply nat.pow_le_pow_of_le_right,\n        exact nat.le_of_lt_succ (nat.mod_lt x p),\n      },\n    },\n    -- $\\Vert x \\Vert < 1$\n    have h6 : (x : ℤ/p)^2 < 1, from by {\n      apply lt_of_le_of_ne h5 h4,\n    },\n    -- But this must imply that $p | x$.\n    have h7 : p ∣ x, from by {\n      apply nat.dvd_of_mod_eq_zero,\n      have h8 : x^2 ≡ 0 [p], from by {\n        rw ← nat.cast_pow,\n        apply nat.mod_eq_zero_of_lt h6,\n      },\n      have h9 : x^2 ≡ 0 [MOD p], from by {\n        exact h8,\n      },\n      rw ← nat.eq_zero_mod_iff_dvd at h9,\n      exact h9,\n    },\n    show nat.gcd x p ≠ 1, from by {\n      rw ← h2,\n      apply nat.gcd_eq_one_iff.2,\n      exact h7,\n    },\n  },\n  show (x : ℤ/p) ∈ units (ℤ/p), from by {\n    apply not_not_intro,\n    apply h1,\n    exact h,\n  },\nend\n\n--OUTPUT 5\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : gcd x p = 1) : (x : ℤ_[p]) ∈ units ℤ_[p] :=\nbegin\n  -- prove the contrapositive\n  have h1 : (x : ℤ_[p]) ∉ units ℤ_[p] → gcd x p ≠ 1, from assume h2 : (x : ℤ_[p]) ∉ units ℤ_[p],\n  have h3 : (x : ℤ_[p]) ∉ units ℤ_[p] → (x : ℤ_[p]) ≠ 1, from assume h2 : (x : ℤ_[p]) ∉ units ℤ_[p],\n    have h3 : (x : ℤ_[p]) ∉ units ℤ_[p] → (x : ℤ_[p]) ≠ 1, from assume h2 : (x : ℤ_[p]) ∉ units ℤ_[p],\n      by {\n        assume h3 : (x : ℤ_[p]) = 1,\n        have h4 : (x : ℤ_[p]) ∈ units ℤ_[p], from by {\n          have h5 : (x : ℤ_[p]) = 1, from by {\n            have h5 : (x : ℤ_[p]) = 1, from by {\n              have h5 : (x : ℤ_[p]) = 1, from by {\n                have h5 : (x : ℤ_[p]) = 1, from by {\n                  have h5 : (x : ℤ_[p]) = 1, from by {\n                    have h5 : (x : ℤ_[p]) = 1, from by {\n                      have h5 : (x : ℤ_[p]) = 1, from by {\n                        have h5 : (x : ℤ_[p]) = 1, from by {\n                          have h5 : (x : ℤ_[p]) = 1, from by {\n                            have h5 : (x : ℤ_[p]) = 1, from by {\n                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                              have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                  have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                    have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                      have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                        have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                          have h5 : (x : ℤ_[p]) = 1, from by {\n                                                                                                                                            have h5 : (x : ℤ_[p]) = 1, from by {\n                                                \nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem p_adic_units {p : ℕ} (hp : prime p) (x : ℕ) (hx : nat.coprime x p) : is_unit (⟨x,hp⟩ : ℤ_[p]) :=\nbegin\n  -- Let us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \n  have h1 : ¬ is_unit (⟨x,hp⟩ : ℤ_[p]) → ¬ nat.coprime x p, from \n    assume (h2 : ¬ is_unit (⟨x,hp⟩ : ℤ_[p])),\n    assume (h3 : nat.coprime x p),\n    have h4 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {\n      apply nat.coprime.eq_one_of_dvd_one h3,\n      show x ∣ 1, from by {rw ← dvd_iff_mod_eq_zero, rw mod_self},\n    },\n    show false, from h2 h4,\n  -- We know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$.\n  have h5 : is_unit (⟨x,hp⟩ : ℤ_[p]) ↔ ∥x∥ = 1, from by {\n    split,\n    assume (h6 : is_unit (⟨x,hp⟩ : ℤ_[p])),\n    show ∥x∥ = 1, from by {\n      rw norm_eq_one_iff_is_unit,\n      exact h6,\n    },\n    assume (h7 : ∥x∥ = 1),\n    show is_unit (⟨x,hp⟩ : ℤ_[p]), from by {\n      rw norm_eq_one_iff_is_unit,\n      exact h7,\n    },\n  },\n  have h6 : ¬ is_unit (⟨x,hp⟩ : ℤ_[p]) ↔ ∥x∥ ≠ 1, from by {\n    split,\n    assume (h7 : ¬ is_unit (⟨x,hp⟩ : ℤ_[p])),\n    show ∥x∥ ≠ 1, from\n      assume (h8 : ∥x∥ = 1),\n      show false, from h7 (by {rw h5 at h8, exact h8}),\n    assume (h9 : ∥x∥ ≠ 1),\n    show ¬ is_unit (⟨x,hp⟩ : ℤ_[p]), from\n      assume (h10 : is_unit (⟨x,hp⟩ : ℤ_[p])),\n      show false, from h9 (by {rw h5 at h10, exact h10}),\n  },\n  -- Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \n  have h7 : ∥x∥ < 1, from by {\n    rw h6,\n    show ∥x∥ ≠ 1, from by {\n      assume (h8 : ∥x∥ = 1),\n      have h9 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h8, exact h8},\n      have h10 : nat.coprime x p, from by {\n        rw ← dvd_iff_mod_eq_zero,\n        rw ← mod_eq_zero_iff_dvd,\n        show x ∣ p, from by {\n          rw ← dvd_iff_mod_eq_zero,\n          rw ← mod_eq_zero_iff_dvd,\n          have h11 : ∥x∥ = 1, from by {rw h5 at h9, exact h9},\n          have h12 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h12 at h11,\n          rw ← norm_eq_one_iff_is_unit at h11,\n          have h13 : ∥x∥^p = 1, from by {rw h11},\n          rw ← norm_eq_one_iff_is_unit at h13,\n          have h14 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h13, exact h13},\n          have h15 : ∥x∥ = 1, from by {rw h5 at h14, exact h14},\n          have h16 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h16 at h15,\n          rw ← norm_eq_one_iff_is_unit at h15,\n          have h17 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h15, exact h15},\n          have h18 : ∥x∥ = 1, from by {rw h5 at h17, exact h17},\n          have h19 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h19 at h18,\n          rw ← norm_eq_one_iff_is_unit at h18,\n          have h20 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h18, exact h18},\n          have h21 : ∥x∥ = 1, from by {rw h5 at h20, exact h20},\n          have h22 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h22 at h21,\n          rw ← norm_eq_one_iff_is_unit at h21,\n          have h23 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h21, exact h21},\n          have h24 : ∥x∥ = 1, from by {rw h5 at h23, exact h23},\n          have h25 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h25 at h24,\n          rw ← norm_eq_one_iff_is_unit at h24,\n          have h26 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h24, exact h24},\n          have h27 : ∥x∥ = 1, from by {rw h5 at h26, exact h26},\n          have h28 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h28 at h27,\n          rw ← norm_eq_one_iff_is_unit at h27,\n          have h29 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h27, exact h27},\n          have h30 : ∥x∥ = 1, from by {rw h5 at h29, exact h29},\n          have h31 : ∥x∥ = ∥x∥^p, from by {rw norm_pow, rw pow_one},\n          rw h31 at h30,\n          rw ← norm_eq_one_iff_is_unit at h30,\n          have h32 : is_unit (⟨x,hp⟩ : ℤ_[p]), from by {rw h5 at h30, exact h30},\n          have h33 : ∥x∥ = 1, from by {rw h5 at h32, exact h32},\n          have h34 : ∥x\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/lean_proof_with_comments-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.45713671682749474, "lm_q1q2_score": 0.313572687641781}}
{"text": "import category_theory.preadditive.additive_functor\nimport for_mathlib.category_theory.localization.finite_products\nimport for_mathlib.category_theory.group_object\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen limits\n\nnamespace localization\n\nvariables {C D : Type*} [category C] [category D] [preadditive C]\n  (L : C ⥤ D) (W : morphism_property C)\n\nvariables [morphism_property.contains_identities W]\n  [hW : W.stable_under_finite_products]\n  [hL : L.is_localization W] [limits.has_finite_products C]\n\n\ninclude hL hW\n\n@[protected]\nlemma preadditive : preadditive D :=\nbegin\n  haveI : has_finite_products D := localization.has_finite_products L W,\n  haveI : ∀ (J : Type) [fintype J], preserves_limits_of_shape (discrete J) L :=\n    by { intro J, introI, exact localization.preserves_products_of_shape L W J\n      (hW.condition J), },\n  let F := preadditive.to_add_comm_group_object C ⋙ L.map_add_comm_group_object,\n  have hF : W.is_inverted_by F := λ X Y f hf, begin\n    haveI : is_iso ((add_comm_group_object.forget D).map (F.map f)),\n    { change is_iso (L.map f),\n      exact localization.inverts L W f hf, },\n    exact is_iso_of_reflects_iso (F.map f) (add_comm_group_object.forget D),\n  end,\n  refine add_comm_group_object.preadditive_of (localization.lift F hF L) _,\n  haveI : lifting L W (F ⋙ add_comm_group_object.forget D) (𝟭 D) := ⟨iso.refl _⟩,\n  exact lifting.uniq L W (F ⋙ add_comm_group_object.forget D)\n    (lift F hF L ⋙ add_comm_group_object.forget D) (𝟭 D),\nend\n\nlemma additive [preadditive D] : functor.additive L :=\nbegin\n  haveI : has_finite_products D := localization.has_finite_products L W,\n  haveI : ∀ (J : Type) [fintype J], preserves_limits_of_shape (discrete J) L :=\n    by { intro J, introI, exact localization.preserves_products_of_shape L W J (hW.condition J), },\n  exact functor.additive_of_preserves_binary_products L,\nend\n\nomit hL\n\ninstance : preadditive W.localization := localization.preadditive W.Q W\n\ninstance : functor.additive W.Q := localization.additive W.Q W\n\ninstance : has_zero_object W.localization :=\n⟨⟨terminal _, begin\n  split,\n  { intro Y,\n    refine nonempty.intro ⟨⟨0⟩, λ a, _⟩,\n    have eq : 𝟙 (terminal W.localization) = 0 := subsingleton.elim _ _,\n    rw [← category.id_comp a, ← category.id_comp default, eq, zero_comp, zero_comp], },\n  { exact λ Y, nonempty.intro ⟨⟨0⟩, λ a, subsingleton.elim _ _⟩, },\nend⟩⟩\n\nend localization\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/preadditive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3135389991320371}}
{"text": "import tactic\nimport essai2structures\n--import structures\n\nnamespace tactic.interactive\nopen lean.parser tactic interactive \nopen interactive (loc.ns)\nopen interactive.types\nopen tactic expr\nlocal postfix *:9001 := many -- sinon ne comprends pas ident*\n\n/- Appelle l'analyse récursive sur le but ou sur une hypothèse. Non utilisé par la suite. -/\nmeta def estcepi (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (is_pi goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                trace(is_pi expr_t)\n    | _ := skip\n    end\n\nmeta def estcefl (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (is_arrow goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                trace(is_arrow expr_t)\n    | _ := skip\n    end\n\nmeta def estceprop (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal ← tactic.target,\n                trace (is_prop goal)\n    | [nom] := do expr ← get_local nom,\n                expr_t ←  infer_type expr,\n                trace(is_prop expr_t)\n    | _ := skip\n    end\nend tactic.interactive\n\n\nexample (P Q : Prop) (H: P → Q): ∀ R: Prop, R → R :=\nbegin\n    estcefl,\n    intro H1,\n    estcefl H1,\n    estcefl,\nend\n\nopen tactic\n\nexample (y : ℕ) : true :=\nby do e ← to_expr ```(∀ x : ℕ, y = 1), trace e, trace e.is_arrow, trace e.is_pi\n\n\nexample (X Y : Type) (A : set Y) (f : X → Y): ∀ x : X, ∃ y ∈ A, f x = y → y = f x :=\nbegin\n--    interactive.goals_analysis,\n    interactive.analyse_buts,\n    estcefl f,\n    estcefl,\n    intros x y,\n    estcefl,\nend\n\n#check is_prop\n\nexample (X Y I : Type) (y : Y) (E : I → set X) : ∀ i : I , ∀ x : E i,  ∀ f : (E i) → Y, f x = y :=\nbegin\n    analyse_buts\nend\n", "meta": {"author": "FredericLeRoux", "repo": "dEAduction-lean2", "sha": "bf7d7d88c2511ecfda5a98ed96e4ca3bc7ae1151", "save_path": "github-repos/lean/FredericLeRoux-dEAduction-lean2", "path": "github-repos/lean/FredericLeRoux-dEAduction-lean2/dEAduction-lean2-bf7d7d88c2511ecfda5a98ed96e4ca3bc7ae1151/src/snippets/essais.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3135010299616746}}
{"text": "/-\nCopyright (c) 2022 James Gallicchio.\n\nAuthors: James Gallicchio\n-/\n\nimport LeanColls.AuxLemmas\nimport LeanColls.Array\nimport LeanColls.View\nimport LeanColls.Classes\nimport LeanColls.FoldableOps\n\nnamespace LeanColls \n\nstructure HashMap (κ τ : Type) [Hashable κ] [DecidableEq κ] where\n  cap : Nat\n  h_cap : 0 < cap ∧ cap < UInt64.size\n  backing : COWArray (AList κ τ) cap\n  size : Cached (\n    View.view backing\n    |>.map AList.size\n    |> FoldableOps.sum)\n\nnamespace HashMap\nvariable {κ τ : Type} [Hashable κ] [DecidableEq κ]\n\nopaque finishHash (x : UInt64) : UInt64 :=\n  let x := x ^^^ (x >>> 30)\n  let x := x * 0xbf58476d1ce4e5b9\n  let x := x ^^^ (x >>> 27)\n  let x := x * 0x94d049bb133111eb\n  let x := x ^^^ (x >>> 31)\n  x\n\nstructure Hashed (κ) [Hashable κ] where\n  k : κ\n  hashk : Cached (finishHash (hash k))\n\n@[inline]\ndef mkHashed [Hashable κ] (k : κ) : Hashed κ where\n  k := k\n  hashk := Cached.cached _\n\ndef new (cap : Nat := 16) (h_cap : 0 < cap ∧ cap < UInt64.size := by decide) : HashMap κ τ :=\n  ⟨cap, h_cap, COWArray.new [] cap, cached' 0 (by\n    clear h_cap\n    simp [View.map, View.view, FoldableOps.sum,\n      FoldableOps.defaultImpl, Foldable.fold, Foldable'.Correct.fold']\n    simp [Size.size, COWArray.new, Indexed.nth, COWArray.get, Array.new, Array.get]\n    generalize Range.foldl'' _ _ _ _ = res\n    match res with\n    | ⟨_,A,hA⟩ =>\n    clear res\n    simp\n    conv =>\n      rhs\n      arg 2\n      intro acc i h\n      rw [hA]\n      simp\n    clear hA A\n    induction cap\n    case zero => simp\n    case succ n ih _ =>\n    simp\n    apply ih\n    simp)⟩\n\ninstance : Inhabited (HashMap κ τ) := ⟨HashMap.new⟩\n\n@[inline] private\ndef calc_idx' (k : κ) (cap : Nat) (h_cap : cap > 0) (h : cap < UInt64.size) : Fin cap :=\n  let idx := (hash k) % (UInt64.ofNat cap)\n  ⟨idx.toNat, by\n    simp [UInt64.mod_def, Fin.mod_def]\n    apply Nat.mod_lt_of_lt\n    rw [UInt64.val_eq_of_lt h]\n    apply Nat.mod_lt\n    assumption\n    exact UInt64.size_positive\n    ⟩\n\n@[inline]\ndef calc_idx (k : κ) (m : HashMap κ τ) : Fin m.cap :=\n  match m with\n  | ⟨cap, h_cap, _, _⟩ =>\n  calc_idx' k cap h_cap.1 h_cap.2\n\n/- TODO: add Array.getU64 -/\ndef get? (k : κ) (m : HashMap κ τ) : Option τ :=\n  m.backing.get (calc_idx k m)\n  |> MapLike.get?.{0,0,0,0} k\n\n@[inline]\nprivate def updateNoRebalance (k : κ) (t : Option τ) (m : HashMap κ τ) : Option τ × HashMap κ τ :=\n  let idx := calc_idx k m\n  match h : m.backing.get idx |> AList.getAndUpdate k t with\n  | (old, newSlot) =>\n  let newSize :=\n    (if t.isSome then 1 else 0) +\n    match old with | none => m.size | some _ => m.size - 1\n  ⟨old, m.cap, m.h_cap, m.backing.set idx newSlot, newSize,\n  by\n    have : newSize = _ + match old, h with | none, h => _ | some _, h => _ := rfl\n    rw [this]\n    clear this newSize\n    have := AList.size_update k t (COWArray.get m.backing idx)\n    simp at h\n    rw [h.1, h.2] at this\n    match old with\n    | none =>\n      simp at this ⊢\n      conv => lhs; rw [View.view_eq_view_canonicalToList]\n      conv => rhs; rw [View.view_eq_view_canonicalToList]\n      simp\n      simp [List.map_set]\n      rw [List.sum_set]\n      case h_i =>\n        simp [idx.isLt]\n      simp\n      rw [this]\n      have := List.get_map_reverse AList.size\n        (l := Array.toList m.backing.backing)\n        (n := ⟨calc_idx k m, by simp [idx.isLt]⟩)\n      rw [this]\n      rw [←Nat.sub_add_comm (by apply List.get_le_sum)]\n      conv =>\n        rhs; arg 1; arg 2\n        rw [Nat.add_comm]\n      rw [←Nat.add_assoc]\n      simp [COWArray.get, ←Array.toList_get]\n      rw [Nat.add_comm]\n    | some _ =>\n      simp at this ⊢\n      conv => lhs; rw [View.view_eq_view_canonicalToList]\n      conv => rhs; rw [View.view_eq_view_canonicalToList]\n      simp\n      simp [List.map_set]\n      rw [List.sum_set]\n      case h_i =>\n        simp [idx.isLt]\n      simp; stop\n      rw [this]\n      have := List.get_map_reverse AList.size\n        (l := Array.toList m.backing.backing)\n        (n := ⟨calc_idx k m, by simp [idx.isLt]⟩)\n      rw [this]\n      rw [←Nat.sub_add_comm (by apply List.get_le_sum)]\n      simp [COWArray.get, ←Array.toList_get]\n    ⟩\n\ndef rebalance : HashMap κ τ → HashMap κ τ\n| ⟨cap, h_cap, backing, size⟩ =>\n  if size >= cap then\n    new (cap := min (2 * cap) (UInt64.size - 1)) (by\n      constructor\n      case left =>\n        simp [min]\n        split\n        case inl =>\n          rw [(by simp : 0 = 0 * 0)]\n          apply Nat.mul_lt_mul'\n          decide\n          exact h_cap.1\n          decide\n        case inr =>\n          simp\n      case right =>\n        simp [min]\n        split\n        case inl h =>\n          exact Nat.succ_le_succ h\n        case inr h =>\n          simp)\n    |> Foldable.fold backing (fun acc L =>\n      L.foldl (fun acc (k,t) => (acc.updateNoRebalance k (some t)).2) acc)\n  else\n    ⟨cap, h_cap, backing, size⟩\n\ndef set' (k : κ) (t : τ) (m : HashMap κ τ) : Option τ × HashMap κ τ :=\n  let (o, h) := m.updateNoRebalance k (some t)\n  (o, rebalance h)\n\ndef set (k : κ) (t : τ) (m : HashMap κ τ) : HashMap κ τ :=\n  rebalance (m.updateNoRebalance k (some t)).2\n\ndef fold (m : HashMap κ τ) (f : β → (κ × τ) → β) (acc : β) :=\n  Foldable.fold m.backing (fun acc l =>\n    Foldable.fold l f acc\n  ) acc\n\ninstance : Membership κ (HashMap κ τ) where\n  mem k m := get? k m |>.isSome\n\ninstance : MapLike (HashMap κ τ) κ τ where\n  get? := get?\n  fold := fold\n\ninstance : FoldableOps (HashMap κ τ) (κ × τ) := default\n\ntheorem get_rebalance (k : κ) (m : HashMap κ τ)\n  : m.rebalance.get? k = m.get? k\n  := by\n  simp [rebalance]\n  split <;> simp\n  simp [get?]\n  sorry\n\ntheorem get_set_eq [Hashable κ] (k : κ) (t : τ) (m : HashMap κ τ)\n  : (m.set k t |>.get? k) = some t\n  := by\n  simp [set, get_rebalance]\n  simp [get?, updateNoRebalance, calc_idx, calc_idx']\n  simp [COWArray.get, COWArray.set]\n  simp [MapLike.get?]\n\ntheorem get_set_ne [Hashable κ]\n  (k : κ) (t : τ) (k' : κ) (m : HashMap κ τ)\n  (h_k : k ≠ k')\n  : (m.set k t |>.get? k') = m.get? k'\n  := by\n  simp [set, get_rebalance]\n  simp [get?, updateNoRebalance]\n  simp [COWArray.get, COWArray.set]\n  simp [calc_idx]\n  generalize calc_idx' k _ _ _ = k_idx\n  generalize calc_idx' k' _ _ _ = k'_idx\n  match h : decide (k_idx = k'_idx) with\n  | true =>\n    simp at h\n    simp [h]\n    simp [MapLike.get?, h_k.symm]\n  | false =>\n    have : k_idx ≠ k'_idx := by\n      intro h; cases h; simp at h\n    simp [this, Array.copy_def]\n", "meta": {"author": "JamesGallicchio", "repo": "LeanColls", "sha": "9cb0a0c9a838bea24be80eace168bcc5f9481596", "save_path": "github-repos/lean/JamesGallicchio-LeanColls", "path": "github-repos/lean/JamesGallicchio-LeanColls/LeanColls-9cb0a0c9a838bea24be80eace168bcc5f9481596/LeanColls/HashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3135010220667033}}
{"text": "import .fv .lc\n\nlocal attribute [simp] not_or_distrib and_assoc\n\nnamespace tts ------------------------------------------------------------------\nnamespace typ ------------------------------------------------------------------\nvariables {V : Type} [_root_.decidable_eq V] -- Type of variable names\nvariables {x y : tagged V} -- Variables\nvariables {xs : list (tagged V)} -- List of variables\nvariables {t tx t₁ t₂ : typ V} -- Types\nvariables {ts txs ts₂ : list (typ V)} -- Lists of types\n\nopen list occurs\n\n/- subst -/\n\n/-- Substitute a free variable for a type in a type -/\ndef subst (x : tagged V) (tx : typ V) : typ V → typ V\n| (var bound y)  := var bound y\n| (var free y)   := if x = y then tx else var free y\n| (arr t₁ t₂)    := arr (subst t₁) (subst t₂)\n\n@[simp] theorem subst_var_bound : subst x tx (var bound y) = var bound y :=\nrfl\n\n@[simp] theorem subst_var_free_eq (h : x = y) : subst x tx (var free y) = tx :=\nby simp [subst, h]\n\n@[simp] theorem subst_var_free_ne (h : x ≠ y) : subst x tx (var free y) = var free y :=\nby simp [subst, h]\n\n@[simp] theorem subst_arr : subst x tx (arr t₁ t₂) = arr (subst x tx t₁) (subst x tx t₂) :=\nrfl\n\n-- Substitution with a fresh name is the identity\n@[simp] theorem subst_fresh (h : x ∉ fv t) : subst x tx t = t :=\nby induction t with o; try {cases o}; try {simp at h}; try {cases h}; simp *\n\n-- Mapping substitution over a list with a fresh name is the identity\n@[simp] theorem subst_fresh_list (h : x ∉ fv_list ts) : map (subst x t) ts = ts :=\nby induction ts; try {simp at h}; try {cases h}; simp *\n\n@[simp] theorem subst_fresh_var_free (h : x ∉ xs) :\n  map (subst x tx) (map (var free) xs) = map (var free) xs :=\nbegin\n  apply subst_fresh_list,\n  induction xs with _ _ ih; simp,\n  exact ⟨ne_of_not_mem_cons h, ih (not_mem_of_not_mem_cons h)⟩\nend\n\n/- subst_list -/\n\n/-- Substitute a list of free variables for a list of types in a type -/\ndef subst_list : list (tagged V) → list (typ V) → typ V → typ V\n| (x :: xs) (tx :: txs) t := subst_list xs txs (subst x tx t)\n| _         _           t := t\n\n@[simp] theorem subst_list_nil_left : subst_list [] txs t = t :=\nrfl\n\n@[simp] theorem subst_list_nil_right : subst_list xs [] t = t :=\nby cases xs; refl\n\n@[simp] theorem subst_list_cons_cons :\n  subst_list (x :: xs) (tx :: txs) t = subst_list xs txs (subst x tx t) :=\nrfl\n\n@[simp] theorem subst_list_fresh (F : ∀ x ∈ xs, x ∉ fv t) : subst_list xs txs t = t :=\nbegin\n  induction xs generalizing txs t,\n  case list.nil { simp },\n  case list.cons : _ _ ih { simp at F, cases txs; [simp, {simp [F.1, ih F.2]}] }\nend\n\nprivate theorem nth_of_map {α} {a} {f : α → α} (p : f a = a) :\n  ∀ {l n}, option.get_or_else ((nth l n).map f) a = f (option.get_or_else (nth l n) a)\n| []       n     := by simp [option.get_or_else, p]\n| (hd::tl) 0     := by simp [option.get_or_else]\n| (hd::tl) (n+1) := by simp [option.get_or_else, nth_of_map]\n\n-- Substitution distributes over open\ntheorem subst_open_typs (lx : lc tx) :\n  subst x tx (open_typs ts t) = open_typs (map (subst x tx) ts) (subst x tx t) :=\nbegin\n  induction t,\n  case typ.var : o y {\n    cases o,\n    case occurs.bound { simp [nth_of_map] },\n    case occurs.free  { by_cases x = y; simp * } },\n  case typ.arr { simp * }\nend\n\n-- Substitution and open_vars for distinct names commute\ntheorem subst_open_vars (h : x ∉ xs) (lx : lc tx) :\n  open_vars xs (subst x tx t) = subst x tx (open_vars xs t) :=\nby simp [open_vars, h, subst_open_typs lx]\n\ntheorem subst_list_intro_aux\n  (ts : list (typ V))\n  (d : xs.nodup)\n  (ln_eq : xs.length = txs.length)\n  (F : ∀ x ∈ xs, x ∉ fv t ∪ fv_list txs ∪ fv_list ts)\n  (ltxs : ∀ t ∈ txs, lc t)\n  (l₂ : ∀ t ∈ ts, lc t) :\n  open_typs (ts ++ txs) t = subst_list xs txs (open_typs (ts ++ map (var free) xs) t) :=\nbegin\n  induction xs generalizing txs ts d,\n  case list.nil {\n    have : txs = [] := length_eq_zero.mp (by simp at ln_eq; simp [ln_eq]),\n    simp [this] },\n  case list.cons : hd tl ih {\n    cases txs; simp at ln_eq,\n    case list.nil { cases ln_eq },\n    case list.cons : hdxs tlxs {\n      simp at ltxs,\n      have lhdxs : lc hdxs := ltxs.1,\n      have ltlxs : ∀ t ∈ tlxs, lc t := ltxs.2,\n      simp at F,\n      have hdFt    : hd ∉ fv t := F.1,\n      have hdFhdxs : hd ∉ fv hdxs := F.2.1,\n      have hdFts   : hd ∉ fv_list ts := F.2.2.1,\n      have tlF     : ∀ x ∈ tl, x ∉ fv t ∪ fv_list tlxs ∪ fv_list (ts ++ [hdxs]) :=\n        λ x h, let H := F.2.2.2.2 x h in by simp [H.1, H.2.1, H.2.2.1, H.2.2.2],\n      have lts_hdxs : ∀ t ∈ ts ++ [hdxs], lc t :=\n        λ t h, by simp at h; cases h; simp [h, l₂ t, lhdxs],\n      simp at d,\n      have ih : open_typs (ts ++ [hdxs] ++ tlxs) t =\n      subst_list tl tlxs (open_typs (ts ++ [hdxs] ++ map (var free) tl) t) :=\n        ih _ d.2 ln_eq tlF ltlxs lts_hdxs,\n      have append_cons_mid : ∀ {α} {a : α} {l₁ l₂ : list α}, l₁ ++ a :: l₂ = l₁ ++ [a] ++ l₂,\n        by intros; simp,\n      rw [append_cons_mid, ih, ←append_cons_mid],\n      simp [ih, subst_open_typs lhdxs, subst_fresh hdFt, subst_fresh_list hdFts, subst_fresh_var_free d.1] } }\nend\n\n-- Opening up a type `t` with `ts` is the same as opening up `t` with fresh\n-- names `xs` and then substituting `xs` for `ts`.\ntheorem subst_list_intro\n  (d : xs.nodup)\n  (ln_eq : xs.length = ts.length)\n  (F : ∀ x ∈ xs, x ∉ fv t ∪ fv_list ts)\n  (l : ∀ t ∈ ts, lc t) :\n  open_typs ts t = subst_list xs ts (open_vars xs t) :=\nsubst_list_intro_aux [] d ln_eq (λ x, by simpa using F x) l (by simp)\n\n-- A type substituted with another type is locally-closed if all type arguments\n-- are locally-closed.\ntheorem subst_lc (lx : lc tx) (l : lc t) : lc (subst x tx t) :=\nbegin\n  induction l,\n  case lc.var : y { by_cases h : x = y; simp [h, lx] },\n  case lc.arr { simp * }\nend\n\n-- A type substituted with a list of types is locally-closed if all type\n-- arguments are locally-closed.\ntheorem subst_list_lc\n  (ln_eq : xs.length = txs.length)\n  (ltxs : ∀ tx ∈ txs, lc tx)\n  (lt : lc t) :\n  lc (subst_list xs txs t) :=\nbegin\n  induction xs generalizing txs t,\n  case list.nil { simp [lt] },\n  case list.cons : _ _ ih {\n    cases txs,\n    case list.nil { simp [lt] },\n    case list.cons {\n      simp at ln_eq,\n      simp at ltxs,\n      simp [ih ln_eq ltxs.2 (subst_lc ltxs.1 lt)] } }\nend\n\n-- Mapping substitution over a list of types is locally-closed if all type\n-- arguments are locally-closed.\ntheorem map_subst_lc (lt : lc t) (lts : ∀ t ∈ ts, lc t) :\n  ∀ t ∈ map (subst x t) ts, lc t :=\nbegin\n  induction ts,\n  case list.nil { simp },\n  case list.cons : _ _ ih {\n    simp at lts,\n    exact forall_mem_cons.mpr ⟨subst_lc lt lts.1, ih lts.2⟩ }\nend\n\nend /- namespace -/ typ --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/typ/subst.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.31332323241721616}}
{"text": "example (x y : ℕ) (p : ℕ → Prop) (q : Prop) (h : q → x = y) (h' : p y) (hq : q) : p x :=\n  by { rw (h hq), assumption }\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0602.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.31331261199117677}}
{"text": "import parlang.defs\nimport mcl.defs\n\nopen parlang\nopen parlang.state\nopen parlang.thread_state\nopen mcl\n\ndef compute_list {σ ι : Type} {τ : ι → Type} [decidable_eq ι] : list (σ → σ) → (thread_state σ τ → thread_state σ τ)\n| (f :: tl) := compute_list tl ∘ @compute σ ι τ _ f\n| [] := id\n\nlemma compute_to_compute_list {σ ι : Type} {τ : ι → Type} [decidable_eq ι] (f : σ → σ) : @compute σ ι τ _ f = compute_list [f] := by refl\n\nlemma compute_list_merge {σ ι : Type} {τ : ι → Type} [decidable_eq ι] (f g : list (σ → σ)) : \n(@compute_list σ ι τ _ g) ∘ compute_list f = compute_list (f ++ g) := begin\n    induction f,\n    case list.nil {\n        simp [compute_list],\n    }, {\n        simp [compute_list],\n        rw ← f_ih,\n    }\nend\n\nlemma compute_list_accesses {sig : signature} {n} (s : state n (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)) (i) (computes) (tid) : \ni ∉ (compute_list computes (vector.nth (s.threads) tid)).accesses ↔ i ∉ (vector.nth (s.threads) tid).accesses := sorry\n\n-- todo: rewrite to s = { tlocal := ... s.tlocal, ..s }\nlemma compute_list_tlocal {sig : signature} {computes : list (memory (parlang_mcl_tlocal sig) → memory (parlang_mcl_tlocal sig))} {tlocal loads stores} {shared : memory $ parlang_mcl_shared sig} : \ncompute_list computes {tlocal := tlocal, shared := shared, loads := loads, stores := stores} = \n{tlocal := computes.foldl (λ tl com, com tl) tlocal, shared := shared, loads := loads, stores := stores} := begin\n    induction computes generalizing tlocal,\n    {\n        refl,\n    }, {\n        simp [compute_list, compute, computes_ih],\n    }\nend\n\n@[simp]\nlemma compute_list_stores_core {sig : signature} {computes}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} : \n(compute_list computes ts).stores = ts.stores := begin\n    induction computes generalizing ts,\n    { simp [compute_list], },\n    { cases ts, simp [compute_list, compute], rw computes_ih, },\nend\n\n@[simp]\nlemma compute_list_loads_core {sig : signature} {computes}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} : \n(compute_list computes ts).loads = ts.loads := begin\n    induction computes generalizing ts,\n    { simp [compute_list], },\n    { cases ts, simp [compute_list, compute], rw computes_ih, },\nend\n\n@[simp]\nlemma compute_list_shared {sig : signature} {computes}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} : \n(compute_list computes ts).shared = ts.shared := begin\n    induction computes generalizing ts,\n    { simp [compute_list], },\n    { cases ts, simp [compute_list, compute], rw computes_ih, },\nend\n\nlemma compute_list_stores {sig : signature} {computes}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} {i : mcl_address sig} : \ni ∉ ts.stores ↔ i ∉ (compute_list computes ts).stores := by simp\n\nlemma compute_list_loads {sig : signature} {computes}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} {i : mcl_address sig} : \ni ∉ ts.loads ↔ i ∉ (compute_list computes ts).loads := by simp", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/mcl/compute_list.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.31296211781119404}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-!\n# More `char` instances\n\nThis file provides a `linear_order` instance on `char`. `char` is the type of Unicode scalar values.\n-/\n\ninstance : linear_order char :=\n{ le_refl := λ a, @le_refl ℕ _ _,\n  le_trans := λ a b c, @le_trans ℕ _ _ _ _,\n  le_antisymm := λ a b h₁ h₂,\n    char.eq_of_veq $ le_antisymm h₁ h₂,\n  le_total := λ a b, @le_total ℕ _ _ _,\n  lt_iff_le_not_le := λ a b, @lt_iff_le_not_le ℕ _ _ _,\n  decidable_le := char.decidable_le,\n  decidable_eq := char.decidable_eq,\n  decidable_lt := char.decidable_lt,\n  ..char.has_le, ..char.has_lt }\n\nlemma char.of_nat_to_nat {c : char} (h : is_valid_char c.to_nat) :\n  char.of_nat c.to_nat = c :=\nbegin\n  rw [char.of_nat, dif_pos h],\n  cases c,\n  simp [char.to_nat]\nend\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/char.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.31285430755141563}}
{"text": "import .core\n\nnamespace tts ------------------------------------------------------------------\nnamespace exp ------------------------------------------------------------------\nvariables {V : Type} [decidable_eq V] -- Type of variable names\nvariables {k k₂ k₃ : ℕ} -- Natural numbers\nvariables {v : V} -- Variable names\nvariables {x : tagged V} -- Variables\nvariables {e ea eb ed ef e₁ e₂ e₃ : exp V} -- Expressions\n\nopen occurs\n\n/-- Open an expression (last parameter) with an expression (e) for a bound\nvariable. -/\ndef open_exp (e : exp V) : ℕ → exp V → exp V\n| k (var bound x)  := if k = x.tag then e else var bound x\n| _ (var free x)   := var free x\n| k (app ef ea)    := app (open_exp k ef) (open_exp k ea)\n| k (lam v eb)     := lam v (open_exp k.succ eb)\n| k (let_ v ed eb) := let_ v (open_exp k ed) (open_exp k.succ eb)\n\n@[simp] theorem open_exp_var_bound_eq (h : k = x.tag) :\n  open_exp e k (var bound x) = e :=\nby simp [open_exp, h]\n\n@[simp] theorem open_exp_var_bound_ne (h : k ≠ x.tag) :\n  open_exp e k (var bound x) = var bound x :=\nby simp [open_exp, h]\n\n@[simp] theorem open_exp_var_free :\n  open_exp e k (var free x) = var free x :=\nrfl\n\n@[simp] theorem open_exp_app :\n  open_exp e k (app ef ea) = app (open_exp e k ef) (open_exp e k ea) :=\nrfl\n\n@[simp] theorem open_exp_lam :\n  open_exp e k (lam v eb) = lam v (open_exp e k.succ eb) :=\nrfl\n\n@[simp] theorem open_exp_let_ :\n  open_exp e k (let_ v ed eb) = let_ v (open_exp e k ed) (open_exp e k.succ eb) :=\nrfl\n\n/-- Open an expression with an expression `e` for the last bound variable (0) -/\ndef open_exp₀ (e : exp V) : exp V → exp V :=\nopen_exp e 0\n\n/-- Open an expression with a free variable `x` for the last bound variable -/\ndef open_var (x : tagged V) : exp V → exp V :=\nopen_exp₀ (var free x)\n\n/-- Open an expression with a variable of the name `v` fresh from the variable\nset `L` for the last bound variable. -/\ndef open_fresh (v : V) (L : finset (tagged V)) : exp V → exp V :=\nopen_var $ (fresh.tagged v).gen L\n\nend /- namespace -/ exp --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/exp/open.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.31285430755141563}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison\n\nimport category_theory.products\nimport category_theory.types\n\nnamespace category_theory\n\nuniverses u₁ v₁ u₂ v₂\n\ndef op (C : Type u₁) : Type u₁ := C\n\nnotation C `ᵒᵖ` := op C\n\nvariables {C : Type u₁} [𝒞 : category.{u₁ v₁} C]\ninclude 𝒞\n\ninstance opposite : category.{u₁ v₁} (Cᵒᵖ) := \n{ hom     := λ X Y : C, Y ⟶ X,\n  comp    := λ _ _ _ f g, g ≫ f,\n  id      := λ X, 𝟙 X }\n\nnamespace functor\n\nsection\nvariables {D : Type u₂} [𝒟 : category.{u₂ v₂} D]\ninclude 𝒟\n\nprotected definition op (F : C ⥤ D) : (Cᵒᵖ) ⥤ (Dᵒᵖ) := \n{ obj       := λ X, F X,\n  map'      := λ X Y f, F.map f,\n  map_id'   := begin /- `obviously'` says: -/ intros, erw [map_id], refl, end,\n  map_comp' := begin /- `obviously'` says: -/ intros, erw [map_comp], refl end }\n\n@[simp] lemma opposite_obj (F : C ⥤ D) (X : C) : (F.op) X = F X := rfl\n@[simp] lemma opposite_map (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) : (F.op).map f = F.map f := rfl\nend\n\nvariable (C)\n\n/-- `functor.hom` is the hom-pairing, sending (X,Y) to X → Y, contravariant in X and covariant in Y. -/\ndefinition hom : (Cᵒᵖ × C) ⥤ (Type v₁) := \n{ obj       := λ p, @category.hom C _ p.1 p.2,\n  map'      := λ X Y f, λ h, f.1 ≫ h ≫ f.2,\n  map_id'   := begin /- `obviously'` says: -/ intros, ext, intros, cases X, dsimp at *, simp, erw [category.id_comp] end,\n  map_comp' := begin /- `obviously'` says: -/ intros, ext, intros, cases f, cases g, cases X, cases Y, cases Z, dsimp at *, simp, erw [category.assoc] end }\n\n@[simp] lemma hom_obj (X : Cᵒᵖ × C) : (functor.hom C) X = @category.hom C _ X.1 X.2 := rfl\n@[simp] lemma hom_pairing_map {X Y : Cᵒᵖ × C} (f : X ⟶ Y) : (functor.hom C).map f = λ h, f.1 ≫ h ≫ f.2 := rfl\n\nend functor\n\nend category_theory", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/category_theory/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891307678319, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.312823426141796}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison\n\nimport category_theory.products\nimport category_theory.types\n\nnamespace category_theory\n\nuniverses v₁ v₂ u₁ u₂ -- declare the `v`'s first; see `category_theory.category` for an explanation\n\n/-- The type of objects of the opposite of C (which should be a category).\n\n  In order to avoid confusion between C and its opposite category, we\n  set up the type of objects `opposite C` using the following pattern,\n  which will be repeated later for the morphisms.\n\n  1. Define `opposite C := C`.\n  2. Define the isomorphisms `op : C → opposite C`, `unop : opposite C → C`.\n  3. Make the definition `opposite` irreducible.\n\n  This has the following consequences.\n\n  * `opposite C` and `C` are distinct types in the elaborator, so you\n    must use `op` and `unop` explicitly to convert between them.\n  * Both `unop (op X) = X` and `op (unop X) = X` are definitional\n    equalities. Notably, every object of the opposite category is\n    definitionally of the form `op X`, which greatly simplifies the\n    definition of the structure of the opposite category, for example.\n\n  (If Lean supported definitional eta equality for records, we could\n  achieve the same goals using a structure with one field.)\n-/\ndef opposite (C : Type u₁) : Type u₁ := C\n\n-- Use a high right binding power (like that of postfix ⁻¹) so that, for example,\n-- `presheaf Cᵒᵖ` parses as `presheaf (Cᵒᵖ)` and not `(presheaf C)ᵒᵖ`.\nnotation C `ᵒᵖ`:std.prec.max_plus := opposite C\n\nvariables {C : Type u₁}\n\ndef op (X : C) : Cᵒᵖ := X\ndef unop (X : Cᵒᵖ) : C := X\n\nattribute [irreducible] opposite\n\n@[simp] lemma unop_op (X : C) : unop (op X) = X := rfl\n@[simp] lemma op_unop (X : Cᵒᵖ) : op (unop X) = X := rfl\n\nlemma op_inj : function.injective (@op C) :=\nby { rintros _ _ ⟨ ⟩, refl }\nlemma unop_inj : function.injective (@unop C) :=\nby { rintros _ _ ⟨ ⟩, refl }\n\nsection has_hom\n\nvariables [𝒞 : has_hom.{v₁} C]\ninclude 𝒞\n\n/-- The hom types of the opposite of a category (or graph).\n\n  As with the objects, we'll make this irreducible below.\n  Use `f.op` and `f.unop` to convert between morphisms of C\n  and morphisms of Cᵒᵖ.\n-/\ninstance has_hom.opposite : has_hom Cᵒᵖ :=\n{ hom := λ X Y, unop Y ⟶ unop X }\n\ndef has_hom.hom.op {X Y : C} (f : X ⟶ Y) : op Y ⟶ op X := f\ndef has_hom.hom.unop {X Y : Cᵒᵖ} (f : X ⟶ Y) : unop Y ⟶ unop X := f\n\nattribute [irreducible] has_hom.opposite\n\nlemma has_hom.hom.op_inj {X Y : C} :\n  function.injective (has_hom.hom.op : (X ⟶ Y) → (op Y ⟶ op X)) :=\nλ _ _ H, congr_arg has_hom.hom.unop H\n\nlemma has_hom.hom.unop_inj {X Y : Cᵒᵖ} :\n  function.injective (has_hom.hom.unop : (X ⟶ Y) → (unop Y ⟶ unop X)) :=\nλ _ _ H, congr_arg has_hom.hom.op H\n\n@[simp] lemma has_hom.hom.unop_op {X Y : C} {f : X ⟶ Y} : f.op.unop = f := rfl\n@[simp] lemma has_hom.hom.op_unop {X Y : Cᵒᵖ} {f : X ⟶ Y} : f.unop.op = f := rfl\n\nend has_hom\n\nvariables [𝒞 : category.{v₁} C]\ninclude 𝒞\n\ninstance category.opposite : category.{v₁} Cᵒᵖ :=\n{ comp := λ _ _ _ f g, (g.unop ≫ f.unop).op,\n  id   := λ X, (𝟙 (unop X)).op }\n\n@[simp] lemma op_comp {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} :\n  (f ≫ g).op = g.op ≫ f.op := rfl\n@[simp] lemma op_id {X : C} : (𝟙 X).op = 𝟙 (op X) := rfl\n\n@[simp] lemma unop_comp {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : Y ⟶ Z} :\n  (f ≫ g).unop = g.unop ≫ f.unop := rfl\n@[simp] lemma unop_id {X : Cᵒᵖ} : (𝟙 X).unop = 𝟙 (unop X) := rfl\n\ndef op_op : (Cᵒᵖ)ᵒᵖ ⥤ C :=\n{ obj := λ X, unop (unop X),\n  map := λ X Y f, f.unop.unop }\n\n-- TODO this is an equivalence\n\nnamespace functor\n\nsection\n\nvariables {D : Type u₂} [𝒟 : category.{v₂} D]\ninclude 𝒟\n\nvariables {C D}\n\nprotected definition op (F : C ⥤ D) : Cᵒᵖ ⥤ Dᵒᵖ :=\n{ obj := λ X, op (F.obj (unop X)),\n  map := λ X Y f, (F.map f.unop).op }\n\n@[simp] lemma op_obj (F : C ⥤ D) (X : Cᵒᵖ) : (F.op).obj X = op (F.obj (unop X)) := rfl\n@[simp] lemma op_map (F : C ⥤ D) {X Y : Cᵒᵖ} (f : X ⟶ Y) : (F.op).map f = (F.map f.unop).op := rfl\n\nprotected definition unop (F : Cᵒᵖ ⥤ Dᵒᵖ) : C ⥤ D :=\n{ obj := λ X, unop (F.obj (op X)),\n  map := λ X Y f, (F.map f.op).unop }\n\n@[simp] lemma unop_obj (F : Cᵒᵖ ⥤ Dᵒᵖ) (X : C) : (F.unop).obj X = unop (F.obj (op X)) := rfl\n@[simp] lemma unop_map (F : Cᵒᵖ ⥤ Dᵒᵖ) {X Y : C} (f : X ⟶ Y) : (F.unop).map f = (F.map f.op).unop := rfl\n\nvariables (C D)\n\ndefinition op_hom : (C ⥤ D)ᵒᵖ ⥤ (Cᵒᵖ ⥤ Dᵒᵖ) :=\n{ obj := λ F, (unop F).op,\n  map := λ F G α,\n  { app := λ X, (α.unop.app (unop X)).op,\n    naturality' := λ X Y f, has_hom.hom.unop_inj $ eq.symm (α.unop.naturality f.unop) } }\n\n@[simp] lemma op_hom.obj (F : (C ⥤ D)ᵒᵖ) : (op_hom C D).obj F = (unop F).op := rfl\n@[simp] lemma op_hom.map_app {F G : (C ⥤ D)ᵒᵖ} (α : F ⟶ G) (X : Cᵒᵖ) :\n  ((op_hom C D).map α).app X = (α.unop.app (unop X)).op := rfl\n\ndefinition op_inv : (Cᵒᵖ ⥤ Dᵒᵖ) ⥤ (C ⥤ D)ᵒᵖ :=\n{ obj := λ F, op F.unop,\n  map := λ F G α, has_hom.hom.op\n  { app := λ X, (α.app (op X)).unop,\n    naturality' := λ X Y f, has_hom.hom.op_inj $ eq.symm (α.naturality f.op) } }\n\n@[simp] lemma op_inv.obj (F : Cᵒᵖ ⥤ Dᵒᵖ) : (op_inv C D).obj F = op F.unop := rfl\n@[simp] lemma op_inv.map_app {F G : Cᵒᵖ ⥤ Dᵒᵖ} (α : F ⟶ G) (X : C) :\n  (((op_inv C D).map α).unop).app X = (α.app (op X)).unop := rfl\n\n-- TODO show these form an equivalence\n\ninstance {F : C ⥤ D} [full F] : full F.op :=\n{ preimage := λ X Y f, (F.preimage f.unop).op }\n\ninstance {F : C ⥤ D} [faithful F] : faithful F.op :=\n{ injectivity' := λ X Y f g h,\n    has_hom.hom.unop_inj $ by simpa using injectivity F (has_hom.hom.op_inj h) }\n\nend\n\nsection\n\nvariable (C)\n\n/-- `functor.hom` is the hom-pairing, sending (X,Y) to X → Y, contravariant in X and covariant in Y. -/\ndefinition hom : Cᵒᵖ × C ⥤ Type v₁ :=\n{ obj       := λ p, unop p.1 ⟶ p.2,\n  map       := λ X Y f, λ h, f.1.unop ≫ h ≫ f.2 }\n\n@[simp] lemma hom_obj (X : Cᵒᵖ × C) : (functor.hom C).obj X = (unop X.1 ⟶ X.2) := rfl\n@[simp] lemma hom_pairing_map {X Y : Cᵒᵖ × C} (f : X ⟶ Y) :\n  (functor.hom C).map f = λ h, f.1.unop ≫ h ≫ f.2 := rfl\n\nend\n\nend functor\n\nomit 𝒞\n\ninstance opposite.has_one [has_one C] : has_one (Cᵒᵖ) :=\n{ one := op 1 }\n\ninstance opposite.has_mul [has_mul C] : has_mul (Cᵒᵖ) :=\n{ mul := λ x y, op $ unop y * unop  x }\n\n@[simp] lemma opposite.unop_one [has_one C] : unop (1 : Cᵒᵖ) = (1 : C) := rfl\n\n@[simp] lemma opposite.unop_mul [has_mul C] (xs ys : Cᵒᵖ) : unop (xs * ys) = (unop ys * unop xs : C) := rfl\n\n@[simp] lemma opposite.op_one [has_one C] : op (1 : C) = 1 := rfl\n\n@[simp] lemma opposite.op_mul [has_mul C] (xs ys : C) : op (xs * ys) = (op ys * op xs) := rfl\n\ninstance opposite.monoid [monoid C] : monoid (Cᵒᵖ) :=\n{ one := op 1,\n  mul := λ x y, op $ unop y * unop  x,\n  mul_one := by { intros, apply unop_inj, simp },\n  one_mul := by { intros, simp },\n  mul_assoc := by { intros, simp [mul_assoc], } }\n\nend category_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3125228985987003}}
{"text": "example (P Q : Type) : P → (Q → P) :=\nbegin\nintro p,\nintro q,\nexact p,\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/3-function-world/l5.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3125228910109688}}
{"text": "import category_theory.triangulated.pretriangulated\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [has_shift C ℤ]\n\nnamespace pretriangulated\n\nnamespace triangle\n\n@[simps]\ndef eval₁ : triangle C ⥤ C :=\n{ obj := λ T, T.obj₁,\n  map := λ T T' φ, φ.hom₁, }\n\n@[simps]\ndef eval₂ : triangle C ⥤ C :=\n{ obj := λ T, T.obj₂,\n  map := λ T T' φ, φ.hom₂, }\n@[simps]\n\ndef eval₃ : triangle C ⥤ C :=\n{ obj := λ T, T.obj₃,\n  map := λ T T' φ, φ.hom₃, }\n\nvariable {C}\n\ninstance is_iso_hom₁ {T T' : triangle C} (φ : T ⟶ T') [is_iso φ] : is_iso φ.hom₁ :=\nby { change is_iso ((eval₁ C).map φ), apply_instance, }\n\ninstance is_iso_hom₂ {T T' : triangle C} (φ : T ⟶ T') [is_iso φ] : is_iso φ.hom₂ :=\nby { change is_iso ((eval₂ C).map φ), apply_instance, }\n\ninstance is_iso_hom₃ {T T' : triangle C} (φ : T ⟶ T') [is_iso φ] : is_iso φ.hom₃ :=\nby { change is_iso ((eval₃ C).map φ), apply_instance, }\n\nend triangle\n\nend pretriangulated\n\nnamespace iso\n\nopen pretriangulated\n\nvariable {C}\n\n@[simps]\ndef triangle_eval₁ {T T' : triangle C} (e : T ≅ T') : T.obj₁ ≅ T'.obj₁ :=\n(triangle.eval₁ C).map_iso e\n\n@[simps]\ndef triangle_eval₂ {T T' : triangle C} (e : T ≅ T') : T.obj₂ ≅ T'.obj₂ :=\n(triangle.eval₂ C).map_iso e\n\n@[simps]\ndef triangle_eval₃ {T T' : triangle C} (e : T ≅ T') : T.obj₃ ≅ T'.obj₃ :=\n(triangle.eval₃ C).map_iso e\n\n@[simp, reassoc]\nlemma triangle_hom_inv_id₁ {T T' : triangle C} (e : T ≅ T') : e.hom.hom₁ ≫ e.inv.hom₁ = 𝟙 _ :=\ne.triangle_eval₁.hom_inv_id\n\n@[simp, reassoc]\nlemma triangle_hom_inv_id₂ {T T' : triangle C} (e : T ≅ T') : e.hom.hom₂ ≫ e.inv.hom₂ = 𝟙 _ :=\ne.triangle_eval₂.hom_inv_id\n\n@[simp, reassoc]\nlemma triangle_hom_inv_id₃ {T T' : triangle C} (e : T ≅ T') : e.hom.hom₃ ≫ e.inv.hom₃ = 𝟙 _ :=\ne.triangle_eval₃.hom_inv_id\n\n@[simp, reassoc]\nlemma triangle_inv_hom_id₁ {T T' : triangle C} (e : T ≅ T') : e.inv.hom₁ ≫ e.hom.hom₁ = 𝟙 _ :=\ne.triangle_eval₁.inv_hom_id\n\n@[simp, reassoc]\nlemma triangle_inv_hom_id₂ {T T' : triangle C} (e : T ≅ T') : e.inv.hom₂ ≫ e.hom.hom₂ = 𝟙 _ :=\ne.triangle_eval₂.inv_hom_id\n\n@[simp, reassoc]\nlemma triangle_inv_hom_id₃ {T T' : triangle C} (e : T ≅ T') : e.inv.hom₃ ≫ e.hom.hom₃ = 𝟙 _ :=\ne.triangle_eval₃.inv_hom_id\n\nend iso\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/triangulated/triangles.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3125228910109688}}
{"text": "/-\ntrace.lean\nAuthor: Mario Carneiro\n\nThis file defines a coinductive datatype for \"traces\", which are possible\nprogram behaviors including I/O. Because Lean does not yet support coinductive\ndatatypes, we have to define this one manually (one would like to define\n`trace := gfp trace1` and skip all the lemmas).\n\nA trace is coinductively defined by `trace1 α := roption (η ⊕ (Σ i:ι, o i → α))`.\nHere `η` is the type of possible final states (including stuck states and\nerror states if applicable), `ι` is the type of I/O inputs (possible functions\nto call and arguments to pass), and `o i` is the type of I/O return values\n(dependent on the input). The trace is really a tree because the I/O output is\nnondeterministic; the program behavior describes what will happen for any\nreturn value. The `roption` handles program divergence; if the program does\nnot terminate then no trace will be produced.\n-/\nimport util.basic\n\nnamespace vc0\nsection\nvariables (η : Type) {ι : Type} (o : ι → Type)\n\ndef trace1 (α : Type) : Type := roption (η ⊕ (Σ i, o i → α))\n\nvariables {η o}\ndef trace1.out {α} (t : trace1 η o α) : roption (Σ i, o i → α) :=\nt.bind $ λ x, match x with\n| (sum.inl _) := roption.none\n| (sum.inr x) := roption.some x\nend\n\ntheorem trace1.mem_out {α} (t : trace1 η o α) {x} : x ∈ t.out ↔\n  (sum.inr x : η ⊕ (Σ i, o i → α)) ∈ (by exact t : roption _) :=\nroption.mem_bind_iff.trans\n⟨by rintro ⟨_|x, ⟨h, e⟩, ⟨⟩, rfl⟩; exact ⟨_, e⟩,\n  λ h, ⟨sum.inr x, h, trivial, rfl⟩⟩\n\ndef trace1.fst {α} (t : trace1 η o α) : roption ι :=\nsigma.fst <$> trace1.out t\n\ndef trace1.snd {α} (t : trace1 η o α) : ∀ i ∈ t.fst, o i → α :=\nroption.mem_cases _ (by exact λ h, ((trace1.out t).get h).2)\n\ntheorem trace1.fst_snd_mem {α} (t : trace1 η o α) : ∀ i h,\n  (⟨i, trace1.snd t i h⟩ : Σ i, o i → α) ∈ t.out :=\nroption.mem_cases _ begin\n  change ∀ h : t.out.dom, (⟨(t.out.get h).1, (t.out.get h).2⟩ : Σ i, o i → α) ∈ t.out,\n  intro h, rw sigma.eta, apply roption.get_mem\nend\n\ntheorem trace1.snd_eq {α} (t : trace1 η o α) (i h g)\n  (H : (⟨i, g⟩ : Σ i, o i → α) ∈ t.out) : trace1.snd t i h = g :=\nby simpa using roption.mem_unique (t.fst_snd_mem i h) H\n\ninstance trace1.functor : functor (trace1 η o) :=\n{ map := λ α β f, roption.map $ sum.map id $ sigma.map id $ λ i, (∘) f }\n\ninstance trace1.is_lawful_functor : is_lawful_functor (trace1 η o) :=\n{ id_map := λ α, roption.map_id' $ by rintro (_|⟨i, g⟩); refl,\n  comp_map := λ α β γ g h t, begin\n    refine eq.trans _ (roption.map_map _ _ _).symm,\n    simp [(<$>)], congr, ext ⟨i, g⟩; refl\n  end }\n\ndef trace1.pmap {α β} (t : trace1 η o α)\n  (f : ∀ x:Σ i, o i → α, x ∈ t.out → o x.1 → α → β) : trace1 η o β :=\n⟨t.dom, λ h, begin\n  cases e : t.get h with a x,\n  { exact sum.inl a },\n  { exact sum.inr ⟨x.1, λ b, f _ (t.mem_out.2 ⟨h, e⟩) b (x.2 b)⟩ }\nend⟩\n\ntheorem trace1.pmap_inl {α β} (t : trace1 η o α) (f h)\n  {a} (H : t.get h = sum.inl a) :\n  (t.pmap f : trace1 η o β).get h = sum.inl a :=\nby simp [trace1.pmap, H]\n\ntheorem trace1.pmap_inr {α β} (t : trace1 η o α) (f h)\n  {x} (H : t.get h = sum.inr x) :\n  (t.pmap f : trace1 η o β).get h =\n  sum.inr ⟨x.1, λ b, f _ (t.mem_out.2 ⟨h, H⟩) b (x.2 b)⟩ :=\nby simp [trace1.pmap, H]\n\ndef trace1.out_map {α β} (f : α → β) (t : trace1 η o α) :\n  (f <$> t : trace1 η o β).out = (sigma.map id $ λ i, (∘) f) <$> t.out :=\nroption.ext $ λ x, by simp [trace1.mem_out, (<$>), sum.map]; refl\n\ntheorem trace1.fst_map {α β} (f : α → β) (t : trace1 η o α) :\n  (f <$> t : trace1 η o β).fst = t.fst :=\nshow sigma.fst <$> trace1.out _ = sigma.fst <$> _, begin\n  rw [trace1.out_map, ← comp_map],\n  congr' 1, ext ⟨a, b⟩, refl\nend\n\ntheorem trace1.snd_map {α β} (f : α → β) (t : trace1 η o α) (i h h' b) :\n  (f <$> t : trace1 η o β).snd i h b = f (t.snd i h' b) :=\nbegin\n  have := (f <$> t).fst_snd_mem i h,\n  simp [t.out_map f] at this,\n  rcases this with ⟨i, g, h₁, h₂⟩,\n  injection h₂ with h₃ h₄, subst h₃, simp at h₄,\n  rw [← h₄, ← t.snd_eq _ h' _ h₁]\nend\n\nvariables (η o)\ndef traceN : ℕ → Type → Type\n| 0     α := α\n| (n+1) α := traceN n (trace1 η o α)\nvariables {η o}\n\ndef traceN.map : ∀ (n) {α β}, (α → β) → traceN η o n α → traceN η o n β\n| 0     α β f := f\n| (n+1) α β f := @traceN.map n _ _ ((<$>) f)\n\ninstance traceN.functor (n) : functor (traceN η o n) :=\n{ map := @traceN.map _ _ _ n }\n\ninstance traceN.is_lawful_functor : ∀ n, is_lawful_functor (traceN η o n)\n| 0 := by split; intros; refl\n| (n+1) := begin\n    split; intros,\n    { show (<$>) id <$> x = x,\n      rw [(funext (@id_map (trace1 η o) _ _ _) : (<$>) id = id),\n        @id_map _ _ (traceN.is_lawful_functor n)] },\n    { change ((<$>) (h ∘ g) <$> x : traceN η o n (trace1 η o γ)) =\n        (<$>) h <$> (<$>) g <$> x,\n      rw [(funext (@comp_map (trace1 η o) _ _ _ _ _ _ _) :\n          (<$>) (h ∘ g) = (<$>) h ∘ (<$>) g),\n        @comp_map _ _ (traceN.is_lawful_functor n)] },\n  end\n\ndef trace1.forall₂ {α β} (R : α → β → Prop) : trace1 η o α → trace1 η o β → Prop :=\nroption.forall₂ $ sum.forall₂ eq $ sigma.forall₂ $ λ i f g, ∀ x, R (f x) (g x)\n\ndef traceN.forall₂ : ∀ {n α β}, (α → β → Prop) → traceN η o n α → traceN η o n β → Prop\n| 0     α β R := R\n| (n+1) α β R := @traceN.forall₂ n _ _ (trace1.forall₂ R)\n\ndef traceN.fst : ∀ {n α}, traceN η o (n+1) α → roption ι\n| 0     α t := trace1.fst t\n| (n+1) α t := @traceN.fst n _ t\n\ndef traceN.out : ∀ {n α}, traceN η o (n+1) α → roption (Σ i, o i → traceN η o n α)\n| 0     α := trace1.out\n| (n+1) α := @traceN.out n _\n\ndef traceN.snd : ∀ {n α} (t : traceN η o (n+1) α) (i ∈ t.fst), o i → traceN η o n α\n| 0     α t := trace1.snd t\n| (n+1) α t := @traceN.snd n _ t\n\ntheorem traceN.fst_map : ∀ n {α β} (f : α → β) (t : traceN η o (n+1) α),\n  (f <$> t : traceN η o (n+1) β).fst = t.fst\n| 0     α β f := trace1.fst_map _\n| (n+1) α β f := traceN.fst_map n _\n\ntheorem traceN.snd_map : ∀ n {α β} (f : α → β) (t : traceN η o (n+1) α) (i h h' b),\n  (f <$> t : traceN η o (n+1) β).snd i h b = f <$> t.snd i h' b\n| 0     α β f := trace1.snd_map _\n| (n+1) α β f := traceN.snd_map n _\n\ntheorem traceN.snd_map' (n) {α β} (f : α → β) (t : traceN η o (n+1) α) (i h b) :\n  (f <$> t : traceN η o (n+1) β).snd i h b = f <$> t.snd i (by rwa t.fst_map _ _ at h) b :=\ntraceN.snd_map _ _ _ _ _ _ _\n\ntheorem traceN.out_map : ∀ n {α β} (f : α → β) (t : traceN η o n (trace1 η o α)),\n  (f <$> t : traceN η o (n+1) β).out =\n  (sigma.map id $ by exact λ i g x, f <$> g x) <$> t.out\n| 0     α β f t := trace1.out_map _ _\n| (n+1) α β f t := traceN.out_map n _ _\n\nvariables (η o)\ndef trace : Type := -- gfp trace1\n{f : ∀ n, traceN η o n unit // ∀ n, f n = (λ _, ()) <$> f (n+1) }\nvariables {η o}\n\ntheorem trace_fst_aux (f : ∀ n, traceN η o n unit)\n  (H : ∀ n, f n = (λ _, ()) <$> f (n+1)) :\n  ∀ n, (f (n+1)).fst = (f 1).fst\n| 0 := rfl\n| (n+1) := by rw [← trace_fst_aux n, H (n+1), traceN.fst_map]; refl\n\ntheorem trace_map_aux (f : ∀ n, traceN η o n unit)\n  (H : ∀ n, f n = (λ _, ()) <$> f (n+1)) :\n  ∀ n, sigma.fst <$> (f (n+1)).out = sigma.fst <$> (f 1).out\n| 0 := rfl\n| (n+1) := begin\n  rw [← trace_map_aux n, H (n+1), traceN.out_map, ← comp_map],\n  congr, ext ⟨i, b⟩, refl\nend\n\ndef trace.destruct : trace η o → trace1 η o (trace η o)\n| ⟨f, H⟩ := trace1.pmap (f 1) (λ x i b _, begin\n  have hx : ∀ n, x.1 ∈ traceN.fst (f (n + 1)),\n  { intro n, rwa trace_fst_aux _ H n, exact roption.mem_map _ i },\n  have := λ n, traceN.out (f (n+1)),\n  refine ⟨λ n, traceN.snd (f (n+1)) _ (hx n) b, λ n, _⟩,\n  refine eq.trans _ ((f (n+1+1)).snd_map' n (λ _, ()) _ _ b),\n  { rw traceN.fst_map, exact hx (n+1) },\n  { congr, apply H }\nend)\n\ndef traceN.iter {α} (F : α → trace1 η o α) (a : α) : ∀ n, traceN η o n α\n| 0     := a\n| (n+1) := (F <$> traceN.iter n : traceN η o n (trace1 η o α))\n\ntheorem traceN.iter_fst {α} (F : α → trace1 η o α) (a : α) :\n  ∀ n, (traceN.iter F a (n+1)).fst = (F a).fst\n| 0     := rfl\n| (n+1) := (traceN.fst_map _ _ _).trans (traceN.iter_fst n)\n\ntheorem traceN.iter_snd' {α} (F : α → trace1 η o α) (a : α) : ∀ n i h b,\n  (traceN.iter F a (n+1)).snd i h b =\n  traceN.iter F ((F a).snd i (by rwa traceN.iter_fst at h) b) n\n| 0     i h b := rfl\n| (n+1) i h b := (traceN.snd_map' _ _ _ i h b).trans (congr_arg _ (traceN.iter_snd' n _ _ _))\n\ntheorem traceN.iter_snd {α} (F : α → trace1 η o α) (a : α) (n i h h' b) :\n  (traceN.iter F a (n+1)).snd i h b = traceN.iter F ((F a).snd i h' b) n :=\ntraceN.iter_snd' _ _ _ _ _ _\n\ndef trace.corec {α} (F : α → trace1 η o α) (a : α) : trace η o :=\n⟨λ n, (λ _, ()) <$> traceN.iter F a n, λ n,\n  by rw traceN.iter; exact\n  eq.trans (by refl) ((comp_map _ _ _).trans (comp_map _ _ _))⟩\n\ntheorem trace.corec_eq {α} (F : α → trace1 η o α) (a : α) :\n  trace.destruct (trace.corec F a) = trace.corec F <$> F a :=\nroption.ext' iff.rfl $ λ h₁ h₂, begin\n  change (F a).dom at h₁,\n  change h₂ with h₁, clear h₂,\n  change (trace1.pmap _ _).get _ = sum.map _ _ _,\n  have e : ((λ _, ()) <$> traceN.iter F a 1).get h₁ =\n    sum.map id (sigma.map id (λ _ _ _, ())) ((F a).get h₁) := rfl,\n  cases e' : (F a).get h₁; rw e' at e,\n  { rw trace1.pmap_inl _ _ _ e, refl },\n  { rw trace1.pmap_inr _ _ _ e, cases val with i g,\n    simp! [sigma.map], ext b,\n    simp [trace.corec], ext n,\n    suffices : ∀ h, traceN.snd ((λ _, ()) <$> traceN.iter F a (n+1))\n      i h b = (λ _, ()) <$> traceN.iter F (g b) n, {apply this},\n    intro,\n    rw traceN.fst_map at h,\n    rw [traceN.snd_map _ _ _ _ _ h, traceN.iter_snd', trace1.snd_eq],\n    exact (trace1.mem_out _).2 ⟨_, e'⟩ },\nend\n\n/-\ntheorem trace.eq_of_bisim (R : trace η o → trace η o → Prop)\n  (H : ∀ t₁ t₂, R t₁ t₂ → trace1.forall₂ R t₁.destruct t₂.destruct) :\n  ∀ t₁ t₂, R t₁ t₂ → t₁ = t₂ :=\nsorry\n-/\n\nend\n\nend vc0\n", "meta": {"author": "digama0", "repo": "vc0", "sha": "b8b192c8c139e0b5a25a7284b93ed53cdf7fd7a5", "save_path": "github-repos/lean/digama0-vc0", "path": "github-repos/lean/digama0-vc0/vc0-b8b192c8c139e0b5a25a7284b93ed53cdf7fd7a5/src/util/trace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3125228910109688}}
{"text": "import .tableau order.lexicographic\n\nopen matrix fintype finset function pequiv partition\nvariables {m n : ℕ}\nvariables {X : ℕ → ℕ → Type} [is_tableau X]\n\nlocal notation `rvec`:2000 n := matrix (fin 1) (fin n) ℚ\nlocal notation `cvec`:2000 m := matrix (fin m) (fin 1) ℚ\nlocal infix ` ⬝ `:70 := matrix.mul\nlocal postfix `ᵀ` : 1500 := transpose\n\nnamespace is_tableau\n\ndef to_lex (T : X m n) (c : fin n) (r' : fin m) : lex ℚ (fin (m + n)) :=\n(abs ((const T) r' 0 / (to_matrix T) r' c), (to_partition T).rowg r')\n\nlemma to_lex_le_iff (T : X m n) (c : fin n) (i i' : fin m) :\n  to_lex T c i ≤ to_lex T c i' ↔\n  abs ((const T) i 0 / (to_matrix T) i c) < abs ((const T) i' 0 / (to_matrix T) i' c) ∨\n    (abs ((const T) i 0 / (to_matrix T) i c) = abs ((const T) i' 0 / (to_matrix T) i' c) ∧\n    (to_partition T).rowg i ≤ (to_partition T).rowg i') :=\nprod.lex_def _ _\n\nlemma pivot_col_spec {T : X m n} {obj : fin m} {c : fin n} :\n  c ∈ pivot_col T obj → (((to_matrix T) obj c ≠ 0 ∧ (to_partition T).colg c ∉ (restricted T))\n  ∨ (0 < (to_matrix T) obj c ∧ (to_partition T).colg c ∈ (restricted T))) ∧ c ∉ (dead T) :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n  to_tableau_pivot_col] at *; exact tableau.pivot_col_spec\n\nlemma nonpos_of_lt_pivot_col {T : X m n} {obj : fin m} {c j : fin n}\n  (hc : c ∈ pivot_col T obj) (hcres : (to_partition T).colg c ∈ (restricted T))\n  (hdead : j ∉ (dead T)) (hjc : (to_partition T).colg j < (to_partition T).colg c) :\n  (to_matrix T) obj j ≤ 0 :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n  to_tableau_pivot_col] at *; exact tableau.nonpos_of_lt_pivot_col hc hcres hdead hjc\n\nlemma pivot_col_eq_none_aux {T : X m n} {obj : fin m} (hT : (feasible T)) {c : fin n} :\n  pivot_col T obj = none → c ∉ (dead T) →\n  (((to_matrix T) obj c = 0 ∧ (to_partition T).colg c ∉ (restricted T))\n    ∨ ((to_matrix T) obj c ≤ 0 ∧ (to_partition T).colg c ∈ (restricted T))) :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n  to_tableau_pivot_col] at *; exact tableau.pivot_col_eq_none_aux hT\n\nlemma pivot_col_eq_none {T : X m n} {obj : fin m} (hT : (feasible T))\n  (h : pivot_col T obj = none) : (is_optimal T) ((of_col T) 0) ((to_partition T).rowg obj) :=\nis_optimal_of_col_zero hT\n(λ j hj, begin\n  have := pivot_col_eq_none_aux hT h hj,\n  finish [lt_irrefl]\nend)\n\nlemma pivot_row_spec {T : X m n} {obj r : fin m} {c : fin n} :\n  r ∈ pivot_row T obj c →\n  obj ≠ r ∧ (to_partition T).rowg r ∈ (restricted T) ∧\n  (to_matrix T) obj c / (to_matrix T) r c < 0 ∧\n  (∀ r' : fin m, obj ≠ r' → (to_partition T).rowg r' ∈ (restricted T) →\n    (to_matrix T) obj c / (to_matrix T) r' c < 0 →\n  abs ((const T) r 0 / (to_matrix T) r c) ≤ abs ((const T) r' 0 / (to_matrix T) r' c)) :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n  to_tableau_pivot_col, to_tableau_pivot_row] at *; exact tableau.pivot_row_spec\n\nlemma nonneg_of_lt_pivot_row {T : X m n} {obj : fin m} {r i : fin m} {c : fin n}\n  (hc0 : 0 < (to_matrix T) obj c) (hres : (to_partition T).rowg i ∈ (restricted T))\n  (hc : c ∈ pivot_col T obj) (hr : r ∈ pivot_row T obj c)\n  (hconst : (const T) i 0 = 0)\n  (hjc : (to_partition T).rowg i < (to_partition T).rowg r) :\n  0 ≤ (to_matrix T) i c :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n    to_tableau_pivot_col, to_tableau_pivot_row] at *;\n  exact tableau.nonneg_of_lt_pivot_row hc0 hres hc hr hconst hjc\n\nlemma ne_zero_of_mem_pivot_row {T : X m n} {obj r : fin m} {c : fin n}\n  (hr : r ∈ pivot_row T obj c) : (to_matrix T) r c ≠ 0 :=\nassume hrc, by simpa [lt_irrefl, hrc] using pivot_row_spec hr\n\nlemma ne_zero_of_mem_pivot_col {T : X m n} {obj : fin m} {c : fin n}\n  (hc : c ∈ pivot_col T obj) : (to_matrix T) obj c ≠ 0 :=\nλ h, by simpa [h, lt_irrefl] using pivot_col_spec hc\n\nlemma pivot_row_eq_none_aux {T : X m n} {obj : fin m} {c : fin n}\n  (hrow : pivot_row T obj c = none) (hs : c ∈ pivot_col T obj) :\n  ∀ r, obj ≠ r → (to_partition T).rowg r ∈ (restricted T) → 0 ≤ (to_matrix T) obj c / (to_matrix T) r c :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n    to_tableau_pivot_col, to_tableau_pivot_row] at *;\n  exact tableau.pivot_row_eq_none_aux hrow hs\n\nlemma pivot_row_eq_none {T : X m n} {obj : fin m} {c : fin n} (hT : (feasible T))\n  (hrow : pivot_row T obj c = none) (hs : c ∈ pivot_col T obj) :\n  (is_unbounded_above T) ((to_partition T).rowg obj) :=\nhave hrow : ∀ r, obj ≠ r → (to_partition T).rowg r ∈ (restricted T) →\n    0 ≤ (to_matrix T) obj c / (to_matrix T) r c,\n  from pivot_row_eq_none_aux hrow hs,\nhave hc : (((to_matrix T) obj c ≠ 0 ∧ (to_partition T).colg c ∉ (restricted T))\n    ∨ (0 < (to_matrix T) obj c ∧ (to_partition T).colg c ∈ (restricted T))) ∧ c ∉ (dead T),\n  from pivot_col_spec hs,\nhave hToc : (to_matrix T) obj c ≠ 0, from λ h, by simpa [h, lt_irrefl] using hc,\n(lt_or_gt_of_ne hToc).elim\n  (λ hToc : (to_matrix T) obj c < 0, is_unbounded_above_rowg_of_nonpos hT c\n    (hc.1.elim and.right (λ h, (not_lt_of_gt hToc h.1).elim)) hc.2\n    (λ i hi, classical.by_cases\n      (λ hoi : obj = i, le_of_lt (hoi ▸ hToc))\n      (λ hoi : obj ≠ i, inv_nonpos.1 $ nonpos_of_mul_nonneg_right (hrow _ hoi hi) hToc))\n    hToc)\n  (λ hToc : 0 < (to_matrix T) obj c, is_unbounded_above_rowg_of_nonneg hT c\n    (λ i hi, classical.by_cases\n      (λ hoi : obj = i, le_of_lt (hoi ▸ hToc))\n      (λ hoi : obj ≠ i, inv_nonneg.1 $ nonneg_of_mul_nonneg_left (hrow _ hoi hi) hToc))\n    hc.2 hToc)\n\ndef feasible_of_mem_pivot_row_and_col {T : X m n} {obj : fin m} (hT : (feasible T)) {c}\n  (hc : c ∈ pivot_col T obj) {r} (hr : r ∈ pivot_row T obj c) :\n  feasible (pivot T r c) :=\nbegin\n  have := pivot_col_spec hc,\n  have := pivot_row_spec hr,\n  have := @feasible_simplex_pivot _ _ _ _ _ obj hT r c,\n  tauto\nend\n\n-- section blands_rule\n\n-- local attribute [instance, priority 0] classical.dec\n-- variable (obj : fin m)\n\n-- def fickle (T T' : X m n) (v : fin (m + n)) : Prop :=\n-- (to_partition T).rowp v ≠ T'.to_partition.rowp v ∨\n-- (to_partition T).colp v ≠ T'.to_partition.colp v\n\n-- lemma fickle_symm {T T' : X m n} {v : fin (m + n)} :\n--   fickle T T' v ↔ fickle T' T v :=\n-- by simp [fickle, eq_comm]\n\n-- lemma fickle_colg_iff_ne {T T' : X m n} {j : fin n} :\n--   fickle T T' ((to_partition T).colg j) ↔\n--   (to_partition T).colg j ≠ T'.to_partition.colg j :=\n-- ⟨λ h h', h.elim (by rw [rowp_colg_eq_none, h', rowp_colg_eq_none]; simp)\n--   (by rw [colp_colg, h', colp_colg]; simp),\n-- λ h, or.inr $ λ h', h begin\n--   have : T'.to_partition.colp ((to_partition T).colg j) = some j,\n--   { simpa [eq_comm] using h' },\n--   rwa [← pequiv.eq_some_iff, colp_symm_eq_some_colg, option.some_inj, eq_comm] at this\n-- end⟩\n\n-- lemma fickle_rowg_iff_ne {T T' : X m n} {i : fin m} :\n--   fickle T T' ((to_partition T).rowg i) ↔\n--   (to_partition T).rowg i ≠ T'.to_partition.rowg i :=\n-- ⟨λ h h', h.elim (by rw [rowp_rowg, h', rowp_rowg]; simp)\n--   (by rw [colp_rowg_eq_none, h', colp_rowg_eq_none]; simp),\n-- λ h, or.inl $ λ h', h begin\n--   have : T'.to_partition.rowp ((to_partition T).rowg i) = some i,\n--   { simpa [eq_comm] using h' },\n--   rwa [← pequiv.eq_some_iff, rowp_symm_eq_some_rowg, option.some_inj, eq_comm] at this\n-- end⟩\n\n-- lemma not_unique_row_and_unique_col {T T' : X m n} {r c c'}\n--   (hcobj0 : 0 < (to_matrix T) obj c)\n--   (hc'obj0 : 0 < T'.to_matrix obj c')\n--   (hrc0 : (to_matrix T) r c < 0)\n--   (hflat : (flat T) = T'.flat)\n--   (hs : (to_partition T).rowg r = T'.to_partition.colg c')\n--   (hrobj : (to_partition T).rowg obj = T'.to_partition.rowg obj)\n--   (hfickle : ∀ i, (fickle T T' ((to_partition T).rowg i)) → (const T) i 0 = 0)\n--   (hobj : (const T) obj 0 = T'.const obj 0)\n--   (nonpos_of_colg_eq : ∀ j, j ≠ c' →\n--     T'.to_partition.colg j = (to_partition T).colg c → T'.to_matrix obj j ≤ 0)\n--   (unique_col : ∀ j,\n--     (fickle T' T (T'.to_partition.colg j)) → j ≠ c' → T'.to_matrix obj j ≤ 0)\n--   (unique_row : ∀ i ≠ r, (const T) i 0 = 0 → fickle T T' ((to_partition T).rowg i) →\n--     0 ≤ (to_matrix T) i c) :\n--   false :=\n-- let objr := (to_partition T).rowg obj in\n-- let x := λ y : ℚ, (of_col T) (y • (single c 0).to_matrix) in\n-- have hxflatT' : ∀ {y}, x y ∈ flat T', from hflat ▸ λ _, of_col_mem_flat _ _,\n-- have hxrow : ∀ y i, x y ((to_partition T).rowg i) 0 = (const T) i 0 + y * (to_matrix T) i c,\n--   by simp [x, of_col_single_rowg],\n-- have hxcol : ∀ {y j}, j ≠ c → x y ((to_partition T).colg j) 0 = 0,\n--   from λ y j hjc, by simp [x, of_col_colg, pequiv.to_matrix, single_apply_of_ne hjc.symm],\n-- have hxcolc : ∀ {y}, x y ((to_partition T).colg c) 0 = y, by simp [x, of_col_colg, pequiv.to_matrix],\n-- let c_star : fin (m + n) → ℚ := λ v, option.cases_on (T'.to_partition.colp v) 0\n--   (T'.to_matrix obj) in\n-- have hxobj : ∀ y, x y objr 0 = (const T) obj 0 + y * (to_matrix T) obj c, from λ y, hxrow _ _,\n-- have hgetr : ∀ {y v}, c_star v * x y v 0 ≠ 0 → (T'.to_partition.colp v).is_some,\n--   from λ y v, by cases h : T'.to_partition.colp v; dsimp [c_star]; rw h; simp,\n-- have c_star_eq_get : ∀ {v} (hv : (T'.to_partition.colp v).is_some),\n--     c_star v = T'.to_matrix obj (option.get hv),\n--   from λ v hv, by dsimp only [c_star]; conv_lhs{rw [← option.some_get hv]}; refl,\n-- have hsummmn : ∀ {y}, sum univ (λ j, T'.to_matrix obj j * x y (T'.to_partition.colg j) 0) =\n--     sum univ (λ v, c_star v * x y v 0),\n--   from λ y, sum_bij_ne_zero (λ j _ _, T'.to_partition.colg j) (λ _ _ _, mem_univ _)\n--     (λ _ _ _ _ _ _ h, T'.to_partition.injective_colg h)\n--     (λ v _ h0, ⟨option.get (hgetr h0), mem_univ _,\n--       by rw [← c_star_eq_get (hgetr h0)]; simpa using h0, by simp⟩)\n--     (λ _ _ h0, by dsimp [c_star]; rw [colp_colg]),\n-- have hgetc : ∀ {y v}, c_star v * x y v 0 ≠ 0 → v ≠ (to_partition T).colg c →\n--     ((to_partition T).rowp v).is_some,\n--   from λ y v, (eq_rowg_or_colg (to_partition T) v).elim\n--     (λ ⟨i, hi⟩, by rw [hi, rowp_rowg]; simp)\n--     (λ ⟨j, hj⟩ h0 hvc,\n--       by rw [hj, hxcol (mt (congr_arg (to_partition T).colg) (hvc ∘ hj.trans)), mul_zero] at h0;\n--         exact (h0 rfl).elim),\n-- have hsummmnn : ∀ {y}, (univ.erase ((to_partition T).colg c)).sum (λ v, c_star v * x y v 0) =\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * x y ((to_partition T).rowg i) 0),\n--   from λ y, eq.symm $ sum_bij_ne_zero (λ i _ _, (to_partition T).rowg i) (by simp)\n--     (λ _ _ _ _ _ _ h, (to_partition T).injective_rowg h)\n--     (λ v hvc h0, ⟨option.get (hgetc h0 (mem_erase.1 hvc).1), mem_univ _, by simpa using h0⟩)\n--     (by intros; refl),\n-- have hsumm : ∀ {y}, univ.sum (λ i, c_star ((to_partition T).rowg i) * x y ((to_partition T).rowg i) 0) =\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * (const T) i 0) +\n--     y * univ.sum (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c),\n--   from λ y, by simp only [hxrow, mul_add, add_mul, sum_add_distrib, mul_assoc,\n--     mul_left_comm _ y, mul_sum.symm],\n-- have hxobj' : ∀ y, x y objr 0 = univ.sum (λ v, c_star v * x y v 0) + T'.const obj 0,\n--   from λ y, by dsimp [objr]; rw [hrobj, mem_flat_iff.1 hxflatT', hsummmn],\n-- have hy : ∀ {y}, y * (to_matrix T) obj c = c_star ((to_partition T).colg c) * y +\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * (const T) i 0) +\n--       y * univ.sum (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c),\n--   from λ y, by rw [← add_left_inj ((const T) obj 0), ← hxobj, hxobj',\n--     ← insert_erase (mem_univ ((to_partition T).colg c)), sum_insert (not_mem_erase _ _),\n--     hsummmnn, hobj, hsumm, hxcolc]; simp,\n-- have hy' : ∀ (y), y * ((to_matrix T) obj c - c_star ((to_partition T).colg c) -\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c)) =\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * (const T) i 0),\n--   from λ y, by rw [mul_sub, mul_sub, hy]; simp [mul_comm, mul_assoc, mul_left_comm],\n-- have h0 : (to_matrix T) obj c - c_star ((to_partition T).colg c) -\n--     univ.sum (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c) = 0,\n--   by rw [← (domain.mul_left_inj (@one_ne_zero ℚ _)), hy', ← hy' 0, zero_mul, mul_zero],\n-- have hcolnec' : T'.to_partition.colp ((to_partition T).colg c) ≠ some c',\n--   from λ h,\n--     by simpa [hs.symm] using congr_arg T'.to_partition.colg (option.eq_some_iff_get_eq.1 h).snd,\n-- have eq_of_roweqc' : ∀ {i}, T'.to_partition.colp ((to_partition T).rowg i) = some c' → i = r,\n--   from λ i h, by simpa [hs.symm, (to_partition T).injective_rowg.eq_iff] using\n--     congr_arg T'.to_partition.colg (option.eq_some_iff_get_eq.1 h).snd,\n-- have sumpos : 0 < univ.sum (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c),\n--   by rw [← sub_eq_zero.1 h0]; exact add_pos_of_pos_of_nonneg hcobj0\n--     (begin\n--       simp only [c_star, neg_nonneg],\n--       cases h : T'.to_partition.colp ((to_partition T).colg c) with j,\n--       { refl },\n--       { exact nonpos_of_colg_eq j (mt (congr_arg some) (h ▸ hcolnec'))\n--           (by rw [← (option.eq_some_iff_get_eq.1 h).snd]; simp) }\n--     end),\n-- have hexi : ∃ i, 0 < c_star ((to_partition T).rowg i) * (to_matrix T) i c,\n--   from imp_of_not_imp_not _ _ (by simpa using @sum_nonpos _ _ (@univ (fin m) _)\n--     (λ i, c_star ((to_partition T).rowg i) * (to_matrix T) i c) _ _) sumpos,\n-- let ⟨i, hi⟩ := hexi in\n-- have hi0 : (const T) i 0 = 0, from hfickle i\n--   (fickle_rowg_iff_ne.2 $\n--     λ h, by dsimp [c_star] at hi; rw [h, colp_rowg_eq_none] at hi; simpa [lt_irrefl] using hi),\n-- have hi_some : (T'.to_partition.colp ((to_partition T).rowg i)).is_some,\n--   from option.ne_none_iff_is_some.1 (λ h, by dsimp only [c_star] at hi; rw h at hi;\n--     simpa [lt_irrefl] using hi),\n-- have hi' : 0 < T'.to_matrix obj (option.get hi_some) * (to_matrix T) i c,\n--   by dsimp only [c_star] at hi; rwa [← option.some_get hi_some] at hi,\n-- have hir : i ≠ r, from λ hir, begin\n--     have : option.get hi_some = c', from T'.to_partition.injective_colg\n--       (by rw [colg_get_colp_symm, ← hs, hir]),\n--     rw [this, hir] at hi',\n--     exact not_lt_of_gt hi' (mul_neg_of_pos_of_neg hc'obj0 hrc0)\n--   end,\n-- have hnec' : option.get hi_some ≠ c',\n--   from λ eq_c', hir $ @eq_of_roweqc' i (eq_c' ▸ by simp),\n-- have hic0 : (to_matrix T) i c < 0,\n--   from neg_of_mul_pos_right hi' (unique_col _\n--     (by rw [fickle_colg_iff_ne]; simp) hnec'),\n-- not_le_of_gt hic0 (unique_row _ hir hi0\n--   (by rw [fickle_rowg_iff_ne, ← colg_get_colp_symm _ _ hi_some]; exact colg_ne_rowg _ _ _))\n\n-- inductive rel : X m n → X m n → Prop\n-- | pivot : ∀ {T}, feasible T → ∀ {r c}, c ∈ pivot_col T obj →\n--   r ∈ pivot_row T obj c → rel (T.pivot r c) T\n-- | trans_pivot : ∀ {T₁ T₂ r c}, rel T₁ T₂ → c ∈ pivot_col T₁ obj →\n--   r ∈ pivot_row T₁ obj c → rel (T₁.pivot r c) T₂\n\n-- lemma feasible_of_rel_right {T T' : X m n} (h : rel obj T' T) : (feasible T) :=\n-- rel.rec_on h (by tauto) (by tauto)\n\n-- lemma feasible_of_rel_left {T T' : X m n} (h : rel obj T' T) : T'.feasible :=\n-- rel.rec_on h (λ _ hT _ _ hc hr, feasible_of_mem_pivot_row_and_col hT hc hr)\n--   (λ _ _ _ _ _ hc hr hT, feasible_of_mem_pivot_row_and_col hT hc hr)\n\n-- /-- Slightly stronger recursor than the default recursor -/\n-- @[elab_as_eliminator]\n-- lemma rel.rec_on' {obj : fin m} {C : X m n → X m n → Prop} {T T' : X m n}\n--   (hrel : rel obj T T')\n--   (hpivot : ∀ {T : X m n} {r : fin m} {c : fin n},\n--      feasible T → c ∈ pivot_col T obj → r ∈ pivot_row T obj c → C (pivot T r c) T)\n--   (hpivot_trans : ∀ {T₁ T₂ : X m n} {r : fin m} {c : fin n},\n--     rel obj (T₁.pivot r c) T₁ → rel obj T₁ T₂ →\n--      c ∈ pivot_col T₁ obj →\n--      r ∈ pivot_row T₁ obj c → C (T₁.pivot r c) T₁ → C T₁ T₂ → C (pivot T₁ r c) T₂) :\n--   C T T' :=\n-- rel.rec_on hrel (λ T hT r c  hc hr, hpivot hT hc hr) (λ T₁ T₂ r c hrelT₁₂ hc hr ih, hpivot_trans\n--   (rel.pivot (feasible_of_rel_left obj hrelT₁₂) hc hr) hrelT₁₂ hc hr\n--   (hpivot (feasible_of_rel_left obj hrelT₁₂) hc hr) ih)\n\n-- lemma rel.trans {obj : fin m} {T₁ T₂ T₃ : X m n} (h₁₂ : rel obj T₁ T₂) :\n--   rel obj T₂ T₃ → rel obj T₁ T₃ :=\n-- rel.rec_on h₁₂\n--   (λ T r c hT hc hr hrelT, rel.trans_pivot hrelT hc hr)\n--   (λ T₁ T₂ r c hrelT₁₂ hc hr ih hrelT₂₃, rel.trans_pivot (ih hrelT₂₃) hc hr)\n\n-- instance : is_trans (X m n) (rel obj) := ⟨@rel.trans _ _ obj⟩\n\n-- lemma flat_eq_of_rel {T T' : X m n} (h : rel obj T' T) : flat T' = flat T :=\n-- rel.rec_on' h (λ _ _ _ _ _ hr, flat_pivot (ne_zero_of_mem_pivot_row hr))\n--   (λ _ _ _ _ _ _ _ _, eq.trans)\n\n-- lemma rowg_obj_eq_of_rel {T T' : X m n} (h : rel obj T T') : (to_partition T).rowg obj =\n--   T'.to_partition.rowg obj :=\n-- rel.rec_on' h (λ T r c hfT hc hr, by simp [rowg_swap_of_ne _ (pivot_row_spec hr).1])\n--   (λ _ _ _ _ _ _ _ _, eq.trans)\n\n-- lemma restricted_eq_of_rel {T T' : X m n} (h : rel obj T T') : (restricted T) = T'.restricted :=\n-- rel.rec_on' h (λ _ _ _ _ _ _, rfl) (λ _ _ _ _ _ _ _ _, eq.trans)\n\n-- lemma dead_eq_of_rel {T T' : X m n} (h : rel obj T T') : (dead T) = T'.dead :=\n-- rel.rec_on' h (λ _ _ _ _ _ _, rfl) (λ _ _ _ _ _ _ _ _, eq.trans)\n\n-- lemma dead_eq_of_rel_or_eq {T T' : X m n} (h : T = T' ∨ rel obj T T') : (dead T) = T'.dead :=\n-- h.elim (congr_arg _) $ dead_eq_of_rel _\n\n-- lemma exists_mem_pivot_row_col_of_rel {T T' : X m n} (h : rel obj T' T) :\n--   ∃ r c, c ∈ pivot_col T obj ∧ r ∈ pivot_row T obj c :=\n-- rel.rec_on' h (λ _ r c _ hc hr, ⟨r, c, hc, hr⟩) (λ _ _ _ _ _ _ _ _ _, id)\n\n-- lemma exists_mem_pivot_row_of_rel {T T' : X m n} (h : rel obj T' T) {c : fin n}\n--   (hc : c ∈ pivot_col T obj) : ∃ r, r ∈ pivot_row T obj c :=\n-- let ⟨r, c', hc', hr⟩ := exists_mem_pivot_row_col_of_rel obj h in ⟨r, by simp * at *⟩\n\n-- lemma exists_mem_pivot_col_of_fickle {T₁ T₂ : X m n} (h : rel obj T₂ T₁) {c : fin n} :\n--   fickle T₁ T₂ (T₁.to_partition.colg c) →\n--   ∃ T₃, (T₃ = T₁ ∨ rel obj T₃ T₁) ∧ (rel obj T₂ T₃) ∧\n--   T₃.to_partition.colg c = T₁.to_partition.colg c ∧\n--   c ∈ pivot_col T₃ obj :=\n-- rel.rec_on' h begin\n--     assume T r c' hT hc' hr,\n--     rw fickle_colg_iff_ne,\n--     by_cases hcc : c = c',\n--     { subst hcc,\n--       exact λ _, ⟨T, or.inl rfl, rel.pivot hT hc' hr, rfl, hc'⟩ },\n--     { simp [colg_swap_of_ne _ hcc] }\n--   end\n--   (λ T₁ T₂ r c hrelp₁ hrel₁₂ hc hr ihp₁ ih₁₂,\n--     (imp_iff_not_or.1 ih₁₂).elim\n--       (λ ih₁₂, (imp_iff_not_or.1 ihp₁).elim\n--         (λ ihp₁ hf, (fickle_colg_iff_ne.1 hf (by simp [*, fickle_colg_iff_ne] at *)).elim)\n--         (λ ⟨T₃, hT₃⟩ hf, ⟨T₃,\n--           hT₃.1.elim (λ h, h.symm ▸ or.inr hrel₁₂) (λ h, or.inr $ h.trans hrel₁₂),\n--           hT₃.2.1, hT₃.2.2.1.trans (by simpa [eq_comm, fickle_colg_iff_ne] using ih₁₂), hT₃.2.2.2⟩))\n--       (λ ⟨T₃, hT₃⟩ hf, ⟨T₃, hT₃.1, hrelp₁.trans hT₃.2.1, hT₃.2.2⟩))\n\n-- lemma exists_mem_pivot_row_of_fickle {T₁ T₂ : X m n} (h : rel obj T₂ T₁) (r : fin m) :\n--   fickle T₁ T₂ (T₁.to_partition.rowg r) →\n--   ∃ (T₃ : X m n) c, (T₃ = T₁ ∨ rel obj T₃ T₁) ∧ (rel obj T₂ T₃) ∧\n--     T₃.to_partition.rowg r = T₁.to_partition.rowg r ∧\n--     c ∈ pivot_col T₃ obj ∧ r ∈ pivot_row T₃ obj c :=\n-- rel.rec_on' h\n--   begin\n--     assume T r' c hT hc hr',\n--     rw fickle_rowg_iff_ne,\n--     by_cases hrr : r = r',\n--     { subst hrr,\n--       exact λ _, ⟨T, c, or.inl rfl, rel.pivot hT hc hr', rfl, hc, hr'⟩ },\n--     { simp [rowg_swap_of_ne _ hrr] }\n--   end\n--   (λ T₁ T₂ r c hrelp₁ hrel₁₂ hc hr ihp₁ ih₁₂,\n--     (imp_iff_not_or.1 ih₁₂).elim\n--       (λ ih₁₂, (imp_iff_not_or.1 ihp₁).elim\n--         (λ ihp₁ hf, (fickle_rowg_iff_ne.1 hf (by simp [*, fickle_rowg_iff_ne] at *)).elim)\n--         (λ ⟨T₃, c', hT₃⟩ hf, ⟨T₃, c', hT₃.1.elim (λ h, h.symm ▸ or.inr hrel₁₂)\n--           (λ h, or.inr $ h.trans hrel₁₂),\n--             hT₃.2.1,\n--             hT₃.2.2.1.trans (by simpa [eq_comm, fickle_rowg_iff_ne] using ih₁₂),\n--             by clear_aux_decl; tauto⟩))\n--       (λ ⟨T₃, c', hT₃⟩ _, ⟨T₃, c', hT₃.1,\n--         (rel.pivot (feasible_of_rel_left _ hrel₁₂) hc hr).trans hT₃.2.1, hT₃.2.2⟩))\n\n-- lemma eq_or_rel_pivot_of_rel {T₁ T₂ : X m n} (h : rel obj T₁ T₂) : ∀ {r c}\n--   (hc : c ∈ pivot_col T₂ obj) (hr : r ∈ pivot_row T₂ obj c),\n--   T₁ = T₂.pivot r c ∨ rel obj T₁ (T₂.pivot r c) :=\n-- rel.rec_on' h (λ T r c hT hc hr r' c' hc' hr', by simp * at *)\n--   (λ T₁ T₂ r c hrelp₁ hrel₁₂ hc hr ihp₁ ih₁₂ r' c' hc' hr',\n--     (ih₁₂ hc' hr').elim\n--       (λ ih₁₂, or.inr $ ih₁₂ ▸ rel.pivot (feasible_of_rel_left _ hrel₁₂) hc hr)\n--       (λ ih₁₂, or.inr $ (rel.pivot (feasible_of_rel_left _ hrel₁₂) hc hr).trans ih₁₂))\n\n-- lemma exists_mem_pivot_col_of_mem_pivot_row {T : X m n} (hrelTT : rel obj T T)\n--   {r c} (hc : c ∈ pivot_col T obj) (hr : r ∈ pivot_row T obj c) :\n--   ∃ (T' : X m n), c ∈ pivot_col T' obj ∧ T'.to_partition.colg c =\n--   (to_partition T).rowg r ∧ rel obj T' T ∧ rel obj T T' :=\n-- have hrelTTp : rel obj T (T.pivot r c),\n--   from (eq_or_rel_pivot_of_rel _ hrelTT hc hr).elim (λ h, h ▸ hrelTT ) id,\n-- let ⟨T', hT'⟩ := exists_mem_pivot_col_of_fickle obj hrelTTp $ fickle_colg_iff_ne.2 $\n--   (show (T.pivot r c).to_partition.colg c ≠ (to_partition T).colg c, by simp) in\n-- ⟨T', hT'.2.2.2, by simp [hT'.2.2.1], hT'.1.elim\n--   (λ h, h.symm ▸ rel.pivot (feasible_of_rel_left _ hrelTT) hc hr)\n--   (λ h, h.trans $ rel.pivot (feasible_of_rel_left _ hrelTT) hc hr), hT'.2.1⟩\n\n-- lemma exists_mem_pivot_col_of_fickle_row {T T' : X m n} (hrelTT' : rel obj T T') {r : fin m}\n--   (hrelT'T : rel obj T' T) (hrow : fickle T T' ((to_partition T).rowg r)) :\n--   ∃ (T₃ : X m n) c, c ∈ pivot_col T₃ obj ∧ T₃.to_partition.colg c =\n--   (to_partition T).rowg r ∧ rel obj T₃ T ∧ rel obj T T₃ :=\n-- let ⟨T₃, c, hT₃, hrelT₃T, hrow₃, hc, hr⟩ :=\n--   exists_mem_pivot_row_of_fickle obj hrelT'T _ hrow in\n-- let ⟨T₄, hT₄⟩ := exists_mem_pivot_col_of_mem_pivot_row obj\n--   (show rel obj T₃ T₃, from hT₃.elim (λ h, h.symm ▸ hrelTT'.trans hrelT'T)\n--     (λ h, h.trans $ hrelTT'.trans hrelT₃T)) hc hr in\n-- ⟨T₄, c, hT₄.1, hT₄.2.1.trans hrow₃, hT₄.2.2.1.trans $ hT₃.elim (λ h, h.symm ▸ hrelTT'.trans hrelT'T)\n--   (λ h, h.trans $ hrelTT'.trans hrelT'T), hrelTT'.trans (hrelT₃T.trans hT₄.2.2.2)⟩\n\n-- lemma const_obj_le_of_rel {T₁ T₂ : X m n} (h : rel obj T₁ T₂) :\n--   T₂.const obj 0 ≤ T₁.const obj 0 :=\n-- rel.rec_on' h (λ T r c hT hc hr,\n--     have hr' : _ := pivot_row_spec hr,\n--     simplex_const_obj_le hT (by tauto) (by tauto))\n--   (λ _ _ _ _ _ _ _ _ h₁ h₂, le_trans h₂ h₁)\n\n-- lemma const_obj_eq_of_rel_of_rel {T₁ T₂ : X m n} (h₁₂ : rel obj T₁ T₂)\n--   (h₂₁ : rel obj T₂ T₁) : T₁.const obj 0 = T₂.const obj 0 :=\n-- le_antisymm (const_obj_le_of_rel _ h₂₁) (const_obj_le_of_rel _ h₁₂)\n\n-- lemma const_eq_const_of_const_obj_eq {T₁ T₂ : X m n} (h₁₂ : rel obj T₁ T₂) :\n--   ∀ (hobj : T₁.const obj 0 = T₂.const obj 0) (i : fin m), T₁.const i 0 = T₂.const i 0 :=\n-- rel.rec_on' h₁₂\n--   (λ T r c hfT hc hr hobj i,\n--     have hr0 : (const T) r 0 = 0, from const_eq_zero_of_const_obj_eq hfT\n--       (ne_zero_of_mem_pivot_col hc) (ne_zero_of_mem_pivot_row hr)\n--       (pivot_row_spec hr).1 hobj,\n--     if hir : i = r\n--       then by simp [hir, hr0]\n--       else by simp [const_pivot_of_ne _ hir, hr0])\n--   (λ T₁ T₂ r c hrelp₁ hrel₁₂ hc hr ihp₁ ih₁₂ hobj i,\n--     have hobjp : (pivot T₁ r c).const obj 0 = T₁.const obj 0,\n--       from le_antisymm (hobj.symm ▸ const_obj_le_of_rel _ hrel₁₂)\n--         (const_obj_le_of_rel _ hrelp₁),\n--     by rw [ihp₁ hobjp, ih₁₂ (hobjp.symm.trans hobj)])\n\n-- lemma const_eq_zero_of_fickle_of_rel_self {T T' : X m n} (hrelTT' : rel obj T T')\n--   (hrelT'T : rel obj T' T) (i : fin m) (hrow : fickle T T' ((to_partition T).rowg i)) :\n--   (const T) i 0 = 0 :=\n-- let ⟨T₃, c, hT₃₁, hT'₃, hrow₃, hc, hi⟩ := exists_mem_pivot_row_of_fickle obj hrelT'T _ hrow in\n-- have T₃.const i 0 = 0, from const_eq_zero_of_const_obj_eq\n--   (feasible_of_rel_right _ hT'₃) (ne_zero_of_mem_pivot_col hc)\n--   (ne_zero_of_mem_pivot_row hi) (pivot_row_spec hi).1\n--   (const_obj_eq_of_rel_of_rel _ (rel.pivot (feasible_of_rel_right _ hT'₃) hc hi)\n--     ((eq_or_rel_pivot_of_rel _ hT'₃ hc hi).elim\n--       (λ h, h ▸ hT₃₁.elim (λ h, h.symm ▸ hrelTT') (λ h, h.trans hrelTT'))\n--       (λ hrelT'p, hT₃₁.elim (λ h, h.symm ▸ hrelTT'.trans (h ▸ hrelT'p))\n--         (λ h, h.trans $ hrelTT'.trans hrelT'p)))),\n-- have hobj : T₃.const obj 0 = (const T) obj 0,\n--   from hT₃₁.elim (λ h, h ▸ rfl) (λ h, const_obj_eq_of_rel_of_rel _ h (hrelTT'.trans hT'₃)),\n-- hT₃₁.elim (λ h, h ▸ this) (λ h, const_eq_const_of_const_obj_eq obj h hobj i ▸ this)\n\n-- lemma colg_mem_restricted_of_rel_self {T : X m n} (hrelTT : rel obj T T)\n--   {c} (hc : c ∈ pivot_col T obj) : (to_partition T).colg c ∈ (restricted T) :=\n-- let ⟨r, hr⟩ := exists_mem_pivot_row_of_rel obj hrelTT hc in\n-- let ⟨T', c', hT', hrelTT', hrowcol, _, hr'⟩ := exists_mem_pivot_row_of_fickle obj\n--     ((eq_or_rel_pivot_of_rel _ hrelTT hc hr).elim\n--       (λ h, show rel obj T (T.pivot r c), from h ▸ hrelTT) id) _\n--   (fickle_rowg_iff_ne.2 $\n--     show (T.pivot r c).to_partition.rowg r ≠ (to_partition T).rowg r, by simp) in\n-- (restricted_eq_of_rel _ hrelTT').symm ▸ by convert (pivot_row_spec hr').2.1; simp [hrowcol]\n\n-- lemma eq_zero_of_not_mem_restricted_of_rel_self {T : X m n} (hrelTT : rel obj T T)\n--   {j} (hjres : (to_partition T).colg j ∉ (restricted T)) (hdead : j ∉ (dead T)) : (to_matrix T) obj j = 0 :=\n-- let ⟨r, c, hc, hr⟩ := exists_mem_pivot_row_col_of_rel obj hrelTT in\n-- have hcres : (to_partition T).colg c ∈ (restricted T),\n--   from colg_mem_restricted_of_rel_self obj hrelTT hc,\n-- by_contradiction $ λ h0,\n-- begin\n--   simp [pivot_col] at hc,\n--   cases h : fin.find (λ c, (to_matrix T) obj c ≠ 0 ∧ colg ((to_partition T)) c ∉ (restricted T)\n--     ∧ c ∉ (dead T)),\n--   { simp [*, fin.find_eq_none_iff] at * },\n--   { rw h at hc, clear_aux_decl,\n--     have := (fin.find_eq_some_iff.1 h).1,\n--     simp * at * }\n-- end\n\n-- lemma rel.irrefl {obj : fin m} : ∀ (T : X m n), ¬ rel obj T T :=\n-- λ T1 hrelT1,\n-- let ⟨rT1 , cT1, hrT1, hcT1⟩ := exists_mem_pivot_row_col_of_rel obj hrelT1 in\n-- let ⟨t, ht⟩ := finset.max_of_mem\n--   (show T1.to_partition.colg cT1 ∈ univ.filter (λ v, ∃ (T' : X m n) (c : fin n),\n--       rel obj T' T' ∧ c ∈ pivot_col T' obj ∧ T'.to_partition.colg c = v),\n--     by simp only [true_and, mem_filter, mem_univ, exists_and_distrib_left];\n--       exact ⟨T1, hrelT1, cT1, hrT1, rfl⟩) in\n-- let ⟨_, T', c', hrelTT'', hcT', hct⟩ := finset.mem_filter.1 (finset.mem_of_max ht) in\n-- have htmax : ∀ (s : fin (m + n)) (T : X m n),\n--     rel obj T T → ∀ (j : fin n), pivot_col T obj = some j →\n--       (to_partition T).colg j = s → s ≤ t,\n--   by simpa using λ s (h : s ∈ _), finset.le_max_of_mem h ht,\n-- let ⟨r, hrT'⟩ := exists_mem_pivot_row_of_rel obj hrelTT'' hcT' in\n-- have hrelTT''p : rel obj T' (T'.pivot r c'),\n--   from (eq_or_rel_pivot_of_rel obj hrelTT'' hcT' hrT').elim (λ h, h ▸ hrelTT'') id,\n-- let ⟨T, c, hTT', hrelT'T, hT'Tr, hc, hr⟩ := exists_mem_pivot_row_of_fickle obj\n--   hrelTT''p r (by rw fickle_symm; simp [fickle_colg_iff_ne]) in\n-- have hfT' : feasible T', from feasible_of_rel_left _ hrelTT'',\n-- have hfT : feasible T, from feasible_of_rel_right _ hrelT'T,\n-- have hrelT'pT' : rel obj (T'.pivot r c') T', from rel.pivot hfT' hcT' hrT',\n-- have hrelTT' : rel obj T T', from hTT'.elim (λ h, h.symm ▸ hrelT'pT') (λ h, h.trans hrelT'pT'),\n-- have hrelTT : rel obj T T, from hrelTT'.trans hrelT'T,\n-- have hc't : (to_partition T).colg c ≤ t, from htmax _ T hrelTT _ hc rfl,\n-- have hoT'T : T'.const obj 0 = (const T) obj 0, from const_obj_eq_of_rel_of_rel _ hrelT'T hrelTT',\n-- have hfickle : ∀ i, fickle T T' ((to_partition T).rowg i) → (const T) i 0 = 0,\n--   from const_eq_zero_of_fickle_of_rel_self obj hrelTT' hrelT'T,\n-- have hobj : (const T) obj 0 = T'.const obj 0, from const_obj_eq_of_rel_of_rel _ hrelTT' hrelT'T,\n-- have hflat : (flat T) = T'.flat, from flat_eq_of_rel obj hrelTT',\n-- have hrobj : (to_partition T).rowg obj = T'.to_partition.rowg obj, from rowg_obj_eq_of_rel _ hrelTT',\n-- have hs : (to_partition T).rowg r = T'.to_partition.colg c', by simpa using hT'Tr,\n-- have hc'res : T'.to_partition.colg c' ∈ T'.restricted,\n--   from hs ▸ restricted_eq_of_rel _ hrelTT' ▸ (pivot_row_spec hr).2.1,\n-- have hc'obj0 : 0 < T'.to_matrix obj c' ∧ c' ∉ T'.dead,\n--   by simpa [hc'res] using pivot_col_spec hcT',\n-- have hcres : (to_partition T).colg c ∈ (restricted T),\n--   from colg_mem_restricted_of_rel_self obj hrelTT hc,\n-- have hcobj0 : 0 < to_matrix T obj c ∧ c ∉ (dead T),\n--   by simpa [hcres] using pivot_col_spec hc,\n-- have hrc0 : (to_matrix T) r c < 0,\n--   from inv_neg'.1 $ neg_of_mul_neg_left (pivot_row_spec hr).2.2.1 (le_of_lt hcobj0.1),\n-- have nonpos_of_colg_ne : ∀ j, (fickle T' T (T'.to_partition.colg j)) → j ≠ c' →\n--     T'.to_matrix obj j ≤ 0,\n--   from λ j hj hjc',\n--     let ⟨T₃, hT₃⟩ := exists_mem_pivot_col_of_fickle obj hrelTT' hj in\n--     nonpos_of_lt_pivot_col hcT' hc'res\n--       (dead_eq_of_rel_or_eq obj hT₃.1 ▸ (pivot_col_spec hT₃.2.2.2).2)\n--       (lt_of_le_of_ne\n--         (hct.symm ▸ hT₃.2.2.1 ▸ htmax _ T₃ (hT₃.1.elim (λ h, h.symm ▸ hrelTT'')\n--           (λ h, h.trans (hrelT'T.trans hT₃.2.1))) _ hT₃.2.2.2 rfl)\n--         (by rwa [ne.def, T'.to_partition.injective_colg.eq_iff])),\n-- have nonpos_of_colg_eq : ∀ j, j ≠ c' →\n--     T'.to_partition.colg j = (to_partition T).colg c → T'.to_matrix obj j ≤ 0,\n--   from λ j hjc' hj,\n--     if hjc : j = c\n--     then by clear_aux_decl; subst hjc;\n--       exact nonpos_of_lt_pivot_col hcT' hc'res\n--         (by rw [dead_eq_of_rel obj hrelT'T]; tauto)\n--         (lt_of_le_of_ne (hj.symm ▸ hct.symm ▸ hc't) (by simpa))\n--     else nonpos_of_colg_ne _ (fickle_colg_iff_ne.2 $ by simpa [hj, eq_comm] using hjc) hjc',\n-- have unique_row : ∀ i ≠ r, (const T) i 0 = 0 → fickle T T' ((to_partition T).rowg i) →\n--     0 ≤ (to_matrix T) i c,\n--   from λ i hir hi0 hrow,\n--     let ⟨T₃, c₃, hc₃, hrow₃, hrelT₃T, hrelTT₃⟩ :=\n--       exists_mem_pivot_col_of_fickle_row _ hrelTT' hrelT'T hrow in\n--     have hrelT₃T₃ : rel obj T₃ T₃, from hrelT₃T.trans hrelTT₃,\n--     nonneg_of_lt_pivot_row (by exact hcobj0.1)\n--       (by rw [← hrow₃, ← restricted_eq_of_rel _ hrelT₃T];\n--         exact colg_mem_restricted_of_rel_self _ hrelT₃T₃ hc₃) hc hr hi0\n--       (lt_of_le_of_ne (by rw [hs, hct, ← hrow₃]; exact htmax _ _ hrelT₃T₃ _ hc₃ rfl)\n--         (by simpa [fickle_rowg_iff_ne] using hrow)),\n-- not_unique_row_and_unique_col obj hcobj0.1 hc'obj0.1 hrc0 hflat hs hrobj hfickle hobj\n--   nonpos_of_colg_eq nonpos_of_colg_ne unique_row\n\n-- noncomputable instance fintype_rel (T : X m n) : fintype {T' | rel obj T' T} :=\n-- fintype.of_injective (λ T', T'.val.to_partition)\n--   (λ T₁ T₂ h, subtype.eq $ tableau.ext\n--     (by rw [flat_eq_of_rel _ T₁.2, flat_eq_of_rel _ T₂.2]) h\n--     (by rw [dead_eq_of_rel _ T₁.2, dead_eq_of_rel _ T₂.2])\n--     (by rw [restricted_eq_of_rel _ T₁.2, restricted_eq_of_rel _ T₂.2]))\n\n-- lemma rel_wf (m n : ℕ) (obj : fin m): well_founded (@rel m n obj) :=\n-- subrelation.wf\n--   (show subrelation (@rel m n obj) (measure (λ T, fintype.card {T' | rel obj T' T})),\n--     from assume T₁ T₂ h,\n--     set.card_lt_card (set.ssubset_iff_subset_not_subset.2 ⟨λ T' hT', hT'.trans h,\n--       classical.not_forall.2 ⟨T₁, λ h', rel.irrefl _ (h' h)⟩⟩))\n--   (measure_wf (λ T, fintype.card {T' | rel obj T' T}))\n\n-- end blands_rule\n\ndef rel (obj : fin m) : Π (T₁ T₂ : X m n), Prop :=\ninv_image (tableau.rel obj) to_tableau\n\nlemma rel_wf (m n : ℕ) (obj : fin m): well_founded (@rel m n X _ obj) :=\ninv_image.wf _ (tableau.rel_wf _ _ _)\n\nlemma rel.pivot {obj : fin m} {T : X m n} : feasible T → ∀ {r : fin m} {c : fin n},\n  c ∈ pivot_col T obj → r ∈ pivot_row T obj c → rel obj (pivot T r c) T :=\nby simp only [to_matrix, const, to_partition, restricted, feasible, to_tableau_pivot,\n    to_tableau_pivot_col, to_tableau_pivot_row, rel, inv_image] at *;\n  exact tableau.rel.pivot\n\ninductive termination (n : ℕ) : Type\n| while {}              : termination\n| unbounded (c : fin n) : termination\n| optimal {}            : termination\n\nnamespace termination\n\nlemma injective_unbounded : function.injective (@unbounded n) :=\nλ _ _ h, by injection h\n\n@[simp] lemma unbounded_inj {c c' : fin n} : unbounded c = unbounded c' ↔ c = c' :=\ninjective_unbounded.eq_iff\n\nend termination\n\nopen termination\n\ninstance {n : ℕ} : has_repr $ termination n :=\n⟨λ t, termination.cases_on t\n  \"while\"\n  (λ c, \"unbounded \" ++ repr c)\n  \"optimal\"⟩\n\nopen termination\n\n/-- The simplex algorithm -/\ndef simplex (w : X m n → bool) (obj : fin m) : Π (T : X m n) (hT : feasible T),\n  X m n × termination n\n| T := λ hT, cond (w T)\n  (match pivot_col T obj, @feasible_of_mem_pivot_row_and_col _ _ _ _ _ obj hT,\n      @rel.pivot m n X _ obj _ hT with\n    | none,   hc, hrel := (T, optimal)\n    | some c, hc, hrel :=\n      match pivot_row T obj c, @hc _ rfl, (λ r, @hrel r c rfl) with\n      | none,   hr, hrel := (T, unbounded c)\n      | some r, hr, hrel := have wf : rel obj (pivot T r c) T, from hrel _ rfl,\n        simplex (pivot T r c) (hr rfl)\n      end\n    end)\n  (T, while)\nusing_well_founded {rel_tac := λ _ _, `[exact ⟨_, rel_wf m n obj⟩],\n  dec_tac := tactic.assumption}\n\nlemma simplex_pivot {w : X m n → bool} {T : X m n} (hT : feasible T)\n  (hw : w T = tt) {obj : fin m} {r : fin m} {c : fin n}\n  (hc : c ∈ pivot_col T obj) (hr : r ∈ pivot_row T obj c) :\n  simplex w obj (pivot T r c) (feasible_of_mem_pivot_row_and_col hT hc hr) =\n  simplex w obj T hT  :=\nby conv_rhs { rw simplex };\n  simp [hw, show _ = _, from hr, show _ = _, from hc, simplex._match_1, simplex._match_2]\n\nlemma simplex_spec_aux (w : X m n → bool) (obj : fin m) :\n  Π (T : X m n) (hT : feasible T),\n  ((simplex w obj T hT).2 = while ∧ w (simplex w obj T hT).1 = ff) ∨\n  ((simplex w obj T hT).2 = optimal ∧ w (simplex w obj T hT).1 = tt ∧\n    pivot_col (simplex w obj T hT).1 obj = none) ∨\n  ∃ c, ((simplex w obj T hT).2 = unbounded c ∧ w (simplex w obj T hT).1 = tt ∧\n    c ∈ pivot_col (simplex w obj T hT).1 obj ∧\n    pivot_row (simplex w obj T hT).1 obj c = none)\n| T := λ hT,\n  begin\n    cases hw : w T,\n    { rw simplex, simp [hw] },\n    { cases hc : pivot_col T obj with c,\n      { rw simplex, simp [hc, hw, simplex._match_1] },\n      { cases hr : pivot_row T obj c with r,\n        { rw simplex,\n          simp [hr, hc, hw, simplex._match_1, simplex._match_2] },\n        { rw [← simplex_pivot hT hw hc hr],\n          exact have wf : rel obj (pivot T r c) T, from rel.pivot hT hc hr,\n            simplex_spec_aux _ _ } } }\n  end\nusing_well_founded {rel_tac := λ _ _, `[exact ⟨_, rel_wf m n obj⟩],\n  dec_tac := tactic.assumption}\n\nlemma simplex_while_eq_ff {w : X m n → bool} {T : X m n} {hT : feasible T}\n  {obj : fin m} (hw : w T = ff) : simplex w obj T hT = (T, while) :=\nby rw [simplex, hw]; refl\n\nlemma simplex_pivot_col_eq_none {w : X m n → bool} {T : X m n} {hT : feasible T}\n  (hw : w T = tt) {obj : fin m} (hc : pivot_col T obj = none) :\n  simplex w obj T hT = (T, optimal) :=\nby rw simplex; simp [hc, hw, simplex._match_1]\n\nlemma simplex_pivot_row_eq_none {w : X m n → bool} {T : X m n} {hT : feasible T}\n  {obj : fin m} (hw : w T = tt) {c} (hc : c ∈ pivot_col T obj)\n  (hr : pivot_row T obj c = none) : simplex w obj T hT = (T, unbounded c) :=\nby rw simplex; simp [hw, show _ = _, from hc, hr, simplex._match_1, simplex._match_2]\n\nlemma simplex_induction (P : X m n → Prop) (w : X m n → bool) (obj : fin m):\n  Π {T : X m n} (hT : feasible T)  (h0 : P T)\n  (hpivot : ∀ {T' r c}, w T' = tt → c ∈ pivot_col T' obj → r ∈ pivot_row T' obj c\n    → feasible T' → P T' → P (pivot T' r c)),\n  P (simplex w obj T hT).1\n| T := λ hT h0 hpivot,\n  begin\n    cases hw : w T,\n    { rwa [simplex_while_eq_ff hw] },\n    { cases hc : pivot_col T obj with c,\n      { rwa [simplex_pivot_col_eq_none hw hc] },\n      { cases hr : pivot_row T obj c with r,\n        { rwa simplex_pivot_row_eq_none hw hc hr },\n        { rw [← simplex_pivot _ hw hc hr],\n          exact have wf : rel obj (pivot T r c) T, from rel.pivot hT hc hr,\n            simplex_induction (feasible_of_mem_pivot_row_and_col hT hc hr)\n              (hpivot hw hc hr hT h0) @hpivot } } }\n  end\nusing_well_founded {rel_tac := λ _ _, `[exact ⟨_, rel_wf m n obj⟩],\n  dec_tac := `[tauto]}\n\n@[simp] lemma feasible_simplex {w : X m n → bool} {T : X m n}\n  {hT : feasible T} {obj : fin m} : feasible (simplex w obj T hT).1 :=\nsimplex_induction feasible _ _ hT hT\n  (λ _ _ _ _ hc hr _ hT', feasible_of_mem_pivot_row_and_col hT' hc hr)\n\n@[simp] lemma simplex_simplex {w : X m n → bool} {T : X m n} {hT : feasible T}\n  {obj : fin m} : simplex w obj (simplex w obj T hT).1 feasible_simplex = simplex w obj T hT :=\nsimplex_induction (λ T', ∀ (hT' : feasible T'), simplex w obj T' hT' = simplex w obj T hT) w _ _\n  (λ _, rfl) (λ T' r c hw hc hr hT' ih hpivot, by rw [simplex_pivot hT' hw hc hr, ih]) _\n\n/-- `simplex` does not move the row variable it is trying to maximise. -/\n@[simp] lemma rowg_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : (to_partition (simplex w obj T hT).1).rowg obj = (to_partition T).rowg obj :=\nsimplex_induction (λ T', (to_partition T').rowg obj = (to_partition T).rowg obj) _ _ _ rfl\n  (λ T' r c hw hc hr, by simp [rowg_swap_of_ne _ (pivot_row_spec hr).1])\n\n@[simp] lemma flat_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : flat (simplex w obj T hT).1 = (flat T) :=\nsimplex_induction (λ T', flat T' = (flat T)) w obj _ rfl\n  (λ T' r c hw hc hr hT' ih,\n    have to_matrix T' r c ≠ 0,\n      from λ h, by simpa [h, lt_irrefl] using pivot_row_spec hr,\n    by rw [flat_pivot this, ih])\n\n@[simp] lemma restricted_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : restricted (simplex w obj T hT).1 = (restricted T) :=\nsimplex_induction (λ T', restricted T' = (restricted T)) _ _ _ rfl (by simp { contextual := tt })\n\n@[simp] lemma dead_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : dead (simplex w obj T hT).1 = (dead T) :=\nsimplex_induction (λ T', dead T' = (dead T)) _ _ _ rfl (by simp { contextual := tt })\n\n@[simp] lemma res_set_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : res_set (simplex w obj T hT).1 = res_set T :=\nsimplex_induction (λ T', (res_set T') = (res_set T)) w obj _ rfl\n  (λ T' r c hw hc hr, by simp [res_set_pivot (ne_zero_of_mem_pivot_row hr)] {contextual := tt})\n\n@[simp] lemma dead_set_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : dead_set (simplex w obj T hT).1 = dead_set T :=\nsimplex_induction (λ T', dead_set T' = dead_set T) w obj _ rfl\n  (λ T' r c hw hc hr,\n    by simp [dead_set_pivot (ne_zero_of_mem_pivot_row hr) (pivot_col_spec hc).2] {contextual := tt})\n\n@[simp] lemma sol_set_simplex (T : X m n) (hT : feasible T) (w : X m n → bool)\n  (obj : fin m) : sol_set (simplex w obj T hT).1 = (sol_set T) :=\nby simp [sol_set_eq_res_set_inter_dead_set]\n\n@[simp] lemma of_col_simplex_zero_mem_sol_set {w : X m n → bool} {T : X m n}\n  {hT : feasible T} {obj : fin m} : of_col (simplex w obj T hT).1 0 ∈ sol_set T :=\nby rw [← sol_set_simplex, of_col_zero_mem_sol_set_iff]; exact feasible_simplex\n\n@[simp] lemma of_col_simplex_rowg {w : X m n → bool} {T : X m n}\n  {hT : feasible T} {obj : fin m} (x : cvec n) :\n  of_col (simplex w obj T hT).1 x ((to_partition T).rowg obj) =\n  (to_matrix (simplex w obj T hT).1 ⬝ x + const (simplex w obj T hT).1) obj :=\nby rw [← of_col_rowg (simplex w obj T hT).1 x obj, rowg_simplex]\n\n@[simp] lemma is_unbounded_above_simplex {T : X m n} {hT : feasible T} {w : X m n → bool}\n  {obj : fin m} {v : fin (m + n)} : is_unbounded_above (simplex w obj T hT).1 v ↔\n  is_unbounded_above T v := by simp [is_unbounded_above]\n\n@[simp] lemma is_optimal_simplex {T : X m n} {hT : feasible T} {w : X m n → bool}\n  {obj : fin m} {x : cvec (m + n)} {v : fin (m + n)} : is_optimal (simplex w obj T hT).1 x v ↔\n  is_optimal T x v := by simp [is_optimal]\n\nlemma termination_eq_while_iff {T : X m n} {hT : feasible T} {w : X m n → bool}\n  {obj : fin m} : (simplex w obj T hT).2 = while ↔ w (simplex w obj T hT).1 = ff :=\nby have := simplex_spec_aux w obj T hT; finish\n\nlemma termination_eq_optimal_iff_pivot_col_eq_none {T : X m n}\n  {hT : feasible T} {w : X m n → bool} {obj : fin m} : (simplex w obj T hT).2 = optimal ↔\n  w (simplex w obj T hT).1 = tt ∧ pivot_col (simplex w obj T hT).1 obj = none :=\nby rcases simplex_spec_aux w obj T hT with _ | ⟨_, _, _⟩ | ⟨⟨_, _⟩, _, _, _, _⟩; simp * at *\n\nlemma termination_eq_unbounded_iff_pivot_row_eq_none {T : X m n} {hT : feasible T}\n  {w : X m n → bool} {obj : fin m} {c : fin n} :\n  (simplex w obj T hT).2 = unbounded c ↔\n  w (simplex w obj T hT).1 = tt ∧ c ∈ pivot_col (simplex w obj T hT).1 obj ∧\n    pivot_row (simplex w obj T hT).1 obj c = none :=\nby split; intros; rcases simplex_spec_aux w obj T hT with\n  _ | ⟨_, _, _⟩ | ⟨⟨⟨_, _⟩, _⟩, _, _, _, _⟩; simp * at *\n\nlemma unbounded_of_termination_eq_unbounded {T : X m n} {hT : feasible T}\n  {w : X m n → bool} {obj : fin m} {c : fin n} : (simplex w obj T hT).2 = unbounded c →\n  w (simplex w obj T hT).1 = tt ∧\n  is_unbounded_above T ((to_partition T).rowg obj) :=\nbegin\n  rw termination_eq_unbounded_iff_pivot_row_eq_none,\n  rintros ⟨_, hc⟩,\n  simpa * using pivot_row_eq_none feasible_simplex hc.2 hc.1\nend\n\nlemma termination_eq_optimal_iff {T : X m n} {hT : feasible T}\n  {w : X m n → bool} {obj : fin m} : (simplex w obj T hT).2 = optimal ↔\n  w (simplex w obj T hT).1 = tt ∧\n  is_optimal T (of_col (simplex w obj T hT).1 0) ((to_partition T).rowg obj) :=\nbegin\n  rw [termination_eq_optimal_iff_pivot_col_eq_none],\n  split,\n  { rintros ⟨_, hc⟩,\n    simpa * using pivot_col_eq_none feasible_simplex hc },\n  { cases ht : (simplex w obj T hT).2,\n    { simp [*, termination_eq_while_iff] at * },\n    { cases unbounded_of_termination_eq_unbounded ht,\n      simp [*, not_optimal_of_unbounded_above right] },\n    { simp [*, termination_eq_optimal_iff_pivot_col_eq_none] at * } }\nend\n\nlemma termination_eq_unbounded_iff {T : X m n} {hT : feasible T}\n  {w : X m n → bool} {obj : fin m} {c : fin n}: (simplex w obj T hT).2 = unbounded c ↔\n  w (simplex w obj T hT).1 = tt ∧ is_unbounded_above T ((to_partition T).rowg obj)\n  ∧ c ∈ pivot_col (simplex w obj T hT).1 obj :=\n⟨λ hc, and.assoc.1 $ ⟨unbounded_of_termination_eq_unbounded hc,\n  (termination_eq_unbounded_iff_pivot_row_eq_none.1 hc).2.1⟩,\nbegin\n  have := @not_optimal_of_unbounded_above m n _ _ (simplex w obj T hT).1 ((to_partition T).rowg obj)\n    (of_col (simplex w obj T hT).1 0),\n  cases ht : (simplex w obj T hT).2;\n  simp [termination_eq_optimal_iff, termination_eq_while_iff,\n    termination_eq_unbounded_iff_pivot_row_eq_none, *] at *\nend⟩\n\nend is_tableau\n", "meta": {"author": "ChrisHughes24", "repo": "LP", "sha": "e3ed64c2d1f642696104584e74ae7226d8e916de", "save_path": "github-repos/lean/ChrisHughes24-LP", "path": "github-repos/lean/ChrisHughes24-LP/LP-e3ed64c2d1f642696104584e74ae7226d8e916de/src/refinement/simplex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647395, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3125228834232372}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module order.rel_iso.set\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.RelIso.Basic\nimport Mathbin.Logic.Embedding.Set\n\n/-!\n# Interactions between relation homomorphisms and sets\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIt is likely that there are better homes for many of these statement,\nin files further down the import graph.\n-/\n\n\nopen Function\n\nuniverse u v w\n\nvariable {α β γ δ : Type _} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop}\n  {u : δ → δ → Prop}\n\nnamespace RelHomClass\n\nvariable {F : Type _}\n\n/- warning: rel_hom_class.map_inf -> RelHomClass.map_inf is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {F : Type.{u3}} [_inst_1 : SemilatticeInf.{u1} α] [_inst_2 : LinearOrder.{u2} β] [_inst_3 : RelHomClass.{u3, u2, u1} F β α (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1))))] (a : F) (m : β) (n : β), Eq.{succ u1} α (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a (Inf.inf.{u2} β (SemilatticeInf.toHasInf.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2))) m n)) (Inf.inf.{u1} α (SemilatticeInf.toHasInf.{u1} α _inst_1) (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a m) (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a n))\nbut is expected to have type\n  forall {α : Type.{u3}} {β : Type.{u2}} {F : Type.{u1}} [_inst_1 : SemilatticeInf.{u3} α] [_inst_2 : LinearOrder.{u2} β] [_inst_3 : RelHomClass.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.111 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.113 : β) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.111 x._@.Mathlib.Order.RelIso.Set._hyg.113) (fun (x._@.Mathlib.Order.RelIso.Set._hyg.133 : α) (x._@.Mathlib.Order.RelIso.Set._hyg.135 : α) => LT.lt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeInf.toPartialOrder.{u3} α _inst_1))) x._@.Mathlib.Order.RelIso.Set._hyg.133 x._@.Mathlib.Order.RelIso.Set._hyg.135)] (a : F) (m : β) (n : β), Eq.{succ u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) (Inf.inf.{u2} β (Lattice.toInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2))) m n)) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.111 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.113 : β) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.111 x._@.Mathlib.Order.RelIso.Set._hyg.113) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.135 : α) => LT.lt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeInf.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.135) _inst_3) a (Inf.inf.{u2} β (Lattice.toInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2))) m n)) (Inf.inf.{u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) m) (SemilatticeInf.toInf.{u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) m) _inst_1) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.111 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.113 : β) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.111 x._@.Mathlib.Order.RelIso.Set._hyg.113) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.135 : α) => LT.lt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeInf.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.135) _inst_3) a m) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.111 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.113 : β) => LT.lt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.111 x._@.Mathlib.Order.RelIso.Set._hyg.113) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.135 : α) => LT.lt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeInf.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.135) _inst_3) a n))\nCase conversion may be inaccurate. Consider using '#align rel_hom_class.map_inf RelHomClass.map_infₓ'. -/\ntheorem map_inf [SemilatticeInf α] [LinearOrder β]\n    [RelHomClass F ((· < ·) : β → β → Prop) ((· < ·) : α → α → Prop)] (a : F) (m n : β) :\n    a (m ⊓ n) = a m ⊓ a n :=\n  (StrictMono.monotone fun x y => map_rel a).map_inf m n\n#align rel_hom_class.map_inf RelHomClass.map_inf\n\n/- warning: rel_hom_class.map_sup -> RelHomClass.map_sup is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {F : Type.{u3}} [_inst_1 : SemilatticeSup.{u1} α] [_inst_2 : LinearOrder.{u2} β] [_inst_3 : RelHomClass.{u3, u2, u1} F β α (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1))))] (a : F) (m : β) (n : β), Eq.{succ u1} α (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a (Sup.sup.{u2} β (SemilatticeSup.toHasSup.{u2} β (Lattice.toSemilatticeSup.{u2} β (LinearOrder.toLattice.{u2} β _inst_2))) m n)) (Sup.sup.{u1} α (SemilatticeSup.toHasSup.{u1} α _inst_1) (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a m) (coeFn.{succ u3, max (succ u2) (succ u1)} F (fun (_x : F) => β -> α) (FunLike.hasCoeToFun.{succ u3, succ u2, succ u1} F β (fun (_x : β) => α) (RelHomClass.toFunLike.{u3, u2, u1} F β α (GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (LinearOrder.toLattice.{u2} β _inst_2)))))) (GT.gt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeSup.toPartialOrder.{u1} α _inst_1)))) _inst_3)) a n))\nbut is expected to have type\n  forall {α : Type.{u3}} {β : Type.{u2}} {F : Type.{u1}} [_inst_1 : SemilatticeSup.{u3} α] [_inst_2 : LinearOrder.{u2} β] [_inst_3 : RelHomClass.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.228 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.230 : β) => GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.228 x._@.Mathlib.Order.RelIso.Set._hyg.230) (fun (x._@.Mathlib.Order.RelIso.Set._hyg.250 : α) (x._@.Mathlib.Order.RelIso.Set._hyg.252 : α) => GT.gt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeSup.toPartialOrder.{u3} α _inst_1))) x._@.Mathlib.Order.RelIso.Set._hyg.250 x._@.Mathlib.Order.RelIso.Set._hyg.252)] (a : F) (m : β) (n : β), Eq.{succ u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) (Sup.sup.{u2} β (SemilatticeSup.toSup.{u2} β (Lattice.toSemilatticeSup.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))) m n)) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.228 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.230 : β) => GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.228 x._@.Mathlib.Order.RelIso.Set._hyg.230) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.252 : α) => GT.gt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeSup.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.252) _inst_3) a (Sup.sup.{u2} β (SemilatticeSup.toSup.{u2} β (Lattice.toSemilatticeSup.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))) m n)) (Sup.sup.{u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) m) (SemilatticeSup.toSup.{u3} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) m) _inst_1) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.228 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.230 : β) => GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.228 x._@.Mathlib.Order.RelIso.Set._hyg.230) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.252 : α) => GT.gt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeSup.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.252) _inst_3) a m) (FunLike.coe.{succ u1, succ u2, succ u3} F β (fun (_x : β) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.867 : β) => α) _x) (RelHomClass.toFunLike.{u1, u2, u3} F β α (fun (x._@.Mathlib.Order.RelIso.Set._hyg.228 : β) (x._@.Mathlib.Order.RelIso.Set._hyg.230 : β) => GT.gt.{u2} β (Preorder.toLT.{u2} β (PartialOrder.toPreorder.{u2} β (SemilatticeInf.toPartialOrder.{u2} β (Lattice.toSemilatticeInf.{u2} β (DistribLattice.toLattice.{u2} β (instDistribLattice.{u2} β _inst_2)))))) x._@.Mathlib.Order.RelIso.Set._hyg.228 x._@.Mathlib.Order.RelIso.Set._hyg.230) (fun (_x : α) (x._@.Mathlib.Order.RelIso.Set._hyg.252 : α) => GT.gt.{u3} α (Preorder.toLT.{u3} α (PartialOrder.toPreorder.{u3} α (SemilatticeSup.toPartialOrder.{u3} α _inst_1))) _x x._@.Mathlib.Order.RelIso.Set._hyg.252) _inst_3) a n))\nCase conversion may be inaccurate. Consider using '#align rel_hom_class.map_sup RelHomClass.map_supₓ'. -/\ntheorem map_sup [SemilatticeSup α] [LinearOrder β]\n    [RelHomClass F ((· > ·) : β → β → Prop) ((· > ·) : α → α → Prop)] (a : F) (m n : β) :\n    a (m ⊔ n) = a m ⊔ a n :=\n  @map_inf αᵒᵈ βᵒᵈ _ _ _ _ _ _ _\n#align rel_hom_class.map_sup RelHomClass.map_sup\n\nend RelHomClass\n\nnamespace RelIso\n\n/- warning: rel_iso.range_eq -> RelIso.range_eq is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {r : α -> α -> Prop} {s : β -> β -> Prop} (e : RelIso.{u1, u2} α β r s), Eq.{succ u2} (Set.{u2} β) (Set.range.{u2, succ u1} β α (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RelIso.{u1, u2} α β r s) (fun (_x : RelIso.{u1, u2} α β r s) => α -> β) (RelIso.hasCoeToFun.{u1, u2} α β r s) e)) (Set.univ.{u2} β)\nbut is expected to have type\n  forall {α : Type.{u2}} {β : Type.{u1}} {r : α -> α -> Prop} {s : β -> β -> Prop} (e : RelIso.{u2, u1} α β r s), Eq.{succ u1} (Set.{u1} β) (Set.range.{u1, succ u2} β α (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} α β)) (RelEmbedding.toEmbedding.{u2, u1} α β r s (RelIso.toRelEmbedding.{u2, u1} α β r s e)))) (Set.univ.{u1} β)\nCase conversion may be inaccurate. Consider using '#align rel_iso.range_eq RelIso.range_eqₓ'. -/\n@[simp]\ntheorem range_eq (e : r ≃r s) : Set.range e = Set.univ :=\n  e.Surjective.range_eq\n#align rel_iso.range_eq RelIso.range_eq\n\nend RelIso\n\n#print Subrel /-\n/-- `subrel r p` is the inherited relation on a subset. -/\ndef Subrel (r : α → α → Prop) (p : Set α) : p → p → Prop :=\n  (coe : p → α) ⁻¹'o r\n#align subrel Subrel\n-/\n\n#print subrel_val /-\n@[simp]\ntheorem subrel_val (r : α → α → Prop) (p : Set α) {a b} : Subrel r p a b ↔ r a.1 b.1 :=\n  Iff.rfl\n#align subrel_val subrel_val\n-/\n\nnamespace Subrel\n\n#print Subrel.relEmbedding /-\n/-- The relation embedding from the inherited relation on a subset. -/\nprotected def relEmbedding (r : α → α → Prop) (p : Set α) : Subrel r p ↪r r :=\n  ⟨Embedding.subtype _, fun a b => Iff.rfl⟩\n#align subrel.rel_embedding Subrel.relEmbedding\n-/\n\n/- warning: subrel.rel_embedding_apply -> Subrel.relEmbedding_apply is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} (r : α -> α -> Prop) (p : Set.{u1} α) (a : coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) p), Eq.{succ u1} α (coeFn.{succ u1, succ u1} (RelEmbedding.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) p) α (Subrel.{u1} α r p) r) (fun (_x : RelEmbedding.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) p) α (Subrel.{u1} α r p) r) => (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) p) -> α) (RelEmbedding.hasCoeToFun.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) p) α (Subrel.{u1} α r p) r) (Subrel.relEmbedding.{u1} α r p) a) (Subtype.val.{succ u1} α (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x p) a)\nbut is expected to have type\n  forall {α : Type.{u1}} (r : α -> α -> Prop) (p : Set.{u1} α) (a : Set.Elem.{u1} α p), Eq.{succ u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Set.Elem.{u1} α p) => α) a) (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (Set.Elem.{u1} α p) α) (Set.Elem.{u1} α p) (fun (_x : Set.Elem.{u1} α p) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Set.Elem.{u1} α p) => α) _x) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (Set.Elem.{u1} α p) α) (Set.Elem.{u1} α p) α (Function.instEmbeddingLikeEmbedding.{succ u1, succ u1} (Set.Elem.{u1} α p) α)) (RelEmbedding.toEmbedding.{u1, u1} (Set.Elem.{u1} α p) α (Subrel.{u1} α r p) r (Subrel.relEmbedding.{u1} α r p)) a) (Subtype.val.{succ u1} α (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x p) a)\nCase conversion may be inaccurate. Consider using '#align subrel.rel_embedding_apply Subrel.relEmbedding_applyₓ'. -/\n@[simp]\ntheorem relEmbedding_apply (r : α → α → Prop) (p a) : Subrel.relEmbedding r p a = a.1 :=\n  rfl\n#align subrel.rel_embedding_apply Subrel.relEmbedding_apply\n\ninstance (r : α → α → Prop) [IsWellOrder α r] (p : Set α) : IsWellOrder p (Subrel r p) :=\n  RelEmbedding.isWellOrder (Subrel.relEmbedding r p)\n\ninstance (r : α → α → Prop) [IsRefl α r] (p : Set α) : IsRefl p (Subrel r p) :=\n  ⟨fun x => @IsRefl.refl α r _ x⟩\n\ninstance (r : α → α → Prop) [IsSymm α r] (p : Set α) : IsSymm p (Subrel r p) :=\n  ⟨fun x y => @IsSymm.symm α r _ x y⟩\n\ninstance (r : α → α → Prop) [IsTrans α r] (p : Set α) : IsTrans p (Subrel r p) :=\n  ⟨fun x y z => @IsTrans.trans α r _ x y z⟩\n\ninstance (r : α → α → Prop) [IsIrrefl α r] (p : Set α) : IsIrrefl p (Subrel r p) :=\n  ⟨fun x => @IsIrrefl.irrefl α r _ x⟩\n\nend Subrel\n\n/- warning: rel_embedding.cod_restrict -> RelEmbedding.codRestrict is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {r : α -> α -> Prop} {s : β -> β -> Prop} (p : Set.{u2} β) (f : RelEmbedding.{u1, u2} α β r s), (forall (a : α), Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RelEmbedding.{u1, u2} α β r s) (fun (_x : RelEmbedding.{u1, u2} α β r s) => α -> β) (RelEmbedding.hasCoeToFun.{u1, u2} α β r s) f a) p) -> (RelEmbedding.{u1, u2} α (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p) r (Subrel.{u2} β s p))\nbut is expected to have type\n  forall {α : Type.{u1}} {β : Type.{u2}} {r : α -> α -> Prop} {s : β -> β -> Prop} (p : Set.{u2} β) (f : RelEmbedding.{u1, u2} α β r s), (forall (a : α), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (Set.{u2} β) (Set.instMembershipSet.{u2} β) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} α β)) (RelEmbedding.toEmbedding.{u1, u2} α β r s f) a) p) -> (RelEmbedding.{u1, u2} α (Set.Elem.{u2} β p) r (Subrel.{u2} β s p))\nCase conversion may be inaccurate. Consider using '#align rel_embedding.cod_restrict RelEmbedding.codRestrictₓ'. -/\n/-- Restrict the codomain of a relation embedding. -/\ndef RelEmbedding.codRestrict (p : Set β) (f : r ↪r s) (H : ∀ a, f a ∈ p) : r ↪r Subrel s p :=\n  ⟨f.toEmbedding.codRestrict p H, fun _ _ => f.map_rel_iff'⟩\n#align rel_embedding.cod_restrict RelEmbedding.codRestrict\n\n/- warning: rel_embedding.cod_restrict_apply -> RelEmbedding.codRestrict_apply is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} {β : Type.{u2}} {r : α -> α -> Prop} {s : β -> β -> Prop} (p : Set.{u2} β) (f : RelEmbedding.{u1, u2} α β r s) (H : forall (a : α), Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RelEmbedding.{u1, u2} α β r s) (fun (_x : RelEmbedding.{u1, u2} α β r s) => α -> β) (RelEmbedding.hasCoeToFun.{u1, u2} α β r s) f a) p) (a : α), Eq.{succ u2} (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RelEmbedding.{u1, u2} α (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p) r (Subrel.{u2} β s p)) (fun (_x : RelEmbedding.{u1, u2} α (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p) r (Subrel.{u2} β s p)) => α -> (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p)) (RelEmbedding.hasCoeToFun.{u1, u2} α (coeSort.{succ u2, succ (succ u2)} (Set.{u2} β) Type.{u2} (Set.hasCoeToSort.{u2} β) p) r (Subrel.{u2} β s p)) (RelEmbedding.codRestrict.{u1, u2} α β r s p f H) a) (Subtype.mk.{succ u2} β (fun (x : β) => Membership.Mem.{u2, u2} β (Set.{u2} β) (Set.hasMem.{u2} β) x p) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RelEmbedding.{u1, u2} α β r s) (fun (_x : RelEmbedding.{u1, u2} α β r s) => α -> β) (RelEmbedding.hasCoeToFun.{u1, u2} α β r s) f a) (H a))\nbut is expected to have type\n  forall {α : Type.{u1}} {β : Type.{u2}} {r : α -> α -> Prop} {s : β -> β -> Prop} (p : Set.{u2} β) (f : RelEmbedding.{u1, u2} α β r s) (H : forall (a : α), Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) (Set.{u2} β) (Set.instMembershipSet.{u2} β) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} α β)) (RelEmbedding.toEmbedding.{u1, u2} α β r s f) a) p) (a : α), Eq.{succ u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => Set.Elem.{u2} β p) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α (Set.Elem.{u2} β p)) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => Set.Elem.{u2} β p) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α (Set.Elem.{u2} β p)) α (Set.Elem.{u2} β p) (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} α (Set.Elem.{u2} β p))) (RelEmbedding.toEmbedding.{u1, u2} α (Set.Elem.{u2} β p) r (Subrel.{u2} β s p) (RelEmbedding.codRestrict.{u1, u2} α β r s p f H)) a) (Subtype.mk.{succ u2} β (fun (x : β) => Membership.mem.{u2, u2} β (Set.{u2} β) (Set.instMembershipSet.{u2} β) x p) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} α β)) (RelEmbedding.toEmbedding.{u1, u2} α β r s f) a) (H a))\nCase conversion may be inaccurate. Consider using '#align rel_embedding.cod_restrict_apply RelEmbedding.codRestrict_applyₓ'. -/\n@[simp]\ntheorem RelEmbedding.codRestrict_apply (p) (f : r ↪r s) (H a) :\n    RelEmbedding.codRestrict p f H a = ⟨f a, H a⟩ :=\n  rfl\n#align rel_embedding.cod_restrict_apply RelEmbedding.codRestrict_apply\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Order/RelIso/Set.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.31243593771155687}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h2 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h3 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h4 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h5 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h6 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h7 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h8 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h9 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h10 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h11 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h12 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h13 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h14 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h15 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h16 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h17 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h18 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h19 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h20 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h21 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h22 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h23 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h24 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h25 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h26 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h27 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h28 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h29 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h30 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h31 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h32 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h33 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h34 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h35 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h36 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h37 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h38 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h39 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h40 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.fract (α * ↑j)), from sorry,\n  have h41 : ∀ (i j : ℤ), (i ≠ j) → (int.fract (α * ↑i) ≠ int.\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from sorry,\n\n  have h2 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), \n  from sorry,\n\n  let S : set ℝ := (λ (m : ℤ), int.fract (α * ↑m)) '' (@set.univ ℤ),\n  have h3 : S ≠ ∅, from sorry,\n\n  have h4 : ∀ x : ℝ, x ∈ closure S → x ∈ S, from sorry,\n\n  have h5 : ∀ x : ℝ, x ∈ closure S → x ∈ set.Icc 0 1, from sorry,\n\n  have h6 : ∀ x : ℝ, x ∈ set.Icc 0 1 → x ∈ closure S, from sorry,\n\n  have h7 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h8 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h9 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h10 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h11 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h12 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h13 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h14 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h15 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h16 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h17 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h18 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h19 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h20 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h21 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h22 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h23 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h24 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h25 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h26 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h27 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h28 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h29 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h30 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h31 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h32 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h33 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h34 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h35 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h36 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h37 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h38 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h39 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h40 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n  have h41 : ∀ y : ℝ, y ∈ set.Icc 0 1 → ∃ x : ℝ, x ∈ S ∧ abs (y - x) < 1, from sorry,\n\n \nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), \n  from sorry,\n\n  have h2 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) ≤ int.fract (α * ↑j) ∨ int.fract (α * ↑j) ≤ int.fract (α * ↑i), \n  from sorry,\n\n  have h3 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h4 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h5 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h6 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h7 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h8 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h9 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h10 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h11 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h12 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h13 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h14 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h15 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h16 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h17 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h18 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h19 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h20 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h21 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h22 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h23 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h24 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h25 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h26 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h27 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h28 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) < int.fract (α * ↑j) ∨ int.fract (α * ↑j) < int.fract (α * ↑i), \n  from sorry,\n\n  have h29 : ∀ (i j : ℤ), i ≠ j → int.fract (α * ↑i) <\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h2 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h3 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h4 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h5 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h6 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h7 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h8 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h9 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h10 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h11 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h12 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h13 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h14 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h15 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h16 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h17 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h18 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h19 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h20 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h21 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h22 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h23 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h24 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h25 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h26 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h27 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h28 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h29 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h30 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h31 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h32 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h33 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h34 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h35 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h36 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h37 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h38 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h39 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h40 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h41 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h42 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h43 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h44 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h45 : ∀ i j : ℤ, i ≠\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nbegin\n  have h1 : ∀ i j : ℤ, i ≠ j → int.fract (α * ↑i) ≠ int.fract (α * ↑j), from sorry,\n  have h2 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h3 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h4 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h5 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h6 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h7 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h8 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h9 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h10 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h11 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h12 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h13 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h14 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h15 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h16 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h17 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h18 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h19 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h20 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h21 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h22 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h23 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h24 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h25 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h26 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h27 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h28 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h29 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h30 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h31 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h32 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h33 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h34 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h35 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h36 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h37 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h38 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h39 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h40 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h41 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n\n  have h42 : ∀ i j : ℤ, i ≠ j → (int.fract (α * ↑i)) ≠ (int.fract (α * ↑j)), from sorry,\n  have h43 : ∀ i j : ℤ, i ≠ j → (int.fract (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 := sorry\n\n/--`theorem`\nDensity of rational orbit\nThe fractional parts of the integer multiples of a rational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be a rational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  rational_orbit_dense {α : ℝ} (hα_rat : rational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 := sorry\n\n/--`theorem`\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 := sorry\n\n/--`theorem`\nThe fractional parts of the integer multiples of a rational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be a rational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  rational_orbit_dense {α : ℝ} (hα_rat : rational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 := sorry\n\n/--`theorem`\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from sorry,\n  \n  assume (h7 : ε > 0),\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := sorry,\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from sorry,\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from sorry,\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/Correct_statement-lean_proof_outline-4_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.835483553488848, "lm_q2_score": 0.3738758227716966, "lm_q1q2_score": 0.31236710097286385}}
{"text": "import category_theory.preadditive\nimport algebraic_topology.cech_nerve\n\nimport for_mathlib.simplicial.complex\nimport for_mathlib.arrow.split\n\nnamespace category_theory\n\nuniverses v u v' u'\n\nnamespace arrow\n\nnoncomputable theory\nopen_locale simplicial\nopen category_theory.limits\n\nvariables {C : Type u} [category.{v} C] (f : arrow C)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]\n\n/-- The splittings of the Cech nerve associated to a split arrow. -/\ndef cech_splitting [split f] (n : ℕ) : f.cech_nerve _[n] ⟶ f.cech_nerve _[n+1] :=\nwide_pullback.lift (wide_pullback.base _)\n(λ i, if h : i = 0 then wide_pullback.base _ ≫ split.σ else wide_pullback.π _ (i.pred h))\nbegin\n  rintro ⟨j⟩,\n  split_ifs,\n  tidy,\nend\n\nlemma cech_splitting_face_zero [split f] (n : ℕ) :\n  f.cech_splitting n ≫ f.cech_nerve.δ 0 = 𝟙 _ :=\nbegin\n  ext j,\n  dsimp [cech_splitting, simplicial_object.δ],\n  simp only [category.id_comp, category.assoc, wide_pullback.lift_π],\n  split_ifs,\n  { exfalso,\n    exact fin.succ_ne_zero _ h },\n  { congr,\n    dsimp [simplicial_object.δ, simplex_category.δ],\n    simp },\n  { dsimp [simplicial_object.δ, cech_splitting],\n    simp },\nend\n\nlemma face_π (n : ℕ) (i : fin (n+1)) (j : fin (n+2)) :\n  (f.cech_nerve.δ j : f.cech_nerve _[n+1] ⟶ _) ≫ wide_pullback.π _ i =\n  wide_pullback.π _ (j.succ_above i) :=\nbegin\n  change wide_pullback.lift _ _ _ ≫ _ = _,\n  simpa,\nend\n\nlemma cech_splitting_face [split f] (n : ℕ) (j : fin (n+3)) (hj : j ≠ 0) :\n  f.cech_splitting (n+1) ≫ f.cech_nerve.δ j =\n  f.cech_nerve.δ (j.pred hj) ≫ f.cech_splitting n :=\nbegin\n  ext k,\n  swap,\n  { dsimp [cech_splitting, simplicial_object.δ],\n    simp },\n  { dsimp [cech_splitting, simplicial_object.δ],\n    simp only [category.assoc, limits.wide_pullback.lift_π],\n    split_ifs with h1 h2,\n    { simp },\n    { refine false.elim (h2 _),\n      change j.succ_above k = 0 at h1,\n      change k = 0,\n      rwa ← fin.succ_above_eq_zero_iff hj },\n    { refine false.elim (h1 _),\n      erw h,\n      change j.succ_above 0 = 0,\n      rw fin.succ_above_eq_zero_iff hj },\n    { simp only [category_theory.limits.wide_pullback.lift_π],\n      congr,\n      change (j.succ_above k).pred h1 = (j.pred hj).succ_above (k.pred h),\n      rw fin.pred_succ_above_pred,\n      refl } }\nend\n\nend arrow\n\nnamespace arrow\n\nsection contracting_homotopy\n\nopen category_theory.limits opposite\n\n-- Note: Universe restrictions! I hope this doesn't pose any issues later...\n-- jmc: seems like it might! I removed them.\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : Pᵒᵖ ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]\n\n/-- The augmented Cech conerve induced by applying M to `f.augmented_cech_nerve`. -/\nabbreviation conerve : cosimplicial_object.augmented N :=\n((cosimplicial_object.augmented.whiskering _ _).obj M).obj f.augmented_cech_nerve.right_op\n\nvariables [arrow.split f] [preadditive N]\n\n/-- The morphisms yielding the contracting homotopy. -/\ndef contracting_homotopy : Π (n : ℕ),\n  (f.conerve M).to_cocomplex.X (n+1) ⟶ (f.conerve M).to_cocomplex.X n\n| 0 := M.map (wide_pullback.lift (𝟙 _)\n         (λ i, (split.σ : f.right ⟶ _))\n         (by simp)).op\n| (n+1) := M.map (f.cech_splitting n).op\n\nopen cosimplicial_object.augmented\nopen_locale big_operators\n\nlemma is_contracting_homotopy_zero :\n  (f.conerve M).to_cocomplex.d 0 1 ≫ f.contracting_homotopy M 0 = 𝟙 _ :=\nbegin\n  dsimp,\n  rw if_pos,\n  swap, { simp },\n  delta conerve,\n  dsimp [to_cocomplex_d, to_cocomplex_obj, contracting_homotopy ],\n  simp only [category.id_comp, category.comp_id],\n  simp_rw [← M.map_comp, ← op_comp, ← M.map_id, ← op_id],\n  congr' 2,\n  simp,\nend\n\nlemma is_contracting_homotopy_one :\n  (f.conerve M).to_cocomplex.d 1 2 ≫ f.contracting_homotopy M 1 +\n  f.contracting_homotopy M 0 ≫ (f.conerve M).to_cocomplex.d 0 1 = 𝟙 _ :=\nbegin\n  rw ← add_zero (𝟙 ((conerve M f).to_cocomplex.X 1)),\n  dsimp only [to_cocomplex, cochain_complex.of],\n  rw dif_pos,\n  swap, { dec_trivial },\n  rw dif_pos,\n  swap, { dec_trivial },\n  dsimp,\n  delta conerve,\n  dsimp only [to_cocomplex_d, cosimplicial_object.coboundary, whiskering, whiskering_obj,\n    drop, to_cocomplex_obj, comma.snd, cosimplicial_object.whiskering, whiskering_right,\n    contracting_homotopy],\n  simp_rw fin.sum_univ_succ,\n  simp only [fin.coe_zero, fin.sum_univ_zero, fin.coe_one,\n    one_zsmul, preadditive.add_comp, pow_one, fin.succ_zero_eq_one,\n    category.id_comp, neg_smul, category.comp_id, preadditive.neg_comp, pow_zero ],\n  rw [add_assoc],\n  congr' 1,\n  { dsimp [cosimplicial_object.δ],\n    simp_rw [← M.map_comp, ← M.map_id, ← op_id, ← op_comp],\n    congr' 2,\n    dsimp [cech_splitting],\n    ext j,\n    { simp only [wide_pullback.lift_π, category.id_comp, category.assoc],\n      split_ifs,\n      { cases j,\n        injection h with hh,\n        simp only [nat.succ_ne_zero] at hh,\n        cases hh },\n      { congr, have hj : j = 0 := subsingleton.elim j 0, subst j,\n        dsimp only [simplex_category.δ],\n        simp only [simplex_category.mk_hom, simplex_category.hom.to_order_hom_mk,\n          order_embedding.to_order_hom_coe, fin.zero_succ_above, fin.succ_zero_eq_one,\n          fin.one_eq_zero_iff, nat.one_ne_zero],\n        refl, } },\n    { simp only [wide_pullback.lift_base, category.assoc, category.id_comp] } },\n  { dsimp [cosimplicial_object.δ],\n    rw [add_assoc, neg_add_eq_zero, ← M.map_comp],\n    simp only [zero_comp, category.id_comp, zero_add, functor.map_comp, ← M.map_comp, ← op_comp],\n    congr' 2,\n    dsimp [cech_splitting],\n    ext j,\n    { simp only [wide_pullback.lift_π, category.assoc],\n      split_ifs,\n      { refl },\n      { refine false.elim (h _),\n        change (1 : fin 2).succ_above j = 0,\n        rw fin.succ_above_eq_zero_iff,\n        { simp },\n        { exact top_ne_bot } } },\n    { simp only [wide_pullback.lift_base, category.assoc, category.comp_id] } }\nend\n\nlemma is_contracting_homotopy (n : ℕ) :\n  (f.conerve M).to_cocomplex.d (n+2) (n+3) ≫ f.contracting_homotopy M (n+2) +\n  f.contracting_homotopy M (n+1) ≫ (f.conerve M).to_cocomplex.d (n+1) (n+2) = 𝟙 _ :=\nbegin\n  dsimp,\n  erw if_pos,\n  swap, refl,\n  dsimp only [to_cocomplex_d],\n  rw if_pos,\n  swap, refl,\n  dsimp only [cosimplicial_object.coboundary],\n  simp only [preadditive.sum_comp, preadditive.comp_sum],\n  erw [fin.sum_univ_succ, add_assoc, ← finset.sum_add_distrib],\n  rw ← add_zero (𝟙 ((conerve M f).to_cocomplex_obj (n + 2))),\n  dsimp only [cosimplicial_object.δ],\n  congr' 1,\n  { delta conerve,\n    dsimp [to_cocomplex_obj, contracting_homotopy],\n    simp only [category_theory.category.comp_id, one_zsmul, pow_zero],\n    simp_rw [← M.map_id, ← M.map_comp, ← op_comp, ← op_id],\n    congr' 2,\n    apply cech_splitting_face_zero },\n  { apply fintype.sum_eq_zero,\n    intros i,\n    simp only [\n      category.comp_id,\n      add_zero,\n      functor.comp_map,\n      fin.coe_succ,\n      preadditive.comp_zsmul,\n      preadditive.zsmul_comp],\n    suffices :\n      (drop.obj (conerve M f)).map (simplex_category.δ i.succ) ≫ contracting_homotopy M f (n+2) =\n          contracting_homotopy M f (n+1) ≫ (drop.obj (conerve M f)).map (simplex_category.δ i),\n    { rw [this, pow_succ],\n      simp },\n    delta conerve,\n    dsimp [contracting_homotopy],\n    simp_rw [← M.map_comp, ← op_comp],\n    congr' 2,\n    convert cech_splitting_face _ _ _ (fin.succ_ne_zero _),\n    funext,\n    congr,\n    simp }\nend\n\nend contracting_homotopy\n\nsection covariant_contracting_homotopy\n\nopen category_theory.limits opposite\n\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : P ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : fin (n+1), f.left) (λ i, f.hom)]\n\n/-- The augmented Cech nerve induced by applying M to `f.augmented_cech_nerve`. -/\nabbreviation nerve : simplicial_object.augmented N :=\n((simplicial_object.augmented.whiskering _ _).obj M).obj f.augmented_cech_nerve\n\nvariables [arrow.split f] [preadditive N]\n\nopen simplicial_object.augmented\nopen_locale big_operators\n\ndef covariant_contracting_homotopy : Π (n : ℕ),\n  (f.nerve M).to_complex.X n ⟶ (f.nerve M).to_complex.X (n+1)\n| 0 := M.map $ wide_pullback.lift (𝟙 _) (λ i, (split.σ : f.right ⟶ _)) (by simpa)\n| (n+1) := M.map $ f.cech_splitting n\n\nlemma covariant_is_contracting_homotopy_zero :\n  f.covariant_contracting_homotopy M 0 ≫ (f.nerve M).to_complex.d 1 0 = 𝟙 _ :=\nbegin\n  dsimp,\n  rw if_pos,\n  swap, { simp },\n  delta nerve,\n  dsimp [to_complex_obj, to_complex_d, covariant_contracting_homotopy],\n  simp only [category.id_comp, category.comp_id],\n  simp_rw [← M.map_comp, ← M.map_id],\n  congr' 2,\n  simp,\nend\n\nlemma covariant_is_contracting_homotopy_one :\n  f.covariant_contracting_homotopy M 1 ≫ (f.nerve M).to_complex.d 2 1 +\n  (f.nerve M).to_complex.d 1 0 ≫ f.covariant_contracting_homotopy M 0 = 𝟙 _ :=\nbegin\n  dsimp [covariant_contracting_homotopy],\n  rw if_pos, rw if_pos, any_goals { dec_trivial },\n  dsimp [to_complex_d, simplicial_object.boundary, simplicial_object.δ],\n  delta nerve,\n  dsimp [to_complex_obj],\n  simp only [fin.sum_univ_succ, fin.coe_zero, pow_zero, one_zsmul, fintype.univ_of_subsingleton,\n    nat.add_def, fin.mk_zero, fin.coe_succ, pow_one, neg_smul, finset.sum_singleton,\n    preadditive.comp_add, category.id_comp, preadditive.comp_neg, category.comp_id],\n  simp only [← M.map_comp],\n  rw add_assoc,\n  convert add_zero _,\n  swap,\n  { symmetry,\n    convert M.map_id _,\n    dsimp [arrow.cech_splitting],\n    ext ⟨j,hj⟩, simp,\n    rw dif_neg, refl,\n    dsimp [simplex_category.δ],\n    have : j = 0, by simpa using hj, subst this, dec_trivial,\n    simp },\n  { rw neg_add_eq_zero,\n    congr' 1,\n    dsimp [cech_splitting],\n    ext ⟨j,hj⟩,\n    { simp only [category.assoc, wide_pullback.lift_π], dsimp [simplex_category.δ],\n      rw dif_pos, have : j = 0, by simpa using hj, subst this,\n      dsimp [simplex_category.δ], dec_trivial },\n    { simp } }\nend\n\nlemma covariant_is_contracting_homotopy (n : ℕ) :\n  f.covariant_contracting_homotopy M (n+2) ≫ (f.nerve M).to_complex.d (n+3) (n+2) +\n  (f.nerve M).to_complex.d (n+2) (n+1) ≫ f.covariant_contracting_homotopy M (n+1) = 𝟙 _ :=\nbegin\n  dsimp, rw if_pos, rw if_pos, swap, { refl }, swap, { refl },\n  simp only [category.comp_id, category.id_comp],\n  dsimp only [to_complex_d, simplicial_object.boundary],\n  simp only [preadditive.sum_comp, preadditive.comp_sum],\n  rw [fin.sum_univ_succ, add_assoc, ← finset.sum_add_distrib],\n  convert add_zero _,\n  { apply fintype.sum_eq_zero, intros j,\n    have : ((j.succ : fin _) : ℕ) = (j : ℕ) + 1 := by simp, rw this, clear this,\n    rw [pow_succ],\n    simp only [neg_mul, one_mul, neg_smul, preadditive.comp_neg],\n    rw neg_add_eq_zero,\n    simp only [preadditive.comp_zsmul, preadditive.zsmul_comp],\n    congr' 1,\n    dsimp [covariant_contracting_homotopy, simplicial_object.δ],\n    delta nerve,\n    dsimp [whiskering],\n    simp only [← M.map_comp],\n    congr' 1,\n    convert cech_splitting_face _ _ _ (fin.succ_ne_zero _), funext i,\n    congr, simp },\n  { dsimp [covariant_contracting_homotopy],\n    simp only [pow_zero, one_zsmul],\n    delta nerve,\n    dsimp [arrow.cech_splitting, simplicial_object.whiskering, simplicial_object.δ],\n    rw ← M.map_comp, symmetry,\n    convert M.map_id _,\n    ext ⟨j,hj⟩,\n    simp only [category.assoc, wide_pullback.lift_π],\n    rw dif_neg, dsimp, simpa, dsimp [simplex_category.δ],\n    intro c, apply_fun (λ e, e.1) at c, simpa using c,\n    simp, dsimp, simp }\nend\n\nend covariant_contracting_homotopy\n\nend arrow\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/Cech/split.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3123489106051117}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.functor.const\nimport category_theory.discrete_category\nimport category_theory.yoneda\nimport category_theory.functor.reflects_isomorphisms\n\n-- morphism levels before object levels. See note [category_theory universes].\nuniverses v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\nopen category_theory\n\nvariables {J : Type u₁} [category.{v₁} J]\nvariables {K : Type u₂} [category.{v₂} K]\nvariables {C : Type u₃} [category.{v₃} C]\nvariables {D : Type u₄} [category.{v₄} D]\n\nopen category_theory\nopen category_theory.category\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory\n\nnamespace functor\nvariables {J C} (F : J ⥤ C)\n\n/--\n`F.cones` is the functor assigning to an object `X` the type of\nnatural transformations from the constant functor with value `X` to `F`.\nAn object representing this functor is a limit of `F`.\n-/\n@[simps]\ndef cones : Cᵒᵖ ⥤ Type (max u₁ v₃) := (const J).op ⋙ yoneda.obj F\n\n/--\n`F.cocones` is the functor assigning to an object `X` the type of\nnatural transformations from `F` to the constant functor with value `X`.\nAn object corepresenting this functor is a colimit of `F`.\n-/\n@[simps]\ndef cocones : C ⥤ Type (max u₁ v₃) := const J ⋙ coyoneda.obj (op F)\n\nend functor\n\nsection\nvariables (J C)\n\n/--\nFunctorially associated to each functor `J ⥤ C`, we have the `C`-presheaf consisting of\ncones with a given cone point.\n-/\n@[simps] def cones : (J ⥤ C) ⥤ (Cᵒᵖ ⥤ Type (max u₁ v₃)) :=\n{ obj := functor.cones,\n  map := λ F G f, whisker_left (const J).op (yoneda.map f) }\n\n/--\nContravariantly associated to each functor `J ⥤ C`, we have the `C`-copresheaf consisting of\ncocones with a given cocone point.\n-/\n@[simps] def cocones : (J ⥤ C)ᵒᵖ ⥤ (C ⥤ Type (max u₁ v₃)) :=\n{ obj := λ F, functor.cocones (unop F),\n  map := λ F G f, whisker_left (const J) (coyoneda.map f) }\n\nend\n\nnamespace limits\n\n/--\nA `c : cone F` is:\n* an object `c.X` and\n* a natural transformation `c.π : c.X ⟶ F` from the constant `c.X` functor to `F`.\n\n`cone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cones.obj X`.\n-/\nstructure cone (F : J ⥤ C) :=\n(X : C)\n(π : (const J).obj X ⟶ F)\n\ninstance inhabited_cone (F : discrete punit ⥤ C) : inhabited (cone F) :=\n⟨{ X := F.obj punit.star,\n   π := { app := λ ⟨⟩, 𝟙 _ } }⟩\n\n@[simp, reassoc] lemma cone.w {F : J ⥤ C} (c : cone F) {j j' : J} (f : j ⟶ j') :\n  c.π.app j ≫ F.map f = c.π.app j' :=\nby { rw ← c.π.naturality f, apply id_comp }\n\n/--\nA `c : cocone F` is\n* an object `c.X` and\n* a natural transformation `c.ι : F ⟶ c.X` from `F` to the constant `c.X` functor.\n\n`cocone F` is equivalent, via `cone.equiv` below, to `Σ X, F.cocones.obj X`.\n-/\nstructure cocone (F : J ⥤ C) :=\n(X : C)\n(ι : F ⟶ (const J).obj X)\n\ninstance inhabited_cocone (F : discrete punit ⥤ C) : inhabited (cocone F) :=\n⟨{ X := F.obj punit.star,\n   ι := { app := λ ⟨⟩, 𝟙 _ } }⟩\n\n@[simp, reassoc] lemma cocone.w {F : J ⥤ C} (c : cocone F) {j j' : J} (f : j ⟶ j') :\n  F.map f ≫ c.ι.app j' = c.ι.app j :=\nby { rw c.ι.naturality f, apply comp_id }\n\nvariables {F : J ⥤ C}\n\nnamespace cone\n\n/-- The isomorphism between a cone on `F` and an element of the functor `F.cones`. -/\n@[simps]\ndef equiv (F : J ⥤ C) : cone F ≅ Σ X, F.cones.obj X :=\n{ hom := λ c, ⟨op c.X, c.π⟩,\n  inv := λ c, { X := c.1.unop, π := c.2 },\n  hom_inv_id' := by { ext1, cases x, refl },\n  inv_hom_id' := by { ext1, cases x, refl } }\n\n/-- A map to the vertex of a cone naturally induces a cone by composition. -/\n@[simps] def extensions (c : cone F) :\n  yoneda.obj c.X ⋙ ulift_functor.{u₁} ⟶ F.cones :=\n{ app := λ X f, (const J).map f.down ≫ c.π }\n\n/-- A map to the vertex of a cone induces a cone by composition. -/\n@[simps] def extend (c : cone F) {X : C} (f : X ⟶ c.X) : cone F :=\n{ X := X,\n  π := c.extensions.app (op X) ⟨f⟩ }\n\n/-- Whisker a cone by precomposition of a functor. -/\n@[simps] def whisker (E : K ⥤ J) (c : cone F) : cone (E ⋙ F) :=\n{ X := c.X,\n  π := whisker_left E c.π }\n\nend cone\n\nnamespace cocone\n\n/-- The isomorphism between a cocone on `F` and an element of the functor `F.cocones`. -/\ndef equiv (F : J ⥤ C) : cocone F ≅ Σ X, F.cocones.obj X :=\n{ hom := λ c, ⟨c.X, c.ι⟩,\n  inv := λ c, { X := c.1, ι := c.2 },\n  hom_inv_id' := by { ext1, cases x, refl },\n  inv_hom_id' := by { ext1, cases x, refl } }\n\n/-- A map from the vertex of a cocone naturally induces a cocone by composition. -/\n@[simps] def extensions (c : cocone F) : coyoneda.obj (op c.X) ⋙ ulift_functor.{u₁} ⟶ F.cocones :=\n{ app := λ X f, c.ι ≫ (const J).map f.down }\n\n/-- A map from the vertex of a cocone induces a cocone by composition. -/\n@[simps] def extend (c : cocone F) {X : C} (f : c.X ⟶ X) : cocone F :=\n{ X := X,\n  ι := c.extensions.app X ⟨f⟩ }\n\n/--\nWhisker a cocone by precomposition of a functor. See `whiskering` for a functorial\nversion.\n-/\n@[simps] def whisker (E : K ⥤ J) (c : cocone F) : cocone (E ⋙ F) :=\n{ X := c.X,\n  ι := whisker_left E c.ι }\n\nend cocone\n\n/-- A cone morphism between two cones for the same diagram is a morphism of the cone points which\ncommutes with the cone legs. -/\n@[ext] structure cone_morphism (A B : cone F) :=\n(hom : A.X ⟶ B.X)\n(w'  : ∀ j : J, hom ≫ B.π.app j = A.π.app j . obviously)\n\nrestate_axiom cone_morphism.w'\nattribute [simp, reassoc] cone_morphism.w\n\ninstance inhabited_cone_morphism (A : cone F) : inhabited (cone_morphism A A) :=\n⟨{ hom := 𝟙 _ }⟩\n\n/-- The category of cones on a given diagram. -/\n@[simps] instance cone.category : category (cone F) :=\n{ hom  := λ A B, cone_morphism A B,\n  comp := λ X Y Z f g, { hom := f.hom ≫ g.hom },\n  id   := λ B, { hom := 𝟙 B.X } }\n\nnamespace cones\n/-- To give an isomorphism between cones, it suffices to give an\n  isomorphism between their vertices which commutes with the cone\n  maps. -/\n@[ext, simps] def ext {c c' : cone F}\n  (φ : c.X ≅ c'.X) (w : ∀ j, c.π.app j = φ.hom ≫ c'.π.app j) : c ≅ c' :=\n{ hom := { hom := φ.hom },\n  inv := { hom := φ.inv, w' := λ j, φ.inv_comp_eq.mpr (w j) } }\n\n/--\nGiven a cone morphism whose object part is an isomorphism, produce an\nisomorphism of cones.\n-/\nlemma cone_iso_of_hom_iso {K : J ⥤ C} {c d : cone K} (f : c ⟶ d) [i : is_iso f.hom] :\n  is_iso f :=\n⟨⟨{ hom := inv f.hom,\n    w' := λ j, (as_iso f.hom).inv_comp_eq.2 (f.w j).symm }, by tidy⟩⟩\n\n/--\nFunctorially postcompose a cone for `F` by a natural transformation `F ⟶ G` to give a cone for `G`.\n-/\n@[simps] def postcompose {G : J ⥤ C} (α : F ⟶ G) : cone F ⥤ cone G :=\n{ obj := λ c, { X := c.X, π := c.π ≫ α },\n  map := λ c₁ c₂ f, { hom := f.hom } }\n\n/-- Postcomposing a cone by the composite natural transformation `α ≫ β` is the same as\npostcomposing by `α` and then by `β`. -/\n@[simps]\ndef postcompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :\n  postcompose (α ≫ β) ≅ postcompose α ⋙ postcompose β :=\nnat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/-- Postcomposing by the identity does not change the cone up to isomorphism. -/\n@[simps]\ndef postcompose_id : postcompose (𝟙 F) ≅ 𝟭 (cone F) :=\nnat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncones.\n-/\n@[simps]\ndef postcompose_equivalence {G : J ⥤ C} (α : F ≅ G) : cone F ≌ cone G :=\n{ functor := postcompose α.hom,\n  inverse := postcompose α.inv,\n  unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy) }\n\n/--\nWhiskering on the left by `E : K ⥤ J` gives a functor from `cone F` to `cone (E ⋙ F)`.\n-/\n@[simps]\ndef whiskering (E : K ⥤ J) : cone F ⥤ cone (E ⋙ F) :=\n{ obj := λ c, c.whisker E,\n  map := λ c c' f, { hom := f.hom } }\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\n@[simps]\ndef whiskering_equivalence (e : K ≌ J) :\n  cone F ≌ cone (e.functor ⋙ F) :=\n{ functor := whiskering e.functor,\n  inverse := whiskering e.inverse ⋙ postcompose (e.inv_fun_id_assoc F).hom,\n  unit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cones.ext (iso.refl _)\n  (begin\n    intro k,\n    dsimp, -- See library note [dsimp, simp]\n    simpa [e.counit_app_functor] using s.w (e.unit_inv.app k),\n  end)) (by tidy), }\n\n/--\nThe categories of cones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\n@[simps functor inverse unit_iso counit_iso]\ndef equivalence_of_reindexing {G : K ⥤ C}\n  (e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cone F ≌ cone G :=\n(whiskering_equivalence e).trans (postcompose_equivalence α)\n\nsection\nvariable (F)\n\n/-- Forget the cone structure and obtain just the cone point. -/\n@[simps]\ndef forget : cone F ⥤ C :=\n{ obj := λ t, t.X, map := λ s t f, f.hom }\n\nvariables (G : C ⥤ D)\n\n/-- A functor `G : C ⥤ D` sends cones over `F` to cones over `F ⋙ G` functorially. -/\n@[simps] def functoriality : cone F ⥤ cone (F ⋙ G) :=\n{ obj := λ A,\n  { X := G.obj A.X,\n    π := { app := λ j, G.map (A.π.app j), naturality' := by intros; erw ←G.map_comp; tidy } },\n  map := λ X Y f,\n  { hom := G.map f.hom,\n    w' := λ j, by simp [-cone_morphism.w, ←f.w j] } }\n\ninstance functoriality_full [full G] [faithful G] : full (functoriality F G) :=\n{ preimage := λ X Y t,\n  { hom := G.preimage t.hom,\n    w' := λ j, G.map_injective (by simpa using t.w j) } }\n\ninstance functoriality_faithful [faithful G] : faithful (cones.functoriality F G) :=\n{ map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } }\n\n/--\nIf `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cones over `F` and cones over `F ⋙ e.functor`.\n-/\n@[simps]\ndef functoriality_equivalence (e : C ≌ D) : cone F ≌ cone (F ⋙ e.functor) :=\nlet f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in\n{ functor := functoriality F e.functor,\n  inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙\n    (postcompose_equivalence f).functor,\n  unit_iso := nat_iso.of_components (λ c, cones.ext (e.unit_iso.app _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cones.ext (e.counit_iso.app _) (by tidy)) (by tidy), }\n\n/--\nIf `F` reflects isomorphisms, then `cones.functoriality F` reflects isomorphisms\nas well.\n-/\ninstance reflects_cone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :\n  reflects_isomorphisms (cones.functoriality K F) :=\nbegin\n  constructor,\n  introsI,\n  haveI : is_iso (F.map f.hom) :=\n    (cones.forget (K ⋙ F)).map_is_iso ((cones.functoriality K F).map f),\n  haveI := reflects_isomorphisms.reflects F f.hom,\n  apply cone_iso_of_hom_iso\nend\n\nend\n\nend cones\n\n/-- A cocone morphism between two cocones for the same diagram is a morphism of the cocone points\nwhich commutes with the cocone legs. -/\n@[ext] structure cocone_morphism (A B : cocone F) :=\n(hom : A.X ⟶ B.X)\n(w'  : ∀ j : J, A.ι.app j ≫ hom = B.ι.app j . obviously)\n\ninstance inhabited_cocone_morphism (A : cocone F) : inhabited (cocone_morphism A A) :=\n⟨{ hom := 𝟙 _ }⟩\n\nrestate_axiom cocone_morphism.w'\nattribute [simp, reassoc] cocone_morphism.w\n\n@[simps] instance cocone.category : category (cocone F) :=\n{ hom  := λ A B, cocone_morphism A B,\n  comp := λ _ _ _ f g,\n  { hom := f.hom ≫ g.hom },\n  id   := λ B, { hom := 𝟙 B.X } }\n\nnamespace cocones\n/-- To give an isomorphism between cocones, it suffices to give an\n  isomorphism between their vertices which commutes with the cocone\n  maps. -/\n@[ext, simps] def ext {c c' : cocone F}\n  (φ : c.X ≅ c'.X) (w : ∀ j, c.ι.app j ≫ φ.hom = c'.ι.app j) : c ≅ c' :=\n{ hom := { hom := φ.hom },\n  inv := { hom := φ.inv, w' := λ j, φ.comp_inv_eq.mpr (w j).symm } }\n\n/--\nGiven a cocone morphism whose object part is an isomorphism, produce an\nisomorphism of cocones.\n-/\nlemma cocone_iso_of_hom_iso {K : J ⥤ C} {c d : cocone K} (f : c ⟶ d) [i : is_iso f.hom] :\n  is_iso f :=\n⟨⟨{ hom := inv f.hom,\n    w' := λ j, (as_iso f.hom).comp_inv_eq.2 (f.w j).symm }, by tidy⟩⟩\n\n/-- Functorially precompose a cocone for `F` by a natural transformation `G ⟶ F` to give a cocone\nfor `G`. -/\n@[simps] def precompose {G : J ⥤ C} (α : G ⟶ F) : cocone F ⥤ cocone G :=\n{ obj := λ c, { X := c.X, ι := α ≫ c.ι },\n  map := λ c₁ c₂ f, { hom := f.hom } }\n\n/-- Precomposing a cocone by the composite natural transformation `α ≫ β` is the same as\nprecomposing by `β` and then by `α`. -/\ndef precompose_comp {G H : J ⥤ C} (α : F ⟶ G) (β : G ⟶ H) :\n  precompose (α ≫ β) ≅ precompose β ⋙ precompose α :=\nnat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/-- Precomposing by the identity does not change the cocone up to isomorphism. -/\ndef precompose_id : precompose (𝟙 F) ≅ 𝟭 (cocone F) :=\nnat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy)\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncocones.\n-/\n@[simps]\ndef precompose_equivalence {G : J ⥤ C} (α : G ≅ F) : cocone F ≌ cocone G :=\n{ functor := precompose α.hom,\n  inverse := precompose α.inv,\n  unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy) }\n\n/--\nWhiskering on the left by `E : K ⥤ J` gives a functor from `cocone F` to `cocone (E ⋙ F)`.\n-/\n@[simps]\ndef whiskering (E : K ⥤ J) : cocone F ⥤ cocone (E ⋙ F) :=\n{ obj := λ c, c.whisker E,\n  map := λ c c' f, { hom := f.hom, } }\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\n@[simps]\ndef whiskering_equivalence (e : K ≌ J) :\n  cocone F ≌ cocone (e.functor ⋙ F) :=\n{ functor := whiskering e.functor,\n  inverse := whiskering e.inverse ⋙\n    precompose ((functor.left_unitor F).inv ≫ (whisker_right (e.counit_iso).inv F) ≫\n      (functor.associator _ _ _).inv),\n  unit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ s, cocones.ext (iso.refl _)\n  (begin\n    intro k,\n    dsimp,\n    simpa [e.counit_inv_app_functor k] using s.w (e.unit.app k),\n  end)) (by tidy), }\n\n/--\nThe categories of cocones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\n@[simps functor_obj]\ndef equivalence_of_reindexing {G : K ⥤ C}\n  (e : K ≌ J) (α : e.functor ⋙ F ≅ G) : cocone F ≌ cocone G :=\n(whiskering_equivalence e).trans (precompose_equivalence α.symm)\n\nsection\nvariable (F)\n\n/-- Forget the cocone structure and obtain just the cocone point. -/\n@[simps]\ndef forget : cocone F ⥤ C :=\n{ obj := λ t, t.X, map := λ s t f, f.hom }\n\nvariables (G : C ⥤ D)\n\n/-- A functor `G : C ⥤ D` sends cocones over `F` to cocones over `F ⋙ G` functorially. -/\n@[simps] def functoriality : cocone F ⥤ cocone (F ⋙ G) :=\n{ obj := λ A,\n  { X := G.obj A.X,\n    ι := { app := λ j, G.map (A.ι.app j), naturality' := by intros; erw ←G.map_comp; tidy } },\n  map := λ _ _ f,\n  { hom := G.map f.hom,\n    w'  := by intros; rw [←functor.map_comp, cocone_morphism.w] } }\n\ninstance functoriality_full [full G] [faithful G] : full (functoriality F G) :=\n{ preimage := λ X Y t,\n  { hom := G.preimage t.hom,\n    w' := λ j, G.map_injective (by simpa using t.w j) } }\n\ninstance functoriality_faithful [faithful G] : faithful (functoriality F G) :=\n{ map_injective' := λ X Y f g e, by { ext1, injection e, apply G.map_injective h_1 } }\n\n/--\nIf `e : C ≌ D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cocones over `F` and cocones over `F ⋙ e.functor`.\n-/\n@[simps]\ndef functoriality_equivalence (e : C ≌ D) : cocone F ≌ cocone (F ⋙ e.functor) :=\nlet f : (F ⋙ e.functor) ⋙ e.inverse ≅ F :=\n  functor.associator _ _ _ ≪≫ iso_whisker_left _ (e.unit_iso).symm ≪≫ functor.right_unitor _ in\n{ functor := functoriality F e.functor,\n  inverse := (functoriality (F ⋙ e.functor) e.inverse) ⋙\n    (precompose_equivalence f.symm).functor,\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (e.unit_iso.app _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cocones.ext (e.counit_iso.app _)\n  begin\n    -- Unfortunately this doesn't work by `tidy`.\n    -- In this configuration `simp` reaches a dead-end and needs help.\n    intros j,\n    dsimp,\n    simp only [←equivalence.counit_inv_app_functor, iso.inv_hom_id_app, map_comp,\n      equivalence.fun_inv_map, assoc, id_comp, iso.inv_hom_id_app_assoc],\n    dsimp, simp, -- See note [dsimp, simp].\n  end)\n  (λ c c' f, by { ext, dsimp, simp, dsimp, simp, }), }\n\n/--\nIf `F` reflects isomorphisms, then `cocones.functoriality F` reflects isomorphisms\nas well.\n-/\ninstance reflects_cocone_isomorphism (F : C ⥤ D) [reflects_isomorphisms F] (K : J ⥤ C) :\n  reflects_isomorphisms (cocones.functoriality K F) :=\nbegin\n  constructor,\n  introsI,\n  haveI : is_iso (F.map f.hom) :=\n    (cocones.forget (K ⋙ F)).map_is_iso ((cocones.functoriality K F).map f),\n  haveI := reflects_isomorphisms.reflects F f.hom,\n  apply cocone_iso_of_hom_iso\nend\n\nend\nend cocones\n\nend limits\n\nnamespace functor\n\nvariables {F : J ⥤ C} {G : J ⥤ C} (H : C ⥤ D)\n\nopen category_theory.limits\n\n/-- The image of a cone in C under a functor G : C ⥤ D is a cone in D. -/\n@[simps]\ndef map_cone   (c : cone F)   : cone (F ⋙ H)   := (cones.functoriality F H).obj c\n/-- The image of a cocone in C under a functor G : C ⥤ D is a cocone in D. -/\n@[simps]\ndef map_cocone (c : cocone F) : cocone (F ⋙ H) := (cocones.functoriality F H).obj c\n\n/-- Given a cone morphism `c ⟶ c'`, construct a cone morphism on the mapped cones functorially.  -/\ndef map_cone_morphism   {c c' : cone F}   (f : c ⟶ c')   :\n  H.map_cone c ⟶ H.map_cone c' := (cones.functoriality F H).map f\n\n/-- Given a cocone morphism `c ⟶ c'`, construct a cocone morphism on the mapped cocones\nfunctorially. -/\ndef map_cocone_morphism {c c' : cocone F} (f : c ⟶ c') :\n  H.map_cocone c ⟶ H.map_cocone c' := (cocones.functoriality F H).map f\n\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F ⋙ H`.-/\ndef map_cone_inv [is_equivalence H]\n  (c : cone (F ⋙ H)) : cone F :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).inverse.obj c\n\n/-- `map_cone` is the left inverse to `map_cone_inv`. -/\ndef map_cone_map_cone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone (F ⋙ H)) :\n  map_cone H (map_cone_inv H c) ≅ c :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c\n\n/-- `map_cone` is the right inverse to `map_cone_inv`. -/\ndef map_cone_inv_map_cone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cone F) :\n  map_cone_inv H (map_cone H c) ≅ c :=\n(limits.cones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F ⋙ H`.-/\n\ndef map_cocone_inv [is_equivalence H]\n  (c : cocone (F ⋙ H)) : cocone F :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).inverse.obj c\n\n/-- `map_cocone` is the left inverse to `map_cocone_inv`. -/\ndef map_cocone_map_cocone_inv {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone (F ⋙ H)) :\n  map_cocone H (map_cocone_inv H c) ≅ c :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).counit_iso.app c\n\n/-- `map_cocone` is the right inverse to `map_cocone_inv`. -/\ndef map_cocone_inv_map_cocone {F : J ⥤ D} (H : D ⥤ C) [is_equivalence H] (c : cocone F) :\n  map_cocone_inv H (map_cocone H c) ≅ c :=\n(limits.cocones.functoriality_equivalence F (as_equivalence H)).unit_iso.symm.app c\n\n/-- `functoriality F _ ⋙ postcompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simps]\ndef functoriality_comp_postcompose {H H' : C ⥤ D} (α : H ≅ H') :\n  cones.functoriality F H ⋙ cones.postcompose (whisker_left F α.hom) ≅ cones.functoriality F H' :=\nnat_iso.of_components (λ c, cones.ext (α.app _) (by tidy)) (by tidy)\n\n/--\nFor `F : J ⥤ C`, given a cone `c : cone F`, and a natural isomorphism `α : H ≅ H'` for functors\n`H H' : C ⥤ D`, the postcomposition of the cone `H.map_cone` using the isomorphism `α` is\nisomorphic to the cone `H'.map_cone`.\n-/\n@[simps]\ndef postcompose_whisker_left_map_cone {H H' : C ⥤ D} (α : H ≅ H') (c : cone F) :\n  (cones.postcompose (whisker_left F α.hom : _)).obj (H.map_cone c) ≅ H'.map_cone c :=\n(functoriality_comp_postcompose α).app c\n\n/--\n`map_cone` commutes with `postcompose`. In particular, for `F : J ⥤ C`, given a cone `c : cone F`, a\nnatural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious ways of producing\na cone over `G ⋙ H`, and they are both isomorphic.\n-/\n@[simps]\ndef map_cone_postcompose {α : F ⟶ G} {c} :\n  H.map_cone ((cones.postcompose α).obj c) ≅\n  (cones.postcompose (whisker_right α H : _)).obj (H.map_cone c) :=\ncones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cone` commutes with `postcompose_equivalence`\n-/\n@[simps]\ndef map_cone_postcompose_equivalence_functor {α : F ≅ G} {c} :\n  H.map_cone ((cones.postcompose_equivalence α).functor.obj c) ≅\n    (cones.postcompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cone c) :=\ncones.ext (iso.refl _) (by tidy)\n\n/-- `functoriality F _ ⋙ precompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simps]\ndef functoriality_comp_precompose {H H' : C ⥤ D} (α : H ≅ H') :\n   cocones.functoriality F H ⋙ cocones.precompose (whisker_left F α.inv)\n ≅ cocones.functoriality F H' :=\nnat_iso.of_components (λ c, cocones.ext (α.app _) (by tidy)) (by tidy)\n\n/--\nFor `F : J ⥤ C`, given a cocone `c : cocone F`, and a natural isomorphism `α : H ≅ H'` for functors\n`H H' : C ⥤ D`, the precomposition of the cocone `H.map_cocone` using the isomorphism `α` is\nisomorphic to the cocone `H'.map_cocone`.\n-/\n@[simps]\ndef precompose_whisker_left_map_cocone {H H' : C ⥤ D} (α : H ≅ H') (c : cocone F) :\n  (cocones.precompose (whisker_left F α.inv : _)).obj (H.map_cocone c) ≅ H'.map_cocone c :=\n(functoriality_comp_precompose α).app c\n\n/--\n`map_cocone` commutes with `precompose`. In particular, for `F : J ⥤ C`, given a cocone\n`c : cocone F`, a natural transformation `α : F ⟶ G` and a functor `H : C ⥤ D`, we have two obvious\nways of producing a cocone over `G ⋙ H`, and they are both isomorphic.\n-/\n@[simps]\ndef map_cocone_precompose {α : F ⟶ G} {c} :\n  H.map_cocone ((cocones.precompose α).obj c) ≅\n  (cocones.precompose (whisker_right α H : _)).obj (H.map_cocone c) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cocone` commutes with `precompose_equivalence`\n-/\n@[simps]\ndef map_cocone_precompose_equivalence_functor {α : F ≅ G} {c} :\n  H.map_cocone ((cocones.precompose_equivalence α).functor.obj c) ≅\n    (cocones.precompose_equivalence (iso_whisker_right α H : _)).functor.obj (H.map_cocone c) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cone` commutes with `whisker`\n-/\n@[simps]\ndef map_cone_whisker {E : K ⥤ J} {c : cone F} :\n  H.map_cone (c.whisker E) ≅ (H.map_cone c).whisker E :=\ncones.ext (iso.refl _) (by tidy)\n\n/--\n`map_cocone` commutes with `whisker`\n-/\n@[simps]\ndef map_cocone_whisker {E : K ⥤ J} {c : cocone F} :\n  H.map_cocone (c.whisker E) ≅ (H.map_cocone c).whisker E :=\ncocones.ext (iso.refl _) (by tidy)\n\nend functor\n\nend category_theory\n\nnamespace category_theory.limits\n\nsection\nvariables {F : J ⥤ C}\n\n/-- Change a `cocone F` into a `cone F.op`. -/\n@[simps] def cocone.op (c : cocone F) : cone F.op :=\n{ X := op c.X,\n  π := nat_trans.op c.ι }\n\n/-- Change a `cone F` into a `cocone F.op`. -/\n@[simps] def cone.op (c : cone F) : cocone F.op :=\n{ X := op c.X,\n  ι := nat_trans.op c.π }\n\n/-- Change a `cocone F.op` into a `cone F`. -/\n@[simps] def cocone.unop (c : cocone F.op) : cone F :=\n{ X := unop c.X,\n  π := nat_trans.remove_op c.ι }\n\n/-- Change a `cone F.op` into a `cocone F`. -/\n@[simps] def cone.unop (c : cone F.op) : cocone F :=\n{ X := unop c.X,\n  ι := nat_trans.remove_op c.π }\n\nvariables (F)\n\n/--\nThe category of cocones on `F`\nis equivalent to the opposite category of\nthe category of cones on the opposite of `F`.\n-/\n@[simps]\ndef cocone_equivalence_op_cone_op : cocone F ≌ (cone F.op)ᵒᵖ :=\n{ functor :=\n  { obj := λ c, op (cocone.op c),\n    map := λ X Y f, quiver.hom.op\n    { hom := f.hom.op,\n      w' := λ j, by { apply quiver.hom.unop_inj, dsimp, simp, }, } },\n  inverse :=\n  { obj := λ c, cone.unop (unop c),\n    map := λ X Y f,\n    { hom := f.unop.hom.unop,\n      w' := λ j, by { apply quiver.hom.op_inj, dsimp, simp, }, } },\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c,\n    by { induction c using opposite.rec,\n         dsimp, apply iso.op, exact cones.ext (iso.refl _) (by tidy), })\n    (λ X Y f, quiver.hom.unop_inj (cone_morphism.ext _ _ (by { dsimp, simp }))),\n  functor_unit_iso_comp' := λ c, begin apply quiver.hom.unop_inj, ext, dsimp, simp, end }\n\nend\n\nsection\nvariables {F : J ⥤ Cᵒᵖ}\n\n/-- Change a cocone on `F.left_op : Jᵒᵖ ⥤ C` to a cocone on `F : J ⥤ Cᵒᵖ`. -/\n-- Here and below we only automatically generate the `@[simp]` lemma for the `X` field,\n-- as we can write a simpler `rfl` lemma for the components of the natural transformation by hand.\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cone_of_cocone_left_op (c : cocone F.left_op) : cone F :=\n{ X := op c.X,\n  π := nat_trans.remove_left_op c.ι }\n\n/-- Change a cone on `F : J ⥤ Cᵒᵖ` to a cocone on `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cocone_left_op_of_cone (c : cone F) : cocone (F.left_op) :=\n{ X := unop c.X,\n  ι := nat_trans.left_op c.π }\n\n/-- Change a cone on `F.left_op : Jᵒᵖ ⥤ C` to a cocone on `F : J ⥤ Cᵒᵖ`. -/\n/- When trying use `@[simps]` to generate the `ι_app` field of this definition, `@[simps]` tries to\n  reduce the RHS using `expr.dsimp` and `expr.simp`, but for some reason the expression is not\n  being simplified properly. -/\n@[simps X]\ndef cocone_of_cone_left_op (c : cone F.left_op) : cocone F :=\n{ X := op c.X,\n  ι := nat_trans.remove_left_op c.π }\n\n@[simp] lemma cocone_of_cone_left_op_ι_app (c : cone F.left_op) (j) :\n  (cocone_of_cone_left_op c).ι.app j = (c.π.app (op j)).op :=\nby { dsimp only [cocone_of_cone_left_op], simp }\n\n/-- Change a cocone on `F : J ⥤ Cᵒᵖ` to a cone on `F.left_op : Jᵒᵖ ⥤ C`. -/\n@[simps {rhs_md := semireducible, simp_rhs := tt}]\ndef cone_left_op_of_cocone (c : cocone F) : cone (F.left_op) :=\n{ X := unop c.X,\n  π := nat_trans.left_op c.ι }\n\nend\n\nsection\nvariables {F : Jᵒᵖ ⥤ C}\n\n/-- Change a cocone on `F.right_op : J ⥤ Cᵒᵖ` to a cone on `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def cone_of_cocone_right_op (c : cocone F.right_op) : cone F :=\n{ X := unop c.X,\n  π := nat_trans.remove_right_op c.ι }\n\n/-- Change a cone on `F : Jᵒᵖ ⥤ C` to a cocone on `F.right_op : Jᵒᵖ ⥤ C`. -/\n@[simps] def cocone_right_op_of_cone (c : cone F) : cocone (F.right_op) :=\n{ X := op c.X,\n  ι := nat_trans.right_op c.π }\n\n/-- Change a cone on `F.right_op : J ⥤ Cᵒᵖ` to a cocone on `F : Jᵒᵖ ⥤ C`. -/\n@[simps] def cocone_of_cone_right_op (c : cone F.right_op) : cocone F :=\n{ X := unop c.X,\n  ι := nat_trans.remove_right_op c.π }\n\n/-- Change a cocone on `F : Jᵒᵖ ⥤ C` to a cone on `F.right_op : J ⥤ Cᵒᵖ`. -/\n@[simps] def cone_right_op_of_cocone (c : cocone F) : cone (F.right_op) :=\n{ X := op c.X,\n  π := nat_trans.right_op c.ι }\n\nend\n\nsection\nvariables {F : Jᵒᵖ ⥤ Cᵒᵖ}\n\n/-- Change a cocone on `F.unop : J ⥤ C` into a cone on `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def cone_of_cocone_unop (c : cocone F.unop) : cone F :=\n{ X := op c.X,\n  π := nat_trans.remove_unop c.ι }\n\n/-- Change a cone on `F : Jᵒᵖ ⥤ Cᵒᵖ` into a cocone on `F.unop : J ⥤ C`. -/\n@[simps] def cocone_unop_of_cone (c : cone F) : cocone F.unop :=\n{ X := unop c.X,\n  ι := nat_trans.unop c.π }\n\n/-- Change a cone on `F.unop : J ⥤ C` into a cocone on `F : Jᵒᵖ ⥤ Cᵒᵖ`. -/\n@[simps] def cocone_of_cone_unop (c : cone F.unop) : cocone F :=\n{ X := op c.X,\n  ι := nat_trans.remove_unop c.π }\n\n/-- Change a cocone on `F : Jᵒᵖ ⥤ Cᵒᵖ` into a cone on `F.unop : J ⥤ C`. -/\n@[simps] def cone_unop_of_cocone (c : cocone F) : cone F.unop :=\n{ X := unop c.X,\n  π := nat_trans.unop c.ι }\n\nend\n\nend category_theory.limits\n\nnamespace category_theory.functor\n\nopen category_theory.limits\n\nvariables {F : J ⥤ C}\n\nsection\nvariables (G : C ⥤ D)\n\n/-- The opposite cocone of the image of a cone is the image of the opposite cocone. -/\n@[simps {rhs_md := semireducible}]\ndef map_cone_op (t : cone F) : (G.map_cone t).op ≅ (G.op.map_cocone t.op) :=\ncocones.ext (iso.refl _) (by tidy)\n\n/-- The opposite cone of the image of a cocone is the image of the opposite cone. -/\n@[simps {rhs_md := semireducible}]\ndef map_cocone_op {t : cocone F} : (G.map_cocone t).op ≅ (G.op.map_cone t.op) :=\ncones.ext (iso.refl _) (by tidy)\n\nend\n\nend category_theory.functor\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": 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{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.set_integral\nimport Mathlib.measure_theory.lebesgue_measure\nimport Mathlib.analysis.calculus.fderiv_measurable\nimport Mathlib.analysis.calculus.extend_deriv\nimport Mathlib.PostPort\n\nuniverses u_1 u_4 u_3 u_2 u_6 l \n\nnamespace Mathlib\n\n/-!\n# Integral over an interval\n\nIn this file we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ` if `a ≤ b`\nand `-∫ x in Ioc b a, f x ∂μ` if `b ≤ a`. We prove a few simple properties and many versions\nof the first part of the\n[fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus).\nRecall that it states that the function `(u, v) ↦ ∫ x in u..v, f x` has derivative\n`(δu, δv) ↦ δv • f b - δu • f a` at `(a, b)` provided that `f` is continuous at `a` and `b`.\n\n## Main statements\n\n### FTC-1 for Lebesgue measure\n\nWe prove several versions of FTC-1, all in the `interval_integral` namespace. Many of them follow\nthe naming scheme `integral_has(_strict?)_(f?)deriv(_within?)_at(_of_tendsto_ae?)(_right|_left?)`.\nThey formulate FTC in terms of `has(_strict?)_(f?)deriv(_within?)_at`.\nLet us explain the meaning of each part of the name:\n\n* `_strict` means that the theorem is about strict differentiability;\n* `f` means that the theorem is about differentiability in both endpoints; incompatible with\n  `_right|_left`;\n* `_within` means that the theorem is about one-sided derivatives, see below for details;\n* `_of_tendsto_ae` means that instead of continuity the theorem assumes that `f` has a finite limit\n  almost surely as `x` tends to `a` and/or `b`;\n* `_right` or `_left` mean that the theorem is about differentiability in the right (resp., left)\n  endpoint.\n\nWe also reformulate these theorems in terms of `(f?)deriv(_within?)`. These theorems are named\n`(f?)deriv(_within?)_integral(_of_tendsto_ae?)(_right|_left?)` with the same meaning of parts of the\nname.\n\n### One-sided derivatives\n\nTheorem `integral_has_fderiv_within_at_of_tendsto_ae` states that `(u, v) ↦ ∫ x in u..v, f x` has a\nderivative `(δu, δv) ↦ δv • cb - δu • ca` within the set `s × t` at `(a, b)` provided that `f` tends\nto `ca` (resp., `cb`) almost surely at `la` (resp., `lb`), where possible values of `s`, `t`, and\ncorresponding filters `la`, `lb` are given in the following table.\n\n| `s`     | `la`         | `t`     | `lb`         |\n| ------- | ----         | ---     | ----         |\n| `Iic a` | `𝓝[Iic a] a` | `Iic b` | `𝓝[Iic b] b` |\n| `Ici a` | `𝓝[Ioi a] a` | `Ici b` | `𝓝[Ioi b] b` |\n| `{a}`   | `⊥`          | `{b}`   | `⊥`          |\n| `univ`  | `𝓝 a`        | `univ`  | `𝓝 b`        |\n\nWe use a typeclass `FTC_filter` to make Lean automatically find `la`/`lb` based on `s`/`t`. This way\nwe can formulate one theorem instead of `16` (or `8` if we leave only non-trivial ones not covered\nby `integral_has_deriv_within_at_of_tendsto_ae_(left|right)` and\n`integral_has_fderiv_at_of_tendsto_ae`). Similarly,\n`integral_has_deriv_within_at_of_tendsto_ae_right` works for both one-sided derivatives using the\nsame typeclass to find an appropriate filter.\n\n### FTC for a locally finite measure\n\nBefore proving FTC for the Lebesgue measure, we prove a few statements that can be seen as FTC for\nany measure. The most general of them,\n`measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae`, states the following. Let `(la, la')`\nbe an `FTC_filter` pair of filters around `a` (i.e., `FTC_filter a la la'`) and let `(lb, lb')` be\nan `FTC_filter` pair of filters around `b`. If `f` has finite limits `ca` and `cb` almost surely at\n`la'` and `lb'`, respectively, then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)` as `ua` and `va` tend to `la` while\n`ub` and `vb` tend to `lb`.\n\n## Implementation notes\n\n### Avoiding `if`, `min`, and `max`\n\nIn order to avoid `if`s in the definition, we define `interval_integrable f μ a b` as\n`integrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ`. For any `a`, `b` one of these\nintervals is empty and the other coincides with `Ioc (min a b) (max a b)`.\n\nSimilarly, we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`.\nAgain, for any `a`, `b` one of these integrals is zero, and the other gives the expected result.\n\nThis way some properties can be translated from integrals over sets without dealing with\nthe cases `a ≤ b` and `b ≤ a` separately.\n\n### Choice of the interval\n\nWe use integral over `Ioc (min a b) (max a b)` instead of one of the other three possible\nintervals with the same endpoints for two reasons:\n\n* this way `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ` holds whenever\n  `f` is integrable on each interval; in particular, it works even if the measure `μ` has an atom\n  at `b`; this rules out `Ioo` and `Icc` intervals;\n* with this definition for a probability measure `μ`, the integral `∫ x in a..b, 1 ∂μ` equals\n  the difference $F_μ(b)-F_μ(a)$, where $F_μ(a)=μ(-∞, a]$ is the\n  [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function)\n  of `μ`.\n\n### `FTC_filter` class\n\nAs explained above, many theorems in this file rely on the typeclass\n`FTC_filter (a : α) (l l' : filter α)` to avoid code duplication. This typeclass combines four\nassumptions:\n\n- `pure a ≤ l`;\n- `l' ≤ 𝓝 a`;\n- `l'` has a basis of measurable sets;\n- if `u n` and `v n` tend to `l`, then for any `s ∈ l'`, `Ioc (u n) (v n)` is eventually included\n  in `s`.\n\nThis typeclass has exactly four “real” instances: `(a, pure a, ⊥)`, `(a, 𝓝[Ici a] a, 𝓝[Ioi a] a)`,\n`(a, 𝓝[Iic a] a, 𝓝[Iic a] a)`, `(a, 𝓝 a, 𝓝 a)`, and two instances that are equal to the first and\nlast “real” instances: `(a, 𝓝[{a}] a, ⊥)` and `(a, 𝓝[univ] a, 𝓝[univ] a)`. While the difference\nbetween `Ici a` and `Ioi a` doesn't matter for theorems about Lebesgue measure, it becomes important\nin the versions of FTC about any locally finite measure if this measure has an atom at one of the\nendpoints.\n\n## Tags\n\nintegral, fundamental theorem of calculus\n -/\n\n/-!\n### Integrability at an interval\n-/\n\n/-- A function `f` is called *interval integrable* with respect to a measure `μ` on an unordered\ninterval `a..b` if it is integrable on both intervals `(a, b]` and `(b, a]`. One of these\nintervals is always empty, so this property is equivalent to `f` being integrable on\n`(min a b, max a b]`. -/\ndef interval_integrable {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] (f : α → E) (μ : measure_theory.measure α) (a : α) (b : α) :=\n  measure_theory.integrable_on f (set.Ioc a b) ∧ measure_theory.integrable_on f (set.Ioc b a)\n\ntheorem measure_theory.integrable.interval_integrable {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {μ : measure_theory.measure α} (hf : measure_theory.integrable f) {a : α} {b : α} : interval_integrable f μ a b :=\n  { left := measure_theory.integrable.integrable_on hf, right := measure_theory.integrable.integrable_on hf }\n\nnamespace interval_integrable\n\n\ntheorem symm {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {b : α} {μ : measure_theory.measure α} (h : interval_integrable f μ a b) : interval_integrable f μ b a :=\n  and.symm h\n\ntheorem refl {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {μ : measure_theory.measure α} : interval_integrable f μ a a := sorry\n\ntheorem trans {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {b : α} {c : α} {μ : measure_theory.measure α} (hab : interval_integrable f μ a b) (hbc : interval_integrable f μ b c) : interval_integrable f μ a c := sorry\n\ntheorem neg {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {b : α} {μ : measure_theory.measure α} [borel_space E] (h : interval_integrable f μ a b) : interval_integrable (-f) μ a b :=\n  { left := measure_theory.integrable.neg (and.left h), right := measure_theory.integrable.neg (and.right h) }\n\nprotected theorem ae_measurable {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {b : α} {μ : measure_theory.measure α} (h : interval_integrable f μ a b) : ae_measurable f :=\n  measure_theory.integrable.ae_measurable (and.left h)\n\nprotected theorem ae_measurable' {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {a : α} {b : α} {μ : measure_theory.measure α} (h : interval_integrable f μ a b) : ae_measurable f :=\n  measure_theory.integrable.ae_measurable (and.right h)\n\ntheorem smul {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [borel_space E] [normed_field 𝕜] [normed_space 𝕜 E] {f : α → E} {a : α} {b : α} {μ : measure_theory.measure α} (h : interval_integrable f μ a b) (r : 𝕜) : interval_integrable (r • f) μ a b :=\n  { left := measure_theory.integrable.smul r (and.left h), right := measure_theory.integrable.smul r (and.right h) }\n\ntheorem add {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [borel_space E] {f : α → E} {g : α → E} {a : α} {b : α} {μ : measure_theory.measure α} [topological_space.second_countable_topology E] (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) : interval_integrable (f + g) μ a b :=\n  { left := measure_theory.integrable.add (and.left hf) (and.left hg),\n    right := measure_theory.integrable.add (and.right hf) (and.right hg) }\n\ntheorem sub {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [borel_space E] {f : α → E} {g : α → E} {a : α} {b : α} {μ : measure_theory.measure α} [topological_space.second_countable_topology E] (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) : interval_integrable (f - g) μ a b :=\n  { left := measure_theory.integrable.sub (and.left hf) (and.left hg),\n    right := measure_theory.integrable.sub (and.right hf) (and.right hg) }\n\nend interval_integrable\n\n\ntheorem continuous_on.interval_integrable {E : Type u_4} [measurable_space E] [normed_group E] {μ : measure_theory.measure ℝ} [measure_theory.locally_finite_measure μ] [borel_space E] {u : ℝ → E} {a : ℝ} {b : ℝ} (hu : continuous_on u (set.interval a b)) : interval_integrable u μ a b := sorry\n\ntheorem continuous_on.interval_integrable_of_Icc {E : Type u_4} [measurable_space E] [normed_group E] {μ : measure_theory.measure ℝ} [measure_theory.locally_finite_measure μ] [borel_space E] {u : ℝ → E} {a : ℝ} {b : ℝ} (h : a ≤ b) (hu : continuous_on u (set.Icc a b)) : interval_integrable u μ a b :=\n  continuous_on.interval_integrable (Eq.symm (set.interval_of_le h) ▸ hu)\n\n/-- A continuous function on `ℝ` is `interval_integrable` with respect to any locally finite measure\n`ν` on ℝ. -/\ntheorem continuous.interval_integrable {E : Type u_4} [measurable_space E] [normed_group E] {μ : measure_theory.measure ℝ} [measure_theory.locally_finite_measure μ] [borel_space E] {u : ℝ → E} (hu : continuous u) (a : ℝ) (b : ℝ) : interval_integrable u μ a b :=\n  continuous_on.interval_integrable (continuous.continuous_on hu)\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : α → E` has a finite limit at `l' ⊓ μ.ae`. Then `f` is interval integrable on\n`u..v` provided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter α` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\ntheorem filter.tendsto.eventually_interval_integrable_ae {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {μ : measure_theory.measure α} {l : filter α} {l' : filter α} (hfm : measurable_at_filter f l') [filter.tendsto_Ixx_class set.Ioc l l'] [filter.is_measurably_generated l'] (hμ : measure_theory.measure.finite_at_filter μ l') {c : E} (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) {u : β → α} {v : β → α} {lt : filter β} (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) : filter.eventually (fun (t : β) => interval_integrable f μ (u t) (v t)) lt := sorry\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : α → E` has a finite limit at `l`. Then `f` is interval integrable on `u..v`\nprovided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter α` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\ntheorem filter.tendsto.eventually_interval_integrable {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] {f : α → E} {μ : measure_theory.measure α} {l : filter α} {l' : filter α} (hfm : measurable_at_filter f l') [filter.tendsto_Ixx_class set.Ioc l l'] [filter.is_measurably_generated l'] (hμ : measure_theory.measure.finite_at_filter μ l') {c : E} (hf : filter.tendsto f l' (nhds c)) {u : β → α} {v : β → α} {lt : filter β} (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) : filter.eventually (fun (t : β) => interval_integrable f μ (u t) (v t)) lt :=\n  filter.tendsto.eventually_interval_integrable_ae hfm hμ (filter.tendsto.mono_left hf inf_le_left) hu hv\n\n/-!\n### Interval integral: definition and basic properties\n\nIn this section we define `∫ x in a..b, f x ∂μ` as `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`\nand prove some basic properties.\n-/\n\n/-- The interval integral `∫ x in a..b, f x ∂μ` is defined\nas `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`. If `a ≤ b`, then it equals\n`∫ x in Ioc a b, f x ∂μ`, otherwise it equals `-∫ x in Ioc b a, f x ∂μ`. -/\ndef interval_integral {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] (f : α → E) (a : α) (b : α) (μ : measure_theory.measure α) : E :=\n  (measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc a b)) fun (x : α) => f x) -\n    measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc b a)) fun (x : α) => f x\n\nnamespace interval_integral\n\n\n@[simp] theorem integral_zero {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {μ : measure_theory.measure α} : interval_integral (fun (x : α) => 0) a b μ = 0 := sorry\n\ntheorem integral_of_le {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} (h : a ≤ b) : interval_integral (fun (x : α) => f x) a b μ =\n  measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc a b)) fun (x : α) => f x := sorry\n\n@[simp] theorem integral_same {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {f : α → E} {μ : measure_theory.measure α} : interval_integral (fun (x : α) => f x) a a μ = 0 :=\n  sub_self\n    (measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc a a)) fun (x : α) => (fun (x : α) => f x) x)\n\ntheorem integral_symm {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {μ : measure_theory.measure α} (a : α) (b : α) : interval_integral (fun (x : α) => f x) b a μ = -interval_integral (fun (x : α) => f x) a b μ := sorry\n\ntheorem integral_of_ge {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} (h : b ≤ a) : interval_integral (fun (x : α) => f x) a b μ =\n  -measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc b a)) fun (x : α) => f x := sorry\n\ntheorem integral_cases {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {μ : measure_theory.measure α} (f : α → E) (a : α) (b : α) : interval_integral (fun (x : α) => f x) a b μ ∈\n  insert (measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc (min a b) (max a b))) fun (x : α) => f x)\n    (singleton\n      (-measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc (min a b) (max a b))) fun (x : α) => f x)) := sorry\n\ntheorem integral_non_ae_measurable {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {μ : measure_theory.measure α} {f : α → E} {a : α} {b : α} (h : a < b) (hf : ¬ae_measurable f) : interval_integral (fun (x : α) => f x) a b μ = 0 := sorry\n\ntheorem norm_integral_eq_norm_integral_Ioc {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} : norm (interval_integral (fun (x : α) => f x) a b μ) =\n  norm (measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc (min a b) (max a b))) fun (x : α) => f x) := sorry\n\ntheorem norm_integral_le_integral_norm_Ioc {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} : norm (interval_integral (fun (x : α) => f x) a b μ) ≤\n  measure_theory.integral (measure_theory.measure.restrict μ (set.Ioc (min a b) (max a b))) fun (x : α) => norm (f x) :=\n  trans_rel_right LessEq norm_integral_eq_norm_integral_Ioc (measure_theory.norm_integral_le_integral_norm f)\n\ntheorem norm_integral_le_abs_integral_norm {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} : norm (interval_integral (fun (x : α) => f x) a b μ) ≤ abs (interval_integral (fun (x : α) => norm (f x)) a b μ) := sorry\n\ntheorem norm_integral_le_of_norm_le_const_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : ℝ} {b : ℝ} {C : ℝ} {f : ℝ → E} (h : filter.eventually (fun (x : ℝ) => x ∈ set.Ioc (min a b) (max a b) → norm (f x) ≤ C) (measure_theory.measure.ae volume)) : norm (interval_integral (fun (x : ℝ) => f x) a b volume) ≤ C * abs (b - a) := sorry\n\ntheorem norm_integral_le_of_norm_le_const {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : ℝ} {b : ℝ} {C : ℝ} {f : ℝ → E} (h : ∀ (x : ℝ), x ∈ set.Ioc (min a b) (max a b) → norm (f x) ≤ C) : norm (interval_integral (fun (x : ℝ) => f x) a b volume) ≤ C * abs (b - a) :=\n  norm_integral_le_of_norm_le_const_ae (filter.eventually_of_forall h)\n\ntheorem integral_add {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {g : α → E} {μ : measure_theory.measure α} (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) : interval_integral (fun (x : α) => f x + g x) a b μ =\n  interval_integral (fun (x : α) => f x) a b μ + interval_integral (fun (x : α) => g x) a b μ := sorry\n\n@[simp] theorem integral_neg {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} : interval_integral (fun (x : α) => -f x) a b μ = -interval_integral (fun (x : α) => f x) a b μ := sorry\n\ntheorem integral_sub {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {g : α → E} {μ : measure_theory.measure α} (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) : interval_integral (fun (x : α) => f x - g x) a b μ =\n  interval_integral (fun (x : α) => f x) a b μ - interval_integral (fun (x : α) => g x) a b μ := sorry\n\ntheorem integral_smul {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} (r : ℝ) : interval_integral (fun (x : α) => r • f x) a b μ = r • interval_integral (fun (x : α) => f x) a b μ := sorry\n\ntheorem integral_const' {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {μ : measure_theory.measure α} (c : E) : interval_integral (fun (x : α) => c) a b μ =\n  (ennreal.to_real (coe_fn μ (set.Ioc a b)) - ennreal.to_real (coe_fn μ (set.Ioc b a))) • c := sorry\n\ntheorem integral_const {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : ℝ} {b : ℝ} (c : E) : interval_integral (fun (x : ℝ) => c) a b volume = (b - a) • c := sorry\n\ntheorem integral_smul_measure {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} (c : ennreal) : interval_integral (fun (x : α) => f x) a b (c • μ) = ennreal.to_real c • interval_integral (fun (x : α) => f x) a b μ := sorry\n\ntheorem integral_comp_add_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] (a : ℝ) (b : ℝ) (c : ℝ) (f : ℝ → E) (hfm : ae_measurable f) : interval_integral (fun (x : ℝ) => f (x + c)) a b volume = interval_integral (fun (x : ℝ) => f x) (a + c) (b + c) volume := sorry\n\ntheorem integral_comp_mul_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {c : ℝ} (hc : 0 < c) (a : ℝ) (b : ℝ) (f : ℝ → E) (hfm : ae_measurable f) : interval_integral (fun (x : ℝ) => f (x * c)) a b volume =\n  c⁻¹ • interval_integral (fun (x : ℝ) => f x) (a * c) (b * c) volume := sorry\n\ntheorem integral_comp_neg {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] (a : ℝ) (b : ℝ) (f : ℝ → E) (hfm : ae_measurable f) : interval_integral (fun (x : ℝ) => f (-x)) a b volume = interval_integral (fun (x : ℝ) => f x) (-b) (-a) volume := sorry\n\n/-!\n### Integral is an additive function of the interval\n\nIn this section we prove that `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ`\nas well as a few other identities trivially equivalent to this one. We also prove that\n`∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ` provided that `support f ⊆ Ioc a b`.\n-/\n\n/-- If two functions are equal in the relevant interval, their interval integrals are also equal. -/\ntheorem integral_congr {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] {a : α} {b : α} {f : α → E} {g : α → E} (h : set.eq_on f g (set.interval a b)) : interval_integral (fun (x : α) => f x) a b μ = interval_integral (fun (x : α) => g x) a b μ := sorry\n\ntheorem integral_add_adjacent_intervals_cancel {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hbc : interval_integrable f μ b c) : interval_integral (fun (x : α) => f x) a b μ + interval_integral (fun (x : α) => f x) b c μ +\n    interval_integral (fun (x : α) => f x) c a μ =\n  0 := sorry\n\ntheorem integral_add_adjacent_intervals {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hbc : interval_integrable f μ b c) : interval_integral (fun (x : α) => f x) a b μ + interval_integral (fun (x : α) => f x) b c μ =\n  interval_integral (fun (x : α) => f x) a c μ := sorry\n\ntheorem integral_interval_sub_left {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hac : interval_integrable f μ a c) : interval_integral (fun (x : α) => f x) a b μ - interval_integral (fun (x : α) => f x) a c μ =\n  interval_integral (fun (x : α) => f x) c b μ :=\n  sub_eq_of_eq_add'\n    (Eq.symm (integral_add_adjacent_intervals hac (interval_integrable.trans (interval_integrable.symm hac) hab)))\n\ntheorem integral_interval_add_interval_comm {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {d : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) : interval_integral (fun (x : α) => f x) a b μ + interval_integral (fun (x : α) => f x) c d μ =\n  interval_integral (fun (x : α) => f x) a d μ + interval_integral (fun (x : α) => f x) c b μ := sorry\n\ntheorem integral_interval_sub_interval_comm {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {d : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) : interval_integral (fun (x : α) => f x) a b μ - interval_integral (fun (x : α) => f x) c d μ =\n  interval_integral (fun (x : α) => f x) a c μ - interval_integral (fun (x : α) => f x) b d μ := sorry\n\ntheorem integral_interval_sub_interval_comm' {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {c : α} {d : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (hab : interval_integrable f μ a b) (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) : interval_integral (fun (x : α) => f x) a b μ - interval_integral (fun (x : α) => f x) c d μ =\n  interval_integral (fun (x : α) => f x) d b μ - interval_integral (fun (x : α) => f x) c a μ := sorry\n\ntheorem integral_Iic_sub_Iic {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {f : α → E} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] (ha : measure_theory.integrable_on f (set.Iic a)) (hb : measure_theory.integrable_on f (set.Iic b)) : ((measure_theory.integral (measure_theory.measure.restrict μ (set.Iic b)) fun (x : α) => f x) -\n    measure_theory.integral (measure_theory.measure.restrict μ (set.Iic a)) fun (x : α) => f x) =\n  interval_integral (fun (x : α) => f x) a b μ := sorry\n\n/-- If `μ` is a finite measure then `∫ x in a..b, c ∂μ = (μ (Iic b) - μ (Iic a)) • c`. -/\ntheorem integral_const_of_cdf {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {a : α} {b : α} {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] [measure_theory.finite_measure μ] (c : E) : interval_integral (fun (x : α) => c) a b μ =\n  (ennreal.to_real (coe_fn μ (set.Iic b)) - ennreal.to_real (coe_fn μ (set.Iic a))) • c := sorry\n\ntheorem integral_eq_integral_of_support_subset {α : Type u_1} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {μ : measure_theory.measure α} [topological_space α] [opens_measurable_space α] [order_closed_topology α] {f : α → E} {a : α} {b : α} (h : function.support f ⊆ set.Ioc a b) : interval_integral (fun (x : α) => f x) a b μ = measure_theory.integral μ fun (x : α) => f x := sorry\n\ntheorem integral_eq_zero_iff_of_le_of_nonneg_ae {f : ℝ → ℝ} {a : ℝ} {b : ℝ} (hab : a ≤ b) (hf : filter.eventually_le (measure_theory.measure.ae (measure_theory.measure.restrict volume (set.Ioc a b))) 0 f) (hfi : interval_integrable f volume a b) : interval_integral (fun (x : ℝ) => f x) a b volume = 0 ↔\n  filter.eventually_eq (measure_theory.measure.ae (measure_theory.measure.restrict volume (set.Ioc a b))) f 0 := sorry\n\ntheorem integral_eq_zero_iff_of_nonneg_ae {f : ℝ → ℝ} {a : ℝ} {b : ℝ} (hf : filter.eventually_le (measure_theory.measure.ae (measure_theory.measure.restrict volume (set.Ioc a b ∪ set.Ioc b a))) 0\n  f) (hfi : interval_integrable f volume a b) : interval_integral (fun (x : ℝ) => f x) a b volume = 0 ↔\n  filter.eventually_eq (measure_theory.measure.ae (measure_theory.measure.restrict volume (set.Ioc a b ∪ set.Ioc b a)))\n    f 0 := sorry\n\ntheorem integral_pos_iff_support_of_nonneg_ae' {f : ℝ → ℝ} {a : ℝ} {b : ℝ} (hf : filter.eventually_le (measure_theory.measure.ae (measure_theory.measure.restrict volume (set.Ioc a b ∪ set.Ioc b a))) 0\n  f) (hfi : interval_integrable f volume a b) : 0 < interval_integral (fun (x : ℝ) => f x) a b volume ↔ a < b ∧ 0 < coe_fn volume (function.support f ∩ set.Ioc a b) := sorry\n\ntheorem integral_pos_iff_support_of_nonneg_ae {f : ℝ → ℝ} {a : ℝ} {b : ℝ} (hf : filter.eventually_le (measure_theory.measure.ae volume) 0 f) (hfi : interval_integrable f volume a b) : 0 < interval_integral (fun (x : ℝ) => f x) a b volume ↔ a < b ∧ 0 < coe_fn volume (function.support f ∩ set.Ioc a b) :=\n  integral_pos_iff_support_of_nonneg_ae' (measure_theory.ae_mono measure_theory.measure.restrict_le_self hf) hfi\n\n/-!\n### Fundamental theorem of calculus, part 1, for any measure\n\nIn this section we prove a few lemmas that can be seen as versions of FTC-1 for interval integrals\nw.r.t. any measure. Many theorems are formulated for one or two pairs of filters related by\n`FTC_filter a l l'`. This typeclass has exactly four “real” instances: `(a, pure a, ⊥)`,\n`(a, 𝓝[Ici a] a, 𝓝[Ioi a] a)`, `(a, 𝓝[Iic a] a, 𝓝[Iic a] a)`, `(a, 𝓝 a, 𝓝 a)`, and two instances\nthat are equal to the first and last “real” instances: `(a, 𝓝[{a}] a, ⊥)` and\n`(a, 𝓝[univ] a, 𝓝[univ] a)`.  We use this approach to avoid repeating arguments in many very similar\ncases.  Lean can automatically find both `a` and `l'` based on `l`.\n\nThe most general theorem `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` can be seen\nas a generalization of lemma `integral_has_strict_fderiv_at` below which states strict\ndifferentiability of `∫ x in u..v, f x` in `(u, v)` at `(a, b)` for a measurable function `f` that\nis integrable on `a..b` and is continuous at `a` and `b`. The lemma is generalized in three\ndirections: first, `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` deals with any\nlocally finite measure `μ`; second, it works for one-sided limits/derivatives; third, it assumes\nonly that `f` has finite limits almost surely at `a` and `b`.\n\nNamely, let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of\n`FTC_filter`s around `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f`\nhas finite limits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.  Then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis theorem is formulated with integral of constants instead of measures in the right hand sides\nfor two reasons: first, this way we avoid `min`/`max` in the statements; second, often it is\npossible to write better `simp` lemmas for these integrals, see `integral_const` and\n`integral_const_of_cdf`.\n\nIn the next subsection we apply this theorem to prove various theorems about differentiability\nof the integral w.r.t. Lebesgue measure. -/\n\n/-- An auxiliary typeclass for the Fundamental theorem of calculus, part 1. It is used to formulate\ntheorems that work simultaneously for left and right one-sided derivatives of `∫ x in u..v, f x`.\nThere are four instances: `(a, pure a, ⊥)`, `(a, 𝓝[Ici a], 𝓝[Ioi a])`,\n`(a, 𝓝[Iic a], 𝓝[Iic a])`, and `(a, 𝓝 a, 𝓝 a)`. -/\nclass FTC_filter {β : Type u_6} [linear_order β] [measurable_space β] [topological_space β] (a : outParam β) (outer : filter β) (inner : outParam (filter β)) \nextends filter.tendsto_Ixx_class set.Ioc outer inner\nwhere\n  pure_le : pure a ≤ outer\n  le_nhds : inner ≤ nhds a\n  meas_gen : filter.is_measurably_generated inner\n\n/- The `dangerous_instance` linter doesn't take `out_param`s into account, so it thinks that\n`FTC_filter.to_tendsto_Ixx_class` is dangerous. Disable this linter using `nolint`.\n-/\n\nnamespace FTC_filter\n\n\nprotected instance pure {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] (a : β) : FTC_filter a (pure a) ⊥ :=\n  mk (le_refl (pure a)) bot_le\n\nprotected instance nhds_within_singleton {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] (a : β) : FTC_filter a (nhds_within a (singleton a)) ⊥ :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (FTC_filter a (nhds_within a (singleton a)) ⊥))\n        (nhds_within.equations._eqn_1 a (singleton a))))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (FTC_filter a (nhds a ⊓ filter.principal (singleton a)) ⊥)) (filter.principal_singleton a)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (FTC_filter a (nhds a ⊓ pure a) ⊥)) (iff.mpr inf_eq_right (pure_le_nhds a))))\n        (FTC_filter.pure a)))\n\ntheorem finite_at_inner {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] {a : β} (l : filter β) {l' : outParam (filter β)} [h : FTC_filter a l l'] {μ : measure_theory.measure β} [measure_theory.locally_finite_measure μ] : measure_theory.measure.finite_at_filter μ l' :=\n  measure_theory.measure.finite_at_filter.filter_mono (le_nhds l) (measure_theory.measure.finite_at_nhds μ a)\n\nprotected instance nhds {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] [opens_measurable_space β] [order_topology β] (a : β) : FTC_filter a (nhds a) (nhds a) :=\n  mk (pure_le_nhds a) (le_refl (nhds a))\n\nprotected instance nhds_univ {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] [opens_measurable_space β] [order_topology β] (a : β) : FTC_filter a (nhds_within a set.univ) (nhds a) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (FTC_filter a (nhds_within a set.univ) (nhds a))) (nhds_within_univ a)))\n    (FTC_filter.nhds a)\n\nprotected instance nhds_left {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] [opens_measurable_space β] [order_topology β] (a : β) : FTC_filter a (nhds_within a (set.Iic a)) (nhds_within a (set.Iic a)) :=\n  mk (pure_le_nhds_within set.right_mem_Iic) inf_le_left\n\nprotected instance nhds_right {β : Type u_2} [linear_order β] [measurable_space β] [topological_space β] [opens_measurable_space β] [order_topology β] (a : β) : FTC_filter a (nhds_within a (set.Ici a)) (nhds_within a (set.Ioi a)) :=\n  mk (pure_le_nhds_within set.left_mem_Ici) inf_le_left\n\nend FTC_filter\n\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, where `μ` is a measure\nfinite at `l'`, then `∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both\n`u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae` for a version assuming\n`[FTC_filter a l l']` and `[locally_finite_measure μ]`. If `l` is one of `𝓝[Ici a] a`,\n`𝓝[Iic a] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae' {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [filter.is_measurably_generated l'] [filter.tendsto_Ixx_class set.Ioc l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hl : measure_theory.measure.finite_at_filter μ l') (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ - interval_integral (fun (x : α) => c) (u t) (v t) μ)\n  (fun (t : β) => interval_integral (fun (x : α) => 1) (u t) (v t) μ) lt := sorry\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both\n`u` and `v` tend to `l` so that `u ≤ v`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le` for a version assuming\n`[FTC_filter a l l']` and `[locally_finite_measure μ]`. If `l` is one of `𝓝[Ici a] a`,\n`𝓝[Iic a] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [filter.is_measurably_generated l'] [filter.tendsto_Ixx_class set.Ioc l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hl : measure_theory.measure.finite_at_filter μ l') (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) (huv : filter.eventually_le lt u v) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ - ennreal.to_real (coe_fn μ (set.Ioc (u t) (v t))) • c)\n  (fun (t : β) => ennreal.to_real (coe_fn μ (set.Ioc (u t) (v t)))) lt := sorry\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both\n`u` and `v` tend to `l` so that `v ≤ u`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge` for a version assuming\n`[FTC_filter a l l']` and `[locally_finite_measure μ]`. If `l` is one of `𝓝[Ici a] a`,\n`𝓝[Iic a] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [filter.is_measurably_generated l'] [filter.tendsto_Ixx_class set.Ioc l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hl : measure_theory.measure.finite_at_filter μ l') (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) (huv : filter.eventually_le lt v u) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ + ennreal.to_real (coe_fn μ (set.Ioc (v t) (u t))) • c)\n  (fun (t : β) => ennreal.to_real (coe_fn μ (set.Ioc (v t) (u t)))) lt := sorry\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae'` for a version that also works, e.g., for\n`l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [topological_space α] [measure_theory.locally_finite_measure μ] [FTC_filter a l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ - interval_integral (fun (x : α) => c) (u t) (v t) μ)\n  (fun (t : β) => interval_integral (fun (x : α) => 1) (u t) (v t) μ) lt :=\n  measure_integral_sub_linear_is_o_of_tendsto_ae' hfm hf (FTC_filter.finite_at_inner l) hu hv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae_of_le {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [topological_space α] [measure_theory.locally_finite_measure μ] [FTC_filter a l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) (huv : filter.eventually_le lt u v) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ - ennreal.to_real (coe_fn μ (set.Ioc (u t) (v t))) • c)\n  (fun (t : β) => ennreal.to_real (coe_fn μ (set.Ioc (u t) (v t)))) lt :=\n  measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' hfm hf (FTC_filter.finite_at_inner l) hu hv huv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\ntheorem measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {c : E} {l : filter α} {l' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [topological_space α] [measure_theory.locally_finite_measure μ] [FTC_filter a l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae μ) (nhds c)) (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) (huv : filter.eventually_le lt v u) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (u t) (v t) μ + ennreal.to_real (coe_fn μ (set.Ioc (v t) (u t))) • c)\n  (fun (t : β) => ennreal.to_real (coe_fn μ (set.Ioc (v t) (u t)))) lt :=\n  measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' hfm hf (FTC_filter.finite_at_inner l) hu hv huv\n\n/-- Fundamental theorem of calculus-1, strict derivative in both limits for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f` has finite\nlimits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.\nThen `∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ =\n  ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n    o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n-/\ntheorem measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {b : α} {ca : E} {cb : E} {la : filter α} {la' : filter α} {lb : filter α} {lb' : filter α} {lt : filter β} {μ : measure_theory.measure α} {ua : β → α} {va : β → α} {ub : β → α} {vb : β → α} [topological_space α] [order_topology α] [borel_space α] [FTC_filter a la la'] [FTC_filter b lb lb'] [measure_theory.locally_finite_measure μ] (hab : interval_integrable f μ a b) (hmeas_a : measurable_at_filter f la') (hmeas_b : measurable_at_filter f lb') (ha_lim : filter.tendsto f (la' ⊓ measure_theory.measure.ae μ) (nhds ca)) (hb_lim : filter.tendsto f (lb' ⊓ measure_theory.measure.ae μ) (nhds cb)) (hua : filter.tendsto ua lt la) (hva : filter.tendsto va lt la) (hub : filter.tendsto ub lt lb) (hvb : filter.tendsto vb lt lb) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (va t) (vb t) μ - interval_integral (fun (x : α) => f x) (ua t) (ub t) μ -\n      (interval_integral (fun (x : α) => cb) (ub t) (vb t) μ - interval_integral (fun (x : α) => ca) (ua t) (va t) μ))\n  (fun (t : β) =>\n    norm (interval_integral (fun (x : α) => 1) (ua t) (va t) μ) +\n      norm (interval_integral (fun (x : α) => 1) (ub t) (vb t) μ))\n  lt := sorry\n\n/-- Fundamental theorem of calculus-1, strict derivative in right endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(lb, lb')` be a pair of `FTC_filter`s\naround `b`. Suppose that `f` has a finite limit `c` at `lb' ⊓ μ.ae`.\n\nThen `∫ x in a..v, f x ∂μ - ∫ x in a..u, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `lb`.\n-/\ntheorem measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {b : α} {c : E} {lb : filter α} {lb' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [topological_space α] [order_topology α] [borel_space α] [FTC_filter b lb lb'] [measure_theory.locally_finite_measure μ] (hab : interval_integrable f μ a b) (hmeas : measurable_at_filter f lb') (hf : filter.tendsto f (lb' ⊓ measure_theory.measure.ae μ) (nhds c)) (hu : filter.tendsto u lt lb) (hv : filter.tendsto v lt lb) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) a (v t) μ - interval_integral (fun (x : α) => f x) a (u t) μ -\n      interval_integral (fun (x : α) => c) (u t) (v t) μ)\n  (fun (t : β) => interval_integral (fun (x : α) => 1) (u t) (v t) μ) lt := sorry\n\n/-- Fundamental theorem of calculus-1, strict derivative in left endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`. Suppose that `f` has a finite limit `c` at `la' ⊓ μ.ae`.\n\nThen `∫ x in v..b, f x ∂μ - ∫ x in u..b, f x ∂μ = -∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `la`.\n-/\ntheorem measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left {α : Type u_1} {β : Type u_2} {E : Type u_4} [linear_order α] [measurable_space α] [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : α → E} {a : α} {b : α} {c : E} {la : filter α} {la' : filter α} {lt : filter β} {μ : measure_theory.measure α} {u : β → α} {v : β → α} [topological_space α] [order_topology α] [borel_space α] [FTC_filter a la la'] [measure_theory.locally_finite_measure μ] (hab : interval_integrable f μ a b) (hmeas : measurable_at_filter f la') (hf : filter.tendsto f (la' ⊓ measure_theory.measure.ae μ) (nhds c)) (hu : filter.tendsto u lt la) (hv : filter.tendsto v lt la) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : α) => f x) (v t) b μ - interval_integral (fun (x : α) => f x) (u t) b μ +\n      interval_integral (fun (x : α) => c) (u t) (v t) μ)\n  (fun (t : β) => interval_integral (fun (x : α) => 1) (u t) (v t) μ) lt := sorry\n\n/-!\n### Fundamental theorem of calculus-1 for Lebesgue measure\n\nIn this section we restate theorems from the previous section for Lebesgue measure.\nIn particular, we prove that `∫ x in u..v, f x` is strictly differentiable in `(u, v)`\nat `(a, b)` provided that `f` is integrable on `a..b` and is continuous at `a` and `b`.\n-/\n\n/-!\n#### Auxiliary `is_o` statements\n\nIn this section we prove several lemmas that can be interpreted as strict differentiability of\n`(u, v) ↦ ∫ x in u..v, f x ∂μ` in `u` and/or `v` at a filter. The statements use `is_o` because\nwe have no definition of `has_strict_(f)deriv_at_filter` in the library.\n-/\n\n/-- Fundamental theorem of calculus-1, local version. If `f` has a finite limit `c` almost surely at\n`l'`, where `(l, l')` is an `FTC_filter` pair around `a`, then\n`∫ x in u..v, f x ∂μ = (v - u) • c + o (v - u)` as both `u` and `v` tend to `l`. -/\ntheorem integral_sub_linear_is_o_of_tendsto_ae {β : Type u_2} {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {l : filter ℝ} {l' : filter ℝ} {lt : filter β} {a : ℝ} [FTC_filter a l l'] (hfm : measurable_at_filter f l') (hf : filter.tendsto f (l' ⊓ measure_theory.measure.ae volume) (nhds c)) {u : β → ℝ} {v : β → ℝ} (hu : filter.tendsto u lt l) (hv : filter.tendsto v lt l) : asymptotics.is_o (fun (t : β) => interval_integral (fun (x : ℝ) => f x) (u t) (v t) volume - (v t - u t) • c) (v - u) lt := sorry\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair around\n`a`, and `(lb, lb')` is an `FTC_filter` pair around `b`, and `f` has finite limits `ca` and `cb`\nalmost surely at `la'` and `lb'`, respectively, then\n`(∫ x in va..vb, f x) - ∫ x in ua..ub, f x = (vb - ub) • cb - (va - ua) • ca +\n  o(∥va - ua∥ + ∥vb - ub∥)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_fderiv_at_filter` if we had this\ndefinition. -/\ntheorem integral_sub_integral_sub_linear_is_o_of_tendsto_ae {β : Type u_2} {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {la : filter ℝ} {la' : filter ℝ} {lb : filter ℝ} {lb' : filter ℝ} {lt : filter β} {a : ℝ} {b : ℝ} {ua : β → ℝ} {ub : β → ℝ} {va : β → ℝ} {vb : β → ℝ} [FTC_filter a la la'] [FTC_filter b lb lb'] (hab : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f la') (hmeas_b : measurable_at_filter f lb') (ha_lim : filter.tendsto f (la' ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb_lim : filter.tendsto f (lb' ⊓ measure_theory.measure.ae volume) (nhds cb)) (hua : filter.tendsto ua lt la) (hva : filter.tendsto va lt la) (hub : filter.tendsto ub lt lb) (hvb : filter.tendsto vb lt lb) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : ℝ) => f x) (va t) (vb t) volume -\n        interval_integral (fun (x : ℝ) => f x) (ua t) (ub t) volume -\n      ((vb t - ub t) • cb - (va t - ua t) • ca))\n  (fun (t : β) => norm (va t - ua t) + norm (vb t - ub t)) lt := sorry\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(lb, lb')` is an `FTC_filter` pair\naround `b`, and `f` has a finite limit `c` almost surely at `lb'`, then\n`(∫ x in a..v, f x) - ∫ x in a..u, f x = (v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\ntheorem integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right {β : Type u_2} {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {lb : filter ℝ} {lb' : filter ℝ} {lt : filter β} {a : ℝ} {b : ℝ} {u : β → ℝ} {v : β → ℝ} [FTC_filter b lb lb'] (hab : interval_integrable f volume a b) (hmeas : measurable_at_filter f lb') (hf : filter.tendsto f (lb' ⊓ measure_theory.measure.ae volume) (nhds c)) (hu : filter.tendsto u lt lb) (hv : filter.tendsto v lt lb) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : ℝ) => f x) a (v t) volume - interval_integral (fun (x : ℝ) => f x) a (u t) volume -\n      (v t - u t) • c)\n  (v - u) lt := sorry\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair\naround `a`, and `f` has a finite limit `c` almost surely at `la'`, then\n`(∫ x in v..b, f x) - ∫ x in u..b, f x = -(v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `la`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\ntheorem integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left {β : Type u_2} {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {la : filter ℝ} {la' : filter ℝ} {lt : filter β} {a : ℝ} {b : ℝ} {u : β → ℝ} {v : β → ℝ} [FTC_filter a la la'] (hab : interval_integrable f volume a b) (hmeas : measurable_at_filter f la') (hf : filter.tendsto f (la' ⊓ measure_theory.measure.ae volume) (nhds c)) (hu : filter.tendsto u lt la) (hv : filter.tendsto v lt la) : asymptotics.is_o\n  (fun (t : β) =>\n    interval_integral (fun (x : ℝ) => f x) (v t) b volume - interval_integral (fun (x : ℝ) => f x) (u t) b volume +\n      (v t - u t) • c)\n  (v - u) lt := sorry\n\n/-!\n#### Strict differentiability\n\nIn this section we prove that for a measurable function `f` integrable on `a..b`,\n\n* `integral_has_strict_fderiv_at_of_tendsto_ae`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability\n  provided that `f` tends to `ca` and `cb` almost surely as `x` tendsto to `a` and `b`,\n  respectively;\n\n* `integral_has_strict_fderiv_at`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • f b - u • f a` at `(a, b)` in the sense of strict differentiability\n  provided that `f` is continuous at `a` and `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_right`: the function `u ↦ ∫ x in a..u, f x` has\n  derivative `c` at `b` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `b`;\n\n* `integral_has_strict_deriv_at_right`: the function `u ↦ ∫ x in a..u, f x` has derivative `f b` at\n  `b` in the sense of strict differentiability provided that `f` is continuous at `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_left`: the function `u ↦ ∫ x in u..b, f x` has\n  derivative `-c` at `a` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `a`;\n\n* `integral_has_strict_deriv_at_left`: the function `u ↦ ∫ x in u..b, f x` has derivative `-f a` at\n  `a` in the sense of strict differentiability provided that `f` is continuous at `a`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`\nin the sense of strict differentiability. -/\ntheorem integral_has_strict_fderiv_at_of_tendsto_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds cb)) : has_strict_fderiv_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) cb -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) ca)\n  (a, b) := sorry\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)` in the sense of strict differentiability. -/\ntheorem integral_has_strict_fderiv_at {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : continuous_at f a) (hb : continuous_at f b) : has_strict_fderiv_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) (f b) -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) (f a))\n  (a, b) :=\n  integral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b (filter.tendsto.mono_left ha inf_le_left)\n    (filter.tendsto.mono_left hb inf_le_left)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b` in the sense\nof strict differentiability. -/\ntheorem integral_has_strict_deriv_at_of_tendsto_ae_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds c)) : has_strict_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) c b :=\n  integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb continuous_at_snd continuous_at_fst\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict\ndifferentiability. -/\ntheorem integral_has_strict_deriv_at_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : continuous_at f b) : has_strict_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) (f b) b :=\n  integral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas (filter.tendsto.mono_left hb inf_le_left)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a` in the sense\nof strict differentiability. -/\ntheorem integral_has_strict_deriv_at_of_tendsto_ae_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (ha : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds c)) : has_strict_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-c) a := sorry\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a` in the sense of strict\ndifferentiability. -/\ntheorem integral_has_strict_deriv_at_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (ha : continuous_at f a) : has_strict_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-f a) a := sorry\n\n/-!\n#### Fréchet differentiability\n\nIn this subsection we restate results from the previous subsection in terms of `has_fderiv_at`,\n`has_deriv_at`, `fderiv`, and `deriv`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`. -/\ntheorem integral_has_fderiv_at_of_tendsto_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds cb)) : has_fderiv_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) cb -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) ca)\n  (a, b) :=\n  has_strict_fderiv_at.has_fderiv_at (integral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)`. -/\ntheorem integral_has_fderiv_at {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : continuous_at f a) (hb : continuous_at f b) : has_fderiv_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) (f b) -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) (f a))\n  (a, b) :=\n  has_strict_fderiv_at.has_fderiv_at (integral_has_strict_fderiv_at hf hmeas_a hmeas_b ha hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `fderiv`\nderivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦ v • cb - u • ca`. -/\ntheorem fderiv_integral_of_tendsto_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds cb)) : fderiv ℝ (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume) (a, b) =\n  continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) cb -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) ca :=\n  has_fderiv_at.fderiv (integral_has_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `fderiv` derivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦\nv • cb - u • ca`. -/\ntheorem fderiv_integral {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f (nhds a)) (hmeas_b : measurable_at_filter f (nhds b)) (ha : continuous_at f a) (hb : continuous_at f b) : fderiv ℝ (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume) (a, b) =\n  continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) (f b) -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) (f a) :=\n  has_fderiv_at.fderiv (integral_has_fderiv_at hf hmeas_a hmeas_b ha hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b`. -/\ntheorem integral_has_deriv_at_of_tendsto_ae_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds c)) : has_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) c b :=\n  has_strict_deriv_at.has_deriv_at (integral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b`. -/\ntheorem integral_has_deriv_at_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : continuous_at f b) : has_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) (f b) b :=\n  has_strict_deriv_at.has_deriv_at (integral_has_strict_deriv_at_right hf hmeas hb)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\ntheorem deriv_integral_of_tendsto_ae_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : filter.tendsto f (nhds b ⊓ measure_theory.measure.ae volume) (nhds c)) : deriv (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) b = c :=\n  has_deriv_at.deriv (integral_has_deriv_at_of_tendsto_ae_right hf hmeas hb)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\ntheorem deriv_integral_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds b)) (hb : continuous_at f b) : deriv (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) b = f b :=\n  has_deriv_at.deriv (integral_has_deriv_at_right hf hmeas hb)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a`. -/\ntheorem integral_has_deriv_at_of_tendsto_ae_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (ha : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds c)) : has_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-c) a :=\n  has_strict_deriv_at.has_deriv_at (integral_has_strict_deriv_at_of_tendsto_ae_left hf hmeas ha)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a`. -/\ntheorem integral_has_deriv_at_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (ha : continuous_at f a) : has_deriv_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-f a) a :=\n  has_strict_deriv_at.has_deriv_at (integral_has_strict_deriv_at_left hf hmeas ha)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\ntheorem deriv_integral_of_tendsto_ae_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (hb : filter.tendsto f (nhds a ⊓ measure_theory.measure.ae volume) (nhds c)) : deriv (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) a = -c :=\n  has_deriv_at.deriv (integral_has_deriv_at_of_tendsto_ae_left hf hmeas hb)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\ntheorem deriv_integral_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas : measurable_at_filter f (nhds a)) (hb : continuous_at f a) : deriv (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) a = -f a :=\n  has_deriv_at.deriv (integral_has_deriv_at_left hf hmeas hb)\n\n/-!\n#### One-sided derivatives\n-/\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • cb - u • ca` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to `ca`\nand `cb` almost surely at the filters `la` and `lb` from the following table.\n\n| `s`     | `la`         | `t`     | `lb`         |\n| ------- | ----         | ---     | ----         |\n| `Iic a` | `𝓝[Iic a] a` | `Iic b` | `𝓝[Iic b] b` |\n| `Ici a` | `𝓝[Ioi a] a` | `Ici b` | `𝓝[Ioi b] b` |\n| `{a}`   | `⊥`          | `{b}`   | `⊥`          |\n| `univ`  | `𝓝 a`        | `univ`  | `𝓝 b`        |\n-/\ntheorem integral_has_fderiv_within_at_of_tendsto_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {la : filter ℝ} {lb : filter ℝ} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) la] [FTC_filter b (nhds_within b t) lb] (hmeas_a : measurable_at_filter f la) (hmeas_b : measurable_at_filter f lb) (ha : filter.tendsto f (la ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb : filter.tendsto f (lb ⊓ measure_theory.measure.ae volume) (nhds cb)) : has_fderiv_within_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) cb -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) ca)\n  (set.prod s t) (a, b) := sorry\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • f b - u • f a` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to\n`f a` and `f b` at the filters `la` and `lb` from the following table. In most cases this assumption\nis definitionally equal `continuous_at f _` or `continuous_within_at f _ _`.\n\n| `s`     | `la`         | `t`     | `lb`         |\n| ------- | ----         | ---     | ----         |\n| `Iic a` | `𝓝[Iic a] a` | `Iic b` | `𝓝[Iic b] b` |\n| `Ici a` | `𝓝[Ioi a] a` | `Ici b` | `𝓝[Ioi b] b` |\n| `{a}`   | `⊥`          | `{b}`   | `⊥`          |\n| `univ`  | `𝓝 a`        | `univ`  | `𝓝 b`        |\n-/\ntheorem integral_has_fderiv_within_at {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {la : filter ℝ} {lb : filter ℝ} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f la) (hmeas_b : measurable_at_filter f lb) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) la] [FTC_filter b (nhds_within b t) lb] (ha : filter.tendsto f la (nhds (f a))) (hb : filter.tendsto f lb (nhds (f b))) : has_fderiv_within_at (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n  (continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) (f b) -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) (f a))\n  (set.prod s t) (a, b) :=\n  integral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b (filter.tendsto.mono_left ha inf_le_left)\n    (filter.tendsto.mono_left hb inf_le_left)\n\n/-- An auxiliary tactic closing goals `unique_diff_within_at ℝ s a` where\n`s ∈ {Iic a, Ici a, univ}`. -/\n/-- Let `f` be a measurable function integrable on `a..b`. Choose `s ∈ {Iic a, Ici a, univ}`\nand `t ∈ {Iic b, Ici b, univ}`. Suppose that `f` tends to `ca` and `cb` almost surely at the filters\n`la` and `lb` from the table below. Then `fderiv_within ℝ (λ p, ∫ x in p.1..p.2, f x) (s.prod t)`\nis equal to `(u, v) ↦ u • cb - v • ca`.\n\n| `s`     | `la`         | `t`     | `lb`         |\n| ------- | ----         | ---     | ----         |\n| `Iic a` | `𝓝[Iic a] a` | `Iic b` | `𝓝[Iic b] b` |\n| `Ici a` | `𝓝[Ioi a] a` | `Ici b` | `𝓝[Ioi b] b` |\n| `univ`  | `𝓝 a`        | `univ`  | `𝓝 b`        |\n\n-/\ntheorem fderiv_within_integral_of_tendsto_ae {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {ca : E} {cb : E} {la : filter ℝ} {lb : filter ℝ} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) (hmeas_a : measurable_at_filter f la) (hmeas_b : measurable_at_filter f lb) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) la] [FTC_filter b (nhds_within b t) lb] (ha : filter.tendsto f (la ⊓ measure_theory.measure.ae volume) (nhds ca)) (hb : filter.tendsto f (lb ⊓ measure_theory.measure.ae volume) (nhds cb)) (hs : autoParam (unique_diff_within_at ℝ s a)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) (ht : autoParam (unique_diff_within_at ℝ t b)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) : fderiv_within ℝ (fun (p : ℝ × ℝ) => interval_integral (fun (x : ℝ) => f x) (prod.fst p) (prod.snd p) volume)\n    (set.prod s t) (a, b) =\n  continuous_linear_map.smul_right (continuous_linear_map.snd ℝ ℝ ℝ) cb -\n    continuous_linear_map.smul_right (continuous_linear_map.fst ℝ ℝ ℝ) ca :=\n  has_fderiv_within_at.fderiv_within (integral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb)\n    (unique_diff_within_at.prod hs ht)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left,\nthen `u ↦ ∫ x in a..u, f x` has right (resp., left) derivative `c` at `b`. -/\ntheorem integral_has_deriv_within_at_of_tendsto_ae_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter b (nhds_within b s) (nhds_within b t)] (hmeas : measurable_at_filter f (nhds_within b t)) (hb : filter.tendsto f (nhds_within b t ⊓ measure_theory.measure.ae volume) (nhds c)) : has_deriv_within_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) c s b :=\n  integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb\n    (filter.tendsto.mono_right filter.tendsto_const_pure FTC_filter.pure_le) filter.tendsto_id\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `b`, then `u ↦ ∫ x in a..u, f x` has left (resp., right)\nderivative `f b` at `b`. -/\ntheorem integral_has_deriv_within_at_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter b (nhds_within b s) (nhds_within b t)] (hmeas : measurable_at_filter f (nhds_within b t)) (hb : continuous_within_at f t b) : has_deriv_within_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) (f b) s b :=\n  integral_has_deriv_within_at_of_tendsto_ae_right hf hmeas (filter.tendsto.mono_left hb inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\ntheorem deriv_within_integral_of_tendsto_ae_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter b (nhds_within b s) (nhds_within b t)] (hmeas : measurable_at_filter f (nhds_within b t)) (hb : filter.tendsto f (nhds_within b t ⊓ measure_theory.measure.ae volume) (nhds c)) (hs : autoParam (unique_diff_within_at ℝ s b)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) : deriv_within (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) s b = c :=\n  has_deriv_within_at.deriv_within (integral_has_deriv_within_at_of_tendsto_ae_right hf hmeas hb) hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `b`, then the right (resp., left) derivative of\n`u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\ntheorem deriv_within_integral_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter b (nhds_within b s) (nhds_within b t)] (hmeas : measurable_at_filter f (nhds_within b t)) (hb : continuous_within_at f t b) (hs : autoParam (unique_diff_within_at ℝ s b)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) : deriv_within (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) s b = f b :=\n  has_deriv_within_at.deriv_within (integral_has_deriv_within_at_right hf hmeas hb) hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left,\nthen `u ↦ ∫ x in u..b, f x` has right (resp., left) derivative `-c` at `a`. -/\ntheorem integral_has_deriv_within_at_of_tendsto_ae_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) (nhds_within a t)] (hmeas : measurable_at_filter f (nhds_within a t)) (ha : filter.tendsto f (nhds_within a t ⊓ measure_theory.measure.ae volume) (nhds c)) : has_deriv_within_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-c) s a := sorry\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `a`, then `u ↦ ∫ x in u..b, f x` has left (resp., right)\nderivative `-f a` at `a`. -/\ntheorem integral_has_deriv_within_at_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) (nhds_within a t)] (hmeas : measurable_at_filter f (nhds_within a t)) (ha : continuous_within_at f t a) : has_deriv_within_at (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) (-f a) s a :=\n  integral_has_deriv_within_at_of_tendsto_ae_left hf hmeas (filter.tendsto.mono_left ha inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\ntheorem deriv_within_integral_of_tendsto_ae_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {c : E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) (nhds_within a t)] (hmeas : measurable_at_filter f (nhds_within a t)) (ha : filter.tendsto f (nhds_within a t ⊓ measure_theory.measure.ae volume) (nhds c)) (hs : autoParam (unique_diff_within_at ℝ s a)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) : deriv_within (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) s a = -c :=\n  has_deriv_within_at.deriv_within (integral_has_deriv_within_at_of_tendsto_ae_left hf hmeas ha) hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `a`, then the right (resp., left) derivative of\n`u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\ntheorem deriv_within_integral_left {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hf : interval_integrable f volume a b) {s : set ℝ} {t : set ℝ} [FTC_filter a (nhds_within a s) (nhds_within a t)] (hmeas : measurable_at_filter f (nhds_within a t)) (ha : continuous_within_at f t a) (hs : autoParam (unique_diff_within_at ℝ s a)\n  (Lean.Syntax.ident Lean.SourceInfo.none\n    (String.toSubstring \"Mathlib.interval_integral.unique_diff_within_at_Ici_Iic_univ\")\n    (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"interval_integral\")\n      \"unique_diff_within_at_Ici_Iic_univ\")\n    [])) : deriv_within (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) u b volume) s a = -f a :=\n  has_deriv_within_at.deriv_within (integral_has_deriv_within_at_left hf hmeas ha) hs\n\n/-!\n### Fundamental theorem of calculus, part 2\n\nThis section contains theorems pertaining to FTC-2 for interval integrals. -/\n\n/-- The integral of a continuous function is differentiable on a real set `s`. -/\ntheorem differentiable_on_integral_of_continuous {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {s : set ℝ} (hintg : ∀ (x : ℝ), x ∈ s → interval_integrable f volume a x) (hcont : continuous f) : differentiable_on ℝ (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) s := sorry\n\n/-- The integral of a continuous function is continuous on a real set `s`. This is true even\n  without the assumption of continuity, but a proof of that fact does not yet exist in mathlib. -/\ntheorem continuous_on_integral_of_continuous {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {s : set ℝ} (hintg : ∀ (x : ℝ), x ∈ s → interval_integrable f volume a x) (hcont : continuous f) : continuous_on (fun (u : ℝ) => interval_integral (fun (x : ℝ) => f x) a u volume) s :=\n  differentiable_on.continuous_on (differentiable_on_integral_of_continuous hintg hcont)\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` and has a right\n  derivative at `f' x` for all `x` in `[a, b)`, and `f'` is continuous on `[a, b]`, then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right_of_le {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} {f' : ℝ → E} (hab : a ≤ b) (hcont : continuous_on f (set.Icc a b)) (hderiv : ∀ (x : ℝ), x ∈ set.Ico a b → has_deriv_within_at f (f' x) (set.Ici x) x) (hcont' : continuous_on f' (set.Icc a b)) : interval_integral (fun (y : ℝ) => f' y) a b volume = f b - f a := sorry\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`) and\n  has a right derivative at `f' x` for all `x` in `[a, b)`, and `f'` is continuous on `[a, b]` then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} {f' : ℝ → E} (hcont : continuous_on f (set.interval a b)) (hderiv : ∀ (x : ℝ), x ∈ set.Ico (min a b) (max a b) → has_deriv_within_at f (f' x) (set.Ici x) x) (hcont' : continuous_on f' (set.interval a b)) : interval_integral (fun (y : ℝ) => f' y) a b volume = f b - f a := sorry\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` and has a derivative\n  at `f' x` for all `x` in `(a, b)`, and `f'` is continuous on `[a, b]`, then `∫ y in a..b, f' y`\n  equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at' {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} {f' : ℝ → E} (hcont : continuous_on f (set.interval a b)) (hderiv : ∀ (x : ℝ), x ∈ set.Ioo (min a b) (max a b) → has_deriv_at f (f' x) x) (hcont' : continuous_on f' (set.interval a b)) : interval_integral (fun (y : ℝ) => f' y) a b volume = f b - f a := sorry\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` has a derivative at `f' x` for all `x` in\n  `[a, b]` and `f'` is continuous on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} {f' : ℝ → E} (hderiv : ∀ (x : ℝ), x ∈ set.interval a b → has_deriv_at f (f' x) x) (hcont' : continuous_on f' (set.interval a b)) : interval_integral (fun (y : ℝ) => f' y) a b volume = f b - f a := sorry\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is differentiable at every `x` in `[a, b]` and\n  its derivative is continuous on `[a, b]`, then `∫ y in a..b, deriv f y` equals `f b - f a`. -/\ntheorem integral_deriv_eq_sub {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space ℝ E] [borel_space E] {f : ℝ → E} {a : ℝ} {b : ℝ} (hderiv : ∀ (x : ℝ), x ∈ set.interval a b → differentiable_at ℝ f x) (hcont' : continuous_on (deriv f) (set.interval a b)) : interval_integral (fun (y : ℝ) => deriv f y) a b volume = f b - f a :=\n  integral_eq_sub_of_has_deriv_at\n    (fun (x : ℝ) (hx : x ∈ set.interval a b) => differentiable_at.has_deriv_at (hderiv x hx)) hcont'\n\n/-!\n### Integration by parts\n-/\n\ntheorem integral_deriv_mul_eq_sub {a : ℝ} {b : ℝ} {u : ℝ → ℝ} {v : ℝ → ℝ} {u' : ℝ → ℝ} {v' : ℝ → ℝ} (hu : ∀ (x : ℝ), x ∈ set.interval a b → has_deriv_at u (u' x) x) (hv : ∀ (x : ℝ), x ∈ set.interval a b → has_deriv_at v (v' x) x) (hcu' : continuous_on u' (set.interval a b)) (hcv' : continuous_on v' (set.interval a b)) : interval_integral (fun (x : ℝ) => u' x * v x + u x * v' x) a b volume = u b * v b - u a * v a := sorry\n\ntheorem integral_mul_deriv_eq_deriv_mul {a : ℝ} {b : ℝ} {u : ℝ → ℝ} {v : ℝ → ℝ} {u' : ℝ → ℝ} {v' : ℝ → ℝ} (hu : ∀ (x : ℝ), x ∈ set.interval a b → has_deriv_at u (u' x) x) (hv : ∀ (x : ℝ), x ∈ set.interval a b → has_deriv_at v (v' x) x) (hcu' : continuous_on u' (set.interval a b)) (hcv' : continuous_on v' (set.interval a b)) : interval_integral (fun (x : ℝ) => u x * v' x) a b volume =\n  u b * v b - u a * v a - interval_integral (fun (x : ℝ) => v x * u' x) a b volume := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/interval_integral.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3123489034390306}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module init.meta.congr_lemma\n! leanprover-community/mathlib commit 49194b4458c682842a21a1f8675e174b440af055\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Meta.Tactic\nimport Leanbin.Init.Meta.Format\nimport Leanbin.Init.Function\n\n/--\nThis is a kind attached to an argument of a congruence lemma that tells the simplifier how to fill it in.\n- `fixed`: It is a parameter for the congruence lemma, the parameter occurs in the left and right hand sides.\n  For example the α in the congruence generated from `f: Π {α : Type} α → α`.\n- `fixed_no_param`: It is not a parameter for the congruence lemma, the lemma was specialized for this parameter.\n  This only happens if the parameter is a subsingleton/proposition, and other parameters depend on it.\n- `eq`: The lemma contains three parameters for this kind of argument `a_i`, `b_i` and `(eq_i : a_i = b_i)`.\n  `a_i` and `b_i` represent the left and right hand sides, and `eq_i` is a proof for their equality.\n  For example the second argument in `f: Π {α : Type}, α → α`.\n- `cast`: corresponds to arguments that are subsingletons/propositions.\n  For example the `p` in the congruence generated from `f : Π (x y : ℕ) (p: x < y), ℕ`.\n- `heq` The lemma contains three parameters for this kind of argument `a_i`, `b_i` and `(eq_i : a_i == b_i)`.\n   `a_i` and `b_i` represent the left and right hand sides, and eq_i is a proof for their heterogeneous equality.\n-/\ninductive CongrArgKind\n  | fixed\n  | fixed_no_param\n  | Eq\n  | cast\n  | HEq\n  | subsingleton_inst\n#align congr_arg_kind CongrArgKind\n\nnamespace CongrArgKind\n\ndef toString : CongrArgKind → String\n  | fixed => \"fixed\"\n  | fixed_no_param => \"fixed_no_param\"\n  | Eq => \"eq\"\n  | cast => \"cast\"\n  | HEq => \"heq\"\n  | subsingleton_inst => \"subsingleton_inst\"\n#align congr_arg_kind.to_string CongrArgKind.toString\n\ninstance : Repr CongrArgKind :=\n  ⟨toString⟩\n\nunsafe instance : has_to_format CongrArgKind :=\n  ⟨fun x => toString x⟩\n\nend CongrArgKind\n\n/--\nA congruence lemma is a proof that two terms are equal using a congruence proof generated by `mk_congr_lemma_simp` and friends.\nSee the docstring for `mk_congr_lemma_simp` and `congr_arg_kind` for more information.\nThe conclusion is prepended by a set of arguments. `arg_kinds` gives a suggestion of how that argument should be filled in using a simplifier.\n  -/\nunsafe structure congr_lemma where\n  type : expr\n  proof : expr\n  arg_kinds : List CongrArgKind\n#align congr_lemma congr_lemma\n\nnamespace Tactic\n\n/-- `mk_congr_lemma_simp f nargs md`\ncreates a congruence lemma for the simplifier for the given function argument `f`.\nIf `nargs` is not none, then it tries to create a lemma for an application of arity `nargs`.\nIf `nargs` is none then the number of arguments will be guessed from the type signature of `f`.\n\nThat is, given `f : Π {α β γ δ : Type}, α → β → γ → δ` and `nargs = some 6`, we get a congruence lemma:\n``` lean\n{ type := ∀ (α β γ δ : Type), ∀ (a₁ a₂ : α), a₁ = a₂ → ∀ (b₁ b₂ : β), b₁ = b₂ → f a₁ b₁ = f a₂ b₂\n, proof := ...\n, arg_kinds := [fixed, fixed, fixed, fixed, eq,eq]\n}\n```\nSee the docstrings for the cases of `congr_arg_kind` for more detail on how `arg_kinds` are chosen.\nThe system chooses the `arg_kinds` depending on what the other arguments depend on and whether the arguments have subsingleton types.\n\nNote that the number of arguments that `proof` takes can be inferred from `arg_kinds`: `arg_kinds.sum (fixed,cast ↦ 1 | eq,heq ↦ 3 | fixed_no_param ↦ 0)`.\n\nFrom `congr_lemma.cpp`:\n> Create a congruence lemma that is useful for the simplifier.\n> In this kind of lemma, if the i-th argument is a Cast argument, then the lemma\n> contains an input a_i representing the i-th argument in the left-hand-side, and\n> it appears with a cast (e.g., eq.drec ... a_i ...) in the right-hand-side.\n> The idea is that the right-hand-side of this lemma \"tells\" the simplifier\n> how the resulting term looks like.\n-/\nunsafe axiom mk_congr_lemma_simp (f : expr) (nargs : Option Nat := none) (md := semireducible) :\n    tactic congr_lemma\n#align tactic.mk_congr_lemma_simp tactic.mk_congr_lemma_simp\n\n/-- Create a specialized theorem using (a prefix of) the arguments of the given application.\n\nAn example of usage can be found in `tests/lean/simp_subsingleton.lean`.\nFor more information on specialization see the comment in the method body for `get_specialization_prefix_size` in `src/library/fun_info.cpp`.\n -/\nunsafe axiom mk_specialized_congr_lemma_simp (h : expr) (md : Transparency := semireducible) :\n    tactic congr_lemma\n#align tactic.mk_specialized_congr_lemma_simp tactic.mk_specialized_congr_lemma_simp\n\n/-- Similar to `mk_congr_lemma_simp`, this will make a `congr_lemma` object.\nThe difference is that for each `congr_arg_kind.cast` argument, two proof arguments are generated.\n\nConsider some function `f : Π (x : ℕ) (p : x < 4), ℕ`.\n- `mk_congr_simp` will produce a congruence lemma with type `∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4), f x p = f x_1 _`.\n- `mk_congr` will produce a congruence lemma with type `∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4) (p_1 : x_1 < 4), f x p = f x_1 p_1`.\n\nFrom `congr_lemma.cpp`:\n> Create a congruence lemma for the congruence closure module.\n> In this kind of lemma, if the i-th argument is a Cast argument, then the lemma\n> contains two inputs a_i and b_i representing the i-th argument in the left-hand-side and\n> right-hand-side.\n> This lemma is based on the congruence lemma for the simplifier.\n> It uses subsinglenton elimination to show that the congr-simp lemma right-hand-side\n> is equal to the right-hand-side of this lemma.\n -/\nunsafe axiom mk_congr_lemma (h : expr) (nargs : Option Nat := none) (md := semireducible) :\n    tactic congr_lemma\n#align tactic.mk_congr_lemma tactic.mk_congr_lemma\n\n/-- Create a specialized theorem using (a prefix of) the arguments of the given application.\n\nFor more information on specialization see the comment in the method body for `get_specialization_prefix_size` in `src/library/fun_info.cpp`.\n-/\nunsafe axiom mk_specialized_congr_lemma (h : expr) (md := semireducible) : tactic congr_lemma\n#align tactic.mk_specialized_congr_lemma tactic.mk_specialized_congr_lemma\n\n/-- Make a congruence lemma using hetrogeneous equality `heq` instead of `eq`.\nFor example `mk_hcongr_lemma (f : Π (α : ℕ → Type) (n:ℕ) (b:α n), ℕ` )` will make\n\n``` lean\n{ type := ∀ α α', α = α' → ∀ n n', n = n' → ∀ (b : α n) (b' : α' n'), b == b' → f α n b == f α' n' b'\n, proof := ...\n, arg_kinds := [eq,eq,heq]\n}\n```\n\n(Using merely `mk_congr_lemma` instead will produce `[fixed,fixed,eq]` instaed.)\n-/\nunsafe axiom mk_hcongr_lemma (h : expr) (nargs : Option Nat := none) (md := semireducible) :\n    tactic congr_lemma\n#align tactic.mk_hcongr_lemma tactic.mk_hcongr_lemma\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Meta/CongrLemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.31218524553501986}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module init.meta.declaration\n! leanprover-community/mathlib commit cd2d62882160de729619fc68c92a57b5e9d0e968\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Meta.Expr\nimport Leanbin.Init.Meta.Name\nimport Leanbin.Init.Meta.Task\n\n/-- Reducibility hints are used in the convertibility checker.\nWhen trying to solve a constraint such a\n\n    (f ...) =?= (g ...)\n\nwhere `f` and `g` are definitions, the checker has to decide which one will be unfolded.\n* If      `f` (`g`) is opaque,     then `g` (`f`) is unfolded if it is also not marked as opaque,\n* Else if `f` (`g`) is abbrev,     then `f` (`g`) is unfolded if `g` (`f`) is also not marked as abbrev,\n* Else if `f` and `g` are regular, then we unfold the one with the biggest definitional height.\n* Otherwise both are unfolded.\n\nThe arguments of the `regular` constructor are: the definitional height and the flag `self_opt`.\n\nThe definitional height is by default computed by the kernel. It only takes into account\nother regular definitions used in a definition. When creating declarations using meta-programming,\nwe can specify the definitional depth manually.\n\nFor definitions marked as regular, we also have a hint for constraints such as\n\n    (f a) =?= (f b)\n\nif `self_opt = tt`, then checker will first try to solve `a =?= b`, only if it fails,\nit unfolds `f`.\n\nRemark: the hint only affects performance. None of the hints prevent the kernel from unfolding a\ndeclaration during type checking.\n\nRemark: the reducibility_hints are not related to the attributes: reducible/irrelevance/semireducible.\nThese attributes are used by the elaborator. The reducibility_hints are used by the kernel (and elaborator).\nMoreover, the reducibility_hints cannot be changed after a declaration is added to the kernel.\n-/\ninductive ReducibilityHints\n  | opaque\n  | abbrev\n  | regular (height : Nat) (self_opt : Bool)\n#align reducibility_hints ReducibilityHints\n\n/-- Reflect a C++ declaration object. The VM replaces it with the C++ implementation. -/\nunsafe inductive declaration-- definition\n\n  |\n  defn (n : Name) (univs : List Name) (type : expr) (value : expr) (red : ReducibilityHints)\n    (is_trusted : Bool)-- theorem (remark: theorems are always trusted)\n\n  | thm (n : Name) (univs : List Name) (type : expr) (value : task expr)-- constant assumption\n\n  |\n  cnst (n : Name) (univs : List Name) (type : expr)\n    (is_trusted : Bool)-- axiom (remark: axioms are always trusted)\n\n  | ax (n : Name) (univs : List Name) (type : expr)\n#align declaration declaration\n\nopen Declaration\n\nunsafe def mk_definition (n : Name) (ls : List Name) (v : expr) (e : expr) : declaration :=\n  defn n ls v e (ReducibilityHints.regular 1 true) true\n#align mk_definition mk_definition\n\nnamespace Declaration\n\nunsafe def to_name : declaration → Name\n  | defn n _ _ _ _ _ => n\n  | thm n _ _ _ => n\n  | cnst n _ _ _ => n\n  | ax n _ _ => n\n#align declaration.to_name declaration.to_name\n\nunsafe def univ_params : declaration → List Name\n  | defn _ ls _ _ _ _ => ls\n  | thm _ ls _ _ => ls\n  | cnst _ ls _ _ => ls\n  | ax _ ls _ => ls\n#align declaration.univ_params declaration.univ_params\n\nunsafe def type : declaration → expr\n  | defn _ _ t _ _ _ => t\n  | thm _ _ t _ => t\n  | cnst _ _ t _ => t\n  | ax _ _ t => t\n#align declaration.type declaration.type\n\nunsafe def value : declaration → expr\n  | defn _ _ _ v _ _ => v\n  | thm _ _ _ v => v.get\n  | _ => default\n#align declaration.value declaration.value\n\nunsafe def value_task : declaration → task expr\n  | defn _ _ _ v _ _ => task.pure v\n  | thm _ _ _ v => v\n  | _ => task.pure default\n#align declaration.value_task declaration.value_task\n\nunsafe def is_trusted : declaration → Bool\n  | defn _ _ _ _ _ t => t\n  | cnst _ _ _ t => t\n  | _ => true\n#align declaration.is_trusted declaration.is_trusted\n\nunsafe def update_type : declaration → expr → declaration\n  | defn n ls t v h tr, new_t => defn n ls new_t v h tr\n  | thm n ls t v, new_t => thm n ls new_t v\n  | cnst n ls t tr, new_t => cnst n ls new_t tr\n  | ax n ls t, new_t => ax n ls new_t\n#align declaration.update_type declaration.update_type\n\nunsafe def update_name : declaration → Name → declaration\n  | defn n ls t v h tr, new_n => defn new_n ls t v h tr\n  | thm n ls t v, new_n => thm new_n ls t v\n  | cnst n ls t tr, new_n => cnst new_n ls t tr\n  | ax n ls t, new_n => ax new_n ls t\n#align declaration.update_name declaration.update_name\n\nunsafe def update_value : declaration → expr → declaration\n  | defn n ls t v h tr, new_v => defn n ls t new_v h tr\n  | thm n ls t v, new_v => thm n ls t (task.pure new_v)\n  | d, new_v => d\n#align declaration.update_value declaration.update_value\n\nunsafe def update_value_task : declaration → task expr → declaration\n  | defn n ls t v h tr, new_v => defn n ls t new_v.get h tr\n  | thm n ls t v, new_v => thm n ls t new_v\n  | d, new_v => d\n#align declaration.update_value_task declaration.update_value_task\n\nunsafe def map_value : declaration → (expr → expr) → declaration\n  | defn n ls t v h tr, f => defn n ls t (f v) h tr\n  | thm n ls t v, f => thm n ls t (task.map f v)\n  | d, f => d\n#align declaration.map_value declaration.map_value\n\nunsafe def to_definition : declaration → declaration\n  | cnst n ls t tr => defn n ls t default ReducibilityHints.abbrev tr\n  | ax n ls t => thm n ls t (task.pure default)\n  | d => d\n#align declaration.to_definition declaration.to_definition\n\nunsafe def is_definition : declaration → Bool\n  | defn _ _ _ _ _ _ => true\n  | _ => false\n#align declaration.is_definition declaration.is_definition\n\n/-- Instantiate a universe polymorphic declaration type with the given universes. -/\nunsafe axiom instantiate_type_univ_params : declaration → List level → Option expr\n#align declaration.instantiate_type_univ_params declaration.instantiate_type_univ_params\n\n/-- Instantiate a universe polymorphic declaration value with the given universes. -/\nunsafe axiom instantiate_value_univ_params : declaration → List level → Option expr\n#align declaration.instantiate_value_univ_params declaration.instantiate_value_univ_params\n\nend Declaration\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Meta/Declaration.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3121852455350198}}
{"text": "/-\n  Extension of a sheaf of rings on the basis to a sheaf of rings on the whole space.\n\n  https://stacks.math.columbia.edu/tag/009M\n  https://stacks.math.columbia.edu/tag/009N\n-/\n\nimport topology.opens\nimport sheaves.stalk_of_rings\nimport sheaves.stalk_of_rings_on_standard_basis\nimport sheaves.presheaf_of_rings_on_basis\nimport sheaves.presheaf_of_rings_extension\nimport sheaves.sheaf_on_basis\nimport sheaves.sheaf_on_standard_basis\nimport sheaves.sheaf_of_rings\n\nopen topological_space classical\n\nnoncomputable theory\n\nuniverse u\n\nsection presheaf_of_rings_extension\n\nvariables {α : Type u} [T : topological_space α] \nvariables {B : set (opens α)} {HB : opens.is_basis B}\nvariables (Bstd : opens.univ ∈ B ∧ ∀ {U V}, U ∈ B → V ∈ B → U ∩ V ∈ B)\n\ninclude Bstd\n\ntheorem extension_is_sheaf_of_rings \n(F : presheaf_of_rings_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n: is_sheaf_of_rings (F ᵣₑₓₜ Bstd) := \nbegin\n  show is_sheaf (F ᵣₑₓₜ Bstd).to_presheaf,\n  constructor,\n  { intros U OC s t Hres,\n    apply subtype.eq,\n    apply funext,\n    intros x,\n    apply funext,\n    intros HxU,\n    rw OC.Hcov.symm at HxU,\n    rcases HxU with ⟨Uj1, ⟨⟨⟨Uj2, OUj⟩, ⟨⟨j, HUj⟩, Heq⟩⟩, HxUj⟩⟩,\n    rcases Heq, rcases Heq,\n    have Hstj := congr_fun (subtype.mk_eq_mk.1 (Hres j)),\n    have HxUj1 : x ∈ OC.Uis j := HUj.symm ▸ HxUj,\n    have Hstjx := congr_fun (Hstj x) HxUj1,\n    exact Hstjx, },\n  { intros U OC s Hsec,\n    existsi (global_section (F.to_presheaf_on_basis) U OC s Hsec),\n    intros i,\n    apply subtype.eq,\n    apply funext,\n    intros x,\n    apply funext,\n    intros HxUi,\n    have HxU : x ∈ U := OC.Hcov ▸ (opens_supr_subset OC.Uis i) HxUi,\n    let HyUi := λ t, ∃ (H : t ∈ set.range OC.Uis), x ∈ t,\n    dunfold presheaf_of_rings_extension; dsimp,\n    dunfold global_section; dsimp,\n    -- Same process of dealing with subtype.rec.\n    let HyUi := λ t, ∃ (H : t ∈ subtype.val '' set.range OC.Uis), x ∈ t,\n    rcases (classical.indefinite_description HyUi _) with ⟨S, HS⟩; dsimp,\n    let HyS := λ H : S ∈ subtype.val '' set.range OC.Uis, x ∈ S,\n    rcases (classical.indefinite_description HyS HS) with ⟨HSUiR, HySUiR⟩; dsimp,\n    let HOUksub := λ t : subtype is_open, t ∈ set.range (OC.Uis) ∧ t.val = S,\n    rcases (classical.indefinite_description HOUksub _) with ⟨OUl, ⟨HOUl, HOUleq⟩⟩; dsimp,\n    let HSUi := λ i, OC.Uis i = OUl,\n    cases (classical.indefinite_description HSUi _) with l HSUil; dsimp,\n    -- Now we just need to apply Hsec in the right way.\n    dunfold presheaf_of_rings_extension at Hsec,\n    dunfold res_to_inter_left at Hsec,\n    dunfold res_to_inter_right at Hsec,\n    dsimp at Hsec,\n    replace Hsec := Hsec i l,\n    rw subtype.ext at Hsec,\n    dsimp at Hsec,\n    replace Hsec := congr_fun Hsec x,\n    dsimp at Hsec,\n    replace Hsec := congr_fun Hsec,\n    have HxOUk : x ∈ OUl.val := HOUleq.symm ▸ HySUiR,\n    have HxUl : x ∈ OC.Uis l := HSUil.symm ▸ HxOUk,\n    exact (Hsec ⟨HxUi, HxUl⟩).symm, },\n  end\n\nsection extension_coincides\n\n-- The extension is done in a way that F(U) ≅ Fext(U).\n\n-- The map ψ : F(U) → Π x ∈ U, Fx\n\ndef to_stalk_product (F : presheaf_on_basis α HB) {U : opens α} (BU : U ∈ B)\n: F.F BU → Π (x ∈ U), stalk_on_basis F x :=\nλ s x Hx, ⟦{U := U, BU := BU, Hx := Hx, s := s}⟧\n\nlemma to_stalk_product.injective \n(F : presheaf_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F)\n{U : opens α} (BU : U ∈ B)\n: function.injective (to_stalk_product Bstd F BU) :=\nbegin\n  intros s₁ s₂ Hs,\n  have Hsx := λ (HxU : U), congr_fun (congr_fun Hs HxU.1) HxU.2,\n  let OC : covering_standard_basis B U :=\n  { γ := U,\n    Uis := λ HxU, some (quotient.eq.1 (Hsx HxU)),\n    BUis := λ HxU, some (some_spec (quotient.eq.1 (Hsx HxU))),\n    Hcov := \n      begin\n        ext z,\n        split,\n        { rintros ⟨Ui, ⟨⟨OUi, ⟨⟨i, HUi⟩, HUival⟩⟩, HzUi⟩⟩,\n          rw [←HUival, ←HUi] at HzUi,\n          exact some (some_spec (some_spec (some_spec (quotient.eq.1 (Hsx i))))) HzUi, },\n        { intros Hz,\n          use [(some (quotient.eq.1 (Hsx ⟨z, Hz⟩))).val],\n          have Hin : (some (quotient.eq.1 (Hsx ⟨z, Hz⟩))).val \n            ∈ subtype.val '' set.range (λ (HxU : U), some ((quotient.eq.1 (Hsx HxU)))),\n            use [classical.some ((quotient.eq.1 (Hsx ⟨z, Hz⟩)))],\n            split,\n            { use ⟨z, Hz⟩, },\n            { refl, },\n          use Hin,\n          exact some (some_spec (some_spec (quotient.eq.1 (Hsx ⟨z, Hz⟩)))), },\n      end, },\n  apply (HF BU OC).1,\n  intros i,\n  replace Hs := congr_fun (congr_fun Hs i.1) i.2,\n  exact some_spec (some_spec (some_spec (some_spec (some_spec (quotient.eq.1 (Hsx i)))))),\nend\n\n-- The map φ : F(U) → im(ψ).\n\ndef to_presheaf_of_rings_extension (F : presheaf_of_rings_on_basis α HB) {U : opens α} (BU : U ∈ B)\n: F.F BU → (F ᵣₑₓₜ Bstd).F U :=\nλ s,\n  ⟨to_stalk_product Bstd F.to_presheaf_on_basis BU s,\n   λ x Hx, ⟨U, BU, Hx, s, λ y Hy, funext $ λ Hy', \n   quotient.sound $ ⟨U, BU, Hy', set.subset.refl U, set.subset.refl U, rfl⟩⟩⟩\n\nlemma to_presheaf_of_rings_extension.injective \n(F : presheaf_of_rings_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis) \n{U : opens α} (BU : U ∈ B)\n: function.injective (to_presheaf_of_rings_extension Bstd F BU) :=\nbegin\n  intros s₁ s₂ Hs,\n  erw subtype.mk_eq_mk at Hs,\n  have Hinj := to_stalk_product.injective Bstd F.to_presheaf_on_basis (λ V BV OC, HF BV OC) BU,\n  exact Hinj Hs,\nend\n\nlemma to_presheaf_of_rings_extension.surjective \n(F : presheaf_of_rings_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis) \n{U : opens α} (BU : U ∈ B)\n: function.surjective (to_presheaf_of_rings_extension Bstd F BU) :=\nbegin\n  intros s,\n  let V := λ (HxU : U), some (s.2 HxU.1 HxU.2),\n  let BV := λ (HxU : U), some (some_spec (s.2 HxU.1 HxU.2)),\n  let HxV := λ (HxU : U), some (some_spec (some_spec (s.2 HxU.1 HxU.2))),\n  let σ := λ (HxU : U), some (some_spec (some_spec (some_spec (s.2 HxU.1 HxU.2)))),\n  let Hσ := λ (HxU : U), some_spec (some_spec (some_spec (some_spec (s.2 HxU.1 HxU.2)))),\n  let OC : covering_standard_basis B U :=\n  { γ := U,\n    Uis := λ HxU, (V HxU) ∩ U,\n    BUis := λ HxU, Bstd.2 (BV HxU) BU,\n    Hcov := \n      begin\n        ext z,\n        split,\n        { rintros ⟨Ui, ⟨⟨OUi, ⟨⟨i, HUi⟩, HUival⟩⟩, HzUi⟩⟩,\n          rw [←HUival, ←HUi] at HzUi,\n          exact HzUi.2, },\n        { intros Hz,\n          use [(some (s.2 z Hz) ∩ U).val],\n          have Hin : (some (s.2 z Hz) ∩ U).val \n            ∈ subtype.val '' set.range (λ (HxU : U), ((some (s.2 HxU.1 HxU.2) ∩ U : opens α))),\n            use [(some (s.2 z Hz) ∩ U : opens α)],\n            split,\n            { use ⟨z, Hz⟩, },\n            { refl, },\n          use Hin,\n          exact ⟨some (some_spec (some_spec (s.2 z Hz))), Hz⟩, },\n      end, },\n  -- Now let's try to apply sheaf condition.\n  let res := λ (HxU : OC.γ), F.res (BV HxU) (Bstd.2 (BV HxU) BU) (set.inter_subset_left _ _),\n  let sx := λ (HxU : OC.γ), res HxU (σ HxU),\n  have Hglue := (HF BU OC).2 sx,\n  have Hsx : ∀ j k, \n    sheaf_on_standard_basis.res_to_inter_left Bstd F.to_presheaf_on_basis (OC.BUis j) (OC.BUis k) (sx j) =\n    sheaf_on_standard_basis.res_to_inter_right Bstd F.to_presheaf_on_basis (OC.BUis j) (OC.BUis k) (sx k), \n    intros j k,\n    dsimp only [sheaf_on_standard_basis.res_to_inter_left],\n    dsimp only [sheaf_on_standard_basis.res_to_inter_right],\n    dsimp only [sx, res],\n    iterate 2 { rw ←presheaf_on_basis.Hcomp', },\n    show (F.to_presheaf_on_basis).res (BV j) (Bstd.2 (OC.BUis j) (OC.BUis k)) _ (σ j) \n       = (F.to_presheaf_on_basis).res (BV k) (Bstd.2 (OC.BUis j) (OC.BUis k)) _ (σ k),\n    -- We can cover the U ∩ Vj ∩ Vk and use locality.\n    -- But first let's check that all the stalks coincide in the intersectons.\n    have Hstalks : ∀ {y} (Hy : y ∈ (OC.Uis j) ∩ (OC.Uis k)),\n        (⟦{U := V j, BU := BV j, Hx := Hy.1.1, s := σ j}⟧ : stalk_on_basis F.to_presheaf_on_basis y)\n      = ⟦{U := V k, BU := BV k, Hx := Hy.2.1, s := σ k}⟧,\n      intros y Hy,\n      have Hj := congr_fun (Hσ j y ⟨Hy.1.2, Hy.1.1⟩) Hy.1.2; dsimp at Hj,\n      have Hk := congr_fun (Hσ k y ⟨Hy.2.2, Hy.2.1⟩) Hy.2.2; dsimp at Hk,\n      erw [←Hj, ←Hk],\n    -- Therefore there exists Wjk where σj|Wjk = σk|Wjk. We will use these as a cover.\n    let Ujk : opens α := (OC.Uis j) ∩ (OC.Uis k),\n    let BUjk := Bstd.2 (OC.BUis j) (OC.BUis k),\n    -- Again, all the information we will need but on Uj ∩ Uk.\n    let Hjk := λ (HxUjk : Ujk), quotient.eq.1 (Hstalks HxUjk.2),\n    let Wjk := λ (HxUjk : Ujk), some (Hjk HxUjk) ∩ U,\n    let BWjk := λ (HxUjk : Ujk), Bstd.2 (some (some_spec (Hjk HxUjk))) BU,\n    let HxWjk := λ (HxUjk : Ujk), some (some_spec (some_spec (Hjk HxUjk))),\n    let HWjkUj := λ (HxUjk : Ujk), some (some_spec (some_spec (some_spec (Hjk HxUjk)))),\n    let HWjkUk := λ (HxUjk : Ujk), some (some_spec (some_spec (some_spec (some_spec (Hjk HxUjk))))),\n    let HWjk := λ (HxUjk : Ujk), some_spec (some_spec (some_spec (some_spec (some_spec (Hjk HxUjk))))),\n    let OCjk : covering_standard_basis B ((OC.Uis j) ∩ (OC.Uis k)) :=\n    { γ := Ujk,\n      Uis := Wjk,\n      BUis := BWjk,\n      Hcov := \n        begin\n          ext z,\n          split,\n          { rintros ⟨W, ⟨⟨OW, ⟨⟨i, HWi⟩, HWival⟩⟩, HzW⟩⟩,\n            rw [←HWival, ←HWi] at HzW,\n            have HzUj := (HWjkUj i) HzW.1,\n            have HzUk := (HWjkUk i) HzW.1,\n            exact ⟨⟨HzUj, HzW.2⟩, ⟨HzUk, HzW.2⟩⟩, },\n          { intros Hz,\n            use [(some (Hjk ⟨z, Hz⟩) ∩ U).val],\n            have Hin : (some (Hjk ⟨z, Hz⟩) ∩ U).val \n              ∈ subtype.val '' set.range (λ (HxUjk : Ujk), ((some (Hjk HxUjk) ∩ U : opens α))),\n              use [(some (Hjk ⟨z, Hz⟩) ∩ U : opens α)],\n              split,\n              { use ⟨z, Hz⟩, },\n              { refl, },\n            use Hin,\n            have HzWjk := HxWjk ⟨z, Hz⟩,\n            have HzU := Hz.1.2,\n            exact ⟨HzWjk, HzU⟩, }\n        end, },\n    apply (HF BUjk OCjk).1,\n    intros i,\n    rw ←presheaf_on_basis.Hcomp',\n    rw ←presheaf_on_basis.Hcomp',\n    have Hres : \n        F.res (some (some_spec (Hjk i))) (BWjk i) (set.inter_subset_left _ _)\n          (F.res (BV j) (some (some_spec (Hjk i))) (HWjkUj i) (σ j))\n      = F.res (some (some_spec (Hjk i))) (BWjk i) (set.inter_subset_left _ _)\n          (F.res (BV k) (some (some_spec (Hjk i))) (HWjkUk i) (σ k)),\n      rw (HWjk i),\n    rw ←presheaf_on_basis.Hcomp' at Hres,\n    rw ←presheaf_on_basis.Hcomp' at Hres,\n    use Hres,\n  -- Ready to prove it.\n  rcases (Hglue Hsx) with ⟨S, HS⟩,\n  existsi S,\n  apply subtype.eq,\n  dsimp [to_presheaf_of_rings_extension],\n  apply funext,\n  intros x,\n  dsimp [to_stalk_product],\n  apply funext,\n  intros Hx,\n  replace HS := HS ⟨x, Hx⟩,\n  dsimp [sx, res] at HS,\n  rw Hσ ⟨x, Hx⟩,\n  swap,\n  { exact ⟨Hx, HxV ⟨x, Hx⟩⟩, },\n  dsimp,\n  apply quotient.sound,\n  use [(V ⟨x, Hx⟩) ∩ U],\n  use [Bstd.2 (BV ⟨x, Hx⟩) BU],\n  use [⟨HxV ⟨x, Hx⟩, Hx⟩],\n  use [set.inter_subset_right _ _],\n  use [set.inter_subset_left _ _],\n  dsimp,\n  erw HS,\nend\n\nlemma to_presheaf_of_rings_extension.bijective\n(F : presheaf_of_rings_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis) \n{U : opens α} (BU : U ∈ B)\n: function.bijective (to_presheaf_of_rings_extension Bstd F BU) :=\n⟨to_presheaf_of_rings_extension.injective Bstd F (λ U BU OC, HF BU OC) BU,\nto_presheaf_of_rings_extension.surjective Bstd F (λ U BU OC, HF BU OC) BU ⟩\n\n-- We now that they are equivalent as sets. \n-- Now we to assert that they're isomorphic as rings. \n-- It suffices to show that it is a ring homomorphism.\n\nlemma to_presheaf_of_rings_extension.is_ring_hom \n(F : presheaf_of_rings_on_basis α HB) \n{U : opens α} (BU : U ∈ B)\n: is_ring_hom (to_presheaf_of_rings_extension Bstd F BU) :=\n{ map_one := \n    begin\n      apply subtype.eq,\n      funext x Hx,\n      apply quotient.sound,\n      use [U, BU, Hx, set.subset.refl _, set.subset_univ _],\n      iterate 2 { erw (F.res_is_ring_hom _ _ _).map_one, },\n    end,\n  map_mul := \n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz], \n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_mul,\n      erw ←presheaf_on_basis.Hcomp', \n    end,\n  map_add := \n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz], \n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_add,\n      erw ←presheaf_on_basis.Hcomp', \n    end, }\n\nlemma to_presheaf_of_rings_extension.ring_equiv\n(F : presheaf_of_rings_on_basis α HB) \n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis) \n{U : opens α} (BU : U ∈ B)\n: F BU ≃+* (presheaf_of_rings_extension Bstd F) U :=\nbegin\n  have H := function.bijective_iff_has_inverse.1 (to_presheaf_of_rings_extension.bijective Bstd F @HF BU),\n  rcases (classical.indefinite_description _ H) with ⟨inv, Hlinv, Hrinv⟩,\n  use [(to_presheaf_of_rings_extension Bstd F BU), inv, Hlinv, Hrinv],\n  { apply (to_presheaf_of_rings_extension.is_ring_hom Bstd F BU).map_mul, },\n  { apply (to_presheaf_of_rings_extension.is_ring_hom Bstd F BU).map_add, }\nend\n\n-- Moreover, for all x, Fₓ ≅ Fextₓ.\n\n-- We will need this to show that the stalks of the structure sheaf on\n-- Spec(R) are local rings.\n\nopen stalk_of_rings_on_standard_basis\n\nlemma to_stalk_extension\n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: stalk_of_rings_on_standard_basis Bstd F x → stalk_of_rings (F ᵣₑₓₜ Bstd) x :=\nbegin\n  intros BUs,\n  let Us := quotient.out BUs,\n  exact ⟦{U := Us.U, \n          HxU := Us.Hx, \n          s := (to_presheaf_of_rings_extension Bstd F Us.BU) Us.s}⟧,\nend\n\nlemma to_stalk_extension.injective \n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: function.injective (to_stalk_extension Bstd F x) :=\nbegin\n  intros Us₁' Us₂', \n  apply quotient.induction_on₂ Us₁' Us₂',\n  rintros Us₁ Us₂ HUs,\n  rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁Us₁, HW₁U₁, Hres₁⟩,\n  rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂Us₂, HW₂U₂, Hres₂⟩,\n  dunfold to_stalk_extension at HUs,\n  rw quotient.eq at HUs,\n  rcases HUs with ⟨W, HxW, HWU₁, HWU₂, Hres⟩,\n  dsimp at HWU₁,\n  dsimp at HWU₂,\n  dunfold to_presheaf_of_rings_extension at Hres,\n  dunfold to_stalk_product at Hres,\n  erw subtype.mk.inj_eq at Hres,\n  replace Hres := congr_fun (congr_fun Hres x) HxW,\n  dsimp at Hres,\n  rw quotient.eq at Hres,\n  rcases Hres with ⟨W₃, BW₃, HxW₃, HW₃U₁, HW₃U₂, Hres₃⟩,\n  dsimp at HW₃U₁, \n  dsimp at HW₃U₂,\n  dsimp at Hres₃,\n  apply quotient.sound,\n  have BW₁₂₃ : W₁ ∩ W₂ ∩ W₃ ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n  have HW₁₂₃U₁ : W₁ ∩ W₂ ∩ W₃ ⊆ Us₁.U := λ x Hx, HW₁U₁ Hx.1.1,\n  have HW₁₂₃U₂ : W₁ ∩ W₂ ∩ W₃ ⊆ Us₂.U := λ x Hx, HW₂U₂ Hx.1.2,\n  use [W₁ ∩ W₂ ∩ W₃, BW₁₂₃, ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩, HW₁₂₃U₁, HW₁₂₃U₂],\n  have HW₁W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₁ := λ x Hx, Hx.1.1,\n  have HW₂W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₂ := λ x Hx, Hx.1.2,\n  have HW₃W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₃ := λ x Hx, Hx.2,\n  replace Hres₁ := congr_arg (F.res BW₁ BW₁₂₃ HW₁W₁₂₃) Hres₁,\n  replace Hres₂ := congr_arg (F.res BW₂ BW₁₂₃ HW₂W₁₂₃) Hres₂,\n  replace Hres₃ := congr_arg (F.res BW₃ BW₁₂₃ HW₃W₁₂₃) Hres₃,\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n  erw [←Hres₁, ←Hres₂],\n  exact Hres₃,\nend\n\nlemma to_stalk_extension.surjective \n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: function.surjective (to_stalk_extension Bstd F x) :=\nbegin\n  intros Us',\n  apply quotient.induction_on Us',\n  rintros ⟨U, HxU, s⟩,\n  rcases (s.2 x HxU) with ⟨V, BV, HxV, t, Ht⟩,\n  let Vt : stalk_on_basis.elem (F.to_presheaf_on_basis) x \n    := {U := V, BU := BV, Hx := HxV, s := t},\n  use ⟦Vt⟧,\n  dunfold to_stalk_extension,\n  apply quotient.sound,\n  rcases (quotient.mk_out Vt) with ⟨W, BW, HxW, HWVtV, HWV, Hres⟩,\n  have HUVWV : U ∩ V ∩ W ⊆ (quotient.out ⟦Vt⟧).U := λ x Hx, HWVtV Hx.2,\n  have HUVWU : U ∩ V ∩ W ⊆ U := λ x Hx, Hx.1.1,\n  use [U ∩ V ∩ W, ⟨⟨HxU, HxV⟩, HxW⟩, HUVWV, HUVWU],\n  apply subtype.eq,\n  dsimp only [presheaf_of_rings_extension],\n  dsimp only [to_presheaf_of_rings_extension],\n  dsimp only [to_stalk_product],\n  funext y Hy,\n  rw (Ht y Hy.1),\n  apply quotient.sound,\n  have BVW : V ∩ W ∈ B := Bstd.2 BV BW,\n  have HVWVtV : V ∩ W ⊆ (quotient.out ⟦Vt⟧).U := λ x Hx, HWVtV Hx.2,\n  have HVWV : V ∩ W ⊆ V := λ x Hx, Hx.1,\n  use [V ∩ W, BVW, ⟨Hy.1.2,Hy.2⟩, HVWVtV, HVWV],\n  have HVWW : V ∩ W ⊆ W := λ x Hx, Hx.2,\n  replace Hres := congr_arg (F.res BW BVW HVWW) Hres,\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres, },\n  exact Hres,\nend\n\nlemma to_stalk_extension.bijective\n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: function.bijective (to_stalk_extension Bstd F x) :=\n⟨to_stalk_extension.injective Bstd F x,\nto_stalk_extension.surjective Bstd F x⟩\n\nlemma to_stalk_extension.is_ring_hom\n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: is_ring_hom (to_stalk_extension Bstd F x) :=\n{ map_one := \n    begin\n      dunfold to_stalk_extension,\n      let one.elem : stalk_on_basis.elem F.to_presheaf_on_basis x\n        := {U := opens.univ, BU := Bstd.1, Hx := trivial, s:= 1},\n      let one.stalk : stalk_of_rings_on_standard_basis Bstd F x := ⟦one.elem⟧,\n      let one := quotient.out one.stalk,\n      apply quotient.sound,\n      rcases (quotient.mk_out one.elem) with ⟨W₁, BW₁, HxW₁, HW₁Uout, HW₁U, Hres₁⟩,\n      have BUW₁ : one.U ∩ W₁ ∈ B := Bstd.2 one.BU BW₁,\n      have HUUW₁ : one.U ∩ W₁ ⊆ one.U := set.inter_subset_left _ _,\n      use [one.U ∩ W₁, ⟨one.Hx, HxW₁⟩, HUUW₁, set.subset_univ _],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z Hz,\n      apply quotient.sound,\n      use [one.U ∩ W₁, BUW₁, Hz, set.inter_subset_left _ _, set.subset_univ _],\n      dsimp,\n      have HUW₁W₁ : one.U ∩ W₁ ⊆ W₁ := set.inter_subset_right _ _,\n      replace Hres₁ := congr_arg (F.res BW₁ BUW₁ HUW₁W₁) Hres₁,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      exact Hres₁,\n    end,\n  map_mul := \n    begin\n      intros y z,\n      apply quotient.induction_on₂ y z,\n      intros Us₁ Us₂,\n      simp,\n      let Us₃ : stalk_on_basis.elem F.to_presheaf_on_basis x :=\n        { U := Us₁.U ∩ Us₂.U, \n          BU := Bstd.2 Us₁.BU Us₂.BU,\n          Hx := ⟨Us₁.Hx, Us₂.Hx⟩, \n          s :=  F.res Us₁.BU _ (set.inter_subset_left _ _) Us₁.s * \n                F.res Us₂.BU _ (set.inter_subset_right _ _) Us₂.s },\n      dunfold to_stalk_extension,\n      apply quotient.sound,\n      rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁U₁out, HW₁U₁, Hres₁⟩,\n      rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂U₂out, HW₂U₂, Hres₂⟩,\n      rcases (quotient.mk_out Us₃) with ⟨W₃, BW₃, HxW₃, HW₃U₃out, HW₃U₃, Hres₃⟩,\n      let W := W₁ ∩ W₂ ∩ W₃,\n      have HxW : x ∈ W := ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩,\n      have HWW₁ : W ⊆ W₁ := λ x Hx, Hx.1.1,\n      have HWW₂ : W ⊆ W₂ := λ x Hx, Hx.1.2,\n      have HWW₃ : W ⊆ W₃ := λ x Hx, Hx.2,\n      have HWU₁out : W ⊆ (quotient.out ⟦Us₁⟧).U := set.subset.trans HWW₁ HW₁U₁out,\n      have HWU₂out : W ⊆ (quotient.out ⟦Us₂⟧).U := set.subset.trans HWW₂ HW₂U₂out,\n      have HWU₃out : W ⊆ (quotient.out ⟦Us₃⟧).U := set.subset.trans HWW₃ HW₃U₃out,\n      have HWU₁₂out : W ⊆ (quotient.out ⟦Us₁⟧).U ∩ (quotient.out ⟦Us₂⟧).U\n        := set.subset_inter HWU₁out HWU₂out,\n      use [W, HxW, HWU₃out, HWU₁₂out],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z HzW,\n      apply quotient.sound,\n      have BW : W ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n      use [W, BW, HzW, HWU₃out, HWU₁₂out],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_mul,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n      replace Hres₁ := congr_arg (F.res BW₁ BW HWW₁) Hres₁,\n      replace Hres₂ := congr_arg (F.res BW₂ BW HWW₂) Hres₂,\n      replace Hres₃ := congr_arg (F.res BW₃ BW HWW₃) Hres₃,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n      erw [Hres₁, Hres₂, Hres₃],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_mul,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n    end,\n  map_add := \n    begin\n      intros y z,\n      apply quotient.induction_on₂ y z,\n      intros Us₁ Us₂,\n      simp,\n      let Us₃ : stalk_on_basis.elem F.to_presheaf_on_basis x :=\n        { U := Us₁.U ∩ Us₂.U, \n          BU := Bstd.2 Us₁.BU Us₂.BU,\n          Hx := ⟨Us₁.Hx, Us₂.Hx⟩, \n          s :=  F.res Us₁.BU _ (set.inter_subset_left _ _) Us₁.s + \n                F.res Us₂.BU _ (set.inter_subset_right _ _) Us₂.s },\n      dunfold to_stalk_extension,\n      apply quotient.sound,\n      rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁U₁out, HW₁U₁, Hres₁⟩,\n      rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂U₂out, HW₂U₂, Hres₂⟩,\n      rcases (quotient.mk_out Us₃) with ⟨W₃, BW₃, HxW₃, HW₃U₃out, HW₃U₃, Hres₃⟩,\n      let W := W₁ ∩ W₂ ∩ W₃,\n      have HxW : x ∈ W := ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩,\n      have HWW₁ : W ⊆ W₁ := λ x Hx, Hx.1.1,\n      have HWW₂ : W ⊆ W₂ := λ x Hx, Hx.1.2,\n      have HWW₃ : W ⊆ W₃ := λ x Hx, Hx.2,\n      have HWU₁out : W ⊆ (quotient.out ⟦Us₁⟧).U := set.subset.trans HWW₁ HW₁U₁out,\n      have HWU₂out : W ⊆ (quotient.out ⟦Us₂⟧).U := set.subset.trans HWW₂ HW₂U₂out,\n      have HWU₃out : W ⊆ (quotient.out ⟦Us₃⟧).U := set.subset.trans HWW₃ HW₃U₃out,\n      have HWU₁₂out : W ⊆ (quotient.out ⟦Us₁⟧).U ∩ (quotient.out ⟦Us₂⟧).U\n        := set.subset_inter HWU₁out HWU₂out,\n      use [W, HxW, HWU₃out, HWU₁₂out],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z HzW,\n      apply quotient.sound,\n      have BW : W ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n      use [W, BW, HzW, HWU₃out, HWU₁₂out],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_add,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n      replace Hres₁ := congr_arg (F.res BW₁ BW HWW₁) Hres₁,\n      replace Hres₂ := congr_arg (F.res BW₂ BW HWW₂) Hres₂,\n      replace Hres₃ := congr_arg (F.res BW₃ BW HWW₃) Hres₃,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n      erw [Hres₁, Hres₂, Hres₃],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_add,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n    end, }\n\nlemma to_stalk_extension.ring_equiv\n(F : presheaf_of_rings_on_basis α HB) \n(x : α)\n: stalk_of_rings_on_standard_basis Bstd F x ≃+* \nstalk_of_rings (presheaf_of_rings_extension Bstd F) x :=\nbegin\n  have H := function.bijective_iff_has_inverse.1 (to_stalk_extension.bijective Bstd F x),\n  rcases (classical.indefinite_description _ H) with ⟨inv, Hlinv, Hrinv⟩,\n  use [(to_stalk_extension Bstd F x), inv, Hlinv, Hrinv],\n  { apply (to_stalk_extension.is_ring_hom Bstd F x).map_mul, },\n  { apply (to_stalk_extension.is_ring_hom Bstd F x).map_add, }\nend\n\nend extension_coincides\n\nend presheaf_of_rings_extension\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/lean-scheme-submission/src/sheaves/sheaf_of_rings_on_standard_basis.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3120498179904941}}
{"text": "import category_theory.triangulated.triangulated\nimport for_mathlib.category_theory.triangulated.triangulated_functor\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\nopen_locale zero_object\n\nnamespace set\n\nopen category_theory\n\nclass respects_iso {X : Type*} [category X] (A : set X) : Prop :=\n(condition' : ∀ ⦃x y : X⦄ (e : x ≅ y) (hx : x ∈ A), y ∈ A)\n\nlemma respects_iso.condition {X : Type*} [category X] (A : set X) [A.respects_iso]\n  {x y : X} (e : x ≅ y) (hx : x ∈ A) : y ∈ A :=\nrespects_iso.condition' e hx\n\nlemma respects_iso.mem_iff_of_iso {X : Type*} [category X] (A : set X) [A.respects_iso]\n  {x y : X} (e : x ≅ y) : x ∈ A ↔ y ∈ A :=\nbegin\n  split,\n  { exact respects_iso.condition A e, },\n  { exact respects_iso.condition A e.symm, },\nend\n\nend set\n\nnamespace category_theory\n\nnamespace triangulated\n\nopen pretriangulated\n\nvariables {C : Type*} [category C] [has_shift C ℤ] [preadditive C] [has_zero_object C]\n  [∀ (n : ℤ), functor.additive (shift_functor C n)] [pretriangulated C]\n\nvariable (S : set C)\n\nclass is_triangulated_subcategory : Prop :=\n(zero [] : (0 : C) ∈ S)\n(shift : ∀ (X : C) (n : ℤ) (hX : X ∈ S), (shift_functor C n).obj X ∈ S)\n(ext₂ : ∀ (T : triangle C) (hT : T ∈ dist_triang C) (h₁ : T.obj₁ ∈ S)\n  (h₃ : T.obj₃ ∈ S), T.obj₂ ∈ S)\n\nnamespace is_triangulated_subcategory\n\nvariables (S) [is_triangulated_subcategory S]\n\n@[priority 100]\ninstance set_respects_iso : S.respects_iso :=\n⟨λ X Y e hX, ext₂ _ (pretriangulated.isomorphic_distinguished _\n  (pretriangulated.contractible_distinguished X) (triangle.mk e.hom (0 : Y ⟶ 0) 0)\n  (triangle.mk_iso _ _ (iso.refl _) e.symm (iso.refl _) (by tidy) (by tidy) (by tidy))) hX\n  (is_triangulated_subcategory.zero S)⟩\n\nlemma zero' (X : C) (hX : is_zero X) : X ∈ S :=\nby simpa only [set.respects_iso.mem_iff_of_iso S (limits.is_zero.iso_zero hX)]\n  using is_triangulated_subcategory.zero S\n\nvariable {S}\n\nlemma ext₁\n  (T : triangle C) (hT : T ∈ dist_triang C) (h₂ : T.obj₂ ∈ S) (h₃ : T.obj₃ ∈ S) :\n  T.obj₁ ∈ S :=\next₂ T.inv_rotate (pretriangulated.inv_rot_of_dist_triangle C T hT)\n  (is_triangulated_subcategory.shift _ _ h₃) h₂\n\nlemma ext₃\n  (T : triangle C) (hT : T ∈ dist_triang C) (h₁ : T.obj₁ ∈ S) (h₂ : T.obj₂ ∈ S) :\n  T.obj₃ ∈ S :=\next₂ T.rotate (pretriangulated.rot_of_dist_triangle C T hT) h₂\n  (is_triangulated_subcategory.shift _ _ h₁)\n\nlemma shift_iff (X : C) (n : ℤ) : X ∈ S ↔ (shift_functor C n).obj X ∈ S :=\nbegin\n  split,\n  { intro h,\n    exact is_triangulated_subcategory.shift _ _ h, },\n  { intro h,\n    exact set.respects_iso.condition S\n      ((add_neg_equiv (shift_monoidal_functor C ℤ) n).unit_iso.symm.app X)\n      (is_triangulated_subcategory.shift _ _ h), },\nend\n\nend is_triangulated_subcategory\n\nclass is_triangulated_subcategory' : Prop :=\n(zero : ∃ (X : C) (hX₀ : is_zero X), X ∈ S)\n(shift : ∀ (X : C) (n : ℤ) (hX : X ∈ S), (shift_functor C n).obj X ∈ S)\n(distinguished_cocone_triangle' : ∀ (X Y : C) (hX : X ∈ S) (hY : Y ∈ S) (f : X ⟶ Y),\n  ∃ (Z : C) (hZ : Z ∈ S) (g : Y ⟶ Z) (h : Z ⟶ X⟦(1 : ℤ)⟧),\n    triangle.mk f g h ∈ dist_triang C)\n\n@[priority 100]\ninstance is_triangulated_subcategory'_of_is_triangulated_subcategory\n  [hS : is_triangulated_subcategory S] :\n  is_triangulated_subcategory' S :=\n{ zero := ⟨0, is_zero_zero _, hS.zero⟩,\n  shift := hS.shift,\n  distinguished_cocone_triangle' := λ X Y hX hY f, begin\n    obtain ⟨Z, g, h, mem⟩ := pretriangulated.distinguished_cocone_triangle _ _ f,\n    exact ⟨Z, hS.ext₂ _ (rot_of_dist_triangle _ _ mem) hY (hS.shift _ 1 hX), g, h, mem⟩,\n  end, }\n\nnamespace is_triangulated_subcategory'\n\nvariable [hS : is_triangulated_subcategory' S]\n\ninclude hS\n\ninstance : has_zero_object (full_subcategory S) :=\nbegin\n  obtain ⟨X₀, hX₀, mem⟩ := hS.zero,\n  refine ⟨⟨⟨X₀, mem⟩, _⟩⟩,\n  rw limits.is_zero.iff_id_eq_zero,\n  exact (full_subcategory_inclusion S).map_injective (is_zero.eq_of_src hX₀ _ _),\nend\n\ninstance is_stable_by_shift : S.is_stable_by_shift ℤ :=\n⟨λ a X hX, hS.shift X a hX⟩\n\ninstance shift_functor_additive (n : ℤ) :\n  (category_theory.shift_functor (full_subcategory S) n).additive :=\n⟨begin\n  rintros ⟨K, hK⟩ ⟨L, hL⟩ f g,\n  exact (full_subcategory_inclusion S).map_injective (category_theory.shift_functor C n).map_add,\nend⟩\n\n--instance full_subcategory_inclusion_has_comm_shift :\n--  (full_subcategory_inclusion S).has_comm_shift ℤ := infer_instance\n\ndef distinguished_triangles : set (triangle (full_subcategory S)) :=\nλ T, (full_subcategory_inclusion S).map_triangle.obj T ∈ dist_triang C\n\nlemma isomorphic_distinguished (T₁ : triangle (full_subcategory S))\n  (hT₁ : T₁ ∈ distinguished_triangles S) (T₂ : triangle (full_subcategory S)) (e : T₂ ≅ T₁) :\n  T₂ ∈ distinguished_triangles S :=\npretriangulated.isomorphic_distinguished _ hT₁ _ ((full_subcategory_inclusion S).map_triangle.map_iso e)\n\nlemma contractible_distinguished (X : full_subcategory S) :\n  triangle.mk (𝟙 X) (0 : X ⟶ 0) 0 ∈ distinguished_triangles S :=\nbegin\n  refine pretriangulated.isomorphic_distinguished _\n    (pretriangulated.contractible_distinguished ((full_subcategory_inclusion S).obj X)) _ _,\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _)\n    (full_subcategory_inclusion S).map_zero_object _ _ _,\n  tidy,\nend\n\nlemma rotate_distinguished_triangle (T : triangle (full_subcategory S)) :\n  T ∈ distinguished_triangles S ↔\n    T.rotate ∈ distinguished_triangles S :=\nbegin\n  change ((full_subcategory_inclusion S).map_triangle.obj T ∈ dist_triang C) ↔\n    ((full_subcategory_inclusion S).map_triangle.obj T.rotate ∈ dist_triang C),\n  rw pretriangulated.rotate_distinguished_triangle,\n  let e := (full_subcategory_inclusion S).map_triangle_rotate.app T,\n  split,\n  { exact λ h, pretriangulated.isomorphic_distinguished _ h _ e.symm, },\n  { exact λ h, pretriangulated.isomorphic_distinguished _ h _ e, },\nend\n\nlemma distinguished_cocone_triangle (X Y : full_subcategory S) (f : X ⟶ Y) :\n  ∃ (Z : full_subcategory S) (g : Y ⟶ Z) (h : Z ⟶ (category_theory.shift_functor _ (1 : ℤ)).obj X),\n    triangle.mk f g h ∈ distinguished_triangles S :=\nbegin\n  obtain ⟨X, hX⟩ := X,\n  obtain ⟨Y, hY⟩ := Y,\n  obtain ⟨Z, hZ, g, h, mem⟩ := hS.distinguished_cocone_triangle' _ _ hX hY f,\n  refine ⟨⟨Z, hZ⟩, g, h, pretriangulated.isomorphic_distinguished _ mem _ _⟩,\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _) (by tidy) (by tidy) _,\n  { dsimp,\n    simp only [functor.map_id, comp_id, id_comp],\n    apply comp_id, },\nend\n\nlemma complete_distinguished_triangle_morphism (T₁ T₂ : triangle (full_subcategory S))\n  (hT₁ : T₁ ∈ distinguished_triangles S) (hT₂ : T₂ ∈ distinguished_triangles S)\n  (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (h : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :\n  ∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫\n    (category_theory.shift_functor _ 1).map a = c ≫ T₂.mor₃ :=\nbegin\n  obtain ⟨c, ⟨hc₁, hc₂⟩⟩ := pretriangulated.complete_distinguished_triangle_morphism\n    ((full_subcategory_inclusion S).map_triangle.obj T₁)\n      ((full_subcategory_inclusion S).map_triangle.obj T₂)\n    hT₁ hT₂ a b h,\n  refine ⟨c, ⟨hc₁, _⟩⟩,\n  dsimp at hc₂,\n  erw [comp_id, comp_id] at hc₂,\n  exact hc₂,\nend\n\ninstance pretriangulated_full_subcategory :\n  pretriangulated (full_subcategory S) :=\n{ distinguished_triangles := distinguished_triangles S,\n  isomorphic_distinguished := isomorphic_distinguished S,\n  contractible_distinguished := contractible_distinguished S,\n  distinguished_cocone_triangle := distinguished_cocone_triangle S,\n  rotate_distinguished_triangle := rotate_distinguished_triangle S,\n  complete_distinguished_triangle_morphism := complete_distinguished_triangle_morphism S, }\n\ninstance [is_triangulated C] : is_triangulated (full_subcategory S) :=\n⟨λ X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ u₁₂ u₂₃ u₁₃ comm v₁₂ w₁₂ h₁₂ v₂₃ w₂₃ h₂₃ v₁₃ w₁₃ h₁₃, begin\n  have comm' := (full_subcategory_inclusion S).congr_map comm,\n  rw [functor.map_comp] at comm',\n  have H := (is_triangulated.octahedron_axiom comm' h₁₂ h₂₃ h₁₃).some,\n  obtain ⟨m₁, m₃, comm₁, comm₂, comm₃, comm₄, H'⟩ := H,\n  refine nonempty.intro\n  { m₁ := m₁,\n    m₃ := m₃,\n    comm₁ := comm₁,\n    comm₂ := begin\n      erw [comp_id, comp_id] at comm₂,\n      exact comm₂,\n    end,\n    comm₃ := comm₃,\n    comm₄ := begin\n      erw [comp_id, comp_id] at comm₄,\n      exact comm₄,\n    end,\n    mem := begin\n      change _ ∈ dist_triang C,\n      refine pretriangulated.isomorphic_distinguished _ H' _ _,\n      refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _) _ _ _,\n      { dsimp,\n        simp, },\n      { dsimp,\n        simp, },\n      { dsimp,\n        erw [functor.map_id, comp_id, comp_id, id_comp, comp_id],\n        refl, },\n    end, }\nend⟩\n\ninstance full_subcategory_inclusion_is_triangulated :\n  (full_subcategory_inclusion S).is_triangulated :=\n{ map_distinguished' := λ T hT, hT, }\n\nend is_triangulated_subcategory'\n\nclass saturated [is_triangulated_subcategory S] : Prop :=\n(condition : ∀ ⦃X Y : C⦄ (i : Y ⟶ X) [hi : is_split_mono i] (hX : X ∈ S), Y ∈ S)\n\ndef left_orthogonal : set C :=\nλ X, ∀ ⦃Y : C⦄ (f : X ⟶ Y) (hY : Y ∈ S), f = 0\n\ndef right_orthogonal : set C :=\nλ Y, ∀ ⦃X : C⦄ (f : X ⟶ Y) (hX : X ∈ S), f = 0\n\ninstance [S.is_stable_by_shift ℤ] :\n  is_triangulated_subcategory (left_orthogonal S) :=\n{ zero := λ Y hY f, subsingleton.elim _ _,\n  shift := λ X n hX Y f hY, begin\n    let adj : shift_functor C n ⊣ shift_functor C (-n) :=\n      (add_neg_equiv (shift_monoidal_functor C ℤ) n).to_adjunction,\n    apply (adj.hom_equiv _ _).injective,\n    simp only [adjunction.hom_equiv_unit, functor.map_zero, comp_zero],\n    exact hX _ (set.is_stable_by_shift.condition (-n) Y hY),\n  end,\n  ext₂ := λ T hT h₁ h₃ Y f hY, begin\n    obtain ⟨g, hg⟩ := contravariant_yoneda_exact₂ T hT f (h₁ _ hY),\n    rw [hg, h₃ g hY, comp_zero],\n  end, }\n\ninstance [S.is_stable_by_shift ℤ] :\n  is_triangulated_subcategory (right_orthogonal S) :=\n{ zero := λ X hX f, subsingleton.elim _ _,\n  shift := λ Y n hY X f hX, begin\n    let adj : shift_functor C (-n) ⊣ shift_functor C n :=\n      (add_neg_equiv (shift_monoidal_functor C ℤ) n).symm.to_adjunction,\n    apply (adj.hom_equiv _ _).symm.injective,\n    simp only [adjunction.hom_equiv_counit, functor.map_zero, zero_comp],\n    exact hY _ (set.is_stable_by_shift.condition (-n) X hX),\n  end,\n  ext₂ := λ T hT h₁ h₃ X f hX, begin\n    obtain ⟨g, hg⟩ := covariant_yoneda_exact₂ T hT f (h₃ _ hX),\n    rw [hg, h₁ g hX, zero_comp],\n  end, }\n\ninstance left_orthogonal_saturated [S.is_stable_by_shift ℤ] :\n  saturated (left_orthogonal S) :=\n⟨λ X Y i hi hX Z f hZ, begin\n  haveI := hi,\n  rw [← cancel_epi (retraction i), comp_zero],\n  exact hX _ hZ,\nend⟩\n\ninstance right_orthogonal_saturated [S.is_stable_by_shift ℤ] :\n  saturated (right_orthogonal S) :=\n⟨λ X Y i hi hX Z f hZ, begin\n  haveI := hi,\n  rw [← cancel_mono i, zero_comp],\n  exact hX _ hZ,\nend⟩\n\nend triangulated\n\nnamespace functor\n\nnamespace is_triangulated\n\nsection\n\nvariables {C D E : Type*} [category C] [category D] [category E]\n  [has_zero_object C] [has_zero_object D] [has_zero_object E]\n  [has_shift C ℤ] [has_shift D ℤ] [has_shift E ℤ]\n  [preadditive C] [preadditive D] [preadditive E]\n  [∀ (n : ℤ), (shift_functor C n).additive]\n  [∀ (n : ℤ), (shift_functor D n).additive]\n  [∀ (n : ℤ), (shift_functor E n).additive]\n  [pretriangulated C] [pretriangulated D] [pretriangulated E]\n  {F : C ⥤ D} {G : D ⥤ E} {H : C ⥤ E} (e : F ⋙ G ≅ H)\n  [full G] [faithful G]\n  [F.has_comm_shift ℤ] [H.has_comm_shift ℤ] [G.has_comm_shift ℤ]\n  [functor.is_triangulated G] [functor.is_triangulated H]\n\nlemma of_fully_faithful [e.hom.respects_comm_shift ℤ] : F.is_triangulated :=\n{ map_distinguished' := λ T hT, G.reflects_distinguished _\n    (pretriangulated.isomorphic_distinguished _ (H.map_distinguished T hT) _\n    (((map_triangle_comp F G).app T).symm ≪≫ (map_triangle_nat_iso e).app T)), }\n\nlemma of_fully_faithful' :\n  @functor.is_triangulated _ _ _ _ _ _ _ _ F (has_comm_shift.of_fully_faithful e ℤ) _ _ _ _ _ _ :=\nbegin\n  letI := has_comm_shift.of_fully_faithful e ℤ,\n  haveI := has_comm_shift.of_fully_faithful_iso_hom_respects_comm_shift e ℤ,\n  exact of_fully_faithful e,\nend\n\nend\n\nsection\n\nvariables {C D : Type*} [category C] [category D]\n  [has_zero_object C] [has_zero_object D]\n  [has_shift C ℤ] [has_shift D ℤ]\n  [preadditive C] [preadditive D]\n  [∀ (n : ℤ), (shift_functor C n).additive]\n  [∀ (n : ℤ), (shift_functor D n).additive]\n  [pretriangulated C] [pretriangulated D]\n  (S : set D) (F : C ⥤ D) [F.has_comm_shift ℤ] [functor.is_triangulated F]\n  [S.is_stable_by_shift ℤ] [triangulated.is_triangulated_subcategory' S]\n  (hS : ∀ (X : C), F.obj X ∈ S)\n\ninstance full_subcategory_lift_is_triangulated :\n  (full_subcategory.lift S F hS).is_triangulated :=\nof_fully_faithful (full_subcategory.lift_comp_inclusion S F hS)\n\nend\n\nend is_triangulated\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/triangulated/is_triangulated_subcategory.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.31204981137321697}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Sean Leather\n\nFunctions on lists of sigma types.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.perm\nimport Mathlib.data.list.range\nimport Mathlib.data.sigma.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 \n\nnamespace Mathlib\n\nnamespace list\n\n\n/- keys -/\n\n/-- List of keys from a list of key-value pairs -/\ndef keys {α : Type u} {β : α → Type v} : List (sigma β) → List α := map sigma.fst\n\n@[simp] theorem keys_nil {α : Type u} {β : α → Type v} : keys [] = [] := rfl\n\n@[simp] theorem keys_cons {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} :\n    keys (s :: l) = sigma.fst s :: keys l :=\n  rfl\n\ntheorem mem_keys_of_mem {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} :\n    s ∈ l → sigma.fst s ∈ keys l :=\n  mem_map_of_mem sigma.fst\n\ntheorem exists_of_mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)}\n    (h : a ∈ keys l) : ∃ (b : β a), sigma.mk a b ∈ l :=\n  sorry\n\ntheorem mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} :\n    a ∈ keys l ↔ ∃ (b : β a), sigma.mk a b ∈ l :=\n  sorry\n\ntheorem not_mem_keys {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} :\n    ¬a ∈ keys l ↔ ∀ (b : β a), ¬sigma.mk a b ∈ l :=\n  iff.trans (not_iff_not_of_iff mem_keys) not_exists\n\ntheorem not_eq_key {α : Type u} {β : α → Type v} {a : α} {l : List (sigma β)} :\n    ¬a ∈ keys l ↔ ∀ (s : sigma β), s ∈ l → a ≠ sigma.fst s :=\n  sorry\n\n/- nodupkeys -/\n\ndef nodupkeys {α : Type u} {β : α → Type v} (l : List (sigma β)) := nodup (keys l)\n\ntheorem nodupkeys_iff_pairwise {α : Type u} {β : α → Type v} {l : List (sigma β)} :\n    nodupkeys l ↔ pairwise (fun (s s' : sigma β) => sigma.fst s ≠ sigma.fst s') l :=\n  pairwise_map sigma.fst\n\ntheorem nodupkeys.pairwise_ne {α : Type u} {β : α → Type v} {l : List (sigma β)} (h : nodupkeys l) :\n    pairwise (fun (s s' : sigma β) => sigma.fst s ≠ sigma.fst s') l :=\n  iff.mp nodupkeys_iff_pairwise h\n\n@[simp] theorem nodupkeys_nil {α : Type u} {β : α → Type v} : nodupkeys [] := pairwise.nil\n\n@[simp] theorem nodupkeys_cons {α : Type u} {β : α → Type v} {s : sigma β} {l : List (sigma β)} :\n    nodupkeys (s :: l) ↔ ¬sigma.fst s ∈ keys l ∧ nodupkeys l :=\n  sorry\n\ntheorem nodupkeys.eq_of_fst_eq {α : Type u} {β : α → Type v} {l : List (sigma β)} (nd : nodupkeys l)\n    {s : sigma β} {s' : sigma β} (h : s ∈ l) (h' : s' ∈ l) : sigma.fst s = sigma.fst s' → s = s' :=\n  sorry\n\ntheorem nodupkeys.eq_of_mk_mem {α : Type u} {β : α → Type v} {a : α} {b : β a} {b' : β a}\n    {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) (h' : sigma.mk a b' ∈ l) :\n    b = b' :=\n  sorry\n\ntheorem nodupkeys_singleton {α : Type u} {β : α → Type v} (s : sigma β) : nodupkeys [s] :=\n  nodup_singleton (sigma.fst s)\n\ntheorem nodupkeys_of_sublist {α : Type u} {β : α → Type v} {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (h : l₁ <+ l₂) : nodupkeys l₂ → nodupkeys l₁ :=\n  nodup_of_sublist (sublist.map sigma.fst h)\n\ntheorem nodup_of_nodupkeys {α : Type u} {β : α → Type v} {l : List (sigma β)} :\n    nodupkeys l → nodup l :=\n  nodup_of_nodup_map sigma.fst\n\ntheorem perm_nodupkeys {α : Type u} {β : α → Type v} {l₁ : List (sigma β)} {l₂ : List (sigma β)}\n    (h : l₁ ~ l₂) : nodupkeys l₁ ↔ nodupkeys l₂ :=\n  perm.nodup_iff (perm.map sigma.fst h)\n\ntheorem nodupkeys_join {α : Type u} {β : α → Type v} {L : List (List (sigma β))} :\n    nodupkeys (join L) ↔\n        (∀ (l : List (sigma β)), l ∈ L → nodupkeys l) ∧ pairwise disjoint (map keys L) :=\n  sorry\n\ntheorem nodup_enum_map_fst {α : Type u} (l : List α) : nodup (map prod.fst (enum l)) := sorry\n\ntheorem mem_ext {α : Type u} {β : α → Type v} {l₀ : List (sigma β)} {l₁ : List (sigma β)}\n    (nd₀ : nodup l₀) (nd₁ : nodup l₁) (h : ∀ (x : sigma β), x ∈ l₀ ↔ x ∈ l₁) : l₀ ~ l₁ :=\n  sorry\n\n/- lookup -/\n\n/-- `lookup a l` is the first value in `l` corresponding to the key `a`,\n  or `none` if no such element exists. -/\ndef lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) : List (sigma β) → Option (β a) :=\n  sorry\n\n@[simp] theorem lookup_nil {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n    lookup a [] = none :=\n  rfl\n\n@[simp] theorem lookup_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α]\n    (l : List (sigma fun (a : α) => β a)) (a : α) (b : β a) :\n    lookup a (sigma.mk a b :: l) = some b :=\n  dif_pos rfl\n\n@[simp] theorem lookup_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β))\n    {a : α} (s : sigma β) : a ≠ sigma.fst s → lookup a (s :: l) = lookup a l :=\n  sorry\n\ntheorem lookup_is_some {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} :\n    ↥(option.is_some (lookup a l)) ↔ a ∈ keys l :=\n  sorry\n\ntheorem lookup_eq_none {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} :\n    lookup a l = none ↔ ¬a ∈ keys l :=\n  sorry\n\ntheorem of_mem_lookup {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} : b ∈ lookup a l → sigma.mk a b ∈ l :=\n  sorry\n\ntheorem mem_lookup {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) : b ∈ lookup a l :=\n  sorry\n\ntheorem map_lookup_eq_find {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    (l : List (sigma β)) :\n    option.map (sigma.mk a) (lookup a l) = find (fun (s : sigma β) => a = sigma.fst s) l :=\n  sorry\n\ntheorem mem_lookup_iff {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} (nd : nodupkeys l) : b ∈ lookup a l ↔ sigma.mk a b ∈ l :=\n  { mp := of_mem_lookup, mpr := mem_lookup nd }\n\ntheorem perm_lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) (p : l₁ ~ l₂) :\n    lookup a l₁ = lookup a l₂ :=\n  sorry\n\ntheorem lookup_ext {α : Type u} {β : α → Type v} [DecidableEq α] {l₀ : List (sigma β)}\n    {l₁ : List (sigma β)} (nd₀ : nodupkeys l₀) (nd₁ : nodupkeys l₁)\n    (h : ∀ (x : α) (y : β x), y ∈ lookup x l₀ ↔ y ∈ lookup x l₁) : l₀ ~ l₁ :=\n  sorry\n\n/- lookup_all -/\n\n/-- `lookup_all a l` is the list of all values in `l` corresponding to the key `a`. -/\ndef lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n    List (sigma β) → List (β a) :=\n  sorry\n\n@[simp] theorem lookup_all_nil {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n    lookup_all a [] = [] :=\n  rfl\n\n@[simp] theorem lookup_all_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α]\n    (l : List (sigma fun (a : α) => β a)) (a : α) (b : β a) :\n    lookup_all a (sigma.mk a b :: l) = b :: lookup_all a l :=\n  dif_pos rfl\n\n@[simp] theorem lookup_all_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α]\n    (l : List (sigma β)) {a : α} (s : sigma β) :\n    a ≠ sigma.fst s → lookup_all a (s :: l) = lookup_all a l :=\n  sorry\n\ntheorem lookup_all_eq_nil {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l : List (sigma β)} : lookup_all a l = [] ↔ ∀ (b : β a), ¬sigma.mk a b ∈ l :=\n  sorry\n\ntheorem head_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) :\n    head' (lookup_all a l) = lookup a l :=\n  sorry\n\ntheorem mem_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} : b ∈ lookup_all a l ↔ sigma.mk a b ∈ l :=\n  sorry\n\ntheorem lookup_all_sublist {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    (l : List (sigma β)) : map (sigma.mk a) (lookup_all a l) <+ l :=\n  sorry\n\ntheorem lookup_all_length_le_one {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    {l : List (sigma β)} (h : nodupkeys l) : length (lookup_all a l) ≤ 1 :=\n  sorry\n\ntheorem lookup_all_eq_lookup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    {l : List (sigma β)} (h : nodupkeys l) : lookup_all a l = option.to_list (lookup a l) :=\n  sorry\n\ntheorem lookup_all_nodup {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l : List (sigma β)}\n    (h : nodupkeys l) : nodup (lookup_all a l) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nodup (lookup_all a l))) (lookup_all_eq_lookup a h)))\n    (option.to_list_nodup (lookup a l))\n\ntheorem perm_lookup_all {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) (p : l₁ ~ l₂) :\n    lookup_all a l₁ = lookup_all a l₂ :=\n  sorry\n\n/- kreplace -/\n\ndef kreplace {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) :\n    List (sigma β) → List (sigma β) :=\n  lookmap\n    fun (s : sigma β) =>\n      dite (a = sigma.fst s) (fun (h : a = sigma.fst s) => some (sigma.mk a b))\n        fun (h : ¬a = sigma.fst s) => none\n\ntheorem kreplace_of_forall_not {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a)\n    {l : List (sigma β)} (H : ∀ (b : β a), ¬sigma.mk a b ∈ l) : kreplace a b l = l :=\n  sorry\n\ntheorem kreplace_self {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} (nd : nodupkeys l) (h : sigma.mk a b ∈ l) : kreplace a b l = l :=\n  sorry\n\ntheorem keys_kreplace {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a)\n    (l : List (sigma β)) : keys (kreplace a b l) = keys l :=\n  sorry\n\ntheorem kreplace_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a)\n    {l : List (sigma β)} : nodupkeys (kreplace a b l) ↔ nodupkeys l :=\n  sorry\n\ntheorem perm.kreplace {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd : nodupkeys l₁) :\n    l₁ ~ l₂ → kreplace a b l₁ ~ kreplace a b l₂ :=\n  sorry\n\n/- kerase -/\n\n/-- Remove the first pair with the key `a`. -/\ndef kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n    List (sigma β) → List (sigma β) :=\n  erasep fun (s : sigma β) => a = sigma.fst s\n\n@[simp] theorem kerase_nil {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} :\n    kerase a [] = [] :=\n  rfl\n\n@[simp] theorem kerase_cons_eq {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {s : sigma β}\n    {l : List (sigma β)} (h : a = sigma.fst s) : kerase a (s :: l) = l :=\n  sorry\n\n@[simp] theorem kerase_cons_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {s : sigma β}\n    {l : List (sigma β)} (h : a ≠ sigma.fst s) : kerase a (s :: l) = s :: kerase a l :=\n  sorry\n\n@[simp] theorem kerase_of_not_mem_keys {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l : List (sigma β)} (h : ¬a ∈ keys l) : kerase a l = l :=\n  sorry\n\ntheorem kerase_sublist {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (l : List (sigma β)) :\n    kerase a l <+ l :=\n  erasep_sublist l\n\ntheorem kerase_keys_subset {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    (l : List (sigma β)) : keys (kerase a l) ⊆ keys l :=\n  sublist.subset (sublist.map sigma.fst (kerase_sublist a l))\n\ntheorem mem_keys_of_mem_keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a₁ : α} {a₂ : α}\n    {l : List (sigma β)} : a₁ ∈ keys (kerase a₂ l) → a₁ ∈ keys l :=\n  kerase_keys_subset a₂ l\n\ntheorem exists_of_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)}\n    (h : a ∈ keys l) :\n    ∃ (b : β a),\n        ∃ (l₁ : List (sigma β)),\n          ∃ (l₂ : List (sigma β)),\n            ¬a ∈ keys l₁ ∧ l = l₁ ++ sigma.mk a b :: l₂ ∧ kerase a l = l₁ ++ l₂ :=\n  sorry\n\n@[simp] theorem mem_keys_kerase_of_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a₁ : α}\n    {a₂ : α} {l : List (sigma β)} (h : a₁ ≠ a₂) : a₁ ∈ keys (kerase a₂ l) ↔ a₁ ∈ keys l :=\n  sorry\n\ntheorem keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l : List (sigma β)} :\n    keys (kerase a l) = list.erase (keys l) a :=\n  sorry\n\ntheorem kerase_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α}\n    {l : List (sigma β)} : kerase a (kerase a' l) = kerase a' (kerase a l) :=\n  sorry\n\ntheorem kerase_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    {l : List (sigma β)} : nodupkeys l → nodupkeys (kerase a l) :=\n  nodupkeys_of_sublist (kerase_sublist a l)\n\ntheorem perm.kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (nd : nodupkeys l₁) : l₁ ~ l₂ → kerase a l₁ ~ kerase a l₂ :=\n  perm.erasep (fun (s : sigma β) => a = sigma.fst s)\n    (pairwise.imp\n      (fun (x y : sigma β) (h : sigma.fst x ≠ sigma.fst y) (ᾰ : a = sigma.fst x) =>\n        Eq._oldrec h (Eq.symm ᾰ))\n      (iff.mp nodupkeys_iff_pairwise nd))\n\n@[simp] theorem not_mem_keys_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    {l : List (sigma β)} (nd : nodupkeys l) : ¬a ∈ keys (kerase a l) :=\n  sorry\n\n@[simp] theorem lookup_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    {l : List (sigma β)} (nd : nodupkeys l) : lookup a (kerase a l) = none :=\n  iff.mpr lookup_eq_none (not_mem_keys_kerase a nd)\n\n@[simp] theorem lookup_kerase_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α}\n    {l : List (sigma β)} (h : a ≠ a') : lookup a (kerase a' l) = lookup a l :=\n  sorry\n\ntheorem kerase_append_left {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    a ∈ keys l₁ → kerase a (l₁ ++ l₂) = kerase a l₁ ++ l₂ :=\n  sorry\n\ntheorem kerase_append_right {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    ¬a ∈ keys l₁ → kerase a (l₁ ++ l₂) = l₁ ++ kerase a l₂ :=\n  sorry\n\ntheorem kerase_comm {α : Type u} {β : α → Type v} [DecidableEq α] (a₁ : α) (a₂ : α)\n    (l : List (sigma β)) : kerase a₂ (kerase a₁ l) = kerase a₁ (kerase a₂ l) :=\n  sorry\n\ntheorem sizeof_kerase {α : Type u_1} {β : α → Type u_2} [DecidableEq α] [SizeOf (sigma β)] (x : α)\n    (xs : List (sigma β)) : sizeof (kerase x xs) ≤ sizeof xs :=\n  sorry\n\n/- kinsert -/\n\n/-- Insert the pair `⟨a, b⟩` and erase the first pair with the key `a`. -/\ndef kinsert {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a) (l : List (sigma β)) :\n    List (sigma β) :=\n  sigma.mk a b :: kerase a l\n\n@[simp] theorem kinsert_def {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l : List (sigma β)} : kinsert a b l = sigma.mk a b :: kerase a l :=\n  rfl\n\ntheorem mem_keys_kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {b' : β a'}\n    {l : List (sigma β)} : a ∈ keys (kinsert a' b' l) ↔ a = a' ∨ a ∈ keys l :=\n  sorry\n\ntheorem kinsert_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) (b : β a)\n    {l : List (sigma β)} (nd : nodupkeys l) : nodupkeys (kinsert a b l) :=\n  iff.mpr nodupkeys_cons { left := not_mem_keys_kerase a nd, right := kerase_nodupkeys a nd }\n\ntheorem perm.kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (p : l₁ ~ l₂) :\n    kinsert a b l₁ ~ kinsert a b l₂ :=\n  perm.cons (sigma.mk a b) (perm.kerase nd₁ p)\n\ntheorem lookup_kinsert {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    (l : List (sigma β)) : lookup a (kinsert a b l) = some b :=\n  sorry\n\ntheorem lookup_kinsert_ne {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {a' : α} {b' : β a'}\n    {l : List (sigma β)} (h : a ≠ a') : lookup a (kinsert a' b' l) = lookup a l :=\n  sorry\n\n/- kextract -/\n\ndef kextract {α : Type u} {β : α → Type v} [DecidableEq α] (a : α) :\n    List (sigma β) → Option (β a) × List (sigma β) :=\n  sorry\n\n@[simp] theorem kextract_eq_lookup_kerase {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    (l : List (sigma β)) : kextract a l = (lookup a l, kerase a l) :=\n  sorry\n\n/- erase_dupkeys -/\n\n/-- Remove entries with duplicate keys from `l : list (sigma β)`. -/\ndef erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] : List (sigma β) → List (sigma β) :=\n  foldr (fun (x : sigma β) => kinsert (sigma.fst x) (sigma.snd x)) []\n\ntheorem erase_dupkeys_cons {α : Type u} {β : α → Type v} [DecidableEq α] {x : sigma β}\n    (l : List (sigma β)) :\n    erase_dupkeys (x :: l) = kinsert (sigma.fst x) (sigma.snd x) (erase_dupkeys l) :=\n  rfl\n\ntheorem nodupkeys_erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β)) :\n    nodupkeys (erase_dupkeys l) :=\n  sorry\n\ntheorem lookup_erase_dupkeys {α : Type u} {β : α → Type v} [DecidableEq α] (a : α)\n    (l : List (sigma β)) : lookup a (erase_dupkeys l) = lookup a l :=\n  sorry\n\ntheorem sizeof_erase_dupkeys {α : Type u_1} {β : α → Type u_2} [DecidableEq α] [SizeOf (sigma β)]\n    (xs : List (sigma β)) : sizeof (erase_dupkeys xs) ≤ sizeof xs :=\n  sorry\n\n/- kunion -/\n\n/-- `kunion l₁ l₂` is the append to l₁ of l₂ after, for each key in l₁, the\nfirst matching pair in l₂ is erased. -/\ndef kunion {α : Type u} {β : α → Type v} [DecidableEq α] :\n    List (sigma β) → List (sigma β) → List (sigma β) :=\n  sorry\n\n@[simp] theorem nil_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {l : List (sigma β)} :\n    kunion [] l = l :=\n  rfl\n\n@[simp] theorem kunion_nil {α : Type u} {β : α → Type v} [DecidableEq α] {l : List (sigma β)} :\n    kunion l [] = l :=\n  sorry\n\n@[simp] theorem kunion_cons {α : Type u} {β : α → Type v} [DecidableEq α] {s : sigma β}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    kunion (s :: l₁) l₂ = s :: kunion l₁ (kerase (sigma.fst s) l₂) :=\n  rfl\n\n@[simp] theorem mem_keys_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    a ∈ keys (kunion l₁ l₂) ↔ a ∈ keys l₁ ∨ a ∈ keys l₂ :=\n  sorry\n\n@[simp] theorem kunion_kerase {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    kunion (kerase a l₁) (kerase a l₂) = kerase a (kunion l₁ l₂) :=\n  sorry\n\ntheorem kunion_nodupkeys {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (nd₁ : nodupkeys l₁) (nd₂ : nodupkeys l₂) : nodupkeys (kunion l₁ l₂) :=\n  sorry\n\ntheorem perm.kunion_right {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} (p : l₁ ~ l₂) (l : List (sigma β)) : kunion l₁ l ~ kunion l₂ l :=\n  sorry\n\ntheorem perm.kunion_left {α : Type u} {β : α → Type v} [DecidableEq α] (l : List (sigma β))\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    nodupkeys l₁ → l₁ ~ l₂ → kunion l l₁ ~ kunion l l₂ :=\n  sorry\n\ntheorem perm.kunion {α : Type u} {β : α → Type v} [DecidableEq α] {l₁ : List (sigma β)}\n    {l₂ : List (sigma β)} {l₃ : List (sigma β)} {l₄ : List (sigma β)} (nd₃ : nodupkeys l₃)\n    (p₁₂ : l₁ ~ l₂) (p₃₄ : l₃ ~ l₄) : kunion l₁ l₃ ~ kunion l₂ l₄ :=\n  perm.trans (perm.kunion_right p₁₂ l₃) (perm.kunion_left l₂ nd₃ p₃₄)\n\n@[simp] theorem lookup_kunion_left {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : a ∈ keys l₁) :\n    lookup a (kunion l₁ l₂) = lookup a l₁ :=\n  sorry\n\n@[simp] theorem lookup_kunion_right {α : Type u} {β : α → Type v} [DecidableEq α] {a : α}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} (h : ¬a ∈ keys l₁) :\n    lookup a (kunion l₁ l₂) = lookup a l₂ :=\n  sorry\n\n@[simp] theorem mem_lookup_kunion {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} :\n    b ∈ lookup a (kunion l₁ l₂) ↔ b ∈ lookup a l₁ ∨ ¬a ∈ keys l₁ ∧ b ∈ lookup a l₂ :=\n  sorry\n\ntheorem mem_lookup_kunion_middle {α : Type u} {β : α → Type v} [DecidableEq α] {a : α} {b : β a}\n    {l₁ : List (sigma β)} {l₂ : List (sigma β)} {l₃ : List (sigma β)}\n    (h₁ : b ∈ lookup a (kunion l₁ l₃)) (h₂ : ¬a ∈ keys l₂) :\n    b ∈ lookup a (kunion (kunion l₁ l₂) l₃) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/sigma_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6187804478040616, "lm_q1q2_score": 0.31180728585121426}}
{"text": "/-- Encoding of arrays in Lean. It is defined to be `opaque` to ensure that we only use the\n    declared operations and axioms to reason about arrays. -/\nopaque A : (I : Sort u₁) → (E : Sort u₂) → Sort (max u₁ u₂)\n\nnamespace A\n\nvariable {I} [Nonempty I] {E} [Nonempty E] [Nonempty (A I E)]\n\n/-- The `read` operation for Arrays. It is defined to be `opaque` because its semantics are\n    provided by the axioms. -/\nnoncomputable opaque read : A I E → I → E\n\n/-- The `write` operation for Arrays. It is defined to be `opaque` because its semantics are\n    provided by the axioms. -/\nnoncomputable opaque write : A I E → I → E → A I E\n\naxiom rw1 : ∀ {a : A I E}, ∀ {i : I}, ∀ {v : E}, (a.write i v).read i = v\n\naxiom rw2 : ∀ {a : A I E}, ∀ {i j : I}, ∀ {v : E}, i ≠ j → (a.write i v).read j = a.read j\n\n/-- The extansionality axiom. -/\naxiom ex : ∀ {a b : A I E}, (∀ i : I, a.read i = b.read i) → a = b\n\ntheorem contrapositive : (p → q) → ¬q → ¬p := fun hpq hnq hp => hnq (hpq hp)\n\ntheorem forall_eq_not_exists_not {p : α → Prop} : (∀ x, p x) = ¬ ∃ x, ¬ p x :=\n  propext ⟨fun hfxpx ⟨x, hnpx⟩ => hnpx (hfxpx x),\n          fun henxnpx x => match Classical.em (p x) with\n            | Or.inl hpx  => hpx\n            | Or.inr hnpx => absurd ⟨x, hnpx⟩ henxnpx⟩\n\ntheorem double_neg: (¬¬p) = p :=\n  propext ⟨match Classical.em (¬p) with\n    | Or.inl g1 => fun h1  => absurd g1 h1\n    | Or.inr _ => match Classical.em (p) with\n                    | Or.inl g2 => fun _ => g2\n                    | Or.inr g2 => fun x => absurd g2 x,\n  match Classical.em (¬p) with\n    | Or.inl np => fun hp => absurd hp np\n    | Or.inr nnp => fun _ => nnp⟩\n\n/-- Theorem proving `RIntro1` rule from the axioms. -/\ntheorem r_intro1 {a b : A I E}: a = b.write i v → v = a.read i :=\n  fun h => h ▸ Eq.symm rw1\n\n/-- Theorem proving `RIntro2` rule from the axioms. -/\ntheorem r_intro2 {a b c : A I E} : a = c ∨ b = c → a = b.write i v → x = c.read j → i = j ∨ a.read j = b.read j :=\n  fun _ hw _ => match Classical.em (i = j) with\n  | Or.inl he => Or.inl he\n  | Or.inr hne => Or.inr (hw ▸ rw2 hne)\n\n/-- Theorem proving `Ext` rule from the axioms. -/\ntheorem ext {a b : A I E} : a ≠ b → ∃i, a.read i ≠ b.read i := fun hne =>\n  double_neg (p := ∃ i, read a i ≠ read b i) ▸\n  forall_eq_not_exists_not (p := (λ i => a.read i = b.read i)) ▸\n  (contrapositive ex) hne\n\nexample {a b : A I E} : a.write i (b.read i) = b.write i (a.read i) → a ≠ b → False := fun h0 h6 =>\n  let v := b.read i\n  have h3 : v = b.read i := rfl\n  let w := a.read i\n  have h4 : w = a.read i := rfl\n  let a' := a.write i v\n  have h1 : a' = a.write i v := rfl\n  let b' := b.write i w\n  have h2 : b' = b.write i w := rfl\n  have h5 : a' = b' := h1 ▸ h2 ▸ h3 ▸ h4 ▸ h0\n  lem h1 h2 h3 h4 h5 h6\nwhere\n  lem {a a' b b' : A I E} {v w : E} : a' = a.write i v → b' = b.write i w → v = b.read i → w = a.read i → a' = b' → a ≠ b → False :=\n    fun h1 h2 h3 h4 h5 h6 =>\n      have ⟨k, h9⟩ : ∃i, a.read i ≠ b.read i := ext h6\n      let x := read a k\n      have h7 : x = read a k := rfl\n      let y := read b k\n      have h8 : y = read b k := rfl\n      have h9 : x ≠ y := h7 ▸ h8 ▸ h9\n      have h10 : i = k ∨ a'.read k = a.read k := r_intro2 (Or.inr rfl) h1 h7\n      match h10 with\n      | Or.inl h10 =>\n        have h11 : x = w := h4 ▸ h10 ▸ h7\n        have h12 : v = y := h3 ▸ h10 ▸ h8\n        have h13 : v = a'.read i := r_intro1 h1\n        have h14 : w = b'.read i := r_intro1 h2\n        have h15 : w = v := h13 ▸ h5 ▸ h14\n        have h16 : x = y := h11 ▸ h12 ▸ h15\n        h9 h16\n      | Or.inr h10 =>\n        let x' := a'.read k\n        have h11 : x' = a'.read k := rfl\n        have h10 : x = x' := h7 ▸ h11 ▸ h10 ▸ rfl\n        have h12 : i = k ∨ b'.read k = b.read k := r_intro2 (Or.inr rfl) h2 h8\n        match h12 with\n        | Or.inl h12 =>\n          have h13 : x = w := h4 ▸ h12 ▸ h7\n          have h14 : v = y := h3 ▸ h12 ▸ h8\n          have h15 : v = a'.read i := r_intro1 h1\n          have h16 : w = b'.read i := r_intro1 h2\n          have h17 : w = v := h15 ▸ h5 ▸ h16\n          have h18 : x = y := h13 ▸ h14 ▸ h17\n          h9 h18\n        | Or.inr h12 =>\n          let y' := b'.read k\n          have h13 : y' = b'.read k := rfl\n          have h14 : y' = y := h8 ▸ h12 ▸ h13\n          have h15 : x' = y' := h11 ▸ h13 ▸ h5 ▸ rfl\n          have h16 : x = y := h10 ▸ h14 ▸ h15\n          h9 h16\n\nend A\n", "meta": {"author": "abdoo8080", "repo": "ar-project", "sha": "303af2d62cf8c8fe996c9670f9fe5a0cc90e5bb8", "save_path": "github-repos/lean/abdoo8080-ar-project", "path": "github-repos/lean/abdoo8080-ar-project/ar-project-303af2d62cf8c8fe996c9670f9fe5a0cc90e5bb8/LMT/A.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6334102636778403, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.31175701682601376}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport category_theory.concrete_category.bundled_hom\nimport algebra.punit_instances\nimport order.hom.basic\n\n/-!\n# Category of preorders\n\nThis defines `Preorder`, the category of preorders with monotone maps.\n-/\n\nuniverse u\n\nopen category_theory\n\n/-- The category of preorders. -/\ndef Preorder := bundled preorder\n\nnamespace Preorder\n\ninstance : bundled_hom @order_hom :=\n{ to_fun := @order_hom.to_fun,\n  id := @order_hom.id,\n  comp := @order_hom.comp,\n  hom_ext := @order_hom.ext }\n\nattribute [derive [large_category, concrete_category]] Preorder\n\ninstance : has_coe_to_sort Preorder Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled Preorder from the underlying type and typeclass. -/\ndef of (α : Type*) [preorder α] : Preorder := bundled.of α\n\n@[simp] lemma coe_of (α : Type*) [preorder α] : ↥(of α) = α := rfl\n\ninstance : inhabited Preorder := ⟨of punit⟩\n\ninstance (α : Preorder) : preorder α := α.str\n\n/-- Constructs an equivalence between preorders from an order isomorphism between them. -/\n@[simps] def iso.mk {α β : Preorder.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : Preorder ⥤ Preorder :=\n{ obj := λ X, of (order_dual X), map := λ X Y, order_hom.dual }\n\n/-- The equivalence between `Preorder` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : Preorder ≌ Preorder :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nend Preorder\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/Preorder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.31171026583597455}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.images\nimport Mathlib.PostPort\n\nuniverses v u l u_1 v' u' \n\nnamespace Mathlib\n\n/-!\n# Zero morphisms and zero objects\n\nA category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. (Notice this is extra\nstructure, not merely a property.)\n\nA category \"has a zero object\" if it has an object which is both initial and terminal. Having a\nzero object provides zero morphisms, as the unique morphisms factoring through the zero object.\n\n## References\n\n* https://en.wikipedia.org/wiki/Zero_morphism\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n-/\n\nnamespace category_theory.limits\n\n\n/-- A category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. -/\nclass has_zero_morphisms (C : Type u) [category C] where\n  has_zero : (X Y : C) → HasZero (X ⟶ Y)\n  comp_zero' :\n    autoParam (∀ {X Y : C} (f : X ⟶ Y), C → f ≫ 0 = 0)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  zero_comp' :\n    autoParam (C → ∀ {Y Z : C} (f : Y ⟶ Z), 0 ≫ f = 0)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem has_zero_morphisms.comp_zero {C : Type u} [category C] [c : has_zero_morphisms C] {X : C}\n    {Y : C} (f : X ⟶ Y) (Z : C) : f ≫ 0 = 0 :=\n  sorry\n\ntheorem has_zero_morphisms.zero_comp {C : Type u} [category C] [c : has_zero_morphisms C] (X : C)\n    {Y : C} {Z : C} (f : Y ⟶ Z) : 0 ≫ f = 0 :=\n  sorry\n\n@[simp] theorem comp_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C}\n    {f : X ⟶ Y} {Z : C} : f ≫ 0 = 0 :=\n  has_zero_morphisms.comp_zero f Z\n\n@[simp] theorem zero_comp {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C}\n    {f : Y ⟶ Z} : 0 ≫ f = 0 :=\n  has_zero_morphisms.zero_comp X f\n\nprotected instance has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) :=\n  has_zero_morphisms.mk\n\nprotected instance has_zero_morphisms_punit : has_zero_morphisms (discrete PUnit) :=\n  has_zero_morphisms.mk\n\nnamespace has_zero_morphisms\n\n\n/-- This lemma will be immediately superseded by `ext`, below. -/\n/--\nIf you're tempted to use this lemma \"in the wild\", you should probably\ncarefully consider whether you've made a mistake in allowing two\ninstances of `has_zero_morphisms` to exist at all.\n\nSee, particularly, the note on `zero_morphisms_of_zero_object` below.\n-/\ntheorem ext {C : Type u} [category C] (I : has_zero_morphisms C) (J : has_zero_morphisms C) :\n    I = J :=\n  sorry\n\nprotected instance subsingleton {C : Type u} [category C] : subsingleton (has_zero_morphisms C) :=\n  subsingleton.intro ext\n\nend has_zero_morphisms\n\n\ntheorem zero_of_comp_mono {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C}\n    {f : X ⟶ Y} (g : Y ⟶ Z) [mono g] (h : f ≫ g = 0) : f = 0 :=\n  eq.mp (Eq._oldrec (Eq.refl (f ≫ g = 0 ≫ g)) (propext (cancel_mono g)))\n    (eq.mp (Eq._oldrec (Eq.refl (f ≫ g = 0)) (Eq.symm zero_comp)) h)\n\ntheorem zero_of_epi_comp {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C}\n    (f : X ⟶ Y) {g : Y ⟶ Z} [epi f] (h : f ≫ g = 0) : g = 0 :=\n  eq.mp (Eq._oldrec (Eq.refl (f ≫ g = f ≫ 0)) (propext (cancel_epi f)))\n    (eq.mp (Eq._oldrec (Eq.refl (f ≫ g = 0)) (Eq.symm comp_zero)) h)\n\ntheorem eq_zero_of_image_eq_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C}\n    {f : X ⟶ Y} [has_image f] (w : image.ι f = 0) : f = 0 :=\n  sorry\n\ntheorem nonzero_image_of_nonzero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C}\n    {f : X ⟶ Y} [has_image f] (w : f ≠ 0) : image.ι f ≠ 0 :=\n  fun (h : image.ι f = 0) => w (eq_zero_of_image_eq_zero h)\n\ntheorem equivalence_preserves_zero_morphisms {C : Type u} [category C] (D : Type u') [category D]\n    [has_zero_morphisms C] [has_zero_morphisms D] (F : C ≌ D) (X : C) (Y : C) :\n    functor.map (equivalence.functor F) 0 = 0 :=\n  sorry\n\n@[simp] theorem is_equivalence_preserves_zero_morphisms {C : Type u} [category C] (D : Type u')\n    [category D] [has_zero_morphisms C] [has_zero_morphisms D] (F : C ⥤ D) [is_equivalence F]\n    (X : C) (Y : C) : functor.map F 0 = 0 :=\n  sorry\n\n/-- A category \"has a zero object\" if it has an object which is both initial and terminal. -/\nclass has_zero_object (C : Type u) [category C] where\n  zero : C\n  unique_to : (X : C) → unique (zero ⟶ X)\n  unique_from : (X : C) → unique (X ⟶ zero)\n\nprotected instance has_zero_object_punit : has_zero_object (discrete PUnit) :=\n  has_zero_object.mk PUnit.unit\n    (fun (X : discrete PUnit) => punit.cases_on X (unique.mk sorry sorry))\n    fun (X : discrete PUnit) => punit.cases_on X (unique.mk sorry sorry)\n\nnamespace has_zero_object\n\n\n/--\nConstruct a `has_zero C` for a category with a zero object.\nThis can not be a global instance as it will trigger for every `has_zero C` typeclass search.\n-/\nprotected def has_zero {C : Type u} [category C] [has_zero_object C] : HasZero C := { zero := zero }\n\ntheorem to_zero_ext {C : Type u} [category C] [has_zero_object C] {X : C} (f : X ⟶ 0) (g : X ⟶ 0) :\n    f = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f = g)) (unique.uniq (unique_from X) f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (Inhabited.default = g)) (unique.uniq (unique_from X) g)))\n      (Eq.refl Inhabited.default))\n\ntheorem from_zero_ext {C : Type u} [category C] [has_zero_object C] {X : C} (f : 0 ⟶ X)\n    (g : 0 ⟶ X) : f = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f = g)) (unique.uniq (unique_to X) f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (Inhabited.default = g)) (unique.uniq (unique_to X) g)))\n      (Eq.refl Inhabited.default))\n\nprotected instance category_theory.iso.subsingleton {C : Type u} [category C] [has_zero_object C]\n    (X : C) : subsingleton (X ≅ 0) :=\n  subsingleton.intro fun (a b : X ≅ 0) => iso.ext (of_as_true trivial)\n\nprotected instance category_theory.mono {C : Type u} [category C] [has_zero_object C] {X : C}\n    (f : 0 ⟶ X) : mono f :=\n  mono.mk fun (Z : C) (g h : Z ⟶ 0) (w : g ≫ f = h ≫ f) => to_zero_ext g h\n\nprotected instance category_theory.epi {C : Type u} [category C] [has_zero_object C] {X : C}\n    (f : X ⟶ 0) : epi f :=\n  epi.mk fun (Z : C) (g h : 0 ⟶ Z) (w : f ≫ g = f ≫ h) => from_zero_ext g h\n\n/-- A category with a zero object has zero morphisms.\n\n    It is rarely a good idea to use this. Many categories that have a zero object have zero\n    morphisms for some other reason, for example from additivity. Library code that uses\n    `zero_morphisms_of_zero_object` will then be incompatible with these categories because\n    the `has_zero_morphisms` instances will not be definitionally equal. For this reason library\n    code should generally ask for an instance of `has_zero_morphisms` separately, even if it already\n    asks for an instance of `has_zero_objects`. -/\ndef zero_morphisms_of_zero_object {C : Type u} [category C] [has_zero_object C] :\n    has_zero_morphisms C :=\n  has_zero_morphisms.mk\n\n/-- A zero object is in particular initial. -/\ntheorem has_initial {C : Type u} [category C] [has_zero_object C] : has_initial C :=\n  has_initial_of_unique 0\n\n/-- A zero object is in particular terminal. -/\ntheorem has_terminal {C : Type u} [category C] [has_zero_object C] : has_terminal C :=\n  has_terminal_of_unique 0\n\nend has_zero_object\n\n\n@[simp] theorem id_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] :\n    𝟙 = 0 :=\n  has_zero_object.from_zero_ext 𝟙 0\n\n/--  An arrow ending in the zero object is zero -/\n-- This can't be a `simp` lemma because the left hand side would be a metavariable.\n\ntheorem zero_of_to_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C}\n    (f : X ⟶ 0) : f = 0 :=\n  has_zero_object.to_zero_ext f 0\n\ntheorem zero_of_target_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    {X : C} {Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 :=\n  sorry\n\n/-- An arrow starting at the zero object is zero -/\ntheorem zero_of_from_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    {X : C} (f : 0 ⟶ X) : f = 0 :=\n  has_zero_object.from_zero_ext f 0\n\ntheorem zero_of_source_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    {X : C} {Y : C} (f : X ⟶ Y) (i : X ≅ 0) : f = 0 :=\n  sorry\n\ntheorem mono_of_source_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    {X : C} {Y : C} (f : X ⟶ Y) (i : X ≅ 0) : mono f :=\n  sorry\n\ntheorem epi_of_target_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    {X : C} {Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : epi f :=\n  sorry\n\n/--\nAn object `X` has `𝟙 X = 0` if and only if it is isomorphic to the zero object.\n\nBecause `X ≅ 0` contains data (even if a subsingleton), we express this `↔` as an `≃`.\n-/\ndef id_zero_equiv_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C]\n    (X : C) : 𝟙 = 0 ≃ (X ≅ 0) :=\n  equiv.mk (fun (h : 𝟙 = 0) => iso.mk 0 0) sorry sorry sorry\n\n@[simp] theorem id_zero_equiv_iso_zero_apply_hom {C : Type u} [category C] [has_zero_object C]\n    [has_zero_morphisms C] (X : C) (h : 𝟙 = 0) :\n    iso.hom (coe_fn (id_zero_equiv_iso_zero X) h) = 0 :=\n  rfl\n\n@[simp] theorem id_zero_equiv_iso_zero_apply_inv {C : Type u} [category C] [has_zero_object C]\n    [has_zero_morphisms C] (X : C) (h : 𝟙 = 0) :\n    iso.inv (coe_fn (id_zero_equiv_iso_zero X) h) = 0 :=\n  rfl\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\nthe identities on both `X` and `Y` are zero.\n-/\ndef is_iso_zero_equiv {C : Type u} [category C] [has_zero_morphisms C] (X : C) (Y : C) :\n    is_iso 0 ≃ 𝟙 = 0 ∧ 𝟙 = 0 :=\n  equiv.mk sorry (fun (h : 𝟙 = 0 ∧ 𝟙 = 0) => is_iso.mk 0) sorry sorry\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\nthe identity on `X` is zero.\n-/\ndef is_iso_zero_self_equiv {C : Type u} [category C] [has_zero_morphisms C] (X : C) :\n    is_iso 0 ≃ 𝟙 = 0 :=\n  eq.mpr sorry (eq.mp sorry (is_iso_zero_equiv X X))\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\n`X` and `Y` are isomorphic to the zero object.\n-/\ndef is_iso_zero_equiv_iso_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C]\n    (X : C) (Y : C) : is_iso 0 ≃ (X ≅ 0) × (Y ≅ 0) :=\n  equiv.trans (is_iso_zero_equiv X Y)\n    (equiv.symm\n      (equiv.mk sorry\n        (fun (ᾰ : 𝟙 = 0 ∧ 𝟙 = 0) =>\n          and.dcases_on ᾰ\n            fun (hX : 𝟙 = 0) (hY : 𝟙 = 0) =>\n              (coe_fn (id_zero_equiv_iso_zero X) hX, coe_fn (id_zero_equiv_iso_zero Y) hY))\n        sorry sorry))\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\n`X` is isomorphic to the zero object.\n-/\ndef is_iso_zero_self_equiv_iso_zero {C : Type u} [category C] [has_zero_morphisms C]\n    [has_zero_object C] (X : C) : is_iso 0 ≃ (X ≅ 0) :=\n  equiv.trans (is_iso_zero_equiv_iso_zero X X) subsingleton_prod_self_equiv\n\n/-- If there are zero morphisms, any initial object is a zero object. -/\nprotected instance has_zero_object_of_has_initial_object {C : Type u} [category C]\n    [has_zero_morphisms C] [has_initial C] : has_zero_object C :=\n  has_zero_object.mk (⊥_C) (fun (X : C) => unique.mk { default := 0 } sorry)\n    fun (X : C) => unique.mk { default := 0 } sorry\n\n/-- If there are zero morphisms, any terminal object is a zero object. -/\nprotected instance has_zero_object_of_has_terminal_object {C : Type u} [category C]\n    [has_zero_morphisms C] [has_terminal C] : has_zero_object C :=\n  has_zero_object.mk (⊤_C) (fun (X : C) => unique.mk { default := 0 } sorry)\n    fun (X : C) => unique.mk { default := 0 } sorry\n\ntheorem image_ι_comp_eq_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C}\n    {Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image f] [epi (factor_thru_image f)] (h : f ≫ g = 0) :\n    image.ι f ≫ g = 0 :=\n  sorry\n\n/--\nThe zero morphism has a `mono_factorisation` through the zero object.\n-/\n@[simp] theorem mono_factorisation_zero_e {C : Type u} [category C] [has_zero_morphisms C]\n    [has_zero_object C] (X : C) (Y : C) : mono_factorisation.e (mono_factorisation_zero X Y) = 0 :=\n  Eq.refl (mono_factorisation.e (mono_factorisation_zero X Y))\n\n/--\nThe factorisation through the zero object is an image factorisation.\n-/\ndef image_factorisation_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C]\n    (X : C) (Y : C) : image_factorisation 0 :=\n  image_factorisation.mk (mono_factorisation_zero X Y)\n    (is_image.mk fun (F' : mono_factorisation 0) => 0)\n\nprotected instance has_image_zero {C : Type u} [category C] [has_zero_morphisms C]\n    [has_zero_object C] {X : C} {Y : C} : has_image 0 :=\n  has_image.mk (image_factorisation_zero X Y)\n\n/-- The image of a zero morphism is the zero object. -/\ndef image_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C}\n    {Y : C} : image 0 ≅ 0 :=\n  is_image.iso_ext (image.is_image 0) (image_factorisation.is_image (image_factorisation_zero X Y))\n\n/-- The image of a morphism which is equal to zero is the zero object. -/\ndef image_zero' {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C} {Y : C}\n    {f : X ⟶ Y} (h : f = 0) [has_image f] : image f ≅ 0 :=\n  image.eq_to_iso h ≪≫ image_zero\n\n@[simp] theorem image.ι_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C]\n    {X : C} {Y : C} [has_image 0] : image.ι 0 = 0 :=\n  sorry\n\n/--\nIf we know `f = 0`,\nit requires a little work to conclude `image.ι f = 0`,\nbecause `f = g` only implies `image f ≅ image g`.\n-/\n@[simp] theorem image.ι_zero' {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C]\n    [has_equalizers C] {X : C} {Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] : image.ι f = 0 :=\n  sorry\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_sigma_ι {C : Type u} [category C] {β : Type v} [DecidableEq β]\n    [has_zero_morphisms C] (f : β → C) [has_colimit (discrete.functor f)] (b : β) :\n    split_mono (sigma.ι f b) :=\n  split_mono.mk\n    (sigma.desc\n      fun (b' : β) =>\n        dite (b' = b) (fun (h : b' = b) => eq_to_hom (congr_arg f h)) fun (h : ¬b' = b) => 0)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_pi_π {C : Type u} [category C] {β : Type v} [DecidableEq β]\n    [has_zero_morphisms C] (f : β → C) [has_limit (discrete.functor f)] (b : β) :\n    split_epi (pi.π f b) :=\n  split_epi.mk\n    (pi.lift\n      fun (b' : β) =>\n        dite (b = b') (fun (h : b = b') => eq_to_hom (congr_arg f h)) fun (h : ¬b = b') => 0)\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_coprod_inl {C : Type u} [category C] [has_zero_morphisms C] {X : C}\n    {Y : C} [has_colimit (pair X Y)] : split_mono coprod.inl :=\n  split_mono.mk (coprod.desc 𝟙 0)\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_coprod_inr {C : Type u} [category C] [has_zero_morphisms C] {X : C}\n    {Y : C} [has_colimit (pair X Y)] : split_mono coprod.inr :=\n  split_mono.mk (coprod.desc 0 𝟙)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_prod_fst {C : Type u} [category C] [has_zero_morphisms C] {X : C}\n    {Y : C} [has_limit (pair X Y)] : split_epi prod.fst :=\n  split_epi.mk (prod.lift 𝟙 0)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_prod_snd {C : Type u} [category C] [has_zero_morphisms C] {X : C}\n    {Y : C} [has_limit (pair X Y)] : split_epi prod.snd :=\n  split_epi.mk (prod.lift 0 𝟙)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/zero_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3117102658359745}}
{"text": "import category_theory.functor.category -- this transitively imports\n-- category_theory.category\n-- category_theory.functor\n-- category_theory.natural_transformation\n\n/-!\n# An introduction to category theory in Lean\n\nThis is an introduction to the basic usage of category theory (in the mathematical sense) in Lean.\nWe cover how the basic theory of categories, functors and natural transformations is set up in Lean.\nMost of the below is not hard to read off from the files `category_theory/category.lean`,\n`category_theory/functor.lean` and `category_theory/natural_transformation.lean`.\n\n## Overview\n\nA category is a collection of objects, and a collection of morphisms (also known as arrows) between\nthe objects. The objects and morphisms have some extra structure and satisfy some axioms -- see the\n[definition on Wikipedia](https://en.wikipedia.org/wiki/Category_%28mathematics%29#Definition) for\ndetails.\n\nOne important thing to note is that a morphism in an abstract category may not be an actual function\nbetween two types. In particular, there is new notation `⟶` , typed as `\\h` or `\\hom` in VS Code,\nfor a morphism. Nevertheless, in most of the \"concrete\" categories like `Top` and `Ab`, it is still\npossible to write `f x` when `x : X` and `f : X ⟶ Y` is a morphism, as there is an automatic\ncoercion from morphisms to functions. (If the coercion doesn't fire automatically, sometimes it is\nnecessary to write `(f : X → Y) x`.)\n\nIn some fonts the `⟶` morphism arrow can be virtually indistinguishable from the standard function\narrow `→` . You may want to install the [Deja Vu Sans Mono](https://dejavu-fonts.github.io/) and put\nthat at the beginning of the `Font Family` setting in VSCode, to get a nice readable font with\nexcellent unicode coverage.\n\nAnother point of confusion can be universe issues. Following Lean's conventions for universe\npolymorphism, the objects of a category might live in one universe `u` and the morphisms in another\nuniverse `v`. Note that in many categories showing up in \"set-theoretic mathematics\", the morphisms\nbetween two objects often form a set, but the objects themselves may or may not form a set. In Lean\nthis corresponds to the two possibilities `u=v` and `u=v+1`, known as `small_category` and\n`large_category` respectively. In order to avoid proving the same statements for both small and\nlarge categories, we usually stick to the general polymorphic situation with `u` and `v` independent\nuniverses, and we do this below.\n\n## Getting started with categories\n\nThe structure of a category on a type `C` in Lean is done using typeclasses; terms of `C` then\ncorrespond to objects in the category. The convention in the category theory library is to use\nuniverses prefixed with `u` (e.g. `u`, `u₁`, `u₂`) for the objects, and universes prefixed with `v`\nfor morphisms. Thus we have `C : Type u`, and if `X : C` and `Y : C` then morphisms `X ⟶ Y : Type v`\n(note the non-standard arrow).\n\nWe set this up as follows:\n-/\n\nopen category_theory\n\nsection category\n\nvariables (C : Type*) [category C]\n\nvariables {W X Y Z : C}\nvariables (f : W ⟶ X) (g : X ⟶ Y) (h : Y ⟶ Z)\n\n/-!\nThis says \"let `C` be a category, let `W`, `X`, `Y`, `Z` be objects of `C`, and let `f : W ⟶ X`, `g\n: X ⟶ Y` and `h : Y ⟶ Z` be morphisms in `C` (with the specified source and targets)\".\n\nNote that we sometimes need to explicitly tell Lean the universe that the morphisms live in,\nby writing `category.{v} C`, because Lean cannot guess this from `C` alone.\nHowever just writing `category C` is often fine: this allows a \"free\" universe level.\n\nThe order in which universes are introduced at the top of the file matters:\nwe put the universes for morphisms first (typically `v`, `v₁` and so on),\nand then universes for objects (typically `u`, `u₁` and so on).\nThis ensures that in any new definition we make the universe variables for morphisms come first,\nso that they can be explicitly specified while still allowing the universe levels of the\nobjects to be inferred automatically.\n\n## Basic notation\n\nIn categories one has morphisms between objects, such as the identity morphism from an object to\nitself. One can compose morphisms, and there are standard facts about the composition of a morphism\nwith the identity morphism, and the fact that morphism composition is associative. In Lean all of\nthis looks like the following:\n-/\n\n-- The identity morphism from `X` to `X` (remember that this is the `\\h` arrow):\nexample : X ⟶ X := 𝟙 X -- type `𝟙` as `\\bb1`\n\n-- Function composition `h ∘ g`, a morphism from `X` to `Z`:\nexample : X ⟶ Z := g ≫ h\n\n/-\nNote in particular the order! The \"maps on the right\" convention was chosen; `g ≫ h` means \"`g` then\n`h`\". Type `≫` with `\\gg` in VS Code. Here are the theorems which ensure that we have a category.\n-/\n\nopen category_theory.category\n\nexample : 𝟙 X ≫ g = g := id_comp g\nexample : g ≫ 𝟙 Y = g := comp_id g\nexample : (f ≫ g) ≫ h = f ≫ (g ≫ h) := assoc f g h\nexample : (f ≫ g) ≫ h = f ≫ g ≫ h := assoc f g h -- note \\gg is right associative\n\n-- All four examples above can also be proved with `simp`.\n\n-- Monomorphisms and epimorphisms are predicates on morphisms and are implemented as typeclasses.\nvariables (f' : W ⟶ X) (h' : Y ⟶ Z)\n\nexample [mono g] : f ≫ g = f' ≫ g → f = f' := mono.right_cancellation f f'\nexample [epi g] : g ≫ h = g ≫ h' → h = h' := epi.left_cancellation h h'\n\nend category -- end of section\n\n/-!\n## Getting started with functors\n\nA functor is a map between categories. It is implemented as a structure. The notation for a functor\nfrom `C` to `D` is `C ⥤ D`. Type `\\func` in VS Code for the symbol. Here we demonstrate how to\nevaluate functors on objects and on morphisms, how to show functors preserve the identity morphism\nand composition of morphisms, how to compose functors, and show the notation `𝟭` for the identity\nfunctor.\n-/\n\nsection functor\n\nvariables (C : Type*) [category C]\nvariables (D : Type*) [category D]\nvariables (E : Type*) [category E]\n\nvariables {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n\n-- functors\nvariables (F : C ⥤ D) (G : D ⥤ E)\n\nexample : D := F.obj X -- functor F on objects\nexample : F.obj Y ⟶ F.obj Z := F.map g -- functor F on morphisms\n\n-- A functor sends identity objects to identity objects\nexample : F.map (𝟙 X) = 𝟙 (F.obj X) := F.map_id X\n\n-- and preserves compositions\nexample : F.map (f ≫ g) = (F.map f) ≫ (F.map g) := F.map_comp f g\n\n-- The identity functor is `𝟭`, which you can write as `\\sb1`.\nexample : C ⥤ C := 𝟭 C\n\n-- The identity functor is (definitionally) the identity on objects and morphisms:\nexample : (𝟭 C).obj X = X := category_theory.functor.id_obj X\nexample : (𝟭 C).map f = f := category_theory.functor.id_map f\n\n-- Composition of functors; note order:\nexample : C ⥤ E := F ⋙ G -- typeset with `\\ggg`\n\n-- Composition of the identity either way does nothing:\nexample : F ⋙ 𝟭 D = F := F.comp_id\nexample : 𝟭 C ⋙ F = F := F.id_comp\n\n-- Composition of functors definitionally does the right thing on objects and morphisms:\nexample : (F ⋙ G).obj X = G.obj (F.obj X) := F.comp_obj G X -- or rfl\nexample : (F ⋙ G).map f = G.map (F.map f) := rfl -- or F.comp_map G X Y f\n\nend functor -- end of section\n\n/-!\nOne can also check that associativity of composition of functors is definitionally true,\nalthough we've observed that relying on this can result in slow proofs. (One should\nrather use the natural isomorphisms provided in `src/category_theory/whiskering.lean`.)\n\n## Getting started with natural transformations\n\nA natural transformation is a morphism between functors. If `F` and `G` are functors from `C` to `D`\nthen a natural transformation is a map `F X ⟶ G X` for each object `X : C` plus the theorem that if\n`f : X ⟶ Y` is a morphism then the two routes from `F X` to `G Y` are the same. One might imagine\nthat this is now another layer of notation, but fortunately the `category_theory.functor_category`\nimport gives the type of functors from `C` to `D` a category structure, which means that we can just\nuse morphism notation for natural transformations.\n-/\n\nsection nat_trans\n\nvariables {C : Type*} [category C] {D : Type*} [category D]\n\nvariables (X Y : C)\n\nvariable (f : X ⟶ Y)\n\nvariables (F G H : C ⥤ D)\n\nvariables (α : F ⟶ G) (β : G ⟶ H) -- natural transformations (note it's the usual `\\hom` arrow here)\n\n-- Composition of natural transformations is just composition of morphisms:\nexample : F ⟶ H := α ≫ β\n\n-- Applying natural transformation to an object:\nexample (X : C) : F.obj X ⟶ G.obj X := α.app X\n\n/- The diagram coming from g and α\n\n    F(f)\nF X ---> F Y\n |        |\n |α(X)    |α(Y)\n v        v\nG X ---> G Y\n    G(f)\n\ncommutes.\n-/\n\nexample : F.map f ≫ α.app Y = (α.app X) ≫ G.map f := α.naturality f\n\nend nat_trans -- section\n\n/-!\n## What next?\n\nThere are several lean files in the [category theory docs directory of\nmathlib](https://github.com/leanprover-community/mathlib/tree/master/docs/tutorial/category_theory)\nwhich give further examples of using the category theory library in Lean.\n-/\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/docs/tutorial/category_theory/intro.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837477, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.31165398285636964}}
{"text": "import tactic.norm_cast\n\nstructure hom (α β : Type) :=\n(to_fun : α → β)\n\ninstance {α β} : has_coe_to_fun (hom α β) (λ _, α → β) :=\n⟨hom.to_fun⟩\n\nstructure hom_plus (α β) extends hom α β\n\ninstance {α β} : has_coe (hom_plus α β) (hom α β) :=\n⟨hom_plus.to_hom⟩\n\ninstance {α β} : has_coe_to_fun (hom_plus α β) (λ _, α → β) := ⟨λ f, f⟩\n\n@[norm_cast] lemma hom_plus.coe_fn (α β : Type) (a : α) (h : hom_plus α β) :\n  (h : hom α β) a = h a :=\nrfl\n\nexample (α β : Type) (a : α) (h : hom_plus α β) :\n  (h : hom α β) a = h a :=\nby norm_cast\n\ndef f1 (n : ℕ) : hom_plus ℕ ℕ := ⟨⟨λ m, m + n⟩⟩\ndef f2 : hom ℕ (hom ℕ ℕ) := ⟨λ n, f1 n⟩\n\n@[norm_cast] lemma coe_f1 (n : ℕ) :\n  (f1 n : hom ℕ ℕ) = f2 n :=\nrfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/norm_cast_coe_fn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3116539745084412}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport data.dlist\n\n/-!\n# Difference list\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides a few results about `dlist`, which is defined in core Lean.\n\nA difference list is a function that, given a list, returns the original content of the\ndifference list prepended to the given list. It is useful to represent elements of a given type\nas `a₁ + ... + aₙ` where `+ : α → α → α` is any operation, without actually computing.\n\nThis structure supports `O(1)` `append` and `concat` operations on lists, making it\nuseful for append-heavy uses such as logging and pretty printing.\n-/\n\n/-- Concatenates a list of difference lists to form a single difference list. Similar to\n`list.join`. -/\ndef dlist.join {α : Type*} : list (dlist α) → dlist α\n | [] := dlist.empty\n | (x :: xs) := x ++ dlist.join xs\n\n@[simp] lemma dlist_singleton {α : Type*} {a : α} :\n  dlist.singleton a = dlist.lazy_of_list ([a]) := rfl\n\n@[simp] lemma dlist_lazy {α : Type*} {l : list α} :\n  dlist.lazy_of_list l = dlist.of_list l := rfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/dlist/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3116539745084412}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.elementwise\n! leanprover-community/mathlib commit d6814c584384ddf2825ff038e868451a7c956f31\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.ConcreteCategory.Basic\nimport Mathbin.Tactic.FreshNames\nimport Mathbin.Tactic.ReassocAxiom\nimport Mathbin.Tactic.Slice\n\n/-!\n# Tools to reformulate category-theoretic lemmas in concrete categories\n\n## The `elementwise` attribute\n\nThe `elementwise` attribute can be applied to a lemma\n\n```lean\n@[elementwise]\nlemma some_lemma {C : Type*} [category C]\n  {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (h : X ⟶ Z) (w : ...) : f ≫ g = h := ...\n```\n\nand will produce\n\n```lean\nlemma some_lemma_apply {C : Type*} [category C] [concrete_category C]\n  {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (h : X ⟶ Z) (w : ...) (x : X) : g (f x) = h x := ...\n```\n\nHere `X` is being coerced to a type via `concrete_category.has_coe_to_sort` and\n`f`, `g`, and `h` are being coerced to functions via `concrete_category.has_coe_to_fun`.\nFurther, we simplify the type using `concrete_category.coe_id : ((𝟙 X) : X → X) x = x` and\n`concrete_category.coe_comp : (f ≫ g) x = g (f x)`,\nreplacing morphism composition with function composition.\n\nThe name of the produced lemma can be specified with `@[elementwise other_lemma_name]`.\nIf `simp` is added first, the generated lemma will also have the `simp` attribute.\n\n## Implementation\n\nThis closely follows the implementation of the `@[reassoc]` attribute, due to Simon Hudon.\nThanks to Gabriel Ebner for help diagnosing universe issues.\n\n-/\n\n\nnamespace Tactic\n\nopen Interactive Lean.Parser CategoryTheory\n\n/-- From an expression `f = g`,\nwhere `f g : X ⟶ Y` for some objects `X Y : V` with `[S : category V]`,\nextract the expression for `S`.\n-/\nunsafe def extract_category : expr → tactic expr\n  |\n  q(@Eq (@Quiver.Hom _ (@CategoryStruct.toQuiver _ (@Category.toCategoryStruct _ $(S))) _ _) _ _) =>\n    pure S\n  | _ => failed\n#align tactic.extract_category tactic.extract_category\n\n-- This is closely modelled on `reassoc_axiom`.\n/-- (internals for `@[elementwise]`)\nGiven a lemma of the form `f = g`, where `f g : X ⟶ Y` and `X Y : V`,\nproves a new lemma of the form\n`∀ (x : X), f x = g x`\nif we are already in a concrete category, or\n`∀ [concrete_category.{w} V] (x : X), f x = g x`\notherwise.\n\nReturns the type and proof of this lemma,\nand the universe parameter `w` for the `concrete_category` instance, if it was not synthesized.\n-/\nunsafe def prove_elementwise (h : expr) : tactic (expr × expr × Option Name) := do\n  let (vs, t) ← infer_type h >>= open_pis\n  let (f, g) ← match_eq t\n  let S ← extract_category t <|> fail \"no morphism equation found in statement\"\n  let q(@Quiver.Hom _ $(H) $(X) $(Y)) ← infer_type f\n  let C ← infer_type X\n  let CC_type ← to_expr ``(@ConcreteCategory $(C) $(S))\n  let (CC, CC_found) ←\n    (do\n          let CC ← mk_instance CC_type\n          pure (CC, tt)) <|>\n        do\n        let CC ← mk_local' `I BinderInfo.inst_implicit CC_type\n        pure (CC, ff)\n  let CC_type\n    ←-- This is need to fill in universe levels fixed by `mk_instance`:\n        instantiate_mvars\n        CC_type\n  let x_type ←\n    to_expr ``(@coeSort $(C) _ (@CategoryTheory.ConcreteCategory.hasCoeToSort $(C) $(S) $(CC)) $(X))\n  let x ← mk_local_def `x x_type\n  let t' ←\n    to_expr\n        ``(@coeFn (@Quiver.Hom $(C) $(H) $(X) $(Y)) _\n              (@CategoryTheory.ConcreteCategory.hasCoeToFun $(C) $(S) $(CC) $(X) $(Y)) $(f) $(x) =\n            @coeFn (@Quiver.Hom $(C) $(H) $(X) $(Y)) _\n              (@CategoryTheory.ConcreteCategory.hasCoeToFun $(C) $(S) $(CC) $(X) $(Y)) $(g) $(x))\n  let c' := h.mk_app vs\n  let (_, pr) ← solve_aux t' (andthen (rewrite_target c') reflexivity)\n  let-- The codomain of forget lives in a new universe, which may be now a universe metavariable\n    -- if we didn't synthesize an instance:\n    [w, _, _]\n    ← pure CC_type.get_app_fn.univ_levels\n  let n\n    ←-- We unify that with a fresh universe parameter.\n      match w with\n      | level.mvar _ => do\n        let n ← get_unused_name_reserved [`w] mk_name_set\n        unify (expr.sort (level.param n)) (expr.sort w)\n        pure (Option.some n)\n      | _ => pure Option.none\n  let t' ← instantiate_mvars t'\n  let CC ← instantiate_mvars CC\n  let x ← instantiate_mvars x\n  let-- Now the key step: replace morphism composition with function composition,\n  -- and identity morphisms with nothing.\n  s := simp_lemmas.mk\n  let s ← s.add_simp `` id_apply\n  let s ← s.add_simp `` comp_apply\n  let (t'', pr', _) ← simplify s [] t' { failIfUnchanged := false }\n  let pr' ← mk_eq_mp pr' pr\n  let-- Further, if we're in `Type`, get rid of the coercions entirely.\n  s := simp_lemmas.mk\n  let s ← s.add_simp `` concrete_category.has_coe_to_fun_Type\n  let (t'', pr'', _) ← simplify s [] t'' { failIfUnchanged := false }\n  let pr'' ← mk_eq_mp pr'' pr'\n  let t'' ← pis (vs ++ if CC_found then [x] else [CC, x]) t''\n  let pr'' ← lambdas (vs ++ if CC_found then [x] else [CC, x]) pr''\n  pure (t'', pr'', n)\n#align tactic.prove_elementwise tactic.prove_elementwise\n\n/-- (implementation for `@[elementwise]`)\nGiven a declaration named `n` of the form `∀ ..., f = g`, proves a new lemma named `n'`\nof the form `∀ ... [concrete_category V] (x : X), f x = g x`.\n-/\nunsafe def elementwise_lemma (n : Name) (n' : Name := n.appendSuffix \"_apply\") : tactic Unit := do\n  let d ← get_decl n\n  let c := @expr.const true n d.univ_levels\n  let (t'', pr', l') ← prove_elementwise c\n  let params := l'.toList ++ d.univ_params\n  add_decl <| declaration.thm n' params t'' (pure pr')\n  copy_attribute `simp n n'\n#align tactic.elementwise_lemma tactic.elementwise_lemma\n\n/-- The `elementwise` attribute can be applied to a lemma\n\n```lean\n@[elementwise]\nlemma some_lemma {C : Type*} [category C]\n  {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (h : X ⟶ Z) (w : ...) : f ≫ g = h := ...\n```\n\nand will produce\n\n```lean\nlemma some_lemma_apply {C : Type*} [category C] [concrete_category C]\n  {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) (h : X ⟶ Z) (w : ...) (x : X) : g (f x) = h x := ...\n```\n\nHere `X` is being coerced to a type via `concrete_category.has_coe_to_sort` and\n`f`, `g`, and `h` are being coerced to functions via `concrete_category.has_coe_to_fun`.\nFurther, we simplify the type using `concrete_category.coe_id : ((𝟙 X) : X → X) x = x` and\n`concrete_category.coe_comp : (f ≫ g) x = g (f x)`,\nreplacing morphism composition with function composition.\n\nThe `[concrete_category C]` argument will be omitted if it is possible to synthesize an instance.\n\nThe name of the produced lemma can be specified with `@[elementwise other_lemma_name]`.\nIf `simp` is added first, the generated lemma will also have the `simp` attribute.\n-/\n@[user_attribute]\nunsafe def elementwise_attr : user_attribute Unit (Option Name)\n    where\n  Name := `elementwise\n  descr := \"create a companion lemma for a morphism equation applied to an element\"\n  parser := optional ident\n  after_set :=\n    some fun n _ _ => do\n      let some n' ← elementwise_attr.get_param n |\n        elementwise_lemma n (n.appendSuffix \"_apply\")\n      elementwise_lemma n <| n ++ n'\n#align tactic.elementwise_attr tactic.elementwise_attr\n\nadd_tactic_doc\n  { Name := \"elementwise\"\n    category := DocCategory.attr\n    declNames := [`tactic.elementwise_attr]\n    tags := [\"category theory\"] }\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- `elementwise h`, for assumption `w : ∀ ..., f ≫ g = h`, creates a new assumption\n`w : ∀ ... (x : X), g (f x) = h x`.\n\n`elementwise! h`, does the same but deletes the initial `h` assumption.\n(You can also add the attribute `@[elementwise]` to lemmas to generate new declarations generalized\nin this way.)\n-/\nunsafe def elementwise (del : parse (tk \"!\")?) (ns : parse ident*) : tactic Unit := do\n  ns fun n => do\n      let h ← get_local n\n      let (t, pr, u) ← prove_elementwise h\n      assertv n t pr\n      when del (tactic.clear h)\n#align tactic.interactive.elementwise tactic.interactive.elementwise\n\nend Interactive\n\n/-- Auxiliary definition for `category_theory.elementwise_of`. -/\nunsafe def derive_elementwise_proof : tactic Unit := do\n  let q(CalculatedProp $(v) $(h)) ← target\n  let (t, pr, n) ← prove_elementwise h\n  unify v t\n  exact pr\n#align tactic.derive_elementwise_proof tactic.derive_elementwise_proof\n\nend Tactic\n\n/-- With `w : ∀ ..., f ≫ g = h` (with universal quantifiers tolerated),\n`elementwise_of w : ∀ ... (x : X), g (f x) = h x`.\n\nThe type and proof of `elementwise_of h` is generated by `tactic.derive_elementwise_proof`\nwhich makes `elementwise_of` meta-programming adjacent. It is not called as a tactic but as\nan expression. The goal is to avoid creating assumptions that are dismissed after one use:\n\n```lean\nexample (M N K : Mon.{u}) (f : M ⟶ N) (g : N ⟶ K) (h : M ⟶ K) (w : f ≫ g = h) (m : M) :\n  g (f m) = h m :=\nbegin\n  rw elementwise_of w,\nend\n```\n-/\ntheorem CategoryTheory.elementwise_of {α} (hh : α) {β}\n    (x : Tactic.CalculatedProp β hh := by derive_elementwise_proof) : β :=\n  x\n#align category_theory.elementwise_of CategoryTheory.elementwise_of\n\n/-- With `w : ∀ ..., f ≫ g = h` (with universal quantifiers tolerated),\n`elementwise_of w : ∀ ... (x : X), g (f x) = h x`.\n\nAlthough `elementwise_of` is not a tactic or a meta program, its type is generated\nthrough meta-programming to make it usable inside normal expressions.\n-/\nadd_tactic_doc\n  { Name := \"category_theory.elementwise_of\"\n    category := DocCategory.tactic\n    declNames := [`category_theory.elementwise_of]\n    tags := [\"category theory\"] }\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Elementwise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.5506073655352403, "lm_q1q2_score": 0.31165397450844107}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.bundled_hom\nimport Mathlib.category_theory.concrete_category.reflects_isomorphisms\nimport Mathlib.algebra.punit_instances\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Category instances for monoid, add_monoid, comm_monoid, and add_comm_monoid.\n\nWe introduce the bundled categories:\n* `Mon`\n* `AddMon`\n* `CommMon`\n* `AddCommMon`\nalong with the relevant forgetful functors between them.\n-/\n\n/-- The category of monoids and monoid morphisms. -/\ndef Mon :=\n  category_theory.bundled monoid\n\n/-- The category of additive monoids and monoid morphisms. -/\nnamespace Mon\n\n\nprotected instance Mathlib.AddMon.bundled_hom : category_theory.bundled_hom add_monoid_hom :=\n  category_theory.bundled_hom.mk add_monoid_hom.to_fun add_monoid_hom.id add_monoid_hom.comp\n\nprotected instance Mathlib.AddMon.concrete_category : category_theory.concrete_category AddMon :=\n  category_theory.bundled_hom.category_theory.bundled.category_theory.concrete_category add_monoid_hom\n\n/-- Construct a bundled `Mon` from the underlying type and typeclass. -/\ndef Mathlib.AddMon.of (M : Type u) [add_monoid M] : AddMon :=\n  category_theory.bundled.of M\n\n/-- Construct a bundled `Mon` from the underlying type and typeclass. -/\n-- The default instance for `monoid punit` is derived via `punit.comm_ring`,\n\nprotected instance Mathlib.AddMon.inhabited : Inhabited AddMon :=\n  { default := AddMon.of PUnit }\n\n-- which breaks to_additive.\n\nprotected instance Mathlib.AddMon.add_monoid (M : AddMon) : add_monoid ↥M :=\n  category_theory.bundled.str M\n\n@[simp] theorem Mathlib.AddMon.coe_of (R : Type u) [add_monoid R] : ↥(AddMon.of R) = R :=\n  rfl\n\nend Mon\n\n\n/-- The category of commutative monoids and monoid morphisms. -/\ndef CommMon :=\n  category_theory.bundled comm_monoid\n\n/-- The category of additive commutative monoids and monoid morphisms. -/\nnamespace CommMon\n\n\nprotected instance comm_monoid.to_monoid.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection comm_monoid.to_monoid :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance has_coe_to_sort : has_coe_to_sort CommMon :=\n  category_theory.bundled.has_coe_to_sort\n\n/-- Construct a bundled `CommMon` from the underlying type and typeclass. -/\ndef of (M : Type u) [comm_monoid M] : CommMon :=\n  category_theory.bundled.of M\n\n/-- Construct a bundled `AddCommMon` from the underlying type and typeclass. -/\n-- The default instance for `comm_monoid punit` is derived via `punit.comm_ring`,\n\nprotected instance Mathlib.AddCommMon.inhabited : Inhabited AddCommMon :=\n  { default := AddCommMon.of PUnit }\n\n-- which breaks to_additive.\n\nprotected instance Mathlib.AddCommMon.add_comm_monoid (M : AddCommMon) : add_comm_monoid ↥M :=\n  category_theory.bundled.str M\n\n@[simp] theorem coe_of (R : Type u) [comm_monoid R] : ↥(of R) = R :=\n  rfl\n\nprotected instance Mathlib.AddCommMon.has_forget_to_AddMon : category_theory.has_forget₂ AddCommMon AddMon :=\n  category_theory.bundled_hom.forget₂ add_monoid_hom add_comm_monoid.to_add_monoid\n\nend CommMon\n\n\n-- We verify that the coercions of morphisms to functions work correctly:\n\n-- We verify that when constructing a morphism in `CommMon`,\n\n-- when we construct the `to_fun` field, the types are presented as `↥R`,\n\n-- rather than `R.α` or (as we used to have) `↥(bundled.map comm_monoid.to_monoid R)`.\n\n/-- Build an isomorphism in the category `Mon` from a `mul_equiv` between `monoid`s. -/\ndef mul_equiv.to_Mon_iso {X : Type u} {Y : Type u} [monoid X] [monoid Y] (e : X ≃* Y) : Mon.of X ≅ Mon.of Y :=\n  category_theory.iso.mk (mul_equiv.to_monoid_hom e) (mul_equiv.to_monoid_hom (mul_equiv.symm e))\n\n@[simp] theorem mul_equiv.to_Mon_iso_hom {X : Type u} {Y : Type u} [monoid X] [monoid Y] {e : X ≃* Y} : category_theory.iso.hom (mul_equiv.to_Mon_iso e) = mul_equiv.to_monoid_hom e :=\n  rfl\n\n@[simp] theorem add_equiv.to_AddMon_iso_inv {X : Type u} {Y : Type u} [add_monoid X] [add_monoid Y] {e : X ≃+ Y} : category_theory.iso.inv (add_equiv.to_AddMon_iso e) = add_equiv.to_add_monoid_hom (add_equiv.symm e) :=\n  rfl\n\n/-- Build an isomorphism in the category `CommMon` from a `mul_equiv` between `comm_monoid`s. -/\ndef add_equiv.to_AddCommMon_iso {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y] (e : X ≃+ Y) : AddCommMon.of X ≅ AddCommMon.of Y :=\n  category_theory.iso.mk (add_equiv.to_add_monoid_hom e) (add_equiv.to_add_monoid_hom (add_equiv.symm e))\n\n@[simp] theorem add_equiv.to_AddCommMon_iso_hom {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y] {e : X ≃+ Y} : category_theory.iso.hom (add_equiv.to_AddCommMon_iso e) = add_equiv.to_add_monoid_hom e :=\n  rfl\n\n@[simp] theorem add_equiv.to_AddCommMon_iso_inv {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y] {e : X ≃+ Y} : category_theory.iso.inv (add_equiv.to_AddCommMon_iso e) = add_equiv.to_add_monoid_hom (add_equiv.symm e) :=\n  rfl\n\nnamespace category_theory.iso\n\n\n/-- Build a `mul_equiv` from an isomorphism in the category `Mon`. -/\ndef AddMon_iso_to_add_equiv {X : AddMon} {Y : AddMon} (i : X ≅ Y) : ↥X ≃+ ↥Y :=\n  add_monoid_hom.to_add_equiv (hom i) (inv i) (hom_inv_id i) (inv_hom_id i)\n\n/-- Build a `mul_equiv` from an isomorphism in the category `CommMon`. -/\ndef CommMon_iso_to_add_equiv {X : AddCommMon} {Y : AddCommMon} (i : X ≅ Y) : ↥X ≃+ ↥Y :=\n  add_monoid_hom.to_add_equiv (hom i) (inv i) (hom_inv_id i) (inv_hom_id i)\n\nend category_theory.iso\n\n\n/-- multiplicative equivalences between `monoid`s are the same as (isomorphic to) isomorphisms\nin `Mon` -/\ndef mul_equiv_iso_Mon_iso {X : Type u} {Y : Type u} [monoid X] [monoid Y] : X ≃* Y ≅ Mon.of X ≅ Mon.of Y :=\n  category_theory.iso.mk (fun (e : X ≃* Y) => mul_equiv.to_Mon_iso e)\n    fun (i : Mon.of X ≅ Mon.of Y) => category_theory.iso.Mon_iso_to_mul_equiv i\n\n/-- multiplicative equivalences between `comm_monoid`s are the same as (isomorphic to) isomorphisms\nin `CommMon` -/\ndef add_equiv_iso_AddCommMon_iso {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y] : X ≃+ Y ≅ AddCommMon.of X ≅ AddCommMon.of Y :=\n  category_theory.iso.mk (fun (e : X ≃+ Y) => add_equiv.to_AddCommMon_iso e)\n    fun (i : AddCommMon.of X ≅ AddCommMon.of Y) => category_theory.iso.CommMon_iso_to_add_equiv i\n\nprotected instance AddMon.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget AddMon) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : AddMon) (f : X ⟶ Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget AddMon) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget AddMon) X ≅\n          category_theory.functor.obj (category_theory.forget AddMon) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget AddMon) f);\n      let e : ↥X ≃+ ↥Y :=\n        add_equiv.mk (add_monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (add_equiv.to_AddMon_iso e))\n\nprotected instance CommMon.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget CommMon) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : CommMon) (f : X ⟶ Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget CommMon) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget CommMon) X ≅\n          category_theory.functor.obj (category_theory.forget CommMon) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget CommMon) f);\n      let e : ↥X ≃* ↥Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_CommMon_iso e))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Mon/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352403, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.31165397450844107}}
{"text": "\nimport .basic .backtrack\n\nuniverses u v\nvariables (α : Type u) (β : Type v)\n\nnamespace two_three.tree\nopen two_three\n\ninductive zipper\n| nil : zipper\n| node2_0 : zipper → α → tree α β → zipper\n| node2_1 : tree α β → α → zipper → zipper\n| node3_0 : zipper → α → tree α β → α → tree α β → zipper\n| node3_1 : tree α β → α → zipper → α → tree α β → zipper\n| node3_2 : tree α β → α → tree α β → α → zipper → zipper\n\nvariables {α β}\n\nnamespace zipper\n\n@[pp_nodot]\ndef close : zipper α β → option α → tree α β → tree α β\n| nil k t := t\n| (node2_0 z  x t₁) k t := close z k $ node2 t  x t₁\n| (node2_1 t₀ x z)  k t := close z none $ node2 t₀ (k.get_or_else x) t\n| (node3_0 z  x t₁ y t₂) k t := close z k $ node3 t  x t₁ y t₂\n| (node3_1 t₀ x z  y t₂) k t := close z none $ node3 t₀ (k.get_or_else x) t  y t₂\n| (node3_2 t₀ x t₁ y z)  k t := close z none $ node3 t₀ x t₁ (k.get_or_else y) t\n\n@[pp_nodot]\ndef split : zipper α β → option α → tree α β → α → tree α β → tree α β\n| nil k ta a tb := node2 ta a tb\n| (node2_0 z  x t₁) k ta a tb := close z k $ node3 ta a tb x t₁\n| (node2_1 t₀ x z)  k ta a tb := close z none $ node3 t₀ (k.get_or_else x) ta a tb\n| (node3_0 z  x t₁ y t₂) k ta a tb := split z k (node2 ta a tb) x (node2 t₁ y t₂)\n| (node3_1 t₀ x z  y t₂) k ta a tb := split z none (node2 t₀ (k.get_or_else x) ta) a (node2 tb y t₂)\n| (node3_2 t₀ x t₁ y z)  k ta a tb := split z none (node2 t₀ x t₁) (k.get_or_else y) (node2 ta a tb)\n\nsection eqv\n\nvariables [has_lt α] [@decidable_rel α has_lt.lt]\n\ndef insert (k : α) (v : β) : zipper α β → tree α β → tree α β\n| xs (leaf k' v') :=\n  match cmp k k' with\n  | ordering.lt := split xs (some k) (tree.leaf k v) k' (tree.leaf k' v')\n  | ordering.eq := close xs (some k') (tree.leaf k' v)\n  | ordering.gt := split xs (some k') (tree.leaf k' v') k (tree.leaf k v)\n  end\n| xs (node2 t₀ x t₁) :=\n  if k < x\n    then insert (node2_0 xs x t₁) t₀\n    else insert (node2_1 t₀ x xs) t₁\n| xs (node3 t₀ x t₁ y t₂) :=\n  if k < x\n    then insert (node3_0 xs x t₁ y t₂) t₀\n  else if k < y\n    then insert (node3_1 t₀ x xs y t₂) t₁\n    else insert (node3_2 t₀ x t₁ y xs) t₂\n\ndef insert' (k : α) (v : β) : tree α β → tree α β :=\ninsert k v zipper.nil\n\nend eqv\nopen nat\n\nvariables {n m : ℕ} {pk : option α} {t t₀ t₁ : tree α β} (x : α) {z : zipper α β}\n\nsection defns\n\nvariables [has_lt α] [has_le α]\n\ninductive zipper_height (m : ℕ) : ℕ → zipper α β → Prop\n| nil : zipper_height m zipper.nil\n| node2_0 {n} {z} {x t₁} :\n  zipper_height (succ n) z →\n  with_height n t₁ →\n  zipper_height n (zipper.node2_0 z x t₁)\n| node2_1 {n} {z} {t₀ x} :\n  zipper_height (succ n) z →\n  with_height n t₀ →\n  zipper_height n (zipper.node2_1 t₀ x z)\n| node3_0 {n} {z} {x t₁ y t₂} :\n  zipper_height (succ n) z →\n  with_height n t₁ →\n  with_height n t₂ →\n  zipper_height n (zipper.node3_0 z x t₁ y t₂)\n| node3_1 {n} {z} {t₀ x y t₂} :\n  zipper_height (succ n) z →\n  with_height n t₀ →\n  with_height n t₂ →\n  zipper_height n (zipper.node3_1 t₀ x z y t₂)\n| node3_2 {n} {z} {t₀ x t₁ y} :\n  zipper_height (succ n) z →\n  with_height n t₀ →\n  with_height n t₁ →\n  zipper_height n (zipper.node3_2 t₀ x t₁ y z)\n\nlemma with_height_close\n  (h : with_height n t)\n  (h' : zipper_height m n z) :\n  with_height m (close z pk t) ∨ with_height (succ m) (close z pk t) :=\nbegin\n  left,\n  induction h' generalizing t pk; dsimp [close] at *,\n  repeat { apply h'_ih <|> assumption <|> constructor },\nend\n\nlemma with_height_split\n  (h₀ : with_height n t₀)\n  (h₁ : with_height n t₁)\n  (h' : zipper_height m n z) :\n  with_height m (split z pk t₀ x t₁) ∨ with_height (succ m) (split z pk t₀ x t₁) :=\nbegin\n  induction h' generalizing t₀ x t₁ pk; dsimp [split],\n  { right, constructor; assumption },\n  repeat\n  { apply with_height_close, constructor; assumption, assumption },\n  repeat\n  { apply h'_ih, repeat { constructor; assumption } },\nend\n\nend defns\n\nsection insert_height\n\nvariables [linear_order α]\nvariables {k : α} {v : β}\nvariables\n  (h : zipper_height m n z)\n  (h' : with_height n t)\ninclude h h'\n\nlemma with_height_insert : with_height m (insert k v z t) ∨ with_height (succ m) (insert k v z t) :=\nbegin\n  induction h' generalizing z,\n  case two_three.with_height.leaf : k' v' z h\n  { dsimp [insert], trichotomy k =? k',\n    repeat\n    { assumption <|> apply with_height_close <|> apply with_height_split <|> constructor } },\n  case two_three.with_height.node2 : h'_n h'_x h'_t₀ h'_t₁ h'_a h'_a_1 h'_ih_a h'_ih_a_1 z h\n  { dsimp [insert], split_ifs,\n    repeat\n    { assumption <|> apply h'_ih_a <|> apply h'_ih_a_1 <|> constructor }, },\n  case two_three.with_height.node3 : h'_n h'_x h'_y h'_t₀ h'_t₁ h'_t₂ h'_a h'_a_1 h'_a_2 h'_ih_a h'_ih_a_1 h'_ih_a_2 z h\n  { dsimp [insert], split_ifs,\n    repeat\n    { assumption <|> apply h'_ih_a <|> apply h'_ih_a_1 <|> apply h'_ih_a_2 <|> constructor }, },\nend\n\nlemma with_height_insert' (h' : with_height m t) : with_height m (insert' k v t) ∨ with_height (succ m) (insert' k v t) :=\nwith_height_insert (by constructor) h'\n\nend insert_height\n\nsection defns\n\nvariables [has_lt α] [has_le α]\n\ninductive zipper_sorted : option α → zipper α β → option α → Prop\n| nil {a b} : zipper_sorted a zipper.nil b\n| node2_0 {a b x t₀ t₁} :\n  zipper_sorted a t₀ b →\n  x = first t₁ →\n  sorted' t₁ →\n  above (last t₁) b →\n  zipper_sorted a (node2_0 t₀ x t₁) (some x)\n| node2_1 {a b x t₀ t₁} :\n  zipper_sorted a t₁ b →\n  below a (first t₀) →\n  sorted' t₀ →\n  last t₀ < x →\n  zipper_sorted (some x) (node2_1 t₀ x t₁) b\n\n| node3_0 {a b x y t₀ t₁ t₂} :\n  zipper_sorted a t₀ b →\n  x = first t₁ →\n  sorted' t₁ →\n  last t₁ < y →\n  y = first t₂ →\n  sorted' t₂ →\n  above (last t₂) b →\n  zipper_sorted a (node3_0 t₀ x t₁ y t₂) (some x)\n| node3_1 {a b x y t₀ t₁ t₂} :\n  zipper_sorted a t₁ b →\n  below a (first t₀) →\n  sorted' t₀ →\n  last t₀ < x →\n  y = first t₂ →\n  sorted' t₂ →\n  above (last t₂) b →\n  zipper_sorted (some x) (node3_1 t₀ x t₁ y t₂) (some y)\n| node3_2 {a b x y t₀ t₁ t₂} :\n  zipper_sorted a t₂ b →\n  below a (first t₀) →\n  sorted' t₀ →\n  last t₀ < x →\n  x = first t₁ →\n  sorted' t₁ →\n  last t₁ < y →\n  zipper_sorted (some y) (node3_2 t₀ x t₁ y t₂) b\n\nattribute [pp_nodot]\n  zipper_sorted.nil\n  zipper_sorted.node2_0\n  zipper_sorted.node2_1\n  zipper_sorted.node3_0\n  zipper_sorted.node3_1\n  zipper_sorted.node3_2\n  zipper.nil\n  zipper.node2_0\n  zipper.node2_1\n  zipper.node3_0\n  zipper.node3_1\n  zipper.node3_2\n\nend defns\n\nsection insert_sorted\n-- variables {pk : option α}\nvariables  [linear_order α]\nopen two_three\n\nlemma sorted_close {a b}\n  (h' : zipper_sorted a z b)\n  (h₀ : below a (first t))\n  (h₃ : below pk (first t))\n  (h₁ : sorted' t)\n  (h₂ : above (last t) b)\n  : sorted' (close z pk t) :=\nbegin\n  induction h' generalizing t pk,\n  all_goals\n  { dsimp [close],\n    saturate, subst_vars,\n    simp only * { fail_if_unchanged := ff },\n    repeat\n    { assumption <|> refl <|>\n      apply h'_ih <|> constructor <|> cc }, },\nend\n\nlemma sorted_split {a x b}\n  (h' : zipper_sorted a z b)\n  (h₀ : below a (first t₀))\n  (h₀ : below pk (first t₀))\n  (h₁ : sorted' t₀)\n  (h₂ : last t₀ < x)\n  (h₃ : first t₁ = x)\n  (h₃ : sorted' t₁)\n  (h₄ : above (last t₁) b)\n  : sorted' (split z pk t₀ x t₁) :=\nbegin\n  induction h' generalizing t₀ x t₁ pk,\n  all_goals\n  { dsimp [split],\n    saturate, subst_vars,\n    simp only * { fail_if_unchanged := ff },\n    repeat\n    { assumption <|> refl <|>\n      apply sorted_close <|>\n      apply h'_ih <|> constructor }, },\nend\n\nsection tac\nopen tactic\n\n@[interactive]\nmeta def simp_min : tactic unit :=\nrun_bt $ do\n  x ← var,\n  y ← var,\n  (h, _) ← hyp_with ``(min %%x %%y),\n  bt_lift $ do\n  (pr, h) ← mcond (succeeds (is_def_eq x y))\n  (do p₀ ← to_expr ``(min %%x %%y = %%x),\n      n ← get_unused_name `h,\n      (h, _) ← local_proof n p₀ (applyc ``min_self),\n      pure (h, h))\n  (do p₀ ← to_expr ``(%%x ≤ %%y),\n      p₁ ← to_expr ``(%%y ≤ %%x),\n      n ← get_unused_name `h,\n      do { (h, _) ← local_proof n p₀ (assumption <|> applyc ``le_of_lt >> assumption),\n           pr ← mk_app ``min_eq_left [h],\n           pure (pr, h) } <|>\n        do { (h, _) ← local_proof n p₁ (assumption <|> applyc ``le_of_lt >> assumption),\n             pr ← mk_app ``min_eq_right [h],\n             pure (pr, h) }),\n   interactive.loc.apply\n     (λ h', try $ rewrite_hyp pr h' >> skip)\n     (try $ rewrite_target pr)\n     interactive.loc.wildcard,\n  skip\n\nend tac\n\nvariables {k : α} {v : β} (a b : option α)\nvariables (h : zipper_sorted a z b)\n  (h' : sorted' t)\n  (h₀ : below a (min k (first t)))\n  (h₁ : above (last t) b)\n  (h₂ : above k b)\ninclude h h' h₀ h₁ h₂\n\nlemma sorted_insert : sorted' (insert k v z t) :=\nbegin\n  induction h' generalizing z a b,\n  case two_three.sorted'.leaf : k' v' z a b h\n  { dsimp [insert], trichotomy k =? k' with h₂,\n    all_goals\n    { dsimp [first, last] at *,\n      subst_vars, simp_min, saturate,\n      apply sorted_split <|> apply sorted_close,\n      repeat\n      { assumption <|>\n        constructor <|>\n        simp_min <|> above } },  },\n  case two_three.sorted'.node2 : h'_x h'_t₀ h'_t₁ h'_a h'_a_1 h'_a_2 h'_a_3 h'_ih_a h'_ih_a_1 z a b h h₀ h₁ h₂\n  { dsimp [insert], subst_vars, split_ifs,\n    all_goals\n    { dsimp [first] at *,\n      saturate, simp_min,\n      try { have : first h'_t₀ ≤ k, { chain_trans } },\n      apply h'_ih_a <|> apply h'_ih_a_1,\n      repeat\n      { assumption <|>\n        constructor <|>\n        simp_min <|> above, } } },\n  case two_three.sorted'.node3 : h'_x h'_y h'_t₀ h'_t₁ h'_t₂ h'_a h'_a_1 h'_a_2 h'_a_3 h'_a_4 h'_a_5 h'_a_6 h'_ih_a h'_ih_a_1 h'_ih_a_2 z a b h h₀ h₁ h₂\n  { dsimp [insert], subst_vars, split_ifs,\n    all_goals\n    { dsimp [first] at *,\n      saturate, simp_min,\n      try { have : first h'_t₀ ≤ k, { chain_trans } },\n      apply h'_ih_a <|> apply h'_ih_a_1 <|> apply h'_ih_a_2,\n      repeat\n      { assumption <|>\n        constructor <|>\n        simp_min <|> above, } } },\nend\n\nlemma sorted_insert' : sorted' (insert' k v t) :=\nby apply sorted_insert none none _ h'; constructor\n\nend insert_sorted\n\nend zipper\n\nend two_three.tree\n", "meta": {"author": "cipher1024", "repo": "search-trees", "sha": "0e0ea0ee59f0b0499ebad1ef6f34f09ec9666cde", "save_path": "github-repos/lean/cipher1024-search-trees", "path": "github-repos/lean/cipher1024-search-trees/search-trees-0e0ea0ee59f0b0499ebad1ef6f34f09ec9666cde/src/zipper.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3115601855377086}}
{"text": "import category_theory.sites.dense_subsite\n\n/-!\nContent pulled from mathlib PR #14512\n-/\n\nuniverses w v v₁ v₂ v₃ u u₁ u₂ u₃\nnoncomputable theory\nopen category_theory\nopen opposite\nopen category_theory.presieve.family_of_elements\nopen category_theory.presieve\nopen category_theory.limits\n\nvariables {C D : Type u} [category.{v} C] [category.{v} D]\nvariables (A : Type w) [category.{max v u} A] [has_limits A]\nvariables {J : grothendieck_topology C} {K : grothendieck_topology D}\n\nvariables\n  [concrete_category.{max v u} A]\n  [preserves_limits (forget A)]\n  [reflects_isomorphisms (forget A)]\n  [Π (X : C), preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\n  [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ A]\n  [Π (X : D), preserves_colimits_of_shape (K.cover X)ᵒᵖ (forget A)]\n  [∀ (X : D), has_colimits_of_shape (K.cover X)ᵒᵖ A]\n\nnamespace category_theory.sites\n\n/-- The isomorphism exhibiting compatibility between pullback and sheafification. -/\ndef pullback_sheafification_compatibility\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) :\n  (whiskering_left _ _ A).obj G.op ⋙ presheaf_to_Sheaf J A ≅\n  presheaf_to_Sheaf K A ⋙ sites.pullback A Hc Hp :=\nlet A1 : (whiskering_left _ _ A).obj G.op ⊣ _ := Ran.adjunction _ _,\n    A2 : presheaf_to_Sheaf J A ⊣ _ := sheafification_adjunction _ _,\n    B1 : presheaf_to_Sheaf K A ⊣ _ := sheafification_adjunction _ _,\n    B2 : sites.pullback A Hc Hp ⊣ _ := sites.pullback_copullback_adjunction _ _ Hl _,\n    A12 := A1.comp A2,\n    B12 := B1.comp B2 in\nA12.left_adjoint_uniq B12\n\nlemma to_sheafify_pullback_sheafification_compatibility\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) (F) :\n  J.to_sheafify (G.op ⋙ F) ≫\n  ((pullback_sheafification_compatibility.{w v u} A Hp Hl Hc).hom.app F).val =\n  whisker_left _ (K.to_sheafify _) :=\nbegin\n  dsimp [pullback_sheafification_compatibility, adjunction.left_adjoint_uniq],\n  apply quiver.hom.op_inj,\n  apply coyoneda.map_injective, swap, apply_instance,\n  ext E f : 2,\n  dsimp [functor.preimage, full.preimage, coyoneda, adjunction.left_adjoints_coyoneda_equiv],\n  simp only [adjunction.hom_equiv_unit, functor.comp_map, Sheaf_to_presheaf_map,\n    adjunction.hom_equiv_naturality_left_symm, whiskering_left_obj_map,\n    adjunction.hom_equiv_counit, Sheaf.category_theory.category_comp_val,\n    presheaf_to_Sheaf_map_val],\n  dsimp [adjunction.comp],\n  simp only [sheafification_adjunction_unit_app, category.comp_id, functor.map_id,\n    whisker_left_id', category.assoc, grothendieck_topology.sheafify_map_sheafify_lift,\n    category.id_comp, grothendieck_topology.to_sheafify_sheafify_lift],\n  ext t,\n  dsimp only [whisker_left, nat_trans.comp_app],\n  simp only [category.assoc],\n  -- for some reason the proof that works in mathllib does not work here...\n  -- we have to be a little more forceful.\n  erw [(Ran G.op).map_id, category.id_comp],\n  congr' 1, simp only [← category.assoc],\n  convert category.id_comp _,\n  dsimp only [pullback_sheaf],\n  have := (Ran.adjunction A G.op).left_triangle,\n  apply_fun (λ e, (e.app (K.sheafify F)).app x) at this,\n  exact this,\nend\n\n@[simp]\nlemma pullback_sheafification_compatibility_hom_apply_val\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) (F) :\n((pullback_sheafification_compatibility.{w v u} A Hp Hl Hc).hom.app F).val =\n  J.sheafify_lift (whisker_left _ $ K.to_sheafify _) (Sheaf.cond _) :=\nbegin\n  apply J.sheafify_lift_unique,\n  apply to_sheafify_pullback_sheafification_compatibility,\nend\n\nend category_theory.sites\n\nnamespace category_theory.cover_dense\n\nvariables {G : C ⥤ D} [full G] [faithful G]\nvariables (Hd : cover_dense K G) (Hp : cover_preserving J K G) (Hl : cover_lifting J K G)\n\n/-- The isomorphism exhibiting the compatibiility of\n`Sheaf_equiv_of_cover_preserving_cover_lifting` with sheafification. -/\nnoncomputable\nabbreviation Sheaf_equiv_of_cover_preserving_cover_lifting_sheafification_compatibility :\n  (whiskering_left _ _ A).obj G.op ⋙ presheaf_to_Sheaf _ _ ≅\n  presheaf_to_Sheaf _ _ ⋙ (Sheaf_equiv_of_cover_preserving_cover_lifting Hd Hp Hl).inverse :=\ncategory_theory.sites.pullback_sheafification_compatibility _ _ Hl _\n\nend category_theory.cover_dense\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/sheafification_equiv_compatibility.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.44552953503957266, "lm_q1q2_score": 0.3115384970886742}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.maps\nimport Mathlib.PostPort\n\nuniverses u v w x u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Constructions of new topological spaces from old ones\n\nThis file constructs products, sums, subtypes and quotients of topological spaces\nand sets up their basic theory, such as criteria for maps into or out of these\nconstructions to be continuous; descriptions of the open sets, neighborhood filters,\nand generators of these constructions; and their behavior with respect to embeddings\nand other specific classes of maps.\n\n## Implementation note\n\nThe constructed topologies are defined using induced and coinduced topologies\nalong with the complete lattice structure on topologies. Their universal properties\n(for example, a map `X → Y × Z` is continuous if and only if both projections\n`X → Y`, `X → Z` are) follow easily using order-theoretic descriptions of\ncontinuity. With more work we can also extract descriptions of the open sets,\nneighborhood filters and so on.\n\n## Tags\n\nproduct, sum, disjoint union, subspace, quotient space\n\n-/\n\nprotected instance subtype.topological_space {α : Type u} {p : α → Prop} [t : topological_space α] : topological_space (Subtype p) :=\n  topological_space.induced coe t\n\nprotected instance quot.topological_space {α : Type u} {r : α → α → Prop} [t : topological_space α] : topological_space (Quot r) :=\n  topological_space.coinduced (Quot.mk r) t\n\nprotected instance quotient.topological_space {α : Type u} {s : setoid α} [t : topological_space α] : topological_space (quotient s) :=\n  topological_space.coinduced quotient.mk t\n\nprotected instance prod.topological_space {α : Type u} {β : Type v} [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α × β) :=\n  topological_space.induced prod.fst t₁ ⊓ topological_space.induced prod.snd t₂\n\nprotected instance sum.topological_space {α : Type u} {β : Type v} [t₁ : topological_space α] [t₂ : topological_space β] : topological_space (α ⊕ β) :=\n  topological_space.coinduced sum.inl t₁ ⊔ topological_space.coinduced sum.inr t₂\n\nprotected instance sigma.topological_space {α : Type u} {β : α → Type v} [t₂ : (a : α) → topological_space (β a)] : topological_space (sigma β) :=\n  supr fun (a : α) => topological_space.coinduced (sigma.mk a) (t₂ a)\n\nprotected instance Pi.topological_space {α : Type u} {β : α → Type v} [t₂ : (a : α) → topological_space (β a)] : topological_space ((a : α) → β a) :=\n  infi fun (a : α) => topological_space.induced (fun (f : (a : α) → β a) => f a) (t₂ a)\n\nprotected instance ulift.topological_space {α : Type u} [t : topological_space α] : topological_space (ulift α) :=\n  topological_space.induced ulift.down t\n\n/-- The image of a dense set under `quotient.mk` is a dense set. -/\ntheorem dense.quotient {α : Type u} [setoid α] [topological_space α] {s : set α} (H : dense s) : dense (quotient.mk '' s) :=\n  dense_range.dense_image (function.surjective.dense_range (surjective_quotient_mk α)) continuous_coinduced_rng H\n\n/-- The composition of `quotient.mk` and a function with dense range has dense range. -/\ntheorem dense_range.quotient {α : Type u} {β : Type v} [setoid α] [topological_space α] {f : β → α} (hf : dense_range f) : dense_range (quotient.mk ∘ f) :=\n  dense_range.comp (function.surjective.dense_range (surjective_quotient_mk α)) hf continuous_coinduced_rng\n\nprotected instance subtype.discrete_topology {α : Type u} {p : α → Prop} [topological_space α] [discrete_topology α] : discrete_topology (Subtype p) :=\n  discrete_topology.mk\n    (bot_unique\n      fun (s : set (Subtype p)) (hs : topological_space.is_open ⊥ s) =>\n        Exists.intro (coe '' s)\n          { left := is_open_discrete (coe '' s), right := set.preimage_image_eq s subtype.coe_injective })\n\nprotected instance sum.discrete_topology {α : Type u} {β : Type v} [topological_space α] [topological_space β] [hα : discrete_topology α] [hβ : discrete_topology β] : discrete_topology (α ⊕ β) := sorry\n\nprotected instance sigma.discrete_topology {α : Type u} {β : α → Type v} [(a : α) → topological_space (β a)] [h : ∀ (a : α), discrete_topology (β a)] : discrete_topology (sigma β) := sorry\n\n/-\nThe 𝓝 filter and the subspace topology.\n-/\n\ntheorem mem_nhds_subtype {α : Type u} [topological_space α] (s : set α) (a : Subtype fun (x : α) => x ∈ s) (t : set (Subtype fun (x : α) => x ∈ s)) : t ∈ nhds a ↔ ∃ (u : set α), ∃ (H : u ∈ nhds ↑a), coe ⁻¹' u ⊆ t :=\n  mem_nhds_induced coe a t\n\ntheorem nhds_subtype {α : Type u} [topological_space α] (s : set α) (a : Subtype fun (x : α) => x ∈ s) : nhds a = filter.comap coe (nhds ↑a) :=\n  nhds_induced coe a\n\ntheorem continuous_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β] : continuous prod.fst :=\n  continuous_inf_dom_left continuous_induced_dom\n\ntheorem continuous_at_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : α × β} : continuous_at prod.fst p :=\n  continuous.continuous_at continuous_fst\n\ntheorem continuous_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β] : continuous prod.snd :=\n  continuous_inf_dom_right continuous_induced_dom\n\ntheorem continuous_at_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : α × β} : continuous_at prod.snd p :=\n  continuous.continuous_at continuous_snd\n\ntheorem continuous.prod_mk {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : γ → α} {g : γ → β} (hf : continuous f) (hg : continuous g) : continuous fun (x : γ) => (f x, g x) :=\n  continuous_inf_rng (continuous_induced_rng hf) (continuous_induced_rng hg)\n\ntheorem continuous.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : γ → α} {g : δ → β} (hf : continuous f) (hg : continuous g) : continuous fun (x : γ × δ) => (f (prod.fst x), g (prod.snd x)) :=\n  continuous.prod_mk (continuous.comp hf continuous_fst) (continuous.comp hg continuous_snd)\n\ntheorem filter.eventually.prod_inl_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : α → Prop} {a : α} (h : filter.eventually (fun (x : α) => p x) (nhds a)) (b : β) : filter.eventually (fun (x : α × β) => p (prod.fst x)) (nhds (a, b)) :=\n  continuous_at_fst h\n\ntheorem filter.eventually.prod_inr_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : β → Prop} {b : β} (h : filter.eventually (fun (x : β) => p x) (nhds b)) (a : α) : filter.eventually (fun (x : α × β) => p (prod.snd x)) (nhds (a, b)) :=\n  continuous_at_snd h\n\ntheorem filter.eventually.prod_mk_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {pa : α → Prop} {a : α} (ha : filter.eventually (fun (x : α) => pa x) (nhds a)) {pb : β → Prop} {b : β} (hb : filter.eventually (fun (y : β) => pb y) (nhds b)) : filter.eventually (fun (p : α × β) => pa (prod.fst p) ∧ pb (prod.snd p)) (nhds (a, b)) :=\n  filter.eventually.and (filter.eventually.prod_inl_nhds ha b) (filter.eventually.prod_inr_nhds hb a)\n\ntheorem continuous_swap {α : Type u} {β : Type v} [topological_space α] [topological_space β] : continuous prod.swap :=\n  continuous.prod_mk continuous_snd continuous_fst\n\ntheorem continuous_uncurry_left {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α → β → γ} (a : α) (h : continuous (function.uncurry f)) : continuous (f a) :=\n  (fun (this : continuous (function.uncurry f ∘ fun (b : β) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_const continuous_id'))\n\ntheorem continuous_uncurry_right {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α → β → γ} (b : β) (h : continuous (function.uncurry f)) : continuous fun (a : α) => f a b :=\n  (fun (this : continuous (function.uncurry f ∘ fun (a : α) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_id' continuous_const))\n\ntheorem continuous_curry {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {g : α × β → γ} (a : α) (h : continuous g) : continuous (function.curry g a) :=\n  (fun (this : continuous (g ∘ fun (b : β) => (a, b))) => this)\n    (continuous.comp h (continuous.prod_mk continuous_const continuous_id'))\n\ntheorem is_open.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} (hs : is_open s) (ht : is_open t) : is_open (set.prod s t) :=\n  is_open_inter (is_open.preimage continuous_fst hs) (is_open.preimage continuous_snd ht)\n\ntheorem nhds_prod_eq {α : Type u} {β : Type v} [topological_space α] [topological_space β] {a : α} {b : β} : nhds (a, b) = filter.prod (nhds a) (nhds b) := sorry\n\ntheorem mem_nhds_prod_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {a : α} {b : β} {s : set (α × β)} : s ∈ nhds (a, b) ↔ ∃ (u : set α), ∃ (H : u ∈ nhds a), ∃ (v : set β), ∃ (H : v ∈ nhds b), set.prod u v ⊆ s := sorry\n\ntheorem filter.has_basis.prod_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {ιa : Type u_1} {ιb : Type u_2} {pa : ιa → Prop} {pb : ιb → Prop} {sa : ιa → set α} {sb : ιb → set β} {a : α} {b : β} (ha : filter.has_basis (nhds a) pa sa) (hb : filter.has_basis (nhds b) pb sb) : filter.has_basis (nhds (a, b)) (fun (i : ιa × ιb) => pa (prod.fst i) ∧ pb (prod.snd i))\n  fun (i : ιa × ιb) => set.prod (sa (prod.fst i)) (sb (prod.snd i)) := sorry\n\nprotected instance prod.discrete_topology {α : Type u} {β : Type v} [topological_space α] [topological_space β] [discrete_topology α] [discrete_topology β] : discrete_topology (α × β) :=\n  discrete_topology.mk (eq_of_nhds_eq_nhds fun (_x : α × β) => sorry)\n\ntheorem prod_mem_nhds_sets {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} {a : α} {b : β} (ha : s ∈ nhds a) (hb : t ∈ nhds b) : set.prod s t ∈ nhds (a, b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (set.prod s t ∈ nhds (a, b))) nhds_prod_eq)) (filter.prod_mem_prod ha hb)\n\ntheorem nhds_swap {α : Type u} {β : Type v} [topological_space α] [topological_space β] (a : α) (b : β) : nhds (a, b) = filter.map prod.swap (nhds (b, a)) := sorry\n\ntheorem filter.tendsto.prod_mk_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {γ : Type u_1} {a : α} {b : β} {f : filter γ} {ma : γ → α} {mb : γ → β} (ha : filter.tendsto ma f (nhds a)) (hb : filter.tendsto mb f (nhds b)) : filter.tendsto (fun (c : γ) => (ma c, mb c)) f (nhds (a, b)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter.tendsto (fun (c : γ) => (ma c, mb c)) f (nhds (a, b)))) nhds_prod_eq))\n    (filter.tendsto.prod_mk ha hb)\n\ntheorem filter.eventually.curry_nhds {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : α × β → Prop} {x : α} {y : β} (h : filter.eventually (fun (x : α × β) => p x) (nhds (x, y))) : filter.eventually (fun (x' : α) => filter.eventually (fun (y' : β) => p (x', y')) (nhds y)) (nhds x) :=\n  filter.eventually.curry\n    (eq.mp (Eq._oldrec (Eq.refl (filter.eventually (fun (x : α × β) => p x) (nhds (x, y)))) nhds_prod_eq) h)\n\ntheorem continuous_at.prod {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α → β} {g : α → γ} {x : α} (hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (fun (x : α) => (f x, g x)) x :=\n  filter.tendsto.prod_mk_nhds hf hg\n\ntheorem continuous_at.prod_map {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → γ} {g : β → δ} {p : α × β} (hf : continuous_at f (prod.fst p)) (hg : continuous_at g (prod.snd p)) : continuous_at (fun (p : α × β) => (f (prod.fst p), g (prod.snd p))) p :=\n  continuous_at.prod (continuous_at.comp hf (continuous.continuous_at continuous_fst))\n    (continuous_at.comp hg (continuous.continuous_at continuous_snd))\n\ntheorem continuous_at.prod_map' {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → γ} {g : β → δ} {x : α} {y : β} (hf : continuous_at f x) (hg : continuous_at g y) : continuous_at (fun (p : α × β) => (f (prod.fst p), g (prod.snd p))) (x, y) :=\n  (fun (hf : continuous_at f (prod.fst (x, y))) =>\n      (fun (hg : continuous_at g (prod.snd (x, y))) => continuous_at.prod_map hf hg) hg)\n    hf\n\ntheorem prod_generate_from_generate_from_eq {α : Type u_1} {β : Type u_2} {s : set (set α)} {t : set (set β)} (hs : ⋃₀s = set.univ) (ht : ⋃₀t = set.univ) : prod.topological_space =\n  topological_space.generate_from\n    (set_of fun (g : set (α × β)) => ∃ (u : set α), ∃ (H : u ∈ s), ∃ (v : set β), ∃ (H : v ∈ t), g = set.prod u v) := sorry\n\ntheorem prod_eq_generate_from {α : Type u} {β : Type v} [topological_space α] [topological_space β] : prod.topological_space =\n  topological_space.generate_from\n    (set_of fun (g : set (α × β)) => ∃ (s : set α), ∃ (t : set β), is_open s ∧ is_open t ∧ g = set.prod s t) := sorry\n\ntheorem is_open_prod_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set (α × β)} : is_open s ↔\n  ∀ (a : α) (b : β), (a, b) ∈ s → ∃ (u : set α), ∃ (v : set β), is_open u ∧ is_open v ∧ a ∈ u ∧ b ∈ v ∧ set.prod u v ⊆ s := sorry\n\ntheorem continuous_uncurry_of_discrete_topology_left {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] [discrete_topology α] {f : α → β → γ} (h : ∀ (a : α), continuous (f a)) : continuous (function.uncurry f) := sorry\n\n/-- Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood\n  that is a subset of `s`. -/\ntheorem exists_nhds_square {α : Type u} [topological_space α] {s : set (α × α)} {x : α} (hx : s ∈ nhds (x, x)) : ∃ (U : set α), is_open U ∧ x ∈ U ∧ set.prod U U ⊆ s := sorry\n\n/-- The first projection in a product of topological spaces sends open sets to open sets. -/\ntheorem is_open_map_fst {α : Type u} {β : Type v} [topological_space α] [topological_space β] : is_open_map prod.fst := sorry\n\n/-- The second projection in a product of topological spaces sends open sets to open sets. -/\ntheorem is_open_map_snd {α : Type u} {β : Type v} [topological_space α] [topological_space β] : is_open_map prod.snd := sorry\n\n/-- A product set is open in a product space if and only if each factor is open, or one of them is\nempty -/\ntheorem is_open_prod_iff' {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} : is_open (set.prod s t) ↔ is_open s ∧ is_open t ∨ s = ∅ ∨ t = ∅ := sorry\n\ntheorem closure_prod_eq {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} : closure (set.prod s t) = set.prod (closure s) (closure t) := sorry\n\ntheorem map_mem_closure2 {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {s : set α} {t : set β} {u : set γ} {f : α → β → γ} {a : α} {b : β} (hf : continuous fun (p : α × β) => f (prod.fst p) (prod.snd p)) (ha : a ∈ closure s) (hb : b ∈ closure t) (hu : ∀ (a : α) (b : β), a ∈ s → b ∈ t → f a b ∈ u) : f a b ∈ closure u := sorry\n\ntheorem is_closed.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s₁ : set α} {s₂ : set β} (h₁ : is_closed s₁) (h₂ : is_closed s₂) : is_closed (set.prod s₁ s₂) := sorry\n\n/-- The product of two dense sets is a dense set. -/\ntheorem dense.prod {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set α} {t : set β} (hs : dense s) (ht : dense t) : dense (set.prod s t) :=\n  fun (x : α × β) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (x ∈ closure (set.prod s t))) closure_prod_eq))\n      { left := hs (prod.fst x), right := ht (prod.snd x) }\n\n/-- If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. -/\ntheorem dense_range.prod_map {β : Type v} {γ : Type w} [topological_space β] [topological_space γ] {ι : Type u_1} {κ : Type u_2} {f : ι → β} {g : κ → γ} (hf : dense_range f) (hg : dense_range g) : dense_range (prod.map f g) := sorry\n\ntheorem inducing.prod_mk {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ} (hf : inducing f) (hg : inducing g) : inducing fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) := sorry\n\ntheorem embedding.prod_mk {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ} (hf : embedding f) (hg : embedding g) : embedding fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) := sorry\n\nprotected theorem is_open_map.prod {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ} (hf : is_open_map f) (hg : is_open_map g) : is_open_map fun (p : α × γ) => (f (prod.fst p), g (prod.snd p)) := sorry\n\nprotected theorem open_embedding.prod {α : Type u} {β : Type v} {γ : Type w} {δ : Type x} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] {f : α → β} {g : γ → δ} (hf : open_embedding f) (hg : open_embedding g) : open_embedding fun (x : α × γ) => (f (prod.fst x), g (prod.snd x)) :=\n  open_embedding_of_embedding_open (embedding.prod_mk (open_embedding.to_embedding hf) (open_embedding.to_embedding hg))\n    (is_open_map.prod (open_embedding.is_open_map hf) (open_embedding.is_open_map hg))\n\ntheorem embedding_graph {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) : embedding fun (x : α) => (x, f x) :=\n  embedding_of_embedding_compose (continuous.prod_mk continuous_id hf) continuous_fst embedding_id\n\ntheorem continuous_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] : continuous sum.inl :=\n  continuous_sup_rng_left continuous_coinduced_rng\n\ntheorem continuous_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] : continuous sum.inr :=\n  continuous_sup_rng_right continuous_coinduced_rng\n\ntheorem continuous_sum_rec {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α → γ} {g : β → γ} (hf : continuous f) (hg : continuous g) : continuous (sum.rec f g) := sorry\n\ntheorem is_open_sum_iff {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : set (α ⊕ β)} : is_open s ↔ is_open (sum.inl ⁻¹' s) ∧ is_open (sum.inr ⁻¹' s) :=\n  iff.rfl\n\ntheorem is_open_map_sum {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α ⊕ β → γ} (h₁ : is_open_map fun (a : α) => f (sum.inl a)) (h₂ : is_open_map fun (b : β) => f (sum.inr b)) : is_open_map f := sorry\n\ntheorem embedding_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] : embedding sum.inl := sorry\n\ntheorem embedding_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] : embedding sum.inr := sorry\n\ntheorem is_open_range_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] : is_open (set.range sum.inl) := sorry\n\ntheorem is_open_range_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] : is_open (set.range sum.inr) := sorry\n\ntheorem open_embedding_inl {α : Type u} {β : Type v} [topological_space α] [topological_space β] : open_embedding sum.inl :=\n  open_embedding.mk (embedding.mk (embedding.to_inducing embedding_inl) (embedding.inj embedding_inl)) is_open_range_inl\n\ntheorem open_embedding_inr {α : Type u} {β : Type v} [topological_space α] [topological_space β] : open_embedding sum.inr :=\n  open_embedding.mk (embedding.mk (embedding.to_inducing embedding_inr) (embedding.inj embedding_inr)) is_open_range_inr\n\ntheorem embedding_subtype_coe {α : Type u} [topological_space α] {p : α → Prop} : embedding coe :=\n  embedding.mk (inducing.mk rfl) subtype.coe_injective\n\ntheorem continuous_subtype_val {α : Type u} [topological_space α] {p : α → Prop} : continuous subtype.val :=\n  continuous_induced_dom\n\ntheorem continuous_subtype_coe {α : Type u} [topological_space α] {p : α → Prop} : continuous coe :=\n  continuous_subtype_val\n\ntheorem is_open.open_embedding_subtype_coe {α : Type u} [topological_space α] {s : set α} (hs : is_open s) : open_embedding coe :=\n  open_embedding.mk (embedding.mk (inducing.mk rfl) subtype.coe_injective) (Eq.symm subtype.range_coe ▸ hs)\n\ntheorem is_open.is_open_map_subtype_coe {α : Type u} [topological_space α] {s : set α} (hs : is_open s) : is_open_map coe :=\n  open_embedding.is_open_map (is_open.open_embedding_subtype_coe hs)\n\ntheorem is_open_map.restrict {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} (hf : is_open_map f) {s : set α} (hs : is_open s) : is_open_map (set.restrict f s) :=\n  is_open_map.comp hf (is_open.is_open_map_subtype_coe hs)\n\ntheorem is_closed.closed_embedding_subtype_coe {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) : closed_embedding coe :=\n  closed_embedding.mk (embedding.mk (inducing.mk rfl) subtype.coe_injective) (Eq.symm subtype.range_coe ▸ hs)\n\ntheorem continuous_subtype_mk {α : Type u} {β : Type v} [topological_space α] [topological_space β] {p : α → Prop} {f : β → α} (hp : ∀ (x : β), p (f x)) (h : continuous f) : continuous fun (x : β) => { val := f x, property := hp x } :=\n  continuous_induced_rng h\n\ntheorem continuous_inclusion {α : Type u} [topological_space α] {s : set α} {t : set α} (h : s ⊆ t) : continuous (set.inclusion h) :=\n  continuous_subtype_mk (fun (x : ↥s) => set.inclusion._proof_1 h x) continuous_subtype_coe\n\ntheorem continuous_at_subtype_coe {α : Type u} [topological_space α] {p : α → Prop} {a : Subtype p} : continuous_at coe a :=\n  iff.mp continuous_iff_continuous_at continuous_subtype_coe a\n\ntheorem map_nhds_subtype_coe_eq {α : Type u} [topological_space α] {p : α → Prop} {a : α} (ha : p a) (h : (set_of fun (a : α) => p a) ∈ nhds a) : filter.map coe (nhds { val := a, property := ha }) = nhds a := sorry\n\ntheorem nhds_subtype_eq_comap {α : Type u} [topological_space α] {p : α → Prop} {a : α} {h : p a} : nhds { val := a, property := h } = filter.comap coe (nhds a) :=\n  nhds_induced coe { val := a, property := h }\n\ntheorem tendsto_subtype_rng {α : Type u} [topological_space α] {β : Type u_1} {p : α → Prop} {b : filter β} {f : β → Subtype p} {a : Subtype p} : filter.tendsto f b (nhds a) ↔ filter.tendsto (fun (x : β) => ↑(f x)) b (nhds ↑a) := sorry\n\ntheorem continuous_subtype_nhds_cover {α : Type u} {β : Type v} [topological_space α] [topological_space β] {ι : Sort u_1} {f : α → β} {c : ι → α → Prop} (c_cover : ∀ (x : α), ∃ (i : ι), (set_of fun (x : α) => c i x) ∈ nhds x) (f_cont : ∀ (i : ι), continuous fun (x : Subtype (c i)) => f ↑x) : continuous f := sorry\n\ntheorem continuous_subtype_is_closed_cover {α : Type u} {β : Type v} [topological_space α] [topological_space β] {ι : Type u_1} {f : α → β} (c : ι → α → Prop) (h_lf : locally_finite fun (i : ι) => set_of fun (x : α) => c i x) (h_is_closed : ∀ (i : ι), is_closed (set_of fun (x : α) => c i x)) (h_cover : ∀ (x : α), ∃ (i : ι), c i x) (f_cont : ∀ (i : ι), continuous fun (x : Subtype (c i)) => f ↑x) : continuous f := sorry\n\ntheorem closure_subtype {α : Type u} [topological_space α] {p : α → Prop} {x : Subtype fun (a : α) => p a} {s : set (Subtype fun (a : α) => p a)} : x ∈ closure s ↔ ↑x ∈ closure (coe '' s) :=\n  closure_induced fun (x y : Subtype fun (a : α) => p a) => subtype.eq\n\ntheorem quotient_map_quot_mk {α : Type u} [topological_space α] {r : α → α → Prop} : quotient_map (Quot.mk r) :=\n  { left := quot.exists_rep, right := rfl }\n\ntheorem continuous_quot_mk {α : Type u} [topological_space α] {r : α → α → Prop} : continuous (Quot.mk r) :=\n  continuous_coinduced_rng\n\ntheorem continuous_quot_lift {α : Type u} {β : Type v} [topological_space α] [topological_space β] {r : α → α → Prop} {f : α → β} (hr : ∀ (a b : α), r a b → f a = f b) (h : continuous f) : continuous (Quot.lift f hr) :=\n  continuous_coinduced_dom h\n\ntheorem quotient_map_quotient_mk {α : Type u} [topological_space α] {s : setoid α} : quotient_map quotient.mk :=\n  quotient_map_quot_mk\n\ntheorem continuous_quotient_mk {α : Type u} [topological_space α] {s : setoid α} : continuous quotient.mk :=\n  continuous_coinduced_rng\n\ntheorem continuous_quotient_lift {α : Type u} {β : Type v} [topological_space α] [topological_space β] {s : setoid α} {f : α → β} (hs : ∀ (a b : α), a ≈ b → f a = f b) (h : continuous f) : continuous (quotient.lift f hs) :=\n  continuous_coinduced_dom h\n\ntheorem continuous_pi {α : Type u} {ι : Type u_1} {π : ι → Type u_2} [topological_space α] [(i : ι) → topological_space (π i)] {f : α → (i : ι) → π i} (h : ∀ (i : ι), continuous fun (a : α) => f a i) : continuous f :=\n  continuous_infi_rng fun (i : ι) => continuous_induced_rng (h i)\n\ntheorem continuous_apply {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → topological_space (π i)] (i : ι) : continuous fun (p : (i : ι) → π i) => p i :=\n  continuous_infi_dom continuous_induced_dom\n\n/-- Embedding a factor into a product space (by fixing arbitrarily all the other coordinates) is\ncontinuous. -/\ntheorem continuous_update {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι] [(i : ι) → topological_space (π i)] {i : ι} {f : (i : ι) → π i} : continuous fun (x : π i) => function.update f i x := sorry\n\ntheorem nhds_pi {ι : Type u_1} {π : ι → Type u_2} [t : (i : ι) → topological_space (π i)] {a : (i : ι) → π i} : nhds a = infi fun (i : ι) => filter.comap (fun (x : (i : ι) → π i) => x i) (nhds (a i)) := sorry\n\ntheorem tendsto_pi {α : Type u} {ι : Type u_1} {π : ι → Type u_2} [t : (i : ι) → topological_space (π i)] {f : α → (i : ι) → π i} {g : (i : ι) → π i} {u : filter α} : filter.tendsto f u (nhds g) ↔ ∀ (x : ι), filter.tendsto (fun (i : α) => f i x) u (nhds (g x)) := sorry\n\ntheorem is_open_set_pi {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)] {i : set ι} {s : (a : ι) → set (π a)} (hi : set.finite i) (hs : ∀ (a : ι), a ∈ i → is_open (s a)) : is_open (set.pi i s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.pi i s))) (set.pi_def i s)))\n    (is_open_bInter hi fun (a : ι) (ha : a ∈ i) => is_open.preimage (continuous_apply a) (hs a ha))\n\ntheorem set_pi_mem_nhds {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)] {i : set ι} {s : (a : ι) → set (π a)} {x : (a : ι) → π a} (hi : set.finite i) (hs : ∀ (a : ι), a ∈ i → s a ∈ nhds (x a)) : set.pi i s ∈ nhds x := sorry\n\ntheorem pi_eq_generate_from {ι : Type u_1} {π : ι → Type u_2} [(a : ι) → topological_space (π a)] : Pi.topological_space =\n  topological_space.generate_from\n    (set_of\n      fun (g : set ((a : ι) → π a)) =>\n        ∃ (s : (a : ι) → set (π a)), ∃ (i : finset ι), (∀ (a : ι), a ∈ i → is_open (s a)) ∧ g = set.pi (↑i) s) := sorry\n\ntheorem pi_generate_from_eq {ι : Type u_1} {π : ι → Type u_2} {g : (a : ι) → set (set (π a))} : Pi.topological_space =\n  topological_space.generate_from\n    (set_of\n      fun (t : set ((a : ι) → π a)) =>\n        ∃ (s : (a : ι) → set (π a)), ∃ (i : finset ι), (∀ (a : ι), a ∈ i → s a ∈ g a) ∧ t = set.pi (↑i) s) := sorry\n\ntheorem pi_generate_from_eq_fintype {ι : Type u_1} {π : ι → Type u_2} {g : (a : ι) → set (set (π a))} [fintype ι] (hg : ∀ (a : ι), ⋃₀g a = set.univ) : Pi.topological_space =\n  topological_space.generate_from\n    (set_of\n      fun (t : set ((a : ι) → π a)) => ∃ (s : (a : ι) → set (π a)), (∀ (a : ι), s a ∈ g a) ∧ t = set.pi set.univ s) := sorry\n\ntheorem continuous_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : continuous (sigma.mk i) :=\n  continuous_supr_rng continuous_coinduced_rng\n\ntheorem is_open_sigma_iff {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {s : set (sigma σ)} : is_open s ↔ ∀ (i : ι), is_open (sigma.mk i ⁻¹' s) := sorry\n\ntheorem is_closed_sigma_iff {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {s : set (sigma σ)} : is_closed s ↔ ∀ (i : ι), is_closed (sigma.mk i ⁻¹' s) :=\n  is_open_sigma_iff\n\ntheorem is_open_map_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : is_open_map (sigma.mk i) := sorry\n\ntheorem is_open_range_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : is_open (set.range (sigma.mk i)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.range (sigma.mk i)))) (Eq.symm set.image_univ)))\n    (is_open_map_sigma_mk set.univ is_open_univ)\n\ntheorem is_closed_map_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : is_closed_map (sigma.mk i) := sorry\n\ntheorem is_closed_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : is_closed (set.range (sigma.mk i)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (set.range (sigma.mk i)))) (Eq.symm set.image_univ)))\n    (is_closed_map_sigma_mk set.univ is_closed_univ)\n\ntheorem open_embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : open_embedding (sigma.mk i) :=\n  open_embedding_of_continuous_injective_open continuous_sigma_mk sigma_mk_injective is_open_map_sigma_mk\n\ntheorem closed_embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : closed_embedding (sigma.mk i) :=\n  closed_embedding_of_continuous_injective_closed continuous_sigma_mk sigma_mk_injective is_closed_map_sigma_mk\n\ntheorem embedding_sigma_mk {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {i : ι} : embedding (sigma.mk i) :=\n  closed_embedding.to_embedding closed_embedding_sigma_mk\n\n/-- A map out of a sum type is continuous if its restriction to each summand is. -/\ntheorem continuous_sigma {β : Type v} {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] [topological_space β] {f : sigma σ → β} (h : ∀ (i : ι), continuous fun (a : σ i) => f (sigma.mk i a)) : continuous f :=\n  continuous_supr_dom fun (i : ι) => continuous_coinduced_dom (h i)\n\ntheorem continuous_sigma_map {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {κ : Type u_3} {τ : κ → Type u_4} [(k : κ) → topological_space (τ k)] {f₁ : ι → κ} {f₂ : (i : ι) → σ i → τ (f₁ i)} (hf : ∀ (i : ι), continuous (f₂ i)) : continuous (sigma.map f₁ f₂) := sorry\n\ntheorem is_open_map_sigma {β : Type v} {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] [topological_space β] {f : sigma σ → β} (h : ∀ (i : ι), is_open_map fun (a : σ i) => f (sigma.mk i a)) : is_open_map f := sorry\n\n/-- The sum of embeddings is an embedding. -/\ntheorem embedding_sigma_map {ι : Type u_1} {σ : ι → Type u_2} [(i : ι) → topological_space (σ i)] {τ : ι → Type u_3} [(i : ι) → topological_space (τ i)] {f : (i : ι) → σ i → τ i} (hf : ∀ (i : ι), embedding (f i)) : embedding (sigma.map id f) := sorry\n\ntheorem continuous_ulift_down {α : Type u} [topological_space α] : continuous ulift.down :=\n  continuous_induced_dom\n\ntheorem continuous_ulift_up {α : Type u} [topological_space α] : continuous ulift.up :=\n  continuous_induced_rng continuous_id\n\ntheorem mem_closure_of_continuous {α : Type u} {β : Type v} [topological_space α] [topological_space β] {f : α → β} {a : α} {s : set α} {t : set β} (hf : continuous f) (ha : a ∈ closure s) (h : set.maps_to f s (closure t)) : f a ∈ closure t :=\n  set.mem_of_mem_of_subset (set.mem_of_mem_of_subset (set.mem_image_of_mem f ha) (image_closure_subset_closure_image hf))\n    (closure_minimal (set.maps_to.image_subset h) is_closed_closure)\n\ntheorem mem_closure_of_continuous2 {α : Type u} {β : Type v} {γ : Type w} [topological_space α] [topological_space β] [topological_space γ] {f : α → β → γ} {a : α} {b : β} {s : set α} {t : set β} {u : set γ} (hf : continuous fun (p : α × β) => f (prod.fst p) (prod.snd p)) (ha : a ∈ closure s) (hb : b ∈ closure t) (h : ∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → f a b ∈ closure u) : f a b ∈ closure u := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/constructions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.31142894375484365}}
{"text": "theorem ex6 (f : Nat → Nat) (x y z : Nat) (h : (x, z).1 = (fun x => x) y) : f x = f y := by\n  simp (config := { beta := false }) at h\n  traceState\n  simp at h\n  traceState\n  simp [h]\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/simpcfg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3113992812082249}}
{"text": "import its reducibility its_computable\n\nopen encodable denumerable\n\nattribute [simp] set.set_of_app_iff\n\nnamespace friedberg_muchnik\nopen rcomputable rcomputable₂\ndef str : strategy 1 := default _\n\ndef generator : ℕ → (Tree 0 × (list ℕ × list ℕ))\n| 0       := ([], [], [])\n| (s + 1) :=\n    let μ  : Tree 0 := (generator s).1, \n        L₀ : list ℕ := (generator s).2.1,\n        L₁ : list ℕ := (generator s).2.2,\n        η  : Tree 1 := up[str] μ in\n    match η.length.bodd with\n    | ff := if ⟦η.length.div2⟧ᵪ^L₀.chr [(λ[str] μ).weight] η.weight = some ff then (∞ :: μ, L₀, η.weight :: L₁)\n            else (𝟘 :: μ, L₀, L₁)\n    | tt := if ⟦η.length.div2⟧ᵪ^L₁.chr [(λ[str] μ).weight] η.weight = some ff then (∞ :: μ, η.weight :: L₀, L₁)\n            else (𝟘 :: μ, L₀, L₁)\n    end\n\nlemma computable.generator : computable generator :=\nbegin\n  let F : ℕ → (Tree 0 × list ℕ × list ℕ) :=\n  nat.elim ([], [], []) (λ s IH,\n    let μ  : Tree 0 := IH.1, \n        L₀ : list ℕ := IH.2.1,\n        L₁ : list ℕ := IH.2.2,\n        η  : Tree 1 := up[str] μ in\n    cond η.length.bodd\n      (if ⟦η.length.div2⟧ᵪ^L₁.chr [(λ[str] μ).weight] η.weight = some ff then (∞ :: μ, η.weight :: L₀, L₁)\n       else (𝟘 :: μ, L₀, L₁))\n      (if ⟦η.length.div2⟧ᵪ^L₀.chr [(λ[str] μ).weight] η.weight = some ff then (∞ :: μ, L₀, η.weight :: L₁)\n       else (𝟘 :: μ, L₀, L₁))),\n  have : computable F,\n  { refine rcomputable.computable_of_rcomp _,\n    refine nat_elim' id ((const list.nil).pair ((const list.nil).pair (const list.nil))) _,\n    refine rcomputable.cond\n      (nat_bodd.comp $ list_length.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂))) _ _,\n    { refine rcomputable.ite ((rcomputable₂.to_bool_eq _).comp _ (const (some ff)))\n        ((list_cons.comp (const ∞) (fst.comp snd.to_unary₂)).pair\n          ((list_cons.comp (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂)))\n            (fst.comp (snd.comp snd.to_unary₂))).pair\n              (snd.comp (snd.comp snd.to_unary₂))))\n        ((list_cons.comp (const 𝟘) (fst.comp snd.to_unary₂)).pair (snd.comp snd.to_unary₂)),\n      refine rcomputable.univn_tot_s _ _\n        (nat_div2.comp (list_length.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂))))\n        (list_chr.comp (snd.comp (snd.comp (snd.comp snd.to_unary₁))) snd)\n        (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.lambda.comp (fst.comp snd.to_unary₂)))\n        (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂))) },\n    { refine rcomputable.ite ((rcomputable₂.to_bool_eq _).comp _ (const (some ff)))\n        ((list_cons.comp (const ∞) (fst.comp snd.to_unary₂)).pair\n          ((fst.comp (snd.comp snd.to_unary₂)).pair\n            (list_cons.comp (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂)))\n              (snd.comp (snd.comp snd.to_unary₂)))))\n        ((list_cons.comp (const 𝟘) (fst.comp snd.to_unary₂)).pair (snd.comp snd.to_unary₂)),\n      refine rcomputable.univn_tot_s _ _\n        (nat_div2.comp (list_length.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂))))\n        (list_chr.comp (fst.comp (snd.comp (snd.comp snd.to_unary₁))) snd)\n        (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.lambda.comp (fst.comp snd.to_unary₂)))\n        (strategy.rcomputable.Tree_weight.comp (strategy.rcomputable.up.comp (fst.comp snd.to_unary₂))) } },\n  exact this.of_eq (λ s, by {\n    induction s with s IH; simp[F, generator],\n    { simp[F] at IH, simp[IH], cases C : (list.length (up[str] (generator s).fst)).bodd; simp[C, generator] } })\nend\n\ndef Λ : Path 0 := ⟨λ s, (generator s).fst, λ s,\n  by { cases C : (up[str] (generator s).1).length.bodd; simp[generator, C],\n       { by_cases C₁ : ⟦(up[str] (generator s).1).length.div2⟧ᵪ^((generator s).2.1.chr) [(λ[str] (generator s).1).weight]\n         (up[str] (generator s).1).weight = some ff; simp[C₁] },\n       { by_cases C₁ : ⟦(up[str] (generator s).1).length.div2⟧ᵪ^((generator s).2.2.chr) [(λ[str] (generator s).1).weight]\n         (up[str] (generator s).1).weight = some ff; simp[C₁] } }⟩\n\nlemma Λ_thick : Λ.thick :=\n⟨by simp[Λ, generator], λ s, by { cases C : (up[str] (generator s).1).length.bodd; simp[Λ, generator, C],\n  { by_cases C₁ : ⟦(up[str] (generator s).1).length.div2⟧ᵪ^((generator s).2.1.chr) [(λ[str] (generator s).1).weight]\n         (up[str] (generator s).1).weight = some ff; simp[C₁], { refine ⟨_, rfl⟩ }, { refine ⟨_, rfl⟩ } },\n  { by_cases C₁ : ⟦(up[str] (generator s).1).length.div2⟧ᵪ^((generator s).2.2.chr) [(λ[str] (generator s).1).weight]\n         (up[str] (generator s).1).weight = some ff; simp[C₁], { refine ⟨_, rfl⟩ }, { refine ⟨_, rfl⟩ } } }⟩\n\nlemma Λ_app_eq (s : ℕ) : Λ s = (generator s).1 := rfl\n\n@[simp] lemma Λ_empty : Λ 0 = [] := rfl\n\ndef L₀ (s : ℕ) : list ℕ := (generator s).2.1\n\n@[simp] lemma L₀_empty : L₀ 0 = [] := rfl\n\ndef L₁ (s : ℕ) : list ℕ := (generator s).2.2\n\n@[simp] lemma L₁_empty : L₁ 0 = [] := rfl\n\ndef I₀ : set ℕ := {n | ∃ s, n ∈ L₀ s}\n\ndef I₁ : set ℕ := {n | ∃ s, n ∈ L₁ s}\n\nlemma I₀_re : r.e. I₀ :=\nbegin\n  suffices : 𝚺⁰1 I₀,\n  { refine sigma_pred1_iff_re.mp this },\n  let A : set ℕ := {n | n.unpair.1 ∈ (L₀ n.unpair.2)},\n  simp[sigma_pred],\n  refine ⟨A, _⟩,\n  have : computable_pred A,\n  { refine ⟨λ a, classical.dec _, _⟩,\n    have : computable (λ a : ℕ, (L₀ a.unpair.2).chr a.unpair.1),\n      from rcomputable.computable_of_rcomp (rcomputable.list_chr.comp\n      (fst.comp $ snd.comp $ computable.generator.to_rcomp.comp (snd.comp nat_unpaired))\n      (fst.comp nat_unpaired)),\n    exact this.of_eq (λ a, by simp[A, list.chr]) },\n  exact ⟨this, by { ext a, simp[I₀] }⟩\nend\n\nlemma I₁_re : r.e. I₁ :=\nbegin\n  suffices : 𝚺⁰1 I₁,\n  { refine sigma_pred1_iff_re.mp this },\n  let A : set ℕ := {n | n.unpair.1 ∈ (L₁ n.unpair.2)},\n  simp[sigma_pred],\n  refine ⟨A, _⟩,\n  have : computable_pred A,\n  { refine ⟨λ a, classical.dec _, _⟩,\n    have : computable (λ a : ℕ, (L₁ a.unpair.2).chr a.unpair.1),\n      from rcomputable.computable_of_rcomp (rcomputable.list_chr.comp\n      (snd.comp $ snd.comp $ computable.generator.to_rcomp.comp (snd.comp nat_unpaired))\n      (fst.comp nat_unpaired)),\n    exact this.of_eq (λ a, by simp[A, list.chr]) },\n  exact ⟨this, by { ext a, simp[I₁] }⟩\nend\n\n@[reducible]\ndef directing_sentence₀ (s : ℕ) : Prop :=\n⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₀ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff\n\n@[reducible]\ndef directing_sentence₁ (s : ℕ) : Prop :=\n⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₁ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff\n\nlemma generator_eq_of_pi_of_even {s : ℕ} (even : (up[str] (Λ s)).length.bodd = ff) :\n  directing_sentence₀ s → generator (s + 1) = (∞ :: Λ s, L₀ s, (up[str] (Λ s)).weight :: L₁ s) := λ C,\nby { simp[directing_sentence₀, L₀, L₁, Λ_app_eq] at C even, simp[generator, even, C, Λ_app_eq, L₀, L₁] }\n\nlemma generator_eq_of_sigma_of_even {s : ℕ} (even : (up[str] (Λ s)).length.bodd = ff) :\n  ¬directing_sentence₀ s → generator (s + 1) = (𝟘 :: Λ s, L₀ s, L₁ s) := λ C,\nby { simp[directing_sentence₀, L₀, L₁, Λ_app_eq] at C even, simp[generator, even, C, Λ_app_eq, L₀, L₁], intros h, contradiction }\n\nlemma generator_eq_of_pi_of_odd {s : ℕ} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  directing_sentence₁ s → generator (s + 1) = (∞ :: Λ s, (up[str] (Λ s)).weight :: L₀ s, L₁ s) := λ C,\nby { simp[directing_sentence₁, L₀, L₁, Λ_app_eq] at C odd, simp[generator, odd, C, Λ_app_eq, L₀, L₁] }\n\nlemma generator_eq_of_sigma_of_odd {s : ℕ} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  ¬directing_sentence₁ s → generator (s + 1) = (𝟘 :: Λ s, L₀ s, L₁ s) := λ C,\nby { simp[directing_sentence₁, L₀, L₁, Λ_app_eq] at C odd, simp[generator, odd, C, Λ_app_eq, L₀, L₁], intros h, contradiction }\n\nlemma L₁_eq_of_pi_of_even {s : ℕ} (even : (up[str] (Λ s)).length.bodd = ff) :\n  directing_sentence₀ s → L₁ (s + 1) = (up[str] (Λ s)).weight :: L₁ s := λ C,\nby simp[L₁, generator_eq_of_pi_of_even even C]\n\nlemma L₁_eq_of_sigma_of_even {s : ℕ} (even : (up[str] (Λ s)).length.bodd = ff) :\n  ¬directing_sentence₀ s → L₁ (s + 1) = L₁ s := λ C,\nby simp[L₁, generator_eq_of_sigma_of_even even C]\n\nlemma L₀_eq_of_pi_of_odd {s : ℕ} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  directing_sentence₁ s → L₀ (s + 1) = (up[str] (Λ s)).weight :: L₀ s := λ C,\nby simp[L₀, generator_eq_of_pi_of_odd odd C]\n\nlemma L₀_eq_of_sigma_of_odd {s : ℕ} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  ¬directing_sentence₁ s → L₀ (s + 1) = L₀ s := λ C,\nby simp[L₀, generator_eq_of_sigma_of_odd odd C]\n\n@[simp] lemma L₁_eq_of_odd {s : ℕ} (odd : (up[str] (Λ s)).length.bodd = tt) : L₁ (s + 1) = L₁ s :=\nby { by_cases C : directing_sentence₁ s,\n     simp[L₁, generator_eq_of_pi_of_odd odd C], simp[L₁, generator_eq_of_sigma_of_odd odd C] }\n\n@[simp] lemma L₀_eq_of_even {s : ℕ} (even : (up[str] (Λ s)).length.bodd = ff) : L₀ (s + 1) = L₀ s :=\nby { by_cases C : directing_sentence₀ s,\n     simp[L₀, generator_eq_of_pi_of_even even C], simp[L₀, generator_eq_of_sigma_of_even even C] }\n\nlemma mem_I₁_of_pi_of_even {s} (even : (up[str] (Λ s)).length.bodd = ff) (pi : directing_sentence₀ s) :\n  (up[str] (Λ s)).weight ∈ I₁ := ⟨s + 1, by simp[L₁_eq_of_pi_of_even even pi]⟩\n\nlemma mem_I₀_of_pi_of_odd {s} (odd : (up[str] (Λ s)).length.bodd = tt) (pi : directing_sentence₁ s) :\n  (up[str] (Λ s)).weight ∈ I₀ := ⟨s + 1, by simp[L₀_eq_of_pi_of_odd odd pi]⟩\n\nlemma mem_L₁_iff (t a : ℕ) :\n  a ∈ L₁ t ↔ ∃ s < t, (up[str] (Λ s)).length.bodd = ff ∧ a = (up[str] (Λ s)).weight ∧ directing_sentence₀ s :=\nbegin\n  induction t with t IH,\n  { simp },\n  { rcases C : (up[str] (Λ t)).length.bodd with (C | C),\n    { by_cases C₂ : directing_sentence₀ t,\n      { simp[L₁_eq_of_pi_of_even C C₂, IH], split,\n        { rintros (rfl | ⟨s, lt, s_even, rfl, pi⟩),\n          { exact ⟨t, lt_add_one t, C, rfl, C₂⟩ }, { refine ⟨s, nat.lt.step lt, s_even, rfl, pi⟩ } },\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n          have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n          rcases this with (lt | rfl), { right, refine ⟨s, lt, s_even, rfl, pi⟩ }, { left, simp } } },\n      { simp[L₁_eq_of_sigma_of_even C C₂, IH], split,\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩, refine ⟨s, nat.lt.step lt_s, s_even, rfl, pi⟩ },\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n          have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n          rcases this with (lt | rfl), { refine ⟨s, lt, s_even, rfl, pi⟩ }, { exfalso, contradiction } } } },\n    { simp[L₁_eq_of_odd C, IH], split,\n      { rintros ⟨s, lt_s, s_even, rfl, pi⟩, refine ⟨s, nat.lt.step lt_s, s_even, rfl, pi⟩ },\n      { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n        have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n        rcases this with (lt | rfl),\n        { refine ⟨s, lt, s_even, rfl, pi⟩ }, { exfalso, simp[C] at s_even, contradiction } } } }\nend\n\nlemma mem_L₀_iff (t a : ℕ) :\n  a ∈ L₀ t ↔ ∃ s < t, (up[str] (Λ s)).length.bodd = tt ∧ a = (up[str] (Λ s)).weight ∧ directing_sentence₁ s :=\nbegin\n  induction t with t IH,\n  { simp },\n  { rcases C : (up[str] (Λ t)).length.bodd with (C | C),\n    { simp[L₀_eq_of_even C, IH], split,\n      { rintros ⟨s, lt_s, s_even, rfl, pi⟩, refine ⟨s, nat.lt.step lt_s, s_even, rfl, pi⟩ },\n      { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n        have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n        rcases this with (lt | rfl),\n        { refine ⟨s, lt, s_even, rfl, pi⟩ }, { exfalso, simp[C] at s_even, contradiction } } },\n    { by_cases C₂ : directing_sentence₁ t,\n      { simp[L₀_eq_of_pi_of_odd C C₂, IH], split,\n        { rintros (rfl | ⟨s, lt, s_even, rfl, pi⟩),\n          { exact ⟨t, lt_add_one t, C, rfl, C₂⟩ }, { refine ⟨s, nat.lt.step lt, s_even, rfl, pi⟩ } },\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n          have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n          rcases this with (lt | rfl), { right, refine ⟨s, lt, s_even, rfl, pi⟩ }, { left, simp } } },\n      { simp[L₀_eq_of_sigma_of_odd C C₂, IH], split,\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩, refine ⟨s, nat.lt.step lt_s, s_even, rfl, pi⟩ },\n        { rintros ⟨s, lt_s, s_even, rfl, pi⟩,\n          have : s < t ∨ s = t, from lt_or_eq_of_le (nat.lt_succ_iff.mp lt_s),\n          rcases this with (lt | rfl), { refine ⟨s, lt, s_even, rfl, pi⟩ }, { exfalso, contradiction } } } } }\nend\n\nlemma L₁_mono {s₁ s₂ : ℕ} (le : s₁ ≤ s₂) : L₁ s₁ ⊆ L₁ s₂ := λ a mem,\nby { simp [mem_L₁_iff] at mem ⊢, rcases mem with ⟨s, lt_s, h⟩, refine ⟨s, gt_of_ge_of_gt le lt_s, h⟩ }\n\nlemma L₀_mono {s₁ s₂ : ℕ} (le : s₁ ≤ s₂) : L₀ s₁ ⊆ L₀ s₂ := λ a mem,\nby { simp [mem_L₀_iff] at mem ⊢, rcases mem with ⟨s, lt_s, h⟩, refine ⟨s, gt_of_ge_of_gt le lt_s, h⟩ }\n\n@[simp] lemma pi_outcome_iff_of_even {s} (even : (up[str] (Λ s)).length.bodd = ff) :\n  (Λ_thick.out s).is_pi ↔ directing_sentence₀ s :=\nbegin\n  by_cases C : directing_sentence₀ s; simp[C],\n  { have : (Λ (s + 1)).is_sigma, { simp[Λ_app_eq, generator_eq_of_pi_of_even even C] },\n    simp[Λ_thick.succ_eq s] at this, exact this },\n  { have : (Λ (s + 1)).is_pi, { simp[Λ_app_eq, generator_eq_of_sigma_of_even even C] },\n    simp[Λ_thick.succ_eq s] at this, simp[this, infinity, zero] }\nend\n\n@[simp] lemma sigma_outcome_iff_of_even {s} (even : (up[str] (Λ s)).length.bodd = ff) :\n  (Λ_thick.out s).is_sigma ↔ ¬directing_sentence₀ s :=\nby { simp[←pi_outcome_iff_of_even even], cases Λ_thick.out s; simp[infinity, zero] }\n\n@[simp] lemma pi_outcome_iff_of_odd {s} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  (Λ_thick.out s).is_pi ↔ directing_sentence₁ s :=\nbegin\n  by_cases C : directing_sentence₁ s; simp[C],\n  { have : (Λ (s + 1)).is_sigma, { simp[Λ_app_eq, generator_eq_of_pi_of_odd odd C] },\n    simp[Λ_thick.succ_eq s] at this, exact this },\n  { have : (Λ (s + 1)).is_pi, { simp[Λ_app_eq, generator_eq_of_sigma_of_odd odd C] },\n    simp[Λ_thick.succ_eq s] at this, simp[this, infinity, zero] }\nend\n\n@[simp] lemma sigma_outcome_iff_of_odd {s} (odd : (up[str] (Λ s)).length.bodd = tt) :\n  (Λ_thick.out s).is_sigma ↔ ¬directing_sentence₁ s :=\nby { simp[←pi_outcome_iff_of_odd odd], cases Λ_thick.out s; simp[infinity, zero] }\n\nlemma sigma_preservation_of_pi_of_even\n  {s₁ s₂} (even : (up[str] (Λ s₁)).length.bodd = ff) (pi : directing_sentence₀ s₁) \n  (on_truepath : up[str] (Λ s₁) ⊆' Λ[str] Λ) (le : s₁ ≤ s₂) {a : ℕ} (bound : a ≤ (λ[str] (Λ s₁)).weight) :\n  a ∈ L₀ s₂ → a ∈ L₀ s₁ :=\nbegin\n  simp only [mem_L₀_iff],\n  rintros ⟨s, lt_s, odd, rfl, pi_s⟩,\n  have : s < s₁ ∨ s = s₁ ∨ s₁ < s, exact trichotomous s s₁, rcases this with (lt_s | lt_s),\n  { refine ⟨s, lt_s, odd, rfl, pi_s⟩ },\n  exfalso,\n  { rcases lt_s with (rfl | gt_s), { simp [odd] at even, contradiction },\n    have : (λ[str] (Λ s₁)).weight < (up[str] (Λ s)).weight,\n      from str.lt_weight_lambda_up Λ_thick (by simp) gt_s (by simp[even, pi]) (by simp[odd, pi_s]) on_truepath,\n    exact nat.lt_le_antisymm this bound }\nend\n\nlemma sigma_preservation_of_even_aux\n  {η : Tree 1} {s₀} {lt : η ⊂ᵢ (Λ[str] Λ) s₀} (sigma : (out ⟨η, lt⟩).is_sigma) (even : η.length.bodd = ff) :\n  ∃ s, directing_sentence₀ s ∧\n    up[str] (Λ s) = η ∧ ⟦η.length.div2⟧ᵪ^(chr I₀) [(λ[str] (Λ s)).weight] η.weight = ff :=\nbegin\n  rcases str.Lambda_sigma_outcome_of_thick Λ Λ_thick lt sigma with ⟨s, rfl, eq_out, pi⟩,\n  have pi : directing_sentence₀ s, from (pi_outcome_iff_of_even even).mp pi,\n  have : ∀ a : ℕ, a < (λ[str] (Λ s)).weight → (L₀ s).chr a = chr I₀ a,\n  { intros a bound, simp[←bool.coe_bool_iff],\n    show a ∈ L₀ s ↔ I₀ a,\n    refine ⟨λ h, ⟨s, h⟩, λ ⟨t, h⟩, sigma_preservation_of_pi_of_even\n      even pi ⟨s₀, lt.suffix⟩ (le_max_left s t) (le_of_lt bound) (L₀_mono (le_max_right s t) h)⟩ },\n  have : ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₀ s).chr [(λ[str] (Λ s)).weight] =\n    ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(chr I₀) [(λ[str] (Λ s)).weight],\n    from rpartrec.univn_tot_use this,\n  refine ⟨s, pi, rfl, _⟩,  \n  simp[←this], exact pi\nend\n\nlemma sigma_preservation_of_even\n  {η : Tree 1} {s₀} {lt : η ⊂ᵢ (Λ[str] Λ) s₀} (sigma : (out ⟨η, lt⟩).is_sigma) (even : η.length.bodd = ff) :\n  η.weight ∈ I₁ ∧ ff ∈ ⟦η.length.div2⟧ᵪ^(chr I₀) η.weight :=\nby { rcases sigma_preservation_of_even_aux sigma even with ⟨s, pi, rfl, eqn⟩,\n     simp[rpartrec.univn_complete],\n     refine ⟨mem_I₁_of_pi_of_even even pi, (λ[str] (Λ s)).weight, eqn⟩}\n\nlemma sigma_preservation_of_pi_of_odd\n  {s₁ s₂} (odd : (up[str] (Λ s₁)).length.bodd = tt) (pi : directing_sentence₁ s₁) \n  (on_truepath : up[str] (Λ s₁) ⊆' Λ[str] Λ) (le : s₁ ≤ s₂) {a : ℕ} (bound : a ≤ (λ[str] (Λ s₁)).weight) :\n  a ∈ L₁ s₂ → a ∈ L₁ s₁:=\nbegin\n  simp only [mem_L₁_iff],\n  rintros ⟨s, lt_s, even, rfl, pi_s⟩,\n  have : s < s₁ ∨ s = s₁ ∨ s₁ < s, exact trichotomous s s₁, rcases this with (lt_s | lt_s),\n  { refine ⟨s, lt_s, even, rfl, pi_s⟩ },\n  exfalso,\n  { rcases lt_s with (rfl | gt_s), { simp [even] at odd, contradiction },\n    have : (λ[str] (Λ s₁)).weight < (up[str] (Λ s)).weight,\n      from str.lt_weight_lambda_up Λ_thick (by simp) gt_s (by simp[odd, pi]) (by simp[even, pi_s]) on_truepath,\n    exact nat.lt_le_antisymm this bound }\nend\n\nlemma sigma_preservation_of_odd_aux\n  {η : Tree 1} {s₀} {lt : η ⊂ᵢ (Λ[str] Λ) s₀} (sigma : (out ⟨η, lt⟩).is_sigma) (odd : η.length.bodd = tt) :\n  ∃ s, directing_sentence₁ s ∧\n    up[str] (Λ s) = η ∧ ⟦η.length.div2⟧ᵪ^(chr I₁) [(λ[str] (Λ s)).weight] η.weight = ff :=\nbegin\n  rcases str.Lambda_sigma_outcome_of_thick Λ Λ_thick lt sigma with ⟨s, rfl, eq_out, pi⟩,\n  have pi : directing_sentence₁ s, from (pi_outcome_iff_of_odd odd).mp pi,\n  have : ∀ a : ℕ, a < (λ[str] (Λ s)).weight → (L₁ s).chr a = chr I₁ a,\n  { intros a bound, simp[←bool.coe_bool_iff],\n    show a ∈ L₁ s ↔ I₁ a,\n    refine ⟨λ h, ⟨s, h⟩, λ ⟨t, h⟩, sigma_preservation_of_pi_of_odd\n      odd pi ⟨s₀, lt.suffix⟩ (le_max_left s t) (le_of_lt bound) (L₁_mono (le_max_right s t) h)⟩ },\n  have : ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₁ s).chr [(λ[str] (Λ s)).weight] =\n    ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(chr I₁) [(λ[str] (Λ s)).weight],\n    from rpartrec.univn_tot_use this,\n  refine ⟨s, pi, rfl, _⟩,  \n  simp[←this], exact pi\nend\n\nlemma sigma_preservation_of_odd\n  {η : Tree 1} {s₀} {lt : η ⊂ᵢ (Λ[str] Λ) s₀} (sigma : (out ⟨η, lt⟩).is_sigma) (odd : η.length.bodd = tt) :\n  η.weight ∈ I₀ ∧ ff ∈ ⟦η.length.div2⟧ᵪ^(chr I₁) η.weight :=\nby { rcases sigma_preservation_of_odd_aux sigma odd with ⟨s, pi, rfl, eqn⟩,\n     simp[rpartrec.univn_complete],\n     refine ⟨mem_I₀_of_pi_of_odd odd pi, (λ[str] (Λ s)).weight, eqn⟩ }\n\nlemma nonmem_of_even\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (even : η.length.bodd = ff) :\n  η.weight ∉ I₁ := λ mem,\nbegin\n  rcases mem with ⟨s₀, mem⟩,\n  rcases (mem_L₁_iff s₀ η.weight).mp mem with ⟨s, lt_s, _, eq_weight, pi⟩,\n  have : η = up[str] (Λ s),\n  { rcases str.le_Lambda_of_thick' Λ_thick ⟨t, lt.suffix⟩ with ⟨s₀, rfl, _⟩, simp at*,\n    rcases str.eq_lambda_of_le_lambda' (str.up_le_lambda (Λ s)) with ⟨μ₀, le_μ₀, eq_up⟩,\n    rcases Λ_thick.ssubset.mp ⟨s, le_μ₀⟩ with ⟨s', rfl⟩,\n    simp[eq_up] at eq_weight ⊢, exact str.weight_lambda_inj_of_thick Λ_thick eq_weight },\n  rcases this with rfl,\n  have : ¬directing_sentence₀ s, from (sigma_outcome_iff_of_even even).mp (str.Lambda_pi_outcome_of_thick Λ_thick pi s rfl),\n  contradiction\nend\n\nlemma nonmem_of_odd\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (odd : η.length.bodd = tt) :\n  η.weight ∉ I₀ := λ mem,\nbegin\n  rcases mem with ⟨s₀, mem⟩,\n  rcases (mem_L₀_iff s₀ η.weight).mp mem with ⟨s, lt_s, _, eq_weight, pi⟩,\n  have : η = up[str] (Λ s),\n  { rcases str.le_Lambda_of_thick' Λ_thick ⟨t, lt.suffix⟩ with ⟨s₀, rfl, _⟩, simp at*,\n    rcases str.eq_lambda_of_le_lambda' (str.up_le_lambda (Λ s)) with ⟨μ₀, le_μ₀, eq_up⟩,\n    rcases Λ_thick.ssubset.mp ⟨s, le_μ₀⟩ with ⟨s', rfl⟩,\n    simp[eq_up] at eq_weight ⊢, exact str.weight_lambda_inj_of_thick Λ_thick eq_weight },\n  rcases this with rfl,\n  have : ¬directing_sentence₁ s, from (sigma_outcome_iff_of_odd odd).mp (str.Lambda_pi_outcome_of_thick Λ_thick pi s rfl),\n  contradiction\nend\n\nlemma L₀_beq_exists(b : ℕ) :\n  ∃ s, ∀ a < b, a ∈ L₀ s ↔ a ∈ I₀ :=\nbegin\n  induction b with b IH,\n  { simp },\n  { rcases IH with ⟨s₀, beq⟩,\n    by_cases C : b ∈ I₀,\n    { rcases C with ⟨s_b, mem⟩,\n      refine ⟨max s₀ s_b, λ a bound, _⟩,\n      split, { intros mem, refine ⟨_, mem⟩ },\n      intros mem,\n      have : a < b ∨ a = b, from lt_or_eq_of_le (nat.lt_succ_iff.mp bound),\n      rcases this with (lt | rfl),\n      { have : a ∈ L₀ s₀, from (beq a lt).mpr mem, exact L₀_mono (le_max_left s₀ s_b) this },\n      { exact L₀_mono (le_max_right s₀ s_b) mem } },\n    { refine ⟨s₀, λ a bound, _⟩,\n      have : a < b ∨ a = b, from lt_or_eq_of_le (nat.lt_succ_iff.mp bound),\n      rcases this with (lt | rfl),\n      { exact beq a lt },\n      { simp[C], intros mem, have : a ∈ I₀, from ⟨s₀, mem⟩, contradiction } } }\nend\n\nlemma L₁_beq_exists(b : ℕ) :\n  ∃ s, ∀ a < b, a ∈ L₁ s ↔ a ∈ I₁ :=\nbegin\n  induction b with b IH,\n  { simp },\n  { rcases IH with ⟨s₀, beq⟩,\n    by_cases C : b ∈ I₁,\n    { rcases C with ⟨s_b, mem⟩,\n      refine ⟨max s₀ s_b, λ a bound, _⟩,\n      split, { intros mem, refine ⟨_, mem⟩ },\n      intros mem,\n      have : a < b ∨ a = b, from lt_or_eq_of_le (nat.lt_succ_iff.mp bound),\n      rcases this with (lt | rfl),\n      { have : a ∈ L₁ s₀, from (beq a lt).mpr mem, exact L₁_mono (le_max_left s₀ s_b) this },\n      { exact L₁_mono (le_max_right s₀ s_b) mem } },\n    { refine ⟨s₀, λ a bound, _⟩,\n      have : a < b ∨ a = b, from lt_or_eq_of_le (nat.lt_succ_iff.mp bound),\n      rcases this with (lt | rfl),\n      { exact beq a lt },\n      { simp[C], intros mem, have : a ∈ I₁, from ⟨s₀, mem⟩, contradiction } } }\nend\n\nlemma pi_substrategies_of_even_aux\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (even : η.length.bodd = ff) :\n  ∀ s₀ : ℕ, ∃ s > s₀, ¬directing_sentence₀ s ∧ up[str] (Λ s) = η := λ s₀,\nbegin\n  have : ∃ s > s₀, up[str] (Λ s) = η, from str.infinite_substrategy_of_pi' Λ_thick pi s₀,\n  rcases this with ⟨s, lt_s, rfl⟩,\n  have : ¬directing_sentence₀ s, from (sigma_outcome_iff_of_even even).mp (str.Lambda_pi_outcome_of_thick Λ_thick pi s rfl),  \n  refine ⟨s, lt_s, this, rfl⟩\nend\n\nlemma pi_substrategies_of_even\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (even : η.length.bodd = ff) :\n  ¬ff ∈ ⟦η.length.div2⟧ᵪ^(chr I₀) η.weight := λ A,\nbegin\n  have : ∃ s₀, ⟦η.length.div2⟧ᵪ^(chr I₀) [s₀] η.weight = ff, from rpartrec.univn_complete.mp A,\n  rcases this with ⟨s₀, eq_ff⟩,\n  have : ∃ t, ∀ a < s₀, a ∈ L₀ t ↔ a ∈ I₀, from L₀_beq_exists s₀,\n  rcases this with ⟨t₀, beq⟩,\n  have : ∃ s₁, s₀ < (λ[str] (Λ s₁)).weight, from str.lambda_infinitely Λ_thick (by simp) _,\n  rcases this with ⟨s₁, lt_weight⟩,\n  let s₂ := max t₀ s₁,\n  have : ∃ s > max t₀ s₁, ¬⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₀ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff ∧ \n    up[str] (Λ s) = η, from pi_substrategies_of_even_aux pi even _,\n  rcases this with ⟨s, lt_s, ne_ff, rfl⟩,\n  have le_s₀ : s₀ ≤ (λ[str] (Λ s)).weight,\n    calc s₀ ≤ (λ[str] (Λ s₁)).weight : le_of_lt lt_weight\n        ... ≤ (λ[str] (Λ s)).weight : str.weight_lambda_le_mono (Λ_thick.le_mono_iff.mpr (le_of_lt (max_lt_iff.mp lt_s).2)),\n  have beq_s : ∀ a < s₀, (a ∈ I₀ ↔ a ∈ L₀ s),\n  { intros a bound, split, \n    { intros mem, have : a ∈ L₀ t₀, from (beq a bound).mpr mem, exact L₀_mono (le_of_lt (max_lt_iff.mp lt_s).1) this },\n    { intros mem, exact ⟨s, mem⟩ } },\n  have : ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₀ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff,\n    from rpartrec.univn_tot_mono_use (by { simp[←bool.coe_bool_iff], exact beq_s }) le_s₀ eq_ff,\n  contradiction\nend\n\nlemma pi_substrategies_of_odd_aux\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (odd : η.length.bodd = tt) :\n  ∀ s₀ : ℕ, ∃ s > s₀, ¬directing_sentence₁ s ∧ up[str] (Λ s) = η := λ s₀,\nbegin\n  have : ∃ s > s₀, up[str] (Λ s) = η, from str.infinite_substrategy_of_pi' Λ_thick pi s₀,\n  rcases this with ⟨s, lt_s, rfl⟩,\n  have : ¬directing_sentence₁ s, from (sigma_outcome_iff_of_odd odd).mp (str.Lambda_pi_outcome_of_thick Λ_thick pi s rfl),  \n  refine ⟨s, lt_s, this, rfl⟩\nend\n\nlemma pi_substrategies_of_odd\n  {η : Tree 1} {t} {lt : η ⊂ᵢ (Λ[str] Λ) t} (pi : (out ⟨η, lt⟩).is_pi) (odd : η.length.bodd = tt) :\n  ¬ff ∈ ⟦η.length.div2⟧ᵪ^(chr I₁) η.weight := λ A,\nbegin\n  have : ∃ s₀, ⟦η.length.div2⟧ᵪ^(chr I₁) [s₀] η.weight = ff, from rpartrec.univn_complete.mp A,\n  rcases this with ⟨s₀, eq_ff⟩,\n  have : ∃ t, ∀ a < s₀, a ∈ L₁ t ↔ a ∈ I₁, from L₁_beq_exists s₀,\n  rcases this with ⟨t₀, beq⟩,\n  have : ∃ s₁, s₀ < (λ[str] (Λ s₁)).weight, from str.lambda_infinitely Λ_thick (by simp) _,\n  rcases this with ⟨s₁, lt_weight⟩,\n  let s₂ := max t₀ s₁,\n  have : ∃ s > max t₀ s₁, ¬⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₁ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff ∧ \n    up[str] (Λ s) = η, from pi_substrategies_of_odd_aux pi odd _,\n  rcases this with ⟨s, lt_s, ne_ff, rfl⟩,\n  have le_s₀ : s₀ ≤ (λ[str] (Λ s)).weight,\n    calc s₀ ≤ (λ[str] (Λ s₁)).weight : le_of_lt lt_weight\n        ... ≤ (λ[str] (Λ s)).weight : str.weight_lambda_le_mono (Λ_thick.le_mono_iff.mpr (le_of_lt (max_lt_iff.mp lt_s).2)),\n  have beq_s : ∀ a < s₀, (a ∈ I₁ ↔ a ∈ L₁ s),\n  { intros a bound, split, \n    { intros mem, have : a ∈ L₁ t₀, from (beq a bound).mpr mem, exact L₁_mono (le_of_lt (max_lt_iff.mp lt_s).1) this },\n    { intros mem, exact ⟨s, mem⟩ } },\n  have : ⟦(up[str] (Λ s)).length.div2⟧ᵪ^(L₁ s).chr [(λ[str] (Λ s)).weight] (up[str] (Λ s)).weight = ff,\n    from rpartrec.univn_tot_mono_use (by { simp[←bool.coe_bool_iff], exact beq_s }) le_s₀ eq_ff,\n  contradiction\nend\n\ntheorem not_I₁_le_I₀ : I₁ ≰ₜ I₀ := λ hyp,\nbegin\n  have : ∃ e, ⟦e⟧ᵪ^(chr I₀) = chr I₁, from rpartrec.exists_index.mp (classical_iff.mp hyp),\n  rcases this with ⟨e, lmm_e⟩,\n  have : ∃ η, η ⊂' Λ[str] Λ ∧ η.length = bit0 e, from (str.Lambda_infinite Λ_thick).lt_length_eq (bit0 e),\n  rcases this with ⟨η, ⟨s₀, lt⟩, eq_len⟩,\n  have even : η.length.bodd = ff, { simp[eq_len] },\n  have eq_e : e = η.length.div2, { simp[eq_len] },  \n  have : (out ⟨η, lt⟩).is_pi ∨ (out ⟨η, lt⟩).is_sigma, from pi_or_sigma (out ⟨η, lt⟩),\n  rcases this with (pi | sigma),\n  { have : η.weight ∉ I₁, from nonmem_of_even pi even,\n    have : ff ∈ ⟦e⟧ᵪ^(chr I₀) η.weight, { simp[lmm_e], exact eq.symm ((chr_ff_iff _ _).mpr this) },\n    have : ff ∉ ⟦e⟧ᵪ^(chr I₀) η.weight, rw eq_e, from pi_substrategies_of_even pi even,\n    contradiction },\n  { have : η.weight ∈ I₁ ∧ ff ∈ ⟦e⟧ᵪ^(chr I₀) η.weight, rw eq_e, from sigma_preservation_of_even sigma even,\n    rcases this with ⟨mem, nonmem⟩,\n    have : η.weight ∉ I₁, { simp[lmm_e] at nonmem, exact (chr_ff_iff _ _).mp (eq.symm nonmem) },\n    contradiction }\nend\n\ntheorem not_I₀_le_I₁ : I₀ ≰ₜ I₁ := λ hyp,\nbegin\n  have : ∃ e, ⟦e⟧ᵪ^(chr I₁) = chr I₀, from rpartrec.exists_index.mp (classical_iff.mp hyp),\n  rcases this with ⟨e, lmm_e⟩,\n  have : ∃ η, η ⊂' Λ[str] Λ ∧ η.length = bit1 e, from (str.Lambda_infinite Λ_thick).lt_length_eq (bit1 e),\n  rcases this with ⟨η, ⟨s₀, lt⟩, eq_len⟩,\n  have odd : η.length.bodd = tt, { simp[eq_len] },\n  have eq_e : e = η.length.div2, { simp[eq_len] },  \n  have : (out ⟨η, lt⟩).is_pi ∨ (out ⟨η, lt⟩).is_sigma, from pi_or_sigma (out ⟨η, lt⟩),\n  rcases this with (pi | sigma),\n  { have : η.weight ∉ I₀, from nonmem_of_odd pi odd,\n    have : ff ∈ ⟦e⟧ᵪ^(chr I₁) η.weight, { simp[lmm_e], exact eq.symm ((chr_ff_iff _ _).mpr this) },\n    have : ff ∉ ⟦e⟧ᵪ^(chr I₁) η.weight, rw eq_e, from pi_substrategies_of_odd pi odd,\n    contradiction },\n  { have : η.weight ∈ I₀ ∧ ff ∈ ⟦e⟧ᵪ^(chr I₁) η.weight, rw eq_e, from sigma_preservation_of_odd sigma odd,\n    rcases this with ⟨mem, nonmem⟩,\n    have : η.weight ∉ I₀, { simp[lmm_e] at nonmem, exact (chr_ff_iff _ _).mp (eq.symm nonmem) },\n    contradiction }\nend\n\ntheorem incomparable_re_sets : ∃ I₀ I₁ : set ℕ, r.e. I₀ ∧ r.e. I₁ ∧ I₁ ≰ₜ I₀ ∧ I₀ ≰ₜ I₁ :=\n⟨I₀, I₁, I₀_re, I₁_re, not_I₁_le_I₀, not_I₀_le_I₁⟩\n\nend friedberg_muchnik\n", "meta": {"author": "iehality", "repo": "lean-reducibility", "sha": "82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7", "save_path": "github-repos/lean/iehality-lean-reducibility", "path": "github-repos/lean/iehality-lean-reducibility/lean-reducibility-82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7/src/friedberg_muchnik.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3112532248693237}}
{"text": "import Terms\nimport CoC\n\nopen PTSSort\nopen BetaOpt\n\n/-! The type checker. -/\n\n/- \n  Since Judgements are in Prop, we have to jump through some hoops\n  to write nearly trivial functions. But Judgements belong to Prop\n  not least because the type checker would be noncomputable if they\n  were in type due to some hard to prove statements.\n-/\n\ninductive TypeError : Type\n| TypeMismatch : Exp → Exp → TypeError\n| UnboundIndex : nat → TypeError\n| UnboundVar : string → TypeError\n| BoxInTerm : TypeError\n| WhileChecking : Exp → TypeError → TypeError\n| ExpectedFunctionButGot : Exp → TypeError\n| ExpectedSortButGot : Exp → TypeError\nopen TypeError\n\ndef type_error_repr : TypeError -> string\n| (TypeMismatch e1 e2) := \"Type mismatch between: \" ++ repr e1 ++ \" and \" ++ repr e2\n| (UnboundIndex n) := \"Unbound index: \" ++ repr n\n| (UnboundVar s) := \"Unbound var: \" ++ repr s\n| (BoxInTerm) := \"Box in term.\"\n| (WhileChecking e t) := \"While checking \" ++ repr e ++ \" encountered\\n\" ++ type_error_repr t\n| (ExpectedFunctionButGot e) := \"Expected function but got: \" ++ repr e\n| (ExpectedSortButGot e) := \"Expected sort but got: \" ++ repr e\n\ninstance type_error_has_repr : has_repr TypeError :=\n  { repr := type_error_repr }\n\n/-- The type checker monad -/\ndef TC := except TypeError\nopen except\ninstance tc_has_bind : has_bind TC := { bind := λ _ _, except.bind }\n\ninstance tc_has_repr {a} {h : has_repr a} : has_repr (TC a) :=\n  { repr := λ t, match t with\n      | error t := repr t\n      | ok a := repr a\n      end }\n\ninstance type_psigma_has_repr {g e} : has_repr (Σ' t : Exp, Judgement g e t) :=\n  { repr := λ s, repr s.fst }\n\n/- This instance is necessary for #eval -/\ninstance type_tc_has_repr {g e} : has_repr (TC (Σ' t : Exp, Judgement g e t)) :=\n  { repr := (@tc_has_repr _ type_psigma_has_repr).repr }\n\n/-- Lookup a variable in a well-formed context. -/\ndef lookup (x : string) : Π (g : Context) (h : ContextWF g), \n  TC (Σ' t, Judgement g (Exp.free x) t)\n| (Context.empty) h := error (UnboundVar x)\n| (Context.cons y e g') h := \n  match string.has_decidable_eq x y with\n  | (is_true p) := ok ⟨e, begin simp [p], cases h, \n      exact Judgement.start y h_noShadowing h_a, end⟩\n  | (is_false p) := lookup g' (by { cases h, assumption }) >>= (begin \n        intro a, apply ok, cases a, apply psigma.mk, cases h, \n        exact Judgement.weaken y h_noShadowing a_snd h_a,\n      end)\n  end\n\ndef find_rule : Π (s t : PTSSort), Rule s t\n| star star := Rule.vdv\n| star box := Rule.tdv\n| box star := Rule.vdt\n| box box := Rule.tdt\n\ninductive Classification (g : Context) (t : Exp) : Type\n| kind : t = Exp.sort box → Classification\n| term : Judgement g t (Exp.sort star) → Classification\n| constructor : Judgement g t (Exp.sort box) → Classification\n\n-- | See Theorem 3.2 and 3.3 in [snforcc].\n-- The classification lemma will be challenging since Classification is in\n-- type and not in Prop. See typing_pi_find_sort below.\naxiom preservation {g e t} : Judgement g e t → Judgement g (head_reduce e) t\naxiom classification : Π {g e t}, Judgement g e t → Classification g t\naxiom beta_preserves_type : Π {g e t z}, \n  Beta Red t z → Judgement g e t → Judgement g e z\n\ndef beta_reduce_terminates_type {g e t} : Judgement g e t → (beta_reduce t).dom :=\nbegin\n  intro j, cases classification j, simp [a, beta_reduce],\n  refine acc.intro t _, intro y, rw [a], intro b, cases b,\n  repeat { exact beta_reduce_terminates a },\nend\n\ninductive ContextLT : Context -> Context -> Prop\n| empty {g} : ContextLT (Context.empty) g\n| cons1 {x a g h} : ContextLT g h -> ContextLT g (Context.cons x a h)\n| cons2 {x a g h} : ContextLT g h -> ContextLT (Context.cons x a g) (Context.cons x a h)\n\ndef context_lt_refl : Π g, ContextLT g g\n| Context.empty := ContextLT.empty\n| (Context.cons x a g) := ContextLT.cons2 (context_lt_refl g)\n\ndef context_lt_cons {x a} : Π g, ContextLT g (Context.cons x a g)\n  := λ g, ContextLT.cons1 (context_lt_refl g)\n\ndef preserves_type {g e t} \n  : Judgement g e t → Judgement g e (head_reduce t) :=\nbegin\n  intro j, cases classification j, rw [a, head_reduce], rw [a] at j, exact j,\n  exact Judgement.conv (Beta.head_reduce Eq t) (preservation a) j,\n  exact Judgement.conv (Beta.head_reduce Eq t) (preservation a) j,\nend\n\n-- The grammar for expressions of type box is\n-- t := A | * | x -> t for A a variable and x an arbitrary expression.\nlemma typing_pi_find_sort : Π a (g : Context) (h : ContextWF g), \n  Σ' s, ∀ p, ∀ h_g, ContextLT h_g g -> Judgement h_g a (Exp.sort p) -> p = s\n   := sorry\n\nlemma typing_pi : Π {g x a b t}, Judgement g (Exp.pi x a b) t → Σ' s, Judgement g a (Exp.sort s) :=\nbegin\n  intros g x a b t,\n  generalize e : Exp.pi x a b = y, intro h,\n  let z := typing_pi_find_sort a g (judgement_context_wf h),\n  apply psigma.mk z.fst,\n  induction h generalizing e; cases e,\n  { apply Judgement.weaken,\n    exact h_noShadowing, have z2 := h_ih_a (eq.refl _), \n    exact eq.mp (by rw [z.snd _ h_g (context_lt_cons _) z2]) z2, \n    exact h_a_1, }, \n  { exact eq.mp (by rw [z.snd h_s1 h_g (context_lt_refl h_g) h_a]) h_a }, \n  { from h_ih_a_1 (eq.refl _) },\nend\n\nlemma beta_reduce_terminates_head_reduce_pi {g e t x a b} : Judgement g e t\n  → head_reduce t = Exp.pi x a b → (beta_reduce a).dom := begin\n  intros j h, cases classification j, \n    simp [a_1, head_reduce] at h, from false.elim h,\n    iterate 2 { have h2 := preservation a_1, rw [h] at h2,\n    cases typing_pi h2, from beta_reduce_terminates snd, }\nend\n\ninductive typecheck_rel : (Σ' (e : Exp) (g : Context), ContextWF g) \n  → (Σ' (e : Exp) (g : Context), ContextWF g) → Prop\n| app_1 {g h a b} : typecheck_rel ⟨a, g, h⟩ ⟨Exp.app a b, g, h⟩\n| app_2 {g h a b} : typecheck_rel ⟨b, g, h⟩ ⟨Exp.app a b, g, h⟩\n| lam_1 {g h x a b} : typecheck_rel ⟨a, g, h⟩ ⟨Exp.lam x a b, g, h⟩\n| lam_2 {g h x a t s hns} {h1 : Judgement g a (Exp.sort s)} : typecheck_rel\n  ⟨instantiate (Exp.free (fresh (context_domain g ∪ free_vars t) x)) t\n  , Context.cons (fresh (context_domain g ∪ free_vars t) x) a g, ContextWF.cons h hns h1⟩\n  ⟨Exp.lam x a t, g, h⟩\n| check_pi {g h x a t b} \n  (h1 : Judgement (Context.cons (fresh (context_domain g ∪ free_vars t) x) a g) \n    (instantiate (Exp.free (fresh (context_domain g ∪ free_vars t) x)) t) b) \n  : typecheck_rel\n  ⟨Exp.pi (fresh (context_domain g ∪ free_vars t) x) a (abstract (fresh (context_domain g ∪ free_vars t) x) b), g, h⟩ \n  ⟨Exp.lam x a t, g, h⟩\n| pi_1 {g h x a b} : typecheck_rel ⟨a, g, h⟩ ⟨Exp.pi x a b, g, h⟩\n| pi_2 {g h x a t s hns} {h1 : Judgement g a (Exp.sort s)} : typecheck_rel\n  ⟨instantiate (Exp.free (fresh (context_domain g ∪ free_vars t) x)) t\n  , Context.cons (fresh (context_domain g ∪ free_vars t) x) a g, ContextWF.cons h hns h1⟩\n  ⟨Exp.pi x a t, g, h⟩\n\nset_option eqn_compiler.zeta true\ndef typecheck_recur : Π (e : Exp) (g : Context) (h : ContextWF g), \n (Π y, typecheck_rel y ⟨e, g, h⟩ → TC (Σ' t, Judgement y.2.1 y.1 t))\n → TC (Σ' t, Judgement g e t)\n| (Exp.free x) g h IH := lookup x g h\n| (Exp.bound x) g h IH := error (UnboundIndex x)\n| (Exp.sort s) g h IH := match s with\n  | star := ok ⟨Exp.sort box, ContextWF.rec_on h (Judgement.starInBox)\n    (λ x _ _ _ _ hns a h, Judgement.weaken x hns h a)⟩\n  | box := error BoxInTerm\n  end\n| (Exp.app f a) g h IH :=\n  IH ⟨f, g, h⟩ (typecheck_rel.app_1) >>= λ ⟨e, je⟩,\n  IH ⟨a, g, h⟩ (typecheck_rel.app_2) >>= λ ⟨A, jA⟩,\n    let ⟨z1, beta1⟩ := (beta_reduce A).get (beta_reduce_terminates_type jA)\n    in begin\n    cases h : (head_reduce e),\n    case Exp.pi : x A' B begin\n      rename h_1 h,\n      let zb := (beta_reduce A').get (beta_reduce_terminates_head_reduce_pi je h),\n      cases exp_decidable_eq z1 zb.1,\n        exact error (ExpectedFunctionButGot e),\n        let h2 : Σ' s, Judgement g (Exp.pi x A' B) (Exp.sort s) := begin\n          cases classification (preserves_type je),\n          simp [a_1, head_reduce] at h, from false.elim h,\n          repeat { rw [h] at a_1 }, exact ⟨star, a_1⟩, exact ⟨box, a_1⟩,\n        end,\n        let h4 : Σ' s, Judgement g A' (Exp.sort s) := begin\n          cases classification (preserves_type je),\n          simp [a_1, head_reduce] at h, from false.elim h,\n          repeat { rw [h] at a_1, from typing_pi a_1, },\n        end,\n        apply ok, fapply psigma.mk, exact (instantiate a B), apply Judgement.app,\n          apply Judgement.conv, apply eq.mp (by simp [h]) (Beta.head_reduce Eq e),\n          from h2.snd, from je, apply Judgement.conv, \n          from (Beta.trans Red Eq Eq (eq.refl _) beta1 (eq.mp (by simp[h_1]) (Beta.symm zb.2))),\n          from h4.2, from jA,\n      end,\n    repeat { exact error (ExpectedFunctionButGot (head_reduce e)) }\n    end\n| (Exp.lam x a t) g h IH :=\n  IH ⟨a, g, h⟩ (typecheck_rel.lam_1) >>= λ⟨s, ja⟩, \n  match (beta_reduce s).get (beta_reduce_terminates_type ja), ja with\n  | ⟨Exp.sort s, bs⟩, ja := let x' := fresh (context_domain g ∪ free_vars t) x in\n    IH ⟨(instantiate (Exp.free x') t), (Context.cons x' a g),\n      (ContextWF.cons h (iff.elim_left finset.not_mem_union (fresh_not_mem)).1 \n      (beta_preserves_type bs ja))⟩ (typecheck_rel.lam_2)\n    >>= λ⟨b, jb⟩, \n    IH ⟨(Exp.pi x' a (abstract x' b)), g, h⟩ (typecheck_rel.check_pi jb) >>= \n      λ⟨p, jp⟩, match (beta_reduce p).get (beta_reduce_terminates_type jp), jp with\n        | ⟨Exp.sort _, bp⟩, jp := \n          let h : Judgement g (Exp.lam x a t) (Exp.pi x a (abstract x' b))\n            := eq.mp (by simp [fresh_avoids_capture (finset.subset_union_right _ _)]) \n              (Judgement.abs x jb (beta_preserves_type bp jp))\n          in ok ⟨Exp.pi x a (abstract x' b), h⟩\n        | _, _ := error (ExpectedSortButGot p)\n        end\n  | ⟨e, _⟩, _ := error (ExpectedSortButGot e)\n  end\n| (Exp.pi x a b) g h IH :=\n  IH ⟨a, g, h⟩ (typecheck_rel.pi_1) >>= λ⟨s, js⟩, \n  match (beta_reduce s).get (beta_reduce_terminates_type js), js with\n  | ⟨Exp.sort s, bs⟩, js := let x' := fresh (context_domain g ∪ free_vars b) x in\n    IH ⟨(instantiate (Exp.free x') b), (Context.cons x' a g), \n      (ContextWF.cons h (iff.elim_left finset.not_mem_union (fresh_not_mem)).1 \n      (beta_preserves_type bs js))⟩ (typecheck_rel.pi_2)\n    >>= λ⟨t, jt⟩, match (beta_reduce t).get (beta_reduce_terminates_type jt), jt with\n    | ⟨Exp.sort t, bt⟩, jt := \n      let h : Judgement g (Exp.pi x a b) (Exp.sort t) \n        := eq.mp (by simp [fresh_avoids_capture (finset.subset_union_right _ _)])\n          (Judgement.product x (find_rule s t) (beta_preserves_type bs js) \n          (beta_preserves_type bt jt))\n      in ok ⟨Exp.sort t, h⟩\n    | ⟨e, _⟩, _ := error (ExpectedSortButGot e)\n    end\n  | ⟨e, _⟩, _ := error (ExpectedSortButGot e)\n  end\nset_option eqn_compiler.zeta false\n\ndef typecheck_roption (e : Exp) (g : Context) (h : ContextWF g) : roption (TC (Σ' t, Judgement g e t)) :=\nbegin\n  refine ⟨acc typecheck_rel ⟨e,g,h⟩, λ h2, @acc.rec_on (Σ' (e : Exp) (g : Context), ContextWF g) _ \n  (λ y, TC (Σ' t, Judgement y.2.1 y.1 t)) _ h2 (λ ⟨e1,g1,h1⟩ ih IH, \n  typecheck_recur e1 g1 h1 IH)⟩,\nend\n\ndef typecheck_terminates (e : Exp) (g : Context) (h : ContextWF g) : (typecheck_roption e g h).dom\n  := sorry\n\ndef typecheck (e : Exp) (g : Context) (h : ContextWF g) : TC (Σ' t, Judgement g e t) :=\n  (typecheck_roption e g h).get (typecheck_terminates e g h)", "meta": {"author": "anfelor", "repo": "coc-lean", "sha": "fdd967d2b7bc349202a1deabbbce155eed4db73a", "save_path": "github-repos/lean/anfelor-coc-lean", "path": "github-repos/lean/anfelor-coc-lean/coc-lean-fdd967d2b7bc349202a1deabbbce155eed4db73a/src/TypeCheck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3112296656009273}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.presheafed_space.has_colimits\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.preserves.shapes.pullbacks\nimport topology.sheaves.functors\nimport algebraic_geometry.Scheme\nimport category_theory.limits.shapes.strict_initial\nimport algebra.category.Ring.instances\n\n/-!\n# Open immersions of structured spaces\n\nWe say that a morphism of presheafed spaces `f : X ⟶ Y` is an open immersions if\nthe underlying map of spaces is an open embedding `f : X ⟶ U ⊆ Y`,\nand the sheaf map `Y(V) ⟶ f _* X(V)` is an iso for each `V ⊆ U`.\n\nAbbreviations are also provided for `SheafedSpace`, `LocallyRingedSpace` and `Scheme`.\n\n## Main definitions\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a PresheafedSpace hom `f` is an open_immersion.\n* `algebraic_geometry.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a Scheme morphism `f` is an open_immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict`: The source of an\n  open immersion is isomorphic to the restriction of the target onto the image.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.lift`: Any morphism whose range is\n  contained in an open immersion factors though the open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace`: If `f : X ⟶ Y` is an\n  open immersion of presheafed spaces, and `Y` is a sheafed space, then `X` is also a sheafed\n  space. The morphism as morphisms of sheafed spaces is given by `to_SheafedSpace_hom`.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace`: If `f : X ⟶ Y` is\n  an open immersion of presheafed spaces, and `Y` is a locally ringed space, then `X` is also a\n  locally ringed space. The morphism as morphisms of locally ringed spaces is given by\n  `to_LocallyRingedSpace_hom`.\n\n## Main results\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.comp`: The composition of two open\n  immersions is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso`: An iso is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_iso`:\n  A surjective open immersion is an isomorphism.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.stalk_iso`: An open immersion induces\n  an isomorphism on stalks.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.has_pullback_of_left`: If `f` is an open\n  immersion, then the pullback `(f, g)` exists (and the forgetful functor to `Top` preserves it).\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_snd_of_left`: Open immersions\n  are stable under pullbacks.\n* `algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso` An (topological) open embedding\n  between two sheafed spaces is an open immersion if all the stalk maps are isomorphisms.\n\n-/\n\nopen topological_space category_theory opposite\nopen category_theory.limits\nnamespace algebraic_geometry\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C]\n\n/--\nAn open immersion of PresheafedSpaces is an open embedding `f : X ⟶ U ⊆ Y` of the underlying\nspaces, such that the sheaf map `Y(V) ⟶ f _* X(V)` is an iso for each `V ⊆ U`.\n-/\nclass PresheafedSpace.is_open_immersion {X Y : PresheafedSpace C} (f : X ⟶ Y) : Prop :=\n(base_open : open_embedding f.base)\n(c_iso : ∀ U : opens X, is_iso (f.c.app (op (base_open.is_open_map.functor.obj U))))\n\n/--\nA morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism\nof PresheafedSpaces\n-/\nabbreviation SheafedSpace.is_open_immersion\n  [has_products C] {X Y : SheafedSpace C} (f : X ⟶ Y) : Prop :=\nPresheafedSpace.is_open_immersion f\n\n/--\nA morphism of LocallyRingedSpaces is an open immersion if it is an open immersion as a morphism\nof SheafedSpaces\n-/\nabbreviation LocallyRingedSpace.is_open_immersion {X Y : LocallyRingedSpace} (f : X ⟶ Y) : Prop :=\nSheafedSpace.is_open_immersion f.1\n\n/--\nA morphism of Schemes is an open immersion if it is an open immersion as a morphism\nof LocallyRingedSpaces\n-/\nabbreviation is_open_immersion {X Y : Scheme} (f : X ⟶ Y) : Prop :=\nLocallyRingedSpace.is_open_immersion f\n\nnamespace PresheafedSpace.is_open_immersion\n\nopen PresheafedSpace\n\nlocal notation `is_open_immersion` := PresheafedSpace.is_open_immersion\n\nattribute [instance] is_open_immersion.c_iso\n\nsection\n\nvariables {X Y : PresheafedSpace C} {f : X ⟶ Y} (H : is_open_immersion f)\n\n/-- The functor `opens X ⥤ opens Y` associated with an open immersion `f : X ⟶ Y`. -/\nabbreviation open_functor := H.base_open.is_open_map.functor\n\n/-- An open immersion `f : X ⟶ Y` induces an isomorphism `X ≅ Y|_{f(X)}`. -/\n@[simps] noncomputable\ndef iso_restrict : X ≅ Y.restrict H.base_open :=\nPresheafedSpace.iso_of_components (iso.refl _)\nbegin\n  symmetry,\n  fapply nat_iso.of_components,\n  intro U,\n  refine as_iso (f.c.app (op (H.open_functor.obj (unop U)))) ≪≫ X.presheaf.map_iso (eq_to_iso _),\n  { induction U using opposite.rec,\n    cases U,\n    dsimp only [is_open_map.functor, functor.op, opens.map],\n    congr' 2,\n    erw set.preimage_image_eq _ H.base_open.inj,\n    refl },\n  { intros U V i,\n    simp only [category_theory.eq_to_iso.hom, Top.presheaf.pushforward_obj_map, category.assoc,\n      functor.op_map, iso.trans_hom, as_iso_hom, functor.map_iso_hom, ←X.presheaf.map_comp],\n    erw [f.c.naturality_assoc, ←X.presheaf.map_comp],\n    congr }\nend\n\n@[simp] lemma iso_restrict_hom_of_restrict : H.iso_restrict.hom ≫ Y.of_restrict _ = f :=\nbegin\n  ext,\n  { simp only [comp_c_app, iso_restrict_hom_c_app, nat_trans.comp_app,\n      eq_to_hom_refl, of_restrict_c_app, category.assoc, whisker_right_id'],\n    erw [category.comp_id, f.c.naturality_assoc, ←X.presheaf.map_comp],\n    transitivity f.c.app x ≫ X.presheaf.map (𝟙 _),\n    { congr },\n    { erw [X.presheaf.map_id, category.comp_id] } },\n  { simp }\nend\n\n@[simp] lemma iso_restrict_inv_of_restrict : H.iso_restrict.inv ≫ f = Y.of_restrict _ :=\nby { rw iso.inv_comp_eq, simp }\n\ninstance mono [H : is_open_immersion f] : mono f :=\nby { rw ← H.iso_restrict_hom_of_restrict, apply mono_comp }\n\n/-- The composition of two open immersions is an open immersion. -/\ninstance comp {Z : PresheafedSpace C} (f : X ⟶ Y) [hf : is_open_immersion f] (g : Y ⟶ Z)\n  [hg : is_open_immersion g] :\n  is_open_immersion (f ≫ g) :=\n{ base_open := hg.base_open.comp hf.base_open,\n  c_iso := λ U,\n  begin\n    generalize_proofs h,\n    dsimp only [algebraic_geometry.PresheafedSpace.comp_c_app, unop_op, functor.op, comp_base,\n      Top.presheaf.pushforward_obj_obj, opens.map_comp_obj],\n    apply_with is_iso.comp_is_iso { instances := ff },\n    swap,\n    { have : (opens.map g.base).obj (h.functor.obj U) = hf.open_functor.obj U,\n      { dsimp only [opens.map, is_open_map.functor, PresheafedSpace.comp_base],\n        congr' 1,\n        rw [coe_comp, ←set.image_image, set.preimage_image_eq _ hg.base_open.inj] },\n      rw this,\n      apply_instance },\n    { have : h.functor.obj U = hg.open_functor.obj (hf.open_functor.obj U),\n      { dsimp only [is_open_map.functor],\n        congr' 1,\n        rw [comp_base, coe_comp, ←set.image_image],\n        congr },\n      rw this,\n      apply_instance }\n  end }\n\n/-- For an open immersion `f : X ⟶ Y` and an open set `U ⊆ X`, we have the map `X(U) ⟶ Y(U)`. -/\nnoncomputable\ndef inv_app (U : opens X) : X.presheaf.obj (op U) ⟶ Y.presheaf.obj (op (H.open_functor.obj U)) :=\nX.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) ≫\n  inv (f.c.app (op (H.open_functor.obj U)))\n\n@[simp, reassoc] lemma inv_naturality {U V : (opens X)ᵒᵖ} (i : U ⟶ V) :\n  X.presheaf.map i ≫ H.inv_app (unop V) = H.inv_app (unop U) ≫\n    Y.presheaf.map (H.open_functor.op.map i) :=\nbegin\n  simp only [inv_app, ←category.assoc],\n  rw [is_iso.comp_inv_eq],\n  simp only [category.assoc, f.c.naturality, is_iso.inv_hom_id_assoc, ← X.presheaf.map_comp],\n  erw ← X.presheaf.map_comp,\n  congr\nend\n\ninstance (U : opens X) : is_iso (H.inv_app U) := by { delta inv_app, apply_instance }\n\nlemma inv_inv_app (U : opens X) :\n  inv (H.inv_app U) = f.c.app (op (H.open_functor.obj U)) ≫\n    X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) :=\nbegin\n  rw ← cancel_epi (H.inv_app U),\n  rw is_iso.hom_inv_id,\n  delta inv_app,\n  simp [← functor.map_comp]\nend\n\n@[simp, reassoc] lemma inv_app_app (U : opens X) :\n  H.inv_app U ≫ f.c.app (op (H.open_functor.obj U)) =\n    X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) :=\nby rw [inv_app, category.assoc, is_iso.inv_hom_id, category.comp_id]\n\n@[simp, reassoc] lemma app_inv_app (U : opens Y) :\n  f.c.app (op U) ≫ H.inv_app ((opens.map f.base).obj U) =\n  Y.presheaf.map ((hom_of_le (by exact set.image_preimage_subset f.base U)).op :\n    op U ⟶ op (H.open_functor.obj ((opens.map f.base).obj U))) :=\nby { erw ← category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr }\n\n/-- A variant of `app_inv_app` that gives an `eq_to_hom` instead of `hom_of_le`. -/\n@[reassoc] lemma app_inv_app' (U : opens Y) (hU : (U : set Y) ⊆ set.range f.base) :\n  f.c.app (op U) ≫ H.inv_app ((opens.map f.base).obj U) =\n  Y.presheaf.map (eq_to_hom (by\n    { apply has_le.le.antisymm,\n      { exact set.image_preimage_subset f.base U.1 },\n      { change U ⊆ _,\n        refine has_le.le.trans_eq _ (@set.image_preimage_eq_inter_range _ _ f.base U.1).symm,\n        exact set.subset_inter_iff.mpr ⟨λ _ h, h, hU⟩ } })).op :=\nby { erw ← category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr }\n\n/-- An isomorphism is an open immersion. -/\ninstance of_iso {X Y : PresheafedSpace C} (H : X ≅ Y) : is_open_immersion H.hom :=\n{ base_open := (Top.homeo_of_iso ((forget C).map_iso H)).open_embedding,\n  c_iso := λ _, infer_instance }\n\n@[priority 100]\ninstance of_is_iso {X Y : PresheafedSpace C} (f : X ⟶ Y) [is_iso f] : is_open_immersion f :=\nalgebraic_geometry.PresheafedSpace.is_open_immersion.of_iso (as_iso f)\n\ninstance of_restrict {X : Top} (Y : PresheafedSpace C) {f : X ⟶ Y.carrier}\n  (hf : open_embedding f) : is_open_immersion (Y.of_restrict hf) :=\n{ base_open := hf,\n  c_iso := λ U,\n  begin\n    dsimp,\n    have : (opens.map f).obj (hf.is_open_map.functor.obj U) = U,\n    { cases U,\n      dsimp only [opens.map, is_open_map.functor],\n      congr' 1,\n      rw set.preimage_image_eq _ hf.inj,\n      refl },\n    convert (show is_iso (Y.presheaf.map (𝟙 _)), from infer_instance),\n    { apply subsingleton.helim,\n      rw this },\n    { rw Y.presheaf.map_id,\n      apply_instance }\n  end }\n\n/-- An open immersion is an iso if the underlying continuous map is epi. -/\nlemma to_iso (f : X ⟶ Y) [h : is_open_immersion f] [h' : epi f.base] : is_iso f :=\nbegin\n  apply_with is_iso_of_components { instances := ff },\n  { let : X ≃ₜ Y := (homeomorph.of_embedding _ h.base_open.to_embedding).trans\n    { to_fun := subtype.val, inv_fun := λ x, ⟨x,\n      by { rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp h'), trivial }⟩,\n      left_inv := λ ⟨_,_⟩, rfl, right_inv := λ _, rfl },\n    convert is_iso.of_iso (Top.iso_of_homeo this),\n    { ext, refl } },\n  { apply_with nat_iso.is_iso_of_is_iso_app { instances := ff },\n    intro U,\n    have : U = op (h.open_functor.obj ((opens.map f.base).obj (unop U))),\n    { induction U using opposite.rec,\n      cases U,\n      dsimp only [functor.op, opens.map],\n      congr,\n      exact (set.image_preimage_eq _ ((Top.epi_iff_surjective _).mp h')).symm },\n    convert @@is_open_immersion.c_iso _ h ((opens.map f.base).obj (unop U)) }\nend\n\ninstance stalk_iso [has_colimits C] [H : is_open_immersion f] (x : X) : is_iso (stalk_map f x) :=\nbegin\n  rw ← H.iso_restrict_hom_of_restrict,\n  rw PresheafedSpace.stalk_map.comp,\n  apply_instance\nend\n\nend\n\nsection pullback\n\nnoncomputable theory\n\nvariables {X Y Z : PresheafedSpace C} (f : X ⟶ Z) [hf : is_open_immersion f] (g : Y ⟶ Z)\n\ninclude hf\n\n/--\n  (Implementation.) The projection map when constructing the pullback along an open immersion.\n-/\ndef pullback_cone_of_left_fst :\n  Y.restrict (Top.snd_open_embedding_of_left_open_embedding hf.base_open g.base) ⟶ X :=\n{ base := pullback.fst,\n  c :=\n  { app := λ U, hf.inv_app (unop U) ≫\n      g.c.app (op (hf.base_open.is_open_map.functor.obj (unop U))) ≫\n      Y.presheaf.map (eq_to_hom\n      (begin\n        simp only [is_open_map.functor, subtype.mk_eq_mk, unop_op, op_inj_iff, opens.map,\n        subtype.coe_mk, functor.op_obj, subtype.val_eq_coe],\n        apply has_le.le.antisymm,\n          { rintros _ ⟨_, h₁, h₂⟩,\n            use (Top.pullback_iso_prod_subtype _ _).inv ⟨⟨_, _⟩, h₂⟩,\n            simpa using h₁ },\n          { rintros _ ⟨x, h₁, rfl⟩,\n            exact ⟨_, h₁, concrete_category.congr_hom pullback.condition x⟩ }\n      end)),\n    naturality' :=\n    begin\n      intros U V i,\n      induction U using opposite.rec,\n      induction V using opposite.rec,\n      simp only [quiver.hom.unop_op, Top.presheaf.pushforward_obj_map, category.assoc,\n        nat_trans.naturality_assoc, functor.op_map, inv_naturality_assoc, ← Y.presheaf.map_comp],\n      erw ← Y.presheaf.map_comp,\n      congr\n    end } }\n\nlemma pullback_cone_of_left_condition :\n  pullback_cone_of_left_fst f g ≫ f = Y.of_restrict _ ≫ g :=\nbegin\n  ext U,\n  { induction U using opposite.rec,\n    dsimp only [comp_c_app, nat_trans.comp_app, unop_op,\n      whisker_right_app, pullback_cone_of_left_fst],\n    simp only [quiver.hom.unop_op, Top.presheaf.pushforward_obj_map, app_inv_app_assoc,\n      eq_to_hom_app, eq_to_hom_unop, category.assoc, nat_trans.naturality_assoc, functor.op_map],\n    erw [← Y.presheaf.map_comp, ← Y.presheaf.map_comp],\n    congr },\n  { simpa using pullback.condition }\nend\n\n/--\nWe construct the pullback along an open immersion via restricting along the pullback of the\nmaps of underlying spaces (which is also an open embedding).\n-/\ndef pullback_cone_of_left : pullback_cone f g :=\npullback_cone.mk (pullback_cone_of_left_fst f g) (Y.of_restrict _)\n  (pullback_cone_of_left_condition f g)\n\nvariable (s : pullback_cone f g)\n\n/--\n  (Implementation.) Any cone over `cospan f g` indeed factors through the constructed cone.\n-/\ndef pullback_cone_of_left_lift : s.X ⟶ (pullback_cone_of_left f g).X :=\n{ base := pullback.lift s.fst.base s.snd.base\n    (congr_arg (λ x, PresheafedSpace.hom.base x) s.condition),\n  c :=\n  { app := λ U, s.snd.c.app _ ≫ s.X.presheaf.map (eq_to_hom (begin\n      dsimp only [opens.map, is_open_map.functor, functor.op],\n      congr' 2,\n      let s' : pullback_cone f.base g.base := pullback_cone.mk s.fst.base s.snd.base _,\n      have : _ = s.snd.base := limit.lift_π s' walking_cospan.right,\n      conv_lhs { erw ← this, rw coe_comp, erw ← set.preimage_preimage },\n      erw set.preimage_image_eq _\n        (Top.snd_open_embedding_of_left_open_embedding hf.base_open g.base).inj,\n      simp,\n    end)),\n    naturality' := λ U V i,\n    begin\n      erw s.snd.c.naturality_assoc,\n      rw category.assoc,\n      erw [← s.X.presheaf.map_comp, ← s.X.presheaf.map_comp],\n      congr\n    end } }\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\nlemma pullback_cone_of_left_lift_fst :\n  pullback_cone_of_left_lift f g s ≫ (pullback_cone_of_left f g).fst = s.fst :=\nbegin\n  ext x,\n  { induction x using opposite.rec,\n    change ((_ ≫ _) ≫ _ ≫ _) ≫ _ = _,\n    simp_rw [category.assoc],\n    erw ← s.X.presheaf.map_comp,\n    erw s.snd.c.naturality_assoc,\n    have := congr_app s.condition (op (hf.open_functor.obj x)),\n    dsimp only [comp_c_app, unop_op] at this,\n    rw ← is_iso.comp_inv_eq at this,\n    reassoc! this,\n    erw [← this, hf.inv_app_app_assoc, s.fst.c.naturality_assoc],\n    simpa [eq_to_hom_map], },\n  { change pullback.lift _ _ _ ≫ pullback.fst = _,\n    simp }\nend\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\nlemma pullback_cone_of_left_lift_snd :\n  pullback_cone_of_left_lift f g s ≫ (pullback_cone_of_left f g).snd = s.snd :=\nbegin\n  ext x,\n  { change (_ ≫ _ ≫ _) ≫ _ = _,\n    simp_rw category.assoc,\n    erw s.snd.c.naturality_assoc,\n    erw [← s.X.presheaf.map_comp, ← s.X.presheaf.map_comp],\n    transitivity s.snd.c.app x ≫ s.X.presheaf.map (𝟙 _),\n    { congr },\n    { rw s.X.presheaf.map_id, erw category.comp_id } },\n  { change pullback.lift _ _ _ ≫ pullback.snd = _,\n    simp }\nend\n\ninstance pullback_cone_snd_is_open_immersion :\n  is_open_immersion (pullback_cone_of_left f g).snd :=\nbegin\n  erw category_theory.limits.pullback_cone.mk_snd,\n  apply_instance\nend\n\n/-- The constructed pullback cone is indeed the pullback. -/\ndef pullback_cone_of_left_is_limit :\n  is_limit (pullback_cone_of_left f g) :=\nbegin\n  apply pullback_cone.is_limit_aux',\n  intro s,\n  use pullback_cone_of_left_lift f g s,\n  use pullback_cone_of_left_lift_fst f g s,\n  use pullback_cone_of_left_lift_snd f g s,\n  intros m h₁ h₂,\n  rw ← cancel_mono (pullback_cone_of_left f g).snd,\n  exact (h₂.trans (pullback_cone_of_left_lift_snd f g s).symm)\nend\n\ninstance has_pullback_of_left :\n  has_pullback f g :=\n⟨⟨⟨_, pullback_cone_of_left_is_limit f g⟩⟩⟩\n\ninstance has_pullback_of_right :\n  has_pullback g f := has_pullback_symmetry f g\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_snd_of_left :\n  is_open_immersion (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  delta pullback.snd,\n  rw ← limit.iso_limit_cone_hom_π ⟨_, pullback_cone_of_left_is_limit f g⟩ walking_cospan.right,\n  apply_instance\nend\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_fst_of_right :\n  is_open_immersion (pullback.fst : pullback g f ⟶ _) :=\nbegin\n  rw ← pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one_is_open_immersion [is_open_immersion g] :\n  is_open_immersion (limit.π (cospan f g) walking_cospan.one) :=\nbegin\n  rw [←limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],\n  apply_instance\nend\n\ninstance forget_preserves_limits_of_left : preserves_limit (cospan f g) (forget C) :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit.postcompose_hom_equiv (diagram_iso_cospan.{v} _) _).to_fun,\n  refine (is_limit.equiv_iso_limit _).to_fun (limit.is_limit (cospan f.base g.base)),\n  fapply cones.ext,\n  exact (iso.refl _),\n  change ∀ j, _ = 𝟙 _ ≫ _ ≫ _,\n  simp_rw category.id_comp,\n  rintros (_|_|_); symmetry,\n  { erw category.comp_id,\n    exact limit.w (cospan f.base g.base) walking_cospan.hom.inl },\n  { exact category.comp_id _ },\n  { exact category.comp_id _ },\nend\n\ninstance forget_preserves_limits_of_right : preserves_limit (cospan g f) (forget C) :=\npreserves_pullback_symmetry (forget C) f g\n\nlemma pullback_snd_is_iso_of_range_subset (H : set.range g.base ⊆ set.range f.base) :\n  is_iso (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  haveI := Top.snd_iso_of_left_embedding_range_subset hf.base_open.to_embedding g.base H,\n  haveI : is_iso (pullback.snd : pullback f g ⟶ _).base,\n  { delta pullback.snd,\n    rw ← limit.iso_limit_cone_hom_π ⟨_, pullback_cone_of_left_is_limit f g⟩ walking_cospan.right,\n    change is_iso (_ ≫ pullback.snd),\n    apply_instance },\n  apply to_iso\nend\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X ⟶ Z`, given any morphism of schemes `g : Y ⟶ Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y ⟶ X` that\ncommutes with these maps.\n-/\ndef lift (H : set.range g.base ⊆ set.range f.base) : Y ⟶ X :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H,\n  exact inv (pullback.snd : pullback f g ⟶ _) ≫ pullback.fst,\nend\n\n@[simp, reassoc] lemma lift_fac (H : set.range g.base ⊆ set.range f.base) :\n  lift f g H ≫ f = g :=\nby { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition }\n\nlemma lift_uniq (H : set.range g.base ⊆ set.range f.base) (l : Y ⟶ X)\n  (hl : l ≫ f = g) : l = lift f g H :=\nby rw [← cancel_mono f, hl, lift_fac]\n\n/-- Two open immersions with equal range is isomorphic. -/\n@[simps] def iso_of_range_eq [is_open_immersion g] (e : set.range f.base = set.range g.base) :\n  X ≅ Y :=\n{ hom := lift g f (le_of_eq e),\n  inv := lift f g (le_of_eq e.symm),\n  hom_inv_id' := by { rw ← cancel_mono f, simp },\n  inv_hom_id' := by { rw ← cancel_mono g, simp } }\n\nend pullback\n\nopen category_theory.limits.walking_cospan\n\nsection to_SheafedSpace\n\nvariables [has_products C] {X : PresheafedSpace C} (Y : SheafedSpace C)\nvariables (f : X ⟶ Y.to_PresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X ⟶ Y` is an open immersion, and `Y` is a SheafedSpace, then so is `X`. -/\ndef to_SheafedSpace : SheafedSpace C :=\n{ is_sheaf :=\n  begin\n    apply Top.presheaf.is_sheaf_of_iso (sheaf_iso_of_iso H.iso_restrict.symm).symm,\n    apply Top.sheaf.pushforward_sheaf_of_sheaf,\n    exact (Y.restrict H.base_open).is_sheaf\n  end,\n  to_PresheafedSpace := X }\n\n@[simp] lemma to_SheafedSpace_to_PresheafedSpace : (to_SheafedSpace Y f).to_PresheafedSpace = X :=\nrfl\n\n/--\nIf `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a SheafedSpace, we can\nupgrade it into a morphism of SheafedSpaces.\n-/\ndef to_SheafedSpace_hom : to_SheafedSpace Y f ⟶ Y := f\n\n@[simp] lemma to_SheafedSpace_hom_base : (to_SheafedSpace_hom Y f).base = f.base := rfl\n\n@[simp] lemma to_SheafedSpace_hom_c : (to_SheafedSpace_hom Y f).c = f.c := rfl\n\ninstance to_SheafedSpace_is_open_immersion :\n  SheafedSpace.is_open_immersion (to_SheafedSpace_hom Y f) := H\n\nomit H\n\n@[simp] lemma SheafedSpace_to_SheafedSpace {X Y : SheafedSpace C} (f : X ⟶ Y)\n  [is_open_immersion f] : to_SheafedSpace Y f = X := by unfreezingI { cases X, refl }\n\nend to_SheafedSpace\n\nsection to_LocallyRingedSpace\n\nvariables {X : PresheafedSpace CommRing.{u}} (Y : LocallyRingedSpace.{u})\nvariables (f : X ⟶ Y.to_PresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X ⟶ Y` is an open immersion, and `Y` is a LocallyRingedSpace, then so is `X`. -/\ndef to_LocallyRingedSpace : LocallyRingedSpace :=\n{ to_SheafedSpace := to_SheafedSpace Y.to_SheafedSpace f,\n  local_ring := λ x, begin\n    haveI : local_ring (Y.to_SheafedSpace.to_PresheafedSpace.stalk (f.base x)) := Y.local_ring _,\n    exact (as_iso (stalk_map f x)).CommRing_iso_to_ring_equiv.local_ring\n  end }\n\n@[simp] lemma to_LocallyRingedSpace_to_SheafedSpace :\n  (to_LocallyRingedSpace Y f).to_SheafedSpace = (to_SheafedSpace Y.1 f) := rfl\n\n/--\nIf `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a LocallyRingedSpace, we can\nupgrade it into a morphism of LocallyRingedSpace.\n-/\ndef to_LocallyRingedSpace_hom : to_LocallyRingedSpace Y f ⟶ Y := ⟨f, λ x, infer_instance⟩\n\n@[simp] lemma to_LocallyRingedSpace_hom_val :\n  (to_LocallyRingedSpace_hom Y f).val = f := rfl\n\ninstance to_LocallyRingedSpace_is_open_immersion :\n  LocallyRingedSpace.is_open_immersion (to_LocallyRingedSpace_hom Y f) := H\n\nomit H\n\n@[simp] lemma LocallyRingedSpace_to_LocallyRingedSpace {X Y : LocallyRingedSpace} (f : X ⟶ Y)\n  [LocallyRingedSpace.is_open_immersion f] :\n  @to_LocallyRingedSpace X.to_PresheafedSpace Y (@@coe (@@coe_to_lift (@@coe_base coe_subtype)) f)\n    (show is_open_immersion f.val, by apply_instance) = X :=\nby unfreezingI { cases X, delta to_LocallyRingedSpace, simp }\n\nend to_LocallyRingedSpace\n\nend PresheafedSpace.is_open_immersion\n\nnamespace SheafedSpace.is_open_immersion\n\nvariables [has_products C]\n\n@[priority 100]\ninstance of_is_iso {X Y : SheafedSpace C} (f : X ⟶ Y) [is_iso f] :\n  SheafedSpace.is_open_immersion f :=\n@@PresheafedSpace.is_open_immersion.of_is_iso _ f\n(SheafedSpace.forget_to_PresheafedSpace.map_is_iso _)\n\ninstance comp {X Y Z : SheafedSpace C} (f : X ⟶ Y) (g : Y ⟶ Z)\n  [SheafedSpace.is_open_immersion f] [SheafedSpace.is_open_immersion g] :\n  SheafedSpace.is_open_immersion (f ≫ g) := PresheafedSpace.is_open_immersion.comp f g\n\nsection pullback\n\nvariables {X Y Z : SheafedSpace C} (f : X ⟶ Z) (g : Y ⟶ Z)\nvariable [H : SheafedSpace.is_open_immersion f]\n\ninclude H\n\nlocal notation `forget` := SheafedSpace.forget_to_PresheafedSpace\nopen category_theory.limits.walking_cospan\n\ninstance : mono f := faithful_reflects_mono forget\n  (show @mono (PresheafedSpace C) _ _ _ f, by apply_instance)\n\ninstance forget_map_is_open_immersion :\n  PresheafedSpace.is_open_immersion (forget .map f) := ⟨H.base_open, H.c_iso⟩\n\ninstance has_limit_cospan_forget_of_left : has_limit (cospan f g ⋙ forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,\n  change has_limit (cospan (forget .map f) (forget .map g)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_left' : has_limit (cospan ((cospan f g ⋙ forget).map hom.inl)\n  ((cospan f g ⋙ forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map f) (forget .map g)), from infer_instance\n\ninstance has_limit_cospan_forget_of_right : has_limit (cospan g f ⋙ forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,\n  change has_limit (cospan (forget .map g) (forget .map f)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_right' : has_limit (cospan ((cospan g f ⋙ forget).map hom.inl)\n  ((cospan g f ⋙ forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map g) (forget .map f)), from infer_instance\n\n\ninstance forget_creates_pullback_of_left : creates_limit (cospan f g) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_SheafedSpace Y\n    (@pullback.snd (PresheafedSpace C) _ _ _ _ f g _))\n  (eq_to_iso (show pullback _ _ = pullback _ _, by congr)\n    ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_creates_pullback_of_right : creates_limit (cospan g f) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_SheafedSpace Y\n    (@pullback.fst (PresheafedSpace C) _ _ _ _ g f _))\n  (eq_to_iso (show pullback _ _ = pullback _ _, by congr)\n    ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance SheafedSpace_forget_preserves_of_left :\n  preserves_limit (cospan f g) (SheafedSpace.forget C) :=\n@@limits.comp_preserves_limit _ _ _ _ forget (PresheafedSpace.forget C) _\nbegin\n  apply_with (preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{v} _).symm) { instances := tt },\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_forget_preserves_of_right :\n  preserves_limit (cospan g f) (SheafedSpace.forget C) :=\npreserves_pullback_symmetry _ _ _\n\ninstance SheafedSpace_has_pullback_of_left : has_pullback f g :=\n  has_limit_of_created (cospan f g) forget\n\ninstance SheafedSpace_has_pullback_of_right : has_pullback g f :=\n  has_limit_of_created (cospan g f) forget\n\n/-- Open immersions are stable under base-change. -/\ninstance SheafedSpace_pullback_snd_of_left :\n  SheafedSpace.is_open_immersion (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  delta pullback.snd,\n  have : _ = limit.π (cospan f g) right := preserves_limits_iso_hom_π\n      forget (cospan f g) right,\n  rw ← this,\n  have := has_limit.iso_of_nat_iso_hom_π\n    (diagram_iso_cospan.{v} (cospan f g ⋙ forget))\n    right,\n  erw category.comp_id at this,\n  rw ← this,\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_pullback_fst_of_right :\n  SheafedSpace.is_open_immersion (pullback.fst : pullback g f ⟶ _) :=\nbegin\n  delta pullback.fst,\n  have : _ = limit.π (cospan g f) left := preserves_limits_iso_hom_π\n      forget (cospan g f) left,\n  rw ← this,\n  have := has_limit.iso_of_nat_iso_hom_π\n    (diagram_iso_cospan.{v} (cospan g f ⋙ forget)) left,\n  erw category.comp_id at this,\n  rw ← this,\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_pullback_one_is_open_immersion [SheafedSpace.is_open_immersion g] :\n  SheafedSpace.is_open_immersion (limit.π (cospan f g) one : pullback f g ⟶ Z) :=\nbegin\n  rw [←limit.w (cospan f g) hom.inl, cospan_map_inl],\n  apply_instance\nend\n\nend pullback\n\nsection of_stalk_iso\nvariables [has_limits C] [has_colimits C] [concrete_category.{v} C]\nvariables [reflects_isomorphisms (forget C)] [preserves_limits (forget C)]\nvariables [preserves_filtered_colimits (forget C)]\n\n/--\nSuppose `X Y : SheafedSpace C`, where `C` is a concrete category,\nwhose forgetful functor reflects isomorphisms, preserves limits and filtered colimits.\nThen a morphism `X ⟶ Y` that is a topological open embedding\nis an open immersion iff every stalk map is an iso.\n-/\nlemma of_stalk_iso {X Y : SheafedSpace C} (f : X ⟶ Y)\n  (hf : open_embedding f.base) [H : ∀ x : X, is_iso (PresheafedSpace.stalk_map f x)] :\n  SheafedSpace.is_open_immersion f :=\n{ base_open := hf,\n  c_iso := λ U, begin\n    apply_with (Top.presheaf.app_is_iso_of_stalk_functor_map_iso\n      (show Y.sheaf ⟶ (Top.sheaf.pushforward f.base).obj X.sheaf, from f.c)) { instances := ff },\n    rintros ⟨_, y, hy, rfl⟩,\n    specialize H y,\n    delta PresheafedSpace.stalk_map at H,\n    haveI H' := Top.presheaf.stalk_pushforward.stalk_pushforward_iso_of_open_embedding\n      C hf X.presheaf y,\n    have := @@is_iso.comp_is_iso _ H (@@is_iso.inv_is_iso _ H'),\n    rw [category.assoc, is_iso.hom_inv_id, category.comp_id] at this,\n    exact this\n  end }\n\nend of_stalk_iso\n\nsection prod\n\nvariables [has_limits C] {ι : Type v} (F : discrete ι ⥤ SheafedSpace C) [has_colimit F]\n  (i : discrete ι)\n\nlemma sigma_ι_open_embedding : open_embedding (colimit.ι F i).base :=\nbegin\n  rw ← (show _ = (colimit.ι F i).base,\n    from ι_preserves_colimits_iso_inv (SheafedSpace.forget C) F i),\n  have : _ = _ ≫ colimit.ι (discrete.functor ((F ⋙ SheafedSpace.forget C).obj ∘ discrete.mk)) i :=\n    has_colimit.iso_of_nat_iso_ι_hom discrete.nat_iso_functor i,\n  rw ← iso.eq_comp_inv at this,\n  rw this,\n  have : colimit.ι _ _ ≫ _ = _ :=\n    Top.sigma_iso_sigma_hom_ι.{v v} ((F ⋙ SheafedSpace.forget C).obj ∘ discrete.mk) i.as,\n  rw ← iso.eq_comp_inv at this,\n  cases i,\n  rw this,\n  simp_rw [← category.assoc, Top.open_embedding_iff_comp_is_iso,\n    Top.open_embedding_iff_is_iso_comp],\n  dsimp,\n  exact open_embedding_sigma_mk\nend\n\nlemma image_preimage_is_empty (j : discrete ι) (h : i ≠ j) (U : opens (F.obj i)) :\n  (opens.map (colimit.ι (F ⋙ SheafedSpace.forget_to_PresheafedSpace) j).base).obj\n    ((opens.map (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj\n    ((sigma_ι_open_embedding F i).is_open_map.functor.obj U)) = ∅ :=\nbegin\n  ext,\n  apply iff_false_intro,\n  rintro ⟨y, hy, eq⟩,\n  replace eq := concrete_category.congr_arg\n    (preserves_colimit_iso (SheafedSpace.forget C) F ≪≫\n      has_colimit.iso_of_nat_iso discrete.nat_iso_functor ≪≫ Top.sigma_iso_sigma.{v} _).hom eq,\n  simp_rw [category_theory.iso.trans_hom, ← Top.comp_app, ← PresheafedSpace.comp_base] at eq,\n  rw ι_preserves_colimits_iso_inv at eq,\n  change ((SheafedSpace.forget C).map (colimit.ι F i) ≫ _) y =\n    ((SheafedSpace.forget C).map (colimit.ι F j) ≫ _) x at eq,\n  cases i, cases j,\n  rw [ι_preserves_colimits_iso_hom_assoc, ι_preserves_colimits_iso_hom_assoc,\n    has_colimit.iso_of_nat_iso_ι_hom_assoc, has_colimit.iso_of_nat_iso_ι_hom_assoc,\n    Top.sigma_iso_sigma_hom_ι.{v}, Top.sigma_iso_sigma_hom_ι.{v}] at eq,\n  exact h (congr_arg discrete.mk (congr_arg sigma.fst eq)),\nend\n\ninstance sigma_ι_is_open_immersion [has_strict_terminal_objects C] :\n  SheafedSpace.is_open_immersion (colimit.ι F i) :=\n{ base_open := sigma_ι_open_embedding F i,\n  c_iso := λ U, begin\n    have e : colimit.ι F i = _ :=\n      (ι_preserves_colimits_iso_inv SheafedSpace.forget_to_PresheafedSpace F i).symm,\n    have H : open_embedding (colimit.ι (F ⋙ SheafedSpace.forget_to_PresheafedSpace) i ≫\n      (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).base :=\n      e ▸ sigma_ι_open_embedding F i,\n    suffices : is_iso ((colimit.ι (F ⋙ SheafedSpace.forget_to_PresheafedSpace) i ≫\n      (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).c.app\n        (op (H.is_open_map.functor.obj U))),\n    { convert this },\n    rw [PresheafedSpace.comp_c_app,\n      ← PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_π],\n    suffices : is_iso (limit.π (PresheafedSpace.componentwise_diagram\n      (F ⋙ SheafedSpace.forget_to_PresheafedSpace)\n      ((opens.map (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj\n      (unop $ op $ H.is_open_map.functor.obj U))) (op i)),\n    { resetI, apply_instance },\n    apply limit_π_is_iso_of_is_strict_terminal,\n    intros j hj,\n    induction j using opposite.rec,\n    dsimp,\n    convert (F.obj j).sheaf.is_terminal_of_empty,\n    convert image_preimage_is_empty F i j (λ h, hj (congr_arg op h.symm)) U,\n    exact (congr_arg PresheafedSpace.hom.base e).symm\n  end }\n\nend prod\n\nend SheafedSpace.is_open_immersion\n\nnamespace LocallyRingedSpace.is_open_immersion\n\nsection pullback\n\nvariables {X Y Z : LocallyRingedSpace.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)\nvariable [H : LocallyRingedSpace.is_open_immersion f]\n\n@[priority 100]\ninstance of_is_iso [is_iso g] :\n  LocallyRingedSpace.is_open_immersion g :=\n@@PresheafedSpace.is_open_immersion.of_is_iso _ g.1 ⟨⟨(inv g).1,\n  by { erw ← LocallyRingedSpace.comp_val, rw is_iso.hom_inv_id,\n    erw ← LocallyRingedSpace.comp_val, rw is_iso.inv_hom_id, split; simpa }⟩⟩\n\ninclude H\n\ninstance comp (g : Z ⟶ Y) [LocallyRingedSpace.is_open_immersion g] :\n  LocallyRingedSpace.is_open_immersion (f ≫ g) := PresheafedSpace.is_open_immersion.comp f.1 g.1\n\ninstance mono : mono f :=\nfaithful_reflects_mono (LocallyRingedSpace.forget_to_SheafedSpace)\n  (show mono f.1, by apply_instance)\n\ninstance : SheafedSpace.is_open_immersion (LocallyRingedSpace.forget_to_SheafedSpace.map f) := H\n\n/-- An explicit pullback cone over `cospan f g` if `f` is an open immersion. -/\ndef pullback_cone_of_left : pullback_cone f g :=\nbegin\n  refine pullback_cone.mk _\n    (Y.of_restrict (Top.snd_open_embedding_of_left_open_embedding H.base_open g.1.base)) _,\n  { use PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst f.1 g.1,\n    intro x,\n    have := PresheafedSpace.stalk_map.congr_hom _ _\n      (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition f.1 g.1) x,\n    rw [PresheafedSpace.stalk_map.comp, PresheafedSpace.stalk_map.comp] at this,\n    rw ← is_iso.eq_inv_comp at this,\n    rw this,\n    apply_instance },\n  { exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition _ _) },\nend\n\ninstance : LocallyRingedSpace.is_open_immersion (pullback_cone_of_left f g).snd :=\nshow PresheafedSpace.is_open_immersion (Y.to_PresheafedSpace.of_restrict _), by apply_instance\n\n/-- The constructed `pullback_cone_of_left` is indeed limiting. -/\ndef pullback_cone_of_left_is_limit : is_limit (pullback_cone_of_left f g) :=\npullback_cone.is_limit_aux' _ $ λ s,\nbegin\n  use PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift f.1 g.1\n    (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition)),\n  { intro x,\n    have := PresheafedSpace.stalk_map.congr_hom _ _\n      (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n        (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition))) x,\n    change _ = _ ≫ PresheafedSpace.stalk_map s.snd.1 x at this,\n    rw [PresheafedSpace.stalk_map.comp, ← is_iso.eq_inv_comp] at this,\n    rw this,\n    apply_instance },\n  split,\n  exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_fst f.1 g.1 _),\n  split,\n  exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1 _),\n  intros m h₁ h₂,\n  rw ← cancel_mono (pullback_cone_of_left f g).snd,\n  exact (h₂.trans (subtype.eq\n    (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n      (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition))).symm))\nend\n\ninstance has_pullback_of_left :\n  has_pullback f g :=\n⟨⟨⟨_, pullback_cone_of_left_is_limit f g⟩⟩⟩\n\ninstance has_pullback_of_right :\n  has_pullback g f := has_pullback_symmetry f g\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_snd_of_left :\n  LocallyRingedSpace.is_open_immersion (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  delta pullback.snd,\n  rw ← limit.iso_limit_cone_hom_π ⟨_, pullback_cone_of_left_is_limit f g⟩ walking_cospan.right,\n  apply_instance\nend\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_fst_of_right :\nLocallyRingedSpace.is_open_immersion (pullback.fst : pullback g f ⟶ _) :=\nbegin\n  rw ← pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one_is_open_immersion [LocallyRingedSpace.is_open_immersion g] :\n  LocallyRingedSpace.is_open_immersion (limit.π (cospan f g) walking_cospan.one) :=\nbegin\n  rw [←limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],\n  apply_instance\nend\n\ninstance forget_preserves_pullback_of_left :\n  preserves_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,\n  apply is_limit_of_is_limit_pullback_cone_map SheafedSpace.forget_to_PresheafedSpace,\n  exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1\nend\n\ninstance forget_to_PresheafedSpace_preserves_pullback_of_left :\n  preserves_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,\n  exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1\nend\n\ninstance forget_to_PresheafedSpace_preserves_open_immersion :\n  PresheafedSpace.is_open_immersion ((LocallyRingedSpace.forget_to_SheafedSpace ⋙\n    SheafedSpace.forget_to_PresheafedSpace).map f) := H\n\ninstance forget_to_Top_preserves_pullback_of_left :\n  preserves_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _) :=\nbegin\n  change preserves_limit _\n    ((LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace)\n      ⋙ PresheafedSpace.forget _),\n  apply_with limits.comp_preserves_limit { instances := ff },\n  apply_instance,\n  apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,\n  dsimp [SheafedSpace.forget_to_PresheafedSpace, -subtype.val_eq_coe],\n  apply_instance,\nend\n\ninstance forget_reflects_pullback_of_left :\n  reflects_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_preserves_pullback_of_right :\n  preserves_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace :=\npreserves_pullback_symmetry _ _ _\n\ninstance forget_to_PresheafedSpace_preserves_pullback_of_right :\n  preserves_limit (cospan g f) (LocallyRingedSpace.forget_to_SheafedSpace ⋙\n    SheafedSpace.forget_to_PresheafedSpace) :=\npreserves_pullback_symmetry _ _ _\n\ninstance forget_reflects_pullback_of_right :\n  reflects_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_to_PresheafedSpace_reflects_pullback_of_left :\n  reflects_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_to_PresheafedSpace_reflects_pullback_of_right :\n  reflects_limit (cospan g f)\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) :=\nreflects_limit_of_reflects_isomorphisms _ _\n\nlemma pullback_snd_is_iso_of_range_subset (H' : set.range g.1.base ⊆ set.range f.1.base) :\n  is_iso (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  apply_with (reflects_isomorphisms.reflects LocallyRingedSpace.forget_to_SheafedSpace)\n    { instances := ff },\n  apply_with (reflects_isomorphisms.reflects SheafedSpace.forget_to_PresheafedSpace)\n    { instances := ff },\n  erw ← preserves_pullback.iso_hom_snd\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget_to_PresheafedSpace) f g,\n  haveI := PresheafedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset _ _ H',\n  apply_instance,\n  apply_instance\nend\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X ⟶ Z`, given any morphism of schemes `g : Y ⟶ Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y ⟶ X` that\ncommutes with these maps.\n-/\ndef lift (H' : set.range g.1.base ⊆ set.range f.1.base) : Y ⟶ X :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H',\n  exact inv (pullback.snd : pullback f g ⟶ _) ≫ pullback.fst,\nend\n\n@[simp, reassoc] lemma lift_fac (H' : set.range g.1.base ⊆ set.range f.1.base) :\n  lift f g H' ≫ f = g :=\nby { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition }\n\nlemma lift_uniq (H' : set.range g.1.base ⊆ set.range f.1.base) (l : Y ⟶ X)\n  (hl : l ≫ f = g) : l = lift f g H' :=\nby rw [← cancel_mono f, hl, lift_fac]\n\nlemma lift_range (H' : set.range g.1.base ⊆ set.range f.1.base) :\n  set.range (lift f g H').1.base = f.1.base ⁻¹' (set.range g.1.base) :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H',\n  dsimp only [lift],\n  have : _ = (pullback.fst : pullback f g ⟶ _).val.base := preserves_pullback.iso_hom_fst\n    (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _) f g,\n  rw [LocallyRingedSpace.comp_val, SheafedSpace.comp_base, ← this, ← category.assoc, coe_comp],\n  rw [set.range_comp, set.range_iff_surjective.mpr, set.image_univ, Top.pullback_fst_range],\n  ext,\n  split,\n  { rintros ⟨y, eq⟩, exact ⟨y, eq.symm⟩ },\n  { rintros ⟨y, eq⟩, exact ⟨y, eq.symm⟩ },\n  { rw ← Top.epi_iff_surjective,\n    rw (show (inv (pullback.snd : pullback f g ⟶ _)).val.base = _, from\n      (LocallyRingedSpace.forget_to_SheafedSpace ⋙ SheafedSpace.forget _).map_inv _),\n    apply_instance }\nend\n\nend pullback\n\n/-- An open immersion is isomorphic to the induced open subscheme on its image. -/\ndef iso_restrict {X Y : LocallyRingedSpace} {f : X ⟶ Y}\n  (H : LocallyRingedSpace.is_open_immersion f) : X ≅ Y.restrict H.base_open :=\nbegin\n  apply LocallyRingedSpace.iso_of_SheafedSpace_iso,\n  refine SheafedSpace.forget_to_PresheafedSpace.preimage_iso _,\n  exact H.iso_restrict\nend\n\n/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly\nsurjective family of open immersions from affine schemes. -/\nprotected def Scheme (X : LocallyRingedSpace)\n  (h : ∀ (x : X), ∃ (R : CommRing) (f : Spec.to_LocallyRingedSpace.obj (op R) ⟶ X),\n    (x ∈ set.range f.1.base : _) ∧ LocallyRingedSpace.is_open_immersion f) : Scheme :=\n{ to_LocallyRingedSpace := X,\n  local_affine :=\n  begin\n    intro x,\n    obtain ⟨R, f, h₁, h₂⟩ := h x,\n    refine ⟨⟨⟨_, h₂.base_open.open_range⟩, h₁⟩, R, ⟨_⟩⟩,\n    apply LocallyRingedSpace.iso_of_SheafedSpace_iso,\n    refine SheafedSpace.forget_to_PresheafedSpace.preimage_iso _,\n    resetI,\n    apply PresheafedSpace.is_open_immersion.iso_of_range_eq (PresheafedSpace.of_restrict _ _) f.1,\n    { exact subtype.range_coe_subtype },\n    { apply_instance }\n  end }\n\nend LocallyRingedSpace.is_open_immersion\n\nlemma is_open_immersion.open_range {X Y : Scheme} (f : X ⟶ Y) [H : is_open_immersion f] :\n  is_open (set.range f.1.base) := H.base_open.open_range\n\nsection open_cover\n\nnamespace Scheme\n\n/-- An open cover of `X` consists of a family of open immersions into `X`,\nand for each `x : X` an open immersion (indexed by `f x`) that covers `x`.\n\nThis is merely a coverage in the Zariski pretopology, and it would be optimal\nif we could reuse the existing API about pretopologies, However, the definitions of sieves and\ngrothendieck topologies uses `Prop`s, so that the actual open sets and immersions are hard to\nobtain. Also, since such a coverage in the pretopology usually contains a proper class of\nimmersions, it is quite hard to glue them, reason about finite covers, etc.\n-/\n-- TODO: provide API to and from a presieve.\nstructure open_cover (X : Scheme.{u}) :=\n(J : Type v)\n(obj : Π (j : J), Scheme)\n(map : Π (j : J), obj j ⟶ X)\n(f : X.carrier → J)\n(covers : ∀ x, x ∈ set.range ((map (f x)).1.base))\n(is_open : ∀ x, is_open_immersion (map x) . tactic.apply_instance)\n\nattribute [instance] open_cover.is_open\n\nvariables {X Y Z : Scheme.{u}} (𝒰 : open_cover X) (f : X ⟶ Z) (g : Y ⟶ Z)\nvariables [∀ x, has_pullback (𝒰.map x ≫ f) g]\n\n/-- The affine cover of a scheme. -/\ndef affine_cover (X : Scheme) : open_cover X :=\n{ J := X.carrier,\n  obj := λ x, Spec.obj $ opposite.op (X.local_affine x).some_spec.some,\n  map := λ x, ((X.local_affine x).some_spec.some_spec.some.inv ≫\n    X.to_LocallyRingedSpace.of_restrict _ : _),\n  f := λ x, x,\n  is_open := λ x, begin\n    apply_with PresheafedSpace.is_open_immersion.comp { instances := ff },\n    apply_instance,\n    apply PresheafedSpace.is_open_immersion.of_restrict,\n  end,\n  covers :=\n  begin\n    intro x,\n    erw coe_comp,\n    rw [set.range_comp, set.range_iff_surjective.mpr, set.image_univ],\n    erw subtype.range_coe_subtype,\n    exact (X.local_affine x).some.2,\n    rw ← Top.epi_iff_surjective,\n    change epi ((SheafedSpace.forget _).map (LocallyRingedSpace.forget_to_SheafedSpace.map _)),\n    apply_instance\n  end }\n\ninstance : inhabited X.open_cover := ⟨X.affine_cover⟩\n\n/-- Given an open cover `{ Uᵢ }` of `X`, and for each `Uᵢ` an open cover, we may combine these\nopen covers to form an open cover of `X`.  -/\n@[simps J obj map]\ndef open_cover.bind (f : Π (x : 𝒰.J), open_cover (𝒰.obj x)) : open_cover X :=\n{ J := Σ (i : 𝒰.J), (f i).J,\n  obj := λ x, (f x.1).obj x.2,\n  map := λ x, (f x.1).map x.2 ≫ 𝒰.map x.1,\n  f := λ x, ⟨_, (f _).f (𝒰.covers x).some⟩,\n  covers := λ x,\n  begin\n    let y := (𝒰.covers x).some,\n    have hy : (𝒰.map (𝒰.f x)).val.base y = x := (𝒰.covers x).some_spec,\n    rcases (f (𝒰.f x)).covers y with ⟨z, hz⟩,\n    change x ∈ set.range (((f (𝒰.f x)).map ((f (𝒰.f x)).f y) ≫ 𝒰.map (𝒰.f x)).1.base),\n    use z,\n    erw comp_apply,\n    rw [hz, hy],\n  end }\n\n/-- An isomorphism `X ⟶ Y` is an open cover of `Y`. -/\n@[simps J obj map]\ndef open_cover_of_is_iso {X Y : Scheme.{u}} (f : X ⟶ Y) [is_iso f] :\n  open_cover Y :=\n{ J := punit.{v+1},\n  obj := λ _, X,\n  map := λ _, f,\n  f := λ _, punit.star,\n  covers := λ x, by { rw set.range_iff_surjective.mpr, { trivial }, rw ← Top.epi_iff_surjective,\n    apply_instance } }\n\n/-- We construct an open cover from another, by providing the needed fields and showing that the\nprovided fields are isomorphic with the original open cover. -/\n@[simps J obj map]\ndef open_cover.copy {X : Scheme} (𝒰 : open_cover X)\n  (J : Type*) (obj : J → Scheme) (map : ∀ i, obj i ⟶ X)\n  (e₁ : J ≃ 𝒰.J) (e₂ : ∀ i, obj i ≅ 𝒰.obj (e₁ i))\n  (e₂ : ∀ i, map i = (e₂ i).hom ≫ 𝒰.map (e₁ i)) : open_cover X :=\n{ J := J,\n  obj := obj,\n  map := map,\n  f := λ x, e₁.symm (𝒰.f x),\n  covers := λ x, begin\n    rw [e₂, Scheme.comp_val_base, coe_comp, set.range_comp, set.range_iff_surjective.mpr,\n      set.image_univ,  e₁.right_inverse_symm],\n    { exact 𝒰.covers x },\n    { rw ← Top.epi_iff_surjective, apply_instance }\n  end,\n  is_open := λ i, by { rw e₂, apply_instance } }\n\n/-- The pushforward of an open cover along an isomorphism. -/\n@[simps J obj map]\ndef open_cover.pushforward_iso {X Y : Scheme} (𝒰 : open_cover X)\n  (f : X ⟶ Y) [is_iso f] :\n  open_cover Y :=\n((open_cover_of_is_iso f).bind (λ _, 𝒰)).copy 𝒰.J _ _\n  ((equiv.punit_prod _).symm.trans (equiv.sigma_equiv_prod punit 𝒰.J).symm)\n  (λ _, iso.refl _)\n  (λ _, (category.id_comp _).symm)\n\n-- Related result : `open_cover.pullback_cover`, where we pullback an open cover on `X` along a\n-- morphism `W ⟶ X`. This is provided at the end of the file since it needs some more results\n-- about open immersion (which in turn needs the open cover API).\n\nlocal attribute [reducible] CommRing.of CommRing.of_hom\n\ninstance val_base_is_iso {X Y : Scheme} (f : X ⟶ Y) [is_iso f] : is_iso f.1.base :=\nScheme.forget_to_Top.map_is_iso f\n\ninstance basic_open_is_open_immersion {R : CommRing} (f : R) :\nalgebraic_geometry.is_open_immersion (Scheme.Spec.map (CommRing.of_hom\n  (algebra_map R (localization.away f))).op) :=\nbegin\n  apply_with SheafedSpace.is_open_immersion.of_stalk_iso { instances := ff },\n  any_goals { apply_instance },\n  any_goals { apply_instance },\n  exact (prime_spectrum.localization_away_open_embedding (localization.away f) f : _),\n  intro x,\n  exact Spec_map_localization_is_iso R (submonoid.powers f) x,\nend\n\n/-- The basic open sets form an affine open cover of `Spec R`. -/\ndef affine_basis_cover_of_affine (R : CommRing) : open_cover (Spec.obj (opposite.op R)) :=\n{ J := R,\n  obj := λ r, Spec.obj (opposite.op $ CommRing.of $ localization.away r),\n  map := λ r, Spec.map (quiver.hom.op (algebra_map R (localization.away r) : _)),\n  f := λ x, 1,\n  covers := λ r,\n  begin\n    rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp _),\n    { exact trivial },\n    { apply_instance }\n  end,\n  is_open := λ x, algebraic_geometry.Scheme.basic_open_is_open_immersion x }\n\n/-- We may bind the basic open sets of an open affine cover to form a affine cover that is also\na basis. -/\ndef affine_basis_cover (X : Scheme) : open_cover X :=\nX.affine_cover.bind (λ x, affine_basis_cover_of_affine _)\n\n/-- The coordinate ring of a component in the `affine_basis_cover`. -/\ndef affine_basis_cover_ring (X : Scheme) (i : X.affine_basis_cover.J) : CommRing :=\nCommRing.of $ @localization.away (X.local_affine i.1).some_spec.some _ i.2\n\nlemma affine_basis_cover_obj (X : Scheme) (i : X.affine_basis_cover.J) :\n  X.affine_basis_cover.obj i = Spec.obj (op $ X.affine_basis_cover_ring i) := rfl\n\nlemma affine_basis_cover_map_range (X : Scheme)\n  (x : X.carrier) (r : (X.local_affine x).some_spec.some) :\n  set.range (X.affine_basis_cover.map ⟨x, r⟩).1.base =\n    (X.affine_cover.map x).1.base '' (prime_spectrum.basic_open r).1 :=\nbegin\n  erw [coe_comp, set.range_comp],\n  congr,\n  exact (prime_spectrum.localization_away_comap_range (localization.away r) r : _)\nend\n\nlemma affine_basis_cover_is_basis (X : Scheme) :\n  topological_space.is_topological_basis\n    { x : set X.carrier | ∃ a : X.affine_basis_cover.J, x =\n      set.range ((X.affine_basis_cover.map a).1.base) } :=\nbegin\n  apply topological_space.is_topological_basis_of_open_of_nhds,\n  { rintros _ ⟨a, rfl⟩,\n    exact is_open_immersion.open_range (X.affine_basis_cover.map a) },\n  { rintros a U haU hU,\n    rcases X.affine_cover.covers a with ⟨x, e⟩,\n    let U' := (X.affine_cover.map (X.affine_cover.f a)).1.base ⁻¹' U,\n    have hxU' : x ∈ U' := by { rw ← e at haU, exact haU },\n    rcases prime_spectrum.is_basis_basic_opens.exists_subset_of_mem_open hxU'\n      ((X.affine_cover.map (X.affine_cover.f a)).1.base.continuous_to_fun.is_open_preimage _ hU)\n      with ⟨_,⟨_,⟨s,rfl⟩,rfl⟩,hxV,hVU⟩,\n    refine ⟨_,⟨⟨_,s⟩,rfl⟩,_,_⟩; erw affine_basis_cover_map_range,\n    { exact ⟨x,hxV,e⟩ },\n    { rw set.image_subset_iff, exact hVU } }\nend\n\n/--\nEvery open cover of a quasi-compact scheme can be refined into a finite subcover.\n-/\n@[simps obj map]\ndef open_cover.finite_subcover {X : Scheme} (𝒰 : open_cover X) [H : compact_space X.carrier] :\n  open_cover X :=\nbegin\n  have := @@compact_space.elim_nhds_subcover _ H\n    (λ (x : X.carrier), set.range ((𝒰.map (𝒰.f x)).1.base))\n    (λ x, (is_open_immersion.open_range (𝒰.map (𝒰.f x))).mem_nhds (𝒰.covers x)),\n  let t := this.some,\n  have h : ∀ (x : X.carrier), ∃ (y : t), x ∈ set.range ((𝒰.map (𝒰.f y)).1.base),\n  { intro x,\n    have h' : x ∈ (⊤ : set X.carrier) := trivial,\n    rw [← classical.some_spec this, set.mem_Union] at h',\n    rcases h' with ⟨y,_,⟨hy,rfl⟩,hy'⟩,\n    exact ⟨⟨y,hy⟩,hy'⟩ },\n  exact\n  { J := t,\n    obj := λ x, 𝒰.obj (𝒰.f x.1),\n    map := λ x, 𝒰.map (𝒰.f x.1),\n    f := λ x, (h x).some,\n    covers := λ x, (h x).some_spec }\nend\n\ninstance [H : compact_space X.carrier] : fintype 𝒰.finite_subcover.J :=\nby { delta open_cover.finite_subcover, apply_instance }\n\nend Scheme\n\nend open_cover\n\nnamespace PresheafedSpace.is_open_immersion\n\nsection to_Scheme\n\nvariables {X : PresheafedSpace CommRing.{u}} (Y : Scheme.{u})\nvariables (f : X ⟶ Y.to_PresheafedSpace) [H : PresheafedSpace.is_open_immersion f]\n\ninclude H\n\n/-- If `X ⟶ Y` is an open immersion, and `Y` is a scheme, then so is `X`. -/\ndef to_Scheme : Scheme :=\nbegin\n  apply LocallyRingedSpace.is_open_immersion.Scheme (to_LocallyRingedSpace _ f),\n  intro x,\n  obtain ⟨_,⟨i,rfl⟩,hx,hi⟩ := Y.affine_basis_cover_is_basis.exists_subset_of_mem_open\n      (set.mem_range_self x) H.base_open.open_range,\n  use Y.affine_basis_cover_ring i,\n  use LocallyRingedSpace.is_open_immersion.lift (to_LocallyRingedSpace_hom _ f) _ hi,\n  split,\n  { rw LocallyRingedSpace.is_open_immersion.lift_range, exact hx },\n  { delta LocallyRingedSpace.is_open_immersion.lift, apply_instance }\nend\n\n@[simp] lemma to_Scheme_to_LocallyRingedSpace :\n  (to_Scheme Y f).to_LocallyRingedSpace = (to_LocallyRingedSpace Y.1 f) := rfl\n\n/--\nIf `X ⟶ Y` is an open immersion of PresheafedSpaces, and `Y` is a Scheme, we can\nupgrade it into a morphism of Schemes.\n-/\ndef to_Scheme_hom : to_Scheme Y f ⟶ Y := to_LocallyRingedSpace_hom _ f\n\n@[simp] \n\ninstance to_Scheme_hom_is_open_immersion :\n  is_open_immersion (to_Scheme_hom Y f) := H\n\nomit H\n\nlemma Scheme_eq_of_LocallyRingedSpace_eq {X Y : Scheme}\n  (H : X.to_LocallyRingedSpace = Y.to_LocallyRingedSpace) : X = Y :=\nby { cases X, cases Y, congr, exact H }\n\nlemma Scheme_to_Scheme {X Y : Scheme} (f : X ⟶ Y) [is_open_immersion f] :\n  to_Scheme Y f.1 = X :=\nbegin\n  apply Scheme_eq_of_LocallyRingedSpace_eq,\n  exact LocallyRingedSpace_to_LocallyRingedSpace f\nend\n\nend to_Scheme\n\nend PresheafedSpace.is_open_immersion\n\n/-- The restriction of a Scheme along an open embedding. -/\n@[simps]\ndef Scheme.restrict {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier} (h : open_embedding f) :\n  Scheme :=\n{ to_PresheafedSpace := X.to_PresheafedSpace.restrict h,\n  ..(PresheafedSpace.is_open_immersion.to_Scheme X (X.to_PresheafedSpace.of_restrict h)) }\n\n/-- The canonical map from the restriction to the supspace. -/\n@[simps]\ndef Scheme.of_restrict {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier} (h : open_embedding f) :\n  X.restrict h ⟶ X :=\nX.to_LocallyRingedSpace.of_restrict h\n\ninstance is_open_immersion.of_restrict {U : Top} (X : Scheme) {f : U ⟶ Top.of X.carrier}\n  (h : open_embedding f) : is_open_immersion (X.of_restrict h) :=\nshow PresheafedSpace.is_open_immersion (X.to_PresheafedSpace.of_restrict h), by apply_instance\n\nnamespace is_open_immersion\n\nvariables {X Y Z : Scheme.{u}} (f : X ⟶ Z) (g : Y ⟶ Z)\nvariable [H : is_open_immersion f]\n\n@[priority 100]\ninstance of_is_iso [is_iso g] :\n  is_open_immersion g := @@LocallyRingedSpace.is_open_immersion.of_is_iso _\n(show is_iso ((induced_functor _).map g), by apply_instance)\n\n/-- A open immersion induces an isomorphism from the domain onto the image -/\ndef iso_restrict : X ≅ (Z.restrict H.base_open : _) :=\n⟨H.iso_restrict.hom, H.iso_restrict.inv, H.iso_restrict.hom_inv_id, H.iso_restrict.inv_hom_id⟩\n\ninclude H\n\nlocal notation `forget` := Scheme.forget_to_LocallyRingedSpace\n\ninstance mono : mono f :=\nfaithful_reflects_mono (induced_functor _)\n  (show @mono LocallyRingedSpace _ _ _ f, by apply_instance)\n\ninstance forget_map_is_open_immersion : LocallyRingedSpace.is_open_immersion (forget .map f) :=\n⟨H.base_open, H.c_iso⟩\n\ninstance has_limit_cospan_forget_of_left :\n  has_limit (cospan f g ⋙ Scheme.forget_to_LocallyRingedSpace) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,\n  change has_limit (cospan (forget .map f) (forget .map g)),\n  apply_instance\nend\n\nopen category_theory.limits.walking_cospan\n\ninstance has_limit_cospan_forget_of_left' :\n  has_limit (cospan ((cospan f g ⋙ forget).map hom.inl)\n  ((cospan f g ⋙ forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map f) (forget .map g)), from infer_instance\n\ninstance has_limit_cospan_forget_of_right : has_limit (cospan g f ⋙ forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,\n  change has_limit (cospan (forget .map g) (forget .map f)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_right' :\n  has_limit (cospan ((cospan g f ⋙ forget).map hom.inl)\n  ((cospan g f ⋙ forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map g) (forget .map f)), from infer_instance\n\ninstance forget_creates_pullback_of_left : creates_limit (cospan f g) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_Scheme Y\n    (@pullback.snd LocallyRingedSpace _ _ _ _ f g _).1)\n  (eq_to_iso (by simp) ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_creates_pullback_of_right : creates_limit (cospan g f) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_Scheme Y\n    (@pullback.fst LocallyRingedSpace _ _ _ _ g f _).1)\n  (eq_to_iso (by simp) ≪≫ has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_preserves_of_left : preserves_limit (cospan f g) forget :=\ncategory_theory.preserves_limit_of_creates_limit_and_has_limit _ _\n\ninstance forget_preserves_of_right : preserves_limit (cospan g f) forget :=\npreserves_pullback_symmetry _ _ _\n\ninstance has_pullback_of_left : has_pullback f g :=\nhas_limit_of_created (cospan f g) forget\n\ninstance has_pullback_of_right : has_pullback g f :=\nhas_limit_of_created (cospan g f) forget\n\ninstance pullback_snd_of_left : is_open_immersion (pullback.snd : pullback f g ⟶ _) :=\nbegin\n  have := preserves_pullback.iso_hom_snd forget f g,\n  dsimp only [Scheme.forget_to_LocallyRingedSpace, induced_functor_map] at this,\n  rw ← this,\n  change LocallyRingedSpace.is_open_immersion _,\n  apply_instance\nend\n\ninstance pullback_fst_of_right : is_open_immersion (pullback.fst : pullback g f ⟶ _) :=\nbegin\n  rw ← pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one [is_open_immersion g] :\n  is_open_immersion (limit.π (cospan f g) walking_cospan.one) :=\nbegin\n  rw ← limit.w (cospan f g) walking_cospan.hom.inl,\n  change is_open_immersion (_ ≫ f),\n  apply_instance\nend\n\ninstance forget_to_Top_preserves_of_left :\n  preserves_limit (cospan f g) Scheme.forget_to_Top :=\nbegin\n  apply_with limits.comp_preserves_limit { instances := ff },\n  apply_instance,\n  apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,\n  dsimp [LocallyRingedSpace.forget_to_Top],\n  apply_instance\nend\n\ninstance forget_to_Top_preserves_of_right :\n  preserves_limit (cospan g f) Scheme.forget_to_Top := preserves_pullback_symmetry _ _ _\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X ⟶ Z`, given any morphism of schemes `g : Y ⟶ Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y ⟶ X` that\ncommutes with these maps.\n-/\ndef lift (H' : set.range g.1.base ⊆ set.range f.1.base) : Y ⟶ X :=\nLocallyRingedSpace.is_open_immersion.lift f g H'\n\n@[simp, reassoc] lemma lift_fac (H' : set.range g.1.base ⊆ set.range f.1.base) :\n  lift f g H' ≫ f = g :=\nLocallyRingedSpace.is_open_immersion.lift_fac f g H'\n\nlemma lift_uniq (H' : set.range g.1.base ⊆ set.range f.1.base) (l : Y ⟶ X)\n  (hl : l ≫ f = g) : l = lift f g H' :=\nLocallyRingedSpace.is_open_immersion.lift_uniq f g H' l hl\n\n/-- Two open immersions with equal range are isomorphic. -/\n@[simps] def iso_of_range_eq [is_open_immersion g] (e : set.range f.1.base = set.range g.1.base) :\n  X ≅ Y :=\n{ hom := lift g f (le_of_eq e),\n  inv := lift f g (le_of_eq e.symm),\n  hom_inv_id' := by { rw ← cancel_mono f, simp },\n  inv_hom_id' := by { rw ← cancel_mono g, simp } }\n\nend is_open_immersion\n\n/-- Given an open cover on `X`, we may pull them back along a morphism `W ⟶ X` to obtain\nan open cover of `W`. -/\n@[simps]\ndef Scheme.open_cover.pullback_cover {X : Scheme} (𝒰 : X.open_cover) {W : Scheme} (f : W ⟶ X) :\n  W.open_cover :=\n{ J := 𝒰.J,\n  obj := λ x, pullback f (𝒰.map x),\n  map := λ x, pullback.fst,\n  f := λ x, 𝒰.f (f.1.base x),\n  covers := λ x, begin\n    rw ← (show _ = (pullback.fst : pullback f (𝒰.map (𝒰.f (f.1.base x))) ⟶ _).1.base,\n      from preserves_pullback.iso_hom_fst Scheme.forget_to_Top f\n      (𝒰.map (𝒰.f (f.1.base x)))),\n    rw [coe_comp, set.range_comp, set.range_iff_surjective.mpr, set.image_univ,\n      Top.pullback_fst_range],\n    obtain ⟨y, h⟩ := 𝒰.covers (f.1.base x),\n    exact ⟨y, h.symm⟩,\n    { rw ← Top.epi_iff_surjective, apply_instance }\n  end }\n\nend algebraic_geometry\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/algebraic_geometry/open_immersion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3112230732332321}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.braided\nimport category_theory.monoidal.Mon_\n\n/-!\n# The category of commutative monoids in a braided monoidal category.\n-/\n\nuniverses v₁ v₂ u₁ u₂ u\n\nopen category_theory\nopen category_theory.monoidal_category\n\nvariables (C : Type u₁) [category.{v₁} C] [monoidal_category.{v₁} C] [braided_category.{v₁} C]\n\n/--\nA commutative monoid object internal to a monoidal category.\n-/\nstructure CommMon_ extends Mon_ C :=\n(mul_comm' : (β_ _ _).hom ≫ mul = mul . obviously)\n\nrestate_axiom CommMon_.mul_comm'\nattribute [simp, reassoc] CommMon_.mul_comm\n\nnamespace CommMon_\n\n/--\nThe trivial commutative monoid object. We later show this is initial in `CommMon_ C`.\n-/\n@[simps]\ndef trivial : CommMon_ C :=\n{ mul_comm' := begin dsimp, rw [braiding_left_unitor, unitors_equal], end\n  ..Mon_.trivial C }\n\ninstance : inhabited (CommMon_ C) := ⟨trivial C⟩\n\nvariables {C} {M : CommMon_ C}\n\ninstance : category (CommMon_ C) :=\ninduced_category.category CommMon_.to_Mon_\n\n@[simp] lemma id_hom (A : CommMon_ C) : Mon_.hom.hom (𝟙 A) = 𝟙 A.X := rfl\n@[simp] lemma comp_hom {R S T : CommMon_ C} (f : R ⟶ S) (g : S ⟶ T) :\n  Mon_.hom.hom (f ≫ g) = f.hom ≫ g.hom := rfl\n\nsection\nvariables (C)\n\n/-- The forgetful functor from commutative monoid objects to monoid objects. -/\n@[derive [full, faithful]]\ndef forget₂_Mon_ : CommMon_ C ⥤ Mon_ C :=\ninduced_functor CommMon_.to_Mon_\n\n@[simp] lemma forget₂_Mon_obj_one (A : CommMon_ C) : ((forget₂_Mon_ C).obj A).one = A.one := rfl\n@[simp] lemma forget₂_Mon_obj_mul (A : CommMon_ C) : ((forget₂_Mon_ C).obj A).mul = A.mul := rfl\n@[simp] lemma forget₂_Mon_map_hom {A B : CommMon_ C} (f : A ⟶ B) :\n  ((forget₂_Mon_ C).map f).hom = f.hom := rfl\n\nend\n\ninstance unique_hom_from_trivial (A : CommMon_ C) : unique (trivial C ⟶ A) :=\nMon_.unique_hom_from_trivial A.to_Mon_\n\nopen category_theory.limits\n\ninstance : has_initial (CommMon_ C) :=\nhas_initial_of_unique (trivial C)\n\nend CommMon_\n\nnamespace category_theory.lax_braided_functor\n\nvariables {C} {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D] [braided_category.{v₂} D]\n\n/--\nA lax braided functor takes commutative monoid objects to commutative monoid objects.\n\nThat is, a lax braided functor `F : C ⥤ D` induces a functor `CommMon_ C ⥤ CommMon_ D`.\n-/\n@[simps]\ndef map_CommMon (F : lax_braided_functor C D) : CommMon_ C ⥤ CommMon_ D :=\n{ obj := λ A,\n  { mul_comm' :=\n    begin\n      dsimp,\n      have := F.braided,\n      slice_lhs 1 2 { rw ←this, },\n      slice_lhs 2 3 { rw [←category_theory.functor.map_comp, A.mul_comm], },\n    end,\n    ..F.to_lax_monoidal_functor.map_Mon.obj A.to_Mon_ },\n  map := λ A B f, F.to_lax_monoidal_functor.map_Mon.map f, }\n\nvariables (C) (D)\n\n/-- `map_CommMon` is functorial in the lax braided functor. -/\ndef map_CommMon_functor : (lax_braided_functor C D) ⥤ (CommMon_ C ⥤ CommMon_ D) :=\n{ obj := map_CommMon,\n  map := λ F G α,\n  { app := λ A,\n    { hom := α.app A.X, } } }\n\nend category_theory.lax_braided_functor\n\nnamespace CommMon_\n\nopen category_theory.lax_braided_functor\n\nnamespace equiv_lax_braided_functor_punit\n\n/-- Implementation of `CommMon_.equiv_lax_braided_functor_punit`. -/\n@[simps]\ndef lax_braided_to_CommMon : lax_braided_functor (discrete punit.{u+1}) C ⥤ CommMon_ C :=\n{ obj := λ F, (F.map_CommMon : CommMon_ _ ⥤ CommMon_ C).obj (trivial (discrete punit)),\n  map := λ F G α, ((map_CommMon_functor (discrete punit) C).map α).app _ }\n\n/-- Implementation of `CommMon_.equiv_lax_braided_functor_punit`. -/\n@[simps]\ndef CommMon_to_lax_braided : CommMon_ C ⥤ lax_braided_functor (discrete punit.{u+1}) C :=\n{ obj := λ A,\n  { obj := λ _, A.X,\n    map := λ _ _ _, 𝟙 _,\n    ε := A.one,\n    μ := λ _ _, A.mul,\n    map_id' := λ _, rfl,\n    map_comp' := λ _ _ _ _ _, (category.id_comp (𝟙 A.X)).symm, },\n  map := λ A B f,\n  { app := λ _, f.hom,\n    naturality' := λ _ _ _, by { dsimp, rw [category.id_comp, category.comp_id], },\n    unit' := f.one_hom,\n    tensor' := λ _ _, f.mul_hom, }, }\n\nlocal attribute [tidy] tactic.discrete_cases\nlocal attribute [simp] eq_to_iso_map\n\n/-- Implementation of `CommMon_.equiv_lax_braided_functor_punit`. -/\n@[simps]\ndef unit_iso :\n  𝟭 (lax_braided_functor (discrete punit.{u+1}) C) ≅\n    lax_braided_to_CommMon C ⋙ CommMon_to_lax_braided C :=\nnat_iso.of_components (λ F, lax_braided_functor.mk_iso\n  (monoidal_nat_iso.of_components\n    (λ _, F.to_lax_monoidal_functor.to_functor.map_iso (eq_to_iso (by ext)))\n    (by tidy) (by tidy) (by tidy)))\n  (by tidy)\n\n/-- Implementation of `CommMon_.equiv_lax_braided_functor_punit`. -/\n@[simps]\ndef counit_iso : CommMon_to_lax_braided C ⋙ lax_braided_to_CommMon C ≅ 𝟭 (CommMon_ C) :=\nnat_iso.of_components (λ F, { hom := { hom := 𝟙 _, }, inv := { hom := 𝟙 _, } })\n  (by tidy)\n\nend equiv_lax_braided_functor_punit\n\nopen equiv_lax_braided_functor_punit\nlocal attribute [simp] eq_to_iso_map\n\n/--\nCommutative monoid objects in `C` are \"just\" braided lax monoidal functors from the trivial\nbraided monoidal category to `C`.\n-/\n@[simps]\ndef equiv_lax_braided_functor_punit : lax_braided_functor (discrete punit.{u+1}) C ≌ CommMon_ C :=\n{ functor := lax_braided_to_CommMon C,\n  inverse := CommMon_to_lax_braided C,\n  unit_iso := unit_iso C,\n  counit_iso := counit_iso C, }\n\nend CommMon_\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/CommMon_.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3109513340706579}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n\nimport data.fun_like.basic\n\n/-!\n# Typeclass for a type `F` with an injective map to `A ↪ B`\n\nThis typeclass is primarily for use by embeddings such as `rel_embedding`.\n\n## Basic usage of `embedding_like`\n\nA typical type of embedding should be declared as:\n```\nstructure my_embedding (A B : Type*) [my_class A] [my_class B] :=\n(to_fun : A → B)\n(injective' : function.injective to_fun)\n(map_op' : ∀ {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))\n\nnamespace my_embedding\n\nvariables (A B : Type*) [my_class A] [my_class B]\n\n-- This instance is optional if you follow the \"Embedding class\" design below:\ninstance : embedding_like (my_embedding A B) A B :=\n{ coe := my_embedding.to_fun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective' }\n\n/-- Helper instance for when there's too many metavariables to directly\napply `fun_like.to_coe_fn`. -/\ninstance : has_coe_to_fun (my_embedding A B) (λ _, A → B) := ⟨my_embedding.to_fun⟩\n\n@[simp] lemma to_fun_eq_coe {f : my_embedding A B} : f.to_fun = (f : A → B) := rfl\n\n@[ext] theorem ext {f g : my_embedding A B} (h : ∀ x, f x = g x) : f = g := fun_like.ext f g h\n\n/-- Copy of a `my_embedding` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : my_embedding A B) (f' : A → B) (h : f' = ⇑f) : my_embedding A B :=\n{ to_fun := f',\n  injective' := h.symm ▸ f.injective',\n  map_op' := h.symm ▸ f.map_op' }\n\nend my_embedding\n```\n\nThis file will then provide a `has_coe_to_fun` instance and various\nextensionality and simp lemmas.\n\n## Embedding classes extending `embedding_like`\n\nThe `embedding_like` design provides further benefits if you put in a bit more work.\nThe first step is to extend `embedding_like` to create a class of those types satisfying\nthe axioms of your new type of morphisms.\nContinuing the example above:\n\n```\n/-- `my_embedding_class F A B` states that `F` is a type of `my_class.op`-preserving embeddings.\nYou should extend this class when you extend `my_embedding`. -/\nclass my_embedding_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]\n  extends embedding_like F A B :=\n(map_op : ∀ (f : F) (x y : A), f (my_class.op x y) = my_class.op (f x) (f y))\n\n@[simp] lemma map_op {F A B : Type*} [my_class A] [my_class B] [my_embedding_class F A B]\n  (f : F) (x y : A) : f (my_class.op x y) = my_class.op (f x) (f y) :=\nmy_embedding_class.map_op\n\n-- You can replace `my_embedding.embedding_like` with the below instance:\ninstance : my_embedding_class (my_embedding A B) A B :=\n{ coe := my_embedding.to_fun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective',\n  map_op := my_embedding.map_op' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThe second step is to add instances of your new `my_embedding_class` for all types extending\n`my_embedding`.\nTypically, you can just declare a new class analogous to `my_embedding_class`:\n\n```\nstructure cooler_embedding (A B : Type*) [cool_class A] [cool_class B]\n  extends my_embedding A B :=\n(map_cool' : to_fun cool_class.cool = cool_class.cool)\n\nclass cooler_embedding_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]\n  extends my_embedding_class F A B :=\n(map_cool : ∀ (f : F), f cool_class.cool = cool_class.cool)\n\n@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_embedding_class F A B]\n  (f : F) : f cool_class.cool = cool_class.cool :=\nmy_embedding_class.map_op\n\n-- You can also replace `my_embedding.embedding_like` with the below instance:\ninstance : cool_embedding_class (cool_embedding A B) A B :=\n{ coe := cool_embedding.to_fun,\n  coe_injective' := λ f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective',\n  map_op := cool_embedding.map_op',\n  map_cool := cool_embedding.map_cool' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThen any declaration taking a specific type of morphisms as parameter can instead take the\nclass you just defined:\n```\n-- Compare with: lemma do_something (f : my_embedding A B) : sorry := sorry\nlemma do_something {F : Type*} [my_embedding_class F A B] (f : F) : sorry := sorry\n```\n\nThis means anything set up for `my_embedding`s will automatically work for `cool_embedding_class`es,\nand defining `cool_embedding_class` only takes a constant amount of effort,\ninstead of linearly increasing the work per `my_embedding`-related declaration.\n\n-/\n\n/-- The class `embedding_like F α β` expresses that terms of type `F` have an\ninjective coercion to injective functions `α ↪ β`.\n-/\nclass embedding_like (F : Sort*) (α β : out_param Sort*)\n  extends fun_like F α (λ _, β) :=\n(injective' : ∀ (f : F), @function.injective α β (coe f))\n\nnamespace embedding_like\n\nvariables {F α β γ : Sort*} [i : embedding_like F α β]\n\ninclude i\n\nprotected lemma injective (f : F) : function.injective f := injective' f\n\n@[simp] lemma apply_eq_iff_eq (f : F) {x y : α} : f x = f y ↔ x = y :=\n(embedding_like.injective f).eq_iff\n\nomit i\n\n@[simp] lemma comp_injective {F : Sort*} [embedding_like F β γ] (f : α → β) (e : F) :\n  function.injective (e ∘ f) ↔ function.injective f :=\n(embedding_like.injective e).of_comp_iff f\n\nend embedding_like\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/fun_like/embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.31081121748456}}
{"text": "import Smt\n\ntheorem cong (x y : Nat) (f : Nat → Nat) : x = y → f x = f y := by\n  smt\n  simp_all\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Nat/Cong.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3108112097982301}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : nat.coprime x p) : (x : ℤ_[p]) ∈ units ℤ_[p] :=\nbegin\n  have h1 : ¬ (x : ℤ_[p]) ∈ units ℤ_[p], from by {\n    assume h2 : (x : ℤ_[p]) ∈ units ℤ_[p],\n    have h3 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = 1, from by {\n      exact h2.right,\n    },\n    have h4 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h3,\n    },\n    have h5 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h4,\n    },\n    have h6 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h5,\n    },\n    have h7 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h6,\n    },\n    have h8 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h7,\n    },\n    have h9 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h8,\n    },\n    have h10 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h9,\n    },\n    have h11 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h10,\n    },\n    have h12 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h11,\n    },\n    have h13 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h12,\n    },\n    have h14 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h13,\n    },\n    have h15 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h14,\n    },\n    have h16 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h15,\n    },\n    have h17 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h16,\n    },\n    have h18 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h17,\n    },\n    have h19 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h18,\n    },\n    have h20 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h19,\n    },\n    have h21 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h20,\n    },\n    have h22 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h21,\n    },\n    have h23 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h22,\n    },\n    have h24 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h23,\n    },\n    have h25 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h24,\n    },\n    have h26 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h25,\n    },\n    have h27 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h26,\n    },\n    have h28 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h27,\n    },\n    have h29 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h28,\n    },\n    have h30 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h29,\n    },\n    have h31 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h30,\n    },\n    have h32 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h31,\n    },\n    have h33 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h32,\n    },\n    have h34 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h33,\n    },\n    have h35 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h34,\n    },\n    have h36 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h35,\n    },\n    have h37 : (x : ℤ_[p]) * (x : ℤ_[p])⁻¹ = (1 : ℤ_[p]), from by {\n      exact h36,\n    },\n    have h38\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : nat.coprime x p) : is_unit (p_adic_int.mk p x) :=\nbegin\n  have h1 : ¬ is_unit (p_adic_int.mk p x), from by {\n    assume h2 : is_unit (p_adic_int.mk p x),\n    have h3 : p_adic_int.norm (p_adic_int.mk p x) = 1, from by {\n      rw is_unit_iff_norm_eq_one at h2,\n      exact h2,\n    },\n    have h4 : p_adic_int.norm (p_adic_int.mk p x) < 1, from by {\n      rw p_adic_int.norm_mk,\n      rw nat.mod_eq_of_lt,\n      apply nat.lt_succ_self,\n    },\n    have h5 : p ∣ x, from by {\n      rw ← h3,\n      apply p_adic_int.norm_mk_dvd,\n      exact h4,\n    },\n    have h6 : ¬ nat.coprime x p, from by {\n      apply nat.not_coprime_of_dvd_of_dvd,\n      exact h5,\n      apply nat.dvd_refl p,\n    },\n    exact h6 h,\n  },\n  show is_unit (p_adic_int.mk p x), from by {\n    apply classical.by_contradiction,\n    exact h1,\n  },\nend\n\n--OUTPUT 3\ntheorem p_adic_units (p : ℕ) (x : ℕ) (h : nat.coprime x p) : (x : ℤ_[p]) ∈ units ℤ_[p] :=\nbegin\n  have h1 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h2 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h3 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h4 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h5 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h6 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h7 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h8 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h9 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h10 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h11 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h12 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h13 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h14 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h15 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h16 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h17 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h18 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h19 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h20 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h21 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h22 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h23 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h24 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h25 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h26 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h27 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h28 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h29 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h30 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h31 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h32 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h33 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h34 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h35 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h36 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h37 : (x : ℤ_[p]) ∈ units ℤ_[p] ↔ (x : ℤ_[p]) ∈ units ℤ_[p], from by obviously,\n  have h38 : (x : ℤ_[p]) ∈ units ℤ\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A ∩ B) ⊆ A, from by apply set.inter_subset_left,\n  have h3 : (A ∩ B) ⊆ S, from by {apply set.subset.trans h2 h1.left},\n  show (A ∩ B) ∈  𝒫 S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw ← sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by {\n    assume a b : G, use a⁻¹ * b, obviously, },\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by {\n    assume a b : G, use b * a⁻¹, obviously, }, \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (hident : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : ∀ a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact ⟨by obviously, h7⟩,\n  }\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/lean_proof-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8152324713956854, "lm_q2_score": 0.38121956625614994, "lm_q1q2_score": 0.31078256914339236}}
{"text": "/-\nCopyright (c) 2014 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Yaël Dillies, Patrick Stevens\n\n! This file was ported from Lean 3 source module data.nat.cast.field\n! leanprover-community/mathlib commit acee671f47b8e7972a1eb6f4eed74b4b3abce829\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Order.Field.Basic\nimport Mathbin.Algebra.Order.Ring.CharZero\nimport Mathbin.Data.Nat.Cast.Basic\n\n/-!\n# Cast of naturals into fields\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file concerns the canonical homomorphism `ℕ → F`, where `F` is a field.\n\n## Main results\n\n * `nat.cast_div`: if `n` divides `m`, then `↑(m / n) = ↑m / ↑n`\n * `nat.cast_div_le`: in all cases, `↑(m / n) ≤ ↑m / ↑ n`\n-/\n\n\nnamespace Nat\n\nvariable {α : Type _}\n\n/- warning: nat.cast_div -> Nat.cast_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} α] {m : Nat} {n : Nat}, (Dvd.Dvd.{0} Nat Nat.hasDvd n m) -> (Ne.{succ u1} α ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) n) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1))))))))) -> (Eq.{succ u1} α ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) m n)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) m) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) n)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} α] {m : Nat} {n : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat n m) -> (Ne.{succ u1} α (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) n) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (MonoidWithZero.toZero.{u1} α (Semiring.toMonoidWithZero.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))) -> (Eq.{succ u1} α (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) m n)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivisionSemiring.toDiv.{u1} α _inst_1)) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) m) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) n)))\nCase conversion may be inaccurate. Consider using '#align nat.cast_div Nat.cast_divₓ'. -/\n@[simp]\ntheorem cast_div [DivisionSemiring α] {m n : ℕ} (n_dvd : n ∣ m) (n_nonzero : (n : α) ≠ 0) :\n    ((m / n : ℕ) : α) = m / n := by\n  rcases n_dvd with ⟨k, rfl⟩\n  have : n ≠ 0 := by\n    rintro rfl\n    simpa using n_nonzero\n  rw [Nat.mul_div_cancel_left _ this.bot_lt, mul_comm n k, cast_mul, mul_div_cancel _ n_nonzero]\n#align nat.cast_div Nat.cast_div\n\n/- warning: nat.cast_div_div_div_cancel_right -> Nat.cast_div_div_div_cancel_right is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} α] [_inst_2 : CharZero.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1))))] {m : Nat} {n : Nat} {d : Nat}, (Dvd.Dvd.{0} Nat Nat.hasDvd d n) -> (Dvd.Dvd.{0} Nat Nat.hasDvd d m) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) m d)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) n d))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α _inst_1)))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) m) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)))))))) n)))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} α] [_inst_2 : CharZero.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1))))] {m : Nat} {n : Nat} {d : Nat}, (Dvd.dvd.{0} Nat Nat.instDvdNat d n) -> (Dvd.dvd.{0} Nat Nat.instDvdNat d m) -> (Eq.{succ u1} α (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivisionSemiring.toDiv.{u1} α _inst_1)) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) m d)) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) n d))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivisionSemiring.toDiv.{u1} α _inst_1)) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) m) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (DivisionSemiring.toSemiring.{u1} α _inst_1)) n)))\nCase conversion may be inaccurate. Consider using '#align nat.cast_div_div_div_cancel_right Nat.cast_div_div_div_cancel_rightₓ'. -/\ntheorem cast_div_div_div_cancel_right [DivisionSemiring α] [CharZero α] {m n d : ℕ} (hn : d ∣ n)\n    (hm : d ∣ m) : (↑(m / d) : α) / (↑(n / d) : α) = (m : α) / n :=\n  by\n  rcases eq_or_ne d 0 with (rfl | hd); · simp [zero_dvd_iff.mp hm]\n  replace hd : (d : α) ≠ 0;\n  · norm_cast\n    assumption\n  simp [hd, hm, hn, div_div_div_cancel_right _ hd]\n#align nat.cast_div_div_div_cancel_right Nat.cast_div_div_div_cancel_right\n\nsection LinearOrderedSemifield\n\nvariable [LinearOrderedSemifield α]\n\n/- warning: nat.cast_div_le -> Nat.cast_div_le is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {m : Nat} {n : Nat}, LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) m n)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) m) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) n))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {m : Nat} {n : Nat}, LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (StrictOrderedSemiring.toPartialOrder.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) m n)) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) m) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) n))\nCase conversion may be inaccurate. Consider using '#align nat.cast_div_le Nat.cast_div_leₓ'. -/\n/-- Natural division is always less than division in the field. -/\ntheorem cast_div_le {m n : ℕ} : ((m / n : ℕ) : α) ≤ m / n :=\n  by\n  cases n\n  · rw [cast_zero, div_zero, Nat.div_zero, cast_zero]\n  rwa [le_div_iff, ← Nat.cast_mul]\n  exact Nat.cast_le.2 (Nat.div_mul_le_self m n.succ)\n  · exact Nat.cast_pos.2 n.succ_pos\n#align nat.cast_div_le Nat.cast_div_le\n\n/- warning: nat.inv_pos_of_nat -> Nat.inv_pos_of_nat is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat}, LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (Inv.inv.{u1} α (DivInvMonoid.toHasInv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat}, LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (StrictOrderedSemiring.toPartialOrder.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CommGroupWithZero.toCommMonoidWithZero.{u1} α (Semifield.toCommGroupWithZero.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (Inv.inv.{u1} α (LinearOrderedSemifield.toInv.{u1} α _inst_1) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))))\nCase conversion may be inaccurate. Consider using '#align nat.inv_pos_of_nat Nat.inv_pos_of_natₓ'. -/\ntheorem inv_pos_of_nat {n : ℕ} : 0 < ((n : α) + 1)⁻¹ :=\n  inv_pos.2 <| add_pos_of_nonneg_of_pos n.cast_nonneg zero_lt_one\n#align nat.inv_pos_of_nat Nat.inv_pos_of_nat\n\n/- warning: nat.one_div_pos_of_nat -> Nat.one_div_pos_of_nat is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat}, LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (OfNat.ofNat.{u1} α 0 (OfNat.mk.{u1} α 0 (Zero.zero.{u1} α (MulZeroClass.toHasZero.{u1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat}, LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (StrictOrderedSemiring.toPartialOrder.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 0 (Zero.toOfNat0.{u1} α (CommMonoidWithZero.toZero.{u1} α (CommGroupWithZero.toCommMonoidWithZero.{u1} α (Semifield.toCommGroupWithZero.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))))))\nCase conversion may be inaccurate. Consider using '#align nat.one_div_pos_of_nat Nat.one_div_pos_of_natₓ'. -/\ntheorem one_div_pos_of_nat {n : ℕ} : 0 < 1 / ((n : α) + 1) :=\n  by\n  rw [one_div]\n  exact inv_pos_of_nat\n#align nat.one_div_pos_of_nat Nat.one_div_pos_of_nat\n\n/- warning: nat.one_div_le_one_div -> Nat.one_div_le_one_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat} {m : Nat}, (LE.le.{0} Nat Nat.hasLe n m) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) m) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat} {m : Nat}, (LE.le.{0} Nat instLENat n m) -> (LE.le.{u1} α (Preorder.toLE.{u1} α (PartialOrder.toPreorder.{u1} α (StrictOrderedSemiring.toPartialOrder.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) m) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))\nCase conversion may be inaccurate. Consider using '#align nat.one_div_le_one_div Nat.one_div_le_one_divₓ'. -/\ntheorem one_div_le_one_div {n m : ℕ} (h : n ≤ m) : 1 / ((m : α) + 1) ≤ 1 / ((n : α) + 1) :=\n  by\n  refine' one_div_le_one_div_of_le _ _\n  exact Nat.cast_add_one_pos _\n  simpa\n#align nat.one_div_le_one_div Nat.one_div_le_one_div\n\n/- warning: nat.one_div_lt_one_div -> Nat.one_div_lt_one_div is a dubious translation:\nlean 3 declaration is\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat} {m : Nat}, (LT.lt.{0} Nat Nat.hasLt n m) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (OrderedCancelAddCommMonoid.toPartialOrder.{u1} α (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) m) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (DivInvMonoid.toHasDiv.{u1} α (GroupWithZero.toDivInvMonoid.{u1} α (DivisionSemiring.toGroupWithZero.{u1} α (Semifield.toDivisionSemiring.{u1} α (LinearOrderedSemifield.toSemifield.{u1} α _inst_1)))))) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toHasAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat α (HasLiftT.mk.{1, succ u1} Nat α (CoeTCₓ.coe.{1, succ u1} Nat α (Nat.castCoe.{u1} α (AddMonoidWithOne.toNatCast.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))) n) (OfNat.ofNat.{u1} α 1 (OfNat.mk.{u1} α 1 (One.one.{u1} α (AddMonoidWithOne.toOne.{u1} α (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} α (NonAssocSemiring.toAddCommMonoidWithOne.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))))))\nbut is expected to have type\n  forall {α : Type.{u1}} [_inst_1 : LinearOrderedSemifield.{u1} α] {n : Nat} {m : Nat}, (LT.lt.{0} Nat instLTNat n m) -> (LT.lt.{u1} α (Preorder.toLT.{u1} α (PartialOrder.toPreorder.{u1} α (StrictOrderedSemiring.toPartialOrder.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) m) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (HDiv.hDiv.{u1, u1, u1} α α α (instHDiv.{u1} α (LinearOrderedSemifield.toDiv.{u1} α _inst_1)) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))) (HAdd.hAdd.{u1, u1, u1} α α α (instHAdd.{u1} α (Distrib.toAdd.{u1} α (NonUnitalNonAssocSemiring.toDistrib.{u1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} α (Semiring.toNonAssocSemiring.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))) (Nat.cast.{u1} α (Semiring.toNatCast.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))) n) (OfNat.ofNat.{u1} α 1 (One.toOfNat1.{u1} α (Semiring.toOne.{u1} α (StrictOrderedSemiring.toSemiring.{u1} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u1} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u1} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u1} α _inst_1))))))))))\nCase conversion may be inaccurate. Consider using '#align nat.one_div_lt_one_div Nat.one_div_lt_one_divₓ'. -/\ntheorem one_div_lt_one_div {n m : ℕ} (h : n < m) : 1 / ((m : α) + 1) < 1 / ((n : α) + 1) :=\n  by\n  refine' one_div_lt_one_div_of_lt _ _\n  exact Nat.cast_add_one_pos _\n  simpa\n#align nat.one_div_lt_one_div Nat.one_div_lt_one_div\n\nend LinearOrderedSemifield\n\nend Nat\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Nat/Cast/Field.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.45713671682749485, "lm_q1q2_score": 0.31047752068020795}}
{"text": "-- SOUNDNESS\n\nimport syntax\nimport tableau\n\nopen classical\nlocal attribute [instance, priority 10] prop_decidable\n\nopen has_sat\n\n-- Combine a collection of pointed models with one new world using the given valuation.\n-- TODO: rewrite to term mode?\ndef combinedModel { β : Type } (collection : β → Σ (W : Type), kripkeModel W × W) (newVal : char → Prop) :\n  kripkeModel (sum unit Σ k : β, (collection k).fst) × (sum unit Σ k : β, (collection k).fst) :=\nbegin\n  split,\n  split,\n  { -- making the valuation function:\n    intro world,\n    cases world,\n    { -- the one new world:\n      cases world,\n      exact newVal, -- use new given valuation here!!\n    },\n    { -- world in one of the given models:\n      cases world with R w,\n      exact (collection R).snd.fst.val w,\n    }\n  },\n  { -- defining relations:\n    intros worldOne worldTwo,\n    cases worldOne; cases worldTwo,\n    -- four cases about two new or old worlds:\n    { exact false, }, -- no reflexive loop at the new world.\n    { exact (worldTwo.snd == (collection worldTwo.fst).snd.snd), }, -- conect new world to given points.\n    { exact false }, -- no connection from models to new world\n    { -- connect two old worlds iff they are from the same model and were connected there already:\n      cases worldOne with kOne wOne,\n      cases worldTwo with kTwo wTwo,\n      have help : kOne = kTwo → Prop, {\n        intro same,\n        have sameCol : (collection kOne = collection kTwo), {\n          rw ← same,\n        },\n        rw ← sameCol at wTwo,\n        exact (collection kOne).snd.fst.rel wOne wTwo,\n      },\n      exact dite (kOne = kTwo)\n        (λ same, help same)\n        (λ _, false)\n    },\n  },\n  { -- point at the new world:\n    left,\n    exact (),\n  }\nend\n\n-- The combined model preserves all truths at the old worlds.\nlemma combMo_preserves_truth_at_oldWOrld { β : Type }\n  (collection : β → Σ (W : Type), kripkeModel W × W) (newVal : char → Prop)\n  : (∀ (f : formula) (R : β) (oldWorld : (collection R).fst),\n    evaluate ((combinedModel collection newVal).fst, (sum.inr ⟨R, oldWorld⟩)) f ↔ evaluate ((collection R).snd.fst, oldWorld) f) :=\nbegin\n  intro f,\n  induction f ; intros R oldWorld,\n  case bottom : {\n    finish,\n  },\n  case atom_prop : c {\n    unfold combinedModel,\n    unfold evaluate,\n  },\n  case neg : f f_IH {\n    unfold evaluate,\n    rw f_IH R oldWorld,\n  },\n  case and : f g f_IH g_IH {\n    unfold evaluate,\n    rw f_IH R oldWorld,\n    rw g_IH R oldWorld,\n  },\n  case box : f f_IH {\n    unfold evaluate,\n    split,\n    {\n      intro true_in_combo,\n      intros otherWorld rel_in_old_model,\n      specialize f_IH R otherWorld,\n      rw ← f_IH,\n      specialize true_in_combo (sum.inr ⟨R, otherWorld⟩),\n      apply true_in_combo,\n      unfold combinedModel,\n      simp,\n      exact rel_in_old_model,\n    },\n    {\n      intro true_in_old,\n      simp,\n      split,\n      {\n        intro newWorld,\n        unfold combinedModel,\n        tauto, -- the new world is never reachable, trivial case\n      },\n      {\n        intros otherR otherWorld,\n        intro rel_in_new_model,\n        specialize f_IH otherR otherWorld,\n        unfold combinedModel at rel_in_new_model,\n        have sameR : R = otherR, {\n          by_contradiction,\n          classical,\n          finish,\n        },\n        subst sameR,\n        rw f_IH,\n        apply true_in_old,\n        -- remains to show that related in old model\n        simp at *,\n        exact rel_in_new_model,\n      },\n    },\n  },\nend\n\n-- The combined model for X satisfies X.\nlemma combMo_sat_X { X : finset formula } { β : set formula } { beta_def : β = { R : formula | ~R.box ∈ X } }\n  (simple_X : simple X)\n  (not_closed_X : ¬ closed X)\n  (collection : β → Σ (W : Type), kripkeModel W × W)\n  (all_pro_sat : ∀ R : β, ∀ f ∈ projection X ∪ {~R}, evaluate ((collection R).snd.fst, (collection R).snd.snd) f)\n  : (∀ f ∈ X, evaluate ((combinedModel collection (λ c, (formula.atom_prop c ∈ X))).fst, (combinedModel collection (λ c, (formula.atom_prop c ∈ X))).snd) f) :=\nbegin\n  intros f f_in_X,\n  cases f, -- no induction because X is simple\n  case formula.bottom: {\n    unfold closed at not_closed_X,\n    tauto,\n  },\n  case atom_prop: {\n    unfold combinedModel,\n    unfold evaluate,\n    simp,\n    exact f_in_X,\n  },\n  case neg: {\n    -- subcases :-/\n    cases f,\n    case atom_prop: {\n      unfold combinedModel,\n      unfold evaluate,\n      unfold closed at not_closed_X,\n      rw not_or_distrib at not_closed_X,\n      simp at *,\n      tauto,\n    },\n    case box: newf {\n      -- set coMo := ,\n      --unfold combinedModel,\n      change (evaluate ((combinedModel collection (λ c, (·c) ∈ X)).fst, (combinedModel collection (λ (c : char), (·c) ∈ X)).snd) (~□newf)),\n      unfold evaluate,\n      rw not_forall,\n      -- need a reachable world where newf holds, choose the witness\n      let h : newf ∈ β, {\n        rw beta_def,\n        use f_in_X,\n      },\n      let oldWorld : unit ⊕ Σ (k : β), (collection k).fst :=\n        sum.inr ⟨⟨newf, h⟩, (collection ⟨newf, h⟩).snd.snd⟩,\n      use oldWorld,\n      simp,\n      split,\n      { -- show that worlds are related in combined model (def above, case 2)\n        unfold combinedModel, simp,\n      },\n      { -- show that f is false at old world\n        have coMoLemma := combMo_preserves_truth_at_oldWOrld collection (λ (c : char), (·c) ∈ X) newf ⟨newf, h⟩ (collection ⟨newf, h⟩).snd.snd,\n        rw coMoLemma,\n        specialize all_pro_sat ⟨newf, h⟩ (~newf),\n        unfold evaluate at all_pro_sat,\n        simp at *,\n        exact all_pro_sat,\n      },\n    },\n    case bottom: {\n      tauto,\n    },\n    all_goals {\n      unfold simple at simple_X,\n      by_contradiction,\n      exact simple_X _ f_in_X,\n    },\n  },\n  case and: fa fb {\n    unfold simple at simple_X,\n    by_contradiction,\n    exact simple_X (fa ⋏ fb) f_in_X,\n  },\n  case box: f {\n    unfold evaluate,\n    intros otherWorld is_rel,\n    cases otherWorld,\n    { cases is_rel, }, -- otherWorld cannot be the (unreachable) new world\n    have coMoLemma := combMo_preserves_truth_at_oldWOrld collection (λ c, (·c) ∈ X) f otherWorld.fst otherWorld.snd,\n    simp at coMoLemma,\n    rw coMoLemma,\n    specialize all_pro_sat otherWorld.fst f,\n    simp at all_pro_sat,\n    rw or_imp_distrib at all_pro_sat,\n    cases all_pro_sat with _ all_pro_sat_right,\n    rw ← proj at f_in_X,\n    specialize all_pro_sat_right f_in_X,\n    have sameWorld : otherWorld.snd = (collection otherWorld.fst).snd.snd, {\n      rw (heq_iff_eq.mp (heq.symm is_rel)),\n    },\n    rw sameWorld,\n    simp,\n    exact all_pro_sat_right,\n  },\nend\n\n-- Lemma 1 (page 16)\n-- A simple set of formulas X is satisfiable if and only if\n-- it is not closed  and  for all ¬[A]R ∈ X also XA; ¬R is satisfiable.\nlemma Lemma1_simple_sat_iff_all_projections_sat { X : finset formula } :\n  simple X → (satisfiable X ↔ (¬ closed X ∧ ∀ R, (~□R) ∈ X → satisfiable (projection X ∪ {~R}))) :=\nbegin\n  intro X_is_simple,\n  split,\n  { -- left to right\n    intro sat_X,\n    unfold satisfiable at *,\n    rcases sat_X with ⟨ W, M, w, w_sat_X ⟩,\n    split,\n    { -- show that X is not closed:\n      by_contradiction hyp,\n      unfold closed at hyp,\n      cases hyp with bot_in_X f_and_notf_in_X,\n      {\n        exact w_sat_X ⊥ bot_in_X,\n      },\n      {\n        rcases f_and_notf_in_X with ⟨ f, f_in_X, notf_in_X ⟩,\n        let w_sat_f :=  w_sat_X f f_in_X,\n        let w_sat_notf :=  w_sat_X (~f) notf_in_X,\n        unfold evaluate at *,\n        exact absurd w_sat_f w_sat_notf,\n      }\n    },\n    { -- show that for each ~[]R ∈ X the projection with ~R is satisfiable:\n      intros R notboxr_in_X,\n      let w_sat_notboxr := w_sat_X (~□R) notboxr_in_X,\n      unfold evaluate at w_sat_notboxr,\n      simp at w_sat_notboxr,\n      rcases w_sat_notboxr with ⟨ v, w_rel_v, v_sat_notr ⟩,\n      use [W, M, v],\n      intro g,\n      simp at *,\n      rw or_imp_distrib,\n      split,\n      {\n        intro g_is_notR,\n        rw g_is_notR,\n        exact v_sat_notr,\n      },\n      {\n        intro boxg_in_X,\n        rw proj at boxg_in_X,\n        specialize w_sat_X (□g) boxg_in_X,\n        unfold evaluate at w_sat_X,\n        exact w_sat_X v w_rel_v,\n      },\n    },\n  },\n  { -- right to left\n    intro rhs,\n    cases rhs with not_closed_X all_pro_sat,\n    unfold satisfiable at *,\n    -- Let's build a new Kripke model!\n    let β := { R : formula | ~□R ∈ X },\n    -- beware, using Axioms of Choice here!\n    choose typeFor this_pro_sat using all_pro_sat,\n    choose modelFor this_pro_sat using this_pro_sat,\n    choose worldFor this_pro_sat using this_pro_sat,\n    -- define the collection:\n    let collection : β → (Σ (W : Type), (kripkeModel W × W)) := begin\n      intro k,\n      cases k with R notboxr_in_X,\n      simp at notboxr_in_X,\n      use [typeFor R notboxr_in_X, modelFor R notboxr_in_X, worldFor R notboxr_in_X],\n    end,\n    let newVal := λ c, formula.atom_prop c ∈ X,\n    let BigM := combinedModel collection newVal,\n    use unit ⊕ Σ k : β, (collection k).fst,\n    use [BigM.fst, BigM.snd],\n    -- apply Lemma, missing last argument \"all_pro_sat\"\n    -- we need to use that X_is_simple (to restrict cases what phi can be)\n    -- and that X is not closed (to ensure that it is locally consistent)\n    apply combMo_sat_X X_is_simple not_closed_X collection,\n    -- it remains to show that the new big model satisfies X\n    intros R f f_inpro_or_notr,\n    cases R with R notrbox_in_X,\n    simp only [finset.mem_union, finset.mem_insert, finset.mem_singleton, subtype.coe_mk] at *,\n    specialize this_pro_sat R notrbox_in_X,\n    cases f_inpro_or_notr with f_inpro f_is_notboxR,\n    { -- if f is in the projection\n      specialize this_pro_sat f,\n      rw or_imp_distrib at this_pro_sat,\n      cases this_pro_sat with this_pro_sat_l  this_pro_sat_r,\n      exact this_pro_sat_l f_inpro,\n    },\n    { -- case where f is ~[]R\n      cases f_is_notboxR,\n      specialize this_pro_sat (~R),\n      rw or_imp_distrib at this_pro_sat,\n      cases this_pro_sat with this_pro_sat_l  this_pro_sat_r,\n      exact this_pro_sat_r f_is_notboxR,\n    },\n    simp, -- to check β\n  },\nend\n\n-- Each rule is sound and preserves satisfiability \"downwards\"\nlemma localRuleSoundness {α : finset formula} { B : finset (finset formula) } :\n  localRule α B → (satisfiable α → ∃ β ∈ B, satisfiable β) :=\nbegin\n  intro r,\n  intro a_sat,\n  unfold satisfiable at a_sat,\n  rcases a_sat with ⟨ W, M, w, w_sat_a ⟩,\n  cases r,\n  case localRule.bot : a bot_in_a {\n    by_contradiction,\n    let w_sat_bot := w_sat_a ⊥ bot_in_a  ,\n    unfold evaluate at w_sat_bot,\n    exact w_sat_bot,\n  },\n  case localRule.not : a f hyp {\n    by_contradiction,\n    have w_sat_f : evaluate (M, w) f , {\n      apply w_sat_a,\n      exact hyp.left,\n    },\n    have w_sat_not_f : evaluate (M, w) (~f) , {\n      apply w_sat_a (~f),\n      exact hyp.right,\n    },\n    unfold evaluate at *,\n    exact absurd w_sat_f w_sat_not_f,\n  },\n  case localRule.neg : a f hyp {\n    have w_sat_f : evaluate (M, w) f, {\n      specialize w_sat_a (~~f) hyp,\n      unfold evaluate at *,\n      finish,\n    },\n    use (a \\ {~~f} ∪ {f}),\n    simp only [true_and, eq_self_iff_true, sdiff_singleton_is_erase, multiset.mem_singleton, finset.mem_mk],\n    unfold satisfiable,\n    use [W, M, w],\n    intro g,\n    simp only [ne.def, union_singleton_is_insert, finset.mem_insert, finset.mem_erase],\n    rw or_imp_distrib,\n    split,\n    {\n      intro g_is_f, rw g_is_f, apply w_sat_f,\n    },\n    {\n      rw and_imp,\n      intros g_neq_notnotf g_in_a,\n      apply w_sat_a,\n      exact g_in_a,\n    },\n  },\n  case localRule.con : a f g hyp {\n    use ( (a \\ {f ⋏ g}) ∪ { f, g } ),\n    split,\n    simp,\n    unfold satisfiable,\n    use [W, M, w],\n    simp only [and_imp, forall_eq_or_imp, sdiff_singleton_is_erase, ne.def, finset.union_insert, finset.mem_insert, finset.mem_erase],\n    split,\n    { -- f\n      specialize w_sat_a (f⋏g) hyp,\n      unfold evaluate at *,\n      exact w_sat_a.left,\n    },\n    {\n      intros h hhyp,\n      simp at hhyp,\n      cases hhyp,\n      { -- h = g\n        specialize w_sat_a (f⋏g) hyp,\n        unfold evaluate at *,\n        rw hhyp,\n        exact w_sat_a.right,\n      },\n      { -- h in a\n        exact w_sat_a h hhyp.right,\n      },\n    },\n  },\n  case localRule.nCo : a f g notfandg_in_a {\n    unfold satisfiable,\n    simp,\n    let w_sat_phi := w_sat_a (~(f⋏g)) notfandg_in_a,\n    unfold evaluate at w_sat_phi,\n    rw not_and_distrib at w_sat_phi,\n    cases w_sat_phi with not_w_f not_w_g,\n    { use (a \\ { ~ (f ⋏ g) } ∪ {~f}),\n      split,\n      { simp at *, },\n      { use [W, M, w],\n        intro psi,\n        simp,\n        intro psi_def,\n        cases psi_def,\n        { rw psi_def,\n          unfold evaluate,\n          exact not_w_f,\n        },\n        { cases psi_def with psi_in_a,\n          exact w_sat_a psi psi_def_right,\n        },\n      },\n    },\n    { use (a \\ { ~ (f ⋏ g) } ∪ {~g}),\n      split,\n      { simp at *, },\n      { use [W, M, w],\n        intro psi,\n        simp,\n        intro psi_def,\n        cases psi_def,\n        { rw psi_def,\n          unfold evaluate,\n          exact not_w_g,\n        },\n        { cases psi_def with psi_in_a,\n          exact w_sat_a psi psi_def_right,\n        },\n      },\n    },\n  },\nend\n\n-- The critical roule is sound and preserves satisfiability \"downwards\".\n-- TODO: is this the same as (one of the directions of) Lemma 1 ??\nlemma atmSoundness {α : finset formula} {f} (not_box_f_in_a : ~□f ∈ α) :\n  simple α → satisfiable α → satisfiable (projection α ∪ {~f}) :=\nbegin\n  intro s,\n  intro aSat,\n  unfold satisfiable at aSat,\n  rcases aSat with ⟨W, M, w, w_sat_a⟩,\n  split,\n  simp,\n  -- get the other reachable world:\n  let w_sat_not_box_f := w_sat_a (~f.box) not_box_f_in_a,\n  unfold evaluate at w_sat_not_box_f,\n  simp at w_sat_not_box_f,\n  rcases w_sat_not_box_f with ⟨ v,  w_rel_v, v_not_sat_f ⟩,\n  -- show that the projection is satisfiable:\n  use [M, v],\n  split,\n  { unfold evaluate,\n    exact v_not_sat_f,\n  },\n  intros phi phi_in_proj,\n  rw proj at phi_in_proj,\n  {\n    specialize w_sat_a phi.box _,\n    exact phi_in_proj,\n    unfold evaluate at w_sat_a,\n    exact w_sat_a v w_rel_v,\n  },\nend\n\nlemma localTableauAndEndNodesUnsatThenNotSat {Z} (ltZ : localTableau Z) :\n  (Π (Y), Y ∈ endNodesOf ⟨Z, ltZ⟩ → ¬ satisfiable Y) → ¬satisfiable Z :=\nbegin\n  intro endsOfXnotSat,\n  induction ltZ,\n  case byLocalRule : X YS lr next IH {\n    by_contradiction satX,\n    rcases localRuleSoundness lr satX with ⟨Y,Y_in_YS,satY⟩,\n    specialize IH Y Y_in_YS,\n    set ltY := next Y Y_in_YS,\n    have endNodesInclusion : ∀ W, W ∈ endNodesOf ⟨Y, ltY⟩ → W ∈ endNodesOf ⟨X, localTableau.byLocalRule lr next⟩ , {\n      rw endNodesOf,\n      intros W W_endOF_Y,\n      simp only [endNodesOf, finset.mem_bUnion, finset.mem_attach, exists_true_left, subtype.exists],\n      use [Y, Y_in_YS],\n      assumption,\n    },\n    have endsOfYnotSat : (∀ (Y_1 : finset formula), Y_1 ∈ endNodesOf ⟨Y, ltY⟩ → ¬ satisfiable Y_1), {\n      intros W W_is_endOf_Y,\n      apply endsOfXnotSat W (endNodesInclusion W W_is_endOf_Y),\n    },\n    finish,\n  },\n  case sim : X X_is_simple {\n    apply endsOfXnotSat,\n    unfold endNodesOf,\n    simp,\n  },\nend\n\nlemma tableauThenNotSat : ∀ X, closedTableau X → ¬ satisfiable X :=\nbegin\n  intros X t,\n  induction t,\n  case loc: Y ltY next IH {\n    apply localTableauAndEndNodesUnsatThenNotSat ltY,\n    intros Z ZisEndOfY,\n    exact IH Z ZisEndOfY,\n  },\n  case atm: Y ϕ notBoxPhiInY Y_is_simple ltProYnPhi {\n    rw Lemma1_simple_sat_iff_all_projections_sat Y_is_simple,\n    simp,\n    intro Ynotclosed,\n    use ϕ,\n    use notBoxPhiInY,\n    finish,\n  },\nend\n\n-- Theorem 2, page 30\ntheorem correctness : ∀ X, satisfiable X → consistent X :=\nbegin\n  intro X,\n  contrapose,\n  unfold consistent,\n  simp,\n  unfold inconsistent,\n  intro hyp,\n  cases hyp with t,\n  exact tableauThenNotSat X t,\nend\n\nlemma soundTableau : ∀ φ, provable φ → ¬ satisfiable ({~φ} : finset formula) :=\nbegin\n  intro phi,\n  intro prov,\n  cases prov with _ tabl,\n  apply tableauThenNotSat,\n  exact tabl,\nend\n\ntheorem soundness : ∀ φ, provable φ → tautology φ :=\nbegin\n  intros φ prov,\n  apply notsatisfnotThenTaut,\n  rw ← singletonSat_iff_sat,\n  apply soundTableau,\n  exact prov,\nend\n", "meta": {"author": "m4lvin", "repo": "tablean", "sha": "836202612fc2bfacb5545696412e7d27f7704141", "save_path": "github-repos/lean/m4lvin-tablean", "path": "github-repos/lean/m4lvin-tablean/tablean-836202612fc2bfacb5545696412e7d27f7704141/src/soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.31046702683668564}}
{"text": "-- Copyright (c) Microsoft Corporation. All rights reserved.\n-- Licensed under the MIT license.\n\nimport ..smtexpr\nimport ..smtcompile\nimport ..bitvector\nimport .spec\nimport .lemmas\nimport .lemmas_basic\nimport .irstate\nimport .freevar\nimport .equiv\nimport .closed\nimport smt2.syntax\nimport system.io\nimport init.meta.tactic\nimport init.meta.interactive\n\nnamespace spec\n\nopen irsem\nopen freevar\n\n-- get_free_*_name\n\nlemma get_free_name_diff: ∀ n,\n  get_free_sbitvec_name n ≠ get_free_sbool_name n\n:= begin\n  intros,\n  intros H,\n  unfold get_free_sbitvec_name at H,\n  unfold get_free_sbool_name at H,\n  rw string.eq_list at H,\n  rw string.append_to_list at H,\n  rw string.append_to_list at H,\n  have H' := list.append_eq2 H,\n  cases H'\nend\n\n\nlemma closed_regfile_apply_add_b_bv: ∀ (rf:regfile irsem_smt) (η:freevar.env)\n    vname vz bname vb\n    (HC:closed_regfile (rf.apply_to_values irsem_smt (env.replace_valty η))),\n  closed_regfile (rf.apply_to_values irsem_smt\n      (env.replace_valty ((η.add_bv vname vz).add_b bname vb)))\n:= begin\n  intros,\n  revert HC,\n  apply regfile.induction rf,\n  {\n    unfold closed_regfile,\n    intros,\n    rw regfile.empty_apply_empty,\n    apply closed_regfile_empty\n  },\n  {\n    intros rf IH,\n    intros,\n    unfold closed_regfile,\n    intros,\n    rw regfile.apply_update_comm at HC,\n    rw regfile.apply_update_comm,\n    unfold closed_regfile at IH,\n    rw closed_regfile_update_split at HC,\n    cases HC with HC HCval,\n    have HC := IH HC,\n    rw closed_regfile_update_split,\n    split,\n    {\n      assumption\n    },\n    {\n      cases v,\n      unfold freevar.env.replace_valty at HCval,\n      rw closed_ival_split at HCval,\n      cases HCval with HCval1 HCval2,\n      unfold freevar.env.replace_valty,\n      rw closed_ival_split,\n      split,\n      {\n        have H := closed_b_add_bv vname vz HCval1,\n        have H := closed_b_add_b bname vb H,\n        assumption\n      },\n      {\n        have H := closed_bv_add_bv vname vz HCval2,\n        have H := closed_bv_add_b bname vb H,\n        assumption\n      }\n    }\n  }\nend\n\nlemma regfile_update_ival_closed: ∀ rf rf' (η:freevar.env) regn sz\n    vn pn bvn p\n    (HCRF: closed_regfile (regfile.apply_to_values irsem_smt rf\n           (env.replace_valty η)))\n    (HRF': rf' = regfile.update irsem_smt rf regn\n           (valty.ival sz (sbitvec.var sz vn) (sbool.var pn))),\n  closed_regfile (regfile.apply_to_values irsem_smt rf'\n    (env.replace_valty (env.add_b (env.add_bv η vn bvn) pn p)))\n:= begin\n  intros,\n  have H1 : closed_regfile (regfile.apply_to_values irsem_smt\n            rf (env.replace_valty (env.add_b (env.add_bv η vn bvn) pn p))),\n  {\n    apply closed_regfile_apply_add_b_bv,\n    assumption\n  },\n  have H2 : closed_valty\n            ((env.add_b (env.add_bv η vn bvn) pn p)\n              ⟦valty.ival sz (sbitvec.var sz vn) (sbool.var pn)⟧),\n  { apply ival_closed },\n  rw HRF',\n  rw regfile.apply_update_comm,\n  rw closed_regfile_update_split,\n  split, assumption, assumption\nend\n\nlemma updatereg_closed: ∀ (ss ss':irstate_smt) (η:freevar.env)\n    regn sz vn pn bvn p\n    (HC:closed_irstate (η⟦ss⟧))\n    (HNOTIN1: vn ∉ η)\n    (HNOTIN2: pn ∉ η)\n    (HNOTEQ: vn ≠ pn)\n    (HS:ss' = irstate.updatereg irsem_smt ss regn\n        (irsem.valty.ival sz (sbitvec.var sz vn) (sbool.var pn))),\n  closed_irstate (((η.add_bv vn bvn).add_b pn p)⟦ss'⟧)\n:= begin\n  intros,\n  cases ss with sub srf,\n  cases ss' with sub' srf',\n  unfold freevar.env.replace at *,\n  rw ← irstate.setub_apply_to_values at *,\n  unfold irstate.getub at *,\n  simp at *,\n  unfold irstate.setub at *,\n  unfold irstate.apply_to_values at *,\n  rw closed_irstate_split,\n  rw closed_irstate_split at HC,\n  cases HC with HCUB HCRF,\n  unfold irstate.updatereg at HS,\n  simp at *,\n  injection HS,\n  subst h_1,\n  split,\n  {\n    have H0: closed_b ((env.add_bv η vn bvn)⟦sub'⟧),\n    {\n      apply closed_b_add_bv,\n      apply HCUB\n    },\n    apply closed_b_add_b,\n    { assumption }\n  },\n  {\n    apply regfile_update_ival_closed, assumption, assumption\n  }\nend\n\n-- encode\n\n\n-- Note that `irstate_equiv η⟦iss⟧ ise` does not imply\n-- closed_irstate η⟦iss⟧. It is because, for example,\n-- `b_equiv (sbool.and (sbool.var _) (sbool.ff)) ff` holds.\n-- Then why `encode iss ise` is needed? -> encode is\n-- the only way to relate ise and iss.\nlemma init_var_encode_intty: ∀ ise iss ise' iss' (sg sg':std_gen) η n t\n    (HENC: encode iss ise η) (HCLOSED: closed_irstate (η⟦iss⟧))\n    (HNOTIN1: get_free_sbitvec_name n ∉ η)\n    (HNOTIN2: get_free_sbool_name n ∉ η)\n    (HIE:(ise', sg') = create_init_var_exec n t (ise, sg))\n    (HIS:iss' = create_init_var_smt n t iss),\n  ∃ η', (encode iss' ise' η' ∧ closed_irstate (η'⟦iss'⟧) ∧\n         env.added2 η (get_free_sbitvec_name n)\n          (get_free_sbool_name n) η')\n:= begin\n  intros,\n  \n  unfold create_init_var_smt at HIS,\n  simp at HIS,\n\n  unfold create_init_var_exec at HIE,\n  simp at HIE,\n  generalize Hrbv':(get_rand_bv (get_sz_from_ty t) sg) = rbv',\n  cases rbv' with rbv' sg'',\n  rw Hrbv' at *,\n  unfold create_init_var_exec._match_2 at HIE,\n  generalize Hrb':(get_rand_bool sg'') = rb',\n  cases rb' with rb' sg''',\n  rw Hrb' at *,\n  unfold create_init_var_exec._match_1 at HIE,\n  injection HIE with HIE HIE_sg,\n  simp at HIE,\n\n  existsi ((η.add_b (get_free_sbool_name n) rb')\n             .add_bv (get_free_sbitvec_name n) rbv'.to_int),\n  split,\n  {\n    unfold encode,\n    rw HIS,\n    rw replace_updatereg,\n    rw HIE,\n    rw env.not_in_add_bv_irstate_comm,\n    rw env.not_in_add_b_irstate_comm,\n    rw HCLOSED, rw HCLOSED,\n    rw env.not_in_add_bv_valty_comm,\n    rw env.not_in_add_b_valty_comm,\n\n    unfold freevar.env.replace_valty,\n    -- making value..\n    unfold get_free_sbitvec,\n    rw env.not_in_replace_sbv,\n    rw env.add_b_replace_sbv,\n    rw env.empty_replace_sbv,\n    rw env.add_bv_replace_match,\n    -- making poison..\n    unfold get_free_sbool,\n    rw env.not_in_replace_sb,\n    rw env.add_b_replace_match,\n    rw env.replace_sb_of_bool,\n    apply irstate.updatereg_equiv,\n    {\n      intros,\n      cases rb',\n      { -- poison\n        apply val_equiv.poison_intty,\n        { constructor, constructor },\n        { refl },\n        { refl }\n      },\n      {\n        apply val_equiv.concrete_intty,\n        { constructor, constructor },\n        {\n          cases rbv',\n          rw sbitvec_of_int_const,\n          constructor\n        },\n        { refl }\n      }\n    },\n    { rw sbitvec_of_int_const, unfold equals_size, simp },\n    { apply HENC },\n    any_goals { assumption },\n    any_goals {\n      apply env.not_in_add_b,\n      apply get_free_name_diff,\n      assumption\n    },\n  },\n  split,\n  {\n    unfold closed_irstate,\n    intros,\n    rw HIS,\n    rw replace_updatereg,\n    rw env.not_in_add_bv_irstate_comm,\n    rw env.not_in_add_b_irstate_comm,\n    rw HCLOSED, rw HCLOSED,\n    rw env.not_in_add_bv_valty_comm,\n    rw env.not_in_add_b_valty_comm,\n\n    unfold freevar.env.replace_valty,\n    -- making value..\n    unfold get_free_sbitvec,\n    rw env.not_in_replace_sbv,\n    rw env.add_b_replace_sbv,\n    rw env.empty_replace_sbv,\n    rw env.add_bv_replace_match,\n    -- making poison..\n    unfold get_free_sbool,\n    rw env.not_in_replace_sb,\n    rw env.add_b_replace_match,\n    rw env.replace_sb_of_bool,\n\n    rw replace_updatereg,\n    unfold freevar.env.replace_valty,\n    rw env.replace_sbv_of_int,\n    rw env.replace_sb_of_bool,\n    rw HCLOSED,\n    any_goals { assumption },\n    apply env.not_in_add_b, apply get_free_name_diff, assumption,\n    apply env.not_in_add_b, apply get_free_name_diff, assumption\n  },\n  {\n    unfold env.added2,\n    split, {\n      intros n_1 H1 H2,\n      cases H2,\n      apply env.not_in_add_bv,\n      assumption,\n      apply env.not_in_add_b,\n      assumption,\n      rw env.not_in_split at H1,\n      rw env.not_in_split,\n      assumption\n    },\n    {\n      intros n_1 H,\n      unfold env.add_b,\n      unfold env.add_bv,\n      unfold has_mem.mem, simp,\n      cases H,\n      {\n        rw if_neg, rw if_neg, unfold has_mem.mem at H,\n        cases H, left, assumption, right, assumption,\n        intros H', rw H' at H, apply HNOTIN1, assumption,\n        intros H', rw H' at H, apply HNOTIN2, assumption,\n      },\n      {\n        cases H,\n        { right, rw if_pos, intros H0, cases H0, assumption },\n        { left, rw if_pos, intros H0, cases H0, assumption }\n      }\n    }\n  }\nend\n\ndef fv_smt_names (fvnames:list string) :=\n  fvnames.map get_free_sbitvec_name ++\n    fvnames.map get_free_sbool_name\n\nlemma init_state_encode_strong: ∀ (freevars:list (string × ty)) (sg sg':std_gen) ise iss\n    (HUNQ: list.unique $ freevars.map prod.fst)\n    (HIE:(ise, sg') = create_init_state_exec freevars sg)\n    (HIS:iss = create_init_state_smt freevars),\n  ∃ η, (encode iss ise η ∧ closed_irstate (η⟦iss⟧)\n      ∧ env.has_only η (fv_smt_names $ freevars.map prod.fst))\n:= begin\n  intros,\n  revert ise iss sg sg',\n  induction freevars,\n  {\n    intros,\n    unfold create_init_state_exec at HIE,\n    unfold create_init_state_smt at HIS,\n    simp at HIE,simp at HIS,\n    injection HIE with HIE _,\n    rw [HIS, HIE],\n    existsi (freevar.env.empty),\n    unfold encode, rw empty_replace_st,\n    constructor, constructor, constructor,\n    any_goals { constructor },\n    {\n      apply closed_irstate_empty\n    },\n    {\n      unfold fv_smt_names, simp,\n      unfold env.has_only, intros, split,\n      { intros H, cases H },\n      { intros H, have H := (env.not_in_empty name) H, cases H }\n    }\n  },\n  {\n    intros,\n    rename freevars_tl tl,\n    cases freevars_hd with vname vty,\n    have HEtmp: ∀ h t, create_init_state_exec (h::t) sg\n              = create_init_var_exec h.1 h.2 (create_init_state_exec t sg),\n    { intros, refl },\n    rw HEtmp at HIE,\n    clear HEtmp,\n    have HStmp: ∀ h t, create_init_state_smt (h::t)\n              = create_init_var_smt h.1 h.2 (create_init_state_smt t),\n    { intros, refl },\n    rw HStmp at HIS,\n    clear HStmp,\n    generalize HE0: create_init_state_exec tl sg = ise_sg0,\n    generalize HS0: create_init_state_smt tl = iss0,\n    rw HE0 at *,\n    rw HS0 at *,\n    cases ise_sg0 with ise0 sg0,\n    simp at HIE HIS,\n    have HEX: (∃ (η0 : env), encode iss0 ise0 η0 ∧ closed_irstate (η0⟦iss0⟧)\n                ∧ env.has_only η0 (fv_smt_names $ tl.map prod.fst)),\n    {\n      apply freevars_ih,\n      {\n        simp at HUNQ, cases HUNQ, assumption\n      }, apply (eq.symm HE0), refl\n    },\n    cases HEX with η0 HEX,\n    cases HEX with HEX1 HEX2,\n    cases HEX2 with HEX2 HEX3,\n    -- Now add a new variable to each irstate\n    have HUPDATED: ∃ η', (encode iss ise η' ∧ closed_irstate (η'⟦iss⟧) ∧\n         env.added2 η0 (get_free_sbitvec_name vname)\n                            (get_free_sbool_name vname) η'),\n    {\n      apply init_var_encode_intty,\n      apply HEX1,\n      apply HEX2,\n      { -- get_free_sbitvec_name vname ∉ η0\n        simp at HUNQ, cases HUNQ,\n        apply env.has_only_not_in,\n        { apply HEX3 },\n        { unfold fv_smt_names,\n          unfold get_free_sbitvec_name,\n          apply list.not_mem_append,\n          apply slist_prefix_notin, assumption,\n          apply slist_prefix_notin2 \"v_\" \"b_\" 'v' 'b', assumption,\n          { intros H0, cases H0 }, refl, refl\n        }\n      },\n      { -- get_free_sbool_name vname ∉ η0\n        simp at HUNQ, cases HUNQ,\n        apply env.has_only_not_in,\n        { apply HEX3 },\n        { unfold fv_smt_names,\n          unfold get_free_sbool_name,\n          apply list.not_mem_append,\n          apply slist_prefix_notin2 \"b_\" \"v_\" 'b' 'v', assumption,\n          { intros H0, cases H0 }, refl, refl,\n          apply slist_prefix_notin, assumption,\n        }\n      },\n      assumption, assumption\n    },\n    cases HUPDATED with η HUPDATED,\n    cases HUPDATED with HUPDATED Htmp, -- env.has_only_added2 (Honly) (Hadd2)\n    cases Htmp with HUPDATED2 HUPDATED3,\n    have Hη := env.has_only_added2 HEX3 HUPDATED3,\n    existsi η,\n    split, assumption, split, assumption,\n    {\n      unfold fv_smt_names, simp,\n      unfold fv_smt_names at Hη, simp at Hη,\n      rw ← list.cons_append,\n      rw ← env.has_only_shuffle (get_free_sbool_name vname),\n      simp,\n      rw env.has_only_shuffle2,\n      apply Hη\n    }\n  }\nend\n\ntheorem init_state_encode_prf: init_state_encode\n:= begin\n  unfold init_state_encode,\n  intros,\n  have H := init_state_encode_strong freevars sg sg' ise iss HUNQ\n            HIE HIS,\n  cases H with η H,\n  cases H, existsi η, assumption\nend\n\n-- Future work: theorem that `freevars.get` correctly returns all\n-- free variables.\n\nend spec", "meta": {"author": "microsoft", "repo": "AliveInLean", "sha": "34370c2c15aa69f010d97b8d38e9e1955e9e387d", "save_path": "github-repos/lean/microsoft-AliveInLean", "path": "github-repos/lean/microsoft-AliveInLean/AliveInLean-34370c2c15aa69f010d97b8d38e9e1955e9e387d/src/spec/initialstate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3104670196452017}}
{"text": "import geometry.manifold.instances.sphere\nimport to_mathlib.linear_algebra.finite_dimensional\nimport to_mathlib.analysis.inner_product_space.rotation\nimport global.gromov\nimport global.twist_one_jet_sec\n-- import interactive_expr\n-- set_option trace.filter_inst_type true\n\nnoncomputable theory\n\nopen metric finite_dimensional set function linear_map filter\nopen_locale manifold topology\n\nsection general\n\nvariables\n{E : Type*} [normed_add_comm_group E] [normed_space ℝ E]\n{H : Type*} [topological_space H] (I : model_with_corners ℝ E H)\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n\n{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E']\n{H' : Type*} [topological_space H'] (I' : model_with_corners ℝ E' H')\n{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n\n{F : Type*} [normed_add_comm_group F] [normed_space ℝ F]\n{G : Type*} [topological_space G] (J : model_with_corners ℝ F G)\n(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]\n\nlocal notation `TM` := tangent_space I\nlocal notation `TM'` := tangent_space I'\nlocal notation `HJ` := model_prod (model_prod H H') (E →L[ℝ] E')\nlocal notation `ψJ` := chart_at HJ\n\n/-- A map between manifolds is an immersion if it is differentiable and its differential at\nany point is injective. Note the formalized definition doesn't require differentiability.\nIf `f` is not differentiable at `m` then, by definition, `mfderiv I I' f m` is zero, which\nis not injective unless the source dimension is zero, which implies differentiability. -/\ndef immersion (f : M → M') : Prop := ∀ m, injective (mfderiv I I' f m)\n\nvariables (M M')\n\n/-- The relation of immersions for maps between two manifolds. -/\ndef immersion_rel : rel_mfld I M I' M' := {σ | injective σ.2}\n\nvariables {M M'}\n\n@[simp] lemma mem_immersion_rel_iff {σ : one_jet_bundle I M I' M'} :\n  σ ∈ immersion_rel I M I' M' ↔ injective (σ.2 : tangent_space I _ →L[ℝ] tangent_space I' _) :=\niff.rfl\n\n/-- A characterisation of the immersion relation in terms of a local chart. -/\nlemma mem_immersion_rel_iff' {σ σ' : one_jet_bundle I M I' M'} (hσ' : σ' ∈ (ψJ σ).source) :\n  σ' ∈ immersion_rel I M I' M' ↔ injective (ψJ σ σ').2 :=\nbegin\n  simp only [fiber_bundle.charted_space_chart_at] with mfld_simps at hσ',\n  simp_rw [mem_immersion_rel_iff],\n  rw [one_jet_bundle_chart_at_apply', in_coordinates'_eq],\n  simp_rw [continuous_linear_map.coe_comp', continuous_linear_equiv.coe_coe,\n    equiv_like.comp_injective, equiv_like.injective_comp],\n  exacts [hσ'.1.1, hσ'.1.2],\nend\n\nlemma chart_at_image_immersion_rel_eq {σ : one_jet_bundle I M I' M'} :\n  (ψJ σ) '' ((ψJ σ).source ∩ immersion_rel I M I' M') = (ψJ σ).target ∩ {q : HJ | injective q.2} :=\nlocal_equiv.is_image.image_eq $ λ σ' hσ', (mem_immersion_rel_iff' I I' hσ').symm\n\nvariables [finite_dimensional ℝ E] [finite_dimensional ℝ E']\n\nlemma immersion_rel_open :\n  is_open (immersion_rel I M I' M') :=\nbegin\n  simp_rw [charted_space.is_open_iff HJ (immersion_rel I M I' M'), chart_at_image_immersion_rel_eq],\n  refine λ σ, (ψJ σ).open_target.inter _,\n  convert is_open_univ.prod continuous_linear_map.is_open_injective,\n  { ext, simp, },\n  { apply_instance, },\n  { apply_instance, },\nend\n\n@[simp] lemma immersion_rel_slice_eq {m : M} {m' : M'} {p : dual_pair $ tangent_space I m}\n  {φ : tangent_space I m →L[ℝ] tangent_space I' m'} (hφ : injective φ) :\n  (immersion_rel I M I' M').slice ⟨(m, m'), φ⟩ p = ((ker p.π).map φ)ᶜ :=\nset.ext_iff.mpr $ λ w, p.injective_update_iff hφ\n\nlemma immersion_rel_ample (h : finrank ℝ E < finrank ℝ E') :\n  (immersion_rel I M I' M').ample :=\nbegin\n  rw [rel_mfld.ample_iff],\n  rintros ⟨⟨m, m'⟩, φ : tangent_space I m →L[ℝ] tangent_space I' m'⟩\n          (p : dual_pair (tangent_space I m)) (hφ : injective φ),\n  haveI : finite_dimensional ℝ (tangent_space I m) := (by apply_instance : finite_dimensional ℝ E),\n  have hcodim := two_le_rank_of_rank_lt_rank p.ker_pi_ne_top h φ.to_linear_map,\n  rw [immersion_rel_slice_eq I I' hφ],\n  exact ample_of_two_le_codim hcodim,\nend\n\n/-- This is lemma `lem:open_ample_immersion` from the blueprint. -/\nlemma immersion_rel_open_ample (h : finrank ℝ E < finrank ℝ E') :\n  is_open (immersion_rel I M I' M') ∧ (immersion_rel I M I' M').ample :=\n⟨immersion_rel_open I I', immersion_rel_ample I I' h⟩\n\nend general\n\nsection generalbis\n\nvariables\n{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E]\n{H : Type*} [topological_space H] (I : model_with_corners ℝ E H) [model_with_corners.boundaryless I]\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n\n{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E'] [finite_dimensional ℝ E']\n{H' : Type*} [topological_space H'] (I' : model_with_corners ℝ E' H') [model_with_corners.boundaryless I']\n{M' : Type*} [metric_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n\nvariables [finite_dimensional ℝ E] [finite_dimensional ℝ E']\n\nvariables\n  {EP : Type*} [normed_add_comm_group EP] [normed_space ℝ EP] [finite_dimensional ℝ EP]\n  {HP : Type*} [topological_space HP] {IP : model_with_corners ℝ EP HP} [model_with_corners.boundaryless IP]\n  {P : Type*} [topological_space P] [charted_space HP P] [smooth_manifold_with_corners IP P]\n  {C : set (P × M)} {ε : M → ℝ}\n\ninclude I I' M' IP\n\nvariables (I M I' M' IP P)\n\n/-- parametric h-principle for immersions. -/\ntheorem immersion_rel_satisfies_h_principle_with\n  [nonempty P] [t2_space P] [sigma_compact_space P] [locally_compact_space P]\n  [nonempty M] [t2_space M] [sigma_compact_space M] [locally_compact_space M]\n  [nonempty M'] [t2_space M'] [locally_compact_space M'] [sigma_compact_space M']\n  (h : finrank ℝ E < finrank ℝ E') (hC : is_closed C)\n  (hε_pos : ∀ x, 0 < ε x) (hε_cont : continuous ε) :\n  (immersion_rel I M I' M').satisfies_h_principle_with IP C ε :=\nby exact (immersion_rel_ample I I' h).satisfies_h_principle_with (immersion_rel_open I I')\n     hC hε_pos hε_cont\n\nend generalbis\n\nsection sphere_eversion\n\nvariables (E : Type*) [normed_add_comm_group E] [inner_product_space ℝ E] [fact (finrank ℝ E = 3)]\n\nlocal attribute [instance] fact_finite_dimensional_of_finrank_eq_succ\n\nlocal notation `𝕊²` := sphere (0 : E) 1\n\n/- Maybe the next two lemmas won't be used directly, but they should be done first as\nsanity checks. -/\n\nlemma immersion_inclusion_sphere : immersion (𝓡 2) 𝓘(ℝ, E) (λ x : 𝕊², (x : E)) :=\nmfderiv_coe_sphere_injective\n\nlemma immersion_antipodal_sphere : immersion (𝓡 2) 𝓘(ℝ, E) (λ x : 𝕊², -(x : E)) :=\nbegin\n  intros x,\n  change injective (mfderiv (𝓡 2) 𝓘(ℝ, E) (-(λ x : 𝕊², (x:E))) x),\n  rw mfderiv_neg,\n  exact neg_injective.comp (mfderiv_coe_sphere_injective x),\nend\n\n/- The relation of immersion of a two-sphere into its ambient Euclidean space. -/\nlocal notation `𝓡_imm` := immersion_rel (𝓡 2) 𝕊² 𝓘(ℝ, E) E\n\nvariables (ω : orientation ℝ E (fin 3))\n\nlemma smooth_bs :\n  smooth (𝓘(ℝ, ℝ).prod (𝓡 2)) 𝓘(ℝ, E)\n    (λ p : ℝ × 𝕊², ((1-p.1) • p.2 + p.1 • (-p.2) : E)) :=\nbegin\n  refine (cont_mdiff.smul _ _).add (cont_mdiff_fst.smul _),\n  { exact (cont_diff_const.sub cont_diff_id).cont_mdiff.comp cont_mdiff_fst },\n  { exact cont_mdiff_coe_sphere.comp cont_mdiff_snd },\n  { exact (cont_diff_neg.cont_mdiff.comp cont_mdiff_coe_sphere).comp cont_mdiff_snd },\nend\n\ndef formal_eversion_aux : family_one_jet_sec (𝓡 2) 𝕊² 𝓘(ℝ, E) E 𝓘(ℝ, ℝ) ℝ :=\nfamily_join\n  (smooth_bs E) $\n  family_twist\n    (drop (one_jet_ext_sec ⟨(coe : 𝕊² → E), cont_mdiff_coe_sphere⟩))\n    (λ p : ℝ × 𝕊², ω.rot (p.1, p.2))\n    begin\n      intros p,\n      have : smooth_at (𝓘(ℝ, ℝ × E)) 𝓘(ℝ, E →L[ℝ] E) ω.rot (p.1, p.2),\n      { refine (ω.cont_diff_rot _).cont_mdiff_at,\n        exact ne_zero_of_mem_unit_sphere p.2 },\n      refine this.comp p (smooth.smooth_at _),\n      exact smooth_fst.prod_mk (cont_mdiff_coe_sphere.comp smooth_snd),\n    end\n\n/-- A formal eversion of a two-sphere into its ambient Euclidean space. -/\ndef formal_eversion_aux2 : htpy_formal_sol 𝓡_imm :=\n{ is_sol' := λ t x, (ω.isometry_rot t x).injective.comp (mfderiv_coe_sphere_injective x),\n  .. formal_eversion_aux E ω }\n\ndef formal_eversion : htpy_formal_sol 𝓡_imm :=\n(formal_eversion_aux2 E ω).reindex ⟨smooth_step, cont_mdiff_iff_cont_diff.mpr smooth_step.smooth⟩\n\n@[simp]\nlemma formal_eversion_bs (t : ℝ) : (formal_eversion E ω t).bs =\n  λ x : 𝕊², (1 - smooth_step t : ℝ) • (x : E) + (smooth_step t : ℝ) • (-x : E) :=\nrfl\n\nlemma formal_eversion_zero (x : 𝕊²) : (formal_eversion E ω 0).bs x = x :=\nby simp\n\nlemma formal_eversion_one (x : 𝕊²) : (formal_eversion E ω 1).bs x = -x :=\nby simp\n\nlemma formal_eversion_hol_at_zero {t : ℝ} (ht : t < 1/4) :\n  (formal_eversion E ω t).to_one_jet_sec.is_holonomic :=\nbegin\n  intros x,\n  change mfderiv (𝓡 2) 𝓘(ℝ, E) (λ y : 𝕊², ((1 : ℝ) - smooth_step t) • (y:E) + smooth_step t • -y) x\n    = (ω.rot (smooth_step t, x)).comp (mfderiv (𝓡 2) 𝓘(ℝ, E) (λ y : 𝕊², (y:E)) x),\n  simp_rw [smooth_step.of_lt ht, ω.rot_zero, continuous_linear_map.id_comp],\n  congr,\n  ext y,\n  simp [smooth_step.of_lt ht],\nend\n\nlemma formal_eversion_hol_at_one {t : ℝ} (ht : 3/4 < t) :\n  (formal_eversion E ω t).to_one_jet_sec.is_holonomic :=\nbegin\n  intros x,\n  change mfderiv (𝓡 2) 𝓘(ℝ, E) (λ y : 𝕊², ((1:ℝ) - smooth_step t) • (y:E) + smooth_step t • -y) x\n    = (ω.rot (smooth_step t, x)).comp (mfderiv (𝓡 2) 𝓘(ℝ, E) (λ y : 𝕊², (y:E)) x),\n  transitivity mfderiv (𝓡 2) 𝓘(ℝ, E) (-(λ y : 𝕊², (y:E))) x,\n  { congr' 2,\n    ext y,\n    simp [smooth_step.of_gt ht], },\n  ext v,\n  simp_rw [mfderiv_neg, continuous_linear_map.coe_comp', comp_app, continuous_linear_map.neg_apply,\n    smooth_step.of_gt ht],\n  rw [ω.rot_one],\n  rw [← range_mfderiv_coe_sphere x],\n  exact linear_map.mem_range_self _ _,\nend\n\nlemma formal_eversion_hol_near_zero_one : ∀ᶠ (s : ℝ × 𝕊²) near {0, 1} ×ˢ univ,\n  (formal_eversion E ω s.1).to_one_jet_sec.is_holonomic_at s.2 :=\nbegin\n  have : (Iio (1/4 : ℝ) ∪ Ioi (3/4)) ×ˢ (univ : set 𝕊²) ∈ 𝓝ˢ (({0, 1} : set ℝ) ×ˢ univ),\n  { refine ((is_open_Iio.union is_open_Ioi).prod is_open_univ).mem_nhds_set.mpr _,\n    rintro ⟨s, x⟩ ⟨hs, hx⟩,\n    refine ⟨_, mem_univ _⟩,\n    simp_rw [mem_insert_iff, mem_singleton_iff] at hs,\n    rcases hs with rfl|rfl,\n    { exact or.inl (show (0 : ℝ) < 1 / 4, by norm_num) },\n    { exact or.inr (show (3 / 4 : ℝ) < 1, by norm_num) } },\n  refine eventually_of_mem this _,\n  rintro ⟨t, x⟩ ⟨ht|ht, hx⟩,\n  { exact formal_eversion_hol_at_zero E ω ht x },\n  { exact formal_eversion_hol_at_one E ω ht x }\nend\n\ntheorem sphere_eversion : ∃ f : ℝ → 𝕊² → E,\n  (cont_mdiff (𝓘(ℝ, ℝ).prod (𝓡 2)) 𝓘(ℝ, E) ∞ (uncurry f)) ∧\n  (f 0 = λ x, x) ∧\n  (f 1 = λ x, -x) ∧\n  ∀ t, immersion (𝓡 2) 𝓘(ℝ, E) (f t) :=\nbegin\n  classical,\n  let ω : orientation ℝ E (fin 3) :=\n    ((std_orthonormal_basis _ _).reindex $ fin_congr (fact.out _ : finrank ℝ E = 3)).to_basis.orientation,\n  have rankE := fact.out (finrank ℝ E = 3),\n  haveI : finite_dimensional ℝ E :=\n    finite_dimensional_of_finrank_eq_succ rankE,\n  have ineq_rank : finrank ℝ (euclidean_space ℝ (fin 2)) < finrank ℝ E := by simp [rankE],\n  let ε : 𝕊² → ℝ := λ x, 1,\n  have hε_pos : ∀ x, 0 < ε x,\n    from λ x, zero_lt_one,\n  have hε_cont : continuous ε := continuous_const,\n  haveI : nontrivial E := nontrivial_of_finrank_eq_succ (fact.out _ : finrank ℝ E = 3),\n  haveI : nonempty ↥(sphere 0 1 : set E) :=\n    (normed_space.sphere_nonempty.mpr zero_le_one).to_subtype,\n  rcases (immersion_rel_satisfies_h_principle_with (𝓡 2) 𝕊² 𝓘(ℝ, E) E 𝓘(ℝ, ℝ) ℝ ineq_rank\n    ((finite.is_closed (by simp : ({0, 1} : set ℝ).finite)).prod is_closed_univ) hε_pos hε_cont).bs\n    (formal_eversion E ω) (formal_eversion_hol_near_zero_one E ω) with ⟨f, h₁, h₂, -, h₅⟩,\n  have := h₂.nhds_set_forall_mem,\n  refine ⟨f, h₁, _, _, h₅⟩,\n  { ext x,\n    rw [this (0, x) (by simp)],\n    convert formal_eversion_zero E ω x },\n  { ext x,\n    rw [this (1, x) (by simp)],\n    convert formal_eversion_one E ω x },\nend\n\n-- The next instance will be used in the main file\ninstance (n : ℕ) : fact (finrank ℝ (euclidean_space ℝ $ fin n) = n) :=\n⟨finrank_euclidean_space_fin⟩\n\n-- The next notation will be used in the main file\nnotation `ℝ^`n:max := euclidean_space ℝ (fin n)\nend sphere_eversion\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/global/immersion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7217431943271998, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.3104559213997218}}
{"text": "\nimport tactic\nimport logic.basic\nimport category.serial\nimport category.liftable\nimport category.liftable.serial\n\nuniverses u v\n\ninductive put_m' (α : Type u)\n| pure : α → put_m'\n| write : unsigned → (unit → put_m') → put_m'\n\ndef put_m'.bind {α β} : put_m' α → (α → put_m' β) → put_m' β\n| (put_m'.pure x)   f := f x\n| (put_m'.write w f) g := put_m'.write w $ λ u, put_m'.bind (f u) g\n\ninstance : monad put_m' :=\n{ pure := λ α, put_m'.pure\n, bind := @put_m'.bind }\n\ninstance : is_lawful_monad put_m' :=\nby { refine { .. }; intros;\n       try { refl };\n       induction x;\n       try { refl };\n     { dsimp [put_m'.bind] at *, congr, simp * } }\n\n@[reducible]\ndef put_m := put_m' punit\n\ndef put_m.eval : put_m → list unsigned\n| (put_m'.pure x) := []\n| (put_m'.write w f) := w :: put_m.eval (f punit.star)\n\ninductive get_m : Type u → Type (u+1)\n| fail {α} : get_m α\n| pure {α} : α → get_m α\n| read {α} : (unsigned → get_m α) → get_m α\n| loop {α β γ : Type u} : (β → unsigned → get_m (α ⊕ β)) → (α → get_m γ) → β → get_m γ\n\ndef get_m.bind : Π {α β}, get_m α → (α → get_m β) → get_m β\n| _ _ (get_m.fail) _ := get_m.fail\n| _ _ (get_m.pure x)   f := f x\n| _ _ (get_m.read f) g := get_m.read $ λ w, get_m.bind (f w) g\n| _ _ (get_m.loop f g x₀) h := get_m.loop f (λ r, get_m.bind (g r) h) x₀\n\ndef get_m.map : Π {α β : Type u}, (α → β) → get_m α → get_m β\n| _ _ _ (get_m.fail) := get_m.fail\n| _ _ f (get_m.pure x) := get_m.pure $ f x\n| _ _ f (get_m.read g) := get_m.read $ λ w, get_m.map f (g w)\n| _ _ h (get_m.loop f g x₀) := get_m.loop f (λ r, get_m.map h (g r)) x₀\n\ninstance : functor get_m.{u} :=\n{ map := @get_m.map }\n\ndef get_m.seq {α β : Type u} : Π (f : get_m (α → β)) (x : get_m α), get_m β :=\nλ (f : get_m (α → β)) (x : get_m α), get_m.bind f (λ f, f <$> x)\n\n-- instance : applicative get_m :=\n-- { to_functor := get_m.functor\n-- , pure := λ α, get_m.pure\n-- , seq := @get_m.seq }\nopen function\n\ninstance : is_lawful_functor.{u} get_m :=\nby { constructor; intros;\n     dsimp [(<$>),get_m.seq];\n     induction x;\n     try { refl };\n     simp [get_m.map,*]; ext }\n\ninstance : monad get_m :=\n{ to_functor := get_m.functor\n, pure := @get_m.pure\n, bind := @get_m.bind }\n\ninstance : is_lawful_monad get_m.{u} :=\n{ to_is_lawful_functor := get_m.is_lawful_functor,\n  bind_assoc := by { intros, dsimp [(>>=)],\n                     induction x; try { refl }; simp [get_m.bind,*], },\n  bind_pure_comp_eq_map := by { intros, dsimp [(>>=),(<$>)],\n                                induction x; try {refl}; simp [get_m.bind,get_m.map,*], },\n  map_pure := by intros; refl,\n  pure_seq_eq_map := by { intros, dsimp [(>>=),(<$>)],\n                          induction x; try {refl}; simp [get_m.bind,get_m.map,*], },\n  pure_bind := by intros; refl }\n\ndef get_m.eval {α} : list unsigned → get_m α → option α\n| [] (get_m.pure x) := pure x\n| [] _  := none\n| (w :: ws) (get_m.read f) := get_m.eval ws (f w)\n| (w :: ws) (get_m.loop f g x₀) :=\n  get_m.eval ws $\n  f x₀ w >>= @sum.rec _ _ (λ _, get_m α) g (get_m.loop f g)\n| (w :: ws) _ := none\n\n\ndef read_write : Π {α}, get_m.{u} α → put_m.{u} → option α\n| ._ (get_m.pure x) (put_m'.pure _) := some x\n| _ _ (put_m'.pure _) := none\n| ._ (get_m.read f) (put_m'.write w g) := read_write (f w) (g punit.star)\n| α (@get_m.loop α' β γ f g x₀) (put_m'.write w h) :=\n  read_write\n    (f x₀ w >>= @sum.rec α' β (λ _, get_m α) g (get_m.loop f g))\n    (h punit.star)\n| _ _ (put_m'.write w g) := none\n\ndef read_write' : Π {α}, get_m α → put_m → option (α × put_m)\n| _ (get_m.read f) (put_m'.write w g) := read_write' (f w) (g punit.star)\n| α (@get_m.loop α' β γ f g x₀) (put_m'.write w h) :=\n  read_write'\n    (f x₀ w >>= @sum.rec α' β (λ _, get_m α) g (get_m.loop f g))\n    (h punit.star)\n-- | _ (get_m.pure x) m@(put_m'.write w g) := some (x,m)\n| _ (get_m.pure x) m := some (x,m)\n| _ _ (put_m'.pure _) := none\n| _ (get_m.fail) (put_m'.write _ _) := none\n-- | _ _ m := none\n\nlemma read_read_write_write {α} (x : get_m α) (m : put_m) (i : α) :\n  read_write x m = some i ↔ read_write' x m = some (i,pure punit.star) :=\nbegin\n  induction m generalizing x;\n  cases x; casesm* punit; simp [read_write,read_write',prod.ext_iff,pure,*],\nend\n\ndef pipeline {α} (x : get_m α) (y : α → put_m) (i : α) : option α :=\nread_write x (y i)\n\ninfix ` -<< `:60  := read_write\ninfix ` -<<< `:60  := read_write'\ninfix ` <-< `:60  := pipeline\n\nlemma eq_star (x : punit) : x = punit.star :=\nby cases x; refl\n\n-- inductive agree : Π {α} (x : α), get_m α → put_m → put_m → Prop\n-- | pure {α} (x : α) (m : put_m) : agree x (get_m.pure x) m m\n-- | read_write {α} (x : α) (w : unsigned)\n--   (f : unsigned → get_m α) (g : punit → put_m) (m : put_m) :\n--   agree x (f w) (g punit.star) m →\n--   agree x (get_m.read f) (put_m'.write w g) m\n-- | loop_write {α} (x : α) {β γ} (σ₀ σ₁ : β) (w : unsigned)\n--   (f : β → unsigned → get_m (γ ⊕ β)) (f' : γ → get_m α)\n--   (g : punit → put_m) (m m' : put_m) :\n--   agree (sum.inr σ₁) (f σ₀ w) (g punit.star) m' →\n--   agree x (get_m.loop σ₁ f f') m' m →\n--   agree x (get_m.loop σ₀ f f') (put_m'.write w g) m\n-- | loop_exit_write {α} (x : α) {β γ} (σ₀ : β) (r : γ) (w : unsigned)\n--   (f : β → unsigned → get_m (γ ⊕ β)) (f' : γ → get_m α)\n--   (g : punit → put_m) (m m' : put_m) :\n--   agree (sum.inl r) (f σ₀ w) (g punit.star) m' →\n--   agree x (f' r) m' m →\n--   agree x (get_m.loop σ₀ f f') (put_m'.write w g) m\n\n-- lemma agree_spec {α} (g : get_m α) (m : put_m) (x : α) :\n--   agree x g m (put_m'.pure punit.star) ↔ g -<< m = some x :=\n-- begin\n--   split; intro h,\n--   { cases h,\n--     refl, simp [read_write], }\n-- end\n\n-- lemma loop_bind {α β γ} (i : β)\n--       (body : β → unsigned → get_m (α ⊕ β)) (f₀ : α → get_m γ) :\n--   get_m.loop i body f₀ = get_m.read (body i) >>= _ := _\n\nlemma read_write_loop_bind {α β γ φ : Type u} (i : α)\n      (body : α → unsigned → get_m (φ ⊕ α))\n      (f₀ : φ → get_m β) (f₁ : β → get_m γ)\n      (m : punit → put_m) (w : unsigned) :\n  (get_m.loop body f₀ i >>= f₁) -<<< put_m'.write w m =\n  (body i w >>= @sum.rec φ α (λ _, get_m β) f₀ (get_m.loop body f₀) >>= f₁) -<<< m punit.star :=\nbegin\n  rw bind_assoc,\n  simp [(>>=),get_m.bind,read_write'],\n  congr, ext, cases x; simp; refl,\nend\n\n-- lemma read_write_left_overs_bind {α} (i : α)\n--       (x₀ : get_m α)\n--       (x₁ x₂ : put_m) :\n-- x₀ -<<< x₁ = some (i,x₂) →\n\nlemma fail_read_write {α} (x₁ : put_m) :\n  get_m.fail -<<< x₁ = @none (α × put_m) :=\nby cases x₁; refl\n\nlemma pure_read_write {α} (x₁ : put_m) (i : α) :\n  get_m.pure i -<<< x₁ = some (i, x₁) :=\nby cases x₁; refl\n\nlemma read_write_left_overs_bind {α} (f : punit → put_m) (i : α)\n      (x₀ : get_m α)\n      (x₁ x₂ : put_m) :\n  x₀ -<<< x₁ = some (i,x₂) → x₀ -<<< (x₁ >>= f) = some (i,x₂ >>= f) :=\nbegin\n  induction x₁ generalizing x₀ x₂,\n  cases x₀; simp [(>>=),put_m'.bind,read_write',pure_read_write],\n  { intros, subst x₂, tauto },\n  cases x₀; simp [(>>=),put_m'.bind,read_write'],\n  { intros, substs x₂ i, split; refl },\n  { apply x₁_ih, },\n  { apply x₁_ih, },\nend\n\nlemma option_eq_forall_some {α} (x y : option α) :\n  x = y ↔ ∀ z, x = some z ↔ y = some z :=\nbegin\n  split; intro h, { rw h; intro, refl },\n  { cases y, cases x, refl,\n    symmetry, rw ← h, rw h, },\nend\n\nlemma read_write_weakening {α}\n  (x₀ x₁ : put_m) (y₀ y₁ : get_m α)\n  (h : y₀ -<<< x₀ = y₁ -<<< x₁) :\n  y₀ -<< x₀ = y₁ -<< x₁ :=\nbegin\n  rw option_eq_forall_some,\n  intro, simp [read_read_write_write,h],\nend\n\nlemma read_write_mono' {α β} (i : α)\n      (x₀ : get_m α) (f₀ : α → get_m β)\n      (x₁ x₂ : put_m)\n      (h : x₀ -<<< x₁ = some (i,x₂)) :\n  (x₀ >>= f₀) -<<< x₁ = f₀ i -<<< x₂ :=\nbegin\n  -- simp [(>>=)],\n  induction x₁ generalizing x₀ f₀;\n    try { cases x₀; cases h },\n  { simp [(>>=),read_write',get_m.bind] },\n  { cases x₀; try { cases h },\n    simp [(>>=),read_write',get_m.bind] at h ⊢,\n    simp [(>>=),read_write',get_m.bind] at h ⊢,\n    { apply x₁_ih, assumption },\n    simp [read_write_loop_bind,x₁_ih],\n    rw [x₁_ih _ _ _ h], }\nend\n\nlemma read_write_mono {α β} {i : α}\n      {x₀ : get_m α} {f₀ : α → get_m β}\n      {x₁ : put_m} {f₁ : punit → put_m}\n      (h : x₀ -<< x₁ = some i) :\n  (x₀ >>= f₀) -<< (x₁ >>= f₁) = f₀ i -<< f₁ punit.star :=\nbegin\n  apply read_write_weakening,\n  apply read_write_mono',\n  rw [read_read_write_write] at h,\n  replace h := read_write_left_overs_bind f₁ _ _ _ _ h,\n  simp [h],\nend\n\nlemma read_write_mono_left {α β} {i : α}\n      {x₀ : get_m α} {f₀ : α → get_m β}\n      {x₁ : put_m}\n      (h : x₀ -<< x₁ = some i) :\n  (x₀ >>= f₀) -<< x₁ = f₀ i -<< pure punit.star :=\nby rw ← read_write_mono h; simp\n\nlemma read_write_eq_eval_eval {α}\n      (x₀ : get_m α) (x₁ : put_m)  :\n  x₀ -<< x₁ = x₀.eval x₁.eval :=\nbegin\n  induction x₁ generalizing x₀; cases x₀; try { refl },\n  simp! [*,read_write],\n  simp! [*,read_write],\nend\nopen ulift\n\nlemma get_m.fold_bind {α β} (x : get_m α) (f : α → get_m β) :\n  get_m.bind x f = x >>= f := rfl\n\nlemma map_read_write {α β} (f : α → β) (x : get_m α) (y : put_m) :\n  (f <$> x) -<< y = f <$> (x -<< y) :=\nbegin\n  rw [← bind_pure_comp_eq_map,← bind_pure_comp_eq_map],\n  symmetry,\n  simp [(>>=)],\n  induction y generalizing x,\n  { cases x; refl },\n  { cases x; simp [read_write]; try { refl };\n    simp [get_m.bind,read_write,y_ih],\n    congr' 1, cases h : x_a x_a_2 y_a, refl,\n    cases a; refl,\n    dsimp [(>>=),get_m.bind],\n    congr, ext, simp [get_m.fold_bind],\n    rw bind_assoc, congr, ext z, cases z; refl,\n    simp [get_m.fold_bind], rw bind_assoc, congr, ext x,\n    cases x; refl, }\nend\n\ndef sum_ulift (α β : Type u) : (α ⊕ β) ≃ (ulift.{v} α ⊕ ulift.{v} β) :=\n(equiv.sum_congr equiv.ulift.symm equiv.ulift.symm)\n\n-- def get_m.up : Π {α : Type u} {β : Type.{max u v}} (Heq : α ≃ β), get_m α → get_m β\n-- | _ _ Heq (get_m.pure x) := get_m.pure $ Heq x\n-- | _ _ Heq (get_m.fail) := get_m.fail\n-- | _ _ Heq (get_m.read f) := get_m.read (λ w, get_m.up Heq (f w))\n-- | _ β' Heq (@get_m.loop α β γ f g x) :=\n--   get_m.loop\n--     (λ a b, get_m.up (sum_ulift α β) (f (down.{v} a) b))\n--     (λ w, get_m.up Heq (g $ down w))\n--     (up.{v} x)\n\ndef get_m.up : Π {α : Type u} {β : Type.{max u v}} (Heq : α → β), get_m α → get_m β :=\nλ α β f x, (@get_m.rec_on (λ α _, Π β, (α → β) → get_m β) α x\n(λ α β f, get_m.fail)\n(λ α x β f, get_m.pure $ f x)\n(λ α next get_m_up β f, get_m.read $ λ w, get_m_up w _ f)\n(λ α β γ body rest x₀ get_m_up₀ get_m_up₁ β' f,\n  get_m.loop\n    (λ a b, get_m_up₀ (down a) b (ulift.{v} α ⊕ ulift.{v} β)\n                     (sum_ulift α β))\n    (λ r, get_m_up₁ (down r) _ f)\n    (up x₀)) β f)\n\nsection eqns\n\nvariables {α β' γ : Type u} {β : Type.{max u v}} (Heq : α → β) (x : get_m α)\n\nvariables {i : α} {f : unsigned → get_m α}\nvariables {f' : β' → unsigned → get_m (γ ⊕ β')}\nvariables {g' : γ → get_m α} {j : β'}\n\n@[simp] lemma get_m.up.eqn_1 : get_m.up Heq (get_m.pure i) = get_m.pure (Heq i) := rfl\n@[simp] lemma get_m.up.eqn_2 : get_m.up Heq (get_m.fail) = get_m.fail := rfl\n@[simp] lemma get_m.up.eqn_3 : get_m.up Heq (get_m.read f) = get_m.read (λ w, get_m.up Heq (f w)) := rfl\n@[simp] lemma get_m.up.eqn_4 :\n  get_m.up Heq (get_m.loop f' g' j) =\n  get_m.loop\n    (λ a b, get_m.up (sum_ulift γ β') (f' (down.{v} a) b))\n    (λ w, get_m.up Heq (g' $ down w))\n    (up.{v} j) := rfl\n\nend eqns\n\ndef put_m.up {α : Type u} {β : Type v} (Heq : α → β) : put_m' α → put_m' β\n| (put_m'.pure x) := put_m'.pure $ Heq x\n| (put_m'.write w f) := put_m'.write w $ λ u, put_m.up $ f u\n\ninstance : liftable1 put_m'.{u} put_m'.{v} :=\n{ up := λ α β (eq : α ≃ β) x, put_m.up eq x\n, down := λ α β (eq : α ≃ β) x, put_m.up eq.symm x\n, down_up := by intros; induction x; simp [put_m.up,*]\n, up_down := by intros; induction x; simp [put_m.up,*]  }\n\nopen pliftable (up')\n\nlemma up_bind {α β : Type u} {β' : Type (max u v)} (x : get_m α) (g : α → get_m β) (f : β → β') :\n  (x >>= g).up f = x.up up.{v} >>= (λ i : ulift α, (g $ down i).up f) :=\nbegin\n  dsimp [(>>=)],\n  induction x generalizing f g; try { refl };\n    simp [get_m.bind,*]\nend\n\nlemma equiv_bind {m} [monad m] [is_lawful_monad m] {α α' β}\n  (Heq : α ≃ α') (x : m α) (f : α → m β) :\n  x >>= f = (Heq <$> x) >>= f ∘ Heq.symm :=\nby simp [(∘)] with functor_norm\n\ndef sum.map {α α' β β'} (f : α → α') (g : β → β') : α ⊕ β → α' ⊕ β'\n| (sum.inr x) := sum.inr $ g x\n| (sum.inl x) := sum.inl $ f x\n\ndef equiv.ulift_sum {α β} : (ulift $ α ⊕ β) ≃ (ulift α ⊕ ulift β) :=\n{ to_fun := λ x, sum.map up up (down x),\n  inv_fun := λ x, up $ sum.map down down x,\n  right_inv := by intro; casesm* [_ ⊕ _,ulift _]; refl,\n  left_inv := by intro; casesm* [_ ⊕ _,ulift _]; refl }\n\nlemma map_get_m_up {α : Type u} {β γ} (x : get_m α) (f : α → β) (g : β → γ) :\n  g <$> get_m.up f x = get_m.up (g ∘ f) x :=\nbegin\n  dsimp [(<$>)],\n  induction x; simp [get_m.map,*]; refl,\nend\n\nlemma up_read_write {α : Type u} {α' : Type (max u v)} (x : get_m α) (y : put_m) (f : α ≃ α') :\n  x.up f -<< up' y = liftable1.up f (x -<< y) :=\nbegin\n  dsimp [up',liftable1.up],\n  induction y generalizing x f,\n  cases x; simp; refl,\n  cases x; simp [up',liftable.up',liftable1.up,read_write,put_m.up,*], refl, refl, refl,\n  rw [read_write,← y_ih,up_bind],\n    apply congr,\n    { apply congr_arg, rw equiv_bind (@equiv.ulift_sum.{u u v v v} x_α x_β) ,\n      congr,\n      { rw map_get_m_up, congr, ext, cases x; refl },\n      simp [(∘)], ext, cases x;\n      dsimp [equiv.ulift_sum,sum.map], refl,\n      cases x, refl, apply_instance },\n    congr,\nend\n\nlemma up_read_write' {α : Type u} {α' : Type (max u v)}\n  {x : get_m α} {y : put_m} (f : α → α') (f' : α ≃ α')\n  (h : ∀ i, f i = f' i) :\n  x.up f -<< up' y = liftable1.up f' (x -<< y) :=\nbegin\n  rw ← up_read_write, congr, ext, apply h\nend\n", "meta": {"author": "cipher1024", "repo": "serialean", "sha": "47881e4a6bc0a62cd68520564610b75f8a4fef2c", "save_path": "github-repos/lean/cipher1024-serialean", "path": "github-repos/lean/cipher1024-serialean/serialean-47881e4a6bc0a62cd68520564610b75f8a4fef2c/src/data/serial/medium.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.31031124935820775}}
{"text": "/-\nCopyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Seul Baek\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.omega.coeffs\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-\nNormalized linear integer arithmetic terms.\n-/\n\nnamespace omega\n\n\n/-- Shadow syntax of normalized terms. The first element\n    represents the constant term and the list represents\n    the coefficients. -/\ndef term :=\n  ℤ × List ℤ\n\nnamespace term\n\n\n/-- Evaluate a term using the valuation v. -/\n@[simp] def val (v : ℕ → ℤ) : term → ℤ :=\n  sorry\n\n@[simp] def neg : term → term :=\n  sorry\n\n@[simp] def add : term → term → term :=\n  sorry\n\n@[simp] def sub : term → term → term :=\n  sorry\n\n@[simp] def mul (i : ℤ) : term → term :=\n  sorry\n\n@[simp] def div (i : ℤ) : term → term :=\n  sorry\n\ntheorem val_neg {v : ℕ → ℤ} {t : term} : val v (neg t) = -val v t := sorry\n\n@[simp] theorem val_sub {v : ℕ → ℤ} {t1 : term} {t2 : term} : val v (sub t1 t2) = val v t1 - val v t2 := sorry\n\n@[simp] theorem val_add {v : ℕ → ℤ} {t1 : term} {t2 : term} : val v (add t1 t2) = val v t1 + val v t2 := sorry\n\n@[simp] theorem val_mul {v : ℕ → ℤ} {i : ℤ} {t : term} : val v (mul i t) = i * val v t := sorry\n\ntheorem val_div {v : ℕ → ℤ} {i : ℤ} {b : ℤ} {as : List ℤ} : i ∣ b → (∀ (x : ℤ), x ∈ as → i ∣ x) → val v (div i (b, as)) = val v (b, as) / i := sorry\n\n/-- Fresh de Brujin index not used by any variable ocurring in the term -/\ndef fresh_index (t : term) : ℕ :=\n  list.length (prod.snd t)\n\ndef to_string (t : term) : string :=\n  list.foldr (fun (_x : ℕ × ℤ) => sorry) (to_string (prod.fst t)) (list.enum (prod.snd t))\n\nprotected instance has_to_string : has_to_string term :=\n  has_to_string.mk to_string\n\nend term\n\n\n/-- Fresh de Brujin index not used by any variable ocurring in the list of terms -/\ndef terms.fresh_index : List term → ℕ :=\n  sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/omega/term.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.4455295350395727, "lm_q1q2_score": 0.31006998895907395}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  -- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.\n  assume h1 : G.colorable 2,\n  -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.\n  have h2 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h1],\n  -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.\n  have h3 : ∀ (v w : V), v ∈ A ∧ w ∈ A → ¬ G.adj v w, from by auto [h2],\n  have h4 : ∀ (v w : V), v ∈ B ∧ w ∈ B → ¬ G.adj v w, from by auto [h2],\n  -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n  have h5 : ∀ (v w : V), G.adj v w → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A), from by auto [h2, h3, h4],\n  have h6 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h7 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h8 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h9 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h10 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h11 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h12 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h13 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h14 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h15 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h16 : ∃ (A B : set V), (∀ (v : V), v ∈ A ∨ v ∈ B) ∧ (∀ (v : V), v ∈ A → ¬ v ∈ B) ∧ (∀ (v : V), v ∈ B → ¬ v ∈ A) ∧ (∀ (v w : V), v ∈ A ∧ w ∈ B → G.adj v w) ∧ (∀ (v w : V), v ∈ B ∧ w ∈ A → G.adj v w), from by auto [h2],\n  have h17 : ∃ (A B : set V), (∀ (v : V), v\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  -- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.\n  have h1 : (G.colorable 2) → (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from\n  begin\n    assume h2 : (G.colorable 2),\n    -- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.\n    have h3 : ∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B), from\n    begin\n      -- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.\n      have h4 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (∀ (a b : A), ¬ (G.adj a b)), from\n      begin\n        assume (A B : Type*) (h : (A ⊕ B) = V) (h5 : (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B))),\n        assume (a b : A),\n        assume h6 : (G.adj a b),\n        have h7 : (cast (congr_arg _ h) (complete_bipartite_graph A B)).adj a b, from by auto [h5, h6],\n        have h8 : (complete_bipartite_graph A B).adj a b, from by auto [h7, cast_adj],\n        have h9 : (complete_bipartite_graph A B).adj a b = ff, from by auto [complete_bipartite_graph.adj_iff],\n        have h10 : ff = ff, from by auto [h9, h8],\n        show false, from by auto [h10],\n      end,\n      have h11 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (∀ (a b : B), ¬ (G.adj a b)), from\n      begin\n        assume (A B : Type*) (h : (A ⊕ B) = V) (h12 : (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B))),\n        assume (a b : B),\n        assume h13 : (G.adj a b),\n        have h14 : (cast (congr_arg _ h) (complete_bipartite_graph A B)).adj a b, from by auto [h12, h13],\n        have h15 : (complete_bipartite_graph A B).adj a b, from by auto [h14, cast_adj],\n        have h16 : (complete_bipartite_graph A B).adj a b = ff, from by auto [complete_bipartite_graph.adj_iff],\n        have h17 : ff = ff, from by auto [h16, h15],\n        show false, from by auto [h17],\n      end,\n      -- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n      have h18 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (∀ (a b : V), (G.adj a b) → (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b'))), from\n      begin\n        assume (A B : Type*) (h : (A ⊕ B) = V) (h19 : (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B))),\n        assume (a b : V),\n        assume h20 : (G.adj a b),\n        have h21 : (cast (congr_arg _ h) (complete_bipartite_graph A B)).adj a b, from by auto [h19, h20],\n        have h22 : (complete_bipartite_graph A B).adj a b, from by auto [h21, cast_adj],\n        have h23 : (complete_bipartite_graph A B).adj a b = tt, from by auto [complete_bipartite_graph.adj_iff],\n        have h24 : tt = tt, from by auto [h23, h22],\n        have h25 : (∃ (a' b' : A), (a = a') ∧ (b = b')), from by auto [h24, complete_bipartite_graph.adj_iff],\n        have h26 : (∃ (a' b' : B), (a = a') ∧ (b = b')), from by auto [h24, complete_bipartite_graph.adj_iff],\n        show (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b')), from by auto [h25, h26],\n      end,\n      have h27 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (∀ (a b : V), (G.adj a b) → (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b'))), from\n      begin\n        assume (A B : Type*) (h : (A ⊕ B) = V) (h28 : (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B))),\n        assume (a b : V),\n        assume h29 : (G.adj a b),\n        have h30 : (cast (congr_arg _ h) (complete_bipartite_graph A B)).adj a b, from by auto [h28, h29],\n        have h31 : (complete_bipartite_graph A B).adj a b, from by auto [h30, cast_adj],\n        have h32 : (complete_bipartite_graph A B).adj a b = tt, from by auto [complete_bipartite_graph.adj_iff],\n        have h33 : tt = tt, from by auto [h32, h31],\n        have h34 : (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b')), from by auto [h33, complete_bipartite_graph.adj_iff],\n        show (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b')), from by auto [h34],\n      end,\n      have h35 : ∀ (A B : Type*) (h : (A ⊕ B) = V), (G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) → (∀ (a b : V), (G.adj a b) → (∃ (a' b' : A), (a = a') ∧ (b = b')) ∨ (∃ (a' b' : B), (a = a') ∧ (b = b'))), from\n      begin\n        assume (A B : Type*) (h : (A ⊕ B) = V) (h36 : (G ≤ cast (congr_arg _ h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  -- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.\n  have h1 : (G.colorable 2) → (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)), from by auto [colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def, simple_graph.colorable_def,\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) ↔ (∃ (A B : Type*) (h : (A ⊕ B) = V), G ≤ cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7826624688140728, "lm_q2_score": 0.39606816627404173, "lm_q1q2_score": 0.3099876888347042}}
{"text": "import Mathlib.Algebra.Group.Basic\n\nsection UnitOperations\n\n  instance : Add PUnit := ⟨λ x y => PUnit.unit⟩\n  instance : Sub PUnit := ⟨λ x y => PUnit.unit⟩\n  instance : Mul PUnit := ⟨λ x y => PUnit.unit⟩\n  instance : Neg PUnit := ⟨λ x => PUnit.unit⟩\n  \n  instance : Zero PUnit := ⟨PUnit.unit⟩\n\n  -- Multiplication on `α` is not necessary but we do not want for example `ℕ*()` to compile\n  instance [Mul α] : HMul α PUnit PUnit := ⟨λ a x => PUnit.unit⟩\n  instance [Mul α] : SMul α PUnit := ⟨λ a x => PUnit.unit⟩\n \nend UnitOperations\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Mathlib/Data/PUnit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.3099561945852113}}
{"text": "/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\nPorted by: Scott Morrison\n-/\n\nimport Mathlib.Tactic.Linarith.Elimination\nimport Mathlib.Tactic.Linarith.Parsing\n\n/-!\n# Deriving a proof of false\n\n`linarith` uses an untrusted oracle to produce a certificate of unsatisfiability.\nIt needs to do some proof reconstruction work to turn this into a proof term.\nThis file implements the reconstruction.\n\n## Main declarations\n\nThe public facing declaration in this file is `proveFalseByLinarith`.\n-/\n\nopen Lean Elab Tactic Meta\n\nnamespace Qq\n\n/-- Typesafe conversion of `n : ℕ` to `Q($α)`. -/\ndef ofNatQ (α : Q(Type $u)) (_ : Q(Semiring $α)) (n : ℕ) : Q($α) :=\n  match n with\n  | 0 => q(0 : $α)\n  | 1 => q(1 : $α)\n  | k+2 =>\n    have lit : Q(ℕ) := mkRawNatLit n\n    let k : Q(ℕ) := mkRawNatLit k\n    let _x : Q(Nat.AtLeastTwo $lit) :=\n      (q(instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (n := $k)) : Expr)\n    q(OfNat.ofNat $lit)\n\n/-- Analogue of `inferTypeQ`, but that gets universe levels right for our application. -/\n-- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Using.20.60QQ.60.20when.20you.20only.20have.20an.20.60Expr.60/near/303349037\ndef inferTypeQ' (e : Expr) : MetaM ((u : Level) × (α : Q(Type $u)) × Q($α)) := do\n  let α ← inferType e\n  let .sort (.succ u) ← instantiateMVars (← whnf (← inferType α))\n    | throwError \"not a type{indentExpr α}\"\n  pure ⟨u, α, e⟩\n\nend Qq\n\nnamespace Linarith\n\nopen Ineq\nopen Qq\n\n/-! ### Auxiliary functions for assembling proofs -/\n\n/-- A typesafe version of `mulExpr`. -/\ndef mulExpr' (n : ℕ) {α : Q(Type $u)} (inst : Q(Semiring $α)) (e : Q($α)) : Q($α) :=\nif n = 1 then e else\n  let n := ofNatQ α inst n\n  q($n * $e)\n\n/--\n`mulExpr n e` creates an `Expr` representing `n*e`.\nWhen elaborated, the coefficient will be a native numeral of the same type as `e`.\n-/\ndef mulExpr (n : ℕ) (e : Expr) : MetaM Expr := do\n  let ⟨_, α, e⟩ ← inferTypeQ' e\n  let inst : Q(Semiring $α) ← synthInstanceQ q(Semiring $α)\n  return mulExpr' n inst e\n\n/-- A type-safe analogue of `addExprs`. -/\ndef addExprs' {α : Q(Type $u)} (_inst : Q(AddMonoid $α)) : List Q($α) → Q($α)\n| []   => q(0)\n| h::t => go h t\n  where\n  /-- Inner loop for `addExprs'`. -/\n  go (p : Q($α)) : List Q($α) → Q($α)\n  | [] => p\n  | [q] => q($p + $q)\n  | q::t => go q($p + $q) t\n\n/-- `addExprs L` creates an `Expr` representing the sum of the elements of `L`, associated left. -/\ndef addExprs : List Expr → MetaM Expr\n| [] => return q(0) -- This may not be of the intended type; use with caution.\n| L@(h::_) => do\n  let ⟨_, α, _⟩ ← inferTypeQ' h\n  let inst : Q(AddMonoid $α) ← synthInstanceQ q(AddMonoid $α)\n  -- This is not type safe; we just assume all the `Expr`s in the tail have the same type:\n  return addExprs' inst L\n\n/--\nIf our goal is to add together two inequalities `t1 R1 0` and `t2 R2 0`,\n`addIneq R1 R2` produces the strength of the inequality in the sum `R`,\nalong with the name of a lemma to apply in order to conclude `t1 + t2 R 0`.\n-/\ndef addIneq : Ineq → Ineq → (Name × Ineq)\n| eq, eq => (``Linarith.eq_of_eq_of_eq, eq)\n| eq, le => (``Linarith.le_of_eq_of_le, le)\n| eq, lt => (``Linarith.lt_of_eq_of_lt, lt)\n| le, eq => (``Linarith.le_of_le_of_eq, le)\n| le, le => (``add_nonpos, le)\n| le, lt => (``add_lt_of_le_of_neg, lt)\n| lt, eq => (``Linarith.lt_of_lt_of_eq, lt)\n| lt, le => (``add_lt_of_neg_of_le, lt)\n| lt, lt => (``Left.add_neg, lt)\n\n/--\n`mkLTZeroProof coeffs pfs` takes a list of proofs of the form `tᵢ Rᵢ 0`,\npaired with coefficients `cᵢ`.\nIt produces a proof that `∑cᵢ * tᵢ R 0`, where `R` is as strong as possible.\n-/\ndef mkLTZeroProof : List (Expr × ℕ) → MetaM Expr\n| [] => throwError \"no linear hypotheses found\"\n| [(h, c)] => do\n     let (_, t) ← mkSingleCompZeroOf c h\n     return t\n| ((h, c)::t) => do\n     let (iq, h') ← mkSingleCompZeroOf c h\n     let (_, t) ← t.foldlM (λ pr ce => step pr.1 pr.2 ce.1 ce.2) (iq, h')\n     return t\nwhere\n  /--\n  `step c pf npf coeff` assumes that `pf` is a proof of `t1 R1 0` and `npf` is a proof\n  of `t2 R2 0`. It uses `mkSingleCompZeroOf` to prove `t1 + coeff*t2 R 0`, and returns `R`\n  along with this proof.\n  -/\n  step (c : Ineq) (pf npf : Expr) (coeff : ℕ) : MetaM (Ineq × Expr) := do\n    let (iq, h') ← mkSingleCompZeroOf coeff npf\n    let (nm, niq) := addIneq c iq\n    return (niq, ←mkAppM nm #[pf, h'])\n\n/-- If `prf` is a proof of `t R s`, `leftOfIneqProof prf` returns `t`. -/\ndef leftOfIneqProof (prf : Expr) : MetaM Expr := do\n  let (t, _) ← getRelSides (← inferType prf)\n  return t\n\n/-- If `prf` is a proof of `t R s`, `typeOfIneqProof prf` returns the type of `t`. -/\ndef typeOfIneqProof (prf : Expr) : MetaM Expr := do\n  inferType (← leftOfIneqProof prf)\n\n/--\n`mkNegOneLtZeroProof tp` returns a proof of `-1 < 0`,\nwhere the numerals are natively of type `tp`.\n-/\ndef mkNegOneLtZeroProof (tp : Expr) : MetaM Expr := do\n  let zero_lt_one ← mkAppOptM ``zero_lt_one #[tp, none, none, none, none, none]\n  mkAppM `neg_neg_of_pos #[zero_lt_one]\n\n/--\n`addNegEqProofs l` inspects the list of proofs `l` for proofs of the form `t = 0`. For each such\nproof, it adds a proof of `-t = 0` to the list.\n-/\ndef addNegEqProofs : List Expr → MetaM (List Expr)\n| [] => return []\n| (h::tl) => do\n  let (iq, t) ← parseCompAndExpr (← inferType h)\n  match iq with\n  | Ineq.eq => do\n    let nep := mkAppN (← mkAppM `Iff.mpr #[← mkAppOptM ``neg_eq_zero #[none, none, t]]) #[h]\n    let tl ← addNegEqProofs tl\n    return h::nep::tl\n  | _ => return h :: (← addNegEqProofs tl)\n\n/--\n`proveEqZeroUsing tac e` tries to use `tac` to construct a proof of `e = 0`.\n-/\ndef proveEqZeroUsing (tac : TacticM Unit) (e : Expr) : MetaM Expr := do\n  let ⟨u, α, e⟩ ← inferTypeQ' e\n  let _h : Q(Zero $α) ← synthInstanceQ q(Zero $α)\n  synthesizeUsing q($e = 0) tac\n\n/-! #### The main method -/\n\n/--\n`proveFalseByLinarith` is the main workhorse of `linarith`.\nGiven a list `l` of proofs of `tᵢ Rᵢ 0`,\nit tries to derive a contradiction from `l` and use this to produce a proof of `False`.\n\nAn oracle is used to search for a certificate of unsatisfiability.\nIn the current implementation, this is the Fourier Motzkin elimination routine in\n`Elimination.lean`, but other oracles could easily be swapped in.\n\nThe returned certificate is a map `m` from hypothesis indices to natural number coefficients.\nIf our set of hypotheses has the form  `{tᵢ Rᵢ 0}`,\nthen the elimination process should have guaranteed that\n1.\\ `∑ (m i)*tᵢ = 0`,\nwith at least one `i` such that `m i > 0` and `Rᵢ` is `<`.\n\nWe have also that\n2.\\ `∑ (m i)*tᵢ < 0`,\nsince for each `i`, `(m i)*tᵢ ≤ 0` and at least one is strictly negative.\nSo we conclude a contradiction `0 < 0`.\n\nIt remains to produce proofs of (1) and (2). (1) is verified by calling the `discharger` tactic\nof the `LinarithConfig` object, which is typically `ring`. We prove (2) by folding over the\nset of hypotheses.\n-/\ndef proveFalseByLinarith (cfg : LinarithConfig) : MVarId → List Expr → MetaM Expr\n| _, [] => throwError \"no args to linarith\"\n| g, l@(h::_) => do\n    trace[linarith.detail] \"Beginning work in `proveFalseByLinarith`.\"\n    -- for the elimination to work properly, we must add a proof of `-1 < 0` to the list,\n    -- along with negated equality proofs.\n    let l' ← addNegEqProofs l\n    trace[linarith.detail] \"... finished `addNegEqProofs`.\"\n    let inputs := (← mkNegOneLtZeroProof (← typeOfIneqProof h))::l'.reverse\n    trace[linarith.detail] \"... finished `mkNegOneLtZeroProof`.\"\n    let (comps, max_var) ← linearFormsAndMaxVar cfg.transparency inputs\n    trace[linarith.detail] \"... finished `linearFormsAndMaxVar`.\"\n    trace[linarith.detail] \"{comps}\"\n    let oracle := cfg.oracle.getD FourierMotzkin.produceCertificate\n    -- perform the elimination and fail if no contradiction is found.\n    let certificate : Std.HashMap Nat Nat ← try\n      oracle comps max_var\n    catch e =>\n      trace[linarith] m!\"{e.toMessageData}\"\n      throwError \"linarith failed to find a contradiction\"\n    trace[linarith] m!\"linarith has found a contradiction: {certificate.toList}\"\n    let enum_inputs := inputs.enum\n    -- construct a list pairing nonzero coeffs with the proof of their corresponding comparison\n    let zip := enum_inputs.filterMap fun ⟨n, e⟩ => (certificate.find? n).map (e, ·)\n    let mls ← zip.mapM fun ⟨e, n⟩ => do mulExpr n (← leftOfIneqProof e)\n    -- `sm` is the sum of input terms, scaled to cancel out all variables.\n    let sm ← addExprs mls\n    -- let sm ← instantiateMVars sm\n    trace[linarith] \"The expression\\n  {sm}\\nshould be both 0 and negative\"\n    -- we prove that `sm = 0`, typically with `ring`.\n    let sm_eq_zero ← proveEqZeroUsing cfg.discharger sm\n    -- we also prove that `sm < 0`\n    let sm_lt_zero ← mkLTZeroProof zip\n    -- this is a contradiction.\n    let pftp ← inferType sm_lt_zero\n    let ⟨_, nep, _⟩ ← g.rewrite pftp sm_eq_zero\n    let pf' ← mkAppM ``Eq.mp #[nep, sm_lt_zero]\n    mkAppM ``Linarith.lt_irrefl #[pf']\n\nend Linarith\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/Linarith/Verification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3099465658120455}}
{"text": "import Lean\n\ndef checkWithMkMatcherInput (matcher : Lean.Name) : Lean.MetaM Unit :=\n  Lean.Meta.Match.withMkMatcherInput matcher fun input => do\n  let res ← Lean.Meta.Match.mkMatcher input\n  let origMatcher ← Lean.getConstInfo matcher\n  if not <| input.matcherName == matcher then\n    throwError \"matcher name not reconstructed correctly: {matcher} ≟ {input.matcherName}\"\n  \n  let lCtx ← Lean.getLCtx\n  let fvars ← Lean.collectFVars {} res.matcher\n  let closure ← Lean.Meta.Closure.mkLambda (fvars.fvarSet.toList.toArray.map lCtx.get!) res.matcher\n\n  let origTy := origMatcher.value! \n  let newTy ← closure\n  if not <| ←Lean.Meta.isDefEq origTy newTy then\n    throwError \"matcher {matcher} does not round-trip correctly:\\n{origTy} ≟ {newTy}\"\n  \ndef f (xs : List Nat) : List Bool :=\nxs.map fun\n  | 0 => true\n  | _ => false\n#eval checkWithMkMatcherInput ``f.match_1\n\n#eval f [1, 2, 0, 2]\n\ntheorem ex1 : f [1, 0, 2] = [false, true, false] :=\nrfl\n\n#check f\n\nset_option pp.raw true\nset_option pp.raw.maxDepth 10\nset_option trace.Elab.step true in\ndef g (xs : List Nat) : List Bool :=\nxs.map <| by {\n  intro\n    | 0 => exact true\n    | _ => exact false\n}\n\ntheorem ex2 : g [1, 0, 2] = [false, true, false] :=\nrfl\n\ntheorem ex3 {p q r : Prop} : p ∨ q → r → (q ∧ r) ∨ (p ∧ r) :=\nby intro\n | Or.inl hp, h => { apply Or.inr; apply And.intro; assumption; assumption }\n | Or.inr hq, h => { apply Or.inl; exact ⟨hq, h⟩ }\n#eval checkWithMkMatcherInput ``ex3.match_1\n\ninductive C\n| mk₁ : Nat → C\n| mk₂ : Nat → Nat → C\n\ndef C.x : C → Nat\n| C.mk₁ x   => x\n| C.mk₂ x _ => x\n#eval checkWithMkMatcherInput ``C.x.match_1\n\ndef head : {α : Type} → List α → Option α\n| _, a::as => some a\n| _, _     => none\n#eval checkWithMkMatcherInput ``head.match_1\n\ntheorem ex4 : head [1, 2] = some 1 :=\nrfl\n\ndef head2 : {α : Type} → List α → Option α :=\n@fun\n  | _, a::as => some a\n  | _, _     => none\n\ntheorem ex5 : head2 [1, 2] = some 1 :=\nrfl\n\ndef head3 {α : Type} (xs : List α) : Option α :=\nlet rec aux : {α : Type} → List α → Option α\n  | _, a::as => some a\n  | _, _     => none;\naux xs\n\ntheorem ex6 : head3 [1, 2] = some 1 :=\nrfl\n\ninductive Vec.{u} (α : Type u) : Nat → Type u\n| nil : Vec α 0\n| cons {n} (head : α) (tail : Vec α n) : Vec α (n+1)\n\ndef Vec.mapHead1 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _,   nil,       nil,       f => none\n| _, cons a as, cons b bs,   f => some (f a b)\n#eval checkWithMkMatcherInput ``Vec.mapHead1.match_1\n\ndef Vec.mapHead2 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, @cons _ n a as, cons b bs,   f => some (f a b)\n#eval checkWithMkMatcherInput ``Vec.mapHead2.match_1\n\ndef Vec.mapHead3 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, cons (tail := as) (head := a), cons b bs,   f => some (f a b)\n\ninductive Foo\n| mk₁ (x y z w : Nat)\n| mk₂ (x y z w : Nat)\n\ndef Foo.z : Foo → Nat\n| mk₁ (z := z) .. => z\n| mk₂ (z := z) .. => z\n#eval checkWithMkMatcherInput ``Foo.z.match_1\n\n#eval (Foo.mk₁ 10 20 30 40).z\n\ntheorem ex7 : (Foo.mk₁ 10 20 30 40).z = 30 :=\nrfl\n\ndef Foo.addY? : Foo × Foo → Option Nat\n| (mk₁ (y := y₁) .., mk₁ (y := y₂) ..) => some (y₁ + y₂)\n| _ => none\n#eval checkWithMkMatcherInput ``Foo.addY?.match_1\n\n#eval Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40)\n\ntheorem ex8 : Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40) = some 22 :=\nrfl\n\ninstance {α} : Inhabited (Sigma fun m => Vec α m) :=\n⟨⟨0, Vec.nil⟩⟩\n\npartial def filter {α} (p : α → Bool) : {n : Nat} → Vec α n → Sigma fun m => Vec α m\n| _, Vec.nil        => ⟨0, Vec.nil⟩\n| _, Vec.cons x xs  => match p x, filter p xs with\n  | true,  ⟨_, ys⟩ => ⟨_, Vec.cons x ys⟩\n  | false, ys      => ys\n#eval checkWithMkMatcherInput ``filter.match_1\n\ninductive Bla\n| ofNat  (x : Nat)\n| ofBool (x : Bool)\n\ndef Bla.optional? : Bla → Option Nat\n| ofNat x  => some x\n| ofBool _ => none\n#eval checkWithMkMatcherInput ``Bla.optional?.match_1\n\ndef Bla.isNat? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch b.optional? with\n| some y => some ⟨y, rfl⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat?.match_1\n\ndef foo (b : Bla) : Option Nat := b.optional?\ntheorem fooEq (b : Bla) : foo b = b.optional? :=\nrfl\n\ndef Bla.isNat2? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch h:foo b with\n| some y => some ⟨y, Eq.trans (fooEq b).symm h⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat2?.match_1\n\ndef foo2 (x : Nat) : Nat :=\nmatch x, rfl : (y : Nat) → x = y → Nat with\n| 0,   h => 0\n| x+1, h => 1\n#eval checkWithMkMatcherInput ``foo2.match_1\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/match1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213368305399, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3098597454774432}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.opposites\nimport category_theory.limits.lattice\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.terminal\nimport category_theory.full_subcategory\nimport category_theory.limits.shapes.regular_mono\nimport category_theory.closed.cartesian\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.over\nimport category_theory.monad.adjunction\nimport category_theory.currying\nimport category_theory.adjunction.fully_faithful\nimport category_theory.skeletal\nimport over\n\nuniverses v v₂ u u₂\n\nnoncomputable theory\nnamespace category_theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u} [category.{v} C] {X Y Z : C}\nvariables {D : Type u₂} [category.{v₂} D]\n\n/--\nThe subobject category as a full subcategory of the over category. In particular this isn't\nskeletal, so it's not a partial order. The quotient is taken in `subq` instead, it's useful to be\nable to work with both.\nIt's reducible for now to get instances to happen quickly, marked semireducible again later.\n-/\n@[derive [category, λ t, has_coe t (over X)]]\ndef sub (X : C) := {f : over X // mono f.hom}\n/-- The inclusion arrow from subobjects to the over category. -/\ndef forget_sub (X : C) : sub X ⥤ over X := full_subcategory_inclusion _\n@[simp]\nlemma forget_sub_obj_left {f} : ((forget_sub X).obj f).left = f.val.left := rfl\n/-- Convenience notation for the underlying arrow of a subobject. -/\nabbreviation sub.arrow (f : sub X) : _ ⟶ X := ((forget_sub X).obj f).hom\n@[simp]\nlemma forget_sub_obj_hom {f} : ((forget_sub X).obj f).hom = f.arrow := rfl\n\ninstance : full (forget_sub X) := full_subcategory.full _\ninstance : faithful (forget_sub X) := full_subcategory.faithful _\n\ninstance sub_mono (f : sub X) : mono f.arrow := f.property\n\n/-- The subobject category is a thin category, which makes defining its skeleton easy. -/\ninstance is_thin {X : C} (f g : sub X) : subsingleton (f ⟶ g) :=\n⟨begin\n  intros h₁ h₂,\n  ext1,\n  erw [← cancel_mono g.arrow, over.w h₁, over.w h₂],\nend⟩\n\n@[reassoc] lemma sub.w {f g : sub X} (k : f ⟶ g) : k.left ≫ g.arrow = f.arrow := over.w _\n\n/-- Convience constructor for a morphism in the subobject category. -/\nabbreviation sub.hom_mk {f g : sub X} (h : f.val.left ⟶ g.val.left) (w : h ≫ g.arrow = f.arrow) : f ⟶ g :=\nover.hom_mk h w\n\ndef sub.iso_mk {f g : sub X} (h : f.val.left ≅ g.val.left) (w : h.hom ≫ g.arrow = f.arrow) : f ≅ g :=\n{ hom := sub.hom_mk h.hom w,\n  inv := sub.hom_mk h.inv (by rw [h.inv_comp_eq, w]) }\n\n@[derive [partial_order, category]]\ndef subq (X : C) := thin_skeleton (sub X)\n\n@[simps]\ndef sub.mk' {X A : C} (f : A ⟶ X) [hf : mono f] : sub X := { val := over.mk f, property := hf }\n@[simp] lemma sub_mk'_arrow {X A : C} (f : A ⟶ X) [hf : mono f] : (sub.mk' f).arrow = f := rfl\n\nabbreviation subq.mk {X A : C} (f : A ⟶ X) [mono f] : subq X := (to_thin_skeleton _).obj (sub.mk' f)\n\n@[simps]\ndef restrict_to_sub {Y : D} (F : over Y ⥤ over X)\n  (h : ∀ (f : sub Y), mono (F.obj ((forget_sub Y).obj f)).hom) : sub Y ⥤ sub X :=\n{ obj := λ f, ⟨_, h f⟩,\n  map := λ _ _ k, (forget_sub X).preimage ((forget_sub Y ⋙ F).map k), }\n\ndef restrict_to_sub_iso {Y : D} {F₁ F₂ : over Y ⥤ over X} (h₁ h₂) (i : F₁ ≅ F₂) :\n  restrict_to_sub F₁ h₁ ≅ restrict_to_sub F₂ h₂ :=\nfully_faithful_cancel_right (forget_sub X) (iso_whisker_left (forget_sub Y) i)\n\ndef restrict_to_sub_comp {X Z : C} {Y : D} (F : over X ⥤ over Y) (G : over Y ⥤ over Z) (h₁ h₂) :\n  restrict_to_sub F h₁ ⋙ restrict_to_sub G h₂ ≅ restrict_to_sub (F ⋙ G) (λ f, h₂ ⟨_, h₁ f⟩) :=\nfully_faithful_cancel_right (forget_sub _) (iso.refl _)\n\ndef restrict_to_sub_id :\n  restrict_to_sub (𝟭 (over X)) (λ f, f.2) ≅ 𝟭 _ :=\nfully_faithful_cancel_right (forget_sub _) (iso.refl _)\n\n@[simp]\nlemma restrict_comm (F : over Y ⥤ over X)\n  (h : ∀ (f : sub Y), mono (F.obj ((forget_sub Y).obj f)).hom) :\n  restrict_to_sub F h ⋙ forget_sub X = forget_sub Y ⋙ F :=\nrfl\n\ndef lower_sub {Y : D} (F : sub Y ⥤ sub X) : subq Y ⥤ subq X := thin_skeleton.map F\n\nlemma lower_sub_iso (F₁ F₂ : sub X ⥤ sub Y) (h : F₁ ≅ F₂) : lower_sub F₁ = lower_sub F₂ :=\nthin_skeleton.map_iso_eq h\n\ndef lower_sub₂ (F : sub X ⥤ sub Y ⥤ sub Z) : subq X ⥤ subq Y ⥤ subq Z :=\nthin_skeleton.map₂ F\n\n@[simp]\nlemma lower_comm (F : sub Y ⥤ sub X) :\n  to_thin_skeleton _ ⋙ lower_sub F = F ⋙ to_thin_skeleton _ :=\nrfl\n\ndef sub.pullback [has_pullbacks.{v} C] (f : X ⟶ Y) : sub Y ⥤ sub X :=\nrestrict_to_sub (real_pullback f)\nbegin\n  intro g,\n  apply @pullback.snd_of_mono _ _ _ _ _ _ _ _ _,\n  change mono g.arrow,\n  apply_instance,\nend\n\ndef sub.pullback_comp [has_pullbacks.{v} C] (f : X ⟶ Y) (g : Y ⟶ Z) :\n  sub.pullback (f ≫ g) ≅ sub.pullback g ⋙ sub.pullback f :=\nrestrict_to_sub_iso _ _ (pullback_comp _ _) ≪≫ (restrict_to_sub_comp _ _ _ _).symm\n\ndef sub.pullback_id [has_pullbacks.{v} C] :\n  sub.pullback (𝟙 X) ≅ 𝟭 _ :=\nrestrict_to_sub_iso _ _ pullback_id ≪≫ restrict_to_sub_id\n\n@[simp] lemma sub.pullback_obj_left [has_pullbacks.{v} C] (f : X ⟶ Y) (g : sub Y) :\n(↑((sub.pullback f).obj g) : over X).left = pullback g.arrow f :=\nrfl\n\n@[simp] lemma sub.pullback_obj_arrow [has_pullbacks.{v} C] (f : X ⟶ Y) (g : sub Y) :\n((sub.pullback f).obj g).arrow = pullback.snd :=\nrfl\n\ndef subq.pullback [has_pullbacks.{v} C] (f : X ⟶ Y) : subq Y ⥤ subq X :=\nlower_sub (sub.pullback f)\n\nlemma subq.pullback_id [has_pullbacks.{v} C] (x : subq X) : (subq.pullback (𝟙 X)).obj x = x :=\nbegin\n  apply quotient.induction_on' x,\n  intro f,\n  apply quotient.sound,\n  exact ⟨sub.pullback_id.app f⟩,\nend\nlemma subq.pullback_comp [has_pullbacks.{v} C] (f : X ⟶ Y) (g : Y ⟶ Z) (x : subq Z) :\n  (subq.pullback (f ≫ g)).obj x = (subq.pullback f).obj ((subq.pullback g).obj x) :=\nbegin\n  apply quotient.induction_on' x,\n  intro t,\n  apply quotient.sound,\n  refine ⟨(sub.pullback_comp _ _).app t⟩,\nend\n\nattribute [instance] mono_comp\n\ndef sub.post (f : X ⟶ Y) [mono f] : sub X ⥤ sub Y :=\nrestrict_to_sub (over.map f)\n(λ g, by apply mono_comp g.arrow f)\n\ndef sub.post_comp (f : X ⟶ Y) (g : Y ⟶ Z) [mono f] [mono g] :\n  sub.post (f ≫ g) ≅ sub.post f ⋙ sub.post g :=\nrestrict_to_sub_iso _ _ (over_map_comp _ _) ≪≫ (restrict_to_sub_comp _ _ _ _).symm\n\ndef sub.post_id : sub.post (𝟙 X) ≅ 𝟭 _ :=\nrestrict_to_sub_iso _ _ over_map_id ≪≫ restrict_to_sub_id\n\n@[simp] lemma sub.post_obj_left (f : X ⟶ Y) [mono f] (g : sub X) :\n(↑((sub.post f).obj g) : over Y).left = g.val.left :=\nrfl\n\n@[simp]\nlemma sub.post_obj_arrow (f : X ⟶ Y) [mono f] (g : sub X) :\n((sub.post f).obj g).arrow = g.arrow ≫ f := rfl\n\ninstance sub.full_post (f : X ⟶ Y) [mono f] : full (sub.post f) :=\n{ preimage := λ g h e,\n  begin\n    refine sub.hom_mk e.left _,\n    rw [← cancel_mono f, assoc],\n    apply sub.w e,\n  end }\n\ninstance sub.faithful_post (f : X ⟶ Y) [mono f] : faithful (sub.post f) := {}.\n\ndef subq.post (f : X ⟶ Y) [mono f] : subq X ⥤ subq Y :=\nlower_sub (sub.post f)\n\nlemma subq.post_id (x : subq X) : (subq.post (𝟙 X)).obj x = x :=\nbegin\n  apply quotient.induction_on' x,\n  intro f,\n  apply quotient.sound,\n  exact ⟨sub.post_id.app f⟩,\nend\nlemma subq.post_comp (f : X ⟶ Y) (g : Y ⟶ Z) [mono f] [mono g] (x : subq X) :\n  (subq.post (f ≫ g)).obj x = (subq.post g).obj ((subq.post f).obj x) :=\nbegin\n  apply quotient.induction_on' x,\n  intro t,\n  apply quotient.sound,\n  refine ⟨(sub.post_comp _ _).app t⟩,\nend\n\n@[simps]\ndef sub.image [has_images C] : over X ⥤ sub X :=\n{ obj := λ f, sub.mk' (image.ι f.hom),\n  map := λ f g k,\n  begin\n    apply (forget_sub X).preimage _,\n    apply over.hom_mk _ _,\n    refine image.lift {I := image _, m := image.ι g.hom, e := k.left ≫ factor_thru_image g.hom},\n    apply image.lift_fac,\n  end }\n\ndef image_forget_adj [has_images C] : sub.image ⊣ forget_sub X :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ f g,\n  { to_fun := λ k,\n    begin\n      apply over.hom_mk (factor_thru_image f.hom ≫ k.left) _,\n      change (factor_thru_image f.hom ≫ k.left) ≫ _ = f.hom,\n      rw [assoc, over.w k],\n      apply image.fac\n    end,\n    inv_fun := λ k,\n    begin\n      refine over.hom_mk _ _,\n      refine image.lift {I := g.val.left, m := g.arrow, e := k.left, fac' := over.w k},\n      apply image.lift_fac,\n    end,\n    left_inv := λ k, subsingleton.elim _ _,\n    right_inv := λ k,\n    begin\n      ext1,\n      change factor_thru_image _ ≫ image.lift _ = _,\n      rw [← cancel_mono g.arrow, assoc, image.lift_fac, image.fac f.hom],\n      exact (over.w k).symm,\n    end } }\n\ninstance [has_images C] : is_right_adjoint (forget_sub X) :=\n{ left := sub.image, adj := image_forget_adj }\n\ninstance sub.reflective [has_images C] : reflective (forget_sub X) := {}.\n\ndef forget_image [has_images C] : forget_sub X ⋙ sub.image ≅ 𝟭 (sub X) :=\nas_iso (adjunction.counit image_forget_adj)\n\ndef sub.exists [has_images C] (f : X ⟶ Y) : sub X ⥤ sub Y :=\nforget_sub _ ⋙ over.map f ⋙ sub.image\n\ndef subq.exists [has_images C] (f : X ⟶ Y) : subq X ⥤ subq Y :=\nlower_sub (sub.exists f)\n\ninstance sub.faithful_pullback (f : X ⟶ Y) [has_pullbacks C] : faithful (sub.pullback f) := {}.\n\ninstance sub.faithful_exists (f : X ⟶ Y) [has_images C] : faithful (sub.exists f) := {}.\n\ndef exists_iso_post [has_images C] (f : X ⟶ Y) [mono f] : sub.exists f ≅ sub.post f :=\nnat_iso.of_components\nbegin\n  intro Z,\n  suffices : (forget_sub _).obj ((sub.exists f).obj Z) ≅ (forget_sub _).obj ((sub.post f).obj Z),\n    apply preimage_iso this,\n  apply over_iso _ _,\n  apply image_mono_iso_source (Z.arrow ≫ f),\n  apply image_mono_iso_source_hom_self,\nend\nbegin\n  intros Z₁ Z₂ g,\n  ext1,\n  change image.lift ⟨_, _, _, _⟩ ≫ (image_mono_iso_source (Z₂.arrow ≫ f)).hom =\n         (image_mono_iso_source (Z₁.arrow ≫ f)).hom ≫ g.left,\n  rw [← cancel_mono (Z₂.arrow ≫ f), assoc, assoc, sub.w_assoc g, image_mono_iso_source_hom_self,\n      image_mono_iso_source_hom_self],\n  apply image.lift_fac,\nend\n\n/-- post is adjoint to pullback for monos -/\ndef sub.pull_post_adj (f : X ⟶ Y) [mono f] [has_pullbacks C] : sub.post f ⊣ sub.pullback f :=\nadjunction.restrict_fully_faithful (forget_sub X) (forget_sub Y) (radj f) (iso.refl _) (iso.refl _)\n\ndef thin_skeleton.lower_adjunction\n  [∀ (X Y : C), subsingleton (X ⟶ Y)] [∀ (X Y : D), subsingleton (X ⟶ Y)]\n  (R : D ⥤ C) (L : C ⥤ D) (h : L ⊣ R) :\n  thin_skeleton.map L ⊣ thin_skeleton.map R :=\nadjunction.mk_of_unit_counit\n{ unit :=\n  { app := λ X,\n    begin\n      letI := is_isomorphic_setoid C,\n      refine quotient.rec_on_subsingleton X (λ x, hom_of_le ⟨h.unit.app x⟩),\n      -- TODO: make quotient.rec_on_subsingleton' so the letI isn't needed\n    end },\n  counit :=\n  { app := λ X,\n    begin\n      letI := is_isomorphic_setoid D,\n      refine quotient.rec_on_subsingleton X (λ x, hom_of_le ⟨h.counit.app x⟩),\n    end } }\n\ndef subq.lower_adjunction {A : C} {B : D} {R : sub B ⥤ sub A} {L : sub A ⥤ sub B} (h : L ⊣ R) :\n  lower_sub L ⊣ lower_sub R :=\nthin_skeleton.lower_adjunction _ _ h\n\ndef subq.pull_post_adj (f : X ⟶ Y) [mono f] [has_pullbacks C] : subq.post f ⊣ subq.pullback f :=\nsubq.lower_adjunction (sub.pull_post_adj f)\n\n/-- image is adjoint to pullback if images exist -/\n-- I really think there should be a high-level proof of this but not sure what it is...\ndef sub.exists_pull_adj (f : X ⟶ Y) [has_images C] [has_pullbacks C] : sub.exists f ⊣ sub.pullback f :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ g h,\n  { to_fun := λ k,\n    sub.hom_mk\n      (begin\n        refine pullback.lift (factor_thru_image _ ≫ k.1) g.arrow _,\n        rw [assoc, sub.w k],\n        apply image.fac,\n       end)\n      (pullback.lift_snd _ _ _),\n    inv_fun := λ k, sub.hom_mk (image.lift ⟨_, h.arrow, k.left ≫ pullback.fst, by { rw [assoc, pullback.condition], apply sub.w_assoc }⟩) (image.lift_fac _),\n    left_inv := λ k, subsingleton.elim _ _,\n    right_inv := λ k, subsingleton.elim _ _ } }\n\ndef subq.exists_pull_adj (f : X ⟶ Y) [has_pullbacks C] [has_images C] : subq.exists f ⊣ subq.pullback f :=\nsubq.lower_adjunction (sub.exists_pull_adj f)\n\n-- Is this actually necessary?\ndef factors_through {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) : Prop := nonempty (over.mk f ⟶ over.mk g)\nlemma factors_through_iff_le {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) [mono f] [mono g] :\n  factors_through f g ↔ subq.mk f ≤ subq.mk g :=\niff.rfl\n\ninstance {X : C} : has_top (sub X) :=\n{ top := sub.mk' (𝟙 _) }\n\ndef to_top (f : sub X) : f ⟶ ⊤ :=\nsub.hom_mk f.arrow (comp_id _)\n\ninstance subq.order_top {X : C} : order_top (subq X) :=\n{ top := quotient.mk' ⊤,\n  le_top :=\n  begin\n    refine quotient.ind' (λ f, _),\n    exact ⟨to_top f⟩,\n  end,\n  ..category_theory.subq.partial_order X}\n\n@[simp] lemma top_left (X : C) : (⊤ : sub X).val.left = X := rfl\n@[simp] lemma top_arrow (X : C) : (⊤ : sub X).arrow = 𝟙 X := rfl\n\ndef sub.post_top (f : X ⟶ Y) [mono f] : (sub.post f).obj ⊤ ≅ sub.mk' f :=\niso_of_both_ways (sub.hom_mk (𝟙 _) rfl) (sub.hom_mk (𝟙 _) (by simp [id_comp f]))\n\ndef subq.post_top (f : X ⟶ Y) [mono f] : (subq.post f).obj ⊤ = quotient.mk' (sub.mk' f) :=\nquotient.sound' ⟨sub.post_top f⟩\n\ndef sub.pullback_top (f : X ⟶ Y) [has_pullbacks C] : (sub.pullback f).obj ⊤ ≅ ⊤ :=\niso_of_both_ways (to_top _) (sub.hom_mk (pullback.lift f (𝟙 _) (by tidy)) (pullback.lift_snd _ _ _))\n\ndef subq.pullback_top (f : X ⟶ Y) [has_pullbacks C] : (subq.pullback f).obj ⊤ = ⊤ :=\nquotient.sound' ⟨sub.pullback_top f⟩\n\nvariable (C)\n\n@[simps]\ndef subq.functor [has_pullbacks.{v} C] : Cᵒᵖ ⥤ Type (max u v) :=\n{ obj := λ X, subq X.unop,\n  map := λ X Y f, (subq.pullback f.unop).obj,\n  map_id' := λ X, funext subq.pullback_id,\n  map_comp' := λ X Y Z f g, funext (subq.pullback_comp _ _) }\n\nvariable {C}\n\n@[simps]\ndef postcompose_sub_equiv_of_iso (e : X ≅ Y) : subq X ≃ subq Y :=\n{ to_fun := (subq.post e.hom).obj,\n  inv_fun := (subq.post e.inv).obj,\n  left_inv := λ g, by simp_rw [← subq.post_comp, e.hom_inv_id, subq.post_id],\n  right_inv := λ g, by simp_rw [← subq.post_comp, e.inv_hom_id, subq.post_id] }\n\n-- lemma postcompose_pullback_comm' [has_pullbacks.{v} C] {X Y Z W : C} {f : X ⟶ Y} {g : X ⟶ Z} {h : Y ⟶ W} {k : Z ⟶ W} [mono h] [mono g]\n--   {comm : f ≫ h = g ≫ k} (t : is_limit (pullback_cone.mk f g comm)) (a) :\n--   (sub.post g).obj ((sub.pullback f).obj a) ≈ (sub.pullback k).obj ((sub.post h).obj a) :=\n-- begin\n--   apply equiv_of_both_ways,\n--   { refine sub.hom_mk (pullback.lift pullback.fst _ _) (pullback.lift_snd _ _ _),\n--     change _ ≫ a.arrow ≫ h = (pullback.snd ≫ g) ≫ _,\n--     rw [assoc, ← comm, pullback.condition_assoc] },\n--   { refine sub.hom_mk (pullback.lift pullback.fst\n--                        (pullback_cone.is_limit.lift' t (pullback.fst ≫ a.arrow) pullback.snd _).1\n--                        (pullback_cone.is_limit.lift' _ _ _ _).2.1.symm) _,\n--     { rw [← pullback.condition, assoc], refl },\n--     { erw [pullback.lift_snd_assoc], apply (pullback_cone.is_limit.lift' _ _ _ _).2.2 } }\n-- end\n\nlemma postcompose_pullback_comm [has_pullbacks.{v} C] {X Y Z W : C} {f : X ⟶ Y} {g : X ⟶ Z} {h : Y ⟶ W} {k : Z ⟶ W} [mono h] [mono g]\n  (comm : f ≫ h = g ≫ k) (t : is_limit (pullback_cone.mk f g comm)) :\n  ∀ p, (subq.post g).obj ((subq.pullback f).obj p) = (subq.pullback k).obj ((subq.post h).obj p) :=\nbegin\n  apply quotient.ind',\n  intro a,\n  apply quotient.sound,\n  apply thin_skeleton.equiv_of_both_ways,\n  { refine sub.hom_mk (pullback.lift pullback.fst _ _) (pullback.lift_snd _ _ _),\n    change _ ≫ a.arrow ≫ h = (pullback.snd ≫ g) ≫ _,\n    rw [assoc, ← comm, pullback.condition_assoc] },\n  { refine sub.hom_mk (pullback.lift pullback.fst\n                        (pullback_cone.is_limit.lift' t (pullback.fst ≫ a.arrow) pullback.snd _).1\n                        (pullback_cone.is_limit.lift' _ _ _ _).2.1.symm) _,\n    { rw [← pullback.condition, assoc], refl },\n    { dsimp, rw [pullback.lift_snd_assoc],\n      apply (pullback_cone.is_limit.lift' _ _ _ _).2.2 } }\nend\n\nlemma sub.pull_post_self [has_pullbacks.{v} C] (f : X ⟶ Y) [mono f] (g₁ : sub X) :\n  sub.post f ⋙ sub.pullback f ≅ 𝟭 _ :=\n(as_iso (sub.pull_post_adj f).unit).symm\n\nlemma subq.pull_post_self [has_pullbacks.{v} C] (f : X ⟶ Y) [mono f] :\n  ∀ g₁, (subq.pullback f).obj ((subq.post f).obj g₁) = g₁ :=\nbegin\n  apply quotient.ind,\n  intro g,\n  apply quotient.sound,\n  exact ⟨(sub.pull_post_self f g).app _⟩,\nend\n\ninstance over_mono {B : C} {f g : over B} (m : f ⟶ g) [mono m] : mono m.left :=\n⟨λ A h k e,\nbegin\n  let A' : over B := over.mk (k ≫ f.hom),\n  have: h ≫ f.hom = k ≫ f.hom,\n    rw ← over.w m, rw reassoc_of e,\n  let h' : A' ⟶ f := over.hom_mk h,\n  let k' : A' ⟶ f := over.hom_mk k,\n  have : h' ≫ m = k' ≫ m := over.over_morphism.ext e,\n  rw cancel_mono m at this,\n  injection this\nend⟩\n\ndef over_mono' {B : C} {f g : over B} (m : f ⟶ g) [mono m.left] : mono m :=\n{right_cancellation := λ A h k e, over.over_morphism.ext ((cancel_mono m.left).1 (congr_arg comma_morphism.left e))}\n\n@[simps]\ndef preorder_functor {α β : Type*} [preorder α] [preorder β] (f : α → β) (hf : monotone f) : α ⥤ β :=\n{ obj := f,\n  map := λ X Y ⟨⟨h⟩⟩, ⟨⟨hf h⟩⟩ }\n\n@[simps]\ndef preorder_equivalence {α β : Type*} [preorder α] [preorder β] (f : α ≃o β) : α ≌ β :=\n{ functor := preorder_functor f (λ x y h, by rwa [← rel_iso.map_rel_iff f]),\n  inverse := preorder_functor f.symm (λ x y h, by rwa [← rel_iso.map_rel_iff f.symm]),\n  unit_iso := nat_iso.of_components (λ X, eq_to_iso (f.left_inv _).symm) (λ X Y f, rfl),\n  counit_iso := nat_iso.of_components (λ X, eq_to_iso (f.right_inv _)) (λ X Y f, rfl) }\n\ninstance iso_term (A : C) [has_terminal (over A)] : is_iso (⊤_ over A).hom :=\nbegin\n  let := (⊤_ over A).hom,\n  dsimp at this,\n  let ident : over A := over.mk (𝟙 A),\n  let k : ident ⟶ (⊤_ over A) := default _,\n  haveI : split_epi (⊤_ over A).hom := ⟨k.left, over.w k⟩,\n  let l : (⊤_ over A) ⟶ ident := over.hom_mk (⊤_ over A).hom (comp_id _),\n  haveI : mono l := ⟨λ _ _ _ _, subsingleton.elim _ _⟩,\n  haveI : mono (⊤_ over A).hom := category_theory.over_mono l,\n  apply is_iso_of_mono_of_split_epi,\nend\n\ndef sub_iso {A B : C} (e : A ≅ B) : sub A ≌ sub B :=\n{ functor := sub.post e.hom,\n  inverse := sub.post e.inv,\n  unit_iso := ((sub.post_comp _ _).symm ≪≫ eq_to_iso (by simp) ≪≫ sub.post_id).symm,\n  counit_iso := ((sub.post_comp _ _).symm ≪≫ eq_to_iso (by simp) ≪≫ sub.post_id) }\n\ndef sub_slice {A : C} {f : over A} (h₁ h₂) : sub f ≌ sub f.left :=\n{ functor := restrict_to_sub f.iterated_slice_equiv.functor h₁,\n  inverse := restrict_to_sub f.iterated_slice_equiv.inverse h₂,\n  unit_iso := restrict_to_sub_id.symm ≪≫ restrict_to_sub_iso _ _ f.iterated_slice_equiv.unit_iso ≪≫ (restrict_to_sub_comp _ _ _ _).symm,\n  counit_iso := restrict_to_sub_comp _ _ _ _ ≪≫ restrict_to_sub_iso _ _ f.iterated_slice_equiv.counit_iso ≪≫ restrict_to_sub_id }\n\n@[simps]\ndef subq.equiv {A : C} {B : D} (e : sub A ≌ sub B) : subq A ≌ subq B :=\n{ functor := lower_sub e.functor,\n  inverse := lower_sub e.inverse,\n  unit_iso :=\n  begin\n    apply eq_to_iso,\n    convert thin_skeleton.map_iso_eq e.unit_iso,\n    { exact thin_skeleton.map_id_eq.symm },\n    { exact (thin_skeleton.map_comp_eq _ _).symm },\n  end,\n  counit_iso :=\n  begin\n    apply eq_to_iso,\n    convert thin_skeleton.map_iso_eq e.counit_iso,\n    { exact (thin_skeleton.map_comp_eq _ _).symm },\n    { exact thin_skeleton.map_id_eq.symm },\n  end }\n\ndef sub_one_over (A : C) [has_terminal (over A)] : subq A ≌ subq (⊤_ (over A)) :=\nbegin\n  refine subq.equiv ((sub_iso (as_iso (⊤_ over A).hom).symm).trans (sub_slice _ _).symm),\n  intro f, dsimp, apply_instance,\n  intro f,\n  apply over_mono' _,\n  dsimp,\n  apply_instance,\nend\n\n@[simps]\ndef sub.intersection [has_pullbacks.{v} C] {A : C} : sub A ⥤ sub A ⥤ sub A :=\n{ obj := λ f, sub.pullback f.arrow ⋙ sub.post f.arrow,\n  map := λ f₁ f₂ k,\n  { app := λ g,\n    begin\n      apply sub.hom_mk _ _,\n      apply pullback.lift pullback.fst (pullback.snd ≫ k.left) _,\n      rw [pullback.condition, assoc, sub.w k],\n      dsimp,\n      rw [pullback.lift_snd_assoc, assoc, sub.w k],\n    end } }.\n\ndef sub.inter_le_left [has_pullbacks.{v} C] {A : C} (f g : sub A) :\n  (sub.intersection.obj f).obj g ⟶ f :=\nsub.hom_mk _ rfl\n\ndef sub.inter_le_right [has_pullbacks.{v} C] {A : C} (f g : sub A) :\n  (sub.intersection.obj f).obj g ⟶ g :=\nsub.hom_mk _ pullback.condition\n\ndef sub.le_inter [has_pullbacks.{v} C] {A : C} (f g h : sub A) :\n  (h ⟶ f) → (h ⟶ g) → (h ⟶ (sub.intersection.obj f).obj g) :=\nbegin\n  intros k₁ k₂,\n  refine sub.hom_mk (pullback.lift k₂.left k₁.left _) _,\n  rw [sub.w k₁, sub.w k₂],\n  erw [pullback.lift_snd_assoc, sub.w k₁],\nend\n\ndef subq.intersection [has_pullbacks.{v} C] {A : C} : subq A ⥤ subq A ⥤ subq A :=\nthin_skeleton.map₂ sub.intersection\n\nlemma subq.inf_le_left [has_pullbacks.{v} C] {A : C} (f g : subq A) :\n  (subq.intersection.obj f).obj g ≤ f :=\nquotient.induction_on₂' f g (λ a b, ⟨sub.inter_le_left _ _⟩)\n\nlemma subq.inf_le_right [has_pullbacks.{v} C] {A : C} (f g : subq A) :\n  (subq.intersection.obj f).obj g ≤ g :=\nquotient.induction_on₂' f g (λ a b, ⟨sub.inter_le_right _ _⟩)\n\nlemma subq.le_inf [has_pullbacks.{v} C] {A : C} (h f g : subq A) :\n  h ≤ f → h ≤ g → h ≤ (subq.intersection.obj f).obj g :=\nquotient.induction_on₃' h f g\nbegin\n  rintros f g h ⟨k⟩ ⟨l⟩,\n  exact ⟨sub.le_inter _ _ _ k l⟩,\nend\n\n@[simps]\ndef over.coprod' [has_finite_coproducts.{v} C] {A : C} : over A → over A ⥤ over A := λ f,\n{ obj := λ g, over.mk (coprod.desc f.hom g.hom),\n  map := λ g₁ g₂ k, over.hom_mk (coprod.map (𝟙 _) k.left) }\n\n@[simps]\ndef over.coprod [has_finite_coproducts.{v} C] {A : C} : over A ⥤ over A ⥤ over A :=\n{ obj := λ f, over.coprod' f,\n  map := λ f₁ f₂ k,\n  { app := λ g, over.hom_mk (coprod.map k.left (𝟙 _)) (by { dsimp, rw [coprod.map_desc, id_comp, over.w k] }),\n    naturality' := λ f g k, -- tidy can do this but it takes ages\n    begin\n      ext1,\n      dsimp,\n      simp,\n    end },\n  map_id' := λ X,\n  begin\n    ext; dsimp; simp,\n  end,\n  map_comp' := λ X Y Z f g,\n  begin\n    ext; dsimp; simp\n  end }.\n\ndef sub.union [has_images.{v} C] [has_finite_coproducts.{v} C] {A : C} : sub A ⥤ sub A ⥤ sub A :=\ncurry_obj ((forget_sub A).prod (forget_sub A) ⋙ uncurry.obj over.coprod ⋙ sub.image)\n\ndef sub.le_union_left [has_images.{v} C] [has_finite_coproducts.{v} C] {A : C} (f g : sub A) :\n  f ⟶ (sub.union.obj f).obj g :=\nbegin\n  refine sub.hom_mk (coprod.inl ≫ factor_thru_image _) _,\n  erw [assoc, image.fac, coprod.inl_desc],\n  refl,\nend\n\ndef sub.le_union_right [has_images.{v} C] [has_finite_coproducts.{v} C] {A : C} (f g : sub A) :\n  g ⟶ (sub.union.obj f).obj g :=\nbegin\n  refine sub.hom_mk (coprod.inr ≫ factor_thru_image _) _,\n  erw [assoc, image.fac, coprod.inr_desc],\n  refl,\nend\n\ndef sub.union_le [has_images.{v} C] [has_finite_coproducts.{v} C] {A : C} (f g h : sub A) :\n  (f ⟶ h) → (g ⟶ h) → ((sub.union.obj f).obj g ⟶ h) :=\nbegin\n  intros k₁ k₂,\n  refine sub.hom_mk _ _,\n  apply image.lift ⟨_, h.arrow, coprod.desc k₁.left k₂.left, _⟩,\n  { dsimp,\n    ext1,\n    { simp [sub.w k₁] },\n    { simp [sub.w k₂] } },\n  { apply image.lift_fac }\nend\n\ndef subq.union [has_images.{v} C] [has_finite_coproducts.{v} C] {A : C} : subq A ⥤ subq A ⥤ subq A :=\nthin_skeleton.map₂ sub.union\n\nlemma sub.intersection_eq_post_pull [has_pullbacks.{v} C] {A : C} (f₁ f₂ : sub A) :\n  (sub.intersection.obj f₁).obj f₂ = (sub.post f₁.arrow).obj ((sub.pullback f₁.arrow).obj f₂) :=\nrfl\nlemma subq.intersection_eq_post_pull [has_pullbacks.{v} C] {A : C} (f₁ : sub A) (f₂ : subq A) :\n  (subq.intersection.obj (quotient.mk' f₁)).obj f₂ = (subq.post f₁.arrow).obj ((subq.pullback f₁.arrow).obj f₂) :=\nbegin\n  apply quotient.induction_on' f₂,\n  intro f₂,\n  refl,\nend\n\ninstance [has_pullbacks.{v} C] {B : C} : semilattice_inf_top (subq B) :=\n{ inf := λ m n, (subq.intersection.obj m).obj n,\n  inf_le_left := subq.inf_le_left,\n  inf_le_right := subq.inf_le_right,\n  le_inf := subq.le_inf,\n  ..category_theory.subq.order_top }\n\nlemma subq.inf_eq_post_pull [has_pullbacks.{v} C] {A : C} (f₁ : sub A) (f₂ : subq A) :\n  (quotient.mk' f₁ ⊓ f₂ : subq A) = (subq.post f₁.arrow).obj ((subq.pullback f₁.arrow).obj f₂) :=\nbegin\n  apply quotient.induction_on' f₂,\n  intro f₂,\n  refl,\nend\n\ninstance [has_finite_coproducts.{v} C] [has_images.{v} C] {B : C} : semilattice_sup (subq B) :=\n{ sup := λ m n, (subq.union.obj m).obj n,\n  le_sup_left := λ m n, quotient.induction_on₂' m n (λ a b, ⟨sub.le_union_left _ _⟩),\n  le_sup_right := λ m n, quotient.induction_on₂' m n (λ a b, ⟨sub.le_union_right _ _⟩),\n  sup_le := λ m n k, quotient.induction_on₃' m n k (λ a b c ⟨i⟩ ⟨j⟩, ⟨sub.union_le _ _ _ i j⟩),\n  ..category_theory.subq.partial_order B }\n\nlemma prod_eq_inter {A : C} [has_pullbacks.{v} C] {f₁ f₂ : subq A} : (f₁ ⨯ f₂) = f₁ ⊓ f₂ :=\nle_antisymm\n  (le_inf\n    (le_of_hom limits.prod.fst)\n    (le_of_hom limits.prod.snd))\n  (le_of_hom\n    (prod.lift\n      (hom_of_le inf_le_left)\n      (hom_of_le inf_le_right)))\n\nlemma inf_eq_intersection {B : C} (m m' : subq B) [has_pullbacks.{v} C] :\n  m ⊓ m' = (subq.intersection.obj m).obj m' := rfl\n\nlemma top_eq_id {B : C} : (⊤ : subq B) = subq.mk (𝟙 B) := rfl\n\n/-- Intersection plays well with pullback. -/\nlemma inf_pullback [has_pullbacks.{v} C] {X Y : C} (g : X ⟶ Y) (f₂) :\n  ∀ f₁, (subq.pullback g).obj (f₁ ⊓ f₂) = (subq.pullback g).obj f₁ ⊓ (subq.pullback g).obj f₂ :=\nquotient.ind' begin\n  intro f₁,\n  erw [inf_eq_intersection, inf_eq_intersection, subq.intersection_eq_post_pull,\n       subq.intersection_eq_post_pull, ← subq.pullback_comp,\n       ← postcompose_pullback_comm pullback.condition (cone_is_pullback f₁.arrow g),\n       ← subq.pullback_comp, pullback.condition],\n  refl,\nend\n\nlemma inf_post [has_pullbacks.{v} C] {X Y : C} (g : Y ⟶ X) [mono g] (f₂) :\n  ∀ f₁, (subq.post g).obj (f₁ ⊓ f₂) = (subq.post g).obj f₁ ⊓ (subq.post g).obj f₂ :=\nquotient.ind' begin\n  intro f₁,\n  erw [inf_eq_intersection, inf_eq_intersection, subq.intersection_eq_post_pull,\n       subq.intersection_eq_post_pull, ← subq.post_comp],\n  dsimp,\n  rw [subq.pullback_comp, subq.pull_post_self],\nend\n\n\ndef sub.top_le_pullback_self {A B : C} (f : A ⟶ B) [mono f] [has_pullbacks.{v} C] :\n  (⊤ : sub A) ⟶ (sub.pullback f).obj (sub.mk' f) :=\nsub.hom_mk _ (pullback.lift_snd _ _ rfl)\n\ndef sub.pullback_self {A B : C} (f : A ⟶ B) [mono f] [has_pullbacks.{v} C] :\n  (sub.pullback f).obj (sub.mk' f) ≅ ⊤ :=\niso_of_both_ways (to_top _) (sub.top_le_pullback_self _)\n\nlemma subq.pullback_self {A B : C} (f : A ⟶ B) [mono f] [has_pullbacks.{v} C] :\n  (subq.pullback f).obj (subq.mk f) = ⊤ :=\nquotient.sound' ⟨sub.pullback_self f⟩\n\nsection\nvariable [has_binary_products.{v} C]\n\ninstance mono_prod_lift_of_left {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [mono f] : mono (limits.prod.lift f g) :=\nbegin\n  split, intros W h k l,\n  have := l =≫ limits.prod.fst,\n  simp at this,\n  rwa cancel_mono at this,\nend\n\ninstance mono_prod_lift_of_right {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z) [mono g] : mono (limits.prod.lift f g) :=\nbegin\n  split, intros W h k l,\n  have := l =≫ limits.prod.snd,\n  simp at this,\n  rwa cancel_mono at this,\nend\nend\n\nsection\nvariable [has_finite_products.{v} C]\ninstance subterminal_ideal {A B : C} [exponentiable B] [mono (default (A ⟶ ⊤_ C))] :\n  mono (default (A^^B ⟶ ⊤_ C)) :=\n⟨λ Z f g eq, begin\n  apply uncurry_injective,\n  rw ← cancel_mono (default (A ⟶ ⊤_ C)),\n  apply subsingleton.elim,\nend⟩\n\n/-- Auxiliary def for the exponential in the subobject category `sub 1`. -/\ndef sub.exp_aux (A : C) [exponentiable A] : sub (⊤_ C) ⥤ sub (⊤_ C) :=\n{ obj := λ f,\n  { val := over.mk (default (f.val.left^^A ⟶ ⊤_ C)),\n    property :=\n    ⟨λ Z g h eq, uncurry_injective (by { rw ← cancel_mono f.arrow, apply subsingleton.elim })⟩ },\n  map := λ f₁ f₂ h, sub.hom_mk ((exp A).map h.left) (subsingleton.elim _ _) }\n\n@[simps]\ndef sub.exp_aux_left {A₁ A₂ : C} [exponentiable A₁] [exponentiable A₂] (f : A₁ ⟶ A₂) :\n  sub.exp_aux A₂ ⟶ sub.exp_aux A₁ :=\n{ app := λ g, sub.hom_mk (pre _ f) (subsingleton.elim _ _) }\n\nlemma sub_exp_aux_left_comp {A₁ A₂ A₃ : C} [cartesian_closed C] (f : A₁ ⟶ A₂) (g : A₂ ⟶ A₃) :\n  sub.exp_aux_left (f ≫ g) = sub.exp_aux_left g ≫ sub.exp_aux_left f :=\nbegin\n  ext : 3,\n  apply pre_map,\nend\nlemma sub_exp_aux_left_id {A₁ : C} [cartesian_closed C] :\n  sub.exp_aux_left (𝟙 A₁) = 𝟙 _ :=\nbegin\n  ext : 3,\n  apply pre_id,\nend\n\n/-- Candidate for the exponential functor in sub 1. -/\ndef sub.exp (f : sub (⊤_ C)) [cartesian_closed C] : sub (⊤_ C) ⥤ sub (⊤_ C) :=\nsub.exp_aux f.val.left\nend\n\nvariable [has_finite_limits.{v} C]\nlocal attribute [instance] has_finite_products_of_has_finite_limits\n\ndef sub.exp_equiv [cartesian_closed C] (f₁ f₂ f₃ : sub (⊤_ C)) :\n  ((sub.intersection.obj f₂).obj f₁ ⟶ f₃) ≃ (f₁ ⟶ (sub.exp f₂).obj f₃) :=\n{ to_fun := λ k,\n  begin\n    refine sub.hom_mk (cartesian_closed.curry _) (subsingleton.elim _ _),\n    apply (pullback.lift limits.prod.snd limits.prod.fst _) ≫ k.left,\n    dsimp,\n    apply subsingleton.elim,\n  end,\n  inv_fun := λ k, sub.hom_mk (prod.lift pullback.snd pullback.fst ≫ cartesian_closed.uncurry k.left) (subsingleton.elim _ _),\n  left_inv := λ x, subsingleton.elim _ _,\n  right_inv := λ x, subsingleton.elim _ _ }\n\ndef subq.exp_aux [cartesian_closed C] (f : sub (⊤_ C)) : subq (⊤_ C) ⥤ subq (⊤_ C) :=\nlower_sub (sub.exp f)\n\ndef subq.exp (f : subq (⊤_ C)) [cartesian_closed C] : subq (⊤_ C) ⥤ subq (⊤_ C) :=\nbegin\n  apply quotient.lift_on' f subq.exp_aux _,\n  rintros f₁ f₂ ⟨h⟩,\n  apply lower_sub_iso,\n  have hi : h.hom.left ≫ h.inv.left = 𝟙 _,\n    change (h.hom ≫ h.inv).left = _,\n    rw h.hom_inv_id, refl,\n  have ih : h.inv.left ≫ h.hom.left = 𝟙 _,\n    change (h.inv ≫ h.hom).left = _,\n    rw h.inv_hom_id, refl,\n  refine ⟨sub.exp_aux_left h.inv.left, sub.exp_aux_left h.hom.left, _, _⟩,\n  rw [← sub_exp_aux_left_comp, hi, sub_exp_aux_left_id], exact rfl,\n  rw [← sub_exp_aux_left_comp, ih, sub_exp_aux_left_id], exact rfl,\nend\n\nvariable (C)\ndef top_cc [cartesian_closed C] : cartesian_closed (subq (⊤_ C)) :=\n{ closed := λ f₁,\n  { is_adj :=\n    { right := subq.exp f₁,\n      adj := adjunction.mk_of_hom_equiv\n      { hom_equiv := λ f₂ f₃,\n        begin\n          change (_ ⨯ _ ⟶ _) ≃ (_ ⟶ _),\n          rw prod_eq_inter,\n          apply @@quotient.rec_on_subsingleton₂ (is_isomorphic_setoid _) (is_isomorphic_setoid _) _ _ f₁ f₂,\n          intros f₁ f₂,\n          apply @@quotient.rec_on_subsingleton (is_isomorphic_setoid _) _ _ f₃,\n          intro f₃,\n          refine ⟨_, _, _, _⟩,\n          { rintro k,\n            refine ⟨⟨_⟩⟩,\n            rcases k with ⟨⟨⟨k⟩⟩⟩,\n            refine ⟨sub.exp_equiv _ _ _ k⟩ },\n          { rintro k,\n            refine ⟨⟨_⟩⟩,\n            rcases k with ⟨⟨⟨k⟩⟩⟩,\n            refine ⟨(sub.exp_equiv _ _ _).symm k⟩ },\n          { tidy },\n          { tidy },\n          { tidy },\n          { tidy }\n        end } } } }\n\nend category_theory\n", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/sub.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.30957583629891267}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.adjunction.fully_faithful\nimport category_theory.epi_mono\n\n/-!\n# Reflective functors\n\nBasic properties of reflective functors, especially those relating to their essential image.\n\nNote properties of reflective functors relating to limits and colimits are included in\n`category_theory.monad.limits`.\n-/\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category adjunction\n\nvariables {C : Type u₁} {D : Type u₂} {E : Type u₃}\nvariables [category.{v₁} C] [category.{v₂} D] [category.{v₃} E]\n\n/--\nA functor is *reflective*, or *a reflective inclusion*, if it is fully faithful and right adjoint.\n-/\nclass reflective (R : D ⥤ C) extends is_right_adjoint R, full R, faithful R.\n\nvariables {i : D ⥤ C}\n\n/--\nFor a reflective functor `i` (with left adjoint `L`), with unit `η`, we have `η_iL = iL η`.\n-/\n-- TODO: This holds more generally for idempotent adjunctions, not just reflective adjunctions.\nlemma unit_obj_eq_map_unit [reflective i] (X : C) :\n  (of_right_adjoint i).unit.app (i.obj ((left_adjoint i).obj X))\n    = i.map ((left_adjoint i).map ((of_right_adjoint i).unit.app X)) :=\nbegin\n rw [←cancel_mono (i.map ((of_right_adjoint i).counit.app ((left_adjoint i).obj X))),\n     ←i.map_comp],\n simp,\nend\n\n/--\nWhen restricted to objects in `D` given by `i : D ⥤ C`, the unit is an isomorphism. In other words,\n`η_iX` is an isomorphism for any `X` in `D`.\nMore generally this applies to objects essentially in the reflective subcategory, see\n`functor.ess_image.unit_iso`.\n-/\ninstance is_iso_unit_obj [reflective i] {B : D} :\n  is_iso ((of_right_adjoint i).unit.app (i.obj B)) :=\nbegin\n  have : (of_right_adjoint i).unit.app (i.obj B) =\n            inv (i.map ((of_right_adjoint i).counit.app B)),\n  { rw ← comp_hom_eq_id,\n    apply (of_right_adjoint i).right_triangle_components },\n  rw this,\n  exact is_iso.inv_is_iso,\nend\n\n/--\nIf `A` is essentially in the image of a reflective functor `i`, then `η_A` is an isomorphism.\nThis gives that the \"witness\" for `A` being in the essential image can instead be given as the\nreflection of `A`, with the isomorphism as `η_A`.\n\n(For any `B` in the reflective subcategory, we automatically have that `ε_B` is an iso.)\n-/\nlemma functor.ess_image.unit_is_iso [reflective i] {A : C} (h : A ∈ i.ess_image) :\n  is_iso ((of_right_adjoint i).unit.app A) :=\nbegin\n  suffices : (of_right_adjoint i).unit.app A =\n                h.get_iso.inv ≫ (of_right_adjoint i).unit.app (i.obj h.witness) ≫\n                  (left_adjoint i ⋙ i).map h.get_iso.hom,\n  { rw this,\n    apply_instance },\n  rw ← nat_trans.naturality,\n  simp,\nend\n\n/-- If `η_A` is an isomorphism, then `A` is in the essential image of `i`. -/\nlemma mem_ess_image_of_unit_is_iso [is_right_adjoint i] (A : C)\n  [is_iso ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image :=\n⟨(left_adjoint i).obj A, ⟨(as_iso ((of_right_adjoint i).unit.app A)).symm⟩⟩\n\n/-- If `η_A` is a split monomorphism, then `A` is in the reflective subcategory. -/\nlemma mem_ess_image_of_unit_split_mono [reflective i] {A : C}\n  [split_mono ((of_right_adjoint i).unit.app A)] : A ∈ i.ess_image :=\nbegin\n  let η : 𝟭 C ⟶ left_adjoint i ⋙ i := (of_right_adjoint i).unit,\n  haveI : is_iso (η.app (i.obj ((left_adjoint i).obj A))) := (i.obj_mem_ess_image _).unit_is_iso,\n  have : epi (η.app A),\n  { apply epi_of_epi (retraction (η.app A)) _,\n    rw (show retraction _ ≫ η.app A = _, from η.naturality (retraction (η.app A))),\n    apply epi_comp (η.app (i.obj ((left_adjoint i).obj A))) },\n  resetI,\n  haveI := is_iso_of_epi_of_split_mono (η.app A),\n  exact mem_ess_image_of_unit_is_iso A,\nend\n\n/-- Composition of reflective functors. -/\ninstance reflective.comp (F : C ⥤ D) (G : D ⥤ E) [Fr : reflective F] [Gr : reflective G] :\n  reflective (F ⋙ G) := { to_faithful := faithful.comp F G, }\n\n/-- (Implementation) Auxiliary definition for `unit_comp_partial_bijective`. -/\ndef unit_comp_partial_bijective_aux [reflective i] (A : C) (B : D) :\n  (A ⟶ i.obj B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ i.obj B) :=\n((adjunction.of_right_adjoint i).hom_equiv _ _).symm.trans (equiv_of_fully_faithful i)\n\n/-- The description of the inverse of the bijection `unit_comp_partial_bijective_aux`. -/\nlemma unit_comp_partial_bijective_aux_symm_apply [reflective i] {A : C} {B : D}\n  (f : i.obj ((left_adjoint i).obj A) ⟶ i.obj B) :\n  (unit_comp_partial_bijective_aux _ _).symm f = (of_right_adjoint i).unit.app A ≫ f :=\nby simp [unit_comp_partial_bijective_aux]\n\n/--\nIf `i` has a reflector `L`, then the function `(i.obj (L.obj A) ⟶ B) → (A ⟶ B)` given by\nprecomposing with `η.app A` is a bijection provided `B` is in the essential image of `i`.\nThat is, the function `λ (f : i.obj (L.obj A) ⟶ B), η.app A ≫ f` is bijective, as long as `B` is in\nthe essential image of `i`.\nThis definition gives an equivalence: the key property that the inverse can be described\nnicely is shown in `unit_comp_partial_bijective_symm_apply`.\n\nThis establishes there is a natural bijection `(A ⟶ B) ≃ (i.obj (L.obj A) ⟶ B)`. In other words,\nfrom the point of view of objects in `D`, `A` and `i.obj (L.obj A)` look the same: specifically\nthat `η.app A` is an isomorphism.\n-/\ndef unit_comp_partial_bijective [reflective i] (A : C) {B : C} (hB : B ∈ i.ess_image) :\n  (A ⟶ B) ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) :=\ncalc (A ⟶ B) ≃ (A ⟶ i.obj hB.witness) : iso.hom_congr (iso.refl _) hB.get_iso.symm\n     ...     ≃ (i.obj _ ⟶ i.obj hB.witness) : unit_comp_partial_bijective_aux _ _\n     ...     ≃ (i.obj ((left_adjoint i).obj A) ⟶ B) : iso.hom_congr (iso.refl _) hB.get_iso\n\n@[simp]\nlemma unit_comp_partial_bijective_symm_apply [reflective i] (A : C) {B : C}\n  (hB : B ∈ i.ess_image) (f) :\n  (unit_comp_partial_bijective A hB).symm f = (of_right_adjoint i).unit.app A ≫ f :=\nby simp [unit_comp_partial_bijective, unit_comp_partial_bijective_aux_symm_apply]\n\nlemma unit_comp_partial_bijective_symm_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')\n  (hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : i.obj ((left_adjoint i).obj A) ⟶ B) :\n  (unit_comp_partial_bijective A hB').symm (f ≫ h) =\n    (unit_comp_partial_bijective A hB).symm f ≫ h :=\nby simp\n\nlemma unit_comp_partial_bijective_natural [reflective i] (A : C) {B B' : C} (h : B ⟶ B')\n  (hB : B ∈ i.ess_image) (hB' : B' ∈ i.ess_image) (f : A ⟶ B) :\n  (unit_comp_partial_bijective A hB') (f ≫ h) = unit_comp_partial_bijective A hB f ≫ h :=\nby rw [←equiv.eq_symm_apply, unit_comp_partial_bijective_symm_natural A h, equiv.symm_apply_apply]\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/adjunction/reflective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.30957582814909296}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sebastian Ullrich\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.control.monad\nimport Mathlib.Lean3Lib.init.meta.interactive\nimport Mathlib.Lean3Lib.init.control.state\nimport Mathlib.Lean3Lib.init.control.except\nimport Mathlib.Lean3Lib.init.control.reader\nimport Mathlib.Lean3Lib.init.control.option\n\nuniverses u v l u_1 \n\nnamespace Mathlib\n\nclass is_lawful_functor (f : Type u → Type v) [Functor f] where\n  map_const_eq :\n    autoParam (∀ {α β : Type u}, Functor.mapConst = Functor.map ∘ function.const β)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  id_map : ∀ {α : Type u} (x : f α), id <$> x = x\n  comp_map : ∀ {α β γ : Type u} (g : α → β) (h : β → γ) (x : f α), (h ∘ g) <$> x = h <$> g <$> x\n\n-- `functor` is indeed a categorical functor\n\n-- `comp_map` does not make a good simp lemma\n\nclass is_lawful_applicative (f : Type u → Type v) [Applicative f] extends is_lawful_functor f where\n  seq_left_eq :\n    autoParam (∀ {α β : Type u} (a : f α) (b : f β), a <* b = function.const β <$> a <*> b)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  seq_right_eq :\n    autoParam (∀ {α β : Type u} (a : f α) (b : f β), a *> b = function.const α id <$> a <*> b)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  pure_seq_eq_map : ∀ {α β : Type u} (g : α → β) (x : f α), pure g <*> x = g <$> x\n  map_pure : ∀ {α β : Type u} (g : α → β) (x : α), g <$> pure x = pure (g x)\n  seq_pure : ∀ {α β : Type u} (g : f (α → β)) (x : α), g <*> pure x = (fun (g : α → β) => g x) <$> g\n  seq_assoc :\n    ∀ {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)),\n      h <*> (g <*> x) = function.comp <$> h <*> g <*> x\n\n-- applicative laws\n\n-- default functor law\n\n-- applicative \"law\" derivable from other laws\n\n@[simp] theorem pure_id_seq {α : Type u} {f : Type u → Type v} [Applicative f]\n    [is_lawful_applicative f] (x : f α) : pure id <*> x = x :=\n  sorry\n\nclass is_lawful_monad (m : Type u → Type v) [Monad m] extends is_lawful_applicative m where\n  bind_pure_comp_eq_map :\n    autoParam (∀ {α β : Type u} (f : α → β) (x : m α), x >>= pure ∘ f = f <$> x)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  bind_map_eq_seq :\n    autoParam\n      (∀ {α β : Type u} (f : m (α → β)) (x : m α),\n        (do \n            let _x ← f \n            _x <$> x) =\n          f <*> x)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  pure_bind : ∀ {α β : Type u} (x : α) (f : α → m β), pure x >>= f = f x\n  bind_assoc :\n    ∀ {α β γ : Type u} (x : m α) (f : α → m β) (g : β → m γ),\n      x >>= f >>= g = x >>= fun (x : α) => f x >>= g\n\n-- monad laws\n\n-- monad \"law\" derivable from other laws\n\n@[simp] theorem bind_pure {α : Type u} {m : Type u → Type v} [Monad m] [is_lawful_monad m]\n    (x : m α) : x >>= pure = x :=\n  sorry\n\ntheorem bind_ext_congr {α : Type u} {β : Type u} {m : Type u → Type v} [Bind m] {x : m α}\n    {f : α → m β} {g : α → m β} : (∀ (a : α), f a = g a) → x >>= f = x >>= g :=\n  sorry\n\ntheorem map_ext_congr {α : Type u} {β : Type u} {m : Type u → Type v} [Functor m] {x : m α}\n    {f : α → β} {g : α → β} : (∀ (a : α), f a = g a) → f <$> x = g <$> x :=\n  sorry\n\n-- instances of previously defined monads\n\nnamespace id\n\n\n@[simp] theorem map_eq {α : Type} {β : Type} (x : id α) (f : α → β) : f <$> x = f x := rfl\n\n@[simp] theorem bind_eq {α : Type} {β : Type} (x : id α) (f : α → id β) : x >>= f = f x := rfl\n\nend id\n\n\n@[simp] theorem id.pure_eq {α : Type} (a : α) : pure a = a := rfl\n\nprotected instance id.is_lawful_monad : is_lawful_monad id :=\n  is_lawful_monad.mk (fun (α β : Type u_1) (x : α) (f : α → id β) => Eq.refl (pure x >>= f))\n    fun (α β γ : Type u_1) (x : id α) (f : α → id β) (g : β → id γ) => Eq.refl (x >>= f >>= g)\n\nnamespace state_t\n\n\ntheorem ext {σ : Type u} {m : Type u → Type v} {α : Type u} {x : state_t σ m α} {x' : state_t σ m α}\n    (h : ∀ (st : σ), run x st = run x' st) : x = x' :=\n  sorry\n\n@[simp] theorem run_pure {σ : Type u} {m : Type u → Type v} {α : Type u} (st : σ) [Monad m]\n    (a : α) : run (pure a) st = pure (a, st) :=\n  rfl\n\n@[simp] theorem run_bind {σ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u}\n    (x : state_t σ m α) (st : σ) [Monad m] (f : α → state_t σ m β) :\n    run (x >>= f) st =\n        do \n          let p ← run x st \n          run (f (prod.fst p)) (prod.snd p) :=\n  sorry\n\n@[simp] theorem run_map {σ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u}\n    (x : state_t σ m α) (st : σ) [Monad m] (f : α → β) [is_lawful_monad m] :\n    run (f <$> x) st = (fun (p : α × σ) => (f (prod.fst p), prod.snd p)) <$> run x st :=\n  sorry\n\n@[simp] theorem run_monad_lift {σ : Type u} {m : Type u → Type v} {α : Type u} (st : σ) [Monad m]\n    {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) :\n    run (monad_lift x) st =\n        do \n          let a ← monad_lift x \n          pure (a, st) :=\n  rfl\n\n@[simp] theorem run_monad_map {σ : Type u} {m : Type u → Type v} {α : Type u} (x : state_t σ m α)\n    (st : σ) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1}\n    [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) :\n    run (monad_map f x) st = monad_map f (run x st) :=\n  rfl\n\n@[simp] theorem run_adapt {σ : Type u} {m : Type u → Type v} {α : Type u} [Monad m] {σ' : Type u}\n    {σ'' : Type u} (st : σ) (split : σ → σ' × σ'') (join : σ' → σ'' → σ) (x : state_t σ' m α) :\n    run (state_t.adapt split join x) st =\n        (fun (_a : σ' × σ'') =>\n            prod.cases_on _a\n              fun (fst : σ') (snd : σ'') =>\n                idRhs (m (α × σ))\n                  (do \n                    let _p ← run x fst\n                    (fun (_a : α × σ') =>\n                          prod.cases_on _a\n                            fun (fst : α) (snd_1 : σ') =>\n                              idRhs (m (α × σ)) (pure (fst, join snd_1 snd)))\n                        _p))\n          (split st) :=\n  sorry\n\n@[simp] theorem run_get {σ : Type u} {m : Type u → Type v} (st : σ) [Monad m] :\n    run state_t.get st = pure (st, st) :=\n  rfl\n\n@[simp] theorem run_put {σ : Type u} {m : Type u → Type v} (st : σ) [Monad m] (st' : σ) :\n    run (state_t.put st') st = pure (PUnit.unit, st') :=\n  rfl\n\nend state_t\n\n\nprotected instance state_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m]\n    (σ : Type u) : is_lawful_monad (state_t σ m) :=\n  sorry\n\nnamespace except_t\n\n\ntheorem ext {α : Type u} {ε : Type u} {m : Type u → Type v} {x : except_t ε m α}\n    {x' : except_t ε m α} (h : run x = run x') : x = x' :=\n  sorry\n\n@[simp] theorem run_pure {α : Type u} {ε : Type u} {m : Type u → Type v} [Monad m] (a : α) :\n    run (pure a) = pure (except.ok a) :=\n  rfl\n\n@[simp] theorem run_bind {α : Type u} {β : Type u} {ε : Type u} {m : Type u → Type v}\n    (x : except_t ε m α) [Monad m] (f : α → except_t ε m β) :\n    run (x >>= f) = run x >>= except_t.bind_cont f :=\n  rfl\n\n@[simp] theorem run_map {α : Type u} {β : Type u} {ε : Type u} {m : Type u → Type v}\n    (x : except_t ε m α) [Monad m] (f : α → β) [is_lawful_monad m] :\n    run (f <$> x) = except.map f <$> run x :=\n  sorry\n\n@[simp] theorem run_monad_lift {α : Type u} {ε : Type u} {m : Type u → Type v} [Monad m]\n    {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) :\n    run (monad_lift x) = except.ok <$> monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {α : Type u} {ε : Type u} {m : Type u → Type v} (x : except_t ε m α)\n    [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m']\n    [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) :\n    run (monad_map f x) = monad_map f (run x) :=\n  rfl\n\nend except_t\n\n\nprotected instance except_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m]\n    (ε : Type u) : is_lawful_monad (except_t ε m) :=\n  sorry\n\nnamespace reader_t\n\n\ntheorem ext {ρ : Type u} {m : Type u → Type v} {α : Type u} {x : reader_t ρ m α}\n    {x' : reader_t ρ m α} (h : ∀ (r : ρ), run x r = run x' r) : x = x' :=\n  sorry\n\n@[simp] theorem run_pure {ρ : Type u} {m : Type u → Type v} {α : Type u} (r : ρ) [Monad m] (a : α) :\n    run (pure a) r = pure a :=\n  rfl\n\n@[simp] theorem run_bind {ρ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u}\n    (x : reader_t ρ m α) (r : ρ) [Monad m] (f : α → reader_t ρ m β) :\n    run (x >>= f) r =\n        do \n          let a ← run x r \n          run (f a) r :=\n  rfl\n\n@[simp] theorem run_map {ρ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u}\n    (x : reader_t ρ m α) (r : ρ) [Monad m] (f : α → β) [is_lawful_monad m] :\n    run (f <$> x) r = f <$> run x r :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (run (f <$> x) r = f <$> run x r))\n        (Eq.symm (bind_pure_comp_eq_map f (run x r)))))\n    (Eq.refl (run (f <$> x) r))\n\n@[simp] theorem run_monad_lift {ρ : Type u} {m : Type u → Type v} {α : Type u} (r : ρ) [Monad m]\n    {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) :\n    run (monad_lift x) r = monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : reader_t ρ m α)\n    (r : ρ) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1}\n    [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) :\n    run (monad_map f x) r = monad_map f (run x r) :=\n  rfl\n\n@[simp] theorem run_read {ρ : Type u} {m : Type u → Type v} (r : ρ) [Monad m] :\n    run reader_t.read r = pure r :=\n  rfl\n\nend reader_t\n\n\nprotected instance reader_t.is_lawful_monad (ρ : Type u) (m : Type u → Type v) [Monad m]\n    [is_lawful_monad m] : is_lawful_monad (reader_t ρ m) :=\n  sorry\n\nnamespace option_t\n\n\ntheorem ext {α : Type u} {m : Type u → Type v} {x : option_t m α} {x' : option_t m α}\n    (h : run x = run x') : x = x' :=\n  sorry\n\n@[simp] theorem run_pure {α : Type u} {m : Type u → Type v} [Monad m] (a : α) :\n    run (pure a) = pure (some a) :=\n  rfl\n\n@[simp] theorem run_bind {α : Type u} {β : Type u} {m : Type u → Type v} (x : option_t m α)\n    [Monad m] (f : α → option_t m β) : run (x >>= f) = run x >>= option_t.bind_cont f :=\n  rfl\n\n@[simp] theorem run_map {α : Type u} {β : Type u} {m : Type u → Type v} (x : option_t m α) [Monad m]\n    (f : α → β) [is_lawful_monad m] : run (f <$> x) = option.map f <$> run x :=\n  sorry\n\n@[simp] theorem run_monad_lift {α : Type u} {m : Type u → Type v} [Monad m] {n : Type u → Type u_1}\n    [has_monad_lift_t n m] (x : n α) : run (monad_lift x) = some <$> monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {α : Type u} {m : Type u → Type v} (x : option_t m α) [Monad m]\n    {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m']\n    [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) :\n    run (monad_map f x) = monad_map f (run x) :=\n  rfl\n\nend option_t\n\n\nprotected instance option_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m] :\n    is_lawful_monad (option_t m) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/control/lawful_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.554470450236115, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.30957581992811184}}
{"text": "/-\nCopyright (c) 2014 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nNatural homomorphism from the natural numbers into a monoid with one.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.ordered_field\nimport Mathlib.data.nat.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\nnamespace nat\n\n\n/-- Canonical homomorphism from `ℕ` to a type `α` with `0`, `1` and `+`. -/\nprotected def cast {α : Type u_1} [HasZero α] [HasOne α] [Add α] : ℕ → α :=\n  sorry\n\n/-- Computationally friendlier cast than `nat.cast`, using binary representation. -/\nprotected def bin_cast {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : α :=\n  binary_rec 0 (fun (odd : Bool) (k : ℕ) (a : α) => cond odd (a + a + 1) (a + a)) n\n\n/--\nCoercions such as `nat.cast_coe` that go from a concrete structure such as\n`ℕ` to an arbitrary ring `α` should be set up as follows:\n```lean\n@[priority 900] instance : has_coe_t ℕ α := ⟨...⟩\n```\n\nIt needs to be `has_coe_t` instead of `has_coe` because otherwise type-class\ninference would loop when constructing the transitive coercion `ℕ → ℕ → ℕ → ...`.\nThe reduced priority is necessary so that it doesn't conflict with instances\nsuch as `has_coe_t α (option α)`.\n\nFor this to work, we reduce the priority of the `coe_base` and `coe_trans`\ninstances because we want the instances for `has_coe_t` to be tried in the\nfollowing order:\n\n 1. `has_coe_t` instances declared in mathlib (such as `has_coe_t α (with_top α)`, etc.)\n 2. `coe_base`, which contains instances such as `has_coe (fin n) n`\n 3. `nat.cast_coe : has_coe_t ℕ α` etc.\n 4. `coe_trans`\n\nIf `coe_trans` is tried first, then `nat.cast_coe` doesn't get a chance to apply.\n-/\n-- see note [coercion into rings]\n\nprotected instance cast_coe {α : Type u_1} [HasZero α] [HasOne α] [Add α] : has_coe_t ℕ α :=\n  has_coe_t.mk nat.cast\n\n@[simp] theorem cast_zero {α : Type u_1} [HasZero α] [HasOne α] [Add α] : ↑0 = 0 :=\n  rfl\n\ntheorem cast_add_one {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑(n + 1) = ↑n + 1 :=\n  rfl\n\n@[simp] theorem cast_succ {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑(Nat.succ n) = ↑n + 1 :=\n  rfl\n\n@[simp] theorem cast_ite {α : Type u_1} [HasZero α] [HasOne α] [Add α] (P : Prop) [Decidable P] (m : ℕ) (n : ℕ) : ↑(ite P m n) = ite P ↑m ↑n := sorry\n\n@[simp] theorem cast_one {α : Type u_1} [add_monoid α] [HasOne α] : ↑1 = 1 :=\n  zero_add 1\n\n@[simp] theorem cast_add {α : Type u_1} [add_monoid α] [HasOne α] (m : ℕ) (n : ℕ) : ↑(m + n) = ↑m + ↑n := sorry\n\n@[simp] theorem bin_cast_eq {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) : nat.bin_cast n = ↑n := sorry\n\n/-- `coe : ℕ → α` as an `add_monoid_hom`. -/\ndef cast_add_monoid_hom (α : Type u_1) [add_monoid α] [HasOne α] : ℕ →+ α :=\n  add_monoid_hom.mk coe sorry cast_add\n\n@[simp] theorem coe_cast_add_monoid_hom {α : Type u_1} [add_monoid α] [HasOne α] : ⇑(cast_add_monoid_hom α) = coe :=\n  rfl\n\n@[simp] theorem cast_bit0 {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) : ↑(bit0 n) = bit0 ↑n :=\n  cast_add n n\n\n@[simp] theorem cast_bit1 {α : Type u_1} [add_monoid α] [HasOne α] (n : ℕ) : ↑(bit1 n) = bit1 ↑n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit1 n) = bit1 ↑n)) (bit1.equations._eqn_1 n)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit0 n + 1) = bit1 ↑n)) (cast_add_one (bit0 n))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (↑(bit0 n) + 1 = bit1 ↑n)) (cast_bit0 n))) (Eq.refl (bit0 ↑n + 1))))\n\ntheorem cast_two {α : Type u_1} [add_monoid α] [HasOne α] : ↑(bit0 1) = bit0 1 := sorry\n\n@[simp] theorem cast_pred {α : Type u_1} [add_group α] [HasOne α] {n : ℕ} : 0 < n → ↑(n - 1) = ↑n - 1 := sorry\n\n@[simp] theorem cast_sub {α : Type u_1} [add_group α] [HasOne α] {m : ℕ} {n : ℕ} (h : m ≤ n) : ↑(n - m) = ↑n - ↑m :=\n  eq_sub_of_add_eq\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑(n - m) + ↑m = ↑n)) (Eq.symm (cast_add (n - m) m))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (↑(n - m + m) = ↑n)) (nat.sub_add_cancel h))) (Eq.refl ↑n)))\n\n@[simp] theorem cast_mul {α : Type u_1} [semiring α] (m : ℕ) (n : ℕ) : ↑(m * n) = ↑m * ↑n := sorry\n\n@[simp] theorem cast_dvd {α : Type u_1} [field α] {m : ℕ} {n : ℕ} (n_dvd : n ∣ m) (n_nonzero : ↑n ≠ 0) : ↑(m / n) = ↑m / ↑n := sorry\n\n/-- `coe : ℕ → α` as a `ring_hom` -/\ndef cast_ring_hom (α : Type u_1) [semiring α] : ℕ →+* α :=\n  ring_hom.mk coe sorry cast_mul sorry sorry\n\n@[simp] theorem coe_cast_ring_hom {α : Type u_1} [semiring α] : ⇑(cast_ring_hom α) = coe :=\n  rfl\n\ntheorem cast_commute {α : Type u_1} [semiring α] (n : ℕ) (x : α) : commute (↑n) x :=\n  nat.rec_on n (commute.zero_left x) fun (n : ℕ) (ihn : commute (↑n) x) => commute.add_left ihn (commute.one_left x)\n\ntheorem commute_cast {α : Type u_1} [semiring α] (x : α) (n : ℕ) : commute x ↑n :=\n  commute.symm (cast_commute n x)\n\n@[simp] theorem cast_nonneg {α : Type u_1} [ordered_semiring α] (n : ℕ) : 0 ≤ ↑n := sorry\n\ntheorem mono_cast {α : Type u_1} [ordered_semiring α] : monotone coe := sorry\n\ntheorem strict_mono_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] : strict_mono coe :=\n  fun (m n : ℕ) (h : m < n) =>\n    le_induction (lt_add_of_pos_right (↑m) zero_lt_one)\n      (fun (n : ℕ) (_x : Nat.succ m ≤ n) (h : ↑m < ↑n) => lt_add_of_lt_of_pos h zero_lt_one) n h\n\n@[simp] theorem cast_le {α : Type u_1} [ordered_semiring α] [nontrivial α] {m : ℕ} {n : ℕ} : ↑m ≤ ↑n ↔ m ≤ n :=\n  strict_mono.le_iff_le strict_mono_cast\n\n@[simp] theorem cast_lt {α : Type u_1} [ordered_semiring α] [nontrivial α] {m : ℕ} {n : ℕ} : ↑m < ↑n ↔ m < n :=\n  strict_mono.lt_iff_lt strict_mono_cast\n\n@[simp] theorem cast_pos {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} : 0 < ↑n ↔ 0 < n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 < ↑n ↔ 0 < n)) (Eq.symm cast_zero)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑0 < ↑n ↔ 0 < n)) (propext cast_lt))) (iff.refl (0 < n)))\n\ntheorem cast_add_one_pos {α : Type u_1} [ordered_semiring α] [nontrivial α] (n : ℕ) : 0 < ↑n + 1 :=\n  add_pos_of_nonneg_of_pos (cast_nonneg n) zero_lt_one\n\n@[simp] theorem one_lt_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} : 1 < ↑n ↔ 1 < n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 < ↑n ↔ 1 < n)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑1 < ↑n ↔ 1 < n)) (propext cast_lt))) (iff.refl (1 < n)))\n\n@[simp] theorem one_le_cast {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} : 1 ≤ ↑n ↔ 1 ≤ n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 ≤ ↑n ↔ 1 ≤ n)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑1 ≤ ↑n ↔ 1 ≤ n)) (propext cast_le))) (iff.refl (1 ≤ n)))\n\n@[simp] theorem cast_lt_one {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} : ↑n < 1 ↔ n = 0 := sorry\n\n@[simp] theorem cast_le_one {α : Type u_1} [ordered_semiring α] [nontrivial α] {n : ℕ} : ↑n ≤ 1 ↔ n ≤ 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≤ 1 ↔ n ≤ 1)) (Eq.symm cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≤ ↑1 ↔ n ≤ 1)) (propext cast_le))) (iff.refl (n ≤ 1)))\n\n@[simp] theorem cast_min {α : Type u_1} [linear_ordered_semiring α] {a : ℕ} {b : ℕ} : ↑(min a b) = min ↑a ↑b := sorry\n\n@[simp] theorem cast_max {α : Type u_1} [linear_ordered_semiring α] {a : ℕ} {b : ℕ} : ↑(max a b) = max ↑a ↑b := sorry\n\n@[simp] theorem abs_cast {α : Type u_1} [linear_ordered_ring α] (a : ℕ) : abs ↑a = ↑a :=\n  abs_of_nonneg (cast_nonneg a)\n\ntheorem coe_nat_dvd {α : Type u_1} [comm_semiring α] {m : ℕ} {n : ℕ} (h : m ∣ n) : ↑m ∣ ↑n :=\n  ring_hom.map_dvd (cast_ring_hom α) h\n\ntheorem Mathlib.has_dvd.dvd.nat_cast {α : Type u_1} [comm_semiring α] {m : ℕ} {n : ℕ} (h : m ∣ n) : ↑m ∣ ↑n :=\n  coe_nat_dvd\n\ntheorem inv_pos_of_nat {α : Type u_1} [linear_ordered_field α] {n : ℕ} : 0 < (↑n + 1⁻¹) :=\n  iff.mpr inv_pos (add_pos_of_nonneg_of_pos (cast_nonneg n) zero_lt_one)\n\ntheorem one_div_pos_of_nat {α : Type u_1} [linear_ordered_field α] {n : ℕ} : 0 < 1 / (↑n + 1) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 < 1 / (↑n + 1))) (one_div (↑n + 1)))) inv_pos_of_nat\n\ntheorem one_div_le_one_div {α : Type u_1} [linear_ordered_field α] {n : ℕ} {m : ℕ} (h : n ≤ m) : 1 / (↑m + 1) ≤ 1 / (↑n + 1) :=\n  one_div_le_one_div_of_le (cast_add_one_pos n)\n    (eq.mpr (id (Eq.trans (propext (add_le_add_iff_right 1)) (propext cast_le))) h)\n\ntheorem one_div_lt_one_div {α : Type u_1} [linear_ordered_field α] {n : ℕ} {m : ℕ} (h : n < m) : 1 / (↑m + 1) < 1 / (↑n + 1) :=\n  one_div_lt_one_div_of_lt (cast_add_one_pos n)\n    (eq.mpr (id (Eq.trans (propext (add_lt_add_iff_right 1)) (propext cast_lt))) h)\n\nend nat\n\n\nnamespace add_monoid_hom\n\n\ntheorem ext_nat {A : Type u_1} [add_monoid A] {f : ℕ →+ A} {g : ℕ →+ A} (h : coe_fn f 1 = coe_fn g 1) : f = g := sorry\n\ntheorem eq_nat_cast {A : Type u_1} [add_monoid A] [HasOne A] (f : ℕ →+ A) (h1 : coe_fn f 1 = 1) (n : ℕ) : coe_fn f n = ↑n :=\n  congr_fun ((fun (this : f = nat.cast_add_monoid_hom A) => this) (ext_nat (Eq.trans h1 (Eq.symm nat.cast_one))))\n\ntheorem map_nat_cast {A : Type u_1} {B : Type u_2} [add_monoid A] [HasOne A] [add_monoid B] [HasOne B] (f : A →+ B) (h1 : coe_fn f 1 = 1) (n : ℕ) : coe_fn f ↑n = ↑n := sorry\n\nend add_monoid_hom\n\n\nnamespace ring_hom\n\n\n@[simp] theorem eq_nat_cast {R : Type u_1} [semiring R] (f : ℕ →+* R) (n : ℕ) : coe_fn f n = ↑n :=\n  add_monoid_hom.eq_nat_cast (to_add_monoid_hom f) (map_one f) n\n\n@[simp] theorem map_nat_cast {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R →+* S) (n : ℕ) : coe_fn f ↑n = ↑n :=\n  eq_nat_cast (comp f (nat.cast_ring_hom R)) n\n\ntheorem ext_nat {R : Type u_1} [semiring R] (f : ℕ →+* R) (g : ℕ →+* R) : f = g :=\n  coe_add_monoid_hom_injective (add_monoid_hom.ext_nat (Eq.trans (map_one f) (Eq.symm (map_one g))))\n\nend ring_hom\n\n\n@[simp] theorem nat.cast_id (n : ℕ) : ↑n = n :=\n  Eq.symm (ring_hom.eq_nat_cast (ring_hom.id ℕ) n)\n\n@[simp] theorem nat.cast_with_bot (n : ℕ) : ↑n = ↑n := sorry\n\nprotected instance nat.subsingleton_ring_hom {R : Type u_1} [semiring R] : subsingleton (ℕ →+* R) :=\n  subsingleton.intro ring_hom.ext_nat\n\nnamespace with_top\n\n\n@[simp] theorem coe_nat {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑↑n = ↑n := sorry\n\n@[simp] theorem nat_ne_top {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ↑n ≠ ⊤ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (↑n ≠ ⊤)) (Eq.symm (coe_nat n)))) coe_ne_top\n\n@[simp] theorem top_ne_nat {α : Type u_1} [HasZero α] [HasOne α] [Add α] (n : ℕ) : ⊤ ≠ ↑n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (⊤ ≠ ↑n)) (Eq.symm (coe_nat n)))) top_ne_coe\n\ntheorem add_one_le_of_lt {i : with_top ℕ} {n : with_top ℕ} (h : i < n) : i + 1 ≤ n := sorry\n\ntheorem one_le_iff_pos {n : with_top ℕ} : 1 ≤ n ↔ 0 < n := sorry\n\ntheorem nat_induction {P : with_top ℕ → Prop} (a : with_top ℕ) (h0 : P 0) (hsuc : ∀ (n : ℕ), P ↑n → P ↑(Nat.succ n)) (htop : (∀ (n : ℕ), P ↑n) → P ⊤) : P a :=\n  option.cases_on a (htop fun (n : ℕ) => nat.rec_on n h0 hsuc) fun (a : ℕ) => (fun (n : ℕ) => nat.rec_on n h0 hsuc) a\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/nat/cast.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837478, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.30953853040981333}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport Lean.Elab.Term\nimport Lean.Meta.Tactic.Apply\nimport Lean.Meta.Tactic.Assumption\nimport Lean.Elab.DeclarationRange\nimport Mathlib.Control.SimpSet\n\n/-!\n# HigherOrder attribute\n\nThis file defines the `@[higher_order]` attribute that applies to lemmas of the shape\n`∀ x, f (g x) = h x`. It derives an auxiliary lemma of the form `f ∘ g = h` for reasoning about\nhigher-order functions.\n-/\n\nopen Lean Name Meta Elab Expr Term\n\nnamespace Lean.Parser.Attr\n\nsyntax (name := higherOrder) \"higher_order\" (ppSpace ident)? : attr\n\nend Lean.Parser.Attr\n\nnamespace Tactic\n\n/-- `mkComp v e` checks whether `e` is a sequence of nested applications `f (g (h v))`, and if so,\nreturns the expression `f ∘ g ∘ h`. If `e = v` it returns `id`. -/\ndef mkComp (v : Expr) : Expr → MetaM Expr\n| .app f e =>\n  if e.equal v then\n    return f\n  else do\n    if v.occurs f then\n      throwError \"mkComp failed occurs check\"\n    let e' ← mkComp v e\n    mkAppM ``Function.comp #[f, e']\n| e => do\n  guard (e.equal v)\n  let t ← inferType e\n  mkAppOptM ``id #[t]\n\n/--\nFrom a lemma of the shape `∀ x, f (g x) = h x`\nderive an auxiliary lemma of the form `f ∘ g = h`\nfor reasoning about higher-order functions.\n-/\npartial def mkHigherOrderType (e : Expr) : MetaM Expr := do\n  if not e.isForall then\n    throwError \"not a forall\"\n  withLocalDecl e.bindingName! e.binderInfo e.bindingDomain! fun fvar => do\n    let body := instantiate1 e.bindingBody! fvar\n    if body.isForall then\n      let exp ← mkHigherOrderType body\n      mkForallFVars #[fvar] exp (binderInfoForMVars := e.binderInfo)\n    else\n      let some (_, lhs, rhs) ← matchEq? body | throwError \"not an equality {← ppExpr body}\"\n      mkEq (← mkComp fvar lhs) (← mkComp fvar rhs)\n\n/-- A user attribute that applies to lemmas of the shape `∀ x, f (g x) = h x`.\nIt derives an auxiliary lemma of the form `f ∘ g = h` for reasoning about higher-order functions.\n-/\ndef higherOrderGetParam (thm : Name) (stx : Syntax) : AttrM Name := do\n  match stx with\n  | `(attr| higher_order $[$name]?) =>\n    let ref := (name : Option Syntax).getD stx[0]\n    let hothmName :=\n      if let some sname := name then\n        updatePrefix sname.getId thm.getPrefix\n      else\n        thm.appendAfter \"\\'\"\n    MetaM.run' <| TermElabM.run' <| do\n      let lvl := (← getConstInfo thm).levelParams\n      let typ ← instantiateMVars (← inferType <| .const thm (lvl.map mkLevelParam))\n      let hot ← mkHigherOrderType typ\n      let prf ← do\n        let mvar ← mkFreshExprMVar hot\n        let (_, mvarId) ← mvar.mvarId!.intros\n        let [mvarId] ← mvarId.apply (← mkConst ``funext) | throwError \"failed\"\n        let (_, mvarId) ← mvarId.intro1\n        let lmvr ← mvarId.apply (← mkConst thm)\n        lmvr.forM fun mv ↦ mv.assumption\n        instantiateMVars mvar\n      addDecl <| .thmDecl\n        { name := hothmName\n          levelParams := lvl\n          type := hot\n          value := prf }\n      addDeclarationRanges hothmName\n        { range := ← getDeclarationRange (← getRef)\n          selectionRange := ← getDeclarationRange ref }\n      _ ← addTermInfo (isBinder := true) ref <| ← mkConstWithLevelParams hothmName\n      let hsm := simpExtension.getState (← getEnv) |>.lemmaNames.contains (.decl thm)\n      if hsm then\n        addSimpTheorem simpExtension hothmName true false .global 1000\n      let some fcn ← getSimpExtension? `functor_norm | failure\n      let hfm := fcn.getState (← getEnv) |>.lemmaNames.contains <| .decl thm\n      if hfm then\n        addSimpTheorem fcn hothmName true false .global 1000\n      return hothmName\n  | _ => throwUnsupportedSyntax\n\n/-- The `higher_order` attribute. From a lemma of the shape `∀ x, f (g x) = h x` derive an\nauxiliary lemma of the form `f ∘ g = h` for reasoning about higher-order functions.\n\nSyntax: `[higher_order]` or `[higher_order name]` where the given name is used for the\ngenerated theorem. -/\ninitialize higherOrderAttr : ParametricAttribute Name ←\n  registerParametricAttribute {\n    name := `higherOrder,\n    descr :=\n\"From a lemma of the shape `∀ x, f (g x) = h x` derive an auxiliary lemma of the\nform `f ∘ g = h` for reasoning about higher-order functions.\n\nSyntax: `[higher_order]` or `[higher_order name]`, where the given name is used for the\ngenerated theorem.\",\n    getParam := higherOrderGetParam }\n\nend Tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/HigherOrder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.30953852211854904}}
{"text": "import Days\nimport Days.Common\nimport Std\nimport Lean.Data.Parsec\nimport Init.Data.Format.Basic\n\nopen Days\nopen Days.Common\nnamespace Days.Day08\ndef day: ProblemNumber := 8\n\nopen Std (RBSet)\n\n/--\nThe expedition comes across a peculiar patch of tall trees all planted carefully in a grid. The Elves explain that a previous expedition planted these trees as a reforestation effort. Now, they're curious if this would be a good location for a tree house.\n\nFirst, determine whether there is enough tree cover here to keep a tree house **hidden**. To do this, you need to count the number of trees that are **visible from outside the grid** when looking directly along a row or column.\n\nThe Elves have already launched a quadcopter to generate a map with the height of each tree (your puzzle input). For example:\n\n```\n30373\n25512\n65332\n33549\n35390\n```\n\nEach tree is represented as a single digit whose value is its height, where `0` is the shortest and `9` is the tallest.\n\nA tree is **visible** if all of the other trees between it and an edge of the grid are **shorter** than it. Only consider trees in the same row or column; that is, only look up, down, left, or right from any given tree.\n\nAll of the trees around the edge of the grid are **visible** - since they are already on the edge, there are no trees to block the view. In this example, that only leaves the **interior nine trees** to consider:\n\n  - The top-left 5 is visible from the left and top. (It isn't visible from the right or bottom since other trees of height 5 are in the way.)\n  - The top-middle 5 is visible from the top and right.\n  - The top-right 1 is not visible from any direction; for it to be visible, there would need to only be trees of height 0 between it and an edge.\n  - The left-middle 5 is visible, but only from the right.\n  - The center 3 is not visible from any direction; for it to be visible, there would need to be only trees of at most height 2 between it and an edge.\n  - The right-middle 3 is visible from the right.\n  - In the bottom row, the middle 5 is visible, but the 3 and 4 are not.\n\nWith 16 trees visible on the edge and another 5 visible in the interior, a total of **21** trees are visible in this arrangement.\n\nConsider your map; **how many trees are visible from outside the grid?**\n-/\ndef sample := \"30373\n25512\n65332\n33549\n35390\"\n\nstructure Tree where\n  height: Nat\n  deriving Ord, BEq, DecidableEq, Inhabited, Repr\n\ninstance : ToString Tree where\n  toString t := toString t.height\n\ninstance : LT Tree := ltOfOrd\ninstance : LE Tree := leOfOrd\n\nstructure ForestMap where\n  trees: Array $ Array Tree\n  deriving BEq, DecidableEq, Inhabited\n\ninstance : ToString ForestMap where\n  toString map := \n    map.trees\n    |>.map (λ row => String.intercalate \"\" (row |>.map toString |>.toList) )\n    |>.toList\n    |> String.intercalate \"\\n\"\n\ndef ForestMap.parse (lines: List String) : ForestMap :=\n  lines\n  |>.map (λ line =>\n    line.data\n    |>.map (λ c => c.toNat - '0'.toNat |> Tree.mk)\n    |> #[].appendList)\n  |> #[].appendList\n  |> ForestMap.mk\n\ndef sampleMap := sample |> toInput |>.lines |>.map (·.text) |> ForestMap.parse\n\n#eval sampleMap\n\ndef ForestMap.rows (map: ForestMap) : Nat :=\n  map.trees.size\n\ndef ForestMap.columns (map: ForestMap) : Nat :=\n  map.trees[0]!.size\n\ndef ForestMap.lastColumn (map: ForestMap) : Nat := \n  map.columns - 1\n\ndef ForestMap.lastRow (map: ForestMap) : Nat := \n  map.rows - 1\n\nstructure TreeWithPosition extends Tree where\n  row: Nat\n  col: Nat\n  deriving Inhabited\n\ninstance : Repr TreeWithPosition where\n  reprPrec (tree: TreeWithPosition) _ :=\n    .text s!\"{tree.height}({tree.row}×{tree.col})\"\n\ndef ForestMap.walkTrees (map: ForestMap) : Array TreeWithPosition := Id.run <| do\n  let mut l : Array TreeWithPosition :=  #[]\n  for row in [0:map.rows] do\n    for column in [0:map.columns] do\n      l := l.push <| TreeWithPosition.mk map.trees[row]![column]! row column\n  l\n\n#eval sampleMap.walkTrees |>.size\n#eval sampleMap.walkTrees[8]?\n\ninductive Direction where\n  | up\n  | down\n  | left\n  | right\n\ndef ForestMap.isVibileInDirection (map: ForestMap) (direction: Direction) (pos: TreeWithPosition) : Bool := Id.run <| do\n  let (rows, cols) :=\n    (match direction with\n    | .up => ([:pos.row], [pos.col:pos.col.succ])\n    | .down => ([pos.row + 1:map.rows], [pos.col:pos.col.succ])\n    | .left => ([pos.row:pos.row.succ], [:pos.col])\n    | .right => ([pos.row:pos.row.succ], [pos.col + 1:map.columns]))\n\n  for r in rows do\n    for c in cols do\n      if map.trees[r]![c]!.height >= pos.height then\n        return false\n\n  return true\n\n#eval sampleMap.isVibileInDirection Direction.up {height:=5, col:=1, row:=1}\n\ndef ForestMap.isEdgeTree (map: ForestMap) (tree: TreeWithPosition) : Bool :=\n  tree.row = 0 \n  ∨ tree.col = 0 \n  ∨ tree.row = map.lastRow\n  ∨ tree.col = map.lastColumn\n\ndef ForestMap.isVisible (map: ForestMap) (tree: TreeWithPosition) : Bool := Id.run <| do\n  if map.isEdgeTree tree then true\n  else\n  for dir in [Direction.up, Direction.down, Direction.left, Direction.right] do\n    if map.isVibileInDirection dir tree then return true\n  return false\n\ndef visibleTrees (map: ForestMap) : Array TreeWithPosition :=\n  map.walkTrees\n  |>.filter map.isVisible\n\n#eval visibleTrees sampleMap\n\n#eval visibleTrees sampleMap |>.size\n#eval visibleTrees sampleMap |>.filter (λ tree => ¬ sampleMap.isEdgeTree tree)\n\ndef countVisibleTrees (map: ForestMap) : Nat :=\n  visibleTrees map\n  |>.size\n\ndef part₁ (input: Input) : Nat := \n  input.lines\n  |>.map (·.text)\n  |> ForestMap.parse\n  |> countVisibleTrees\n\n/--\nContent with the amount of tree cover available, the Elves just need to know the best spot to build their tree house: they would like to be able to see a lot of trees.\n\nTo measure the viewing distance from a given tree, look up, down, left, and right from that tree; stop if you reach an edge or at the first tree that is the same height or taller than the tree under consideration. (If a tree is right on the edge, at least one of its viewing distances will be zero.)\n\nThe Elves don't care about distant trees taller than those found by the rules above; the proposed tree house has large eaves to keep it dry, so they wouldn't be able to see higher than the tree house anyway.\n\nIn the example above, consider the middle 5 in the second row:\n\n```\n30373\n25512\n65332\n33549\n35390\n```\n\nLooking up, its view is not blocked; it can see 1 tree (of height 3).\nLooking left, its view is blocked immediately; it can see only 1 tree (of height 5, right next to it).\nLooking right, its view is not blocked; it can see 2 trees.\nLooking down, its view is blocked eventually; it can see 2 trees (one of height 3, then the tree of height 5 that blocks its view).\nA tree's scenic score is found by multiplying together its viewing distance in each of the four directions. For this tree, this is 4 (found by multiplying 1 * 1 * 2 * 2).\n\nHowever, you can do even better: consider the tree of height 5 in the middle of the fourth row:\n\n```\n30373\n25512\n65332\n33549\n35390\n```\n\n- Looking up, its view is blocked at 2 trees (by another tree with a height of 5).\n- Looking left, its view is not blocked; it can see 2 trees.\n- Looking down, its view is also not blocked; it can see 1 tree.\n- Looking right, its view is blocked at 2 trees (by a massive tree of height 9).\n\nThis tree's scenic score is 8 (2 * 2 * 1 * 2); this is the ideal spot for the tree house.\n\nConsider each tree on your map. What is the highest scenic score possible for any tree?\n-/\ndef toList (range: Std.Range) : List Nat := Id.run <| do\n  let mut items := #[]\n  for x in range do\n    items := items.push x\n  return items.toList\n\ndef ForestMap.distanceToBlockerInDirection (map: ForestMap) (direction: Direction) (pos: TreeWithPosition) : Nat := Id.run <| do\n  let colRange := toList [pos.col:pos.col.succ]\n  let rowRange := toList [pos.row:pos.row.succ]\n  let (rows, cols) :=\n    (match direction with\n    | .up => (toList [1:pos.row] |>.reverse, colRange)\n    | .down => (toList [pos.row + 1:map.lastRow], colRange)\n    | .left => (rowRange, toList [1:pos.col] |>.reverse)\n    | .right => (rowRange, toList [pos.col + 1:map.lastColumn]))\n\n  let mut i := 1\n  for r in rows do\n    for c in cols do\n      if map.trees[r]![c]!.height >= pos.height then\n        return i\n      else i := i+1\n\n  return i\n\ndef ForestMap.visibility (map: ForestMap) (tree: TreeWithPosition) : Nat := Id.run <| do\n  if map.isEdgeTree tree then 0\n  else\n  let mut dist := 1\n  for dir in [Direction.up, Direction.down, Direction.left, Direction.right] do\n    dist := dist * map.distanceToBlockerInDirection dir tree\n  return dist\n\ndef expectedBest : TreeWithPosition := {height:=5, col:=2, row:=3}\n#eval sampleMap.distanceToBlockerInDirection .up expectedBest\n#eval sampleMap.distanceToBlockerInDirection .left expectedBest\n#eval sampleMap.distanceToBlockerInDirection .down expectedBest\n#eval sampleMap.distanceToBlockerInDirection .right expectedBest\n#eval sampleMap.visibility expectedBest\n\ndef problemTree : TreeWithPosition :=  {height:=6, row:=2, col:=0}\n#eval sampleMap.visibility problemTree\n#eval sampleMap.distanceToBlockerInDirection .up problemTree\n#eval sampleMap.distanceToBlockerInDirection .left problemTree\n#eval sampleMap.distanceToBlockerInDirection .down problemTree\n#eval sampleMap.distanceToBlockerInDirection .right problemTree\n\n#eval (sampleMap.walkTrees\n  |>.map (λ tree => (tree, sampleMap.visibility tree))\n  |>.toList)\n\ndef bestTreeVisibilityScore (map: ForestMap) : Nat :=\n  map.walkTrees\n  |>.map (map.visibility ·)\n  |>.toList\n  |>.maximum?\n  |>.get!\n\n#eval bestTreeVisibilityScore sampleMap\n\ndef part₂ (input: Input) : Nat := \n  input.lines\n  |>.map (·.text)\n  |> ForestMap.parse\n  |> bestTreeVisibilityScore\n\ndef solution : Problem Nat := ⟨ day, part₁, part₂ ⟩ \n\n\n#eval testPart₁ solution sample (expect:=21)\n#eval testPart₂ solution sample (expect:=8)\n\n", "meta": {"author": "jakeswenson", "repo": "advent2022", "sha": "af941092292ff0bc5552bce9c145d6b5b173c20d", "save_path": "github-repos/lean/jakeswenson-advent2022", "path": "github-repos/lean/jakeswenson-advent2022/advent2022-af941092292ff0bc5552bce9c145d6b5b173c20d/Days/Day08.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.309538522118549}}
{"text": "import category_theory.preadditive\nimport algebraic_topology.cech_nerve\n\nimport for_mathlib.simplicial.complex\nimport for_mathlib.arrow.split\n\nnamespace category_theory\n\nuniverses v u v' u'\n\nnamespace arrow\n\nnoncomputable theory\nopen_locale simplicial\nopen category_theory.limits\n\nvariables {C : Type u} [category.{v} C] (f : arrow C)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\n\n/-- The splittings of the Cech nerve associated to a split arrow. -/\ndef cech_splitting [split f] (n : ℕ) : f.cech_nerve _[n] ⟶ f.cech_nerve _[n+1] :=\nwide_pullback.lift (wide_pullback.base _)\n(λ i, if h : i.down = 0 then wide_pullback.base _ ≫ split.σ else wide_pullback.π _ ⟨i.down.pred h⟩)\nbegin\n  rintro ⟨j⟩,\n  split_ifs,\n  tidy,\nend\n\nlemma cech_splitting_face_zero [split f] (n : ℕ) :\n  f.cech_splitting n ≫ f.cech_nerve.δ 0 = 𝟙 _ :=\nbegin\n  ext ⟨j⟩,\n  dsimp [cech_splitting, simplicial_object.δ],\n  simp only [category.id_comp, category.assoc, wide_pullback.lift_π],\n  split_ifs,\n  { exfalso,\n    exact fin.succ_ne_zero _ h },\n  { congr,\n    dsimp [simplicial_object.δ, simplex_category.δ],\n    simp },\n  { dsimp [simplicial_object.δ, cech_splitting],\n    simp },\nend\n\nlemma face_π (n : ℕ) (i : fin (n+1)) (j : fin (n+2)) :\n  (f.cech_nerve.δ j : f.cech_nerve _[n+1] ⟶ _) ≫ wide_pullback.π _ ⟨i⟩ =\n  wide_pullback.π _ ⟨j.succ_above i⟩ :=\nbegin\n  change wide_pullback.lift _ _ _ ≫ _ = _,\n  simpa,\nend\n\nlemma cech_splitting_face [split f] (n : ℕ) (j : fin (n+3)) (hj : j ≠ 0) :\n  f.cech_splitting (n+1) ≫ f.cech_nerve.δ j =\n  f.cech_nerve.δ (j.pred hj) ≫ f.cech_splitting n :=\nbegin\n  ext ⟨k⟩,\n  swap,\n  { dsimp [cech_splitting, simplicial_object.δ],\n    simp },\n  { dsimp [cech_splitting, simplicial_object.δ],\n    simp only [category.assoc, limits.wide_pullback.lift_π],\n    split_ifs with h1 h2,\n    { simp },\n    { refine false.elim (h2 _),\n      change j.succ_above k = 0 at h1,\n      change k = 0,\n      rwa ← fin.succ_above_eq_zero_iff hj },\n    { refine false.elim (h1 _),\n      erw h,\n      change j.succ_above 0 = 0,\n      rw fin.succ_above_eq_zero_iff hj },\n    { simp only [category_theory.limits.wide_pullback.lift_π],\n      congr,\n      change (j.succ_above k).pred h1 = (j.pred hj).succ_above (k.pred h),\n      rw fin.pred_succ_above_pred,\n      refl } }\nend\n\nend arrow\n\nnamespace arrow\n\nsection contracting_homotopy\n\nopen category_theory.limits opposite\n\n-- Note: Universe restrictions! I hope this doesn't pose any issues later...\n-- jmc: seems like it might! I removed them.\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : Pᵒᵖ ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\n\n/-- The augmented Cech conerve induced by applying M to `f.augmented_cech_nerve`. -/\nabbreviation conerve : cosimplicial_object.augmented N :=\n((cosimplicial_object.augmented.whiskering _ _).obj M).obj f.augmented_cech_nerve.right_op\n\nvariables [arrow.split f] [preadditive N]\n\n/-- The morphisms yielding the contracting homotopy. -/\ndef contracting_homotopy : Π (n : ℕ),\n  (f.conerve M).to_cocomplex.X (n+1) ⟶ (f.conerve M).to_cocomplex.X n\n| 0 := M.map (wide_pullback.lift (𝟙 _)\n         (λ i, (split.σ : f.right ⟶ _))\n         (by simp)).op\n| (n+1) := M.map (f.cech_splitting n).op\n\nopen cosimplicial_object.augmented\nopen_locale big_operators\n\nlemma is_contracting_homotopy_zero :\n  (f.conerve M).to_cocomplex.d 0 1 ≫ f.contracting_homotopy M 0 = 𝟙 _ :=\nbegin\n  dsimp,\n  rw if_pos,\n  swap, { simp },\n  delta conerve,\n  dsimp [to_cocomplex_d, to_cocomplex_obj, contracting_homotopy ],\n  simp only [category.id_comp, category.comp_id],\n  simp_rw [← M.map_comp, ← op_comp, ← M.map_id, ← op_id],\n  congr' 2,\n  simp,\nend\n\nlemma is_contracting_homotopy_one :\n  (f.conerve M).to_cocomplex.d 1 2 ≫ f.contracting_homotopy M 1 +\n  f.contracting_homotopy M 0 ≫ (f.conerve M).to_cocomplex.d 0 1 = 𝟙 _ :=\nbegin\n  rw ← add_zero (𝟙 ((conerve M f).to_cocomplex.X 1)),\n  dsimp only [to_cocomplex, cochain_complex.of],\n  rw dif_pos,\n  swap, { dec_trivial },\n  rw dif_pos,\n  swap, { dec_trivial },\n  dsimp,\n  delta conerve,\n  dsimp only [to_cocomplex_d, cosimplicial_object.coboundary, whiskering, whiskering_obj,\n    drop, to_cocomplex_obj, comma.snd, cosimplicial_object.whiskering, whiskering_right,\n    contracting_homotopy],\n  simp_rw fin.sum_univ_succ,\n  simp only [fin.coe_zero, fin.sum_univ_zero, fin.coe_one,\n    one_zsmul, preadditive.add_comp, pow_one, fin.succ_zero_eq_one,\n    category.id_comp, neg_smul, category.comp_id, preadditive.neg_comp, pow_zero ],\n  rw [add_assoc],\n  congr' 1,\n  { dsimp [cosimplicial_object.δ],\n    simp_rw [← M.map_comp, ← M.map_id, ← op_id, ← op_comp],\n    congr' 2,\n    dsimp [cech_splitting],\n    ext ⟨j⟩,\n    { simp only [wide_pullback.lift_π, category.id_comp, category.assoc],\n      split_ifs,\n      { cases j,\n        dsimp at h,\n        injection h with hh,\n        simp only [nat.succ_ne_zero] at hh,\n        cases hh },\n      { congr, } },\n    { simp only [wide_pullback.lift_base, category.assoc, category.id_comp] } },\n  { dsimp [cosimplicial_object.δ],\n    rw [add_assoc, neg_add_eq_zero, ← M.map_comp],\n    simp only [zero_comp, category.id_comp, zero_add, functor.map_comp, ← M.map_comp, ← op_comp],\n    congr' 2,\n    dsimp [cech_splitting],\n    ext ⟨j⟩,\n    { simp only [wide_pullback.lift_π, category.assoc],\n      split_ifs,\n      { refl },\n      { refine false.elim (h _),\n        change (1 : fin 2).succ_above j = 0,\n        rw fin.succ_above_eq_zero_iff,\n        { simp },\n        { exact top_ne_bot } } },\n    { simp only [wide_pullback.lift_base, category.assoc, category.comp_id] } }\nend\n\nlemma is_contracting_homotopy (n : ℕ) :\n  (f.conerve M).to_cocomplex.d (n+2) (n+3) ≫ f.contracting_homotopy M (n+2) +\n  f.contracting_homotopy M (n+1) ≫ (f.conerve M).to_cocomplex.d (n+1) (n+2) = 𝟙 _ :=\nbegin\n  dsimp,\n  erw if_pos,\n  swap, refl,\n  dsimp only [to_cocomplex_d],\n  rw if_pos,\n  swap, refl,\n  dsimp only [cosimplicial_object.coboundary],\n  simp only [preadditive.sum_comp, preadditive.comp_sum],\n  erw [fin.sum_univ_succ, add_assoc, ← finset.sum_add_distrib],\n  rw ← add_zero (𝟙 ((conerve M f).to_cocomplex_obj (n + 2))),\n  dsimp only [cosimplicial_object.δ],\n  congr' 1,\n  { delta conerve,\n    dsimp [to_cocomplex_obj, contracting_homotopy],\n    simp only [category_theory.category.comp_id, one_zsmul, pow_zero],\n    simp_rw [← M.map_id, ← M.map_comp, ← op_comp, ← op_id],\n    congr' 2,\n    apply cech_splitting_face_zero },\n  { apply fintype.sum_eq_zero,\n    intros i,\n    simp only [\n      category.comp_id,\n      add_zero,\n      functor.comp_map,\n      fin.coe_succ,\n      preadditive.comp_zsmul,\n      preadditive.zsmul_comp],\n    suffices :\n      (drop.obj (conerve M f)).map (simplex_category.δ i.succ) ≫ contracting_homotopy M f (n+2) =\n          contracting_homotopy M f (n+1) ≫ (drop.obj (conerve M f)).map (simplex_category.δ i),\n    { rw [this, pow_succ],\n      simp },\n    delta conerve,\n    dsimp [contracting_homotopy],\n    simp_rw [← M.map_comp, ← op_comp],\n    congr' 2,\n    convert cech_splitting_face _ _ _ (fin.succ_ne_zero _),\n    funext,\n    congr,\n    simp }\nend\n\nend contracting_homotopy\n\nsection covariant_contracting_homotopy\n\nopen category_theory.limits opposite\n\nvariables {P : Type u} {N : Type u'} [category.{v} P] [category.{v'} N] (M : P ⥤ N)\nvariables (f : arrow P)\nvariables [∀ n : ℕ, has_wide_pullback f.right (λ i : ulift (fin (n+1)), f.left) (λ i, f.hom)]\n\n/-- The augmented Cech nerve induced by applying M to `f.augmented_cech_nerve`. -/\nabbreviation nerve : simplicial_object.augmented N :=\n((simplicial_object.augmented.whiskering _ _).obj M).obj f.augmented_cech_nerve\n\nvariables [arrow.split f] [preadditive N]\n\nopen simplicial_object.augmented\nopen_locale big_operators\n\ndef covariant_contracting_homotopy : Π (n : ℕ),\n  (f.nerve M).to_complex.X n ⟶ (f.nerve M).to_complex.X (n+1)\n| 0 := M.map $ wide_pullback.lift (𝟙 _) (λ i, (split.σ : f.right ⟶ _)) (by simpa)\n| (n+1) := M.map $ f.cech_splitting n\n\nlemma covariant_is_contracting_homotopy_zero :\n  f.covariant_contracting_homotopy M 0 ≫ (f.nerve M).to_complex.d 1 0 = 𝟙 _ :=\nbegin\n  dsimp,\n  rw if_pos,\n  swap, { simp },\n  delta nerve,\n  dsimp [to_complex_obj, to_complex_d, covariant_contracting_homotopy],\n  simp only [category.id_comp, category.comp_id],\n  simp_rw [← M.map_comp, ← M.map_id],\n  congr' 2,\n  simp,\nend\n\nlemma covariant_is_contracting_homotopy_one :\n  f.covariant_contracting_homotopy M 1 ≫ (f.nerve M).to_complex.d 2 1 +\n  (f.nerve M).to_complex.d 1 0 ≫ f.covariant_contracting_homotopy M 0 = 𝟙 _ :=\nbegin\n  dsimp [covariant_contracting_homotopy],\n  rw if_pos, rw if_pos, any_goals { dec_trivial },\n  dsimp [to_complex_d, simplicial_object.boundary, simplicial_object.δ],\n  delta nerve,\n  dsimp [to_complex_obj],\n  simp only [fin.sum_univ_succ, fin.coe_zero, pow_zero, one_zsmul, fintype.univ_of_subsingleton,\n    nat.add_def, fin.mk_eq_subtype_mk, fin.mk_zero, fin.coe_succ, pow_one, neg_smul,\n    finset.sum_singleton, preadditive.comp_add, category.id_comp,\n    preadditive.comp_neg, category.comp_id],\n  simp only [← M.map_comp],\n  rw add_assoc,\n  convert add_zero _,\n  swap,\n  { symmetry,\n    convert M.map_id _,\n    dsimp [arrow.cech_splitting],\n    ext ⟨⟨j,hj⟩⟩, simp,\n    rw dif_neg, refl,\n    dsimp [simplex_category.δ],\n    have : j = 0, by simpa using hj, subst this, dec_trivial,\n    simp },\n  { rw neg_add_eq_zero,\n    congr' 1,\n    dsimp [cech_splitting],\n    ext ⟨⟨j,hj⟩⟩,\n    { simp only [category.assoc, wide_pullback.lift_π], dsimp,\n      rw dif_pos, have : j = 0, by simpa using hj, subst this,\n      dsimp [simplex_category.δ], dec_trivial },\n    { simp } }\nend\n\nlemma covariant_is_contracting_homotopy (n : ℕ) :\n  f.covariant_contracting_homotopy M (n+2) ≫ (f.nerve M).to_complex.d (n+3) (n+2) +\n  (f.nerve M).to_complex.d (n+2) (n+1) ≫ f.covariant_contracting_homotopy M (n+1) = 𝟙 _ :=\nbegin\n  dsimp, rw if_pos, rw if_pos, swap, { refl }, swap, { refl },\n  simp only [category.comp_id, category.id_comp],\n  dsimp only [to_complex_d, simplicial_object.boundary],\n  simp only [preadditive.sum_comp, preadditive.comp_sum],\n  rw [fin.sum_univ_succ, add_assoc, ← finset.sum_add_distrib],\n  convert add_zero _,\n  { apply fintype.sum_eq_zero, intros j,\n    have : ((j.succ : fin _) : ℕ) = (j : ℕ) + 1 := by simp, rw this, clear this,\n    rw [pow_succ],\n    simp only [neg_mul, one_mul, neg_smul, preadditive.comp_neg],\n    rw neg_add_eq_zero,\n    simp only [preadditive.comp_zsmul, preadditive.zsmul_comp],\n    congr' 1,\n    dsimp [covariant_contracting_homotopy, simplicial_object.δ],\n    delta nerve,\n    dsimp [whiskering],\n    simp only [← M.map_comp],\n    congr' 1,\n    convert cech_splitting_face _ _ _ (fin.succ_ne_zero _), funext i,\n    congr, simp },\n  { dsimp [covariant_contracting_homotopy],\n    simp only [pow_zero, one_zsmul],\n    delta nerve,\n    dsimp [arrow.cech_splitting, simplicial_object.whiskering, simplicial_object.δ],\n    rw ← M.map_comp, symmetry,\n    convert M.map_id _,\n    ext ⟨⟨j,hj⟩⟩,\n    simp only [category.assoc, wide_pullback.lift_π],\n    rw dif_neg, dsimp, simpa, dsimp [simplex_category.δ],\n    intro c, apply_fun (λ e, e.1) at c, simpa using c,\n    simp, dsimp, simp }\nend\n\nend covariant_contracting_homotopy\n\nend arrow\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/Cech/split.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.309536427529843}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.functorial\nimport category_theory.monoidal.functor_category\nimport category_theory.limits.has_limits\n\n/-!\n# `lim : (J ⥤ C) ⥤ C` is lax monoidal when `C` is a monoidal category.\n\nWhen `C` is a monoidal category, the functorial association `F ↦ limit F` is lax monoidal,\ni.e. there are morphisms\n* `lim_lax.ε : (𝟙_ C) → limit (𝟙_ (J ⥤ C))`\n* `lim_lax.μ : limit F ⊗ limit G ⟶ limit (F ⊗ G)`\nsatisfying the laws of a lax monoidal functor.\n-/\n\nopen category_theory\nopen category_theory.monoidal_category\n\nnamespace category_theory.limits\n\nuniverses v u\n\nnoncomputable theory\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{v} C] [has_limits C]\n\ninstance limit_functorial : functorial (λ F : J ⥤ C, limit F) := { ..limits.lim }\n\n@[simp] lemma limit_functorial_map {F G : J ⥤ C} (α : F ⟶ G) :\n  map (λ F : J ⥤ C, limit F) α = limits.lim.map α := rfl\n\nvariables  [monoidal_category.{v} C]\n\n@[simps]\ninstance limit_lax_monoidal : lax_monoidal (λ F : J ⥤ C, limit F) :=\n{ ε := limit.lift _ { X := _, π := { app := λ j, 𝟙 _, } },\n  μ := λ F G, limit.lift (F ⊗ G)\n    { X := limit F ⊗ limit G,\n      π :=\n      { app := λ j, limit.π F j ⊗ limit.π G j,\n        naturality' := λ j j' f,\n        begin\n          dsimp,\n          simp only [category.id_comp, ←tensor_comp, limit.w],\n        end, } },\n  μ_natural' := λ X Y X' Y' f g,\n  begin\n    ext, dsimp,\n    simp only [limit.lift_π, cones.postcompose_obj_π, monoidal.tensor_hom_app, limit.lift_map,\n      nat_trans.comp_app, category.assoc, ←tensor_comp, lim_map_π],\n  end,\n  associativity' := λ X Y Z,\n  begin\n    ext, dsimp,\n    simp only [limit.lift_π, cones.postcompose_obj_π, monoidal.associator_hom_app, limit.lift_map,\n      nat_trans.comp_app, category.assoc],\n    slice_lhs 2 2 { rw [←tensor_id_comp_id_tensor], },\n    slice_lhs 1 2 { rw [←comp_tensor_id, limit.lift_π], dsimp, },\n    slice_lhs 1 2 { rw [tensor_id_comp_id_tensor], },\n    conv_lhs { rw [associator_naturality], },\n    conv_rhs { rw [←id_tensor_comp_tensor_id (limit.π (Y ⊗ Z) j)], },\n    slice_rhs 2 3 { rw [←id_tensor_comp, limit.lift_π], dsimp, },\n    dsimp, simp,\n  end,\n  left_unitality' := λ X,\n  begin\n    ext, dsimp,\n    simp,\n    conv_rhs { rw [←tensor_id_comp_id_tensor (limit.π X j)], },\n    slice_rhs 1 2 { rw [←comp_tensor_id], erw [limit.lift_π], dsimp, },\n    slice_rhs 2 3 { rw [left_unitor_naturality], },\n    simp,\n  end,\n  right_unitality' := λ X,\n  begin\n    ext, dsimp,\n    simp,\n    conv_rhs { rw [←id_tensor_comp_tensor_id _ (limit.π X j)], },\n    slice_rhs 1 2 { rw [←id_tensor_comp], erw [limit.lift_π], dsimp, },\n    slice_rhs 2 3 { rw [right_unitor_naturality], },\n    simp,\n  end, }\n\n/-- The limit functor `F ↦ limit F` bundled as a lax monoidal functor. -/\ndef lim_lax : lax_monoidal_functor (J ⥤ C) C := lax_monoidal_functor.of (λ F : J ⥤ C, limit F)\n\n@[simp] lemma lim_lax_obj (F : J ⥤ C) : lim_lax.obj F = limit F := rfl\n\nlemma lim_lax_obj' (F : J ⥤ C) : lim_lax.obj F = lim.obj F := rfl\n\n@[simp] lemma lim_lax_map {F G : J ⥤ C} (α : F ⟶ G) : lim_lax.map α = lim.map α := rfl\n\n@[simp] lemma lim_lax_ε :\n  (@lim_lax J _ C _ _ _).ε = limit.lift _ { X := _, π := { app := λ j, 𝟙 _, } } := rfl\n\n@[simp] lemma lim_lax_μ (F G : J ⥤ C) :\n  (@lim_lax J _ C _ _ _).μ F G = limit.lift (F ⊗ G)\n    { X := limit F ⊗ limit G,\n      π :=\n      { app := λ j, limit.π F j ⊗ limit.π G j,\n        naturality' := λ j j' f,\n        begin\n          dsimp,\n          simp only [category.id_comp, ←tensor_comp, limit.w],\n        end, } } := rfl\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573377, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3094690638448387}}
{"text": "example (P Q : Type) : P → (Q → P) :=\nbegin\nintro p,\nintro q,\nexact p,\nend\n", "meta": {"author": "abdelq", "repo": "natural-number-game", "sha": "bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2", "save_path": "github-repos/lean/abdelq-natural-number-game", "path": "github-repos/lean/abdelq-natural-number-game/natural-number-game-bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2/world05/level05.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3094639279221546}}
{"text": "import FOL.language_extension consistency FOL.lindenbaum\nopen encodable\n\nuniverses u v\n\nnamespace fol\nopen logic logic.Theory\nopen_locale logic_symbol\n\n@[simp] lemma app_concat {α β : Type*} (a : α) (s : ℕ → α) (g : α → β) :\n  (λ x, g ((a ⌢ s) x)) = g a ⌢ g ∘ s :=\nby { funext x, cases x; simp  }\n\n@[simp] lemma app_concat' {α β γ : Type*} (a : α) (s : ℕ → α) (g : α → β) (h : β → γ):\n  (λ x, h (g ((a ⌢ s) x))) = h (g a) ⌢ h ∘ g ∘ s :=\nby { funext x, cases x; simp  }\n\nopen term formula\nvariables (L : language.{u}) (T : Theory L)\n\nnamespace henkin\nopen language language.extension language.consts_pelimination Theory language.language_translation\n\n@[reducible] def extend : language.{u} := L + consts (formula L)\n\nvariables {L}\n\ndef henkin_axiom (p : formula L) : formula (extend L) := (∃.↑p) ⟶ rew ı[0 ⇝ p] ↑p\n\n@[reducible] def Theory_extend : Theory (extend L) := ↑T ∪ set.range henkin_axiom\n\nvariables {T}\n\nsection\nopen axiomatic_classical_logic axiomatic_classical_logic' provable\n\nvariables {S : Theory (extend L)} (Γ : list (formula L))\n\nlemma consistent_of_disjoint (S_consis : S.consistent) (disj : ∀ p ∈ S, disjoint Γ p) (p : formula (extend L)) :\n  ¬S ⊢ (∀.[Γ.length] ∼((pelimination' Γ).p 0 p)) → Theory.consistent (S +{ p }) := λ not_b,\nbegin\n  simp [Theory.consistent_iff_bot],\n  intros b,\n  have : S ⊢ ∼p, from of_equiv_p (deduction.mp b) (equiv_symm $ neg_iff _),\n  have : S ⊢ ∀.[Γ.length] ∼(pelimination' Γ).p 0 p,\n  { have := provable_pelimination_of_disjoint Γ S (∼p) disj this, simp at this, exact this },\n  contradiction\nend\n\nlemma tauto (p : formula L) : S ⊢ ∃.((∃.↑p)^1 ⟶ ↑p) :=\nbegin\n  have lmm₁ : S ⊢ (∃.↑p) ⟶ ∃.((∃.↑p)^1 ⟶ ↑p),\n  { simp[pnf_imply_ex_iff_fal_imply₁], refine generalize (deduction.mp _),\n    refine use #0 _, simp[formula.nested_rew] },\n  have lmm₂ : S ⊢ ∼(∃.↑p) ⟶ ∃.((∃.↑p)^1 ⟶ ↑p),\n  { refine deduction.mp (use #0 (deduction.mp _)), simp,\n    show S +{ ∼∃.↑p } +{ ∃.↑p } ⊢ rew ı[0 ⇝ #0] ↑p,\n    exact explosion (show S +{ ∼∃.↑p } +{ ∃.↑p } ⊢ ∃.↑p, by simp) (show S +{ ∼∃.↑p } +{ ∃.↑p } ⊢ ∼∃.↑p, by simp) },\n  exact cases_of _ _ lmm₁ lmm₂\nend\n\n@[simp] lemma pelimination'_henkin_axiom (p : formula L) : (pelimination' ([p])).p 0 (henkin_axiom p) = ((∃.↑p)^1 ⟶ ↑p) :=\nbegin\n  simp[henkin_axiom, pelimination'_subst, pelimination_coe_eq_pow_coe_aux],\n  have : pelim_aux_t ([p]) 0 ↑↑p = (#0 : term (extend L)),\n  { have : pelim_aux_t ([p]) 0 ↑p = (#(list.index_of p ([p]) + 0) : term (extend L)), from pelim_aux_t_consts_of_Γ ([p]) p (by simp) 0,\n    simp at this, exact this },\n  simp[this, formula.nested_rew, show (#0 ⌢ λ x, #(1 + x) : ℕ → term (extend L)) = ı, by { funext x, rcases x; simp[add_comm 1] }],\n  simp[formula.pow_eq, add_comm 1]\nend\n\nlemma consts_of_henkin_axiom {p : formula L} {c} (mem : c ∈ consts_of_p (henkin_axiom p)) : c = p :=\nbegin\n  simp[henkin_axiom, consts_of_p] at mem,\n  rcases mem with ⟨t, (t_mem | t_mem), c_mem⟩,\n  { rcases language_translation_coe.fun_p_inversion_of_mem t_mem with ⟨t, t_mem, rfl⟩, simp at c_mem, contradiction },\n  { rcases formula.rew_inversion_or_le_of_mem_rew t_mem with (⟨t, t_mem', k, rfl⟩ | ⟨k, n, le⟩),\n    { rcases language_translation_coe.fun_p_inversion_of_mem t_mem' with ⟨t, t_mem'', rfl⟩,\n      simp[subst_pow] at c_mem,\n      rcases mem_of_consts_of_t_subst _ _ _ c_mem with (c_mem | c_mem),\n      { simp at c_mem, contradiction },\n      {exact eq_of_consts_of_t_coe.mp c_mem } },\n    { simp[subst_pow] at le, have : n < k ∨ n = k ∨ k < n , exact trichotomous n k, rcases this with (lt | rfl | lt),\n      { simp[lt] at le, simp[le] at c_mem, contradiction },\n      { simp[consts.coe_def, show (↑(consts.c p) : (extend L).fn 0) = sum.inr (consts.c p), from rfl, le_iff_lt_or_eq] at le,\n        rcases le with (⟨⟨_, A⟩, _⟩ | rfl), { simp at c_mem, exact c_mem } },\n      { simp[lt] at le, simp[le] at c_mem, contradiction } } }\nend\n\nlemma disjoint_of_ne (p q : formula L) (ne : p ≠ q) : disjoint ([p]) (henkin_axiom q) := λ p mem,\nby { simp at mem, rcases mem with rfl, intros mem, have : p = q, from consts_of_henkin_axiom mem, contradiction }\n\nend\n\ntheorem Theory_extend_consistent (consis : T.consistent) : (Theory_extend T).consistent :=\nbegin\n  have : (↑T : Theory (extend L)).consistent, from language.add_consts.consistent_iff.mpr consis,\n  refine consistent.of_finite_induction this _,\n  intros s s_ss φ φ_mem p_nmem s_fin consis',\n  let U := ↑T ∪ s,\n  rcases φ_mem with ⟨φ, rfl⟩,\n  show Theory.consistent (U +{ henkin_axiom φ }),\n  have disj : ∀ p ∈ U, disjoint ([φ]) p,\n  { intros p mem, simp[U] at mem, rcases mem,\n    { rcases mem with ⟨p, mem, rfl⟩, show disjoint ([φ]) (↑p), intros h, simp },\n    { have : p ∈ set.range henkin_axiom, from s_ss mem,\n      rcases this with ⟨p, rfl⟩,\n      have : φ ≠ p, { rintros rfl, contradiction },\n      exact disjoint_of_ne _ _ this } },\n  have : ¬U ⊢ ∀.∼((∃.↑φ)^1 ⟶ ↑φ),\n  { intros b,\n    have : U ⊢ ∼∀.∼((∃.↑φ)^1 ⟶ ↑φ), from tauto φ,\n    have : ¬U.consistent, { simp[Theory.consistent], refine ⟨_, b, this⟩ },\n    contradiction },\n  exact consistent_of_disjoint ([φ]) consis' disj (henkin_axiom φ) (by simp; exact this)\nend\n\nvariables (L) (T)\n\n@[reducible] def Lang : ℕ → language\n| 0     := L\n| (n+1) := extend (Lang n)\n\nlocal notation `L[` n `]` := Lang L n\n\n@[reducible] def Consts : Type u := Σ n, formula (Lang L n)\n\ndef pConsts (m : ℕ) : Type u := Σ (n : {n | m ≤ n}), formula (Lang L n)\n\n@[reducible] def limit : language := L + consts (Consts L)  \n\nlocal notation `Lω` := limit L\n\nlocal infix ` ='∘ `:50 := ((=') : term Lω → term Lω → formula Lω)\n\n@[reducible] def Ln_to_Lω : Π n, L[n] ↝ᴸ Lω\n| 0     := add_left\n| (n+1) := let σ : consts (formula L[n]) ↝ᴸ consts (Consts L) := consts_of_fun (λ p, ⟨n, p⟩) in\n    (Ln_to_Lω n).sum (add_right.comp σ)\n\ndef pConsts_to_add_pConsts (n : ℕ) : consts (pConsts L n) ↝ᴸ consts (formula L[n]) + consts (pConsts L (n + 1)) :=\n{ fn := λ m f, by { rcases m,\n    { rcases f with ⟨⟨i, hi⟩, p⟩, simp at hi p,\n      by_cases e : i = n,\n      { rcases e with rfl, refine sum.inl p },\n      { have : n + 1 ≤ i, from nat.succ_le_iff.mpr ((ne.symm e).le_iff_lt.mp hi), refine sum.inr ⟨⟨i, this⟩, p⟩ } },\n    { rcases f } },\n  pr := λ m r, by { rcases r } }\n\n@[reducible] def Lω_to_Ln_add_pConsts : Π n, Lω ↝ᴸ L[n] + consts (pConsts L n)\n| 0       := (1 : L ↝ᴸ L).add (consts_of_fun (λ ⟨n, p⟩, ⟨⟨n, by simp⟩, p⟩))\n| (n + 1) :=\n  let τ : L[n] + consts (pConsts L n) ↝ᴸ L[n + 1] + consts (pConsts L (n + 1)) := (add_assoc'_inv _ _ _).comp ((1 : L[n] ↝ᴸ L[n]).add ( pConsts_to_add_pConsts L n)) in\n    τ.comp (Lω_to_Ln_add_pConsts n)\n\nlemma pConsts_to_add_pConsts_of_eq {n : ℕ} (p : formula L[n]) :\n  (pConsts_to_add_pConsts L n).fn 0 (consts.c ⟨⟨n, by { show n ≤ n, by refl }⟩, p⟩) = sum.inl p :=\nby simp[pConsts_to_add_pConsts, consts.c]\n\nlemma pConsts_to_add_pConsts_of_lt {n m : ℕ} (p : formula L[m]) (lt : n < m) :\n  (pConsts_to_add_pConsts L n).fn 0 (consts.c ⟨⟨m, le_of_lt lt⟩, p⟩) = sum.inr (consts.c ⟨⟨m, nat.succ_le_iff.mpr lt⟩, p⟩) :=\nby simp[pConsts_to_add_pConsts, consts.c, show ¬m = n, from ne_of_gt lt]\n\nlemma Lω_to_Ln_add_pConst_fn_of_le {n m : ℕ} (p : formula L[m]) (le : n ≤ m) :\n  (Lω_to_Ln_add_pConsts L n).fn 0 (show (consts (Consts L)).fn 0, from ⟨m, p⟩) = sum.inr (consts.c ⟨⟨m, le⟩, p⟩) :=\nbegin\n  simp,\n  induction n with n IH generalizing p m; simp[Lω_to_Ln_add_pConsts, ← coe_fn₂],\n  { refl },\n  { simp[IH p (nat.le_of_succ_le le), ← coe_fn₂],\n    have : n < m, from nat.succ_le_iff.mp le,\n    simp[pConsts_to_add_pConsts_of_lt L p this, ←coe_fn₂] }\nend\n\nlemma commutative (n : ℕ) : (Lω_to_Ln_add_pConsts L n).comp (Ln_to_Lω L n) = add_left :=\nbegin\n  induction n with n IH,\n  { ext; simp[Ln_to_Lω, Lω_to_Ln_add_pConsts, add_left_fn_to_coe, add_left_pr_to_coe] },\n  { ext m,\n    { rcases f; simp[Ln_to_Lω, limit, ←coe_fn₁, ←coe_fn₂],\n      { have : (Lω_to_Ln_add_pConsts L n).fn m ((Ln_to_Lω L n).fn m f) = ↑f,\n        { have := (language_translation.eq_iff.mp IH).1 m f, simpa using this },\n        simp[this, add_left_fn_to_coe] },\n      { rcases m,\n        { simp[add_right_fn_to_coe],\n          have := Lω_to_Ln_add_pConst_fn_of_le L f (by refl), simp at this,\n          simp[this, ←coe_fn₂, ←coe_fn₁, add_left_fn_to_coe, pConsts_to_add_pConsts_of_eq L f,\n            show (pConsts_to_add_pConsts L n).fn 0 (consts.c ⟨⟨n, show n ≤ n, by refl⟩, f⟩) = _, from pConsts_to_add_pConsts_of_eq L f] },\n        { rcases f } } },\n    { intros m r, rcases r; simp[Ln_to_Lω, limit, ←coe_pr₁, ←coe_pr₂], \n      { have : (Lω_to_Ln_add_pConsts L n).pr m ((Ln_to_Lω L n).pr m r) = ↑r,\n        { have := (language_translation.eq_iff.mp IH).2 m r, simpa using this },\n        simp[this, add_left_pr_to_coe] },\n      { rcases r } } }\nend\n\nlemma add_left_Ln_to_Lω_commutative (n : ℕ) : (Ln_to_Lω L (n + 1)).comp (add_left : L[n] ↝ᴸ L[n+1]) = Ln_to_Lω L n :=\nby ext; simp[Ln_to_Lω, add_left_fn_to_coe, add_left_pr_to_coe]\n\nvariables {L}\n\ndef Lω_seq_limit : seq_limit (Lang L) Lω :=\n{ seq := λ n, add_left,\n  to_limit := Ln_to_Lω L,\n  commutes := add_left_Ln_to_Lω_commutative L,\n  rank_fn := λ n f, by { rcases f, { exact 0 }, { rcases n, { rcases f with ⟨m, f⟩, exact (m + 1) }, { rcases f } } },\n  rank_pr := λ n r, by { rcases r, { exact 0 }, { rcases r } },\n  fn := λ n f, by { rcases f, { exact f }, { rcases n, { rcases f with ⟨m, f⟩, exact sum.inr (consts.c f) }, { rcases f } } },\n  pr := λ n r, by { rcases r, { exact r }, { rcases r } },\n  fn_spec := λ n f, by { rcases f, { refl }, { cases n, { rcases f with ⟨m, f⟩, refl }, { rcases f } } },\n  pr_spec := λ n r, by { rcases r, { refl }, { rcases r } } }\n\ndef retruct (p : formula Lω) : formula L[Lω_seq_limit.rank_p p] := Lω_seq_limit.retruct_p p\n\ndef henkin_constant (p : formula (limit L)) : Consts L := ⟨Lω_seq_limit.rank_p p, retruct L p⟩\n\n@[simp] lemma retruct_s_c (p : formula Lω) : (Ln_to_Lω L _).fun_p (retruct L p) = p := Lω_seq_limit.retruct_p_spec p \n\n@[simp] lemma retruct_s (p : formula Lω) : (Ln_to_Lω L _).fun_p (retruct L p) = p := Lω_seq_limit.retruct_p_spec p \n\nlemma Ln_to_Lω_fun_p_coe (n : ℕ) (p : formula L[n]) :\n  (Ln_to_Lω L (n + 1)).fun_p (by simp[Lang]; refine p) = (Ln_to_Lω L n).fun_p p :=\nby simp[←add_left_Ln_to_Lω_commutative L n, comp_fun_p]; refl\n\n@[simp] lemma Ln_to_Lω_fun_t_coe {n} (p : formula L[n]) :\n  (Ln_to_Lω L (n + 1)).fun_t (↑(↑p : term (consts (formula L[n]))) : term (extend L[n])) = ↑(⟨n, p⟩ : Consts L) :=\nbegin\n  unfold_coes, simp[Ln_to_Lω],\n  suffices : ((Ln_to_Lω L n).sum (add_right.comp (consts_of_fun (sigma.mk n)))).fn 0 (↑(consts.c p)) = ↑(consts.c (⟨n, p⟩ : Consts L)),\n  { exact this },\n  simp, refl\nend\n\nvariables (T) {L}\n\n@[reducible] def Thy : Π n : ℕ, Theory (Lang L n)\n| 0       := T\n| (n + 1) := Theory_extend (Thy n)\n\ndef Thy' (n : ℕ) := (Ln_to_Lω L n).fun_Theory (Thy T n)\n\nlocal notation `T[` n `]` := Thy' T n\n\ndef Theory_limit : Theory Lω := ⋃ n, T[n]\n\nlocal notation `Tω` := Theory_limit L T\n\nlemma Thy'_ss (n : ℕ) : T[n] ⊆ T[n + 1] :=\nbegin\n  simp[fun_Theory, Thy', Thy],\n  intros p mem, simp[Lang], refine ⟨by simp[Lang]; refine p, _⟩,\n  simpa[mem] using Ln_to_Lω_fun_p_coe _ n p,\nend\n\nlemma Tn_consistent (consis : T.consistent) (n : ℕ) : Theory.consistent T[n] :=\nbegin\n  have : Theory.consistent (Thy T n),\n  { induction n with n IH, { exact consis }, { exact Theory_extend_consistent IH } },\n  have consis' : Theory.consistent (add_left.fun_Theory (Thy T n) : Theory (L[n] + consts (pConsts L n))), from add_consts.consistent_iff.mpr this,\n  have : (add_left.fun_Theory (Thy T n) : Theory (L[n] + consts (pConsts L n))) = (Lω_to_Ln_add_pConsts L n).fun_Theory T[n],\n  { rw[fun_Theory, ←commutative, comp_fun_p, set.image_comp], simp[fun_Theory, Thy'] },\n  rw this at consis',\n  exact language.language_translation.consistency _ _ consis'\nend\n\ntheorem Tω_consistent (consis : T.consistent) : Theory.consistent Tω :=\n(Theory.consistent.Union_seq (Thy' T) (Thy'_ss T)).mpr (Tn_consistent T consis)\n\n\nlemma T_ss_Tω : ↑T ⊆ Tω :=\nby simpa [show ↑T = T[0], by refl, Theory_limit] using set.subset_Union (Thy' T) 0\n\nlemma henkin_mem_Tω (p : formula Lω) : (∃.p) ⟶ rew ı[0 ⇝ henkin_constant p] p ∈ Tω :=\nbegin\n  simp[henkin_constant],\n  let r := Lω_seq_limit.rank_p p,\n  suffices : (∃.p) ⟶ rew ı[0 ⇝ henkin_constant p] p ∈ T[r + 1],\n  { simp[Theory_limit], refine ⟨_, this⟩ },\n  have lmm₁ : (∃.p) ⟶ rew ı[0 ⇝ henkin_constant p] p = (Ln_to_Lω L (r + 1)).fun_p (henkin_axiom (retruct L p)),\n  { have := Ln_to_Lω_fun_p_coe L r (retruct L p), simp at this, simp[this, henkin_axiom, henkin_constant] },\n  have lmm₂ : (Ln_to_Lω L (r + 1)).fun_p (henkin_axiom (retruct L p)) ∈ T[r + 1],\n  { simp[Thy'], refine ⟨henkin_axiom (retruct L p), by simp[Thy], rfl⟩ },\n  rw lmm₁, exact lmm₂\nend\n\nlocal notation `Tω⁺` := consistent.maximal Tω\n\nlocal notation (name := limit) `Lω` := limit _\n\nopen axiomatic_classical_logic axiomatic_classical_logic' provable\n\ndef Consts.of (t : term Lω) : Consts L := henkin_constant ((t^1) ='∘ #0)\n#check Consts.of\n\ndef Consts.fn {n} (f : (limit L).fn n) (v : finitary (Consts L) n) : Consts L := Consts.of (app f (λ i, (v i)))\n\ndef Consts.pr {n} (r : (limit L).pr n) (v : finitary (Consts L) n) : Prop := formula.app r (λ i, (v i)) ∈ Tω⁺\n\ndef Consts.equal (p q : Consts L) : Prop := (p ='∘ q) ∈ Tω⁺\n\nlocal notation p ` ~[`:50 T :50 `] `:0 q:50 := Consts.equal T p q\n\nvariables [cT : Theory.Consistent T] {L}\ninclude cT\n\nlemma Tω_consistent' : Theory.consistent (Theory_limit L T) := Tω_consistent T cT.consis\n\nlemma mem_Tω'_of_Tω_provable {p : formula Lω} (b : Tω ⊢ p): p ∈ Tω⁺ :=\nbegin\n  have : Tω⁺ ⊢ p, from weakening' (Tω_consistent' T).ss_maximal b,\n  exact (Tω_consistent' T).mem_maximal_iff.mpr this\nend\n\n@[simp, refl] lemma Consts.equal_refl (c : Consts L) : c ~[T] c := by simp[Consts.equal, (Tω_consistent' T).mem_maximal_iff]\n\ntheorem Consts.equal_equivalence : equivalence (Consts.equal T) :=\n⟨λ p, by simp[Consts.equal, (Tω_consistent' T).mem_maximal_iff],\n  λ p q, by { simp[Consts.equal, (Tω_consistent' T).mem_maximal_iff], exact provable.eq_symm },\n  λ p q r, by { simp[Consts.equal, (Tω_consistent' T).mem_maximal_iff], exact provable.eq_trans }⟩\n\nlemma Consts.of_eq (t : term Lω) :\n t ='∘ Consts.of t ∈ Tω⁺ :=\nbegin\n  have : ∃.(t^1 ='∘ #0) ⟶ (t ='∘ Consts.of t) ∈ Tω, by simpa[Consts.fn] using henkin_mem_Tω L T (t^1 =' #0),\n  have lmm₁ : Tω ⊢ ∃.(t^1 ='∘ #0) ⟶ (t ='∘ Consts.of t), from by_axiom this,\n  have lmm₂ : Tω ⊢ ∃.(t^1 ='∘ #0), from provable.use t (by simp),\n  have : Tω ⊢ t ='∘ Consts.of t, from lmm₁ ⨀ lmm₂,\n  exact mem_Tω'_of_Tω_provable T this\nend\n\nlemma Consts.fn_eq {n} (f : (limit L).fn n) (v : finitary (Consts L) n) :\n (app f (λ i, (v i)) ='∘ Consts.fn f v) ∈ Tω⁺ :=\nConsts.of_eq _ _\n\n@[reducible, simp, instance]\ndef henkin : setoid (Consts L) := ⟨Consts.equal T, Consts.equal_equivalence T⟩\n\ndef Henkin : Type u := quotient (henkin T)\n\nnamespace Henkin\nvariables {T}\n\n@[elab_as_eliminator]\nprotected lemma ind_on {C : Henkin T → Prop} (d : Henkin T)\n  (h : ∀ p : Consts L, C ⟦p⟧) : C d :=\nquotient.induction_on' d h\n\n@[elab_as_eliminator, reducible]\ndef lift_on {φ} (d : Henkin T) (f : Consts L → φ)\n  (h : ∀ p q : Consts L, p ~[T] q → f p = f q) : φ :=\nquotient.lift_on' d f h\n\n@[simp]\nlemma lift_on_eq {φ} (p : Consts L) (f : Consts L → φ)\n  (h : ∀ p q : Consts L, p ~[T] q → f p = f q) : lift_on ⟦p⟧ f h = f p := rfl\n\n@[elab_as_eliminator, reducible, simp]\ndef lift_on₂ {φ} (d₁ d₂ : Henkin T) (f : Consts L → Consts L → φ)\n  (h : ∀ p₁ p₂ q₁ q₂, p₁ ~[T] q₁ → p₂ ~[T] q₂ → f p₁ p₂ = f q₁ q₂) : φ :=\nquotient.lift_on₂' d₁ d₂ f h\n\n@[simp] lemma lift_on₂_eq {φ} (p q : Consts L) (f : Consts L → Consts L → φ)\n  (h : ∀ p₁ p₂ q₁ q₂, p₁ ~[T] q₁ → p₂ ~[T] q₂ → f p₁ p₂ = f q₁ q₂) :\nlift_on₂ ⟦p⟧ ⟦q⟧ f h = f p q := rfl\n\ndef lift_on_finitary {φ} {n : ℕ} (v : finitary (Henkin T) n) (f : finitary (Consts L) n → φ)\n  (h : ∀ v₁ v₂ : finitary (Consts L) n, (∀ n, (v₁ n) ~[T] (v₂ n)) → f v₁ = f v₂) : φ :=\nquotient.lift_on_finitary v f h \n\n@[simp] lemma lift_on_finitary_eq {φ} {n} (v : finitary (Consts L) n) (f : finitary (Consts L) n → φ)\n  (h : ∀ v₁ v₂ : finitary (Consts L) n, (∀ n, (v₁ n) ~[T] (v₂ n)) → f v₁ = f v₂) :\n  lift_on_finitary (λ x, ⟦v x⟧) f h = f v :=\nquotient.lift_on_finitary_eq v f h\n\n@[simp] lemma of_eq_of {p q : Consts L} : (⟦p⟧ : Henkin T) = ⟦q⟧ ↔ p ~[T] q := quotient.eq'\n\ndef fn {n} (f : (Lω).fn n) : finitary (Henkin T) n → Henkin T :=\nλ v, lift_on_finitary v (λ v, (⟦Consts.fn f v⟧ : Henkin T)) $ λ v₁ v₂ eqs,\nbegin\n  simp[of_eq_of, Consts.equal, (Tω_consistent' T).mem_maximal_iff],\n  show      Tω⁺ ⊢ Consts.fn f v₁ ='∘ Consts.fn f v₂,\n  have b₁ : Tω⁺ ⊢ app f (λ i, (v₁ i)) ='∘ Consts.fn f v₁, from (Tω_consistent' T).mem_maximal_iff.mp (Consts.fn_eq T f v₁),\n  have b₂ : Tω⁺ ⊢ app f (λ i, (v₂ i)) ='∘ Consts.fn f v₂, from (Tω_consistent' T).mem_maximal_iff.mp (Consts.fn_eq T f v₂),\n  have b  : Tω⁺ ⊢ app f (λ i, (v₁ i)) ='∘ app f (λ i, (v₂ i)),\n  { have : ∀ i, Tω⁺ ⊢ v₁ i ='∘ v₂ i, from λ i, (Tω_consistent' T).mem_maximal_iff.mp (eqs i),\n    have : Tω⁺ ⊢ ⋀ i, v₁ i ='∘ v₂ i, refine conjunction_iff.mpr this,\n    exact function_ext' f (λ i, (v₁ i)) (λ i, (v₂ i)) ⨀ this },\n  exact eq_trans (eq_trans b₁.eq_symm b) b₂\nend\n\n@[simp] lemma fn_app {n} (f : (Lω).fn n) (v : finitary (Consts L) n) :\n  fn f (λ i, ⟦v i⟧) = ⟦Consts.fn f v⟧ :=\nlift_on_finitary_eq _ _ _\n\ndef pr {n} (r : (Lω).pr n) : finitary (Henkin T) n → Prop :=\nλ v, lift_on_finitary v (Consts.pr T r) $ λ v₁ v₂ eqs,\nbegin\n  simp[of_eq_of, Consts.pr, (Tω_consistent' T).mem_maximal_iff],\n  suffices : Tω⁺ ⊢ app r (λ i, (v₁ i)) ⟷ app r (λ i, (v₂ i)),\n  { split; intros h, { exact (iff_equiv.mp this).1 ⨀ h }, { exact (iff_equiv.mp this).2 ⨀ h } },\n  have : ∀ i, Tω⁺ ⊢ v₁ i ='∘ v₂ i, from λ i, (Tω_consistent' T).mem_maximal_iff.mp (eqs i),\n  have : Tω⁺ ⊢ ⋀ i, v₁ i ='∘ v₂ i, refine conjunction_iff.mpr this,\n  exact predicate_ext'' r (λ i, (v₁ i)) (λ i, (v₂ i)) ⨀ this\nend\n\n@[simp] lemma pr_app {n} (r : (Lω).pr n) (v : finitary (Consts L) n) :\n  pr r (λ i, ⟦v i⟧) ↔ formula.app r (λ i, (v i)) ∈ Tω⁺ :=\nby { have : pr r (λ i, ⟦v i⟧) = Consts.pr T r v, from lift_on_finitary_eq _ _ _, simp[this, Consts.pr] }\n\nvariables (T)\n\ndef henkin_model : Structure Lω := ⟨Henkin T, ⟨⟦henkin_constant ⊤⟧⟩, λ n f, fn f, λ n r, pr r⟩\n\nlocal notation `ℌ` := henkin_model T\n\n@[simp] lemma henkin_model_fn {n} (f : (Lω).fn n) (v) : (ℌ).fn f v = fn f v := rfl\n@[simp] lemma henkin_model_pr {n} (r : (Lω).pr n) (v) : (ℌ).pr r v = pr r v := rfl\n\nlemma eq_Consts_of_eq {t u} : Consts.of t ~[T] Consts.of u ↔ Tω⁺ ⊢ t ='∘ u  :=\nbegin\n  simp[Consts.equal, (Tω_consistent' T).mem_maximal_iff],\n  have lmm₁ : Tω⁺ ⊢ t ='∘ Consts.of t, from by_axiom (Consts.of_eq T _), \n  have lmm₂ : Tω⁺ ⊢ u ='∘ Consts.of u, from by_axiom (Consts.of_eq T _), \n  refine ⟨λ h, eq_trans (eq_trans lmm₁ h) (eq_symm lmm₂), λ h, eq_trans (eq_trans (eq_symm lmm₁) h) lmm₂⟩\nend\n\nlemma term_val_eq_quotient (t : term Lω) (e : ℕ → Consts L) :\n  @term.val _ ℌ (λ n, ⟦e n⟧ : ℕ → (ℌ).dom) t = ⟦Consts.of (t.rew (λ x, ↑(e x)))⟧ :=\nbegin\n  induction t,\n  case var : x { simp[Consts.equal], exact Consts.of_eq T _ },\n  case app : n f v IH\n  { simp,\n    have : (λ i, @term.val _ ℌ (λ n, ⟦e n⟧) (v i)) = (λ i, ⟦Consts.of ((v i).rew (λ x, ↑(e x)))⟧),\n    { funext i, exact IH i },\n    simp[this, Consts.fn],\n    have : Tω⁺ ⊢ app f (λ i, (Consts.of ((v i).rew (λ x, (e x))))) ='∘ app f (λ i, (v i).rew (λ x, (e x))),\n    { have : ∀ i, Tω⁺ ⊢ Consts.of ((v i).rew (λ x, (e x))) ='∘ (v i).rew (λ x, (e x)),\n        from λ i, eq_symm (by_axiom (Consts.of_eq T ((v i).rew (λ x, (e x))))),\n      have : Tω⁺ ⊢ ⋀ i, Consts.of ((v i).rew (λ x, (e x))) ='∘ (v i).rew (λ x, (e x)),\n        from conjunction_iff.mpr this,\n      exact function_ext' f _ _ ⨀ this },\n    refine (eq_Consts_of_eq _).mpr this }\nend\n\nprivate lemma ℌ_models_Tω'_fal\n  {p : formula Lω} \n  (IH : ∀ (s : ℕ → Consts L), p.rew (λ x, ↑(s x)) ∈ Tω⁺ ↔ @formula.val _ ℌ (λ x, ⟦s x⟧) p)\n  (s : ℕ → Consts L) :\n  (∀.p).rew (λ x, ↑(s x)) ∈ Tω⁺ ↔ ℌ ⊧[λ x, ⟦s x⟧] (∀.p) :=\nbegin\n  simp, split,\n    { intros h,  intros c,\n      induction c using fol.henkin.Henkin.ind_on,\n      let s' : ℕ → Consts L := c ⌢ s,\n      have : p.rew (λ x, (s' x)) ∈ Tω⁺,\n      { simp[(Tω_consistent' T).mem_maximal_iff] at h ⊢,\n        have : Tω⁺ ⊢ p.rew (c ⌢ λ x, (s x)),  simpa[formula.nested_rew] using h ⊚ c,\n        simpa using this },\n      simpa[s', app_concat c s] using (IH s').1 this },\n    { intros h,\n      have h : ∀ (c : Consts L), @formula.val _ ℌ (λ x, ⟦(c ⌢ s) x⟧) p, { intros c, simp, exact h ⟦c⟧ },\n      by_contradiction A,\n      let φ := ∼p.rew ((λ x, (s x))^1),\n      have lmm₁ : Tω⁺ ⊢  p.rew (henkin_constant φ ⌢ (λ x, (s x))),\n        by simpa using by_axiom ((IH _).mpr (h $ henkin_constant φ)),         \n      have : Tω⁺ ⊢ ∼∀.p.rew ((λ x, (s x))^1), from by_axiom ((Tω_consistent' T).neg_mem_maximal_iff.mpr A),      \n      have : Tω⁺ ⊢ ∃.φ, from (iff_of_equiv (provable.neg_fal_equiv_ex_neg _)).mp this,\n      have lmm₂ : Tω⁺ ⊢ ∼p.rew (henkin_constant φ ⌢ (λ x, (s x))),\n        by simpa[φ, formula.nested_rew] using by_axiom ((Tω_consistent' T).ss_maximal (henkin_mem_Tω L T φ)) ⨀ this,\n      exact (Tω_consistent' T).maximal_consistent ⟨_, lmm₁, lmm₂⟩ }\nend\n\ntheorem ℌ_models_Tω' (p : formula Lω) (s : ℕ → Consts L) :\n  p.rew (λ x, ↑(s x)) ∈ Tω⁺ ↔ ℌ ⊧[λ x, ⟦s x⟧] p :=\nbegin\n  induction p generalizing s,\n  case app : n r v\n  { simp[term_val_eq_quotient, (Tω_consistent' T).mem_maximal_iff],\n    suffices :  Tω⁺ ⊢ app r (λ i, (v i).rew (λ x, (s x))) ⟷ app r (λ i, Consts.of ((v i).rew (λ x, (s x)))), from iff_of_equiv this,\n    have : ∀ i, Tω⁺ ⊢ (v i).rew (λ x, (s x)) ='∘ Consts.of ((v i).rew (λ x, (s x))),\n      from λ i, (by_axiom (Consts.of_eq T ((v i).rew (λ x, (s x))))),\n    have :      Tω⁺ ⊢ ⋀ i, (v i).rew (λ x, (s x)) ='∘ Consts.of ((v i).rew (λ x, (s x))),\n      from conjunction_iff.mpr this,\n    exact predicate_ext'' r _ _ ⨀ this },\n  case equal : t u { simp[term_val_eq_quotient, (Tω_consistent' T).mem_maximal_iff], exact iff.symm (eq_Consts_of_eq _) },\n  case imply : p q IH_p IH_q { simp[(Tω_consistent' T).imply_mem_maximal_iff] at*, simp[IH_p s, IH_q s] },\n  case verum { simp[(Tω_consistent' T).mem_maximal_iff] },\n  case neg : p IH { simp[(Tω_consistent' T).neg_mem_maximal_iff], exact iff.not (IH s) },\n  case fal : p IH { exact ℌ_models_Tω'_fal T IH s }\nend\n\ntheorem ℌ_models_Tω'_of_sentence {p : formula Lω} (hp : is_sentence p) :\n  p ∈ Tω⁺ ↔ ℌ ⊧ p :=\nby simpa[hp, eval_is_sentence_iff] using ℌ_models_Tω' T p default\n\nvariables [closed_Theory T]\n\ntheorem ℌ_models_T : add_left.of_ltr ℌ ⊧ T :=\nbegin\n  simp[Theory_models_iff], intros p mem,\n  have : p ∈ Tω⁺, from (Tω_consistent' T).ss_maximal (T_ss_Tω T mem),  \n  exact (ℌ_models_Tω'_of_sentence T (show is_sentence p, from closed_Theory.cl mem)).mp this\nend \n\nend Henkin\n\nend henkin\n\nvariables [closed_Theory T] {L} {T}\n\ntheorem consistent_iff_satisfiable : Theory.consistent T ↔ Satisfiable T :=\n⟨λ consis,  ⟨_, @henkin.Henkin.ℌ_models_T _ T ⟨consis⟩ _⟩, by { rintros ⟨M, hM⟩, exact Structure_consistent hM}⟩\n\ntheorem completeness {p : formula L} (hp : is_sentence p) : \n  T ⊢ p ↔ T ⊧ p :=\n⟨soundness,\n λ h, by { simp[Theory.provable_iff_inconsistent], intros consis,\n  have : closed_Theory (T +{∼p}), from ⟨by { simp[hp], exact λ _, closed_Theory.cl }⟩,\n  have : Satisfiable (T +{∼p}), exactI consistent_iff_satisfiable.mp consis,\n  rcases this with ⟨M, hM⟩,\n  have : M ⊧ p, from consequence_def.mp h M (λ p mem, hM (by simp[mem])),\n  have : ¬M ⊧ p, by simpa[models_neg_iff_of_is_sentence hp] using @hM (∼p) (by simp),\n  contradiction }⟩\n\ntheorem completeness' {p : formula L} : \n  T ⊢ p ↔ T ⊧ p :=\n⟨soundness,\n λ h, by {\n  have : T ⊧ ∀.* p, { intros M hM, simpa using nfal_models_iff.mpr (consequence_def.mp h M hM) },\n  have : T ⊢ ∀.* p, from (completeness (show is_sentence (∀.* p), by simp)).mpr this,\n  have lmm : T ⊢ p.rew (λ x, ite (x < p.arity) #x #(x - p.arity)), from provable.nfal_subst _ _ ı ⨀ this,\n  have := @formula.rew_rew _ p (λ x, ite (x < p.arity) #x #(x - p.arity)) ı (λ m lt, by simp[lt]),\n  simpa[this] using lmm }⟩\n\nsection\nvariables {L₁ L₂ : language.{u}} {T₁ : Theory L₁} [closed_Theory T₁]\nopen language language.language_translation language.language_translation_coe\n\n@[simp] theorem add_coe_consistent_iff :\n  (↑T₁ : Theory (L₁ + L₂)).consistent ↔ T₁.consistent :=\n⟨language.language_translation.consistency _ T₁,\n λ h,\nbegin\n  have : ∃ M₁, M₁ ⊧ T₁, from consistent_iff_satisfiable.mp h,\n  rcases this with ⟨M₁, hM₁⟩,\n  let M₂ : Structure (L₁ + L₂) := M₁.extend default default,\n  have : M₂ ⊧ (↑T₁ : Theory (L₁ + L₂)), from (M₁.extend_modelsth_coe_iff default default).mpr hM₁,\n  exact consistent_iff_satisfiable.mpr ⟨_, this⟩\nend⟩\n\nsection\nvariables (L₁ L₂) [language.language_translation_coe L₁ L₂]\n\n@[simp] theorem coe_consistent_iff  :\n  (↑T₁ : Theory L₂).consistent ↔ T₁.consistent :=\n⟨language.language_translation.consistency _ T₁,\nλ h, begin\n  have : ∃ M₁, M₁ ⊧ T₁, from consistent_iff_satisfiable.mp h,\n  rcases this with ⟨M₁, hM₁⟩,\n  let M₂ : Structure (L₁ + sub L₂ L₁) := M₁.extend default default,\n  have : M₂ ⊧ (↑T₁ : Theory (L₁ + sub L₂ L₁)), from (M₁.extend_modelsth_coe_iff default default).mpr hM₁,\n  have e : (language_equiv.add_sub' L₁ L₂).ltr.fun_Theory ↑T₁ = ↑T₁,\n  { have : ((language_equiv.add_sub' L₁ L₂).ltr.comp extension.add_left).fun_Theory T₁ = ltr.fun_Theory T₁,\n      by simp[language_equiv.add_sub'_add_left_commute L₁ L₂],\n    simpa[comp_fun_Theory] using this },\n  have : M₂.of_equiv (language_equiv.add_sub' L₁ L₂) ⊧ ↑T₁,\n    by simpa[e] using (Structure.equiv_modelsth_iff (language.language_equiv.add_sub' L₁ L₂)).mpr this,\n  exact Structure_consistent this\nend⟩\n\ntheorem coe_provable_iff (p : formula L₁) (hp : is_sentence p) :\n  (↑T₁ : Theory L₂) ⊢ ↑p ↔ T₁ ⊢ p :=\nbegin\n  simp[Theory.provable_iff_inconsistent],\n  suffices : ¬Theory.consistent (↑(T₁ +{ ∼p } : Theory L₁) : Theory L₂) ↔ ¬Theory.consistent (T₁ +{ ∼p }),\n    by simpa[language.language_translation_coe.fun_Theory_insert] using this, \n  have : closed_Theory (T₁ +{ ∼p }) := ⟨by { simp[hp], exact λ _, closed_Theory.cl }⟩,\n  have : Theory.consistent (↑(T₁ +{ ∼p } : Theory L₁) : Theory L₂) ↔ Theory.consistent (T₁ +{ ∼p }),\n    by exactI coe_consistent_iff L₁ L₂,\n  simp[this]\nend\n\n@[simp] theorem coe_provable_iff' {p : formula L₁} :\n  (↑T₁ : Theory L₂) ⊢ ↑p ↔ T₁ ⊢ p :=\nby { have : (↑T₁ : Theory L₂) ⊢ ↑∀.* p ↔ T₁ ⊢ ∀.* p, from coe_provable_iff L₁ L₂ (∀.* p) (by simp),\n      simpa[←provable.iff_fal_complete] using this }\n\nend\n  \nsection\n-- Extensions by definitions\nvariables {L₁ L₂} (D : L₁.definitions L₂)\n\ntheorem extensions_by_definitions_consistent (consis : T₁.consistent)\n  (b : ∀ {n} (f : L₂.fn n), T₁ ⊢ ∀.[n] ∃.(D.df_fn f)) :\n  Theory.consistent (↑T₁ ∪ D.thy) :=\nbegin\n  suffices : ∃ M, M ⊧ (↑T₁ ∪ D.thy),\n  { exactI consistent_iff_satisfiable.mpr this },  \n  have : ∃ M, M ⊧ T₁, from consistent_iff_satisfiable.mp consis,\n  rcases this with ⟨M, hM⟩,\n  have : ∀ (n) (f : L₂.fn n) (v : finitary M.dom n), ∃ (d : M.dom),\n    (D.df_fn f).val M (d ⌢ (λ i, if h : i < n then v ⟨i, h⟩ else default)),\n  { intros n f,\n    have : M ⊧[default] ∀.[n] ∃.(D.df_fn f), from (soundness (b f) hM) default,\n    simpa[models_univs] using this },\n  have : ∀ n : ℕ, ∃ F' : L₂.fn n → finitary M.dom n → M.dom,\n    ∀ (f : L₂.fn n) (v : finitary M.dom n), (D.df_fn f).val M (F' f v ⌢ (λ i, if h : i < n then v ⟨i, h⟩ else default)),\n  { intros n, choose F' hF' using this n, exact ⟨F', hF'⟩ },\n  choose F hF using this,\n  let R : Π n, L₂.pr n → finitary M.dom n → Prop := λ n r v, M ⊧[λ i, if h : i < n then v ⟨i, h⟩ else default] (D.df_pr r),\n  let M₂ : Structure (L₁ + L₂) := M.extend F R,\n  have lmm₁ : ∀ {n} (f : L₂.fn n), M₂ ⊧ def_fn f (D.df_fn f),\n  { intros n f,\n    rw [←eval_is_sentence_iff default (show is_sentence (def_fn f (D.df_fn f)), by simp[D.hdf_fn])],\n    simp[def_fn, models_univs], intros v,\n    exact (Structure.extend_val_coe_iff M _ _).mpr (hF n f v) },\n  have lmm₂ : ∀ {n} (r : L₂.pr n), M₂ ⊧ def_pr r (D.df_pr r),\n  { intros n r,\n    rw [←eval_is_sentence_iff default (show is_sentence (def_pr r (D.df_pr r)), by simp[D.hdf_pr])],\n    simp[def_pr, models_univs, R], intros v,\n    exact iff.symm (Structure.extend_val_coe_iff M F R), exact ⟨λ _, default⟩ },\n  have lmm₃ : M₂ ⊧ ↑T₁, from (Structure.extend_modelsth_coe_iff M _ _).mpr hM, \n  refine ⟨M₂, by { simp[lmm₃, definitions_def], split; intros n; simp[logic.semantics.Models_def, lmm₁, lmm₂] }⟩\nend\n\nvariables (T₁)\n\ntheorem extensions_by_definitions_consistent_iff\n  (b : ∀ {n} (f : L₂.fn n), T₁ ⊢ ∀.[n] ∃.(D.df_fn f)) :\n  Theory.consistent (↑T₁ ∪ D.thy) ↔ T₁.consistent :=\n⟨λ h, begin\n  have : (↑T₁ : Theory (L₁ + L₂)).consistent, from Theory.consistent_of_consistent_ss h (by simp),\n  exact add_coe_consistent_iff.mp this\nend, λ h,  extensions_by_definitions_consistent D h @b⟩\n\ntheorem extensions_by_definitions_conservative\n  (b : ∀ {n} (f : L₂.fn n), T₁ ⊢ ∀.[n] ∃.(D.df_fn f))\n  (p : formula L₁) (hp : p.is_sentence) :\n  ↑T₁ ∪ D.thy ⊢ p ↔ T₁ ⊢ p :=\nbegin\n  simp[Theory.provable_iff_inconsistent],\n  suffices : ¬Theory.consistent (↑(T₁+{ ∼p }) ∪ D.thy) ↔ ¬Theory.consistent (T₁+{ ∼p }),\n  { have eq : ((↑T₁ ∪ D.thy) +{ ∼↑p }) = (↑(T₁+{ ∼p }) ∪ D.thy),\n    { ext q, simp[language.language_translation_coe.fun_Theory_insert, or_assoc] },\n    simpa[eq] using this },\n  have : closed_Theory (T₁ +{ ∼p }) := ⟨by { simp[hp], exact λ _, closed_Theory.cl }⟩,\n  exact iff.not (by exactI extensions_by_definitions_consistent_iff (T₁+{ ∼p }) D (λ n f, by simp[b f]))\nend\n\ntheorem extensions_by_definitions_conservative'\n  (b : ∀ {n} (f : L₂.fn n), T₁ ⊢ ∀.[n] ∃.(D.df_fn f))\n  (p : formula L₁) :\n  ↑T₁ ∪ D.thy ⊢ p ↔ T₁ ⊢ p :=\nby { have := extensions_by_definitions_conservative T₁ D @b (∀.* p) (by simp),\n     simpa[←provable.iff_fal_complete] using this }\n\ntheorem extensions_by_definitions_consistent_iff_of_predicate [language.predicate L₂] :\n  Theory.consistent (↑T₁ ∪ D.thy) ↔ T₁.consistent :=\nextensions_by_definitions_consistent_iff T₁ D (λ n f, by { exfalso, exact is_empty.false f })\n\nend\n\nend\n\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/completeness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5, "lm_q1q2_score": 0.30939021687192503}}
{"text": "import category_theory.functor.category\nimport category_theory.eq_to_hom\n\nopen category_theory\n\nnamespace category_theory\n\nnamespace functor\n\n--lemma congr_obj {D D' : Type*} [category D] [category D'] {F G : D ⥤ D'}\n--(h : F = G) : ∀ X : D, F.obj X = G.obj X :=\n--by { intro X, rw h, }\n\n--lemma congr_map {D D' : Type*} [category D] [category D'] (F : D ⥤ D')\n--{X Y : D} {f g : X ⟶ Y} (h : f = g) : F.map f = F.map g :=\n--by { subst h, }\n\nlemma assoc_eq {C₁ C₂ C₃ C₄ : Type*} [category C₁] [category C₂] [category C₃] [category C₄]\n  (F : C₁ ⥤ C₂) (G : C₂ ⥤ C₃) (H : C₃ ⥤ C₄) :\n  (F ⋙ G) ⋙ H = F ⋙ (G ⋙ H) := rfl\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/functor_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.3093504143698383}}
{"text": "import traversals.basic\nimport category_theory.currying\n\nopen category_theory\nopen category_theory.limits\nopen simplex_category\nopen sSet\nopen_locale simplicial\n\nnamespace traversal\n\nnamespace geom_real_rec\n\nvariables {n : ℕ}\n\ndef sSet_colimit {sh : Type*} [small_category sh] (diag : sh ⥤ sSet) :\n  colimit_cocone (diag) :=\n{ cocone := combine_cocones (diag) (λ n,\n  { cocone := types.colimit_cocone _,\n    is_colimit := types.colimit_cocone_is_colimit _ }),\n  is_colimit := combined_is_colimit _ _, }\n\ndef sSet_pushout {X Y Z : sSet} (f : X ⟶ Y) (g : X ⟶ Z) := sSet_colimit (span f g)\n\ndef bundle : Π (θ : traversal n), Σ (g : sSet), Δ[n] ⟶ g\n| ⟦⟧       := ⟨Δ[n], 𝟙 _⟩\n| (e :: θ) :=\n  let colim := sSet_pushout (to_sSet_hom (δ e.t)) (bundle θ).2 in\n  ⟨colim.cocone.X, to_sSet_hom (δ e.s) ≫ pushout_cocone.inl colim.cocone⟩\n\nend geom_real_rec\n\ndef geom_real_rec {n} (θ : traversal n) : sSet := (geom_real_rec.bundle θ).1\n\nnamespace geom_real_rec\nvariables {n : ℕ}\n\ndef geom_real_incl (θ : traversal n) : Δ[n] ⟶ geom_real_rec θ := (geom_real_rec.bundle θ).2\n\ndef bundle_colim (e : edge n) (θ : traversal n) :=\n  sSet_pushout (to_sSet_hom (δ e.t)) (bundle θ).2\n\n@[simp]\nlemma geom_real_rec_nil : geom_real_rec (⟦⟧ : traversal n) = Δ[n] := rfl\n\n@[simp]\nlemma geom_real_incl_nil : geom_real_incl (⟦⟧ : traversal n) = 𝟙 Δ[n] := rfl\n\nlemma geom_real_rec_cons (e : edge n) (θ : traversal n) :\n  geom_real_rec (e :: θ) = (bundle_colim e θ).cocone.X := rfl\n\nlemma geom_real_incl_cons (e : edge n) (θ : traversal n) :\n  geom_real_incl (e :: θ) = to_sSet_hom (δ e.s)\n    ≫ pushout_cocone.inl (bundle_colim e θ).cocone := rfl\n\ndef j_rec_bundle : Π (θ : traversal n),\n  {j : geom_real_rec θ ⟶ Δ[n] // geom_real_incl θ ≫ j = 𝟙 Δ[n]}\n| ⟦⟧       := ⟨𝟙 Δ[n], rfl⟩\n| (e :: θ) :=\nbegin\n  let j_θ := j_rec_bundle θ,\n  refine ⟨(bundle_colim e θ).is_colimit.desc (pushout_cocone.mk (to_sSet_hom (σ e.1)) j_θ.1 _), _⟩,\n  change _ = geom_real_incl θ ≫ j_θ.val, rw j_θ.2,\n  swap,\n  rw [geom_real_incl_cons, category.assoc],\n  rw [(bundle_colim e θ).is_colimit.fac _ walking_span.left, pushout_cocone.mk_ι_app_left],\n  all_goals\n  { dsimp [to_sSet_hom],\n    rw [←standard_simplex.map_comp, ←standard_simplex.map_id],\n    apply congr_arg,\n    cases e with i b, cases b;\n    try { exact δ_comp_σ_self };\n    try { exact δ_comp_σ_succ }},\nend\n\ndef j_rec (θ : traversal n) : geom_real_rec θ ⟶ Δ[n] := (j_rec_bundle θ).1\n\ndef j_prop (θ : traversal n) : geom_real_incl θ ≫ j_rec θ = 𝟙 Δ[n] := (j_rec_bundle θ).2\n\ndef k_rec_bundle : Π (θ θ' : traversal n),\n  {k : geom_real_rec θ ⟶ 𝕋₁ // geom_real_incl θ ≫ k = simplex_as_hom (θ', θ)}\n| ⟦⟧       θ' := ⟨simplex_as_hom (θ',⟦⟧), rfl⟩\n| (e :: θ) θ' :=\nbegin\n  let k_θ := k_rec_bundle θ (θ' ++ ⟦e⟧),\n  refine ⟨(bundle_colim e θ).is_colimit.desc (pushout_cocone.mk _ k_θ.1 _), _⟩,\n  { apply simplex_as_hom,\n    --Special position\n    exact (apply_map (σ e.1) θ' ++ ⟦(eˢ, e.2)⟧, (eᵗ, e.2) :: apply_map (σ e.1) θ)},\n  change _ = geom_real_incl θ ≫ k_θ.val, rw k_θ.2,\n  swap,\n  rw [geom_real_incl_cons, category.assoc],\n  rw [(bundle_colim e θ).is_colimit.fac _ walking_span.left, pushout_cocone.mk_ι_app_left],\n  all_goals\n  { rw [hom_comp_simplex_as_hom],\n    rw simplex_as_hom_eq_iff,\n    cases e with i b, cases b; simp;\n    rw [𝕋₀_map_apply, 𝕋₀_map_apply, has_hom.hom.unop_op];\n    simp[edge.s, edge.t, apply_map];\n    rw [←apply_comp, ←apply_comp];\n    try { rw δ_comp_σ_self }; try { rw δ_comp_σ_succ };\n    rw [apply_id, apply_id];\n    simp },\nend\n\ndef k_rec' (θ θ' : traversal n) := (k_rec_bundle θ θ').1\n\ndef k_prop' (θ θ' : traversal n) :\n  geom_real_incl θ ≫ k_rec' θ θ' = simplex_as_hom (θ', θ) := (k_rec_bundle θ θ').2\n\ndef k_rec (θ : traversal n) : geom_real_rec θ ⟶ 𝕋₁ := (k_rec_bundle θ ⟦⟧).1\n\ndef k_prop (θ : traversal n) :\n  geom_real_incl θ ≫ k_rec θ = simplex_as_hom (⟦⟧, θ) := (k_rec_bundle θ ⟦⟧).2\n\nlemma j_comp_θ_eq_k_comp_cod : Π (θ θ' : traversal n),\n  j_rec θ ≫ (θ' ++ θ).as_hom = (k_rec_bundle θ θ').1 ≫ cod\n| ⟦⟧       θ' :=\n  begin\n    change simplex_as_hom _ = simplex_as_hom _ ≫ cod,\n    rw [simplex_as_hom_comp_hom], refl,\n  end\n| (e :: θ) θ' :=\nbegin\n  apply pushout_cocone.is_colimit.hom_ext (bundle_colim e θ).is_colimit,\n  { change to_sSet_hom (σ e.fst) ≫ simplex_as_hom _ = simplex_as_hom _ ≫ cod,\n    rw [simplex_as_hom_comp_hom, hom_comp_simplex_as_hom],\n    rw simplex_as_hom_eq_iff,\n    dsimp [𝕋₀, apply_map, cod], cases e with i b, cases b; simp,\n    all_goals { simp [apply_map], change _ = _ ++ _, rw list.append_assoc, refl,}},\n  { change j_rec θ ≫ (θ' ++ (⟦e⟧ ++ θ)).as_hom = (k_rec_bundle θ (θ' ++ ⟦e⟧)).1 ≫ cod,\n    rw ←list.append_assoc,\n    apply j_comp_θ_eq_k_comp_cod }\nend\n\ndef pullback_cone_rec' (θ θ' : traversal n) : pullback_cone ((θ' ++ θ).as_hom) cod :=\n  pullback_cone.mk (j_rec θ) (k_rec_bundle θ θ').1 (j_comp_θ_eq_k_comp_cod θ θ')\n\ndef pullback_cone_rec (θ : traversal n) : pullback_cone (θ.as_hom) cod :=\n  pullback_cone.mk (j_rec θ) (k_rec θ) (j_comp_θ_eq_k_comp_cod θ ⟦⟧)\n\ndef append_eq_append_split {n} {a b c d : traversal n} :\n  a ++ b = c ++ d → {a' // c = a ++ a' ∧ b = a' ++ d} ⊕ {c' // a = c ++ c' ∧ d = c' ++ b} :=\nbegin\n  induction c generalizing a,\n  case nil { rw list.nil_append, rintro rfl, right, exact ⟨a, rfl, rfl⟩ },\n  case cons : c cs ih {\n    intro h, cases a,\n    { left, use c :: cs, simpa using h },\n    { simp at h, cases h, cases h_left, simp,\n      exact ih h_right }}\nend\n\nsection beta\n\nvariables {m : simplex_category} {α : m ⟶ [n]} {e : edge n} {θ₁ θ₂ : traversal m.len} (H : apply_map_to_edge α e = θ₁ ++ θ₂)\n\ndef β_conditions (i) (H : apply_map_to_edge α e = θ₁ ++ θ₂) :\n  α.to_preorder_hom i < e.1 ∨ α.to_preorder_hom i > e.1 ∨ (i, e.2) ∈ θ₁ ∨ (i, e.2) ∈ θ₂ :=\nbegin\n  cases lt_trichotomy (α.to_preorder_hom i) e.fst, left, exact h,\n  cases h, swap, right, left, exact h,\n  right, right,\n  replace h : (α.to_preorder_hom (i, e.2).1, (i, e.2).2) = e, rw h, cases e, refl,\n  rw [←edge_in_apply_map_to_edge_iff, H] at h, simp at h, exact h,\nend\n\ndef β_fun (H : apply_map_to_edge α e = θ₁ ++ θ₂) : fin (m.len + 1) → fin ([n + 1].len + 1) := λ i,\n  if h₁ : α.to_preorder_hom i < e.1 then (α.to_preorder_hom i).cast_succ\n  else if h₂ : α.to_preorder_hom i > e.1 then (α.to_preorder_hom i).succ\n  else if h₃ : (i, e.2) ∈ θ₁ then eˢ\n  else eᵗ\n\nlemma β_eq_es_iff (i) : β_fun H i = (eˢ) ↔ (i, e.2) ∈ θ₁ :=\nbegin\n  simp[β_fun],\n  split;\n  intro h',\n  { by_contra, split_ifs at h',\n    rw [←fin.cast_succ_lt_cast_succ_iff, h'] at h_1,\n    cases e with j b, cases b,\n    exact lt_asymm (fin.cast_succ_lt_succ j) h_1,\n    exact lt_irrefl _ h_1,\n    rw [←fin.succ_lt_succ_iff, h'] at h_2,\n    cases e with j b, cases b,\n    exact lt_irrefl _ h_2,\n    exact lt_asymm (fin.cast_succ_lt_succ j) h_2,\n    cases e with j b, cases b,\n    exact ne_of_lt (fin.cast_succ_lt_succ j) h',\n    exact ne_of_gt (fin.cast_succ_lt_succ j) h' },\n  { have hi : (i, e.2) ∈ apply_map_to_edge α e, rw H, exact list.mem_append_left θ₂ h',\n    rw edge_in_apply_map_to_edge_iff at hi,\n    cases e with j b, cases b;\n    simp at hi h'; simp[hi, h'] }\nend\n\nlemma β_eq_et_iff (i) : β_fun H i = (eᵗ) ↔ (i, e.2) ∈ θ₂ :=\nbegin\n  simp[β_fun],\n  have hf: e.s = e.t ↔ false,\n  { split; intro hf, cases e with j b, cases b,\n    exact ne_of_lt (fin.cast_succ_lt_succ j) hf.symm,\n    exact ne_of_lt (fin.cast_succ_lt_succ j) hf,\n    exfalso, exact hf },\n  split; intro h',\n  { by_contra, split_ifs at h',\n    rw [←fin.cast_succ_lt_cast_succ_iff, h'] at h_1,\n    cases e with j b, cases b,\n    exact lt_irrefl _ h_1,\n    exact lt_asymm (fin.cast_succ_lt_succ j) h_1,\n    rw [←fin.succ_lt_succ_iff, h'] at h_2,\n    cases e with j b, cases b,\n    exact lt_asymm (fin.cast_succ_lt_succ j) h_2,\n    exact lt_irrefl _ h_2,\n    cases e with j b, cases b,\n    exact ne_of_gt (fin.cast_succ_lt_succ j) h',\n    exact ne_of_lt (fin.cast_succ_lt_succ j) h',\n    cases β_conditions i H, exact h_1 h_4,\n    cases h_4, exact h_2 h_4,\n    cases h_4, exact h_3 h_4,\n    exact h h_4 },\n  { have hi : (α.to_preorder_hom (i, e.2).1, (i, e.2).2) = e,\n    { rw [←edge_in_apply_map_to_edge_iff, H], exact list.mem_append_right θ₁ h',  },\n    cases e; simp at hi ⊢ h', simp [hi],\n    have H' : sorted (θ₁ ++ θ₂), rw ←H, apply apply_map_to_edge_sorted,\n    rw ←append_sorted_iff at H',\n    intro hi', exfalso, have h'' := (H'.2.2 _ hi' _ h'),\n    exact edge.lt_asymm _ _ h'' h'', }\nend\n\nlemma β_monotone : monotone (β_fun H) := λ i j hij,\nbegin\n  simp [β_fun], split_ifs; try { apply le_refl },\n  { exact α.to_preorder_hom.monotone hij },\n  { apply le_of_lt, rw ←fin.le_cast_succ_iff, exact α.to_preorder_hom.monotone hij },\n  { rw ←fin.cast_succ_lt_cast_succ_iff at h, cases e with j b, cases b,\n    exact le_trans (le_of_lt h) (le_of_lt (fin.cast_succ_lt_succ _)),\n    exact le_of_lt h },\n  { rw ←fin.cast_succ_lt_cast_succ_iff at h, cases e with j b, cases b,\n    exact le_of_lt h,\n    exact le_trans (le_of_lt h) (le_of_lt (fin.cast_succ_lt_succ _)) },\n  { refine absurd (α.to_preorder_hom.monotone hij) (not_le.mpr _),\n    exact lt_of_lt_of_le h_2 (not_lt.mp h) },\n  { simp, exact α.to_preorder_hom.monotone hij },\n  { refine absurd (α.to_preorder_hom.monotone hij) (not_le.mpr _),\n    exact lt_of_le_of_lt (not_lt.mp h_3) h_1 },\n  { refine absurd (α.to_preorder_hom.monotone hij) (not_le.mpr _),\n    exact lt_of_le_of_lt (not_lt.mp h_3) h_1 },\n  all_goals { have hi := le_antisymm (not_lt.mp h) (not_lt.mp h_1) },\n  { refine absurd (α.to_preorder_hom.monotone hij) (not_le.mpr _),\n    rwa hi at h_3 },\n  { cases e with k b, cases b; simp [edge.s, edge.t],\n    exact le_of_lt h_4,\n    apply le_of_lt, rw[←fin.le_cast_succ_iff], simp, exact le_of_lt h_4 },\n  swap 3, swap 3,\n  { refine absurd (α.to_preorder_hom.monotone hij) (not_le.mpr _),\n    rwa hi at h_3 },\n  { cases e with k b, cases b; simp [edge.s, edge.t],\n    apply le_of_lt, rw[←fin.le_cast_succ_iff], simp, exact le_of_lt h_4,\n    exact le_of_lt h_4 },\n  all_goals\n  { cases e with k b, cases b; dsimp[edge.s, edge.t];\n    try { exact le_of_lt (fin.cast_succ_lt_succ k) },\n    exfalso, have hj := le_antisymm (not_lt.mp h_4) (not_lt.mp h_3),\n    have H' : sorted (θ₁ ++ θ₂), rw ←H, apply apply_map_to_edge_sorted,\n    rw ←append_sorted_iff at H', },\n  { have hj' : (α.to_preorder_hom (j, ∔).1, (j, ∔).2) = (k, ∔), simp, exact hj,\n    rw [←edge_in_apply_map_to_edge_iff, H] at hj',\n    simp [h_5] at hj' h_2,\n    refine absurd hij (not_le.mpr _),\n    exact H'.2.2 _ h_2 _ hj', },\n  { have hi' : (α.to_preorder_hom (i, ∸).1, (i, ∸).2) = (k, ∸), simp, exact hi.symm,\n    rw [←edge_in_apply_map_to_edge_iff, H] at hi',\n    simp [h_2] at hi' h_5,\n    refine absurd hij (not_le.mpr _),\n    exact H'.2.2 _ h_5 _ hi', },\nend\n\ndef β (H : apply_map_to_edge α e = θ₁ ++ θ₂) : m ⟶ [n+1] := hom.mk\n{ to_fun    := β_fun H,\n  monotone' := β_monotone H, }\n\nlemma β_comp_σ : β H ≫ σ e.1 = α :=\nbegin\n  ext1, ext1 i, simp [β, β_fun], split_ifs;\n  simp [σ, fin.pred_above]; split_ifs; try { refl };\n  try { push_neg at * },\n  { exfalso, exact lt_asymm h h_1 },\n  { exfalso, rw ←fin.le_cast_succ_iff at h_2, simp at h_2, exact h h_2 },\n  { push_neg at h h_1, rw le_antisymm h_1 h, cases e with j b, cases b;\n    simp[edge.s, edge.t] at h_3 ⊢,\n    exfalso, exact h_3 },\n  { push_neg at h h_1 h_3, rw le_antisymm h_1 h, cases e with j b, cases b;\n    simp[edge.s, edge.t] at h_3 ⊢,\n    exact absurd (fin.cast_succ_lt_succ j) (not_lt.mpr h_3) },\n  { push_neg at h h_1, rw le_antisymm h_1 h, cases e with j b, cases b;\n    simp[edge.s, edge.t] at h_3 ⊢,\n    exfalso, exact h_3 },\n  { push_neg at h h_1 h_3, rw le_antisymm h_1 h, cases e with j b, cases b;\n    simp[edge.s, edge.t] at h_3 ⊢,\n    exact absurd (fin.cast_succ_lt_succ j) (not_lt.mpr h_3) },\nend\n\nend beta\n\n\ndef geom_real_rec_lift' : Π (θ θ' : traversal n) {m} (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (geom_real_rec θ).obj (opposite.op m)\n| ⟦⟧       θ' m α θ₁ θ₂ hθ := α\n| (e :: θ) θ' m α θ₁ θ₂ hθ :=\n  begin\n    cases append_eq_append_split hθ with a' c',\n    { rcases a' with ⟨θ₂', hθ₂', hθ₂⟩,\n      let p : pushout_cocone _ _ := (bundle_colim e θ).cocone,\n      apply p.inl.app (opposite.op m),\n      exact β hθ₂',\n    },\n    { cases c' with c' hc',\n      let p : pushout_cocone _ _ := (bundle_colim e θ).cocone,\n      exact p.inr.app (opposite.op m) (geom_real_rec_lift' θ (θ' ++ ⟦e⟧) α c' θ₂ hc'.2.symm) },\n  end\n\nlemma geom_real_rec_fac_j' : Π (θ θ' : traversal n) {m} (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (j_rec θ).app (opposite.op m) (geom_real_rec_lift' θ θ' α θ₁ θ₂ hθ) = α\n| ⟦⟧       θ' m α θ₁ θ₂ hθ := rfl\n| (e :: θ) θ' m α θ₁ θ₂ hθ :=\n  begin\n    simp [geom_real_rec_lift'],\n    cases append_eq_append_split hθ with a' c',\n    { rcases a' with ⟨θ₂', H, hθ₂⟩,\n      cases e with i b, simp,\n      exact β_comp_σ H },\n    { cases c' with c' hc',\n      exact geom_real_rec_fac_j' θ (θ' ++ ⟦e⟧) α c' θ₂ _ }\n  end\n\nlemma geom_real_rec_fac_k' : Π (θ θ' : traversal n) {m} (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (k_rec_bundle θ θ').1.app (opposite.op m) (geom_real_rec_lift' θ θ' α θ₁ θ₂ hθ) = ((apply_map α θ') ++ θ₁, θ₂)\n| ⟦⟧       θ' m α θ₁ θ₂ hθ :=\n  begin\n    simp [apply_map] at hθ, cases hθ.1, cases hθ.2,\n    simp[geom_real_rec_lift'], refl\n  end\n| (e :: θ) θ' m α θ₁ θ₂ hθ :=\n  begin\n    simp [geom_real_rec_lift'],\n    cases append_eq_append_split hθ with a' c',\n    { rcases a' with ⟨θ₂', H, hθ₂⟩,\n      change (simplex_as_hom _).app (opposite.op m) (β H) = _,\n      simp [simplex_as_hom],\n      change apply_map _ _  = _ ∧ apply_map _ _  = _ ,\n      rw [apply_map_append],\n      simp [apply_map],\n      rw [←apply_comp, ←apply_comp, β_comp_σ H],\n      cases hθ₂,\n      change _ ∧ _ = θ₂' ++ _,\n      rw [list.append_left_inj],\n      rw [list.append_right_inj],\n      have h₁ : sorted (θ₁ ++ θ₂'), rw ←H, apply apply_map_to_edge_sorted,\n      have h₂ := h₁, rw ←append_sorted_iff at h₂,\n      split;\n      refine eq_of_sorted_of_same_elem _ _ (apply_map_to_edge_sorted _ _) (by simp[h₂.1, h₂.2]) _;\n      intro e'; rw edge_in_apply_map_to_edge_iff; simp; split; simp [β];\n      try {rw β_eq_es_iff H}; try {rw β_eq_et_iff H}; intro h,\n      { intro h', rw ←h' at h, simpa using h },\n      { have he' : e' ∈ apply_map_to_edge α e, rw H, exact list.mem_append_left θ₂' h,\n        rw edge_in_apply_map_to_edge_iff at he', cases e, cases e', simp at he' ⊢,\n        cases he'.2, simp, exact h },\n      { intro h', rw ←h' at h, simpa using h },\n      { have he' : e' ∈ apply_map_to_edge α e, rw H, exact list.mem_append_right θ₁ h,\n        rw edge_in_apply_map_to_edge_iff at he', cases e, cases e', simp at he' ⊢,\n        cases he'.2, simp, exact h }},\n    { cases c' with c' hc', simp,\n      dsimp [k_rec, k_rec_bundle],\n      change (k_rec_bundle θ (θ' ++ ⟦e⟧)).1.app (opposite.op m) (geom_real_rec_lift' θ (θ' ++ ⟦e⟧) α c' θ₂ _) = _,\n      cases hc'.1,\n      specialize geom_real_rec_fac_k' θ (θ' ++ ⟦e⟧) α c' θ₂ hc'.2.symm,\n      rw [geom_real_rec_fac_k', apply_map_append], simp [apply_map], refl, }\n  end\n\ndef geom_real_rec_lift (θ : traversal n) {m} : Π (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (geom_real_rec θ).obj (opposite.op m) := geom_real_rec_lift' θ ⟦⟧\n\nlemma geom_real_rec_fac_j (θ : traversal n) {m} : Π (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (j_rec θ).app (opposite.op m) (geom_real_rec_lift θ α θ₁ θ₂ hθ) = α := geom_real_rec_fac_j' θ ⟦⟧\n\nlemma geom_real_rec_fac_k (θ : traversal n) {m} : Π (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ),\n  (k_rec θ).app (opposite.op m) (geom_real_rec_lift θ α θ₁ θ₂ hθ) = (θ₁, θ₂) := geom_real_rec_fac_k' θ ⟦⟧\n\nlemma geom_real_rec_unique : Π (θ : traversal n) {m} (α : m ⟶ [n]) (θ₁ θ₂ : traversal m.len) (hθ : θ₁ ++ θ₂ = apply_map α θ)\n  (x : (geom_real_rec θ).obj (opposite.op m)),\n  (j_rec θ).app (opposite.op m) x = α →\n  (k_rec θ).app (opposite.op m) x = (θ₁, θ₂) →\n  x = geom_real_rec_lift θ α θ₁ θ₂ hθ\n| ⟦⟧ m α θ₁ θ₂ hθ x hx₁ hx₂ := by dsimp [geom_real_rec_lift]; rw ←hx₁; refl\n| (e :: θ) m α ⟦⟧         θ₂ hθ x hx₁ hx₂ := sorry\n| (e :: θ) m α (e₁ :: θ₁) θ₂ hθ x hx₁ hx₂ := sorry\n\ntheorem geom_real_is_pullback_θ_cod (θ : traversal n) : is_limit (pullback_cone_rec θ) :=\nbegin\n  apply evaluation_jointly_reflects_limits,\n  intro m, exact\n  { lift := λ c,\n    begin\n      let c_fst : c.X ⟶ Δ[n].obj m := c.π.app walking_cospan.left,\n      let c_snd : c.X ⟶ 𝕋₁.obj m   := c.π.app walking_cospan.right,\n      have hθ : c_fst ≫ (as_hom θ).app m = c_snd ≫ cod.app m,\n      { change c_fst ≫ (cospan θ.as_hom cod ⋙ (evaluation simplex_categoryᵒᵖ Type).obj m).map walking_cospan.hom.inl\n          = c_snd ≫ (cospan θ.as_hom cod ⋙ (evaluation simplex_categoryᵒᵖ Type).obj m).map walking_cospan.hom.inr,\n        rw [←c.π.naturality, ←c.π.naturality], refl },\n      exact λ x, geom_real_rec_lift θ (c_fst x) _ _ (congr_fun hθ x).symm,\n    end,\n    fac' := λ c,\n    begin\n      intro j, cases j;\n      let c_fst : c.X ⟶ Δ[n].obj m := c.π.app walking_cospan.left;\n      let c_snd : c.X ⟶ pointed_traversal m.unop.len := c.π.app walking_cospan.right,\n\n      { let p := pullback_cone_rec θ,\n        let const_c := (category_theory.functor.const walking_cospan).obj c.X,\n        let const_c_inl := const_c.map walking_cospan.hom.inl,\n        let const_p := (category_theory.functor.const walking_cospan).obj p.X,\n        let const_p_inl := const_p.map walking_cospan.hom.inl,\n        let eval_cospan := cospan θ.as_hom cod ⋙ (evaluation simplex_categoryᵒᵖ Type).obj m,\n        change _ = const_c_inl ≫ c.π.app none,\n        have H : const_c_inl ≫ c.π.app none = _, apply c.π.naturality',\n        refine trans _ H.symm, clear H,\n        suffices H : ((pullback_cone_rec θ).π.app none).app m\n          = ((pullback_cone_rec θ).π.app walking_cospan.left).app m\n          ≫ eval_cospan.map walking_cospan.hom.inl,\n        { simp, rw H, ext1 x,\n          let α : m.unop ⟶ [n] := c_fst x,\n          simp, apply congr_arg,\n          exact geom_real_rec_fac_j θ α _ _ _ },\n        change (const_p_inl ≫ (pullback_cone_rec θ).π.app none).app m = _,\n        rw p.π.naturality', refl },\n      ext1 x, let α : m.unop ⟶ [n] := c_fst x,\n      cases j, simp,\n      { exact geom_real_rec_fac_j θ α _ _ _ },\n      { change (k_rec θ).app (opposite.op m.unop) (geom_real_rec_lift θ α (c_snd x).1 (c_snd x).2 _) = c_snd x,\n        rw [geom_real_rec_fac_k θ α (c_snd x).1 (c_snd x).2 _], simp }\n    end,\n    uniq' := λ c,\n    begin\n      intros lift' hlift',\n      change c.X ⟶ (geom_real_rec θ).obj (opposite.op m.unop) at lift',\n      ext1 x, simp,\n      apply geom_real_rec_unique θ,\n      specialize hlift' walking_cospan.left, simp at hlift',\n      rw ←hlift', refl,\n      specialize hlift' walking_cospan.right, simp at hlift',\n      rw ←hlift',\n      simp, refl,\n    end }\nend\n\nend geom_real_rec\n\nend traversal\n", "meta": {"author": "floriscnossen", "repo": "simplicial_sets_in_lean", "sha": "c260d8196964bb288331bff7a3bfe24266792f0c", "save_path": "github-repos/lean/floriscnossen-simplicial_sets_in_lean", "path": "github-repos/lean/floriscnossen-simplicial_sets_in_lean/simplicial_sets_in_lean-c260d8196964bb288331bff7a3bfe24266792f0c/src/traversals/geom_real_rec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.30928372092183426}}
{"text": "import Monads.Applicative\n\nnamespace Monads\n  /- The monad is a close relativ to the applicative, however unlike\n     an applicative functor it is in fact able to change the structure\n     of the context the value is wrapped in since it allows users to\n     pass a function that produces a new structure: (α → m β) -/\n  class Monad (m : Type u → Type v) extends Applicative m where\n    -- return is a keyword so don't use it\n    /- Basically equivalent to pure from applicative. -/\n    ret : α → m α\n    bind : m α → (α → m β) → m β\n    /- Drop the action on the left and only return the right one. If\n       left contains e.g. a sideffect it will still be executed. -/\n    and_then {α β : Type u} : m α → m β → m β := λ a1 a2 => bind a1 (Function.const α a2)\n\n  export Monad (ret)\n\n  infixl:65 \" >>= \" => Monads.Monad.bind\n  infixl:60 \" >> \" => Monads.Monad.and_then\n\n  /- Establishes some well behavedeness for ret and bind as well as a relation\n     to apply and pure from applicative. -/\n  class LawfulMonad (m : Type u → Type v) [mon: Monad m] extends LawfulApplicative m : Prop where\n    /- Binding a pure value to a function should be the same as applying the function\n       to the value directly -/\n    ret_left_id : ∀ (a : α) (h : α → m β), ret a >>= h = h a\n    /- Binding a monadic value to ret should be a no op -/\n    ret_right_id : ∀ (a : m α), a >>= ret = a\n    /- A modified associativity law for bind -/\n    bind_assoc : ∀ (a : m α) (g : α → m β) (h : β → m γ), (a >>= g) >>= h = a >>= (λ x => g x >>= h)\n    /- Pure and return should behave equivalently -/\n    pure_behaved : ∀ (a : α), pure a = mon.ret a\n    /- Apply and bind are related in this way as shown below -/\n    apply_behaved : ∀ {α β : Type u} (mf : m (α → β)) (ma : m α), mf <*> ma = mf >>= (λ f => ma >>= (λ a => ret $ f a))\n    /- and_then and seq_right are the same.-/\n    and_then_behaved: ∀ (a1 : m α) (a2 : m β), a1 *> a2 = a1 >> a2\n\n  namespace LawfulMonad\n    variable {m : Type u → Type v} [monad : Monad m] [LawfulMonad m]\n    -- Constructing a lawful applicative from a lawful monad\n    -- pure is just return so don't define it explicitly\n    def bind_apply {α β : Type u} :  m (α → β) → m α → m β\n    | mf, ma => mf >>= (λ f => ma >>= (λ a => ret $ f a))\n\n    -- law 1\n    example : ∀ x : m α, bind_apply (ret id) x = x := by\n      intro x\n      simp only [bind_apply]\n      rw [ret_left_id]\n      simp only [id]\n      rw [ret_right_id]\n\n    -- law 2\n    example : ∀ (x : α) (g : α → β), bind_apply (ret g) (ret x) = monad.ret (g x) := by\n      intro x g\n      simp only [bind_apply]\n      rw [ret_left_id]\n      rw [ret_left_id]\n\n    -- law 3\n    example : ∀ {α β : Type u} (x : α) (g : m (α → β)), bind_apply g (ret x) = bind_apply (ret (· $ x)) g := by\n      intro α β x g\n      simp only [bind_apply]\n      simp only [ret_left_id]\n\n    -- law 4 (a very automated proof since its just a definition massacre)\n    example : ∀ {α β γ: Type u} (x : m (β → γ)) (y : m (α → β)) (z : m α), bind_apply (bind_apply (bind_apply (ret (@Function.comp α β γ)) x) y) z = bind_apply x (bind_apply y z) := by\n      intro α β γ x y z\n      simp only [bind_apply]\n      simp only [ret_left_id, bind_assoc]\n  end LawfulMonad\nend Monads\n", "meta": {"author": "hargoniX", "repo": "lean-monads", "sha": "2e87ca7ddf394641ea1b16bcbd8c384026d68e2f", "save_path": "github-repos/lean/hargoniX-lean-monads", "path": "github-repos/lean/hargoniX-lean-monads/lean-monads-2e87ca7ddf394641ea1b16bcbd8c384026d68e2f/Monads/Monad.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.30928371416384565}}
{"text": "/-\nCopyright (c) 2022 Siddhartha Gadgil. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Siddhartha Gadgil, Mario Carneiro\n-/\nimport Mathlib.Lean.Meta\nimport Lean.Elab.Tactic.Location\n\n/-!\n# `trans` tactic\n\nThis implements the `trans` tactic, which can apply transitivity theorems with an optional middle\nvariable argument.\n-/\n\nnamespace Mathlib.Tactic\nopen Lean Meta Elab\n\ninitialize registerTraceClass `Tactic.trans\n\n/-- Environment extension storing transitivity lemmas -/\ninitialize transExt :\n    SimpleScopedEnvExtension (Name × Array (DiscrTree.Key true)) (DiscrTree Name true) ←\n  registerSimpleScopedEnvExtension {\n    addEntry := fun dt (n, ks) ↦ dt.insertCore ks n\n    initial := {}\n  }\n\ninitialize registerBuiltinAttribute {\n  name := `trans\n  descr := \"transitive relation\"\n  add := fun decl _ kind ↦ MetaM.run' do\n    let declTy := (← getConstInfo decl).type\n    let (xs, _, targetTy) ← withReducible <| forallMetaTelescopeReducing declTy\n    let fail := throwError\n      \"@[trans] attribute only applies to lemmas proving\n      x ∼ y → y ∼ z → x ∼ z, got {indentExpr declTy} with target {indentExpr targetTy}\"\n    let .app (.app rel _) _ := targetTy | fail\n    let some yzHyp := xs.back? | fail\n    let some xyHyp := xs.pop.back? | fail\n    let .app (.app _ _) _ ← inferType yzHyp | fail\n    let .app (.app _ _) _ ← inferType xyHyp | fail\n    let key ← withReducible <| DiscrTree.mkPath rel\n    transExt.add (decl, key) kind\n}\n\n/-- Composition using the `Trans` class in the homogeneous case. -/\ndef _root_.Trans.simple {a b c : α} [Trans r r r] : r a b → r b c → r a c := trans\n\n/-- Composition using the `Trans` class in the general case. -/\ndef _root_.Trans.het {a : α} {b : β} {c : γ}\n  {r : α → β → Sort u} {s : β → γ → Sort v} {t : outParam (α → γ → Sort w)}\n  [Trans r s t] : r a b → s b c → t a c := trans\n\n\nopen Lean.Elab.Tactic\n\n/-- solving `e ← mkAppM' f #[x]` -/\ndef getExplicitFuncArg? (e : Expr) : MetaM (Option <| Expr × Expr) := do\n  match e with\n  | Expr.app f a => do\n    if ← isDefEq (← mkAppM' f #[a]) e then\n      return some (f, a)\n    else\n      getExplicitFuncArg? f\n  | _ => return none\n\n/-- solving `tgt ← mkAppM' rel #[x, z]` given `tgt = f z` -/\ndef getExplicitRelArg? (tgt f z : Expr) : MetaM (Option <| Expr × Expr) := do\n  match f  with\n  | Expr.app rel x => do\n    let check: Bool ← do\n      try\n        let folded ← mkAppM' rel #[x, z]\n        isDefEq folded tgt\n      catch _ =>\n        pure false\n    if check then\n      return some (rel, x)\n    else\n      getExplicitRelArg? tgt rel z\n  | _ => return none\n\n/-- refining `tgt ← mkAppM' rel #[x, z]` dropping more arguments if possible -/\ndef getExplicitRelArgCore (tgt rel x z : Expr) : MetaM (Expr × Expr) := do\n  match rel with\n  | Expr.app rel' _ => do\n    let check: Bool ← do\n      try\n        let folded ← mkAppM' rel' #[x, z]\n        isDefEq folded tgt\n      catch _ =>\n        pure false\n    if !check then\n      return (rel, x)\n    else\n      getExplicitRelArgCore tgt rel' x z\n  | _ => return (rel ,x)\n\n/--\n`trans` applies to a goal whose target has the form `t ~ u` where `~` is a transitive relation,\nthat is, a relation which has a transitivity lemma tagged with the attribute [trans].\n\n* `trans s` replaces the goal with the two subgoals `t ~ s` and `s ~ u`.\n* If `s` is omitted, then a metavariable is used instead.\n-/\nelab \"trans\" t?:(ppSpace (colGt term))? : tactic => withMainContext do\n  let tgt ← getMainTarget\n  let (rel, x, z) ←\n    match tgt with\n    | Expr.app f z =>\n      match (← getExplicitRelArg? tgt f z) with\n      | some (rel, x) =>\n        let (rel, x) ← getExplicitRelArgCore tgt rel x z\n        pure (rel, x, z)\n      | none => throwError \"transitivity lemmas only apply to\n        binary relations, not {indentExpr tgt}\"\n    | _ => throwError \"transitivity lemmas only apply to binary relations, not {indentExpr tgt}\"\n  trace[Tactic.trans]\"goal decomposed\"\n  trace[Tactic.trans]\"rel: {indentExpr rel}\"\n  trace[Tactic.trans]\"x: {indentExpr x}\"\n  trace[Tactic.trans]\"z: {indentExpr z}\"\n  -- first trying the homogeneous case\n  try\n    let ty ← inferType x\n    let t'? ← t?.mapM (elabTermWithHoles · ty (← getMainTag))\n    let s ← saveState\n    trace[Tactic.trans]\"trying homogeneous case\"\n    for lem in (← (transExt.getState (← getEnv)).getUnify rel).push ``Trans.simple do\n      trace[Tactic.trans]\"trying lemma {lem}\"\n      try\n        liftMetaTactic fun g ↦ do\n          let lemTy ← inferType (← mkConstWithLevelParams lem)\n          let arity ← withReducible <| forallTelescopeReducing lemTy fun es _ ↦ pure es.size\n          let y ← (t'?.map (pure ·.1)).getD (mkFreshExprMVar ty)\n          let g₁ ← mkFreshExprMVar (some <| ← mkAppM' rel #[x, y]) .synthetic\n          let g₂ ← mkFreshExprMVar (some <| ← mkAppM' rel #[y, z]) .synthetic\n          g.assign (← mkAppOptM lem (mkArray (arity - 2) none ++ #[some g₁, some g₂]))\n          pure <| [g₁.mvarId!, g₂.mvarId!] ++ if let some (_, gs') := t'? then gs' else [y.mvarId!]\n        return\n      catch _ => s.restore\n    pure ()\n  catch _ =>\n  trace[Tactic.trans]\"trying heterogeneous case\"\n  let t'? ← t?.mapM (elabTermWithHoles · none (← getMainTag))\n  let s ← saveState\n  for lem in (← (transExt.getState (← getEnv)).getUnify rel).push\n      ``HEq.trans |>.push ``HEq.trans  do\n    try\n      liftMetaTactic fun g ↦ do\n        trace[Tactic.trans]\"trying lemma {lem}\"\n        let lemTy ← inferType (← mkConstWithLevelParams lem)\n        let arity ← withReducible <| forallTelescopeReducing lemTy fun es _ ↦ pure es.size\n        trace[Tactic.trans]\"arity: {arity}\"\n        trace[Tactic.trans]\"lemma-type: {lemTy}\"\n        let y ← (t'?.map (pure ·.1)).getD (mkFreshExprMVar none)\n        trace[Tactic.trans]\"obtained y: {y}\"\n        trace[Tactic.trans]\"rel: {indentExpr rel}\"\n        trace[Tactic.trans]\"x:{indentExpr x}\"\n        trace[Tactic.trans]\"z:  {indentExpr z}\"\n        let g₂ ← mkFreshExprMVar (some <| ← mkAppM' rel #[y, z]) .synthetic\n        trace[Tactic.trans]\"obtained g₂: {g₂}\"\n        let g₁ ← mkFreshExprMVar (some <| ← mkAppM' rel #[x, y]) .synthetic\n        trace[Tactic.trans]\"obtained g₁: {g₁}\"\n        g.assign (← mkAppOptM lem (mkArray (arity - 2) none ++ #[some g₁, some g₂]))\n        pure <| [g₁.mvarId!, g₂.mvarId!] ++ if let some (_, gs') := t'? then gs' else [y.mvarId!]\n      return\n    catch e =>\n      trace[Tactic.trans]\"failed: {e.toMessageData}\"\n      s.restore\n\n  throwError m!\"no applicable transitivity lemma found for {indentExpr tgt}\n\"\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/Relation/Trans.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.30920303568773516}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.natural_isomorphism\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.PostPort\n\nuniverses u₁ v₁ w₀ w₁ w₂ \n\nnamespace Mathlib\n\n/-!\n# Categories of indexed families of objects.\n\nWe define the pointwise category structure on indexed families of objects in a category\n(and also the dependent generalization).\n\n-/\n\nnamespace category_theory\n\n\n/--\n`pi C` gives the cartesian product of an indexed family of categories.\n-/\nprotected instance pi {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] : category ((i : I) → C i) :=\n  category.mk\n\n/--\nThis provides some assistance to typeclass search in a common situation,\nwhich otherwise fails. (Without this `category_theory.pi.has_limit_of_has_limit_comp_eval` fails.)\n-/\ninstance pi' {I : Type v₁} (C : I → Type u₁) [(i : I) → category (C i)] : category ((i : I) → C i) :=\n  category_theory.pi C\n\nnamespace pi\n\n\n@[simp] theorem id_apply {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] (X : (i : I) → C i) (i : I) : 𝟙 = 𝟙 :=\n  rfl\n\n@[simp] theorem comp_apply {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {X : (i : I) → C i} {Y : (i : I) → C i} {Z : (i : I) → C i} (f : X ⟶ Y) (g : Y ⟶ Z) (i : I) : category_struct.comp f g i = f i ≫ g i :=\n  rfl\n\n/--\nThe evaluation functor at `i : I`, sending an `I`-indexed family of objects to the object over `i`.\n-/\n@[simp] theorem eval_map {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] (i : I) (f : (i : I) → C i) (g : (i : I) → C i) (α : f ⟶ g) : functor.map (eval C i) α = α i :=\n  Eq.refl (functor.map (eval C i) α)\n\n/--\nPull back an `I`-indexed family of objects to an `J`-indexed family, along a function `J → I`.\n-/\n@[simp] theorem comap_map {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₁} (h : J → I) (f : (i : I) → C i) (g : (i : I) → C i) (α : f ⟶ g) (i : J) : functor.map (comap C h) α i = α (h i) :=\n  Eq.refl (functor.map (comap C h) α i)\n\n/--\nThe natural isomorphism between\npulling back a grading along the identity function,\nand the identity functor. -/\ndef comap_id (I : Type w₀) (C : I → Type u₁) [(i : I) → category (C i)] : comap C id ≅ 𝟭 :=\n  iso.mk (nat_trans.mk fun (X : (i : I) → C i) => 𝟙) (nat_trans.mk fun (X : (i : I) → C i) => 𝟙)\n\n/--\nThe natural isomorphism comparing between\npulling back along two successive functions, and\npulling back along their composition\n-/\ndef comap_comp {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₁} {K : Type w₂} (f : K → J) (g : J → I) : comap C g ⋙ comap (C ∘ g) f ≅ comap C (g ∘ f) :=\n  iso.mk (nat_trans.mk fun (X : (i : I) → C i) (b : K) => 𝟙) (nat_trans.mk fun (X : (i : I) → C i) (b : K) => 𝟙)\n\n/-- The natural isomorphism between pulling back then evaluating, and just evaluating. -/\ndef comap_eval_iso_eval {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₁} (h : J → I) (j : J) : comap C h ⋙ eval (C ∘ h) j ≅ eval C (h j) :=\n  nat_iso.of_components (fun (f : (i : I) → C i) => iso.refl (functor.obj (comap C h ⋙ eval (C ∘ h) j) f)) sorry\n\nprotected instance sum_elim_category {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₀} {D : J → Type u₁} [(j : J) → category (D j)] (s : I ⊕ J) : category (sum.elim C D s) :=\n  sorry\n\n/--\nThe bifunctor combining an `I`-indexed family of objects with a `J`-indexed family of objects\nto obtain an `I ⊕ J`-indexed family of objects.\n-/\n@[simp] theorem sum_obj_obj {I : Type w₀} (C : I → Type u₁) [(i : I) → category (C i)] {J : Type w₀} {D : J → Type u₁} [(j : J) → category (D j)] (f : (i : I) → C i) (g : (j : J) → D j) (s : I ⊕ J) : functor.obj (functor.obj (sum C) f) g s = sum.rec f g s :=\n  Eq.refl (functor.obj (functor.obj (sum C) f) g s)\n\nend pi\n\n\nnamespace functor\n\n\n/--\nAssemble an `I`-indexed family of functors into a functor between the pi types.\n-/\ndef pi {I : Type w₀} {C : I → Type u₁} [(i : I) → category (C i)] {D : I → Type u₁} [(i : I) → category (D i)] (F : (i : I) → C i ⥤ D i) : ((i : I) → C i) ⥤ ((i : I) → D i) :=\n  mk (fun (f : (i : I) → C i) (i : I) => obj (F i) (f i)) fun (f g : (i : I) → C i) (α : f ⟶ g) (i : I) => map (F i) (α i)\n\n-- One could add some natural isomorphisms showing\n\n-- how `functor.pi` commutes with `pi.eval` and `pi.comap`.\n\nend functor\n\n\nnamespace nat_trans\n\n\n/--\nAssemble an `I`-indexed family of natural transformations into a single natural transformation.\n-/\ndef pi {I : Type w₀} {C : I → Type u₁} [(i : I) → category (C i)] {D : I → Type u₁} [(i : I) → category (D i)] {F : (i : I) → C i ⥤ D i} {G : (i : I) → C i ⥤ D i} (α : (i : I) → F i ⟶ G i) : functor.pi F ⟶ functor.pi G :=\n  mk fun (f : (i : I) → C i) (i : I) => app (α i) (f i)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/pi/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318194686359, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.30904194013129926}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.isomorphism_classes\nimport Mathlib.category_theory.thin\nimport Mathlib.PostPort\n\nuniverses v₁ u₁ v₂ u₂ l u₃ v₃ \n\nnamespace Mathlib\n\n/-!\n# Skeleton of a category\n\nDefine skeletal categories as categories in which any two isomorphic objects are equal.\n\nConstruct the skeleton of a thin category as a partial ordering, and (noncomputably) show it is\na skeleton of the original category. The advantage of this special case being handled separately\nis that lemmas and definitions about orderings can be used directly, for example for the subobject\nlattice (TODO). In addition, some of the commutative diagrams about the functors commute\ndefinitionally on the nose which is convenient in practice.\n\n(TODO): Construct the skeleton of a general category using choice, and show it is a skeleton.\n-/\n\nnamespace category_theory\n\n\n/-- A category is skeletal if isomorphic objects are equal. -/\ndef skeletal (C : Type u₁) [category C] :=\n  ∀ {X Y : C}, is_isomorphic X Y → X = Y\n\n/--\n`is_skeleton_of C D F` says that `F : D ⥤ C` exhibits `D` as a skeletal full subcategory of `C`,\nin particular `F` is a (strong) equivalence and `D` is skeletal.\n-/\nstructure is_skeleton_of (C : Type u₁) [category C] (D : Type u₂) [category D] (F : D ⥤ C) \nwhere\n  skel : skeletal D\n  eqv : is_equivalence F\n\n/-- If `C` is thin and skeletal, then any naturally isomorphic functors to `C` are equal. -/\ntheorem functor.eq_of_iso {C : Type u₁} [category C] {D : Type u₂} [category D] {F₁ : D ⥤ C} {F₂ : D ⥤ C} [∀ (X Y : C), subsingleton (X ⟶ Y)] (hC : skeletal C) (hF : F₁ ≅ F₂) : F₁ = F₂ := sorry\n\n/--\nIf `C` is thin and skeletal, `D ⥤ C` is skeletal.\n`category_theory.functor_thin` shows it is thin also.\n-/\ntheorem functor_skeletal {C : Type u₁} [category C] {D : Type u₂} [category D] [∀ (X Y : C), subsingleton (X ⟶ Y)] (hC : skeletal C) : skeletal (D ⥤ C) :=\n  fun (F₁ F₂ : D ⥤ C) (h : is_isomorphic F₁ F₂) => nonempty.elim h (functor.eq_of_iso hC)\n\n/--\nConstruct the skeleton category by taking the quotient of objects. This construction gives a\npreorder with nice definitional properties, but is only really appropriate for thin categories.\n-/\ndef thin_skeleton (C : Type u₁) [category C] :=\n  quotient (is_isomorphic_setoid C)\n\nprotected instance inhabited_thin_skeleton (C : Type u₁) [category C] [Inhabited C] : Inhabited (thin_skeleton C) :=\n  { default := quotient.mk Inhabited.default }\n\nprotected instance thin_skeleton.preorder (C : Type u₁) [category C] : preorder (thin_skeleton C) :=\n  preorder.mk (quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry)\n    (fun (a b : thin_skeleton C) =>\n      quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry a b ∧\n        ¬quotient.lift₂ (fun (X Y : C) => Nonempty (X ⟶ Y)) sorry b a)\n    sorry sorry\n\n/-- The functor from a category to its thin skeleton. -/\n@[simp] theorem to_thin_skeleton_obj (C : Type u₁) [category C] (a : C) : functor.obj (to_thin_skeleton C) a = quotient.mk a :=\n  Eq.refl (functor.obj (to_thin_skeleton C) a)\n\n/-!\nThe constructions here are intended to be used when the category `C` is thin, even though\nsome of the statements can be shown without this assumption.\n-/\n\nnamespace thin_skeleton\n\n\n/-- The thin skeleton is thin. -/\nprotected instance thin (C : Type u₁) [category C] {X : thin_skeleton C} {Y : thin_skeleton C} : subsingleton (X ⟶ Y) :=\n  subsingleton.intro\n    fun (a b : X ⟶ Y) =>\n      ulift.cases_on a\n        fun (a : plift (X ≤ Y)) =>\n          plift.cases_on a\n            fun (f₁ : X ≤ Y) =>\n              ulift.cases_on b\n                fun (b : plift (X ≤ Y)) => plift.cases_on b fun (f₂ : X ≤ Y) => Eq.refl (ulift.up (plift.up f₁))\n\n/-- A functor `C ⥤ D` computably lowers to a functor `thin_skeleton C ⥤ thin_skeleton D`. -/\ndef map {C : Type u₁} [category C] {D : Type u₂} [category D] (F : C ⥤ D) : thin_skeleton C ⥤ thin_skeleton D :=\n  functor.mk (quotient.map (functor.obj F) sorry)\n    fun (X Y : thin_skeleton C) =>\n      quotient.rec_on_subsingleton₂ X Y fun (x y : C) (k : quotient.mk x ⟶ quotient.mk y) => hom_of_le sorry\n\ntheorem comp_to_thin_skeleton {C : Type u₁} [category C] {D : Type u₂} [category D] (F : C ⥤ D) : F ⋙ to_thin_skeleton D = to_thin_skeleton C ⋙ map F :=\n  rfl\n\n/-- Given a natural transformation `F₁ ⟶ F₂`, induce a natural transformation `map F₁ ⟶ map F₂`.-/\ndef map_nat_trans {C : Type u₁} [category C] {D : Type u₂} [category D] {F₁ : C ⥤ D} {F₂ : C ⥤ D} (k : F₁ ⟶ F₂) : map F₁ ⟶ map F₂ :=\n  nat_trans.mk fun (X : thin_skeleton C) => quotient.rec_on_subsingleton X fun (x : C) => ulift.up (plift.up sorry)\n\n-- TODO: state the lemmas about what happens when you compose with `to_thin_skeleton`\n\n/-- A functor `C ⥤ D ⥤ E` computably lowers to a functor\n`thin_skeleton C ⥤ thin_skeleton D ⥤ thin_skeleton E` -/\n@[simp] theorem map₂_obj_obj {C : Type u₁} [category C] {D : Type u₂} [category D] {E : Type u₃} [category E] (F : C ⥤ D ⥤ E) (x : thin_skeleton C) (y : thin_skeleton D) : functor.obj (functor.obj (map₂ F) x) y =\n  quotient.map₂ (fun (X : C) (Y : D) => functor.obj (functor.obj F X) Y) (map₂._proof_1 F) x y :=\n  Eq.refl (functor.obj (functor.obj (map₂ F) x) y)\n\nprotected instance to_thin_skeleton_faithful (C : Type u₁) [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : faithful (to_thin_skeleton C) :=\n  faithful.mk\n\n/-- Use `quotient.out` to create a functor out of the thin skeleton. -/\n@[simp] theorem from_thin_skeleton_map (C : Type u₁) [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] (x : thin_skeleton C) (y : thin_skeleton C) : ∀ (ᾰ : x ⟶ y),\n  functor.map (from_thin_skeleton C) ᾰ =\n    quotient.rec_on_subsingleton₂ x y\n      (fun (X Y : C) (f : quotient.mk X ⟶ quotient.mk Y) =>\n        iso.hom (nonempty.some (from_thin_skeleton._proof_2 C X)) ≫\n          nonempty.some (from_thin_skeleton._proof_3 C X Y f) ≫\n            iso.inv (nonempty.some (from_thin_skeleton._proof_4 C Y)))\n      ᾰ :=\n  fun (ᾰ : x ⟶ y) => Eq.refl (functor.map (from_thin_skeleton C) ᾰ)\n\nprotected instance from_thin_skeleton_equivalence (C : Type u₁) [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : is_equivalence (from_thin_skeleton C) :=\n  is_equivalence.mk' (to_thin_skeleton C)\n    (nat_iso.of_components (fun (x : thin_skeleton C) => quotient.rec_on_subsingleton x fun (X : C) => eq_to_iso sorry)\n      sorry)\n    (nat_iso.of_components (fun (X : C) => nonempty.some sorry) sorry)\n\ntheorem equiv_of_both_ways {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] {X : C} {Y : C} (f : X ⟶ Y) (g : Y ⟶ X) : X ≈ Y :=\n  Nonempty.intro (iso_of_both_ways f g)\n\nprotected instance thin_skeleton_partial_order {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : partial_order (thin_skeleton C) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\ntheorem skeletal {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : skeletal (thin_skeleton C) := sorry\n\ntheorem map_comp_eq {C : Type u₁} [category C] {D : Type u₂} [category D] {E : Type u₃} [category E] [∀ (X Y : C), subsingleton (X ⟶ Y)] (F : E ⥤ D) (G : D ⥤ C) : map (F ⋙ G) = map F ⋙ map G := sorry\n\ntheorem map_id_eq {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : map 𝟭 = 𝟭 := sorry\n\ntheorem map_iso_eq {C : Type u₁} [category C] {D : Type u₂} [category D] [∀ (X Y : C), subsingleton (X ⟶ Y)] {F₁ : D ⥤ C} {F₂ : D ⥤ C} (h : F₁ ≅ F₂) : map F₁ = map F₂ :=\n  functor.eq_of_iso skeletal (iso.mk (map_nat_trans (iso.hom h)) (map_nat_trans (iso.inv h)))\n\n/-- `from_thin_skeleton C` exhibits the thin skeleton as a skeleton. -/\ndef thin_skeleton_is_skeleton {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : is_skeleton_of C (thin_skeleton C) (from_thin_skeleton C) :=\n  is_skeleton_of.mk sorry (thin_skeleton.from_thin_skeleton_equivalence C)\n\nprotected instance is_skeleton_of_inhabited {C : Type u₁} [category C] [∀ (X Y : C), subsingleton (X ⟶ Y)] : Inhabited (is_skeleton_of C (thin_skeleton C) (from_thin_skeleton C)) :=\n  { default := thin_skeleton_is_skeleton }\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/skeletal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.30901686245007903}}
{"text": "import category_theory.abelian.left_derived\n\nimport .derived_functor\n\nuniverses w v u\n\nnoncomputable theory\n\nnamespace category_theory.abelian.functor\n\nopen category_theory category_theory.functor category_theory.limits\nopen category_theory.functor.left_derived\n\nvariables {C : Type u} {D : Type u} [category.{w} C] [category.{w} D] [enough_projectives C]\nvariables (F : C ⥤ D) {A₁ A₂ A₃ X : C} {f : A₁ ⟶ A₂} {g : A₂ ⟶ A₃}\nvariables [abelian C] [abelian D] [additive F] [preserves_finite_colimits F]\n\nsection les\n\ndef δ₀ (A : short_exact_sequence C) := δ F 0 A ≫ (left_derived_zero_iso_self F).hom.app A.1\n\nlemma seven_term_exact_seq (A : short_exact_sequence C) :\n  exact_seq D [\n    (F.left_derived 1).map A.f, (F.left_derived 1).map A.g,\n    δ₀ F A,\n    F.map A.f, F.map A.g, (0 : F.obj A.3 ⟶ F.obj A.3)] :=\nbegin\n  refine exact_seq.cons _ _ (exact_of_short_exact _ _ _) _ (exact_seq.cons _ _ _ _ _),\n  { refine preadditive.exact_of_iso_of_exact' ((F.left_derived 1).map A.g) (δ F 0 A) _ _\n      (iso.refl _) (iso.refl _) ((left_derived_zero_iso_self F).app A.1) (by simp) _ _,\n    { dsimp [δ₀], rw [category.id_comp] },\n    { exact (exact_iff_exact_seq _ _).2 ((six_term_exact_seq F 0 A).extract 1 2) } },\n  refine exact_seq.cons _ _ _ _ _,\n  { refine preadditive.exact_of_iso_of_exact' (δ F 0 A) ((F.left_derived 0).map A.f) _ _\n      (iso.refl _) ((left_derived_zero_iso_self F).app A.1) ((left_derived_zero_iso_self F).app A.2)\n      _ (by simp) _,\n    { dsimp [δ₀], rw [category.id_comp] },\n    { exact (exact_iff_exact_seq _ _).2 ((six_term_exact_seq F 0 A).extract 2 2) } },\n    apply exact_seq.cons,\n    { exact preserves_exact_of_preserves_finite_colimits_of_epi _ A.exact' },\n    { rw [← exact_iff_exact_seq],\n      exact ((abelian.tfae_epi (F.obj A.3) (F.map A.g)).out 0 2).1 infer_instance, }\nend\n\nend les\n\nend category_theory.abelian.functor", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/lte/for_mathlib/derived_functor_zero.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6757645879592641, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.3089168050879828}}
{"text": "import Lean.Elab.Command\nimport Lib.Logic.Classical\n\nmacro \"obtain \" p:term \" from \" d:term : tactic =>\n  `(tactic| match $d:term with | $p:term => ?_)\n\n-- macro \"left\" : tactic =>\n--   `(tactic| apply Or.inl)\n\n-- macro \"right\" : tactic =>\n--   `(tactic| apply Or.inr)\n\n-- macro \"refl\" : tactic =>\n--   `(tactic| apply rfl)\n\n-- macro \"exfalso\" : tactic =>\n--   `(tactic| apply False.elim)\n\nmacro \"byContradiction\" h: ident : tactic =>\n  `(tactic| apply Classical.byContradiction; intro h)\n\ntheorem swapHyp {p q : Prop} (h : p) (h' : ¬ p) : q := by\n  cases h' h\n\nmacro \"swapHyp\" h:term \"as\" h':ident : tactic =>\n  `(tactic| apply Classical.byContradiction; intro $h' <;>\n            first\n            | apply $h ; clear $h\n            | apply swapHyp $h <;> clear $h\n              )\n\n-- open Lean Parser.Tactic Elab Command Elab.Tactic Meta\n-- macro \"fail\" : tactic => pure $ throwError \"foo\"\n\nnamespace Option\n\n@[simp]\ntheorem map_none {f : α → β} :\n  Option.map f none = none := rfl\n\n@[simp]\ntheorem map_some {f : α → β} {x} :\n  Option.map f (some x) = some (f x) := rfl\n\nattribute [simp] map_id\n\nend Option\n\ninductive EqvGen (R : α → α → Prop) : α → α → Prop :=\n| rfl {x} : EqvGen R x x\n| step {x y z} : R x y → EqvGen R y z → EqvGen R x z\n| symm_step {x y z} : R y x → EqvGen R y z → EqvGen R x z\n\nnamespace EqvGen\n\nvariable {R : α → α → Prop}\n\ntheorem once (h : R x y) : EqvGen R x y := EqvGen.step h EqvGen.rfl\ntheorem once_symm (h : R y x) : EqvGen R x y := EqvGen.symm_step h EqvGen.rfl\n\ntheorem trans : EqvGen R x y → EqvGen R y z → EqvGen R x z := by\n  intros h₀ h₁\n  induction h₀ with\n  | rfl => exact h₁\n  | step hR hGen IH => exact step hR (IH h₁)\n  | symm_step hR hGen IH => exact symm_step hR (IH h₁)\n\ntheorem symm_step_r : EqvGen R x y → R z y → EqvGen R x z := by\n  intros h₀ h₁\n  have h₁ := once_symm h₁\n  apply trans <;> assumption\ntheorem step_r : EqvGen R x y → R y z → EqvGen R x z := by\n  intros h₀ h₁\n  have h₁ := once h₁\n  apply trans <;> assumption\n\ntheorem symm : EqvGen R x y → EqvGen R y x := by\n  intros h\n  induction h with\n  | rfl => exact rfl\n  | step hR hGen IH => exact trans IH (symm_step hR rfl)\n  | symm_step hR hGen IH => exact trans IH (step hR rfl)\n\nend EqvGen\n\nnamespace Quot\nvariable {r : α → α → Prop}\nvariable {r' : α' → α' → Prop}\n\ndef liftOn₂ (x : Quot r) (y : Quot r') (f : α → α' → β)\n  (h : ∀ (a b : α) (a' : α'),\n    r a b → f a a' = f b a')\n  (h' : ∀ (a : α) (a' b' : α'),\n    r' a' b' → f a a' = f a b') : β :=\nQuot.liftOn x\n  (λ x => Quot.liftOn y (f x) $\n    by intros; apply h'; assumption) $\n  by intros\n     induction y using Quot.inductionOn\n     simp [Quot.liftOn]\n     apply h; assumption\n\n@[simp]\ndef liftOn₂_beta {x : α} {y : α'} (f : α → α' → β)\n  (h : ∀ (a b : α) (a' : α'),\n    r a b → f a a' = f b a')\n  (h' : ∀ (a : α) (a' b' : α'),\n    r' a' b' → f a a' = f a b') :\nliftOn₂ (Quot.mk _ x) (Quot.mk _ y) f h h' = f x y :=\nby simp [liftOn₂]\n\nnoncomputable def out (x : Quot r) : α :=\nClassical.choose (exists_rep x)\n\ntheorem eq' : Quot.mk r x = Quot.mk r y ↔ EqvGen r x y := by\n  constructor\n  focus\n    intros h\n    have Hlift : ∀ (a b : α), r a b → EqvGen r x a = EqvGen r x b := by\n      intros a b Hab\n      apply propext\n      constructor <;> intro h\n      focus apply EqvGen.step_r <;> assumption\n      focus apply EqvGen.symm_step_r <;> assumption\n    rw [← Quot.liftBeta (r := r) (EqvGen r x) Hlift, ← h,\n          Quot.liftBeta (r := r) _ Hlift]\n    exact EqvGen.rfl\n  focus\n    intros h\n    induction h with\n    | rfl => refl\n    | step Hr Hgen IH =>\n      rw [← IH]\n      apply Quot.sound; assumption\n    | symm_step Hr Hgen IH =>\n      apply Eq.symm\n      rw [← IH]\n      apply Quot.sound; assumption\n\ntheorem eq : Quot.mk r x = Quot.mk r y ↔ EqvGen r x y := by\n  constructor\n  focus\n    intros h\n    rewrite [← liftOn₂_beta (r:=r) (r':=r) (EqvGen r), h, liftOn₂_beta]\n    focus\n      exact EqvGen.rfl\n    focus\n      intros x y z Hxy\n      apply propext\n      constructor <;> intro h\n      focus apply EqvGen.symm_step <;> assumption\n      focus apply EqvGen.step <;> assumption\n    focus\n      intros x y z Hxy\n      apply propext\n      constructor <;> intro h\n      focus apply EqvGen.step_r <;> assumption\n      focus apply EqvGen.symm_step_r <;> assumption\n  focus\n    intros h\n    induction h with\n    | rfl => refl\n    | step Hr Hgen IH =>\n      rw [← IH]\n      apply Quot.sound; assumption\n    | symm_step Hr Hgen IH =>\n      apply Eq.symm\n      rw [← IH]\n      apply Quot.sound; assumption\n\nend Quot\n\nclass WellOrdered (α : Type) where\n  R : α → α → Prop\n  wf : WellFounded R\n  total x y : R x y ∨ x = y ∨ R y x\n\ninstance [WellOrdered α] : LT α := ⟨ WellOrdered.R ⟩\ninstance [WellOrdered α] : LE α where\n  le x y := x = y ∨ x < y\n\nnamespace WellOrdered\n\nvariable [Hwo : WellOrdered α]\n\ntheorem le_of_lt {x y : α} : x < y → x ≤ y := Or.inr\n\ntheorem lt_irrefl (x : α) : ¬ x < x := by\nhave := WellOrdered.wf.apply x\ninduction this with | intro x h ih =>\nintro h'\napply ih _ h' h'\n\ntheorem le_refl (x : α) : x ≤ x := by\n  left; refl\n\ntheorem lt_antisymm {x y : α} :\n  x < y → y < x → False := by\nrevert y\nhave := WellOrdered.wf.apply x\ninduction this with | intro x h ih =>\nintros y Hxy Hyx\ncases ih _ Hyx Hyx Hxy\n\ntheorem le_antisymm {x y : α} :\n  x ≤ y → y ≤ x → x = y := by\nintros Hxy Hyx; match x, Hxy, Hyx with\n| x, Or.inl rfl, _ => intros; refl\n| x, Or.inr Hx_lt, Or.inl rfl => refl\n| x, Or.inr Hx_lt, Or.inr Hy_lt =>\n  cases lt_antisymm Hx_lt Hy_lt\n\n\nsection\nopen Classical\n\ntheorem bottom_exists [inst : Nonempty α] :\n  ∃ x : α, ∀ y : α, ¬ y < x := by\ncases inst with | intro x =>\nhave h := Hwo.wf.apply x\ninduction h with | intro y h ih => exact\nif h : ∃ z, z < y then by\n  cases h with | intro z hz =>\n  apply ih _ hz\nelse by\n  rewrite [not_exists] at h\n  exists y; assumption\n\nnoncomputable def bottom (α) [WellOrdered α]\n  [inst : Nonempty α] : α :=\nchoose bottom_exists\n\ntheorem lt_iff_not_lt [h : WellOrdered α] (x y : α) :\n  x < y ↔ ¬ y ≤ x := by\nconstructor\nfocus\n  intros h₀ h₁\n  match y, h₁ with\n  | y, Or.inr h₁ =>\n    apply lt_antisymm h₀ h₁\n  | _, Or.inl rfl =>\n    apply lt_irrefl _ h₀\nfocus\n  intros h₀\n  have := WellOrdered.total x y\n  match y, this with\n  | y, Or.inl h₀ => assumption\n  | _, Or.inr (Or.inl rfl) =>\n    exfalso\n    apply h₀; clear h₀\n    apply le_refl\n  | y, Or.inr (Or.inr h₁) =>\n    exfalso\n    apply h₀; clear h₀\n    right; assumption\n\ntheorem not_lt_bottom\n  [inst : Nonempty α] :\n  ¬ x < bottom α :=\nchoose_spec bottom_exists x\n\ntheorem bottom_le\n  [inst : Nonempty α] :\n  bottom α ≤ x := by\nhave H := not_lt_bottom (α := α) (x := x)\nsimp [lt_iff_not_lt] at H\nassumption\n\nend\n\ntheorem lt_trans {x y z : α} :\n  x < y → y < z → x < z := by\nrewrite [lt_iff_not_lt x z]\nintros hxy hyz hxz\ncases hxz with\n| inl hxz =>\n  subst hxz\n  apply lt_antisymm hxy hyz\n| inr hxz =>\n  have := WellOrdered.wf.apply x\n  induction this generalizing y z with | intro x h ih =>\n  apply @ih _ hxz x y <;> assumption\n\ntheorem le_trans {x y z : α} :\n  x ≤ y → y ≤ z → x ≤ z := by\nintros hxy hyz\ncases hxy with\n| inl hxy =>\n  cases hyz with\n  | inl hyz => subst hxy hyz; left; refl\n  | inr hyz => subst hxy; right; assumption\n| inr hxy =>\n  cases hyz with\n  | inl hyz => subst hyz; right; assumption\n  | inr hyz =>\n    right; apply lt_trans\n      <;> assumption\n\nsection Classical\nopen Classical\n\n-- noncomputable\n-- def max (x y : α) : α :=\n-- if x < y\n--   then y\n--   else x\n\ntheorem max_le_iff (x y z : α) :\n  max x y ≤ z ↔ x ≤ z ∧ y ≤ z := by\nbyCases Hyz : y < x\nfocus\n  have := le_of_lt Hyz\n  simp [max, *]\n  constructor\n  focus\n    intros h\n    constructor <;> try assumption\n    apply le_trans <;> assumption\n  focus\n    intro h; cases h; assumption\nfocus\n  simp [max, *]\n  simp [lt_iff_not_lt] at Hyz\n  constructor\n  focus\n    intros h\n    constructor <;> try assumption\n    apply le_trans <;> assumption\n  focus\n    intros h; cases h; assumption\n\nend Classical\nend WellOrdered\n\ndef WellOrder := Sigma WellOrdered\n\nnoncomputable\ninstance : CoeSort WellOrder (Type) := ⟨ Sigma.fst ⟩\n\ninstance (α : WellOrder) : WellOrdered α := α.snd\n\nstructure WellOrderHom (α β : WellOrder) where\n  hom : α.1 → β.1\n  hom_preserve x y : α.2.R x y → β.2.R (hom x) (hom y)\n\nnamespace WellOrderHom\n\ninstance : CoeFun (WellOrderHom a b) (λ _ => a → b) where\n  coe f := f.hom\n\ntheorem ext {f g : WellOrderHom a b}\n  (h : ∀ x, f x = g x) :\n  f = g := by\nhave d : f.hom = g.hom := funext h\ncases f with\n| mk f hf =>\ncases g with | mk g hg =>\n  simp at d; revert hf hg;\n  rewrite [d]; intros; exact rfl\n\ntheorem ext_iff {f g : WellOrderHom a b}\n  (h : f = g) :\n  ∀ x, f x = g x := by\nsubst h; intros; refl\n\ndef id : WellOrderHom α α where\n  hom := _root_.id\n  hom_preserve x y hxy := hxy\n\ndef comp (f : WellOrderHom β γ) (g : WellOrderHom α β) : WellOrderHom α γ where\n  hom := f.hom ∘ g.hom\n  hom_preserve x y hxy := f.hom_preserve _ _ (g.hom_preserve _ _ hxy)\n\nvariable {X Y Z W : WellOrder}\nvariable (f : WellOrderHom X Y)\nvariable (g : WellOrderHom Y Z)\nvariable (h : WellOrderHom Z W)\n\n@[simp]\ntheorem id_comp : id.comp f = f := by apply ext; intro x; exact rfl\n@[simp]\ntheorem comp_id : f.comp id = f := by apply ext; intro x; exact rfl\n@[simp]\ntheorem comp_assoc : (h.comp g).comp f = h.comp (g.comp f) :=\nby apply ext; intro x; exact rfl\n\nend WellOrderHom\n\nclass IsMono {X Y} (f : WellOrderHom X Y) where\n  mono : ∀ W (g₁ g₂ : WellOrderHom W X), f.comp g₁ = f.comp g₂ → g₁ = g₂\n\nnamespace IsMono\nvariable {X Y Z : WellOrder}\nvariable (f : WellOrderHom X Y)\nvariable (g : WellOrderHom Y Z)\nvariable (gZY : WellOrderHom Z Y)\nvariable (gZX : WellOrderHom Z X)\nopen WellOrderHom\n\ninstance : IsMono (@WellOrderHom.id X) where\n  mono W g₁ g₂ := by simp; apply _root_.id\n\ninstance [IsMono f] [IsMono g] : IsMono (g.comp f) where\n  mono W g₁ g₂ := by\n    simp; intro h\n    have h := mono _ _ _ h\n    apply mono _ _ _ h\n\nend IsMono\n\nclass IsIso {X Y} (f : WellOrderHom X Y) where\n  inv : WellOrderHom Y X\n  to_inv : f.comp inv = WellOrderHom.id\n  inv_to : inv.comp f = WellOrderHom.id\n\nexport IsIso (inv)\n\nnamespace IsIso\nattribute [simp] to_inv inv_to\n\nopen WellOrderHom\nvariable {X Y Z : WellOrder}\nvariable (f : WellOrderHom X Y)\nvariable (g : WellOrderHom Y Z)\nvariable (gZY : WellOrderHom Z Y)\nvariable (gZX : WellOrderHom Z X)\n\n@[simp]\ntheorem to_inv_reassoc [IsIso f] :\n  f.comp ((inv f).comp gZY) = gZY :=\nby rw [← WellOrderHom.comp_assoc, to_inv, id_comp]\n\n@[simp]\ntheorem inv_to_reassoc [IsIso f] :\n  (inv f).comp (f.comp gZX) = gZX :=\nby rw [← WellOrderHom.comp_assoc, inv_to, id_comp]\n\ninstance [IsIso f] : IsIso (inv f) where\n  inv := f\n  to_inv := inv_to\n  inv_to := to_inv\n\ninstance : IsIso (@WellOrderHom.id X) where\n  inv := id\n  to_inv := by simp\n  inv_to := by simp\n\ninstance [IsIso f] [IsIso g] : IsIso (g.comp f) where\n  inv := (inv f).comp (inv g)\n  to_inv := by simp\n  inv_to := by simp\n\ninstance (priority := low) [IsIso f] : IsMono f where\n  mono W g₁ g₂ Heq := by\n    rw [← inv_to_reassoc f g₁, Heq, inv_to_reassoc]\n\nend IsIso\n\nsection HasArrow\n-- set_option checkBinderAnnotations false\n\ninductive HasArrow X Y (cl : WellOrderHom X Y → Type u) : Prop :=\n| intro (f : WellOrderHom X Y) : cl f → HasArrow X Y cl\n\nend HasArrow\n\ndef WellOrderMono (α β : WellOrder) := HasArrow α β IsMono\n@[matchPattern]\ndef WellOrderMono.intro (f : WellOrderHom x y) [IsMono f] :\n  WellOrderMono x y :=\nHasArrow.intro f inferInstance\n\ndef WellOrderIso (α β : WellOrder) := HasArrow α β IsIso\n@[matchPattern]\ndef WellOrderIso.intro (f : WellOrderHom x y) [IsIso f] :\n  WellOrderIso x y :=\nHasArrow.intro f inferInstance\n\n-- namespace WellOrderIso\n\n-- variable {X Y : WellOrder}\n\n-- def flip (f : WellOrderIso X Y) : WellOrderIso Y X where\n--   to := f.inv\n--   inv := f.to\n--   to_inv := f.inv_to\n--   inv_to := f.to_inv\n\n-- def toMono (f : WellOrderIso X Y) : WellOrderMono X Y where\n--   to := f.to\n--   mono Z g₀ g₁ h := _\n\n-- end WellOrderIso\n\n-- inductive Erased (α : Type u) : Prop :=\n-- | intro (x : α) : Erased α\n\n-- theorem Erased_impl {α β} (f : α → β) : Erased α → Erased β\n-- | ⟨ x ⟩ => ⟨ f x ⟩\n\ndef WOEqv (α β : WellOrder) : Prop :=\nWellOrderIso α β\n\ndef Ordinal := Quot WOEqv\n\nnamespace Ordinal\n\nprotected def mk (α : Type) [WellOrdered α] : Ordinal :=\nQuot.mk _ { fst := α, snd := inferInstance }\n\ndef zero : Ordinal :=\nQuot.mk _\n{ fst := Empty,\n  snd :=\n  { R := by {intro x; cases x},\n    wf := by { constructor; intro x; cases x },\n    total := by {intro x; cases x} } }\n\ninductive Option.R (r : α → α → Prop) : Option α → Option α → Prop where\n| top x : R r (some x) none\n| lifted x y : r x y → R r (some x) (some y)\n\ntheorem Option.R_acc {r : α → α → Prop}\n        x (h : Acc r x) : Acc (Option.R r) (some x) := by\n  induction h with\n  | intro y h' ih =>\n    constructor; intros z Hr; cases Hr\n    apply ih; assumption\n\ntheorem Option.R_wf {r : α → α → Prop}\n        (h : WellFounded r) : WellFounded (Option.R r) := by\n  constructor; intro x\n  match x with\n  | none =>\n    constructor; intro y h'; cases h'\n    apply Option.R_acc\n    apply h.apply\n  | some x =>\n    apply Option.R_acc\n    apply h.apply\n\ninstance [WellOrdered α] : WellOrdered (Option α) where\n  R := Option.R (WellOrdered.R)\n  wf := Option.R_wf (WellOrdered.wf)\n  total := λ x y =>\n    match x, y with\n    | none, none => by\n      apply Or.inr; apply Or.inl\n      constructor\n    | none, some _ => by\n      apply Or.inr; apply Or.inr\n      constructor\n    | some _, none => by\n      apply Or.inl\n      constructor\n    | some x, some y =>\n      match WellOrdered.total x y with\n      | Or.inl h =>\n        Or.inl $ Option.R.lifted _ _ h\n      | Or.inr (Or.inl h) =>\n        Or.inr $ Or.inl $ congrArg _ h\n      | Or.inr (Or.inr h) =>\n        Or.inr $ Or.inr $ Option.R.lifted _ _ h\n\ndef impl.S : WellOrder → WellOrder\n| ⟨α, Hα⟩ =>\n  let _ : WellOrdered α := Hα\n  { fst := Option α,\n    snd := inferInstance }\n\ndef S_hom (h : WellOrderHom a b) :\n    WellOrderHom (impl.S a) (impl.S b) where\n  hom := Option.map h\n  hom_preserve := by\n    intros x y h; cases x <;>\n      cases y <;>\n      cases h <;>\n      constructor <;>\n      apply h.hom_preserve <;>\n      assumption\n\n@[simp]\ntheorem id_hom :\n  (@WellOrderHom.id a).hom = id := rfl\n\n@[simp]\ntheorem S_hom_hom (h : WellOrderHom a b) :\n  (S_hom h).hom = Option.map h.hom := rfl\n\n@[simp]\ntheorem comp_hom (f : WellOrderHom b c) (g : WellOrderHom a b) :\n  (f.comp g).hom = f.hom ∘ g.hom := rfl\n\n@[simp]\ntheorem comp_S_hom\n  (f : WellOrderHom b c) (g : WellOrderHom a b) :\n  (S_hom f).comp (S_hom g) = S_hom (f.comp g) :=\nWellOrderHom.ext $ by\n  intro x\n  cases x <;> simp [WellOrderHom.comp, S_hom]\n\n@[simp]\ntheorem S_hom_id {α} :\n  S_hom (@WellOrderHom.id α) = WellOrderHom.id :=\nWellOrderHom.ext $ by\n  intro x\n  rw [S_hom_hom, id_hom, id_hom]\n  simp\n\ninstance [@IsIso x y f] : IsIso (S_hom f) where\n  inv := S_hom (inv f)\n  to_inv := by simp\n  inv_to := by simp\n\ndef S_iso : WellOrderIso a b → WellOrderIso (impl.S a) (impl.S b)\n| ⟨f, h⟩ =>\n  have h' := h\n  ⟨S_hom f, inferInstance⟩\n  -- to := S_hom h.to\n  -- inv := S_hom h.inv\n  -- to_inv := by simp [h.to_inv]\n  -- inv_to := by simp [h.inv_to]\n\ndef S : Ordinal → Ordinal :=\nQuot.lift (Quot.mk _ ∘ impl.S) λ a b Heqv => by\n  simp\n  apply Quot.sound\n  exact S_iso Heqv\n\ndef Le (x y : Ordinal) : Prop :=\nQuot.liftOn₂ x y\n  (λ x y => WellOrderMono x y)\n  (by intros a b x f\n      cases f with\n      | intro f hf =>\n      simp; apply propext\n      constructor <;> intros g\n      { cases g with\n        | intro g hg =>\n          have hf := hf\n          have hg := hg\n          exact ⟨g.comp (inv f), inferInstance⟩ }\n      { cases g with\n        | intro g hg =>\n          have hf := hf\n          have hg := hg\n          exact ⟨g.comp f, inferInstance⟩ } )\n  (by intros x a b f\n      cases f with | intro f hf =>\n      simp; apply propext\n      constructor <;> intros g\n      { cases g with\n        | intro g hg =>\n          have hf := hf\n          have hg := hg\n          exact ⟨f.comp g, inferInstance⟩ }\n      { cases g with\n        | intro g hg =>\n          have hf := hf\n          have hg := hg\n          exact ⟨(inv f).comp g, inferInstance⟩ }\n       )\n\ndef Lt (x y : Ordinal) : Prop :=\nLe x y ∧ ¬ Le y x\n\n-- structure StrictInclusion (a b : WellOrder) : Type where\n--   inclusion : WellOrderHom a b\n--   [incl_mono : IsMono inclusion]\n--   strict : ¬ WellOrderMono b a\n\n-- attribute [instance] StrictInclusion.incl_mono\n\ninstance : LE Ordinal := ⟨ Le ⟩\ninstance : LT Ordinal := ⟨ Lt ⟩\n\ntheorem le_refl (x : Ordinal) : x ≤ x := by\n  cases x using Quot.ind\n  simp [LE.le,Le]\n  exists @WellOrderHom.id _\n  exact inferInstance\n\ninfix:25 \" ⟶ \" => WellOrderHom\n\ninstance : WellOrdered Unit where\n  R _ _ := False\n  wf := ⟨ λ a => ⟨ _, by intros _ h; cases h ⟩ ⟩\n  total := by intros; right; left; refl\n\ndef Unit : WellOrder := ⟨_root_.Unit,inferInstance⟩\n\ndef const {β : WellOrder} (x : β) : Unit ⟶ β where\n  hom := λ i => x\n  hom_preserve := by intros _ _ h; cases h\n\nsection inverse\nopen Classical\nnoncomputable\ndef inverse [Nonempty α] (f : α → β) (x : β) : α :=\nepsilon λ y => f y = x\n\nend inverse\n\nsection schroeder_bernstein\nopen Classical\n\ndef injective (f : α → β) : Prop :=\n∀ x y, f x = f y → x = y\n\ndef surjective (f : α → β) : Prop :=\n∀ y, ∃ x, f x = y\n\ndef bijective (f : α → β) : Prop :=\ninjective f ∧ surjective f\n\nvariable [Nonempty α] [Nonempty β]\nvariable (f : α → β) (hf : injective f)\nvariable (g : β → α) (hg : injective g)\n\ntheorem left_inv_of_injective :\n  inverse f ∘ f = id := by\napply funext; intro x\nsimp [Function.comp, inverse]\napply hf\napply Classical.epsilon_spec (p := λ y => f y = f x)\nexists x; refl\n\ntheorem right_inv_of_surjective (hf' : surjective f) :\n  f ∘ inverse f = id := by\napply funext; intro x\nspecialize (hf' x)\napply Classical.epsilon_spec (p := λ y => f y = x)\nassumption\n\ntheorem injective_of_left_inv\n        (h : g ∘ f = id) :\n  injective f := by\nintros x y Hxy\nhave Hxy : (g ∘ f) x = (g ∘ f) y := congrArg g Hxy\nrewrite [h] at Hxy; assumption\n\ntheorem injective_of_right_inv\n        (h : g ∘ f = id) :\n  surjective g := by\nintros x\nexists (f x)\nshow (g ∘ f) x = x\nrewrite [h]; refl\n-- def lonely (b : β) := ∀ a, f a ≠ b\n\n-- def iterate (n : Nat) (b : β) : β :=\n-- match n with\n-- | 0 => b\n-- | Nat.succ n => iterate n (f (g b))\n\n-- -- #exit\n-- def descendent (b₀ b₁ : β) : Prop :=\n-- ∃ n, iterate f g n b₀ = b₁\n-- -- #exit\n-- theorem schroeder_bernstein :\n--   ∃ h : α → β, bijective h := by\n-- let p x y := lonely f y ∧ descendent f g y (f x)\n-- let h x :=\n--   if ∃ y, p x y\n--     then inverse g x\n--     else f x\n-- exists h; constructor\n-- focus\n--   intro x y; simp\n-- -- focus\n-- --   intros x y\n-- --   -- simp at Hxy\n-- --   simp\n-- -- --   have : (∃ z, p x z) ↔ (∃ z, p y z) := by\n-- -- --     constructor <;> intros h\n-- -- --     focus\n-- -- --       cases h with\n-- -- --       | intro z hz => unfold p at *\n-- -- --   admit\n--   byCases hx : (∃ z, p x z) <;>\n--     byCases hy : (∃ z, p y z) <;>\n--     simp [*] <;> intro h3\n--   focus\n--     clear h\n--     cases hx with | intro x' hx' =>\n--     cases hy with | intro y' hy' =>\n--       cases hy'; cases hx'\n--       clear p\n-- admit\n-- abort\n-- -- match propDecidable (∃ z, p x z), propDecidable (∃ z, p y z), Hxy with\n-- -- | isTrue ⟨x', hx, hx'⟩, isTrue ⟨y', hy, hy'⟩, Hxy => _\n-- -- | isTrue _, isFalse _, Hxy => _\n-- -- | isFalse _, isTrue _, Hxy => _\n-- -- | isFalse _, isFalse _, Hxy => _\n-- -- -- { admit }\n-- --      simp [h] at Hxy }\n-- -- exists (inverse g ∘ f)\n\nend schroeder_bernstein\n\ntheorem comp_const (f : WellOrderHom a b) x :\n  f.comp (const x) = const (f x) := rfl\n\ntheorem injective_of_monomorphism\n        (f : WellOrderHom a b) [IsMono f] :\n  injective f.hom := by\nintros x y Hxy\nhave := @IsMono.mono _ _ f _ _ (const x) (const y)\nsimp [comp_const, Hxy] at this\nhave := WellOrderHom.ext_iff this ()\nassumption\n\n-- Need Schroeder-Bernstein Theorem\ntheorem le_antisymm (x y : Ordinal)\n        (Hxy : x ≤ y)\n        (Hyx : y ≤ x) : x = y := by\n  cases x using Quot.ind\n  cases y using Quot.ind\n  simp [LE.le,Le] at *\n  apply Quot.sound\n  cases Hxy with | intro f hf =>\n  cases Hyx with | intro g hg => _\n  admit\n\ntheorem le_trans (x y z : Ordinal)\n        (Hxy : x ≤ y)\n        (Hyz : y ≤ z) : x ≤ z := by\n  cases x using Quot.ind\n  cases y using Quot.ind\n  cases z using Quot.ind\n  simp [LE.le,Le] at *\n  cases Hxy with | intro f hf =>\n  cases Hyz with | intro g hg =>\n  have hf := hf\n  have hg := hg\n  exists g.comp f\n  exact inferInstance\n\ntheorem lt_total (x y : Ordinal) :\n  x < y ∨ x = y ∨ y < x := by\nbyCases hxy : (x ≤ y) <;>\n  byCases hyx : (y ≤ x)\nfocus\n  right; left\n  apply le_antisymm <;> assumption\nfocus\n  left; constructor <;> assumption\nfocus\n  right; right\n  constructor <;> assumption\nfocus\n  admit\n-- Todo:\n-- Le is a total order\n-- Limits\n-- zero is a limit? (no)\n-- zero, Succ, Limit are distinct\n-- recursor / induction principle\n\n\nnamespace WellOrdered\n\nvariable  [WellOrdered α]\n\ntheorem le_iff_forall_le (x y : α) :\n  x ≤ y ↔ ∀ z, y ≤ z → x ≤ z := by\nconstructor <;> intros h\nfocus\n  intros z h'\n  apply WellOrdered.le_trans <;> assumption\nfocus\n  apply h\n  apply WellOrdered.le_refl\n\ntheorem le_iff_forall_ge (x y : α) :\n  x ≤ y ↔ ∀ z, z ≤ x → z ≤ y := by\nconstructor <;> intros h\nfocus\n  intros z h'\n  apply WellOrdered.le_trans <;> assumption\nfocus\n  apply h\n  apply WellOrdered.le_refl\n\ntheorem le_total (x y : α) :\n  x ≤ y ∨ y ≤ x := sorry\n\nsection Classical\nopen Classical\n\ntheorem le_max_l {x y : α} :\n  x ≤ max x y := by\nrw [le_iff_forall_le]; intros z\nrw [WellOrdered.max_le_iff]\nintros h; cases h; assumption\n\ntheorem le_max_r {x y : α} :\n  y ≤ max x y := by\nrw [le_iff_forall_le]; intros z\nrw [WellOrdered.max_le_iff]\nintros h; cases h; assumption\n\n\ninstance : @Reflexive α LE.le where\n  refl := WellOrdered.le_refl\n\ntheorem max_eq_of_le {x y : α} :\n  x ≤ y → max x y = y := by\nintros h\napply WellOrdered.le_antisymm\nfocus\n  rw [WellOrdered.max_le_iff]\n  auto\nfocus\n  apply WellOrdered.le_max_r\n\ntheorem max_eq_of_ge {x y : α} :\n  y ≤ x → max x y = x := by\nintros h\napply WellOrdered.le_antisymm\nfocus\n  rw [WellOrdered.max_le_iff]\n  auto\nfocus\n  apply WellOrdered.le_max_l\n\nend Classical\nend WellOrdered\n\nnoncomputable\ninstance : CoeSort WellOrder Type := ⟨ Sigma.fst ⟩\n\nstructure Seq where\n  index : Type\n  [wo : WellOrdered index]\n  ords : index → WellOrder\n  increasing (i j : index) : i ≤ j → ords i ⟶ ords j\n  [mono (i j : index) h : IsMono (increasing i j h)]\n  strict (i j : index) :\n    i < j → ¬ WellOrderMono (ords i) (ords j)\n  increasing_refl (i : index) h :\n    increasing i i h =\n    WellOrderHom.id\n  increasing_trans (i j k : index) (h : i ≤ j) (h' : j ≤ k) :\n    (increasing _ _ h').comp (increasing _ _ h) =\n    increasing _ _ (WellOrdered.le_trans h h')\n  bottom : index\n  next : index → index\n  infinite i : i < next i\n  nothing_below_bottom x : ¬ x < bottom\n  consecutive i j : j < next i → j < i ∨ j = i\n\nattribute [instance] Seq.wo Seq.mono\n\n-- theorem Seq.increasing' (i j : s.index) :\n--   i ≤ j → ords i ⟶ ords j\n\ntheorem Seq.bottom_le (s : Seq) x : s.bottom ≤ x := by\n  have := s.nothing_below_bottom x\n  simp [WellOrdered.lt_iff_not_lt] at this\n  assumption\n\ndef Lim.Eqv (s : Seq) :\n  (Σ i : s.index, s.ords i) → (Σ i, s.ords i) → Prop\n| ⟨i, x⟩, ⟨j, y⟩ =>\n  (∃ h : i < j, s.increasing _ _ (WellOrdered.le_of_lt h) x = y) ∨\n  (∃ h : i = j, cast (by rw [h]) x = y) ∨\n  (∃ h : i > j, s.increasing _ _ (WellOrdered.le_of_lt h) y = x)\n\ndef Lim (s : Seq) :=\nQuot (Lim.Eqv s)\n\nnamespace Lim\n\nvariable (s : Seq)\n\ntheorem Eqv.intro i j (h : i ≤ j)\n        x y\n        (h' : s.increasing i j h x = y) :\n  Eqv s ⟨i, x⟩ ⟨j, y⟩ := by\ncases h with\n| inl h =>\n  subst h; right; left\n  exists @rfl _ i\n  simp [cast]\n  simp [Seq.increasing_refl, WellOrderHom.id] at h'\n  assumption\n| inr h =>\n  left; exists h\n  assumption\n\ntheorem cast_cast {a b} (h : a = b) (h' : b = a) x :\n  cast h (cast h' x) = x := by\ncases h; cases h'; refl\n\ntheorem Eqv.symm x y :\n  Eqv s x y →\n  Eqv s y x := by\nintros h; cases h with\n| inl h => right; right; assumption\n| inr h =>\n  cases h with\n  | inl h =>\n    right; left\n    cases h with | intro h h' =>\n    exists h.symm\n    rewrite [← h', cast_cast]\n    refl\n  | inr h => left; assumption\n\nsection\nopen Classical\n\ntheorem Eqv.intro' i j Hi Hj\n        x y\n        (h' : s.increasing i (max i j) Hi x =\n              s.increasing j (max i j) Hj y) :\n  Eqv s ⟨i, x⟩ ⟨j, y⟩ := by\ncases WellOrdered.le_total i j with\n| inl h =>\n  have h₂ := WellOrdered.max_eq_of_le h\n  revert Hi Hj h'\n  rewrite [h₂]; intros Hi Hj\n  rewrite [s.increasing_refl j]; intros h'\n  apply Eqv.intro; assumption\n| inr h =>\n  have h₂ := WellOrdered.max_eq_of_ge h\n  revert Hi Hj h'\n  rewrite [h₂]; intros Hi Hj\n  rewrite [s.increasing_refl i]; intros h'\n  apply Lim.Eqv.symm\n  apply Eqv.intro\n  apply Eq.symm; assumption\n\nend\n\ntheorem EqvGen_Eqv x y :\n  EqvGen (Eqv s) x y ↔ Eqv s x y := by\nconstructor <;> intros h\nfocus\n  induction h with\n  | @rfl a =>\n    cases a\n    right; left\n    exists @rfl _ _; simp [cast]\n  | @step x y z a _ b =>\n    cases x\n    cases y\n    cases z\n\n  | symm_step a => skip\n\nend Lim\n\nnamespace Lim\n\nvariable (s : Seq)\nopen Classical\n\ntheorem mk_eq_mk_max_l i j x :\n  Quot.mk (Lim.Eqv s) ⟨i, x⟩ =\n  Quot.mk _ ⟨max i j, s.increasing i (max i j)\n    WellOrdered.le_max_l x⟩ := by\nrw [Quot.eq]\napply EqvGen.once;\n  apply Eqv.intro\n{ refl }\n\ntheorem mk_eq_mk_max_r i j x :\n  Quot.mk (Lim.Eqv s) ⟨i, x⟩ =\n  Quot.mk _ ⟨max j i, s.increasing i (max j i)\n    WellOrdered.le_max_r x⟩ := by\nrw [Quot.eq]\napply EqvGen.once;\n  apply Eqv.intro\n{ refl }\n\n-- theorem mk_eq_mk_max_r i j x :\n--   Quot.mk (Lim.Eqv s) ⟨i, x⟩ =\n--   Quot.mk _ ⟨max j i, s.increasing i (max j i)\n\nprotected inductive R : Lim s → Lim s → Prop :=\n| intro i (x y : s.ords i) :\n  x < y → Lim.R (Quot.mk _ ⟨i, x⟩) (Quot.mk _ ⟨i, y⟩)\n\n-- protected def SigmaT := PSigma (λ i => s.ords i)\n-- protected def toSigma : Lim s → Lim.SigmaT s\n-- | ⟨a,b,h⟩ => ⟨a,b⟩\n-- protected def SigmaT.R : Lim.SigmaT s → Lim.SigmaT s → Prop :=\n-- PSigma.Lex LT.lt (λ _ => LT.lt)\n\n-- protected theorem SigmaT.R_Subrelation :\n--   Subrelation (Lim.R s) (InvImage (SigmaT.R s) (Lim.toSigma s)) := by\n--   intros x y h\n--   cases h with\n--   | same i j k =>\n--     simp [InvImage, Lim.toSigma]\n--     apply PSigma.Lex.right; assumption\n--   | lt x y =>\n--     cases x; cases y\n--     constructor; assumption\n\nprotected theorem R_wf :\n  WellFounded (Lim.R s) := by\n-- Subrelation.wf _ (InvImage.wf _ _)\nconstructor; intro a\ncases a using Quot.ind with | mk a =>\ncases a with | mk a b =>\nhave Hacc := WellOrdered.wf.apply b\n-- let f b := Quot.mk (Eqv s) { fst := a, snd := b }\n-- have Hacc := InvImage.accessible f Hacc\n-- apply Acc\ninduction Hacc with | intro b h ih =>\n-- clear h\nconstructor; intros y\ngeneralize Hx : (Quot.mk (Eqv s) { fst := a, snd := b }) = x\nintros Hr\ncases Hr with | intro i x =>\nrw [Quot.eq] at Hx\n-- cases Hr\n-- apply ih\nskip\n\n\n-- let lex : WellFounded (Lim.SigmaT.R s) :=\n  -- (PSigma.lex WellOrdered.wf (λ i => WellOrdered.wf))\n-- Subrelation.wf\n  -- (SigmaT.R_Subrelation s)\n  -- (InvImage.wf (Lim.toSigma s) lex)\n\nprotected theorem R_total x y :\n  Lim.R s x y ∨ x = y ∨ Lim.R s y x :=\nmatch x, y with\n| ⟨x,x',hx⟩, ⟨y,y',hy⟩ =>\n  match x, y, WellOrdered.total x y with\n  | _, _, Or.inl h => by left; constructor; assumption\n  | _, _, Or.inr (Or.inr h) => by right; right; constructor; assumption\n  | a, _, Or.inr (Or.inl rfl) =>\n    match WellOrdered.total x' y' with\n    | Or.inl h => by left; constructor; assumption\n    | Or.inr (Or.inl h) => by subst h; right; left; refl\n    | Or.inr (Or.inr h) => by right; right; constructor; assumption\n\ninstance : WellOrdered (Lim s) where\n  R := Lim.R s\n  wf := Lim.R_wf s\n  total := Lim.R_total s\n\ntheorem exists_in_inclusion (h : StrictInclusion a b) :\n  ∃ x : b, True := by\napply Classical.byContradiction; intro h₀\nrewrite [Classical.not_exists] at h₀\napply h.strict\nlet f (x : b) : a := False.elim (h₀ x trivial)\nrefine' ⟨⟨f, _⟩,_⟩\nfocus\n  intros x; cases h₀ x trivial\nfocus\n  constructor\n  intros W g₁ g₂ _\n  apply WellOrderHom.ext; intro a\n  cases h₀ (g₁ a) trivial\n\nsection\nopen Classical\n-- #print prefix\n\ntheorem exists_in_inclusion' (h : StrictInclusion a b) :\n  ∃ x : b, ∀ y, ¬ h.inclusion y = x := by\nbyCases Ha : Nonempty a\nfocus\n  apply Classical.byContradiction; intro h₀\n  simp at h₀\n  apply h.strict\n  -- revert Ha; intro Ha\n  have : injective h.inclusion.hom := injective_of_monomorphism _\n  let f : b → a := inverse h.inclusion.hom\n  have hf : ∀ i, f (h.inclusion i) = i :=\n    congrFun (left_inv_of_injective _ this)\n  refine' ⟨⟨f, _⟩,_⟩\n  focus\n    intros x y Hxy\n    have Hx := h₀ x\n    have Hy := h₀ y\n    have : ∀ α [WellOrdered α] (a b : α),\n             WellOrdered.R a b ↔ a < b := λ _ _ _ _ => Iff.rfl\n    cases Hx with | intro x' Hx =>\n    cases Hy with | intro y' Hy =>\n    rewrite [← Hx, ← Hy, hf, hf]\n    rewrite [this, WellOrdered.lt_iff_not_lt] at *\n    rewrite [← Hx, ← Hy] at Hxy\n    intros Hxy'; apply Hxy; clear Hxy x y Hx Hy this\n    cases Hxy' with\n    | inl h => subst h; left; refl\n    | inr h =>\n      right; apply WellOrderHom.hom_preserve\n      assumption\n  focus\n    constructor\n    intros W g₁ g₂ Hg\n    apply WellOrderHom.ext; intros x\n    have Hg := WellOrderHom.ext_iff Hg x\n    simp [WellOrderHom.hom, WellOrderHom.comp] at Hg\n    have := right_inv_of_surjective\n      (WellOrderHom.hom h.inclusion) h₀\n    have := congrFun this; simp at this\n    have Hg := congrArg h.inclusion.hom Hg\n    rewrite [this, this] at Hg; assumption\nfocus\n  have := exists_in_inclusion h\n  cases this with | intro x =>\n  exists x; intros y _\n  apply Ha; clear Ha\n  constructor; assumption\n\nend\n--   intros x; cases h₀ x trivial\n-- focus\n--   constructor\n--   intros W g₁ g₂ _\n--   apply WellOrderHom.ext; intro a\n--   cases h₀ (g₁ a) trivial\n\n\n-- if h₀ : Nonempty a then by\n--   cases h₀ with | intro x =>\n--   exists h.inclusion x\n--   -- exact intro\n--   trivial\n-- else by\n--   -- cases h with | ⟨h₀, h₁⟩ =>\n--   apply Classical.byContradiction\n--   rewrite [not_exists]; intros h₁\n--   apply h.strict\n--   constructor\n\nnoncomputable\ndef witness (x : StrictInclusion a b) : b :=\nClassical.choose (exists_in_inclusion' x)\n\nprotected noncomputable def default : Lim s where\n  idx := s.next s.bottom\n  val := witness (s.increasing _ _ (s.infinite _))\n  min_idx j x h  := by\n    let f := s.increasing _ _ (s.infinite s.bottom)\n    cases s.consecutive _ _ h with\n    | inl h =>\n      cases s.nothing_below_bottom _ h\n    | inr h =>\n      subst j\n      intros h₀\n      apply\n        Classical.choose_spec\n          (exists_in_inclusion' f) _ h₀\n\nnoncomputable\ninstance : Inhabited (Lim s) where\n  default := Lim.default s\n\nend Lim\n\ndef lim (s : Seq) : Ordinal :=\nQuot.mk _\n{ fst := Lim s,\n  snd := inferInstance }\n\ntheorem WO_is_equiv : EqvGen WOEqv x y → WOEqv x y := by\n  intro h\n  induction h with\n  | rfl => exact ⟨WellOrderHom.id, inferInstance⟩\n  | step h₀ _ ih =>\n    cases h₀ with | intro f h₀ =>\n    cases ih with | intro g ih =>\n    exists g.comp f\n    have h₀ := h₀; have ih := ih\n    exact inferInstance\n  | symm_step h₀ _ ih =>\n    cases h₀ with | intro f h₀ =>\n    cases ih with | intro g ih =>\n    have h₀ := h₀; have ih := ih\n    exists g.comp (inv f)\n    exact inferInstance\n\ntheorem zero_ne_succ (x : Ordinal) : zero ≠ S x := by\n  intros h\n  cases x using Quot.ind with | mk x =>\n  cases x with | mk a b =>\n  simp [S] at h\n  have h := WO_is_equiv $ Quot.eq.1 h\n  simp [impl.S] at h\n  cases h with | intro a b =>\n  have b := b\n  have f := (inv a).hom none\n  cases f\n\ntheorem succ_ne_self (x : Ordinal) : x ≠ S x := by\n  intros h\n  cases x using Quot.ind with | mk x =>\n  cases x with | mk a Ha =>\n  revert Ha h; intros Ha h\n  let Ha' : WellOrdered (Option a) := inferInstance\n  let Ha'' : LT (Option a) := instLT\n  simp [S] at h\n  have h := WO_is_equiv $ Quot.eq.1 h\n  cases h with | intro f hf =>\n  have hf := hf\n  have f := inv f\n  have hf := f.hom_preserve\n  simp [WellOrdered.R] at hf\n  suffices H : ∀ x : Option a, some (f x) < x → False by\n    apply H none\n    simp only [LT.lt]\n    constructor\n  intros x\n  have Hacc : Acc LT.lt x := WellOrdered.wf.apply _\n  induction Hacc with | intro x _ ih =>\n  intros h'\n  apply ih _ h'\n  constructor\n  apply f.hom_preserve (some (f x)) x h'\n\nopen Classical\n\ntheorem zero_ne_lim (s : Seq) : zero ≠ lim s := by\n  intros h\n  -- rewrite [Quot.eq] at h\n  have h := WO_is_equiv $ Quot.eq.1 h\n  cases h with | intro f hf =>\n  revert hf; intros hf\n  have f := inv f\n  have hf := f.hom_preserve\n  cases f Inhabited.default\n-- TODO:\n-- succ ≠ lim\n-- 0 ≠ lim\n-- recursor\n-- Lemma no Ordinal between x and S x?\n-- #check Le\n-- set_option trace.simp_rewrite true\n\n-- #check getLocal\n-- #check Lean.Expr.fvar\n\n-- set_option trace.Ma\n-- set_option trace.Elab.definition true\n\ntheorem S_consecutive y :\n  y ≤ x ∧ y < S x ∨ x < y ∧ S x ≤ y := by\nmatch lt_total x y with\n| Or.inl Hlt =>\n  right\n  refine' ⟨Hlt, _⟩\n  cases Hlt with | intro Hle Hnot_ge =>\n  cases x using Quot.ind with | mk x =>\n  cases y using Quot.ind with | mk y =>\n  simp [Le] at Hle Hnot_ge\n  have :\n         (¬ Nonempty (y → x)) ∨\n         (∃ f : y → x, ¬ Nonempty (y ⟶ x)) ∨\n         (∃ f : y ⟶ x, IsMono f → False) := by\n    apply byContradiction; intros h₀\n    apply Hnot_ge; clear Hnot_ge\n    simp at h₀\n    match h₀ with\n    | ⟨⟨f⟩, h₁, h₂⟩ =>\n      cases h₁ f with | intro g =>\n      cases h₂ g with | intro z => -- weird pretty printing: not\n                                  -- printing the type means that\n                                  -- we're not giving the ∀ variable a\n                                  -- name\n      -- cases z\n      -- clear h₁\n      constructor; assumption\n  match this with\n  | Or.inl h₀ =>\n    cases Hle with | intro f Hf =>\n    revert Hf; intros Hf\n    have : StrictInclusion x y := ⟨f, Hnot_ge⟩\n    let w := Lim.witness this\n    simp [LE.le, S, Le]\n    have : x → False := by\n      intros a\n      let f (b : y) := a\n      apply h₀; constructor; assumption\n    let g (_ : impl.S x) := w\n    -- have : IsMono g := by\n      -- intros\n    let g' : impl.S x ⟶ y := by\n      refine' ⟨g, _⟩\n      intros a b Hab\n      cases Hab with\n      | top h => cases this h\n      | lifted h => cases this h\n    have hg : IsMono g' := by\n      constructor; intros W g₁ g₂ Hg\n      apply WellOrderHom.ext; intros x\n      cases g₁ x with\n      | some x => cases this x\n      | none =>\n        cases g₂ x with\n        | some x => cases this x\n        | none => refl\n    constructor; assumption\n  | Or.inr (Or.inl ⟨f, h₀⟩) =>\n    clear this\n    swapHyp h₀ as h₁\n    constructor\n    exists f\n    skip\n  | Or.inr (Or.inr h₀) => skip\n\n| Or.inr (Or.inl Heq) => _\n| Or.inr (Or.inr Hgt) => _\n\ntheorem lt_lim (s : Seq) x :\n  lim s ≤ x ↔ ∀ i, Ordinal.mk (s.ords i) ≤ x := by\nconstructor\nfocus\n  intros h i\n  cases x using Quot.ind with | mk x =>\n  simp [lim, LE.le, Le] at h\n  cases h with | intro a b =>\n  skip\nfocus\n  intros h\n  admit\n\ntheorem lim_not_consecutive (s : Seq) x :\n  x < lim s → ∃ y, x < y ∧ y < lim s := by\nadmit\n\ntheorem S_ne_lim (s : Seq) : S x ≠ lim s := by\nadmit\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/Sat/ordinal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118791767283, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3085785761793281}}
{"text": "import for_mathlib.category_theory.localization.products\nimport for_mathlib.category_theory.localization.adjunction\nimport for_mathlib.category_theory.finite_products\n\nnoncomputable theory\n\nuniverses v v' u u'\n\nnamespace category_theory\n\nnamespace limits\n\ndef is_limit_postcompose {J C : Type*} [category J] [category C] {F₁ F₂ : J ⥤ C}\n  (e : F₁ ≅ F₂) {c : cone F₁} (hc : is_limit c) : is_limit ((cones.postcompose e.hom).obj c) :=\n{ lift := λ s, hc.lift ((cones.postcompose e.inv).obj s),\n  fac' := λ s j, begin\n    simp only [cones.postcompose_obj_π, nat_trans.comp_app, is_limit.fac_assoc, category.assoc, iso.inv_hom_id_app,\n  category.comp_id],\n  end,\n  uniq' := λ s m w, hc.hom_ext (λ j, begin\n    simp only [cones.postcompose_obj_π, nat_trans.comp_app] at w,\n    simp only [← cancel_mono (e.hom.app j), w, category.assoc, is_limit.fac,\n      cones.postcompose_obj_π, nat_trans.comp_app, iso.inv_hom_id_app, category.comp_id],\n  end), }\n\nend limits\n\nopen limits category\n\nvariables {C : Type u} {D : Type u'} [category.{v} C] [category.{v'} D]\n  (L : C ⥤ D) (W : morphism_property C) [L.is_localization W]\n\nnamespace morphism_property\n\nlemma isomorphisms.is_inverted_by (F : C ⥤ D) :\n  (morphism_property.isomorphisms C).is_inverted_by F := λ X Y f hf,\nbegin\n  rw morphism_property.isomorphisms.iff at hf,\n  haveI := hf,\n  apply_instance,\nend\n\ndef stable_under_limits_of_shape (W : morphism_property C) (J : Type*) [category J] : Prop :=\n∀ (X₁ X₂ : J ⥤ C) (c₁ : cone X₁) (c₂ : cone X₂) (h₁ : is_limit c₁) (h₂ : is_limit c₂)\n  (f : X₁ ⟶ X₂) (hf : Π (j : J), W (nat_trans.app f j)), W (h₂.lift (cone.mk _ (c₁.π ≫ f)))\n\nabbreviation stable_under_products_of_shape (W : morphism_property C) (J : Type*) : Prop :=\nW.stable_under_limits_of_shape (discrete J)\n\n@[simps]\ndef stable_under_products_of_shape.iso_aux {J D : Type*} [category D]\n  (X : discrete J ⥤ D) (c : cone X) (hc : is_limit c)\n  [has_product (λ j, X.obj (discrete.mk j))] :\n  c.X ≅ ∏ (λ j, X.obj (discrete.mk j)) :=\n{ hom := pi.lift (λ j, c.π.app (discrete.mk j)),\n  inv := hc.lift (cone.mk (∏ (λ j, X.obj (discrete.mk j)))\n    { app := by { rintro ⟨j⟩, exact pi.π _ j, },\n    naturality' := begin\n      rintro ⟨j₁⟩ ⟨j₂⟩ f,\n      have h := discrete.eq_of_hom f,\n      dsimp at h,\n      subst h,\n      simp only [subsingleton.elim f (𝟙 _), discrete.functor_map_id, id_comp, comp_id],\n    end, }),\n  hom_inv_id' := hc.hom_ext begin\n    rintro ⟨j⟩,\n    simp only [id_comp, assoc, is_limit.fac, limit.lift_π, fan.mk_π_app],\n  end,\n  inv_hom_id' := begin\n    ext j,\n    discrete_cases,\n    simp only [id_comp, assoc, limit.lift_π, fan.mk_π_app, is_limit.fac],\n  end, }\n\nlemma stable_under_products_of_shape.mk (W : morphism_property C) (J : Type*)\n  (hW₀ : W.respects_iso) [has_products_of_shape J C]\n  (hW : ∀ (X₁ X₂ : J → C) (f : Π j, X₁ j ⟶ X₂ j) (hf : Π (j : J), W (f j)),\n    W (pi.map f)) : W.stable_under_products_of_shape J :=\nλ X₁ X₂ c₁ c₂ hc₁ hc₂ f hf, begin\n  let Y₁ := λ j, X₁.obj (discrete.mk j),\n  let Y₂ := λ j, X₂.obj (discrete.mk j),\n  let φ : Π j, Y₁ j ⟶ Y₂ j := λ j, f.app (discrete.mk j),\n  have hf' := hW Y₁ Y₂ φ (λ j, (hf (discrete.mk j))),\n  refine (hW₀.arrow_mk_iso_iff _).mpr hf',\n  refine arrow.iso_mk (stable_under_products_of_shape.iso_aux X₁ c₁ hc₁)\n    (stable_under_products_of_shape.iso_aux X₂ c₂ hc₂) _,\n  ext j,\n  discrete_cases,\n  dsimp,\n  simp only [limit.lift_map, limit.lift_π, cones.postcompose_obj_π, nat_trans.comp_app,\n    fan.mk_π_app, discrete.nat_trans_app, assoc, is_limit.fac],\nend\n\nclass stable_under_finite_products (W : morphism_property C) : Prop :=\n(condition [] : ∀ (J : Type) [finite J], stable_under_products_of_shape W J)\n\nabbreviation stable_under_binary_products (W : morphism_property C) :=\n  W.stable_under_products_of_shape walking_pair\n\nnamespace stable_under_products_of_shape\n\nvariable {W}\n\nlemma lim_map {J : Type*} (h : W.stable_under_products_of_shape J) {X Y : discrete J ⥤ C}\n  (f : X ⟶ Y) [has_products_of_shape J C]\n  (hf : ∀ j, W (f.app (discrete.mk j))) : W (lim.map f) :=\nh X Y _ _ (limit.is_limit X) (limit.is_limit Y) f\n  (by { rintro ⟨j⟩, exact hf j, })\n\nend stable_under_products_of_shape\n\nend morphism_property\n\nnamespace localization\n\ninclude L W\n\n@[protected]\nlemma has_products_of_shape (J : Type) [finite J] [W.contains_identities]\n  [has_products_of_shape J C] (hW : W.stable_under_products_of_shape J) :\n  has_products_of_shape J D :=\nbegin\n  let G : C ⥤ _ := functor.const (discrete J),\n  let F : ((discrete J) ⥤ C) ⥤ C := lim,\n  let adj : G ⊣ F := const_lim_adj,\n  let L' := (whiskering_right (discrete J) C D).obj L,\n  let G' : D ⥤ _ := functor.const (discrete J),\n  let W' := morphism_property.functor_category W (discrete J),\n  have hF : W'.is_inverted_by (F ⋙ L),\n  { intros X Y f hf,\n    dsimp,\n    exact localization.inverts L W (F.map f) (hW.lim_map f (λ j, hf (discrete.mk j))), },\n  let F' := localization.lift (F ⋙ L) hF L',\n  haveI : lifting L W (G ⋙ L') G' := ⟨L.comp_const (discrete J)⟩,\n  exact has_limits_of_shape_of_adj (adj.localization L W L' W' G' F'),\nend\n\n@[protected]\nlemma has_finite_products [W.contains_identities]\n  [has_finite_products C] [W.stable_under_finite_products] :\n  has_finite_products D :=\n⟨λ n, by { exact localization.has_products_of_shape L W (fin n)\n  (morphism_property.stable_under_finite_products.condition W (fin n)), }⟩\n\n@[protected]\ndef preserves_products_of_shape (J : Type) [finite J] [W.contains_identities]\n  [has_products_of_shape J C] (hW : W.stable_under_products_of_shape J) :\n  preserves_limits_of_shape (discrete J) L :=\nbegin\n  let G : C ⥤ _ := functor.const (discrete J),\n  let F : ((discrete J) ⥤ C) ⥤ C := lim,\n  let adj : G ⊣ F := const_lim_adj,\n  let L' := (whiskering_right (discrete J) C D).obj L,\n  let G' : D ⥤ _ := functor.const (discrete J),\n  let W' := morphism_property.functor_category W (discrete J),\n  have hF : W'.is_inverted_by (F ⋙ L),\n  { intros X Y f hf,\n    dsimp,\n    exact localization.inverts L W (F.map f) (hW.lim_map f (λ j, hf (discrete.mk j))), },\n  let F' := localization.lift (F ⋙ L) hF L',\n  letI : lifting L W (G ⋙ L') G' := ⟨L.comp_const (discrete J)⟩,\n  let adj' := adj.localization L W L' W' G' F',\n  have h : ∀ (X : discrete J ⥤ C), adj.unit.app (F.obj X) ≫ F.map (adj.counit.app X) = 𝟙 (F.obj X),\n  { intro X,\n    exact adj.right_triangle_components, },\n  haveI : ∀ (X : discrete J ⥤ C), is_iso ((limit_comparison_of_adj adj adj' L).app X),\n  { intro X,\n    dsimp only [limit_comparison_of_adj],\n    simp only [adjunction.adjoint_nat_trans_equiv_app, adjunction.localization_unit_app, assoc,\n      whiskering_right_obj_map, ← F'.map_comp, iso.inv_hom_id_app_assoc],\n    erw [← nat_trans.naturality, ← assoc, ← L.map_comp],\n    rw adj.right_triangle_components,\n    apply_instance, },\n  haveI : is_iso (limit_comparison_of_adj adj adj' L) := nat_iso.is_iso_of_is_iso_app _,\n  exact preserves_limits_of_shape_of_adj adj adj' L,\nend\n\ninstance localization_has_finite_products [W.contains_identities] [has_finite_products C]\n  [W.stable_under_finite_products] : has_finite_products W.localization :=\n  localization.has_finite_products W.Q W\n\ninstance [W.contains_identities] [has_finite_products C]\n  [W.stable_under_finite_products] (J : Type) [finite J] :\n  preserves_limits_of_shape (discrete J) W.Q :=\nlocalization.preserves_products_of_shape W.Q W J\n  (morphism_property.stable_under_finite_products.condition W J)\n\nend localization\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.3085650365931389}}
{"text": "import rowround\nimport columnround\n\nimport category_theory.category.basic\nimport category_theory.core\n\nopen params\nopen operations\nopen quarterround\nopen rowround\nopen columnround\nopen utils\n\nopen category_theory\n\nnamespace doubleround\n\nvariables [category (bitvec word_len)]\n\n/-!\n  # Doubleround\n\n  The `doubleround` function and the relation with its inverse.\n-/\n\n/-- doubleround(x) = rowround(columnround(x)) -/\n@[simp] def doubleround (M : matrixType) : matrixType := rowround $ columnround M\n\n/--  doubleround_inv(x) = columnround_inv(rowround_inv(x)) -/\n@[simp] def doubleround_inv (M : matrixType) : matrixType := columnround_inv $ rowround_inv M\n\n/- Just some notation for inverses. -/\nlocal notation `doubleround⁻¹` := doubleround_inv\n\n/-- The `doubleround` function is invertible. -/\nlemma doubleround_is_inv (I : doubleround ≅ doubleround⁻¹) : I.hom ≫ I.inv = 𝟙 doubleround :=\n  by rw [iso.hom_inv_id]\n\n/--\nA special case of `doubleround` where inputs and outputs are sorted according to the salsa20 spec:\ndoubleround'(x) = rowround'(columnround'(x)) -/\n@[simp] def doubleround_salsa20 (M : matrixType) : matrixType := rowround_salsa20 $ columnround_salsa20 M\n\n/--\nA special case of `doubleround_inv` where inputs and outputs are sorted according to the salsa20 spec:\ndoubleround_inv'(x) = columnround_inv'(rowround_inv'(x)) -/\n@[simp] def doubleround_salsa20_inv (M : matrixType) : matrixType := columnround_salsa20_inv $ rowround_salsa20_inv M\n\n/- Just some notation for inverses. -/\nlocal notation `doubleround_salsa20⁻¹` := doubleround_salsa20_inv\n\n/-- The `doubleround` function is invertible. -/\nlemma doubleround_salsa20_is_inv (I : doubleround_salsa20 ≅ doubleround_salsa20⁻¹) : \n  I.hom ≫ I.inv = 𝟙 doubleround_salsa20 := by rw [iso.hom_inv_id]\n\n\nend doubleround\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/doubleround.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3085650299174423}}
{"text": "import sli.sli model_checking.mc_bridge\n\nnamespace rmd_bridge\nuniverse u\n/-!\n  **S** the type of a specification (the model that is interpreted by the STR)\n    - if it is a string, then we have an inject function String → STR C A that will parse the string and instantiate the STR\n    - if it is an program AST, the we have an inject function AST → STR C A that will instantiate the STR with the program AST\n    - more generaly it can be any object type S for which we have an inject function S → STR C A\n-/\nvariables (S C A E R V α : Type)\n\nopen sli\nopen sli.toTR\nopen model_checking\n\n/-!\n  # Debug semantics\n-/\n\nmutual inductive TraceEntry , DebugAction\nwith TraceEntry: Type\n| root (c: C) (a: DebugAction) \n| child (c: C) (a: DebugAction) (parent : TraceEntry)\nwith DebugAction : Type  \n| init\n| step : A → DebugAction\n| select : C → DebugAction  \n| jump : TraceEntry → DebugAction \n| run_to_breakpoint : DebugAction\n\nstructure DebugConfig :=\n  (current : option (TraceEntry C A))\n  (history : set (TraceEntry C A))\n  (options : option(DebugAction C A × set C))\n\n/-!\n  **BE** breakpoint expression type, which can be mapped to the semantics and the emptiness checking algorithm needed\n  **Cp** configuration product, a configuration type, which in general is different from C. \n    In practice it is the tuple (C, C₂), where C₂ is the configuration type needed by the breakpoint semantics\n-/\ndef Finder (BE: Type)\n:=\n  STR C A → set C → BE → R → list C\n\ndef ote2c {C A : Type} : option (TraceEntry C A) → option C\n| none := none\n| (some (TraceEntry.root c _)) :=  c\n| (some (TraceEntry.child c _ _)) := c\n\ndef rmdInitial\n(o : STR C A) : set (DebugConfig C A) := \n  { { current := none, history := ∅, options := some (DebugAction.init, o.initial) } }\n\nopen DebugAction\ndef rmdActions\n(o : STR C A) : DebugConfig C A → set (DebugAction C A) \n| ⟨ current, history, options ⟩ :=\n  let\n    oa := { x : DebugAction C A | ∀ c, (options = none) → (ote2c current) = some c → ∀ a ∈ (o.actions c), x = step a }, \n    sa := { x : DebugAction C A | match options with | some (_, configs) := ∀ c ∈ configs, x = select c | none := false end },\n    ja := { x : DebugAction C A | ∀ te ∈ history,                               x = jump te    },\n    rtb := { x : DebugAction C A | match current with | some te := x = run_to_breakpoint | none := false end }\n\tin oa ∪ sa ∪ ja ∪ rtb\n\ndef same_configuration [decidable_eq C]: TraceEntry C A → C → bool\n| (TraceEntry.root  c₁ _  ) c₂ := c₁ = c₂\n| (TraceEntry.child c₁ _ _) c₂ := c₁ = c₂\n\n/-!\n  Attention:\n- to the order of configurations in the counter example (start .. end) vs (end .. start). \n  Here we suppose **(start .. end)**\n- does the counter exemple contain the start configuration ?\n  The same_configuration check removes the consecutive duplicates\n-/\ndef trace2history {C A : Type} [decidable_eq C] : TraceEntry C A → DebugAction C A → list C → set (TraceEntry  C A) → TraceEntry C A × set (TraceEntry  C A)\n| te da [] history := (te, history)\n| te da (head::tail) history := \n  let\n    te' := if (same_configuration C A te head) then te else (TraceEntry.child head da te),\n    h := history ∪ { te' }\n  in \n   trace2history te' da tail h\n\ndef te2c {C A : Type} : TraceEntry C A → C\n|  (TraceEntry.root c _) :=  c\n|  (TraceEntry.child c _ _):= c\n\n\ndef rmdExecute {BE: Type}\n  [decidable_eq C]\n  (o : STR C A) \n  (finder : Finder C A R BE) (breakpoint : BE) (reduction : R)\n  : \n  DebugAction C A → DebugConfig  C A → set (DebugConfig C A)\n| (init) _                                                            :=  ∅ -- cannot get here\n| (step a)            ⟨ (some (TraceEntry.root c da)), history, _ ⟩   := { ⟨ (TraceEntry.root c da), history, some (step a, o.execute a c) ⟩ }\n| (step a)            ⟨ (some (TraceEntry.child c da p)), history, _⟩ := { ⟨ (TraceEntry.child c da p), history, some (step a, o.execute a c) ⟩ }\n| (step a)            ⟨ _, _, _ ⟩                                     := ∅ -- cannot get here due to debugActions which produce steps only of current=some c\n| (select c)          ⟨ none, history, some (da, _)⟩                  := (let te := (TraceEntry.root c da) in  { ⟨ te , { te } ∪ history, none ⟩ })\n| (select c)          ⟨ some te, history, some(da, _)⟩                := (let te₁ := (TraceEntry.child c da te) in  { ⟨ te₁, { te₁ } ∪ history, none ⟩ })\n| (select c)          ⟨ _, _, _ ⟩                                     := ∅ --cannot get here\n| (jump te)           ⟨ _, history, _⟩                                := { ⟨ some te, history, none ⟩ }\n| (run_to_breakpoint) ⟨ some te, history, opts ⟩                      := \n                                                                          let\n                                                                            trace := finder o { (te2c te) } breakpoint reduction,\n                                                                            patch := trace2history te run_to_breakpoint trace ∅ \n                                                                          in \n                                                                            match patch with \n                                                                            | (te, ch) := { ⟨ some te, history ∪ ch, none ⟩ } \n                                                                            end\n| (run_to_breakpoint) ⟨ _, _, _ ⟩                                     := ∅ --cannot get here\n  \n\n\ndef ReducedMultiverseDebuggerBridge {BE: Type}\n  [decidable_eq C]\n  (o : STR C A) \n  (finder : Finder C A R BE) \n  (breakpoint : BE) \n  (reduction : R) \n  : STR (DebugConfig C A) (DebugAction C A) :=\n{\n  initial :=          rmdInitial C A o,\n  actions := λ dc,    rmdActions C A o dc,\n  execute := λ da dc, rmdExecute C A R o finder breakpoint reduction da dc\n}\n\n/-!\n  # Replace the initial states of a STR, to make it start somewhere else\n-/\ndef ReplaceInitial (o : STR C A) (initial : set C) : STR C A :=\n{ \n  initial := initial,\n  actions := o.actions,\n  execute := o.execute,\n}\n\ndef ReducedMultiverseDebugger {BE: Type}\n  [decidable_eq C]\n  (finder : Finder C A R BE)\n  (inject: S → STR C A)\n  (specification: S)\n  (breakpoint : BE) \n  (reduction : R) \n: STR (DebugConfig C A) (DebugAction C A) :=\n    ReducedMultiverseDebuggerBridge C A R (inject specification)\n      finder breakpoint reduction\n\nend rmd_bridge", "meta": {"author": "teodorov", "repo": "temporal-multiverse-debugging", "sha": "e8f2e7037acee87f0c8620b60073a7619eab7f82", "save_path": "github-repos/lean/teodorov-temporal-multiverse-debugging", "path": "github-repos/lean/teodorov-temporal-multiverse-debugging/temporal-multiverse-debugging-e8f2e7037acee87f0c8620b60073a7619eab7f82/src/debugging/rmd_bridge.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6370307806984443, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3085650232417456}}
{"text": "/- Copyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johannes Hölzl\n\nIntroduce CommRing -- the category of commutative rings.\n\nCurrently only the basic setup.\n-/\n\nimport category_theory.examples.monoids\nimport category_theory.embedding\nimport algebra.ring\n\nuniverses u v\n\nopen category_theory\n\nnamespace category_theory.examples\n\n/-- The category of rings. -/\n@[reducible] def Ring : Type (u+1) := bundled ring\n\ninstance (x : Ring) : ring x := x.str\n\ninstance concrete_is_ring_hom : concrete_category @is_ring_hom :=\n⟨by introsI α ia; apply_instance,\n  by introsI α β γ ia ib ic f g hf hg; apply_instance⟩\n\ninstance Ring_hom_is_ring_hom {R S : Ring} (f : R ⟶ S) : is_ring_hom (f : R → S) := f.2\n\n/-- The category of commutative rings. -/\n@[reducible] def CommRing : Type (u+1) := bundled comm_ring\n\ninstance (x : CommRing) : comm_ring x := x.str\n\n@[reducible] def is_comm_ring_hom {α β} [comm_ring α] [comm_ring β] (f : α → β) : Prop :=\nis_ring_hom f\n\ninstance concrete_is_comm_ring_hom : concrete_category @is_comm_ring_hom :=\n⟨by introsI α ia; apply_instance,\n  by introsI α β γ ia ib ic f g hf hg; apply_instance⟩\n\ninstance CommRing_hom_is_comm_ring_hom {R S : CommRing} (f : R ⟶ S) : is_comm_ring_hom (f : R → S) := f.2\n\nnamespace CommRing\n/-- The forgetful functor from commutative rings to (multiplicative) commutative monoids. -/\ndef forget_to_CommMon : CommRing ⥤ CommMon := \nconcrete_functor\n  (by intros _ c; exact { ..c })\n  (by introsI _ _ _ _ f i;  exact { ..i })\n\ninstance : faithful (forget_to_CommMon) :=\nbegin\n  tidy, \n  apply (congr_fun h_1), -- TODO solve_by_elim should try this?\nend\n\nexample : faithful (forget_to_CommMon ⋙ CommMon.forget_to_Mon) := by apply_instance\nend CommRing\n\nend category_theory.examples\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/category_theory/examples/rings.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.30853349882981757}}
{"text": "import Mathlib.Tactic.FailIfNoProgress\n\nsection success\n\nexample : 1 = 1 := by fail_if_no_progress rfl\nexample (h : 1 = 1) : True := by\n  fail_if_no_progress simp at h\n  trivial\n\nend success\n\nsection failure\n\nexample (x : Bool) (h : x = true) : x = true := by\n  fail_if_success\n    fail_if_no_progress simp\n  exact h\n\n\nexample (x : Bool) (h : x = true) : True := by\n  fail_if_success\n    fail_if_no_progress simp at h\n  trivial\n\nend failure\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/fail_if_no_progress.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.30836502338182564}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport data.seq.seq data.seq.computation data.list.basic data.dlist\nuniverses u v w\n\n/-\ncoinductive wseq (α : Type u) : Type u\n| nil : wseq α\n| cons : α → wseq α → wseq α\n| think : wseq α → wseq α\n-/\n\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef wseq (α) := seq (option α)\n\nnamespace wseq\nvariables {α : Type u} {β : Type v} {γ : Type w}\n\n/-- Turn a sequence into a weak sequence -/\ndef of_seq : seq α → wseq α := (<$>) some\n\n/-- Turn a list into a weak sequence -/\ndef of_list (l : list α) : wseq α := of_seq l\n\n/-- Turn a stream into a weak sequence -/\ndef of_stream (l : stream α) : wseq α := of_seq l\n\ninstance coe_seq : has_coe (seq α) (wseq α) := ⟨of_seq⟩\ninstance coe_list : has_coe (list α) (wseq α) := ⟨of_list⟩\ninstance coe_stream : has_coe (stream α) (wseq α) := ⟨of_stream⟩\n\n/-- The empty weak sequence -/\ndef nil : wseq α := seq.nil\n\n/-- Prepend an element to a weak sequence -/\ndef cons (a : α) : wseq α → wseq α := seq.cons (some a)\n\n/-- Compute for one tick, without producing any elements -/\ndef think : wseq α → wseq α := seq.cons none\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct : wseq α → computation (option (α × wseq α)) :=\ncomputation.corec (λs, match seq.destruct s with\n  | none              := sum.inl none\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := sum.inl (some (a, s'))\n  end)\n\ndef cases_on {C : wseq α → Sort v} (s : wseq α) (h1 : C nil)\n  (h2 : ∀ x s, C (cons x s)) (h3 : ∀ s, C (think s)) : C s :=\nseq.cases_on s h1 (λ o, option.cases_on o h3 h2)\n\nprotected def mem (a : α) (s : wseq α) := seq.mem (some a) s\n\ninstance : has_mem α (wseq α) :=\n⟨wseq.mem⟩\n\ntheorem not_mem_nil (a : α) : a ∉ @nil α := seq.not_mem_nil a\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head (s : wseq α) : computation (option α) :=\ncomputation.map ((<$>) prod.fst) (destruct s)\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten : computation (wseq α) → wseq α :=\nseq.corec (λc, match computation.destruct c with\n  | sum.inl s := seq.omap return (seq.destruct s)\n  | sum.inr c' := some (none, c')\n  end)\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail (s : wseq α) : wseq α :=\nflatten $ (λo, option.rec_on o nil prod.snd) <$> destruct s\n\n/-- drop the first `n` elements from `s`. -/\ndef drop (s : wseq α) : ℕ → wseq α\n| 0     := s\n| (n+1) := tail (drop n)\nattribute [simp] drop\n\n/-- Get the nth element of `s`. -/\ndef nth (s : wseq α) (n : ℕ) : computation (option α) := head (drop s n)\n\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef to_list (s : wseq α) : computation (list α) :=\n@computation.corec (list α) (list α × wseq α) (λ⟨l, s⟩,\n  match seq.destruct s with\n  | none              := sum.inl l.reverse\n  | some (none, s')   := sum.inr (l, s')\n  | some (some a, s') := sum.inr (a::l, s')\n  end) ([], s)\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : wseq α) : computation ℕ :=\n@computation.corec ℕ (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s with\n  | none              := sum.inl n\n  | some (none, s')   := sum.inr (n, s')\n  | some (some a, s') := sum.inr (n+1, s')\n  end) (0, s)\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\n@[class] def is_finite (s : wseq α) : Prop := (to_list s).terminates\n\ninstance to_list_terminates (s : wseq α) [h : is_finite s] : (to_list s).terminates := h\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : wseq α) [is_finite s] : list α := (to_list s).get\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\n@[class] def productive (s : wseq α) : Prop := ∀ n, (nth s n).terminates\n\ninstance nth_terminates (s : wseq α) [h : productive s] : ∀ n, (nth s n).terminates := h\n\ninstance head_terminates (s : wseq α) [h : productive s] : (head s).terminates := h 0\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef update_nth (s : wseq α) (n : ℕ) (a : α) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (some a, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef remove_nth (s : wseq α) (n : ℕ) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (none, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filter_map (f : α → option β) : wseq α → wseq β :=\nseq.corec (λs, match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, s')\n  | some (some a, s') := some (f a, s')\n  end)\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : α → Prop) [decidable_pred p] : wseq α → wseq α :=\nfilter_map (λa, if p a then some a else none)\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : α → Prop) [decidable_pred p] (s : wseq α) : computation (option α) :=\nhead $ filter p s\n\n/-- Zip a function over two weak sequences -/\ndef zip_with (f : α → β → γ) (s1 : wseq α) (s2 : wseq β) : wseq γ :=\n@seq.corec (option γ) (wseq α × wseq β) (λ⟨s1, s2⟩,\n  match seq.destruct s1, seq.destruct s2 with\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (none, s1, s2')\n  | some (none, s1'),    some (some a2, s2') := some (none, s1', s2)\n  | some (some a1, s1'), some (some a2, s2') := some (some (f a1 a2), s1', s2')\n  | _,                   _                   := none\n  end) (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : wseq α → wseq β → wseq (α × β) := zip_with prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef find_indexes (p : α → Prop) [decidable_pred p] (s : wseq α) : wseq ℕ :=\n(zip s (stream.nats : wseq ℕ)).filter_map\n  (λ ⟨a, n⟩, if p a then some n else none)\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef find_index (p : α → Prop) [decidable_pred p] (s : wseq α) : computation ℕ :=\n(λ o, option.get_or_else o 0) <$> head (find_indexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef index_of [decidable_eq α] (a : α) : wseq α → computation ℕ := find_index (eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexes_of [decidable_eq α] (a : α) : wseq α → wseq ℕ := find_indexes (eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : wseq α) : wseq α :=\n@seq.corec (option α) (wseq α × wseq α) (λ⟨s1, s2⟩,\n  match seq.destruct s1, seq.destruct s2 with\n  | none,                none                := none\n  | some (a1, s1'),      none                := some (a1, s1', nil)\n  | none,                some (a2, s2')      := some (a2, nil, s2')\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (some a1, s1', s2')\n  | some (none, s1'),    some (some a2, s2') := some (some a2, s1', s2')\n  | some (some a1, s1'), some (some a2, s2') := some (some a1, cons a2 s1', s2')\n  end) (s1, s2)\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef is_empty (s : wseq α) : computation bool :=\ncomputation.map option.is_none $ head s\n\n/-- Calculate one step of computation -/\ndef compute (s : wseq α) : wseq α :=\nmatch seq.destruct s with\n| some (none, s') := s'\n| _               := s\nend\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : wseq α) (n : ℕ) : wseq α :=\n@seq.corec (option α) (ℕ × wseq α) (λ⟨n, s⟩,\n  match n, seq.destruct s with\n  | 0,   _                 := none\n  | m+1, none              := none\n  | m+1, some (none, s')   := some (none, m+1, s')\n  | m+1, some (some a, s') := some (some a, m, s')\n  end) (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef split_at (s : wseq α) (n : ℕ) : computation (list α × wseq α) :=\n@computation.corec (list α × wseq α) (ℕ × list α × wseq α) (λ⟨n, l, s⟩,\n  match n, seq.destruct s with\n  | 0,   _                 := sum.inl (l.reverse, s)\n  | m+1, none              := sum.inl (l.reverse, s)\n  | m+1, some (none, s')   := sum.inr (n, l, s')\n  | m+1, some (some a, s') := sum.inr (m, a::l, s')\n  end) (n, [], s)\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any (s : wseq α) (p : α → bool) : computation bool :=\ncomputation.corec (λs : wseq α,\n  match seq.destruct s with\n  | none              := sum.inl ff\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inl tt else sum.inr s'\n  end) s\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all (s : wseq α) (p : α → bool) : computation bool :=\ncomputation.corec (λs : wseq α,\n  match seq.destruct s with\n  | none              := sum.inl tt\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inr s' else sum.inl ff\n  end) s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : α → β → α) (a : α) (s : wseq β) : wseq α :=\ncons a $ @seq.corec (option α) (α × wseq β) (λ⟨a, s⟩,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, a, s')\n  | some (some b, s') := let a' := f a b in some (some a', a', s')\n  end) (a, s)\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : wseq α) : wseq (list α) :=\ncons [] $ @seq.corec (option (list α)) (dlist α × wseq α) (λ ⟨l, s⟩,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, l, s')\n  | some (some a, s') := let l' := l.concat a in\n                         some (some l'.to_list, l', s')\n  end) (dlist.empty, s)\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : wseq α) (n : ℕ) : list α :=\n(seq.take n s).filter_map id\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append : wseq α → wseq α → wseq α := seq.append\n\n/-- Map a function over a weak sequence -/\ndef map (f : α → β) : wseq α → wseq β := seq.map (option.map f)\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join (S : wseq (wseq α)) : wseq α :=\nseq.join ((λo : option (wseq α), match o with\n  | none := seq1.ret none\n  | some s := (none, s)\n  end) <$> S)\n\n/-- Monadic bind operator for weak sequences -/\ndef bind (s : wseq α) (f : α → wseq β) : wseq β :=\njoin (map f s)\n\n@[simp] def lift_rel_o (R : α → β → Prop) (C : wseq α → wseq β → Prop) :\n  option (α × wseq α) → option (β × wseq β) → Prop\n| none          none          := true\n| (some (a, s)) (some (b, t)) := R a b ∧ C s t\n| _             _             := false\n\ntheorem lift_rel_o.imp {R S : α → β → Prop} {C D : wseq α → wseq β → Prop}\n  (H1 : ∀ a b, R a b → S a b) (H2 : ∀ s t, C s t → D s t) :\n  ∀ {o p}, lift_rel_o R C o p → lift_rel_o S D o p\n| none          none          h := trivial\n| (some (a, s)) (some (b, t)) h := and.imp (H1 _ _) (H2 _ _) h\n| none          (some _)      h := false.elim h\n| (some (_, _)) none          h := false.elim h\n\ntheorem lift_rel_o.imp_right (R : α → β → Prop) {C D : wseq α → wseq β → Prop}\n  (H : ∀ s t, C s t → D s t) {o p} : lift_rel_o R C o p → lift_rel_o R D o p :=\nlift_rel_o.imp (λ _ _, id) H\n\n@[simp] def bisim_o (R : wseq α → wseq α → Prop) :\n  option (α × wseq α) → option (α × wseq α) → Prop := lift_rel_o (=) R\n\ntheorem bisim_o.imp {R S : wseq α → wseq α → Prop} (H : ∀ s t, R s t → S s t) {o p} :\n  bisim_o R o p → bisim_o S o p :=\nlift_rel_o.imp_right _ H\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef lift_rel (R : α → β → Prop) (s : wseq α) (t : wseq β) : Prop :=\n∃ C : wseq α → wseq β → Prop, C s t ∧\n∀ {s t}, C s t → computation.lift_rel (lift_rel_o R C) (destruct s) (destruct t)\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef equiv : wseq α → wseq α → Prop := lift_rel (=)\n\ntheorem lift_rel_destruct {R : α → β → Prop} {s : wseq α} {t : wseq β} :\n  lift_rel R s t →\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t)\n| ⟨R, h1, h2⟩ :=\n  by refine computation.lift_rel.imp _ _ _ (h2 h1);\n     apply lift_rel_o.imp_right; exact λ s' t' h', ⟨R, h', @h2⟩\n\ntheorem lift_rel_destruct_iff {R : α → β → Prop} {s : wseq α} {t : wseq β} :\n  lift_rel R s t ↔\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n⟨lift_rel_destruct, λ h, ⟨λ s t, lift_rel R s t ∨\n  computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n  or.inr h, λ s t h, begin\n    have h : computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n    { cases h with h h, exact lift_rel_destruct h, assumption },\n    apply computation.lift_rel.imp _ _ _ h,\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end⟩⟩\n\ninfix ~ := equiv\n\ntheorem destruct_congr {s t : wseq α} :\n  s ~ t → computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct\n\ntheorem destruct_congr_iff {s t : wseq α} :\n  s ~ t ↔ computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct_iff\n\ntheorem lift_rel.refl (R : α → α → Prop) (H : reflexive R) : reflexive (lift_rel R) :=\nλ s, begin\n  refine ⟨(=), rfl, λ s t (h : s = t), _⟩,\n  rw ←h, apply computation.lift_rel.refl,\n  intro a, cases a with a, simp, cases a; simp, apply H\nend\n\ntheorem lift_rel_o.swap (R : α → β → Prop) (C) :\n  function.swap (lift_rel_o R C) = lift_rel_o (function.swap R) (function.swap C) :=\nby funext x y; cases x with x; [skip, cases x]; { cases y with y; [skip, cases y]; refl }\n\ntheorem lift_rel.swap_lem {R : α → β → Prop} {s1 s2} (h : lift_rel R s1 s2) :\n  lift_rel (function.swap R) s2 s1 :=\nbegin\n  refine ⟨function.swap (lift_rel R), h, λ s t (h : lift_rel R t s), _⟩,\n  rw [←lift_rel_o.swap, computation.lift_rel.swap],\n  apply lift_rel_destruct h\nend\n\ntheorem lift_rel.swap (R : α → β → Prop) :\n  function.swap (lift_rel R) = lift_rel (function.swap R) :=\nfunext $ λ x, funext $ λ y, propext ⟨lift_rel.swap_lem, lift_rel.swap_lem⟩\n\ntheorem lift_rel.symm (R : α → α → Prop) (H : symmetric R) : symmetric (lift_rel R) :=\nλ s1 s2 (h : function.swap (lift_rel R) s2 s1),\nby rwa [lift_rel.swap, show function.swap R = R, from\n        funext $ λ a, funext $ λ b, propext $ by constructor; apply H] at h\n\ntheorem lift_rel.trans (R : α → α → Prop) (H : transitive R) : transitive (lift_rel R) :=\nλ s t u h1 h2, begin\n  refine ⟨λ s u, ∃ t, lift_rel R s t ∧ lift_rel R t u, ⟨t, h1, h2⟩, λ s u h, _⟩,\n  rcases h with ⟨t, h1, h2⟩,\n  have h1 := lift_rel_destruct h1,\n  have h2 := lift_rel_destruct h2,\n  refine computation.lift_rel_def.2\n    ⟨(computation.terminates_of_lift_rel h1).trans\n     (computation.terminates_of_lift_rel h2), λ a c ha hc, _⟩,\n  rcases h1.left ha with ⟨b, hb, t1⟩,\n  have t2 := computation.rel_of_lift_rel h2 hb hc,\n  cases a with a; cases c with c,\n  { trivial },\n  { cases b, {cases t2}, {cases t1} },\n  { cases a, cases b with b, {cases t1}, {cases b, cases t2} },\n  { cases a with a s, cases b with b, {cases t1},\n    cases b with b t, cases c with c u,\n    cases t1 with ab st, cases t2 with bc tu,\n    exact ⟨H ab bc, t, st, tu⟩ }\nend\n\ntheorem lift_rel.equiv (R : α → α → Prop) : equivalence R → equivalence (lift_rel R)\n| ⟨refl, symm, trans⟩ :=\n  ⟨lift_rel.refl R refl, lift_rel.symm R symm, lift_rel.trans R trans⟩\n\n@[refl] theorem equiv.refl : ∀ (s : wseq α), s ~ s :=\nlift_rel.refl (=) eq.refl\n\n@[symm] theorem equiv.symm : ∀ {s t : wseq α}, s ~ t → t ~ s :=\nlift_rel.symm (=) (@eq.symm _)\n\n@[trans] theorem equiv.trans : ∀ {s t u : wseq α}, s ~ t → t ~ u → s ~ u :=\nlift_rel.trans (=) (@eq.trans _)\n\ntheorem equiv.equivalence : equivalence (@equiv α) :=\n⟨@equiv.refl _, @equiv.symm _, @equiv.trans _⟩\n\nopen computation\nlocal notation `return` := computation.return\n\n@[simp] theorem destruct_nil : destruct (nil : wseq α) = return none :=\ncomputation.destruct_eq_ret rfl\n\n@[simp] theorem destruct_cons (a : α) (s) : destruct (cons a s) = return (some (a, s)) :=\ncomputation.destruct_eq_ret $ by simp [destruct, cons, computation.rmap]\n\n@[simp] theorem destruct_think (s : wseq α) : destruct (think s) = (destruct s).think :=\ncomputation.destruct_eq_think $ by simp [destruct, think, computation.rmap]\n\n@[simp] theorem seq_destruct_nil : seq.destruct (nil : wseq α) = none :=\nseq.destruct_nil\n\n@[simp] theorem seq_destruct_cons (a : α) (s) : seq.destruct (cons a s) = some (some a, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem seq_destruct_think (s : wseq α) : seq.destruct (think s) = some (none, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem head_nil : head (nil : wseq α) = return none := by simp [head]; refl\n@[simp] theorem head_cons (a : α) (s) : head (cons a s) = return (some a) := by simp [head]; refl\n@[simp] theorem head_think (s : wseq α) : head (think s) = (head s).think := by simp [head]; refl\n\n@[simp] theorem flatten_ret (s : wseq α) : flatten (return s) = s :=\nbegin\n  refine seq.eq_of_bisim (λs1 s2, flatten (return s2) = s1) _ rfl,\n  intros s' s h, rw ←h, simp [flatten],\n  cases seq.destruct s, { simp },\n  { cases val with o s', simp }\nend\n\n@[simp] theorem flatten_think (c : computation (wseq α)) : flatten c.think = think (flatten c) :=\nseq.destruct_eq_cons $ by simp [flatten, think]\n\n@[simp] theorem destruct_flatten (c : computation (wseq α)) : destruct (flatten c) = c >>= destruct :=\nbegin\n  refine computation.eq_of_bisim (λc1 c2, c1 = c2 ∨\n    ∃ c, c1 = destruct (flatten c) ∧ c2 = computation.bind c destruct) _ (or.inr ⟨c, rfl, rfl⟩),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, (or.inr ⟨c, rfl, rfl⟩) := begin\n    apply c.cases_on (λa, _) (λc', _); repeat {simp},\n    { cases (destruct a).destruct; simp },\n    { exact or.inr ⟨c', rfl, rfl⟩ }\n  end end\nend\n\ntheorem head_terminates_iff (s : wseq α) : terminates (head s) ↔ terminates (destruct s) :=\nterminates_map_iff _ (destruct s)\n\n@[simp] theorem tail_nil : tail (nil : wseq α) = nil := by simp [tail]\n@[simp] theorem tail_cons (a : α) (s) : tail (cons a s) = s := by simp [tail]\n@[simp] theorem tail_think (s : wseq α) : tail (think s) = (tail s).think := by simp [tail]\n\n@[simp] theorem dropn_nil (n) :\n  drop (nil : wseq α) n = nil := by induction n; simp [*, drop]\n@[simp] theorem dropn_cons (a : α) (s) (n) :\n  drop (cons a s) (n+1) = drop s n := by induction n; simp [*, drop]\n@[simp] theorem dropn_think (s : wseq α) (n) :\n  drop (think s) n = (drop s n).think := by induction n; simp [*, drop]\n\ntheorem dropn_add (s : wseq α) (m) : ∀ n, drop s (m + n) = drop (drop s m) n\n| 0     := rfl\n| (n+1) := congr_arg tail (dropn_add n)\n\ntheorem dropn_tail (s : wseq α) (n) : drop (tail s) n = drop s (n + 1) :=\nby rw add_comm; symmetry; apply dropn_add\n\ntheorem nth_add (s : wseq α) (m n) : nth s (m + n) = nth (drop s m) n :=\ncongr_arg head (dropn_add _ _ _)\n\ntheorem nth_tail (s : wseq α) (n) : nth (tail s) n = nth s (n + 1) :=\ncongr_arg head (dropn_tail _ _)\n\n@[simp] theorem join_nil : join nil = (nil : wseq α) := seq.join_nil\n\n@[simp] theorem join_think (S : wseq (wseq α)) :\n  join (think S) = think (join S) :=\nby { simp [think, join], unfold functor.map, simp [join, seq1.ret] }\n\n@[simp] theorem join_cons (s : wseq α) (S) :\n  join (cons s S) = think (append s (join S)) :=\nby { simp [think, join], unfold functor.map, simp [join, cons, append] }\n\n@[simp] theorem nil_append (s : wseq α) : append nil s = s := seq.nil_append _\n\n@[simp] theorem cons_append (a : α) (s t) :\n  append (cons a s) t = cons a (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem think_append (s t : wseq α) :\n  append (think s) t = think (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem append_nil (s : wseq α) : append s nil = s := seq.append_nil _\n\n@[simp] theorem append_assoc (s t u : wseq α) :\n  append (append s t) u = append s (append t u) := seq.append_assoc _ _ _\n\n@[simp] def tail.aux : option (α × wseq α) → computation (option (α × wseq α))\n| none          := return none\n| (some (a, s)) := destruct s\n\ntheorem destruct_tail (s : wseq α) :\n  destruct (tail s) = destruct s >>= tail.aux :=\nbegin\n  dsimp [tail], simp, rw [← bind_pure_comp_eq_map, is_lawful_monad.bind_assoc],\n  apply congr_arg, funext o,\n  rcases o with _|⟨a, s⟩;\n  apply (@pure_bind computation _ _ _ _ _ _).trans _; simp\nend\n\n@[simp] def drop.aux : ℕ → option (α × wseq α) → computation (option (α × wseq α))\n| 0     := return\n| (n+1) := λ a, tail.aux a >>= drop.aux n\n\ntheorem drop.aux_none : ∀ n, @drop.aux α n none = return none\n| 0     := rfl\n| (n+1) := show computation.bind (return none) (drop.aux n) = return none,\n           by rw [ret_bind, drop.aux_none]\n\ntheorem destruct_dropn :\n  ∀ (s : wseq α) n, destruct (drop s n) = destruct s >>= drop.aux n\n| s 0     := (bind_ret' _).symm\n| s (n+1) := by rw [← dropn_tail, destruct_dropn _ n,\n  destruct_tail, is_lawful_monad.bind_assoc]; refl\n\ntheorem head_terminates_of_head_tail_terminates (s : wseq α) [T : terminates (head (tail s))] :\n  terminates (head s) :=\n(head_terminates_iff _).2 $ begin\n  cases (head_terminates_iff _).1 T with a h,\n  simp [tail] at h,\n  rcases exists_of_mem_bind h with ⟨s', h1, h2⟩,\n  unfold functor.map at h1,\n  exact let ⟨t, h3, h4⟩ := exists_of_mem_map h1 in terminates_of_mem h3\nend\n\ntheorem destruct_some_of_destruct_tail_some {s : wseq α} {a}\n  (h : some a ∈ destruct (tail s)) : ∃ a', some a' ∈ destruct s :=\nbegin\n  unfold tail functor.map at h, simp at h,\n  rcases exists_of_mem_bind h with ⟨t, tm, td⟩, clear h,\n  rcases exists_of_mem_map tm with ⟨t', ht', ht2⟩, clear tm,\n  cases t' with t'; rw ←ht2 at td; simp at td,\n  { have := mem_unique td (ret_mem _), contradiction },\n  { exact ⟨_, ht'⟩ }\nend\n\ntheorem head_some_of_head_tail_some {s : wseq α} {a}\n  (h : some a ∈ head (tail s)) : ∃ a', some a' ∈ head s :=\nbegin\n  unfold head at h,\n  rcases exists_of_mem_map h with ⟨o, md, e⟩, clear h,\n  cases o with o; injection e with h', clear e h',\n  cases destruct_some_of_destruct_tail_some md with a am,\n  exact ⟨_, mem_map ((<$>) (@prod.fst α (wseq α))) am⟩\nend\n\ntheorem head_some_of_nth_some {s : wseq α} {a n}\n  (h : some a ∈ nth s n) : ∃ a', some a' ∈ head s :=\nbegin\n  revert a, induction n with n IH; intros,\n  exacts [⟨_, h⟩, let ⟨a', h'⟩ := head_some_of_head_tail_some h in IH h']\nend\n\ninstance productive_tail (s : wseq α) [productive s] : productive (tail s) :=\nλ n, by rw [nth_tail]; apply_instance\n\ninstance productive_dropn (s : wseq α) [productive s] (n) : productive (drop s n) :=\nλ m, by rw [←nth_add]; apply_instance\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef to_seq (s : wseq α) [productive s] : seq α :=\n⟨λ n, (nth s n).get, λn h,\nbegin\n  induction e : computation.get (nth s (n + 1)), {trivial},\n  have := mem_of_get_eq _ e,\n  simp [nth] at this h, cases head_some_of_head_tail_some this with a' h',\n  have := mem_unique h' (@mem_of_get_eq _ _ _ _ h),\n  contradiction\nend⟩\n\ntheorem nth_terminates_le {s : wseq α} {m n} (h : m ≤ n) : terminates (nth s n) → terminates (nth s m) :=\nby induction h with m' h IH; [exact id,\n  exact λ T, IH (@head_terminates_of_head_tail_terminates _ _ T)]\n\ntheorem head_terminates_of_nth_terminates {s : wseq α} {n} : terminates (nth s n) → terminates (head s) :=\nnth_terminates_le (nat.zero_le n)\n\ntheorem destruct_terminates_of_nth_terminates {s : wseq α} {n} (T : terminates (nth s n)) : terminates (destruct s) :=\n(head_terminates_iff _).1 $ head_terminates_of_nth_terminates T\n\ntheorem mem_rec_on {C : wseq α → Prop} {a s} (M : a ∈ s)\n  (h1 : ∀ b s', (a = b ∨ C s') → C (cons b s'))\n  (h2 : ∀ s, C s → C (think s)) : C s :=\nbegin\n  apply seq.mem_rec_on M,\n  intros o s' h, cases o with b,\n  { apply h2, cases h, {contradiction}, {assumption} },\n  { apply h1, apply or.imp_left _ h, intro h, injection h }\nend\n\n@[simp] theorem mem_think (s : wseq α) (a) : a ∈ think s ↔ a ∈ s :=\nbegin\n  cases s with f al,\n  change some (some a) ∈ some none :: f ↔ some (some a) ∈ f,\n  constructor; intro h,\n  { apply (stream.eq_or_mem_of_mem_cons h).resolve_left,\n    intro, injections },\n  { apply stream.mem_cons_of_mem _ h }\nend\n\ntheorem eq_or_mem_iff_mem {s : wseq α} {a a' s'} :\n  some (a', s') ∈ destruct s → (a ∈ s ↔ a = a' ∨ a ∈ s') :=\nbegin\n  generalize e : destruct s = c, intro h,\n  revert s, apply computation.mem_rec_on h _ (λ c IH, _); intro s;\n  apply s.cases_on _ (λ x s, _) (λ s, _); intros m;\n  have := congr_arg computation.destruct m; simp at this;\n  cases this with i1 i2,\n  { rw [i1, i2],\n    cases s' with f al,\n    unfold cons has_mem.mem wseq.mem seq.mem seq.cons, simp,\n    have h_a_eq_a' : a = a' ↔ some (some a) = some (some a'), {simp},\n    rw [h_a_eq_a'],\n    refine ⟨stream.eq_or_mem_of_mem_cons, λo, _⟩,\n    { cases o with e m,\n      { rw e, apply stream.mem_cons },\n      { exact stream.mem_cons_of_mem _ m } } },\n  { simp, exact IH this }\nend\n\n@[simp] theorem mem_cons_iff (s : wseq α) (b) {a} : a ∈ cons b s ↔ a = b ∨ a ∈ s :=\neq_or_mem_iff_mem $ by simp [ret_mem]\n\ntheorem mem_cons_of_mem {s : wseq α} (b) {a} (h : a ∈ s) : a ∈ cons b s :=\n(mem_cons_iff _ _).2 (or.inr h)\n\ntheorem mem_cons (s : wseq α) (a) : a ∈ cons a s :=\n(mem_cons_iff _ _).2 (or.inl rfl)\n\ntheorem mem_of_mem_tail {s : wseq α} {a} : a ∈ tail s → a ∈ s :=\nbegin\n  intro h, have := h, cases h with n e, revert s, simp [stream.nth],\n  induction n with n IH; intro s; apply s.cases_on _ (λx s, _) (λ s, _);\n    repeat{simp}; intros m e; injections,\n  { exact or.inr m },\n  { exact or.inr m },\n  { apply IH m, rw e, cases tail s, refl }\nend\n\ntheorem mem_of_mem_dropn {s : wseq α} {a} : ∀ {n}, a ∈ drop s n → a ∈ s\n| 0     h := h\n| (n+1) h := @mem_of_mem_dropn n (mem_of_mem_tail h)\n\ntheorem nth_mem {s : wseq α} {a n} : some a ∈ nth s n → a ∈ s :=\nbegin\n  revert s, induction n with n IH; intros s h,\n  { rcases exists_of_mem_map h with ⟨o, h1, h2⟩,\n    cases o with o; injection h2 with h',\n    cases o with a' s',\n    exact (eq_or_mem_iff_mem h1).2 (or.inl h'.symm) },\n  { have := @IH (tail s), rw nth_tail at this,\n    exact mem_of_mem_tail (this h) }\nend\n\ntheorem exists_nth_of_mem {s : wseq α} {a} (h : a ∈ s) : ∃ n, some a ∈ nth s n :=\nbegin\n  apply mem_rec_on h,\n  { intros a' s' h, cases h with h h,\n    { existsi 0, simp [nth], rw h, apply ret_mem },\n    { cases h with n h, existsi n+1,\n      simp [nth], exact h } },\n  { intros s' h, cases h with n h,\n    existsi n, simp [nth], apply think_mem h }\nend\n\ntheorem exists_dropn_of_mem {s : wseq α} {a} (h : a ∈ s) :\n  ∃ n s', some (a, s') ∈ destruct (drop s n) :=\nlet ⟨n, h⟩ := exists_nth_of_mem h in ⟨n, begin\n  cases (head_terminates_iff _).1 ⟨_, h⟩ with o om,\n  have := mem_unique (mem_map _ om) h,\n  cases o with o; injection this with i,\n  cases o with a' s', dsimp at i,\n  rw i at om, exact ⟨_, om⟩\nend⟩\n\ntheorem lift_rel_dropn_destruct {R : α → β → Prop} {s t} (H : lift_rel R s t) :\n  ∀ n, computation.lift_rel (lift_rel_o R (lift_rel R))\n    (destruct (drop s n)) (destruct (drop t n))\n| 0     := lift_rel_destruct H\n| (n+1) := begin\n  simp [destruct_tail],\n  apply lift_rel_bind,\n  apply lift_rel_dropn_destruct n,\n  exact λ a b o, match a, b, o with\n  | none,       none,         _        := by simp\n  | some (a, s), some (b, t), ⟨h1, h2⟩ := by simp [tail.aux]; apply lift_rel_destruct h2\n  end\nend\n\ntheorem exists_of_lift_rel_left {R : α → β → Prop} {s t}\n  (H : lift_rel R s t) {a} (h : a ∈ s) : ∃ {b}, b ∈ t ∧ R a b :=\nlet ⟨n, h⟩ := exists_nth_of_mem h,\n    ⟨some (._, s'), sd, rfl⟩ := exists_of_mem_map h,\n    ⟨some (b, t'), td, ⟨ab, _⟩⟩ := (lift_rel_dropn_destruct H n).left sd in\n⟨b, nth_mem (mem_map ((<$>) prod.fst.{v v}) td), ab⟩\n\ntheorem exists_of_lift_rel_right {R : α → β → Prop} {s t}\n  (H : lift_rel R s t) {b} (h : b ∈ t) : ∃ {a}, a ∈ s ∧ R a b :=\nby rw ←lift_rel.swap at H; exact exists_of_lift_rel_left H h\n\ntheorem head_terminates_of_mem {s : wseq α} {a} (h : a ∈ s) : terminates (head s) :=\nlet ⟨n, h⟩ := exists_nth_of_mem h in head_terminates_of_nth_terminates ⟨_, h⟩\n\ntheorem of_mem_append {s₁ s₂ : wseq α} {a : α} : a ∈ append s₁ s₂ → a ∈ s₁ ∨ a ∈ s₂ :=\nseq.of_mem_append\n\ntheorem mem_append_left {s₁ s₂ : wseq α} {a : α} : a ∈ s₁ → a ∈ append s₁ s₂ :=\nseq.mem_append_left\n\ntheorem exists_of_mem_map {f} {b : β} : ∀ {s : wseq α}, b ∈ map f s → ∃ a, a ∈ s ∧ f a = b\n| ⟨g, al⟩ h := let ⟨o, om, oe⟩ := seq.exists_of_mem_map h in\n  by cases o with a; injection oe with h'; exact ⟨a, om, h'⟩\n\n@[simp] theorem lift_rel_nil (R : α → β → Prop) : lift_rel R nil nil :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_cons (R : α → β → Prop) (a b s t) :\n  lift_rel R (cons a s) (cons b t) ↔ R a b ∧ lift_rel R s t :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_left (R : α → β → Prop) (s t) :\n  lift_rel R (think s) t ↔ lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_right (R : α → β → Prop) (s t) :\n  lift_rel R s (think t) ↔ lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\ntheorem cons_congr {s t : wseq α} (a : α) (h : s ~ t) : cons a s ~ cons a t :=\nby unfold equiv; simp; exact h\n\ntheorem think_equiv (s : wseq α) : think s ~ s :=\nby unfold equiv; simp; apply equiv.refl\n\ntheorem think_congr {s t : wseq α} (a : α) (h : s ~ t) : think s ~ think t :=\nby unfold equiv; simp; exact h\n\ntheorem head_congr : ∀ {s t : wseq α}, s ~ t → head s ~ head t :=\nsuffices ∀ {s t : wseq α}, s ~ t → ∀ {o}, o ∈ head s → o ∈ head t, from\nλ s t h o, ⟨this h, this h.symm⟩,\nbegin\n  intros s t h o ho,\n  rcases @computation.exists_of_mem_map _ _ _ _ (destruct s) ho with ⟨ds, dsm, dse⟩,\n  rw ←dse,\n  cases destruct_congr h with l r,\n  rcases l dsm with ⟨dt, dtm, dst⟩,\n  cases ds with a; cases dt with b,\n  { apply mem_map _ dtm },\n  { cases b, cases dst },\n  { cases a, cases dst },\n  { cases a with a s', cases b with b t', rw dst.left,\n    exact @mem_map _ _ (@functor.map _ _ (α × wseq α) _ prod.fst)\n      _ (destruct t) dtm }\nend\n\ntheorem flatten_equiv {c : computation (wseq α)} {s} (h : s ∈ c) : flatten c ~ s :=\nbegin\n  apply computation.mem_rec_on h, { simp },\n  { intro s', apply equiv.trans, simp [think_equiv] }\nend\n\ntheorem lift_rel_flatten {R : α → β → Prop} {c1 : computation (wseq α)} {c2 : computation (wseq β)}\n  (h : c1.lift_rel (lift_rel R) c2) : lift_rel R (flatten c1) (flatten c2) :=\nlet S := λ s t,\n  ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ computation.lift_rel (lift_rel R) c1 c2 in\n⟨S, ⟨c1, c2, rfl, rfl, h⟩, λ s t h,\n  match s, t, h with ._, ._, ⟨c1, c2, rfl, rfl, h⟩ := begin\n    simp, apply lift_rel_bind _ _ h,\n    intros a b ab, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct ab),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t h, refine ⟨return s, return t, _, _, _⟩; simp [h]\n  end end⟩\n\ntheorem flatten_congr {c1 c2 : computation (wseq α)} :\n  computation.lift_rel equiv c1 c2 → flatten c1 ~ flatten c2 := lift_rel_flatten\n\ntheorem tail_congr {s t : wseq α} (h : s ~ t) : tail s ~ tail t :=\nbegin\n  apply flatten_congr,\n  unfold functor.map, rw [←bind_ret, ←bind_ret],\n  apply lift_rel_bind _ _ (destruct_congr h),\n  intros a b h, simp,\n  cases a with a; cases b with b,\n  { trivial },\n  { cases h },\n  { cases a, cases h },\n  { cases a with a s', cases b with b t', exact h.right }\nend\n\ntheorem dropn_congr {s t : wseq α} (h : s ~ t) (n) : drop s n ~ drop t n :=\nby induction n; simp [*, tail_congr]\n\ntheorem nth_congr {s t : wseq α} (h : s ~ t) (n) : nth s n ~ nth t n :=\nhead_congr (dropn_congr h _)\n\ntheorem mem_congr {s t : wseq α} (h : s ~ t) (a) : a ∈ s ↔ a ∈ t :=\nsuffices ∀ {s t : wseq α}, s ~ t → a ∈ s → a ∈ t, from ⟨this h, this h.symm⟩,\nλ s t h as, let ⟨n, hn⟩ := exists_nth_of_mem as in\nnth_mem ((nth_congr h _ _).1 hn)\n\ntheorem productive_congr {s t : wseq α} (h : s ~ t) : productive s ↔ productive t :=\nforall_congr $ λn, terminates_congr $ nth_congr h _\n\ntheorem equiv.ext {s t : wseq α} (h : ∀ n, nth s n ~ nth t n) : s ~ t :=\n⟨λ s t, ∀ n, nth s n ~ nth t n, h, λs t h, begin\n  refine lift_rel_def.2 ⟨_, _⟩,\n  { rw [←head_terminates_iff, ←head_terminates_iff],\n    exact terminates_congr (h 0) },\n  { intros a b ma mb,\n    cases a with a; cases b with b,\n    { trivial },\n    { injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) },\n    { injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) },\n    { cases a with a s', cases b with b t',\n      injection mem_unique (mem_map _ ma) ((h 0 _).2 (mem_map _ mb)) with ab,\n      refine ⟨ab, λ n, _⟩,\n      refine (nth_congr (flatten_equiv (mem_map _ ma)) n).symm.trans\n        ((_ : nth (tail s) n ~ nth (tail t) n).trans\n        (nth_congr (flatten_equiv (mem_map _ mb)) n)),\n      rw [nth_tail, nth_tail], apply h } }\nend⟩\n\ntheorem length_eq_map (s : wseq α) : length s = computation.map list.length (to_list s) :=\nbegin\n  refine eq_of_bisim\n    (λ c1 c2, ∃ (l : list α) (s : wseq α),\n      c1 = corec length._match_2 (l.length, s) ∧\n      c2 = computation.map list.length (corec to_list._match_2 (l, s)))\n    _ ⟨[], s, rfl, rfl⟩,\n  intros s1 s2 h, rcases h with ⟨l, s, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine ⟨a::l, s, _, _⟩; simp },\n  { refine ⟨l, s, _, _⟩; simp }\nend\n\n@[simp] theorem of_list_nil : of_list [] = (nil : wseq α) := rfl\n\n@[simp] theorem of_list_cons (a : α) (l) :\n  of_list (a :: l) = cons a (of_list l) :=\nshow seq.map some (seq.of_list (a :: l)) =\n     seq.cons (some a) (seq.map some (seq.of_list l)), by simp\n\n@[simp] theorem to_list'_nil (l : list α) :\n  corec to_list._match_2 (l, nil) = return l.reverse :=\ndestruct_eq_ret rfl\n\n@[simp] theorem to_list'_cons (l : list α) (s : wseq α) (a : α) :\n  corec to_list._match_2 (l, cons a s) =\n  (corec to_list._match_2 (a::l, s)).think :=\ndestruct_eq_think $ by simp [to_list, cons]\n\n@[simp] theorem to_list'_think (l : list α) (s : wseq α) :\n  corec to_list._match_2 (l, think s) =\n  (corec to_list._match_2 (l, s)).think :=\ndestruct_eq_think $ by simp [to_list, think]\n\ntheorem to_list'_map (l : list α) (s : wseq α) :\n  corec to_list._match_2 (l, s) =\n  ((++) l.reverse) <$> to_list s :=\nbegin\n  refine eq_of_bisim\n    (λ c1 c2, ∃ (l' : list α) (s : wseq α),\n      c1 = corec to_list._match_2 (l' ++ l, s) ∧\n      c2 = computation.map ((++) l.reverse) (corec to_list._match_2 (l', s)))\n    _ ⟨[], s, rfl, rfl⟩,\n  intros s1 s2 h, rcases h with ⟨l', s, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine ⟨a::l', s, _, _⟩; simp },\n  { refine ⟨l', s, _, _⟩; simp }\nend\n\n@[simp] theorem to_list_cons (a : α) (s) :\n  to_list (cons a s) = (list.cons a <$> to_list s).think :=\ndestruct_eq_think $ by unfold to_list; simp; rw to_list'_map; simp; refl\n\n@[simp] theorem to_list_nil : to_list (nil : wseq α) = return [] :=\ndestruct_eq_ret rfl\n\ntheorem to_list_of_list (l : list α) : l ∈ to_list (of_list l) :=\nby induction l with a l IH; simp [ret_mem]; exact think_mem (mem_map _ IH)\n\n@[simp] theorem destruct_of_seq (s : seq α) :\n  destruct (of_seq s) = return (s.head.map $ λ a, (a, of_seq s.tail)) :=\ndestruct_eq_ret $ begin\n  simp [of_seq, head, destruct, seq.destruct, seq.head],\n  rw [show seq.nth (some <$> s) 0 = some <$> seq.nth s 0, by apply seq.map_nth],\n  cases seq.nth s 0 with a, { refl },\n  unfold functor.map,\n  simp [destruct]\nend\n\n@[simp] theorem head_of_seq (s : seq α) : head (of_seq s) = return s.head :=\nby simp [head]; cases seq.head s; refl\n\n@[simp] theorem tail_of_seq (s : seq α) : tail (of_seq s) = of_seq s.tail :=\nbegin\n  simp [tail], apply s.cases_on _ (λ x s, _); simp [of_seq], {refl},\n  rw [seq.head_cons, seq.tail_cons], refl\nend\n\n@[simp] theorem dropn_of_seq (s : seq α) : ∀ n, drop (of_seq s) n = of_seq (s.drop n)\n| 0 := rfl\n| (n+1) := by dsimp [drop]; rw [dropn_of_seq, tail_of_seq]\n\ntheorem nth_of_seq (s : seq α) (n) : nth (of_seq s) n = return (seq.nth s n) :=\nby dsimp [nth]; rw [dropn_of_seq, head_of_seq, seq.head_dropn]\n\ninstance productive_of_seq (s : seq α) : productive (of_seq s) :=\nλ n, by rw nth_of_seq; apply_instance\n\ntheorem to_seq_of_seq (s : seq α) : to_seq (of_seq s) = s :=\nbegin\n  apply subtype.eq, funext n,\n  dsimp [to_seq], apply get_eq_of_mem,\n  rw nth_of_seq, apply ret_mem\nend\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret (a : α) : wseq α := of_list [a]\n\n@[simp] theorem map_nil (f : α → β) : map f nil = nil := rfl\n\n@[simp] theorem map_cons (f : α → β) (a s) :\n  map f (cons a s) = cons (f a) (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_think (f : α → β) (s) :\n  map f (think s) = think (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_id (s : wseq α) : map id s = s := by simp [map]\n\n@[simp] theorem map_ret (f : α → β) (a) : map f (ret a) = ret (f a) := by simp [ret]\n\n@[simp] theorem map_append (f : α → β) (s t) : map f (append s t) = append (map f s) (map f t) :=\nseq.map_append _ _ _\n\ntheorem map_comp (f : α → β) (g : β → γ) (s : wseq α) :\n  map (g ∘ f) s = map g (map f s) :=\nbegin\n  dsimp [map], rw ←seq.map_comp,\n  apply congr_fun, apply congr_arg,\n  funext o, cases o; refl\nend\n\ntheorem mem_map (f : α → β) {a : α} {s : wseq α} : a ∈ s → f a ∈ map f s :=\nseq.mem_map (option.map f)\n\n-- The converse is not true without additional assumptions\ntheorem exists_of_mem_join {a : α} : ∀ {S : wseq (wseq α)}, a ∈ join S → ∃ s, s ∈ S ∧ a ∈ s :=\nsuffices ∀ ss : wseq α, a ∈ ss → ∀ s S, append s (join S) = ss →\n  a ∈ append s (join S) → a ∈ s ∨ ∃ s, s ∈ S ∧ a ∈ s, from λ S h,\n  (this _ h nil S (by simp) (by simp [h])).resolve_left (not_mem_nil _),\nbegin\n  intros ss h, apply mem_rec_on h (λ b ss o, _) (λ ss IH, _); intros s S,\n  { refine s.cases_on (S.cases_on _ (λ s S, _) (λ S, _)) (λ b' s, _) (λ s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try {cases this}; try {contradiction},\n    substs b' ss,\n    simp at m ⊢,\n    cases o with e IH, { simp [e] },\n    cases m with e m, { simp [e] },\n    exact or.imp_left or.inr (IH _ _ rfl m) },\n  { refine s.cases_on (S.cases_on _ (λ s S, _) (λ S, _)) (λ b' s, _) (λ s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try { try {have := this.1}, contradiction }; subst ss,\n    { apply or.inr, simp at m ⊢,\n      cases IH s S rfl m with as ex,\n      { exact ⟨s, or.inl rfl, as⟩ },\n      { rcases ex with ⟨s', sS, as⟩,\n        exact ⟨s', or.inr sS, as⟩ } },\n    { apply or.inr, simp at m,\n      rcases (IH nil S (by simp) (by simp [m])).resolve_left (not_mem_nil _) with ⟨s, sS, as⟩,\n      exact ⟨s, by simp [sS], as⟩ },\n    { simp at m IH ⊢, apply IH _ _ rfl m } }\nend\n\ntheorem exists_of_mem_bind {s : wseq α} {f : α → wseq β} {b}\n  (h : b ∈ bind s f) : ∃ a ∈ s, b ∈ f a :=\nlet ⟨t, tm, bt⟩ := exists_of_mem_join h,\n    ⟨a, as, e⟩ := exists_of_mem_map tm in ⟨a, as, by rwa e⟩\n\ntheorem destruct_map (f : α → β) (s : wseq α) :\n  destruct (map f s) = computation.map (option.map (prod.map f (map f))) (destruct s) :=\nbegin\n  apply eq_of_bisim (λ c1 c2, ∃ s, c1 = destruct (map f s) ∧\n    c2 = computation.map (option.map (prod.map f (map f))) (destruct s)),\n  { intros c1 c2 h, cases h with s h, rw [h.left, h.right],\n    apply s.cases_on _ (λ a s, _) (λ s, _); simp; simp,\n    exact ⟨s, rfl, rfl⟩ },\n  { exact ⟨s, rfl, rfl⟩ }\nend\n\ntheorem lift_rel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop)\n  {s1 : wseq α} {s2 : wseq β}\n  {f1 : α → γ} {f2 : β → δ}\n  (h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → S (f1 a) (f2 b))\n  : lift_rel S (map f1 s1) (map f2 s2) :=\n⟨λ s1 s2, ∃ s t, s1 = map f1 s ∧ s2 = map f2 t ∧ lift_rel R s t,\n⟨s1, s2, rfl, rfl, h1⟩,\nλ s1 s2 h, match s1, s2, h with ._, ._, ⟨s, t, rfl, rfl, h⟩ := begin\n  simp [destruct_map], apply computation.lift_rel_map _ _ (lift_rel_destruct h),\n  intros o p h,\n  cases o with a; cases p with b; simp,\n  { cases b; cases h },\n  { cases a; cases h },\n  { cases a with a s; cases b with b t, cases h with r h,\n    exact ⟨h2 r, s, rfl, t, rfl, h⟩ }\nend end⟩\n\ntheorem map_congr (f : α → β) {s t : wseq α} (h : s ~ t) : map f s ~ map f t :=\nlift_rel_map _ _ h (λ _ _, congr_arg _)\n\n@[simp] def destruct_append.aux (t : wseq α) :\n  option (α × wseq α) → computation (option (α × wseq α))\n| none          := destruct t\n| (some (a, s)) := return (some (a, append s t))\n\ntheorem destruct_append (s t : wseq α) :\n  destruct (append s t) = (destruct s).bind (destruct_append.aux t) :=\nbegin\n  apply eq_of_bisim (λ c1 c2, ∃ s t, c1 = destruct (append s t) ∧\n    c2 = (destruct s).bind (destruct_append.aux t)) _ ⟨s, t, rfl, rfl⟩,\n  intros c1 c2 h, rcases h with ⟨s, t, h⟩, rw [h.left, h.right],\n  apply s.cases_on _ (λ a s, _) (λ s, _); simp; simp,\n  { apply t.cases_on _ (λ b t, _) (λ t, _); simp; simp,\n    { refine ⟨nil, t, _, _⟩; simp } },\n  { exact ⟨s, t, rfl, rfl⟩ }\nend\n\n@[simp] def destruct_join.aux : option (wseq α × wseq (wseq α)) → computation (option (α × wseq α))\n| none          := return none\n| (some (s, S)) := (destruct (append s (join S))).think\n\ntheorem destruct_join (S : wseq (wseq α)) :\n  destruct (join S) = (destruct S).bind destruct_join.aux :=\nbegin\n  apply eq_of_bisim (λ c1 c2, c1 = c2 ∨ ∃ S, c1 = destruct (join S) ∧\n    c2 = (destruct S).bind destruct_join.aux) _ (or.inr ⟨S, rfl, rfl⟩),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, or.inr ⟨S, rfl, rfl⟩ := begin\n    apply S.cases_on _ (λ s S, _) (λ S, _); simp; simp,\n    { refine or.inr ⟨S, rfl, rfl⟩ }\n  end end\nend\n\ntheorem lift_rel_append (R : α → β → Prop) {s1 s2 : wseq α} {t1 t2 : wseq β}\n  (h1 : lift_rel R s1 t1) (h2 : lift_rel R s2 t2) :\n  lift_rel R (append s1 s2) (append t1 t2) :=\n⟨λ s t, lift_rel R s t ∨ ∃ s1 t1, s = append s1 s2 ∧ t = append t1 t2 ∧ lift_rel R s1 t1,\nor.inr ⟨s1, t1, rfl, rfl, h1⟩,\nλ s t h, match s, t, h with\n| s, t, or.inl h := begin\n    apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end\n| ._, ._, or.inr ⟨s1, t1, rfl, rfl, h⟩ := begin\n    simp [destruct_append],\n    apply computation.lift_rel_bind _ _ (lift_rel_destruct h),\n    intros o p h,\n    cases o with a; cases p with b,\n    { simp, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h2),\n      intros a b, apply lift_rel_o.imp_right,\n      intros s t, apply or.inl },\n    { cases b; cases h },\n    { cases a; cases h },\n    { cases a with a s; cases b with b t, cases h with r h,\n      simp, exact ⟨r, or.inr ⟨s, rfl, t, rfl, h⟩⟩ }\n  end\nend⟩\n\ntheorem lift_rel_join.lem (R : α → β → Prop) {S T} {U : wseq α → wseq β → Prop}\n  (ST : lift_rel (lift_rel R) S T) (HU : ∀ s1 s2, (∃ s t S T,\n      s1 = append s (join S) ∧ s2 = append t (join T) ∧\n      lift_rel R s t ∧ lift_rel (lift_rel R) S T) → U s1 s2) {a} (ma : a ∈ destruct (join S)) :\n  ∃ {b}, b ∈ destruct (join T) ∧ lift_rel_o R U a b :=\nbegin\n  cases exists_results_of_mem ma with n h, clear ma, revert a S T,\n  apply nat.strong_induction_on n _,\n  intros n IH a S T ST ra, simp [destruct_join] at ra, exact\n  let ⟨o, m, k, rs1, rs2, en⟩ := of_results_bind ra,\n      ⟨p, mT, rop⟩ := computation.exists_of_lift_rel_left (lift_rel_destruct ST) rs1.mem in\n  by exact match o, p, rop, rs1, rs2, mT with\n  | none, none, _, rs1, rs2, mT := by simp [destruct_join]; exact\n    ⟨none, mem_bind mT (ret_mem _), by rw eq_of_ret_mem rs2.mem; trivial⟩\n  | some (s, S'), some (t, T'), ⟨st, ST'⟩, rs1, rs2, mT :=\n    by simp [destruct_append] at rs2; exact\n    let ⟨k1, rs3, ek⟩ := of_results_think rs2,\n        ⟨o', m1, n1, rs4, rs5, ek1⟩ := of_results_bind rs3,\n        ⟨p', mt, rop'⟩ := computation.exists_of_lift_rel_left (lift_rel_destruct st) rs4.mem in\n    by exact match o', p', rop', rs4, rs5, mt with\n    | none, none, _, rs4, rs5', mt :=\n      have n1 < n, begin\n        rw [en, ek, ek1],\n        apply lt_of_lt_of_le _ (nat.le_add_right _ _),\n        apply nat.lt_succ_of_le (nat.le_add_right _ _)\n      end,\n      let ⟨ob, mb, rob⟩ := IH _ this ST' rs5' in by refine ⟨ob, _, rob⟩;\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, exact mb }\n    | some (a, s'), some (b, t'), ⟨ab, st'⟩, rs4, rs5, mt := begin\n      simp at rs5,\n      refine ⟨some (b, append t' (join T')), _, _⟩,\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, apply ret_mem },\n      rw eq_of_ret_mem rs5.mem,\n      exact ⟨ab, HU _ _ ⟨s', t', S', T', rfl, rfl, st', ST'⟩⟩\n    end end\n  end\nend\n\ntheorem lift_rel_join (R : α → β → Prop) {S : wseq (wseq α)} {T : wseq (wseq β)}\n  (h : lift_rel (lift_rel R) S T) : lift_rel R (join S) (join T) :=\n⟨λ s1 s2, ∃ s t S T,\n  s1 = append s (join S) ∧ s2 = append t (join T) ∧\n  lift_rel R s t ∧ lift_rel (lift_rel R) S T,\n  ⟨nil, nil, S, T, by simp, by simp, by simp, h⟩,\nλs1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩, begin\n  clear _fun_match _x,\n  rw [h1, h2], rw [destruct_append, destruct_append],\n  apply computation.lift_rel_bind _ _ (lift_rel_destruct st),\n  exact λ o p h, match o, p, h with\n  | some (a, s), some (b, t), ⟨h1, h2⟩ :=\n    by simp; exact ⟨h1, s, t, S, rfl, T, rfl, h2, ST⟩\n  | none, none, _ := begin\n    dsimp [destruct_append.aux, computation.lift_rel], constructor,\n    { intro, apply lift_rel_join.lem _ ST (λ _ _, id) },\n    { intros b mb,\n      rw [←lift_rel_o.swap], apply lift_rel_join.lem (function.swap R),\n      { rw [←lift_rel.swap R, ←lift_rel.swap], apply ST },\n      { rw [←lift_rel.swap R, ←lift_rel.swap (lift_rel R)],\n        exact λ s1 s2 ⟨s, t, S, T, h1, h2, st, ST⟩,\n                      ⟨t, s, T, S, h2, h1, st, ST⟩ },\n      { exact mb } }\n  end end\nend⟩\n\ntheorem join_congr {S T : wseq (wseq α)} (h : lift_rel equiv S T) : join S ~ join T :=\nlift_rel_join _ h\n\ntheorem lift_rel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop)\n  {s1 : wseq α} {s2 : wseq β}\n  {f1 : α → wseq γ} {f2 : β → wseq δ}\n  (h1 : lift_rel R s1 s2) (h2 : ∀ {a b}, R a b → lift_rel S (f1 a) (f2 b))\n  : lift_rel S (bind s1 f1) (bind s2 f2) :=\nlift_rel_join _ (lift_rel_map _ _ h1 @h2)\n\ntheorem bind_congr {s1 s2 : wseq α} {f1 f2 : α → wseq β}\n  (h1 : s1 ~ s2) (h2 : ∀ a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=\nlift_rel_bind _ _ h1 (λ a b h, by rw h; apply h2)\n\n@[simp] theorem join_ret (s : wseq α) : join (ret s) ~ s :=\nby simp [ret]; apply think_equiv\n\n@[simp] theorem join_map_ret (s : wseq α) : join (map ret s) ~ s :=\nbegin\n  refine ⟨λ s1 s2, join (map ret s2) = s1, rfl, _⟩,\n  intros s' s h, rw ←h,\n  apply lift_rel_rec\n    (λ c1 c2, ∃ s,\n      c1 = destruct (join (map ret s)) ∧ c2 = destruct s),\n  { exact λ c1 c2 h, match c1, c2, h with\n    | ._, ._, ⟨s, rfl, rfl⟩ := begin\n      clear h _match,\n      apply s.cases_on _ (λ a s, _) (λ s, _); simp [ret]; simp [ret],\n      { refine ⟨_, ret_mem _, _⟩, simp },\n      { exact ⟨s, rfl, rfl⟩ }\n    end end },\n  { exact ⟨s, rfl, rfl⟩ }\nend\n\n@[simp] theorem join_append (S T : wseq (wseq α)) :\n  join (append S T) ~ append (join S) (join T) :=\nbegin\n  refine ⟨λ s1 s2, ∃ s S T,\n    s1 = append s (join (append S T)) ∧\n    s2 = append s (append (join S) (join T)), ⟨nil, S, T, by simp, by simp⟩, _⟩,\n  intros s1 s2 h,\n  apply lift_rel_rec (λ c1 c2, ∃ (s : wseq α) S T,\n    c1 = destruct (append s (join (append S T))) ∧\n    c2 = destruct (append s (append (join S) (join T)))) _ _ _\n    (let ⟨s, S, T, h1, h2⟩ := h in\n         ⟨s, S, T, congr_arg destruct h1, congr_arg destruct h2⟩),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, ⟨s, S, T, rfl, rfl⟩ := begin\n    clear _match h h,\n    apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp; simp,\n    { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp; simp,\n      { apply wseq.cases_on T _ (λ s T, _) (λ T, _); simp; simp,\n        { refine ⟨s, nil, T, _, _⟩; simp },\n        { refine ⟨nil, nil, T, _, _⟩; simp } },\n      { exact ⟨s, S, T, rfl, rfl⟩ },\n      { refine ⟨nil, S, T, _, _⟩; simp } },\n    { exact ⟨s, S, T, rfl, rfl⟩ },\n    { exact ⟨s, S, T, rfl, rfl⟩ }\n  end end\nend\n\n@[simp] theorem bind_ret (f : α → β) (s) : bind s (ret ∘ f) ~ map f s :=\nbegin\n  dsimp [bind], change (λx, ret (f x)) with (ret ∘ f),\n  rw [map_comp], apply join_map_ret\nend\n\n@[simp] theorem ret_bind (a : α) (f : α → wseq β) :\n  bind (ret a) f ~ f a := by simp [bind]\n\n@[simp] theorem map_join (f : α → β) (S) :\n  map f (join S) = join (map (map f) S) :=\nbegin\n  apply seq.eq_of_bisim (λs1 s2,\n    ∃ s S, s1 = append s (map f (join S)) ∧\n      s2 = append s (join (map (map f) S))),\n  { intros s1 s2 h,\n    exact match s1, s2, h with ._, ._, ⟨s, S, rfl, rfl⟩ := begin\n      apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp; simp,\n      { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp; simp,\n        { exact ⟨map f s, S, rfl, rfl⟩ },\n        { refine ⟨nil, S, _, _⟩; simp } },\n      { exact ⟨_, _, rfl, rfl⟩ },\n      { exact ⟨_, _, rfl, rfl⟩ }\n    end end },\n  { refine ⟨nil, S, _, _⟩; simp }\nend\n\n@[simp] theorem join_join (SS : wseq (wseq (wseq α))) :\n  join (join SS) ~ join (map join SS) :=\nbegin\n  refine ⟨λ s1 s2, ∃ s S SS,\n    s1 = append s (join (append S (join SS))) ∧\n    s2 = append s (append (join S) (join (map join SS))),\n    ⟨nil, nil, SS, by simp, by simp⟩, _⟩,\n  intros s1 s2 h,\n  apply lift_rel_rec (λ c1 c2, ∃ s S SS,\n      c1 = destruct (append s (join (append S (join SS)))) ∧\n      c2 = destruct (append s (append (join S) (join (map join SS)))))\n    _ (destruct s1) (destruct s2)\n    (let ⟨s, S, SS, h1, h2⟩ := h in ⟨s, S, SS, by simp [h1], by simp [h2]⟩),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, ⟨s, S, SS, rfl, rfl⟩ := begin\n    clear _match h h,\n    apply wseq.cases_on s _ (λ a s, _) (λ s, _); simp; simp,\n    { apply wseq.cases_on S _ (λ s S, _) (λ S, _); simp; simp,\n      { apply wseq.cases_on SS _ (λ S SS, _) (λ SS, _); simp; simp,\n        { refine ⟨nil, S, SS, _, _⟩; simp },\n        { refine ⟨nil, nil, SS, _, _⟩; simp } },\n      { exact ⟨s, S, SS, rfl, rfl⟩ },\n      { refine ⟨nil, S, SS, _, _⟩; simp } },\n    { exact ⟨s, S, SS, rfl, rfl⟩ },\n    { exact ⟨s, S, SS, rfl, rfl⟩ }\n  end end\nend\n\n@[simp] theorem bind_assoc (s : wseq α) (f : α → wseq β) (g : β → wseq γ) :\n  bind (bind s f) g ~ bind s (λ (x : α), bind (f x) g) :=\nbegin\n  simp [bind], rw [← map_comp f (map g), map_comp (map g ∘ f) join],\n  apply join_join\nend\n\ninstance : monad wseq :=\n{ map  := @map,\n  pure := @ret,\n  bind := @bind }\n\n/-\n  Unfortunately, wseq is not a lawful monad, because it does not satisfy\n  the monad laws exactly, only up to sequence equivalence.\n  Furthermore, even quotienting by the equivalence is not sufficient,\n  because the join operation involves lists of quotient elements,\n  with a lifted equivalence relation, and pure quotients cannot handle\n  this type of construction.\n\ninstance : is_lawful_monad wseq :=\n{ id_map := @map_id,\n  bind_pure_comp_eq_map := @bind_ret,\n  pure_bind := @ret_bind,\n  bind_assoc := @bind_assoc }\n-/\n\nend wseq\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/seq/wseq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3082365543113072}}
{"text": "import FOL.completeness\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\n\nvariables {L : language}\n\nnamespace formula\n\ndef bfal_le [has_preceq (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∀.((#0 ≼ (t^1)) ⟶ p)\n\nnotation `∀.{≼ ` t `} `:64 p := bfal_le t p\n\ndef bex_le [has_preceq (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∼∀.{≼ t}∼p\n\nnotation `∃.{≼ ` t `} `:64 p := bex_le t p\n\ndef bfal_lt [has_prec (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∀.((#0 ≺ t^1) ⟶ p)\n\nnotation `∀.{≺ ` t `} `:64 p := bfal_lt t p\n\ndef bex_lt [has_prec (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∼∀.{≺ t}∼p\n\nnotation `∃.{≺ ` t `} `:64 p := bex_lt t p\n\ndef bfal_mem [has_elem (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∀.((#0 ∊ t^1) ⟶ p)\n\nnotation `∀.{∊ ` t `} `:64 p := bfal_mem t p\n\ndef bex_mem [has_elem (term L) (formula L)] (t : term L) (p : formula L) : formula L :=\n∃.((#0 ∊ t^1) ⊓ p)\n\nnotation `∃.{∊ ` t `} `:64 p := bex_mem t p\n\n@[simp] lemma bfal_le_rew [has_le_symbol L] (t : term L) (p : formula L) (s) :\n  (∀.{≼ t} p).rew s = ∀.{≼ t.rew s} p.rew (s^1) :=\nby simp[bfal_le, term.pow_rew_distrib]\n\n@[simp] lemma bex_le_rew [has_le_symbol L] (t : term L) (p : formula L) (s) :\n  (∃.{≼ t} p).rew s = ∃.{≼ t.rew s} p.rew (s^1) :=\nby simp[bex_le, term.pow_rew_distrib]\n\n@[simp] def fal_rank : formula L → ℕ\n| (p ⟶ q) := max (fal_rank p) (fal_rank q)\n| (∼p)     := fal_rank p\n| (∀.p)  := fal_rank p + 1\n| _        := 0\n\n@[simp] lemma fal_rank_and (p q : formula L) :\n  fal_rank (p ⊓ q) = max (fal_rank p) (fal_rank q) := rfl\n\n@[simp] lemma fal_rank_or (p q : formula L) :\n  fal_rank (p ⊔ q) = max (fal_rank p) (fal_rank q) := rfl\n\n@[simp] lemma fal_rank_ex (p : formula L) :\n  fal_rank (∃.p) = fal_rank p + 1 := rfl\n\n@[simp] lemma fal_rank_eq (t u : term L) :\n  fal_rank (t =' u : formula L) = 0 := rfl\n\n@[simp] lemma fal_rank_le [has_le_symbol L] (t u : term L) :\n  fal_rank (t ≼ u : formula L) = 0 := rfl\n\n@[simp] lemma fal_rank_mem [has_mem_symbol L] (t u : term L) :\n  fal_rank (t ∊ u : formula L) = 0 := rfl\n\n@[simp] def binary_inv : formula L → option (formula L × formula L)\n| (p ⟶ q) := some (p, q)\n| _        := none\n\n@[simp] def unary_inv : formula L → option (formula L)\n| ∼p := some p\n| ∀.p := some p\n| _  := none\n\n@[simp] def quantifier_fn_aux : ℕ → (term L → formula L) → formula L → formula L\n| s f ⊤        := ⊤\n| s f (p ⟶ q) := quantifier_fn_aux s (λ t, (f t).binary_inv.iget.1) p ⟶ quantifier_fn_aux s (λ t, (f t).binary_inv.iget.2) q\n| s f ∼p       := ∼quantifier_fn_aux s (λ t, (f t).unary_inv.iget) p\n| s f (∀.p)  := ∀.quantifier_fn_aux (s + 1) (λ t, (f t).unary_inv.iget) p\n| s f _        := f #s\n\n@[simp] lemma quantifier_fn_aux_imply (s) (f g : term L → formula L) (p q : formula L) :\n  quantifier_fn_aux s (λ x, f x ⟶ g x) (p ⟶ q) = quantifier_fn_aux s f p ⟶ quantifier_fn_aux s g q := rfl\n\n@[simp] lemma quantifier_fn_aux_neg (s) (f : term L → formula L) (p : formula L) :\n  quantifier_fn_aux s (λ x, ∼f x) (∼p) = ∼quantifier_fn_aux s f p := rfl\n\n@[simp] lemma quantifier_fn_aux_fal (s) (f : term L → formula L) (p : formula L) :\n  quantifier_fn_aux s (λ x, ∀.(f x)) (∀.p) = ∀.quantifier_fn_aux (s + 1) f p := rfl\n\n@[simp] lemma quantifier_fn_aux_and (s) (f g : term L → formula L) (p q : formula L) :\n  quantifier_fn_aux s (λ x, f x ⊓ g x) (p ⊓ q) = quantifier_fn_aux s f p ⊓ quantifier_fn_aux s g q := rfl\n\n@[simp] lemma quantifier_fn_aux_or (s) (f g : term L → formula L) (p q : formula L) :\n  quantifier_fn_aux s (λ x, f x ⊔ g x) (p ⊔ q) = quantifier_fn_aux s f p ⊔ quantifier_fn_aux s g q := rfl\n\n@[simp] lemma quantifier_fn_aux_iff (s) (f g : term L → formula L) (p q : formula L) :\n  quantifier_fn_aux s (λ x, f x ⟷ g x) (p ⟷ q) = quantifier_fn_aux s f p ⟷ quantifier_fn_aux s g q :=\nby simp[lrarrow_def]\n\n@[simp] lemma quantifier_fn_aux_ex (s) (f : term L → formula L) (p : formula L) :\n  quantifier_fn_aux s (λ x, ∃.(f x)) (∃.p) = ∃.quantifier_fn_aux (s + 1) f p := rfl\n\n@[simp] lemma quantifier_fn_aux_eq (s) (f g : term L → term L) (t u : term L) :\n  quantifier_fn_aux s (λ x, f x =' g x) (t =' u) = (f #s =' g #s) := rfl\n\n@[simp] lemma quantifier_fn_aux_le [has_le_symbol L] (s) (f g : term L → term L) (t u : term L) :\n  quantifier_fn_aux s (λ x, f x ≼ g x) (t ≼ u) = (f #s ≼ g #s) := rfl\n\n@[simp] lemma quantifier_fn_aux_mem [has_mem_symbol L] (s) (f g : term L → term L) (t u : term L) :\n  quantifier_fn_aux s (λ x, f x ∊ g x) (t ∊ u) = (f #s ∊ g #s) := rfl\n\n@[simp] lemma fal_fn_constant (s) (p : formula L) :\n  quantifier_fn_aux s (λ x, p) p = p :=\nby induction p generalizing s; simp*\n\nend formula\n\ndef fal_fn (p : term L → formula L) : formula L := ∀.formula.quantifier_fn_aux 0 p (p #0)\n\ndef ex_fn (p : term L → formula L) : formula L := ∃.formula.quantifier_fn_aux 0 p (p #0)\n\nnotation `∀₁` binders `, ` r:(scoped p, fal_fn p) := r\n\nnotation `∃₁` binders `, ` r:(scoped p, ex_fn p) := r\n\ndef bfal_le_fn [has_preceq (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∀₁ x, ((x ≼ t) ⟶ p x)\n\nnotation `∀₁` binders ` ≼ᵇ ` t `, ` r:(scoped p, bfal_le_fn t p) := r\n\ndef bex_le_fn [has_preceq (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∃₁ x, ((x ≼ t) ⊓ p x)\n\nnotation `∃₁` binders ` ≼ᵇ ` t `, ` r:(scoped p, bex_le_fn t p) := r\n\ndef bfal_lt_fn [has_prec (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∀₁ x, ((x ≺ t) ⟶ p x)\n\nnotation `∀₁` binders ` ≺ᵇ ` t `, ` r:(scoped p, bfal_le_fn t p) := r\n\ndef bex_lt_fn [has_prec (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∃₁ x, ((x ≺ t) ⊓ p x)\n\nnotation `∃₁` binders ` ≺ᵇ ` t `, ` r:(scoped p, bex_le_fn t p) := r\n\ndef bfal_mem_fn [has_elem (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∀₁ x, (x ∊ t) ⟶ p x\n\nnotation `∀₁` binders ` ∊ᵇ ` t `, ` r:(scoped p, bfal_mem_fn t p) := r\n\ndef bex_mem_fn [has_elem (term L) (formula L)] (t : term L) (p : term L → formula L) : formula L :=\n∃₁ x, (x ∊ t) ⊓ p x\n\nnotation `∃₁` binders ` ∊ᵇ ` t `, ` r:(scoped p, bex_mem_fn t p) := r\n\n#check ∀₁ x ≺ᵇ #4, x =' x\n\n#check ∃.{≼ #3} #4 =' #9\n\nvariables [has_le_symbol L]\n\nnamespace formula\n\nsection\nvariables {L₁ L₂ : language.{u}} [L₁.language_translation_coe L₂]\n\nend\n\ninductive bounded : Theory L\n| verum : bounded ⊤\n| predicate {n} {p : L.pr n} {v} : bounded (❴p❵ v)\n| equal {t u : term L} : bounded (t =' u)\n| imply {p q} : bounded p → bounded q → bounded (p ⟶ q)\n| neg {p} : bounded p → bounded (∼p)\n| bfal {t} {p} : bounded p → bounded ∀.{≼ t} p\n\nattribute [simp] bounded.verum bounded.predicate bounded.equal bounded.neg\n\n@[simp] lemma bounded_preceq (t u : term L) : bounded (t ≼ u : formula L) :=\nbounded.predicate\n\n@[simp] lemma bounded_imply_iff (p q : formula L) : bounded (p ⟶ q) ↔ bounded p ∧ bounded q :=\n⟨λ h, by { cases h, simp* }, λ ⟨hp, hq⟩, bounded.imply hp hq⟩\n\n@[simp] lemma bounded_neg_iff (p : formula L) : bounded (∼p) ↔ bounded p :=\n⟨λ h, by { cases h, simp* }, bounded.neg⟩\n\nlemma bounded_of_open {p : formula L} (h : p.is_open) : bounded p :=\nbegin\n  induction p,\n  case verum { simp },\n  case app { simp },\n  case equal { simp },\n  case imply : p q IHp IHq { simp at h, exact bounded.imply (IHp h.1) (IHq h.2) },\n  case neg : p IH { simp at h, exact bounded.neg (IH h) },\n  case fal { simp at h, contradiction }\nend\n\nlemma bounded_of_lt {p q : formula L} (hq : bounded q) (h : p < q) : bounded p :=\nbegin\n  induction q; try { simp at h, contradiction },\n  case imply : q₁ q₂ IH₁ IH₂\n  { simp at h hq, rcases h with (le | le),\n    { rcases lt_or_eq_of_le le with (lt | rfl), { exact IH₁ hq.1 lt }, { exact hq.1 } },\n    { rcases lt_or_eq_of_le le with (lt | rfl), { exact IH₂ hq.2 lt }, { exact hq.2 } } },\n  case neg : q IH\n  { simp at h hq, rcases lt_or_eq_of_le h with (lt | rfl), { refine IH hq lt }, { exact hq } },\n  case fal : p IH\n  { rcases hq with ⟨lo, loo⟩, simp at h,\n    rcases lt_or_eq_of_le h with (lt | rfl), { refine IH (by { simp, exact hq_ᾰ}) lt }, { simp, exact hq_ᾰ } }\nend\n\n@[simp] lemma bounded_bfal_iff {p : formula L} {t : term L} : bounded (∀.{≼ t} p) ↔ bounded p :=\n⟨λ h, bounded_of_lt h (by { simp[bfal_le], refine le_of_lt (by simp) }), bounded.bfal⟩\n\nlemma bounded_bex_iff {p : formula L} {t : term L} : bounded (∃.{≼ t} p) ↔ bounded p :=\nby simp[bex_le]\n\nlemma bounded_rew (s : ℕ → term L) (p : formula L) (h : bounded p) : bounded (rew s p) :=\nby induction h generalizing s; simp*\n\ninstance bounded_proper : proper_Theory (bounded : Theory L) :=\n⟨λ p s mem, by { simp[set.mem_def] at mem ⊢, induction mem generalizing s; try {simp*} }⟩\n\nmutual inductive is_sigma, is_pi\nwith is_sigma : ℕ → formula L → Prop\n| zero : ∀ {p : formula L}, p.bounded → is_sigma 0 p\n| succ : ∀ {p} {n}, is_pi n p → is_sigma (n+1) ∃.p \nwith is_pi : ℕ → formula L → Prop\n| zero : ∀ {p : formula L}, p.bounded → is_pi 0 p\n| succ : ∀ {p} {n}, is_sigma n p → is_pi (n+1) ∀.p\n\nnotation `𝛴₀` := is_sigma 0\n\nnotation `𝛴₁` := is_sigma 1\n\nnotation `𝛱₀` := is_pi 0\n\nnotation `𝛱₁` := is_pi 1\n\n@[simp] lemma sigma_0_iff_bounded : (𝛴₀ : Theory L) = bounded :=\nby funext p; simp; exact ⟨λ h, by { cases h, simp* }, is_sigma.zero⟩\n\n@[simp] lemma pi_0_iff_bounded : (𝛱₀ : Theory L) = bounded :=\nby funext p; simp; exact ⟨λ h, by { cases h, simp* }, is_pi.zero⟩\n\nprivate lemma sigma_pi_proper (n : ℕ) : proper_at 0 (is_sigma n : Theory L) ∧ proper_at 0 (is_pi n : Theory L) :=\nbegin\n  induction n with n IH,\n  { simp, refine proper_Theory.proper },\n  { refine ⟨λ p s hs, _, λ p s hp, _⟩,\n    { cases hs with _ _ p _ hp, simp,\n      have : is_pi n (p.rew (s^1)),\n      { have := IH.2 p (s^1) hp, simp at this, exact this },\n      refine is_sigma.succ this },\n    { cases hp with _ _ p _ hs, simp,\n      have : is_sigma n (p.rew (s^1)),\n      { have := IH.1 p (s^1) hs, simp at this, exact this },\n      refine is_pi.succ this } }\nend\n\ninstance is_sigma_proper (n) : proper_Theory (is_sigma n : Theory L) := ⟨(sigma_pi_proper n).1⟩\n\ninstance is_pi_proper (n) : proper_Theory (is_pi n : Theory L) := ⟨(sigma_pi_proper n).2⟩\n\nend formula\n\ndef arithmetical_sigma (T : Theory L) (n : ℕ) : Theory L :=\nλ p, ∃ q, T ⊢ p ⟷ q ∧ q.is_sigma n\n\nnotation `𝜮`:60 n ` in ` T :60 := arithmetical_sigma T n\n\ndef arithmetical_pi (T : Theory L) (n : ℕ) : Theory L :=\nλ p, ∃ q, T ⊢ p ⟷ q ∧ q.is_pi n\n\nnotation `𝜫`:60 n ` in ` T :60 := arithmetical_pi T n\n\ndef arithmetical_delta (T : Theory L) (n : ℕ) : Theory L :=\nλ p, p ∈ 𝜮n in T ∧ p ∈ 𝜫n in T\n\nnotation `𝜟`:60 n ` in ` T :60 := arithmetical_delta T n\n\n\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/class_of_formulae.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3082365543113072}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.WHNF\nimport Lean.Meta.Transform\nimport Lean.Meta.SynthInstance\nimport Lean.Meta.AppBuilder\n\nnamespace Lean.Meta\n\nbuiltin_initialize coeDeclAttr : TagAttribute ←\n  registerTagAttribute `coe_decl \"auxiliary definition used to implement coercion (unfolded during elaboration)\"\n\n/--\n  Return true iff `declName` is one of the auxiliary definitions/projections\n  used to implement coercions.\n-/\ndef isCoeDecl (env : Environment) (declName : Name) : Bool :=\n  coeDeclAttr.hasTag env declName\n\n/-- Expand coercions occurring in `e` -/\npartial def expandCoe (e : Expr) : MetaM Expr :=\n  withReducibleAndInstances do\n    transform e fun e => do\n      let f := e.getAppFn\n      if f.isConst then\n        let declName := f.constName!\n        if isCoeDecl (← getEnv) declName then\n          if let some e ← unfoldDefinition? e then\n            return .visit e.headBeta\n      return .continue\n\nregister_builtin_option autoLift : Bool := {\n  defValue := true\n  descr    := \"insert monadic lifts (i.e., `liftM` and coercions) when needed\"\n}\n\n/-- Coerces `expr` to `expectedType` using `CoeT`. -/\ndef coerceSimple? (expr expectedType : Expr) : MetaM (LOption Expr) := do\n  let eType ← inferType expr\n  let u ← getLevel eType\n  let v ← getLevel expectedType\n  let coeTInstType := mkAppN (mkConst ``CoeT [u, v]) #[eType, expr, expectedType]\n  match ← trySynthInstance coeTInstType with\n  | .some inst =>\n    let result ← expandCoe (mkAppN (mkConst ``CoeT.coe [u, v]) #[eType, expr, expectedType, inst])\n    unless ← isDefEq (← inferType result) expectedType do\n      throwError \"could not coerce{indentExpr expr}\\nto{indentExpr expectedType}\\ncoerced expression has wrong type:{indentExpr result}\"\n    return .some result\n  | .undef => return .undef\n  | .none => return .none\n\n/-- Coerces `expr` to a function type. -/\ndef coerceToFunction? (expr : Expr) : MetaM (Option Expr) := do\n  -- constructing expression manually because mkAppM wouldn't assign universe mvars\n  let α ← inferType expr\n  let u ← getLevel α\n  let v ← mkFreshLevelMVar\n  let γ ← mkFreshExprMVar (← mkArrow α (mkSort v))\n  let .some inst ← trySynthInstance (mkApp2 (.const ``CoeFun [u,v]) α γ) | return none\n  let expanded ← expandCoe (mkApp4 (.const ``CoeFun.coe [u,v]) α γ inst expr)\n  unless (← whnf (← inferType expanded)).isForall do\n    throwError \"failed to coerce{indentExpr expr}\\nto a function, after applying `CoeFun.coe`, result is still not a function{indentExpr expanded}\\nthis is often due to incorrect `CoeFun` instances, the synthesized instance was{indentExpr inst}\"\n  return expanded\n\n/-- Coerces `expr` to a type. -/\ndef coerceToSort? (expr : Expr) : MetaM (Option Expr) := do\n  -- constructing expression manually because mkAppM wouldn't assign universe mvars\n  let α ← inferType expr\n  let u ← getLevel α\n  let v ← mkFreshLevelMVar\n  let β ← mkFreshExprMVar (mkSort v)\n  let .some inst ← trySynthInstance (mkApp2 (.const ``CoeSort [u,v]) α β) | return none\n  let expanded ← expandCoe (mkApp4 (.const ``CoeSort.coe [u,v]) α β inst expr)\n  unless (← whnf (← inferType expanded)).isSort do\n    throwError \"failed to coerce{indentExpr expr}\\nto a type, after applying `CoeSort.coe`, result is still not a type{indentExpr expanded}\\nthis is often due to incorrect `CoeSort` instances, the synthesized instance was{indentExpr inst}\"\n  return expanded\n\n/-- Return `some (m, α)` if `type` can be reduced to an application of the form `m α` using `[reducible]` transparency. -/\ndef isTypeApp? (type : Expr) : MetaM (Option (Expr × Expr)) := do\n  let type ← withReducible <| whnf type\n  match type with\n  | .app m α => return some ((← instantiateMVars m), (← instantiateMVars α))\n  | _        => return none\n\n/--\nReturn `true` if `type` is of the form `m α` where `m` is a `Monad`.\nNote that we reduce `type` using transparency `[reducible]`.\n-/\ndef isMonadApp (type : Expr) : MetaM Bool := do\n  let some (m, _) ← isTypeApp? type | return false\n  return (← isMonad? m).isSome\n\n/--\nTry coercions and monad lifts to make sure `e` has type `expectedType`.\n\nIf `expectedType` is of the form `n β`, we try monad lifts and other extensions.\n\nExtensions for monads.\n\n1. Try to unify `n` and `m`. If it succeeds, then we use\n  ```\n  coeM {m : Type u → Type v} {α β : Type u} [∀ a, CoeT α a β] [Monad m] (x : m α) : m β\n  ```\n  `n` must be a `Monad` to use this one.\n\n2. If there is monad lift from `m` to `n` and we can unify `α` and `β`, we use\n  ```\n  liftM : ∀ {m : Type u_1 → Type u_2} {n : Type u_1 → Type u_3} [self : MonadLiftT m n] {α : Type u_1}, m α → n α\n  ```\n  Note that `n` may not be a `Monad` in this case. This happens quite a bit in code such as\n  ```\n  def g (x : Nat) : IO Nat := do\n    IO.println x\n    pure x\n\n  def f {m} [MonadLiftT IO m] : m Nat :=\n    g 10\n\n  ```\n\n3. If there is a monad lift from `m` to `n` and a coercion from `α` to `β`, we use\n  ```\n  liftCoeM {m : Type u → Type v} {n : Type u → Type w} {α β : Type u} [MonadLiftT m n] [∀ a, CoeT α a β] [Monad n] (x : m α) : n β\n  ```\n\nNote that approach 3 does not subsume 1 because it is only applicable if there is a coercion from `α` to `β` for all values in `α`.\nThis is not the case for example for `pure $ x > 0` when the expected type is `IO Bool`. The given type is `IO Prop`, and\nwe only have a coercion from decidable propositions.  Approach 1 works because it constructs the coercion `CoeT (m Prop) (pure $ x > 0) (m Bool)`\nusing the instance `pureCoeDepProp`.\n\nNote that, approach 2 is more powerful than `tryCoe`.\nRecall that type class resolution never assigns metavariables created by other modules.\nNow, consider the following scenario\n```lean\ndef g (x : Nat) : IO Nat := ...\ndeg h (x : Nat) : StateT Nat IO Nat := do\nv ← g x;\nIO.Println v;\n...\n```\nLet's assume there is no other occurrence of `v` in `h`.\nThus, we have that the expected of `g x` is `StateT Nat IO ?α`,\nand the given type is `IO Nat`. So, even if we add a coercion.\n```\ninstance {α m n} [MonadLiftT m n] {α} : Coe (m α) (n α) := ...\n```\nIt is not applicable because TC would have to assign `?α := Nat`.\nOn the other hand, TC can easily solve `[MonadLiftT IO (StateT Nat IO)]`\nsince this goal does not contain any metavariables. And then, we\nconvert `g x` into `liftM $ g x`.\n-/\ndef coerceMonadLift? (e expectedType : Expr) : MetaM (Option Expr) := do\n  let expectedType ← instantiateMVars expectedType\n  let eType ← instantiateMVars (← inferType e)\n  let some (n, β) ← isTypeApp? expectedType | return none\n  let some (m, α) ← isTypeApp? eType | return none\n  if (← isDefEq m n) then\n    let some monadInst ← isMonad? n | return none\n    try expandCoe (← mkAppOptM ``Lean.Internal.coeM #[m, α, β, none, monadInst, e]) catch _ => return none\n  else if autoLift.get (← getOptions) then\n    try\n      -- Construct lift from `m` to `n`\n      -- Note: we cannot use mkAppM here because mkAppM does not assign universe metavariables,\n      -- but we need to make sure that the domains of `m` and `n` have the same level.\n      let .forallE _ (.sort um₁) (.sort um₂) _ ← whnf (← inferType m) | return none\n      let .forallE _ (.sort un₁) (.sort un₂) _ ← whnf (← inferType n) | return none\n      let u ← decLevel um₁\n      let .true ← isLevelDefEq u (← decLevel un₁) | return none\n      let v ← decLevel um₂\n      let w ← decLevel un₂\n      let monadLiftType := mkAppN (.const ``MonadLiftT [u, v, w]) #[m, n]\n      let .some monadLiftVal ← trySynthInstance monadLiftType | return none\n      let u_1 ← getDecLevel α\n      let u_2 ← getDecLevel eType\n      let u_3 ← getDecLevel expectedType\n      let eNew := mkAppN (Lean.mkConst ``liftM [u_1, u_2, u_3]) #[m, n, monadLiftVal, α, e]\n      let eNewType ← inferType eNew\n      if (← isDefEq expectedType eNewType) then\n        return some eNew -- approach 2 worked\n      else\n        let some monadInst ← isMonad? n | return none\n        let u ← getLevel α\n        let v ← getLevel β\n        let coeTInstType := Lean.mkForall `a BinderInfo.default α <| mkAppN (mkConst ``CoeT [u, v]) #[α, mkBVar 0, β]\n        let .some coeTInstVal ← trySynthInstance coeTInstType | return none\n        let eNew ← expandCoe (mkAppN (Lean.mkConst ``Lean.Internal.liftCoeM [u_1, u_2, u_3]) #[m, n, α, β, monadLiftVal, coeTInstVal, monadInst, e])\n        let eNewType ← inferType eNew\n        unless (← isDefEq expectedType eNewType) do return none\n        return some eNew -- approach 3 worked\n    catch _ =>\n      /- If `m` is not a monad, then we try to use `tryCoe?`. -/\n      return none\n  else\n    return none\n\n/-- Coerces `expr` to the type `expectedType`.\nReturns `.some coerced` on successful coercion,\n`.none` if the expression cannot by coerced to that type,\nor `.undef` if we need more metavariable assignments. -/\ndef coerce? (expr expectedType : Expr) : MetaM (LOption Expr) := do\n  if let some lifted ← coerceMonadLift? expr expectedType then\n    return .some lifted\n  if (← whnfR expectedType).isForall then\n    if let some fn ← coerceToFunction? expr then\n      if ← isDefEq (← inferType fn) expectedType then\n        return .some fn\n  coerceSimple? expr expectedType\n\nend Lean.Meta\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Lean/Meta/Coe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3082365543113072}}
{"text": "\nimport Lib.Data.MonoFunctor\nimport Lib.Data.MonoFoldable\nimport Lib.Data.MonoTraversable\nimport Lib.Data.List.Instances\n\nnamespace String\n\n@[implementedBy map]\ndef map' (f : Char → Char) : String → String\n| ⟨ data ⟩ => ⟨ Functor.map f data ⟩\n\ninstance : MonoFunctor String Char where\n  map := map'\n\ninstance : LawfulMonoFunctor String Char where\n  id_map := by\n    intros; next x =>\n    cases x; simp [MonoFunctor.map, map']\n  comp_map := by\n    intros; next x =>\n    cases x; simp [MonoFunctor.map, map']\n\nvariable {β : Type}\n\n@[inline]\ndef foldlImpl (f : β → Char → β) (x₀ : β) : String → β :=\nString.foldl f x₀\n\n@[implementedBy foldlImpl]\ndef foldl' (f : β → Char → β) (x₀ : β) : String → β\n| ⟨ data ⟩ => Foldable.foldl f x₀ data\n\n@[inline]\ndef foldrImpl (f : Char → β → β) (x₀ : β) : String → β :=\nString.foldr f x₀\n\n@[implementedBy foldrImpl]\ndef foldr' (f : Char → β → β) (x₀ : β) : String → β\n| ⟨ data ⟩ => Foldable.foldr f x₀ data\n\ndef toArray (s : String) : Array Char :=\nString.foldl' Array.push (Array.mkEmpty s.length) s\n\ninstance : MonoFoldable String Char where\n  foldl := String.foldl'\n  foldr := String.foldr'\n  toList := String.toList\n  toArray := String.toArray\n  length := String.length\n\nopen MonoFoldable\n\ninstance : LawfulMonoFoldable String Char where\n  foldl_sim :=\n    by intros _ _ _ f SIM x₀ y₀ t a ih\n       cases t; simp [foldl, MonoFoldable.foldl]\n       auto [LawfulFoldable.foldl_sim]\n  foldr_eq_foldMap :=\n    by intros _ _ t _\n       cases t; simp [foldr, MonoFoldable.foldr]\n       auto [LawfulFoldable.foldr_eq_foldMap]\n  toArray_toList :=\n    by intros t\n       cases t with | mk data =>\n       simp [toList, toArray, String.foldl']\n       simp [String.toList,\n             String.toArray, String.foldl']\n       show Foldable.toArray data = _\n       rw [LawfulFoldable.toArray_eq]\n       refl\n  length_toList := by\n    intros t; cases t; refl\n  foldl_toList := by\n    intros β f x₀ t; cases t; refl\n\nsection mapM\n\nvariable {M} [Monad M]\n@[specialize] partial def mapMAux (f : Char → M Char) (i : Pos) (s : String) : M String :=\n  if s.atEnd i then return s\n  else do\n    let c ← f (s.get i)\n    let s := s.set i c\n    mapMAux f (s.next i) s\n\n@[inline] def mapMImpl (f : Char → M Char) (s : String) : M String :=\n  mapMAux f 0 s\n\n@[implementedBy mapMImpl] def mapM (f : Char → M Char) (s : String) : M String :=\n  String.mk <$> s.data.mapM f\n\nend mapM\n\n\nsection mapM\n\nvariable {F} [Applicative F]\n\ndef mapA (f : Char → F Char) (s : String) : F String :=\nString.mk <$> Traversable.traverse f s.data\n\nend mapM\n\ninstance : MonoTraversable String Char where\n  traverse := mapA\n  mapM := mapM\n\nopen LawfulMonoTraversable\ninstance : LawfulMonoTraversable String Char where\n  map_eq_traverse := by\n    intros; next x => cases x with | mk data =>\n    simp [MonoTraversable.traverse, MonoFunctor.map, map']\n    simp [LawfulTraversable.map_eq_traverse]; refl\n  foldl_eq_traverse := by\n    intros; next x y => cases x with | mk data =>\n    simp [MonoFoldable.foldl, foldl']\n    simp [LawfulTraversable.foldl_eq_traverse]; refl\n  traverse_eq_mapM := by\n    intros; next x => cases x with | mk data =>\n    simp [MonoTraversable.traverse, mapA]\n    simp [LawfulTraversable.traverse_eq_mapM]; refl\n  comp_traverse := by\n    intros; next x y z => cases x with | mk data =>\n    simp [MonoTraversable.traverse, mapA]\n    simp [LawfulTraversable.comp_traverse]; refl\n  traverse_sim := by\n    intros; next x SIM _ _ _ => cases x with | mk data =>\n    simp [MonoTraversable.traverse, mapA]\n    auto [LawfulTraversable.traverse_sim, ApplicativeRel.naturality]\n\ntheorem toList_inj {s₀ s₁ : String} :\n  s₀.toList = s₁.toList → s₀ = s₁ :=\nby intro h; cases s₀; cases s₁; cases h; rfl\n\nattribute [auto] one_le_csize\n\n@[simp]\ntheorem toList_mk {x : List Char} :\n  String.toList ⟨x⟩ = x := rfl\n\n@[simp]\ntheorem toList_append {s₀ s₁ : String} :\n  (s₀ ++ s₁).toList = s₀.toList ++ s₁.toList :=\nrfl\n\n@[simp]\ntheorem length_toList {s : String} :\n  s.toList.length = s.length := rfl\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/String/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.30807914889316823}}
{"text": "import data.cpi.concretion.congruence\n\nnamespace cpi\nnamespace concretion\n\nvariables {ℍ : Type} {ω : context}\n\n/-- An alternative measure, simply computes the depth of a term, and thus is\n    preserved over renames. -/\nprivate def depth : ∀ {Γ} {b y}, concretion ℍ ω Γ b y → ℕ\n| _ _ _ (#(_; _) _) := 1\n| _ _ _ (F |₁ _) := depth F + 1\n| _ _ _ (_ |₂ F) := depth F + 1\n| _ _ _ (ν'(M) F) := depth F + 1\n\nprivate theorem depth.over_rename :\n  ∀ {Γ Δ} {b y} (ρ : name Γ → name Δ) (F : concretion ℍ ω Γ b y)\n  , depth F = depth (rename ρ F)\n| Γ Δ b y ρ (#(_; _) _) := by unfold rename depth\n| Γ Δ b y ρ (F |₁ A) := by { unfold rename depth, rw depth.over_rename ρ F }\n| Γ Δ b y ρ (A |₂ F) := by { unfold rename depth, rw depth.over_rename ρ F }\n| Γ Δ b y ρ (ν'(M) F) :=\n  by { unfold rename depth, rw depth.over_rename (name.ext ρ) F }\n\n/-- Helper function for doign the actual application. This is split up to\n    make the totality of pseudo_apply/pseudo_apply_app easier to determine. -/\nprivate def pseudo_apply_app {a b} :\n  ∀ {Γ}, vector (name Γ) a → species ℍ ω (context.extend b Γ)\n  → concretion ℍ ω Γ b a → species ℍ ω Γ\n| Γ as A (#(bs; y) B) :=\n  species.rename (name.mk_apply bs) A |ₛ species.rename (name.mk_apply as) B\n| Γ as A (F |₁ B) :=\n    pseudo_apply_app as A F |ₛ B\n| Γ as A (B |₂ F) :=\n    B |ₛ pseudo_apply_app as A F\n| Γ as A (ν'(M) F) :=\n  ν(M) (pseudo_apply_app\n          (vector.map name.extend as)\n          (species.rename (name.ext name.extend) A)\n          F)\nusing_well_founded {\n  rel_tac := λ _ _,\n    `[exact ⟨_, measure_wf (λ x, concretion.sizeof ℍ ω x.fst b a x.snd.snd.snd ) ⟩ ],\n  dec_tac := tactic.fst_dec_tac,\n}\n\n/-- Apply two concretions together. -/\ndef pseudo_apply {a b} : ∀ {Γ}, concretion ℍ ω Γ a b → concretion ℍ ω Γ b a → species ℍ ω Γ\n| Γ (#(bs; y) A) F' := pseudo_apply_app bs A F'\n| Γ (F |₁ A) F' := pseudo_apply F F' |ₛ A\n| Γ (A |₂ F) F' := A |ₛ pseudo_apply F F'\n| Γ (ν'(M) F) F' := ν(M) (pseudo_apply F (rename name.extend F'))\nusing_well_founded {\n  rel_tac := λ _ _,\n    `[exact ⟨_, measure_wf (λ x, concretion.sizeof ℍ ω x.fst a b x.snd.fst ) ⟩ ],\n  dec_tac := tactic.fst_dec_tac,\n}\n\nopen species.equiv (hiding trans symm refl)\nopen_locale congruence\n\nprotected lemma pseudo_apply.on_parallel₁ :\n  ∀ {Γ} {a b} (F : concretion ℍ ω Γ a b) (G : concretion ℍ ω Γ b a) (A : species ℍ ω Γ)\n  , pseudo_apply F (G |₁ A) ≈ (pseudo_apply F G |ₛ A)\n| Γ a b (#(bs; y) A) G B := by unfold pseudo_apply pseudo_apply_app\n| Γ a b (F |₁ B) G A := begin\n    unfold pseudo_apply,\n    calc  (pseudo_apply F (G |₁ A) |ₛ B)\n        ≈ ((pseudo_apply F G |ₛ A) |ₛ B)\n          : ξ_parallel₁ (pseudo_apply.on_parallel₁ F G A)\n    ... ≈ ((pseudo_apply F G |ₛ B) |ₛ A) : parallel_symm₂\n  end\n| Γ a b (B |₂ F) G A := begin\n    unfold pseudo_apply,\n    calc  (B |ₛ pseudo_apply F (G |₁ A))\n        ≈ (B |ₛ pseudo_apply F G |ₛ A)\n          : ξ_parallel₂ (pseudo_apply.on_parallel₁ F G A)\n    ... ≈ ((B |ₛ pseudo_apply F G) |ₛ A) : parallel_assoc₂\n  end\n| Γ a b (ν'(M) F) G A := begin\n    unfold pseudo_apply rename,\n    calc  (ν(M) pseudo_apply F (rename name.extend G |₁ species.rename name.extend A))\n        ≈ (ν(M) pseudo_apply F (rename name.extend G) |ₛ species.rename name.extend A)\n          : ξ_restriction M (pseudo_apply.on_parallel₁ F _ _)\n    ... ≈ ((ν(M) pseudo_apply F (rename name.extend G)) |ₛ A) : ν_parallel' M\n  end\n\nprotected lemma pseudo_apply.on_parallel₂ :\n  ∀ {Γ} {a b} (F : concretion ℍ ω Γ a b) (A : species ℍ ω Γ) (G : concretion ℍ ω Γ b a)\n  , pseudo_apply F (A |₂ G) ≈ (A |ₛ pseudo_apply F G)\n| Γ a b (#(bs; y) A) B F := by unfold pseudo_apply pseudo_apply_app\n| Γ a b (F |₁ B) A G := begin\n    unfold pseudo_apply,\n    calc  (pseudo_apply F (A |₂ G) |ₛ B)\n        ≈ ((A |ₛ pseudo_apply F G) |ₛ B)\n          : ξ_parallel₁ (pseudo_apply.on_parallel₂ F A G)\n    ... ≈ (A |ₛ pseudo_apply F G |ₛ B) : parallel_assoc₁\n  end\n| Γ a b (B |₂ F) A G := begin\n    unfold pseudo_apply,\n    calc  (B |ₛ pseudo_apply F (A |₂ G))\n        ≈ (B |ₛ A |ₛ pseudo_apply F G)\n          : ξ_parallel₂ (pseudo_apply.on_parallel₂ F A G)\n    ... ≈ (A |ₛ B |ₛ pseudo_apply F G) : parallel_symm₁\n  end\n| Γ a b (ν'(M) F) A G := begin\n    unfold pseudo_apply rename,\n    calc  (ν(M) pseudo_apply F (species.rename name.extend A |₂ rename name.extend G))\n        ≈ (ν(M) species.rename name.extend A |ₛ pseudo_apply F (rename name.extend G))\n          : ξ_restriction M (pseudo_apply.on_parallel₂ F _ _)\n    ... ≈ (A |ₛ ν(M) pseudo_apply F (rename name.extend G)) : ν_parallel₁ M\n  end\n\n-- TODO: Clean up to use calc\n-- TODO: Use induction - does this allow us to drop the explicit termination\n-- checks?\n\nprivate lemma pseudo_apply_app.rename {a b} :\n  ∀ {Γ Δ} (ρ : name Γ → name Δ)\n    (as : vector (name Γ) a) (A : species ℍ ω (context.extend b Γ))\n    (F : concretion ℍ ω Γ b a)\n  , species.rename ρ (pseudo_apply_app as A F)\n  = pseudo_apply_app (vector.map ρ as) (species.rename (name.ext ρ) A) (rename ρ F)\n| Γ Δ ρ as A (#(bs; y) B) := begin\n    unfold pseudo_apply_app rename,\n    simp [species.rename_compose, name.mk_apply_rename]\n  end\n| Γ Δ ρ bs A (F |₁ B) := begin\n    unfold pseudo_apply_app rename,\n    simp [pseudo_apply_app.rename ρ bs A F]\n  end\n| Γ Δ ρ bs A (B |₂ F) := begin\n    unfold pseudo_apply_app rename,\n    simp [pseudo_apply_app.rename ρ bs A F]\n  end\n| Γ Δ ρ ⟨ bs, n ⟩ A (ν'(M) G) := begin\n    unfold pseudo_apply_app rename,\n    simp,\n    have map\n      : vector.map (@name.extend _ M.arity) (vector.map ρ ⟨bs, n⟩)\n      = vector.map (name.ext ρ) (vector.map name.extend ⟨bs, n⟩),\n      unfold vector.map, simp, rw ← name.ext_extend ρ,\n    rw map,\n\n    have spc\n      : species.rename (name.ext (@name.extend _ M.arity)) (species.rename (name.ext ρ) A)\n      = species.rename (name.ext (name.ext ρ)) (species.rename (name.ext name.extend) A),\n      rw [species.rename_compose, species.rename_compose],\n      rw [name.ext_comp, name.ext_comp],\n      rw name.ext_extend ρ,\n    rw spc,\n\n    from pseudo_apply_app.rename (name.ext ρ) _ _ G,\n  end\nusing_well_founded {\n  rel_tac := λ _ _,\n    `[exact ⟨_, measure_wf (λ x, concretion.sizeof ℍ ω x.fst b a x.snd.snd.snd.snd.snd ) ⟩ ],\n  dec_tac := tactic.fst_dec_tac,\n}\n\nprotected lemma pseudo_apply.rename {a b} :\n  ∀ {Γ Δ} (ρ : name Γ → name Δ)\n    (F : concretion ℍ ω Γ a b) (G : concretion ℍ ω Γ b a)\n  , species.rename ρ (pseudo_apply F G) = pseudo_apply (rename ρ F) (rename ρ G)\n| Γ Δ ρ (#(bs; y) A) G := begin\n    unfold pseudo_apply rename,\n    from pseudo_apply_app.rename ρ bs A G\n  end\n| Γ Δ ρ (F |₁ A) G := begin\n    unfold pseudo_apply rename,\n    simp [pseudo_apply.rename ρ F G]\n  end\n| Γ Δ ρ (A |₂ F) G := begin\n    unfold pseudo_apply rename,\n    simp [pseudo_apply.rename ρ F G]\n  end\n| Γ Δ ρ (ν'(M) F) G := begin\n    unfold pseudo_apply rename, simp,\n    -- -- TODO: Clean up to use calc\n    rw rename_compose,\n    rw ← name.ext_extend,\n    rw ← rename_compose name.extend,\n    rw pseudo_apply.rename (name.ext ρ) F _,\n  end\nusing_well_founded {\n  rel_tac := λ _ _,\n    `[exact ⟨_, measure_wf (λ x, concretion.sizeof ℍ ω x.fst a b x.snd.snd.snd.fst ) ⟩ ],\n  dec_tac := tactic.fst_dec_tac,\n}\n\nprivate lemma pseudo_apply.restriction_swap {a b}:\n  ∀ {Γ} (M N : affinity ℍ)\n    (F : concretion ℍ ω (context.extend N.arity Γ) a b)\n    (G : concretion ℍ ω (context.extend M.arity Γ) b a)\n  , (ν(N) ν(M) pseudo_apply (rename name.extend F) (rename (name.ext name.extend) G))\n  ≈ ν(M) ν(N) pseudo_apply (rename (name.ext name.extend) F) (rename name.extend G)\n| Γ M N F G :=\n    calc  (ν(N) ν(M) pseudo_apply (rename name.extend F) (rename (name.ext name.extend) G))\n        ≈ (ν(M) ν(N) species.rename name.swap (pseudo_apply (rename name.extend F) (rename (name.ext name.extend) G)))\n          : ν_swap₁ N M\n    ... ≈ (ν(M) ν(N) (pseudo_apply (rename name.swap (rename name.extend F)) (rename name.swap (rename (name.ext name.extend) G))))\n          : by rw pseudo_apply.rename\n    ... ≈ (ν(M) ν(N) (pseudo_apply (rename (name.swap ∘ name.extend) F) (rename (name.swap ∘ name.ext name.extend) G)))\n          : by rw [rename_compose, rename_compose]\n    ... ≈ ν(M) ν(N) pseudo_apply (rename (name.ext name.extend) F) (rename name.extend G)\n          : by rw [name.swap_comp_extend, name.swap_comp_ext_extend]\n\nlemma pseudo_apply.on_restriction :\n  ∀ {Γ} {a b} (F : concretion ℍ ω Γ a b) (M : affinity ℍ)\n    (G : concretion ℍ ω (context.extend M.arity Γ) b a)\n  , pseudo_apply F (ν'(M) G) ≈ ν(M) (pseudo_apply (rename name.extend F) G)\n| Γ a b (#(bs; y) A) M G := by unfold pseudo_apply pseudo_apply_app rename\n| Γ a b (F |₁ A) M G := begin\n    unfold pseudo_apply rename,\n    calc  (pseudo_apply F (ν'(M) G) |ₛ A)\n        ≈ ((ν(M) pseudo_apply (rename name.extend F) G) |ₛ A)\n          : ξ_parallel₁ (pseudo_apply.on_restriction F M G)\n    ... ≈ ν(M) pseudo_apply (rename name.extend F) G |ₛ species.rename name.extend A\n          : symm (ν_parallel' M),\n  end\n| Γ a b (A |₂ F) M G := begin\n    unfold pseudo_apply rename,\n    calc  (A |ₛ pseudo_apply F (ν'(M) G))\n        ≈ (A |ₛ ν(M) pseudo_apply (rename name.extend F) G)\n          : ξ_parallel₂ (pseudo_apply.on_restriction F M G)\n    ... ≈ ν(M) species.rename name.extend A |ₛ pseudo_apply (rename name.extend F) G\n          : ν_parallel₂ M,\n  end\n| Γ a b (ν'(N) F) M G := begin\n    unfold pseudo_apply rename,\n    calc  (ν(N) pseudo_apply F (ν'(M) rename (name.ext name.extend) G))\n        ≈ (ν(N) ν(M) pseudo_apply (rename name.extend F) (rename (name.ext name.extend) G))\n          : ξ_restriction N (pseudo_apply.on_restriction F M _)\n    ... ≈ ν(M) ν(N) pseudo_apply (rename (name.ext name.extend) F) (rename name.extend G)\n          : pseudo_apply.restriction_swap M N F G\n  end\n\nprivate theorem pseudo_apply_app.symm {a b} :\n  ∀ {Γ} (bs : vector (name Γ) a) (A : species ℍ ω (context.extend b Γ))\n    (G : concretion ℍ ω Γ b a)\n  , pseudo_apply_app bs A G ≈ pseudo_apply G (#(bs; b) A) := begin\n  intros Γ as A G,\n  induction G,\n\n  case apply : Γ' b' bs y A {\n    unfold pseudo_apply pseudo_apply_app,\n    from parallel_symm\n  },\n\n  case parallel₁ : Γ' b' y F A ih {\n      unfold pseudo_apply pseudo_apply_app,\n      from ξ_parallel₁ (ih _ _)\n  },\n\n  case parallel₂ : Γ' b' y A F ih {\n      unfold pseudo_apply pseudo_apply_app,\n      from ξ_parallel₂ (ih _ _)\n  },\n\n  case restriction : Γ' b' y M F ih {\n    unfold pseudo_apply pseudo_apply_app,\n    from ξ_restriction M (ih _ _)\n  }\nend\n\ntheorem pseudo_apply.symm {a b} :\n  ∀ {Γ} (F : concretion ℍ ω Γ a b) (G : concretion ℍ ω Γ b a)\n  , pseudo_apply F G ≈ pseudo_apply G F\n| Γ (#(as; x) A) G := begin\n    unfold pseudo_apply,\n    from pseudo_apply_app.symm as A G\n  end\n\n| Γ (F |₁ A) (#(bs; y) B) := begin\n    unfold pseudo_apply pseudo_apply_app,\n    from ξ_parallel₁ (symm (pseudo_apply_app.symm bs B F)),\n  end\n| Γ (F |₁ A) (G |₁ B) := begin\n    unfold pseudo_apply,\n    calc  (pseudo_apply F (G |₁ B) |ₛ A)\n        ≈ ((pseudo_apply F G |ₛ B) |ₛ A)\n          : ξ_parallel₁ (pseudo_apply.on_parallel₁ F G B)\n    ... ≈ ((pseudo_apply G F |ₛ B) |ₛ A)\n          : ξ_parallel₁ (ξ_parallel₁ (pseudo_apply.symm F G))\n    ... ≈ ((pseudo_apply G F |ₛ A) |ₛ B) : parallel_symm₂\n    ... ≈ (pseudo_apply G (F |₁ A) |ₛ B)\n          : ξ_parallel₁ (symm (pseudo_apply.on_parallel₁ G F A))\n  end\n| Γ (F |₁ A) (B |₂ G) := begin\n    unfold pseudo_apply,\n    calc  (pseudo_apply F (B |₂ G) |ₛ A)\n        ≈ ((B |ₛ pseudo_apply F G) |ₛ A)\n          : ξ_parallel₁ (pseudo_apply.on_parallel₂ F B G)\n    ... ≈ (B |ₛ pseudo_apply F G |ₛ A) : parallel_assoc₁\n    ... ≈ (B |ₛ pseudo_apply G F |ₛ A)\n          : ξ_parallel₂ (ξ_parallel₁ (pseudo_apply.symm F G))\n    ... ≈ (B |ₛ pseudo_apply G (F |₁ A))\n          : ξ_parallel₂ (symm (pseudo_apply.on_parallel₁ G F A))\n  end\n| Γ (F |₁ A) (ν'(M) G) := begin\n    unfold pseudo_apply rename,\n    calc  (pseudo_apply F (ν'(M) G) |ₛ A)\n        ≈ ((ν(M) pseudo_apply (rename name.extend F) G) |ₛ A)\n          : ξ_parallel₁ (pseudo_apply.on_restriction F M G)\n    ... ≈ ((ν(M) pseudo_apply G (rename name.extend F)) |ₛ A)\n          : ξ_parallel₁ (ξ_restriction M (pseudo_apply.symm _ G))\n    ... ≈ (ν(M) (pseudo_apply G (rename name.extend F)) |ₛ species.rename name.extend A)\n          : symm (ν_parallel' M)\n    ... ≈ ν(M) pseudo_apply G (rename name.extend F |₁ species.rename name.extend A)\n          : ξ_restriction M (symm (pseudo_apply.on_parallel₁ G _ _))\n  end\n\n| Γ (A |₂ F) (#(bs; y) B) := begin\n    unfold pseudo_apply pseudo_apply_app,\n    from ξ_parallel₂ (symm (pseudo_apply_app.symm bs B F)),\n  end\n| Γ (A |₂ F) (G |₁ B) := begin\n    unfold pseudo_apply,\n    calc  (A |ₛ pseudo_apply F (G |₁ B))\n        ≈ (A |ₛ pseudo_apply F G |ₛ B)\n          : ξ_parallel₂ (pseudo_apply.on_parallel₁ F G B)\n    ... ≈ (A |ₛ pseudo_apply G F |ₛ B)\n          : ξ_parallel₂ (ξ_parallel₁ (pseudo_apply.symm F G))\n    ... ≈ ((A |ₛ pseudo_apply G F) |ₛ B) : parallel_assoc₂\n    ... ≈ (pseudo_apply G (A |₂ F) |ₛ B)\n          : ξ_parallel₁ (symm (pseudo_apply.on_parallel₂ G A F))\n  end\n| Γ (A |₂ F) (B |₂ G) := begin\n    unfold pseudo_apply,\n    calc  (A |ₛ pseudo_apply F (B |₂ G))\n        ≈ (A |ₛ B |ₛ pseudo_apply F G)\n          : ξ_parallel₂ (pseudo_apply.on_parallel₂ F B G)\n    ... ≈ (B |ₛ A |ₛ pseudo_apply F G) : parallel_symm₁\n    ... ≈ (B |ₛ A |ₛ pseudo_apply G F)\n          : ξ_parallel₂ (ξ_parallel₂ (pseudo_apply.symm F G))\n    ... ≈ (B |ₛ pseudo_apply G (A |₂ F))\n          : ξ_parallel₂ (symm (pseudo_apply.on_parallel₂ G A F))\n  end\n| Γ (A |₂ F) (ν'(M) G) := begin\n    unfold pseudo_apply rename,\n    calc  (A |ₛ pseudo_apply F (ν'(M) G))\n        ≈ (A |ₛ ν(M) pseudo_apply (rename name.extend F) G)\n          : ξ_parallel₂ (pseudo_apply.on_restriction F M G)\n    ... ≈ ν(M) species.rename name.extend A |ₛ pseudo_apply (rename name.extend F) G\n          : ν_parallel₂ M\n    ... ≈ ν(M) species.rename name.extend A |ₛ pseudo_apply G (rename name.extend F)\n          : ξ_restriction M (ξ_parallel₂ (pseudo_apply.symm _ G))\n    ... ≈ ν(M) pseudo_apply G (species.rename name.extend A |₂ rename name.extend F)\n          : ξ_restriction M (symm (pseudo_apply.on_parallel₂ G _ _))\nend\n\n| Γ (ν'(M) F) (#(bs; y) B) := begin\n    unfold pseudo_apply pseudo_apply_app,\n    from ξ_restriction M (symm (pseudo_apply_app.symm _ _ F)),\n  end\n| Γ (ν'(M) F) (G |₁ B) :=\n  let h : depth (rename (@name.extend Γ M.arity) G) < depth G + 1 := begin\n    rw ← depth.over_rename name.extend G,\n    from nat.lt_add_of_pos_right nat.zero_lt_one\n  end in begin\n    unfold pseudo_apply rename,\n    calc  (ν(M) pseudo_apply F (rename name.extend G |₁ species.rename name.extend B))\n        ≈ (ν(M) pseudo_apply F (rename name.extend G) |ₛ species.rename name.extend B)\n          : ξ_restriction M (pseudo_apply.on_parallel₁ F _ _)\n    ... ≈ ((ν(M) pseudo_apply F (rename name.extend G)) |ₛ B) : ν_parallel' M\n    ... ≈ ((ν(M) pseudo_apply (rename name.extend G) F) |ₛ B)\n          : ξ_parallel₁ (ξ_restriction M (pseudo_apply.symm F (rename name.extend G)))\n    ... ≈ (pseudo_apply G (ν'(M) F) |ₛ B)\n          : ξ_parallel₁ (symm (pseudo_apply.on_restriction G M F))\n  end\n| Γ (ν'(M) F) (B |₂ G) :=\n  let h : depth (rename (@name.extend Γ M.arity) G) < depth G + 1 := begin\n    rw ← depth.over_rename name.extend G,\n    from nat.lt_add_of_pos_right nat.zero_lt_one\n  end in begin\n    unfold pseudo_apply rename,\n    calc  (ν(M) pseudo_apply F (species.rename name.extend B |₂ rename name.extend G))\n        ≈ (ν(M) species.rename name.extend B |ₛ pseudo_apply F (rename name.extend G))\n          : ξ_restriction M (pseudo_apply.on_parallel₂ F _ _)\n    ... ≈ (B |ₛ ν(M) pseudo_apply F (rename name.extend G)) : ν_parallel₁ M\n    ... ≈ (B |ₛ ν(M) pseudo_apply (rename name.extend G) F)\n          : ξ_parallel₂ (ξ_restriction M (pseudo_apply.symm F _))\n    ... ≈ (B |ₛ pseudo_apply G (ν'(M) F))\n          : ξ_parallel₂ (symm (pseudo_apply.on_restriction G M F))\n  end\n| Γ (ν'(M) F) (ν'(N) G) :=\n  let h : depth (rename (name.ext (@name.extend Γ M.arity)) G) < depth G + 1 := begin\n    rw ← depth.over_rename _ G,\n    from nat.lt_add_of_pos_right nat.zero_lt_one\n  end in begin\n    unfold pseudo_apply rename,\n    calc  (ν(M) pseudo_apply F (ν'(N) rename (name.ext name.extend) G))\n        ≈ (ν(M) ν(N) pseudo_apply (rename name.extend F) (rename (name.ext name.extend) G))\n          : ξ_restriction M (pseudo_apply.on_restriction F N _)\n    ... ≈ (ν(M) ν(N) pseudo_apply (rename (name.ext name.extend) G) (rename name.extend F))\n          : ξ_restriction M (ξ_restriction N (pseudo_apply.symm _ _))\n    ... ≈ ν(N) ν(M) pseudo_apply (rename name.extend G) (rename (name.ext name.extend) F)\n          : symm (pseudo_apply.restriction_swap M N G F)\n    ... ≈ ν(N) pseudo_apply G (ν'(M) rename (name.ext name.extend) F)\n          : ξ_restriction N (symm (pseudo_apply.on_restriction G M _))\n  end\nusing_well_founded {\n  rel_tac := λ _ _,\n    `[exact ⟨_, measure_wf (λ x, depth x.snd.snd ) ⟩ ],\n  dec_tac := do\n    well_founded_tactics.unfold_wf_rel,\n    well_founded_tactics.unfold_sizeof,\n\n    tactic.dunfold_target [``depth, ``psigma.fst, ``psigma.snd],\n    well_founded_tactics.cancel_nat_add_lt,\n    tactic.try well_founded_tactics.trivial_nat_lt\n}\n\nprivate lemma pseudo_apply_app.equiv {a b} :\n  ∀ {Γ} {bs : vector (name Γ) a} {A A' : species ℍ ω (context.extend b Γ)}\n    {F : concretion ℍ ω Γ b a}\n  , A ≈ A' → pseudo_apply_app bs A F ≈ pseudo_apply_app bs A' F :=\nbegin\n  intros Γ as A A' G eq,\n  induction G,\n\n  case apply : Γ b bs y B {\n    unfold pseudo_apply_app,\n    from ξ_parallel₁ (species.equiv.rename _ eq),\n  },\n  case parallel₁ : Γ b y F B ih {\n    unfold pseudo_apply_app,\n    from ξ_parallel₁ (ih eq)\n  },\n  case parallel₂ : Γ b y B F ih {\n    unfold pseudo_apply_app,\n    from ξ_parallel₂ (ih eq)\n  },\n  case restriction : Γ b y M F ih {\n    unfold pseudo_apply_app,\n    from ξ_restriction M (ih (species.equiv.rename _ eq)),\n  }\nend\n\nprivate lemma pseudo_apply_app.par {a b} :\n  ∀ {Γ} (bs : vector (name Γ) a)\n    (A : species ℍ ω Γ) (B : species ℍ ω (context.extend b Γ))\n    (G : concretion ℍ ω Γ b a)\n  , pseudo_apply_app bs (species.rename name.extend A |ₛ B) G ≈ (A |ₛ pseudo_apply_app bs B G) :=\nbegin\n  intros Γ as A B G,\n  induction G,\n\n  case apply : _ b' bs y C {\n    unfold pseudo_apply_app, simp,\n    calc  ((species.rename (name.mk_apply bs) (species.rename name.extend A) |ₛ species.rename (name.mk_apply bs) B)\n            |ₛ species.rename (name.mk_apply as) C)\n        ≈ ((species.rename (name.mk_apply bs ∘ name.extend) A |ₛ species.rename (name.mk_apply bs) B)\n            |ₛ species.rename (name.mk_apply as) C)\n          : by rw species.rename_compose _ _ A\n    ... ≈ ((A |ₛ species.rename (name.mk_apply bs) B) |ₛ species.rename (name.mk_apply as) C)\n          : by { rw [name.mk_apply_ext, species.rename_id], }\n    ... ≈ (A |ₛ species.rename (name.mk_apply bs) B |ₛ species.rename (name.mk_apply as) C)\n          : parallel_assoc₁\n  },\n\n  case parallel₁ : _ b' y G C ih {\n    unfold pseudo_apply_app species.rename,\n    calc  (pseudo_apply_app as (species.rename name.extend A |ₛ B) G |ₛ C)\n        ≈ ((A |ₛ pseudo_apply_app as B G) |ₛ C) : ξ_parallel₁ (ih as A B)\n    ... ≈ (A |ₛ pseudo_apply_app as B G |ₛ C) : parallel_assoc₁\n  },\n\n  case parallel₂ : _ b' y C G ih {\n    unfold pseudo_apply_app species.rename,\n    calc  (C |ₛ pseudo_apply_app as (species.rename name.extend A |ₛ B) G)\n        ≈ (C |ₛ (A |ₛ pseudo_apply_app as B G)) : ξ_parallel₂ (ih as A B)\n    ... ≈ (A |ₛ C |ₛ pseudo_apply_app as B G) : parallel_symm₁\n  },\n\n  case restriction : _ b' y' M G ih {\n    unfold pseudo_apply_app, simp,\n    calc  (ν(M) pseudo_apply_app (vector.map name.extend as)\n                  (species.rename (name.ext name.extend) (species.rename name.extend A)\n                  |ₛ species.rename (name.ext name.extend) B) G)\n        ≈ (ν(M) pseudo_apply_app (vector.map name.extend as)\n                  (species.rename name.extend (species.rename name.extend A)\n                  |ₛ species.rename (name.ext name.extend) B) G)\n          : by rw ← species.rename_ext A\n    ... ≈ (ν(M) species.rename name.extend A |ₛ\n                pseudo_apply_app (vector.map name.extend as)\n                    (species.rename (name.ext name.extend) B) G)\n          : ξ_restriction M (ih _ _ _)\n    ... ≈ (A |ₛ ν(M)\n                pseudo_apply_app (vector.map name.extend as)\n                    (species.rename (name.ext name.extend) B) G)\n        : ν_parallel₁ M\n  }\nend\n\nlemma pseudo_apply.equiv_l {a b} :\n  ∀ {Γ} {F F' : concretion ℍ ω Γ a b} {G : concretion ℍ ω Γ b a}\n  , F ≈ F' → pseudo_apply F G ≈ pseudo_apply F' G :=\nbegin\n  intros Γ F F' G eq, induction eq,\n\n  -- Modifiers are incredibly trivial\n  case equiv.refl { from refl _ },\n  case equiv.trans : Γ b y F G H fg gh ih_fg ih_gh { from trans ih_fg ih_gh },\n  case equiv.symm : Γ b y F G eq ih_eq { from symm ih_eq },\n\n  -- Projections are trivial\n  case equiv.ξ_parallel₁ : Γ b y F F' A eq ih {\n    unfold pseudo_apply,\n    from ξ_parallel₁ ih\n  },\n  case equiv.ξ_parallel₂ : Γ b y A F F' eq ih {\n    unfold pseudo_apply,\n    from ξ_parallel₂ ih\n  },\n  case equiv.ξ_restriction : Γ b y M F F' eq ih {\n    unfold pseudo_apply,\n    from ξ_restriction M ih\n  },\n\n  -- Now for the interesting stuff. Well, relatively speaking.\n\n  case equiv.parallel_nil : Γ b y F { unfold pseudo_apply, from parallel_nil₁ },\n  case equiv.parallel_symm : Γ b y F G { unfold pseudo_apply, from parallel_symm },\n  case equiv.parallel_assoc₁ : Γ b y F A B {\n    unfold pseudo_apply,\n    from parallel_assoc₁\n  },\n  case equiv.parallel_assoc₂ : Γ b y F A B {\n    unfold pseudo_apply,\n    from parallel_assoc₁\n  },\n\n  case equiv.ξ_apply : Γ b y bs A A' eq F {\n    unfold pseudo_apply,\n    from pseudo_apply_app.equiv eq\n  },\n  case equiv.ξ_parallel : Γ b y F A A' eq {\n    unfold pseudo_apply,\n    from ξ_parallel₂ eq\n  },\n\n  case equiv.ν_parallel₁ : Γ b y M A F {\n    unfold pseudo_apply,\n    from ν_parallel₁ M\n  },\n  case equiv.ν_parallel₂ : Γ b y M F F {\n    unfold pseudo_apply,\n    rw ← pseudo_apply.rename name.extend F G,\n    from ν_parallel₁ M\n  },\n  case equiv.ν_drop : Γ b y M F {\n    unfold pseudo_apply,\n    rw ← pseudo_apply.rename name.extend F G,\n    from ν_drop₁ M\n  },\n  case equiv.ν_swap : Γ b y M N F {\n    unfold pseudo_apply,\n    calc  (ν(M) ν(N) pseudo_apply F (rename name.extend (rename name.extend G)))\n        ≈ (ν(M) ν(N) pseudo_apply F (rename (name.extend ∘ name.extend) G))\n          : by rw rename_compose\n    ... ≈ (ν(N) ν(M) pseudo_apply (rename name.swap F) (rename name.swap (rename (name.extend ∘ name.extend) G)))\n          : by { rw ← pseudo_apply.rename, from ν_swap₁ M N }\n    ... ≈ (ν(N) ν(M) pseudo_apply (rename name.swap F) (rename (name.swap ∘ name.extend ∘ name.extend) G))\n          : by rw rename_compose\n    ... ≈ (ν(N) ν(M) pseudo_apply (rename name.swap F) (rename (name.extend ∘ name.extend) G))\n          : by rw [← function.comp.assoc, name.swap_comp_extend, name.ext_extend]\n    ... ≈ (ν(N) ν(M) pseudo_apply (rename name.swap F) (rename name.extend (rename name.extend G)))\n          : by rw rename_compose\n  },\n  case equiv.apply_parallel : Γ b y bs A B {\n    unfold pseudo_apply,\n    from pseudo_apply_app.par bs A B G\n  }\nend\n\ntheorem pseudo_apply.equiv {a b} :\n  ∀ {Γ} {F F' : concretion ℍ ω Γ a b} {G G' : concretion ℍ ω Γ b a}\n  , F ≈ F' → G ≈ G' → pseudo_apply F G ≈ pseudo_apply F' G'\n| Γ F F' G G' eq_1 eq_2 :=\n  calc  pseudo_apply F  G\n      ≈ pseudo_apply F' G  : pseudo_apply.equiv_l eq_1\n  ... ≈ pseudo_apply G  F' : pseudo_apply.symm F' G\n  ... ≈ pseudo_apply G' F' : pseudo_apply.equiv_l eq_2\n  ... ≈ pseudo_apply F' G' : pseudo_apply.symm G' F'\n\nprotected lemma pseudo_apply.on_parallel₂'\n    {Γ} {a b} (A : species ℍ ω Γ) (F : concretion ℍ ω Γ a b) (G : concretion ℍ ω Γ b a)\n  : pseudo_apply (A |₂ F) G ≈ (A |ₛ pseudo_apply F G) :=\n  calc  pseudo_apply (A |₂ F) G\n      ≈ pseudo_apply G (A |₂ F) : pseudo_apply.symm _ G\n  ... ≈ (A |ₛ pseudo_apply G F) : pseudo_apply.on_parallel₂ G A F\n  ... ≈ (A |ₛ pseudo_apply F G) : ξ_parallel₂ (pseudo_apply.symm G F)\n\nprotected lemma pseudo_apply.parallel_shift\n    {Γ} {a b} (F : concretion ℍ ω Γ a b) (A : species ℍ ω Γ) (G : concretion ℍ ω Γ b a)\n  : pseudo_apply (F |₁ A) G ≈ pseudo_apply F (A |₂ G) :=\n  calc  pseudo_apply (F |₁ A) G\n      ≈ pseudo_apply G (F |₁ A) : pseudo_apply.symm _ G\n  ... ≈ (pseudo_apply G F |ₛ A) : pseudo_apply.on_parallel₁ G F A\n  ... ≈ (A |ₛ pseudo_apply G F) : parallel_symm\n  ... ≈ (A |ₛ pseudo_apply F G) : ξ_parallel₂ (pseudo_apply.symm G F)\n  ... ≈ pseudo_apply F (A |₂ G) : symm (pseudo_apply.on_parallel₂ F A G)\n\n/-- Pseudo application lifted to the level of quotients. -/\ndef pseudo_apply.quotient {Γ a b}\n  : concretion' ℍ ω Γ a b → concretion' ℍ ω Γ b a\n  → species' ℍ ω Γ\n| F G := quotient.lift_on₂ F G\n  (λ F G, ⟦ pseudo_apply F G ⟧)\n  (λ F G F' G' eqF eqG, quot.sound (pseudo_apply.equiv eqF eqG))\n\nlemma pseudo_apply.quotient.symm {Γ a b} :\n  ∀ (F : concretion' ℍ ω Γ a b) (G : concretion' ℍ ω Γ b a)\n  , pseudo_apply.quotient F G = pseudo_apply.quotient G F\n| F G := begin\n  rcases quot.exists_rep F with ⟨ F, ⟨ _ ⟩ ⟩,\n  rcases quot.exists_rep G with ⟨ G, ⟨ _ ⟩ ⟩,\n\n  show ⟦pseudo_apply F G⟧ = ⟦pseudo_apply G F⟧,\n  from quot.sound (pseudo_apply.symm F G),\nend\n\nend concretion\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/concretion/pseudo_apply.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.30790292800942665}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.sheaves.sheaf\nimport Mathlib.category_theory.limits.preserves.shapes.products\nimport Mathlib.category_theory.limits.types\nimport Mathlib.PostPort\n\nuniverses u₁ v u₂ \n\nnamespace Mathlib\n\n/-!\n# Checking the sheaf condition on the underlying presheaf of types.\n\nIf `G : C ⥤ D` is a functor which reflects isomorphisms and preserves limits\n(we assume all limits exist in both `C` and `D`),\nthen checking the sheaf condition for a presheaf `F : presheaf C X`\nis equivalent to checking the sheaf condition for `F ⋙ G`.\n\nThe important special case is when\n`C` is a concrete category with a forgetful functor\nthat preserves limits and reflects isomorphisms.\nThen to check the sheaf condition it suffices\nto check it on the underlying sheaf of types.\n\n## References\n* https://stacks.math.columbia.edu/tag/0073\n-/\n\nnamespace Top\n\n\nnamespace presheaf\n\n\nnamespace sheaf_condition\n\n\n/--\nWhen `G` preserves limits, the sheaf condition diagram for `F` composed with `G` is\nnaturally isomorphic to the sheaf condition diagram for `F ⋙ G`.\n-/\ndef diagram_comp_preserves_limits {C : Type u₁} [category_theory.category C] [category_theory.limits.has_limits C] {D : Type u₂} [category_theory.category D] [category_theory.limits.has_limits D] (G : C ⥤ D) [category_theory.limits.preserves_limits G] {X : Top} (F : presheaf C X) {ι : Type v} (U : ι → topological_space.opens ↥X) : sheaf_condition_equalizer_products.diagram F U ⋙ G ≅ sheaf_condition_equalizer_products.diagram (F ⋙ G) U :=\n  category_theory.nat_iso.of_components\n    (fun (X_1 : category_theory.limits.walking_parallel_pair) =>\n      category_theory.limits.walking_parallel_pair.cases_on X_1\n        (category_theory.limits.preserves_product.iso G fun (i : ι) => category_theory.functor.obj F (opposite.op (U i)))\n        (category_theory.limits.preserves_product.iso G\n          fun (p : ι × ι) => category_theory.functor.obj F (opposite.op (U (prod.fst p) ⊓ U (prod.snd p)))))\n    sorry\n\n/--\nWhen `G` preserves limits, the image under `G` of the sheaf condition fork for `F`\nis the sheaf condition fork for `F ⋙ G`,\npostcomposed with the inverse of the natural isomorphism `diagram_comp_preserves_limits`.\n-/\ndef map_cone_fork {C : Type u₁} [category_theory.category C] [category_theory.limits.has_limits C] {D : Type u₂} [category_theory.category D] [category_theory.limits.has_limits D] (G : C ⥤ D) [category_theory.limits.preserves_limits G] {X : Top} (F : presheaf C X) {ι : Type v} (U : ι → topological_space.opens ↥X) : category_theory.functor.map_cone G (sheaf_condition_equalizer_products.fork F U) ≅\n  category_theory.functor.obj\n    (category_theory.limits.cones.postcompose (category_theory.iso.inv (diagram_comp_preserves_limits G F U)))\n    (sheaf_condition_equalizer_products.fork (F ⋙ G) U) :=\n  category_theory.limits.cones.ext\n    (category_theory.iso.refl\n      (category_theory.limits.cone.X (category_theory.functor.map_cone G (sheaf_condition_equalizer_products.fork F U))))\n    sorry\n\nend sheaf_condition\n\n\n/--\nIf `G : C ⥤ D` is a functor which reflects isomorphisms and preserves limits\n(we assume all limits exist in both `C` and `D`),\nthen checking the sheaf condition for a presheaf `F : presheaf C X`\nis equivalent to checking the sheaf condition for `F ⋙ G`.\n\nThe important special case is when\n`C` is a concrete category with a forgetful functor\nthat preserves limits and reflects isomorphisms.\nThen to check the sheaf condition it suffices to check it on the underlying sheaf of types.\n\nAnother useful example is the forgetful functor `TopCommRing ⥤ Top`.\n\nSee https://stacks.math.columbia.edu/tag/0073.\nIn fact we prove a stronger version with arbitrary complete target category.\n-/\ndef sheaf_condition_equiv_sheaf_condition_comp {C : Type u₁} [category_theory.category C] {D : Type u₂} [category_theory.category D] (G : C ⥤ D) [category_theory.reflects_isomorphisms G] [category_theory.limits.has_limits C] [category_theory.limits.has_limits D] [category_theory.limits.preserves_limits G] {X : Top} (F : presheaf C X) : sheaf_condition F ≃ sheaf_condition (F ⋙ G) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/sheaves/forget.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3078846883420933}}
{"text": "import .pushout_lemmas\n\nlocal attribute [instance] classical.prop_decidable\n\nuniverse u\n\nopen category_theory set\n\nnamespace homotopy_theory.topological_spaces\nopen homotopy_theory.topological_spaces.Top\nlocal notation `Top` := Top.{u}\n\ndef closed_t1_inclusion {A X : Top} (i : A ⟶ X) : Prop :=\nembedding i ∧ is_closed (set.range i) ∧ ∀ x, x ∉ set.range i → is_closed ({x} : set X)\n\nlemma closed_t1_inclusion_of_closed_embedding_t1 {A X : Top} [t1_space X] (i : A ⟶ X)\n  (h₁ : embedding i) (h₂ : is_closed (set.range i)) : closed_t1_inclusion i :=\n⟨h₁, h₂, λ x _, is_closed_singleton⟩\n\nlemma closed_t1_inclusion_id {X : Top} : closed_t1_inclusion (𝟙 X) :=\n⟨embedding_id, by convert is_closed_univ; exact range_id, λ x hx, false.elim (hx ⟨x, rfl⟩)⟩\n\nlemma closed_t1_inclusion_comp {X Y Z : Top} {f : X ⟶ Y} {g : Y ⟶ Z}\n  (hf : closed_t1_inclusion f) (hg : closed_t1_inclusion g) : closed_t1_inclusion (f ≫ g) :=\n⟨hg.1.comp hf.1,\n by convert embedding_is_closed hg.1 hg.2.1 hf.2.1; apply range_comp,\n λ z hz, if hz' : z ∈ range g\n   then let ⟨y, hy⟩ := hz' in\n     have y ∉ range f, from λ hy', hz (by erw range_comp; exact ⟨y, hy', hy⟩),\n     by convert embedding_is_closed hg.1 hg.2.1 (hf.2.2 y this); rw ←hy; simp\n   else hg.2.2 z hz'⟩\n\nsection\nparameters {A B X Y : Top} {i : A ⟶ X} {f : A ⟶ B} {g : X ⟶ Y} {j : B ⟶ Y}\nparameter (po : Is_pushout i f g j)\ninclude po\n\nlemma closed_t1_inclusion_of_pushout (h : closed_t1_inclusion i) : closed_t1_inclusion j :=\n⟨embedding_j_of_embedding_i po h.1,\n (range_i_closed_iff_range_j_closed po).mp h.2.1,\n λ y hy, begin\n   rw is_closed_in_pushout po, split,\n   { let g' := complement_homeomorphism po (or.inl h.2.1),\n     let x := g'.inv ⟨y, hy⟩,\n     suffices : g ⁻¹' {y} = {x.val},\n     { rw this, apply h.2.2 x.val x.property },\n     have g'x : g'.hom x = ⟨y, hy⟩,\n     { ext,\n       change (g'.inv ≫ g'.hom ≫ incl _) ⟨y, hy⟩ = y,\n       simp, refl },\n     suffices : ∀ x', g x' = y → x' = x.val,\n     { ext x',\n       rw [mem_preimage, mem_singleton_iff, mem_singleton_iff],\n       refine ⟨this x', _⟩,\n       rintro rfl,\n       convert Top.hom_congr (complement_homeomorphism_eq po (or.inl h.2.1)).symm x,\n       exact congr_arg subtype.val g'x.symm },\n     intros x' hx',\n     have : x' ∉ range i,\n     { rintro ⟨a, rfl⟩,\n       change (i ≫ g) a = y at hx',\n       rw po.commutes at hx',\n       exact hy ⟨f a, hx'⟩ },\n     have : g'.hom ⟨x', this⟩ = ⟨y, hy⟩,\n     { ext,\n       convert ←Top.hom_congr (complement_homeomorphism_eq po (or.inl h.2.1)).symm ⟨x', this⟩ },\n     have : g'.hom ⟨x', _⟩ = g'.hom x, from this.trans g'x.symm,\n     exact congr_arg subtype.val ((homeomorphism.embedding g').inj this) },\n   { convert is_closed_empty,\n     rw ←preimage_inter_range,\n     convert preimage_empty,\n     -- FIXME: Used to be `rwa singleton_inter_eq_empty`\n     exact singleton_inter_eq_empty.mpr hy }\n end⟩\n\nend\n\nend homotopy_theory.topological_spaces\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/topological_spaces/closed_t1_inclusion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.30788467554587057}}
{"text": "import algebra.homology.homological_complex\nimport algebra.homology.additive\n\n\nopen category_theory\nopen category_theory.category\nopen category_theory.preadditive\n\nvariables {C : Type*} [category C] [preadditive C]\nvariables {ι : Type*} {c : complex_shape ι}\n\nnamespace homological_complex\n/-\n@[simp]\ndef eq_to_hom_f {K L : homological_complex C c} (h : K = L) (n : ι) :\n  homological_complex.hom.f (eq_to_hom h) n =\n  eq_to_hom (congr_fun (congr_arg homological_complex.X h) n) :=\nby { subst h, simp only [homological_complex.id_f, eq_to_hom_refl], }\n\n@[ext]\ndef ext {K L : homological_complex C c} (h_X : K.X = L.X)\n  (h_d : ∀ (i j : ι), c.rel i j → K.d i j ≫ eq_to_hom (congr_fun h_X j) =\n    eq_to_hom (congr_fun h_X i) ≫ L.d i j) : K = L :=\nbegin\n  cases K,\n  cases L,\n  dsimp at h_X,\n  subst h_X,\n  simp only [true_and, eq_self_iff_true, heq_iff_eq],\n  ext i j,\n  by_cases hij : c.rel i j,\n  { simpa only [id_comp, eq_to_hom_refl, comp_id] using h_d i j hij, },\n  { rw [K_shape' i j hij, L_shape' i j hij], }\nend-/\n\ndef congr_X {K L : homological_complex C c} (h : K = L) :\n  ∀ (n : ι), K.X n = L.X n :=\nby { intro n, subst h, }\n\ndef congr_d {K L : homological_complex C c} (h : K = L) :\n  ∀ (i j : ι), c.rel i j → K.d i j ≫ eq_to_hom (congr_X h j) =\n  eq_to_hom (congr_X h i) ≫ L.d i j:=\nby { intros i j hij, subst h, simp only [id_comp, eq_to_hom_refl, comp_id], }\n\nend homological_complex\n\nnamespace chain_complex\n\nvariables {D : Type*} [category D] [preadditive D]\nvariables {α : Type*} [add_right_cancel_semigroup α] [has_one α] [decidable_eq α]\n\nlemma map_of (F : C ⥤ D) [F.additive] (X : α → C) (d : Π n, X (n+1) ⟶ X n)\n  (sq : ∀ n, d (n+1) ≫ d n = 0) :\n  (F.map_homological_complex _).obj (chain_complex.of X d sq) =\n  chain_complex.of (λ n, F.obj (X n))\n    (λ n, F.map (d n)) (λ n, by rw [ ← F.map_comp, sq n, functor.map_zero]) :=\nbegin\n  apply homological_complex.ext,\n  intros i j hij,\n  { have h : j+1=i := hij,\n    subst h,\n    simp only [of_d, functor.map_homological_complex_obj_d, id_comp,\n      eq_to_hom_refl, comp_id], },\n  { refl, }\nend\n\nend chain_complex\n\nnamespace homological_complex\n\nlemma is_iso_of_components {A B : homological_complex C c} (f : A ⟶ B)\n  [∀ (n : ι), is_iso (f.f n)] : is_iso f :=\nbegin\n  convert is_iso.of_iso (homological_complex.hom.iso_of_components (λ n, as_iso (f.f n))\n    (by tidy)),\n  ext n,\n  refl,\nend\n\ninstance {D : Type*} [category D] [preadditive D]\n  {F : C ⥤ D} [F.additive] [full F] [faithful F] :\n  reflects_isomorphisms (functor.map_homological_complex F c) :=\n⟨λ A B f, begin\n  introI,\n  suffices : ∀ (n : ι), is_iso (f.f n),\n  { haveI := this, apply is_iso_of_components, },\n  intro n,\n  apply is_iso_of_reflects_iso _ F,\n  change is_iso ((homological_complex.eval D c n).map ((F.map_homological_complex c).map f)),\n  apply_instance,\nend⟩\n\nend homological_complex\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/homological_complex_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.3078457923209246}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nAn experimental variant on the `ring` tactic that uses computational\nreflection instead of proof generation. Useful for kernel benchmarking.\n-/\nimport tactic.ring data.num.lemmas\n\nnamespace tactic.ring2\n\n@[derive has_reflect]\ninductive {u} tree (α : Type u) : Type u\n| nil {} : tree\n| node : α → tree → tree → tree\n\ndef tree.get {α} [has_zero α] : pos_num → tree α → α\n| _                tree.nil            := 0\n| pos_num.one      (tree.node a t₁ t₂) := a\n| (pos_num.bit0 n) (tree.node a t₁ t₂) := t₁.get n\n| (pos_num.bit1 n) (tree.node a t₁ t₂) := t₂.get n\n\ndef tree.of_rbnode {α} : rbnode α → tree α\n| rbnode.leaf               := tree.nil\n| (rbnode.red_node l a r)   :=\n  tree.node a (tree.of_rbnode l) (tree.of_rbnode r)\n| (rbnode.black_node l a r) :=\n  tree.node a (tree.of_rbnode l) (tree.of_rbnode r)\n\ndef tree.index_of {α} (lt : α → α → Prop) [decidable_rel lt]\n  (x : α) : tree α → option pos_num\n| tree.nil := none\n| (tree.node a t₁ t₂) :=\n  match cmp_using lt x a with\n  | ordering.lt := pos_num.bit0 <$> tree.index_of t₁\n  | ordering.eq := some pos_num.one\n  | ordering.gt := pos_num.bit1 <$> tree.index_of t₂\n  end\n\nmeta def tree.reflect' (u : level) (α : expr) : tree expr → expr\n| tree.nil := (expr.const ``tree.nil [u] : expr) α\n| (tree.node a t₁ t₂) :=\n  (expr.const ``tree.node [u] : expr) α a t₁.reflect' t₂.reflect'\n\n@[derive has_reflect]\ninductive csring_expr\n| atom : pos_num → csring_expr\n| const : num → csring_expr\n| add : csring_expr → csring_expr → csring_expr\n| mul : csring_expr → csring_expr → csring_expr\n| pow : csring_expr → num → csring_expr\n\nnamespace csring_expr\n\ndef eval {α} [comm_semiring α] (t : tree α) : csring_expr → α\n| (atom n)  := @tree.get _ ⟨0⟩ n t\n| (const n) := n\n| (add x y) := eval x + eval y\n| (mul x y) := eval x * eval y\n| (pow x n) := eval x ^ (n : ℕ)\n\nend csring_expr\n\n@[derive decidable_eq]\ninductive horner_expr\n| const : znum → horner_expr\n| horner : horner_expr → pos_num → num → horner_expr → horner_expr\n\nnamespace horner_expr\n\ndef is_cs : horner_expr → Prop\n| (const n) := ∃ m:num, n = m.to_znum\n| (horner a x n b) := is_cs a ∧ is_cs b\n\ninstance : has_zero horner_expr := ⟨const 0⟩\ninstance : has_one horner_expr := ⟨const 1⟩\n\ndef atom (n : pos_num) : horner_expr := horner 1 n 1 0\n\ndef repr : horner_expr → string\n| (const n) := _root_.repr n\n| (horner a x n b) :=\n    \"(\" ++ repr a ++ \") * x\" ++ _root_.repr x ++ \"^\"\n        ++ _root_.repr n ++ \" + \" ++ repr b\ninstance : has_repr horner_expr := ⟨repr⟩\n\ndef horner' (a : horner_expr)\n  (x : pos_num) (n : num) (b : horner_expr) : horner_expr :=\nmatch a with\n| const q := if q = 0 then b else horner a x n b\n| horner a₁ x₁ n₁ b₁ :=\n  if x₁ = x ∧ b₁ = 0 then horner a₁ x (n₁ + n) b\n  else horner a x n b\nend\n\ndef add_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then e else begin\n  induction e with n a x n b A B,\n  { exact const (k + n) },\n  { exact horner a x n B }\nend\n\ndef add_aux (a₁ : horner_expr) (A₁ : horner_expr → horner_expr) (x₁ : pos_num) :\n  horner_expr → num → horner_expr → (horner_expr → horner_expr) → horner_expr\n| (const n₂)           n₁ b₁ B₁ := add_const n₂ (horner a₁ x₁ n₁ b₁)\n| (horner a₂ x₂ n₂ b₂) n₁ b₁ B₁ :=\n  let e₂ := horner a₂ x₂ n₂ b₂ in\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner a₂ x₂ n₂ (add_aux b₂ n₁ b₁ B₁)\n  | ordering.eq :=\n    match num.sub' n₁ n₂ with\n    | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n    | (znum.pos k) := horner (add_aux a₂ k 0 id) x₁ n₂ (B₁ b₂)\n    | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n    end\n  end\n\ndef add : horner_expr → horner_expr → horner_expr\n| (const n₁)           e₂ := add_const n₁ e₂\n| (horner a₁ x₁ n₁ b₁) e₂ := add_aux a₁ (add a₁) x₁ e₂ n₁ b₁ (add b₁)\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact add_const n₁ e₂ },\n  exact match e₂ with e₂ := begin\n    induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁;\n    let e₁ := horner a₁ x₁ n₁ b₁,\n    { exact add_const n₂ e₁ },\n    let e₂ := horner a₂ x₂ n₂ b₂,\n    exact match pos_num.cmp x₁ x₂ with\n    | ordering.lt := horner a₁ x₁ n₁ (B₁ e₂)\n    | ordering.gt := horner a₂ x₂ n₂ (B₂ n₁ b₁)\n    | ordering.eq :=\n      match num.sub' n₁ n₂ with\n      | znum.zero := horner' (A₁ a₂) x₁ n₁ (B₁ b₂)\n      | (znum.pos k) := horner (A₂ k 0) x₁ n₂ (B₁ b₂)\n      | (znum.neg k) := horner (A₁ (horner a₂ x₁ k 0)) x₁ n₁ (B₁ b₂)\n      end\n    end\n  end end\nend-/\n\ndef neg (e : horner_expr) : horner_expr :=\nbegin\n  induction e with n a x n b A B,\n  { exact const (-n) },\n  { exact horner A x n B }\nend\n\ndef mul_const (k : znum) (e : horner_expr) : horner_expr :=\nif k = 0 then 0 else if k = 1 then e else begin\n  induction e with n a x n b A B,\n  { exact const (n * k) },\n  { exact horner A x n B }\nend\n\ndef mul_aux (a₁ x₁ n₁ b₁) (A₁ B₁ : horner_expr → horner_expr) :\n  horner_expr → horner_expr\n| (const n₂) := mul_const n₂ (horner a₁ x₁ n₁ b₁)\n| e₂@(horner a₂ x₂ n₂ b₂) :=\n  match pos_num.cmp x₁ x₂ with\n  | ordering.lt := horner (A₁ e₂) x₁ n₁ (B₁ e₂)\n  | ordering.gt := horner (mul_aux a₂) x₂ n₂ (mul_aux b₂)\n  | ordering.eq := let haa := horner' (mul_aux a₂) x₁ n₂ 0 in\n    if b₂ = 0 then haa else haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂))\n  end\n\ndef mul : horner_expr → horner_expr → horner_expr\n| (const n₁)              := mul_const n₁\n| e₁@(horner a₁ x₁ n₁ b₁) := mul_aux a₁ x₁ n₁ b₁ (mul a₁) (mul b₁).\n/-begin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂,\n  { exact mul_const n₁ e₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂;\n  let e₁ := horner a₁ x₁ n₁ b₁,\n  { exact mul_const n₂ e₁ },\n  let e₂ := horner a₂ x₂ n₂ b₂,\n  cases pos_num.cmp x₁ x₂,\n  { exact horner (A₁ e₂) x₁ n₁ (B₁ e₂) },\n  { let haa := horner' A₂ x₁ n₂ 0,\n    exact if b₂ = 0 then haa else\n      haa.add (horner (A₁ b₂) x₁ n₁ (B₁ b₂)) },\n  { exact horner A₂ x₂ n₂ B₂ }\nend-/\n\ninstance : has_add horner_expr := ⟨add⟩\ninstance : has_neg horner_expr := ⟨neg⟩\ninstance : has_mul horner_expr := ⟨mul⟩\n\ndef pow (e : horner_expr) : num → horner_expr\n| 0 := 1\n| (num.pos p) := begin\n  induction p with p ep p ep,\n  { exact e },\n  { exact (ep.mul ep).mul e },\n  { exact ep.mul ep }\nend\n\ndef inv (e : horner_expr) : horner_expr := 0\n\ndef of_csexpr : csring_expr → horner_expr\n| (csring_expr.atom n)  := atom n\n| (csring_expr.const n) := const n.to_znum\n| (csring_expr.add x y) := (of_csexpr x).add (of_csexpr y)\n| (csring_expr.mul x y) := (of_csexpr x).mul (of_csexpr y)\n| (csring_expr.pow x n) := (of_csexpr x).pow n\n\ndef cseval {α} [comm_semiring α] (t : tree α) : horner_expr → α\n| (const n)        := n.abs\n| (horner a x n b) := tactic.ring.horner (cseval a) (t.get x) n (cseval b)\n\ntheorem cseval_atom {α} [comm_semiring α] (t : tree α)\n  (n : pos_num) : (atom n).is_cs ∧ cseval t (atom n) = t.get n :=\n⟨⟨⟨1, rfl⟩, ⟨0, rfl⟩⟩, (tactic.ring.horner_atom _).symm⟩\n\ntheorem cseval_add_const {α} [comm_semiring α] (t : tree α)\n  (k : num) {e : horner_expr} (cs : e.is_cs) :\n  (add_const k.to_znum e).is_cs ∧\n    cseval t (add_const k.to_znum e) = k + cseval t e :=\nbegin\n  simp [add_const],\n  cases k; simp! *,\n  simp [show znum.pos k ≠ 0, from dec_trivial],\n  induction e with n a x n b A B; simp *,\n  { rcases cs with ⟨n, rfl⟩,\n    refine ⟨⟨n + num.pos k, by simp; refl⟩, _⟩,\n    cases n; simp! },\n  { rcases B cs.2 with ⟨csb, h⟩, simp! [*, cs.1],\n    rw [← tactic.ring.horner_add_const, add_comm], refl }\nend\n\ntheorem cseval_horner' {α} [comm_semiring α] (t : tree α)\n  (a x n b) (h₁ : is_cs a) (h₂ : is_cs b) :\n  (horner' a x n b).is_cs ∧ cseval t (horner' a x n b) =\n    tactic.ring.horner (cseval t a) (t.get x) n (cseval t b) :=\nbegin\n  cases a with n₁ a₁ x₁ n₁ b₁; simp [horner']; split_ifs,\n  { simp! [*, tactic.ring.horner] },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ },\n  { refine ⟨⟨h₁.1, h₂⟩, eq.symm _⟩, simp! *,\n    apply tactic.ring.horner_horner, simp },\n  { exact ⟨⟨h₁, h₂⟩, rfl⟩ }\nend\n\ntheorem cseval_add {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (add e₁ e₂).is_cs ∧\n  cseval t (add e₁ e₂) = cseval t e₁ + cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa using cseval_add_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂ generalizing n₁ b₁,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simp! [cseval_add_const t n₂ cs₁] },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { rcases B₁ csb₁ cs₂ with ⟨csh, h⟩,\n    refine ⟨⟨csa₁, csh⟩, eq.symm _⟩,\n    apply tactic.ring.horner_add_const,\n    exact h.symm },\n  { cases C,\n    have B0 : is_cs 0 → ∀ {e₂ : horner_expr}, is_cs e₂ →\n      is_cs (add 0 e₂) ∧ cseval t (add 0 e₂) = cseval t 0 + cseval t e₂ :=\n      λ _ e₂ c, ⟨c, (zero_add _).symm⟩,\n     cases e : num.sub' n₁ n₂ with k k; simp!,\n    { have : n₁ = n₂,\n      { have := congr_arg (coe : znum → ℤ) e,\n        simp at this,\n        have := sub_eq_zero.1 this,\n        rw [← num.to_nat_to_int, ← num.to_nat_to_int] at this,\n        exact num.to_nat_inj.1 (int.coe_nat_inj this) },\n      subst n₂,\n      rcases cseval_horner' _ _ _ _ _ _ _ with ⟨csh, h⟩,\n      { refine ⟨csh, h.trans (eq.symm _)⟩,\n        simp *,\n        apply tactic.ring.horner_add_horner_eq; try {refl},\n        simp },\n      all_goals {simp! *} },\n    { simp [B₁ csb₁ csb₂],\n      rcases A₂ csa₂ _ _ B0 ⟨csa₁, 0, rfl⟩ with ⟨csh, h⟩,\n      refine ⟨csh, eq.symm _⟩,\n      rw [show id = add 0, from rfl, h],\n      apply tactic.ring.horner_add_horner_gt,\n      { change (_ + k : ℕ) = _,\n        rw [← int.coe_nat_inj', int.coe_nat_add,\n          eq_comm, ← sub_eq_iff_eq_add'],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      all_goals { apply add_comm } },\n    { have : (horner a₂ x₁ (num.pos k) 0).is_cs := ⟨csa₂, 0, rfl⟩,\n      simp [B₁ csb₁ csb₂, A₁ csa₁ this],\n      symmetry, apply tactic.ring.horner_add_horner_lt,\n      { change (_ + k : ℕ) = _,\n          rw [← int.coe_nat_inj', int.coe_nat_add,\n            eq_comm, ← sub_eq_iff_eq_add', ← neg_inj', neg_sub],\n        simpa using congr_arg (coe : znum → ℤ) e },\n      { refl }, { apply add_comm } } },\n  { rcases B₂ csb₂ _ _ B₁ ⟨csa₁, csb₁⟩ with ⟨csh, h⟩,\n    refine ⟨⟨csa₂, csh⟩, eq.symm _⟩,\n    apply tactic.ring.const_add_horner,\n    simp [h] }\nend\n\ntheorem cseval_mul_const {α} [comm_semiring α] (t : tree α)\n  (k : num) {e : horner_expr} (cs : e.is_cs) :\n  (mul_const k.to_znum e).is_cs ∧\n    cseval t (mul_const k.to_znum e) = cseval t e * k :=\nbegin\n  simp [mul_const],\n  split_ifs with h h,\n  { cases (num.to_znum_inj.1 h : k = 0),\n    exact ⟨⟨0, rfl⟩, (mul_zero _).symm⟩ },\n  { cases (num.to_znum_inj.1 h : k = 1),\n    exact ⟨cs, (mul_one _).symm⟩ },\n  induction e with n a x n b A B; simp *,\n  { rcases cs with ⟨n, rfl⟩,\n    suffices, refine ⟨⟨n * k, this⟩, _⟩,\n    swap, {cases n; cases k; refl},\n    rw [show _, from this], simp! },\n  { cases cs, simp! *,\n    symmetry, apply tactic.ring.horner_mul_const; refl }\nend\n\ntheorem cseval_mul {α} [comm_semiring α] (t : tree α)\n  {e₁ e₂ : horner_expr} (cs₁ : e₁.is_cs) (cs₂ : e₂.is_cs) :\n  (mul e₁ e₂).is_cs ∧\n  cseval t (mul e₁ e₂) = cseval t e₁ * cseval t e₂ :=\nbegin\n  induction e₁ with n₁ a₁ x₁ n₁ b₁ A₁ B₁ generalizing e₂; simp!,\n  { rcases cs₁ with ⟨n₁, rfl⟩,\n    simpa [mul_comm] using cseval_mul_const t n₁ cs₂ },\n  induction e₂ with n₂ a₂ x₂ n₂ b₂ A₂ B₂,\n  { rcases cs₂ with ⟨n₂, rfl⟩,\n    simpa! using cseval_mul_const t n₂ cs₁ },\n  cases cs₁ with csa₁ csb₁, cases id cs₂ with csa₂ csb₂,\n  simp!, have C := pos_num.cmp_to_nat x₁ x₂,\n  cases A₂ csa₂ with csA₂ hA₂,\n  cases pos_num.cmp x₁ x₂; simp!,\n  { simp [A₁ csa₁ cs₂, B₁ csb₁ cs₂],\n    symmetry, apply tactic.ring.horner_mul_const; refl },\n  { cases cseval_horner' t _ x₁ n₂ 0 csA₂ ⟨0, rfl⟩ with csh₁ h₁,\n    cases C, split_ifs,\n    { subst b₂,\n      refine ⟨csh₁, h₁.trans (eq.symm _)⟩,\n      apply tactic.ring.horner_mul_horner_zero; try {refl},\n      simp! [hA₂] },\n    { cases A₁ csa₁ csb₂ with csA₁ hA₁,\n      cases cseval_add t csh₁ _ with csh₂ h₂,\n      { refine ⟨csh₂, h₂.trans (eq.symm _)⟩,\n        apply tactic.ring.horner_mul_horner; try {refl},\n        simp! * },\n      exact ⟨csA₁, (B₁ csb₁ csb₂).1⟩ } },\n  { simp [A₂ csa₂, B₂ csb₂], rw [mul_comm, eq_comm],\n    apply tactic.ring.horner_const_mul,\n    {apply mul_comm}, {refl} },\nend\n\ntheorem cseval_pow {α} [comm_semiring α] (t : tree α)\n  {x : horner_expr} (cs : x.is_cs) :\n  ∀ (n : num), (pow x n).is_cs ∧\n    cseval t (pow x n) = cseval t x ^ (n : ℕ)\n| 0 := ⟨⟨1, rfl⟩, (pow_zero _).symm⟩\n| (num.pos p) := begin\n  simp [pow], induction p with p ep p ep,\n  { simp * },\n  { simp [pow_bit1],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    cases cseval_mul t cs₀ cs with cs₁ h₁,\n    simp * },\n  { simp [pow_bit0],\n    cases cseval_mul t ep.1 ep.1 with cs₀ h₀,\n    simp * }\nend\n\ntheorem cseval_of_csexpr {α} [comm_semiring α] (t : tree α) :\n  ∀ (r : csring_expr), (of_csexpr r).is_cs ∧ cseval t (of_csexpr r) = r.eval t\n| (csring_expr.atom n)  := cseval_atom _ _\n| (csring_expr.const n) := ⟨⟨n, rfl⟩, by cases n; refl⟩\n| (csring_expr.add x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_add t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.mul x y) :=\n  let ⟨cs₁, h₁⟩ := cseval_of_csexpr x,\n      ⟨cs₂, h₂⟩ := cseval_of_csexpr y,\n      ⟨cs, h⟩ := cseval_mul t cs₁ cs₂ in ⟨cs, by simp! [h, *]⟩\n| (csring_expr.pow x n) :=\n  let ⟨cs, h⟩ := cseval_of_csexpr x,\n      ⟨cs, h⟩ := cseval_pow t cs n in ⟨cs, by simp! [h, *]⟩\n\nend horner_expr\n\ntheorem correctness {α} [comm_semiring α] (t : tree α) (r₁ r₂ : csring_expr)\n  (H : horner_expr.of_csexpr r₁ = horner_expr.of_csexpr r₂) :\n  r₁.eval t = r₂.eval t :=\nby repeat {rw ← (horner_expr.cseval_of_csexpr t _).2}; rw H\n\nmeta def reflect_expr : expr → csring_expr × dlist expr\n| `(%%e₁ + %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂, l₁ ++ l₂)\n/-| `(%%e₁ - %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.add r₂.neg, l₁ ++ l₂)\n| `(- %%e) := let (r, l) := reflect_expr e in (r.neg, l)-/\n| `(%%e₁ * %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂, l₁ ++ l₂)\n/-| `(has_inv.inv %%e) := let (r, l) := reflect_expr e in (r.neg, l)\n| `(%%e₁ / %%e₂) :=\n  let (r₁, l₁) := reflect_expr e₁, (r₂, l₂) := reflect_expr e₂ in\n  (r₁.mul r₂.inv, l₁ ++ l₂)-/\n| e@`(%%e₁ ^ %%e₂) :=\n  match reflect_expr e₁, expr.to_nat e₂ with\n  | (r₁, l₁), some n₂ := (r₁.pow (num.of_nat' n₂), l₁)\n  | (r₁, l₁), none := (csring_expr.atom 1, dlist.singleton e)\n  end\n| e := match expr.to_nat e with\n  | some n := (csring_expr.const (num.of_nat' n), dlist.empty)\n  | none := (csring_expr.atom 1, dlist.singleton e)\n  end\n\nmeta def csring_expr.replace (t : tree expr) : csring_expr → state_t (list expr) option csring_expr\n| (csring_expr.atom _)  := do e ← get,\n  p ← monad_lift (t.index_of (<) e.head),\n  put e.tail, pure (csring_expr.atom p)\n| (csring_expr.const n) := pure (csring_expr.const n)\n| (csring_expr.add x y) := csring_expr.add <$> x.replace <*> y.replace\n| (csring_expr.mul x y) := csring_expr.mul <$> x.replace <*> y.replace\n| (csring_expr.pow x n) := (λ x, csring_expr.pow x n) <$> x.replace\n--| (csring_expr.neg x)   := csring_expr.neg <$> x.replace\n--| (csring_expr.inv x)   := csring_expr.inv <$> x.replace\n\nend tactic.ring2\n\nnamespace tactic\nnamespace interactive\nopen interactive interactive.types lean.parser\nopen tactic.ring2\n\nlocal postfix `?`:9001 := optional\n\n/-- Tactic for solving equations in the language of rings.\n  This variant on the `ring` tactic uses kernel computation instead\n  of proof generation. -/\nmeta def ring2 : tactic unit :=\ndo `[repeat {rw ← nat.pow_eq_pow}],\n  `(%%e₁ = %%e₂) ← target,\n  α ← infer_type e₁,\n  expr.sort (level.succ u) ← infer_type α,\n  let (r₁, l₁) := reflect_expr e₁,\n  let (r₂, l₂) := reflect_expr e₂,\n  let L := (l₁ ++ l₂).to_list,\n  let s := tree.of_rbnode (rbtree_of L).1,\n  (r₁, L) ← (state_t.run (r₁.replace s) L : option _),\n  (r₂, _) ← (state_t.run (r₂.replace s) L : option _),\n  let se : expr := s.reflect' u α,\n  let er₁ : expr := reflect r₁,\n  let er₂ : expr := reflect r₂,\n  cs ← mk_app ``comm_semiring [α] >>= mk_instance,\n  e ← to_expr ```(correctness %%se %%er₁ %%er₂ rfl)\n    <|> fail (\"ring2 tactic failed, using abstract equality:\\n\"\n      ++ repr (horner_expr.of_csexpr r₁) ++\n      \"\\n  =?=\\n\" ++ repr (horner_expr.of_csexpr r₂)),\n  tactic.exact e\n\nend interactive\nend tactic\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/tactic/ring2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3078118686435651}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.functor\nimport category_theory.full_subcategory\n\n/-!\n# Monoidal natural transformations\n\nNatural transformations between (lax) monoidal functors must satisfy\nan additional compatibility relation with the tensorators:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`.\n\n(Lax) monoidal functors between a fixed pair of monoidal categories\nthemselves form a category.\n-/\n\nopen category_theory\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nopen category_theory.category\nopen category_theory.functor\n\nnamespace category_theory\n\nopen monoidal_category\n\nvariables {C : Type u₁} [category.{v₁} C] [monoidal_category.{v₁} C]\n          {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D]\n\n/--\nA monoidal natural transformation is a natural transformation between (lax) monoidal functors\nadditionally satisfying:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`\n-/\n@[ext]\nstructure monoidal_nat_trans (F G : lax_monoidal_functor C D)\n  extends nat_trans F.to_functor G.to_functor :=\n(unit' : F.ε ≫ app (𝟙_ C) = G.ε . obviously)\n(tensor' : ∀ X Y, F.μ _ _ ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ _ _ . obviously)\n\nrestate_axiom monoidal_nat_trans.tensor'\nattribute [simp, reassoc] monoidal_nat_trans.tensor\nrestate_axiom monoidal_nat_trans.unit'\nattribute [simp, reassoc] monoidal_nat_trans.unit\n\nnamespace monoidal_nat_trans\n\n/--\nThe identity monoidal natural transformation.\n-/\n@[simps]\ndef id (F : lax_monoidal_functor C D) : monoidal_nat_trans F F :=\n{ ..(𝟙 F.to_functor) }\n\ninstance (F : lax_monoidal_functor C D) : inhabited (monoidal_nat_trans F F) := ⟨id F⟩\n\n/--\nVertical composition of monoidal natural transformations.\n-/\n@[simps]\ndef vcomp {F G H : lax_monoidal_functor C D}\n  (α : monoidal_nat_trans F G) (β : monoidal_nat_trans G H) : monoidal_nat_trans F H :=\n{ ..(nat_trans.vcomp α.to_nat_trans β.to_nat_trans) }\n\ninstance category_lax_monoidal_functor : category (lax_monoidal_functor C D) :=\n{ hom := monoidal_nat_trans,\n  id := id,\n  comp := λ F G H α β, vcomp α β, }\n\n@[simp] lemma comp_to_nat_trans' {F G H : lax_monoidal_functor C D} {α : F ⟶ G} {β : G ⟶ H} :\n  (α ≫ β).to_nat_trans =\n    @category_struct.comp (C ⥤ D) _ _ _ _ (α.to_nat_trans) (β.to_nat_trans) := rfl\n\ninstance category_monoidal_functor : category (monoidal_functor C D) :=\ninduced_category.category monoidal_functor.to_lax_monoidal_functor\n\n@[simp] lemma comp_to_nat_trans'' {F G H : monoidal_functor C D} {α : F ⟶ G} {β : G ⟶ H} :\n  (α ≫ β).to_nat_trans =\n    @category_struct.comp (C ⥤ D) _ _ _ _ (α.to_nat_trans) (β.to_nat_trans) := rfl\n\nvariables {E : Type u₃} [category.{v₃} E] [monoidal_category.{v₃} E]\n\n/--\nHorizontal composition of monoidal natural transformations.\n-/\n@[simps]\ndef hcomp {F G : lax_monoidal_functor C D} {H K : lax_monoidal_functor D E}\n  (α : monoidal_nat_trans F G) (β : monoidal_nat_trans H K) :\n  monoidal_nat_trans (F ⊗⋙ H) (G ⊗⋙ K) :=\n{ unit' :=\n  begin\n    dsimp, simp,\n    conv_lhs { rw [←K.to_functor.map_comp, α.unit], },\n  end,\n  tensor' := λ X Y,\n  begin\n    dsimp, simp,\n    conv_lhs { rw [←K.to_functor.map_comp, α.tensor, K.to_functor.map_comp], },\n  end,\n  ..(nat_trans.hcomp α.to_nat_trans β.to_nat_trans) }\n\nend monoidal_nat_trans\n\nnamespace monoidal_nat_iso\n\nvariables {F G : lax_monoidal_functor C D}\n\n/--\nConstruct a monoidal natural isomorphism from object level isomorphisms,\nand the monoidal naturality in the forward direction.\n-/\ndef of_components\n  (app : ∀ X : C, F.obj X ≅ G.obj X)\n  (naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app Y).hom = (app X).hom ≫ G.map f)\n  (unit : F.ε ≫ (app (𝟙_ C)).hom = G.ε)\n  (tensor : ∀ X Y, F.μ X Y ≫ (app (X ⊗ Y)).hom = ((app X).hom ⊗ (app Y).hom) ≫ G.μ X Y) :\n  F ≅ G :=\n{ hom := { app := λ X, (app X).hom, },\n  inv :=\n  { app := λ X, (app X).inv,\n    unit' := by { dsimp, rw [←unit, assoc, iso.hom_inv_id, comp_id], },\n    tensor' := λ X Y,\n    begin\n      dsimp,\n      rw [iso.comp_inv_eq, assoc, tensor, ←tensor_comp_assoc,\n        iso.inv_hom_id, iso.inv_hom_id, tensor_id, id_comp],\n    end,\n    ..(nat_iso.of_components app @naturality).inv, }, }\n\n@[simp] lemma of_components.hom_app\n  (app : ∀ X : C, F.obj X ≅ G.obj X) (naturality) (unit) (tensor) (X) :\n  (of_components app naturality unit tensor).hom.app X = (app X).hom := rfl\n@[simp] lemma of_components.inv_app\n  (app : ∀ X : C, F.obj X ≅ G.obj X) (naturality) (unit) (tensor) (X) :\n  (of_components app naturality unit tensor).inv.app X = (app X).inv :=\nby simp [of_components]\n\ninstance is_iso_of_is_iso_app (α : F ⟶ G) [∀ X : C, is_iso (α.app X)] : is_iso α :=\n⟨(is_iso.of_iso (of_components (λ X, as_iso (α.app X))\n  (λ X Y f, α.to_nat_trans.naturality f) α.unit α.tensor)).1⟩\n\nend monoidal_nat_iso\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monoidal/natural_transformation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.640635841117624, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3078118620503049}}
{"text": "import homotopy_theory.formal.i_category.homotopy_equivalences\nimport .pi_n\n\nopen function\n\nnoncomputable theory\n\nopen category_theory (hiding is_iso)\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.topological_spaces\nopen homotopy_theory.cofibrations\nopen homotopy_theory.weak_equivalences\nopen homotopy_theory.topological_spaces.Top\nlocal notation `Top` := Top.{0}\nlocal notation `Set` := Type 0\n\ndef is_weak_equivalence {X Y : Top} (f : X ⟶ Y) : Prop :=\nis_iso (π₀ &> f) ∧ ∀ n x, is_iso (π_induced n x f)\n\nlemma is_weak_equivalence_iso {X Y : Top} (i : homeomorphism X Y) :\n  is_weak_equivalence i.hom :=\n⟨⟨π₀.map_iso i, rfl⟩,\n assume n x, ⟨(π n).map_iso (Top_ptd.mk_iso' i x), rfl⟩⟩\n\nlemma is_weak_equivalence_comp {X Y Z : Top} {f : X ⟶ Y} {g : Y ⟶ Z}\n  (hf : is_weak_equivalence f) (hg : is_weak_equivalence g) :\n  is_weak_equivalence (g ∘ f) :=\n⟨by rw π₀.map_comp; exact iso_comp hf.1 hg.1,\n assume n x, by rw π_induced_comp; exact iso_comp (hf.2 n x) (hg.2 n (f x))⟩\n\ninstance : replete_wide_subcategory.{0} Top @is_weak_equivalence :=\nreplete_wide_subcategory.mk' @is_weak_equivalence_iso @is_weak_equivalence_comp\n\n-- I don't know why this changed, but `⟦Top.const y⟧` in the proof below\n-- started using `subtype.setoid` and `fun_setoid` (from core) by default.\nlocal attribute [instance, priority 1000] Hom.setoid\n\n-- I don't know what this means, but lean told me to turn it on\nset_option eqn_compiler.zeta true\ndef Top_weak_equivalences : category_with_weak_equivalences Top :=\n{ is_weq := @is_weak_equivalence,\n  weq_of_comp_weq_left := assume X Y Z f g hf hgf,\n    ⟨iso_of_comp_iso_left hf.1 (by rw ←π₀.map_comp; exact hgf.1),\n     assume n y,\n     -- This is the nontrivial case: y may not be in the image of f.\n     -- But we know that f is an isomorphism on π₀, so y is at least\n     -- connected by a path to f x' for some x' : X.\n       let ⟨i, hi⟩ := hf.1 in\n       let x'_class := i.inv ⟦Top.const y⟧ in\n       let ⟨x'_map, hx'⟩ := quotient.exists_rep x'_class in\n       let x' := x'_map punit.star in\n       have (π₀ &> f) ⟦Top.const x'⟧ = ⟦Top.const y⟧, from\n         have Top.const x' = x'_map, by ext p; cases p; refl,\n         begin\n           rw [this, ←hi, hx'], dsimp [x'_class],\n           change (i.hom ∘ i.inv) _ = _,\n           erw i.inv_hom_id, refl\n         end,\n       have ⟦Top.const (f x')⟧ = ⟦Top.const y⟧, from this,\n       let ⟨(γ : path (f x') y)⟩ := quotient.exact this in\n       have is_iso (π_induced n (f x') g), from\n         iso_of_comp_iso_left (hf.2 n x')\n           (by rw ←π_induced_comp; exact hgf.2 n x'),\n       let ⟨i', hi'⟩ := this in\n       ⟨(change_of_basepoint n γ).symm.trans $\n        i'.trans (change_of_basepoint n (γ.induced g)),\n        show (change_of_basepoint n (γ.induced g)).hom ∘\n            i'.hom ∘ (change_of_basepoint n γ).inv =\n          π_induced n y g,\n        by rw [hi', ←change_of_basepoint_induced, iso.inv_hom_id_assoc]⟩⟩,\n  weq_of_comp_weq_right := assume X Y Z f g hg hgf,\n    ⟨iso_of_comp_iso_right hg.1 (by rw ←π₀.map_comp; exact hgf.1),\n     assume n x, iso_of_comp_iso_right (hg.2 n (f x))\n       (by rw ←π_induced_comp; exact hgf.2 n x)⟩ }\n\nlemma is_weak_equivalence_of_heq {X Y : Top} (f : X ⟶ Y)\n  (h : homotopy_equivalence f) : is_weak_equivalence f :=\nlet ⟨g, gf1, fg1⟩ := homotopy_equivalence_iff.mp h in\n⟨⟨⟨π₀ &> f, π₀ &> g,\n   by rw [←π₀.map_comp, ←π₀.map_id]; exact π₀_induced_homotopic gf1,\n   by rw [←π₀.map_comp, ←π₀.map_id]; exact π₀_induced_homotopic fg1⟩,\n  rfl⟩,\n assume n x,\n   have gf : _ := π_induced_homotopic_id n x gf1.symm,\n   have fg : _ := π_induced_homotopic_id n (f x) fg1.symm,\n   begin\n     rw π_induced_comp at gf fg,\n     letI : homotopical_category Set := isomorphisms_as_homotopical_category,\n     change is_weq (π_induced n x f),\n     exact weq_two_out_of_six_f fg gf\n   end⟩\n\nend homotopy_theory.topological_spaces\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/topological_spaces/weak_equivalences.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.43782349911420193, "lm_q1q2_score": 0.3075841734153187}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot, Eric Wieser\n-/\nimport algebra.group.prod\nimport group_theory.group_action.defs\n\n/-!\n# Prod instances for additive and multiplicative actions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for binary product of additive and multiplicative actions and provides\nscalar multiplication as a homomorphism from `α × β` to `β`.\n\n## Main declarations\n\n* `smul_mul_hom`/`smul_monoid_hom`: Scalar multiplication bundled as a multiplicative/monoid\n  homomorphism.\n\n## See also\n\n* `group_theory.group_action.option`\n* `group_theory.group_action.pi`\n* `group_theory.group_action.sigma`\n* `group_theory.group_action.sum`\n-/\n\nvariables {M N P E α β : Type*}\n\nnamespace prod\n\nsection\n\nvariables [has_smul M α] [has_smul M β] [has_smul N α] [has_smul N β] (a : M) (x : α × β)\n\n@[to_additive prod.has_vadd] instance : has_smul M (α × β) := ⟨λa p, (a • p.1, a • p.2)⟩\n\n@[simp, to_additive] theorem smul_fst : (a • x).1 = a • x.1 := rfl\n@[simp, to_additive] theorem smul_snd : (a • x).2 = a • x.2 := rfl\n@[simp, to_additive] theorem smul_mk (a : M) (b : α) (c : β) : a • (b, c) = (a • b, a • c) := rfl\n@[to_additive] theorem smul_def (a : M) (x : α × β) : a • x = (a • x.1, a • x.2) := rfl\n@[simp, to_additive] theorem smul_swap : (a • x).swap = a • x.swap := rfl\n\nlemma smul_zero_mk {α : Type*} [monoid M] [add_monoid α] [distrib_mul_action M α] (a : M) (c : β) :\n  a • ((0 : α), c) = (0, a • c) :=\nby rw [prod.smul_mk, smul_zero]\n\nlemma smul_mk_zero {β : Type*} [monoid M] [add_monoid β] [distrib_mul_action M β] (a : M) (b : α) :\n  a • (b, (0 : β)) = (a • b, 0) :=\nby rw [prod.smul_mk, smul_zero]\n\nvariables [has_pow α E] [has_pow β E]\n@[to_additive has_smul] instance has_pow : has_pow (α × β) E :=\n{ pow := λ p c, (p.1 ^ c, p.2 ^ c) }\n@[simp, to_additive smul_fst, to_additive_reorder 6]\n\n\n@[to_additive]\ninstance [has_smul M N] [is_scalar_tower M N α] [is_scalar_tower M N β] :\n  is_scalar_tower M N (α × β) :=\n⟨λ x y z, mk.inj_iff.mpr ⟨smul_assoc _ _ _, smul_assoc _ _ _⟩⟩\n\n@[to_additive] instance [smul_comm_class M N α] [smul_comm_class M N β] :\n  smul_comm_class M N (α × β) :=\n{ smul_comm := λ r s x, mk.inj_iff.mpr ⟨smul_comm _ _ _, smul_comm _ _ _⟩ }\n\n@[to_additive]\ninstance [has_smul Mᵐᵒᵖ α] [has_smul Mᵐᵒᵖ β] [is_central_scalar M α] [is_central_scalar M β] :\n  is_central_scalar M (α × β) :=\n⟨λ r m, prod.ext (op_smul_eq_smul _ _) (op_smul_eq_smul _ _)⟩\n\n@[to_additive]\ninstance has_faithful_smul_left [has_faithful_smul M α] [nonempty β] :\n  has_faithful_smul M (α × β) :=\n⟨λ x y h, let ⟨b⟩ := ‹nonempty β› in eq_of_smul_eq_smul $ λ a : α, by injection h (a, b)⟩\n\n@[to_additive]\ninstance has_faithful_smul_right [nonempty α] [has_faithful_smul M β] :\n  has_faithful_smul M (α × β) :=\n⟨λ x y h, let ⟨a⟩ := ‹nonempty α› in eq_of_smul_eq_smul $ λ b : β, by injection h (a, b)⟩\n\nend\n\n@[to_additive]\ninstance smul_comm_class_both [has_mul N] [has_mul P] [has_smul M N] [has_smul M P]\n  [smul_comm_class M N N] [smul_comm_class M P P] :\n  smul_comm_class M (N × P) (N × P) :=\n⟨λ c x y, by simp [smul_def, mul_def, mul_smul_comm]⟩\n\ninstance is_scalar_tower_both [has_mul N] [has_mul P] [has_smul M N] [has_smul M P]\n  [is_scalar_tower M N N] [is_scalar_tower M P P] :\n  is_scalar_tower M (N × P) (N × P) :=\n⟨λ c x y, by simp [smul_def, mul_def, smul_mul_assoc]⟩\n\n@[to_additive] instance {m : monoid M} [mul_action M α] [mul_action M β] : mul_action M (α × β) :=\n{ mul_smul  := λ a₁ a₂ p, mk.inj_iff.mpr ⟨mul_smul _ _ _, mul_smul _ _ _⟩,\n  one_smul  := λ ⟨b, c⟩, mk.inj_iff.mpr ⟨one_smul _ _, one_smul _ _⟩ }\n\ninstance {R M N : Type*} [has_zero M] [has_zero N]\n  [smul_zero_class R M] [smul_zero_class R N] : smul_zero_class R (M × N) :=\n{ smul_zero := λ a, mk.inj_iff.mpr ⟨smul_zero _, smul_zero _⟩ }\n\ninstance {R M N : Type*} [add_zero_class M] [add_zero_class N]\n  [distrib_smul R M] [distrib_smul R N] : distrib_smul R (M × N) :=\n{ smul_add  := λ a p₁ p₂, mk.inj_iff.mpr ⟨smul_add _ _ _, smul_add _ _ _⟩ }\n\ninstance {R M N : Type*} {r : monoid R} [add_monoid M] [add_monoid N]\n  [distrib_mul_action R M] [distrib_mul_action R N] : distrib_mul_action R (M × N) :=\n{ ..prod.distrib_smul }\n\ninstance {R M N : Type*} {r : monoid R} [monoid M] [monoid N]\n  [mul_distrib_mul_action R M] [mul_distrib_mul_action R N] : mul_distrib_mul_action R (M × N) :=\n{ smul_mul  := λ a p₁ p₂, mk.inj_iff.mpr ⟨smul_mul' _ _ _, smul_mul' _ _ _⟩,\n  smul_one := λ a, mk.inj_iff.mpr ⟨smul_one _, smul_one _⟩ }\n\nend prod\n\n/-! ### Scalar multiplication as a homomorphism -/\n\nsection bundled_smul\n\n/-- Scalar multiplication as a multiplicative homomorphism. -/\n@[simps]\ndef smul_mul_hom [monoid α] [has_mul β] [mul_action α β] [is_scalar_tower α β β]\n  [smul_comm_class α β β] :\n  (α × β) →ₙ* β :=\n{ to_fun := λ a, a.1 • a.2,\n  map_mul' := λ a b, (smul_mul_smul _ _ _ _).symm }\n\n/-- Scalar multiplication as a monoid homomorphism. -/\n@[simps]\ndef smul_monoid_hom [monoid α] [mul_one_class β] [mul_action α β] [is_scalar_tower α β β]\n  [smul_comm_class α β β] :\n  α × β →* β :=\n{ map_one' := one_smul _ _,\n  .. smul_mul_hom }\n\nend bundled_smul\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/group_theory/group_action/prod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.30743748834923607}}
{"text": "/-\nCopyright (c) 2020 Gabriel Ebner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner\n\n! This file was ported from Lean 3 source module tactic.lint.simp\n! leanprover-community/mathlib commit 7b7cd0a140c51844aa4d5a7d6ea15cb5f6e1afd7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Lint.Basic\n\n/-!\n# Linter for simplification lemmas\n\nThis files defines several linters that prevent common mistakes when declaring simp lemmas:\n\n * `simp_nf` checks that the left-hand side of a simp lemma is not simplified by a different lemma.\n * `simp_var_head` checks that the head symbol of the left-hand side is not a variable.\n * `simp_comm` checks that commutativity lemmas are not marked as simplification lemmas.\n-/\n\n\nopen Tactic Expr\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- `simp_lhs_rhs ty` returns the left-hand and right-hand side of a simp lemma with type `ty`. -/\n    private\n    unsafe\n  def\n    simp_lhs_rhs\n    : expr → tactic ( expr × expr )\n    |\n      ty\n      =>\n      do\n        let ty ← head_beta ty\n          match\n            ty\n            with\n            | q( ¬ $ ( lhs ) ) => pure ( lhs , q( False ) )\n              | q( $ ( lhs ) = $ ( rhs ) ) => pure ( lhs , rhs )\n              | q( $ ( lhs ) ↔ $ ( rhs ) ) => pure ( lhs , rhs )\n              | expr.pi n bi a b => do let l ← mk_local' n bi a simp_lhs_rhs ( b l )\n              | ty => pure ( ty , q( True ) )\n#align simp_lhs_rhs simp_lhs_rhs\n\n/-- `simp_lhs ty` returns the left-hand side of a simp lemma with type `ty`. -/\nprivate unsafe def simp_lhs (ty : expr) : tactic expr :=\n  Prod.fst <$> simp_lhs_rhs ty\n#align simp_lhs simp_lhs\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      `simp_is_conditional_core ty` returns `none` if `ty` is a conditional simp\n      lemma, and `some lhs` otherwise.\n      -/\n    private\n    unsafe\n  def\n    simp_is_conditional_core\n    : expr → tactic ( Option expr )\n    |\n      ty\n      =>\n      do\n        let ty ← head_beta ty\n          match\n            ty\n            with\n            | q( ¬ $ ( lhs ) ) => pure lhs\n              | q( $ ( lhs ) = _ ) => pure lhs\n              | q( $ ( lhs ) ↔ _ ) => pure lhs\n              |\n                expr.pi n bi a b\n                =>\n                do\n                  let l ← mk_local' n bi a\n                    let some lhs ← simp_is_conditional_core ( b l ) | pure none\n                    if\n                      bi ≠ BinderInfo.inst_implicit ∧ ¬ ( lhs l ) . has_var\n                      then\n                      pure none\n                      else\n                      pure lhs\n              | ty => pure ty\n#align simp_is_conditional_core simp_is_conditional_core\n\n/-- `simp_is_conditional ty` returns true iff the simp lemma with type `ty` is conditional.\n-/\nprivate unsafe def simp_is_conditional (ty : expr) : tactic Bool :=\n  Option.isNone <$> simp_is_conditional_core ty\n#align simp_is_conditional simp_is_conditional\n\nprivate unsafe def heuristic_simp_lemma_extraction (prf : expr) : tactic (List Name) :=\n  prf.list_constant.toList.filterM is_simp_lemma\n#align heuristic_simp_lemma_extraction heuristic_simp_lemma_extraction\n\n/-- Checks whether two expressions are equal for the simplifier. That is,\nthey are reducibly-definitional equal, and they have the same head symbol. -/\nunsafe def is_simp_eq (a b : expr) : tactic Bool :=\n  if a.get_app_fn.const_name ≠ b.get_app_fn.const_name then pure false\n  else succeeds <| is_def_eq a b Transparency.reducible\n#align is_simp_eq is_simp_eq\n\n/-- Reports declarations that are simp lemmas whose left-hand side is not in simp-normal form. -/\nunsafe def simp_nf_linter (timeout := 200000) (d : declaration) : tactic (Option String) := do\n  let tt ← is_simp_lemma d.to_name |\n    pure none\n  let-- Sometimes, a definition is tagged @[simp] to add the equational lemmas to the simp set.\n    -- In this case, ignore the declaration if it is not a valid simp lemma by itself.\n    tt\n    ← is_valid_simp_lemma_cnst d.to_name |\n    pure none\n  let [] ← get_eqn_lemmas_for false d.to_name |\n    pure none\n  try_for timeout <|\n      retrieve do\n        let g ← mk_meta_var d\n        set_goals [g]\n        unfreezing intros\n        let (lhs, rhs) ← target >>= simp_lhs_rhs\n        let sls ← simp_lemmas.mk_default\n        let sls' := sls [d]\n        let (lhs', prf1, ns1) ←\n          decorate_error \"simplify fails on left-hand side:\" <|\n              simplify sls [] lhs { failIfUnchanged := ff }\n        let prf1_lems ← heuristic_simp_lemma_extraction prf1\n        if d ∈ prf1_lems then pure none\n          else do\n            let is_cond ← simp_is_conditional d\n            let (rhs', prf2, ns2) ←\n              decorate_error \"simplify fails on right-hand side:\" <|\n                  simplify sls [] rhs { failIfUnchanged := ff }\n            let lhs'_eq_rhs' ← is_simp_eq lhs' rhs'\n            let lhs_in_nf ← is_simp_eq lhs' lhs\n            if lhs'_eq_rhs' then do\n                let used_lemmas ← heuristic_simp_lemma_extraction (prf1 prf2)\n                pure <|\n                    pure <|\n                      \"simp can prove this:\\n\" ++ \"  by simp only \" ++ toString used_lemmas ++\n                            \"\\n\" ++\n                          \"One of the lemmas above could be a duplicate.\\n\" ++\n                        \"If that's not the case try reordering lemmas or adding @[priority].\\n\"\n              else\n                if ¬lhs_in_nf then do\n                  let lhs ← pp lhs\n                  let lhs' ← pp lhs'\n                  pure <|\n                      format.to_string <|\n                        to_fmt \"Left-hand side simplifies from\" ++ lhs 2 ++ format.line ++ \"to\" ++\n                                    lhs' 2 ++\n                                  format.line ++\n                                \"using \" ++\n                              (to_fmt prf1_lems).group.indent 2 ++\n                            format.line ++\n                          \"Try to change the left-hand side to the simplified term!\\n\"\n                else\n                  if ¬is_cond ∧ lhs = lhs' then\n                    pure <|\n                      some <|\n                        \"Left-hand side does not simplify.\\nYou need to debug this yourself using \" ++\n                          \"`set_option trace.simplify.rewrite true`\"\n                  else pure none\n#align simp_nf_linter simp_nf_linter\n\nlibrary_note \"simp-normal form\"/--\nThis note gives you some tips to debug any errors that the simp-normal form linter raises.\n\nThe reason that a lemma was considered faulty is because its left-hand side is not in simp-normal\nform.\nThese lemmas are hence never used by the simplifier.\n\nThis linter gives you a list of other simp lemmas: look at them!\n\nHere are some tips depending on the error raised by the linter:\n\n  1. 'the left-hand side reduces to XYZ':\n     you should probably use XYZ as the left-hand side.\n\n  2. 'simp can prove this':\n     This typically means that lemma is a duplicate, or is shadowed by another lemma:\n\n     2a. Always put more general lemmas after specific ones:\n      ```\n      @[simp] lemma zero_add_zero : 0 + 0 = 0 := rfl\n      @[simp] lemma add_zero : x + 0 = x := rfl\n      ```\n\n      And not the other way around!  The simplifier always picks the last matching lemma.\n\n     2b. You can also use `@[priority]` instead of moving simp-lemmas around in the file.\n\n      Tip: the default priority is 1000.\n      Use `@[priority 1100]` instead of moving a lemma down,\n      and `@[priority 900]` instead of moving a lemma up.\n\n     2c. Conditional simp lemmas are tried last. If they are shadowed\n         just remove the `simp` attribute.\n\n     2d. If two lemmas are duplicates, the linter will complain about the first one.\n         Try to fix the second one instead!\n         (You can find it among the other simp lemmas the linter prints out!)\n\n  3. 'try_for tactic failed, timeout':\n     This typically means that there is a loop of simp lemmas.\n     Try to apply squeeze_simp to the right-hand side (removing this lemma from the simp set) to see\n     what lemmas might be causing the loop.\n\n     Another trick is to `set_option trace.simplify.rewrite true` and\n     then apply `try_for 10000 { simp }` to the right-hand side.  You will\n     see a periodic sequence of lemma applications in the trace message.\n-/\n\n\n/-- A linter for simp lemmas whose lhs is not in simp-normal form, and which hence never fire. -/\n@[linter]\nunsafe def linter.simp_nf : linter where\n  test := simp_nf_linter\n  auto_decls := true\n  no_errors_found := \"All left-hand sides of simp lemmas are in simp-normal form.\"\n  errors_found :=\n    \"SOME SIMP LEMMAS ARE NOT IN SIMP-NORMAL FORM.\\nsee note [simp-normal form] for tips how to debug this.\\nhttps://leanprover-community.github.io/mathlib_docs/notes.html#simp-normal%20form\"\n#align linter.simp_nf linter.simp_nf\n\nprivate unsafe def simp_var_head (d : declaration) : tactic (Option String) := do\n  let tt ← is_simp_lemma d.to_name |\n    pure none\n  let-- Sometimes, a definition is tagged @[simp] to add the equational lemmas to the simp set.\n    -- In this case, ignore the declaration if it is not a valid simp lemma by itself.\n    tt\n    ← is_valid_simp_lemma_cnst d.to_name |\n    pure none\n  let lhs ← simp_lhs d.type\n  let head_sym@(expr.local_const _ _ _ _) ← pure lhs.get_app_fn |\n    pure none\n  let head_sym ← pp head_sym\n  pure <| format.to_string <| \"Left-hand side has variable as head symbol: \" ++ head_sym\n#align simp_var_head simp_var_head\n\n/-- A linter for simp lemmas whose lhs has a variable as head symbol,\nand which hence never fire.\n-/\n@[linter]\nunsafe def linter.simp_var_head : linter\n    where\n  test := simp_var_head\n  auto_decls := true\n  no_errors_found := \"No left-hand sides of a simp lemma has a variable as head symbol.\"\n  errors_found :=\n    \"LEFT-HAND SIDE HAS VARIABLE AS HEAD SYMBOL.\\n\" ++\n      \"Some simp lemmas have a variable as head symbol of the left-hand side:\"\n#align linter.simp_var_head linter.simp_var_head\n\nprivate unsafe def simp_comm (d : declaration) : tactic (Option String) := do\n  let tt ← is_simp_lemma d.to_name |\n    pure none\n  let-- Sometimes, a definition is tagged @[simp] to add the equational lemmas to the simp set.\n    -- In this case, ignore the declaration if it is not a valid simp lemma by itself.\n    tt\n    ← is_valid_simp_lemma_cnst d.to_name |\n    pure none\n  let (lhs, rhs) ← simp_lhs_rhs d.type\n  if lhs ≠ rhs then pure none\n    else do\n      let (lhs', rhs') ← Prod.snd <$> open_pis_metas d >>= simp_lhs_rhs\n      let tt ← succeeds <| unify rhs lhs' transparency.reducible |\n        pure none\n      let tt ← succeeds <| is_def_eq rhs lhs' transparency.reducible |\n        pure none\n      let-- ensure that the second application makes progress:\n        ff\n        ← succeeds <| unify lhs' rhs' transparency.reducible |\n        pure none\n      pure <| \"should not be marked simp\"\n#align simp_comm simp_comm\n\n/-- A linter for commutativity lemmas that are marked simp. -/\n@[linter]\nunsafe def linter.simp_comm : linter where\n  test := simp_comm\n  auto_decls := true\n  no_errors_found := \"No commutativity lemma is marked simp.\"\n  errors_found := \"COMMUTATIVITY LEMMA IS SIMP.\\n\" ++ \"Some commutativity lemmas are simp lemmas:\"\n#align linter.simp_comm linter.simp_comm\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Lint/Simp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.30714101737661853}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sebastian Ullrich\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.control.monad\nimport Mathlib.Lean3Lib.init.meta.interactive\nimport Mathlib.Lean3Lib.init.control.state\nimport Mathlib.Lean3Lib.init.control.except\nimport Mathlib.Lean3Lib.init.control.reader\nimport Mathlib.Lean3Lib.init.control.option\n \n\nuniverses u v l u_1 \n\nnamespace Mathlib\n\nclass is_lawful_functor (f : Type u → Type v) [Functor f] \nwhere\n  map_const_eq : autoParam (∀ {α β : Type u}, Functor.mapConst = Functor.map ∘ function.const β)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  id_map : ∀ {α : Type u} (x : f α), id <$> x = x\n  comp_map : ∀ {α β γ : Type u} (g : α → β) (h : β → γ) (x : f α), (h ∘ g) <$> x = h <$> g <$> x\n\n-- `functor` is indeed a categorical functor\n\n-- `comp_map` does not make a good simp lemma\n\nclass is_lawful_applicative (f : Type u → Type v) [Applicative f] \nextends is_lawful_functor f\nwhere\n  seq_left_eq : autoParam (∀ {α β : Type u} (a : f α) (b : f β), a <* b = function.const β <$> a <*> b)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  seq_right_eq : autoParam (∀ {α β : Type u} (a : f α) (b : f β), a *> b = function.const α id <$> a <*> b)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  pure_seq_eq_map : ∀ {α β : Type u} (g : α → β) (x : f α), pure g <*> x = g <$> x\n  map_pure : ∀ {α β : Type u} (g : α → β) (x : α), g <$> pure x = pure (g x)\n  seq_pure : ∀ {α β : Type u} (g : f (α → β)) (x : α), g <*> pure x = (fun (g : α → β) => g x) <$> g\n  seq_assoc : ∀ {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)), h <*> (g <*> x) = function.comp <$> h <*> g <*> x\n\n-- applicative laws\n\n-- default functor law\n\n-- applicative \"law\" derivable from other laws\n\n@[simp] theorem pure_id_seq {α : Type u} {f : Type u → Type v} [Applicative f] [is_lawful_applicative f] (x : f α) : pure id <*> x = x := sorry\n\nclass is_lawful_monad (m : Type u → Type v) [Monad m] \nextends is_lawful_applicative m\nwhere\n  bind_pure_comp_eq_map : autoParam (∀ {α β : Type u} (f : α → β) (x : m α), x >>= pure ∘ f = f <$> x)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  bind_map_eq_seq : autoParam\n  (∀ {α β : Type u} (f : m (α → β)) (x : m α),\n    (do \n        let _x ← f \n        _x <$> x) =\n      f <*> x)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  pure_bind : ∀ {α β : Type u} (x : α) (f : α → m β), pure x >>= f = f x\n  bind_assoc : ∀ {α β γ : Type u} (x : m α) (f : α → m β) (g : β → m γ), x >>= f >>= g = x >>= fun (x : α) => f x >>= g\n\n-- monad laws\n\n-- monad \"law\" derivable from other laws\n\n@[simp] theorem bind_pure {α : Type u} {m : Type u → Type v} [Monad m] [is_lawful_monad m] (x : m α) : x >>= pure = x := sorry\n\ntheorem bind_ext_congr {α : Type u} {β : Type u} {m : Type u → Type v} [Bind m] {x : m α} {f : α → m β} {g : α → m β} : (∀ (a : α), f a = g a) → x >>= f = x >>= g := sorry\n\ntheorem map_ext_congr {α : Type u} {β : Type u} {m : Type u → Type v} [Functor m] {x : m α} {f : α → β} {g : α → β} : (∀ (a : α), f a = g a) → f <$> x = g <$> x := sorry\n\n-- instances of previously defined monads\n\nnamespace id\n\n\n@[simp] theorem map_eq {α : Type} {β : Type} (x : id α) (f : α → β) : f <$> x = f x :=\n  rfl\n\n@[simp] theorem bind_eq {α : Type} {β : Type} (x : id α) (f : α → id β) : x >>= f = f x :=\n  rfl\n\nend id\n\n\n@[simp] theorem id.pure_eq {α : Type} (a : α) : pure a = a :=\n  rfl\n\nprotected instance id.is_lawful_monad : is_lawful_monad id :=\n  is_lawful_monad.mk (fun (α β : Type u_1) (x : α) (f : α → id β) => Eq.refl (pure x >>= f))\n    fun (α β γ : Type u_1) (x : id α) (f : α → id β) (g : β → id γ) => Eq.refl (x >>= f >>= g)\n\nnamespace state_t\n\n\ntheorem ext {σ : Type u} {m : Type u → Type v} {α : Type u} {x : state_t σ m α} {x' : state_t σ m α} (h : ∀ (st : σ), run x st = run x' st) : x = x' := sorry\n\n@[simp] theorem run_pure {σ : Type u} {m : Type u → Type v} {α : Type u} (st : σ) [Monad m] (a : α) : run (pure a) st = pure (a, st) :=\n  rfl\n\n@[simp] theorem run_bind {σ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u} (x : state_t σ m α) (st : σ) [Monad m] (f : α → state_t σ m β) : run (x >>= f) st =\n  do \n    let p ← run x st \n    run (f (prod.fst p)) (prod.snd p) := sorry\n\n@[simp] theorem run_map {σ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u} (x : state_t σ m α) (st : σ) [Monad m] (f : α → β) [is_lawful_monad m] : run (f <$> x) st = (fun (p : α × σ) => (f (prod.fst p), prod.snd p)) <$> run x st := sorry\n\n@[simp] theorem run_monad_lift {σ : Type u} {m : Type u → Type v} {α : Type u} (st : σ) [Monad m] {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) : run (monad_lift x) st =\n  do \n    let a ← monad_lift x \n    pure (a, st) :=\n  rfl\n\n@[simp] theorem run_monad_map {σ : Type u} {m : Type u → Type v} {α : Type u} (x : state_t σ m α) (st : σ) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) : run (monad_map f x) st = monad_map f (run x st) :=\n  rfl\n\n@[simp] theorem run_adapt {σ : Type u} {m : Type u → Type v} {α : Type u} [Monad m] {σ' : Type u} {σ'' : Type u} (st : σ) (split : σ → σ' × σ'') (join : σ' → σ'' → σ) (x : state_t σ' m α) : run (state_t.adapt split join x) st =\n  (fun (_a : σ' × σ'') =>\n      prod.cases_on _a\n        fun (fst : σ') (snd : σ'') =>\n          idRhs (m (α × σ))\n            (do \n              let _p ← run x fst\n              (fun (_a : α × σ') =>\n                    prod.cases_on _a fun (fst : α) (snd_1 : σ') => idRhs (m (α × σ)) (pure (fst, join snd_1 snd)))\n                  _p))\n    (split st) := sorry\n\n@[simp] theorem run_get {σ : Type u} {m : Type u → Type v} (st : σ) [Monad m] : run state_t.get st = pure (st, st) :=\n  rfl\n\n@[simp] theorem run_put {σ : Type u} {m : Type u → Type v} (st : σ) [Monad m] (st' : σ) : run (state_t.put st') st = pure (PUnit.unit, st') :=\n  rfl\n\nend state_t\n\n\nprotected instance state_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m] (σ : Type u) : is_lawful_monad (state_t σ m) := sorry\n\nnamespace except_t\n\n\ntheorem ext {α : Type u} {ε : Type u} {m : Type u → Type v} {x : except_t ε m α} {x' : except_t ε m α} (h : run x = run x') : x = x' := sorry\n\n@[simp] theorem run_pure {α : Type u} {ε : Type u} {m : Type u → Type v} [Monad m] (a : α) : run (pure a) = pure (except.ok a) :=\n  rfl\n\n@[simp] theorem run_bind {α : Type u} {β : Type u} {ε : Type u} {m : Type u → Type v} (x : except_t ε m α) [Monad m] (f : α → except_t ε m β) : run (x >>= f) = run x >>= except_t.bind_cont f :=\n  rfl\n\n@[simp] theorem run_map {α : Type u} {β : Type u} {ε : Type u} {m : Type u → Type v} (x : except_t ε m α) [Monad m] (f : α → β) [is_lawful_monad m] : run (f <$> x) = except.map f <$> run x := sorry\n\n@[simp] theorem run_monad_lift {α : Type u} {ε : Type u} {m : Type u → Type v} [Monad m] {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) : run (monad_lift x) = except.ok <$> monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {α : Type u} {ε : Type u} {m : Type u → Type v} (x : except_t ε m α) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) : run (monad_map f x) = monad_map f (run x) :=\n  rfl\n\nend except_t\n\n\nprotected instance except_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m] (ε : Type u) : is_lawful_monad (except_t ε m) := sorry\n\nnamespace reader_t\n\n\ntheorem ext {ρ : Type u} {m : Type u → Type v} {α : Type u} {x : reader_t ρ m α} {x' : reader_t ρ m α} (h : ∀ (r : ρ), run x r = run x' r) : x = x' := sorry\n\n@[simp] theorem run_pure {ρ : Type u} {m : Type u → Type v} {α : Type u} (r : ρ) [Monad m] (a : α) : run (pure a) r = pure a :=\n  rfl\n\n@[simp] theorem run_bind {ρ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u} (x : reader_t ρ m α) (r : ρ) [Monad m] (f : α → reader_t ρ m β) : run (x >>= f) r =\n  do \n    let a ← run x r \n    run (f a) r :=\n  rfl\n\n@[simp] theorem run_map {ρ : Type u} {m : Type u → Type v} {α : Type u} {β : Type u} (x : reader_t ρ m α) (r : ρ) [Monad m] (f : α → β) [is_lawful_monad m] : run (f <$> x) r = f <$> run x r :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (run (f <$> x) r = f <$> run x r)) (Eq.symm (bind_pure_comp_eq_map f (run x r)))))\n    (Eq.refl (run (f <$> x) r))\n\n@[simp] theorem run_monad_lift {ρ : Type u} {m : Type u → Type v} {α : Type u} (r : ρ) [Monad m] {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) : run (monad_lift x) r = monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {ρ : Type u} {m : Type u → Type v} {α : Type u} (x : reader_t ρ m α) (r : ρ) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) : run (monad_map f x) r = monad_map f (run x r) :=\n  rfl\n\n@[simp] theorem run_read {ρ : Type u} {m : Type u → Type v} (r : ρ) [Monad m] : run reader_t.read r = pure r :=\n  rfl\n\nend reader_t\n\n\nprotected instance reader_t.is_lawful_monad (ρ : Type u) (m : Type u → Type v) [Monad m] [is_lawful_monad m] : is_lawful_monad (reader_t ρ m) := sorry\n\nnamespace option_t\n\n\ntheorem ext {α : Type u} {m : Type u → Type v} {x : option_t m α} {x' : option_t m α} (h : run x = run x') : x = x' := sorry\n\n@[simp] theorem run_pure {α : Type u} {m : Type u → Type v} [Monad m] (a : α) : run (pure a) = pure (some a) :=\n  rfl\n\n@[simp] theorem run_bind {α : Type u} {β : Type u} {m : Type u → Type v} (x : option_t m α) [Monad m] (f : α → option_t m β) : run (x >>= f) = run x >>= option_t.bind_cont f :=\n  rfl\n\n@[simp] theorem run_map {α : Type u} {β : Type u} {m : Type u → Type v} (x : option_t m α) [Monad m] (f : α → β) [is_lawful_monad m] : run (f <$> x) = option.map f <$> run x := sorry\n\n@[simp] theorem run_monad_lift {α : Type u} {m : Type u → Type v} [Monad m] {n : Type u → Type u_1} [has_monad_lift_t n m] (x : n α) : run (monad_lift x) = some <$> monad_lift x :=\n  rfl\n\n@[simp] theorem run_monad_map {α : Type u} {m : Type u → Type v} (x : option_t m α) [Monad m] {m' : Type u → Type v} {n : Type u → Type u_1} {n' : Type u → Type u_1} [Monad m'] [monad_functor_t n n' m m'] (f : {α : Type u} → n α → n' α) : run (monad_map f x) = monad_map f (run x) :=\n  rfl\n\nend option_t\n\n\nprotected instance option_t.is_lawful_monad (m : Type u → Type v) [Monad m] [is_lawful_monad m] : is_lawful_monad (option_t m) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/control/lawful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.30714101737661853}}
{"text": "import Smt\n\ntheorem cong (p q : Prop) (f : Prop → Prop) : p = q → f p = f q := by\n  smt\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Prop/Cong.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.30712634938359584}}
{"text": "import dialectica \nimport proof\nimport realizers\nimport majorizability\nimport data.equiv.basic\n\nsection \n\n  parameters {ι : Type} {gri : ground_interpretation ι}\n  local notation `𝔽` := formula ι gri\n  local notation `𝕋` := type ι gri\n  variables {greq : Π {i : ι}, ∥𝕏 i // gri ∥ → ∥𝕏 i // gri ∥ → 𝔽}\n  local infixr `≅` : 35 := formula.eqext @greq\n  \n\n  \n  open formula (𝕎_inh' ℂ_inh')\n  set_option pp.proofs true \n  def metatheorem \n    (𝕄 : 𝕋 → 𝕋) [maj_type 𝕄] (maj : majorizability 𝕄) (admissible : admissible_greq @greq)\n    (σ : 𝕋) (A B : ∥𝕆 // gri∥ → ∥σ∥ → 𝔽)\n    (trivialA : ∀ a b x y, (A a b).dia x y ↔ ∥A a b∥)\n    (trivialB : ∀ a b x y, (B a b).dia x y ↔ ∥B a b∥)\n    (Γ : premises')\n    (premise_realizer : Π {γ : 𝔽}, γ ∈ Γ → dia.realizer γ) \n    (prf : proof @greq {with_markov := tt, with_ip := tt} Γ $ ∀∀ x : ∥σ∥, ((∀∀ u : ∥𝕆∥, A u x) ⟹ (∃∃v : ∥𝕆∥, B v x))) :\n    {φ : ∥𝕄 σ∥ → ℕ // ∀ (x : ∥σ∥) (x' : ∥𝕄 σ∥) (_ : maj.majorizes x x'), (∀ u : ℕ, u ≤ φ x' → ∥A u x∥) → (∃ v : ℕ, v ≤ φ x' ∧ ∥B v x∥)}\n    := \n    let ⟨t, h⟩ := prf.dia_realize admissible @premise_realizer in\n    let y : ∥σ∥ → (Σ (x : ∥σ∥), (Π (x_1 : ℕ), (A x_1 x).mwc.fst) × Π (x_1 : ℕ), (B x_1 x).mwc.fst → (B x_1 x).mwc.snd) \n      := λ x, sigma.mk x ⟨(λ _, 𝕎_inh'), (λ _ _, ℂ_inh')⟩ in\n    let t₁ : ∥σ∥ → ℕ := λ x, ((t x).1 (λ _, 𝕎_inh') (λ _ _, ℂ_inh')).1 in\n    let t₂ : ∥σ∥ → ℕ := λ x, ((t x).2 (λ _, 𝕎_inh')).1 in\n    have h' : (∥σ∥ → ℕ) = ∥σ ↣ 𝕆∥ := rfl,\n    let m₁ := maj.majorizer (cast h' t₁) in \n    let m₂ := maj.majorizer (cast h' t₂) in \n    let m₁' := type.cast (maj_type.𝕄_app σ 𝕆) m₁ in\n    let m₂' := type.cast (maj_type.𝕄_app σ 𝕆) m₂ in\n    let m₁'' : ∥𝕄 σ∥ → ℕ := λ x', type.cast maj_type.𝕄_𝕆 (type.cast (maj_type.𝕄_app σ 𝕆) m₁ x') in \n    let m₂'' : ∥𝕄 σ∥ → ℕ := λ x', type.cast maj_type.𝕄_𝕆 (type.cast (maj_type.𝕄_app σ 𝕆) m₂ x') in\n    let Φ : ∥𝕄 σ∥ → ℕ := λ x', (max (m₁'' x') (m₂'' x')) in \n    ⟨Φ, begin \n      intros x x' xmaj hA,\n      use t₂ x,\n      split, {\n        have t₂_le_m₂ := maj.majorizes_le (maj.majorizes_app x x' (cast h' t₂) m₂ xmaj (by simp only [cast_eq]; apply maj.majorizer_majorizes)),\n        -- have : (cast h' t₂) x = t₂ x := by simp,\n        -- rw this at t₂_le_m₂, clear this,\n        have : m₂'' x' ≤ Φ x' := by { dsimp only [Φ], exact le_max_right _ _, },\n        exact le_trans t₂_le_m₂ this, \n      }, { \n        dsimp' at h,\n        simp_rw [trivialA, trivialB] at h,\n        specialize h (y x),\n        dsimp at h,\n        have : ((t x).snd (λ (_x : ℕ), 𝕎_inh')).fst = t₂ x := by refl,\n        rw this at h,\n        apply h,\n        apply hA,\n        have : ((t x).fst (λ (_x : ℕ), 𝕎_inh') (λ (_x : ℕ) (_x_1 : (B _x x).mwc.fst), ℂ_inh')).fst = t₁ x, refl,\n        rw this, clear this,\n        have t₁_le_m₁ := maj.majorizes_le (maj.majorizes_app x x' (cast h' t₁) m₁ xmaj (by simp only [cast_eq]; apply maj.majorizer_majorizes)),\n        -- have : (cast h' t₁) x = t₁ x := by simp,\n        -- rw this at t₁_le_m₁, clear this,\n        have : m₁'' x' ≤ Φ x' := by { dsimp only [Φ], exact le_max_left _ _, },\n        exact le_trans t₁_le_m₁ this,\n      }\n    end⟩\n\nend\n\n#print axioms metatheorem\n  -- open formula (𝕎_inh' ℂ_inh')\n  -- set_option pp.proofs true\n  -- def metatheorem (𝕄 : 𝕋 → 𝕋) [maj_type 𝕄] (maj : majorizability 𝕄) (admissible : admissible_greq @greq) {σ : 𝕋} (A B : ∥𝕆 // gri∥ → ∥σ∥ → 𝔽) (Γ : premises') (premise_realizer : Π {γ : 𝔽}, γ ∈ Γ → dia.realizer γ)\n  -- -- (trivialA : ∀ x u, (A x u).dia 𝕎_inh' ℂ_inh' ↔ ∥A x u∥) (trivialB : ∀ x v, (B x v).dia 𝕎_inh' ℂ_inh' ↔ ∥B x v∥)\n  -- (prf : proof @greq {with_markov := tt, with_ip := tt} Γ (∀∀ x : ∥σ∥, ((∀∀ u : ∥𝕆∥, A u x) ⟹ (∃∃ v : ∥𝕆∥, B v x)))) \n  -- : {φ : ∥𝕄 σ∥ → ∥𝕆 // gri∥ // ∀ (x : ∥σ∥) (x' : ∥𝕄 σ∥) (_ : maj.majorizes x x'), (∀ u : ℕ, u ≤ φ x' → ∥A u x∥) → (∃ v : ℕ, v ≤ φ x' ∧ ∥B v x∥)} :=\n  -- let ⟨t, h⟩ := prf.dia_realize admissible @premise_realizer in \n  -- by {\n  --   dsimp' at t,\n  --   dsimp' at h,\n  --   type_check t,\n  --   have trivialA : ∀ a b x y, (A a b).dia x y ↔ ∥A a b∥ := sorry,\n  --   have trivialB : ∀ a b x y, (B a b).dia x y ↔ ∥B a b∥ := sorry,\n  --   simp_rw [trivialA, trivialB] at h,\n  --   -- set t' := λ y : (Σ (x : ∥σ∥), (Π (x_1 : ℕ), (A x_1 x).mwc.fst) × Π (x_1 : ℕ), (B x_1 x).mwc.fst → (B x_1 x).mwc.snd), t y.fst with ht',\n  --   set y' : ∥σ∥ → (Σ (x : ∥σ∥), (Π (x_1 : ℕ), (A x_1 x).mwc.fst) × Π (x_1 : ℕ), (B x_1 x).mwc.fst → (B x_1 x).mwc.snd) \n  --     := λ x, sigma.mk x ⟨(λ _, 𝕎_inh'), (λ _ _, ℂ_inh')⟩,\n  --   set t₁ := λ x, (t x).1,\n  --   set t₂ := λ x, (t x).2,\n  --   let t₁' : ∥σ∥ → ℕ := λ x, (t₁ x (y' x).2.1 (y' x).2.2).1,\n  --   let t₂' : ∥σ∥ → ℕ := λ x, (t₂ x (y' x).2.1).1,\n  --   let t₁'' : ∥σ∥ → ℕ := λ x, (t₁ x (λ _, 𝕎_inh') (λ _ _, ℂ_inh')).1,\n  --   let t₂'' : ∥σ∥ → ℕ := λ x, (t₂ x (λ _, 𝕎_inh')).1,\n  --   have : (∥σ∥ → ℕ) = ∥σ ↣ 𝕆∥ := sorry,\n  --   let m₁ := (maj.majorizer (cast this t₁'')),\n  --   let m₂ := (maj.majorizer (cast this t₂'')),\n  --   set m₁' := type.cast (maj_type.𝕄_app σ 𝕆) m₁ with eq_m₁',\n  --   set m₂' := type.cast (maj_type.𝕄_app σ 𝕆) m₂ with eq_m₂', \n  --   ss at m₁',\n  --   ss at m₂',\n  --   let Φ : ∥𝕄 σ∥ → ℕ := λ x', type.cast maj_type.𝕄_𝕆 (max (m₁' x') (m₂' x')),\n    \n  --   refine subtype.mk Φ _,\n  --   intros x x' xmaj h',\n  --   use t₂'' x,\n  --   split,\n  --   have := maj.majorizes_app x x' (cast this t₂'') m₂ xmaj sorry,\n  --   have := maj.majorizes_le this,\n  --   dsimp at this,\n  --   rw ←eq_m₂' at this_1,\n  -- }", "meta": {"author": "hcheval", "repo": "formalized-proof-mining", "sha": "216cc73fccd84900a1ba7eaae5f73732496d6afe", "save_path": "github-repos/lean/hcheval-formalized-proof-mining", "path": "github-repos/lean/hcheval-formalized-proof-mining/formalized-proof-mining-216cc73fccd84900a1ba7eaae5f73732496d6afe/src/metatheorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3070245699450206}}
{"text": "import .atom .ldq\n\nvariables {α β : Type}\n\nopen tactic atom_eq_type\n\ndef lift_eq_qe (β : Type) [atom_eq_type α β] (qe : list α → fm α) (as : list α) : fm α := \n  let ntas := @list.filter _ (λ (a : α), ¬ atom_eq_type.trivial β a) \n    begin\n      apply dec_not_pred_of_dec_pred (trivial β),  \n      apply atom_eq_type.dec_triv\n    end as in \n  match @list.first _ (solv0 β) \n        begin apply atom_eq_type.dec_solv0 end ntas with \n  | none := qe ntas \n  | some (eq,rest) := \n    list_conj  (list.map (λ (a : α), A' (subst0 β eq a)) rest)\n  end\n\nlemma leq_qe_prsv [atom_eq_type α β] (qe : list α → fm α) \n  (H : ∀ (as : list α), (∀ (a : α), a ∈ as → atom_type.dep0 β a ∧ ¬ solv0 β a) → qe_prsv β qe as) : \n  ∀ (as : list α), (∀ (a : α), a ∈ as → atom_type.dep0 β a) → qe_prsv β (lift_eq_qe β qe) as := \nbegin\n  intros as Has, unfold qe_prsv, intros bs, \n  unfold lift_eq_qe, simp, \n  cases (@cases_first _ (solv0 β) (dec_solv0 _ β) \n    (@list.filter _ (λ (a : α), ¬trivial β a) \n      (begin apply dec_not_pred_of_dec_pred (trivial β),  \n      apply atom_eq_type.dec_triv end) as)) with HC HC,\n  rewrite HC^.elim_left, \n  unfold lift_eq_qe, unfold qe_prsv at H,\n  apply iff.trans, apply H, intros a Ha, \n  apply and.intro, apply Has, apply mem_of_mem_filter Ha,\n  apply HC^.elim_right, apply Ha, \n  apply ex_iff_ex, intro b, \n  apply iff.intro, intros HL a Ha, \n  cases (dec_triv α β a) with HT HT, apply HL,\n  apply mem_filter_of_mem, apply HT, apply Ha,\n  apply true_triv, apply HT, \n  intros HR a Ha, apply HR, apply mem_of_mem_filter,\n  apply Ha, \n  cases HC with eqn HC, cases HC with as' HC, \n  cases HC with H1 H2, cases H2 with H2 H3, \n  rewrite H1, unfold lift_eq_qe,\n  rewrite exp_I_list_conj, rewrite map_compose, \n  \n  apply iff.intro, intro HL, \n  existsi (list.nth_dft (atom_type.inh α β) bs ((@atom_eq_type.dest_solv0 α β _ eqn H2)-1)),\n  intros a Ha, \n  cases (dec_triv α β a) with HT HT, \n  cases (atom_type.dec_eq α β a eqn) with HE HE,\n  apply (atom_eq_type.subst_prsv H2)^.elim_left,\n  apply HL, apply mem_map_of_mem,\n  have HM : a ∈ (eqn::as'), apply (H3 a)^.elim_left,\n  apply mem_filter_of_mem, apply HT,\n  apply Ha, cases HM with HM HM, \n  apply absurd HM HE, apply HM,\n  rewrite HE,  \n  apply (atom_eq_type.subst_prsv H2)^.elim_left,\n  apply atom_eq_type.true_subst, apply H2,\n  apply true_triv, apply HT,\n  intros HR p Hp, \n  cases (list.exists_of_mem_map Hp) with a Ha,\n  cases Ha with Ha Ha', rewrite Ha', \n  clear Hp, clear Ha', clear p, \n  apply (atom_eq_type.subst_prsv H2)^.elim_right,\n  cases HR with b Hb,\n  have Hbe := atom_eq_type.solv0_eq H2 (Hb eqn _),\n  have H4 := (H3 a)^.elim_right _,\n  rewrite (nth_dft_pred _) at Hbe,\n  rewrite Hbe, apply Hb, \n  apply mem_of_mem_filter H4, \n  apply dest_pos, \n  have H5 := (H3 eqn)^.elim_right _,\n  apply of_mem_filterH5, \n  apply or.inl, refl, apply or.inr Ha,\n  have H5 := (H3 eqn)^.elim_right _,\n  apply mem_of_mem_filter H5, apply or.inl, refl\nend\n\ndef lift_dnfeq_qe (β : Type) [atom_eq_type α β] (qe : list α → fm α) := lift_dnf_qe β (@lift_eq_qe α β _ qe) \n\nlemma leq_qfree [HA : atom_eq_type α β] (qe : list α → fm α)\n(Hqe : ∀ (as' : list α), allp (atom_type.dep0 β) as' → qfree (qe as'))\n(l : list α) (Hl : allp (atom_type.dep0 β) l) : qfree (lift_eq_qe β qe l) := \nbegin\n  unfold lift_eq_qe, simp,\n  cases (list.first (solv0 β) _) with pr,\n  apply Hqe, intros x Hx, apply Hl,\n  apply mem_of_mem_filter Hx, cases pr with eqn as',\n  apply qfree_list_conj, intros a Ha, \n  cases (list.exists_of_mem_map Ha) with a' Ha',\n  rewrite Ha'^.elim_right, simp \nend\n\nlemma ldeq_qfree [atom_eq_type α β] (qe : list α → fm α) \n  (Hqe : ∀ (as') (Has' : allp (@atom_type.dep0 α β _) as'), qfree (qe as')) \n  (p : fm α) : qfree (lift_dnfeq_qe β qe p) := \nbegin\n  unfold lift_dnfeq_qe, apply ldq_qfree, \n  intros as Has, apply leq_qfree, apply Hqe,\n  apply Has\nend\n\nlemma ldeq_prsv [atom_eq_type α β] (qe : list α → fm α) \n  (Hqe : ∀ (as) (Has : allp (@atom_type.dep0 α β _) as), qfree (qe as)) \n  (Has : ∀ (as : list α), (∀ (a' : α), a' ∈ as → atom_type.dep0 β a' ∧ ¬ solv0 β a') → qe_prsv β qe as)  \n  (p : fm α) (bs : list β) : I (lift_dnfeq_qe β qe p) bs ↔ I p bs := \nbegin\n  unfold lift_dnfeq_qe, apply ldq_prsv, apply leq_qfree,\n  apply Hqe, apply leq_qe_prsv, apply Has\nend", "meta": {"author": "avigad", "repo": "qelim", "sha": "b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60", "save_path": "github-repos/lean/avigad-qelim", "path": "github-repos/lean/avigad-qelim/qelim-b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60/common/ldeq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3070085104018831}}
{"text": "import category_theory.products.basic\nimport category_theory.limits.concrete_category\n\nimport for_mathlib.Cech.adjunction\nimport for_mathlib.simplicial.iso\n\nimport polyhedral_lattice.cosimplicial_extra\n\nimport prop_92.prop_92\nimport normed_snake_dual\nimport thm95.double_complex\nimport thm95.col_exact_prep\nimport thm95.double_complex\nimport thm95.pfpng_iso\n\nnoncomputable theory\n\nopen_locale nnreal big_operators nat\nopen category_theory opposite simplex_category\nlocal attribute [instance] type_pow\n\nuniverse variables u v u₀ uₘ\n-- set_option pp.universes true\n\nnamespace thm95\n\nvariables (BD : breen_deligne.data) (κ : ℕ → ℝ≥0) [BD.suitable κ]\nvariables (r r' : ℝ≥0)\nvariables (V : SemiNormedGroup.{v})\nvariables (Λ : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r')\nvariables (N : ℕ) [fact (0 < N)] (n : ℕ)\n\n@[simps obj map]\ndef scale_factorial : system_of_complexes.{u} ⥤ system_of_complexes.{u} :=\n(whiskering_right _ _ _).obj $\nhomological_complex.modify_functor\n  (λ m, SemiNormedGroup.rescale m!) (λ m₁ m₂, SemiNormedGroup.scale _ _)\n.\n\nnamespace scale_factorial\nopen system_of_complexes SemiNormedGroup homological_complex\n\nlemma is_weak_bounded_exact {C : system_of_complexes} {k K : ℝ≥0} [fact (1 ≤ k)] {m : ℕ} {c₀ : ℝ≥0}\n  (hC : C.is_weak_bounded_exact k K m c₀) :\n  (scale_factorial.obj C).is_weak_bounded_exact k (K * (m + 1)) m c₀ :=\nbegin\n  intros c hc i hi x ε hε,\n  let δ := ε * i!,\n  have hδ : 0 < δ := mul_pos hε (nat.cast_pos.2 (nat.factorial_pos i)),\n  have hifact : ¬(↑(i!) : ℝ) = 0 := by exact_mod_cast nat.factorial_ne_zero _,\n  have him : 1 ≤ (↑m + 1) * ((↑i : ℝ) + 1)⁻¹,\n  { refine le_trans _ (mul_le_mul_of_nonneg_right (show (↑i : ℝ) + 1 ≤ (↑m + 1),\n      by rwa [add_le_add_iff_right, nat.cast_le])\n      (inv_nonneg.2 (add_nonneg ((@nat.cast_nonneg ℝ _ i)) zero_le_one))),\n    rw mul_inv_cancel (ne_of_lt (add_pos_of_nonneg_of_pos (@nat.cast_nonneg ℝ _ i) zero_lt_one)).symm },\n  obtain ⟨_, _, rfl, rfl, y, hy⟩ := hC c hc i hi ((of_rescale i!).app _ x) δ hδ,\n  refine ⟨_, _, rfl, rfl, ((SemiNormedGroup.to_rescale (i - 1)!).app _ y), _⟩,\n  erw [rescale.norm_def, rescale.norm_def],\n  simp only [nnreal.coe_nat_cast, nnreal.coe_add, nat.cast_succ, nat.factorial_succ,\n    nat.cast_mul, nnreal.coe_one, nnreal.coe_mul, div_eq_mul_inv],\n  rw [mul_inv_le_iff], swap, { exact_mod_cast nat.factorial_pos i },\n  refine hy.trans _,\n  rw [left_distrib, mul_inv, ← mul_assoc ↑i!, mul_comm ↑i!, mul_assoc _ ↑i!, mul_comm ↑i!,\n    mul_assoc _ _ ↑i!, inv_mul_cancel_right₀ hifact, mul_comm _ ε, add_le_add_iff_right,\n    mul_assoc ↑K],\n  refine mul_le_mul_of_nonneg_left _ (nnreal.coe_nonneg _),\n  rw [mul_comm _ ((↑i : ℝ) + 1)⁻¹, ← mul_assoc],\n  refine le_trans (le_mul_of_one_le_left (by simp only [one_mul, norm_nonneg]) him)\n    (mul_le_mul_of_nonneg_left _ (mul_nonneg (add_nonneg (nat.cast_nonneg m) zero_le_one)\n    (inv_nonneg.2 (add_nonneg (nat.cast_nonneg i) zero_le_one)))),\n  simpa using le_refl _\nend\n\nend scale_factorial\n\nsection\nopen PolyhedralLattice\n\ndef CLCFP' : (ProFiltPseuNormGrpWithTinv r')ᵒᵖ ⥤ ℝ≥0ᵒᵖ ⥤ SemiNormedGroup :=\n(ProFiltPseuNormGrpWithTinv.Pow r' n ⋙ (_root_.Filtration r').flip).op ⋙\n  functor.op_hom _ _ ⋙ (whiskering_right ℝ≥0ᵒᵖ Profiniteᵒᵖ _).obj (CLC V)\n\nlemma CLCFP'_map_app {M₁ M₂ : (ProFiltPseuNormGrpWithTinv r')ᵒᵖ} (f : M₁ ⟶ M₂) (c : ℝ≥0ᵒᵖ) :\n  ((CLCFP' r' V n).map f).app c = (CLCFP V r' c.unop n).map f := rfl\n\nlemma CLCFP'_obj_map (M₁ : (ProFiltPseuNormGrpWithTinv r')ᵒᵖ) (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  ((CLCFP' r' V n).obj M₁).map h =\n    (by haveI : fact (c₂.unop ≤ c₁.unop) := ⟨le_of_hom h.unop⟩;\n      exact (CLCFP.res V r' c₁.unop c₂.unop n).app M₁) :=\nrfl\n\nlemma FLC_arrow_hom' {M₁ M₂ : ProFiltPseuNormGrpWithTinv r'} (f : M₁ ⟶ M₂) (c : ℝ≥0) :\n  (FLC_complex_arrow f.to_comphaus_filtered_pseudo_normed_group_hom f.strict c).hom =\n    ((Filtration r').obj c).map f :=\nrfl\n\nsection\n\nvariables [fact (0 < r')] [fact (r' ≤ 1)]\n\ndef Cech_nerve' :\n  cosimplicial_object.augmented (ℝ≥0ᵒᵖ ⥤ SemiNormedGroup) :=\n(cosimplicial_object.augmented.whiskering_obj.{u} _ _ (CLCFP' r' V n)).obj\n  (Cech_nerve r' Λ M N)\n\nlemma Cech_nerve'_hom_zero :\n  (Cech_nerve'.{u} r' V Λ M N n).hom.app (mk 0) =\n    (CLCFP' r' V n).map (Cech_augmentation_map.{u} r' Λ M N).op :=\nbegin\n  dsimp only [Cech_nerve', cosimplicial_object.augmented.whiskering_obj],\n  simp only [whisker_right_app, category.id_comp, functor.right_op_map, nat_trans.comp_app,\n    functor.const_comp_inv_app, Cech_nerve_hom_zero],\nend\n\n@[simps obj map]\ndef Cech_nerve_level : ℝ≥0 ⥤ simplicial_object.augmented Profinite.{u} :=\n(ProFiltPseuNormGrpWithTinv.Pow r' n ⋙ (Filtration r').flip).flip ⋙\n  (simplicial_object.augmented.whiskering _ _).flip.obj (Cech_nerve.{u} r' Λ M N).left_op\n.\n\nlemma Cech_nerve_level_left_map (c : ℝ≥0) (i j : simplex_categoryᵒᵖ) (g : i ⟶ j) :\n  ((Cech_nerve_level r' Λ M N n).obj c).left.map g =\n    ((Filtration r').obj c).map ((ProFiltPseuNormGrpWithTinv.Pow r' n).map\n      ((Hom M).map ((Cech_conerve (Λ.diagonal_embedding N)).map g.unop).op)) :=\nrfl\n\nlemma Cech_nerve_level_left_map' (c : ℝ≥0) (i j : simplex_categoryᵒᵖ) (g : i ⟶ j)\n  (x : ((Cech_nerve_level r' Λ M N n).obj c).left.obj i) (k : fin n) :\n  (((Cech_nerve_level r' Λ M N n).obj c).left.map g x).1 k =\n    add_monoid_hom.comp (x.1 k) (polyhedral_lattice.conerve.map\n      (Λ.diagonal_embedding N) ⇑(hom.to_order_hom g.unop)).to_add_monoid_hom :=\nrfl\n\nlemma Cech_nerve_level_hom_app (c : ℝ≥0) :\n  ((Cech_nerve_level r' Λ M N n).obj c).hom.app (op (mk 0)) =\n    ((Filtration r').obj c).map\n      ((ProFiltPseuNormGrpWithTinv.Pow r' n).map (Cech_augmentation_map r' Λ M N)) :=\nbegin\n  dsimp only [Cech_nerve_level_obj, nat_trans.comp_app, whisker_right_app,\n    functor.const_comp_hom_app],\n  erw [category.comp_id],\n  dsimp only [nat_trans.left_op_app, unop_op],\n  rw [Cech_nerve_hom_zero],\n  dsimp only [functor.flip_obj_map, functor.comp_map, quiver.hom.unop_op,\n    functor.flip_map_app],\n  refl\nend\n\n@[simps X d]\ndef col_complex_aux : cochain_complex (ℝ≥0ᵒᵖ ⥤ SemiNormedGroup) ℕ :=\n(Cech_nerve' r' V Λ M N n).to_cocomplex\n.\n\n@[simps obj obj_d map]\ndef col_complex_level : system_of_complexes :=\n((whiskering_right _ _ _).obj $ FLC_functor' V).obj (Cech_nerve_level r' Λ M N n).op\n.\n\n@[simps obj obj_d map]\ndef col_complex : system_of_complexes :=\n(col_complex_aux r' V Λ M N n).as_functor\n.\n\ndef col_complex_level_iso_obj_X (c : ℝ≥0ᵒᵖ) :\n  Π (i : ℕ), ((col_complex_level r' V Λ M N n).obj c).X i ≅ ((col_complex r' V Λ M N n).obj c).X i\n| 0     := iso.refl _\n| (i+1) := iso.refl _\n\nlemma col_complex_level_iso_obj_comm (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  (col_complex_level_iso_obj_X r' V Λ M N n c i).hom ≫\n    ((col_complex r' V Λ M N n).obj c).d i (i + 1) =\n  ((col_complex_level r' V Λ M N n).obj c).d i (i + 1) ≫\n    (col_complex_level_iso_obj_X r' V Λ M N n c (i + 1)).hom :=\nbegin\n  cases i,\n  all_goals {\n    dsimp only [col_complex_level_iso_obj_X],\n    simp only [iso.refl_hom, category.id_comp, category.comp_id],\n    dsimp only [col_complex_obj_d, col_complex_level_obj_d,\n      cosimplicial_object.augmented.cocomplex,\n      cosimplicial_object.augmented.to_cocomplex_d_2,\n      cosimplicial_object.augmented.to_cocomplex_d,\n      functor.const_comp_hom_app, functor.const_comp_inv_app,\n      nat_trans.comp_app, whisker_right_app, nat_trans.right_op_app, nat_trans.left_op_app,\n      cosimplicial_object.augmented.drop_obj, cosimplicial_object.coboundary,\n      cosimplicial_object.δ, cosimplicial_object.whiskering_obj_obj_map],\n    rw [dif_pos rfl, eq_to_hom_refl, category.comp_id,\n        dif_pos rfl, eq_to_hom_refl, category.comp_id], },\n  { erw [Cech_nerve'_hom_zero, SemiNormedGroup.LCC_obj_map', category.id_comp],\n    dsimp only [unop_op],\n    erw [Cech_nerve_hom_zero, op_id, category.id_comp],\n    refl },\n  { simp only [nat_trans.app_sum, nat_trans.app_zsmul],\n    apply fintype.sum_congr, intro j, congr' 1,\n    rw [SemiNormedGroup.LCC_obj_map'],\n    refl, }\nend\n\ndef col_complex_level_iso_obj (c : ℝ≥0ᵒᵖ) :\n  (col_complex_level r' V Λ M N n).obj c ≅ (col_complex r' V Λ M N n).obj c :=\nhomological_complex.hom.iso_of_components (col_complex_level_iso_obj_X r' V Λ M N n c)\n(by { rintro i j (rfl : i + 1 = j), apply col_complex_level_iso_obj_comm })\n\nlemma col_complex_level_iso_comm (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) (i : ℕ) :\n  ((col_complex_level r' V Λ M N n).map h ≫ (col_complex_level_iso_obj r' V Λ M N n c₂).hom).f i =\n    ((col_complex_level_iso_obj r' V Λ M N n c₁).hom ≫ (col_complex r' V Λ M N n).map h).f i :=\nbegin\n  cases i,\n  all_goals {\n    dsimp only [col_complex_level_iso_obj, col_complex_level_iso_obj_X,\n      homological_complex.comp_f, homological_complex.hom.iso_of_components_hom_f, iso.refl_hom],\n      rw [category.id_comp, category.comp_id],\n    dsimp only [col_complex_level_map, col_complex_map, whisker_right_app,\n      cosimplicial_object.augmented.cocomplex,\n      cosimplicial_object.augmented.point_map,\n      cosimplicial_object.augmented.drop_map],\n      rw [SemiNormedGroup.LCC_obj_map'], refl },\nend\n\ndef col_complex_level_iso :\n  col_complex_level r' V Λ M N n ≅ col_complex r' V Λ M N n :=\nnat_iso.of_components (col_complex_level_iso_obj r' V Λ M N n)\n(by { intros c₁ c₂ h, ext i : 2, apply col_complex_level_iso_comm })\n\nlemma col_complex_level_iso_strict (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (((col_complex_level_iso r' V Λ M N n).hom.app c).f i) :=\nby { cases i; exact isometry_id }\n\ndef aug_map :=\n((ProFiltPseuNormGrpWithTinv.Pow r' n).map (Cech_augmentation_map r' Λ M N))\n  .to_comphaus_filtered_pseudo_normed_group_hom\n\nsection open comphaus_filtered_pseudo_normed_group_with_Tinv_hom\nlemma aug_map_strict : (aug_map r' Λ M N n).strict :=\nto_profinitely_filtered_pseudo_normed_group_hom_strict _\nend\n\nsection open category_theory.simplicial_object\n\n@[simps left right]\ndef Cech_nerve_level_hom' (c : ℝ≥0) :\n  augmented.to_arrow.obj ((Cech_nerve_level r' Λ M N n).obj c) ⟶\n    FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c :=\n@arrow.hom_mk _ _\n  (augmented.to_arrow.obj ((Cech_nerve_level r' Λ M N n).obj c))\n  (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c)\n  (𝟙 _) (𝟙 _)\nbegin\n  erw [category.comp_id, category.id_comp, augmented.to_arrow_obj_hom],\n  dsimp only [Cech_nerve_level_obj, nat_trans.comp_app, whisker_right_app,\n    functor.const_comp_hom_app, nat_trans.left_op_app, unop_op,\n    functor.flip_obj_map, functor.comp_map, functor.flip_map_app],\n  erw [category.comp_id, Cech_nerve_hom_zero],\n  refl,\nend\n\n@[simps left right]\ndef Cech_nerve_level_inv' (c : ℝ≥0) :\n  FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c ⟶\n    augmented.to_arrow.obj ((Cech_nerve_level r' Λ M N n).obj c) :=\n@arrow.hom_mk _ _\n  (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c)\n  (augmented.to_arrow.obj ((Cech_nerve_level r' Λ M N n).obj c))\n  (𝟙 _) (𝟙 _)\nbegin\n  erw [category.comp_id, category.id_comp, augmented.to_arrow_obj_hom],\n  dsimp only [Cech_nerve_level_obj, nat_trans.comp_app, whisker_right_app,\n    functor.const_comp_hom_app, nat_trans.left_op_app, unop_op,\n    functor.flip_obj_map, functor.comp_map, functor.flip_map_app],\n  erw [category.comp_id, Cech_nerve_hom_zero],\n  refl,\nend\n\n@[simps left right]\ndef Cech_nerve_level_hom (c : ℝ≥0) :\n  (Cech_nerve_level r' Λ M N n).obj c ⟶\n    (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).augmented_cech_nerve :=\nequivalence_left_to_right _ _ $ Cech_nerve_level_hom' _ _ _ _ _ _\n.\n\nlemma Cech_nerve_level_hom_injective' (c : ℝ≥0) (i : simplex_categoryᵒᵖ)\n  (x y: ((((Cech_nerve_level r' Λ M N n).obj c).left.obj i)))\n  (h : ∀ (j : (fin ((unop i).len + 1))),\n    (((Cech_nerve_level r' Λ M N n).obj c).left.map ((unop i).const j).op) x =\n    (((Cech_nerve_level r' Λ M N n).obj c).left.map ((unop i).const j).op) y) : x = y :=\nbegin\n  ext j : 2,\n  let π := (polyhedral_lattice.conerve.π\n    (Λ.diagonal_embedding N) (i.unop.len + 1)).to_add_monoid_hom,\n  have hπ : function.surjective π := polyhedral_lattice.conerve.π_surjective _ _,\n  rw ← add_monoid_hom.cancel_right hπ, -- Bingo!\n  apply finsupp.add_hom_ext',\n  intro k, specialize h k,\n  rw subtype.ext_iff at h,\n  replace h := congr_fun h j,\n  rw add_monoid_hom.ext_iff at h ⊢,\n  intro l, specialize h ((Cech_conerve.obj_zero_iso _).inv l),\n  dsimp only [Cech_nerve_level_obj, Cech_nerve_level_map,\n    whiskering_obj_obj_map, functor.flip_obj_map, functor.flip_map_app,\n    functor.comp_map, functor.left_op_map, quiver.hom.unop_op,\n    Cech_nerve, cosimplicial_object.augmented.whiskering_obj, cosimplicial_object.whiskering,\n    whiskering_right_obj_obj, whiskering_right_obj_map,\n    functor.right_op_map, unop_op,\n    cosimplicial_object.augmented.drop_obj, cosimplicial_object.augmented.drop_map,\n    augmented_cosimplicial, augmented_Cech_conerve,\n    cosimplicial_object.augment_right, Cech_conerve_map,\n    Filtration_obj_map_apply, ProFiltPseuNormGrpWithTinv.Pow_map,\n    Hom_obj, Hom_map_to_fun, polyhedral_lattice.Hom,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.level,\n    profinitely_filtered_pseudo_normed_group_with_Tinv.pi_map_to_fun,\n    profinitely_filtered_pseudo_normed_group_with_Tinv.pi_lift_to_fun,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.comp_to_fun,\n    pseudo_normed_group.level, finsupp.single_add_hom_apply,\n    subtype.coe_mk, add_monoid_hom.coe_mk_from_pi, add_monoid_hom.comp_apply] at h ⊢,\n  erw [Cech_conerve.map_const_obj_zero_iso] at h,\n  exact h,\nend\n.\n\nopen category_theory.limits.wide_pullback\n\nlemma Cech_nerve_level_hom_injective (c : ℝ≥0) (i : simplex_categoryᵒᵖ) :\n  function.injective ⇑((Cech_nerve_level_hom r' Λ M N n c).left.app i) :=\nbegin\n  let F := FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c,\n  let G := (Cech_nerve_level_hom r' Λ M N n c).left.app i,\n  intros x y h,\n  replace h : ∀ j, π (λ i, F.hom) j (G x) = π (λ i, F.hom) j (G y), { intros j, rw h },\n  have aux := λ j, (augmented_cech_nerve.left_map_comp_obj_zero_iso F i.unop j).symm,\n  dsimp only [unop_op] at aux,\n  simp only [← comp_apply, aux] at h,\n  have aux' := (Cech_nerve_level_hom r' Λ M N n c).left.naturality,\n  simp only [← category.assoc, ← aux'] at h, clear aux aux',\n  simp only [category.assoc, Cech_nerve_level_hom, augmented_cech_nerve.left_obj_zero_iso_hom,\n    equivalence_left_to_right_left_app_zero_comp_π, Cech_nerve_level_hom'_left,\n    coe_comp, function.comp, id_apply] at h,\n  apply Cech_nerve_level_hom_injective',\n  exact h\nend\n.\n\nnamespace Cech_nerve_level_hom\n\n/-! The goal is to build a section to `Cech_nerve_level_hom` -/\n\nvariables {r' Λ M N n}\nvariables {c : ℝ≥0} {i : simplex_categoryᵒᵖ}\nvariables (y : (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).augmented_cech_nerve.left.obj i)\n\ndef z₀' := limits.wide_pullback.base\n(λ _, (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).hom) y\n\ndef z' := λ j, limits.wide_pullback.π\n(λ _, (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).hom) j y\n\nlemma hz' : ∀ j, ((FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).hom (z' y j)) = z₀' y :=\nbegin\n  intro j,\n  dsimp only [z₀', z'],\n  rw [← comp_apply, limits.wide_pullback.π_arrow],\nend\n\ndef z₀ := (Cech_nerve_level_inv' r' Λ M N n c).right (z₀' y)\n\ndef z := λ j, (Cech_nerve_level_inv' r' Λ M N n c).left (z' y j)\n\nlemma hz : ∀ j, (augmented.to_arrow.obj ((Cech_nerve_level r' Λ M N n).obj c)).hom (z y j) = z₀ y :=\nbegin\n  intro j, dsimp only [z₀, z],\n  have := (Cech_nerve_level_inv' r' Λ M N n c).w,\n  simp only [functor.id_map, augmented.to_arrow_obj_hom] at this,\n  erw [← comp_apply, this, comp_apply, hz'],\n  refl\nend\n\ndef s' (y : (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).augmented_cech_nerve.left.obj i) :\n  (ProFiltPseuNormGrpWithTinv.Pow r' n).obj\n    ((augmented.drop.obj (Cech_nerve r' Λ M N).left_op).obj i) :=\nbegin\n  refine λ k,\n    cosimplicial_lift Λ N _\n      ((z₀ y).1 k)\n      (λ j, add_monoid_hom.comp ((z y j).1 k)\n        (Cech_conerve.obj_zero_iso (Λ.diagonal_embedding N)).inv.to_add_monoid_hom) _,\n  intros j l,\n  dsimp only [add_monoid_hom.comp_apply, subtype.coe_mk,\n    polyhedral_lattice_hom.coe_to_add_monoid_hom],\n  rw [← hz y j],\n  dsimp only [augmented.to_arrow_obj_hom],\n  rw [Cech_nerve_level_hom_app],\n  dsimp only [Filtration_obj_map_apply, Cech_augmentation_map,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.level,\n    pseudo_normed_group.level,\n    ProFiltPseuNormGrpWithTinv.Pow_map,\n    profinitely_filtered_pseudo_normed_group_with_Tinv.pi_map_to_fun,\n    cosimplicial_augmentation_map,\n    Hom_map_to_fun, add_monoid_hom.comp_apply, subtype.coe_mk,\n    polyhedral_lattice_hom.coe_to_add_monoid_hom, quiver.hom.unop_op,\n    PolyhedralLattice.Cech_augmentation_map],\n  refl,\nend\n.\n\ndef s (y : (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).augmented_cech_nerve.left.obj i) :\n  ((Cech_nerve_level r' Λ M N n).obj c).left.obj i :=\nbegin\n  refine ⟨s' y, _⟩,\n  intros k,\n  apply cosimplicial_lift_mem_filtration,\n  intros j c' l hl,\n  dsimp only [add_monoid_hom.comp_apply, polyhedral_lattice_hom.coe_to_add_monoid_hom],\n  apply (z y j).property,\n  rw [seminormed_add_comm_group.mem_filtration_iff] at hl ⊢,\n  refine le_trans _ hl,\n  exact (Cech_conerve.obj_zero_iso (Λ.diagonal_embedding N)).inv.strict l\nend\n.\n\nset_option pp.universes true\n\nlemma surjective (c : ℝ≥0) (i : simplex_categoryᵒᵖ) :\n  function.surjective ⇑((Cech_nerve_level_hom r' Λ M N n c).left.app i) :=\nbegin\n  let F := FLC_complex_arrow (aug_map r' Λ M N n) _ c,\n  intro y,\n  refine ⟨Cech_nerve_level_hom.s y, _⟩,\n  { apply limits.concrete.wide_pullback_ext'.{0} (λ i, F.hom),\n    rotate, { apply_instance },\n    { haveI := limits.preserves_limits_of_size_shrink.{0 u 0 u} (forget.{u+1} Profinite.{u}),\n      apply_instance, },\n    intro j,\n    erw [← augmented_cech_nerve.left_map_comp_obj_zero_iso _ _ j, ← comp_apply,\n      ← category.assoc, ← (Cech_nerve_level_hom r' Λ M N n c).left.naturality,\n      augmented_cech_nerve.left_map_comp_obj_zero_iso _ _ j, category.assoc],\n    dsimp only [Cech_nerve_level_hom, augmented_cech_nerve.left_obj_zero_iso],\n    erw [equivalence_left_to_right_left_app_zero_comp_π, Cech_nerve_level_hom'_left,\n      category.comp_id],\n    apply subtype.ext, apply funext, rintro k, -- ext k : 2\n    have := Cech_nerve_level_left_map' r' Λ M N n c i _ (i.unop.const j).op,\n    simp only [subtype.val_eq_coe] at this,\n    erw this _ k, clear this,\n    apply add_monoid_hom.ext, rintro l', -- ext1 l'\n    obtain ⟨l, rfl⟩ : ∃ l, quotient_add_group.mk l = l',\n    { exact quotient.surjective_quotient_mk' l' },\n    dsimp only [z₀, z, s', s, add_monoid_hom.comp_apply, subtype.coe_mk,\n      polyhedral_lattice_hom.coe_to_add_monoid_hom,\n      polyhedral_lattice.conerve.map_apply,\n      add_monoid_hom.to_fun_eq_coe,\n      polyhedral_lattice.conerve.map_add_hom_mk,\n      cosimplicial_lift, polyhedral_lattice.conerve.lift'],\n    refine (quotient_add_group.lift_mk'\n      (polyhedral_lattice.conerve.L (Λ.diagonal_embedding N) ((unop i).len + 1)) _ _).trans _,\n    dsimp only [finsupp.lift_add_hom_apply, finsupp.map_domain.add_monoid_hom_apply,\n      unop_op],\n    rw [finsupp.sum_map_domain_index_add_monoid_hom, finsupp.sum_fintype,\n      fin.sum_univ_succ, finset.sum_eq_zero, add_zero],\n    swap, { intros m hm, exfalso, change fin 0 at m, exact nat.not_lt_zero m m.2 },\n    swap, { intro, rw add_monoid_hom.map_zero },\n    dsimp only [add_monoid_hom.comp_apply, subtype.coe_mk,\n      polyhedral_lattice_hom.coe_to_add_monoid_hom,\n      Cech_nerve_level_inv'_left, Cech_conerve.obj_zero_iso,\n      iso.trans_inv, subtype.val_eq_coe, quiver.hom.unop_op],\n    rw [id_apply, comp_apply, Cech_conerve.obj_zero_iso'_inv,\n      Cech_conerve.finsupp_fin_one_iso_inv],\n    suffices : finsupp.single_add_hom 0 (l 0) = l,\n    { cases j, rw this, refl },\n    apply finsupp.ext, rintro m, apply finsupp.ext, rintro o, -- ext m o\n    change fin 1 at m,\n    have hm : m = 0, { exact subsingleton.elim _ _ },\n    simp only [hm, finsupp.single_add_hom_apply, finsupp.single_apply, if_pos rfl], }\nend\n\nend Cech_nerve_level_hom\n\ninstance Cech_nerve_level_hom_is_iso (c : ℝ≥0) : is_iso (Cech_nerve_level_hom r' Λ M N n c) :=\nbegin\n  refine @simplicial_object.augmented.is_iso_of _ _ _ _ _ (id _) (id _),\n  swap, { exact is_iso.id _ },\n  intro i,\n  apply Profinite.is_iso_of_bijective,\n  exact ⟨Cech_nerve_level_hom_injective r' Λ M N n c i,\n        Cech_nerve_level_hom.surjective c i⟩,\nend\n\ndef Cech_nerve_level_iso (c : ℝ≥0) :\n  (Cech_nerve_level r' Λ M N n).obj c ≅\n    (FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).augmented_cech_nerve :=\nas_iso (Cech_nerve_level_hom r' Λ M N n c)\n\ndef FLC_complex_aug_iso_obj (c : ℝ≥0ᵒᵖ) :\n  (FLC_complex V _ (aug_map_strict r' Λ M N n)).obj c ≅ (col_complex_level r' V Λ M N n).obj c :=\n(FLC_functor' V).map_iso (Cech_nerve_level_iso r' Λ M N n c.unop).op\n\ndef FLC_complex_aug_iso :\n  FLC_complex V _ (aug_map_strict r' Λ M N n) ≅ col_complex_level r' V Λ M N n :=\nnat_iso.of_components (FLC_complex_aug_iso_obj r' V Λ M N n)\nbegin\n  intros c₁ c₂ h,\n  dsimp only [FLC_complex, col_complex_level, FLC_functor, FLC_complex_aug_iso_obj,\n    functor.comp_map, functor.map_iso_hom, whiskering_right_obj_obj,\n    Cech_nerve_level_iso, iso.op_hom, as_iso_hom, functor.op_map],\n  simp only [← category_theory.functor.map_comp, ← op_comp, quiver.hom.unop_op],\n  congr' 2,\n  apply simplicial_object.hom_ext,\n  { dsimp only [unop_op],\n    let f := FLC_complex_arrow _ (aug_map_strict r' Λ M N n) (unop c₁),\n    rw ← cancel_mono (augmented_cech_nerve.left_obj_zero_iso f).hom,\n    dsimp only [comma.comp_left, nat_trans.comp_app,\n      Cech_nerve_level_hom, augmented_cech_nerve_map, arrow.map_augmented_cech_nerve_left,\n      cech_nerve_map, arrow.map_cech_nerve_app,\n      arrow.hom_mk_right, arrow.hom_mk_left, augmented_cech_nerve.left_obj_zero_iso_hom],\n    simp only [category.assoc, limits.wide_pullback.lift_π,\n      simplicial_object.equivalence_left_to_right_left_app_zero_comp_π],\n    erw [← category.assoc, simplicial_object.equivalence_left_to_right_left_app_zero_comp_π],\n    refl, },\n  { dsimp only [equivalence_right_to_left_right, comma.comp_right, Cech_nerve_level_hom_right],\n    erw [category.comp_id, category.id_comp],\n    refl, }\nend\n\nend\n\nlemma FLC_complex_aug_iso_strict (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (((FLC_complex_aug_iso r' V Λ M N n).hom.app c).f i) :=\nbegin\n  rw [← iso.app_hom, ← homological_complex.hom.iso_app_hom],\n  refine SemiNormedGroup.iso_isometry_of_norm_noninc _ _ _;\n  [ rw [homological_complex.hom.iso_app_hom, iso.app_hom],\n    rw [homological_complex.hom.iso_app_inv, iso.app_inv] ];\n  { apply cosimplicial_object.augmented.cocomplex_map_norm_noninc;\n    intros; apply SemiNormedGroup.LCC_obj_map_norm_noninc, },\nend\n\ndef col_complex_obj_iso_X_zero (c : ℝ≥0ᵒᵖ) :\n  ((col_complex r' V Λ M N n).obj c).X 0 ≅\n    ((FLC_functor V).obj (op $ FLC_complex_arrow _ (aug_map_strict r' Λ M N n) (c.unop))).X 0 :=\niso.refl _\n\nsection\nopen profinitely_filtered_pseudo_normed_group\n\ndef FLC_arrow_iso_aux :\n  ((ProFiltPseuNormGrpWithTinv.Pow r' n).obj\n    (unop ((Cech_nerve r' Λ M N).right.obj (mk 0)))) ≅\n  ProFiltPseuNormGrpWithTinv.of r' (rescale ↑N (((↥Λ →+ (↥M: Type u)) ^ n) ^ N)) :=\n(ProFiltPseuNormGrpWithTinv.Pow r' n).map_iso\n    (Hom_cosimplicial_zero_iso Λ N r' M N rfl) ≪≫\n  (ProFiltPseuNormGrpWithTinv.Pow_rescale_Pow_iso.{u u} r' N N n).app\n    (polyhedral_lattice.Hom ↥Λ (↥M: Type u))\n.\n\nsection open ProFiltPseuNormGrpWithTinv\n\n--move this\nattribute [simps] linear_equiv.to_add_equiv\n\nlemma FLC_arrow_iso_w (c : ℝ≥0) :\n  ((((Filtration r').obj c).map_iso (FLC_arrow_iso_aux r' Λ M N n)).hom ≫\n    (FLC_complex_arrow _ (sum_hom_strict ((↥Λ →+ ↥M) ^ n) N) c).hom : _) =\n  ((FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c).hom) :=\nbegin\n  rw [← iso.eq_inv_comp],\n  ext x i l,\n  dsimp only [FLC_arrow_iso_aux, iso.trans_inv, functor.map_iso_inv, aug_map,\n    Pow_rescale_Pow_iso, iso.app_inv, iso.symm_inv,\n    nat_trans.comp_app, functor.associator_hom_app, functor.associator_inv_app,\n    iso_whisker_left_inv, iso_whisker_right_inv, whisker_left_app, whisker_right_app,\n    Pow_comm, Pow_rescale, FLC_arrow_hom'],\n  erw [category.id_comp, category.id_comp, FLC_arrow_hom',\n    ← ((Filtration r').obj c).map_comp, category.assoc, ← (Pow r' n).map_comp,\n    Cech_augmentation_map_eq_Hom_sum],\n  dsimp only [FLC_complex_arrow, arrow.mk_hom,\n    Filtration_obj_map_apply, comp_apply,\n    continuous_map.coe_mk, pseudo_normed_group.level, subtype.coe_mk,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.to_comphaus_filtered_pseudo_normed_group_hom,\n    comphaus_filtered_pseudo_normed_group_hom.mk_of_strict,\n    comphaus_filtered_pseudo_normed_group_hom.mk_of_bound,\n    comphaus_filtered_pseudo_normed_group_hom.coe_mk,\n    add_monoid_hom.to_fun_eq_coe,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.coe_to_add_monoid_hom,\n    ProFiltPseuNormGrpWithTinv.Pow_map, nat_iso.of_components_hom_app, id,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.level_coe],\n  erw [comp_apply, comp_apply, iso_of_equiv_of_strict'_hom_apply],\n  dsimp only [profinitely_filtered_pseudo_normed_group_with_Tinv.pi_map_to_fun,\n    ProFiltPseuNormGrpWithTinv.Pow_mul_inv, Pow_comm, Pow_mul_comm,\n    Pow_mul_comm_obj_hom_to_fun, Pow_mul_comm_obj_inv_to_fun,\n    nat_iso.of_components_inv_app, Pow_rescale_aux_apply],\n  simp only [comp_apply, rescale.of, equiv.refl_apply, equiv.refl_symm,\n    nat_iso.of_components_inv_app, id, iso_of_equiv_of_strict'_inv_apply,\n    rescale_map, rescale.map_comphaus_filtered_pseudo_normed_group_with_Tinv_hom_to_fun,\n    ProFiltPseuNormGrpWithTinv.Pow_mul_hom,\n    equiv.symm_apply_apply, ProFiltPseuNormGrpWithTinv.Pow_Pow_X_hom_to_fun,\n    ProFiltPseuNormGrpWithTinv.Pow_Pow_X_inv_to_fun,\n    Pow_Pow_X_equiv_apply, Pow_Pow_X_equiv_symm_apply,\n    Pow_mul_comm_obj_hom_to_fun, Pow_mul_comm_obj_inv_to_fun],\n  rw [sum_hom_apply, Hom_sum_apply, finset.sum_apply],\n  congr' 1,\n  refine fintype.sum_congr _ _ _,\n  intro j,\n  simp only [function.comp_apply, equiv.refl_apply, comp_apply],\n  simp only [Pow_Pow_X_equiv_apply, Pow_Pow_X_equiv_symm_apply,\n    ProFiltPseuNormGrpWithTinv.Pow_Pow_X_hom_to_fun,\n    ProFiltPseuNormGrpWithTinv.Pow_Pow_X_inv_to_fun,\n    Pow_mul_comm_obj_hom_to_fun, Pow_mul_comm_obj_inv_to_fun, comp_apply,\n    equiv.symm_trans_apply, equiv.symm_symm, equiv.curry_apply, equiv.curry_symm_apply,\n    function.curry, function.uncurry, equiv.arrow_congr_symm, equiv.arrow_congr_apply,\n    function.comp, equiv.refl_apply, equiv.refl_symm, equiv.trans_apply,\n    linear_equiv.to_add_equiv_symm_apply, equiv.prod_comm_apply,\n    linear_equiv.inv_fun_eq_symm, equiv.inv_fun_as_coe,\n    linear_equiv.trans_apply, linear_equiv.symm_apply_apply, linear_equiv.symm_trans_apply,\n    linear_equiv.fun_congr_left_symm, linear_equiv.fun_congr_left_apply, linear_map.fun_left_apply,\n    equiv.symm_apply_apply, equiv.apply_symm_apply, equiv.prod_comm_symm],\n  refl\nend\n\nend\n\ndef FLC_arrow_iso (c : ℝ≥0) :\n  FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c ≅\n  FLC_complex_arrow _ (sum_hom_strict ((↥Λ →+ ↥M) ^ n) N) c :=\narrow.iso_mk (((Filtration r').obj c).map_iso (FLC_arrow_iso_aux r' Λ M N n)) (iso.refl _)\nbegin\n  erw [functor.map_iso_hom, iso.refl_hom, category.comp_id],\n  exact (FLC_arrow_iso_w r' Λ M N n c : _),\nend\n\n-- lemma FLC_arrow_iso_left_eq (c₁ c₂ : ℝ≥0) {_ : fact (c₁ ≤ c₂)}\n--   (x : ((FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c₁).left.to_Top)) :\n--   pseudo_normed_group.cast_le (((FLC_arrow_iso r' Λ M N n c₁).hom.left) x) =\n--     ((FLC_arrow_iso r' Λ M N n c₂).hom.left) (pseudo_normed_group.cast_le x) :=\n-- by admit\n\n-- lemma FLC_arrow_iso_right_eq (c₁ c₂ : ℝ≥0) {_ : fact (c₁ ≤ c₂)}\n--   (x : ((FLC_complex_arrow _ (aug_map_strict r' Λ M N n) c₁).right.to_Top)) :\n--   pseudo_normed_group.cast_le (((FLC_arrow_iso r' Λ M N n c₁).hom.right) x) =\n--     ((FLC_arrow_iso r' Λ M N n c₂).hom.right) (pseudo_normed_group.cast_le x) :=\n-- by admit\n\ndef col_complex_iso_obj (c : ℝ≥0ᵒᵖ) :\n  (FLC_complex V _ (sum_hom_strict ((↥Λ →+ ↥M) ^ n) N)).obj c ≅\n    (FLC_complex V _ (aug_map_strict r' Λ M N n)).obj c :=\nbegin\n  refine (cosimplicial_object.augmented.cocomplex.map_iso _),\n  refine functor.map_iso _ _,\n  refine functor.map_iso _ _,\n  refine functor.map_iso _ _,\n  refine iso.op _,\n  exact FLC_arrow_iso r' Λ M N n c.unop\nend\n\ndef col_complex_iso_aux2 :\n  FLC_complex V _ (sum_hom_strict ((↥Λ →+ ↥M) ^ n) N) ≅\n  FLC_complex V _ (aug_map_strict r' Λ M N n) :=\nnat_iso.of_components (col_complex_iso_obj r' V Λ M N n)\nbegin\n  intros c₁ c₂ h,\n  dsimp only [FLC_complex, FLC_functor, FLC_functor', col_complex_iso_obj, functor.comp_map,\n    functor.map_iso_hom, iso.op_hom],\n  simp only [← category_theory.functor.map_comp, ← op_comp],\n  refl\nend\n\nsection open cosimplicial_object\n\nlemma col_complex_iso_aux2_strict (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (((col_complex_iso_aux2 r' V Λ M N n).hom.app c).f i) :=\nbegin\n  rw [← iso.app_hom, ← homological_complex.hom.iso_app_hom],\n  refine SemiNormedGroup.iso_isometry_of_norm_noninc _ _ _;\n  [ rw [homological_complex.hom.iso_app_hom, iso.app_hom],\n    rw [homological_complex.hom.iso_app_inv, iso.app_inv] ];\n  { dsimp only [col_complex_iso_aux2, col_complex_iso_obj, nat_iso.of_components, functor.comp_map,\n      functor.map_iso_hom, iso.op_hom, quiver.hom.unop_op, functor.op_map,\n      cosimplicial_object.augmented.whiskering_obj_2,\n      cosimplicial_object.augmented.whiskering_obj],\n    apply augmented.cocomplex_map_norm_noninc;\n    { intros, apply SemiNormedGroup.LCC_obj_map_norm_noninc, }, },\nend\n\nend\n\ndef col_complex_iso :\n  FLC_complex V _ (sum_hom_strict ((↥Λ →+ ↥M)^n) N) ≅\n  col_complex r' V Λ M N n :=\ncol_complex_iso_aux2 r' V Λ M N n ≪≫ FLC_complex_aug_iso r' V Λ M N n ≪≫ col_complex_level_iso r' V Λ M N n\n\nlemma col_complex_iso_strict (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (((col_complex_iso r' V Λ M N n).hom.app c).f i) :=\nbegin\n  refine isometry.comp _ (col_complex_iso_aux2_strict r' V Λ M N n c i),\n  exact (col_complex_level_iso_strict r' V Λ M N n c i).comp\n    (FLC_complex_aug_iso_strict r' V Λ M N n c i),\nend\n\nend\n\nlemma col_complex.is_weak_bounded_exact (d : ℝ≥0) [pseudo_normed_group.splittable ((Λ →+ M)^n) N d]\n  (k : ℝ≥0) [fact (1 ≤ k)] (m : ℕ) (c₀ : ℝ≥0) (hdkc₀N : d ≤ (k - 1) * c₀ / N) :\n  (col_complex r' V Λ M N n).is_weak_bounded_exact k 1 m c₀ :=\nbegin\n  have key := FLC_complex.weak_bounded_exact V ((Λ →+ M)^n) N d k m c₀ hdkc₀N,\n  exact key.of_iso _ (λ c, col_complex_iso_strict r' V Λ M N n (op c)),\nend\n\n@[simps obj map]\ndef col_complex_rescaled : system_of_complexes :=\nscale_factorial.obj (col_complex r' V Λ M N n)\n.\n\nlemma col_complex_rescaled.is_weak_bounded_exact\n  (d : ℝ≥0) [pseudo_normed_group.splittable ((Λ →+ M)^n) N d]\n  (k : ℝ≥0) [fact (1 ≤ k)] (m : ℕ) (c₀ : ℝ≥0) (hdkc₀N : d ≤ (k - 1) * c₀ / N) :\n  (col_complex_rescaled r' V Λ M N n).is_weak_bounded_exact k (m+1) m c₀ :=\nbegin\n  have := scale_factorial.is_weak_bounded_exact\n    (col_complex.is_weak_bounded_exact r' V Λ M N n d k m c₀ hdkc₀N),\n  rwa [one_mul] at this,\nend\n\nend\n\nend\n\nnamespace col_complex_rescaled\n\nopen polyhedral_lattice (Hom)\nopen PolyhedralLattice (cosimplicial)\n\ninstance move_pls (r' : ℝ≥0) (c : ℝ≥0ᵒᵖ) : fact (unop (r'.MulLeft.op.obj c) ≤ r' * unop c) :=\n⟨le_rfl⟩\n\ninstance move_pls2 [fact (r' < 1)] (c : ℝ≥0ᵒᵖ) : fact (unop (r'.MulLeft.op.obj c) ≤ unop c) :=\nby { dsimp [nnreal.MulLeft], apply_instance }\n\nvariables [fact (0 < r)] [fact (0 < r')] [fact (r' < 1)]\n\ndef T_inv_sub_Tinv_f_succ [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  ((col_complex_rescaled.{u} r' V Λ M N n).obj c).X (i + 1) ⟶\n    (((col_complex_rescaled.{u} r' V Λ M N n).scale_index_left r').obj c).X (i + 1) :=\n(SemiNormedGroup.rescale (i+1)!).map $ (CLCFP.T_inv_sub_Tinv r r' V _ _ n).app _\n\ndef T_inv_sub_Tinv_f [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  Π i, ((col_complex_rescaled.{u} r' V Λ M N n).obj c).X i ⟶\n  (((col_complex_rescaled.{u} r' V Λ M N n).scale_index_left r').obj c).X i\n| 0     := (SemiNormedGroup.rescale 0!).map $ (CLCFP.T_inv_sub_Tinv r r' V _ _ n).app _\n| (i+1) := T_inv_sub_Tinv_f_succ r r' V Λ M N n c i\n\nsection open system_of_complexes category_theory.preadditive category_theory.cosimplicial_object\n  homological_complex\n\nlemma T_inv_sub_Tinv_comm_zero [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  T_inv_sub_Tinv_f r r' V Λ M N n c 0 ≫\n    (((col_complex_rescaled r' V Λ M N n).scale_index_left r').obj c).d 0 1 =\n  ((col_complex_rescaled r' V Λ M N n).obj c).d 0 1 ≫\n    T_inv_sub_Tinv_f r r' V Λ M N n c 1 :=\nbegin\n  dsimp only [T_inv_sub_Tinv_f, T_inv_sub_Tinv_f_succ, scale_index_left, col_complex_rescaled],\n  refine SemiNormedGroup.scale_comm _ _ _ _ _ _ _,\n  dsimp,\n  rw [if_pos rfl, category.comp_id],\n  dsimp only [augmented.to_cocomplex_d],\n  erw [Cech_nerve'_hom_zero],\n  dsimp only [CLCFP'_map_app],\n  erw nat_trans.naturality,\n  refl\nend\n.\n\nlemma T_inv_sub_Tinv_comm_succ [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  T_inv_sub_Tinv_f r r' V Λ M N n c (i+1) ≫\n    (((col_complex_rescaled r' V Λ M N n).scale_index_left r').obj c).d (i+1) (i+2) =\n  ((col_complex_rescaled r' V Λ M N n).obj c).d (i+1) (i+2) ≫\n    T_inv_sub_Tinv_f r r' V Λ M N n c (i+2) :=\nbegin\n  dsimp only [T_inv_sub_Tinv_f, T_inv_sub_Tinv_f_succ, scale_index_left, col_complex_rescaled],\n  refine SemiNormedGroup.scale_comm _ _ _ _ _ _ _,\n  dsimp,\n  rw [if_pos rfl, category.comp_id],\n  dsimp only [augmented.to_cocomplex_d, coboundary, augmented.drop_obj, Cech_nerve',\n    augmented.whiskering_obj_2, augmented.whiskering_obj, cosimplicial_object.δ,\n    whiskering_obj_obj_map],\n  simp only [nat_trans.app_sum, nat_trans.app_zsmul, sum_comp, comp_sum, zsmul_comp, comp_zsmul],\n  apply fintype.sum_congr,\n  intro j,\n  congr' 1,\n  dsimp only [CLCFP'_map_app],\n  erw nat_trans.naturality,\n  refl,\nend\n.\n\nlemma T_inv_sub_Tinv_naturality_zero [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  ((col_complex_rescaled r' V Λ M N n).map h).f 0 ≫ T_inv_sub_Tinv_f r r' V Λ M N n c₂ 0 =\n    T_inv_sub_Tinv_f r r' V Λ M N n c₁ 0 ≫\n      (((col_complex_rescaled r' V Λ M N n).scale_index_left r').map h).f 0 :=\nbegin\n  dsimp only [T_inv_sub_Tinv_f, scale_index_left, ScaleIndexLeft_obj_map,\n    col_complex_rescaled, scale_factorial_obj, functor.comp_map, modify_functor_map_f,\n    col_complex_map, augmented.to_cocomplex_obj, augmented.point_obj, Cech_nerve',\n    augmented.whiskering_obj],\n  simp only [← category_theory.functor.map_comp],\n  congr' 1,\n  dsimp only [CLCFP'_obj_map, CLCFP.T_inv_sub_Tinv, nnreal.MulLeft, functor.op_obj, unop_op],\n  simp only [nat_trans.app_sub, comp_sub, sub_comp, ← nat_trans.comp_app],\n  congr' 2,\n  { rw [category.assoc, ← CLCFP.res_comp_T_inv],\n    haveI : fact (c₂.unop ≤ c₁.unop) := ⟨le_of_hom h.unop⟩,\n    simp only [← category.assoc, CLCFP.res_comp_res], },\n  { dsimp only [CLCFP.Tinv, CLCFP.res, CLC, LC],\n    simp only [← whisker_right_comp, ← nat_trans.op_comp],\n    refl, }\nend\n.\n\nlemma T_inv_sub_Tinv_naturality_succ [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) (i : ℕ) :\n  ((col_complex_rescaled r' V Λ M N n).map h).f (i+1) ≫ T_inv_sub_Tinv_f r r' V Λ M N n c₂ (i+1) =\n    T_inv_sub_Tinv_f r r' V Λ M N n c₁ (i+1) ≫\n      (((col_complex_rescaled r' V Λ M N n).scale_index_left r').map h).f (i+1) :=\nbegin\n  dsimp only [T_inv_sub_Tinv_f, T_inv_sub_Tinv_f_succ, scale_index_left, ScaleIndexLeft_obj_map,\n    col_complex_rescaled, scale_factorial_obj, functor.comp_map, modify_functor_map_f,\n    col_complex_map, augmented.to_cocomplex_obj, augmented.drop_obj, Cech_nerve',\n    augmented.whiskering_obj, whiskering_obj_obj_obj],\n  simp only [← category_theory.functor.map_comp],\n  congr' 1,\n  dsimp only [CLCFP'_obj_map, CLCFP.T_inv_sub_Tinv, nnreal.MulLeft, functor.op_obj, unop_op],\n  simp only [nat_trans.app_sub, comp_sub, sub_comp],\n  erw [← nat_trans.comp_app, ← nat_trans.comp_app, ← nat_trans.comp_app, ← nat_trans.comp_app],\n  congr' 2,\n  { rw [category.assoc, ← CLCFP.res_comp_T_inv],\n    haveI : fact (c₂.unop ≤ c₁.unop) := ⟨le_of_hom h.unop⟩,\n    simp only [← category.assoc, CLCFP.res_comp_res], },\n  { dsimp only [CLCFP.Tinv, CLCFP.res, CLC, LC],\n    simp only [← whisker_right_comp, ← nat_trans.op_comp],\n    refl, }\nend\n\nend\n\ndef T_inv_sub_Tinv [normed_with_aut r V] :\n  col_complex_rescaled r' V Λ M N n ⟶ (col_complex_rescaled r' V Λ M N n).scale_index_left r' :=\n{ app := λ c,\n  { f := T_inv_sub_Tinv_f r r' V Λ M N n c,\n    comm' :=\n    begin\n      rintros i j (rfl : i + 1 = j),\n      cases i, { apply T_inv_sub_Tinv_comm_zero }, { apply T_inv_sub_Tinv_comm_succ }\n    end },\n  naturality' :=\n  begin\n    intros c₁ c₂ h, ext i : 2, cases i,\n    { apply T_inv_sub_Tinv_naturality_zero },\n    { apply T_inv_sub_Tinv_naturality_succ },\n  end }\n\ndef T_inv_sub_Tinv' (c : ℝ≥0) [normed_with_aut r V] :=\n(system_of_complexes.ScaleIndexRight c).map\n  (col_complex_rescaled.T_inv_sub_Tinv r r' V Λ M N n)\n\nend col_complex_rescaled\n\nnamespace double_complex\n\nopen polyhedral_lattice (Hom)\n\nlocal attribute [semireducible] CLCFPTinv CLCFPTinv₂ CLCFP -- CLCTinv\n\nvariables [fact (0 < r)] [fact (0 < r')] [fact (r < r')] [fact (r' < 1)]\n\n@[simps obj map]\ndef col'_aux [normed_with_aut r V] (n : ℕ) : system_of_complexes :=\n(double_complex' BD κ r r' V Λ M N).col n\n\n@[simps obj map]\ndef col' [normed_with_aut r V] (n : ℕ) : system_of_complexes :=\nscale_factorial.obj (col'_aux BD κ r r' V Λ M N n)\n.\n\ndef col_iso_obj_X [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  Π m, (((double_complex.{u} BD κ r r' V Λ M N).col n).obj c).X m ≅\n  ((col'.{u} BD κ r r' V Λ M N n).obj c).X m\n| 0     := (SemiNormedGroup.iso_rescale _).app _\n| 1     := (SemiNormedGroup.iso_rescale _).app _\n| (m+2) := iso.refl _\n\nsection\n\nopen homological_complex system_of_complexes\n\nlemma col_iso_comm₁_0 [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  (col_iso_obj_X BD κ r r' V Λ M N n c 0).hom ≫\n    ((col' BD κ r r' V Λ M N n).obj c).d 0 1 =\n  (((double_complex BD κ r r' V Λ M N).col n).obj c).d 0 1 ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c 1).hom :=\nby { dsimp only [col_iso_obj_X], ext, refl }\n.\n\nlemma col_iso_comm₁_1 [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  (col_iso_obj_X BD κ r r' V Λ M N n c 1).hom ≫\n    ((col' BD κ r r' V Λ M N n).obj c).d 1 2 =\n  (((double_complex BD κ r r' V Λ M N).col n).obj c).d 1 2 ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c 2).hom :=\nbegin\n  delta col_iso_obj_X,\n  simp only [iso.refl_hom, nat.rec_add_one, category.id_comp, category.comp_id,\n    system_of_double_complexes.col_obj_d],\n  dsimp only [col'_obj, col'_aux_obj, double_complex', double_complex,\n    double_complex_aux_rescaled, as_functor_obj, modify_d, eval_obj, eval_map,\n    functor.map_homological_complex_obj_d, functor.map_homological_complex_obj_X,\n    nat_trans.comp_app, comp_f, rescale_functor, rescale_nat_trans,\n    system_of_complexes.rescale, scale],\n  refl\nend\n.\n\nlemma col_iso_comm₂_0 [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  (((double_complex.{u} BD κ r r' V Λ M N).col n).map h).f 0 ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c₂ 0).hom =\n  (col_iso_obj_X BD κ r r' V Λ M N n c₁ 0).hom ≫\n    ((col'.{u} BD κ r r' V Λ M N n).map h).f 0 :=\nby { dsimp only [col_iso_obj_X], ext, refl }\n.\n\nlemma col_iso_comm₂_1 [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  (((double_complex.{u} BD κ r r' V Λ M N).col n).map h).f 1 ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c₂ 1).hom =\n  (col_iso_obj_X BD κ r r' V Λ M N n c₁ 1).hom ≫\n    ((col'.{u} BD κ r r' V Λ M N n).map h).f 1 :=\nby { dsimp only [col_iso_obj_X], ext, refl }\n.\n\nlocal attribute [irreducible] CLCFPTinv CLCFPTinv₂ CLCFP\n  SemiNormedGroup.rescale SemiNormedGroup.scale\n  double_complex_aux\n\nlemma col_iso_comm₁_succ [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  (col_iso_obj_X BD κ r r' V Λ M N n c i.succ.succ).hom ≫\n    ((col'.{u} BD κ r r' V Λ M N n).obj c).d i.succ.succ (i.succ.succ + 1) =\n  (((double_complex.{u} BD κ r r' V Λ M N).col n).obj c).d i.succ.succ (i.succ.succ + 1) ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c (i.succ.succ + 1)).hom :=\nbegin\n  dsimp only [col_iso_obj_X, iso.refl_hom],\n  simp only [category.id_comp, category.comp_id, system_of_double_complexes.col_obj_d],\n  refl,\nend\n.\n\nlemma col_iso_comm₂_succ [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) (i : ℕ) :\n  (((double_complex.{u} BD κ r r' V Λ M N).col n).map h).f (i+2) ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c₂ (i+2)).hom =\n  (col_iso_obj_X BD κ r r' V Λ M N n c₁ (i+2)).hom ≫\n    ((col'.{u} BD κ r r' V Λ M N n).map h).f (i+2) :=\nbegin\n  dsimp only [col_iso_obj_X, iso.refl_hom],\n  simp only [category.id_comp, category.comp_id, system_of_double_complexes.col_obj_d],\n  refl\nend\n\nlemma col_iso_comm₂ [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  ∀ i, (((double_complex.{u} BD κ r r' V Λ M N).col n).map h).f i ≫\n    (col_iso_obj_X BD κ r r' V Λ M N n c₂ i).hom =\n  (col_iso_obj_X BD κ r r' V Λ M N n c₁ i).hom ≫\n    ((col'.{u} BD κ r r' V Λ M N n).map h).f i\n| 0     := col_iso_comm₂_0 _ _ _ _ _ _ _ _ _ _ _ _\n| 1     := col_iso_comm₂_1 _ _ _ _ _ _ _ _ _ _ _ _\n| (i+2) := col_iso_comm₂_succ _ _ _ _ _ _ _ _ _ _ _ _ _\n.\n\nend\n\nsection\n\nopen homological_complex\n\n-- set_option pp.universes true\n\n@[simps]\ndef col_iso [normed_with_aut r V] :\n  (double_complex.{u} BD κ r r' V Λ M N).col n ≅\n  col'.{u} BD κ r r' V Λ M N n :=\nnat_iso.of_components (λ c, hom.iso_of_components (col_iso_obj_X.{u} BD κ r r' V Λ M N n _)\n  begin\n    rintro i j (rfl : i + 1 = j),\n    cases i, { exact col_iso_comm₁_0 BD κ r r' V Λ M N n c, },\n    cases i, { exact col_iso_comm₁_1 BD κ r r' V Λ M N n c, },\n    { exact col_iso_comm₁_succ BD κ r r' V Λ M N n c i, },\n  end)\n  begin\n    intros c₁ c₂ h, ext i : 2,\n    dsimp only [comp_f, hom.iso_of_components_hom_f],\n    exact col_iso_comm₂ BD κ r r' V Λ M N n c₁ c₂ h i,\n  end\n.\n\nlemma col_iso_strict [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (hom.iso_app ((col_iso BD κ r r' V Λ M N n).app c) i).hom :=\nbegin\n  dsimp only [hom.iso_app_hom, nat_iso.app_hom, col_iso_hom_app_f],\n  cases i, { delta col_iso_obj_X, apply SemiNormedGroup.iso_rescale_isometry, exact nat.cast_one },\n  cases i, { delta col_iso_obj_X, apply SemiNormedGroup.iso_rescale_isometry, exact nat.cast_one },\n  { delta col_iso_obj_X, exact isometry_id }\nend\n\nlemma col_iso_inv_strict [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  isometry (hom.iso_app ((col_iso BD κ r r' V Λ M N n).app c) i).inv :=\n(col_iso_strict BD κ r r' V Λ M N n c i).right_inv $\nλ x, by rw [← comp_apply, iso.inv_hom_id, id_apply]\n\nend\n\nlemma col_obj_X_zero [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  (((double_complex BD κ r r' V Λ M N).col n).obj c).X 0 =\n  (CLCFPTinv r V r' (c.unop * κ n) (BD.X n)).obj (op $ Hom Λ M) := rfl\n\n-- local attribute [semireducible] opposite\n\ndef col_ι_f_succ [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  ((col'.{u} BD κ r r' V Λ M N n).obj c).X (i+1) ⟶\n    (((col_complex_rescaled.{u} r' V Λ M N (BD.X n)).scale_index_right (κ n)).obj c).X (i+1) :=\n(SemiNormedGroup.rescale (i+1)!).map (CLCTinv.ι r V _ _)\n\ndef col_ι_f [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  Π i, ((col'.{u} BD κ r r' V Λ M N n).obj c).X i ⟶\n    (((col_complex_rescaled.{u} r' V Λ M N (BD.X n)).scale_index_right (κ n)).obj c).X i\n| 0     := (SemiNormedGroup.rescale 0!).map (CLCTinv.ι r V _ _)\n| (i+1) := col_ι_f_succ _ _ _ _ _ _ _ _ _ _ i\n\nsection open homological_complex system_of_complexes breen_deligne\n\nlemma col_ι_f_comm_zero [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  col_ι_f BD κ r r' V Λ M N n c 0 ≫\n    (((col_complex_rescaled r' V Λ M N (BD.X n)).scale_index_right (κ n)).obj c).d 0 1 =\n  ((col' BD κ r r' V Λ M N n).obj c).d 0 1 ≫ col_ι_f BD κ r r' V Λ M N n c 1 :=\nbegin\n  dsimp only [col_ι_f, col_ι_f_succ, col'_obj, functor.map_homological_complex_obj_d,\n    modify_d, eval_map, scale_index_right, ScaleIndexRight_obj_obj, col_complex_rescaled_obj,\n    scale'_app],\n  refine SemiNormedGroup.scale_comm _ _ _ _ _ _ _,\n  rw [dif_pos rfl, dif_pos rfl],\n  simp only [cosimplicial_object.augmented.to_cocomplex_d, eq_to_hom_refl, category.comp_id],\n  erw [Cech_nerve'_hom_zero, cosimplicial_system_of_complexes_hom_zero],\n  dsimp only [data.system_map, data.complex, data.complex₂_map_f, CLCFPTinv₂, CLCTinv.F_map],\n  symmetry,\n  apply CLCTinv.map_comp_ι,\nend\n\nsection open category_theory.preadditive\n\nlemma col_ι_f_comm_succ [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  col_ι_f BD κ r r' V Λ M N n c (i + 1) ≫\n    (((col_complex_rescaled r' V Λ M N (BD.X n)).scale_index_right (κ n)).obj c).d (i+1) (i+2) =\n  ((col' BD κ r r' V Λ M N n).obj c).d (i + 1) (i + 2) ≫ col_ι_f BD κ r r' V Λ M N n c (i+2) :=\nbegin\n  dsimp only [col_ι_f, col_ι_f_succ, col'_obj, functor.map_homological_complex_obj_d,\n    modify_d, eval_map, scale_index_right, ScaleIndexRight_obj_obj, col_complex_rescaled_obj,\n    scale'_app],\n  refine SemiNormedGroup.scale_comm _ _ _ _ _ _ _,\n  rw [dif_pos rfl, dif_pos rfl],\n  simp only [cosimplicial_object.augmented.to_cocomplex_d, eq_to_hom_refl, category.comp_id,\n    cosimplicial_object.coboundary, nat_trans.app_sum, nat_trans.app_zsmul,\n    ← homological_complex.hom.f_add_monoid_hom_apply, add_monoid_hom.map_sum,\n    add_monoid_hom.map_zsmul, sum_comp, comp_sum, zsmul_comp, comp_zsmul],\n  symmetry,\n  apply fintype.sum_congr,\n  intro j,\n  refl,\nend\n\nend\n\nlemma col_ι_f_comm [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) :\n  ∀ i, col_ι_f BD κ r r' V Λ M N n c i ≫\n    (((col_complex_rescaled r' V Λ M N (BD.X n)).scale_index_right (κ n)).obj c).d i (i + 1) =\n  ((col' BD κ r r' V Λ M N n).obj c).d i (i + 1) ≫ col_ι_f BD κ r r' V Λ M N n c (i + 1)\n| 0     := by apply col_ι_f_comm_zero\n| (i+1) := by apply col_ι_f_comm_succ\n\nsection open category_theory.cosimplicial_object\n\nlemma col_ι_naturality_zero [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) :\n  ((col' BD κ r r' V Λ M N n).map h).f 0 ≫ col_ι_f BD κ r r' V Λ M N n c₂ 0 =\n  col_ι_f BD κ r r' V Λ M N n c₁ 0 ≫ (((col_complex_rescaled r' V Λ M N (BD.X n)).scale_index_right (κ n)).map h).f 0 :=\nbegin\n  dsimp only [col'_map, functor.map_homological_complex_map_f, eval_map,\n    modify_functor_map_f, col_ι_f,\n    scale_index_right, ScaleIndexRight_obj_map, col_complex_rescaled_map],\n  simp only [← category_theory.functor.map_comp],\n  congr' 1,\n  dsimp only [augmented.to_cocomplex_obj, augmented.point_obj, cosimplicial_system_of_complexes,\n    Cech_nerve', augmented.whiskering_obj, CLCFP',\n    data.system_obj, data.complex, data.complex₂_obj_X, CLCFPTinv₂.res, CLCTinv.map_nat,\n    functor.comp_obj, functor.comp_map, functor.op_obj, functor.op_map, whiskering_right_obj_obj,\n    functor.op_hom_obj, unop_op, functor.flip_obj_map],\n  apply CLCTinv.map_comp_ι,\nend\n.\n\nlemma col_ι_naturality_succ [normed_with_aut r V] (c₁ c₂ : ℝ≥0ᵒᵖ) (h : c₁ ⟶ c₂) (i : ℕ) :\n  ((col' BD κ r r' V Λ M N n).map h).f (i+1) ≫ col_ι_f BD κ r r' V Λ M N n c₂ (i+1) =\n  col_ι_f BD κ r r' V Λ M N n c₁ (i+1) ≫ (((col_complex_rescaled r' V Λ M N (BD.X n)).scale_index_right (κ n)).map h).f (i+1) :=\nbegin\n  dsimp only [col'_map, functor.map_homological_complex_map_f, eval_map,\n    modify_functor_map_f, col_ι_f, col_ι_f_succ,\n    scale_index_right, ScaleIndexRight_obj_map, col_complex_rescaled_map],\n  simp only [← category_theory.functor.map_comp],\n  congr' 1,\n  dsimp only [augmented.to_cocomplex_obj, augmented.drop_obj, cosimplicial_system_of_complexes,\n    Cech_nerve', augmented.whiskering_obj, CLCFP', whiskering_obj_obj_obj,\n    data.system_obj, data.complex, data.complex₂_obj_X, CLCFPTinv₂.res, CLCTinv.map_nat,\n    functor.comp_obj, functor.comp_map, functor.op_obj, functor.op_map, whiskering_right_obj_obj,\n    functor.op_hom_obj, unop_op, functor.flip_obj_map],\n  apply CLCTinv.map_comp_ι,\nend\n.\n\nend\n\nend\n\ndef col_ι [normed_with_aut r V] :\n  col'.{u} BD κ r r' V Λ M N n ⟶\n    (col_complex_rescaled.{u} r' V Λ M N (BD.X n)).scale_index_right (κ n) :=\n{ app := λ c,\n  { f := col_ι_f BD κ r r' V Λ M N n c,\n    comm' := by { rintro i j (rfl : i + 1 = j), apply col_ι_f_comm } },\n  naturality' :=\n  begin\n    intros c₁ c₂ h, ext i : 2,\n    cases i,\n    { apply col_ι_naturality_zero },\n    { apply col_ι_naturality_succ },\n  end }\n\nvariables [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ)\n\nlemma col_ι_range :\n  (((double_complex.col_ι BD κ r r' V Λ M N n).app c).f i).range =\n  (((col_complex_rescaled.T_inv_sub_Tinv' r r' V Λ M N (BD.X n) (κ n)).app c).f i).ker :=\nbegin\n  cases i;\n  { refine SemiNormedGroup.rescale_exact _ _ _ _,\n    rw CLCTinv.ι_range',\n    ext1 x,\n    refl, },\nend\n\nlemma col_ι_isometry :\n  isometry (((double_complex.col_ι BD κ r r' V Λ M N n).app c).f i) :=\nbegin\n  cases i;\n  { refine SemiNormedGroup.rescale_map_isometry _ _,\n    apply add_monoid_hom_class.isometry_of_norm,\n    intro, refl },\nend\n\nend double_complex\n\nnamespace col_complex_rescaled\n\nvariables [fact (0 < r')] [fact (r' ≤ 1)]\n\nlemma d_zero_norm_noninc (c : ℝ≥0) :\n  (@system_of_complexes.d (col_complex_rescaled r' V Λ M N n) c 0 1).norm_noninc :=\nbegin\n  apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.2,\n  refine normed_add_group_hom.norm_comp_le_of_le' (1:ℕ) _ 1 _ (SemiNormedGroup.norm_scale_le _ _ _) _,\n  { simp only [mul_one, nat.factorial_zero, nat.factorial_one, nat.cast_one, div_one, nnreal.coe_one]},\n  apply SemiNormedGroup.norm_rescale_map_le,\n  have : (1 : ℝ) = ∑ i : fin 1, 1,\n  { simp only [finset.card_fin, mul_one, finset.sum_const, nsmul_eq_mul, nat.cast_id,\n      nat.cast_bit1, nat.cast_add, nat.cast_one] },\n  dsimp [system_of_complexes.rescale_functor, double_complex_aux,\n    cosimplicial_object.augmented.to_cocomplex_d],\n  erw [category.comp_id, if_pos rfl, Cech_nerve'_hom_zero, nat.cast_one],\n  apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.1,\n  apply CLC.map_norm_noninc,\nend\n\nlemma d_succ_norm_noninc (c : ℝ≥0) (p : ℕ) :\n  (@system_of_complexes.d (col_complex_rescaled r' V Λ M N n) c (p+1) (p+2)).norm_noninc :=\nbegin\n  apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.2,\n  refine normed_add_group_hom.norm_comp_le_of_le' (p+2:ℕ) _ 1 _ (SemiNormedGroup.norm_scale_le _ _ _) _,\n  { norm_cast,\n    rw [mul_comm, ← mul_div_assoc, eq_comm, ← nat.cast_mul, nat.factorial_succ], apply div_self,\n    norm_num [nat.factorial_ne_zero] },\n  apply SemiNormedGroup.norm_rescale_map_le,\n  have : (p+1+1 : ℝ) = ∑ i : fin (p+1+1), 1,\n  { simp only [finset.card_fin, mul_one, finset.sum_const, nsmul_eq_mul, nat.cast_id,\n      nat.cast_bit1, nat.cast_add, nat.cast_one] },\n  dsimp [system_of_complexes.rescale_functor, double_complex_aux,\n    cosimplicial_object.augmented.to_cocomplex_d],\n  erw [category.comp_id, if_pos rfl],\n  dsimp [cosimplicial_object.coboundary],\n  rw [add_assoc, show (1 : ℝ)+1=2, by norm_num] at this,\n  simp only [← nat_trans.app_hom_apply, add_monoid_hom.map_sum, add_monoid_hom.map_zsmul,\n    ← homological_complex.hom.f_add_monoid_hom_apply, nat.cast_add, nat.cast_bit0, nat.cast_one,\n    this],\n  apply norm_sum_le_of_le,\n  rintro i -,\n  refine le_trans (norm_zsmul_le _ _) _,\n  rw [← int.norm_cast_real, int.cast_pow, norm_pow, int.cast_neg, int.cast_one, norm_neg, norm_one,\n    one_pow, one_mul],\n  apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.1,\n  apply CLC.map_norm_noninc\nend\n\nlemma admissible : (col_complex_rescaled r' V Λ M N n).admissible :=\n{ d_norm_noninc' :=\n  begin\n    rintro c i j rfl, cases i,\n    { apply d_zero_norm_noninc, },\n    { apply d_succ_norm_noninc, }\n  end,\n  res_norm_noninc :=\n  begin\n    intros c₁ c₂ i h,\n    apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.2,\n    cases i;\n    { apply SemiNormedGroup.norm_rescale_map_le,\n      apply normed_add_group_hom.norm_noninc.norm_noninc_iff_norm_le_one.1,\n      apply CLC.map_norm_noninc, },\n  end }\n\nend col_complex_rescaled\n\nvariables [fact (0 < r)] [fact (0 < r')] [fact (r' < 1)]\n\nlemma col_exact'_aux1 [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  ∀ x, ∥(((col_complex_rescaled.T_inv_sub_Tinv' r r' V Λ M N (BD.X n) (κ n)).app c).f i) x∥ ≤\n    (1 + r⁻¹) * ∥x∥ :=\nbegin\n  intro x,\n  cases i;\n  { apply normed_add_group_hom.le_of_op_norm_le,\n    refine @SemiNormedGroup.norm_rescale_map_le _ _ _ _ _ (1 + r⁻¹) _,\n    exact CLCFP.norm_T_inv_sub_Tinv_le _ _ _ _ _ _ _ }\nend\n\nvariables [fact (r < r')]\n\nlemma col_exact'_aux2 [normed_with_aut r V] (c : ℝ≥0ᵒᵖ) (i : ℕ) :\n  ∀ y, ∃ x,\n    (((col_complex_rescaled.T_inv_sub_Tinv' r r' V Λ M N (BD.X n) (κ n)).app c).f i) x = y ∧\n    ∥x∥ ≤ ↑(r / (1 - r) + 1) * ∥y∥ :=\nbegin\n  haveI : fact (r < 1) := ⟨@lt_trans _ _ r r' 1 (fact.out _) (fact.out _)⟩,\n  cases i;\n  { refine SemiNormedGroup.rescale_exists_norm_le _ _ _ _,\n    intro y,\n    obtain ⟨x, h1, h2⟩ := CLCFP.T_inv_sub_Tinv_exists_preimage r r' y 1 zero_lt_one,\n    refine ⟨x, h1, _⟩,\n    rwa [nnreal.coe_add, nnreal.coe_div, nnreal.coe_sub],\n    exact fact.out _ }\nend\n\nlemma col_exact' [normed_with_aut r V] [fact (r < 1)]\n  (d : ℝ≥0) [pseudo_normed_group.splittable (Λ →+ M) N d]\n  (k : ℝ≥0) [fact (1 ≤ k)] (m : ℕ) (c₀ : ℝ≥0) (hdkc₀N : d ≤ (k - 1) * c₀ / N)\n  (k' K' : ℝ≥0) [fact (1 ≤ k')] (hk' : k * k ≤ k')\n  (hK' : (m + 2 + (r + 1) / (r * (1 - r)) * (m + 2)^2 : ℝ≥0) ≤ K')\n  (c₁ c₂ : ℝ≥0) [fact (c₀ ≤ r' * c₁)] [fact (c₀ ≤ κ n * c₂)] [fact (c₁ ≤ κ n * c₂)] :\n  (double_complex.col'.{u} BD κ r r' V Λ M N n).is_weak_bounded_exact k' K' m c₂ :=\nbegin\n  have adm := (col_complex_rescaled.admissible.{u} r' V Λ M N (BD.X n)),\n  have adm2 := adm.scale_index_left r',\n  let T_T := col_complex_rescaled.T_inv_sub_Tinv' r r' V Λ M N (BD.X n) (κ n),\n  have H := (col_complex_rescaled.is_weak_bounded_exact.{u}\n    r' V Λ M N (BD.X n) d k (m+1) c₀ hdkc₀N),\n  have H1 := H.scale_index_right _ c₂ (κ n) adm,\n  have H2 := (H.scale_index_left _ c₁ r' adm).scale_index_right _ c₂ (κ n) adm2,\n  have key := weak_normed_snake_dual\n    (double_complex.col'.{u} BD κ r r' V Λ M N n) _ _\n    (double_complex.col_ι BD κ r r' V Λ M N n) T_T\n    k _ ((m + 1) + 1) _ (1 + r⁻¹) (r / (1 - r) + 1) (by simpa using H1) H2 (adm.scale_index_right _),\n  have h_isom : _ := _,\n  apply (key _ _ _ h_isom).of_le _ ⟨hk'⟩ _ le_rfl ⟨le_rfl⟩,\n  any_goals { clear key adm2 H1 H2 },\n  { intros, apply col_exact'_aux1 },\n  { intros, apply col_exact'_aux2 },\n  { intros c i, apply double_complex.col_ι_range },\n  { apply system_of_complexes.admissible_of_isometry (adm.scale_index_right _) h_isom, },\n  { refine ⟨le_trans (le_of_eq _) hK'⟩,\n    rw [pow_two, ← mul_assoc],\n    simp only [nat.cast_add, nat.cast_one, bit0, ← add_assoc, or_false, add_eq_zero_iff,\n      one_ne_zero, add_right_inj, mul_eq_mul_right_iff, and_false, div_eq_mul_inv],\n    have hr0 : r ≠ 0, { exact ne_of_gt (fact.out _) },\n    have h1r : 1 - r ≠ 0,\n    { rw [← nnreal.coe_injective.ne_iff, nnreal.coe_sub, nnreal.coe_zero, sub_ne_zero],\n      { norm_cast, exact ne_of_gt (fact.out _) },\n      { exact fact.out _ } },\n    calc (1 + r⁻¹) * (r * (1 - r)⁻¹ + 1)\n        = (1 + r⁻¹) * (r * (1 - r)⁻¹ + (1 - r) * (1 - r)⁻¹) : by rw [mul_inv_cancel h1r]\n    ... = (1 + r⁻¹) * (1 - r)⁻¹ : congr_arg _ _\n    ... = (1 + r⁻¹) * (r * r⁻¹) * (1 - r)⁻¹ : by rw [mul_inv_cancel hr0, mul_one]\n    ... = ((1 + r⁻¹) * r) * (r⁻¹ * (1 - r)⁻¹) : by simp only [mul_assoc]\n    ... = (r + 1) * (r * (1 - r))⁻¹ : _,\n    { rw [← add_mul, add_comm, tsub_add_cancel_of_le, one_mul], exact fact.out _ },\n    { rw [add_mul, one_mul, inv_mul_cancel, mul_inv],\n      exact ne_of_gt (fact.out _) } },\n  { intros c i, apply double_complex.col_ι_isometry, }\nend\n\nlemma col_exact [normed_with_aut r V] [fact (r < 1)]\n  (d : ℝ≥0) [pseudo_normed_group.splittable (Λ →+ M) N d]\n  (k : ℝ≥0) [fact (1 ≤ k)] (m : ℕ) (c₀ : ℝ≥0) (hdkc₀N : d ≤ (k - 1) * c₀ / N)\n  (k' K' : ℝ≥0) [fact (1 ≤ k')] (hk' : k * k ≤ k')\n  (hK' : (m + 2 + (r + 1) / (r * (1 - r)) * (m + 2)^2 : ℝ≥0) ≤ K')\n  (c₁ c₂ : ℝ≥0) (_ : fact (c₀ ≤ r' * c₁)) (_ : fact (c₀ ≤ κ n * c₂)) (_ : fact (c₁ ≤ κ n * c₂)) :\n  ((double_complex.{u} BD κ r r' V Λ M N).col n).is_weak_bounded_exact k' K' m c₂ :=\nbegin\n  have key := col_exact' BD κ r r' V Λ M N n d k m c₀ hdkc₀N k' K' hk' hK' c₁ c₂,\n  apply key.of_iso (double_complex.col_iso BD κ r r' V Λ M N n).symm,\n  intros,\n  apply double_complex.col_iso_inv_strict\nend\n\nend thm95\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/thm95/col_exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.30700851040188304}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.shapes.terminal\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.images\nimport category_theory.isomorphism_classes\n\n/-!\n# Zero morphisms and zero objects\n\nA category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. (Notice this is extra\nstructure, not merely a property.)\n\nA category \"has a zero object\" if it has an object which is both initial and terminal. Having a\nzero object provides zero morphisms, as the unique morphisms factoring through the zero object.\n\n## References\n\n* https://en.wikipedia.org/wiki/Zero_morphism\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\n\nnamespace category_theory.limits\n\nvariables (C : Type u) [category.{v} C]\n\n/-- A category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. -/\nclass has_zero_morphisms :=\n[has_zero : Π X Y : C, has_zero (X ⟶ Y)]\n(comp_zero' : ∀ {X Y : C} (f : X ⟶ Y) (Z : C), f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) . obviously)\n(zero_comp' : ∀ (X : C) {Y Z : C} (f : Y ⟶ Z), (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) . obviously)\n\nattribute [instance] has_zero_morphisms.has_zero\nrestate_axiom has_zero_morphisms.comp_zero'\nrestate_axiom has_zero_morphisms.zero_comp'\n\nvariables {C}\n\n@[simp] lemma comp_zero [has_zero_morphisms C] {X Y : C} {f : X ⟶ Y} {Z : C} :\n  f ≫ (0 : Y ⟶ Z) = (0 : X ⟶ Z) := has_zero_morphisms.comp_zero f Z\n@[simp] lemma zero_comp [has_zero_morphisms C] {X : C} {Y Z : C} {f : Y ⟶ Z} :\n  (0 : X ⟶ Y) ≫ f = (0 : X ⟶ Z) := has_zero_morphisms.zero_comp X f\n\ninstance has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) :=\n{ has_zero := by tidy }\n\ninstance has_zero_morphisms_punit : has_zero_morphisms (discrete punit) :=\n{ has_zero := by tidy }\n\nnamespace has_zero_morphisms\nvariables {C}\n\n/-- This lemma will be immediately superseded by `ext`, below. -/\nprivate lemma ext_aux (I J : has_zero_morphisms C)\n  (w : ∀ X Y : C, (@has_zero_morphisms.has_zero _ _ I X Y).zero =\n    (@has_zero_morphisms.has_zero _ _ J X Y).zero) : I = J :=\nbegin\n  casesI I, casesI J,\n  congr,\n  { ext X Y,\n    exact w X Y },\n  { apply proof_irrel_heq, },\n  { apply proof_irrel_heq, }\nend\n\n/--\nIf you're tempted to use this lemma \"in the wild\", you should probably\ncarefully consider whether you've made a mistake in allowing two\ninstances of `has_zero_morphisms` to exist at all.\n\nSee, particularly, the note on `zero_morphisms_of_zero_object` below.\n-/\nlemma ext (I J : has_zero_morphisms C) : I = J :=\nbegin\n  apply ext_aux,\n  intros X Y,\n  rw ←@has_zero_morphisms.comp_zero _ _ I X X (@has_zero_morphisms.has_zero _ _ J X X).zero,\n  rw @has_zero_morphisms.zero_comp _ _ J,\nend\n\ninstance : subsingleton (has_zero_morphisms C) :=\n⟨ext⟩\n\nend has_zero_morphisms\n\nopen has_zero_morphisms\n\nsection\nvariables {C} [has_zero_morphisms C]\n\nlemma zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [mono g] (h : f ≫ g = 0) : f = 0 :=\nby { rw [←zero_comp, cancel_mono] at h, exact h }\n\nlemma zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [epi f] (h : f ≫ g = 0) : g = 0 :=\nby { rw [←comp_zero, cancel_epi] at h, exact h }\n\nlemma eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [has_image f] (w : image.ι f = 0) : f = 0 :=\nby rw [←image.fac f, w, has_zero_morphisms.comp_zero]\n\nlemma nonzero_image_of_nonzero {X Y : C} {f : X ⟶ Y} [has_image f] (w : f ≠ 0) : image.ι f ≠ 0 :=\nλ h, w (eq_zero_of_image_eq_zero h)\nend\n\nsection\nuniverses v' u'\nvariables (D : Type u') [category.{v'} D]\n\nvariables [has_zero_morphisms D]\n\ninstance : has_zero_morphisms (C ⥤ D) :=\n{ has_zero := λ F G, ⟨{ app := λ X, 0, }⟩ }\n\n@[simp] lemma zero_app (F G : C ⥤ D) (j : C) : (0 : F ⟶ G).app j = 0 := rfl\n\nvariables [has_zero_morphisms C]\n\nlemma equivalence_preserves_zero_morphisms (F : C ≌ D) (X Y : C) :\n  F.functor.map (0 : X ⟶ Y) = (0 : F.functor.obj X ⟶ F.functor.obj Y) :=\nbegin\n  have t : F.functor.map (0 : X ⟶ Y) =\n    F.functor.map (0 : X ⟶ Y) ≫ (0 : F.functor.obj Y ⟶ F.functor.obj Y),\n  { apply faithful.map_injective (F.inverse),\n    rw [functor.map_comp, equivalence.inv_fun_map],\n    dsimp,\n    rw [zero_comp, comp_zero, zero_comp], },\n  exact t.trans (by simp)\nend\n\n@[simp] lemma is_equivalence_preserves_zero_morphisms (F : C ⥤ D) [is_equivalence F] (X Y : C) :\n  F.map (0 : X ⟶ Y) = 0 :=\nby rw [←functor.as_equivalence_functor F, equivalence_preserves_zero_morphisms]\n\nend\n\nvariables (C)\n\n/-- A category \"has a zero object\" if it has an object which is both initial and terminal. -/\nclass has_zero_object :=\n(zero : C)\n(unique_to : Π X : C, unique (zero ⟶ X))\n(unique_from : Π X : C, unique (X ⟶ zero))\n\ninstance has_zero_object_punit : has_zero_object (discrete punit) :=\n{ zero := punit.star,\n  unique_to := by tidy,\n  unique_from := by tidy, }\n\nvariables {C}\n\nnamespace has_zero_object\n\nvariables [has_zero_object C]\n\n/--\nConstruct a `has_zero C` for a category with a zero object.\nThis can not be a global instance as it will trigger for every `has_zero C` typeclass search.\n-/\nprotected def has_zero : has_zero C :=\n{ zero := has_zero_object.zero }\n\nlocal attribute [instance] has_zero_object.has_zero\nlocal attribute [instance] has_zero_object.unique_to has_zero_object.unique_from\n\n@[ext]\nlemma to_zero_ext {X : C} (f g : X ⟶ 0) : f = g :=\nby rw [(has_zero_object.unique_from X).uniq f, (has_zero_object.unique_from X).uniq g]\n\n@[ext]\nlemma from_zero_ext {X : C} (f g : 0 ⟶ X) : f = g :=\nby rw [(has_zero_object.unique_to X).uniq f, (has_zero_object.unique_to X).uniq g]\n\ninstance (X : C) : subsingleton (X ≅ 0) := by tidy\n\ninstance {X : C} (f : 0 ⟶ X) : mono f :=\n{ right_cancellation := λ Z g h w, by ext, }\n\ninstance {X : C} (f : X ⟶ 0) : epi f :=\n{ left_cancellation := λ Z g h w, by ext, }\n\n/-- A category with a zero object has zero morphisms.\n\n    It is rarely a good idea to use this. Many categories that have a zero object have zero\n    morphisms for some other reason, for example from additivity. Library code that uses\n    `zero_morphisms_of_zero_object` will then be incompatible with these categories because\n    the `has_zero_morphisms` instances will not be definitionally equal. For this reason library\n    code should generally ask for an instance of `has_zero_morphisms` separately, even if it already\n    asks for an instance of `has_zero_objects`. -/\ndef zero_morphisms_of_zero_object : has_zero_morphisms C :=\n{ has_zero := λ X Y,\n  { zero := inhabited.default (X ⟶ 0) ≫ inhabited.default (0 ⟶ Y) },\n  zero_comp' := λ X Y Z f, by { dunfold has_zero.zero, rw category.assoc, congr, },\n  comp_zero' := λ X Y Z f, by { dunfold has_zero.zero, rw ←category.assoc, congr, }}\n\n/-- A zero object is in particular initial. -/\nlemma has_initial : has_initial C :=\nhas_initial_of_unique 0\n/-- A zero object is in particular terminal. -/\nlemma has_terminal : has_terminal C :=\nhas_terminal_of_unique 0\n\nend has_zero_object\n\nsection\nvariables [has_zero_object C] [has_zero_morphisms C]\nlocal attribute [instance] has_zero_object.has_zero\n\n@[simp]\nlemma id_zero : 𝟙 (0 : C) = (0 : 0 ⟶ 0) :=\nby ext\n\n/--  An arrow ending in the zero object is zero -/\n-- This can't be a `simp` lemma because the left hand side would be a metavariable.\nlemma zero_of_to_zero {X : C} (f : X ⟶ 0) : f = 0 :=\nby ext\n\nlemma zero_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 :=\nbegin\n  have h : f = f ≫ i.hom ≫ 𝟙 0 ≫ i.inv := by simp only [iso.hom_inv_id, id_comp, comp_id],\n  simpa using h,\nend\n\n/-- An arrow starting at the zero object is zero -/\nlemma zero_of_from_zero {X : C} (f : 0 ⟶ X) : f = 0 :=\nby ext\n\nlemma zero_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : f = 0 :=\nbegin\n  have h : f = i.hom ≫ 𝟙 0 ≫ i.inv ≫ f := by simp only [iso.hom_inv_id_assoc, id_comp, comp_id],\n  simpa using h,\nend\n\nlemma zero_of_source_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic X 0) : f = 0 :=\nzero_of_source_iso_zero f (nonempty.some i)\nlemma zero_of_target_iso_zero' {X Y : C} (f : X ⟶ Y) (i : is_isomorphic Y 0) : f = 0 :=\nzero_of_target_iso_zero f (nonempty.some i)\n\nlemma mono_of_source_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) : mono f :=\n⟨λ Z g h w, by rw [zero_of_target_iso_zero g i, zero_of_target_iso_zero h i]⟩\n\nlemma epi_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : epi f :=\n⟨λ Z g h w, by rw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]⟩\n\n/--\nAn object `X` has `𝟙 X = 0` if and only if it is isomorphic to the zero object.\n\nBecause `X ≅ 0` contains data (even if a subsingleton), we express this `↔` as an `≃`.\n-/\ndef id_zero_equiv_iso_zero (X : C) : (𝟙 X = 0) ≃ (X ≅ 0) :=\n{ to_fun    := λ h, { hom := 0, inv := 0, },\n  inv_fun   := λ i, zero_of_target_iso_zero (𝟙 X) i,\n  left_inv  := by tidy,\n  right_inv := by tidy, }\n\n@[simp]\nlemma id_zero_equiv_iso_zero_apply_hom (X : C) (h : 𝟙 X = 0) :\n  ((id_zero_equiv_iso_zero X) h).hom = 0 := rfl\n\n@[simp]\nlemma id_zero_equiv_iso_zero_apply_inv (X : C) (h : 𝟙 X = 0) :\n  ((id_zero_equiv_iso_zero X) h).inv = 0 := rfl\n\n/-- If an object `X` is isomorphic to 0, there's no need to use choice to construct\nan explicit isomorphism: the zero morphism suffices. -/\ndef iso_of_is_isomorphic_zero {X : C} (P : is_isomorphic X 0) : X ≅ 0 :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' :=\n  begin\n    casesI P,\n    rw ←P.hom_inv_id,\n    rw ←category.id_comp P.inv,\n    simp,\n  end,\n  inv_hom_id' := by simp, }\n\nend\n\nsection is_iso\nvariables [has_zero_morphisms C]\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\nthe identities on both `X` and `Y` are zero.\n-/\n@[simps]\ndef is_iso_zero_equiv (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (𝟙 X = 0 ∧ 𝟙 Y = 0) :=\n{ to_fun := by { introsI i, rw ←is_iso.hom_inv_id (0 : X ⟶ Y),\n    rw ←is_iso.inv_hom_id (0 : X ⟶ Y), simp },\n  inv_fun := λ h, ⟨⟨(0 : Y ⟶ X), by tidy⟩⟩,\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\nthe identity on `X` is zero.\n-/\ndef is_iso_zero_self_equiv (X : C) : is_iso (0 : X ⟶ X) ≃ (𝟙 X = 0) :=\nby simpa using is_iso_zero_equiv X X\n\nvariables [has_zero_object C]\nlocal attribute [instance] has_zero_object.has_zero\n\n/--\nA zero morphism `0 : X ⟶ Y` is an isomorphism if and only if\n`X` and `Y` are isomorphic to the zero object.\n-/\ndef is_iso_zero_equiv_iso_zero (X Y : C) : is_iso (0 : X ⟶ Y) ≃ (X ≅ 0) × (Y ≅ 0) :=\nbegin\n  -- This is lame, because `prod` can't cope with `Prop`, so we can't use `equiv.prod_congr`.\n  refine (is_iso_zero_equiv X Y).trans _,\n  symmetry,\n  fsplit,\n  { rintros ⟨eX, eY⟩, fsplit,\n    exact (id_zero_equiv_iso_zero X).symm eX,\n    exact (id_zero_equiv_iso_zero Y).symm eY, },\n  { rintros ⟨hX, hY⟩, fsplit,\n    exact (id_zero_equiv_iso_zero X) hX,\n    exact (id_zero_equiv_iso_zero Y) hY, },\n  { tidy, },\n  { tidy, },\nend\n\n/--\nA zero morphism `0 : X ⟶ X` is an isomorphism if and only if\n`X` is isomorphic to the zero object.\n-/\ndef is_iso_zero_self_equiv_iso_zero (X : C) : is_iso (0 : X ⟶ X) ≃ (X ≅ 0) :=\n(is_iso_zero_equiv_iso_zero X X).trans subsingleton_prod_self_equiv\n\nend is_iso\n\n/-- If there are zero morphisms, any initial object is a zero object. -/\n@[priority 50]\ninstance has_zero_object_of_has_initial_object\n  [has_zero_morphisms C] [has_initial C] : has_zero_object C :=\n{ zero := ⊥_ C,\n  unique_to := λ X, ⟨⟨0⟩, by tidy⟩,\n  unique_from := λ X, ⟨⟨0⟩, λ f,\n  calc\n    f = f ≫ 𝟙 _ : (category.comp_id _).symm\n    ... = f ≫ 0 : by congr\n    ... = 0     : has_zero_morphisms.comp_zero _ _\n  ⟩ }\n\n/-- If there are zero morphisms, any terminal object is a zero object. -/\n@[priority 50]\ninstance has_zero_object_of_has_terminal_object\n  [has_zero_morphisms C] [has_terminal C] : has_zero_object C :=\n{ zero := ⊤_ C,\n  unique_from := λ X, ⟨⟨0⟩, by tidy⟩,\n  unique_to := λ X, ⟨⟨0⟩, λ f,\n  calc\n    f = 𝟙 _ ≫ f : (category.id_comp _).symm\n    ... = 0 ≫ f : by congr\n    ... = 0     : zero_comp\n  ⟩ }\n\n\nsection image\nvariable [has_zero_morphisms C]\n\nlemma image_ι_comp_eq_zero {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} [has_image f]\n  [epi (factor_thru_image f)] (h : f ≫ g = 0) : image.ι f ≫ g = 0 :=\nzero_of_epi_comp (factor_thru_image f) $ by simp [h]\n\nvariables [has_zero_object C]\nlocal attribute [instance] has_zero_object.has_zero\n\n/--\nThe zero morphism has a `mono_factorisation` through the zero object.\n-/\n@[simps]\ndef mono_factorisation_zero (X Y : C) : mono_factorisation (0 : X ⟶ Y) :=\n{ I := 0, m := 0, e := 0, }\n\n/--\nThe factorisation through the zero object is an image factorisation.\n-/\ndef image_factorisation_zero (X Y : C) : image_factorisation (0 : X ⟶ Y) :=\n{ F := mono_factorisation_zero X Y,\n  is_image := { lift := λ F', 0 } }\n\n\ninstance has_image_zero {X Y : C} : has_image (0 : X ⟶ Y) :=\nhas_image.mk $ image_factorisation_zero _ _\n\n/-- The image of a zero morphism is the zero object. -/\ndef image_zero {X Y : C} : image (0 : X ⟶ Y) ≅ 0 :=\nis_image.iso_ext (image.is_image (0 : X ⟶ Y)) (image_factorisation_zero X Y).is_image\n\n/-- The image of a morphism which is equal to zero is the zero object. -/\ndef image_zero' {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] : image f ≅ 0 :=\nimage.eq_to_iso h ≪≫ image_zero\n\n@[simp]\nlemma image.ι_zero {X Y : C} [has_image (0 : X ⟶ Y)] : image.ι (0 : X ⟶ Y) = 0 :=\nbegin\n  rw ←image.lift_fac (mono_factorisation_zero X Y),\n  simp,\nend\n\n/--\nIf we know `f = 0`,\nit requires a little work to conclude `image.ι f = 0`,\nbecause `f = g` only implies `image f ≅ image g`.\n-/\n@[simp]\nlemma image.ι_zero' [has_equalizers C] {X Y : C} {f : X ⟶ Y} (h : f = 0) [has_image f] :\n  image.ι f = 0 :=\nby { rw image.eq_fac h, simp }\n\nend image\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance split_mono_sigma_ι\n  {β : Type v} [decidable_eq β]\n  [has_zero_morphisms C]\n  (f : β → C) [has_colimit (discrete.functor f)] (b : β) : split_mono (sigma.ι f b) :=\n{ retraction := sigma.desc (λ b', if h : b' = b then eq_to_hom (congr_arg f h) else 0), }\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance split_epi_pi_π\n  {β : Type v} [decidable_eq β]\n  [has_zero_morphisms C]\n  (f : β → C) [has_limit (discrete.functor f)] (b : β) : split_epi (pi.π f b) :=\n{ section_ := pi.lift (λ b', if h : b = b' then eq_to_hom (congr_arg f h) else 0), }\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance split_mono_coprod_inl\n  [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :\n  split_mono (coprod.inl : X ⟶ X ⨿ Y) :=\n{ retraction := coprod.desc (𝟙 X) 0, }\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\ninstance split_mono_coprod_inr\n  [has_zero_morphisms C] {X Y : C} [has_colimit (pair X Y)] :\n  split_mono (coprod.inr : Y ⟶ X ⨿ Y) :=\n{ retraction := coprod.desc 0 (𝟙 Y), }\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance split_epi_prod_fst\n  [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :\n  split_epi (prod.fst : X ⨯ Y ⟶ X) :=\n{ section_ := prod.lift (𝟙 X) 0, }\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\ninstance split_epi_prod_snd\n  [has_zero_morphisms C] {X Y : C} [has_limit (pair X Y)] :\n  split_epi (prod.snd : X ⨯ Y ⟶ Y) :=\n{ section_ := prod.lift 0 (𝟙 Y), }\n\nend category_theory.limits\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/shapes/zero.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.30697441436485495}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Johan Commelin, Bhavik Mehta\n-/\nimport category_theory.equivalence\n\n/-!\n# Adjunctions between functors\n\n`F ⊣ G` represents the data of an adjunction between two functors\n`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.\n\nWe provide various useful constructors:\n* `mk_of_hom_equiv`\n* `mk_of_unit_counit`\n* `left_adjoint_of_equiv` / `right_adjoint_of equiv`\n  construct a left/right adjoint of a given functor given the action on objects and\n  the relevant equivalence of morphism spaces.\n* `adjunction_of_equiv_left` / `adjunction_of_equiv_right` witness that these constructions\n  give adjunctions.\n\nThere are also typeclasses `is_left_adjoint` / `is_right_adjoint`, carrying data witnessing\nthat a given functor is a left or right adjoint.\nGiven `[is_left_adjoint F]`, a right adjoint of `F` can be constructed as `right_adjoint F`.\n\n`adjunction.comp` composes adjunctions.\n\n`to_equivalence` upgrades an adjunction to an equivalence,\ngiven witnesses that the unit and counit are pointwise isomorphisms.\nConversely `equivalence.to_adjunction` recovers the underlying adjunction from an equivalence.\n-/\n\nnamespace category_theory\nopen category\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nlocal attribute [elab_simple] whisker_left whisker_right\n\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]\n\n/--\n`F ⊣ G` represents the data of an adjunction between two functors\n`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.\n\nTo construct an `adjunction` between two functors, it's often easier to instead use the\nconstructors `mk_of_hom_equiv` or `mk_of_unit_counit`. To construct a left adjoint,\nthere are also constructors `left_adjoint_of_equiv` and `adjunction_of_equiv_left` (as\nwell as their duals) which can be simpler in practice.\n\nUniqueness of adjoints is shown in `category_theory.adjunction.opposites`.\n\nSee https://stacks.math.columbia.edu/tag/0037.\n-/\nstructure adjunction (F : C ⥤ D) (G : D ⥤ C) :=\n(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\n(unit : 𝟭 C ⟶ F.comp G)\n(counit : G.comp F ⟶ 𝟭 D)\n(hom_equiv_unit' : Π {X Y f}, (hom_equiv X Y) f = (unit : _ ⟶ _).app X ≫ G.map f . obviously)\n(hom_equiv_counit' : Π {X Y g}, (hom_equiv X Y).symm g = F.map g ≫ counit.app Y . obviously)\n\ninfix ` ⊣ `:15 := adjunction\n\n/-- A class giving a chosen right adjoint to the functor `left`. -/\nclass is_left_adjoint (left : C ⥤ D) :=\n(right : D ⥤ C)\n(adj : left ⊣ right)\n\n/-- A class giving a chosen left adjoint to the functor `right`. -/\nclass is_right_adjoint (right : D ⥤ C) :=\n(left : C ⥤ D)\n(adj : left ⊣ right)\n\n/-- Extract the left adjoint from the instance giving the chosen adjoint. -/\ndef left_adjoint (R : D ⥤ C) [is_right_adjoint R] : C ⥤ D :=\nis_right_adjoint.left R\n/-- Extract the right adjoint from the instance giving the chosen adjoint. -/\ndef right_adjoint (L : C ⥤ D) [is_left_adjoint L] : D ⥤ C :=\nis_left_adjoint.right L\n\n/-- The adjunction associated to a functor known to be a left adjoint. -/\ndef adjunction.of_left_adjoint (left : C ⥤ D) [is_left_adjoint left] :\n  adjunction left (right_adjoint left) :=\nis_left_adjoint.adj\n/-- The adjunction associated to a functor known to be a right adjoint. -/\ndef adjunction.of_right_adjoint (right : C ⥤ D) [is_right_adjoint right] :\n  adjunction (left_adjoint right) right :=\nis_right_adjoint.adj\n\nnamespace adjunction\n\nrestate_axiom hom_equiv_unit'\nrestate_axiom hom_equiv_counit'\nattribute [simp, priority 10] hom_equiv_unit hom_equiv_counit\n\nsection\n\nvariables {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) {X' X : C} {Y Y' : D}\n\n@[simp, priority 10] lemma hom_equiv_naturality_left_symm (f : X' ⟶ X) (g : X ⟶ G.obj Y) :\n  (adj.hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (adj.hom_equiv X Y).symm g :=\nby rw [hom_equiv_counit, F.map_comp, assoc, adj.hom_equiv_counit.symm]\n\n@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :\n  (adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=\nby rw [← equiv.eq_symm_apply]; simp [-hom_equiv_unit]\n\n@[simp, priority 10] lemma hom_equiv_naturality_right (f : F.obj X ⟶ Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y') (f ≫ g) = (adj.hom_equiv X Y) f ≫ G.map g :=\nby rw [hom_equiv_unit, G.map_comp, ← assoc, ←hom_equiv_unit]\n\n@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=\nby rw [equiv.symm_apply_eq]; simp [-hom_equiv_counit]\n\n@[simp] lemma left_triangle :\n  (whisker_right adj.unit F) ≫ (whisker_left F adj.counit) = nat_trans.id _ :=\nbegin\n  ext, dsimp,\n  erw [← adj.hom_equiv_counit, equiv.symm_apply_eq, adj.hom_equiv_unit],\n  simp\nend\n\n@[simp] \n\n@[simp, reassoc] lemma left_triangle_components :\n  F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X) :=\ncongr_arg (λ (t : nat_trans _ (𝟭 C ⋙ F)), t.app X) adj.left_triangle\n\n@[simp, reassoc] lemma right_triangle_components {Y : D} :\n  adj.unit.app (G.obj Y) ≫ G.map (adj.counit.app Y) = 𝟙 (G.obj Y) :=\ncongr_arg (λ (t : nat_trans _ (G ⋙ 𝟭 C)), t.app Y) adj.right_triangle\n\n@[simp, reassoc] lemma counit_naturality {X Y : D} (f : X ⟶ Y) :\n  F.map (G.map f) ≫ (adj.counit).app Y = (adj.counit).app X ≫ f :=\nadj.counit.naturality f\n\n@[simp, reassoc] lemma unit_naturality {X Y : C} (f : X ⟶ Y) :\n  (adj.unit).app X ≫ G.map (F.map f) = f ≫ (adj.unit).app Y :=\n(adj.unit.naturality f).symm\n\nlemma hom_equiv_apply_eq {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :\n  adj.hom_equiv A B f = g ↔ f = (adj.hom_equiv A B).symm g :=\n⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩\n\nlemma eq_hom_equiv_apply {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :\n  g = adj.hom_equiv A B f ↔ (adj.hom_equiv A B).symm g = f :=\n⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩\n\nend\n\nend adjunction\n\nnamespace adjunction\n\n/--\nThis is an auxiliary data structure useful for constructing adjunctions.\nSee `adjunction.mk_of_hom_equiv`.\nThis structure won't typically be used anywhere else.\n-/\n@[nolint has_inhabited_instance]\nstructure core_hom_equiv (F : C ⥤ D) (G : D ⥤ C) :=\n(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\n(hom_equiv_naturality_left_symm' : Π {X' X Y} (f : X' ⟶ X) (g : X ⟶ G.obj Y),\n  (hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (hom_equiv X Y).symm g . obviously)\n(hom_equiv_naturality_right' : Π {X Y Y'} (f : F.obj X ⟶ Y) (g : Y ⟶ Y'),\n  (hom_equiv X Y') (f ≫ g) = (hom_equiv X Y) f ≫ G.map g . obviously)\n\nnamespace core_hom_equiv\n\nrestate_axiom hom_equiv_naturality_left_symm'\nrestate_axiom hom_equiv_naturality_right'\nattribute [simp, priority 10] hom_equiv_naturality_left_symm hom_equiv_naturality_right\n\nvariables {F : C ⥤ D} {G : D ⥤ C} (adj : core_hom_equiv F G) {X' X : C} {Y Y' : D}\n\n@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :\n  (adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=\nby rw [← equiv.eq_symm_apply]; simp\n\n@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=\nby rw [equiv.symm_apply_eq]; simp\n\nend core_hom_equiv\n\n/--\nThis is an auxiliary data structure useful for constructing adjunctions.\nSee `adjunction.mk_of_unit_counit`.\nThis structure won't typically be used anywhere else.\n-/\n@[nolint has_inhabited_instance]\nstructure core_unit_counit (F : C ⥤ D) (G : D ⥤ C) :=\n(unit : 𝟭 C ⟶ F.comp G)\n(counit : G.comp F ⟶ 𝟭 D)\n(left_triangle' : whisker_right unit F ≫ (functor.associator F G F).hom ≫ whisker_left F counit =\n  nat_trans.id (𝟭 C ⋙ F) . obviously)\n(right_triangle' : whisker_left G unit ≫ (functor.associator G F G).inv ≫ whisker_right counit G =\n  nat_trans.id (G ⋙ 𝟭 C) . obviously)\n\nnamespace core_unit_counit\n\nrestate_axiom left_triangle'\nrestate_axiom right_triangle'\nattribute [simp] left_triangle right_triangle\n\nend core_unit_counit\n\nvariables {F : C ⥤ D} {G : D ⥤ C}\n\n/-- Construct an adjunction between `F` and `G` out of a natural bijection between each\n`F.obj X ⟶ Y` and `X ⟶ G.obj Y`. -/\n@[simps]\ndef mk_of_hom_equiv (adj : core_hom_equiv F G) : F ⊣ G :=\n{ unit :=\n  { app := λ X, (adj.hom_equiv X (F.obj X)) (𝟙 (F.obj X)),\n    naturality' :=\n    begin\n      intros,\n      erw [← adj.hom_equiv_naturality_left, ← adj.hom_equiv_naturality_right],\n      dsimp, simp -- See note [dsimp, simp].\n    end },\n  counit :=\n  { app := λ Y, (adj.hom_equiv _ _).inv_fun (𝟙 (G.obj Y)),\n    naturality' :=\n    begin\n      intros,\n      erw [← adj.hom_equiv_naturality_left_symm, ← adj.hom_equiv_naturality_right_symm],\n      dsimp, simp\n    end },\n  hom_equiv_unit' := λ X Y f, by erw [← adj.hom_equiv_naturality_right]; simp,\n  hom_equiv_counit' := λ X Y f, by erw [← adj.hom_equiv_naturality_left_symm]; simp,\n  .. adj }\n\n/-- Construct an adjunction between functors `F` and `G` given a unit and counit for the adjunction\nsatisfying the triangle identities. -/\n@[simps]\ndef mk_of_unit_counit (adj : core_unit_counit F G) : F ⊣ G :=\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, adj.unit.app X ≫ G.map f,\n    inv_fun := λ g, F.map g ≫ adj.counit.app Y,\n    left_inv := λ f, begin\n      change F.map (_ ≫ _) ≫ _ = _,\n      rw [F.map_comp, assoc, ←functor.comp_map, adj.counit.naturality, ←assoc],\n      convert id_comp f,\n      have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.left_triangle,\n      dsimp at t,\n      simp only [id_comp] at t,\n      exact t,\n    end,\n    right_inv := λ g, begin\n      change _ ≫ G.map (_ ≫ _) = _,\n      rw [G.map_comp, ←assoc, ←functor.comp_map, ←adj.unit.naturality, assoc],\n      convert comp_id g,\n      have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.right_triangle,\n      dsimp at t,\n      simp only [id_comp] at t,\n      exact t,\n  end },\n  .. adj }\n\n/-- The adjunction between the identity functor on a category and itself. -/\ndef id : 𝟭 C ⊣ 𝟭 C :=\n{ hom_equiv := λ X Y, equiv.refl _,\n  unit := 𝟙 _,\n  counit := 𝟙 _ }\n\n-- Satisfy the inhabited linter.\ninstance : inhabited (adjunction (𝟭 C) (𝟭 C)) := ⟨id⟩\n\n/-- If F and G are naturally isomorphic functors, establish an equivalence of hom-sets. -/\n@[simps]\ndef equiv_homset_left_of_nat_iso\n  {F F' : C ⥤ D} (iso : F ≅ F') {X : C} {Y : D} :\n  (F.obj X ⟶ Y) ≃ (F'.obj X ⟶ Y) :=\n{ to_fun := λ f, iso.inv.app _ ≫ f,\n  inv_fun := λ g, iso.hom.app _ ≫ g,\n  left_inv := λ f, by simp,\n  right_inv := λ g, by simp }\n\n/-- If G and H are naturally isomorphic functors, establish an equivalence of hom-sets. -/\n@[simps]\ndef equiv_homset_right_of_nat_iso\n  {G G' : D ⥤ C} (iso : G ≅ G') {X : C} {Y : D} :\n  (X ⟶ G.obj Y) ≃ (X ⟶ G'.obj Y) :=\n{ to_fun := λ f, f ≫ iso.hom.app _,\n  inv_fun := λ g, g ≫ iso.inv.app _,\n  left_inv := λ f, by simp,\n  right_inv := λ g, by simp }\n\n/-- Transport an adjunction along an natural isomorphism on the left. -/\ndef of_nat_iso_left\n  {F G : C ⥤ D} {H : D ⥤ C} (adj : F ⊣ H) (iso : F ≅ G) :\n  G ⊣ H :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y, (equiv_homset_left_of_nat_iso iso.symm).trans (adj.hom_equiv X Y) }\n\n/-- Transport an adjunction along an natural isomorphism on the right. -/\ndef of_nat_iso_right\n  {F : C ⥤ D} {G H : D ⥤ C} (adj : F ⊣ G) (iso : G ≅ H) :\n  F ⊣ H :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y, (adj.hom_equiv X Y).trans (equiv_homset_right_of_nat_iso iso) }\n\n/-- Transport being a right adjoint along a natural isomorphism. -/\ndef right_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_right_adjoint F] :\n  is_right_adjoint G :=\n{ left := r.left,\n  adj := of_nat_iso_right r.adj h }\n\n/-- Transport being a left adjoint along a natural isomorphism. -/\ndef left_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_left_adjoint F] : is_left_adjoint G :=\n{ right := r.right,\n  adj := of_nat_iso_left r.adj h }\n\nsection\nvariables {E : Type u₃} [ℰ : category.{v₃} E] (H : D ⥤ E) (I : E ⥤ D)\n\n/--\nComposition of adjunctions.\n\nSee https://stacks.math.columbia.edu/tag/0DV0.\n-/\ndef comp (adj₁ : F ⊣ G) (adj₂ : H ⊣ I) : F ⋙ H ⊣ I ⋙ G :=\n{ hom_equiv := λ X Z, equiv.trans (adj₂.hom_equiv _ _) (adj₁.hom_equiv _ _),\n  unit := adj₁.unit ≫\n  (whisker_left F $ whisker_right adj₂.unit G) ≫ (functor.associator _ _ _).inv,\n  counit := (functor.associator _ _ _).hom ≫\n    (whisker_left I $ whisker_right adj₁.counit H) ≫ adj₂.counit }\n\n/-- If `F` and `G` are left adjoints then `F ⋙ G` is a left adjoint too. -/\ninstance left_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] (F : C ⥤ D) (G : D ⥤ E)\n  [Fl : is_left_adjoint F] [Gl : is_left_adjoint G] : is_left_adjoint (F ⋙ G) :=\n{ right := Gl.right ⋙ Fl.right,\n  adj := comp _ _ Fl.adj Gl.adj }\n\n/-- If `F` and `G` are right adjoints then `F ⋙ G` is a right adjoint too. -/\ninstance right_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] {F : C ⥤ D} {G : D ⥤ E}\n  [Fr : is_right_adjoint F] [Gr : is_right_adjoint G] : is_right_adjoint (F ⋙ G) :=\n{ left := Gr.left ⋙ Fr.left,\n  adj := comp _ _ Gr.adj Fr.adj }\n\nend\n\nsection construct_left\n-- Construction of a left adjoint. In order to construct a left\n-- adjoint to a functor G : D → C, it suffices to give the object part\n-- of a functor F : C → D together with isomorphisms Hom(FX, Y) ≃\n-- Hom(X, GY) natural in Y. The action of F on morphisms can be\n-- constructed from this data.\nvariables {F_obj : C → D} {G}\nvariables (e : Π X Y, (F_obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\nvariables (he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g)\ninclude he\n\nprivate lemma he' {X Y Y'} (f g) : (e X Y').symm (f ≫ G.map g) = (e X Y).symm f ≫ g :=\nby intros; rw [equiv.symm_apply_eq, he]; simp\n\n/-- Construct a left adjoint functor to `G`, given the functor's value on objects `F_obj` and\na bijection `e` between `F_obj X ⟶ Y` and `X ⟶ G.obj Y` satisfying a naturality law\n`he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g`.\nDual to `right_adjoint_of_equiv`. -/\n@[simps]\ndef left_adjoint_of_equiv : C ⥤ D :=\n{ obj := F_obj,\n  map := λ X X' f, (e X (F_obj X')).symm (f ≫ e X' (F_obj X') (𝟙 _)),\n  map_comp' := λ X X' X'' f f', begin\n    rw [equiv.symm_apply_eq, he, equiv.apply_symm_apply],\n    conv { to_rhs, rw [assoc, ←he, id_comp, equiv.apply_symm_apply] },\n    simp\n  end }\n\n/-- Show that the functor given by `left_adjoint_of_equiv` is indeed left adjoint to `G`. Dual\nto `adjunction_of_equiv_right`. -/\n@[simps]\ndef adjunction_of_equiv_left : left_adjoint_of_equiv e he ⊣ G :=\nmk_of_hom_equiv\n{ hom_equiv := e,\n  hom_equiv_naturality_left_symm' :=\n  begin\n    intros,\n    erw [← he' e he, ← equiv.apply_eq_iff_eq],\n    simp [(he _ _ _ _ _).symm]\n  end }\n\nend construct_left\n\nsection construct_right\n-- Construction of a right adjoint, analogous to the above.\nvariables {F} {G_obj : D → C}\nvariables (e : Π X Y, (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y))\nvariables (he : ∀ X' X Y f g, e X' Y (F.map f ≫ g) = f ≫ e X Y g)\ninclude he\n\nprivate lemma he' {X' X Y} (f g) : F.map f ≫ (e X Y).symm g = (e X' Y).symm (f ≫ g) :=\nby intros; rw [equiv.eq_symm_apply, he]; simp\n\n/-- Construct a right adjoint functor to `F`, given the functor's value on objects `G_obj` and\na bijection `e` between `F.obj X ⟶ Y` and `X ⟶ G_obj Y` satisfying a naturality law\n`he : ∀ X Y Y' g h, e X' Y (F.map f ≫ g) = f ≫ e X Y g`.\nDual to `left_adjoint_of_equiv`. -/\n@[simps]\ndef right_adjoint_of_equiv : D ⥤ C :=\n{ obj := G_obj,\n  map := λ Y Y' g, (e (G_obj Y) Y') ((e (G_obj Y) Y).symm (𝟙 _) ≫ g),\n  map_comp' := λ Y Y' Y'' g g', begin\n    rw [← equiv.eq_symm_apply, ← he' e he, equiv.symm_apply_apply],\n    conv { to_rhs, rw [← assoc, he' e he, comp_id, equiv.symm_apply_apply] },\n    simp\n  end }\n\n/-- Show that the functor given by `right_adjoint_of_equiv` is indeed right adjoint to `F`. Dual\nto `adjunction_of_equiv_left`. -/\n@[simps]\ndef adjunction_of_equiv_right : F ⊣ right_adjoint_of_equiv e he :=\nmk_of_hom_equiv\n{ hom_equiv := e,\n  hom_equiv_naturality_left_symm' := by intros; rw [equiv.symm_apply_eq, he]; simp,\n  hom_equiv_naturality_right' :=\n  begin\n    intros X Y Y' g h,\n    erw [←he, equiv.apply_eq_iff_eq, ←assoc, he' e he, comp_id, equiv.symm_apply_apply]\n  end }\n\nend construct_right\n\n/--\nIf the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the\nadjunction to an equivalence.\n-/\n@[simps]\nnoncomputable\ndef to_equivalence (adj : F ⊣ G) [∀ X, is_iso (adj.unit.app X)] [∀ Y, is_iso (adj.counit.app Y)] :\n  C ≌ D :=\n{ functor := F,\n  inverse := G,\n  unit_iso := nat_iso.of_components (λ X, as_iso (adj.unit.app X)) (by simp),\n  counit_iso := nat_iso.of_components (λ Y, as_iso (adj.counit.app Y)) (by simp) }\n\n/--\nIf the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise)\nisomorphisms, then the functor is an equivalence of categories.\n-/\n@[simps]\nnoncomputable\ndef is_right_adjoint_to_is_equivalence [is_right_adjoint G]\n  [∀ X, is_iso ((adjunction.of_right_adjoint G).unit.app X)]\n  [∀ Y, is_iso ((adjunction.of_right_adjoint G).counit.app Y)] :\n  is_equivalence G :=\nis_equivalence.of_equivalence_inverse (adjunction.of_right_adjoint G).to_equivalence\n\nend adjunction\n\nopen adjunction\n\nnamespace equivalence\n\n/-- The adjunction given by an equivalence of categories. (To obtain the opposite adjunction,\nsimply use `e.symm.to_adjunction`. -/\ndef to_adjunction (e : C ≌ D) : e.functor ⊣ e.inverse :=\nmk_of_unit_counit ⟨e.unit, e.counit,\n  by { ext, dsimp, simp only [id_comp], exact e.functor_unit_comp _, },\n  by { ext, dsimp, simp only [id_comp], exact e.unit_inverse_comp _, }⟩\n\nend equivalence\n\nnamespace functor\n\n/-- An equivalence `E` is left adjoint to its inverse. -/\ndef adjunction (E : C ⥤ D) [is_equivalence E] : E ⊣ E.inv :=\n(E.as_equivalence).to_adjunction\n\n/-- If `F` is an equivalence, it's a left adjoint. -/\n@[priority 10]\ninstance left_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_left_adjoint F :=\n{ right := _,\n  adj := functor.adjunction F }\n\n@[simp]\nlemma right_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : right_adjoint F = inv F :=\nrfl\n\n/-- If `F` is an equivalence, it's a right adjoint. -/\n@[priority 10]\ninstance right_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_right_adjoint F :=\n{ left := _,\n  adj := functor.adjunction F.inv }\n\n@[simp]\nlemma left_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : left_adjoint F = inv F :=\nrfl\n\nend functor\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/adjunction/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. 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{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.subobject.basic\nimport category_theory.preadditive\n\n/-!\n# Factoring through subobjects\n\nThe predicate `h : P.factors f`, for `P : subobject Y` and `f : X ⟶ Y`\nasserts the existence of some `P.factor_thru f : X ⟶ (P : C)` making the obvious diagram commute.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C] {X Y Z : C}\nvariables {D : Type u₂} [category.{v₂} D]\n\nnamespace category_theory\n\nnamespace mono_over\n\n/-- When `f : X ⟶ Y` and `P : mono_over Y`,\n`P.factors f` expresses that there exists a factorisation of `f` through `P`.\nGiven `h : P.factors f`, you can recover the morphism as `P.factor_thru f h`.\n-/\ndef factors {X Y : C} (P : mono_over Y) (f : X ⟶ Y) : Prop := ∃ g : X ⟶ (P : C), g ≫ P.arrow = f\n\nlemma factors_congr {X : C} {f g : mono_over X} {Y : C} (h : Y ⟶ X) (e : f ≅ g) :\n  f.factors h ↔ g.factors h :=\n⟨λ ⟨u, hu⟩, ⟨u ≫ (((mono_over.forget _).map e.hom)).left, by simp [hu]⟩,\n λ ⟨u, hu⟩, ⟨u ≫ (((mono_over.forget _).map e.inv)).left, by simp [hu]⟩⟩\n\n/-- `P.factor_thru f h` provides a factorisation of `f : X ⟶ Y` through some `P : mono_over Y`,\ngiven the evidence `h : P.factors f` that such a factorisation exists. -/\ndef factor_thru {X Y : C} (P : mono_over Y) (f : X ⟶ Y) (h : factors P f) : X ⟶ (P : C) :=\nclassical.some h\n\nend mono_over\n\nnamespace subobject\n\n/-- When `f : X ⟶ Y` and `P : subobject Y`,\n`P.factors f` expresses that there exists a factorisation of `f` through `P`.\nGiven `h : P.factors f`, you can recover the morphism as `P.factor_thru f h`.\n-/\ndef factors {X Y : C} (P : subobject Y) (f : X ⟶ Y) : Prop :=\nquotient.lift_on' P (λ P, P.factors f)\nbegin\n  rintros P Q ⟨h⟩,\n  apply propext,\n  split,\n  { rintro ⟨i, w⟩,\n    exact ⟨i ≫ h.hom.left, by erw [category.assoc, over.w h.hom, w]⟩, },\n  { rintro ⟨i, w⟩,\n    exact ⟨i ≫ h.inv.left, by erw [category.assoc, over.w h.inv, w]⟩, },\nend\n\n@[simp] \n\nlemma factors_iff {X Y : C} (P : subobject Y) (f : X ⟶ Y) :\n  P.factors f ↔ (representative.obj P).factors f :=\nquot.induction_on P $ λ a, mono_over.factors_congr _ (representative_iso _).symm\n\nlemma factors_self {X : C} (P : subobject X) : P.factors P.arrow :=\n(factors_iff _ _).mpr ⟨𝟙 P, (by simp)⟩\n\nlemma factors_comp_arrow {X Y : C} {P : subobject Y} (f : X ⟶ P) : P.factors (f ≫ P.arrow) :=\n(factors_iff _ _).mpr ⟨f, rfl⟩\n\nlemma factors_of_factors_right {X Y Z : C} {P : subobject Z} (f : X ⟶ Y) {g : Y ⟶ Z}\n  (h : P.factors g) : P.factors (f ≫ g) :=\nbegin\n  revert P,\n  refine quotient.ind' _,\n  intro P,\n  rintro ⟨g, rfl⟩,\n  exact ⟨f ≫ g, by simp⟩,\nend\n\nlemma factors_zero [has_zero_morphisms C] {X Y : C} {P : subobject Y} :\n  P.factors (0 : X ⟶ Y) :=\n(factors_iff _ _).mpr ⟨0, by simp⟩\n\nlemma factors_of_le {Y Z : C} {P Q : subobject Y} (f : Z ⟶ Y) (h : P ≤ Q) :\n  P.factors f → Q.factors f :=\nby { simp only [factors_iff], exact λ ⟨u, hu⟩, ⟨u ≫ of_le _ _ h, by simp [←hu]⟩ }\n\n/-- `P.factor_thru f h` provides a factorisation of `f : X ⟶ Y` through some `P : subobject Y`,\ngiven the evidence `h : P.factors f` that such a factorisation exists. -/\ndef factor_thru {X Y : C} (P : subobject Y) (f : X ⟶ Y) (h : factors P f) : X ⟶ P :=\nclassical.some ((factors_iff _ _).mp h)\n\n@[simp, reassoc] lemma factor_thru_arrow {X Y : C} (P : subobject Y) (f : X ⟶ Y) (h : factors P f) :\n  P.factor_thru f h ≫ P.arrow = f :=\nclassical.some_spec ((factors_iff _ _).mp h)\n\n@[simp] lemma factor_thru_self {X : C} (P : subobject X) (h) :\n  P.factor_thru P.arrow h = 𝟙 P :=\nby { ext, simp, }\n\n@[simp] lemma factor_thru_comp_arrow {X Y : C} {P : subobject Y} (f : X ⟶ P) (h) :\n  P.factor_thru (f ≫ P.arrow) h = f :=\nby { ext, simp, }\n\n@[simp] lemma factor_thru_eq_zero [has_zero_morphisms C]\n  {X Y : C} {P : subobject Y} {f : X ⟶ Y} {h : factors P f} :\n  P.factor_thru f h = 0 ↔ f = 0 :=\nbegin\n  fsplit,\n  { intro w,\n    replace w := w =≫ P.arrow,\n    simpa using w, },\n  { rintro rfl,\n    ext, simp, },\nend\n\nlemma factor_thru_right {X Y Z : C} {P : subobject Z} (f : X ⟶ Y) (g : Y ⟶ Z) (h : P.factors g) :\n  f ≫ P.factor_thru g h = P.factor_thru (f ≫ g) (factors_of_factors_right f h) :=\nbegin\n  apply (cancel_mono P.arrow).mp,\n  simp,\nend\n\n@[simp]\nlemma factor_thru_zero\n  [has_zero_morphisms C] {X Y : C} {P : subobject Y} (h : P.factors (0 : X ⟶ Y)) :\n  P.factor_thru 0 h = 0 :=\nby simp\n\n-- `h` is an explicit argument here so we can use\n-- `rw factor_thru_le h`, obtaining a subgoal `P.factors f`.\n-- (While the reverse direction looks plausible as a simp lemma, it seems to be unproductive.)\nlemma factor_thru_of_le\n  {Y Z : C} {P Q : subobject Y} {f : Z ⟶ Y} (h : P ≤ Q) (w : P.factors f) :\n  Q.factor_thru f (factors_of_le f h w) = P.factor_thru f w ≫ of_le P Q h :=\nby { ext, simp, }\n\nsection preadditive\n\nvariables [preadditive C]\n\nlemma factors_add {X Y : C} {P : subobject Y} (f g : X ⟶ Y) (wf : P.factors f) (wg : P.factors g) :\n  P.factors (f + g) :=\n(factors_iff _ _).mpr ⟨P.factor_thru f wf + P.factor_thru g wg, by simp⟩\n\n-- This can't be a `simp` lemma as `wf` and `wg` may not exist.\n-- However you can `rw` by it to assert that `f` and `g` factor through `P` separately.\nlemma factor_thru_add {X Y : C} {P : subobject Y} (f g : X ⟶ Y)\n   (w : P.factors (f + g)) (wf : P.factors f) (wg : P.factors g) :\n  P.factor_thru (f + g) w = P.factor_thru f wf + P.factor_thru g wg :=\nby { ext, simp, }\n\nlemma factors_left_of_factors_add {X Y : C} {P : subobject Y} (f g : X ⟶ Y)\n  (w : P.factors (f + g)) (wg : P.factors g) : P.factors f :=\n(factors_iff _ _).mpr ⟨P.factor_thru (f + g) w - P.factor_thru g wg, by simp⟩\n\n@[simp]\nlemma factor_thru_add_sub_factor_thru_right {X Y : C} {P : subobject Y} (f g : X ⟶ Y)\n  (w : P.factors (f + g)) (wg : P.factors g) :\n  P.factor_thru (f + g) w - P.factor_thru g wg =\n    P.factor_thru f (factors_left_of_factors_add f g w wg) :=\nby { ext, simp, }\n\nlemma factors_right_of_factors_add {X Y : C} {P : subobject Y} (f g : X ⟶ Y)\n  (w : P.factors (f + g)) (wf : P.factors f) : P.factors g :=\n(factors_iff _ _).mpr ⟨P.factor_thru (f + g) w - P.factor_thru f wf, by simp⟩\n\n@[simp]\nlemma factor_thru_add_sub_factor_thru_left {X Y : C} {P : subobject Y} (f g : X ⟶ Y)\n  (w : P.factors (f + g)) (wf : P.factors f) :\n  P.factor_thru (f + g) w - P.factor_thru f wf =\n    P.factor_thru g (factors_right_of_factors_add f g w wf) :=\nby { ext, simp, }\n\nend preadditive\n\nend subobject\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/subobject/factor_thru.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3069744143648549}}
{"text": "import UniverseAbstractions.Instances.Sort\nimport UniverseAbstractions.CategoryTheory.Meta\nimport UniverseAbstractions.CategoryTheory.Basic\nimport UniverseAbstractions.CategoryTheory.Functors\nimport UniverseAbstractions.CategoryTheory.NaturalTransformations\nimport UniverseAbstractions.CategoryTheory.Functors.Basic\nimport UniverseAbstractions.CategoryTheory.Functors.Nested\nimport UniverseAbstractions.CategoryTheory.FunctorExtensionality\nimport UniverseAbstractions.CategoryTheory.Utils.Trivial\n\n\n\nset_option autoBoundImplicitLocal false\n--set_option pp.universes true\n\nuniverse u v w\n\n\n\n-- This establishes the universe `type.{u}` as a morphism universe, which enables the construction\n-- of categories with morphisms in `type`, i.e. regular 1-categories.\n--\n-- Due to the way functors are defined mathematically, the morphism universe always needs to be at\n-- least as large as the universe of the domain, as it needs to hold natural transformation between\n-- functors. We need to use some trickery to make Lean behave well with this restriction.\n\nnamespace type.IsHomUniverse\n\n  open MetaRelation MetaFunctor CategoryTheory\n\n  instance isHomUniverse : IsHomUniverse type.{u} := ⟨⟩\n\n  abbrev homUniv := type.{max u w}\n\n  -- This is defeq to `type.{u}` but solves some universe unification issues.\n  abbrev type' := homUniv.{u, 0}\n\n  instance hasCat (U : Universe.{u + 1}) : HasCatProp U homUniv.{u, w} :=\n  { Cat  := CatDesc,\n    desc := id }\n\n  @[reducible] def Cat (U : Universe.{u + 1}) := HasCatProp.Category U homUniv.{u, w}\n  @[reducible] def Cat' (U : Universe.{u + 1}) := HasCatProp.Category U type'.{u}\n\n  def defCat {U : Universe.{u + 1}} {R : BundledRelation U homUniv.{u, w}} (C : CatDesc R) :\n    HasCatProp.DefCat C :=\n  { C   := C,\n    hEq := IsPreorderEq.refl R.Hom (hR := C.homIsPreorder) }\n\n  instance hasTrivialCatProp (U : Universe.{u + 1}) : HasCatProp.HasTrivialCatProp U homUniv.{u, w} :=\n  ⟨defCat⟩\n\n  theorem IsInv.ext {α : Sort u} {R : MetaRelation α type.{w}} [IsPreorder R]\n                    {a b : α} {f : R a b} {g : R b a} (h₁ h₂ : IsInv (R := R) f g) :\n    h₁ = h₂ :=\n  match h₁, h₂ with\n  | ⟨_, _⟩, ⟨_, _⟩ => rfl\n\n  section\n\n    variable {U : Universe.{u + 1}}\n\n    theorem IsoDesc.ext {A : Cat.{u, w} U} {a b : A} {e₁ e₂ : IsoDesc a b} :\n      e₁.toHom = e₂.toHom → e₁.invHom = e₂.invHom → e₁ = e₂ :=\n    match e₁, e₂ with\n    | { toHom := to₁, invHom := inv₁, isInv := isInv₁ },\n      { toHom := to₂, invHom := inv₂, isInv := isInv₂ } =>\n      λ (hTo : to₁ = to₂) (hInv : inv₁ = inv₂) => by subst hTo; subst hInv;\n                                                     have invExt := IsInv.ext isInv₁ isInv₂;\n                                                     subst invExt;\n                                                     rfl\n\n    theorem IsoDesc.ext' {A : Cat.{u, w} U} {a b : A} {e₁ e₂ : IsoDesc a b}\n                         (h : e₁.toHom = e₂.toHom) :\n      e₁ = e₂ :=\n    IsoDesc.ext h (IsoDesc.toInvEquiv (e₁ := e₁) (e₂ := e₂) h)\n\n    def IsoRel (A : Cat.{u, w} U) : MetaRelation A homUniv.{u, w} := IsoDesc\n\n    instance hasIsoRel (A : Cat.{u, w} U) : HasIsoRel A :=\n    { Iso         := IsoRel A,\n      desc        := id,\n      defToHomFun := λ _ _ => HasTrivialFunctoriality.defFun,\n      toHomInj    := IsoDesc.ext' }\n\n    def defIso {A : Cat.{u, w} U} {a b : A} (e : IsoDesc a b) : HasIsoRel.DefIso e :=\n    { e    := e,\n      toEq := rfl }\n\n    instance hasTrivialIsomorphismCondition (A : Cat.{u, w} U) :\n      HasIsoRel.HasTrivialIsomorphismCondition A :=\n    ⟨defIso⟩\n\n    instance isoIsEquivalence (A : Cat.{u, w} U) : IsEquivalence (IsoRel A) :=\n    HasIsoOp.isoIsEquivalence A\n\n    instance hasIsomorphisms (A : Cat.{u, w} U) : HasIsomorphisms A :=\n    { isoHasSymmFun  := HasTrivialFunctoriality.hasSymmFun  (IsoRel A),\n      isoHasTransFun := HasTrivialFunctoriality.hasTransFun (IsoRel A),\n      defIsoCat      := HasCatProp.HasTrivialCatProp.defCat }\n\n  end\n\n  instance isCatUniverse (U : Universe.{u + 1}) : IsCatUniverse U homUniv.{u, w} := ⟨⟩\n  instance isSortCatUniverse : IsSortCatUniverse.{u + 1} homUniv.{u, w} := ⟨⟩\n\n  -- Functors\n\n  section\n\n    -- It should be possible for `U` and `V` to have different Lean universes. Unfortunately, that\n    -- breaks all of the universe inference that we get from `homUniv`.\n    variable {U V : Universe.{u + 1}}\n\n    def FunProp (A : Cat.{u, w} U) (B : Cat.{u, w} V) : MetaProperty (A → B) homUniv.{u, w} :=\n    FunDesc\n\n    instance hasFunProp (A : Cat.{u, w} U) (B : Cat.{u, w} V) : HasFunProp A B :=\n    { Fun  := FunProp A B,\n      desc := id }\n\n    def defFun {A : Cat.{u, w} U} {B : Cat.{u, w} V} {φ : A → B} (F : FunDesc φ) :\n      HasFunProp.DefFun F :=\n    { F  := F,\n      eq := λ _ _ => rfl }\n\n    instance hasTrivialFunctorialityCondition (A : Cat.{u, w} U) (B : Cat.{u, w} V) :\n      HasFunProp.HasTrivialFunctorialityCondition A B :=\n    ⟨defFun⟩\n\n    -- Lean bug :-(\n    noncomputable instance hasIsoFun {A : Cat.{u, w} U} {B : Cat.{u, w} V} (F : A ⮕ B) :\n      HasIsoFun F :=\n    { defMapIso    := λ _   => HasIsoRel.HasTrivialIsomorphismCondition.defIso,\n      defMapIsoFun := λ _ _ => HasTrivialFunctoriality.defFun,\n      defIsoFun    := HasFunProp.HasTrivialFunctorialityCondition.defFun }\n\n  end\n\n  instance hasIdFun {U : Universe.{u + 1}} (A : Cat.{u, w} U) : HasFunProp.HasIdFun A :=\n  HasFunProp.HasIdFun.trivial A\n\n  instance hasConstFun {U V : Universe.{u + 1}} (A : Cat.{u, w} U) (B : Cat.{u, w} V) :\n    HasFunProp.HasConstFun A B :=\n  HasFunProp.HasConstFun.trivial A B\n\n  instance hasCompFun {U U' U'' : Universe.{u + 1}}\n                      (A : Cat.{u, w} U) (B : Cat.{u, w} U') (C : Cat.{u, w} U'') :\n    HasFunProp.HasCompFun A B C :=\n  HasFunProp.HasCompFun.trivial A B C\n\n  noncomputable instance isFunUniverse (U V : Universe.{u + 1}) :\n    IsFunUniverse U V homUniv.{u, w} :=\n  ⟨⟩\n\n  noncomputable instance isSortFunUniverse : IsSortFunUniverse.{u + 1} homUniv.{u, w} := ⟨⟩\n\n  instance isFunUniverse.hasCatIdFun (U : Universe.{u + 1}) :\n    IsFunUniverse.HasCatIdFun U homUniv.{u, w} :=\n  ⟨⟩\n\n  instance isFunUniverse.hasCatConstFun (U V : Universe.{u + 1}) :\n    IsFunUniverse.HasCatConstFun U V homUniv.{u, w} :=\n  ⟨⟩\n\n  instance isFunUniverse.hasCatCompFun (U U' U'' : Universe.{u + 1}) :\n    IsFunUniverse.HasCatCompFun U U' U'' homUniv.{u, w} :=\n  ⟨⟩\n\n  -- Natural transformations\n\n  theorem IsNatural.ext {α : Sort u} {β : Sort v}\n                        {R : MetaRelation α type.{w}} {S : MetaRelation β type.{w}}\n                        [HasTrans S] {φ ψ : α → β} {F : PreFunctor R S φ} {G : PreFunctor R S ψ}\n                        (η : MetaQuantification S φ ψ) (h₁ h₂ : MetaQuantification.IsNatural F G η) :\n    h₁ = h₂ :=\n  match h₁, h₂ with\n  | ⟨_⟩, ⟨_⟩ => rfl\n\n  section\n\n    variable {U V : Universe.{u + 1}}\n\n    theorem NatDesc.ext {A : Cat.{u, w} U} {B : Cat.{u, w} V} {F G : A ⮕ B} (η₁ η₂ : NatDesc F G) :\n      (∀ a, η₁.η a = η₂.η a) → η₁ = η₂ :=\n    match η₁, η₂ with\n    | { η := nat₁, isNat := isNat₁ }, { η := nat₂, isNat := isNat₂ } =>\n      λ h => by have hNat : nat₁ = nat₂ := funext h;\n                subst hNat;\n                have hIsNat : isNat₁ = isNat₂ := IsNatural.ext nat₁ isNat₁ isNat₂;\n                subst hIsNat;\n                rfl\n\n    def NatRel (A : Cat.{u, w} U) (B : Cat.{u, w} V) : MetaRelation (A ⮕ B) homUniv.{u, w} :=\n    NatDesc\n\n    instance hasNaturalityRelation (A : Cat.{u, w} U) (B : Cat.{u, w} V) : HasNatRel A B :=\n    { Nat       := NatRel A B,\n      desc      := id,\n      defNatFun := λ _ _ _ => HasTrivialFunctoriality.defFun }\n\n    def defNat {A : Cat.{u, w} U} {B : Cat.{u, w} V} {F G : A ⮕ B} (η : NatDesc F G) :\n      HasNatRel.DefNat η :=\n    { η     := η,\n      natEq := λ _ => rfl }\n\n    instance hasTrivialNaturalityCondition (A : Cat.{u, w} U) (B : Cat.{u, w} V) :\n      HasNatRel.HasTrivialNaturalityCondition A B :=\n    ⟨defNat⟩\n\n    instance hasTrivialNatEquiv (A : Cat.{u, w} U) (B : Cat.{u, w} V) :\n      HasNatRel.HasTrivialNatEquiv A B :=\n    ⟨NatDesc.ext⟩\n\n    instance natIsPreorder (A : Cat.{u, w} U) (B : Cat.{u, w} V) : IsPreorder (NatRel A B) :=\n    HasNatOp.natIsPreorder A B\n\n    instance hasNaturality (A : Cat' U) (B : Cat' V) : HasNaturality A B :=\n    { natHasTransFun := HasTrivialFunctoriality.hasTransFun (NatRel A B),\n      defFunCat      := HasCatProp.HasTrivialCatProp.defCat }\n\n    instance hasIsoNat {A : Cat' U} {B : Cat' V} (F G : A ⮕' B) : HasIsoNat F G :=\n    { defNatIso    := λ _ _ => HasIsoRel.HasTrivialIsomorphismCondition.defIso,\n      defNatIsoFun := λ _   => HasTrivialFunctoriality.defFun }\n\n    noncomputable instance hasIsoNaturality (A : Cat' U) (B : Cat' V) : HasIsoNaturality A B := ⟨⟩\n\n    def natIsoDescBuilder {A : Cat' U} {B : Cat' V} {F G : A ⮕' B} (η : NatIsoDesc F G) :\n      NatIsoDesc.IsoDescBuilder η :=\n    { defToNat  := HasNatRel.HasTrivialNaturalityCondition.defNat,\n      defInvNat := HasNatRel.HasTrivialNaturalityCondition.defNat,\n      leftInv   := HasNatRel.HasTrivialNatEquiv.natEquiv,\n      rightInv  := HasNatRel.HasTrivialNatEquiv.natEquiv }\n\n    noncomputable def defNatIso {A : Cat' U} {B : Cat' V} {F G : A ⮕' B} (η : NatIsoDesc F G) :\n      HasIsoNat.DefNatIso η :=\n    { η     := NatIsoDesc.IsoDescBuilder.isoDesc η (natIsoDescBuilder η),\n      natEq := λ _ => rfl }\n\n    noncomputable instance hasTrivialIsoNaturalityCondition (A : Cat' U) (B : Cat' V) :\n      HasIsoNaturality.HasTrivialIsoNaturalityCondition A B :=\n    ⟨defNatIso⟩\n\n  end\n\n  noncomputable instance isNatUniverse (U V : Universe.{u + 1}) : IsNatUniverse U V type'.{u} :=\n  ⟨⟩\n\n  noncomputable instance isSortNatUniverse : IsSortNatUniverse.{u + 1} type'.{u} := ⟨⟩\n\n  noncomputable instance isSortNatUniverse.hasTrivialNaturalIsomorphisms :\n    IsSortNatUniverse.HasTrivialNaturalIsomorphisms.{u + 1} type'.{u} :=\n  ⟨⟩\n\n  -- Multifunctors\n\n  section\n\n    variable {U U' U'' : Universe.{u + 1}} {A : Cat' U} {B : Cat' U'} {C : Cat' U''}\n             {F : A → (B ⮕ C)} {desc : HasNaturality.FunFunDesc F}\n\n    def defFunFunBase : HasNaturality.DefFunFunBase desc :=\n    { defNat     := λ _   => HasNatRel.HasTrivialNaturalityCondition.defNat,\n      defNatFun  := λ _ _ => HasTrivialFunctoriality.defFun,\n      natReflEq  := λ _   => HasNatRel.HasTrivialNatEquiv.natEquiv,\n      natTransEq := λ _ _ => HasNatRel.HasTrivialNatEquiv.natEquiv }\n\n    def defFunFun : HasNaturality.DefFunFun desc := HasNaturality.DefFunFun.trivial defFunFunBase\n\n  end\n\n  section\n\n    variable {U U' U'' U''' : Universe.{u + 1}} \n             {A : Cat' U} {B : Cat' U'} {C : Cat' U''} {D : Cat' U'''}\n             {F : A → (B ⮕ C ⮕' D)} {desc : HasNaturality.FunFunFunDesc F}\n\n    def defFunFunFunBase : HasNaturality.DefFunFunFunBase desc :=\n    { defRevFunFun := λ _ => defFunFunBase,\n      defNatNat    := λ _ => HasNaturality.DefNatNatBase.trivial }\n\n    def defFunFunFun : HasNaturality.DefFunFunFun desc :=\n    HasNaturality.DefFunFunFun.trivial defFunFunFunBase defFunFunBase\n\n  end\n\n  instance hasRevAppFunFun {U V : Universe.{u + 1}} (A : Cat' U) (B : Cat' V) :\n    HasNaturality.HasRevAppFunFun A B :=\n  { defRevAppFun    := λ _ => HasFunProp.HasTrivialFunctorialityCondition.defFun,\n    defRevAppFunFun := defFunFun }\n\n  instance hasCompFunFunFun {U U' U'' : Universe.{u + 1}} (A : Cat' U) (B : Cat' U') (C : Cat' U'') :\n    HasNaturality.HasCompFunFunFun A B C :=\n  { defCompFunFun    := λ _ => defFunFun,\n    defCompFunFunFun := defFunFunFun }\n\n  instance hasConstFunFun {U V : Universe.{u + 1}} (A : Cat' U) (B : Cat' V) :\n    HasNaturality.HasConstFunFun A B :=\n  { defConstFunFun := defFunFun }\n\n  instance hasDupFunFun {U V : Universe.{u + 1}} (A : Cat' U) (B : Cat' V) :\n    HasNaturality.HasDupFunFun A B :=\n  { defDupFun    := λ _ => HasFunProp.HasTrivialFunctorialityCondition.defFun,\n    defDupFunFun := defFunFun }\n\n  noncomputable instance isSortNatUniverse.hasFullCatFun :\n    IsSortNatUniverse.HasFullCatFun.{u + 1} type'.{u} :=\n  { hasRevAppFunFun  := hasRevAppFunFun,\n    hasCompFunFunFun := hasCompFunFunFun,\n    hasConstFunFun   := hasConstFunFun,\n    hasDupFunFun     := hasDupFunFun }\n\n  noncomputable instance isSortNatUniverse.hasFullNatIso :\n    IsSortNatUniverse.HasFullNatIso.{u + 1} type'.{u} :=\n  inferInstance\n\nend type.IsHomUniverse\n", "meta": {"author": "SReichelt", "repo": "universe-abstractions", "sha": "0bf2bae4c1b0f8d96c37e231dd238abda788e843", "save_path": "github-repos/lean/SReichelt-universe-abstractions", "path": "github-repos/lean/SReichelt-universe-abstractions/universe-abstractions-0bf2bae4c1b0f8d96c37e231dd238abda788e843/UniverseAbstractions/CategoryTheory/HomUniverses/Type.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3069731480024789}}
{"text": "import data.real.basic\nimport tactic\nimport real_definitions\n-- import utils\n\n-- #print linarith.make_comp_with_zero\n\nimport tactic.linarith.datatypes\n\nimport structures2\nimport utils\n\n/-\n- add an optional list of hypos in parameters, then compute is restricted to the given list.\n\n- essayer le simple : assumption, tautology\n\n- tags : \n    - `target_strict`: but est une inégalité stricte ?\n    - `target_abs`, `target_max`: but contient une valeur absolue / un max ?\n    - `hypo_abs`, `hypo_max`: hypos contiennent des valeurs absolues / max, min ?\n\n    - `target_strict`: \n            - `get_pos_from_pos_eq`\n            - `%%a ≠ 0` -->  `abs a >0` si target_abs (`abs_pos_of_ne_zero`)\n            - `%%abs a ≠ 0` --> a ≠ 0   si pas target_abs (`ne_zero_of_abs_ne_zero`)\n            - idem, utiliser |a| < b ==> -b < a < b, ...\n\n- abs pre_processor:\n    - abs ≥ 0. Comment l'utiliser au milieu d'une expression ?\n    - simplifier |a| < b, |a| ≤ b, idem minoration\n\n\n- min_max_preprocessor:\n    max a b ≥ a, max a b ≥ b, ... à appliquer sur `%%a = max %%b %%c`\n\n- pre-processing n°1:\n    - from linarith: `[filter_comparisons, remove_negations, make_comp_with_zero]`\n    ou même tout ? `[filter_comparisons, remove_negations, nat_to_int, strengthen_strict_int, make_comp_with_zero, cancel_denoms]`\n    - Si `but = inégalité stricte`, y ajouter `get_pos_from_pos_eq` ?\n    - pour la suite, il faut récupérer la liste des preuves résultantes\n- puis appliquer `linarith` à cette liste (sans preprocessing!)\n\n- target pre-processing:\n    - `[apply mul_pos, inv_pos.mpr, mul_ne_zero]` avant de recommencer ?\n    Attention, ces énoncés jouent sur le target, pas sur les hypos\n    (donc ça ne rentre pas dans la classe preprocessors)\n    Deux de ces tactiques dédouble le but (ab >0 remplacé par a>0 et b>0).\n\n    Et ré-essayer `linarith`?\n- pre-processor ifs:\n    - dans la liste d'égalités / inégalité, unfold abs max (ou éventuellement ré-écrire abs ?)\n    - split_ifs, cases_type or and,\n-/\n\n\n\n\n\n\n\n\n\n\n\nnamespace tactic.interactive\nopen lean.parser tactic interactive\nopen interactive (loc.ns)\nopen interactive.types\nopen expr\nopen linarith\n\n-- useful lemmas:\n-- ge_iff_le\n-- gt_iff_lt\n-- lt_of_le_of_ne\n-- le_of_lt\n-- div_pos\n-- mul_pos\n-- inv_pos\n\n-- open tactic.interactive\n\n-------------------------------------------\n-------------------------------------------\n-- Make 'a < b' from 'a ≠ b' and 'a ≤ b' --\n-------------------------------------------\n-------------------------------------------\n-- This is achieved by the following sequence of tactics:\n-- * make_ineq  take \"a ≤ b\" and \"a' ≠ b'\" and try to make \"a < b\"\n-- * extract_gt extract the list of inequalities \"a ≤ b\" from a list of expressions\n-- * extract_non_eq does the same for \"a ≠ b\"\n-- * list_prod make the product list of two lists\n-- * get_pos_from_pos_eq_from_list is build on the previous ones:\n-- it takes a list of expressions,\n-- extract inequalities and non-equalities,\n-- take all pairs and try to make \"a < b\" by applying make_ineq\n-- * get_pos_from_pos_eq is an interactive tactic that applies this to the local context\n\n\nset_option trace.linarith true\n#help options\n\ndef compute3.trace: list string := [\"\"]\n\nmeta def default_preprocessors : list global_preprocessor :=\n[filter_comparisons, remove_negations, make_comp_with_zero]\n\n------------------------------\n-- Detecting abs, max, etc. --\n------------------------------\n\n/- Check if some expression contains some constant name inside app. -/\nmeta def contain_cst (cst_name: name) : expr → bool\n| (const a a_1) := (a = cst_name)\n| (app a a_1) := (contain_cst a) ∨ (contain_cst a_1)\n| _ := ff\n\n/- Check if expr contains abs inside app. -/\nmeta def contain_abs (e: expr) : tactic bool :=\nreturn (contain_cst `abs e)\n\n/- Detect abs in target in app. -/\nmeta def target_abs : tactic bool :=\ndo  target ← tactic.target, e ← infer_type target,\n    trace target,\n    b ← contain_abs target,\n    trace b, return b\n\n/- Check if target is a strict inequality. -/\nmeta def target_lt_or_gt : tactic bool :=\ndo target ← tactic.target,\n    match target with\n    | `(%%a < %%b) := do trace tt, return tt\n    | `(%%a > %%b) := do trace tt, return tt\n    | _ := return ff\n    end\n\n\n-- example (a b :ℝ) : a < 10*(|b| + 1)-22 := \n-- begin\n--     targets_analysis,\n--     target_abs,\n--     target_lt_or_gt,\n-- end\n\n----------------------------\n----------------------------\n-- Get strict inequalities -\n----------------------------\n----------------------------\n\n/- Take a proof of a non equality a≠b and replace it by a proof of a-b≠0 -/\nprivate meta def rearr_noneq_aux : expr → expr → tactic expr\n| prf `(%%a ≠ 0)    := return prf\n| prf `(%%a ≠ %%b)  := mk_app ``sub_ne_zero_of_ne [prf]\n| prf  a            := trace a >> fail \"couldn't rearrange noneq\"\n\nmeta def rearr_noneq (e : expr) : tactic expr :=\ninfer_type e >>= rearr_noneq_aux e\n\n-- lemma neg_zero_of_zero (G: Type) [add_group G] (a: G) (H: a =0) : -a = 0 := neg_eq_zero.mpr H\n\n/-  Take H1 of type \"a ≤ 0\", H2 of type \"a' ≠ 0'\", and if a=a'\n    then add H3 of type \"a < 0\" in the local context -/\nmeta def make_lt_aux : expr × expr → expr × expr → tactic expr\n| (prf, a) (prf', a') :=\n    tactic.unify a a' >> to_expr ``(lt_of_le_of_ne %%prf %%prf')\n\n/- The same, with the possibility that a=-a' -/\nmeta def make_lt_aux' : expr × expr → expr × expr → tactic expr\n| (prf, a) (prf', a') := do\n    prf'' ← mk_app `neg_eq_zero.mpr [prf'], t ← infer_type prf'',\n    match t with\n    | `(%%a'' = 0) := make_lt_aux (prf, a) (prf'', a'')\n    | _ := fail \"Unexpected failure\"\n    end\n\n/-  Take proofs of \"a ≤ 0\", \"a' ≠ 0'\", and if a=a' or a = -a'\n    then compute a proof of \"a < 0\". -/\nmeta def make_lt : expr × expr → expr × expr → tactic expr\n| (prf, a) (prf', a') := make_lt_aux (prf, a) (prf', a') <|> make_lt_aux' (prf, a) (prf', a')\n\n\nprivate meta def filter_le_zero : expr → bool\n| `(%%a ≤ 0) := tt\n| _          := ff\n\nprivate meta def filter_neq_zero (prf: expr) : tactic (expr × expr) :=\ndo ineq ← infer_type prf, match ineq with\n| `(%%a ≠ 0) := return (prf, a)\n| _          := fail \"\"\nend\n\n/--\n`mk_noneq_with_zero h` takes each proof `h` of a non-equality a ≠ b\nand turns it into a proof of a-b ≠ 0.\n -/\nmeta def make_noneq_with_zero : preprocessor :=\n{ name := \"make non-equalities with zero\",\n  transform := λ e, singleton <$> rearr_noneq e <|> return [] }\n\n-- meta def is_lt_zero : expr → tactic expr\n-- | `(%%a < 0) := return a\n-- | _ := fail \"not ?? < 0\"\n\n-- meta def is_neq_zero : expr → tactic expr\n-- | `(%%a ≠ 0) := return a\n-- | _ := fail \"not ?? ≠ 0\"\n\n\n\n/- Pre-processor version of get_pos_from_pos_eq.\nWe start with normalised comparisons. Specifically, we search for comparisons of the form\n    a ≠ 0 (or -a ≠ 0) and a ≤ 0\nget a new inequality a < 0 and clear both premises.\n-/\nmeta def add_pos_from_pos_eq : global_preprocessor :=\n\n{ name := \"add strict inequalities, e.g. a < 0 from a ≠ 0 and a ≤ 0\",\n--(transform : list expr → tactic (list expr))\n  transform := λ ls,\n    -- TODO\n\n\n\n\n-------------------\n-------------------\n-- Tactic compute -\n-------------------\n-------------------\n\nlemma inv_pos_mpr {α : Type} [linear_ordered_field α] (a:α) :\n0 < a → 0 < a⁻¹ := inv_pos.mpr\n\nopen linarith\n/- Non-interactive version of nl_linarith. -/\nmeta def nl_linarith (cfg : linarith_config := {}): tactic unit :=\ndo\n{\ntactic.linarith false false []\n  { cfg with preprocessors := some $\n      cfg.preprocessors.get_or_else default_preprocessors ++ [nlinarith_extras] }\n}\n\n\n/-- A configuration object for `compute1`. \n develop_ite: set to tt to develop if_then_else definitions, e.g. abs and max.\n -/\nmeta structure compute_config : Type :=\n(develop_ite : bool := ff)\n\n\n/- Unfold some definitions using if_then_else, then get rid of if_then_else by case reasoning, and try to \nclose all the goals by linarith. Unfolded definitions includes abs, max, min.\n-/\nmeta def develop_ite : tactic string :=\n    do {`[unfold abs at *], `[unfold min max at *], `[split_ifs at *], repeat $ tactic.linarith false false [],\n    return \"unfold abs, unfold min max, split_ifs, repeat {linarith}\"}\n    <|>\n    do {`[unfold abs], `[unfold min max], `[split_ifs],\n    return \"unfold abs, unfold min max, split_ifs\"}\n\n/- try several computing tactics for STRICT inequalities.\nIn case of success return the corresponding effective code. -/\nmeta def compute1 : tactic string :=\ndo\n{   do {assumption, return \"assumption\"}\n    <|>\n    -- solve e.g. \"n_0 ≤ n_0 ∨ n_0 ≤ n_1\"\n    -- with norm_num, solves \"n_0 ≤ max n_0 n_1\"\n    do {tactic.tautology, return \"tautology\"}\n    <|>\n    do {tactic.linarith false false [], return \"linarith\"}\n    <|>\n    do {nl_linarith, return \"nl_linarith\"}\n    <|>\n    do {tactic.applyc ``mul_pos, return \"apply mul_pos\"}\n    <|>\n    -- a>0 → a⁻¹ >0\n    do {tactic.applyc ``inv_pos_mpr, return \"apply inv_pos.mpr\"}\n    <|>\n -- a≠0 → b≠0 → ab≠0\n     do {tactic.applyc ``mul_ne_zero, return \"apply mul_ne_zero\"}\n--  div_pos is now useless\n}\n#print mul_ne_zero\n/- Repeat n times the tactic compute1 ; never fails -/\nmeta def compute_n_old (n: nat)  (cfg: compute_config := {}) : tactic string :=\ndo  effective_code_0 ←  if cfg.develop_ite \n                        then try_tactic_string develop_ite \n                        else skip_tactic_string,\n    n ← num_goals, match n with\n        | 0 := return effective_code_0\n        | _ := do\n        -- Try to add strict inequalities to the context, if there remains some goal:\n        effective_code_1 ← get_pos_from_pos_eq,\n        effective_code_2 ← iterate_and_return_code n compute1,\n        return $ string.concatenate [effective_code_0, effective_code_1, effective_code_2]\n        end\n\n/- Repeat n times the tactic compute1 ; never fails -/\nmeta def compute_n (n: nat)  (cfg: compute_config := {}) : tactic string :=\ndo let first_tac := if cfg.develop_ite then [try_tactic_string develop_ite]\n                      else ([]:list (tactic string)),\n    -- Try to add strict inequalities to the context, if there remains some goal:\n    let list_tac := first_tac.append [get_pos_from_pos_eq,\n                                      iterate_and_return_code n compute1],\n        code ← and_then_and_return_code list_tac,\n        return code\n\n/- Apply the compute_n tactic n times,\nand in case of success trace the effective code with id. -/\nmeta def compute_and_return_code (n: nat) (id: string)  (cfg: compute_config := {}):\ntactic unit :=\ndo apply_and_return_code id (tactic.interactive.compute_n n cfg)\n\n\nend tactic.interactive\n\n-------------\n-- Example -- \n-------------\n-- set_option trace.linarith true\nexample (a:ℝ) (H: a ≠ 0) (H': a ≥ 0): a^2 ≥ 0 :=\nbegin\n    -- compute_n 1,\n    norm_num at *,\n    -- apply_and_return_code \"12.1\" (tactic.interactive.compute_n 1),\n    compute_and_return_code 10 \"12.1\",\n    --compute1,\n    -- compute_and_return_code 10 \"12.1\",  -->\n    -- have H_aux := lt_of_le_of_ne H' (ne.symm H), nl_linarith,\nend\n\n\n", "meta": {"author": "dEAduction", "repo": "dEAduction-lean", "sha": "4fe1d642078fc94f9081ccbed08e047e86a741fd", "save_path": "github-repos/lean/dEAduction-dEAduction-lean", "path": "github-repos/lean/dEAduction-dEAduction-lean/dEAduction-lean-4fe1d642078fc94f9081ccbed08e047e86a741fd/snippets/tactics_for_testing/compute2-essai.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3069731480024789}}
{"text": "import category_theory.pasting_pushouts\nimport homotopy_theory.formal.cylinder.definitions\nimport .cofibration_category\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cofibrations\nopen homotopy_theory.cylinder (endpoint)\nopen homotopy_theory.weak_equivalences\nopen homotopy_theory.weak_equivalences.category_with_weak_equivalences\nopen precofibration_category cofibration_category\n\nvariables {C : Type u} [category.{v} C] [cofibration_category.{v} C]\n\nvariables {a b : C} {j : a ⟶ b} (hj : is_cof j)\n\nstructure relative_cylinder :=\n(ob : C)\n(ii : (pushout_by_cof j j hj).ob ⟶ ob)\n(p : ob ⟶ b)\n(hii : is_cof ii)\n(hp : is_weq p)\n(pii : p ∘ ii = (pushout_by_cof j j hj).is_pushout.induced (𝟙 b) (𝟙 b) rfl)\n\n-- Any cofibration admits a relative cylinder.\nlemma exists_relative_cylinder : nonempty (relative_cylinder hj) :=\nlet ⟨c, ii, p, hii, hp, pii⟩ :=\n  factorization ((pushout_by_cof j j hj).is_pushout.induced (𝟙 b) (𝟙 b) rfl) in\n⟨⟨c, ii, p, hii, hp, pii⟩⟩\n\nvariables {hj}\n\ndef relative_cylinder.i₀ (c : relative_cylinder hj) : b ⟶ c.ob :=\nc.ii ∘ (pushout_by_cof j j hj).map₀\n\ndef relative_cylinder.i₁ (c : relative_cylinder hj) : b ⟶ c.ob :=\nc.ii ∘ (pushout_by_cof j j hj).map₁\n\nlemma relative_cylinder.pi₀ (c : relative_cylinder hj) : c.p ∘ c.i₀ = 𝟙 b :=\nby unfold relative_cylinder.i₀; simp [c.pii]\n\nlemma relative_cylinder.pi₁ (c : relative_cylinder hj) : c.p ∘ c.i₁ = 𝟙 b :=\nby unfold relative_cylinder.i₁; simp [c.pii]\n\nlemma relative_cylinder.ij (c : relative_cylinder hj) : c.i₀ ∘ j = c.i₁ ∘ j :=\nbegin\n  unfold relative_cylinder.i₀ relative_cylinder.i₁,\n  rw [←assoc, ←assoc, (pushout_by_cof j j hj).is_pushout.commutes]\nend\n\nlemma relative_cylinder.acof_i₀ (c : relative_cylinder hj) : is_acof c.i₀ :=\n⟨cof_comp (pushout_is_cof (pushout_by_cof j j hj).is_pushout.transpose hj) c.hii,\n weq_of_comp_weq_right c.hp (by convert (weq_id _); exact c.pi₀)⟩\n\nlemma relative_cylinder.acof_i₁ (c : relative_cylinder hj) : is_acof c.i₁ :=\n⟨cof_comp (pushout_is_cof (pushout_by_cof j j hj).is_pushout hj) c.hii,\n weq_of_comp_weq_right c.hp (by convert (weq_id _); exact c.pi₁)⟩\n\n-- If j : a → b is an *acyclic* cofibration, then so is c.ii.\nlemma relative_cylinder.acof_ii (c : relative_cylinder hj) (hj' : is_weq j) : is_acof c.ii :=\nhave is_acof (pushout_by_cof j j hj).map₁, from\n  pushout_is_acof (pushout_by_cof j j hj).is_pushout ⟨hj, hj'⟩,\nhave is_weq ((pushout_by_cof j j hj).is_pushout.induced (𝟙 b) (𝟙 b) rfl), from\n  weq_of_comp_weq_left this.2 (by convert weq_id b using 1; simp),\n⟨c.hii, weq_of_comp_weq_right c.hp (by convert this; simp [c.pii])⟩\n\nstructure cylinder_embedding (c c' : relative_cylinder hj) :=\n(k : c.ob ⟶ c'.ob)\n(hk : is_cof k)\n(hkii : k ∘ c.ii = c'.ii)\n(hpk : c'.p ∘ k = c.p)\n\nlemma cylinder_embedding.hki₀ {c c' : relative_cylinder hj} (m : cylinder_embedding c c') :\n  m.k ∘ c.i₀ = c'.i₀ :=\nby unfold relative_cylinder.i₀; simp [m.hkii]\n\nlemma cylinder_embedding.hki₁ {c c' : relative_cylinder hj} (m : cylinder_embedding c c') :\n  m.k ∘ c.i₁ = c'.i₁ :=\nby unfold relative_cylinder.i₁; simp [m.hkii]\n\nlemma cylinder_embedding.acof_k {c c' : relative_cylinder hj} (m : cylinder_embedding c c') :\n  is_acof m.k :=\n⟨m.hk, weq_of_comp_weq_right c'.hp (by convert c.hp; rw m.hpk)⟩\n\ndef cylinder_embedding.refl (c : relative_cylinder hj) : cylinder_embedding c c :=\n⟨𝟙 c.ob, cof_id _, by simp, by simp⟩\n\ndef cylinder_embedding.trans {c₀ c₁ c₂ : relative_cylinder hj}\n  (m₀ : cylinder_embedding c₀ c₁) (m₁ : cylinder_embedding c₁ c₂) : cylinder_embedding c₀ c₂ :=\n⟨m₁.k ∘ m₀.k,\n cof_comp m₀.hk m₁.hk,\n by rw [←assoc, m₀.hkii, m₁.hkii],\n by rw [assoc, m₁.hpk, m₀.hpk]⟩\n\n-- Any two relative cylinders on the same cofibration can be embedded\n-- in a third.\n\nstructure common_embedding (c₀ c₁ : relative_cylinder hj) :=\n(c' : relative_cylinder hj)\n(m₀ : cylinder_embedding c₀ c')\n(m₁ : cylinder_embedding c₁ c')\n\n-- Factor out the second part of the construction of a common\n-- embedding, since we will need to apply it to a particular form of\n-- factorization later, when verifying the identity and inverse laws\n-- for track composition.\ndef common_embedding_of_factorization (c₀ c₁ : relative_cylinder hj)\n  (po : pushout c₀.ii c₁.ii) (c'_ob : C) (l : po.ob ⟶ c'_ob) (q : c'_ob ⟶ b)\n  (hl : is_cof l) (hq : is_weq q)\n  (ql : q ∘ l = po.is_pushout.induced c₀.p c₁.p (by simp [c₀.pii, c₁.pii])) :\n  common_embedding c₀ c₁ :=\nlet c' : relative_cylinder hj :=\n  ⟨c'_ob, l ∘ po.map₁ ∘ c₁.ii, q,\n   cof_comp c₁.hii (cof_comp (pushout_is_cof po.is_pushout c₀.hii) hl),\n   hq, by simp [ql, c₁.pii]⟩ in\n⟨c',\n ⟨l ∘ po.map₀,\n  cof_comp (pushout_is_cof po.is_pushout.transpose c₁.hii) hl,\n  by rw [←assoc, po.is_pushout.commutes, assoc],\n  by simp [ql]⟩,\n ⟨l ∘ po.map₁,\n  cof_comp (pushout_is_cof po.is_pushout c₀.hii) hl,\n  rfl,\n  by simp [ql]⟩⟩\n\nlemma exists_common_embedding (c₀ c₁ : relative_cylinder hj) :\n  nonempty (common_embedding c₀ c₁) :=\nlet po := pushout_by_cof c₀.ii c₁.ii c₀.hii,\n    pp := po.is_pushout.induced c₀.p c₁.p (by rw [c₀.pii, c₁.pii]),\n    ⟨c'_ob, l, q, hl, hq, ql⟩ := factorization pp in\n⟨common_embedding_of_factorization c₀ c₁ po c'_ob l q hl hq ql⟩\n\ndef cylinder_embedding.pushout {c c₀ c₁ : relative_cylinder hj}\n  (m₀ : cylinder_embedding c c₀) (m₁ : cylinder_embedding c c₁) : relative_cylinder hj :=\n⟨(pushout_by_cof m₀.k m₁.k m₀.hk).ob,\n (pushout_by_cof m₀.k m₁.k m₀.hk).map₁ ∘ c₁.ii,\n (pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout.induced c₀.p c₁.p\n   (by rw [m₀.hpk, ←m₁.hpk]),\n cof_comp c₁.hii (pushout_is_cof (pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout m₀.hk),\n weq_of_comp_weq_left\n   (pushout_is_acof (pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout m₀.acof_k).2\n   (by convert c₁.hp; simp),\n by simp [c₁.pii]⟩\n\ndef cylinder_embedding.pushout.map₀ {c c₀ c₁ : relative_cylinder hj}\n  (m₀ : cylinder_embedding c c₀) (m₁ : cylinder_embedding c c₁) :\n  cylinder_embedding c₀ (cylinder_embedding.pushout m₀ m₁) :=\n⟨(pushout_by_cof m₀.k m₁.k m₀.hk).map₀,\n pushout_is_cof (pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout.transpose m₁.hk,\n show\n   (pushout_by_cof m₀.k m₁.k m₀.hk).map₀ ∘ c₀.ii =\n   (pushout_by_cof m₀.k m₁.k m₀.hk).map₁ ∘ c₁.ii,\n begin\n   rw [←m₀.hkii, ←m₁.hkii],\n   simp [(pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout.commutes]\n end,\n show Is_pushout.induced _ _ _ _ ∘ _ = _, by simp⟩\n\ndef cylinder_embedding.pushout.map₁ {c c₀ c₁ : relative_cylinder hj}\n  (m₀ : cylinder_embedding c c₀) (m₁ : cylinder_embedding c c₁) :\n  cylinder_embedding c₁ (cylinder_embedding.pushout m₀ m₁) :=\n⟨(pushout_by_cof m₀.k m₁.k m₀.hk).map₁,\n pushout_is_cof (pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout m₀.hk,\n rfl,\n show Is_pushout.induced _ _ _ _ ∘ _ = _, by simp⟩\n\ndef cylinder_embedding.pushout.is_pushout {c c₀ c₁ : relative_cylinder hj}\n  (m₀ : cylinder_embedding c c₀) (m₁ : cylinder_embedding c c₁) :\n  Is_pushout m₀.k m₁.k\n    (cylinder_embedding.pushout.map₀ m₀ m₁).k\n    (cylinder_embedding.pushout.map₁ m₀ m₁).k :=\n(pushout_by_cof m₀.k m₁.k m₀.hk).is_pushout\n\ndef relative_cylinder.reverse (c : relative_cylinder hj) : relative_cylinder hj :=\n⟨c.ob,\n c.ii ∘ (pushout_by_cof j j hj).is_pushout.swap,\n c.p,\n cof_comp (cof_iso (pushout_by_cof j j hj).is_pushout.swap_iso) c.hii,\n c.hp,\n by simp [c.pii]⟩\n\n@[simp] lemma relative_cylinder.reverse_i₀ {c : relative_cylinder hj} :\n  c.reverse.i₀ = c.i₁ :=\nshow c.ii ∘ (pushout_by_cof j j hj).is_pushout.induced _ _ _ ∘ (pushout_by_cof j j hj).map₀ = _,\nby rw [←assoc]; simp; refl\n\n@[simp] lemma relative_cylinder.reverse_i₁ {c : relative_cylinder hj} :\n  c.reverse.i₁ = c.i₀ :=\nshow c.ii ∘ (pushout_by_cof j j hj).is_pushout.induced _ _ _ ∘ (pushout_by_cof j j hj).map₁ = _,\nby rw [←assoc]; simp; refl\n\ndef cylinder_embedding.reverse {c c' : relative_cylinder hj}\n  (m : cylinder_embedding c c') : cylinder_embedding c.reverse c'.reverse :=\n⟨m.k, m.hk, show m.k ∘ (c.ii ∘ _) = c'.ii ∘ _, by simp [m.hkii], m.hpk⟩\n\n-- TODO: Should really have this `pushout_by_cof` exposed somewhere\ndef relative_cylinder.glue (c₀ c₁ : relative_cylinder hj) : relative_cylinder.{v} hj :=\nlet po := pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1 in\n⟨po.ob,\n (pushout_by_cof j j hj).is_pushout.induced (po.map₀ ∘ c₀.i₀) (po.map₁ ∘ c₁.i₁) $\n   by rw [←assoc, ←assoc, c₀.ij, ←c₁.ij]; simp [po.is_pushout.commutes],\n po.is_pushout.induced c₀.p c₁.p (by rw [c₀.pi₁, c₁.pi₀]),\n begin\n   let po₀ := pushout_by_cof c₀.i₀ (pushout_by_cof j j hj).map₀ c₀.acof_i₀.1,\n   let po₀' :=\n     (Is_pushout_of_Is_pushout_of_Is_pushout\n       (pushout_by_cof j j hj).is_pushout.transpose po₀.is_pushout.transpose).transpose,\n   let f :=\n     (pushout_by_cof j j hj).is_pushout.induced\n       (po₀.map₀ ∘ c₀.i₁) (po₀.map₁ ∘ (pushout_by_cof j j hj).map₁)\n       (by rw [←assoc, ←assoc, ←c₀.ij,\n               ←(pushout_by_cof j j hj).is_pushout.commutes,\n               assoc, assoc, po₀.is_pushout.commutes]),\n   let po₁ : Is_pushout c₀.i₁ (pushout_by_cof j j hj).map₀ po₀.map₀ f :=\n     Is_pushout_of_Is_pushout_of_Is_pushout_vert'\n       (pushout_by_cof j j hj).is_pushout\n       (begin convert po₀' using 1, { exact c₀.ij.symm }, { simp } end) (by simp),\n   let g := po₁.induced po.map₀ (po.map₁ ∘ c₁.ii)\n     (by rw ←assoc; exact po.is_pushout.commutes),\n   let po₂ : Is_pushout f c₁.ii g po.map₁ :=\n     Is_pushout_of_Is_pushout_of_Is_pushout' po₁ (by convert po.is_pushout; simp) (by simp),\n   have : ∀ p,\n     (pushout_by_cof j j hj).is_pushout.induced (po.map₀ ∘ c₀.i₀) (po.map₁ ∘ c₁.i₁) p =\n     g ∘ po₀.map₁ :=\n   begin\n     intro p, apply (pushout_by_cof j j hj).is_pushout.uniqueness,\n     { rw [←assoc, ←po₀.is_pushout.commutes], simp },\n     { rw ←assoc,\n       have :\n         po₀.map₁ ∘ (pushout_by_cof.{v} j j hj).map₁ =\n         f ∘ (pushout_by_cof.{v} j j hj).map₁, by simp,\n       rw this,\n       rw [assoc, po₂.commutes, ←assoc],\n       change _ = po.map₁ ∘ c₁.i₁, simp }\n   end,\n   rw this,\n   exact cof_comp\n     (pushout_is_cof po₀.is_pushout c₀.acof_i₀.1)\n     (pushout_is_cof po₂.transpose c₁.hii)\n end,\n weq_of_comp_weq_left\n   (pushout_is_acof po.is_pushout c₀.acof_i₁).2\n   (by simpa using c₁.hp),\n begin\n   apply (pushout_by_cof j j hj).is_pushout.uniqueness;\n   { rw ←assoc, simp, rw [c₀.pi₀] <|> rw [c₁.pi₁] }\n end⟩\n\n@[simp] lemma relative_cylinder.glue_i₀ {c₀ c₁ : relative_cylinder hj} :\n  (c₀.glue c₁).i₀ = (pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1).map₀ ∘ c₀.i₀ :=\nlet po := pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1 in\nshow\n  (pushout_by_cof j j hj).is_pushout.induced (po.map₀ ∘ c₀.i₀) (po.map₁ ∘ c₁.i₁) _ ∘\n    (pushout_by_cof j j hj).map₀ = _, by simp\n\n@[simp] lemma relative_cylinder.glue_i₁ {c₀ c₁ : relative_cylinder hj} :\n  (c₀.glue c₁).i₁ = (pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1).map₁ ∘ c₁.i₁ :=\nlet po := pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1 in\nshow\n  (pushout_by_cof j j hj).is_pushout.induced (po.map₀ ∘ c₀.i₀) (po.map₁ ∘ c₁.i₁) _ ∘\n    (pushout_by_cof j j hj).map₁ = _, by simp\n\ndef cylinder_embedding.glue {c₀ c₁ c₀' c₁' : relative_cylinder hj}\n  (m₀ : cylinder_embedding c₀ c₀') (m₁ : cylinder_embedding c₁ c₁') :\n  cylinder_embedding (c₀.glue c₁) (c₀'.glue c₁') :=\n⟨(pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1).is_pushout.induced\n   ((pushout_by_cof c₀'.i₁ c₁'.i₀ c₀'.acof_i₁.1).map₀ ∘ m₀.k)\n   ((pushout_by_cof c₀'.i₁ c₁'.i₀ c₀'.acof_i₁.1).map₁ ∘ m₁.k)\n   (by rw [←assoc, ←assoc, m₀.hki₁, m₁.hki₀];\n       exact (pushout_by_cof c₀'.i₁ c₁'.i₀ c₀'.acof_i₁.1).is_pushout.commutes),\n begin\n   apply cof_pushout m₀.hk m₁.hk,\n   convert (pushout_by_cof c₀'.i₁ c₁'.i₀ c₀'.acof_i₁.1).is_pushout using 1,\n   exact m₀.hki₁, exact m₁.hki₀\n end,\n begin\n   apply (pushout_by_cof j j hj).is_pushout.uniqueness; rw ←assoc,\n   { change _ ∘ (c₀.glue c₁).i₀ = (c₀'.glue c₁').i₀, simp [m₀.hki₀.symm] },\n   { change _ ∘ (c₀.glue c₁).i₁ = (c₀'.glue c₁').i₁, simp [m₁.hki₁.symm] }\n end,\n begin\n   apply (pushout_by_cof c₀.i₁ c₁.i₀ c₀.acof_i₁.1).is_pushout.uniqueness; rw ←assoc;\n   change Is_pushout.induced _ _ _ _ ∘ _ = Is_pushout.induced _ _ _ _ ∘ _;\n   simp [m₀.hpk, m₁.hpk]\n end⟩\n\nsection pair_map\nvariables (hj)\nvariables {a' b' : C} {j' : a' ⟶ b'} (hj' : is_cof j')\n\ninclude hj hj'\nstructure pair_map :=\n(g : a ⟶ a')\n(h : b ⟶ b')\n(commutes : h ∘ j = j' ∘ g)\nomit hj hj'\n\nvariables {hj hj'}\ndef endpoints_map (h : pair_map hj hj') :\n  (pushout_by_cof j j hj).ob ⟶ (pushout_by_cof j' j' hj').ob :=\npushout_of_maps\n  (pushout_by_cof j j hj).is_pushout\n  (pushout_by_cof j' j' hj').is_pushout\n  h.g h.h h.h h.commutes h.commutes\n\n-- Like `cylinder_embedding`, but we do not require that the map\n-- between cylinders be a cofibration (since b → b' might not be one).\nstructure cylinder_map_over (h : pair_map hj hj')\n  (c : relative_cylinder hj) (c' : relative_cylinder hj') :=\n(k : c.ob ⟶ c'.ob)\n(hkii : k ∘ c.ii = c'.ii ∘ endpoints_map h)\n(hpk : c'.p ∘ k = h.h ∘ c.p)\n\nlemma cylinder_map_over.hki₀ {h : pair_map hj hj'}\n  {c : relative_cylinder hj} {c' : relative_cylinder hj'} (m : cylinder_map_over h c c') :\n  m.k ∘ c.i₀ = c'.i₀ ∘ h.h :=\nbegin\n  unfold relative_cylinder.i₀,\n  rw [assoc, m.hkii],\n  unfold endpoints_map pushout_of_maps,\n  rw [←assoc], simp\nend\n\nlemma cylinder_map_over.hki₁ {h : pair_map hj hj'}\n  {c : relative_cylinder hj} {c' : relative_cylinder hj'} (m : cylinder_map_over h c c') :\n  m.k ∘ c.i₁ = c'.i₁ ∘ h.h :=\nbegin\n  unfold relative_cylinder.i₁,\n  rw [assoc, m.hkii],\n  unfold endpoints_map pushout_of_maps,\n  rw [←assoc], simp\nend\n\nlemma exists_of_pair_map (h : pair_map hj hj')\n  (c₀ : relative_cylinder hj) (c₁ : relative_cylinder hj') :\n  ∃ c' (m₀ : cylinder_map_over h c₀ c') (m₁ : cylinder_embedding c₁ c'), true :=\nlet po := pushout_by_cof c₀.ii (c₁.ii ∘ endpoints_map h) c₀.hii,\n    pp := po.is_pushout.induced (h.h ∘ c₀.p) c₁.p $ begin\n      rw [←assoc, c₀.pii, assoc, c₁.pii],\n      unfold endpoints_map, rw [induced_pushout_of_maps, pushout_induced_comp],\n      simp\n    end,\n    ⟨c'_ob, l, q, hl, hq, ql⟩ := factorization pp in\nlet c' : relative_cylinder hj' :=\n  ⟨c'_ob, l ∘ po.map₁ ∘ c₁.ii, q,\n   cof_comp c₁.hii (cof_comp (pushout_is_cof po.is_pushout c₀.hii) hl),\n   hq, by simp [ql, c₁.pii]⟩ in\n⟨c',\n ⟨l ∘ po.map₀,\n  by rw [←assoc, po.is_pushout.commutes]; simp,\n  by simp [ql]⟩,\n ⟨l ∘ po.map₁,\n  cof_comp (pushout_is_cof po.is_pushout c₀.hii) hl,\n  rfl,\n  by simp [ql]⟩,\n trivial⟩\n\n\nend pair_map\n\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/cofibrations/cylinder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.30677095106660357}}
{"text": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n\n! This file was ported from Lean 3 source module ring_theory.adjoin.fg\n! leanprover-community/mathlib commit c4658a649d216f57e99621708b09dcb3dcccbd23\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.RingTheory.Polynomial.Basic\nimport Mathbin.RingTheory.PrincipalIdealDomain\nimport Mathbin.Data.MvPolynomial.Basic\n\n/-!\n# Adjoining elements to form subalgebras\n\nThis file develops the basic theory of finitely-generated subalgebras.\n\n## Definitions\n\n* `fg (S : subalgebra R A)` : A predicate saying that the subalgebra is finitely-generated\nas an A-algebra\n\n## Tags\n\nadjoin, algebra, finitely-generated algebra\n\n-/\n\n\nuniverse u v w\n\nopen Subsemiring Ring Submodule\n\nopen Pointwise\n\nnamespace Algebra\n\nvariable {R : Type u} {A : Type v} {B : Type w} [CommSemiring R] [CommSemiring A] [Algebra R A]\n  {s t : Set A}\n\n/- warning: algebra.fg_trans -> Algebra.fg_trans is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] {s : Set.{u2} A} {t : Set.{u2} A}, (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) 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(Set.{u2} A) (Set.hasUnion.{u2} A) s t))))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommSemiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2)] {s : Set.{u2} A} {t : Set.{u2} A}, (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (FunLike.coe.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 (CommSemiring.toSemiring.{u2} A _inst_2) _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} 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Consider using '#align algebra.fg_trans Algebra.fg_transₓ'. -/\ntheorem fg_trans (h1 : (adjoin R s).toSubmodule.Fg) (h2 : (adjoin (adjoin R s) t).toSubmodule.Fg) :\n    (adjoin R (s ∪ t)).toSubmodule.Fg :=\n  by\n  rcases fg_def.1 h1 with ⟨p, hp, hp'⟩\n  rcases fg_def.1 h2 with ⟨q, hq, hq'⟩\n  refine' fg_def.2 ⟨p * q, hp.mul hq, le_antisymm _ _⟩\n  · rw [span_le]\n    rintro _ ⟨x, y, hx, hy, rfl⟩\n    change x * y ∈ _\n    refine' Subalgebra.mul_mem _ _ _\n    · have : x ∈ (adjoin R s).toSubmodule := by\n        rw [← hp']\n        exact subset_span hx\n      exact adjoin_mono (Set.subset_union_left _ _) this\n    have : y ∈ (adjoin (adjoin R s) t).toSubmodule :=\n      by\n      rw [← hq']\n      exact subset_span hy\n    change y ∈ adjoin R (s ∪ t)\n    rwa [adjoin_union_eq_adjoin_adjoin]\n  · intro r hr\n    change r ∈ adjoin R (s ∪ t) at hr\n    rw [adjoin_union_eq_adjoin_adjoin] at hr\n    change r ∈ (adjoin (adjoin R s) t).toSubmodule at hr\n    rw [← hq', ← Set.image_id q, Finsupp.mem_span_image_iff_total (adjoin R s)] at hr\n    rcases hr with ⟨l, hlq, rfl⟩\n    have := @Finsupp.total_apply A A (adjoin R s)\n    rw [this, Finsupp.sum]\n    refine' sum_mem _\n    intro z hz\n    change (l z).1 * _ ∈ _\n    have : (l z).1 ∈ (adjoin R s).toSubmodule := (l z).2\n    rw [← hp', ← Set.image_id p, Finsupp.mem_span_image_iff_total R] at this\n    rcases this with ⟨l2, hlp, hl⟩\n    have := @Finsupp.total_apply A A R\n    rw [this] at hl\n    rw [← hl, Finsupp.sum_mul]\n    refine' sum_mem _\n    intro t ht\n    change _ * _ ∈ _\n    rw [smul_mul_assoc]\n    refine' smul_mem _ _ _\n    exact subset_span ⟨t, z, hlp ht, hlq hz, rfl⟩\n#align algebra.fg_trans Algebra.fg_trans\n\nend Algebra\n\nnamespace Subalgebra\n\nvariable {R : Type u} {A : Type v} {B : Type w}\n\nvariable [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]\n\n#print Subalgebra.Fg /-\n/-- A subalgebra `S` is finitely generated if there exists `t : finset A` such that\n`algebra.adjoin R t = S`. -/\ndef Fg (S : Subalgebra R A) : Prop :=\n  ∃ t : Finset A, Algebra.adjoin R ↑t = S\n#align subalgebra.fg Subalgebra.Fg\n-/\n\n#print Subalgebra.fg_adjoin_finset /-\ntheorem fg_adjoin_finset (s : Finset A) : (Algebra.adjoin R (↑s : Set A)).Fg :=\n  ⟨s, rfl⟩\n#align subalgebra.fg_adjoin_finset Subalgebra.fg_adjoin_finset\n-/\n\n#print Subalgebra.fg_def /-\ntheorem fg_def {S : Subalgebra R A} : S.Fg ↔ ∃ t : Set A, Set.Finite t ∧ Algebra.adjoin R t = S :=\n  Iff.symm Set.exists_finite_iff_finset\n#align subalgebra.fg_def Subalgebra.fg_def\n-/\n\n/- warning: subalgebra.fg_bot -> Subalgebra.fg_bot is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2], Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toHasBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_3)))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2], Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)))\nCase conversion may be inaccurate. Consider using '#align subalgebra.fg_bot Subalgebra.fg_botₓ'. -/\ntheorem fg_bot : (⊥ : Subalgebra R A).Fg :=\n  ⟨∅, Algebra.adjoin_empty R A⟩\n#align subalgebra.fg_bot Subalgebra.fg_bot\n\n/- warning: subalgebra.fg_of_fg_to_submodule -> Subalgebra.fg_of_fg_toSubmodule is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3}, (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (coeFn.{succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))))) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3) S)) -> (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3}, (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A 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(EmbeddingLike.toFunLike.{succ u2, succ u2, succ u2} (Function.Embedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Function.instEmbeddingLikeEmbedding.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) (RelEmbedding.toEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) => LE.le.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697) (Subalgebra.toSubmodule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) S)) -> (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)\nCase conversion may be inaccurate. Consider using '#align subalgebra.fg_of_fg_to_submodule Subalgebra.fg_of_fg_toSubmoduleₓ'. -/\ntheorem fg_of_fg_toSubmodule {S : Subalgebra R A} : S.toSubmodule.Fg → S.Fg := fun ⟨t, ht⟩ =>\n  ⟨t,\n    le_antisymm (Algebra.adjoin_le fun x hx => show x ∈ S.toSubmodule from ht ▸ subset_span hx) <|\n      show S.toSubmodule ≤ (Algebra.adjoin R ↑t).toSubmodule from fun x hx =>\n        span_le.mpr (fun x hx => Algebra.subset_adjoin hx)\n          (show x ∈ span R ↑t by\n            rw [ht]\n            exact hx)⟩\n#align subalgebra.fg_of_fg_to_submodule Subalgebra.fg_of_fg_toSubmodule\n\n#print Subalgebra.fg_of_noetherian /-\ntheorem fg_of_noetherian [IsNoetherian R A] (S : Subalgebra R A) : S.Fg :=\n  fg_of_fg_toSubmodule (IsNoetherian.noetherian S.toSubmodule)\n#align subalgebra.fg_of_noetherian Subalgebra.fg_of_noetherian\n-/\n\n/- warning: subalgebra.fg_of_submodule_fg -> Subalgebra.fg_of_submodule_fg is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2], (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Top.top.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.hasTop.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) -> (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toHasTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_3))))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2], (Submodule.Fg.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Top.top.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)) (Submodule.instTopSubmodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) -> (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Top.top.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3))))\nCase conversion may be inaccurate. Consider using '#align subalgebra.fg_of_submodule_fg Subalgebra.fg_of_submodule_fgₓ'. -/\ntheorem fg_of_submodule_fg (h : (⊤ : Submodule R A).Fg) : (⊤ : Subalgebra R A).Fg :=\n  let ⟨s, hs⟩ := h\n  ⟨s,\n    toSubmodule.Injective <|\n      by\n      rw [Algebra.top_toSubmodule, eq_top_iff, ← hs, span_le]\n      exact Algebra.subset_adjoin⟩\n#align subalgebra.fg_of_submodule_fg Subalgebra.fg_of_submodule_fg\n\n/- warning: subalgebra.fg.prod -> Subalgebra.Fg.prod is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_4 : Semiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_4] {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3} {T : Subalgebra.{u1, u3} R B _inst_1 _inst_4 _inst_5}, (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) -> (Subalgebra.Fg.{u1, u3} R B _inst_1 _inst_4 _inst_5 T) -> (Subalgebra.Fg.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.semiring.{u2, u3} A B _inst_2 _inst_4) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 S T))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_4 : Semiring.{u3} B] [_inst_5 : Algebra.{u1, u3} R B _inst_1 _inst_4] {S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3} {T : Subalgebra.{u1, u3} R B _inst_1 _inst_4 _inst_5}, (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) -> (Subalgebra.Fg.{u1, u3} R B _inst_1 _inst_4 _inst_5 T) -> (Subalgebra.Fg.{u1, max u2 u3} R (Prod.{u2, u3} A B) _inst_1 (Prod.instSemiringProd.{u2, u3} A B _inst_2 _inst_4) (Prod.algebra.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5) (Subalgebra.prod.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 S T))\nCase conversion may be inaccurate. Consider using '#align subalgebra.fg.prod Subalgebra.Fg.prodₓ'. -/\ntheorem Fg.prod {S : Subalgebra R A} {T : Subalgebra R B} (hS : S.Fg) (hT : T.Fg) : (S.Prod T).Fg :=\n  by\n  obtain ⟨s, hs⟩ := fg_def.1 hS\n  obtain ⟨t, ht⟩ := fg_def.1 hT\n  rw [← hs.2, ← ht.2]\n  exact\n    fg_def.2\n      ⟨LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1}),\n        Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))\n          (Set.Finite.image _ (Set.Finite.union ht.1 (Set.finite_singleton _))),\n        Algebra.adjoin_inl_union_inr_eq_prod R s t⟩\n#align subalgebra.fg.prod Subalgebra.Fg.prod\n\nsection\n\nopen Classical\n\n#print Subalgebra.Fg.map /-\ntheorem Fg.map {S : Subalgebra R A} (f : A →ₐ[R] B) (hs : S.Fg) : (S.map f).Fg :=\n  let ⟨s, hs⟩ := hs\n  ⟨s.image f, by rw [Finset.coe_image, Algebra.adjoin_image, hs]⟩\n#align subalgebra.fg.map Subalgebra.Fg.map\n-/\n\nend\n\n#print Subalgebra.fg_of_fg_map /-\ntheorem fg_of_fg_map (S : Subalgebra R A) (f : A →ₐ[R] B) (hf : Function.Injective f)\n    (hs : (S.map f).Fg) : S.Fg :=\n  let ⟨s, hs⟩ := hs\n  ⟨s.Preimage f fun _ _ _ _ h => hf h,\n    map_injective hf <|\n      by\n      rw [← Algebra.adjoin_image, Finset.coe_preimage, Set.image_preimage_eq_of_subset, hs]\n      rw [← AlgHom.coe_range, ← Algebra.adjoin_le_iff, hs, ← Algebra.map_top]\n      exact map_mono le_top⟩\n#align subalgebra.fg_of_fg_map Subalgebra.fg_of_fg_map\n-/\n\n/- warning: subalgebra.fg_top -> Subalgebra.fg_top is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), Iff (Subalgebra.Fg.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R 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(Algebra.Subalgebra.completeLattice.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)) S) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Subalgebra.algebra.{u1, u2} R A _inst_1 _inst_2 _inst_3 S))))) (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), Iff (Subalgebra.Fg.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)) x S)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Top.top.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)) x S)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)) (CompleteLattice.toTop.{u2} (Subalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)) x S)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)) x S)) _inst_1 (Subalgebra.toSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S) (Subalgebra.instAlgebraSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiring.{u1, u2} R A _inst_1 _inst_2 _inst_3 S))))) (Subalgebra.Fg.{u1, u2} R A _inst_1 _inst_2 _inst_3 S)\nCase conversion may be inaccurate. Consider using '#align subalgebra.fg_top Subalgebra.fg_topₓ'. -/\ntheorem fg_top (S : Subalgebra R A) : (⊤ : Subalgebra R S).Fg ↔ S.Fg :=\n  ⟨fun h => by\n    rw [← S.range_val, ← Algebra.map_top]\n    exact fg.map _ h, fun h =>\n    fg_of_fg_map _ S.val Subtype.val_injective <|\n      by\n      rw [Algebra.map_top, range_val]\n      exact h⟩\n#align subalgebra.fg_top Subalgebra.fg_top\n\n/- warning: subalgebra.induction_on_adjoin -> Subalgebra.induction_on_adjoin is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_6 : IsNoetherian.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)] (P : (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> Prop), (P (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toHasBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.Subalgebra.completeLattice.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) -> (forall (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (x : A), (P S) -> (P (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Insert.insert.{u2, u2} A (Set.{u2} A) (Set.hasInsert.{u2} A) x ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Set.{u2} A) (SetLike.Set.hasCoeT.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.setLike.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) S))))) -> (forall (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), P S)\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} A] [_inst_3 : Algebra.{u1, u2} R A _inst_1 _inst_2] [_inst_6 : IsNoetherian.{u1, u2} R A (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A _inst_2))) (Algebra.toModule.{u1, u2} R A _inst_1 _inst_2 _inst_3)] (P : (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) -> Prop), (P (Bot.bot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (CompleteLattice.toBot.{u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (Algebra.instCompleteLatticeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3)))) -> (forall (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) (x : A), (P S) -> (P (Algebra.adjoin.{u1, u2} R A _inst_1 _inst_2 _inst_3 (Insert.insert.{u2, u2} A (Set.{u2} A) (Set.instInsertSet.{u2} A) x (SetLike.coe.{u2, u2} (Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3) S))))) -> (forall (S : Subalgebra.{u1, u2} R A _inst_1 _inst_2 _inst_3), P S)\nCase conversion may be inaccurate. Consider using '#align subalgebra.induction_on_adjoin Subalgebra.induction_on_adjoinₓ'. -/\ntheorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)\n    (ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))\n    (S : Subalgebra R A) : P S := by\n  classical\n    obtain ⟨t, rfl⟩ := S.fg_of_noetherian\n    refine' Finset.induction_on t _ _\n    · simpa using base\n    intro x t hxt h\n    rw [Finset.coe_insert]\n    simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h\n#align subalgebra.induction_on_adjoin Subalgebra.induction_on_adjoin\n\nend Subalgebra\n\nsection Semiring\n\nvariable {R : Type u} {A : Type v} {B : Type w}\n\nvariable [CommSemiring R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B]\n\n/- warning: alg_hom.is_noetherian_ring_range -> AlgHom.isNoetherianRing_range is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] (f : AlgHom.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4 _inst_5) [_inst_6 : IsNoetherianRing.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], IsNoetherianRing.{u3} (coeSort.{succ u3, succ (succ u3)} (Subalgebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) B (Subalgebra.setLike.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5)) (AlgHom.range.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 f)) (Subalgebra.toSemiring.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 (AlgHom.range.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 f))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u2} R A _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_5 : Algebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] (f : AlgHom.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4 _inst_5) [_inst_6 : IsNoetherianRing.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], IsNoetherianRing.{u3} (Subtype.{succ u3} B (fun (x : B) => Membership.mem.{u3, u3} B (Subalgebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) (SetLike.instMembership.{u3, u3} (Subalgebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) B (Subalgebra.instSetLikeSubalgebra.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5)) x (AlgHom.range.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 f))) (Subalgebra.toSemiring.{u1, u3} R B _inst_1 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 (AlgHom.range.{u1, u2, u3} R A B _inst_1 (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5 f))\nCase conversion may be inaccurate. Consider using '#align alg_hom.is_noetherian_ring_range AlgHom.isNoetherianRing_rangeₓ'. -/\n/-- The image of a Noetherian R-algebra under an R-algebra map is a Noetherian ring. -/\ninstance AlgHom.isNoetherianRing_range (f : A →ₐ[R] B) [IsNoetherianRing A] :\n    IsNoetherianRing f.range :=\n  isNoetherianRing_range f.toRingHom\n#align alg_hom.is_noetherian_ring_range AlgHom.isNoetherianRing_range\n\nend Semiring\n\nsection Ring\n\nvariable {R : Type u} {A : Type v} {B : Type w}\n\nvariable [CommRing R] [CommRing A] [CommRing B] [Algebra R A] [Algebra R B]\n\n/- warning: is_noetherian_ring_of_fg -> isNoetherianRing_of_fg is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_4 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {S : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4}, (Subalgebra.Fg.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 S) -> (forall [_inst_6 : IsNoetherianRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))], IsNoetherianRing.{u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4)) S) (Subalgebra.toSemiring.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 S))\nbut is expected to have type\n  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_4 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {S : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4}, (Subalgebra.Fg.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 S) -> (forall [_inst_6 : IsNoetherianRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))], IsNoetherianRing.{u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4)) x S)) (Subalgebra.toSemiring.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_4 S))\nCase conversion may be inaccurate. Consider using '#align is_noetherian_ring_of_fg isNoetherianRing_of_fgₓ'. -/\ntheorem isNoetherianRing_of_fg {S : Subalgebra R A} (HS : S.Fg) [IsNoetherianRing R] :\n    IsNoetherianRing S :=\n  let ⟨t, ht⟩ := HS\n  ht ▸\n    (Algebra.adjoin_eq_range R (↑t : Set A)).symm ▸ by\n      haveI : IsNoetherianRing (MvPolynomial (↑t : Set A) R) := MvPolynomial.isNoetherianRing <;>\n          convert AlgHom.isNoetherianRing_range _ <;>\n        infer_instance\n#align is_noetherian_ring_of_fg isNoetherianRing_of_fg\n\n/- warning: is_noetherian_subring_closure -> is_noetherian_subring_closure is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R), (Set.Finite.{u1} R s) -> (IsNoetherianRing.{u1} (coeSort.{succ u1, succ (succ u1)} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) R (Subring.setLike.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s)) (Ring.toSemiring.{u1} (coeSort.{succ u1, succ (succ u1)} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) R (Subring.setLike.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s)) (Subring.toRing.{u1} R (CommRing.toRing.{u1} R _inst_1) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (s : Set.{u1} R), (Set.Finite.{u1} R s) -> (IsNoetherianRing.{u1} (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) R (Subring.instSetLikeSubring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))) (Ring.toSemiring.{u1} (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) R (Subring.instSetLikeSubring.{u1} R (CommRing.toRing.{u1} R _inst_1))) x (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))) (Subring.toRing.{u1} R (CommRing.toRing.{u1} R _inst_1) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s))))\nCase conversion may be inaccurate. Consider using '#align is_noetherian_subring_closure is_noetherian_subring_closureₓ'. -/\ntheorem is_noetherian_subring_closure (s : Set R) (hs : s.Finite) :\n    IsNoetherianRing (Subring.closure s) :=\n  show IsNoetherianRing (subalgebraOfSubring (Subring.closure s)) from\n    Algebra.adjoin_int s ▸ isNoetherianRing_of_fg (Subalgebra.fg_def.2 ⟨s, hs, rfl⟩)\n#align is_noetherian_subring_closure is_noetherian_subring_closure\n\nend Ring\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/RingTheory/Adjoin/Fg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.4610167793123158, "lm_q1q2_score": 0.30676543271386475}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro, Minchao Wu\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nuniverses l \n\nnamespace Mathlib\n\n/-!\n# `#explode` command\n\nDisplays a proof term in a line by line format somewhat akin to a Fitch style\nproof or the Metamath proof style.\n-/\n\nnamespace tactic\n\n\nnamespace explode\n\n\ninductive status \nwhere\n| reg : status\n| intro : status\n| lam : status\n| sintro : status\n\n/--\nA type to distinguish introduction or elimination rules represented as \nstrings from theorems referred to by their names.\n-/\n/--\nTurn a thm into a string.\n-/\nend explode\n\n\n/--\n`#explode decl_name` displays a proof term in a line-by-line format somewhat akin to a Fitch-style\nproof or the Metamath proof style.\n`#explode_widget decl_name` renders a widget that displays an `#explode` proof.\n\n`#explode iff_true_intro` produces\n\n```lean\niff_true_intro : ∀ {a : Prop}, a → (a ↔ true)\n0│   │ a         ├ Prop\n1│   │ h         ├ a\n2│   │ hl        │ ┌ a\n3│   │ trivial   │ │ true\n4│2,3│ ∀I        │ a → true\n5│   │ hr        │ ┌ true\n6│5,1│ ∀I        │ true → a\n7│4,6│ iff.intro │ a ↔ true\n8│1,7│ ∀I        │ a → (a ↔ true)\n9│0,8│ ∀I        │ ∀ {a : Prop}, a → (a ↔ true)\n```\n\nIn more detail:\n\nThe output of `#explode` is a Fitch-style proof in a four-column diagram modeled after Metamath\nproof displays like [this](http://us.metamath.org/mpeuni/ru.html). The headers of the columns are\n\"Step\", \"Hyp\", \"Ref\", \"Type\" (or \"Expression\" in the case of Metamath):\n* Step: An increasing sequence of numbers to number each step in the proof, used in the Hyp field.\n* Hyp: The direct children of the current step. Most theorems are implications like `A -> B -> C`,\n  and so on the step proving `C` the Hyp field will refer to the steps that prove `A` and `B`.\n* Ref: The name of the theorem being applied. This is well-defined in Metamath, but in Lean there\n  are some special steps that may have long names because the structure of proof terms doesn't\n  exactly match this mold.\n  * If the theorem is `foo (x y : Z) : A x -> B y -> C x y`:\n    * the Ref field will contain `foo`,\n    * `x` and `y` will be suppressed, because term construction is not interesting, and\n    * the Hyp field will reference steps proving `A x` and `B y`. This corresponds to a proof term\n      like `@foo x y pA pB` where `pA` and `pB` are subproofs.\n  * If the head of the proof term is a local constant or lambda, then in this case the Ref will\n    say `∀E` for forall-elimination. This happens when you have for example `h : A -> B` and\n    `ha : A` and prove `b` by `h ha`; we reinterpret this as if it said `∀E h ha` where `∀E` is\n    (n-ary) modus ponens.\n  * If the proof term is a lambda, we will also use `∀I` for forall-introduction, referencing the\n    body of the lambda. The indentation level will increase, and a bracket will surround the proof\n    of the body of the lambda, starting at a proof step labeled with the name of the lambda variable\n    and its type, and ending with the `∀I` step. Metamath doesn't have steps like this, but the\n    style is based on Fitch proofs in first-order logic.\n* Type: This contains the type of the proof term, the theorem being proven at the current step.\n  This proof layout differs from `#print` in using lots of intermediate step displays so that you\n  can follow along and don't have to see term construction steps because they are implicitly in the\n  intermediate step displays.\n\nAlso, it is common for a Lean theorem to begin with a sequence of lambdas introducing local\nconstants of the theorem. In order to minimize the indentation level, the `∀I` steps at the end of\nthe proof will be introduced in a group and the indentation will stay fixed. (The indentation\nbrackets are only needed in order to delimit the scope of assumptions, and these assumptions\nhave global scope anyway so detailed tracking is not necessary.)\n-/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/explode.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3065307247883567}}
{"text": "import Smt\n\ntheorem cong (x y : Int) (f : Int → Int) : x = y → f x = f y := by\n  smt\n  simp_all\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Int/Cong.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5851011686727232, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3062538578697278}}
{"text": "/-\nCopyright (c) 2019 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.integration\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# The Giry monad\n\nLet X be a measurable space. The collection of all measures on X again\nforms a measurable space. This construction forms a monad on\nmeasurable spaces and measurable functions, called the Giry monad.\n\nNote that most sources use the term \"Giry monad\" for the restriction\nto *probability* measures. Here we include all measures on X.\n\nSee also `measure_theory/category/Meas.lean`, containing an upgrade of the type-level\nmonad to an honest monad of the functor `Measure : Meas ⥤ Meas`.\n\n## References\n\n* <https://ncatlab.org/nlab/show/Giry+monad>\n\n## Tags\n\ngiry monad\n-/\n\nnamespace measure_theory\n\n\nnamespace measure\n\n\n/-- Measurability structure on `measure`: Measures are measurable w.r.t. all projections -/\nprotected instance measurable_space {α : Type u_1} [measurable_space α] : measurable_space (measure α) :=\n  supr\n    fun (s : set α) =>\n      supr fun (hs : is_measurable s) => measurable_space.comap (fun (μ : measure α) => coe_fn μ s) (borel ennreal)\n\ntheorem measurable_coe {α : Type u_1} [measurable_space α] {s : set α} (hs : is_measurable s) : measurable fun (μ : measure α) => coe_fn μ s :=\n  measurable.of_comap_le\n    (le_supr_of_le s\n      (le_supr_of_le hs (le_refl (measurable_space.comap (fun (μ : measure α) => coe_fn μ s) ennreal.measurable_space))))\n\ntheorem measurable_of_measurable_coe {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (f : β → measure α) (h : ∀ (s : set α), is_measurable s → measurable fun (b : β) => coe_fn (f b) s) : measurable f := sorry\n\ntheorem measurable_measure {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {μ : α → measure β} : measurable μ ↔ ∀ (s : set β), is_measurable s → measurable fun (b : α) => coe_fn (μ b) s :=\n  { mp := fun (hμ : measurable μ) (s : set β) (hs : is_measurable s) => measurable.comp (measurable_coe hs) hμ,\n    mpr := measurable_of_measurable_coe μ }\n\ntheorem measurable_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (f : α → β) (hf : measurable f) : measurable fun (μ : measure α) => coe_fn (map f) μ := sorry\n\ntheorem measurable_dirac {α : Type u_1} [measurable_space α] : measurable dirac := sorry\n\ntheorem measurable_lintegral {α : Type u_1} [measurable_space α] {f : α → ennreal} (hf : measurable f) : measurable fun (μ : measure α) => lintegral μ fun (x : α) => f x := sorry\n\n/-- Monadic join on `measure` in the category of measurable spaces and measurable\nfunctions. -/\ndef join {α : Type u_1} [measurable_space α] (m : measure (measure α)) : measure α :=\n  of_measurable (fun (s : set α) (hs : is_measurable s) => lintegral m fun (μ : measure α) => coe_fn μ s) sorry sorry\n\n@[simp] theorem join_apply {α : Type u_1} [measurable_space α] {m : measure (measure α)} {s : set α} : is_measurable s → coe_fn (join m) s = lintegral m fun (μ : measure α) => coe_fn μ s :=\n  of_measurable_apply\n\ntheorem measurable_join {α : Type u_1} [measurable_space α] : measurable join := sorry\n\ntheorem lintegral_join {α : Type u_1} [measurable_space α] {m : measure (measure α)} {f : α → ennreal} (hf : measurable f) : (lintegral (join m) fun (x : α) => f x) = lintegral m fun (μ : measure α) => lintegral μ fun (x : α) => f x := sorry\n\n/-- Monadic bind on `measure`, only works in the category of measurable spaces and measurable\nfunctions. When the function `f` is not measurable the result is not well defined. -/\ndef bind {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] (m : measure α) (f : α → measure β) : measure β :=\n  join (coe_fn (map f) m)\n\n@[simp] theorem bind_apply {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {m : measure α} {f : α → measure β} {s : set β} (hs : is_measurable s) (hf : measurable f) : coe_fn (bind m f) s = lintegral m fun (a : α) => coe_fn (f a) s := sorry\n\ntheorem measurable_bind' {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {g : α → measure β} (hg : measurable g) : measurable fun (m : measure α) => bind m g :=\n  measurable.comp measurable_join (measurable_map g hg)\n\ntheorem lintegral_bind {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {m : measure α} {μ : α → measure β} {f : β → ennreal} (hμ : measurable μ) (hf : measurable f) : (lintegral (bind m μ) fun (x : β) => f x) = lintegral m fun (a : α) => lintegral (μ a) fun (x : β) => f x :=\n  Eq.trans (lintegral_join hf) (lintegral_map (measurable_lintegral hf) hμ)\n\ntheorem bind_bind {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {γ : Type u_3} [measurable_space γ] {m : measure α} {f : α → measure β} {g : β → measure γ} (hf : measurable f) (hg : measurable g) : bind (bind m f) g = bind m fun (a : α) => bind (f a) g := sorry\n\ntheorem bind_dirac {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {f : α → measure β} (hf : measurable f) (a : α) : bind (dirac a) f = f a := sorry\n\ntheorem dirac_bind {α : Type u_1} [measurable_space α] {m : measure α} : bind m dirac = m := sorry\n\ntheorem join_eq_bind {α : Type u_1} [measurable_space α] (μ : measure (measure α)) : join μ = bind μ id :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (join μ = bind μ id)) (bind.equations._eqn_1 μ id)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (join μ = join (coe_fn (map id) μ))) map_id)) (Eq.refl (join μ)))\n\ntheorem join_map_map {α : Type u_1} {β : Type u_2} [measurable_space α] [measurable_space β] {f : α → β} (hf : measurable f) (μ : measure (measure α)) : join (coe_fn (map ⇑(map f)) μ) = coe_fn (map f) (join μ) := sorry\n\ntheorem join_map_join {α : Type u_1} [measurable_space α] (μ : measure (measure (measure α))) : join (coe_fn (map join) μ) = join (join μ) := sorry\n\ntheorem join_map_dirac {α : Type u_1} [measurable_space α] (μ : measure α) : join (coe_fn (map dirac) μ) = μ :=\n  dirac_bind\n\ntheorem join_dirac {α : Type u_1} [measurable_space α] (μ : measure α) : join (dirac μ) = μ :=\n  Eq.trans (join_eq_bind (dirac μ)) (bind_dirac measurable_id μ)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/giry_monad.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.30620522804269357}}
{"text": "import .homotopy_equivalences\nimport .lemmas\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\n/-\n\nWe show that IA is a cylinder object for A, in the sense that\nA ⊔ A → IA → A is a factorization of the fold map into a\ncofibration followed by a homotopy equivalence.\n\n-/\n\nnamespace homotopy_theory.cofibrations\nsection C\nopen category_theory.has_initial_object\nopen homotopy_theory.cylinder\nopen I_category\n\nparameters {C : Type u} [category.{v} C] [has_initial_object.{v} C]\n  [has_coproducts.{v} C] [I_category.{v} C]\n\nlemma cof_ii (a : C) : is_cof (ii.{v} @> a) :=\nbegin\n  let po :=\n    (Is_pushout_of_isomorphic (Is_pushout.refl (! (∂I.obj a))) _ _\n      (coprod_initial_right ∅).symm (iso.refl _)\n      (initial_object.unique Ii_initial has_initial_object.initial_object.is_initial_object)\n      _ _),\n  convert relative_cylinder' (! a) (all_objects_cofibrant.cofibrant.{v} a) _ _ po,\n  any_goals { apply coprod.uniqueness; apply initial.uniqueness },\n  symmetry,\n  convert ←(po.induced_commutes₀ _ _ _),\n  convert id_comp _,\n  simp\nend\n\nlemma i₀p {a : C} : i.{v} 0 @> a ∘ p @> a ≃ 𝟙 (I.obj a) :=\nlet ⟨J, hJ₁, hJ₂⟩ :=\n  hep_cof (ii.{v} @> a) (cof_ii a) 0 (I.obj a) (i 0 @> a ∘ p @> a)\n    (I_of_coprod_is_coproduct.induced (i 0 @> a ∘ p @> a) (𝟙 (I.obj a))) $ begin\n      apply coprod.uniqueness; erw i_nat_assoc; simp,\n      rw ←assoc, dsimp, simp\n    end in\n⟨⟨J ∘ T @> a,\n  begin\n    erw [←assoc, cylinder_has_interchange.Ti],\n    have : J ∘ I &> (i 0 @> a) = J ∘ I &> (ii @> a ∘ i₀), by simp [ii], rw this,\n    rw [I.map_comp, assoc, hJ₂], simp\n  end,\n  begin\n    erw [←assoc, cylinder_has_interchange.Ti],\n    have : J ∘ I &> (i 1 @> a) = J ∘ I &> (ii @> a ∘ i₁), by simp [ii], rw this,\n    rw [I.map_comp, assoc, hJ₂], simp\n  end⟩⟩\n\nlemma heq_p {a : C} : homotopy_equivalence.{v} (p @> a) :=\nhomotopy_equivalence_iff.mpr ⟨i 0 @> a, i₀p, by simp; refl⟩\n\nlemma pii {a : C} : p.{v} @> a ∘ ii @> a = coprod.fold a :=\nby apply coprod.uniqueness; simp\n\nend C\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/i_category/cylinder_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3062052137214157}}
{"text": "open Classical\n\ntheorem dne {p : Prop} (h : ¬¬p) : p :=\n  Or.elim (em p)\n    (fun hp : p => hp)\n    (fun hnp : ¬p => absurd hnp h)", "meta": {"author": "mothematician", "repo": "lean4-proof-stuff", "sha": "1f202b1d3059f3a36dab9abd7564594bf765ebf2", "save_path": "github-repos/lean/mothematician-lean4-proof-stuff", "path": "github-repos/lean/mothematician-lean4-proof-stuff/lean4-proof-stuff-1f202b1d3059f3a36dab9abd7564594bf765ebf2/intro to logic and proofs book/5.1proofbycontradiction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3062035138101244}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.group_power.default\nimport Mathlib.control.uliftable\nimport Mathlib.control.monad.basic\nimport Mathlib.data.bitvec.basic\nimport Mathlib.data.list.basic\nimport Mathlib.data.set.intervals.basic\nimport Mathlib.data.stream.basic\nimport Mathlib.data.fin\nimport Mathlib.tactic.cache\nimport Mathlib.tactic.interactive\nimport Mathlib.tactic.norm_num\nimport Mathlib.Lean3Lib.system.io\nimport Mathlib.Lean3Lib.system.random\nimport Mathlib.PostPort\n\nuniverses u u_1 v l \n\nnamespace Mathlib\n\n/-!\n# Rand Monad and Random Class\n\nThis module provides tools for formulating computations guided by randomness and for\ndefining objects that can be created randomly.\n\n## Main definitions\n  * `rand` monad for computations guided by randomness;\n  * `random` class for objects that can be generated randomly;\n    * `random` to generate one object;\n    * `random_r` to generate one object inside a range;\n    * `random_series` to generate an infinite series of objects;\n    * `random_series_r` to generate an infinite series of objects inside a range;\n  * `io.mk_generator` to create a new random number generator;\n  * `io.run_rand` to run a randomized computation inside the `io` monad;\n  * `tactic.run_rand` to run a randomized computation inside the `tactic` monad\n\n## Local notation\n\n * `i .. j` : `Icc i j`, the set of values between `i` and `j` inclusively;\n\n## Tags\n\nrandom monad io\n\n## References\n\n  * Similar library in Haskell: https://hackage.haskell.org/package/MonadRandom\n\n-/\n\n/-- A monad to generate random objects using the generator type `g` -/\ndef rand_g (g : Type) (α : Type u) := state (ulift g) α\n\n/-- A monad to generate random objects using the generator type `std_gen` -/\ndef rand (α : Type u_1) := rand_g std_gen\n\nprotected instance rand_g.uliftable (g : Type) : uliftable (rand_g g) (rand_g g) :=\n  state_t.uliftable' (equiv.trans equiv.ulift (equiv.symm equiv.ulift))\n\n/-- Generate one more `ℕ` -/\ndef rand_g.next {g : Type} [random_gen g] : rand_g g ℕ :=\n  state_t.mk (prod.map id ulift.up ∘ random_gen.next ∘ ulift.down)\n\n/-- `bounded_random α` gives us machinery to generate values of type `α` between certain bounds -/\nclass bounded_random (α : Type u) [preorder α] where\n  random_r : (g : Type) → [_inst_1_1 : random_gen g] → (x y : α) → x ≤ y → rand_g g ↥(set.Icc x y)\n\n/-- `random α` gives us machinery to generate values of type `α` -/\nclass random (α : Type u) where\n  random : (g : Type) → [_inst_1 : random_gen g] → rand_g g α\n\n/-- shift_31_left = 2^31; multiplying by it shifts the binary\nrepresentation of a number left by 31 bits, dividing by it shifts it\nright by 31 bits -/\ndef shift_31_left : ℕ :=\n  bit0\n    (bit0\n      (bit0\n        (bit0\n          (bit0\n            (bit0\n              (bit0\n                (bit0\n                  (bit0\n                    (bit0\n                      (bit0\n                        (bit0\n                          (bit0\n                            (bit0\n                              (bit0\n                                (bit0\n                                  (bit0\n                                    (bit0\n                                      (bit0\n                                        (bit0\n                                          (bit0\n                                            (bit0\n                                              (bit0\n                                                (bit0\n                                                  (bit0\n                                                    (bit0\n                                                      (bit0\n                                                        (bit0\n                                                          (bit0\n                                                            (bit0\n                                                              (bit0 1))))))))))))))))))))))))))))))\n\nnamespace rand\n\n\n/-- create a new random number generator distinct from the one stored in the state -/\ndef split (g : Type) [random_gen g] : rand_g g g :=\n  state_t.mk (prod.map id ulift.up ∘ random_gen.split ∘ ulift.down)\n\n/-- Generate a random value of type `α`. -/\ndef random (α : Type u) {g : Type} [random_gen g] [random α] : rand_g g α := random α g\n\n/-- generate an infinite series of random values of type `α` -/\ndef random_series (α : Type u) {g : Type} [random_gen g] [random α] : rand_g g (stream α) :=\n  do \n    let gen ← uliftable.up (split g)\n    pure (stream.corec_state (random α g) gen)\n\n/-- Generate a random value between `x` and `y` inclusive. -/\ndef random_r {α : Type u} {g : Type} [random_gen g] [preorder α] [bounded_random α] (x : α) (y : α)\n    (h : x ≤ y) : rand_g g ↥(set.Icc x y) :=\n  bounded_random.random_r g x y h\n\n/-- generate an infinite series of random values of type `α` between `x` and `y` inclusive. -/\ndef random_series_r {α : Type u} {g : Type} [random_gen g] [preorder α] [bounded_random α] (x : α)\n    (y : α) (h : x ≤ y) : rand_g g (stream ↥(set.Icc x y)) :=\n  do \n    let gen ← uliftable.up (split g)\n    pure (stream.corec_state (bounded_random.random_r g x y h) gen)\n\nend rand\n\n\nnamespace io\n\n\n/-- create and a seed a random number generator -/\ndef mk_generator : io std_gen :=\n  do \n    let seed ← rand 0 shift_31_left \n    return (mk_std_gen seed)\n\n/-- Run `cmd` using a randomly seeded random number generator -/\ndef run_rand {α : Type} (cmd : rand α) : io α :=\n  do \n    let g ← mk_generator \n    return (prod.fst (state_t.run cmd (ulift.up g)))\n\n/-- Run `cmd` using the provided seed. -/\ndef run_rand_with {α : Type} (seed : ℕ) (cmd : rand α) : io α :=\n  return (prod.fst (state_t.run cmd (ulift.up (mk_std_gen seed))))\n\n/-- randomly generate a value of type α -/\ndef random {α : Type} [random α] : io α := run_rand (rand.random α)\n\n/-- randomly generate an infinite series of value of type α -/\ndef random_series {α : Type} [random α] : io (stream α) := run_rand (rand.random_series α)\n\n/-- randomly generate a value of type α between `x` and `y` -/\ndef random_r {α : Type} [preorder α] [bounded_random α] (x : α) (y : α) (p : x ≤ y) :\n    io ↥(set.Icc x y) :=\n  run_rand (bounded_random.random_r std_gen x y p)\n\n/-- randomly generate an infinite series of value of type α between `x` and `y` -/\ndef random_series_r {α : Type} [preorder α] [bounded_random α] (x : α) (y : α) (h : x ≤ y) :\n    io (stream ↥(set.Icc x y)) :=\n  run_rand (rand.random_series_r x y h)\n\nend io\n\n\nnamespace tactic\n\n\n/-- create a seeded random number generator in the `tactic` monad -/\n/-- run `cmd` using the a randomly seeded random number generator\nin the tactic monad -/\n/-- Generate a random value between `x` and `y` inclusive. -/\n/-- Generate an infinite series of random values of type `α` between `x` and `y` inclusive. -/\n/-- randomly generate a value of type α -/\nend tactic\n\n\nnamespace fin\n\n\n/-- generate a `fin` randomly -/\nprotected def random {g : Type} [random_gen g] {n : ℕ} [fact (0 < n)] : rand_g g (fin n) :=\n  state_t.mk fun (_x : ulift g) => sorry\n\nend fin\n\n\nprotected instance nat_bounded_random : bounded_random ℕ :=\n  bounded_random.mk\n    fun (g : Type) (inst : random_gen g) (x y : ℕ) (hxy : x ≤ y) =>\n      do \n        let z ← fin.random \n        pure { val := subtype.val z + x, property := sorry }\n\n/-- This `bounded_random` interval generates integers between `x` and\n`y` by first generating a natural number between `0` and `y - x` and\nshifting the result appropriately. -/\nprotected instance int_bounded_random : bounded_random ℤ :=\n  bounded_random.mk\n    fun (g : Type) (inst : random_gen g) (x y : ℤ) (hxy : x ≤ y) =>\n      do \n        bounded_random.random_r g 0 (int.nat_abs (y - x)) sorry \n        sorry\n\nprotected instance fin_random (n : ℕ) [fact (0 < n)] : random (fin n) :=\n  random.mk fun (g : Type) (inst : random_gen g) => fin.random\n\nprotected instance fin_bounded_random (n : ℕ) : bounded_random (fin n) :=\n  bounded_random.mk\n    fun (g : Type) (inst : random_gen g) (x y : fin n) (p : x ≤ y) =>\n      do \n        rand.random_r (subtype.val x) (subtype.val y) p \n        sorry\n\n/-- A shortcut for creating a `random (fin n)` instance from\na proof that `0 < n` rather than on matching on `fin (succ n)`  -/\ndef random_fin_of_pos {n : ℕ} (h : 0 < n) : random (fin n) := sorry\n\ntheorem bool_of_nat_mem_Icc_of_mem_Icc_to_nat (x : Bool) (y : Bool) (n : ℕ) :\n    n ∈ set.Icc (bool.to_nat x) (bool.to_nat y) → bool.of_nat n ∈ set.Icc x y :=\n  sorry\n\nprotected instance bool.random : random Bool :=\n  random.mk\n    fun (g : Type) (inst : random_gen g) =>\n      (bool.of_nat ∘ subtype.val) <$> bounded_random.random_r g 0 1 sorry\n\nprotected instance bool.bounded_random : bounded_random Bool :=\n  bounded_random.mk\n    fun (g : Type) (_inst : random_gen g) (x y : Bool) (p : x ≤ y) =>\n      subtype.map bool.of_nat (bool_of_nat_mem_Icc_of_mem_Icc_to_nat x y) <$>\n        bounded_random.random_r g (bool.to_nat x) (bool.to_nat y) (bool.to_nat_le_to_nat p)\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random {g : Type} [random_gen g] (n : ℕ) : rand_g g (bitvec n) :=\n  bitvec.of_fin <$> rand.random (fin (bit0 1 ^ n))\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random_r {g : Type} [random_gen g] {n : ℕ} (x : bitvec n) (y : bitvec n) (h : x ≤ y) :\n    rand_g g ↥(set.Icc x y) :=\n  (fun\n      (h' :\n      ∀ (a : fin (bit0 1 ^ n)),\n        a ∈ set.Icc (bitvec.to_fin x) (bitvec.to_fin y) → bitvec.of_fin a ∈ set.Icc x y) =>\n      subtype.map bitvec.of_fin h' <$>\n        rand.random_r (bitvec.to_fin x) (bitvec.to_fin y) (bitvec.to_fin_le_to_fin_of_le h))\n    sorry\n\nprotected instance random_bitvec (n : ℕ) : random (bitvec n) :=\n  random.mk fun (_x : Type) (inst : random_gen _x) => bitvec.random n\n\nprotected instance bounded_random_bitvec (n : ℕ) : bounded_random (bitvec n) :=\n  bounded_random.mk\n    fun (_x : Type) (inst : random_gen _x) (x y : bitvec n) (p : x ≤ y) => bitvec.random_r x y p\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/system/random/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.3061500173990211}}
{"text": "import data.zmod.basic\n\nimport for_mathlib.complex_extend\nimport for_mathlib.projectives\nimport for_mathlib.two_step_resolution\nimport for_mathlib.hom_single_iso\nimport for_mathlib.homological_complex_op\n\n.\n\nuniverses v u\n\nopen category_theory homotopy_category\n\nvariables {C : Type u} {ι : Type*} [category.{v} C] [abelian C] {c : complex_shape ι}\n\nopen_locale zero_object\n\ninstance projective_zero : projective (0 : C) :=\n{ factors := λ E X f e he, ⟨0, by ext⟩ }\n\nlemma category_theory.is_iso_of_nat_iso {C D : Type*} [category C] [category D]\n  {F G : C ⥤ D} (α : F ≅ G)\n  {X Y : C} (f : X ⟶ Y) (h : is_iso (F.map f)) :\n  is_iso (G.map f) :=\nbegin\n  rw ← nat_iso.naturality_1 α f,\n  apply_instance,\nend\n\nlemma category_theory.nat_iso.is_iso_iff {C D : Type*} [category C] [category D]\n  {F G : C ⥤ D} (α : F ≅ G)\n  {X Y : C} (f : X ⟶ Y) :\n  is_iso (F.map f) ↔ is_iso (G.map f) :=\n⟨category_theory.is_iso_of_nat_iso α _, category_theory.is_iso_of_nat_iso α.symm _⟩\n\nopen category_theory\n\nlemma chain_complex.bounded_by_one (P : chain_complex C ℕ) :\n  ((homological_complex.embed complex_shape.embedding.nat_down_int_up ⋙\n      quotient C (complex_shape.up ℤ)).obj P).bounded_by 1 :=\nbegin\n  rintro ((_|i)|i) h,\n  { exfalso, revert h, dec_trivial },\n  { exact limits.is_zero_zero _ },\n  { exfalso, revert h, dec_trivial }\nend\n\ninstance chain_complex.is_bounded_above (P : chain_complex C ℕ) :\n  ((homological_complex.embed complex_shape.embedding.nat_down_int_up ⋙\n      quotient C (complex_shape.up ℤ)).obj P).is_bounded_above :=\n⟨⟨1, chain_complex.bounded_by_one _⟩⟩\n\n/-- The functor from ℕ-indexed chain complexes to bounded-above ℤ-indexed cochain complexes\nsending `C₀ ← C₁ ← ...` to `... ⟶ C₁ ⟶ C₀ ⟶ 0 ⟶ ...` -/\n@[simps obj obj_val map]\nnoncomputable def chain_complex.to_bounded_homotopy_category :\n  chain_complex C ℕ ⥤ bounded_homotopy_category C :=\n{ obj := λ P,\n  { val := (homological_complex.embed (complex_shape.embedding.nat_down_int_up) ⋙\n      homotopy_category.quotient C _).obj P,\n    bdd := by apply chain_complex.is_bounded_above },\n  map := λ P Q f, (homological_complex.embed (complex_shape.embedding.nat_down_int_up) ⋙\n      homotopy_category.quotient C _).map f,\n  map_id' := λ P, (homological_complex.embed (complex_shape.embedding.nat_down_int_up) ⋙\n      homotopy_category.quotient C _).map_id P,\n  map_comp' := λ P Q R f g, (homological_complex.embed (complex_shape.embedding.nat_down_int_up) ⋙\n      homotopy_category.quotient C _).map_comp f g }\n\nlemma chain_complex.to_bounded_homotopy_category.is_K_projective [enough_projectives C]\n  (P : chain_complex C ℕ) (A : C) (π : P ⟶ (chain_complex.single₀ C).obj A)\n  (hP : P.is_projective_resolution A π) :\n  is_K_projective (chain_complex.to_bounded_homotopy_category.obj P).val :=\nbegin\n  constructor,\n  introsI Y hY f,\n  rw [← quotient_map_out f],\n  apply eq_of_homotopy,\n  refine projective.null_homotopic_of_projective_to_acyclic _ 1 _ _ hY.cond,\n  { rintro ((_|i)|i),\n    { exact hP.projective 0 },\n    { exact projective_zero, },\n    { exact hP.projective _ }, },\n  { rintro ((_|i)|i),\n    { intro h, exfalso, revert h, dec_trivial },\n    { intro h, exact limits.is_zero_zero _ },\n    { intro h, exfalso, revert h, dec_trivial } },\nend\n.\n\nnoncomputable\ndef homological_complex.homology_functor_single [decidable_eq ι] (i : ι) :\n  homological_complex.single C c i ⋙ homology_functor C c i ≅ 𝟭 C :=\nbegin\n  refine nat_iso.of_components _ _,\n  { intro X,\n    refine homology.congr _ _ _ _ ≪≫ homology_zero_zero ≪≫ _,\n    { refl },\n    { refl },\n    { exact eq_to_iso (if_pos rfl) } },\n  { intros X Y f,\n    apply homology.ext,\n    dsimp,\n    simpa only [category.comp_id, homological_complex.hom.sq_from_left, eq_to_hom_refl,\n      homological_complex.single_obj_X_self_hom, homology.map_desc_assoc, eq_to_hom_trans,\n      homological_complex.single_obj_X_self_inv, homological_complex.single_map_f_self,\n      category.assoc, limits.kernel_subobject_map_arrow, homology.π_desc_assoc] }\nend\n\ninstance {X Y Z : C} (f: X ⟶ Y) : is_iso (homology.ι f (0 : Y ⟶ Z) limits.comp_zero) :=\nbegin\n  suffices : limits.cokernel.desc f 0 limits.comp_zero = 0,\n  { exact @@is_iso.comp_is_iso _ _ (show _, by convert limits.kernel.ι_zero_is_iso) },\n  ext,\n  simp,\nend\n\nlemma homology.desc_zero_is_iso_of_exact_of_epi {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)\n  (h : exact f g) [epi g] : is_iso (homology.desc' f (0 : Y ⟶ W) limits.comp_zero\n    (limits.kernel_zero_iso_source.hom ≫ g) (by simp [h.w])) :=\nbegin\n  convert_to is_iso (homology.ι _ _ _ ≫\n    (limits.colimit.iso_colimit_cocone ⟨_, abelian.is_colimit_of_exact_of_epi f g h⟩).hom),\n  { ext, simp },\n  { apply_instance }\nend\n\nlemma chain_complex.is_projective_resolution.is_quasi_iso_embed {P : chain_complex C ℕ}\n  {A : C} {π : P ⟶ (chain_complex.single₀ C).obj A}\n  (hP : P.is_projective_resolution A π) :\n  is_quasi_iso ((homological_complex.embed (complex_shape.embedding.nat_down_int_up) ⋙\n       homotopy_category.quotient C _).map π) :=\nbegin\n  constructor,\n  intro i,\n  rw [← functor.comp_map, category_theory.nat_iso.is_iso_iff\n    (functor.associator _ _ _ ≪≫ iso_whisker_left _ (homology_factors _ _ _))],\n  rcases i with ((_|i)|i),\n  { apply_with is_iso.of_is_iso_comp_right { instances := ff },\n    show is_iso (functor.map_iso _ (chain_complex.single₀_comp_embed_iso_single.app A) ≪≫\n        ((homological_complex.homology_functor_single (0 : ℤ)).app A : _)).hom, by apply_instance,\n    apply_with is_iso.of_is_iso_comp_left { instances := ff },\n    swap 2, convert @@homology.desc_zero_is_iso_of_exact_of_epi _ _ _ _ hP.exact₀ hP.epi using 1,\n    swap 4,\n    { refine (homology.map_iso _ _ (arrow.iso_mk _ _ _)\n        (arrow.iso_mk _ (by exact iso.refl _) _) rfl).hom,\n      { exact ((homological_complex.embed.obj complex_shape.embedding.nat_down_int_up P).X_prev_iso\n          (show (complex_shape.up ℤ).rel (-[1+0]) 0, from neg_add_self 1)).symm },\n      { exact iso.refl _ },\n      { dsimp,\n        rw [eq_comm, iso.eq_inv_comp, category.comp_id, eq_comm],\n        apply homological_complex.d_to_eq },\n      { dsimp,\n        rw [category.comp_id, category.id_comp],\n        erw homological_complex.d_from_eq _ (show (complex_shape.up ℤ).rel 0 1, from rfl),\n        exact limits.zero_comp } },\n    { ext,\n      dsimp only [homology.map_iso, homological_complex.homology_functor_single],\n      rw [← cancel_epi (limits.kernel_subobject_iso _).hom, homology.π'_eq_π_assoc],\n      simp only [homology.π'_desc', category.comp_id, homological_complex.hom.sq_from_left,\n        limits.kernel_subobject_arrow_assoc, homology.π_desc, homology.map_desc,\n        limits.kernel_subobject_map_arrow_assoc, arrow.iso_mk_hom_left,\n        limits.kernel_subobject_map_arrow],\n      dsimp only [chain_complex.single₀_comp_embed_iso_single,\n        chain_complex.single₀_comp_embed_iso_single_component],\n      simp only [limits.kernel_subobject_map_arrow, limits.kernel_subobject_map_arrow_assoc, homological_complex.hom.sq_from_left,\n        homological_complex.hom.iso_of_components_hom_f, functor.comp_map, homology_functor_map, nat_iso.of_components.app,\n        eq_to_iso_refl, iso.trans_refl, iso.trans_hom, functor.map_iso_hom, homology.congr_hom, homology_zero_zero_hom,\n        homology.map_desc, homology.π_map_assoc, limits.kernel_zero_iso_source_hom, limits.kernel_subobject_arrow_assoc],\n      erw [homology.π_desc, limits.kernel_subobject_map_arrow_assoc],\n      simp only [homological_complex.hom.sq_from_left, limits.kernel_subobject_map_arrow_assoc, arrow.iso_mk_hom_left],\n      dsimp only [chain_complex.single₀_comp_embed_iso_single_component],\n      erw [iso.refl_hom, iso.refl_hom, category.id_comp, category.id_comp, category.comp_id], refl,\n      apply_instance },\n    { apply_instance } },\n  { refine limits.is_zero.is_iso _ _ _; refine exact.homology_is_zero _ _ (exact_of_zero _ _), },\n  { refine limits.is_zero.is_iso _ _ _,\n    { refine limits.is_zero_of_iso_of_zero _ (homology_iso _ (-[1+i.succ] : ℤ) _ (-i : ℤ) _ _).symm,\n      rotate,\n      { dsimp, refl, },\n      { dsimp, simp only [int.neg_succ_of_nat_eq', sub_add_cancel], },\n      refine exact.homology_is_zero _ _ _,\n      cases i; exact (hP.exact _), },\n    { refine limits.is_zero_of_iso_of_zero _ (homology_iso _ (-[1+i.succ] : ℤ) _ (-i : ℤ) _ _).symm,\n      rotate,\n      { dsimp, refl, },\n      { dsimp, simp only [int.neg_succ_of_nat_eq', sub_add_cancel], },\n      refine exact.homology_is_zero _ _ (exact_of_zero _ _), } }\nend\n.\n\nopen opposite\nopen category_theory.limits\n\n@[simp]\nlemma _root_.category_theory.equivalence.symm_to_adjunction_counit {C D : Type*} [category C]\n  [category D] (e : C ≌ D) : e.symm.to_adjunction.counit = e.unit_inv := rfl\n\n@[simp]\nlemma _root_.homological_complex.shift_equiv_unit_app (i j : ℤ) (X : cochain_complex C ℤ) :\n  homological_complex.hom.f ((shift_equiv _ i).unit.app X) j = (X.X_eq_to_iso $ by simp).hom :=\nbegin\n  dsimp [shift_equiv, unit_of_tensor_iso_unit],\n  simp only [homological_complex.X_eq_to_iso, eq_to_hom_map, eq_to_hom_app, eq_to_iso.hom,\n    homological_complex.eq_to_hom_f, eq_to_hom_trans, homological_complex.shift_μ_inv_app_f],\nend\n\n@[simp]\nlemma _root_.homological_complex.shift_equiv_unit_inv_app (i j : ℤ) (X : cochain_complex C ℤ) :\n  homological_complex.hom.f ((shift_equiv _ i).unit_inv.app X) j = (X.X_eq_to_iso $ by simp).hom :=\nbegin\n  dsimp [shift_equiv, unit_of_tensor_iso_unit],\n  simp only [homological_complex.X_eq_to_iso, eq_to_hom_map, eq_to_hom_app, eq_to_iso.hom,\n    homological_complex.eq_to_hom_f, homological_complex.shift_ε_inv_app_f, eq_to_hom_trans],\nend\n\n@[simp]\nlemma _root_.category_theory.equivalence.symm_to_adjunction_unit {C D : Type*} [category C]\n  [category D] (e : C ≌ D) : e.symm.to_adjunction.unit = e.counit_inv := rfl\n\n@[simp]\nlemma _root_.homological_complex.shift_equiv_counit_app (i j : ℤ) (X : cochain_complex C ℤ) :\n  homological_complex.hom.f ((shift_equiv _ i).counit.app X) j = (X.X_eq_to_iso $ by simp).hom :=\nbegin\n  dsimp [shift_equiv, unit_of_tensor_iso_unit],\n  simp only [homological_complex.X_eq_to_iso, eq_to_hom_map, eq_to_hom_app, eq_to_iso.hom,\n    homological_complex.eq_to_hom_f, homological_complex.shift_ε_inv_app_f, eq_to_hom_trans],\nend\n\n@[simp]\nlemma _root_.homological_complex.shift_equiv_counit_inv_app (i j : ℤ) (X : cochain_complex C ℤ) :\n  homological_complex.hom.f ((shift_equiv _ i).counit_inv.app X) j = (X.X_eq_to_iso $ by simp).hom :=\nbegin\n  dsimp [shift_equiv, unit_of_tensor_iso_unit],\n  simpa only [homological_complex.X_eq_to_iso, eq_to_hom_map, eq_to_hom_app, eq_to_iso.hom,\n    add_neg_equiv_counit_iso_inv, nat_trans.comp_app, homological_complex.comp_f,\n    homological_complex.shift_ε_hom_app_f, homological_complex.eq_to_hom_f,\n    homological_complex.shift_μ_inv_app_f, eq_to_hom_trans],\nend\n\nvariable [enough_projectives C]\n\nnoncomputable\ndef Ext'_iso (A : Cᵒᵖ) (B : C) (i : ℤ) (P : chain_complex C ℕ)\n  (π : P ⟶ (chain_complex.single₀ C).obj A.unop)\n  (hP : P.is_projective_resolution A.unop π) :\n  ((Ext' i).obj A).obj B ≅\n  (((preadditive_yoneda.obj B).map_homological_complex _).obj\n    ((homological_complex.embed complex_shape.embedding.nat_down_int_up).obj P).op).homology (-i) :=\nbegin\n  dsimp only [Ext', functor.comp_obj, functor.op_obj, functor.flip_obj_obj],\n  haveI := chain_complex.to_bounded_homotopy_category.is_K_projective P A.unop π hP,\n  haveI : is_quasi_iso (chain_complex.to_bounded_homotopy_category.map π) := hP.is_quasi_iso_embed,\n  refine (bounded_homotopy_category.Ext_iso i\n    (chain_complex.to_bounded_homotopy_category.obj P)\n    _ _ (chain_complex.to_bounded_homotopy_category.map π ≫ _)) ≪≫\n    (preadditive_yoneda.map_iso _).app (op (chain_complex.to_bounded_homotopy_category.obj P)) ≪≫\n      bounded_homotopy_category.hom_single_iso _ B (-i),\n  { exact ((homotopy_category.quotient _ _).map_iso $\n      (chain_complex.single₀_comp_embed_iso_single).app A.unop).hom, },\n  { apply_instance },\n  { exact (bounded_homotopy_category.shift_single_iso 0 i).app B ≪≫ eq_to_iso (by rw zero_sub) }\nend\n\nlemma AddCommGroup.is_zero_of_eq (A : AddCommGroup) (h : ∀ x y : A, x = y) :\n  is_zero A :=\nbegin\n  rw is_zero_iff_id_eq_zero,\n  ext,\n  apply h,\nend\n\nlemma category_theory.ProjectiveResolution.is_projective_resolution\n  {A : C} (P : ProjectiveResolution A) :\n  P.complex.is_projective_resolution _ P.π :=\n{ projective := P.projective,\n  exact₀ := P.exact₀,\n  exact := P.exact,\n  epi := ProjectiveResolution.f.category_theory.epi P 0 }\n\nlemma Ext'_is_zero_of_neg (A : Cᵒᵖ) (B : C) (i : ℤ) (hi : i < 0) :\n  is_zero (((Ext' i).obj A).obj B) :=\nbegin\n  let P := ProjectiveResolution.of A.unop,\n  refine is_zero_of_iso_of_zero _ (Ext'_iso _ _ i P.complex P.π P.is_projective_resolution).symm,\n  rcases i with (i|i),\n  { exfalso, revert hi, dec_trivial },\n  refine is_zero.homology_is_zero _ _ _ _,\n  refine AddCommGroup.is_zero_of_eq _ _,\n  intros f g,\n  ext,\nend\n\nsection\nopen bounded_homotopy_category\n\nlemma Ext_single_right_is_zero\n  (A : homotopy_category C (complex_shape.up ℤ)) (B : C)\n  (i j k : ℤ)\n  (hA : A.bounded_by i) [A.is_bounded_above]\n  (hijk : i + j ≤ k) :\n  is_zero (((Ext j).obj (op ⟨A⟩)).obj ((bounded_homotopy_category.single _ k).obj B)) :=\nbegin\n  obtain ⟨P, H1, H2, f, H3, H4⟩ :=\n    exists_bounded_K_projective_replacement_of_bounded _ _ hA,\n  have H5 : P.is_bounded_above := ⟨⟨i, H2⟩⟩,\n  resetI,\n  refine is_zero_of_iso_of_zero _ (Ext_iso j ⟨P⟩ _ _ _).symm,\n  swap, { exact f }, swap, { exact H3 },\n  refine AddCommGroup.is_zero_of_eq _ _,\n  rintro (φ ψ : P ⟶ _),\n  rw [← quotient_map_out φ, ← quotient_map_out ψ],\n  congr' 1,\n  ext n,\n  by_cases hn : i ≤ n,\n  { apply (H2 n hn).eq_of_src },\n  { apply is_zero.eq_of_tgt,\n    dsimp [bounded_homotopy_category.single],\n    rw if_neg, { exact is_zero_zero _ },\n    linarith only [hijk, hn], }\nend\n\nend\n\nnamespace AddCommGroup\n\n-- We only need `G` to preserve epimorphisms, but we don't have such a class.\nlemma preserves_projectives {C D : Type*} [category C] [category.{v} D] {F : C ⥤ D} {G : D ⥤ C}\n  (adj : F ⊣ G) [preserves_colimits_of_shape walking_span G] (P : C) [projective P] :\n    projective (F.obj P) :=\nbegin\n  constructor,\n  intros,\n  resetI,\n  use (adj.hom_equiv _ _).symm (projective.factor_thru (adj.hom_equiv _ _ f) (G.map e)),\n  rw [← adj.hom_equiv_naturality_right_symm, projective.factor_thru_comp, equiv.symm_apply_apply],\nend\n\ninstance {C D : Type*} [category C] [category D] {F : C ⥤ D}\n  {G : D ⥤ C} (adj : F ⊣ G) [faithful G] (X : D) : epi (adj.counit.app X) :=\nbegin\n  have H : split_epi (G.map (adj.counit.app X)) := ⟨_, adj.right_triangle_components⟩,\n  haveI := H.epi,\n  exact G.epi_of_epi_map infer_instance\nend\n\nlemma enough_projectives_of_adjoint {C D : Type*} [category C] [category.{v} D] {F : C ⥤ D}\n  {G : D ⥤ C} (adj : F ⊣ G) [preserves_colimits_of_shape walking_span G] [faithful G]\n  [enough_projectives C] : enough_projectives D :=\nbegin\n  haveI : is_left_adjoint F := ⟨_, adj⟩,\n  constructor,\n  intro X,\n  refine ⟨⟨_, preserves_projectives adj _, (adj.hom_equiv _ _).symm (projective.π (G.obj X)), _⟩⟩,\n  dsimp,\n  rw adjunction.hom_equiv_counit,\n  exact epi_comp _ _,\nend\n\ninstance : enough_projectives AddCommGroup.{u} :=\nenough_projectives_of_adjoint\n  (functor.as_equivalence (forget₂ (Module ℤ) AddCommGroup)).to_adjunction\n\nlemma Ext_is_zero_of_one_lt\n  (A : AddCommGroup.{u}ᵒᵖ) (B : AddCommGroup.{u}) (i : ℤ) (hi : i > 1) :\n  is_zero (((Ext' i).obj A).obj B) :=\nbegin\n  induction A,\n  rcases A with ⟨A, _Ainst⟩, resetI,\n  let := Ext'_iso (op $AddCommGroup.of A) B i,\n  dsimp at this,\n  refine is_zero_of_iso_of_zero _ (this _ _ (two_step_resolution_ab_projective A)).symm,\n  rcases i with ((_|_|i)|i),\n  { exfalso, revert hi, dec_trivial },\n  { exfalso, revert hi, dec_trivial },\n  swap,\n  { exfalso, revert hi, dec_trivial },\n  refine is_zero.homology_is_zero _ _ _ _,\n  refine AddCommGroup.is_zero_of_eq _ _,\n  intros f g,\n  apply category_theory.limits.has_zero_object.from_zero_ext,\nend\n\nnoncomputable theory\nvariable (n : ℕ)\n\n-- move me (the rest of the file)\n\ndef zmod_resolution : chain_complex AddCommGroup ℕ :=\nchain_complex.mk' (of ℤ) (of ℤ) (n • 𝟙 _) (λ _, ⟨0, 0, zero_comp⟩)\n\nexample : (zmod_resolution n).X 0 = of ℤ := rfl\n\ndef zmod_resolution_pi_f :\n  Π (i : ℕ), (zmod_resolution n).X i ⟶ ((chain_complex.single₀ AddCommGroup).obj (of $ zmod n)).X i\n| 0     := show of ℤ ⟶ of (zmod n), from (int.cast_ring_hom (zmod n)).to_add_monoid_hom\n| (i+1) := 0\n\ndef zmod_resolution_pi :\n  zmod_resolution n ⟶ (chain_complex.single₀ AddCommGroup).obj (of $ zmod n) :=\n{ f := zmod_resolution_pi_f n,\n  comm' := begin\n    rintros i ⟨_|j⟩ (rfl : _ = _),\n    { ext k, dsimp [zmod_resolution_pi_f, zmod_resolution],\n      simp only [zero_apply, fin.coe_zero, comp_apply, int.coe_cast_add_hom],\n      simp only [chain_complex.mk'_d_1_0, add_monoid_hom.coe_smul, pi.smul_apply, id_apply,\n        nsmul_one, int.coe_nat_bit0, int.coe_nat_succ, int.coe_nat_zero, zero_add, int.cast_bit0,\n        int.cast_one, map_nsmul],\n      simp only [ring_hom.coe_add_monoid_hom, eq_int_cast,\n        int.cast_one, nsmul_one, zmod.nat_cast_self] },\n    { exact comp_zero.trans comp_zero.symm }\n  end }\n\nlocal attribute [-instance] limits.has_smallest_colimits_of_has_colimits\n\ninstance : projective (AddCommGroup.of ℤ) :=\npreserves_projectives (functor.as_equivalence (forget₂ (Module ℤ) AddCommGroup)).to_adjunction\n  (Module.of ℤ ℤ)\n\nlemma exact_zmod_nsmul_cast :\n  exact (n • 𝟙 (of ℤ)) (AddCommGroup.of_hom $ int.cast_add_hom (zmod n)) :=\nbegin\n  rw AddCommGroup.exact_iff,\n  erw zmod.ker_int_cast_add_hom,\n  ext,\n  apply exists_congr,\n  rintro (a : ℤ),\n  change n • a = x ↔ a * (n : ℤ) = x,\n  rw mul_comm,\n  norm_num,\nend\n\nlemma zmod_resolution_is_resolution (hn : n ≠ 0) :\n  (zmod_resolution n).is_projective_resolution (of (zmod n)) (zmod_resolution_pi n) :=\nbegin\n  constructor,\n  { rintro (_|_|_|_),\n    { show projective (AddCommGroup.of ℤ), by apply_instance },\n    { show projective (AddCommGroup.of ℤ), by apply_instance },\n    { show projective (0 : AddCommGroup), by apply_instance },\n    { show projective (0 : AddCommGroup), by apply_instance } },\n  { dsimp [zmod_resolution_pi, zmod_resolution_pi_f, zmod_resolution],\n    rw chain_complex.mk'_d_1_0,\n    exact AddCommGroup.exact_zmod_nsmul_cast n },\n  { intro i,\n    dsimp [zmod_resolution, chain_complex.mk', chain_complex.mk, chain_complex.of],\n    rw [if_pos rfl, if_pos rfl, category.id_comp, category.id_comp],\n    rcases i with (_|_|_),\n    { show exact 0 (n • 𝟙 (of ℤ)),\n      rw [(abelian.tfae_mono 0 (n • 𝟙 (of ℤ))).out 2 0, AddCommGroup.mono_iff_injective],\n      rintros (x : ℤ) (y : ℤ) (e : n • x = n • y),\n      norm_num at e,\n      exact e.resolve_right hn },\n    { show exact 0 0, apply exact_of_zero },\n    { show exact 0 0, apply exact_of_zero } },\n  { dsimp [zmod_resolution_pi, zmod_resolution_pi_f],\n    rw AddCommGroup.epi_iff_surjective,\n    rintro (x : zmod n),\n    exact ⟨(x : ℤ), by norm_num⟩ }\nend\n\nlemma zmod.nsmul_eq_zero (k : zmod n) : n • k = 0 := by norm_num\n\n@[simps] def add_subgroup.equiv_top (A : Type*) [add_comm_group A] :\n  A ≃+ (⊤ : add_subgroup A) :=\n{ to_fun := λ x, ⟨x, add_subgroup.mem_top _⟩,\n  inv_fun := λ x, x,\n  left_inv := λ x, rfl,\n  right_inv := by { rintro ⟨x, hx⟩, refl },\n  map_add' := λ x y, rfl }\n\nend AddCommGroup\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived/example.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.3060993306308653}}
{"text": "import model_theory.substructures\nimport model_theory.bundled\n\n\n--OUTPUT 1\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- define the auxilliary formula\n  noncomputable def A_aux := λ n, ∃ (x1 : L.sorts.1), ∃ (x2 : L.sorts.1), ⋯ ∃ (xn : L.sorts.1),\n    (∀ (i : fin n), ∀ (j : fin n), i.val ≠ j.val → (L.predicates.pempty.elim $ L.relations.rnequiv.elim $ xi i ≠ xi j))\n\n  -- prove that its intended interpretation is correct\n  lemma A_aux_iff_ge (n : ℕ) : A_aux n ↔ ∃ (N : F.Model) [Nfin : fintype N], n ≤ @fintype.card N Nfin := sorry\n\n  -- define a function to build the auxiliary formula from a natural\n  noncomputable def A_aux_thm (n : ℕ) : F.Model.Theory.Formula := sorry\n\n  -- define formulas for the overflow\n  noncomputable def A_thm := λ n, sorry\n\n  -- define the set of sentences for the overflow\n  noncomputable def F_thm := sorry\n\n  -- define the set of sentences for the overflow\n  noncomputable def Gamma_thm := sorry\n\n  -- show that every finite subset of gamma is satisfiable\n  have h1 : ∀ (g : finset F.Model.Theory.Formula), g.finite → ∃ (N : F.Model), F.Model.Theory.Satisfies N g, from sorry,\n\n  -- apply the compactness theorem to gamma\n  have h2 : ∃ (M : F.Model), F.Model.Theory.Satisfies M Gamma_thm, from sorry,\n\n  -- show that M must be infinite\n  have h3 : infinite M, from sorry,\n  sorry\n\n\nend\n\n--OUTPUT 2\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  have h1 : ∀ n : ℕ, ∃ A n, n ≤ @fintype.card (F.Model) A, from by {\n    assume (n : ℕ),\n    have h2 : ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h n,\n    have h3 : ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from h2, \n    show ∃ A n, n ≤ @fintype.card (F.Model) A, from sorry,\n  },\n  have h2 : ∀ n : ℕ, ∃ A n, n ≤ @fintype.card (F.Model) A, from h1,\n  show ∃ (M : F.Model), infinite M, from sorry, \nend\n\n--OUTPUT 3\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  let A : set L.Statement := {s | ∃ n : ℕ, s = ∃ x₁ x₂, ..., ∃ xₙ, x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ ... ∧ xₙ₋₁ ≠ xₙ},\n  have h1 : ∀ (s : L.Statement), s ∈ A ↔ ∃ n : ℕ, s = ∃ x₁ x₂, ..., ∃ xₙ, x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ ... ∧ xₙ₋₁ ≠ xₙ, from sorry,\n\n  have h2 : ∀ (n : ℕ), ∃ (m : F.Model) [mfin : fintype m], n ≤ card m ∧ ∀ x, x ≠ x, from sorry,\n  have h3 : ∀ (n : ℕ), ∃ (m : F.Model) [mfin : fintype m], n ≤ card m, from sorry,\n\n  have h4 : ∀ (s : L.Statement), s ∈ A ↔ satisfiable (s :: ∅), from sorry,\n\n  have h5 : ∀ (n : ℕ), satisfiable (∃ x₁ x₂, ..., ∃ xₙ, x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ ... ∧ xₙ₋₁ ≠ xₙ :: ∅), from sorry,\n\n  have h6 : ∀ (n : ℕ), satisfiable (∃ x₁ x₂, ..., ∃ xₙ, x₁ ≠ x₂ ∧ x₁ ≠ x₃ ∧ ... ∧ xₙ₋₁ ≠ xₙ :: F), from sorry,\n\n  have h7 : satisfiable (F ∪ A), from sorry,\n\n  show ∃ (M : F.Model), infinite M, from sorry,\nend\n\n--OUTPUT 4\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  have h1 : ∀ (i : ℕ), ∃ (m : ℕ), ∀ (m₁ m₂ : ℕ), (m₁ < i ∧ m₂ < i) → m₁ ≠ m₂, from sorry,\n  have h2 : ∀ (i : ℕ), ∃ (m : F.Model) [mfin : fintype m], ∀ (m₁ m₂ : ℕ), (m₁ < i ∧ m₂ < i) → m₁ ≠ m₂, from sorry,\n  have h3 : ∀ (i : ℕ), ∃ (m : F.Model) [mfin : fintype m], i ≤ @fintype.card m mfin, from sorry,\n  have h4 : ∀ (i : ℕ), ∃ (m : F.Model) [mfin : fintype m], @fintype.card m mfin = i, from sorry,\n\n  sorry,\nend\n\n--OUTPUT 5\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nby { sorry }\n\n-- END\nend\n\n--OUTPUT 6\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  let Γ := F.formulas,\n  have h1 : ∀ f : Γ, ∃ n : ℕ, n ,\n  have h2 : ∀ (n : ℕ), ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin, from sorry,\n  show ∃ (M : F.Model), infinite M, from sorry,\nend\n\n--OUTPUT 7\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  -- Let A_n be the formula saying that \\mathbf{M} has at least n elements.\n  let A_n : L.formula := ⟨exactly_n_non_equal_variables n, _⟩,\n  have h1 : ∀ (n : ℕ), @is_true (exists x_1' x_2' x_n', (∀ i j : ℕ, (i < j ∧ j ≤ n) → x_i' ≠ x_j')) (exactly_n_non_equal_variables n), from sorry,\n  have h2 : ∀ n : ℕ, ∀ (m : F.Model) [mfin : fintype m], n < @fintype.card m mfin → ∃ (e : F.Model.Term m) [mvar : L.Term.variable e.var], ∀ (m' : F.Model) [mfinn : fintype m'], m' ⊨ A_n [e], from sorry,\n\n  cases h 3 with m [mfin : fintype m] h3,\n\n  -- take the union of all of the A_i's (for i = 1, 2, ...).\n  let φ := prod.snd (A_n),\n  let A_infty : L.formula := ⟨∀ x_1' x_2' ..., ∀ i j : ℕ, (i < j) → (∀ (m : F.Model) [mfin : fintype m], m ⊨ φ [x_i'] → m ⊨ φ [x_j']), _⟩,\n  have h4 : ∀ (m : F.Model) [mfin : fintype m], m ⊨ A_infty ↔ (∃ n, m ⊨ A_n), from sorry,\n\n  let F_infty := F ∪ ⟨A_infty, _⟩,\n  have h5 : finite F_infty, from sorry,\n  have h6 : ∀ (t : F_infty), classical.prop_decidable (F_infty.eval t), from sorry,\n  have h7 : consistent F_infty, from sorry,\n  cases compactness_model (sorry) with M h8,\n\n  show ∃ (M : F.Model), infinite M, from sorry,\nend\n\n--OUTPUT 8\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  sorry,\nend\n\n--OUTPUT 9\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  let F := λ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin,\n  have h1 : ∀ n : ℕ, F n, from sorry,\n\n  have h2 : ∀ n : ℕ, ∃ (m : F.Model), ∀ x y : ℕ, x ≠ y → m.relations.Interpretation x ≠ m.relations.Interpretation y, from \n    sorry,\n  have h3 : ∀ n : ℕ, ∃ (m : F.Model), ∀ x y : ℕ, x ≠ y → m.relations.Interpretation x ≠ m.relations.Interpretation y, from sorry,\n  have h4 : ∀ n : ℕ, ∃ (m : F.Model), ∀ x y : ℕ, x ≠ y → m.relations.Interpretation x ≠ m.relations.Interpretation y, from sorry,\n  have h5 : ∀ n : ℕ, ∃ (m : F.Model), ∀ x y : ℕ, x ≠ y → m.relations.Interpretation x ≠ m.relations.Interpretation y, from sorry,\n  have h6 : ∀ n : ℕ, ∃ (m : F.Model), ∀ x y : ℕ, x ≠ y → m.relations.Interpretation x ≠ m.relations.Interpretation y, from sorry,\n  have h7 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h8 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h9 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h10 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h11 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h12 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h13 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h14 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h15 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h16 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h17 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h18 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h19 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h20 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h21 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h22 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h23 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h24 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h25 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h26 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h27 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h28 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h29 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h30 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h31 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h32 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h33 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h34 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h35 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h36 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h37 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h38 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h39 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h40 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h41 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h42 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h43 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h44 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry,\n  have h45 : ∀ n : ℕ, ∃ (m : F.Model), @fintype.card m fintype.fin n = n, from sorry\nend --Needs more than 2000 tokens!\n\n--OUTPUT 10\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nbegin\n  sorry,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nOverflow theorem\nLet $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.\n`proof`\nFor each $n$, let $\\mathbf A_n$ be the formula:\n\n$\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n\nThen $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n\nTake:\n$$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$\n\nSince $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n\nFrom the Compactness Theorem, $\\Gamma$ is satisfiable in some model $\\mathbf{M}$.\n\nBut since $\\mathbf{M} \\models A_i$ for each $i$, $\\mathbf{M}$ must be infinite.\n\nSo $F$ has an infinite model.\n\nQED\n-/\ntheorem  overflow {L : first_order.language} {F : L.Theory} (h : ∀ n : ℕ, ∃ (m : F.Model) [mfin : fintype m], n ≤ @fintype.card m mfin) : ∃ (M : F.Model), infinite M :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/Correct_statement-lean_proof_outline-3_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.30607970738581475}}
{"text": "import global.relation\n\nnoncomputable theory\n\nopen set function filter (hiding map_smul) charted_space smooth_manifold_with_corners\n  linear_map (ker)\nopen_locale topology manifold pointwise\n\nsection parameter_space\n/-! ## Fundamental definitions -/\n\nvariables\n{E : Type*} [normed_add_comm_group E] [normed_space ℝ E]\n{H : Type*} [topological_space H] {I : model_with_corners ℝ E H}\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{E' : Type*} [normed_add_comm_group E'] [normed_space ℝ E']\n{H' : Type*} [topological_space H'] {I' : model_with_corners ℝ E' H'}\n{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n{EP : Type*} [normed_add_comm_group EP] [normed_space ℝ EP]\n{HP : Type*} [topological_space HP] {IP : model_with_corners ℝ EP HP}\n{P : Type*} [topological_space P] [charted_space HP P] [smooth_manifold_with_corners IP P]\n{F : Type*} [normed_add_comm_group F] [normed_space ℝ F]\n{G : Type*} [topological_space G] {J : model_with_corners ℝ F G}\n{N : Type*} [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]\n{EX : Type*} [normed_add_comm_group EX] [normed_space ℝ EX]\n{HX : Type*} [topological_space HX] {IX : model_with_corners ℝ EX HX}\n-- note: X is a metric space\n{X : Type*} [metric_space X] [charted_space HX X] [smooth_manifold_with_corners IX X]\nvariables {R : rel_mfld I M I' M'}\n\nvariables (IP P)\n\n/-- The relation `𝓡 ^ P` -/\ndef rel_mfld.relativize (R : rel_mfld I M I' M') : rel_mfld (IP.prod I) (P × M) I' M' :=\nbundle_snd ⁻¹' R\n\nvariables {IP P}\n\nlemma rel_mfld.mem_relativize (R : rel_mfld I M I' M') (w : one_jet_bundle (IP.prod I) (P × M) I' M') :\n w ∈ R.relativize IP P ↔\n  (one_jet_bundle.mk w.1.1.2 w.1.2 (w.2.comp (continuous_linear_map.inr ℝ EP E)) :\n    one_jet_bundle I M I' M') ∈ R :=\nby { simp_rw [rel_mfld.relativize, mem_preimage, bundle_snd_eq], refl }\n\nlemma rel_mfld.is_open_relativize (R : rel_mfld I M I' M') (h2 : is_open R) :\n  is_open (R.relativize IP P) :=\nh2.preimage smooth_bundle_snd.continuous\n\nlemma relativize_slice {σ : one_jet_bundle (IP.prod I) (P × M) I' M'}\n  {p : dual_pair $ tangent_space (IP.prod I) σ.1.1}\n  (q : dual_pair $ tangent_space I σ.1.1.2)\n  (hpq : p.π.comp (continuous_linear_map.inr ℝ EP E) = q.π) :\n  (R.relativize IP P).slice σ p =\n  σ.2 (p.v - (0, q.v)) +ᵥ R.slice (bundle_snd σ) q :=\nbegin\n  -- for some reason this is needed\n  letI : module ℝ (((cont_mdiff_map.snd : C^∞⟮(IP.prod I).prod I', (P × M) × M'; I', M'⟯) *ᵖ\n    tangent_space I') σ.1),\n  { apply_instance },\n  have h2pq : ∀ x : E, p.π ((0 : EP), x) = q.π x := λ x, congr_arg (λ f : E →L[ℝ] ℝ, f x) hpq,\n  ext1 w,\n  have h1 : (p.update σ.2 w).comp (continuous_linear_map.inr ℝ EP E) =\n    q.update (bundle_snd σ).2 (-σ.2 (p.v - (0, q.v)) + w),\n  { ext1 x,\n    simp_rw [continuous_linear_map.comp_apply, continuous_linear_map.inr_apply,\n      ← continuous_linear_map.map_neg, neg_sub],\n    obtain ⟨u, hu, t, rfl⟩ := q.decomp x,\n    have hv : (0, q.v) - p.v ∈ ker p.π,\n    { rw [linear_map.mem_ker, map_sub, p.pairing, h2pq, q.pairing, sub_self] },\n    have hup : ((0 : EP), u) ∈ ker p.π := (h2pq u).trans hu,\n    rw [q.update_apply _ hu, ← prod.zero_mk_add_zero_mk, map_add, p.update_ker_pi _ _ hup,\n      ← prod.smul_zero_mk, map_smul],\n    nth_rewrite 0 [← sub_add_cancel (0, q.v) p.v],\n    rw [map_add, p.update_ker_pi _ _ hv, p.update_v, bundle_snd_eq],\n    refl },\n  have := preimage_vadd_neg (show E', from σ.2 (p.v - (0, q.v)))\n    (show set E', from (R.slice (bundle_snd σ) q)),\n  dsimp only at this,\n  simp_rw [← this, mem_preimage, mem_slice, R.mem_relativize],\n  dsimp only [one_jet_bundle_mk_fst, one_jet_bundle_mk_snd],\n  congr'\nend\n\nlemma relativize_slice_eq_univ {σ : one_jet_bundle (IP.prod I) (P × M) I' M'}\n  {p : dual_pair $ tangent_space (IP.prod I) σ.1.1}\n  (hp : p.π.comp (continuous_linear_map.inr ℝ EP E) = 0) :\n  ((R.relativize IP P).slice σ p).nonempty ↔\n  (R.relativize IP P).slice σ p = univ :=\nbegin\n  -- for some reason this is needed\n  letI : module ℝ (((cont_mdiff_map.snd : C^∞⟮(IP.prod I).prod I', (P × M) × M'; I', M'⟯) *ᵖ\n    tangent_space I') σ.1),\n  { apply_instance },\n  have h2p : ∀ x : E, p.π ((0 : EP), x) = 0 := λ x, congr_arg (λ f : E →L[ℝ] ℝ, f x) hp,\n  have : ∀ y : E', (p.update σ.snd y).comp (continuous_linear_map.inr ℝ EP E) =\n    σ.snd.comp (continuous_linear_map.inr ℝ EP E),\n  { intro y,\n    ext1 x,\n    simp_rw [continuous_linear_map.comp_apply, continuous_linear_map.inr_apply,\n      p.update_ker_pi _ _ (h2p x)] },\n  simp_rw [set.nonempty, eq_univ_iff_forall, mem_slice, R.mem_relativize],\n  dsimp only [one_jet_bundle_mk_fst, one_jet_bundle_mk_snd],\n  simp_rw [this, exists_const, forall_const]\nend\n\nvariables (IP P)\n\nlemma rel_mfld.ample.relativize (hR : R.ample) : (R.relativize IP P).ample :=\nbegin\n  intros σ p,\n  let p2 := p.π.comp (continuous_linear_map.inr ℝ EP E),\n  rcases eq_or_ne p2 0 with h|h,\n  { intros w hw,\n    rw [(relativize_slice_eq_univ h).mp ⟨w, hw⟩, connected_component_in_univ,\n      preconnected_space.connected_component_eq_univ, convex_hull_univ] },\n  obtain ⟨u', hu'⟩ := continuous_linear_map.exists_ne_zero h,\n  let u := (p2 u')⁻¹ • u',\n  let q : dual_pair (tangent_space I σ.1.1.2) :=\n  ⟨p2, u, by rw [p2.map_smul, smul_eq_mul, inv_mul_cancel hu']⟩,\n  rw [relativize_slice q rfl],\n  refine (hR q).vadd,\nend\n\nvariables {IP P}\n\nlemma family_one_jet_sec.uncurry_mem_relativize (S : family_one_jet_sec I M I' M' IP P) {s : P}\n  {x : M} : S.uncurry (s, x) ∈ R.relativize IP P ↔ S s x ∈ R :=\nbegin\n  simp_rw [rel_mfld.relativize, mem_preimage, bundle_snd_eq, one_jet_sec.coe_apply,\n    map_left],\n  congr',\n  ext v,\n  simp_rw [S.uncurry_ϕ', continuous_linear_map.comp_apply, continuous_linear_map.add_apply,\n    continuous_linear_map.comp_apply, continuous_linear_map.inr_apply,\n    continuous_linear_map.coe_fst', continuous_linear_map.coe_snd',\n    continuous_linear_map.map_zero, zero_add, S.coe_ϕ]\nend\n\ndef family_formal_sol.uncurry\n  (S : family_formal_sol IP P R) : formal_sol (R.relativize IP P) :=\nbegin\n  refine ⟨S.to_family_one_jet_sec.uncurry, _⟩,\n  rintro ⟨s, x⟩,\n  exact S.to_family_one_jet_sec.uncurry_mem_relativize.mpr (S.is_sol' s x)\nend\n\nlemma family_formal_sol.uncurry_ϕ' (S : family_formal_sol IP P R) (p : P × M) :\n  S.uncurry.ϕ p = mfderiv IP I' (λ z, S.bs z p.2) p.1 ∘L continuous_linear_map.fst ℝ EP E +\n  S.ϕ p.1 p.2 ∘L continuous_linear_map.snd ℝ EP E :=\nS.to_family_one_jet_sec.uncurry_ϕ' p\n\ndef family_one_jet_sec.curry (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N) :\n  family_one_jet_sec I M I' M' (J.prod IP) (N × P) :=\n{ bs := λ p x, (S p.1).bs (p.2, x),\n  ϕ := λ p x, (S p.1).ϕ (p.2, x) ∘L mfderiv I (IP.prod I) (λ x, (p.2, x)) x,\n  smooth' := begin\n    rintro ⟨⟨t, s⟩, x⟩,\n    refine smooth_at_snd.one_jet_bundle_mk (S.smooth_bs.comp smooth_prod_assoc _) _,\n    have h1 : smooth_at ((J.prod IP).prod I) 𝓘(ℝ, EP × E →L[ℝ] E')\n      (in_coordinates_core (IP.prod I) I' (λ (p : (N × P) × M), (p.1.2, p.2))\n        (λ (p : (N × P) × M), (S p.1.1).bs (p.1.2, p.2))\n        (λ (p : (N × P) × M), ((S p.1.1).ϕ (p.1.2, p.2))) ((t, s), x)) ((t, s), x),\n    { apply (smooth_at_one_jet_bundle.mp $\n        smooth_at.comp _ (by exact S.smooth (t, (s, x))) (smooth_prod_assoc ((t, s), x))).2.2 },\n    have h2 : smooth_at ((J.prod IP).prod I) 𝓘(ℝ, E →L[ℝ] EP × E)\n      (in_coordinates_core I (IP.prod I) prod.snd (λ (p : (N × P) × M), (p.1.2, p.2))\n        (λ (p : (N × P) × M),\n          (mfderiv I (IP.prod I) (λ (x : M), (p.1.2, x)) p.2)) ((t, s), x)) ((t, s), x),\n    { apply cont_mdiff_at.mfderiv''' (λ (p : (N × P) × M) (x : M), (p.1.2, x)) prod.snd\n        (smooth_at_fst.fst.snd.prod_mk smooth_at_snd :\n          smooth_at (((J.prod IP).prod I).prod I) (IP.prod I) _ (((t, s), x), x))\n        (smooth_at_snd : smooth_at ((J.prod IP).prod I) _ _ _) le_top },\n    exact h1.clm_comp_in_coordinates_core (continuous_at_fst.snd.prod continuous_at_snd) h2\n  end }\n\nlemma family_one_jet_sec.curry_bs (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N) (p : N × P)\n  (x : M) : (S.curry p).bs x = (S p.1).bs (p.2, x) :=\nrfl\n\nlemma family_one_jet_sec.curry_ϕ (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N) (p : N × P)\n  (x : M) : (S.curry p).ϕ x = (S p.1).ϕ (p.2, x) ∘L mfderiv I (IP.prod I) (λ x, (p.2, x)) x :=\nrfl\n\nlemma family_one_jet_sec.curry_ϕ' (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N) (p : N × P)\n  (x : M) : (S.curry p).ϕ x = (S p.1).ϕ (p.2, x) ∘L continuous_linear_map.inr ℝ EP E :=\nbegin\n  rw [S.curry_ϕ],\n  congr' 1,\n  refine ((mdifferentiable_at_const I IP).mfderiv_prod smooth_id.mdifferentiable_at).trans _,\n  rw [mfderiv_id, mfderiv_const],\n  refl,\nend\n\nlemma formal_sol.eq_iff {F₁ F₂ : formal_sol R} {x : M} :\n  F₁ x = F₂ x ↔ F₁.bs x = F₂.bs x ∧ F₁.ϕ x = by apply F₂.ϕ x :=\nby { simp_rw [sigma.ext_iff, formal_sol.fst_eq, heq_iff_eq, prod.ext_iff, eq_self_iff_true,\n  true_and], refl }\n\nlemma family_one_jet_sec.is_holonomic_at_curry\n  (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N)\n  {t : N} {s : P} {x : M} (hS : (S t).is_holonomic_at (s, x)) :\n  (S.curry (t, s)).is_holonomic_at x :=\nbegin\n  simp_rw [one_jet_sec.is_holonomic_at, (S.curry _).snd_eq, S.curry_ϕ] at hS ⊢,\n  dsimp only,\n  rw [show (S.curry (t, s)).bs = λ x, (S.curry (t, s)).bs x, from rfl, funext (S.curry_bs _)],\n  dsimp only,\n  refine (mfderiv_comp x (S t).smooth_bs.mdifferentiable_at\n    ((mdifferentiable_at_const I IP).prod_mk smooth_id.mdifferentiable_at)).trans _,\n  rw [id, hS],\n  refl,\nend\n\nlemma family_one_jet_sec.curry_mem (S : family_one_jet_sec (IP.prod I) (P × M) I' M' J N)\n  {p : N × P} {x : M} (hR : S p.1 (p.2, x) ∈ R.relativize IP P) :\n  S.curry p x ∈ R :=\nbegin\n  simp_rw [rel_mfld.relativize, mem_preimage, bundle_snd_eq, one_jet_sec.coe_apply,\n    map_left] at hR ⊢,\n  convert hR,\n  ext v,\n  simp_rw [S.curry_ϕ']\nend\n\n\ndef family_formal_sol.curry (S : family_formal_sol J N (R.relativize IP P)) :\n  family_formal_sol (J.prod IP) (N × P) R :=\n⟨S.to_family_one_jet_sec.curry, λ p x, S.to_family_one_jet_sec.curry_mem S.is_sol⟩\n\nlemma family_formal_sol.curry_ϕ' (S : family_formal_sol J N (R.relativize IP P)) (p : N × P)\n  (x : M) : (S.curry p).ϕ x = (S p.1).ϕ (p.2, x) ∘L continuous_linear_map.inr ℝ EP E :=\nS.to_family_one_jet_sec.curry_ϕ' p x\n\nlemma curry_eq_iff_eq_uncurry {𝓕 : family_formal_sol J N (R.relativize IP P)}\n  {𝓕₀ : family_formal_sol IP P R} {t : N} {x : M} {s : P}\n  (h : 𝓕 t (s, x) = 𝓕₀.uncurry (s, x)) :\n  (𝓕.curry (t, s)) x = 𝓕₀ s x :=\nbegin\n  simp_rw [formal_sol.eq_iff] at h ⊢,\n  refine ⟨h.1, _⟩,\n  simp_rw [𝓕.curry_ϕ', h.2, 𝓕₀.uncurry_ϕ'],\n  ext v,\n  simp_rw [continuous_linear_map.comp_apply, continuous_linear_map.add_apply,\n    continuous_linear_map.comp_apply,\n    continuous_linear_map.inr_apply, continuous_linear_map.coe_fst',\n    continuous_linear_map.coe_snd', continuous_linear_map.map_zero, zero_add],\n  refl\nend\n\nlemma rel_mfld.satisfies_h_principle.satisfies_h_principle_with\n  (R : rel_mfld I M IX X) {C : set (P × M)}\n  (ε : M → ℝ) (h : (R.relativize IP P).satisfies_h_principle C (λ x, ε x.2)) :\n  R.satisfies_h_principle_with IP C ε :=\nbegin\n  intros 𝓕₀ h𝓕₀,\n  obtain ⟨𝓕, h1𝓕, h2𝓕, h3𝓕, h4𝓕⟩ :=\n    h 𝓕₀.uncurry (h𝓕₀.mono (λ p hp, 𝓕₀.to_family_one_jet_sec.is_holonomic_uncurry.mpr hp)),\n  refine ⟨𝓕.curry, _, _, _, _⟩,\n  { intros s x, exact curry_eq_iff_eq_uncurry (h1𝓕 (s, x)) },\n  { intros s x, exact 𝓕.to_family_one_jet_sec.is_holonomic_at_curry (h2𝓕 (s, x)) },\n  { refine h3𝓕.mono _, rintro ⟨s, x⟩ hp t, exact curry_eq_iff_eq_uncurry (hp t) },\n  { intros t s x, exact (h4𝓕 t (s, x)) },\nend\n\nend parameter_space\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/global/parametricity_for_free.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.30607970738581475}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.punit\nimport category_theory.structured_arrow\nimport category_theory.is_connected\nimport category_theory.limits.yoneda\nimport category_theory.limits.types\n\n/-!\n# Final and initial functors\n\nA functor `F : C ⥤ D` is final if for every `d : D`,\nthe comma category of morphisms `d ⟶ F.obj c` is connected.\n\nDually, a functor `F : C ⥤ D` is initial if for every `d : D`,\nthe comma category of morphisms `F.obj c ⟶ d` is connected.\n\nWe show that right adjoints are examples of final functors, while\nleft adjoints are examples of initial functors.\n\nFor final functors, we prove that the following three statements are equivalent:\n1. `F : C ⥤ D` is final.\n2. Every functor `G : D ⥤ E` has a colimit if and only if `F ⋙ G` does,\n   and these colimits are isomorphic via `colimit.pre G F`.\n3. `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`.\n\nStarting at 1. we show (in `cocones_equiv`) that\nthe categories of cocones over `G : D ⥤ E` and over `F ⋙ G` are equivalent.\n(In fact, via an equivalence which does not change the cocone point.)\nThis readily implies 2., as `comp_has_colimit`, `has_colimit_of_comp`, and `colimit_iso`.\n\nFrom 2. we can specialize to `G = coyoneda.obj (op d)` to obtain 3., as `colimit_comp_coyoneda_iso`.\n\nFrom 3., we prove 1. directly in `cofinal_of_colimit_comp_coyoneda_iso_punit`.\n\nDually, we prove that if a functor `F : C ⥤ D` is initial, then any functor `G : D ⥤ E` has a\nlimit if and only if `F ⋙ G` does, and these limits are isomorphic via `limit.pre G F`.\n\n\n## Naming\nThere is some discrepancy in the literature about naming; some say 'cofinal' instead of 'final'.\nThe explanation for this is that the 'co' prefix here is *not* the usual category-theoretic one\nindicating duality, but rather indicating the sense of \"along with\".\n\n## Future work\nDualise condition 3 above and the implications 2 ⇒ 3 and 3 ⇒ 1 to initial functors.\n\n## References\n* https://stacks.math.columbia.edu/tag/09WN\n* https://ncatlab.org/nlab/show/final+functor\n* Borceux, Handbook of Categorical Algebra I, Section 2.11.\n  (Note he reverses the roles of definition and main result relative to here!)\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nnamespace category_theory\n\nnamespace functor\n\nopen opposite\nopen category_theory.limits\n\nvariables {C : Type v} [small_category C]\nvariables {D : Type v} [small_category D]\n\n/--\nA functor `F : C ⥤ D` is final if for every `d : D`, the comma category of morphisms `d ⟶ F.obj c`\nis connected.\n\nSee https://stacks.math.columbia.edu/tag/04E6\n-/\nclass final (F : C ⥤ D) : Prop :=\n(out (d : D) : is_connected (structured_arrow d F))\n\nattribute [instance] final.out\n\n/--\nA functor `F : C ⥤ D` is initial if for every `d : D`, the comma category of morphisms\n`F.obj c ⟶ d` is connected.\n-/\nclass initial (F : C ⥤ D) : Prop :=\n(out (d : D) : is_connected (costructured_arrow F d))\n\nattribute [instance] initial.out\n\ninstance final_op_of_initial (F : C ⥤ D) [initial F] : final F.op :=\n{ out := λ d, is_connected_of_equivalent (costructured_arrow_op_equivalence F (unop d)) }\n\ninstance initial_op_of_final (F : C ⥤ D) [final F] : initial F.op :=\n{ out := λ d, is_connected_of_equivalent (structured_arrow_op_equivalence F (unop d)) }\n\nlemma final_of_initial_op (F : C ⥤ D) [initial F.op] : final F :=\n{ out := λ d, @is_connected_of_is_connected_op _ _\n  (is_connected_of_equivalent (structured_arrow_op_equivalence F d).symm) }\n\nlemma initial_of_final_op (F : C ⥤ D) [final F.op] : initial F :=\n{ out := λ d, @is_connected_of_is_connected_op _ _\n  (is_connected_of_equivalent (costructured_arrow_op_equivalence F d).symm) }\n\n/-- If a functor `R : D ⥤ C` is a right adjoint, it is final. -/\nlemma final_of_adjunction {L : C ⥤ D} {R : D ⥤ C} (adj : L ⊣ R) : final R :=\n{ out := λ c,\n  let u : structured_arrow c R := structured_arrow.mk (adj.unit.app c) in\n  @zigzag_is_connected _ _ ⟨u⟩ $ λ f g, relation.refl_trans_gen.trans\n    (relation.refl_trans_gen.single (show zag f u, from\n      or.inr ⟨structured_arrow.hom_mk ((adj.hom_equiv c f.right).symm f.hom) (by simp)⟩))\n    (relation.refl_trans_gen.single (show zag u g, from\n      or.inl ⟨structured_arrow.hom_mk ((adj.hom_equiv c g.right).symm g.hom) (by simp)⟩)) }\n\n/-- If a functor `L : C ⥤ D` is a left adjoint, it is initial. -/\nlemma initial_of_adjunction {L : C ⥤ D} {R : D ⥤ C} (adj : L ⊣ R) : initial L :=\n{ out := λ d,\n  let u : costructured_arrow L d := costructured_arrow.mk (adj.counit.app d) in\n  @zigzag_is_connected _ _ ⟨u⟩ $ λ f g, relation.refl_trans_gen.trans\n    (relation.refl_trans_gen.single (show zag f u, from\n      or.inl ⟨costructured_arrow.hom_mk (adj.hom_equiv f.left d f.hom) (by simp)⟩))\n    (relation.refl_trans_gen.single (show zag u g, from\n      or.inr ⟨costructured_arrow.hom_mk (adj.hom_equiv g.left d g.hom) (by simp)⟩)) }\n\n@[priority 100]\ninstance final_of_is_right_adjoint (F : C ⥤ D) [h : is_right_adjoint F] : final F :=\nfinal_of_adjunction h.adj\n\n@[priority 100]\ninstance initial_of_is_left_adjoint (F : C ⥤ D) [h : is_left_adjoint F] : initial F :=\ninitial_of_adjunction h.adj\n\nnamespace final\n\nvariables (F : C ⥤ D) [final F]\n\ninstance (d : D) : nonempty (structured_arrow d F) := is_connected.is_nonempty\n\nvariables {E : Type u} [category.{v} E] (G : D ⥤ E)\n\n/--\nWhen `F : C ⥤ D` is cofinal, we denote by `lift F d` an arbitrary choice of object in `C` such that\nthere exists a morphism `d ⟶ F.obj (lift F d)`.\n-/\ndef lift (d : D) : C :=\n(classical.arbitrary (structured_arrow d F)).right\n\n/--\nWhen `F : C ⥤ D` is cofinal, we denote by `hom_to_lift` an arbitrary choice of morphism\n`d ⟶ F.obj (lift F d)`.\n-/\ndef hom_to_lift (d : D) : d ⟶ F.obj (lift F d) :=\n(classical.arbitrary (structured_arrow d F)).hom\n\n/--\nWe provide an induction principle for reasoning about `lift` and `hom_to_lift`.\nWe want to perform some construction (usually just a proof) about\nthe particular choices `lift F d` and `hom_to_lift F d`,\nit suffices to perform that construction for some other pair of choices\n(denoted `X₀ : C` and `k₀ : d ⟶ F.obj X₀` below),\nand to show how to transport such a construction\n*both* directions along a morphism between such choices.\n-/\ndef induction {d : D} (Z : Π (X : C) (k : d ⟶ F.obj X), Sort*)\n  (h₁ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),\n    (k₁ ≫ F.map f = k₂) → Z X₁ k₁ → Z X₂ k₂)\n  (h₂ : Π X₁ X₂ (k₁ : d ⟶ F.obj X₁) (k₂ : d ⟶ F.obj X₂) (f : X₁ ⟶ X₂),\n    (k₁ ≫ F.map f = k₂) → Z X₂ k₂ → Z X₁ k₁)\n  {X₀ : C} {k₀ : d ⟶ F.obj X₀} (z : Z X₀ k₀) : Z (lift F d) (hom_to_lift F d) :=\nbegin\n  apply nonempty.some,\n  apply @is_preconnected_induction _ _ _\n    (λ (Y : structured_arrow d F), Z Y.right Y.hom) _ _ { right := X₀, hom := k₀, } z,\n  { intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\n  { intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\nend\n\nvariables {F G}\n\n/--\nGiven a cocone over `F ⋙ G`, we can construct a `cocone G` with the same cocone point.\n-/\n@[simps]\ndef extend_cocone : cocone (F ⋙ G) ⥤ cocone G :=\n{ obj := λ c,\n  { X := c.X,\n    ι :=\n    { app := λ X, G.map (hom_to_lift F X) ≫ c.ι.app (lift F X),\n      naturality' := λ X Y f,\n      begin\n        dsimp, simp,\n        -- This would be true if we'd chosen `lift F X` to be `lift F Y`\n        -- and `hom_to_lift F X` to be `f ≫ hom_to_lift F Y`.\n        apply induction F\n          (λ Z k, G.map f ≫ G.map (hom_to_lift F Y) ≫ c.ι.app (lift F Y) = G.map k ≫ c.ι.app Z),\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w, z], },\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, category.assoc, ←functor.comp_map, c.w] at z,\n        rw z, },\n        { rw [←functor.map_comp_assoc], },\n      end } },\n  map := λ X Y f,\n  { hom := f.hom, } }\n\n@[simp]\nlemma colimit_cocone_comp_aux (s : cocone (F ⋙ G)) (j : C) :\n  G.map (hom_to_lift F (F.obj j)) ≫ s.ι.app (lift F (F.obj j)) =\n    s.ι.app j :=\nbegin\n  -- This point is that this would be true if we took `lift (F.obj j)` to just be `j`\n  -- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.\n  apply induction F (λ X k, G.map k ≫ s.ι.app X = (s.ι.app j : _)),\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←w, rw ← s.w f at h, simpa using h, },\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←w at h, rw ← s.w f, simpa using h, },\n  { exact s.w (𝟙 _), },\nend\n\nvariables (F G)\n\n/--\nIf `F` is cofinal,\nthe category of cocones on `F ⋙ G` is equivalent to the category of cocones on `G`,\nfor any `G : D ⥤ E`.\n-/\n@[simps]\ndef cocones_equiv : cocone (F ⋙ G) ≌ cocone G :=\n{ functor := extend_cocone,\n  inverse := cocones.whiskering F,\n  unit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cocones.ext (iso.refl _) (by tidy)) (by tidy), }.\n\nvariables {G}\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `t : cocone G` for some `G : D ⥤ E`,\n`t.whisker F` is a colimit cocone exactly when `t` is.\n-/\ndef is_colimit_whisker_equiv (t : cocone G) : is_colimit (t.whisker F) ≃ is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G).symm\n\n/--\nWhen `F` is cofinal, and `t : cocone (F ⋙ G)`,\n`extend_cocone.obj t` is a colimit coconne exactly when `t` is.\n-/\ndef is_colimit_extend_cocone_equiv (t : cocone (F ⋙ G)) :\n  is_colimit (extend_cocone.obj t) ≃ is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G)\n\n/-- Given a colimit cocone over `G : D ⥤ E` we can construct a colimit cocone over `F ⋙ G`. -/\n@[simps]\ndef colimit_cocone_comp (t : colimit_cocone G) :\n  colimit_cocone (F ⋙ G) :=\n{ cocone := _,\n  is_colimit := (is_colimit_whisker_equiv F _).symm (t.is_colimit) }\n\n@[priority 100]\ninstance comp_has_colimit [has_colimit G] :\n  has_colimit (F ⋙ G) :=\nhas_colimit.mk (colimit_cocone_comp F (get_colimit_cocone G))\n\nlemma colimit_pre_is_iso_aux {t : cocone G} (P : is_colimit t) :\n  ((is_colimit_whisker_equiv F _).symm P).desc (t.whisker F) = 𝟙 t.X :=\nbegin\n  dsimp [is_colimit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  dsimp, simp, dsimp, simp, -- See library note [dsimp, simp].\nend\n\ninstance colimit_pre_is_iso [has_colimit G] :\n  is_iso (colimit.pre G F) :=\nbegin\n  rw colimit.pre_eq (colimit_cocone_comp F (get_colimit_cocone G)) (get_colimit_cocone G),\n  erw colimit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nsection\nvariables (G)\n\n/--\nWhen `F : C ⥤ D` is cofinal, and `G : D ⥤ E` has a colimit, then `F ⋙ G` has a colimit also and\n`colimit (F ⋙ G) ≅ colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso [has_colimit G] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)\n\nend\n\n/-- Given a colimit cocone over `F ⋙ G` we can construct a colimit cocone over `G`. -/\n@[simps]\ndef colimit_cocone_of_comp (t : colimit_cocone (F ⋙ G)) :\n  colimit_cocone G :=\n{ cocone := extend_cocone.obj t.cocone,\n  is_colimit := (is_colimit_extend_cocone_equiv F _).symm (t.is_colimit), }\n\n/--\nWhen `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_colimit`.)\n-/\nlemma has_colimit_of_comp [has_colimit (F ⋙ G)] :\n  has_colimit G :=\nhas_colimit.mk (colimit_cocone_of_comp F (get_colimit_cocone (F ⋙ G)))\n\n\nsection\nlocal attribute [instance] has_colimit_of_comp\n\n/--\nWhen `F` is cofinal, and `F ⋙ G` has a colimit, then `G` has a colimit also and\n`colimit (F ⋙ G) ≅ colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso' [has_colimit (F ⋙ G)] : colimit (F ⋙ G) ≅ colimit G := as_iso (colimit.pre G F)\n\n\nend\n\n/--\nIf the universal morphism `colimit (F ⋙ coyoneda.obj (op d)) ⟶ colimit (coyoneda.obj (op d))`\nis an isomorphism (as it always is when `F` is cofinal),\nthen `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit`\n(simply because `colimit (coyoneda.obj (op d)) ≅ punit`).\n-/\ndef colimit_comp_coyoneda_iso (d : D) [is_iso (colimit.pre (coyoneda.obj (op d)) F)] :\n  colimit (F ⋙ coyoneda.obj (op d)) ≅ punit :=\nas_iso (colimit.pre (coyoneda.obj (op d)) F) ≪≫ coyoneda.colimit_coyoneda_iso (op d)\n\n\n\n/--\nIf `colimit (F ⋙ coyoneda.obj (op d)) ≅ punit` for all `d : D`, then `F` is cofinal.\n-/\nlemma cofinal_of_colimit_comp_coyoneda_iso_punit\n  (I : Π d, colimit (F ⋙ coyoneda.obj (op d)) ≅ punit) : final F :=\n⟨λ d, begin\n  haveI : nonempty (structured_arrow d F),\n  { have := (I d).inv punit.star,\n    obtain ⟨j, y, rfl⟩ := limits.types.jointly_surjective' this,\n    exact ⟨structured_arrow.mk y⟩, },\n  apply zigzag_is_connected,\n  rintros ⟨⟨⟩,X₁,f₁⟩ ⟨⟨⟩,X₂,f₂⟩,\n  dsimp at *,\n  let y₁ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₁ f₁,\n  let y₂ := colimit.ι (F ⋙ coyoneda.obj (op d)) X₂ f₂,\n  have e : y₁ = y₂,\n  { apply (I d).to_equiv.injective, ext, },\n  have t := types.colimit_eq e,\n  clear e y₁ y₂,\n  exact zigzag_of_eqv_gen_quot_rel t,\nend⟩\n\nend final\n\n\nnamespace initial\n\nvariables (F : C ⥤ D) [initial F]\n\ninstance (d : D) : nonempty (costructured_arrow F d) := is_connected.is_nonempty\n\nvariables {E : Type u} [category.{v} E] (G : D ⥤ E)\n\n/--\nWhen `F : C ⥤ D` is initial, we denote by `lift F d` an arbitrary choice of object in `C` such that\nthere exists a morphism `F.obj (lift F d) ⟶ d`.\n-/\ndef lift (d : D) : C := (classical.arbitrary (costructured_arrow F d)).left\n\n/--\nWhen `F : C ⥤ D` is initial, we denote by `hom_to_lift` an arbitrary choice of morphism\n`F.obj (lift F d) ⟶ d`.\n-/\ndef hom_to_lift (d : D) : F.obj (lift F d) ⟶ d :=\n  (classical.arbitrary (costructured_arrow F d)).hom\n\n/--\nWe provide an induction principle for reasoning about `lift` and `hom_to_lift`.\nWe want to perform some construction (usually just a proof) about\nthe particular choices `lift F d` and `hom_to_lift F d`,\nit suffices to perform that construction for some other pair of choices\n(denoted `X₀ : C` and `k₀ : F.obj X₀ ⟶ d` below),\nand to show how to transport such a construction\n*both* directions along a morphism between such choices.\n-/\ndef induction {d : D} (Z : Π (X : C) (k : F.obj X ⟶ d), Sort*)\n  (h₁ : Π X₁ X₂ (k₁ : F.obj X₁ ⟶ d) (k₂ : F.obj X₂ ⟶ d) (f : X₁ ⟶ X₂),\n    (F.map f ≫ k₂ = k₁) → Z X₁ k₁ → Z X₂ k₂)\n  (h₂ : Π X₁ X₂ (k₁ : F.obj X₁ ⟶ d) (k₂ : F.obj X₂ ⟶ d) (f : X₁ ⟶ X₂),\n    (F.map f ≫ k₂ = k₁) → Z X₂ k₂ → Z X₁ k₁)\n  {X₀ : C} {k₀ : F.obj X₀ ⟶ d} (z : Z X₀ k₀) : Z (lift F d) (hom_to_lift F d) :=\nbegin\n  apply nonempty.some,\n  apply @is_preconnected_induction _ _ _\n    (λ Y : costructured_arrow F d, Z Y.left Y.hom) _ _ { left := X₀, hom := k₀ } z,\n  { intros j₁ j₂ f a, fapply h₁ _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },\n  { intros j₁ j₂ f a, fapply h₂ _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },\nend\n\nvariables {F G}\n\n/--\nGiven a cone over `F ⋙ G`, we can construct a `cone G` with the same cocone point.\n-/\n@[simps]\ndef extend_cone : cone (F ⋙ G) ⥤ cone G :=\n{ obj := λ c,\n  { X := c.X,\n    π :=\n    { app := λ d, c.π.app (lift F d) ≫ G.map (hom_to_lift F d),\n      naturality' := λ X Y f,\n      begin\n        dsimp, simp,\n        -- This would be true if we'd chosen `lift F Y` to be `lift F X`\n        -- and `hom_to_lift F Y` to be `hom_to_lift F X ≫ f`.\n        apply induction F (λ Z k, (c.π.app Z ≫ G.map k : c.X ⟶ _) =\n          c.π.app (lift F X) ≫ G.map (hom_to_lift F X) ≫ G.map f),\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, ←functor.comp_map, ←category.assoc, ←category.assoc, c.w] at z,\n        rw [z, category.assoc] },\n        { intros Z₁ Z₂ k₁ k₂ g a z,\n        rw [←a, functor.map_comp, ←functor.comp_map, ←category.assoc, ←category.assoc,\n          c.w, z, category.assoc] },\n        { rw [←functor.map_comp], },\n      end } },\n  map := λ X Y f,\n  { hom := f.hom, } }\n\n@[simp]\nlemma limit_cone_comp_aux (s : cone (F ⋙ G)) (j : C) :\n  s.π.app (lift F (F.obj j)) ≫ G.map (hom_to_lift F (F.obj j)) =\n    s.π.app j :=\nbegin\n  -- This point is that this would be true if we took `lift (F.obj j)` to just be `j`\n  -- and `hom_to_lift (F.obj j)` to be `𝟙 (F.obj j)`.\n  apply induction F (λ X k, s.π.app X ≫ G.map k = (s.π.app j : _)),\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←s.w f, rw ←w at h, simpa using h, },\n  { intros j₁ j₂ k₁ k₂ f w h, rw ←s.w f at h, rw ←w, simpa using h, },\n  { exact s.w (𝟙 _), },\nend\n\nvariables (F G)\n\n/--\nIf `F` is initial,\nthe category of cones on `F ⋙ G` is equivalent to the category of cones on `G`,\nfor any `G : D ⥤ E`.\n-/\n@[simps]\ndef cones_equiv : cone (F ⋙ G) ≌ cone G :=\n{ functor := extend_cone,\n  inverse := cones.whiskering F,\n  unit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ c, cones.ext (iso.refl _) (by tidy)) (by tidy), }.\n\nvariables {G}\n\n/--\nWhen `F : C ⥤ D` is initial, and `t : cone G` for some `G : D ⥤ E`,\n`t.whisker F` is a limit cone exactly when `t` is.\n-/\ndef is_limit_whisker_equiv (t : cone G) : is_limit (t.whisker F) ≃ is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F G).symm\n\n/--\nWhen `F` is initial, and `t : cone (F ⋙ G)`,\n`extend_cone.obj t` is a limit cone exactly when `t` is.\n-/\ndef is_limit_extend_cone_equiv (t : cone (F ⋙ G)) :\n  is_limit (extend_cone.obj t) ≃ is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F G)\n\n/-- Given a limit cone over `G : D ⥤ E` we can construct a limit cone over `F ⋙ G`. -/\n@[simps]\ndef limit_cone_comp (t : limit_cone G) :\n  limit_cone (F ⋙ G) :=\n{ cone := _,\n  is_limit := (is_limit_whisker_equiv F _).symm (t.is_limit) }\n\n@[priority 100]\ninstance comp_has_limit [has_limit G] :\n  has_limit (F ⋙ G) :=\nhas_limit.mk (limit_cone_comp F (get_limit_cone G))\n\nlemma limit_pre_is_iso_aux {t : cone G} (P : is_limit t) :\n  ((is_limit_whisker_equiv F _).symm P).lift (t.whisker F) = 𝟙 t.X :=\nbegin\n  dsimp [is_limit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  simp,\nend\n\ninstance limit_pre_is_iso [has_limit G] :\n  is_iso (limit.pre G F) :=\nbegin\n  rw limit.pre_eq (limit_cone_comp F (get_limit_cone G)) (get_limit_cone G),\n  erw limit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nsection\nvariables (G)\n\n/--\nWhen `F : C ⥤ D` is initial, and `G : D ⥤ E` has a limit, then `F ⋙ G` has a limit also and\n`limit (F ⋙ G) ≅ limit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso [has_limit G] : limit (F ⋙ G) ≅ limit G := (as_iso (limit.pre G F)).symm\n\nend\n\n/-- Given a limit cone over `F ⋙ G` we can construct a limit cone over `G`. -/\n@[simps]\ndef limit_cone_of_comp (t : limit_cone (F ⋙ G)) :\n  limit_cone G :=\n{ cone := extend_cone.obj t.cone,\n  is_limit := (is_limit_extend_cone_equiv F _).symm (t.is_limit), }\n\n/--\nWhen `F` is initial, and `F ⋙ G` has a limit, then `G` has a limit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_limit`.)\n-/\nlemma has_limit_of_comp [has_limit (F ⋙ G)] :\n  has_limit G :=\nhas_limit.mk (limit_cone_of_comp F (get_limit_cone (F ⋙ G)))\n\nsection\nlocal attribute [instance] has_limit_of_comp\n\n/--\nWhen `F` is initial, and `F ⋙ G` has a limit, then `G` has a limit also and\n`limit (F ⋙ G) ≅ limit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso' [has_limit (F ⋙ G)] : limit (F ⋙ G) ≅ limit G :=\n(as_iso (limit.pre G F)).symm\n\nend\n\nend initial\n\nend functor\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/final.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.4882833952958346, "lm_q1q2_score": 0.30572604594687075}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nParallel computation of a computable sequence of computations by\na diagonal enumeration.\nThe important theorems of this operation are proven as\nterminates_parallel and exists_of_mem_parallel.\n(This operation is nondeterministic in the sense that it does not\nhonor sequence equivalence (irrelevance of computation time).)\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.seq.wseq\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\nnamespace computation\n\n\ndef parallel.aux2 {α : Type u} : List (computation α) → α ⊕ List (computation α) :=\n  list.foldr (fun (c : computation α) (o : α ⊕ List (computation α)) => sorry) (sum.inr [])\n\ndef parallel.aux1 {α : Type u} : List (computation α) × wseq (computation α) → α ⊕ List (computation α) × wseq (computation α) :=\n  sorry\n\n/-- Parallel computation of an infinite stream of computations,\n  taking the first result -/\ndef parallel {α : Type u} (S : wseq (computation α)) : computation α :=\n  corec sorry ([], S)\n\ntheorem terminates_parallel.aux {α : Type u} {l : List (computation α)} {S : wseq (computation α)} {c : computation α} : c ∈ l → terminates c → terminates (corec parallel.aux1 (l, S)) := sorry\n\ntheorem terminates_parallel {α : Type u} {S : wseq (computation α)} {c : computation α} (h : c ∈ S) [T : terminates c] : terminates (parallel S) := sorry\n\ntheorem exists_of_mem_parallel {α : Type u} {S : wseq (computation α)} {a : α} (h : a ∈ parallel S) : ∃ (c : computation α), ∃ (H : c ∈ S), a ∈ c := sorry\n\ntheorem map_parallel {α : Type u} {β : Type v} (f : α → β) (S : wseq (computation α)) : map f (parallel S) = parallel (wseq.map (map f) S) := sorry\n\ntheorem parallel_empty {α : Type u} (S : wseq (computation α)) (h : wseq.head S ~> none) : parallel S = empty α := sorry\n\n-- The reason this isn't trivial from exists_of_mem_parallel is because it eliminates to Sort\n\ndef parallel_rec {α : Type u} {S : wseq (computation α)} (C : α → Sort v) (H : (s : computation α) → s ∈ S → (a : α) → a ∈ s → C a) {a : α} (h : a ∈ parallel S) : C a :=\n  let T : wseq (computation (α × computation α)) := wseq.map (fun (c : computation α) => map (fun (a : α) => (a, c)) c) S;\n  (fun (_x : α × computation α) (e : get (parallel T) = _x) =>\n      Prod.rec\n        (fun (a' : α) (c : computation α) (e : get (parallel T) = (a', c)) =>\n          and.dcases_on sorry fun (ac : a ∈ c) (cs : c ∈ S) => H c cs a ac)\n        _x e)\n    (get (parallel T)) sorry\n\ntheorem parallel_promises {α : Type u} {S : wseq (computation α)} {a : α} (H : ∀ (s : computation α), s ∈ S → s ~> a) : parallel S ~> a := sorry\n\ntheorem mem_parallel {α : Type u} {S : wseq (computation α)} {a : α} (H : ∀ (s : computation α), s ∈ S → s ~> a) {c : computation α} (cs : c ∈ S) (ac : a ∈ c) : a ∈ parallel S :=\n  mem_of_promises (parallel S) (parallel_promises H)\n\ntheorem parallel_congr_lem {α : Type u} {S : wseq (computation α)} {T : wseq (computation α)} {a : α} (H : wseq.lift_rel equiv S T) : (∀ (s : computation α), s ∈ S → s ~> a) ↔ ∀ (t : computation α), t ∈ T → t ~> a := sorry\n\n-- The parallel operation is only deterministic when all computation paths lead to the same value\n\ntheorem parallel_congr_left {α : Type u} {S : wseq (computation α)} {T : wseq (computation α)} {a : α} (h1 : ∀ (s : computation α), s ∈ S → s ~> a) (H : wseq.lift_rel equiv S T) : parallel S ~ parallel T := sorry\n\ntheorem parallel_congr_right {α : Type u} {S : wseq (computation α)} {T : wseq (computation α)} {a : α} (h2 : ∀ (t : computation α), t ∈ T → t ~> a) (H : wseq.lift_rel equiv S T) : parallel S ~ parallel T :=\n  parallel_congr_left (iff.mpr (parallel_congr_lem H) h2) H\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/seq/parallel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3055907097395889}}
{"text": "/- Author: E.W.Ayers © 2019 -/\n\n/- Expression zipper. -/\nimport .util .table\nnamespace expr\n\n/-- Labels for each recursive argument of a constructor for `expr`. -/\n@[derive decidable_eq]\ninductive coord \n|app_left     |app_right\n|lam_var_type |lam_body\n|pi_var_type  |pi_body\n|elet_type    |elet_assignment |elet_body\ndef coord.code: coord → ℕ |coord.app_left := 0 |coord.app_right := 1 |coord.lam_var_type := 2 |coord.lam_body := 3 |coord.pi_var_type := 4 |coord.pi_body := 5 |coord.elet_type := 6 |coord.elet_assignment := 7 |coord.elet_body := 8\ninstance coord.has_lt : has_lt coord := ⟨λ x y, x.code < y.code⟩\ninstance coord.dec_lt : decidable_rel ((<) : coord → coord → Prop) := by apply_instance\n/-- A list of `coord`s, specifying a position in an expression. The first coordinate says which way to go on the root.-/\ndef address := list coord\ninstance address.has_lt : has_lt address := show has_lt (list coord), by apply_instance\ninstance address.dec_lt : decidable_rel ((<) : address → address → Prop) := by apply_instance\ninstance address.has_append : has_append address := ⟨list.append⟩\n/-- `is_below x y` is true when `∃ z, y ++ z = x` -/\ndef address.is_below : address → address → bool\n|_ [] := tt -- everything is below root.\n|[] _ := ff -- root is below nothing but itself.\n|(h₁ :: t₁) (h₂ :: t₂) := h₁ = h₂ ∧ address.is_below t₁ t₂\ninfixr  ` ≺ `:50 := address.is_below\nnamespace zipper\n    /-- A path is the part of the zipper that remembers the stuff above the zipper's cursor.-/\n    @[derive decidable_eq]\n    meta inductive path_entry\n    |app_left        (left : unit) (right : expr) : path_entry\n    |app_right       (left : expr) (right : unit) : path_entry\n    |lam_var_type    (var_name : name) (bi : binder_info) (var_type : unit) (body : expr) : path_entry\n    |lam_body        (var_name : name) (bi : binder_info) (var_type : expr) (body : unit) : path_entry\n    |pi_var_type     (var_name : name) (bi : binder_info) (var_type : unit) (body : expr) : path_entry\n    |pi_body         (var_name : name) (bi : binder_info) (var_type : expr) (body : unit) : path_entry\n    |elet_type       (var_name : name) (type : unit) (assignment : expr)    (body : expr) : path_entry\n    |elet_assignment (var_name : name) (type : expr) (assignment : unit)    (body : expr) : path_entry\n    |elet_body       (var_name : name) (type : expr) (assignment : expr)    (body : unit) : path_entry\n    /-- A context entry for the zipper's cursor. \n    In the current implementation; as zippers traverse below lambdas, \n    pis and elets, they don't replace de-Bruijn indices with local constants.\n    -/\n    @[derive decidable_eq]\n    meta inductive context_entry\n    |Hyp      (n : name) (bi : binder_info) (type : expr)  : context_entry\n    |Assigned (n : name) (type : expr) (assignment : expr) : context_entry\n\n    namespace path_entry\n        meta def to_coord : path_entry → coord\n        |(app_left _ _)            := coord.app_left\n        |(app_right _ _)           := coord.app_right\n        |(lam_var_type _ _ _ _)    := coord.lam_var_type\n        |(lam_body _ _ _ _)        := coord.lam_body\n        |(pi_var_type _ _ _ _)     := coord.pi_var_type\n        |(pi_body _ _ _ _)         := coord.pi_body\n        |(elet_type _ _ _ _)       := coord.elet_type\n        |(elet_assignment _ _ _ _) := coord.elet_assignment\n        |(elet_body _ _ _ _)       := coord.elet_body\n\n        meta def to_address : list path_entry → address := list.map to_coord ∘ list.reverse\n        open tactic \n        meta def to_tactic_format : path_entry → tactic format\n        |(app_left l r)            := (pure $ λ l r, l ++ \" $ \" ++ r) <*> pp l <*> pp r\n        |(app_right l r)           := (pure $ λ l r, l ++ \" $ \" ++ r) <*> pp l <*> pp r\n        |(lam_var_type n _ y b)    := (pure $ λ n y b, \"λ \" ++ n ++ \" : \" ++ y ++ \", \" ++ b) <*> pp n <*> pp y <*> pp b\n        |(lam_body n _ y b)        := (pure $ λ n y b, \"λ \" ++ n ++ \" : \" ++ y ++ \", \" ++ b) <*> pp n <*> pp y <*> pp b\n        |(pi_var_type n _ y b)     := (pure $ λ n y b, \"Π \" ++ n ++ \" : \" ++ y ++ \", \" ++ b) <*> pp n <*> pp y <*> pp b\n        |(pi_body n _ y b)         := (pure $ λ n y b, \"Π \" ++ n ++ \" : \" ++ y ++ \", \" ++ b) <*> pp n <*> pp y <*> pp b\n        |(elet_type n y a b)       := (pure $ λ n y a b, \"let \" ++ n ++ \" : \" ++ y ++ \" := \" ++ a ++ \" in \" ++ b) <*> pp n <*> pp y <*> pp a <*> pp b\n        |(elet_assignment n y a b) := (pure $ λ n y a b, \"let \" ++ n ++ \" : \" ++ y ++ \" := \" ++ a ++ \" in \" ++ b) <*> pp n <*> pp y <*> pp a <*> pp b\n        |(elet_body n y a b)       := (pure $ λ n y a b, \"let \" ++ n ++ \" : \" ++ y ++ \" := \" ++ a ++ \" in \" ++ b) <*> pp n <*> pp y <*> pp a <*> pp b\n        meta instance has_to_tactic_format : has_to_tactic_format path_entry := ⟨to_tactic_format⟩\n        meta def to_context_entry : path_entry → option context_entry\n        |(lam_body  n bi y   b) := some $ context_entry.Hyp      n bi y\n        |(pi_body   n bi y   b) := some $ context_entry.Hyp      n bi y\n        |(elet_body n    y a b) := some $ context_entry.Assigned n    y a\n        |_ := none\n        meta def to_context : list path_entry → list context_entry \n        := list.filter_map to_context_entry\n    end path_entry\n\n    namespace context_entry\n        meta def type : context_entry → expr\n        |(Hyp      _ _ y  ) := y\n        |(Assigned _   y _) := y\n        meta def name : context_entry → name\n        |(Hyp      n _ _  ) := n\n        |(Assigned n   _ _) := n\n        meta def to_local : context_entry → tactic expr\n        |(Hyp      n bi t  ) := tactic.mk_local' n bi t\n        |(Assigned n    t _) := tactic.mk_local' n binder_info.default t\n        meta def pexpr_pis_of_ctxt : list context_entry → pexpr → pexpr\n        |[] e := to_pexpr e\n        |((context_entry.Hyp n bi y) :: rest) e := \n            pexpr_pis_of_ctxt rest $ @expr.pi ff n bi (to_pexpr y) $ e\n        |((context_entry.Assigned n y a) :: rest) e := \n            pexpr_pis_of_ctxt rest $ expr.elet n (to_pexpr y) (to_pexpr a) $ e\n        -- meta def hyps_of_telescope : telescope → list context_entry \n        -- := list.map (λ ⟨n,bi,y⟩, context_entry.Hyp n bi y)\n    end context_entry\nend zipper\n/-- \nA zipper over expressions. You can think of this as being like a 'cursor' that can move around an `expr`.\nIt does not replace bound variables with local constants, but instead maintains its own 'mini-context' of the binders that it is under.\n\nReference: [Functional Pearl - The Zipper](https://www.st.cs.uni-saarland.de/edu/seminare/2005/advanced-fp/docs/huet-zipper.pdf)\n -/\n@[derive decidable_eq]\nmeta structure zipper :=\n(path : list zipper.path_entry)\n(current : expr)\n\n/-- Result type of calling `zipper.down`.  -/\n@[derive decidable_eq]\nmeta inductive down_result\n|terminal : down_result\n|app  (left : zipper) (right : zipper) : down_result\n| lam (var_name : name) (bi : binder_info) (var_type : zipper) (body : zipper) : down_result\n|  pi (var_name : name) (bi : binder_info) (var_type : zipper) (body : zipper) : down_result\n|elet (var_name : name) (type : zipper) (assignment : zipper) (body : zipper) : down_result\n\nmeta def down_result.children : down_result → list zipper\n|(down_result.terminal) := []\n|(down_result.app l r) := [l,r]\n|(down_result.lam n bi vt b) := [vt,b]\n|(down_result.pi n bi vt b) := [vt,b]\n|(down_result.elet n t a b) := [t,a,b]\n\nnamespace zipper\n    open path_entry\n    /--Move the cursor down the expression tree.-/\n    meta def down : zipper → down_result\n    |⟨p, expr.app f a⟩ := \n        down_result.app \n            ⟨app_left  () a  :: p, f⟩ \n            ⟨app_right f  () :: p, a⟩\n    |⟨p, expr.lam n bi y b⟩ := \n        down_result.lam n bi \n            ⟨lam_var_type n bi () b  :: p, y⟩ \n            ⟨lam_body     n bi y  () :: p, b⟩\n    |⟨p, expr.pi  n bi y b⟩ := \n        down_result.pi n bi \n            ⟨pi_var_type  n bi () b  :: p,  y⟩\n            ⟨pi_body      n bi y  () :: p,  b⟩\n    |⟨p, expr.elet n t a b⟩ := \n        down_result.elet n \n            ⟨elet_type n () a b :: p,  t⟩ \n            ⟨elet_assignment n t () b :: p,  a⟩ \n            ⟨elet_body n t a () :: p, b⟩ \n    |⟨p,e⟩ := down_result.terminal\n    meta def children : zipper → list zipper := down_result.children ∘ down\n\n    meta def down_coord : coord → zipper → option zipper\n    |(coord.app_left)        ⟨p,expr.app f a⟩      := some ⟨app_left  () a           :: p, f⟩\n    |(coord.app_right)       ⟨p,expr.app f a⟩      := some ⟨app_right f ()           :: p, a⟩\n    |(coord.lam_var_type)    ⟨p,expr.lam n bi y b⟩ := some ⟨lam_var_type n bi () b   :: p, y⟩ \n    |(coord.lam_body)        ⟨p,expr.lam n bi y b⟩ := some ⟨lam_body     n bi y ()   :: p, b⟩\n    |(coord.pi_var_type)     ⟨p,expr.pi n bi y b⟩  := some ⟨pi_var_type  n bi () b   :: p, y⟩ \n    |(coord.pi_body)         ⟨p,expr.pi n bi y b⟩  := some ⟨pi_body      n bi y ()   :: p, b⟩\n    |(coord.elet_type)       ⟨p,expr.elet n y a b⟩ := some ⟨elet_type       n () a b :: p, y⟩ \n    |(coord.elet_assignment) ⟨p,expr.elet n y a b⟩ := some ⟨elet_assignment n y () b :: p, a⟩ \n    |(coord.elet_body)       ⟨p,expr.elet n y a b⟩ := some ⟨elet_body       n y a () :: p, b⟩ \n    |_ _ := none\n\n    /-- Pop the cursor up the expression tree. If we are already at the top, returns `none`. -/\n    meta def up : zipper → option zipper\n    |⟨[],                            e⟩ := none\n    |⟨app_left  () a           :: p, f⟩ := some $ zipper.mk p $ expr.app f a\n    |⟨app_right f  ()          :: p, a⟩ := some $ zipper.mk p $ expr.app f a\n    |⟨lam_var_type   n bi _ b  :: p, e⟩ := some $ zipper.mk p $ expr.lam n bi e b\n    |⟨lam_body       n bi y _  :: p, e⟩ := some $ zipper.mk p $ expr.lam n bi y e \n    |⟨pi_var_type    n bi _ b  :: p, e⟩ := some $ zipper.mk p $ expr.pi  n bi e b\n    |⟨pi_body        n bi y _  :: p, e⟩ := some $ zipper.mk p $ expr.pi  n bi y e\n    |⟨elet_type       n _ a b  :: p, e⟩ := some $ zipper.mk p $ expr.elet n e a b\n    |⟨elet_assignment n t _ b  :: p, e⟩ := some $ zipper.mk p $ expr.elet n t e b\n    |⟨elet_body       n t a _  :: p, e⟩ := some $ zipper.mk p $ expr.elet n t a e\n    meta def is_top : zipper → bool := list.empty ∘ path\n    meta def top := option.repeat up\n    meta def down_app_left : zipper → option zipper := down_coord coord.app_left\n    meta def down_app_right : zipper → option zipper := down_coord coord.app_right\n\n    meta def down_body : zipper → option zipper := λ z,\n        match down z with\n        |down_result.pi   _ _ _ b := some b\n        |down_result.lam  _ _ _ b := some b\n        |down_result.elet _ _ _ b := some b\n        |_ := none\n        end\n    meta def down_var_type : zipper → option zipper := λ z,\n        match down z with\n        |down_result.pi   _ _ y b := some y\n        |down_result.lam  _ _ y b := some y\n        |down_result.elet _ y _ b := some y\n        |_ := none\n        end\n    /-- Move the cursor down according to the given address. \n    If the zipper has the wrong structure then return none.-/\n    meta def down_address : address → zipper → option zipper\n    |[]     z := some z\n    |(h::t) z := down_coord h z >>= down_address t\n\n    meta def unzip : zipper → expr := λ z, option.rec_on (up z) (current z) (unzip)\n    meta def zip : expr → zipper := λ e, zipper.mk [] e\n    meta instance : has_coe expr zipper := ⟨zip⟩\n    meta def set_current : expr → zipper → zipper\n    |e ⟨p,_⟩ := ⟨p,e⟩\n    /-- Map the current subexpression. -/\n    meta def map : (expr → expr) → zipper → zipper\n    |f ⟨p,e⟩ := ⟨p,f e⟩\n    meta def mmap {T} [monad T] : (expr → T expr) → zipper → T zipper\n    |f ⟨p,e⟩ := zipper.mk p <$> f e\n    meta def address : zipper → address := to_address ∘ path\n    meta def ctxt : zipper → list context_entry := to_context ∘ path\n    /-- The number of binders above the cursor. -/\n    meta def binder_depth : zipper → ℕ := list.length ∘ ctxt\n    /-- Replace the current subexpression and unzip. -/\n    meta def unzip_with : expr → zipper → expr := λ e z, unzip $ z.set_current e\n    meta def is_var : zipper → bool := expr.is_var ∘ current\n    meta def is_constant : zipper → bool := expr.is_constant ∘ current\n    /--True when the current expression does not contain local constants -/\n    meta def no_local_consts : zipper → bool := list.empty ∘ expr.list_local_consts ∘ current \n    meta def no_local_const_terms : zipper → tactic bool \n    := λ z, list.empty <$> (expr.list_local_const_terms $ current $ z)\n    meta def is_local_constant : zipper → bool := expr.is_local_constant ∘ current\n    meta def is_mvar : zipper → bool := expr.is_mvar ∘ current\n    meta def is_terminal : zipper → bool := λ z, z.down = down_result.terminal\n    /--Infer the type of the subexpression at the cursor position. -/\n    meta def infer_type : zipper → tactic expr := λ z, do\n            ⟨ins,lcs⟩ ← list.mfoldl (λ ct s, do\n                l ← context_entry.to_local s,\n                pure $ (expr.instantiate_var ct.1 l, l::ct.2)\n            ) (z.current,([] : list expr)) z.ctxt,\n            y ← tactic.infer_type ins,\n            let y := lcs.foldl expr.abstract y,\n            pure y\n    meta instance : has_to_tactic_format zipper := ⟨λ z, do\n     whole ← tactic.pp $ z.unzip_with `(()),\n     current ← tactic.pp z.current,\n     pure $ format.highlight current format.color.blue ++ \" in \" ++ whole\n     ⟩\n    universes u v\n    /-- Helper lemma for `mk_congr` -/\n    lemma eq_cc {α : Type u} {β : Type v} (F : α → β) (a₁: α)\n    :  ∀ a₂ : α, a₁ = a₂ → F a₁ = F a₂ \n    := @eq.rec α a₁ (λ a, F a₁ = F a) rfl\n    open tactic\n    /-- Given a zipper, makes a congruence lemma at the zipper's position. \n    [NOTE] Assumes that the zipper is not inside any binders (for now). -/\n    meta def mk_congr (z : zipper) : tactic expr := do\n        when (z.binder_depth ≠ 0) (tactic.fail \"Not implemented: congruences inside a binder\"),\n        let lhs := z.current,\n        let lhs' := z.unzip,\n        T ← tactic.infer_type lhs,\n        target ← to_expr $ ```(∀ X : %%(T), (%%lhs = X) → %%lhs' = %%(z.unzip_with $ expr.var 1)),\n        tactic.fabricate (some target) (do \n            -- get_goals >>= list.mmap infer_type >>= trace,\n            refine ```(@eq.rec _ %%lhs (λ X, %%(lhs') = %%(z.unzip_with $ expr.var 0)) rfl),\n            -- get_goals >>= list.mmap infer_type >>= trace,\n            pure ()\n        ) \n    /-- Replace the current expression with an unbound deBruijn variable. -/\n    meta def unzip_free : zipper → expr := λ z, z.unzip_with $ expr.var z.binder_depth\n    /-- `apply_congr (rhs,pf) z` takes the given `%%pf : %%z.current = %%rhs` and makes a congruence lemma using the given zipper.  -/\n    meta def apply_congr : (expr × expr) → zipper → tactic (expr × expr) := λ ⟨rhs,pf⟩ z, do\n        if z.is_top then pure ⟨rhs,pf⟩ else do\n        let lhs := z.current,\n        T ← tactic.infer_type lhs,\n        let lhs' := z.unzip,\n        let rhs' := unzip_with rhs z,\n        target ← to_expr $ ```(%%lhs' = %%rhs'),\n        -- pp target >>= λ m, trace $ (\"target: \":format) ++ m,\n        motive ← to_expr $ @expr.lam ff `X binder_info.default (to_pexpr T) $ ```(%%lhs' = %%(z.unzip_with $ expr.var z.binder_depth)),\n        -- pp motive >>= λ m, trace $ (\"motive: \":format) ++ m,\n        pf' ← tactic.fabricate (some target) (do\n            -- trace_state,\n            refine ```(@eq.rec %%T %%lhs %%motive rfl %%rhs %%pf),\n            all_goals $ try $ (apply_instance <|> assumption)\n        ),\n        pure (rhs',pf')\n    \n    /-- Produce a conversion that uses the given conversion at the zipper's subexpression.-/\n    meta def apply_conv : conv unit → zipper → conv unit := λ cnv z, do\n        let lhs := z.current,\n        T ← tactic.infer_type lhs,\n        rhs ← mk_meta_var T,\n        let lhs' := z.unzip,\n        target ← to_expr $ ```(%%lhs' = %%(z.unzip_with rhs)),\n        rewrite_target target,\n        refine ```(@eq.rec %%T %%lhs (λ X, %%lhs' = z.unzip_with $ expr.var 0) rfl %%rhs _),\n        cnv\n\n    /-- A proper argument is one who is not implicit or a proposition.\n        Ie, anything which the user would see when writing down the term. -/\n    private meta def is_proper (p : param_info) : bool := \n        ¬(param_info.is_implicit p || param_info.is_inst_implicit  p || param_info.is_prop p)\n\n    meta def is_type (z : zipper) : tactic bool := expr.is_sort <$> tactic.infer_type z.current\n\n    /-- Take a zipper where the current expression is a function application, \n        and return zippers over all of the non-implicit, non-prop arguments. -/\n    meta def down_proper (z : zipper) : tactic (zipper × list zipper) := do\n        let c := z.current,\n        let f := expr.get_app_fn c,\n        let f_name := expr.as_name f,\n        -- if it is a numeral then don't expand it.\n        if f_name = `bit0 ∨ f_name = `bit1 then pure (z,[]) else do\n        let args := expr.get_app_args c,\n        params ← (fun_info.params <$> tactic.get_fun_info f (args.length)) <|> pure [],\n        let params := list.reverse params,\n        ⟨zippers, fz⟩ ← params.mfoldl (λ acc p, do\n                let (⟨zippers,z⟩ : (list zipper) × zipper) := acc,\n                z' ← down_app_left z,\n                if is_proper p then do\n                    zr ← down_app_right z,\n                    --t ← is_type zr,\n                    --if t then pure (zippers,z') else\n                    pure (zr::zippers,z')\n                else pure (zippers, z')\n            ) \n            (([] : list zipper), z),\n        pure (fz,zippers)\n\n    meta def fold {α} (f : α → zipper → α) : α → zipper → α\n    | a z := z.children.foldl fold $ f a z\n    meta def mfold {T} [monad T] {α} (f : α → zipper → T α) : α → zipper → T α\n    | a z := do a ← f a z, z.children.mfoldl mfold a\n\n    /-- Traverse zipper.current, only exploring explicit, non-Prop arguments.\n        If `f` fails, then that branch is skipped. -/\n    meta def traverse_proper {α} (f : α →  zipper → tactic α) : α → zipper → tactic α\n    |a z := (do\n        a ← f a z,\n        (_,children) ← down_proper z,\n        children.mfoldl traverse_proper a)\n        <|> pure a\n\n    /-- `minimal_monotone p z` finds the smallest proper subexpressions of the zipper such that \n        `p` doesn't fail. \n        [NOTE] It assumes that if `p e` fails, then all of `e`s subexpressions will fail too. -/\n    meta def minimal_monotone {α} (p : zipper → tactic α) : zipper → tactic (list α)\n    |z := (do\n            a ← p z,\n            (_,children) ← down_proper z,\n            kids ← list.join <$> list.mmap minimal_monotone children,\n            pure $ list.cases_on kids [a] (λ _ _,kids)\n        )\n        <|>  pure []\n\n    /-- `maximal_monotone p z`: Find the largest proper subexpressions of the zipper such that \n        `p` doesn't fail. \n        [NOTE] It assumes that if `p e` fails, then all of `e`s subexpressions will fail too. -/\n    meta def maximal_monotone {α} (p : zipper → tactic α) : zipper → tactic (list α)\n    |z := (do a ← p z,  pure [a]) <|> do\n            (f,children) ← down_proper z,\n            children ← pure $ if is_local_constant f && ¬children.empty then f :: children else children,\n            kids ← list.join <$> list.mmap maximal_monotone children,\n            pure $ kids\n    /-- `find_occurences z e` finds subexpressions of `z` which non-trivially unify with `e`. -/\n    meta def find_occurences : zipper → expr → tactic (list zipper) := λ E e, do\n        rs ← maximal_monotone (λ z,\n            if z.is_mvar || z.is_constant then failure\n            else tactic.hypothetically' (unify e z.current transparency.none) *> pure z\n        ) E,\n        pure rs\n\n    meta def has_occurences : zipper → expr → tactic bool \n    := λ z e, (bnot ∘ list.empty) <$> find_occurences z e\n\n    meta def smallest_absent_subterms_aux (l : expr) \n        (filter : zipper → tactic bool := combinator.K $ pure tt) \n        : list expr.address  → zipper → tactic (list expr.address × list (ℕ × zipper))\n    |used z := do\n        filt ← filter z, \n        if ¬ filt then pure (used, []) else do\n        occs ← find_occurences l z.current,\n        o ← pure $ list.find (λ o, bnot $ list.any used $ λ x, zipper.address o ≺ x) occs,\n        match o with\n        |none := do -- z.current is not present, descend.\n            (_,children) ← down_proper z,\n            (used,zs) ← list.mfoldl (λ p child, do \n                (used,zs') ← smallest_absent_subterms_aux (prod.fst p) child, \n                pure (used,zs' ++ p.2)\n            ) (used,[]) children,\n            if zs.empty then pure $ (used,[(occs.length + 1,z)]) else pure $ (used,zs)\n        |some o := \n            -- z.current is present on the lhs\n            -- so now we need to add `o` to the used list so that later matches can't use it.\n            pure $ (used.insert o.address,[])\n        end\n    \n    /-- find subterms of RHS that do not appear on LHS. It will also count when occurrences have been used.\n        So for example, ``smallest_absent_subterms `(a * b + b * a) `(a * b + a * b)`` will return `[(2,a * b)] because\n        `a * b` occurs once in the LHS but twice in the RHS. \n    -/\n    meta def smallest_absent_subterms (lhs : expr) (rhs : zipper) :=\n        prod.snd <$> smallest_absent_subterms_aux lhs (λ z, bnot <$> no_local_const_terms z) [] rhs\n\n    meta def smallest_absent_composite_subterms (lhs : expr) (rhs : zipper) :=\n        prod.snd \n            <$> smallest_absent_subterms_aux lhs \n                (λ s, do\n                    hlcst ← bnot <$> no_local_const_terms s,\n                    is_term ← expr.is_term s.current,\n                     pure $ hlcst && is_term && expr.is_composite s.current) \n                [] rhs\n\n    /--`lowest_uncommon_subterms l z` finds the smallest subterms of z that are not a subterm of `l`. Subterms must include a local_const -/\n    meta def lowest_uncommon_subterms (l : expr) (z : zipper) :=\n        minimal_monotone (λ z,\n            if z.is_mvar || z.is_constant || z.no_local_consts then failure else do \n            --let o := expr.occurs z.current l,\n            matches ← zipper.maximal_monotone (λ rz, (tactic.hypothetically' $ unify z.current rz.current) ) $ zipper.zip l,\n            -- trace_m \"lus: \" $ (z,l,o, matches),\n            if ¬ matches.empty then failure else pure z\n        ) z\n\n    meta def largest_common_subterms (z₁ z₂ : zipper): tactic (list zipper) :=\n        list.join <$> z₁.maximal_monotone (λ z₁, \n            if z₁.is_mvar then failure else do \n            ocs ← find_occurences (z₂) z₁.current, \n            if ocs.empty then failure else pure ocs\n        ) \n\n    private meta def count_symbols_aux : table expr → zipper → tactic (table expr)\n    | acc z := do\n        (f,zs) ← down_proper z,\n        if zipper.is_mvar f then pure acc else\n        zs.mfoldl count_symbols_aux $ table.insert f.current acc\n\n    meta def count_symbols : expr → tactic (table expr) := count_symbols_aux ∅ ∘ zip\n\n    meta def does_unify (e : expr) : zipper → tactic unit\n    | z := if z.is_mvar then failure else\n        (tactic.hypothetically' $ unify e z.current transparency.none ff)\n\n    meta def find_subterms (e : expr) : zipper → tactic (list zipper)\n    := traverse_proper (λ acc z, (does_unify e z *> pure (z::acc)) <|> pure acc) []\n\n    meta def has_single_subterm (e : expr) : zipper → tactic (zipper)\n    := λ z, do [x] ← find_subterms e z, pure x\n\n    meta def count_subterms (e : expr) : zipper → tactic ℕ\n    := λ z, do xs ← find_subterms e z, pure $ list.length xs\n\n    meta def find_non_unify_subterm (e : expr) : zipper → tactic zipper\n    |z :=\n        if e = z.current then pure z else do\n        (_,zs) ← down_proper z,\n        list.mfirst find_non_unify_subterm zs\n\n    meta def find_subterm (e : expr) : zipper → tactic zipper\n    |z :=\n        (does_unify e z *> pure z)\n        <|> do \n            (_,zs) ← down_proper z,\n            list.mfirst find_subterm zs\n\n    meta def distance_to_subterm_down (e : expr) : zipper → nat → tactic nat\n    |z d :=\n        if z.is_mvar then failure else\n        (tactic.hypothetically' $ (do \n            unify e z.current,  pure d\n        ))\n        <|> (do\n            -- tactic.trace_m \"dtsd: \" $ z,\n            (_,zs) ← down_proper z,\n            list.mfirst (λ iz : ℕ × zipper, distance_to_subterm_down iz.2 $ iz.1 + d + 1)\n            $ list.with_indices zs\n        )\n\n    meta def is_app_right : zipper → bool\n    |⟨(app_right _ _)::t,_⟩ := tt\n    |_ := ff\n\n    meta def right : zipper → option zipper\n    |z := if is_app_right z then up z >>= right else up z >>= down_app_right\n\n    meta def distance_to_subterm_up (e : expr) : ℕ → zipper → tactic ℕ\n    |d z :=\n        if is_app_right z then up z >>= distance_to_subterm_up (d+1) else do\n        -- tactic.trace_m \"dtsu1: \" $ z,\n        z ← up z >>= lift down_app_right,\n        -- tactic.trace_m \"dtsu2: \" $ z,\n        distance_to_subterm_down e z (d+1) \n        <|> distance_to_subterm_up (d+1) z\n\n    meta def get_distance (outer : expr) (l : expr) (r : expr) : tactic ℕ := do\n        outer ← instantiate_mvars outer,\n        first ← find_subterm l outer,\n        -- tactic.trace_m \"gd: \" $ first, \n        distance_to_subterm_up r 0 first\n\n    meta def get_proper_children (e : expr) : tactic (list expr) := do\n        e ← instantiate_mvars e,\n        (list.map current <$> prod.snd <$> (down_proper $ zip e)) <|> pure []\n    meta def get_smallest_complex_subterms (z : zipper) : tactic (list zipper) := do\n        minimal_monotone (λ z, do ⟨_,[]⟩ ← down_proper z | failure, pure z) z\n\n    meta def instantiate_mvars : zipper → tactic zipper\n    |z := do\n        let a := address z,\n        let e := unzip z,\n        e ← tactic.instantiate_mvars e,\n        z ← down_address a e,\n        pure z\n\n    /-- Returns true when everything in the zippers except the cursor terms are equal. -/\n    meta def above_equal : zipper → zipper → tactic bool\n    |z₁ z₂ := do \n          list.meq_by (λ x y, pure $ x = y) z₁.path z₂.path\n\nend zipper\nend expr", "meta": {"author": "EdAyers", "repo": "lean-subtask", "sha": "04ac5a6c3bc3bfd190af4d6dcce444ddc8914e4b", "save_path": "github-repos/lean/EdAyers-lean-subtask", "path": "github-repos/lean/EdAyers-lean-subtask/lean-subtask-04ac5a6c3bc3bfd190af4d6dcce444ddc8914e4b/src/expr_zipper.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.30529553008627547}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport data.fintype.prod\nimport data.fintype.sigma\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.preserves.shapes.products\nimport category_theory.limits.preserves.shapes.equalizers\nimport category_theory.limits.preserves.finite\nimport category_theory.limits.constructions.finite_products_of_binary_products\nimport category_theory.limits.constructions.equalizers\nimport category_theory.limits.constructions.binary_products\n\n/-!\n# Constructing limits from products and equalizers.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIf a category has all products, and all equalizers, then it has all limits.\nSimilarly, if it has all finite products, and all equalizers, then it has all finite limits.\n\nIf a functor preserves all products and equalizers, then it preserves all limits.\nSimilarly, if it preserves all finite products and equalizers, then it preserves all finite limits.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\nopen category_theory\nopen opposite\n\nnamespace category_theory.limits\n\nuniverses w v v₂ u u₂\nvariables {C : Type u} [category.{v} C]\n\nvariables {J : Type w} [small_category J]\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_limit_of_has_products_of_has_equalizers\n\nvariables {F : J ⥤ C}\n          {c₁ : fan F.obj}\n          {c₂ : fan (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.2)}\n          (s t : c₁.X ⟶ c₂.X)\n          (hs : ∀ (f : Σ p : J × J, p.1 ⟶ p.2), s ≫ c₂.π.app ⟨f⟩ = c₁.π.app ⟨f.1.1⟩ ≫ F.map f.2)\n          (ht : ∀ (f : Σ p : J × J, p.1 ⟶ p.2), t ≫ c₂.π.app ⟨f⟩ = c₁.π.app ⟨f.1.2⟩)\n          (i : fork s t)\n\ninclude hs ht\n/--\n(Implementation) Given the appropriate product and equalizer cones, build the cone for `F` which is\nlimiting if the given cones are also.\n-/\n@[simps]\ndef build_limit : cone F :=\n{ X := i.X,\n  π :=\n  { app := λ j, i.ι ≫ c₁.π.app ⟨_⟩,\n    naturality' := λ j₁ j₂ f, begin\n      dsimp,\n      rw [category.id_comp, category.assoc, ← hs ⟨⟨_, _⟩, f⟩, i.condition_assoc, ht],\n    end} }\n\nvariable {i}\n/--\n(Implementation) Show the cone constructed in `build_limit` is limiting, provided the cones used in\nits construction are.\n-/\ndef build_is_limit (t₁ : is_limit c₁) (t₂ : is_limit c₂) (hi : is_limit i) :\n  is_limit (build_limit s t hs ht i) :=\n{ lift := λ q,\n  begin\n    refine hi.lift (fork.of_ι _ _),\n    { refine t₁.lift (fan.mk _ (λ j, _)),\n      apply q.π.app j },\n    { apply t₂.hom_ext,\n      intro j, discrete_cases,\n      simp [hs, ht] },\n  end,\n  uniq' := λ q m w, hi.hom_ext (i.equalizer_ext (t₁.hom_ext\n    (λ j, by { cases j, simpa using w j }))) }\n\nend has_limit_of_has_products_of_has_equalizers\n\nopen has_limit_of_has_products_of_has_equalizers\n\n/--\nGiven the existence of the appropriate (possibly finite) products and equalizers,\nwe can construct a limit cone for `F`.\n(This assumes the existence of all equalizers, which is technically stronger than needed.)\n-/\nnoncomputable\ndef limit_cone_of_equalizer_and_product (F : J ⥤ C)\n  [has_limit (discrete.functor F.obj)]\n  [has_limit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.2))]\n  [has_equalizers C] : limit_cone F :=\n{ cone := _,\n  is_limit :=\n    build_is_limit\n      (pi.lift (λ f, limit.π (discrete.functor F.obj) ⟨_⟩ ≫ F.map f.2))\n      (pi.lift (λ f, limit.π (discrete.functor F.obj) ⟨f.1.2⟩))\n      (by simp)\n      (by simp)\n      (limit.is_limit _)\n      (limit.is_limit _)\n      (limit.is_limit _) }\n\n/--\nGiven the existence of the appropriate (possibly finite) products and equalizers, we know a limit of\n`F` exists.\n(This assumes the existence of all equalizers, which is technically stronger than needed.)\n-/\nlemma has_limit_of_equalizer_and_product (F : J ⥤ C)\n  [has_limit (discrete.functor F.obj)]\n  [has_limit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.2))]\n  [has_equalizers C] : has_limit F :=\nhas_limit.mk (limit_cone_of_equalizer_and_product F)\n\n/-- A limit can be realised as a subobject of a product. -/\nnoncomputable\ndef limit_subobject_product [has_limits_of_size.{w w} C] (F : J ⥤ C) :\n  limit F ⟶ ∏ (λ j, F.obj j) :=\n(limit.iso_limit_cone (limit_cone_of_equalizer_and_product F)).hom ≫ equalizer.ι _ _\n\ninstance limit_subobject_product_mono [has_limits_of_size.{w w} C] (F : J ⥤ C) :\n  mono (limit_subobject_product F) :=\nmono_comp _ _\n\n/--\nAny category with products and equalizers has all limits.\n\nSee <https://stacks.math.columbia.edu/tag/002N>.\n-/\nlemma has_limits_of_has_equalizers_and_products\n  [has_products.{w} C] [has_equalizers C] : has_limits_of_size.{w w} C :=\n{ has_limits_of_shape := λ J 𝒥,\n  { has_limit := λ F, by exactI has_limit_of_equalizer_and_product F } }\n\n/--\nAny category with finite products and equalizers has all finite limits.\n\nSee <https://stacks.math.columbia.edu/tag/002O>.\n-/\nlemma has_finite_limits_of_has_equalizers_and_finite_products\n  [has_finite_products C] [has_equalizers C] : has_finite_limits C :=\n⟨λ J _ _, { has_limit := λ F, by exactI has_limit_of_equalizer_and_product F }⟩\n\nvariables {D : Type u₂} [category.{v₂} D]\nnoncomputable theory\n\nsection\n\nvariables [has_limits_of_shape (discrete J) C]\n          [has_limits_of_shape (discrete (Σ p : J × J, p.1 ⟶ p.2)) C]\n          [has_equalizers C]\nvariables (G : C ⥤ D)\n          [preserves_limits_of_shape walking_parallel_pair G]\n          [preserves_limits_of_shape (discrete.{w} J) G]\n          [preserves_limits_of_shape (discrete.{w} (Σ p : J × J, p.1 ⟶ p.2)) G]\n\n/-- If a functor preserves equalizers and the appropriate products, it preserves limits. -/\ndef preserves_limit_of_preserves_equalizers_and_product :\n  preserves_limits_of_shape J G :=\n{ preserves_limit := λ K,\n  begin\n    let P := ∏ K.obj,\n    let Q := ∏ (λ (f : (Σ (p : J × J), p.fst ⟶ p.snd)), K.obj f.1.2),\n    let s : P ⟶ Q := pi.lift (λ f, limit.π (discrete.functor K.obj) ⟨_⟩ ≫ K.map f.2),\n    let t : P ⟶ Q := pi.lift (λ f, limit.π (discrete.functor K.obj) ⟨f.1.2⟩),\n    let I := equalizer s t,\n    let i : I ⟶ P := equalizer.ι s t,\n    apply preserves_limit_of_preserves_limit_cone\n      (build_is_limit s t (by simp) (by simp)\n        (limit.is_limit _)\n        (limit.is_limit _)\n        (limit.is_limit _)),\n    refine is_limit.of_iso_limit (build_is_limit _ _ _ _ _ _ _) _,\n    { exact fan.mk _ (λ j, G.map (pi.π _ j)) },\n    { exact fan.mk (G.obj Q) (λ f, G.map (pi.π _ f)) },\n    { apply G.map s },\n    { apply G.map t },\n    { intro f,\n      dsimp,\n      simp only [←G.map_comp, limit.lift_π, fan.mk_π_app] },\n    { intro f,\n      dsimp,\n      simp only [←G.map_comp, limit.lift_π, fan.mk_π_app] },\n    { apply fork.of_ι (G.map i) _,\n      simp only [← G.map_comp, equalizer.condition] },\n    { apply is_limit_of_has_product_of_preserves_limit },\n    { apply is_limit_of_has_product_of_preserves_limit },\n    { apply is_limit_fork_map_of_is_limit,\n      apply equalizer_is_equalizer },\n    refine cones.ext (iso.refl _) _,\n    intro j,\n    dsimp,\n    simp, -- See note [dsimp, simp].\n  end }\nend\n\n/-- If G preserves equalizers and finite products, it preserves finite limits. -/\ndef preserves_finite_limits_of_preserves_equalizers_and_finite_products\n  [has_equalizers C] [has_finite_products C]\n  (G : C ⥤ D) [preserves_limits_of_shape walking_parallel_pair G]\n  [∀ (J : Type) [fintype J], preserves_limits_of_shape (discrete J) G] :\n  preserves_finite_limits G :=\n⟨λ _ _ _, by exactI preserves_limit_of_preserves_equalizers_and_product G⟩\n\n/-- If G preserves equalizers and products, it preserves all limits. -/\ndef preserves_limits_of_preserves_equalizers_and_products\n  [has_equalizers C] [has_products.{w} C]\n  (G : C ⥤ D) [preserves_limits_of_shape walking_parallel_pair G]\n  [∀ J, preserves_limits_of_shape (discrete.{w} J) G] :\npreserves_limits_of_size.{w w} G :=\n{ preserves_limits_of_shape := λ J 𝒥,\n  by exactI preserves_limit_of_preserves_equalizers_and_product G }\n\nlemma has_finite_limits_of_has_terminal_and_pullbacks [has_terminal C] [has_pullbacks C] :\n  has_finite_limits C :=\n@@has_finite_limits_of_has_equalizers_and_finite_products _\n  (@@has_finite_products_of_has_binary_and_terminal _\n    (has_binary_products_of_has_terminal_and_pullbacks C) infer_instance)\n  (@@has_equalizers_of_has_pullbacks_and_binary_products _\n    (has_binary_products_of_has_terminal_and_pullbacks C) infer_instance)\n\n/-- If G preserves terminal objects and pullbacks, it preserves all finite limits. -/\ndef preserves_finite_limits_of_preserves_terminal_and_pullbacks\n  [has_terminal C] [has_pullbacks C] (G : C ⥤ D)\n  [preserves_limits_of_shape (discrete.{0} pempty) G]\n  [preserves_limits_of_shape walking_cospan G] :\npreserves_finite_limits G :=\nbegin\n  haveI : has_finite_limits C := has_finite_limits_of_has_terminal_and_pullbacks,\n  haveI : preserves_limits_of_shape (discrete walking_pair) G :=\n    preserves_binary_products_of_preserves_terminal_and_pullbacks G,\n  exact @@preserves_finite_limits_of_preserves_equalizers_and_finite_products _ _ _ _ G\n    (preserves_equalizers_of_preserves_pullbacks_and_binary_products G)\n    (preserves_finite_products_of_preserves_binary_and_terminal G),\nend\n\n/-!\nWe now dualize the above constructions, resorting to copy-paste.\n-/\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_colimit_of_has_coproducts_of_has_coequalizers\n\nvariables {F : J ⥤ C}\n          {c₁ : cofan (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.1)}\n          {c₂ : cofan F.obj}\n          (s t : c₁.X ⟶ c₂.X)\n          (hs : ∀ (f : Σ p : J × J, p.1 ⟶ p.2), c₁.ι.app ⟨f⟩ ≫ s = F.map f.2 ≫ c₂.ι.app ⟨f.1.2⟩)\n          (ht : ∀ (f : Σ p : J × J, p.1 ⟶ p.2), c₁.ι.app ⟨f⟩ ≫ t = c₂.ι.app ⟨f.1.1⟩)\n          (i : cofork s t)\n\ninclude hs ht\n/--\n(Implementation) Given the appropriate coproduct and coequalizer cocones,\nbuild the cocone for `F` which is colimiting if the given cocones are also.\n-/\n@[simps]\ndef build_colimit : cocone F :=\n{ X := i.X,\n  ι :=\n  { app := λ j, c₂.ι.app ⟨_⟩ ≫ i.π,\n    naturality' := λ j₁ j₂ f, begin\n      dsimp,\n      rw [category.comp_id, ←reassoc_of (hs ⟨⟨_, _⟩, f⟩), i.condition, ←category.assoc, ht],\n    end} }\n\nvariable {i}\n/--\n(Implementation) Show the cocone constructed in `build_colimit` is colimiting,\nprovided the cocones used in its construction are.\n-/\ndef build_is_colimit (t₁ : is_colimit c₁) (t₂ : is_colimit c₂) (hi : is_colimit i) :\n  is_colimit (build_colimit s t hs ht i) :=\n{ desc := λ q,\n  begin\n    refine hi.desc (cofork.of_π _ _),\n    { refine t₂.desc (cofan.mk _ (λ j, _)),\n      apply q.ι.app j },\n    { apply t₁.hom_ext,\n      intro j, discrete_cases,\n      simp [reassoc_of hs, reassoc_of ht] },\n  end,\n  uniq' := λ q m w, hi.hom_ext (i.coequalizer_ext (t₂.hom_ext\n    (λ j, by { cases j, simpa using w j }))) }\n\nend has_colimit_of_has_coproducts_of_has_coequalizers\n\nopen has_colimit_of_has_coproducts_of_has_coequalizers\n\n/--\nGiven the existence of the appropriate (possibly finite) coproducts and coequalizers,\nwe can construct a colimit cocone for `F`.\n(This assumes the existence of all coequalizers, which is technically stronger than needed.)\n-/\nnoncomputable\ndef colimit_cocone_of_coequalizer_and_coproduct (F : J ⥤ C)\n  [has_colimit (discrete.functor F.obj)]\n  [has_colimit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.1))]\n  [has_coequalizers C] : colimit_cocone F :=\n{ cocone := _,\n  is_colimit :=\n    build_is_colimit\n      (sigma.desc (λ f, F.map f.2 ≫ colimit.ι (discrete.functor F.obj) ⟨f.1.2⟩))\n      (sigma.desc (λ f, colimit.ι (discrete.functor F.obj) ⟨f.1.1⟩))\n      (by simp)\n      (by simp)\n      (colimit.is_colimit _)\n      (colimit.is_colimit _)\n      (colimit.is_colimit _) }\n\n\n/--\nGiven the existence of the appropriate (possibly finite) coproducts and coequalizers,\nwe know a colimit of `F` exists.\n(This assumes the existence of all coequalizers, which is technically stronger than needed.)\n-/\nlemma has_colimit_of_coequalizer_and_coproduct (F : J ⥤ C)\n  [has_colimit (discrete.functor F.obj)]\n  [has_colimit (discrete.functor (λ f : (Σ p : J × J, p.1 ⟶ p.2), F.obj f.1.1))]\n  [has_coequalizers C] : has_colimit F :=\nhas_colimit.mk (colimit_cocone_of_coequalizer_and_coproduct F)\n\n/-- A colimit can be realised as a quotient of a coproduct. -/\nnoncomputable\ndef colimit_quotient_coproduct [has_colimits_of_size.{w w} C] (F : J ⥤ C) :\n  ∐ (λ j, F.obj j) ⟶ colimit F :=\ncoequalizer.π _ _ ≫ (colimit.iso_colimit_cocone (colimit_cocone_of_coequalizer_and_coproduct F)).inv\n\ninstance colimit_quotient_coproduct_epi [has_colimits_of_size.{w w} C] (F : J ⥤ C) :\n  epi (colimit_quotient_coproduct F) :=\nepi_comp _ _\n\n/--\nAny category with coproducts and coequalizers has all colimits.\n\nSee <https://stacks.math.columbia.edu/tag/002P>.\n-/\nlemma has_colimits_of_has_coequalizers_and_coproducts\n  [has_coproducts.{w} C] [has_coequalizers C] : has_colimits_of_size.{w w} C :=\n{ has_colimits_of_shape := λ J 𝒥,\n  { has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F } }\n\n/--\nAny category with finite coproducts and coequalizers has all finite colimits.\n\nSee <https://stacks.math.columbia.edu/tag/002Q>.\n-/\nlemma has_finite_colimits_of_has_coequalizers_and_finite_coproducts\n  [has_finite_coproducts C] [has_coequalizers C] : has_finite_colimits C :=\n⟨λ J _ _, { has_colimit := λ F, by exactI has_colimit_of_coequalizer_and_coproduct F }⟩\n\nnoncomputable theory\n\nsection\n\nvariables [has_colimits_of_shape (discrete.{w} J) C]\n          [has_colimits_of_shape (discrete.{w} (Σ p : J × J, p.1 ⟶ p.2)) C]\n          [has_coequalizers C]\nvariables (G : C ⥤ D)\n          [preserves_colimits_of_shape walking_parallel_pair G]\n          [preserves_colimits_of_shape (discrete.{w} J) G]\n          [preserves_colimits_of_shape (discrete.{w} (Σ p : J × J, p.1 ⟶ p.2)) G]\n\n/-- If a functor preserves coequalizers and the appropriate coproducts, it preserves colimits. -/\ndef preserves_colimit_of_preserves_coequalizers_and_coproduct :\n  preserves_colimits_of_shape J G :=\n{ preserves_colimit := λ K,\n  begin\n    let P := ∐ K.obj,\n    let Q := ∐ (λ (f : (Σ (p : J × J), p.fst ⟶ p.snd)), K.obj f.1.1),\n    let s : Q ⟶ P := sigma.desc (λ f, K.map f.2 ≫ colimit.ι (discrete.functor K.obj) ⟨_⟩),\n    let t : Q ⟶ P := sigma.desc (λ f, colimit.ι (discrete.functor K.obj) ⟨f.1.1⟩),\n    let I := coequalizer s t,\n    let i : P ⟶ I := coequalizer.π s t,\n    apply preserves_colimit_of_preserves_colimit_cocone\n      (build_is_colimit s t (by simp) (by simp)\n        (colimit.is_colimit _)\n        (colimit.is_colimit _)\n        (colimit.is_colimit _)),\n    refine is_colimit.of_iso_colimit (build_is_colimit _ _ _ _ _ _ _) _,\n    { exact cofan.mk (G.obj Q) (λ j, G.map (sigma.ι _ j)) },\n    { exact cofan.mk _ (λ f, G.map (sigma.ι _ f)) },\n    { apply G.map s },\n    { apply G.map t },\n    { intro f,\n      dsimp,\n      simp only [←G.map_comp, colimit.ι_desc, cofan.mk_ι_app] },\n    { intro f,\n      dsimp,\n      simp only [←G.map_comp, colimit.ι_desc, cofan.mk_ι_app] },\n    { apply cofork.of_π (G.map i) _,\n      simp only [← G.map_comp, coequalizer.condition] },\n    { apply is_colimit_of_has_coproduct_of_preserves_colimit },\n    { apply is_colimit_of_has_coproduct_of_preserves_colimit },\n    { apply is_colimit_cofork_map_of_is_colimit,\n      apply coequalizer_is_coequalizer },\n    refine cocones.ext (iso.refl _) _,\n    intro j,\n    dsimp,\n    simp, -- See note [dsimp, simp].\n  end }\nend\n\n/-- If G preserves coequalizers and finite coproducts, it preserves finite colimits. -/\ndef preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts\n  [has_coequalizers C] [has_finite_coproducts C]\n  (G : C ⥤ D) [preserves_colimits_of_shape walking_parallel_pair G]\n  [∀ J [fintype J], preserves_colimits_of_shape (discrete.{0} J) G] :\n  preserves_finite_colimits G :=\n⟨λ _ _ _, by exactI preserves_colimit_of_preserves_coequalizers_and_coproduct G⟩\n\n/-- If G preserves coequalizers and coproducts, it preserves all colimits. -/\ndef preserves_colimits_of_preserves_coequalizers_and_coproducts\n  [has_coequalizers C] [has_coproducts.{w} C]\n  (G : C ⥤ D) [preserves_colimits_of_shape walking_parallel_pair G]\n  [∀ J, preserves_colimits_of_shape (discrete.{w} J) G] :\npreserves_colimits_of_size.{w} G :=\n{ preserves_colimits_of_shape := λ J 𝒥,\n  by exactI preserves_colimit_of_preserves_coequalizers_and_coproduct G }\n\nlemma has_finite_colimits_of_has_initial_and_pushouts [has_initial C] [has_pushouts C] :\n  has_finite_colimits C :=\n@@has_finite_colimits_of_has_coequalizers_and_finite_coproducts _\n  (@@has_finite_coproducts_of_has_binary_and_initial _\n    (has_binary_coproducts_of_has_initial_and_pushouts C) infer_instance)\n  (@@has_coequalizers_of_has_pushouts_and_binary_coproducts _\n    (has_binary_coproducts_of_has_initial_and_pushouts C) infer_instance)\n\n/-- If G preserves initial objects and pushouts, it preserves all finite colimits. -/\ndef preserves_finite_colimits_of_preserves_initial_and_pushouts\n  [has_initial C] [has_pushouts C] (G : C ⥤ D)\n  [preserves_colimits_of_shape (discrete.{0} pempty) G]\n  [preserves_colimits_of_shape walking_span G] :\npreserves_finite_colimits G :=\nbegin\n  haveI : has_finite_colimits C := has_finite_colimits_of_has_initial_and_pushouts,\n  haveI : preserves_colimits_of_shape (discrete walking_pair) G :=\n    preserves_binary_coproducts_of_preserves_initial_and_pushouts G,\n  exact @@preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts _ _ _ _ G\n    (preserves_coequalizers_of_preserves_pushouts_and_binary_coproducts G)\n    (preserves_finite_coproducts_of_preserves_binary_and_initial G),\nend\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.30529552197896065}}
{"text": "/-\n  Copyright (c) 2022 Arthur Paulino. All rights reserved.\n  Released under Apache 2.0 license as described in the file LICENSE.\n  Authors: Arthur Paulino\n-/\n\nimport FxyLang.Reasoning.Defs\n\ntheorem State.doneLoop : done c k v^[n] = done c k v := by\n  induction n with\n  | zero      => rw [stepN]\n  | succ _ hi => rw [stepN, step]; exact hi\n\ntheorem State.errorLoop : error c k t v^[n] = error c k t v := by\n  induction n with\n  | zero      => rw [stepN]\n  | succ _ hi => rw [stepN, step]; exact hi\n\nopen Lean.Parser.Tactic in\nsyntax \"step_induction \" ident (\" using \" term,+)?\n  (\" with \" simpLemma)? : tactic\n\nopen Lean.Elab.Tactic in\nset_option hygiene false in\nelab_rules : tactic\n  | `(tactic| step_induction $hi $[using $ts,*]? $[with $h]?) => do\n    match ts with\n    | none    => pure ()\n    | some ts => evalTactic $ ← `(tactic| have $hi:ident := @$hi:ident $ts*)\n    evalTactic $ ← `(tactic| cases $hi:ident with | intro n $hi:ident => ?_)\n    match h with\n    | none   => evalTactic $\n      ← `(tactic| exact ⟨n + 1, by simp only [stepN, step]; exact $hi:ident⟩)\n    | some h => evalTactic $\n      ← `(tactic| exact ⟨n + 1, by simp only [stepN, step, $h];exact $hi:ident⟩)\n\ntheorem State.retProgression :\n    ∃ n, (ret c k v^[n]).isEnd ∨ (ret c k v^[n]).isProg := by\n  induction k generalizing c v with\n  | exit => exact ⟨1, by simp [stepN, step, isEnd]⟩\n  | seq  => exact ⟨1, by simp [stepN, step, isProg]⟩\n  | decl nm _ hi => step_induction hi using (c.insert nm v), .nil\n  | fork =>\n    cases v with\n    | lit l =>\n      cases l with\n      | bool b => cases b <;> exact ⟨1, by simp [stepN, step, isProg]⟩\n      | _ => exact ⟨1, by simp [stepN, step, isEnd]⟩\n    | _ => exact ⟨1, by simp [stepN, step, isEnd]⟩\n  | loop _ _ _ hi =>\n    cases v with\n    | lit l =>\n      cases l with\n      | bool b =>\n        cases b with\n        | true  => exact ⟨1, by simp [stepN, step, isProg]⟩\n        | false => step_induction hi using c, .nil\n      | _ => exact ⟨1, by simp [stepN, step, isEnd]⟩\n    | _ => exact ⟨1, by simp [stepN, step, isEnd]⟩\n  | unOp o _ hi =>\n    cases h : v.unOp o with\n    | error => exact ⟨1, by simp [stepN, step, h, isEnd]⟩\n    | ok v' => step_induction hi using c, v' with h\n  | block c' _ hi => step_induction hi using c', v\n  | print _ hi => step_induction hi using c, .nil with dbgTrace\n  | binOp₂ o v₂ _ hi =>\n    cases h : v₂.binOp v o with\n    | error => exact ⟨1, by simp [stepN, step, h, isEnd]⟩\n    | ok v' => step_induction hi using c, v' with h\n  | binOp₁ o e k hi =>\n    sorry\n    -- cases @exprProgression e c (Continuation.binOp₂ o v k) with | intro n h =>\n    -- exact ⟨n + 1, by simp only [stepN, step]; exact h⟩\n  | app e es k hi =>\n    cases v with\n    | lam lm =>\n      cases lm with\n      | mk ns h p =>\n        cases h' : consume p ns es with\n        | none =>\n          exists 1\n          simp only [stepN, step]\n          split\n          next h'' => simp only [h'] at h''\n          next h'' => simp only [h'] at h''\n          simp [isEnd]\n        | some x =>\n          match x with\n          | (some l, p') =>\n            cases @hi c (.lam $ .mk l (noDupOfConsumeNoDup h h') p') with\n            | intro n hi =>\n              exists n + 1\n              simp [stepN, step]\n              split\n              next h'' =>\n                simp [h'] at h''\n                simp [← h'', hi]\n              all_goals { next h'' => simp [h''] at h' }\n          | (none, _) =>\n            exists 1\n            simp only [stepN, step]\n            split\n            next h'' => simp [h'] at h''\n            simp [isProg]\n            simp [isEnd]\n    | _ => exact ⟨1, by simp [stepN, step, isEnd]⟩\n\nopen Lean.Parser.Tactic in\nsyntax \"step_ret \" num \" using \" term \", \" term \", \" term\n  (\" with \" simpLemma)? : tactic\n\nopen Lean.Elab.Tactic in\nset_option hygiene false in\nelab_rules : tactic\n  | `(tactic| step_ret $n using $v, $c, $k $[with $h]?) => do\n    evalTactic $\n      ← `(tactic| cases @retProgression $v $c $k with | intro n_ h_ => ?_)\n    match h with\n    | none => evalTactic $\n      ← `(tactic| exact ⟨n_ + $n, by simp only [stepN, step]; exact h_⟩)\n    | some h => evalTactic $\n      ← `(tactic| exact ⟨n_ + $n, by simp only [stepN, step, $h]; exact h_⟩)\n\ntheorem State.exprProgression :\n    ∃ n, (expr c k e^[n]).isEnd ∨ (expr c k e^[n]).isProg := by\n  cases e with\n  | lit l =>\n    cases k with\n    | exit => exact ⟨2, by simp [stepN, step, isEnd]⟩\n    | seq  => exact ⟨2, by simp [stepN, step, isProg]⟩\n    | decl nm k => step_ret 2 using c.insert nm (.lit l), k, .nil\n    | print k => step_ret 2 using c, k, .nil\n    | fork e pT pF k => step_ret 1 using c, .fork e pT pF k, .lit l\n    | loop e p k => step_ret 1 using c, .loop e p k, .lit l\n    | unOp o k => step_ret 1 using c, .unOp o k, .lit l\n    | binOp₁ o e₂ k => step_ret 1 using c, .binOp₁ o e₂ k, .lit l\n    | binOp₂ o v₁ k => step_ret 1 using c, .binOp₂ o v₁ k, .lit l\n    | app e es k => step_ret 1 using c, .app e es k, .lit l\n    | block c' k => step_ret 1 using c, .block c' k, .lit l\n  | list l =>\n    cases k with\n    | exit => exact ⟨2, by simp [stepN, step, isEnd]⟩\n    | seq  => exact ⟨2, by simp [stepN, step, isProg]⟩\n    | decl nm k => step_ret 2 using c.insert nm (.list l), k, .nil\n    | print k => step_ret 2 using c, k, .nil\n    | fork e pT pF k => step_ret 1 using c, .fork e pT pF k, .list l\n    | loop e p k => step_ret 1 using c, .loop e p k, .list l\n    | unOp o k => step_ret 1 using c, .unOp o k, .list l\n    | binOp₁ o e₂ k => step_ret 1 using c, .binOp₁ o e₂ k, .list l\n    | binOp₂ o v₁ k => step_ret 1 using c, .binOp₂ o v₁ k, .list l\n    | app e es k => step_ret 1 using c, .app e es k, .list l\n    | block c' k => step_ret 1 using c, .block c' k, .list l\n  | var nm =>\n    cases h' : c[nm] with\n    | none => exact ⟨1, by simp [stepN, step, h', isEnd]⟩\n    | some v =>\n      cases k with\n      | exit => exact ⟨2, by simp [stepN, step, h', isEnd]⟩\n      | seq  => exact ⟨2, by simp [stepN, step, h', isProg]⟩\n      | decl nm k => step_ret 2 using c.insert nm v, k, .nil with h'\n      | print k => step_ret 2 using c, k, .nil with h'\n      | fork e pT pF k => step_ret 1 using c, .fork e pT pF k, v with h'\n      | loop e p k => step_ret 1 using c, .loop e p k, v with h'\n      | unOp o k => step_ret 1 using c, .unOp o k, v with h'\n      | binOp₁ o e₂ k => step_ret 1 using c, .binOp₁ o e₂ k, v with h'\n      | binOp₂ o v₁ k => step_ret 1 using c, .binOp₂ o v₁ k, v with h'\n      | app e es k => step_ret 1 using c, .app e es k, v with h'\n      | block c' k => step_ret 1 using c, .block c' k, v with h'\n  | lam l =>\n    cases k with\n    | exit => exact ⟨2, by simp [stepN, step, isEnd]⟩\n    | seq  => exact ⟨2, by simp [stepN, step, isProg]⟩\n    | decl nm k => step_ret 2 using c.insert nm (.lam l), k, .nil\n    | print k => step_ret 2 using c, k, .nil\n    | fork e pT pF k => step_ret 1 using c, .fork e pT pF k, .lam l\n    | loop e p k => step_ret 1 using c, .loop e p k, .lam l\n    | unOp o k => step_ret 1 using c, .unOp o k, .lam l\n    | binOp₁ o e₂ k => step_ret 1 using c, .binOp₁ o e₂ k, .lam l\n    | binOp₂ o v₁ k => step_ret 1 using c, .binOp₂ o v₁ k, .lam l\n    | app e es k => step_ret 1 using c, .app e es k, .lam l\n    | block c' k => step_ret 1 using c, .block c' k, .lam l\n  | app e es =>\n    sorry\n    -- cases @exprProgression e c (.app e es k) with | intro n h =>\n    -- exact ⟨n + 1, by rw [stepN, step]; exact h⟩\n  | unOp o e =>\n    sorry\n    -- cases @exprProgression e c (.unOp o e k) with | intro n h =>\n    -- exact ⟨n + 1, by rw [stepN, step]; exact h⟩\n  | binOp o e₁ e₂ =>\n    sorry\n    -- cases @exprProgression e₁ c (.binOp₁ o e₂ k) with | intro n h =>\n    -- exact ⟨n + 1, by rw [stepN, step]; exact h⟩\n\ntheorem State.Progression : ∃ n, (s^[n]).isEnd ∨ (s^[n]).isProg := by\n  cases s with\n  | prog  => exact ⟨0, by simp [stepN, isProg]⟩\n  | done  => exact ⟨0, by simp [stepN,  isEnd]⟩\n  | error => exact ⟨0, by simp [stepN,  isEnd]⟩\n  | ret  v c k => exact retProgression\n  | expr e c k => exact exprProgression\n", "meta": {"author": "arthurpaulino", "repo": "FxyLang", "sha": "fe1c1df2af522bb3f5c7f4e5b895715e27671df0", "save_path": "github-repos/lean/arthurpaulino-FxyLang", "path": "github-repos/lean/arthurpaulino-FxyLang/FxyLang-fe1c1df2af522bb3f5c7f4e5b895715e27671df0/FxyLang/Reasoning/Progression.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.30529552197896065}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (h1 : ¬ (α ∈ ℚ)) :\nlet subset_of_unit : set ℝ := {a | ∃ a', (a : ℝ) = a' % 1 ∧  a' ∈ ℤ} in\nlet dense_subset : set ℝ → Prop := λ (S : set ℝ), ∀ x, x ∈ S → ∃ y, y ∈ S ∧ |y - x| < (1 : ℝ) in\nsubset_of_unit ⊆ (Icc 0 1) ∧ dense_subset subset_of_unit :=\nbegin\n  have h2 : ∀ (i j : ℤ), i ≠ j → {i * α % 1} ≠ {j * α % 1},\n  from sorry,\n\n  have h3 : ∀ (i j : ℤ), i ≠ j → {i * α % 1} ≠ {j * α % 1},\n  from sorry,\n\n  have h4 : ∀ (i j : ℤ), i ≠ j → {i * α % 1} ≠ {j * α % 1},\n  from sorry,\n  have h5 : {a | ∃ a', (a : ℝ) = a' % 1 ∧ a' ∈ ℤ} ⊆ (Icc 0 1), \n  from sorry,\n  have h6 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h7 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h8 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h9 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h10 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h11 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h12 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h13 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h14 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h15 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h16 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h17 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h18 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h19 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h20 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h21 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h22 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h23 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h24 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h25 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h26 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h27 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h28 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h29 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h30 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h31 : ∀ (S : set ℝ), dense_subset S → ∀ x, x ∈ S → ∃ y : ℝ, y ∈ S ∧ |y - x| < (1 : ℝ),\n  from sorry,\n  have h32 : ∀ (S : set ℝ), dense_subset S → ∀ x\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit : ∀ (α : ℝ) (α_irrational : ¬ ∃ (i j : ℤ), α = i / j), ∃ (S : set ℝ) (S_infinite : infinite S) (S_dense : is_dense S),\n(∀ (n : ℤ), ∃ (a : ℝ), a ∈ S ∧ a = n / α) ∧\n(∀ (a : ℝ), a ∈ S → ∀ (b : ℝ), b ∈ S → ∀ (n : ℤ), ∃ (a_b : ℝ), a_b ∈ S ∧ a_b = |a - b| ∧ a_b = n / α) :=\nbegin\n  assume (α : ℝ) (α_irrational : ¬ ∃ (i j : ℤ), α = i / j),\n  let S := {x : ℝ | ∃ (i : ℤ), x = i / α},\n  have S_infinite : infinite S, from sorry,\n  have S_dense : is_dense S, from sorry,\n  have S_nonempty : nonempty S, from sorry,\n  have S_in_unit_interval : ∀ (a : ℝ), a ∈ S → ∃ (i : ℤ), a = i / α ∧ a ∈ (Icc (0 : ℝ) 1), from sorry,\n  have S_bounded : bounded S, from sorry,\n\n  have h1 : ∀ (a b : ℝ), a ∈ S ∧ b ∈ S → (∃ (i j : ℤ), a = i / α ∧ b = j / α) ∧ ¬ (a = b), from sorry,\n  have h2 : ∀ (a : ℝ), a ∈ S → (∃ (i : ℤ), a = i / α) ∧ ∀ (b : ℝ), b ∈ S → ¬ (a = b), from sorry,\n  have h3 : ∀ (a : ℝ), a ∈ S → (∃ (i : ℤ), a = i / α) ∧ ∀ (b : ℝ), b ∈ S → |a - b| ∈ S, from sorry,\n\n  use S,\n  have h4 : (∀ (n : ℤ), ∃ (a : ℝ), a ∈ S ∧ a = n / α) ∧\n    (∀ (a : ℝ), a ∈ S → ∀ (b : ℝ), b ∈ S → ∀ (n : ℤ), ∃ (a_b : ℝ), a_b ∈ S ∧ a_b = |a - b| ∧ a_b = n / α), from sorry,\n\n  show (∀ (n : ℤ), ∃ (a : ℝ), a ∈ S ∧ a = n / α) ∧\n    (∀ (a : ℝ), a ∈ S → ∀ (b : ℝ), b ∈ S → ∀ (n : ℤ), ∃ (a_b : ℝ), a_b ∈ S ∧ a_b = |a - b| ∧ a_b = n / α), from sorry,\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) (h1 : irrational α) : ∃ S, dense S ∧ S ⊆ {x : ℝ | 0 ≤ x ∧ x ≤ 1} :=\nbegin\n  have h2 : ∀ (i : ℤ) (j : ℤ), i ≠ j → ¬(i * α) = (j * α), \n  from sorry,\n  have h3 : ∃ S, infinite S ∧ S ⊆ {x : ℝ | 0 ≤ x ∧ x ≤ 1}, \n  from sorry,\n  have h4 : ∃ S, dense S ∧ S ⊆ {x : ℝ | 0 ≤ x ∧ x ≤ 1}, \n  from sorry,\n\n  show ∃ S, dense S ∧ S ⊆ {x : ℝ | 0 ≤ x ∧ x ≤ 1}, from sorry,\nend\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_irrational_orbit (α : ℝ) (hα : α ∉ ℚ) : \n∀ ε > 0, ∃ N ∈ ℤ, ∀ i > N, ∃ j ∈ ℤ, |(i*α) - j| < ε :=\nbegin\n  assume (ε : ℝ) (hε : ε > 0),\n  have h1 : ∃ N ∈ ℤ, ∀ i > N, ∃ j ∈ ℤ, |i*α - j| < ε, from sorry,\n  show ∃ N ∈ ℤ, ∀ i > N, ∃ j ∈ ℤ, |(i*α) - j| < ε, from sorry,\nend\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit {α : Type*} [linear_ordered_field α] (α_irr : ¬ is_rat α) :\n  ∀ (a : α), \n  let \n    S : set α := {(n : ℤ) ↦ n * a | n : ℤ}\n  in \n  ∀ (y : α), ∀ (ε : α), ε > 0 → ∃ (x : α), x ∈ S ∧ abs(y - x) < ε :=\nbegin\n  assume (a : α) (y : α) (ε : α),\n  assume (h1 : ε > 0),\n  have h2 : ∀ (a b : ℤ), a ≠ b → ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b, from by {\n    assume (a b : ℤ) (h2 : a ≠ b),\n    have h3 : (a - b) ≠ 0, from sorry,\n    have h4 : ∃ (n : ℤ), (n : α) * (a - b) ≠ 0, from by {\n      have h5 : ∃ (n : ℤ), (n : α) * (a - b) = 1, from sorry,\n      have h6 : ∃ (n : ℤ), (n : α) * (a - b) ≠ 0, from sorry,\n      show ∃ (n : ℤ), (n : α) * (a - b) ≠ 0, from sorry,\n    },\n    cases h4 with (n : ℤ) (h4 : (n : α) * (a - b) ≠ 0),\n    use n,\n    show (n : α) * a ≠ (n : α) * b, from sorry,\n  },\n  have h3 : ∀ (a b : ℤ), a ≠ b → ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S, from by {\n    assume (a b : ℤ) (h3 : a ≠ b),\n    have h4 : ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b, from sorry,\n    cases h4 with (n : ℤ) (h4 : (n : α) * a ≠ (n : α) * b),\n    use n,\n    show (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S, from sorry,\n  },\n  have h4 : ∀ (a b : ℤ), a ≠ b → ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε, from by {\n    assume (a b : ℤ) (h4 : a ≠ b),\n    have h5 : ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S, from sorry,\n    cases h5 with (n : ℤ) (h5 : (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S),\n    have h6 : ∃ (n : ℤ), abs((n : α) * a - (n : α) * b) < ε, from sorry,\n    cases h6 with (n : ℤ) (h6 : abs((n : α) * a - (n : α) * b) < ε),\n    use n,\n    show (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε, from sorry,\n  },\n  have h5 : ∀ (a b : ℤ), a ≠ b → ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε ∧ abs((n : α) * a - y) < ε, from by {\n    assume (a b : ℤ) (h5 : a ≠ b),\n    have h6 : ∃ (n : ℤ), (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε, from sorry,\n    cases h6 with (n : ℤ) (h6 : (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε),\n    have h7 : ∃ (n : ℤ), abs((n : α) * a - y) < ε, from sorry,\n    cases h7 with (n : ℤ) (h7 : abs((n : α) * a - y) < ε),\n    use n,\n    show (n : α) * a ≠ (n : α) * b ∧ (n : α) * a ∈ S ∧ (n : α) * b ∈ S ∧ abs((n : α) * a - (n : α) * b) < ε ∧ abs((n : α) * a - y) < ε, from sorry,\n  },\n  show ∃ (x : α), x ∈ S ∧ abs(y - x) < ε, from sorry,\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (h : α ∉ ℚ) : ∃ S : set ℝ,\n  (∀ (i j : ℤ), i ≠ j → {i • α} ≠ {j • α}) ∧\n  ∀ (y : ℝ), y ∈ [0,1] → ∃ (x : ℝ), x ∈ S ∧ |y - x| < 1 :=\nsorry\n\n\n/--`theorem`\nProduct of two irrational numbers is irrational\n`proof`\nLet $\\alpha$ and $\\beta$ be irrational numbers. We will show that $\\alpha \\beta$ is irrational.\n\nSuppose $\\alpha \\beta \\in \\mathbb{Q}$. Then there exist $m, n \\in \\mathbb{Z}$, $m>0$, $n>0$ such that\n$$\n\\alpha \\beta=\\frac{m}{n}.\n$$\n\nWe know that $\\alpha \\notin \\mathbb{Q}$, so $\\frac{m}{n} \\neq \\alpha$. But, since $\\alpha$ is irrational, we must have $\\frac{m}{n}>\\alpha$.\n\nSince $\\frac{m}{n}>\\alpha$ and $\\alpha \\beta=\\frac{m}{n}$, we have $\\alpha \\beta>\\alpha^2$.\n\nMultiplying by $\\alpha$, we get $\\alpha^2 \\beta>\\alpha^3$.\n\nSince $\\alpha \\beta$ is irrational, we have $\\alpha^2 \\beta \\neq \\alpha^3$. But, since $\\alpha^2 \\beta>\\alpha^3$, we must have $\\alpha^2 \\beta>\\alpha^3$.\n\nThus, we have $\\alpha^2 \\beta>\\alpha^3$ and $\\alpha^2 \\beta \\neq \\alpha^3$.\n\nThis is a contradiction, since if $a>b$ and $a \\neq b$, then $a$ is not less than $b$.\n\nHence, $\\alpha \\beta$ must be irrational.\n\nQED\n-/\ntheorem product_of_two_irrationals (α β : ℝ) (h1 : α ∉ ℚ) (h2 : β ∉ ℚ) : α * β ∉ ℚ :=\nsorry\n\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) (h : irrational α) : ∀ x ∈ set.Icc 0 1, ∃ y ∈ set.Icc 0 1, y ≠ x ∧ y ∈ {n : ℤ | n * α} :=\nbegin\n  assume (x : ℝ) (hx : x ∈ set.Icc 0 1),\n  let S := {n : ℤ | n * α},\n  have h1 : ∀ i j : ℤ, (i : ℝ) * α ≠ (j : ℝ) * α, \n  from begin\n    assume (i j : ℤ) (h1 : (i : ℝ) * α = (j : ℝ) * α),\n    have h2 : (i : ℝ) * α - ⌊(i : ℝ) * α⌋ = (j : ℝ) * α - ⌊(j : ℝ) * α⌋, from sorry,\n    have h3 : (i : ℝ) * α - ⌊(i : ℝ) * α⌋ = (j : ℝ) * α - ⌊(j : ℝ) * α⌋, from sorry,\n    have h4 : α = (⌊(i : ℝ) * α⌋ - ⌊(j : ℝ) * α⌋)/(i - j), from sorry,\n    have h5 : α ∈ ℚ, from sorry,\n    have h6 : irrational α, from sorry,\n    have h7 : ¬ (α ∈ ℚ), from sorry,\n    show false, from sorry,\n  end,\n  have h2 : infinite (S : set ℝ), from sorry,\n  have h3 : ∃ x ∈ S, ∀ ε > 0, ∃ y ∈ S, y ≠ x ∧ |y - x| < ε, from sorry,\n  obtain x0 hx0, from h3,\n  have h4 : ∀ ε > 0, ∃ y ∈ S, y ≠ x0 ∧ |y - x0| < ε, from sorry,\n  have h5 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ y ∈ S, y ≠ x0 ∧ ε' < ε, from sorry,\n  have h6 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ y ∈ S, y ≠ x0 ∧ ε' < ε ∧ ε'' < ε, from sorry,\n  have h7 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0, from sorry,\n  have h8 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S, from sorry,\n  have h9 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1, from sorry,\n  have h10 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ≠ x, from sorry,\n  have h11 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ≠ x ∧ (n : ℝ) * α ∈ {n : ℤ | n * α}, from sorry,\n  have h12 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ≠ x ∧ (n : ℝ) * α ∈ {n : ℤ | n * α} ∧ (n : ℝ) * α ∈ set.Icc 0 1, from sorry,\n  have h13 : ∀ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ≠ x ∧ (n : ℝ) * α ∈ {n : ℤ | n * α} ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ∈ S, from sorry,\n  have h14 : ∃ ε > 0, ∃ (ε' : ℝ) > 0, ∃ (ε'' : ℝ) > 0, ∃ n : ℤ, n ≠ 0 ∧ ε' < ε ∧ ε'' < ε ∧ (n : ℝ) * α ≠ x0 ∧ (n : ℝ) * α ∈ S ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ≠ x ∧ (n : ℝ) * α ∈ {n : ℤ | n * α} ∧ (n : ℝ) * α ∈ set.Icc 0 1 ∧ (n : ℝ) * α ∈ S, from sorry,\n  obtain (ε : ℝ) (h15 : ε > 0) (ε' : ℝ) (h16 : ε' > 0) (ε'' : ℝ) (h17 : ε'' > 0) (n : ℤ) (hn : n ≠ 0) (hn0 : (n : ℝ) * α ≠ x0) (hn1 : (n : ℝ) * α ∈ S) (hn2 : (n : ℝ) * α ∈ set.Icc 0 1) (hn3 : (n : ℝ) * α ≠ x) (hn4 : (n : ℝ) * α ∈ {n : ℤ | n * α}) (hn5 : (n : ℝ) * α ∈ set.Icc 0 1) (hn6 : (n :\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=dense_irrational_orbit (α : ℝ) [irrational α] : ∃ S : set ℝ, ∀ x ∈ S, x ∈ Icc 0 1 ∧ ∀ y ∈ S, x ≠ y → x ≠ y ∧ dense S :=\nbegin\n  have h1 : ∀ (i j : ℤ), i ≠ j → {i * α} ≠ {j * α}, from sorry,\n  have h2 : ∀ i : ℤ, {i * α} ∈ Icc 0 1, from sorry,\n  have h3 : ∀ i : ℤ, {i * α} ∈ set.range (λ (i : ℤ), {i * α}), from sorry,\n  have h4 : set.range (λ (i : ℤ), {i * α}) ⊆ Icc 0 1, from sorry,\n  have h5 : dense (set.range (λ (i : ℤ), {i * α})), from sorry,\n  have h6 : set.range (λ (i : ℤ), {i * α}) ≠ ∅, from sorry,\n  \n  show ∃ S : set ℝ, ∀ x ∈ S, x ∈ Icc 0 1 ∧ ∀ y ∈ S, x ≠ y → x ≠ y ∧ dense S,\n  from ⟨set.range (λ (i : ℤ), {i * α}), \n       sorry, \n       sorry, \n       sorry, \n       sorry, \n       sorry⟩,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from sorry,\n  \n  assume (h7 : ε > 0),\n  cases h2 ε h7 with N1 h8,\n  cases h3 ε h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := sorry,\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from sorry,\n\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from sorry,\n\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-4_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.793105951184112, "lm_q2_score": 0.3849121444839335, "lm_q1q2_score": 0.30527611247324643}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Sean Leather\n-/\nimport algebra.free_monoid\nimport algebra.opposites\nimport control.traversable.instances\nimport control.traversable.lemmas\nimport category_theory.category\nimport category_theory.endomorphism\nimport category_theory.types\nimport category_theory.category.Kleisli\n\n/-!\n\n# List folds generalized to `traversable`\n\nInformally, we can think of `foldl` as a special case of `traverse` where we do not care about the\nreconstructed data structure and, in a state monad, we care about the final state.\n\nThe obvious way to define `foldl` would be to use the state monad but it\nis nicer to reason about a more abstract interface with `fold_map` as a\nprimitive and `fold_map_hom` as a defining property.\n\n```\ndef fold_map {α ω} [has_one ω] [has_mul ω] (f : α → ω) : t α → ω := ...\n\nlemma fold_map_hom (α β)\n  [monoid α] [monoid β] (f : α → β) [is_monoid_hom f]\n  (g : γ → α) (x : t γ) :\n  f (fold_map g x) = fold_map (f ∘ g) x :=\n...\n```\n\n`fold_map` uses a monoid ω to accumulate a value for every element of\na data structure and `fold_map_hom` uses a monoid homomorphism to\nsubstitute the monoid used by `fold_map`. The two are sufficient to\ndefine `foldl`, `foldr` and `to_list`. `to_list` permits the\nformulation of specifications in terms of operations on lists.\n\nEach fold function can be defined using a specialized\nmonoid. `to_list` uses a free monoid represented as a list with\nconcatenation while `foldl` uses endofunctions together with function\ncomposition.\n\nThe definition through monoids uses `traverse` together with the\napplicative functor `const m` (where `m` is the monoid). As an\nimplementation, `const` guarantees that no resource is spent on\nreconstructing the structure during traversal.\n\nA special class could be defined for `foldable`, similarly to Haskell,\nbut the author cannot think of instances of `foldable` that are not also\n`traversable`.\n-/\n\nuniverses u v\n\nopen ulift category_theory opposite\n\nnamespace monoid\nvariables {m : Type u → Type u} [monad m]\nvariables {α β : Type u}\n/--\nFor a list, foldl f x [y₀,y₁] reduces as follows:\n\n```\ncalc  foldl f x [y₀,y₁]\n    = foldl f (f x y₀) [y₁]      : rfl\n... = foldl f (f (f x y₀) y₁) [] : rfl\n... = f (f x y₀) y₁              : rfl\n```\nwith\n```\nf : α → β → α\nx : α\n[y₀,y₁] : list β\n```\n\nWe can view the above as a composition of functions:\n```\n... = f (f x y₀) y₁              : rfl\n... = flip f y₁ (flip f y₀ x)    : rfl\n... = (flip f y₁ ∘ flip f y₀) x  : rfl\n```\n\nWe can use traverse and const to construct this composition:\n```\ncalc   const.run (traverse (λ y, const.mk' (flip f y)) [y₀,y₁]) x\n     = const.run ((::) <$> const.mk' (flip f y₀) <*> traverse (λ y, const.mk' (flip f y)) [y₁]) x\n...  = const.run ((::) <$> const.mk' (flip f y₀) <*>\n         ( (::) <$> const.mk' (flip f y₁) <*> traverse (λ y, const.mk' (flip f y)) [] )) x\n...  = const.run ((::) <$> const.mk' (flip f y₀) <*>\n         ( (::) <$> const.mk' (flip f y₁) <*> pure [] )) x\n...  = const.run ( ((::) <$> const.mk' (flip f y₁) <*> pure []) ∘\n         ((::) <$> const.mk' (flip f y₀)) ) x\n...  = const.run ( const.mk' (flip f y₁) ∘ const.mk' (flip f y₀) ) x\n...  = const.run ( flip f y₁ ∘ flip f y₀ ) x\n...  = f (f x y₀) y₁\n```\n\nAnd this is how `const` turns a monoid into an applicative functor and\nhow the monoid of endofunctions define `foldl`.\n-/\n@[reducible] def foldl (α : Type u) : Type u := (End α)ᵒᵖ\ndef foldl.mk (f : α → α) : foldl α := op f\ndef foldl.get (x : foldl α) : α → α := unop x\ndef foldl.of_free_monoid (f : β → α → β) (xs : free_monoid α) : monoid.foldl β :=\nop $ flip (list.foldl f) xs\n\n@[reducible] def foldr (α : Type u) : Type u := End α\ndef foldr.mk (f : α → α) : foldr α := f\ndef foldr.get (x : foldr α) : α → α := x\ndef foldr.of_free_monoid (f : α → β → β) (xs : free_monoid α) : monoid.foldr β :=\nflip (list.foldr f) xs\n\n@[reducible] def mfoldl (m : Type u → Type u) [monad m] (α : Type u) : Type u :=\nopposite $ End $ Kleisli.mk m α\ndef mfoldl.mk (f : α → m α) : mfoldl m α := op f\ndef mfoldl.get (x : mfoldl m α) : α → m α := unop x\ndef mfoldl.of_free_monoid (f : β → α → m β) (xs : free_monoid α) : monoid.mfoldl m β :=\nop $ flip (list.mfoldl f) xs\n\n@[reducible] def mfoldr (m : Type u → Type u) [monad m] (α : Type u) : Type u :=\nEnd $ Kleisli.mk m α\ndef mfoldr.mk (f : α → m α) : mfoldr m α := f\ndef mfoldr.get (x : mfoldr m α) : α → m α := x\ndef mfoldr.of_free_monoid (f : α → β → m β) (xs : free_monoid α) : monoid.mfoldr m β :=\nflip (list.mfoldr f) xs\n\nend monoid\n\nnamespace traversable\nopen monoid functor\n\nsection defs\nvariables {α β : Type u} {t : Type u → Type u} [traversable t]\n\ndef fold_map {α ω} [has_one ω] [has_mul ω] (f : α → ω) : t α → ω :=\ntraverse (const.mk' ∘ f)\n\ndef foldl (f : α → β → α) (x : α) (xs : t β) : α :=\n(fold_map (foldl.mk ∘ flip f) xs).get x\n\ndef foldr (f : α → β → β) (x : β) (xs : t α) : β :=\n(fold_map (foldr.mk ∘ f) xs).get x\n\n/--\nConceptually, `to_list` collects all the elements of a collection\nin a list. This idea is formalized by\n\n  `lemma to_list_spec (x : t α) : to_list x = fold_map free_monoid.mk x`.\n\nThe definition of `to_list` is based on `foldl` and `list.cons` for\nspeed. It is faster than using `fold_map free_monoid.mk` because, by\nusing `foldl` and `list.cons`, each insertion is done in constant\ntime. As a consequence, `to_list` performs in linear.\n\nOn the other hand, `fold_map free_monoid.mk` creates a singleton list\naround each element and concatenates all the resulting lists. In\n`xs ++ ys`, concatenation takes a time proportional to `length xs`. Since\nthe order in which concatenation is evaluated is unspecified, nothing\nprevents each element of the traversable to be appended at the end\n`xs ++ [x]` which would yield a `O(n²)` run time. -/\ndef to_list : t α → list α :=\nlist.reverse ∘ foldl (flip list.cons) []\n\ndef length (xs : t α) : ℕ :=\ndown $ foldl (λ l _, up $ l.down + 1) (up 0) xs\n\nvariables {m : Type u → Type u} [monad m]\n\ndef mfoldl (f : α → β → m α) (x : α) (xs : t β) : m α :=\n(fold_map (mfoldl.mk ∘ flip f) xs).get x\n\ndef mfoldr (f : α → β → m β) (x : β) (xs : t α) : m β :=\n(fold_map (mfoldr.mk ∘ f) xs).get x\n\nend defs\n\nsection applicative_transformation\nvariables {α β γ : Type u}\n\nopen function (hiding const) is_monoid_hom\n\ndef map_fold [monoid α] [monoid β] (f : α → β) [is_monoid_hom f] :\n  applicative_transformation (const α) (const β) :=\n{ app := λ x, f,\n  preserves_seq'  := by { intros, simp only [map_mul f, (<*>)], },\n  preserves_pure' := by { intros, simp only [map_one f, pure] } }\n\ndef free.mk : α → free_monoid α := list.ret\n\ndef free.map (f : α → β) : free_monoid α → free_monoid β := list.map f\n\nlemma free.map_eq_map (f : α → β) (xs : list α) :\n  f <$> xs = free.map f xs := rfl\n\ninstance (f : α → β) : is_monoid_hom (free.map f) :=\n{ map_mul := λ x y,\n    by simp only [free.map, free_monoid.mul_def, list.map_append, free_add_monoid.add_def],\n  map_one := by simp only [free.map, free_monoid.one_def, list.map, free_add_monoid.zero_def] }\n\ninstance fold_foldl (f : β → α → β) :\n  is_monoid_hom (foldl.of_free_monoid f) :=\n{ map_one := rfl,\n  map_mul := by intros; simp only [free_monoid.mul_def, foldl.of_free_monoid, flip, unop_op,\n    list.foldl_append, op_inj_iff]; refl }\n\nlemma foldl.unop_of_free_monoid  (f : β → α → β) (xs : free_monoid α) (a : β) :\n  unop (foldl.of_free_monoid f xs) a = list.foldl f a xs := rfl\n\ninstance fold_foldr (f : α → β → β) :\n  is_monoid_hom (foldr.of_free_monoid f) :=\n{ map_one := rfl,\n  map_mul :=\n    begin\n      intros,\n      simp only [free_monoid.mul_def, foldr.of_free_monoid, list.foldr_append, flip],\n      refl\n    end }\n\nvariables (m : Type u → Type u) [monad m] [is_lawful_monad m]\n\n@[simp]\nlemma mfoldl.unop_of_free_monoid  (f : β → α → m β) (xs : free_monoid α) (a : β) :\n  unop (mfoldl.of_free_monoid f xs) a = list.mfoldl f a xs := rfl\n\ninstance fold_mfoldl (f : β → α → m β) :\n  is_monoid_hom (mfoldl.of_free_monoid f) :=\n{ map_one := rfl,\n  map_mul := by intros; apply unop_injective; ext; apply list.mfoldl_append }\n\ninstance fold_mfoldr (f : α → β → m β) :\n  is_monoid_hom (mfoldr.of_free_monoid f) :=\n{ map_one := rfl,\n  map_mul := by intros; ext; apply list.mfoldr_append }\n\nvariables {t : Type u → Type u} [traversable t] [is_lawful_traversable t]\nopen is_lawful_traversable\n\nlemma fold_map_hom\n  [monoid α] [monoid β] (f : α → β) [is_monoid_hom f]\n  (g : γ → α) (x : t γ) :\n  f (fold_map g x) = fold_map (f ∘ g) x :=\ncalc  f (fold_map g x)\n    = f (traverse (const.mk' ∘ g) x)  : rfl\n... = (map_fold f).app _ (traverse (const.mk' ∘ g) x) : rfl\n... = traverse ((map_fold f).app _ ∘ (const.mk' ∘ g)) x : naturality (map_fold f) _ _\n... = fold_map (f ∘ g) x : rfl\n\nlemma fold_map_hom_free\n  [monoid β] (f : free_monoid α → β) [is_monoid_hom f] (x : t α) :\n  f (fold_map free.mk x) = fold_map (f ∘ free.mk) x :=\nfold_map_hom _ _ x\n\nvariable {m}\n\nlemma fold_mfoldl_cons (f : α → β → m α) (x : β) (y : α) :\n  list.mfoldl f y (free.mk x) = f y x :=\nby simp only [free.mk, list.ret, list.mfoldl, bind_pure]\n\nlemma fold_mfoldr_cons (f : β → α → m α) (x : β) (y : α) :\n  list.mfoldr f y (free.mk x) = f x y :=\nby simp only [free.mk, list.ret, list.mfoldr, pure_bind]\n\nend applicative_transformation\n\nsection equalities\nopen is_lawful_traversable list (cons)\nvariables {α β γ : Type u}\nvariables {t : Type u → Type u} [traversable t] [is_lawful_traversable t]\n\n@[simp]\nlemma foldl.of_free_monoid_comp_free_mk (f : α → β → α) :\n  foldl.of_free_monoid f ∘ free.mk = foldl.mk ∘ flip f := rfl\n\n@[simp]\nlemma foldr.of_free_monoid_comp_free_mk (f : β → α → α) :\n  foldr.of_free_monoid f ∘ free.mk = foldr.mk ∘ f := rfl\n\n@[simp]\nlemma mfoldl.of_free_monoid_comp_free_mk {m} [monad m] [is_lawful_monad m] (f : α → β → m α) :\n  mfoldl.of_free_monoid f ∘ free.mk = mfoldl.mk ∘ flip f :=\nby ext; simp only [(∘), mfoldl.of_free_monoid, mfoldl.mk, flip, fold_mfoldl_cons]; refl\n\n@[simp]\nlemma mfoldr.of_free_monoid_comp_free_mk {m} [monad m] [is_lawful_monad m] (f : β → α → m α) :\n  mfoldr.of_free_monoid f ∘ free.mk = mfoldr.mk ∘ f :=\nby { ext, simp only [(∘), mfoldr.of_free_monoid, mfoldr.mk, flip, fold_mfoldr_cons] }\n\nlemma to_list_spec (xs : t α) :\n  to_list xs = (fold_map free.mk xs : free_monoid _) :=\neq.symm $\ncalc  fold_map free.mk xs\n    = (fold_map free.mk xs).reverse.reverse : by simp only [list.reverse_reverse]\n... = (list.foldr cons [] (fold_map free.mk xs).reverse).reverse\n                 : by simp only [list.foldr_eta]\n... = (unop (foldl.of_free_monoid (flip cons) (fold_map free.mk xs)) []).reverse\n                 : by simp only [flip,list.foldr_reverse,foldl.of_free_monoid, unop_op]\n... = to_list xs : begin\n                     rw fold_map_hom_free (foldl.of_free_monoid (flip cons)),\n                     simp only [to_list, foldl, list.reverse_inj, foldl.get,\n                       foldl.of_free_monoid_comp_free_mk],\n                     all_goals { apply_instance }\n                   end\n\nlemma fold_map_map [monoid γ]  (f : α → β) (g : β → γ) (xs : t α) :\n  fold_map g (f <$> xs) = fold_map (g ∘ f) xs :=\nby simp only [fold_map,traverse_map]\n\nlemma foldl_to_list (f : α → β → α) (xs : t β) (x : α) :\n  foldl f x xs = list.foldl f x (to_list xs) :=\nbegin\n  rw ← foldl.unop_of_free_monoid,\n  simp only [foldl, to_list_spec, fold_map_hom_free (foldl.of_free_monoid f),\n    foldl.of_free_monoid_comp_free_mk, foldl.get]\nend\n\nlemma foldr_to_list (f : α → β → β) (xs : t α) (x : β) :\n  foldr f x xs = list.foldr f x (to_list xs) :=\nbegin\n  change _ = foldr.of_free_monoid _ _ _,\n  simp only [foldr, to_list_spec, fold_map_hom_free (foldr.of_free_monoid f),\n    foldr.of_free_monoid_comp_free_mk, foldr.get]\nend\n\nlemma to_list_map (f : α → β) (xs : t α) :\n  to_list (f <$> xs) = f <$> to_list xs :=\nby simp only [to_list_spec,free.map_eq_map,fold_map_hom (free.map f), fold_map_map]; refl\n\n@[simp] theorem foldl_map (g : β → γ) (f : α → γ → α) (a : α) (l : t β) :\n  foldl f a (g <$> l) = foldl (λ x y, f x (g y)) a l :=\nby simp only [foldl, fold_map_map, (∘), flip]\n\n@[simp] theorem foldr_map (g : β → γ) (f : γ → α → α) (a : α) (l : t β) :\n  foldr f a (g <$> l) = foldr (f ∘ g) a l :=\nby simp only [foldr, fold_map_map, (∘), flip]\n\n@[simp] theorem to_list_eq_self {xs : list α} : to_list xs = xs :=\nbegin\n  simp only [to_list_spec, fold_map, traverse],\n  induction xs,\n  case list.nil { refl },\n  case list.cons : _ _ ih { unfold list.traverse list.ret, rw ih, refl }\nend\n\ntheorem length_to_list {xs : t α} : length xs = list.length (to_list xs) :=\nbegin\n  unfold length,\n  rw foldl_to_list,\n  generalize : to_list xs = ys,\n  let f := λ (n : ℕ) (a : α), n + 1,\n  transitivity list.foldl f 0 ys,\n  { generalize : 0 = n,\n    induction ys with _ _ ih generalizing n,\n    { simp only [list.foldl_nil] },\n    { simp only [list.foldl, ih (n+1)] } },\n  { induction ys with _ tl ih,\n    { simp only [list.length, list.foldl_nil] },\n    { simp only [list.foldl, list.length],\n      rw [← ih],\n      exact tl.foldl_hom (λx, x+1) f f 0 (λ n x, rfl) } }\nend\n\nvariables {m : Type u → Type u} [monad m] [is_lawful_monad m]\n\nsection\nlocal attribute [semireducible] opposite\nlemma mfoldl_to_list {f : α → β → m α} {x : α} {xs : t β} :\n  mfoldl f x xs = list.mfoldl f x (to_list xs) :=\nbegin\n  change _ = unop (mfoldl.of_free_monoid f (to_list xs)) x,\n  simp only [mfoldl, to_list_spec, fold_map_hom_free (mfoldl.of_free_monoid f),\n    mfoldl.of_free_monoid_comp_free_mk, mfoldl.get]\nend\nend\n\nlemma mfoldr_to_list (f : α → β → m β) (x : β) (xs : t α) :\n  mfoldr f x xs = list.mfoldr f x (to_list xs) :=\nbegin\n  change _ = mfoldr.of_free_monoid f (to_list xs) x,\n  simp only [mfoldr, to_list_spec, fold_map_hom_free (mfoldr.of_free_monoid f),\n    mfoldr.of_free_monoid_comp_free_mk, mfoldr.get]\nend\n\n@[simp] theorem mfoldl_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β) :\n  mfoldl f a (g <$> l) = mfoldl (λ x y, f x (g y)) a l :=\nby simp only [mfoldl, fold_map_map, (∘), flip]\n\n@[simp] theorem mfoldr_map (g : β → γ) (f : γ → α → m α) (a : α) (l : t β) :\n  mfoldr f a (g <$> l) = mfoldr (f ∘ g) a l :=\nby simp only [mfoldr, fold_map_map, (∘), flip]\n\nend equalities\n\nend traversable\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/fold.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.30520474453955704}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.computability.primrec\nimport Mathlib.data.nat.psub\nimport Mathlib.data.pfun\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# The partial recursive functions\n\nThe partial recursive functions are defined similarly to the primitive\nrecursive functions, but now all functions are partial, implemented\nusing the `roption` monad, and there is an additional operation, called\nμ-recursion, which performs unbounded minimization.\n\n## References\n\n* [Mario Carneiro, *Formalizing computability theory via partial recursive functions*][carneiro2019]\n-/\n\nnamespace nat\n\n\ndef rfind_x (p : ℕ →. Bool) (H : ∃ (n : ℕ), tt ∈ p n ∧ ∀ (k : ℕ), k < n → roption.dom (p k)) : Subtype fun (n : ℕ) => tt ∈ p n ∧ ∀ (m : ℕ), m < n → false ∈ p m :=\n  (fun\n      (this :\n      (k : ℕ) → (∀ (n : ℕ), n < k → false ∈ p n) → Subtype fun (n : ℕ) => tt ∈ p n ∧ ∀ (m : ℕ), m < n → false ∈ p m) =>\n      this 0 sorry)\n    (well_founded.fix (wf_lbp p H)\n      fun (m : ℕ)\n        (IH :\n        (y : ℕ) →\n          lbp p y m → (∀ (n : ℕ), n < y → false ∈ p n) → Subtype fun (n : ℕ) => tt ∈ p n ∧ ∀ (m : ℕ), m < n → false ∈ p m)\n        (al : ∀ (n : ℕ), n < m → false ∈ p n) =>\n        (fun (_x : Bool) (e : roption.get (p m) sorry = _x) =>\n            bool.cases_on _x (fun (e : roption.get (p m) sorry = false) => IH (m + 1) sorry sorry)\n              (fun (e : roption.get (p m) sorry = tt) => { val := m, property := sorry }) e)\n          (roption.get (p m) sorry) sorry)\n\ndef rfind (p : ℕ →. Bool) : roption ℕ :=\n  roption.mk (∃ (n : ℕ), tt ∈ p n ∧ ∀ (k : ℕ), k < n → roption.dom (p k))\n    fun (h : ∃ (n : ℕ), tt ∈ p n ∧ ∀ (k : ℕ), k < n → roption.dom (p k)) => subtype.val (rfind_x p h)\n\ntheorem rfind_spec {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) : tt ∈ p n :=\n  Exists.snd h ▸ and.left (subtype.property (rfind_x p (Exists.fst h)))\n\ntheorem rfind_min {p : ℕ →. Bool} {n : ℕ} (h : n ∈ rfind p) {m : ℕ} : m < n → false ∈ p m :=\n  Exists.snd h ▸ and.right (subtype.property (rfind_x p (Exists.fst h)))\n\n@[simp] theorem rfind_dom {p : ℕ →. Bool} : roption.dom (rfind p) ↔ ∃ (n : ℕ), tt ∈ p n ∧ ∀ {m : ℕ}, m < n → roption.dom (p m) :=\n  iff.rfl\n\ntheorem rfind_dom' {p : ℕ →. Bool} : roption.dom (rfind p) ↔ ∃ (n : ℕ), tt ∈ p n ∧ ∀ {m : ℕ}, m ≤ n → roption.dom (p m) := sorry\n\n@[simp] theorem mem_rfind {p : ℕ →. Bool} {n : ℕ} : n ∈ rfind p ↔ tt ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m := sorry\n\ntheorem rfind_min' {p : ℕ → Bool} {m : ℕ} (pm : ↥(p m)) : ∃ (n : ℕ), ∃ (H : n ∈ rfind ↑p), n ≤ m := sorry\n\ntheorem rfind_zero_none (p : ℕ →. Bool) (p0 : p 0 = roption.none) : rfind p = roption.none := sorry\n\ndef rfind_opt {α : Type u_1} (f : ℕ → Option α) : roption α :=\n  roption.bind (rfind ↑fun (n : ℕ) => option.is_some (f n)) fun (n : ℕ) => ↑(f n)\n\ntheorem rfind_opt_spec {α : Type u_1} {f : ℕ → Option α} {a : α} (h : a ∈ rfind_opt f) : ∃ (n : ℕ), a ∈ f n := sorry\n\ntheorem rfind_opt_dom {α : Type u_1} {f : ℕ → Option α} : roption.dom (rfind_opt f) ↔ ∃ (n : ℕ), ∃ (a : α), a ∈ f n := sorry\n\ntheorem rfind_opt_mono {α : Type u_1} {f : ℕ → Option α} (H : ∀ {a : α} {m n : ℕ}, m ≤ n → a ∈ f m → a ∈ f n) {a : α} : a ∈ rfind_opt f ↔ ∃ (n : ℕ), a ∈ f n := sorry\n\ninductive partrec : (ℕ →. ℕ) → Prop\nwhere\n| zero : partrec (pure 0)\n| succ : partrec ↑Nat.succ\n| left : partrec ↑fun (n : ℕ) => prod.fst (unpair n)\n| right : partrec ↑fun (n : ℕ) => prod.snd (unpair n)\n| pair : ∀ {f g : ℕ →. ℕ}, partrec f → partrec g → partrec fun (n : ℕ) => mkpair <$> f n <*> g n\n| comp : ∀ {f g : ℕ →. ℕ}, partrec f → partrec g → partrec fun (n : ℕ) => g n >>= f\n| prec : ∀ {f g : ℕ →. ℕ},\n  partrec f →\n    partrec g →\n      partrec\n        (unpaired\n          fun (a n : ℕ) =>\n            elim (f a)\n              (fun (y : ℕ) (IH : roption ℕ) =>\n                do \n                  let i ← IH \n                  g (mkpair a (mkpair y i)))\n              n)\n| rfind : ∀ {f : ℕ →. ℕ},\n  partrec f → partrec fun (a : ℕ) => rfind fun (n : ℕ) => (fun (m : ℕ) => to_bool (m = 0)) <$> f (mkpair a n)\n\nnamespace partrec\n\n\ntheorem of_eq {f : ℕ →. ℕ} {g : ℕ →. ℕ} (hf : partrec f) (H : ∀ (n : ℕ), f n = g n) : partrec g :=\n  funext H ▸ hf\n\ntheorem of_eq_tot {f : ℕ →. ℕ} {g : ℕ → ℕ} (hf : partrec f) (H : ∀ (n : ℕ), g n ∈ f n) : partrec ↑g :=\n  of_eq hf fun (n : ℕ) => iff.mpr roption.eq_some_iff (H n)\n\ntheorem of_primrec {f : ℕ → ℕ} (hf : primrec f) : partrec ↑f := sorry\n\nprotected theorem some : partrec roption.some :=\n  of_primrec primrec.id\n\ntheorem none : partrec fun (n : ℕ) => roption.none := sorry\n\ntheorem prec' {f : ℕ →. ℕ} {g : ℕ →. ℕ} {h : ℕ →. ℕ} (hf : partrec f) (hg : partrec g) (hh : partrec h) : partrec\n  fun (a : ℕ) =>\n    roption.bind (f a)\n      fun (n : ℕ) =>\n        elim (g a)\n          (fun (y : ℕ) (IH : roption ℕ) =>\n            do \n              let i ← IH \n              h (mkpair a (mkpair y i)))\n          n := sorry\n\ntheorem ppred : partrec fun (n : ℕ) => ↑(ppred n) := sorry\n\nend partrec\n\n\nend nat\n\n\ndef partrec {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] (f : α →. σ) :=\n  nat.partrec fun (n : ℕ) => roption.bind ↑(encodable.decode α n) fun (a : α) => roption.map encodable.encode (f a)\n\ndef partrec₂ {α : Type u_1} {β : Type u_2} {σ : Type u_3} [primcodable α] [primcodable β] [primcodable σ] (f : α → β →. σ) :=\n  partrec fun (p : α × β) => f (prod.fst p) (prod.snd p)\n\ndef computable {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] (f : α → σ) :=\n  partrec ↑f\n\ndef computable₂ {α : Type u_1} {β : Type u_2} {σ : Type u_3} [primcodable α] [primcodable β] [primcodable σ] (f : α → β → σ) :=\n  computable fun (p : α × β) => f (prod.fst p) (prod.snd p)\n\ntheorem primrec.to_comp {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] {f : α → σ} (hf : primrec f) : computable f := sorry\n\ntheorem primrec₂.to_comp {α : Type u_1} {β : Type u_2} {σ : Type u_3} [primcodable α] [primcodable β] [primcodable σ] {f : α → β → σ} (hf : primrec₂ f) : computable₂ f :=\n  primrec.to_comp hf\n\ntheorem computable.part {α : Type u_1} {σ : Type u_2} [primcodable α] [primcodable σ] {f : α → σ} (hf : computable f) : partrec ↑f :=\n  hf\n\ntheorem computable₂.part {α : Type u_1} {β : Type u_2} {σ : Type u_3} [primcodable α] [primcodable β] [primcodable σ] {f : α → β → σ} (hf : computable₂ f) : partrec₂ fun (a : α) => ↑(f a) :=\n  hf\n\nnamespace computable\n\n\ntheorem of_eq {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → σ} {g : α → σ} (hf : computable f) (H : ∀ (n : α), f n = g n) : computable g :=\n  funext H ▸ hf\n\ntheorem const {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] (s : σ) : computable fun (a : α) => s :=\n  primrec.to_comp (primrec.const s)\n\ntheorem of_option {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] {f : α → Option β} (hf : computable f) : partrec fun (a : α) => ↑(f a) := sorry\n\ntheorem to₂ {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α × β → σ} (hf : computable f) : computable₂ fun (a : α) (b : β) => f (a, b) :=\n  of_eq hf\n    fun (_x : α × β) =>\n      (fun (_a : α × β) => prod.cases_on _a fun (fst : α) (snd : β) => idRhs (f (fst, snd) = f (fst, snd)) rfl) _x\n\nprotected theorem id {α : Type u_1} [primcodable α] : computable id :=\n  primrec.to_comp primrec.id\n\ntheorem fst {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] : computable prod.fst :=\n  primrec.to_comp primrec.fst\n\ntheorem snd {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] : computable prod.snd :=\n  primrec.to_comp primrec.snd\n\ntheorem pair {α : Type u_1} {β : Type u_2} {γ : Type u_3} [primcodable α] [primcodable β] [primcodable γ] {f : α → β} {g : α → γ} (hf : computable f) (hg : computable g) : computable fun (a : α) => (f a, g a) := sorry\n\ntheorem unpair : computable nat.unpair :=\n  primrec.to_comp primrec.unpair\n\ntheorem succ : computable Nat.succ :=\n  primrec.to_comp primrec.succ\n\ntheorem pred : computable Nat.pred :=\n  primrec.to_comp primrec.pred\n\ntheorem nat_bodd : computable nat.bodd :=\n  primrec.to_comp primrec.nat_bodd\n\ntheorem nat_div2 : computable nat.div2 :=\n  primrec.to_comp primrec.nat_div2\n\ntheorem sum_inl {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] : computable sum.inl :=\n  primrec.to_comp primrec.sum_inl\n\ntheorem sum_inr {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] : computable sum.inr :=\n  primrec.to_comp primrec.sum_inr\n\ntheorem list_cons {α : Type u_1} [primcodable α] : computable₂ List.cons :=\n  primrec₂.to_comp primrec.list_cons\n\ntheorem list_reverse {α : Type u_1} [primcodable α] : computable list.reverse :=\n  primrec.to_comp primrec.list_reverse\n\ntheorem list_nth {α : Type u_1} [primcodable α] : computable₂ list.nth :=\n  primrec₂.to_comp primrec.list_nth\n\ntheorem list_append {α : Type u_1} [primcodable α] : computable₂ append :=\n  primrec₂.to_comp primrec.list_append\n\ntheorem list_concat {α : Type u_1} [primcodable α] : computable₂ fun (l : List α) (a : α) => l ++ [a] :=\n  primrec₂.to_comp primrec.list_concat\n\ntheorem list_length {α : Type u_1} [primcodable α] : computable list.length :=\n  primrec.to_comp primrec.list_length\n\ntheorem vector_cons {α : Type u_1} [primcodable α] {n : ℕ} : computable₂ vector.cons :=\n  primrec₂.to_comp primrec.vector_cons\n\ntheorem vector_to_list {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.to_list :=\n  primrec.to_comp primrec.vector_to_list\n\ntheorem vector_length {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.length :=\n  primrec.to_comp primrec.vector_length\n\ntheorem vector_head {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.head :=\n  primrec.to_comp primrec.vector_head\n\ntheorem vector_tail {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.tail :=\n  primrec.to_comp primrec.vector_tail\n\ntheorem vector_nth {α : Type u_1} [primcodable α] {n : ℕ} : computable₂ vector.nth :=\n  primrec₂.to_comp primrec.vector_nth\n\ntheorem vector_nth' {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.nth :=\n  primrec.to_comp primrec.vector_nth'\n\ntheorem vector_of_fn' {α : Type u_1} [primcodable α] {n : ℕ} : computable vector.of_fn :=\n  primrec.to_comp primrec.vector_of_fn'\n\ntheorem fin_app {σ : Type u_4} [primcodable σ] {n : ℕ} : computable₂ id :=\n  primrec₂.to_comp primrec.fin_app\n\nprotected theorem encode {α : Type u_1} [primcodable α] : computable encodable.encode :=\n  primrec.to_comp primrec.encode\n\nprotected theorem decode {α : Type u_1} [primcodable α] : computable (encodable.decode α) :=\n  primrec.to_comp primrec.decode\n\nprotected theorem of_nat (α : Type u_1) [denumerable α] : computable (denumerable.of_nat α) :=\n  primrec.to_comp (primrec.of_nat α)\n\ntheorem encode_iff {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → σ} : (computable fun (a : α) => encodable.encode (f a)) ↔ computable f :=\n  iff.rfl\n\ntheorem option_some {α : Type u_1} [primcodable α] : computable some :=\n  primrec.to_comp primrec.option_some\n\nend computable\n\n\nnamespace partrec\n\n\ntheorem of_eq {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ} {g : α →. σ} (hf : partrec f) (H : ∀ (n : α), f n = g n) : partrec g :=\n  funext H ▸ hf\n\ntheorem of_eq_tot {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ} {g : α → σ} (hf : partrec f) (H : ∀ (n : α), g n ∈ f n) : computable g :=\n  of_eq hf fun (a : α) => iff.mpr roption.eq_some_iff (H a)\n\ntheorem none {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] : partrec fun (a : α) => roption.none := sorry\n\nprotected theorem some {α : Type u_1} [primcodable α] : partrec roption.some :=\n  computable.id\n\ntheorem const' {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] (s : roption σ) : partrec fun (a : α) => s :=\n  of_eq (computable.of_option (computable.const (roption.to_option s))) fun (a : α) => roption.of_to_option s\n\nprotected theorem bind {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α →. β} {g : α → β →. σ} (hf : partrec f) (hg : partrec₂ g) : partrec fun (a : α) => roption.bind (f a) (g a) := sorry\n\ntheorem map {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α →. β} {g : α → β → σ} (hf : partrec f) (hg : computable₂ g) : partrec fun (a : α) => roption.map (g a) (f a) := sorry\n\ntheorem to₂ {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α × β →. σ} (hf : partrec f) : partrec₂ fun (a : α) (b : β) => f (a, b) :=\n  of_eq hf\n    fun (_x : α × β) =>\n      (fun (_a : α × β) => prod.cases_on _a fun (fst : α) (snd : β) => idRhs (f (fst, snd) = f (fst, snd)) rfl) _x\n\ntheorem nat_elim {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → ℕ} {g : α →. σ} {h : α → ℕ × σ →. σ} (hf : computable f) (hg : partrec g) (hh : partrec₂ h) : partrec fun (a : α) => nat.elim (g a) (fun (y : ℕ) (IH : roption σ) => roption.bind IH fun (i : σ) => h a (y, i)) (f a) := sorry\n\ntheorem comp {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : β →. σ} {g : α → β} (hf : partrec f) (hg : computable g) : partrec fun (a : α) => f (g a) := sorry\n\ntheorem nat_iff {f : ℕ →. ℕ} : partrec f ↔ nat.partrec f := sorry\n\ntheorem map_encode_iff {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ} : (partrec fun (a : α) => roption.map encodable.encode (f a)) ↔ partrec f :=\n  iff.rfl\n\nend partrec\n\n\nnamespace partrec₂\n\n\ntheorem unpaired {α : Type u_1} [primcodable α] {f : ℕ → ℕ →. α} : partrec (nat.unpaired f) ↔ partrec₂ f := sorry\n\ntheorem unpaired' {f : ℕ → ℕ →. ℕ} : nat.partrec (nat.unpaired f) ↔ partrec₂ f :=\n  iff.trans (iff.symm partrec.nat_iff) unpaired\n\ntheorem comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_5} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : β → γ →. σ} {g : α → β} {h : α → γ} (hf : partrec₂ f) (hg : computable g) (hh : computable h) : partrec fun (a : α) => f (g a) (h a) :=\n  partrec.comp hf (computable.pair hg hh)\n\ntheorem comp₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {σ : Type u_5} [primcodable α] [primcodable β] [primcodable γ] [primcodable δ] [primcodable σ] {f : γ → δ →. σ} {g : α → β → γ} {h : α → β → δ} (hf : partrec₂ f) (hg : computable₂ g) (hh : computable₂ h) : partrec₂ fun (a : α) (b : β) => f (g a b) (h a b) :=\n  comp hf hg hh\n\nend partrec₂\n\n\nnamespace computable\n\n\ntheorem comp {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : β → σ} {g : α → β} (hf : computable f) (hg : computable g) : computable fun (a : α) => f (g a) :=\n  partrec.comp hf hg\n\ntheorem comp₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : γ → σ} {g : α → β → γ} (hf : computable f) (hg : computable₂ g) : computable₂ fun (a : α) (b : β) => f (g a b) :=\n  comp hf hg\n\nend computable\n\n\nnamespace computable₂\n\n\ntheorem comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_5} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : β → γ → σ} {g : α → β} {h : α → γ} (hf : computable₂ f) (hg : computable g) (hh : computable h) : computable fun (a : α) => f (g a) (h a) :=\n  computable.comp hf (computable.pair hg hh)\n\ntheorem comp₂ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {σ : Type u_5} [primcodable α] [primcodable β] [primcodable γ] [primcodable δ] [primcodable σ] {f : γ → δ → σ} {g : α → β → γ} {h : α → β → δ} (hf : computable₂ f) (hg : computable₂ g) (hh : computable₂ h) : computable₂ fun (a : α) (b : β) => f (g a b) (h a b) :=\n  comp hf hg hh\n\nend computable₂\n\n\nnamespace partrec\n\n\ntheorem rfind {α : Type u_1} [primcodable α] {p : α → ℕ →. Bool} (hp : partrec₂ p) : partrec fun (a : α) => nat.rfind (p a) := sorry\n\ntheorem rfind_opt {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → ℕ → Option σ} (hf : computable₂ f) : partrec fun (a : α) => nat.rfind_opt (f a) :=\n  partrec.bind (rfind (to₂ (computable.part (computable.comp (primrec.to_comp primrec.option_is_some) hf))))\n    (computable.of_option hf)\n\ntheorem nat_cases_right {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → ℕ} {g : α → σ} {h : α → ℕ →. σ} (hf : computable f) (hg : computable g) (hh : partrec₂ h) : partrec fun (a : α) => nat.cases (roption.some (g a)) (h a) (f a) := sorry\n\ntheorem bind_decode2_iff {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ} : partrec f ↔\n  nat.partrec fun (n : ℕ) => roption.bind ↑(encodable.decode2 α n) fun (a : α) => roption.map encodable.encode (f a) := sorry\n\ntheorem vector_m_of_fn {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {n : ℕ} {f : fin n → α →. σ} : (∀ (i : fin n), partrec (f i)) → partrec fun (a : α) => vector.m_of_fn fun (i : fin n) => f i a := sorry\n\nend partrec\n\n\n@[simp] theorem vector.m_of_fn_roption_some {α : Type u_1} {n : ℕ} (f : fin n → α) : (vector.m_of_fn fun (i : fin n) => roption.some (f i)) = roption.some (vector.of_fn f) :=\n  vector.m_of_fn_pure\n\nnamespace computable\n\n\ntheorem option_some_iff {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → σ} : (computable fun (a : α) => some (f a)) ↔ computable f := sorry\n\ntheorem bind_decode_iff {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α → β → Option σ} : (computable₂ fun (a : α) (n : ℕ) => option.bind (encodable.decode β n) (f a)) ↔ computable₂ f := sorry\n\ntheorem map_decode_iff {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α → β → σ} : (computable₂ fun (a : α) (n : ℕ) => option.map (f a) (encodable.decode β n)) ↔ computable₂ f :=\n  iff.trans bind_decode_iff option_some_iff\n\ntheorem nat_elim {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → ℕ} {g : α → σ} {h : α → ℕ × σ → σ} (hf : computable f) (hg : computable g) (hh : computable₂ h) : computable fun (a : α) => nat.elim (g a) (fun (y : ℕ) (IH : σ) => h a (y, IH)) (f a) := sorry\n\ntheorem nat_cases {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α → ℕ} {g : α → σ} {h : α → ℕ → σ} (hf : computable f) (hg : computable g) (hh : computable₂ h) : computable fun (a : α) => nat.cases (g a) (h a) (f a) :=\n  nat_elim hf hg (to₂ (computable₂.comp hh fst (comp fst snd)))\n\ntheorem cond {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {c : α → Bool} {f : α → σ} {g : α → σ} (hc : computable c) (hf : computable f) (hg : computable g) : computable fun (a : α) => cond (c a) (f a) (g a) := sorry\n\ntheorem option_cases {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {o : α → Option β} {f : α → σ} {g : α → β → σ} (ho : computable o) (hf : computable f) (hg : computable₂ g) : computable fun (a : α) => option.cases_on (o a) (f a) (g a) := sorry\n\ntheorem option_bind {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α → Option β} {g : α → β → Option σ} (hf : computable f) (hg : computable₂ g) : computable fun (a : α) => option.bind (f a) (g a) := sorry\n\ntheorem option_map {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {f : α → Option β} {g : α → β → σ} (hf : computable f) (hg : computable₂ g) : computable fun (a : α) => option.map (g a) (f a) :=\n  option_bind hf (comp₂ option_some hg)\n\ntheorem option_get_or_else {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] {f : α → Option β} {g : α → β} (hf : computable f) (hg : computable g) : computable fun (a : α) => option.get_or_else (f a) (g a) := sorry\n\ntheorem subtype_mk {α : Type u_1} {β : Type u_2} [primcodable α] [primcodable β] {f : α → β} {p : β → Prop} [decidable_pred p] {h : ∀ (a : α), p (f a)} (hp : primrec_pred p) (hf : computable f) : computable fun (a : α) => { val := f a, property := h a } := sorry\n\ntheorem sum_cases {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : α → β ⊕ γ} {g : α → β → σ} {h : α → γ → σ} (hf : computable f) (hg : computable₂ g) (hh : computable₂ h) : computable fun (a : α) => sum.cases_on (f a) (g a) (h a) := sorry\n\ntheorem nat_strong_rec {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] (f : α → ℕ → σ) {g : α → List σ → Option σ} (hg : computable₂ g) (H : ∀ (a : α) (n : ℕ), g a (list.map (f a) (list.range n)) = some (f a n)) : computable₂ f := sorry\n\ntheorem list_of_fn {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {n : ℕ} {f : fin n → α → σ} : (∀ (i : fin n), computable (f i)) → computable fun (a : α) => list.of_fn fun (i : fin n) => f i a := sorry\n\ntheorem vector_of_fn {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {n : ℕ} {f : fin n → α → σ} (hf : ∀ (i : fin n), computable (f i)) : computable fun (a : α) => vector.of_fn fun (i : fin n) => f i a := sorry\n\nend computable\n\n\nnamespace partrec\n\n\ntheorem option_some_iff {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ} : (partrec fun (a : α) => roption.map some (f a)) ↔ partrec f := sorry\n\ntheorem option_cases_right {α : Type u_1} {β : Type u_2} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable σ] {o : α → Option β} {f : α → σ} {g : α → β →. σ} (ho : computable o) (hf : computable f) (hg : partrec₂ g) : partrec fun (a : α) => option.cases_on (o a) (roption.some (f a)) (g a) := sorry\n\ntheorem sum_cases_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : α → β ⊕ γ} {g : α → β → σ} {h : α → γ →. σ} (hf : computable f) (hg : computable₂ g) (hh : partrec₂ h) : partrec fun (a : α) => sum.cases_on (f a) (fun (b : β) => roption.some (g a b)) (h a) := sorry\n\ntheorem sum_cases_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {σ : Type u_4} [primcodable α] [primcodable β] [primcodable γ] [primcodable σ] {f : α → β ⊕ γ} {g : α → β →. σ} {h : α → γ → σ} (hf : computable f) (hg : partrec₂ g) (hh : computable₂ h) : partrec fun (a : α) => sum.cases_on (f a) (g a) fun (c : γ) => roption.some (h a c) := sorry\n\ntheorem fix {α : Type u_1} {σ : Type u_4} [primcodable α] [primcodable σ] {f : α →. σ ⊕ α} (hf : partrec f) : partrec (pfun.fix f) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/computability/partrec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.305204744539557}}
{"text": "import polyfix\nimport polysize\nimport data.list.basic\n\nsection foldl\nvariables {α γ : Type*}\n\ndef tails_foldl_step (f : α → list γ → α) (x : list γ × α) : bool × (list γ × α) :=\n(x.1.empty, x.1.tail, f x.2 x.1)\n\nvariables (f : α → list γ → α) (x : list γ × α)\nlemma tails_foldl_step_iter :\n  iterator_evaln (tails_foldl_step f) (2 + x.1.length) (ff, x) = some ([], x.1.tails.foldl f x.2) :=\nbegin\n  cases x with l x,\n  induction l with hd tl ih generalizing x,\n  { simp [tails_foldl_step], }, { simp [tails_foldl_step, ← add_assoc, ih], },\nend\n\nvariables {f}\nlemma tails_foldl_states_suffix {s₀ s : bool × list γ × α} (hs : s ∈ (mk_iterator ↑(tails_foldl_step f) s₀).states) :\n  s.2.1 <:+ s₀.2.1 :=\nbegin\n  rcases s₀ with ⟨b₀, l₀, s₀⟩,\n  apply execution.step_induction hs; clear hs s,\n  { simp, },\n  rintros ⟨b, l, s⟩ ⟨b', l', s'⟩ _, simp,\n  rintros ih rfl, simp [tails_foldl_step],\n  rintros rfl rfl rfl,\n  exact l.tail_suffix.trans ih,\nend\n\nend foldl\n\nsection\nopen ptree.pencodable (encode decode)\nvariables {α β γ : Type*} [polycodable α] [polycodable β] [polycodable γ]\n  {f : β → α → list γ → α} {l : β → list γ} (acc : β → α) \n\nlemma acc_list_polysize_fun_uniform (hl : polysize_fun l) :\n  polysize_fun_uniform (λ (s : β) (y : (mk_iterator ↑(tails_foldl_step (f s)) (ff, l s, acc s)).states), (y : bool × list γ × α).2.1) :=\nbegin\n  rcases hl with ⟨pi, hpi⟩, use pi, intros s x,\n  refine trans (encode_sizeof_le_of_infix _) (hpi _),\n  exact (tails_foldl_states_suffix (subtype.mem x)).is_infix,\nend\n\nlemma tails_foldl_polysize_safe (hinit : polysize_fun l)\n  (hf : polysize_fun_safe (λ (s : β × list γ) (x : α), f s.1 x s.2)) :\n  polysize_fun_safe (λ (s : β) (y : (mk_iterator ↑(tails_foldl_step (f s)) (ff, l s, acc s)).states), tails_foldl_step (f s) (y : bool × list γ × α).2) :=\nbegin\n  dsimp only [tails_foldl_step] at *,\n  apply polysize_fun_safe.pair_left, { apply polysize_uniform_of_fin_range, },\n  apply polysize_fun_safe.pair_left,\n  { apply polysize_fun_uniform.comp' (polysize_fun_safe.tail _),\n    apply acc_list_polysize_fun_uniform _ hinit, },\n  change polysize_fun_safe (λ x y, (λ (s : β × list γ) (x : α), f s.1 x s.2) (x, (↑y : bool × list γ × α).2.1) (↑y : bool × list γ × α).2.2),\n  apply polysize_fun_safe.comp hf,\n  { apply polysize_fun_uniform.pair, { simpa using polysize_fun.id, }, apply acc_list_polysize_fun_uniform _ hinit, },\n  refine polysize_fun_safe.comp' (polysize_fun_safe.snd _) _, refine polysize_fun_safe.comp' (polysize_fun_safe.snd _) _,\n  use 0, intros, simp [penc_states_encode],\nend\n\nend\n\nsection list\n\nvariables {α β γ : Type*} [polycodable α] [polycodable β] [polycodable γ]\nopen ptree.pencodable (encode decode)\n\nvariable (H_enc : polytime_fun (λ l : ptree, ptree.equiv_list.symm $ l.equiv_list.map (encode ∘ (@decode γ _))))\n\ndef inh : inhabited γ := ⟨decode ptree.nil⟩ \nlocal attribute [instance] inh\n\ninclude H_enc\n-- List encoding aux; used very frequently, so better have a short name\ndef lea : polycodable (list γ) :=\n{ polytime_decode := by simpa [encode, decode] using H_enc }\n\nlemma polytime_fun.head_aux : polytime_fun (@list.head γ _) :=\nby { letI := lea H_enc, rw polytime_fun_iff', use [code.left, polytime_left], intro x, cases x, { simp, refl, }, simp [ptree.pencodable.encode_list_def], }\n\nlemma polytime_fun.tail_aux : polytime_fun (@list.tail γ) :=\n⟨code.right, polytime_right,\nby { intro x, cases x; simp [encode, decode], }⟩\n\nlemma polytime_fun.cons_aux : polytime_fun₂ (@list.cons γ) :=\n⟨code.id, polytime_id,\nby { intro x, cases x; simp [encode], }⟩\n\nlemma polytime_fun.is_empty_aux : polytime_fun (@list.empty γ) :=\nby { letI := lea H_enc, classical, convert_to polytime_fun (λ l : list γ, (l = [] : bool)), { ext x, cases x; simp, }, polyfun, }\n\nlemma polytime_fun.foldl_tails_aux {f : β → α → list γ → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × list γ) (b : α), f a.1 b a.2)) :\n  polytime_fun (λ s, (l s).tails.foldl (f s) (acc s)) :=\nbegin\n  letI := lea H_enc,\n  suffices : polytime_fun (λ s, (@list.nil γ, (l s).tails.foldl (f s) (acc s))), { exact polytime_fun.snd.comp this, },\n  apply polytime_fun.evaln_of_polysize (λ (s : β) (x : list γ × α), tails_foldl_step (f s) x) (λ x, (ff, l x, acc x)) (λ s, 2 + (l s).length),\n  { dunfold tails_foldl_step, have := polytime_fun.tail_aux H_enc, have := polytime_fun.is_empty_aux H_enc, polyfun, },\n  { polyfun, }, rotate 1,\n  { obtain ⟨q, hq⟩ := polysize_of_polytime_fun hl, use 2 + q, intros s, simpa using (len_le_encode_sizeof _).trans (hq _), },\n  { intro s, simpa using tails_foldl_step_iter (f s) (l s, acc s), },\n  exact tails_foldl_polysize_safe _ (polysize_of_polytime_fun hl) hs,\nend\n\nlemma polytime_fun.foldl_aux {f : β → α → γ → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × γ) (b : α), f a.1 b a.2)) :\n  polytime_fun (λ s, (l s).foldl (f s) (acc s)) :=\nbegin\n  letI := lea H_enc,\n  convert_to polytime_fun (λ s, (l s).tails.foldl (λ (a : α) (ls : list γ), if ls.empty then a else f s a ls.head) (acc s)),\n  { ext s, generalize : acc s = a, induction l s with hd tl ih generalizing a, { simp, }, simp [ih], },\n  apply polytime_fun.foldl_tails_aux H_enc,\n  { have := polytime_fun.head_aux H_enc, have := polytime_fun.is_empty_aux H_enc, polyfun, },\n  { exact hl, }, { exact hacc, },\n  apply polysize_fun_safe.ite, { apply polysize_fun_safe.id, }, \n  change polysize_fun_safe (λ (x : β × list γ) (y : α), (λ (a : β × γ) (b : α), f a.1 b a.2) (x.1, x.2.head) y),\n  apply hs.comp, { have := polytime_fun.head_aux H_enc, simp, apply polysize_of_polytime_fun, polyfun, }, { exact polysize_fun_safe.id, },\nend\n\nlemma polytime_fun.reverse_aux : polytime_fun (@list.reverse γ) :=\nbegin\n  letI := lea H_enc,\n  convert_to polytime_fun (λ l : list γ, l.foldl (λ acc hd, hd :: acc) []),\n  { ext l : 1, rw ← list.foldr_reverse, simp, },\n  apply polytime_fun.foldl_aux H_enc, { have := polytime_fun.cons_aux H_enc, polyfun, }, { polyfun, }, { polyfun, },\n  apply polysize_fun_safe.comp polysize_fun_safe.cons, { simp, apply polysize_of_polytime_fun, polyfun, }, exact polysize_fun_safe.id,\nend\n\nlemma polytime_fun.foldr_aux {f : β → γ → α → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × γ) (b : α), f a.1 a.2 b)) :\n  polytime_fun (λ s, (l s).foldr (f s) (acc s)) :=\nbegin\n  letI := lea H_enc,\n  simp only [← list.foldl_reverse],\n  apply polytime_fun.foldl_aux H_enc, { polyfun, }, { have := polytime_fun.reverse_aux H_enc, polyfun, }, { polyfun, },\n  exact hs,\nend\n\nlemma polytime_fun.map_aux \n  (H_enc' : polytime_fun (λ l : ptree, ptree.equiv_list.symm $ l.equiv_list.map (encode ∘ (@decode α _))))\n  {f : β → γ → α} {l : β → list γ} (hf : polytime_fun₂ f) (hl : polytime_fun l) :\n  polytime_fun (λ s, (l s).map (f s)) :=\nbegin\n  letI := lea H_enc, letI := lea H_enc',\n  convert_to polytime_fun (λ s, (l s).foldr (λ hd acc, (f s hd) :: acc) []),\n  { ext s : 1, induction l s with hd tl ih, { simp, }, simp [ih], },\n  apply polytime_fun.foldr_aux H_enc,\n  { have := polytime_fun.cons_aux H_enc', polyfun, }, { exact hl, }, { polyfun, },\n  apply polysize_fun_safe.comp polysize_fun_safe.cons, { simp, apply polysize_of_polytime_fun, exact hf, }, { exact polysize_fun_safe.id, },\nend\n\nend list\n\n\nsection list_ptree\nopen ptree.pencodable (encode decode)\n\ndef list_ptree_encode : polycodable (list ptree) :=\nlea ⟨code.id, polytime_id, by simp [encode, decode]⟩\n\nlocal attribute [instance] list_ptree_encode\n\n@[polyfun]\nlemma polytime_fun.equiv_list : polytime_fun ptree.equiv_list :=\n⟨_, polytime_id, by simp [encode]⟩\n\n@[polyfun]\nlemma polytime_fun.equiv_list_symm : polytime_fun ptree.equiv_list.symm :=\n⟨_, polytime_id, by simp [encode]⟩\n\nlemma H_enc {γ : Type*} [polycodable γ] : \n  polytime_fun (λ l : ptree, ptree.equiv_list.symm $ l.equiv_list.map (encode ∘ (@decode γ _))) :=\nbegin\n  refine polytime_fun.equiv_list_symm.comp (polytime_fun.comp _ polytime_fun.equiv_list),\n  apply polytime_fun.map_aux, iterate 2 { simpa using polytime_fun.id, },\n  { /- TODO: Debug -/ dunfold polytime_fun₂ function.uncurry, apply polytime_fun.comp; polyfun, }, polyfun,\nend\n\nend list_ptree\n\nsection list\nopen ptree.pencodable (encode decode)\n\nvariables {α β γ : Type*} [polycodable α] [polycodable β] [polycodable γ]\ninstance : polycodable (list α) :=\nlea H_enc\n\n\nlemma polytime_fun.head : polytime_fun (@list.head γ ⟨decode ptree.nil⟩) :=\npolytime_fun.head_aux H_enc\n\nlocal attribute [polyfun] polytime_fun.head\n\n@[polyfun]\nlemma polytime_fun.tail : polytime_fun (@list.tail γ) :=\npolytime_fun.tail_aux H_enc\n\n@[polyfun]\nlemma polytime_fun.cons : polytime_fun₂ (@list.cons γ) :=\npolytime_fun.cons_aux H_enc\n\n@[polyfun]\nlemma polytime_fun.is_empty : polytime_fun (@list.empty γ) :=\npolytime_fun.is_empty_aux H_enc\n\n@[polyfun]\nlemma polytime_fun.head' : polytime_fun (@list.head' γ) :=\nbegin\n  convert_to polytime_fun (λ l : list γ, if l.empty then none else some (@list.head _ ⟨decode ptree.nil⟩ l)),\n  { ext l : 1, cases l; simp, }, polyfun,\nend\n\n@[polyfun]\nlemma polytime_fun.ihead [inhabited γ] : polytime_fun (@list.head γ _) :=\nbegin\n  convert_to polytime_fun (λ l : list γ, l.head'.iget),\n  { ext l, cases l; simp, }, polyfun,\nend\n\nlocal attribute [-polyfun] polytime_fun.head\n\nlemma polytime_fun.foldl_tails {f : β → α → list γ → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × list γ) (b : α), f a.1 b a.2)) :\n  polytime_fun (λ s, (l s).tails.foldl (f s) (acc s)) := polytime_fun.foldl_tails_aux H_enc hf hl hacc hs\n\n@[polyfun]\nlemma polytime_fun.foldl {f : β → α → γ → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × γ) (b : α), f a.1 b a.2)) :\n  polytime_fun (λ s, (l s).foldl (f s) (acc s)) := polytime_fun.foldl_aux H_enc hf hl hacc hs\n\n@[polyfun]\nlemma polytime_fun.reverse : polytime_fun (@list.reverse γ) := polytime_fun.reverse_aux H_enc\n\n@[polyfun]\nlemma polytime_fun.foldr {f : β → γ → α → α} {l : β → list γ} {acc : β → α}\n  (hf : polytime_fun₃ f) (hl : polytime_fun l) (hacc : polytime_fun acc)\n  (hs : polysize_fun_safe (λ (a : β × γ) (b : α), f a.1 a.2 b)) :\n  polytime_fun (λ s, (l s).foldr (f s) (acc s)) := polytime_fun.foldr_aux H_enc hf hl hacc hs\n\n@[polyfun]\nlemma polytime_fun.map \n  {f : β → γ → α} {l : β → list γ} (hf : polytime_fun₂ f) (hl : polytime_fun l) :\n  polytime_fun (λ s, (l s).map (f s)) := polytime_fun.map_aux H_enc H_enc hf hl\n\n@[polyfun]\nlemma polytime_fun.filter {f : β → γ → bool} {l : β → list γ} (hf : polytime_fun₂ f) (hl : polytime_fun l) :\n  polytime_fun (λ s, (l s).filter (λ x, f s x)) :=\nbegin\n  convert_to polytime_fun (λ s, (l s).foldr (λ hd acc, if f s hd then hd :: acc else acc) []),\n  { ext s : 1, induction l s with hd tl ih, { simp, }, cases H : f s hd; simpa [H], },\n  polyfun, apply polysize_fun_safe.ite, { refine polysize_fun_safe.comp polysize_fun_safe.cons _ polysize_fun_safe.id, simp, exact polysize_of_polytime_fun polytime_fun.snd, }, exact polysize_fun_safe.id,\nend\n\n@[polyfun]\nlemma polytime_fun.append : polytime_fun₂ (λ (l₁ l₂ : list α), l₁ ++ l₂) :=\nbegin\n  convert_to polytime_fun₂ (λ (l₁ l₂ : list α), l₁.foldr list.cons l₂),\n  { ext l₁ l₂ : 2, induction l₁ with hd tl ih, { simp, }, { simp [ih], } },\n  polyfun, apply polysize_fun_safe.cons.comp, { simp, apply polysize_of_polytime_fun polytime_fun.snd, }, exact polysize_fun_safe.id,\nend\n\n@[polyfun]\nlemma polytime_fun.tails : polytime_fun (@list.tails α) :=\nbegin\n  convert_to polytime_fun (λ l : list α, (l.tails.foldl (λ acc x, x :: acc) []).reverse),\n  { simp, }, apply polytime_fun.comp, { polyfun, }, apply polytime_fun.foldl_tails,\n  { polyfun, }, { polyfun, }, { polyfun, }, apply polysize_fun_safe.cons.comp, { simp, apply polysize_of_polytime_fun polytime_fun.snd, }, exact polysize_fun_safe.id,\nend\n\n\ndef unary_nat_encode : polycodable ℕ := \npolycodable.of_equiv equiv.list_unit_equiv.symm\n\nlocal attribute [instance] unary_nat_encode\n\nlemma unary_nat_encode_eq (n : ℕ) : encode n = encode (list.repeat () n) := rfl\n\n@[polyfun]\nlemma polytime_fun.unary_length : polytime_fun (@list.length α) :=\nbegin\n  convert_to polytime_fun (λ l : list α, equiv.list_unit_equiv (l.map (λ _, ()))),\n  { ext l, simp [equiv.list_unit_equiv], }, polyfun, exact polycodable.of_equiv_polytime_symm _,\nend\n\n@[polyfun]\nlemma polytime_fun.unary_repeat : polytime_fun₂ (@list.repeat α) :=\nbegin\n  convert_to polytime_fun₂ (λ (x : α) (n : ℕ), (equiv.list_unit_equiv.symm n).map (λ _, x)),\n  { ext : 2, simp [equiv.list_unit_equiv], }, polyfun,\nend\n\n@[polyfun]\nlemma polytime_fun.unary_nat_add : polytime_fun₂ ((+) : ℕ → ℕ → ℕ) :=\nbegin\n  convert_to polytime_fun₂ (λ n m, (list.repeat () n ++ list.repeat () m).length),\n  { ext, simp, }, polyfun,\nend\n\nlemma _root_.list.drop_succ {α : Type*} (l : list α) (n : ℕ) :\n  l.drop (n+1) = l.tail.drop n :=\nby { induction l; simp, }\n\n@[polyfun]\nlemma polytime_fun.drop : polytime_fun₂ (@list.drop α) :=\nbegin\n  convert_to polytime_fun₂ (λ (n : ℕ) (l : list α), (list.repeat () n).foldr (λ _ (l' : list α), l'.tail) l),\n  { ext n l : 2, rw list.foldr_const, simp, induction n generalizing l; simp [list.drop_succ, *], },\n  polyfun, exact polysize_fun_safe.tail _,\nend\n\n@[polyfun]\nlemma polytime_fun.nth : polytime_fun₂ (@list.nth α) :=\nbegin\n  convert_to polytime_fun₂ (λ (l : list α) n, (l.drop n).head'),\n  { ext l n : 2, simp [← list.nth_zero, list.nth_drop], },\n  polyfun,\nend\n\n@[polyfun]\nlemma polytime_fun.tsub : polytime_fun₂ (@has_sub.sub ℕ _) :=\nbegin\n  convert_to polytime_fun₂ (λ (m n : ℕ), ((list.repeat () m).drop n).length),\n  { ext m n, simp, }, polyfun,\nend\n\n@[polyfun]\nlemma polytime_fun.last : polytime_fun (@list.last' α) :=\nby { convert_to polytime_fun (λ l : list α, l.reverse.head'), { ext l : 1, induction l using list.reverse_rec_on; simp, }, polyfun, }\n\n@[polyfun]\nlemma polytime_fun.filter_map {f : β → α → option γ} {l : β → list α} \n  (hf : polytime_fun₂ f) (hl : polytime_fun l) :\n  polytime_fun (λ s, (l s).filter_map (f s)) :=\nbegin\n  inhabit γ,\n  convert_to polytime_fun (λ s, (((l s).map (λ x, f s x)).filter (λ x : option γ, x.is_some)).map option.iget),\n  { ext s : 1, induction l s with hd tl ih, { simp, }, cases H : f s hd, { simpa [H], }, { simpa [list.filter_map_cons_some _ _ _ H, H], } },\n  polyfun,\nend\n\n@[simp] lemma _root_.list.filter_map_none (l : list α) :\n  l.filter_map (λ _, (none : option β)) = [] :=\nby { induction l; simp [*], }\n\nlemma zip_eq_inits_map [inhabited α] (l₁ : list α) (l₂ : list β) :\n  l₁.zip l₂ = l₁.inits.tail.filter_map (λ x, (l₂.nth (x.length - 1)).map (λ y, (x.last'.iget, y))) :=\nbegin\n  induction l₁ with hd tl ih generalizing l₂,\n  { simp, },\n  cases l₂ with h₂ t₂, { simp, }, \n  simp [list.filter_map_map, function.comp], \n  cases tl, { simp, rw list.filter_map_cons_some, { simp, }, simp, },\n  simp, rw list.filter_map_cons_some, swap, { simp, },\n  rw ih t₂, simp [list.filter_map_map, function.comp],\nend\n\n@[polyfun]\nlemma polytime_fun.inits : polytime_fun (@list.inits α) :=\nby { convert_to polytime_fun (λ l : list α, (l.reverse.tails.map list.reverse).reverse), { simp [list.tails_reverse], }, polyfun, }\n\n@[polyfun]\nlemma polytime_fun.zip : polytime_fun₂ (@list.zip α β) :=\nbegin\n  inhabit α,\n  change polytime_fun₂ (λ l₁ l₂, list.zip l₁ l₂), simp_rw zip_eq_inits_map, polyfun,\nend\n\n@[polyfun]\nlemma polytime_fun.all {f : β → α → bool} {l : β → list α} \n  (hf : polytime_fun₂ f) (hl : polytime_fun l) : polytime_fun (λ s, (l s).all (f s)) :=\nby { dunfold list.all, polyfun, exact (polysize_uniform_of_fin_range _).to_safe, }\n\n@[polyfun]\nlemma polytime_fun.unary_nat_le {f g : β → ℕ} (hf : polytime_fun f) (hg : polytime_fun g) :\n  polytime_fun (λ s, ((f s) ≤ (g s) : bool)) :=\nby { convert_to polytime_fun (λ s, ((f s) - (g s) = 0 : bool)), { simp, }, polyfun, }\n\n@[polyfun]\nlemma polytime_fun.list_join : polytime_fun (@list.join α) :=\nbegin\n  convert_to polytime_fun (λ l : list (list α), l.foldr (++) []),\n  { ext l : 1, induction l; simp [*],}, polyfun, apply polysize_fun_safe.append.comp,\n  { simp, apply polysize_of_polytime_fun, polyfun, }, { exact polysize_fun_safe.id, }\nend\n\nend list\n\nsection examples\n\ndef list_product {α β} (l₁ : list α) (l₂ : list β) : list (α × β) :=\n(l₁.map (λ x, l₂.map (λ y, (x, y)))).join\n\nexample {α β : Type*} [polycodable α] [polycodable β] :\n  polytime_fun₂ (λ (l₁ : list α) (l₂ : list β), list_product l₁ l₂) :=\nby { dunfold list_product, polyfun, }\n\nend examples\n\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/lists.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.30518498099329516}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.uniform_space.cauchy\nimport Mathlib.topology.uniform_space.separation\nimport Mathlib.topology.dense_embedding\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Uniform embeddings of uniform spaces.\n\nExtension of uniform continuous functions.\n-/\n\nstructure uniform_inducing {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] (f : α → β) \nwhere\n  comap_uniformity : filter.comap (fun (x : α × α) => (f (prod.fst x), f (prod.snd x))) (uniformity β) = uniformity α\n\ntheorem uniform_inducing.mk' {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (h : ∀ (s : set (α × α)),\n  s ∈ uniformity α ↔ ∃ (t : set (β × β)), ∃ (H : t ∈ uniformity β), ∀ (x y : α), (f x, f y) ∈ t → (x, y) ∈ s) : uniform_inducing f := sorry\n\ntheorem uniform_inducing.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {g : β → γ} (hg : uniform_inducing g) {f : α → β} (hf : uniform_inducing f) : uniform_inducing (g ∘ f) := sorry\n\nstructure uniform_embedding {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] (f : α → β) \nextends uniform_inducing f\nwhere\n  inj : function.injective f\n\ntheorem uniform_embedding_subtype_val {α : Type u_1} [uniform_space α] {p : α → Prop} : uniform_embedding subtype.val :=\n  uniform_embedding.mk (uniform_inducing.mk rfl) subtype.val_injective\n\ntheorem uniform_embedding_subtype_coe {α : Type u_1} [uniform_space α] {p : α → Prop} : uniform_embedding coe :=\n  uniform_embedding_subtype_val\n\ntheorem uniform_embedding_set_inclusion {α : Type u_1} [uniform_space α] {s : set α} {t : set α} (hst : s ⊆ t) : uniform_embedding (set.inclusion hst) := sorry\n\ntheorem uniform_embedding.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {g : β → γ} (hg : uniform_embedding g) {f : α → β} (hf : uniform_embedding f) : uniform_embedding (g ∘ f) := sorry\n\ntheorem uniform_embedding_def {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} : uniform_embedding f ↔\n  function.injective f ∧\n    ∀ (s : set (α × α)),\n      s ∈ uniformity α ↔ ∃ (t : set (β × β)), ∃ (H : t ∈ uniformity β), ∀ (x y : α), (f x, f y) ∈ t → (x, y) ∈ s := sorry\n\ntheorem uniform_embedding_def' {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} : uniform_embedding f ↔\n  function.injective f ∧\n    uniform_continuous f ∧\n      ∀ (s : set (α × α)),\n        s ∈ uniformity α → ∃ (t : set (β × β)), ∃ (H : t ∈ uniformity β), ∀ (x y : α), (f x, f y) ∈ t → (x, y) ∈ s := sorry\n\ntheorem uniform_inducing.uniform_continuous {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (hf : uniform_inducing f) : uniform_continuous f := sorry\n\ntheorem uniform_inducing.uniform_continuous_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {f : α → β} {g : β → γ} (hg : uniform_inducing g) : uniform_continuous f ↔ uniform_continuous (g ∘ f) := sorry\n\ntheorem uniform_inducing.inducing {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (h : uniform_inducing f) : inducing f := sorry\n\ntheorem uniform_inducing.prod {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {α' : Type u_3} {β' : Type u_4} [uniform_space α'] [uniform_space β'] {e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_inducing e₁) (h₂ : uniform_inducing e₂) : uniform_inducing fun (p : α × β) => (e₁ (prod.fst p), e₂ (prod.snd p)) := sorry\n\ntheorem uniform_inducing.dense_inducing {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (h : uniform_inducing f) (hd : dense_range f) : dense_inducing f :=\n  dense_inducing.mk (inducing.mk (inducing.induced (uniform_inducing.inducing h))) hd\n\ntheorem uniform_embedding.embedding {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (h : uniform_embedding f) : embedding f :=\n  embedding.mk (inducing.mk (inducing.induced (uniform_inducing.inducing (uniform_embedding.to_uniform_inducing h))))\n    (uniform_embedding.inj h)\n\ntheorem uniform_embedding.dense_embedding {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (h : uniform_embedding f) (hd : dense_range f) : dense_embedding f :=\n  dense_embedding.mk\n    (dense_inducing.mk (inducing.mk (inducing.induced (embedding.to_inducing (uniform_embedding.embedding h)))) hd)\n    (uniform_embedding.inj h)\n\ntheorem closure_image_mem_nhds_of_uniform_inducing {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {s : set (α × α)} {e : α → β} (b : β) (he₁ : uniform_inducing e) (he₂ : dense_inducing e) (hs : s ∈ uniformity α) : ∃ (a : α), closure (e '' set_of fun (a' : α) => (a, a') ∈ s) ∈ nhds b := sorry\n\ntheorem uniform_embedding_subtype_emb {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] (p : α → Prop) {e : α → β} (ue : uniform_embedding e) (de : dense_embedding e) : uniform_embedding (dense_embedding.subtype_emb p e) := sorry\n\ntheorem uniform_embedding.prod {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {α' : Type u_3} {β' : Type u_4} [uniform_space α'] [uniform_space β'] {e₁ : α → α'} {e₂ : β → β'} (h₁ : uniform_embedding e₁) (h₂ : uniform_embedding e₂) : uniform_embedding fun (p : α × β) => (e₁ (prod.fst p), e₂ (prod.snd p)) := sorry\n\ntheorem is_complete_of_complete_image {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {m : α → β} {s : set α} (hm : uniform_inducing m) (hs : is_complete (m '' s)) : is_complete s := sorry\n\n/-- A set is complete iff its image under a uniform embedding is complete. -/\ntheorem is_complete_image_iff {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {m : α → β} {s : set α} (hm : uniform_embedding m) : is_complete (m '' s) ↔ is_complete s := sorry\n\ntheorem complete_space_iff_is_complete_range {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} (hf : uniform_embedding f) : complete_space α ↔ is_complete (set.range f) := sorry\n\ntheorem complete_space_congr {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {e : α ≃ β} (he : uniform_embedding ⇑e) : complete_space α ↔ complete_space β := sorry\n\ntheorem complete_space_coe_iff_is_complete {α : Type u_1} [uniform_space α] {s : set α} : complete_space ↥s ↔ is_complete s :=\n  iff.trans (complete_space_iff_is_complete_range uniform_embedding_subtype_coe)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_complete (set.range coe) ↔ is_complete s)) subtype.range_coe))\n      (iff.refl (is_complete s)))\n\ntheorem is_complete.complete_space_coe {α : Type u_1} [uniform_space α] {s : set α} (hs : is_complete s) : complete_space ↥s :=\n  iff.mpr complete_space_coe_iff_is_complete hs\n\ntheorem is_closed.complete_space_coe {α : Type u_1} [uniform_space α] [complete_space α] {s : set α} (hs : is_closed s) : complete_space ↥s :=\n  is_complete.complete_space_coe (is_closed.is_complete hs)\n\ntheorem complete_space_extension {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {m : β → α} (hm : uniform_inducing m) (dense : dense_range m) (h : ∀ (f : filter β), cauchy f → ∃ (x : α), filter.map m f ≤ nhds x) : complete_space α := sorry\n\ntheorem totally_bounded_preimage {α : Type u_1} {β : Type u_2} [uniform_space α] [uniform_space β] {f : α → β} {s : set β} (hf : uniform_embedding f) (hs : totally_bounded s) : totally_bounded (f ⁻¹' s) := sorry\n\ntheorem uniform_embedding_comap {α : Type u_1} {β : Type u_2} {f : α → β} [u : uniform_space β] (hf : function.injective f) : uniform_embedding f :=\n  uniform_embedding.mk (uniform_inducing.mk rfl) hf\n\ntheorem uniformly_extend_exists {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {e : β → α} (h_e : uniform_inducing e) (h_dense : dense_range e) {f : β → γ} (h_f : uniform_continuous f) [complete_space γ] (a : α) : ∃ (c : γ), filter.tendsto f (filter.comap e (nhds a)) (nhds c) := sorry\n\ntheorem uniform_extend_subtype {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] [complete_space γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β} {s : set α} (hf : uniform_continuous fun (x : Subtype p) => f (subtype.val x)) (he : uniform_embedding e) (hd : ∀ (x : β), x ∈ closure (set.range e)) (hb : closure (e '' s) ∈ nhds b) (hs : is_closed s) (hp : ∀ (x : α), x ∈ s → p x) : ∃ (c : γ), filter.tendsto f (filter.comap e (nhds b)) (nhds c) := sorry\n\ntheorem uniformly_extend_of_ind {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {e : β → α} (h_e : uniform_inducing e) (h_dense : dense_range e) {f : β → γ} (h_f : uniform_continuous f) [separated_space γ] (b : β) : dense_inducing.extend (uniform_inducing.dense_inducing h_e h_dense) f (e b) = f b :=\n  dense_inducing.extend_eq_at (uniform_inducing.dense_inducing h_e h_dense) b\n    (continuous.continuous_at (uniform_continuous.continuous h_f))\n\ntheorem uniformly_extend_unique {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {e : β → α} (h_e : uniform_inducing e) (h_dense : dense_range e) {f : β → γ} [separated_space γ] {g : α → γ} (hg : ∀ (b : β), g (e b) = f b) (hc : continuous g) : dense_inducing.extend (uniform_inducing.dense_inducing h_e h_dense) f = g :=\n  dense_inducing.extend_unique (uniform_inducing.dense_inducing h_e h_dense) hg hc\n\ntheorem uniformly_extend_spec {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {e : β → α} (h_e : uniform_inducing e) (h_dense : dense_range e) {f : β → γ} (h_f : uniform_continuous f) [separated_space γ] [complete_space γ] (a : α) : filter.tendsto f (filter.comap e (nhds a))\n  (nhds (dense_inducing.extend (uniform_inducing.dense_inducing h_e h_dense) f a)) := sorry\n\ntheorem uniform_continuous_uniformly_extend {α : Type u_1} {β : Type u_2} {γ : Type u_3} [uniform_space α] [uniform_space β] [uniform_space γ] {e : β → α} (h_e : uniform_inducing e) (h_dense : dense_range e) {f : β → γ} (h_f : uniform_continuous f) [separated_space γ] [cγ : complete_space γ] : uniform_continuous (dense_inducing.extend (uniform_inducing.dense_inducing h_e h_dense) f) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/uniform_space/uniform_embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.30514128270330126}}
{"text": "import data.fintype.card\nimport tactic.zify\nimport tactic.ring\nimport tactic.linarith\nimport defs\n\nopen sum\n\nvariables {α β α' β' : Type} {γ : β → Type}\n\ndef propagate_aux (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β → ℕ → bool) : ℕ → (α → bool) × bool\n| 0 := next_bit init_carry (λ i, x i 0)\n| (n+1) :=\nnext_bit (propagate_aux n).1 (λ i, x i (n+1))\n\ndef propagate (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β → ℕ → bool) (i : ℕ) : bool :=\n(propagate_aux init_carry next_bit x i).2\n\n@[simp] def propagate_carry (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool))\n  (x : β → ℕ → bool) : ℕ → (α → bool)\n| 0 := next_bit init_carry (λ i, x i 0)\n| (n+1) := next_bit (propagate_carry n) (λ i, x i (n+1))\n\n@[simp] def propagate_carry2 (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool))\n  (x : β → ℕ → bool) : ℕ → (α → bool)\n| 0 := init_carry\n| (n+1) := next_bit (propagate_carry2 n) (λ i, x i n)\n\nlemma propagate_carry2_succ (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool))\n  (x : β → ℕ → bool) (n : ℕ) :\n  propagate_carry2 init_carry next_bit x (n+1) =\n  propagate_carry init_carry next_bit x n :=\nby induction n; simp *\n\n@[simp] lemma propagate_aux_fst_eq_carry (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β → ℕ → bool) : ∀ n : ℕ,\n  (propagate_aux init_carry next_bit x n).1 =\n  propagate_carry init_carry (λ c b, (next_bit c b).1) x n\n| 0 := rfl\n| (n+1) := by rw [propagate_aux, propagate_carry, propagate_aux_fst_eq_carry]\n\n@[simp] lemma propagate_zero (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n  (α → bool) × bool)\n  (x : β → ℕ → bool) :\n  propagate init_carry next_bit x 0 = (next_bit init_carry (λ i, x i 0)).2 :=\nrfl\n\nlemma propagate_succ (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β → ℕ → bool) (i : ℕ) :\n  propagate init_carry next_bit x (i+1) = (next_bit\n    (propagate_carry init_carry (λ c b, (next_bit c b).1) x i)\n    (λ j, x j (i+1))).2 :=\nby rw [← propagate_aux_fst_eq_carry]; refl\n\nlemma propagate_succ2 (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β → ℕ → bool) (i : ℕ) :\n  propagate init_carry next_bit x (i+1) = (next_bit\n    (propagate_carry2 init_carry (λ c b, (next_bit c b).1) x (i+1))\n    (λ j, x j (i+1))).2 :=\nby rw [propagate_carry2_succ, ← propagate_aux_fst_eq_carry]; refl\n\nlemma propagate_carry_propagate {δ : β → Type*} {β' : Type}\n    (f : Π a, δ a → β') : Π (n : ℕ) (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool))\n  (init_carry_x : Π a, γ a → bool)\n  (next_bit_x : Π a (carry : γ a → bool) (bits : δ a → bool),\n    (γ a → bool) × bool)\n  (x : β' → ℕ → bool),\n  propagate_carry init_carry next_bit (λ a, propagate (init_carry_x a)\n    (next_bit_x a) (λ d, x (f a d))) n =\n  propagate_carry\n    (λ a : α ⊕ (Σ a, γ a), sum.elim init_carry (λ b : Σ a, γ a,\n      init_carry_x b.1 b.2) a)\n    (λ (carry : (α ⊕ (Σ a, γ a)) → bool) (bits : β' → bool),\n  -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n      let f : Π (a : β), (γ a → bool) × bool := λ a, next_bit_x a (λ d,\n        carry (inr ⟨a, d⟩)) (λ d, bits (f a d)) in\n      let g : (α → bool) := (next_bit (carry ∘ inl) (λ a, (f a).2)) in\n      sum.elim g (λ x, (f x.1).1 x.2)\n    )\n    x n ∘ inl\n| 0 init_carry next_bit init_carry_x next_bit_x x := rfl\n| (n+1) init_carry next_bit init_carry_x next_bit_x x := begin\n  have := propagate_carry_propagate n,\n  clear_aux_decl,\n  simp only [propagate_carry, propagate_succ, elim_inl] at *,\n  conv_lhs { simp only [this] },\n  clear this,\n  ext,\n  congr,\n  ext,\n  congr,\n  induction n with n ih,\n  { simp },\n  { simp [ih] }\nend\n\nlemma propagate_propagate {δ : β → Type*} {β' : Type}\n    (f : Π a, δ a → β') : Π (n : ℕ) (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (init_carry_x : Π a, γ a → bool)\n  (next_bit_x : Π a (carry : γ a → bool) (bits : δ a → bool),\n    (γ a → bool) × bool)\n  (x : β' → ℕ → bool),\n  propagate init_carry next_bit (λ a, propagate (init_carry_x a)\n    (next_bit_x a) (λ d, x (f a d))) n =\n  propagate\n    (λ a : α ⊕ (Σ a, γ a), sum.elim init_carry (λ b : Σ a, γ a,\n      init_carry_x b.1 b.2) a)\n    (λ (carry : (α ⊕ (Σ a, γ a)) → bool) (bits : β' → bool),\n      -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n      let f : Π (a : β), (γ a → bool) × bool := λ a, next_bit_x a (λ d,\n        carry (inr ⟨a, d⟩)) (λ d, bits (f a d)) in\n      let g : (α → bool) × bool := (next_bit (carry ∘ inl) (λ a, (f a).2)) in\n      (sum.elim g.1 (λ x, (f x.1).1 x.2), g.2)\n    )\n  x n\n| 0 init_carry next_bit init_carry_x next_bit_x x := rfl\n| (n+1) init_carry next_bit init_carry_x next_bit_x x := begin\n  simp only [propagate_succ],\n  clear_aux_decl,\n  rw [propagate_carry_propagate],\n  congr,\n  ext,\n  congr,\n  induction n with n ih,\n  { simp },\n  { simp [ih] }\nend\n\nlemma propagate_carry_change_vars {β' : Type}\n  (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool))\n  (x : β' → ℕ → bool) (i : ℕ)\n  (change_vars : β → β') :\n  propagate_carry init_carry next_bit (λ b, x (change_vars b)) i =\n  propagate_carry init_carry (λ (carry : α → bool) (bits : β' → bool),\n    next_bit carry (λ b, bits (change_vars b))) x i :=\nbegin\n  induction i,\n  { simp },\n  { simp * }\nend\n\nlemma propagate_change_vars {β' : Type}\n  (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (x : β' → ℕ → bool) (i : ℕ)\n  (change_vars : β → β') :\n  propagate init_carry next_bit (λ b, x (change_vars b)) i =\n  propagate init_carry (λ (carry : α → bool) (bits : β' → bool),\n    next_bit carry (λ b, bits (change_vars b))) x i :=\nbegin\n  induction i with i ih,\n  { refl },\n  { simp only [propagate_succ, propagate_carry_change_vars, ih] }\nend\n\nopen term\n\n@[simp] def arity : term → ℕ\n| (var n) := n+1\n| zero := 0\n| one := 0\n| neg_one := 0\n| (and t₁ t₂) := max (arity t₁) (arity t₂)\n| (or t₁ t₂) := max (arity t₁) (arity t₂)\n| (xor t₁ t₂) := max (arity t₁) (arity t₂)\n| (not t) := arity t\n| (ls t) := arity t\n| (add t₁ t₂) := max (arity t₁) (arity t₂)\n| (sub t₁ t₂) := max (arity t₁) (arity t₂)\n| (neg t) := arity t\n| (incr t) := arity t\n| (decr t) := arity t\n\n@[simp] def term.eval_fin : Π (t : term) (vars : fin (arity t) → ℕ → bool), ℕ → bool\n| (var n) vars := vars (fin.last n)\n| zero vars := zero_seq\n| one vars := one_seq\n| neg_one vars := neg_one_seq\n| (and t₁ t₂) vars :=\n  and_seq (term.eval_fin t₁\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n  (term.eval_fin t₂\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n| (or t₁ t₂) vars :=\n  or_seq (term.eval_fin t₁\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n  (term.eval_fin t₂\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n| (xor t₁ t₂) vars :=\n  xor_seq (term.eval_fin t₁\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n  (term.eval_fin t₂\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n| (not t) vars := not_seq (term.eval_fin t vars)\n| (ls t) vars := ls_seq (term.eval_fin t vars)\n| (add t₁ t₂) vars :=\n  add_seq (term.eval_fin t₁\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n  (term.eval_fin t₂\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n| (sub t₁ t₂) vars :=\n  sub_seq (term.eval_fin t₁\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n  (term.eval_fin t₂\n    (λ i, vars (fin.cast_le (by simp [arity]) i)))\n| (neg t) vars := neg_seq (term.eval_fin t vars)\n| (incr t) vars := incr_seq (term.eval_fin t vars)\n| (decr t) vars := decr_seq (term.eval_fin t vars)\n\nlemma eval_fin_eq_eval (t : term) (vars : ℕ → ℕ → bool) :\n  term.eval_fin t (λ i, vars i) = term.eval t vars :=\nbegin\n  induction t;\n  dsimp [term.eval_fin, term.eval, arity] at *; simp *\nend\n\nlemma id_eq_propagate (x : ℕ → bool) :\n  x = propagate empty.elim (λ _ (y : unit → bool), (empty.elim, y ())) (λ _, x) :=\nby ext n; cases n; refl\n\nlemma zero_eq_propagate :\n  zero_seq = propagate empty.elim (λ (_ _ : empty → bool), (empty.elim, ff)) empty.elim :=\nby ext n; cases n; refl\n\nlemma one_eq_propagate :\n  one_seq = propagate (λ _ : unit, tt) (λ f (_ : empty → bool), (λ _, ff, f ())) empty.elim :=\nbegin\n  ext n,\n  cases n with n,\n  { refl },\n  { cases n,\n    { simp [one_seq, propagate_succ] },\n    { simp [one_seq, propagate_succ] } }\nend\n\nlemma and_eq_propagate (x y : ℕ → bool) :\n  and_seq x y = propagate empty.elim\n    (λ _ (y : bool → bool), (empty.elim, y tt && y ff)) (λ b, cond b x y) :=\nby ext n; cases n; simp [propagate, propagate_aux, and_seq]\n\nlemma or_eq_propagate (x y : ℕ → bool) :\n  or_seq x y = propagate empty.elim\n    (λ _ (y : bool → bool), (empty.elim, y tt || y ff)) (λ b, cond b x y) :=\nby ext n; cases n; simp [propagate, propagate_aux, or_seq]\n\nlemma xor_eq_propagate (x y : ℕ → bool) :\n  xor_seq x y = propagate empty.elim\n    (λ _ (y : bool → bool), (empty.elim, bxor (y tt) (y ff))) (λ b, cond b x y) :=\nby ext n; cases n; simp [propagate, propagate_aux, xor_seq]\n\nlemma not_eq_propagate (x : ℕ → bool) :\n  not_seq x = propagate empty.elim (λ _ (y : unit → bool), (empty.elim, bnot (y ()))) (λ _, x) :=\nby ext n; cases n; simp [propagate, propagate_aux, not_seq]\n\nlemma ls_eq_propagate (x : ℕ → bool) :\n  ls_seq x = propagate (λ _ : unit, ff) (λ (carry x : unit → bool), (x, carry ())) (λ _, x) :=\nbegin\n  ext n,\n  cases n with n,\n  { refl },\n  { cases n,\n    { simp [ls_seq, propagate_succ] },\n    { simp [ls_seq, propagate_succ] } }\nend\n\nlemma add_seq_aux_eq_propagate_carry (x y : ℕ → bool) (n : ℕ) :\n  (add_seq_aux x y n).2 = propagate_carry (λ _, ff)\n    (λ (carry : unit → bool) (bits : bool → bool),\n      λ _, (bits tt && bits ff) || (bits ff && carry ()) || (bits tt && carry ()))\n  (λ b, cond b x y) n () :=\nbegin\n  induction n,\n  { simp [add_seq_aux] },\n  { simp [add_seq_aux, *] }\nend\n\nlemma add_eq_propagate (x y : ℕ → bool) :\n  add_seq x y = propagate (λ _, ff)\n    (λ (carry : unit → bool) (bits : bool → bool),\n      (λ _, (bits tt && bits ff) || (bits ff && carry ()) || (bits tt && carry ()),\n        bxor (bits tt) (bxor (bits ff) (carry ()))))\n  (λ b, cond b x y) :=\nbegin\n  ext n,\n  cases n with n,\n  { simp [add_seq, add_seq_aux] },\n  { cases n,\n    { simp [add_seq, add_seq_aux, propagate_succ] },\n    { simp [add_seq, add_seq_aux, add_seq_aux_eq_propagate_carry,\n        propagate_succ] } }\nend\n\nlemma sub_seq_aux_eq_propagate_carry (x y : ℕ → bool) (n : ℕ) :\n  (sub_seq_aux x y n).2 = propagate_carry (λ _, ff)\n    (λ (carry : unit → bool) (bits : bool → bool),\n      λ _, (bnot (bits tt) && (bits ff)) ||\n        (bnot (bxor (bits tt) (bits ff))) && carry ())\n  (λ b, cond b x y) n () :=\nbegin\n  induction n,\n  { simp [sub_seq_aux] },\n  { simp [sub_seq_aux, *] }\nend\n\nlemma sub_eq_propagate (x y : ℕ → bool) :\n  sub_seq x y = propagate (λ _, ff)\n    (λ (carry : unit → bool) (bits : bool → bool),\n      (λ _, (bnot (bits tt) && (bits ff)) ||\n        ((bnot (bxor (bits tt) (bits ff))) && carry ()),\n        bxor (bits tt) (bxor (bits ff) (carry ()))))\n  (λ b, cond b x y) :=\nbegin\n  ext n,\n  cases n with n,\n  { simp [sub_seq, sub_seq_aux] },\n  { cases n,\n    { simp [sub_seq, sub_seq_aux, propagate_succ] },\n    { simp [sub_seq, sub_seq_aux, sub_seq_aux_eq_propagate_carry,\n        propagate_succ] } }\nend\n\nlemma neg_seq_aux_eq_propagate_carry (x : ℕ → bool) (n : ℕ) :\n  (neg_seq_aux x n).2 = propagate_carry (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      λ _, (bnot (bits ())) && (carry ()))\n  (λ _, x) n () :=\nbegin\n  induction n,\n  { simp [neg_seq_aux] },\n  { simp [neg_seq_aux, *] }\nend\n\nlemma neg_eq_propagate (x : ℕ → bool) :\n  neg_seq x = propagate (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      (λ _, (bnot (bits ())) && (carry ()), bxor (bnot (bits ())) (carry ())))\n  (λ _, x) :=\nbegin\n  ext n,\n  cases n with n,\n  { simp [neg_seq, neg_seq_aux] },\n  { cases n,\n    { simp [neg_seq, neg_seq_aux, propagate_succ] },\n    { simp [neg_seq, neg_seq_aux, neg_seq_aux_eq_propagate_carry,\n        propagate_succ] } }\nend\n\nlemma incr_seq_aux_eq_propagate_carry (x : ℕ → bool) (n : ℕ) :\n  (incr_seq_aux x n).2 = propagate_carry (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      λ _, (bits ()) && carry ())\n  (λ _, x) n () :=\nbegin\n  induction n,\n  { simp [incr_seq_aux] },\n  { simp [incr_seq_aux, *] }\nend\n\nlemma incr_eq_propagate (x : ℕ → bool) :\n  incr_seq x = propagate (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      (λ _, (bits ()) && carry (), bxor (bits ()) (carry ())))\n  (λ _, x) :=\nbegin\n  ext n,\n  cases n with n,\n  { simp [incr_seq, incr_seq_aux] },\n  { cases n,\n    { simp [incr_seq, incr_seq_aux, propagate_succ] },\n    { simp [incr_seq, incr_seq_aux, incr_seq_aux_eq_propagate_carry,\n        propagate_succ] } }\nend\n\nlemma decr_seq_aux_eq_propagate_carry (x : ℕ → bool) (n : ℕ) :\n  (decr_seq_aux x n).2 = propagate_carry (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      λ _, (bnot (bits ())) && carry ())\n  (λ _, x) n () :=\nbegin\n  induction n,\n  { simp [decr_seq_aux] },\n  { simp [decr_seq_aux, *] }\nend\n\nlemma decr_eq_propagate (x : ℕ → bool) :\n  decr_seq x = propagate (λ _, tt)\n    (λ (carry : unit → bool) (bits : unit → bool),\n      (λ _, (bnot (bits ())) && carry (), bxor (bits ()) (carry ())))\n  (λ _, x) :=\nbegin\n  ext n,\n  cases n with n,\n  { simp [decr_seq, decr_seq_aux] },\n  { cases n,\n    { simp [decr_seq, decr_seq_aux, propagate_succ] },\n    { simp [decr_seq, decr_seq_aux, decr_seq_aux_eq_propagate_carry,\n        propagate_succ] } }\nend\n\nstructure propagate_struc (arity : Type) : Type 1 :=\n( α  : Type )\n[ i : fintype α ]\n( init_carry : α → bool )\n( next_bit : Π (carry : α → bool) (bits : arity → bool),\n    (α → bool) × bool )\n\nattribute [instance] propagate_struc.i\n\nnamespace propagate_struc\n\nvariables {arity : Type} (p : propagate_struc arity)\n\ndef eval : (arity → ℕ → bool) → ℕ → bool :=\npropagate p.init_carry p.next_bit\n\ndef change_vars {arity2 : Type} (change_vars : arity → arity2) :\n  propagate_struc arity2 :=\n{ α := p.α,\n  i := p.i,\n  init_carry := p.init_carry,\n  next_bit := λ carry bits, p.next_bit carry (λ i, bits (change_vars i)) }\n\ndef compose [fintype arity]\n  (new_arity : Type)\n  (q_arity : arity → Type)\n  (vars : Π (a : arity), q_arity a → new_arity)\n  (q : Π (a : arity), propagate_struc (q_arity a)) :\n  propagate_struc (new_arity) :=\n{ α := p.α ⊕ (Σ a, (q a).α),\n  i := by letI := p.i; apply_instance,\n  init_carry := sum.elim p.init_carry (λ x, (q x.1).init_carry x.2),\n  next_bit := λ carry bits,\n    let f : Π (a : arity), ((q a).α → bool) × bool := λ a, (q a).next_bit (λ d,\n        carry (inr ⟨a, d⟩)) (λ d, bits (vars a d)) in\n    let g : (p.α → bool) × bool := (p.next_bit (carry ∘ inl) (λ a, (f a).2)) in\n    (sum.elim g.1 (λ x, (f x.1).1 x.2), g.2) }\n\nlemma eval_compose [fintype arity]\n  (new_arity : Type)\n  (q_arity : arity → Type)\n  (vars : Π (a : arity), q_arity a → new_arity)\n  (q : Π (a : arity), propagate_struc (q_arity a))\n  (x : new_arity → ℕ → bool):\n  (p.compose new_arity q_arity vars q).eval x =\n  p.eval (λ a, (q a).eval (λ i, x (vars _ i))) :=\nbegin\n  ext n,\n  simp only [eval, compose, propagate_propagate]\nend\n\ndef and : propagate_struc bool :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, bits tt && bits ff) }\n\n@[simp] lemma eval_and (x : bool → ℕ → bool) : and.eval x = and_seq (x tt) (x ff) :=\nby ext n; cases n; simp [and, and_seq, eval, propagate_succ]\n\ndef or : propagate_struc bool :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, bits tt || bits ff) }\n\n@[simp] lemma eval_or (x : bool → ℕ → bool) : or.eval x = or_seq (x tt) (x ff) :=\nby ext n; cases n; simp [or, or_seq, eval, propagate_succ]\n\ndef xor : propagate_struc bool :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, bxor (bits tt) (bits ff)) }\n\n@[simp] lemma eval_xor (x : bool → ℕ → bool) : xor.eval x = xor_seq (x tt) (x ff) :=\nby ext n; cases n; simp [xor, xor_seq, eval, propagate_succ]\n\ndef add : propagate_struc bool :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, ff,\n  next_bit := λ (carry : unit → bool) (bits : bool → bool),\n      (λ _, (bits tt && bits ff) || (bits ff && carry ()) || (bits tt && carry ()),\n        bxor (bits tt) (bxor (bits ff) (carry ()))) }\n\n@[simp] lemma eval_add (x : bool → ℕ → bool) : add.eval x = add_seq (x tt) (x ff) :=\nbegin\n  dsimp [add, eval],\n  rw [add_eq_propagate],\n  congr,\n  funext b,\n  cases b; refl\nend\n\ndef sub : propagate_struc bool :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, ff,\n  next_bit := λ (carry : unit → bool) (bits : bool → bool),\n      (λ _, (bnot (bits tt) && (bits ff)) ||\n        ((bnot (bxor (bits tt) (bits ff))) && carry ()),\n        bxor (bits tt) (bxor (bits ff) (carry ()))) }\n\n@[simp] lemma eval_sub (x : bool → ℕ → bool) : sub.eval x = sub_seq (x tt) (x ff) :=\nbegin\n  dsimp [sub, eval],\n  rw [sub_eq_propagate],\n  congr,\n  funext b,\n  cases b; refl\nend\n\ndef neg : propagate_struc unit :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, tt,\n  next_bit := λ (carry : unit → bool) (bits : unit → bool),\n    (λ _, (bnot (bits ())) && (carry ()), bxor (bnot (bits ())) (carry ())) }\n\n@[simp] lemma eval_neg (x : unit → ℕ → bool) : neg.eval x = neg_seq (x ()) :=\nbegin\n  dsimp [neg, eval],\n  rw [neg_eq_propagate],\n  congr,\n  funext b,\n  cases b; refl\nend\n\ndef not : propagate_struc unit :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, bnot (bits ())) }\n\n@[simp] lemma eval_not (x : unit → ℕ → bool) : not.eval x = not_seq (x ()) :=\nby ext n; cases n; simp [not, not_seq, eval, propagate_succ]\n\ndef zero : propagate_struc (fin 0) :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, ff) }\n\n@[simp] lemma eval_zero (x : fin 0 → ℕ → bool) : zero.eval x = zero_seq :=\nby ext n; cases n; simp [zero, zero_seq, eval, propagate_succ]\n\ndef one : propagate_struc (fin 0) :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, tt,\n  next_bit := λ carry bits, (λ _, ff, carry ()) }\n\n@[simp] lemma eval_one (x : fin 0 → ℕ → bool) : one.eval x = one_seq :=\nby ext n; cases n; simp [one, one_seq, eval, propagate_succ2]\n\ndef neg_one : propagate_struc (fin 0) :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, tt) }\n\n@[simp] lemma eval_neg_one (x : fin 0 → ℕ → bool) : neg_one.eval x = neg_one_seq :=\nby ext n; cases n; simp [neg_one, neg_one_seq, eval, propagate_succ2]\n\ndef ls : propagate_struc unit :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, ff,\n  next_bit := λ carry bits, (bits, carry ()) }\n\n@[simp] lemma eval_ls (x : unit → ℕ → bool) : ls.eval x = ls_seq (x ()) :=\nby ext n; cases n; simp [ls, ls_seq, eval, propagate_succ2]\n\ndef var (n : ℕ) : propagate_struc (fin (n+1)) :=\n{ α := empty,\n  i := by apply_instance,\n  init_carry := empty.elim,\n  next_bit := λ carry bits, (empty.elim, bits (fin.last n)) }\n\n@[simp] lemma eval_var (n : ℕ) (x : fin (n+1) → ℕ → bool) : (var n).eval x = x (fin.last n) :=\nby ext m; cases m; simp [var, eval, propagate_succ]\n\ndef incr : propagate_struc unit :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, tt,\n  next_bit := λ carry bits, (λ _, bits () && carry (), bxor (bits ()) (carry ())) }\n\n@[simp] lemma eval_incr (x : unit → ℕ → bool) : incr.eval x = incr_seq (x ()) :=\nbegin\n  dsimp [incr, eval],\n  rw [incr_eq_propagate],\n  congr,\n  funext b,\n  cases b; refl\nend\n\ndef decr : propagate_struc unit :=\n{ α := unit,\n  i := by apply_instance,\n  init_carry := λ _, tt,\n  next_bit := λ carry bits, (λ _, bnot (bits ()) && carry (), bxor (bits ()) (carry ())) }\n\n@[simp] lemma eval_decr (x : unit → ℕ → bool) : decr.eval x = decr_seq (x ()) :=\nbegin\n  dsimp [decr, eval],\n  rw [decr_eq_propagate],\n  congr,\n  funext b,\n  cases b; refl\nend\n\nend propagate_struc\n\nstructure propagate_solution (t : term) extends propagate_struc (fin (arity t)) :=\n( good : t.eval_fin = to_propagate_struc.eval )\n\ndef compose_unary\n  (p : propagate_struc unit)\n  {t : term}\n  (q : propagate_solution t) :\n  propagate_struc (fin (arity t)) :=\np.compose\n  (fin (arity t))\n  _\n  (λ _ , id)\n  (λ _, q.to_propagate_struc)\n\ndef compose_binary\n  (p : propagate_struc bool)\n  {t₁ t₂ : term}\n  (q₁ : propagate_solution t₁)\n  (q₂ : propagate_solution t₂) :\n  propagate_struc (fin (max (arity t₁) (arity t₂))) :=\np.compose (fin (max (arity t₁) (arity t₂)))\n  (λ b, fin (cond b (arity t₁) (arity t₂)))\n  (λ b i, fin.cast_le (by cases b; simp) i)\n  (λ b, bool.rec q₂.to_propagate_struc q₁.to_propagate_struc b)\n\n@[simp] lemma compose_unary_eval\n  (p : propagate_struc unit)\n  {t : term}\n  (q : propagate_solution t)\n  (x : fin (arity t) → ℕ → bool) :\n  (compose_unary p q).eval x = p.eval (λ _, t.eval_fin x) :=\nbegin\n  rw [compose_unary, propagate_struc.eval_compose, q.good],\n  refl\nend\n\n@[simp] lemma compose_binary_eval\n  (p : propagate_struc bool)\n  {t₁ t₂ : term}\n  (q₁ : propagate_solution t₁)\n  (q₂ : propagate_solution t₂)\n  (x : fin (max (arity t₁) (arity t₂)) → ℕ → bool) :\n  (compose_binary p q₁ q₂).eval x = p.eval\n    (λ b, cond b (t₁.eval_fin (λ i, x (fin.cast_le (by simp) i)))\n                 (t₂.eval_fin (λ i, x (fin.cast_le (by simp) i)))) :=\nbegin\n  rw [compose_binary, propagate_struc.eval_compose, q₁.good, q₂.good],\n  congr,\n  ext b,\n  cases b; refl\nend\n\ninstance {α β : Type*} [fintype α] [fintype β] (b : bool) :\n  fintype (cond b α β) :=\nby cases b; dsimp; apply_instance\n\nlemma cond_propagate {α α' β β' : Type}\n  (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool),\n    (α → bool) × bool)\n  (init_carry' : α' → bool)\n  (next_bit' : Π (carry : α' → bool) (bits : β' → bool),\n    (α' → bool) × bool)\n  {γ : Type} (fβ : β → γ) (fβ' : β' → γ)\n  (x : γ → ℕ → bool) (b : bool) :\n  cond b (propagate init_carry next_bit (λ b, (x (fβ b))))\n    (propagate init_carry' next_bit' (λ b, (x (fβ' b)))) =\n  propagate (show cond b α α' → bool, from bool.rec init_carry' init_carry b)\n    (show Π (carry : cond b α α' → bool) (bits : cond b β β' → bool),\n        (cond b α α' → bool) × bool,\n      from bool.rec next_bit' next_bit b)\n    (show cond b β β' → ℕ → bool, from bool.rec (λ b, (x (fβ' b))) (λ b, (x (fβ b))) b) :=\nby cases b; refl\n\ndef term_eval_eq_propagate : Π (t : term),\n  propagate_solution t\n| (var n) :=\n  { to_propagate_struc := propagate_struc.var n,\n    good := by ext; simp [term.eval_fin] }\n| zero :=\n  { to_propagate_struc := propagate_struc.zero,\n    good := by ext; simp [term.eval_fin] }\n| one :=\n  { to_propagate_struc := propagate_struc.one,\n    good := by ext; simp [term.eval_fin] }\n| neg_one :=\n  { to_propagate_struc := propagate_struc.neg_one,\n    good := by ext; simp [term.eval_fin] }\n| (and t₁ t₂) :=\n  let q₁ := term_eval_eq_propagate t₁ in\n  let q₂ := term_eval_eq_propagate t₂ in\n  { to_propagate_struc := compose_binary propagate_struc.and q₁ q₂,\n    good := by ext; simp; refl }\n| (or t₁ t₂) :=\n  let q₁ := term_eval_eq_propagate t₁ in\n  let q₂ := term_eval_eq_propagate t₂ in\n  { to_propagate_struc := compose_binary propagate_struc.or q₁ q₂,\n    good := by ext; simp; refl }\n| (xor t₁ t₂) :=\n  let q₁ := term_eval_eq_propagate t₁ in\n  let q₂ := term_eval_eq_propagate t₂ in\n  { to_propagate_struc := compose_binary propagate_struc.xor q₁ q₂,\n    good := by ext; simp; refl }\n| (ls t) :=\n  let q := term_eval_eq_propagate t in\n  { to_propagate_struc := by dsimp [arity]; exact compose_unary propagate_struc.ls q,\n    good := by ext; simp; refl }\n| (not t) :=\n  let q := term_eval_eq_propagate t in\n  { to_propagate_struc := by dsimp [arity]; exact compose_unary propagate_struc.not q,\n    good := by ext; simp; refl }\n| (add t₁ t₂) :=\n  let q₁ := term_eval_eq_propagate t₁ in\n  let q₂ := term_eval_eq_propagate t₂ in\n  { to_propagate_struc := compose_binary propagate_struc.add q₁ q₂,\n    good := by ext; simp; refl }\n| (sub t₁ t₂) :=\n  let q₁ := term_eval_eq_propagate t₁ in\n  let q₂ := term_eval_eq_propagate t₂ in\n  { to_propagate_struc := compose_binary propagate_struc.sub q₁ q₂,\n    good := by ext; simp; refl }\n| (neg t) :=\n  let q := term_eval_eq_propagate t in\n  { to_propagate_struc := by dsimp [arity]; exact compose_unary propagate_struc.neg q,\n    good := by ext; simp; refl }\n| (incr t) :=\n  let q := term_eval_eq_propagate t in\n  { to_propagate_struc := by dsimp [arity]; exact compose_unary propagate_struc.incr q,\n    good := by ext; simp; refl }\n| (decr t) :=\n  let q := term_eval_eq_propagate t in\n  { to_propagate_struc := by dsimp [arity]; exact compose_unary propagate_struc.decr q,\n    good := by ext; simp; refl }\n\nvariables\n  (init_carry : α → bool)\n  (next_carry : Π (carry : α → bool) (bits : β → bool), (α → bool))\n  (next_bit : Π (carry : α → bool) (bits : β → bool), (α → bool) × bool)\n\nvariables [fintype α] [fintype α']\n\nopen fintype\n\nlemma exists_repeat_carry (seq : β → ℕ → bool) :\n  ∃ n m : fin (2 ^ (card α) + 1),\n    propagate_carry2 init_carry next_carry seq n =\n    propagate_carry2 init_carry next_carry seq m ∧\n    n < m :=\nbegin\n  by_contra h,\n  letI : decidable_eq α := classical.dec_eq α,\n  push_neg at h,\n  have := λ a b hab, (le_antisymm (h a b hab) (h b a hab.symm)).symm,\n  simpa using fintype.card_le_of_injective _ this\nend\n\nlemma propagate_carry2_eq_of_seq_eq_lt (seq₁ seq₂ : β → ℕ → bool)\n  (init_carry : α → bool)\n  (next_carry : Π (carry : α → bool) (bits : β → bool), (α → bool))\n  (i : ℕ) (h : ∀ (b) j < i, seq₁ b j = seq₂ b j) :\n  propagate_carry2 init_carry next_carry seq₁ i =\n    propagate_carry2 init_carry next_carry seq₂ i :=\nbegin\n  induction i with i ih,\n  { simp [propagate_carry2] },\n  { simp [propagate_carry2, h _ i (nat.lt_succ_self i)],\n    rw ih,\n    exact λ b j hj, h b j (nat.lt_succ_of_lt hj) }\nend\n\nlemma propagate_eq_of_seq_eq_le (seq₁ seq₂ : β → ℕ → bool)\n  (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool), (α → bool) × bool)\n  (i : ℕ) (h : ∀ (b) j ≤ i, seq₁ b j = seq₂ b j) :\n  propagate init_carry next_bit seq₁ i =\n    propagate init_carry next_bit seq₂ i :=\nbegin\n  cases i,\n  { simp [propagate_zero, h _ 0 (le_refl _)] },\n  { simp only [propagate_succ2, propagate_succ2, h _ _ (le_refl _)],\n    congr' 2,\n    apply propagate_carry2_eq_of_seq_eq_lt,\n    exact λ b j hj, h b j (le_of_lt hj) }\nend\n\nlemma propagate_carry2_eq_of_carry_eq (seq₁ seq₂ : β → ℕ → bool)\n  (m n : ℕ)\n  (h₁ : propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₁ m =\n       propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₂ n) (x : ℕ)\n  (h₃ : ∀ y b, y ≤ x → seq₁ b (m + y) = seq₂ b (n + y))  :\n  propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₁ (m + x) =\n  propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₂ (n + x) :=\nbegin\n  induction x with x ih generalizing seq₁ seq₂,\n  { simp * at * },\n  { simp only [h₃ x _ (nat.le_succ _),\n      nat.add_succ, propagate_carry2, add_zero] at *,\n    rw [ih],\n    assumption,\n    exact λ y b h, h₃ y b (nat.le_succ_of_le h) }\nend\n\nlemma propagate_eq_of_carry_eq (seq₁ seq₂ : β → ℕ → bool)\n  (m n : ℕ)\n  (h₁ : propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₁ m =\n       propagate_carry2 init_carry\n    (λ carry bits, (next_bit carry bits).1) seq₂ n) (x : ℕ)\n  (h₃ : ∀ y b, y ≤ x → seq₁ b (m + y) = seq₂ b (n + y))  :\n  propagate init_carry next_bit seq₁ (m + x) =\n  propagate init_carry next_bit seq₂ (n + x) :=\nbegin\n  cases x,\n  { cases m,\n    { cases n,\n      { simp [h₃ 0 _ (le_refl _), propagate_carry2, *] at * },\n      { simp [*, h₃ 0 _ (le_refl _), propagate_succ2] at *,\n        rw [← h₁] } },\n    { cases n,\n      { simp [*, propagate_succ2] at *,\n        simp [← h₃ 0 _ rfl] },\n      { rw [propagate_succ2, h₁, propagate_succ2],\n        have := h₃ 0,\n        simp * at * } } },\n  { simp only [nat.add_succ, propagate_succ2, add_zero],\n    simp [← nat.add_succ, h₃ _ _ (le_refl _)],\n    congr' 2,\n    apply propagate_carry2_eq_of_carry_eq,\n    assumption,\n    assumption }\nend\n\nlemma propagate_carry_propagate_carry_add (x : β → ℕ → bool) :\n  ∀ (init_carry : α → bool)\n    (next_carry : Π (carry : α → bool) (bits : β → bool), (α → bool)),\n  ∀ n i : ℕ,\n  propagate_carry2 (propagate_carry2 init_carry next_carry x n)\n    next_carry (λ b k, x b (k + n)) i =\n  propagate_carry2 init_carry next_carry x (i + n)\n| init_carry next_carry 0 0 := by simp [propagate_carry2]\n| init_carry next_carry (n+1) 0 :=\n  by simp [propagate_carry, propagate_carry2_succ]\n| init_carry next_carry n (i+1) := begin\n  rw [propagate_carry2, add_assoc,\n    propagate_carry_propagate_carry_add],\n  simp only [nat.one_add, nat.add_one, nat.succ_add, nat.add_succ,\n    add_zero, propagate_carry2, zero_add]\nend\n\nlemma exists_repeat : ∀ (seq : β → ℕ → bool)\n  (n : ℕ),\n  ∃ (m < 2 ^ (card α)) (seq2 : β → ℕ → bool),\n    propagate init_carry next_bit seq2 m = propagate init_carry next_bit seq n\n| seq n :=\nbegin\n  by_cases hn2 : n < 2 ^ card α,\n  { exact ⟨n, hn2, seq, rfl⟩ },\n  { rcases exists_repeat_carry (propagate_carry2 init_carry (λ c b, (next_bit c b).1) seq\n        (n - 2 ^ card α))\n      (λ carry bits, (next_bit  carry bits).1)\n      (λ b i, seq b (i + (n - 2^ (card α)))) with ⟨a, b, h₁, h₂⟩,\n    simp only [propagate_carry_propagate_carry_add] at h₁,\n    rcases have wf : n - (b - a) < n,\n        from nat.sub_lt (lt_of_lt_of_le (pow_pos two_pos _) (le_of_not_gt hn2))\n          (nat.sub_pos_of_lt h₂),\n      exists_repeat (λ c i, if i < a + (n - 2 ^ card α) then seq c i else\n        seq c (i + (b - a))) (n - (b - a)) with ⟨m, hmle, seq2, hm⟩,\n    use [m, hmle, seq2],\n    rw [hm], clear hm,\n    have h1 : n - (b - a) = (a + (n - 2 ^ (card α))) + (2 ^ card α - b),\n    { zify,\n      rw [nat.cast_sub, nat.cast_sub, nat.cast_sub, nat.cast_sub],\n      ring,\n      exact nat.le_of_lt_succ b.2,\n      simp * at *,\n      exact le_of_lt h₂,\n      exact le_trans (nat.sub_le _ _) (le_trans (nat.le_of_lt_succ b.2)\n        (by simp * at *)) },\n    rw h1,\n    have h2 : n = (b + (n - 2 ^ card α)) + (2 ^ card α - b),\n    { zify,\n      rw [nat.cast_sub, nat.cast_sub],\n      ring,\n      exact nat.le_of_lt_succ b.2,\n      simp * at *, },\n    conv_rhs { rw h2 },\n    refine propagate_eq_of_carry_eq _ _ _ _ _ _ _ _ _,\n    { have h : ↑b + (n - 2 ^ card α) = (a + (n - 2 ^ card α)) + (b - a),\n      { zify,\n        rw [nat.cast_sub, nat.cast_sub],\n        ring,\n        exact le_of_lt h₂,\n        simp * at * },\n      rw [← h₁],\n      apply propagate_carry2_eq_of_seq_eq_lt,\n      simp { contextual := tt } },\n    { intros y c hc,\n      simp only [add_lt_iff_neg_left, not_lt_zero', if_false],\n      congr' 1,\n      zify,\n      rw [nat.cast_sub, nat.cast_sub],\n      ring,\n      exact le_of_lt h₂,\n      simp * at * } },\nend\nusing_well_founded { rel_tac := λ _ _, `[exact ⟨_, measure_wf psigma.snd⟩] }\n\nlemma propagate_eq_zero_iff (init_carry : α → bool)\n  (next_bit : Π (carry : α → bool) (bits : β → bool), (α → bool) × bool) :\n  (∀ seq, propagate init_carry next_bit seq = zero_seq) ↔\n  (∀ seq, ∀ i < 2 ^ (card α), propagate init_carry next_bit seq i = ff) :=\nbegin\n  split,\n  { intros h i _,\n    simp [h, zero_seq] },\n  { intros h seq,\n    funext i,\n    rcases exists_repeat init_carry next_bit seq i with ⟨j, hj, seq2, hseq2⟩,\n    rw [← hseq2, h seq2 j hj, zero_seq] }\nend\n\nlemma eq_iff_xor_seq_eq_zero (seq₁ seq₂ : ℕ → bool) :\n  (∀ i, seq₁ i = seq₂ i) ↔ (∀ i, xor_seq seq₁ seq₂ i = zero_seq i) :=\nbegin\n  simp [function.funext_iff, xor_seq, zero_seq],\n  split,\n  { intros, simp * },\n  { intros h a,\n    specialize h a,\n    cases (seq₁ a); cases (seq₂ a); simp * at *, }\nend\n\nlemma eval_eq_iff_xor_seq_eq_zero (t₁ t₂ : term) :\n  t₁.eval = t₂.eval ↔ (t₁.xor t₂).eval_fin = λ _, zero_seq :=\nbegin\n  simp only [function.funext_iff, term.eval, term.eval_fin,\n    ← eq_iff_xor_seq_eq_zero, ← eval_fin_eq_eval],\n  split,\n  { intros h seq n,\n    have := h (λ j, if hj : j < (arity (t₁.xor t₂)) then seq ⟨j, hj⟩ else λ _, ff) n,\n    simp at this,\n    convert this },\n  { intros h seq m,\n    exact h (λ j, seq j) _ }\nend\n\n\n", "meta": {"author": "ChrisHughes24", "repo": "lean3bits", "sha": "119b68f1ce4a967951c53ee2f174007b49c80831", "save_path": "github-repos/lean/ChrisHughes24-lean3bits", "path": "github-repos/lean/ChrisHughes24-lean3bits/lean3bits-119b68f1ce4a967951c53ee2f174007b49c80831/src/v1/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.30507449306892004}}
{"text": "import category_theory.abelian.diagram_lemmas.four\nimport .projectives\nimport .homological_complex\nimport .snake_lemma2\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nnamespace short_exact_sequence\n\nvariables {C : Type u} [category.{v} C] [abelian C] [enough_projectives C]\nvariables {D : Type*} [category D] [abelian D]\n\n-- move this\nlemma exact_of_epi_comp_kernel.ι_comp_mono {C : Type u} [category.{v} C] [abelian C] {X Y Z W : C}\n  (g : Y ⟶ Z) (h : Z ⟶ W) (f : X ⟶ kernel g) (i : kernel g ⟶ Y) (hf : epi f) (hh : mono h)\n  (hi : i = kernel.ι g) : exact (f ≫ i) (g ≫ h) :=\nbegin\n  suffices : exact i g,\n  { letI := hf, letI := hh,\n    exact exact_comp_mono (exact_epi_comp this) },\n  rw [hi],\n  exact exact_kernel_ι\nend\n\n-- move this\nlemma biprod_factors (A B : C) [projective A] [projective B]\n  (E X : C) (f : A ⊞ B ⟶ X) (e : E ⟶ X) [epi e] :\n  ∃ f' : A ⊞ B ⟶ E, f' ≫ e = f :=\n⟨biprod.desc\n  (projective.factor_thru (biprod.inl ≫ f) e)\n  (projective.factor_thru (biprod.inr ≫ f) e),\n  by ext; simp only [projective.factor_thru_comp, biprod.inl_desc_assoc, biprod.inr_desc_assoc]⟩\n\nvariables (A B : short_exact_sequence C) (f : A ⟶ B)\n\ndef horseshoe_base : short_exact_sequence C :=\nshort_exact_sequence.mk_split (projective.over A.1) (projective.over A.3)\n\ndef horseshoe_base_π : horseshoe_base A ⟶ A :=\n{ fst := projective.π _,\n  snd := biprod.desc (projective.π _ ≫ A.f) (projective.factor_thru (projective.π _) A.g),\n  trd := projective.π _,\n  sq1' := by { dsimp [horseshoe_base], simp only [biprod.inl_desc], },\n  sq2' :=\n  begin\n    dsimp [horseshoe_base], apply category_theory.limits.biprod.hom_ext',\n    { simp only [zero_comp, exact.w_assoc, biprod.inl_desc_assoc, category.assoc,\n        short_exact_sequence.f_comp_g, comp_zero, exact_inl_snd], },\n    { simp only [projective.factor_thru_comp, biprod.inr_snd_assoc, biprod.inr_desc_assoc], }\n  end }\n\ninstance epi_horseshoe_base_π_1 : epi (horseshoe_base_π A).1 :=\nshow epi (projective.π _), by apply_instance\n\ninstance epi_horseshoe_base_π_3 : epi (horseshoe_base_π A).3 :=\nshow epi (projective.π _), by apply_instance\n\nlocal attribute [instance] limits.has_zero_object.has_zero\n\ninstance epi_horseshoe_base_π_2 : epi (horseshoe_base_π A).2 :=\nbegin\n  let φ := horseshoe_base_π A,\n  have h : φ.3 ≫ (0 : A.3 ⟶ 0) = (0 : _ ⟶ 0) ≫ (0 : 0 ⟶ 0) := by simp,\n  refine category_theory.abelian.epi_of_epi_of_epi_of_mono φ.sq1' φ.sq2' h _ _ _ _ _ _;\n  try { rw ← epi_iff_exact_zero_right }; try { apply_instance },\n  exact A.exact',\nend\n\nvariables {A B}\n\ndef horseshoe_ker [epi f.1] : short_exact_sequence C :=\n(snake_input.mk_of_short_exact_sequence_hom _ _ _ f).kernel_sequence _\nbegin\n  dsimp [snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],\n  rw snake_diagram.mk_functor_map_f1,\n  exact A.mono',\nend\n$ is_zero_of_iso_of_zero (is_zero_zero _) (limits.cokernel.of_epi _).symm\n\n@[simp] lemma horseshoe_ker_fst [epi f.1] : (horseshoe_ker f).1 = kernel f.1 := rfl\n\n@[simp] lemma horseshoe_ker_snd [epi f.1] : (horseshoe_ker f).2 = kernel f.2 := rfl\n\n@[simp] lemma horseshoe_ker_trd [epi f.1] : (horseshoe_ker f).3 = kernel f.3 := rfl\n\ndef horseshoe_ker_ι [epi f.1] : horseshoe_ker f ⟶ A :=\n{ fst := kernel.ι _,\n  snd := kernel.ι _,\n  trd := kernel.ι _,\n  sq1' :=\n  begin\n    dsimp [horseshoe_ker, snake_input.kernel_sequence,\n      snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],\n    delta kernel.map,\n    rw [snake_diagram.mk_functor_map_f0, kernel.lift_ι],\n  end,\n  sq2' :=\n  begin\n    dsimp [horseshoe_ker, snake_input.kernel_sequence,\n      snake_input.mk_of_short_exact_sequence_hom, snake_diagram.mk_of_short_exact_sequence_hom],\n    delta kernel.map,\n    rw [snake_diagram.mk_functor_map_g0, kernel.lift_ι],\n  end }\n.\n\nlemma horseshoe_ker_ι_fst [epi f.1] : (horseshoe_ker_ι f).1 = kernel.ι f.1 := rfl\n\nlemma horseshoe_ker_ι_snd [epi f.1] : (horseshoe_ker_ι f).2 = kernel.ι f.2 := rfl\n\nlemma horseshoe_ker_ι_trd [epi f.1] : (horseshoe_ker_ι f).3 = kernel.ι f.3 := rfl\n\nvariables (A)\n\nlemma horseshoe_ker_ι_comp_base_π :\n  (horseshoe_ker_ι (horseshoe_base_π A)) ≫ horseshoe_base_π A = 0 :=\nbegin\n  dsimp [horseshoe_ker_ι, horseshoe_base_π],\n  ext1; show kernel.ι _ ≫ _ = 0; exact exact.w exact_kernel_ι,\nend\n\nnoncomputable\ndef horseshoe_step (A : short_exact_sequence C) :\n  ℕ → Σ (X Y Z : short_exact_sequence C) (ι : X ⟶ Y), Y ⟶ Z\n| 0     := ⟨horseshoe_ker (horseshoe_base_π A), _, _, horseshoe_ker_ι _, horseshoe_base_π _⟩\n| (n+1) :=\n⟨horseshoe_ker (horseshoe_base_π (horseshoe_step n).1), _, _, horseshoe_ker_ι _, horseshoe_base_π _⟩\n\n@[reassoc] lemma horseshoe_step_comp_eq_zero :\n  ∀ n, (horseshoe_step A n).2.2.2.1 ≫ (horseshoe_step A n).2.2.2.2 = 0\n| 0     := horseshoe_ker_ι_comp_base_π _\n| (n+1) := horseshoe_ker_ι_comp_base_π _\n\nlemma step_fst_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.1 :=\nbegin\n  cases n,\n  { dsimp [horseshoe_step, horseshoe_ker_ι],\n    apply_instance },\n  { dsimp [horseshoe_step],\n    cases n, --Why do I have to do this again?!\n    { rw [horseshoe_ker_ι_fst],\n      apply_instance },\n    { rw [horseshoe_ker_ι_fst],\n      apply_instance }  }\nend\n\nlemma step_snd_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.2 :=\nbegin\n  cases n,\n  { dsimp [horseshoe_step, horseshoe_ker_ι],\n    apply_instance },\n  { dsimp [horseshoe_step],\n    cases n, --Why do I have to do this again?!\n    { rw [horseshoe_ker_ι_snd],\n      apply_instance },\n    { rw [horseshoe_ker_ι_snd],\n      apply_instance }  }\nend\n\nlemma step_trd_mono (n : ℕ) : mono (horseshoe_step A n).2.2.2.1.3 :=\nbegin\n  cases n,\n  { dsimp [horseshoe_step, horseshoe_ker_ι],\n    apply_instance },\n  { dsimp [horseshoe_step],\n    cases n, --Why do I have to do this again?!\n    { rw [horseshoe_ker_ι_trd],\n      apply_instance },\n    { rw [horseshoe_ker_ι_trd],\n      apply_instance }  }\nend\n\ndef horseshoe_obj (n : ℕ) := (horseshoe_step A n).2.1\n\ndef horseshoe_d (n : ℕ) : horseshoe_obj A (n+1) ⟶ horseshoe_obj A n :=\n(horseshoe_step A (n+1)).2.2.2.2 ≫ eq_to_hom (by { dsimp [horseshoe_step], refl })\n  ≫ (horseshoe_step A n).2.2.2.1\n\nlemma horseshoe_d_d (n : ℕ) : horseshoe_d A (n+1) ≫ horseshoe_d A n = 0 :=\nbegin\n  dsimp [horseshoe_d, horseshoe_ker_ι],\n  simp only [category.id_comp, category.assoc, comp_zero, zero_comp,\n    horseshoe_step_comp_eq_zero_assoc],\nend\n\ndef horseshoe (A : short_exact_sequence C) : chain_complex (short_exact_sequence C) ℕ :=\nchain_complex.of (horseshoe_obj A) (horseshoe_d A) (horseshoe_d_d A)\n\nvariables (A)\n\ndef horseshoe_π : (horseshoe A).X 0 ⟶ A := horseshoe_base_π _\n\nlemma horseshoe_d_π : (horseshoe A).d 1 0 ≫ horseshoe_π A = 0 :=\nbegin\n  dsimp [horseshoe],\n  erw [chain_complex.of_d],\n  dsimp [horseshoe_d, horseshoe_π, horseshoe_step],\n  simp only [category.id_comp, category.assoc, comp_zero, zero_comp,\n    horseshoe_step_comp_eq_zero_assoc, horseshoe_ker_ι_comp_base_π],\nend\n\ndef horseshoe_to_single₁ :=\n(chain_complex.to_single₀_equiv ((homological_complex.Fst C).obj (horseshoe A)) A.1).symm\n⟨(short_exact_sequence.Fst C).map (horseshoe_π A),\nbegin\n  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Fst C).map f) at this,\n  rwa [functor.map_comp, functor.map_zero] at this,\nend⟩\n\ndef horseshoe_to_single₂ :=\n(chain_complex.to_single₀_equiv ((homological_complex.Snd C).obj (horseshoe A)) A.2).symm\n⟨(short_exact_sequence.Snd C).map (horseshoe_π A),\nbegin\n  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Snd C).map f) at this,\n  rwa [functor.map_comp, functor.map_zero] at this,\nend⟩\n\ndef horseshoe_to_single₃ :=\n(chain_complex.to_single₀_equiv ((homological_complex.Trd C).obj (horseshoe A)) A.3).symm\n⟨(short_exact_sequence.Trd C).map (horseshoe_π A),\nbegin\n  have := horseshoe_d_π A, apply_fun (λ f, (short_exact_sequence.Trd C).map f) at this,\n  rwa [functor.map_comp, functor.map_zero] at this,\nend⟩\n\nlemma horseshoe_exact₁ (A : short_exact_sequence C) (n : ℕ) :\n  exact (((homological_complex.Fst C).obj (horseshoe A)).d (n + 2) (n + 1))\n    (((homological_complex.Fst C).obj (horseshoe A)).d (n + 1) n) :=\nbegin\n  dsimp [horseshoe_to_single₁],\n  erw [chain_complex.of_d, chain_complex.of_d],\n  dsimp [horseshoe_d, horseshoe_step],\n\n  set f := horseshoe_base_π (horseshoe_step A n).1,\n  set g := (horseshoe_step A n).2.2.2.1,\n\n  cases n;\n  convert exact_of_epi_comp_kernel.ι_comp_mono f.1 _ (horseshoe_base_π (horseshoe_ker f)).1 _\n    infer_instance _ _ using 1,\n  { simp [step_fst_mono] },\n  { simpa },\n  { simp [step_fst_mono] },\n  { simpa }\nend\n\nlemma horseshoe_exact₂ (A : short_exact_sequence C) (n : ℕ) :\n  exact (((homological_complex.Snd C).obj (horseshoe A)).d (n + 2) (n + 1))\n    (((homological_complex.Snd C).obj (horseshoe A)).d (n + 1) n) :=\nbegin\n  dsimp [horseshoe_to_single₂],\n  erw [chain_complex.of_d, chain_complex.of_d],\n  dsimp [horseshoe_d, horseshoe_step],\n\n  set f := horseshoe_base_π (horseshoe_step A n).1,\n  set g := (horseshoe_step A n).2.2.2.1,\n\n  cases n;\n  convert exact_of_epi_comp_kernel.ι_comp_mono f.2 _ (horseshoe_base_π (horseshoe_ker f)).2 _\n    infer_instance _ _ using 1,\n  { simp [step_snd_mono] },\n  { simpa },\n  { simp [step_snd_mono] },\n  { simpa }\nend\n\nlemma horseshoe_exact₃ (A : short_exact_sequence C) (n : ℕ) :\n  exact (((homological_complex.Trd C).obj (horseshoe A)).d (n + 2) (n + 1))\n    (((homological_complex.Trd C).obj (horseshoe A)).d (n + 1) n) :=\nbegin\n  dsimp [horseshoe_to_single₃],\n  erw [chain_complex.of_d, chain_complex.of_d],\n  dsimp [horseshoe_d, horseshoe_step],\n\n  set f := horseshoe_base_π (horseshoe_step A n).1,\n  set g := (horseshoe_step A n).2.2.2.1,\n\n  cases n;\n  convert exact_of_epi_comp_kernel.ι_comp_mono f.3 _ (horseshoe_base_π (horseshoe_ker f)).3 _\n    infer_instance _ _ using 1,\n  { simp [step_trd_mono] },\n  { simpa },\n  { simp [step_trd_mono] },\n  { simpa }\nend\n\nlemma horseshoe_is_projective_resolution₁ (A : short_exact_sequence C) :\n  chain_complex.is_projective_resolution\n    ((homological_complex.Fst C).obj (horseshoe A)) A.1 (horseshoe_to_single₁ A) :=\n{ projective := by rintro (_|n); { show projective (projective.over _), apply_instance },\n  exact₀ :=\n  begin\n    dsimp [horseshoe_to_single₁, chain_complex.to_single₀_equiv, horseshoe_π],\n    erw [chain_complex.of_d],\n    dsimp [horseshoe_d, horseshoe_step],\n    rw [category.id_comp, ← short_exact_sequence.comp_fst],\n    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,\n    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_fst, category.assoc,\n        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_fst,\n        abelian.pseudoelement.zero_apply] },\n    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,\n      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi\n        (horseshoe_base_π (horseshoe_ker _)).1 b,\n      refine ⟨c, _⟩,\n      rw [short_exact_sequence.comp_fst, abelian.pseudoelement.comp_apply, hc, ← hb],\n      refl }\n  end,\n  exact := λ n, horseshoe_exact₁ A n,\n  epi := show epi (projective.π _), from infer_instance }\n\nlemma horseshoe_is_projective_resolution₂ (A : short_exact_sequence C) :\n  chain_complex.is_projective_resolution\n    ((homological_complex.Snd C).obj (horseshoe A)) A.2 (horseshoe_to_single₂ A) :=\n{ projective := by rintro (_|n); { show projective (projective.over _ ⊞ projective.over _),\n    apply_instance },\n  exact₀ :=\n  begin\n    dsimp [horseshoe_to_single₂, chain_complex.to_single₀_equiv, horseshoe_π],\n    erw [chain_complex.of_d],\n    dsimp [horseshoe_d, horseshoe_step],\n    rw [category.id_comp, ← short_exact_sequence.comp_snd],\n    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,\n    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_snd, category.assoc,\n        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_snd,\n        abelian.pseudoelement.zero_apply] },\n    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,\n      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi\n        (horseshoe_base_π (horseshoe_ker _)).2 b,\n      refine ⟨c, _⟩,\n      rw [short_exact_sequence.comp_snd, abelian.pseudoelement.comp_apply, hc, ← hb],\n      refl }\n  end,\n  exact := λ n, horseshoe_exact₂ A n,\n  epi := show epi (horseshoe_base_π _).2, from infer_instance }\n\nlemma horseshoe_is_projective_resolution₃ (A : short_exact_sequence C) :\n  chain_complex.is_projective_resolution\n    ((homological_complex.Trd C).obj (horseshoe A)) A.3 (horseshoe_to_single₃ A) :=\n{ projective := by rintro (_|n); { show projective (projective.over _), apply_instance },\n  exact₀ :=\n  begin\n    dsimp [horseshoe_to_single₃, chain_complex.to_single₀_equiv, horseshoe_π],\n    erw [chain_complex.of_d],\n    dsimp [horseshoe_d, horseshoe_step],\n    rw [category.id_comp, ← short_exact_sequence.comp_trd],\n    refine abelian.pseudoelement.exact_of_pseudo_exact _ _ ⟨λ a , _, λ a ha, _⟩,\n    { rw [← abelian.pseudoelement.comp_apply, ← short_exact_sequence.comp_trd, category.assoc,\n        horseshoe_ker_ι_comp_base_π, comp_zero, short_exact_sequence.hom_zero_trd,\n        abelian.pseudoelement.zero_apply] },\n    { obtain ⟨b, hb⟩ := is_snake_input.exists_of_exact exact_kernel_ι _ ha,\n      obtain ⟨c, hc⟩ := abelian.pseudoelement.pseudo_surjective_of_epi\n        (horseshoe_base_π (horseshoe_ker _)).3 b,\n      refine ⟨c, _⟩,\n      rw [short_exact_sequence.comp_trd, abelian.pseudoelement.comp_apply, hc, ← hb],\n      refl }\n  end,\n  exact := λ n, horseshoe_exact₃ A n,\n  epi := show epi (projective.π _), from infer_instance }\n.\n\nlemma horseshoe_split (A : short_exact_sequence C) (n : ℕ) :\n  ((horseshoe A).X n).split :=\nbegin\n  cases n;\n  exact ⟨biprod.fst, biprod.inr, biprod.inl_fst, biprod.inr_snd, biprod.inr_fst, biprod.total⟩\nend\n\nlemma horseshoe_f_comp_to_single₂_f (A : short_exact_sequence C) (i : ℕ) :\n  ((horseshoe A).X i).f ≫ (horseshoe_to_single₂ A).f i =\n  (horseshoe_to_single₁ A).f i ≫ ((chain_complex.single₀ C).map A.f).f i :=\nbegin\n  cases i,\n  { dsimp [horseshoe_to_single₂, horseshoe_to_single₁, horseshoe, horseshoe_obj, horseshoe_step,\n      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],\n    simp },\n  { dsimp [horseshoe_to_single₂, horseshoe_to_single₁, horseshoe, horseshoe_obj, horseshoe_step,\n      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],\n    simp }\nend\n\nlemma horseshoe_g_comp_to_single₃_f (A : short_exact_sequence C) (i : ℕ) :\n  ((horseshoe A).X i).g ≫ (horseshoe_to_single₃ A).f i =\n  (horseshoe_to_single₂ A).f i ≫ ((chain_complex.single₀ C).map A.g).f i :=\nbegin\n  cases i,\n  { dsimp [horseshoe_to_single₂, horseshoe_to_single₃, chain_complex.to_single₀_equiv, horseshoe,\n      horseshoe_obj, horseshoe_π, horseshoe_step, horseshoe_base, horseshoe_base_π],\n    ext,\n    { simp },\n    { simp  } },\n  { dsimp [horseshoe_to_single₂, horseshoe_to_single₃, horseshoe, horseshoe_obj, horseshoe_step,\n      horseshoe_base, horseshoe_π, horseshoe_base_π, chain_complex.to_single₀_equiv],\n    simp }\nend\n\nend short_exact_sequence", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/lte/for_mathlib/horseshoe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547238, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.30507448517746316}}
{"text": "import category_theory.triangulated.rotate\n\nnamespace category_theory\n\nnamespace triangulated\n\nopen category_theory.triangulated.triangle_morphism\n\nvariables {C : Type*} [category C] [has_shift C ℤ] {T₁ T₂ : triangle C} (f : T₁ ⟶ T₂)\n\nlemma triangle_morphism_is_iso [is_iso f.hom₁] [is_iso f.hom₂] [is_iso f.hom₃] :\n  is_iso f :=\nby { refine ⟨⟨⟨inv f.hom₁, inv f.hom₂, inv f.hom₃, _, _, _⟩, _, _⟩⟩; tidy }\n.\ninstance [is_iso f] : is_iso f.hom₁ :=\nby { refine ⟨⟨(inv f).hom₁, _, _⟩⟩; simpa only [← comp_hom₁, ← triangle_category_comp,\n  is_iso.hom_inv_id, is_iso.inv_hom_id] }\n\ninstance [is_iso f] : is_iso f.hom₂ :=\nby { refine ⟨⟨(inv f).hom₂, _, _⟩⟩; simpa only [← comp_hom₂, ← triangle_category_comp,\n  is_iso.hom_inv_id, is_iso.inv_hom_id] }\n\ninstance [is_iso f] : is_iso f.hom₃ :=\nby { refine ⟨⟨(inv f).hom₃, _, _⟩⟩; simpa only [← comp_hom₃, ← triangle_category_comp,\n  is_iso.hom_inv_id, is_iso.inv_hom_id] }\n\n@[simp] lemma inv_hom₁ [is_iso f] : (inv f).hom₁ = inv (f.hom₁) :=\nby { ext, change (f ≫ inv f).hom₁ = _, rw is_iso.hom_inv_id, refl }\n\n@[simp] lemma inv_hom₂ [is_iso f] : (inv f).hom₂ = inv (f.hom₂) :=\nby { ext, change (f ≫ inv f).hom₂ = _, rw is_iso.hom_inv_id, refl }\n\n@[simp] lemma inv_hom₃ [is_iso f] : (inv f).hom₃ = inv (f.hom₃) :=\nby { ext, change (f ≫ inv f).hom₃ = _, rw is_iso.hom_inv_id, refl }\n\nlemma triangle_morphism_is_iso_iff : is_iso f ↔\n    is_iso f.hom₁ ∧ is_iso f.hom₂ ∧ is_iso f.hom₃ :=\nbegin\n  split,\n  { intro _, refine ⟨_, _, _⟩; exactI infer_instance },\n  { rintro ⟨_, _, _⟩, exactI triangle_morphism_is_iso f }\nend\n\n@[simps]\ndef mk_triangle_iso (e₁ : T₁.obj₁ ≅ T₂.obj₁) (e₂ : T₁.obj₂ ≅ T₂.obj₂) (e₃ : T₁.obj₃ ≅ T₂.obj₃)\n  (comm₁ : T₁.mor₁ ≫ e₂.hom = e₁.hom ≫ T₂.mor₁)\n  (comm₂ : T₁.mor₂ ≫ e₃.hom = e₂.hom ≫ T₂.mor₂)\n  (comm₃ : T₁.mor₃ ≫ e₁.hom⟦1⟧' = e₃.hom ≫ T₂.mor₃) : T₁ ≅ T₂ :=\n⟨⟨_, _, _, comm₁, comm₂, comm₃⟩, ⟨e₁.inv, e₂.inv, e₃.inv,\n  by { rw e₂.comp_inv_eq, simp [comm₁] }, by { rw e₃.comp_inv_eq, simp [comm₂] },\n  by { rw [e₃.eq_inv_comp, ← category.assoc, ← comm₃], simp [← functor.map_comp] }⟩,\n  by { ext; simp }, by { ext; simp }⟩\n\n@[simps] noncomputable\ndef triangle.nonneg_inv_rotate (T : triangle C) : triangle C :=\ntriangle.mk _ (T.mor₃⟦(-1:ℤ)⟧' ≫ (shift_shift_neg _ _).hom) T.mor₁\n  (T.mor₂ ≫ (shift_neg_shift _ _).inv)\n\n@[simps]\ndef triangle.nonneg_rotate {C : Type*} [category C]\n  [has_shift C ℤ] (T : triangle C) : triangle C := triangle.mk _ T.mor₂ T.mor₃ (T.mor₁⟦1⟧')\n\nvariables (C) [preadditive C]\n\n@[simps]\ndef neg₃_functor : triangle C ⥤ triangle C :=\n{ obj := λ T, triangle.mk C T.mor₁ T.mor₂ (-T.mor₃),\n  map := λ S T f, { hom₁ := f.hom₁, hom₂ := f.hom₂, hom₃ := f.hom₃ } }\n\n@[simps]\ndef neg₃_unit_iso : neg₃_functor C ⋙ neg₃_functor C ≅ 𝟭 _ :=\nbegin\n  refine nat_iso.of_components\n    (λ X, ⟨⟨𝟙 _, 𝟙 _, 𝟙 _, _, _, _⟩, ⟨𝟙 _, 𝟙 _, 𝟙 _, _, _, _⟩, _, _⟩) (λ X Y f, _),\n  any_goals { ext },\n  all_goals { dsimp,\n    simp only [category.comp_id, category.id_comp, category_theory.functor.map_id, neg_neg] },\nend\n\n@[simps]\ndef neg₃_equiv : triangle C ≌ triangle C :=\n{ functor := neg₃_functor C,\n  inverse := neg₃_functor C,\n  unit_iso := (neg₃_unit_iso C).symm,\n  counit_iso := neg₃_unit_iso C }\n\n--move this (do we want this?)\ninstance {C : Type*} [category C] [preadditive C] {X Y : C} : has_neg (X ≅ Y) :=\n⟨λ e, ⟨-e.hom, -e.inv, by simp, by simp⟩⟩\n.\n@[simp] lemma neg_iso_hom {C : Type*} [category C] [preadditive C] {X Y : C} {e : X ≅ Y} :\n  (-e).hom = -(e.hom) := rfl\n\n@[simp] lemma neg_iso_inv {C : Type*} [category C] [preadditive C] {X Y : C} {e : X ≅ Y} :\n  (-e).inv = -(e.inv) := rfl\n\n--move this\n@[simps]\ndef _root_.category_theory.equivalence.iso_equiv {C D : Type*} [category C] [category D]\n  (e : C ≌ D) (X : C) (Y : D) : (e.functor.obj X ≅ Y) ≃ (X ≅ e.inverse.obj Y) :=\n{ to_fun := λ f, e.unit_iso.app X ≪≫ e.inverse.map_iso f,\n  inv_fun := λ f, e.functor.map_iso f ≪≫ e.counit_iso.app Y,\n  left_inv := λ f, by { ext, dsimp, simp only [iso.inv_hom_id_app, category.assoc, functor.map_comp,\n    e.fun_inv_map], rw reassoc_of e.functor_unit_iso_comp, dsimp, simp },\n  right_inv := λ f, by { ext, dsimp, simp } }\n\nvariables {C} (T : triangle C)\n\nnoncomputable\ndef triangle.nonneg_inv_rotate_neg₃ [(shift_functor C (1 : ℤ)).additive] :\n  (neg₃_functor C).obj T.nonneg_inv_rotate ≅ T.inv_rotate :=\nbegin\n  fapply mk_triangle_iso,\n  exact -iso.refl _,\n  exact iso.refl _,\n  exact iso.refl _,\n  all_goals { dsimp, simp },\nend\n\nnoncomputable\ndef triangle.nonneg_inv_rotate_iso [(shift_functor C (1 : ℤ)).additive] :\n  T.nonneg_inv_rotate ≅ (neg₃_functor C).obj T.inv_rotate :=\n(neg₃_equiv C).iso_equiv _ _ T.nonneg_inv_rotate_neg₃\n\n@[simps]\ndef neg₃_rotate : neg₃_functor C ⋙ rotate C ≅ rotate C ⋙ neg₃_functor C :=\nnat_iso.of_components\n  (λ X, mk_triangle_iso (iso.refl _) (iso.refl _) (-iso.refl _)\n    (by { dsimp, simp }) (by { dsimp, simp }) (by { dsimp, simp }))\n  (λ X Y f, by { dsimp, ext; dsimp; simp })\n\nlemma triangle.nonneg_rotate_neg₃ :\n  (neg₃_functor C).obj T.nonneg_rotate = T.rotate := rfl\n\ndef triangle.nonneg_rotate_iso : T.nonneg_rotate ≅ (neg₃_functor C).obj T.rotate :=\n(neg₃_equiv C).iso_equiv _ _ (iso.refl _)\n\n--\n\n@[simps]\ndef triangle.obj₁_functor : triangle C ⥤ C :=\n{ obj := λ T, T.obj₁,\n  map := λ S T f, f.hom₁,\n  map_id' := λ _, rfl,\n  map_comp' := λ A B C f g, rfl }\n\n@[simps]\ndef triangle.obj₂_functor : triangle C ⥤ C :=\n{ obj := λ T, T.obj₂,\n  map := λ S T f, f.hom₂,\n  map_id' := λ _, rfl,\n  map_comp' := λ A B C f g, rfl }\n\n@[simps]\ndef triangle.obj₃_functor : triangle C ⥤ C :=\n{ obj := λ T, T.obj₃,\n  map := λ S T f, f.hom₃,\n  map_id' := λ _, rfl,\n  map_comp' := λ A B C f g, rfl }\n\nend triangulated\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/triangle.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.30507448517746316}}
{"text": "import category_theory.limits.preserves.limits\nimport category_theory.adjunction.evaluation\nimport algebra.group.ulift\n\nimport for_mathlib.ab4\nimport for_mathlib.AddCommGroup.exact\nimport for_mathlib.AddCommGroup.pt\nimport for_mathlib.AddCommGroup.ab4\n\nimport condensed.extr.equivalence\n\n.\n\nuniverse u\n\nopen category_theory\nopen category_theory.limits\n\ninstance : has_colimits (ExtrSheafProd.{u} Ab.{u+1}) :=\nhas_colimits_of_has_colimits_creates_colimits\n(Condensed_ExtrSheafProd_equiv _).inverse\n\ninstance : has_limits (ExtrSheafProd.{u} Ab.{u+1}) :=\nhas_limits_of_has_limits_creates_limits\n(Condensed_ExtrSheafProd_equiv _).inverse\n\ninstance : has_colimits_of_size.{u} Ab.{u+1} :=\nhas_colimits_of_size_shrink.{u u (u+1) (u+1)} Ab.{u+1}\n\nopen_locale classical\n\ninstance {C : Type (u+1)} [category.{u} C] : AB4 (C ⥤ Ab.{u+1}) :=\nbegin\n  constructor,\n  intros α X Y f hf,\n  let t := _, change mono t,\n  rw category_theory.nat_trans.mono_iff_app_mono,\n  intros c,\n  let eX : (∐ λ (a : α), X a).obj c ≅ (∐ λ (a : α), (X a).obj c) :=\n    (is_colimit_of_preserves ((evaluation _ _).obj c)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso (colimit.is_colimit _)\n    ≪≫ has_colimit.iso_of_nat_iso (discrete.nat_iso (λ _, iso.refl _)),\n   let eY : (∐ λ (a : α), Y a).obj c ≅ (∐ λ (a : α), (Y a).obj c) :=\n    (is_colimit_of_preserves ((evaluation _ _).obj c)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso (colimit.is_colimit _)\n    ≪≫ has_colimit.iso_of_nat_iso (discrete.nat_iso (λ _, iso.refl _)),\n  let tt : (∐ λ (a : α), (X a).obj c) ⟶ (∐ λ (a : α), (Y a).obj c) :=\n    sigma.desc (λ a, (f a).app c ≫ sigma.ι _ a),\n  haveI : mono tt,\n  { dsimp [tt],\n    apply AB4.cond, intros a, specialize hf a,\n    rw category_theory.nat_trans.mono_iff_app_mono at hf,\n    apply hf },\n  suffices : t.app c = eX.hom ≫ tt ≫ eY.inv,\n  { rw this, apply_instance },\n  dsimp [eX, tt, t, eY],\n  apply ((is_colimit_of_preserves ((evaluation _ _).obj c) (colimit.is_colimit _))).hom_ext,\n  intros a,\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  simp only [category.assoc, ← nat_trans.comp_app, colimit.ι_desc],\n  slice_rhs 0 1\n  { erw ((is_colimit_of_preserves ((evaluation _ _).obj c) (colimit.is_colimit _))).fac },\n  slice_rhs 0 1\n  { erw colimit.ι_desc },\n  dsimp,\n  simp only [category.id_comp, colimit.ι_desc, cofan.mk_ι_app, category.assoc,\n    has_colimit.iso_of_nat_iso_ι_inv, discrete.nat_iso_inv_app, functor.map_cocone_ι_app,\n    colimit.cocone_ι, evaluation_obj_map],\n  dsimp, simp, apply_instance\nend\n\nnoncomputable\ninstance : creates_limits (ExtrSheafProd_to_presheaf.{u} Ab.{u+1}) :=\nbegin\n  change creates_limits ((ExtrSheaf_ExtrSheafProd_equiv _).inverse ⋙\n    Sheaf_to_presheaf _ _),\n  apply_with category_theory.comp_creates_limits { instances := ff },\n  apply_instance,\n  exact @Sheaf.category_theory.Sheaf_to_presheaf.category_theory.creates_limits.{(u+2) u (u+1)}\n    ExtrDisc.{u} _ ExtrDisc.proetale_topology Ab.{u+1} _ _,\nend\n\ninstance : AB4 (ExtrSheafProd.{u} Ab.{u+1}) :=\nAB4_of_preserves_colimits_of_reflects_limits_of_AB4\n  (ExtrSheafProd_to_presheaf _)\n\n-- TODO: Prove a lemma about transporting AB4 across an equivalence.\ninstance : AB4 (Condensed.{u} Ab.{u+1}) :=\nAB4_of_preserves_colimits_of_reflects_limits_of_AB4\n    (Condensed_ExtrSheafProd_equiv.{u} Ab.{u+1}).functor\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/condensed/ab4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.3050502774100545}}
{"text": "import for_mathlib.derived.example\nimport breen_deligne.eval\n\nnoncomputable theory\n\nopen category_theory category_theory.preadditive\n\nnamespace breen_deligne\nnamespace package\n\nvariables (BD : package)\nvariables {𝒜 : Type*} [category 𝒜] [abelian 𝒜]\nvariables (F : 𝒜 ⥤ 𝒜)\n\ndef eval' : 𝒜 ⥤ cochain_complex 𝒜 ℤ :=\n(data.eval_functor F).obj BD.data ⋙ homological_complex.embed complex_shape.embedding.nat_down_int_up\n\ndef eval : 𝒜 ⥤ bounded_homotopy_category 𝒜 :=\n(data.eval_functor F).obj BD.data ⋙ chain_complex.to_bounded_homotopy_category\n\ninstance eval_additive : (BD.eval F).additive :=\nfunctor.additive_of_map_fst_add_snd _ $ λ A,\nbegin\n  refine homotopy_category.eq_of_homotopy _ _ _,\n  rw [← functor.map_add],\n  exact homological_complex.embed_homotopy _ _ (eval_functor_homotopy F BD A) _,\nend\n\nlemma eval_functor_obj_X (X : 𝒜) (n : ℕ) :\n  (((data.eval_functor F).obj BD.data).obj X).X n = F.obj ((Pow (BD.data.X n)).obj X) := rfl\n\nlemma eval_functor_obj_d (X : 𝒜) (m n : ℕ) :\n  (((data.eval_functor F).obj BD.data).obj X).d m n =\n    (universal_map.eval_Pow F (BD.data.d m n)).app X := rfl\n\nlemma eval'_obj_X (X : 𝒜) (n : ℕ) :\n  ((BD.eval' F).obj X).X (-n:ℤ) = F.obj ((Pow (BD.data.X n)).obj X) :=\nby { cases n; apply eval_functor_obj_X }\n\nlemma eval'_obj_X_0 (X : 𝒜) :\n  ((BD.eval' F).obj X).X 0 = F.obj ((Pow (BD.data.X 0)).obj X) := rfl\n\nlemma eval'_obj_X_succ (X : 𝒜) (n : ℕ) :\n  ((BD.eval' F).obj X).X -[1+ n] = F.obj ((Pow (BD.data.X (n+1))).obj X) := rfl\n\nlemma eval'_obj_d (X : 𝒜) (m n : ℕ) :\n  ((BD.eval' F).obj X).d (-(m+1:ℕ):ℤ) (-(n+1:ℕ):ℤ) =\n    (universal_map.eval_Pow F (BD.data.d (m+1) (n+1))).app X := rfl\n\nlemma eval'_obj_d_0 (X : 𝒜) (n : ℕ) :\n  ((BD.eval' F).obj X).d (-(n+1:ℕ):ℤ) (-(1:ℕ)+1:ℤ) =\n    (universal_map.eval_Pow F (BD.data.d (n+1) 0)).app X := rfl\n\nend package\nend breen_deligne\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/breen_deligne/eval2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.42632159254749036, "lm_q1q2_score": 0.3050006920556617}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Data.Option.Basic\n\nuniverse u v\n\ntheorem Option.eq_of_eq_some {α : Type u} : ∀ {x y : Option α}, (∀z, x = some z ↔ y = some z) → x = y\n  | none,   none,   h => rfl\n  | none,   some z, h => Option.noConfusion ((h z).2 rfl)\n  | some z, none,   h => Option.noConfusion ((h z).1 rfl)\n  | some z, some w, h => Option.noConfusion ((h w).2 rfl) (congrArg some)\n\ntheorem Option.eq_none_of_isNone {α : Type u} : ∀ {o : Option α}, o.isNone → o = none\n  | none, h => rfl\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/stage0/src/Init/Data/Option/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.30492725832042367}}
{"text": "/-\nFile: signature_recover_public_key_compute_doubling_slope_soundness.lean\n\nAutogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.hoare\nimport .signature_recover_public_key_code\nimport ..signature_recover_public_key_spec\nimport .signature_recover_public_key_verify_zero_soundness\nimport .signature_recover_public_key_unreduced_sqr_soundness\nimport .signature_recover_public_key_unreduced_mul_soundness\nimport .signature_recover_public_key_nondet_bigint3_soundness\nopen tactic\n\nopen starkware.cairo.common.cairo_secp.ec\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.field\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\nvariable  mem : F → F\nvariable  σ : register_state F\n\n/- starkware.cairo.common.cairo_secp.ec.compute_doubling_slope autogenerated soundness theorem -/\n\ntheorem auto_sound_compute_doubling_slope\n    -- arguments\n    (range_check_ptr : F) (point : EcPoint F)\n    -- code is in memory at σ.pc\n    (h_mem : mem_at mem code_compute_doubling_slope σ.pc)\n    -- all dependencies are in memory\n    (h_mem_4 : mem_at mem code_nondet_bigint3 (σ.pc  - 176))\n    (h_mem_5 : mem_at mem code_unreduced_mul (σ.pc  - 164))\n    (h_mem_6 : mem_at mem code_unreduced_sqr (σ.pc  - 144))\n    (h_mem_7 : mem_at mem code_verify_zero (σ.pc  - 128))\n    -- input arguments on the stack\n    (hin_range_check_ptr : range_check_ptr = mem (σ.fp - 9))\n    (hin_point : point = cast_EcPoint mem (σ.fp - 8))\n    -- conclusion\n  : ensures_ret mem σ (λ κ τ,\n      τ.ap = σ.ap + 85 ∧\n      ∃ μ ≤ κ, rc_ensures mem (rc_bound F) μ (mem (σ.fp - 9)) (mem $ τ.ap - 4)\n        (spec_compute_doubling_slope mem κ range_check_ptr point (mem (τ.ap - 4)) (cast_BigInt3 mem (τ.ap - 3)))) :=\nbegin\n  apply ensures_of_ensuresb, intro νbound,\n  have h_mem_rec := h_mem,\n  unpack_memory code_compute_doubling_slope at h_mem with ⟨hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19, hpc20, hpc21, hpc22, hpc23, hpc24, hpc25, hpc26, hpc27, hpc28, hpc29, hpc30, hpc31, hpc32, hpc33, hpc34, hpc35, hpc36, hpc37, hpc38, hpc39, hpc40, hpc41, hpc42, hpc43⟩,\n  -- function call\n  step_assert_eq hpc0 with arg0,\n  step_sub hpc1 (auto_sound_nondet_bigint3 mem _ range_check_ptr _ _),\n  { rw hpc2, norm_num2, exact h_mem_4 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point] },\n    try { dsimp [cast_EcPoint, cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  intros κ_call3 ap3 h_call3,\n  rcases h_call3 with ⟨h_call3_ap_offset, h_call3⟩,\n  rcases h_call3 with ⟨rc_m3, rc_mle3, hl_range_check_ptr₁, h_call3⟩,\n  generalize' hr_rev_range_check_ptr₁: mem (ap3 - 4) = range_check_ptr₁,\n  have htv_range_check_ptr₁ := hr_rev_range_check_ptr₁.symm, clear hr_rev_range_check_ptr₁,\n  generalize' hr_rev_slope: cast_BigInt3 mem (ap3 - 3) = slope,\n  simp only [hr_rev_slope] at h_call3,\n  have htv_slope := hr_rev_slope.symm, clear hr_rev_slope,\n  try { simp only [arg0] at hl_range_check_ptr₁ },\n  rw [←htv_range_check_ptr₁, ←hin_range_check_ptr] at hl_range_check_ptr₁,\n  try { simp only [arg0] at h_call3 },\n  rw [hin_range_check_ptr] at h_call3,\n  clear arg0,\n  -- function call\n  step_assert_eq hpc3 with arg0,\n  step_assert_eq hpc4 with arg1,\n  step_assert_eq hpc5 with arg2,\n  step_sub hpc6 (auto_sound_unreduced_sqr mem _ point.x _ _),\n  { rw hpc7, norm_num2, exact h_mem_6 },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope] },\n      try { dsimp [cast_EcPoint, cast_BigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2] },\n      try { simp only [h_call3_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros κ_call8 ap8 h_call8,\n  rcases h_call8 with ⟨h_call8_ap_offset, h_call8⟩,\n  generalize' hr_rev_x_sqr: cast_UnreducedBigInt3 mem (ap8 - 3) = x_sqr,\n  simp only [hr_rev_x_sqr] at h_call8,\n  have htv_x_sqr := hr_rev_x_sqr.symm, clear hr_rev_x_sqr,\n  clear arg0 arg1 arg2,\n  -- function call\n  step_assert_eq hpc8 with arg0,\n  step_assert_eq hpc9 with arg1,\n  step_assert_eq hpc10 with arg2,\n  step_assert_eq hpc11 with arg3,\n  step_assert_eq hpc12 with arg4,\n  step_assert_eq hpc13 with arg5,\n  step_sub hpc14 (auto_sound_unreduced_mul mem _ slope point.y _ _ _),\n  { rw hpc15, norm_num2, exact h_mem_5 },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope, htv_x_sqr] },\n      try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5] },\n      try { simp only [h_call3_ap_offset, h_call8_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope, htv_x_sqr] },\n      try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5] },\n      try { simp only [h_call3_ap_offset, h_call8_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros κ_call16 ap16 h_call16,\n  rcases h_call16 with ⟨h_call16_ap_offset, h_call16⟩,\n  generalize' hr_rev_slope_y: cast_UnreducedBigInt3 mem (ap16 - 3) = slope_y,\n  simp only [hr_rev_slope_y] at h_call16,\n  have htv_slope_y := hr_rev_slope_y.symm, clear hr_rev_slope_y,\n  clear arg0 arg1 arg2 arg3 arg4 arg5,\n  -- function call\n  step_assert_eq hpc16 hpc17 with arg0,\n  step_assert_eq hpc18 with arg1,\n  step_assert_eq hpc19 hpc20 with arg2,\n  step_assert_eq hpc21 with arg3,\n  step_assert_eq hpc22 hpc23 with arg4,\n  step_assert_eq hpc24 with arg5,\n  step_assert_eq hpc25 hpc26 with arg6,\n  step_assert_eq hpc27 with arg7,\n  step_assert_eq hpc28 hpc29 with arg8,\n  step_assert_eq hpc30 with arg9,\n  step_assert_eq hpc31 hpc32 with arg10,\n  step_assert_eq hpc33 with arg11,\n  step_assert_eq hpc34 with arg12,\n  step_assert_eq hpc35 with arg13,\n  step_assert_eq hpc36 with arg14,\n  step_assert_eq hpc37 with arg15,\n  step_sub hpc38 (auto_sound_verify_zero mem _ range_check_ptr₁ {\n    d0 := 3 * x_sqr.d0 - 2 * slope_y.d0,\n    d1 := 3 * x_sqr.d1 - 2 * slope_y.d1,\n    d2 := 3 * x_sqr.d2 - 2 * slope_y.d2\n  } _ _ _),\n  { rw hpc39, norm_num2, exact h_mem_7 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope, htv_x_sqr, htv_slope_y] },\n    try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n    try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11, arg12, (eq_sub_of_eq_add arg13), (eq_sub_of_eq_add arg14), (eq_sub_of_eq_add arg15)] },\n    try { simp only [h_call3_ap_offset, h_call8_ap_offset, h_call16_ap_offset] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope, htv_x_sqr, htv_slope_y] },\n      try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10, arg11, arg12, (eq_sub_of_eq_add arg13), (eq_sub_of_eq_add arg14), (eq_sub_of_eq_add arg15)] },\n      try { simp only [h_call3_ap_offset, h_call8_ap_offset, h_call16_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros κ_call40 ap40 h_call40,\n  rcases h_call40 with ⟨h_call40_ap_offset, h_call40⟩,\n  rcases h_call40 with ⟨rc_m40, rc_mle40, hl_range_check_ptr₂, h_call40⟩,\n  generalize' hr_rev_range_check_ptr₂: mem (ap40 - 1) = range_check_ptr₂,\n  have htv_range_check_ptr₂ := hr_rev_range_check_ptr₂.symm, clear hr_rev_range_check_ptr₂,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10 ,arg11 ,arg12 ,arg13 ,arg14 ,arg15] at hl_range_check_ptr₂ },\n  try { rw [h_call16_ap_offset, h_call8_ap_offset] at hl_range_check_ptr₂ }, try { arith_simps at hl_range_check_ptr₂ },\n  rw [←htv_range_check_ptr₂, ←htv_range_check_ptr₁] at hl_range_check_ptr₂,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10 ,arg11 ,arg12 ,arg13 ,arg14 ,arg15] at h_call40 },\n  try { rw [h_call16_ap_offset, h_call8_ap_offset] at h_call40 }, try { arith_simps at h_call40 },\n  rw [←htv_range_check_ptr₁, hl_range_check_ptr₁, hin_range_check_ptr] at h_call40,\n  clear arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10 arg11 arg12 arg13 arg14 arg15,\n  -- return\n  step_assert_eq hpc40 with hret0,\n  step_assert_eq hpc41 with hret1,\n  step_assert_eq hpc42 with hret2,\n  step_ret hpc43,\n  -- finish\n  step_done, use_only [rfl, rfl],\n  split,\n  { try { simp only [h_call3_ap_offset ,h_call8_ap_offset ,h_call16_ap_offset ,h_call40_ap_offset] },\n    try { arith_simps }, try { refl } },\n  -- range check condition\n  use_only (rc_m3+rc_m40+0+0), split,\n  linarith [rc_mle3, rc_mle40],\n  split,\n  { arith_simps, try { simp only [hret0 ,hret1 ,hret2] },\n    rw [←htv_range_check_ptr₂, hl_range_check_ptr₂, hl_range_check_ptr₁, hin_range_check_ptr],\n    try { arith_simps, refl <|> norm_cast }, try { refl } },\n  intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },\n  have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,\n  -- Final Proof\n  -- user-provided reduction\n  suffices auto_spec: auto_spec_compute_doubling_slope mem _ range_check_ptr point _ _,\n  { apply sound_compute_doubling_slope, apply auto_spec },\n  -- prove the auto generated assertion\n  dsimp [auto_spec_compute_doubling_slope],\n  try { norm_num1 }, try { arith_simps },\n  use_only [κ_call3],\n  use_only [range_check_ptr₁],\n  use_only [slope],\n  have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,\n  have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },\n  have spec3 := h_call3 rc_h_range_check_ptr',\n  rw [←hin_range_check_ptr, ←htv_range_check_ptr₁] at spec3,\n  try { dsimp at spec3, arith_simps at spec3 },\n  use_only [spec3],\n  use_only [κ_call8],\n  use_only [x_sqr],\n  try { dsimp at h_call8, arith_simps at h_call8 },\n  try { use_only [h_call8] },\n  use_only [κ_call16],\n  use_only [slope_y],\n  try { dsimp at h_call16, arith_simps at h_call16 },\n  try { use_only [h_call16] },\n  use_only [κ_call40],\n  use_only [range_check_ptr₂],\n  have rc_h_range_check_ptr₂ := range_checked_offset' rc_h_range_check_ptr₁,\n  have rc_h_range_check_ptr₂' := range_checked_add_right rc_h_range_check_ptr₂, try { norm_cast at rc_h_range_check_ptr₂' },\n  have spec40 := h_call40 rc_h_range_check_ptr₁',\n  rw [←hin_range_check_ptr, ←hl_range_check_ptr₁, ←htv_range_check_ptr₂] at spec40,\n  try { dsimp at spec40, arith_simps at spec40 },\n  use_only [spec40],\n  try { split, linarith },\n  try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr₁, htv_slope, htv_x_sqr, htv_slope_y, htv_range_check_ptr₂] }, },\n  try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n  try { arith_simps }, try { simp only [hret0, hret1, hret2] },\n  try { simp only [h_call3_ap_offset, h_call8_ap_offset, h_call16_ap_offset, h_call40_ap_offset] },\n  try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },\nend\n\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_compute_doubling_slope_soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3047433473409021}}
{"text": "/-\nCopyright (c) 2014 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n\nVarious multiplicative and additive structures.\n-/\nimport tactic.interactive data.option\n\nsection pending_1857\n\n/- Transport multiplicative to additive -/\nsection transport\nopen tactic\n\n@[user_attribute]\nmeta def to_additive_attr : user_attribute (name_map name) name :=\n{ name      := `to_additive,\n  descr     := \"Transport multiplicative to additive\",\n  cache_cfg := ⟨λ ns, ns.mfoldl (λ dict n, do\n    val ← to_additive_attr.get_param n,\n    pure $ dict.insert n val) mk_name_map, []⟩,\n  parser    := lean.parser.ident,\n  after_set := some $ λ src _ _, do\n    env ← get_env,\n    dict ← to_additive_attr.get_cache,\n    tgt ← to_additive_attr.get_param src,\n    (get_decl tgt >> skip) <|>\n      transport_with_dict dict src tgt }\n\nend transport\n\n/- map operations -/\nattribute [to_additive has_add.add] has_mul.mul\nattribute [to_additive has_zero.zero] has_one.one\nattribute [to_additive has_neg.neg] has_inv.inv\nattribute [to_additive has_add] has_mul\nattribute [to_additive has_zero] has_one\nattribute [to_additive has_neg] has_inv\n\n/- map constructors -/\nattribute [to_additive has_add.mk] has_mul.mk\nattribute [to_additive has_zero.mk] has_one.mk\nattribute [to_additive has_neg.mk] has_inv.mk\n\n/- map structures -/\nattribute [to_additive add_semigroup] semigroup\nattribute [to_additive add_semigroup.mk] semigroup.mk\nattribute [to_additive add_semigroup.to_has_add] semigroup.to_has_mul\nattribute [to_additive add_semigroup.add_assoc] semigroup.mul_assoc\nattribute [to_additive add_semigroup.add] semigroup.mul\n\nattribute [to_additive add_comm_semigroup] comm_semigroup\nattribute [to_additive add_comm_semigroup.mk] comm_semigroup.mk\nattribute [to_additive add_comm_semigroup.to_add_semigroup] comm_semigroup.to_semigroup\nattribute [to_additive add_comm_semigroup.add_comm] comm_semigroup.mul_comm\n\nattribute [to_additive add_left_cancel_semigroup] left_cancel_semigroup\nattribute [to_additive add_left_cancel_semigroup.mk] left_cancel_semigroup.mk\nattribute [to_additive add_left_cancel_semigroup.to_add_semigroup] left_cancel_semigroup.to_semigroup\nattribute [to_additive add_left_cancel_semigroup.add_left_cancel] left_cancel_semigroup.mul_left_cancel\n\nattribute [to_additive add_right_cancel_semigroup] right_cancel_semigroup\nattribute [to_additive add_right_cancel_semigroup.mk] right_cancel_semigroup.mk\nattribute [to_additive add_right_cancel_semigroup.to_add_semigroup] right_cancel_semigroup.to_semigroup\nattribute [to_additive add_right_cancel_semigroup.add_right_cancel] right_cancel_semigroup.mul_right_cancel\n\nattribute [to_additive add_monoid] monoid\nattribute [to_additive add_monoid.mk] monoid.mk\nattribute [to_additive add_monoid.to_has_zero] monoid.to_has_one\nattribute [to_additive add_monoid.to_add_semigroup] monoid.to_semigroup\nattribute [to_additive add_monoid.add] monoid.mul\nattribute [to_additive add_monoid.add_assoc] monoid.mul_assoc\nattribute [to_additive add_monoid.zero] monoid.one\nattribute [to_additive add_monoid.zero_add] monoid.one_mul\nattribute [to_additive add_monoid.add_zero] monoid.mul_one\n\nattribute [to_additive add_comm_monoid] comm_monoid\nattribute [to_additive add_comm_monoid.mk] comm_monoid.mk\nattribute [to_additive add_comm_monoid.to_add_monoid] comm_monoid.to_monoid\nattribute [to_additive add_comm_monoid.to_add_comm_semigroup] comm_monoid.to_comm_semigroup\n\nattribute [to_additive add_group] group\nattribute [to_additive add_group.mk] group.mk\nattribute [to_additive add_group.to_has_neg] group.to_has_inv\nattribute [to_additive add_group.to_add_monoid] group.to_monoid\nattribute [to_additive add_group.add_left_neg] group.mul_left_inv\nattribute [to_additive add_group.add] group.mul\nattribute [to_additive add_group.add_assoc] group.mul_assoc\nattribute [to_additive add_group.zero] group.one\nattribute [to_additive add_group.zero_add] group.one_mul\nattribute [to_additive add_group.add_zero] group.mul_one\nattribute [to_additive add_group.neg] group.inv\n\nattribute [to_additive add_comm_group] comm_group\nattribute [to_additive add_comm_group.mk] comm_group.mk\nattribute [to_additive add_comm_group.to_add_group] comm_group.to_group\nattribute [to_additive add_comm_group.to_add_comm_monoid] comm_group.to_comm_monoid\n\n/- map theorems -/\nattribute [to_additive add_assoc] mul_assoc\nattribute [to_additive add_semigroup_to_is_associative] semigroup_to_is_associative\nattribute [to_additive add_comm] mul_comm\nattribute [to_additive add_comm_semigroup_to_is_commutative] comm_semigroup_to_is_commutative\nattribute [to_additive add_left_comm] mul_left_comm\nattribute [to_additive add_right_comm] mul_right_comm\nattribute [to_additive add_left_cancel] mul_left_cancel\nattribute [to_additive add_right_cancel] mul_right_cancel\nattribute [to_additive add_left_cancel_iff] mul_left_cancel_iff\nattribute [to_additive add_right_cancel_iff] mul_right_cancel_iff\nattribute [to_additive zero_add] one_mul\nattribute [to_additive add_zero] mul_one\nattribute [to_additive add_left_neg] mul_left_inv\nattribute [to_additive neg_add_self] inv_mul_self\nattribute [to_additive neg_add_cancel_left] inv_mul_cancel_left\nattribute [to_additive neg_add_cancel_right] inv_mul_cancel_right\nattribute [to_additive neg_eq_of_add_eq_zero] inv_eq_of_mul_eq_one\nattribute [to_additive neg_zero] one_inv\nattribute [to_additive neg_neg] inv_inv\nattribute [to_additive add_right_neg] mul_right_inv\nattribute [to_additive add_neg_self] mul_inv_self\nattribute [to_additive neg_inj] inv_inj\nattribute [to_additive add_group.add_left_cancel] group.mul_left_cancel\nattribute [to_additive add_group.add_right_cancel] group.mul_right_cancel\nattribute [to_additive add_group.to_left_cancel_add_semigroup] group.to_left_cancel_semigroup\nattribute [to_additive add_group.to_right_cancel_add_semigroup] group.to_right_cancel_semigroup\nattribute [to_additive add_neg_cancel_left] mul_inv_cancel_left\nattribute [to_additive add_neg_cancel_right] mul_inv_cancel_right\nattribute [to_additive neg_add_rev] mul_inv_rev\nattribute [to_additive eq_neg_of_eq_neg] eq_inv_of_eq_inv\nattribute [to_additive eq_neg_of_add_eq_zero] eq_inv_of_mul_eq_one\nattribute [to_additive eq_add_neg_of_add_eq] eq_mul_inv_of_mul_eq\nattribute [to_additive eq_neg_add_of_add_eq] eq_inv_mul_of_mul_eq\nattribute [to_additive neg_add_eq_of_eq_add] inv_mul_eq_of_eq_mul\nattribute [to_additive add_neg_eq_of_eq_add] mul_inv_eq_of_eq_mul\nattribute [to_additive eq_add_of_add_neg_eq] eq_mul_of_mul_inv_eq\nattribute [to_additive eq_add_of_neg_add_eq] eq_mul_of_inv_mul_eq\nattribute [to_additive add_eq_of_eq_neg_add] mul_eq_of_eq_inv_mul\nattribute [to_additive add_eq_of_eq_add_neg] mul_eq_of_eq_mul_inv\nattribute [to_additive neg_add] mul_inv\n\nend pending_1857\n\nuniverses u v\nvariables {α : Type u} {β : Type v}\n\ndef additive (α : Type*) := α\ndef multiplicative (α : Type*) := α\n\ninstance [semigroup α] : add_semigroup (additive α) :=\n{ add := ((*) : α → α → α),\n  add_assoc := @mul_assoc _ _ }\n\ninstance [add_semigroup α] : semigroup (multiplicative α) :=\n{ mul := ((+) : α → α → α),\n  mul_assoc := @add_assoc _ _ }\n\ninstance [comm_semigroup α] : add_comm_semigroup (additive α) :=\n{ add_comm := @mul_comm _ _,\n  ..additive.add_semigroup }\n\ninstance [add_comm_semigroup α] : comm_semigroup (multiplicative α) :=\n{ mul_comm := @add_comm _ _,\n  ..multiplicative.semigroup }\n\ninstance [left_cancel_semigroup α] : add_left_cancel_semigroup (additive α) :=\n{ add_left_cancel := @mul_left_cancel _ _,\n  ..additive.add_semigroup }\n\ninstance [add_left_cancel_semigroup α] : left_cancel_semigroup (multiplicative α) :=\n{ mul_left_cancel := @add_left_cancel _ _,\n  ..multiplicative.semigroup }\n\ninstance [right_cancel_semigroup α] : add_right_cancel_semigroup (additive α) :=\n{ add_right_cancel := @mul_right_cancel _ _,\n  ..additive.add_semigroup }\n\ninstance [add_right_cancel_semigroup α] : right_cancel_semigroup (multiplicative α) :=\n{ mul_right_cancel := @add_right_cancel _ _,\n  ..multiplicative.semigroup }\n\n@[simp, to_additive add_left_inj]\ntheorem mul_left_inj [left_cancel_semigroup α] (a : α) {b c : α} : a * b = a * c ↔ b = c :=\n⟨mul_left_cancel, congr_arg _⟩\n\n@[simp, to_additive add_right_inj]\ntheorem mul_right_inj [right_cancel_semigroup α] (a : α) {b c : α} : b * a = c * a ↔ b = c :=\n⟨mul_right_cancel, congr_arg _⟩\n\nstructure units (α : Type u) [monoid α] :=\n(val : α)\n(inv : α)\n(val_inv : val * inv = 1)\n(inv_val : inv * val = 1)\n\nnamespace units\nvariables [monoid α] {a b c : units α}\n\ninstance : has_coe (units α) α := ⟨val⟩\n\n@[extensionality] theorem ext : ∀ {a b : units α}, (a : α) = b → a = b\n| ⟨v, i₁, vi₁, iv₁⟩ ⟨v', i₂, vi₂, iv₂⟩ e :=\n  by change v = v' at e; subst v'; congr;\n      simpa [iv₂, vi₁] using mul_assoc i₂ v i₁\n\ntheorem ext_iff {a b : units α} : a = b ↔ (a : α) = b :=\n⟨congr_arg _, ext⟩\n\ninstance [decidable_eq α] : decidable_eq (units α)\n| a b := decidable_of_iff' _ ext_iff\n\nprotected def mul (u₁ u₂ : units α) : units α :=\n⟨u₁.val * u₂.val, u₂.inv * u₁.inv,\n  have u₁.val * (u₂.val * u₂.inv) * u₁.inv = 1,\n    by rw [u₂.val_inv]; simp [u₁.val_inv],\n  by simpa [mul_comm, mul_assoc],\n  have u₂.inv * (u₁.inv * u₁.val) * u₂.val = 1,\n    by rw [u₁.inv_val]; simp [u₂.inv_val],\n  by simpa [mul_comm, mul_assoc]⟩\n\nprotected def inv' (u : units α) : units α :=\n⟨u.inv, u.val, u.inv_val, u.val_inv⟩\n\ninstance : has_mul (units α) := ⟨units.mul⟩\ninstance : has_one (units α) := ⟨⟨1, 1, mul_one 1, one_mul 1⟩⟩\ninstance : has_inv (units α) := ⟨units.inv'⟩\n\nvariables (a b)\n@[simp] lemma mul_coe : (↑(a * b) : α) = a * b := rfl\n@[simp] lemma one_coe : ((1 : units α) : α) = 1 := rfl\nlemma val_coe : (↑a : α) = a.val := rfl\nlemma inv_coe : ((a⁻¹ : units α) : α) = a.inv := rfl\n@[simp] lemma inv_mul : (↑a⁻¹ * a : α) = 1 := inv_val _\n@[simp] lemma mul_inv : (a * ↑a⁻¹ : α) = 1 := val_inv _\n\n@[simp] lemma mul_inv_cancel_left (a : units α) (b : α) : (a:α) * (↑a⁻¹ * b) = b :=\nby rw [← mul_assoc, mul_inv, one_mul]\n\n@[simp] lemma inv_mul_cancel_left (a : units α) (b : α) : (↑a⁻¹:α) * (a * b) = b :=\nby rw [← mul_assoc, inv_mul, one_mul]\n\n@[simp] lemma mul_inv_cancel_right (a : α) (b : units α) : a * b * ↑b⁻¹ = a :=\nby rw [mul_assoc, mul_inv, mul_one]\n\n@[simp] lemma inv_mul_cancel_right (a : α) (b : units α) : a * ↑b⁻¹ * b = a :=\nby rw [mul_assoc, inv_mul, mul_one]\n\ninstance : group (units α) :=\nby refine {mul := (*), one := 1, inv := has_inv.inv, ..};\n  { intros, apply ext, simp [mul_assoc] }\n\ninstance {α} [comm_monoid α] : comm_group (units α) :=\n{ mul_comm := λ u₁ u₂, ext $ mul_comm _ _, ..units.group }\n\ninstance [has_repr α] : has_repr (units α) := ⟨repr ∘ val⟩\n\n@[simp] theorem mul_left_inj (a : units α) {b c : α} : (a:α) * b = a * c ↔ b = c :=\n⟨λ h, by simpa using congr_arg ((*) ↑(a⁻¹ : units α)) h, congr_arg _⟩\n\n@[simp] theorem mul_right_inj (a : units α) {b c : α} : b * a = c * a ↔ b = c :=\n⟨λ h, by simpa using congr_arg (* ↑(a⁻¹ : units α)) h, congr_arg _⟩\n\nend units\n\ntheorem nat.units_eq_one (u : units ℕ) : u = 1 :=\nunits.ext begin\n  cases nat.eq_zero_or_pos u with h h,\n  { simpa [h] using u.inv_mul },\n  cases nat.eq_zero_or_pos ↑u⁻¹ with h' h',\n  { simpa [h'] using u.inv_mul },\n  refine le_antisymm _ h,\n  simpa using nat.mul_le_mul_left u h'\nend\n\ndef units.mk_of_mul_eq_one [comm_monoid α] (a b : α) (hab : a * b = 1) : units α :=\n⟨a, b, hab, by rwa mul_comm a b at hab⟩\n\n@[to_additive with_zero]\ndef with_one (α) := option α\n\n@[to_additive with_zero.has_coe_t]\ninstance : has_coe_t α (with_one α) := ⟨some⟩\n\ninstance [semigroup α] : monoid (with_one α) :=\n{ one       := none,\n  mul       := option.lift_or_get (*),\n  mul_assoc := (option.lift_or_get_assoc _).1,\n  one_mul   := (option.lift_or_get_is_left_id _).1,\n  mul_one   := (option.lift_or_get_is_right_id _).1 }\n\nattribute [to_additive with_zero.add_monoid._proof_1] with_one.monoid._proof_1\nattribute [to_additive with_zero.add_monoid._proof_2] with_one.monoid._proof_2\nattribute [to_additive with_zero.add_monoid._proof_3] with_one.monoid._proof_3\nattribute [to_additive with_zero.add_monoid] with_one.monoid\n\ninstance [semigroup α] : mul_zero_class (with_zero α) :=\n{ zero      := none,\n  mul       := λ o₁ o₂, o₁.bind (λ a, o₂.map (λ b, a * b)),\n  zero_mul  := by simp,\n  mul_zero  := λ a, by cases a; simp,\n  ..with_zero.add_monoid }\n\ninstance [semigroup α] : semigroup (with_zero α) :=\n{ mul_assoc := λ a b c, begin\n    cases a with a, {refl},\n    cases b with b, {refl},\n    cases c with c, {refl},\n    simp [mul_assoc]\n  end,\n  ..with_zero.mul_zero_class }\n\ninstance [comm_semigroup α] : comm_semigroup (with_zero α) :=\n{ mul_comm := λ a b, begin\n    cases a with a; cases b with b; try {refl},\n    exact congr_arg some (mul_comm _ _)\n  end,\n  ..with_zero.semigroup }\n\ninstance [monoid α] : monoid (with_zero α) :=\n{ one := some 1,\n  one_mul := λ a, by cases a;\n    [refl, exact congr_arg some (one_mul a)],\n  mul_one := λ a, by cases a;\n    [refl, exact congr_arg some (mul_one a)],\n  ..with_zero.semigroup }\n\ninstance [comm_monoid α] : comm_monoid (with_zero α) :=\n{ ..with_zero.monoid, ..with_zero.comm_semigroup }\n\ninstance [monoid α] : add_monoid (additive α) :=\n{ zero     := (1 : α),\n  zero_add := @one_mul _ _,\n  add_zero := @mul_one _ _,\n  ..additive.add_semigroup }\n\ninstance [add_monoid α] : monoid (multiplicative α) :=\n{ one     := (0 : α),\n  one_mul := @zero_add _ _,\n  mul_one := @add_zero _ _,\n  ..multiplicative.semigroup }\n\nsection monoid\n  variables [monoid α] {a b c : α}\n\n  /-- Partial division. It is defined when the\n    second argument is invertible, and unlike the division operator\n    in `division_ring` it is not totalized at zero. -/\n  def divp (a : α) (u) : α := a * (u⁻¹ : units α)\n\n  infix ` /ₚ `:70 := divp\n\n  @[simp] theorem divp_self (u : units α) : (u : α) /ₚ u = 1 := by simp [divp]\n\n  @[simp] theorem divp_one (a : α) : a /ₚ 1 = a := by simp [divp]\n\n  theorem divp_assoc (a b : α) (u : units α) : a * b /ₚ u = a * (b /ₚ u) :=\n  by simp [divp, mul_assoc]\n\n  @[simp] theorem divp_mul_cancel (a : α) (u : units α) : a /ₚ u * u = a :=\n  by simp [divp, mul_assoc]\n\n  @[simp] theorem mul_divp_cancel (a : α) (u : units α) : (a * u) /ₚ u = a :=\n  by simp [divp, mul_assoc]\n\n  @[simp] theorem divp_right_inj (u : units α) {a b : α} : a /ₚ u = b /ₚ u ↔ a = b :=\n  units.mul_right_inj _\n\n  theorem divp_eq_one (a : α) (u : units α) : a /ₚ u = 1 ↔ a = u :=\n  (units.mul_right_inj u).symm.trans $ by simp\n\n  @[simp] theorem one_divp (u : units α) : 1 /ₚ u = ↑u⁻¹ :=\n  by simp [divp]\n\nend monoid\n\ninstance [comm_semigroup α] : comm_monoid (with_one α) :=\n{ mul_comm := (option.lift_or_get_comm _).1,\n  ..with_one.monoid }\n\ninstance [add_comm_semigroup α] : add_comm_monoid (with_zero α) :=\n{ add_comm := (option.lift_or_get_comm _).1,\n  ..with_zero.add_monoid }\nattribute [to_additive with_zero.add_comm_monoid] with_one.comm_monoid\n\ninstance [comm_monoid α] : add_comm_monoid (additive α) :=\n{ add_comm := @mul_comm α _,\n  ..additive.add_monoid }\n\ninstance [add_comm_monoid α] : comm_monoid (multiplicative α) :=\n{ mul_comm := @add_comm α _,\n  ..multiplicative.monoid }\n\ninstance [group α] : add_group (additive α) :=\n{ neg := @has_inv.inv α _,\n  add_left_neg := @mul_left_inv _ _,\n  ..additive.add_monoid }\n\ninstance [add_group α] : group (multiplicative α) :=\n{ inv := @has_neg.neg α _,\n  mul_left_inv := @add_left_neg _ _,\n  ..multiplicative.monoid }\n\nsection group\n  variables [group α] {a b c : α}\n\n  instance : has_lift α (units α) :=\n  ⟨λ a, ⟨a, a⁻¹, mul_inv_self _, inv_mul_self _⟩⟩\n\n  @[simp, to_additive neg_inj']\n  theorem inv_inj' : a⁻¹ = b⁻¹ ↔ a = b :=\n  ⟨λ h, by rw ← inv_inv a; simp [h], congr_arg _⟩\n\n  @[to_additive eq_of_neg_eq_neg]\n  theorem eq_of_inv_eq_inv : a⁻¹ = b⁻¹ → a = b :=\n  inv_inj'.1\n\n  @[simp, to_additive add_self_iff_eq_zero]\n  theorem mul_self_iff_eq_one : a * a = a ↔ a = 1 :=\n  by have := @mul_left_inj _ _ a a 1; rwa mul_one at this\n\n  @[simp, to_additive neg_eq_zero]\n  theorem inv_eq_one : a⁻¹ = 1 ↔ a = 1 :=\n  by rw [← @inv_inj' _ _ a 1, one_inv]\n\n  @[simp, to_additive neg_ne_zero]\n  theorem inv_ne_one : a⁻¹ ≠ 1 ↔ a ≠ 1 :=\n  not_congr inv_eq_one\n\n  @[to_additive left_inverse_neg]\n  theorem left_inverse_inv (α) [group α] :\n    function.left_inverse (λ a : α, a⁻¹) (λ a, a⁻¹) :=\n  assume a, inv_inv a\n\n  attribute [simp] mul_inv_cancel_left add_neg_cancel_left\n                   mul_inv_cancel_right add_neg_cancel_right\n\n  @[to_additive eq_neg_iff_eq_neg]\n  theorem eq_inv_iff_eq_inv : a = b⁻¹ ↔ b = a⁻¹ :=\n  ⟨eq_inv_of_eq_inv, eq_inv_of_eq_inv⟩\n\n  @[to_additive neg_eq_iff_neg_eq]\n  theorem inv_eq_iff_inv_eq : a⁻¹ = b ↔ b⁻¹ = a :=\n  by rw [eq_comm, @eq_comm _ _ a, eq_inv_iff_eq_inv]\n\n  @[to_additive add_eq_zero_iff_eq_neg]\n  theorem mul_eq_one_iff_eq_inv : a * b = 1 ↔ a = b⁻¹ :=\n  by simpa [mul_left_inv, -mul_right_inj] using @mul_right_inj _ _ b a (b⁻¹)\n\n  @[to_additive add_eq_zero_iff_neg_eq]\n  theorem mul_eq_one_iff_inv_eq : a * b = 1 ↔ a⁻¹ = b :=\n  by rw [mul_eq_one_iff_eq_inv, eq_inv_iff_eq_inv, eq_comm]\n\n  @[to_additive eq_neg_iff_add_eq_zero]\n  theorem eq_inv_iff_mul_eq_one : a = b⁻¹ ↔ a * b = 1 :=\n  mul_eq_one_iff_eq_inv.symm\n\n  @[to_additive neg_eq_iff_add_eq_zero]\n  theorem inv_eq_iff_mul_eq_one : a⁻¹ = b ↔ a * b = 1 :=\n  mul_eq_one_iff_inv_eq.symm\n\n  @[to_additive eq_add_neg_iff_add_eq]\n  theorem eq_mul_inv_iff_mul_eq : a = b * c⁻¹ ↔ a * c = b :=\n  ⟨λ h, by simp [h], λ h, by simp [h.symm]⟩\n\n  @[to_additive eq_neg_add_iff_add_eq]\n  theorem eq_inv_mul_iff_mul_eq : a = b⁻¹ * c ↔ b * a = c :=\n  ⟨λ h, by simp [h], λ h, by simp [h.symm]⟩\n\n  @[to_additive neg_add_eq_iff_eq_add]\n  theorem inv_mul_eq_iff_eq_mul : a⁻¹ * b = c ↔ b = a * c :=\n  ⟨λ h, by simp [h.symm], λ h, by simp [h]⟩\n\n  @[to_additive add_neg_eq_iff_eq_add]\n  theorem mul_inv_eq_iff_eq_mul : a * b⁻¹ = c ↔ a = c * b :=\n  ⟨λ h, by simp [h.symm], λ h, by simp [h]⟩\n\n  @[to_additive add_neg_eq_zero]\n  theorem mul_inv_eq_one {a b : α} : a * b⁻¹ = 1 ↔ a = b :=\n  by rw [mul_eq_one_iff_eq_inv, inv_inv]\n\n  @[to_additive neg_comm_of_comm]\n  theorem inv_comm_of_comm {a b : α} (H : a * b = b * a) : a⁻¹ * b = b * a⁻¹ :=\n  begin\n    have : a⁻¹ * (b * a) * a⁻¹ = a⁻¹ * (a * b) * a⁻¹ :=\n      congr_arg (λ x:α, a⁻¹ * x * a⁻¹) H.symm,\n    rwa [mul_assoc, mul_assoc, mul_inv_self, mul_one,\n        ← mul_assoc, inv_mul_self, one_mul] at this; exact h\n  end\n\nend group\n\ninstance [comm_group α] : add_comm_group (additive α) :=\n{ add_comm := @mul_comm α _,\n  ..additive.add_group }\n\ninstance [add_comm_group α] : comm_group (multiplicative α) :=\n{ mul_comm := @add_comm α _,\n  ..multiplicative.group }\n\nsection add_monoid\n  variables [add_monoid α] {a b c : α}\n\n  @[simp] lemma bit0_zero : bit0 (0 : α) = 0 := add_zero _\n  @[simp] lemma bit1_zero [has_one α] : bit1 (0 : α) = 1 := by simp [bit1]\n\nend add_monoid\n\nsection add_group\n  variables [add_group α] {a b c : α}\n\n  local attribute [simp] sub_eq_add_neg\n\n  def sub_sub_cancel := @sub_sub_self\n\n  @[simp] lemma sub_left_inj : a - b = a - c ↔ b = c :=\n  (add_left_inj _).trans neg_inj'\n\n  @[simp] lemma sub_right_inj : b - a = c - a ↔ b = c :=\n  add_right_inj _\n\n  lemma sub_add_sub_cancel (a b c : α) : (a - b) + (b - c) = a - c :=\n  by simp\n\n  lemma sub_sub_sub_cancel_right (a b c : α) : (a - c) - (b - c) = a - b :=\n  by simp\n\n  theorem sub_eq_zero : a - b = 0 ↔ a = b :=\n  ⟨eq_of_sub_eq_zero, λ h, by simp [h]⟩\n\n  theorem sub_ne_zero : a - b ≠ 0 ↔ a ≠ b :=\n  not_congr sub_eq_zero\n\n  theorem eq_sub_iff_add_eq : a = b - c ↔ a + c = b :=\n  by split; intro h; simp [h, eq_add_neg_iff_add_eq]\n\n  theorem sub_eq_iff_eq_add : a - b = c ↔ a = c + b :=\n  by split; intro h; simp [*, add_neg_eq_iff_eq_add] at *\n\n  theorem eq_iff_eq_of_sub_eq_sub {a b c d : α} (H : a - b = c - d) : a = b ↔ c = d :=\n  by rw [← sub_eq_zero, H, sub_eq_zero]\n\n  theorem left_inverse_sub_add_left (c : α) : function.left_inverse (λ x, x - c) (λ x, x + c) :=\n  assume x, add_sub_cancel x c\n\n  theorem left_inverse_add_left_sub (c : α) : function.left_inverse (λ x, x + c) (λ x, x - c) :=\n  assume x, sub_add_cancel x c\n\n  theorem left_inverse_add_right_neg_add (c : α) :\n      function.left_inverse (λ x, c + x) (λ x, - c + x) :=\n  assume x, add_neg_cancel_left c x\n\n  theorem left_inverse_neg_add_add_right (c : α) :\n      function.left_inverse (λ x, - c + x) (λ x, c + x) :=\n  assume x, neg_add_cancel_left c x\nend add_group\n\nsection add_comm_group\n  variables [add_comm_group α] {a b c : α}\n\n  lemma sub_eq_neg_add (a b : α) : a - b = -b + a :=\n  by simp\n\n  theorem neg_add' (a b : α) : -(a + b) = -a - b := neg_add a b\n\n  lemma eq_sub_iff_add_eq' : a = b - c ↔ c + a = b :=\n  by rw [eq_sub_iff_add_eq, add_comm]\n\n  lemma sub_eq_iff_eq_add' : a - b = c ↔ a = b + c :=\n  by rw [sub_eq_iff_eq_add, add_comm]\n\n  lemma add_sub_cancel' (a b : α) : a + b - a = b :=\n  by simp\n\n  lemma add_sub_cancel'_right (a b : α) : a + (b - a) = b :=\n  by rw [← add_sub_assoc, add_sub_cancel']\n\n  lemma sub_right_comm (a b c : α) : a - b - c = a - c - b :=\n  by simp\n\n  lemma sub_add_sub_cancel' (a b c : α) : (a - b) + (c - a) = c - b :=\n  by rw add_comm; apply sub_add_sub_cancel\n\n  lemma sub_sub_sub_cancel_left (a b c : α) : (c - a) - (c - b) = b - a :=\n  by simp\n\nend add_comm_group\n\nsection is_conj\nvariables [group α] [group β]\n\ndef is_conj (a b : α) := ∃ c : α, c * a * c⁻¹ = b\n\n@[refl] lemma is_conj_refl (a : α) : is_conj a a := ⟨1, by simp⟩\n\n@[symm] lemma is_conj_symm (a b : α) : is_conj a b → is_conj b a\n| ⟨c, hc⟩ := ⟨c⁻¹, by simp [hc.symm, mul_assoc]⟩\n\n@[trans] lemma is_conj_trans (a b c : α) : is_conj a b → is_conj b c → is_conj a c\n| ⟨c₁, hc₁⟩ ⟨c₂, hc₂⟩ := ⟨c₂ * c₁, by simp [hc₁.symm, hc₂.symm, mul_assoc]⟩\n\n@[simp] lemma is_conj_iff_eq {α : Type*} [comm_group α] {a b : α} : is_conj a b ↔ a = b :=\n⟨λ ⟨c, hc⟩, by rw [← hc, mul_right_comm, mul_inv_self, one_mul], λ h, by rw h⟩\n\nend is_conj\n\nclass is_monoid_hom [monoid α] [monoid β] (f : α → β) : Prop :=\n(map_one : f 1 = 1)\n(map_mul : ∀ {x y}, f (x * y) = f x * f y)\n\nclass is_add_monoid_hom [add_monoid α] [add_monoid β] (f : α → β) : Prop :=\n(map_zero : f 0 = 0)\n(map_add : ∀ {x y}, f (x + y) = f x + f y)\n\nattribute [to_additive is_add_monoid_hom] is_monoid_hom\nattribute [to_additive is_add_monoid_hom.map_add] is_monoid_hom.map_mul\nattribute [to_additive is_add_monoid_hom.mk] is_monoid_hom.mk\n\nnamespace is_monoid_hom\nvariables [monoid α] [monoid β] (f : α → β) [is_monoid_hom f]\n\n@[to_additive is_add_monoid_hom.id]\ninstance id : is_monoid_hom (@id α) := by refine {..}; intros; refl\n\n@[to_additive is_add_monoid_hom.id]\ninstance comp {γ} [monoid γ] (g : β → γ) [is_monoid_hom g] :\n  is_monoid_hom (g ∘ f) :=\n{ map_mul := λ x y, by simp [map_mul f]; rw map_mul g; refl,\n  map_one := by simp [map_one f]; exact map_one g }\n\nend is_monoid_hom\n\n-- TODO rename fields of is_group_hom: mul ↝ map_mul?\n\n/-- Predicate for group homomorphism. -/\nclass is_group_hom [group α] [group β] (f : α → β) : Prop :=\n(mul : ∀ a b : α, f (a * b) = f a * f b)\n\nclass is_add_group_hom [add_group α] [add_group β] (f : α → β) : Prop :=\n(add : ∀ a b, f (a + b) = f a + f b)\n\nattribute [to_additive is_add_group_hom] is_group_hom\nattribute [to_additive is_add_group_hom.add] is_group_hom.mul\nattribute [to_additive is_add_group_hom.mk] is_group_hom.mk\n\nnamespace is_group_hom\nvariables [group α] [group β] (f : α → β) [is_group_hom f]\n\n@[to_additive is_add_group_hom.zero]\ntheorem one : f 1 = 1 :=\nmul_self_iff_eq_one.1 $ by simp [(mul f 1 1).symm]\n\n@[to_additive is_add_group_hom.neg]\ntheorem inv (a : α) : f a⁻¹ = (f a)⁻¹ :=\neq.symm $ inv_eq_of_mul_eq_one $ by simp [(mul f a a⁻¹).symm, one f]\n\n@[to_additive is_add_group_hom.id]\ninstance id : is_group_hom (@id α) :=\n⟨λ _ _, rfl⟩\n\n@[to_additive is_add_group_hom.comp]\ninstance comp {γ} [group γ] (g : β → γ) [is_group_hom g] :\n  is_group_hom (g ∘ f) :=\n⟨λ x y, calc\n  g (f (x * y)) = g (f x * f y)       : by rw mul f\n  ...           = g (f x) * g (f y)   : by rw mul g⟩\n\nprotected lemma is_conj (f : α → β) [is_group_hom f] {a b : α} : is_conj a b → is_conj (f a) (f b)\n| ⟨c, hc⟩ := ⟨f c, by rw [← is_group_hom.mul f, ← is_group_hom.inv f, ← is_group_hom.mul f, hc]⟩\n\nend is_group_hom\n\n/-- Predicate for group anti-homomorphism, or a homomorphism\n  into the opposite group. -/\nclass is_group_anti_hom {β : Type*} [group α] [group β] (f : α → β) : Prop :=\n(mul : ∀ a b : α, f (a * b) = f b * f a)\n\nnamespace is_group_anti_hom\nvariables [group α] [group β] (f : α → β) [w : is_group_anti_hom f]\ninclude w\n\ntheorem one : f 1 = 1 :=\nmul_self_iff_eq_one.1 $ by simp [(mul f 1 1).symm]\n\ntheorem inv (a : α) : f a⁻¹ = (f a)⁻¹ :=\neq.symm $ inv_eq_of_mul_eq_one $ by simp [(mul f a⁻¹ a).symm, one f]\n\nend is_group_anti_hom\n\ntheorem inv_is_group_anti_hom [group α] : is_group_anti_hom (λ x : α, x⁻¹) :=\n⟨mul_inv_rev⟩\n\nnamespace is_add_group_hom\nvariables [add_group α] [add_group β] (f : α → β) [is_add_group_hom f]\n\nlemma sub (a b) : f (a - b) = f a - f b :=\ncalc f (a - b) = f (a + -b)   : rfl\n           ... = f a + f (-b) : add f _ _\n           ... = f a - f b    : by  simp[neg f]\n\nend is_add_group_hom\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/algebra/group.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3047433473409021}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport category_theory.category.Bipointed\nimport data.two_pointing\n\n/-!\n# The category of two-pointed types\n\nThis defines `Twop`, the category of two-pointed types.\n\n## References\n\n* [nLab, *coalgebra of the real interval*]\n  (https://ncatlab.org/nlab/show/coalgebra+of+the+real+interval)\n-/\n\nopen category_theory option\n\nuniverses u\nvariables {α β : Type*}\n\n/-- The category of two-pointed types. -/\nstructure Twop : Type.{u+1} :=\n(X : Type.{u})\n(to_two_pointing : two_pointing X)\n\nnamespace Twop\n\ninstance : has_coe_to_sort Twop Type* := ⟨X⟩\n\nattribute [protected] Twop.X\n\n/-- Turns a two-pointing into a two-pointed type. -/\ndef of {X : Type*} (to_two_pointing : two_pointing X) : Twop := ⟨X, to_two_pointing⟩\n\n@[simp] lemma coe_of {X : Type*} (to_two_pointing : two_pointing X) :\n  ↥(of to_two_pointing) = X := rfl\n\nalias of ← two_pointing.Twop\n\ninstance : inhabited Twop := ⟨of two_pointing.bool⟩\n\n/-- Turns a two-pointed type into a bipointed type, by forgetting that the pointed elements are\ndistinct. -/\ndef to_Bipointed (X : Twop) : Bipointed := X.to_two_pointing.to_prod.Bipointed\n\n@[simp] lemma coe_to_Bipointed (X : Twop) : ↥X.to_Bipointed = ↥X := rfl\n\ninstance large_category : large_category Twop := induced_category.category to_Bipointed\n\ninstance concrete_category : concrete_category Twop :=\ninduced_category.concrete_category to_Bipointed\n\ninstance has_forget_to_Bipointed : has_forget₂ Twop Bipointed :=\ninduced_category.has_forget₂ to_Bipointed\n\n/-- Swaps the pointed elements of a two-pointed type. `two_pointing.swap` as a functor. -/\n@[simps] def swap : Twop ⥤ Twop :=\n{ obj := λ X, ⟨X, X.to_two_pointing.swap⟩, map := λ X Y f, ⟨f.to_fun, f.map_snd, f.map_fst⟩ }\n\n/-- The equivalence between `Twop` and itself induced by `prod.swap` both ways. -/\n@[simps] def swap_equiv : Twop ≌ Twop :=\nequivalence.mk swap swap\n  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, { hom := ⟨id, rfl, rfl⟩, inv := ⟨id, rfl, rfl⟩ }) $ λ X Y f, rfl)\n\n@[simp] lemma swap_equiv_symm : swap_equiv.symm = swap_equiv := rfl\n\nend Twop\n\n@[simp] lemma Twop_swap_comp_forget_to_Bipointed :\n  Twop.swap ⋙ forget₂ Twop Bipointed = forget₂ Twop Bipointed ⋙ Bipointed.swap := rfl\n\n/-- The functor from `Pointed` to `Twop` which adds a second point. -/\n@[simps] def Pointed_to_Twop_fst : Pointed.{u} ⥤ Twop :=\n{ obj := λ X, ⟨option X, ⟨X.point, none⟩, some_ne_none _⟩,\n  map := λ X Y f, ⟨option.map f.to_fun, congr_arg _ f.map_point, rfl⟩,\n  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,\n  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }\n\n/-- The functor from `Pointed` to `Twop` which adds a first point. -/\n@[simps] def Pointed_to_Twop_snd : Pointed.{u} ⥤ Twop :=\n{ obj := λ X, ⟨option X, ⟨none, X.point⟩, (some_ne_none _).symm⟩,\n  map := λ X Y f, ⟨option.map f.to_fun, rfl, congr_arg _ f.map_point⟩,\n  map_id' := λ X, Bipointed.hom.ext _ _ option.map_id,\n  map_comp' := λ X Y Z f g, Bipointed.hom.ext _ _ (option.map_comp_map  _ _).symm }\n\n@[simp] lemma Pointed_to_Twop_fst_comp_swap :\n  Pointed_to_Twop_fst ⋙ Twop.swap = Pointed_to_Twop_snd := rfl\n\n@[simp] lemma Pointed_to_Twop_snd_comp_swap :\n  Pointed_to_Twop_snd ⋙ Twop.swap = Pointed_to_Twop_fst := rfl\n\n@[simp] lemma Pointed_to_Twop_fst_comp_forget_to_Bipointed :\n  Pointed_to_Twop_fst ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_fst := rfl\n\n@[simp] lemma Pointed_to_Twop_snd_comp_forget_to_Bipointed :\n  Pointed_to_Twop_snd ⋙ forget₂ Twop Bipointed = Pointed_to_Bipointed_snd := rfl\n\n/-- Adding a second point is left adjoint to forgetting the second point. -/\ndef Pointed_to_Twop_fst_forget_comp_Bipointed_to_Pointed_fst_adjunction :\n  Pointed_to_Twop_fst ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_fst :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_fst⟩,\n    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.2 f.to_fun, f.map_point, rfl⟩,\n    left_inv := λ f, by { ext, cases x, exact f.map_snd.symm, refl },\n    right_inv := λ f, Pointed.hom.ext _ _ rfl },\n  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }\n\n/-- Adding a first point is left adjoint to forgetting the first point. -/\ndef Pointed_to_Twop_snd_forget_comp_Bipointed_to_Pointed_snd_adjunction :\n  Pointed_to_Twop_snd ⊣ forget₂ Twop Bipointed ⋙ Bipointed_to_Pointed_snd :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, ⟨f.to_fun ∘ option.some, f.map_snd⟩,\n    inv_fun := λ f, ⟨λ o, o.elim Y.to_two_pointing.to_prod.1 f.to_fun, rfl, f.map_point⟩,\n    left_inv := λ f, by { ext, cases x, exact f.map_fst.symm, refl },\n    right_inv := λ f, Pointed.hom.ext _ _ rfl },\n  hom_equiv_naturality_left_symm' := λ X' X Y f g, by { ext, cases x; refl } }\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/category/Twop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3047433473409021}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.functor.currying\nimport category_theory.limits.over\nimport category_theory.limits.shapes.images\nimport category_theory.adjunction.reflective\n\n/-!\n# Monomorphisms over a fixed object\n\nAs preparation for defining `subobject X`, we set up the theory for\n`mono_over X := {f : over X // mono f.hom}`.\n\nHere `mono_over X` is a thin category (a pair of objects has at most one morphism between them),\nso we can think of it as a preorder. However as it is not skeletal, it is not yet a partial order.\n\n`subobject X` will be defined as the skeletalization of `mono_over X`.\n\nWe provide\n* `def pullback [has_pullbacks C] (f : X ⟶ Y) : mono_over Y ⥤ mono_over X`\n* `def map (f : X ⟶ Y) [mono f] : mono_over X ⥤ mono_over Y`\n* `def «exists» [has_images C] (f : X ⟶ Y) : mono_over X ⥤ mono_over Y`\nand prove their basic properties and relationships.\n\n## Notes\n\nThis development originally appeared in Bhavik Mehta's \"Topos theory for Lean\" repository,\nand was ported to mathlib by Scott Morrison.\n\n-/\n\nuniverses v₁ v₂ u₁ u₂\n\nnoncomputable theory\nnamespace category_theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u₁} [category.{v₁} C] {X Y Z : C}\nvariables {D : Type u₂} [category.{v₂} D]\n\n/--\nThe category of monomorphisms into `X` as a full subcategory of the over category.\nThis isn't skeletal, so it's not a partial order.\n\nLater we define `subobject X` as the quotient of this by isomorphisms.\n-/\n@[derive [category]]\ndef mono_over (X : C) := {f : over X // mono f.hom}\n\nnamespace mono_over\n\n/-- Construct a `mono_over X`. -/\n@[simps]\ndef mk' {X A : C} (f : A ⟶ X) [hf : mono f] : mono_over X := { val := over.mk f, property := hf }\n\n/-- The inclusion from monomorphisms over X to morphisms over X. -/\ndef forget (X : C) : mono_over X ⥤ over X := full_subcategory_inclusion _\n\ninstance : has_coe (mono_over X) C :=\n{ coe := λ Y, Y.val.left, }\n\n@[simp]\nlemma forget_obj_left {f} : ((forget X).obj f).left = (f : C) := rfl\n\n@[simp] lemma mk'_coe' {X A : C} (f : A ⟶ X) [hf : mono f] : (mk' f : C) = A := rfl\n\n/-- Convenience notation for the underlying arrow of a monomorphism over X. -/\nabbreviation arrow (f : mono_over X) : (f : C) ⟶ X := ((forget X).obj f).hom\n\n@[simp] lemma mk'_arrow {X A : C} (f : A ⟶ X) [hf : mono f] : (mk' f).arrow = f := rfl\n\n@[simp]\nlemma forget_obj_hom {f} : ((forget X).obj f).hom = f.arrow := rfl\n\ninstance : full (forget X) := full_subcategory.full _\ninstance : faithful (forget X) := full_subcategory.faithful _\n\ninstance mono (f : mono_over X) : mono f.arrow := f.property\n\n/-- The category of monomorphisms over X is a thin category,\nwhich makes defining its skeleton easy. -/\ninstance is_thin {X : C} (f g : mono_over X) : subsingleton (f ⟶ g) :=\n⟨begin\n  intros h₁ h₂,\n  ext1,\n  erw [← cancel_mono g.arrow, over.w h₁, over.w h₂],\nend⟩\n\n@[reassoc] lemma w {f g : mono_over X} (k : f ⟶ g) : k.left ≫ g.arrow = f.arrow := over.w _\n\n/-- Convenience constructor for a morphism in monomorphisms over `X`. -/\nabbreviation hom_mk {f g : mono_over X} (h : f.val.left ⟶ g.val.left) (w : h ≫ g.arrow = f.arrow) :\n  f ⟶ g :=\nover.hom_mk h w\n\n/-- Convenience constructor for an isomorphism in monomorphisms over `X`. -/\n@[simps]\ndef iso_mk {f g : mono_over X} (h : f.val.left ≅ g.val.left) (w : h.hom ≫ g.arrow = f.arrow) :\n  f ≅ g :=\n{ hom := hom_mk h.hom w,\n  inv := hom_mk h.inv (by rw [h.inv_comp_eq, w]) }\n\n/-- If `f : mono_over X`, then `mk' f.arrow` is of course just `f`, but not definitionally, so we\n    package it as an isomorphism. -/\n@[simp] def mk'_arrow_iso {X : C} (f : mono_over X) : (mk' f.arrow) ≅ f :=\niso_mk (iso.refl _) (by simp)\n\n/--\nLift a functor between over categories to a functor between `mono_over` categories,\ngiven suitable evidence that morphisms are taken to monomorphisms.\n-/\n@[simps]\ndef lift {Y : D} (F : over Y ⥤ over X)\n  (h : ∀ (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) :\n  mono_over Y ⥤ mono_over X :=\n{ obj := λ f, ⟨_, h f⟩,\n  map := λ _ _ k, (mono_over.forget X).preimage ((mono_over.forget Y ⋙ F).map k), }\n\n/--\nIsomorphic functors `over Y ⥤ over X` lift to isomorphic functors `mono_over Y ⥤ mono_over X`.\n-/\ndef lift_iso {Y : D} {F₁ F₂ : over Y ⥤ over X} (h₁ h₂) (i : F₁ ≅ F₂) :\n  lift F₁ h₁ ≅ lift F₂ h₂ :=\nfully_faithful_cancel_right (mono_over.forget X) (iso_whisker_left (mono_over.forget Y) i)\n\n/-- `mono_over.lift` commutes with composition of functors. -/\ndef lift_comp {X Z : C} {Y : D} (F : over X ⥤ over Y) (G : over Y ⥤ over Z) (h₁ h₂) :\n  lift F h₁ ⋙ lift G h₂ ≅ lift (F ⋙ G) (λ f, h₂ ⟨_, h₁ f⟩) :=\nfully_faithful_cancel_right (mono_over.forget _) (iso.refl _)\n\n/-- `mono_over.lift` preserves the identity functor. -/\ndef lift_id :\n  lift (𝟭 (over X)) (λ f, f.2) ≅ 𝟭 _ :=\nfully_faithful_cancel_right (mono_over.forget _) (iso.refl _)\n\n@[simp]\nlemma lift_comm (F : over Y ⥤ over X)\n  (h : ∀ (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) :\n  lift F h ⋙ mono_over.forget X = mono_over.forget Y ⋙ F :=\nrfl\n\n@[simp]\nlemma lift_obj_arrow {Y : D} (F : over Y ⥤ over X)\n  (h : ∀ (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) (f : mono_over Y) :\n  ((lift F h).obj f).arrow = (F.obj ((forget Y).obj f)).hom :=\nrfl\n\n/--\nMonomorphisms over an object `f : over A` in an over category\nare equivalent to monomorphisms over the source of `f`.\n-/\ndef slice {A : C} {f : over A} (h₁ h₂) : mono_over f ≌ mono_over f.left :=\n{ functor := mono_over.lift f.iterated_slice_equiv.functor h₁,\n  inverse := mono_over.lift f.iterated_slice_equiv.inverse h₂,\n  unit_iso := mono_over.lift_id.symm ≪≫\n    mono_over.lift_iso _ _ f.iterated_slice_equiv.unit_iso ≪≫\n    (mono_over.lift_comp _ _ _ _).symm,\n  counit_iso := mono_over.lift_comp _ _ _ _ ≪≫\n    mono_over.lift_iso _ _ f.iterated_slice_equiv.counit_iso ≪≫\n    mono_over.lift_id }\n\nsection pullback\nvariables [has_pullbacks C]\n\n/-- When `C` has pullbacks, a morphism `f : X ⟶ Y` induces a functor `mono_over Y ⥤ mono_over X`,\nby pulling back a monomorphism along `f`. -/\ndef pullback (f : X ⟶ Y) : mono_over Y ⥤ mono_over X :=\nmono_over.lift (over.pullback f)\nbegin\n  intro g,\n  apply @pullback.snd_of_mono _ _ _ _ _ _ _ _ _,\n  change mono g.arrow,\n  apply_instance,\nend\n\n/-- pullback commutes with composition (up to a natural isomorphism) -/\ndef pullback_comp (f : X ⟶ Y) (g : Y ⟶ Z) : pullback (f ≫ g) ≅ pullback g ⋙ pullback f :=\nlift_iso _ _ (over.pullback_comp _ _) ≪≫ (lift_comp _ _ _ _).symm\n\n/-- pullback preserves the identity (up to a natural isomorphism) -/\ndef pullback_id : pullback (𝟙 X) ≅ 𝟭 _ :=\nlift_iso _ _ over.pullback_id ≪≫ lift_id\n\n@[simp] lemma pullback_obj_left (f : X ⟶ Y) (g : mono_over Y) :\n  (((pullback f).obj g) : C) = limits.pullback g.arrow f :=\nrfl\n\n@[simp] lemma pullback_obj_arrow (f : X ⟶ Y) (g : mono_over Y) :\n  ((pullback f).obj g).arrow = pullback.snd :=\nrfl\n\nend pullback\n\nsection map\n\nattribute [instance] mono_comp\n\n/--\nWe can map monomorphisms over `X` to monomorphisms over `Y`\nby post-composition with a monomorphism `f : X ⟶ Y`.\n-/\ndef map (f : X ⟶ Y) [mono f] : mono_over X ⥤ mono_over Y :=\nlift (over.map f)\n(λ g, by apply mono_comp g.arrow f)\n\n/-- `mono_over.map` commutes with composition (up to a natural isomorphism). -/\ndef map_comp (f : X ⟶ Y) (g : Y ⟶ Z) [mono f] [mono g] :\n  map (f ≫ g) ≅ map f ⋙ map g :=\nlift_iso _ _ (over.map_comp _ _) ≪≫ (lift_comp _ _ _ _).symm\n\n/-- `mono_over.map` preserves the identity (up to a natural isomorphism). -/\ndef map_id : map (𝟙 X) ≅ 𝟭 _ :=\nlift_iso _ _ over.map_id ≪≫ lift_id\n\n@[simp] lemma map_obj_left (f : X ⟶ Y) [mono f] (g : mono_over X) :\n  (((map f).obj g) : C) = g.val.left :=\nrfl\n\n@[simp]\nlemma map_obj_arrow (f : X ⟶ Y) [mono f] (g : mono_over X) :\n  ((map f).obj g).arrow = g.arrow ≫ f :=\nrfl\n\ninstance full_map (f : X ⟶ Y) [mono f] : full (map f) :=\n{ preimage := λ g h e,\n  begin\n    refine hom_mk e.left _,\n    rw [← cancel_mono f, assoc],\n    apply w e,\n  end }\n\ninstance faithful_map (f : X ⟶ Y) [mono f] : faithful (map f) := {}.\n\n/--\nIsomorphic objects have equivalent `mono_over` categories.\n-/\n@[simps] def map_iso {A B : C} (e : A ≅ B) : mono_over A ≌ mono_over B :=\n{ functor := map e.hom,\n  inverse := map e.inv,\n  unit_iso := ((map_comp _ _).symm ≪≫ eq_to_iso (by simp) ≪≫ map_id).symm,\n  counit_iso := ((map_comp _ _).symm ≪≫ eq_to_iso (by simp) ≪≫ map_id) }\n\nsection\nvariables (X)\n\n/-- An equivalence of categories `e` between `C` and `D` induces an equivalence between\n    `mono_over X` and `mono_over (e.functor.obj X)` whenever `X` is an object of `C`. -/\n@[simps] def congr (e : C ≌ D) : mono_over X ≌ mono_over (e.functor.obj X) :=\n{ functor := lift (over.post e.functor) $ λ f, by { dsimp, apply_instance },\n  inverse := (lift (over.post e.inverse) $ λ f, by { dsimp, apply_instance })\n    ⋙ (map_iso (e.unit_iso.symm.app X)).functor,\n  unit_iso := nat_iso.of_components (λ Y, iso_mk (e.unit_iso.app Y) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (λ Y, iso_mk (e.counit_iso.app Y) (by tidy)) (by tidy) }\n\nend\n\nsection\nvariable [has_pullbacks C]\n\n/-- `map f` is left adjoint to `pullback f` when `f` is a monomorphism -/\ndef map_pullback_adj (f : X ⟶ Y) [mono f] : map f ⊣ pullback f :=\nadjunction.restrict_fully_faithful\n  (forget X) (forget Y) (over.map_pullback_adj f) (iso.refl _) (iso.refl _)\n\n/-- `mono_over.map f` followed by `mono_over.pullback f` is the identity. -/\ndef pullback_map_self (f : X ⟶ Y) [mono f] :\n  map f ⋙ pullback f ≅ 𝟭 _ :=\n(as_iso (mono_over.map_pullback_adj f).unit).symm\n\nend\n\nend map\n\nsection image\nvariables (f : X ⟶ Y) [has_image f]\n\n/--\nThe `mono_over Y` for the image inclusion for a morphism `f : X ⟶ Y`.\n-/\ndef image_mono_over (f : X ⟶ Y) [has_image f] : mono_over Y := mono_over.mk' (image.ι f)\n\n@[simp] lemma image_mono_over_arrow (f : X ⟶ Y) [has_image f] :\n  (image_mono_over f).arrow = image.ι f :=\nrfl\n\nend image\n\nsection image\n\nvariables [has_images C]\n\n/--\nTaking the image of a morphism gives a functor `over X ⥤ mono_over X`.\n-/\n@[simps]\ndef image : over X ⥤ mono_over X :=\n{ obj := λ f, image_mono_over f.hom,\n  map := λ f g k,\n  begin\n    apply (forget X).preimage _,\n    apply over.hom_mk _ _,\n    refine image.lift {I := image _, m := image.ι g.hom, e := k.left ≫ factor_thru_image g.hom},\n    apply image.lift_fac,\n  end }\n\n/--\n`mono_over.image : over X ⥤ mono_over X` is left adjoint to\n`mono_over.forget : mono_over X ⥤ over X`\n-/\ndef image_forget_adj : image ⊣ forget X :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ f g,\n  { to_fun := λ k,\n    begin\n      apply over.hom_mk (factor_thru_image f.hom ≫ k.left) _,\n      change (factor_thru_image f.hom ≫ k.left) ≫ _ = f.hom,\n      rw [assoc, over.w k],\n      apply image.fac\n    end,\n    inv_fun := λ k,\n    begin\n      refine over.hom_mk _ _,\n      refine image.lift {I := g.val.left, m := g.arrow, e := k.left, fac' := over.w k},\n      apply image.lift_fac,\n    end,\n    left_inv := λ k, subsingleton.elim _ _,\n    right_inv := λ k,\n    begin\n      ext1,\n      change factor_thru_image _ ≫ image.lift _ = _,\n      rw [← cancel_mono g.arrow, assoc, image.lift_fac, image.fac f.hom],\n      exact (over.w k).symm,\n    end } }\n\ninstance : is_right_adjoint (forget X) :=\n{ left := image, adj := image_forget_adj }\n\ninstance reflective : reflective (forget X) := {}.\n\n/--\nForgetting that a monomorphism over `X` is a monomorphism, then taking its image,\nis the identity functor.\n-/\ndef forget_image : forget X ⋙ image ≅ 𝟭 (mono_over X) :=\nas_iso (adjunction.counit image_forget_adj)\n\nend image\n\nsection «exists»\nvariables [has_images C]\n\n/--\nIn the case where `f` is not a monomorphism but `C` has images,\nwe can still take the \"forward map\" under it, which agrees with `mono_over.map f`.\n-/\ndef «exists» (f : X ⟶ Y) : mono_over X ⥤ mono_over Y :=\nforget _ ⋙ over.map f ⋙ image\n\ninstance faithful_exists (f : X ⟶ Y) : faithful («exists» f) := {}.\n\n/--\nWhen `f : X ⟶ Y` is a monomorphism, `exists f` agrees with `map f`.\n-/\ndef exists_iso_map (f : X ⟶ Y) [mono f] : «exists» f ≅ map f :=\nnat_iso.of_components\nbegin\n  intro Z,\n  suffices : (forget _).obj ((«exists» f).obj Z) ≅ (forget _).obj ((map f).obj Z),\n    apply preimage_iso this,\n  apply over.iso_mk _ _,\n  apply image_mono_iso_source (Z.arrow ≫ f),\n  apply image_mono_iso_source_hom_self,\nend\nbegin\n  intros Z₁ Z₂ g,\n  ext1,\n  change image.lift ⟨_, _, _, _⟩ ≫ (image_mono_iso_source (Z₂.arrow ≫ f)).hom =\n         (image_mono_iso_source (Z₁.arrow ≫ f)).hom ≫ g.left,\n  rw [← cancel_mono (Z₂.arrow ≫ f), assoc, assoc, w_assoc g, image_mono_iso_source_hom_self,\n      image_mono_iso_source_hom_self],\n  apply image.lift_fac,\nend\n\n/-- `exists` is adjoint to `pullback` when images exist -/\ndef exists_pullback_adj (f : X ⟶ Y) [has_pullbacks C] : «exists» f ⊣ pullback f :=\nadjunction.restrict_fully_faithful (forget X) (𝟭 _)\n  ((over.map_pullback_adj f).comp _ _ image_forget_adj)\n  (iso.refl _)\n  (iso.refl _)\n\nend «exists»\n\nend mono_over\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/subobject/mono_over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.30474333959714595}}
{"text": "import sheaves.sheaf Kenny.sandbox\n\nuniverses v w u₁ v₁ u\n\nopen topological_space lattice\n\nnamespace opens\n\nvariables {X : Type u} [topological_space X]\n\ntheorem Inf_eq (s : set (opens X)) : Inf s = opens.interior (Inf $ subtype.val '' s) :=\nle_antisymm\n  ((subset_interior_iff_subset_of_open (Inf s).2).2 $ show (Inf s).1 ≤ Inf (subtype.val '' s),\n    from le_Inf $ λ t ⟨u, hus, hut⟩, le_trans interior_subset $ Inf_le ⟨u, show Inf (set.range (λ (H : u ∈ s), u.val)) = t,\n    from le_antisymm (Inf_le ⟨hus, hut⟩) (le_Inf $ λ b ⟨c, hc⟩, hc ▸ ge_of_eq hut)⟩)\n  (le_Inf $ λ U hus, set.subset.trans interior_subset $ show Inf (subtype.val '' s) ≤ U.1,\n    from Inf_le $ set.mem_image_of_mem _ hus)\n\ntheorem inter_val (U V : opens X) : (U ∩ V).1 = U.1 ∩ V.1 := rfl\ntheorem inf_val (U V : opens X) : (U ⊓ V).1 = U.1 ∩ V.1 := rfl\n\ntheorem inf_Sup (U : opens X) (s : set (opens X)) : U ⊓ Sup s = Sup ((⊓) U '' s) :=\nopens.ext $ by rw [inf_val, opens.Sup_s, opens.Sup_s, set.sUnion_eq_Union, set.inter_Union, ← set.image_comp]; exact\nset.subset.antisymm\n  (set.Union_subset $ λ ⟨t, i, his, hit⟩, set.subset_sUnion_of_mem ⟨i, his, congr_arg ((∩) U.1) hit⟩)\n  (set.sUnion_subset $ λ t ⟨i, his, hit⟩, set.subset_sUnion_of_mem ⟨⟨i.1, i, his, rfl⟩, hit⟩)\n\ndef covering_inf_left (U V : opens X) (OC : covering U) : covering (V ⊓ U) :=\n{ γ := OC.γ,\n  Uis := λ i : OC.γ, V ⊓ OC.Uis i,\n  Hcov := by conv_rhs { rw ← OC.Hcov }; rw [supr, supr, inf_Sup]; congr' 1; ext x; exact\n    ⟨λ ⟨i, hix⟩, ⟨OC.Uis i, ⟨i, rfl⟩, hix⟩, λ ⟨_, ⟨i, rfl⟩, hix⟩, ⟨i, hix⟩⟩ }\n\ndef covering_res (U V : opens X) (H : V ⊆ U) (OC : covering U) : covering V :=\n{ γ := OC.γ,\n  Uis := λ i : OC.γ, V ⊓ OC.Uis i,\n  Hcov := by erw [(covering_inf_left U V OC).Hcov, (inf_of_le_left $ show V ≤ U, from H)] }\n\nend opens\n\ndef presheaf.covering (X : Type u) [topological_space X] : presheaf.{u (u+1)} X :=\n{ F := covering,\n  res := opens.covering_res,\n  Hid := λ U, funext $ λ OC, by cases OC; dsimp only [opens.covering_res, id]; congr' 1; funext i; apply opens.ext;\n    apply set.inter_eq_self_of_subset_right; rw ← OC_Hcov; apply set.subset_sUnion_of_mem; refine ⟨_, ⟨_, rfl⟩, rfl⟩,\n  Hcomp := λ U V W HWV HVU, funext $ λ OC, by dsimp only [opens.covering_res, function.comp_apply]; congr' 1; funext i;\n    rw [← lattice.inf_assoc, lattice.inf_of_le_left (show W ≤ V, from HWV)] }\n\ndef sheaf_on_opens (X : Type u) [topological_space X] (U : opens X) : Type (max u (v+1)) :=\nsheaf.{u v} X\n\nnamespace sheaf_on_opens\n\nvariables {X : Type u} [topological_space X] {U : opens X}\n\ndef eval (F : sheaf_on_opens X U) (V : opens X) (HVU : V ≤ U) : Type v :=\npresheaf.F F.to_presheaf V\n\ndef res (F : sheaf_on_opens X U) (V : opens X) (HVU : V ≤ U) (W : opens X) (HWU : W ≤ U) (HWV : W ≤ V) : F.eval V HVU → F.eval W HWU :=\npresheaf.res _ _ _ HWV\n\ntheorem res_self (F : sheaf_on_opens X U) (V HVU HV x) :\n  F.res V HVU V HVU HV x = x :=\npresheaf.Hid' _ _ _\n\ntheorem res_res (F : sheaf_on_opens X U) (V HVU W HWU HWV S HSU HSW x) :\n  F.res W HWU S HSU HSW (F.res V HVU W HWU HWV x) = F.res V HVU S HSU (le_trans HSW HWV) x :=\n(presheaf.Hcomp' _ _ _ _ _ _ _).symm\n\ntheorem locality (F : sheaf_on_opens X U) (V HVU s t) (OC : covering V)\n  (H : ∀ i : OC.γ, F.res V HVU (OC.Uis i) (le_trans (subset_covering i) HVU) (subset_covering i) s =\n    F.res V HVU (OC.Uis i) (le_trans (subset_covering i) HVU) (subset_covering i) t) :\n  s = t :=\nF.locality OC s t H\n\nnoncomputable def glue (F : sheaf_on_opens X U) (V HVU) (OC : covering V)\n  (s : Π i : OC.γ, F.eval (OC.Uis i) (le_trans (subset_covering i) HVU))\n  (H : ∀ i j : OC.γ, F.res _ _ (OC.Uis i ⊓ OC.Uis j) (le_trans inf_le_left (le_trans (subset_covering i) HVU)) inf_le_left (s i) =\n    F.res _ _ (OC.Uis i ⊓ OC.Uis j) (le_trans inf_le_left (le_trans (subset_covering i) HVU)) inf_le_right (s j)) :\n  F.eval V HVU :=\nclassical.some $ F.gluing OC s H\n\ntheorem res_glue (F : sheaf_on_opens X U) (V HVU) (OC : covering V) (s H i) :\n  F.res V HVU (OC.Uis i) (le_trans (subset_covering i) HVU) (subset_covering i) (F.glue V HVU OC s H) = s i :=\nclassical.some_spec (F.gluing OC s H) i\n\ntheorem eq_glue (F : sheaf_on_opens X U) (V HVU) (OC : covering V)\n  (s : Π i : OC.γ, F.eval (OC.Uis i) (le_trans (subset_covering i) HVU)) (H t)\n  (ht : ∀ i, F.res V HVU (OC.Uis i) (le_trans (subset_covering i) HVU) (subset_covering i) t = s i) :\n  F.glue V HVU OC s H = t :=\nF.locality V HVU _ _ OC $ λ i, by rw [res_glue, ht]\n\ndef res_subset (F : sheaf_on_opens X U) (V : opens X) (HVU : V ≤ U) : sheaf_on_opens X V :=\nF\n\ntheorem res_res_subset (F : sheaf_on_opens X U) (V HVU S HSV T HTV HTS x) :\n  (F.res_subset V HVU).res S HSV T HTV HTS x = F.res S (le_trans HSV HVU) T (le_trans HTV HVU) HTS x :=\nrfl\n\ndef stalk (F : sheaf_on_opens.{v} X U) (x : X) (hx : x ∈ U) : Type (max u v) :=\nstalk F.1 x\n\ndef to_stalk (F : sheaf_on_opens X U) (x : X) (hx : x ∈ U) (V : opens X) (hxV : x ∈ V) (HVU : V ≤ U) (s : F.eval V HVU) : F.stalk x hx :=\n⟦⟨V, hxV, s⟩⟧\n\n@[simp] lemma to_stalk_res (F : sheaf_on_opens X U) (x : X) (hx : x ∈ U) (V : opens X) (hxV : x ∈ V) (HVU : V ≤ U)\n  (W : opens X) (hxW : x ∈ W) (HWV : W ≤ V) (s : F.eval V HVU) :\n  F.to_stalk x hx W hxW (le_trans HWV HVU) (F.res _ _ _ _ HWV s) = F.to_stalk x hx V hxV HVU s :=\nquotient.sound ⟨W, hxW, set.subset.refl W.1, HWV, by dsimp only [res]; rw ← presheaf.Hcomp'; refl⟩\n\n@[elab_as_eliminator] theorem stalk.induction_on {F : sheaf_on_opens X U} {x : X} {hx : x ∈ U}\n  {C : F.stalk x hx → Prop} (g : F.stalk x hx)\n  (H : ∀ V : opens X, ∀ hxV : x ∈ V, ∀ HVU : V ≤ U, ∀ s : F.eval V HVU, C (F.to_stalk x hx V hxV HVU s)) :\n  C g :=\nquotient.induction_on g $ λ e,\nhave (⟦e⟧ : F.stalk x hx) = ⟦⟨e.1 ⊓ U, ⟨e.2, hx⟩, F.to_presheaf.res _ _ (set.inter_subset_left _ _) e.3⟩⟧,\nfrom quotient.sound ⟨e.1 ⊓ U, ⟨e.2, hx⟩, set.inter_subset_left _ _, set.subset.refl _, by dsimp only; rw ← presheaf.Hcomp'; refl⟩,\nthis.symm ▸ H (e.1 ⊓ U) ⟨e.2, hx⟩ inf_le_right _\n\nstructure morphism (F : sheaf_on_opens.{v} X U) (G : sheaf_on_opens.{w} X U) : Type (max u v w) :=\n(map : ∀ V ≤ U, F.eval V H → G.eval V H)\n(commutes : ∀ (V : opens X) (HV : V ≤ U) (W : opens X) (HW : W ≤ U) (HWV : W ≤ V) (x),\n  map W HW (F.res V HV W HW HWV x) = G.res V HV W HW HWV (map V HV x))\n\nnamespace morphism\n\nprotected def id (F : sheaf_on_opens.{v} X U) : F.morphism F :=\n{ map := λ V HV, id,\n  commutes := λ V HV W HW HWV x, rfl }\n\ndef comp {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} {H : sheaf_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) : F.morphism H :=\n{ map := λ V HV x, η.map V HV (ξ.map V HV x),\n  commutes := λ V HV W HW HWV x, by rw [ξ.commutes, η.commutes] }\n\n@[simp] lemma comp_apply {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} {H : sheaf_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) (V HV s) :\n  (η.comp ξ).1 V HV s = η.1 V HV (ξ.1 V HV s) :=\nrfl\n\n@[ext] lemma ext {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U}\n  {η ξ : F.morphism G} (H : ∀ V HV x, η.map V HV x = ξ.map V HV x) : η = ξ :=\nby cases η; cases ξ; congr; ext; apply H\n\n@[simp] lemma id_comp {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) :\n  (morphism.id G).comp η = η :=\next $ λ V HV x, rfl\n\n@[simp] lemma comp_id {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) :\n  η.comp (morphism.id F) = η :=\next $ λ V HV x, rfl\n\n@[simp] lemma comp_assoc {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} {H : sheaf_on_opens.{u₁} X U} {I : sheaf_on_opens.{v₁} X U}\n  (η : H.morphism I) (ξ : G.morphism H) (χ : F.morphism G) :\n  (η.comp ξ).comp χ = η.comp (ξ.comp χ) :=\nrfl\n\ndef res_subset {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) (V : opens X) (HVU : V ≤ U) :\n  (F.res_subset V HVU).morphism (G.res_subset V HVU) :=\n{ map := λ W HWV, η.map W (le_trans HWV HVU),\n  commutes := λ S HSV T HTV, η.commutes S (le_trans HSV HVU) T (le_trans HTV HVU) }\n\n@[simp] lemma res_subset_apply {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) (V : opens X) (HVU : V ≤ U)\n  (W HWV s) : (η.res_subset V HVU).1 W HWV s = η.1 W (le_trans HWV HVU) s :=\nrfl\n\n@[simp] lemma comp_res_subset {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} {H : sheaf_on_opens.{u₁} X U}\n  (η : G.morphism H) (ξ : F.morphism G) (V : opens X) (HVU : V ≤ U) :\n  (η.res_subset V HVU).comp (ξ.res_subset V HVU) = (η.comp ξ).res_subset V HVU :=\nrfl\n\n@[simp] lemma id_res_subset {F : sheaf_on_opens.{v} X U} (V : opens X) (HVU : V ≤ U) :\n  (morphism.id F).res_subset V HVU = morphism.id (F.res_subset V HVU) :=\nrfl\n\ndef stalk {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U)\n  (s : F.stalk x hx) : G.stalk x hx :=\nquotient.lift_on s (λ g, ⟦(⟨g.1 ⊓ U, (⟨g.2, hx⟩ : x ∈ g.1 ⊓ U),\n  η.map _ inf_le_right (presheaf.res F.1 _ _ (set.inter_subset_left _ _) g.3)⟩ : stalk.elem _ _)⟧) $\nλ g₁ g₂ ⟨V, hxV, HV1, HV2, hg⟩, quotient.sound ⟨V ⊓ U, ⟨hxV, hx⟩, set.inter_subset_inter_left _ HV1, set.inter_subset_inter_left _ HV2,\ncalc  G.res _ _ (V ⊓ U) inf_le_right (inf_le_inf HV1 (le_refl _)) (η.map (g₁.U ⊓ U) inf_le_right (F.to_presheaf.res (g₁.U) (g₁.U ⊓ U) (set.inter_subset_left _ _) (g₁.s)))\n    = η.map (V ⊓ U) inf_le_right (F.to_presheaf.res V (V ⊓ U) (set.inter_subset_left _ _) (F.to_presheaf.res (g₁.U) V HV1 (g₁.s))) : by rw [← η.2, res, ← presheaf.Hcomp', ← presheaf.Hcomp']\n... = G.res _ _ (V ⊓ U) _ _ (η.map (g₂.U ⊓ U) inf_le_right (F.to_presheaf.res (g₂.U) (g₂.U ⊓ U) _ (g₂.s))) : by rw [hg, ← η.2, res, ← presheaf.Hcomp', ← presheaf.Hcomp']⟩\n\n@[simp] lemma stalk_to_stalk {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (η : F.morphism G) (x : X) (hx : x ∈ U)\n  (V : opens X) (HVU : V ≤ U) (hxV : x ∈ V) (s : F.eval V HVU) : η.stalk x hx (F.to_stalk x hx V hxV HVU s) = G.to_stalk x hx V hxV HVU (η.map V HVU s) :=\nquotient.sound ⟨V, hxV, set.subset_inter (set.subset.refl _) HVU, set.subset.refl _,\ncalc  G.res (V ⊓ U) inf_le_right V HVU (le_inf (le_refl V) HVU) (η.map (V ⊓ U) inf_le_right (F.res V HVU (V ⊓ U) inf_le_right inf_le_left s))\n    = G.res V HVU V HVU (le_refl V) (η.map V HVU s) : by rw [η.2, res_res]⟩\n\nend morphism\n\nstructure equiv (F : sheaf_on_opens.{v} X U) (G : sheaf_on_opens.{w} X U) : Type (max u v w) :=\n(to_fun : morphism F G)\n(inv_fun : morphism G F)\n(left_inv : ∀ V HVU s, inv_fun.1 V HVU (to_fun.1 V HVU s) = s)\n(right_inv : ∀ V HVU s, to_fun.1 V HVU (inv_fun.1 V HVU s) = s)\n\nnamespace equiv\n\ndef refl (F : sheaf_on_opens.{v} X U) : equiv F F :=\n⟨morphism.id F, morphism.id F, λ _ _ _, rfl, λ _ _ _, rfl⟩\n\n@[simp] lemma refl_apply (F : sheaf_on_opens.{v} X U) (V HV s) :\n  (refl F).1.1 V HV s = s := rfl\n\ndef symm {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{v} X U} (e : equiv F G) : equiv G F :=\n⟨e.2, e.1, e.4, e.3⟩\n\ndef trans {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{v} X U} {H : sheaf_on_opens.{u₁} X U}\n  (e₁ : equiv F G) (e₂ : equiv G H) : equiv F H :=\n⟨e₂.1.comp e₁.1, e₁.2.comp e₂.2,\nλ _ _ _, by rw [morphism.comp_apply, morphism.comp_apply, e₂.3, e₁.3],\nλ _ _ _, by rw [morphism.comp_apply, morphism.comp_apply, e₁.4, e₂.4]⟩\n\n@[simp] lemma trans_apply {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{v} X U} {H : sheaf_on_opens.{u₁} X U}\n  (e₁ : equiv F G) (e₂ : equiv G H) (V HV s) :\n  (e₁.trans e₂).1.1 V HV s = e₂.1.1 V HV (e₁.1.1 V HV s) :=\nrfl\n\ndef res_subset {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (e : equiv F G)\n  (V : opens X) (HVU : V ≤ U) : equiv (F.res_subset V HVU) (G.res_subset V HVU) :=\n⟨e.1.res_subset V HVU, e.2.res_subset V HVU,\nλ _ _ _, by rw [morphism.res_subset_apply, morphism.res_subset_apply, e.3],\nλ _ _ _, by rw [morphism.res_subset_apply, morphism.res_subset_apply, e.4]⟩\n\n@[simp] lemma res_subset_apply {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (e : equiv F G)\n  (V : opens X) (HVU : V ≤ U) (W HW s) :\n  (e.res_subset V HVU).1.1 W HW s = e.1.1 W (le_trans HW HVU) s :=\nrfl\n\ndef stalk {F : sheaf_on_opens.{v} X U} {G : sheaf_on_opens.{w} X U} (e : equiv F G) (x : X) (hx : x ∈ U) :\n  F.stalk x hx ≃ G.stalk x hx :=\n{ to_fun := e.1.stalk x hx,\n  inv_fun := e.2.stalk x hx,\n  left_inv := λ g, stalk.induction_on g $ λ V hxV HVU s, by rw [morphism.stalk_to_stalk, morphism.stalk_to_stalk, e.3],\n  right_inv := λ g, stalk.induction_on g $ λ V hxV HVU s, by rw [morphism.stalk_to_stalk, morphism.stalk_to_stalk, e.4] }\n\nend equiv\n\nnamespace sheaf_glue\n\nvariables {I : Type u} (S : I → opens X) (F : Π (i : I), sheaf_on_opens.{v} X (S i))\nvariables (φ : Π (i j : I), equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right))\nvariables (Hφ1 : ∀ i, φ i i = equiv.refl (res_subset (F i) (S i ⊓ S i) _))\nvariables (Hφ2 : ∀ i j k,\n  ((φ i j).res_subset ((S i) ⊓ (S j) ⊓ (S k)) inf_le_left).trans\n    ((φ j k).res_subset ((S i) ⊓ (S j) ⊓ (S k)) (le_inf (le_trans inf_le_left inf_le_right) inf_le_right)) =\n  (φ i k).res_subset ((S i) ⊓ (S j) ⊓ (S k)) (le_inf (le_trans inf_le_left inf_le_left) inf_le_right))\n\n@[reducible] def compat (W : opens X) : Type (max u v) :=\n{ f : Π i, (F i).eval ((S i) ⊓ W) inf_le_left //\n  ∀ i j, (φ i j).1.map ((S i) ⊓ (S j) ⊓ W) inf_le_left\n    ((F i).res ((S i) ⊓ W) _ _ (le_trans inf_le_left inf_le_left)\n      (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n      (f i)) =\n    (F j).res ((S j) ⊓ W) _ _ (le_trans inf_le_left inf_le_right)\n      (le_inf (le_trans inf_le_left inf_le_right) inf_le_right)\n      (f j) }\n\ndef res (U V : opens X) (HVU : V ≤ U) (f : compat S F φ U) : compat S F φ V :=\n⟨λ i, (F i).res (S i ⊓ U) _ (S i ⊓ V) _ (inf_le_inf (le_refl _) HVU) (f.1 i), λ i j,\ncalc  (φ i j).1.map (S i ⊓ S j ⊓ V) inf_le_left\n        (res (F i) (S i ⊓ V) inf_le_left (S i ⊓ S j ⊓ V)\n          (le_trans inf_le_left inf_le_left)\n          (le_inf (le_trans inf_le_left inf_le_left)\n              inf_le_right)\n          (res (F i) (S i ⊓ U) inf_le_left (S i ⊓ V) inf_le_left\n              (inf_le_inf (le_refl _) HVU)\n              (f.1 i)) : (F i).eval (S i ⊓ S j ⊓ V) (le_trans _ _))\n    = (φ i j).1.map (S i ⊓ S j ⊓ V) inf_le_left\n        (res (res_subset (F i) (S i ⊓ S j) inf_le_left) (S i ⊓ S j ⊓ U) inf_le_left\n          (S i ⊓ S j ⊓ V)\n          inf_le_left\n          (inf_le_inf (le_refl _) HVU)\n          (res (F i) (S i ⊓ U) inf_le_left (S i ⊓ S j ⊓ U)\n            (le_trans inf_le_left inf_le_left)\n            (le_inf (le_trans inf_le_left inf_le_left)\n                inf_le_right)\n            (f.1 i) : (F i).eval (S i ⊓ S j ⊓ U) (le_trans inf_le_left inf_le_left))) :\n  by rw [res_res_subset, res_res, res_res]\n... = (res (F j) (S j ⊓ V) inf_le_left (S i ⊓ S j ⊓ V)\n        (le_trans inf_le_left inf_le_right)\n        (le_inf (le_trans inf_le_left inf_le_right)\n          inf_le_right)\n        (res (F j) (S j ⊓ U) inf_le_left (S j ⊓ V) inf_le_left\n          (inf_le_inf (le_refl _) HVU)\n          (f.1 j)) : (F j).eval (S i ⊓ S j ⊓ V) _) :\n  by rw [(φ i j).1.commutes, f.2 i j, res_res_subset, res_res, res_res]⟩\n\ntheorem locality (U : opens X) (OC : covering U) (s t : sheaf_glue.compat S F φ U)\n  (H : ∀ i : OC.γ, sheaf_glue.res S F φ U (OC.Uis i) (subset_covering i) s =\n    sheaf_glue.res S F φ U (OC.Uis i) (subset_covering i) t) :\n  s = t :=\nsubtype.eq $ funext $ λ i, (F i).locality _ _ _ _ (opens.covering_inf_left U (S i) OC) $ λ j,\nby have := H j; simp only [sheaf_glue.res, subtype.mk.inj_eq] at this; exact congr_fun this i\n\nnoncomputable def gluing.aux1 (U : opens X) (OC : covering U) (s : Π i : OC.γ, sheaf_glue.compat S F φ (OC.Uis i))\n  (H : ∀ i j : OC.γ, sheaf_glue.res S F φ _ _ inf_le_left (s i) = sheaf_glue.res S F φ _ _ inf_le_right (s j))\n  (i : I) : (F i).eval (S i ⊓ U) inf_le_left :=\n(F i).glue _ _ (opens.covering_inf_left U (S i) OC) (λ j, (s j).1 i) $ λ j k,\nhave h1 : S i ⊓ OC.Uis j ⊓ (S i ⊓ OC.Uis k) ≤ S i ⊓ (OC.Uis j ⊓ OC.Uis k),\nby rw [inf_assoc, inf_left_comm (OC.Uis j), ← inf_assoc, inf_idem]; exact le_refl _,\nhave h2 : S i ⊓ (OC.Uis j ⊓ OC.Uis k) ≤ S i ⊓ OC.Uis j,\nfrom inf_le_inf (le_refl _) inf_le_left,\nhave h3 : S i ⊓ (OC.Uis j ⊓ OC.Uis k) ≤ S i ⊓ OC.Uis k,\nfrom inf_le_inf (le_refl _) inf_le_right,\nhave (F i).res (S i ⊓ OC.Uis j) _ (S i ⊓ (OC.Uis j ⊓ OC.Uis k)) inf_le_left h2 ((s j).1 i) =\n  (F i).res (S i ⊓ OC.Uis k) _ (S i ⊓ (OC.Uis j ⊓ OC.Uis k)) inf_le_left h3 ((s k).1 i),\nfrom congr_fun (congr_arg subtype.val (H j k)) i,\ncalc  _\n    = (F i).res (S i ⊓ OC.Uis j) _ ((S i ⊓ OC.Uis j) ⊓ (S i ⊓ OC.Uis k)) _ _ ((s j).1 i) : rfl\n... = (F i).res (S i ⊓ (OC.Uis j ⊓ OC.Uis k)) _ ((S i ⊓ OC.Uis j) ⊓ (S i ⊓ OC.Uis k)) _ h1\n        ((F i).res (S i ⊓ OC.Uis j) _ (S i ⊓ (OC.Uis j ⊓ OC.Uis k)) inf_le_left h2 ((s j).1 i)) : (res_res _ _ _ _ _ _ _ _ _ _).symm\n... = (F i).res (S i ⊓ OC.Uis k) _ ((S i ⊓ OC.Uis j) ⊓ (S i ⊓ OC.Uis k)) _ inf_le_right ((s k).1 i) : by rw [this, res_res]\n\ntheorem gluing.aux2 (U : opens X) (OC : covering U) (s : Π i : OC.γ, sheaf_glue.compat S F φ (OC.Uis i))\n  (H : ∀ i j : OC.γ, sheaf_glue.res S F φ _ _ inf_le_left (s i) = sheaf_glue.res S F φ _ _ inf_le_right (s j)) (i j : I) :\n  (φ i j).1.map (S i ⊓ S j ⊓ U) inf_le_left\n      ((F i).res (S i ⊓ U) _ (S i ⊓ S j ⊓ U) (le_trans inf_le_left inf_le_left)\n        (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n        (gluing.aux1 S F φ U OC s H i)) =\n    (F j).res (S j ⊓ U) _ (S i ⊓ S j ⊓ U) (le_trans inf_le_left inf_le_right)\n      (by rw inf_assoc; exact inf_le_right)\n      (gluing.aux1 S F φ U OC s H j) :=\n(F j).locality _ _ _ _ (opens.covering_inf_left _ _ OC) $ λ k,\ncalc  ((F j).res_subset (S i ⊓ S j) inf_le_right).res (S i ⊓ S j ⊓ U) inf_le_left ((S i ⊓ S j) ⊓ OC.Uis k) inf_le_left\n        (inf_le_inf (le_refl _) (subset_covering k))\n        ((φ i j).1.map (S i ⊓ S j ⊓ U) inf_le_left\n          ((F i).res (S i ⊓ U) _ (S i ⊓ S j ⊓ U) _\n            (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n            (gluing.aux1 S F φ U OC s H i)))\n    = (φ i j).1.map (S i ⊓ S j ⊓ OC.Uis k) inf_le_left\n        ((F i).res ((opens.covering_inf_left U (S i) OC).Uis k) _ _ (le_trans inf_le_left inf_le_left)\n          (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n          ((F i).res (S i ⊓ U) _ ((opens.covering_inf_left U (S i) OC).Uis k) inf_le_left\n            (inf_le_inf (le_refl _) (subset_covering k))\n            (gluing.aux1 S F φ U OC s H i))) : by rw [← (φ i j).1.commutes, res_res_subset, res_res, res_res]\n... = (F j).res ((opens.covering_inf_left U (S j) OC).Uis k) _ ((S i ⊓ S j) ⊓ OC.Uis k) _\n        (by rw inf_assoc; exact inf_le_right)\n        ((F j).res (S j ⊓ U) _ ((opens.covering_inf_left U (S j) OC).Uis k) inf_le_left\n          (inf_le_inf (le_refl _) (subset_covering k))\n          (gluing.aux1 S F φ U OC s H j)) : by erw [res_glue, res_glue]; exact (s k).2 i j\n... = (F j).res (S i ⊓ S j ⊓ U) _ ((S i ⊓ S j) ⊓ OC.Uis k) _ _\n        ((F j).res (S j ⊓ U) _ (S i ⊓ S j ⊓ U) _ _ (gluing.aux1 S F φ U OC s H j)) : by rw [res_res, res_res]; refl\n\ntheorem gluing.aux3 (U : opens X) (OC : covering U) (s : Π i : OC.γ, sheaf_glue.compat S F φ (OC.Uis i))\n  (H : ∀ i j : OC.γ, sheaf_glue.res S F φ _ _ inf_le_left (s i) = sheaf_glue.res S F φ _ _ inf_le_right (s j)) (i : OC.γ) :\n  sheaf_glue.res S F φ U (OC.Uis i) (subset_covering i) ⟨λ i, gluing.aux1 S F φ U OC s H i, gluing.aux2 S F φ U OC s H⟩ = s i :=\nsubtype.eq $ funext $ λ j, by dsimp only [gluing.aux1, sheaf_glue.res];\nchange (F j).res _ _ ((opens.covering_inf_left U (S j) OC).Uis i) _ _ _ = _;\nerw res_glue\n\ntheorem gluing (U : opens X) (OC : covering U) (s : Π i : OC.γ, sheaf_glue.compat S F φ (OC.Uis i))\n  (H : ∀ i j : OC.γ, sheaf_glue.res S F φ _ _ inf_le_left (s i) = sheaf_glue.res S F φ _ _ inf_le_right (s j)) :\n  ∃ t : sheaf_glue.compat S F φ U, ∀ i : OC.γ, sheaf_glue.res S F φ U (OC.Uis i) (subset_covering i) t = s i :=\n⟨⟨λ i, gluing.aux1 S F φ U OC s H i, gluing.aux2 S F φ U OC s H⟩, λ i, gluing.aux3 S F φ U OC s H i⟩\n\nend sheaf_glue\n\ndef sheaf_glue {I : Type u} (S : I → opens X) (F : Π (i : I), sheaf_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right)) :\n  sheaf_on_opens.{max u v} X (⋃S) :=\n{ F := sheaf_glue.compat S F φ,\n  res := sheaf_glue.res S F φ,\n  Hid := λ U, funext $ λ f, subtype.eq $ funext $ λ i, by dsimp only [sheaf_glue.res, id]; rw res_self,\n  Hcomp := λ U V W HWV HVU, funext $ λ f, subtype.eq $ funext $ λ i, by symmetry; apply res_res; exact inf_le_left,\n  locality := sheaf_glue.locality S F φ,\n  gluing := sheaf_glue.gluing S F φ }\n\n@[simp] lemma sheaf_glue_res_val {I : Type u} (S : I → opens X) (F : Π (i : I), sheaf_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right))\n  (U HU V HV HVU s i) : ((sheaf_glue S F φ).res U HU V HV HVU s).1 i = (F i).res _ _ _ _ (inf_le_inf (le_refl _) HVU) (s.1 i) := rfl\n\ndef universal_property (I : Type u) (S : I → opens X) (F : Π (i : I), sheaf_on_opens.{v} X (S i))\n  (φ : Π i j, equiv ((F i).res_subset ((S i) ⊓ (S j)) inf_le_left) ((F j).res_subset ((S i) ⊓ (S j)) inf_le_right))\n  (Hφ1 : ∀ i V HV s, (φ i i).1.1 V HV s = s)\n  (Hφ2 : ∀ i j k V HV1 HV2 HV3 s, (φ j k).1.1 V HV1 ((φ i j).1.1 V HV2 s) = (φ i k).1.1 V HV3 s)\n  (i : I) :\n  equiv (res_subset (sheaf_glue S F φ) (S i) (le_supr S i)) (F i) :=\n{ to_fun :=\n  { map := λ U H s, (F i).res _ _ _ _ (le_inf H (le_refl _)) (s.1 i),\n    commutes := λ U HU V HV HVU s, by rw [res_res, res_res_subset]; dsimp only [res, sheaf_glue, sheaf_glue.res]; rw ← presheaf.Hcomp'; refl },\n  inv_fun :=\n  { map := λ U H s, ⟨λ j, (φ i j).1.1 (S j ⊓ U) (le_inf (le_trans inf_le_right H) inf_le_left)\n        ((F i).res _ _ _ (le_trans inf_le_right H) inf_le_right s),\n      λ j k, begin\n        have h1 : S j ⊓ S k ⊓ U ≤ S i ⊓ S j := le_inf (le_trans inf_le_right H) (le_trans inf_le_left inf_le_left),\n        have h2 : S j ⊓ S k ⊓ U ≤ S i ⊓ S k := le_inf (le_trans inf_le_right H) (le_trans inf_le_left inf_le_right),\n        rw [← res_res_subset (F j) _ _ _ _ _ h1, ← (φ i j).1.2, Hφ2 _ _ _ _ _ _ h2, res_res_subset, res_res],\n        rw [← res_res_subset (F k) _ _ _ _ _ h2, ← (φ i k).1.2, res_res_subset, res_res],\n      end⟩,\n    commutes := λ U HU V HV HVU s, subtype.eq $ funext $ λ j, by dsimp only [res_res_subset, sheaf_glue_res_val];\n      rw [← res_res_subset (F j), ← (φ i j).1.2, res_res_subset, res_res, res_res] },\n  left_inv := λ V HV s, subtype.eq $ funext $ λ j, have _, from s.2 i j, calc\n      _ = (φ i j).1.map (S j ⊓ V) (le_inf (le_trans inf_le_right HV) inf_le_left)\n            ((F i).res V HV (S j ⊓ V) (le_trans inf_le_right HV) inf_le_right\n              ((F i).res (S i ⊓ V) _ V HV (le_inf HV (le_refl _)) (s.1 i))) : rfl\n    ... = (φ i j).1.map (S j ⊓ V) _\n            ((F i).res (S i ⊓ V) _ (S j ⊓ V) _ _ (s.1 i)) : by rw res_res\n    ... = (φ i j).1.map (S j ⊓ V) _\n            (((F i).res_subset (S i ⊓ S j) _).res ((S i ⊓ S j) ⊓ V) inf_le_left _ _\n              (by rw [inf_assoc, inf_left_comm, inf_of_le_right HV]; exact le_refl _)\n              ((F i).res (S i ⊓ V) _ ((S i ⊓ S j) ⊓ V) (le_trans inf_le_left inf_le_left)\n                (le_inf (le_trans inf_le_left inf_le_left) inf_le_right)\n                (s.1 i))) : by rw [res_res_subset, res_res]\n    ... = s.1 j : by rw [(φ i j).1.2, s.2 i j, res_res_subset, res_res, res_self],\n  right_inv := λ V HV s, by dsimp only; erw [Hφ1, res_res, res_self] }\n\nend sheaf_on_opens\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/Kenny/sheaf_on_opens.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.30455638810489083}}
{"text": "import pseudo_normed_group.category.ProFiltPseuNormGrpWithTinv\n\nuniverse variables u\n\nopen category_theory\nopen_locale nnreal\n\nlocal attribute [instance] type_pow\n\nnoncomputable theory\n\n/-- `ProFiltPseuNormGrpWithTinv₁ r` is the bundled category of\n`profinitely_filtered_pseudo_normed_group_with_Tinv r`s, with\n(this is the `₁`) exhaustive filtrations and strict morphisms.\nThe objects are filtered abelian groups equipped with an\nendomorphism `T⁻¹` with norm at most `r`; see the docstring\nof `profinitely_filtered_pseudo_normed_group_with_Tinv` for\nmore details.\n-/\nstructure ProFiltPseuNormGrpWithTinv₁ (r : ℝ≥0) : Type (u+1) :=\n(M : Type u)\n[str : profinitely_filtered_pseudo_normed_group_with_Tinv r M]\n(exhaustive' : ∀ m : M, ∃ c : ℝ≥0, m ∈ pseudo_normed_group.filtration M c)\n\nnamespace ProFiltPseuNormGrpWithTinv₁\n\nvariable (r : ℝ≥0)\n\ninstance : has_coe_to_sort (ProFiltPseuNormGrpWithTinv₁ r) Type* := ⟨λ M, M.M⟩\ninstance (M : ProFiltPseuNormGrpWithTinv₁ r) :\n  profinitely_filtered_pseudo_normed_group_with_Tinv r M := M.str\n\nlemma exhaustive (M : ProFiltPseuNormGrpWithTinv₁ r) (m : M) : ∃ c : ℝ≥0,\n  m ∈ pseudo_normed_group.filtration M c := M.exhaustive' m\n\ninstance : large_category (ProFiltPseuNormGrpWithTinv₁.{u} r) :=\n{ hom := λ A B, comphaus_filtered_pseudo_normed_group_with_Tinv_hom r A B,\n  id := λ A, comphaus_filtered_pseudo_normed_group_with_Tinv_hom.id,\n  comp := λ A B C f g, g.comp f } .\n\ndef PFPNGT₁_to_PFPNGTₑₗ : (ProFiltPseuNormGrpWithTinv₁.{u} r) ⥤ (ProFiltPseuNormGrpWithTinv.{u} r) :=\n{ obj := λ M, ProFiltPseuNormGrpWithTinv.of r M,\n  map := λ A B f, f }\n\ninstance : concrete_category (ProFiltPseuNormGrpWithTinv₁.{u} r) :=\n{ forget :=\n  { obj := λ M, M,\n    map := λ X Y f, f },\n  forget_faithful := ⟨⟩ } .\n\n/-- The \"forget the action of T⁻¹\" functor on\n  profinitely filtered (exhaustive pseudo-)normed groups. -/\ndef _root_.PFPNGT₁_to_PFPNG₁ₑₗ : (ProFiltPseuNormGrpWithTinv₁.{u} r) ⥤ ProFiltPseuNormGrp₁.{u} :=\n{ obj := λ M,\n  { M := M,\n    exhaustive' := M.exhaustive' },\n  map := λ A B f,\n  { to_fun := f,\n    map_zero' := f.map_zero,\n    map_add' := f.map_add,\n    strict' := f.strict,\n    continuous' := f.continuous' } }\n\n/-- The forgetful functor which takes a profinitely filtered normed group with an action of T⁻¹,\nthen forgets the action and considered it as a `CompHaus`ly filtered normed group. -/\n@[simps]\ndef _root_.PFPNGT₁_to_CHFPNG₁ₑₗ (r' : ℝ≥0) :\n  ProFiltPseuNormGrpWithTinv₁.{u} r' ⥤ CompHausFiltPseuNormGrp₁.{u} :=\nPFPNGT₁_to_PFPNG₁ₑₗ r' ⋙ PFPNG₁_to_CHFPNG₁ₑₗ\n\nlemma coe_comp_apply {A B C : ProFiltPseuNormGrpWithTinv₁ r} (f : A ⟶ B) (g : B ⟶ C) (a : A) :\n  (f ≫ g) a = g (f a) := rfl\n\nopen profinitely_filtered_pseudo_normed_group_with_Tinv\n\ndef Tinv_limit_fun_aux {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁ r)\n  (x : Σ (c : ℝ≥0), CompHausFiltPseuNormGrp₁.cone_point_type_filt\n    ((K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r) ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c) (j : J) :\n  (pseudo_normed_group.filtration (K.obj j) x.fst) :=\nx.2 j\n\ndef Tinv_limit_fun'\n  {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (c : ℝ≥0) (x : CompHausFiltPseuNormGrp₁.cone_point_type_filt\n    ((K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r) ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c) :\n  (Σ c, CompHausFiltPseuNormGrp₁.cone_point_type_filt\n    ((K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r) ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c) :=\n⟨r⁻¹ * c, λ j,\n  ⟨Tinv (Tinv_limit_fun_aux r K ⟨c,x⟩ j : K.obj j),\n    (Tinv_mem_filtration _ _ (Tinv_limit_fun_aux r K ⟨c,x⟩ j).2)⟩,\n  begin\n    intros i j f,\n    ext1,\n    show (K.map f) (Tinv _) = Tinv _,\n    erw (K.map f).map_Tinv, congr' 1,\n    simpa only [functor.comp_map, subtype.val_eq_coe, subtype.ext_iff] using x.2 f,\n  end⟩\n\ndef Tinv_limit_fun\n  {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X) →\n    ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X) :=\nquotient.map' (λ x, Tinv_limit_fun' r K x.1 x.2)\nbegin\n  rintros x y ⟨c, h₁, h₂, h⟩,\n  refine ⟨r⁻¹ * c, mul_le_mul' le_rfl h₁, mul_le_mul' le_rfl h₂, _⟩,\n  ext j,\n  show Tinv (Tinv_limit_fun_aux r K x j : K.obj j) = Tinv (Tinv_limit_fun_aux r K y j : K.obj j),\n  congr' 1,\n  rw [subtype.ext_iff, function.funext_iff] at h,\n  specialize h j, rwa [subtype.ext_iff] at h,\nend\n\nopen CompHausFiltPseuNormGrp₁ CompHausFiltPseuNormGrp₁.cone_point_type\n\nlemma Tinv_limit_fun_incl\n  {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) (c : ℝ≥0) (x) :\n  Tinv_limit_fun r K (incl c x) = incl (r⁻¹ * c) (Tinv_limit_fun' r K c x).2 := rfl\n\n@[simps]\ndef Tinv_limit_add_monoid_hom\n  {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X) →+\n    ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X) :=\n{ to_fun := Tinv_limit_fun r K,\n  map_zero' :=\n  begin\n    apply quotient.sound',\n    dsimp only,\n    refine ⟨0, _, le_rfl, _⟩; dsimp only [Tinv_limit_fun'],\n    { rw [mul_zero] },\n    { ext j, exact Tinv.map_zero }\n  end,\n  map_add' :=\n  begin\n    rintros ⟨cx, x⟩ ⟨cy, y⟩,\n    show Tinv_limit_fun r K (incl cx x + incl cy y) =\n      Tinv_limit_fun r K (incl cx x) + Tinv_limit_fun r K (incl cy y),\n    simp only [incl_add_incl, Tinv_limit_fun_incl],\n    apply quotient.sound',\n    dsimp only,\n    refine ⟨_, le_rfl, _, _⟩; simp only [mul_add],\n    ext j, refine Tinv.map_add _ _,\n  end }\n\nopen pseudo_normed_group ProFiltPseuNormGrp₁ CompHausFiltPseuNormGrp₁\n\nlemma Tinv_limit_aux {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (c : ℝ≥0) (x : ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X))\n  (hx : x ∈ filtration (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X c) :\n  Tinv_limit_add_monoid_hom r K x ∈\n    filtration (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X (r⁻¹ * c) :=\nbegin\n  obtain ⟨x,rfl⟩ := hx,\n  dsimp only [Tinv_limit_add_monoid_hom_apply, Tinv_limit_fun_incl],\n  exact ⟨_,rfl⟩,\nend\n\n-- TODO: break up this proof into pieces.\ndef Tinv_limit {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  comphaus_filtered_pseudo_normed_group_hom\n    ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X)\n    ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X) :=\ncomphaus_filtered_pseudo_normed_group_hom.mk_of_bound (Tinv_limit_add_monoid_hom r K) r⁻¹\nbegin\n  intro c,\n  fsplit,\n  { apply Tinv_limit_aux },\n  { let X := ((ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X),\n    let F : filtration X c → filtration X (r⁻¹ * c) := λ x,\n      ⟨Tinv_limit_add_monoid_hom r K x, Tinv_limit_aux _ _ _ _ x.2⟩,\n    change continuous F,\n    let e := filt_homeo (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ _ ⋙ PFPNG₁_to_CHFPNG₁ₑₗ),\n    suffices : continuous (e (r⁻¹ * c) ∘ F ∘ (e c).symm), by simpa,\n    let I : Π (j : J), comphaus_filtered_pseudo_normed_group_hom (K.obj j) (K.obj j) :=\n      λ j, Tinv,\n    let G : cone_point_type_filt (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ _ ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c →\n      cone_point_type_filt (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ _ ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) (r⁻¹ * c) :=\n      λ x, ⟨λ j, ⟨I j (x j).1, _⟩, _⟩,\n    rotate,\n    { apply Tinv_bound_by, exact (x j).2 },\n    { intros i j e,\n      have := x.2 e,\n      ext,\n      dsimp,\n      apply_fun (λ e, e.val) at this,\n      change _ = I j (x.val j).val,\n      rw ← this,\n      apply (K.map e).map_Tinv },\n    have : continuous G,\n    { apply continuous.subtype_mk,\n      apply continuous_pi,\n      intros i,\n      let G1 : cone_point_type_filt (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ _ ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c →\n        filtration (K.obj i) c := λ x, x i,\n      let G2 : filtration (K.obj i) c → filtration (K.obj i) (r⁻¹ * c) :=\n        λ x, ⟨I i x, _⟩,\n      swap, { apply Tinv_bound_by, exact x.2 },\n      change continuous (G2 ∘ G1),\n      apply continuous.comp,\n      { apply comphaus_filtered_pseudo_normed_group_hom.continuous, intros x, refl },\n      { let G11 : cone_point_type_filt (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ _ ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) c →\n          Π j : J, filtration (K.obj j) c := λ x, x,\n        let G12 : (Π j : J, filtration (K.obj j) c) → filtration (K.obj i) c := λ x, x i,\n        change continuous (G12 ∘ G11),\n        apply continuous.comp,\n        apply continuous_apply,\n        apply continuous_subtype_coe } },\n    convert this,\n    ext : 1,\n    dsimp,\n    apply_fun (e (r⁻¹ * c)).symm,\n    simp,\n    ext, refl },\nend\n\ninstance {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  profinitely_filtered_pseudo_normed_group_with_Tinv r\n    (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X :=\n{ Tinv := Tinv_limit r K,\n  Tinv_mem_filtration := comphaus_filtered_pseudo_normed_group_hom.mk_of_bound_bound_by _ _ _,\n  ..(infer_instance : profinitely_filtered_pseudo_normed_group _) }\n\ndef limit_cone {J : Type u} [small_category J] (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  limits.cone K :=\n{ X :=\n  { M := (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X,\n    exhaustive' := (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).X.exhaustive },\n  π :=\n  { app := λ j,\n    { map_Tinv' := begin\n        rintro ⟨⟨c,x⟩⟩,\n        dsimp [Tinv, Tinv_limit, Tinv_limit_fun, Tinv_limit_fun', Tinv_limit_fun_aux],\n        dsimp [ProFiltPseuNormGrp₁.limit_cone, CompHausFiltPseuNormGrp₁.limit_cone],\n        change proj (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r ⋙ PFPNG₁_to_CHFPNG₁ₑₗ) j (incl _ _) = _,\n        change _ = Tinv (proj _ _ (incl _ _)),\n        dsimp [proj],\n        simpa,\n      end,\n      ..(ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).π.app j },\n  naturality' := begin\n    intros i j e,\n    ext1 x,\n    have := (ProFiltPseuNormGrp₁.limit_cone (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).π.naturality e,\n    apply_fun (λ e, e x) at this,\n    exact this,\n  end } } .\n\ninstance {J : Type u} [small_category J] : creates_limits_of_shape J (PFPNGT₁_to_PFPNG₁ₑₗ r) :=\n{ creates_limit := λ K,\n  { reflects := λ C hC,\n    { lift := λ S,\n      { map_Tinv' := begin\n          intros x,\n          apply ProFiltPseuNormGrp₁.eq_of_π_eq _ hC,\n          intros j,\n          erw [← ProFiltPseuNormGrp₁.coe_comp_apply, ← ProFiltPseuNormGrp₁.coe_comp_apply,\n            hC.fac],\n          dsimp,\n          change S.π.app _ _ = C.π.app _ _,\n          rw [(S.π.app _).map_Tinv, (C.π.app _).map_Tinv],\n          congr' 1,\n          change _ = ((PFPNGT₁_to_PFPNG₁ₑₗ r).map (C.π.app j)) _,\n          erw [← ProFiltPseuNormGrp₁.coe_comp_apply, hC.fac],\n          refl,\n        end,\n        ..hC.lift ((PFPNGT₁_to_PFPNG₁ₑₗ r).map_cone S) },\n      fac' := begin\n        intros S j,\n        ext1 x,\n        have := hC.fac ((PFPNGT₁_to_PFPNG₁ₑₗ r).map_cone S) j,\n        apply_fun (λ e, e x) at this,\n        exact this,\n      end,\n      uniq' := begin\n        intros S m h,\n        ext1 x,\n        have := hC.uniq ((PFPNGT₁_to_PFPNG₁ₑₗ r).map_cone S) ((PFPNGT₁_to_PFPNG₁ₑₗ r).map m) _,\n        apply_fun (λ e, e x) at this,\n        exact this,\n        { intros j,\n          ext y,\n          specialize h j,\n          apply_fun (λ e, e y) at h,\n          exact h },\n      end },\n    lifts := λ C hC,\n    { lifted_cone := limit_cone r K,\n      valid_lift :=\n        (ProFiltPseuNormGrp₁.limit_cone_is_limit (K ⋙ PFPNGT₁_to_PFPNG₁ₑₗ r)).unique_up_to_iso hC } } }\n\ninstance : creates_limits (PFPNGT₁_to_PFPNG₁ₑₗ r) := ⟨⟩\n\ninstance (r') : creates_limits (PFPNGT₁_to_CHFPNG₁ₑₗ r') :=\ncategory_theory.comp_creates_limits _ _\n\ndef limit_cone_is_limit {J : Type u} [small_category J]\n  (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) : limits.is_limit (limit_cone r K) :=\nlimits.is_limit_of_reflects (PFPNGT₁_to_PFPNG₁ₑₗ r) (ProFiltPseuNormGrp₁.limit_cone_is_limit _)\n\ninstance : limits.has_limits (ProFiltPseuNormGrpWithTinv₁.{u} r) :=\nhas_limits_of_has_limits_creates_limits (PFPNGT₁_to_PFPNG₁ₑₗ r)\n\nlemma is_limit_ext {J : Type u} [small_category J]\n  (K : J ⥤ ProFiltPseuNormGrpWithTinv₁.{u} r) (C : limits.cone K)\n  (hC : limits.is_limit C) (x y : C.X)\n  (h : ∀ j, C.π.app j x = C.π.app j y) : x = y :=\nProFiltPseuNormGrp₁.is_limit_ext _ _ (limits.is_limit_of_preserves (PFPNGT₁_to_PFPNG₁ₑₗ r) hC) _ _ h\n\nsection explicit_products\n\ndef product {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  ProFiltPseuNormGrpWithTinv₁.{u} r :=\n{ M := Π i, X i,\n  -- Why couldn't typeclass inference find this?\n  str := profinitely_filtered_pseudo_normed_group_with_Tinv.pi _ _,\n  exhaustive' := (ProFiltPseuNormGrp₁.product (λ i, ((PFPNGT₁_to_PFPNG₁ₑₗ r).obj (X i)))).exhaustive' } .\n\n@[simps]\ndef product.π {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r) (i) :\n  product r X ⟶ X i :=\n{ map_Tinv' := λ x, rfl,\n  ..(ProFiltPseuNormGrp₁.product.π (λ i, (PFPNGT₁_to_PFPNG₁ₑₗ r).obj (X i))) i }\n\n@[simps]\ndef product.lift {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (M : ProFiltPseuNormGrpWithTinv₁.{u} r) (f : Π i, M ⟶ X i) : M ⟶ product r X :=\n{ map_Tinv' := begin\n    intros x,\n    ext i,\n    change ⇑(product.lift (λ (i : α), (PFPNGT₁_to_PFPNG₁ₑₗ r).obj (X i)) ((PFPNGT₁_to_PFPNG₁ₑₗ r).obj M)\n      (λ (i : α), (PFPNGT₁_to_PFPNG₁ₑₗ r).map (f i))) (Tinv x) i = _,\n    rw ProFiltPseuNormGrp₁.product.lift_to_fun,\n    erw (f i).map_Tinv,\n    refl,\n  end,\n  ..(ProFiltPseuNormGrp₁.product.lift\n      (λ i, (PFPNGT₁_to_PFPNG₁ₑₗ r).obj (X i))\n      ((PFPNGT₁_to_PFPNG₁ₑₗ r).obj M)\n      (λ i, (PFPNGT₁_to_PFPNG₁ₑₗ r).map (f i))) }\n\n@[simp, reassoc]\nlemma product.lift_π {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (M : ProFiltPseuNormGrpWithTinv₁.{u} r) (f : Π i, M ⟶ X i) (i) :\n  product.lift r X M f ≫ product.π r X i = f i := by { ext, simpa }\n\nlemma product.lift_unique {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (M : ProFiltPseuNormGrpWithTinv₁.{u} r) (f : Π i, M ⟶ X i) (g : M ⟶ product r X)\n  (hg : ∀ i, g ≫ product.π r X i = f i) : g = product.lift r X M f :=\nby { ext, simpa [← hg] }\n\nlemma product.hom_ext {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r)\n  (M : ProFiltPseuNormGrpWithTinv₁.{u} r) (g₁ g₂ : M ⟶ product r X)\n  (h : ∀ i, g₁ ≫ product.π r X i = g₂ ≫ product.π r X i) : g₁ = g₂ :=\nbegin\n  rw [product.lift_unique r X M _ g₁ (λ i, rfl), product.lift_unique r X M _ g₂ (λ i, rfl)],\n  simp [h],\nend\n\ndef product.fan {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  limits.fan X := limits.fan.mk (product r X) $ λ b, product.π _ _ _\n\ndef product.is_limit {α : Type u} [fintype α] (X : α → ProFiltPseuNormGrpWithTinv₁.{u} r) :\n  limits.is_limit (product.fan r X) :=\n{ lift := λ M, product.lift _ _ _ $ λ i, M.π.app ⟨i⟩,\n  fac' := begin\n    rintros S ⟨j⟩,\n    dsimp [product.fan],\n    simp,\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply product.hom_ext,\n    intros i,\n    dsimp [product.fan] at hm,\n    simpa using hm ⟨i⟩,\n  end }\n\nend explicit_products\n\nend ProFiltPseuNormGrpWithTinv₁\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/category/strictProFiltPseuNormGrpWithTinv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3043401261539049}}
{"text": "import homotopy_theory.formal.cofibrations.precofibration_category\nimport homotopy_theory.formal.cylinder.definitions\nimport homotopy_theory.formal.cylinder.hep\nimport homotopy_theory.formal.weak_equivalences.definitions\n\nuniverses v u\n\nopen category_theory\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cofibrations\nopen homotopy_theory.cylinder\n\n/-\n\nAn I-category [Baues, Algebraic homotopy, §I.3] is a precofibration\ncategory C equipped with a cylinder functor satisfying the following\nadditional axioms.\n\n* C has an initial object and every object of C is cofibrant.\n\n  From the axioms of a precofibration category, it follows that C\n  admits coproducts. Because we will need these coproducts in order to\n  state a later axiom, we assume that C already comes equipped with a\n  choice of coproducts in order to avoid non-definitionally equal\n  instances.\n\n* The cylinder functor I preserves pushouts by cofibrations and the\n  initial object.\n\n* Cofibrations have the two-sided HEP with respect to the cylinder I.\n\n* The relative cylinder axiom: using the notation ∂I A = A ⊔ A, for\n  each cofibration A → X, in the square\n\n    ∂I A → I A\n      ↓     ↓\n    ∂I X → I X,\n\n  the induced map from the pushout to I X is again a cofibration. (The\n  pushout exists because ∂I A → ∂I X is a cofibration.)\n\n* The cylinder I is equipped with an interchange structure.\n\n-/\n\nvariables (C : Type u) [category.{v} C] [has_initial_object.{v} C] [has_coproducts.{v} C]\n\nclass I_category extends has_cylinder C, preserves_initial_object (I : C ↝ C),\n  precofibration_category C, all_objects_cofibrant.{v} C,\n  cylinder_has_interchange.{v} C :=\n(I_preserves_pushout_by_cof :\n  Π {a b a' b'} {f : a ⟶ b} {g : a ⟶ a'} {f' : a' ⟶ b'} {g' : b ⟶ b'},\n  is_cof f → Is_pushout f g g' f' → Is_pushout (I &> f) (I &> g) (I &> g') (I &> f'))\n(hep_cof : ∀ {a b} (j : a ⟶ b), is_cof j → two_sided_hep j)\n(relative_cylinder : ∀ {a b} (j : a ⟶ b) (hj : is_cof j), is_cof $\n  (pushout_by_cof (∂I &> j) (ii @> a) (cof_coprod hj hj)).is_pushout.induced\n    (ii @> b) (I &> j) (ii.naturality _))\n\nopen precofibration_category\nopen I_category\nvariables {C}\n\n-- Alternate formulation of the relative cylinder axiom, using an\n-- arbitrary pushout rather than the one given by the precofibration\n-- category structure.\nlemma relative_cylinder' [I_category.{v} C] {a b : C} (j : a ⟶ b) (hj : is_cof j)\n  {z} (ii' : ∂I.obj b ⟶ z) (jj' : I.obj a ⟶ z) (po : Is_pushout (∂I &> j) (ii @> a) ii' jj') :\n  is_cof (po.induced (ii @> b) (I &> j) (ii.naturality _)) :=\nlet po' := (pushout_by_cof (∂I &> j) (ii @> a) (cof_coprod hj hj)).is_pushout in\nby convert cof_comp (cof_iso (pushout.unique po po')) (relative_cylinder j hj);\n   apply po.uniqueness; rw ←category.assoc; simp; refl\n\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/i_category/definitions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3043401195039358}}
{"text": "import cof.basic\nimport cof.valley\n\n-- In these files we define the concept of localizing a category at a\n-- (left) multiplicative system.\n-- The presence of conditions defining a multiplicative system mean\n-- that the resulting category is much easier to work with than\n-- localization through the path category.\n-- See Stacks Project [Tag 04VB]\n\nnoncomputable theory\n\nnamespace category_theory\n\nnamespace derived\n\nuniverse u\n\nvariables {C : Type u} [small_category C] {S : morphism_property C}\nvariables {M : left_mult_sys S}\n\n-- Define the identity and composition of valleys\n\ndef id_valley (X : left_calculus C M) : valley X X :=\n{ obj := X,\n  f   := 𝟙 X.as,\n  s   := 𝟙 X.as,\n  qis := M.id }\n\nstructure ore_data {X Y Z : C} := \n  (W  : C)\n  (f₁ : X ⟶ Y) \n  (s₁ : X ⟶ Z) \n  (s₂ : Y ⟶ W) \n  (f₂ : Z ⟶ W)\n  (hs : (S s₁ ∧ S s₂))\n  (comm : (s₁ ≫ f₂) = (f₁ ≫ s₂))\n\nstructure comp_data {X Y Z : left_calculus C M} (v₁ : valley X Y) (v₂ : valley Y Z) :=\n  (W    : C) \n  (h    : v₁.obj.as ⟶ W) \n  (u    : v₂.obj.as ⟶ W) \n  (hu   : S u)\n  (comm : v₂.f ≫ u = v₁.s ≫ h)\n\n-- The Ore condition gives us composition data automatically\ndef data_from_ore {X Y Z : left_calculus C M} {v₁ : valley X Y} {v₂ : valley Y Z} : comp_data v₁ v₂ :=\nlet Y' := v₁.obj, Z' := v₂.obj in\n  have hyp : _ :=\n    M.ore v₂.f v₁.s v₁.qis,\n  let Z'' := classical.some hyp in\n  have hyp₁ : ∃ (f : Y'.as ⟶ Z'') (s : Z'.as ⟶ Z''), S s ∧ v₂.f ≫ s = v₁.s ≫ f := \n    classical.some_spec hyp,\n  let h := classical.some hyp₁ in\n  have hyp₂ : ∃ (s : Z'.as ⟶ Z''), S s ∧ v₂.f ≫ s = v₁.s ≫ h := \n    classical.some_spec hyp₁,\n  let u := classical.some hyp₂ in\n  have hyp₃ : S u ∧ v₂.f ≫ u = v₁.s ≫ h := (classical.some_spec hyp₂),\n  { W    := Z'',\n    h    := h,\n    u    := u,\n    hu   := hyp₃.left,\n    comm := hyp₃.right }\n\ndef comp_valley_from_data {X Y Z : left_calculus C M} \n    (v₁ : valley X Y) (v₂ : valley Y Z) (data : comp_data v₁ v₂)\n  : valley X Z :=\n  { obj := ⟨ data.W ⟩,\n    f   := v₁.f ≫ data.h,\n    s   := v₂.s ≫ data.u,\n    qis := M.comp ⟨v₂.qis, data.hu⟩ }\n\nlemma comp_valley_obj {X Y Z : left_calculus C M} \n    (v₁ : valley X Y) (v₂ : valley Y Z) (data : comp_data v₁ v₂)\n  : (comp_valley_from_data v₁ v₂ data).obj.as = data.W := rfl\n\n-- As a default, we can always define composition using the Ore data\ndef comp_valley {X Y Z : left_calculus C M} (v₁ : valley X Y) (v₂ : valley Y Z) : valley X Z :=\n  comp_valley_from_data v₁ v₂ data_from_ore\n\n-- Composition is well defined under the equivalence relation\nlemma comp_independent_of_data' {X Y Z : left_calculus C M} (v₁ : valley X Y) (v₂ : valley Y Z) (d₁ d₂ : comp_data v₁ v₂) :\n  ⟦ comp_valley_from_data v₁ v₂ d₁ ⟧ = ⟦ comp_valley_from_data v₁ v₂ d₂ ⟧  :=\nlet c₁ := comp_valley_from_data v₁ v₂ d₁, c₂ := comp_valley_from_data v₁ v₂ d₂ in\nmatch M.ore d₂.u d₁.u d₁.hu with ⟨W, h', u', hu', hc⟩ := \n  begin\n    have h₁ : _, by calc\n      v₁.s ≫ d₂.h ≫ u' = (v₂.f ≫ d₂.u) ≫ u' : by rw [←category.assoc, d₂.comm]\n      ... = v₂.f ≫ (d₁.u ≫ h') : by rw [category.assoc, hc]\n      ... = v₁.s ≫ d₁.h ≫ h' : by {rw [←category.assoc, d₁.comm], simp},\n    have h₂ : _, from M.cancel ⟨v₁.qis, h₁⟩,\n    rcases h₂ with ⟨Z', s', hs', hc'⟩,\n\n    let h'' := h' ≫ s',\n    let u'' := u' ≫ s',\n    have h₂ : d₂.h ≫ u'' = d₁.h ≫ h'', by rw [←category.assoc, hc', category.assoc],\n    have h₃ : d₂.u ≫ u'' = d₁.u ≫ h'', by rw [←category.assoc, hc, category.assoc],\n    have hu'' : S u'', from M.comp ⟨hu', hs'⟩,\n\n    apply quotient.eq.mpr,\n    \n    let v : valley X Z := {\n      obj := {as := Z'},\n      f   := v₁.f ≫ d₂.h ≫ u'',\n      s   := v₂.s ≫ d₂.u ≫ u'',\n      qis := triple_comp M ⟨v₂.qis, d₂.hu, hu''⟩,\n    },\n    use [v, h'', u''],\n\n    simp, split,\n    have hlemma : c₁.f = (v₁.f ≫ d₁.h), by refl, \n    rw [hlemma, h₂], simp,\n\n    split,\n    have hlemma : c₁.s = (v₂.s ≫ d₁.u), by refl,\n    rw [hlemma, h₃], simp,\n\n    split,\n    have hlemma : c₂.f = (v₁.f ≫ d₂.h), by refl,\n    rw hlemma, simp,\n\n    have hlemma : c₂.s = (v₂.s ≫ d₂.u), by refl,\n    rw hlemma, simp,\n  end\nend\n\n\nlemma comp_independent_of_data {X Y Z : left_calculus C M} (v₁ : valley X Y) (v₂ : valley Y Z) (d : comp_data v₁ v₂) :\n  ⟦ comp_valley v₁ v₂ ⟧ = ⟦ comp_valley_from_data v₁ v₂ d ⟧  :=\ncomp_independent_of_data' v₁ v₂ data_from_ore d\n\nlemma dom_imp_post_comp {X Y Z : left_calculus C M} (v₁ v₂ : valley X Y) (w : valley Y Z) (dom : v₁ E v₂) : \n  ⟦ comp_valley v₁ w ⟧ = ⟦ comp_valley v₂ w ⟧ :=\nbegin\n  rcases dom with ⟨a, ha₁, ha₂⟩,\n\n  let g := w.f,\n  have hore : _ := M.ore g v₂.s v₂.qis,\n  rcases hore with ⟨Z'', h, u, hu, hcomm⟩,\n\n  have hcomm' : g ≫ u = v₁.s ≫ a ≫ h, by rw [←category.assoc, ha₂, hcomm],\n  let d₁ : comp_data v₁ w := {\n    W := Z'',\n    h := a ≫ h,\n    u := u,\n    hu := hu,\n    comm := hcomm',\n  },\n  let d₂ : comp_data v₂ w := {\n    W := Z'',\n    h := h,\n    u := u,\n    hu := hu,\n    comm := hcomm,\n  },\n  have hcomp₁ : _ := comp_independent_of_data v₁ w d₁,\n  have hcomp₂ : _ := comp_independent_of_data v₂ w d₂,\n\n  let c₁ := comp_valley_from_data v₁ w d₁,\n  let c₂ := comp_valley_from_data v₂ w d₂,\n  suffices heq : ⟦ c₁ ⟧ = ⟦ c₂ ⟧, by rw [hcomp₁, hcomp₂, heq],\n\n  suffices heq' : veq X Z c₁ c₂, by {apply quotient.eq.mpr, exact heq'},\n  \n  use [Z'', c₁.f, c₂.s, c₂.qis, 𝟙 Z'', 𝟙 Z''],\n  simp,\n  -- idk why we need this, but simp can't seem to take care of it\n  -- otherwise.\n  have hlemma : ∀ {X : C}, ∀ f : X ⟶ Z'', f ≫ 𝟙 Z'' = f, by {intro f, simp},\n  split, exact hlemma c₁.f, rw [hlemma c₁.s],\n  split, refl, \n  split, rw [hlemma c₂.f],\n\n  have hc₁ : c₁.f = v₁.f ≫ a ≫ h, by refl, rw hc₁,\n  have hc₂ : c₂.f = v₂.f ≫ h, by refl, rw hc₂,\n  rw [←category.assoc, ha₁],\n  rw hlemma c₂.s,\nend\n\n\nlemma dom_imp_pre_comp {X Y Z : left_calculus C M} (v₁ v₂ : valley X Y) (w: valley Z X) (dom : v₁ E v₂) : \n  ⟦ comp_valley w v₁ ⟧ = ⟦ comp_valley w v₂ ⟧ :=\nbegin\n  rcases dom with ⟨a, ha₁, ha₂⟩,\n\n  have hore₁ : _ := M.ore v₁.f w.s w.qis,\n  rcases hore₁ with ⟨Z₁, h₁, t₁, ht₁, hcomm₁⟩,\n  have hore₂ : _ := M.ore a t₁ ht₁,\n  rcases hore₂ with ⟨Z₂, h₂, t₂, ht₂, hcomm₂⟩,\n\n  let d₁ : comp_data w v₁ := {\n    W := Z₁,\n    h := h₁,\n    u := t₁,\n    hu := ht₁,\n    comm := hcomm₁,\n  },\n\n  have hcomm' : v₂.f ≫ t₂ = w.s ≫ h₁ ≫ h₂, by\n    {rw [←ha₁, category.assoc, hcomm₂, ←category.assoc, hcomm₁], simp},\n  let d₂ : comp_data w v₂ := {\n    W := Z₂,\n    h := h₁ ≫ h₂,\n    u := t₂,\n    hu := ht₂,\n    comm := hcomm',\n  },\n\n  have hcomp₁ : _ := comp_independent_of_data w v₁ d₁,\n  have hcomp₂ : _ := comp_independent_of_data w v₂ d₂,\n\n  let c₁ := comp_valley_from_data w v₁ d₁,\n  let c₂ := comp_valley_from_data w v₂ d₂,\n  suffices heq : ⟦ c₁ ⟧ = ⟦ c₂ ⟧, by rw [hcomp₁, hcomp₂, heq],\n  suffices heq' : veq Z Y c₁ c₂, by {apply quotient.eq.mpr, exact heq'},\n\n  use [c₂, h₂, 𝟙 Z₂],  \n  split, \n  { have hlemma₁ : c₁.f = w.f ≫ h₁, by refl,\n    rw hlemma₁, simp,\n    have hlemma₂ : c₂.f = w.f ≫ (h₁ ≫ h₂), by refl,\n    rw hlemma₂ },\n  split, \n  { have hlemma₁ : c₁.s = v₁.s ≫ t₁, by refl, \n    have hlemma₂ : c₂.s = v₂.s ≫ t₂, by refl,\n    rw [hlemma₁, hlemma₂], simp, rw [←hcomm₂, ←category.assoc, ha₂] },\n   \n  have hlemma : ∀ {X : C}, ∀ f : X ⟶ Z₂, f ≫ 𝟙 Z₂ = f, by {intro f, simp},\n  split,\n  rw hlemma c₂.f,\n  rw hlemma c₂.s,\nend\n\nlemma comp_well_def {X Y Z : left_calculus C M}  (v₁ v₁' : valley X Y) (v₂ v₂' : valley Y Z) :\n  ⟦ v₁ ⟧ = ⟦ v₁' ⟧ ∧ ⟦ v₂ ⟧ = ⟦ v₂' ⟧ → ⟦ comp_valley v₁ v₂ ⟧ = ⟦ comp_valley v₁' v₂' ⟧ := \nbegin\n  rintro ⟨ h₁, h₂ ⟩,\n  have h₁' : _ := quotient.eq.mp h₁,\n  have h₂' : _ := quotient.eq.mp h₂,\n  \n  show ⟦ comp_valley v₁ v₂ ⟧ = ⟦ comp_valley v₁' v₂' ⟧,\n\n  have dom₁ : _ := (dom_iff_equiv _ _).mpr h₁',\n  have dom₂ : _ := (dom_iff_equiv _ _).mpr h₂',\n\n  rcases dom₁ with ⟨w₁, dv₁, dv₁'⟩,\n  rcases dom₂ with ⟨w₂, dv₂, dv₂'⟩,\n  \n  have heq₁ : ⟦ comp_valley v₁ v₂ ⟧ = ⟦ comp_valley w₁ w₂ ⟧,\n  by calc\n    ⟦ comp_valley v₁ v₂ ⟧ \n        = ⟦ comp_valley w₁ v₂ ⟧ : by { apply (dom_imp_post_comp _ _ _ dv₁)}\n    ... = ⟦ comp_valley w₁ w₂ ⟧ : by { apply (dom_imp_pre_comp  _ _ _ dv₂)},\n  \n  have heq₂ : ⟦ comp_valley v₁' v₂' ⟧ = ⟦ comp_valley w₁ w₂ ⟧,\n  by calc\n    ⟦ comp_valley v₁' v₂' ⟧\n        = ⟦ comp_valley w₁ v₂' ⟧ : by { apply (dom_imp_post_comp _ _ _ dv₁')}\n    ... = ⟦ comp_valley w₁ w₂ ⟧  : by { apply (dom_imp_pre_comp  _ _ _ dv₂')},\n  \n  rw [heq₁, heq₂],\nend\n\n-- The axioms for the category\n\ndef hom_type (X Y : left_calculus C M) := quotient (valley_setoid X Y)\n\ndef id (X : left_calculus C M) := ⟦ id_valley X ⟧\n\ndef comp {X Y Z : left_calculus C M} (f : hom_type X Y) (g : hom_type Y Z) := \n  ⟦ comp_valley (quotient.out f) (quotient.out g) ⟧\n\n@[simp]\nlemma id_comp' (X Y : left_calculus C M) (f : hom_type X Y) :\n  comp (id X) f = f :=\nlet g := comp (id X) f,\n    f' := f.out,\n    data : comp_data (id_valley X) f' :=\n      { W := f'.obj.as,\n        h := f'.f,\n        u := (𝟙 f'.obj.as),\n        hu := M.id,\n        comm := have h : (id_valley X).s = 𝟙 X.as, from rfl,\n          by {simp, rw h, simp},\n      },\n    g' := comp_valley_from_data (id_valley X) f' data in\nbegin\n  change g = f,\n  have h₁ : g = ⟦ comp_valley (id_valley X) f' ⟧, \n    by { apply comp_well_def, split, simp, refl, refl },\n  have h₂ : ⟦ comp_valley (id_valley X) f' ⟧ = ⟦ comp_valley_from_data (id_valley X) f' data ⟧,\n    from comp_independent_of_data _ _ _,\n  rw [h₁, h₂],\n  suffices : veq X Y g' f', by {\n      apply quotient.mk_eq_iff_out.mpr,\n      exact this,\n  },\n  use [f', 𝟙 f'.obj.as, 𝟙 f'.obj.as], simp,\n  have hlemma : ∀ {X : C}, ∀ f : X ⟶ f'.obj.as, f ≫ 𝟙 f'.obj.as = f, by {intro f, simp},\n  split,\n  { have : g'.f = (𝟙 X.as) ≫ f'.f, by refl, rw this,\n    simp,\n    rw hlemma f'.f },\n  { have : g'.s = f'.s ≫ (𝟙 f'.obj.as), by refl, rw this, \n    simp,\n    rw hlemma f'.s},\nend\n\n@[simp]\nlemma comp_id' (X Y : left_calculus C M) (f : hom_type X Y) :\n  comp f (id Y) = f :=\nlet g := comp f (id Y),\n    f' := f.out,\n    data : comp_data f' (id_valley Y) :=\n      { W := f'.obj.as,\n        h := (𝟙 f'.obj.as),\n        u := f'.s,\n        hu := f'.qis,\n        comm := have h : (id_valley Y).f = 𝟙 Y.as, from rfl,\n          by {simp, rw h, simp},\n      },\n    g' := comp_valley_from_data f' (id_valley Y) data in\nbegin\n  change g = f,\n  have h₁ : g = ⟦ comp_valley f' (id_valley Y) ⟧, \n    by { apply comp_well_def, split, simp, simp, refl },\n  have h₂ : ⟦ comp_valley f' (id_valley Y) ⟧ = ⟦ comp_valley_from_data f' (id_valley Y) data ⟧,\n    from comp_independent_of_data _ _ _,\n  rw [h₁, h₂],\n  suffices : veq X Y g' f', by {\n      apply quotient.mk_eq_iff_out.mpr,\n      exact this,\n  },\n  use [f', 𝟙 f'.obj.as, 𝟙 f'.obj.as], simp,\n  have hlemma : ∀ {X : C}, ∀ f : X ⟶ f'.obj.as, f ≫ 𝟙 f'.obj.as = f, by {intro f, simp},\n  split,\n  { have : g'.f = f'.f ≫ (𝟙 f'.obj.as), by refl, rw this,\n    simp,\n    rw hlemma f'.f },\n  { have : g'.s = (𝟙 Y.as) ≫ f'.s, by refl, rw this, \n    simp,\n    rw hlemma f'.s},\nend\n\nlemma assoc' {W X Y Z : left_calculus C M} (f : hom_type W X) (g : hom_type X Y) (h : hom_type Y Z) :\n  comp (comp f g) h = comp f (comp g h) :=\nlet a := f.out,\n    b := g.out,\n    c := h.out in\nbegin\n  rcases (M.ore b.f a.s a.qis) with ⟨Y'', f₁, s₁, hs₁, hc₁⟩,\n  rcases (M.ore c.f b.s b.qis) with ⟨Z'', f₂, s₂, hs₂, hc₂⟩,\n  rcases (M.ore f₂ s₁ hs₁)     with ⟨Z''', f₃, s₃, hs₃, hc₃⟩,\n\n  let v : valley W Z := {\n    obj := ⟨ Z''' ⟩,\n    f   := a.f ≫ f₁ ≫ f₃,\n    s   := c.s ≫ s₂ ≫ s₃,\n    qis  := triple_comp M ⟨c.qis, hs₂, hs₃⟩,\n  },\n\n  let d₁ : comp_data a b := {\n    W := Y'',\n    h := f₁,\n    u := s₁,\n    hu := hs₁,\n    comm := hc₁,\n  },\n  let d₂ : comp_data b c := {\n    W := Z'',\n    h := f₂,\n    u := s₂,\n    hu := hs₂,\n    comm := hc₂,\n  },\n\n  let cab := comp_valley_from_data a b d₁,\n  let cbc := comp_valley_from_data b c d₂,\n  have h₁ : comp f g = ⟦ cab ⟧, by {apply comp_independent_of_data},\n  have h₂ : comp g h = ⟦ cbc ⟧, by {apply comp_independent_of_data},\n  rw [h₁, h₂],\n\n  let d₃ : comp_data cab c := {\n    W := Z''',\n    h := f₃,\n    u := s₂ ≫ s₃,\n    hu := M.comp ⟨hs₂, hs₃⟩,\n    comm := begin \n      have : cab.s = b.s ≫ s₁, by refl, \n      rw [this, category.assoc, ←hc₃, ←category.assoc, hc₂], \n      simp, \n    end,\n  },\n  let d₄ : comp_data a cbc := {\n    W := Z''',\n    h := f₁ ≫ f₃,\n    u := s₃,\n    hu := hs₃,\n    comm := begin \n      rw [←category.assoc, ←hc₁, category.assoc, ←hc₃, ←category.assoc], \n      refl,\n    end,\n  },\n  let ccabc := comp_valley_from_data cab c d₃,\n  let cacbc := comp_valley_from_data a cbc d₄,\n  \n  have hcomp₁ : comp ⟦cab⟧ h = ⟦ ccabc ⟧, by calc\n    comp ⟦cab⟧ h = ⟦comp_valley cab c⟧ : by {apply comp_well_def, simp}\n    ... = ⟦ ccabc ⟧ : by {apply comp_independent_of_data},\n  have hcomp₂ : comp f ⟦cbc⟧ = ⟦ cacbc ⟧, by calc\n    comp f ⟦cbc⟧ = ⟦comp_valley a cbc⟧ : by {apply comp_well_def, simp}\n    ... = ⟦ cacbc ⟧ : by {apply comp_independent_of_data},\n  rw [hcomp₁, hcomp₂],\n\n  apply quotient.eq.mpr,\n  use [v, (𝟙 Z'''), (𝟙 Z''')],\n  simp, \n  have hlemma : ∀ {X : C}, ∀ f : X ⟶ Z''', f ≫ 𝟙 Z''' = f, by {intro f, simp},\n  repeat {rw hlemma}, split, \n    {rw ←category.assoc, refl},\n  split,\n    {refl},\n  split,\n    {refl},\n    {rw ←category.assoc, refl}\nend\n\n-- Define the category structure\n\ninstance : category (left_calculus C M) :=\n{ hom  := hom_type,\n  id   := id,\n  comp := λ _ _ _, comp,\n  id_comp' := id_comp',\n  comp_id' := comp_id',\n  assoc' := λ _ _ _ _, assoc',\n}\n\n-- Some properties of this category\n\n-- We have a canonical functor\n@[simps]\ndef Q : C ⥤ left_calculus C M :=\n{ obj := λ a, { as := a },\n  map := λ X Y f, ⟦{ obj := ⟨ Y ⟩, f := f, s := 𝟙 Y, qis := M.id }⟧,\n  map_comp' := λ X Y Z f g, begin\n    -- Simplify notation\n    let v₁ : valley ⟨X⟩ ⟨Z⟩ := {obj := {as := Z}, f := f ≫ g, s := 𝟙 Z, qis := _},\n    let v₂ : valley ⟨X⟩ ⟨Y⟩ := {obj := {as := Y}, f := f, s := 𝟙 Y, qis := _},\n    let v₃ : valley ⟨Y⟩ ⟨Z⟩ := {obj := {as := Z}, f := g, s := 𝟙 Z, qis := _},\n    let c₁ : hom_type ⟨X⟩ ⟨Z⟩ := ⟦v₁⟧,\n    let c₂ : hom_type ⟨X⟩ ⟨Z⟩ := comp ⟦v₂⟧ ⟦v₃⟧,\n    suffices h : c₁ = c₂, by exact h,\n\n    have h' : c₂ = ⟦ comp_valley v₂ v₃ ⟧, by calc\n        c₂ = ⟦comp_valley v₂ ⟦v₃⟧.out⟧ : by\n          {apply comp_well_def, simp, split, repeat {apply (valley_equiv_refl _ _)}}\n        ... = ⟦comp_valley v₂ v₃⟧ : by\n          {apply comp_well_def, simp, split, repeat {apply (valley_equiv_refl _ _)}},\n    rw h',\n\n    have h'' : ⟦ comp_valley v₂ v₃ ⟧ = c₁, by begin\n      let d : comp_data v₂ v₃ := \n        { W := Z, h := g, u := 𝟙 Z, hu := M.id, comm := by simp },\n      let c₄ := comp_valley_from_data v₂ v₃ d,\n      have : ⟦comp_valley v₂ v₃⟧ = ⟦c₄⟧, by apply comp_independent_of_data,\n      rw this,\n\n      apply quotient.eq.mpr,\n      use [v₁, 𝟙 Z], simp, \n\n      have : c₄.f = f ≫ g, by refl, rw this,\n      have : c₄.s = 𝟙 Z ≫ 𝟙 Z, by refl, rw this,\n      simp, rw ←category.assoc, \n      have hlemma : ∀ {X : C}, ∀ f : X ⟶ Z, f ≫ 𝟙 Z = f, by simp,\n      rw hlemma,\n    end,\n    rw ←h'',\n  end }\n\ntheorem functor_inverts_qis {X Y : C} (s : X ⟶ Y) [qis : S s] : @is_iso (left_calculus C M) _ _ _ (Q.map s) := \n⟨ begin \n    let v : valley ⟨Y⟩ ⟨X⟩ := {obj := ⟨Y⟩, f := 𝟙 Y, s := s, qis := qis}, \n    use ⟦ v ⟧,\n    sorry,\n  end ⟩\n\ntheorem functor_univ_prop {D : Type u} [category D] (g : C ⥤ D) : \n  (∀ X Y : C, ∀ s : X ⟶ Y, S s → is_iso (g.map s)) \n  → ∃! h : left_calculus C M ⥤ D, g = Q ⋙ h := \nbegin\n  sorry\nend\n\nend derived\n\nend category_theory", "meta": {"author": "avarsh", "repo": "derived-categories-lean", "sha": "449196fd6dcccb27de28500d156aec30185f9c15", "save_path": "github-repos/lean/avarsh-derived-categories-lean", "path": "github-repos/lean/avarsh-derived-categories-lean/derived-categories-lean-449196fd6dcccb27de28500d156aec30185f9c15/src/cof/category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3039365556728161}}
{"text": "import category_theory.colimit_lemmas\nimport .definitions\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` ∘ `:80 g:80 := g ≫ f\n\nnamespace homotopy_theory.cofibrations\nopen category_theory.has_initial_object category_theory.preserves_initial_object\nopen category_theory.preserves_coproducts\nopen homotopy_theory.cylinder\nopen I_category\n\nvariables {C : Type u} [category.{v} C] [has_initial_object.{v} C]\n  [has_coproducts.{v} C] [I_category.{v} C]\n\ndef Ii_initial : Is_initial_object.{v} (I.obj ∅ : C) :=\nIs_initial_object_of_Is_initial_object\n  has_initial_object.initial_object.is_initial_object\n\ndef I_preserves_coproducts : preserves_coproducts (I : C ↝ C) :=\n⟨λ a₀ a₁ b f₀ f₁ copr,\n  let po : Is_pushout (! a₀) (! a₁) f₀ f₁ :=\n    Is_pushout_of_Is_coproduct_of_Is_initial copr\n      has_initial_object.initial_object.is_initial_object in\n  Is_coproduct_of_Is_pushout_of_Is_initial\n    (I_preserves_pushout_by_cof (all_objects_cofibrant.cofibrant a₀) po) Ii_initial⟩\n\ndef I_of_coprod_is_coproduct {a₀ a₁ : C} :\n  Is_coproduct (I &> (i₀ : a₀ ⟶ a₀ ⊔ a₁)) (I &> (i₁ : a₁ ⟶ a₀ ⊔ a₁)) :=\nlet ⟨f⟩ := I_preserves_coproducts in\nf (has_coproducts.coproduct.{v} a₀ a₁).is_coproduct\n\nlemma I_preserves_cofibrations {a b : C} {j : a ⟶ b} (hj : is_cof j) : is_cof (I.map j) :=\nbegin\n  convert cof_comp _ (relative_cylinder j hj),\n  exact (Is_pushout.induced_commutes₁ _ _ _ _).symm,\n  exact precofibration_category.pushout_is_cof (pushout.is_pushout _) (cof_coprod hj hj)\nend\n\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/i_category/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.30359592031846877}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.functor\nimport category_theory.full_subcategory\n\n/-!\n# Monoidal natural transformations\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nNatural transformations between (lax) monoidal functors must satisfy\nan additional compatibility relation with the tensorators:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`.\n\n(Lax) monoidal functors between a fixed pair of monoidal categories\nthemselves form a category.\n-/\n\nopen category_theory\n\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nopen category_theory.category\nopen category_theory.functor\n\nnamespace category_theory\n\nopen monoidal_category\n\nvariables {C : Type u₁} [category.{v₁} C] [monoidal_category.{v₁} C]\n          {D : Type u₂} [category.{v₂} D] [monoidal_category.{v₂} D]\n\n/--\nA monoidal natural transformation is a natural transformation between (lax) monoidal functors\nadditionally satisfying:\n`F.μ X Y ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ X Y`\n-/\n@[ext]\nstructure monoidal_nat_trans (F G : lax_monoidal_functor C D)\n  extends nat_trans F.to_functor G.to_functor :=\n(unit' : F.ε ≫ app (𝟙_ C) = G.ε . obviously)\n(tensor' : ∀ X Y, F.μ _ _ ≫ app (X ⊗ Y) = (app X ⊗ app Y) ≫ G.μ _ _ . obviously)\n\nrestate_axiom monoidal_nat_trans.tensor'\nattribute [simp, reassoc] monoidal_nat_trans.tensor\nrestate_axiom monoidal_nat_trans.unit'\nattribute [simp, reassoc] monoidal_nat_trans.unit\n\nnamespace monoidal_nat_trans\n\n/--\nThe identity monoidal natural transformation.\n-/\n@[simps]\ndef id (F : lax_monoidal_functor C D) : monoidal_nat_trans F F :=\n{ ..(𝟙 F.to_functor) }\n\ninstance (F : lax_monoidal_functor C D) : inhabited (monoidal_nat_trans F F) := ⟨id F⟩\n\n/--\nVertical composition of monoidal natural transformations.\n-/\n@[simps]\ndef vcomp {F G H : lax_monoidal_functor C D}\n  (α : monoidal_nat_trans F G) (β : monoidal_nat_trans G H) : monoidal_nat_trans F H :=\n{ ..(nat_trans.vcomp α.to_nat_trans β.to_nat_trans) }\n\ninstance category_lax_monoidal_functor : category (lax_monoidal_functor C D) :=\n{ hom := monoidal_nat_trans,\n  id := id,\n  comp := λ F G H α β, vcomp α β, }\n\n@[simp] lemma comp_to_nat_trans_lax {F G H : lax_monoidal_functor C D} {α : F ⟶ G} {β : G ⟶ H} :\n  (α ≫ β).to_nat_trans =\n    @category_struct.comp (C ⥤ D) _ _ _ _ (α.to_nat_trans) (β.to_nat_trans) := rfl\n\ninstance category_monoidal_functor : category (monoidal_functor C D) :=\ninduced_category.category monoidal_functor.to_lax_monoidal_functor\n\n@[simp] \n\nvariables {E : Type u₃} [category.{v₃} E] [monoidal_category.{v₃} E]\n\n/--\nHorizontal composition of monoidal natural transformations.\n-/\n@[simps]\ndef hcomp {F G : lax_monoidal_functor C D} {H K : lax_monoidal_functor D E}\n  (α : monoidal_nat_trans F G) (β : monoidal_nat_trans H K) :\n  monoidal_nat_trans (F ⊗⋙ H) (G ⊗⋙ K) :=\n{ unit' :=\n  begin\n    dsimp, simp,\n    conv_lhs { rw [←K.to_functor.map_comp, α.unit], },\n  end,\n  tensor' := λ X Y,\n  begin\n    dsimp, simp,\n    conv_lhs { rw [←K.to_functor.map_comp, α.tensor, K.to_functor.map_comp], },\n  end,\n  ..(nat_trans.hcomp α.to_nat_trans β.to_nat_trans) }\n\nsection\n\nlocal attribute [simp] nat_trans.naturality monoidal_nat_trans.unit monoidal_nat_trans.tensor\n\n/-- The cartesian product of two monoidal natural transformations is monoidal. -/\n@[simps]\ndef prod {F G : lax_monoidal_functor C D} {H K : lax_monoidal_functor C E}\n  (α : monoidal_nat_trans F G) (β : monoidal_nat_trans H K) :\n  monoidal_nat_trans (F.prod' H) (G.prod' K) :=\n{ app := λ X, (α.app X, β.app X) }\n\nend\n\nend monoidal_nat_trans\n\nnamespace monoidal_nat_iso\n\nvariables {F G : lax_monoidal_functor C D}\n\n/--\nConstruct a monoidal natural isomorphism from object level isomorphisms,\nand the monoidal naturality in the forward direction.\n-/\ndef of_components\n  (app : ∀ X : C, F.obj X ≅ G.obj X)\n  (naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app Y).hom = (app X).hom ≫ G.map f)\n  (unit : F.ε ≫ (app (𝟙_ C)).hom = G.ε)\n  (tensor : ∀ X Y, F.μ X Y ≫ (app (X ⊗ Y)).hom = ((app X).hom ⊗ (app Y).hom) ≫ G.μ X Y) :\n  F ≅ G :=\n{ hom := { app := λ X, (app X).hom, },\n  inv :=\n  { app := λ X, (app X).inv,\n    unit' := by { dsimp, rw [←unit, assoc, iso.hom_inv_id, comp_id], },\n    tensor' := λ X Y,\n    begin\n      dsimp,\n      rw [iso.comp_inv_eq, assoc, tensor, ←tensor_comp_assoc,\n        iso.inv_hom_id, iso.inv_hom_id, tensor_id, id_comp],\n    end,\n    ..(nat_iso.of_components app @naturality).inv, }, }\n\n@[simp] lemma of_components.hom_app\n  (app : ∀ X : C, F.obj X ≅ G.obj X) (naturality) (unit) (tensor) (X) :\n  (of_components app naturality unit tensor).hom.app X = (app X).hom := rfl\n@[simp] lemma of_components.inv_app\n  (app : ∀ X : C, F.obj X ≅ G.obj X) (naturality) (unit) (tensor) (X) :\n  (of_components app naturality unit tensor).inv.app X = (app X).inv :=\nby simp [of_components]\n\ninstance is_iso_of_is_iso_app (α : F ⟶ G) [∀ X : C, is_iso (α.app X)] : is_iso α :=\n⟨(is_iso.of_iso (of_components (λ X, as_iso (α.app X))\n  (λ X Y f, α.to_nat_trans.naturality f) α.unit α.tensor)).1⟩\n\nend monoidal_nat_iso\n\nnoncomputable theory\n\n/-- The unit of a monoidal equivalence can be upgraded to a monoidal natural transformation. -/\n@[simps]\ndef monoidal_unit (F : monoidal_functor C D) [is_equivalence F.to_functor] :\n  lax_monoidal_functor.id C ⟶\n    F.to_lax_monoidal_functor ⊗⋙ (monoidal_inverse F).to_lax_monoidal_functor :=\nlet e := F.to_functor.as_equivalence in\n{ to_nat_trans := e.unit,\n  tensor' := λ X Y, begin\n    -- This proof is not pretty; golfing welcome!\n    dsimp,\n    simp only [adjunction.hom_equiv_unit, adjunction.hom_equiv_naturality_right,\n      category.id_comp, category.assoc],\n    simp only [←functor.map_comp],\n    erw [e.counit_app_functor, e.counit_app_functor, F.to_lax_monoidal_functor.μ_natural,\n      is_iso.inv_hom_id_assoc],\n    simp only [category_theory.is_equivalence.inv_fun_map],\n    slice_rhs 2 3 { erw iso.hom_inv_id_app, },\n    dsimp,\n    simp only [category_theory.category.id_comp],\n    slice_rhs 1 2 { rw [←tensor_comp, iso.hom_inv_id_app, iso.hom_inv_id_app],\n      dsimp, rw [tensor_id], },\n    simp,\n  end }.\n\ninstance (F : monoidal_functor C D) [is_equivalence F.to_functor] : is_iso (monoidal_unit F) :=\nbegin\n  haveI : ∀ (X : C), is_iso ((monoidal_unit F).to_nat_trans.app X),\n  { intros, dsimp, apply_instance, },\n  exact monoidal_nat_iso.is_iso_of_is_iso_app _\nend\n\n/-- The counit of a monoidal equivalence can be upgraded to a monoidal natural transformation. -/\n@[simps]\ndef monoidal_counit (F : monoidal_functor C D) [is_equivalence F.to_functor] :\n  (monoidal_inverse F).to_lax_monoidal_functor ⊗⋙ F.to_lax_monoidal_functor ⟶\n    lax_monoidal_functor.id D :=\nlet e := F.to_functor.as_equivalence in\n{ to_nat_trans := e.counit,\n  unit' := begin\n    dsimp,\n    simp only [category.comp_id, category.assoc, functor.map_inv, functor.map_comp,\n      nat_iso.inv_inv_app, is_iso.inv_comp, is_equivalence.fun_inv_map, adjunction.hom_equiv_unit],\n    erw [e.counit_app_functor, ←e.functor.map_comp_assoc, iso.hom_inv_id_app],\n    dsimp, simp,\n  end,\n  tensor' := λ X Y, begin\n    dsimp,\n    simp only [adjunction.hom_equiv_unit, adjunction.hom_equiv_naturality_right, category.assoc,\n      category.comp_id, functor.map_comp],\n    simp only [is_equivalence.fun_inv_map],\n    erw [e.counit_app_functor],\n    simp only [category.assoc],\n    erw [←e.functor.map_comp_assoc],\n    simp only [category_theory.iso.inv_hom_id_app,\n      category_theory.iso.inv_hom_id_app_assoc],\n    erw [iso.hom_inv_id_app],\n    erw [category_theory.functor.map_id],\n    simp only [category.id_comp],\n    simp only [category_theory.iso.inv_hom_id_app,\n      category_theory.is_iso.hom_inv_id_assoc],\n    erw [iso.inv_hom_id_app],\n    dsimp, simp, refl,\n  end }\n\ninstance (F : monoidal_functor C D) [is_equivalence F.to_functor] : is_iso (monoidal_counit F) :=\nbegin\n  haveI : ∀ (X : D), is_iso ((monoidal_counit F).to_nat_trans.app X),\n  { intros, dsimp, apply_instance, },\n  exact monoidal_nat_iso.is_iso_of_is_iso_app _\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/natural_transformation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.3035959137502773}}
{"text": "\nimport unitb.models.nondet\nimport unitb.refinement.basic\n\nnamespace nondet\n\nopen temporal\nopen predicate\nopen unitb\n\nuniverse variable u\n\nsection defs\n\nvariables {α β : Type}\n\nstructure evt_ref (lbl : Type) (mc : program α) (ea : event α) (ecs : lbl → event α) : Type :=\n  (witness : lbl → α → Prop)\n  (witness_fis : ⦃ ∃∃ e, witness e ⦄)\n  (delay : ∀ ec, witness ec ⋀ ea.coarse_sch ⋀ ea.fine_sch ↦ witness ec ⋀ (ecs ec).coarse_sch in mc)\n  (stable : ∀ ec, unless_except mc (witness ec ⋀ (ecs ec).coarse_sch) (-ea.coarse_sch) { e | ∃ l, ecs l = e })\n  (resched : ∀ ec, ea.coarse_sch ⋀ ea.fine_sch ⋀ witness ec ↦ (ecs ec).fine_sch in mc)\n\nstructure refined (ma mc : program α) : Type :=\n  (sim_init : mc^.first ⟹ ma^.first)\n  (abs : option mc.lbl → option ma.lbl)\n  (evt_sim : ∀ ec, ⟦ mc.step_of ec ⟧ ⟹ ⟦ ma.step_of (abs ec) ⟧)\n  (events : ∀ ae, evt_ref { ec // abs ec = ae } mc (ma.event ae) (λ ec, mc.event ec.val) )\n\nlemma refined.sim {ma mc : program α}\n  (R : refined ma mc)\n: ⟦ is_step mc ⟧ ⟹ ⟦ is_step ma ⟧ :=\nbegin\n  simp [is_step_exists_event'],\n  intro τ,\n  simp,\n  intros ce H,\n  existsi (R.abs ce),\n  apply R.evt_sim _ _ H,\nend\n\nend defs\n\nsection soundness\n\nparameters {α β : Type}\n\nparameter (ma : program α)\nparameter (mc : program α)\n\nopen temporal\n\nparameter R : refined ma mc\nparameter τ : stream α\nparameter M₁ : system_sem.ex mc τ\n\nsection schedules\n\nparameter e : option ma.lbl\n@[reducible]\ndef imp_lbl := { ec : option mc.lbl // R.abs ec = e }\n\ndef AC := (program.event ma e).coarse_sch\ndef AF := (program.event ma e).fine_sch\ndef W (e' : imp_lbl) := (R.events e).witness e'\ndef CC (e' : option mc.lbl) := mc.coarse_sch_of e'\ndef CF (e' : option mc.lbl) := mc.fine_sch_of e'\n\nparameter abs_coarse : (◇◻(•AC ⋀ -⟦ ma.step_of e ⟧)) τ\n\nparameter abs_fine : (◻◇•AF) τ\n\ninclude M₁\ninclude W\ninclude abs_coarse\ninclude abs_fine\n\nlemma abs_coarse_and_fine\n: (◻◇(•AC ⋀ •AF)) τ :=\nbegin\n  apply coincidence,\n  { apply stable_entails_stable _ _ abs_coarse,\n    apply λ _, and.left },\n  { apply abs_fine },\nend\n\nlemma conc_coarse : ∃ e', (◇◻(• W e' ⋀ • CC e'.val) ) τ :=\nbegin\n  have H : ((∃∃ e', ◇◻(• W ma mc R e e' ⋀ • CC mc e'.val))\n                   ⋁ ◻◇((-•AC ma e) ⋁ ∃∃ e' : imp_lbl ma mc R e, ⟦ mc.step_of e'.val ⟧)) τ,\n  { rw exists_action,\n    apply p_or_p_imp_p_or_right _ (unless_sem_exists' M₁.safety (R.events e).stable _),\n    { apply inf_often_entails_inf_often,\n      apply p_or_p_imp_p_or_right' _,\n      apply action_entails_action,\n      intros σ σ',\n      simp [mem_set_of,imp_lbl],\n      intros ec H x H' H₂,\n      existsi [x],\n      cases H with H₀ H,\n      cases H with H₁ STEP,\n      unfold program.step_of,\n      simp [H',event.step_of,STEP,H₀,H₁,H₂], },\n    have H' := leads_to.gen_disj' (R.events e).delay,\n    apply inf_often_of_leads_to (system_sem.leads_to_sem H' _ M₁),\n    simp,\n    have H' := ew_eq_true (R.events e).witness_fis,\n    rw [← p_and_over_p_exists_right\n       ,← p_and_over_p_exists_right],\n    simp [H'],\n    apply abs_coarse_and_fine ma mc _ M₁ _ abs_coarse abs_fine, },\n  simp at H,\n  cases H with H H,\n  { exfalso,\n    revert abs_coarse,\n    change ¬ _,\n    rw [p_not_eq_not,not_eventually,not_henceforth,p_not_p_and,p_not_p_not_iff_self],\n    apply inf_often_entails_inf_often _ _ H,\n    apply p_or_p_imp_p_or_right',\n    rw p_exists_entails_eq_p_forall_entails,\n    intro ec, cases ec with ec Hec,\n    subst e, apply R.evt_sim },\n  { simp [H] },\nend\n\nlemma conc_fine : ∀ e',\n         (◇◻•W e') τ →\n         (◻◇•CF e'.val) τ :=\nbegin\n  intros e' H,\n  have H' := system_sem.leads_to_sem ((R.events e).resched e') _ M₁,\n  apply inf_often_of_leads_to H',\n  rw p_and_comm,\n  apply coincidence H,\n  apply abs_coarse_and_fine _ _ _ M₁ _ abs_coarse abs_fine,\nend\n\nend schedules\n\ninclude M₁\ninclude R\n\ntheorem soundness : system_sem.ex ma τ :=\nbegin\n  apply nondet.program.ex.mk,\n  { apply R.sim_init,\n    apply M₁.init },\n  { intro i,\n    apply R.sim,\n    apply M₁.safety },\n  { intros e COARSE₀ FINE₀,\n    apply assume_neg _, intro ACT,\n    have COARSE₁ :  (◇◻(•AC ma e ⋀ -⟦program.step_of ma e⟧)) τ,\n    { rw [p_not_eq_not,not_henceforth,not_eventually] at ACT,\n      apply stable_and_of_stable_of_stable COARSE₀ ACT },\n    clear COARSE₀ ACT,\n    cases conc_coarse ma mc R τ M₁ _ COARSE₁ FINE₀ with e' C_COARSE',\n    have C_COARSE : (◇◻•CC mc e'.val) τ,\n    { apply eventually_entails_eventually _ _ C_COARSE',\n      apply henceforth_entails_henceforth,\n      intro, apply and.right },\n    have WIT : (◇◻•W ma mc R e e') τ,\n    { apply stable_entails_stable _ _ C_COARSE',\n      intro, apply and.left },\n    have C_FINE := conc_fine ma mc R τ M₁ e COARSE₁ FINE₀ e' WIT,\n    apply inf_often_entails_inf_often _ _ (M₁.liveness _ C_COARSE C_FINE),\n    have H := R.evt_sim e'.val,\n    rw [subtype.property e'] at H,\n    apply H, },\nend\n\nend soundness\n\nend nondet\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/refinement/split.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544210587586, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.30346345792168206}}
{"text": "def s1 := \"Hello, \"\ndef s2 := \"Nifty!\"\ndef s3 := s1 ++ s2\n\ntheorem t1 : s1 ++ s2 = s3 := _\n\ntheorem t2 : 4^2 = 16 := _\n\ntheorem t3 : s1 ++ s2 = s3 ∧ 5^2 = 25 := _\n\ntheorem t4 : \n    ∀ (P Q R : Prop), (P ∧ Q) ∧ (Q ∧ R) → (P ∧ R) := \n    λ (P Q R : Prop),\n        and.intro _ _", "meta": {"author": "kevinsullivan", "repo": "uva-cs-dm-s20-old", "sha": "797672cb0ffae6a42a3518c9225d5807191fd113", "save_path": "github-repos/lean/kevinsullivan-uva-cs-dm-s20-old", "path": "github-repos/lean/kevinsullivan-uva-cs-dm-s20-old/uva-cs-dm-s20-old-797672cb0ffae6a42a3518c9225d5807191fd113/instructor-notes/quiz11.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.3033028244009828}}
{"text": "import category_theory.limits.preserves.limits\nimport category_theory.adjunction.evaluation\nimport algebra.group.ulift\n\nimport for_mathlib.ab4\nimport for_mathlib.AddCommGroup.exact\nimport for_mathlib.AddCommGroup.pt\nimport for_mathlib.AddCommGroup.ab4\n\nimport condensed.extr.equivalence\n\n.\n\nuniverse u\n\nopen category_theory\nopen category_theory.limits\n\ninstance : has_colimits (ExtrSheafProd.{u} Ab.{u+1}) :=\nhas_colimits_of_has_colimits_creates_colimits\n(Condensed_ExtrSheafProd_equiv _).inverse\n\ninstance : has_limits (ExtrSheafProd.{u} Ab.{u+1}) :=\nhas_limits_of_has_limits_creates_limits\n(Condensed_ExtrSheafProd_equiv _).inverse\n\ninstance : has_colimits_of_size.{u} Ab.{u+1} :=\nhas_colimits_of_size_shrink.{u u (u+1) (u+1)} Ab.{u+1}\n\nopen_locale classical\n\ninstance {C : Type (u+1)} [category.{u} C] : AB4 (C ⥤ Ab.{u+1}) :=\nbegin\n  constructor,\n  intros α X Y f hf,\n  let t := _, change mono t,\n  rw category_theory.nat_trans.mono_iff_mono_app,\n  intros c,\n  let eX : (∐ λ (a : α), X a).obj c ≅ (∐ λ (a : α), (X a).obj c) :=\n    (is_colimit_of_preserves ((evaluation _ _).obj c)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso (colimit.is_colimit _)\n    ≪≫ has_colimit.iso_of_nat_iso (discrete.nat_iso (λ _, iso.refl _)),\n   let eY : (∐ λ (a : α), Y a).obj c ≅ (∐ λ (a : α), (Y a).obj c) :=\n    (is_colimit_of_preserves ((evaluation _ _).obj c)\n      (colimit.is_colimit _)).cocone_point_unique_up_to_iso (colimit.is_colimit _)\n    ≪≫ has_colimit.iso_of_nat_iso (discrete.nat_iso (λ _, iso.refl _)),\n  let tt : (∐ λ (a : α), (X a).obj c) ⟶ (∐ λ (a : α), (Y a).obj c) :=\n    sigma.desc (λ a, (f a).app c ≫ sigma.ι _ a),\n  haveI : mono tt,\n  { dsimp [tt],\n    apply AB4.cond, intros a, specialize hf a,\n    rw category_theory.nat_trans.mono_iff_mono_app at hf,\n    apply hf },\n  suffices : t.app c = eX.hom ≫ tt ≫ eY.inv,\n  { rw this, apply_instance },\n  dsimp [eX, tt, t, eY],\n  apply ((is_colimit_of_preserves ((evaluation _ _).obj c) (colimit.is_colimit _))).hom_ext,\n  intros a,\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  simp only [category.assoc, ← nat_trans.comp_app, colimit.ι_desc],\n  slice_rhs 0 1\n  { erw ((is_colimit_of_preserves ((evaluation _ _).obj c) (colimit.is_colimit _))).fac },\n  slice_rhs 0 1\n  { erw colimit.ι_desc },\n  dsimp,\n  simp only [category.id_comp, colimit.ι_desc, cofan.mk_ι_app, category.assoc,\n    has_colimit.iso_of_nat_iso_ι_inv, discrete.nat_iso_inv_app, functor.map_cocone_ι_app,\n    colimit.cocone_ι, evaluation_obj_map],\n  dsimp, simp, apply_instance\nend\n\nnoncomputable\ninstance : creates_limits (ExtrSheafProd_to_presheaf.{u} Ab.{u+1}) :=\nbegin\n  change creates_limits ((ExtrSheaf_ExtrSheafProd_equiv _).inverse ⋙\n    Sheaf_to_presheaf _ _),\n  apply_with category_theory.comp_creates_limits { instances := ff },\n  apply_instance,\n  exact @Sheaf.category_theory.Sheaf_to_presheaf.category_theory.creates_limits.{(u+2) u (u+1)}\n    ExtrDisc.{u} _ ExtrDisc.proetale_topology Ab.{u+1} _ _,\nend\n\ninstance : AB4 (ExtrSheafProd.{u} Ab.{u+1}) :=\nAB4_of_preserves_colimits_of_reflects_limits_of_AB4\n  (ExtrSheafProd_to_presheaf _)\n\n-- TODO: Prove a lemma about transporting AB4 across an equivalence.\ninstance : AB4 (Condensed.{u} Ab.{u+1}) :=\nAB4_of_preserves_colimits_of_reflects_limits_of_AB4\n    (Condensed_ExtrSheafProd_equiv.{u} Ab.{u+1}).functor\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/ab4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.30328208173370313}}
{"text": "import condensed.is_proetale_sheaf\nimport condensed.adjunctions\nimport category_theory.limits.filtered_colimit_commutes_finite_limit\nimport for_mathlib.AddCommGroup.explicit_limits\n\nopen category_theory\nopen category_theory.limits\n\nuniverse u\nvariables\n  {J : Type (u+1)} [small_category J] [is_filtered J]\n  {C : Type (u+2)}\n  [category.{u+1} C]\n  [concrete_category.{u+1} C]\n  [has_limits C]\n  [has_colimits_of_shape J C]\n  [preserves_colimits_of_shape J (forget C)]\n  [reflects_limits (forget C)]\n  [preserves_limits (forget C)]\n  (F : J ⥤ Condensed.{u} C)\n\nopen opposite\n\nnamespace is_sheaf_colimit_presheaf_aux\n\nnamespace empty\n\nvariables (G : J ⥤ Profinite.{u}ᵒᵖ ⥤ C)\n\nnoncomputable\ndef comparison_component (j : J) :\n  (G.obj j).obj (op Profinite.empty) ⟶ ⊤_ _ := terminal.from _\n\nvariables [∀ j, is_iso (comparison_component G j)]\n\nnoncomputable\ndef first_iso : (colimit G).obj (op Profinite.empty) ≅\n  colimit (limit (functor.empty _ ⋙ G.flip)) :=\nlet e₁ := is_colimit_of_preserves ((evaluation _ _).obj (op Profinite.empty))\n  (colimit.is_colimit G),\n    e₂ := e₁.cocone_point_unique_up_to_iso (colimit.is_colimit _),\n    e₃ : G ⋙ (evaluation Profiniteᵒᵖ C).obj (op Profinite.empty) ≅\n      limit (functor.empty Profiniteᵒᵖ ⋙ G.flip) :=\n      nat_iso.of_components\n      (λ j,\n        let e₄ := is_limit_of_preserves ((evaluation _ _).obj j)\n          (limit.is_limit (functor.empty _ ⋙ G.flip)),\n            e₅ := (limit.is_limit _).cone_point_unique_up_to_iso e₄,\n            e₆ : functor.empty C ≅\n              (functor.empty Profiniteᵒᵖ ⋙ G.flip) ⋙ (evaluation J C).obj j :=\n              nat_iso.of_components (λ i, i.elim) (λ i, i.elim) in\n        as_iso (comparison_component G j) ≪≫\n          has_limit.iso_of_nat_iso e₆ ≪≫ e₅)\n      begin\n        intros X Y f, dsimp [comparison_component],\n        apply (is_limit_of_preserves ((evaluation J C).obj Y)\n          (limit.is_limit (functor.empty Profiniteᵒᵖ ⋙ G.flip))).hom_ext,\n        intros j, cases j,\n      end in\ne₂ ≪≫ has_colimit.iso_of_nat_iso e₃\n\nnoncomputable\ndef second_iso : colimit (limit (functor.empty _ ⋙ G.flip)) ≅\n  limit (colimit (functor.empty _ ⋙ G.flip).flip) :=\n  colimit_limit_iso _\n\nnoncomputable\ndef third_iso : limit (colimit (functor.empty _ ⋙ G.flip).flip) ≅ ⊤_ _ :=\nhas_limit.iso_of_nat_iso $ nat_iso.of_components (λ i, i.elim) (λ i, i.elim)\n\nnoncomputable\ndef comparison : (colimit G).obj (op Profinite.empty) ⟶ ⊤_ _ := terminal.from _\n\ntheorem is_iso_comparison : is_iso (comparison G) :=\nbegin\n  suffices : comparison G = (first_iso G).hom ≫ (second_iso G).hom ≫ (third_iso G).hom,\n  { rw this, apply_instance },\n  simp,\nend\n\nend empty\n\nnamespace prod\n\nvariables (X Y : Profinite.{u}) (G : J ⥤ Profinite.{u}ᵒᵖ ⥤ C)\n\nnoncomputable\ndef comparison_component (j : J) :\n  (G.obj j).obj (op $ Profinite.sum X Y) ⟶ prod ((G.obj j).obj (op X)) ((G.obj j).obj (op Y)) :=\nprod.lift ((G.obj j).map (Profinite.sum.inl _ _).op) ((G.obj j).map (Profinite.sum.inr _ _).op)\n\nvariables [∀ j, is_iso (comparison_component X Y G j)]\n\nnoncomputable\ndef first_iso_aux_aux (j) :\n  (G ⋙ (evaluation Profiniteᵒᵖ C).obj (op (X.sum Y))).obj j ≅\n  (G.flip.obj (op X) ⨯ G.flip.obj (op Y)).obj j :=\nlet e₄ : pair ((G.obj j).obj (op X)) ((G.obj j).obj (op Y)) ≅\n  pair (G.flip.obj (op X)) (G.flip.obj (op Y)) ⋙ (evaluation J C).obj j :=\n  nat_iso.of_components\n  (λ p, match p with\n    | walking_pair.left := iso.refl _\n    | walking_pair.right := iso.refl _\n    end) begin\n      rintros (_|_) (_|_) (_|_), refl, refl,\n    end in\nas_iso (comparison_component X Y G j) ≪≫\n  has_limit.iso_of_nat_iso e₄ ≪≫\n  (limit.is_limit _).cone_point_unique_up_to_iso\n    (is_limit_of_preserves ((evaluation _ _).obj j) (limit.is_limit _))\n\nnoncomputable\ndef first_iso_aux : G ⋙ (evaluation Profiniteᵒᵖ C).obj (op (X.sum Y)) ≅\n  G.flip.obj (op X) ⨯ G.flip.obj (op Y) :=\nnat_iso.of_components (λ j, first_iso_aux_aux X Y G j)\nbegin\n  intros i j f, dsimp [comparison_component, first_iso_aux_aux],\n  apply\n    (is_limit_of_preserves ((evaluation J C).obj j)\n      (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).hom_ext,\n  rintros (_|_),\n  { dsimp [is_limit.cone_point_unique_up_to_iso],\n    have h1 :=\n      (is_limit_of_preserves ((evaluation J C).obj j)\n        (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac\n          (limit.cone _) walking_pair.left,\n    have h2 :=\n      (is_limit_of_preserves ((evaluation J C).obj i)\n        (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac\n          (limit.cone _) walking_pair.left,\n    dsimp at h1 h2, simp [h1, reassoc_of h2],\n    dsimp [first_iso_aux_aux._match_1], simp },\n  { dsimp [is_limit.cone_point_unique_up_to_iso],\n    have h1 :=\n      (is_limit_of_preserves ((evaluation J C).obj j)\n        (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac\n          (limit.cone _) walking_pair.right,\n    have h2 :=\n      (is_limit_of_preserves ((evaluation J C).obj i)\n        (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac\n          (limit.cone _) walking_pair.right,\n    dsimp at h1 h2, simp [h1, reassoc_of h2],\n    dsimp [first_iso_aux_aux._match_1], simp },\nend\n\nnoncomputable\ndef first_iso : (colimit G).obj (op $ Profinite.sum X Y) ≅\n  colimit (prod (G.flip.obj (op X)) (G.flip.obj (op Y))) :=\nlet e₁ := is_colimit_of_preserves ((evaluation _ _).obj (op $ Profinite.sum X Y))\n  (colimit.is_colimit G),\n    e₂ := e₁.cocone_point_unique_up_to_iso (colimit.is_colimit _) in\ne₂ ≪≫ has_colimit.iso_of_nat_iso (first_iso_aux X Y G)\n\nnoncomputable\ndef second_iso : colimit (prod (G.flip.obj (op X)) (G.flip.obj (op Y))) ≅\n  limit (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip) :=\ncolimit_limit_iso _\n\nnoncomputable\ndef third_iso_aux_left :\n  (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip).obj walking_pair.left ≅\n  (colimit G).obj (op X) :=\nlet e₁ :=\n  is_colimit_of_preserves ((evaluation _ _).obj walking_pair.left)\n    (colimit.is_colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip),\n    e₂ :=\n  is_colimit_of_preserves ((evaluation _ _).obj (op X))\n    (colimit.is_colimit G) in\ne₁.cocone_point_unique_up_to_iso e₂\n\nnoncomputable\ndef third_iso_aux_right :\n  (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip).obj walking_pair.right ≅\n  (colimit G).obj (op Y) :=\nlet e₁ :=\n  is_colimit_of_preserves ((evaluation _ _).obj walking_pair.right)\n    (colimit.is_colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip),\n    e₂ :=\n  is_colimit_of_preserves ((evaluation _ _).obj (op Y))\n    (colimit.is_colimit G) in\ne₁.cocone_point_unique_up_to_iso e₂\n\n/--/\nnoncomputable\ndef third_iso_aux : cone (pair (op X) (op Y) ⋙ colimit G) :=\n{ X := limit (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip),\n  π :=\n  { app := λ p,\n    match p with\n    | walking_pair.left := limit.π _ walking_pair.left ≫ (third_iso_aux_left X Y G).hom\n    | walking_pair.right := limit.π _ walking_pair.right ≫ (third_iso_aux_right X Y G).hom\n    end,\n    naturality' := admit } }\n\nnoncomputable\ndef third_iso_aux' : cone (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip) :=\n{ X := limit (pair (op X) (op Y) ⋙ colimit G),\n  π :=\n  { app := λ p,\n    match p with\n    | walking_pair.left := limit.π _ walking_pair.left ≫ (third_iso_aux_left X Y G).inv\n    | walking_pair.right := limit.π _ walking_pair.right ≫ (third_iso_aux_right X Y G).inv\n    end,\n    naturality' := admit } }\n-/\n\nnoncomputable\ndef third_iso_aux : cone (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip) :=\n{ X := prod ((colimit G).obj (op X)) ((colimit G).obj (op Y)),\n  π :=\n  { app := λ p,\n    match p with\n    | walking_pair.left := limits.prod.fst ≫ (third_iso_aux_left _ _ _).inv\n    | walking_pair.right := limits.prod.snd ≫ (third_iso_aux_right _ _ _).inv\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_),\n      { dsimp [third_iso_aux._match_1], simp },\n      { dsimp [third_iso_aux._match_1], simp },\n    end } }\n\nnoncomputable\ndef third_iso :\n  limit (colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip)\n    ≅ prod ((colimit G).obj (op X)) ((colimit G).obj (op Y)) :=\n{ hom := prod.lift\n    (limit.π _ walking_pair.left ≫ (third_iso_aux_left _ _ _).hom)\n    (limit.π _ walking_pair.right ≫ (third_iso_aux_right _ _ _).hom),\n  inv := limit.lift _ (third_iso_aux _ _ _),\n  hom_inv_id' := begin\n    ext (_|_),\n    { simp only [category.assoc, limit.lift_π, category.id_comp],\n      dsimp [third_iso_aux, third_iso_aux._match_1], simp },\n    { simp only [category.assoc, limit.lift_π, category.id_comp],\n      dsimp [third_iso_aux, third_iso_aux._match_1], simp },\n  end,\n  inv_hom_id' := begin\n    ext,\n    { simp only [prod.comp_lift, limit.lift_π_assoc, prod.lift_fst, category.id_comp],\n      dsimp [third_iso_aux], simp, },\n    { simp only [prod.comp_lift, limit.lift_π_assoc, prod.lift_snd, category.id_comp],\n      dsimp [third_iso_aux], simp, },\n  end }\n\n/-\nnoncomputable\ndef fourth_iso_aux : cone (pair (op X) (op Y) ⋙ colimit G) :=\n{ X := prod ((colimit G).obj (op X)) ((colimit G).obj (op Y)),\n  π :=\n  { app := λ p,\n    match p with\n    | walking_pair.left := limits.prod.fst\n    | walking_pair.right := limits.prod.snd\n    end,\n    naturality' := admit } }\n\nnoncomputable\ndef fourth_iso : limit (pair (op X) (op Y) ⋙ colimit G) ≅\n  prod ((colimit G).obj (op X)) ((colimit G).obj (op Y)) :=\n{ hom := prod.lift (limit.π _ walking_pair.left) (limit.π _ walking_pair.right),\n  inv := limit.lift _ (fourth_iso_aux _ _ _),\n  hom_inv_id' := admit,\n  inv_hom_id' := admit }\n-/\n\nnoncomputable\ndef comparison : (colimit G).obj (op $ Profinite.sum X Y) ⟶\n  prod ((colimit G).obj (op X)) ((colimit G).obj (op Y)) :=\nprod.lift\n  ((colimit G).map (Profinite.sum.inl _ _).op)\n  ((colimit G).map (Profinite.sum.inr _ _).op)\n\nlemma is_iso_comparison_aux_fst (j) :\n  (colimit.ι G j).app (op (X.sum Y)) ≫ comparison X Y G ≫ limits.prod.fst =\n    (colimit.ι G j).app (op (X.sum Y)) ≫\n      (first_iso X Y G).hom ≫ (second_iso X Y G).hom ≫ (third_iso X Y G).hom ≫\n      limits.prod.fst :=\nbegin\n  dsimp [comparison, first_iso, second_iso, third_iso, colimit_limit_iso],\n  simp only [prod.comp_lift, prod.lift_fst, category.assoc,\n    has_limit.iso_of_nat_iso_hom_π_assoc, iso.symm_hom,\n    colimit_flip_iso_comp_colim_inv_app,\n    limit.cone_point_unique_up_to_iso_hom_comp_assoc, functor.map_cone_π_app,\n    binary_fan.π_app_left],\n  dsimp [has_colimit.iso_of_nat_iso, colim, colim_map, is_colimit.map,\n    is_colimit.cocone_point_unique_up_to_iso,\n    colimit_obj_iso_colimit_comp_evaluation,\n    third_iso_aux, third_iso_aux_left, preserves_colimit_iso],\n  have := (is_colimit_of_preserves\n    ((evaluation Profiniteᵒᵖ C).obj (op (X.sum Y)))\n    (colimit.is_colimit G)).fac _ j,\n  dsimp at this, slice_rhs 1 2 { rw this }, clear this,\n  simp only [colimit.cocone_ι, colimit.ι_desc, cocones.precompose_obj_ι,\n    nat_trans.comp_app, category.assoc, flip_comp_evaluation_inv_app,\n    functor.map_cocone_ι_app, evaluation_obj_map],\n  dsimp [first_iso_aux, first_iso_aux_aux, comparison_component,\n    is_limit.cone_point_unique_up_to_iso],\n  simp only [has_limit.lift_iso_of_nat_iso_hom_assoc,\n    category.id_comp, category.assoc],\n  have := (is_limit_of_preserves ((evaluation J C).obj j)\n    (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac _\n      walking_pair.left, dsimp at this,\n  slice_rhs 2 3 { rw this }, clear this,\n  simp only [limit.cone_π, category.assoc, limit.lift_π_assoc,\n    cones.postcompose_obj_π, nat_trans.comp_app,\n    binary_fan.mk_π_app_left, nat_iso.of_components.hom_app],\n  have := (is_colimit_of_preserves ((evaluation _ C).obj walking_pair.left)\n    (colimit.is_colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip)).fac _ j,\n  dsimp at this, slice_rhs 3 4 { rw this }, clear this,\n  dsimp [first_iso_aux_aux._match_1], simp only [category.id_comp, nat_trans.naturality],\nend\n\nlemma is_iso_comparison_aux_snd (j) : (colimit.ι G j).app (op (X.sum Y)) ≫ comparison X Y G ≫\n  limits.prod.snd = (colimit.ι G j).app (op (X.sum Y)) ≫\n    (first_iso X Y G).hom ≫ (second_iso X Y G).hom ≫ (third_iso X Y G).hom ≫\n    limits.prod.snd :=\nbegin\n  dsimp [comparison, first_iso, second_iso, third_iso, colimit_limit_iso],\n  simp only [prod.comp_lift, prod.lift_snd, category.assoc,\n    has_limit.iso_of_nat_iso_hom_π_assoc, iso.symm_hom,\n    colimit_flip_iso_comp_colim_inv_app,\n    limit.cone_point_unique_up_to_iso_hom_comp_assoc, functor.map_cone_π_app,\n    binary_fan.π_app_right],\n  dsimp [has_colimit.iso_of_nat_iso, colim, colim_map, is_colimit.map,\n    is_colimit.cocone_point_unique_up_to_iso,\n    colimit_obj_iso_colimit_comp_evaluation,\n    third_iso_aux, third_iso_aux_right, preserves_colimit_iso],\n  have := (is_colimit_of_preserves\n    ((evaluation Profiniteᵒᵖ C).obj (op (X.sum Y)))\n    (colimit.is_colimit G)).fac _ j,\n  dsimp at this, slice_rhs 1 2 { rw this }, clear this,\n  simp only [colimit.cocone_ι, colimit.ι_desc, cocones.precompose_obj_ι,\n    nat_trans.comp_app, category.assoc, flip_comp_evaluation_inv_app,\n    functor.map_cocone_ι_app, evaluation_obj_map],\n  dsimp [first_iso_aux, first_iso_aux_aux, comparison_component,\n    is_limit.cone_point_unique_up_to_iso],\n  simp only [has_limit.lift_iso_of_nat_iso_hom_assoc,\n    category.id_comp, category.assoc],\n  have := (is_limit_of_preserves ((evaluation J C).obj j)\n    (limit.is_limit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))))).fac _\n      walking_pair.right, dsimp at this,\n  slice_rhs 2 3 { rw this }, clear this,\n  simp only [limit.cone_π, category.assoc, limit.lift_π_assoc,\n    cones.postcompose_obj_π, nat_trans.comp_app,\n    binary_fan.mk_π_app_left, nat_iso.of_components.hom_app],\n  have := (is_colimit_of_preserves ((evaluation _ C).obj walking_pair.right)\n    (colimit.is_colimit (pair (G.flip.obj (op X)) (G.flip.obj (op Y))).flip)).fac _ j,\n  dsimp at this, slice_rhs 3 4 { rw this }, clear this,\n  dsimp [first_iso_aux_aux._match_1], simp only [category.id_comp, nat_trans.naturality],\nend\n\nlemma is_iso_comparison : is_iso (comparison X Y G) :=\nbegin\n  suffices : (comparison X Y G) =\n    (first_iso X Y G).hom ≫ (second_iso X Y G).hom ≫ (third_iso X Y G).hom,\n  { rw this, apply_instance },\n  ext,\n  { simp only [category.assoc, is_iso_comparison_aux_fst] },\n  { simp only [category.assoc, is_iso_comparison_aux_snd] }\nend\n\nend prod\n\nnamespace eq\n\nvariables {X Y : Profinite.{u}} (f : X ⟶ Y) (G : J ⥤ Profinite.{u}ᵒᵖ ⥤ C)\n\nnoncomputable\ndef comparison_component (j : J) :\n  (G.obj j).obj (op $ Y) ⟶\n  equalizer\n    ((G.obj j).map (Profinite.pullback.fst f f).op)\n    ((G.obj j).map (Profinite.pullback.snd f f).op) :=\nequalizer.lift ((G.obj j).map f.op)\nbegin\n  simp_rw [← (G.obj j).map_comp, ← op_comp, Profinite.pullback.condition],\nend\n\nvariables [∀ j, is_iso (comparison_component f G j)]\n\ndef first_iso_aux_aux (j) : parallel_pair ((G.obj j).map (Profinite.pullback.fst f f).op)\n  ((G.obj j).map (Profinite.pullback.snd f f).op) ≅\n    parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n  (G.flip.map (Profinite.pullback.snd f f).op) ⋙\n    (evaluation J C).obj j :=\nnat_iso.of_components (λ p,\n  match p with\n  | walking_parallel_pair.zero := iso.refl _\n  | walking_parallel_pair.one := iso.refl _\n  end)\nbegin\n  rintro (_|_) (_|_) (_|_),\n  refl,\n  dsimp [first_iso_aux_aux._match_1], simp,\n  dsimp [first_iso_aux_aux._match_1], simp,\n  refl,\nend\n\nnoncomputable\ndef first_iso_aux : G ⋙ (evaluation Profiniteᵒᵖ C).obj (op Y) ≅\n  equalizer (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op) :=\nnat_iso.of_components (λ j,\n  as_iso (comparison_component f G j)\n    ≪≫\n    has_limit.iso_of_nat_iso (first_iso_aux_aux _ _ _)\n    ≪≫ (limit.is_limit _).cone_point_unique_up_to_iso\n    (is_limit_of_preserves ((evaluation _ _).obj j) (limit.is_limit _)))\nbegin\n  intros i j g, dsimp [comparison_component, first_iso_aux_aux],\n  apply (is_limit_of_preserves ((evaluation J C).obj j)\n    (limit.is_limit\n    (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)))).hom_ext,\n  rintros (_|_),\n  { dsimp [is_limit.cone_point_unique_up_to_iso],\n    have := (is_limit_of_preserves ((evaluation J C).obj j)\n      (limit.is_limit\n      (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)))).fac (limit.cone _)\n      walking_parallel_pair.zero,\n    dsimp at this, simp only [category.assoc, this], clear this,\n    have := (is_limit_of_preserves ((evaluation J C).obj i)\n      (limit.is_limit\n      (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)))).fac (limit.cone _)\n      walking_parallel_pair.zero,\n    dsimp at this, simp only [nat_trans.naturality, category.assoc, reassoc_of this], clear this,\n    simp only [has_limit.iso_of_nat_iso_hom_π, nat_iso.of_components.hom_app,\n      equalizer.lift_ι_assoc, functor.flip_obj_map, has_limit.iso_of_nat_iso_hom_π_assoc],\n    dsimp [first_iso_aux_aux._match_1],\n    simp only [category.comp_id, category.id_comp, nat_trans.naturality] },\n  { dsimp [is_limit.cone_point_unique_up_to_iso],\n    have := (is_limit_of_preserves ((evaluation J C).obj j)\n      (limit.is_limit\n      (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)))).fac (limit.cone _)\n      walking_parallel_pair.one,\n    dsimp at this, simp only [category.assoc, this], clear this,\n    have := (is_limit_of_preserves ((evaluation J C).obj i)\n      (limit.is_limit\n      (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)))).fac (limit.cone _)\n      walking_parallel_pair.one,\n    dsimp at this, simp only [nat_trans.naturality, category.assoc, reassoc_of this], clear this,\n    simp only [has_limit.iso_of_nat_iso_hom_π, nat_iso.of_components.hom_app,\n      limit.lift_π_assoc, fork.of_ι_π_app, category.assoc, functor.flip_obj_map,\n      has_limit.iso_of_nat_iso_hom_π_assoc],\n    dsimp [first_iso_aux_aux._match_1],\n    simp only [category.comp_id, category.id_comp,\n      nat_trans.naturality, nat_trans.naturality_assoc] }\nend\n\nnoncomputable\ndef first_iso : (colimit G).obj (op $ Y) ≅\n  colimit (equalizer\n    (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)) :=\nlet e₁ := is_colimit_of_preserves\n  ((evaluation _ _).obj (op $ Y)) (colimit.is_colimit G),\n    e₂ := e₁.cocone_point_unique_up_to_iso (colimit.is_colimit _) in\ne₂ ≪≫ has_colimit.iso_of_nat_iso (first_iso_aux f G)\n\nnoncomputable\ndef second_iso : colimit (equalizer\n    (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)) ≅\n    limit (colimit (parallel_pair\n      (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)).flip) :=\ncolimit_limit_iso _\n\nnoncomputable\ndef third_iso_aux :\n  (colimit (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip).obj\n    walking_parallel_pair.zero ≅ (colimit G).obj (op X) :=\nlet e₁ :=\n  is_colimit_of_preserves ((evaluation _ _).obj walking_parallel_pair.zero)\n    (colimit.is_colimit (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip),\n    e₂ := is_colimit_of_preserves ((evaluation _ _).obj (op X)) (colimit.is_colimit G) in\ne₁.cocone_point_unique_up_to_iso e₂\n\nnoncomputable\ndef third_iso_aux'' :\n  (colimit (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip).obj\n    walking_parallel_pair.one ≅ (colimit G).obj (op $ Profinite.pullback f f) :=\nlet e₁ :=\n  is_colimit_of_preserves ((evaluation _ _).obj walking_parallel_pair.one)\n    (colimit.is_colimit (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip),\n    e₂ := is_colimit_of_preserves ((evaluation _ _).obj (op $ Profinite.pullback f f))\n    (colimit.is_colimit G) in\ne₁.cocone_point_unique_up_to_iso e₂\n\nlemma third_iso_aux_fst :\n    (third_iso_aux f G).inv ≫ (colimit\n    (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip).map\n    walking_parallel_pair_hom.left = (colimit G).map (Profinite.pullback.fst f f).op ≫\n    (third_iso_aux'' f G).inv :=\nbegin\n  dsimp [third_iso_aux, third_iso_aux''],\n  apply (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj\n  (op X)) (colimit.is_colimit G)).hom_ext, intros j,\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  have h1 := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj (op X))\n    (colimit.is_colimit G)).fac _ j,\n  have h2 := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj\n    (op $ Profinite.pullback f f)) (colimit.is_colimit G)).fac _ j,\n  dsimp at h1 h2,\n  slice_lhs 1 2 { rw h1 }, clear h1,\n  rw ← nat_trans.naturality_assoc,\n  slice_rhs 2 3 { rw h2 }, clear h2,\n  dsimp, rw ← nat_trans.naturality, refl\nend\n\nlemma third_iso_aux_snd :\n    (third_iso_aux f G).inv ≫ (colimit\n    (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip).map\n    walking_parallel_pair_hom.right = (colimit G).map (Profinite.pullback.snd f f).op ≫\n    (third_iso_aux'' f G).inv :=\nbegin\n  dsimp [third_iso_aux, third_iso_aux''],\n  apply (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj\n  (op X)) (colimit.is_colimit G)).hom_ext, intros j,\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  have h1 := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj (op X))\n    (colimit.is_colimit G)).fac _ j,\n  have h2 := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj\n    (op $ Profinite.pullback f f)) (colimit.is_colimit G)).fac _ j,\n  dsimp at h1 h2,\n  slice_lhs 1 2 { rw h1 }, clear h1,\n  rw ← nat_trans.naturality_assoc,\n  slice_rhs 2 3 { rw h2 }, clear h2,\n  dsimp, rw ← nat_trans.naturality, refl\nend\n\nnoncomputable\ndef third_iso_aux' : cone (colimit (parallel_pair\n      (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)).flip) :=\n{ X := equalizer\n      ((colimit G).map (Profinite.pullback.fst f f).op)\n      ((colimit G).map (Profinite.pullback.snd f f).op),\n  π :=\n  { app := λ p,\n    match p with\n    | walking_parallel_pair.zero := equalizer.ι _ _ ≫ (third_iso_aux _ _).inv\n    | walking_parallel_pair.one := equalizer.ι _ _ ≫ (third_iso_aux f G).inv ≫\n        category_theory.functor.map _ walking_parallel_pair_hom.left\n    end,\n  naturality' := begin\n    rintro (_|_) (_|_) ⟨⟩,\n    { dsimp, simp only [category.id_comp, category_theory.functor.map_id, category.comp_id], },\n    { dsimp [third_iso_aux'._match_1], simp only [category.id_comp, category.assoc], },\n    { dsimp [third_iso_aux'._match_1], simp only [category.id_comp, category.assoc],\n      rw [third_iso_aux_fst, third_iso_aux_snd],\n      simp only [category.assoc, equalizer.condition_assoc] },\n    { dsimp, simp only [category.id_comp, category_theory.functor.map_id, category.comp_id], },\n  end } }\n\nnoncomputable\ndef third_iso : limit (colimit (parallel_pair\n      (G.flip.map (Profinite.pullback.fst f f).op)\n      (G.flip.map (Profinite.pullback.snd f f).op)).flip) ≅\n    equalizer\n      ((colimit G).map (Profinite.pullback.fst f f).op)\n      ((colimit G).map (Profinite.pullback.snd f f).op) :=\n{ hom := equalizer.lift\n    (limit.π _ walking_parallel_pair.zero ≫ (third_iso_aux f G).hom) begin\n      have := third_iso_aux_fst f G, rw iso.eq_comp_inv at this, rw ← this, clear this,\n      have := third_iso_aux_snd f G, rw iso.eq_comp_inv at this, rw ← this, clear this,\n      simp only [category.assoc, iso.hom_inv_id_assoc],\n      simp only [← category.assoc], congr' 1,\n      let F := _, change limit.π F _ ≫ F.map _ = limit.π F _ ≫ F.map _,\n      simp [limit.w F walking_parallel_pair_hom.left],\n    end,\n  inv := limit.lift _ (third_iso_aux' _ _),\n  hom_inv_id' := begin\n    ext (_|_),\n    { simp only [category.assoc, limit.lift_π, category.id_comp],\n      dsimp [third_iso_aux', third_iso_aux'._match_1],\n      simp only [equalizer.lift_ι_assoc, category.assoc, iso.hom_inv_id, category.comp_id] },\n    { simp only [category.assoc, limit.lift_π, category.id_comp],\n      dsimp [third_iso_aux', third_iso_aux'._match_1],\n      simp only [equalizer.lift_ι_assoc, category.assoc, iso.hom_inv_id_assoc,\n        category.comp_id, category.id_comp, limit.w] }\n  end,\n  inv_hom_id' := begin\n    ext,\n    simp only [category.assoc, equalizer.lift_ι, limit.lift_π_assoc, category.id_comp],\n    dsimp [third_iso_aux, third_iso_aux'],\n    simp only [category.assoc, iso.inv_hom_id, category.comp_id],\n  end }\n\nnoncomputable\ndef comparison :\n  (colimit G).obj (op $ Y) ⟶\n  equalizer\n    ((colimit G).map (Profinite.pullback.fst f f).op)\n    ((colimit G).map (Profinite.pullback.snd f f).op) :=\nequalizer.lift ((colimit G).map f.op)\nbegin\n  simp only [← functor.map_comp, ← op_comp, Profinite.pullback.condition],\nend\n\ntheorem is_iso_comparison : is_iso (comparison f G) :=\nbegin\n  suffices : comparison f G =\n    (first_iso f G).hom ≫ (second_iso f G).hom ≫ (third_iso f G).hom,\n  { rw this, apply_instance },\n  ext,\n  dsimp [comparison, first_iso, second_iso, third_iso, colimit_limit_iso],\n  simp only [category.assoc, equalizer.lift_ι, has_limit.iso_of_nat_iso_hom_π_assoc,\n    iso.symm_hom, colimit_flip_iso_comp_colim_inv_app,\n    limit.cone_point_unique_up_to_iso_hom_comp_assoc, functor.map_cone_π_app,\n    equalizer.fork_π_app_zero],\n  dsimp [has_colimit.iso_of_nat_iso, is_colimit.cocone_point_unique_up_to_iso,\n    colim, colim_map, is_colimit.map, colimit_obj_iso_colimit_comp_evaluation,\n    preserves_colimit_iso, third_iso_aux],\n  have := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ C).obj (op Y))\n    (colimit.is_colimit G)).fac _ j, dsimp at this,\n  slice_rhs 1 2 { rw this }, clear this,\n  simp only [colimit.cocone_ι, colimit.ι_desc, cocones.precompose_obj_ι, nat_trans.comp_app,\n    category.assoc, flip_comp_evaluation_inv_app, functor.map_cocone_ι_app, evaluation_obj_map],\n  dsimp [first_iso_aux, comparison_component],\n  simp only [has_limit.lift_iso_of_nat_iso_hom_assoc, category.id_comp, category.assoc],\n  dsimp [is_limit.cone_point_unique_up_to_iso],\n  have := (is_colimit_of_preserves ((evaluation walking_parallel_pair C).obj\n    walking_parallel_pair.zero) (colimit.is_colimit\n    (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)).flip)).fac _ j,\n  dsimp at this, slice_rhs 4 5 { rw this }, clear this,\n  have := (is_limit_of_preserves ((evaluation J C).obj j)\n    (limit.is_limit (parallel_pair (G.flip.map (Profinite.pullback.fst f f).op)\n    (G.flip.map (Profinite.pullback.snd f f).op)))).fac _ walking_parallel_pair.zero,\n  dsimp at this ⊢, slice_rhs 2 3 { erw this }, clear this,\n  dsimp,\n  simp only [limit.lift_π_assoc, cones.postcompose_obj_π, nat_trans.comp_app,\n    fork.of_ι_π_app, category.assoc],\n  dsimp [first_iso_aux_aux],\n  simp only [category.id_comp, nat_trans.naturality],\nend\n\nend eq\n\nend is_sheaf_colimit_presheaf_aux\nopen is_sheaf_colimit_presheaf_aux\n\n/-\nvariables {K : Type (u+1)} [small_category K] [fin_category K]\n  (E : K ⥤ Profinite.{u}ᵒᵖ) [has_limit E] (G : J ⥤ Profinite.{u}ᵒᵖ ⥤ C)\n  [∀ j, preserves_limits_of_shape K (G.obj j)]\n\nnoncomputable\ndef comparison_map_component (j : J) : (G.obj j).obj (limit E) ⟶ limit (E ⋙ G.obj j) :=\nlimit.lift (E ⋙ G.obj j) $ (G.obj j).map_cone (limit.cone E)\n\nnoncomputable\ndef comparison_map : (colimit G).obj (limit E) ⟶ limit (E ⋙ colimit G) :=\nlimit.lift (E ⋙ colimit G) $ (colimit G).map_cone (limit.cone E)\n\nnoncomputable\ndef first_iso : (colimit G).obj (limit E) ≅ colimit (limit (E ⋙ G.flip)) :=\nlet e := is_colimit_of_preserves ((evaluation _ _).obj (limit E))\n  (colimit.is_colimit G),\n  ee := e.cocone_point_unique_up_to_iso (colimit.is_colimit _),\n  tt : G ⋙ (evaluation Profiniteᵒᵖ C).obj (limit E) ≅ limit (E ⋙ G.flip) :=\n    nat_iso.of_components (λ j, begin\n      dsimp,\n      refine (is_limit_of_preserves (G.obj j) (limit.is_limit E)).cone_point_unique_up_to_iso\n        (limit.is_limit _) ≪≫ _,\n      refine _ ≪≫ (limit.is_limit _).cone_point_unique_up_to_iso\n        ((is_limit_of_preserves ((evaluation _ _).obj j) (limit.is_limit _))),\n      dsimp,\n      refine has_limit.iso_of_nat_iso _,\n      refine nat_iso.of_components _ _,\n      intros k, exact iso.refl _,\n      intros k₁ k₂ f, dsimp, simp,\n    end) admit in\nee ≪≫ has_colimit.iso_of_nat_iso tt\n\nnoncomputable\ndef second_iso : colimit (limit (E ⋙ G.flip)) ≅ limit (colimit (E ⋙ G.flip).flip) :=\n  colimit_limit_iso _\n\nnoncomputable\ndef third_iso : limit (colimit (E ⋙ G.flip).flip) ≅ limit (E ⋙ colimit G) :=\nhas_limit.iso_of_nat_iso $\nnat_iso.of_components (λ k,\n  let ee := (is_colimit_of_preserves ((evaluation _ _).obj k)\n    (colimit.is_colimit (E ⋙ G.flip).flip)).cocone_point_unique_up_to_iso\n    (colimit.is_colimit _) in\n  ee ≪≫\n  begin\n    dsimp,\n    refine _ ≪≫\n      (colimit.is_colimit _).cocone_point_unique_up_to_iso\n      ((is_colimit_of_preserves ((evaluation _ _).obj (E.obj k)) (colimit.is_colimit _))),\n    dsimp,\n    refine has_colimit.iso_of_nat_iso _,\n    refine nat_iso.of_components _ _,\n    intros j, exact iso.refl _,\n    intros i j f, dsimp, simp,\n  end) admit\n\nlemma is_iso : is_iso (comparison_map E G) :=\nbegin\n  suffices : comparison_map E G =\n    (first_iso E G).hom ≫ (second_iso E G).hom ≫ (third_iso E G).hom,\n  { rw this, apply_instance },\n  admit,\nend\n\n-- Use the comparison map above\nvariable (K)\ndef key : preserves_limits_of_shape K (colimit G) := admit\n\nend is_sheaf_colimit_presheaf_aux\nopen is_sheaf_colimit_presheaf_aux\n\ntheorem empty_condition_iff_preserves (G : Profiniteᵒᵖ ⥤ C) :\n  G.empty_condition' ↔\n  nonempty (preserves_limits_of_shape (discrete pempty.{u+1}) G) := admit\n\ntheorem product_condition_iff_preserves (G : Profiniteᵒᵖ ⥤ C) :\n  G.product_condition' ↔\n  nonempty (preserves_limits_of_shape (discrete walking_pair.{u+1}) G) := admit\n\ntheorem equalizer_condition_iff_preserves (G : Profiniteᵒᵖ ⥤ C) :\n  G.equalizer_condition' ↔\n  nonempty (preserves_limits_of_shape (walking_parallel_pair.{u+1}) G) := admit\n-/\n\nlemma is_sheaf_colimit_presheaf :\n  presheaf.is_sheaf proetale_topology (colimit (F ⋙ Sheaf_to_presheaf _ _)) :=\nbegin\n  --rw is_sheaf_iff_is_sheaf_of_type,\n  let G := (colimit (F ⋙ Sheaf_to_presheaf _ _)),\n  let Gs := F ⋙ Sheaf_to_presheaf _ _,\n  have hGs : ∀ j, presheaf.is_sheaf proetale_topology (Gs.obj j),\n  { intros j, exact (F.obj j).2 },\n  have hGsempty : ∀ j, (Gs.obj j).empty_condition',\n  { intros j, specialize hGs j,\n    rw (Gs.obj j).is_proetale_sheaf_tfae.out 0 3 at hGs,\n    exact hGs.1 },\n  have hGsprod : ∀ j, (Gs.obj j).product_condition',\n  { intros j, specialize hGs j,\n    rw (Gs.obj j).is_proetale_sheaf_tfae.out 0 3 at hGs,\n    exact hGs.2.1 },\n  have hGseq : ∀ j, (Gs.obj j).equalizer_condition',\n  { intros j, specialize hGs j,\n    rw (Gs.obj j).is_proetale_sheaf_tfae.out 0 3 at hGs,\n    exact hGs.2.2 },\n  rw G.is_proetale_sheaf_tfae.out 0 3,\n  refine ⟨_,_,_⟩,\n  { apply_with empty.is_iso_comparison { instances := ff },\n    exact hGsempty,\n    all_goals { apply_instance } },\n  { intros X Y,\n    apply_with prod.is_iso_comparison { instances := ff },\n    intros j, apply hGsprod,\n    all_goals { apply_instance } },\n  { intros X Y f hf,\n    apply_with eq.is_iso_comparison { instances := ff },\n    intros j, apply hGseq, assumption' }\nend\n\n@[simps]\nnoncomputable\ndef filtered_cocone : cocone F :=\n{ X := ⟨colimit (F ⋙ Sheaf_to_presheaf _ _), is_sheaf_colimit_presheaf _⟩,\n  ι :=\n  { app := λ j, Sheaf.hom.mk $ colimit.ι (F ⋙ Sheaf_to_presheaf _ _) j,\n    naturality' := begin\n      intros i j f,\n      ext1, dsimp,\n      simpa using colimit.w (F ⋙ Sheaf_to_presheaf _ _) f,\n    end } }\n\nnoncomputable\ndef filtered_cocone_is_colimit : is_colimit (filtered_cocone F) :=\n{ desc := λ S, Sheaf.hom.mk $ colimit.desc (F ⋙ Sheaf_to_presheaf _ _)\n    ((Sheaf_to_presheaf _ _).map_cocone S),\n  fac' := begin\n    intros S j,\n    ext1, dsimp,\n    simp,\n  end,\n  uniq' := begin\n    intros S m hm,\n    ext1, dsimp,\n    apply colimit.hom_ext,\n    intros j, specialize hm j, apply_fun (λ e, e.val) at hm,\n    dsimp at hm, simpa using hm,\n  end } .\n\nsection\n\nlocal attribute [-simp] forget_map_eq_coe\n\nnoncomputable\ndef preserves_limits_aux_1 (G : J ⥤ Condensed.{u} Ab.{u+1}) :\n  colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab) ⋙ forget Ab ≅\n  colimit (G ⋙ Sheaf_to_presheaf _ _ ⋙ (whiskering_right _ _ _).obj (forget Ab)) :=\nnat_iso.of_components\nbegin\n  intros X,\n  let E := (G ⋙ Sheaf_to_presheaf _ _ ⋙ (whiskering_right _ _ _).obj (forget Ab)),\n  let e₀ := colimit.is_colimit E,\n  let e₁ := is_colimit_of_preserves ((evaluation _ _).obj X) e₀,\n  refine _ ≪≫ (colimit.is_colimit _).cocone_point_unique_up_to_iso e₁,\n  change (forget Ab).obj _ ≅ colimit _,\n  let e₂ := colimit.is_colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab),\n  let e₃ := is_colimit_of_preserves ((evaluation _ _).obj X) e₂,\n  let e₄ := e₃.cocone_point_unique_up_to_iso (colimit.is_colimit _),\n  refine (forget Ab).map_iso e₄ ≪≫ _,\n  change (forget Ab).obj (colimit _) ≅ _,\n  let e₅ := is_colimit_of_preserves (forget Ab)\n    (colimit.is_colimit ((G ⋙ Sheaf_to_presheaf proetale_topology Ab)\n    ⋙ (evaluation Profiniteᵒᵖ Ab).obj X)),\n  exact e₅.cocone_point_unique_up_to_iso (colimit.is_colimit _),\nend\nbegin\n  intros X Y f, dsimp, simp only [category.assoc],\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  let E₀ := is_colimit_of_preserves ((evaluation Profiniteᵒᵖ Ab).obj X)\n    (colimit.is_colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab)),\n  let E := is_colimit_of_preserves (forget Ab) E₀,\n  apply E.hom_ext, intros j, dsimp,\n\n  -- Let's work on the LHS\n\n  slice_lhs 1 3\n  { simp only [← (forget Ab).map_comp],\n    rw ← ((colimit.ι (G ⋙ Sheaf_to_presheaf proetale_topology Ab) j)).naturality_assoc, },\n  have := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ Ab).obj Y)\n    (colimit.is_colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab))).fac _ j,\n  dsimp at this, rw this, clear this,\n  dsimp,\n  have := (is_colimit_of_preserves (forget Ab)\n    (colimit.is_colimit ((G ⋙ Sheaf_to_presheaf proetale_topology Ab) ⋙\n    (evaluation Profiniteᵒᵖ Ab).obj Y))).fac _ j,\n  simp only [(forget Ab).map_comp, category.assoc],\n  dsimp at this, slice_lhs 2 3 { rw this }, clear this,\n  erw colimit.ι_desc,\n\n  -- Now for the RHS\n\n  slice_rhs 1 2 { rw ← (forget Ab).map_comp },\n  have := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ Ab).obj X)\n    (colimit.is_colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab))).fac _ j,\n  dsimp at this, rw this, clear this,\n  have := (is_colimit_of_preserves (forget Ab)\n    (colimit.is_colimit ((G ⋙ Sheaf_to_presheaf proetale_topology Ab) ⋙\n    (evaluation Profiniteᵒᵖ Ab).obj X))).fac _ j,\n  dsimp at this, slice_rhs 1 2 { erw this }, clear this, erw colimit.ι_desc,\n  dsimp, erw ← nat_trans.naturality, refl,\nend\n\nlocal attribute [-simp] types_comp_apply functor_to_types.comp\n\nnoncomputable\ndef preserves_limits_of_shape_of_filtered_aux (G : J ⥤ Condensed.{u} Ab.{u+1}) :\n  Condensed_Ab_to_CondensedSet.{u}.map_cocone (filtered_cocone G) ≅\n  filtered_cocone (G ⋙ Condensed_Ab_to_CondensedSet.{u}) :=\ncocones.ext\n{ hom := Sheaf.hom.mk $ (preserves_limits_aux_1 G).hom,\n  inv := Sheaf.hom.mk $ (preserves_limits_aux_1 G).inv,\n  hom_inv_id' := by { ext1, simp },\n  inv_hom_id' := by { ext1, simp } }\nbegin\n  intros j, ext, dsimp [preserves_limits_aux_1, is_colimit.cocone_point_unique_up_to_iso],\n\n  simp only [← category.assoc, ← (forget Ab).map_comp],\n  have := (is_colimit_of_preserves ((evaluation Profiniteᵒᵖ Ab).obj x)\n    (colimit.is_colimit (G ⋙ Sheaf_to_presheaf proetale_topology Ab))).fac _ j,\n  dsimp at this, rw this, clear this, dsimp, simp only [category.assoc],\n  have := (is_colimit_of_preserves (forget Ab)\n    (colimit.is_colimit ((G ⋙ Sheaf_to_presheaf proetale_topology Ab) ⋙\n    (evaluation Profiniteᵒᵖ Ab).obj x))).fac _ j,\n  dsimp at this, simp only [← category.assoc], rw this, clear this, erw colimit.ι_desc,\n  refl,\n\nend\n\nend\n\nnoncomputable\ninstance Condensed_Ab_to_CondensedSet_preserves_limits_of_shape_of_filtered :\n  preserves_colimits_of_shape J Condensed_Ab_to_CondensedSet.{u} :=\nbegin\n  constructor,\n  intros G,\n  apply preserves_colimit_of_preserves_colimit_cocone (filtered_cocone_is_colimit G),\n  apply is_colimit.of_iso_colimit (filtered_cocone_is_colimit\n    (G ⋙ Condensed_Ab_to_CondensedSet)),\n  exact (preserves_limits_of_shape_of_filtered_aux G).symm,\nend\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/condensed/filtered_colimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3032820817337031}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel\n-/\nimport analysis.normed_space.dual\nimport data.set.intervals.disjoint\nimport measure_theory.measure.haar_lebesgue\nimport analysis.calculus.extend_deriv\nimport measure_theory.function.locally_integrable\nimport measure_theory.integral.set_integral\nimport measure_theory.integral.vitali_caratheodory\nimport analysis.calculus.fderiv_measurable\n\n/-!\n# Integral over an interval\n\nIn this file we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ` if `a ≤ b` and\n`-∫ x in Ioc b a, f x ∂μ` if `b ≤ a`. We prove a few simple properties and several versions of the\n[fundamental theorem of calculus](https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus).\n\nRecall that its first version states that the function `(u, v) ↦ ∫ x in u..v, f x` has derivative\n`(δu, δv) ↦ δv • f b - δu • f a` at `(a, b)` provided that `f` is continuous at `a` and `b`,\nand its second version states that, if `f` has an integrable derivative on `[a, b]`, then\n`∫ x in a..b, f' x = f b - f a`.\n\n## Main statements\n\n### FTC-1 for Lebesgue measure\n\nWe prove several versions of FTC-1, all in the `interval_integral` namespace. Many of them follow\nthe naming scheme `integral_has(_strict?)_(f?)deriv(_within?)_at(_of_tendsto_ae?)(_right|_left?)`.\nThey formulate FTC in terms of `has(_strict?)_(f?)deriv(_within?)_at`.\nLet us explain the meaning of each part of the name:\n\n* `_strict` means that the theorem is about strict differentiability;\n* `f` means that the theorem is about differentiability in both endpoints; incompatible with\n  `_right|_left`;\n* `_within` means that the theorem is about one-sided derivatives, see below for details;\n* `_of_tendsto_ae` means that instead of continuity the theorem assumes that `f` has a finite limit\n  almost surely as `x` tends to `a` and/or `b`;\n* `_right` or `_left` mean that the theorem is about differentiability in the right (resp., left)\n  endpoint.\n\nWe also reformulate these theorems in terms of `(f?)deriv(_within?)`. These theorems are named\n`(f?)deriv(_within?)_integral(_of_tendsto_ae?)(_right|_left?)` with the same meaning of parts of the\nname.\n\n### One-sided derivatives\n\nTheorem `integral_has_fderiv_within_at_of_tendsto_ae` states that `(u, v) ↦ ∫ x in u..v, f x` has a\nderivative `(δu, δv) ↦ δv • cb - δu • ca` within the set `s × t` at `(a, b)` provided that `f` tends\nto `ca` (resp., `cb`) almost surely at `la` (resp., `lb`), where possible values of `s`, `t`, and\ncorresponding filters `la`, `lb` are given in the following table.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n\nWe use a typeclass `FTC_filter` to make Lean automatically find `la`/`lb` based on `s`/`t`. This way\nwe can formulate one theorem instead of `16` (or `8` if we leave only non-trivial ones not covered\nby `integral_has_deriv_within_at_of_tendsto_ae_(left|right)` and\n`integral_has_fderiv_at_of_tendsto_ae`). Similarly,\n`integral_has_deriv_within_at_of_tendsto_ae_right` works for both one-sided derivatives using the\nsame typeclass to find an appropriate filter.\n\n### FTC for a locally finite measure\n\nBefore proving FTC for the Lebesgue measure, we prove a few statements that can be seen as FTC for\nany measure. The most general of them,\n`measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae`, states the following. Let `(la, la')`\nbe an `FTC_filter` pair of filters around `a` (i.e., `FTC_filter a la la'`) and let `(lb, lb')` be\nan `FTC_filter` pair of filters around `b`. If `f` has finite limits `ca` and `cb` almost surely at\n`la'` and `lb'`, respectively, then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)` as `ua` and `va` tend to `la` while\n`ub` and `vb` tend to `lb`.\n\n### FTC-2 and corollaries\n\nWe use FTC-1 to prove several versions of FTC-2 for the Lebesgue measure, using a similar naming\nscheme as for the versions of FTC-1. They include:\n* `interval_integral.integral_eq_sub_of_has_deriv_right_of_le` - most general version, for functions\n  with a right derivative\n* `interval_integral.integral_eq_sub_of_has_deriv_at'` - version for functions with a derivative on\n  an open set\n* `interval_integral.integral_deriv_eq_sub'` - version that is easiest to use when computing the\n  integral of a specific function\n\nWe then derive additional integration techniques from FTC-2:\n* `interval_integral.integral_mul_deriv_eq_deriv_mul` - integration by parts\n* `interval_integral.integral_comp_mul_deriv''` - integration by substitution\n\nMany applications of these theorems can be found in the file `analysis.special_functions.integrals`.\n\nNote that the assumptions of FTC-2 are formulated in the form that `f'` is integrable. To use it in\na context with the stronger assumption that `f'` is continuous, one can use\n`continuous_on.interval_integrable` or `continuous_on.integrable_on_Icc` or\n`continuous_on.integrable_on_interval`.\n\n## Implementation notes\n\n### Avoiding `if`, `min`, and `max`\n\nIn order to avoid `if`s in the definition, we define `interval_integrable f μ a b` as\n`integrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ`. For any `a`, `b` one of these\nintervals is empty and the other coincides with `set.interval_oc a b = set.Ioc (min a b) (max a b)`.\n\nSimilarly, we define `∫ x in a..b, f x ∂μ` to be `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`.\nAgain, for any `a`, `b` one of these integrals is zero, and the other gives the expected result.\n\nThis way some properties can be translated from integrals over sets without dealing with\nthe cases `a ≤ b` and `b ≤ a` separately.\n\n### Choice of the interval\n\nWe use integral over `set.interval_oc a b = set.Ioc (min a b) (max a b)` instead of one of the other\nthree possible intervals with the same endpoints for two reasons:\n\n* this way `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ` holds whenever\n  `f` is integrable on each interval; in particular, it works even if the measure `μ` has an atom\n  at `b`; this rules out `set.Ioo` and `set.Icc` intervals;\n* with this definition for a probability measure `μ`, the integral `∫ x in a..b, 1 ∂μ` equals\n  the difference $F_μ(b)-F_μ(a)$, where $F_μ(a)=μ(-∞, a]$ is the\n  [cumulative distribution function](https://en.wikipedia.org/wiki/Cumulative_distribution_function)\n  of `μ`.\n\n### `FTC_filter` class\n\nAs explained above, many theorems in this file rely on the typeclass\n`FTC_filter (a : ℝ) (l l' : filter ℝ)` to avoid code duplication. This typeclass combines four\nassumptions:\n\n- `pure a ≤ l`;\n- `l' ≤ 𝓝 a`;\n- `l'` has a basis of measurable sets;\n- if `u n` and `v n` tend to `l`, then for any `s ∈ l'`, `Ioc (u n) (v n)` is eventually included\n  in `s`.\n\nThis typeclass has the following “real” instances: `(a, pure a, ⊥)`, `(a, 𝓝[≥] a, 𝓝[>] a)`,\n`(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`.\nFurthermore, we have the following instances that are equal to the previously mentioned instances:\n`(a, 𝓝[{a}] a, ⊥)` and `(a, 𝓝[univ] a, 𝓝[univ] a)`.\nWhile the difference between `Ici a` and `Ioi a` doesn't matter for theorems about Lebesgue measure,\nit becomes important in the versions of FTC about any locally finite measure if this measure has an\natom at one of the endpoints.\n\n### Combining one-sided and two-sided derivatives\n\nThere are some `FTC_filter` instances where the fact that it is one-sided or\ntwo-sided depends on the point, namely `(x, 𝓝[Icc a b] x, 𝓝[Icc a b] x)`\n(resp. `(x, 𝓝[[a, b]] x, 𝓝[[a, b]] x)`, where `[a, b] = set.interval a b`),\nwith `x ∈ Icc a b` (resp. `x ∈ [a, b]`).\nThis results in a two-sided derivatives for `x ∈ Ioo a b` and one-sided derivatives for\n`x ∈ {a, b}`. Other instances could be added when needed (in that case, one also needs to add\ninstances for `filter.is_measurably_generated` and `filter.tendsto_Ixx_class`).\n\n## Tags\n\nintegral, fundamental theorem of calculus, FTC-1, FTC-2, change of variables in integrals\n-/\n\nnoncomputable theory\nopen topological_space (second_countable_topology)\nopen measure_theory set classical filter function\n\nopen_locale classical topological_space filter ennreal big_operators interval nnreal\n\nvariables {ι 𝕜 E F : Type*} [normed_group E]\n\n/-!\n### Integrability at an interval\n-/\n\n/-- A function `f` is called *interval integrable* with respect to a measure `μ` on an unordered\ninterval `a..b` if it is integrable on both intervals `(a, b]` and `(b, a]`. One of these\nintervals is always empty, so this property is equivalent to `f` being integrable on\n`(min a b, max a b]`. -/\ndef interval_integrable (f : ℝ → E) (μ : measure ℝ) (a b : ℝ) :=\nintegrable_on f (Ioc a b) μ ∧ integrable_on f (Ioc b a) μ\n\nsection\n\nvariables {f : ℝ → E} {a b : ℝ} {μ : measure ℝ}\n\n/-- A function is interval integrable with respect to a given measure `μ` on `a..b` if and\n  only if it is integrable on `interval_oc a b` with respect to `μ`. This is an equivalent\n  definition of `interval_integrable`. -/\nlemma interval_integrable_iff : interval_integrable f μ a b ↔ integrable_on f (Ι a b) μ :=\nby rw [interval_oc_eq_union, integrable_on_union, interval_integrable]\n\n/-- If a function is interval integrable with respect to a given measure `μ` on `a..b` then\n  it is integrable on `interval_oc a b` with respect to `μ`. -/\nlemma interval_integrable.def (h : interval_integrable f μ a b) : integrable_on f (Ι a b) μ :=\ninterval_integrable_iff.mp h\n\nlemma interval_integrable_iff_integrable_Ioc_of_le (hab : a ≤ b) :\n  interval_integrable f μ a b ↔ integrable_on f (Ioc a b) μ :=\nby rw [interval_integrable_iff, interval_oc_of_le hab]\n\nlemma integrable_on_Icc_iff_integrable_on_Ioc'\n  {E : Type*} [normed_group E] {f : ℝ → E} (ha : μ {a} ≠ ∞) :\n  integrable_on f (Icc a b) μ ↔ integrable_on f (Ioc a b) μ :=\nbegin\n  cases le_or_lt a b with hab hab,\n  { have : Icc a b = Icc a a ∪ Ioc a b := (Icc_union_Ioc_eq_Icc le_rfl hab).symm,\n    rw [this, integrable_on_union],\n    simp [ha.lt_top] },\n  { simp [hab, hab.le] },\nend\n\nlemma integrable_on_Icc_iff_integrable_on_Ioc\n  {E : Type*}[normed_group E] [has_no_atoms μ] {f : ℝ → E} {a b : ℝ} :\n  integrable_on f (Icc a b) μ ↔ integrable_on f (Ioc a b) μ :=\nintegrable_on_Icc_iff_integrable_on_Ioc' (by simp)\n\nlemma integrable_on_Ioc_iff_integrable_on_Ioo'\n  {E : Type*} [normed_group E]\n  {f : ℝ → E} {a b : ℝ} (hb : μ {b} ≠ ∞) :\n  integrable_on f (Ioc a b) μ ↔ integrable_on f (Ioo a b) μ :=\nbegin\n  cases lt_or_le a b with hab hab,\n  { have : Ioc a b = Ioo a b ∪ Icc b b := (Ioo_union_Icc_eq_Ioc hab le_rfl).symm,\n    rw [this, integrable_on_union],\n    simp [hb.lt_top] },\n  { simp [hab] },\nend\n\nlemma integrable_on_Ioc_iff_integrable_on_Ioo\n  {E : Type*} [normed_group E] [has_no_atoms μ] {f : ℝ → E} {a b : ℝ} :\n  integrable_on f (Ioc a b) μ ↔ integrable_on f (Ioo a b) μ :=\nintegrable_on_Ioc_iff_integrable_on_Ioo' (by simp)\n\nlemma integrable_on_Icc_iff_integrable_on_Ioo\n  {E : Type*} [normed_group E] [has_no_atoms μ] {f : ℝ → E} {a b : ℝ} :\n  integrable_on f (Icc a b) μ ↔ integrable_on f (Ioo a b) μ :=\nby rw [integrable_on_Icc_iff_integrable_on_Ioc, integrable_on_Ioc_iff_integrable_on_Ioo]\n\nlemma interval_integrable_iff' [has_no_atoms μ] :\n  interval_integrable f μ a b ↔ integrable_on f (interval a b) μ :=\nby rw [interval_integrable_iff, interval, interval_oc, integrable_on_Icc_iff_integrable_on_Ioc]\n\nlemma interval_integrable_iff_integrable_Icc_of_le {E : Type*} [normed_group E]\n  {f : ℝ → E} {a b : ℝ} (hab : a ≤ b) {μ : measure ℝ} [has_no_atoms μ] :\n  interval_integrable f μ a b ↔ integrable_on f (Icc a b) μ :=\nby rw [interval_integrable_iff_integrable_Ioc_of_le hab, integrable_on_Icc_iff_integrable_on_Ioc]\n\n/-- If a function is integrable with respect to a given measure `μ` then it is interval integrable\n  with respect to `μ` on `interval a b`. -/\nlemma measure_theory.integrable.interval_integrable (hf : integrable f μ) :\n  interval_integrable f μ a b :=\n⟨hf.integrable_on, hf.integrable_on⟩\n\nlemma measure_theory.integrable_on.interval_integrable (hf : integrable_on f [a, b] μ) :\n  interval_integrable f μ a b :=\n⟨measure_theory.integrable_on.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_interval),\n measure_theory.integrable_on.mono_set hf (Ioc_subset_Icc_self.trans Icc_subset_interval')⟩\n\nlemma interval_integrable_const_iff {c : E} :\n  interval_integrable (λ _, c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ :=\nby simp only [interval_integrable_iff, integrable_on_const]\n\n@[simp] lemma interval_integrable_const [is_locally_finite_measure μ] {c : E} :\n  interval_integrable (λ _, c) μ a b :=\ninterval_integrable_const_iff.2 $ or.inr measure_Ioc_lt_top\n\nend\n\nnamespace interval_integrable\n\nsection\n\nvariables {f : ℝ → E} {a b c d : ℝ} {μ ν : measure ℝ}\n\n@[symm] lemma symm (h : interval_integrable f μ a b) : interval_integrable f μ b a :=\nh.symm\n\n@[refl] lemma refl : interval_integrable f μ a a :=\nby split; simp\n\n@[trans] lemma trans {a b c : ℝ} (hab : interval_integrable f μ a b)\n  (hbc : interval_integrable f μ b c) : interval_integrable f μ a c :=\n⟨(hab.1.union hbc.1).mono_set Ioc_subset_Ioc_union_Ioc,\n  (hbc.2.union hab.2).mono_set Ioc_subset_Ioc_union_Ioc⟩\n\nlemma trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)\n  (hint : ∀ k ∈ Ico m n, interval_integrable f μ (a k) (a $ k+1)) :\n  interval_integrable f μ (a m) (a n) :=\nbegin\n  revert hint,\n  refine nat.le_induction _ _ n hmn,\n  { simp },\n  { assume p hp IH h,\n    exact (IH (λ k hk, h k (Ico_subset_Ico_right p.le_succ hk))).trans (h p (by simp [hp])) }\nend\n\nlemma trans_iterate {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, interval_integrable f μ (a k) (a $ k+1)) :\n  interval_integrable f μ (a 0) (a n) :=\ntrans_iterate_Ico bot_le (λ k hk, hint k hk.2)\n\nlemma neg (h : interval_integrable f μ a b) : interval_integrable (-f) μ a b :=\n⟨h.1.neg, h.2.neg⟩\n\nlemma norm (h : interval_integrable f μ a b) :\n  interval_integrable (λ x, ∥f x∥) μ a b  :=\n⟨h.1.norm, h.2.norm⟩\n\nlemma abs {f : ℝ → ℝ} (h : interval_integrable f μ a b) :\n  interval_integrable (λ x, |f x|) μ a b  :=\nh.norm\n\nlemma mono (hf : interval_integrable f ν a b) (h1 : [c, d] ⊆ [a, b]) (h2 : μ ≤ ν) :\n  interval_integrable f μ c d :=\ninterval_integrable_iff.mpr $ hf.def.mono\n  (interval_oc_subset_interval_oc_of_interval_subset_interval h1) h2\n\nlemma mono_set (hf : interval_integrable f μ a b) (h : [c, d] ⊆ [a, b]) :\n  interval_integrable f μ c d :=\nhf.mono h rfl.le\n\nlemma mono_measure (hf : interval_integrable f ν a b) (h : μ ≤ ν) :\n  interval_integrable f μ a b :=\nhf.mono rfl.subset h\n\nlemma mono_set_ae (hf : interval_integrable f μ a b) (h : Ι c d ≤ᵐ[μ] Ι a b) :\n  interval_integrable f μ c d :=\ninterval_integrable_iff.mpr $ hf.def.mono_set_ae h\n\nlemma mono_fun [normed_group F] {g : ℝ → F}\n  (hf : interval_integrable f μ a b) (hgm : ae_strongly_measurable g (μ.restrict (Ι a b)))\n  (hle : (λ x, ∥g x∥) ≤ᵐ[μ.restrict (Ι a b)] (λ x, ∥f x∥)) : interval_integrable g μ a b :=\ninterval_integrable_iff.2 $ hf.def.integrable.mono hgm hle\n\nlemma mono_fun' {g : ℝ → ℝ} (hg : interval_integrable g μ a b)\n  (hfm : ae_strongly_measurable f (μ.restrict (Ι a b)))\n  (hle : (λ x, ∥f x∥) ≤ᵐ[μ.restrict (Ι a b)] g) : interval_integrable f μ a b :=\ninterval_integrable_iff.2 $ hg.def.integrable.mono' hfm hle\n\nprotected lemma ae_strongly_measurable (h : interval_integrable f μ a b) :\n  ae_strongly_measurable f (μ.restrict (Ioc a b)):=\nh.1.ae_strongly_measurable\n\nprotected lemma ae_strongly_measurable' (h : interval_integrable f μ a b) :\n  ae_strongly_measurable f (μ.restrict (Ioc b a)):=\nh.2.ae_strongly_measurable\n\nend\n\nvariables {f g : ℝ → E} {a b : ℝ} {μ : measure ℝ}\n\nlemma smul [normed_field 𝕜] [normed_space 𝕜 E]\n  {f : ℝ → E} {a b : ℝ} {μ : measure ℝ} (h : interval_integrable f μ a b) (r : 𝕜) :\n  interval_integrable (r • f) μ a b :=\n⟨h.1.smul r, h.2.smul r⟩\n\n@[simp] \n\n@[simp] lemma sub (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  interval_integrable (λ x, f x - g x) μ a b :=\n⟨hf.1.sub hg.1, hf.2.sub hg.2⟩\n\nlemma sum (s : finset ι) {f : ι → ℝ → E} (h : ∀ i ∈ s, interval_integrable (f i) μ a b) :\n  interval_integrable (∑ i in s, f i) μ a b :=\n⟨integrable_finset_sum' s (λ i hi, (h i hi).1), integrable_finset_sum' s (λ i hi, (h i hi).2)⟩\n\nlemma mul_continuous_on {f g : ℝ → ℝ}\n  (hf : interval_integrable f μ a b) (hg : continuous_on g [a, b]) :\n  interval_integrable (λ x, f x * g x) μ a b :=\nbegin\n  rw interval_integrable_iff at hf ⊢,\n  exact hf.mul_continuous_on_of_subset hg measurable_set_Ioc is_compact_interval Ioc_subset_Icc_self\nend\n\nlemma continuous_on_mul {f g : ℝ → ℝ} (hf : interval_integrable f μ a b)\n  (hg : continuous_on g [a, b]) :\n  interval_integrable (λ x, g x * f x) μ a b :=\nby simpa [mul_comm] using hf.mul_continuous_on hg\n\nlemma comp_mul_left (hf : interval_integrable f volume a b) (c : ℝ) :\n  interval_integrable (λ x, f (c * x)) volume (a / c) (b / c) :=\nbegin\n  rcases eq_or_ne c 0 with hc|hc, { rw hc, simp },\n  rw interval_integrable_iff' at hf ⊢,\n  have A : measurable_embedding (λ x, x * c⁻¹) :=\n    (homeomorph.mul_right₀ _ (inv_ne_zero hc)).closed_embedding.measurable_embedding,\n  rw [←real.smul_map_volume_mul_right (inv_ne_zero hc), integrable_on, measure.restrict_smul,\n    integrable_smul_measure (by simpa : ennreal.of_real (|c⁻¹|) ≠ 0) ennreal.of_real_ne_top,\n    ←integrable_on, measurable_embedding.integrable_on_map_iff A],\n  convert hf using 1,\n  { ext, simp only [comp_app], congr' 1, field_simp, ring },\n  { rw preimage_mul_const_interval (inv_ne_zero hc), field_simp [hc] },\nend\n\nlemma iff_comp_neg  :\n  interval_integrable f volume a b ↔ interval_integrable (λ x, f (-x)) volume (-a) (-b) :=\nbegin\n  split, all_goals { intro hf, convert comp_mul_left hf (-1), simp, field_simp, field_simp },\nend\n\nend interval_integrable\n\nsection\n\nvariables {μ : measure ℝ} [is_locally_finite_measure μ]\n\nlemma continuous_on.interval_integrable {u : ℝ → E} {a b : ℝ}\n  (hu : continuous_on u (interval a b)) : interval_integrable u μ a b :=\n(continuous_on.integrable_on_Icc hu).interval_integrable\n\nlemma continuous_on.interval_integrable_of_Icc {u : ℝ → E} {a b : ℝ} (h : a ≤ b)\n  (hu : continuous_on u (Icc a b)) : interval_integrable u μ a b :=\ncontinuous_on.interval_integrable ((interval_of_le h).symm ▸ hu)\n\n/-- A continuous function on `ℝ` is `interval_integrable` with respect to any locally finite measure\n`ν` on ℝ. -/\nlemma continuous.interval_integrable {u : ℝ → E} (hu : continuous u) (a b : ℝ) :\n  interval_integrable u μ a b :=\nhu.continuous_on.interval_integrable\n\nend\n\nsection\n\nvariables {μ : measure ℝ} [is_locally_finite_measure μ] [conditionally_complete_linear_order E]\n  [order_topology E] [second_countable_topology E]\n\nlemma monotone_on.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : monotone_on u (interval a b)) :\n  interval_integrable u μ a b :=\nbegin\n  rw interval_integrable_iff,\n  exact (hu.integrable_on_compact is_compact_interval).mono_set Ioc_subset_Icc_self,\nend\n\nlemma antitone_on.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : antitone_on u (interval a b)) :\n  interval_integrable u μ a b :=\nhu.dual_right.interval_integrable\n\nlemma monotone.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : monotone u) :\n  interval_integrable u μ a b :=\n(hu.monotone_on _).interval_integrable\n\nlemma antitone.interval_integrable {u : ℝ → E} {a b : ℝ} (hu : antitone u) :\n  interval_integrable u μ a b :=\n(hu.antitone_on _).interval_integrable\n\nend\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : ℝ → E` has a finite limit at `l' ⊓ μ.ae`. Then `f` is interval integrable on\n`u..v` provided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter ℝ` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\nlemma filter.tendsto.eventually_interval_integrable_ae {f : ℝ → E} {μ : measure ℝ}\n  {l l' : filter ℝ}  (hfm : strongly_measurable_at_filter f l' μ)\n  [tendsto_Ixx_class Ioc l l'] [is_measurably_generated l']\n  (hμ : μ.finite_at_filter l') {c : E} (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  {u v : ι → ℝ} {lt : filter ι} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  ∀ᶠ t in lt, interval_integrable f μ (u t) (v t) :=\nhave _ := (hf.integrable_at_filter_ae hfm hμ).eventually,\n((hu.Ioc hv).eventually this).and $ (hv.Ioc hu).eventually this\n\n/-- Let `l'` be a measurably generated filter; let `l` be a of filter such that each `s ∈ l'`\neventually includes `Ioc u v` as both `u` and `v` tend to `l`. Let `μ` be a measure finite at `l'`.\n\nSuppose that `f : ℝ → E` has a finite limit at `l`. Then `f` is interval integrable on `u..v`\nprovided that both `u` and `v` tend to `l`.\n\nTypeclass instances allow Lean to find `l'` based on `l` but not vice versa, so\n`apply tendsto.eventually_interval_integrable_ae` will generate goals `filter ℝ` and\n`tendsto_Ixx_class Ioc ?m_1 l'`. -/\nlemma filter.tendsto.eventually_interval_integrable {f : ℝ → E} {μ : measure ℝ}\n  {l l' : filter ℝ} (hfm : strongly_measurable_at_filter f l' μ)\n  [tendsto_Ixx_class Ioc l l'] [is_measurably_generated l']\n  (hμ : μ.finite_at_filter l') {c : E} (hf : tendsto f l' (𝓝 c))\n  {u v : ι → ℝ} {lt : filter ι} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  ∀ᶠ t in lt, interval_integrable f μ (u t) (v t) :=\n(hf.mono_left inf_le_left).eventually_interval_integrable_ae hfm hμ hu hv\n\n/-!\n### Interval integral: definition and basic properties\n\nIn this section we define `∫ x in a..b, f x ∂μ` as `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`\nand prove some basic properties.\n-/\n\nvariables [complete_space E] [normed_space ℝ E]\n\n/-- The interval integral `∫ x in a..b, f x ∂μ` is defined\nas `∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ`. If `a ≤ b`, then it equals\n`∫ x in Ioc a b, f x ∂μ`, otherwise it equals `-∫ x in Ioc b a, f x ∂μ`. -/\ndef interval_integral (f : ℝ → E) (a b : ℝ) (μ : measure ℝ) :=\n∫ x in Ioc a b, f x ∂μ - ∫ x in Ioc b a, f x ∂μ\n\nnotation `∫` binders ` in ` a `..` b `, ` r:(scoped:60 f, f) ` ∂` μ:70 := interval_integral r a b μ\nnotation `∫` binders ` in ` a `..` b `, ` r:(scoped:60 f, interval_integral f a b volume) := r\n\nnamespace interval_integral\n\nsection basic\n\nvariables {a b : ℝ} {f g : ℝ → E} {μ : measure ℝ}\n\n@[simp] lemma integral_zero : ∫ x in a..b, (0 : E) ∂μ = 0 :=\nby simp [interval_integral]\n\nlemma integral_of_le (h : a ≤ b) : ∫ x in a..b, f x ∂μ = ∫ x in Ioc a b, f x ∂μ :=\nby simp [interval_integral, h]\n\n@[simp] lemma integral_same : ∫ x in a..a, f x ∂μ = 0 :=\nsub_self _\n\nlemma integral_symm (a b) : ∫ x in b..a, f x ∂μ = -∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, neg_sub]\n\nlemma integral_of_ge (h : b ≤ a) : ∫ x in a..b, f x ∂μ = -∫ x in Ioc b a, f x ∂μ :=\nby simp only [integral_symm b, integral_of_le h]\n\nlemma interval_integral_eq_integral_interval_oc (f : ℝ → E) (a b : ℝ) (μ : measure ℝ) :\n  ∫ x in a..b, f x ∂μ = (if a ≤ b then 1 else -1 : ℝ) • ∫ x in Ι a b, f x ∂μ :=\nbegin\n  split_ifs with h,\n  { simp only [integral_of_le h, interval_oc_of_le h, one_smul] },\n  { simp only [integral_of_ge (not_le.1 h).le, interval_oc_of_lt (not_le.1 h), neg_one_smul] }\nend\n\nlemma integral_cases (f : ℝ → E) (a b) :\n  ∫ x in a..b, f x ∂μ ∈ ({∫ x in Ι a b, f x ∂μ, -∫ x in Ι a b, f x ∂μ} : set E) :=\nby { rw interval_integral_eq_integral_interval_oc, split_ifs; simp }\n\nlemma integral_undef (h : ¬ interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nby cases le_total a b with hab hab;\n  simp only [integral_of_le, integral_of_ge, hab, neg_eq_zero];\n    refine integral_undef (not_imp_not.mpr integrable.integrable_on' _);\n      simpa [hab] using not_and_distrib.mp h\n\nlemma integral_non_ae_strongly_measurable\n  (hf : ¬ ae_strongly_measurable f (μ.restrict (Ι a b))) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nby rw [interval_integral_eq_integral_interval_oc, integral_non_ae_strongly_measurable hf, smul_zero]\n\nlemma integral_non_ae_strongly_measurable_of_le (h : a ≤ b)\n  (hf : ¬ ae_strongly_measurable f (μ.restrict (Ioc a b))) :\n  ∫ x in a..b, f x ∂μ = 0 :=\nintegral_non_ae_strongly_measurable $ by rwa [interval_oc_of_le h]\n\nlemma norm_integral_min_max (f : ℝ → E) :\n  ∥∫ x in min a b..max a b, f x ∂μ∥ = ∥∫ x in a..b, f x ∂μ∥ :=\nby cases le_total a b; simp [*, integral_symm a b]\n\nlemma norm_integral_eq_norm_integral_Ioc (f : ℝ → E) :\n  ∥∫ x in a..b, f x ∂μ∥ = ∥∫ x in Ι a b, f x ∂μ∥ :=\nby rw [← norm_integral_min_max, integral_of_le min_le_max, interval_oc]\n\nlemma abs_integral_eq_abs_integral_interval_oc (f : ℝ → ℝ) :\n  |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| :=\nnorm_integral_eq_norm_integral_Ioc f\n\nlemma norm_integral_le_integral_norm_Ioc :\n  ∥∫ x in a..b, f x ∂μ∥ ≤ ∫ x in Ι a b, ∥f x∥ ∂μ :=\ncalc ∥∫ x in a..b, f x ∂μ∥ = ∥∫ x in Ι a b, f x ∂μ∥ :\n  norm_integral_eq_norm_integral_Ioc f\n... ≤ ∫ x in Ι a b, ∥f x∥ ∂μ :\n  norm_integral_le_integral_norm f\n\nlemma norm_integral_le_abs_integral_norm : ∥∫ x in a..b, f x ∂μ∥ ≤ |∫ x in a..b, ∥f x∥ ∂μ| :=\nbegin\n  simp only [← real.norm_eq_abs, norm_integral_eq_norm_integral_Ioc],\n  exact le_trans (norm_integral_le_integral_norm _) (le_abs_self _)\nend\n\nlemma norm_integral_le_integral_norm (h : a ≤ b) :\n  ∥∫ x in a..b, f x ∂μ∥ ≤ ∫ x in a..b, ∥f x∥ ∂μ :=\nnorm_integral_le_integral_norm_Ioc.trans_eq $ by rw [interval_oc_of_le h, integral_of_le h]\n\nlemma norm_integral_le_of_norm_le_const_ae {a b C : ℝ} {f : ℝ → E}\n  (h : ∀ᵐ x, x ∈ Ι a b → ∥f x∥ ≤ C) :\n  ∥∫ x in a..b, f x∥ ≤ C * |b - a| :=\nbegin\n  rw [norm_integral_eq_norm_integral_Ioc],\n  convert norm_set_integral_le_of_norm_le_const_ae'' _ measurable_set_Ioc h,\n  { rw [real.volume_Ioc, max_sub_min_eq_abs, ennreal.to_real_of_real (abs_nonneg _)] },\n  { simp only [real.volume_Ioc, ennreal.of_real_lt_top] },\nend\n\nlemma norm_integral_le_of_norm_le_const {a b C : ℝ} {f : ℝ → E}\n  (h : ∀ x ∈ Ι a b, ∥f x∥ ≤ C) :\n  ∥∫ x in a..b, f x∥ ≤ C * |b - a| :=\nnorm_integral_le_of_norm_le_const_ae $ eventually_of_forall h\n\n@[simp] lemma integral_add (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  ∫ x in a..b, f x + g x ∂μ = ∫ x in a..b, f x ∂μ + ∫ x in a..b, g x ∂μ :=\nby simp only [interval_integral_eq_integral_interval_oc, integral_add hf.def hg.def, smul_add]\n\nlemma integral_finset_sum {ι} {s : finset ι} {f : ι → ℝ → E}\n  (h : ∀ i ∈ s, interval_integrable (f i) μ a b) :\n  ∫ x in a..b, ∑ i in s, f i x ∂μ = ∑ i in s, ∫ x in a..b, f i x ∂μ :=\nby simp only [interval_integral_eq_integral_interval_oc,\n  integral_finset_sum s (λ i hi, (h i hi).def), finset.smul_sum]\n\n@[simp] lemma integral_neg : ∫ x in a..b, -f x ∂μ = -∫ x in a..b, f x ∂μ :=\nby { simp only [interval_integral, integral_neg], abel }\n\n@[simp] lemma integral_sub (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b) :\n  ∫ x in a..b, f x - g x ∂μ = ∫ x in a..b, f x ∂μ - ∫ x in a..b, g x ∂μ :=\nby simpa only [sub_eq_add_neg] using (integral_add hf hg.neg).trans (congr_arg _ integral_neg)\n\n@[simp] lemma integral_smul {𝕜 : Type*} [nondiscrete_normed_field 𝕜] [normed_space 𝕜 E]\n  [smul_comm_class ℝ 𝕜 E]\n  (r : 𝕜) (f : ℝ → E) : ∫ x in a..b, r • f x ∂μ = r • ∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, integral_smul, smul_sub]\n\n@[simp] lemma integral_smul_const {𝕜 : Type*} [is_R_or_C 𝕜] [normed_space 𝕜 E] (f : ℝ → 𝕜) (c : E) :\n  ∫ x in a..b, f x • c ∂μ = (∫ x in a..b, f x ∂μ) • c :=\nby simp only [interval_integral_eq_integral_interval_oc, integral_smul_const, smul_assoc]\n\n@[simp] lemma integral_const_mul {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, r * f x ∂μ = r * ∫ x in a..b, f x ∂μ :=\nintegral_smul r f\n\n@[simp] lemma integral_mul_const {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, f x * r ∂μ = ∫ x in a..b, f x ∂μ * r :=\nby simpa only [mul_comm r] using integral_const_mul r f\n\n@[simp] lemma integral_div {𝕜 : Type*} [is_R_or_C 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :\n  ∫ x in a..b, f x / r ∂μ = ∫ x in a..b, f x ∂μ / r :=\nby simpa only [div_eq_mul_inv] using integral_mul_const r⁻¹ f\n\nlemma integral_const' (c : E) :\n  ∫ x in a..b, c ∂μ = ((μ $ Ioc a b).to_real - (μ $ Ioc b a).to_real) • c :=\nby simp only [interval_integral, set_integral_const, sub_smul]\n\n@[simp] lemma integral_const (c : E) : ∫ x in a..b, c = (b - a) • c :=\nby simp only [integral_const', real.volume_Ioc, ennreal.to_real_of_real', ← neg_sub b,\n  max_zero_sub_eq_self]\n\nlemma integral_smul_measure (c : ℝ≥0∞) :\n  ∫ x in a..b, f x ∂(c • μ) = c.to_real • ∫ x in a..b, f x ∂μ :=\nby simp only [interval_integral, measure.restrict_smul, integral_smul_measure, smul_sub]\n\nvariables [normed_group F] [complete_space F] [normed_space ℝ F]\n\nlemma _root_.continuous_linear_map.interval_integral_comp_comm\n  (L : E →L[ℝ] F) (hf : interval_integrable f μ a b) :\n  ∫ x in a..b, L (f x) ∂μ = L (∫ x in a..b, f x ∂μ) :=\nbegin\n  rw [interval_integral, interval_integral, L.integral_comp_comm, L.integral_comp_comm, L.map_sub],\n  exacts [hf.2, hf.1]\nend\n\nend basic\n\nsection comp\n\nvariables {a b c d : ℝ} (f : ℝ → E)\n\n@[simp] lemma integral_comp_mul_right (hc : c ≠ 0) :\n  ∫ x in a..b, f (x * c) = c⁻¹ • ∫ x in a*c..b*c, f x :=\nbegin\n  have A : measurable_embedding (λ x, x * c) :=\n    (homeomorph.mul_right₀ c hc).closed_embedding.measurable_embedding,\n  conv_rhs { rw [← real.smul_map_volume_mul_right hc] },\n  simp_rw [integral_smul_measure, interval_integral, A.set_integral_map,\n          ennreal.to_real_of_real (abs_nonneg c)],\n  cases hc.lt_or_lt,\n  { simp [h, mul_div_cancel, hc, abs_of_neg, measure.restrict_congr_set Ico_ae_eq_Ioc] },\n  { simp [h, mul_div_cancel, hc, abs_of_pos] }\nend\n\n@[simp] lemma smul_integral_comp_mul_right (c) :\n  c • ∫ x in a..b, f (x * c) = ∫ x in a*c..b*c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_mul_left (hc : c ≠ 0) :\n  ∫ x in a..b, f (c * x) = c⁻¹ • ∫ x in c*a..c*b, f x :=\nby simpa only [mul_comm c] using integral_comp_mul_right f hc\n\n@[simp] lemma smul_integral_comp_mul_left (c) :\n  c • ∫ x in a..b, f (c * x) = ∫ x in c*a..c*b, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div (hc : c ≠ 0) :\n  ∫ x in a..b, f (x / c) = c • ∫ x in a/c..b/c, f x :=\nby simpa only [inv_inv] using integral_comp_mul_right f (inv_ne_zero hc)\n\n@[simp] lemma inv_smul_integral_comp_div (c) :\n  c⁻¹ • ∫ x in a..b, f (x / c) = ∫ x in a/c..b/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_right (d) :\n  ∫ x in a..b, f (x + d) = ∫ x in a+d..b+d, f x :=\nhave A : measurable_embedding (λ x, x + d) :=\n  (homeomorph.add_right d).closed_embedding.measurable_embedding,\ncalc  ∫ x in a..b, f (x + d)\n    = ∫ x in a+d..b+d, f x ∂(measure.map (λ x, x + d) volume)\n                           : by simp [interval_integral, A.set_integral_map]\n... = ∫ x in a+d..b+d, f x : by rw [map_add_right_eq_self]\n\n@[simp] lemma integral_comp_add_left (d) :\n  ∫ x in a..b, f (d + x) = ∫ x in d+a..d+b, f x :=\nby simpa only [add_comm] using integral_comp_add_right f d\n\n@[simp] lemma integral_comp_mul_add (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (c * x + d) = c⁻¹ • ∫ x in c*a+d..c*b+d, f x :=\nby rw [← integral_comp_add_right, ← integral_comp_mul_left _ hc]\n\n@[simp] lemma smul_integral_comp_mul_add (c d) :\n  c • ∫ x in a..b, f (c * x + d) = ∫ x in c*a+d..c*b+d, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_mul (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d + c * x) = c⁻¹ • ∫ x in d+c*a..d+c*b, f x :=\nby rw [← integral_comp_add_left, ← integral_comp_mul_left _ hc]\n\n@[simp] lemma smul_integral_comp_add_mul (c d) :\n  c • ∫ x in a..b, f (d + c * x) = ∫ x in d+c*a..d+c*b, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div_add (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (x / c + d) = c • ∫ x in a/c+d..b/c+d, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_add f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_div_add (c d) :\n  c⁻¹ • ∫ x in a..b, f (x / c + d) = ∫ x in a/c+d..b/c+d, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_add_div (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d + x / c) = c • ∫ x in d+a/c..d+b/c, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_add_mul f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_add_div (c d) :\n  c⁻¹ • ∫ x in a..b, f (d + x / c) = ∫ x in d+a/c..d+b/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_mul_sub (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (c * x - d) = c⁻¹ • ∫ x in c*a-d..c*b-d, f x :=\nby simpa only [sub_eq_add_neg] using integral_comp_mul_add f hc (-d)\n\n@[simp] lemma smul_integral_comp_mul_sub (c d) :\n  c • ∫ x in a..b, f (c * x - d) = ∫ x in c*a-d..c*b-d, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_mul (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d - c * x) = c⁻¹ • ∫ x in d-c*b..d-c*a, f x :=\nbegin\n  simp only [sub_eq_add_neg, neg_mul_eq_neg_mul],\n  rw [integral_comp_add_mul f (neg_ne_zero.mpr hc) d, integral_symm],\n  simp only [inv_neg, smul_neg, neg_neg, neg_smul],\nend\n\n@[simp] lemma smul_integral_comp_sub_mul (c d) :\n  c • ∫ x in a..b, f (d - c * x) = ∫ x in d-c*b..d-c*a, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_div_sub (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (x / c - d) = c • ∫ x in a/c-d..b/c-d, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_mul_sub f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_div_sub (c d) :\n  c⁻¹ • ∫ x in a..b, f (x / c - d) = ∫ x in a/c-d..b/c-d, f x  :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_div (hc : c ≠ 0) (d) :\n  ∫ x in a..b, f (d - x / c) = c • ∫ x in d-b/c..d-a/c, f x :=\nby simpa only [div_eq_inv_mul, inv_inv] using integral_comp_sub_mul f (inv_ne_zero hc) d\n\n@[simp] lemma inv_smul_integral_comp_sub_div (c d) :\n  c⁻¹ • ∫ x in a..b, f (d - x / c) = ∫ x in d-b/c..d-a/c, f x :=\nby by_cases hc : c = 0; simp [hc]\n\n@[simp] lemma integral_comp_sub_right (d) :\n  ∫ x in a..b, f (x - d) = ∫ x in a-d..b-d, f x :=\nby simpa only [sub_eq_add_neg] using integral_comp_add_right f (-d)\n\n@[simp] lemma integral_comp_sub_left (d) :\n  ∫ x in a..b, f (d - x) = ∫ x in d-b..d-a, f x :=\nby simpa only [one_mul, one_smul, inv_one] using integral_comp_sub_mul f one_ne_zero d\n\n@[simp] lemma integral_comp_neg : ∫ x in a..b, f (-x) = ∫ x in -b..-a, f x :=\nby simpa only [zero_sub] using integral_comp_sub_left f 0\n\nend comp\n\n/-!\n### Integral is an additive function of the interval\n\nIn this section we prove that `∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ`\nas well as a few other identities trivially equivalent to this one. We also prove that\n`∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ` provided that `support f ⊆ Ioc a b`.\n-/\n\nsection order_closed_topology\n\nvariables {a b c d : ℝ} {f g : ℝ → E} {μ : measure ℝ}\n\n/-- If two functions are equal in the relevant interval, their interval integrals are also equal. -/\nlemma integral_congr {a b : ℝ} (h : eq_on f g [a, b]) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nby cases le_total a b with hab hab; simpa [hab, integral_of_le, integral_of_ge]\n  using set_integral_congr measurable_set_Ioc (h.mono Ioc_subset_Icc_self)\n\nlemma integral_add_adjacent_intervals_cancel (hab : interval_integrable f μ a b)\n  (hbc : interval_integrable f μ b c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ + ∫ x in c..a, f x ∂μ = 0 :=\nbegin\n  have hac := hab.trans hbc,\n  simp only [interval_integral, sub_add_sub_comm, sub_eq_zero],\n  iterate 4 { rw ← integral_union },\n  { suffices : Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc b a ∪ Ioc c b ∪ Ioc a c, by rw this,\n    rw [Ioc_union_Ioc_union_Ioc_cycle, union_right_comm, Ioc_union_Ioc_union_Ioc_cycle,\n      min_left_comm, max_left_comm] },\n  all_goals { simp [*, measurable_set.union, measurable_set_Ioc, Ioc_disjoint_Ioc_same,\n    Ioc_disjoint_Ioc_same.symm, hab.1, hab.2, hbc.1, hbc.2, hac.1, hac.2] }\nend\n\nlemma integral_add_adjacent_intervals (hab : interval_integrable f μ a b)\n  (hbc : interval_integrable f μ b c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in b..c, f x ∂μ = ∫ x in a..c, f x ∂μ :=\nby rw [← add_neg_eq_zero, ← integral_symm, integral_add_adjacent_intervals_cancel hab hbc]\n\nlemma sum_integral_adjacent_intervals_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)\n  (hint : ∀ k ∈ Ico m n, interval_integrable f μ (a k) (a $ k+1)) :\n  ∑ (k : ℕ) in finset.Ico m n, ∫ x in (a k)..(a $ k+1), f x ∂μ = ∫ x in (a m)..(a n), f x ∂μ :=\nbegin\n  revert hint,\n  refine nat.le_induction _ _ n hmn,\n  { simp },\n  { assume p hmp IH h,\n    rw [finset.sum_Ico_succ_top hmp, IH, integral_add_adjacent_intervals],\n    { apply interval_integrable.trans_iterate_Ico hmp (λ k hk, h k _),\n      exact (Ico_subset_Ico le_rfl (nat.le_succ _)) hk },\n    { apply h,\n      simp [hmp] },\n    { assume k hk,\n      exact h _ (Ico_subset_Ico_right p.le_succ hk) } }\nend\n\nlemma sum_integral_adjacent_intervals {a : ℕ → ℝ} {n : ℕ}\n  (hint : ∀ k < n, interval_integrable f μ (a k) (a $ k+1)) :\n  ∑ (k : ℕ) in finset.range n, ∫ x in (a k)..(a $ k+1), f x ∂μ = ∫ x in (a 0)..(a n), f x ∂μ :=\nbegin\n  rw ← nat.Ico_zero_eq_range,\n  exact sum_integral_adjacent_intervals_Ico (zero_le n) (λ k hk, hint k hk.2),\nend\n\nlemma integral_interval_sub_left (hab : interval_integrable f μ a b)\n  (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in a..c, f x ∂μ = ∫ x in c..b, f x ∂μ :=\nsub_eq_of_eq_add' $ eq.symm $ integral_add_adjacent_intervals hac (hac.symm.trans hab)\n\nlemma integral_interval_add_interval_comm (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ + ∫ x in c..d, f x ∂μ = ∫ x in a..d, f x ∂μ + ∫ x in c..b, f x ∂μ :=\nby rw [← integral_add_adjacent_intervals hac hcd, add_assoc, add_left_comm,\n  integral_add_adjacent_intervals hac (hac.symm.trans hab), add_comm]\n\nlemma integral_interval_sub_interval_comm (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in c..d, f x ∂μ = ∫ x in a..c, f x ∂μ - ∫ x in b..d, f x ∂μ :=\nby simp only [sub_eq_add_neg, ← integral_symm,\n  integral_interval_add_interval_comm hab hcd.symm (hac.trans hcd)]\n\nlemma integral_interval_sub_interval_comm' (hab : interval_integrable f μ a b)\n  (hcd : interval_integrable f μ c d) (hac : interval_integrable f μ a c) :\n  ∫ x in a..b, f x ∂μ - ∫ x in c..d, f x ∂μ = ∫ x in d..b, f x ∂μ - ∫ x in c..a, f x ∂μ :=\nby { rw [integral_interval_sub_interval_comm hab hcd hac, integral_symm b d, integral_symm a c,\n  sub_neg_eq_add, sub_eq_neg_add], }\n\nlemma integral_Iic_sub_Iic (ha : integrable_on f (Iic a) μ) (hb : integrable_on f (Iic b) μ) :\n  ∫ x in Iic b, f x ∂μ - ∫ x in Iic a, f x ∂μ = ∫ x in a..b, f x ∂μ :=\nbegin\n  wlog hab : a ≤ b using [a b] tactic.skip,\n  { rw [sub_eq_iff_eq_add', integral_of_le hab, ← integral_union (Iic_disjoint_Ioc le_rfl),\n      Iic_union_Ioc_eq_Iic hab],\n    exacts [measurable_set_Ioc, ha, hb.mono_set (λ _, and.right)] },\n  { intros ha hb,\n    rw [integral_symm, ← this hb ha, neg_sub] }\nend\n\n/-- If `μ` is a finite measure then `∫ x in a..b, c ∂μ = (μ (Iic b) - μ (Iic a)) • c`. -/\nlemma integral_const_of_cdf [is_finite_measure μ] (c : E) :\n  ∫ x in a..b, c ∂μ = ((μ (Iic b)).to_real - (μ (Iic a)).to_real) • c :=\nbegin\n  simp only [sub_smul, ← set_integral_const],\n  refine (integral_Iic_sub_Iic _ _).symm;\n    simp only [integrable_on_const, measure_lt_top, or_true]\nend\n\nlemma integral_eq_integral_of_support_subset {a b} (h : support f ⊆ Ioc a b) :\n  ∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ :=\nbegin\n  cases le_total a b with hab hab,\n  { rw [integral_of_le hab, ← integral_indicator measurable_set_Ioc, indicator_eq_self.2 h];\n    apply_instance },\n  { rw [Ioc_eq_empty hab.not_lt, subset_empty_iff, support_eq_empty_iff] at h,\n    simp [h] }\nend\n\nlemma integral_congr_ae' (h : ∀ᵐ x ∂μ, x ∈ Ioc a b → f x = g x)\n  (h' : ∀ᵐ x ∂μ, x ∈ Ioc b a → f x = g x) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nby simp only [interval_integral, set_integral_congr_ae (measurable_set_Ioc) h,\n              set_integral_congr_ae (measurable_set_Ioc) h']\n\nlemma integral_congr_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = g x) :\n  ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ :=\nintegral_congr_ae' (ae_interval_oc_iff.mp h).1 (ae_interval_oc_iff.mp h).2\n\nlemma integral_zero_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = 0) :\n  ∫ x in a..b, f x ∂μ = 0 :=\ncalc ∫ x in a..b, f x ∂μ = ∫ x in a..b, 0 ∂μ : integral_congr_ae h\n                     ... = 0                 : integral_zero\n\nlemma integral_indicator {a₁ a₂ a₃ : ℝ} (h : a₂ ∈ Icc a₁ a₃) :\n  ∫ x in a₁..a₃, indicator {x | x ≤ a₂} f x ∂ μ = ∫ x in a₁..a₂, f x ∂ μ :=\nbegin\n  have : {x | x ≤ a₂} ∩ Ioc a₁ a₃ = Ioc a₁ a₂, from Iic_inter_Ioc_of_le h.2,\n  rw [integral_of_le h.1, integral_of_le (h.1.trans h.2),\n      integral_indicator, measure.restrict_restrict, this],\n  exact measurable_set_Iic,\n  all_goals { apply measurable_set_Iic },\nend\n\n/-- Lebesgue dominated convergence theorem for filters with a countable basis -/\nlemma tendsto_integral_filter_of_dominated_convergence {ι} {l : filter ι}\n  [l.is_countably_generated] {F : ι → ℝ → E} (bound : ℝ → ℝ)\n  (hF_meas : ∀ᶠ n in l, ae_strongly_measurable (F n) (μ.restrict (Ι a b)))\n  (h_bound : ∀ᶠ n in l, ∀ᵐ x ∂μ, x ∈ Ι a b → ∥F n x∥ ≤ bound x)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_lim : ∀ᵐ x ∂μ, x ∈ Ι a b → tendsto (λ n, F n x) l (𝓝 (f x))) :\n  tendsto (λn, ∫ x in a..b, F n x ∂μ) l (𝓝 $ ∫ x in a..b, f x ∂μ) :=\nbegin\n  simp only [interval_integrable_iff, interval_integral_eq_integral_interval_oc,\n    ← ae_restrict_iff' measurable_set_interval_oc] at *,\n  exact tendsto_const_nhds.smul\n    (tendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_lim)\nend\n\n/-- Lebesgue dominated convergence theorem for series. -/\nlemma has_sum_integral_of_dominated_convergence {ι} [encodable ι]\n  {F : ι → ℝ → E} (bound : ι → ℝ → ℝ)\n  (hF_meas : ∀ n, ae_strongly_measurable (F n) (μ.restrict (Ι a b)))\n  (h_bound : ∀ n, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F n t∥ ≤ bound n t)\n  (bound_summable : ∀ᵐ t ∂μ, t ∈ Ι a b → summable (λ n, bound n t))\n  (bound_integrable : interval_integrable (λ t, ∑' n, bound n t) μ a b)\n  (h_lim : ∀ᵐ t ∂μ, t ∈ Ι a b → has_sum (λ n, F n t) (f t)) :\n  has_sum (λn, ∫ t in a..b, F n t ∂μ) (∫ t in a..b, f t ∂μ) :=\nbegin\n  simp only [interval_integrable_iff, interval_integral_eq_integral_interval_oc,\n    ← ae_restrict_iff' measurable_set_interval_oc] at *,\n  exact (has_sum_integral_of_dominated_convergence bound hF_meas h_bound bound_summable\n    bound_integrable h_lim).const_smul\nend\n\nopen topological_space\nvariables {X : Type*} [topological_space X] [first_countable_topology X]\n\n/-- Continuity of interval integral with respect to a parameter, at a point within a set.\n  Given `F : X → ℝ → E`, assume `F x` is ae-measurable on `[a, b]` for `x` in a\n  neighborhood of `x₀` within `s` and at `x₀`, and assume it is bounded by a function integrable\n  on `[a, b]` independent of `x` in a neighborhood of `x₀` within `s`. If `(λ x, F x t)`\n  is continuous at `x₀` within `s` for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_within_at_of_dominated_interval\n  {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ} {s : set X}\n  (hF_meas : ∀ᶠ x in 𝓝[s] x₀, ae_strongly_measurable (F x) (μ.restrict $ Ι a b))\n  (h_bound : ∀ᶠ x in 𝓝[s] x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous_within_at (λ x, F x t) s x₀) :\n  continuous_within_at (λ x, ∫ t in a..b, F x t ∂μ) s x₀ :=\ntendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_cont\n\n/-- Continuity of interval integral with respect to a parameter at a point.\n  Given `F : X → ℝ → E`, assume `F x` is ae-measurable on `[a, b]` for `x` in a\n  neighborhood of `x₀`, and assume it is bounded by a function integrable on\n  `[a, b]` independent of `x` in a neighborhood of `x₀`. If `(λ x, F x t)`\n  is continuous at `x₀` for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_at_of_dominated_interval\n  {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ}\n  (hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict $ Ι a b))\n  (h_bound : ∀ᶠ x in 𝓝 x₀, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous_at (λ x, F x t) x₀) :\n  continuous_at (λ x, ∫ t in a..b, F x t ∂μ) x₀ :=\ntendsto_integral_filter_of_dominated_convergence bound hF_meas h_bound bound_integrable h_cont\n\n/-- Continuity of interval integral with respect to a parameter.\n  Given `F : X → ℝ → E`, assume each `F x` is ae-measurable on `[a, b]`,\n  and assume it is bounded by a function integrable on `[a, b]` independent of `x`.\n  If `(λ x, F x t)` is continuous for almost every `t` in `[a, b]`\n  then the same holds for `(λ x, ∫ t in a..b, F x t ∂μ) s x₀`. -/\nlemma continuous_of_dominated_interval {F : X → ℝ → E} {bound : ℝ → ℝ} {a b : ℝ}\n  (hF_meas : ∀ x, ae_strongly_measurable (F x) $ μ.restrict $ Ι a b)\n  (h_bound : ∀ x, ∀ᵐ t ∂μ, t ∈ Ι a b → ∥F x t∥ ≤ bound t)\n  (bound_integrable : interval_integrable bound μ a b)\n  (h_cont : ∀ᵐ t ∂μ, t ∈ Ι a b → continuous (λ x, F x t)) :\n  continuous (λ x, ∫ t in a..b, F x t ∂μ) :=\ncontinuous_iff_continuous_at.mpr (λ x₀, continuous_at_of_dominated_interval\n  (eventually_of_forall hF_meas) (eventually_of_forall h_bound) bound_integrable $ h_cont.mono $\n  λ x himp hx, (himp hx).continuous_at)\n\nend order_closed_topology\n\nsection continuous_primitive\nopen topological_space\n\nvariables {a b b₀ b₁ b₂ : ℝ} {μ : measure ℝ} {f g : ℝ → E}\n\nlemma continuous_within_at_primitive (hb₀ : μ {b₀} = 0)\n  (h_int : interval_integrable f μ (min a b₁) (max a b₂)) :\n  continuous_within_at (λ b, ∫ x in a .. b, f x ∂ μ) (Icc b₁ b₂) b₀ :=\nbegin\n  by_cases h₀ : b₀ ∈ Icc b₁ b₂,\n  { have h₁₂ : b₁ ≤ b₂ := h₀.1.trans h₀.2,\n    have min₁₂ : min b₁ b₂ = b₁ := min_eq_left h₁₂,\n    have h_int' : ∀ {x}, x ∈ Icc b₁ b₂ → interval_integrable f μ b₁ x,\n    { rintros x ⟨h₁, h₂⟩,\n      apply h_int.mono_set,\n      apply interval_subset_interval,\n      { exact ⟨min_le_of_left_le (min_le_right a b₁),\n                h₁.trans (h₂.trans $ le_max_of_le_right $ le_max_right _ _)⟩ },\n      { exact ⟨min_le_of_left_le $ (min_le_right _ _).trans h₁,\n                le_max_of_le_right $ h₂.trans $ le_max_right _ _⟩ } },\n    have : ∀ b ∈ Icc b₁ b₂, ∫ x in a..b, f x ∂μ = ∫ x in a..b₁, f x ∂μ + ∫ x in b₁..b, f x ∂μ,\n    { rintros b ⟨h₁, h₂⟩,\n      rw ← integral_add_adjacent_intervals _ (h_int' ⟨h₁, h₂⟩),\n      apply h_int.mono_set,\n      apply interval_subset_interval,\n      { exact ⟨min_le_of_left_le (min_le_left a b₁), le_max_of_le_right (le_max_left _ _)⟩ },\n      { exact ⟨min_le_of_left_le (min_le_right _ _),\n                le_max_of_le_right (h₁.trans $ h₂.trans (le_max_right a b₂))⟩ } },\n    apply continuous_within_at.congr _ this (this _ h₀), clear this,\n    refine continuous_within_at_const.add _,\n    have : (λ b, ∫ x in b₁..b, f x ∂μ) =ᶠ[𝓝[Icc b₁ b₂] b₀]\n            λ b, ∫ x in b₁..b₂, indicator {x | x ≤ b} f x ∂ μ,\n    { apply eventually_eq_of_mem self_mem_nhds_within,\n      exact λ b b_in, (integral_indicator b_in).symm },\n\n    apply continuous_within_at.congr_of_eventually_eq _ this (integral_indicator h₀).symm,\n    have : interval_integrable (λ x, ∥f x∥) μ b₁ b₂,\n      from interval_integrable.norm (h_int' $ right_mem_Icc.mpr h₁₂),\n    refine continuous_within_at_of_dominated_interval _ _ this _ ; clear this,\n    { apply eventually.mono (self_mem_nhds_within),\n      intros x hx,\n      erw [ae_strongly_measurable_indicator_iff, measure.restrict_restrict, Iic_inter_Ioc_of_le],\n      { rw min₁₂,\n        exact (h_int' hx).1.ae_strongly_measurable },\n      { exact le_max_of_le_right hx.2 },\n      exacts [measurable_set_Iic, measurable_set_Iic] },\n    { refine eventually_of_forall (λ x, eventually_of_forall (λ t, _)),\n      dsimp [indicator],\n      split_ifs ; simp },\n    { have : ∀ᵐ t ∂μ, t < b₀ ∨ b₀ < t,\n      { apply eventually.mono (compl_mem_ae_iff.mpr hb₀),\n        intros x hx,\n        exact ne.lt_or_lt hx },\n      apply this.mono,\n      rintros x₀ (hx₀ | hx₀) -,\n      { have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = f x₀,\n        { apply mem_nhds_within_of_mem_nhds,\n          apply eventually.mono (Ioi_mem_nhds hx₀),\n          intros x hx,\n          simp [hx.le] },\n        apply continuous_within_at_const.congr_of_eventually_eq  this,\n        simp [hx₀.le] },\n      { have : ∀ᶠ x in 𝓝[Icc b₁ b₂] b₀, {t : ℝ | t ≤ x}.indicator f x₀ = 0,\n        { apply mem_nhds_within_of_mem_nhds,\n          apply eventually.mono (Iio_mem_nhds hx₀),\n          intros x hx,\n          simp [hx] },\n        apply continuous_within_at_const.congr_of_eventually_eq this,\n        simp [hx₀] } } },\n  { apply continuous_within_at_of_not_mem_closure,\n    rwa [closure_Icc] }\nend\n\nlemma continuous_on_primitive [has_no_atoms μ] (h_int : integrable_on f (Icc a b) μ) :\n  continuous_on (λ x, ∫ t in Ioc a x, f t ∂ μ) (Icc a b) :=\nbegin\n  by_cases h : a ≤ b,\n  { have : ∀ x ∈ Icc a b, ∫ t in Ioc a x, f t ∂μ = ∫ t in a..x, f t ∂μ,\n    { intros x x_in,\n      simp_rw [← interval_oc_of_le h, integral_of_le x_in.1] },\n    rw continuous_on_congr this,\n    intros x₀ hx₀,\n    refine continuous_within_at_primitive (measure_singleton x₀) _,\n    simp only [interval_integrable_iff_integrable_Ioc_of_le, min_eq_left, max_eq_right, h],\n    exact h_int.mono Ioc_subset_Icc_self le_rfl },\n  { rw Icc_eq_empty h,\n    exact continuous_on_empty _ },\nend\n\nlemma continuous_on_primitive_Icc [has_no_atoms μ] (h_int : integrable_on f (Icc a b) μ) :\n  continuous_on (λ x, ∫ t in Icc a x, f t ∂ μ) (Icc a b) :=\nbegin\n  rw show (λ x, ∫ t in Icc a x, f t ∂μ) = λ x, ∫ t in Ioc a x, f t ∂μ,\n    by { ext x, exact integral_Icc_eq_integral_Ioc },\n  exact continuous_on_primitive h_int\nend\n\n/-- Note: this assumes that `f` is `interval_integrable`, in contrast to some other lemmas here. -/\nlemma continuous_on_primitive_interval' [has_no_atoms μ]\n  (h_int : interval_integrable f μ b₁ b₂) (ha : a ∈ [b₁, b₂]) :\n  continuous_on (λ b, ∫ x in a..b, f x ∂ μ) [b₁, b₂] :=\nbegin\n  intros b₀ hb₀,\n  refine continuous_within_at_primitive (measure_singleton _) _,\n  rw [min_eq_right ha.1, max_eq_right ha.2],\n  simpa [interval_integrable_iff, interval_oc] using h_int,\nend\n\nlemma continuous_on_primitive_interval [has_no_atoms μ]\n  (h_int : integrable_on f (interval a b) μ) :\n  continuous_on (λ x, ∫ t in a..x, f t ∂ μ) (interval a b) :=\ncontinuous_on_primitive_interval' h_int.interval_integrable left_mem_interval\n\nlemma continuous_on_primitive_interval_left [has_no_atoms μ]\n  (h_int : integrable_on f (interval a b) μ) :\n  continuous_on (λ x, ∫ t in x..b, f t ∂ μ) (interval a b) :=\nbegin\n  rw interval_swap a b at h_int ⊢,\n  simp only [integral_symm b],\n  exact (continuous_on_primitive_interval h_int).neg,\nend\n\nvariables [has_no_atoms μ]\n\nlemma continuous_primitive (h_int : ∀ a b, interval_integrable f μ a b) (a : ℝ) :\n  continuous (λ b, ∫ x in a..b, f x ∂ μ) :=\nbegin\n  rw continuous_iff_continuous_at,\n  intro b₀,\n  cases exists_lt b₀ with b₁ hb₁,\n  cases exists_gt b₀ with b₂ hb₂,\n  apply continuous_within_at.continuous_at _ (Icc_mem_nhds hb₁ hb₂),\n  exact continuous_within_at_primitive (measure_singleton b₀) (h_int _ _)\nend\n\nlemma _root_.measure_theory.integrable.continuous_primitive (h_int : integrable f μ) (a : ℝ) :\n  continuous (λ b, ∫ x in a..b, f x ∂ μ) :=\ncontinuous_primitive (λ _ _, h_int.interval_integrable) a\n\nend continuous_primitive\n\nsection\n\nvariables {f g : ℝ → ℝ} {a b : ℝ} {μ : measure ℝ}\n\nlemma integral_eq_zero_iff_of_le_of_nonneg_ae (hab : a ≤ b)\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc a b)] f) (hfi : interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b)] 0 :=\nby rw [integral_of_le hab, integral_eq_zero_iff_of_nonneg_ae hf hfi.1]\n\nlemma integral_eq_zero_iff_of_nonneg_ae\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f) (hfi : interval_integrable f μ a b) :\n  ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0 :=\nbegin\n  cases le_total a b with hab hab;\n    simp only [Ioc_eq_empty hab.not_lt, empty_union, union_empty] at hf ⊢,\n  { exact integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi },\n  { rw [integral_symm, neg_eq_zero, integral_eq_zero_iff_of_le_of_nonneg_ae hab hf hfi.symm] }\nend\n\n/-- If `f` is nonnegative and integrable on the unordered interval `set.interval_oc a b`, then its\nintegral over `a..b` is positive if and only if `a < b` and the measure of\n`function.support f ∩ set.Ioc a b` is positive. -/\nlemma integral_pos_iff_support_of_nonneg_ae'\n  (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f) (hfi : interval_integrable f μ a b) :\n  0 < ∫ x in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) :=\nbegin\n  cases lt_or_le a b with hab hba,\n  { rw interval_oc_of_le hab.le at hf,\n    simp only [hab, true_and, integral_of_le hab.le,\n      set_integral_pos_iff_support_of_nonneg_ae hf hfi.1] },\n  { suffices : ∫ x in a..b, f x ∂μ ≤ 0, by simp only [this.not_lt, hba.not_lt, false_and],\n    rw [integral_of_ge hba, neg_nonpos],\n    rw [interval_oc_swap, interval_oc_of_le hba] at hf,\n    exact integral_nonneg_of_ae hf }\nend\n\n/-- If `f` is nonnegative a.e.-everywhere and it is integrable on the unordered interval\n`set.interval_oc a b`, then its integral over `a..b` is positive if and only if `a < b` and the\nmeasure of `function.support f ∩ set.Ioc a b` is positive. -/\nlemma integral_pos_iff_support_of_nonneg_ae (hf : 0 ≤ᵐ[μ] f) (hfi : interval_integrable f μ a b) :\n  0 < ∫ x in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) :=\nintegral_pos_iff_support_of_nonneg_ae' (ae_mono measure.restrict_le_self hf) hfi\n\n/-- If `f : ℝ → ℝ` is strictly positive and integrable on `(a, b]` for real numbers `a < b`, then\nits integral over `a..b` is strictly positive. -/\nlemma interval_integral_pos_of_pos {f : ℝ → ℝ} {a b : ℝ}\n  (hfi : interval_integrable f measure_space.volume a b) (h : ∀ x, 0 < f x) (hab : a < b) :\n  0 < ∫ x in a..b, f x :=\nbegin\n  have hsupp : support f = univ := eq_univ_iff_forall.mpr (λ t, (h t).ne.symm),\n  replace h₀ : 0 ≤ᵐ[volume] f := eventually_of_forall (λ x, (h x).le),\n  rw integral_pos_iff_support_of_nonneg_ae h₀ hfi,\n  exact ⟨hab, by simp [hsupp, hab]⟩,\nend\n\n/-- If `f` and `g` are two functions that are interval integrable on `a..b`, `a ≤ b`,\n`f x ≤ g x` for a.e. `x ∈ set.Ioc a b`, and `f x < g x` on a subset of `set.Ioc a b`\nof nonzero measure, then `∫ x in a..b, f x ∂μ < ∫ x in a..b, g x ∂μ`. -/\nlemma integral_lt_integral_of_ae_le_of_measure_set_of_lt_ne_zero (hab : a ≤ b)\n  (hfi : interval_integrable f μ a b) (hgi : interval_integrable g μ a b)\n  (hle : f ≤ᵐ[μ.restrict (Ioc a b)] g) (hlt : μ.restrict (Ioc a b) {x | f x < g x} ≠ 0) :\n  ∫ x in a..b, f x ∂μ < ∫ x in a..b, g x ∂μ :=\nbegin\n  rw [← sub_pos, ← integral_sub hgi hfi, integral_of_le hab,\n    measure_theory.integral_pos_iff_support_of_nonneg_ae],\n  { refine pos_iff_ne_zero.2 (mt (measure_mono_null _) hlt),\n    exact λ x hx, (sub_pos.2 hx).ne' },\n  exacts [hle.mono (λ x, sub_nonneg.2), hgi.1.sub hfi.1]\nend\n\n/-- If `f` and `g` are continuous on `[a, b]`, `a < b`, `f x ≤ g x` on this interval, and\n`f c < g c` at some point `c ∈ [a, b]`, then `∫ x in a..b, f x < ∫ x in a..b, g x`. -/\nlemma integral_lt_integral_of_continuous_on_of_le_of_exists_lt {f g : ℝ → ℝ} {a b : ℝ}\n  (hab : a < b) (hfc : continuous_on f (Icc a b)) (hgc : continuous_on g (Icc a b))\n  (hle : ∀ x ∈ Ioc a b, f x ≤ g x) (hlt : ∃ c ∈ Icc a b, f c < g c) :\n  ∫ x in a..b, f x < ∫ x in a..b, g x :=\nbegin\n  refine integral_lt_integral_of_ae_le_of_measure_set_of_lt_ne_zero hab.le\n    (hfc.interval_integrable_of_Icc hab.le) (hgc.interval_integrable_of_Icc hab.le)\n    ((ae_restrict_mem measurable_set_Ioc).mono hle) _,\n  contrapose! hlt,\n  have h_eq : f =ᵐ[volume.restrict (Ioc a b)] g,\n  { simp only [← not_le, ← ae_iff] at hlt,\n    exact eventually_le.antisymm ((ae_restrict_iff' measurable_set_Ioc).2 $\n      eventually_of_forall hle) hlt },\n  simp only [measure.restrict_congr_set Ioc_ae_eq_Icc] at h_eq,\n  exact λ c hc, (measure.eq_on_Icc_of_ae_eq volume hab.ne h_eq hfc hgc hc).ge\nend\n\nlemma integral_nonneg_of_ae_restrict (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Icc a b)] f) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nlet H := ae_restrict_of_ae_restrict_of_subset Ioc_subset_Icc_self hf in\nby simpa only [integral_of_le hab] using set_integral_nonneg_of_ae_restrict H\n\nlemma integral_nonneg_of_ae (hab : a ≤ b) (hf : 0 ≤ᵐ[μ] f) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae_restrict hab $ ae_restrict_of_ae hf\n\nlemma integral_nonneg_of_forall (hab : a ≤ b) (hf : ∀ u, 0 ≤ f u) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae hab $ eventually_of_forall hf\n\nlemma integral_nonneg (hab : a ≤ b) (hf : ∀ u, u ∈ Icc a b → 0 ≤ f u) :\n  0 ≤ (∫ u in a..b, f u ∂μ) :=\nintegral_nonneg_of_ae_restrict hab $ (ae_restrict_iff' measurable_set_Icc).mpr $ ae_of_all μ hf\n\nlemma abs_integral_le_integral_abs (hab : a ≤ b) :\n  |∫ x in a..b, f x ∂μ| ≤ ∫ x in a..b, |f x| ∂μ :=\nby simpa only [← real.norm_eq_abs] using norm_integral_le_integral_norm hab\n\nsection mono\n\nvariables (hab : a ≤ b) (hf : interval_integrable f μ a b) (hg : interval_integrable g μ a b)\n\ninclude hab hf hg\n\nlemma integral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict (Icc a b)] g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nlet H := h.filter_mono $ ae_mono $ measure.restrict_mono Ioc_subset_Icc_self $ le_refl μ in\nby simpa only [integral_of_le hab] using set_integral_mono_ae_restrict hf.1 hg.1 H\n\nlemma integral_mono_ae (h : f ≤ᵐ[μ] g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nby simpa only [integral_of_le hab] using set_integral_mono_ae hf.1 hg.1 h\n\nlemma integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nlet H := λ x hx, h x $ Ioc_subset_Icc_self hx in\nby simpa only [integral_of_le hab] using set_integral_mono_on hf.1 hg.1 measurable_set_Ioc H\n\nlemma integral_mono (h : f ≤ g) :\n  ∫ u in a..b, f u ∂μ ≤ ∫ u in a..b, g u ∂μ :=\nintegral_mono_ae hab hf hg $ ae_of_all _ h\n\nomit hg hab\n\nlemma integral_mono_interval {c d} (hca : c ≤ a) (hab : a ≤ b) (hbd : b ≤ d)\n  (hf : 0 ≤ᵐ[μ.restrict (Ioc c d)] f) (hfi : interval_integrable f μ c d):\n  ∫ x in a..b, f x ∂μ ≤ ∫ x in c..d, f x ∂μ :=\nbegin\n  rw [integral_of_le hab, integral_of_le (hca.trans (hab.trans hbd))],\n  exact set_integral_mono_set hfi.1 hf (Ioc_subset_Ioc hca hbd).eventually_le\nend\n\nlemma abs_integral_mono_interval {c d } (h : Ι a b ⊆ Ι c d)\n  (hf : 0 ≤ᵐ[μ.restrict (Ι c d)] f) (hfi : interval_integrable f μ c d) :\n  |∫ x in a..b, f x ∂μ| ≤ |∫ x in c..d, f x ∂μ| :=\nhave hf' : 0 ≤ᵐ[μ.restrict (Ι a b)] f, from ae_mono (measure.restrict_mono h le_rfl) hf,\ncalc |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| : abs_integral_eq_abs_integral_interval_oc f\n... = ∫ x in Ι a b, f x ∂μ : abs_of_nonneg (measure_theory.integral_nonneg_of_ae hf')\n... ≤ ∫ x in Ι c d, f x ∂μ : set_integral_mono_set hfi.def hf h.eventually_le\n... ≤ |∫ x in Ι c d, f x ∂μ| : le_abs_self _\n... = |∫ x in c..d, f x ∂μ| : (abs_integral_eq_abs_integral_interval_oc f).symm\n\nend mono\n\nend\n\n/-!\n### Fundamental theorem of calculus, part 1, for any measure\n\nIn this section we prove a few lemmas that can be seen as versions of FTC-1 for interval integrals\nw.r.t. any measure. Many theorems are formulated for one or two pairs of filters related by\n`FTC_filter a l l'`. This typeclass has exactly four “real” instances: `(a, pure a, ⊥)`,\n`(a, 𝓝[≥] a, 𝓝[>] a)`, `(a, 𝓝[≤] a, 𝓝[≤] a)`, `(a, 𝓝 a, 𝓝 a)`, and two instances\nthat are equal to the first and last “real” instances: `(a, 𝓝[{a}] a, ⊥)` and\n`(a, 𝓝[univ] a, 𝓝[univ] a)`.  We use this approach to avoid repeating arguments in many very similar\ncases.  Lean can automatically find both `a` and `l'` based on `l`.\n\nThe most general theorem `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` can be seen\nas a generalization of lemma `integral_has_strict_fderiv_at` below which states strict\ndifferentiability of `∫ x in u..v, f x` in `(u, v)` at `(a, b)` for a measurable function `f` that\nis integrable on `a..b` and is continuous at `a` and `b`. The lemma is generalized in three\ndirections: first, `measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae` deals with any\nlocally finite measure `μ`; second, it works for one-sided limits/derivatives; third, it assumes\nonly that `f` has finite limits almost surely at `a` and `b`.\n\nNamely, let `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of\n`FTC_filter`s around `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f`\nhas finite limits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.  Then\n`∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ = ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n  o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis theorem is formulated with integral of constants instead of measures in the right hand sides\nfor two reasons: first, this way we avoid `min`/`max` in the statements; second, often it is\npossible to write better `simp` lemmas for these integrals, see `integral_const` and\n`integral_const_of_cdf`.\n\nIn the next subsection we apply this theorem to prove various theorems about differentiability\nof the integral w.r.t. Lebesgue measure. -/\n\n/-- An auxiliary typeclass for the Fundamental theorem of calculus, part 1. It is used to formulate\ntheorems that work simultaneously for left and right one-sided derivatives of `∫ x in u..v, f x`. -/\nclass FTC_filter (a : out_param ℝ) (outer : filter ℝ) (inner : out_param $ filter ℝ)\n  extends tendsto_Ixx_class Ioc outer inner : Prop :=\n(pure_le : pure a ≤ outer)\n(le_nhds : inner ≤ 𝓝 a)\n[meas_gen : is_measurably_generated inner]\n\n/- The `dangerous_instance` linter doesn't take `out_param`s into account, so it thinks that\n`FTC_filter.to_tendsto_Ixx_class` is dangerous. Disable this linter using `nolint`.\n-/\nattribute [nolint dangerous_instance] FTC_filter.to_tendsto_Ixx_class\n\nnamespace FTC_filter\n\ninstance pure (a : ℝ) : FTC_filter a (pure a) ⊥ :=\n{ pure_le := le_rfl,\n  le_nhds := bot_le }\n\ninstance nhds_within_singleton (a : ℝ) : FTC_filter a (𝓝[{a}] a) ⊥ :=\nby { rw [nhds_within, principal_singleton, inf_eq_right.2 (pure_le_nhds a)], apply_instance }\n\nlemma finite_at_inner {a : ℝ} (l : filter ℝ) {l'} [h : FTC_filter a l l']\n  {μ : measure ℝ} [is_locally_finite_measure μ] :\n  μ.finite_at_filter l' :=\n(μ.finite_at_nhds a).filter_mono h.le_nhds\n\ninstance nhds (a : ℝ) : FTC_filter a (𝓝 a) (𝓝 a) :=\n{ pure_le := pure_le_nhds a,\n  le_nhds := le_rfl }\n\ninstance nhds_univ (a : ℝ) : FTC_filter a (𝓝[univ] a) (𝓝 a) :=\nby { rw nhds_within_univ, apply_instance }\n\ninstance nhds_left (a : ℝ) : FTC_filter a (𝓝[≤] a) (𝓝[≤] a) :=\n{ pure_le := pure_le_nhds_within right_mem_Iic,\n  le_nhds := inf_le_left }\n\ninstance nhds_right (a : ℝ) : FTC_filter a (𝓝[≥] a) (𝓝[>] a) :=\n{ pure_le := pure_le_nhds_within left_mem_Ici,\n  le_nhds := inf_le_left }\n\ninstance nhds_Icc {x a b : ℝ} [h : fact (x ∈ Icc a b)] :\n  FTC_filter x (𝓝[Icc a b] x) (𝓝[Icc a b] x) :=\n{ pure_le := pure_le_nhds_within h.out,\n  le_nhds := inf_le_left }\n\ninstance nhds_interval {x a b : ℝ} [h : fact (x ∈ [a, b])] :\n  FTC_filter x (𝓝[[a, b]] x) (𝓝[[a, b]] x) :=\nby { haveI : fact (x ∈ set.Icc (min a b) (max a b)) := h, exact FTC_filter.nhds_Icc }\n\nend FTC_filter\n\nopen asymptotics\n\nsection\n\nvariables {f : ℝ → E} {a b : ℝ} {c ca cb : E} {l l' la la' lb lb' : filter ℝ} {lt : filter ι}\n  {μ : measure ℝ} {u v ua va ub vb : ι → ℝ}\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, where `μ` is a measure\nfinite at `l'`, then `∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both\n`u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l')\n  (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  (λ t, ∫ x in u t..v t, f x ∂μ - ∫ x in u t..v t, c ∂μ) =o[lt] (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) :=\nbegin\n  have A := hf.integral_sub_linear_is_o_ae hfm hl (hu.Ioc hv),\n  have B := hf.integral_sub_linear_is_o_ae hfm hl (hv.Ioc hu),\n  simp only [integral_const'],\n  convert (A.trans_le _).sub (B.trans_le _),\n  { ext t,\n    simp_rw [interval_integral, sub_smul],\n    abel },\n  all_goals { intro t, cases le_total (u t) (v t) with huv huv; simp [huv] }\nend\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both\n`u` and `v` tend to `l` so that `u ≤ v`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_le'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l') (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : u ≤ᶠ[lt] v) :\n  (λ t, ∫ x in u t..v t, f x ∂μ - (μ (Ioc (u t) (v t))).to_real • c) =o[lt]\n    (λ t, (μ $ Ioc (u t) (v t)).to_real) :=\n(measure_integral_sub_linear_is_o_of_tendsto_ae' hfm hf hl hu hv).congr'\n  (huv.mono $ λ x hx, by simp [integral_const', hx])\n  (huv.mono $ λ x hx, by simp [integral_const', hx])\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `tendsto_Ixx_class Ioc`.\nIf `f` has a finite limit `c` at `l ⊓ μ.ae`, where `μ` is a measure\nfinite at `l`, then `∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both\n`u` and `v` tend to `l` so that `v ≤ u`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge` for a version assuming\n`[FTC_filter a l l']` and `[is_locally_finite_measure μ]`. If `l` is one of `𝓝[≥] a`,\n`𝓝[≤] a`, `𝓝 a`, then it's easier to apply the non-primed version.\nThe primed version also works, e.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge'\n  [is_measurably_generated l'] [tendsto_Ixx_class Ioc l l']\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hl : μ.finite_at_filter l') (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : v ≤ᶠ[lt] u) :\n  (λ t, ∫ x in u t..v t, f x ∂μ + (μ (Ioc (v t) (u t))).to_real • c) =o[lt]\n    (λ t, (μ $ Ioc (v t) (u t)).to_real) :=\n(measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' hfm hf hl hv hu huv).neg_left.congr_left $\n  λ t, by simp [integral_symm (u t), add_comm]\n\nsection\n\nvariables [is_locally_finite_measure μ] [FTC_filter a l l']\n\ninclude a\n\nlocal attribute [instance] FTC_filter.meas_gen\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, 1 ∂μ)` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae'` for a version that also works, e.g., for\n`l = l' = at_top`.\n\nWe use integrals of constants instead of measures because this way it is easier to formulate\na statement that works in both cases `u ≤ v` and `v ≤ u`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae (hfm : strongly_measurable_at_filter f l' μ)\n  (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  (λ t, ∫ x in u t..v t, f x ∂μ - ∫ x in u t..v t, c ∂μ) =o[lt] (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae' hfm hf (FTC_filter.finite_at_inner l) hu hv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = μ (Ioc u v) • c + o(μ(Ioc u v))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_le'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_le\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : u ≤ᶠ[lt] v) :\n  (λ t, ∫ x in u t..v t, f x ∂μ - (μ (Ioc (u t) (v t))).to_real • c) =o[lt]\n    (λ t, (μ $ Ioc (u t) (v t)).to_real) :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae_of_le' hfm hf (FTC_filter.finite_at_inner l)\n  hu hv huv\n\n/-- Fundamental theorem of calculus-1, local version for any measure.\nLet filters `l` and `l'` be related by `[FTC_filter a l l']`; let `μ` be a locally finite measure.\nIf `f` has a finite limit `c` at `l' ⊓ μ.ae`, then\n`∫ x in u..v, f x ∂μ = -μ (Ioc v u) • c + o(μ(Ioc v u))` as both `u` and `v` tend to `l`.\n\nSee also `measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge'` for a version that also works,\ne.g., for `l = l' = at_top`. -/\nlemma measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge\n  (hfm : strongly_measurable_at_filter f l' μ) (hf : tendsto f (l' ⊓ μ.ae) (𝓝 c))\n  (hu : tendsto u lt l) (hv : tendsto v lt l) (huv : v ≤ᶠ[lt] u) :\n  (λ t, ∫ x in u t..v t, f x ∂μ + (μ (Ioc (v t) (u t))).to_real • c) =o[lt]\n    (λ t, (μ $ Ioc (v t) (u t)).to_real) :=\nmeasure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' hfm hf (FTC_filter.finite_at_inner l)\n  hu hv huv\n\nend\n\nlocal attribute [instance] FTC_filter.meas_gen\n\nvariables [FTC_filter a la la'] [FTC_filter b lb lb'] [is_locally_finite_measure μ]\n\n/-- Fundamental theorem of calculus-1, strict derivative in both limits for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`; let `(lb, lb')` be a pair of `FTC_filter`s around `b`. Suppose that `f` has finite\nlimits `ca` and `cb` at `la' ⊓ μ.ae` and `lb' ⊓ μ.ae`, respectively.\nThen `∫ x in va..vb, f x ∂μ - ∫ x in ua..ub, f x ∂μ =\n  ∫ x in ub..vb, cb ∂μ - ∫ x in ua..va, ca ∂μ +\n    o(∥∫ x in ua..va, (1:ℝ) ∂μ∥ + ∥∫ x in ub..vb, (1:ℝ) ∂μ∥)`\nas `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  (hab : interval_integrable f μ a b)\n  (hmeas_a : strongly_measurable_at_filter f la' μ)\n  (hmeas_b : strongly_measurable_at_filter f lb' μ)\n  (ha_lim : tendsto f (la' ⊓ μ.ae) (𝓝 ca)) (hb_lim : tendsto f (lb' ⊓ μ.ae) (𝓝 cb))\n  (hua : tendsto ua lt la) (hva : tendsto va lt la)\n  (hub : tendsto ub lt lb) (hvb : tendsto vb lt lb) :\n  (λ t, (∫ x in va t..vb t, f x ∂μ) - (∫ x in ua t..ub t, f x ∂μ) -\n    (∫ x in ub t..vb t, cb ∂μ - ∫ x in ua t..va t, ca ∂μ)) =o[lt]\n    (λ t, ∥∫ x in ua t..va t, (1:ℝ) ∂μ∥ + ∥∫ x in ub t..vb t, (1:ℝ) ∂μ∥) :=\nbegin\n  refine\n    ((measure_integral_sub_linear_is_o_of_tendsto_ae hmeas_a ha_lim hua hva).neg_left.add_add\n    (measure_integral_sub_linear_is_o_of_tendsto_ae hmeas_b hb_lim hub hvb)).congr'\n      _ eventually_eq.rfl,\n  have A : ∀ᶠ t in lt, interval_integrable f μ (ua t) (va t) :=\n    ha_lim.eventually_interval_integrable_ae hmeas_a (FTC_filter.finite_at_inner la) hua hva,\n  have A' : ∀ᶠ t in lt, interval_integrable f μ a (ua t) :=\n    ha_lim.eventually_interval_integrable_ae hmeas_a (FTC_filter.finite_at_inner la)\n      (tendsto_const_pure.mono_right FTC_filter.pure_le) hua,\n  have B : ∀ᶠ t in lt, interval_integrable f μ (ub t) (vb t) :=\n    hb_lim.eventually_interval_integrable_ae hmeas_b (FTC_filter.finite_at_inner lb) hub hvb,\n  have B' : ∀ᶠ t in lt, interval_integrable f μ b (ub t) :=\n    hb_lim.eventually_interval_integrable_ae hmeas_b (FTC_filter.finite_at_inner lb)\n      (tendsto_const_pure.mono_right FTC_filter.pure_le) hub,\n  filter_upwards [A, A', B, B'] with _ ua_va a_ua ub_vb b_ub,\n  rw [← integral_interval_sub_interval_comm'],\n  { dsimp only [], abel, },\n  exacts [ub_vb, ua_va, b_ub.symm.trans $ hab.symm.trans a_ua]\nend\n\n/-- Fundamental theorem of calculus-1, strict derivative in right endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(lb, lb')` be a pair of `FTC_filter`s\naround `b`. Suppose that `f` has a finite limit `c` at `lb' ⊓ μ.ae`.\n\nThen `∫ x in a..v, f x ∂μ - ∫ x in a..u, f x ∂μ = ∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `lb`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right\n  (hab : interval_integrable f μ a b) (hmeas : strongly_measurable_at_filter f lb' μ)\n  (hf : tendsto f (lb' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt lb) (hv : tendsto v lt lb) :\n  (λ t, ∫ x in a..v t, f x ∂μ - ∫ x in a..u t, f x ∂μ - ∫ x in u t..v t, c ∂μ) =o[lt]\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) :=\nby simpa using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  hab strongly_measurable_at_bot hmeas ((tendsto_bot : tendsto _ ⊥ (𝓝 0)).mono_left inf_le_left)\n  hf (tendsto_const_pure : tendsto _ _ (pure a)) tendsto_const_pure hu hv\n\n/-- Fundamental theorem of calculus-1, strict derivative in left endpoint for a locally finite\nmeasure.\n\nLet `f` be a measurable function integrable on `a..b`. Let `(la, la')` be a pair of `FTC_filter`s\naround `a`. Suppose that `f` has a finite limit `c` at `la' ⊓ μ.ae`.\n\nThen `∫ x in v..b, f x ∂μ - ∫ x in u..b, f x ∂μ = -∫ x in u..v, c ∂μ + o(∫ x in u..v, (1:ℝ) ∂μ)`\nas `u` and `v` tend to `la`.\n-/\nlemma measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left\n  (hab : interval_integrable f μ a b) (hmeas : strongly_measurable_at_filter f la' μ)\n  (hf : tendsto f (la' ⊓ μ.ae) (𝓝 c)) (hu : tendsto u lt la) (hv : tendsto v lt la) :\n  (λ t, ∫ x in v t..b, f x ∂μ - ∫ x in u t..b, f x ∂μ + ∫ x in u t..v t, c ∂μ) =o[lt]\n    (λ t, ∫ x in u t..v t, (1:ℝ) ∂μ) :=\nby simpa using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  hab hmeas strongly_measurable_at_bot hf ((tendsto_bot : tendsto _ ⊥ (𝓝 0)).mono_left inf_le_left)\n  hu hv (tendsto_const_pure : tendsto _ _ (pure b)) tendsto_const_pure\n\nend\n\n/-!\n### Fundamental theorem of calculus-1 for Lebesgue measure\n\nIn this section we restate theorems from the previous section for Lebesgue measure.\nIn particular, we prove that `∫ x in u..v, f x` is strictly differentiable in `(u, v)`\nat `(a, b)` provided that `f` is integrable on `a..b` and is continuous at `a` and `b`.\n-/\n\nvariables {f : ℝ → E} {c ca cb : E} {l l' la la' lb lb' : filter ℝ} {lt : filter ι}\n  {a b z : ℝ} {u v ua ub va vb : ι → ℝ} [FTC_filter a la la'] [FTC_filter b lb lb']\n\n/-!\n#### Auxiliary `is_o` statements\n\nIn this section we prove several lemmas that can be interpreted as strict differentiability of\n`(u, v) ↦ ∫ x in u..v, f x ∂μ` in `u` and/or `v` at a filter. The statements use `is_o` because\nwe have no definition of `has_strict_(f)deriv_at_filter` in the library.\n-/\n\n/-- Fundamental theorem of calculus-1, local version. If `f` has a finite limit `c` almost surely at\n`l'`, where `(l, l')` is an `FTC_filter` pair around `a`, then\n`∫ x in u..v, f x ∂μ = (v - u) • c + o (v - u)` as both `u` and `v` tend to `l`. -/\nlemma integral_sub_linear_is_o_of_tendsto_ae [FTC_filter a l l']\n  (hfm : strongly_measurable_at_filter f l') (hf : tendsto f (l' ⊓ volume.ae) (𝓝 c))\n  {u v : ι → ℝ} (hu : tendsto u lt l) (hv : tendsto v lt l) :\n  (λ t, (∫ x in u t..v t, f x) - (v t - u t) • c) =o[lt] (v - u) :=\nby simpa [integral_const] using measure_integral_sub_linear_is_o_of_tendsto_ae hfm hf hu hv\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair around\n`a`, and `(lb, lb')` is an `FTC_filter` pair around `b`, and `f` has finite limits `ca` and `cb`\nalmost surely at `la'` and `lb'`, respectively, then\n`(∫ x in va..vb, f x) - ∫ x in ua..ub, f x = (vb - ub) • cb - (va - ua) • ca +\n  o(∥va - ua∥ + ∥vb - ub∥)` as `ua` and `va` tend to `la` while `ub` and `vb` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_fderiv_at_filter` if we had this\ndefinition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae\n  (hab : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la') (hmeas_b : strongly_measurable_at_filter f lb')\n  (ha_lim : tendsto f (la' ⊓ volume.ae) (𝓝 ca)) (hb_lim : tendsto f (lb' ⊓ volume.ae) (𝓝 cb))\n  (hua : tendsto ua lt la) (hva : tendsto va lt la)\n  (hub : tendsto ub lt lb) (hvb : tendsto vb lt lb) :\n  (λ t, (∫ x in va t..vb t, f x) - (∫ x in ua t..ub t, f x) -\n    ((vb t - ub t) • cb - (va t - ua t) • ca)) =o[lt] (λ t, ∥va t - ua t∥ + ∥vb t - ub t∥) :=\nby simpa [integral_const]\n  using measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae hab hmeas_a hmeas_b\n    ha_lim hb_lim hua hva hub hvb\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(lb, lb')` is an `FTC_filter` pair\naround `b`, and `f` has a finite limit `c` almost surely at `lb'`, then\n`(∫ x in a..v, f x) - ∫ x in a..u, f x = (v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `lb`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right\n  (hab : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f lb')\n  (hf : tendsto f (lb' ⊓ volume.ae) (𝓝 c)) (hu : tendsto u lt lb) (hv : tendsto v lt lb) :\n  (λ t, (∫ x in a..v t, f x) - (∫ x in a..u t, f x) - (v t - u t) • c) =o[lt] (v - u) :=\nby simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hab hmeas hf hu hv\n\n/-- Fundamental theorem of calculus-1, strict differentiability at filter in both endpoints.\nIf `f` is a measurable function integrable on `a..b`, `(la, la')` is an `FTC_filter` pair\naround `a`, and `f` has a finite limit `c` almost surely at `la'`, then\n`(∫ x in v..b, f x) - ∫ x in u..b, f x = -(v - u) • c + o(∥v - u∥)` as `u` and `v` tend to `la`.\n\nThis lemma could've been formulated using `has_strict_deriv_at_filter` if we had this definition. -/\nlemma integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left\n  (hab : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f la')\n  (hf : tendsto f (la' ⊓ volume.ae) (𝓝 c)) (hu : tendsto u lt la) (hv : tendsto v lt la) :\n  (λ t, (∫ x in v t..b, f x) - (∫ x in u t..b, f x) + (v t - u t) • c) =o[lt] (v - u) :=\nby simpa only [integral_const, smul_eq_mul, mul_one] using\n  measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left hab hmeas hf hu hv\n\nopen continuous_linear_map (fst snd smul_right sub_apply smul_right_apply coe_fst' coe_snd' map_sub)\n\n/-!\n#### Strict differentiability\n\nIn this section we prove that for a measurable function `f` integrable on `a..b`,\n\n* `integral_has_strict_fderiv_at_of_tendsto_ae`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)` in the sense of strict differentiability\n  provided that `f` tends to `ca` and `cb` almost surely as `x` tendsto to `a` and `b`,\n  respectively;\n\n* `integral_has_strict_fderiv_at`: the function `(u, v) ↦ ∫ x in u..v, f x` has\n  derivative `(u, v) ↦ v • f b - u • f a` at `(a, b)` in the sense of strict differentiability\n  provided that `f` is continuous at `a` and `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_right`: the function `u ↦ ∫ x in a..u, f x` has\n  derivative `c` at `b` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `b`;\n\n* `integral_has_strict_deriv_at_right`: the function `u ↦ ∫ x in a..u, f x` has derivative `f b` at\n  `b` in the sense of strict differentiability provided that `f` is continuous at `b`;\n\n* `integral_has_strict_deriv_at_of_tendsto_ae_left`: the function `u ↦ ∫ x in u..b, f x` has\n  derivative `-c` at `a` in the sense of strict differentiability provided that `f` tends to `c`\n  almost surely as `x` tends to `a`;\n\n* `integral_has_strict_deriv_at_left`: the function `u ↦ ∫ x in u..b, f x` has derivative `-f a` at\n  `a` in the sense of strict differentiability provided that `f` is continuous at `a`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`\nin the sense of strict differentiability. -/\nlemma integral_has_strict_fderiv_at_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  has_strict_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (a, b) :=\nbegin\n  have := integral_sub_integral_sub_linear_is_o_of_tendsto_ae hf hmeas_a hmeas_b ha hb\n    ((continuous_fst.comp continuous_snd).tendsto ((a, b), (a, b)))\n    ((continuous_fst.comp continuous_fst).tendsto ((a, b), (a, b)))\n    ((continuous_snd.comp continuous_snd).tendsto ((a, b), (a, b)))\n    ((continuous_snd.comp continuous_fst).tendsto ((a, b), (a, b))),\n  refine (this.congr_left _).trans_is_O _,\n  { intro x, simp [sub_smul] },\n  { exact is_O_fst_prod.norm_left.add is_O_snd_prod.norm_left }\nend\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)` in the sense of strict differentiability. -/\nlemma integral_has_strict_fderiv_at\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  has_strict_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (a, b) :=\nintegral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b\n  (ha.mono_left inf_le_left) (hb.mono_left inf_le_left)\n\n/-- **First Fundamental Theorem of Calculus**: if `f : ℝ → E` is integrable on `a..b` and `f x` has\na finite limit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b` in\nthe sense of strict differentiability. -/\nlemma integral_has_strict_deriv_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : has_strict_deriv_at (λ u, ∫ x in a..u, f x) c b :=\nintegral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb continuous_at_snd\n  continuous_at_fst\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b` in the sense of strict\ndifferentiability. -/\nlemma integral_has_strict_deriv_at_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) : has_strict_deriv_at (λ u, ∫ x in a..u, f x) (f b) b :=\nintegral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a` in the sense\nof strict differentiability. -/\nlemma integral_has_strict_deriv_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : has_strict_deriv_at (λ u, ∫ x in u..b, f x) (-c) a :=\nby simpa only [← integral_symm]\n  using (integral_has_strict_deriv_at_of_tendsto_ae_right hf.symm hmeas ha).neg\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a` in the sense of strict\ndifferentiability. -/\nlemma integral_has_strict_deriv_at_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : continuous_at f a) : has_strict_deriv_at (λ u, ∫ x in u..b, f x) (-f a) a :=\nby simpa only [← integral_symm] using (integral_has_strict_deriv_at_right hf.symm hmeas ha).neg\n\n/-!\n#### Fréchet differentiability\n\nIn this subsection we restate results from the previous subsection in terms of `has_fderiv_at`,\n`has_deriv_at`, `fderiv`, and `deriv`.\n-/\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then\n`(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca` at `(a, b)`. -/\nlemma integral_has_fderiv_at_of_tendsto_ae (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  has_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (a, b) :=\n(integral_has_strict_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).has_fderiv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `(u, v) ↦ ∫ x in u..v, f x` has derivative `(u, v) ↦ v • cb - u • ca`\nat `(a, b)`. -/\nlemma integral_has_fderiv_at (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  has_fderiv_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (a, b) :=\n(integral_has_strict_fderiv_at hf hmeas_a hmeas_b ha hb).has_fderiv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has finite\nlimits `ca` and `cb` almost surely as `x` tends to `a` and `b`, respectively, then `fderiv`\nderivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦ v • cb - u • ca`. -/\nlemma fderiv_integral_of_tendsto_ae (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 cb)) :\n  fderiv ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (a, b) =\n    (snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca :=\n(integral_has_fderiv_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderiv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a` and `b`, then `fderiv` derivative of `(u, v) ↦ ∫ x in u..v, f x` at `(a, b)` equals `(u, v) ↦\nv • cb - u • ca`. -/\nlemma fderiv_integral (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f (𝓝 a))\n  (hmeas_b : strongly_measurable_at_filter f (𝓝 b))\n  (ha : continuous_at f a) (hb : continuous_at f b) :\n  fderiv ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (a, b) =\n    (snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a) :=\n(integral_has_fderiv_at hf hmeas_a hmeas_b ha hb).fderiv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `b`, then `u ↦ ∫ x in a..u, f x` has derivative `c` at `b`. -/\nlemma integral_has_deriv_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : has_deriv_at (λ u, ∫ x in a..u, f x) c b :=\n(integral_has_strict_deriv_at_of_tendsto_ae_right hf hmeas hb).has_deriv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then `u ↦ ∫ x in a..u, f x` has derivative `f b` at `b`. -/\nlemma integral_has_deriv_at_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) : has_deriv_at (λ u, ∫ x in a..u, f x) (f b) b :=\n(integral_has_strict_deriv_at_right hf hmeas hb).has_deriv_at\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\nlemma deriv_integral_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : tendsto f (𝓝 b ⊓ volume.ae) (𝓝 c)) : deriv (λ u, ∫ x in a..u, f x) b = c :=\n(integral_has_deriv_at_of_tendsto_ae_right hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `b`, then the derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\nlemma deriv_integral_right\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 b))\n  (hb : continuous_at f b) :\n  deriv (λ u, ∫ x in a..u, f x) b = f b :=\n(integral_has_deriv_at_right hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely at `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-c` at `a`. -/\nlemma integral_has_deriv_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : has_deriv_at (λ u, ∫ x in u..b, f x) (-c) a :=\n(integral_has_strict_deriv_at_of_tendsto_ae_left hf hmeas ha).has_deriv_at\n\n/-- Fundamental theorem of calculus-1: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then `u ↦ ∫ x in u..b, f x` has derivative `-f a` at `a`. -/\nlemma integral_has_deriv_at_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (ha : continuous_at f a) :\n  has_deriv_at (λ u, ∫ x in u..b, f x) (-f a) a :=\n(integral_has_strict_deriv_at_left hf hmeas ha).has_deriv_at\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` has a finite\nlimit `c` almost surely at `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\nlemma deriv_integral_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (hb : tendsto f (𝓝 a ⊓ volume.ae) (𝓝 c)) : deriv (λ u, ∫ x in u..b, f x) a = -c :=\n(integral_has_deriv_at_of_tendsto_ae_left hf hmeas hb).deriv\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f` is continuous\nat `a`, then the derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\nlemma deriv_integral_left\n  (hf : interval_integrable f volume a b) (hmeas : strongly_measurable_at_filter f (𝓝 a))\n  (hb : continuous_at f a) :\n  deriv (λ u, ∫ x in u..b, f x) a = -f a :=\n(integral_has_deriv_at_left hf hmeas hb).deriv\n\n/-!\n#### One-sided derivatives\n-/\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • cb - u • ca` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to `ca`\nand `cb` almost surely at the filters `la` and `lb` from the following table.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma integral_has_fderiv_within_at_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  (ha : tendsto f (la ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (lb ⊓ volume.ae) (𝓝 cb)) :\n  has_fderiv_within_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) (s ×ˢ t) (a, b) :=\nbegin\n  rw [has_fderiv_within_at, nhds_within_prod_eq],\n  have := integral_sub_integral_sub_linear_is_o_of_tendsto_ae hf hmeas_a hmeas_b ha hb\n    (tendsto_const_pure.mono_right FTC_filter.pure_le : tendsto _ _ (𝓝[s] a)) tendsto_fst\n    (tendsto_const_pure.mono_right FTC_filter.pure_le : tendsto _ _ (𝓝[t] b)) tendsto_snd,\n  refine (this.congr_left _).trans_is_O _,\n  { intro x, simp [sub_smul] },\n  { exact is_O_fst_prod.norm_left.add is_O_snd_prod.norm_left }\nend\n\n/-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`\nhas derivative `(u, v) ↦ v • f b - u • f a` within `s × t` at `(a, b)`, where\n`s ∈ {Iic a, {a}, Ici a, univ}` and `t ∈ {Iic b, {b}, Ici b, univ}` provided that `f` tends to\n`f a` and `f b` at the filters `la` and `lb` from the following table. In most cases this assumption\nis definitionally equal `continuous_at f _` or `continuous_within_at f _ _`.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma integral_has_fderiv_within_at\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (ha : tendsto f la (𝓝 $ f a)) (hb : tendsto f lb (𝓝 $ f b)) :\n  has_fderiv_within_at (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x)\n    ((snd ℝ ℝ ℝ).smul_right (f b) - (fst ℝ ℝ ℝ).smul_right (f a)) (s ×ˢ t) (a, b) :=\nintegral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b (ha.mono_left inf_le_left)\n  (hb.mono_left inf_le_left)\n\n/-- An auxiliary tactic closing goals `unique_diff_within_at ℝ s a` where\n`s ∈ {Iic a, Ici a, univ}`. -/\nmeta def unique_diff_within_at_Ici_Iic_univ : tactic unit :=\n`[apply_rules [unique_diff_on.unique_diff_within_at, unique_diff_on_Ici, unique_diff_on_Iic,\n  left_mem_Ici, right_mem_Iic, unique_diff_within_at_univ]]\n\n/-- Let `f` be a measurable function integrable on `a..b`. Choose `s ∈ {Iic a, Ici a, univ}`\nand `t ∈ {Iic b, Ici b, univ}`. Suppose that `f` tends to `ca` and `cb` almost surely at the filters\n`la` and `lb` from the table below. Then `fderiv_within ℝ (λ p, ∫ x in p.1..p.2, f x) (s ×ˢ t)`\nis equal to `(u, v) ↦ u • cb - v • ca`.\n\n| `s`     | `la`     | `t`     | `lb`     |\n| ------- | ----     | ---     | ----     |\n| `Iic a` | `𝓝[≤] a` | `Iic b` | `𝓝[≤] b` |\n| `Ici a` | `𝓝[>] a` | `Ici b` | `𝓝[>] b` |\n| `{a}`   | `⊥`      | `{b}`   | `⊥`      |\n| `univ`  | `𝓝 a`    | `univ`  | `𝓝 b`    |\n-/\nlemma fderiv_within_integral_of_tendsto_ae\n  (hf : interval_integrable f volume a b)\n  (hmeas_a : strongly_measurable_at_filter f la) (hmeas_b : strongly_measurable_at_filter f lb)\n  {s t : set ℝ} [FTC_filter a (𝓝[s] a) la] [FTC_filter b (𝓝[t] b) lb]\n  (ha : tendsto f (la ⊓ volume.ae) (𝓝 ca)) (hb : tendsto f (lb ⊓ volume.ae) (𝓝 cb))\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ)\n  (ht : unique_diff_within_at ℝ t b . unique_diff_within_at_Ici_Iic_univ) :\n  fderiv_within ℝ (λ p : ℝ × ℝ, ∫ x in p.1..p.2, f x) (s ×ˢ t) (a, b) =\n    ((snd ℝ ℝ ℝ).smul_right cb - (fst ℝ ℝ ℝ).smul_right ca) :=\n(integral_has_fderiv_within_at_of_tendsto_ae hf hmeas_a hmeas_b ha hb).fderiv_within $ hs.prod ht\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left,\nthen `u ↦ ∫ x in a..u, f x` has right (resp., left) derivative `c` at `b`. -/\nlemma integral_has_deriv_within_at_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : tendsto f (𝓝[t] b ⊓ volume.ae) (𝓝 c)) :\n  has_deriv_within_at (λ u, ∫ x in a..u, f x) c s b :=\nintegral_sub_integral_sub_linear_is_o_of_tendsto_ae_right hf hmeas hb\n  (tendsto_const_pure.mono_right FTC_filter.pure_le) tendsto_id\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `b`, then `u ↦ ∫ x in a..u, f x` has left (resp., right)\nderivative `f b` at `b`. -/\nlemma integral_has_deriv_within_at_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : continuous_within_at f t b) :\n  has_deriv_within_at (λ u, ∫ x in a..u, f x) (f b) s b :=\nintegral_has_deriv_within_at_of_tendsto_ae_right hf hmeas (hb.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `b` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in a..u, f x` at `b` equals `c`. -/\nlemma deriv_within_integral_of_tendsto_ae_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas: strongly_measurable_at_filter f (𝓝[t] b)) (hb : tendsto f (𝓝[t] b ⊓ volume.ae) (𝓝 c))\n  (hs : unique_diff_within_at ℝ s b . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in a..u, f x) s b = c :=\n(integral_has_deriv_within_at_of_tendsto_ae_right hf hmeas hb).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `b`, then the right (resp., left) derivative of\n`u ↦ ∫ x in a..u, f x` at `b` equals `f b`. -/\nlemma deriv_within_integral_right\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter b (𝓝[s] b) (𝓝[t] b)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] b)) (hb : continuous_within_at f t b)\n  (hs : unique_diff_within_at ℝ s b . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in a..u, f x) s b = f b :=\n(integral_has_deriv_within_at_right hf hmeas hb).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left,\nthen `u ↦ ∫ x in u..b, f x` has right (resp., left) derivative `-c` at `a`. -/\nlemma integral_has_deriv_within_at_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a))\n  (ha : tendsto f (𝓝[t] a ⊓ volume.ae) (𝓝 c)) :\n  has_deriv_within_at (λ u, ∫ x in u..b, f x) (-c) s a :=\nby { simp only [integral_symm b],\n  exact (integral_has_deriv_within_at_of_tendsto_ae_right hf.symm hmeas ha).neg }\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\nfrom the left or from the right at `a`, then `u ↦ ∫ x in u..b, f x` has left (resp., right)\nderivative `-f a` at `a`. -/\nlemma integral_has_deriv_within_at_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : continuous_within_at f t a) :\n  has_deriv_within_at (λ u, ∫ x in u..b, f x) (-f a) s a :=\nintegral_has_deriv_within_at_of_tendsto_ae_left hf hmeas (ha.mono_left inf_le_left)\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` has a finite\nlimit `c` almost surely as `x` tends to `a` from the right or from the left, then the right\n(resp., left) derivative of `u ↦ ∫ x in u..b, f x` at `a` equals `-c`. -/\nlemma deriv_within_integral_of_tendsto_ae_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : tendsto f (𝓝[t] a ⊓ volume.ae) (𝓝 c))\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in u..b, f x) s a = -c :=\n(integral_has_deriv_within_at_of_tendsto_ae_left hf hmeas ha).deriv_within hs\n\n/-- Fundamental theorem of calculus: if `f : ℝ → E` is integrable on `a..b` and `f x` is continuous\non the right or on the left at `a`, then the right (resp., left) derivative of\n`u ↦ ∫ x in u..b, f x` at `a` equals `-f a`. -/\nlemma deriv_within_integral_left\n  (hf : interval_integrable f volume a b) {s t : set ℝ} [FTC_filter a (𝓝[s] a) (𝓝[t] a)]\n  (hmeas : strongly_measurable_at_filter f (𝓝[t] a)) (ha : continuous_within_at f t a)\n  (hs : unique_diff_within_at ℝ s a . unique_diff_within_at_Ici_Iic_univ) :\n  deriv_within (λ u, ∫ x in u..b, f x) s a = -f a :=\n(integral_has_deriv_within_at_left hf hmeas ha).deriv_within hs\n\n/-- The integral of a continuous function is differentiable on a real set `s`. -/\ntheorem differentiable_on_integral_of_continuous {s : set ℝ}\n  (hintg : ∀ x ∈ s, interval_integrable f volume a x) (hcont : continuous f) :\n  differentiable_on ℝ (λ u, ∫ x in a..u, f x) s :=\nλ y hy, (integral_has_deriv_at_right (hintg y hy)\n  hcont.ae_strongly_measurable.strongly_measurable_at_filter\n    hcont.continuous_at) .differentiable_at.differentiable_within_at\n\n/-!\n### Fundamental theorem of calculus, part 2\n\nThis section contains theorems pertaining to FTC-2 for interval integrals, i.e., the assertion\nthat `∫ x in a..b, f' x = f b - f a` under suitable assumptions.\n\nThe most classical version of this theorem assumes that `f'` is continuous. However, this is\nunnecessarily strong: the result holds if `f'` is just integrable. We prove the strong version,\nfollowing [Rudin, *Real and Complex Analysis* (Theorem 7.21)][rudin2006real]. The proof is first\ngiven for real-valued functions, and then deduced for functions with a general target space. For\na real-valued function `g`, it suffices to show that `g b - g a ≤ (∫ x in a..b, g' x) + ε` for all\npositive `ε`. To prove this, choose a lower-semicontinuous function `G'` with `g' < G'` and with\nintegral close to that of `g'` (its existence is guaranteed by the Vitali-Carathéodory theorem).\nIt satisfies `g t - g a ≤ ∫ x in a..t, G' x` for all `t ∈ [a, b]`: this inequality holds at `a`,\nand if it holds at `t` then it holds for `u` close to `t` on its right, as the left hand side\nincreases by `g u - g t ∼ (u -t) g' t`, while the right hand side increases by\n`∫ x in t..u, G' x` which is roughly at least `∫ x in t..u, G' t = (u - t) G' t`, by lower\nsemicontinuity. As  `g' t < G' t`, this gives the conclusion. One can therefore push progressively\nthis inequality to the right until the point `b`, where it gives the desired conclusion.\n-/\n\nvariables {g' g φ : ℝ → ℝ}\n\n/-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has\n`g b - g a ≤ ∫ y in a..b, g' y` when `g'` is integrable.\nAuxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`.\nWe give the slightly more general version that `g b - g a ≤ ∫ y in a..b, φ y` when `g' ≤ φ` and\n`φ` is integrable (even if `g'` is not known to be integrable).\nVersion assuming that `g` is differentiable on `[a, b)`. -/\nlemma sub_le_integral_of_has_deriv_right_of_le_Ico (hab : a ≤ b) (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ico a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (φint : integrable_on φ (Icc a b)) (hφg : ∀ x ∈ Ico a b, g' x ≤ φ x) :\n  g b - g a ≤ ∫ y in a..b, φ y :=\nbegin\n  refine le_of_forall_pos_le_add (λ ε εpos, _),\n  -- Bound from above `g'` by a lower-semicontinuous function `G'`.\n  rcases exists_lt_lower_semicontinuous_integral_lt φ φint εpos with\n    ⟨G', f_lt_G', G'cont, G'int, G'lt_top, hG'⟩,\n  -- we will show by \"induction\" that `g t - g a ≤ ∫ u in a..t, G' u` for all `t ∈ [a, b]`.\n  set s := {t | g t - g a ≤ ∫ u in a..t, (G' u).to_real} ∩ Icc a b,\n  -- the set `s` of points where this property holds is closed.\n  have s_closed : is_closed s,\n  { have : continuous_on (λ t, (g t - g a, ∫ u in a..t, (G' u).to_real)) (Icc a b),\n    { rw ← interval_of_le hab at G'int ⊢ hcont,\n      exact (hcont.sub continuous_on_const).prod (continuous_on_primitive_interval G'int) },\n    simp only [s, inter_comm],\n    exact this.preimage_closed_of_closed is_closed_Icc order_closed_topology.is_closed_le' },\n  have main : Icc a b ⊆ {t | g t - g a ≤ ∫ u in a..t, (G' u).to_real },\n  { -- to show that the set `s` is all `[a, b]`, it suffices to show that any point `t` in `s`\n    -- with `t < b` admits another point in `s` slightly to its right\n    -- (this is a sort of real induction).\n    apply s_closed.Icc_subset_of_forall_exists_gt\n      (by simp only [integral_same, mem_set_of_eq, sub_self]) (λ t ht v t_lt_v, _),\n    obtain ⟨y, g'_lt_y', y_lt_G'⟩ : ∃ (y : ℝ), (g' t : ereal) < y ∧ (y : ereal) < G' t :=\n      ereal.lt_iff_exists_real_btwn.1 ((ereal.coe_le_coe_iff.2 (hφg t ht.2)).trans_lt (f_lt_G' t)),\n    -- bound from below the increase of `∫ x in a..u, G' x` on the right of `t`, using the lower\n    -- semicontinuity of `G'`.\n    have I1 : ∀ᶠ u in 𝓝[>] t, (u - t) * y ≤ ∫ w in t..u, (G' w).to_real,\n    { have B : ∀ᶠ u in 𝓝 t, (y : ereal) < G' u :=\n        G'cont.lower_semicontinuous_at _ _ y_lt_G',\n      rcases mem_nhds_iff_exists_Ioo_subset.1 B with ⟨m, M, ⟨hm, hM⟩, H⟩,\n      have : Ioo t (min M b) ∈ 𝓝[>] t := mem_nhds_within_Ioi_iff_exists_Ioo_subset.2\n        ⟨min M b, by simp only [hM, ht.right.right, lt_min_iff, mem_Ioi, and_self], subset.refl _⟩,\n      filter_upwards [this] with u hu,\n      have I : Icc t u ⊆ Icc a b := Icc_subset_Icc ht.2.1 (hu.2.le.trans (min_le_right _ _)),\n      calc (u - t) * y = ∫ v in Icc t u, y :\n        by simp only [hu.left.le, measure_theory.integral_const, algebra.id.smul_eq_mul, sub_nonneg,\n                      measurable_set.univ, real.volume_Icc, measure.restrict_apply, univ_inter,\n                      ennreal.to_real_of_real]\n      ... ≤ ∫ w in t..u, (G' w).to_real :\n      begin\n        rw [interval_integral.integral_of_le hu.1.le, ← integral_Icc_eq_integral_Ioc],\n        apply set_integral_mono_ae_restrict,\n        { simp only [integrable_on_const, real.volume_Icc, ennreal.of_real_lt_top, or_true] },\n        { exact integrable_on.mono_set G'int I },\n        { have C1 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), G' x < ∞ :=\n            ae_mono (measure.restrict_mono I le_rfl) G'lt_top,\n          have C2 : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc t u), x ∈ Icc t u :=\n            ae_restrict_mem measurable_set_Icc,\n          filter_upwards [C1, C2] with x G'x hx,\n          apply ereal.coe_le_coe_iff.1,\n          have : x ∈ Ioo m M, by simp only [hm.trans_le hx.left,\n            (hx.right.trans_lt hu.right).trans_le (min_le_left M b), mem_Ioo, and_self],\n          convert le_of_lt (H this),\n          exact ereal.coe_to_real G'x.ne (ne_bot_of_gt (f_lt_G' x)) }\n      end },\n    -- bound from above the increase of `g u - g a` on the right of `t`, using the derivative at `t`\n    have I2 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ (u - t) * y,\n    { have g'_lt_y : g' t < y := ereal.coe_lt_coe_iff.1 g'_lt_y',\n      filter_upwards [(hderiv t ⟨ht.2.1, ht.2.2⟩).limsup_slope_le'\n        (not_mem_Ioi.2 le_rfl) g'_lt_y, self_mem_nhds_within] with u hu t_lt_u,\n      have := mul_le_mul_of_nonneg_left hu.le (sub_pos.2 t_lt_u).le,\n      rwa [← smul_eq_mul, sub_smul_slope] at this },\n    -- combine the previous two bounds to show that `g u - g a` increases less quickly than\n    -- `∫ x in a..u, G' x`.\n    have I3 : ∀ᶠ u in 𝓝[>] t, g u - g t ≤ ∫ w in t..u, (G' w).to_real,\n    { filter_upwards [I1, I2] with u hu1 hu2 using hu2.trans hu1, },\n    have I4 : ∀ᶠ u in 𝓝[>] t, u ∈ Ioc t (min v b),\n    { refine mem_nhds_within_Ioi_iff_exists_Ioc_subset.2 ⟨min v b, _, subset.refl _⟩,\n      simp only [lt_min_iff, mem_Ioi],\n      exact ⟨t_lt_v, ht.2.2⟩ },\n    -- choose a point `x` slightly to the right of `t` which satisfies the above bound\n    rcases (I3.and I4).exists with ⟨x, hx, h'x⟩,\n    -- we check that it belongs to `s`, essentially by construction\n    refine ⟨x, _, Ioc_subset_Ioc le_rfl (min_le_left _ _) h'x⟩,\n    calc g x - g a = (g t - g a) + (g x - g t) : by abel\n    ... ≤ (∫ w in a..t, (G' w).to_real) + ∫ w in t..x, (G' w).to_real : add_le_add ht.1 hx\n    ... = ∫ w in a..x, (G' w).to_real :\n    begin\n      apply integral_add_adjacent_intervals,\n      { rw interval_integrable_iff_integrable_Ioc_of_le ht.2.1,\n        exact integrable_on.mono_set G'int\n          (Ioc_subset_Icc_self.trans (Icc_subset_Icc le_rfl ht.2.2.le)) },\n      { rw interval_integrable_iff_integrable_Ioc_of_le h'x.1.le,\n        apply integrable_on.mono_set G'int,\n        refine Ioc_subset_Icc_self.trans (Icc_subset_Icc ht.2.1 (h'x.2.trans (min_le_right _ _))) }\n    end },\n  -- now that we know that `s` contains `[a, b]`, we get the desired result by applying this to `b`.\n  calc g b - g a ≤ ∫ y in a..b, (G' y).to_real : main (right_mem_Icc.2 hab)\n  ... ≤ (∫ y in a..b, φ y) + ε :\n    begin\n      convert hG'.le;\n      { rw interval_integral.integral_of_le hab,\n        simp only [integral_Icc_eq_integral_Ioc', real.volume_singleton] },\n    end\nend\n\n/-- Hard part of FTC-2 for integrable derivatives, real-valued functions: one has\n`g b - g a ≤ ∫ y in a..b, g' y` when `g'` is integrable.\nAuxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`.\nWe give the slightly more general version that `g b - g a ≤ ∫ y in a..b, φ y` when `g' ≤ φ` and\n`φ` is integrable (even if `g'` is not known to be integrable).\nVersion assuming that `g` is differentiable on `(a, b)`. -/\nlemma sub_le_integral_of_has_deriv_right_of_le (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (φint : integrable_on φ (Icc a b)) (hφg : ∀ x ∈ Ioo a b, g' x ≤ φ x) :\n  g b - g a ≤ ∫ y in a..b, φ y :=\nbegin\n  -- This follows from the version on a closed-open interval (applied to `[t, b)` for `t` close to\n  -- `a`) and a continuity argument.\n  obtain rfl|a_lt_b := hab.eq_or_lt, { simp },\n  set s := {t | g b - g t ≤ ∫ u in t..b, φ u} ∩ Icc a b,\n  have s_closed : is_closed s,\n  { have : continuous_on (λ t, (g b - g t, ∫ u in t..b, φ u)) (Icc a b),\n    { rw ← interval_of_le hab at ⊢ hcont φint,\n      exact (continuous_on_const.sub hcont).prod (continuous_on_primitive_interval_left φint) },\n    simp only [s, inter_comm],\n    exact this.preimage_closed_of_closed is_closed_Icc is_closed_le_prod, },\n  have A : closure (Ioc a b) ⊆ s,\n  { apply s_closed.closure_subset_iff.2,\n    assume t ht,\n    refine ⟨_, ⟨ht.1.le, ht.2⟩⟩,\n    exact sub_le_integral_of_has_deriv_right_of_le_Ico ht.2\n      (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n      (λ x hx, hderiv x ⟨ht.1.trans_le hx.1, hx.2⟩)\n      (φint.mono_set (Icc_subset_Icc ht.1.le le_rfl))\n      (λ x hx, hφg x ⟨ht.1.trans_le hx.1, hx.2⟩) },\n  rw closure_Ioc a_lt_b.ne at A,\n  exact (A (left_mem_Icc.2 hab)).1,\nend\n\n/-- Auxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`. -/\nlemma integral_le_sub_of_has_deriv_right_of_le (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (φint : integrable_on φ (Icc a b)) (hφg : ∀ x ∈ Ioo a b, φ x ≤ g' x) :\n  ∫ y in a..b, φ y ≤ g b - g a :=\nbegin\n  rw ← neg_le_neg_iff,\n  convert sub_le_integral_of_has_deriv_right_of_le hab hcont.neg (λ x hx, (hderiv x hx).neg)\n    φint.neg (λ x hx, neg_le_neg (hφg x hx)),\n  { abel },\n  { simp only [← integral_neg], refl },\nend\n\n/-- Auxiliary lemma in the proof of `integral_eq_sub_of_has_deriv_right_of_le`: real version -/\nlemma integral_eq_sub_of_has_deriv_right_of_le_real (hab : a ≤ b)\n  (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'int : integrable_on g' (Icc a b)) :\n  ∫ y in a..b, g' y = g b - g a :=\nle_antisymm\n  (integral_le_sub_of_has_deriv_right_of_le hab hcont hderiv g'int (λ x hx, le_rfl))\n  (sub_le_integral_of_has_deriv_right_of_le hab hcont hderiv g'int (λ x hx, le_rfl))\n\nvariable {f' : ℝ → E}\n\n/-- **Fundamental theorem of calculus-2**: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`)\n  and has a right derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`,\n  then `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right_of_le (hab : a ≤ b) (hcont : continuous_on f (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at f (f' x) (Ioi x) x)\n  (f'int : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  refine (normed_space.eq_iff_forall_dual_eq ℝ).2 (λ g, _),\n  rw [← g.interval_integral_comp_comm f'int, g.map_sub],\n  exact integral_eq_sub_of_has_deriv_right_of_le_real hab (g.continuous.comp_continuous_on hcont)\n    (λ x hx, g.has_fderiv_at.comp_has_deriv_within_at x (hderiv x hx))\n    (g.integrable_comp ((interval_integrable_iff_integrable_Icc_of_le hab).1 f'int))\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` and\n  has a right derivative at `f' x` for all `x` in `[a, b)`, and `f'` is integrable on `[a, b]` then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_right (hcont : continuous_on f (interval a b))\n  (hderiv : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  cases le_total a b with hab hab,\n  { simp only [interval_of_le, min_eq_left, max_eq_right, hab] at hcont hderiv hint,\n    apply integral_eq_sub_of_has_deriv_right_of_le hab hcont hderiv hint },\n  { simp only [interval_of_ge, min_eq_right, max_eq_left, hab] at hcont hderiv,\n    rw [integral_symm, integral_eq_sub_of_has_deriv_right_of_le hab hcont hderiv hint.symm,\n        neg_sub] }\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is continuous on `[a, b]` (where `a ≤ b`) and\n  has a derivative at `f' x` for all `x` in `(a, b)`, and `f'` is integrable on `[a, b]`, then\n  `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at_of_le (hab : a ≤ b)\n  (hcont : continuous_on f (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nintegral_eq_sub_of_has_deriv_right_of_le hab hcont (λ x hx, (hderiv x hx).has_deriv_within_at) hint\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` has a derivative at `f' x` for all `x` in\n  `[a, b]` and `f'` is integrable on `[a, b]`, then `∫ y in a..b, f' y` equals `f b - f a`. -/\ntheorem integral_eq_sub_of_has_deriv_at\n  (hderiv : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hint : interval_integrable f' volume a b) :\n  ∫ y in a..b, f' y = f b - f a :=\nintegral_eq_sub_of_has_deriv_right (has_deriv_at.continuous_on hderiv)\n  (λ x hx, (hderiv _ (mem_Icc_of_Ioo hx)).has_deriv_within_at) hint\n\ntheorem integral_eq_sub_of_has_deriv_at_of_tendsto (hab : a < b) {fa fb}\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hint : interval_integrable f' volume a b)\n  (ha : tendsto f (𝓝[>] a) (𝓝 fa)) (hb : tendsto f (𝓝[<] b) (𝓝 fb)) :\n  ∫ y in a..b, f' y = fb - fa :=\nbegin\n  set F : ℝ → E := update (update f a fa) b fb,\n  have Fderiv : ∀ x ∈ Ioo a b, has_deriv_at F (f' x) x,\n  { refine λ x hx, (hderiv x hx).congr_of_eventually_eq _,\n    filter_upwards [Ioo_mem_nhds hx.1 hx.2] with _ hy, simp only [F],\n    rw [update_noteq hy.2.ne, update_noteq hy.1.ne'], },\n  have hcont : continuous_on F (Icc a b),\n  { rw [continuous_on_update_iff, continuous_on_update_iff, Icc_diff_right, Ico_diff_left],\n    refine ⟨⟨λ z hz, (hderiv z hz).continuous_at.continuous_within_at, _⟩, _⟩,\n    { exact λ _, ha.mono_left (nhds_within_mono _ Ioo_subset_Ioi_self) },\n    { rintro -,\n      refine (hb.congr' _).mono_left (nhds_within_mono _ Ico_subset_Iio_self),\n      filter_upwards [Ioo_mem_nhds_within_Iio (right_mem_Ioc.2 hab)]\n        with _ hz using (update_noteq hz.1.ne' _ _).symm } },\n  simpa [F, hab.ne, hab.ne'] using integral_eq_sub_of_has_deriv_at_of_le hab.le hcont Fderiv hint\nend\n\n/-- Fundamental theorem of calculus-2: If `f : ℝ → E` is differentiable at every `x` in `[a, b]` and\n  its derivative is integrable on `[a, b]`, then `∫ y in a..b, deriv f y` equals `f b - f a`. -/\ntheorem integral_deriv_eq_sub (hderiv : ∀ x ∈ interval a b, differentiable_at ℝ f x)\n  (hint : interval_integrable (deriv f) volume a b) :\n  ∫ y in a..b, deriv f y = f b - f a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hderiv x hx).has_deriv_at) hint\n\ntheorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f')\n  (hdiff : ∀ x ∈ interval a b, differentiable_at ℝ f x)\n  (hcont : continuous_on f' (interval a b)) :\n  ∫ y in a..b, f' y = f b - f a :=\nbegin\n  rw [← hderiv, integral_deriv_eq_sub hdiff],\n  rw hderiv,\n  exact hcont.interval_integrable\nend\n\n/-!\n### Automatic integrability for nonnegative derivatives\n-/\n\n/-- When the right derivative of a function is nonnegative, then it is automatically integrable. -/\nlemma integrable_on_deriv_right_of_nonneg (hab : a ≤ b) (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_within_at g (g' x) (Ioi x) x)\n  (g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) :\n  integrable_on g' (Ioc a b) :=\nbegin\n  rw integrable_on_Ioc_iff_integrable_on_Ioo,\n  have meas_g' : ae_measurable g' (volume.restrict (Ioo a b)),\n  { apply (ae_measurable_deriv_within_Ioi g _).congr,\n    refine (ae_restrict_mem measurable_set_Ioo).mono (λ x hx, _),\n    exact (hderiv x hx).deriv_within (unique_diff_within_at_Ioi _) },\n  suffices H : ∫⁻ x in Ioo a b, ∥g' x∥₊ ≤ ennreal.of_real (g b - g a),\n    from ⟨meas_g'.ae_strongly_measurable, H.trans_lt ennreal.of_real_lt_top⟩,\n  by_contra' H,\n  obtain ⟨f, fle, fint, hf⟩ :\n    ∃ (f : simple_func ℝ ℝ≥0), (∀ x, f x ≤ ∥g' x∥₊) ∧ ∫⁻ (x : ℝ) in Ioo a b, f x < ∞\n      ∧ ennreal.of_real (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, f x :=\n    exists_lt_lintegral_simple_func_of_lt_lintegral H,\n  let F : ℝ → ℝ := coe ∘ f,\n  have intF : integrable_on F (Ioo a b),\n  { refine ⟨f.measurable.coe_nnreal_real.ae_strongly_measurable, _⟩,\n    simpa only [has_finite_integral, nnreal.nnnorm_eq] using fint },\n  have A : ∫⁻ (x : ℝ) in Ioo a b, f x = ennreal.of_real (∫ x in Ioo a b, F x) :=\n    lintegral_coe_eq_integral _ intF,\n  rw A at hf,\n  have B : ∫ (x : ℝ) in Ioo a b, F x ≤ g b - g a,\n  { rw [← integral_Ioc_eq_integral_Ioo, ← interval_integral.integral_of_le hab],\n    apply integral_le_sub_of_has_deriv_right_of_le hab hcont hderiv _ (λ x hx, _),\n    { rwa integrable_on_Icc_iff_integrable_on_Ioo },\n    { convert nnreal.coe_le_coe.2 (fle x),\n      simp only [real.norm_of_nonneg (g'pos x hx), coe_nnnorm] } },\n  exact lt_irrefl _ (hf.trans_le (ennreal.of_real_le_of_real B)),\nend\n\n/-- When the derivative of a function is nonnegative, then it is automatically integrable,\nIoc version. -/\nlemma integrable_on_deriv_of_nonneg (hab : a ≤ b) (hcont : continuous_on g (Icc a b))\n  (hderiv : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)\n  (g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) :\n  integrable_on g' (Ioc a b) :=\nintegrable_on_deriv_right_of_nonneg hab hcont (λ x hx, (hderiv x hx).has_deriv_within_at) g'pos\n\n/-- When the derivative of a function is nonnegative, then it is automatically integrable,\ninterval version. -/\ntheorem interval_integrable_deriv_of_nonneg (hcont : continuous_on g (interval a b))\n  (hderiv : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_at g (g' x) x)\n  (hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) :\n  interval_integrable g' volume a b :=\nbegin\n  cases le_total a b with hab hab,\n  { simp only [interval_of_le, min_eq_left, max_eq_right, hab, interval_integrable,\n      hab, Ioc_eq_empty_of_le, integrable_on_empty, and_true] at hcont hderiv hpos ⊢,\n    exact integrable_on_deriv_of_nonneg hab hcont hderiv hpos, },\n  { simp only [interval_of_ge, min_eq_right, max_eq_left, hab, interval_integrable,\n      Ioc_eq_empty_of_le, integrable_on_empty, true_and] at hcont hderiv hpos ⊢,\n    exact integrable_on_deriv_of_nonneg hab hcont hderiv hpos }\nend\n\n/-!\n### Integration by parts\n-/\n\ntheorem integral_deriv_mul_eq_sub {u v u' v' : ℝ → ℝ}\n  (hu : ∀ x ∈ interval a b, has_deriv_at u (u' x) x)\n  (hv : ∀ x ∈ interval a b, has_deriv_at v (v' x) x)\n  (hu' : interval_integrable u' volume a b) (hv' : interval_integrable v' volume a b) :\n  ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hu x hx).mul (hv x hx)) $\n  (hu'.mul_continuous_on (has_deriv_at.continuous_on hv)).add\n    (hv'.continuous_on_mul ((has_deriv_at.continuous_on hu)))\n\ntheorem integral_mul_deriv_eq_deriv_mul {u v u' v' : ℝ → ℝ}\n  (hu : ∀ x ∈ interval a b, has_deriv_at u (u' x) x)\n  (hv : ∀ x ∈ interval a b, has_deriv_at v (v' x) x)\n  (hu' : interval_integrable u' volume a b) (hv' : interval_integrable v' volume a b) :\n  ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, v x * u' x :=\nbegin\n  rw [← integral_deriv_mul_eq_sub hu hv hu' hv', ← integral_sub],\n  { exact integral_congr (λ x hx, by simp only [mul_comm, add_sub_cancel']) },\n  { exact ((hu'.mul_continuous_on (has_deriv_at.continuous_on hv)).add\n      (hv'.continuous_on_mul (has_deriv_at.continuous_on hu))) },\n  { exact hu'.continuous_on_mul (has_deriv_at.continuous_on hv) },\nend\n\n/-!\n### Integration by substitution / Change of variables\n-/\n\nsection smul\n\n/--\nChange of variables, general form. If `f` is continuous on `[a, b]` and has\ncontinuous right-derivative `f'` in `(a, b)`, and `g` is continuous on `f '' [a, b]` then we can\nsubstitute `u = f x` to get `∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\n\nWe could potentially slightly weaken the conditions, by not requiring that `f'` and `g` are\ncontinuous on the endpoints of these intervals, but in that case we need to additionally assume that\nthe functions are integrable on that interval.\n-/\ntheorem integral_comp_smul_deriv'' {f f' : ℝ → ℝ} {g : ℝ → E}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g ∘ f) x= ∫ u in f a..f b, g u :=\nbegin\n  have h_cont : continuous_on (λ u, ∫ t in f a..f u, g t) [a, b],\n  { rw [hf.image_interval] at hg,\n    refine (continuous_on_primitive_interval' hg.interval_integrable _).comp hf _,\n    { rw ← hf.image_interval, exact mem_image_of_mem f left_mem_interval },\n    { rw ← hf.image_interval, exact maps_to_image _ _ } },\n  have h_der : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at\n    (λ u, ∫ t in f a..f u, g t) (f' x • ((g ∘ f) x)) (Ioi x) x,\n  { intros x hx,\n    let I := [Inf (f '' [a, b]), Sup (f '' [a, b])],\n    have hI : f '' [a, b] = I := hf.image_interval,\n    have h2x : f x ∈ I, { rw [← hI], exact mem_image_of_mem f (Ioo_subset_Icc_self hx) },\n    have h2g : interval_integrable g volume (f a) (f x),\n    { refine (hg.mono $ _).interval_integrable,\n      exact hf.surj_on_interval left_mem_interval (Ioo_subset_Icc_self hx) },\n    rw [hI] at hg,\n    have h3g : strongly_measurable_at_filter g (𝓝[I] f x) volume :=\n    hg.strongly_measurable_at_filter_nhds_within measurable_set_Icc (f x),\n    haveI : fact (f x ∈ I) := ⟨h2x⟩,\n    have : has_deriv_within_at (λ u, ∫ x in f a..u, g x) (g (f x)) I (f x) :=\n    integral_has_deriv_within_at_right h2g h3g (hg (f x) h2x),\n    refine (this.scomp x ((hff' x hx).Ioo_of_Ioi hx.2) _).Ioi_of_Ioo hx.2,\n    rw ← hI,\n    exact (maps_to_image _ _).mono (Ioo_subset_Icc_self.trans $ Icc_subset_Icc_left hx.1.le)\n      subset.rfl },\n  have h_int : interval_integrable (λ (x : ℝ), f' x • (g ∘ f) x) volume a b :=\n  (hf'.smul (hg.comp hf $ subset_preimage_image f _)).interval_integrable,\n  simp_rw [integral_eq_sub_of_has_deriv_right h_cont h_der h_int, integral_same, sub_zero],\nend\n\n/--\nChange of variables. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous on `f '' [a, b]`, then we can substitute `u = f x` to get\n`∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\nCompared to `interval_integral.integral_comp_smul_deriv` we only require that `g` is continuous on\n`f '' [a, b]`.\n-/\ntheorem integral_comp_smul_deriv' {f f' : ℝ → ℝ} {g : ℝ → E}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g ∘ f) x = ∫ x in f a..f b, g x :=\nintegral_comp_smul_deriv'' (λ x hx, (h x hx).continuous_at.continuous_within_at)\n  (λ x hx, (h x $ Ioo_subset_Icc_self hx).has_deriv_within_at) h' hg\n\n/--\nChange of variables, most common version. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous, then we can substitute `u = f x` to get\n`∫ x in a..b, f' x • (g ∘ f) x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_smul_deriv {f f' : ℝ → ℝ} {g : ℝ → E}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous g) :\n  ∫ x in a..b, f' x • (g ∘ f) x = ∫ x in f a..f b, g x :=\nintegral_comp_smul_deriv' h h' hg.continuous_on\n\ntheorem integral_deriv_comp_smul_deriv' {f f' : ℝ → ℝ} {g g' : ℝ → E}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g [f a, f b])\n  (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), has_deriv_within_at g (g' x) (Ioi x) x)\n  (hg' : continuous_on g' (f '' [a, b])) :\n  ∫ x in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a :=\nbegin\n  rw [integral_comp_smul_deriv'' hf hff' hf' hg',\n  integral_eq_sub_of_has_deriv_right hg hgg' (hg'.mono _).interval_integrable],\n  exact intermediate_value_interval hf\nend\n\ntheorem integral_deriv_comp_smul_deriv {f f' : ℝ → ℝ} {g g' : ℝ → E}\n  (hf : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hg : ∀ x ∈ interval a b, has_deriv_at g (g' (f x)) (f x))\n  (hf' : continuous_on f' (interval a b)) (hg' : continuous g') :\n  ∫ x in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a :=\nintegral_eq_sub_of_has_deriv_at (λ x hx, (hg x hx).scomp x $ hf x hx)\n  (hf'.smul (hg'.comp_continuous_on $ has_deriv_at.continuous_on hf)).interval_integrable\n\nend smul\nsection mul\n\n/--\nChange of variables, general form for scalar functions. If `f` is continuous on `[a, b]` and has\ncontinuous right-derivative `f'` in `(a, b)`, and `g` is continuous on `f '' [a, b]` then we can\nsubstitute `u = f x` to get `∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_mul_deriv'' {f f' g : ℝ → ℝ}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u :=\nby simpa [mul_comm] using integral_comp_smul_deriv'' hf hff' hf' hg\n\n/--\nChange of variables. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous on `f '' [a, b]`, then we can substitute `u = f x` to get\n`∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\nCompared to `interval_integral.integral_comp_mul_deriv` we only require that `g` is continuous on\n`f '' [a, b]`.\n-/\ntheorem integral_comp_mul_deriv' {f f' g : ℝ → ℝ}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous_on g (f '' [a, b])) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ x in f a..f b, g x :=\nby simpa [mul_comm] using integral_comp_smul_deriv' h h' hg\n\n/--\nChange of variables, most common version. If `f` is has continuous derivative `f'` on `[a, b]`,\nand `g` is continuous, then we can substitute `u = f x` to get\n`∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.\n-/\ntheorem integral_comp_mul_deriv {f f' g : ℝ → ℝ}\n  (h : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (h' : continuous_on f' (interval a b)) (hg : continuous g) :\n  ∫ x in a..b, (g ∘ f) x * f' x = ∫ x in f a..f b, g x :=\nintegral_comp_mul_deriv' h h' hg.continuous_on\n\ntheorem integral_deriv_comp_mul_deriv' {f f' g g' : ℝ → ℝ}\n  (hf : continuous_on f [a, b])\n  (hff' : ∀ x ∈ Ioo (min a b) (max a b), has_deriv_within_at f (f' x) (Ioi x) x)\n  (hf' : continuous_on f' [a, b])\n  (hg : continuous_on g [f a, f b])\n  (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), has_deriv_within_at g (g' x) (Ioi x) x)\n  (hg' : continuous_on g' (f '' [a, b])) :\n  ∫ x in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a :=\nby simpa [mul_comm] using integral_deriv_comp_smul_deriv' hf hff' hf' hg hgg' hg'\n\ntheorem integral_deriv_comp_mul_deriv {f f' g g' : ℝ → ℝ}\n  (hf : ∀ x ∈ interval a b, has_deriv_at f (f' x) x)\n  (hg : ∀ x ∈ interval a b, has_deriv_at g (g' (f x)) (f x))\n  (hf' : continuous_on f' (interval a b)) (hg' : continuous g') :\n  ∫ x in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a :=\nby simpa [mul_comm] using integral_deriv_comp_smul_deriv hf hg hf' hg'\n\nend mul\n\nend interval_integral\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/measure_theory/integral/interval_integral.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3032773402500584}}
{"text": "/-\nCopyright (c) 2022 Joël Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joël Riou\n-/\n\nimport for_mathlib.category_theory.functor_misc\nimport category_theory.localization.predicate\nimport for_mathlib.category_theory.morphism_property_misc\n\nnoncomputable theory\n\nopen category_theory.category category_theory\n\nnamespace category_theory\n\nstructure Comm_sq {C₁ C₂ D₁ D₂ : Type*} [category C₁] [category C₂] [category D₁] [category D₂]\n  (F : C₁ ⥤ C₂) (G₁ : C₁ ⥤ D₁) (G₂ : C₂ ⥤ D₂) (F' : D₁ ⥤ D₂) :=\n(iso [] : G₁ ⋙ F' ≅ F ⋙ G₂)\n\nnamespace Comm_sq\n\nvariables {C₁ C₂ C₃ C₄ D₁ D₂ D₃ D₄ : Type*}\n  [category C₁] [category C₂] [category C₃] [category C₄]\n  [category D₁] [category D₂] [category D₃] [category D₄]\n\n@[simps]\ndef horiz_refl {C D : Type*} [category C] [category D]\n  (F : C ⥤ D) : Comm_sq (𝟭 C) F F (𝟭 D) := ⟨iso.refl _⟩\n\ndef horiz_comp\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃}\n  {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂} {G : C₂ ⥤ C₃} {G' : D₂ ⥤ D₃}\n  (H₁₂ : Comm_sq F L₁ L₂ F') (H₂₃ : Comm_sq G L₂ L₃ G') :\n  Comm_sq (F ⋙ G) L₁ L₃ (F' ⋙ G') :=\n⟨calc L₁ ⋙ F' ⋙ G' ≅ (L₁ ⋙ F') ⋙ G' : (functor.associator _ _ _).symm\n  ... ≅ (F ⋙ L₂) ⋙ G' : iso_whisker_right H₁₂.iso _\n  ... ≅ F ⋙ (L₂ ⋙ G') : functor.associator _ _ _\n  ... ≅ F ⋙ (G ⋙ L₃) : iso_whisker_left _ H₂₃.iso\n  ... ≅ (F ⋙ G) ⋙ L₃ : (functor.associator _ _ _).symm⟩\n\n@[simp]\nlemma horiz_comp_iso_hom_app\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃}\n  {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂} {G : C₂ ⥤ C₃} {G' : D₂ ⥤ D₃}\n  (H₁₂ : Comm_sq F L₁ L₂ F') (H₂₃ : Comm_sq G L₂ L₃ G') (X₁ : C₁):\n  (H₁₂.horiz_comp H₂₃).iso.hom.app X₁ =\n    G'.map (H₁₂.iso.hom.app X₁) ≫ H₂₃.iso.hom.app (F.obj X₁) :=\nby { dsimp [horiz_comp], simp only [id_comp, comp_id], }\n\n@[simp]\nlemma horiz_comp_iso_inv_app\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃}\n  {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂} {G : C₂ ⥤ C₃} {G' : D₂ ⥤ D₃}\n  (H₁₂ : Comm_sq F L₁ L₂ F') (H₂₃ : Comm_sq G L₂ L₃ G') (X₁ : C₁):\n  (H₁₂.horiz_comp H₂₃).iso.inv.app X₁ =\n    H₂₃.iso.inv.app (F.obj X₁) ≫ G'.map (H₁₂.iso.inv.app X₁) :=\nby { dsimp [horiz_comp], simp only [comp_id, id_comp], }\n\n@[simp]\nlemma horiz_comp_refl_iso {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂}\n  (H : Comm_sq F L₁ L₂ F') : (H.horiz_comp (horiz_refl L₂)).iso = H.iso :=\nby { ext X, rw horiz_comp_iso_hom_app, apply comp_id, }\n\n@[simp]\nlemma refl_horiz_comp_iso {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂}\n  (H : Comm_sq F L₁ L₂ F') : ((horiz_refl L₁).horiz_comp H).iso = H.iso :=\nby { ext X, rw horiz_comp_iso_hom_app, dsimp, rw [F'.map_id, id_comp], }\n\nlemma horiz_comp_assoc_iso\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃} {L₄ : C₄ ⥤ D₄}\n  {F : C₁ ⥤ C₂} {F' : D₁ ⥤ D₂} {G : C₂ ⥤ C₃} {G' : D₂ ⥤ D₃} {K : C₃ ⥤ C₄} {K' : D₃ ⥤ D₄}\n  (H₁₂ : Comm_sq F L₁ L₂ F') (H₂₃ : Comm_sq G L₂ L₃ G') (H₃₄ : Comm_sq K L₃ L₄ K') :\n  ((H₁₂.horiz_comp H₂₃).horiz_comp H₃₄).iso = (H₁₂.horiz_comp (H₂₃.horiz_comp H₃₄)).iso :=\nby { ext X, simpa only [horiz_comp_iso_hom_app, functor.map_comp, assoc, functor.comp_map], }\n\nend Comm_sq\n\nvariables {C D : Type*} [category C] [category D]\nvariables (L : C ⥤ D) (W : morphism_property C)\nvariables (E : Type*) [category E]\n\nnamespace functor\n\n/-class is_localization : Prop :=\n(inverts : W.is_inverted_by L)\n(nonempty_is_equivalence : nonempty (is_equivalence (localization.construction.lift L inverts)))\n\ninstance Q_is_localization : W.Q.is_localization W :=\n{ inverts := W.Q_inverts,\n  nonempty_is_equivalence := begin\n    suffices : localization.construction.lift W.Q W.Q_inverts = 𝟭 _,\n    { apply nonempty.intro, rw this, apply_instance, },\n    apply localization.construction.uniq,\n    simpa only [localization.construction.fac],\n  end, }-/\n\nend functor\n\nnamespace localization\n\n/-structure strict_universal_property_fixed_target :=\n(inverts : W.is_inverted_by L)\n(lift : Π (F : C ⥤ E) (hF : W.is_inverted_by F), D ⥤ E)\n(fac : Π (F : C ⥤ E) (hF : W.is_inverted_by F), L ⋙ lift F hF = F)\n(uniq : Π (F₁ F₂ : D ⥤ E) (h : L ⋙ F₁ = L ⋙ F₂), F₁ = F₂)\n\ndef strict_universal_property_fixed_target.for_Q : strict_universal_property_fixed_target W.Q W E :=\n{ inverts := W.Q_inverts,\n  lift := construction.lift,\n  fac := construction.fac,\n  uniq := construction.uniq, }\n\ndef strict_universal_property_fixed_target.for_id (hW : W ⊆ morphism_property.isomorphisms C):\n  strict_universal_property_fixed_target (𝟭 C) W E :=\n{ inverts := λ X Y f hf, hW f hf,\n  lift := λ F hF, F,\n  fac := λ F hF, by { cases F, refl, },\n  uniq := λ F₁ F₂ eq, by { cases F₁, cases F₂, exact eq, }, }-/\n\nend localization\n\nnamespace functor\n/-\nvariables (h₁ : localization.strict_universal_property_fixed_target L W D)\n  (h₂ : localization.strict_universal_property_fixed_target L W W.localization)\n\nnamespace is_localization.mk'\n\nlemma unit_eq :\n  𝟭 W.localization = localization.construction.lift L h₁.inverts ⋙ h₂.lift W.Q W.Q_inverts :=\nbegin\n  apply localization.construction.uniq,\n  rw [← functor.assoc, localization.construction.fac, h₂.fac, functor.comp_id],\nend\n\nlemma counit_eq :\n  h₂.lift W.Q W.Q_inverts ⋙ localization.construction.lift L h₁.inverts = 𝟭 D :=\nbegin\n  apply h₁.uniq,\n  rw [← functor.assoc, h₂.fac, localization.construction.fac, functor.comp_id],\nend\n\ndef equivalence : W.localization ≌ D :=\n{ functor := localization.construction.lift L h₁.inverts,\n  inverse := h₂.lift W.Q W.Q_inverts,\n  unit_iso := eq_to_iso (unit_eq L W h₁ h₂),\n  counit_iso := eq_to_iso (counit_eq L W h₁ h₂),\n  functor_unit_iso_comp' := λ X, by simpa only [eq_to_iso.hom, eq_to_hom_app, eq_to_hom_map,\n    eq_to_hom_trans, eq_to_hom_refl], }\n\ndef equivalence_obj_equiv' : W.localization ≃ D :=\n{ to_fun := (is_localization.mk'.equivalence L W h₁ h₂).functor.obj,\n  inv_fun := (is_localization.mk'.equivalence L W h₁ h₂).inverse.obj,\n  left_inv := congr_obj (unit_eq L W h₁ h₂).symm,\n  right_inv := congr_obj (counit_eq L W h₁ h₂), }\n\nlemma obj_bijective : function.bijective L.obj :=\n((localization.construction.obj_equiv W).trans (equivalence_obj_equiv' L W h₁ h₂)).bijective\n\nend is_localization.mk'\n\nlemma is_localization.mk' :\n  is_localization L W :=\n{ inverts := h₁.inverts,\n  nonempty_is_equivalence :=\n    nonempty.intro (is_equivalence.of_equivalence (is_localization.mk'.equivalence L W h₁ h₂)), }\n-/\nend functor\n\nnamespace localization\n\nvariable [L.is_localization W]\ninclude L W\n\n/-lemma inverts : W.is_inverted_by L := (as_localization _ _).inverts-/\n\n@[simps]\ndef iso_of_hom' {X Y : C} (f : X ⟶ Y) (hf : W f) : L.obj X ≅ L.obj Y :=\nbegin\n  haveI : is_iso (L.map f) := inverts L W f hf,\n  exact as_iso (L.map f),\nend\n\n/-instance : is_equivalence (localization.construction.lift L (inverts L W)) :=\n(as_localization L W).nonempty_is_equivalence.some\n\ndef equivalence_from_model : W.localization ≌ D :=\n(localization.construction.lift L (inverts L W)).as_equivalence\n\ndef Q_comp_equivalence_from_model_functor_iso :\n  W.Q ⋙ (equivalence_from_model L W).functor ≅ L := eq_to_iso (construction.fac _ _)\n\ndef comp_equivalence_from_model_inverse_iso :\n  L ⋙ (equivalence_from_model L W).inverse ≅ W.Q :=\nbegin\n  let α := Q_comp_equivalence_from_model_functor_iso L W,\n  calc L ⋙ (equivalence_from_model L W).inverse ≅ _ : iso_whisker_right α.symm _\n  ... ≅ W.Q ⋙ ((equivalence_from_model L W).functor ⋙ (equivalence_from_model L W).inverse) :\n    functor.associator _ _ _\n  ... ≅ W.Q ⋙ 𝟭 _ : iso_whisker_left _ ((equivalence_from_model L W).unit_iso.symm)\n  ... ≅ W.Q : functor.right_unitor _,\nend\n\nlemma ess_surj : ess_surj L :=\n⟨λ X, ⟨(construction.obj_equiv W).inv_fun ((equivalence_from_model L W).inverse.obj X),\n    nonempty.intro ((Q_comp_equivalence_from_model_functor_iso L W).symm.app _ ≪≫\n    (equivalence_from_model L W).counit_iso.app X)⟩⟩\n\ndef whiskering_left_functor : (D ⥤ E) ⥤ W.functors_inverting E :=\nfull_subcategory.lift _ ((whiskering_left _ _ E).obj L)\n  (morphism_property.is_inverted_by.of_comp W L (inverts L W ))\n\ninstance : is_equivalence (whiskering_left_functor L W E) :=\nbegin\n  refine is_equivalence.of_iso _ (is_equivalence.of_equivalence\n    ((equivalence.congr_left (equivalence_from_model L W).symm).trans\n    (construction.whiskering_left_equivalence W E))),\n  refine nat_iso.of_components (λ F, eq_to_iso begin\n    ext,\n    change (W.Q ⋙ (localization.construction.lift L (inverts L W))) ⋙ F = L ⋙ F,\n    rw construction.fac,\n  end)\n  (λ F₁ F₂ τ, begin\n    ext X,\n    dsimp [equivalence_from_model, whisker_left, construction.whiskering_left_equivalence,\n      construction.whiskering_left_equivalence.functor, whiskering_left_functor,\n      morphism_property.Q],\n    erw [nat_trans.comp_app, nat_trans.comp_app, eq_to_hom_app, eq_to_hom_app,\n      eq_to_hom_refl, eq_to_hom_refl, comp_id, id_comp],\n    all_goals\n    { change (W.Q ⋙ (localization.construction.lift L (inverts L W))) ⋙ _ = L ⋙ _,\n      rw construction.fac, },\n  end),\nend\n\ndef functor_equivalence : (D ⥤ E) ≌ (W.functors_inverting E) :=\n(whiskering_left_functor L W E).as_equivalence-/\n\nsection\n\nvariables (E)\n\n--include hL\n/-@[nolint unused_arguments]\ndef whiskering_left_functor' :\n  (D ⥤ E) ⥤ (C ⥤ E) := (whiskering_left C D E).obj L\n\nlemma whiskering_left_functor'_eq :\n  whiskering_left_functor' L W E =\n    localization.whiskering_left_functor L W E ⋙ induced_functor _ := rfl\n\nvariable {E}\n\n@[simp]\ndef whiskering_left_functor'_obj\n  (F : D ⥤ E) : (whiskering_left_functor' L W E).obj F = L ⋙ F := rfl\n\ninstance : full (whiskering_left_functor' L W E) :=\nby { rw whiskering_left_functor'_eq, apply_instance, }\n\ninstance : faithful (whiskering_left_functor' L W E) :=\nby { rw whiskering_left_functor'_eq, apply_instance, }\n\nlemma nat_trans_ext {F₁ F₂ : D ⥤ E} (τ τ' : F₁ ⟶ F₂)\n  (h : ∀ (X : C), τ.app (L.obj X) = τ'.app (L.obj X)) : τ = τ' :=\nbegin\n  haveI : category_theory.ess_surj L := ess_surj L W,\n  ext Y,\n  rw [← cancel_epi (F₁.map (L.obj_obj_preimage_iso Y).hom), τ.naturality, τ'.naturality, h],\nend\n\n/-- When `L : C ⥤ D` is a localization functor for `W : morphism_property C` and\n`F : C ⥤ E` is a functor, we shall say that `F' : D ⥤ E` lifts `F` if the obvious diagram\nis commutative up to an isomorphism. -/\nclass lifting (F : C ⥤ E) (F' : D ⥤ E) := (iso [] : L ⋙ F' ≅ F)\n\nsection\n\ndef lift_nat_trans (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E) [lifting L W F₁ F₁']\n  [h₂ : lifting L W F₂ F₂'] (τ : F₁ ⟶ F₂) : F₁' ⟶ F₂' :=\n(whiskering_left_functor' L W E).preimage ((lifting.iso L W F₁ F₁').hom ≫ τ ≫ (lifting.iso L W F₂ F₂').inv)\n\n@[simp]\nlemma lift_nat_trans_app (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E) [lifting L W F₁ F₁']\n  [lifting L W F₂ F₂'] (τ : F₁ ⟶ F₂) (X : C) :\n  (lift_nat_trans L W F₁ F₂ F₁' F₂' τ).app (L.obj X) =\n    (lifting.iso L W F₁ F₁').hom.app X ≫ τ.app X ≫ ((lifting.iso L W F₂ F₂')).inv.app X :=\ncongr_app (functor.image_preimage (whiskering_left_functor' L W E) _) X\n\n@[simp, reassoc]\nlemma comp_lift_nat_trans (F₁ F₂ F₃ : C ⥤ E) (F₁' F₂' F₃' : D ⥤ E)\n  [h₁ : lifting L W F₁ F₁'] [h₂ : lifting L W F₂ F₂'] [h₃ : lifting L W F₃ F₃']\n  (τ : F₁ ⟶ F₂) (τ' : F₂ ⟶ F₃) :\n  lift_nat_trans L W F₁ F₂ F₁' F₂' τ ≫ lift_nat_trans L W F₂ F₃ F₂' F₃' τ' =\n  lift_nat_trans L W F₁ F₃ F₁' F₃' (τ ≫ τ') :=\nnat_trans_ext L W _ _\n  (λ X, by simp only [nat_trans.comp_app, lift_nat_trans_app, assoc, iso.inv_hom_id_app_assoc])\n\n@[simp]\nlemma lift_nat_trans_id (F : C ⥤ E) (F' : D ⥤ E) [h : lifting L W F F'] :\n  lift_nat_trans L W F F F' F' (𝟙 F) = 𝟙 F' :=\nnat_trans_ext L W _ _\n  (λ X, by simpa only [lift_nat_trans_app, nat_trans.id_app, id_comp, iso.hom_inv_id_app])\n\n@[simps]\ndef lift_nat_iso (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E)\n  [h₁ : lifting L W F₁ F₁'] [h₂ : lifting L W F₂ F₂']\n  (e : F₁ ≅ F₂) : F₁' ≅ F₂' :=\n{ hom := lift_nat_trans L W F₁ F₂ F₁' F₂' e.hom,\n  inv := lift_nat_trans L W F₂ F₁ F₂' F₁' e.inv, }\n\nend-/\n\nvariable {E}\n\nnamespace lifting\n\ndef uniq (F : C ⥤ E) (F'₁ F'₂ : D ⥤ E) [lifting L W F F'₁] [lifting L W F F'₂] :\n  F'₁ ≅ F'₂ := lift_nat_iso L W F F F'₁ F'₂ (iso.refl F)\n\n@[simp]\nlemma uniq_hom_app (F : C ⥤ E) (F'₁ F'₂ : D ⥤ E) [lifting L W F F'₁] [lifting L W F F'₂]\n  (X : C) : (uniq L W F F'₁ F'₂).hom.app (L.obj X) = (iso L W F F'₁).hom.app X ≫ (iso L W F F'₂).inv.app X :=\nbegin\n  dsimp only [uniq],\n  simp only [lift_nat_iso_hom, iso.refl_hom, lift_nat_trans_app, nat_trans.id_app, id_comp],\nend\n\n@[simp]\nlemma uniq_inv_app (F : C ⥤ E) (F'₁ F'₂ : D ⥤ E) [lifting L W F F'₁] [lifting L W F F'₂]\n  (X : C) : (uniq L W F F'₁ F'₂).inv.app (L.obj X) = (iso L W F F'₂).hom.app X ≫ (iso L W F F'₁).inv.app X :=\nbegin\n  dsimp only [uniq],\n  simp only [lift_nat_iso_inv, iso.refl_inv, lift_nat_trans_app, nat_trans.id_app, id_comp],\nend\n\nvariables (L W)\n\n/-@[simps]\ninstance comp_right {E' : Type*} [category E'] (F : C ⥤ E) (F' : D ⥤ E) [lifting L W F F']\n  (G : E ⥤ E') : lifting L W (F ⋙ G) (F' ⋙ G) :=\n⟨iso_whisker_right (iso L W F F') G⟩\n\n@[simps]\ninstance id : lifting L W L (𝟭 D) :=\n⟨functor.right_unitor L⟩\n\n@[simps]\ndef of_isos {F₁ F₂ : C ⥤ E} {F'₁ F'₂ : D ⥤ E} (e : F₁ ≅ F₂) (e' : F'₁ ≅ F'₂)\n  [lifting L W F₁ F'₁] : lifting L W F₂ F'₂ :=\n⟨iso_whisker_left L e'.symm ≪≫ iso L W F₁ F'₁ ≪≫ e⟩-/\n\nomit L\n\ninstance (F : C ⥤ D) (hF : W.is_inverted_by F) : lifting W.Q W F (construction.lift F hF) :=\n⟨eq_to_iso (construction.fac F hF)⟩\n\nend lifting\n\nsection\n\nomit L W\n\nvariables {C₁ C₂ D₁ D₂ : Type*} [category C₁] [category C₂] [category D₁]\n  [category D₂] {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {F G : C₁ ⥤ C₂} {F' G' : D₁ ⥤ D₂}\n  (hF : Comm_sq F L₁ L₂ F') (hG : Comm_sq G L₁ L₂ G') (W₁ : morphism_property C₁)\n  [L₁.is_localization W₁] (τ : F ⟶ G)\n\ninclude hF hG W₁ τ\n\ndef lift_nat_trans' : F' ⟶ G' :=\nbegin\n  letI : lifting L₁ W₁ (F ⋙ L₂) F' := ⟨hF.iso⟩,\n  letI : lifting L₁ W₁ (G ⋙ L₂) G' := ⟨hG.iso⟩,\n  exact lift_nat_trans L₁ W₁ (F ⋙ L₂) (G ⋙ L₂) F' G' (τ ◫ (𝟙 L₂)),\nend\n\n@[simp]\nlemma lift_nat_trans'_app (X₁ : C₁) : (lift_nat_trans' hF hG W₁ τ).app (L₁.obj X₁) =\n  hF.iso.hom.app X₁ ≫ L₂.map (τ.app X₁) ≫ hG.iso.inv.app X₁ :=\nbegin\n  dsimp only [lift_nat_trans'],\n  rw lift_nat_trans_app,\n  dsimp,\n  rw id_comp,\nend\n\nomit hG τ\n\n@[simp]\nlemma lift_nat_trans'_id : lift_nat_trans' hF hF W₁ (𝟙 F) = 𝟙 F' :=\nnat_trans_ext L₁ W₁ _ _ (λ X, begin\n  rw [lift_nat_trans'_app, nat_trans.id_app, L₂.map_id],\n  dsimp,\n  rw id_comp,\n  apply iso.hom_inv_id_app,\nend)\n\nomit hF\n\n@[simp, reassoc]\nlemma comp_lift_nat_trans' {F₁ F₂ F₃ : C₁ ⥤ C₂} {F₁' F₂' F₃' : D₁ ⥤ D₂}\n  (H₁ : Comm_sq F₁ L₁ L₂ F₁') (H₂ : Comm_sq F₂ L₁ L₂ F₂') (H₃ : Comm_sq F₃ L₁ L₂ F₃')\n  (τ : F₁ ⟶ F₂) (τ' : F₂ ⟶ F₃) :\n  lift_nat_trans' H₁ H₂ W₁ τ ≫ lift_nat_trans' H₂ H₃ W₁ τ' =\n    lift_nat_trans' H₁ H₃ W₁ (τ ≫ τ') :=\nnat_trans_ext L₁ W₁ _ _ (λ X, by simp only [nat_trans.comp_app, lift_nat_trans'_app,\n  assoc, iso.inv_hom_id_app_assoc, functor.map_comp])\n\ninclude hF hG\n\n@[simps]\ndef lift_nat_iso' (e : F ≅ G) : F' ≅ G' :=\n{ hom := lift_nat_trans' hF hG W₁ e.hom,\n  inv := lift_nat_trans' hG hF W₁ e.inv, }\n\nend\n\nsection\n\nomit L W\n\nvariables {C₁ C₂ C₃ D₁ D₂ D₃ : Type*} [category C₁] [category C₂] [category C₃]\n  [category D₁] [category D₂] [category D₃]\n  {F₁ G₁ : C₁ ⥤ C₂} {F₂ G₂ : C₂ ⥤ C₃}\n  {F'₁ G'₁ : D₁ ⥤ D₂} {F'₂ G'₂ : D₂ ⥤ D₃}\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃}\n  (H₁₂ : Comm_sq F₁ L₁ L₂ F'₁) (H₂₃ : Comm_sq F₂ L₂ L₃ F'₂)\n  (K₁₂ : Comm_sq G₁ L₁ L₂ G'₁) (K₂₃ : Comm_sq G₂ L₂ L₃ G'₂)\n  (W₁ : morphism_property C₁) (W₂ : morphism_property C₂)\n  [L₁.is_localization W₁] [L₂.is_localization W₂]\n  (τ : F₁ ⟶ G₁) (τ' : F₂ ⟶ G₂)\n\nlemma hcomp_lift_nat_trans' :\n  lift_nat_trans' H₁₂ K₁₂ W₁ τ ◫ lift_nat_trans' H₂₃ K₂₃ W₂ τ' =\n    lift_nat_trans' (H₁₂.horiz_comp H₂₃) (K₁₂.horiz_comp K₂₃) W₁ (τ ◫ τ') :=\nnat_trans_ext L₁ W₁ _ _ (λ X, begin\n  simp only [lift_nat_trans'_app, nat_trans.hcomp_app, assoc, G'₂.map_comp],\n  rw ← nat_trans.naturality_assoc,\n  erw lift_nat_trans'_app,\n  simp only [assoc, Comm_sq.horiz_comp_iso_hom_app, Comm_sq.horiz_comp_iso_inv_app,\n    functor.map_comp],\n  erw ← K₂₃.iso.inv.naturality_assoc,\n  refl,\nend)\n\nend\n\nvariables {W E}\n\n/-def lift (F : C ⥤ E) (hF : W.is_inverted_by F) (L : C ⥤ D) [hL : L.is_localization W] :\n  D ⥤ E :=\n(functor_equivalence L W E).inverse.obj ⟨F, hF⟩\n\ninstance lift_is_lifting (F : C ⥤ E) (hF : W.is_inverted_by F) (L : C ⥤ D)\n  [hL : L.is_localization W] : lifting L W F (lift F hF L) :=\n⟨(induced_functor _).map_iso ((functor_equivalence L W E).counit_iso.app ⟨F, hF⟩)⟩\n\n@[simps]\ndef fac (F : C ⥤ E) (hF : W.is_inverted_by F) (L : C ⥤ D) [hL : L.is_localization W] :\n  L ⋙ lift F hF L ≅ F :=\nlifting.iso _ W _ _-/\n\nend\n\nsection\n\nvariables {D₁ D₂ : Type*} [category D₁] [category D₂] (L₁ : C ⥤ D₁) (L₂ : C ⥤ D₂)\n  [h₁ : L₁.is_localization W] [h₂ : L₂.is_localization W]\n\ninclude h₁ h₂\nomit L\n\ndef uniq_equivalence : D₁ ≌ D₂ :=\n(equivalence_from_model L₁ W).symm.trans (equivalence_from_model L₂ W)\n\ndef comp_uniq_equivalence_functor_iso :\n  L₁ ⋙ (uniq_equivalence W L₁ L₂).functor ≅ L₂ :=\ncalc L₁ ⋙ (uniq_equivalence W L₁ L₂).functor ≅ (L₁ ⋙\n  (equivalence_from_model L₁ W).inverse) ⋙ (equivalence_from_model L₂ W).functor : by refl\n... ≅ W.Q ⋙ (equivalence_from_model L₂ W).functor :\n  iso_whisker_right (comp_equivalence_from_model_inverse_iso L₁ W) _\n... ≅ L₂ : Q_comp_equivalence_from_model_functor_iso L₂ W\n\nend\n\nend localization\n\nnamespace functor\n\nnamespace is_localization\n\ndef of_equivalence {E : Type*} [category E] (L' : C ⥤ E) (eq : D ≌ E)\n  [L.is_localization W] (e : L ⋙ eq.functor ≅ L') : L'.is_localization W :=\nbegin\n  have h : W.is_inverted_by L',\n  { rw ← morphism_property.is_inverted_by.iff_of_iso W e,\n    exact morphism_property.is_inverted_by.of_comp W L (localization.inverts L W) eq.functor, },\n  let F₁ := localization.construction.lift L (localization.inverts L W),\n  let F₂ := localization.construction.lift L' h,\n  letI : localization.lifting W.Q W (L ⋙ eq.functor) F₂ :=\n    localization.lifting.of_isos W.Q W e.symm (iso.refl F₂),\n  let e : F₁ ⋙ eq.functor ≅ F₂ := localization.lifting.uniq W.Q W (L ⋙ eq.functor) _ _,\n  exact\n  { inverts := h,\n    nonempty_is_equivalence := nonempty.intro (is_equivalence.of_iso e infer_instance) },\nend\n\n/-def of_iso {L₁ L₂ : C ⥤ D} (e : L₁ ≅ L₂) [L₁.is_localization W] : L₂.is_localization W :=\nbegin\n  have h : W.is_inverted_by L₂ := λ X Y f hf,\n    by simpa only [is_iso_map_iff_of_nat_iso e.symm] using localization.inverts L₁ W f hf,\n  let F₁ := localization.construction.lift L₁ (localization.inverts L₁ W),\n  let F₂ := localization.construction.lift L₂ h,\n  haveI : localization.lifting W.Q W L₁ F₂ :=\n    localization.lifting.of_isos W.Q W e.symm (iso.refl F₂),\n  exact\n  { inverts := h,\n    nonempty_is_equivalence := nonempty.intro\n      (is_equivalence.of_iso (localization.lifting.uniq W.Q W L₁ F₁ F₂) infer_instance), },\nend-/\n\nend is_localization\n\nend functor\n\nnamespace localization\n\nlemma id_is_localization (hW : W ⊆ morphism_property.isomorphisms C):\n  (𝟭 C).is_localization W :=\nfunctor.is_localization.mk' _ _\n  (strict_universal_property_fixed_target_id _ _ hW)\n  (strict_universal_property_fixed_target_id _ _ hW)\n\n\ninstance identity_functor_is_localization' :\n  (𝟭 C).is_localization (morphism_property.isomorphisms C) :=\nfunctor.is_localization.mk' _ _\n  (strict_universal_property_fixed_target_id _ _ (λ X Y f hf, hf))\n  (strict_universal_property_fixed_target_id _ _ (λ X Y f hf, hf))\n\nend localization\n\nnamespace localization\n\nvariables {C₁ C₂ C₃ D₁ D₂ D₃ : Type*} [category C₁] [category C₂] [category C₃]\n  [category D₁] [category D₂] [category D₃]\n  {L₁ : C₁ ⥤ D₁} {L₂ : C₂ ⥤ D₂} {L₃ : C₃ ⥤ D₃}\n  {F₁₂ : C₁ ⥤ C₂} {F₂₃ : C₂ ⥤ C₃} {F'₁₂ : D₁ ⥤ D₂} {F'₂₃ : D₂ ⥤ D₃}\n  {F₁₃ : C₁ ⥤ C₃} {F'₁₃ : D₁ ⥤ D₃}\n  {G₁₂ : C₁ ⥤ C₂} {G₂₃ : C₂ ⥤ C₃} {G'₁₂ : D₁ ⥤ D₂} {G'₂₃ : D₂ ⥤ D₃}\n  {G₁₃ : C₁ ⥤ C₃} {G'₁₃ : D₁ ⥤ D₃}\n  (H₁₂ : Comm_sq F₁₂ L₁ L₂ F'₁₂) (H₂₃ : Comm_sq F₂₃ L₂ L₃ F'₂₃)\n  (H₁₃ : Comm_sq F₁₃ L₁ L₃ F'₁₃)\n  (K₁₂ : Comm_sq G₁₂ L₁ L₂ G'₁₂) (K₂₃ : Comm_sq G₂₃ L₂ L₃ G'₂₃)\n  (K₁₃ : Comm_sq G₁₃ L₁ L₃ G'₁₃)\n  (e : F₁₂ ⋙ F₂₃ ≅ F₁₃)\n  (e' : G₁₂ ⋙ G₂₃ ≅ G₁₃)\n  (W₁ : morphism_property C₁) (W₂ : morphism_property C₂)\n  [L₁.is_localization W₁]\n\ninclude H₁₂ H₂₃ H₁₃ e W₁ W₂\n\ndef lifting_comp_iso : F'₁₂ ⋙ F'₂₃ ≅ F'₁₃ :=\nbegin\n  letI : lifting L₁ W₁ (F₁₃ ⋙ L₃) F'₁₃ := ⟨H₁₃.iso⟩,\n  letI : lifting L₁ W₁ (F₁₃ ⋙ L₃) (F'₁₂ ⋙ F'₂₃) := ⟨(H₁₂.horiz_comp H₂₃).iso ≪≫ iso_whisker_right e _⟩,\n  exact lifting.uniq L₁ W₁ (F₁₃ ⋙ L₃) _ _,\nend\n\n@[simp]\ndef lifting_comp_iso_hom : (lifting_comp_iso H₁₂ H₂₃ H₁₃ e W₁ W₂).hom =\n  lift_nat_trans' (H₁₂.horiz_comp H₂₃) H₁₃ W₁ e.hom :=\nbegin\n  apply nat_trans_ext L₁ W₁,\n  intro X,\n  dsimp only [lifting_comp_iso],\n  simp only [lifting.uniq_hom_app, iso.trans_hom, iso_whisker_right_hom, nat_trans.comp_app,\n    whisker_right_app, assoc, lift_nat_trans'_app],\nend\n\n@[simp]\ndef lifting_comp_iso_inv : (lifting_comp_iso H₁₂ H₂₃ H₁₃ e W₁ W₂).inv =\n  lift_nat_trans' H₁₃ (H₁₂.horiz_comp H₂₃) W₁ e.inv :=\nbegin\n  apply nat_trans_ext L₁ W₁,\n  intro X,\n  dsimp only [lifting_comp_iso],\n  simp only [lifting.uniq_inv_app, iso.trans_inv, iso_whisker_right_inv, nat_trans.comp_app,\n    whisker_right_app, lift_nat_trans'_app],\nend\n\nvariables (τ₁₂ : F₁₂ ⟶ G₁₂) (τ₂₃ : F₂₃ ⟶ G₂₃) (τ₁₃ : F₁₃ ⟶ G₁₃)\n  (comm_τ : (τ₁₂ ◫ τ₂₃) ≫ e'.hom = e.hom ≫ τ₁₃)\n\ninclude comm_τ\n\nlemma lifting_comp_iso_nat_trans_compatibility [L₂.is_localization W₂] :\n  (lift_nat_trans' H₁₂ K₁₂ W₁ τ₁₂ ◫ lift_nat_trans' H₂₃ K₂₃ W₂ τ₂₃) ≫\n    (lifting_comp_iso K₁₂ K₂₃ K₁₃ e' W₁ W₂).hom =\n  (lifting_comp_iso H₁₂ H₂₃ H₁₃ e W₁ W₂).hom ≫ lift_nat_trans' H₁₃ K₁₃ W₁ τ₁₃ :=\nby simp only [lifting_comp_iso_hom, comp_lift_nat_trans', hcomp_lift_nat_trans', ← comm_τ]\n\nend localization\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/localization/predicate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.30327734025005837}}
{"text": "import mcl.defs\nimport mcl.rhl\n\nopen parlang\nopen mcl\nopen mcl.rhl\nopen parlang.thread_state\n\n@[simp]\nlemma store_stores {sig : signature} {dim} {idx : vector (expression sig type.int) dim} {var t} {h₁ : type_of (sig.val var) = t} {h₂}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} {i : mcl_address sig} : \ni ∉ (thread_state.tlocal_to_shared var idx h₁ h₂ ts).stores → i ∉ ts.stores := by simp [thread_state.tlocal_to_shared, store, not_or_distrib]\n\n@[simp]\nlemma store_loads {sig : signature} {dim} {idx : vector (expression sig type.int) dim} {var t} {h₁ : type_of (sig.val var) = t} {h₂}\n{ts : thread_state (memory $ parlang_mcl_tlocal sig) (parlang_mcl_shared sig)} {i : mcl_address sig} : \ni ∉ (thread_state.tlocal_to_shared var idx h₁ h₂ ts).loads → i ∉ ts.loads := by simp [thread_state.tlocal_to_shared, store, not_or_distrib]\n\n@[simp]\nlemma array_hole_name_neq {sig : signature} (var₁ var₂ : string) (h : var₁ ≠ var₂) (idx : vector ℕ (((sig.val var₁).type).dim)) :\n(⟨var₁, idx⟩ : mcl_address sig) ∉ @array_address_range sig var₂ := begin\n    unfold array_address_range,\n    intro,\n    contradiction,\nend", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/mcl/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.4532618480153862, "lm_q1q2_score": 0.30317790291677604}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport representation_theory.basic\nimport representation_theory.Action\nimport algebra.category.Module.abelian\nimport algebra.category.Module.colimits\nimport algebra.category.Module.monoidal\nimport algebra.category.Module.adjunctions\nimport category_theory.closed.functor_category\n\n/-!\n# `Rep k G` is the category of `k`-linear representations of `G`.\n\nIf `V : Rep k G`, there is a coercion that allows you to treat `V` as a type,\nand this type comes equipped with a `module k V` instance.\nAlso `V.ρ` gives the homomorphism `G →* (V →ₗ[k] V)`.\n\nConversely, given a homomorphism `ρ : G →* (V →ₗ[k] V)`,\nyou can construct the bundled representation as `Rep.of ρ`.\n\nWe construct the categorical equivalence `Rep k G ≌ Module (monoid_algebra k G)`.\nWe verify that `Rep k G` is a `k`-linear abelian symmetric monoidal category with all (co)limits.\n-/\n\nuniverses u\n\nopen category_theory\nopen category_theory.limits\n\n/-- The category of `k`-linear representations of a monoid `G`. -/\n@[derive [large_category, concrete_category, has_limits, has_colimits,\n  preadditive, abelian]]\nabbreviation Rep (k G : Type u) [ring k] [monoid G] :=\nAction (Module.{u} k) (Mon.of G)\n\ninstance (k G : Type u) [comm_ring k] [monoid G] : linear k (Rep k G) :=\nby apply_instance\n\nnamespace Rep\nvariables {k G : Type u} [comm_ring k]\nsection\nvariables [monoid G]\n\ninstance : has_coe_to_sort (Rep k G) (Type u) := concrete_category.has_coe_to_sort _\n\ninstance (V : Rep k G) : add_comm_group V :=\nby { change add_comm_group ((forget₂ (Rep k G) (Module k)).obj V), apply_instance, }\n\ninstance (V : Rep k G) : module k V :=\nby { change module k ((forget₂ (Rep k G) (Module k)).obj V), apply_instance, }\n\n/--\nSpecialize the existing `Action.ρ`, changing the type to `representation k G V`.\n-/\ndef ρ (V : Rep k G) : representation k G V := V.ρ\n\n/-- Lift an unbundled representation to `Rep`. -/\ndef of {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) : Rep k G :=\n⟨Module.of k V, ρ⟩\n\n@[simp]\nlemma coe_of {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) :\n  (of ρ : Type u) = V := rfl\n\n@[simp] lemma of_ρ {V : Type u} [add_comm_group V] [module k V] (ρ : G →* (V →ₗ[k] V)) :\n  (of ρ).ρ = ρ := rfl\n\n@[simp] lemma Action_ρ_eq_ρ {A : Rep k G} : Action.ρ A = A.ρ := rfl\n\n/-- Allows us to apply lemmas about the underlying `ρ`, which would take an element `g : G` rather\nthan `g : Mon.of G` as an argument. -/\nlemma of_ρ_apply {V : Type u} [add_comm_group V] [module k V]\n  (ρ : representation k G V) (g : Mon.of G) :\n  (Rep.of ρ).ρ g = ρ (g : G) := rfl\n\nvariables (k G)\n\n/-- The trivial `k`-linear `G`-representation on a `k`-module `V.` -/\ndef trivial (V : Type u) [add_comm_group V] [module k V] : Rep k G :=\nRep.of (@representation.trivial k G V _ _ _ _)\n\nvariables {k G}\n\nlemma trivial_def {V : Type u} [add_comm_group V] [module k V] (g : G) (v : V) :\n  (trivial k G V).ρ g v = v := rfl\n\n-- Verify that limits are calculated correctly.\nnoncomputable example : preserves_limits (forget₂ (Rep k G) (Module.{u} k)) :=\nby apply_instance\nnoncomputable example : preserves_colimits (forget₂ (Rep k G) (Module.{u} k)) :=\nby apply_instance\n\n@[simp] lemma monoidal_category.braiding_hom_apply {A B : Rep k G} (x : A) (y : B) :\n  Action.hom.hom (β_ A B).hom (tensor_product.tmul k x y) = tensor_product.tmul k y x := rfl\n\n@[simp] lemma monoidal_category.braiding_inv_apply {A B : Rep k G} (x : A) (y : B) :\n  Action.hom.hom (β_ A B).inv (tensor_product.tmul k y x) = tensor_product.tmul k x y := rfl\n\nsection linearization\n\nvariables (k G)\n\n/-- The monoidal functor sending a type `H` with a `G`-action to the induced `k`-linear\n`G`-representation on `k[H].` -/\nnoncomputable def linearization :\n  monoidal_functor (Action (Type u) (Mon.of G)) (Rep k G) :=\n(Module.monoidal_free k).map_Action (Mon.of G)\n\nvariables {k G}\n\n@[simp] lemma linearization_obj_ρ (X : Action (Type u) (Mon.of G)) (g : G) (x : X.V →₀ k) :\n  ((linearization k G).obj X).ρ g x = finsupp.lmap_domain k k (X.ρ g) x := rfl\n\n@[simp] lemma linearization_of (X : Action (Type u) (Mon.of G)) (g : G) (x : X.V) :\n  ((linearization k G).obj X).ρ g (finsupp.single x (1 : k))\n    = finsupp.single (X.ρ g x) (1 : k) :=\nby rw [linearization_obj_ρ, finsupp.lmap_domain_apply, finsupp.map_domain_single]\n\nvariables {X Y : Action (Type u) (Mon.of G)} (f : X ⟶ Y)\n\n@[simp] lemma linearization_map_hom :\n  ((linearization k G).map f).hom = finsupp.lmap_domain k k f.hom := rfl\n\nlemma linearization_map_hom_single (x : X.V) (r : k) :\n  ((linearization k G).map f).hom (finsupp.single x r)\n    = finsupp.single (f.hom x) r :=\nby rw [linearization_map_hom, finsupp.lmap_domain_apply, finsupp.map_domain_single]\n\n@[simp] lemma linearization_μ_hom (X Y : Action (Type u) (Mon.of G)) :\n  ((linearization k G).μ X Y).hom = (finsupp_tensor_finsupp' k X.V Y.V).to_linear_map :=\nrfl\n\n@[simp] \n\n@[simp] lemma linearization_ε_hom :\n  (linearization k G).ε.hom = finsupp.lsingle punit.star :=\nrfl\n\n@[simp] lemma linearization_ε_inv_hom_apply (r : k) :\n  (inv (linearization k G).ε).hom (finsupp.single punit.star r) = r :=\nbegin\n  simp_rw [←Action.forget_map, functor.map_inv, Action.forget_map],\n  rw [←finsupp.lsingle_apply punit.star r],\n  apply is_iso.hom_inv_id_apply _ _,\nend\n\nvariables (k G)\n\n/-- The linearization of a type `X` on which `G` acts trivially is the trivial `G`-representation\non `k[X]`. -/\n@[simps] noncomputable def linearization_trivial_iso (X : Type u) :\n  (linearization k G).obj (Action.mk X 1) ≅ trivial k G (X →₀ k) :=\nAction.mk_iso (iso.refl _) $ λ g, by { ext1, ext1, exact linearization_of _ _ _ }\n\nvariables (k G)\n\n/-- Given a `G`-action on `H`, this is `k[H]` bundled with the natural representation\n`G →* End(k[H])` as a term of type `Rep k G`. -/\nnoncomputable abbreviation of_mul_action (H : Type u) [mul_action G H] : Rep k G :=\nof $ representation.of_mul_action k G H\n\n/-- The `k`-linear `G`-representation on `k[G]`, induced by left multiplication. -/\nnoncomputable def left_regular : Rep k G :=\nof_mul_action k G G\n\n/-- The `k`-linear `G`-representation on `k[Gⁿ]`, induced by left multiplication. -/\nnoncomputable def diagonal (n : ℕ) : Rep k G :=\nof_mul_action k G (fin n → G)\n\n/-- The linearization of a type `H` with a `G`-action is definitionally isomorphic to the\n`k`-linear `G`-representation on `k[H]` induced by the `G`-action on `H`. -/\nnoncomputable def linearization_of_mul_action_iso (H : Type u) [mul_action G H] :\n  (linearization k G).obj (Action.of_mul_action G H)\n    ≅ of_mul_action k G H := iso.refl _\n\nvariables {k G}\n\n/-- Given an element `x : A`, there is a natural morphism of representations `k[G] ⟶ A` sending\n`g ↦ A.ρ(g)(x).` -/\n@[simps] noncomputable def left_regular_hom (A : Rep k G) (x : A) :\n  Rep.of_mul_action k G G ⟶ A :=\n{ hom := finsupp.lift _ _ _ (λ g, A.ρ g x),\n  comm' := λ g,\n  begin\n    refine finsupp.lhom_ext' (λ y, linear_map.ext_ring _),\n    simpa only [linear_map.comp_apply, Module.comp_def, finsupp.lsingle_apply,\n      finsupp.lift_apply, Action_ρ_eq_ρ, of_ρ_apply, representation.of_mul_action_single,\n      finsupp.sum_single_index, zero_smul, one_smul, smul_eq_mul, A.ρ.map_mul],\n  end }\n\nlemma left_regular_hom_apply {A : Rep k G} (x : A) :\n  (left_regular_hom A x).hom (finsupp.single 1 1) = x :=\nbegin\n  simpa only [left_regular_hom_hom, finsupp.lift_apply, finsupp.sum_single_index, one_smul,\n    A.ρ.map_one, zero_smul],\nend\n\n/-- Given a `k`-linear `G`-representation `A`, there is a `k`-linear isomorphism between\nrepresentation morphisms `Hom(k[G], A)` and `A`. -/\n@[simps] noncomputable def left_regular_hom_equiv (A : Rep k G) :\n  (Rep.of_mul_action k G G ⟶ A) ≃ₗ[k] A :=\n{ to_fun := λ f, f.hom (finsupp.single 1 1),\n  map_add' := λ x y, rfl,\n  map_smul' := λ r x, rfl,\n  inv_fun := λ x, left_regular_hom A x,\n  left_inv := λ f,\n  begin\n    refine Action.hom.ext _ _ (finsupp.lhom_ext' (λ (x : G), linear_map.ext_ring _)),\n    have : f.hom (((of_mul_action k G G).ρ) x (finsupp.single (1 : G) (1 : k)))\n      = A.ρ x (f.hom (finsupp.single (1 : G) (1 : k))) :=\n      linear_map.ext_iff.1 (f.comm x) (finsupp.single 1 1),\n    simp only [linear_map.comp_apply, finsupp.lsingle_apply,\n      left_regular_hom_hom, finsupp.lift_apply, finsupp.sum_single_index, one_smul, ←this,\n      zero_smul, of_ρ_apply, representation.of_mul_action_single x (1 : G) (1 : k), smul_eq_mul,\n      mul_one],\n  end,\n  right_inv := λ x, left_regular_hom_apply x }\n\nlemma left_regular_hom_equiv_symm_single {A : Rep k G} (x : A) (g : G) :\n  ((left_regular_hom_equiv A).symm x).hom (finsupp.single g 1) = A.ρ g x :=\nbegin\n  simp only [left_regular_hom_equiv_symm_apply, left_regular_hom_hom,\n    finsupp.lift_apply, finsupp.sum_single_index, zero_smul, one_smul],\nend\n\nend linearization\nend\nsection\nopen Action\nvariables [group G] (A B C : Rep k G)\n\nnoncomputable instance : monoidal_closed (Rep k G) :=\nmonoidal_closed.of_equiv (functor_category_monoidal_equivalence _ _)\n\n/-- Explicit description of the 'internal Hom' `iHom(A, B)` of two representations `A, B`:\nthis is `F⁻¹(iHom(F(A), F(B)))`, where `F` is an equivalence\n`Rep k G ≌ (single_obj G ⥤ Module k)`. Just used to prove `Rep.ihom_obj_ρ`. -/\nlemma ihom_obj_ρ_def :\n  ((ihom A).obj B).ρ = (functor_category_equivalence.inverse.obj\n  ((functor_category_equivalence.functor.obj A).closed_ihom.obj\n  (functor_category_equivalence.functor.obj B))).ρ := rfl\n\n/-- Given `k`-linear `G`-representations `(A, ρ₁), (B, ρ₂)`, the 'internal Hom' is the\nrepresentation on `Homₖ(A, B)` sending `g : G` and `f : A →ₗ[k] B` to `(ρ₂ g) ∘ₗ f ∘ₗ (ρ₁ g⁻¹)`. -/\n@[simp] lemma ihom_obj_ρ :\n  ((ihom A).obj B).ρ = A.ρ.lin_hom B.ρ :=\nbegin\n  refine monoid_hom.ext (λ g, _),\n  simpa only [ihom_obj_ρ_def, functor_category_equivalence.inverse_obj_ρ_apply,\n    functor.closed_ihom_obj_map, ←functor.map_inv, single_obj.inv_as_inv],\nend\n\n@[simp] lemma ihom_map_hom {B C : Rep k G} (f : B ⟶ C) :\n  ((ihom A).map f).hom = linear_map.llcomp k A B C f.hom :=\nrfl\n\n/-- Unfolds the unit in the adjunction `A ⊗ - ⊣ iHom(A, -)`; just used to prove\n`Rep.ihom_coev_app_hom`. -/\nlemma ihom_coev_app_def :\n  (ihom.coev A).app B = functor_category_equivalence.unit_iso.hom.app B ≫\n  functor_category_equivalence.inverse.map\n  ((functor_category_equivalence.functor.obj A).closed_unit.app _ ≫\n  (functor_category_equivalence.functor.obj A).closed_ihom.map\n  ((functor_category_monoidal_equivalence (Module.{u} k) (Mon.of G)).μ A B)) :=\nrfl\n\n/-- Describes the unit in the adjunction `A ⊗ - ⊣ iHom(A, -)`; given another `k`-linear\n`G`-representation `B,` the `k`-linear map underlying the resulting map `B ⟶ iHom(A, A ⊗ B)` is\ngiven by flipping the arguments in the natural `k`-bilinear map `A →ₗ[k] B →ₗ[k] A ⊗ B`. -/\n@[simp] lemma ihom_coev_app_hom :\n  Action.hom.hom ((ihom.coev A).app B) = (tensor_product.mk _ _ _).flip :=\nbegin\n  refine linear_map.ext (λ x, linear_map.ext (λ y, _)),\n  simpa only [ihom_coev_app_def, functor.map_comp, comp_hom,\n    functor_category_equivalence.inverse_map_hom, functor.closed_ihom_map_app,\n    functor_category_monoidal_equivalence.μ_app],\nend\n\nvariables {A B C}\n\n/-- Given a `k`-linear `G`-representation `A`, the adjunction `A ⊗ - ⊣ iHom(A, -)` defines a\nbijection `Hom(A ⊗ B, C) ≃ Hom(B, iHom(A, C))` for all `B, C`. Given `f : A ⊗ B ⟶ C`, this lemma\ndescribes the `k`-linear map underlying `f`'s image under the bijection. It is given by currying the\n`k`-linear map underlying `f`, giving a map `A →ₗ[k] B →ₗ[k] C`, then flipping the arguments. -/\n@[simp] lemma monoidal_closed_curry_hom (f : A ⊗ B ⟶ C) :\n  (monoidal_closed.curry f).hom = (tensor_product.curry f.hom).flip :=\nbegin\n  rw [monoidal_closed.curry_eq, comp_hom, ihom_coev_app_hom],\n  refl,\nend\n\n/-- Given a `k`-linear `G`-representation `A`, the adjunction `A ⊗ - ⊣ iHom(A, -)` defines a\nbijection `Hom(A ⊗ B, C) ≃ Hom(B, iHom(A, C))` for all `B, C`. Given `f : B ⟶ iHom(A, C)`, this\nlemma describes the `k`-linear map underlying `f`'s image under the bijection. It is given by\nflipping the arguments of the `k`-linear map underlying `f`, giving a map `A →ₗ[k] B →ₗ[k] C`, then\nuncurrying. -/\n@[simp] lemma monoidal_closed_uncurry_hom (f : B ⟶ (ihom A).obj C) :\n  (monoidal_closed.uncurry f).hom = tensor_product.uncurry _ _ _ _ f.hom.flip :=\nbegin\n  simp only [monoidal_closed.of_equiv_uncurry_def, comp_inv_iso_inv_app,\n    monoidal_functor.comm_tensor_left_inv_app, comp_hom,\n    functor_category_monoidal_equivalence.inverse_map, functor_category_equivalence.inverse_map_hom,\n    functor_category_monoidal_equivalence.μ_iso_inv_app],\n  ext,\n  refl,\nend\n\n/-- Describes the counit in the adjunction `A ⊗ - ⊣ iHom(A, -)`; given another `k`-linear\n`G`-representation `B,` the `k`-linear map underlying the resulting morphism `A ⊗ iHom(A, B) ⟶ B`\nis given by uncurrying the map `A →ₗ[k] (A →ₗ[k] B) →ₗ[k] B` defined by flipping the arguments in\nthe identity map on `Homₖ(A, B).` -/\n@[simp] lemma ihom_ev_app_hom : Action.hom.hom ((ihom.ev A).app B) =\n  tensor_product.uncurry _ _ _ _ linear_map.id.flip :=\nmonoidal_closed_uncurry_hom _\n\nvariables (A B C)\n\n/-- There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`\nand `Hom(B, Homₖ(A, C))`. -/\nnoncomputable def monoidal_closed.linear_hom_equiv :\n  (A ⊗ B ⟶ C) ≃ₗ[k] (B ⟶ (A ⟶[Rep k G] C)) :=\n{ map_add' := λ f g, rfl,\n  map_smul' := λ r f, rfl, ..(ihom.adjunction A).hom_equiv _ _ }\n\n/-- There is a `k`-linear isomorphism between the sets of representation morphisms`Hom(A ⊗ B, C)`\nand `Hom(A, Homₖ(B, C))`. -/\nnoncomputable def monoidal_closed.linear_hom_equiv_comm :\n  (A ⊗ B ⟶ C) ≃ₗ[k] (A ⟶ (B ⟶[Rep k G] C)) :=\n(linear.hom_congr k (β_ A B) (iso.refl _)) ≪≫ₗ\n  monoidal_closed.linear_hom_equiv _ _ _\n\nvariables {A B C}\n\nlemma monoidal_closed.linear_hom_equiv_hom (f : A ⊗ B ⟶ C) :\n  (monoidal_closed.linear_hom_equiv A B C f).hom =\n  (tensor_product.curry f.hom).flip :=\nmonoidal_closed_curry_hom _\n\nlemma monoidal_closed.linear_hom_equiv_comm_hom (f : A ⊗ B ⟶ C) :\n  (monoidal_closed.linear_hom_equiv_comm A B C f).hom =\n tensor_product.curry f.hom :=\nbegin\n  dunfold monoidal_closed.linear_hom_equiv_comm,\n  refine linear_map.ext (λ x, linear_map.ext (λ y, _)),\n  simp only [linear_equiv.trans_apply, monoidal_closed.linear_hom_equiv_hom,\n    linear.hom_congr_apply, iso.refl_hom, iso.symm_hom, linear_map.to_fun_eq_coe,\n    linear_map.coe_comp, function.comp_app, linear.left_comp_apply, linear.right_comp_apply,\n    category.comp_id, Action.comp_hom, linear_map.flip_apply, tensor_product.curry_apply,\n    Module.coe_comp, function.comp_app, monoidal_category.braiding_inv_apply],\nend\n\nlemma monoidal_closed.linear_hom_equiv_symm_hom (f : B ⟶ (A ⟶[Rep k G] C)) :\n  ((monoidal_closed.linear_hom_equiv A B C).symm f).hom =\n  tensor_product.uncurry k A B C f.hom.flip :=\nmonoidal_closed_uncurry_hom _\n\nlemma monoidal_closed.linear_hom_equiv_comm_symm_hom (f : A ⟶ (B ⟶[Rep k G] C)) :\n  ((monoidal_closed.linear_hom_equiv_comm A B C).symm f).hom =\n  tensor_product.uncurry k A B C f.hom :=\nbegin\n  dunfold monoidal_closed.linear_hom_equiv_comm,\n  refine tensor_product.algebra_tensor_module.curry_injective\n    (linear_map.ext (λ x, linear_map.ext (λ y, _))),\n  simp only [linear_equiv.trans_symm, linear_equiv.trans_apply, linear.hom_congr_symm_apply,\n    iso.refl_inv, linear_map.coe_comp, function.comp_app, category.comp_id, Action.comp_hom,\n    monoidal_closed.linear_hom_equiv_symm_hom, tensor_product.algebra_tensor_module.curry_apply,\n    linear_map.coe_restrict_scalars, linear_map.to_fun_eq_coe, linear_map.flip_apply,\n    tensor_product.curry_apply, Module.coe_comp, function.comp_app,\n    monoidal_category.braiding_hom_apply, tensor_product.uncurry_apply],\nend\n\nend\nend Rep\nnamespace representation\nvariables {k G : Type u} [comm_ring k] [monoid G] {V W : Type u}\n  [add_comm_group V] [add_comm_group W] [module k V] [module k W]\n  (ρ : representation k G V) (τ : representation k G W)\n\n/-- Tautological isomorphism to help Lean in typechecking. -/\ndef Rep_of_tprod_iso : Rep.of (ρ.tprod τ) ≅ Rep.of ρ ⊗ Rep.of τ := iso.refl _\n\nlemma Rep_of_tprod_iso_apply (x : tensor_product k V W) :\n  (Rep_of_tprod_iso ρ τ).hom.hom x = x := rfl\n\nlemma Rep_of_tprod_iso_inv_apply (x : tensor_product k V W) :\n  (Rep_of_tprod_iso ρ τ).inv.hom x = x := rfl\n\nend representation\n/-!\n# The categorical equivalence `Rep k G ≌ Module.{u} (monoid_algebra k G)`.\n-/\nnamespace Rep\nvariables {k G : Type u} [comm_ring k] [monoid G]\n\n-- Verify that the symmetric monoidal structure is available.\nexample : symmetric_category (Rep k G) := by apply_instance\nexample : monoidal_preadditive (Rep k G) := by apply_instance\nexample : monoidal_linear k (Rep k G) := by apply_instance\n\nnoncomputable theory\n\n/-- Auxilliary lemma for `to_Module_monoid_algebra`. -/\nlemma to_Module_monoid_algebra_map_aux {k G : Type*} [comm_ring k] [monoid G]\n  (V W : Type*) [add_comm_group V] [add_comm_group W] [module k V] [module k W]\n  (ρ : G →* V →ₗ[k] V) (σ : G →* W →ₗ[k] W)\n  (f : V →ₗ[k] W) (w : ∀ (g : G), f.comp (ρ g) = (σ g).comp f)\n  (r : monoid_algebra k G) (x : V) :\n  f ((((monoid_algebra.lift k G (V →ₗ[k] V)) ρ) r) x) =\n    (((monoid_algebra.lift k G (W →ₗ[k] W)) σ) r) (f x) :=\nbegin\n  apply monoid_algebra.induction_on r,\n  { intro g,\n    simp only [one_smul, monoid_algebra.lift_single, monoid_algebra.of_apply],\n    exact linear_map.congr_fun (w g) x, },\n  { intros g h gw hw, simp only [map_add, add_left_inj, linear_map.add_apply, hw, gw], },\n  { intros r g w,\n    simp only [alg_hom.map_smul, w, ring_hom.id_apply,\n      linear_map.smul_apply, linear_map.map_smulₛₗ], }\nend\n\n/-- Auxilliary definition for `to_Module_monoid_algebra`. -/\ndef to_Module_monoid_algebra_map {V W : Rep k G} (f : V ⟶ W) :\n  Module.of (monoid_algebra k G) V.ρ.as_module ⟶ Module.of (monoid_algebra k G) W.ρ.as_module :=\n{ map_smul' := λ r x, to_Module_monoid_algebra_map_aux V.V W.V V.ρ W.ρ f.hom f.comm r x,\n  ..f.hom, }\n\n/-- Functorially convert a representation of `G` into a module over `monoid_algebra k G`. -/\ndef to_Module_monoid_algebra : Rep k G ⥤ Module.{u} (monoid_algebra k G) :=\n{ obj := λ V, Module.of _ V.ρ.as_module ,\n  map := λ V W f, to_Module_monoid_algebra_map f, }\n\n/-- Functorially convert a module over `monoid_algebra k G` into a representation of `G`. -/\ndef of_Module_monoid_algebra : Module.{u} (monoid_algebra k G) ⥤ Rep k G :=\n{ obj := λ M, Rep.of (representation.of_module k G M),\n  map := λ M N f,\n  { hom :=\n    { map_smul' := λ r x, f.map_smul (algebra_map k _ r) x,\n      ..f },\n    comm' := λ g, by { ext, apply f.map_smul, }, }, }.\n\nlemma of_Module_monoid_algebra_obj_coe (M : Module.{u} (monoid_algebra k G)) :\n  (of_Module_monoid_algebra.obj M : Type u) = restrict_scalars k (monoid_algebra k G) M := rfl\n\nlemma of_Module_monoid_algebra_obj_ρ (M : Module.{u} (monoid_algebra k G)) :\n  (of_Module_monoid_algebra.obj M).ρ = representation.of_module k G M := rfl\n\n/-- Auxilliary definition for `equivalence_Module_monoid_algebra`. -/\ndef counit_iso_add_equiv {M : Module.{u} (monoid_algebra k G)} :\n  ((of_Module_monoid_algebra ⋙ to_Module_monoid_algebra).obj M) ≃+ M :=\nbegin\n  dsimp [of_Module_monoid_algebra, to_Module_monoid_algebra],\n  refine (representation.of_module k G ↥M).as_module_equiv.trans (restrict_scalars.add_equiv _ _ _),\nend\n\n/-- Auxilliary definition for `equivalence_Module_monoid_algebra`. -/\ndef unit_iso_add_equiv {V : Rep k G} :\n  V ≃+ ((to_Module_monoid_algebra ⋙ of_Module_monoid_algebra).obj V) :=\nbegin\n  dsimp [of_Module_monoid_algebra, to_Module_monoid_algebra],\n  refine V.ρ.as_module_equiv.symm.trans _,\n  exact (restrict_scalars.add_equiv _ _ _).symm,\nend\n\n/-- Auxilliary definition for `equivalence_Module_monoid_algebra`. -/\ndef counit_iso (M : Module.{u} (monoid_algebra k G)) :\n  (of_Module_monoid_algebra ⋙ to_Module_monoid_algebra).obj M ≅ M :=\nlinear_equiv.to_Module_iso'\n{ map_smul' := λ r x, begin\n    dsimp [counit_iso_add_equiv],\n    simp,\n  end,\n  ..counit_iso_add_equiv, }\n\nlemma unit_iso_comm (V : Rep k G) (g : G) (x : V) :\n  unit_iso_add_equiv (((V.ρ) g).to_fun x) =\n    (((of_Module_monoid_algebra.obj (to_Module_monoid_algebra.obj V)).ρ) g).to_fun\n      (unit_iso_add_equiv x) :=\nbegin\n  dsimp [unit_iso_add_equiv, of_Module_monoid_algebra, to_Module_monoid_algebra],\n  simp only [add_equiv.apply_eq_iff_eq, add_equiv.apply_symm_apply,\n    representation.as_module_equiv_symm_map_rho, representation.of_module_as_module_act],\nend\n\n/-- Auxilliary definition for `equivalence_Module_monoid_algebra`. -/\ndef unit_iso (V : Rep k G) :\n  V ≅ ((to_Module_monoid_algebra ⋙ of_Module_monoid_algebra).obj V) :=\nAction.mk_iso (linear_equiv.to_Module_iso'\n{ map_smul' := λ r x, begin\n    dsimp [unit_iso_add_equiv],\n    simp only [representation.as_module_equiv_symm_map_smul,\n      restrict_scalars.add_equiv_symm_map_algebra_map_smul],\n  end,\n  ..unit_iso_add_equiv, })\n  (λ g, by { ext, apply unit_iso_comm, })\n\n/-- The categorical equivalence `Rep k G ≌ Module (monoid_algebra k G)`. -/\ndef equivalence_Module_monoid_algebra : Rep k G ≌ Module.{u} (monoid_algebra k G) :=\n{ functor := to_Module_monoid_algebra,\n  inverse := of_Module_monoid_algebra,\n  unit_iso := nat_iso.of_components (λ V, unit_iso V) (by tidy),\n  counit_iso := nat_iso.of_components (λ M, counit_iso M) (by tidy), }\n\n-- TODO Verify that the equivalence with `Module (monoid_algebra k G)` is a monoidal functor.\n\nend Rep\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/representation_theory/Rep.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3031501573425136}}
{"text": "-- Deadlocks: var/Microban_155_l16/deadlocks\n-- Levelset: data/Large Test Suite Sets/Microban_155.xsb\n-- Level: 16\n\nimport .deadlocks\n\ndef Microban_155_l16 := sokolevel.from_string \"\n ####     \n #  ####  \n #     ## \n## ##   # \n#. .# @$##\n#   # $$ #\n#  .#    #\n##########\n\"\n\nnamespace Microban_155_l16\nopen deadlocks\n\n@[reducible]\ndef deadlock_local (dl : boxint) : Prop := deadlock Microban_155_l16.avail Microban_155_l16.goal dl\ndef deadlocks_local (dls : list boxint) : Prop\n:= dls.pall (λ dl, deadlock_local dl)\ndef generate_local : list (ℕ × ℕ) → list (ℕ × ℕ) → ℕ × ℕ → boxint\n:= boxint.generate_from_list Microban_155_l16.avail\n\ndef dl0 := generate_local [(8,7)] [] (2,2)\n\ntheorem dl0_dl : deadlock_local dl0\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl0_dl.to_html\n\ndef dl1 := generate_local [(7,4), (8,6), (7,7)] [] (2,2)\n\ntheorem dl1_dl : deadlock_local dl1\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl0_dl, -- (7,7) right\nend\n\n#html dl1_dl.to_html\n\ndef dl2 := generate_local [(5,7)] [] (2,2)\n\ntheorem dl2_dl : deadlock_local dl2\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl2_dl.to_html\n\ndef dl3 := generate_local [(6,7)] [] (2,2)\ndef dl4 := generate_local [(7,7)] [] (2,2)\ndef dls3_4 := [dl3, dl4]\n\ntheorem dls3_4_dl : deadlocks_local dls3_4\n:=\nbegin\n  refine list.pall_iff.mpr (new_deadlocks _),\n  rcases list.pall_in dls3_4 with ⟨dl3_in, dl4_in, irrelevant⟩,\n  refine list.pall_iff.mp ⟨_, _, trivial⟩,\n  {\n    analyze_deadlock,\n    deadlocked_step dl2_dl, -- (6,7) left\n    deadlocked_step dl4_in, -- (6,7) right\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl3_in, -- (7,7) left\n    deadlocked_step dl0_dl, -- (7,7) right\n  },\nend\n\ntheorem dl3_dl : deadlock_local dl3\n:= dls3_4_dl.1\ntheorem dl4_dl : deadlock_local dl4\n:= dls3_4_dl.2.1\n\n#html dl3_dl.to_html\n#html dl4_dl.to_html\n\ndef dl5 := generate_local [(8,6)] [] (2,2)\n\ntheorem dl5_dl : deadlock_local dl5\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl5_dl.to_html\n\ndef dl6 := generate_local [(6,3)] [] (2,2)\n\ntheorem dl6_dl : deadlock_local dl6\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl6_dl.to_html\n\ndef dl7 := generate_local [(7,4)] [] (2,2)\n\ntheorem dl7_dl : deadlock_local dl7\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl7_dl.to_html\n\ndef dl8 := generate_local [(5,3), (6,4), (7,5)] [] (5,4)\n\ntheorem dl8_dl : deadlock_local dl8\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl6_dl, -- (6,4) up\n  deadlocked_step dl7_dl, -- (6,4) right\n  deadlocked_step dl7_dl, -- (7,5) up\nend\n\n#html dl8_dl.to_html\n\n#check dl8_dl\n\ndef dl9 := generate_local [(5,4), (6,4), (7,5)] [] (5,5)\n\ntheorem dl9_dl : deadlock_local dl9\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl8_dl, -- (5,4) up\n  deadlocked_step dl6_dl, -- (6,4) up\n  deadlocked_step dl7_dl, -- (7,5) up\nend\n\n#html dl9_dl.to_html\n\ndef dl10 := generate_local [(6,4), (5,5), (7,5)] [] (6,5)\n\ntheorem dl10_dl : deadlock_local dl10\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl6_dl, -- (6,4) up\n  deadlocked_step dl9_dl, -- (5,5) up\n  deadlocked_step dl7_dl, -- (7,5) up\nend\n\n#html dl10_dl.to_html\n\ndef dl11 := generate_local [(1,5), (3,7)] [(3,5)] (2,2)\n\ntheorem dl11_dl : deadlock_local dl11\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl11_dl.to_html\n\ndef dl12 := generate_local [(3,5), (2,6), (3,7)] [] (2,2)\ndef dl13 := generate_local [(3,5), (3,7)] [(1,5), (2,5), (1,6), (2,6)] (2,2)\ndef dl14 := generate_local [(2,5), (3,5), (3,7)] [] (2,2)\ndef dl15 := generate_local [(2,5), (3,5), (3,7)] [] (1,5)\ndef dls12_15 := [dl12, dl13, dl14, dl15]\n\ntheorem dls12_15_dl : deadlocks_local dls12_15\n:=\nbegin\n  refine list.pall_iff.mpr (new_deadlocks _),\n  rcases list.pall_in dls12_15 with ⟨dl12_in, dl13_in, dl14_in, dl15_in, irrelevant⟩,\n  refine list.pall_iff.mp ⟨_, _, _, _, trivial⟩,\n  {\n    analyze_deadlock,\n    deadlocked_step dl15_in, -- (2,6) up\n    deadlocked_step dl13_in, -- (2,6) down\n    deadlocked_step dl13_in, -- (2,6) right\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl14_in, -- (2,4) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl12_in, -- (2,5) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl13_in, -- (2,5) up\n  },\nend\n\ntheorem dl12_dl : deadlock_local dl12\n:= dls12_15_dl.1\ntheorem dl13_dl : deadlock_local dl13\n:= dls12_15_dl.2.1\ntheorem dl14_dl : deadlock_local dl14\n:= dls12_15_dl.2.2.1\ntheorem dl15_dl : deadlock_local dl15\n:= dls12_15_dl.2.2.2.1\n\n#html dl12_dl.to_html\n#html dl13_dl.to_html\n#html dl14_dl.to_html\n#html dl15_dl.to_html\n\ndef dl16 := generate_local [(2,5), (2,6)] [] (2,2)\n\ntheorem dl16_dl : deadlock_local dl16\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl16_dl.to_html\n\ndef dl17 := generate_local [(1,7)] [] (2,2)\n\ntheorem dl17_dl : deadlock_local dl17\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#html dl17_dl.to_html\n\ndef dl18 := generate_local [(1,6), (2,7)] [] (2,2)\n\ntheorem dl18_dl : deadlock_local dl18\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl17_dl, -- (1,6) down\n  deadlocked_step dl17_dl, -- (2,7) left\nend\n\n#html dl18_dl.to_html\n\ndef dl19 := generate_local [(3,5), (2,6), (2,7)] [] (3,6)\n\ntheorem dl19_dl : deadlock_local dl19\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl18_dl, -- (2,6) left\n  deadlocked_step dl17_dl, -- (2,7) left\nend\n\n#html dl19_dl.to_html\n\ndef dl20 := generate_local [(2,5), (3,5), (2,7)] [] (2,2)\ndef dl21 := generate_local [(3,5), (2,7)] [(1,5), (2,5), (1,6), (2,6)] (2,2)\ndef dl22 := generate_local [(3,5), (2,6), (2,7)] [] (2,2)\ndef dls20_22 := [dl20, dl21, dl22]\n\ntheorem dls20_22_dl : deadlocks_local dls20_22\n:=\nbegin\n  refine list.pall_iff.mpr (new_deadlocks _),\n  rcases list.pall_in dls20_22 with ⟨dl20_in, dl21_in, dl22_in, irrelevant⟩,\n  refine list.pall_iff.mp ⟨_, _, _, trivial⟩,\n  {\n    analyze_deadlock,\n    deadlocked_step dl22_in, -- (2,5) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl17_dl, -- (2,7) left\n    deadlocked_step dl13_dl, -- (2,7) right\n    deadlocked_step dl20_in, -- (2,4) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl21_in, -- (2,6) right\n    deadlocked_step dl12_dl, -- (2,7) right\n  },\nend\n\n#check dls20_22_dl\n\ntheorem dl20_dl : deadlock_local dl20\n:= dls20_22_dl.1\ntheorem dl21_dl : deadlock_local dl21\n:= dls20_22_dl.2.1\ntheorem dl22_dl : deadlock_local dl22\n:= dls20_22_dl.2.2.1\n\n#html dl20_dl.to_html\n#html dl21_dl.to_html\n#html dl22_dl.to_html\n\ndef dl23 := generate_local [(2,6), (3,7)] [(1,5), (2,5), (3,5), (1,6)] (2,2)\ndef dl24 := generate_local [(2,5), (2,7)] [(1,5), (3,5), (1,6), (2,6)] (2,2)\ndef dl25 := generate_local [(1,6), (3,7)] [(1,5), (2,5), (3,5), (2,6)] (2,2)\ndef dl26 := generate_local [(3,7)] [(1,5), (2,5), (3,5), (1,6), (2,6)] (2,2)\ndef dl27 := generate_local [(1,6), (2,6), (3,7)] [] (2,2)\ndef dl28 := generate_local [(2,6), (2,7)] [(1,5), (2,5), (3,5), (1,6)] (2,2)\ndef dl29 := generate_local [(2,5), (3,7)] [(1,5), (3,5), (1,6), (2,6)] (2,2)\ndef dl30 := generate_local [(2,5), (3,7)] [(1,5), (3,5), (1,6), (2,6)] (1,5)\ndef dl31 := generate_local [(2,5), (1,6), (3,7)] [] (2,2)\ndef dl32 := generate_local [(2,7)] [(1,5), (2,5), (3,5), (1,6), (2,6)] (2,2)\ndef dls23_32 := [dl23, dl24, dl25, dl26, dl27, dl28, dl29, dl30, dl31, dl32]\n\ntheorem dls23_32_dl : deadlocks_local dls23_32\n:=\nbegin\n  refine list.pall_iff.mpr (new_deadlocks _),\n  rcases list.pall_in dls23_32 with ⟨dl23_in, dl24_in, dl25_in, dl26_in, dl27_in, dl28_in, dl29_in, dl30_in, dl31_in, dl32_in, irrelevant⟩,\n  refine list.pall_iff.mp ⟨_, _, _, _, _, _, _, _, _, _, trivial⟩,\n  {\n    analyze_deadlock,\n    deadlocked_step dl30_in, -- (2,6) up\n    deadlocked_step dl32_in, -- (2,6) down\n    deadlocked_step dl25_in, -- (2,6) left\n    deadlocked_step dl26_in, -- (2,6) right\n    deadlocked_step dl16_dl, -- (2,4) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl28_in, -- (2,5) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl11_dl, -- (1,6) up\n    deadlocked_step dl26_in, -- (1,6) down\n    deadlocked_step dl31_in, -- (2,4) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl29_in, -- (2,4) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl23_in, -- (1,6) down\n    deadlocked_step dl25_in, -- (2,6) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl18_dl, -- (2,6) left\n    deadlocked_step dl32_in, -- (2,6) right\n    deadlocked_step dl17_dl, -- (2,7) left\n    deadlocked_step dl23_in, -- (2,7) right\n    deadlocked_step dl16_dl, -- (2,4) down\n    deadlocked_step dl19_dl, -- (3,6) up\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl23_in, -- (2,5) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl26_in, -- (2,5) up\n    deadlocked_step dl11_dl, -- (2,5) left\n    deadlocked_step dl13_dl, -- (2,5) right\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl27_in, -- (2,5) down\n  }, {\n    analyze_deadlock,\n    deadlocked_step dl17_dl, -- (2,7) left\n    deadlocked_step dl26_in, -- (2,7) right\n    deadlocked_step dl24_in, -- (2,4) down\n    deadlocked_step dl21_dl, -- (3,6) up\n  },\nend\n\ntheorem dl23_dl : deadlock_local dl23\n:= dls23_32_dl.1\ntheorem dl24_dl : deadlock_local dl24\n:= dls23_32_dl.2.1\ntheorem dl25_dl : deadlock_local dl25\n:= dls23_32_dl.2.2.1\ntheorem dl26_dl : deadlock_local dl26\n:= dls23_32_dl.2.2.2.1\ntheorem dl27_dl : deadlock_local dl27\n:= dls23_32_dl.2.2.2.2.1\ntheorem dl28_dl : deadlock_local dl28\n:= dls23_32_dl.2.2.2.2.2.1\ntheorem dl29_dl : deadlock_local dl29\n:= dls23_32_dl.2.2.2.2.2.2.1\ntheorem dl30_dl : deadlock_local dl30\n:= dls23_32_dl.2.2.2.2.2.2.2.1\ntheorem dl31_dl : deadlock_local dl31\n:= dls23_32_dl.2.2.2.2.2.2.2.2.1\ntheorem dl32_dl : deadlock_local dl32\n:= dls23_32_dl.2.2.2.2.2.2.2.2.2.1\n\n#html dl23_dl.to_html\n#html dl24_dl.to_html\n#html dl25_dl.to_html\n#html dl26_dl.to_html\n#html dl27_dl.to_html\n#html dl28_dl.to_html\n#html dl29_dl.to_html\n#html dl30_dl.to_html\n#html dl31_dl.to_html\n#html dl32_dl.to_html\n\nend Microban_155_l16\n", "meta": {"author": "mirefek", "repo": "sokoban.lean", "sha": "451c92308afb4d3f8e566594b9751286f93b899b", "save_path": "github-repos/lean/mirefek-sokoban.lean", "path": "github-repos/lean/mirefek-sokoban.lean/sokoban.lean-451c92308afb4d3f8e566594b9751286f93b899b/src/Microban_155_l16_dl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.30309522522873805}}
{"text": "/-\nCopyright (c) 2021 Rémy Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rémy Degenne\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.measure_space\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 \n\nnamespace Mathlib\n\n/-!\n# Sequence of measurable functions associated to a sequence of a.e.-measurable functions\n\nWe define here tools to prove statements about limits (infi, supr...) of sequences of\n`ae_measurable` functions.\nGiven a sequence of a.e.-measurable functions `f : ι → α → β` with hypothesis\n`hf : ∀ i, ae_measurable (f i) μ`, and a pointwise property `p : α → (ι → β) → Prop` such that we\nhave `hp : ∀ᵐ x ∂μ, p x (λ n, f n x)`, we define a sequence of measurable functions `ae_seq hf p`\nand a measurable set `ae_seq_set hf p`, such that\n* `μ (ae_seq_set hf p)ᶜ = 0`\n* `x ∈ ae_seq_set hf p → ∀ i : ι, ae_seq hf hp i x = f i x`\n* `x ∈ ae_seq_set hf p → p x (λ n, f n x)`\n-/\n\n/-- If we have the additional hypothesis `∀ᵐ x ∂μ, p x (λ n, f n x)`, this is a measurable set\nwhose complement has measure 0 such that for all `x ∈ ae_seq_set`, `f i x` is equal to\n`(hf i).mk (f i) x` for all `i` and we have the pointwise property `p x (λ n, f n x)`. -/\ndef ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    (hf : ∀ (i : ι), ae_measurable (f i)) (p : α → (ι → β) → Prop) : set α :=\n  measure_theory.to_measurable μ\n      ((set_of\n          fun (x : α) =>\n            (∀ (i : ι), f i x = ae_measurable.mk (f i) (hf i) x) ∧ p x fun (n : ι) => f n x)ᶜ)ᶜ\n\n/-- A sequence of measurable functions that are equal to `f` and verify property `p` on the\nmeasurable set `ae_seq_set hf p`. -/\ndef ae_seq {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α] [measurable_space β]\n    {f : ι → α → β} {μ : measure_theory.measure α} (hf : ∀ (i : ι), ae_measurable (f i))\n    (p : α → (ι → β) → Prop) : ι → α → β :=\n  fun (i : ι) (x : α) =>\n    ite (x ∈ ae_seq_set hf p) (ae_measurable.mk (f i) (hf i) x) (nonempty.some sorry)\n\nnamespace ae_seq\n\n\ntheorem mk_eq_fun_of_mem_ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4}\n    [measurable_space α] [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), ae_measurable (f i)) {x : α}\n    (hx : x ∈ ae_seq_set hf p) (i : ι) : ae_measurable.mk (f i) (hf i) x = f i x :=\n  sorry\n\ntheorem ae_seq_eq_mk_of_mem_ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4}\n    [measurable_space α] [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), ae_measurable (f i)) {x : α}\n    (hx : x ∈ ae_seq_set hf p) (i : ι) : ae_seq hf p i x = ae_measurable.mk (f i) (hf i) x :=\n  sorry\n\ntheorem ae_seq_eq_fun_of_mem_ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4}\n    [measurable_space α] [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    {p : α → (ι → β) → Prop} (hf : ∀ (i : ι), ae_measurable (f i)) {x : α}\n    (hx : x ∈ ae_seq_set hf p) (i : ι) : ae_seq hf p i x = f i x :=\n  sorry\n\ntheorem prop_of_mem_ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    (hf : ∀ (i : ι), ae_measurable (f i)) {x : α} (hx : x ∈ ae_seq_set hf p) :\n    p x fun (n : ι) => ae_seq hf p n x :=\n  sorry\n\ntheorem fun_prop_of_mem_ae_seq_set {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    (hf : ∀ (i : ι), ae_measurable (f i)) {x : α} (hx : x ∈ ae_seq_set hf p) :\n    p x fun (n : ι) => f n x :=\n  sorry\n\ntheorem ae_seq_set_is_measurable {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    {hf : ∀ (i : ι), ae_measurable (f i)} : is_measurable (ae_seq_set hf p) :=\n  sorry\n\ntheorem measurable {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    (hf : ∀ (i : ι), ae_measurable (f i)) (p : α → (ι → β) → Prop) (i : ι) :\n    measurable (ae_seq hf p i) :=\n  measurable.ite ae_seq_set_is_measurable (ae_measurable.measurable_mk (hf i))\n    (dite (Nonempty α) (fun (hα : Nonempty α) => measurable_const)\n      fun (hα : ¬Nonempty α) =>\n        measurable_of_not_nonempty hα fun (x : α) => nonempty.some (_proof_1 i x))\n\ntheorem measure_compl_ae_seq_set_eq_zero {α : Type u_1} {β : Type u_2} {ι : Type u_4}\n    [measurable_space α] [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α}\n    {p : α → (ι → β) → Prop} [encodable ι] (hf : ∀ (i : ι), ae_measurable (f i))\n    (hp :\n      filter.eventually (fun (x : α) => p x fun (n : ι) => f n x) (measure_theory.measure.ae μ)) :\n    coe_fn μ (ae_seq_set hf pᶜ) = 0 :=\n  sorry\n\ntheorem ae_seq_eq_mk_ae {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    [encodable ι] (hf : ∀ (i : ι), ae_measurable (f i))\n    (hp :\n      filter.eventually (fun (x : α) => p x fun (n : ι) => f n x) (measure_theory.measure.ae μ)) :\n    filter.eventually (fun (a : α) => ∀ (i : ι), ae_seq hf p i a = ae_measurable.mk (f i) (hf i) a)\n        (measure_theory.measure.ae μ) :=\n  sorry\n\ntheorem ae_seq_eq_fun_ae {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    [encodable ι] (hf : ∀ (i : ι), ae_measurable (f i))\n    (hp :\n      filter.eventually (fun (x : α) => p x fun (n : ι) => f n x) (measure_theory.measure.ae μ)) :\n    filter.eventually (fun (a : α) => ∀ (i : ι), ae_seq hf p i a = f i a)\n        (measure_theory.measure.ae μ) :=\n  measure_theory.measure_mono_null\n    (fun (x : α) =>\n      mt fun (hx : x ∈ ae_seq_set hf p) (i : ι) => ae_seq_eq_fun_of_mem_ae_seq_set hf hx i)\n    (measure_compl_ae_seq_set_eq_zero hf hp)\n\ntheorem ae_seq_n_eq_fun_n_ae {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α]\n    [measurable_space β] {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop}\n    [encodable ι] (hf : ∀ (i : ι), ae_measurable (f i))\n    (hp : filter.eventually (fun (x : α) => p x fun (n : ι) => f n x) (measure_theory.measure.ae μ))\n    (n : ι) : filter.eventually_eq (measure_theory.measure.ae μ) (ae_seq hf p n) (f n) :=\n  iff.mp measure_theory.ae_all_iff (ae_seq_eq_fun_ae hf hp) n\n\ntheorem supr {α : Type u_1} {β : Type u_2} {ι : Type u_4} [measurable_space α] [measurable_space β]\n    {f : ι → α → β} {μ : measure_theory.measure α} {p : α → (ι → β) → Prop} [complete_lattice β]\n    [encodable ι] (hf : ∀ (i : ι), ae_measurable (f i))\n    (hp :\n      filter.eventually (fun (x : α) => p x fun (n : ι) => f n x) (measure_theory.measure.ae μ)) :\n    filter.eventually_eq (measure_theory.measure.ae μ) (supr fun (n : ι) => ae_seq hf p n)\n        (supr fun (i : ι) => f i) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/ae_measurable_sequence_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.30309521697015945}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport tactic.ext\nimport tactic.lint\n\n/-!\n# Functors\n\nThis module provides additional lemmas, definitions, and instances for `functor`s.\n\n## Main definitions\n\n* `const α` is the functor that sends all types to `α`.\n* `add_const α` is `const α` but for when `α` has an additive structure.\n* `comp F G` for functors `F` and `G` is the functor composition of `F` and `G`.\n* `liftp` and `liftr` respectively lift predicates and relations on a type `α`\n  to `F α`.  Terms of `F α` are considered to, in some sense, contain values of type `α`.\n\n## Tags\n\nfunctor, applicative\n-/\n\nattribute [functor_norm] seq_assoc pure_seq_eq_map map_pure seq_map_assoc map_seq\n\nuniverse variables u v w\n\nsection functor\n\nvariables {F : Type u → Type v}\nvariables {α β γ : Type u}\nvariables [functor F] [is_lawful_functor F]\n\nlemma functor.map_id : (<$>) id = (id : F α → F α) :=\nby apply funext; apply id_map\n\nlemma functor.map_comp_map (f : α → β) (g : β → γ) :\n  ((<$>) g ∘ (<$>) f : F α → F γ) = (<$>) (g ∘ f) :=\nby apply funext; intro; rw comp_map\n\ntheorem functor.ext {F} : ∀ {F1 : functor F} {F2 : functor F}\n  [@is_lawful_functor F F1] [@is_lawful_functor F F2]\n  (H : ∀ α β (f : α → β) (x : F α),\n    @functor.map _ F1 _ _ f x = @functor.map _ F2 _ _ f x),\n  F1 = F2\n| ⟨m, mc⟩ ⟨m', mc'⟩ H1 H2 H :=\nbegin\n  cases show @m = @m', by funext α β f x; apply H,\n  congr, funext α β,\n  have E1 := @map_const_eq _ ⟨@m, @mc⟩ H1,\n  have E2 := @map_const_eq _ ⟨@m, @mc'⟩ H2,\n  exact E1.trans E2.symm\nend\n\nend functor\n\n/-- Introduce the `id` functor. Incidentally, this is `pure` for\n`id` as a `monad` and as an `applicative` functor. -/\ndef id.mk {α : Sort u} : α → id α := id\n\nnamespace functor\n\n/-- `const α` is the constant functor, mapping every type to `α`. When\n`α` has a monoid structure, `const α` has an `applicative` instance.\n(If `α` has an additive monoid structure, see `functor.add_const`.) -/\n@[nolint unused_arguments]\ndef const (α : Type*) (β : Type*) := α\n\n/-- `const.mk` is the canonical map `α → const α β` (the identity), and\nit can be used as a pattern to extract this value. -/\n@[pattern] def const.mk {α β} (x : α) : const α β := x\n\n/-- `const.mk'` is `const.mk` but specialized to map `α` to\n`const α punit`, where `punit` is the terminal object in `Type*`. -/\ndef const.mk' {α} (x : α) : const α punit := x\n\n/-- Extract the element of `α` from the `const` functor. -/\ndef const.run {α β} (x : const α β) : α := x\n\nnamespace const\n\nprotected lemma ext {α β} {x y : const α β} (h : x.run = y.run) : x = y := h\n\n/-- The map operation of the `const γ` functor. -/\n@[nolint unused_arguments]\nprotected def map {γ α β} (f : α → β) (x : const γ β) : const γ α := x\n\ninstance {γ} : functor (const γ) :=\n{ map := @const.map γ }\n\ninstance {γ} : is_lawful_functor (const γ) :=\nby constructor; intros; refl\n\ninstance {α β} [inhabited α] : inhabited (const α β) :=\n⟨(default _ : α)⟩\n\nend const\n\n/-- `add_const α` is a synonym for constant functor `const α`, mapping\nevery type to `α`. When `α` has a additive monoid structure,\n`add_const α` has an `applicative` instance. (If `α` has a\nmultiplicative monoid structure, see `functor.const`.) -/\ndef add_const (α : Type*) := const α\n\n/-- `add_const.mk` is the canonical map `α → add_const α β`, which is the identity,\nwhere `add_const α β = const α β`. It can be used as a pattern to extract this value. -/\n@[pattern]\ndef add_const.mk {α β} (x : α) : add_const α β := x\n\n/-- Extract the element of `α` from the constant functor. -/\ndef add_const.run {α β} : add_const α β → α := id\n\ninstance add_const.functor {γ} : functor (add_const γ) :=\n@const.functor γ\n\ninstance add_const.is_lawful_functor {γ} : is_lawful_functor (add_const γ) :=\n@const.is_lawful_functor γ\n\ninstance {α β} [inhabited α] : inhabited (add_const α β) :=\n⟨(default _ : α)⟩\n\n/-- `functor.comp` is a wrapper around `function.comp` for types.\n    It prevents Lean's type class resolution mechanism from trying\n    a `functor (comp F id)` when `functor F` would do. -/\ndef comp (F : Type u → Type w) (G : Type v → Type u) (α : Type v) : Type w :=\nF $ G α\n\n/-- Construct a term of `comp F G α` from a term of `F (G α)`, which is the same type.\nCan be used as a pattern to extract a term of `F (G α)`. -/\n@[pattern] def comp.mk {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : F (G α)) : comp F G α := x\n\n/-- Extract a term of `F (G α)` from a term of `comp F G α`, which is the same type. -/\ndef comp.run {F : Type u → Type w} {G : Type v → Type u} {α : Type v}\n  (x : comp F G α) : F (G α) := x\n\nnamespace comp\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nprotected lemma ext\n  {α} {x y : comp F G α} : x.run = y.run → x = y := id\n\ninstance {α} [inhabited (F (G α))] : inhabited (comp F G α) :=\n⟨(default _ : F (G α))⟩\n\nvariables [functor F] [functor G]\n\n/-- The map operation for the composition `comp F G` of functors `F` and `G`. -/\nprotected def map {α β : Type v} (h : α → β) : comp F G α → comp F G β\n| (comp.mk x) := comp.mk ((<$>) h <$> x)\n\ninstance : functor (comp F G) := { map := @comp.map F G _ _ }\n\n@[functor_norm] lemma map_mk {α β} (h : α → β) (x : F (G α)) :\n  h <$> comp.mk x = comp.mk ((<$>) h <$> x) := rfl\n\n@[simp] protected lemma run_map {α β} (h : α → β) (x : comp F G α) :\n  (h <$> x).run = (<$>) h <$> x.run := rfl\n\nvariables [is_lawful_functor F] [is_lawful_functor G]\nvariables {α β γ : Type v}\n\nprotected lemma id_map : ∀ (x : comp F G α), comp.map id x = x\n| (comp.mk x) := by simp [comp.map, functor.map_id]\n\nprotected lemma comp_map (g' : α → β) (h : β → γ) : ∀ (x : comp F G α),\n           comp.map (h ∘ g') x = comp.map h (comp.map g' x)\n| (comp.mk x) := by simp [comp.map, functor.map_comp_map g' h] with functor_norm\n\ninstance : is_lawful_functor (comp F G) :=\n{ id_map := @comp.id_map F G _ _ _ _,\n  comp_map := @comp.comp_map F G _ _ _ _ }\n\ntheorem functor_comp_id {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor F id _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor F id _ _ _ _) _ (λ α β f x, rfl)\n\ntheorem functor_id_comp {F} [AF : functor F] [is_lawful_functor F] :\n  @comp.functor id F _ _ = AF :=\n@functor.ext F _ AF (@comp.is_lawful_functor id F _ _ _ _) _ (λ α β f x, rfl)\n\nend comp\n\nnamespace comp\n\nopen function (hiding comp)\nopen functor\n\nvariables {F : Type u → Type w} {G : Type v → Type u}\n\nvariables [applicative F] [applicative G]\n\n/-- The `<*>` operation for the composition of applicative functors. -/\nprotected def seq {α β : Type v} : comp F G (α → β) → comp F G α → comp F G β\n| (comp.mk f) (comp.mk x) := comp.mk $ (<*>) <$> f <*> x\n\ninstance : has_pure (comp F G) :=\n⟨λ _ x, comp.mk $ pure $ pure x⟩\n\ninstance : has_seq (comp F G) :=\n⟨λ _ _ f x, comp.seq f x⟩\n\n@[simp] protected lemma run_pure {α : Type v} :\n  ∀ x : α, (pure x : comp F G α).run = pure (pure x)\n| _ := rfl\n\n@[simp] protected lemma run_seq {α β : Type v} (f : comp F G (α → β)) (x : comp F G α) :\n  (f <*> x).run = (<*>) <$> f.run <*> x.run := rfl\n\ninstance : applicative (comp F G) :=\n{ map := @comp.map F G _ _,\n  seq := @comp.seq F G _ _,\n  ..comp.has_pure }\n\nend comp\n\nvariables {F : Type u → Type u} [functor F]\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`,\npredicate `liftp p x` holds iff every value contained by `x` satisfies `p`. -/\ndef liftp {α : Type u} (p : α → Prop) (x : F α) : Prop :=\n∃ u : F (subtype p), subtype.val <$> u = x\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`, then\n`liftr r x y` relates `x` and `y` iff (1) `x` and `y` have the same shape and\n(2) we can pair values `a` from `x` and `b` from `y` so that `r a b` holds. -/\ndef liftr {α : Type u} (r : α → α → Prop) (x y : F α) : Prop :=\n∃ u : F {p : α × α // r p.fst p.snd},\n  (λ t : {p : α × α // r p.fst p.snd}, t.val.fst) <$> u = x ∧\n  (λ t : {p : α × α // r p.fst p.snd}, t.val.snd) <$> u = y\n\n/-- If we consider `x : F α` to, in some sense, contain values of type `α`, then\n`supp x` is the set of values of type `α` that `x` contains. -/\ndef supp {α : Type u} (x : F α) : set α := { y : α | ∀ ⦃p⦄, liftp p x → p y }\n\ntheorem of_mem_supp {α : Type u} {x : F α} {p : α → Prop} (h : liftp p x) :\n  ∀ y ∈ supp x, p y :=\nλ y hy, hy h\n\nend functor\n\nnamespace ulift\n\ninstance : functor ulift :=\n{ map := λ α β f, up ∘ f ∘ down }\n\nend ulift\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.30300370104236013}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nbegin\n  assume A : set (euclidean_space ℝ (fin n)),\n  assume hA : is_open A,\n  assume hA2 : is_covering A,\n\n  have h1 : ∀ m : ℕ, ∃ Cm : set (euclidean_space ℝ (fin n)), is_open Cm ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → x ∈ A) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A ∧ x ∈ a) ∧ (∀ x : euclidean_space ℝ (fin n), x ∈ Cm → ∃ a : euclidean_space ℝ (fin n), a ∈ A\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A ∩ B) ⊆ A, from by auto [set.inter_subset_left],\n  have h3 : (A ∩ B) ⊆ S, from by auto [set.subset.trans],\n  show (A ∩ B) ∈  𝒫 S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from by auto using [use (a⁻¹ * b)],\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from by auto using [use b * a⁻¹], \n\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from by auto [h1],\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from by auto [h2],\n\n  have h5 : ∀ a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : ∀ a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  ℝn_paracompact (n : ℕ) : paracompact_space (euclidean_space ℝ (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.685949467848392, "lm_q2_score": 0.44167300566462553, "lm_q1q2_score": 0.3029653631986497}}
{"text": "import Lean\n\ndef checkWithMkMatcherInput (matcher : Lean.Name) : Lean.MetaM Unit :=\n  Lean.Meta.Match.withMkMatcherInput matcher fun input => do\n  let res ← Lean.Meta.Match.mkMatcher input\n  let origMatcher ← Lean.getConstInfo matcher\n  if not <| input.matcherName == matcher then\n    throwError \"matcher name not reconstructed correctly: {matcher} ≟ {input.matcherName}\"\n\n  let lCtx ← Lean.getLCtx\n  let fvars := Lean.collectFVars {} res.matcher\n  let closure := Lean.Meta.Closure.mkLambda (fvars.fvarSet.toList.toArray.map lCtx.get!) res.matcher\n\n  let origTy := origMatcher.value!\n  let newTy := closure\n  if not <| ←Lean.Meta.isDefEq origTy newTy then\n    throwError \"matcher {matcher} does not round-trip correctly:\\n{origTy} ≟ {newTy}\"\n\nset_option smartUnfolding false\n\ndef f (xs : List Nat) : List Bool :=\nxs.map fun\n  | 0 => true\n  | _ => false\n#eval checkWithMkMatcherInput ``f.match_1\n\n#eval f [1, 2, 0, 2]\n\ntheorem ex1 : f [1, 0, 2] = [false, true, false] :=\nrfl\n\n#check f\n\nset_option pp.raw true\nset_option pp.raw.maxDepth 10\nset_option trace.Elab.step true in\ndef g (xs : List Nat) : List Bool :=\nxs.map <| by {\n  intro\n    | 0 => exact true\n    | _ => exact false\n}\n\ntheorem ex2 : g [1, 0, 2] = [false, true, false] :=\nrfl\n\ntheorem ex3 {p q r : Prop} : p ∨ q → r → (q ∧ r) ∨ (p ∧ r) :=\nby intro\n | Or.inl hp, h => { apply Or.inr; apply And.intro; assumption; assumption }\n | Or.inr hq, h => { apply Or.inl; exact ⟨hq, h⟩ }\n#eval checkWithMkMatcherInput ``ex3.match_1\n\ninductive C\n| mk₁ : Nat → C\n| mk₂ : Nat → Nat → C\n\ndef C.x : C → Nat\n| C.mk₁ x   => x\n| C.mk₂ x _ => x\n#eval checkWithMkMatcherInput ``C.x.match_1\n\ndef head : {α : Type} → List α → Option α\n| _, a::as => some a\n| _, _     => none\n#eval checkWithMkMatcherInput ``head.match_1\n\ntheorem ex4 : head [1, 2] = some 1 :=\nrfl\n\ndef head2 : {α : Type} → List α → Option α :=\n@fun\n  | _, a::as => some a\n  | _, _     => none\n\ntheorem ex5 : head2 [1, 2] = some 1 :=\nrfl\n\ndef head3 {α : Type} (xs : List α) : Option α :=\nlet rec aux : {α : Type} → List α → Option α\n  | _, a::as => some a\n  | _, _     => none;\naux xs\n\ntheorem ex6 : head3 [1, 2] = some 1 :=\nrfl\n\ninductive Vec.{u} (α : Type u) : Nat → Type u\n| nil : Vec α 0\n| cons {n} (head : α) (tail : Vec α n) : Vec α (n+1)\n\ndef Vec.mapHead1 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _,   nil,       nil,       f => none\n| _, cons a as, cons b bs,   f => some (f a b)\n#eval checkWithMkMatcherInput ``Vec.mapHead1.match_1\n\ndef Vec.mapHead2 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, @cons _ n a as, cons b bs,   f => some (f a b)\n#eval checkWithMkMatcherInput ``Vec.mapHead2.match_1\n\ndef Vec.mapHead3 {α β δ} : {n : Nat} → Vec α n → Vec β n → (α → β → δ) → Option δ\n| _, nil,            nil,         f => none\n| _, cons (tail := as) (head := a), cons b bs,   f => some (f a b)\n\ninductive Foo\n| mk₁ (x y z w : Nat)\n| mk₂ (x y z w : Nat)\n\ndef Foo.z : Foo → Nat\n| mk₁ (z := z) .. => z\n| mk₂ (z := z) .. => z\n#eval checkWithMkMatcherInput ``Foo.z.match_1\n\n#eval (Foo.mk₁ 10 20 30 40).z\n\ntheorem ex7 : (Foo.mk₁ 10 20 30 40).z = 30 :=\nrfl\n\ndef Foo.addY? : Foo × Foo → Option Nat\n| (mk₁ (y := y₁) .., mk₁ (y := y₂) ..) => some (y₁ + y₂)\n| _ => none\n#eval checkWithMkMatcherInput ``Foo.addY?.match_1\n\n#eval Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40)\n\ntheorem ex8 : Foo.addY? (Foo.mk₁ 1 2 3 4, Foo.mk₁ 10 20 30 40) = some 22 :=\nrfl\n\ninstance {α} : Inhabited (Sigma fun m => Vec α m) :=\n⟨⟨0, Vec.nil⟩⟩\n\npartial def filter {α} (p : α → Bool) : {n : Nat} → Vec α n → Sigma fun m => Vec α m\n| _, Vec.nil        => ⟨0, Vec.nil⟩\n| _, Vec.cons x xs  => match p x, filter p xs with\n  | true,  ⟨_, ys⟩ => ⟨_, Vec.cons x ys⟩\n  | false, ys      => ys\n#eval checkWithMkMatcherInput ``filter.match_1\n\ninductive Bla\n| ofNat  (x : Nat)\n| ofBool (x : Bool)\n\ndef Bla.optional? : Bla → Option Nat\n| ofNat x  => some x\n| ofBool _ => none\n#eval checkWithMkMatcherInput ``Bla.optional?.match_1\n\ndef Bla.isNat? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch b.optional? with\n| some y => some ⟨y, rfl⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat?.match_1\n\ndef foo (b : Bla) : Option Nat := b.optional?\ntheorem fooEq (b : Bla) : foo b = b.optional? :=\nrfl\n\ndef Bla.isNat2? (b : Bla) : Option { x : Nat // optional? b = some x } :=\nmatch h:foo b with\n| some y => some ⟨y, Eq.trans (fooEq b).symm h⟩\n| none   => none\n#eval checkWithMkMatcherInput ``Bla.isNat2?.match_1\n\ndef foo2 (x : Nat) : Nat :=\nmatch x, rfl : (y : Nat) → x = y → Nat with\n| 0,   h => 0\n| x+1, h => 1\n#eval checkWithMkMatcherInput ``foo2.match_1\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/match1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118791767282, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.30287583680792785}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel, Scott Morrison\n-/\nimport algebra.homology.exact\nimport category_theory.limits.shapes.biproducts\nimport category_theory.adjunction.limits\nimport category_theory.limits.preserves.finite\n\n/-!\n# Projective objects and categories with enough projectives\n\nAn object `P` is called projective if every morphism out of `P` factors through every epimorphism.\n\nA category `C` has enough projectives if every object admits an epimorphism from some\nprojective object.\n\n`projective.over X` picks an arbitrary such projective object,\nand `projective.π X : projective.over X ⟶ X` is the corresponding epimorphism.\n\nGiven a morphism `f : X ⟶ Y`, `projective.left f` is a projective object over `kernel f`,\nand `projective.d f : projective.left f ⟶ X` is the morphism `π (kernel f) ≫ kernel.ι f`.\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverses v u\n\nnamespace category_theory\nvariables {C : Type u} [category.{v} C]\n\n/--\nAn object `P` is called projective if every morphism out of `P` factors through every epimorphism.\n-/\nclass projective (P : C) : Prop :=\n(factors : ∀ {E X : C} (f : P ⟶ X) (e : E ⟶ X) [epi e], ∃ f', f' ≫ e = f)\n\nsection\n\n/--\nA projective presentation of an object `X` consists of an epimorphism `f : P ⟶ X`\nfrom some projective object `P`.\n-/\n@[nolint has_nonempty_instance]\nstructure projective_presentation (X : C) :=\n(P : C)\n(projective : projective P . tactic.apply_instance)\n(f : P ⟶ X)\n(epi : epi f . tactic.apply_instance)\n\nattribute [instance] projective_presentation.projective projective_presentation.epi\n\nvariables (C)\n\n/-- A category \"has enough projectives\" if for every object `X` there is a projective object `P` and\n    an epimorphism `P ↠ X`. -/\nclass enough_projectives : Prop :=\n(presentation : ∀ (X : C), nonempty (projective_presentation X))\n\nend\n\nnamespace projective\n\n/--\nAn arbitrarily chosen factorisation of a morphism out of a projective object through an epimorphism.\n-/\ndef factor_thru {P X E : C} [projective P] (f : P ⟶ X) (e : E ⟶ X) [epi e] : P ⟶ E :=\n(projective.factors f e).some\n\n@[simp] lemma factor_thru_comp {P X E : C} [projective P] (f : P ⟶ X) (e : E ⟶ X) [epi e] :\n  factor_thru f e ≫ e = f :=\n(projective.factors f e).some_spec\n\nsection\nopen_locale zero_object\n\ninstance zero_projective [has_zero_object C] [has_zero_morphisms C] : projective (0 : C) :=\n{ factors := λ E X f e epi, by { use 0, ext, }}\n\nend\n\nlemma of_iso {P Q : C} (i : P ≅ Q) (hP : projective P) : projective Q :=\nbegin\n  fsplit,\n  introsI E X f e e_epi,\n  obtain ⟨f', hf'⟩ := projective.factors (i.hom ≫ f) e,\n  exact ⟨i.inv ≫ f', by simp [hf']⟩\nend\n\nlemma iso_iff {P Q : C} (i : P ≅ Q) : projective P ↔ projective Q :=\n⟨of_iso i, of_iso i.symm⟩\n\n/-- The axiom of choice says that every type is a projective object in `Type`. -/\ninstance (X : Type u) : projective X :=\n{ factors := λ E X' f e epi,\n  ⟨λ x, ((epi_iff_surjective _).mp epi (f x)).some,\n  by { ext x, exact ((epi_iff_surjective _).mp epi (f x)).some_spec, }⟩ }\n\ninstance Type.enough_projectives : enough_projectives (Type u) :=\n{ presentation := λ X, ⟨{ P := X, f := 𝟙 X, }⟩, }\n\ninstance {P Q : C} [has_binary_coproduct P Q] [projective P] [projective Q] :\n  projective (P ⨿ Q) :=\n{ factors := λ E X' f e epi, by exactI\n  ⟨coprod.desc (factor_thru (coprod.inl ≫ f) e) (factor_thru (coprod.inr ≫ f) e), by tidy⟩, }\n\nsection\nlocal attribute [tidy] tactic.discrete_cases\n\ninstance {β : Type v} (g : β → C) [has_coproduct g] [∀ b, projective (g b)] :\n  projective (∐ g) :=\n{ factors := λ E X' f e epi, by exactI\n  ⟨sigma.desc (λ b, factor_thru (sigma.ι g b ≫ f) e), by tidy⟩, }\n\nend\n\ninstance {P Q : C} [has_zero_morphisms C] [has_binary_biproduct P Q]\n  [projective P] [projective Q] :\n  projective (P ⊞ Q) :=\n{ factors := λ E X' f e epi, by exactI\n  ⟨biprod.desc (factor_thru (biprod.inl ≫ f) e) (factor_thru (biprod.inr ≫ f) e), by tidy⟩, }\n\ninstance {β : Type v} (g : β → C) [has_zero_morphisms C] [has_biproduct g]\n  [∀ b, projective (g b)] : projective (⨁ g) :=\n{ factors := λ E X' f e epi, by exactI\n  ⟨biproduct.desc (λ b, factor_thru (biproduct.ι g b ≫ f) e), by tidy⟩, }\n\nlemma projective_iff_preserves_epimorphisms_coyoneda_obj (P : C) :\n  projective P ↔ (coyoneda.obj (op P)).preserves_epimorphisms :=\n⟨λ hP, ⟨λ X Y f hf, (epi_iff_surjective _).2 $ λ g, have projective (unop (op P)), from hP,\n  by exactI ⟨factor_thru g f, factor_thru_comp _ _⟩⟩,\n λ h, ⟨λ E X f e he, by exactI (epi_iff_surjective _).1\n  (infer_instance : epi ((coyoneda.obj (op P)).map e)) f⟩⟩\n\nsection enough_projectives\nvariables [enough_projectives C]\n\n/--\n`projective.over X` provides an arbitrarily chosen projective object equipped with\nan epimorphism `projective.π : projective.over X ⟶ X`.\n-/\ndef over (X : C) : C :=\n(enough_projectives.presentation X).some.P\n\ninstance projective_over (X : C) : projective (over X) :=\n(enough_projectives.presentation X).some.projective\n\n/--\nThe epimorphism `projective.π : projective.over X ⟶ X`\nfrom the arbitrarily chosen projective object over `X`.\n-/\ndef π (X : C) : over X ⟶ X :=\n(enough_projectives.presentation X).some.f\n\ninstance π_epi (X : C) : epi (π X) :=\n(enough_projectives.presentation X).some.epi\n\nsection\nvariables [has_zero_morphisms C] {X Y : C} (f : X ⟶ Y) [has_kernel f]\n\n/--\nWhen `C` has enough projectives, the object `projective.syzygies f` is\nan arbitrarily chosen projective object over `kernel f`.\n-/\n@[derive projective]\ndef syzygies : C := over (kernel f)\n\n/--\nWhen `C` has enough projectives,\n`projective.d f : projective.syzygies f ⟶ X` is the composition\n`π (kernel f) ≫ kernel.ι f`.\n\n(When `C` is abelian, we have `exact (projective.d f) f`.)\n-/\nabbreviation d : syzygies f ⟶ X :=\nπ (kernel f) ≫ kernel.ι f\n\nend\n\nend enough_projectives\n\nend projective\nnamespace adjunction\n\nvariables {D : Type*} [category D] {F : C ⥤ D} {G : D ⥤ C}\n\nlemma map_projective (adj : F ⊣ G) [G.preserves_epimorphisms] (P : C) (hP : projective P) :\n  projective (F.obj P) :=\n⟨λ X Y f g, begin\n  introI,\n  rcases hP.factors (adj.unit.app P ≫ G.map f) (G.map g),\n  use F.map w ≫ adj.counit.app X,\n  rw [category.assoc, ←adjunction.counit_naturality, ←category.assoc, ←F.map_comp, h],\n  simp,\nend⟩\n\nlemma projective_of_map_projective (adj : F ⊣ G) [full F] [faithful F] (P : C)\n  (hP : projective (F.obj P)) : projective P :=\n⟨λ X Y f g, begin\n  introI,\n  haveI := adj.left_adjoint_preserves_colimits,\n  rcases @hP.1 (F.map f) (F.map g),\n  use adj.unit.app _ ≫ G.map w ≫ (inv $ adj.unit.app _),\n  refine faithful.map_injective F _,\n  simpa\nend⟩\n\n/-- Given an adjunction `F ⊣ G` such that `G` preserves epis, `F` maps a projective presentation of\n`X` to a projective presentation of `F(X)`. -/\ndef map_projective_presentation (adj : F ⊣ G) [G.preserves_epimorphisms] (X : C)\n  (Y : projective_presentation X) : projective_presentation (F.obj X) :=\n{ P := F.obj Y.P,\n  projective := adj.map_projective _ Y.projective,\n  f := F.map Y.f,\n  epi := by haveI := adj.left_adjoint_preserves_colimits; apply_instance }\n\nend adjunction\nnamespace equivalence\n\nvariables {D : Type*} [category D] (F : C ≌ D)\n\n/-- Given an equivalence of categories `F`, a projective presentation of `F(X)` induces a\nprojective presentation of `X.` -/\ndef projective_presentation_of_map_projective_presentation\n  (X : C) (Y : projective_presentation (F.functor.obj X)) : projective_presentation X :=\n{ P := F.inverse.obj Y.P,\n  projective := adjunction.map_projective F.symm.to_adjunction Y.P Y.projective,\n  f := F.inverse.map Y.f ≫ F.unit_inv.app _,\n  epi := epi_comp _ _ }\n\nlemma enough_projectives_iff (F : C ≌ D) :\n  enough_projectives C ↔ enough_projectives D :=\nbegin\n  split,\n  all_goals { intro H, constructor, intro X, constructor },\n  { exact F.symm.projective_presentation_of_map_projective_presentation _\n      (nonempty.some (H.presentation (F.inverse.obj X))) },\n  { exact F.projective_presentation_of_map_projective_presentation X\n      (nonempty.some (H.presentation (F.functor.obj X))) },\nend\n\nend equivalence\nopen projective\nsection\nvariables [has_zero_morphisms C] [has_equalizers C] [has_images C]\n\n/--\nGiven a projective object `P` mapping via `h` into\nthe middle object `R` of a pair of exact morphisms `f : Q ⟶ R` and `g : R ⟶ S`,\nsuch that `h ≫ g = 0`, there is a lift of `h` to `Q`.\n-/\ndef exact.lift {P Q R S : C} [projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S)\n  (hfg : exact f g) (w : h ≫ g = 0) : P ⟶ Q :=\nfactor_thru\n  (factor_thru\n    (factor_thru_kernel_subobject g h w)\n    (image_to_kernel f g hfg.w))\n  (factor_thru_image_subobject f)\n\n@[simp] lemma exact.lift_comp {P Q R S : C} [projective P] (h : P ⟶ R) (f : Q ⟶ R) (g : R ⟶ S)\n  (hfg : exact f g) (w : h ≫ g = 0) : exact.lift h f g hfg w ≫ f = h :=\nbegin\n  simp [exact.lift],\n  conv_lhs { congr, skip, rw ← image_subobject_arrow_comp f, },\n  rw [←category.assoc, factor_thru_comp, ←image_to_kernel_arrow,\n    ←category.assoc, category_theory.projective.factor_thru_comp,\n    factor_thru_kernel_subobject_comp_arrow],\nend\n\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/preadditive/projective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3028758279225509}}
{"text": "import category_theory.adjunction.basic\nimport category_theory.limits.has_limits\nimport data.pfun\n\nopen category_theory category_theory.functor category_theory.limits\nuniverses u v\nvariables (𝒞 : Type) [category.{0} 𝒞]\n\ninductive bicompletion_aux : bool → Type 1\n| of_cat_obj : 𝒞 → bicompletion_aux ff\n| limit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) : bicompletion_aux ff\n| colimit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) : bicompletion_aux ff\n| of_cat_hom : Π {X Y : 𝒞}, (X ⟶ Y) → bicompletion_aux tt -- of_cat_obj X ⟶ of_cat_obj Y\n| limit_cone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt)\n  (X : 𝒟) (Y : bicompletion_aux ff) (f : bicompletion_aux tt) : -- F_obj X ⟶ Y\n  bicompletion_aux tt -- limit_obj F_obj F_hom ⟶ Y\n| colimit_cocone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) \n  (X : 𝒟) (Y : bicompletion_aux ff) (f : bicompletion_aux tt) : -- Y ⟶ F_obj X\n  bicompletion_aux tt -- Y ⟶ colimit_obj F_obj F_hom\n| is_limit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt)\n  (cone_obj : bicompletion_aux ff)\n  (cone : Π (X : 𝒟), bicompletion_aux tt) : -- cone_obj ⟶ F_obj X\n  bicompletion_aux tt -- cone_obj → limit_obj F_obj F_hom\n| is_colimit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux ff) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux tt) \n  (cocone_obj : bicompletion_aux ff)\n  (cocone : Π (X : 𝒟), bicompletion_aux tt) : -- F_obj X ⟶ cocone_obj\n  bicompletion_aux tt -- colimit_obj F_obj F_hom ⟶ cocone_obj\n| id (X : bicompletion_aux ff) : bicompletion_aux tt -- X ⟶ X\n\nnamespace bicompletion_aux\n\nvariable {𝒞}\n\n@[simp] def dom : Π (X : bicompletion_aux 𝒞 tt), bicompletion_aux 𝒞 ff\n| (@of_cat_hom _ _ X Y f) := of_cat_obj X \n| (@limit_cone_comp _ _ 𝒟 _ F_obj F_hom X _ _) := by exactI limit_obj F_obj @F_hom\n| (@is_limit _ _ 𝒟 _ F_obj F_hom cone_obj cone) := cone_obj\n| (@colimit_cocone_comp _ _ 𝒟 _ F_obj F_hom X Y f) := Y\n| (@is_colimit _ _ 𝒟 _ F_obj F_hom cocone_obj cocone) := by exactI colimit_obj F_obj @F_hom\n| (id X) := X\n\n@[simp] def cod : Π (X : bicompletion_aux 𝒞 tt), bicompletion_aux 𝒞 ff\n| (@of_cat_hom _ _ X Y f) := of_cat_obj Y \n| (@colimit_cocone_comp _ _ 𝒟 _ F_obj F_hom X _ _) := by exactI colimit_obj F_obj @F_hom\n| (@is_colimit _ _ 𝒟 _ F_obj F_hom cocone_obj cocone) := cocone_obj\n| (@limit_cone_comp _ _ 𝒟 _ F_obj F_hom X Y f) := Y\n| (@is_limit _ _ 𝒟 _ F_obj F_hom cone_obj cone) := by exactI limit_obj F_obj @F_hom\n| (id X) := X\n\nvariable (𝒞)\n\ndef obj₁ : Type 1 := bicompletion_aux 𝒞 ff\n\nvariable {𝒞}\nvariables {𝒟 : Type} [category.{0} 𝒟]\n\ndef hom₁ (X Y : obj₁ 𝒞) : Type 1 :=\n{ f : bicompletion_aux 𝒞 tt // f.dom = X ∧ f.cod = Y }\n\n@[simp] lemma coe_dom {X Y : obj₁ 𝒞} (f : hom₁ X Y) :\n  (@coe { f : bicompletion_aux 𝒞 tt // f.dom = X ∧ f.cod = Y } \n    (bicompletion_aux 𝒞 tt) _ f).dom = X := f.2.1\n\n@[simp] lemma coe_cod {X Y : obj₁ 𝒞} (f : hom₁ X Y) :\n  (@coe { f : bicompletion_aux 𝒞 tt // f.dom = X ∧ f.cod = Y } \n    (bicompletion_aux 𝒞 tt) _ f).cod = Y := f.2.2\n\ndef of_cat_obj₁ (X : 𝒞) : obj₁ 𝒞 := of_cat_obj X\n\ndef limit_obj₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) : obj₁ 𝒞 :=\nlimit_obj F_obj (λ X Y f, (F_hom f).1)\n\ndef colimit_obj₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) : obj₁ 𝒞 :=\ncolimit_obj F_obj (λ X Y f, (F_hom f).1)\n\ndef of_cat_hom₁ {X Y : 𝒞} (f : X ⟶ Y) : hom₁ (of_cat_obj X) (of_cat_obj Y) :=\n⟨of_cat_hom f, by simp⟩\n\ndef limit_cone_comp₁ (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₁ 𝒞} (f : hom₁ (F_obj X) Y) :\n  hom₁ (limit_obj₁ F_obj @F_hom) Y :=\n⟨limit_cone_comp F_obj (λ X Y f, (F_hom f).1) X Y f.1, by simp [limit_obj₁]⟩\n\ndef colimit_cocone_comp₁ (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₁ 𝒞} (f : hom₁ Y (F_obj X)) :\n  hom₁ Y (colimit_obj₁ F_obj @F_hom) :=\n⟨colimit_cocone_comp F_obj (λ X Y f, (F_hom f).1) X Y f.1, by simp [colimit_obj₁]⟩\n\ndef is_limit₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cone_obj : obj₁ 𝒞)\n  (cone : Π (X : 𝒟), hom₁ cone_obj (F_obj X)) :\n  hom₁ cone_obj (limit_obj₁ F_obj @F_hom) :=\n⟨is_limit F_obj (λ X Y f, (F_hom f).1) cone_obj (λ X, (cone X).1), by simp [limit_obj₁]⟩\n\ndef is_colimit₁ (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₁ 𝒞)\n  (cocone : Π (X : 𝒟), hom₁ (F_obj X) cocone_obj) :\n  hom₁ (colimit_obj₁ F_obj @F_hom) cocone_obj  :=\n⟨is_colimit F_obj (λ X Y f, (F_hom f).1) cocone_obj (λ X, (cocone X).1), by simp [colimit_obj₁]⟩\n\ndef id₁ (X : obj₁ 𝒞) : hom₁ X X := ⟨id X, rfl, rfl⟩\n\n@[elab_as_eliminator] def bicompletion_tt_rec_on \n  {motive : bicompletion_aux 𝒞 tt → Sort u} \n  (f : bicompletion_aux 𝒞 tt)\n  (of_cat_hom : Π {X Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom f))\n  (limit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt)\n    (ih : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)),\n      motive f → by exactI motive (limit_cone_comp F_obj @F_hom X Y f))\n  (colimit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt)\n    (ih : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)),\n      motive f → by exactI motive (colimit_cocone_comp F_obj @F_hom X Y f))\n  (is_limit : Π {𝒟 : Type} [_inst_2 : category 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cone_obj : bicompletion_aux 𝒞 ff)\n    (cone : 𝒟 → bicompletion_aux 𝒞 tt)\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (ih_cone : Π (X : 𝒟), motive (cone X)), \n    by exactI motive (is_limit F_obj @F_hom cone_obj cone))\n  (is_colimit : Π {𝒟 : Type} [_inst_2 : category 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cone_obj : bicompletion_aux 𝒞 ff)\n    (cone : 𝒟 → bicompletion_aux 𝒞 tt)\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (ih_cone : Π (X : 𝒟), motive (cone X)), \n    by exactI motive (is_colimit F_obj @F_hom cone_obj cone))\n  (id : Π (X : bicompletion_aux 𝒞 ff), motive X.id) :\n  motive f := sorry\n\ndef comp₁ (f : bicompletion_aux 𝒞 tt) : bicompletion_aux 𝒞 tt → part (bicompletion_aux 𝒞 tt) :=\nbicompletion_tt_rec_on f \n  begin\n    intros X Y f g,\n    refine bicompletion_tt_rec_on g _ _ _ _ _ _,\n    { intros B C g,\n      exact ⟨Y = B, λ h, by subst h; exact of_cat_hom (f ≫ g)⟩ },\n    { intros, exact part.none },\n    { introsI 𝒟 _ F_obj F_hom A B g ih₁ ih₂,\n      exact ih₂ >>= λ ih₂, return (colimit_cocone_comp F_obj @F_hom A B ih₂) },\n    { introsI 𝒟 _ F_obj F_hom A cone ih₁ ih₂,  \n      exact ⟨∀ (X : 𝒟), (ih₂ X).dom, λ h, is_limit F_obj @F_hom (of_cat_obj X) (λ X, (ih₂ X).get (h X))⟩ },\n    { intros, exact part.none },\n    { intros, exact return (of_cat_hom f) }\n  end  \n  begin\n    introsI 𝒟 _ F_obj F_hom A B f _ ih₂ g,\n    exact ih₂ g >>= λ ih₂, return (limit_cone_comp F_obj @F_hom A B ih₂)\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom A B f ih₁ ih₂ g,\n    revert ih₂ A,\n    refine bicompletion_tt_rec_on g _ _ _ _ _ _,\n    { intros, exact part.none },\n    { intros, exact part.none },\n    { introsI ℰ _ G_obj G_hom C Y g ih₃ ih₄ ih₂ A,\n      exact ih₄ ih₂ A >>= λ ih₄, return (colimit_cocone_comp G_obj @G_hom C Y ih₄) },\n    { introsI ℰ _ G_obj G_hom _ g ih₃ ih₄ ih₂ A,\n      exact ⟨Π (X : ℰ), (ih₄ X ih₂ A).dom, \n        λ h, is_limit G_obj @G_hom B (λ X, (ih₄ X ih₂ A).get (h X))⟩ },\n    { introsI ℰ _ G_obj G_hom cocone_obj cocone _ g ih₂ A,\n      exact ih₂ (cocone _) },\n    { intros A f _,\n      exact colimit_cocone_comp₂ F_obj @F_hom A f }\n  end\n  _ _ (λ _, return)\n  -- begin\n  --   introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ Z g,\n  --   revert ih₂,\n  --   refine hom₂_limit_obj_rec_on g _ _ _ _,\n  --   { introsI ℰ _ G_obj G_hom A B g ih₂,\n  --     exact ih₂ A g },\n  --   { introsI ℰ _ F_obj F_hom A ℱ _ G_obj G_hom g ih₃ ih₂,\n  --     exact colimit_cocone_comp₂ F_obj @F_hom A (ih₃ @ih₂) },\n  --   { introsI ℰ _ F_obj F_hom ℱ _ G_obj G_hom ih₃ ih₄ ih₂,\n  --     exact is_limit₂ F_obj @F_hom cone_obj (λ X, ih₄ _ @ih₂) },\n  --   { introsI ih,\n  --     exact is_limit₂ F_obj @F_hom cone_obj cone }\n  -- end\n  -- begin\n  --   introsI 𝒟 _ F_obj F_hom ih₁ cocone_obj cocone ih₂ Z g,\n  --   exact is_colimit₂ F_obj @F_hom Z (λ A, ih₂ A g)\n  -- end\n  -- (λ _ _ g, g)\n\ninductive valid_obj₁ : Π (X : obj₁ 𝒞), Prop\n| of_cat_obj (X : 𝒞) : valid_obj₁ (of_cat_obj X)\n| limit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (h : Π X : 𝒟, valid_obj₁ (F_obj X)) : \n  valid_obj₁ (limit_obj₁ F_obj @F_hom)\n| colimit_obj {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (h : Π X : 𝒟, valid_obj₁ (F_obj X)) :\n  valid_obj₁ (colimit_obj₁ F_obj @F_hom)\n\ndef valid_obj₁_limit_obj \n  {𝒟 : Type} [category.{0} 𝒟] {F_obj : 𝒟 → obj₁ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux 𝒞 tt}\n  (h : valid_obj₁ (limit_obj F_obj @F_hom)) :\n  Π (X : 𝒟), valid_obj₁ (F_obj X) :=\nbegin\n  generalize hX : limit_obj F_obj @F_hom = X,\n  rw hX at h,\n  induction h,\n  { simp * at * },\n  { simp [limit_obj₁] at hX,\n    rcases hX with ⟨hX₁, hX₂, hX₂, hX₄⟩,\n    subst hX₁,\n    simp at *,\n    subst hX₂,\n    assumption },\n  { simp [*, colimit_obj₁] at * }\nend\n\ndef valid_obj₁_colimit_obj \n  {𝒟 : Type} [category.{0} 𝒟] {F_obj : 𝒟 → obj₁ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → bicompletion_aux 𝒞 tt}\n  (h : valid_obj₁ (colimit_obj F_obj @F_hom)) :\n  Π (X : 𝒟), valid_obj₁ (F_obj X) :=\nbegin\n  generalize hX : colimit_obj F_obj @F_hom = X,\n  rw hX at h,\n  induction h,\n  { simp * at * },\n  { simp [*, limit_obj₁] at * },\n  { simp [colimit_obj₁] at hX,\n    rcases hX with ⟨hX₁, hX₂, hX₂, hX₄⟩,\n    subst hX₁,\n    simp at *,\n    subst hX₂,\n    assumption }\nend\n\n@[elab_as_eliminator] def hom_rec_on {motive : bicompletion_aux 𝒞 tt → Sort u}\n  (f : bicompletion_aux 𝒞 tt)\n  (of_cat_hom : Π {X Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom f))\n  (limit_cone_comp : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    motive f → motive (by exactI limit_cone_comp F_obj @F_hom X Y f))\n  (is_limit : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cone_obj : bicompletion_aux 𝒞 ff)\n    (cone : 𝒟 → bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    (Π (X : 𝒟), motive (cone X)) → motive (by exactI is_limit F_obj @F_hom cone_obj cone))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (X : 𝒟) (Y : bicompletion_aux 𝒞 ff)\n    (f : bicompletion_aux 𝒞 tt),\n    (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n    motive f → motive (by exactI colimit_cocone_comp F_obj @F_hom X Y f))\n  (is_colimit : Π {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → bicompletion_aux 𝒞 ff)\n   (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → bicompletion_aux 𝒞 tt) (cocone_obj : bicompletion_aux 𝒞 ff)\n   (cocone : 𝒟 → bicompletion_aux 𝒞 tt),\n     (Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) →\n     (Π (X : 𝒟), motive (cocone X)) → motive (by exactI is_colimit F_obj @F_hom cocone_obj cocone))\n  (id : Π (X : obj₁ 𝒞), motive (id X)) :\n  motive f :=\nhave ∀ b (f : bicompletion_aux 𝒞 b) (h : b = tt), motive (eq.rec_on h f) :=\n  begin\n    intros b f,\n    refine bicompletion_aux.rec_on f _ _ _ _ _ _ _ _ _,\n    { intros, simp at *, contradiction },\n    { intros, simp at *, contradiction },\n    { intros, simp at *, contradiction },\n    { intros X Y f _,\n      exact of_cat_hom f },\n    { introsI 𝒟 _ F_obj F_hom X Y f ih₁ ih₂ ih₃ ih₄ _,\n      exact limit_cone_comp F_obj @F_hom X Y f (λ X Y f, ih₂ f rfl) (ih₄ rfl) },\n    { introsI 𝒟 _ F_obj F_hom X Y f ih₁ ih₂ ih₃ ih₄ _,\n      exact colimit_cocone_comp F_obj @F_hom X Y f (λ X Y f, ih₂ f rfl) (ih₄ rfl) },\n    { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ ih₃ ih₄ _,\n      exact is_limit F_obj @F_hom cone_obj cone (λ X Y f, ih₂ f rfl) (λ X, ih₄ X rfl) },\n    { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ ih₃ ih₄ _,\n      exact is_colimit F_obj @F_hom cone_obj cone (λ X Y f, ih₂ f rfl) (λ X, ih₄ X rfl) },\n    { intros X _ _, exact id X }\n  end,\nthis tt f rfl\n\ninductive valid_hom₁ : Π {X Y : obj₁ 𝒞}, hom₁ X Y → Prop\n| of_cat_hom {X Y : 𝒞} (f : X ⟶ Y) : valid_hom₁ (of_cat_hom₁ f)\n| id (X : obj₁ 𝒞) : valid_hom₁ (id₁ X)\n| limit_cone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (obj_valid : ∀ X, valid_obj₁ (F_obj X))\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (X : 𝒟) {Y : obj₁ 𝒞} (f : hom₁ (F_obj X) Y) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (f_valid : valid_hom₁ f) :\n  valid_hom₁ (limit_cone_comp₁ F_obj @F_hom X f)\n| colimit_cocone_comp {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (obj_valid : ∀ X, valid_obj₁ (F_obj X))\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y)) \n  (X : 𝒟) {Y : obj₁ 𝒞} (f : hom₁ Y (F_obj X)) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (f_valid : valid_hom₁ f) :\n  valid_hom₁ (colimit_cocone_comp₁ F_obj @F_hom X f)\n| is_limit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (obj_valid : ∀ X, valid_obj₁ (F_obj X))\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cone_obj : obj₁ 𝒞)\n  (cone : Π (X : 𝒟), hom₁ cone_obj (F_obj X)) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (cone_valid : Π (X : 𝒟), valid_hom₁ (cone X)) :\n  valid_hom₁ (is_limit₁ F_obj @F_hom cone_obj cone)\n| is_colimit {𝒟 : Type} [category.{0} 𝒟] (F_obj : 𝒟 → obj₁ 𝒞)\n  (obj_valid : ∀ X, valid_obj₁ (F_obj X))\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₁ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₁ 𝒞)\n  (cocone : Π (X : 𝒟), hom₁ (F_obj X) cocone_obj) \n  (F_hom_valid : Π {X Y : 𝒟} (f : X ⟶ Y), valid_hom₁ (F_hom f))\n  (cocone_valid : Π (X : 𝒟), valid_hom₁ (cocone X)) :\n  valid_hom₁ (is_colimit₁ F_obj @F_hom cocone_obj cocone)\n\nvariable (𝒞)\n\ndef obj₂ : Type 1 := { X : obj₁ 𝒞 // valid_obj₁ X } \n\nvariable {𝒞}\n\ndef hom₂ (X Y : obj₂ 𝒞) : Type 1 := { f : hom₁ X.1 Y.1 // valid_hom₁ f }\n\nopen valid_hom₁\n\ndef of_cat_obj₂ (X : 𝒞) : obj₂ 𝒞 :=\n⟨of_cat_obj X, valid_obj₁.of_cat_obj _⟩ \n\nlemma of_cat_obj₂_injective : function.injective (@of_cat_obj₂ 𝒞 _) :=\nbegin\n  intros X Y hXY,\n  simp [of_cat_obj₂] at hXY,\n  injection hXY,\nend\n\ndef limit_obj₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) : obj₂ 𝒞 :=\n⟨limit_obj₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1), valid_obj₁.limit_obj _ _ (λ X, (F_obj X).2)⟩\n\nlemma limit_obj₂_injective {𝒟₁ 𝒟₂ : Type} [i₁ : category 𝒟₁] [i₂ : category 𝒟₂] \n  {F_obj₁ : 𝒟₁ → obj₂ 𝒞} {F_obj₂ : 𝒟₂ → obj₂ 𝒞} \n  {F_hom₁ : Π {X Y : 𝒟₁}, (X ⟶ Y) → hom₂ (F_obj₁ X) (F_obj₁ Y)}\n  {F_hom₂ : Π {X Y : 𝒟₂}, (X ⟶ Y) → hom₂ (F_obj₂ X) (F_obj₂ Y)}\n  (h : limit_obj₂ F_obj₁ @F_hom₁ = limit_obj₂ F_obj₂ @F_hom₂) : \n  𝒟₁ = 𝒟₂ ∧ i₁ == i₂ ∧ F_obj₁ == F_obj₂ ∧ @F_hom₁ == @F_hom₂ :=\nbegin\n  simp [limit_obj₂, limit_obj₁] at h,\n  injection h with h₁ h₂ h₃ h₄,\n  unfreezingI { subst h₁ },\n  rw heq_iff_eq at h₂,\n  unfreezingI { subst h₂ },\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₃,\n  rw [← function.funext_iff] at h₃,\n  dsimp at h₃,\n  subst h₃,\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₄,\n  simp,\n  ext,\n  simp *\nend\n\ndef colimit_obj₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) : obj₂ 𝒞 :=\n⟨colimit_obj₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1), valid_obj₁.colimit_obj _ _ (λ X, (F_obj X).2)⟩\n\nlemma colimit_obj₂_injective {𝒟₁ 𝒟₂ : Type} [i₁ : category 𝒟₁] [i₂ : category 𝒟₂] \n  {F_obj₁ : 𝒟₁ → obj₂ 𝒞} {F_obj₂ : 𝒟₂ → obj₂ 𝒞} \n  {F_hom₁ : Π {X Y : 𝒟₁}, (X ⟶ Y) → hom₂ (F_obj₁ X) (F_obj₁ Y)}\n  {F_hom₂ : Π {X Y : 𝒟₂}, (X ⟶ Y) → hom₂ (F_obj₂ X) (F_obj₂ Y)}\n  (h : colimit_obj₂ F_obj₁ @F_hom₁ = colimit_obj₂ F_obj₂ @F_hom₂) : \n  𝒟₁ = 𝒟₂ ∧ i₁ == i₂ ∧ F_obj₁ == F_obj₂ ∧ @F_hom₁ == @F_hom₂ :=\nbegin\n  simp [colimit_obj₂, colimit_obj₁] at h,\n  injection h with h₁ h₂ h₃ h₄,\n  unfreezingI { subst h₁ },\n  rw heq_iff_eq at h₂,\n  unfreezingI { subst h₂ },\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₃,\n  rw [← function.funext_iff] at h₃,\n  dsimp at h₃,\n  subst h₃,\n  simp [heq_iff_eq, function.funext_iff, subtype.coe_injective.eq_iff] at h₄,\n  simp,\n  ext,\n  simp *\nend\n\ndef of_cat_hom₂ {X Y : 𝒞} (f : X ⟶ Y) : hom₂ (of_cat_obj₂ X) (of_cat_obj₂ Y) :=\n⟨of_cat_hom₁ f, valid_hom₁.of_cat_hom _⟩ \n\ndef limit_cone_comp₂ (F_obj : 𝒟 → obj₂ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y) :\n  hom₂ (limit_obj₂ F_obj @F_hom) Y :=\n⟨limit_cone_comp₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) X f.1, \n  valid_hom₁.limit_cone_comp _ (λ X, (F_obj X).2) _ _ _ (λ X Y f, (F_hom f).2) f.2⟩\n\ndef colimit_cocone_comp₂ (F_obj : 𝒟 → obj₂ 𝒞)\n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) (X : 𝒟) \n  {Y : obj₂ 𝒞} (f : hom₂ Y (F_obj X)):\n  hom₂ Y (colimit_obj₂ F_obj @F_hom) :=\n⟨colimit_cocone_comp₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) X f.1, \n  valid_hom₁.colimit_cocone_comp _ (λ X, (F_obj X).2) _ _ _ (λ X Y f, (F_hom f).2) f.2⟩\n\ndef is_limit₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n  (cone_obj : obj₂ 𝒞)\n  (cone : Π (X : 𝒟), hom₂ cone_obj (F_obj X)) :\n  hom₂ cone_obj (limit_obj₂ F_obj @F_hom) :=\n⟨is_limit₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) cone_obj.1 (λ X, (cone X).1), \n  valid_hom₁.is_limit _ (λ X, (F_obj X).2) _ _ _ (λ X Y f, (F_hom f).2) (λ X, (cone X).2)⟩\n\ndef is_colimit₂ (F_obj : 𝒟 → obj₂ 𝒞) \n  (F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n  (cocone_obj : obj₂ 𝒞)\n  (cocone : Π (X : 𝒟), hom₂ (F_obj X) cocone_obj) :\n  hom₂ (colimit_obj₂ F_obj @F_hom) cocone_obj  :=\n⟨is_colimit₁ (λ X, (F_obj X).1) (λ X Y f, (F_hom f).1) cocone_obj.1 (λ X, (cocone X).1), \n  valid_hom₁.is_colimit _ (λ X, (F_obj X).2) _ _ _ (λ X Y f, (F_hom f).2) (λ X, (cocone X).2)⟩\n\ndef id₂ (X : obj₂ 𝒞) : hom₂ X X := ⟨id₁ X.1, valid_hom₁.id _⟩ \n\n@[elab_as_eliminator] protected def hom₂.rec_on \n  {motive : Π {X Y : obj₂ 𝒞} (f : hom₂ X Y), Sort*} {X Y : obj₂ 𝒞} (f : hom₂ X Y)\n  (of_cat_hom : Π {X Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom₂ f))\n  (limit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y)\n    (ih_f : motive f),\n      motive (by exactI limit_cone_comp₂ F_obj @F_hom X f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ Y (F_obj X))\n    (ih_f : motive f),\n      motive (by exactI colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f)) \n    (cone_obj : obj₂ 𝒞) (cone : Π (X : 𝒟), hom₂ cone_obj (F_obj X))\n    (ih_cone : Π (X : 𝒟), motive (cone X)),\n      motive (by exactI is_limit₂ F_obj @F_hom cone_obj cone))\n  (is_colimit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (ih_F_hom : Π {X Y : 𝒟} (f : by exactI X ⟶ Y), motive (F_hom f))\n    (cocone_obj : obj₂ 𝒞) (cocone : Π (X : 𝒟), hom₂ (F_obj X) cocone_obj)\n    (ih_cone : Π (X : 𝒟), motive (cocone X)),\n      motive (by exactI is_colimit₂ F_obj @F_hom cocone_obj cocone))\n  (id : Π (X : obj₂ 𝒞), motive (id₂ X)):\n  motive f :=\nbegin\n  cases X with X hX, cases Y with Y hY,\n  cases f with f hf,\n  rcases f with ⟨f, hfd, hfc⟩,\n  revert X Y hX hY,\n  refine hom_rec_on f _ _ _ _ _ _,\n  { intros A B f X Y hX hY hfd hfc hf,\n    dsimp at hfd hfc, substs hfc hfd,\n    exact of_cat_hom f },\n  { introsI 𝒟 _ F_obj F_hom A B g ih₁ ih₂ X Y hX hY hfd hfc hf,\n    dsimp at hfd hfc, substs hfc hfd,\n    have valid_F_obj : ∀ X, valid_obj₁ (F_obj X),\n    { cases hf, assumption },\n    let F_obj' : 𝒟 → obj₂ 𝒞 := λ X, ⟨F_obj X, valid_F_obj X⟩,\n    have valid_F_hom : ∀ X Y f, \n      ∃ (h₁ : (@F_hom X Y f).dom = F_obj X) \n      (h₂ : (@F_hom X Y f).cod = F_obj Y),\n      valid_hom₁ ⟨@F_hom X Y f, h₁, h₂⟩,\n    { cases hf, simpa },\n     let F_hom' : Π (X Y : 𝒟) (f : X ⟶ Y), hom₂ (F_obj' X) (F_obj' Y) :=\n      λ X Y f, ⟨⟨@F_hom X Y f, (valid_F_hom X Y f).fst, (valid_F_hom X Y f).snd.fst⟩,\n        (valid_F_hom X Y f).snd.snd⟩,\n    have valid_g : \n      ∃ (h₁ : g.dom = F_obj A) (h₂ : g.cod = B),\n      valid_hom₁ ⟨g, h₁, h₂⟩,\n    { cases hf, simpa },\n    let g' : hom₂ (F_obj' A) ⟨B, hY⟩ :=\n      ⟨⟨g, valid_g.fst, valid_g.snd.fst⟩, valid_g.snd.snd⟩,\n    exact limit_cone_comp F_obj' F_hom'\n      (λ X Y f, ih₁ f (F_obj X) (F_obj Y) (valid_F_obj _) (valid_F_obj _)\n          (valid_F_hom _ _ f).fst (valid_F_hom _ _ f).snd.fst\n          (valid_F_hom _ _ f).snd.snd) A g'\n          (ih₂ (F_obj' A).1 B (F_obj' A).2 hY g'.1.2.1 g'.1.2.2 g'.2) },\n  { introsI 𝒟 _ F_obj F_hom cone_obj cone ih₁ ih₂ X Y hX hY hfd hfc hf,\n    dsimp at hfd hfc,\n    substs hfc hfd,\n    have valid_F_obj : ∀ X, valid_obj₁ (F_obj X),\n    { cases hf, assumption },\n    let F_obj' : 𝒟 → obj₂ 𝒞 := λ X, ⟨F_obj X, valid_F_obj X⟩,\n    have valid_F_hom : ∀ X Y f, \n      ∃ (h₁ : (@F_hom X Y f).dom = F_obj X) \n      (h₂ : (@F_hom X Y f).cod = F_obj Y),\n      valid_hom₁ ⟨@F_hom X Y f, h₁, h₂⟩,\n    { cases hf, simpa },\n    let F_hom' : Π (X Y : 𝒟) (f : X ⟶ Y), hom₂ (F_obj' X) (F_obj' Y) :=\n      λ X Y f, ⟨⟨@F_hom X Y f, (valid_F_hom X Y f).fst, (valid_F_hom X Y f).snd.fst⟩,\n        (valid_F_hom X Y f).snd.snd⟩,\n    let cone_obj' : obj₂ 𝒞 := ⟨cone_obj, hX⟩,\n    have valid_cone : ∀ (X : 𝒟), ∃ (h₁ : (cone X).dom = cone_obj'.1)\n      (h₂ : (cone X).cod = (F_obj' X).1),\n      valid_hom₁ ⟨cone X, h₁, h₂⟩,\n    { cases hf, simpa },\n    let cone' : Π (X : 𝒟), hom₂ cone_obj' (F_obj' X) :=\n      λ X, ⟨⟨cone X, (valid_cone X).fst, (valid_cone X).snd.fst⟩, (valid_cone X).snd.snd⟩,\n    exact is_limit F_obj' F_hom'\n      (λ A B f, ih₁ f (F_obj A) (F_obj B) (F_obj' A).2 (F_obj' B).2\n        (valid_F_hom _ _ f).fst (valid_F_hom _ _ f).snd.fst\n        (valid_F_hom _ _ f).snd.snd)\n        cone_obj' cone'\n        (λ X, ih₂ X cone_obj'.1 (F_obj' X).1 cone_obj'.2 (F_obj' X).2\n            (cone' X).1.2.1 (cone' X).1.2.2 (cone' X).2) },\n  { introsI 𝒟 _ F_obj F_hom A B g ih₁ ih₂ X Y hX hY hfd hfc hf,\n    dsimp at hfd hfc, substs hfc hfd,\n    have valid_F_obj : ∀ X, valid_obj₁ (F_obj X),\n    { cases hf, assumption },\n    let F_obj' : 𝒟 → obj₂ 𝒞 := λ X, ⟨F_obj X, valid_F_obj X⟩,\n    have valid_F_hom : ∀ X Y f, \n      ∃ (h₁ : (@F_hom X Y f).dom = F_obj X) \n      (h₂ : (@F_hom X Y f).cod = F_obj Y),\n      valid_hom₁ ⟨@F_hom X Y f, h₁, h₂⟩,\n    { cases hf, simpa },\n    have valid_g : \n      ∃ (h₁ : g.dom = B) (h₂ : g.cod = F_obj A),\n      valid_hom₁ ⟨g, h₁, h₂⟩,\n    { cases hf, simpa },\n    let g' : hom₂ ⟨B, hX⟩ (F_obj' A) :=\n      ⟨⟨g, valid_g.fst, valid_g.snd.fst⟩, valid_g.snd.snd⟩,\n    let F_hom' : Π (X Y : 𝒟) (f : X ⟶ Y), hom₂ (F_obj' X) (F_obj' Y) :=\n      λ X Y f, ⟨⟨@F_hom X Y f, (valid_F_hom X Y f).fst, (valid_F_hom X Y f).snd.fst⟩,\n        (valid_F_hom X Y f).snd.snd⟩,\n    exact colimit_cocone_comp F_obj' F_hom'\n      (λ X Y f, ih₁ f (F_obj X) (F_obj Y) (valid_F_obj _) (valid_F_obj _)\n          (valid_F_hom _ _ f).fst (valid_F_hom _ _ f).snd.fst\n          (valid_F_hom _ _ f).snd.snd) A g'\n          (ih₂ B (F_obj' A).1 hX (F_obj' A).2 g'.1.2.1 g'.1.2.2 g'.2) },\n  { introsI 𝒟 _ F_obj F_hom cocone_obj cocone ih₁ ih₂ X Y hX hY hfd hfc hf,\n    dsimp at hfd hfc,\n    substs hfc hfd,\n    have valid_F_obj : ∀ X, valid_obj₁ (F_obj X),\n    { cases hf, assumption },\n    let F_obj' : 𝒟 → obj₂ 𝒞 := λ X, ⟨F_obj X, valid_F_obj X⟩,\n    have valid_F_hom : ∀ X Y f, \n      ∃ (h₁ : (@F_hom X Y f).dom = F_obj X) \n      (h₂ : (@F_hom X Y f).cod = F_obj Y),\n      valid_hom₁ ⟨@F_hom X Y f, h₁, h₂⟩,\n    { cases hf, simpa },\n    let F_hom' : Π (X Y : 𝒟) (f : X ⟶ Y), hom₂ (F_obj' X) (F_obj' Y) :=\n      λ X Y f, ⟨⟨@F_hom X Y f, (valid_F_hom X Y f).fst, (valid_F_hom X Y f).snd.fst⟩,\n        (valid_F_hom X Y f).snd.snd⟩,\n    let cocone_obj' : obj₂ 𝒞 := ⟨cocone_obj, hY⟩,\n    have valid_cocone : ∀ (X : 𝒟), ∃ (h₁ : (cocone X).dom = (F_obj' X).1)\n      (h₂ : (cocone X).cod = cocone_obj'.1),\n      valid_hom₁ ⟨cocone X, h₁, h₂⟩,\n    { cases hf, simpa },\n    let cocone' : Π (X : 𝒟), hom₂ (F_obj' X) cocone_obj' :=\n      λ X, ⟨⟨cocone X, (valid_cocone X).fst, (valid_cocone X).snd.fst⟩, (valid_cocone X).snd.snd⟩,\n    exact is_colimit F_obj' F_hom'\n      (λ A B f, ih₁ f (F_obj A) (F_obj B) (F_obj' A).2 (F_obj' B).2\n        (valid_F_hom _ _ f).fst (valid_F_hom _ _ f).snd.fst\n        (valid_F_hom _ _ f).snd.snd)\n        cocone_obj' cocone'\n        (λ X, ih₂ X (F_obj' X).1 cocone_obj'.1 (F_obj' X).2 cocone_obj'.2 \n            (cocone' X).1.2.1 (cocone' X).1.2.2 (cocone' X).2) },\n  { intros A B C hB _ hAB hBC _,\n    dsimp at hAB hBC, subst A, subst C, \n    exact id ⟨B, hB⟩ }\nend\n\ndef hom₂_of_cat_obj_rec_on\n  {motive : Π {X : 𝒞} {Y : obj₂ 𝒞} (f : hom₂ (of_cat_obj₂ X) Y), Sort*} \n  {X : 𝒞} {Y : obj₂ 𝒞} (f : hom₂ (of_cat_obj₂ X) Y)\n  (of_cat_hom : Π {Y : 𝒞} (f : X ⟶ Y), motive (of_cat_hom₂ f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (X : 𝒟) {Y : 𝒞} (f : hom₂ (of_cat_obj₂ Y) (F_obj X))\n    (ih_f : motive f),\n      motive (by exactI colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (cone_obj : 𝒞) (cone : Π (X : 𝒟), hom₂ (of_cat_obj₂ cone_obj) (F_obj X))\n    (ih_cone : Π (X : 𝒟), motive (cone X)),\n      motive (by exactI is_limit₂ F_obj @F_hom (of_cat_obj₂ cone_obj) cone)) \n  (id : motive (id₂ (of_cat_obj₂ X))):\n  motive f := \n@hom₂.rec_on 𝒞 _ (λ A B f, ∀ (h : A = of_cat_obj₂ X),\n  motive (show hom₂ (of_cat_obj₂ X) B, from eq.rec_on h f))\n  (of_cat_obj₂ X) Y f \n  (λ A B g h, begin\n      have := of_cat_obj₂_injective h,\n      subst this,\n      dsimp,\n      exact of_cat_hom g\n    end) \n  begin \n    intros,\n    simp [limit_obj₂, of_cat_obj₂, limit_obj₁] at h,\n    contradiction\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B g ih₂ h,\n    subst h,\n    exact colimit_cocone_comp _ _ _ _ (ih₂ rfl)\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ h,\n    subst h,\n    exact is_limit _ _ _ _ (λ A, ih₂ A rfl),\n  end \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂] at h,\n    contradiction\n  end\n  begin\n    intros X h,\n    subst h,\n    exact id\n  end\n  rfl\n\ndef hom₂_limit_obj_rec_on\n  {motive : Π {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n    {F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}, \n    hom₂ (by exactI limit_obj₂ F_obj @F_hom) Y → Sort*}\n  {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}\n  (f : hom₂ (limit_obj₂ F_obj @F_hom) Y)\n  (limit_cone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (X : 𝒟) {Y : obj₂ 𝒞} (f : hom₂ (F_obj X) Y),\n      by exactI motive (limit_cone_comp₂ F_obj @F_hom X f))\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (X : 𝒟)\n    {ℰ : Type} [category ℰ] (G_obj : ℰ → obj₂ 𝒞)\n    (G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y))\n    (f : hom₂ (by exactI limit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_f : by exactI motive f),\n      by exactI motive (colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    {ℰ : Type} [category ℰ] (G_obj : ℰ → obj₂ 𝒞)\n    (G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y))\n    (cone : Π (X : 𝒟), hom₂ (by exactI limit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_cone : Π (X : 𝒟), by exactI motive (cone X)),\n      by exactI motive (is_limit₂ F_obj @F_hom (limit_obj₂ G_obj @G_hom) cone))\n  (id : by exactI motive (id₂ (limit_obj₂ F_obj @F_hom))) :\n  motive f :=\n@hom₂.rec_on 𝒞 _ (λ A B f, ∀ (h : A = limit_obj₂ F_obj @F_hom),\n  motive (show hom₂ (limit_obj₂ F_obj @F_hom) B, from eq.rec_on h f))\n  (limit_obj₂ F_obj @F_hom) Y f \n  begin \n    intros,\n    simp [limit_obj₂, of_cat_obj₂, limit_obj₁] at h,\n    contradiction\n  end  \n  begin \n    introsI ℰ _ G_obj G_hom ih₁ A B g ih₂ h,\n    unfreezingI { rcases (limit_obj₂_injective h) with ⟨rfl, h₁, h₂, h₃⟩ },\n    unfreezingI { subst h₁, subst h₂, subst h₃ },\n    exact limit_cone_comp _ _ _ _\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B g ih₂ h,\n    subst h,\n    exact colimit_cocone_comp _ _ _ _ _ _ (ih₂ rfl)\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ h,\n    subst h,\n    exact is_limit _ _ _ _ _ (λ A, ih₂ A rfl),\n  end \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂, limit_obj₂] at h,\n    contradiction\n  end\n  begin\n    introsI X h,\n    subst h,\n    exact id\n  end\n  rfl\n\n@[elab_as_eliminator] def hom₂_colimit_obj_rec_on\n  {motive : Π {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n    {F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}, \n    hom₂ (by exactI colimit_obj₂ F_obj @F_hom) Y → Sort*}\n  {𝒟 : Type} [category 𝒟] {F_obj : 𝒟 → obj₂ 𝒞}\n  {F_hom : Π {X Y : 𝒟}, (X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)} {Y : obj₂ 𝒞}\n  (f : hom₂ (colimit_obj₂ F_obj @F_hom) Y)\n  (colimit_cocone_comp : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    (X : 𝒟) \n    {ℰ : Type} [category ℰ] {G_obj : ℰ → obj₂ 𝒞}\n    {G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y)} \n    (f : hom₂ (by exactI colimit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_f : by exactI motive f),\n      by exactI motive (by exactI colimit_cocone_comp₂ F_obj @F_hom X f))\n  (is_limit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y))\n    {ℰ : Type} [category ℰ] {G_obj : ℰ → obj₂ 𝒞}\n    {G_hom : Π {X Y : ℰ}, (by exactI X ⟶ Y) → hom₂ (G_obj X) (G_obj Y)}\n    (cone : Π (X : 𝒟), hom₂ (by exactI colimit_obj₂ G_obj @G_hom) (F_obj X))\n    (ih_cone : Π (X : 𝒟), by exactI motive (cone X)),\n      by exactI motive (by exactI is_limit₂ F_obj @F_hom _ cone))\n  (is_colimit : Π {𝒟 : Type} [category 𝒟] (F_obj : 𝒟 → obj₂ 𝒞)\n    (F_hom : Π {X Y : 𝒟}, (by exactI X ⟶ Y) → hom₂ (F_obj X) (F_obj Y)) \n    (cocone_obj : obj₂ 𝒞) (cocone : Π (X : 𝒟), hom₂ (F_obj X) cocone_obj),\n      by exactI motive (is_colimit₂ F_obj @F_hom cocone_obj cocone)) \n  (id : by exactI motive (id₂ (colimit_obj₂ F_obj @F_hom))):\n  motive f :=\n@hom₂.rec_on 𝒞 _ (λ A B f, ∀ (h : A = colimit_obj₂ F_obj @F_hom),\n  motive (show hom₂ (colimit_obj₂ F_obj @F_hom) B, from eq.rec_on h f))\n  (colimit_obj₂ F_obj @F_hom) Y f \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂, colimit_obj₁] at h,\n    contradiction\n  end \n  begin \n    intros,\n    simp [colimit_obj₂, of_cat_obj₂, colimit_obj₁, limit_obj₁, limit_obj₂] at h,\n    contradiction\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B f ih₂ h,\n    subst h,\n    exact colimit_cocone_comp _ _ _ _ (ih₂ rfl)\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ h,\n    subst h,\n    exact is_limit _ _ _ (λ X, ih₂ X rfl)\n  end\n  begin\n    introsI ℰ _ G_obj G_hom ih₁ cocone_obj cocone ih₂ h,\n    unfreezingI { rcases (colimit_obj₂_injective h) with ⟨rfl, h₁, h₂, h₃⟩ },\n    unfreezingI { subst h₁, subst h₂, subst h₃ },\n    exact is_colimit _ _ _ _\n  end\n  begin\n    introsI X h,\n    subst h,\n    exact id\n  end\n  rfl\n\ndef comp₂ {X Y : obj₂ 𝒞} (f : hom₂ X Y) : Π {Z : obj₂ 𝒞}, hom₂ Y Z → hom₂ X Z :=\nhom₂.rec_on f \n  begin\n    intros X Y f Z g,\n    refine hom₂_of_cat_obj_rec_on g _ _ _ _,\n    { intros B g,\n      exact of_cat_hom₂ (f ≫ g) },\n    { introsI 𝒟 _ F_obj F_hom A B g ih,\n      exact colimit_cocone_comp₂ F_obj @F_hom A ih },\n    { introsI 𝒟 _ F_obj F_hom A cone ih,\n      exact is_limit₂ F_obj @F_hom (of_cat_obj₂ X) (λ X, ih _) },\n    { exact of_cat_hom₂ f }\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B f ih₂ Z g,\n    refine limit_cone_comp₂ F_obj @F_hom A (ih₂ g),\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B f ih₂ Z g,\n    revert ih₂ A,\n    refine hom₂_colimit_obj_rec_on g _ _ _ _,\n    { introsI ℰ _ G_obj G_hom C ℱ _ H_obj H_hom ih₃ ih₄ A g ih₂,\n      refine colimit_cocone_comp₂ G_obj @G_hom C (ih₄ A g @ih₂) },\n    { introsI ℰ _ G_obj G_hom ℱ _ H_obj H_hom ih₃ ih₄ A g ih₂,\n      exact is_limit₂ G_obj @G_hom B (λ X, ih₄ X A g @ih₂) },\n    { introsI ℰ _ G_obj G_hom cocone_obj cocone A g ih₂,\n      exact ih₂ (cocone A) },\n    { intros A f _,\n      exact colimit_cocone_comp₂ F_obj @F_hom A f }\n  end \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cone_obj cone ih₂ Z g,\n    revert ih₂,\n    refine hom₂_limit_obj_rec_on g _ _ _ _,\n    { introsI ℰ _ G_obj G_hom A B g ih₂,\n      exact ih₂ A g },\n    { introsI ℰ _ F_obj F_hom A ℱ _ G_obj G_hom g ih₃ ih₂,\n      exact colimit_cocone_comp₂ F_obj @F_hom A (ih₃ @ih₂) },\n    { introsI ℰ _ F_obj F_hom ℱ _ G_obj G_hom ih₃ ih₄ ih₂,\n      exact is_limit₂ F_obj @F_hom cone_obj (λ X, ih₄ _ @ih₂) },\n    { introsI ih,\n      exact is_limit₂ F_obj @F_hom cone_obj cone }\n  end\n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ cocone_obj cocone ih₂ Z g,\n    exact is_colimit₂ F_obj @F_hom Z (λ A, ih₂ A g)\n  end\n  (λ _ _ g, g)\n#print has_limits_of_size\nvariables {ℰ : Type u} [category ℰ] [has_limits_of_size.{0} ℰ] [has_colimits_of_size.{0} ℰ] (F : 𝒞 ⥤ ℰ)\n\ndef UMP_obj {X Y : obj₂ 𝒞} (f : hom₂ X Y) : Σ A B : ℰ, A ⟶ B :=\nhom₂.rec_on f \n  (λ X Y f, ⟨F.obj X, F.obj Y, F.map f⟩) \n  begin\n    introsI 𝒟 _ F_obj F_hom ih₁ A B g ih₂,\n\n  end\n  _ \n  _ \n  _ \n  _\n\n\nend bicompletion_aux\n", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/bicompletion/inductive4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.3028179895997181}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : ∀ ε > 0, ∃ N : ℤ, ∀ n : ℤ, n > N → |n * α % 1 - 0| < ε :=\nbegin\n  assume ε hε,\n  -- Let $\\alpha$ be an irrational number. \n  -- Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. \n  -- If this were not true, then\n  have h1 : ∀ i j : ℤ, i ≠ j → (i * α % 1 ≠ j * α % 1), from sorry,\n\n  -- Hence, $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  have h2 : ∀ i : ℤ, ∃ j : ℤ, i * α % 1 = j * α % 1, from sorry,\n  have h3 : ∀ i : ℤ, i * α % 1 ∈ Icc 0 1, from sorry,\n  have h4 : ∀ i : ℤ, i * α % 1 ∈ set.range (λ (i : ℤ), i * α % 1), from sorry,\n  have h5 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h6 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h7 : ∀ i : ℤ, i * α % 1 ∈ set.range (λ (i : ℤ), i * α % 1), from sorry,\n  have h8 : ∀ i : ℤ, i * α % 1 ∈ set.range (λ (i : ℤ), i * α % 1), from sorry,\n  have h9 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h10 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h11 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h12 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h13 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h14 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h15 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h16 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h17 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h18 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h19 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h20 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h21 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h22 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h23 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h24 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h25 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h26 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h27 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h28 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h29 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h30 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h31 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h32 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h33 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h34 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h35 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h36 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h37 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h38 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h39 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h40 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h41 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h42 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h43 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h44 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h45 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h46 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h47 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h48 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h49 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h50 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h51 : set.range (λ (i : ℤ), i * α % 1) ⊆ Icc 0 1, from sorry,\n  have h52 : set.range (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=irrational_orbit_dense (α : ℝ) (h : ¬ is_rat α) : ∃ S : set ℝ, ∀ y ∈ set.Icc 0 1, ∃ x ∈ S, |y - x| < 1 :=\nbegin\n  --Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  have h1 : ∀ i j : ℤ, i ≠ j → (i : ℝ) * α - i * α ≠ j * α - j * α, from sorry,\n  \n  --$S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  let S := {(i : ℝ) * α - i * α | i : ℤ},\n  have h2 : ∀ i : ℤ, (i : ℝ) * α - i * α ∈ S, from sorry,\n  have h3 : ∀ i j : ℤ, i ≠ j → (i : ℝ) * α - i * α ≠ j * α - j * α, from sorry,\n  have h4 : ∀ i : ℤ, (i : ℝ) * α - i * α ∈ set.Icc 0 1, from sorry,\n  have h5 : ∀ i j : ℤ, i ≠ j → (i : ℝ) * α - i * α ≠ j * α - j * α, from sorry,\n  \n  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.\n  have h6 : ∃ x ∈ set.Icc 0 1, ∃ s ∈ S, ∀ ε > 0, ∃ n : ℤ, n ≠ 0 ∧ |x - s - n| < ε, from sorry,\n\n  --One can thus find pairs of elements of $S$ that are arbitrarily close.\n  have h7 : ∃ x ∈ set.Icc 0 1, ∃ s ∈ S, ∀ ε > 0, ∃ n : ℤ, n ≠ 0 ∧ |x - s - n| < ε, from sorry,\n\n  --Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n  have h8 : ∃ x ∈ set.Icc 0 1, ∃ s ∈ S, ∀ ε > 0, ∃ n : ℤ, n ≠ 0 ∧ |x - s - n| < ε, from sorry,\n\n  --To show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n  have h9 : ∃ x ∈ set.Icc 0 1, ∃ s ∈ S, ∀ ε > 0, ∃ n : ℤ, n ≠ 0 ∧ |x - s - n| < ε, from sorry,\n  \n  show ∃ S : set ℝ, ∀ y ∈ set.Icc 0 1, ∃ x ∈ S, |y - x| < 1, from sorry,\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) : ¬ is_rat α → ∀ ε > 0, ∃ (i : ℤ), 0 ≤ i ∧ i < 1 / ε ∧ ∀ (j : ℤ), 0 ≤ j ∧ j < 1 / ε → abs (α * i - α * j) < ε :=\nbegin\n  assume h1 (ε),\n  have h2 : ∀ (i j : ℤ), i ≠ j → (abs (α * i - α * j) = abs (α * (i - j))) ∧ (abs (α * (i - j)) = abs (α) * abs (i - j)) ∧ (abs (α) * abs (i - j) = abs (α) * (abs (i) + abs (j))), \n  from sorry,\n  have h3 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) = abs (α) * (abs (i) + abs (j)), \n  from sorry,\n  have h4 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ abs (α), \n  from sorry,\n  have h5 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 0, \n  from sorry,\n  have h6 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1, \n  from sorry,\n  have h7 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ ε, \n  from sorry,\n  have h8 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h9 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h10 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h11 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h12 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h13 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h14 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h15 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h16 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h17 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h18 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h19 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h20 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h21 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h22 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h23 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h24 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h25 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h26 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h27 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h28 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h29 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h30 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h31 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h32 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h33 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h34 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h35 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h36 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h37 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h38 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h39 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h40 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h41 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h42 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h43 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε, \n  from sorry,\n  have h44 : ∀ (i j : ℤ), i ≠ j → abs (α * i - α * j) ≥ 1 / ε\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (h : ¬ is_rat α) : ∀ y ∈ Icc 0 1, ∃ x ∈ (set.range (λ (i : ℤ), i*α) : set ℝ), abs (y - x) < 1 :=\nbegin\n  -- Let $\\alpha$ be an irrational number.\n  assume (h2 : ¬ is_rat α),\n  -- Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$.\n  have h3 : ∀ (i j : ℤ), i ≠ j → (i*α) % 1 ≠ (j*α) % 1, from sorry,\n  -- If this were not true, then\n  have h4 : ∀ (i j : ℤ), i ≠ j → (i*α) % 1 = (j*α) % 1 → false, from sorry,\n  -- Hence, $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  have h5 : ∀ (i : ℤ), (i*α) % 1 ∈ Icc 0 1, from sorry,\n  have h6 : set.range (λ (i : ℤ), (i*α) % 1) = set.range (λ (i : ℤ), i*α) : set ℝ, from sorry,\n  have h7 : set.range (λ (i : ℤ), i*α) : set ℝ = {(i*α) % 1 | i : ℤ}, from sorry,\n  have h8 : {(i*α) % 1 | i : ℤ} ⊆ Icc 0 1, from sorry,\n  have h9 : set.range (λ (i : ℤ), i*α) : set ℝ ⊆ Icc 0 1, from sorry,\n  have h10 : set.range (λ (i : ℤ), i*α) : set ℝ ≠ ∅, from sorry,\n  have h11 : set.range (λ (i : ℤ), i*α) : set ℝ ⊆ Icc 0 1, from sorry,\n  have h12 : set.range (λ (i : ℤ), i*α) : set ℝ ≠ ∅, from sorry,\n  have h13 : ∀ (y : ℝ), y ∈ Icc 0 1 → ∃ x ∈ (set.range (λ (i : ℤ), i*α) : set ℝ), abs (y - x) < 1, from sorry,\n  show ∀ (y : ℝ), y ∈ Icc 0 1 → ∃ x ∈ (set.range (λ (i : ℤ), i*α) : set ℝ), abs (y - x) < 1, from sorry,\nend\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : ¬ is_rat α) : \n∀ (ε : ℝ) (hε : ε > 0), ∃ (i : ℤ) (x : ℝ), x ∈ {i | i ∈ ℤ} ∧ |x - α| < ε :=\nbegin\n  assume (ε : ℝ) (hε : ε > 0),\n  -- then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$.\n  have h1 : ∀ (i j : ℤ), i ≠ j → (i * α) % 1 ≠ (j * α) % 1, from sorry,\n  -- If this were not true, then\n  -- $i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor$,\n  -- which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$.\n  have h2 : ∀ (i j : ℤ), i ≠ j → ¬ is_rat α, from sorry,\n  -- Hence,\n  -- $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$\n  -- is an infinite subset of $\\left[0,1\\right]$.\n  have h3 : ∀ (i : ℤ), (i * α) % 1 ∈ {i | i ∈ ℤ}, from sorry,\n  have h4 : ∀ (i : ℤ), (i * α) % 1 ∈ set.Ico (0 : ℝ) (1 : ℝ), from sorry,\n  have h5 : ∞ = set.card {i | i ∈ ℤ}, from sorry,\n  have h6 : ∞ = set.card (set.Ico (0 : ℝ) (1 : ℝ)), from sorry,\n  have h7 : set.finite (set.Ico (0 : ℝ) (1 : ℝ)), from sorry,\n  have h8 : set.finite {i | i ∈ ℤ}, from sorry,\n  have h9 : set.finite {i | i ∈ ℤ} ∧ ∞ = set.card {i | i ∈ ℤ}, from sorry,\n  have h10 : set.finite (set.Ico (0 : ℝ) (1 : ℝ)) ∧ ∞ = set.card (set.Ico (0 : ℝ) (1 : ℝ)), from sorry,\n\n  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$.\n  have h11 : ∃ (x : ℝ), x ∈ set.Ico (0 : ℝ) (1 : ℝ) ∧ ∀ (ε : ℝ), ε > 0 → ∃ (y : ℝ), y ∈ {i | i ∈ ℤ} ∧ |x - y| < ε, \n  from sorry,\n  \n  --One can thus find pairs of elements of $S$ that are arbitrarily close.\n  have h12 : ∀ (ε : ℝ), ε > 0 → ∃ (x y : ℝ), x ∈ {i | i ∈ ℤ} ∧ y ∈ {i | i ∈ ℤ} ∧ |x - y| < ε, \n  from sorry,\n  \n  --Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n  have h13 : ∀ (x y : ℝ), x ∈ {i | i ∈ ℤ} ∧ y ∈ {i | i ∈ ℤ} → |x - y| ∈ {i | i ∈ ℤ}, from sorry,\n  have h14 : ∀ (ε : ℝ), ε > 0 → ∃ (y : ℝ), y ∈ {i | i ∈ ℤ} ∧ |0 - y| < ε, from sorry,\n  \n  --To show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$.\n  --Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n  have h15 : ∀ (y : ℝ) (ε : ℝ), y ∈ set.Ico (0 : ℝ) (1 : ℝ) ∧ ε > 0 → ∃ (i : ℤ) (x : ℝ), x ∈ {i | i ∈ ℤ} ∧ |x - y| < ε, \n  from sorry,\n  show ∃ (i : ℤ) (x : ℝ), x ∈ {i | i ∈ ℤ} ∧ |x - α| < ε, from sorry,\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {α : ℝ} (hα_irrat : irrational α) : closure ((λ m : ℤ, int.fract (α * ↑m)) '' (@set.univ ℤ)) = set.Icc 0 1 :=density_of_irrational_orbit (α : ℝ) (hα : irrational α) : \n∀ y ∈ Icc 0 1, ∃ N : ℤ, ∃ x ∈ Icc 0 1, |y - N * x| < 1 :=\nbegin\n  assume (y : ℝ) (h1 : y ∈ Icc 0 1),\n  --By the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. \n  --Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n  have h2 : ∃ (x : ℝ) (h2 : x ∈ Icc 0 1), ∃ (N : ℤ), ∃ (ε : ℝ) (hε : ε > 0), ∀ (ε' : ℝ) (hε' : ε' > 0), ∃ (N' : ℤ), ∃ (x' : ℝ) (hx' : x' ∈ Icc 0 1), |x' - x| < ε', from sorry,\n\n  --To show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. \n  --Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n  cases h2 with x hx,\n  cases hx with N hN,\n  cases hN with ε hε,\n  cases hε with hε hε',\n  use N,\n  use x,\n  sorry,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),\n  -- Then $A ⊆ S$ and $B ⊆ S$, by power set definition\n  have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,\n  -- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset\n  have h2 : (A ∩ B) ⊆ A, from sorry,\n  -- Then $(A ∩ B) ⊆ S$, by subset relation is transitive \n  have h3 : (A ∩ B) ⊆ S, from sorry,\n  -- Hence $(A ∩ B) ∈  𝒫 S$, by power set definition\n  show (A ∩ B) ∈  𝒫 S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by sorry\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by sorry\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,\n  have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,\n\n  -- Setting $b = a$, this becomes:\n  have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,\n  have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {\n    use (1 : G),\n    have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {\n      assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),\n      have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,\n      have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : ℕ → ℝ) (l : ℝ) : \nlet seq_limit : (ℕ → ℝ) → ℝ → Prop :=  λ (u : ℕ → ℝ) (l : ℝ), ∀ ε > 0, ∃ N, ∀ n > N, |u n - l| < ε in\n seq_limit y l → seq_limit z l →  (∀ n : ℕ, (y n) ≤ (x n) ∧ (x n) ≤ (z n)) → seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : ∀ (n : ℕ), y n ≤ x n ∧ x n ≤ z n) (ε), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : ∀ x, |x - l| < ε ↔ (((l - ε) < x) ∧ (x < (l + ε))), \n  from sorry,\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : ε > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 ε h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 ε h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : ∀ n > N, n > N1 ∧ n > N2 := sorry,\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : ∀ n > N, (((l - ε) < (y n)) ∧ ((y n) ≤ (x n))) ∧ (((x n) ≤ (z n)) ∧ ((z n) < l+ε)), \n  from sorry,\n    \n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : ∀ n > N, ((l - ε) < (x n)) ∧ ((x n) < (l+ε)), \n  from sorry,\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  ∀ (n : ℕ), n > N → |x n - l| < ε, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline_with_comments-Natural-Language-Proof-Translation/lean_proof_outline_with_comments-4_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8175744761936437, "lm_q2_score": 0.3702253925955866, "lm_q1q2_score": 0.3026868314249228}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nComputational realization of topological spaces (experimental).\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.bases\nimport Mathlib.data.analysis.filter\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 u_4 u_5 u_6 \n\nnamespace Mathlib\n\n/-- A `ctop α σ` is a realization of a topology (basis) on `α`,\n  represented by a type `σ` together with operations for the top element and\n  the intersection operation. -/\nstructure ctop (α : Type u_1) (σ : Type u_2) where\n  f : σ → set α\n  top : α → σ\n  top_mem : ∀ (x : α), x ∈ f (top x)\n  inter : (a b : σ) → (x : α) → x ∈ f a ∩ f b → σ\n  inter_mem : ∀ (a b : σ) (x : α) (h : x ∈ f a ∩ f b), x ∈ f (inter a b x h)\n  inter_sub : ∀ (a b : σ) (x : α) (h : x ∈ f a ∩ f b), f (inter a b x h) ⊆ f a ∩ f b\n\nnamespace ctop\n\n\nprotected instance has_coe_to_fun {α : Type u_1} {σ : Type u_3} : has_coe_to_fun (ctop α σ) :=\n  has_coe_to_fun.mk (fun (x : ctop α σ) => σ → set α) f\n\n@[simp] theorem coe_mk {α : Type u_1} {σ : Type u_3} (f : σ → set α) (T : α → σ)\n    (h₁ : ∀ (x : α), x ∈ f (T x)) (I : (a b : σ) → (x : α) → x ∈ f a ∩ f b → σ)\n    (h₂ : ∀ (a b : σ) (x : α) (h : x ∈ f a ∩ f b), x ∈ f (I a b x h))\n    (h₃ : ∀ (a b : σ) (x : α) (h : x ∈ f a ∩ f b), f (I a b x h) ⊆ f a ∩ f b) (a : σ) :\n    coe_fn (mk f T h₁ I h₂ h₃) a = f a :=\n  rfl\n\n/-- Map a ctop to an equivalent representation type. -/\ndef of_equiv {α : Type u_1} {σ : Type u_3} {τ : Type u_4} (E : σ ≃ τ) : ctop α σ → ctop α τ := sorry\n\n@[simp] theorem of_equiv_val {α : Type u_1} {σ : Type u_3} {τ : Type u_4} (E : σ ≃ τ) (F : ctop α σ)\n    (a : τ) : coe_fn (of_equiv E F) a = coe_fn F (coe_fn (equiv.symm E) a) :=\n  sorry\n\n/-- Every `ctop` is a topological space. -/\ndef to_topsp {α : Type u_1} {σ : Type u_3} (F : ctop α σ) : topological_space α :=\n  topological_space.generate_from (set.range (f F))\n\ntheorem to_topsp_is_topological_basis {α : Type u_1} {σ : Type u_3} (F : ctop α σ) :\n    topological_space.is_topological_basis (set.range (f F)) :=\n  sorry\n\n@[simp] theorem mem_nhds_to_topsp {α : Type u_1} {σ : Type u_3} (F : ctop α σ) {s : set α} {a : α} :\n    s ∈ nhds a ↔ ∃ (b : σ), a ∈ coe_fn F b ∧ coe_fn F b ⊆ s :=\n  sorry\n\nend ctop\n\n\n/-- A `ctop` realizer for the topological space `T` is a `ctop`\n  which generates `T`. -/\nstructure ctop.realizer (α : Type u_5) [T : topological_space α] where\n  σ : Type u_6\n  F : ctop α σ\n  eq : ctop.to_topsp F = T\n\nprotected def ctop.to_realizer {α : Type u_1} {σ : Type u_3} (F : ctop α σ) : ctop.realizer α :=\n  ctop.realizer.mk σ F sorry\n\nnamespace ctop.realizer\n\n\nprotected theorem is_basis {α : Type u_1} [T : topological_space α] (F : realizer α) :\n    topological_space.is_topological_basis (set.range (f (F F))) :=\n  eq.mp (Eq._oldrec (Eq.refl (topological_space.is_topological_basis (set.range (f (F F))))) (eq F))\n    (to_topsp_is_topological_basis (F F))\n\nprotected theorem mem_nhds {α : Type u_1} [T : topological_space α] (F : realizer α) {s : set α}\n    {a : α} : s ∈ nhds a ↔ ∃ (b : σ F), a ∈ coe_fn (F F) b ∧ coe_fn (F F) b ⊆ s :=\n  eq.mp\n    (Eq._oldrec (Eq.refl (s ∈ nhds a ↔ ∃ (b : σ F), a ∈ coe_fn (F F) b ∧ coe_fn (F F) b ⊆ s))\n      (eq F))\n    (mem_nhds_to_topsp (F F))\n\ntheorem is_open_iff {α : Type u_1} [topological_space α] (F : realizer α) {s : set α} :\n    is_open s ↔ ∀ (a : α), a ∈ s → ∃ (b : σ F), a ∈ coe_fn (F F) b ∧ coe_fn (F F) b ⊆ s :=\n  iff.trans is_open_iff_mem_nhds (ball_congr fun (a : α) (h : a ∈ s) => realizer.mem_nhds F)\n\ntheorem is_closed_iff {α : Type u_1} [topological_space α] (F : realizer α) {s : set α} :\n    is_closed s ↔\n        ∀ (a : α), (∀ (b : σ F), a ∈ coe_fn (F F) b → ∃ (z : α), z ∈ coe_fn (F F) b ∩ s) → a ∈ s :=\n  sorry\n\ntheorem mem_interior_iff {α : Type u_1} [topological_space α] (F : realizer α) {s : set α} {a : α} :\n    a ∈ interior s ↔ ∃ (b : σ F), a ∈ coe_fn (F F) b ∧ coe_fn (F F) b ⊆ s :=\n  iff.trans mem_interior_iff_mem_nhds (realizer.mem_nhds F)\n\nprotected theorem is_open {α : Type u_1} [topological_space α] (F : realizer α) (s : σ F) :\n    is_open (coe_fn (F F) s) :=\n  sorry\n\ntheorem ext' {α : Type u_1} [T : topological_space α] {σ : Type u_2} {F : ctop α σ}\n    (H : ∀ (a : α) (s : set α), s ∈ nhds a ↔ ∃ (b : σ), a ∈ coe_fn F b ∧ coe_fn F b ⊆ s) :\n    to_topsp F = T :=\n  sorry\n\ntheorem ext {α : Type u_1} [T : topological_space α] {σ : Type u_2} {F : ctop α σ}\n    (H₁ : ∀ (a : σ), is_open (coe_fn F a))\n    (H₂ : ∀ (a : α) (s : set α), s ∈ nhds a → ∃ (b : σ), a ∈ coe_fn F b ∧ coe_fn F b ⊆ s) :\n    to_topsp F = T :=\n  sorry\n\nprotected def id {α : Type u_1} [topological_space α] : realizer α :=\n  mk (Subtype fun (x : set α) => is_open x)\n    (mk subtype.val (fun (_x : α) => { val := set.univ, property := is_open_univ }) set.mem_univ\n      (fun (_x : Subtype fun (x : set α) => is_open x) => sorry) sorry sorry)\n    sorry\n\ndef of_equiv {α : Type u_1} {τ : Type u_4} [topological_space α] (F : realizer α) (E : σ F ≃ τ) :\n    realizer α :=\n  mk τ (of_equiv E (F F)) sorry\n\n@[simp] theorem of_equiv_σ {α : Type u_1} {τ : Type u_4} [topological_space α] (F : realizer α)\n    (E : σ F ≃ τ) : σ (of_equiv F E) = τ :=\n  rfl\n\n@[simp] theorem of_equiv_F {α : Type u_1} {τ : Type u_4} [topological_space α] (F : realizer α)\n    (E : σ F ≃ τ) (s : τ) : coe_fn (F (of_equiv F E)) s = coe_fn (F F) (coe_fn (equiv.symm E) s) :=\n  sorry\n\nprotected def nhds {α : Type u_1} [topological_space α] (F : realizer α) (a : α) :\n    filter.realizer (nhds a) :=\n  filter.realizer.mk (Subtype fun (s : σ F) => a ∈ coe_fn (F F) s)\n    (cfilter.mk\n      (fun (s : Subtype fun (s : σ F) => a ∈ coe_fn (F F) s) => coe_fn (F F) (subtype.val s))\n      { val := top (F F) a, property := sorry }\n      (fun (_x : Subtype fun (s : σ F) => a ∈ coe_fn (F F) s) => sorry) sorry sorry)\n    sorry\n\n@[simp] theorem nhds_σ {α : Type u_1} {β : Type u_2} [topological_space α] (m : α → β)\n    (F : realizer α) (a : α) :\n    filter.realizer.σ (realizer.nhds F a) = Subtype fun (s : σ F) => a ∈ coe_fn (F F) s :=\n  rfl\n\n@[simp] theorem nhds_F {α : Type u_1} {β : Type u_2} [topological_space α] (m : α → β)\n    (F : realizer α) (a : α) (s : filter.realizer.σ (realizer.nhds F a)) :\n    coe_fn (filter.realizer.F (realizer.nhds F a)) s = coe_fn (F F) (subtype.val s) :=\n  rfl\n\ntheorem tendsto_nhds_iff {α : Type u_1} {β : Type u_2} [topological_space α] {m : β → α}\n    {f : filter β} (F : filter.realizer f) (R : realizer α) {a : α} :\n    filter.tendsto m f (nhds a) ↔\n        ∀ (t : σ R),\n          a ∈ coe_fn (F R) t →\n            ∃ (s : filter.realizer.σ F),\n              ∀ (x : β), x ∈ coe_fn (filter.realizer.F F) s → m x ∈ coe_fn (F R) t :=\n  iff.trans (filter.realizer.tendsto_iff m F (realizer.nhds R a)) subtype.forall\n\nend ctop.realizer\n\n\nstructure locally_finite.realizer {α : Type u_1} {β : Type u_2} [topological_space α]\n    (F : ctop.realizer α) (f : β → set α)\n    where\n  bas : (a : α) → Subtype fun (s : ctop.realizer.σ F) => a ∈ coe_fn (ctop.realizer.F F) s\n  sets :\n    (x : α) →\n      fintype ↥(set_of fun (i : β) => set.nonempty (f i ∩ coe_fn (ctop.realizer.F F) ↑(bas x)))\n\ntheorem locally_finite.realizer.to_locally_finite {α : Type u_1} {β : Type u_2}\n    [topological_space α] {F : ctop.realizer α} {f : β → set α} (R : locally_finite.realizer F f) :\n    locally_finite f :=\n  sorry\n\ntheorem locally_finite_iff_exists_realizer {α : Type u_1} {β : Type u_2} [topological_space α]\n    (F : ctop.realizer α) {f : β → set α} :\n    locally_finite f ↔ Nonempty (locally_finite.realizer F f) :=\n  sorry\n\ndef compact.realizer {α : Type u_1} [topological_space α] (R : ctop.realizer α) (s : set α) :=\n  {f : filter α} →\n    (F : filter.realizer f) →\n      (x : filter.realizer.σ F) →\n        f ≠ ⊥ → coe_fn (filter.realizer.F F) x ⊆ s → Subtype fun (a : α) => a ∈ s ∧ nhds a ⊓ f ≠ ⊥\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/analysis/topology_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.30251010335578427}}
{"text": "example (P : Prop) : ∀ {p : P}, P := by\n  exact fun {p} => p\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  intro h; exact h\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact @id _\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact no_implicit_lambda% id\n\nmacro \"exact'\" x:term : tactic => `(exact no_implicit_lambda% $x)\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact' id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  apply id\n\nexample (P : Prop) : ∀ p : P, P := by\n  have : _ := 1\n  apply id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  refine no_implicit_lambda% (have : _ := 1; ?_)\n  apply id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  have : _ := 1\n  apply id\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/impLambdaTac.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.30251010335578415}}
{"text": "import data.list.basic\nimport data.equiv.basic\nimport tactic\nimport meta.expr\nimport tactic.refine\n\nuniverses u v w\n\nopen function\n\nclass canonical_equiv (α : Sort*) (β : Sort*) extends equiv α β.\n\ndef equiv.injective {α : Sort*} {β : Sort*} (eq : α ≃ β) :\n  injective eq :=\ninjective_of_left_inverse eq.left_inv\n\n#print prefix equiv.injective\n#print equiv.injective.equations._eqn_1\n\nvariables  (f : Type u → Prop)\nvariables (on_equiv : Π {α β : Type u} (e : equiv α β), equiv (f α) (f β))\n\n\nsection\nopen tactic.interactive\nmeta def prove_on_equiv_law' (n := `on_equiv) :=\ndo intros [],\n   eqv ← tactic.resolve_name n,\n   `[dsimp [equiv.refl,equiv.trans]],\n   generalize none () (``(%%eqv _),`k),\n   `[cases k, refl]\n\nmeta def prove_on_equiv_law := prove_on_equiv_law'\nend\n\nclass transportable (f : Type u → Type v) :=\n(on_equiv : Π {α β : Type u} (e : equiv α β), equiv (f α) (f β))\n(on_refl  : Π (α : Type u), on_equiv (equiv.refl α) = equiv.refl (f α) . prove_on_equiv_law)\n(on_trans : Π {α β γ : Type u} (d : equiv α β) (e : equiv β γ), on_equiv (equiv.trans d e) = equiv.trans (on_equiv d) (on_equiv e) . prove_on_equiv_law)\n\n-- Finally a command like: `initialize_transport group` would create the next two declarations automagically:\n\nopen transportable\n\ndefinition equiv_mul {α β : Type u} : equiv α β → equiv (has_mul α) (has_mul β) := λ E,\n{ to_fun :=  λ αmul,⟨λ b1 b2, E.to_fun (@has_mul.mul α αmul (E.inv_fun b1) (E.inv_fun b2))⟩,\n  inv_fun := λ βmul,⟨λ a1 a2, E.inv_fun (@has_mul.mul β βmul (E.to_fun a1) (E.to_fun a2))⟩, -- didn't I just write that?\n                                                                      -- should we introduce E-dual?\n  left_inv := λ f, begin\n    cases f, simp, -- aargh why do I struggle\n    congr,\n    -- suffices :  (λ (a1 a2 : α), E.inv_fun (E.to_fun (f _ _))) = (λ a1 a2, f a1 a2),\n    --   by rw this,\n    funext,\n    simp [E.left_inv _,E.right_inv _], -- got there in the end\n  end,\n  right_inv := -- I can't even do this in term mode so I just copy out the entire tactic mode proof again.\n λ g, begin\n    cases g, simp, -- aargh why do I struggle\n    suffices :  (λ (b1 b2 : β), E.to_fun (E.inv_fun (g _ _))) = (λ b1 b2, g b1 b2),\n      by rw this,\n    funext,\n    simp [E.left_inv _,E.right_inv _], -- got there in the end\n  end, -- didn't I just write that?\n}\n\nnamespace tactic\n\nnamespace interactive\n\nopen interactive function\n#check @rfl\nmeta def eqn_lemma_type : expr → expr → expr → tactic (expr × expr)\n| (expr.pi n bi d b) fn arg :=\ndo v ← mk_local' n bi d,\n   arg ← head_beta (arg v),\n   prod.map (expr.pis [v]) (expr.lambdas [v]) <$> eqn_lemma_type (b.instantiate_var v) (fn v) arg\n| _ fn arg :=\nprod.mk <$> to_expr ``(%%fn = %%arg) <*> to_expr ``(@rfl _ %%arg)\n\nmeta def eq_lemma_name (n : name) := n <.> \"equations\" <.> \"_eqn_1\"\n\nmeta def abstract_def (suffix : name) (t : expr) (tac : tactic unit) : tactic name :=\ndo (_,v) ← solve_aux t tac,\n   n ← (++ suffix) <$> decl_name,\n   v ← instantiate_mvars v,\n   t ← instantiate_mvars t,\n   add_decl $ mk_definition n t.collect_univ_params t v,\n   defn ← mk_const n,\n   (t,p) ← eqn_lemma_type t defn v,\n   add_decl $ declaration.thm (eq_lemma_name n) t.collect_univ_params t (return p),\n   set_basic_attribute `_refl_lemma (eq_lemma_name n),\n   return n\n\nmeta def build_aux_decl_with (c : name) (type : pexpr) (is_lemma : bool) (tac : tactic unit) : tactic expr :=\ndo type' ← to_expr type,\n   ((),v) ← solve_aux type' tac,\n   add_aux_decl c type' v is_lemma\n\nopen functor\n\ndef strip_prefix (n : name) : name :=\nname.update_prefix n name.anonymous\n-- #check @equiv.injective\n-- #print injective\n\nmeta def all_field_names : tactic (list name) :=\ndo gs ← get_goals,\n   gs.mmap (λ g, set_goals [g] >> get_current_field)\n   <* set_goals gs\n\n-- meta def mk_app' (e : expr) (es : list expr) : tactic expr :=\n\nmeta def mk_to_fun (n : name) : tactic name :=\ndo env  ← get_env,\n   decl ← env.get n,\n   let univs := decl.univ_params,\n   e ← resolve_name n,\n   fields ← qualified_field_list n,\n   t ← to_expr ``(Π α β : Type u, α ≃ β → %%e α → %%e β),\n   abstract_def (n <.> \"transport\" <.> \"to_fun\") t\n     (do α  ← tactic.intro `α, β  ← tactic.intro `β, eq_ ← tactic.intro `eqv,\n         eqv ← to_expr ``(@coe_fn _ equiv.has_coe_to_fun (%%eq_)),\n         eqv_symm ← to_expr ``(@coe_fn _ equiv.has_coe_to_fun (%%eq_).symm),\n         x ← tactic.intro `x,\n         [v] ← get_goals,\n         refine_struct ``( { .. } ),\n         let unfold_more := [`has_one.one,`has_zero.zero,`has_mul.mul,`has_add.add\n                            ,`semigroup.mul,`group.mul],\n         fs ← all_field_names >>= mmap resolve_name ∘ list.append unfold_more,\n         all_goals (do\n           fl ← get_current_field,\n           xs ← tactic.intros,\n           p ← mk_const fl >>= infer_type >>= pp,\n           xs' ← mmap (λ x, eqv_symm <$>\n             mcond (is_proof x)\n                 (mk_mvar)\n                 (pure x)) xs,\n           e ← mk_app fl (map eqv_symm xs) <|>\n               ((flip expr.mk_app xs' <$> mk_mapp fl [none,some x]) >>= to_expr ∘ to_pexpr) <|>\n               fail format!\"{fl} - {xs'} :: {p}\",\n           infer_type e >>= trace,\n           trace \"a\",\n           tactic.exact (eqv e) <|>\n           (do refine ``((equiv.apply_eq_iff_eq (%%eq_).symm _ _).1 _) <|>\n                      refine ``((not_iff_not_of_iff $ equiv.apply_eq_iff_eq (%%eq_).symm _ _).1 _),\n               trace \"b\",\n               g ← mk_mvar,\n               refine ``(iff.elim_left (eq.to_iff %%g) %%e),\n               -- tactic.type_check e,\n               num_goals >>= trace,\n               gs ← get_goals, set_goals $ g :: gs.diff [g],\n               num_goals >>= trace,\n               solve1 (do\n               -- simp (some ()) ff (map simp_arg_type.expr $ ``(@equiv.inverse_apply_apply) :: fs) [] (loc.ns [none]),\n               -- simp (some ()) ff (map simp_arg_type.expr $ fs) [] (loc.ns [none]),\n               -- repeat $ dsimp ff (map simp_arg_type.expr $ fs ++ [``(@equiv.inverse_apply_apply)]) [] (loc.ns [none]),\n                 trace \"c\",\n                 -- infer_type e >>= trace,\n                 simp (some ()) tt (map simp_arg_type.expr $ fs ++ [``(@equiv.inverse_apply_apply),``(id)]) [] (loc.ns [none]),\n               -- try $ simp (some ()) ff (map simp_arg_type.expr $ fs ++ [``(@has_mul.mul),``(@equiv.inverse_apply_apply)]) [] (loc.ns [none]),\n               -- repeat `[rw [equiv.inverse_apply_apply]],\n                 trace_state,\n                 trace \"d\",\n                 done <|> refl <|> (congr; done <|> refl)),\n               trace \"e\",\n               done <|> solve1 (do\n                 trace \"f\",\n                 target >>= infer_type >>= trace,\n                 target >>= instantiate_mvars >>= tactic.change,\n                 target >>= trace,\n                 done )) <|>\n           (do refine ``((not_iff_not_of_iff $ equiv.apply_eq_iff_eq (%%eq_).symm _ _).1 _),\n               refine ``(iff.elim_left _ %%e),\n               refine ``(not_iff_not_of_iff _),\n               refine ``(eq.to_iff _),\n               simp (some ()) tt (map simp_arg_type.expr $ fs ++ [``(@equiv.inverse_apply_apply),``(id)]) [] (loc.ns [none]),\n               done) <|>\n           (do -- trace_state,\n               infer_type e >>= trace,\n               trace \"dude\",\n               done)))\n\n\n-- set_option pp.universes true\n-- set_option pp.notation false\n-- set_option pp.implicit true\n\n-- run_cmd add_interactive [`mk_to_fun]\n\n-- run_cmd mk_to_fun `group\n-- #print prefix _run_command.group\n\nmeta def mk_on_equiv (n to_fun : name) : tactic name :=\ndo f ← resolve_name n,\n   fn ← resolve_name to_fun,\n   t ← to_expr ``(Π (α β : Type u), α ≃ β → %%f α ≃ %%f β),\n   abstract_def (n ++ `on_eqv) t $\n     do tactic.intron 2,\n        eqv ← tactic.intro1,\n        refine ``( { to_fun := %%fn _ _ %%eqv,\n                     inv_fun := %%fn _ _ (%%eqv).symm,\n                     .. } );\n        abstract none (\n        dunfold [`function.left_inverse,`function.right_inverse] (loc.ns [none]);\n        (do x ← tactic.intro1, tactic.cases x,\n            dunfold [to_fun] (loc.ns [none]),\n            congr; tactic.funext;\n            `[ simp only [equiv.inverse_apply_apply,equiv.apply_inverse_apply,id] ],\n            return ()))\n\nmeta def mk_transportable_instance  (n : name) : tactic unit :=\ndo d ← decl_name,\n   -- let to_fun := d ++ `group.transport.to_fun,\n   to_fun ← mk_to_fun n,\n   to_fun_lmm ← resolve_name $ eq_lemma_name to_fun,\n   on_eqv_n ← mk_on_equiv n to_fun,\n   on_eqv ← resolve_name on_eqv_n,\n   on_eqv_lmm ← resolve_name $ eq_lemma_name on_eqv_n,\n   fs ← expanded_field_list n,\n   env  ← get_env,\n   decl ← env.get n,\n   let univs := decl.univ_params,\n   let e := @expr.const tt n $ univs.map level.param,\n   t ← to_expr ``(transportable %%e),\n   let goal := (loc.ns [none]),\n   (_,d) ← solve_aux t\n     (do refine ``( { on_equiv := %%on_eqv, on_refl := _, on_trans := _ } ),\n         -- cleanup,\n         abstract (some $ n ++ `on_refl) (do\n           intro1,\n           fs' ← fs.mmap (resolve_name ∘ uncurry (flip name.update_prefix)),\n           -- simp (some ()) ff (map simp_arg_type.expr $ [``(equiv.refl),``(equiv.symm),on_eqv_lmm,to_fun_lmm]) [] goal,\n           dsimp ff (map simp_arg_type.expr $\n             [``(equiv.refl),``(equiv.symm),on_eqv_lmm,to_fun_lmm]) [] goal,\n           -- done,\n           -- dunfold [``equiv.refl,on_eqv_n,to_fun] goal,\n         -- num_goals >>= trace,\n           -- done,\n           try congr; tactic.funext; cases (none,```(x)) [];\n           dunfold [`id,to_fun] goal;refl ),\n         abstract (some $ n ++ `on_trans) (do\n           intro1,\n           fs' ← fs.mmap (resolve_name ∘ uncurry (flip name.update_prefix)),\n           dsimp ff (map simp_arg_type.expr $\n             [``(function.comp),``(equiv.to_fun),``(equiv.inv_fun),``(equiv.trans),\n              ``(equiv.symm),on_eqv_lmm,to_fun_lmm  ]) [] goal,\n         -- num_goals >>= trace,\n           -- dunfold [on_eqv_n,`function.comp,`equiv.to_fun,`equiv.inv_fun,``equiv.trans] goal,\n           -- dunfold [`id] goal,\n         -- num_goals >>= trace,\n           intron 4,\n           try congr; tactic.funext; cases (none,```(x)) [];\n           dunfold [`id,to_fun] goal;refl ),\n         -- num_goals >>= trace,\n         -- dunfold [`id,to_fun] goal,\n         -- num_goals >>= trace,\n         -- target >>= trace,\n         -- num_goals >>= trace,\n         -- dsimp ff (map simp_arg_type.expr fs') [] goal,\n         -- num_goals >>= trace,\n         -- trace_state,\n         -- target >>= trace,\n         -- prove_on_equiv_law,\n         -- prove_on_equiv_law,\n         done),\n   d ← instantiate_mvars d,\n   let def_name := n <.> \"transportable\",\n   add_decl $ mk_definition def_name univs t d,\n   set_basic_attribute `instance def_name tt\n\nset_option profiler true\nset_option formatter.hide_full_terms false\nset_option pp.delayed_abstraction false\nset_option pp.universes true\n\n\n-- run_cmd do\n--  mk_transportable_instance `group\n\nset_option profiler false\n\n-- example  : Π {α β : Type u} (e : equiv α β), equiv (group α) (group β) :=\n-- begin\n--   introv eqv,\n--   refine_struct { to_fun  := @group.transport.to_fun _ _ eqv\n--                 , inv_fun := @group.transport.to_fun _ _ eqv.symm\n--                 , .. };\n--   dsimp [left_inverse,function.right_inverse];\n--   intro x ; dunfold group.transport.to_fun;\n--   cases x; congr; funext;\n--   simp only [equiv.inverse_apply_apply,equiv.apply_inverse_apply,id],\n-- end\n\n#check congr_arg\n\n\n-- #print group.transport.to_fun\n\n-- meta def mk_transportable (n : name) (e : expr) : tactic unit :=\n-- do [v] ← get_goals,\n--    trace \"begin mk_transportable\",\n--    trace \"-- | TO_FUN\",\n--    fields ← qualified_field_list n,\n--    to_fun ← to_expr ``(Π α β : Type u, α ≃ β → %%e α → %%e β)\n--      >>= define ( `to_fun),\n--    solve1\n--      (do α  ← tactic.intro `α,\n--          β  ← tactic.intro `β,\n--          eq ← tactic.intro `eqv,\n--          eqv ← to_expr ``(@coe_fn _ equiv.has_coe_to_fun (%%eq)),\n--          eqv_symm ← to_expr ``(@coe_fn _ equiv.has_coe_to_fun (%%eq).symm),\n--          x ← tactic.intro `x,\n--          [v] ← get_goals,\n--          refine_struct ``( { .. } ),\n--          -- trace_state,\n--          all_goals (do\n--            tgt ← target,\n--            p ← is_prop tgt,\n--            if p then do\n--              trace \"A\",\n--              current ← get_current_field <|> fail \"get_current_field\",\n--              vs ← tactic.intros,\n--              h ← mk_mapp current ( [α,x] ++ vs.map (some ∘ eqv_symm) ) >>= note `h none\n--                <|> fail \"mk_mapp\",\n--              unfold (fields) (loc.ns [none,h.local_pp_name]), -- h.local_pp_name]),\n--              h ← get_local h.local_pp_name,\n--              infer_type h >>= trace,\n--              -- tactic.revert h,\n--              fs ← mmap (resolve_name ∘ strip_prefix) fields,\n--              simp (some ()) tt (map simp_arg_type.expr $ [``(equiv.apply_inverse_apply),``(equiv.inverse_apply_apply)] ++ fs) [] (loc.ns [none,h.local_pp_name]), -- ,h.local_pp_name\n--              -- h ← get_local h.local_pp_name,\n--              -- infer_type h >>= trace,\n--              trace_state,\n--              -- target >>= trace,\n--              -- tactic.exact h,\n--              done,\n--              trace \"C\",\n--              return ()\n--            else do\n--              trace \"B\",\n--              -- target >>= trace,\n--              -- [v] ← get_goals,\n--              current ← get_current_field,\n--              trace current,\n--              vs ← tactic.intros,\n--              -- infer_type eqv_symm >>= trace,\n--              -- refine ``(%%eqv _) <|> fail \"refine\",\n--              trace eqv_symm,\n--              -- trace vs,\n--              -- instantiate_mvars v >>= trace,\n--              let vs' := map (some ∘ eqv_symm) vs,\n--              e ← mk_mapp current ( [α,x] ++ vs' ),\n--              e ← to_expr ``(@coe_fn _ equiv.has_coe_to_fun %%eq %%e),\n--              trace \"to_expr\",\n--              trace e,\n--              tactic.exact e,\n--              trace \"C\",\n--              -- trace_state,\n--              return () ),\n--          -- instantiate_mvars v >>= trace,\n--          trace_state,\n--          return () ),\n--    -- inv_fun ← build_aux_decl_with ( (`inv_fun).update_prefix n)\n--    --   ``(Π {α β}, %%e β → %%e α) ff\n--    --   (do trace_state >> admit),\n--    trace \"-- | EQUIV\",\n--    is_inv ← to_expr ``(∀ α β (eqv : equiv α β),\n--             left_inverse (%%to_fun β α eqv.symm) (%%to_fun α β eqv))\n--           >>= assert `is_inv,\n--    solve1 (do\n--           α  ← tactic.intro `α,\n--           β  ← tactic.intro `β,\n--           eq ← tactic.intro `eqv,\n--           x ← tactic.intro `x,\n--           tactic.cases x,\n--           -- trace_state,\n--           `[simp only [to_fun]],\n--           congr ; funext [] ; dunfold fields (loc.ns [none]) ;\n--           `[simp! only [_root_.eq.mpr,equiv.apply_inverse_apply,equiv.inverse_apply_apply]],\n--           -- trace_state,\n--           return () ),\n--    fn ← to_expr ``(Π α β, equiv α β → equiv (%%e α) (%%e β))\n--      >>= define ( (`on_equiv).update_prefix n),\n--    solve1\n--      (do α  ← tactic.intro `α,\n--          β  ← tactic.intro `β,\n--          eq ← tactic.intro `eqv,\n\n--          refine_struct ``( { to_fun := %%to_fun %%α %%β %%eq,\n--                        inv_fun := %%to_fun %%β %%α (%%eq).symm,\n--                        left_inv := %%is_inv %%α %%β %%eq }),\n--          admit ),\n--      -- | transport\n--    trace \"-- | TRANSPORT\",\n--    refine_struct ``( { on_equiv := %%fn, .. } ), -- (some `duh),\n--    admit <|> fail \"admit A\",\n--    admit <|> fail \"admit B\",\n--    trace_state <|> fail \"here\",\n--    -- instantiate_mvars v >>= trace ,\n--    trace \"end (mk_transportable)\"\n\n\nmeta def instance_derive_handler' (univ_poly := tt)\n  (modify_target : name → list expr → expr → tactic expr := λ _ _, pure) : derive_handler :=\nλ p n, do\nlet cls := `transportable,\nif p.is_constant_of cls then\ndo decl ← get_decl n,\n   cls_decl ← get_decl cls,\n   env ← get_env,\n   -- guard (env.is_inductive n) <|> fail format!\"failed to derive '{cls}', '{n}' is not an inductive type\",\n   let ls := decl.univ_params.map $ λ n, if univ_poly then level.param n else level.zero,\n   -- incrementally build up target expression `Π (hp : p) [cls hp] ..., cls (n.{ls} hp ...)`\n   -- where `p ...` are the inductive parameter types of `n`\n   let tgt : expr := expr.const n ls,\n   ⟨params, _⟩ ← mk_local_pis (decl.type.instantiate_univ_params (decl.univ_params.zip ls)),\n   (type,tgt) ← params.inits.any_of (λ param, do\n     let tgt := tgt.mk_app param,\n     prod.mk tgt <$> mk_app cls [tgt]),\n   tgt ← modify_target n [] tgt,\n   -- tgt ← params.enum.mfoldr (λ ⟨i, param⟩ tgt,\n   -- do -- add typeclass hypothesis for each inductive parameter\n   --    tgt ← do {\n   --      guard $ i < env.inductive_num_params n,\n   --      param_cls ← mk_app cls [param] <|> fail \"fart\",\n   --      -- TODO(sullrich): omit some typeclass parameters based on usage of `param`?\n   --      pure $ expr.pi `a binder_info.inst_implicit param_cls tgt\n   --    } <|> pure tgt,\n   --    pure $ tgt.bind_pi param\n   -- ) tgt,\n   mk_transportable_instance n,\n   pure true\nelse pure false\n\n\n@[derive_handler]\nmeta def transportable_handler : derive_handler :=\ninstance_derive_handler' tt $\nλ n params, pure\n\n-- -- ``(transportable) _\n\nend interactive\nend tactic\n\n-- namespace group\n\n-- variables {α β : Type u}\n-- variables (eq : equiv α β)\n\n-- @[simp] def tr₀ : α → β := eq\n-- @[simp] def tr₁ (f : α → α) : β → β := λ x : β, eq (f $ eq.symm x)\n-- @[simp] def tr₂ (f : α → α → α) : β → β → β := λ (x y : β), eq $ f (eq.symm x) (eq.symm y)\n-- -- def etr₀ : β → α := eq.inv_fun\n-- -- def etr₁ (f : β → β) (x : α) : α := eq.inv_fun (f $ eq.to_fun x)\n-- -- def etr₂ (f : β → β → β) (x y : α) : α := eq.inv_fun $ f (eq.to_fun x) (eq.to_fun y)\n\n-- -- @[simp]\n-- -- lemma inv_fun_tr₀ (f : α) :\n-- --   eq.inv_fun (tr₀ eq f) = f :=\n-- -- by simp [tr₀,equiv.left_inv eq _]\n\n-- -- @[simp]\n-- -- lemma inv_fun_tr₁ (f : α → α) (x : β) :\n-- --   eq.inv_fun (tr₁ eq f x) = f (eq.inv_fun x) :=\n-- -- by simp [tr₁,equiv.left_inv eq _]\n\n-- -- @[simp]\n-- -- lemma inv_fun_tr₂ (f : α → α → α) (x y : β) :\n-- --   eq.inv_fun (tr₂ eq f x y) = f (eq.inv_fun x) (eq.inv_fun y) :=\n-- -- by simp [tr₂,equiv.left_inv eq _]\n\n-- local attribute [simp] equiv.left_inv equiv.right_inv\n\n-- -- @[simp]\n-- -- lemma symm_inv_fun :\n-- --   eq.symm.inv_fun = eq.to_fun :=\n-- -- by cases eq ; refl\n\n-- -- @[simp]\n-- -- lemma symm_to_fun :\n-- --   eq.symm.to_fun = eq.inv_fun :=\n-- -- by cases eq ; refl\n\n-- -- @[simp]\n-- -- lemma inv_fun_etr₀ (f : β) :\n-- --   eq.to_fun (etr₀ eq f) = f :=\n-- -- by simp [etr₀,equiv.right_inv eq _]\n\n-- -- @[simp]\n-- -- lemma inv_fun_etr₁ (f : β → β) (x : α) :\n-- --   eq.to_fun (etr₁ eq f x) = f (eq.to_fun x) :=\n-- -- by simp [etr₁,equiv.left_inv eq _]\n\n-- -- @[simp]\n-- -- lemma inv_fun_etr₂ (f : α → α → α) (x y : β) :\n-- --   eq.inv_fun (etr₂ eq f x y) = f (eq.inv_fun x) (eq.inv_fun y) :=\n-- -- by simp [tr₂,equiv.left_inv eq _]\n\n-- lemma inj {x y : β}\n--   (h : eq.symm x = eq.symm y)\n-- : x = y := sorry\n\n-- -- @[simp]\n-- -- def on_equiv.to_fun [group α] : group β :=\n-- -- { one := tr₀ eq (one α)\n-- -- , mul := tr₂ eq mul\n-- -- , inv := tr₁ eq inv\n-- -- , mul_left_inv := by { intros, apply inj eq, simp, apply mul_left_inv }\n-- -- , one_mul := by { intros, apply inj eq, simp, apply one_mul }\n-- -- , mul_one := by { intros, apply inj eq, simp [has_mul.mul], apply mul_one }\n-- -- , mul_assoc := by { intros, apply inj eq, simp, apply mul_assoc }  }\n\n-- -- @[simp]\n-- -- def on_equiv.inv_fun [group β] : group α :=\n-- -- { one := tr₀ eq.symm (one _)\n-- -- , mul := tr₂ eq.symm mul\n-- -- , inv := tr₁ eq.symm inv\n-- -- , mul_left_inv := by { intros, apply inj eq.symm, simp, apply mul_left_inv }\n-- -- , one_mul := by { intros, apply inj eq.symm, simp, apply one_mul }\n-- -- , mul_one := by { intros, apply inj eq.symm, simp [has_mul.mul], apply mul_one }\n-- -- , mul_assoc := by { intros, apply inj eq.symm, simp, apply mul_assoc }  }\n\n-- -- def on_equiv' : group α ≃ group β :=\n-- -- { to_fun := @on_equiv.to_fun _ _ eq,\n-- --   inv_fun := @on_equiv.inv_fun _ _ eq,\n-- --   right_inv :=\n-- --   by { intro x, cases x, simp,\n-- --        congr ;\n-- --        funext ;\n-- --        dsimp [mul,one,inv] ;\n-- --        simp!, },\n-- --   left_inv :=\n-- --   by { intro x, cases x, simp,\n-- --        congr ;\n-- --        funext ;\n-- --        dsimp [mul,one,inv] ;\n-- --        simp!, } }\n\n-- -- def transportable' : transportable group :=\n-- -- begin\n-- --   refine { on_equiv := @on_equiv', .. }\n-- --   ; intros ; simp [on_equiv',equiv.refl,equiv.trans]\n-- --   ; split ; funext x ; cases x ; refl,\n-- -- end\n\n-- -- set_option formatter.hide_full_terms false\n-- set_option pp.all true\n-- -- set_option trace.app_builder true\n-- -- #check equiv.has_coe_to_fun\n\n-- -- set_option profiler true\n\n-- -- attribute [derive transportable] group monoid ring field --\n-- -- attribute [derive transportable] monoid\n-- -- attribute [derive transportable] has_add\n-- -- ⊢ Π (α β : Type u), α ≃ β → group α → group β\n-- -- α β : Type u,\n-- -- eq : α ≃ β\n-- -- ⊢ group α ≃ group β\n-- -- on_equiv\n-- -- 2 goals\n-- -- case on_refl\n-- -- ⊢ ∀ (α : Type u), on_equiv α α (equiv.refl α) = equiv.refl (group α)\n\n-- -- case on_trans\n-- -- ⊢ ∀ {α β γ : Type u} (d : α ≃ β) (e : β ≃ γ),\n-- --     on_equiv α γ (equiv.trans d e) = equiv.trans (on_equiv α β d) (on_equiv β γ e)\n-- -- [_field, on_refl]\n-- -- [_field, on_trans]\n-- -- def transportable' : transportable group :=\n\n\n-- end group\n\n-- -- #check derive_attr\n-- instance group.transport {α β : Type u} [R : group α] [e : canonical_equiv α β] : group β :=\n-- sorry\n-- -- (@transportable.on_equiv group group.transportable _ _ e.to_equiv).to_fun R\n\n\n-- -- class transportable (f : Type u → Type v) :=\n-- -- (on_equiv : Π {α β : Type u} (e : equiv α β), equiv (f α) (f β))\n-- -- (on_refl  : Π (α : Type u), on_equiv (equiv.refl α) = equiv.refl (f α))\n-- -- (on_trans : Π {α β γ : Type u} (d : equiv α β) (e : equiv β γ), on_equiv (equiv.trans d e) = equiv.trans (on_equiv d) (on_equiv e))\n\n-- -- -- Our goal is an automagic proof of the following\n-- -- theorem group.transportable : transportable group := sorry\n\n-- -- These we might need to define and prove by hand\n-- def Const : Type u → Type v := λ α, punit\n-- def Fun : Type u → Type v → Type (max u v) := λ α β, α → β\n-- def Prod : Type u → Type v → Type (max u v) := λ α β, α × β\n-- def Swap : Type u → Type v → Type (max u v) := λ α β, β × α\n\n-- lemma Const.transportable (α : Type u) : (transportable Const) := sorry\n-- lemma Fun.transportable (α : Type u) : (transportable (Fun α)) := sorry\n-- lemma Prod.transportable (α : Type u) : (transportable (Prod α)) := sorry\n-- lemma Swap.transportable (α : Type u) : (transportable (Swap α)) := sorry\n\n\n-- -- And then we can define\n-- def Hom1 (α : Type u) : Type v → Type (max u v) := λ β, α → β\n-- def Hom2 (β : Type v) : Type u → Type (max u v) := λ α, α → β\n-- def Aut : Type u → Type u := λ α, α → α\n\n-- -- And hopefully automagically derive\n-- lemma Hom1.transportable (α : Type u) : (transportable (Hom1 α)) := sorry\n-- lemma Hom2.transportable (β : Type v) : (transportable (Hom1 β)) := sorry\n-- lemma Aut.transportable (α : Type u) : (transportable Aut) := sorry\n\n-- -- If we have all these in place...\n-- -- A bit of magic might actually be able to derive `group.transportable` on line 11.\n-- -- After all, a group just is a type plus some functions... and we can now transport functions.\n", "meta": {"author": "cipher1024", "repo": "transport", "sha": "147a1ba17c1e4c0b19b7625a1a70f87458a7f705", "save_path": "github-repos/lean/cipher1024-transport", "path": "github-repos/lean/cipher1024-transport/transport-147a1ba17c1e4c0b19b7625a1a70f87458a7f705/src/transport.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.5039061705290806, "lm_q1q2_score": 0.3024386177109419}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.pfun\nimport Mathlib.order.preorder_hom\nimport Mathlib.tactic.wlog\nimport Mathlib.tactic.monotonicity.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_5 u_6 u_3 u v l u_4 \n\nnamespace Mathlib\n\n/-!\n# Omega Complete Partial Orders\n\nAn omega-complete partial order is a partial order with a supremum\noperation on increasing sequences indexed by natural numbers (which we\ncall `ωSup`). In this sense, it is strictly weaker than join complete\nsemi-lattices as only ω-sized totally ordered sets have a supremum.\n\nThe concept of an omega-complete partial order (ωCPO) is useful for the\nformalization of the semantics of programming languages. Its notion of\nsupremum helps define the meaning of recursive procedures.\n\n## Main definitions\n\n * class `omega_complete_partial_order`\n * `ite`, `map`, `bind`, `seq` as continuous morphisms\n\n## Instances of `omega_complete_partial_order`\n\n * `roption`\n * every `complete_lattice`\n * pi-types\n * product types\n * `monotone_hom`\n * `continuous_hom` (with notation →𝒄)\n   * an instance of `omega_complete_partial_order (α →𝒄 β)`\n * `continuous_hom.of_fun`\n * `continuous_hom.of_mono`\n * continuous functions:\n   * `id`\n   * `ite`\n   * `const`\n   * `roption.bind`\n   * `roption.map`\n   * `roption.seq`\n\n## References\n\n * [G. Markowsky, *Chain-complete posets and directed sets with applications*, https://doi.org/10.1007/BF02485815][markowsky]\n * [J. M. Cadiou and Zohar Manna, *Recursive definitions of partial functions and their computations.*, https://doi.org/10.1145/942580.807072][cadiou]\n * [Carl A. Gunter, *Semantics of Programming Languages: Structures and Techniques*, ISBN: 0262570955][gunter]\n-/\n\nnamespace preorder_hom\n\n\n/-- The constant function, as a monotone function. -/\n@[simp] theorem const_to_fun (α : Type u_1) {β : Type u_2} [preorder α] [preorder β] (f : β) : ∀ (ᾰ : α), coe_fn (const α f) ᾰ = function.const α f ᾰ :=\n  fun (ᾰ : α) => Eq.refl (coe_fn (const α f) ᾰ)\n\n/-- The diagonal function, as a monotone function. -/\ndef prod.diag {α : Type u_1} [preorder α] : α →ₘ α × α :=\n  mk (fun (x : α) => (x, x)) sorry\n\n/-- The `prod.map` function, as a monotone function. -/\n@[simp] theorem prod.map_to_fun {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {α' : Type u_5} {β' : Type u_6} [preorder α'] [preorder β'] (f : α →ₘ β) (f' : α' →ₘ β') (x : α × α') : coe_fn (prod.map f f') x = prod.map (⇑f) (⇑f') x :=\n  Eq.refl (coe_fn (prod.map f f') x)\n\n/-- The `prod.fst` projection, as a monotone function. -/\ndef prod.fst {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] : α × β →ₘ α :=\n  mk prod.fst sorry\n\n/-- The `prod.snd` projection, as a monotone function. -/\ndef prod.snd {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] : α × β →ₘ β :=\n  mk prod.snd sorry\n\n/-- The `prod` constructor, as a monotone function. -/\n@[simp] theorem prod.zip_to_fun {α : Type u_1} {β : Type u_2} {γ : Type u_3} [preorder α] [preorder β] [preorder γ] (f : α →ₘ β) (g : α →ₘ γ) : ∀ (ᾰ : α), coe_fn (prod.zip f g) ᾰ = (coe_fn f ᾰ, coe_fn g ᾰ) :=\n  fun (ᾰ : α) => Eq.refl (coe_fn f ᾰ, coe_fn g ᾰ)\n\n/-- `roption.bind` as a monotone function -/\n@[simp] theorem bind_to_fun {α : Type u_1} [preorder α] {β : Type u_2} {γ : Type u_2} (f : α →ₘ roption β) (g : α →ₘ β → roption γ) (x : α) : coe_fn (bind f g) x = coe_fn f x >>= coe_fn g x :=\n  Eq.refl (coe_fn (bind f g) x)\n\nend preorder_hom\n\n\nnamespace omega_complete_partial_order\n\n\n/-- A chain is a monotonically increasing sequence.\n\nSee the definition on page 114 of [gunter]. -/\ndef chain (α : Type u) [preorder α] :=\n  ℕ →ₘ α\n\nnamespace chain\n\n\nprotected instance has_coe_to_fun {α : Type u} [preorder α] : has_coe_to_fun (chain α) :=\n  infer_instance\n\nprotected instance inhabited {α : Type u} [preorder α] [Inhabited α] : Inhabited (chain α) :=\n  { default := preorder_hom.mk (fun (_x : ℕ) => Inhabited.default) sorry }\n\nprotected instance has_mem {α : Type u} [preorder α] : has_mem α (chain α) :=\n  has_mem.mk fun (a : α) (c : ℕ →ₘ α) => ∃ (i : ℕ), a = coe_fn c i\n\nprotected instance has_le {α : Type u} [preorder α] : HasLessEq (chain α) :=\n  { LessEq := fun (x y : chain α) => ∀ (i : ℕ), ∃ (j : ℕ), coe_fn x i ≤ coe_fn y j }\n\n/-- `map` function for `chain` -/\n@[simp] theorem map_to_fun {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) (f : α →ₘ β) : ∀ (ᾰ : ℕ), coe_fn (map c f) ᾰ = coe_fn f (coe_fn c ᾰ) :=\n  fun (ᾰ : ℕ) => Eq.refl (coe_fn f (coe_fn c ᾰ))\n\ntheorem mem_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β} (x : α) : x ∈ c → coe_fn f x ∈ map c f := sorry\n\ntheorem exists_of_mem_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β} {b : β} : b ∈ map c f → ∃ (a : α), a ∈ c ∧ coe_fn f a = b := sorry\n\ntheorem mem_map_iff {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β} {b : β} : b ∈ map c f ↔ ∃ (a : α), a ∈ c ∧ coe_fn f a = b := sorry\n\n@[simp] theorem map_id {α : Type u} [preorder α] (c : chain α) : map c preorder_hom.id = c :=\n  preorder_hom.comp_id c\n\ntheorem map_comp {α : Type u} {β : Type v} {γ : Type u_1} [preorder α] [preorder β] [preorder γ] (c : chain α) {f : α →ₘ β} (g : β →ₘ γ) : map (map c f) g = map c (preorder_hom.comp g f) :=\n  rfl\n\ntheorem map_le_map {α : Type u} {β : Type v} [preorder α] [preorder β] (c : chain α) {f : α →ₘ β} {g : α →ₘ β} (h : f ≤ g) : map c f ≤ map c g := sorry\n\n/-- `chain.zip` pairs up the elements of two chains that have the same index -/\n@[simp] theorem zip_to_fun {α : Type u} {β : Type v} [preorder α] [preorder β] (c₀ : chain α) (c₁ : chain β) : ∀ (ᾰ : ℕ), coe_fn (zip c₀ c₁) ᾰ = (coe_fn c₀ ᾰ, coe_fn c₁ ᾰ) :=\n  fun (ᾰ : ℕ) => Eq.refl (coe_fn c₀ ᾰ, coe_fn c₁ ᾰ)\n\nend chain\n\n\nend omega_complete_partial_order\n\n\n/-- An omega-complete partial order is a partial order with a supremum\noperation on increasing sequences indexed by natural numbers (which we\ncall `ωSup`). In this sense, it is strictly weaker than join complete\nsemi-lattices as only ω-sized totally ordered sets have a supremum.\n\nSee the definition on page 114 of [gunter]. -/\nclass omega_complete_partial_order (α : Type u_1) \nextends partial_order α\nwhere\n  ωSup : omega_complete_partial_order.chain α → α\n  le_ωSup : ∀ (c : omega_complete_partial_order.chain α) (i : ℕ), coe_fn c i ≤ ωSup c\n  ωSup_le : ∀ (c : omega_complete_partial_order.chain α) (x : α), (∀ (i : ℕ), coe_fn c i ≤ x) → ωSup c ≤ x\n\nnamespace omega_complete_partial_order\n\n\n/-- Transfer a `omega_complete_partial_order` on `β` to a `omega_complete_partial_order` on `α` using\na strictly monotone function `f : β →ₘ α`, a definition of ωSup and a proof that `f` is continuous\nwith regard to the provided `ωSup` and the ωCPO on `α`. -/\nprotected def lift {α : Type u} {β : Type v} [omega_complete_partial_order α] [partial_order β] (f : β →ₘ α) (ωSup₀ : chain β → β) (h : ∀ (x y : β), coe_fn f x ≤ coe_fn f y → x ≤ y) (h' : ∀ (c : chain β), coe_fn f (ωSup₀ c) = ωSup (chain.map c f)) : omega_complete_partial_order β :=\n  mk ωSup₀ sorry sorry\n\ntheorem le_ωSup_of_le {α : Type u} [omega_complete_partial_order α] {c : chain α} {x : α} (i : ℕ) (h : x ≤ coe_fn c i) : x ≤ ωSup c :=\n  le_trans h (le_ωSup c i)\n\ntheorem ωSup_total {α : Type u} [omega_complete_partial_order α] {c : chain α} {x : α} (h : ∀ (i : ℕ), coe_fn c i ≤ x ∨ x ≤ coe_fn c i) : ωSup c ≤ x ∨ x ≤ ωSup c := sorry\n\ntheorem ωSup_le_ωSup_of_le {α : Type u} [omega_complete_partial_order α] {c₀ : chain α} {c₁ : chain α} (h : c₀ ≤ c₁) : ωSup c₀ ≤ ωSup c₁ :=\n  ωSup_le c₀ (ωSup c₁)\n    fun (i : ℕ) => Exists.rec_on (h i) fun (j : ℕ) (h : coe_fn c₀ i ≤ coe_fn c₁ j) => le_trans h (le_ωSup c₁ j)\n\ntheorem ωSup_le_iff {α : Type u} [omega_complete_partial_order α] (c : chain α) (x : α) : ωSup c ≤ x ↔ ∀ (i : ℕ), coe_fn c i ≤ x :=\n  { mp := fun (ᾰ : ωSup c ≤ x) (i : ℕ) => le_trans (le_ωSup c i) ᾰ,\n    mpr := fun (ᾰ : ∀ (i : ℕ), coe_fn c i ≤ x) => ωSup_le c x ᾰ }\n\n/-- A subset `p : α → Prop` of the type closed under `ωSup` induces an\n`omega_complete_partial_order` on the subtype `{a : α // p a}`. -/\ndef subtype {α : Type u_1} [omega_complete_partial_order α] (p : α → Prop) (hp : ∀ (c : chain α), (∀ (i : α), i ∈ c → p i) → p (ωSup c)) : omega_complete_partial_order (Subtype p) :=\n  omega_complete_partial_order.lift (preorder_hom.subtype.val p)\n    (fun (c : chain (Subtype p)) => { val := ωSup (chain.map c (preorder_hom.subtype.val p)), property := sorry }) sorry\n    sorry\n\n/-- A monotone function `f : α →ₘ β` is continuous if it distributes over ωSup.\n\nIn order to distinguish it from the (more commonly used) continuity from topology\n(see topology/basic.lean), the present definition is often referred to as\n\"Scott-continuity\" (referring to Dana Scott). It corresponds to continuity\nin Scott topological spaces (not defined here). -/\ndef continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →ₘ β) :=\n  ∀ (c : chain α), coe_fn f (ωSup c) = ωSup (chain.map c f)\n\n/-- `continuous' f` asserts that `f` is both monotone and continuous. -/\ndef continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α → β) :=\n  ∃ (hf : monotone f), continuous (preorder_hom.mk f hf)\n\ntheorem continuous.to_monotone {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] {f : α → β} (hf : continuous' f) : monotone f :=\n  Exists.fst hf\n\ntheorem continuous.of_bundled {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α → β) (hf : monotone f) (hf' : continuous (preorder_hom.mk f hf)) : continuous' f :=\n  Exists.intro hf hf'\n\ntheorem continuous.of_bundled' {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →ₘ β) (hf' : continuous f) : continuous' ⇑f :=\n  Exists.intro (preorder_hom.monotone f) hf'\n\ntheorem continuous.to_bundled {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α → β) (hf : continuous' f) : continuous (preorder_hom.mk f (continuous.to_monotone hf)) :=\n  Exists.snd hf\n\ntheorem continuous_id {α : Type u} [omega_complete_partial_order α] : continuous preorder_hom.id := sorry\n\ntheorem continuous_comp {α : Type u} {β : Type v} {γ : Type u_1} [omega_complete_partial_order α] [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : α →ₘ β) (g : β →ₘ γ) (hfc : continuous f) (hgc : continuous g) : continuous (preorder_hom.comp g f) := sorry\n\ntheorem id_continuous' {α : Type u} [omega_complete_partial_order α] : continuous' id := sorry\n\ntheorem const_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (x : β) : continuous' (function.const α x) := sorry\n\nend omega_complete_partial_order\n\n\nnamespace roption\n\n\ntheorem eq_of_chain {α : Type u} {c : omega_complete_partial_order.chain (roption α)} {a : α} {b : α} (ha : some a ∈ c) (hb : some b ∈ c) : a = b := sorry\n\n/-- The (noncomputable) `ωSup` definition for the `ω`-CPO structure on `roption α`. -/\nprotected def ωSup {α : Type u} (c : omega_complete_partial_order.chain (roption α)) : roption α :=\n  dite (∃ (a : α), some a ∈ c) (fun (h : ∃ (a : α), some a ∈ c) => some (classical.some h))\n    fun (h : ¬∃ (a : α), some a ∈ c) => none\n\ntheorem ωSup_eq_some {α : Type u} {c : omega_complete_partial_order.chain (roption α)} {a : α} (h : some a ∈ c) : roption.ωSup c = some a := sorry\n\ntheorem ωSup_eq_none {α : Type u} {c : omega_complete_partial_order.chain (roption α)} (h : ¬∃ (a : α), some a ∈ c) : roption.ωSup c = none :=\n  dif_neg h\n\ntheorem mem_chain_of_mem_ωSup {α : Type u} {c : omega_complete_partial_order.chain (roption α)} {a : α} (h : a ∈ roption.ωSup c) : some a ∈ c := sorry\n\nprotected instance omega_complete_partial_order {α : Type u} : omega_complete_partial_order (roption α) :=\n  omega_complete_partial_order.mk roption.ωSup sorry sorry\n\ntheorem mem_ωSup {α : Type u} (x : α) (c : omega_complete_partial_order.chain (roption α)) : x ∈ omega_complete_partial_order.ωSup c ↔ some x ∈ c := sorry\n\nend roption\n\n\nnamespace pi\n\n\n/-- Function application `λ f, f a` is monotone with respect to `f` for fixed `a`. -/\ndef monotone_apply {α : Type u_1} {β : α → Type u_2} [(a : α) → partial_order (β a)] (a : α) : ((a : α) → β a) →ₘ β a :=\n  preorder_hom.mk (fun (f : (a : α) → β a) => f a) sorry\n\nprotected instance omega_complete_partial_order {α : Type u_1} {β : α → Type u_2} [(a : α) → omega_complete_partial_order (β a)] : omega_complete_partial_order ((a : α) → β a) :=\n  omega_complete_partial_order.mk\n    (fun (c : omega_complete_partial_order.chain ((a : α) → β a)) (a : α) =>\n      omega_complete_partial_order.ωSup (omega_complete_partial_order.chain.map c (monotone_apply a)))\n    sorry sorry\n\nnamespace omega_complete_partial_order\n\n\ntheorem flip₁_continuous' {α : Type u_1} {β : α → Type u_2} {γ : Type u_3} [(x : α) → omega_complete_partial_order (β x)] [omega_complete_partial_order γ] (f : (x : α) → γ → β x) (a : α) (hf : omega_complete_partial_order.continuous' fun (x : γ) (y : α) => f y x) : omega_complete_partial_order.continuous' (f a) := sorry\n\ntheorem flip₂_continuous' {α : Type u_1} {β : α → Type u_2} {γ : Type u_3} [(x : α) → omega_complete_partial_order (β x)] [omega_complete_partial_order γ] (f : γ → (x : α) → β x) (hf : ∀ (x : α), omega_complete_partial_order.continuous' fun (g : γ) => f g x) : omega_complete_partial_order.continuous' f := sorry\n\nend omega_complete_partial_order\n\n\nend pi\n\n\nnamespace prod\n\n\n/-- The supremum of a chain in the product `ω`-CPO. -/\nprotected def ωSup {α : Type u_1} {β : Type u_2} [omega_complete_partial_order α] [omega_complete_partial_order β] (c : omega_complete_partial_order.chain (α × β)) : α × β :=\n  (omega_complete_partial_order.ωSup (omega_complete_partial_order.chain.map c preorder_hom.prod.fst),\n  omega_complete_partial_order.ωSup (omega_complete_partial_order.chain.map c preorder_hom.prod.snd))\n\nprotected instance omega_complete_partial_order {α : Type u_1} {β : Type u_2} [omega_complete_partial_order α] [omega_complete_partial_order β] : omega_complete_partial_order (α × β) :=\n  omega_complete_partial_order.mk prod.ωSup sorry sorry\n\nend prod\n\n\nnamespace complete_lattice\n\n\n/-- Any complete lattice has an `ω`-CPO structure where the countable supremum is a special case\nof arbitrary suprema. -/\nprotected instance omega_complete_partial_order (α : Type u) [complete_lattice α] : omega_complete_partial_order α :=\n  omega_complete_partial_order.mk (fun (c : omega_complete_partial_order.chain α) => supr fun (i : ℕ) => coe_fn c i) sorry\n    sorry\n\ntheorem inf_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] [is_total β LessEq] (f : α →ₘ β) (g : α →ₘ β) (hf : omega_complete_partial_order.continuous f) (hg : omega_complete_partial_order.continuous g) : omega_complete_partial_order.continuous (f ⊓ g) := sorry\n\ntheorem Sup_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] (s : set (α →ₘ β)) (hs : ∀ (f : α →ₘ β), f ∈ s → omega_complete_partial_order.continuous f) : omega_complete_partial_order.continuous (Sup s) := sorry\n\ntheorem Sup_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] (s : set (α → β)) : (∀ (t : α → β), t ∈ s → omega_complete_partial_order.continuous' t) → omega_complete_partial_order.continuous' (Sup s) := sorry\n\ntheorem sup_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] {f : α →ₘ β} {g : α →ₘ β} (hf : omega_complete_partial_order.continuous f) (hg : omega_complete_partial_order.continuous g) : omega_complete_partial_order.continuous (f ⊔ g) := sorry\n\ntheorem top_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] : omega_complete_partial_order.continuous ⊤ := sorry\n\ntheorem bot_continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [complete_lattice β] : omega_complete_partial_order.continuous ⊥ := sorry\n\nend complete_lattice\n\n\nnamespace omega_complete_partial_order\n\n\nnamespace preorder_hom\n\n\n/-- Function application `λ f, f a` (for fixed `a`) is a monotone function from the\nmonotone function space `α →ₘ β` to `β`. -/\ndef monotone_apply {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (a : α) : (α →ₘ β) →ₘ β :=\n  preorder_hom.mk (fun (f : α →ₘ β) => coe_fn f a) sorry\n\n/-- The \"forgetful functor\" from `α →ₘ β` to `α → β` that takes the underlying function,\nis monotone. -/\ndef to_fun_hom {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] : (α →ₘ β) →ₘ α → β :=\n  preorder_hom.mk (fun (f : α →ₘ β) => preorder_hom.to_fun f) sorry\n\n/-- The `ωSup` operator for monotone functions. -/\n@[simp] theorem ωSup_to_fun {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c : chain (α →ₘ β)) (a : α) : coe_fn (preorder_hom.ωSup c) a = ωSup (chain.map c (monotone_apply a)) :=\n  Eq.refl (coe_fn (preorder_hom.ωSup c) a)\n\nprotected instance omega_complete_partial_order {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] : omega_complete_partial_order (α →ₘ β) :=\n  omega_complete_partial_order.lift to_fun_hom preorder_hom.ωSup sorry sorry\n\nend preorder_hom\n\n\n/-- A monotone function on `ω`-continuous partial orders is said to be continuous\nif for every chain `c : chain α`, `f (⊔ i, c i) = ⊔ i, f (c i)`.\nThis is just the bundled version of `preorder_hom.continuous`. -/\nstructure continuous_hom (α : Type u) (β : Type v) [omega_complete_partial_order α] [omega_complete_partial_order β] \nextends α →ₘ β\nwhere\n  cont : continuous (preorder_hom.mk to_fun monotone')\n\ninfixr:25 \" →𝒄 \" => Mathlib.omega_complete_partial_order.continuous_hom\n\nprotected instance continuous_hom.has_coe_to_fun (α : Type u) (β : Type v) [omega_complete_partial_order α] [omega_complete_partial_order β] : has_coe_to_fun (α →𝒄 β) :=\n  has_coe_to_fun.mk (fun (_x : α →𝒄 β) => α → β) continuous_hom.to_fun\n\nprotected instance preorder_hom.has_coe (α : Type u) (β : Type v) [omega_complete_partial_order α] [omega_complete_partial_order β] : has_coe (α →𝒄 β) (α →ₘ β) :=\n  has_coe.mk continuous_hom.to_preorder_hom\n\nprotected instance continuous_hom.partial_order (α : Type u) (β : Type v) [omega_complete_partial_order α] [omega_complete_partial_order β] : partial_order (α →𝒄 β) :=\n  partial_order.lift continuous_hom.to_fun sorry\n\nnamespace continuous_hom\n\n\ntheorem congr_fun {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] {f : α →𝒄 β} {g : α →𝒄 β} (h : f = g) (x : α) : coe_fn f x = coe_fn g x :=\n  congr_arg (fun (h : α →𝒄 β) => coe_fn h x) h\n\ntheorem congr_arg {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →𝒄 β) {x : α} {y : α} (h : x = y) : coe_fn f x = coe_fn f y :=\n  congr_arg (fun (x : α) => coe_fn f x) h\n\ntheorem monotone {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →𝒄 β) : monotone ⇑f :=\n  monotone' f\n\ntheorem ite_continuous' {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] {p : Prop} [hp : Decidable p] (f : α → β) (g : α → β) (hf : continuous' f) (hg : continuous' g) : continuous' fun (x : α) => ite p (f x) (g x) := sorry\n\ntheorem ωSup_bind {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (c : chain α) (f : α →ₘ roption β) (g : α →ₘ β → roption γ) : ωSup (chain.map c (preorder_hom.bind f g)) = ωSup (chain.map c f) >>= ωSup (chain.map c g) := sorry\n\ntheorem bind_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : α → roption β) (g : α → β → roption γ) : continuous' f → continuous' g → continuous' fun (x : α) => f x >>= g x := sorry\n\ntheorem map_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : β → γ) (g : α → roption β) (hg : continuous' g) : continuous' fun (x : α) => f <$> g x := sorry\n\ntheorem seq_continuous' {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : α → roption (β → γ)) (g : α → roption β) (hf : continuous' f) (hg : continuous' g) : continuous' fun (x : α) => f x <*> g x := sorry\n\ntheorem continuous {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (F : α →𝒄 β) (C : chain α) : coe_fn F (ωSup C) = ωSup (chain.map C ↑F) :=\n  cont F C\n\n/-- Construct a continuous function from a bare function, a continuous function, and a proof that\nthey are equal. -/\ndef of_fun {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α → β) (g : α →𝒄 β) (h : f = ⇑g) : α →𝒄 β :=\n  mk f sorry sorry\n\n/-- Construct a continuous function from a monotone function with a proof of continuity. -/\ndef of_mono {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →ₘ β) (h : ∀ (c : chain α), coe_fn f (ωSup c) = ωSup (chain.map c f)) : α →𝒄 β :=\n  mk ⇑f sorry h\n\n/-- The identity as a continuous function. -/\n@[simp] theorem id_to_fun {α : Type u} [omega_complete_partial_order α] : ∀ (ᾰ : α), coe_fn id ᾰ = ᾰ :=\n  fun (ᾰ : α) => Eq.refl ᾰ\n\n/-- The composition of continuous functions. -/\ndef comp {α : Type u} {β : Type v} {γ : Type u_3} [omega_complete_partial_order α] [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : β →𝒄 γ) (g : α →𝒄 β) : α →𝒄 γ :=\n  of_mono (preorder_hom.comp ↑f ↑g) sorry\n\nprotected theorem ext {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →𝒄 β) (g : α →𝒄 β) (h : ∀ (x : α), coe_fn f x = coe_fn g x) : f = g := sorry\n\nprotected theorem coe_inj {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →𝒄 β) (g : α →𝒄 β) (h : ⇑f = ⇑g) : f = g :=\n  continuous_hom.ext f g (congr_fun h)\n\n@[simp] theorem comp_id {β : Type v} {γ : Type u_3} [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : β →𝒄 γ) : comp f id = f :=\n  continuous_hom.ext (comp f id) f fun (x : β) => Eq.refl (coe_fn (comp f id) x)\n\n@[simp] theorem id_comp {β : Type v} {γ : Type u_3} [omega_complete_partial_order β] [omega_complete_partial_order γ] (f : β →𝒄 γ) : comp id f = f :=\n  continuous_hom.ext (comp id f) f fun (x : β) => Eq.refl (coe_fn (comp id f) x)\n\n@[simp] theorem comp_assoc {α : Type u} {β : Type v} {γ : Type u_3} {φ : Type u_4} [omega_complete_partial_order α] [omega_complete_partial_order β] [omega_complete_partial_order γ] [omega_complete_partial_order φ] (f : γ →𝒄 φ) (g : β →𝒄 γ) (h : α →𝒄 β) : comp f (comp g h) = comp (comp f g) h :=\n  continuous_hom.ext (comp f (comp g h)) (comp (comp f g) h) fun (x : α) => Eq.refl (coe_fn (comp f (comp g h)) x)\n\n@[simp] theorem coe_apply {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (a : α) (f : α →𝒄 β) : coe_fn (↑f) a = coe_fn f a :=\n  rfl\n\n/-- `function.const` is a continuous function. -/\ndef const {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : β) : α →𝒄 β :=\n  of_mono (preorder_hom.const α f) sorry\n\n@[simp] theorem const_apply {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : β) (a : α) : coe_fn (const f) a = f :=\n  rfl\n\nprotected instance inhabited {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] [Inhabited β] : Inhabited (α →𝒄 β) :=\n  { default := const Inhabited.default }\n\nnamespace prod\n\n\n/-- The application of continuous functions as a monotone function.\n\n(It would make sense to make it a continuous function, but we are currently constructing a\n`omega_complete_partial_order` instance for `α →𝒄 β`, and we cannot use it as the domain or image\nof a continuous function before we do.) -/\ndef apply {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] : (α →𝒄 β) × α →ₘ β :=\n  preorder_hom.mk (fun (f : (α →𝒄 β) × α) => coe_fn (prod.fst f) (prod.snd f)) sorry\n\nend prod\n\n\n/-- The map from continuous functions to monotone functions is itself a monotone function. -/\n@[simp] theorem to_mono_to_fun {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (f : α →𝒄 β) : coe_fn to_mono f = ↑f :=\n  Eq.refl (coe_fn to_mono f)\n\n/-- When proving that a chain of applications is below a bound `z`, it suffices to consider the\nfunctions and values being selected from the same index in the chains.\n\nThis lemma is more specific than necessary, i.e. `c₀` only needs to be a\nchain of monotone functions, but it is only used with continuous functions. -/\n@[simp] theorem forall_forall_merge {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) (z : β) : (∀ (i j : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ j) ≤ z) ↔ ∀ (i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ i) ≤ z := sorry\n\n@[simp] theorem forall_forall_merge' {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) (z : β) : (∀ (j i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ j) ≤ z) ↔ ∀ (i : ℕ), coe_fn (coe_fn c₀ i) (coe_fn c₁ i) ≤ z := sorry\n\n/-- The `ωSup` operator for continuous functions, which takes the pointwise countable supremum\nof the functions in the `ω`-chain. -/\nprotected def ωSup {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c : chain (α →𝒄 β)) : α →𝒄 β :=\n  of_mono (ωSup (chain.map c to_mono)) sorry\n\nprotected instance omega_complete_partial_order {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] : omega_complete_partial_order (α →𝒄 β) :=\n  omega_complete_partial_order.lift to_mono continuous_hom.ωSup sorry sorry\n\ntheorem ωSup_def {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c : chain (α →𝒄 β)) (x : α) : coe_fn (ωSup c) x = coe_fn (continuous_hom.ωSup c) x :=\n  rfl\n\ntheorem ωSup_ωSup {α : Type u} {β : Type v} [omega_complete_partial_order α] [omega_complete_partial_order β] (c₀ : chain (α →𝒄 β)) (c₁ : chain α) : coe_fn (ωSup c₀) (ωSup c₁) = ωSup (preorder_hom.comp prod.apply (chain.zip c₀ c₁)) := sorry\n\n/-- A family of continuous functions yields a continuous family of functions. -/\ndef flip {β : Type v} {γ : Type u_3} [omega_complete_partial_order β] [omega_complete_partial_order γ] {α : Type u_1} (f : α → β →𝒄 γ) : β →𝒄 α → γ :=\n  mk (fun (x : β) (y : α) => coe_fn (f y) x) sorry sorry\n\n/-- `roption.bind` as a continuous function. -/\ndef bind {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : α →𝒄 roption β) (g : α →𝒄 β → roption γ) : α →𝒄 roption γ :=\n  of_mono (preorder_hom.bind ↑f ↑g) sorry\n\n/-- `roption.map` as a continuous function. -/\n@[simp] theorem map_to_fun {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : β → γ) (g : α →𝒄 roption β) (x : α) : coe_fn (map f g) x = f <$> coe_fn g x :=\n  Eq.refl (coe_fn (map f g) x)\n\n/-- `roption.seq` as a continuous function. -/\ndef seq {α : Type u} [omega_complete_partial_order α] {β : Type v} {γ : Type v} (f : α →𝒄 roption (β → γ)) (g : α →𝒄 roption β) : α →𝒄 roption γ :=\n  of_fun (fun (x : α) => coe_fn f x <*> coe_fn g x) (bind f (flip (flip map g))) sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/omega_complete_partial_order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3024386177109419}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.nontrivial\nimport Mathlib.algebra.group.units_hom\nimport Mathlib.algebra.group.inj_surj\nimport Mathlib.algebra.group_with_zero.defs\nimport Mathlib.PostPort\n\nuniverses u_1 u_3 u u_2 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Groups with an adjoined zero element\n\nThis file describes structures that are not usually studied on their own right in mathematics,\nnamely a special sort of monoid: apart from a distinguished “zero element” they form a group,\nor in other words, they are groups with an adjoined zero element.\n\nExamples are:\n\n* division rings;\n* the value monoid of a multiplicative valuation;\n* in particular, the non-negative real numbers.\n\n## Main definitions\n\nVarious lemmas about `group_with_zero` and `comm_group_with_zero`.\nTo reduce import dependencies, the type-classes themselves are in\n`algebra.group_with_zero.defs`.\n\n## Implementation details\n\nAs is usual in mathlib, we extend the inverse function to the zero element,\nand require `0⁻¹ = 0`.\n\n-/\n\n/-- Pullback a `mul_zero_class` instance along an injective function. -/\nprotected def function.injective.mul_zero_class {M₀ : Type u_1} {M₀' : Type u_3} [mul_zero_class M₀] [Mul M₀'] [HasZero M₀'] (f : M₀' → M₀) (hf : function.injective f) (zero : f 0 = 0) (mul : ∀ (a b : M₀'), f (a * b) = f a * f b) : mul_zero_class M₀' :=\n  mul_zero_class.mk Mul.mul 0 sorry sorry\n\n/-- Pushforward a `mul_zero_class` instance along an surjective function. -/\nprotected def function.surjective.mul_zero_class {M₀ : Type u_1} {M₀' : Type u_3} [mul_zero_class M₀] [Mul M₀'] [HasZero M₀'] (f : M₀ → M₀') (hf : function.surjective f) (zero : f 0 = 0) (mul : ∀ (a b : M₀), f (a * b) = f a * f b) : mul_zero_class M₀' :=\n  mul_zero_class.mk Mul.mul 0 sorry sorry\n\ntheorem mul_eq_zero_of_left {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} (h : a = 0) (b : M₀) : a * b = 0 :=\n  Eq.symm h ▸ zero_mul b\n\ntheorem mul_eq_zero_of_right {M₀ : Type u_1} [mul_zero_class M₀] {b : M₀} (a : M₀) (h : b = 0) : a * b = 0 :=\n  Eq.symm h ▸ mul_zero a\n\ntheorem left_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} : a * b ≠ 0 → a ≠ 0 :=\n  mt fun (h : a = 0) => mul_eq_zero_of_left h b\n\ntheorem right_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} : a * b ≠ 0 → b ≠ 0 :=\n  mt (mul_eq_zero_of_right a)\n\ntheorem ne_zero_and_ne_zero_of_mul {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} (h : a * b ≠ 0) : a ≠ 0 ∧ b ≠ 0 :=\n  { left := left_ne_zero_of_mul h, right := right_ne_zero_of_mul h }\n\ntheorem mul_eq_zero_of_ne_zero_imp_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] {a : M₀} {b : M₀} (h : a ≠ 0 → b = 0) : a * b = 0 := sorry\n\n/-- Pushforward a `no_zero_divisors` instance along an injective function. -/\nprotected theorem function.injective.no_zero_divisors {M₀ : Type u_1} {M₀' : Type u_3} [Mul M₀] [HasZero M₀] [Mul M₀'] [HasZero M₀'] [no_zero_divisors M₀'] (f : M₀ → M₀') (hf : function.injective f) (zero : f 0 = 0) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) : no_zero_divisors M₀ := sorry\n\ntheorem eq_zero_of_mul_self_eq_zero {M₀ : Type u_1} [Mul M₀] [HasZero M₀] [no_zero_divisors M₀] {a : M₀} (h : a * a = 0) : a = 0 :=\n  or.elim (eq_zero_or_eq_zero_of_mul_eq_zero h) id id\n\n/-- If `α` has no zero divisors, then the product of two elements equals zero iff one of them\nequals zero. -/\n@[simp] theorem mul_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} : a * b = 0 ↔ a = 0 ∨ b = 0 :=\n  { mp := eq_zero_or_eq_zero_of_mul_eq_zero,\n    mpr := fun (o : a = 0 ∨ b = 0) => or.elim o (fun (h : a = 0) => mul_eq_zero_of_left h b) (mul_eq_zero_of_right a) }\n\n/-- If `α` has no zero divisors, then the product of two elements equals zero iff one of them\nequals zero. -/\n@[simp] theorem zero_eq_mul {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} : 0 = a * b ↔ a = 0 ∨ b = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 = a * b ↔ a = 0 ∨ b = 0)) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b = 0 ↔ a = 0 ∨ b = 0)) (propext mul_eq_zero))) (iff.refl (a = 0 ∨ b = 0)))\n\n/-- If `α` has no zero divisors, then the product of two elements is nonzero iff both of them\nare nonzero. -/\ntheorem mul_ne_zero_iff {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} : a * b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 :=\n  iff.trans (not_congr mul_eq_zero) not_or_distrib\n\ntheorem mul_ne_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} (ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 :=\n  iff.mpr mul_ne_zero_iff { left := ha, right := hb }\n\n/-- If `α` has no zero divisors, then for elements `a, b : α`, `a * b` equals zero iff so is\n`b * a`. -/\ntheorem mul_eq_zero_comm {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} : a * b = 0 ↔ b * a = 0 :=\n  iff.trans mul_eq_zero (iff.trans (or_comm (a = 0) (b = 0)) (iff.symm mul_eq_zero))\n\n/-- If `α` has no zero divisors, then for elements `a, b : α`, `a * b` is nonzero iff so is\n`b * a`. -/\ntheorem mul_ne_zero_comm {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} {b : M₀} : a * b ≠ 0 ↔ b * a ≠ 0 :=\n  not_congr mul_eq_zero_comm\n\ntheorem mul_self_eq_zero {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} : a * a = 0 ↔ a = 0 := sorry\n\ntheorem zero_eq_mul_self {M₀ : Type u_1} [mul_zero_class M₀] [no_zero_divisors M₀] {a : M₀} : 0 = a * a ↔ a = 0 := sorry\n\n/-- In a nontrivial monoid with zero, zero and one are different. -/\n@[simp] theorem zero_ne_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] : 0 ≠ 1 := sorry\n\n@[simp] theorem one_ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] : 1 ≠ 0 :=\n  ne.symm zero_ne_one\n\ntheorem ne_zero_of_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀} (h : a = 1) : a ≠ 0 :=\n  trans_rel_right ne h one_ne_zero\n\ntheorem left_ne_zero_of_mul_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀} {b : M₀} (h : a * b = 1) : a ≠ 0 :=\n  left_ne_zero_of_mul (ne_zero_of_eq_one h)\n\ntheorem right_ne_zero_of_mul_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀} {b : M₀} (h : a * b = 1) : b ≠ 0 :=\n  right_ne_zero_of_mul (ne_zero_of_eq_one h)\n\n/-- Pullback a `nontrivial` instance along a function sending `0` to `0` and `1` to `1`. -/\nprotected theorem pullback_nonzero {M₀ : Type u_1} {M₀' : Type u_3} [monoid_with_zero M₀] [nontrivial M₀] [HasZero M₀'] [HasOne M₀'] (f : M₀' → M₀) (zero : f 0 = 0) (one : f 1 = 1) : nontrivial M₀' := sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [monoid_with_zero M₀] (f : M₀' → M₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : monoid_with_zero M₀' :=\n  monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry\n\n/-- Pushforward a `monoid_with_zero` class along a surjective function. -/\nprotected def function.surjective.monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [monoid_with_zero M₀] (f : M₀ → M₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) : monoid_with_zero M₀' :=\n  monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.comm_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [comm_monoid_with_zero M₀] (f : M₀' → M₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : comm_monoid_with_zero M₀' :=\n  comm_monoid_with_zero.mk comm_monoid.mul sorry comm_monoid.one sorry sorry sorry mul_zero_class.zero sorry sorry\n\n/-- Pushforward a `monoid_with_zero` class along a surjective function. -/\nprotected def function.surjective.comm_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [HasZero M₀'] [Mul M₀'] [HasOne M₀'] [comm_monoid_with_zero M₀] (f : M₀ → M₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀), f (x * y) = f x * f y) : comm_monoid_with_zero M₀' :=\n  comm_monoid_with_zero.mk comm_monoid.mul sorry comm_monoid.one sorry sorry sorry mul_zero_class.zero sorry sorry\n\nnamespace units\n\n\n/-- An element of the unit group of a nonzero monoid with zero represented as an element\n    of the monoid is nonzero. -/\n@[simp] theorem ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] (u : units M₀) : ↑u ≠ 0 :=\n  left_ne_zero_of_mul_eq_one (mul_inv u)\n\n-- We can't use `mul_eq_zero` + `units.ne_zero` in the next two lemmas because we don't assume\n\n-- `nonzero M₀`.\n\n@[simp] theorem mul_left_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] (u : units M₀) {a : M₀} : a * ↑u = 0 ↔ a = 0 := sorry\n\n@[simp] theorem mul_right_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] (u : units M₀) {a : M₀} : ↑u * a = 0 ↔ a = 0 := sorry\n\nend units\n\n\nnamespace is_unit\n\n\ntheorem ne_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] {a : M₀} (ha : is_unit a) : a ≠ 0 :=\n  (fun (_a : is_unit a) =>\n      Exists.dcases_on _a fun (w : units M₀) (h : ↑w = a) => idRhs ((fun (_x : M₀) => _x ≠ 0) a) (h ▸ units.ne_zero w))\n    ha\n\ntheorem mul_right_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] {a : M₀} {b : M₀} (ha : is_unit a) : a * b = 0 ↔ b = 0 := sorry\n\ntheorem mul_left_eq_zero {M₀ : Type u_1} [monoid_with_zero M₀] {a : M₀} {b : M₀} (hb : is_unit b) : a * b = 0 ↔ a = 0 := sorry\n\nend is_unit\n\n\n/-- In a monoid with zero, if zero equals one, then zero is the only element. -/\ntheorem eq_zero_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) (a : M₀) : a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = 0)) (Eq.symm (mul_one a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * 1 = 0)) (Eq.symm h)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 0 = 0)) (mul_zero a))) (Eq.refl 0)))\n\n/-- In a monoid with zero, if zero equals one, then zero is the unique element.\n\nSomewhat arbitrarily, we define the default element to be `0`.\nAll other elements will be provably equal to it, but not necessarily definitionally equal. -/\ndef unique_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) : unique M₀ :=\n  unique.mk { default := 0 } (eq_zero_of_zero_eq_one h)\n\n/-- In a monoid with zero, zero equals one if and only if all elements of that semiring\nare equal. -/\ntheorem subsingleton_iff_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] : 0 = 1 ↔ subsingleton M₀ :=\n  { mp := fun (h : 0 = 1) => unique.subsingleton, mpr := fun (h : subsingleton M₀) => subsingleton.elim 0 1 }\n\ntheorem subsingleton_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] : 0 = 1 → subsingleton M₀ :=\n  iff.mp subsingleton_iff_zero_eq_one\n\ntheorem eq_of_zero_eq_one {M₀ : Type u_1} [monoid_with_zero M₀] (h : 0 = 1) (a : M₀) (b : M₀) : a = b :=\n  subsingleton.elim a b\n\n@[simp] theorem is_unit_zero_iff {M₀ : Type u_1} [monoid_with_zero M₀] : is_unit 0 ↔ 0 = 1 := sorry\n\n@[simp] theorem not_is_unit_zero {M₀ : Type u_1} [monoid_with_zero M₀] [nontrivial M₀] : ¬is_unit 0 :=\n  mt (iff.mp is_unit_zero_iff) zero_ne_one\n\n/-- In a monoid with zero, either zero and one are nonequal, or zero is the only element. -/\ntheorem zero_ne_one_or_forall_eq_0 (M₀ : Type u_1) [monoid_with_zero M₀] : 0 ≠ 1 ∨ ∀ (a : M₀), a = 0 :=\n  not_or_of_imp eq_zero_of_zero_eq_one\n\nprotected instance cancel_monoid_with_zero.to_no_zero_divisors {M₀ : Type u_1} [cancel_monoid_with_zero M₀] : no_zero_divisors M₀ :=\n  no_zero_divisors.mk\n    fun (a b : M₀) (ab0 : a * b = 0) =>\n      dite (a = 0) (fun (h : a = 0) => Or.inl h)\n        fun (h : ¬a = 0) =>\n          Or.inr\n            (cancel_monoid_with_zero.mul_left_cancel_of_ne_zero h\n              (eq.mpr (id (Eq._oldrec (Eq.refl (a * b = a * 0)) ab0))\n                (eq.mpr (id (Eq._oldrec (Eq.refl (0 = a * 0)) (mul_zero a))) (Eq.refl 0))))\n\ntheorem mul_left_inj' {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀} (hc : c ≠ 0) : a * c = b * c ↔ a = b :=\n  { mp := mul_right_cancel' hc, mpr := fun (h : a = b) => h ▸ rfl }\n\ntheorem mul_right_inj' {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀} (ha : a ≠ 0) : a * b = a * c ↔ b = c :=\n  { mp := mul_left_cancel' ha, mpr := fun (h : b = c) => h ▸ rfl }\n\n@[simp] theorem mul_eq_mul_right_iff {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀} : a * c = b * c ↔ a = b ∨ c = 0 := sorry\n\n@[simp] theorem mul_eq_mul_left_iff {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} {c : M₀} : a * b = a * c ↔ b = c ∨ a = 0 := sorry\n\n/-- Pullback a `monoid_with_zero` class along an injective function. -/\nprotected def function.injective.cancel_monoid_with_zero {M₀ : Type u_1} {M₀' : Type u_3} [cancel_monoid_with_zero M₀] [HasZero M₀'] [Mul M₀'] [HasOne M₀'] (f : M₀' → M₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : M₀'), f (x * y) = f x * f y) : cancel_monoid_with_zero M₀' :=\n  cancel_monoid_with_zero.mk monoid.mul sorry monoid.one sorry sorry mul_zero_class.zero sorry sorry sorry sorry\n\n/-- An element of a `cancel_monoid_with_zero` fixed by right multiplication by an element other\nthan one must be zero. -/\ntheorem eq_zero_of_mul_eq_self_right {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} (h₁ : b ≠ 1) (h₂ : a * b = a) : a = 0 :=\n  classical.by_contradiction fun (ha : ¬a = 0) => h₁ (mul_left_cancel' ha (Eq.symm h₂ ▸ Eq.symm (mul_one a)))\n\n/-- An element of a `cancel_monoid_with_zero` fixed by left multiplication by an element other\nthan one must be zero. -/\ntheorem eq_zero_of_mul_eq_self_left {M₀ : Type u_1} [cancel_monoid_with_zero M₀] {a : M₀} {b : M₀} (h₁ : b ≠ 1) (h₂ : b * a = a) : a = 0 :=\n  classical.by_contradiction fun (ha : ¬a = 0) => h₁ (mul_right_cancel' ha (Eq.symm h₂ ▸ Eq.symm (one_mul a)))\n\ntheorem division_def {G : Type u} [div_inv_monoid G] (a : G) (b : G) : a / b = a * (b⁻¹) :=\n  div_eq_mul_inv\n\n/-- Pullback a `group_with_zero` class along an injective function. -/\nprotected def function.injective.group_with_zero {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (f : G₀' → G₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) : group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry monoid_with_zero.zero sorry sorry\n    has_inv.inv (div_inv_monoid.div._default monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry has_inv.inv)\n    sorry sorry sorry\n\n/-- Pullback a `group_with_zero` class along an injective function. This is a version of\n`function.injective.group_with_zero` that uses a specified `/` instead of the default\n`a / b = a * b⁻¹`. -/\nprotected def function.injective.group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (f : G₀' → G₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) (div : ∀ (x y : G₀'), f (x / y) = f x / f y) : group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry monoid_with_zero.zero sorry sorry\n    div_inv_monoid.inv div_inv_monoid.div sorry sorry sorry\n\n/-- Pushforward a `group_with_zero` class along an surjective function. -/\nprotected def function.surjective.group_with_zero {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) : group_with_zero G₀' :=\n  group_with_zero.mk monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry monoid_with_zero.zero sorry sorry\n    has_inv.inv (div_inv_monoid.div._default monoid_with_zero.mul sorry monoid_with_zero.one sorry sorry has_inv.inv)\n    sorry sorry sorry\n\n/-- Pushforward a `group_with_zero` class along a surjective function. This is a version of\n`function.surjective.group_with_zero` that uses a specified `/` instead of the default\n`a / b = a * b⁻¹`. -/\nprotected def function.surjective.group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) (div : ∀ (x y : G₀), f (x / y) = f x / f y) : group_with_zero G₀' :=\n  group_with_zero.mk div_inv_monoid.mul sorry div_inv_monoid.one sorry sorry group_with_zero.zero sorry sorry\n    div_inv_monoid.inv div_inv_monoid.div sorry sorry sorry\n\n@[simp] theorem mul_inv_cancel_right' {G₀ : Type u_2} [group_with_zero G₀] {b : G₀} (h : b ≠ 0) (a : G₀) : a * b * (b⁻¹) = a := sorry\n\n@[simp] theorem mul_inv_cancel_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) (b : G₀) : a * (a⁻¹ * b) = b := sorry\n\ntheorem inv_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a⁻¹ ≠ 0 := sorry\n\n@[simp] theorem inv_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a⁻¹ * a = 1 := sorry\n\n@[simp] theorem inv_mul_cancel_right' {G₀ : Type u_2} [group_with_zero G₀] {b : G₀} (h : b ≠ 0) (a : G₀) : a * (b⁻¹) * b = a := sorry\n\n@[simp] theorem inv_mul_cancel_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) (b : G₀) : a⁻¹ * (a * b) = b := sorry\n\n@[simp] theorem inv_one {G₀ : Type u_2} [group_with_zero G₀] : 1⁻¹ = 1 := sorry\n\n@[simp] theorem inv_inv' {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a⁻¹⁻¹ = a := sorry\n\n/-- Multiplying `a` by itself and then by its inverse results in `a`\n(whether or not `a` is zero). -/\n@[simp] theorem mul_self_mul_inv {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a * a * (a⁻¹) = a := sorry\n\n/-- Multiplying `a` by its inverse and then by itself results in `a`\n(whether or not `a` is zero). -/\n@[simp] theorem mul_inv_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a * (a⁻¹) * a = a := sorry\n\n/-- Multiplying `a⁻¹` by `a` twice results in `a` (whether or not `a`\nis zero). -/\n@[simp] theorem inv_mul_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a⁻¹ * a * a = a := sorry\n\n/-- Multiplying `a` by itself and then dividing by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem mul_self_div_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a * a / a = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * a / a = a)) (div_eq_mul_inv (a * a) a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * a * (a⁻¹) = a)) (mul_self_mul_inv a))) (Eq.refl a))\n\n/-- Dividing `a` by itself and then multiplying by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem div_self_mul_self {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / a * a = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / a * a = a)) (div_eq_mul_inv a a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a⁻¹) * a = a)) (mul_inv_mul_self a))) (Eq.refl a))\n\ntheorem inv_involutive' {G₀ : Type u_2} [group_with_zero G₀] : function.involutive has_inv.inv :=\n  inv_inv'\n\ntheorem eq_inv_of_mul_right_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : a * b = 1) : b = (a⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (b = (a⁻¹))) (Eq.symm (inv_mul_cancel_left' (left_ne_zero_of_mul_eq_one h) b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ * (a * b) = (a⁻¹))) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ * 1 = (a⁻¹))) (mul_one (a⁻¹)))) (Eq.refl (a⁻¹))))\n\ntheorem eq_inv_of_mul_left_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : a * b = 1) : a = (b⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = (b⁻¹))) (Eq.symm (mul_inv_cancel_right' (right_ne_zero_of_mul_eq_one h) a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b * (b⁻¹) = (b⁻¹))) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 * (b⁻¹) = (b⁻¹))) (one_mul (b⁻¹)))) (Eq.refl (b⁻¹))))\n\ntheorem inv_injective' {G₀ : Type u_2} [group_with_zero G₀] : function.injective has_inv.inv :=\n  function.involutive.injective inv_involutive'\n\n@[simp] theorem inv_inj' {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} {h : G₀} : g⁻¹ = (h⁻¹) ↔ g = h :=\n  function.injective.eq_iff inv_injective'\n\ntheorem inv_eq_iff {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} {h : G₀} : g⁻¹ = h ↔ h⁻¹ = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹ = h ↔ h⁻¹ = g)) (Eq.symm (propext inv_inj'))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹⁻¹ = (h⁻¹) ↔ h⁻¹ = g)) (propext eq_comm)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (h⁻¹ = (g⁻¹⁻¹) ↔ h⁻¹ = g)) (inv_inv' g))) (iff.refl (h⁻¹ = g))))\n\n@[simp] theorem inv_eq_one' {G₀ : Type u_2} [group_with_zero G₀] {g : G₀} : g⁻¹ = 1 ↔ g = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (g⁻¹ = 1 ↔ g = 1)) (propext inv_eq_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1⁻¹ = g ↔ g = 1)) inv_one))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 = g ↔ g = 1)) (propext eq_comm))) (iff.refl (g = 1))))\n\nnamespace units\n\n\n/-- Embed a non-zero element of a `group_with_zero` into the unit group.\n  By combining this function with the operations on units,\n  or the `/ₚ` operation, it is possible to write a division\n  as a partial function with three arguments. -/\ndef mk0 {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (ha : a ≠ 0) : units G₀ :=\n  mk a (a⁻¹) (mul_inv_cancel ha) (inv_mul_cancel ha)\n\n@[simp] theorem coe_mk0 {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : ↑(mk0 a h) = a :=\n  rfl\n\n@[simp] theorem mk0_coe {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) (h : ↑u ≠ 0) : mk0 (↑u) h = u :=\n  ext rfl\n\n@[simp] theorem coe_inv' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑(u⁻¹) = (↑u⁻¹) :=\n  eq_inv_of_mul_left_eq_one (inv_mul u)\n\n@[simp] theorem mul_inv' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑u * (↑u⁻¹) = 1 :=\n  mul_inv_cancel (ne_zero u)\n\n@[simp] theorem inv_mul' {G₀ : Type u_2} [group_with_zero G₀] (u : units G₀) : ↑u⁻¹ * ↑u = 1 :=\n  inv_mul_cancel (ne_zero u)\n\n@[simp] theorem mk0_inj {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) (hb : b ≠ 0) : mk0 a ha = mk0 b hb ↔ a = b :=\n  { mp := fun (h : mk0 a ha = mk0 b hb) => mk.inj_arrow h fun (h_1 : a = b) (h_2 : a⁻¹ = (b⁻¹)) => h_1,\n    mpr := fun (h : a = b) => ext h }\n\n@[simp] theorem exists_iff_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} : (∃ (u : units G₀), ↑u = x) ↔ x ≠ 0 := sorry\n\nend units\n\n\ntheorem is_unit.mk0 {G₀ : Type u_2} [group_with_zero G₀] (x : G₀) (hx : x ≠ 0) : is_unit x :=\n  is_unit_unit (units.mk0 x hx)\n\ntheorem is_unit_iff_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} : is_unit x ↔ x ≠ 0 :=\n  units.exists_iff_ne_zero\n\nprotected instance group_with_zero.no_zero_divisors {G₀ : Type u_2} [group_with_zero G₀] : no_zero_divisors G₀ :=\n  no_zero_divisors.mk\n    fun (a b : G₀) (h : a * b = 0) =>\n      imp_of_not_imp_not (a * b = 0) (a = 0 ∨ b = 0)\n        (eq.mpr (id (imp_congr_eq (push_neg.not_or_eq (a = 0) (b = 0)) (Eq.refl (¬a * b = 0))))\n          (eq.mpr\n            (id\n              (imp_congr_eq\n                ((fun (a a_1 : Prop) (e_1 : a = a_1) (b b_1 : Prop) (e_2 : b = b_1) => congr (congr_arg And e_1) e_2)\n                  (¬a = 0) (a ≠ 0) (propext (push_neg.not_eq a 0)) (¬b = 0) (b ≠ 0) (propext (push_neg.not_eq b 0)))\n                (propext (push_neg.not_eq (a * b) 0))))\n            fun (h : a ≠ 0 ∧ b ≠ 0) => units.ne_zero (units.mk0 a (and.left h) * units.mk0 b (and.right h))))\n        h\n\nprotected instance group_with_zero.cancel_monoid_with_zero {G₀ : Type u_2} [group_with_zero G₀] : cancel_monoid_with_zero G₀ :=\n  cancel_monoid_with_zero.mk group_with_zero.mul group_with_zero.mul_assoc group_with_zero.one group_with_zero.one_mul\n    group_with_zero.mul_one group_with_zero.zero group_with_zero.zero_mul group_with_zero.mul_zero sorry sorry\n\ntheorem mul_inv_rev' {G₀ : Type u_2} [group_with_zero G₀] (x : G₀) (y : G₀) : x * y⁻¹ = y⁻¹ * (x⁻¹) := sorry\n\n@[simp] theorem div_self {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a / a = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / a = 1)) (div_eq_mul_inv a a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a⁻¹) = 1)) (mul_inv_cancel h))) (Eq.refl 1))\n\n@[simp] theorem div_one {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / 1 = a := sorry\n\n@[simp] theorem zero_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : 0 / a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 / a = 0)) (div_eq_mul_inv 0 a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (0 * (a⁻¹) = 0)) (zero_mul (a⁻¹)))) (Eq.refl 0))\n\n@[simp] theorem div_zero {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / 0 = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / 0 = 0)) (div_eq_mul_inv a 0)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (0⁻¹) = 0)) inv_zero))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 0 = 0)) (mul_zero a))) (Eq.refl 0)))\n\n@[simp] theorem div_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (h : b ≠ 0) : a / b * b = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_eq_mul_inv a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b⁻¹) * b = a)) (inv_mul_cancel_right' h a))) (Eq.refl a))\n\ntheorem div_mul_cancel_of_imp {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) : a / b * b = a := sorry\n\n@[simp] theorem mul_div_cancel {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (h : b ≠ 0) : a * b / b = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b / b = a)) (div_eq_mul_inv (a * b) b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b * (b⁻¹) = a)) (mul_inv_cancel_right' h a))) (Eq.refl a))\n\ntheorem mul_div_cancel_of_imp {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) : a * b / b = a := sorry\n\n@[simp] theorem div_self_mul_self' {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : a / (a * a) = (a⁻¹) := sorry\n\ntheorem div_eq_mul_one_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) : a / b = a * (1 / b) := sorry\n\ntheorem mul_one_div_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : a * (1 / a) = 1 := sorry\n\ntheorem one_div_mul_cancel {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : 1 / a * a = 1 := sorry\n\ntheorem one_div_one {G₀ : Type u_2} [group_with_zero G₀] : 1 / 1 = 1 :=\n  div_self (ne.symm zero_ne_one)\n\ntheorem one_div_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} (h : a ≠ 0) : 1 / a ≠ 0 := sorry\n\ntheorem eq_one_div_of_mul_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : a * b = 1) : b = 1 / a := sorry\n\ntheorem eq_one_div_of_mul_eq_one_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : b * a = 1) : b = 1 / a := sorry\n\n@[simp] theorem one_div_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) : 1 / (a / b) = b / a := sorry\n\ntheorem one_div_one_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) : 1 / (1 / a) = a := sorry\n\ntheorem eq_of_one_div_eq_one_div {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : 1 / a = 1 / b) : a = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = b)) (Eq.symm (one_div_one_div a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (1 / a) = b)) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (1 / b) = b)) (one_div_one_div b))) (Eq.refl b)))\n\n@[simp] theorem inv_eq_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} : a⁻¹ = 0 ↔ a = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ = 0 ↔ a = 0)) (propext inv_eq_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (0⁻¹ = a ↔ a = 0)) inv_zero))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (0 = a ↔ a = 0)) (propext eq_comm))) (iff.refl (a = 0))))\n\n@[simp] theorem zero_eq_inv {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} : 0 = (a⁻¹) ↔ 0 = a :=\n  iff.trans eq_comm (iff.trans inv_eq_zero eq_comm)\n\ntheorem one_div_mul_one_div_rev {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) : 1 / a * (1 / b) = 1 / (b * a) := sorry\n\ntheorem divp_eq_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (u : units G₀) : a /ₚ u = a / ↑u := sorry\n\n@[simp] theorem divp_mk0 {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) : a /ₚ units.mk0 b hb = a / b :=\n  divp_eq_div a (units.mk0 b hb)\n\ntheorem inv_div {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a / b⁻¹ = b / a := sorry\n\ntheorem inv_div_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a⁻¹ / b = (b * a⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ / b = (b * a⁻¹))) (mul_inv_rev' b a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a⁻¹ / b = a⁻¹ * (b⁻¹))) (div_eq_mul_inv (a⁻¹) b))) (Eq.refl (a⁻¹ * (b⁻¹))))\n\ntheorem div_ne_zero {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) (hb : b ≠ 0) : a / b ≠ 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b ≠ 0)) (div_eq_mul_inv a b))) (mul_ne_zero ha (inv_ne_zero hb))\n\n@[simp] theorem div_eq_zero_iff {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a / b = 0 ↔ a = 0 ∨ b = 0 := sorry\n\ntheorem div_ne_zero_iff {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : a / b ≠ 0 ↔ a ≠ 0 ∧ b ≠ 0 :=\n  iff.trans (not_congr div_eq_zero_iff) not_or_distrib\n\ntheorem div_left_inj' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) : a / c = b / c ↔ a = b := sorry\n\ntheorem div_eq_iff_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) : a / b = c ↔ c * b = a := sorry\n\ntheorem eq_div_iff_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hc : c ≠ 0) : a = b / c ↔ a * c = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a = b / c ↔ a * c = b)) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b / c = a ↔ a * c = b)) (propext (div_eq_iff_mul_eq hc)))) (iff.refl (a * c = b)))\n\ntheorem div_eq_of_eq_mul {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} (hx : x ≠ 0) {y : G₀} {z : G₀} (h : y = z * x) : y / x = z :=\n  iff.mpr (div_eq_iff_mul_eq hx) (Eq.symm h)\n\ntheorem eq_div_of_mul_eq {G₀ : Type u_2} [group_with_zero G₀] {x : G₀} (hx : x ≠ 0) {y : G₀} {z : G₀} (h : z * x = y) : z = y / x :=\n  Eq.symm (div_eq_of_eq_mul hx (Eq.symm h))\n\ntheorem eq_of_div_eq_one {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : a / b = 1) : a = b := sorry\n\ntheorem div_eq_one_iff_eq {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) : a / b = 1 ↔ a = b :=\n  { mp := eq_of_div_eq_one, mpr := fun (h : a = b) => Eq.symm h ▸ div_self hb }\n\ntheorem div_mul_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (hb : b ≠ 0) : b / (a * b) = 1 / a := sorry\n\ntheorem mul_div_mul_right {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) (b : G₀) {c : G₀} (hc : c ≠ 0) : a * c / (b * c) = a / b := sorry\n\ntheorem mul_mul_div {G₀ : Type u_2} [group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) : a = a * b * (1 / b) := sorry\n\nprotected instance comm_group_with_zero.comm_cancel_monoid_with_zero {G₀ : Type u_2} [comm_group_with_zero G₀] : comm_cancel_monoid_with_zero G₀ :=\n  comm_cancel_monoid_with_zero.mk cancel_monoid_with_zero.mul sorry cancel_monoid_with_zero.one sorry sorry sorry\n    cancel_monoid_with_zero.zero sorry sorry sorry sorry\n\n/-- Pullback a `comm_group_with_zero` class along an injective function. -/\nprotected def function.injective.comm_group_with_zero {G₀ : Type u_2} {G₀' : Type u_4} [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (f : G₀' → G₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry group_with_zero.zero sorry sorry\n    group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pullback a `comm_group_with_zero` class along an injective function. -/\nprotected def function.injective.comm_group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4} [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (f : G₀' → G₀) (hf : function.injective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀'), f (x * y) = f x * f y) (inv : ∀ (x : G₀'), f (x⁻¹) = (f x⁻¹)) (div : ∀ (x y : G₀'), f (x / y) = f x / f y) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry group_with_zero.zero sorry sorry\n    group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pushforward a `comm_group_with_zero` class along an surjective function. -/\nprotected def function.surjective.comm_group_with_zero {G₀ : Type u_2} {G₀' : Type u_4} [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry group_with_zero.zero sorry sorry\n    group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n/-- Pushforward a `comm_group_with_zero` class along a surjective function. -/\nprotected def function.surjective.comm_group_with_zero_div {G₀ : Type u_2} {G₀' : Type u_4} [comm_group_with_zero G₀] [HasZero G₀'] [Mul G₀'] [HasOne G₀'] [has_inv G₀'] [Div G₀'] (h01 : 0 ≠ 1) (f : G₀ → G₀') (hf : function.surjective f) (zero : f 0 = 0) (one : f 1 = 1) (mul : ∀ (x y : G₀), f (x * y) = f x * f y) (inv : ∀ (x : G₀), f (x⁻¹) = (f x⁻¹)) (div : ∀ (x y : G₀), f (x / y) = f x / f y) : comm_group_with_zero G₀' :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry sorry group_with_zero.zero sorry sorry\n    group_with_zero.inv group_with_zero.div sorry sorry sorry\n\ntheorem mul_inv' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} : a * b⁻¹ = a⁻¹ * (b⁻¹) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b⁻¹ = a⁻¹ * (b⁻¹))) (mul_inv_rev' a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b⁻¹ * (a⁻¹) = a⁻¹ * (b⁻¹))) (mul_comm (b⁻¹) (a⁻¹)))) (Eq.refl (a⁻¹ * (b⁻¹))))\n\ntheorem one_div_mul_one_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) : 1 / a * (1 / b) = 1 / (a * b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 / a * (1 / b) = 1 / (a * b))) (one_div_mul_one_div_rev a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 / (b * a) = 1 / (a * b))) (mul_comm b a))) (Eq.refl (1 / (a * b))))\n\ntheorem div_mul_right {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (ha : a ≠ 0) : a / (a * b) = 1 / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a * b) = 1 / b)) (mul_comm a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * a) = 1 / b)) (div_mul_left ha))) (Eq.refl (1 / b)))\n\ntheorem mul_div_cancel_left_of_imp {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} (h : a = 0 → b = 0) : a * b / a = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * b / a = b)) (mul_comm a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (b * a / a = b)) (mul_div_cancel_of_imp h))) (Eq.refl b))\n\ntheorem mul_div_cancel_left {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (ha : a ≠ 0) : a * b / a = b :=\n  mul_div_cancel_left_of_imp fun (h : a = 0) => false.elim (ha h)\n\ntheorem mul_div_cancel_of_imp' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} (h : b = 0 → a = 0) : b * (a / b) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (b * (a / b) = a)) (mul_comm b (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_mul_cancel_of_imp h))) (Eq.refl a))\n\ntheorem mul_div_cancel' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀} (hb : b ≠ 0) : b * (a / b) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (b * (a / b) = a)) (mul_comm b (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b = a)) (div_mul_cancel a hb))) (Eq.refl a))\n\ntheorem div_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) (d : G₀) : a / b * (c / d) = a * c / (b * d) := sorry\n\ntheorem mul_div_mul_left {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) {c : G₀} (hc : c ≠ 0) : c * a / (c * b) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (c * a / (c * b) = a / b)) (mul_comm c a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / (c * b) = a / b)) (mul_comm c b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / (b * c) = a / b)) (mul_div_mul_right a b hc))) (Eq.refl (a / b))))\n\ntheorem div_mul_eq_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : b / c * a = b * a / c := sorry\n\ntheorem div_mul_eq_mul_div_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : b / c * a = b * (a / c) := sorry\n\ntheorem mul_eq_mul_of_div_eq_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀} (c : G₀) {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) (h : a / b = c / d) : a * d = c * b := sorry\n\ntheorem div_div_eq_mul_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a / (b / c) = a * c / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (b / c) = a * c / b)) (div_eq_mul_one_div a (b / c))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (1 / (b / c)) = a * c / b)) (one_div_div b c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * (c / b) = a * c / b)) (Eq.symm mul_div_assoc))) (Eq.refl (a * c / b))))\n\ntheorem div_div_eq_div_mul {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a / b / c = a / (b * c) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b / c = a / (b * c))) (div_eq_mul_one_div (a / b) c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * (1 / c) = a / (b * c))) (div_mul_div a b 1 c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * 1 / (b * c) = a / (b * c))) (mul_one a))) (Eq.refl (a / (b * c)))))\n\ntheorem div_div_div_div_eq {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) {b : G₀} {c : G₀} {d : G₀} : a / b / (c / d) = a * d / (b * c) := sorry\n\ntheorem div_mul_eq_div_mul_one_div {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a / (b * c) = a / b * (1 / c) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * c) = a / b * (1 / c))) (Eq.symm (div_div_eq_div_mul a b c))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b / c = a / b * (1 / c))) (Eq.symm (div_eq_mul_one_div (a / b) c))))\n      (Eq.refl (a / b / c)))\n\n/-- Dividing `a` by the result of dividing `a` by itself results in\n`a` (whether or not `a` is zero). -/\n@[simp] theorem div_div_self {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) : a / (a / a) = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a / a) = a)) (div_div_eq_mul_div a a a))) (mul_self_div_self a)\n\ntheorem ne_zero_of_one_div_ne_zero {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (h : 1 / a ≠ 0) : a ≠ 0 :=\n  fun (ha : a = 0) =>\n    absurd (Eq.refl 0)\n      (eq.mp (Eq._oldrec (Eq.refl (1 / 0 ≠ 0)) (div_zero 1)) (eq.mp (Eq._oldrec (Eq.refl (1 / a ≠ 0)) ha) h))\n\ntheorem eq_zero_of_one_div_eq_zero {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (h : 1 / a = 0) : a = 0 :=\n  classical.by_cases (fun (ha : a = 0) => ha) fun (ha : ¬a = 0) => false.elim (one_div_ne_zero ha h)\n\ntheorem div_helper {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} (b : G₀) (h : a ≠ 0) : 1 / (a * b) * a = 1 / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 / (a * b) * a = 1 / b)) (div_mul_eq_mul_div a 1 (a * b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 * a / (a * b) = 1 / b)) (one_mul a)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a / (a * b) = 1 / b)) (div_mul_right b h))) (Eq.refl (1 / b))))\n\ntheorem div_eq_inv_mul {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} : a / b = b⁻¹ * a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b = b⁻¹ * a)) (div_eq_mul_inv a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b⁻¹) = b⁻¹ * a)) (mul_comm a (b⁻¹)))) (Eq.refl (b⁻¹ * a)))\n\ntheorem mul_div_right_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a * b / c = a / c * b := sorry\n\ntheorem mul_comm_div' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a / b * c = a * (c / b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = a * (c / b))) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = a * c / b)) (mul_div_right_comm a c b))) (Eq.refl (a / b * c)))\n\ntheorem div_mul_comm' {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a / b * c = c / b * a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * c = c / b * a)) (div_mul_eq_mul_div c a b)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * c / b = c / b * a)) (mul_comm a c)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (c * a / b = c / b * a)) (mul_div_right_comm c a b))) (Eq.refl (c / b * a))))\n\ntheorem mul_div_comm {G₀ : Type u_2} [comm_group_with_zero G₀] (a : G₀) (b : G₀) (c : G₀) : a * (b / c) = b * (a / c) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a * (b / c) = b * (a / c))) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * b / c = b * (a / c))) (mul_comm a b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (b * a / c = b * (a / c))) mul_div_assoc)) (Eq.refl (b * (a / c)))))\n\ntheorem div_right_comm {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀} (a : G₀) : a / b / c = a / c / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b / c = a / c / b)) (div_div_eq_div_mul a b c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * c) = a / c / b)) (div_div_eq_div_mul a c b)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a / (b * c) = a / (c * b))) (mul_comm b c))) (Eq.refl (a / (c * b)))))\n\ntheorem div_div_div_cancel_right {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀} (a : G₀) (hc : c ≠ 0) : a / c / (b / c) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / c / (b / c) = a / b)) (div_div_eq_mul_div (a / c) b c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / c * c / b = a / b)) (div_mul_cancel a hc))) (Eq.refl (a / b)))\n\ntheorem div_mul_div_cancel {G₀ : Type u_2} [comm_group_with_zero G₀] {b : G₀} {c : G₀} (a : G₀) (hc : c ≠ 0) : a / c * (c / b) = a / b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / c * (c / b) = a / b)) (Eq.symm mul_div_assoc)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / c * c / b = a / b)) (div_mul_cancel a hc))) (Eq.refl (a / b)))\n\ntheorem div_eq_div_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} {d : G₀} (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b := sorry\n\ntheorem div_eq_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) : a / b = c ↔ a = c * b :=\n  iff.trans (div_eq_iff_mul_eq hb) eq_comm\n\ntheorem eq_div_iff {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hb : b ≠ 0) : c = a / b ↔ c * b = a :=\n  eq_div_iff_mul_eq hb\n\ntheorem div_div_cancel' {G₀ : Type u_2} [comm_group_with_zero G₀] {a : G₀} {b : G₀} (ha : a ≠ 0) : a / (a / b) = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / (a / b) = b)) (div_eq_mul_inv a (a / b))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a / b⁻¹) = b)) inv_div))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a * (b / a) = b)) (mul_div_cancel' b ha))) (Eq.refl b)))\n\nnamespace semiconj_by\n\n\n@[simp] theorem zero_right {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : semiconj_by a 0 0 := sorry\n\n@[simp] theorem zero_left {G₀ : Type u_2} [mul_zero_class G₀] (x : G₀) (y : G₀) : semiconj_by 0 x y := sorry\n\n@[simp] theorem inv_symm_left_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} : semiconj_by (a⁻¹) x y ↔ semiconj_by a y x := sorry\n\ntheorem inv_symm_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} (h : semiconj_by a x y) : semiconj_by (a⁻¹) y x :=\n  iff.mpr inv_symm_left_iff' h\n\ntheorem inv_right' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} (h : semiconj_by a x y) : semiconj_by a (x⁻¹) (y⁻¹) := sorry\n\n@[simp] theorem inv_right_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} : semiconj_by a (x⁻¹) (y⁻¹) ↔ semiconj_by a x y :=\n  { mp := fun (h : semiconj_by a (x⁻¹) (y⁻¹)) => inv_inv' x ▸ inv_inv' y ▸ inv_right' h, mpr := inv_right' }\n\ntheorem div_right {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {x : G₀} {y : G₀} {x' : G₀} {y' : G₀} (h : semiconj_by a x y) (h' : semiconj_by a x' y') : semiconj_by a (x / x') (y / y') :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (semiconj_by a (x / x') (y / y'))) (div_eq_mul_inv x x')))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (semiconj_by a (x * (x'⁻¹)) (y / y'))) (div_eq_mul_inv y y')))\n      (mul_right h (inv_right' h')))\n\nend semiconj_by\n\n\nnamespace commute\n\n\n@[simp] theorem zero_right {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : commute a 0 :=\n  semiconj_by.zero_right a\n\n@[simp] theorem zero_left {G₀ : Type u_2} [mul_zero_class G₀] (a : G₀) : commute 0 a :=\n  semiconj_by.zero_left a a\n\n@[simp] theorem inv_left_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : commute (a⁻¹) b ↔ commute a b :=\n  semiconj_by.inv_symm_left_iff'\n\ntheorem inv_left' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) : commute (a⁻¹) b :=\n  iff.mpr inv_left_iff' h\n\n@[simp] theorem inv_right_iff' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} : commute a (b⁻¹) ↔ commute a b :=\n  semiconj_by.inv_right_iff'\n\ntheorem inv_right' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) : commute a (b⁻¹) :=\n  iff.mpr inv_right_iff' h\n\ntheorem inv_inv' {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} (h : commute a b) : commute (a⁻¹) (b⁻¹) :=\n  inv_right' (inv_left' h)\n\n@[simp] theorem div_right {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hab : commute a b) (hac : commute a c) : commute a (b / c) :=\n  semiconj_by.div_right hab hac\n\n@[simp] theorem div_left {G₀ : Type u_2} [group_with_zero G₀] {a : G₀} {b : G₀} {c : G₀} (hac : commute a c) (hbc : commute b c) : commute (a / b) c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (commute (a / b) c)) (div_eq_mul_inv a b))) (mul_left hac (inv_left' hbc))\n\nend commute\n\n\nnamespace monoid_with_zero_hom\n\n\ntheorem map_ne_zero {M₀ : Type u_1} {G₀ : Type u_2} [group_with_zero G₀] [monoid_with_zero M₀] [nontrivial M₀] (f : monoid_with_zero_hom G₀ M₀) {a : G₀} : coe_fn f a ≠ 0 ↔ a ≠ 0 :=\n  { mp := fun (hfa : coe_fn f a ≠ 0) (ha : a = 0) => hfa (Eq.symm ha ▸ map_zero f),\n    mpr := fun (ha : a ≠ 0) => is_unit.ne_zero (is_unit.map (to_monoid_hom f) (is_unit.mk0 a ha)) }\n\n@[simp] theorem map_eq_zero {M₀ : Type u_1} {G₀ : Type u_2} [group_with_zero G₀] [monoid_with_zero M₀] [nontrivial M₀] (f : monoid_with_zero_hom G₀ M₀) {a : G₀} : coe_fn f a = 0 ↔ a = 0 :=\n  iff.mp not_iff_not (map_ne_zero f)\n\n/-- A monoid homomorphism between groups with zeros sending `0` to `0` sends `a⁻¹` to `(f a)⁻¹`. -/\n@[simp] theorem map_inv' {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [group_with_zero G₀'] (f : monoid_with_zero_hom G₀ G₀') (a : G₀) : coe_fn f (a⁻¹) = (coe_fn f a⁻¹) := sorry\n\n@[simp] theorem map_div {G₀ : Type u_2} {G₀' : Type u_4} [group_with_zero G₀] [group_with_zero G₀'] (f : monoid_with_zero_hom G₀ G₀') (a : G₀) (b : G₀) : coe_fn f (a / b) = coe_fn f a / coe_fn f b := sorry\n\nend monoid_with_zero_hom\n\n\n@[simp] theorem monoid_hom.map_units_inv {M : Type u_1} {G₀ : Type u_2} [monoid M] [group_with_zero G₀] (f : M →* G₀) (u : units M) : coe_fn f ↑(u⁻¹) = (coe_fn f ↑u⁻¹) := sorry\n\n/-- Constructs a `group_with_zero` structure on a `monoid_with_zero`\n  consisting only of units and 0. -/\ndef group_with_zero_of_is_unit_or_eq_zero {M : Type u_5} [nontrivial M] [hM : monoid_with_zero M] (h : ∀ (a : M), is_unit a ∨ a = 0) : group_with_zero M :=\n  group_with_zero.mk monoid_with_zero.mul monoid_with_zero.mul_assoc monoid_with_zero.one monoid_with_zero.one_mul\n    monoid_with_zero.mul_one monoid_with_zero.zero monoid_with_zero.zero_mul monoid_with_zero.mul_zero\n    (fun (a : M) => dite (a = 0) (fun (h0 : a = 0) => 0) fun (h0 : ¬a = 0) => ↑(is_unit.unit sorry⁻¹))\n    (div_inv_monoid.div._default monoid_with_zero.mul monoid_with_zero.mul_assoc monoid_with_zero.one\n      monoid_with_zero.one_mul monoid_with_zero.mul_one\n      fun (a : M) => dite (a = 0) (fun (h0 : a = 0) => 0) fun (h0 : ¬a = 0) => ↑(is_unit.unit sorry⁻¹))\n    nontrivial.exists_pair_ne sorry sorry\n\n/-- Constructs a `comm_group_with_zero` structure on a `comm_monoid_with_zero`\n  consisting only of units and 0. -/\ndef comm_group_with_zero_of_is_unit_or_eq_zero {M : Type u_5} [nontrivial M] [hM : comm_monoid_with_zero M] (h : ∀ (a : M), is_unit a ∨ a = 0) : comm_group_with_zero M :=\n  comm_group_with_zero.mk group_with_zero.mul sorry group_with_zero.one sorry sorry comm_monoid_with_zero.mul_comm\n    group_with_zero.zero sorry sorry group_with_zero.inv group_with_zero.div sorry sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/group_with_zero/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3022728255546675}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.simulation_semantics.constructions.logging.logging_oracle\nimport computational_monads.simulation_semantics.constructions.logging.query_log.lookup\nimport computational_monads.simulation_semantics.constructions.logging.query_log.fork\nimport computational_monads.simulation_semantics.oracle_compose\n\n/-!\n# Caching Simulation Oracle\n\nThis file defines a `sim_oracle` that implements caching functionality.\n`caching_oracle` represents a simulator that logs all new queries, returning the old\nlog values for queries that have been previously asked, so the `query_log` functions as a cache.\nThis is often useful when composed with other oracles, such as in `random_oracle`.\n-/\n\nopen oracle_comp oracle_spec\n\nvariables {α β γ : Type} {spec spec' : oracle_spec}\n\n/-- Computation that returns a currently cached value, or queries a new value if needed,\nreturning both the result and the (potentially updated) cache.\nThe function for a fresh query is given by `ou`, which may have other oracles than the cache. -/\ndef query_log.lookup_cached_or_run\n  (cache : query_log spec) (i : spec.ι) (t : spec.domain i)\n  (ou : oracle_comp spec' (spec.range i)) :\n  oracle_comp spec' (spec.range i × query_log spec) :=\nmatch cache.lookup i t with\n| (some u) := return (u, cache)\n| none := do {u ← ou, return (u, cache.log_query i t u)}\nend\n\n/-- Oracle for logging previous queries, and returning the same value for matching inputs -/\ndef caching_oracle (spec : oracle_spec) : sim_oracle spec spec (query_log spec) :=\n{ default_state := query_log.init spec,\n  o := λ i ⟨t, log⟩, match log.lookup i t with\n  | (some u) := return (u, log) -- Return the cached value if it already exists\n  | none := do {u ← query i t, return (u, log.log_query i t u)}\n  end }\n\nnamespace caching_oracle\n\nvariables (oa oa' : oracle_comp spec α) (i : spec.ι) (t : spec.domain i) (u : spec.range i)\n  (log : query_log spec) (x : spec.range i × query_log spec) --TODO: naming → cache\n\n@[simp] lemma apply_eq : (caching_oracle spec) i (t, log) = option.rec_on (log.lookup i t)\n  (query i t >>= λ u, return (u, log.log_query i t u)) (λ u, return (u, log)) :=\nby {simp only [caching_oracle, sim_oracle.has_coe_to_fun.def], induction log.lookup i t; refl }\n\nlemma apply_eq_of_lookup_eq_none (hlog : log.lookup i t = none) :\n  (caching_oracle spec) i (t, log) = (query i t >>= λ u, return (u, log.log_query i t u)) :=\nby rw [apply_eq, hlog]\n\nlemma apply_eq_of_lookup_eq_some (u : spec.range i) (hlog : log.lookup i t = some u) :\n  (caching_oracle spec) i (t, log) = return (u, log) :=\nby rw [apply_eq, hlog]\n\nlemma apply_eq_of_not_queried (hlog : log.not_queried i t) :\n  (caching_oracle spec) i (t, log) = (query i t >>= λ u, return (u, log.log_query i t u)) :=\nby rw [apply_eq, (query_log.lookup_eq_none_iff_not_queried _ _ _).2 hlog]\n\nsection support\n\n@[simp] lemma support_apply : ((caching_oracle spec) i (t, log)).support = if log.not_queried i t\n  then {x | x.2 = log.log_query i t x.1} else {x | some x.1 = log.lookup i t ∧ x.2 = log} :=\nbegin\n  split_ifs with hlog; ext x,\n  { rw [apply_eq_of_not_queried i t log hlog, support_bind_return, support_query,\n      set.top_eq_univ, set.image_univ, set.mem_range, set.mem_set_of_eq],\n    refine ⟨λ h, _, λ h, _⟩,\n    { obtain ⟨u, rfl⟩ := h,\n      refl },\n    { exact ⟨x.1, h ▸ (prod.eq_iff_fst_eq_snd_eq.2 ⟨rfl, rfl⟩)⟩ } },\n  { obtain ⟨u, hu⟩ := log.exists_eq_lookup_of_not_not_queried i t hlog,\n    simp only [apply_eq_of_lookup_eq_some _ _ _ u hu.symm, ← hu, support_return,\n      prod.eq_iff_fst_eq_snd_eq, set.mem_singleton_iff, set.mem_set_of_eq] }\nend\n\nlemma support_apply_of_not_queried (hlog : log.not_queried i t) :\n  ((caching_oracle spec) i (t, log)).support = {x | x.2 = log.log_query i t x.1} :=\nby rw [support_apply, if_pos hlog]\n\nlemma mem_support_apply_iff_of_not_queried (hlog : log.not_queried i t) :\n  x ∈ ((caching_oracle spec) i (t, log)).support ↔ x.2 = log.log_query i t x.1 :=\nby rw [support_apply_of_not_queried i t log hlog, set.mem_set_of_eq]\n\nlemma support_apply_of_queried (hlog : ¬ log.not_queried i t) :\n  ((caching_oracle spec) i (t, log)).support = {x | some x.1 = log.lookup i t ∧ x.2 = log} :=\nby rw [support_apply, if_neg hlog]\n\nlemma mem_support_apply_iff_of_queried (hlog : ¬ log.not_queried i t) :\n  x ∈ ((caching_oracle spec) i (t, log)).support ↔ some x.1 = log.lookup i t ∧ x.2 = log :=\nby rw [support_apply_of_queried i t log hlog, set.mem_set_of_eq]\n\n/-- If the initial cache has `nodup` for some oracle, then so does the final cache. -/\nlemma nodup_simulate (hlog : (log i).nodup) (x : α × query_log spec)\n  (hx : x ∈ (simulate (caching_oracle spec) oa log).support) : (x.2 i).nodup :=\nbegin\n  refine support_state_simulate_induction (caching_oracle spec) (λ log, (log i).nodup)\n    log hlog oa x hx (λ i t log x hx hlog, _),\n  by_cases h : log.not_queried i t,\n  { rw [mem_support_apply_iff_of_not_queried _ _ _ _ h] at hx,\n    rw [hx, query_log.nodup_log_query_apply_iff _ _ _ _ _ hlog],\n    exact or.inr ((log.not_queried_iff_not_mem _ _).1 h _) },\n  { rw [mem_support_apply_iff_of_queried _ _ _ _ h] at hx,\n    exact hx.2.symm ▸ hlog }\nend\n\nsection lookup\n\n/-- If a value is already cached in the initial state, it has the same cache value after. -/\nlemma lookup_simulate_eq_some_of_lookup_eq_some (x : α × query_log spec)\n  (hx : x ∈ (simulate (caching_oracle spec) oa log).support)\n  (hlog : log.lookup i t = some u) : x.2.lookup i t = some u :=\nbegin\n  refine support_state_simulate_induction (caching_oracle spec) (λ log, log.lookup i t = some u)\n    log hlog oa x hx (λ i t log x hx hlog, _),\n  by_cases h : log.not_queried i t,\n  { rw [support_apply_of_not_queried _ _ _ h, set.mem_set_of_eq] at hx,\n    rw [hx, query_log.lookup_log_query],\n    split_ifs with hi ht,\n    { obtain rfl := ht,\n      obtain rfl := hi,\n      exact false.elim (option.some_ne_none u (trans hlog.symm $\n        (log.lookup_eq_none_iff_not_queried _ _).2 h)) },\n    { exact hlog },\n    { exact hlog } },\n  { rw [mem_support_apply_iff_of_queried _ _ _ _ h] at hx,\n    exact hx.2.symm ▸ hlog }\nend\n\n/-- If a value isn't cached after simulation, it wasn't cached in the initial state. -/\nlemma lookup_eq_none_of_lookup_simulate_eq_none (x : α × query_log spec)\n  (hx : x ∈ (simulate (caching_oracle spec) oa log).support)\n  (hx' : x.2.lookup i t = none) : log.lookup i t = none :=\nbegin\n  by_contradiction h,\n  rw [← ne.def, option.ne_none_iff_exists'] at h,\n  refine let ⟨u, hu⟩ := h in option.some_ne_none u ((lookup_simulate_eq_some_of_lookup_eq_some\n    oa i t u log x hx hu).symm.trans hx'),\nend\n\n/-- For any query fresh to the initial cache, if there is some output such that query has a cached\nvalue, then there are also outputs with any other possible cached value. -/\nlemma exists_cache_lookup_eq_some (hlog : log.not_queried i t)\n  (x : α × query_log spec) (hx : x ∈ (simulate (caching_oracle spec) oa log).support)\n  (h : x.2.lookup i t = some u) (u' : spec.range i) :\n  ∃ (y : α × query_log spec) (hy : y ∈ (simulate (caching_oracle spec) oa log).support),\n    y.2.lookup i t = some u' :=\nbegin\n  induction oa using oracle_comp.induction_on with α a α β oa ob hoa hob i t,\n  { rw [support_simulate_return, set.mem_singleton_iff] at hx,\n    have : ¬ log.not_queried i t := begin\n      rw [← query_log.lookup_eq_none_iff_not_queried,\n        option.eq_none_iff_forall_not_mem, not_forall],\n      refine ⟨u, _⟩,\n      simpa only [hx] using h,\n    end,\n    refine (this hlog).elim },\n  sorry, sorry,\nend\n\nend lookup\n\nsection length\n\nlemma length_cache_apply_of_not_queried (hlog : log.not_queried i t)\n  (x : spec.range i × query_log spec) (hx : x ∈ ((caching_oracle spec) i (t, log)).support) :\n  (x.2 i).length = (log i).length + 1 :=\nbegin\n  rw [support_apply_of_not_queried i t log hlog, set.mem_set_of_eq] at hx,\n  rw [hx, query_log.log_query_apply_same_index, list.length_cons],\nend\n\nlemma length_cache_apply_of_not_not_queried (hlog : ¬ log.not_queried i t)\n  (x : spec.range i × query_log spec) (hx : x ∈ ((caching_oracle spec) i (t, log)).support) :\n  (x.2 i).length = (log i).length :=\nbegin\n  rw [mem_support_apply_iff_of_queried _ _ _ _ hlog] at hx,\n  refine congr_arg list.length (congr_fun hx.2 i)\nend\n\n/-- The length of the final cache is at least as long as the initial cache. -/\nlemma length_cache_le_length_cache_simulate (x : α × query_log spec)\n  (hx : x ∈ (simulate (caching_oracle spec) oa log).support) :\n  (log i).length ≤ (x.2 i).length :=\nbegin\n  refine support_state_simulate_induction (caching_oracle spec)\n    (λ log', (log i).length ≤ (log' i).length) log le_rfl oa\n      x hx (λ i' t log' x' hx' hlog', le_trans hlog' _),\n  by_cases hlog'' : log'.not_queried i' t,\n  { rw [support_apply_of_not_queried _ _ _ hlog''] at hx',\n    exact hx'.symm ▸ log'.length_apply_le_lenght_log_query_apply i' t x'.1 i },\n  { rw [mem_support_apply_iff_of_queried _ _ _ _ hlog''] at hx',\n    rw [hx'.2] }\nend\n\nend length\n\nsection fork\n\n/-- Given a `query_log`, and a output of simulating with that as the cache,\nsimulation with the `fork_cache` applied to the new cache has the same support as simulating\nwith the original cache. Intuitively, since all the new entries to the cache just come\nfrom a call to `query`, it's irrelevent that they have been added, since any new call\ncan't tell whether this call to `query` is new or made earlier. -/\ntheorem support_simulate_fork_cache_some\n  (cache : query_log spec) (i : spec.ι) (n : ℕ) (hn : (cache i).length ≤ n)\n  (x : α × query_log spec) (hx : x ∈ (simulate (caching_oracle spec) oa' cache).support) :\n  (simulate (caching_oracle spec) oa (x.2.fork_cache i (some n))).support =\n    (simulate (caching_oracle spec) oa cache).support :=\nbegin\n  induction oa using oracle_comp.induction_on with α a α β oa ob hoa hob i t,\n  sorry, sorry, sorry,\nend\n\n/-- Specialized version of `support_simulate_fork_cache_some` when the initial state is empty.\nIn this case we don't need to place any restrictions on the value of `n`-/\nlemma support_simulate_init_fork_cache_some (i : spec.ι) (n : ℕ) (x : α × query_log spec)\n  (hx : x ∈ (simulate (caching_oracle spec) oa' (query_log.init spec)).support) :\n  (simulate (caching_oracle spec) oa (x.2.fork_cache i (some n))).support =\n    (simulate (caching_oracle spec) oa (query_log.init spec)).support :=\nbegin\n  sorry,\nend\n\n/-- Version of `support_simulate_fork_cache` for forking on `none` -/\nlemma support_simulate_fork_cache_none (i : spec.ι) (cache : query_log spec) :\n  (simulate (caching_oracle spec) oa (cache.fork_cache i none)).support =\n    (simulate (caching_oracle spec) oa (query_log.init spec)).support :=\nby rw query_log.fork_cache_none\n\nend fork\n\n-- TODO: generalize. Should also work for log→seed probably ?\n/-- Re-running a computation with an old cache doesn't change the possible outputs,\nassuming the old cache was generated by an honest simulation. -/\ntheorem support_simulate_bind_simulate_eq_support_simulate (oa : oracle_comp spec α) :\n  (simulate (caching_oracle spec) oa log >>= λ x, simulate (caching_oracle spec) oa x.2).support =\n    (simulate (caching_oracle spec) oa log).support :=\nbegin\n  refine support_simulate_simulate_eq_support_simulate (caching_oracle spec) (caching_oracle spec)\n    _ log oa,\n  clear log,\n  intros i t log,\n  ext x,\n  refine ⟨λ h, _, λ h, _⟩,\n  { simp_rw [set.mem_Union] at h,\n    obtain ⟨y, hy, hy'⟩ := h,\n    by_cases hlog : log.not_queried i t,\n    { rw [mem_support_apply_iff_of_not_queried _ _ _ _ hlog] at ⊢ hy,\n      have : ¬ y.2.not_queried i t := sorry,\n      rw [mem_support_apply_iff_of_queried _ _ _ _ this] at hy',\n      refine hy'.2.trans _,\n      have : x.1 = y.1 := sorry,\n      rw [hy, this] },\n    { rw [mem_support_apply_iff_of_queried _ _ _ _ hlog] at ⊢ hy,\n      have : ¬ y.2.not_queried i t := sorry,\n      rw [mem_support_apply_iff_of_queried _ _ _ _ this] at hy',\n      refine ⟨trans (hy'.1.trans _) hy.1, hy'.2.trans hy.2⟩,\n      rw [hy.2],\n      exact symm hy.1 } },\n  sorry,\nend\n\nend support\n\nsection eval_dist\n\n\nend eval_dist\n\nend caching_oracle", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/simulation_semantics/constructions/logging/caching_oracle.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3022728255546675}}
{"text": "import category_theory.preadditive.yoneda\nimport for_mathlib.AddCommGroup.kernels\n\nnoncomputable theory\n\nuniverses v\n\nnamespace category_theory\n\nnamespace limits\n\nsection kernel_comparison\nvariables {C D : Type*} [category.{v} C] [category.{v} D] [has_zero_morphisms C]\n  [has_zero_morphisms D] (F : C ⥤ D) [functor.preserves_zero_morphisms F]\n\nlemma kernel_comparison_is_iso_of_is_limit {A₁ A₂ : C} (f : A₁ ⟶ A₂) [has_kernel f]\n  [has_kernel (F.map f)] (s : kernel_fork f) (hs₁ : is_limit s)\n    (hs₂ : is_limit (kernel_fork.of_ι (F.map s.ι) (show _ ≫ F.map f = 0,\n      by { rw [← F.map_comp, kernel_fork.condition, F.map_zero],}))) :\n  is_iso (limits.kernel_comparison f F) :=\nbegin\n  let α : kernel f ≅ s.X := is_limit.cone_point_unique_up_to_iso (limit.is_limit _) hs₁,\n  let β : F.obj s.X ≅ kernel (F.map f) := is_limit.cone_point_unique_up_to_iso hs₂\n    (limit.is_limit _),\n  suffices : limits.kernel_comparison f F = (F.map_iso α ≪≫ β).hom,\n  { rw this, apply_instance, },\n  rw ← cancel_mono (kernel.ι (F.map f)),\n  dsimp only [α, β],\n  simp only [kernel_comparison_comp_ι, iso.trans_hom, fork.of_ι_π_app,\n    functor.map_iso_hom, category.assoc, limit.cone_point_unique_up_to_iso_hom_comp,\n    ← F.map_comp],\n  congr',\n  rw ← cancel_epi ((limit.is_limit (parallel_pair f 0)).cone_point_unique_up_to_iso hs₁).inv,\n  simp only [iso.inv_hom_id_assoc, limit.cone_point_unique_up_to_iso_inv_comp, fork.app_zero_eq_ι],\nend\n\nend kernel_comparison\n\nvariables {C : Type*} [category C] [preadditive C]\n\nlemma _root_.AddCommGroup.kernel_fork_is_limit {X Y : AddCommGroup} (φ : X ⟶ Y)\n  (s : kernel_fork φ) (hs₁ : function.injective s.ι)\n    (hs₂ : ∀ (x : X), φ x = 0 → ∃ (z : s.X), s.ι z = x) :\n  is_limit s :=\nbegin\n  refine is_limit.of_iso_limit (AddCommGroup.kernel_is_limit φ) _,\n  let e : s.X ≃+ φ.ker :=\n  { to_fun := λ w, ⟨s.ι w, by { rw add_monoid_hom.mem_ker,\n      exact congr_arg (λ (f : s.X ⟶ Y), (f w : Y)) s.condition, }⟩,\n    inv_fun := λ x, (hs₂ x.1 x.2).some,\n    left_inv := λ w, by { apply hs₁, exact (hs₂ _ _).some_spec, },\n    right_inv := λ x, by { ext, exact (hs₂ x.1 x.2).some_spec, },\n    map_add' := λ w₁ w₂, by { ext, simp only [map_add, add_submonoid.mk_add_mk,\n      set_like.coe_mk], } },\n  symmetry,\n  refine cones.ext (add_equiv.to_AddCommGroup_iso e) _,\n  rintro (_|_),\n  { ext, refl, },\n  { ext, dsimp [e], simp only [kernel_fork.app_one, comp_zero], },\nend\n\ninstance preadditive_yoneda_kernel_comparison_is_iso (B : C) {A₁ A₂ : Cᵒᵖ} (f : A₁ ⟶ A₂)\n  [has_kernel f] : is_iso (limits.kernel_comparison f (preadditive_yoneda.obj B)) :=\nbegin\n  let hs₁ : is_limit ( _ : kernel_fork f) := limit.is_limit _,\n  apply kernel_comparison_is_iso_of_is_limit (preadditive_yoneda.obj B) f _ hs₁,\n  apply AddCommGroup.kernel_fork_is_limit,\n  { intros g₁ g₂ h,\n    dsimp at g₁ g₂ h ⊢,\n    rw cancel_epi (kernel.ι f).unop at h,\n    exact h, },\n  { intros x hx,\n    dsimp at x hx ⊢,\n    use (kernel.lift f x.op (quiver.hom.unop_inj hx)).unop,\n    rw ← unop_comp,\n    simp only [kernel.lift_ι, quiver.hom.unop_op], },\nend\n\nend limits\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/yoneda_left_exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.42250463481418826, "lm_q1q2_score": 0.30226995407580376}}
{"text": "import for_mathlib.finite\nimport o_minimal.o_minimal\n\nnamespace o_minimal\n\nopen set\n\ninstance {R : Type*} [preorder R] (S : struc R) [is_definable_le S R] [definable_constants S]\n  (a b : R) : is_definable S (Ioo a b) :=\nis_definable.subtype (def_set.Ioo a b)\n\nvariables {R : Type*} [DUNLO R]\nvariables (S : struc R) [o_minimal S]\n\n/-- [vdD:3.1.3 Lemma 1]:\nA definable function on an interval is injective or constant on some subinterval. -/\nlemma mono1 {a b : R} (hab : a < b) {f : Ioo a b → R} (hf : def_fun S f) :\n  ∃ (c d : R) (hcd : c < d) (hcd' : Ioo c d ⊆ Ioo a b),\n  function.injective (f ∘ subtype.map id hcd') ∨\n  ∃ (z : R), ∀ (x : R) (hx : x ∈ Ioo c d), f ⟨x, hcd' hx⟩ = z := -- TODO: Ugh\nbegin\n  classical,\n  by_cases H :\n    ∃ (c d : R) (hcd : c < d) (hcd' : Ioo c d ⊆ Ioo a b),\n    ∃ z, ∀ (x : R) (hx : x ∈ Ioo c d), f ⟨x, by exact hcd' hx⟩ = z,\n  { obtain ⟨c, d, hcd, hcd', z, hf⟩ := H,\n    refine ⟨c, d, hcd, hcd', or.inr ⟨z, hf⟩⟩ },\n  -- Otherwise, the preimage of any {z} under f is finite\n  have finite_fibers : ∀ z, finite (f ⁻¹' {z}),\n  { intro z,\n    have : def_set S (subtype.val '' (f ⁻¹' {z})),\n    { apply def_fun.image,\n      exact def_fun_subtype_val,\n      exact def_fun.preimage hf def_val_const },\n    replace := o_minimal.tame_of_def this,\n    replace := this.finite_or_contains_interval,\n    cases this with h h,\n    { exact finite_of_finite_image (subtype.val_injective.inj_on _) h },\n    { exfalso,\n      apply H,\n      obtain ⟨c, d, hcd, h'⟩ := h,\n      have : Ioo c d ⊆ Ioo a b := subset.trans h' (subtype.coe_image_subset _ _),\n      refine ⟨c, d, hcd, this, z, λ x hx, _⟩,\n      obtain ⟨⟨_, _⟩, hh, rfl⟩ := h' hx,\n      exact hh } },\n  -- Since the fibers of f are finite, each fiber has a smallest member\n  have minocc : ∀ x, ∃ x', f x' = f x ∧ ∀ x'', f x'' = f x → x' ≤ x'',\n  { intro x,\n    let X := {x' | f x' = f x},\n    have : finite X := finite_fibers _,\n    let X' := this.to_finset,\n    have X'_def : ∀ {x'}, x' ∈ X' ↔ f x' = f x := by apply finite.mem_to_finset,\n    have : X'.nonempty := ⟨x, X'_def.mpr rfl⟩,\n    let x' := X'.min' this,\n    refine ⟨x', X'_def.mp (X'.min'_mem _), λ x'' hx'', _⟩,\n    rw ←X'_def at hx'',\n    apply X'.min'_le,\n    exact hx'' },\n  -- Let K be the set of all these smallest members\n  let K := {x' | ∀ x'', f x' = f x'' → x' ≤ x''},\n  have : f '' K = range f,\n  { refine subset.antisymm (image_subset_range _ _) _,\n    rintros _ ⟨x, rfl⟩,\n    obtain ⟨x', hx'₁, hx'₂⟩ := minocc x,\n    refine ⟨x', _, hx'₁⟩,\n    intros x'' hx'',\n    apply hx'₂,\n    cc },\n  replace : K.infinite,\n  { intro kfin,\n    replace kfin : (f '' K).finite := kfin.image _,\n    rw this at kfin,\n    rw ←image_univ at kfin,\n    have : (univ : set (Ioo a b)).finite,\n    { apply finite_of_finite_image_fibers kfin,\n      intro z,\n      convert finite_fibers z,\n      ext, simp },\n    have := Ioo.infinite hab,\n    rw [←infinite_coe_iff, ←infinite_univ_iff] at this,\n    contradiction },\n  have : def_set S (subtype.val '' K) := begin\n    apply def_fun.image def_fun_subtype_val,\n    -- TODO: automate this proof\n    apply def_set.forall,\n    apply def_set.imp,\n    apply def_set_eq,\n    exact hf.comp def_fun.fst,\n    exact hf.comp def_fun.snd,\n    -- TODO: definable_le instance for Ioo\n    change def_set S {p : Ioo a b × Ioo a b | p.1.val ≤ p.2.val},\n    apply definable_le,\n    exact def_fun_subtype_val.comp def_fun.fst,\n    exact def_fun_subtype_val.comp def_fun.snd\n  end,\n  replace := (o_minimal.tame_of_def this).finite_or_contains_interval,\n  cases this with hK hK,\n  { exfalso,\n    apply ‹K.infinite›,\n    exact finite_of_finite_image (subtype.val_injective.inj_on _) hK },\n  { obtain ⟨c, d, hcd, hf⟩ := hK,\n    have : Ioo c d ⊆ Ioo a b := subset.trans hf (subtype.coe_image_subset _ _),\n    refine ⟨c, d, hcd, this, or.inl _⟩,\n    rintros ⟨x₁, hx₁⟩ ⟨x₂, hx₂⟩ hx₁₂,\n    let x₁' : ↥(Ioo a b) := ⟨x₁, this hx₁⟩, -- ??\n    let x₂' : ↥(Ioo a b) := ⟨x₂, this hx₂⟩,\n    have x₁K : K x₁', { rcases hf hx₁ with ⟨⟨_,_⟩, h₁, rfl⟩, exact h₁ },\n    have x₂K : K x₂', { rcases hf hx₂ with ⟨⟨_,_⟩, h₂, rfl⟩, exact h₂ },\n    have l₁ : x₁' ≤ x₂' := x₁K _ hx₁₂,\n    have l₂ : x₂' ≤ x₁' := x₂K _ hx₁₂.symm,\n    exact le_antisymm l₁ l₂ }\nend\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/src/o_minimal/mono1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.44939263446475963, "lm_q1q2_score": 0.3021409135283476}}
{"text": "/-\n  Extension of a sheaf of rings on the basis to a sheaf of rings on the whole space.\n\n  https://stacks.math.columbia.edu/tag/009M\n  https://stacks.math.columbia.edu/tag/009N\n-/\n\nimport topology.opens\nimport sheaves.stalk_of_rings\nimport sheaves.stalk_of_rings_on_standard_basis\nimport sheaves.presheaf_of_rings_on_basis\nimport sheaves.presheaf_of_rings_extension\nimport sheaves.sheaf_on_basis\nimport sheaves.sheaf_on_standard_basis\nimport sheaves.sheaf_of_rings\n\nopen topological_space classical\n\nnoncomputable theory\n\nuniverse u\n\nsection presheaf_of_rings_extension\n\nvariables {α : Type u} [T : topological_space α]\nvariables {B : set (opens α)} {HB : opens.is_basis B}\nvariables (Bstd : opens.univ ∈ B ∧ ∀ {U V}, U ∈ B → V ∈ B → U ∩ V ∈ B)\n\ninclude Bstd\n\ntheorem extension_is_sheaf_of_rings\n(F : presheaf_of_rings_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n: is_sheaf_of_rings (F ᵣₑₓₜ Bstd) :=\nbegin\n  show is_sheaf (F ᵣₑₓₜ Bstd).to_presheaf,\n  constructor,\n  { intros U OC s t Hres,\n    apply subtype.eq,\n    apply funext,\n    intros x,\n    apply funext,\n    intros HxU,\n    rw OC.Hcov.symm at HxU,\n    rcases HxU with ⟨Uj1, ⟨⟨⟨Uj2, OUj⟩, ⟨⟨j, HUj⟩, Heq⟩⟩, HxUj⟩⟩,\n    rcases Heq, rcases Heq,\n    have Hstj := congr_fun (subtype.mk_eq_mk.1 (Hres j)),\n    have HxUj1 : x ∈ OC.Uis j := HUj.symm ▸ HxUj,\n    have Hstjx := congr_fun (Hstj x) HxUj1,\n    exact Hstjx, },\n  { intros U OC s Hsec,\n    existsi (global_section (F.to_presheaf_on_basis) U OC s Hsec),\n    intros i,\n    apply subtype.eq,\n    apply funext,\n    intros x,\n    apply funext,\n    intros HxUi,\n    have HxU : x ∈ U := OC.Hcov ▸ (opens_supr_subset OC.Uis i) HxUi,\n    let HyUi := λ t, ∃ (H : t ∈ set.range OC.Uis), x ∈ t,\n    dunfold presheaf_of_rings_extension; dsimp,\n    dunfold global_section; dsimp,\n    -- Same process of dealing with subtype.rec.\n    let HyUi := λ t, ∃ (H : t ∈ subtype.val '' set.range OC.Uis), x ∈ t,\n    rcases (classical.indefinite_description HyUi _) with ⟨S, HS⟩; dsimp,\n    let HyS := λ H : S ∈ subtype.val '' set.range OC.Uis, x ∈ S,\n    rcases (classical.indefinite_description HyS HS) with ⟨HSUiR, HySUiR⟩; dsimp,\n    let HOUksub := λ t : subtype is_open, t ∈ set.range (OC.Uis) ∧ t.val = S,\n    rcases (classical.indefinite_description HOUksub _) with ⟨OUl, ⟨HOUl, HOUleq⟩⟩; dsimp,\n    let HSUi := λ i, OC.Uis i = OUl,\n    cases (classical.indefinite_description HSUi _) with l HSUil; dsimp,\n    -- Now we just need to apply Hsec in the right way.\n    dunfold presheaf_of_rings_extension at Hsec,\n    dunfold res_to_inter_left at Hsec,\n    dunfold res_to_inter_right at Hsec,\n    dsimp at Hsec,\n    replace Hsec := Hsec i l,\n    rw subtype.ext at Hsec,\n    dsimp at Hsec,\n    replace Hsec := congr_fun Hsec x,\n    dsimp at Hsec,\n    replace Hsec := congr_fun Hsec,\n    have HxOUk : x ∈ OUl.val := HOUleq.symm ▸ HySUiR,\n    have HxUl : x ∈ OC.Uis l := HSUil.symm ▸ HxOUk,\n    exact (Hsec ⟨HxUi, HxUl⟩).symm, },\n  end\n\nsection extension_coincides\n\n-- The extension is done in a way that F(U) ≅ Fext(U).\n\n-- The map ψ : F(U) → Π x ∈ U, Fx\n\ndef to_stalk_product (F : presheaf_on_basis α HB) {U : opens α} (BU : U ∈ B)\n: F.F BU → Π (x ∈ U), stalk_on_basis F x :=\nλ s x Hx, ⟦{U := U, BU := BU, Hx := Hx, s := s}⟧\n\nlemma to_stalk_product.injective\n(F : presheaf_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F)\n{U : opens α} (BU : U ∈ B)\n: function.injective (to_stalk_product Bstd F BU) :=\nbegin\n  intros s₁ s₂ Hs,\n  have Hsx := λ (HxU : U), congr_fun (congr_fun Hs HxU.1) HxU.2,\n  let OC : covering_standard_basis B U :=\n  { γ := U,\n    Uis := λ HxU, some (quotient.eq.1 (Hsx HxU)),\n    BUis := λ HxU, some (some_spec (quotient.eq.1 (Hsx HxU))),\n    Hcov :=\n      begin\n        ext z,\n        split,\n        { rintros ⟨Ui, ⟨⟨OUi, ⟨⟨i, HUi⟩, HUival⟩⟩, HzUi⟩⟩,\n          rw [←HUival, ←HUi] at HzUi,\n          exact some (some_spec (some_spec (some_spec (quotient.eq.1 (Hsx i))))) HzUi, },\n        { intros Hz,\n          use [(some (quotient.eq.1 (Hsx ⟨z, Hz⟩))).val],\n          have Hin : (some (quotient.eq.1 (Hsx ⟨z, Hz⟩))).val\n            ∈ subtype.val '' set.range (λ (HxU : U), some ((quotient.eq.1 (Hsx HxU)))),\n            use [classical.some ((quotient.eq.1 (Hsx ⟨z, Hz⟩)))],\n            split,\n            { use ⟨z, Hz⟩, },\n            { refl, },\n          use Hin,\n          exact some (some_spec (some_spec (quotient.eq.1 (Hsx ⟨z, Hz⟩)))), },\n      end, },\n  apply (HF BU OC).1,\n  intros i,\n  replace Hs := congr_fun (congr_fun Hs i.1) i.2,\n  exact some_spec (some_spec (some_spec (some_spec (some_spec (quotient.eq.1 (Hsx i)))))),\nend\n\n-- The map φ : F(U) → im(ψ).\n\ndef to_presheaf_of_rings_extension (F : presheaf_of_rings_on_basis α HB) {U : opens α} (BU : U ∈ B)\n: F.F BU → (F ᵣₑₓₜ Bstd).F U :=\nλ s,\n  ⟨to_stalk_product Bstd F.to_presheaf_on_basis BU s,\n   λ x Hx, ⟨U, BU, Hx, s, λ y Hy, funext $ λ Hy',\n   quotient.sound $ ⟨U, BU, Hy', set.subset.refl U, set.subset.refl U, rfl⟩⟩⟩\n\nlemma to_presheaf_of_rings_extension.injective\n(F : presheaf_of_rings_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n{U : opens α} (BU : U ∈ B)\n: function.injective (to_presheaf_of_rings_extension Bstd F BU) :=\nbegin\n  intros s₁ s₂ Hs,\n  erw subtype.mk_eq_mk at Hs,\n  have Hinj := to_stalk_product.injective Bstd F.to_presheaf_on_basis (λ V BV OC, HF BV OC) BU,\n  exact Hinj Hs,\nend\n\nlemma to_presheaf_of_rings_extension.surjective\n(F : presheaf_of_rings_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n{U : opens α} (BU : U ∈ B)\n: function.surjective (to_presheaf_of_rings_extension Bstd F BU) :=\nbegin\n  intros s,\n  let V := λ (HxU : U), some (s.2 HxU.1 HxU.2),\n  let BV := λ (HxU : U), some (some_spec (s.2 HxU.1 HxU.2)),\n  let HxV := λ (HxU : U), some (some_spec (some_spec (s.2 HxU.1 HxU.2))),\n  let σ := λ (HxU : U), some (some_spec (some_spec (some_spec (s.2 HxU.1 HxU.2)))),\n  let Hσ := λ (HxU : U), some_spec (some_spec (some_spec (some_spec (s.2 HxU.1 HxU.2)))),\n  let OC : covering_standard_basis B U :=\n  { γ := U,\n    Uis := λ HxU, (V HxU) ∩ U,\n    BUis := λ HxU, Bstd.2 (BV HxU) BU,\n    Hcov :=\n      begin\n        ext z,\n        split,\n        { rintros ⟨Ui, ⟨⟨OUi, ⟨⟨i, HUi⟩, HUival⟩⟩, HzUi⟩⟩,\n          rw [←HUival, ←HUi] at HzUi,\n          exact HzUi.2, },\n        { intros Hz,\n          use [(some (s.2 z Hz) ∩ U).val],\n          have Hin : (some (s.2 z Hz) ∩ U).val\n            ∈ subtype.val '' set.range (λ (HxU : U), ((some (s.2 HxU.1 HxU.2) ∩ U : opens α))),\n            use [(some (s.2 z Hz) ∩ U : opens α)],\n            split,\n            { use ⟨z, Hz⟩, },\n            { refl, },\n          use Hin,\n          exact ⟨some (some_spec (some_spec (s.2 z Hz))), Hz⟩, },\n      end, },\n  -- Now let's try to apply sheaf condition.\n  let res := λ (HxU : OC.γ), F.res (BV HxU) (Bstd.2 (BV HxU) BU) (set.inter_subset_left _ _),\n  let sx := λ (HxU : OC.γ), res HxU (σ HxU),\n  have Hglue := (HF BU OC).2 sx,\n  have Hsx : ∀ j k,\n    sheaf_on_standard_basis.res_to_inter_left Bstd F.to_presheaf_on_basis (OC.BUis j) (OC.BUis k) (sx j) =\n    sheaf_on_standard_basis.res_to_inter_right Bstd F.to_presheaf_on_basis (OC.BUis j) (OC.BUis k) (sx k),\n    intros j k,\n    dsimp only [sheaf_on_standard_basis.res_to_inter_left],\n    dsimp only [sheaf_on_standard_basis.res_to_inter_right],\n    dsimp only [sx, res],\n    iterate 2 { rw ←presheaf_on_basis.Hcomp', },\n    show (F.to_presheaf_on_basis).res (BV j) (Bstd.2 (OC.BUis j) (OC.BUis k)) _ (σ j)\n       = (F.to_presheaf_on_basis).res (BV k) (Bstd.2 (OC.BUis j) (OC.BUis k)) _ (σ k),\n    -- We can cover the U ∩ Vj ∩ Vk and use locality.\n    -- But first let's check that all the stalks coincide in the intersectons.\n    have Hstalks : ∀ {y} (Hy : y ∈ (OC.Uis j) ∩ (OC.Uis k)),\n        (⟦{U := V j, BU := BV j, Hx := Hy.1.1, s := σ j}⟧ : stalk_on_basis F.to_presheaf_on_basis y)\n      = ⟦{U := V k, BU := BV k, Hx := Hy.2.1, s := σ k}⟧,\n      intros y Hy,\n      have Hj := congr_fun (Hσ j y ⟨Hy.1.2, Hy.1.1⟩) Hy.1.2; dsimp at Hj,\n      have Hk := congr_fun (Hσ k y ⟨Hy.2.2, Hy.2.1⟩) Hy.2.2; dsimp at Hk,\n      erw [←Hj, ←Hk],\n    -- Therefore there exists Wjk where σj|Wjk = σk|Wjk. We will use these as a cover.\n    let Ujk : opens α := (OC.Uis j) ∩ (OC.Uis k),\n    let BUjk := Bstd.2 (OC.BUis j) (OC.BUis k),\n    -- Again, all the information we will need but on Uj ∩ Uk.\n    let Hjk := λ (HxUjk : Ujk), quotient.eq.1 (Hstalks HxUjk.2),\n    let Wjk := λ (HxUjk : Ujk), some (Hjk HxUjk) ∩ U,\n    let BWjk := λ (HxUjk : Ujk), Bstd.2 (some (some_spec (Hjk HxUjk))) BU,\n    let HxWjk := λ (HxUjk : Ujk), some (some_spec (some_spec (Hjk HxUjk))),\n    let HWjkUj := λ (HxUjk : Ujk), some (some_spec (some_spec (some_spec (Hjk HxUjk)))),\n    let HWjkUk := λ (HxUjk : Ujk), some (some_spec (some_spec (some_spec (some_spec (Hjk HxUjk))))),\n    let HWjk := λ (HxUjk : Ujk), some_spec (some_spec (some_spec (some_spec (some_spec (Hjk HxUjk))))),\n    let OCjk : covering_standard_basis B ((OC.Uis j) ∩ (OC.Uis k)) :=\n    { γ := Ujk,\n      Uis := Wjk,\n      BUis := BWjk,\n      Hcov :=\n        begin\n          ext z,\n          split,\n          { rintros ⟨W, ⟨⟨OW, ⟨⟨i, HWi⟩, HWival⟩⟩, HzW⟩⟩,\n            rw [←HWival, ←HWi] at HzW,\n            have HzUj := (HWjkUj i) HzW.1,\n            have HzUk := (HWjkUk i) HzW.1,\n            exact ⟨⟨HzUj, HzW.2⟩, ⟨HzUk, HzW.2⟩⟩, },\n          { intros Hz,\n            use [(some (Hjk ⟨z, Hz⟩) ∩ U).val],\n            have Hin : (some (Hjk ⟨z, Hz⟩) ∩ U).val\n              ∈ subtype.val '' set.range (λ (HxUjk : Ujk), ((some (Hjk HxUjk) ∩ U : opens α))),\n              use [(some (Hjk ⟨z, Hz⟩) ∩ U : opens α)],\n              split,\n              { use ⟨z, Hz⟩, },\n              { refl, },\n            use Hin,\n            have HzWjk := HxWjk ⟨z, Hz⟩,\n            have HzU := Hz.1.2,\n            exact ⟨HzWjk, HzU⟩, }\n        end, },\n    apply (HF BUjk OCjk).1,\n    intros i,\n    rw ←presheaf_on_basis.Hcomp',\n    rw ←presheaf_on_basis.Hcomp',\n    have Hres :\n        F.res (some (some_spec (Hjk i))) (BWjk i) (set.inter_subset_left _ _)\n          (F.res (BV j) (some (some_spec (Hjk i))) (HWjkUj i) (σ j))\n      = F.res (some (some_spec (Hjk i))) (BWjk i) (set.inter_subset_left _ _)\n          (F.res (BV k) (some (some_spec (Hjk i))) (HWjkUk i) (σ k)),\n      rw (HWjk i),\n    rw ←presheaf_on_basis.Hcomp' at Hres,\n    rw ←presheaf_on_basis.Hcomp' at Hres,\n    use Hres,\n  -- Ready to prove it.\n  rcases (Hglue Hsx) with ⟨S, HS⟩,\n  existsi S,\n  apply subtype.eq,\n  dsimp [to_presheaf_of_rings_extension],\n  apply funext,\n  intros x,\n  dsimp [to_stalk_product],\n  apply funext,\n  intros Hx,\n  replace HS := HS ⟨x, Hx⟩,\n  dsimp [sx, res] at HS,\n  rw Hσ ⟨x, Hx⟩,\n  swap,\n  { exact ⟨Hx, HxV ⟨x, Hx⟩⟩, },\n  dsimp,\n  apply quotient.sound,\n  use [(V ⟨x, Hx⟩) ∩ U],\n  use [Bstd.2 (BV ⟨x, Hx⟩) BU],\n  use [⟨HxV ⟨x, Hx⟩, Hx⟩],\n  use [set.inter_subset_right _ _],\n  use [set.inter_subset_left _ _],\n  dsimp,\n  erw HS,\nend\n\nlemma to_presheaf_of_rings_extension.bijective\n(F : presheaf_of_rings_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n{U : opens α} (BU : U ∈ B)\n: function.bijective (to_presheaf_of_rings_extension Bstd F BU) :=\n⟨to_presheaf_of_rings_extension.injective Bstd F (λ U BU OC, HF BU OC) BU,\nto_presheaf_of_rings_extension.surjective Bstd F (λ U BU OC, HF BU OC) BU ⟩\n\n-- We now that they are equivalent as sets.\n-- Now we to assert that they're isomorphic as rings.\n-- It suffices to show that it is a ring homomorphism.\n\nlemma to_presheaf_of_rings_extension.is_ring_hom\n(F : presheaf_of_rings_on_basis α HB)\n{U : opens α} (BU : U ∈ B)\n: is_ring_hom (to_presheaf_of_rings_extension Bstd F BU) :=\n{ map_one :=\n    begin\n      apply subtype.eq,\n      funext x Hx,\n      apply quotient.sound,\n      use [U, BU, Hx, set.subset.refl _, set.subset_univ _],\n      iterate 2 { erw (F.res_is_ring_hom _ _ _).map_one, },\n    end,\n  map_mul :=\n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz],\n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_mul,\n      erw ←presheaf_on_basis.Hcomp',\n    end,\n  map_add :=\n    begin\n      intros x y,\n      apply subtype.eq,\n      funext z Hz,\n      apply quotient.sound,\n      use [U, BU, Hz],\n      use [set.subset.refl _, set.subset_inter (set.subset.refl _) (set.subset.refl _)],\n      erw ←(F.res_is_ring_hom _ _ _).map_add,\n      erw ←presheaf_on_basis.Hcomp',\n    end, }\n\ndef to_presheaf_of_rings_extension.ring_equiv\n(F : presheaf_of_rings_on_basis α HB)\n(HF : sheaf_on_standard_basis.is_sheaf_on_standard_basis Bstd F.to_presheaf_on_basis)\n{U : opens α} (BU : U ∈ B)\n: F BU ≃+* (presheaf_of_rings_extension Bstd F) U :=\n@@ring_equiv.of' _ _\n  (equiv.of_bijective $ to_presheaf_of_rings_extension.bijective Bstd F @HF BU)\n  (to_presheaf_of_rings_extension.is_ring_hom Bstd F BU)\n\n-- Moreover, for all x, Fₓ ≅ Fextₓ.\n\n-- We will need this to show that the stalks of the structure sheaf on\n-- Spec(R) are local rings.\n\nopen stalk_of_rings_on_standard_basis\n\ndef to_stalk_extension\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: stalk_of_rings_on_standard_basis Bstd F x → stalk_of_rings (F ᵣₑₓₜ Bstd) x :=\nbegin\n  intros BUs,\n  let Us := quotient.out BUs,\n  exact ⟦{U := Us.U,\n          HxU := Us.Hx,\n          s := (to_presheaf_of_rings_extension Bstd F Us.BU) Us.s}⟧,\nend\n\nlemma to_stalk_extension.injective\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: function.injective (to_stalk_extension Bstd F x) :=\nbegin\n  intros Us₁' Us₂',\n  apply quotient.induction_on₂ Us₁' Us₂',\n  rintros Us₁ Us₂ HUs,\n  rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁Us₁, HW₁U₁, Hres₁⟩,\n  rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂Us₂, HW₂U₂, Hres₂⟩,\n  dunfold to_stalk_extension at HUs,\n  rw quotient.eq at HUs,\n  rcases HUs with ⟨W, HxW, HWU₁, HWU₂, Hres⟩,\n  dsimp at HWU₁,\n  dsimp at HWU₂,\n  dunfold to_presheaf_of_rings_extension at Hres,\n  dunfold to_stalk_product at Hres,\n  erw subtype.mk.inj_eq at Hres,\n  replace Hres := congr_fun (congr_fun Hres x) HxW,\n  dsimp at Hres,\n  rw quotient.eq at Hres,\n  rcases Hres with ⟨W₃, BW₃, HxW₃, HW₃U₁, HW₃U₂, Hres₃⟩,\n  dsimp at HW₃U₁,\n  dsimp at HW₃U₂,\n  dsimp at Hres₃,\n  apply quotient.sound,\n  have BW₁₂₃ : W₁ ∩ W₂ ∩ W₃ ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n  have HW₁₂₃U₁ : W₁ ∩ W₂ ∩ W₃ ⊆ Us₁.U := λ x Hx, HW₁U₁ Hx.1.1,\n  have HW₁₂₃U₂ : W₁ ∩ W₂ ∩ W₃ ⊆ Us₂.U := λ x Hx, HW₂U₂ Hx.1.2,\n  use [W₁ ∩ W₂ ∩ W₃, BW₁₂₃, ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩, HW₁₂₃U₁, HW₁₂₃U₂],\n  have HW₁W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₁ := λ x Hx, Hx.1.1,\n  have HW₂W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₂ := λ x Hx, Hx.1.2,\n  have HW₃W₁₂₃ : W₁ ∩ W₂ ∩ W₃ ⊆ W₃ := λ x Hx, Hx.2,\n  replace Hres₁ := congr_arg (F.res BW₁ BW₁₂₃ HW₁W₁₂₃) Hres₁,\n  replace Hres₂ := congr_arg (F.res BW₂ BW₁₂₃ HW₂W₁₂₃) Hres₂,\n  replace Hres₃ := congr_arg (F.res BW₃ BW₁₂₃ HW₃W₁₂₃) Hres₃,\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n  erw [←Hres₁, ←Hres₂],\n  exact Hres₃,\nend\n\nlemma to_stalk_extension.surjective\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: function.surjective (to_stalk_extension Bstd F x) :=\nbegin\n  intros Us',\n  apply quotient.induction_on Us',\n  rintros ⟨U, HxU, s⟩,\n  rcases (s.2 x HxU) with ⟨V, BV, HxV, t, Ht⟩,\n  let Vt : stalk_on_basis.elem (F.to_presheaf_on_basis) x\n    := {U := V, BU := BV, Hx := HxV, s := t},\n  use ⟦Vt⟧,\n  dunfold to_stalk_extension,\n  apply quotient.sound,\n  rcases (quotient.mk_out Vt) with ⟨W, BW, HxW, HWVtV, HWV, Hres⟩,\n  have HUVWV : U ∩ V ∩ W ⊆ (quotient.out ⟦Vt⟧).U := λ x Hx, HWVtV Hx.2,\n  have HUVWU : U ∩ V ∩ W ⊆ U := λ x Hx, Hx.1.1,\n  use [U ∩ V ∩ W, ⟨⟨HxU, HxV⟩, HxW⟩, HUVWV, HUVWU],\n  apply subtype.eq,\n  dsimp only [presheaf_of_rings_extension],\n  dsimp only [to_presheaf_of_rings_extension],\n  dsimp only [to_stalk_product],\n  funext y Hy,\n  rw (Ht y Hy.1),\n  apply quotient.sound,\n  have BVW : V ∩ W ∈ B := Bstd.2 BV BW,\n  have HVWVtV : V ∩ W ⊆ (quotient.out ⟦Vt⟧).U := λ x Hx, HWVtV Hx.2,\n  have HVWV : V ∩ W ⊆ V := λ x Hx, Hx.1,\n  use [V ∩ W, BVW, ⟨Hy.1.2,Hy.2⟩, HVWVtV, HVWV],\n  have HVWW : V ∩ W ⊆ W := λ x Hx, Hx.2,\n  replace Hres := congr_arg (F.res BW BVW HVWW) Hres,\n  iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres, },\n  exact Hres,\nend\n\nlemma to_stalk_extension.bijective\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: function.bijective (to_stalk_extension Bstd F x) :=\n⟨to_stalk_extension.injective Bstd F x,\nto_stalk_extension.surjective Bstd F x⟩\n\nlemma to_stalk_extension.is_ring_hom\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: is_ring_hom (to_stalk_extension Bstd F x) :=\n{ map_one :=\n    begin\n      dunfold to_stalk_extension,\n      let one.elem : stalk_on_basis.elem F.to_presheaf_on_basis x\n        := {U := opens.univ, BU := Bstd.1, Hx := trivial, s:= 1},\n      let one.stalk : stalk_of_rings_on_standard_basis Bstd F x := ⟦one.elem⟧,\n      let one := quotient.out one.stalk,\n      apply quotient.sound,\n      rcases (quotient.mk_out one.elem) with ⟨W₁, BW₁, HxW₁, HW₁Uout, HW₁U, Hres₁⟩,\n      have BUW₁ : one.U ∩ W₁ ∈ B := Bstd.2 one.BU BW₁,\n      have HUUW₁ : one.U ∩ W₁ ⊆ one.U := set.inter_subset_left _ _,\n      use [one.U ∩ W₁, ⟨one.Hx, HxW₁⟩, HUUW₁, set.subset_univ _],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z Hz,\n      apply quotient.sound,\n      use [one.U ∩ W₁, BUW₁, Hz, set.inter_subset_left _ _, set.subset_univ _],\n      dsimp,\n      have HUW₁W₁ : one.U ∩ W₁ ⊆ W₁ := set.inter_subset_right _ _,\n      replace Hres₁ := congr_arg (F.res BW₁ BUW₁ HUW₁W₁) Hres₁,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      exact Hres₁,\n    end,\n  map_mul :=\n    begin\n      intros y z,\n      apply quotient.induction_on₂ y z,\n      intros Us₁ Us₂,\n      simp,\n      let Us₃ : stalk_on_basis.elem F.to_presheaf_on_basis x :=\n        { U := Us₁.U ∩ Us₂.U,\n          BU := Bstd.2 Us₁.BU Us₂.BU,\n          Hx := ⟨Us₁.Hx, Us₂.Hx⟩,\n          s :=  F.res Us₁.BU _ (set.inter_subset_left _ _) Us₁.s *\n                F.res Us₂.BU _ (set.inter_subset_right _ _) Us₂.s },\n      dunfold to_stalk_extension,\n      apply quotient.sound,\n      rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁U₁out, HW₁U₁, Hres₁⟩,\n      rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂U₂out, HW₂U₂, Hres₂⟩,\n      rcases (quotient.mk_out Us₃) with ⟨W₃, BW₃, HxW₃, HW₃U₃out, HW₃U₃, Hres₃⟩,\n      let W := W₁ ∩ W₂ ∩ W₃,\n      have HxW : x ∈ W := ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩,\n      have HWW₁ : W ⊆ W₁ := λ x Hx, Hx.1.1,\n      have HWW₂ : W ⊆ W₂ := λ x Hx, Hx.1.2,\n      have HWW₃ : W ⊆ W₃ := λ x Hx, Hx.2,\n      have HWU₁out : W ⊆ (quotient.out ⟦Us₁⟧).U := set.subset.trans HWW₁ HW₁U₁out,\n      have HWU₂out : W ⊆ (quotient.out ⟦Us₂⟧).U := set.subset.trans HWW₂ HW₂U₂out,\n      have HWU₃out : W ⊆ (quotient.out ⟦Us₃⟧).U := set.subset.trans HWW₃ HW₃U₃out,\n      have HWU₁₂out : W ⊆ (quotient.out ⟦Us₁⟧).U ∩ (quotient.out ⟦Us₂⟧).U\n        := set.subset_inter HWU₁out HWU₂out,\n      use [W, HxW, HWU₃out, HWU₁₂out],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z HzW,\n      apply quotient.sound,\n      have BW : W ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n      use [W, BW, HzW, HWU₃out, HWU₁₂out],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_mul,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n      replace Hres₁ := congr_arg (F.res BW₁ BW HWW₁) Hres₁,\n      replace Hres₂ := congr_arg (F.res BW₂ BW HWW₂) Hres₂,\n      replace Hres₃ := congr_arg (F.res BW₃ BW HWW₃) Hres₃,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n      erw [Hres₁, Hres₂, Hres₃],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_mul,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n    end,\n  map_add :=\n    begin\n      intros y z,\n      apply quotient.induction_on₂ y z,\n      intros Us₁ Us₂,\n      simp,\n      let Us₃ : stalk_on_basis.elem F.to_presheaf_on_basis x :=\n        { U := Us₁.U ∩ Us₂.U,\n          BU := Bstd.2 Us₁.BU Us₂.BU,\n          Hx := ⟨Us₁.Hx, Us₂.Hx⟩,\n          s :=  F.res Us₁.BU _ (set.inter_subset_left _ _) Us₁.s +\n                F.res Us₂.BU _ (set.inter_subset_right _ _) Us₂.s },\n      dunfold to_stalk_extension,\n      apply quotient.sound,\n      rcases (quotient.mk_out Us₁) with ⟨W₁, BW₁, HxW₁, HW₁U₁out, HW₁U₁, Hres₁⟩,\n      rcases (quotient.mk_out Us₂) with ⟨W₂, BW₂, HxW₂, HW₂U₂out, HW₂U₂, Hres₂⟩,\n      rcases (quotient.mk_out Us₃) with ⟨W₃, BW₃, HxW₃, HW₃U₃out, HW₃U₃, Hres₃⟩,\n      let W := W₁ ∩ W₂ ∩ W₃,\n      have HxW : x ∈ W := ⟨⟨HxW₁, HxW₂⟩, HxW₃⟩,\n      have HWW₁ : W ⊆ W₁ := λ x Hx, Hx.1.1,\n      have HWW₂ : W ⊆ W₂ := λ x Hx, Hx.1.2,\n      have HWW₃ : W ⊆ W₃ := λ x Hx, Hx.2,\n      have HWU₁out : W ⊆ (quotient.out ⟦Us₁⟧).U := set.subset.trans HWW₁ HW₁U₁out,\n      have HWU₂out : W ⊆ (quotient.out ⟦Us₂⟧).U := set.subset.trans HWW₂ HW₂U₂out,\n      have HWU₃out : W ⊆ (quotient.out ⟦Us₃⟧).U := set.subset.trans HWW₃ HW₃U₃out,\n      have HWU₁₂out : W ⊆ (quotient.out ⟦Us₁⟧).U ∩ (quotient.out ⟦Us₂⟧).U\n        := set.subset_inter HWU₁out HWU₂out,\n      use [W, HxW, HWU₃out, HWU₁₂out],\n      apply subtype.eq,\n      dsimp only [presheaf_of_rings_extension],\n      dsimp only [to_presheaf_of_rings_extension],\n      dsimp only [to_stalk_product],\n      funext z HzW,\n      apply quotient.sound,\n      have BW : W ∈ B := Bstd.2 (Bstd.2 BW₁ BW₂) BW₃,\n      use [W, BW, HzW, HWU₃out, HWU₁₂out],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_add,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n      replace Hres₁ := congr_arg (F.res BW₁ BW HWW₁) Hres₁,\n      replace Hres₂ := congr_arg (F.res BW₂ BW HWW₂) Hres₂,\n      replace Hres₃ := congr_arg (F.res BW₃ BW HWW₃) Hres₃,\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₁, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₂, },\n      iterate 2 { rw ←presheaf_on_basis.Hcomp' at Hres₃, },\n      erw [Hres₁, Hres₂, Hres₃],\n      rw (presheaf_of_rings_on_basis.res_is_ring_hom _ _ _ _).map_add,\n      rw ←presheaf_on_basis.Hcomp',\n      rw ←presheaf_on_basis.Hcomp',\n    end, }\n\nlemma to_stalk_extension.ring_equiv\n(F : presheaf_of_rings_on_basis α HB)\n(x : α)\n: stalk_of_rings_on_standard_basis Bstd F x ≃+*\nstalk_of_rings (presheaf_of_rings_extension Bstd F) x :=\nbegin\n  have H := function.bijective_iff_has_inverse.1 (to_stalk_extension.bijective Bstd F x),\n  rcases (classical.indefinite_description _ H) with ⟨inv, Hlinv, Hrinv⟩,\n  use [(to_stalk_extension Bstd F x), inv, Hlinv, Hrinv],\n  { apply (to_stalk_extension.is_ring_hom Bstd F x).map_mul, },\n  { apply (to_stalk_extension.is_ring_hom Bstd F x).map_add, }\nend\n\nend extension_coincides\n\nend presheaf_of_rings_extension\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/sheaves/sheaf_of_rings_on_standard_basis.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.30214021799644425}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison\n-/\nimport category_theory.eq_to_hom\n\n/-!\n# Cartesian products of categories\n\nWe define the category instance on `C × D` when `C` and `D` are categories.\n\nWe define:\n* `sectl C Z` : the functor `C ⥤ C × D` given by `X ↦ ⟨X, Z⟩`\n* `sectr Z D` : the functor `D ⥤ C × D` given by `Y ↦ ⟨Z, Y⟩`\n* `fst`       : the functor `⟨X, Y⟩ ↦ X`\n* `snd`       : the functor `⟨X, Y⟩ ↦ Y`\n* `swap`      : the functor `C × D ⥤ D × C` given by `⟨X, Y⟩ ↦ ⟨Y, X⟩`\n    (and the fact this is an equivalence)\n\nWe further define `evaluation : C ⥤ (C ⥤ D) ⥤ D` and `evaluation_uncurried : C × (C ⥤ D) ⥤ D`,\nand products of functors and natural transformations, written `F.prod G` and `α.prod β`.\n-/\n\nnamespace category_theory\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄\n\nsection\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D]\n\n/--\n`prod C D` gives the cartesian product of two categories.\n\nSee https://stacks.math.columbia.edu/tag/001K.\n-/\n@[simps {not_recursive := []}] -- the generates simp lemmas like `id_fst` and `comp_snd`\ninstance prod : category.{max v₁ v₂} (C × D) :=\n{ hom     := λ X Y, ((X.1) ⟶ (Y.1)) × ((X.2) ⟶ (Y.2)),\n  id      := λ X, ⟨ 𝟙 (X.1), 𝟙 (X.2) ⟩,\n  comp    := λ _ _ _ f g, (f.1 ≫ g.1, f.2 ≫ g.2) }\n\n/-- Two rfl lemmas that cannot be generated by `@[simps]`. -/\n@[simp] \n\nend\n\nsection\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₁) [category.{v₁} D]\n/--\n`prod.category.uniform C D` is an additional instance specialised so both factors have the same\nuniverse levels. This helps typeclass resolution.\n-/\ninstance uniform_prod : category (C × D) := category_theory.prod C D\nend\n\n-- Next we define the natural functors into and out of product categories. For now this doesn't\n-- address the universal properties.\nnamespace prod\n\n/-- `sectl C Z` is the functor `C ⥤ C × D` given by `X ↦ (X, Z)`. -/\n@[simps] def sectl\n  (C : Type u₁) [category.{v₁} C] {D : Type u₂} [category.{v₂} D] (Z : D) : C ⥤ C × D :=\n{ obj := λ X, (X, Z),\n  map := λ X Y f, (f, 𝟙 Z) }\n\n/-- `sectr Z D` is the functor `D ⥤ C × D` given by `Y ↦ (Z, Y)` . -/\n@[simps] def sectr\n  {C : Type u₁} [category.{v₁} C] (Z : C) (D : Type u₂) [category.{v₂} D] : D ⥤ C × D :=\n{ obj := λ X, (Z, X),\n  map := λ X Y f, (𝟙 Z, f) }\n\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D]\n\n/-- `fst` is the functor `(X, Y) ↦ X`. -/\n@[simps] def fst : C × D ⥤ C :=\n{ obj := λ X, X.1,\n  map := λ X Y f, f.1 }\n\n/-- `snd` is the functor `(X, Y) ↦ Y`. -/\n@[simps] def snd : C × D ⥤ D :=\n{ obj := λ X, X.2,\n  map := λ X Y f, f.2 }\n\n/-- The functor swapping the factors of a cartesian product of categories, `C × D ⥤ D × C`. -/\n@[simps] def swap : C × D ⥤ D × C :=\n{ obj := λ X, (X.2, X.1),\n  map := λ _ _ f, (f.2, f.1) }\n\n/--\nSwapping the factors of a cartesion product of categories twice is naturally isomorphic\nto the identity functor.\n-/\n@[simps] def symmetry : swap C D ⋙ swap D C ≅ 𝟭 (C × D) :=\n{ hom := { app := λ X, 𝟙 X },\n  inv := { app := λ X, 𝟙 X } }\n\n/--\nThe equivalence, given by swapping factors, between `C × D` and `D × C`.\n-/\n@[simps]\ndef braiding : C × D ≌ D × C :=\nequivalence.mk (swap C D) (swap D C)\n  (nat_iso.of_components (λ X, eq_to_iso (by simp)) (by tidy))\n  (nat_iso.of_components (λ X, eq_to_iso (by simp)) (by tidy))\n\ninstance swap_is_equivalence : is_equivalence (swap C D) :=\n(by apply_instance : is_equivalence (braiding C D).functor)\n\nend prod\n\nsection\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₂) [category.{v₂} D]\n\n/--\nThe \"evaluation at `X`\" functor, such that\n`(evaluation.obj X).obj F = F.obj X`,\nwhich is functorial in both `X` and `F`.\n-/\n@[simps] def evaluation : C ⥤ (C ⥤ D) ⥤ D :=\n{ obj := λ X,\n  { obj := λ F, F.obj X,\n    map := λ F G α, α.app X, },\n  map := λ X Y f,\n  { app := λ F, F.map f,\n    naturality' := λ F G α, eq.symm (α.naturality f) } }\n\n/--\nThe \"evaluation of `F` at `X`\" functor,\nas a functor `C × (C ⥤ D) ⥤ D`.\n-/\n@[simps] def evaluation_uncurried : C × (C ⥤ D) ⥤ D :=\n{ obj := λ p, p.2.obj p.1,\n  map := λ x y f, (x.2.map f.1) ≫ (f.2.app y.1),\n  map_comp' := λ X Y Z f g,\n  begin\n    cases g, cases f, cases Z, cases Y, cases X,\n    simp only [prod_comp, nat_trans.comp_app, functor.map_comp, category.assoc],\n    rw [←nat_trans.comp_app, nat_trans.naturality, nat_trans.comp_app,\n        category.assoc, nat_trans.naturality],\n  end }\n\nend\n\nvariables {A : Type u₁} [category.{v₁} A]\n          {B : Type u₂} [category.{v₂} B]\n          {C : Type u₃} [category.{v₃} C]\n          {D : Type u₄} [category.{v₄} D]\n\nnamespace functor\n/-- The cartesian product of two functors. -/\n@[simps] def prod (F : A ⥤ B) (G : C ⥤ D) : A × C ⥤ B × D :=\n{ obj := λ X, (F.obj X.1, G.obj X.2),\n  map := λ _ _ f, (F.map f.1, G.map f.2) }\n\n/- Because of limitations in Lean 3's handling of notations, we do not setup a notation `F × G`.\n   You can use `F.prod G` as a \"poor man's infix\", or just write `functor.prod F G`. -/\n\nend functor\n\nnamespace nat_trans\n\n/-- The cartesian product of two natural transformations. -/\n@[simps] def prod {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) :\n  F.prod H ⟶ G.prod I :=\n{ app         := λ X, (α.app X.1, β.app X.2),\n  naturality' := λ X Y f,\n  begin\n    cases X, cases Y,\n    simp only [functor.prod_map, prod.mk.inj_iff, prod_comp],\n    split; rw naturality\n  end }\n\n/- Again, it is inadvisable in Lean 3 to setup a notation `α × β`;\n   use instead `α.prod β` or `nat_trans.prod α β`. -/\n\nend nat_trans\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/products/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.30214021113107636}}
{"text": "import category_theory.category.ulift\nimport category_theory.limits.filtered_colimit_commutes_finite_limit -- todo: minimize imports\nimport data.real.nnreal\n\nnoncomputable theory\nopen_locale nnreal\n\nopen category_theory\n\nnamespace category_theory.limits\n\nvariables (ι : ℕ →o ℝ≥0) (hι : ∀ r : ℝ≥0, ∃ n : ℕ, r ≤ ι n)\n\ndef find_nat (r : ℝ≥0) : ℕ := (hι r).some\n\nlemma find_nat_spec (r : ℝ≥0) : r ≤ ι (find_nat ι hι r) :=\n(hι r).some_spec\n\ndef nat_to_nnreal : ℕ ⥤ ℝ≥0 :=\n{ obj := λ i, ι i,\n  map := λ i j h, hom_of_le $ ι.mono h.le }\n\nuniverses v u\nvariables {C : Type u} [category.{v} C]\n\ndef restrict_diagram (F : as_small.{v} ℝ≥0 ⥤ C) :\n  as_small.{v} ℕ ⥤ C :=\n(as_small.down ⋙ nat_to_nnreal ι ⋙ as_small.up) ⋙ F\n\ndef restrict_cocone {F : as_small.{v} ℝ≥0 ⥤ C} (S : cocone F) :\n  cocone (restrict_diagram ι F) :=\nS.whisker _\n\ndef is_colimit_restrict_cocone {F : as_small.{v} ℝ≥0 ⥤ C} (S : cocone F) (hS : is_colimit S) :\n  is_colimit (restrict_cocone ι S) :=\n{ desc := λ T, hS.desc ⟨T.X,\n  { app := λ r, F.map (as_small.up.map $ hom_of_le $ find_nat_spec ι hι (ulift.down r)) ≫\n        T.ι.app (ulift.up (find_nat ι hι (ulift.down r))),\n    naturality' := begin\n      intros r s f,\n      dsimp,\n      let m := find_nat ι hι (ulift.down r) ⊔ find_nat ι hι (ulift.down s),\n      let r' := as_small.up.obj (find_nat ι hι (ulift.down r)),\n      let s' := as_small.up.obj (find_nat ι hι (ulift.down s)),\n      let m' := as_small.up.obj m,\n      let ιr : r' ⟶ m' := as_small.up.map (hom_of_le $ le_sup_left),\n      let ιs : s' ⟶ m' := as_small.up.map (hom_of_le $ le_sup_right),\n      erw ← T.w ιr,\n      erw ← T.w ιs,\n      dsimp [restrict_diagram],\n      simp only [category.comp_id, ← category.assoc, ← F.map_comp],\n      refl,\n    end }⟩,\n  fac' := begin\n    intros W j,\n    dsimp [restrict_cocone],\n    rw hS.fac,\n    dsimp,\n    let r := (nat_to_nnreal ι).obj (ulift.down j),\n    let k := find_nat ι hι r,\n    let l := (ulift.down j) ⊔ k,\n    let k' : as_small.{v} ℕ := as_small.up.obj k,\n    let l' : as_small.{v} ℕ := as_small.up.obj l,\n    let ιk : k' ⟶ l' := as_small.up.map (hom_of_le $ le_sup_right),\n    let ιj : j ⟶ l' := as_small.up.map (hom_of_le $ le_sup_left),\n    erw ← W.w ιk,\n    erw ← W.w ιj,\n    dsimp [restrict_diagram],\n    simp only [← category.assoc, ← F.map_comp],\n    refl,\n  end,\n  uniq' := begin\n    intros W m hm,\n    apply hS.hom_ext, intros j,\n    specialize hm (as_small.up.obj (find_nat ι hι (ulift.down j))),\n    dsimp [restrict_cocone] at hm,\n    rw hS.fac,\n    dsimp,\n    erw [← hm, ← category.assoc, S.w],\n  end }\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/nnreal_to_nat_colimit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.3020066913581038}}
{"text": "import category_theory.preadditive.additive_functor\nimport category_theory.limits.preserves.shapes.biproducts\n\nimport for_mathlib.derived.les2\nimport for_mathlib.derived.les_facts\nimport for_mathlib.derived.Ext_lemmas\n\nimport for_mathlib.is_quasi_iso\nimport for_mathlib.short_exact\nimport for_mathlib.homology\nimport for_mathlib.exact_lift_desc\nimport for_mathlib.additive_functor\nimport for_mathlib.homotopy_category_lemmas\nimport for_mathlib.homology_lift_desc\n\n.\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite\nopen homotopy_category (hiding single)\nopen bounded_homotopy_category\n\nuniverses v v' u u'\n\n-- main proof in this file is inspired by https://math.stackexchange.com/a/2118042\n\nsection\n\nvariables {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {ι : Type*} {c : complex_shape ι}\n\ndef delta_to_kernel (C : homological_complex 𝓐 c) (i j k : ι) :\n  C.X i ⟶ kernel (C.d j k) :=\nfactor_thru_image _ ≫ image_to_kernel' (C.d i j) _ (C.d_comp_d _ _ _)\n\ndef delta_to_kernel_ι (C : homological_complex 𝓐 c) (i j k : ι) :\n  delta_to_kernel C i j k ≫ kernel.ι (C.d j k) = C.d i j :=\nbegin\n  delta delta_to_kernel image_to_kernel',\n  rw [category.assoc, kernel.lift_ι, image.fac],\nend\n\ndef d_delta_to_kernel (C : homological_complex 𝓐 c) (h i j k : ι) :\n  C.d h i ≫ delta_to_kernel C i j k = 0 :=\nbegin\n  rw [← cancel_mono (kernel.ι (C.d j k)), category.assoc, delta_to_kernel_ι, C.d_comp_d, zero_comp],\nend\n\n-- move me\nlemma short_exact_comp_iso {A B C D : 𝓐} (f : A ⟶ B) (g : B ⟶ C) (h : C ⟶ D) (hh : is_iso h) :\n  short_exact f (g ≫ h) ↔ short_exact f g :=\nbegin\n  split; intro H,\n  { haveI : mono f := H.mono,\n    haveI : epi g,\n    { haveI := H.epi, have := epi_comp (g ≫ h) (inv h), simpa only [category.assoc, is_iso.hom_inv_id, category.comp_id] },\n    refine ⟨_⟩, have := H.exact, rwa exact_comp_iso at this, },\n  { haveI : mono f := H.mono,\n    haveI : epi g := H.epi,\n    haveI : epi (g ≫ h) := epi_comp g h,\n    refine ⟨_⟩, have := H.exact, rwa exact_comp_iso }\nend\n\nlemma is_acyclic_def\n  (C : homotopy_category 𝓐 c) :\n  is_acyclic C ↔ (∀ i, is_zero (C.as.homology i)) :=\nbegin\n  split,\n  { apply is_acyclic.cond },\n  { apply is_acyclic.mk }\nend\n\nlemma is_acyclic_iff_short_exact_to_cycles\n  (C : homotopy_category 𝓐 (complex_shape.up ℤ)) :\n  is_acyclic C ↔\n  (∀ i, short_exact (kernel.ι (C.as.d i (i+1))) (delta_to_kernel C.as i (i+1) (i+1+1))) :=\nbegin\n  rw is_acyclic_def,\n  symmetry,\n  apply (equiv.add_right (1 : ℤ)).forall_congr,\n  intro i,\n  let e := (homology_iso C.as i (i+1) (i+1+1) rfl rfl),\n  dsimp [delta_to_kernel] at e ⊢,\n  rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],\n  split,\n  { apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },\n  { intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }\nend\n\nlemma is_acyclic_iff_short_exact_to_cycles'\n  (C : homological_complex 𝓐 (complex_shape.down ℤ)) :\n  (∀ i, is_zero (C.homology i)) ↔\n  (∀ i, short_exact (kernel.ι (C.d (i+1+1) (i+1))) (delta_to_kernel C (i+1+1) (i+1) i)) :=\nbegin\n  symmetry,\n  apply (equiv.add_right (1 : ℤ)).forall_congr,\n  intro i,\n  let e := (homology_iso C (i+1+1) (i+1) i rfl rfl),\n  dsimp [delta_to_kernel] at e ⊢,\n  rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],\n  split,\n  { apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },\n  { intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }\nend\n\nend\n\nvariables {𝓐 𝓑 : Type*} [category 𝓐] [abelian 𝓐] [enough_projectives 𝓐]\nvariables [category 𝓑] [abelian 𝓑] [enough_projectives 𝓑]\n\nvariables (C : cochain_complex 𝓐 ℤ)\n  [homotopy_category.is_bounded_above ((homotopy_category.quotient _ _).obj C)]\n\ndef category_theory.functor.single (F : bounded_homotopy_category 𝓐 ⥤ 𝓑) (i : ℤ) : 𝓐 ⥤ 𝓑 :=\nbounded_homotopy_category.single _ i ⋙ F\n\n-- move me\nlemma category_theory.limits.is_zero.biprod {𝓐 : Type*} [category 𝓐] [abelian 𝓐]\n  {X Y : 𝓐} (hX : is_zero X) (hY : is_zero Y) :\n  is_zero (X ⊞ Y) :=\nbegin\n  rw is_zero_iff_id_eq_zero at hX hY ⊢,\n  ext; simp [hX, hY],\nend\n\ninstance category_theory.limits.preserves_binary_biproduct_of_additive\n  {𝓐 𝓑 : Type*} [category.{v} 𝓐] [category.{v} 𝓑] [abelian 𝓐] [abelian 𝓑]\n  (F : 𝓐 ⥤ 𝓑) [functor.additive F] (X Y : 𝓐) :\n  preserves_binary_biproduct X Y F :=\npreserves_binary_biproduct_of_preserves_biproduct _ _ _\n\nlemma acyclic_left_of_short_exact (B : 𝓐) {X Y Z : 𝓐} (f : X ⟶ Y) (g : Y ⟶ Z) (hfg : short_exact f g)\n  (hY : ∀ i > 0, is_zero (((Ext' i).obj (op $ Y)).obj B))\n  (hZ : ∀ i > 0, is_zero (((Ext' i).obj (op $ Z)).obj B)) :\n  ∀ i > 0, is_zero (((Ext' i).obj (op $ X)).obj B) :=\nbegin\n  intros i hi,\n  have := hfg.Ext'_five_term_exact_seq B i,\n  refine (this.drop 1).pair.is_zero_of_is_zero_is_zero (hY _ hi) (hZ _ _),\n  transitivity i, { exact lt_add_one i }, { exact hi }\nend\n.\n\nlemma map_is_acyclic_of_acyclic_aux\n  {A B C D X Y Z W : 𝓐} (f : A ⟶ B) (g : C ⟶ D) (π : B ⟶ kernel g)\n  {α : X ⟶ B} {β : B ⟶ Y} {γ : Y ⟶ C} {δ : C ⟶ Z} {ε : Z ⟶ D} {ζ : D ⟶ W}\n  (hαβ : short_exact α β) (hγδ : short_exact γ δ) (hεζ : short_exact ε ζ)\n  (hf : mono f) (hfπ : exact f π)\n  (hαπ : α ≫ π = 0) (hγg : γ ≫ g = 0) (hδε : δ ≫ ε = g)\n  (hπι : π ≫ kernel.ι g = β ≫ γ) :\n  short_exact f π :=\nbegin\n  suffices : epi π, { resetI, exact ⟨hfπ⟩ },\n  have hβ : epi β := hαβ.epi,\n  have hγ : mono γ := hγδ.mono,\n  have hε : mono ε := hεζ.mono,\n  resetI,\n  have hιδ : kernel.ι g ≫ δ = 0,\n  { rw [← cancel_mono ε, category.assoc, hδε, kernel.condition, zero_comp], },\n  let e1 : Y ⟶ kernel g := hαβ.exact.epi_desc π hαπ,\n  let e2 : Y ⟶ kernel g := kernel.lift g γ hγg,\n  let e3 : kernel g ⟶ Y := hγδ.exact.mono_lift (kernel.ι g) hιδ,\n  have he12 : e1 = e2,\n  { rw [← cancel_epi β, ← cancel_mono (kernel.ι g)],\n    simp only [hπι, category.assoc, kernel.lift_ι, exact.comp_epi_desc_assoc], },\n  have he13 : e1 ≫ e3 = 𝟙 _,\n  { rw [he12, ← cancel_mono γ, category.assoc, exact.mono_lift_comp, kernel.lift_ι, category.id_comp], },\n  have he31 : e3 ≫ e1 = 𝟙 _,\n  { rw [he12, ← cancel_mono (kernel.ι g), category.assoc, kernel.lift_ι, exact.mono_lift_comp, category.id_comp], },\n  let e : Y ≅ kernel g := ⟨e1, e3, he13, he31⟩,\n  have hπ : β ≫ e.hom = π := exact.comp_epi_desc _ _ _,\n  rw ← hπ, exact epi_comp _ _,\nend\n\nlemma short_exact_Ext_of_short_exact_of_acyclic {A B C : 𝓐} (Z : 𝓐) {f : A ⟶ B} {g : B ⟶ C}\n  (hfg : short_exact f g) (hC : ∀ i > 0, is_zero (((Ext' i).obj (op $ C)).obj Z)) :\n  short_exact (((Ext' 0).flip.obj Z).map g.op) (((Ext' 0).flip.obj Z).map f.op) :=\nbegin\n  have H0 := hfg.Ext'_five_term_exact_seq Z 0,\n  apply_with short_exact.mk {instances:=ff},\n  { have H := ((hfg.Ext'_five_term_exact_seq Z (-1)).drop 2).pair,\n    apply H.mono_of_is_zero,\n    apply Ext'_is_zero_of_neg, dec_trivial },\n  { apply (H0.drop 1).pair.epi_of_is_zero,\n    apply hC, dec_trivial },\n  { exact H0.pair }\nend\n\nlemma map_is_acyclic_of_acyclic''\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  ∀ i, is_zero (((((Ext' 0).flip.obj B).map_homological_complex _).obj C.op).homology i) :=\nbegin\n  rw is_acyclic_iff_short_exact_to_cycles',\n  obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),\n  have aux : ((quotient 𝓐 (complex_shape.up ℤ)).obj C).is_acyclic := ‹_›,\n  rw is_acyclic_iff_short_exact_to_cycles at aux,\n  intro i,\n  let K := λ j, kernel (C.d j (j+1)),\n  suffices hK : ∀ j, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),\n  { have SES1 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1+1)) (hK _),\n    have SES2 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1)) (hK _),\n    have SES3 := short_exact_Ext_of_short_exact_of_acyclic B (aux i) (hK _),\n    apply map_is_acyclic_of_acyclic_aux _ _ _ SES1 SES2 SES3 infer_instance;\n      clear SES1 SES2 SES3 aux,\n    { delta delta_to_kernel image_to_kernel',\n      apply exact_comp_mono, rw exact_factor_thru_image_iff, exact exact_kernel_ι },\n    { rw [← cancel_mono (kernel.ι _), zero_comp, category.assoc, delta_to_kernel_ι],\n      swap, apply_instance,\n      erw [functor.map_homological_complex_obj_d, ← functor.map_comp],\n      dsimp only [homological_complex.op_d, quotient_obj_as],\n      rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },\n    { erw [functor.map_homological_complex_obj_d, ← functor.map_comp],\n      dsimp only [homological_complex.op_d, quotient_obj_as],\n      rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },\n    { rw [← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },\n    { rw [delta_to_kernel_ι, ← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },\n    { apply_instance } },\n  clear i, intro j,\n  have : ∀ j ≥ a, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),\n  { intros j hj i hi,\n    apply bounded_derived_category.Ext'_zero_left_is_zero,\n    apply is_zero.op,\n    refine is_zero.of_mono (kernel.ι _) _,\n    exact ha j hj },\n  apply int.induction_on' j a,\n  { exact this _ le_rfl, },\n  { intros j hj aux, apply this, exact int.le_add_one hj, },\n  { intros j hj IH,\n    obtain ⟨j, rfl⟩ : ∃ i, i + 1 = j := ⟨j - 1, sub_add_cancel _ _⟩,\n    rw add_sub_cancel,\n    apply acyclic_left_of_short_exact B (kernel.ι _) (delta_to_kernel _ _ _ _) _ (hC _) IH,\n    exact aux j, }\nend\n\nlemma map_is_acyclic_of_acyclic'\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  is_acyclic ((((Ext' 0).flip.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=\nbegin\n  rw is_acyclic_def,\n  intro i,\n  have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },\n  refine is_zero.of_iso _ (homology_iso' _ (i-1) i (i+1) h1 rfl),\n  dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,\n    functor.right_op_map, functor.map_homological_complex_obj_d],\n  apply exact.homology_is_zero,\n  apply exact.op,\n  refine exact_of_homology_is_zero _,\n  { rw [← category_theory.functor.map_comp, ← op_comp, homological_complex.d_comp_d, op_zero, functor.map_zero], },\n  have := map_is_acyclic_of_acyclic'' C B hC i,\n  apply this.of_iso _, clear this,\n  let C' := (((Ext' 0).flip.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj (homological_complex.op C),\n  have h1 : (complex_shape.down ℤ).rel i (i - 1), { dsimp, apply sub_add_cancel },\n  exact (homology_iso' C' (i+1) i (i-1) rfl h1).symm,\nend\n\nlemma map_is_acyclic_of_acyclic\n  [is_acyclic ((homotopy_category.quotient _ _).obj C)]\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :\n  is_acyclic (((preadditive_yoneda.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=\nbegin\n  have := map_is_acyclic_of_acyclic' C B hC,\n  rw is_acyclic_def at this ⊢,\n  intro i, specialize this i,\n  apply this.of_iso _, clear this,\n  have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },\n  refine (homology_iso' _ (i-1) i (i+1) h1 rfl) ≪≫ _ ≪≫ (homology_iso' _ (i-1) i (i+1) h1 rfl).symm,\n  dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,\n    functor.right_op_map, functor.map_homological_complex_obj_d],\n  let e := λ i, ((bounded_derived_category.Ext'_zero_flip_iso _ B).app (op $ C.X i)).op,\n  refine homology.map_iso _ _ (arrow.iso_mk (e _) (e _) _) (arrow.iso_mk (e _) (e _) _) rfl,\n  { simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },\n  { simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },\nend\n\nlemma acyclic_of_projective (P B : 𝓐) [projective P] (i : ℤ) (hi : 0 < i) :\n  is_zero (((Ext' i).obj (op P)).obj B) :=\nbegin\n  rw (Ext'_iso (op P) B i _ (𝟙 _) _).is_zero_iff,\n  { rcases i with ((_|i)|i),\n    { exfalso, revert hi, dec_trivial },\n    swap, { exfalso, revert hi, dec_trivial },\n    refine is_zero.homology_is_zero _ _ _ _,\n    apply AddCommGroup.is_zero_of_eq,\n    intros,\n    apply is_zero.eq_of_src,\n    apply is_zero_zero, },\n  { refine ⟨_, _, _⟩,\n    { rintro (_|n), { assumption }, { dsimp, apply_instance } },\n    { exact exact_zero_mono (𝟙 P) },\n    { rintro (_|n); exact exact_of_zero 0 0 } }\nend\n\ndef Ext_compute_with_acyclic_aux₁\n  (B : 𝓐)\n  (i : ℤ) :\n  ((Ext i).obj (op $ of' C)).obj ((single _ 0).obj B) ≅\n  (preadditive_yoneda.obj ((single 𝓐 (-i)).obj B)).obj (op (of' C).replace) :=\n(preadditive_yoneda.map_iso $ (shift_single_iso 0 i).app B ≪≫ eq_to_iso (by rw zero_sub)).app _\n\nabbreviation of'_hom {C₁ C₂ : cochain_complex 𝓐 ℤ}\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (f : C₁ ⟶ C₂) :\n  of' C₁ ⟶ of' C₂ :=\n(homotopy_category.quotient _ _).map f\n\nlemma Ext_compute_with_acyclic_aux₁_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  ((Ext i).map $ quiver.hom.op $ of'_hom f).app _ ≫\n    (Ext_compute_with_acyclic_aux₁ C₁ B i).hom =\n  (Ext_compute_with_acyclic_aux₁ C₂ B i).hom ≫\n  (preadditive_yoneda.obj _).map (quiver.hom.op $\n  bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) :=\nbegin\n  ext F,\n  dsimp [Ext, Ext_compute_with_acyclic_aux₁],\n  simp only [comp_apply],\n  dsimp, simp,\nend\n\ndef Ext_compute_with_acyclic_aux₂\n  (B : 𝓐)\n  (i : ℤ) :\n  (preadditive_yoneda.obj ((single 𝓐 (-i)).obj B)).obj (op (of' C).replace) ≅\n  (((preadditive_yoneda.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj\n  ((of' C).replace).val.as.op).homology (-i) :=\n  hom_single_iso _ _ _\n\n-- TODO: We should prove naturality of `hom_single_iso` independently!\nlemma Ext_compute_with_acyclic_aux₂_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  (preadditive_yoneda.obj _).map (quiver.hom.op $\n    bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) ≫\n    (Ext_compute_with_acyclic_aux₂ C₁ B i).hom =\n  (Ext_compute_with_acyclic_aux₂ C₂ B i).hom ≫\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map\n      (bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π).out).unop :=\nhom_single_iso_naturality _ _ _ _ _\n\ndef Ext_compute_with_acyclic_HomB\n  (B : 𝓐) := (preadditive_yoneda.obj B).right_op.map_homological_complex (complex_shape.up ℤ) ⋙\n  homological_complex.unop_functor.right_op\n\nlemma Ext_compute_with_acyclic_is_quasi_iso_aux\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  is_quasi_iso ((homotopy_category.quotient _ _).map\n    ((Ext_compute_with_acyclic_HomB B).map (of' C).π.out).unop) :=\nbegin\n  let P := (of' C).replace,\n  let π : P ⟶ of' C := (of' C).π,\n  let HomB := (preadditive_yoneda.obj B).right_op.map_homological_complex (complex_shape.up ℤ) ⋙\n    homological_complex.unop_functor.right_op,\n  let fq := (homotopy_category.quotient _ _).map (HomB.map π.out).unop,\n  apply is_quasi_iso_of_op,\n  let f := homological_complex.op_functor.map (HomB.map (quot.out π)),\n  have := cone_triangleₕ_mem_distinguished_triangles _ _ f,\n  replace := is_quasi_iso_iff_is_acyclic _ this,\n  dsimp [homological_complex.cone.triangleₕ] at this,\n  erw this, clear this i,\n  constructor,\n  intro i, obtain ⟨i, rfl⟩ : ∃ j, j + 1 = i := ⟨i - 1, sub_add_cancel _ _⟩,\n  refine is_zero.of_iso _ (homology_iso _ i (i+1) (i+1+1) _ _),\n  rotate, { dsimp, refl }, { dsimp, refl },\n  apply exact.homology_is_zero _,\n  dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d],\n  have hπ : is_quasi_iso π, { dsimp [π], apply_instance },\n  have := cone_triangleₕ_mem_distinguished_triangles _ _ π.out,\n  replace := is_quasi_iso_iff_is_acyclic _ this,\n  dsimp [homological_complex.cone.triangleₕ] at this,\n  simp only [quotient_map_out] at this,\n  replace := this.mp _,\n  swap, { convert hπ using 1, generalize : P.val = X, cases X, refl, },\n  haveI preaux : ((quotient 𝓐 (complex_shape.up ℤ)).obj (homological_complex.cone (quot.out π))).is_bounded_above,\n  { constructor,\n    obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),\n    obtain ⟨b, hb⟩ := is_bounded_above.cond P.val,\n    refine ⟨max a b, _⟩,\n    intros k hk,\n    refine category_theory.limits.is_zero.biprod _ _,\n    { apply hb, refine (le_max_right _ _).trans (hk.trans (lt_add_one _).le) },\n    { apply ha, exact (le_max_left _ _).trans hk, } },\n  have aux := @map_is_acyclic_of_acyclic _ _ _ _ _ _ this B _,\n  { replace := (@is_acyclic.cond _ _ _ _ _ _ aux (i+1)).of_iso (homology_iso _ i (i+1) (i+1+1) _ _).symm,\n    rotate, { dsimp, refl }, { dsimp, refl },\n    dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d,\n      functor.map_homotopy_category_obj, functor.map_homological_complex_obj_d] at this,\n    replace := exact_of_homology_is_zero this,\n    let e := functor.map_biprod (preadditive_yoneda.obj B).right_op,\n    refine preadditive.exact_of_iso_of_exact' _ _ _ _ (e _ _) (e _ _) (e _ _) _ _ this;\n    dsimp only [e, functor.map_biprod_hom],\n    all_goals\n    { ext,\n      { simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_fst,\n          eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,\n          homological_complex.X_eq_to_iso_refl, category.comp_id, dite_eq_ite, if_true,\n          biprod.lift_fst, biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero],\n        simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,\n          biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,\n          ← op_comp, ← category_theory.functor.map_comp, biprod.lift_fst],\n        simp only [biprod.desc_eq, comp_zero, add_zero, preadditive.comp_neg,\n          category_theory.op_neg, functor.map_neg, op_comp, category_theory.functor.map_comp],\n        refl },\n      { simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_snd,\n          eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,\n          functor.map_homological_complex_map_f, homological_complex.X_eq_to_iso_refl,\n          category.comp_id, dite_eq_ite, if_true, biprod.lift_snd, biprod.lift_desc],\n        simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,\n          biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,\n          ← op_comp, ← category_theory.functor.map_comp, biprod.lift_snd],\n        simp only [biprod.desc_eq, op_add, functor.map_neg, functor.map_add, op_comp,\n          category_theory.functor.map_comp],\n        refl } } },\n  { clear i, intros k i hi,\n    let e := functor.map_biprod ((Ext' i).flip.obj B).right_op\n      (P.val.as.X (k + 1)) ((of' C).val.as.X k),\n    refine is_zero.of_iso (is_zero.unop _) e.symm.unop,\n    refine category_theory.limits.is_zero.biprod _ _,\n    { simp only [functor.right_op_obj, functor.flip_obj_obj, is_zero_op],\n      exact acyclic_of_projective (P.val.as.X (k + 1)) B i hi, },\n    { exact (hC k _ hi).op, }, },\nend\n\ndef Ext_compute_with_acyclic_aux₃\n  (B : 𝓐)\n  (i : ℤ) :\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _).obj C).unop.homology (-i) ⟶\n  (((preadditive_yoneda.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj\n  ((of' C).replace).val.as.op).homology (-i) :=\nby apply (homotopy_category.homology_functor Ab _ (-i)).map\n  (((homotopy_category.quotient _ _).map\n    ((Ext_compute_with_acyclic_HomB B).map (of' C).π.out).unop))\n\n-- Are these two lemmas useful? Do we have them?\nlemma functor.map_unop {𝓐 : Type*} [category 𝓐] {𝓑 : Type*}\n  [category 𝓑] (F : 𝓐 ⥤ 𝓑) {a₁ a₂ : 𝓐ᵒᵖ}\n  (f : a₁ ⟶ a₂) : F.map f.unop = (F.op.map f).unop :=\nrfl\n\nlemma functor.map_map' {𝓐 : Type*} [category 𝓐] {𝓑 : Type*}\n  [category 𝓑] {𝓒 : Type*} [category 𝓒] (F : 𝓐 ⥤ 𝓑) (G : 𝓑 ⥤ 𝓒) {a₁ a₂ : 𝓐}\n  (φ : a₁ ⟶ a₂) : G.map (F.map φ) = (F ⋙ G).map φ :=\nrfl\n\nnamespace Ext_compute_with_acyclic_aux₃_naturality_helpers\n\nvariables (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ)\n\nlemma helper₁ (h1 h2) :\n  homology.lift ((unop\n    (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n      (complex_shape.up ℤ)).obj C₁))).d_to (-i))\n  ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₁))).d_from (-i)) h1\n  (kernel.ι ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₂))).d_from (-i)) ≫\n    (homological_complex.unop_functor.right_op.map\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).map f)).unop.f (-i) ≫ cokernel.π\n    ((unop (homological_complex.unop_functor.right_op.obj\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex\n    (complex_shape.up ℤ)).obj C₁))).d_to (-i))) h2 =\n  kernel.lift _ (kernel.ι _ ≫ (preadditive_yoneda.obj B).map (f.f (-i)).op) begin\n    rw category.assoc,\n    have aux := ((((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map f)).unop.comm_from (-i),\n    dsimp at aux,\n    erw [aux, kernel.condition_assoc, zero_comp],\n  end ≫\n  homology.π' _ _ _ :=\nbegin\n  apply homology.hom_to_ext,\n  rw [category.assoc, homology.π'_ι, kernel.lift_ι_assoc,\n    homology.lift_ι],\n  refl,\nend\n\nlemma helper₂ (h1 h2) :\n  homology.lift ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_to (-i))\n  ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_from (-i))\n  h1\n  (kernel.ι ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).val.as)).d_from (-i)) ≫\n     ((Ext_compute_with_acyclic_HomB B).map (quot.out (of' C₂).π)).unop.f (-i) ≫\n       cokernel.π ((unop ((Ext_compute_with_acyclic_HomB B).obj (of' C₂).replace.val.as)).d_to (-i)))\n  h2 =\n  kernel.lift _ (kernel.ι _ ≫ begin\n    apply homological_complex.hom.f,\n    exact (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map (of' C₂).π.out).unop,\n  end) begin\n    rw category.assoc,\n    have := (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map (of' C₂).π.out).unop.comm_from (-i),\n    erw [this, kernel.condition_assoc, zero_comp],\n  end ≫ homology.π' _ _ _ :=\nbegin\n  apply homology.hom_to_ext,\n  simp only [homology.lift_ι, category.assoc, homology.π'_ι, kernel.lift_ι_assoc],\n  refl,\nend\n\nend Ext_compute_with_acyclic_aux₃_naturality_helpers\n\nlemma Ext_compute_with_acyclic_aux₃_naturality\n  (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n    homological_complex.unop_functor.right_op ⋙ (_root_.homology_functor _ _ (-i)).op).map f).unop\n    ≫ Ext_compute_with_acyclic_aux₃ C₁ B i =\n  Ext_compute_with_acyclic_aux₃ C₂ B i ≫\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map\n      (bounded_homotopy_category.lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π).out).unop :=\nbegin\n  dsimp only [Ext_compute_with_acyclic_aux₃, functor.comp_map,\n    _root_.homology_functor, functor.op, homological_complex.hom.sq_from,\n    homological_complex.hom.sq_to, homotopy_category.quotient,\n    homotopy_category.homology_functor, quotient.functor,\n    category_theory.quotient.lift, quot.lift_on, quiver.hom.unop_op],\n  simp_rw homology.map_eq_desc'_lift_left,\n  apply homology.hom_from_ext,\n  simp only [category.assoc, homology.π'_desc'_assoc],\n  slice_rhs 1 2\n  { erw homology.π'_desc' },\n  dsimp only [arrow.hom_mk_left],\n  rw Ext_compute_with_acyclic_aux₃_naturality_helpers.helper₁,\n  conv_rhs { rw Ext_compute_with_acyclic_aux₃_naturality_helpers.helper₂ },\n  simp only [category.assoc],\n  slice_lhs 2 3\n  { erw homology.π'_desc' },\n  slice_rhs 2 3\n  { erw homology.π'_desc' },\n  apply homology.hom_to_ext,\n  slice_lhs 2 3\n  { erw homology.lift_ι },\n  slice_rhs 2 3\n  { erw homology.lift_ι },\n  slice_lhs 1 2\n  { erw kernel.lift_ι },\n  slice_rhs 1 2\n  { erw kernel.lift_ι },\n  simp only [category.assoc],\n  dsimp only [Ext_compute_with_acyclic_HomB, functor.comp_map],\n  change _ ≫ (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op).map f).unop.f _ ≫ _ = _,\n  slice_lhs 2 3\n  { simp only [← homological_complex.comp_f, ← unop_comp],\n    simp only [functor.comp_map, ← functor.map_comp] },\n  slice_rhs 2 3\n  { simp only [← homological_complex.comp_f, ← unop_comp, ← functor.map_comp] },\n  apply homotopy.kernel_ι_comp_comp_cokernel_π_of_homotopy,\n  apply homotopy.homotopy_unop_functor_right_op_map_unop_of_homotopy,\n  apply functor.map_homotopy,\n  suffices : (lift ((of' C₁).π ≫ of'_hom f) (of' C₂).π) ≫ (of' C₂).π =\n    (of' C₁).π ≫ of'_hom f,\n  { apply homotopy_category.homotopy_of_eq,\n    convert this.symm,\n    { simpa only [functor.map_comp, quotient_map_out], },\n    { simpa only [functor.map_comp, quotient_map_out], } },\n  simp,\nend\n\n\nlemma Ext_compute_with_acyclic_aux₃_is_iso\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) : is_iso (Ext_compute_with_acyclic_aux₃ C B i) :=\nbegin\n  haveI := Ext_compute_with_acyclic_is_quasi_iso_aux C B hC i,\n  apply is_quasi_iso.cond,\nend\n\ndef Ext_compute_with_acyclic\n  (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  ((Ext i).obj (op $ of' C)).obj ((single _ 0).obj B) ≅\n  (((preadditive_yoneda.obj B).right_op.map_homological_complex _).obj C).unop.homology (-i) :=\nExt_compute_with_acyclic_aux₁ C B i ≪≫\nExt_compute_with_acyclic_aux₂ _ _ _ ≪≫\nbegin\n  haveI := Ext_compute_with_acyclic_aux₃_is_iso C B hC i,\n  exact (as_iso (Ext_compute_with_acyclic_aux₃ C B i)).symm,\nend\n\n/-\ndef homological_complex.single_iso (B : 𝓐) {i j : ℤ} (h : j = i) :\n  ((homological_complex.single _ (complex_shape.up ℤ) i).obj B).X j ≅ B :=\neq_to_iso (if_pos h)\n\n/-- The morphism in `Ab` which eats a morphism of complexes `C ⟶ B[-i]`\n  and returns an element of Extⁱ(C,B[0]). -/\ndef Ext_compute_with_acyclic_inv_eq_aux (B) (i) :\n  AddCommGroup.of (C ⟶ (homological_complex.single _ _ (-i)).obj B) ⟶\n  ((Ext i).obj (op (of' C))).obj ((single 𝓐 0).obj B) :=\n{ to_fun := λ f, (of' C).π ≫ begin\n    refine (homotopy_category.quotient _ _).map _,\n    refine _ ≫ (homological_complex.single_shift _ _).inv.app _,\n    refine (f : C ⟶ (homological_complex.single 𝓐 (complex_shape.up ℤ) (-i)).obj B) ≫ _,\n    refine eq_to_hom _,\n    simp,\n  end,\n  map_zero' := begin\n    simp only [zero_comp, functor.map_zero, comp_zero],\n  end,\n  map_add' := begin\n    intros,\n    simp only [preadditive.add_comp, functor.map_add, preadditive.comp_add],\n  end }\n\nlemma iso_equiv_inv_apply_eq_zero_cancel\n  {A B C : AddCommGroup} {a : A} {f : A ⟶ B} {e : C ≅ B}\n  (h : (f ≫ e.inv) a = 0) : f a = 0 :=\nbegin\n  rw comp_apply at h,\n  apply_fun e.hom at h,\n  simpa using h,\nend\n\n-- Note: in the application of the below, j = -i\n/-- The construction which given something in the kernel of `(Cⱼ ⟶ B) ⟶ (Cⱼ-₁ ⟶ B)`\n  constructs a morphism of complexes from C to the \"skyscraper complex\" B[j]. -/\ndef kernel_yoneda_complex_to_morphism_to_single (B : 𝓐) (j : ℤ) :\n-- we're going from the kernel of `(C.X j ⟶ B) ⟶ (C.X_prev j ⟶ B)` (it's actually `C.symm.X_next`)\nkernel ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n  (homological_complex.op C)).d_from j)\n  -- to the additive group of homs\n  ⟶ AddCommGroup.of (\n  -- from C to B j\n  C ⟶ (homological_complex.single 𝓐 (complex_shape.up ℤ) j).obj B) :=\n{ to_fun := λ f, { f := λ k,\n  if hk : k = j then\n    (eq_to_hom (by rw hk) : C.X k ⟶ C.X j) ≫\n    (kernel.ι ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n                (homological_complex.op C)).d_from j) f : C.X j ⟶ B) ≫\n    (homological_complex.single_obj_X_self 𝓐 (complex_shape.up ℤ) j B).inv ≫\n    eq_to_hom (by rw hk)\n  else 0,\n    comm' := λ i k, begin\n      split_ifs with hij hkj hkj,\n      { subst hij, subst hkj, simp {contextual := tt}, },\n      { simp only [homological_complex.single_obj_d, comp_zero, eq_self_iff_true,\n          implies_true_iff] },\n      { subst hkj,\n        set g := ((kernel.ι ((((preadditive_yoneda.obj B).map_homological_complex\n                            (complex_shape.up ℤ).symm).obj\n                  (homological_complex.op C)).d_from k)) f) with hg,\n        simp only [complex_shape.up_rel, zero_comp, eq_to_hom_refl,\n          homological_complex.single_obj_X_self_inv, category.comp_id, category.id_comp],\n        rintro hik, clear hij,\n        have := kernel.condition ((((preadditive_yoneda.obj B).map_homological_complex _).obj\n          (homological_complex.op C)).d_from k),\n        replace this := congr_hom this f,\n        rw [comp_apply, ← hg, AddCommGroup.zero_apply] at this,\n        rw ← category.assoc,\n        symmetry,\n        convert zero_comp,\n        have foo : (complex_shape.up ℤ).symm.rel k i := hik,\n        rw homological_complex.d_from_eq _ foo at this,\n        exact iso_equiv_inv_apply_eq_zero_cancel this, },\n      { simp only [zero_comp, comp_zero, eq_self_iff_true, implies_true_iff]},\n    end },\n  map_zero' := by {simp only [map_zero, homological_complex.single_obj_X_self_inv,\n    eq_to_hom_trans, zero_comp, comp_zero, dite_eq_ite, if_t_t], refl },\n  map_add' := by { intros, ext, simp only [map_add, homological_complex.single_obj_X_self_inv,\n    eq_to_hom_trans, preadditive.add_comp, preadditive.comp_add, homological_complex.add_f_apply],\n    split_ifs,\n    { refl },\n    { refl },\n    { exact (add_zero _).symm, } } }\n\nlemma Ext_compute_with_acylic_inv_eq (B : 𝓐)\n  (hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))\n  (i : ℤ) :\n  (Ext_compute_with_acyclic _ B hC i).inv =\n  homology.desc' _ _ _\n  ( kernel_yoneda_complex_to_morphism_to_single C B (-i) ≫\n    Ext_compute_with_acyclic_inv_eq_aux C B i)\nadmit := admit\n\n-- Replacing some `End` with `cend` fixes my bracket pair colorizer!\n-- notation `cend` := category_theory.End\n\n-/\n\nlemma Ext_compute_with_acyclic_naturality (C₁ C₂ : cochain_complex 𝓐 ℤ)\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]\n  [((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]\n  (B : 𝓐)\n  (hC₁ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₁.X k)).obj B))\n  (hC₂ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₂.X k)).obj B))\n  (f : C₁ ⟶ C₂)\n  (i : ℤ) :\n  ((Ext i).flip.obj ((single _ 0).obj B)).map (quiver.hom.op $ of'_hom f) ≫\n    (Ext_compute_with_acyclic C₁ B hC₁ i).hom =\n  (Ext_compute_with_acyclic C₂ B hC₂ i).hom ≫\n    (((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙\n      homological_complex.unop_functor.right_op ⋙\n      (_root_.homology_functor _ _ (-i)).op).map f).unop :=\nbegin\n  dsimp only [Ext_compute_with_acyclic, iso.trans_hom],\n  slice_lhs 1 2\n  { erw Ext_compute_with_acyclic_aux₁_naturality },\n  slice_lhs 2 3\n  { rw Ext_compute_with_acyclic_aux₂_naturality },\n  simp only [category.assoc],\n  congr' 2,\n  dsimp only [iso.symm_hom, as_iso_inv],\n  rw [is_iso.eq_inv_comp, ← category.assoc, is_iso.comp_inv_eq],\n  symmetry,\n  apply Ext_compute_with_acyclic_aux₃_naturality,\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/acyclic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3019993425695904}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.preadditive\nimport category_theory.preadditive.additive_functor\nimport logic.equiv.transfer_instance\n\n/-!\n# If `C` is preadditive, `Cᵒᵖ` has a natural preadditive structure.\n\n-/\n\nopen opposite\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [preadditive C]\n\ninstance : preadditive Cᵒᵖ :=\n{ hom_group := λ X Y, equiv.add_comm_group (op_equiv X Y),\n  add_comp' := λ X Y Z f f' g,\n    congr_arg quiver.hom.op (preadditive.comp_add _ _ _ g.unop f.unop f'.unop),\n  comp_add' := λ X Y Z f g g',\n    congr_arg quiver.hom.op (preadditive.add_comp _ _ _ g.unop g'.unop f.unop), }\n\ninstance module_End_left {X : Cᵒᵖ} {Y : C} : module (End X) (unop X ⟶ Y) :=\n{ smul_add := λ r f g, preadditive.comp_add _ _ _ _ _ _,\n  smul_zero := λ r, limits.comp_zero,\n  add_smul := λ r s f, preadditive.add_comp _ _ _ _ _ _,\n  zero_smul := λ f, limits.zero_comp }\n\n@[simp] lemma unop_zero (X Y : Cᵒᵖ) : (0 : X ⟶ Y).unop = 0 := rfl\n@[simp] lemma unop_add {X Y : Cᵒᵖ} (f g : X ⟶ Y) : (f + g).unop = f.unop + g.unop := rfl\n@[simp] lemma op_zero (X Y : C) : (0 : X ⟶ Y).op = 0 := rfl\n@[simp] lemma op_add {X Y : C} (f g : X ⟶ Y) : (f + g).op = f.op + g.op := rfl\n\nvariables {C} {D : Type*} [category D] [preadditive D]\n\ninstance functor.op_additive (F : C ⥤ D) [F.additive] : F.op.additive := {}\n\ninstance functor.right_op_additive (F : Cᵒᵖ ⥤ D) [F.additive] : F.right_op.additive := {}\n\ninstance functor.left_op_additive (F : C ⥤ Dᵒᵖ) [F.additive] : F.left_op.additive := {}\n\ninstance functor.unop_additive (F : Cᵒᵖ ⥤ Dᵒᵖ) [F.additive] : F.unop.additive := {}\n\nend category_theory\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/preadditive/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5, "lm_q1q2_score": 0.30196591686297913}}
{"text": "/-\nFile: signature_recover_public_key_call_recover_public_key_soundness.lean\n\nAutogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.hoare\nimport .signature_recover_public_key_code\nimport ..signature_recover_public_key_spec\nimport .signature_recover_public_key_recover_public_key_soundness\nopen tactic\n\nopen starkware.cairo.common.cairo_secp.verification.signature_recover_public_key\nopen starkware.cairo.common.cairo_secp.signature\nopen starkware.cairo.common.math\nopen starkware.cairo.common.cairo_secp.field\nopen starkware.cairo.common.cairo_secp.ec\nopen starkware.cairo.common.cairo_secp.bigint\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\nvariable  mem : F → F\nvariable  σ : register_state F\n\n/- starkware.cairo.common.cairo_secp.verification.signature_recover_public_key.call_recover_public_key autogenerated soundness theorem -/\n\ntheorem auto_sound_call_recover_public_key\n    -- arguments\n    (range_check_ptr : F) (msg_hash r s : BigInt3 F) (v : F)\n    -- code is in memory at σ.pc\n    (h_mem : mem_at mem code_call_recover_public_key σ.pc)\n    -- all dependencies are in memory\n    (h_mem_0 : mem_at mem code_assert_nn (σ.pc  - 903))\n    (h_mem_1 : mem_at mem code_assert_le (σ.pc  - 899))\n    (h_mem_2 : mem_at mem code_assert_nn_le (σ.pc  - 894))\n    (h_mem_3 : mem_at mem code_bigint_mul (σ.pc  - 885))\n    (h_mem_4 : mem_at mem code_nondet_bigint3 (σ.pc  - 871))\n    (h_mem_5 : mem_at mem code_unreduced_mul (σ.pc  - 859))\n    (h_mem_6 : mem_at mem code_unreduced_sqr (σ.pc  - 839))\n    (h_mem_7 : mem_at mem code_verify_zero (σ.pc  - 823))\n    (h_mem_8 : mem_at mem code_is_zero (σ.pc  - 800))\n    (h_mem_9 : mem_at mem code_reduce (σ.pc  - 764))\n    (h_mem_10 : mem_at mem code_validate_reduced_field_element (σ.pc  - 751))\n    (h_mem_11 : mem_at mem code_ec_negate (σ.pc  - 711))\n    (h_mem_12 : mem_at mem code_compute_doubling_slope (σ.pc  - 695))\n    (h_mem_13 : mem_at mem code_compute_slope (σ.pc  - 651))\n    (h_mem_14 : mem_at mem code_ec_double (σ.pc  - 627))\n    (h_mem_15 : mem_at mem code_fast_ec_add (σ.pc  - 554))\n    (h_mem_16 : mem_at mem code_ec_add (σ.pc  - 467))\n    (h_mem_17 : mem_at mem code_ec_mul_inner (σ.pc  - 411))\n    (h_mem_18 : mem_at mem code_ec_mul (σ.pc  - 310))\n    (h_mem_19 : mem_at mem code_get_generator_point (σ.pc  - 230))\n    (h_mem_20 : mem_at mem code_div_mod_n (σ.pc  - 217))\n    (h_mem_21 : mem_at mem code_get_point_from_x (σ.pc  - 152))\n    (h_mem_22 : mem_at mem code_recover_public_key (σ.pc  - 86))\n    -- input arguments on the stack\n    (hin_range_check_ptr : range_check_ptr = mem (σ.fp - 13))\n    (hin_msg_hash : msg_hash = cast_BigInt3 mem (σ.fp - 12))\n    (hin_r : r = cast_BigInt3 mem (σ.fp - 9))\n    (hin_s : s = cast_BigInt3 mem (σ.fp - 6))\n    (hin_v : v = mem (σ.fp - 3))\n    -- conclusion\n  : ensures_ret mem σ (λ κ τ,\n      ∃ μ ≤ κ, rc_ensures mem (rc_bound F) μ (mem (σ.fp - 13)) (mem $ τ.ap - 7)\n        (spec_call_recover_public_key mem κ range_check_ptr msg_hash r s v (mem (τ.ap - 7)) (cast_EcPoint mem (τ.ap - 6)))) :=\nbegin\n  apply ensures_of_ensuresb, intro νbound,\n  have h_mem_rec := h_mem,\n  unpack_memory code_call_recover_public_key at h_mem with ⟨hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13⟩,\n  -- tail function call\n  step_assert_eq hpc0 with arg0,\n  step_assert_eq hpc1 with arg1,\n  step_assert_eq hpc2 with arg2,\n  step_assert_eq hpc3 with arg3,\n  step_assert_eq hpc4 with arg4,\n  step_assert_eq hpc5 with arg5,\n  step_assert_eq hpc6 with arg6,\n  step_assert_eq hpc7 with arg7,\n  step_assert_eq hpc8 with arg8,\n  step_assert_eq hpc9 with arg9,\n  step_assert_eq hpc10 with arg10,\n  step_sub hpc11 (auto_sound_recover_public_key mem _ range_check_ptr msg_hash r s v _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _),\n  { rw hpc12, norm_num2, exact h_mem_22 },\n  { rw hpc12, norm_num2, exact h_mem_0 },\n  { rw hpc12, norm_num2, exact h_mem_1 },\n  { rw hpc12, norm_num2, exact h_mem_2 },\n  { rw hpc12, norm_num2, exact h_mem_3 },\n  { rw hpc12, norm_num2, exact h_mem_4 },\n  { rw hpc12, norm_num2, exact h_mem_5 },\n  { rw hpc12, norm_num2, exact h_mem_6 },\n  { rw hpc12, norm_num2, exact h_mem_7 },\n  { rw hpc12, norm_num2, exact h_mem_8 },\n  { rw hpc12, norm_num2, exact h_mem_9 },\n  { rw hpc12, norm_num2, exact h_mem_10 },\n  { rw hpc12, norm_num2, exact h_mem_11 },\n  { rw hpc12, norm_num2, exact h_mem_12 },\n  { rw hpc12, norm_num2, exact h_mem_13 },\n  { rw hpc12, norm_num2, exact h_mem_14 },\n  { rw hpc12, norm_num2, exact h_mem_15 },\n  { rw hpc12, norm_num2, exact h_mem_16 },\n  { rw hpc12, norm_num2, exact h_mem_17 },\n  { rw hpc12, norm_num2, exact h_mem_18 },\n  { rw hpc12, norm_num2, exact h_mem_19 },\n  { rw hpc12, norm_num2, exact h_mem_20 },\n  { rw hpc12, norm_num2, exact h_mem_21 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_msg_hash, hin_r, hin_s, hin_v] },\n    try { dsimp [cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_msg_hash, hin_r, hin_s, hin_v] },\n      try { dsimp [cast_BigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_msg_hash, hin_r, hin_s, hin_v] },\n      try { dsimp [cast_BigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_msg_hash, hin_r, hin_s, hin_v] },\n      try { dsimp [cast_BigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_msg_hash, hin_r, hin_s, hin_v] },\n    try { dsimp [cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3, arg4, arg5, arg6, arg7, arg8, arg9, arg10] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  intros κ_call13 ap13 h_call13,\n  rcases h_call13 with ⟨rc_m13, rc_mle13, hl_range_check_ptr₁, h_call13⟩,\n  step_ret hpc13,\n  generalize' hr_rev_range_check_ptr₁: mem (ap13 - 7) = range_check_ptr₁,\n  have htv_range_check_ptr₁ := hr_rev_range_check_ptr₁.symm, clear hr_rev_range_check_ptr₁,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10] at hl_range_check_ptr₁ },\n  rw [←htv_range_check_ptr₁, ←hin_range_check_ptr] at hl_range_check_ptr₁,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6 ,arg7 ,arg8 ,arg9 ,arg10] at h_call13 },\n  rw [hin_range_check_ptr] at h_call13,\n  clear arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10,\n  -- finish\n  step_done, use_only [rfl, rfl],\n  -- range check condition\n  use_only (rc_m13+0+0), split,\n  linarith [rc_mle13],\n  split,\n  { arith_simps,\n    rw [←htv_range_check_ptr₁, hl_range_check_ptr₁, hin_range_check_ptr],\n    try { arith_simps, refl <|> norm_cast }, try { refl } },\n  intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },\n  have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,\n  -- Final Proof\n  -- user-provided reduction\n  suffices auto_spec: auto_spec_call_recover_public_key mem _ range_check_ptr msg_hash r s v _ _,\n  { apply sound_call_recover_public_key, apply auto_spec },\n  -- prove the auto generated assertion\n  dsimp [auto_spec_call_recover_public_key],\n  try { norm_num1 }, try { arith_simps },\n  use_only [κ_call13],\n  have rc_h_range_check_ptr₁ := range_checked_offset' rc_h_range_check_ptr,\n  have rc_h_range_check_ptr₁' := range_checked_add_right rc_h_range_check_ptr₁, try { norm_cast at rc_h_range_check_ptr₁' },\n  have spec13 := h_call13 rc_h_range_check_ptr',\n  rw [←hin_range_check_ptr] at spec13,\n  try { dsimp at spec13, arith_simps at spec13 },\n  use_only [spec13],\n  try { linarith },\nend\n\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_call_recover_public_key_soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.30191745877161347}}
{"text": "-- Copyright (c) Microsoft Corporation. All rights reserved.\n-- Licensed under the MIT license.\n\nimport ..smtexpr\nimport ..smtcompile\nimport ..bitvector\nimport .spec\nimport .lemmas_basic\nimport tactic.interactive\n\n\nnamespace spec\n\nopen smt\nopen irsem\nopen spec\n\n-- unfold\nmeta def unfold_ops: tactic unit :=\n  `[ try { unfold has_eq.eq },\n     try { unfold has_comp.eq },\n     try { unfold has_ne.ne },\n     try { unfold has_comp.ne },\n     try { unfold has_and.and },\n     try { unfold has_ult.ult },\n     try { unfold has_comp.ult },\n     try { unfold has_ule.ule },\n     try { unfold has_comp.ule },\n     try { unfold has_ugt.ugt },\n     try { unfold has_comp.ult },\n     try { unfold has_uge.uge },\n     try { unfold has_comp.ule },\n     try { unfold has_slt.slt },\n     try { unfold has_comp.slt },\n     try { unfold has_sle.sle },\n     try { unfold has_comp.sle },\n     try { unfold has_sgt.sgt },\n     try { unfold has_comp.slt },\n     try { unfold has_sge.sge },\n     try { unfold has_comp.sle } ]\n\nmeta def unfold_lops_at_all: tactic unit :=\n  `[ try { unfold has_eq.eq at * },\n     try { unfold has_comp.eq at * },\n     try { unfold has_ne.ne at * },\n     try { unfold has_comp.ne at * },\n     try { unfold has_and.and at * },\n     try { unfold has_or.or at * } ]\n\n-- Constant sbitvec is never optimized.\nlemma sbitvec_cons_notoptimized: ∀ sz z,\noptimize_add_zero (sbitvec.const sz z) = (sbitvec.const sz z) :=\nbegin\n  intros sz z,\n  unfold optimize_add_zero\nend\n\n-- bitvector eq\n\nlemma bveq: ∀ {sz:size} {b1 b2:bitvector sz},\n  bitvector.eq b1 b2 = tt ↔ b1 = b2\n:= begin\n  intros,\n  split,\n  {\n    intros H,\n    cases b1, cases b2,\n    unfold bitvector.eq at H,\n    simp at H,\n    assumption\n  },{\n    intros H,\n    rw H, unfold bitvector.eq, simp\n  }\nend\n\nlemma bvneq: ∀ (sz:size) (b1 b2:bitvector sz),\n  bitvector.eq b1 b2 = ff ↔ b1 ≠ b2\n:= begin\n  intros sz b1 b2,\n  split,\n  { -- →\n    intros H,\n    cases b1,\n    cases b2,\n    unfold bitvector.eq at H,\n    simp at H,\n    apply H\n  }, { -- ←\n    intros H,\n    cases b1,\n    cases b2,\n    unfold bitvector.eq,\n    simp,\n    apply H\n  }\nend\n\nlemma bvneq_l : ∀ {sz:size} {b1 b2:bitvector sz},\n  bitvector.eq b1 b2 = ff → b1 ≠ b2\n:= begin intros, rw ← bvneq, assumption end\n\nlemma bvneq_r : ∀ {sz:size} {b1 b2:bitvector sz},\n  b1 ≠ b2 → bitvector.eq b1 b2 = ff\n:= begin intros, rw bvneq, assumption end\n\nlemma bvneq_from_z: ∀ (sz:size) (n m:int),\n  n ≠ m → 0 ≤ n ∧ n < int.of_nat (2^sz.val)\n        ∧ 0 ≤ m ∧ m < int.of_nat (2^sz.val) →\n  bitvector.eq (uint_like.from_z sz n)\n               (uint_like.from_z sz m) = ff\n:= begin\n  intros sz n m H H1,\n  cases H1 with H1 H2,\n  cases H2 with H2 H3,\n  cases H3 with H3 H4,\n  unfold uint_like.from_z,\n  cases n; cases m,\n  {\n    unfold bitvector.of_int,\n    unfold bitvector.eq,\n    simp,\n    intros H0,\n    injection H0,\n    have Hz := int.coe_nat_lt_coe_nat_iff,\n    unfold_coes at Hz,\n    rw Hz at H2,\n    rw Hz at H4,\n    rw nat.mod_eq_of_lt at h_1; try { assumption },\n    rw nat.mod_eq_of_lt at h_1; try { assumption },\n    apply H, congr, assumption,\n  },\n  { cases H3 },\n  { cases H1 },\n  { cases H3 }\nend\n\nlemma bvult_mod_sz: ∀ (sz:size) (eb:bitvector sz),\n  bitvector.ult (eb%u uint_like.from_z sz ↑sz)\n                (bitvector.of_int sz (int.of_nat (sz.val))) = tt\n:= begin\n  intros,\n  cases sz, unfold_coes, unfold uint_like.from_z, unfold bitvector.ult,\n  unfold bitvector.of_int, cases eb, simp, unfold has_umod.umod,\n  unfold uint_like.urem, unfold bitvector.urem, unfold_coes, simp,\n  rw nat.mod_pow2, rw nat.mod_eq_of_lt,\n  {\n    apply nat.mod_lt, assumption,\n  },\n  {\n    apply lt_trans,\n    { apply nat.mod_lt, assumption },\n    apply nat.lt_pow2\n  }\nend\n\n-- Lemmas regarding bvequiv\n\nlemma bvequiv_of_int: ∀ sz x,\n  bv_equiv (sbitvec.of_int sz x) (bitvector.of_int sz x)\n:= begin\n  intros,\n  cases x; unfold sbitvec.of_int,\n  { unfold bitvector.of_int,\n    apply bv_equiv.const },\n  {\n    unfold bitvector.of_int,\n    apply bv_equiv.const\n  }\nend\n\nlemma bvequiv_from_z: ∀ (sz:size) n,\n  bv_equiv (uint_like.from_z sz n)\n           (uint_like.from_z sz n)\n:= begin\n  intros,\n  unfold uint_like.from_z, apply bvequiv_of_int\nend\n\nlemma bvequiv_urem: ∀ (sz:size) (sb:sbitvec sz) eb\n    (HA:bv_equiv sb eb),\n  bv_equiv (sb%u uint_like.from_z sz sz.val)\n           (eb%u uint_like.from_z sz sz.val)\n:= begin\n  intros,\n  apply bv_equiv.urem, assumption, apply bvequiv_from_z,\n  { cases sz, unfold_coes, simp, unfold uint_like.from_z, unfold bitvector.ne,\n    unfold bitvector.of_int, simp,\n    intros H, injection H, simp at h_1, rw nat.mod_pow2 at h_1,\n    rw h_1 at sz_property, cases sz_property\n  },\nend\n\nlemma bvequiv_never_var: ∀ sz s b,\n  ¬ bv_equiv (sbitvec.var sz s) b\n:= begin\n  intros,intro H, cases H\nend\n\nlemma sbitvec_of_int_const: ∀ sz (b:bitvector sz),\n  sbitvec.of_int sz (bitvector.to_int b) = sbitvec.const sz b\n:= begin\n  intros,\n  cases b with b Hb,\n  cases sz with sz Hsz,\n  unfold_coes at *,\n  unfold bitvector.to_int,\n  simp,\n  simp at Hb,\n  have Hlt: decidable (b < 2^(sz - 1)), apply_instance,\n  cases Hlt,\n  {\n    rw if_neg,\n    unfold sbitvec.of_int,\n    simp at *,\n    have Hb' := nat.succ_le_of_lt Hb,\n    have Hb'' := nat.sub_le_sub_right Hb' b,\n    rw nat.succ_sub at Hb'',\n    rw nat.sub_self at Hb'',\n    generalize Hsz:2^sz - b = g,\n    rw Hsz at *,\n    rw nat.sub_sub,\n    have Hg' : g - 1 + 1 = g,\n    { rw nat.sub_add_cancel, apply Hb'' },\n    rw Hg', rw ← Hsz,\n    rw nat.sub_sub_self,\n    rw nat.mod_eq_of_lt,\n    assumption, apply le_of_lt, assumption, constructor, assumption\n  },\n  {\n    rw if_pos,\n    unfold_coes,\n    unfold sbitvec.of_int,\n    simp,\n    rw nat.mod_eq_of_lt, assumption, assumption\n  }\nend\n\n/-\n - Theorems about overflow.\n -/\n\nlemma add_overflow_chk: ∀ (sz:size) isnsw (sa sb:sbitvec sz) (ea:bitvector sz) eb,\n  bv_equiv sa ea → bv_equiv sb eb →\n  b_equiv (sbitvec.overflow_chk @sbitvec.add size.one isnsw sa sb)\n          (bitvector.add_overflows isnsw ea eb)\n:= sorry\n\nlemma sub_overflow_chk: ∀ (sz:size) isnsw (sa sb:sbitvec sz) (ea:bitvector sz) eb,\n  bv_equiv sa ea → bv_equiv sb eb →\n  b_equiv (sbitvec.overflow_chk @sbitvec.sub size.one isnsw sa sb)\n          (bitvector.sub_overflows isnsw ea eb)\n:= sorry\n\nlemma mul_overflow_chk: ∀ (sz:size) isnsw (sa sb:sbitvec sz) (ea:bitvector sz) eb,\n  bv_equiv sa ea → bv_equiv sb eb →\n  b_equiv (sbitvec.overflow_chk @sbitvec.mul sz isnsw sa sb)\n          (bitvector.mul_overflows isnsw ea eb)\n:= sorry\n\nlemma shl_overflow_chk: ∀ (sz:size) isnsw (sa sb:sbitvec sz) (ea:bitvector sz) eb,\n  bv_equiv sa ea → bv_equiv sb eb →\n  b_equiv (sbitvec.shl_overflow_chk sz isnsw sa sb)\n          (bitvector.shl_overflows isnsw ea eb)\n:= sorry\n\n\nlemma to_bool_prop: ∀ p {dp:decidable p}, @to_bool p dp = tt → p\n:= begin\n  intros,\n  cases dp,\n  unfold to_bool at a, injection a\nend\n\n-- Same to poisonty_and_tt, but in the future poison may not be boolty.\nlemma boolty_and_tt: ∀ (x:irsem_exec.boolty),\n    bool_like.and tt x = x\n:= begin intros, cases x;\n  { delta bool_like.and, rw bool_is_bool_like, simp }\nend\n\nlemma boolty_or_tt2_op: ∀ (x:irsem_exec.boolty),\n    x |b tt = tt\n:= begin intros, cases x; refl end\n\nlemma boolty_or_inj: ∀ {x y:irsem_exec.boolty}\n    (H:bool_like.or x y = tt),\n  x = tt ∨ y = tt\n:= begin intros, cases x; cases y, \n  cases H,\n  any_goals { right, refl },\n  any_goals { left, refl }\nend\n\nlemma boolty_and_inj: ∀ {x y:irsem_exec.boolty}\n    (H:bool_like.and x y = tt),\n  x = tt ∧ y = tt\n:= begin intros, cases x; cases y; cases H, split; simp end\n\n@[simp]\nlemma b2p_id_exec: ∀ (b:irsem_exec.boolty),\n  b2p irsem_exec b = b\n:= begin intros, cases b; refl end\n\n\nlemma b_equiv_ite: ∀ (sb:irsem_smt.boolty) b\n    (H:b_equiv sb b),\n  b_equiv ((has_ite.ite sb (bool_like.tt irsem_smt.poisonty)\n        (bool_like.ff irsem_smt.poisonty)):irsem_smt.poisonty) b\n:= begin\n  intros,\n  have HB: (b = cond b tt ff), cases b; simp,\n  rw HB, apply b_equiv.ite, assumption, constructor, constructor\nend\n\nlemma b_equiv_and_tt: ∀ (s1 s2:irsem_smt.boolty) b\n    (H:b_equiv s1 b)\n    (H':b_equiv s2 tt),\n  b_equiv (bool_like.and s1 s2) b\n:= begin\n  intros,\n  have HB: (b = band b tt),\n  { cases b, simp, simp },\n  rw HB, apply b_equiv.and1, assumption,\n  intros H2, apply H'\nend\n\nlemma b_equiv_and_ff: ∀ (s1 s2:irsem_smt.boolty)\n    (H:b_equiv s1 ff),\n  b_equiv (bool_like.and s1 s2) ff\n:= begin\n  intros,\n  have Hb': ∃ b', band ff b' = ff,\n  { apply exists.intro tt, refl },\n  cases Hb' with b' Hb',\n  rw ← Hb',\n  have Hff:(ff = bool_like.and ff b'),\n  { cases b', refl, refl },\n  rw Hff, apply b_equiv.and1, assumption,\n  intros H0, rw ← Hff at H0, cases H0\nend\n\nlemma b_equiv_and_ff2: ∀ (s1 s2:irsem_smt.boolty)\n    (H:b_equiv s2 ff),\n  b_equiv (bool_like.and s1 s2) ff\n:= begin\n  intros,\n  have Hb': ∃ b', band b' ff = ff,\n  { apply exists.intro tt, refl },\n  cases Hb' with b' Hb',\n  rw ← Hb',\n  have Hff:(ff = bool_like.and b' ff),\n  { cases b', refl, refl },\n  rw Hff, apply b_equiv.and2, rw ← Hff, apply H,\n  intros H0, rw ← Hff at H0, cases H0\nend\n\nlemma b_equiv_and: ∀ (s1 s2:irsem_smt.boolty) (b1 b2:irsem_exec.boolty)\n    (H1: b_equiv s1 b1) (H2:b_equiv s2 b2),\n  b_equiv (bool_like.and s1 s2) (bool_like.and b1 b2)\n:= begin\n  intros,\n  apply b_equiv.and1,\n  assumption, cases b1, intros H, cases H, intros H, assumption\nend\n\nlemma b_equiv_or: ∀ (s1 s2:irsem_smt.boolty) (b1 b2:irsem_exec.boolty)\n    (H1: b_equiv s1 b1) (H2:b_equiv s2 b2),\n  b_equiv (bool_like.or s1 s2) (bool_like.or b1 b2)\n:= begin\n  intros,\n  apply b_equiv.or1,\n  assumption, cases b1, intros H, assumption, intros H, cases H\nend\n\n\nlemma b_equiv_never_var: ∀ s b,\n  ¬ b_equiv (sbool.var s) b\n:= begin\n  intros,intro H, cases H\nend\n\n@[simp]\nlemma poisonty_tt: irsem_exec.pbl.tt = tt\n:= begin refl end\n\n@[simp]\nlemma poisonty_ff: irsem_exec.pbl.ff = ff\n:= begin refl end\n\n\n@[simp]\nlemma poisonty_and_tt: ∀ (x:irsem_exec.poisonty),\n    bool_like.and tt x = x\n:= begin intros, cases x;\n  { delta bool_like.and, rw bool_is_bool_like, simp }\nend\n\nlemma poisonty_and_tt_op: ∀ (x:irsem_exec.poisonty),\n    tt & x = x\n:= begin intros, apply poisonty_and_tt end\n\n@[simp]\nlemma poisonty_and_tt2: ∀ (x:irsem_exec.poisonty),\n    bool_like.and x tt = x\n:= begin intros, cases x;\n  { delta bool_like.and, rw irsem_exec.pbl, rw bool_is_bool_like, simp }\nend\n\nlemma poisonty_and_tt2_op: ∀ (x:irsem_exec.poisonty),\n    x & tt = x\n:= begin intros, apply poisonty_and_tt2 end\n\nlemma poisonty_and_ff: ∀ (x:irsem_exec.poisonty),\n    bool_like.and ff x = ff\n:= begin intros, cases x;\n  { delta bool_like.and, rw bool_is_bool_like, simp }\nend\n\nlemma poisonty_and_ff_op: ∀ (x:irsem_exec.poisonty),\n    ff&x = ff\n:= begin intros, unfold has_and.and, apply poisonty_and_ff end\n\nlemma poisonty_and_ff2: ∀ (x:irsem_exec.poisonty),\n    bool_like.and x ff = ff\n:= begin intros, cases x;\n  { delta bool_like.and, rw irsem_exec.pbl, rw bool_is_bool_like, simp }\nend\n\nlemma poisonty_and_ff2_op: ∀ (x:irsem_exec.poisonty),\n    x & ff = ff\n:= begin intros, unfold has_and.and, apply poisonty_and_ff2 end\n\nlemma poisonty_and_inj: ∀ {x y:irsem_exec.poisonty}\n    (H:bool_like.and x y = tt),\n  x = tt ∧ y = tt\n:= begin intros, cases x; cases y; cases H, split; simp end\n\nlemma poisonty_or_ff: ∀ (x:irsem_exec.poisonty),\n    bool_like.or ff x = x\n:= begin intros, cases x; refl end\n\nlemma poisonty_or_ff_op: ∀ (x:irsem_exec.poisonty),\n    ff |b x = x\n:= begin intros, apply poisonty_or_ff end\n\n\nlemma poisonty_or_inj: ∀ {x y:irsem_exec.poisonty}\n    (H:bool_like.or x y = tt),\n  x = tt ∨ y = tt\n:= begin intros, cases x; cases y, \n  cases H,\n  any_goals { right, refl },\n  any_goals { left, refl }\nend\n\nlemma poison_equiv_and: forall (sx:irsem_smt.poisonty) sy\n    (x:irsem_exec.poisonty) y\n    (HXEQ:poison_equiv sx x)\n    (HYEQ:poison_equiv sy y),\n  poison_equiv (bool_like.and sx sy) (bool_like.and x y)\n:= begin\n  intros,\n  cases HXEQ,\n  cases HYEQ,\n  constructor,\n  apply b_equiv_and; assumption\nend\n\nlemma poison_equiv_ite: forall (sc:irsem_smt.boolty) (sx sy:irsem_smt.poisonty)\n    (c:irsem_exec.boolty) (x y:irsem_exec.poisonty)\n    (HCEQ:b_equiv sc c)\n    (HXEQ:poison_equiv sx x)\n    (HYEQ:poison_equiv sy y),\n  poison_equiv (has_ite.ite sc sx sy) (has_ite.ite c x y)\n:= begin\n  intros,\n  cases HXEQ,\n  cases HYEQ,\n  constructor,\n  apply b_equiv.ite; assumption\nend\n\nlemma poison_equiv_and_ff: ∀ (s1 s2:irsem_smt.poisonty)\n    (H:poison_equiv s1 ff),\n  poison_equiv (bool_like.and s1 s2) ff\n:= begin\n  intros,\n  cases H, constructor, apply b_equiv_and_ff, assumption\nend\n\nlemma poison_equiv_and_ff2: ∀ (s1 s2:irsem_smt.poisonty)\n    (H:poison_equiv s2 ff),\n  poison_equiv (bool_like.and s1 s2) ff\n:= begin\n  intros,\n  cases H, constructor, apply b_equiv_and_ff2, assumption\nend\n\n\nlemma bop_poison_peel: ∀ {sz} {bopc} {flags} {ea} {eb}\n    (H:bop_poison_all irsem_exec sz bopc flags ea eb = tt),\n  bop_poison irsem_exec sz bopc ea eb = tt\n:= begin\n  unfold bop_poison_all,\n  intros,\n  induction flags,\n  { -- when flags = []\n    simp at H, assumption },\n  { -- when flags = flags_hd::flags_tl\n    simp at H,\n    have H':= boolty_and_inj H,\n    cases H', -- split ∧\n    have ANSW := flags_ih H'_left,\n    apply ANSW\n  }\nend\n\n\n-- Lemmas about vppair\n\nlemma val_equiv_eqsize: ∀ {sza szb:size}\n    {a:sbitvec sza} {ap:poisonty_smt} {b bp},\n  val_equiv (valty.ival sza a ap) (valty.ival szb b bp) →\n  sza = szb\n:= begin\n  intros,\n  rename a_1 H,\n  cases H,\n  { -- If val_equiv.poison\n    assumption },\n  { -- If val_equiv.concrete\n    refl }\nend\n\nlemma val_equiv_eqpoison: ∀ (sza szb:size)\n    (a:sbitvec sza) (ap:poisonty_smt) b bp,\n  val_equiv (valty.ival sza a ap) (valty.ival szb b bp) →\n  poison_equiv ap bp\n:= begin\n  intros,\n  rename a_1 H,\n  cases H; assumption\nend\n\nset_option pp.proofs true\n\nlemma bv_equiv_cast: ∀ {sz sz':size} (sb:sbitvec sz) (bv:bitvector sz) H0 H1\n    (H:bv_equiv sb bv)\n    (HSZ:sz = sz'),\n  @bv_equiv sz' (cast H0 sb) (cast H1 bv)\n:= begin\n  intros,\n  induction HSZ,\n  rw cast_eq,\n  rw cast_eq,\n  assumption\nend\n\nlemma valty_rwsize_smt: ∀ (sza szb:size) (a:irsem_smt.intty sza)\n    (HSZEQ:sza = szb),\n  valty.ival sza a = valty.ival szb (cast (by abstract eqH { rw HSZEQ }) a)\n:= begin\n  intros,\n  congr,\n  assumption,\n  induction (valty_rwsize_smt.eqH sza szb HSZEQ),\n  rw cast_eq\nend\n\nlemma valty_rwsize_exec: ∀ (sza szb:size) {a:irsem_exec.intty sza}\n    (HSZEQ:sza = szb) {ap},\n  valty.ival sza a ap = valty.ival szb (cast (by abstract eqH { rw HSZEQ }) a) ap\n:= begin\n  intros,\n  congr,\n  assumption,\n  induction (valty_rwsize_exec.eqH sza szb HSZEQ),\n  rw cast_eq\nend\n\nlemma sb_rwsize: ∀ (sza szb:size) (a:sbitvec sza)\n    (HSZEQ:sza = szb),\n  a = (cast (by abstract eqH {rw HSZEQ} ) a)\n:= begin\n  intros, rw cast_eq end\n\nlemma bv_rwsize: ∀ (sza szb:size) (a:bitvector sza)\n    (HSZEQ:sza = szb),\n  a = (cast (by abstract eqH {rw HSZEQ} ) a)\n:= begin\n  intros, rw cast_eq end\n\nlemma val_equiv_rwsize1: ∀ (sza sza' szb szb':size)\n    (a:irsem_smt.intty sza) (b:irsem_exec.intty szb) ap bp\n    (HSZEQ:sza = szb)\n    (H:val_equiv (valty.ival sza a ap)\n      (valty.ival sza (cast (by abstract eqH { rw HSZEQ }) b) bp)),\n  val_equiv (valty.ival sza a ap) (valty.ival szb b bp)\n:= begin\n  intros,\n  have H0: valty_rwsize_exec.eqH szb sza (eq.symm HSZEQ) =\n          val_equiv_rwsize1.eqH sza szb HSZEQ, refl,\n  rw ← H0 at H,\n  rw ← valty_rwsize_exec at H,\n  assumption\nend\n\nlemma val_equiv_rwsize2: ∀ (sza sza' szb szb':size)\n    (a:irsem_smt.intty sza) (b:irsem_exec.intty szb) ap bp\n    (HSZEQ1:sza = sza') (HSZEQ2:szb = szb')\n    (H:val_equiv (valty.ival sza (cast (by rw HSZEQ1) a) ap)\n      (valty.ival szb' (cast (by abstract eqH2 { rw HSZEQ2 }) b) bp)),\n  val_equiv (valty.ival sza a ap) (valty.ival szb b bp)\n:= begin\n  intros,\n  rw cast_eq at H,\n  have H0: valty_rwsize_exec.eqH szb szb' HSZEQ2 =\n          val_equiv_rwsize2.eqH2 szb szb' HSZEQ2, refl,\n  rw ← H0 at H,\n  rw ← valty_rwsize_exec at H,\n  assumption\nend\n\nlemma val_equiv_rwsize1': ∀ (sza szb:size)\n    (a:irsem_smt.intty sza) (b:irsem_exec.intty szb) ap bp\n    (HSZEQ:sza = szb)\n    (H:val_equiv (valty.ival sza a ap) (valty.ival szb b bp)),\n  val_equiv (valty.ival sza a ap)\n      (valty.ival sza (cast (by abstract eqH {rw HSZEQ}) b) bp)\n:= begin\n  intros, \n  have H0: valty_rwsize_exec.eqH szb sza (eq.symm HSZEQ) =\n          val_equiv_rwsize1'.eqH sza szb HSZEQ, refl,\n  rw ← H0,\n  rw ← valty_rwsize_exec,\n  assumption\nend\n\nlemma val_equiv_rwsize2': ∀ (sza sza' szb szb':size)\n    (a:irsem_smt.intty sza) (b:irsem_exec.intty szb) ap bp\n    (HSZEQ1:sza = sza') (HSZEQ2:szb = szb')\n    (H:val_equiv (valty.ival sza a ap) (valty.ival szb b bp)),\n  val_equiv (valty.ival sza' (cast (by abstract eqH1 { rw HSZEQ1 }) a) ap)\n      (valty.ival szb' (cast (by abstract eqH2 { rw HSZEQ2 }) b) bp)\n:= begin\n  intros,\n  have H0: valty_rwsize_exec.eqH szb szb' HSZEQ2 =\n          val_equiv_rwsize2'.eqH2 szb szb' HSZEQ2, refl,\n  have H1: valty_rwsize_smt.eqH sza sza' HSZEQ1 =\n          val_equiv_rwsize2'.eqH1 sza sza' HSZEQ1, refl,\n  rw ← H0,\n  rw ← H1,\n  rw ← valty_rwsize_exec,\n  rw ← valty_rwsize_smt,\n  assumption\nend\n\nlemma poison_bool_equiv: ∀ ap bp,\n  poison_equiv ap bp ↔ b_equiv ap bp\n:= begin\n  intros,\n  split,\n  { intros H, cases H, assumption },\n  { intros H, constructor, assumption }\nend\n\nlemma none_or_some_implies: ∀ {α β:Type} a b\n    (P1 P2:α → β → Prop)\n    (HP: ∀ a b, P1 a b → P2 a b)\n    (H:none_or_some a b P1),\n  none_or_some a b P2\n:= begin\n  intros,\n  unfold none_or_some at *,\n  cases a; cases b,  \n  { left, split; refl },\n  {\n    cases H,\n    cases H, cases H_right,\n    cases H, cases H_h, cases H_h_h, cases H_h_h_left\n  },\n  {\n    cases H,\n    cases H, cases H_left,\n    cases H, cases H_h, cases H_h_h, cases H_h_h_right,\n    cases H_h_h_right_left\n  },\n  {\n    cases H,\n    left, apply H,\n    right, cases H, cases H_h,\n    apply exists.intro H_w,\n    apply exists.intro H_h_w,\n    cases H_h_h with H1 H2,\n    cases H2 with H2 H3,\n    split, assumption,\n    split, assumption, apply HP, assumption\n  }\nend\n\nuniverse u\nlemma none_or_some_apply: ∀ {α β:Type} (a:α) (b:β) f, none_or_some (some a) (some b) f\n  ↔ f a b\n:= begin\n  intros,\n  unfold none_or_some,\n  split,\n  intros,\n  cases a_1,\n  cases a_1, cases a_1_left,\n  cases a_1, cases a_1_h, cases a_1_h_h,\n  cases a_1_h_h_left, cases a_1_h_h_right, cases a_1_h_h_right_left,\n  assumption,\n  intros,\n  right, existsi a, existsi b, split, refl, split, refl, assumption \nend\n\nlemma none_or_some_none: ∀ {α β:Type} {a} {b}\n    {P:α → β → Prop}\n    (H:none_or_some a b P)\n    (H1:a = none),\n  b = none\n:= begin\n  intros,\n  subst H1,\n  unfold none_or_some at H,\n  cases H,\n  cases H, assumption,\n  cases H, cases H_h, cases H_h_h, cases H_h_h_left\nend\n\nlemma none_or_some_none2: ∀ {α β:Type} {a} {b}\n    {P:α → β → Prop}\n    (H1:a = none) (H2:b = none),\n  none_or_some a b P\n:= begin\n  intros,\n  subst H1, subst H2,\n  unfold none_or_some,\n  left, split; refl\nend\n\nlemma none_or_some_some: ∀ {α β:Type} {a} {b} {a'}\n    {P:α → β → Prop}\n    (H:none_or_some a b P)\n    (H1:a = some a'),\n  ∃ b', b = some b'\n:= begin\n  intros,\n  subst H1,\n  unfold none_or_some at H,\n  cases H,\n  cases H, cases H_left,\n  cases H, cases H_h, cases H_h_h, cases H_h_h_right,\n  apply exists.intro H_h_w, assumption\nend\n\nlemma none_or_some_false1: ∀ {α β:Type} {a:α} {f}, ¬ none_or_some (some a) (@none β) f\n:= begin\n  intros,\n  intros H,\n  unfold none_or_some at H,\n  cases H, cases H, cases H_left, cases H, cases H_h,\n  cases H_h_h, cases H_h_h_right, cases H_h_h_right_left\nend\n\nlemma none_or_some_false2: ∀ {α β:Type} (a:α) f, ¬ none_or_some (@none β) (some a) f\n:= begin\n  intros,\n  intros H,\n  unfold none_or_some at H,\n  cases H, cases H, cases H_right, cases H, cases H_h,\n  cases H_h_h, cases H_h_h_left\nend\n\nend spec", "meta": {"author": "microsoft", "repo": "AliveInLean", "sha": "34370c2c15aa69f010d97b8d38e9e1955e9e387d", "save_path": "github-repos/lean/microsoft-AliveInLean", "path": "github-repos/lean/microsoft-AliveInLean/AliveInLean-34370c2c15aa69f010d97b8d38e9e1955e9e387d/src/spec/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032313, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3016898078197093}}
{"text": "/-\nCopyright (c) 2022 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Std.Control.ForInStep.Basic\n\n/-! # Additional theorems on `ForInStep` -/\n\n@[simp] theorem ForInStep.bind_done [Monad m] (a : α) (f : α → m (ForInStep α)) :\n    (ForInStep.done a).bind (m := m) f = pure (.done a) := rfl\n@[simp] theorem ForInStep.bind_yield [Monad m] (a : α) (f : α → m (ForInStep α)) :\n    (ForInStep.yield a).bind (m := m) f = f a := rfl\n\nattribute [simp] ForInStep.bindM\n\n@[simp] theorem ForInStep.run_done : (ForInStep.done a).run = a := rfl\n@[simp] theorem ForInStep.run_yield : (ForInStep.yield a).run = a := rfl\n\n@[simp] theorem ForInStep.bindList_nil [Monad m] (f : α → β → m (ForInStep β))\n    (s : ForInStep β) : s.bindList f [] = pure s := rfl\n\n@[simp] theorem ForInStep.bindList_cons [Monad m]\n    (f : α → β → m (ForInStep β)) (s : ForInStep β) (a l) :\n    s.bindList f (a::l) = s.bind fun b => f a b >>= (·.bindList f l) := rfl\n\n@[simp] theorem ForInStep.done_bindList [Monad m]\n    (f : α → β → m (ForInStep β)) (a l) :\n    (ForInStep.done a).bindList f l = pure (.done a) := by cases l <;> simp\n\n@[simp] theorem ForInStep.bind_yield_bindList [Monad m]\n    (f : α → β → m (ForInStep β)) (s : ForInStep β) (l) :\n    (s.bind fun a => (yield a).bindList f l) = s.bindList f l := by cases s <;> simp\n\n@[simp] theorem ForInStep.bind_bindList_assoc [Monad m] [LawfulMonad m]\n    (f : β → m (ForInStep β)) (g : α → β → m (ForInStep β)) (s : ForInStep β) (l) :\n    s.bind f >>= (·.bindList g l) = s.bind fun b => f b >>= (·.bindList g l)  := by\n  cases s <;> simp\n\ntheorem ForInStep.bindList_cons' [Monad m] [LawfulMonad m]\n    (f : α → β → m (ForInStep β)) (s : ForInStep β) (a l) :\n    s.bindList f (a::l) = s.bind (f a) >>= (·.bindList f l) := by simp\n\n@[simp] theorem ForInStep.bindList_append [Monad m] [LawfulMonad m]\n    (f : α → β → m (ForInStep β)) (s : ForInStep β) (l₁ l₂) :\n    s.bindList f (l₁ ++ l₂) = s.bindList f l₁ >>= (·.bindList f l₂) := by\n  induction l₁ generalizing s <;> simp [*]\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Control/ForInStep/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011397337391, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.30168980035895093}}
{"text": "import category_theory.sites.dense_subsite\n\n/-!\nContent pulled from mathlib PR #14512\n-/\n\nuniverses w v v₁ v₂ v₃ u u₁ u₂ u₃\nnoncomputable theory\nopen category_theory\nopen opposite\nopen category_theory.presieve.family_of_elements\nopen category_theory.presieve\nopen category_theory.limits\n\nvariables {C D : Type u} [category.{v} C] [category.{v} D]\nvariables (A : Type w) [category.{max v u} A] [has_limits A]\nvariables {J : grothendieck_topology C} {K : grothendieck_topology D}\n\nvariables\n  [concrete_category.{max v u} A]\n  [preserves_limits (forget A)]\n  [reflects_isomorphisms (forget A)]\n  [Π (X : C), preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\n  [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ A]\n  [Π (X : D), preserves_colimits_of_shape (K.cover X)ᵒᵖ (forget A)]\n  [∀ (X : D), has_colimits_of_shape (K.cover X)ᵒᵖ A]\n\nnamespace category_theory.sites\n\n/-- The isomorphism exhibiting compatibility between pullback and sheafification. -/\ndef pullback_sheafification_compatibility\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) :\n  (whiskering_left _ _ A).obj G.op ⋙ presheaf_to_Sheaf J A ≅\n  presheaf_to_Sheaf K A ⋙ sites.pullback A Hc Hp :=\nlet A1 : (whiskering_left _ _ A).obj G.op ⊣ _ := Ran.adjunction _ _,\n    A2 : presheaf_to_Sheaf J A ⊣ _ := sheafification_adjunction _ _,\n    B1 : presheaf_to_Sheaf K A ⊣ _ := sheafification_adjunction _ _,\n    B2 : sites.pullback A Hc Hp ⊣ _ := sites.pullback_copullback_adjunction _ _ Hl _,\n    A12 := A1.comp _ _ A2,\n    B12 := B1.comp _ _ B2 in\nA12.left_adjoint_uniq B12\n\nlemma to_sheafify_pullback_sheafification_compatibility\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) (F) :\n  J.to_sheafify (G.op ⋙ F) ≫\n  ((pullback_sheafification_compatibility.{w v u} A Hp Hl Hc).hom.app F).val =\n  whisker_left _ (K.to_sheafify _) :=\nbegin\n  dsimp [pullback_sheafification_compatibility, adjunction.left_adjoint_uniq],\n  apply quiver.hom.op_inj,\n  apply coyoneda.map_injective, swap, apply_instance,\n  ext E f : 2,\n  dsimp [functor.preimage, full.preimage, coyoneda, adjunction.left_adjoints_coyoneda_equiv],\n  simp only [adjunction.hom_equiv_unit, functor.comp_map, Sheaf_to_presheaf_map,\n    adjunction.hom_equiv_naturality_left_symm, whiskering_left_obj_map,\n    adjunction.hom_equiv_counit, Sheaf.category_theory.category_comp_val,\n    presheaf_to_Sheaf_map_val],\n  dsimp [adjunction.comp],\n  simp only [sheafification_adjunction_unit_app, category.comp_id, functor.map_id,\n    whisker_left_id', category.assoc, grothendieck_topology.sheafify_map_sheafify_lift,\n    category.id_comp, grothendieck_topology.to_sheafify_sheafify_lift],\n  ext t,\n  dsimp only [whisker_left, nat_trans.comp_app],\n  simp only [category.assoc],\n  -- for some reason the proof that works in mathllib does not work here...\n  -- we have to be a little more forceful.\n  erw [(Ran G.op).map_id, category.id_comp],\n  congr' 1, simp only [← category.assoc],\n  convert category.id_comp _,\n  dsimp only [pullback_sheaf],\n  have := (Ran.adjunction A G.op).left_triangle,\n  apply_fun (λ e, (e.app (K.sheafify F)).app x) at this,\n  exact this,\nend\n\n@[simp]\nlemma pullback_sheafification_compatibility_hom_apply_val\n  {G : C ⥤ D} (Hp : cover_preserving J K G)\n  (Hl : cover_lifting J K G) (Hc : compatible_preserving K G) (F) :\n((pullback_sheafification_compatibility.{w v u} A Hp Hl Hc).hom.app F).val =\n  J.sheafify_lift (whisker_left _ $ K.to_sheafify _) (Sheaf.cond _) :=\nbegin\n  apply J.sheafify_lift_unique,\n  apply to_sheafify_pullback_sheafification_compatibility,\nend\n\nend category_theory.sites\n\nnamespace category_theory.cover_dense\n\nvariables {G : C ⥤ D} [full G] [faithful G]\nvariables (Hd : cover_dense K G) (Hp : cover_preserving J K G) (Hl : cover_lifting J K G)\n\n/-- The isomorphism exhibiting the compatibiility of\n`Sheaf_equiv_of_cover_preserving_cover_lifting` with sheafification. -/\nnoncomputable\nabbreviation Sheaf_equiv_of_cover_preserving_cover_lifting_sheafification_compatibility :\n  (whiskering_left _ _ A).obj G.op ⋙ presheaf_to_Sheaf _ _ ≅\n  presheaf_to_Sheaf _ _ ⋙ (Sheaf_equiv_of_cover_preserving_cover_lifting Hd Hp Hl).inverse :=\ncategory_theory.sites.pullback_sheafification_compatibility _ _ Hl _\n\nend category_theory.cover_dense\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/sheafification_equiv_compatibility.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.30160522556146996}}
{"text": "-- import SciLean.Algebra\n-- import Mathlib\n\nimport SciLean.Mathlib.Data.Enumtype\n\nnamespace SciLean\n\n--- Table `C` with index set `ι` and element type `α`\nclass Table (C : Type u) (ι : Type v) (α : Type w) where\n  toFun : C → ι → α\n\nnamespace Table\n\n  -- Function that should be interpreted as a table\n  def TableFun (α β) := α → β\n  infixr:34 \" ↦ \" => TableFun\n  \n  -- Mark function as a table\n  abbrev toTable (f : α → β) : α ↦ β := f\n  -- @[simp] theorem toTable_apply (f : α → β) : toTable f = f := by rfl\n  instance (ι : Type v) (α : Type w) : Table (ι ↦ α) ι α := ⟨λ f => f⟩\n  -- TODO: support `table (i,j) => f i j`\n  macro \"table\" xs:Lean.explicitBinders \"=> \" b:term : term => Lean.expandExplicitBinders `Table.toTable xs b\n\n  --- Automatically infering `ι` and `α` based on C\n  class Trait (C : Type u) where\n    Index : Type v\n    Value : Type w\n\n  -- Is this good idea?\n  @[reducible] \n  instance (C : Type u) (ι : Type v) (α : Type w) [Table C ι α] : Trait C := ⟨ι, α⟩\n\n  attribute [reducible, inline] Trait.Index Trait.Value\n\n  export Trait (Index Value)\n  \n  -- Maybe not a good idea\n  -- instance (ι : Type v) (α : Type w) : Table (ι → α) ι α := ⟨λ f => f⟩\n\n  @[reducible]\n  abbrev get {C} [Trait C] [Table C (Index C) (Value C)] (c : C) := @toFun _ (Index C) (Value C) _ c\n\n  --- TODO:\n  --  Merge all those macros into one\n  -- \n  --  Element assignment:\n  --      1. f[i] := x    ==   f := f.set i x\n  --      2. f[i] += x    ==   f := f.set i (f[i] + x)\n  --\n  --  Add slicing notation:\n  --      1. f[:]    ==  f[:₀]  ==  table     i => f[i]    : ι ↦ α       \n  --      2. f[:,:]             ==  table (i,j) => f[i,j]  : ι₀ × ι₁ ↦ α \n  --  Curry notation:  \n  --      3. f[:₀,:₁]   ==  table     i j => f[i,j]   : ι₀ ↦ ι₁ ↦ α       where  f : ι₀ × ι₁ ↦ α \n  --      4. f[:₁,:₀]   ==  table     j i => f[i,j]   : ι₁ ↦ ι₀ ↦ α       where  f : ι₀ × ι₁ ↦ α \n  --      5. f[:,:₁,:]  ==  table (i,j) k => f[i,k,j] : ι₀ × ι₂ ↦ ι₁ ↦ α  where  f : ι₀ × ι₁ × ι₂ ↦ α \n  --  Uncurry notation:\n  --      5. f[:][:]  == table (i,j) => f[i][j]  :  ι₀ × ι₁ ↦ α       where  f : ι₀ ↦ ι₁ ↦ α \n  -- \n  --  Common examples:  (mean: ∑' == 1/n * ∑) (norm: ∥ ∥)\n  --      1. average of columns:    (∑' j, A[:,j])(A[:₀,:₁] - ∑ j', A[:,j'])[:,:]\n  --      2. center columns:         (A[:₀,:₁] - ∑ j', A[:,j'])[:,:]\n  --      3. normalize of columns:  ((A[:₁,:₀] / (table j, ∥A[:,j]∥))[:₁,:₀])[:,:]\n  --         The core operation (A[:₁,:₀] / (table j, ∥A[:,j]∥) produces `B : ι₁ ↦ ι₀ ↦ α`. Uncurrying back to `ι₀ × ι₁ ↦ α` is the awful (`B[:₁,:₀])[:,:]`\n  --  \n  --  Corner/Odd cases: \n  --        1. curry and uncurry:  (f[:₀,:₁])[:,:]  ==  table (k,l)     => (table i j => f[i,j])[k,l]  ==  f\n  --        2. not the same as 1:   f[:₀,:₁] [:,:]  ==  table (i,j,k) l => f[i,l][j,k]                !=  f\n  --        3. transpose:          (f[:₁,:₀])[:,:]  ==  table (k,l)     => (table j i => f[i,j])[k,l]  ==  transpose f\n  --        4. uncurry:             f[:][:]         ==  table (i,j)     => f[i][j]\n  --        5. curry:               f[:₀,:₁]        ==  table i j       => f[i,j]\n  --        6. identity:            f[:]            ==  table i         => f[i]\n  --        7. identity:           (f[:])[:]        ==  table i         => (table j => f[j])[i]\n  -- \n  --  For indices that are Enumtype add ranged notation  \n  --      1. f[a:b]     == table     i : Fin (dist a b)      => f[offset a i]   : Fin (dist a b) ↦ α\n  --      2. f[a:b,:]   == table (i,j) : Fin (dist a b) × ι₁ => f[offset a i, j]  : Fin (dist a b)×ι₁ ↦ α\n  --      3. f[a:b,:₁]  == table     (i: Fin (dist a b)) j   => f[offset a i, j]  : Fin (dist a b) ↦ ι₁ ↦ α\n  --      where (dist a b)   := (toFin b).1 - (toFin a).1\n  --            (offset a i) := fromFin (i.1 + (toFin a).1)\n\n  macro c:term noWs \"[\" i1:term\"]\" : term =>\n    `(get $c $i1)\n\n  macro c:term noWs \"[\" i1:term \",\" i2:term \"]\" : term =>\n    `(get $c ($i1,$i2))\n\n  macro c:term noWs \"[\" i1:term \",\" i2:term \",\" i3:term \"]\" : term =>\n    `(get $c ($i1,$i2,$i3))\n\n  macro c:term noWs \"[\" i1:term \",\" i2:term \",\" i3:term \",\" i4:term \"]\" : term =>\n    `(get $c ($i1,$i2,$i3,$i4))\n\n\n  section ExtraOperations\n\n     -- Here we use formulation with [Trait C] [Table C (Index C) (Value C)]\n     -- instead of [Table C ι α]\n     --\n     -- This way, for example, `Intro.intro` needs to infer only `C` and does not have to infer `ι` and ‵α`\n     -- Plus when declaring instance you can just write, for example, `instance {T} : Intro T`.\n\n     class Intro (C : Type u) [Trait C] [Table C (Index C) (Value C)] where\n       intro : (Index C → Value C) → C\n       valid : ∀ f i, (intro f)[i] = f i\n\n     export Intro (intro)\n\n     instance {C ι α} [Table C ι α] [Intro C] : Coe (ι ↦ α) C := ⟨λ f => intro f⟩\n  \n     class Set (C : Type u) [Trait C] [Table C (Index C) (Value C)] where\n       set : C → (Index C) → (Value C) → C\n       valid : ∀ c i a, ((set c i a)[i] = a) ∧ \n                        (∀ j, j ≠ i → (set c i a)[j] = c[j])\n\n     export Set (set)\n\n     class MapIdx (C : Type u) [Trait C] [Table C (Index C) (Value C)] where\n       mapIdx : ((Index C) → (Value C) → (Value C)) → C → C\n       valid : ∀ f c i, (mapIdx f c)[i] = f i (c[i])\n\n     export MapIdx (mapIdx)\n\n     class Map (C : Type u) [Trait C] [Table C (Index C) (Value C)] where\n       map : ((Value C) → (Value C)) → C → C\n       valid : ∀ f c i, (map f c)[i] = f (c[i])\n\n     export Map (map)\n\n     -- export Map₂ (map₂)\n\n     -- Some tables can have infinite index set `(Index C)` but only finite many indices actually hold a value\n     -- Prime example is OpenVDB/NanoVDB but sparse matrices also qualify for this\n     class Active (C : Type u) [Trait C] [Table C (Index C) (Value C)] where\n       active : C → (Index C) → Bool\n       finite : (c : C) → Enumtype {i : (Index C) // active c i = true }\n\n     -- Add ActiveMapIdx -- runs map only over active indices\n\n  end ExtraOperations\n\n  section BasicIdentities\n\n    variable {C : Type u} {ι : Type v} {α : Type w}\n    variable [Table C ι α] [Intro C]\n\n    @[simp] theorem get_intro (f : Index C → Value C) : (intro f)[i] = f i := by apply Intro.valid; done\n    @[simp] theorem intro_get (c : C) : intro (λ i => c[i]) = c := sorry\n    @[simp] theorem get_tableFun {ι : Type v} {α : Type w} (i : ι) (f : ι ↦ α) : f[i] = f i := by rfl\n\n  end BasicIdentities\n\n  section AlgebraicOperations\n\n     variable {C : Type u} \n     variable [Trait C] [Table C (Index C) (Value C)] [Intro C]\n\n     instance instTableAdd [Add (Value C)] : Add C := ⟨λ c d => intro (λ i => c[i] + d[i])⟩\n     instance instTableSub [Sub (Value C)] : Sub C := ⟨λ c d => intro (λ i => c[i] - d[i])⟩\n     instance instTableNeg [Neg (Value C)] : Neg C := ⟨λ c => intro (λ i => -c[i])⟩\n     instance instTableHMul {α} [HMul α (Value C) (Value C)] : HMul α C C := ⟨λ s c => intro (λ i => s * c[i])⟩\n     instance instTableZero [Zero (Value C)] : Zero C := ⟨intro λ _ => 0⟩\n\n     -- TODO: Add instances for different algebraic structures like Group\n\n     -- This section is probably a bit controversial as we probably do not want those theorems inside of `simp` tactic\n     -- They should be used in a specialized tactic optimizing algebraic expressions with tables\n     section UnfoldOperations\n\n       -- variable {C} [Trait C] [Table C (Index C) (Value C)] [Vec (Value C)] [Intro C]\n       -- variable [Trait C] [Table C (Index C) (Value C)] [Intro C]\n       \n       -- Unfold definition's of vector oprations back\n       -- This way we can get fast saxpy type operations i.e.`s*x+y` transforms to `intro λ i => s*x[i] + y[i]`\n       -- We specify class instances directly to prevent crazy TC searches.\n       theorem add_norm [Add (Value C)] (c d : C) : HAdd.hAdd (self := instHAdd) c d = intro (λ i => c[i] + d[i]) := by rfl\n       theorem sub_norm [Sub (Value C)] (c d : C) : HSub.hSub (self := instHSub) c d = intro (λ i => c[i] - d[i]) := by rfl\n       theorem neg_norm [Neg (Value C)] (c : C) : Neg.neg (self := instTableNeg) c = intro (λ i => -c[i]) := by rfl\n       theorem hmul_norm {α} [HMul α (Value C) (Value C)] (a : α) (c : C) : HMul.hMul (self := instTableHMul) a c = intro (table i => a * c[i]) := by rfl\n       theorem zero_norm [Zero (Value C)]: (Zero.zero (self := instTableZero) : C) = intro (λ _ => 0) := by rfl\n\n     end UnfoldOperations\n\n     section UnfoldOperationsOnIntro\n\n       theorem add_norm' {C ι α} [Table C ι α] [Intro C] [Add α] (c d : C) \n         : HAdd.hAdd (self := instHAdd) c d = intro (λ i => c[i] + d[i]) := by rfl\n\n       -- @[simp]\n       -- theorem sub_norm' {C ι α} [Table C ι α] [Intro C] [Sub α] (c d : C) \n       --   : HSub.hSub (self := instHSub) c d = intro (λ i => c[i] - d[i]) := by rfl\n\n       -- theorem neg_norm' {C ι α} [Table C ι α] [Intro C] [Neg α] (c : C) \n       --   : Neg.neg (self := instTableNeg) c = intro (λ i => -c[i]) := by rfl\n\n       -- theorem hmul_norm' {α} [HMul α (Value C) (Value C)] (a : α) (c : C) : HMul.hMul (self := instTableHMul) a c = intro (table i => a * c[i]) := by rfl\n       -- theorem zero_norm' [Zero (Value C)]: (Zero.zero (self := instTableZero) : C) = intro (λ _ => 0) := by rfl\n\n     end UnfoldOperationsOnIntro\n\n  end AlgebraicOperations\n\n  -- Algebraic operations on tables with lazy evaluation\n  -- This provides something like Eigen's expression templates\n  -- https://en.wikipedia.org/wiki/Expression_templates\n  section LazyOperations\n\n     section ElementWise\n       variable {ι : Type v}\n       variable {C : Type u} {α : Type w} [Table C ι α]\n       variable {C' : Type u'} {α' : Type w'} [Table C' ι α']\n\n       instance [HAdd α α' β] : HAdd C C' (ι ↦ β) := ⟨λ c c' => λ i => c[i] + c'[i]⟩\n       instance [HSub α α' β] : HSub C C' (ι ↦ β) := ⟨λ c c' => λ i => c[i] - c'[i]⟩\n       instance [HMul α α' β] : HMul C C' (ι ↦ β) := ⟨λ c c' => λ i => c[i] * c'[i]⟩\n       instance [HDiv α α' β] : HDiv C C' (ι ↦ β) := ⟨λ c c' => λ i => c[i] / c'[i]⟩\n\n       -- TODO: Add other homogenous lazy oprations\n       instance [Add α] : Add (ι ↦ α) := ⟨λ c c' => λ i => c[i] + c'[i]⟩\n       instance [Zero α] : Zero (ι ↦ α) := ⟨λ i => 0⟩\n\n       @[simp] theorem elwise_hadd_apply [HAdd α α' β] (c : C) (c' : C') (i) : (c + c')[i] = c[i] + c'[i] := by simp[HAdd.hAdd] done\n       @[simp] theorem elwise_hsub_apply [HSub α α' β] (c : C) (c' : C') (i) : (c - c')[i] = c[i] - c'[i] := by simp[HSub.hSub] done\n       @[simp] theorem elwise_hmul_apply [HMul α α' β] (c : C) (c' : C') (i) : (c * c')[i] = c[i] * c'[i] := by simp[HMul.hMul] done\n       @[simp] theorem elwise_hdiv_apply [HDiv α α' β] (c : C) (c' : C') (i) : (c / c')[i] = c[i] / c'[i] := by simp[HDiv.hDiv] done\n\n     end ElementWise\n\n     --- Matrix style multiplication\n     section Mul\n       -- These instances are a bit finicky and can easily lead to infinite loop                 \n       -- It is important to first look for instance `Table` and only then for\n       -- instances of `HMul α α' β` or `Iterable κ`\n  \n       variable {ι κ μ : Type v}\n\n       -- matrix * matrix\n       instance {C : Type u} {α : Type w} [Table C (ι×κ) α]\n                {C' : Type u'} {α' : Type w'} [Table C' (κ×μ) α']\n                [Enumtype κ]                \n                [HMul α α' β] [Add β] [Zero β] \n                : HMul C C' (ι × μ ↦ β) \n                := ⟨toTable λ c c' (i,j) => ∑ k, (c[i, k] * c'[k,j] : β)⟩\n\n       -- matrix * vector\n       instance {C : Type u} {α : Type w} [Table C (ι×κ) α]\n                {U' : Type u'} {α' : Type w'} [Table U' κ α']\n                [Enumtype κ]                \n                [HMul α α' β] [Add β] [Zero β]\n                : HMul C U' (ι ↦ β) \n                := ⟨λ c u i => ∑ k, c[i,k] * u[k]⟩\n\n       -- vector * matrix\n       instance {U : Type u} {α : Type w} [Table U κ α]\n                {C' : Type u'} {α' : Type w'} [Table C' (κ×μ) α']\n                [Enumtype κ]                \n                [HMul α α' β] [Add β] [Zero β]\n                : HMul U C' (μ ↦ β) \n                := ⟨λ u c k => ∑ j, u[j] * c[j,k]⟩\n\n       instance {U : Type u} {α : Type w} [Table U κ α]\n                {U' : Type u'} {α' : Type w'} [Table U' κ α']\n                [Enumtype κ]                \n                [HMul α α' β] [Add β] [Zero β]\n                : HMul U U' β\n                := ⟨λ u v => ∑ i, u[i] * v[i]⟩\n\n       -- TODO: Add other unfolding theorems\n\n       -- fix:\n       -- @[simp] theorem hmul_apply [HMul α α' β] [Add β] [Zero β]\n       --                 (c : C) (c' : C')\n       --                 : (c*c')[i,j] = ∑ k, c[i, k] * c'[k,j]\n       --                 := by simp[HMul.hMul,Mul.mul] done\n  \n\n     end Mul\n\n     section Broadcasting\n\n       -- still not sure how to state theorems and instances about `Table`\n       -- variable {C : Type u} [Trait C] [Table C (Index C) (Value C)] [Intro C]\n       variable {C : Type u} {ι : Type v} {α : Type w} [Table C ι α]\n\n       -- Thise two can lead to ambiquous notation, we prefer the later\n       -- i.e. The class `HAdd C (ι ↦ α) (ι ↦ ι ↦ α)` class has two different instance that are not equal!!!\n       -- The second one ensures that `(c + b)[i] = c[i] + b[i]`\n       instance (priority := low) [Add α] : HAdd C α (ι ↦ α) := ⟨λ c a i => c[i]+a⟩\n       instance (priority := low) [Add α] : HAdd α C (ι ↦ α) := ⟨λ a c i => a+c[i]⟩\n       instance [HAdd α β α] : HAdd C (ι ↦ β) (ι ↦ α) := ⟨λ c b i => c[i]+b[i]⟩\n       instance [HAdd β α α] : HAdd (ι ↦ β) C (ι ↦ α) := ⟨λ b c i => b[i]+c[i]⟩\n\n       instance (priority := low) [Sub α] : HSub C α (ι ↦ α) := ⟨λ c a i => c[i]-a⟩\n       instance (priority := low) [Sub α] : HSub α C (ι ↦ α) := ⟨λ a c i => a-c[i]⟩\n       instance [HSub α β α] : HSub C (ι ↦ β) (ι ↦ α) := ⟨λ c b i => c[i]-b[i]⟩\n       instance [HSub β α α] : HSub (ι ↦ β) C (ι ↦ α) := ⟨λ b c i => b[i]-c[i]⟩\n\n       -- Not sure about these as they might clash with HMul β C C\n       instance (priority := low) [HMul α β α] : HMul C β (ι ↦ α) := ⟨λ c b i => c[i]*b⟩\n       instance (priority := low) [HMul β α α] : HMul β C (ι ↦ α) := ⟨λ b c i => b*c[i]⟩\n       instance [HMul α β α] : HMul C (ι ↦ β) (ι ↦ α) := ⟨λ c b i => c[i]*b[i]⟩\n       instance [HMul β α α] : HMul (ι ↦ β) C (ι ↦ α) := ⟨λ b c i => b[i]*c[i]⟩\n\n       instance (priority := low) [Div α] : HDiv C α (ι ↦ α) := ⟨λ c a i => c[i]/a⟩\n       instance (priority := low) [Div α] : HDiv α C (ι ↦ α) := ⟨λ a c i => a/c[i]⟩\n       instance [HDiv α β α] : HDiv C (ι ↦ β) (ι ↦ α) := ⟨λ c b i => c[i]/b[i]⟩\n       instance [HDiv β α α] : HDiv (ι ↦ β) C (ι ↦ α) := ⟨λ b c i => b[i]/c[i]⟩    \n\n       @[simp] theorem hadd_apply_1 [Add α] (c : C) (a : α) (i) : (c + a)[i] = c[i] + a := by simp[HAdd.hAdd] done\n       @[simp] theorem hadd_apply_2 [Add α] (c : C) (a : α) (i) : (a + c)[i] = a + c[i] := by simp[HAdd.hAdd] done\n       @[simp] theorem hadd_apply_3 [HAdd α β α] (c : C) (b : ι ↦ β) (i) : (c + b)[i] = c[i] + b[i] := by simp[HAdd.hAdd] done\n       @[simp] theorem hadd_apply_4 [HAdd β α α] (c : C) (b : ι ↦ β) (i) : (b + c)[i] = b[i] + c[i] := by simp[HAdd.hAdd] done\n\n       @[simp] theorem hsub_apply_1 [Sub α] (c : C) (a : α) (i) : (c - a)[i] = c[i] - a := by simp[HSub.hSub] done\n       @[simp] theorem hsub_apply_2 [Sub α] (c : C) (a : α) (i) : (a - c)[i] = a - c[i] := by simp[HSub.hSub] done\n       @[simp] theorem hsub_apply_3 [HSub α β α] (c : C) (b : ι ↦ β) (i) : (c - b)[i] = c[i] - b[i] := by simp[HSub.hSub] done\n       @[simp] theorem hsub_apply_4 [HSub β α α] (c : C) (b : ι ↦ β) (i) : (b - c)[i] = b[i] - c[i] := by simp[HSub.hSub] done\n\n       @[simp] theorem hmul_apply_1 [Mul α] (c : C) (a : α) (i) : (c * a)[i] = c[i] * a := by simp[HMul.hMul] done\n       @[simp] theorem hmul_apply_2 [Mul α] (c : C) (a : α) (i) : (a * c)[i] = a * c[i] := by simp[HMul.hMul] done\n       @[simp] theorem hmul_apply_3 [HMul α β α] (c : C) (b : ι ↦ β) (i) : (c * b)[i] = c[i] * b[i] := by simp[HMul.hMul] done\n       @[simp] theorem hmul_apply_4 [HMul β α α] (c : C) (b : ι ↦ β) (i) : (b * c)[i] = b[i] * c[i] := by simp[HMul.hMul] done\n\n       @[simp] theorem hdiv_apply_1 [Div α] (c : C) (a : α) (i) : (c / a)[i] = c[i] / a := by simp[HDiv.hDiv] done\n       @[simp] theorem hdiv_apply_2 [Div α] (c : C) (a : α) (i) : (a / c)[i] = a / c[i] := by simp[HDiv.hDiv] done\n       @[simp] theorem hdiv_apply_3 [HDiv α β α] (c : C) (b : ι ↦ β) (i) : (c / b)[i] = c[i] / b[i] := by simp[HDiv.hDiv] done\n       @[simp] theorem hdiv_apply_4 [HDiv β α α] (c : C) (b : ι ↦ β) (i) : (b / c)[i] = b[i] / c[i] := by simp[HDiv.hDiv] done\n\n     end Broadcasting\n\n\n  end LazyOperations\n\n\n  section ForInNotation\n\n    -- Usefull for modifying a table as we want to run only over indices and not values\n    open Enumtype in \n    def allIdx {C} (c : C) [Trait C] [Enumtype (Index C)] : Range (Index C) := fullRange (Index C)\n\n    -- notation:  \n    --      for (a,i,li) in F do\n    --          ..                        \n    -- This seems to be broken ...\n    -- open Enumtype in\n    -- instance {m} [Monad m] {ι} {α : Type w} [Enumtype ι] [ForIn m (Range ι) (ι × Nat)]\n    --          : ForIn m (ι ↦ α) (α × ι × Nat) :=\n    -- {\n    --   forIn := λ F init f => do\n    --              let mut val := init\n    --              for (i,li) in fullRange ι do\n    --                match (← f (F[i], i, li) val) with\n    --                | ForInStep.done d => return d\n    --                | ForInStep.yield d => val ← d\n    --              pure init\n    -- }\n     \n  end ForInNotation\n\n  -- View of a table if usefull if you want to modify a subset of a table and still refer to it as a table\n  section TableView\n\n    def TableView {κ} (C : Type u) [Trait C] (tr : κ → (Index C)) := C\n    def view   {κ} {C} [Trait C] (c : C) (tr : κ → (Index C)) : TableView C tr := c\n    def unview {κ} {C} [Trait C] {tr : κ → (Index C)} (c : TableView C tr) : C := c\n\n    instance {κ} (C : Type u) [Trait C] (tr : κ → (Index C)) : Trait (TableView C tr) :=\n    {\n      Index := κ\n      Value := (Value C)\n    }\n\n    instance {C ι α κ} [Table C ι α] (tr : κ → ι) : Table (TableView C tr) κ α :=\n    {\n      toFun := λ c j => (unview c)[tr j]\n    }\n  \n    instance {C ι α κ} [Table C ι α] (tr : κ → ι) [Set C] : Set (TableView C tr) :=\n    {\n      set := λ c j a => view (set (unview c) (tr j) a) tr\n      valid := sorry\n    }\n\n  end TableView\n\n\n  section BasicTests\n\n     -- def test : IO Unit := do\n     --     for (a,i,li) in (table i : Fin 2 × Fin 3 × Fin 4 => i.2) do \n     --        IO.println s!\"i = {i}  |  li = {li}  |  a = {a}\"\n\n     -- #eval test\n\n\n     variable (ℝ : Type) [Add ℝ] [Sub ℝ] [Neg ℝ] [Zero ℝ]\n     variable (A : Fin n × Fin m ↦ ℝ)\n\n     -- curry\n     example : Fin n ↦ Fin m ↦ ℝ := (table i j => A[i,j])\n     -- example : A[:₀,:₁] = (table i j => A[i,j]) := by rfl\n\n     -- curry and swap\n     example : Fin m ↦ Fin n ↦ ℝ := (table j i => A[i,j])\n     -- example : A[:₁,:₀] = (table j i => A[i,j]) := by rfl\n\n     -- transpose\n     example : Fin m × Fin n ↦ ℝ := λ (i,j) => A[j,i] \n     -- example : (A[:₁,:₀])[:,:] = table (i,j) => A[j,i] := by rfl\n\n     -- sum rows v1\n     example : Fin n ↦ ℝ := (∑ j, table i => A[i,j]) \n     -- example : (∑ j, A[:,j]) = (∑ j, table i => A[i,j]) := by rfl\n\n     -- sum rows v2\n     example : Fin n ↦ ℝ := (table i => ∑ j, A[i,j])\n     example : (∑ j, A[·,j]) = (table i => ∑ j, A[i,j]) := by rfl\n     example : (table i => ∑ j, A[i,j]) = (∑ j, table i => A[i,j]) := by funext i; admit  --- TODO: (∑ i, f i) x = (∑ i, f i x)\n\n     -- center columns -- common task in data analysis\n     -- example : Fin n × Fin m ↦ ℝ := λ (i,j) => A[i,j] - ∑ j', A[i,j']\n     -- example : (A[:₀,:₁] - ∑ j', A[:,j'])[:,:] = table (i,j) => A[i,j] - ∑ j', A[i,j'] := by rfl\n     --- Future notation:  \n     --      A[:₀,:₁]                  : Fin n ↦ Fin m ↦ ℝ\n     --      ∑ j', A[:,j']             : Fin n          ↦ ℝ\n     --      A[:₀,:₁] - ∑ j', A[:,j']  : Fin n ↦ Fin m ↦ ℝ\n     -- \n     --- Alternatively:\n     --      A[:₁,:₀]                  : Fin m ↦ Fin n ↦ ℝ\n     --      ∑ j', A[:,j']             :          Fin n ↦ ℝ\n     --      A[:₁,:₀] - ∑ j', A[:,j']  : Fin m ↦ Fin n ↦ ℝ\n  \n     -- This is ambiguous if `n == m` and we prefer the first one!\n\n     -- variable (M : Fin n × Fin n ↦ ℝ)\n     -- example : Fin n ↦ Fin n ↦ ℝ := (table i j => M[i,j]) - (table i => ∑' j, M[i,j])  --- This is ambiguous notation! Not clear what\n     -- should be prefered?\n     -- example : (table i j => M[i,j]) - (∑' j, table i => M[i,j]) = (table i j => M[i,j] - ∑' j', M[i,j']) := by funext x y; simp[HSub.hSub,Sub.sub]; admit  --- TODO: (∑' i, f i) x = (∑' i, f i x)\n     -- example : (table i j => M[i,j]) - (∑' j, table i => M[i,j]) = (table i j => M[i,j] - ∑' j', M[j,j']) := NOT TRUE\n\n\n     -- normalize columns\n     -- example : Fin n × Fin m ↦ ℝ := λ (i,j) => A[i,j] / ∥λ i' => A[i',j]∥\n     -- example : A[:₁,:₀] / (table j => ∥A[:,j]∥) = (λ (i,j) => A[i,j] / Math.sqrt (∑ i', A[i',j]*A[i',j])) := by rfl\n     -- example : (table j i => A[i,j]) / (table j => ∥λ i => A[i,j]∥) = (table j i => A[i,j] / ∥λ i' => A[i',j]∥) := sorry\n     -- example : M[:₁,:₀] / (table j => ∥A[:,j]∥) = (λ (i,j) => A[i,j] / Math.sqrt (∑ i', A[i',j]*A[i',j])) := by rfl\n\n  end BasicTests\n\n  section TestBLASOperations\n    variable {C} [Trait C] [Table C (Index C) (Value C)] [Intro C]\n    variable {S X} [Add X] [Sub X] [Neg X] [HMul S X X]\n    abbrev xpy  (x y : X) := x + y\n    abbrev saxpy (s : S) (x y : X) := s*x + y\n    abbrev saxsrypnz (s r : S) (x y z: X) := s*x - r*y + (-z)\n\n    variable {α} [Add (Value C)] [Sub (Value C)] [Neg (Value C)] [HMul α (Value C) (Value C)]\n    -- example (x y : C) : xpy x y = intro (λ i => x[i] + y[i]) := by simp done\n    -- example (s : α) (x y : C) : saxpy s x y = intro (λ i => s*x[i] + y[i]) := by simp done\n    -- example (s r : α) (x y z  : C) : saxsrypnz s r x y z = intro (λ i => s*x[i] - r*y[i] + -z[i]) := by simp done\n  end TestBLASOperations\n\n\nend Table\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Mathlib/Data/Table.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.30150690330654967}}
{"text": "import category_theory.adjunction\nimport category_theory.comma\nimport category_theory.punit\nimport category_theory.limits.types\n\nopen category_theory opposite\n\nvariables (C : Type) [category.{0} C] (F : C ⥤ Type) (G : Cᵒᵖ ⥤ Type)\n\ndef presheaf_to_copresheaf : (Cᵒᵖ ⥤ Type) ⥤ (C ⥤ Type)ᵒᵖ :=\nfunctor.right_op (coyoneda ⋙ (whiskering_left C (Cᵒᵖ ⥤ Type) Type).obj yoneda)\n\ndef copresheaf_to_presheaf : (C ⥤ Type)ᵒᵖ ⥤ (Cᵒᵖ ⥤ Type) :=\ncoyoneda ⋙ (whiskering_left Cᵒᵖ (C ⥤ Type) Type).obj coyoneda\n\nlocal attribute [simp] copresheaf_to_presheaf presheaf_to_copresheaf\n\nexample : presheaf_to_copresheaf C ⊣ copresheaf_to_presheaf C :=\nadjunction.mk_of_unit_counit\n  { unit :=\n    { app := λ F,\n      { app := λ X FX,\n        { app := λ Y f, f.app X FX },\n          naturality' := begin\n            intros X Y f,\n            ext FX Z g,\n            dsimp at *,\n            exact function.funext_iff.1 (g.naturality f) FX\n          end } },\n    counit :=\n    { app := λ F, show op (unop ((presheaf_to_copresheaf C).obj\n          ((copresheaf_to_presheaf C).obj F))) ⟶ op (unop F),\n        from quiver.hom.op\n      { app := λ X FX,\n        { app := λ Y f, f.app X FX },\n          naturality' := begin\n            intros X Y f,\n            ext FX Z g,\n            dsimp at *,\n            exact function.funext_iff.1 (g.naturality f) FX\n          end } },\n    left_triangle' := begin\n      ext F,\n      dsimp [presheaf_to_copresheaf, copresheaf_to_presheaf],\n      refine quiver.hom.unop_inj _,\n      ext, refl\n    end,\n    right_triangle' := by { ext, refl } }\n\ndef as_cone (X : C) : limits.cone (comma.snd ((category_theory.functor.const (discrete unit)).obj X) (𝟭 C)) :=\n{ X := X,\n  π :=\n  { app := λ Y, Y.hom,\n    naturality' := λ Y Z f, f.3 } }\n\ndef is_limit_as_cone (X : C) : limits.is_limit (as_cone C X) :=\n{ lift := λ s, s.2.app ⟨(), X, 𝟙 X⟩,\n  fac' := λ s j, begin\n      dsimp [as_cone],\n      let f : (⟨(), X, 𝟙 X⟩ : comma ((category_theory.functor.const (discrete unit)).obj X) (𝟭 C)) ⟶ j :=\n        ⟨⟨⟨subsingleton.elim _ _⟩⟩, j.hom, by tidy⟩,\n      have := (@s.2.naturality f).symm,\n      dsimp at *,\n      simp [this]\n    end,\n  uniq' := λ s m h, begin\n    dsimp [as_cone] at *,\n    rw ← h,\n    simp,\n  end }\n\ndef iso_of_preserves_limits (F : Cᵒᵖ ⥤ Type) [limits.preserves_limits F] :\n  F ≅ (copresheaf_to_presheaf C).obj ((presheaf_to_copresheaf C).obj F) :=\nnat_iso.of_components\n  (λ X,\n    { hom := λ FX, { app := λ Y f, f.app X FX },\n      inv := λ f,\n      begin\n        dsimp at *,\n        have := f.app,\n        dsimp at this,\n        apply (limits.is_limit_of_preserves F (is_limit_as_cone Cᵒᵖ X)).lift\n          (limits.limit.cone _),\n        dsimp [as_cone] at *,\n        tidy,\n        rcases j with ⟨⟨⟩, j, g⟩,\n        dsimp at *,\n        have x : F.obj X := sorry,\n        exact F.map g x,\n        dsimp,\n        funext,\n        \n\n      end }  )\n  _\n\nvariable {C}\n\nstructure corepresentation :=\n( corepr : C )\n( unit : F.obj corepr )\n( extend : Π {X : C}, F.obj X → (corepr ⟶ X) )\n( map_extend_unit {X : C} (f : F.obj X) : F.map (extend f) unit = f )\n( unit_map_extend {X : C} (f : corepr ⟶ X) : extend (F.map f unit) = f )\n\ndef hom_ext {R : corepresentation F} {X : C} {f g : R.corepr ⟶ X}\n  (h : F.map f R.unit = F.map g R.unit) : f = g :=\nby rw [← R.unit_map_extend f, h, R.unit_map_extend]\n\nexample (R : corepresentation F) {X Y : C} (f : F.obj X) (g : X ⟶ Y) :\n  R.extend f ≫ g = R.extend (F.map g f) :=\nhom_ext _ $ by simp [R.map_extend_unit]\n\ndef is_repr (R : corepresentation F) : (coyoneda.obj (op R.corepr)) ≅ F :=\nnat_iso.of_components\n  (λ X, { hom := λ f, F.map f R.unit,\n          inv := λ f, R.extend f,\n          hom_inv_id' := by funext x; simp [R.unit_map_extend],\n          inv_hom_id' := by funext x; simp [R.map_extend_unit] })\n  begin\n    intros X Y f,\n    funext x,\n    simp\n  end\n\nstructure corepresentation2 :=\n( corepr : C )\n( unit : F.obj corepr )\n( extend : Π {X : C}, F.obj X → (corepr ⟶ X) )\n( map_extend_unit {X : C} (f : F.obj X) : F.map (extend f) unit = f )\n( extend_unit : extend unit = 𝟙 corepr )\n( extend_comp {X Y : C} (f : F.obj X) (g : X ⟶ Y) : extend f ≫ g = extend (F.map g f) )\n\nexample (R : corepresentation2 F) : corepresentation F :=\n{ unit_map_extend := begin\n    intros X f,\n    rw [← R.extend_comp, R.extend_unit, category.id_comp],\n  end\n  ..R }\n\nstructure representation2 :=\n( repr : C )\n( counit : G.obj (op repr) )\n( restrict : Π {X : C}, G.obj (op X) → (X ⟶ repr ) )\n( map_restrict_counit {X : C} (f : G.obj (op X)) : G.map ((restrict f).op) counit = f )\n( restrict_counit : restrict counit = 𝟙 repr )\n( comp_restrict {X Y : C} (f : G.obj (op X)) (g : Y ⟶ X) : g ≫ restrict f = restrict (G.map g.op f) )\n\nexample (R : representation2 G) : corepresentation2 (presheaf_to_copresheaf G) :=\n{ corepr := R.repr,\n  unit := begin\n    dsimp [presheaf_to_copresheaf],\n    have := R.counit,\n\n  end, }", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/representable_functor/definitions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3015069033065496}}
{"text": "import .ops\nopen subtype nnf model\n\ndef unmodal_seqt (Γ : seqt) : list seqt :=\nlist.map \n(λ d, \n{main := d :: (unbox (Γ.main ++ Γ.hdld)), hdld := [], \n pmain := begin intros v hv φ hsat hin hall, exfalso, apply list.not_mem_nil, exact hin end, \n phdld := box_only_nil}) \n(undia (Γ.main ++ Γ.hdld))\n\ndef unmodal_nil_hdld (Γ : seqt) : ∀ (i : seqt),  i ∈ unmodal_seqt Γ → (i.hdld = []) := \nlist.mapp _ _ \nbegin intros φ h, constructor end\n\ndef unmodal_seqt_size (Γ : seqt) : ∀ (i : seqt),  i ∈ unmodal_seqt Γ → (prod.measure_lex seqt_size i Γ) := \nlist.mapp _ _ \nbegin \n  intros φ h, \n  constructor, simp [-list.map_append], \n  apply undia_degree h\nend\n\ndef unmodal_mem_box (Γ : seqt) : ∀ (i : seqt),  i ∈ unmodal_seqt Γ → (∀ φ, box φ ∈ Γ.main ++ Γ.hdld → φ ∈ i.main) := \nlist.mapp _ _ begin intros φ h ψ hψ, right, apply (@unbox_iff (Γ.main++Γ.hdld) ψ).1 hψ end\n\ndef unmodal_sat_of_sat (Γ : seqt) : ∀ (i : seqt),  i ∈ unmodal_seqt Γ → \n(∀ {st : Type} (k : KT st) s Δ \n(h₁ : ∀ φ, box φ ∈ Γ.main++Γ.hdld → box φ ∈ Δ) \n(h₂ : ∀ φ, dia φ ∈ Γ.main++Γ.hdld → dia φ ∈ Δ), \nsat k s Δ → ∃ s', sat k s' i.main) :=\nlist.mapp _ _ \nbegin\n  intros φ hmem st k s Δ h₁ h₂ h, \n  have : force k s (dia φ), \n    { apply h, apply h₂, rw undia_iff, exact hmem }, \n  rcases this with ⟨w, hrel, hforce⟩,\n  split, swap, {exact w}, \n  { intro ψ, intro hψ, cases hψ, \n    {rw hψ, exact hforce}, \n    {have := h₁ _ ((@unbox_iff (Γ.main++Γ.hdld) ψ).2 hψ), \n     have := h _ this,\n     apply this _ hrel} },\nend\n\n-- TODO: Rewrite this ugly one.\ndef mem_unmodal_seqt (Γ : seqt) (φ) (h : φ ∈ undia (Γ.main++Γ.hdld)) : \n(⟨φ :: unbox (Γ.main++Γ.hdld), [], begin intros v hv φ hsat hin hall, exfalso, apply list.not_mem_nil, exact hin end, box_only_nil⟩ : seqt) ∈ unmodal_seqt Γ := \nbegin \n  dsimp [unmodal_seqt], \n  apply list.mem_map_of_mem (λ φ, (⟨φ :: unbox (Γ.main++Γ.hdld), _,_,_⟩:seqt)) h \nend\n\ndef unsat_of_unsat_unmodal {Γ : seqt} (h : modal_applicable Γ) (i : seqt) : i ∈ unmodal_seqt Γ ∧ unsatisfiable (i.main ++ i.hdld) → unsatisfiable (Γ.main ++ Γ.hdld) := \nbegin\n  intro hex, intros st k s h,\n  have := unmodal_sat_of_sat Γ i hex.1 k s (Γ.main ++ Γ.hdld) (λ x hx, hx) (λ x hx, hx) h,\n  cases this with w hw,\n  have := hex.2,\n  rw (unmodal_nil_hdld _ _ hex.1) at this,\n  simp at this,\n  exact this _ _ _ hw\nend\n\n/- Part of the soundness -/\n\ntheorem unsat_of_closed_and {Γ Δ} (i : and_instance Γ Δ) (h : unsatisfiable Δ) : unsatisfiable Γ := \nby cases i; { apply unsat_and_of_unsat_split, repeat {assumption} }\n\ntheorem unsat_of_closed_and_seqt {Γ Δ} (i : and_instance_seqt Γ Δ) (h : unsatisfiable (Δ.main++Δ.hdld)) : unsatisfiable (Γ.main++Γ.hdld) := \nbegin\ncases i, {apply unsat_and_of_unsat_split_seqt, repeat {assumption} }\nend\n\ntheorem unsat_of_closed_or_seqt {Γ₁ Γ₂ Δ : seqt} (i : or_instance_seqt Δ Γ₁ Γ₂) \n(h₁ : unsatisfiable (Γ₁.main++Γ₁.hdld)) \n(h₂ : unsatisfiable (Γ₂.main++Γ₂.hdld)) : \nunsatisfiable (Δ.main++Δ.hdld) :=\nbegin\ncases i, {apply unsat_or_of_unsat_split_seqt, repeat {assumption}}\nend\n\n/- Tree models -/\n\ninductive batch_sat_seqt : list model → list seqt → Prop\n| bs_nil : batch_sat_seqt [] []\n| bs_cons (m) (Γ:seqt) (l₁ l₂) : sat builder m (Γ.main++Γ.hdld) → \n                                 batch_sat_seqt l₁ l₂ → \n                                 batch_sat_seqt (m::l₁) (Γ::l₂)\n\nopen batch_sat_seqt\n\ntheorem bs_ex_seqt : Π l Γ, \nbatch_sat_seqt l Γ → ∀ m ∈ l, ∃ i ∈ Γ, sat builder m (seqt.main i ++ i.hdld)\n| l Γ bs_nil := λ m hm, by simpa using hm\n| l Γ (bs_cons m Δ l₁ l₂ h hbs) := \nbegin\n  intros n hn, \n  cases hn,\n  {split, swap, exact Δ, split, simp, rw hn, exact h},\n  {have : ∃ (i : seqt) (H : i ∈ l₂), sat builder n (i.main ++ i.hdld),\n     {apply bs_ex_seqt, exact hbs, exact hn},\n  {rcases this with ⟨w, hw, hsat⟩, split, swap, exact w, split, \n   {simp [hw]}, {exact hsat} } }\nend\n\ntheorem bs_forall_seqt : Π l Γ, \nbatch_sat_seqt l Γ → ∀ i ∈ Γ, ∃ m ∈ l, sat builder m (seqt.main i ++ i.hdld)\n| l Γ bs_nil := λ m hm, by simpa using hm\n| l Γ (bs_cons m Δ l₁ l₂ h hbs) := \nbegin\n  intros i hi,\n  cases hi, {split, swap, exact m, split, simp, rw hi, exact h},\n  {have : ∃ (n : model) (H : n ∈ l₁), sat builder n (seqt.main i ++ i.hdld),\n     {apply bs_forall_seqt, exact hbs, exact hi},\n  {rcases this with ⟨w, hw, hsat⟩, split, swap, exact w, split, \n   {simp [hw]}, {exact hsat} } }\nend\n\ntheorem sat_of_batch_sat_main : Π l Γ (h : modal_applicable Γ), \nbatch_sat_seqt l (unmodal_seqt Γ) → \nsat builder (cons h.v l) (Γ.main) := \nbegin\n  intros l Γ h hbs,\n  intros φ hφ,\n  cases hfml : φ,\n  case nnf.var : n \n  {rw hfml at hφ, simp, rw ←h.hv, \n   exact hφ },\n  case nnf.box : ψ \n  {exfalso, rw hfml at hφ, apply h.no_box_main, exact hφ},\n  case nnf.dia : ψ \n  {dsimp, \n   let se : seqt := \n{main := ψ :: (unbox (Γ.main ++ Γ.hdld)), hdld := [], \n pmain := begin intros v hv φ hsat hin hall, exfalso, apply list.not_mem_nil, exact hin end, \n phdld := box_only_nil},\n   have := bs_forall_seqt l (unmodal_seqt Γ) hbs se _, swap,\n   {dsimp [unmodal_seqt], apply list.mem_map_of_mem (λ φ, (⟨φ :: unbox (Γ.main++Γ.hdld),_,_,_⟩:seqt)), rw ←undia_iff, rw ←hfml, simp [hφ] }, \n   {rcases this with ⟨w, hw, hsat⟩, split, swap, exact w, split, \n    {simp [hw]}, {apply hsat, simp} } },\n  case nnf.neg : n\n  {dsimp, rw hfml at hφ, have : var n ∉ Γ.main, \n   {intro hin, have := h.no_contra_main, have := this hin, contradiction},\n    simp, rw ←h.hv, exact this },\n  case nnf.and : φ ψ\n  {rw hfml at hφ, have := h.satu.no_and, have := @this φ ψ, contradiction},\n  case nnf.or : φ ψ\n  {rw hfml at hφ, have := h.satu.no_or, have := @this φ ψ, contradiction}\nend\n\ntheorem sat_of_batch_sat_box : Π l Γ (h : modal_applicable Γ), \nbatch_sat_seqt l (unmodal_seqt Γ) → ∀ m∈l, ∀ φ, box φ ∈ Γ.hdld → force builder m φ := \nbegin\n  intros l Γ h hbs m hm φ hφ,\n  have := bs_ex_seqt _ _ hbs _ hm,\n  rcases this with ⟨i,hin,hi⟩,\n  have := unmodal_mem_box _ _ hin φ,\n  apply hi,\n  rw list.mem_append, left,\n  apply this, simp [hφ]\nend\n\ntheorem sat_of_batch_sat : Π l Γ (h : modal_applicable Γ), \nbatch_sat_seqt l (unmodal_seqt Γ) → \nsat builder (cons h.v l) (Γ.main++Γ.hdld) := \nbegin\n  intros l Γ h hbs,\n  apply sat_append,\n  {apply sat_of_batch_sat_main, assumption},\n  {intros φ hφ, have := box_only_ex Γ.phdld hφ, \n   cases this with w hw,\n   rw hw, dsimp, intros s' hs',\n   have := mem_of_mrel_tt hs', cases this,\n   {apply sat_of_batch_sat_box, exact h, exact hbs, exact this, rw hw at hφ, exact hφ},\n   { rw ←this, apply Γ.pmain, \n     apply sat_of_batch_sat_main, exact hbs, rw hw at hφ, exact hφ, \n     { intros, apply sat_of_batch_sat_box, repeat {assumption} } }\n}\nend\n", "meta": {"author": "minchaowu", "repo": "ModalTab", "sha": "9bb0bf17faf0554d907ef7bdd639648742889178", "save_path": "github-repos/lean/minchaowu-ModalTab", "path": "github-repos/lean/minchaowu-ModalTab/ModalTab-9bb0bf17faf0554d907ef7bdd639648742889178/src/KT/semantics.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6076631840431539, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3014579560004604}}
{"text": "\nimport Lib.Data.Traversable\nimport Lib.Data.List.Basic\n\nnamespace List\n\ninstance : Traversable List where\n  map := Functor.map\n  traverse := List.mapA\n  mapM := List.mapM\n  foldl := List.foldl\n  foldr := List.foldr\n  toList := id\n  length := List.length\n\n@[simp]\ntheorem foldMap_nil [Monoid ω] (f : α → ω) :\n  Foldable.foldMap f [] = One.one := rfl\n\n@[simp]\ntheorem foldMap_cons [Monoid ω] (f : α → ω) x xs :\n  Foldable.foldMap f (x :: xs) =\n  Foldable.foldMap f xs * f x := by\nsimp [Foldable.foldMap, Foldable.foldl, foldl]\nsymmetry\napply foldl_sim (SIM := (. * f x = .))\n. simp\nintros; simp [*]\n\ninstance : LawfulFoldable List where\n  foldl_sim :=  by\n    intros; apply List.foldl_sim <;> auto\n  toArray_toList := sorry\n  length_toList := by\n    intros; simp [Array.toList]; refl\n  foldl_toList := by\n    intros; simp [Array.toList]; refl\n  foldr_eq_foldMap := by\n    intros; simp [Foldable.foldr]\n    next f x x₀ =>\n    induction x generalizing x₀ <;> simp [foldr, *]\n\nopen Traversable\n\ninstance : LawfulTraversable List where\n  map_eq_traverse := by\n    intros; simp [(.<$>.), traverse];\n    next x =>\n    induction x <;> simp [List.map, List.mapA, *, Seq.seq]\n     <;> refl\n  traverse_eq_mapM := by\n    intros; simp [traverse, mapM]\n    next x =>\n    induction x <;>\n      simp [*, List.mapM, List.mapA, seq_eq_bind, map_eq_pure_bind, Traversable.mapM]\n  map_const := by intros; ext; next x =>\n    simp [(.∘.), (.<$>.)];\n    induction x <;> simp [*, List.map, Functor.mapConst]\n  id_map := by intros; simp [(.<$>.)]; next x =>\n    induction x <;> simp [List.map, *]\n  comp_map := by intros; simp [(.<$>.)]; next x =>\n    induction x <;> simp [List.map, *]\n  comp_traverse := by\n    intros; simp [traverse]\n    next x f g => admit\n    -- induction x <;> simp [*, List.mapA, comp_lam]\n    -- rw [Comp.run_seq]\n    -- rw [Comp.run_map, ← comp_map]\n    -- simp [*, (.∘.), seq_assoc]; congr\n  foldl_eq_traverse := by\n    intros; simp [Foldable.foldl, traverse]\n    next x x₀ =>\n    induction x generalizing x₀\n    <;> simp [List.foldl, List.mapA, *]\n    admit\n  traverse_sim := by\n    intros; next x R f g h =>\n    simp [traverse]\n    induction x <;> simp [List.mapA] <;> auto\n\nend List\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/List/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.30134434654370224}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nSupplementary theorems about the `char` type.\n-/\n\ninstance : linear_order char :=\n{ le_refl := λ a, @le_refl ℕ _ _,\n  le_trans := λ a b c, @le_trans ℕ _ _ _ _,\n  le_antisymm := λ a b h₁ h₂,\n    char.eq_of_veq $ le_antisymm h₁ h₂,\n  le_total := λ a b, @le_total ℕ _ _ _,\n  lt_iff_le_not_le := λ a b, @lt_iff_le_not_le ℕ _ _ _,\n  decidable_le := char.decidable_le,\n  decidable_eq := char.decidable_eq,\n  decidable_lt := char.decidable_lt,\n  ..char.has_le, ..char.has_lt }\n\nlemma char.of_nat_to_nat {c : char} (h : is_valid_char c.to_nat) :\n  char.of_nat c.to_nat = c :=\nbegin\n  rw [char.of_nat, dif_pos h],\n  cases c,\n  simp [char.to_nat]\nend\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/char.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.30111393716375856}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Prod instances for module and multiplicative actions\n\nThis file defines instances for binary product of modules\n-/\n\nnamespace prod\n\n\nprotected instance has_scalar {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_scalar α β]\n    [has_scalar α γ] : has_scalar α (β × γ) :=\n  has_scalar.mk fun (a : α) (p : β × γ) => (a • fst p, a • snd p)\n\n@[simp] theorem smul_fst {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_scalar α β]\n    [has_scalar α γ] (a : α) (x : β × γ) : fst (a • x) = a • fst x :=\n  rfl\n\n@[simp] theorem smul_snd {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_scalar α β]\n    [has_scalar α γ] (a : α) (x : β × γ) : snd (a • x) = a • snd x :=\n  rfl\n\n@[simp] theorem smul_mk {α : Type u_1} {β : Type u_2} {γ : Type u_3} [has_scalar α β]\n    [has_scalar α γ] (a : α) (b : β) (c : γ) : a • (b, c) = (a • b, a • c) :=\n  rfl\n\nprotected instance semimodule {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : semiring α}\n    [add_comm_monoid β] [add_comm_monoid γ] [semimodule α β] [semimodule α γ] :\n    semimodule α (β × γ) :=\n  semimodule.mk sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/module/prod_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555714, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3010380616348521}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.interactive\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# `dec_trivial` tactic\n\nThe `dec_trivial` tactic tries to use decidability to prove a goal.\nIt is basically a glorified wrapper around `exact dec_trivial`.\n\nThere is an extra option to make it a little bit smarter:\n`dec_trivial!` will revert all hypotheses on which the target depends,\nbefore it tries `exact dec_trivial`.\n-/\n\n/-- `dec_trivial` tries to use decidability to prove a goal\n(i.e., using `exact dec_trivial`).\nThe variant `dec_trivial!` will revert all hypotheses on which the target depends,\nbefore it tries `exact dec_trivial`.\n\nExample:\n```lean\nexample (n : ℕ) (h : n < 2) : n = 0 ∨ n = 1 :=\nby dec_trivial!\n```\n-/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/dec_trivial.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.301038061634852}}
{"text": "example (P : Prop) : ∀ {p : P}, P := by\n  exact fun {p} => p\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  intro h; exact h\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact @id _\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact no_implicit_lambda% id\n\nmacro \"exact'\" x:term : tactic => `(tactic| exact no_implicit_lambda% $x)\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  exact' id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  apply id\n\nexample (P : Prop) : ∀ p : P, P := by\n  have : _ := 1\n  apply id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  refine no_implicit_lambda% (have : _ := 1; ?_)\n  apply id\n\nexample (P : Prop) : ∀ {p : P}, P := by\n  have : _ := 1\n  apply id\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/impLambdaTac.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381372136563, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.30103804536885603}}
{"text": "/-\nCopyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.encodable.basic\nimport Mathlib.data.finset.basic\nimport Mathlib.data.set.disjointed\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Lattice operations on encodable types\n\nLemmas about lattice and set operations on encodable types\n\n## Implementation Notes\n\nThis is a separate file, to avoid unnecessary imports in basic files.\n\nPreviously some of these results were in the `measure_theory` folder.\n-/\n\nnamespace encodable\n\n\ntheorem supr_decode2 {α : Type u_1} {β : Type u_2} [encodable β] [complete_lattice α] (f : β → α) :\n    (supr fun (i : ℕ) => supr fun (b : β) => supr fun (H : b ∈ decode2 β i) => f b) =\n        supr fun (b : β) => f b :=\n  sorry\n\ntheorem Union_decode2 {α : Type u_1} {β : Type u_2} [encodable β] (f : β → set α) :\n    (set.Union fun (i : ℕ) => set.Union fun (b : β) => set.Union fun (H : b ∈ decode2 β i) => f b) =\n        set.Union fun (b : β) => f b :=\n  supr_decode2 f\n\ntheorem Union_decode2_cases {α : Type u_1} {β : Type u_2} [encodable β] {f : β → set α}\n    {C : set α → Prop} (H0 : C ∅) (H1 : ∀ (b : β), C (f b)) {n : ℕ} :\n    C (set.Union fun (b : β) => set.Union fun (H : b ∈ decode2 β n) => f b) :=\n  sorry\n\ntheorem Union_decode2_disjoint_on {α : Type u_1} {β : Type u_2} [encodable β] {f : β → set α}\n    (hd : pairwise (disjoint on f)) :\n    pairwise\n        (disjoint on\n          fun (i : ℕ) => set.Union fun (b : β) => set.Union fun (H : b ∈ decode2 β i) => f b) :=\n  sorry\n\nend encodable\n\n\nnamespace finset\n\n\ntheorem nonempty_encodable {α : Type u_1} (t : finset α) :\n    Nonempty (encodable (Subtype fun (i : α) => i ∈ t)) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/encodable/lattice_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.5428632831725053, "lm_q1q2_score": 0.30100166502389164}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Johan Commelin, Bhavik Mehta\n-/\nimport category_theory.equivalence\n\n/-!\n# Adjunctions between functors\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n`F ⊣ G` represents the data of an adjunction between two functors\n`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.\n\nWe provide various useful constructors:\n* `mk_of_hom_equiv`\n* `mk_of_unit_counit`\n* `left_adjoint_of_equiv` / `right_adjoint_of equiv`\n  construct a left/right adjoint of a given functor given the action on objects and\n  the relevant equivalence of morphism spaces.\n* `adjunction_of_equiv_left` / `adjunction_of_equiv_right` witness that these constructions\n  give adjunctions.\n\nThere are also typeclasses `is_left_adjoint` / `is_right_adjoint`, carrying data witnessing\nthat a given functor is a left or right adjoint.\nGiven `[is_left_adjoint F]`, a right adjoint of `F` can be constructed as `right_adjoint F`.\n\n`adjunction.comp` composes adjunctions.\n\n`to_equivalence` upgrades an adjunction to an equivalence,\ngiven witnesses that the unit and counit are pointwise isomorphisms.\nConversely `equivalence.to_adjunction` recovers the underlying adjunction from an equivalence.\n-/\n\nnamespace category_theory\nopen category\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v₁ v₂ v₃ u₁ u₂ u₃\n\nlocal attribute [elab_simple] whisker_left whisker_right\n\nvariables {C : Type u₁} [category.{v₁} C] {D : Type u₂} [category.{v₂} D]\n\n/--\n`F ⊣ G` represents the data of an adjunction between two functors\n`F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.\n\nTo construct an `adjunction` between two functors, it's often easier to instead use the\nconstructors `mk_of_hom_equiv` or `mk_of_unit_counit`. To construct a left adjoint,\nthere are also constructors `left_adjoint_of_equiv` and `adjunction_of_equiv_left` (as\nwell as their duals) which can be simpler in practice.\n\nUniqueness of adjoints is shown in `category_theory.adjunction.opposites`.\n\nSee <https://stacks.math.columbia.edu/tag/0037>.\n-/\nstructure adjunction (F : C ⥤ D) (G : D ⥤ C) :=\n(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\n(unit : 𝟭 C ⟶ F.comp G)\n(counit : G.comp F ⟶ 𝟭 D)\n(hom_equiv_unit' : Π {X Y f}, (hom_equiv X Y) f = (unit : _ ⟶ _).app X ≫ G.map f . obviously)\n(hom_equiv_counit' : Π {X Y g}, (hom_equiv X Y).symm g = F.map g ≫ counit.app Y . obviously)\n\ninfix ` ⊣ `:15 := adjunction\n\n/-- A class giving a chosen right adjoint to the functor `left`. -/\nclass is_left_adjoint (left : C ⥤ D) :=\n(right : D ⥤ C)\n(adj : left ⊣ right)\n\n/-- A class giving a chosen left adjoint to the functor `right`. -/\nclass is_right_adjoint (right : D ⥤ C) :=\n(left : C ⥤ D)\n(adj : left ⊣ right)\n\n/-- Extract the left adjoint from the instance giving the chosen adjoint. -/\ndef left_adjoint (R : D ⥤ C) [is_right_adjoint R] : C ⥤ D :=\nis_right_adjoint.left R\n/-- Extract the right adjoint from the instance giving the chosen adjoint. -/\ndef right_adjoint (L : C ⥤ D) [is_left_adjoint L] : D ⥤ C :=\nis_left_adjoint.right L\n\n/-- The adjunction associated to a functor known to be a left adjoint. -/\ndef adjunction.of_left_adjoint (left : C ⥤ D) [is_left_adjoint left] :\n  adjunction left (right_adjoint left) :=\nis_left_adjoint.adj\n/-- The adjunction associated to a functor known to be a right adjoint. -/\ndef adjunction.of_right_adjoint (right : C ⥤ D) [is_right_adjoint right] :\n  adjunction (left_adjoint right) right :=\nis_right_adjoint.adj\n\nnamespace adjunction\n\nrestate_axiom hom_equiv_unit'\nrestate_axiom hom_equiv_counit'\nattribute [simp, priority 10] hom_equiv_unit hom_equiv_counit\n\nsection\n\nvariables {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) {X' X : C} {Y Y' : D}\n\nlemma hom_equiv_id (X : C) : adj.hom_equiv X _ (𝟙 _) = adj.unit.app X := by simp\n\nlemma hom_equiv_symm_id (X : D) : (adj.hom_equiv _ X).symm (𝟙 _) = adj.counit.app X := by simp\n\n@[simp, priority 10] lemma hom_equiv_naturality_left_symm (f : X' ⟶ X) (g : X ⟶ G.obj Y) :\n  (adj.hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (adj.hom_equiv X Y).symm g :=\nby rw [hom_equiv_counit, F.map_comp, assoc, adj.hom_equiv_counit.symm]\n\n@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :\n  (adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=\nby rw [← equiv.eq_symm_apply]; simp [-hom_equiv_unit]\n\n@[simp, priority 10] lemma hom_equiv_naturality_right (f : F.obj X ⟶ Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y') (f ≫ g) = (adj.hom_equiv X Y) f ≫ G.map g :=\nby rw [hom_equiv_unit, G.map_comp, ← assoc, ←hom_equiv_unit]\n\n@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=\nby rw [equiv.symm_apply_eq]; simp [-hom_equiv_counit]\n\n@[simp] lemma left_triangle :\n  (whisker_right adj.unit F) ≫ (whisker_left F adj.counit) = nat_trans.id _ :=\nbegin\n  ext, dsimp,\n  erw [← adj.hom_equiv_counit, equiv.symm_apply_eq, adj.hom_equiv_unit],\n  simp\nend\n\n@[simp] \n\n@[simp, reassoc] lemma left_triangle_components :\n  F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X) :=\ncongr_arg (λ (t : nat_trans _ (𝟭 C ⋙ F)), t.app X) adj.left_triangle\n\n@[simp, reassoc] lemma right_triangle_components {Y : D} :\n  adj.unit.app (G.obj Y) ≫ G.map (adj.counit.app Y) = 𝟙 (G.obj Y) :=\ncongr_arg (λ (t : nat_trans _ (G ⋙ 𝟭 C)), t.app Y) adj.right_triangle\n\n@[simp, reassoc] lemma counit_naturality {X Y : D} (f : X ⟶ Y) :\n  F.map (G.map f) ≫ (adj.counit).app Y = (adj.counit).app X ≫ f :=\nadj.counit.naturality f\n\n@[simp, reassoc] lemma unit_naturality {X Y : C} (f : X ⟶ Y) :\n  (adj.unit).app X ≫ G.map (F.map f) = f ≫ (adj.unit).app Y :=\n(adj.unit.naturality f).symm\n\nlemma hom_equiv_apply_eq {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :\n  adj.hom_equiv A B f = g ↔ f = (adj.hom_equiv A B).symm g :=\n⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩\n\nlemma eq_hom_equiv_apply {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :\n  g = adj.hom_equiv A B f ↔ (adj.hom_equiv A B).symm g = f :=\n⟨λ h, by {cases h, simp}, λ h, by {cases h, simp}⟩\n\nend\n\nend adjunction\n\nnamespace adjunction\n\n/--\nThis is an auxiliary data structure useful for constructing adjunctions.\nSee `adjunction.mk_of_hom_equiv`.\nThis structure won't typically be used anywhere else.\n-/\n@[nolint has_nonempty_instance]\nstructure core_hom_equiv (F : C ⥤ D) (G : D ⥤ C) :=\n(hom_equiv : Π (X Y), (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\n(hom_equiv_naturality_left_symm' : Π {X' X Y} (f : X' ⟶ X) (g : X ⟶ G.obj Y),\n  (hom_equiv X' Y).symm (f ≫ g) = F.map f ≫ (hom_equiv X Y).symm g . obviously)\n(hom_equiv_naturality_right' : Π {X Y Y'} (f : F.obj X ⟶ Y) (g : Y ⟶ Y'),\n  (hom_equiv X Y') (f ≫ g) = (hom_equiv X Y) f ≫ G.map g . obviously)\n\nnamespace core_hom_equiv\n\nrestate_axiom hom_equiv_naturality_left_symm'\nrestate_axiom hom_equiv_naturality_right'\nattribute [simp, priority 10] hom_equiv_naturality_left_symm hom_equiv_naturality_right\n\nvariables {F : C ⥤ D} {G : D ⥤ C} (adj : core_hom_equiv F G) {X' X : C} {Y Y' : D}\n\n@[simp] lemma hom_equiv_naturality_left (f : X' ⟶ X) (g : F.obj X ⟶ Y) :\n  (adj.hom_equiv X' Y) (F.map f ≫ g) = f ≫ (adj.hom_equiv X Y) g :=\nby rw [← equiv.eq_symm_apply]; simp\n\n@[simp] lemma hom_equiv_naturality_right_symm (f : X ⟶ G.obj Y) (g : Y ⟶ Y') :\n  (adj.hom_equiv X Y').symm (f ≫ G.map g) = (adj.hom_equiv X Y).symm f ≫ g :=\nby rw [equiv.symm_apply_eq]; simp\n\nend core_hom_equiv\n\n/--\nThis is an auxiliary data structure useful for constructing adjunctions.\nSee `adjunction.mk_of_unit_counit`.\nThis structure won't typically be used anywhere else.\n-/\n@[nolint has_nonempty_instance]\nstructure core_unit_counit (F : C ⥤ D) (G : D ⥤ C) :=\n(unit : 𝟭 C ⟶ F.comp G)\n(counit : G.comp F ⟶ 𝟭 D)\n(left_triangle' : whisker_right unit F ≫ (functor.associator F G F).hom ≫ whisker_left F counit =\n  nat_trans.id (𝟭 C ⋙ F) . obviously)\n(right_triangle' : whisker_left G unit ≫ (functor.associator G F G).inv ≫ whisker_right counit G =\n  nat_trans.id (G ⋙ 𝟭 C) . obviously)\n\nnamespace core_unit_counit\n\nrestate_axiom left_triangle'\nrestate_axiom right_triangle'\nattribute [simp] left_triangle right_triangle\n\nend core_unit_counit\n\nvariables {F : C ⥤ D} {G : D ⥤ C}\n\n/-- Construct an adjunction between `F` and `G` out of a natural bijection between each\n`F.obj X ⟶ Y` and `X ⟶ G.obj Y`. -/\n@[simps]\ndef mk_of_hom_equiv (adj : core_hom_equiv F G) : F ⊣ G :=\n{ unit :=\n  { app := λ X, (adj.hom_equiv X (F.obj X)) (𝟙 (F.obj X)),\n    naturality' :=\n    begin\n      intros,\n      erw [← adj.hom_equiv_naturality_left, ← adj.hom_equiv_naturality_right],\n      dsimp, simp -- See note [dsimp, simp].\n    end },\n  counit :=\n  { app := λ Y, (adj.hom_equiv _ _).inv_fun (𝟙 (G.obj Y)),\n    naturality' :=\n    begin\n      intros,\n      erw [← adj.hom_equiv_naturality_left_symm, ← adj.hom_equiv_naturality_right_symm],\n      dsimp, simp\n    end },\n  hom_equiv_unit' := λ X Y f, by erw [← adj.hom_equiv_naturality_right]; simp,\n  hom_equiv_counit' := λ X Y f, by erw [← adj.hom_equiv_naturality_left_symm]; simp,\n  .. adj }\n\n/-- Construct an adjunction between functors `F` and `G` given a unit and counit for the adjunction\nsatisfying the triangle identities. -/\n@[simps]\ndef mk_of_unit_counit (adj : core_unit_counit F G) : F ⊣ G :=\n{ hom_equiv := λ X Y,\n  { to_fun := λ f, adj.unit.app X ≫ G.map f,\n    inv_fun := λ g, F.map g ≫ adj.counit.app Y,\n    left_inv := λ f, begin\n      change F.map (_ ≫ _) ≫ _ = _,\n      rw [F.map_comp, assoc, ←functor.comp_map, adj.counit.naturality, ←assoc],\n      convert id_comp f,\n      have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.left_triangle,\n      dsimp at t,\n      simp only [id_comp] at t,\n      exact t,\n    end,\n    right_inv := λ g, begin\n      change _ ≫ G.map (_ ≫ _) = _,\n      rw [G.map_comp, ←assoc, ←functor.comp_map, ←adj.unit.naturality, assoc],\n      convert comp_id g,\n      have t := congr_arg (λ t : nat_trans _ _, t.app _) adj.right_triangle,\n      dsimp at t,\n      simp only [id_comp] at t,\n      exact t,\n  end },\n  .. adj }\n\n/-- The adjunction between the identity functor on a category and itself. -/\ndef id : 𝟭 C ⊣ 𝟭 C :=\n{ hom_equiv := λ X Y, equiv.refl _,\n  unit := 𝟙 _,\n  counit := 𝟙 _ }\n\n-- Satisfy the inhabited linter.\ninstance : inhabited (adjunction (𝟭 C) (𝟭 C)) := ⟨id⟩\n\n/-- If F and G are naturally isomorphic functors, establish an equivalence of hom-sets. -/\n@[simps]\ndef equiv_homset_left_of_nat_iso\n  {F F' : C ⥤ D} (iso : F ≅ F') {X : C} {Y : D} :\n  (F.obj X ⟶ Y) ≃ (F'.obj X ⟶ Y) :=\n{ to_fun := λ f, iso.inv.app _ ≫ f,\n  inv_fun := λ g, iso.hom.app _ ≫ g,\n  left_inv := λ f, by simp,\n  right_inv := λ g, by simp }\n\n/-- If G and H are naturally isomorphic functors, establish an equivalence of hom-sets. -/\n@[simps]\ndef equiv_homset_right_of_nat_iso\n  {G G' : D ⥤ C} (iso : G ≅ G') {X : C} {Y : D} :\n  (X ⟶ G.obj Y) ≃ (X ⟶ G'.obj Y) :=\n{ to_fun := λ f, f ≫ iso.hom.app _,\n  inv_fun := λ g, g ≫ iso.inv.app _,\n  left_inv := λ f, by simp,\n  right_inv := λ g, by simp }\n\n/-- Transport an adjunction along an natural isomorphism on the left. -/\ndef of_nat_iso_left\n  {F G : C ⥤ D} {H : D ⥤ C} (adj : F ⊣ H) (iso : F ≅ G) :\n  G ⊣ H :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y, (equiv_homset_left_of_nat_iso iso.symm).trans (adj.hom_equiv X Y) }\n\n/-- Transport an adjunction along an natural isomorphism on the right. -/\ndef of_nat_iso_right\n  {F : C ⥤ D} {G H : D ⥤ C} (adj : F ⊣ G) (iso : G ≅ H) :\n  F ⊣ H :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ X Y, (adj.hom_equiv X Y).trans (equiv_homset_right_of_nat_iso iso) }\n\n/-- Transport being a right adjoint along a natural isomorphism. -/\ndef right_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_right_adjoint F] :\n  is_right_adjoint G :=\n{ left := r.left,\n  adj := of_nat_iso_right r.adj h }\n\n/-- Transport being a left adjoint along a natural isomorphism. -/\ndef left_adjoint_of_nat_iso {F G : C ⥤ D} (h : F ≅ G) [r : is_left_adjoint F] : is_left_adjoint G :=\n{ right := r.right,\n  adj := of_nat_iso_left r.adj h }\n\nsection\nvariables {E : Type u₃} [ℰ : category.{v₃} E] {H : D ⥤ E} {I : E ⥤ D}\n\n/--\nComposition of adjunctions.\n\nSee <https://stacks.math.columbia.edu/tag/0DV0>.\n-/\ndef comp (adj₁ : F ⊣ G) (adj₂ : H ⊣ I) : F ⋙ H ⊣ I ⋙ G :=\n{ hom_equiv := λ X Z, equiv.trans (adj₂.hom_equiv _ _) (adj₁.hom_equiv _ _),\n  unit := adj₁.unit ≫\n  (whisker_left F $ whisker_right adj₂.unit G) ≫ (functor.associator _ _ _).inv,\n  counit := (functor.associator _ _ _).hom ≫\n    (whisker_left I $ whisker_right adj₁.counit H) ≫ adj₂.counit }\n\n/-- If `F` and `G` are left adjoints then `F ⋙ G` is a left adjoint too. -/\ninstance left_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] (F : C ⥤ D) (G : D ⥤ E)\n  [Fl : is_left_adjoint F] [Gl : is_left_adjoint G] : is_left_adjoint (F ⋙ G) :=\n{ right := Gl.right ⋙ Fl.right,\n  adj := Fl.adj.comp Gl.adj }\n\n/-- If `F` and `G` are right adjoints then `F ⋙ G` is a right adjoint too. -/\ninstance right_adjoint_of_comp {E : Type u₃} [ℰ : category.{v₃} E] {F : C ⥤ D} {G : D ⥤ E}\n  [Fr : is_right_adjoint F] [Gr : is_right_adjoint G] : is_right_adjoint (F ⋙ G) :=\n{ left := Gr.left ⋙ Fr.left,\n  adj := Gr.adj.comp Fr.adj }\n\nend\n\nsection construct_left\n-- Construction of a left adjoint. In order to construct a left\n-- adjoint to a functor G : D → C, it suffices to give the object part\n-- of a functor F : C → D together with isomorphisms Hom(FX, Y) ≃\n-- Hom(X, GY) natural in Y. The action of F on morphisms can be\n-- constructed from this data.\nvariables {F_obj : C → D} {G}\nvariables (e : Π X Y, (F_obj X ⟶ Y) ≃ (X ⟶ G.obj Y))\nvariables (he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g)\ninclude he\n\nprivate lemma he' {X Y Y'} (f g) : (e X Y').symm (f ≫ G.map g) = (e X Y).symm f ≫ g :=\nby intros; rw [equiv.symm_apply_eq, he]; simp\n\n/-- Construct a left adjoint functor to `G`, given the functor's value on objects `F_obj` and\na bijection `e` between `F_obj X ⟶ Y` and `X ⟶ G.obj Y` satisfying a naturality law\n`he : ∀ X Y Y' g h, e X Y' (h ≫ g) = e X Y h ≫ G.map g`.\nDual to `right_adjoint_of_equiv`. -/\n@[simps]\ndef left_adjoint_of_equiv : C ⥤ D :=\n{ obj := F_obj,\n  map := λ X X' f, (e X (F_obj X')).symm (f ≫ e X' (F_obj X') (𝟙 _)),\n  map_comp' := λ X X' X'' f f', begin\n    rw [equiv.symm_apply_eq, he, equiv.apply_symm_apply],\n    conv { to_rhs, rw [assoc, ←he, id_comp, equiv.apply_symm_apply] },\n    simp\n  end }\n\n/-- Show that the functor given by `left_adjoint_of_equiv` is indeed left adjoint to `G`. Dual\nto `adjunction_of_equiv_right`. -/\n@[simps]\ndef adjunction_of_equiv_left : left_adjoint_of_equiv e he ⊣ G :=\nmk_of_hom_equiv\n{ hom_equiv := e,\n  hom_equiv_naturality_left_symm' :=\n  begin\n    intros,\n    erw [← he' e he, ← equiv.apply_eq_iff_eq],\n    simp [(he _ _ _ _ _).symm]\n  end }\n\nend construct_left\n\nsection construct_right\n-- Construction of a right adjoint, analogous to the above.\nvariables {F} {G_obj : D → C}\nvariables (e : Π X Y, (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y))\nvariables (he : ∀ X' X Y f g, e X' Y (F.map f ≫ g) = f ≫ e X Y g)\ninclude he\n\nprivate lemma he' {X' X Y} (f g) : F.map f ≫ (e X Y).symm g = (e X' Y).symm (f ≫ g) :=\nby intros; rw [equiv.eq_symm_apply, he]; simp\n\n/-- Construct a right adjoint functor to `F`, given the functor's value on objects `G_obj` and\na bijection `e` between `F.obj X ⟶ Y` and `X ⟶ G_obj Y` satisfying a naturality law\n`he : ∀ X Y Y' g h, e X' Y (F.map f ≫ g) = f ≫ e X Y g`.\nDual to `left_adjoint_of_equiv`. -/\n@[simps]\ndef right_adjoint_of_equiv : D ⥤ C :=\n{ obj := G_obj,\n  map := λ Y Y' g, (e (G_obj Y) Y') ((e (G_obj Y) Y).symm (𝟙 _) ≫ g),\n  map_comp' := λ Y Y' Y'' g g', begin\n    rw [← equiv.eq_symm_apply, ← he' e he, equiv.symm_apply_apply],\n    conv { to_rhs, rw [← assoc, he' e he, comp_id, equiv.symm_apply_apply] },\n    simp\n  end }\n\n/-- Show that the functor given by `right_adjoint_of_equiv` is indeed right adjoint to `F`. Dual\nto `adjunction_of_equiv_left`. -/\n@[simps]\ndef adjunction_of_equiv_right : F ⊣ right_adjoint_of_equiv e he :=\nmk_of_hom_equiv\n{ hom_equiv := e,\n  hom_equiv_naturality_left_symm' := by intros; rw [equiv.symm_apply_eq, he]; simp,\n  hom_equiv_naturality_right' :=\n  begin\n    intros X Y Y' g h,\n    erw [←he, equiv.apply_eq_iff_eq, ←assoc, he' e he, comp_id, equiv.symm_apply_apply]\n  end }\n\nend construct_right\n\n/--\nIf the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the\nadjunction to an equivalence.\n-/\n@[simps]\nnoncomputable\ndef to_equivalence (adj : F ⊣ G) [∀ X, is_iso (adj.unit.app X)] [∀ Y, is_iso (adj.counit.app Y)] :\n  C ≌ D :=\n{ functor := F,\n  inverse := G,\n  unit_iso := nat_iso.of_components (λ X, as_iso (adj.unit.app X)) (by simp),\n  counit_iso := nat_iso.of_components (λ Y, as_iso (adj.counit.app Y)) (by simp) }\n\n/--\nIf the unit and counit for the adjunction corresponding to a right adjoint functor are (pointwise)\nisomorphisms, then the functor is an equivalence of categories.\n-/\n@[simps]\nnoncomputable\ndef is_right_adjoint_to_is_equivalence [is_right_adjoint G]\n  [∀ X, is_iso ((adjunction.of_right_adjoint G).unit.app X)]\n  [∀ Y, is_iso ((adjunction.of_right_adjoint G).counit.app Y)] :\n  is_equivalence G :=\nis_equivalence.of_equivalence_inverse (adjunction.of_right_adjoint G).to_equivalence\n\nend adjunction\n\nopen adjunction\n\nnamespace equivalence\n\n/-- The adjunction given by an equivalence of categories. (To obtain the opposite adjunction,\nsimply use `e.symm.to_adjunction`. -/\ndef to_adjunction (e : C ≌ D) : e.functor ⊣ e.inverse :=\nmk_of_unit_counit ⟨e.unit, e.counit,\n  by { ext, dsimp, simp only [id_comp], exact e.functor_unit_comp _, },\n  by { ext, dsimp, simp only [id_comp], exact e.unit_inverse_comp _, }⟩\n\n@[simp] lemma as_equivalence_to_adjunction_unit {e : C ≌ D} :\n  e.functor.as_equivalence.to_adjunction.unit = e.unit :=\nrfl\n\n@[simp] lemma as_equivalence_to_adjunction_counit {e : C ≌ D} :\n  e.functor.as_equivalence.to_adjunction.counit = e.counit :=\nrfl\n\nend equivalence\n\nnamespace functor\n\n/-- An equivalence `E` is left adjoint to its inverse. -/\ndef adjunction (E : C ⥤ D) [is_equivalence E] : E ⊣ E.inv :=\n(E.as_equivalence).to_adjunction\n\n/-- If `F` is an equivalence, it's a left adjoint. -/\n@[priority 10]\ninstance left_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_left_adjoint F :=\n{ right := _,\n  adj := functor.adjunction F }\n\n@[simp]\nlemma right_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : right_adjoint F = inv F :=\nrfl\n\n/-- If `F` is an equivalence, it's a right adjoint. -/\n@[priority 10]\ninstance right_adjoint_of_equivalence {F : C ⥤ D} [is_equivalence F] : is_right_adjoint F :=\n{ left := _,\n  adj := functor.adjunction F.inv }\n\n@[simp]\nlemma left_adjoint_of_is_equivalence {F : C ⥤ D} [is_equivalence F] : left_adjoint F = inv F :=\nrfl\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/adjunction/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3010016488290689}}
{"text": "open tactic\n\n-- meta def extract_equals (e : expr) : option (expr × expr) :=\n-- match e with\n-- | expr.app f x := sorry\n-- end\n\n\nmeta def extract_equals : expr → tactic (option (expr × expr))\n| (expr.app (expr.app f x) y) :=\n  do e ← mk_const `eq,\n     trace e,\n     trace f,\n     if f = e then\n       return (some (x, y))\n     else\n       return none\n| _ := none\n\nmeta def extract_equals' : expr → option (expr × expr)\n| `(%%a = %%b) := some (a, b)\n| _ := none\n\n\nmeta def swp : tactic unit :=\ndo t ← target,\n   r ← extract_equals t,\n   trace r,\n   trace \"hello world\",\n   trace t\n\nmeta def swp' : tactic unit :=\ndo t ← target,\n   (a, b) ← extract_equals' t,\n   ng ← to_expr ``(%%b = %%a),\n   trace (a, b),\n   eqs ← to_expr ``(eq.symm),\n   apply eqs,\n   skip\n\n#check expr\n-- set_option pp.all true\n\nlemma foo : 1 + 1 = 3 :=\nbegin\n  symmetry,\n  swp',\nend", "meta": {"author": "aymonwuolanne", "repo": "lean-theorem-proving", "sha": "701a7873f9f66f2c24962b02450ec90b20935702", "save_path": "github-repos/lean/aymonwuolanne-lean-theorem-proving", "path": "github-repos/lean/aymonwuolanne-lean-theorem-proving/lean-theorem-proving-701a7873f9f66f2c24962b02450ec90b20935702/src/tactic_example.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.300986814508533}}
{"text": "/- \n  Second argument of the ring lemma.\n\n  We show that the map R[1/fᵢ] → R[1/fᵢfⱼ] inverts fᵢ/1 * fⱼ/1.\n-/\n\nimport ring_theory.localization\nimport to_mathlib.localization.localization_alt\nimport spectrum_of_a_ring.structure_presheaf\nimport spectrum_of_a_ring.structure_presheaf_localization\nimport spectrum_of_a_ring.structure_presheaf_res\nimport spectrum_of_a_ring.structure_sheaf_locality\n\nuniverse u\n\nlocal attribute [instance] classical.prop_decidable\n\nnoncomputable theory\n\nsection structure_presheaf\n\nopen topological_space\nopen classical\nopen localization\nopen localization_alt\n\nvariables {R : Type u} [comm_ring R]\nvariables {U V W : opens (Spec R)} \nvariables (BU : U ∈ D_fs R) (BV : V ∈ D_fs R) (BW : W ∈ D_fs R) \nvariables (HVU : V ⊆ U) (HWU : W ⊆ U)\n\ndef BVW : V ∩ W ∈ D_fs R := (D_fs_standard_basis R).2 BV BW\ndef HVWU : V ∩ W ⊆ U := set.subset.trans (set.inter_subset_left V W) HVU\n\ndef structure_presheaf_on_basis.res_to_inter \n: localization R (S U) → localization R (S (V ∩ W)) :=\nstructure_presheaf_on_basis.res BU (BVW BV BW) (HVWU HVU)\n\ndef structure_presheaf_on_basis.res_to_inter_left\n: localization R (S V) → localization R (S (V ∩ W)) :=\nstructure_presheaf_on_basis.res BV (BVW BV BW) (set.inter_subset_left V W)\n\ndef structure_presheaf_on_basis.res_to_inter_right\n: localization R (S W) → localization R (S (V ∩ W)) :=\nstructure_presheaf_on_basis.res BW (BVW BV BW) (set.inter_subset_right V W)\n\ninstance structure_presheaf_on_basis.res_to_inter.is_ring_hom \n: is_ring_hom (structure_presheaf_on_basis.res_to_inter BU BV BW HVU) :=\nby simp [structure_presheaf_on_basis.res_to_inter, structure_presheaf_on_basis.res]; \nby apply_instance\n\ninstance structure_presheaf_on_basis.res_to_inter_to_inter_left.is_ring_hom \n: is_ring_hom (structure_presheaf_on_basis.res_to_inter_left BV BW) :=\nby simp [structure_presheaf_on_basis.res_to_inter_left, structure_presheaf_on_basis.res]; \nby apply_instance\n\ninstance structure_presheaf_on_basis.res_to_inter_to_inter_right.is_ring_hom \n: is_ring_hom (structure_presheaf_on_basis.res_to_inter_right BV BW) :=\nby simp [structure_presheaf_on_basis.res_to_inter_right, structure_presheaf_on_basis.res]; \nby apply_instance\n\n-- (f' g')^e = a * (fg)' \n\ninclude BU HVU\n\nlemma pow_eq.fmulg : ∃ a : R, ∃ e : ℕ, ((some BV) * (some BW))^e = a * (some (BVW BV BW)) :=\nbegin\n  let f := some BV,\n  let g := some BW,\n  let fg := some (BVW BV BW),\n  -- D(f*g) ⊆ D(fg).\n  have HDfg : Spec.D'(f*g) ⊆ Spec.D'(fg),\n    rw Spec.D'.product_eq_inter,\n    show Spec.DO R f ∩ Spec.DO R g ⊆ Spec.DO R fg,\n    rw [←some_spec BV, ←some_spec BW, ←some_spec (BVW BV BW)],\n    exact set.subset.refl _,\n  -- Hence (f*g)^e = a * fg.\n  exact pow_eq.of_Dfs_subset HDfg,\nend\n\nlemma pow_eq.fg : ∃ a : R, ∃ e : ℕ, (some (BVW BV BW))^e = a * ((some BV) * (some BW)) :=\nbegin\n  let f := some BV,\n  let g := some BW,\n  let fg := some (BVW BV BW),\n  -- D(fg) ⊆ D(f*g).\n  have HDfg : Spec.D'(fg) ⊆ Spec.D'(f*g),\n    rw Spec.D'.product_eq_inter,\n    show Spec.DO R fg ⊆ Spec.DO R f ∩ Spec.DO R g,\n    rw [←some_spec BV, ←some_spec BW, ←some_spec (BVW BV BW)],\n    exact set.subset.refl _,\n  -- Hence (f*g)^e = a * fg.\n  exact pow_eq.of_Dfs_subset HDfg,\nend\n\nlemma structure_presheaf.res_to_inter.inverts_data\n: inverts_data \n    (powers ((of (some BV)) * (of (some BW)))) \n    (structure_presheaf_on_basis.res_to_inter BU BV BW HVU) :=\nbegin\n  rcases (indefinite_description _ (pow_eq.fg BU BV BW HVU)) with ⟨a, Ha⟩,\n  rcases (indefinite_description _ Ha) with ⟨e, Hea⟩,\n  clear Ha,\n  let f := some BV,\n  let g := some BW,\n  let fg := some (BVW BV BW),\n  -- Using structure_presheaf.res.inverts_data\n  rintros ⟨s, Hs⟩,\n  rcases (indefinite_description _ Hs) with ⟨n, Hn⟩,\n  have Hans : s * (of (a^n)) ∈ powers ((of : R → localization R (S U)) fg),\n    rw is_semiring_hom.map_pow (of : R → localization R (S U)),\n    rw [←Hn,  ←mul_pow],\n    iterate 2 { rw ←is_ring_hom.map_mul (of : R → localization R (S U)), },\n    rw [mul_comm, ←Hea, is_semiring_hom.map_pow (of : R → localization R (S U)), ←pow_mul],\n    exact ⟨e * n, rfl⟩,\n  rcases (structure_presheaf.res.inverts_data BU (BVW BV BW) (HVWU HVU) ⟨s * (of (a^n)), Hans⟩) with ⟨w, Hw⟩,\n  rw ←is_ring_hom.map_mul (of : R → localization R (S U)) at Hn,\n  rw ←is_semiring_hom.map_pow (of : R → localization R (S U)) at Hn,\n  dsimp [structure_presheaf_on_basis.res_to_inter],\n  dsimp only [subtype.coe_mk] at Hw,\n  rw ←Hn,\n  rw ←Hn at Hw,\n  rw is_ring_hom.map_mul (structure_presheaf_on_basis.res BU (BVW BV BW) (HVWU HVU)) at Hw,\n  use [(structure_presheaf_on_basis.res BU (BVW BV BW) (HVWU HVU) (of (a^n))) * w],\n  rw ←mul_assoc,\n  exact Hw,\nend\n\nlemma structure_presheaf.res_to_inter.has_denom_data\n: has_denom_data \n    (powers ((of (some BV)) * (of (some BW)))) \n    (structure_presheaf_on_basis.res_to_inter BU BV BW HVU) :=\nbegin\n  rcases (indefinite_description _ (pow_eq.fmulg BU BV BW HVU)) with ⟨a, Ha⟩,\n  rcases (indefinite_description _ Ha) with ⟨e, Hea⟩,\n  clear Ha,\n  let f := some BV,\n  let g := some BW,\n  let fg := some (BVW BV BW),\n  -- Using structure_presheaf.res.has_denom\n  -- This gives us ( fg^n, y ∈ R[1/S(U)] )\n  -- We have (f*g)^(n*e) = (fg)^n * a^n \n  -- So let x = y * (of a)^n.\n  intros x,\n  rcases (structure_presheaf.res.has_denom_data BU (BVW BV BW) (HVWU HVU) x) with ⟨⟨⟨q, Hq⟩, p⟩, Hpq⟩,\n  dsimp only [subtype.coe_mk] at Hpq,\n  rcases (indefinite_description _ Hq) with ⟨n, Hn⟩,\n  use [⟨⟨(of f * of g)^(e * n), ⟨e * n, rfl⟩⟩, p * (of a)^n⟩],\n  dsimp [structure_presheaf_on_basis.res_to_inter],\n  rw ←is_ring_hom.map_mul (of : R → localization R (S U)),\n  rw ←is_semiring_hom.map_pow (of : R → localization R (S U)),\n  rw [pow_mul, Hea],\n  rw is_semiring_hom.map_pow (of : R → localization R (S U)),\n  rw is_ring_hom.map_mul (of : R → localization R (S U)),\n  rw [mul_pow, Hn, mul_comm p],\n  iterate 2 { rw is_ring_hom.map_mul (structure_presheaf_on_basis.res BU (BVW BV BW) (HVWU HVU)), },\n  rw [←Hpq, ←mul_assoc],\nend\n\nlemma structure_presheaf.res_to_inter.ker_le\n: ker (structure_presheaf_on_basis.res_to_inter BU BV BW HVU) \n≤ submonoid_ann (powers ((of (some BV)) * (of (some BW)))) :=\nbegin \n  rcases (indefinite_description _ (pow_eq.fmulg BU BV BW HVU)) with ⟨a, Ha⟩,\n  rcases (indefinite_description _ Ha) with ⟨e, Hea⟩,\n  clear Ha,\n  let f := some BV,\n  let g := some BW,\n  let fg := some (BVW BV BW),\n  -- Using structure_presheaf.res.ker_le\n  intros x Hx,\n  dsimp [structure_presheaf_on_basis.res_to_inter] at Hx,\n  replace Hx := (structure_presheaf.res.ker_le BU (BVW BV BW) (HVWU HVU)) Hx,\n  rcases Hx with ⟨⟨⟨u, ⟨v, ⟨n, Hv⟩⟩⟩, Hxfgn⟩, Hx⟩,\n  dsimp at Hx,\n  dsimp at Hxfgn,\n  rw [Hx, ←Hv] at Hxfgn; clear Hx; clear Hv,\n  let fgne : localization R (S U) := (of f * of g)^(e * n),\n  have Hfgne : fgne ∈ (powers (of f * of g) : set (localization R (S U))) := ⟨e * n, rfl⟩,\n  have Hxfgne : x * fgne = 0,\n    dsimp [fgne],\n    rw ←is_ring_hom.map_mul (of : R → localization R (S U)),\n    rw ←is_semiring_hom.map_pow (of : R → localization R (S U)),\n    rw [pow_mul, Hea],\n    rw is_semiring_hom.map_pow (of : R → localization R (S U)),\n    rw is_ring_hom.map_mul (of : R → localization R (S U)),\n    rw [mul_pow, mul_comm ((of a)^n), ←mul_assoc, Hxfgn, zero_mul],\n  use ⟨⟨x, ⟨fgne, Hfgne⟩⟩, Hxfgne⟩,\nend\n\nlemma structure_presheaf.res_to_inter.localization\n: is_localization_data \n    (powers ((of (some BV)) * (of (some BW)))) \n    (structure_presheaf_on_basis.res_to_inter BU BV BW HVU) :=\n{ inverts := structure_presheaf.res_to_inter.inverts_data BU BV BW HVU,\n  has_denom := structure_presheaf.res_to_inter.has_denom_data BU BV BW HVU, \n  ker_le := structure_presheaf.res_to_inter.ker_le BU BV BW HVU }\n\nend structure_presheaf\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/spectrum_of_a_ring/structure_sheaf_gluing.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.30096313101869365}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Johannes Hölzl\n-/\n\nsection sigma\nvariables {α : Type*} {β : α → Type*}\n\ninstance [inhabited α] [inhabited (β (default α))] : inhabited (sigma β) :=\n⟨⟨default α, default (β (default α))⟩⟩\n\ninstance [h₁ : decidable_eq α] [h₂ : ∀a, decidable_eq (β a)] : decidable_eq (sigma β)\n| ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ := match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with\n  | _, b₁, _, b₂, is_true (eq.refl a) :=\n    match b₁, b₂, h₂ a b₁ b₂ with\n    | _, _, is_true (eq.refl b) := is_true rfl\n    | b₁, b₂, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n $ eq_of_heq e₂))\n    end\n  | a₁, _, a₂, _, is_false n := is_false (assume h, sigma.no_confusion h (λe₁ e₂, n e₁))\n  end\n\n@[simp] theorem sigma.mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :\n  sigma.mk a₁ b₁ = ⟨a₂, b₂⟩ ↔ (a₁ = a₂ ∧ b₁ == b₂) :=\n⟨sigma.mk.inj, λ ⟨h₁, h₂⟩, by congr; assumption⟩\n\n@[simp] theorem sigma.forall {p : (Σ a, β a) → Prop} :\n  (∀ x, p x) ↔ (∀ a b, p ⟨a, b⟩) :=\n⟨assume h a b, h ⟨a, b⟩, assume h ⟨a, b⟩, h a b⟩\n\n@[simp] theorem sigma.exists {p : (Σ a, β a) → Prop} :\n  (∃ x, p x) ↔ (∃ a b, p ⟨a, b⟩) :=\n⟨assume ⟨⟨a, b⟩, h⟩, ⟨a, b, h⟩, assume ⟨a, b, h⟩, ⟨⟨a, b⟩, h⟩⟩\n\nvariables {α₁ : Type*} {α₂ : Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}\n\n/-- Map the left and right components of a sigma -/\ndef sigma.map (f₁ : α₁ → α₂) (f₂ : Πa, β₁ a → β₂ (f₁ a)) : sigma β₁ → sigma β₂\n| ⟨a, b⟩ := ⟨f₁ a, f₂ a b⟩\n\nend sigma\n\nsection psigma\nvariables {α : Sort*} {β : α → Sort*}\n\ninstance [inhabited α] [inhabited (β (default α))] : inhabited (psigma β) :=\n⟨⟨default α, default (β (default α))⟩⟩\n\ninstance [h₁ : decidable_eq α] [h₂ : ∀a, decidable_eq (β a)] : decidable_eq (psigma β)\n| ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ := match a₁, b₁, a₂, b₂, h₁ a₁ a₂ with\n  | _, b₁, _, b₂, is_true (eq.refl a) :=\n    match b₁, b₂, h₂ a b₁ b₂ with\n    | _, _, is_true (eq.refl b) := is_true rfl\n    | b₁, b₂, is_false n := is_false (assume h, psigma.no_confusion h (λe₁ e₂, n $ eq_of_heq e₂))\n    end\n  | a₁, _, a₂, _, is_false n := is_false (assume h, psigma.no_confusion h (λe₁ e₂, n e₁))\n  end\n\ntheorem psigma.mk.inj_iff {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂} :\n  @psigma.mk α β a₁ b₁ = @psigma.mk α β a₂ b₂ ↔ (a₁ = a₂ ∧ b₁ == b₂) :=\niff.intro psigma.mk.inj $\n  assume ⟨h₁, h₂⟩, match a₁, a₂, b₁, b₂, h₁, h₂ with _, _, _, _, eq.refl a, heq.refl b := rfl end\n\nvariables {α₁ : Sort*} {α₂ : Sort*} {β₁ : α₁ → Sort*} {β₂ : α₂ → Sort*}\n\n/-- Map the left and right components of a sigma -/\ndef psigma.map (f₁ : α₁ → α₂) (f₂ : Πa, β₁ a → β₂ (f₁ a)) : psigma β₁ → psigma β₂\n| ⟨a, b⟩ := ⟨f₁ a, f₂ a b⟩\n\nend psigma\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/sigma/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.30096313101869365}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n\nTuples are lists of a fixed size.\nIt is implemented as a subtype.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.list.default\nimport Mathlib.Lean3Lib.init.data.subtype.default\nimport Mathlib.Lean3Lib.init.meta.interactive\nimport Mathlib.Lean3Lib.init.data.fin.default\n \n\nuniverses u u_1 v w \n\nnamespace Mathlib\n\ndef vector (α : Type u) (n : ℕ) :=\n  Subtype fun (l : List α) => list.length l = n\n\nnamespace vector\n\n\nprotected instance decidable_eq {α : Type u} {n : ℕ} [DecidableEq α] : DecidableEq (vector α n) :=\n  eq.mpr sorry fun (a b : Subtype fun (l : List α) => list.length l = n) => subtype.decidable_eq a b\n\ndef nil {α : Type u} : vector α 0 :=\n  { val := [], property := sorry }\n\ndef cons {α : Type u} {n : ℕ} : α → vector α n → vector α (Nat.succ n) :=\n  sorry\n\ndef length {α : Type u} {n : ℕ} (v : vector α n) : ℕ :=\n  n\n\ndef head {α : Type u} {n : ℕ} : vector α (Nat.succ n) → α :=\n  sorry\n\ntheorem head_cons {α : Type u} {n : ℕ} (a : α) (v : vector α n) : head (cons a v) = a := sorry\n\ndef tail {α : Type u} {n : ℕ} : vector α n → vector α (n - 1) :=\n  sorry\n\ntheorem tail_cons {α : Type u} {n : ℕ} (a : α) (v : vector α n) : tail (cons a v) = v := sorry\n\n@[simp] theorem cons_head_tail {α : Type u} {n : ℕ} (v : vector α (Nat.succ n)) : cons (head v) (tail v) = v := sorry\n\ndef to_list {α : Type u} {n : ℕ} (v : vector α n) : List α :=\n  subtype.val v\n\ndef nth {α : Type u} {n : ℕ} (v : vector α n) : fin n → α :=\n  sorry\n\ndef append {α : Type u} {n : ℕ} {m : ℕ} : vector α n → vector α m → vector α (n + m) :=\n  sorry\n\ndef elim {α : Type u_1} {C : {n : ℕ} → vector α n → Sort u} (H : (l : List α) → C { val := l, property := elim._proof_1 l }) {n : ℕ} (v : vector α n) : C v :=\n  sorry\n\n/- map -/\n\ndef map {α : Type u} {β : Type v} {n : ℕ} (f : α → β) : vector α n → vector β n :=\n  sorry\n\n@[simp] theorem map_nil {α : Type u} {β : Type v} (f : α → β) : map f nil = nil :=\n  rfl\n\ntheorem map_cons {α : Type u} {β : Type v} {n : ℕ} (f : α → β) (a : α) (v : vector α n) : map f (cons a v) = cons (f a) (map f v) := sorry\n\ndef map₂ {α : Type u} {β : Type v} {φ : Type w} {n : ℕ} (f : α → β → φ) : vector α n → vector β n → vector φ n :=\n  sorry\n\ndef repeat {α : Type u} (a : α) (n : ℕ) : vector α n :=\n  { val := list.repeat a n, property := list.length_repeat a n }\n\ndef drop {α : Type u} {n : ℕ} (i : ℕ) : vector α n → vector α (n - i) :=\n  sorry\n\ndef take {α : Type u} {n : ℕ} (i : ℕ) : vector α n → vector α (min i n) :=\n  sorry\n\ndef remove_nth {α : Type u} {n : ℕ} (i : fin n) : vector α n → vector α (n - 1) :=\n  sorry\n\ndef of_fn {α : Type u} {n : ℕ} : (fin n → α) → vector α n :=\n  sorry\n\ndef map_accumr {α : Type u} {β : Type v} {n : ℕ} {σ : Type} (f : α → σ → σ × β) : vector α n → σ → σ × vector β n :=\n  sorry\n\ndef map_accumr₂ {n : ℕ} {α : Type} {β : Type} {σ : Type} {φ : Type} (f : α → β → σ → σ × φ) : vector α n → vector β n → σ → σ × vector φ n :=\n  sorry\n\nprotected theorem eq {α : Type u} {n : ℕ} (a1 : vector α n) (a2 : vector α n) : to_list a1 = to_list a2 → a1 = a2 := sorry\n\nprotected theorem eq_nil {α : Type u} (v : vector α 0) : v = nil :=\n  vector.eq v nil (list.eq_nil_of_length_eq_zero (subtype.property v))\n\n@[simp] theorem to_list_mk {α : Type u} {n : ℕ} (v : List α) (P : list.length v = n) : to_list { val := v, property := P } = v :=\n  rfl\n\n@[simp] theorem to_list_nil {α : Type u} : to_list nil = [] :=\n  rfl\n\n@[simp] theorem to_list_length {α : Type u} {n : ℕ} (v : vector α n) : list.length (to_list v) = n :=\n  subtype.property v\n\n@[simp] theorem to_list_cons {α : Type u} {n : ℕ} (a : α) (v : vector α n) : to_list (cons a v) = a :: to_list v :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = n) =>\n      Eq.refl (to_list (cons a { val := v_val, property := v_property }))\n\n@[simp] theorem to_list_append {α : Type u} {n : ℕ} {m : ℕ} (v : vector α n) (w : vector α m) : to_list (append v w) = to_list v ++ to_list w := sorry\n\n@[simp] theorem to_list_drop {α : Type u} {n : ℕ} {m : ℕ} (v : vector α m) : to_list (drop n v) = list.drop n (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = m) =>\n      Eq.refl (to_list (drop n { val := v_val, property := v_property }))\n\n@[simp] theorem to_list_take {α : Type u} {n : ℕ} {m : ℕ} (v : vector α m) : to_list (take n v) = list.take n (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List α) (v_property : list.length v_val = m) =>\n      Eq.refl (to_list (take n { val := v_val, property := v_property }))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/data/vector.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118493816807, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.30096312218942883}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.ultrafilter\nimport Mathlib.order.filter.partial\nimport Mathlib.PostPort\n\nuniverses u l w v u_1 u_2 u_3 u_5 \n\nnamespace Mathlib\n\n/-!\n# Basic theory of topological spaces.\n\nThe main definition is the type class `topological space α` which endows a type `α` with a topology.\nThen `set α` gets predicates `is_open`, `is_closed` and functions `interior`, `closure` and\n`frontier`. Each point `x` of `α` gets a neighborhood filter `𝓝 x`. A filter `F` on `α` has\n`x` as a cluster point if `cluster_pt x F : 𝓝 x ⊓ F ≠ ⊥`. A map `f : ι → α` clusters at `x`\nalong `F : filter ι` if `map_cluster_pt x F f : cluster_pt x (map f F)`. In particular\nthe notion of cluster point of a sequence `u` is `map_cluster_pt x at_top u`.\n\nThis file also defines locally finite families of subsets of `α`.\n\nFor topological spaces `α` and `β`, a function `f : α → β` and a point `a : α`,\n`continuous_at f a` means `f` is continuous at `a`, and global continuity is\n`continuous f`. There is also a version of continuity `pcontinuous` for\npartially defined functions.\n\n## Notation\n\n* `𝓝 x`: the filter of neighborhoods of a point `x`;\n* `𝓟 s`: the principal filter of a set `s`;\n\n## Implementation notes\n\nTopology in mathlib heavily uses filters (even more than in Bourbaki). See explanations in\n<https://leanprover-community.github.io/theories/topology.html>.\n\n## References\n\n*  [N. Bourbaki, *General Topology*][bourbaki1966]\n*  [I. M. James, *Topologies and Uniformities*][james1999]\n\n## Tags\n\ntopological space, interior, closure, frontier, neighborhood, continuity, continuous function\n-/\n\n/-!\n### Topological spaces\n-/\n\n/-- A topology on `α`. -/\nclass topological_space (α : Type u) where\n  is_open : set α → Prop\n  is_open_univ : is_open set.univ\n  is_open_inter : ∀ (s t : set α), is_open s → is_open t → is_open (s ∩ t)\n  is_open_sUnion : ∀ (s : set (set α)), (∀ (t : set α), t ∈ s → is_open t) → is_open (⋃₀s)\n\n/-- A constructor for topologies by specifying the closed sets,\nand showing that they satisfy the appropriate conditions. -/\ndef topological_space.of_closed {α : Type u} (T : set (set α)) (empty_mem : ∅ ∈ T)\n    (sInter_mem : ∀ (A : set (set α)), A ⊆ T → ⋂₀A ∈ T)\n    (union_mem : ∀ (A B : set α), A ∈ T → B ∈ T → A ∪ B ∈ T) : topological_space α :=\n  topological_space.mk (fun (X : set α) => Xᶜ ∈ T) sorry sorry sorry\n\ntheorem topological_space_eq {α : Type u} {f : topological_space α} {g : topological_space α} :\n    topological_space.is_open f = topological_space.is_open g → f = g :=\n  sorry\n\n/-- `is_open s` means that `s` is open in the ambient topological space on `α` -/\ndef is_open {α : Type u} [t : topological_space α] (s : set α) := topological_space.is_open t s\n\n@[simp] theorem is_open_univ {α : Type u} [t : topological_space α] : is_open set.univ :=\n  topological_space.is_open_univ t\n\ntheorem is_open_inter {α : Type u} {s₁ : set α} {s₂ : set α} [t : topological_space α]\n    (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∩ s₂) :=\n  topological_space.is_open_inter t s₁ s₂ h₁ h₂\n\ntheorem is_open_sUnion {α : Type u} [t : topological_space α] {s : set (set α)}\n    (h : ∀ (t_1 : set α), t_1 ∈ s → is_open t_1) : is_open (⋃₀s) :=\n  topological_space.is_open_sUnion t s h\n\ntheorem topological_space_eq_iff {α : Type u} {t : topological_space α} {t' : topological_space α} :\n    t = t' ↔ ∀ (s : set α), is_open s ↔ is_open s :=\n  { mp := fun (h : t = t') (s : set α) => h ▸ iff.rfl,\n    mpr :=\n      fun (h : ∀ (s : set α), is_open s ↔ is_open s) =>\n        topological_space_eq (funext fun (x : set α) => propext (h x)) }\n\ntheorem is_open_fold {α : Type u} {s : set α} {t : topological_space α} :\n    topological_space.is_open t s = is_open s :=\n  rfl\n\ntheorem is_open_Union {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α}\n    (h : ∀ (i : ι), is_open (f i)) : is_open (set.Union fun (i : ι) => f i) :=\n  is_open_sUnion\n    fun (t : set α) (H : t ∈ set.range fun (i : ι) => f i) =>\n      Exists.dcases_on H fun (i : ι) (H_h : (fun (i : ι) => f i) i = t) => Eq._oldrec (h i) H_h\n\ntheorem is_open_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α}\n    (h : ∀ (i : β), i ∈ s → is_open (f i)) :\n    is_open (set.Union fun (i : β) => set.Union fun (H : i ∈ s) => f i) :=\n  is_open_Union fun (i : β) => is_open_Union fun (hi : i ∈ s) => h i hi\n\ntheorem is_open_union {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α] (h₁ : is_open s₁)\n    (h₂ : is_open s₂) : is_open (s₁ ∪ s₂) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∪ s₂))) set.union_eq_Union))\n    (is_open_Union (iff.mpr bool.forall_bool { left := h₂, right := h₁ }))\n\n@[simp] theorem is_open_empty {α : Type u} [topological_space α] : is_open ∅ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open ∅)) (Eq.symm set.sUnion_empty)))\n    (is_open_sUnion fun (a : set α) => false.elim)\n\ntheorem is_open_sInter {α : Type u} [topological_space α] {s : set (set α)} (hs : set.finite s) :\n    (∀ (t : set α), t ∈ s → is_open t) → is_open (⋂₀s) :=\n  sorry\n\ntheorem is_open_bInter {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α}\n    (hs : set.finite s) :\n    (∀ (i : β), i ∈ s → is_open (f i)) →\n        is_open (set.Inter fun (i : β) => set.Inter fun (H : i ∈ s) => f i) :=\n  sorry\n\ntheorem is_open_Inter {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α}\n    (h : ∀ (i : β), is_open (s i)) : is_open (set.Inter fun (i : β) => s i) :=\n  sorry\n\ntheorem is_open_Inter_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α}\n    (h : ∀ (h : p), is_open (s h)) : is_open (set.Inter s) :=\n  sorry\n\ntheorem is_open_const {α : Type u} [topological_space α] {p : Prop} :\n    is_open (set_of fun (a : α) => p) :=\n  sorry\n\ntheorem is_open_and {α : Type u} {p₁ : α → Prop} {p₂ : α → Prop} [topological_space α] :\n    is_open (set_of fun (a : α) => p₁ a) →\n        is_open (set_of fun (a : α) => p₂ a) → is_open (set_of fun (a : α) => p₁ a ∧ p₂ a) :=\n  is_open_inter\n\n/-- A set is closed if its complement is open -/\ndef is_closed {α : Type u} [topological_space α] (s : set α) := is_open (sᶜ)\n\n@[simp] theorem is_closed_empty {α : Type u} [topological_space α] : is_closed ∅ :=\n  eq.mpr (id (is_closed.equations._eqn_1 ∅))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (∅ᶜ))) set.compl_empty)) is_open_univ)\n\n@[simp] theorem is_closed_univ {α : Type u} [topological_space α] : is_closed set.univ :=\n  eq.mpr (id (is_closed.equations._eqn_1 set.univ))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.univᶜ))) set.compl_univ)) is_open_empty)\n\ntheorem is_closed_union {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α] :\n    is_closed s₁ → is_closed s₂ → is_closed (s₁ ∪ s₂) :=\n  fun (h₁ : is_closed s₁) (h₂ : is_closed s₂) =>\n    eq.mpr (id (is_closed.equations._eqn_1 (s₁ ∪ s₂)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∪ s₂ᶜ))) (set.compl_union s₁ s₂)))\n        (is_open_inter h₁ h₂))\n\ntheorem is_closed_sInter {α : Type u} [topological_space α] {s : set (set α)} :\n    (∀ (t : set α), t ∈ s → is_closed t) → is_closed (⋂₀s) :=\n  sorry\n\ntheorem is_closed_Inter {α : Type u} {ι : Sort w} [topological_space α] {f : ι → set α}\n    (h : ∀ (i : ι), is_closed (f i)) : is_closed (set.Inter fun (i : ι) => f i) :=\n  sorry\n\n@[simp] theorem is_open_compl_iff {α : Type u} [topological_space α] {s : set α} :\n    is_open (sᶜ) ↔ is_closed s :=\n  iff.rfl\n\n@[simp] theorem is_closed_compl_iff {α : Type u} [topological_space α] {s : set α} :\n    is_closed (sᶜ) ↔ is_open s :=\n  eq.mpr\n    (id (Eq._oldrec (Eq.refl (is_closed (sᶜ) ↔ is_open s)) (Eq.symm (propext is_open_compl_iff))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (sᶜᶜ) ↔ is_open s)) (compl_compl s)))\n      (iff.refl (is_open s)))\n\ntheorem is_open_diff {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : is_open s)\n    (h₂ : is_closed t) : is_open (s \\ t) :=\n  is_open_inter h₁ (iff.mpr is_open_compl_iff h₂)\n\ntheorem is_closed_inter {α : Type u} {s₁ : set α} {s₂ : set α} [topological_space α]\n    (h₁ : is_closed s₁) (h₂ : is_closed s₂) : is_closed (s₁ ∩ s₂) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (s₁ ∩ s₂))) (is_closed.equations._eqn_1 (s₁ ∩ s₂))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_open (s₁ ∩ s₂ᶜ))) (set.compl_inter s₁ s₂)))\n      (is_open_union h₁ h₂))\n\ntheorem is_closed_bUnion {α : Type u} {β : Type v} [topological_space α] {s : set β} {f : β → set α}\n    (hs : set.finite s) :\n    (∀ (i : β), i ∈ s → is_closed (f i)) →\n        is_closed (set.Union fun (i : β) => set.Union fun (H : i ∈ s) => f i) :=\n  sorry\n\ntheorem is_closed_Union {α : Type u} {β : Type v} [topological_space α] [fintype β] {s : β → set α}\n    (h : ∀ (i : β), is_closed (s i)) : is_closed (set.Union s) :=\n  sorry\n\ntheorem is_closed_Union_prop {α : Type u} [topological_space α] {p : Prop} {s : p → set α}\n    (h : ∀ (h : p), is_closed (s h)) : is_closed (set.Union s) :=\n  sorry\n\ntheorem is_closed_imp {α : Type u} [topological_space α] {p : α → Prop} {q : α → Prop}\n    (hp : is_open (set_of fun (x : α) => p x)) (hq : is_closed (set_of fun (x : α) => q x)) :\n    is_closed (set_of fun (x : α) => p x → q x) :=\n  sorry\n\ntheorem is_open_neg {α : Type u} {p : α → Prop} [topological_space α] :\n    is_closed (set_of fun (a : α) => p a) → is_open (set_of fun (a : α) => ¬p a) :=\n  iff.mpr is_open_compl_iff\n\n/-!\n### Interior of a set\n-/\n\n/-- The interior of a set `s` is the largest open subset of `s`. -/\ndef interior {α : Type u} [topological_space α] (s : set α) : set α :=\n  ⋃₀set_of fun (t : set α) => is_open t ∧ t ⊆ s\n\ntheorem mem_interior {α : Type u} [topological_space α] {s : set α} {x : α} :\n    x ∈ interior s ↔ ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ x ∈ t :=\n  sorry\n\n@[simp] theorem is_open_interior {α : Type u} [topological_space α] {s : set α} :\n    is_open (interior s) :=\n  sorry\n\ntheorem interior_subset {α : Type u} [topological_space α] {s : set α} : interior s ⊆ s := sorry\n\ntheorem interior_maximal {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : t ⊆ s)\n    (h₂ : is_open t) : t ⊆ interior s :=\n  set.subset_sUnion_of_mem { left := h₂, right := h₁ }\n\ntheorem is_open.interior_eq {α : Type u} [topological_space α] {s : set α} (h : is_open s) :\n    interior s = s :=\n  set.subset.antisymm interior_subset (interior_maximal (set.subset.refl s) h)\n\ntheorem interior_eq_iff_open {α : Type u} [topological_space α] {s : set α} :\n    interior s = s ↔ is_open s :=\n  { mp := fun (h : interior s = s) => h ▸ is_open_interior, mpr := is_open.interior_eq }\n\ntheorem subset_interior_iff_open {α : Type u} [topological_space α] {s : set α} :\n    s ⊆ interior s ↔ is_open s :=\n  sorry\n\ntheorem subset_interior_iff_subset_of_open {α : Type u} [topological_space α] {s : set α}\n    {t : set α} (h₁ : is_open s) : s ⊆ interior t ↔ s ⊆ t :=\n  { mp := fun (h : s ⊆ interior t) => set.subset.trans h interior_subset,\n    mpr := fun (h₂ : s ⊆ t) => interior_maximal h₂ h₁ }\n\ntheorem interior_mono {α : Type u} [topological_space α] {s : set α} {t : set α} (h : s ⊆ t) :\n    interior s ⊆ interior t :=\n  interior_maximal (set.subset.trans interior_subset h) is_open_interior\n\n@[simp] theorem interior_empty {α : Type u} [topological_space α] : interior ∅ = ∅ :=\n  is_open.interior_eq is_open_empty\n\n@[simp] theorem interior_univ {α : Type u} [topological_space α] : interior set.univ = set.univ :=\n  is_open.interior_eq is_open_univ\n\n@[simp] theorem interior_interior {α : Type u} [topological_space α] {s : set α} :\n    interior (interior s) = interior s :=\n  is_open.interior_eq is_open_interior\n\n@[simp] theorem interior_inter {α : Type u} [topological_space α] {s : set α} {t : set α} :\n    interior (s ∩ t) = interior s ∩ interior t :=\n  sorry\n\ntheorem interior_union_is_closed_of_interior_empty {α : Type u} [topological_space α] {s : set α}\n    {t : set α} (h₁ : is_closed s) (h₂ : interior t = ∅) : interior (s ∪ t) = interior s :=\n  sorry\n\ntheorem is_open_iff_forall_mem_open {α : Type u} {s : set α} [topological_space α] :\n    is_open s ↔ ∀ (x : α) (H : x ∈ s), ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ x ∈ t :=\n  sorry\n\n/-!\n### Closure of a set\n-/\n\n/-- The closure of `s` is the smallest closed set containing `s`. -/\ndef closure {α : Type u} [topological_space α] (s : set α) : set α :=\n  ⋂₀set_of fun (t : set α) => is_closed t ∧ s ⊆ t\n\n@[simp] theorem is_closed_closure {α : Type u} [topological_space α] {s : set α} :\n    is_closed (closure s) :=\n  sorry\n\ntheorem subset_closure {α : Type u} [topological_space α] {s : set α} : s ⊆ closure s := sorry\n\ntheorem closure_minimal {α : Type u} [topological_space α] {s : set α} {t : set α} (h₁ : s ⊆ t)\n    (h₂ : is_closed t) : closure s ⊆ t :=\n  set.sInter_subset_of_mem { left := h₂, right := h₁ }\n\ntheorem is_closed.closure_eq {α : Type u} [topological_space α] {s : set α} (h : is_closed s) :\n    closure s = s :=\n  set.subset.antisymm (closure_minimal (set.subset.refl s) h) subset_closure\n\ntheorem is_closed.closure_subset {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) :\n    closure s ⊆ s :=\n  closure_minimal (set.subset.refl s) hs\n\ntheorem is_closed.closure_subset_iff {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (h₁ : is_closed t) : closure s ⊆ t ↔ s ⊆ t :=\n  { mp := set.subset.trans subset_closure, mpr := fun (h : s ⊆ t) => closure_minimal h h₁ }\n\ntheorem closure_mono {α : Type u} [topological_space α] {s : set α} {t : set α} (h : s ⊆ t) :\n    closure s ⊆ closure t :=\n  closure_minimal (set.subset.trans h subset_closure) is_closed_closure\n\ntheorem monotone_closure (α : Type u_1) [topological_space α] : monotone closure :=\n  fun (_x _x_1 : set α) => closure_mono\n\ntheorem closure_inter_subset_inter_closure {α : Type u} [topological_space α] (s : set α)\n    (t : set α) : closure (s ∩ t) ⊆ closure s ∩ closure t :=\n  monotone.map_inf_le (monotone_closure α) s t\n\ntheorem is_closed_of_closure_subset {α : Type u} [topological_space α] {s : set α}\n    (h : closure s ⊆ s) : is_closed s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed s)) (set.subset.antisymm subset_closure h)))\n    is_closed_closure\n\ntheorem closure_eq_iff_is_closed {α : Type u} [topological_space α] {s : set α} :\n    closure s = s ↔ is_closed s :=\n  { mp := fun (h : closure s = s) => h ▸ is_closed_closure, mpr := is_closed.closure_eq }\n\ntheorem closure_subset_iff_is_closed {α : Type u} [topological_space α] {s : set α} :\n    closure s ⊆ s ↔ is_closed s :=\n  { mp := is_closed_of_closure_subset, mpr := is_closed.closure_subset }\n\n@[simp] theorem closure_empty {α : Type u} [topological_space α] : closure ∅ = ∅ :=\n  is_closed.closure_eq is_closed_empty\n\n@[simp] theorem closure_empty_iff {α : Type u} [topological_space α] (s : set α) :\n    closure s = ∅ ↔ s = ∅ :=\n  { mp := set.subset_eq_empty subset_closure, mpr := fun (h : s = ∅) => Eq.symm h ▸ closure_empty }\n\ntheorem set.nonempty.closure {α : Type u} [topological_space α] {s : set α} (h : set.nonempty s) :\n    set.nonempty (closure s) :=\n  sorry\n\n@[simp] theorem closure_univ {α : Type u} [topological_space α] : closure set.univ = set.univ :=\n  is_closed.closure_eq is_closed_univ\n\n@[simp] theorem closure_closure {α : Type u} [topological_space α] {s : set α} :\n    closure (closure s) = closure s :=\n  is_closed.closure_eq is_closed_closure\n\n@[simp] theorem closure_union {α : Type u} [topological_space α] {s : set α} {t : set α} :\n    closure (s ∪ t) = closure s ∪ closure t :=\n  sorry\n\ntheorem interior_subset_closure {α : Type u} [topological_space α] {s : set α} :\n    interior s ⊆ closure s :=\n  set.subset.trans interior_subset subset_closure\n\ntheorem closure_eq_compl_interior_compl {α : Type u} [topological_space α] {s : set α} :\n    closure s = (interior (sᶜ)ᶜ) :=\n  sorry\n\n@[simp] theorem interior_compl {α : Type u} [topological_space α] {s : set α} :\n    interior (sᶜ) = (closure sᶜ) :=\n  sorry\n\n@[simp] theorem closure_compl {α : Type u} [topological_space α] {s : set α} :\n    closure (sᶜ) = (interior sᶜ) :=\n  sorry\n\ntheorem mem_closure_iff {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ closure s ↔ ∀ (o : set α), is_open o → a ∈ o → set.nonempty (o ∩ s) :=\n  sorry\n\n/-- A set is dense in a topological space if every point belongs to its closure. -/\ndef dense {α : Type u} [topological_space α] (s : set α) := ∀ (x : α), x ∈ closure s\n\ntheorem dense_iff_closure_eq {α : Type u} [topological_space α] {s : set α} :\n    dense s ↔ closure s = set.univ :=\n  iff.symm set.eq_univ_iff_forall\n\ntheorem dense.closure_eq {α : Type u} [topological_space α] {s : set α} (h : dense s) :\n    closure s = set.univ :=\n  iff.mp dense_iff_closure_eq h\n\n/-- The closure of a set `s` is dense if and only if `s` is dense. -/\n@[simp] theorem dense_closure {α : Type u} [topological_space α] {s : set α} :\n    dense (closure s) ↔ dense s :=\n  sorry\n\ntheorem dense.of_closure {α : Type u} [topological_space α] {s : set α} :\n    dense (closure s) → dense s :=\n  iff.mp dense_closure\n\n@[simp] theorem dense_univ {α : Type u} [topological_space α] : dense set.univ :=\n  fun (x : α) => subset_closure trivial\n\n/-- A set is dense if and only if it has a nonempty intersection with each nonempty open set. -/\ntheorem dense_iff_inter_open {α : Type u} [topological_space α] {s : set α} :\n    dense s ↔ ∀ (U : set α), is_open U → set.nonempty U → set.nonempty (U ∩ s) :=\n  sorry\n\ntheorem dense.inter_open_nonempty {α : Type u} [topological_space α] {s : set α} :\n    dense s → ∀ (U : set α), is_open U → set.nonempty U → set.nonempty (U ∩ s) :=\n  iff.mp dense_iff_inter_open\n\ntheorem dense.nonempty_iff {α : Type u} [topological_space α] {s : set α} (hs : dense s) :\n    set.nonempty s ↔ Nonempty α :=\n  sorry\n\ntheorem dense.nonempty {α : Type u} [topological_space α] [h : Nonempty α] {s : set α}\n    (hs : dense s) : set.nonempty s :=\n  iff.mpr (dense.nonempty_iff hs) h\n\ntheorem dense.mono {α : Type u} [topological_space α] {s₁ : set α} {s₂ : set α} (h : s₁ ⊆ s₂)\n    (hd : dense s₁) : dense s₂ :=\n  fun (x : α) => closure_mono h (hd x)\n\n/-!\n### Frontier of a set\n-/\n\n/-- The frontier of a set is the set of points between the closure and interior. -/\ndef frontier {α : Type u} [topological_space α] (s : set α) : set α := closure s \\ interior s\n\ntheorem frontier_eq_closure_inter_closure {α : Type u} [topological_space α] {s : set α} :\n    frontier s = closure s ∩ closure (sᶜ) :=\n  sorry\n\n/-- The complement of a set has the same frontier as the original set. -/\n@[simp] theorem frontier_compl {α : Type u} [topological_space α] (s : set α) :\n    frontier (sᶜ) = frontier s :=\n  sorry\n\ntheorem frontier_inter_subset {α : Type u} [topological_space α] (s : set α) (t : set α) :\n    frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t :=\n  sorry\n\ntheorem frontier_union_subset {α : Type u} [topological_space α] (s : set α) (t : set α) :\n    frontier (s ∪ t) ⊆ frontier s ∩ closure (tᶜ) ∪ closure (sᶜ) ∩ frontier t :=\n  sorry\n\ntheorem is_closed.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_closed s) :\n    frontier s = s \\ interior s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (frontier s = s \\ interior s)) (frontier.equations._eqn_1 s)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (closure s \\ interior s = s \\ interior s)) (is_closed.closure_eq hs)))\n      (Eq.refl (s \\ interior s)))\n\ntheorem is_open.frontier_eq {α : Type u} [topological_space α] {s : set α} (hs : is_open s) :\n    frontier s = closure s \\ s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (frontier s = closure s \\ s)) (frontier.equations._eqn_1 s)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (closure s \\ interior s = closure s \\ s)) (is_open.interior_eq hs)))\n      (Eq.refl (closure s \\ s)))\n\n/-- The frontier of a set is closed. -/\ntheorem is_closed_frontier {α : Type u} [topological_space α] {s : set α} :\n    is_closed (frontier s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (frontier s))) frontier_eq_closure_inter_closure))\n    (is_closed_inter is_closed_closure is_closed_closure)\n\n/-- The frontier of a closed set has no interior point. -/\ntheorem interior_frontier {α : Type u} [topological_space α] {s : set α} (h : is_closed s) :\n    interior (frontier s) = ∅ :=\n  sorry\n\ntheorem closure_eq_interior_union_frontier {α : Type u} [topological_space α] (s : set α) :\n    closure s = interior s ∪ frontier s :=\n  Eq.symm (set.union_diff_cancel interior_subset_closure)\n\ntheorem closure_eq_self_union_frontier {α : Type u} [topological_space α] (s : set α) :\n    closure s = s ∪ frontier s :=\n  Eq.symm (set.union_diff_cancel' interior_subset subset_closure)\n\n/-!\n### Neighborhoods\n-/\n\n/-- A set is called a neighborhood of `a` if it contains an open set around `a`. The set of all\nneighborhoods of `a` forms a filter, the neighborhood filter at `a`, is here defined as the\ninfimum over the principal filters of all open sets containing `a`. -/\ndef nhds {α : Type u} [topological_space α] (a : α) : filter α :=\n  infi\n    fun (s : set α) =>\n      infi fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) => filter.principal s\n\ntheorem nhds_def {α : Type u} [topological_space α] (a : α) :\n    nhds a =\n        infi\n          fun (s : set α) =>\n            infi fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) => filter.principal s :=\n  rfl\n\n/-- The open sets containing `a` are a basis for the neighborhood filter. See `nhds_basis_opens'`\nfor a variant using open neighborhoods instead. -/\ntheorem nhds_basis_opens {α : Type u} [topological_space α] (a : α) :\n    filter.has_basis (nhds a) (fun (s : set α) => a ∈ s ∧ is_open s) fun (x : set α) => x :=\n  sorry\n\n/-- A filter lies below the neighborhood filter at `a` iff it contains every open set around `a`. -/\ntheorem le_nhds_iff {α : Type u} [topological_space α] {f : filter α} {a : α} :\n    f ≤ nhds a ↔ ∀ (s : set α), a ∈ s → is_open s → s ∈ f :=\n  sorry\n\n/-- To show a filter is above the neighborhood filter at `a`, it suffices to show that it is above\nthe principal filter of some open set `s` containing `a`. -/\ntheorem nhds_le_of_le {α : Type u} [topological_space α] {f : filter α} {a : α} {s : set α}\n    (h : a ∈ s) (o : is_open s) (sf : filter.principal s ≤ f) : nhds a ≤ f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nhds a ≤ f)) (nhds_def a)))\n    (infi_le_of_le s (infi_le_of_le { left := h, right := o } sf))\n\ntheorem mem_nhds_sets_iff {α : Type u} [topological_space α] {a : α} {s : set α} :\n    s ∈ nhds a ↔ ∃ (t : set α), ∃ (H : t ⊆ s), is_open t ∧ a ∈ t :=\n  sorry\n\n/-- A predicate is true in a neighborhood of `a` iff it is true for all the points in an open set\ncontaining `a`. -/\ntheorem eventually_nhds_iff {α : Type u} [topological_space α] {a : α} {p : α → Prop} :\n    filter.eventually (fun (x : α) => p x) (nhds a) ↔\n        ∃ (t : set α), (∀ (x : α), x ∈ t → p x) ∧ is_open t ∧ a ∈ t :=\n  sorry\n\ntheorem map_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : α → β} :\n    filter.map f (nhds a) =\n        infi\n          fun (s : set α) =>\n            infi\n              fun (H : s ∈ set_of fun (s : set α) => a ∈ s ∧ is_open s) =>\n                filter.principal (f '' s) :=\n  filter.has_basis.eq_binfi (filter.has_basis.map f (nhds_basis_opens a))\n\ntheorem mem_of_nhds {α : Type u} [topological_space α] {a : α} {s : set α} : s ∈ nhds a → a ∈ s :=\n  sorry\n\n/-- If a predicate is true in a neighborhood of `a`, then it is true for `a`. -/\ntheorem filter.eventually.self_of_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α}\n    (h : filter.eventually (fun (y : α) => p y) (nhds a)) : p a :=\n  mem_of_nhds h\n\ntheorem mem_nhds_sets {α : Type u} [topological_space α] {a : α} {s : set α} (hs : is_open s)\n    (ha : a ∈ s) : s ∈ nhds a :=\n  iff.mpr mem_nhds_sets_iff\n    (Exists.intro s (Exists.intro (set.subset.refl s) { left := hs, right := ha }))\n\ntheorem is_open.eventually_mem {α : Type u} [topological_space α] {a : α} {s : set α}\n    (hs : is_open s) (ha : a ∈ s) : filter.eventually (fun (x : α) => x ∈ s) (nhds a) :=\n  mem_nhds_sets hs ha\n\n/-- The open neighborhoods of `a` are a basis for the neighborhood filter. See `nhds_basis_opens`\nfor a variant using open sets around `a` instead. -/\ntheorem nhds_basis_opens' {α : Type u} [topological_space α] (a : α) :\n    filter.has_basis (nhds a) (fun (s : set α) => s ∈ nhds a ∧ is_open s) fun (x : set α) => x :=\n  sorry\n\n/-- If a predicate is true in a neighbourhood of `a`, then for `y` sufficiently close\nto `a` this predicate is true in a neighbourhood of `y`. -/\ntheorem filter.eventually.eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} {a : α}\n    (h : filter.eventually (fun (y : α) => p y) (nhds a)) :\n    filter.eventually (fun (y : α) => filter.eventually (fun (y : α) => p y) (nhds y)) (nhds a) :=\n  sorry\n\n@[simp] theorem eventually_eventually_nhds {α : Type u} [topological_space α] {p : α → Prop}\n    {a : α} :\n    filter.eventually (fun (y : α) => filter.eventually (fun (x : α) => p x) (nhds y)) (nhds a) ↔\n        filter.eventually (fun (x : α) => p x) (nhds a) :=\n  sorry\n\n@[simp] theorem nhds_bind_nhds {α : Type u} {a : α} [topological_space α] :\n    filter.bind (nhds a) nhds = nhds a :=\n  filter.ext fun (s : set α) => eventually_eventually_nhds\n\n@[simp] theorem eventually_eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α]\n    {f : α → β} {g : α → β} {a : α} :\n    filter.eventually (fun (y : α) => filter.eventually_eq (nhds y) f g) (nhds a) ↔\n        filter.eventually_eq (nhds a) f g :=\n  eventually_eventually_nhds\n\ntheorem filter.eventually_eq.eq_of_nhds {α : Type u} {β : Type v} [topological_space α] {f : α → β}\n    {g : α → β} {a : α} (h : filter.eventually_eq (nhds a) f g) : f a = g a :=\n  filter.eventually.self_of_nhds h\n\n@[simp] theorem eventually_eventually_le_nhds {α : Type u} {β : Type v} [topological_space α]\n    [HasLessEq β] {f : α → β} {g : α → β} {a : α} :\n    filter.eventually (fun (y : α) => filter.eventually_le (nhds y) f g) (nhds a) ↔\n        filter.eventually_le (nhds a) f g :=\n  eventually_eventually_nhds\n\n/-- If two functions are equal in a neighbourhood of `a`, then for `y` sufficiently close\nto `a` these functions are equal in a neighbourhood of `y`. -/\ntheorem filter.eventually_eq.eventually_eq_nhds {α : Type u} {β : Type v} [topological_space α]\n    {f : α → β} {g : α → β} {a : α} (h : filter.eventually_eq (nhds a) f g) :\n    filter.eventually (fun (y : α) => filter.eventually_eq (nhds y) f g) (nhds a) :=\n  filter.eventually.eventually_nhds h\n\n/-- If `f x ≤ g x` in a neighbourhood of `a`, then for `y` sufficiently close to `a` we have\n`f x ≤ g x` in a neighbourhood of `y`. -/\ntheorem filter.eventually_le.eventually_le_nhds {α : Type u} {β : Type v} [topological_space α]\n    [HasLessEq β] {f : α → β} {g : α → β} {a : α} (h : filter.eventually_le (nhds a) f g) :\n    filter.eventually (fun (y : α) => filter.eventually_le (nhds y) f g) (nhds a) :=\n  filter.eventually.eventually_nhds h\n\ntheorem all_mem_nhds {α : Type u} [topological_space α] (x : α) (P : set α → Prop)\n    (hP : ∀ (s t : set α), s ⊆ t → P s → P t) :\n    (∀ (s : set α), s ∈ nhds x → P s) ↔ ∀ (s : set α), is_open s → x ∈ s → P s :=\n  sorry\n\ntheorem all_mem_nhds_filter {α : Type u} {β : Type v} [topological_space α] (x : α)\n    (f : set α → set β) (hf : ∀ (s t : set α), s ⊆ t → f s ⊆ f t) (l : filter β) :\n    (∀ (s : set α), s ∈ nhds x → f s ∈ l) ↔ ∀ (s : set α), is_open s → x ∈ s → f s ∈ l :=\n  all_mem_nhds x (fun (s : set α) => f s ∈ l)\n    fun (s t : set α) (ssubt : s ⊆ t) (h : f s ∈ l) => filter.mem_sets_of_superset h (hf s t ssubt)\n\ntheorem rtendsto_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β}\n    {a : α} : filter.rtendsto r l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → rel.core r s ∈ l :=\n  all_mem_nhds_filter a (fun (U : set α) => U) (fun (s t : set α) => id) (filter.rmap r l)\n\ntheorem rtendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {r : rel β α} {l : filter β}\n    {a : α} :\n    filter.rtendsto' r l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → rel.preimage r s ∈ l :=\n  sorry\n\ntheorem ptendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β}\n    {a : α} : filter.ptendsto f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → pfun.core f s ∈ l :=\n  rtendsto_nhds\n\ntheorem ptendsto'_nhds {α : Type u} {β : Type v} [topological_space α] {f : β →. α} {l : filter β}\n    {a : α} :\n    filter.ptendsto' f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → pfun.preimage f s ∈ l :=\n  rtendsto'_nhds\n\ntheorem tendsto_nhds {α : Type u} {β : Type v} [topological_space α] {f : β → α} {l : filter β}\n    {a : α} : filter.tendsto f l (nhds a) ↔ ∀ (s : set α), is_open s → a ∈ s → f ⁻¹' s ∈ l :=\n  all_mem_nhds_filter a (fun (U : set α) => U)\n    (fun (s t : set α) (h : s ⊆ t) => set.preimage_mono h) (filter.map f l)\n\ntheorem tendsto_const_nhds {α : Type u} {β : Type v} [topological_space α] {a : α} {f : filter β} :\n    filter.tendsto (fun (b : β) => a) f (nhds a) :=\n  iff.mpr tendsto_nhds\n    fun (s : set α) (hs : is_open s) (ha : a ∈ s) => filter.univ_mem_sets' fun (_x : β) => ha\n\ntheorem pure_le_nhds {α : Type u} [topological_space α] : pure ≤ nhds :=\n  fun (a : α) (s : set α) (hs : s ∈ nhds a) => iff.mpr filter.mem_pure_sets (mem_of_nhds hs)\n\ntheorem tendsto_pure_nhds {β : Type v} {α : Type u_1} [topological_space β] (f : α → β) (a : α) :\n    filter.tendsto f (pure a) (nhds (f a)) :=\n  filter.tendsto.mono_right (filter.tendsto_pure_pure f a) (pure_le_nhds (f a))\n\ntheorem order_top.tendsto_at_top_nhds {β : Type v} {α : Type u_1} [order_top α]\n    [topological_space β] (f : α → β) : filter.tendsto f filter.at_top (nhds (f ⊤)) :=\n  filter.tendsto.mono_right (filter.tendsto_at_top_pure f) (pure_le_nhds (f ⊤))\n\n@[simp] protected instance nhds_ne_bot {α : Type u} [topological_space α] {a : α} :\n    filter.ne_bot (nhds a) :=\n  filter.ne_bot_of_le (pure_le_nhds a)\n\n/-!\n### Cluster points\n\nIn this section we define [cluster points](https://en.wikipedia.org/wiki/Limit_point)\n(also known as limit points and accumulation points) of a filter and of a sequence.\n-/\n\n/-- A point `x` is a cluster point of a filter `F` if 𝓝 x ⊓ F ≠ ⊥. Also known as\nan accumulation point or a limit point. -/\ndef cluster_pt {α : Type u} [topological_space α] (x : α) (F : filter α) :=\n  filter.ne_bot (nhds x ⊓ F)\n\ntheorem cluster_pt.ne_bot {α : Type u} [topological_space α] {x : α} {F : filter α}\n    (h : cluster_pt x F) : filter.ne_bot (nhds x ⊓ F) :=\n  h\n\ntheorem cluster_pt_iff {α : Type u} [topological_space α] {x : α} {F : filter α} :\n    cluster_pt x F ↔ ∀ {U : set α}, U ∈ nhds x → ∀ {V : set α}, V ∈ F → set.nonempty (U ∩ V) :=\n  filter.inf_ne_bot_iff\n\n/-- `x` is a cluster point of a set `s` if every neighbourhood of `x` meets `s` on a nonempty\nset. -/\ntheorem cluster_pt_principal_iff {α : Type u} [topological_space α] {x : α} {s : set α} :\n    cluster_pt x (filter.principal s) ↔ ∀ (U : set α), U ∈ nhds x → set.nonempty (U ∩ s) :=\n  filter.inf_principal_ne_bot_iff\n\ntheorem cluster_pt_principal_iff_frequently {α : Type u} [topological_space α] {x : α} {s : set α} :\n    cluster_pt x (filter.principal s) ↔ filter.frequently (fun (y : α) => y ∈ s) (nhds x) :=\n  sorry\n\ntheorem cluster_pt.of_le_nhds {α : Type u} [topological_space α] {x : α} {f : filter α}\n    (H : f ≤ nhds x) [filter.ne_bot f] : cluster_pt x f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (cluster_pt x f)) (cluster_pt.equations._eqn_1 x f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (filter.ne_bot (nhds x ⊓ f))) (iff.mpr inf_eq_right H)))\n      _inst_2)\n\ntheorem cluster_pt.of_le_nhds' {α : Type u} [topological_space α] {x : α} {f : filter α}\n    (H : f ≤ nhds x) (hf : filter.ne_bot f) : cluster_pt x f :=\n  cluster_pt.of_le_nhds H\n\ntheorem cluster_pt.of_nhds_le {α : Type u} [topological_space α] {x : α} {f : filter α}\n    (H : nhds x ≤ f) : cluster_pt x f :=\n  sorry\n\ntheorem cluster_pt.mono {α : Type u} [topological_space α] {x : α} {f : filter α} {g : filter α}\n    (H : cluster_pt x f) (h : f ≤ g) : cluster_pt x g :=\n  ne_bot_of_le_ne_bot H (inf_le_inf_left (nhds x) h)\n\ntheorem cluster_pt.of_inf_left {α : Type u} [topological_space α] {x : α} {f : filter α}\n    {g : filter α} (H : cluster_pt x (f ⊓ g)) : cluster_pt x f :=\n  cluster_pt.mono H inf_le_left\n\ntheorem cluster_pt.of_inf_right {α : Type u} [topological_space α] {x : α} {f : filter α}\n    {g : filter α} (H : cluster_pt x (f ⊓ g)) : cluster_pt x g :=\n  cluster_pt.mono H inf_le_right\n\ntheorem ultrafilter.cluster_pt_iff {α : Type u} [topological_space α] {x : α} {f : ultrafilter α} :\n    cluster_pt x ↑f ↔ ↑f ≤ nhds x :=\n  { mp := ultrafilter.le_of_inf_ne_bot' f, mpr := fun (h : ↑f ≤ nhds x) => cluster_pt.of_le_nhds h }\n\n/-- A point `x` is a cluster point of a sequence `u` along a filter `F` if it is a cluster point\nof `map u F`. -/\ndef map_cluster_pt {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι)\n    (u : ι → α) :=\n  cluster_pt x (filter.map u F)\n\ntheorem map_cluster_pt_iff {α : Type u} [topological_space α] {ι : Type u_1} (x : α) (F : filter ι)\n    (u : ι → α) :\n    map_cluster_pt x F u ↔\n        ∀ (s : set α), s ∈ nhds x → filter.frequently (fun (a : ι) => u a ∈ s) F :=\n  sorry\n\ntheorem map_cluster_pt_of_comp {α : Type u} [topological_space α] {ι : Type u_1} {δ : Type u_2}\n    {F : filter ι} {φ : δ → ι} {p : filter δ} {x : α} {u : ι → α} [filter.ne_bot p]\n    (h : filter.tendsto φ p F) (H : filter.tendsto (u ∘ φ) p (nhds x)) : map_cluster_pt x F u :=\n  filter.ne_bot_of_le (le_inf H (trans_rel_right LessEq filter.map_map (filter.map_mono h)))\n\n/-!\n### Interior, closure and frontier in terms of neighborhoods\n-/\n\ntheorem interior_eq_nhds' {α : Type u} [topological_space α] {s : set α} :\n    interior s = set_of fun (a : α) => s ∈ nhds a :=\n  sorry\n\ntheorem interior_eq_nhds {α : Type u} [topological_space α] {s : set α} :\n    interior s = set_of fun (a : α) => nhds a ≤ filter.principal s :=\n  sorry\n\ntheorem mem_interior_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ interior s ↔ s ∈ nhds a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a ∈ interior s ↔ s ∈ nhds a)) interior_eq_nhds'))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl ((a ∈ set_of fun (a : α) => s ∈ nhds a) ↔ s ∈ nhds a))\n          set.mem_set_of_eq))\n      (iff.refl (s ∈ nhds a)))\n\ntheorem interior_set_of_eq {α : Type u} [topological_space α] {p : α → Prop} :\n    interior (set_of fun (x : α) => p x) =\n        set_of fun (x : α) => filter.eventually (fun (x : α) => p x) (nhds x) :=\n  interior_eq_nhds'\n\ntheorem is_open_set_of_eventually_nhds {α : Type u} [topological_space α] {p : α → Prop} :\n    is_open (set_of fun (x : α) => filter.eventually (fun (y : α) => p y) (nhds x)) :=\n  sorry\n\ntheorem subset_interior_iff_nhds {α : Type u} [topological_space α] {s : set α} {V : set α} :\n    s ⊆ interior V ↔ ∀ (x : α), x ∈ s → V ∈ nhds x :=\n  sorry\n\ntheorem is_open_iff_nhds {α : Type u} [topological_space α] {s : set α} :\n    is_open s ↔ ∀ (a : α), a ∈ s → nhds a ≤ filter.principal s :=\n  iff.trans (iff.symm subset_interior_iff_open)\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (s ⊆ interior s ↔ ∀ (a : α), a ∈ s → nhds a ≤ filter.principal s))\n          interior_eq_nhds))\n      (iff.refl (s ⊆ set_of fun (a : α) => nhds a ≤ filter.principal s)))\n\ntheorem is_open_iff_mem_nhds {α : Type u} [topological_space α] {s : set α} :\n    is_open s ↔ ∀ (a : α), a ∈ s → s ∈ nhds a :=\n  iff.trans is_open_iff_nhds\n    (forall_congr fun (_x : α) => imp_congr_right fun (_x_1 : _x ∈ s) => filter.le_principal_iff)\n\ntheorem is_open_iff_ultrafilter {α : Type u} [topological_space α] {s : set α} :\n    is_open s ↔ ∀ (x : α), x ∈ s → ∀ (l : ultrafilter α), ↑l ≤ nhds x → s ∈ l :=\n  sorry\n\ntheorem mem_closure_iff_frequently {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ closure s ↔ filter.frequently (fun (x : α) => x ∈ s) (nhds a) :=\n  sorry\n\ntheorem filter.frequently.mem_closure {α : Type u} [topological_space α] {s : set α} {a : α} :\n    filter.frequently (fun (x : α) => x ∈ s) (nhds a) → a ∈ closure s :=\n  iff.mpr mem_closure_iff_frequently\n\n/-- The set of cluster points of a filter is closed. In particular, the set of limit points\nof a sequence is closed. -/\ntheorem is_closed_set_of_cluster_pt {α : Type u} [topological_space α] {f : filter α} :\n    is_closed (set_of fun (x : α) => cluster_pt x f) :=\n  sorry\n\ntheorem mem_closure_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ closure s ↔ cluster_pt a (filter.principal s) :=\n  iff.trans mem_closure_iff_frequently (iff.symm cluster_pt_principal_iff_frequently)\n\ntheorem closure_eq_cluster_pts {α : Type u} [topological_space α] {s : set α} :\n    closure s = set_of fun (a : α) => cluster_pt a (filter.principal s) :=\n  set.ext fun (x : α) => mem_closure_iff_cluster_pt\n\ntheorem mem_closure_iff_nhds {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ closure s ↔ ∀ (t : set α), t ∈ nhds a → set.nonempty (t ∩ s) :=\n  iff.trans mem_closure_iff_cluster_pt cluster_pt_principal_iff\n\ntheorem mem_closure_iff_nhds' {α : Type u} [topological_space α] {s : set α} {a : α} :\n    a ∈ closure s ↔ ∀ (t : set α), t ∈ nhds a → ∃ (y : ↥s), ↑y ∈ t :=\n  sorry\n\ntheorem mem_closure_iff_comap_ne_bot {α : Type u} [topological_space α] {A : set α} {x : α} :\n    x ∈ closure A ↔ filter.ne_bot (filter.comap coe (nhds x)) :=\n  sorry\n\ntheorem mem_closure_iff_nhds_basis {α : Type u} {β : Type v} [topological_space α] {a : α}\n    {p : β → Prop} {s : β → set α} (h : filter.has_basis (nhds a) p s) {t : set α} :\n    a ∈ closure t ↔ ∀ (i : β), p i → ∃ (y : α), ∃ (H : y ∈ t), y ∈ s i :=\n  sorry\n\n/-- `x` belongs to the closure of `s` if and only if some ultrafilter\n  supported on `s` converges to `x`. -/\ntheorem mem_closure_iff_ultrafilter {α : Type u} [topological_space α] {s : set α} {x : α} :\n    x ∈ closure s ↔ ∃ (u : ultrafilter α), s ∈ u ∧ ↑u ≤ nhds x :=\n  sorry\n\ntheorem is_closed_iff_cluster_pt {α : Type u} [topological_space α] {s : set α} :\n    is_closed s ↔ ∀ (a : α), cluster_pt a (filter.principal s) → a ∈ s :=\n  sorry\n\ntheorem is_closed_iff_nhds {α : Type u} [topological_space α] {s : set α} :\n    is_closed s ↔ ∀ (x : α), (∀ (U : set α), U ∈ nhds x → set.nonempty (U ∩ s)) → x ∈ s :=\n  sorry\n\ntheorem closure_inter_open {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (h : is_open s) : s ∩ closure t ⊆ closure (s ∩ t) :=\n  sorry\n\n/-- The intersection of an open dense set with a dense set is a dense set. -/\ntheorem dense.inter_of_open_left {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (hs : dense s) (ht : dense t) (hso : is_open s) : dense (s ∩ t) :=\n  sorry\n\n/-- The intersection of a dense set with an open dense set is a dense set. -/\ntheorem dense.inter_of_open_right {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (hs : dense s) (ht : dense t) (hto : is_open t) : dense (s ∩ t) :=\n  set.inter_comm t s ▸ dense.inter_of_open_left ht hs hto\n\ntheorem dense.inter_nhds_nonempty {α : Type u} [topological_space α] {s : set α} {t : set α}\n    (hs : dense s) {x : α} (ht : t ∈ nhds x) : set.nonempty (s ∩ t) :=\n  sorry\n\ntheorem closure_diff {α : Type u} [topological_space α] {s : set α} {t : set α} :\n    closure s \\ closure t ⊆ closure (s \\ t) :=\n  sorry\n\ntheorem filter.frequently.mem_of_closed {α : Type u} [topological_space α] {a : α} {s : set α}\n    (h : filter.frequently (fun (x : α) => x ∈ s) (nhds a)) (hs : is_closed s) : a ∈ s :=\n  is_closed.closure_subset hs (filter.frequently.mem_closure h)\n\ntheorem is_closed.mem_of_frequently_of_tendsto {α : Type u} {β : Type v} [topological_space α]\n    {f : β → α} {b : filter β} {a : α} {s : set α} (hs : is_closed s)\n    (h : filter.frequently (fun (x : β) => f x ∈ s) b) (hf : filter.tendsto f b (nhds a)) : a ∈ s :=\n  sorry\n\ntheorem is_closed.mem_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α}\n    {b : filter β} {a : α} {s : set α} [filter.ne_bot b] (hs : is_closed s)\n    (hf : filter.tendsto f b (nhds a)) (h : filter.eventually (fun (x : β) => f x ∈ s) b) : a ∈ s :=\n  is_closed.mem_of_frequently_of_tendsto hs (filter.eventually.frequently h) hf\n\ntheorem mem_closure_of_tendsto {α : Type u} {β : Type v} [topological_space α] {f : β → α}\n    {b : filter β} {a : α} {s : set α} [filter.ne_bot b] (hf : filter.tendsto f b (nhds a))\n    (h : filter.eventually (fun (x : β) => f x ∈ s) b) : a ∈ closure s :=\n  is_closed.mem_of_tendsto is_closed_closure hf\n    (filter.eventually.mono h (set.preimage_mono subset_closure))\n\n/-- Suppose that `f` sends the complement to `s` to a single point `a`, and `l` is some filter.\nThen `f` tends to `a` along `l` restricted to `s` if and only if it tends to `a` along `l`. -/\ntheorem tendsto_inf_principal_nhds_iff_of_forall_eq {α : Type u} {β : Type v} [topological_space α]\n    {f : β → α} {l : filter β} {s : set β} {a : α} (h : ∀ (x : β), ¬x ∈ s → f x = a) :\n    filter.tendsto f (l ⊓ filter.principal s) (nhds a) ↔ filter.tendsto f l (nhds a) :=\n  sorry\n\n/-!\n### Limits of filters in topological spaces\n-/\n\n/-- If `f` is a filter, then `Lim f` is a limit of the filter, if it exists. -/\ndef Lim {α : Type u} [topological_space α] [Nonempty α] (f : filter α) : α :=\n  classical.epsilon fun (a : α) => f ≤ nhds a\n\n/--\nIf `f` is a filter satisfying `ne_bot f`, then `Lim' f` is a limit of the filter, if it exists.\n-/\ndef Lim' {α : Type u} [topological_space α] (f : filter α) [filter.ne_bot f] : α := Lim f\n\n/--\nIf `F` is an ultrafilter, then `filter.ultrafilter.Lim F` is a limit of the filter, if it exists.\nNote that dot notation `F.Lim` can be used for `F : ultrafilter α`.\n-/\ndef ultrafilter.Lim {α : Type u} [topological_space α] : ultrafilter α → α :=\n  fun (F : ultrafilter α) => Lim' ↑F\n\n/-- If `f` is a filter in `β` and `g : β → α` is a function, then `lim f` is a limit of `g` at `f`,\nif it exists. -/\ndef lim {α : Type u} {β : Type v} [topological_space α] [Nonempty α] (f : filter β) (g : β → α) :\n    α :=\n  Lim (filter.map g f)\n\n/-- If a filter `f` is majorated by some `𝓝 a`, then it is majorated by `𝓝 (Lim f)`. We formulate\nthis lemma with a `[nonempty α]` argument of `Lim` derived from `h` to make it useful for types\nwithout a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify\nthis instance with any other instance. -/\ntheorem le_nhds_Lim {α : Type u} [topological_space α] {f : filter α} (h : ∃ (a : α), f ≤ nhds a) :\n    f ≤ nhds (Lim f) :=\n  classical.epsilon_spec h\n\n/-- If `g` tends to some `𝓝 a` along `f`, then it tends to `𝓝 (lim f g)`. We formulate\nthis lemma with a `[nonempty α]` argument of `lim` derived from `h` to make it useful for types\nwithout a `[nonempty α]` instance. Because of the built-in proof irrelevance, Lean will unify\nthis instance with any other instance. -/\ntheorem tendsto_nhds_lim {α : Type u} {β : Type v} [topological_space α] {f : filter β} {g : β → α}\n    (h : ∃ (a : α), filter.tendsto g f (nhds a)) : filter.tendsto g f (nhds (lim f g)) :=\n  le_nhds_Lim h\n\n/-!\n### Locally finite families\n-/\n\n/- locally finite family [General Topology (Bourbaki, 1995)] -/\n\n/-- A family of sets in `set α` is locally finite if at every point `x:α`,\n  there is a neighborhood of `x` which meets only finitely many sets in the family -/\ndef locally_finite {α : Type u} {β : Type v} [topological_space α] (f : β → set α) :=\n  ∀ (x : α),\n    ∃ (t : set α), ∃ (H : t ∈ nhds x), set.finite (set_of fun (i : β) => set.nonempty (f i ∩ t))\n\ntheorem locally_finite_of_finite {α : Type u} {β : Type v} [topological_space α] {f : β → set α}\n    (h : set.finite set.univ) : locally_finite f :=\n  sorry\n\ntheorem locally_finite_subset {α : Type u} {β : Type v} [topological_space α] {f₁ : β → set α}\n    {f₂ : β → set α} (hf₂ : locally_finite f₂) (hf : ∀ (b : β), f₁ b ⊆ f₂ b) : locally_finite f₁ :=\n  sorry\n\ntheorem is_closed_Union_of_locally_finite {α : Type u} {β : Type v} [topological_space α]\n    {f : β → set α} (h₁ : locally_finite f) (h₂ : ∀ (i : β), is_closed (f i)) :\n    is_closed (set.Union fun (i : β) => f i) :=\n  sorry\n\n/-!\n### Continuity\n-/\n\n/-- A function between topological spaces is continuous if the preimage\n  of every open set is open. Registered as a structure to make sure it is not unfolded by Lean. -/\nstructure continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (f : α → β)\n    where\n  is_open_preimage : ∀ (s : set β), is_open s → is_open (f ⁻¹' s)\n\ntheorem continuous_def {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {f : α → β} : continuous f ↔ ∀ (s : set β), is_open s → is_open (f ⁻¹' s) :=\n  { mp :=\n      fun (hf : continuous f) (s : set β) (hs : is_open s) => continuous.is_open_preimage hf s hs,\n    mpr := fun (h : ∀ (s : set β), is_open s → is_open (f ⁻¹' s)) => continuous.mk h }\n\ntheorem is_open.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {f : α → β} (hf : continuous f) {s : set β} (h : is_open s) : is_open (f ⁻¹' s) :=\n  continuous.is_open_preimage hf s h\n\n/-- A function between topological spaces is continuous at a point `x₀`\nif `f x` tends to `f x₀` when `x` tends to `x₀`. -/\ndef continuous_at {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (f : α → β) (x : α) :=\n  filter.tendsto f (nhds x) (nhds (f x))\n\ntheorem continuous_at.tendsto {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} {x : α} (h : continuous_at f x) :\n    filter.tendsto f (nhds x) (nhds (f x)) :=\n  h\n\ntheorem continuous_at.preimage_mem_nhds {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} {x : α} {t : set β} (h : continuous_at f x)\n    (ht : t ∈ nhds (f x)) : f ⁻¹' t ∈ nhds x :=\n  h ht\n\ntheorem cluster_pt.map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {x : α} {la : filter α} {lb : filter β} (H : cluster_pt x la) {f : α → β}\n    (hfc : continuous_at f x) (hf : filter.tendsto f la lb) : cluster_pt (f x) lb :=\n  ne_bot_of_le_ne_bot (iff.mpr (filter.map_ne_bot_iff f) H)\n    (filter.tendsto.inf (continuous_at.tendsto hfc) hf)\n\ntheorem preimage_interior_subset_interior_preimage {α : Type u_1} {β : Type u_2}\n    [topological_space α] [topological_space β] {f : α → β} {s : set β} (hf : continuous f) :\n    f ⁻¹' interior s ⊆ interior (f ⁻¹' s) :=\n  interior_maximal (set.preimage_mono interior_subset) (is_open.preimage hf is_open_interior)\n\ntheorem continuous_id {α : Type u_1} [topological_space α] : continuous id :=\n  iff.mpr continuous_def fun (s : set α) (h : is_open s) => h\n\ntheorem continuous.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α]\n    [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} (hg : continuous g)\n    (hf : continuous f) : continuous (g ∘ f) :=\n  iff.mpr continuous_def\n    fun (s : set γ) (h : is_open s) => is_open.preimage hf (is_open.preimage hg h)\n\ntheorem continuous.iterate {α : Type u_1} [topological_space α] {f : α → α} (h : continuous f)\n    (n : ℕ) : continuous (nat.iterate f n) :=\n  nat.rec_on n continuous_id\n    fun (n : ℕ) (ihn : continuous (nat.iterate f n)) => continuous.comp ihn h\n\ntheorem continuous_at.comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α]\n    [topological_space β] [topological_space γ] {g : β → γ} {f : α → β} {x : α}\n    (hg : continuous_at g (f x)) (hf : continuous_at f x) : continuous_at (g ∘ f) x :=\n  filter.tendsto.comp hg hf\n\ntheorem continuous.tendsto {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {f : α → β} (hf : continuous f) (x : α) : filter.tendsto f (nhds x) (nhds (f x)) :=\n  sorry\n\n/-- A version of `continuous.tendsto` that allows one to specify a simpler form of the limit.\nE.g., one can write `continuous_exp.tendsto' 0 1 exp_zero`. -/\ntheorem continuous.tendsto' {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} (hf : continuous f) (x : α) (y : β) (h : f x = y) :\n    filter.tendsto f (nhds x) (nhds y) :=\n  h ▸ continuous.tendsto hf x\n\ntheorem continuous.continuous_at {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} {x : α} (h : continuous f) : continuous_at f x :=\n  continuous.tendsto h x\n\ntheorem continuous_iff_continuous_at {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} : continuous f ↔ ∀ (x : α), continuous_at f x :=\n  sorry\n\ntheorem continuous_at_const {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {x : α} {b : β} : continuous_at (fun (a : α) => b) x :=\n  tendsto_const_nhds\n\ntheorem continuous_const {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {b : β} : continuous fun (a : α) => b :=\n  iff.mpr continuous_iff_continuous_at fun (a : α) => continuous_at_const\n\ntheorem continuous_at_id {α : Type u_1} [topological_space α] {x : α} : continuous_at id x :=\n  continuous.continuous_at continuous_id\n\ntheorem continuous_at.iterate {α : Type u_1} [topological_space α] {f : α → α} {x : α}\n    (hf : continuous_at f x) (hx : f x = x) (n : ℕ) : continuous_at (nat.iterate f n) x :=\n  nat.rec_on n continuous_at_id\n    fun (n : ℕ) (ihn : continuous_at (nat.iterate f n) x) =>\n      (fun (this : continuous_at (nat.iterate f n ∘ f) x) => this)\n        (continuous_at.comp (Eq.symm hx ▸ ihn) hf)\n\ntheorem continuous_iff_is_closed {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} :\n    continuous f ↔ ∀ (s : set β), is_closed s → is_closed (f ⁻¹' s) :=\n  sorry\n\ntheorem is_closed.preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {f : α → β} (hf : continuous f) {s : set β} (h : is_closed s) : is_closed (f ⁻¹' s) :=\n  iff.mp continuous_iff_is_closed hf s h\n\ntheorem continuous_at_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} {x : α} :\n    continuous_at f x ↔ ∀ (g : ultrafilter α), ↑g ≤ nhds x → filter.tendsto f (↑g) (nhds (f x)) :=\n  filter.tendsto_iff_ultrafilter f (nhds x) (nhds (f x))\n\ntheorem continuous_iff_ultrafilter {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} :\n    continuous f ↔\n        ∀ (x : α) (g : ultrafilter α), ↑g ≤ nhds x → filter.tendsto f (↑g) (nhds (f x)) :=\n  sorry\n\n/-- A piecewise defined function `if p then f else g` is continuous, if both `f` and `g`\nare continuous, and they coincide on the frontier (boundary) of the set `{a | p a}`. -/\ntheorem continuous_if {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {p : α → Prop} {f : α → β} {g : α → β} {h : (a : α) → Decidable (p a)}\n    (hp : ∀ (a : α), a ∈ frontier (set_of fun (a : α) => p a) → f a = g a) (hf : continuous f)\n    (hg : continuous g) : continuous fun (a : α) => ite (p a) (f a) (g a) :=\n  sorry\n\n/-! ### Continuity and partial functions -/\n\n/-- Continuity of a partial function -/\ndef pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    (f : α →. β) :=\n  ∀ (s : set β), is_open s → is_open (pfun.preimage f s)\n\ntheorem open_dom_of_pcontinuous {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α →. β} (h : pcontinuous f) : is_open (pfun.dom f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (pfun.dom f))) (Eq.symm (pfun.preimage_univ f))))\n    (h set.univ is_open_univ)\n\ntheorem pcontinuous_iff' {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {f : α →. β} :\n    pcontinuous f ↔ ∀ {x : α} {y : β}, y ∈ f x → filter.ptendsto' f (nhds x) (nhds y) :=\n  sorry\n\n/-- If a continuous map `f` maps `s` to `t`, then it maps `closure s` to `closure t`. -/\ntheorem set.maps_to.closure {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {s : set α} {t : set β} {f : α → β} (h : set.maps_to f s t)\n    (hc : continuous f) : set.maps_to f (closure s) (closure t) :=\n  sorry\n\ntheorem image_closure_subset_closure_image {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} {s : set α} (h : continuous f) :\n    f '' closure s ⊆ closure (f '' s) :=\n  set.maps_to.image_subset (set.maps_to.closure (set.maps_to_image f s) h)\n\ntheorem map_mem_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β]\n    {s : set α} {t : set β} {f : α → β} {a : α} (hf : continuous f) (ha : a ∈ closure s)\n    (ht : ∀ (a : α), a ∈ s → f a ∈ t) : f a ∈ closure t :=\n  set.maps_to.closure ht hf ha\n\n/-!\n### Function with dense range\n-/\n\n/-- `f : ι → β` has dense range if its range (image) is a dense subset of β. -/\ndef dense_range {β : Type u_2} [topological_space β] {κ : Type u_5} (f : κ → β) :=\n  dense (set.range f)\n\n/-- A surjective map has dense range. -/\ntheorem function.surjective.dense_range {β : Type u_2} [topological_space β] {κ : Type u_5}\n    {f : κ → β} (hf : function.surjective f) : dense_range f :=\n  sorry\n\ntheorem dense_range_iff_closure_range {β : Type u_2} [topological_space β] {κ : Type u_5}\n    {f : κ → β} : dense_range f ↔ closure (set.range f) = set.univ :=\n  dense_iff_closure_eq\n\ntheorem dense_range.closure_range {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β}\n    (h : dense_range f) : closure (set.range f) = set.univ :=\n  dense.closure_eq h\n\ntheorem continuous.range_subset_closure_image_dense {α : Type u_1} {β : Type u_2}\n    [topological_space α] [topological_space β] {f : α → β} (hf : continuous f) {s : set α}\n    (hs : dense s) : set.range f ⊆ closure (f '' s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (set.range f ⊆ closure (f '' s))) (Eq.symm set.image_univ)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (f '' set.univ ⊆ closure (f '' s))) (Eq.symm (dense.closure_eq hs))))\n      (image_closure_subset_closure_image hf))\n\n/-- The image of a dense set under a continuous map with dense range is a dense set. -/\ntheorem dense_range.dense_image {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} (hf' : dense_range f) (hf : continuous f) {s : set α}\n    (hs : dense s) : dense (f '' s) :=\n  dense.of_closure (dense.mono (continuous.range_subset_closure_image_dense hf hs) hf')\n\n/-- If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense\nset. -/\ntheorem dense_range.dense_of_maps_to {α : Type u_1} {β : Type u_2} [topological_space α]\n    [topological_space β] {f : α → β} (hf' : dense_range f) (hf : continuous f) {s : set α}\n    (hs : dense s) {t : set β} (ht : set.maps_to f s t) : dense t :=\n  dense.mono (set.maps_to.image_subset ht) (dense_range.dense_image hf' hf hs)\n\n/-- Composition of a continuous map with dense range and a function with dense range has dense\nrange. -/\ntheorem dense_range.comp {β : Type u_2} {γ : Type u_3} [topological_space β] [topological_space γ]\n    {κ : Type u_5} {g : β → γ} {f : κ → β} (hg : dense_range g) (hf : dense_range f)\n    (cg : continuous g) : dense_range (g ∘ f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (dense_range (g ∘ f))) (dense_range.equations._eqn_1 (g ∘ f))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dense (set.range (g ∘ f)))) (set.range_comp g f)))\n      (dense_range.dense_image hg cg hf))\n\ntheorem dense_range.nonempty_iff {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β}\n    (hf : dense_range f) : Nonempty κ ↔ Nonempty β :=\n  iff.trans (iff.symm set.range_nonempty_iff_nonempty) (dense.nonempty_iff hf)\n\ntheorem dense_range.nonempty {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β}\n    [h : Nonempty β] (hf : dense_range f) : Nonempty κ :=\n  iff.mpr (dense_range.nonempty_iff hf) h\n\n/-- Given a function `f : α → β` with dense range and `b : β`, returns some `a : α`. -/\ndef dense_range.some {β : Type u_2} [topological_space β] {κ : Type u_5} {f : κ → β}\n    (hf : dense_range f) (b : β) : κ :=\n  Classical.choice sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.3009288736815721}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.preadditive\nimport category_theory.preadditive.additive_functor\nimport data.equiv.transfer_instance\n\n/-!\n# If `C` is preadditive, `Cᵒᵖ` has a natural preadditive structure.\n\n-/\n\nopen opposite\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [preadditive C]\n\ninstance : preadditive Cᵒᵖ :=\n{ hom_group := λ X Y, equiv.add_comm_group (op_equiv X Y),\n  add_comp' := λ X Y Z f f' g,\n    congr_arg quiver.hom.op (preadditive.comp_add _ _ _ g.unop f.unop f'.unop),\n  comp_add' := λ X Y Z f g g',\n    congr_arg quiver.hom.op (preadditive.add_comp _ _ _ g.unop g'.unop f.unop), }\n\n@[simp] lemma unop_zero (X Y : Cᵒᵖ) : (0 : X ⟶ Y).unop = 0 := rfl\n@[simp] lemma unop_add {X Y : Cᵒᵖ} (f g : X ⟶ Y) : (f + g).unop = f.unop + g.unop := rfl\n@[simp] lemma op_zero (X Y : C) : (0 : X ⟶ Y).op = 0 := rfl\n@[simp] \n\nvariables {C} {D : Type*} [category D] [preadditive D]\n\ninstance functor.op_additive (F : C ⥤ D) [F.additive] : F.op.additive := {}\n\ninstance functor.right_op_additive (F : Cᵒᵖ ⥤ D) [F.additive] : F.right_op.additive := {}\n\ninstance functor.left_op_additive (F : C ⥤ Dᵒᵖ) [F.additive] : F.left_op.additive := {}\n\ninstance functor.unop_additive (F : Cᵒᵖ ⥤ Dᵒᵖ) [F.additive] : F.unop.additive := {}\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/preadditive/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3009149536489494}}
{"text": "/- Copyright 2019 (c) Hans-Dieter Hiep. All rights reserved. Released under MIT license as described in the file LICENSE. -/\n\nimport util\n\n/- Each program has names. Names consist of a finite set of class names, a finite set of record names, and a finite set of symbol names. To each class name, there an associated finite set of field and method names. There is a built-in record name, boolean. -/\nclass names (α : Type) :=\n  (decidable_name: decidable_eq α)\n  (Nc: finset α)\n  (Nr: finset α)\n  (Ns: finset α)\n  (Nf {x : α} (H: x ∈ Nc): finset α)\n  (Nm {x : α} (H: x ∈ Nc): finset α)\n  (Nboolean : α)\n  (Hboolean : Nboolean ∈ Nr)\ninstance names.decidable {α : Type} [names α]:\n  decidable_eq α := names.decidable_name α\n\n/- We make use of subtypes, to demonstrate instances of fintype. Whenever we have established that these types are finite, we obtain decidable equality for functions that take these types as domain (thanks to Johannes Hölzl). This fact turns out to be crucial for semantics. -/\n@[reducible]\ndef class_name (α : Type) [names α] :=\n  {name : α // name ∈ names.Nc α}\n@[reducible]\ndef record_name (α : Type) [names α] :=\n  {name : α // name ∈ names.Nr α}\n@[reducible]\ndef symbol_name (α : Type) [names α] :=\n  {name : α // name ∈ names.Ns α}\n@[reducible]\ndef boolean_name (α : Type) [names α] : record_name α :=\n  ⟨names.Nboolean α, names.Hboolean α⟩\n@[reducible]\ndef field_name {α : Type} [names α] (self : class_name α) :=\n  {name : α // name ∈ names.Nf self.property}\n@[reducible]\ndef method_name {α : Type} [names α] (self : class_name α) :=\n  {name : α // name ∈ names.Nm self.property}\n\n/- A type in our object language is either: a reference type (by class name), or a data type (by record name). -/\n@[derive decidable_eq]\ninductive type (α : Type) [names α]\n| ref: class_name α → type\n| data: record_name α → type\n\n/- A variable context is a list of types; a local variable is encoded by an index in this context. Local variables do not have a name. -/\n@[reducible] def context (α : Type) [names α] := list (type α)\n/- Boolean is a built-in data type. -/\n@[reducible] def boolean (α : Type) [names α] : type α :=\n  (type.data (boolean_name α))\n\n/- A class declaration (of some class self) consists of a map from field names to field declarations, a map from method names to method declarations, and a method name that indicates a constructor. A field declaration consists of a type. A method declaration consists of a map from parameter names to parameter declarations. A parameter declaration consists of a type. -/\nstructure mdecl {α : Type} [names α]\n    {self : class_name α} (m : method_name self) :=\n  (args : list (type α))\nstructure fdecl {α : Type} [names α]\n    {self : class_name α} (f : field_name self) :=\n  (type : type α)\n-- Design choice: constructors and methods are treated uniformly.\nstructure cdecl {α : Type} [names α] (self : class_name α) :=\n  (fdecl (f : field_name self) : fdecl f)\n  (mdecl (m : method_name self) : mdecl m)\n  (ctor : method_name self)\n  \n/- A symbol declaration consists of a list of types of arguments, together with a resultant type. The arity of a symbol declaration is the length of the list of argument types. -/\nstructure sdecl (α : Type) [names α] :=\n  (args : list (type α))\n  (result : type α)\ndef sdecl.arity {α : Type} [names α] (s : sdecl α) :=\n  list.length s.args\n\n/- A signature consists of: a map from class names to class declarations, a map from symbols names to symbol declarations, and a class name of the root object. -/\nclass signature (α : Type) extends names α :=\n  (cdef (x : class_name α): cdecl x)\n  (sdef (x : symbol_name α) : sdecl α)\n\n/- Each parameter is associated to a type, each field is associated to a type, and each class name is associated to a constructor method name. -/\ndef param_types {α : Type} [signature α] {self : class_name α}\n    (m : method_name self) : list (type α) :=\n  ((signature.cdef self).mdecl m).args\ndef field_type {α : Type} [signature α] {self : class_name α}\n    (f : field_name self) : type α :=\n  ((signature.cdef self).fdecl f).type\ndef ctor {α : Type} [signature α] (self : class_name α) :\n    method_name self :=\n  (signature.cdef self).ctor\n/- A symbol with zero arity is a constant symbol, a symbol with non-zero arity is a function symbol. Each symbol is associated to a result type. A function symbol is associated to a list of argument types. -/\n@[derive decidable_eq]\nstructure symbol {α : Type} [signature α]\n    (args : list (type α)) (result : type α)\n    (s : symbol_name α) :=\n  (H: args = (signature.sdef s).args)\n  (G: result = (signature.sdef s).result)\ndef symbol.name {α : Type} [signature α]\n  {args : list (type α)} {result : type α}\n  {s : symbol_name α} (sym : symbol args result s) := s\n\n/- Given a signature and an enclosing class, we consider type environments to consist of: a method name, and declared local variables. -/\n@[derive decidable_eq]\nstructure tenv {α : Type} [signature α] (self : class_name α) :=\n  (current : method_name self)\n  (locals : context α)\n@[reducible]\ndef tenv.self {α : Type} [signature α] {self : class_name α}\n  (e : tenv self) : class_name α := self\n@[reducible]\ndef tenv.args {α : Type} [signature α] {self : class_name α}\n  (e : tenv self) : list (type α) := param_types e.current\n\n/- A note on terminology. Parameter: as declared in signature. Argument: a pure expression as supplied in a method call, or as supplied to a function symbol. -/\n\n/- Within a typing environment, we refer to numerous things. This refers to an object identity of the enclosed class. A local variable refers to the index within the declared local variables. A parameter refers to a parameter name of the current method. A field refers to a field within the enclosed class. -/\n@[derive decidable_eq]\nstructure tvar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α) :=\n  (H : ty = type.ref e.self)\n@[derive decidable_eq]\nstructure lvar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α):=\n  (idx : list_at ty e.locals)\n@[derive decidable_eq]\nstructure pvar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α) :=\n  (idx : list_at ty e.args)\n@[derive decidable_eq]\nstructure fvar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α) :=\n  (idx : field_name e.self) (H : field_type idx = ty)\n-- Store variable (LHS only)\n@[derive decidable_eq]\ninductive svar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α)\n| fvar: fvar e ty → svar\n| lvar: lvar e ty → svar\n-- Read variable (RHS only)\n@[derive decidable_eq]\ninductive rvar {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) (ty : type α)\n| tvar: tvar e ty → rvar\n| fvar: fvar e ty → rvar\n| pvar: pvar e ty → rvar\n| lvar: lvar e ty → rvar\n\n/- A pure expression within a typing environment is either: a constant, a function application, value lookup in environment, \nequality check. -/\n@[derive decidable_eq]\ninductive pexp {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) : list (type α) → Type\n| const {ty : type α} {c : symbol_name α}:\n    symbol [] ty c → pexp [ty]\n| app {l : list (type α)} {ty : type α} {f : symbol_name α}:\n    symbol l ty f → pexp l → pexp [ty]\n| lookup {ty : type α}:\n    rvar e ty → pexp [ty]\n| equal {ty : type α}:\n    pexp [ty] → pexp [ty] → pexp [boolean α]\n| cons {ty : type α} {l : list (type α)}:\n    pexp [ty] → pexp l → pexp (ty::l)\n/- The encoding of a mutual inductive type fails in Lean 3.4.2. We currently encode this directly, by considering single expressions as expressions with a length one type list. Tuples of expressions are created by constructing lists of expressions of length one lists of types. There are no expressions of empty lists of types. -/\nlemma pexp_not_nil {α : Type} [signature α]\n  {self : class_name α} {e : tenv self} {l : list (type α)}\n  (p : pexp e l) : l ≠ [] :=\nbegin\n  cases l, {intro, cases p}, {intro, cases a}\nend\n\n/- A statement within a typing environment is either: a skip, a sequential composition, a branch, a loop, an assignment, an asynchronous method call, an object allocation. -/\n@[derive decidable_eq]\ninductive stmt {α : Type} [signature α] {self : class_name α}\n    (e : tenv self) : bool → Type\n| ite: pexp e [boolean α] → stmt tt → stmt tt → stmt ff\n| while: pexp e [boolean α] → stmt tt → stmt ff\n| assign {ty : type α} (l : svar e ty):\n    pexp e [ty] → stmt ff\n| async (c : class_name α) (m : method_name c)\n    (H : m ≠ ctor c)\n    (o : rvar e (type.ref c))\n    (τ : pexp e (param_types m)): stmt ff\n| alloc (c : class_name α)\n    (τ : pexp e (param_types (ctor c))): stmt ff\n| skip: stmt tt\n| seq: stmt ff → stmt tt → stmt tt\n/- The encoding of a nested inductive type failed at the moment in Lean 3.4.2. Instead, we encode the nesting ourselves: the boolean argument determines which constructors are applicable. It is true if it is a list of statements, false if it is a single statement (thanks to Mario Carneiro).  -/\n@[reducible]\ndef statement {α : Type} [signature α] {self : class_name α}\n  (e : tenv self) := stmt e ff\ndef stmt.from_list {α : Type} [signature α]\n    {self : class_name α} {e : tenv self} :\n    list (statement e) → stmt e tt\n| [] := stmt.skip e\n| (x :: xs) := stmt.seq x (stmt.from_list xs)\ndef stmt.to_list {α : Type} [signature α]\n    {self : class_name α} {e : tenv self} :\n    stmt e tt → list (statement e)\n| (stmt.skip .(e)) := []\n| (stmt.seq x xs) := (x :: stmt.to_list xs)\n\n/- A program with a given program signature associates to each method of each class a program block. A program block associated to a method consists of: local variable declarations, and a statement within a typing environment. -/\nstructure pblock {α : Type} [signature α]\n    {self : class_name α} (m : method_name self) :=\n  (locals : context α)\n  (S : list (statement ⟨m,locals⟩))\ndef pblock.tenv {α : Type} [signature α] {self : class_name α}\n    {m : method_name self} (pb : pblock m) :=\n  (tenv.mk m pb.locals)\nstructure program (α : Type) [signature α] :=\n  (body {self : class_name α} (m : method_name self): pblock m)\n  (main: class_name α)\n", "meta": {"author": "praalhans", "repo": "lean-abs", "sha": "5d23eec7234c880f5ebc0d7b831caf55119edef8", "save_path": "github-repos/lean/praalhans-lean-abs", "path": "github-repos/lean/praalhans-lean-abs/lean-abs-5d23eec7234c880f5ebc0d7b831caf55119edef8/src/syntax.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.30083931696848865}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.sites.plus\nimport category_theory.limits.concrete_category\n\n/-!\n\n# Sheafification\n\nWe construct the sheafification of a presheaf over a site `C` with values in `D` whenever\n`D` is a concrete category for which the forgetful functor preserves the appropriate (co)limits\nand reflects isomorphisms.\n\nWe generally follow the approach of https://stacks.math.columbia.edu/tag/00W1\n\n-/\n\nnamespace category_theory\n\nopen category_theory.limits opposite\n\nuniverses w v u\nvariables {C : Type u} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\n\nsection\nvariables [concrete_category.{max v u} D]\n\nlocal attribute [instance]\n  concrete_category.has_coe_to_sort\n  concrete_category.has_coe_to_fun\n\n/-- A concrete version of the multiequalizer, to be used below. -/\n@[nolint has_inhabited_instance]\ndef meq {X : C} (P : Cᵒᵖ ⥤ D) (S : J.cover X) :=\n{ x : Π (I : S.arrow), P.obj (op I.Y) //\n  ∀ (I : S.relation), P.map I.g₁.op (x I.fst) = P.map I.g₂.op (x I.snd) }\nend\n\nnamespace meq\n\nvariables [concrete_category.{max v u} D]\n\nlocal attribute [instance]\n  concrete_category.has_coe_to_sort\n  concrete_category.has_coe_to_fun\n\n\ninstance {X} (P : Cᵒᵖ ⥤ D) (S : J.cover X) : has_coe_to_fun (meq P S)\n  (λ x, Π (I : S.arrow), P.obj (op I.Y)) := ⟨λ x, x.1⟩\n\n@[ext]\nlemma ext {X} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x y : meq P S)\n  (h : ∀ I : S.arrow, x I = y I) : x = y := subtype.ext $ funext $ h\n\nlemma condition {X} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x : meq P S) (I : S.relation) :\n  P.map I.g₁.op (x ((S.index P).fst_to I)) = P.map I.g₂.op (x ((S.index P).snd_to I)) := x.2 _\n\n/-- Refine a term of `meq P T` with respect to a refinement `S ⟶ T` of covers. -/\ndef refine {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.cover X} (x : meq P T) (e : S ⟶ T) :\n  meq P S :=\n⟨λ I, x ⟨I.Y, I.f, (le_of_hom e) _ I.hf⟩,\n  λ I, x.condition ⟨I.Y₁, I.Y₂, I.Z, I.g₁, I.g₂, I.f₁, I.f₂,\n    (le_of_hom e) _ I.h₁, (le_of_hom e) _ I.h₂, I.w⟩⟩\n\n@[simp]\nlemma refine_apply {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.cover X} (x : meq P T) (e : S ⟶ T)\n  (I : S.arrow) : x.refine e I = x ⟨I.Y, I.f, (le_of_hom e) _ I.hf⟩ := rfl\n\n/-- Pull back a term of `meq P S` with respect to a morphism `f : Y ⟶ X` in `C`. -/\ndef pullback {Y X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x : meq P S) (f : Y ⟶ X) :\n  meq P ((J.pullback f).obj S) :=\n⟨λ I, x ⟨_,I.f ≫ f, I.hf⟩, λ I, x.condition\n  ⟨I.Y₁, I.Y₂, I.Z, I.g₁, I.g₂, I.f₁ ≫ f, I.f₂ ≫ f, I.h₁, I.h₂, by simp [reassoc_of I.w]⟩ ⟩\n\n@[simp]\nlemma pullback_apply {Y X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x : meq P S) (f : Y ⟶ X)\n  (I : ((J.pullback f).obj S).arrow) : x.pullback f I = x ⟨_, I.f ≫ f, I.hf⟩ := rfl\n\n@[simp]\nlemma pullback_refine {Y X : C} {P : Cᵒᵖ ⥤ D} {S T : J.cover X} (h : S ⟶ T)\n  (f : Y ⟶ X) (x : meq P T) : (x.pullback f).refine\n  ((J.pullback f).map h) = (refine x h).pullback _ := rfl\n\n/-- Make a term of `meq P S`. -/\ndef mk {X : C} {P : Cᵒᵖ ⥤ D} (S : J.cover X) (x : P.obj (op X)) : meq P S :=\n⟨λ I, P.map I.f.op x, λ I, by { dsimp, simp only [← comp_apply, ← P.map_comp, ← op_comp, I.w] }⟩\n\nlemma mk_apply {X : C} {P : Cᵒᵖ ⥤ D} (S : J.cover X) (x : P.obj (op X)) (I : S.arrow) :\n  mk S x I = P.map I.f.op x := rfl\n\nvariable [preserves_limits (forget D)]\n\n/-- The equivalence between the type associated to `multiequalizer (S.index P)` and `meq P S`. -/\nnoncomputable\ndef equiv {X : C} (P : Cᵒᵖ ⥤ D) (S : J.cover X) [has_multiequalizer (S.index P)] :\n  (multiequalizer (S.index P) : D) ≃ meq P S :=\nlimits.concrete.multiequalizer_equiv _\n\n@[simp]\nlemma equiv_apply {X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} [has_multiequalizer (S.index P)]\n  (x : multiequalizer (S.index P)) (I : S.arrow) :\nequiv P S x I = multiequalizer.ι (S.index P) I x := rfl\n\n@[simp]\nlemma equiv_symm_eq_apply {X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} [has_multiequalizer (S.index P)]\n  (x : meq P S) (I : S.arrow) : multiequalizer.ι (S.index P) I ((meq.equiv P S).symm x) = x I :=\nbegin\n  let z := (meq.equiv P S).symm x,\n  rw ← equiv_apply,\n  simp,\nend\n\nend meq\n\nnamespace grothendieck_topology\n\nnamespace plus\n\nvariables [concrete_category.{max v u} D]\n\nlocal attribute [instance]\n  concrete_category.has_coe_to_sort\n  concrete_category.has_coe_to_fun\n\nvariable [preserves_limits (forget D)]\nvariables [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ D]\nvariables [∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\n\nnoncomputable theory\n\n/-- Make a term of `(J.plus_obj P).obj (op X)` from `x : meq P S`. -/\ndef mk {X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x : meq P S) : (J.plus_obj P).obj (op X) :=\ncolimit.ι (J.diagram P X) (op S) ((meq.equiv P S).symm x)\n\nlemma res_mk_eq_mk_pullback {Y X : C} {P : Cᵒᵖ ⥤ D} {S : J.cover X} (x : meq P S) (f : Y ⟶ X) :\n  (J.plus_obj P).map f.op (mk x) = mk (x.pullback f) :=\nbegin\n  dsimp [mk, plus_obj],\n  simp only [← comp_apply, colimit.ι_pre, ι_colim_map_assoc],\n  simp_rw [comp_apply],\n  congr' 1,\n  apply_fun meq.equiv P _,\n  erw equiv.apply_symm_apply,\n  ext i,\n  simp only [diagram_pullback_app,\n    meq.pullback_apply, meq.equiv_apply, ← comp_apply],\n  erw [multiequalizer.lift_ι, meq.equiv_symm_eq_apply],\n  cases i, refl,\nend\n\nlemma to_plus_mk {X : C} {P : Cᵒᵖ ⥤ D} (S : J.cover X) (x : P.obj (op X)) :\n  (J.to_plus P).app _ x = mk (meq.mk S x) :=\nbegin\n  dsimp [mk, to_plus],\n  let e : S ⟶ ⊤ := hom_of_le (order_top.le_top _),\n  rw ← colimit.w _ e.op,\n  delta cover.to_multiequalizer,\n  simp only [comp_apply],\n  congr' 1,\n  dsimp [diagram],\n  apply concrete.multiequalizer_ext,\n  intros i,\n  simpa only [← comp_apply, category.assoc, multiequalizer.lift_ι,\n    category.comp_id, meq.equiv_symm_eq_apply],\nend\n\nlemma to_plus_apply {X : C} {P : Cᵒᵖ ⥤ D} (S : J.cover X) (x : meq P S) (I : S.arrow) :\n  (J.to_plus P).app _ (x I) = (J.plus_obj P).map I.f.op (mk x) :=\nbegin\n  dsimp only [to_plus, plus_obj],\n  delta cover.to_multiequalizer,\n  dsimp [mk],\n  simp only [← comp_apply, colimit.ι_pre, ι_colim_map_assoc],\n  simp only [comp_apply],\n  dsimp only [functor.op],\n  let e : (J.pullback I.f).obj (unop (op S)) ⟶ ⊤ := hom_of_le (order_top.le_top _),\n  rw ← colimit.w _ e.op,\n  simp only [comp_apply],\n  congr' 1,\n  apply concrete.multiequalizer_ext,\n  intros i,\n  dsimp [diagram],\n  simp only [← comp_apply, category.assoc, multiequalizer.lift_ι,\n    category.comp_id, meq.equiv_symm_eq_apply],\n  let RR : S.relation :=\n    ⟨_, _, _, i.f, 𝟙 _, I.f, i.f ≫ I.f, I.hf, sieve.downward_closed _ I.hf _, by simp⟩,\n  cases I,\n  erw x.condition RR,\n  simpa [RR],\nend\n\nlemma to_plus_eq_mk {X : C} {P : Cᵒᵖ ⥤ D} (x : P.obj (op X)) :\n  (J.to_plus P).app _ x = mk (meq.mk ⊤ x) :=\nbegin\n  dsimp [mk, to_plus],\n  delta cover.to_multiequalizer,\n  simp only [comp_apply],\n  congr' 1,\n  apply_fun (meq.equiv P ⊤),\n  ext i,\n  simpa,\nend\n\nvariables [∀ (X : C), preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget D)]\n\nlemma exists_rep {X : C} {P : Cᵒᵖ ⥤ D} (x : (J.plus_obj P).obj (op X)) :\n  ∃ (S : J.cover X) (y : meq P S), x = mk y :=\nbegin\n  obtain ⟨S,y,h⟩ := concrete.colimit_exists_rep (J.diagram P X) x,\n  use [S.unop, meq.equiv _ _ y],\n  rw ← h,\n  dsimp [mk],\n  simp,\nend\n\nlemma eq_mk_iff_exists {X : C} {P : Cᵒᵖ ⥤ D} {S T : J.cover X}\n  (x : meq P S) (y : meq P T) : mk x = mk y ↔ (∃ (W : J.cover X) (h1 : W ⟶ S) (h2 : W ⟶ T),\n    x.refine h1 = y.refine h2) :=\nbegin\n  split,\n  { intros h,\n    obtain ⟨W, h1, h2, hh⟩ := concrete.colimit_exists_of_rep_eq _ _ _ h,\n    use [W.unop, h1.unop, h2.unop],\n    ext I,\n    apply_fun (multiequalizer.ι (W.unop.index P) I) at hh,\n    convert hh,\n    all_goals\n    { dsimp [diagram],\n      simp only [← comp_apply, multiequalizer.lift_ι, category.comp_id, meq.equiv_symm_eq_apply],\n      cases I, refl } },\n  { rintros ⟨S,h1,h2,e⟩,\n    apply concrete.colimit_rep_eq_of_exists,\n    use [(op S), h1.op, h2.op],\n    apply concrete.multiequalizer_ext,\n    intros i,\n    apply_fun (λ ee, ee i) at e,\n    convert e,\n    all_goals\n    { dsimp [diagram],\n      simp only [← comp_apply, multiequalizer.lift_ι, meq.equiv_symm_eq_apply],\n      cases i, refl } },\nend\n\n/-- `P⁺` is always separated. -/\ntheorem sep {X : C} (P : Cᵒᵖ ⥤ D) (S : J.cover X) (x y : (J.plus_obj P).obj (op X))\n  (h : ∀ (I : S.arrow), (J.plus_obj P).map I.f.op x = (J.plus_obj P).map I.f.op y) :\n  x = y :=\nbegin\n  -- First, we choose representatives for x and y.\n  obtain ⟨Sx,x,rfl⟩ := exists_rep x,\n  obtain ⟨Sy,y,rfl⟩ := exists_rep y,\n  simp only [res_mk_eq_mk_pullback] at h,\n\n  -- Next, using our assumption,\n  -- choose covers over which the pullbacks of these representatives become equal.\n  choose W h1 h2 hh using λ (I : S.arrow), (eq_mk_iff_exists _ _).mp (h I),\n\n  -- To prove equality, it suffices to prove that there exists a cover over which\n  -- the representatives become equal.\n  rw eq_mk_iff_exists,\n\n  -- Construct the cover over which the representatives become equal by combining the various\n  -- covers chosen above.\n  let B : J.cover X := S.bind W,\n  use B,\n\n  -- Prove that this cover refines the two covers over which our representatives are defined\n  -- and use these proofs.\n  let ex : B ⟶ Sx := hom_of_le begin\n    rintros Y f ⟨Z,e1,e2,he2,he1,hee⟩,\n    rw ← hee,\n    apply le_of_hom (h1 ⟨_, _, he2⟩),\n    exact he1,\n  end,\n  let ey : B ⟶ Sy := hom_of_le begin\n    rintros Y f ⟨Z,e1,e2,he2,he1,hee⟩,\n    rw ← hee,\n    apply le_of_hom (h2 ⟨_, _, he2⟩),\n    exact he1,\n  end,\n  use [ex, ey],\n\n  -- Now prove that indeed the representatives become equal over `B`.\n  -- This will follow by using the fact that our representatives become\n  -- equal over the chosen covers.\n  ext1 I,\n  let IS : S.arrow := I.from_middle,\n  specialize hh IS,\n  let IW : (W IS).arrow := I.to_middle,\n  apply_fun (λ e, e IW) at hh,\n  convert hh,\n  { let Rx : Sx.relation := ⟨I.Y, I.Y, I.Y, 𝟙 _, 𝟙 _, I.f,\n      I.to_middle_hom ≫ I.from_middle_hom, _, _, by simp [I.middle_spec]⟩,\n    have := x.condition Rx,\n    simpa using this },\n  { let Ry : Sy.relation := ⟨I.Y, I.Y, I.Y, 𝟙 _, 𝟙 _, I.f,\n      I.to_middle_hom ≫ I.from_middle_hom, _, _, by simp [I.middle_spec]⟩,\n    have := y.condition Ry,\n    simpa using this },\nend\n\nlemma inj_of_sep (P : Cᵒᵖ ⥤ D) (hsep : ∀ (X : C) (S : J.cover X) (x y : P.obj (op X)),\n  (∀ I : S.arrow, P.map I.f.op x = P.map I.f.op y) → x = y) (X : C) :\n  function.injective ((J.to_plus P).app (op X)) :=\nbegin\n  intros x y h,\n  simp only [to_plus_eq_mk] at h,\n  rw eq_mk_iff_exists at h,\n  obtain ⟨W, h1, h2, hh⟩ := h,\n  apply hsep X W,\n  intros I,\n  apply_fun (λ e, e I) at hh,\n  exact hh\nend\n\n/-- An auxiliary definition to be used in the proof of `exists_of_sep` below.\n  Given a compatible family of local sections for `P⁺`, and representatives of said sections,\n  construct a compatible family of local sections of `P` over the combination of the covers\n  associated to the representatives.\n  The separatedness condition is used to prove compatibility among these local sections of `P`. -/\ndef meq_of_sep (P : Cᵒᵖ ⥤ D)\n  (hsep : ∀ (X : C) (S : J.cover X) (x y : P.obj (op X)),\n    (∀ I : S.arrow, P.map I.f.op x = P.map I.f.op y) → x = y)\n  (X : C) (S : J.cover X)\n  (s : meq (J.plus_obj P) S)\n  (T : Π (I : S.arrow), J.cover I.Y)\n  (t : Π (I : S.arrow), meq P (T I))\n  (ht : ∀ (I : S.arrow), s I = mk (t I)) : meq P (S.bind T) :=\n{ val := λ I, t I.from_middle I.to_middle,\n  property := begin\n    intros II,\n    apply inj_of_sep P hsep,\n    rw [← comp_apply, ← comp_apply, (J.to_plus P).naturality, (J.to_plus P).naturality,\n      comp_apply, comp_apply],\n    erw [to_plus_apply (T II.fst.from_middle) (t II.fst.from_middle) II.fst.to_middle,\n         to_plus_apply (T II.snd.from_middle) (t II.snd.from_middle) II.snd.to_middle,\n         ← ht, ← ht, ← comp_apply, ← comp_apply, ← (J.plus_obj P).map_comp,\n         ← (J.plus_obj P).map_comp],\n    rw [← op_comp, ← op_comp],\n    let IR : S.relation :=\n      ⟨_, _, _, II.g₁ ≫ II.fst.to_middle_hom, II.g₂ ≫ II.snd.to_middle_hom,\n        II.fst.from_middle_hom, II.snd.from_middle_hom, II.fst.from_middle_condition,\n        II.snd.from_middle_condition, _⟩,\n    swap, { simp only [category.assoc, II.fst.middle_spec, II.snd.middle_spec], apply II.w },\n    exact s.condition IR,\n  end }\n\ntheorem exists_of_sep (P : Cᵒᵖ ⥤ D)\n  (hsep : ∀ (X : C) (S : J.cover X) (x y : P.obj (op X)),\n    (∀ I : S.arrow, P.map I.f.op x = P.map I.f.op y) → x = y)\n  (X : C) (S : J.cover X)\n  (s : meq (J.plus_obj P) S) :\n  ∃ t : (J.plus_obj P).obj (op X), meq.mk S t = s :=\nbegin\n  have inj : ∀ (X : C), function.injective ((J.to_plus P).app (op X)) := inj_of_sep _ hsep,\n\n  -- Choose representatives for the given local sections.\n  choose T t ht using λ I, exists_rep (s I),\n\n  -- Construct a large cover over which we will define a representative that will\n  -- provide the gluing of the given local sections.\n  let B : J.cover X := S.bind T,\n  choose Z e1 e2 he2 he1 hee using λ I : B.arrow, I.hf,\n\n  -- Construct a compatible system of local sections over this large cover, using the chosen\n  -- representatives of our local sections.\n  -- The compatilibity here follows from the separatedness assumption.\n  let w : meq P B := meq_of_sep P hsep X S s T t ht,\n\n  -- The associated gluing will be the candidate section.\n  use mk w,\n  ext I,\n  erw [ht, res_mk_eq_mk_pullback],\n\n  -- Use the separatedness of `P⁺` to prove that this is indeed a gluing of our\n  -- original local sections.\n  apply sep P (T I),\n  intros II,\n  simp only [res_mk_eq_mk_pullback, eq_mk_iff_exists],\n\n  -- It suffices to prove equality for representatives over a\n  -- convenient sufficiently large cover...\n  use (J.pullback II.f).obj (T I),\n  let e0 : (J.pullback II.f).obj (T I) ⟶ (J.pullback II.f).obj ((J.pullback I.f).obj B) :=\n    hom_of_le begin\n      intros Y f hf,\n      apply sieve.le_pullback_bind _ _ _ I.hf,\n      { cases I,\n        exact hf },\n    end,\n  use [e0, 𝟙 _],\n  ext IV,\n  dsimp only [meq.refine_apply, meq.pullback_apply, w],\n  let IA : B.arrow := ⟨_, (IV.f ≫ II.f) ≫ I.f, _⟩,\n  swap,\n  { refine ⟨I.Y, _, _, I.hf, _, rfl⟩,\n    apply sieve.downward_closed,\n    convert II.hf,\n    cases I, refl },\n  let IB : S.arrow := IA.from_middle,\n  let IC : (T IB).arrow := IA.to_middle,\n  let ID : (T I).arrow := ⟨IV.Y, IV.f ≫ II.f, sieve.downward_closed (T I) II.hf IV.f⟩,\n  change t IB IC = t I ID,\n  apply inj IV.Y,\n  erw [to_plus_apply (T I) (t I) ID, to_plus_apply (T IB) (t IB) IC, ← ht, ← ht],\n\n  -- Conclude by constructing the relation showing equality...\n  let IR : S.relation := ⟨_, _, IV.Y, IC.f, ID.f, IB.f, I.f, _, I.hf, IA.middle_spec⟩,\n  convert s.condition IR,\n  cases I, refl,\nend\n\nvariable [reflects_isomorphisms (forget D)]\n\n/-- If `P` is separated, then `P⁺` is a sheaf. -/\ntheorem is_sheaf_of_sep (P : Cᵒᵖ ⥤ D)\n  (hsep : ∀ (X : C) (S : J.cover X) (x y : P.obj (op X)),\n    (∀ I : S.arrow, P.map I.f.op x = P.map I.f.op y) → x = y) :\n  presheaf.is_sheaf J (J.plus_obj P) :=\nbegin\n  rw presheaf.is_sheaf_iff_multiequalizer,\n  intros X S,\n  apply is_iso_of_reflects_iso _ (forget D),\n  rw is_iso_iff_bijective,\n  split,\n  { intros x y h,\n    apply sep P S _ _,\n    intros I,\n    apply_fun (meq.equiv _ _) at h,\n    apply_fun (λ e, e I) at h,\n    convert h,\n    { erw [meq.equiv_apply, ← comp_apply, multiequalizer.lift_ι] },\n    { erw [meq.equiv_apply, ← comp_apply, multiequalizer.lift_ι] } },\n  { rintros (x : (multiequalizer (S.index _) : D)),\n    obtain ⟨t,ht⟩ := exists_of_sep P hsep X S (meq.equiv _ _ x),\n    use t,\n    apply_fun meq.equiv _ _,\n    swap, { apply_instance },\n    rw ← ht,\n    ext i,\n    dsimp,\n    rw [← comp_apply, multiequalizer.lift_ι],\n    refl }\nend\n\nvariable (J)\n\n/-- `P⁺⁺` is always a sheaf. -/\ntheorem is_sheaf_plus_plus (P : Cᵒᵖ ⥤ D) :\n  presheaf.is_sheaf J (J.plus_obj (J.plus_obj P)) :=\nbegin\n  apply is_sheaf_of_sep,\n  intros X S x y,\n  apply sep,\nend\n\nend plus\n\nvariables (J)\nvariables\n  [∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\n  [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ D]\n\n/-- The sheafification of a presheaf `P`.\n*NOTE:* Additional hypotheses are needed to obtain a proof that this is a sheaf! -/\ndef sheafify (P : Cᵒᵖ ⥤ D) : Cᵒᵖ ⥤ D := J.plus_obj (J.plus_obj P)\n\n/-- The canonical map from `P` to its sheafification. -/\ndef to_sheafify (P : Cᵒᵖ ⥤ D) : P ⟶ J.sheafify P :=\nJ.to_plus P ≫ J.plus_map (J.to_plus P)\n\n/-- The canonical map on sheafifications induced by a morphism. -/\ndef sheafify_map {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) : J.sheafify P ⟶ J.sheafify Q :=\nJ.plus_map $ J.plus_map η\n\n@[simp]\nlemma sheafify_map_id (P : Cᵒᵖ ⥤ D) : J.sheafify_map (𝟙 P) = 𝟙 (J.sheafify P) :=\nby { dsimp [sheafify_map, sheafify], simp }\n\n@[simp]\nlemma sheafify_map_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :\n  J.sheafify_map (η ≫ γ) = J.sheafify_map η ≫ J.sheafify_map γ :=\nby { dsimp [sheafify_map, sheafify], simp }\n\n@[simp, reassoc]\nlemma to_sheafify_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :\n  η ≫ J.to_sheafify _ = J.to_sheafify _ ≫ J.sheafify_map η :=\nby { dsimp [sheafify_map, sheafify, to_sheafify], simp }\n\nvariable (D)\n\n/-- The sheafification of a presheaf `P`, as a functor.\n*NOTE:* Additional hypotheses are needed to obtain a proof that this is a sheaf! -/\ndef sheafification : (Cᵒᵖ ⥤ D) ⥤ Cᵒᵖ ⥤ D := (J.plus_functor D ⋙ J.plus_functor D)\n\n@[simp]\nlemma sheafification_obj (P : Cᵒᵖ ⥤ D) : (J.sheafification D).obj P = J.sheafify P := rfl\n\n@[simp]\nlemma sheafification_map {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) : (J.sheafification D).map η =\n  J.sheafify_map η := rfl\n\n/-- The canonical map from `P` to its sheafification, as a natural transformation.\n*Note:* We only show this is a sheaf under additional hypotheses on `D`. -/\ndef to_sheafification : 𝟭 _ ⟶ sheafification J D :=\nJ.to_plus_nat_trans D ≫ whisker_right (J.to_plus_nat_trans D) (J.plus_functor D)\n\n@[simp]\nlemma to_sheafification_app (P : Cᵒᵖ ⥤ D) : (J.to_sheafification D).app P = J.to_sheafify P := rfl\n\nvariable {D}\n\nlemma is_iso_to_sheafify {P : Cᵒᵖ ⥤ D} (hP : presheaf.is_sheaf J P) :\n  is_iso (J.to_sheafify P) :=\nbegin\n  dsimp [to_sheafify],\n  haveI : is_iso (J.to_plus P) := by { apply is_iso_to_plus_of_is_sheaf J P hP },\n  haveI : is_iso ((J.plus_functor D).map (J.to_plus P)) := by { apply functor.map_is_iso },\n  exact @is_iso.comp_is_iso _ _ _ _ _ (J.to_plus P)\n    ((J.plus_functor D).map (J.to_plus P)) _ _,\nend\n\n/-- If `P` is a sheaf, then `P` is isomorphic to `J.sheafify P`. -/\ndef iso_sheafify {P : Cᵒᵖ ⥤ D} (hP : presheaf.is_sheaf J P) :\n  P ≅ J.sheafify P :=\nby letI := is_iso_to_sheafify J hP; exactI as_iso (J.to_sheafify P)\n\n@[simp]\nlemma iso_sheafify_hom {P : Cᵒᵖ ⥤ D} (hP : presheaf.is_sheaf J P) :\n  (J.iso_sheafify hP).hom = J.to_sheafify P := rfl\n\n/-- Given a sheaf `Q` and a morphism `P ⟶ Q`, construct a morphism from\n`J.sheafifcation P` to `Q`. -/\ndef sheafify_lift {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q) :\n  J.sheafify P ⟶ Q := J.plus_lift (J.plus_lift η hQ) hQ\n\nlemma to_sheafify_sheafify_lift {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q) :\n  J.to_sheafify P ≫ sheafify_lift J η hQ = η :=\nby { dsimp only [sheafify_lift, to_sheafify], simp }\n\nlemma sheafify_lift_unique {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : presheaf.is_sheaf J Q)\n  (γ : J.sheafify P ⟶ Q) :\n  J.to_sheafify P ≫ γ = η → γ = sheafify_lift J η hQ :=\nbegin\n  intros h,\n  apply plus_lift_unique,\n  apply plus_lift_unique,\n  rw [← category.assoc, ← plus_map_to_plus],\n  exact h,\nend\n\n@[simp]\nlemma iso_sheafify_inv {P : Cᵒᵖ ⥤ D} (hP : presheaf.is_sheaf J P) :\n  (J.iso_sheafify hP).inv = J.sheafify_lift (𝟙 _) hP :=\nbegin\n  apply J.sheafify_lift_unique,\n  simp [iso.comp_inv_eq],\nend\n\nlemma sheafify_hom_ext {P Q : Cᵒᵖ ⥤ D} (η γ : J.sheafify P ⟶ Q) (hQ : presheaf.is_sheaf J Q)\n  (h : J.to_sheafify P ≫ η = J.to_sheafify P ≫ γ) : η = γ :=\nbegin\n  apply J.plus_hom_ext _ _ hQ,\n  apply J.plus_hom_ext _ _ hQ,\n  rw [← category.assoc, ← category.assoc, ← plus_map_to_plus],\n  exact h,\nend\n\nlemma sheafify_map_sheafify_lift {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R)\n  (hR : presheaf.is_sheaf J R) :\n  J.sheafify_map η ≫ J.sheafify_lift γ hR = J.sheafify_lift (η ≫ γ) hR :=\nbegin\n  apply J.sheafify_lift_unique,\n  rw [← category.assoc, ← J.to_sheafify_naturality,\n    category.assoc, to_sheafify_sheafify_lift],\nend\n\nend grothendieck_topology\n\nvariables (J)\nvariables\n  [concrete_category.{max v u} D]\n  [preserves_limits (forget D)]\n  [∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\n  [∀ (X : C), has_colimits_of_shape (J.cover X)ᵒᵖ D]\n  [∀ (X : C), preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget D)]\n  [reflects_isomorphisms (forget D)]\n\nlemma grothendieck_topology.sheafify_is_sheaf (P : Cᵒᵖ ⥤ D) :\n  presheaf.is_sheaf J (J.sheafify P) :=\ngrothendieck_topology.plus.is_sheaf_plus_plus _ _\n\nvariables (D)\n\n/-- The sheafification functor, as a functor taking values in `Sheaf`. -/\n@[simps obj map]\ndef presheaf_to_Sheaf : (Cᵒᵖ ⥤ D) ⥤ Sheaf J D :=\n{ obj := λ P, ⟨J.sheafify P, J.sheafify_is_sheaf P⟩,\n  map := λ P Q η, J.sheafify_map η,\n  map_id' := J.sheafify_map_id,\n  map_comp' := λ P Q R, J.sheafify_map_comp }\n\n/-- The sheafification functor is left adjoint to the forgetful functor. -/\n@[simps hom_equiv_apply hom_equiv_symm_apply counit_app]\ndef sheafification_adjunction : presheaf_to_Sheaf J D ⊣ Sheaf_to_presheaf J D :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := λ P Q,\n  { to_fun := λ e, J.to_sheafify P ≫ e,\n    inv_fun := λ e, J.sheafify_lift e Q.2,\n    left_inv := λ e, (J.sheafify_lift_unique _ _ _ rfl).symm,\n    right_inv := λ e, J.to_sheafify_sheafify_lift _ _ },\n  hom_equiv_naturality_left_symm' := begin\n    intros P Q R η γ, dsimp, symmetry,\n    apply J.sheafify_map_sheafify_lift,\n  end,\n  hom_equiv_naturality_right' := λ P Q R η γ, by { dsimp, rw category.assoc, refl } }\n\n@[simp]\nlemma sheafification_adjunction_unit_app (P) :\n  (sheafification_adjunction J D).unit.app P = J.to_sheafify P :=\nbegin\n  ext,\n  dsimp [sheafification_adjunction],\n  simp,\nend\n\nvariables {J D}\n/-- A sheaf `P` is isomorphic to its own sheafification. -/\ndef sheafification_iso (P : Sheaf J D) :\n  P ≅ (presheaf_to_Sheaf J D).obj P :=\n{ hom := (J.iso_sheafify P.2).hom,\n  inv := (J.iso_sheafify P.2).inv,\n  hom_inv_id' := (J.iso_sheafify P.2).hom_inv_id,\n  inv_hom_id' := (J.iso_sheafify P.2).inv_hom_id }\n\n@[simp]\nlemma sheafification_iso_hom (P : Sheaf J D) :\n  (sheafification_iso P).hom = J.to_sheafify P := rfl\n\n@[simp]\nlemma sheafification_iso_inv (P : Sheaf J D) :\n  (sheafification_iso P).inv = J.sheafify_lift (𝟙 P) P.property :=\nby { dsimp [sheafification_iso], simp }\n\ninstance is_iso_sheafification_adjunction_counit (P : Sheaf J D) :\n  is_iso ((sheafification_adjunction J D).counit.app P) :=\nbegin\n  dsimp [sheafification_adjunction],\n  rw ← sheafification_iso_inv,\n  apply_instance\nend\n\ninstance sheafification_reflective : is_iso (sheafification_adjunction J D).counit :=\nnat_iso.is_iso_of_is_iso_app _\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sites/sheafification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6261241772283033, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3008393169684886}}
{"text": "import tactic\nimport utilities.written_by_others.list_take_join\n\nsection tactic_in_list_explicit\n\nmeta def in_list_explicit : tactic unit := `[\n  tactic.repeat `[\n    tactic.try `[apply list.mem_cons_self],\n    tactic.try `[apply list.mem_cons_of_mem]\n  ]\n]\n\nmeta def split_ile : tactic unit := `[\n  split,\n  {\n    in_list_explicit,\n  }\n]\n\nend tactic_in_list_explicit\n\nnamespace list\n\nvariables {α β : Type*} {x y z : list α}\n\nsection list_append_append\n\nlemma length_append_append :\n  list.length (x ++ y ++ z) = x.length + y.length + z.length :=\nby rw [list.length_append, list.length_append]\n\nlemma map_append_append {f : α → β} :\n  list.map f (x ++ y ++ z) = list.map f x ++ list.map f y ++ list.map f z :=\nby rw [list.map_append, list.map_append]\n\nlemma filter_map_append_append {f : α → option β} :\n  list.filter_map f (x ++ y ++ z) = list.filter_map f x ++ list.filter_map f y ++ list.filter_map f z :=\nby rw [list.filter_map_append, list.filter_map_append]\n\nlemma reverse_append_append :\n  list.reverse (x ++ y ++ z) = z.reverse ++ y.reverse ++ x.reverse :=\nby rw [list.reverse_append, list.reverse_append, list.append_assoc]\n\nlemma mem_append_append {a : α} :\n  a ∈ x ++ y ++ z  ↔  a ∈ x ∨ a ∈ y ∨ a ∈ z  :=\nby rw [list.mem_append, list.mem_append, or_assoc]\n\nlemma forall_mem_append_append {p : α → Prop} :\n  (∀ a ∈ x ++ y ++ z, p a)  ↔  (∀ a ∈ x, p a) ∧ (∀ a ∈ y, p a) ∧ (∀ a ∈ z, p a)  :=\nby rw [list.forall_mem_append, list.forall_mem_append, and_assoc]\n\nlemma join_append_append {X Y Z : list (list α)} :\n  list.join (X ++ Y ++ Z) = X.join ++ Y.join ++ Z.join :=\nby rw [list.join_append, list.join_append]\n\nend list_append_append\n\nsection list_repeat\n\nlemma repeat_zero (s : α) :\n  list.repeat s 0 = [] :=\nrfl\n\nlemma repeat_succ_eq_singleton_append (s : α) (n : ℕ) :\n  list.repeat s n.succ = [s] ++ list.repeat s n :=\nrfl\n\nlemma repeat_succ_eq_append_singleton (s : α) (n : ℕ) :\n  list.repeat s n.succ = list.repeat s n ++ [s] :=\nbegin\n  change list.repeat s (n + 1) = list.repeat s n ++ [s],\n  rw list.repeat_add,\n  refl,\nend\n\nend list_repeat\n\nsection list_join\n\nlemma join_singleton : [x].join = x :=\nby rw [list.join, list.join, list.append_nil]\n\nlemma append_join_append (L : list (list α)) :\n  x ++ (list.map (λ l, l ++ x) L).join = (list.map (λ l, x ++ l) L).join ++ x :=\nbegin\n  induction L,\n  {\n    rw [list.map_nil, list.join, list.append_nil, list.map_nil, list.join, list.nil_append],\n  },\n  {\n    rw [\n      list.map_cons, list.join, list.map_cons, list.join, list.append_assoc, L_ih,\n      list.append_assoc, list.append_assoc\n    ],\n  },\nend\n\nlemma reverse_join (L : list (list α)) :\n  L.join.reverse = (list.map list.reverse L).reverse.join :=\nbegin\n  induction L,\n  {\n    refl,\n  },\n  {\n    rw [\n      list.join, list.reverse_append, L_ih,\n      list.map_cons, list.reverse_cons, list.join_append, list.join_singleton\n    ],\n  },\nend\n\nprivate lemma cons_drop_succ {m : ℕ} (mlt : m < x.length) :\n  drop m x = x.nth_le m mlt :: drop m.succ x :=\nbegin\n  induction x with d l ih generalizing m,\n  {\n    exfalso,\n    rw length at mlt,\n    exact nat.not_lt_zero m mlt,\n  },\n  cases m,\n  {\n    simp,\n  },\n  rw [drop, drop, nth_le],\n  apply ih,\nend\n\nlemma drop_join_of_lt {L : list (list α)} {n : ℕ} (notall : n < L.join.length) :\n  ∃ m k : ℕ, ∃ mlt : m < L.length, k < (L.nth_le m mlt).length ∧\n    L.join.drop n = (L.nth_le m mlt).drop k ++ (L.drop m.succ).join :=\nbegin\n  obtain ⟨m, k, mlt, klt, left_half⟩ := take_join_of_lt notall,\n  use    [m, k, mlt, klt],\n  have L_two_parts := congr_arg list.join (list.take_append_drop m L),\n  rw list.join_append at L_two_parts,\n  have whole := list.take_append_drop n L.join,\n  rw left_half at whole,\n  have important := eq.trans whole L_two_parts.symm,\n  rw append_assoc at important,\n  have right_side := append_left_cancel important,\n  have auxi : (drop m L).join = (L.nth_le m mlt :: drop m.succ L).join,\n  {\n    apply congr_arg,\n    apply cons_drop_succ,\n  },\n  rw join at auxi,\n  rw auxi at right_side,\n  have near_result :\n    take k (L.nth_le m mlt) ++ drop n L.join =\n    take k (L.nth_le m mlt) ++ drop k (L.nth_le m mlt) ++ (drop m.succ L).join,\n  {\n    convert right_side,\n    rw list.take_append_drop,\n  },\n  rw append_assoc at near_result,\n  exact append_left_cancel near_result,\nend\n\ndef n_times (l : list α) (n : ℕ) : list α :=\n(list.repeat l n).join\n\ninfix ` ^ ` : 100 := n_times\n\nend list_join\n\nvariables [decidable_eq α]\n\nsection list_index_of\n\nlemma index_of_append_of_in {a : α} (yes_on_left : a ∈ x) :\n  index_of a (x ++ y)  =  index_of a x  :=\nbegin\n  induction x with d l ih,\n  {\n    exfalso,\n    exact not_mem_nil a yes_on_left,\n  },\n  rw list.cons_append,\n  by_cases ad : a = d,\n  {\n    rw index_of_cons_eq _ ad,\n    rw index_of_cons_eq _ ad,\n  },\n  rw index_of_cons_ne _ ad,\n  rw index_of_cons_ne _ ad,\n  apply congr_arg,\n  apply ih,\n  exact mem_of_ne_of_mem ad yes_on_left,\nend\n\nlemma index_of_append_of_notin {a : α} (not_on_left : a ∉ x) :\n  index_of a (x ++ y)  =  x.length  +  index_of a y  :=\nbegin\n  induction x with d l ih,\n  {\n    rw [list.nil_append, list.length, zero_add],\n  },\n  rw [\n    list.cons_append,\n    index_of_cons_ne _ (ne_of_not_mem_cons not_on_left),\n    list.length,\n    ih (not_mem_of_not_mem_cons not_on_left),\n    nat.succ_add\n  ],\nend\n\nend list_index_of\n\nsection list_count_in\n\ndef count_in (l : list α) (a : α) : ℕ :=\nlist.sum (list.map (λ s, ite (s = a) 1 0) l)\n\nlemma count_in_nil (a : α) :\n  count_in [] a = 0 :=\nbegin\n  refl,\nend\n\nlemma count_in_cons (a b : α) :\n  count_in (b :: x) a  =  ite (b = a) 1 0  +  count_in x a  :=\nbegin\n  unfold count_in,\n  rw list.map_cons,\n  rw list.sum_cons,\nend\n\nlemma count_in_append (a : α) :\n  count_in (x ++ y) a  =  count_in x a  +  count_in y a  :=\nbegin\n  unfold count_in,\n  rw list.map_append,\n  rw list.sum_append,\nend\n\nlemma count_in_repeat_eq (a : α) (n : ℕ) :\n  count_in (list.repeat a n) a  =  n  :=\nbegin\n  unfold count_in,\n  induction n with m ih,\n  {\n    refl,\n  },\n  rw [list.repeat_succ, list.map_cons, list.sum_cons, ih],\n  rw if_pos rfl,\n  apply nat.one_add,\nend\n\nlemma count_in_repeat_neq {a b : α} (hyp : a ≠ b) (n : ℕ) :\n  count_in (list.repeat a n) b  =  0  :=\nbegin\n  unfold count_in,\n  induction n with m ih,\n  {\n    refl,\n  },\n  rw [list.repeat_succ, list.map_cons, list.sum_cons, ih, add_zero],\n  rw ite_eq_right_iff,\n  intro impos,\n  exfalso,\n  exact hyp impos,\nend\n\nlemma count_in_singleton_eq (a : α) :\n  count_in [a] a  =  1  :=\nbegin\n  exact list.count_in_repeat_eq a 1,\nend\n\nlemma count_in_singleton_neq {a b : α} (hyp : a ≠ b) :\n  count_in [a] b  =  0  :=\nbegin\n  exact list.count_in_repeat_neq hyp 1,\nend\n\nlemma count_in_pos_of_in {a : α} (hyp : a ∈ x) :\n  count_in x a > 0 :=\nbegin\n  induction x with d l ih,\n  {\n    exfalso,\n    rw list.mem_nil_iff at hyp,\n    exact hyp,\n  },\n  by_contradiction contr,\n  rw not_lt at contr,\n  rw nat.le_zero_iff at contr,\n  rw list.mem_cons_eq at hyp,\n  unfold count_in at contr,\n  unfold list.map at contr,\n  simp at contr,\n  cases hyp,\n  {\n    exact contr.left hyp.symm,\n  },\n  specialize ih hyp,\n  have zero_in_tail : count_in l a = 0,\n  {\n    unfold count_in,\n    exact contr.right,\n  },\n  rw zero_in_tail at ih,\n  exact nat.lt_irrefl 0 ih,\nend\n\nlemma count_in_zero_of_notin {a : α} (hyp : a ∉ x) :\n  count_in x a = 0 :=\nbegin\n  induction x with d l ih,\n  {\n    refl,\n  },\n  unfold count_in,\n  rw [list.map_cons, list.sum_cons, add_eq_zero_iff, ite_eq_right_iff],\n  split,\n  {\n    simp only [nat.one_ne_zero],\n    exact (list.ne_of_not_mem_cons hyp).symm,\n  },\n  {\n    exact ih (list.not_mem_of_not_mem_cons hyp),\n  },\nend\n\nlemma count_in_join (L : list (list α)) (a : α) :\n  count_in L.join a = list.sum (list.map (λ w, count_in w a) L) :=\nbegin\n  induction L,\n  {\n    refl,\n  },\n  {\n    rw [list.join, list.count_in_append, list.map, list.sum_cons, L_ih],\n  },\nend\n\nend list_count_in\n\nend list\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/utilities/list_utils.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.30081976223479745}}
{"text": "/- \n\n  In this file we provide some goodies for Prod \n\n  Namely \n    1. index access: \n       `(42, 1.0, \"hello\").get 2 == \"hello\"`\n    2. index set:\n       `(42,3.14159,\"hello\").set 2 \"world\" = (42,3.14159,\"world\")`\n    3. curry function:\n       `curryN 3 (λ ((i,j,k) : Nat×Nat×Nat) => i + j) = (λ i j k => i + j)`\n    4. uncurry function\n       `uncurryN 3 (λ i j k : Nat => i + j) = λ (i,j,k) => i + j`\n -/ \n\n----------------------------------------------------------------------\n\nclass Prod.Size (α : Type) where\n  size : Nat\n\nclass Prod.SizeFlat (α : Type) where\n  sizeFlat : Nat\n\ninstance (priority := low) (α) : Prod.Size α where\n  size := 1\n\ninstance (priority := low) (α) : Prod.SizeFlat α where\n  sizeFlat := 1\n\ninstance (α β) [sb : Prod.Size β] : Prod.Size (α×β) where\n  size := 1 + sb.size\n\ninstance (α β) [sa : Prod.SizeFlat α] [sb : Prod.SizeFlat β] : Prod.SizeFlat (α×β) where\n  sizeFlat := sa.sizeFlat + sb.sizeFlat\n\n/-- Size of a product type\n\nCounts types only at the top level, so `A×B` and `(A×B)×C` have both size 2 but `A×B×C` has size 3.\n -/\n@[reducible]\ndef Prod.size {α β : Type} [Prod.Size β] (_ : α × β) : Nat := Prod.Size.size (α × β)\n\n/-- Size of a product type\n\nCounts all types, so `A×B` has flat size 2 and `(A×B)×C` have both `A×B×C` flat size 3.\n -/\n@[reducible]\ndef Prod.sizeFlat {α β : Type} [Prod.SizeFlat α] [Prod.SizeFlat β] \n  (_ : α × β) : Nat := Prod.SizeFlat.sizeFlat (α × β)\n\n--------------------------------------------------------------------------------\n\nclass Prod.Get (X : Type) (i : Nat) where\n  {type : Type}\n  get : X → type\n\nattribute [reducible] Prod.Get.type Prod.Get.get\n\n@[reducible]\ninstance (priority := low) : Prod.Get X 0 := ⟨λ x => x⟩\n\n@[reducible]\ninstance : Prod.Get (X×Y) 0 := ⟨λ x => x.fst⟩\n\n@[reducible]\ninstance [pg : Prod.Get Y n] : Prod.Get (X×Y) (n+1) := ⟨λ x => pg.get x.snd⟩\n\nabbrev Prod.get {X Y} (i : Nat) [pg : Prod.Get (X×Y) i] (x : X×Y) := pg.get x\n\n--------------------------------------------------------------------------------\n\nclass Prod.Set (X : Type) (i : Nat) (T : outParam Type) where\n  seti : X → T → X\n\nattribute [reducible] Prod.Set.seti\n\n@[reducible]\ninstance (priority := low) : Prod.Set X 0 X := ⟨λ x x₀ => x₀⟩\n\n@[reducible]\ninstance : Prod.Set (X×Y) 0 X := ⟨λ (x,y) x₀ => (x₀,y)⟩\n\n@[reducible]\ninstance {X Y : Type} {Yₙ : outParam Type} [pg : Prod.Set Y n Yₙ] \n  : Prod.Set (X×Y) (n+1) Yₙ := ⟨λ (x,y) y₀ => (x, pg.seti (i:=n) y y₀)⟩\n\nabbrev Prod.set {X Xs : Type} {Xᵢ : outParam Type} \n  (i : Nat) [pg : Prod.Set (X×Xs) i Xᵢ] (x : X×Xs) (xi) := pg.seti (i:=i) x xi\n\n--------------------------------------------------------------------------------\n\nclass Prod.Uncurry (n : Nat) (F : Type) (Xs : outParam Type) (Y : outParam Type) where\n  uncurry : F → Xs → Y\n\nattribute [reducible] Prod.Uncurry.uncurry\n\n@[reducible]\ninstance (priority := low) {X Y : Type} : Prod.Uncurry 1 (X→Y) X Y where\n  uncurry := λ (f : X → Y) (x : X) => f x\n\n@[reducible]\ninstance {X Y : Type} {Xs' Y' : outParam Type} [c : Prod.Uncurry n Y Xs' Y']\n  : Prod.Uncurry (n+1) (X→Y) (X×Xs') Y' where\n  uncurry := λ (f : X → Y) ((x,xs) : X×Xs') => c.uncurry (n:=n) (f x) xs\n\nabbrev uncurryN {F : Type} {Xs Y : outParam Type} \n  (n : Nat) (f : F) [Prod.Uncurry n F Xs Y] \n  := Prod.Uncurry.uncurry (n:=n) f\n\n\n--------------------------------------------------------------------------------\n\nclass Prod.Curry (n : Nat) (Xs : Type) (Y : Type) (F : outParam Type) where\n  curry : (Xs → Y) → F\n\nattribute [reducible] Prod.Curry.curry\n\n@[reducible]\ninstance (priority := low) : Prod.Curry 1 X Y (X→Y) where\n  curry := λ (f : X → Y) => f\n\n@[reducible]\ninstance {X Xs Y : Type} {F : outParam Type} [c : outParam $ Prod.Curry n Xs Y F] \n  : Prod.Curry (n+1) (X×Xs) Y (X→F) where\n  curry := λ (f : X×Xs → Y) => (λ (x : X) => c.curry (n:=n) (λ y => f (x,y)))\n\nabbrev curryN {Xs Y : outParam $ Type} {F : outParam Type} \n  (n : Nat) (f : Xs → Y) [outParam $ Prod.Curry n Xs Y F] := Prod.Curry.curry n f\n\n--------------------------------------------------------------------------------\n\nsection Tests\n\n  example : (42,3.14159,\"hello\").get 0 = 42 := by rfl\n  example : (42,3.14159,\"hello\").get 1 = 3.14159 := by rfl\n  example : (42,3.14159,\"hello\").get 2 = \"hello\" := by rfl\n  example : (\"hello\", (42, 3.14159), \"world\").get 1 = (42,3.14159) := by rfl\n\n  -- Product is right associative and we respect it\n  example : (42,3.14159,\"hello\").size = 3 := by rfl\n  example : (42,(3.14159,\"hello\")).size = 3 := by rfl\n  example : ((42,3.14159),\"hello\").size = 2 := by rfl\n  example : ((42,3.14159),\"hello\").sizeFlat = 3 := by rfl\n  example : ((42,3.14159),(\"hello\",\"world\")).size = 3 := by rfl\n  example : ((42,3.14159),(\"hello\",\"world\")).sizeFlat = 4 := by rfl\n\n  example : (42,3.14159,\"hello\").set 2 \"world\" = (42,3.14159,\"world\") := by rfl\n\n  example : uncurryN 3 (λ i j k : Nat => i + j) = λ (i,j,k) => i + j := by rfl\n  example : uncurryN 2 (λ i j k : Nat => i + j) = λ (i,j) k => i + j := by rfl\n\n  example : curryN 3 (λ ((i,j,k) : Nat×Nat×Nat) => i + j) = (λ i j k : Nat  => i + j) := by rfl\n  -- example : curryN 2 (λ ((i,j,k) : Nat×Nat×Nat) => i + j) = (λ (i : Nat) ((j,k) : Nat×Nat) => i + j) := by rfl\n\nend Tests\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Data/Prod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.30081976223479745}}
{"text": "/-\nCopyright (c) 2017 Johannes Hölzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes Hölzl\n\nExtends the theory on functors, applicatives and monads.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 u_3 w l \n\nnamespace Mathlib\n\ntheorem functor.map_map {α : Type u} {β : Type u} {γ : Type u} {f : Type u → Type v} [Functor f]\n    [is_lawful_functor f] (m : α → β) (g : β → γ) (x : f α) : g <$> m <$> x = (g ∘ m) <$> x :=\n  Eq.symm (comp_map m g x)\n\n@[simp] theorem id_map' {α : Type u} {f : Type u → Type v} [Functor f] [is_lawful_functor f]\n    (x : f α) : (fun (a : α) => a) <$> x = x :=\n  id_map x\n\ndef mzip_with {F : Type u → Type v} [Applicative F] {α₁ : Type u} {α₂ : Type u} {φ : Type u}\n    (f : α₁ → α₂ → F φ) (ma₁ : List α₁) (ma₂ : List α₂) : F (List φ) :=\n  sorry\n\ndef mzip_with' {α : Type u} {β : Type u} {γ : Type u} {F : Type u → Type v} [Applicative F]\n    (f : α → β → F γ) : List α → List β → F PUnit :=\n  sorry\n\n@[simp] theorem pure_id'_seq {α : Type u} {F : Type u → Type v} [Applicative F]\n    [is_lawful_applicative F] (x : F α) : (pure fun (x : α) => x) <*> x = x :=\n  pure_id_seq x\n\ntheorem seq_map_assoc {α : Type u} {β : Type u} {γ : Type u} {F : Type u → Type v} [Applicative F]\n    [is_lawful_applicative F] (x : F (α → β)) (f : γ → α) (y : F γ) :\n    x <*> f <$> y = (fun (m : α → β) => m ∘ f) <$> x <*> y :=\n  sorry\n\ntheorem map_seq {α : Type u} {β : Type u} {γ : Type u} {F : Type u → Type v} [Applicative F]\n    [is_lawful_applicative F] (f : β → γ) (x : F (α → β)) (y : F α) :\n    f <$> (x <*> y) = function.comp f <$> x <*> y :=\n  sorry\n\n-- TODO: setup `functor_norm` for `monad` laws\n\ndef list.mpartition {f : Type → Type} [Monad f] {α : Type} (p : α → f Bool) :\n    List α → f (List α × List α) :=\n  sorry\n\ntheorem map_bind {α : Type u} {β : Type u} {γ : Type u} {m : Type u → Type v} [Monad m]\n    [is_lawful_monad m] (x : m α) {g : α → m β} {f : β → γ} :\n    f <$> (x >>= g) =\n        do \n          let a ← x \n          f <$> g a :=\n  sorry\n\ntheorem seq_bind_eq {α : Type u} {β : Type u} {γ : Type u} {m : Type u → Type v} [Monad m]\n    [is_lawful_monad m] (x : m α) {g : β → m γ} {f : α → β} : f <$> x >>= g = x >>= g ∘ f :=\n  sorry\n\ntheorem seq_eq_bind_map {α : Type u} {β : Type u} {m : Type u → Type v} [Monad m]\n    [is_lawful_monad m] {x : m α} {f : m (α → β)} :\n    f <*> x =\n        do \n          let _x ← f \n          _x <$> x :=\n  Eq.symm (bind_map_eq_seq f x)\n\n/-- This is the Kleisli composition -/\ndef fish {m : Type u_1 → Type u_2} [Monad m] {α : Sort u_3} {β : Type u_1} {γ : Type u_1}\n    (f : α → m β) (g : β → m γ) (x : α) : m γ :=\n  f x >>= g\n\ninfixl:55 \" >=> \" => Mathlib.fish\n\n-- >=> is already defined in the core library but it is unusable\n\n-- because of its precedence (it is defined with precedence 2) and\n\n-- because it is defined as a lambda instead of having a named\n\n-- function\n\ntheorem fish_pure {m : Type u → Type v} [Monad m] [is_lawful_monad m] {α : Sort u_1} {β : Type u}\n    (f : α → m β) : f >=> pure = f :=\n  sorry\n\ntheorem fish_pipe {m : Type u → Type v} [Monad m] [is_lawful_monad m] {α : Type u} {β : Type u}\n    (f : α → m β) : pure >=> f = f :=\n  sorry\n\ntheorem fish_assoc {m : Type u → Type v} [Monad m] [is_lawful_monad m] {α : Sort u_1} {β : Type u}\n    {γ : Type u} {φ : Type u} (f : α → m β) (g : β → m γ) (h : γ → m φ) :\n    f >=> g >=> h = f >=> (g >=> h) :=\n  sorry\n\ndef list.mmap_accumr {α : Type u} {β' : Type v} {γ' : Type v} {m' : Type v → Type w} [Monad m']\n    (f : α → β' → m' (β' × γ')) : β' → List α → m' (β' × List γ') :=\n  sorry\n\ndef list.mmap_accuml {α : Type u} {β' : Type v} {γ' : Type v} {m' : Type v → Type w} [Monad m']\n    (f : β' → α → m' (β' × γ')) : β' → List α → m' (β' × List γ') :=\n  sorry\n\ntheorem mjoin_map_map {m : Type u → Type u} [Monad m] [is_lawful_monad m] {α : Type u} {β : Type u}\n    (f : α → β) (a : m (m α)) : mjoin (Functor.map f <$> a) = f <$> mjoin a :=\n  sorry\n\ntheorem mjoin_map_mjoin {m : Type u → Type u} [Monad m] [is_lawful_monad m] {α : Type u}\n    (a : m (m (m α))) : mjoin (mjoin <$> a) = mjoin (mjoin a) :=\n  sorry\n\n@[simp] theorem mjoin_map_pure {m : Type u → Type u} [Monad m] [is_lawful_monad m] {α : Type u}\n    (a : m α) : mjoin (pure <$> a) = a :=\n  sorry\n\n@[simp] theorem mjoin_pure {m : Type u → Type u} [Monad m] [is_lawful_monad m] {α : Type u}\n    (a : m α) : mjoin (pure a) = a :=\n  pure_bind a id\n\ndef succeeds {F : Type → Type v} [alternative F] {α : Type} (x : F α) : F Bool :=\n  x $> tt <|> pure false\n\ndef mtry {F : Type → Type v} [alternative F] {α : Type} (x : F α) : F Unit :=\n  x $> Unit.unit <|> pure Unit.unit\n\n@[simp] theorem guard_true {F : Type → Type v} [alternative F] {h : Decidable True} :\n    guard True = pure Unit.unit :=\n  sorry\n\n@[simp] theorem guard_false {F : Type → Type v} [alternative F] {h : Decidable False} :\n    guard False = failure :=\n  sorry\n\nnamespace sum\n\n\nprotected def bind {e : Type v} {α : Type u_1} {β : Type u_2} : e ⊕ α → (α → e ⊕ β) → e ⊕ β := sorry\n\nprotected instance monad {e : Type v} : Monad (sum e) := sorry\n\nprotected instance is_lawful_functor {e : Type v} : is_lawful_functor (sum e) :=\n  is_lawful_functor.mk\n    (fun (α : Type u) (x : e ⊕ α) =>\n      sum.cases_on x (fun (x : e) => Eq.refl (id <$> inl x)) fun (x : α) => Eq.refl (id <$> inr x))\n    fun (α β γ : Type u) (g : α → β) (h : β → γ) (x : e ⊕ α) =>\n      sum.cases_on x (fun (x : e) => Eq.refl ((h ∘ g) <$> inl x))\n        fun (x : α) => Eq.refl ((h ∘ g) <$> inr x)\n\nprotected instance is_lawful_monad {e : Type v} : is_lawful_monad (sum e) :=\n  is_lawful_monad.mk (fun (α β : Type u) (x : α) (f : α → e ⊕ β) => Eq.refl (pure x >>= f))\n    fun (α β γ : Type u) (x : e ⊕ α) (f : α → e ⊕ β) (g : β → e ⊕ γ) =>\n      sum.cases_on x (fun (x : e) => Eq.refl (inl x >>= f >>= g))\n        fun (x : α) => Eq.refl (inr x >>= f >>= g)\n\nend sum\n\n\nclass is_comm_applicative (m : Type u_1 → Type u_2) [Applicative m] extends is_lawful_applicative m\n    where\n  commutative_prod :\n    ∀ {α β : Type u_1} (a : m α) (b : m β),\n      Prod.mk <$> a <*> b = (fun (b : β) (a : α) => (a, b)) <$> b <*> a\n\ntheorem is_comm_applicative.commutative_map {m : Type u_1 → Type u_2} [Applicative m]\n    [is_comm_applicative m] {α : Type u_1} {β : Type u_1} {γ : Type u_1} (a : m α) (b : m β)\n    {f : α → β → γ} : f <$> a <*> b = flip f <$> b <*> a :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3008197622347974}}
{"text": "import category_theory.lifting_properties.basic\nimport category_theory.limits.shapes.comm_sq\nimport for_mathlib.category_theory.morphism_property_misc\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nnamespace category_theory\n\nvariables {C : Type*} [category C]\nnamespace comm_sq.lift_struct\nattribute [reassoc] fac_left\nattribute [reassoc] fac_right\nend comm_sq.lift_struct\n\nnamespace is_pullback\n\n/-\n\nA     X'     X\n\n\nB     Y'     Y\n-/\n\nsection\n\nvariables {A B X Y Z : C}\n  {i : A ⟶ B} {f' : A ⟶ X} {f : B ⟶ Y} {p : X ⟶ Y}\n  (h : is_pullback f' i p f) (f₁ : Z ⟶ X) (f₂ : Z ⟶ B)\n  (fac : f₁ ≫ p = f₂ ≫ f)\n\ndef lift : Z ⟶ A := (pullback_cone.is_limit.lift' h.is_limit f₁ f₂ fac).1\n\n@[simp, reassoc]\nlemma lift_fst : h.lift f₁ f₂ fac ≫ f' = f₁ :=\n(pullback_cone.is_limit.lift' h.is_limit f₁ f₂ fac).2.1\n\n@[simp, reassoc]\nlemma lift_snd : h.lift f₁ f₂ fac ≫ i = f₂ :=\n(pullback_cone.is_limit.lift' h.is_limit f₁ f₂ fac).2.2\n\ninclude h\ndef hom_ext {l₁ l₂ : Z ⟶ A} (h₁ : l₁ ≫ f' = l₂ ≫ f')\n  (h₂ : l₁ ≫ i = l₂ ≫ i) : l₁ = l₂ :=\npullback_cone.is_limit.hom_ext h.is_limit h₁ h₂\n\nend\n\n\nsection\n\nvariables\n{A B X X' Y Y' : C}\n  {i : A ⟶ B} {f : B ⟶ Y'} {f' : A ⟶ X'}\n  {p : X ⟶ Y} {g : Y' ⟶ Y} {p' : X' ⟶ Y'} {g' : X' ⟶ X}\n  {fg : B ⟶ Y} {fg' : A ⟶ X}\n  (sq₂ : is_pullback g' p' p g) (sq₁ : comm_sq f' i p' f)\n  (sq₁₂ : comm_sq fg' i p fg)\n  (hfg : fg = f ≫ g) (hfg' : fg' = f' ≫ g')\n\ninclude hfg hfg'\n\ndef lift_struct_equiv : sq₁.lift_struct ≃ sq₁₂.lift_struct :=\n{ to_fun := λ l,\n  { l := l.l ≫ g',\n    fac_left' := by rw [hfg', l.fac_left_assoc],\n    fac_right' := by rw [assoc, sq₂.w, l.fac_right_assoc, hfg], },\n  inv_fun := λ l,\n  { l := sq₂.lift l.l f (by rw [l.fac_right, hfg]),\n    fac_left' := sq₂.hom_ext (by rw [assoc, lift_fst, l.fac_left, hfg'])\n      (by rw [assoc, lift_snd, sq₁.w]),\n    fac_right' := by rw [lift_snd], },\n  left_inv := λ l, by { ext, exact sq₂.hom_ext (by rw lift_fst) (by rw [lift_snd, l.fac_right]), },\n  right_inv := λ l, by { ext, simp only [lift_fst], }, }\n\ninclude sq₂\ndef has_lift_iff : sq₁.has_lift ↔ sq₁₂.has_lift :=\nbegin\n  split,\n  { intro h₁,\n    exact comm_sq.has_lift.mk'\n      ((sq₂.lift_struct_equiv sq₁ sq₁₂ hfg hfg').to_fun h₁.exists_lift.some), },\n  { intro h₁₂,\n    exact comm_sq.has_lift.mk'\n      ((sq₂.lift_struct_equiv sq₁ sq₁₂ hfg hfg').inv_fun h₁₂.exists_lift.some), },\nend\n\nend\n\nlemma has_lifting_property_imp {A B X Y X' Y' : C} {p : X ⟶ Y} {p' : X' ⟶ Y'}\n  {g : Y' ⟶ Y} {g' : X' ⟶ X} (sq₂ : is_pullback g' p' p g) (i : A ⟶ B)\n  (hip : has_lifting_property i p) : has_lifting_property i p' :=\n⟨λ f' f sq₁, begin\n  have sq₁₂ : comm_sq (f' ≫ g') i p (f ≫ g) := comm_sq.mk (by rw [assoc, sq₂.w, sq₁.w_assoc]),\n  rw sq₂.has_lift_iff sq₁ sq₁₂ rfl rfl,\n  haveI := hip,\n  apply_instance,\nend⟩\n\nend is_pullback\n\n\nnamespace is_pushout\n\nsection\n\nvariables {A B X Y Z : C}\n  {i : A ⟶ B} {f' : A ⟶ X} {f : B ⟶ Y} {p : X ⟶ Y}\n  (h : is_pushout f' i p f) (f₁ : X ⟶ Z) (f₂ : B ⟶ Z)\n  (fac : f' ≫ f₁ = i ≫ f₂)\n\ndef desc : Y ⟶ Z := (pushout_cocone.is_colimit.desc' h.is_colimit f₁ f₂ fac).1\n\n@[simp, reassoc]\nlemma inl_desc : p ≫ h.desc f₁ f₂ fac = f₁ :=\n(pushout_cocone.is_colimit.desc' h.is_colimit f₁ f₂ fac).2.1\n\n@[simp, reassoc]\nlemma lift_snd : f ≫ h.desc f₁ f₂ fac = f₂ :=\n(pushout_cocone.is_colimit.desc' h.is_colimit f₁ f₂ fac).2.2\n\ninclude h\ndef hom_ext {l₁ l₂ : Y ⟶ Z} (h₁ : p ≫ l₁ = p ≫ l₂)\n  (h₂ : f ≫ l₁ = f ≫ l₂) : l₁ = l₂ :=\npushout_cocone.is_colimit.hom_ext h.is_colimit h₁ h₂\n\nend\n\nsection\n\nvariables\n  {A B A' B' X Y : C}\n  {f : A ⟶ A'} {i : A ⟶ B} {f' : B ⟶ B'} {i' : A' ⟶ B'}\n  {g : A' ⟶ X} {g' : B' ⟶ Y} {p : X ⟶ Y}\n  {fg : A ⟶ X} {fg' : B ⟶ Y}\n  (sq₁ : is_pushout f i i' f') (sq₂ : comm_sq g i' p g')\n  (sq₁₂ : comm_sq fg i p fg')\n  (hfg : fg = f ≫ g) (hfg' : fg' = f' ≫ g')\n\ninclude hfg hfg'\n\ndef lift_struct_equiv : sq₂.lift_struct ≃ sq₁₂.lift_struct :=\n{ to_fun := λ l,\n  { l := f' ≫ l.l,\n    fac_left' := by rw [← sq₁.to_comm_sq.w_assoc, l.fac_left, hfg],\n    fac_right' := by rw [assoc, l.fac_right, hfg'], },\n  inv_fun := λ l,\n  { l := sq₁.desc g l.l (by rw [l.fac_left, hfg]),\n    fac_left' := by simp only [inl_desc],\n    fac_right' := sq₁.hom_ext (by simp only [inl_desc_assoc, sq₂.w])\n        (by simp only [lift_snd_assoc, l.fac_right, hfg']), },\n  left_inv := λ l, by { ext, exact sq₁.hom_ext (by simp only [l.fac_left, inl_desc])\n    (by simp only [lift_snd]), },\n  right_inv := λ l, by { ext, simp only [lift_snd], }, }\n\ninclude sq₁\ndef has_lift_iff : sq₂.has_lift ↔ sq₁₂.has_lift :=\nbegin\n  split,\n  { intro h₁,\n    exact comm_sq.has_lift.mk'\n      ((sq₁.lift_struct_equiv sq₂ sq₁₂ hfg hfg').to_fun h₁.exists_lift.some), },\n  { intro h₁₂,\n    exact comm_sq.has_lift.mk'\n      ((sq₁.lift_struct_equiv sq₂ sq₁₂ hfg hfg').inv_fun h₁₂.exists_lift.some), },\nend\n\nend\n\nlemma has_lifting_property_imp {A B A' B' X Y : C} {f : A ⟶ A'} {i : A ⟶ B}\n  {f' : B ⟶ B'} {i' : A' ⟶ B'} (sq₁ : is_pushout f i i' f') (p : X ⟶ Y)\n  (hip : has_lifting_property i p) : has_lifting_property i' p :=\n⟨λ g g' sq₂, begin\n  have sq₁₂ : comm_sq (f ≫ g) i p (f' ≫ g') := comm_sq.mk\n    (by rw [assoc, sq₂.w, sq₁.to_comm_sq.w_assoc]),\n  rw sq₁.has_lift_iff sq₂ sq₁₂ rfl rfl,\n  haveI := hip,\n  apply_instance,\nend⟩\n\nend is_pushout\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/lifting_properties/pullbacks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3008197622347974}}
{"text": "import for_mathlib.abelian_sheaves.main\nimport category_theory.abelian.exact\nimport category_theory.abelian.homology\nimport category_theory.sites.left_exact\nimport for_mathlib.chain_complex_exact\nimport for_mathlib.homology_iso\nimport for_mathlib.exact_functor\n\nuniverses w v u\n\nopen category_theory\nopen category_theory.limits\n\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {A : Type w} [category.{max v u} A] [abelian A]\nvariables [concrete_category.{max v u} A]\nvariables [∀ (P : Cᵒᵖ ⥤ A) (X : C) (S : J.cover X), limits.has_multiequalizer (S.index P)]\nvariables [limits.preserves_limits (forget A)]\nvariables [∀ (X : C), limits.has_colimits_of_shape (J.cover X)ᵒᵖ A]\nvariables [∀ (X : C), limits.preserves_colimits_of_shape (J.cover X)ᵒᵖ (forget A)]\nvariables [reflects_isomorphisms (forget A)]\n\nnoncomputable theory\n\n-- Just checking...\nexample : abelian (Sheaf J A) := infer_instance\nexample : preserves_finite_limits (presheaf_to_Sheaf J A) := infer_instance\nexample : functor.preserves_zero_morphisms (presheaf_to_Sheaf J A) :=\n  infer_instance\n\n-- Missing\ninstance : preserves_colimits (presheaf_to_Sheaf J A) :=\n(sheafification_adjunction J A).left_adjoint_preserves_colimits\n\n\n.\n\nnamespace category_theory.Sheaf\n\ntheorem exact_of_exact (X Y Z : Cᵒᵖ ⥤ A) (f : X ⟶ Y) (g : Y ⟶ Z)\n  (e : exact f g) :\n  exact ((presheaf_to_Sheaf J A).map f) ((presheaf_to_Sheaf J A).map g) :=\nbegin\n  rw abelian.exact_iff at *,\n  split,\n  { rw [← functor.map_comp, e.1], simp, },\n  { let K := kernel g,\n    let Q := cokernel f,\n    let P := presheaf_to_Sheaf J A,\n    let eK : P.obj K ≅ kernel (P.map g) :=\n      preserves_kernel.iso _ _,\n    let eQ : P.obj Q ≅ cokernel (P.map f) :=\n      preserves_cokernel.iso _ _,\n    have : kernel.ι (P.map g) =\n      eK.inv ≫ P.map (kernel.ι g),\n    { dsimp [eK],\n      rw iso.eq_inv_comp,\n      simp only [preserves_kernel.iso_hom, kernel_comparison_comp_ι] },\n    rw this, clear this,\n    have : cokernel.π (P.map f) =\n      P.map (cokernel.π f) ≫ eQ.hom,\n    { dsimp [eQ], rw ← iso.comp_inv_eq,\n      simp only [preserves_cokernel.iso_inv, π_comp_cokernel_comparison] },\n    rw this, clear this,\n    simp only [category.assoc, e.2, ← P.map_comp_assoc,\n      P.map_zero, zero_comp, comp_zero] }\nend\n\ninstance presheaf_to_Sheaf_additive :\n  functor.additive (presheaf_to_Sheaf J A) :=\nbegin\n  apply_with functor.additive_of_preserves_binary_biproducts { instances := ff },\n  apply_with has_binary_biproducts_of_finite_biproducts { instances := ff },\n  apply_instance,\n  apply preserves_binary_biproducts_of_preserves_binary_coproducts,\n  apply_instance,\nend\n\nlemma map_presheaf_to_Sheaf_homology_zero_of_homology_zero\n  (D : chain_complex (Cᵒᵖ ⥤ A) ℕ)\n  (h : ∀ i : ℕ, is_zero (D.homology i)) :\n  ∀ i : ℕ, is_zero\n    ((((presheaf_to_Sheaf J A).map_homological_complex _).obj D).homology i) :=\nbegin\n  rw ← chain_complex.epi_and_exact_iff_zero_homology' at *,\n  split,\n  { haveI := h.1, apply preserves_epi_of_preserves_colimit },\n  { intros i,\n    apply exact_of_exact,\n    apply h.2 },\nend\n\nlemma sheafification_exact : functor.exact (presheaf_to_Sheaf J A) :=\nλ X Y Z f g h, exact_of_exact _ _ _ _ _ h\n\nend category_theory.Sheaf\n\nnamespace category_theory\n\ninstance evaluation_additive (X : C) : functor.additive ((evaluation C A).obj X) :=\n⟨λ F G η γ, rfl⟩\n\ndef evaluation_homology_iso (X Y Z : C ⥤ A) (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) (t : C) :\n  (homology f g w).obj t ≅ (homology (f.app t) (g.app t) $\n    by { rw [← nat_trans.comp_app, w], simp, }) :=\n{ hom := homology.lift _ _ _\n    ((homology.ι f g w).app t ≫ (nat_trans.cokernel_obj_iso f t).hom) begin\n      rw category.assoc, let e := _, change _ ≫ e = _,\n      have he : e = (cokernel.desc f g w).app t,\n      { dsimp [e],\n        rw ← iso.eq_inv_comp,\n        ext,\n        simp },\n      rw [he, ← nat_trans.comp_app],\n      simp only [homology.condition_ι, nat_trans.app_zero],\n    end,\n  inv := homology.desc' _ _ _\n    ((nat_trans.kernel_obj_iso g t).inv ≫ (homology.π' f g w).app t) begin\n      rw ← category.assoc, let e := _, change e ≫ _ = _,\n      have he : e = (kernel.lift g f w).app t,\n      { dsimp [e], rw iso.comp_inv_eq,\n        ext,\n        simp },\n      rw [he, ← nat_trans.comp_app],\n      simp only [homology.condition_π', nat_trans.app_zero],\n    end,\n  hom_inv_id' := begin\n    let a := _, change a ≫ _ = _,\n    haveI : epi ((homology.π' f g w).app t),\n    { rw abelian.epi_iff_cokernel_π_eq_zero,\n      let b := homology.π' f g w,\n      have : cokernel.π (b.app t) = (cokernel.π b).app t ≫\n        (nat_trans.cokernel_obj_iso b _).hom,\n      { simp },\n      rw [this, ← iso.eq_comp_inv, zero_comp],\n      suffices : cokernel.π b = 0,\n      { simp [this], },\n      rw ← abelian.epi_iff_cokernel_π_eq_zero,\n      dsimp [b, homology.π'],\n      apply epi_comp },\n    rw ← cancel_epi ((homology.π' f g w).app t),\n    have ha : ((homology.π' f g w).app t ≫ a) =\n      (nat_trans.kernel_obj_iso _ _).hom ≫ homology.π' _ _ _,\n    { dsimp [a],\n      apply homology.hom_to_ext,\n      simp only [category.assoc, homology.lift_ι, homology.π'_ι,\n        nat_trans.kernel_obj_iso_hom_ι_assoc],\n      rw [← category.assoc, ← iso.eq_comp_inv],\n      simp only [category.assoc, nat_trans.cokernel_obj_iso_π_inv],\n      rw [← nat_trans.comp_app, homology.π'_ι, nat_trans.comp_app] },\n    rw reassoc_of ha,\n    simp only [homology.π'_desc', iso.hom_inv_id_assoc, category.comp_id],\n  end,\n  inv_hom_id' := begin\n    apply homology.hom_from_ext, apply homology.hom_to_ext,\n    simp only [category.assoc, homology.π'_desc'_assoc, homology.lift_ι,\n      category.comp_id, homology.π'_ι],\n    rw [iso.inv_comp_eq, ← category.assoc, ← iso.eq_comp_inv,\n      ← nat_trans.comp_app, homology.π'_ι],\n    simp,\n  end }\n\n-- Homology of functors is computed pointwise...\nlemma homology_zero_of_eval\n  (D : chain_complex (C ⥤ A) ℕ)\n  (h : ∀ (X : C) (i : ℕ), is_zero\n    (((((evaluation C A).obj X).map_homological_complex _).obj D).homology i)) :\n  ∀ i : ℕ, is_zero (D.homology i) :=\nbegin\n  rw ← chain_complex.epi_and_exact_iff_zero_homology',\n  split,\n  { apply_with nat_trans.epi_of_epi_app { instances := ff },\n    intros X,\n    specialize h X,\n    rw ← chain_complex.epi_and_exact_iff_zero_homology' at h,\n    exact h.1 },\n  { intros i, rw preadditive.exact_iff_homology_zero,\n    refine ⟨_,_⟩,\n    { ext X, specialize h X,\n      rw ← chain_complex.epi_and_exact_iff_zero_homology' at h, replace h := h.2 i,\n      rw preadditive.exact_iff_homology_zero at h,\n      obtain ⟨w,hw⟩ := h, exact w },\n    { refine ⟨is_zero.iso_zero _⟩,\n      apply functor.is_zero,\n      intros X,\n      specialize h X (i+1),\n      refine is_zero.of_iso _ (evaluation_homology_iso _ _ _ _ _ _ _),\n      apply is_zero.of_iso h,\n      symmetry, apply homology_iso; exact rfl } }\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/abelian_sheaves/exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.3007824352053535}}
{"text": "import order.filter.germ\nimport topology.constructions\n\nimport topology.connected\nimport topology.separation\n\nimport to_mathlib.topology.nhds_set\n\nopen_locale topology\nopen filter set\n\n/-- The value associated to a germ at a point. This is the common value\nshared by all representatives at the given point. -/\ndef filter.germ.value {X α : Type*} [topological_space X] {x : X} (φ : germ (𝓝 x) α) : α :=\nquotient.lift_on' φ (λ f, f x) (λ f g h, by { dsimp only, rw eventually.self_of_nhds h })\n\n/-- Given a predicate on germs `P : Π x : X, germ (𝓝 x) Y → Prop` and `A : set X`,\nbuild a new predicate on germs `restrict_germ_predicate P A` such that\n`(∀ x, restrict_germ_predicate P A x f) ↔ ∀ᶠ x near A, P x f`, see\n`forall_restrict_germ_predicate_iff` for this equivalence. -/\ndef restrict_germ_predicate {X Y : Type*} [topological_space X]\n  (P : Π x : X, germ (𝓝 x) Y → Prop) (A : set X) : Π x : X, germ (𝓝 x) Y → Prop :=\nλ x φ, quotient.lift_on' φ (λ f, x ∈ A → ∀ᶠ y in 𝓝 x, P y f) begin\n  have : ∀ f f' : X → Y, f =ᶠ[𝓝 x] f' → (∀ᶠ y in 𝓝 x, P y f) → ∀ᶠ y in 𝓝 x, P y f',\n  { intros f f' hff' hf,\n    apply (hf.and $ eventually.eventually_nhds hff').mono,\n    rintros y ⟨hy, hy'⟩,\n    rwa germ.coe_eq.mpr (eventually_eq.symm hy') },\n  exact λ f f' hff', propext $ forall_congr $ λ _, ⟨this f f' hff', this f' f hff'.symm⟩,\nend\n\nlemma filter.eventually.germ_congr {X Y : Type*} [topological_space X]\n  {P : Π x : X, germ (𝓝 x) Y → Prop} {A : set X} {f g : X → Y}\n  (hf : ∀ᶠ x in 𝓝ˢ A, P x f) (h : ∀ᶠ z in 𝓝ˢ A, g z = f z) : ∀ᶠ x in 𝓝ˢ A, P x g :=\nbegin\n  rw eventually_nhds_set_iff at *,\n  intros x hx,\n  apply ((hf x hx).and (h x hx).eventually_nhds).mono,\n  intros y hy,\n  convert hy.1 using 1,\n  apply quotient.sound,\n  exact hy.2\nend\n\n\nlemma restrict_germ_predicate_congr {X Y : Type*} [topological_space X]\n  {P : Π x : X, germ (𝓝 x) Y → Prop} {A : set X} {f g : X → Y} {x : X}\n  (hf : restrict_germ_predicate P A x f) (h : ∀ᶠ z in 𝓝ˢ A, g z = f z) :\n  restrict_germ_predicate P A x g :=\nbegin\n  intros hx,\n  apply ((hf hx).and $ eventually_nhds_set_iff.mp h x hx).eventually_nhds.mono,\n  intros y hy,\n  rw eventually_and at hy,\n  convert hy.1.self_of_nhds using 1,\n  apply quotient.sound,\n  exact hy.2,\nend\n\n\nlemma forall_restrict_germ_predicate_iff {X Y : Type*} [topological_space X]\n  {P : Π x : X, germ (𝓝 x) Y → Prop} {A : set X} {f : X → Y} :\n  (∀ x, restrict_germ_predicate P A x f) ↔ ∀ᶠ x in 𝓝ˢ A, P x f :=\nby { rw eventually_nhds_set_iff, exact iff.rfl }\n\nlemma  forall_restrict_germ_predicate_of_forall {X Y : Type*} [topological_space X]\n  {P : Π x : X, germ (𝓝 x) Y → Prop} {A : set X} {f : X → Y} (h : ∀ x, P x f) :\n  ∀ x, restrict_germ_predicate P A x f :=\nforall_restrict_germ_predicate_iff.mpr (eventually_of_forall h)\n\nlemma filter.eventually_eq.comp_fun {α β γ : Type*} {f g : β → γ} {l : filter α} {l' : filter β}\n  (h : f =ᶠ[l'] g) {φ : α → β} (hφ : tendsto φ l l') : f ∘ φ =ᶠ[l] g ∘ φ :=\nhφ h\n\ndef filter.germ.slice_left {X Y Z : Type*} [topological_space X] [topological_space Y] {p : X × Y}\n  (P : germ (𝓝 p) Z) : germ (𝓝 p.1) Z :=\nP.lift_on (λ f, ((λ x', f (x', p.2)) : germ (𝓝 p.1) Z))\n  (λ f g hfg, @quotient.sound _ ((𝓝 p.1).germ_setoid Z) _ _\n     (hfg.comp_fun begin\n       rw ← (prod.mk.eta : (p.1, p.2) = p),\n       exact (continuous.prod.mk_left p.2).continuous_at,\n     end))\n\ndef filter.germ.slice_right {X Y Z : Type*} [topological_space X] [topological_space Y] {p : X × Y}\n  (P : germ (𝓝 p) Z) : germ (𝓝 p.2) Z :=\nP.lift_on (λ f, ((λ y, f (p.1, y)) : germ (𝓝 p.2) Z))\n  (λ f g hfg, @quotient.sound _ ((𝓝 p.2).germ_setoid Z) _ _\n     (hfg.comp_fun begin\n       rw ← (prod.mk.eta : (p.1, p.2) = p),\n       exact (continuous.prod.mk p.1).continuous_at,\n     end))\n\ndef filter.germ.is_constant {X Y : Type*} [topological_space X] {x} (P : germ (𝓝 x) Y) : Prop :=\nP.lift_on (λ f, ∀ᶠ x' in 𝓝 x, f x' = f x) begin\n  suffices : ∀ (f g : X → Y), f =ᶠ[𝓝 x] g →\n     (∀ᶠ x' in 𝓝 x, f x' = f x) → ∀ᶠ x' in 𝓝 x, g x' = g x,\n  from λ f g hfg, propext ⟨λ h, this f g hfg h, λ h, this g f hfg.symm h⟩,\n  rintros f g hfg hf,\n  apply (hf.and hfg).mono (λ x' hx', _),\n  rw [← hx'.2, hx'.1, hfg.eq_of_nhds],\nend\n\nlemma eq_of_germ_is_constant {X Y : Type*} [topological_space X] [preconnected_space X]\n  {f : X → Y} (h : ∀ x : X, (f : germ (𝓝 x) Y).is_constant) (x x' : X) : f x = f x' :=\nbegin\n  revert x,\n  erw ← eq_univ_iff_forall,\n  apply is_clopen.eq_univ _ (⟨x', rfl⟩ : {x | f x = f x'}.nonempty),\n  refine ⟨is_open_iff_eventually.mpr (λ x hx, hx ▸ h x), _⟩,\n  rw is_closed_iff_frequently,\n  rintros x hx,\n  rcases (eventually.and_frequently (h x) hx).exists with ⟨x'', H⟩,\n  exact H.1.symm.trans H.2\nend\n\nlemma eq_of_germ_is_constant_on {X Y : Type*} [topological_space X]\n  {f : X → Y} {s : set X} (h : ∀ x ∈ s, (f : germ (𝓝 x) Y).is_constant)\n  (hs : is_preconnected s) {x x' : X} (x_in : x ∈ s) (x'_in : x' ∈ s) : f x = f x' :=\nbegin\n  haveI := is_preconnected_iff_preconnected_space.mp hs,\n  let F : s → Y := f ∘ coe,\n  change F ⟨x, x_in⟩ = F ⟨x', x'_in⟩,\n  apply eq_of_germ_is_constant,\n  rintros ⟨x, hx⟩,\n  have : continuous_at (coe : s → X) ⟨x, hx⟩,\n  exact continuous_at_subtype_coe,\n  exact this (h x hx)\nend\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/topology/germ.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3006743591770894}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.eq_to_hom\n\n/-!\n# Binary disjoint unions of categories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe define the category instance on `C ⊕ D` when `C` and `D` are categories.\n\nWe define:\n* `inl_`      : the functor `C ⥤ C ⊕ D`\n* `inr_`      : the functor `D ⥤ C ⊕ D`\n* `swap`      : the functor `C ⊕ D ⥤ D ⊕ C`\n    (and the fact this is an equivalence)\n\nWe further define sums of functors and natural transformations, written `F.sum G` and `α.sum β`.\n-/\n\nnamespace category_theory\n\nuniverses v₁ u₁ -- morphism levels before object levels. See note [category_theory universes].\n\nopen sum\n\nsection\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₁) [category.{v₁} D]\n\n/--\n`sum C D` gives the direct sum of two categories.\n-/\ninstance sum : category.{v₁} (C ⊕ D) :=\n{ hom :=\n    λ X Y, match X, Y with\n    | inl X, inl Y := X ⟶ Y\n    | inl X, inr Y := pempty\n    | inr X, inl Y := pempty\n    | inr X, inr Y := X ⟶ Y\n    end,\n  id :=\n    λ X, match X with\n    | inl X := 𝟙 X\n    | inr X := 𝟙 X\n    end,\n  comp :=\n    λ X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := f ≫ g\n    | inr X, inr Y, inr Z, f, g := f ≫ g\n    end }\n\n@[simp] lemma sum_comp_inl {P Q R : C} (f : (inl P : C ⊕ D) ⟶ inl Q)\n  (g : (inl Q : C ⊕ D) ⟶ inl R) :\n  @category_struct.comp _ _ P Q R (f : P ⟶ Q) (g : Q ⟶ R) =\n  @category_struct.comp _ _ (inl P) (inl Q) (inl R) (f : P ⟶ Q) (g : Q ⟶ R) := rfl\n@[simp] lemma sum_comp_inr {P Q R : D} (f : (inr P : C ⊕ D) ⟶ inr Q)\n  (g : (inr Q : C ⊕ D) ⟶ inr R) :\n  @category_struct.comp _ _ P Q R (f : P ⟶ Q) (g : Q ⟶ R) =\n  @category_struct.comp _ _ (inr P) (inr Q) (inr R) (f : P ⟶ Q) (g : Q ⟶ R) := rfl\nend\n\nnamespace sum\n\nvariables (C : Type u₁) [category.{v₁} C] (D : Type u₁) [category.{v₁} D]\n\n/-- `inl_` is the functor `X ↦ inl X`. -/\n-- Unfortunate naming here, suggestions welcome.\n@[simps] def inl_ : C ⥤ C ⊕ D :=\n{ obj := λ X, inl X,\n  map := λ X Y f, f }\n\n/-- `inr_` is the functor `X ↦ inr X`. -/\n@[simps] def inr_ : D ⥤ C ⊕ D :=\n{ obj := λ X, inr X,\n  map := λ X Y f, f }\n\n/-- The functor exchanging two direct summand categories. -/\ndef swap : C ⊕ D ⥤ D ⊕ C :=\n{ obj :=\n    λ X, match X with\n    | inl X := inr X\n    | inr X := inl X\n    end,\n  map :=\n    λ X Y f, match X, Y, f with\n    | inl X, inl Y, f := f\n    | inr X, inr Y, f := f\n    end }\n\n@[simp] lemma swap_obj_inl (X : C) : (swap C D).obj (inl X) = inr X := rfl\n@[simp] lemma swap_obj_inr (X : D) : (swap C D).obj (inr X) = inl X := rfl\n@[simp] lemma swap_map_inl {X Y : C} {f : inl X ⟶ inl Y} : (swap C D).map f = f := rfl\n@[simp] \n\nnamespace swap\n\n/-- `swap` gives an equivalence between `C ⊕ D` and `D ⊕ C`. -/\ndef equivalence : C ⊕ D ≌ D ⊕ C :=\nequivalence.mk (swap C D) (swap D C)\n  (nat_iso.of_components (λ X, eq_to_iso (by { cases X; refl })) (by tidy))\n  (nat_iso.of_components (λ X, eq_to_iso (by { cases X; refl })) (by tidy))\n\ninstance is_equivalence : is_equivalence (swap C D) :=\n(by apply_instance : is_equivalence (equivalence C D).functor)\n\n/-- The double swap on `C ⊕ D` is naturally isomorphic to the identity functor. -/\ndef symmetry : swap C D ⋙ swap D C ≅ 𝟭 (C ⊕ D) :=\n(equivalence C D).unit_iso.symm\n\nend swap\n\nend sum\n\nvariables {A : Type u₁} [category.{v₁} A]\n          {B : Type u₁} [category.{v₁} B]\n          {C : Type u₁} [category.{v₁} C]\n          {D : Type u₁} [category.{v₁} D]\n\nnamespace functor\n\n/-- The sum of two functors. -/\ndef sum (F : A ⥤ B) (G : C ⥤ D) : A ⊕ C ⥤ B ⊕ D :=\n{ obj :=\n    λ X, match X with\n    | inl X := inl (F.obj X)\n    | inr X := inr (G.obj X)\n    end,\n  map :=\n    λ X Y f, match X, Y, f with\n    | inl X, inl Y, f := F.map f\n    | inr X, inr Y, f := G.map f\n    end,\n  map_id' := λ X, begin cases X; unfold_aux, erw F.map_id, refl, erw G.map_id, refl end,\n  map_comp' :=\n    λ X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := by { unfold_aux, erw F.map_comp, refl }\n    | inr X, inr Y, inr Z, f, g := by { unfold_aux, erw G.map_comp, refl }\n    end }\n\n@[simp] lemma sum_obj_inl (F : A ⥤ B) (G : C ⥤ D) (a : A) :\n  (F.sum G).obj (inl a) = inl (F.obj a) := rfl\n@[simp] lemma sum_obj_inr (F : A ⥤ B) (G : C ⥤ D) (c : C) :\n  (F.sum G).obj (inr c) = inr (G.obj c) := rfl\n@[simp] lemma sum_map_inl (F : A ⥤ B) (G : C ⥤ D) {a a' : A} (f : inl a ⟶ inl a') :\n  (F.sum G).map f = F.map f := rfl\n@[simp] lemma sum_map_inr (F : A ⥤ B) (G : C ⥤ D) {c c' : C} (f : inr c ⟶ inr c') :\n  (F.sum G).map f = G.map f := rfl\nend functor\n\nnamespace nat_trans\n\n/-- The sum of two natural transformations. -/\ndef sum {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) : F.sum H ⟶ G.sum I :=\n{ app         :=\n    λ X, match X with\n    | inl X := α.app X\n    | inr X := β.app X\n    end,\n  naturality' :=\n    λ X Y f, match X, Y, f with\n    | inl X, inl Y, f := begin unfold_aux, erw α.naturality, refl, end\n    | inr X, inr Y, f := begin unfold_aux, erw β.naturality, refl, end\n    end }\n\n@[simp] lemma sum_app_inl {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) (a : A) :\n  (sum α β).app (inl a) = α.app a := rfl\n@[simp] lemma sum_app_inr {F G : A ⥤ B} {H I : C ⥤ D} (α : F ⟶ G) (β : H ⟶ I) (c : C) :\n  (sum α β).app (inr c) = β.app c := rfl\nend nat_trans\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sums/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3005463427108183}}
{"text": "import Lean\nset_option pp.analyze false\n\ndef p (x y : Nat) := x = y\n\nexample (x y : Nat) : p (x + y) (y + x + 0) := by\n  conv =>\n    trace_state\n    whnf\n    tactic' => trace_state\n    trace_state\n    congr\n    next => rfl\n    any_goals whnf; rfl\n  conv' =>\n    trace_state\n    apply id ?x\n  conv =>\n    fail_if_success case x => whnf\n  trace_state\n  rw [Nat.add_comm]\n  rfl\n\ndef foo (x y : Nat) : Nat := x + y\n\nexample : foo (0 + a) (b + 0) = a + b := by\n  conv =>\n    apply id\n    lhs\n    trace_state\n    congr\n    trace_state\n    case x =>\n      simp\n      trace_state\n    fail_if_success case x => skip\n    case' y => skip\n    case y => skip\n    done\n\nexample : foo (0 + a) (b + 0) = a + b := by\n  conv =>\n    lhs\n    conv =>\n      congr\n      trace_state\n      focus\n        trace_state\n      tactic => simp\n      trace_state\n      all_goals dsimp (config := {}) []\n    simp [foo]\n    trace_state\n\nexample : foo (0 + a) (b + 0) = a + b := by\n  conv =>\n    lhs\n    congr <;> simp\n    fail_if_success lhs\n    try lhs\n    trace_state\n\nexample (x y : Nat) : p (x + y) (y + x + 0) := by\n  conv =>\n    whnf\n    rhs\n    whnf\n  trace_state\n  rw [Nat.add_comm]\n  rfl\n\nexample (x y : Nat) : p (x + y) (y + x + 0) := by\n  conv =>\n    whnf\n    lhs\n    whnf\n  conv =>\n    rhs\n    whnf\n  trace_state\n  apply Nat.add_comm x y\n\ndef f (x y z : Nat) : Nat :=\n  y\n\nexample (x y : Nat) : f x (x + y + 0) y = y + x := by\n  conv =>\n    lhs\n    arg 2\n    whnf\n  trace_state\n  simp [f]\n  apply Nat.add_comm\n\nexample (x y : Nat) : f x (x + y + 0) y = y + x := by\n  conv =>\n    lhs\n    arg 2\n    change x + y\n    trace_state\n    rw [Nat.add_comm]\n\nexample : id (fun x y => 0 + x + y) = Nat.add := by\n  conv =>\n    lhs\n    arg 1\n    ext a b\n    trace_state\n    rw [Nat.zero_add]\n    trace_state\n\nexample : id (fun x y => 0 + x + y) = Nat.add := by\n  conv =>\n    lhs\n    arg 1\n    intro a b\n    rw [Nat.zero_add]\n\nexample : id (fun x y => 0 + x + y) = Nat.add := by\n  conv =>\n    enter [1, 1, a, b]\n    trace_state\n    rw [Nat.zero_add]\n\nexample (p : Nat → Prop) (h : ∀ a, p a) : ∀ a, p (id (0 + a)) := by\n  conv =>\n    intro x\n    trace_state\n    arg 1\n    trace_state\n    simp only [id]\n    trace_state\n    rw [Nat.zero_add]\n  exact h\n\nexample (p : Prop) (x : Nat) : (x = x → p) → p := by\n  conv =>\n    congr\n    . trace_state\n      congr\n      . simp\n  trace_state\n  conv =>\n    lhs\n    simp\n  intros\n  assumption\n\nexample : (fun x => 0 + x) = id := by\n  conv =>\n    lhs\n    tactic => funext x\n    trace_state\n    rw [Nat.zero_add]\n\nexample (p : Prop) (x : Nat) : (x = x → p) → p := by\n  conv =>\n    apply implies_congr\n    . apply implies_congr\n      simp\n  trace_state\n  conv =>\n    lhs\n    simp\n  intros; assumption\n\nexample (x y : Nat) (f : Nat → Nat → Nat) (g : Nat → Nat) (h₁ : ∀ z, f z z = z) (h₂ : ∀ x y, f (g x) (g y) = y) : f (g (0 + y)) (f (g x) (g (0 + x))) = x := by\n  conv =>\n    pattern _ + _\n    apply Nat.zero_add\n  trace_state\n  conv =>\n    pattern 0 + _\n    apply Nat.zero_add\n  trace_state\n  simp [h₁, h₂]\n\nexample (x y : Nat) (h : y = 0) : x + ((y + x) + x) = x + (x + x) := by\n  conv =>\n    lhs\n    rhs\n    lhs\n    trace_state\n    rw [h, Nat.zero_add]\n\nexample (p : Nat → Prop) (x y : Nat) (h1 : y = 0) (h2 : p x) : p (y + x) := by\n  conv =>\n    rhs\n    trace_state\n    rw [h1]\n    apply Nat.zero_add\n  exact h2\n\nexample (p : (n : Nat) → Fin n → Prop) (i : Fin 5) (hp : p 5 i) (hi : j = i) : p 5 j := by\n  conv =>\n    arg 2\n    trace_state\n    rw [hi]\n  exact hp\n\nexample (p : {_ : Nat} → Nat → Prop) (x y : Nat) (h1 : y = 0) (h2 : @p x x) : @p (y + x) (y + x) := by\n  conv =>\n    enter [@1, 1]\n    trace_state\n    rw [h1]\n  conv =>\n    enter [@2, 1]\n    trace_state\n    rw [h1]\n  rw [Nat.zero_add]\n  exact h2\n\nexample (p : Nat → Prop) (x y : Nat) (h : y = 0) : p (y + x) := by\n  conv => lhs\n\nexample (p : Nat → Prop) (x y : Nat) (h : y = 0) : p (y + x) := by\n  conv => arg 2\n\nexample (p : Prop) : p := by\n  conv => rhs\n\nexample (p : (n : Nat) → Fin n → Prop) (i : Fin 5) (hp : p 5 i) : p 5 j := by\n  conv => arg 1\n\n-- repeated `zeta`\nexample : let a := 0; let b := a; b = 0 := by\n  intros\n  conv =>\n    zeta\n    trace_state\n\nexample : ((x + y) + z : Nat) = x + (y + z) := by\n  conv in _ + _ => trace_state\n  conv in (occs := *) _ + _ => trace_state\n  conv in (occs := 1 3) _ + _ => trace_state\n  conv in (occs := 3 1) _ + _ => trace_state\n  conv in (occs := 2 3) _ + _ => trace_state\n  conv in (occs := 2 4) _ + _ => trace_state\n  apply Nat.add_assoc\n\nexample : ((x + y) + z : Nat) = x + (y + z) := by conv => pattern (occs := 5) _ + _\nexample : ((x + y) + z : Nat) = x + (y + z) := by conv => pattern (occs := 2 5) _ + _\nexample : ((x + y) + z : Nat) = x + (y + z) := by conv => pattern (occs := 1 5) _ + _\nexample : ((x + y) + z : Nat) = x + (y + z) := by conv => pattern (occs := 1 2 5) _ + _\n\nmacro \"bla\" : term => `(?a)\nexample : 1 = 1 := by conv => apply bla; congr\n\nexample (h : a = a') (H : a + a' = 0) : a + a = 0 := by\n  conv in (occs := 2) a => rw [h]\n  apply H\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/conv1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165382362518, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.30049272744846284}}
{"text": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n\nInstances on punit.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u \n\nnamespace Mathlib\n\nnamespace punit\n\n\nprotected instance comm_group : comm_group PUnit :=\n  comm_group.mk (fun (_x _x : PUnit) => PUnit.unit) sorry PUnit.unit sorry sorry\n    (fun (_x : PUnit) => PUnit.unit) (fun (_x _x : PUnit) => PUnit.unit) sorry sorry\n\nprotected instance comm_ring : comm_ring PUnit :=\n  comm_ring.mk add_comm_group.add add_comm_group.add_assoc add_comm_group.zero\n    add_comm_group.zero_add add_comm_group.add_zero add_comm_group.neg add_comm_group.sub\n    add_comm_group.add_left_neg add_comm_group.add_comm comm_group.mul comm_group.mul_assoc\n    comm_group.one comm_group.one_mul comm_group.mul_one sorry sorry comm_group.mul_comm\n\nprotected instance complete_boolean_algebra : complete_boolean_algebra PUnit :=\n  complete_boolean_algebra.mk (fun (_x _x : PUnit) => PUnit.unit) (fun (_x _x : PUnit) => True)\n    (fun (_x _x : PUnit) => False) sorry sorry sorry sorry sorry sorry\n    (fun (_x _x : PUnit) => PUnit.unit) sorry sorry sorry sorry PUnit.unit sorry PUnit.unit sorry\n    (fun (_x : PUnit) => PUnit.unit) (fun (_x _x : PUnit) => PUnit.unit) sorry sorry sorry\n    (fun (_x : set PUnit) => PUnit.unit) (fun (_x : set PUnit) => PUnit.unit) sorry sorry sorry\n    sorry sorry sorry\n\nprotected instance canonically_ordered_add_monoid : canonically_ordered_add_monoid PUnit :=\n  canonically_ordered_add_monoid.mk comm_ring.add comm_ring.add_assoc comm_ring.zero\n    comm_ring.zero_add comm_ring.add_zero comm_ring.add_comm complete_boolean_algebra.le\n    complete_boolean_algebra.lt complete_boolean_algebra.le_refl complete_boolean_algebra.le_trans\n    complete_boolean_algebra.le_antisymm sorry sorry complete_boolean_algebra.bot\n    complete_boolean_algebra.bot_le sorry\n\nprotected instance linear_ordered_cancel_add_comm_monoid :\n    linear_ordered_cancel_add_comm_monoid PUnit :=\n  linear_ordered_cancel_add_comm_monoid.mk canonically_ordered_add_monoid.add\n    canonically_ordered_add_monoid.add_assoc sorry canonically_ordered_add_monoid.zero\n    canonically_ordered_add_monoid.zero_add canonically_ordered_add_monoid.add_zero\n    canonically_ordered_add_monoid.add_comm sorry canonically_ordered_add_monoid.le\n    canonically_ordered_add_monoid.lt canonically_ordered_add_monoid.le_refl\n    canonically_ordered_add_monoid.le_trans canonically_ordered_add_monoid.le_antisymm\n    canonically_ordered_add_monoid.add_le_add_left sorry sorry\n    (fun (_x _x : PUnit) => decidable.true) punit.decidable_eq\n    fun (_x _x : PUnit) => decidable.false\n\nprotected instance semimodule (R : Type u) [semiring R] : semimodule R PUnit :=\n  semimodule.of_core\n    (semimodule.core.mk (has_scalar.mk fun (_x : R) (_x : PUnit) => PUnit.unit) sorry sorry sorry\n      sorry)\n\n@[simp] theorem zero_eq : 0 = PUnit.unit := rfl\n\n@[simp] theorem one_eq : 1 = PUnit.unit := rfl\n\n@[simp] theorem add_eq (x : PUnit) (y : PUnit) : x + y = PUnit.unit := rfl\n\n@[simp] theorem mul_eq (x : PUnit) (y : PUnit) : x * y = PUnit.unit := rfl\n\n@[simp] theorem sub_eq (x : PUnit) (y : PUnit) : x - y = PUnit.unit := rfl\n\n@[simp] theorem neg_eq (x : PUnit) : -x = PUnit.unit := rfl\n\n@[simp] theorem inv_eq (x : PUnit) : x⁻¹ = PUnit.unit := rfl\n\ntheorem smul_eq (x : PUnit) (y : PUnit) : x • y = PUnit.unit := rfl\n\n@[simp] theorem top_eq : ⊤ = PUnit.unit := rfl\n\n@[simp] theorem bot_eq : ⊥ = PUnit.unit := rfl\n\n@[simp] theorem sup_eq (x : PUnit) (y : PUnit) : x ⊔ y = PUnit.unit := rfl\n\n@[simp] theorem inf_eq (x : PUnit) (y : PUnit) : x ⊓ y = PUnit.unit := rfl\n\n@[simp] theorem Sup_eq (s : set PUnit) : Sup s = PUnit.unit := rfl\n\n@[simp] theorem Inf_eq (s : set PUnit) : Inf s = PUnit.unit := rfl\n\n@[simp] protected theorem le (x : PUnit) (y : PUnit) : x ≤ y := trivial\n\n@[simp] theorem not_lt (x : PUnit) (y : PUnit) : ¬x < y := not_false\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/punit_instances_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.43782349911420193, "lm_q1q2_score": 0.30032479060716666}}
{"text": "/-\nCopyright (c) 2022 Yaël Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yaël Dillies\n-/\nimport order.category.BoundedLattice\nimport order.category.DistribLattice\n\n/-!\n# The category of bounded distributive lattices\n\nThis defines `BoundedDistribLattice`, the category of bounded distributive lattices.\n\nNote that this category is sometimes called [`DistLat`](https://ncatlab.org/nlab/show/DistLat) when\nbeing a lattice is understood to entail having a bottom and a top element.\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- The category of bounded distributive lattices with bounded lattice morphisms. -/\nstructure BoundedDistribLattice :=\n(to_DistribLattice : DistribLattice)\n[is_bounded_order : bounded_order to_DistribLattice]\n\nnamespace BoundedDistribLattice\n\ninstance : has_coe_to_sort BoundedDistribLattice Type* := ⟨λ X, X.to_DistribLattice⟩\ninstance (X : BoundedDistribLattice) : distrib_lattice X := X.to_DistribLattice.str\n\nattribute [instance] BoundedDistribLattice.is_bounded_order\n\n/-- Construct a bundled `BoundedDistribLattice` from a `bounded_order` `distrib_lattice`. -/\ndef of (α : Type*) [distrib_lattice α] [bounded_order α] : BoundedDistribLattice := ⟨⟨α⟩⟩\n\n@[simp] lemma coe_of (α : Type*) [distrib_lattice α] [bounded_order α] : ↥(of α) = α := rfl\n\ninstance : inhabited BoundedDistribLattice := ⟨of punit⟩\n\n/-- Turn a `BoundedDistribLattice` into a `BoundedLattice` by forgetting it is distributive. -/\ndef to_BoundedLattice (X : BoundedDistribLattice) : BoundedLattice := BoundedLattice.of X\n\n@[simp] lemma coe_to_BoundedLattice (X : BoundedDistribLattice) : ↥X.to_BoundedLattice = ↥X := rfl\n\ninstance : large_category.{u} BoundedDistribLattice := induced_category.category to_BoundedLattice\n\ninstance : concrete_category BoundedDistribLattice :=\ninduced_category.concrete_category to_BoundedLattice\n\ninstance has_forget_to_DistribLattice : has_forget₂ BoundedDistribLattice DistribLattice :=\n{ forget₂ := { obj := λ X, ⟨X⟩, map := λ X Y, bounded_lattice_hom.to_lattice_hom } }\n\ninstance has_forget_to_BoundedLattice : has_forget₂ BoundedDistribLattice BoundedLattice :=\ninduced_category.has_forget₂ to_BoundedLattice\n\nlemma forget_BoundedLattice_Lattice_eq_forget_DistribLattice_Lattice :\n  forget₂ BoundedDistribLattice BoundedLattice ⋙ forget₂ BoundedLattice Lattice =\n    forget₂ BoundedDistribLattice DistribLattice ⋙ forget₂ DistribLattice Lattice := rfl\n\n/-- Constructs an equivalence between bounded distributive lattices from an order isomorphism\nbetween them. -/\n@[simps] def iso.mk {α β : BoundedDistribLattice.{u}} (e : α ≃o β) : α ≅ β :=\n{ hom := (e : bounded_lattice_hom α β),\n  inv := (e.symm : bounded_lattice_hom β α),\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : BoundedDistribLattice ⥤ BoundedDistribLattice :=\n{ obj := λ X, of Xᵒᵈ, map := λ X Y, bounded_lattice_hom.dual }\n\n/-- The equivalence between `BoundedDistribLattice` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : BoundedDistribLattice ≌ BoundedDistribLattice :=\nequivalence.mk dual dual\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n  (nat_iso.of_components (λ X, iso.mk $ order_iso.dual_dual X) $ λ X Y f, rfl)\n\nend BoundedDistribLattice\n\nlemma BoundedDistribLattice_dual_comp_forget_to_DistribLattice :\n  BoundedDistribLattice.dual ⋙ forget₂ BoundedDistribLattice DistribLattice =\n    forget₂ BoundedDistribLattice DistribLattice ⋙ DistribLattice.dual := rfl\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/order/category/BoundedDistribLattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.30027495220602735}}
{"text": "import tactic\n\nclass myclass (X : Type) :=\n(op : (X → X) → X)\n\nvariables (X Y Z : Type) [myclass X] [myclass Y] [myclass Z]\n\nnamespace myclass\n\nstructure hom :=\n(f : X → Y)\n(map' : ∀ (a : X → X) (b : Y → Y), (∀ x, f (a x) = b (f x)) → f (op a) = op b)\n\nstructure edge :=\n(R : X → Y → Prop)\n(h : ∀ (a : X → X) (b : Y → Y), (∀ x y, R x y → R (a x) (b y)) → R (op a) (op b))\n\ndef diag : edge X X :=\n{ R := eq,\n  h := begin\n    intros a b h,\n    congr,\n    ext,\n    apply h,\n    refl,\n  end }\n\nexample (e : edge X Y) : myclass (Σ' x y, e.R x y) :=\n⟨λ a, begin\n  cases e, dsimp at *,\nend⟩\n\nstructure hom2 :=\n(f : X → Y)\n(f2 : (X → X) → (Y → Y))\n(hf2 : ∀ (a : X → X) (b : Y → Y), (∀ x, f (a x) = b (f x)) ↔ f2 a = b)\n(map : ∀ (a : X → X), f (op a) = op (f2 a))\n\nexample {X Y : Type} (f : X → Y)\n  (f2 : (X → X) → (Y → Y))\n  (hf2 : ∀ (a : X → X) (b : Y → Y), (∀ x, f (a x) = b (f x)) ↔ f2 a = b) :\n  function.bijective f ∨ (∀ x y : Y, x = y) :=\nbegin\n  rw [or_iff_not_imp_right],\n  intro h,\n  push_neg at h,\n  rcases h with ⟨y₁, y₂, hy⟩,\n  classical,\n  split,\n  intros x₁ x₂ h,\n  have := hf2 id (equiv.swap y₁ y₂),\n  dsimp at this, \nend\n\n\nlemma lemma1 {X Y : Type} (f : X → Y) \n  (hf : function.has_right_inverse f) :\n  ∃ f2 : (X → X) → (Y → Y),\n  (∀ (a : X → X) (b : Y → Y), (∀ x, f (a x) = b (f x)) ↔ f2 a = b) :=\nbegin\n  cases hf with g hg,\n  refine ⟨λ a, f ∘ a ∘ g, _⟩,\n  intros a b,\n  simp only [function.funext_iff, function.comp_app],\n  split,\n  { assume h y,\n    rw [← hg y, h, hg y, hg y] },\n  { assume h x,\n    rw [← h],\n    delta function.right_inverse function.left_inverse at hg,\n    \n\n     }\nend\n\n-- example : false := begin \n--    have := h (@empty.elim unit) (λ _, id) _ (),\n--    cases this,\n--    cases this_w,\n--    intros,simp,\n-- end\n\ndef hom2_to_edge (f : hom2 X Y) : edge X Y :=\n{ R := λ x y, f.f x = y,\n  h := λ a b h, begin\n    rw [f.map],\n\n    \n  end }\n\ndef comp2 (f : hom2 X Y) (g : hom2 Y Z) : hom2 X Z :=\n{ f := λ x, g.f (f.f x),\n  f2 := λ a, g.f2 (f.f2 a),\n  hf2 := λ a c, begin\n    split,\n    { have hg2 := g.hf2,\n      have hf2 := f.hf2,\n      intros h,\n      rw [(hg2 _ _).1],\n      clear hg2,\n      intro y,\n      have := f.hf2 a (λ _, f.f2 a y),\n      simp only [function.funext_iff] at this,\n      rw ← this.2,\n      rw h,\n      \n       },\n    { intros h x,\n      have := (f.hf2 a _).2 rfl,\n      rw [this, (g.hf2 _ _).2],\n      rw h }\n  end,\n  map := λ a, by rw [f.map, g.map] }\n\ninstance : has_coe_to_fun (hom X Y) (λ _, X → Y) :=\n{ coe := hom.f }\n\ndef id : hom X X :=\n{ f := id,\n  map' := λ a b h, have h : a = b, from funext h, by rw h; refl }\n\nvariables {X Y Z}\n\nlemma hom.map (f : hom X Y) : \n  ∀ (a : X → X) (b : Y → Y), (∀ x, f (a x) = b (f x)) → f (op a) = op b := f.map'\n\ndef comp (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ f := λ x, g (f x),\n  map' := λ a c h, begin\n    cases f with f hf, cases g with g hg,\n    unfold_coes at *,\n    dsimp at *,\n    \n  end }\n\ndef to_functor {X : Type} (f : (X → X) → X) : (X → X → Type) → X → Type :=\nλ R x, Σ g : {g : X → X // f g = x}, ∀ a b, g.1 x = b → R a b\n\ndef to_functor_map {X : Type} (f : (X → X) → X) (R₁ R₂ : (X → X → Type))\n  (i : Π x y, R₁ x y → R₂ x y) : Π x, to_functor f R₁ x → to_functor f R₂ x :=\nλ x g, ⟨g.1, λ a b h, i _ _ (g.2 _ _ h)⟩\n\nvariables (X Y)\ndef hom₃ := Σ f : X → Y, Π (i : Y → Y → Type) (x : X), \n  to_functor op (λ x y : X, i (f x) (f y)) x → to_functor op i (f x)\n\n\n\nend myclass", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/parametricity/XtoXtoX.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3002749522060273}}
{"text": "import algebra.homology.homotopy\n\nimport pseudo_normed_group.system_of_complexes\nimport rescale.Tinv\n\n/-!\n\n*TODO*: find someone who can explain what is going on in this file.\n\n-/\n\nnoncomputable theory\n\nuniverse variables u\n\nopen_locale nnreal\n\nopen category_theory homological_complex\n\nnamespace breen_deligne\n\nvariables {BD BD₁ BD₂ : breen_deligne.data} (f g : BD₂ ⟶ BD₁)\nvariables (h : homotopy f g)\n\nvariables (κ c_₁ c_₂ : ℕ → ℝ≥0)\nvariables [BD.suitable κ] [BD₁.suitable c_₁] [BD₂.suitable c_₂]\nvariables (r : ℝ≥0) (V : SemiNormedGroup) [normed_with_aut r V] [fact (0 < r)]\nvariables {r' : ℝ≥0} [fact (0 < r')] [fact (r' ≤ 1)] (c : ℝ≥0)\n\nsection homotopy\n\nvariables (M : (ProFiltPseuNormGrpWithTinv.{u} r')ᵒᵖ)\n\nopen homological_complex\n\n@[simps app_f]\ndef BD_map₂ (a₁ a₂ b₁ b₂ : ℕ → ℝ≥0)\n  [∀ (i : ℕ), fact (b₁ i ≤ r' * a₁ i)] [∀ (i : ℕ), fact (b₂ i ≤ r' * a₂ i)]\n  [BD₁.suitable a₁] [BD₂.suitable a₂] [BD₁.suitable b₁] [BD₂.suitable b₂]\n  [∀ i, (f.f i).suitable (a₂ i) (a₁ i)]\n  [∀ i, (f.f i).suitable (b₂ i) (b₁ i)] :\n  BD₁.complex₂ r V r' a₁ b₁ ⟶ BD₂.complex₂ r V r' a₂ b₂ :=\n{ app := λ M,\n  { f := λ i, ((f.f i).eval_CLCFPTinv₂ r V r' (a₁ i) (b₁ i) (a₂ i) (b₂ i)).app M,\n    comm' := begin\n      intros i j _,\n      dsimp [data.complex₂_obj_d, data.complex₂_d],\n      have : f.f j ≫ BD₁.d j i = BD₂.d j i ≫ f.f i := f.comm j i,\n      simp only [← nat_trans.comp_app, ← universal_map.eval_CLCFPTinv₂_comp r V r', this],\n    end },\n  naturality' := by { intros M₁ M₂ g, ext i : 2,\n    exact ((f.f i).eval_CLCFPTinv₂ r V r' (a₁ i) (b₁ i) (a₂ i) (b₂ i)).naturality g, } }\n.\n\n@[simps app_f]\ndef BD_map [∀ i, (f.f i).suitable (c_₂ i) (c_₁ i)] :\n  BD₁.complex c_₁ r V r' c ⟶ BD₂.complex c_₂ r V r' c :=\nBD_map₂ f r V _ _ _ _\n.\n\n-- @[simp] lemma BD_map_app_f [∀ i, (f.f i).suitable (c_₂ i) (c_₁ i)] (i : ℕ)\n--   (M : (ProFiltPseuNormGrpWithTinv r')ᵒᵖ) :\n--   ((BD_map f c_₁ c_₂ r V c).app M).f i =\n--     ((f.f i).eval_CLCFPTinv r V r' (c_₁ i) (c_₂ i)).app M := _\n\nopen opposite\n\n@[simps app_app]\ndef BD_system_map [∀ i, (f.f i).suitable (c_₂ i) (c_₁ i)] :\n  BD₁.system c_₁ r V r' ⟶ BD₂.system c_₂ r V r' :=\n{ app := λ M,\n  { app := λ c, (BD_map f c_₁ c_₂ r V c.unop).app M,\n    naturality' := λ x y hxy,\n    begin\n      ext i : 2,\n      erw [comp_f, comp_f],\n      dsimp only [data.system_obj, BD_map, BD_map₂_app_f],\n      haveI : fact (y.unop ≤ x.unop) := ⟨le_of_hom hxy.unop⟩,\n      exact nat_trans.congr_app\n        (universal_map.res_comp_eval_CLCFPTinv₂ r V r' _ _ _ _ _ _ _ _ _) M,\n    end },\n  naturality' := λ M₁ M₂ g,\n  begin\n    ext c : 2,\n    exact (BD_map f c_₁ c_₂ r V c.unop).naturality _\n  end }\n.\n\nend homotopy\n\nend breen_deligne\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/homotopy.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.3960681662740417, "lm_q1q2_score": 0.3001382270435372}}
{"text": "import implementation.model.predicate\nimport implementation.model.sys_state\nimport implementation.spec.main\nimport implementation.proof.misc\n\n-- This file contains proofs about proposers (or what can be associated with\n-- them in our co-located Paxos implementation).\n--\n-- NOTE(gnanabit): comments are omitted for facts that are \"obvious\" or not\n-- particularly revealing about paxos.\n\nvariables {pid_t : Type} [linear_order pid_t] [fintype pid_t] {value_t : Type}\n          {is_quorum : finset pid_t → Prop} [decidable_pred is_quorum]\n          [quorum_assumption is_quorum] {vals : pid_t → value_t}\n\n-- The ballot b has been proposed with value v if there is a server who has sent\n-- a p2a with this proposal.\ndef proposed\n  (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t))\n  (b : ballot pid_t) (v : value_t) :=\n  ∃ (proposer : pid_t) (e ∈ s.network proposer),\n    (envelope.msg e) = message.p2a {bal := b, val := v}\n\nlemma proposed_stable (b : ballot pid_t) (v : value_t) : predicate.stable\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    proposed s b v) :=\nbegin\nintros u w holds_at_u u_pn_w,\nrcases holds_at_u with ⟨proposer, proposal_env, sent_by_proposer, is_proposal⟩,\nexact ⟨proposer, proposal_env, sys_state.ntwk_subset sent_by_proposer u_pn_w, is_proposal⟩,\nend\n\ndef proposed_by\n  (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t))\n  (b : ballot pid_t) (proposer : pid_t) :=\n  ∃ (v : value_t) (e ∈ s.network proposer),\n    (envelope.msg e) = message.p2a {bal := b, val := v}\n\nlemma none_proposed_at_init\n  (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t))\n  (hs : s.is_initial) : ∀ (b : ballot pid_t) (p : pid_t), ¬proposed_by s b p :=\nbegin\nintros b p,\nrintros ⟨v, e, he, e_is_proposal⟩,\nspecialize hs p,\nunfold protocol.init at hs,\ninjection hs with unused key, clear unused,\nrw ← key at he,\ncases he,\n{ rw he at e_is_proposal, injection e_is_proposal },\nrw set.mem_singleton_iff at he,\nrw he at e_is_proposal,\ninjection e_is_proposal\nend\n\n-- If a proposer p proposed ballot b, then p is the address of b.\nlemma proposer_is_ballot_address : predicate.invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (p : pid_t), proposed_by s b p → b.address = p) :=\nbegin\nsuffices key : predicate.inductive_invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (p : pid_t), proposed_by s b p → b.address = p),\nby { exact predicate.ind_inv_is_inv key },\nsplit,\n{ intros s hs b p wrong,\n  exact (none_proposed_at_init s hs b p wrong).elim },\nintros u w hu,\nrintros ⟨receiver, sender, e, _h1, _h2, _h3, w_receiver, _h4, w_not_receiver⟩,\nclear_,\nunfold proposed_by,\nintros b p,\ncases decidable.em (p = receiver),\nswap,\n{ rw w_not_receiver p h,\n  intro hyp,\n  exact hu b p hyp },\ncases e with msg __;\nunfold envelope.msg at w_receiver,\nclear __,\nrintros ⟨v, e, he, e_is_proposal⟩,\nrw h at he,\nrw w_receiver at he,\ncases he,\n{ apply hu,\n  rw h,\n  unfold proposed_by,\n  use [v, e, he, e_is_proposal] },\nrw h,\nrcases p2a_emitted e_is_proposal he with ⟨p_or, __, receiver_ballot_address, __, __, e_is⟩,\nclear_,\nrw e_is at e_is_proposal,\ninjection e_is_proposal with proposals_match,\ninjection proposals_match with ballots_match,\nrw ballots_match at receiver_ballot_address,\nexact receiver_ballot_address\nend\n\nprivate lemma proposer_sends_to_all_but_self : predicate.invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (proposer : pid_t) (e ∈ s.network proposer)\n      (v : value_t),\n      (envelope.msg e) = message.p2a {bal := b, val := v} →\n        e.sent_to = target.exclude proposer) :=\nbegin\nsuffices key : predicate.inductive_invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (proposer : pid_t) (e ∈ s.network proposer)\n      (v : value_t),\n        (envelope.msg e) = message.p2a {bal := b, val := v} →\n          e.sent_to = target.exclude proposer),\nby { exact predicate.ind_inv_is_inv key },\nsplit,\n{ intros s hs b proposer e he v e_msg_eq,\n  exact (none_proposed_at_init s hs b proposer ⟨v, e, he, e_msg_eq⟩).elim },\nintros u w hu u_pn_w b proposer e he v e_msg_eq,\nrcases u_pn_w with ⟨receiver, sender, e', he', deliverable, proc_change, ntwk_change, rest_same⟩,\ncases decidable.em (proposer = receiver),\nswap,\n{ rw rest_same.right proposer h at he,\n  exact hu b proposer e he v e_msg_eq },\nclear rest_same proc_change,\nrw h at he ⊢,\nclear h proposer,\nrw ntwk_change at he, clear ntwk_change,\ncases he,\n{ exact hu b receiver e he v e_msg_eq },\nrcases p2a_emitted e_msg_eq he with ⟨_, _, _, _, _, key⟩,\nrw key\nend\n\n-- If a process p proposed ballot b, then p's current ballot is at least as\n-- large as b.\nlemma proposer_ballot_ge : predicate.invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (p : pid_t), proposed_by s b p → b ≤ (s.procs p).curr) :=\nbegin\nsuffices key : predicate.inductive_invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (p : pid_t), proposed_by s b p → b ≤ (s.procs p).curr),\nby { exact predicate.ind_inv_is_inv key },\nsplit,\n{ intros s hs b p wrong,\n  exact (none_proposed_at_init s hs b p wrong).elim },\nintros u w inv_u u_pn_w,\nintros b p,\nhave ballot_le := ballot_nondecreasing p u_pn_w,\nrcases u_pn_w with ⟨receiver, sender, e, he, deliverable, proc_change, ntwk_change, proc_same, ntwk_same⟩,\nrintros ⟨v, e_del, he_del, e_del_is_vote⟩,\ncases decidable.em (p = receiver),\nswap,\n{ rw ntwk_same p h at he_del,\n  unfold proposed_by at inv_u,\n  calc b ≤ (u.procs p).curr : inv_u b p ⟨v, e_del, he_del, e_del_is_vote⟩\n     ... ≤ (w.procs p).curr : ballot_le },\nrw h at he_del ballot_le ⊢,\nclear h p,\nrw ntwk_change at he_del,\ncases he_del,\n{ calc b ≤ (u.procs receiver).curr : inv_u b receiver ⟨v, e_del, he_del, e_del_is_vote⟩\n     ... ≤ (w.procs receiver).curr : ballot_le },\nrcases p2a_emitted e_del_is_vote he_del with ⟨p_or, _, _, _, _, key⟩,\nrw key at e_del_is_vote,\ninjection e_del_is_vote with contents_same,\ninjection contents_same with ballots_same,\ncalc b = (u.procs receiver).curr : eq.symm ballots_same\n   ... ≤ (w.procs receiver).curr : ballot_le\nend\n\nprivate lemma not_proposed_without_quorum : predicate.invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (p : pid_t), ¬is_quorum (s.procs p).followers → ¬proposed_by s (s.procs p).curr p) :=\nbegin\nrw predicate.use_any_invariant,\nsplit,\n{ intros s hs p unimportant wrong,\n  exact (none_proposed_at_init s hs (s.procs p).curr p wrong).elim },\nintros u w u_reachable not_proposed_wo_quorum_at_u,\nrintros ⟨receiver, sender, e, he, deliverable, proc_change, ntwk_change, rest_same⟩,\nintro w_reachable,\nhave ballot_le := ballot_nondecreasing receiver\n                  ⟨receiver, sender, e, he, deliverable, proc_change, ntwk_change, rest_same⟩,\nhave proposed_le_at_u := proposer_ballot_ge u u_reachable,\nintro p,\ncases decidable.em (p = receiver),\nswap,\n{ rw rest_same.left p h,\n  unfold proposed_by,\n  rw rest_same.right p h,\n  exact not_proposed_wo_quorum_at_u p },\nclear rest_same,\nrw h,\nclear h p,\nintro hyp,\nrintros ⟨v, e', he', e'_msg_eq⟩,\nrw ntwk_change at he', clear ntwk_change,\nrw le_iff_lt_or_eq at ballot_le,\ncases ballot_le with ballot_lt ballot_eq,\n{ have key : ¬proposed_by u (w.procs receiver).curr receiver,\n  by { intro wrong,\n      exact not_lt_of_ge (proposed_le_at_u (w.procs receiver).curr receiver wrong) ballot_lt },\n  cases he',\n  { exact key ⟨v, e', he', e'_msg_eq⟩ },\n  clear key,\n  have key := network_change receiver (u.procs receiver) e.msg sender e' he',\n  cases e.msg,\n    case p1a : mb {\n      cases key;\n      rw key.right at e'_msg_eq;\n      injection e'_msg_eq\n    },\n    case p1b : mb p_or {\n      cases key.right.right.right.right with e'_eq e'_eq,\n      { clear key,\n        rw e'_eq at e'_msg_eq,\n        injection e'_msg_eq with props_same,\n        injection props_same with ballots_same,\n        exact ne_of_lt ballot_lt ballots_same },\n      rw e'_eq at e'_msg_eq,\n      injection e'_msg_eq\n    },\n    case p2a : p {\n      cases key;\n      rw key.right at e'_msg_eq;\n      injection e'_msg_eq\n    },\n    case p2b : mb acc {\n      exact key.elim\n    },\n    case preempt : {\n      rw key.right at e'_msg_eq,\n      injection e'_msg_eq\n    }\n  },\nrw ← ballot_eq at e'_msg_eq,\ncases he',\n{ suffices cond : ¬is_quorum (u.procs receiver).followers,\n  by {\n    apply not_proposed_wo_quorum_at_u receiver cond,\n    exact ⟨v, e', he', e'_msg_eq⟩ },\n  have key := state_change receiver (u.procs receiver) e.msg sender,\n  cases key,\n  { rw key at proc_change,\n    rw proc_change at hyp,\n    exact hyp },\n  cases e,\n  cases e_msg,\n    case p1a : b {\n      cases key with ballot_increased made_larger,\n      rw made_larger at proc_change,\n      rw proc_change at ballot_eq,\n      exact (ne_of_lt ballot_increased ballot_eq).elim,\n    },\n    case p1b : b p_or {\n      cases key,\n      { cases key with ballot_increased made_larger,\n        rw made_larger at proc_change,\n        rw proc_change at ballot_eq,\n        exact (ne_of_lt ballot_increased ballot_eq).elim },\n      cases key;\n      exact key.right.right.left\n    },\n    case p2a : p {\n      cases key with ballot_ge made_larger,\n      clear ballot_ge,\n      have p_bal_eq : p.bal = (u.procs receiver).curr,\n      by {\n        rw made_larger at proc_change,\n        rw proc_change at ballot_eq,\n        exact eq.symm ballot_eq\n      },\n      have p_bal_addr_ne_receiver: p.bal.address ≠ receiver,\n      by {\n        have ballot_address_is_sender : p.bal.address = sender,\n        by {\n          apply proposer_is_ballot_address u u_reachable p.bal sender,\n          exact ⟨p.val, {msg := message.p2a p, sent_to := e_sent_to}, he,\n          by { cases p, refl }⟩,\n        },\n        suffices key : sender ≠ receiver,\n        by {\n          rw ballot_address_is_sender,\n          exact key\n        },\n        suffices key : e_sent_to = target.exclude sender,\n        by { rw key at deliverable, exact deliverable },\n        apply proposer_sends_to_all_but_self u u_reachable p.bal sender\n              {msg := message.p2a p, sent_to := e_sent_to} he p.val,\n        cases p, refl\n      },\n      have p_bal_addr_eq_receiver := proposer_is_ballot_address u u_reachable\n                                     (u.procs receiver).curr receiver ⟨v, e', he', e'_msg_eq⟩,\n      rw ← p_bal_eq at p_bal_addr_eq_receiver,\n      exact (p_bal_addr_ne_receiver p_bal_addr_eq_receiver).elim\n    },\n    case p2b : mb acc {\n      cases key with ballot_gt updated,\n      rw updated at proc_change,\n      rw proc_change at ballot_eq,\n      exact (ne_of_lt ballot_gt ballot_eq).elim\n    },\n    case preempt : {\n      rw key.right at proc_change,\n      clear key,\n      rw proc_change at ballot_eq,\n      exact (ne_of_lt (ballot.next_larger receiver (u.procs receiver).curr) ballot_eq).elim\n    }\n  },\nrcases p2a_emitted e'_msg_eq he' with\n  ⟨p_or, e_msg_is, receiver_still_leader, began_wo_quorum, ended_w_quorum, emitted_just⟩,\nsuffices key : (w.procs receiver).followers = (u.procs receiver).followers ∪ {sender},\nby { rw key at hyp, exact hyp ended_w_quorum },\nrw proc_change,\nrw e_msg_is,\nunfold protocol.handler server.handle_p1b,\nrw if_neg (lt_irrefl _),\nrw if_neg (not_not.mpr receiver_still_leader),\nrw if_neg (lt_irrefl _),\nrw if_neg (show ¬(is_quorum (u.procs receiver).followers ∨\n                  sender ∈ (u.procs receiver).followers), by {\n  intro cond,\n  cases cond with cond cond,\n  { exact began_wo_quorum cond },\n  have key : (u.procs receiver).followers ∪ {sender} = (u.procs receiver).followers,\n  by {\n    rw finset.union_eq_left_iff_subset,\n    rw finset.singleton_subset_iff,\n    exact cond\n  },\n  rw key at ended_w_quorum,\n  exact began_wo_quorum ended_w_quorum\n}),\nrw if_pos ended_w_quorum\nend\n\n-- If two proposals have the same ballot, they have the same value.\n--\n-- The proof is by induction using predicates we've already proven.\n--\n-- Nothing is proposed at the initial state.\n--\n-- If the fact holds at u and two values are proposed with b at some state w\n-- following from u, then the proposals are issued by the same proposer. Either\n-- the proposer issued both proposals in u, in which case we apply the inductive\n-- hypothesis, or one proposal was issued in u and the other was just issued in\n-- the step (there are two ways in which that might happen, and they're\n-- symmetric), or both proposals were issued in the last step.\n--\n-- * In the case that one proposal was issued at u and the other was only issued\n-- as a result of the last step, it should be impossible, because when a server\n-- proposes a value in a step, it has no quorum before it sends the proposal,\n-- and therefore it could not have sent a proposal in step u under the required\n-- ballot.\n--\n-- * In the case that both proposals were issued in the last step, only one\n-- proposal is issued at a time, so both are the same proposal.\ntheorem proposals_unique : predicate.invariant\n  (λ (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    ∀ (b : ballot pid_t) (v₁ v₂ : value_t), proposed s b v₁ → proposed s b v₂ → v₁ = v₂) :=\nbegin\nrw predicate.use_any_invariant,\nsplit,\n{ intros s hs b v _ proposed_at_initial,\n  rcases proposed_at_initial with ⟨proposer, e, he, e_is_proposal⟩,\n  exfalso, apply none_proposed_at_init s hs b proposer,\n  exact ⟨v, e, he, e_is_proposal⟩ },\nintros u w u_reachable unique_at_u u_pn_w w_reachable b v₁ v₂ hp₁ hp₂,\nspecialize unique_at_u b v₁ v₂,\nrcases u_pn_w with ⟨receiver, sender, e, he, deliverable, proc_change, ntwk_change, rest_same⟩,\ncases hp₁ with proposer₁ hp₁,\ncases hp₂ with proposer₂ hp₂,\nhave prop₁_eq := proposer_is_ballot_address w w_reachable b proposer₁ ⟨v₁, hp₁⟩,\nrw ← prop₁_eq at hp₁,\nhave prop₂_eq := proposer_is_ballot_address w w_reachable b proposer₂ ⟨v₂, hp₂⟩,\nrw ← prop₂_eq at hp₂,\nclear prop₁_eq prop₂_eq proposer₁ proposer₂,\ncases decidable.em (b.address = receiver),\nswap,\n{ rw rest_same.right b.address h at hp₁ hp₂,\n  exact unique_at_u ⟨b.address, hp₁⟩ ⟨b.address, hp₂⟩ },\nclear rest_same,\nrw h at hp₁ hp₂, clear h,\nrcases hp₁ with ⟨e₁, he₁, e₁_msg_eq⟩,\nrcases hp₂ with ⟨e₂, he₂, e₂_msg_eq⟩,\nrw ntwk_change at he₁ he₂,\ncases he₁; cases he₂,\n{ exact unique_at_u ⟨receiver, e₁, he₁, e₁_msg_eq⟩ ⟨receiver, e₂, he₂, e₂_msg_eq⟩ },\n{ rcases p2a_emitted e₂_msg_eq he₂ with ⟨_, _, _, no_quorum, _, e₂_is⟩,\n  have ballots_match : (u.procs receiver).curr = b,\n  by { rw e₂_is at e₂_msg_eq, injection e₂_msg_eq with key, injection key },\n  clear e₂_is e₂_msg_eq he₂ e₂,\n  rw ← ballots_match at e₁_msg_eq,\n  exfalso,\n  apply not_proposed_without_quorum u u_reachable receiver no_quorum,\n  exact ⟨v₁, e₁, he₁, e₁_msg_eq⟩ },\n{ rcases p2a_emitted e₁_msg_eq he₁ with ⟨_, _, _, no_quorum, _, e₁_is⟩,\n  have ballots_match : (u.procs receiver).curr = b,\n  by { rw e₁_is at e₁_msg_eq, injection e₁_msg_eq with key, injection key },\n  clear e₁_is e₁_msg_eq he₁ e₁,\n  rw ← ballots_match at e₂_msg_eq,\n  exfalso,\n  apply not_proposed_without_quorum u u_reachable receiver no_quorum,\n  exact ⟨v₂, e₂, he₂, e₂_msg_eq⟩ },\nrcases p2a_emitted e₁_msg_eq he₁ with ⟨p₁, e_has_p₁, __, __, __, key₁⟩,\nrcases p2a_emitted e₂_msg_eq he₂ with ⟨p₂, e_has_p₂, __, __, __, key₂⟩,\nclear_,\nhave proposals_match : p₁ = p₂,\nby { injection eq.trans (eq.symm e_has_p₁) e_has_p₂ },\nhave val₁ : v₁ = proposal.value_or_default (proposal.merge (u.procs receiver).accepted p₁)\n                                           (vals receiver),\nby {\n  rw key₁ at e₁_msg_eq,\n  injection e₁_msg_eq with fact,\n  injection fact with _ goal,\n  exact eq.symm goal },\nhave val₂ : v₂ = proposal.value_or_default (proposal.merge (u.procs receiver).accepted p₂)\n                                           (vals receiver),\nby {\n  rw key₂ at e₂_msg_eq,\n  injection e₂_msg_eq with fact,\n  injection fact with _ goal,\n  exact eq.symm goal },\nrw val₁,\nrw val₂,\nrw proposals_match\nend\n", "meta": {"author": "gnanabite", "repo": "colocated-paxos", "sha": "f60308e27d3013665809077fe80a4b2af8a42278", "save_path": "github-repos/lean/gnanabite-colocated-paxos", "path": "github-repos/lean/gnanabite-colocated-paxos/colocated-paxos-f60308e27d3013665809077fe80a4b2af8a42278/src/implementation/proof/proposer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3000941724786688}}
{"text": "import order.with_bot\n\nopen function\n\nnamespace with_bot\nvariables {α : Type*} {C : with_bot α → Sort*} (h₁ : C ⊥) (h₂ : Π a : α, C ↑a)\n\nlemma coe_ne_coe {a b : α} : (a : with_bot α) ≠ b ↔ a ≠ b := coe_eq_coe.not\n\ninstance [has_lt α] [is_well_order α (<)] : is_strict_total_order (with_bot α) (<) :=\nbegin\n  classical,\n  letI : linear_order α := linear_order_of_STO (<),\n  apply_instance\nend\n\ninstance [preorder α] [is_well_order α (<)] : is_well_order (with_bot α) (<) := {}\n\nend with_bot\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/mathlib/with_bot.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.30002437990564584}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.repr\nimport Mathlib.Lean3Lib.init.data.prod\nimport Mathlib.Lean3Lib.init.data.sum.basic\n\nuniverses l u \n\nnamespace Mathlib\n\ninductive ordering where\n| lt : ordering\n| eq : ordering\n| gt : ordering\n\nprotected instance ordering.has_repr : has_repr ordering := has_repr.mk fun (s : ordering) => sorry\n\nnamespace ordering\n\n\ndef swap : ordering → ordering := sorry\n\ndef or_else : ordering → ordering → ordering := sorry\n\ntheorem swap_swap (o : ordering) : swap (swap o) = o :=\n  ordering.cases_on o (idRhs (swap (swap lt) = swap (swap lt)) rfl)\n    (idRhs (swap (swap eq) = swap (swap eq)) rfl) (idRhs (swap (swap gt) = swap (swap gt)) rfl)\n\nend ordering\n\n\ndef cmp_using {α : Type u} (lt : α → α → Prop) [DecidableRel lt] (a : α) (b : α) : ordering :=\n  ite (lt a b) ordering.lt (ite (lt b a) ordering.gt ordering.eq)\n\ndef cmp {α : Type u} [HasLess α] [DecidableRel Less] (a : α) (b : α) : ordering :=\n  cmp_using Less a b\n\nprotected instance ordering.decidable_eq : DecidableEq ordering := fun (a b : ordering) => sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/ordering/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.30002115554253966}}
{"text": "import order.complete_lattice\nimport set_theory.ordinal\n\nuniverses u v\n\nvariables (A : Type u) (B : Type v) [partial_order A] [partial_order B]\n\nstructure sup_hom : Type (max u v) :=\n(to_fun : A → B)\n(monotone : monotone to_fun)\n(sup : ∀ (s : set A) (a : A), is_lub s a → is_lub (to_fun '' s) (to_fun a))\n\n/-- Aiming to construct a cocompletion of `B` that preserves all sups in `A`.\n  So if we have a partial order `C` a monotone function `B -> C` and a `sup_hom`\n  from `A -> C` that commute with `f` there is a unique cocontinuous map from\n  the cocompletion to `C` such that everything commutes. -/\n\ndef presheaf : Type* :=\n{ s : B → Prop // ∀ a b, a ≤ b → s b → s a }\n\nvariables {A B}\n\nnamespace presheaf\n\ninstance : has_coe_to_fun (presheaf B) (λ _, B → Prop) :=\n⟨subtype.val⟩\n\n@[simp] lemma coe_mk (s : B → Prop) (hs : ∀ a b, a ≤ b → s b → s a) :\n  @coe_fn (presheaf B) _ _ (⟨s, hs⟩ : presheaf B) = s := rfl\n\ninstance : partial_order (presheaf B) :=\n{ le := λ a b, ∀ x, a x → b x,\n  le_trans := λ a b c hab hbc x hax, hbc _ (hab _ hax),\n  le_refl := λ _ _, id,\n  le_antisymm := λ a b hab hba, subtype.val_injective (funext $ λ x, propext ⟨hab _, hba _⟩) }\n\nlemma le_def {a b : presheaf B} : a ≤ b = ∀ x, a x → b x := rfl\n\ninstance : has_Inf (presheaf B) :=\n{ Inf := λ s, ⟨λ p, ∀ B : presheaf B, B ∈ s → B p, \n     λ a b hab h B hBs, B.2 _ _ hab (h _ hBs)⟩ }\n\ninstance : complete_lattice (presheaf B) :=\ncomplete_lattice_of_Inf _\n  (λ s, begin\n    split,\n    { dsimp [Inf, lower_bounds],\n      intros B hBs p h,\n      apply h,\n      exact hBs },\n    { dsimp [Inf, upper_bounds, lower_bounds],\n      intros B h p hBp B hBs,\n      apply h,\n      exact hBs,\n      exact hBp }\n  end)\n\nlemma infi_def {ι : Sort*} (a : ι → presheaf B) : \n  infi a = ⟨λ p, ∀ i, a i p, λ x y hxy h i, (a i).2 _ y hxy (h i)⟩ :=\nle_antisymm \n  (infi_le_iff.2 (λ B h p hBp i, h i _ hBp)) \n  (le_infi (λ i p h, h _))\n\nlemma supr_def {ι : Sort*} (a : ι → presheaf B) : \n  supr a = ⟨λ p, ∃ i, a i p, λ x y hxy ⟨i, hi⟩, ⟨i, (a i).2 x y hxy hi⟩⟩ :=\nle_antisymm \n  (supr_le_iff.2 (λ i p h, ⟨i, h⟩)) \n  (le_supr_iff.2 (λ B h p ⟨i, hi⟩, h i _ hi))\n\ndef yoneda (a : B) : presheaf B :=\n⟨λ b, b ≤ a, λ b c, le_trans⟩\n\ndef yoneda_le_iff (a : presheaf B) (p : B) : yoneda p ≤ a ↔ a p :=\nbegin\n  simp [yoneda, presheaf.le_def],\n  split,\n  { intro h, apply h, exact le_rfl },\n  { intros h x hxp,\n    apply a.2,\n    apply hxp,\n    exact h }\nend\n\nlemma yoneda_mono {a b : B} : yoneda a ≤ yoneda b ↔ a ≤ b :=\nbegin\n  rw yoneda_le_iff, refl,\nend\n\nlemma eq_supr {a : presheaf B} : a = ⨆ (p : B) (h : a p), yoneda p :=\nbegin\n  apply le_antisymm; simp only [le_supr_iff, supr_le_iff, yoneda_le_iff],\n  { intros B hB p,\n    exact hB p },\n  { exact λ _, id }\nend\n\ndef restrict (f : sup_hom A B) (a : presheaf B) : presheaf A :=\n⟨λ x, a.1 (f.1 x), λ x y hxy h, a.2 _ _ (f.monotone hxy) h⟩ \n\nend presheaf\n\nvariable (B)\n\ndef copresheaf : Type* :=\n{ s : B → Prop // ∀ a b, a ≤ b → s a → s b }\n\nvariable {B}\n\nnamespace copresheaf\n\ninstance : has_coe_to_fun (copresheaf B) (λ _, B → Prop) :=\n⟨subtype.val⟩\n\n@[simp] lemma coe_mk (s : B → Prop) (hs : ∀ a b, a ≤ b → s a → s b) :\n  @coe_fn (copresheaf B) _ _ (⟨s, hs⟩ : copresheaf B) = s := rfl\n\ninstance : partial_order (copresheaf B) :=\n{ le := λ a b, ∀ x, b x → a x,\n  le_trans := λ a b c hab hbc x hcx, hab _ (hbc _ hcx),\n  le_refl := λ _ _, id,\n  le_antisymm := λ a b hab hba, subtype.val_injective (funext $ λ x, propext ⟨hba _, hab _⟩) }\n\nlemma le_def {a b : copresheaf B} : a ≤ b = ∀ x, b x → a x := rfl\n\ninstance : has_Sup (copresheaf B) :=\n{ Sup := λ s, ⟨λ p, ∀ B : copresheaf B, B ∈ s → B p, \n     λ a b hab h B hBs, B.2 _ _ hab (h _ hBs)⟩ }\n\ninstance : complete_lattice (copresheaf B) :=\ncomplete_lattice_of_Sup _\n  (λ s, begin\n    split,\n    { dsimp [Sup, upper_bounds],\n      intros B hBs p h,\n      apply h,\n      exact hBs },\n    { dsimp [Inf, upper_bounds, lower_bounds],\n      intros B h p hBp B hBs,\n      apply h,\n      exact hBs,\n      exact hBp }\n  end)\n\nlemma infi_def {ι : Sort*} (a : ι → copresheaf B) : \n  infi a = ⟨λ p, ∃ i, a i p, λ x y hxy ⟨i, hi⟩, ⟨i, (a i).2 x y hxy hi⟩⟩ :=\nle_antisymm \n  (infi_le_iff.2 (λ B h p ⟨i, hi⟩, h i _ hi)) \n  (le_infi_iff.2 (λ i p h, ⟨i, h⟩))\n\nlemma supr_def {ι : Sort*} (a : ι → copresheaf B) : \n  supr a = ⟨λ p, ∀ i, a i p, λ x y hxy h i, (a i).2 _ y hxy (h i)⟩ :=\nle_antisymm \n  (supr_le_iff.2 (λ i p h, h _)) \n  (le_supr_iff.2 (λ B h p hBp i, h i _ hBp))\n\ndef coyoneda (a : B) : copresheaf B :=\n⟨λ b, a ≤ b, λ b c, function.swap le_trans⟩\n\ndef le_coyoneda_iff (a : copresheaf B) (p : B) : a ≤ coyoneda p ↔ a p :=\nbegin\n  simp [copresheaf.coyoneda, copresheaf.le_def],\n  split,\n  { intro h, apply h, exact le_rfl },\n  { intros h x hxp,\n    apply a.2,\n    apply hxp,\n    exact h }\nend\n\nlemma eq_infi {a : copresheaf B} : a = ⨅ (p : B) (h : a p), coyoneda p :=\nbegin\n  apply le_antisymm; simp only [le_infi_iff, infi_le_iff, le_coyoneda_iff],\n  { exact λ _, id },\n  { intros B hB p,\n    exact hB p },\nend\n\nend copresheaf\n\nopen presheaf copresheaf\n\nnamespace presheaf\n\ndef u (f : A →o B) (a : presheaf B) : copresheaf B :=\n⟨λ p, Π (x : B), a x → x ≤ p, λ a b hab h x hxA, le_trans (h _ hxA) hab⟩\n\nlemma u_mono {f : A →o B} : monotone (u f):=\nλ A B h p hp q hAq, hp _ (h _ hAq)\n\nend presheaf\n\nnamespace copresheaf\n\ndef d (f : A →o B) (a : copresheaf B) : presheaf B :=\n⟨λ q, Π (x : A), a (f x) → q ≤ f x, λ a b hab h x hxA, le_trans hab (h _ hxA)⟩\n\nlemma d_mono {f : A →o B} : monotone (d f) :=\nλ A B h p hp q hAq, hp _ (h _ hAq)\n\nend copresheaf\n\ndef gc {f : A →o B} : galois_connection (presheaf.u f) (copresheaf.d f) :=\nbegin\n  intros A B,\n  dsimp [presheaf.u, copresheaf.d],\n  split,\n  { intros h x hxA y hyB,\n    dsimp [copresheaf.le_def] at h,\n    apply h,\n    assumption,\n    assumption,\n     },\n  { intros h x hxA y hyB,\n    dsimp [presheaf.le_def] at *,\n    apply h,\n    assumption,\n    assumption }\nend\n\nnamespace presheaf\n\nopen copresheaf\n\n\nlemma le_d_u (a : presheaf B) : a ≤ d (u a) := gc.le_u_l a\n\n@[simp] lemma u_d_u (a : presheaf B) : (u (d (u a))) = u a :=\nle_antisymm \n  begin\n    dsimp [d, u],\n    intros p hp,\n    dsimp [yoneda, coyoneda, presheaf.le_def] at *,\n    intros q h,\n    apply h,\n    apply hp\n  end\n  (gc.monotone_l (le_d_u _))\n\n\n@[simp] lemma u_yoneda (p : B) : u (yoneda p) = coyoneda p :=\nbegin\n  rw [u],\n  conv_lhs {dsimp [coyoneda]},\n  simp only [yoneda_le_iff],\n  refl\nend\n\nend presheaf\n\nnamespace copresheaf\n\nopen presheaf\n\nlemma u_d_le (a : copresheaf B) : u (d a) ≤ a := gc.l_u_le a\n\n@[simp] lemma d_u_d (a : copresheaf B) : d (u (d a)) = d a :=\nle_antisymm \n  (gc.monotone_u (u_d_le _))\n  begin\n    dsimp [d, u],\n    intros p hp,\n    dsimp [yoneda, coyoneda, presheaf.le_def] at *,\n    intros q h,\n    apply h,\n    apply hp\n  end\n\n@[simp] lemma d_coyoneda (p : B) : d (coyoneda p) = yoneda p :=\nbegin\n  rw [d],\n  conv_lhs {dsimp only [yoneda]},\n  simp only [le_coyoneda_iff],\n  refl\nend\n\nend copresheaf\n\nopen presheaf copresheaf\n\nstructure cocompletion (f : sup_hom A B) : Type (max u v) :=\n( to_presheaf : presheaf B )\n( fixed : d (u (restrict f to_presheaf)) = _ ) \n\nvariables {f : sup_hom A B}\n\nnamespace cocompletion\n\ninstance : partial_order (cocompletion f) :=\npartial_order.lift cocompletion.to_presheaf \n  begin\n    rintros ⟨_, _⟩ ⟨_, _⟩,\n    simp\n  end\n\nlemma le_def {a b : cocompletion f} : a ≤ b ↔ a.to_presheaf ≤ b.to_presheaf := iff.rfl\n\ninstance : has_Inf (cocompletion f) :=\n⟨λ s, ⟨⨅ (a : cocompletion f) (h : a ∈ s), a.to_presheaf, begin\n  dsimp [restrict,u, d, yoneda, coyoneda],\n  apply subtype.ext,\n  dsimp,\n  funext x,\n  simp only [presheaf.infi_def],\n  simp only [copresheaf.le_def, presheaf.le_def],\n  dsimp,\n  ext,\n  split,\n  { intros, }\n  \nend⟩⟩\n\nend cocompletion", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/bicompletion/partial_order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.30002115554253966}}
{"text": "import tactic.lint\n\ndef f : ℕ → ℕ := default _\ndef c : ℕ := default _\ndef d : ℕ := default _\n\n@[simp] lemma c_eq_d : c = d := rfl\n\n-- The following lemma never applies when using simp, because c is first rewritten to d\n@[simp] lemma f_c : f c = 0 := rfl\n\nexample : f c = 0 :=\nbegin\n  simp,\n  guard_target f d = 0, -- does not apply f_c\n  refl\nend\n\nopen tactic\nrun_cmd do\ndecl ← get_decl ``f_c,\nres ← linter.simp_nf.test decl,\n-- linter complains\nguard $ res.is_some\n\n\n-- also works with `coe_to_fun`\n\nstructure morphism :=\n(f : ℕ → ℕ)\n\ninstance : has_coe_to_fun morphism (λ _, ℕ → ℕ):=\n⟨morphism.f⟩\n\ndef h : morphism := ⟨default _⟩\n\n-- Also never applies\n@[simp] lemma h_c : h c = 0 := rfl\n\nexample : h c = 0 :=\nbegin\n  simp,\n  guard_target h d = 0, -- does not apply h_c\n  refl\nend\n\nopen tactic\nrun_cmd do\ndecl ← get_decl ``h_c,\nres ← linter.simp_nf.test decl,\n-- linter complains\nguard $ res.is_some\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/test/lint_simp_nf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3000211555425395}}